LMFDB - Embedded newform 177.9.c.a.58.67

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,9,Mod(58,177)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("177.58");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	177.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$72.1060139808$
Analytic rank:	$0$
Dimension:	$80$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			58.67
Character	$\chi$	$=$	177.58
Dual form			177.9.c.a.58.14

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+23.2103i q^{2} -46.7654 q^{3} -282.716 q^{4} +684.951 q^{5} -1085.44i q^{6} -2372.31 q^{7} -620.092i q^{8} +2187.00 q^{9} +O(q^{10})$	\(q+23.2103i q^{2} -46.7654 q^{3} -282.716 q^{4} +684.951 q^{5} -1085.44i q^{6} -2372.31 q^{7} -620.092i q^{8} +2187.00 q^{9} +15897.9i q^{10} -9748.41i q^{11} +13221.3 q^{12} +23347.2i q^{13} -55062.0i q^{14} -32032.0 q^{15} -57982.9 q^{16} +154078. q^{17} +50760.8i q^{18} +71059.1 q^{19} -193647. q^{20} +110942. q^{21} +226263. q^{22} -291010. i q^{23} +28998.8i q^{24} +78532.4 q^{25} -541894. q^{26} -102276. q^{27} +670692. q^{28} +820057. q^{29} -743471. i q^{30} +1.57709e6i q^{31} -1.50454e6i q^{32} +455888. i q^{33} +3.57619e6i q^{34} -1.62492e6 q^{35} -618301. q^{36} +1.45000e6i q^{37} +1.64930e6i q^{38} -1.09184e6i q^{39} -424732. i q^{40} +2.20221e6 q^{41} +2.57500e6i q^{42} +68355.0i q^{43} +2.75603e6i q^{44} +1.49799e6 q^{45} +6.75441e6 q^{46} +6.23313e6i q^{47} +2.71159e6 q^{48} -136932. q^{49} +1.82276e6i q^{50} -7.20551e6 q^{51} -6.60063e6i q^{52} +866154. q^{53} -2.37385e6i q^{54} -6.67718e6i q^{55} +1.47105e6i q^{56} -3.32311e6 q^{57} +1.90337e7i q^{58} +(-1.19635e7 + 1.92498e6i) q^{59} +9.05596e6 q^{60} +7.27564e6i q^{61} -3.66047e7 q^{62} -5.18825e6 q^{63} +2.00772e7 q^{64} +1.59917e7i q^{65} -1.05813e7 q^{66} +2.21972e7i q^{67} -4.35603e7 q^{68} +1.36092e7i q^{69} -3.77148e7i q^{70} -1.10994e7 q^{71} -1.35614e6i q^{72} -1.47868e7i q^{73} -3.36550e7 q^{74} -3.67260e6 q^{75} -2.00896e7 q^{76} +2.31263e7i q^{77} +2.53419e7 q^{78} -3.63921e7 q^{79} -3.97154e7 q^{80} +4.78297e6 q^{81} +5.11139e7i q^{82} +1.21628e7i q^{83} -3.13651e7 q^{84} +1.05536e8 q^{85} -1.58654e6 q^{86} -3.83503e7 q^{87} -6.04491e6 q^{88} -8.33425e7i q^{89} +3.47687e7i q^{90} -5.53868e7i q^{91} +8.22732e7i q^{92} -7.37533e7i q^{93} -1.44673e8 q^{94} +4.86720e7 q^{95} +7.03604e7i q^{96} -5.82423e7i q^{97} -3.17822e6i q^{98} -2.13198e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 10240 q^{4} + 160 q^{7} + 174960 q^{9}+O(q^{10}) $ 80 * q - 10240 * q^4 + 160 * q^7 + 174960 * q^9	$ 80 q - 10240 q^{4} + 160 q^{7} + 174960 q^{9} - 22680 q^{12} - 59616 q^{15} + 1199848 q^{16} - 10608 q^{17} - 27516 q^{19} - 146436 q^{20} - 974696 q^{22} + 5718040 q^{25} - 797484 q^{26} - 3133000 q^{28} + 1725924 q^{29} + 4318800 q^{35} - 22394880 q^{36} - 732180 q^{41} + 22752084 q^{46} + 8703936 q^{48} + 55899176 q^{49} - 10373832 q^{51} - 39265944 q^{53} - 11408040 q^{57} - 33575112 q^{59} - 18034488 q^{60} + 13038600 q^{62} + 349920 q^{63} - 241654260 q^{64} - 35711928 q^{66} + 36772608 q^{68} - 235272660 q^{71} - 63050712 q^{74} + 74363184 q^{75} + 9454680 q^{76} - 10865988 q^{78} + 17252580 q^{79} + 318203976 q^{80} + 382637520 q^{81} - 20743128 q^{84} - 27245820 q^{85} + 105666984 q^{86} + 29437992 q^{87} + 82079788 q^{88} + 121215992 q^{94} - 690837276 q^{95}+O(q^{100}) $ 80 * q - 10240 * q^4 + 160 * q^7 + 174960 * q^9 - 22680 * q^12 - 59616 * q^15 + 1199848 * q^16 - 10608 * q^17 - 27516 * q^19 - 146436 * q^20 - 974696 * q^22 + 5718040 * q^25 - 797484 * q^26 - 3133000 * q^28 + 1725924 * q^29 + 4318800 * q^35 - 22394880 * q^36 - 732180 * q^41 + 22752084 * q^46 + 8703936 * q^48 + 55899176 * q^49 - 10373832 * q^51 - 39265944 * q^53 - 11408040 * q^57 - 33575112 * q^59 - 18034488 * q^60 + 13038600 * q^62 + 349920 * q^63 - 241654260 * q^64 - 35711928 * q^66 + 36772608 * q^68 - 235272660 * q^71 - 63050712 * q^74 + 74363184 * q^75 + 9454680 * q^76 - 10865988 * q^78 + 17252580 * q^79 + 318203976 * q^80 + 382637520 * q^81 - 20743128 * q^84 - 27245820 * q^85 + 105666984 * q^86 + 29437992 * q^87 + 82079788 * q^88 + 121215992 * q^94 - 690837276 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/177\mathbb{Z}\right)^\times$.

$n$	$61$	$119$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			23.2103i			1.45064i	0.688411	+	0.725321i	$0.258309\pi$
$2$			23.2103i			1.45064i	−0.688411	+	0.725321i	$0.741691\pi$
$3$	−46.7654			−0.577350
$3$	−46.7654			−0.577350
$4$	−282.716			−1.10436
$5$	684.951			1.09592			0.547961	−	0.836504i	$-0.315404\pi$
$5$	684.951			1.09592			0.547961	+	0.836504i	$0.315404\pi$
$6$		−	1085.44i		−	0.837528i
$7$	−2372.31			−0.988052			−0.494026	−	0.869447i	$-0.664475\pi$
$7$	−2372.31			−0.988052			−0.494026	+	0.869447i	$0.664475\pi$
$8$		−	620.092i		−	0.151390i
$9$	2187.00			0.333333
$10$			15897.9i			1.58979i
$11$		−	9748.41i		−	0.665830i	−0.942957	−	0.332915i	$-0.891968\pi$
$11$		−	9748.41i		−	0.665830i	0.942957	−	0.332915i	$-0.108032\pi$
$12$	13221.3			0.637603
$13$			23347.2i			0.817450i	0.912658	+	0.408725i	$0.134026\pi$
$13$			23347.2i			0.817450i	−0.912658	+	0.408725i	$0.865974\pi$
$14$		−	55062.0i		−	1.43331i
$15$	−32032.0			−0.632730
$16$	−57982.9			−0.884748
$17$	154078.			1.84478			0.922390	−	0.386260i	$-0.126233\pi$
$17$	154078.			1.84478			0.922390	+	0.386260i	$0.126233\pi$
$18$			50760.8i			0.483547i
$19$	71059.1			0.545262			0.272631	−	0.962119i	$-0.412106\pi$
$19$	71059.1			0.545262			0.272631	+	0.962119i	$0.412106\pi$
$20$	−193647.			−1.21029
$21$	110942.			0.570452
$22$	226263.			0.965880
$23$		−	291010.i		−	1.03991i	−0.854193	−	0.519955i	$-0.825949\pi$
$23$		−	291010.i		−	1.03991i	0.854193	−	0.519955i	$-0.174051\pi$
$24$			28998.8i			0.0874048i
$25$	78532.4			0.201043
$26$	−541894.			−1.18583
$27$	−102276.			−0.192450
$28$	670692.			1.09117
$29$	820057.			1.15945			0.579725	−	0.814812i	$-0.303160\pi$
$29$	820057.			1.15945			0.579725	+	0.814812i	$0.303160\pi$
$30$		−	743471.i		−	0.917865i
$31$			1.57709e6i			1.70770i	0.520523	+	0.853848i	$0.325737\pi$
$31$			1.57709e6i			1.70770i	−0.520523	+	0.853848i	$0.674263\pi$
$32$		−	1.50454e6i		−	1.43484i
$33$			455888.i			0.384417i
$34$			3.57619e6i			2.67611i
$35$	−1.62492e6			−1.08283
$36$	−618301.			−0.368120
$37$			1.45000e6i			0.773681i	0.922147	+	0.386841i	$0.126434\pi$
$37$			1.45000e6i			0.773681i	−0.922147	+	0.386841i	$0.873566\pi$
$38$			1.64930e6i			0.790980i
$39$		−	1.09184e6i		−	0.471955i
$40$		−	424732.i		−	0.165911i
$41$	2.20221e6			0.779333			0.389667	−	0.920956i	$-0.372590\pi$
$41$	2.20221e6			0.779333			0.389667	+	0.920956i	$0.372590\pi$
$42$			2.57500e6i			0.827522i
$43$			68355.0i			0.0199939i	0.999950	+	0.00999693i	$0.00318218\pi$
$43$			68355.0i			0.0199939i	−0.999950	+	0.00999693i	$0.996818\pi$
$44$			2.75603e6i			0.735316i
$45$	1.49799e6			0.365307
$46$	6.75441e6			1.50854
$47$			6.23313e6i			1.27737i	0.769470	+	0.638683i	$0.220520\pi$
$47$			6.23313e6i			1.27737i	−0.769470	+	0.638683i	$0.779480\pi$
$48$	2.71159e6			0.510810
$49$	−136932.			−0.0237530
$50$			1.82276e6i			0.291641i
$51$	−7.20551e6			−1.06508
