LMFDB - Embedded newform 31.7.b.c.30.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [31,7,Mod(30,31)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("31.30"); S:= CuspForms(chi, 7); N := Newforms(S);

Level:	$ N $	$=$	$ 31 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	31.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$7.13167659222$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 7208 x^{10} + 19859688 x^{8} + 26566749360 x^{6} + 17884354852944 x^{4} + \cdots + 59\!\cdots\!00 $ x^12 + 7208x^10 + 19859688x^8 + 26566749360x^6 + 17884354852944x^4 + 5570285336959680*x^2 + 590986232936064000
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2}\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			30.2
Root			$-30.2629i$ of defining polynomial
Character	$\chi$	$=$	31.30
Dual form			31.7.b.c.30.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-12.4307 q^{2} +30.2629i q^{3} +90.5217 q^{4} +96.4253 q^{5} -376.188i q^{6} +161.968 q^{7} -329.682 q^{8} -186.843 q^{9} -1198.63 q^{10} +701.001i q^{11} +2739.45i q^{12} +3277.00i q^{13} -2013.37 q^{14} +2918.11i q^{15} -1695.21 q^{16} -2154.64i q^{17} +2322.59 q^{18} -3211.85 q^{19} +8728.58 q^{20} +4901.61i q^{21} -8713.91i q^{22} -3478.24i q^{23} -9977.14i q^{24} -6327.15 q^{25} -40735.3i q^{26} +16407.2i q^{27} +14661.6 q^{28} -3941.36i q^{29} -36274.1i q^{30} +(14473.5 + 26038.8i) q^{31} +42172.3 q^{32} -21214.3 q^{33} +26783.7i q^{34} +15617.8 q^{35} -16913.4 q^{36} +81754.0i q^{37} +39925.4 q^{38} -99171.5 q^{39} -31789.7 q^{40} -72810.2 q^{41} -60930.4i q^{42} -42043.8i q^{43} +63455.8i q^{44} -18016.4 q^{45} +43236.8i q^{46} +154153. q^{47} -51302.1i q^{48} -91415.4 q^{49} +78650.8 q^{50} +65205.7 q^{51} +296640. i q^{52} -200092. i q^{53} -203953. i q^{54} +67594.2i q^{55} -53397.9 q^{56} -97199.8i q^{57} +48993.7i q^{58} +64070.9 q^{59} +264152. i q^{60} +387246. i q^{61} +(-179916. - 323680. i) q^{62} -30262.6 q^{63} -415737. q^{64} +315986. i q^{65} +263708. q^{66} +99363.7 q^{67} -195042. i q^{68} +105262. q^{69} -194140. q^{70} +217702. q^{71} +61598.9 q^{72} -615580. i q^{73} -1.01626e6i q^{74} -191478. i q^{75} -290742. q^{76} +113540. i q^{77} +1.23277e6 q^{78} +102116. i q^{79} -163462. q^{80} -632739. q^{81} +905080. q^{82} +380355. i q^{83} +443702. i q^{84} -207762. i q^{85} +522633. i q^{86} +119277. q^{87} -231108. i q^{88} -450409. i q^{89} +223956. q^{90} +530768. i q^{91} -314856. i q^{92} +(-788011. + 438011. i) q^{93} -1.91623e6 q^{94} -309703. q^{95} +1.27626e6i q^{96} -341949. q^{97} +1.13636e6 q^{98} -130977. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 2 q^{2} + 122 q^{4} - 146 q^{5} + 6 q^{7} + 1142 q^{8} - 5668 q^{9} - 3004 q^{10} - 1312 q^{14} - 1102 q^{16} - 16382 q^{18} + 10430 q^{19} - 8052 q^{20} + 29618 q^{25} + 136504 q^{28} + 90076 q^{31}+ \cdots + 408810 q^{98}+O(q^{100}) $ 12 * q - 2 * q^2 + 122 * q^4 - 146 * q^5 + 6 * q^7 + 1142 * q^8 - 5668 * q^9 - 3004 * q^10 - 1312 * q^14 - 1102 * q^16 - 16382 * q^18 + 10430 * q^19 - 8052 * q^20 + 29618 * q^25 + 136504 * q^28 + 90076 * q^31 + 38406 * q^32 - 166248 * q^33 - 198898 * q^35 + 26982 * q^36 + 448196 * q^38 + 208224 * q^39 - 354932 * q^40 - 116690 * q^41 - 363818 * q^45 - 221120 * q^47 - 580182 * q^49 - 60318 * q^50 + 179784 * q^51 + 750496 * q^56 + 799510 * q^59 - 73070 * q^62 - 469874 * q^63 - 322286 * q^64 - 511884 * q^66 - 13744 * q^67 + 475128 * q^69 - 71312 * q^70 + 640678 * q^71 - 118870 * q^72 + 2044196 * q^76 + 1373868 * q^78 + 1356020 * q^80 - 678724 * q^81 - 488056 * q^82 - 1051200 * q^87 - 2598532 * q^90 - 2260440 * q^93 - 5045848 * q^94 + 3828206 * q^95 + 3708966 * q^97 + 408810 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/31\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

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 $

$2$	−12.4307			−1.55383			−0.776917	−	0.629603i	$-0.783218\pi$
$2$	−12.4307			−1.55383			−0.776917	+	0.629603i	$0.783218\pi$
$3$			30.2629i			1.12085i	0.828206	+	0.560424i	$0.189362\pi$
$3$			30.2629i			1.12085i	−0.828206	+	0.560424i	$0.810638\pi$
$4$	90.5217			1.41440
$5$	96.4253			0.771403			0.385701	−	0.922624i	$-0.373960\pi$
$5$	96.4253			0.771403			0.385701	+	0.922624i	$0.373960\pi$
$6$		−	376.188i		−	1.74161i
$7$	161.968			0.472209			0.236105	−	0.971728i	$-0.424129\pi$
$7$	161.968			0.472209			0.236105	+	0.971728i	$0.424129\pi$
$8$	−329.682			−0.643911
$9$	−186.843			−0.256301
$10$	−1198.63			−1.19863
$11$			701.001i			0.526672i	0.964704	+	0.263336i	$0.0848228\pi$
$11$			701.001i			0.526672i	−0.964704	+	0.263336i	$0.915177\pi$
$12$			2739.45i			1.58533i
$13$			3277.00i			1.49158i	0.666181	+	0.745790i	$0.267928\pi$
$13$			3277.00i			1.49158i	−0.666181	+	0.745790i	$0.732072\pi$
$14$	−2013.37			−0.733735
$15$			2918.11i			0.864625i
$16$	−1695.21			−0.413870
$17$		−	2154.64i		−	0.438560i	−0.975662	−	0.219280i	$-0.929629\pi$
$17$		−	2154.64i		−	0.438560i	0.975662	−	0.219280i	$-0.0703707\pi$
$18$	2322.59			0.398249
$19$	−3211.85			−0.468268			−0.234134	−	0.972204i	$-0.575225\pi$
$19$	−3211.85			−0.468268			−0.234134	+	0.972204i	$0.575225\pi$
$20$	8728.58			1.09107
$21$			4901.61i			0.529275i
$22$		−	8713.91i		−	0.818361i
$23$		−	3478.24i		−	0.285875i	−0.989732	−	0.142937i	$-0.954345\pi$
$23$		−	3478.24i		−	0.285875i	0.989732	−	0.142937i	$-0.0456547\pi$
$24$		−	9977.14i		−	0.721726i
$25$	−6327.15			−0.404938
$26$		−	40735.3i		−	2.31767i
$27$			16407.2i			0.833574i
$28$	14661.6			0.667893
$29$		−	3941.36i		−	0.161604i	−0.996730	−	0.0808019i	$-0.974252\pi$
$29$		−	3941.36i		−	0.161604i	0.996730	−	0.0808019i	$-0.0257481\pi$
$30$		−	36274.1i		−	1.34348i
$31$	14473.5	+	26038.8i	0.485835	+	0.874050i
$31$	14473.5	+	26038.8i	0.485835	+	0.874050i
$32$	42172.3			1.28700
$33$	−21214.3			−0.590320
$34$			26783.7i			0.681449i
$35$	15617.8			0.364263
$36$	−16913.4			−0.362512
$37$			81754.0i			1.61400i	0.590551	+	0.807000i	$0.298910\pi$
$37$			81754.0i			1.61400i	−0.590551	+	0.807000i	$0.701090\pi$
$38$	39925.4			0.727610
$39$	−99171.5			−1.67183
$40$	−31789.7			−0.496715
$41$	−72810.2			−1.05643			−0.528215	−	0.849111i	$-0.677139\pi$
$41$	−72810.2			−1.05643			−0.528215	+	0.849111i	$0.677139\pi$
$42$		−	60930.4i		−	0.822405i
$43$		−	42043.8i		−	0.528807i	−0.964412	−	0.264403i	$-0.914825\pi$
$43$		−	42043.8i		−	0.528807i	0.964412	−	0.264403i	$-0.0851750\pi$
$44$			63455.8i			0.744926i
$45$	−18016.4			−0.197711
$46$			43236.8i			0.444202i
$47$	154153.			1.48477			0.742385	−	0.669974i	$-0.233695\pi$
$47$	154153.			1.48477			0.742385	+	0.669974i	$0.233695\pi$
$48$		−	51302.1i		−	0.463886i
$49$	−91415.4			−0.777018
$50$	78650.8			0.629206
$51$	65205.7			0.491559
$52$			296640.i			2.10969i
$53$		−	200092.i		−	1.34401i	−0.740548	−	0.672003i	$-0.765434\pi$
$53$		−	200092.i		−	1.34401i	0.740548	−	0.672003i	$-0.234566\pi$
$54$		−	203953.i		−	1.29524i
$55$			67594.2i			0.406276i
$56$	−53397.9			−0.304061
$57$		−	97199.8i		−	0.524857i
$58$			48993.7i			0.251106i
$59$	64070.9			0.311964			0.155982	−	0.987760i	$-0.450146\pi$
$59$	64070.9			0.311964			0.155982	+	0.987760i	$0.450146\pi$
$60$			264152.i			1.22293i
$61$			387246.i			1.70607i	0.521852	+	0.853036i	$0.325241\pi$
$61$			387246.i			1.70607i	−0.521852	+	0.853036i	$0.674759\pi$
$62$	−179916.	−	323680.i	−0.754907	−	1.35813i
$63$	−30262.6			−0.121028
$64$	−415737.			−1.58591
$65$			315986.i			1.15061i
$66$	263708.			0.917259
$67$	99363.7			0.330372			0.165186	−	0.986262i	$-0.447178\pi$
$67$	99363.7			0.330372			0.165186	+	0.986262i	$0.447178\pi$
$68$		−	195042.i		−	0.620299i
$69$	105262.			0.320422
$70$	−194140.			−0.566005
$71$	217702.			0.608257			0.304128	−	0.952631i	$-0.401635\pi$
$71$	217702.			0.608257			0.304128	+	0.952631i	$0.401635\pi$
$72$	61598.9			0.165035
$73$		−	615580.i		−	1.58240i	−0.611558	−	0.791200i	$-0.709457\pi$
$73$		−	615580.i		−	1.58240i	0.611558	−	0.791200i	$-0.290543\pi$
$74$		−	1.01626e6i		−	2.50789i
$75$		−	191478.i		−	0.453874i
$76$	−290742.			−0.662318
$77$			113540.i			0.248699i
$78$	1.23277e6			2.59775
$79$			102116.i			0.207116i	0.994623	+	0.103558i	$0.0330227\pi$
$79$			102116.i			0.207116i	−0.994623	+	0.103558i	$0.966977\pi$
$80$	−163462.			−0.319261
$81$	−632739.			−1.19061
$82$	905080.			1.64152
$83$			380355.i			0.665203i	0.943067	+	0.332602i	$0.107926\pi$
$83$			380355.i			0.665203i	−0.943067	+	0.332602i	$0.892074\pi$
$84$			443702.i			0.748607i
$85$		−	207762.i		−	0.338306i
$86$			522633.i			0.821678i
$87$	119277.			0.181133
$88$		−	231108.i		−	0.339130i
$89$		−	450409.i		−	0.638906i	−0.947602	−	0.319453i	$-0.896501\pi$
$89$		−	450409.i		−	0.638906i	0.947602	−	0.319453i	$-0.103499\pi$
$90$	223956.			0.307210
$91$			530768.i			0.704338i
$92$		−	314856.i		−	0.404341i
$93$	−788011.	+	438011.i	−0.979678	+	0.544547i
$94$	−1.91623e6			−2.30709
$95$	−309703.			−0.361223
$96$			1.27626e6i			1.44253i
$97$	−341949.			−0.374667			−0.187334	−	0.982296i	$-0.559985\pi$
$97$	−341949.			−0.374667			−0.187334	+	0.982296i	$0.559985\pi$
$98$	1.13636e6			1.20736
$99$		−	130977.i		−	0.134986i
$100$	−572745.			−0.572745
$101$	−1.11954e6			−1.08661			−0.543306	−	0.839535i	$-0.682828\pi$
$101$	−1.11954e6			−1.08661			−0.543306	+	0.839535i	$0.682828\pi$
$102$	−810551.			−0.763801
$103$	795682.			0.728161			0.364081	−	0.931367i	$-0.381383\pi$
$103$	795682.			0.728161			0.364081	+	0.931367i	$0.381383\pi$
$104$		−	1.08037e6i		−	0.960444i
$105$			472640.i			0.408284i
$106$			2.48727e6i			2.08836i
$107$	1.13029e6			0.922657			0.461329	−	0.887229i	$-0.347373\pi$
$107$	1.13029e6			0.922657			0.461329	+	0.887229i	$0.347373\pi$
$108$			1.48521e6i			1.17901i
$109$	1.98556e6			1.53322			0.766609	−	0.642114i	$-0.221942\pi$
$109$	1.98556e6			1.53322			0.766609	+	0.642114i	$0.221942\pi$
$110$		−	840242.i		−	0.631286i
$111$	−2.47411e6			−1.80905
$112$	−274570.			−0.195433
$113$	2.24367e6			1.55498			0.777488	−	0.628898i	$-0.216494\pi$
$113$	2.24367e6			1.55498			0.777488	+	0.628898i	$0.216494\pi$
$114$			1.20826e6i			0.815541i
$115$		−	335390.i		−	0.220524i
$116$		−	356778.i		−	0.228573i
$117$		−	612285.i		−	0.382293i
$118$	−796444.			−0.484740
$119$		−	348983.i		−	0.207092i
$120$		−	962049.i		−	0.556742i
$121$	1.28016e6			0.722616
$122$		−	4.81373e6i		−	2.65095i
$123$		−	2.20345e6i		−	1.18410i
$124$	1.31017e6	+	2.35708e6i	0.687166	+	1.23626i
$125$	−2.11674e6			−1.08377
$126$	376184.			0.188057
$127$		−	3.28777e6i		−	1.60506i	−0.596614	−	0.802529i	$-0.703488\pi$
$127$		−	3.28777e6i		−	1.60506i	0.596614	−	0.802529i	$-0.296512\pi$
$128$	2.46886e6			1.17724
$129$	1.27237e6			0.592712
$130$		−	3.92792e6i		−	1.78785i
$131$	3.32420e6			1.47868			0.739339	−	0.673333i	$-0.235138\pi$
$131$	3.32420e6			1.47868			0.739339	+	0.673333i	$0.235138\pi$
$132$	−1.92036e6			−0.834949
$133$	−520216.			−0.221120
$134$	−1.23516e6			−0.513344
$135$			1.58207e6i			0.643021i
$136$			710348.i			0.282393i
$137$			1.38654e6i			0.539227i	0.962969	+	0.269613i	$0.0868959\pi$
$137$			1.38654e6i			0.539227i	−0.962969	+	0.269613i	$0.913104\pi$
$138$	−1.30847e6			−0.497883
$139$		−	2.05391e6i		−	0.764781i	−0.924001	−	0.382390i	$-0.875101\pi$
$139$		−	2.05391e6i		−	0.764781i	0.924001	−	0.382390i	$-0.124899\pi$
$140$	1.41375e6			0.515215
$141$			4.66512e6i			1.66420i
$142$	−2.70618e6			−0.945130
$143$	−2.29718e6			−0.785573
$144$	316739.			0.106075
$145$		−	380047.i		−	0.124662i
$146$			7.65208e6i			2.45879i
$147$		−	2.76650e6i		−	0.870920i
$148$			7.40051e6i			2.28284i
$149$	−4.18443e6			−1.26496			−0.632481	−	0.774576i	$-0.717964\pi$
$149$	−4.18443e6			−1.26496			−0.632481	+	0.774576i	$0.717964\pi$
$150$			2.38020e6i			0.705245i
$151$		−	4.51757e6i		−	1.31212i	−0.754708	−	0.656060i	$-0.772222\pi$
$151$		−	4.51757e6i		−	1.31212i	0.754708	−	0.656060i	$-0.227778\pi$
$152$	1.05889e6			0.301522
$153$			402581.i			0.112403i
$154$		−	1.41137e6i		−	0.386438i
$155$	1.39561e6	+	2.51080e6i	0.374775	+	0.674245i
$156$	−8.97717e6			−2.36464
$157$	7.23828e6			1.87041			0.935205	−	0.354108i	$-0.115215\pi$
$157$	7.23828e6			1.87041			0.935205	+	0.354108i	$0.115215\pi$