$52$		−	6.60063e6i		−	0.902759i
$53$	866154.			0.109772			0.0548860	−	0.998493i	$-0.482520\pi$
$53$	866154.			0.109772			0.0548860	+	0.998493i	$0.482520\pi$
$54$		−	2.37385e6i		−	0.279176i
$55$		−	6.67718e6i		−	0.729697i
$56$			1.47105e6i			0.149581i
$57$	−3.32311e6			−0.314807
$58$			1.90337e7i			1.68195i
$59$	−1.19635e7	+	1.92498e6i	−0.987301	+	0.158861i
$59$	−1.19635e7	+	1.92498e6i	−0.987301	+	0.158861i
$60$	9.05596e6			0.698762
$61$			7.27564e6i			0.525475i	0.964867	+	0.262738i	$0.0846253\pi$
$61$			7.27564e6i			0.525475i	−0.964867	+	0.262738i	$0.915375\pi$
$62$	−3.66047e7			−2.47725
$63$	−5.18825e6			−0.329351
$64$	2.00772e7			1.19669
$65$			1.59917e7i			0.895861i
$66$	−1.05813e7			−0.557651
$67$			2.21972e7i			1.10154i	0.834658	+	0.550768i	$0.185665\pi$
$67$			2.21972e7i			1.10154i	−0.834658	+	0.550768i	$0.814335\pi$
$68$	−4.35603e7			−2.03730
$69$			1.36092e7i			0.600393i
$70$		−	3.77148e7i		−	1.57079i
$71$	−1.10994e7			−0.436781			−0.218391	−	0.975861i	$-0.570081\pi$
$71$	−1.10994e7			−0.436781			−0.218391	+	0.975861i	$0.570081\pi$
$72$		−	1.35614e6i		−	0.0504632i
$73$		−	1.47868e7i		−	0.520695i	−0.965515	−	0.260348i	$-0.916163\pi$
$73$		−	1.47868e7i		−	0.520695i	0.965515	−	0.260348i	$-0.0838372\pi$
$74$	−3.36550e7			−1.12233
$75$	−3.67260e6			−0.116072
$76$	−2.00896e7			−0.602166
$77$			2.31263e7i			0.657874i
$78$	2.53419e7			0.684637
$79$	−3.63921e7			−0.934327			−0.467163	−	0.884171i	$-0.654724\pi$
$79$	−3.63921e7			−0.934327			−0.467163	+	0.884171i	$0.654724\pi$
$80$	−3.97154e7			−0.969614
$81$	4.78297e6			0.111111
$82$			5.11139e7i			1.13053i
$83$			1.21628e7i			0.256284i	0.991756	+	0.128142i	$0.0409013\pi$
$83$			1.21628e7i			0.256284i	−0.991756	+	0.128142i	$0.959099\pi$
$84$	−3.13651e7			−0.629985
$85$	1.05536e8			2.02173
$86$	−1.58654e6			−0.0290039
$87$	−3.83503e7			−0.669409
$88$	−6.04491e6			−0.100800
$89$		−	8.33425e7i		−	1.32833i	−0.747585	−	0.664166i	$-0.768787\pi$
$89$		−	8.33425e7i		−	1.32833i	0.747585	−	0.664166i	$-0.231213\pi$
$90$			3.47687e7i			0.529930i
$91$		−	5.53868e7i		−	0.807683i
$92$			8.22732e7i			1.14844i
$93$		−	7.37533e7i		−	0.985938i
$94$	−1.44673e8			−1.85300
$95$	4.86720e7			0.597564
$96$			7.03604e7i			0.828407i
$97$		−	5.82423e7i		−	0.657888i	−0.944349	−	0.328944i	$-0.893307\pi$
$97$		−	5.82423e7i		−	0.657888i	0.944349	−	0.328944i	$-0.106693\pi$
$98$		−	3.17822e6i		−	0.0344571i
$99$		−	2.13198e7i		−	0.221943i
$100$	−2.22024e7			−0.222024
$101$		−	1.47379e8i		−	1.41628i	−0.706072	−	0.708140i	$-0.749534\pi$
$101$		−	1.47379e8i		−	1.41628i	0.706072	−	0.708140i	$-0.250466\pi$
$102$		−	1.67242e8i		−	1.54505i
$103$			1.43420e8i			1.27426i	0.770754	+	0.637132i	$0.219880\pi$
$103$			1.43420e8i			1.27426i	−0.770754	+	0.637132i	$0.780120\pi$
$104$	1.44774e7			0.123753
$105$	7.59899e7			0.625171
$106$			2.01037e7i			0.159240i
$107$	1.63576e8			1.24791			0.623956	−	0.781459i	$-0.285524\pi$
$107$	1.63576e8			1.24791			0.623956	+	0.781459i	$0.285524\pi$
$108$	2.89151e7			0.212534
$109$			4.50277e6i			0.0318988i	0.999873	+	0.0159494i	$0.00507706\pi$
$109$			4.50277e6i			0.0318988i	−0.999873	+	0.0159494i	$0.994923\pi$
$110$	1.54979e8			1.05853
$111$		−	6.78099e7i		−	0.446685i
$112$	1.37554e8			0.874178
$113$			1.52783e8i			0.937049i	0.883450	+	0.468525i	$0.155214\pi$
$113$			1.52783e8i			0.937049i	−0.883450	+	0.468525i	$0.844786\pi$
$114$		−	7.71302e7i		−	0.456673i
$115$		−	1.99327e8i		−	1.13966i
$116$	−2.31843e8			−1.28045
$117$			5.10603e7i			0.272483i
$118$	−4.46793e7	−	2.77676e8i	−0.230451	−	1.43222i
$119$	−3.65521e8			−1.82274
$120$			1.98628e7i			0.0957888i
$121$	1.19327e8			0.556671
$122$	−1.68870e8			−0.762276
$123$	−1.02987e8			−0.449948
$124$		−	4.45870e8i		−	1.88591i
$125$	−2.13768e8			−0.875594
$126$		−	1.20421e8i		−	0.477770i
$127$	−3.50645e8			−1.34788			−0.673942	−	0.738785i	$-0.735400\pi$
$127$	−3.50645e8			−1.34788			−0.673942	+	0.738785i	$0.735400\pi$
$128$			8.08341e7i			0.301130i
$129$		−	3.19665e6i		−	0.0115435i
$130$	−3.71171e8			−1.29957
$131$			1.17842e8i			0.400143i	0.979781	+	0.200071i	$0.0641174\pi$
$131$			1.17842e8i			0.400143i	−0.979781	+	0.200071i	$0.935883\pi$
$132$		−	1.28887e8i		−	0.424535i
$133$	−1.68574e8			−0.538748
$134$	−5.15202e8			−1.59793
$135$	−7.00539e7			−0.210910
$136$		−	9.55424e7i		−	0.279281i
$137$	2.80381e7			0.0795915			0.0397958	−	0.999208i	$-0.487329\pi$
$137$	2.80381e7			0.0795915			0.0397958	+	0.999208i	$0.487329\pi$
$138$	−3.15872e8			−0.870954
$139$	−3.55113e8			−0.951279			−0.475639	−	0.879640i	$-0.657783\pi$
$139$	−3.55113e8			−0.951279			−0.475639	+	0.879640i	$0.657783\pi$
$140$	4.59391e8			1.19583
$141$		−	2.91495e8i		−	0.737487i
$142$		−	2.57619e8i		−	0.633613i
$143$	2.27598e8			0.544282
$144$	−1.26809e8			−0.294916
$145$	5.61699e8			1.27067
$146$	3.43206e8			0.755342
$147$	6.40365e6			0.0137138
$148$		−	4.09940e8i		−	0.854423i
$149$		−	8.61218e7i		−	0.174730i	−0.996176	−	0.0873651i	$-0.972155\pi$
$149$		−	8.61218e7i		−	0.174730i	0.996176	−	0.0873651i	$-0.0278447\pi$
$150$		−	8.52420e7i		−	0.168379i
$151$			1.01980e9i			1.96159i	0.195051	+	0.980793i	$0.437513\pi$
$151$			1.01980e9i			1.96159i	−0.195051	+	0.980793i	$0.562487\pi$
$152$		−	4.40632e7i		−	0.0825471i
$153$	3.36968e8			0.614926
$154$	−5.36767e8			−0.954340
$155$			1.08023e9i			1.87150i
$156$			3.08681e8i			0.521208i
$157$			2.35092e8i			0.386936i	0.981107	+	0.193468i	$0.0619735\pi$
$157$			2.35092e8i			0.386936i	−0.981107	+	0.193468i	$0.938026\pi$
$158$		−	8.44670e8i		−	1.35537i
$159$	−4.05060e7			−0.0633769
$160$		−	1.03054e9i		−	1.57247i
$161$			6.90366e8i			1.02749i
$162$			1.11014e8i			0.161182i
$163$	−1.10007e9			−1.55837			−0.779183	−	0.626796i	$-0.784366\pi$
$163$	−1.10007e9			−1.55837			−0.779183	+	0.626796i	$0.784366\pi$
$164$	−6.22601e8			−0.860665
$165$			3.12261e8i			0.421291i
$166$	−2.82302e8			−0.371776
$167$	1.00055e9			1.28639			0.643197	−	0.765700i	$-0.277607\pi$
$167$	1.00055e9			1.28639			0.643197	+	0.765700i	$0.277607\pi$
$168$		−	6.87943e7i		−	0.0863605i
$169$	2.70640e8			0.331776
$170$			2.44951e9i			2.93281i
$171$	1.55406e8			0.181754
$172$		−	1.93251e7i		−	0.0220804i
$173$			7.61863e8i			0.850535i	0.905068	+	0.425268i	$0.139820\pi$
$173$			7.61863e8i			0.850535i	−0.905068	+	0.425268i	$0.860180\pi$
$174$		−	8.90120e8i		−	0.971072i
$175$	−1.86304e8			−0.198641
$176$			5.65241e8i			0.589092i
$177$	5.59477e8	−	9.00225e7i	0.570018	−	0.0917187i
$178$	1.93440e9			1.92693
$179$			1.40082e9i			1.36449i	0.731125	+	0.682243i	$0.238996\pi$
$179$			1.40082e9i			1.36449i	−0.731125	+	0.682243i	$0.761004\pi$
$180$	−4.23505e8			−0.403431
$181$	−1.44369e9			−1.34512			−0.672559	−	0.740044i	$-0.734805\pi$
$181$	−1.44369e9			−1.34512			−0.672559	+	0.740044i	$0.734805\pi$
$182$	1.28554e9			1.17166
$183$		−	3.40248e8i		−	0.303383i
$184$	−1.80453e8			−0.157432
$185$			9.93181e8i			0.847894i
$186$	1.71183e9			1.43024
$187$		−	1.50201e9i		−	1.22831i
$188$		−	1.76221e9i		−	1.41067i
$189$	2.42630e8			0.190151
$190$			1.12969e9i			0.866852i
$191$			7.40160e8i			0.556150i	0.960559	+	0.278075i	$0.0896964\pi$
$191$			7.40160e8i			0.556150i	−0.960559	+	0.278075i	$0.910304\pi$
$192$	−9.38917e8			−0.690911
$193$	−1.17043e9			−0.843560			−0.421780	−	0.906698i	$-0.638594\pi$
$193$	−1.17043e9			−0.843560			−0.421780	+	0.906698i	$0.638594\pi$
$194$	1.35182e9			0.954359
$195$		−	7.47856e8i		−	0.517225i
$196$	3.87128e7			0.0262319
$197$	−1.17225e7			−0.00778318			−0.00389159	−	0.999992i	$-0.501239\pi$
$197$	−1.17225e7			−0.00778318			−0.00389159	+	0.999992i	$0.501239\pi$
$198$	4.94838e8			0.321960
$199$	−2.61087e9			−1.66484			−0.832420	−	0.554146i	$-0.813045\pi$
$199$	−2.61087e9			−1.66484			−0.832420	+	0.554146i	$0.813045\pi$
$200$		−	4.86973e7i		−	0.0304358i
$201$		−	1.03806e9i		−	0.635972i
$202$	3.42070e9			2.05451
$203$	−1.94543e9			−1.14560
$204$	2.03711e9			1.17624
$205$	1.50841e9			0.854088
$206$	−3.32881e9			−1.84850
$207$		−	6.36438e8i		−	0.346637i
$208$		−	1.35374e9i		−	0.723237i
$209$		−	6.92714e8i		−	0.363052i
$210$			1.76374e9i			0.906898i
$211$			1.45351e9i			0.733310i	0.930357	+	0.366655i	$0.119497\pi$
$211$			1.45351e9i			0.733310i	−0.930357	+	0.366655i	$0.880503\pi$
$212$	−2.44876e8			−0.121228
$213$	5.19065e8			0.252176
$214$			3.79664e9i			1.81027i
$215$			4.68198e7i			0.0219117i
$216$			6.34204e7i			0.0291349i
$217$		−	3.74136e9i		−	1.68729i
$218$	−1.04511e8			−0.0462737
$219$			6.91512e8i			0.300624i
$220$			1.88775e9i			0.805848i
$221$			3.59728e9i			1.50801i
$222$	1.57389e9			0.647980
$223$	−1.00430e9			−0.406110			−0.203055	−	0.979167i	$-0.565087\pi$
$223$	−1.00430e9			−0.406110			−0.203055	+	0.979167i	$0.565087\pi$
$224$			3.56924e9i			1.41770i
$225$	1.71750e8			0.0670143
$226$	−3.54614e9			−1.35932
$227$		−	3.49000e9i		−	1.31438i	−0.753724	−	0.657191i	$-0.771745\pi$
$227$		−	3.49000e9i		−	1.31438i	0.753724	−	0.657191i	$-0.228255\pi$
$228$	9.39496e8			0.347661
$229$			1.94905e9i			0.708730i	0.935107	+	0.354365i	$0.115303\pi$
$229$			1.94905e9i			0.708730i	−0.935107	+	0.354365i	$0.884697\pi$
$230$	4.62644e9			1.65324
$231$		−	1.08151e9i		−	0.379824i
$232$		−	5.08511e8i		−	0.175529i
$233$			2.50740e9i			0.850747i	0.905018	+	0.425374i	$0.139857\pi$
$233$			2.50740e9i			0.850747i	−0.905018	+	0.425374i	$0.860143\pi$
$234$	−1.18512e9			−0.395276
$235$			4.26939e9i			1.39989i
$236$	3.38227e9	−	5.44224e8i	1.09034	−	0.175440i
$237$	1.70189e9			0.539434
$238$		−	8.48383e9i		−	2.64414i
$239$	7.71077e8			0.236323			0.118162	−	0.992994i	$-0.462300\pi$
$239$	7.71077e8			0.236323			0.118162	+	0.992994i	$0.462300\pi$
$240$	1.85731e9			0.559807
$241$	1.97237e9			0.584681			0.292341	−	0.956314i	$-0.405566\pi$
$241$	1.97237e9			0.584681			0.292341	+	0.956314i	$0.405566\pi$
$242$			2.76962e9i			0.807530i
$243$	−2.23677e8			−0.0641500
$244$		−	2.05694e9i		−	0.580314i
$245$	−9.37913e7			−0.0260315
$246$		−	2.39036e9i		−	0.652714i
$247$			1.65903e9i			0.445725i
$248$	9.77942e8			0.258527
$249$		−	5.68798e8i		−	0.147966i
$250$		−	4.96161e9i		−	1.27017i
$251$	4.98630e9			1.25627			0.628136	−	0.778104i	$-0.283818\pi$
$251$	4.98630e9			1.25627			0.628136	+	0.778104i	$0.283818\pi$
$252$	1.46680e9			0.363722
$253$	−2.83688e9			−0.692403
$254$		−	8.13855e9i		−	1.95530i
$255$	−4.93542e9			−1.16725
$256$	3.26358e9			0.759861
$257$	1.39633e9			0.320079			0.160039	−	0.987111i	$-0.448838\pi$
$257$	1.39633e9			0.320079			0.160039	+	0.987111i	$0.448838\pi$
$258$	7.41951e7			0.0167454
$259$		−	3.43986e9i		−	0.764437i
$260$		−	4.52111e9i		−	0.989353i
$261$	1.79346e9			0.386483
$262$	−2.73514e9			−0.580464
$263$	8.93441e9			1.86742			0.933712	−	0.358025i	$-0.116550\pi$
$263$	8.93441e9			1.86742			0.933712	+	0.358025i	$0.116550\pi$
$264$	2.82693e8			0.0581967
$265$	5.93273e8			0.120302
$266$		−	3.91266e9i		−	0.781530i
$267$			3.89754e9i			0.766913i
$268$		−	6.27551e9i		−	1.21649i
$269$			2.31516e9i			0.442153i	0.975257	+	0.221076i	$0.0709570\pi$
$269$			2.31516e9i			0.442153i	−0.975257	+	0.221076i	$0.929043\pi$
$270$		−	1.62597e9i		−	0.305955i
$271$	5.44804e9			1.01010			0.505049	−	0.863091i	$-0.331475\pi$
$271$	5.44804e9			1.01010			0.505049	+	0.863091i	$0.331475\pi$
$272$	−8.93387e9			−1.63217
$273$			2.59019e9i			0.466316i
$274$			6.50773e8i			0.115459i
$275$		−	7.65567e8i		−	0.133860i
$276$		−	3.84753e9i		−	0.663050i
$277$	8.14080e9			1.38276			0.691382	−	0.722489i	$-0.257002\pi$
$277$	8.14080e9			1.38276			0.691382	+	0.722489i	$0.257002\pi$
$278$		−	8.24228e9i		−	1.37996i
$279$			3.44910e9i			0.569232i
$280$			1.00760e9i			0.163929i
$281$	−4.43224e8			−0.0710883			−0.0355441	−	0.999368i	$-0.511316\pi$
$281$	−4.43224e8			−0.0710883			−0.0355441	+	0.999368i	$0.511316\pi$
$282$	6.76567e9			1.06983
$283$		−	1.33751e9i		−	0.208521i	−0.994550	−	0.104261i	$-0.966752\pi$
$283$		−	1.33751e9i		−	0.208521i	0.994550	−	0.104261i	$-0.0332476\pi$
$284$	3.13797e9			0.482364
$285$	−2.27616e9			−0.345004
$286$			5.28261e9i			0.789559i
$287$	−5.22433e9			−0.770022
$288$		−	3.29043e9i		−	0.478281i
$289$	1.67642e10			2.40321
$290$			1.30372e10i			1.84328i
$291$			2.72372e9i			0.379832i
$292$			4.18048e9i			0.575035i
$293$	9.25231e9			1.25539			0.627696	−	0.778458i	$-0.283998\pi$
$293$	9.25231e9			1.25539			0.627696	+	0.778458i	$0.283998\pi$
$294$			1.48630e8i			0.0198938i
$295$	−8.19439e9	+	1.31852e9i	−1.08200	+	0.174100i
$296$	8.99135e8			0.117127
$297$			9.97027e8i			0.128139i
$298$	1.99891e9			0.253471
$299$	6.79426e9			0.850075
$300$	1.03830e9			0.128186
$301$		−	1.62160e8i		−	0.0197550i
$302$	−2.36698e10			−2.84556
$303$			6.89222e9i			0.817690i
$304$	−4.12021e9			−0.482420
$305$			4.98346e9i			0.575879i
$306$			7.82112e9i			0.892038i
$307$	−1.28673e10			−1.44856			−0.724278	−	0.689508i	$-0.757827\pi$
$307$	−1.28673e10			−1.44856			−0.724278	+	0.689508i	$0.757827\pi$
$308$		−	6.53818e9i		−	0.726530i
$309$		−	6.70707e9i		−	0.735697i
$310$	−2.50724e10			−2.71487
$311$	−6.82362e9			−0.729413			−0.364707	−	0.931123i	$-0.618831\pi$
$311$	−6.82362e9			−0.729413			−0.364707	+	0.931123i	$0.618831\pi$
$312$	−6.77041e8			−0.0714491
$313$			8.37173e8i			0.0872244i	0.999049	+	0.0436122i	$0.0138866\pi$
$313$			8.37173e8i			0.0872244i	−0.999049	+	0.0436122i	$0.986113\pi$
$314$	−5.45654e9			−0.561305
$315$	−3.55369e9			−0.360942
$316$	1.02886e10			1.03183
$317$	1.23020e10			1.21825			0.609126	−	0.793074i	$-0.291520\pi$
$317$	1.23020e10			1.21825			0.609126	+	0.793074i	$0.291520\pi$
$318$		−	9.40156e8i		−	0.0919372i
$319$		−	7.99425e9i		−	0.771996i
$320$	1.37519e10			1.31148
$321$	−7.64969e9			−0.720483
$322$	−1.60236e10			−1.49051
$323$	1.09486e10			1.00589
$324$	−1.35222e9			−0.122707
$325$			1.83351e9i			0.164343i
$326$		−	2.55329e10i		−	2.26063i
$327$		−	2.10574e8i		−	0.0184168i
$328$		−	1.36557e9i		−	0.117983i
$329$		−	1.47869e10i		−	1.26210i
$330$	−7.24766e9			−0.611142
$331$	−1.41369e10			−1.17772			−0.588858	−	0.808236i	$-0.700422\pi$
$331$	−1.41369e10			−1.17772			−0.588858	+	0.808236i	$0.700422\pi$
$332$		−	3.43863e9i		−	0.283030i
$333$			3.17116e9i			0.257894i
$334$			2.32231e10i			1.86610i
$335$			1.52040e10i			1.20720i
$336$	−6.43274e9			−0.504707
$337$			1.60557e10i			1.24483i	0.782687	+	0.622416i	$0.213849\pi$
$337$			1.60557e10i			1.24483i	−0.782687	+	0.622416i	$0.786151\pi$