$158$		−	1.26937e6i		−	0.321823i
$159$	6.05536e6			1.50643
$160$	4.06648e6			0.992793
$161$		−	563362.i		−	0.134993i
$162$	7.86538e6			1.85001
$163$	−1.62356e6			−0.374891			−0.187446	−	0.982275i	$-0.560021\pi$
$163$	−1.62356e6			−0.374891			−0.187446	+	0.982275i	$0.560021\pi$
$164$	−6.59090e6			−1.49422
$165$	−2.04560e6			−0.455374
$166$		−	4.72806e6i		−	1.03362i
$167$		−	2.89718e6i		−	0.622051i	−0.950402	−	0.311026i	$-0.899327\pi$
$167$		−	2.89718e6i		−	0.622051i	0.950402	−	0.311026i	$-0.100673\pi$
$168$		−	1.61598e6i		−	0.340806i
$169$	−5.91192e6			−1.22481
$170$			2.58262e6i			0.525671i
$171$	600112.			0.120017
$172$		−	3.80588e6i		−	0.747945i
$173$	−2.98193e6			−0.575916			−0.287958	−	0.957643i	$-0.592976\pi$
$173$	−2.98193e6			−0.575916			−0.287958	+	0.957643i	$0.592976\pi$
$174$	−1.48269e6			−0.281451
$175$	−1.02479e6			−0.191215
$176$		−	1.18835e6i		−	0.217974i
$177$			1.93897e6i			0.349664i
$178$			5.59889e6i			0.992754i
$179$		−	1.79934e6i		−	0.313728i	−0.987620	−	0.156864i	$-0.949862\pi$
$179$		−	1.79934e6i		−	0.313728i	0.987620	−	0.156864i	$-0.0501384\pi$
$180$	−1.63088e6			−0.279643
$181$			1.02341e7i			1.72589i	0.505300	+	0.862944i	$0.331382\pi$
$181$			1.02341e7i			1.72589i	−0.505300	+	0.862944i	$0.668618\pi$
$182$		−	6.59781e6i		−	1.09442i
$183$	−1.17192e7			−1.91225
$184$			1.14671e6i			0.184078i
$185$			7.88315e6i			1.24504i
$186$	9.79551e6	−	5.44477e6i	1.52226	−	0.846137i
$187$	1.51041e6			0.230977
$188$	1.39542e7			2.10006
$189$			2.65744e6i			0.393621i
$190$	3.84982e6			0.561280
$191$	5.96593e6			0.856206			0.428103	−	0.903730i	$-0.359182\pi$
$191$	5.96593e6			0.856206			0.428103	+	0.903730i	$0.359182\pi$
$192$		−	1.25814e7i		−	1.77756i
$193$	329620.			0.0458502			0.0229251	−	0.999737i	$-0.492702\pi$
$193$	329620.			0.0458502			0.0229251	+	0.999737i	$0.492702\pi$
$194$	4.25065e6			0.582171
$195$	−9.56265e6			−1.28966
$196$	−8.27508e6			−1.09902
$197$			3.04005e6i			0.397633i	0.980037	+	0.198816i	$0.0637097\pi$
$197$			3.04005e6i			0.397633i	−0.980037	+	0.198816i	$0.936290\pi$
$198$			1.62814e6i			0.209747i
$199$			1.10648e6i			0.140405i	0.997533	+	0.0702026i	$0.0223646\pi$
$199$			1.10648e6i			0.140405i	−0.997533	+	0.0702026i	$0.977635\pi$
$200$	2.08595e6			0.260744
$201$			3.00704e6i			0.370297i
$202$	1.39166e7			1.68842
$203$		−	638373.i		−	0.0763108i
$204$	5.90253e6			0.695261
$205$	−7.02075e6			−0.814933
$206$	−9.89086e6			−1.13144
$207$			649885.i			0.0732699i
$208$		−	5.55521e6i		−	0.617321i
$209$		−	2.25151e6i		−	0.246623i
$210$		−	5.87523e6i		−	0.634406i
$211$	−6.03041e6			−0.641948			−0.320974	−	0.947088i	$-0.604010\pi$
$211$	−6.03041e6			−0.641948			−0.320974	+	0.947088i	$0.604010\pi$
$212$		−	1.81126e7i		−	1.90096i
$213$			6.58829e6i			0.681764i
$214$	−1.40503e7			−1.43366
$215$		−	4.05409e6i		−	0.407923i
$216$		−	5.40918e6i		−	0.536747i
$217$	2.34424e6	+	4.21745e6i	0.229416	+	0.412735i
$218$	−2.46819e7			−2.38237
$219$	1.86292e7			1.77363
$220$			6.11874e6i			0.574638i
$221$	7.06076e6			0.654146
$222$	3.07549e7			2.81096
$223$			1.57869e7i			1.42358i	0.702392	+	0.711790i	$0.252115\pi$
$223$			1.57869e7i			1.42358i	−0.702392	+	0.711790i	$0.747885\pi$
$224$	6.83056e6			0.607732
$225$	1.18219e6			0.103786
$226$	−2.78903e7			−2.41618
$227$	1.24063e7			1.06063			0.530317	−	0.847800i	$-0.322073\pi$
$227$	1.24063e7			1.06063			0.530317	+	0.847800i	$0.322073\pi$
$228$		−	8.79869e6i		−	0.742358i
$229$		−	1.44833e7i		−	1.20604i	−0.797727	−	0.603019i	$-0.793964\pi$
$229$		−	1.44833e7i		−	1.20604i	0.797727	−	0.603019i	$-0.206036\pi$
$230$			4.16912e6i			0.342658i
$231$	−3.43604e6			−0.278754
$232$			1.29940e6i			0.104058i
$233$	−7.02565e6			−0.555417			−0.277708	−	0.960665i	$-0.589575\pi$
$233$	−7.02565e6			−0.555417			−0.277708	+	0.960665i	$0.589575\pi$
$234$			7.61112e6i			0.594020i
$235$	1.48643e7			1.14536
$236$	5.79980e6			0.441242
$237$	−3.09033e6			−0.232145
$238$			4.33809e6i			0.321786i
$239$		−	1.77017e7i		−	1.29664i	−0.761366	−	0.648322i	$-0.775471\pi$
$239$		−	1.77017e7i		−	1.29664i	0.761366	−	0.648322i	$-0.224529\pi$
$240$		−	4.94682e6i		−	0.357843i
$241$		−	6.76013e6i		−	0.482952i	−0.970407	−	0.241476i	$-0.922368\pi$
$241$		−	6.76013e6i		−	0.482952i	0.970407	−	0.241476i	$-0.0776315\pi$
$242$	−1.59132e7			−1.12283
$243$		−	7.18765e6i		−	0.500920i
$244$			3.50541e7i			2.41307i
$245$	−8.81477e6			−0.599394
$246$			2.73904e7i			1.83989i
$247$		−	1.05252e7i		−	0.698458i
$248$	−4.77166e6	−	8.58454e6i	−0.312834	−	0.562811i
$249$	−1.15106e7			−0.745592
$250$	2.63126e7			1.68400
$251$			2.48616e7i			1.57220i	0.618098	+	0.786101i	$0.287904\pi$
$251$			2.48616e7i			1.57220i	−0.618098	+	0.786101i	$0.712096\pi$
$252$	−2.73942e6			−0.171182
$253$	2.43825e6			0.150562
$254$			4.08692e7i			2.49399i
$255$	6.28749e6			0.379190
$256$	−4.08244e6			−0.243332
$257$	−1.54178e7			−0.908288			−0.454144	−	0.890928i	$-0.650055\pi$
$257$	−1.54178e7			−0.908288			−0.454144	+	0.890928i	$0.650055\pi$
$258$	−1.58164e7			−0.920976
$259$			1.32415e7i			0.762146i
$260$			2.86036e7i			1.62742i
$261$			736416.i			0.0414192i
$262$	−4.13221e7			−2.29762
$263$		−	2.98355e7i		−	1.64008i	−0.572303	−	0.820042i	$-0.693950\pi$
$263$		−	2.98355e7i		−	1.64008i	0.572303	−	0.820042i	$-0.306050\pi$
$264$	6.99398e6			0.380113
$265$		−	1.92939e7i		−	1.03677i
$266$	6.46663e6			0.343584
$267$	1.36307e7			0.716117
$268$	8.99457e6			0.467279
$269$			1.56423e7i			0.803607i	0.915726	+	0.401804i	$0.131617\pi$
$269$			1.56423e7i			0.803607i	−0.915726	+	0.401804i	$0.868383\pi$
$270$		−	1.96662e7i		−	0.999148i
$271$		−	1.75308e7i		−	0.880834i	−0.897793	−	0.440417i	$-0.854831\pi$
$271$		−	1.75308e7i		−	0.880834i	0.897793	−	0.440417i	$-0.145169\pi$
$272$			3.65258e6i			0.181507i
$273$	−1.60626e7			−0.789455
$274$		−	1.72357e7i		−	0.837869i
$275$		−	4.43534e6i		−	0.213269i
$276$	9.52845e6			0.453205
$277$		−	1.54337e7i		−	0.726160i	−0.931758	−	0.363080i	$-0.881725\pi$
$277$		−	1.54337e7i		−	0.726160i	0.931758	−	0.363080i	$-0.118275\pi$
$278$			2.55315e7i			1.18834i
$279$	−2.70428e6	−	4.86518e6i	−0.124520	−	0.224020i
$280$	−5.14891e6			−0.234553
$281$	4.37222e6			0.197053			0.0985266	−	0.995134i	$-0.468587\pi$
$281$	4.37222e6			0.197053			0.0985266	+	0.995134i	$0.468587\pi$
$282$		−	5.79906e7i		−	2.58589i
$283$	1.85465e7			0.818282			0.409141	−	0.912471i	$-0.365828\pi$
$283$	1.85465e7			0.818282			0.409141	+	0.912471i	$0.365828\pi$
$284$	1.97067e7			0.860319
$285$		−	9.37252e6i		−	0.404876i
$286$	2.85555e7			1.22065
$287$	−1.17929e7			−0.498856
$288$	−7.87961e6			−0.329858
$289$	1.94951e7			0.807666
$290$			4.72424e6i			0.193704i
$291$		−	1.03484e7i		−	0.419945i
$292$		−	5.57234e7i		−	2.23815i
$293$	2.21442e7			0.880355			0.440178	−	0.897911i	$-0.354916\pi$
$293$	2.21442e7			0.880355			0.440178	+	0.897911i	$0.354916\pi$
$294$			3.43894e7i			1.35327i
$295$	6.17805e6			0.240650
$296$		−	2.69528e7i		−	1.03927i
$297$	−1.15015e7			−0.439020
$298$	5.20153e7			1.96554
$299$	1.13982e7			0.426405
$300$		−	1.73329e7i		−	0.641960i
$301$		−	6.80975e6i		−	0.249707i
$302$			5.61564e7i			2.03882i
$303$		−	3.38805e7i		−	1.21793i
$304$	5.44477e6			0.193802
$305$			3.73403e7i			1.31607i
$306$		−	5.00435e6i		−	0.174656i
$307$	1.95249e7			0.674796			0.337398	−	0.941362i	$-0.390453\pi$
$307$	1.95249e7			0.674796			0.337398	+	0.941362i	$0.390453\pi$
$308$			1.02778e7i			0.351761i
$309$			2.40796e7i			0.816158i
$310$	−1.73484e7	−	3.12110e7i	−0.582338	−	1.04766i
$311$	−3.81323e7			−1.26769			−0.633843	−	0.773462i	$-0.718523\pi$
$311$	−3.81323e7			−1.26769			−0.633843	+	0.773462i	$0.718523\pi$
$312$	3.26951e7			1.07651
$313$			1.85533e7i			0.605046i	0.953142	+	0.302523i	$0.0978289\pi$
$313$			1.85533e7i			0.605046i	−0.953142	+	0.302523i	$0.902171\pi$
$314$	−8.99768e7			−2.90631
$315$	−2.91808e6			−0.0933610
$316$			9.24372e6i			0.292945i
$317$	3.69576e6			0.116018			0.0580091	−	0.998316i	$-0.481525\pi$
$317$	3.69576e6			0.116018			0.0580091	+	0.998316i	$0.481525\pi$
$318$	−7.52722e7			−2.34074
$319$	2.76289e6			0.0851122
$320$	−4.00876e7			−1.22337
$321$			3.42060e7i			1.03416i
$322$			7.00297e6i			0.209756i
$323$			6.92038e6i			0.205363i
$324$	−5.72766e7			−1.68400
$325$		−	2.07341e7i		−	0.603997i
$326$	2.01819e7			0.582519
$327$			6.00889e7i			1.71850i
$328$	2.40042e7			0.680247
$329$	2.49679e7			0.701122
$330$	2.54282e7			0.707576
$331$		−	4.67507e7i		−	1.28915i	−0.764540	−	0.644576i	$-0.777034\pi$
$331$		−	4.67507e7i		−	1.28915i	0.764540	−	0.644576i	$-0.222966\pi$
$332$			3.44303e7i			0.940864i
$333$		−	1.52752e7i		−	0.413670i
$334$			3.60139e7i			0.966565i
$335$	9.58118e6			0.254850
$336$		−	8.30928e6i		−	0.219051i
$337$		−	7.00838e6i		−	0.183117i	−0.995800	−	0.0915583i	$-0.970815\pi$
$337$		−	7.00838e6i		−	0.183117i	0.995800	−	0.0915583i	$-0.0291848\pi$
$338$	7.34891e7			1.90315
$339$			6.79000e7i			1.74289i
$340$		−	1.88070e7i		−	0.478500i
$341$	−1.82532e7	+	1.01459e7i	−0.460338	+	0.255876i
$342$	−7.45980e6			−0.186487
$343$	−3.38617e7			−0.839125
$344$			1.38611e7i			0.340504i
$345$	1.01499e7			0.247174
$346$	3.70674e7			0.894878
$347$		−	2.47428e7i		−	0.592188i	−0.955159	−	0.296094i	$-0.904316\pi$
$347$		−	2.47428e7i		−	0.592188i	0.955159	−	0.296094i	$-0.0956842\pi$
$348$	1.07971e7			0.256195
$349$	−6.63858e7			−1.56170			−0.780852	−	0.624716i	$-0.785215\pi$
$349$	−6.63858e7			−1.56170			−0.780852	+	0.624716i	$0.785215\pi$
$350$	1.27389e7			0.297117
$351$	−5.37665e7			−1.24334
$352$			2.95628e7i			0.677825i
$353$			2.01840e7i			0.458862i	0.973325	+	0.229431i	$0.0736865\pi$
$353$			2.01840e7i			0.458862i	−0.973325	+	0.229431i	$0.926313\pi$
$354$		−	2.41027e7i		−	0.543320i
$355$	2.09920e7			0.469211
$356$		−	4.07718e7i		−	0.903669i
$357$	1.05612e7			0.232119
$358$			2.23670e7i			0.487481i
$359$	3.50695e7			0.757959			0.378979	−	0.925405i	$-0.376275\pi$
$359$	3.50695e7			0.757959			0.378979	+	0.925405i	$0.376275\pi$
$360$	5.93970e6			0.127308
$361$	−3.67299e7			−0.780726
$362$		−	1.27216e8i		−	2.68174i
$363$			3.87413e7i			0.809943i
$364$			4.80460e7i			0.996216i
$365$		−	5.93575e7i		−	1.22067i
$366$	1.45677e8			2.97132
$367$			8.41476e7i			1.70233i	0.524899	+	0.851165i	$0.324103\pi$
$367$			8.41476e7i			1.70233i	−0.524899	+	0.851165i	$0.675897\pi$
$368$			5.89635e6i			0.118315i
$369$	1.36041e7			0.270764
$370$		−	9.79929e7i		−	1.93459i
$371$		−	3.24084e7i		−	0.634652i
$372$	−7.13321e7	+	3.96495e7i	−1.38566	+	0.770209i
$373$	1.09717e7			0.211421			0.105711	−	0.994397i	$-0.466288\pi$
$373$	1.09717e7			0.211421			0.105711	+	0.994397i	$0.466288\pi$
$374$	−1.87754e7			−0.358900
$375$		−	6.40588e7i		−	1.21474i
$376$	−5.08216e7			−0.956059
$377$	1.29158e7			0.241045
$378$		−	3.30338e7i		−	0.611622i
$379$	−4.92659e7			−0.904960			−0.452480	−	0.891775i	$-0.649461\pi$
$379$	−4.92659e7			−0.904960			−0.452480	+	0.891775i	$0.649461\pi$
$380$	−2.80349e7			−0.510914
$381$	9.94975e7			1.79903
$382$	−7.41606e7			−1.33040
$383$		−	1.01955e7i		−	0.181474i	−0.995875	−	0.0907368i	$-0.971078\pi$
$383$		−	1.01955e7i		−	0.181474i	0.995875	−	0.0907368i	$-0.0289222\pi$
$384$			7.47148e7i			1.31951i
$385$			1.09481e7i			0.191847i
$386$	−4.09740e6			−0.0712437
$387$			7.85561e6i			0.135534i
$388$	−3.09538e7			−0.529930
$389$		−	6.89117e7i		−	1.17070i	−0.810782	−	0.585348i	$-0.800958\pi$
$389$		−	6.89117e7i		−	1.17070i	0.810782	−	0.585348i	$-0.199042\pi$
$390$	1.18870e8			2.00391
$391$	−7.49435e6			−0.125373
$392$	3.01381e7			0.500331
$393$			1.00600e8i			1.65737i
$394$		−	3.77899e7i		−	0.617855i
$395$			9.84658e6i			0.159770i
$396$		−	1.18563e7i		−	0.190925i
$397$	1.03020e8			1.64646			0.823228	−	0.567711i	$-0.192171\pi$
$397$	1.03020e8			1.64646			0.823228	+	0.567711i	$0.192171\pi$
$398$		−	1.37543e7i		−	0.218166i
$399$		−	1.57432e7i		−	0.247842i
$400$	1.07259e7			0.167592
$401$		−	1.09891e8i		−	1.70423i	−0.523354	−	0.852115i	$-0.675320\pi$
$401$		−	1.09891e8i		−	1.70423i	0.523354	−	0.852115i	$-0.324680\pi$
$402$		−	3.73795e7i		−	0.575380i
$403$	−8.53293e7	+	4.74297e7i	−1.30372	+	0.724662i
$404$	−1.01342e8			−1.53691