$338$			6.28162e9i			0.481288i
$339$		−	7.14497e9i		−	0.541006i
$340$	−2.98367e10			−2.23272
$341$	1.53741e10			1.13703
$342$			3.60702e9i			0.263660i
$343$	1.40008e10			1.01152
$344$	4.23864e7			0.00302686
$345$			9.32161e9i			0.657983i
$346$	−1.76830e10			−1.23382
$347$		−	2.47209e10i		−	1.70508i	−0.522658	−	0.852542i	$-0.675060\pi$
$347$		−	2.47209e10i		−	1.70508i	0.522658	−	0.852542i	$-0.324940\pi$
$348$	1.08422e10			0.739269
$349$			1.81322e10i			1.22222i	0.791546	+	0.611109i	$0.209276\pi$
$349$			1.81322e10i			1.22222i	−0.791546	+	0.611109i	$0.790724\pi$
$350$		−	4.32415e9i		−	0.288157i
$351$		−	2.38785e9i		−	0.157318i
$352$	−1.46669e10			−0.955361
$353$			2.46460e10i			1.58726i	0.608402	+	0.793629i	$0.291811\pi$
$353$			2.46460e10i			1.58726i	−0.608402	+	0.793629i	$0.708189\pi$
$354$	2.08945e9	+	1.29856e10i	0.133051	+	0.826892i
$355$	−7.60251e9			−0.478678
$356$			2.35623e10i			1.46696i
$357$	1.70937e10			1.05236
$358$	−3.25133e10			−1.97938
$359$	−2.27075e10			−1.36707			−0.683536	−	0.729917i	$-0.739559\pi$
$359$	−2.27075e10			−1.36707			−0.683536	+	0.729917i	$0.739559\pi$
$360$		−	9.28890e8i		−	0.0553037i
$361$	−1.19342e10			−0.702689
$362$		−	3.35085e10i		−	1.95128i
$363$	−5.58039e9			−0.321394
$364$			1.56588e10i			0.891973i
$365$		−	1.01283e10i		−	0.570641i
$366$	7.89725e9			0.440100
$367$		−	1.30860e10i		−	0.721343i	−0.932693	−	0.360672i	$-0.882548\pi$
$367$		−	1.30860e10i		−	0.721343i	0.932693	−	0.360672i	$-0.117452\pi$
$368$			1.68736e10i			0.920059i
$369$	4.81623e9			0.259778
$370$	−2.30520e10			−1.22999
$371$	−2.05479e9			−0.108461
$372$			2.08513e10i			1.08883i
$373$	1.77480e10			0.916883			0.458442	−	0.888725i	$-0.348408\pi$
$373$	1.77480e10			0.916883			0.458442	+	0.888725i	$0.348408\pi$
$374$	3.48621e10			1.78184
$375$	9.99694e9			0.505524
$376$	3.86512e9			0.193380
$377$			1.91460e10i			0.947792i
$378$			5.63151e9i			0.275841i
$379$	1.19452e10			0.578946			0.289473	−	0.957186i	$-0.406520\pi$
$379$	1.19452e10			0.578946			0.289473	+	0.957186i	$0.406520\pi$
$380$	−1.37604e10			−0.659927
$381$	1.63980e10			0.778201
$382$	−1.71793e10			−0.806774
$383$	2.82331e8			0.0131209			0.00656045	−	0.999978i	$-0.497912\pi$
$383$	2.82331e8			0.0131209			0.00656045	+	0.999978i	$0.497912\pi$
$384$		−	3.78024e9i		−	0.173858i
$385$			1.58404e10i			0.720978i
$386$		−	2.71660e10i		−	1.22370i
$387$			1.49492e8i			0.00666462i
$388$			1.64661e10i			0.726545i
$389$	−6.39640e9			−0.279343			−0.139671	−	0.990198i	$-0.544605\pi$
$389$	−6.39640e9			−0.279343			−0.139671	+	0.990198i	$0.544605\pi$
$390$	1.73579e10			0.750308
$391$		−	4.48381e10i		−	1.91841i
$392$			8.49101e7i			0.00359596i
$393$		−	5.51093e9i		−	0.231023i
$394$		−	2.72083e8i		−	0.0112906i
$395$	−2.49268e10			−1.02395
$396$			6.02745e9i			0.245105i
$397$			5.53926e9i			0.222992i	0.993765	+	0.111496i	$0.0355642\pi$
$397$			5.53926e9i			0.222992i	−0.993765	+	0.111496i	$0.964436\pi$
$398$		−	6.05989e10i		−	2.41508i
$399$	7.88345e9			0.311046
$400$	−4.55354e9			−0.177873
$401$		−	4.19487e10i		−	1.62234i	−0.584813	−	0.811168i	$-0.698832\pi$
$401$		−	4.19487e10i		−	1.62234i	0.584813	−	0.811168i	$-0.301168\pi$
$402$	2.40936e10			0.922567
$403$	−3.68207e10			−1.39596
$404$			4.16664e10i			1.56408i
$405$	3.27610e9			0.121769
$406$		−	4.51540e10i		−	1.66185i
$407$	1.41352e10			0.515140
$408$			4.46808e9i			0.161243i
$409$		−	1.16259e9i		−	0.0415462i	−0.999784	−	0.0207731i	$-0.993387\pi$
$409$		−	1.16259e9i		−	0.0415462i	0.999784	−	0.0207731i	$-0.00661276\pi$
$410$			3.50105e10i			1.23898i
$411$	−1.31121e9			−0.0459522
$412$		−	4.05471e10i		−	1.40725i
$413$	2.83811e10	−	4.56666e9i	0.975505	−	0.156963i
$414$	1.47719e10			0.502846
$415$			8.33093e9i			0.280867i
$416$	3.51268e10			1.17291
$417$	1.66070e10			0.549221
$418$	1.60781e10			0.526658
$419$			4.46299e10i			1.44800i	0.689798	+	0.724002i	$0.257699\pi$
$419$			4.46299e10i			1.44800i	−0.689798	+	0.724002i	$0.742301\pi$
$420$	−2.14836e10			−0.690414
$421$			4.03763e10i			1.28528i	0.766167	+	0.642641i	$0.222161\pi$
$421$			4.03763e10i			1.28528i	−0.766167	+	0.642641i	$0.777839\pi$
$422$	−3.37363e10			−1.06377
$423$			1.36319e10i			0.425788i
$424$		−	5.37095e8i		−	0.0166184i
$425$	1.21001e10			0.370880
$426$			1.20476e10i			0.365817i
$427$		−	1.72601e10i		−	0.519197i
$428$	−4.62456e10			−1.37815
$429$	−1.06437e10			−0.314242
$430$	−1.08670e9			−0.0317860
$431$			1.43649e10i			0.416288i	0.978098	+	0.208144i	$0.0667423\pi$
$431$			1.43649e10i			0.416288i	−0.978098	+	0.208144i	$0.933258\pi$
$432$	5.93025e9			0.170270
$433$	4.76686e9			0.135607			0.0678034	−	0.997699i	$-0.478401\pi$
$433$	4.76686e9			0.135607			0.0678034	+	0.997699i	$0.478401\pi$
$434$	8.68379e10			2.44766
$435$	−2.62680e10			−0.733619
$436$		−	1.27301e9i		−	0.0352277i
$437$		−	2.06789e10i		−	0.567024i
$438$	−1.60502e10			−0.436097
$439$	−1.66181e9			−0.0447428			−0.0223714	−	0.999750i	$-0.507122\pi$
$439$	−1.66181e9			−0.0447428			−0.0223714	+	0.999750i	$0.507122\pi$
$440$	−4.14047e9			−0.110469
$441$	−2.99469e8			−0.00791768
$442$	−8.34939e10			−2.18759
$443$		−	6.40780e9i		−	0.166377i	−0.996534	−	0.0831887i	$-0.973490\pi$
$443$		−	6.40780e9i		−	0.166377i	0.996534	−	0.0831887i	$-0.0265104\pi$
$444$			1.91710e10i			0.493301i
$445$		−	5.70855e10i		−	1.45575i
$446$		−	2.33101e10i		−	0.589121i
$447$			4.02752e9i			0.100880i
$448$	−4.76294e10			−1.18240
$449$	−5.48463e9			−0.134947			−0.0674733	−	0.997721i	$-0.521494\pi$
$449$	−5.48463e9			−0.134947			−0.0674733	+	0.997721i	$0.521494\pi$
$450$			3.98637e9i			0.0972138i
$451$		−	2.14680e10i		−	0.518903i
$452$		−	4.31944e10i		−	1.03484i
$453$		−	4.76913e10i		−	1.13252i
$454$	8.10037e10			1.90670
$455$		−	3.79372e10i		−	0.885157i
$456$			2.06063e9i			0.0476586i
$457$		−	4.57556e10i		−	1.04901i	−0.851407	−	0.524505i	$-0.824250\pi$
$457$		−	4.57556e10i		−	1.04901i	0.851407	−	0.524505i	$-0.175750\pi$
$458$	−4.52379e10			−1.02811
$459$	−1.57584e10			−0.355028
$460$			5.63531e10i			1.25860i
$461$	−4.61834e10			−1.02255			−0.511273	−	0.859419i	$-0.670826\pi$
$461$	−4.61834e10			−1.02255			−0.511273	+	0.859419i	$0.670826\pi$
$462$	2.51021e10			0.550988
$463$			6.84846e10i			1.49028i	0.666906	+	0.745142i	$0.267618\pi$
$463$			6.84846e10i			1.49028i	−0.666906	+	0.745142i	$0.732382\pi$
$464$	−4.75493e10			−1.02582
$465$		−	5.05174e10i		−	1.08051i
$466$	−5.81975e10			−1.23413
$467$		−	4.07709e10i		−	0.857200i	−0.903494	−	0.428600i	$-0.859007\pi$
$467$		−	4.07709e10i		−	0.857200i	0.903494	−	0.428600i	$-0.140993\pi$
$468$		−	1.44356e10i		−	0.300920i
$469$		−	5.26587e10i		−	1.08837i
$470$	−9.90936e10			−2.03074
$471$		−	1.09942e10i		−	0.223397i
$472$	1.19367e9	+	7.41846e9i	0.0240500	+	0.149467i
$473$	6.66353e8			0.0133125
$474$			3.95013e10i			0.782525i
$475$	5.58045e9			0.109621
$476$	1.03339e11			2.01296
$477$	1.89428e9			0.0365907
$478$			1.78969e10i			0.342820i
$479$	3.48252e10			0.661534			0.330767	−	0.943712i	$-0.392693\pi$
$479$	3.48252e10			0.661534			0.330767	+	0.943712i	$0.392693\pi$
$480$			4.81934e10i			0.907868i
$481$	−3.38535e10			−0.632446
$482$			4.57791e10i			0.848163i
$483$		−	3.22852e10i		−	0.593219i
$484$	−3.37358e10			−0.614765
$485$		−	3.98931e10i		−	0.720993i
$486$		−	5.19161e9i		−	0.0930587i
$487$	5.02901e10			0.894060			0.447030	−	0.894519i	$-0.352482\pi$
$487$	5.02901e10			0.894060			0.447030	+	0.894519i	$0.352482\pi$
$488$	4.51157e9			0.0795515
$489$	5.14452e10			0.899724
$490$		−	2.17692e9i		−	0.0377623i
$491$	5.72661e10			0.985308			0.492654	−	0.870225i	$-0.336027\pi$
$491$	5.72661e10			0.985308			0.492654	+	0.870225i	$0.336027\pi$
$492$	2.91161e10			0.496905
$493$	1.26353e11			2.13893
$494$	−3.85065e10			−0.646587
$495$		−	1.46030e10i		−	0.243232i
$496$		−	9.14443e10i		−	1.51088i
$497$	2.63311e10			0.431563
$498$	1.32020e10			0.214645
$499$	6.34009e10			1.02257			0.511285	−	0.859411i	$-0.329170\pi$
$499$	6.34009e10			1.02257			0.511285	+	0.859411i	$0.329170\pi$
$500$	6.04357e10			0.966971
$501$	−4.67912e10			−0.742700
$502$			1.15733e11i			1.82240i
$503$		−	7.08382e10i		−	1.10661i	−0.832978	−	0.553307i	$-0.813366\pi$
$503$		−	7.08382e10i		−	1.10661i	0.832978	−	0.553307i	$-0.186634\pi$
$504$			3.21719e9i			0.0498603i
$505$		−	1.00947e11i		−	1.55213i
$506$		−	6.58448e10i		−	1.00443i
$507$	−1.26566e10			−0.191551
$508$	9.91330e10			1.48855
$509$			8.20459e10i			1.22232i	0.791506	+	0.611161i	$0.209297\pi$
$509$			8.20459e10i			1.22232i	−0.791506	+	0.611161i	$0.790703\pi$
$510$		−	1.14552e11i		−	1.69326i
$511$			3.50790e10i			0.514474i
$512$			9.64420e10i			1.40342i
$513$	−7.26763e9			−0.104936
$514$			3.24093e10i			0.464320i
$515$			9.82354e10i			1.39649i
$516$			9.03745e8i			0.0127481i
$517$	6.07632e10			0.850508
$518$	7.98401e10			1.10892
$519$		−	3.56288e10i		−	0.491057i
$520$	9.91631e9			0.135624
$521$	−4.24014e10			−0.575478			−0.287739	−	0.957709i	$-0.592904\pi$
$521$	−4.24014e10			−0.575478			−0.287739	+	0.957709i	$0.592904\pi$
$522$			4.16268e10i			0.560649i
$523$	−5.41738e10			−0.724074			−0.362037	−	0.932164i	$-0.617919\pi$
$523$	−5.41738e10			−0.724074			−0.362037	+	0.932164i	$0.617919\pi$
$524$		−	3.33159e10i		−	0.441902i
$525$	8.71255e9			0.114685
$526$			2.07370e11i			2.70896i
$527$			2.42995e11i			3.15032i
$528$		−	2.64337e10i		−	0.340112i
$529$	−6.37561e9			−0.0814140
$530$			1.37700e10i			0.174514i
$531$	−2.61641e10	+	4.20994e9i	−0.329100	+	0.0529538i
$532$	4.76588e10			0.594972
$533$			5.14154e10i			0.637066i
$534$	−9.04630e10			−1.11252
$535$	1.12041e11			1.36761
$536$	1.37643e10			0.166761
$537$		−	6.55097e10i		−	0.787787i
$538$	−5.37355e10			−0.641405
$539$			1.33486e9i			0.0158155i
$540$	1.98054e10			0.232921
$541$			4.29112e10i			0.500936i	0.968125	+	0.250468i	$0.0805844\pi$
$541$			4.29112e10i			0.500936i	−0.968125	+	0.250468i	$0.919416\pi$
$542$			1.26450e11i			1.46529i
$543$	6.75148e10			0.776604
$544$		−	2.31816e11i		−	2.64697i
$545$			3.08418e9i			0.0349585i
$546$	−6.01189e10			−0.676457
$547$	1.38075e11			1.54229			0.771145	−	0.636659i	$-0.219684\pi$
$547$	1.38075e11			1.54229			0.771145	+	0.636659i	$0.219684\pi$
$548$	−7.92684e9			−0.0878978
$549$			1.59118e10i			0.175158i
$550$	1.77690e10			0.194183
$551$	5.82725e10			0.632204
$552$	8.43894e9			0.0908932
$553$	8.63335e10			0.923163
$554$			1.88950e11i			2.00589i
$555$		−	4.64465e10i		−	0.489532i
$556$	1.00396e11			1.05055
$557$	1.25469e11			1.30351			0.651757	−	0.758428i	$-0.274032\pi$
$557$	1.25469e11			1.30351			0.651757	+	0.758428i	$0.274032\pi$
$558$	−8.00545e10			−0.825751
$559$	−1.59590e9			−0.0163440
$560$	9.42174e10			0.958030
$561$			7.02422e10i			0.709164i
$562$		−	1.02873e10i		−	0.103124i
$563$		−	4.51344e10i		−	0.449236i	−0.974447	−	0.224618i	$-0.927887\pi$
$563$		−	4.51344e10i		−	0.449236i	0.974447	−	0.224618i	$-0.0721134\pi$
$564$			8.24103e10i			0.814452i
$565$			1.04649e11i			1.02693i
$566$	3.10439e10			0.302489
$567$	−1.13467e10			−0.109784
$568$			6.88262e9i			0.0661242i
$569$			1.85314e11i			1.76790i	0.467579	+	0.883951i	$0.345126\pi$
$569$			1.85314e11i			1.76790i	−0.467579	+	0.883951i	$0.654874\pi$
$570$		−	5.28304e10i		−	0.500477i
$571$		−	1.48590e11i		−	1.39780i	−0.715220	−	0.698899i	$-0.753674\pi$
$571$		−	1.48590e11i		−	1.39780i	0.715220	−	0.698899i	$-0.246326\pi$
$572$	−6.43457e10			−0.601084
$573$		−	3.46138e10i		−	0.321093i
$574$		−	1.21258e11i		−	1.11703i
$575$		−	2.28537e10i		−	0.209067i
$576$	4.39088e10			0.398898
$577$	−1.06355e11			−0.959519			−0.479760	−	0.877400i	$-0.659276\pi$
$577$	−1.06355e11			−0.959519			−0.479760	+	0.877400i	$0.659276\pi$
$578$			3.89102e11i			3.48620i
$579$	5.47355e10			0.487029
$580$	−1.58801e11			−1.40327
$581$		−	2.88540e10i		−	0.253222i
$582$	−6.32184e10			−0.551000
$583$		−	8.44363e9i		−	0.0730895i
$584$	−9.16920e9			−0.0788279
$585$			3.49738e10i			0.298620i
$586$			2.14748e11i			1.82112i
$587$		−	1.42209e11i		−	1.19777i	−0.800833	−	0.598887i	$-0.795610\pi$
$587$		−	1.42209e11i		−	1.19777i	0.800833	−	0.598887i	$-0.204390\pi$
$588$	−1.81042e9			−0.0151450
$589$			1.12067e11i			0.931142i
$590$	−3.06031e10	−	1.90194e11i	−0.252556	−	1.56960i
$591$	5.48209e8			0.00449362
$592$		−	8.40754e10i		−	0.684513i
$593$	−1.71645e11			−1.38808			−0.694038	−	0.719938i	$-0.744170\pi$
$593$	−1.71645e11			−1.38808			−0.694038	+	0.719938i	$0.744170\pi$
$594$	−2.31413e10			−0.185884
$595$	−2.50364e11			−1.99758
$596$			2.43480e10i			0.192965i
$597$	1.22098e11			0.961195
$598$			1.57696e11i			1.23315i
$599$	−3.93918e10			−0.305984			−0.152992	−	0.988227i	$-0.548891\pi$
$599$	−3.93918e10			−0.305984			−0.152992	+	0.988227i	$0.548891\pi$
$600$			2.27735e9i			0.0175721i
$601$		−	1.02104e11i		−	0.782606i	−0.920262	−	0.391303i	$-0.872025\pi$
$601$		−	1.02104e11i		−	0.782606i	0.920262	−	0.391303i	$-0.127975\pi$
$602$	3.76377e9			0.0286574
$603$			4.85452e10i			0.367179i
$604$		−	2.88314e11i		−	2.16630i
$605$	8.17333e10			0.610067
$606$	−1.59970e11			−1.18617
$607$	1.71502e11			1.26332			0.631662	−	0.775244i	$-0.282373\pi$
$607$	1.71502e11			1.26332			0.631662	+	0.775244i	$0.282373\pi$
$608$		−	1.06911e11i		−	0.782365i
$609$	9.09789e10			0.661411
$610$	−1.15667e11			−0.835394
$611$	−1.45526e11			−1.04418
$612$	−9.52664e10			−0.679101
$613$		−	1.49268e11i		−	1.05712i	−0.848896	−	0.528560i	$-0.822732\pi$
$613$		−	1.49268e11i		−	1.05712i	0.848896	−	0.528560i	$-0.177268\pi$
$614$		−	2.98654e11i		−	2.10134i
$615$	−7.05411e10			−0.493108
$616$	1.43404e10			0.0995954
$617$	−1.42930e11			−0.986238			−0.493119	−	0.869962i	$-0.664143\pi$
$617$	−1.42930e11			−0.986238			−0.493119	+	0.869962i	$0.664143\pi$
$618$	1.55673e11			1.06723
$619$	1.45330e11			0.989906			0.494953	−	0.868920i	$-0.335185\pi$
$619$	1.45330e11			0.989906			0.494953	+	0.868920i	$0.335185\pi$
$620$		−	3.05399e11i		−	2.06681i
$621$			2.97633e10i			0.200131i
$622$		−	1.58378e11i		−	1.05812i
$623$			1.97715e11i			1.31246i
$624$			6.33080e10i			0.417561i
$625$	−1.77097e11			−1.16062
$626$	−1.94310e10			−0.126531
$627$			3.23950e10i			0.209608i
$628$		−	6.64643e10i		−	0.427317i
$629$			2.23413e11i			1.42727i
$630$		−	8.24822e10i		−	0.523598i
$631$	1.41515e10			0.0892660			0.0446330	−	0.999003i	$-0.485788\pi$
$631$	1.41515e10			0.0892660			0.0446330	+	0.999003i	$0.485788\pi$