$405$	−6.10121e7			−0.918440
$406$			7.93540e6i			0.118574i
$407$	−5.73096e7			−0.850049
$408$	−2.14972e7			−0.316520
$409$		−	6.05111e6i		−	0.0884434i	−0.999022	−	0.0442217i	$-0.985919\pi$
$409$		−	6.05111e6i		−	0.0884434i	0.999022	−	0.0442217i	$-0.0140808\pi$
$410$	8.72727e7			1.26627
$411$	−4.19608e7			−0.604391
$412$	7.20264e7			1.02991
$413$	1.03774e7			0.147312
$414$		−	8.07851e6i		−	0.113849i
$415$			3.66758e7i			0.513140i
$416$			1.38199e8i			1.91966i
$417$	6.21573e7			0.857203
$418$			2.79877e7i			0.383212i
$419$	−1.73644e7			−0.236058			−0.118029	−	0.993010i	$-0.537658\pi$
$419$	−1.73644e7			−0.236058			−0.118029	+	0.993010i	$0.537658\pi$
$420$			4.27842e7i			0.577477i
$421$	1.03647e7			0.138902			0.0694511	−	0.997585i	$-0.477875\pi$
$421$	1.03647e7			0.138902			0.0694511	+	0.997585i	$0.477875\pi$
$422$	7.49621e7			0.997480
$423$	−2.88025e7			−0.380548
$424$			6.59667e7i			0.865420i
$425$			1.36328e7i			0.177589i
$426$		−	8.18969e7i		−	1.05935i
$427$			6.27213e7i			0.805623i
$428$	1.02316e8			1.30501
$429$		−	6.95193e7i		−	0.880508i
$430$			5.03951e7i			0.633845i
$431$	2.97300e7			0.371332			0.185666	−	0.982613i	$-0.440556\pi$
$431$	2.97300e7			0.371332			0.185666	+	0.982613i	$0.440556\pi$
$432$		−	2.78138e7i		−	0.344992i
$433$			1.20248e6i			0.0148121i	0.999973	+	0.00740603i	$0.00235743\pi$
$433$			1.20248e6i			0.0148121i	−0.999973	+	0.00740603i	$0.997643\pi$
$434$	−2.91405e7	−	5.24258e7i	−0.356474	−	0.641321i
$435$	1.15013e7			0.139727
$436$	1.79736e8			2.16859
$437$			1.11716e7i			0.133866i
$438$	−2.31574e8			−2.75593
$439$	−2.96954e7			−0.350990			−0.175495	−	0.984480i	$-0.556153\pi$
$439$	−2.96954e7			−0.350990			−0.175495	+	0.984480i	$0.556153\pi$
$440$		−	2.22846e7i		−	0.261606i
$441$	1.70804e7			0.199150
$442$	−8.77701e7			−1.01643
$443$	4.05180e7			0.466055			0.233027	−	0.972470i	$-0.425137\pi$
$443$	4.05180e7			0.466055			0.233027	+	0.972470i	$0.425137\pi$
$444$	−2.23961e8			−2.55872
$445$		−	4.34308e7i		−	0.492854i
$446$		−	1.96242e8i		−	2.21201i
$447$		−	1.26633e8i		−	1.41783i
$448$	−6.73359e7			−0.748881
$449$		−	3.78355e6i		−	0.0417985i	−0.999782	−	0.0208993i	$-0.993347\pi$
$449$		−	3.78355e6i		−	0.0417985i	0.999782	−	0.0208993i	$-0.00665292\pi$
$450$	−1.46954e7			−0.161266
$451$		−	5.10400e7i		−	0.556392i
$452$	2.03101e8			2.19936
$453$	1.36715e8			1.47069
$454$	−1.54219e8			−1.64805
$455$			5.11795e7i			0.543328i
$456$			3.20451e7i			0.337961i
$457$			3.10449e6i			0.0325268i	0.999868	+	0.0162634i	$0.00517703\pi$
$457$			3.10449e6i			0.0325268i	−0.999868	+	0.0162634i	$0.994823\pi$
$458$			1.80037e8i			1.87398i
$459$	3.53517e7			0.365572
$460$		−	3.03601e7i		−	0.311910i
$461$		−	6.13002e7i		−	0.625690i	−0.949804	−	0.312845i	$-0.898718\pi$
$461$		−	6.13002e7i		−	0.625690i	0.949804	−	0.312845i	$-0.101282\pi$
$462$	4.27122e7			0.433138
$463$			1.95894e8i			1.97369i	0.161669	+	0.986845i	$0.448312\pi$
$463$			1.95894e8i			1.97369i	−0.161669	+	0.986845i	$0.551688\pi$
$464$			6.68144e6i			0.0668831i
$465$	−7.59842e7	+	4.22353e7i	−0.755726	+	0.420065i
$466$	8.73336e7			0.863026
$467$	−9.08367e7			−0.891889			−0.445945	−	0.895061i	$-0.647132\pi$
$467$	−9.08367e7			−0.891889			−0.445945	+	0.895061i	$0.647132\pi$
$468$		−	5.54251e7i		−	0.540716i
$469$	1.60937e7			0.156005
$470$	−1.84773e8			−1.77969
$471$			2.19051e8i			2.09645i
$472$	−2.11230e7			−0.200877
$473$	2.94728e7			0.278508
$474$	3.84149e7			0.360715
$475$	2.03218e7			0.189619
$476$		−	3.15905e7i		−	0.292911i
$477$			3.73858e7i			0.344470i
$478$			2.20044e8i			2.01477i
$479$	1.45950e8			1.32800			0.664000	−	0.747733i	$-0.268858\pi$
$479$	1.45950e8			1.32800			0.664000	+	0.747733i	$0.268858\pi$
$480$			1.23063e8i			1.11277i
$481$	−2.67908e8			−2.40741
$482$			8.40330e7i			0.750428i
$483$	1.70490e7			0.151306
$484$	1.15882e8			1.02207
$485$	−3.29725e7			−0.289019
$486$			8.93474e7i			0.778347i
$487$			8.85482e7i			0.766642i	0.923615	+	0.383321i	$0.125220\pi$
$487$			8.85482e7i			0.766642i	−0.923615	+	0.383321i	$0.874780\pi$
$488$		−	1.27668e8i		−	1.09856i
$489$		−	4.91336e7i		−	0.420196i
$490$	1.09573e8			0.931359
$491$		−	7.46263e7i		−	0.630445i	−0.949018	−	0.315223i	$-0.897921\pi$
$491$		−	7.46263e7i		−	0.630445i	0.949018	−	0.315223i	$-0.102079\pi$
$492$		−	1.99460e8i		−	1.67479i
$493$	−8.49222e6			−0.0708729
$494$			1.30836e8i			1.08529i
$495$		−	1.26295e7i		−	0.104129i
$496$	−2.45357e7	−	4.41414e7i	−0.201073	−	0.361744i
$497$	3.52607e7			0.287225
$498$	1.43085e8			1.15853
$499$			1.82186e8i			1.46626i	0.680086	+	0.733132i	$0.261942\pi$
$499$			1.82186e8i			1.46626i	−0.680086	+	0.733132i	$0.738058\pi$
$500$	−1.91611e8			−1.53289
$501$	8.76771e7			0.697225
$502$		−	3.09047e8i		−	2.44294i
$503$	8.98350e7			0.705897			0.352949	−	0.935643i	$-0.385179\pi$
$503$	8.98350e7			0.705897			0.352949	+	0.935643i	$0.385179\pi$
$504$	9.97704e6			0.0779310
$505$	−1.07952e8			−0.838216
$506$	−3.03090e7			−0.233949
$507$		−	1.78912e8i		−	1.37282i
$508$		−	2.97615e8i		−	2.27019i
$509$		−	2.08422e8i		−	1.58049i	−0.612793	−	0.790243i	$-0.709954\pi$
$509$		−	2.08422e8i		−	1.58049i	0.612793	−	0.790243i	$-0.290046\pi$
$510$	−7.81577e7			−0.589198
$511$		−	9.97042e7i		−	0.747224i
$512$	−1.07260e8			−0.799146
$513$		−	5.26975e7i		−	0.390336i
$514$	1.91654e8			1.41133
$515$	7.67239e7			0.561706
$516$	1.15177e8			0.838333
$517$			1.08062e8i			0.781987i
$518$		−	1.64601e8i		−	1.18425i
$519$		−	9.02419e7i		−	0.645514i
$520$		−	1.04175e8i		−	0.740889i
$521$	−1.72718e8			−1.22131			−0.610653	−	0.791898i	$-0.709093\pi$
$521$	−1.72718e8			−1.22131			−0.610653	+	0.791898i	$0.709093\pi$
$522$		−	9.15415e6i		−	0.0643586i
$523$			1.77241e7i			0.123896i	0.998079	+	0.0619482i	$0.0197314\pi$
$523$			1.77241e7i			0.123896i	−0.998079	+	0.0619482i	$0.980269\pi$
$524$	3.00913e8			2.09144
$525$		−	3.10133e7i		−	0.214323i
$526$			3.70875e8i			2.54842i
$527$	5.61044e7	−	3.11853e7i	0.383323	−	0.213068i
$528$	3.59628e7			0.244316
$529$	1.35938e8			0.918276
$530$			2.39836e8i			1.61097i