$632$			2.25664e10i			0.141447i
$633$		−	6.79738e10i		−	0.423377i
$634$			2.85532e11i			1.76725i
$635$	−2.40174e11			−1.47717
$636$	1.14517e10			0.0699910
$637$		−	3.19697e9i		−	0.0194169i
$638$	1.85549e11			1.11989
$639$	−2.42743e10			−0.145594
$640$			5.53674e10i			0.330015i
$641$	2.61465e11			1.54875			0.774375	−	0.632727i	$-0.218064\pi$
$641$	2.61465e11			1.54875			0.774375	+	0.632727i	$0.218064\pi$
$642$		−	1.77551e11i		−	1.04516i
$643$	2.87732e10			0.168323			0.0841617	−	0.996452i	$-0.473179\pi$
$643$	2.87732e10			0.168323			0.0841617	+	0.996452i	$0.473179\pi$
$644$		−	1.95178e11i		−	1.13471i
$645$		−	2.18955e9i		−	0.0126507i
$646$			2.54121e11i			1.45918i
$647$	−9.55469e10			−0.545255			−0.272628	−	0.962120i	$-0.587893\pi$
$647$	−9.55469e10			−0.545255			−0.272628	+	0.962120i	$0.587893\pi$
$648$		−	2.96588e9i		−	0.0168211i
$649$	1.87655e10	+	1.16625e11i	0.105775	+	0.657374i
$650$	−4.25563e10			−0.238402
$651$			1.74966e11i			0.974158i
$652$	3.11008e11			1.72100
$653$	−2.11634e11			−1.16394			−0.581972	−	0.813208i	$-0.697719\pi$
$653$	−2.11634e11			−1.16394			−0.581972	+	0.813208i	$0.697719\pi$
$654$	4.88747e9			0.0267161
$655$			8.07160e10i			0.438525i
$656$	−1.27690e11			−0.689514
$657$		−	3.23388e10i		−	0.173565i
$658$	3.43209e11			1.83086
$659$			2.41201e11i			1.27890i	0.768832	+	0.639451i	$0.220838\pi$
$659$			2.41201e11i			1.27890i	−0.768832	+	0.639451i	$0.779162\pi$
$660$		−	8.82812e10i		−	0.465257i
$661$	−1.17302e11			−0.614468			−0.307234	−	0.951634i	$-0.599403\pi$
$661$	−1.17302e11			−0.614468			−0.307234	+	0.951634i	$0.599403\pi$
$662$		−	3.28120e11i		−	1.70844i
$663$		−	1.68228e11i		−	0.870653i
$664$	7.54206e9			0.0387988
$665$	−1.15465e11			−0.590425
$666$	−7.36034e10			−0.374111
$667$		−	2.38644e11i		−	1.20572i
$668$	−2.82873e11			−1.42064
$669$	4.69665e10			0.234468
$670$	−3.52888e11			−1.75121
$671$	7.09260e10			0.349877
$672$		−	1.66917e11i		−	0.818509i
$673$		−	2.43490e11i		−	1.18692i	−0.804864	−	0.593459i	$-0.797762\pi$
$673$		−	2.43490e11i		−	1.18692i	0.804864	−	0.593459i	$-0.202238\pi$
$674$	−3.72658e11			−1.80580
$675$	−8.03197e9			−0.0386908
$676$	−7.65142e10			−0.366400
$677$	3.07432e11			1.46350			0.731752	−	0.681571i	$-0.238703\pi$
$677$	3.07432e11			1.46350			0.731752	+	0.681571i	$0.238703\pi$
$678$	1.65837e11			0.784805
$679$			1.38169e11i			0.650027i
$680$		−	6.54418e10i		−	0.306069i
$681$			1.63211e11i			0.758859i
$682$			3.56838e11i			1.64943i
$683$		−	3.50090e11i		−	1.60878i	−0.594101	−	0.804390i	$-0.702492\pi$
$683$		−	3.50090e11i		−	1.60878i	0.594101	−	0.804390i	$-0.297508\pi$
$684$	−4.39359e10			−0.200722
$685$	1.92047e10			0.0872261
$686$			3.24961e11i			1.46735i
$687$		−	9.11480e10i		−	0.409185i
$688$		−	3.96342e9i		−	0.0176895i
$689$			2.02223e10i			0.0897331i
$690$	−2.16357e11			−0.954497
$691$		−	2.89905e11i		−	1.27158i	−0.771862	−	0.635790i	$-0.780675\pi$
$691$		−	2.89905e11i		−	1.27158i	0.771862	−	0.635790i	$-0.219325\pi$
$692$		−	2.15391e11i		−	0.939298i
$693$			5.05772e10i			0.219291i
$694$	5.73778e11			2.47347
$695$	−2.43235e11			−1.04253
$696$			2.37807e10i			0.101342i
$697$	3.39312e11			1.43770
$698$	−4.20853e11			−1.77300
$699$		−	1.17260e11i		−	0.491179i
$700$	5.26710e10			0.219371
$701$			2.81948e11i			1.16761i	0.811895	+	0.583804i	$0.198436\pi$
$701$			2.81948e11i			1.16761i	−0.811895	+	0.583804i	$0.801564\pi$
$702$	5.54227e10			0.228212
$703$			1.03036e11i			0.421859i
$704$		−	1.95721e11i		−	0.796794i
$705$		−	1.99660e11i		−	0.808228i
$706$	−5.72040e11			−2.30254
$707$			3.49628e11i			1.39936i
$708$	−1.58173e11	+	2.54508e10i	−0.629506	+	0.101291i
$709$	−3.58537e11			−1.41889			−0.709446	−	0.704760i	$-0.751055\pi$
$709$	−3.58537e11			−1.41889			−0.709446	+	0.704760i	$0.751055\pi$
$710$		−	1.76456e11i		−	0.694390i
$711$	−7.95895e10			−0.311442
$712$	−5.16800e10			−0.201096
$713$	4.58949e11			1.77585
$714$			3.96750e11i			1.52659i
$715$	1.55893e11			0.596491
$716$		−	3.96034e11i		−	1.50689i
$717$	−3.60597e10			−0.136441
$718$		−	5.27047e11i		−	1.98313i
$719$		−	4.59130e11i		−	1.71799i	−0.511985	−	0.858995i	$-0.671090\pi$
$719$		−	4.59130e11i		−	1.71799i	0.511985	−	0.858995i	$-0.328910\pi$
$720$	−8.68576e10			−0.323205
$721$		−	3.40236e11i		−	1.25904i
$722$		−	2.76995e11i		−	1.01935i
$723$	−9.22384e10			−0.337566
$724$	4.08155e11			1.48549
$725$	6.44011e10			0.233099
$726$		−	1.29522e11i		−	0.466228i
$727$	−1.26631e11			−0.453319			−0.226660	−	0.973974i	$-0.572780\pi$
$727$	−1.26631e11			−0.453319			−0.226660	+	0.973974i	$0.572780\pi$
$728$	−3.43449e10			−0.122275
$729$	1.04604e10			0.0370370
$730$	2.35079e11			0.827795
$731$			1.05320e10i			0.0368843i
$732$			9.61937e10i			0.335044i
$733$	3.82085e11			1.32356			0.661781	−	0.749697i	$-0.269801\pi$
$733$	3.82085e11			1.32356			0.661781	+	0.749697i	$0.269801\pi$
$734$	3.03729e11			1.04641
$735$	4.38619e9			0.0150293
$736$	−4.37836e11			−1.49211
$737$	2.16387e11			0.733435
$738$			1.11786e11i			0.376844i
$739$			2.11704e11i			0.709826i	0.934899	+	0.354913i	$0.115489\pi$
$739$			2.11704e11i			0.709826i	−0.934899	+	0.354913i	$0.884511\pi$
$740$		−	2.80788e11i		−	0.936380i
$741$		−	7.75852e10i		−	0.257339i
$742$		−	4.76922e10i		−	0.157337i
$743$	−1.54847e11			−0.508099			−0.254049	−	0.967191i	$-0.581763\pi$
$743$	−1.54847e11			−0.508099			−0.254049	+	0.967191i	$0.581763\pi$
$744$	−4.57338e10			−0.149261
$745$		−	5.89892e10i		−	0.191490i
$746$			4.11936e11i			1.33007i
$747$			2.66001e10i			0.0854280i
$748$			4.24644e11i			1.35650i
$749$	−3.88053e11			−1.23300
$750$			2.32032e11i			0.733334i
$751$			2.66071e11i			0.836446i	0.908344	+	0.418223i	$0.137347\pi$
$751$			2.66071e11i			0.836446i	−0.908344	+	0.418223i	$0.862653\pi$
$752$		−	3.61415e11i		−	1.13015i
$753$	−2.33186e11			−0.725309
$754$	−4.44384e11			−1.37491
$755$			6.98513e11i			2.14974i
$756$	−6.85956e10			−0.209995
$757$	−7.24705e10			−0.220688			−0.110344	−	0.993893i	$-0.535195\pi$
$757$	−7.24705e10			−0.220688			−0.110344	+	0.993893i	$0.535195\pi$
$758$			2.77252e11i			0.839843i
$759$	1.32668e11			0.399759
$760$		−	3.01811e10i		−	0.0904651i
$761$	−4.38033e11			−1.30607			−0.653037	−	0.757326i	$-0.726505\pi$
$761$	−4.38033e11			−1.30607			−0.653037	+	0.757326i	$0.726505\pi$
$762$			3.80603e11i			1.12889i
$763$		−	1.06820e10i		−	0.0315176i
$764$		−	2.09255e11i		−	0.614190i
$765$	2.30807e11			0.673911
$766$			6.55298e9i			0.0190337i
$767$	−4.49429e10	−	2.79314e11i	−0.129861	−	0.807069i
$768$	−1.52622e11			−0.438706
$769$			3.09691e11i			0.885570i	0.896628	+	0.442785i	$0.146010\pi$
$769$			3.09691e11i			0.885570i	−0.896628	+	0.442785i	$0.853990\pi$
$770$	−3.67659e11			−1.04588
$771$	−6.53001e10			−0.184798
$772$	3.30899e11			0.931594
$773$			1.94542e11i			0.544873i	0.962174	+	0.272437i	$0.0878295\pi$
$773$			1.94542e11i			0.544873i	−0.962174	+	0.272437i	$0.912170\pi$
$774$	−3.46976e9			−0.00966798
$775$			1.23853e11i			0.343320i
$776$	−3.61156e10			−0.0995974
$777$			1.60866e11i			0.441348i
$778$		−	1.48462e11i		−	0.405226i
$779$	1.56487e11			0.424941
$780$			2.11431e11i			0.571203i
$781$			1.08201e11i			0.290822i
$782$	1.04070e12			2.78292