$531$	−1.19712e7			−0.0799566
$532$	−4.70908e7			−0.312753
$533$		−	2.38599e8i		−	1.57575i
$534$	−1.69439e8			−1.11273
$535$	1.08989e8			0.711740
$536$	−3.27585e7			−0.212730
$537$	5.44531e7			0.351641
$538$		−	1.94444e8i		−	1.24867i
$539$		−	6.40823e7i		−	0.409234i
$540$			1.43212e8i			0.909490i
$541$	−1.69646e8			−1.07140			−0.535700	−	0.844408i	$-0.679952\pi$
$541$	−1.69646e8			−1.07140			−0.535700	+	0.844408i	$0.679952\pi$
$542$			2.17920e8i			1.36867i
$543$	−3.09713e8			−1.93446
$544$		−	9.08663e7i		−	0.564425i
$545$	1.91458e8			1.18273
$546$	1.99669e8			1.22668
$547$	−1.40402e8			−0.857852			−0.428926	−	0.903340i	$-0.641108\pi$
$547$	−1.40402e8			−0.857852			−0.428926	+	0.903340i	$0.641108\pi$
$548$			1.25512e8i			0.762683i
$549$		−	7.23543e7i		−	0.437268i
$550$			5.51343e7i			0.331385i
$551$			1.26590e7i			0.0756738i
$552$	−3.47029e7			−0.206323
$553$			1.65395e7i			0.0978019i
$554$			1.91852e8i			1.12833i
$555$	−2.38567e8			−1.39551
$556$		−	1.85923e8i		−	1.08171i
$557$		−	1.61216e8i		−	0.932914i	−0.884544	−	0.466457i	$-0.845530\pi$
$557$		−	1.61216e8i		−	0.932914i	0.884544	−	0.466457i	$-0.154470\pi$
$558$	3.36160e7	+	6.04775e7i	0.193483	+	0.348090i
$559$	1.37778e8			0.788757
$560$	−2.64755e7			−0.150758
$561$			4.57093e7i			0.258890i
$562$	−5.43497e7			−0.306188
$563$	2.83735e8			1.58997			0.794984	−	0.606631i	$-0.207479\pi$
$563$	2.83735e8			1.58997			0.794984	+	0.606631i	$0.207479\pi$
$564$			4.22295e8i			2.35385i
$565$	2.16347e8			1.19951
$566$	−2.30546e8			−1.27148
$567$	−1.02483e8			−0.562217
$568$	−7.17724e7			−0.391663
$569$			2.13052e7i			0.115651i	0.998327	+	0.0578255i	$0.0184167\pi$
$569$			2.13052e7i			0.115651i	−0.998327	+	0.0578255i	$0.981583\pi$
$570$			1.16507e8i			0.629110i
$571$			1.64435e8i			0.883252i	0.897199	+	0.441626i	$0.145598\pi$
$571$			1.64435e8i			0.883252i	−0.897199	+	0.441626i	$0.854402\pi$
$572$	−2.07944e8			−1.11112
$573$			1.80546e8i			0.959677i
$574$	1.46594e8			0.775140
$575$			2.20073e7i			0.115761i
$576$	7.76776e7			0.406470
$577$	1.12083e8			0.583460			0.291730	−	0.956501i	$-0.405769\pi$
$577$	1.12083e8			0.583460			0.291730	+	0.956501i	$0.405769\pi$
$578$	−2.42337e8			−1.25498
$579$			9.97526e6i			0.0513912i
$580$		−	3.44025e7i		−	0.176322i
$581$			6.16052e7i			0.314115i
$582$			1.28637e8i			0.652525i
$583$	1.40264e8			0.707851
$584$			2.02946e8i			1.01892i
$585$		−	5.90398e7i		−	0.294902i
$586$	−2.75268e8			−1.36793
$587$			4.68196e7i			0.231480i	0.993280	+	0.115740i	$0.0369239\pi$
$587$			4.68196e7i			0.231480i	−0.993280	+	0.115740i	$0.963076\pi$
$588$		−	2.50428e8i		−	1.23183i
$589$	−4.64867e7	−	8.36328e7i	−0.227501	−	0.409289i
$590$	−7.67974e7			−0.373930
$591$	−9.20008e7			−0.445686
$592$		−	1.38590e8i		−	0.667987i
$593$	−9.33451e7			−0.447639			−0.223819	−	0.974631i	$-0.571853\pi$
$593$	−9.33451e7			−0.447639			−0.223819	+	0.974631i	$0.571853\pi$
$594$	1.42971e8			0.682165
$595$		−	3.36508e7i		−	0.159751i
$596$	−3.78782e8			−1.78916
$597$	−3.34852e7			−0.157373
$598$	−1.41687e8			−0.662562
$599$	3.65248e8			1.69945			0.849724	−	0.527228i	$-0.176769\pi$
$599$	3.65248e8			1.69945			0.849724	+	0.527228i	$0.176769\pi$
$600$			6.31269e7i			0.292254i
$601$			2.51694e8i			1.15944i	0.814814	+	0.579722i	$0.196839\pi$
$601$			2.51694e8i			1.15944i	−0.814814	+	0.579722i	$0.803161\pi$
$602$			8.46497e7i			0.388004i
$603$	−1.85654e7			−0.0846747
$604$		−	4.08938e8i		−	1.85586i
$605$	1.23440e8			0.557428
$606$			4.21157e8i			1.89246i
$607$	−1.80742e8			−0.808152			−0.404076	−	0.914725i	$-0.632407\pi$
$607$	−1.80742e8			−0.808152			−0.404076	+	0.914725i	$0.632407\pi$
$608$	−1.35451e8			−0.602659
$609$	1.93190e7			0.0855329
$610$		−	4.64165e8i		−	2.04495i
$611$			5.05160e8i			2.21465i
$612$			3.64423e7i			0.158983i
$613$			3.70931e7i			0.161032i	0.996753	+	0.0805160i	$0.0256568\pi$
$613$			3.70931e7i			0.161032i	−0.996753	+	0.0805160i	$0.974343\pi$
$614$	−2.42707e8			−1.04852
$615$		−	2.12468e8i		−	0.913416i
$616$		−	3.74320e7i		−	0.160140i
$617$	−2.08022e8			−0.885634			−0.442817	−	0.896612i	$-0.646021\pi$
$617$	−2.08022e8			−0.885634			−0.442817	+	0.896612i	$0.646021\pi$
$618$		−	2.99326e8i		−	1.26818i
$619$		−	4.34198e8i		−	1.83069i	−0.402665	−	0.915347i	$-0.631916\pi$
$619$		−	4.34198e8i		−	1.83069i	0.402665	−	0.915347i	$-0.368084\pi$
$620$	1.26333e8	+	2.27282e8i	0.530082	+	0.953653i
$621$	5.70682e7			0.238298
$622$	4.74010e8			1.96977
$623$		−	7.29517e7i		−	0.301697i
$624$	1.68117e8			0.691923
$625$	−1.05246e8			−0.431087
$626$		−	2.30630e8i		−	0.940141i
$627$	6.81371e7			0.276427
$628$	6.55222e8			2.64551
$629$	1.76151e8			0.707835
$630$	3.62737e7			0.145068
$631$			1.26121e8i			0.501996i	0.967988	+	0.250998i	$0.0807587\pi$
$631$			1.26121e8i			0.501996i	−0.967988	+	0.250998i	$0.919241\pi$
$632$		−	3.36659e7i		−	0.133364i
$633$		−	1.82498e8i		−	0.719526i
$634$	−4.59408e7			−0.180273
$635$		−	3.17025e8i		−	1.23815i
$636$	5.48141e8			2.13069
$637$		−	2.99568e8i		−	1.15898i
$638$	−3.43446e7			−0.132250
$639$	−4.06761e7			−0.155897
$640$	2.38061e8			0.908129
$641$			3.12607e6i			0.0118693i	0.999982	+	0.00593465i	$0.00188907\pi$
$641$			3.12607e6i			0.0118693i	−0.999982	+	0.00593465i	$0.998111\pi$
$642$		−	4.25204e8i		−	1.60691i
$643$		−	1.01491e7i		−	0.0381763i	−0.999818	−	0.0190881i	$-0.993924\pi$
$643$		−	1.01491e7i		−	0.0381763i	0.999818	−	0.0190881i	$-0.00607631\pi$
$644$		−	5.09965e7i		−	0.190934i
$645$	1.22689e8			0.457220
$646$		−	8.60250e7i		−	0.319100i
$647$			5.23282e8i			1.93207i	0.258407	+	0.966036i	$0.416802\pi$
$647$			5.23282e8i			1.93207i	−0.258407	+	0.966036i	$0.583198\pi$
$648$	2.08603e8			0.766647
$649$			4.49137e7i			0.164303i
$650$			2.57739e8i			0.938511i
$651$	−1.27632e8	+	7.09436e7i	−0.462613	+	0.257140i
$652$	−1.46967e8			−0.530246
$653$	−3.31293e8			−1.18980			−0.594898	−	0.803801i	$-0.702808\pi$
$653$	−3.31293e8			−1.18980			−0.594898	+	0.803801i	$0.702808\pi$
$654$		−	7.46945e8i		−	2.67027i
$655$	3.20537e8			1.14066
$656$	1.23429e8			0.437225
$657$			1.15017e8i			0.405570i
$658$	−3.10367e8			−1.08943
$659$	−4.26424e8			−1.49000			−0.744999	−	0.667066i	$-0.767550\pi$
$659$	−4.26424e8			−1.49000			−0.744999	+	0.667066i	$0.767550\pi$
$660$	−1.85171e8			−0.644082
$661$	−1.66151e7			−0.0575306			−0.0287653	−	0.999586i	$-0.509158\pi$
$661$	−1.66151e7			−0.0575306			−0.0287653	+	0.999586i	$0.509158\pi$
$662$			5.81143e8i			2.00313i
$663$			2.13679e8i			0.733199i
$664$		−	1.25396e8i		−	0.428332i
$665$	−5.01620e7			−0.170573
$666$			1.89881e8i			0.642774i
$667$	−1.37090e7			−0.0461984
$668$		−	2.62258e8i		−	0.879830i
$669$	−4.77757e8			−1.59562
$670$	−1.19101e8			−0.395995
$671$	−2.71460e8			−0.898540
$672$			2.06712e8i			0.681175i
$673$			5.71367e8i			1.87443i	0.348748	+	0.937217i	$0.386607\pi$
$673$			5.71367e8i			1.87443i	−0.348748	+	0.937217i	$0.613393\pi$
$674$			8.71189e7i			0.284533i
$675$		−	1.03811e8i		−	0.337546i
$676$	−5.35157e8			−1.73237
$677$			3.74084e8i			1.20560i	0.797892	+	0.602800i	$0.205948\pi$
$677$			3.74084e8i			1.20560i	−0.797892	+	0.602800i	$0.794052\pi$
$678$		−	8.44042e8i		−	2.70817i
$679$	−5.53847e7			−0.176921
$680$			6.84955e7i			0.217839i
$681$			3.75451e8i			1.18881i
$682$	2.26900e8	−	1.26121e8i	0.715289	−	0.397589i
$683$	−3.66698e8			−1.15092			−0.575462	−	0.817829i	$-0.695178\pi$
$683$	−3.66698e8			−1.15092			−0.575462	+	0.817829i	$0.695178\pi$
$684$	5.43231e7			0.169753
$685$			1.33698e8i			0.415961i
$686$	4.20924e8			1.30386
$687$	4.38306e8			1.35179
$688$			7.12733e7i			0.218858i
$689$	6.55700e8			2.00469
$690$	−1.26170e8			−0.384068
$691$	−3.65697e8			−1.10838			−0.554189	−	0.832391i	$-0.686971\pi$
$691$	−3.65697e8			−1.10838			−0.554189	+	0.832391i	$0.686971\pi$
$692$	−2.69929e8			−0.814576
$693$		−	2.12141e7i		−	0.0637419i
$694$			3.07569e8i			0.920162i
$695$		−	1.98049e8i		−	0.589954i
$696$	−3.93235e7			−0.116634
$697$			1.56880e8i			0.463308i
$698$	8.25220e8			2.42663
$699$		−	2.12617e8i		−	0.622538i
$700$	−9.27662e7			−0.270455
$701$	−4.28201e8			−1.24307			−0.621533	−	0.783388i	$-0.713490\pi$
$701$	−4.28201e8			−1.24307			−0.621533	+	0.783388i	$0.713490\pi$
$702$	6.68354e8			1.93195
$703$		−	2.62581e8i		−	0.755784i
$704$		−	2.91432e8i		−	0.835254i
$705$			4.49836e8i			1.28377i
$706$		−	2.50900e8i		−	0.712996i
$707$	−1.81329e8			−0.513108
$708$			1.75519e8i			0.494566i
$709$		−	1.43971e8i		−	0.403958i	−0.979390	−	0.201979i	$-0.935263\pi$
$709$		−	1.43971e8i		−	0.403958i	0.979390	−	0.201979i	$-0.0647372\pi$
$710$	−2.60944e8			−0.729076
$711$		−	1.90797e7i		−	0.0530839i
$712$			1.48492e8i			0.411398i
$713$	9.05692e7	−	5.03423e7i	0.249869	−	0.138888i
$714$	−1.31283e8			−0.360674
$715$	−2.21506e8			−0.605993
$716$		−	1.62879e8i		−	0.443737i
$717$	5.35704e8			1.45334
$718$	−4.35937e8			−1.17774
$719$		−	3.14785e8i		−	0.846890i	−0.905922	−	0.423445i	$-0.860821\pi$
$719$		−	3.14785e8i		−	0.846890i	0.905922	−	0.423445i	$-0.139179\pi$
$720$	3.05417e7			0.0818268
$721$	1.28875e8			0.343845
$722$	4.56578e8			1.21312
$723$	2.04581e8			0.541316
$724$			9.26405e8i			2.44110i
$725$			2.49376e7i			0.0654395i
$726$		−	4.81581e8i		−	1.25852i
$727$	2.93434e8			0.763672			0.381836	−	0.924230i	$-0.375292\pi$
$727$	2.93434e8			0.763672			0.381836	+	0.924230i	$0.375292\pi$
$728$		−	1.74985e8i		−	0.453531i
$729$	−2.43748e8			−0.629155
$730$			7.37854e8i			1.89671i
$731$	−9.05895e7			−0.231913
$732$	−1.06084e9			−2.70468
$733$	3.06484e8			0.778209			0.389104	−	0.921194i	$-0.372785\pi$
$733$	3.06484e8			0.778209			0.389104	+	0.921194i	$0.372785\pi$
$734$		−	1.04601e9i		−	2.64514i
$735$		−	2.66760e8i		−	0.671830i
$736$		−	1.46685e8i		−	0.367920i
$737$			6.96540e7i			0.173998i
$738$	−1.69108e8			−0.420722
$739$			2.31247e8i			0.572985i	0.958083	+	0.286492i	$0.0924893\pi$
$739$			2.31247e8i			0.572985i	−0.958083	+	0.286492i	$0.907511\pi$
$740$			7.13596e8i			1.76099i
$741$	3.18524e8			0.782866
$742$			4.02858e8i			0.986145i
$743$		−	2.72360e7i		−	0.0664015i	−0.999449	−	0.0332007i	$-0.989430\pi$
$743$		−	2.72360e7i		−	0.0664015i	0.999449	−	0.0332007i	$-0.0105701\pi$
$744$	2.59793e8	−	1.44404e8i	0.630825	−	0.350640i
$745$	−4.03485e8			−0.975796
$746$	−1.36386e8			−0.328514
$747$		−	7.10667e7i		−	0.170492i
$748$	1.36724e8			0.326694
$749$	1.83071e8			0.435687
$750$			7.96294e8i			1.88751i
$751$	4.15738e8			0.981522			0.490761	−	0.871294i	$-0.336719\pi$
$751$	4.15738e8			0.981522			0.490761	+	0.871294i	$0.336719\pi$
$752$	−2.61323e8			−0.614502
$753$	−7.52385e8			−1.76220
$754$	−1.60552e8			−0.374544
$755$		−	4.35608e8i		−	1.01217i
$756$			2.40556e8i			0.556738i
$757$		−	2.03280e8i		−	0.468606i	−0.972164	−	0.234303i	$-0.924719\pi$
$757$		−	2.03280e8i		−	0.468606i	0.972164	−	0.234303i	$-0.0752807\pi$
$758$	6.12409e8			1.40616
$759$			7.37884e7i			0.168757i
$760$	1.02104e8			0.232595
$761$			3.21760e7i			0.0730092i	0.999333	+	0.0365046i	$0.0116224\pi$
$761$			3.21760e7i			0.0730092i	−0.999333	+	0.0365046i	$0.988378\pi$
$762$	−1.23682e9			−2.79539
$763$	3.21597e8			0.724000
$764$	5.40046e8			1.21102
$765$			3.88190e7i			0.0867081i
$766$			1.26737e8i			0.281980i
$767$			2.09960e8i			0.465319i
$768$		−	1.23546e8i		−	0.272739i
$769$	−2.43944e8			−0.536428			−0.268214	−	0.963359i	$-0.586433\pi$
$769$	−2.43944e8			−0.536428			−0.268214	+	0.963359i	$0.586433\pi$
$770$		−	1.36092e8i		−	0.298099i
$771$		−	4.66588e8i		−	1.01805i
$772$	2.98378e7			0.0648506
$773$			9.60716e7i			0.207997i	0.994577	+	0.103998i	$0.0331637\pi$
$773$			9.60716e7i			0.207997i	−0.994577	+	0.103998i	$0.966836\pi$
$774$		−	9.76505e7i		−	0.210597i
$775$	−9.15762e7	−	1.64752e8i	−0.196733	−	0.353936i
$776$	1.12734e8			0.241252
$777$	−4.00726e8			−0.854250
$778$			8.56619e8i			1.81907i
$779$	2.33855e8			0.494692
$780$	−8.65627e8			−1.82409
$781$			1.52609e8i			0.320352i
$782$	9.31599e7			0.194809
$783$	6.46668e7			0.134709
$784$	1.54969e8			0.321585
$785$	6.97954e8			1.44284
$786$		−	1.25053e9i		−	2.57529i
$787$		−	7.60481e8i		−	1.56014i	−0.625691	−	0.780071i	$-0.715183\pi$
$787$		−	7.60481e8i		−	1.56014i	0.625691	−	0.780071i	$-0.284817\pi$