$783$	−8.38720e10			−0.223136
$784$	7.93968e9			0.0210155
$785$			1.61026e11i			0.424051i
$786$	1.27910e11			0.335131
$787$	9.22644e9			0.0240511			0.0120256	−	0.999928i	$-0.496172\pi$
$787$	9.22644e9			0.0240511			0.0120256	+	0.999928i	$0.496172\pi$
$788$	3.31415e9			0.00859543
$789$	−4.17821e11			−1.07816
$790$		−	5.78557e11i		−	1.48538i
$791$		−	3.62450e11i		−	0.925854i
$792$	−1.32202e10			−0.0335999
$793$	−1.69866e11			−0.429550
$794$	−1.28568e11			−0.323482
$795$	−2.77446e10			−0.0694561
$796$	7.38134e11			1.83858
$797$			3.96123e11i			0.981742i	0.871232	+	0.490871i	$0.163321\pi$
$797$			3.96123e11i			0.981742i	−0.871232	+	0.490871i	$0.836679\pi$
$798$			1.82977e11i			0.451216i
$799$			9.60388e11i			2.35646i
$800$		−	1.18155e11i		−	0.288465i
$801$		−	1.82270e11i		−	0.442777i
$802$	9.73640e11			2.35343
$803$	−1.44148e11			−0.346694
$804$			2.93476e11i			0.702342i
$805$			4.72867e11i			1.12604i
$806$		−	8.54617e11i		−	2.02503i
$807$		−	1.08269e11i		−	0.255277i
$808$	−9.13883e10			−0.214410
$809$		−	2.46882e11i		−	0.576362i	−0.957576	−	0.288181i	$-0.906950\pi$
$809$		−	2.46882e11i		−	0.576362i	0.957576	−	0.288181i	$-0.0930504\pi$
$810$			7.60391e10i			0.176643i
$811$		−	2.13474e11i		−	0.493471i	−0.969083	−	0.246736i	$-0.920642\pi$
$811$		−	2.13474e11i		−	0.493471i	0.969083	−	0.246736i	$-0.0793580\pi$
$812$	5.50005e11			1.26515
$813$	−2.54780e11			−0.583180
$814$			3.28082e11i			0.747283i
$815$	−7.53493e11			−1.70785
$816$	4.17796e11			0.942331
$817$			4.85725e9i			0.0109019i
$818$	2.69839e10			0.0602687
$819$		−	1.21131e11i		−	0.269228i
$820$	−4.26451e11			−0.943221
$821$		−	3.13176e11i		−	0.689313i	−0.938729	−	0.344656i	$-0.887995\pi$
$821$		−	3.13176e11i		−	0.689313i	0.938729	−	0.344656i	$-0.112005\pi$
$822$		−	3.04336e10i		−	0.0666602i
$823$		−	2.45684e11i		−	0.535522i	−0.963485	−	0.267761i	$-0.913716\pi$
$823$		−	2.45684e11i		−	0.535522i	0.963485	−	0.267761i	$-0.0862837\pi$
$824$	8.89333e10			0.192910
$825$			3.58020e10i			0.0772844i
$826$	1.05993e11	+	6.58733e11i	0.227698	+	1.41511i
$827$	4.47802e11			0.957334			0.478667	−	0.877996i	$-0.341120\pi$
$827$	4.47802e11			0.957334			0.478667	+	0.877996i	$0.341120\pi$
$828$			1.79931e11i			0.382812i
$829$	3.36573e11			0.712626			0.356313	−	0.934367i	$-0.384034\pi$
$829$	3.36573e11			0.712626			0.356313	+	0.934367i	$0.384034\pi$
$830$	−1.93363e11			−0.407437
$831$	−3.80708e11			−0.798339
$832$			4.68746e11i			0.978237i
$833$	−2.10981e10			−0.0438191
$834$			3.85453e11i			0.796723i
$835$	6.85330e11			1.40979
$836$			1.95841e11i			0.400940i
$837$		−	1.61298e11i		−	0.328646i
$838$	−1.03587e12			−2.10054
$839$			9.76890e10i			0.197151i	0.995130	+	0.0985753i	$0.0314285\pi$
$839$			9.76890e10i			0.197151i	−0.995130	+	0.0985753i	$0.968571\pi$
$840$		−	4.71207e10i		−	0.0946443i
$841$	1.72247e11			0.344325
$842$	−9.37145e11			−1.86448
$843$	2.07275e10			0.0410428
$844$		−	4.10930e11i		−	0.809838i
$845$	1.85375e11			0.363600
$846$	−3.16399e11			−0.617666
$847$	−2.83082e11			−0.550020
$848$	−5.02221e10			−0.0971206
$849$			6.25490e10i			0.120390i
$850$			2.80847e11i			0.538014i
$851$	4.21965e11			0.804559
$852$	−1.46748e11			−0.278493
$853$	−9.65153e11			−1.82306			−0.911528	−	0.411239i	$-0.865096\pi$
$853$	−9.65153e11			−1.82306			−0.911528	+	0.411239i	$0.865096\pi$
$854$	4.00612e11			0.753168
$855$	1.06446e11			0.199188
$856$		−	1.01432e11i		−	0.188921i
$857$			3.01239e10i			0.0558455i	0.999610	+	0.0279228i	$0.00888925\pi$
$857$			3.01239e10i			0.0558455i	−0.999610	+	0.0279228i	$0.991111\pi$
$858$		−	2.47043e11i		−	0.455852i
$859$		−	5.06865e11i		−	0.930935i	−0.885065	−	0.465468i	$-0.845886\pi$
$859$		−	5.06865e11i		−	0.930935i	0.885065	−	0.465468i	$-0.154114\pi$
$860$		−	1.32367e10i		−	0.0241984i
$861$	2.44318e11			0.444572
$862$	−3.33414e11			−0.603885
$863$		−	4.76954e11i		−	0.859871i	−0.902860	−	0.429935i	$-0.858536\pi$
$863$		−	4.76954e11i		−	0.859871i	0.902860	−	0.429935i	$-0.141464\pi$
$864$			1.53878e11i			0.276136i
$865$			5.21838e11i			0.932120i
$866$			1.10640e11i			0.196717i
$867$	−7.83985e11			−1.38749
$868$			1.05774e12i			1.86338i
$869$			3.54765e11i			0.622102i
$870$		−	6.09688e11i		−	1.06422i
$871$	−5.18242e11			−0.900450
$872$	2.79213e9			0.00482914
$873$		−	1.27376e11i		−	0.219296i
$874$	4.79962e11			0.822549
$875$	5.07125e11			0.865132
$876$		−	1.95502e11i		−	0.331997i
$877$	1.61951e11			0.273770			0.136885	−	0.990587i	$-0.456291\pi$
$877$	1.61951e11			0.273770			0.136885	+	0.990587i	$0.456291\pi$
$878$		−	3.85710e10i		−	0.0649058i
$879$	−4.32688e11			−0.724801
$880$			3.87162e11i			0.645598i
$881$		−	1.54146e11i		−	0.255876i	−0.991782	−	0.127938i	$-0.959164\pi$
$881$		−	1.54146e11i		−	0.255876i	0.991782	−	0.127938i	$-0.0408358\pi$
$882$		−	6.95076e9i		−	0.0114857i
$883$	7.08028e10			0.116468			0.0582341	−	0.998303i	$-0.481453\pi$
$883$	7.08028e10			0.116468			0.0582341	+	0.998303i	$0.481453\pi$
$884$		−	1.01701e12i		−	1.66539i
$885$	3.83214e11	−	6.16610e10i	0.624695	−	0.100516i
$886$	1.48727e11			0.241354
$887$			1.15504e12i			1.86595i	0.359937	+	0.932976i	$0.382798\pi$
$887$			1.15504e12i			1.86595i	−0.359937	+	0.932976i	$0.617202\pi$
$888$	−4.20484e10			−0.0676235
$889$	8.31839e11			1.33178
$890$	1.32497e12			2.11177
$891$		−	4.66264e10i		−	0.0739811i
$892$	2.83932e11			0.448492
$893$			4.42921e11i			0.696499i
$894$	−9.34797e10			−0.146341
$895$			9.59491e11i			1.49537i
$896$		−	1.91764e11i		−	0.297533i
$897$	−3.17736e11			−0.490791
$898$		−	1.27300e11i		−	0.195759i
$899$			1.29331e12i			1.97999i
$900$	−4.85566e10			−0.0740080
$901$	1.33455e11			0.202505
$902$	4.98279e11			0.752743
$903$			7.58345e9i			0.0114055i
$904$	9.47398e10			0.141860
$905$	−9.88857e11			−1.47414
$906$	1.10693e12			1.64288
$907$	1.00066e12			1.47862			0.739310	−	0.673365i	$-0.235152\pi$
$907$	1.00066e12			1.47862			0.739310	+	0.673365i	$0.235152\pi$
$908$			9.86679e11i			1.45155i
$909$		−	3.22317e11i		−	0.472093i
$910$	8.80534e11			1.28405
$911$	−2.56623e11			−0.372582			−0.186291	−	0.982495i	$-0.559647\pi$
$911$	−2.56623e11			−0.372582			−0.186291	+	0.982495i	$0.559647\pi$
$912$	1.92683e11			0.278525
$913$	1.18568e11			0.170642
$914$	1.06200e12			1.52174
$915$		−	2.33053e11i		−	0.332484i
$916$		−	5.51028e11i		−	0.782693i
$917$		−	2.79558e11i		−	0.395362i
$918$		−	3.65758e11i		−	0.515018i
$919$		−	1.17774e12i		−	1.65115i	−0.564291	−	0.825576i	$-0.690850\pi$
$919$		−	1.17774e12i		−	1.65115i	0.564291	−	0.825576i	$-0.309150\pi$
$920$	−1.23601e11			−0.172533
$921$	6.01746e11			0.836325
$922$		−	1.07193e12i		−	1.48335i
$923$		−	2.59139e11i		−	0.357047i
$924$			3.05760e11i			0.419463i
$925$			1.13872e11i			0.155543i
$926$	−1.58955e12			−2.16187
$927$			3.13659e11i			0.424755i
$928$		−	1.23381e12i		−	1.66363i
$929$		−	7.90276e11i		−	1.06100i	−0.847684	−	0.530501i	$-0.822004\pi$
$929$		−	7.90276e11i		−	1.06100i	0.847684	−	0.530501i	$-0.177996\pi$
$930$	1.17252e12			1.56743
$931$	−9.73024e9			−0.0129516
$932$		−	7.08884e11i		−	0.939532i
$933$	3.19109e11			0.421127
$934$	9.46302e11			1.24349
$935$		−	1.02881e12i		−	1.34613i
$936$	3.16621e10			0.0412511
$937$		−	1.08351e12i		−	1.40565i	−0.711364	−	0.702824i	$-0.751922\pi$
$937$		−	1.08351e12i		−	1.40565i	0.711364	−	0.702824i	$-0.248078\pi$
$938$	1.22222e12			1.57884
$939$		−	3.91507e10i		−	0.0503590i
$940$		−	1.20703e12i		−	1.54598i
$941$		−	7.40783e11i		−	0.944784i	−0.881389	−	0.472392i	$-0.843391\pi$
$941$		−	7.40783e11i		−	0.944784i	0.881389	−	0.472392i	$-0.156609\pi$
$942$	2.55177e11			0.324070
$943$		−	6.40864e11i		−	0.810437i
$944$	6.93677e11	−	1.11616e11i	0.873513	−	0.140552i
$945$	1.66190e11			0.208390
$946$			1.54662e10i			0.0193117i
$947$	−1.85011e11			−0.230038			−0.115019	−	0.993363i	$-0.536693\pi$
$947$	−1.85011e11			−0.230038			−0.115019	+	0.993363i	$0.536693\pi$
$948$	−4.81152e11			−0.595729
$949$	3.45231e11			0.425642
$950$			1.29524e11i			0.159021i
$951$	−5.75305e11			−0.703358
$952$			2.26657e11i			0.275944i
$953$	−8.02904e11			−0.973402			−0.486701	−	0.873569i	$-0.661800\pi$
$953$	−8.02904e11			−0.973402			−0.486701	+	0.873569i	$0.661800\pi$
$954$			4.39667e10i			0.0530800i
$955$			5.06973e11i			0.609496i
$956$	−2.17996e11			−0.260986
$957$			3.73854e11i			0.445712i
$958$			8.08303e11i			0.959648i
$959$	−6.65153e10			−0.0786406
$960$	−6.43112e11			−0.757184
$961$	−1.63433e12			−1.91622
$962$		−	7.85748e11i		−	0.917452i
$963$	3.57740e11			0.415971
$964$	−5.57620e11			−0.645699
$965$	−8.01686e11			−0.924475
$966$	7.49348e11			0.860548
$967$		−	9.15048e11i		−	1.04650i	−0.852180	−	0.523248i	$-0.824720\pi$
$967$		−	9.15048e11i		−	1.04650i	0.852180	−	0.523248i	$-0.175280\pi$
$968$		−	7.39939e10i		−	0.0842742i
$969$	−5.12017e11			−0.580750
$970$	9.25930e11			1.04590
$971$	1.31909e12			1.48387			0.741936	−	0.670471i	$-0.233908\pi$
$971$	1.31909e12			1.48387			0.741936	+	0.670471i	$0.233908\pi$
$972$	6.32372e10			0.0708448
$973$	8.42440e11			0.939913
$974$			1.16725e12i			1.29696i
$975$		−	8.57448e10i		−	0.0948833i
$976$		−	4.21863e11i		−	0.464913i
$977$		−	3.30067e11i		−	0.362263i	−0.983459	−	0.181131i	$-0.942024\pi$
$977$		−	3.30067e11i		−	0.362263i	0.983459	−	0.181131i	$-0.0579759\pi$
$978$			1.19406e12i			1.30518i
$979$	−8.12457e11			−0.884443
$980$	2.65163e10			0.0287481
$981$			9.84756e9i			0.0106329i
$982$			1.32916e12i			1.42933i
$983$		−	8.75999e11i		−	0.938188i	−0.883148	−	0.469094i	$-0.844581\pi$
$983$		−	8.75999e11i		−	0.938188i	0.883148	−	0.469094i	$-0.155419\pi$
$984$			6.38615e10i			0.0681175i
$985$	−8.02936e9			−0.00852975
$986$			2.93268e12i			3.10282i
$987$			6.91517e11i			0.728676i
$988$		−	4.69035e11i		−	0.492241i
$989$	1.98920e10			0.0207918
$990$	3.38939e11			0.352843
$991$			2.68895e11i			0.278797i	0.990236	+	0.139399i	$0.0445170\pi$
$991$			2.68895e11i			0.278797i	−0.990236	+	0.139399i	$0.955483\pi$
$992$	2.37280e12			2.45027
$993$	6.61115e11			0.679955
$994$			6.11153e11i			0.626043i
$995$	−1.78831e12			−1.82453
$996$			1.60809e11i			0.163407i
$997$	1.25980e12			1.27503			0.637516	−	0.770437i	$-0.279962\pi$
$997$	1.25980e12			1.27503			0.637516	+	0.770437i	$0.279962\pi$
$998$			1.47155e12i			1.48338i
$999$		−	1.48300e11i		−	0.148895i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.9.c.a.58.67	yes	80
59.58	odd	2	inner	177.9.c.a.58.14	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.9.c.a.58.14	✓	80	59.58	odd	2	inner
177.9.c.a.58.67	yes	80	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	177.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+23.2103i q^{2} -46.7654 q^{3} -282.716 q^{4} +684.951 q^{5} -1085.44i q^{6} -2372.31 q^{7} -620.092i q^{8} +2187.00 q^{9} +O(q^{10})\)	\(q+23.2103i q^{2} -46.7654 q^{3} -282.716 q^{4} +684.951 q^{5} -1085.44i q^{6} -2372.31 q^{7} -620.092i q^{8} +2187.00 q^{9} +15897.9i q^{10} -9748.41i q^{11} +13221.3 q^{12} +23347.2i q^{13} -55062.0i q^{14} -32032.0 q^{15} -57982.9 q^{16} +154078. q^{17} +50760.8i q^{18} +71059.1 q^{19} -193647. q^{20} +110942. q^{21} +226263. q^{22} -291010. i q^{23} +28998.8i q^{24} +78532.4 q^{25} -541894. q^{26} -102276. q^{27} +670692. q^{28} +820057. q^{29} -743471. i q^{30} +1.57709e6i q^{31} -1.50454e6i q^{32} +455888. i q^{33} +3.57619e6i q^{34} -1.62492e6 q^{35} -618301. q^{36} +1.45000e6i q^{37} +1.64930e6i q^{38} -1.09184e6i q^{39} -424732. i q^{40} +2.20221e6 q^{41} +2.57500e6i q^{42} +68355.0i q^{43} +2.75603e6i q^{44} +1.49799e6 q^{45} +6.75441e6 q^{46} +6.23313e6i q^{47} +2.71159e6 q^{48} -136932. q^{49} +1.82276e6i q^{50} -7.20551e6 q^{51} -6.60063e6i q^{52} +866154. q^{53} -2.37385e6i q^{54} -6.67718e6i q^{55} +1.47105e6i q^{56} -3.32311e6 q^{57} +1.90337e7i q^{58} +(-1.19635e7 + 1.92498e6i) q^{59} +9.05596e6 q^{60} +7.27564e6i q^{61} -3.66047e7 q^{62} -5.18825e6 q^{63} +2.00772e7 q^{64} +1.59917e7i q^{65} -1.05813e7 q^{66} +2.21972e7i q^{67} -4.35603e7 q^{68} +1.36092e7i q^{69} -3.77148e7i q^{70} -1.10994e7 q^{71} -1.35614e6i q^{72} -1.47868e7i q^{73} -3.36550e7 q^{74} -3.67260e6 q^{75} -2.00896e7 q^{76} +2.31263e7i q^{77} +2.53419e7 q^{78} -3.63921e7 q^{79} -3.97154e7 q^{80} +4.78297e6 q^{81} +5.11139e7i q^{82} +1.21628e7i q^{83} -3.13651e7 q^{84} +1.05536e8 q^{85} -1.58654e6 q^{86} -3.83503e7 q^{87} -6.04491e6 q^{88} -8.33425e7i q^{89} +3.47687e7i q^{90} -5.53868e7i q^{91} +8.22732e7i q^{92} -7.37533e7i q^{93} -1.44673e8 q^{94} +4.86720e7 q^{95} +7.03604e7i q^{96} -5.82423e7i q^{97} -3.17822e6i q^{98} -2.13198e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 10240 q^{4} + 160 q^{7} + 174960 q^{9}+O(q^{10}) \) 80 * q - 10240 * q^4 + 160 * q^7 + 174960 * q^9	\( 80 q - 10240 q^{4} + 160 q^{7} + 174960 q^{9} - 22680 q^{12} - 59616 q^{15} + 1199848 q^{16} - 10608 q^{17} - 27516 q^{19} - 146436 q^{20} - 974696 q^{22} + 5718040 q^{25} - 797484 q^{26} - 3133000 q^{28} + 1725924 q^{29} + 4318800 q^{35} - 22394880 q^{36} - 732180 q^{41} + 22752084 q^{46} + 8703936 q^{48} + 55899176 q^{49} - 10373832 q^{51} - 39265944 q^{53} - 11408040 q^{57} - 33575112 q^{59} - 18034488 q^{60} + 13038600 q^{62} + 349920 q^{63} - 241654260 q^{64} - 35711928 q^{66} + 36772608 q^{68} - 235272660 q^{71} - 63050712 q^{74} + 74363184 q^{75} + 9454680 q^{76} - 10865988 q^{78} + 17252580 q^{79} + 318203976 q^{80} + 382637520 q^{81} - 20743128 q^{84} - 27245820 q^{85} + 105666984 q^{86} + 29437992 q^{87} + 82079788 q^{88} + 121215992 q^{94} - 690837276 q^{95}+O(q^{100}) \) 80 * q - 10240 * q^4 + 160 * q^7 + 174960 * q^9 - 22680 * q^12 - 59616 * q^15 + 1199848 * q^16 - 10608 * q^17 - 27516 * q^19 - 146436 * q^20 - 974696 * q^22 + 5718040 * q^25 - 797484 * q^26 - 3133000 * q^28 + 1725924 * q^29 + 4318800 * q^35 - 22394880 * q^36 - 732180 * q^41 + 22752084 * q^46 + 8703936 * q^48 + 55899176 * q^49 - 10373832 * q^51 - 39265944 * q^53 - 11408040 * q^57 - 33575112 * q^59 - 18034488 * q^60 + 13038600 * q^62 + 349920 * q^63 - 241654260 * q^64 - 35711928 * q^66 + 36772608 * q^68 - 235272660 * q^71 - 63050712 * q^74 + 74363184 * q^75 + 9454680 * q^76 - 10865988 * q^78 + 17252580 * q^79 + 318203976 * q^80 + 382637520 * q^81 - 20743128 * q^84 - 27245820 * q^85 + 105666984 * q^86 + 29437992 * q^87 + 82079788 * q^88 + 121215992 * q^94 - 690837276 * q^95

Introduction

L-functions

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