$788$			2.75190e8i			0.562412i
$789$	9.02909e8			1.83829
$790$		−	1.22400e8i		−	0.248256i
$791$	3.63402e8			0.734274
$792$			4.31809e7i			0.0869193i
$793$	−1.26900e9			−2.54474
$794$	−1.28061e9			−2.55832
$795$	5.83890e8			1.16206
$796$			1.00160e8i			0.198589i
$797$		−	7.57082e8i		−	1.49544i	−0.664016	−	0.747718i	$-0.731150\pi$
$797$		−	7.57082e8i		−	1.49544i	0.664016	−	0.747718i	$-0.268850\pi$
$798$			1.95699e8i			0.385106i
$799$		−	3.32145e8i		−	0.651160i
$800$	−2.66831e8			−0.521154
$801$			8.41559e7i			0.163752i
$802$			1.36602e9i			2.64809i
$803$	4.31522e8			0.833406
$804$			2.72202e8i			0.523749i
$805$		−	5.43224e7i		−	0.104134i
$806$	1.06070e9	−	5.89583e8i	2.02576	−	1.12600i
$807$	−4.73381e8			−0.900722
$808$	3.69092e8			0.699682
$809$			1.97025e8i			0.372114i	0.982539	+	0.186057i	$0.0595709\pi$
$809$			1.97025e8i			0.372114i	−0.982539	+	0.186057i	$0.940429\pi$
$810$	7.58422e8			1.42710
$811$	6.36301e8			1.19289			0.596444	−	0.802654i	$-0.296580\pi$
$811$	6.36301e8			1.19289			0.596444	+	0.802654i	$0.296580\pi$
$812$		−	5.77866e7i		−	0.107934i
$813$	5.30533e8			0.987281
$814$	7.12397e8			1.32084
$815$	−1.56552e8			−0.289192
$816$	−1.10538e8			−0.203442
$817$			1.35038e8i			0.247623i
$818$			7.52194e7i			0.137426i
$819$		−	9.91705e7i		−	0.180522i
$820$	−6.35530e8			−1.15264
$821$		−	7.06468e8i		−	1.27662i	−0.769778	−	0.638312i	$-0.779633\pi$
$821$		−	7.06468e8i		−	1.27662i	0.769778	−	0.638312i	$-0.220367\pi$
$822$	5.21601e8			0.939124
$823$		−	2.47665e8i		−	0.444288i	−0.975014	−	0.222144i	$-0.928694\pi$
$823$		−	2.47665e8i		−	0.444288i	0.975014	−	0.222144i	$-0.0713056\pi$
$824$	−2.62322e8			−0.468871
$825$	1.34226e8			0.239043
$826$	−1.28998e8			−0.228899
$827$			1.04907e9i			1.85476i	0.374119	+	0.927381i	$0.377945\pi$
$827$			1.04907e9i			1.85476i	−0.374119	+	0.927381i	$0.622055\pi$
$828$			5.88287e7i			0.103633i
$829$		−	4.90830e8i		−	0.861525i	−0.902465	−	0.430762i	$-0.858245\pi$
$829$		−	4.90830e8i		−	0.861525i	0.902465	−	0.430762i	$-0.141755\pi$
$830$		−	4.55905e8i		−	0.797334i
$831$	4.67070e8			0.813915
$832$		−	1.36237e9i		−	2.36551i
$833$			1.96968e8i			0.340769i
$834$	−7.72657e8			−1.33195
$835$		−	2.79362e8i		−	0.479852i
$836$		−	2.03810e8i		−	0.348825i
$837$	−4.27225e8	+	2.37470e8i	−0.728586	+	0.404980i
$838$	2.15852e8			0.366795
$839$	−3.36732e8			−0.570162			−0.285081	−	0.958503i	$-0.592021\pi$
$839$	−3.36732e8			−0.570162			−0.285081	+	0.958503i	$0.592021\pi$
$840$		−	1.55821e8i		−	0.262899i
$841$	5.79289e8			0.973884
$842$	−1.28840e8			−0.215831
$843$			1.32316e8i			0.220867i
$844$	−5.45883e8			−0.907971
$845$	−5.70059e8			−0.944821
$846$	3.58034e8			0.591308
$847$	2.07345e8			0.341226
$848$			3.39198e8i			0.556245i
$849$			5.61272e8i			0.917170i
$850$		−	1.69464e8i		−	0.275944i
$851$	2.84360e8			0.461402
$852$			5.96383e8i			0.964287i
$853$	−2.30588e8			−0.371527			−0.185763	−	0.982595i	$-0.559476\pi$
$853$	−2.30588e8			−0.371527			−0.185763	+	0.982595i	$0.559476\pi$
$854$		−	7.79669e8i		−	1.25180i
$855$	5.78660e7			0.0925817
$856$	−3.72638e8			−0.594109
$857$	−1.21910e9			−1.93685			−0.968426	−	0.249302i	$-0.919799\pi$
$857$	−1.21910e9			−1.93685			−0.968426	+	0.249302i	$0.919799\pi$
$858$			8.64172e8i			1.36816i
$859$		−	1.00776e9i		−	1.58993i	−0.606653	−	0.794967i	$-0.707488\pi$
$859$		−	1.00776e9i		−	1.58993i	0.606653	−	0.794967i	$-0.292512\pi$
$860$		−	3.66983e8i		−	0.576967i
$861$		−	3.56888e8i		−	0.559142i
$862$	−3.69564e8			−0.576989
$863$		−	6.15762e8i		−	0.958033i	−0.877806	−	0.479017i	$-0.840993\pi$
$863$		−	6.15762e8i		−	0.958033i	0.877806	−	0.479017i	$-0.159007\pi$
$864$			6.91931e8i			1.07281i
$865$	−2.87534e8			−0.444263
$866$		−	1.49477e7i		−	0.0230155i
$867$			5.89978e8i			0.905271i
$868$	2.12205e8	+	3.81771e8i	0.324486	+	0.583772i
$869$	−7.15835e7			−0.109082
$870$	−1.42969e8			−0.217112
$871$			3.25615e8i			0.492776i
$872$	−6.54605e8			−0.987256
$873$	6.38908e7			0.0960275
$874$		−	1.38870e8i		−	0.208005i
$875$	−3.42844e8			−0.511768
$876$	1.68635e9			2.50862
$877$	7.94493e7			0.117785			0.0588926	−	0.998264i	$-0.481243\pi$
$877$	7.94493e7			0.117785			0.0588926	+	0.998264i	$0.481243\pi$
$878$	3.69134e8			0.545381
$879$			6.70149e8i			0.986744i
$880$		−	1.14587e8i		−	0.168146i
$881$			1.08793e9i			1.59101i	0.605950	+	0.795503i	$0.292793\pi$
$881$			1.08793e9i			1.59101i	−0.605950	+	0.795503i	$0.707207\pi$
$882$	−2.12320e8			−0.309447
$883$			4.58699e8i			0.666263i	0.942880	+	0.333132i	$0.108105\pi$
$883$			4.58699e8i			0.666263i	−0.942880	+	0.333132i	$0.891895\pi$
$884$	6.39152e8			0.925225
$885$			1.86966e8i			0.269732i
$886$	−5.03666e8			−0.724172
$887$	−1.05661e9			−1.51406			−0.757030	−	0.653380i	$-0.773350\pi$
$887$	−1.05661e9			−1.51406			−0.757030	+	0.653380i	$0.773350\pi$
$888$	8.15671e8			1.16487
$889$		−	5.32513e8i		−	0.757923i
$890$			5.39875e8i			0.765813i
$891$		−	4.43551e8i		−	0.627062i
$892$			1.42906e9i			2.01351i
$893$	−4.95117e8			−0.695269
$894$			1.57413e9i			2.20307i
$895$		−	1.73502e8i		−	0.242011i
$896$	3.99876e8			0.555905
$897$			3.44942e8i			0.477935i
$898$			4.70321e7i			0.0649480i
$899$	1.02628e8	−	5.70453e7i	0.141250	−	0.0785128i
$900$	1.07013e8			0.146795
$901$	−4.31126e8			−0.589427
$902$			6.34462e8i			0.864542i
$903$	2.06083e8			0.279884
$904$	−7.39698e8			−1.00127
$905$			9.86824e8i			1.33135i
$906$	−1.69946e9			−2.28521
$907$	1.36896e9			1.83471			0.917355	−	0.398069i	$-0.130320\pi$
$907$	1.36896e9			1.83471			0.917355	+	0.398069i	$0.130320\pi$
$908$	1.12304e9			1.50016
$909$	2.09178e8			0.278500
$910$		−	6.36196e8i		−	0.844242i
$911$			2.49994e7i			0.0330655i	0.999863	+	0.0165327i	$0.00526277\pi$
$911$			2.49994e7i			0.0330655i	−0.999863	+	0.0165327i	$0.994737\pi$
$912$			1.64774e8i			0.217223i
$913$	−2.66629e8			−0.350344
$914$		−	3.85909e7i		−	0.0505413i
$915$	−1.13003e9			−1.47511
$916$		−	1.31105e9i		−	1.70582i
$917$	5.38414e8			0.698246
$918$	−4.39446e8			−0.568038
$919$	−7.94410e8			−1.02352			−0.511762	−	0.859127i	$-0.671007\pi$
$919$	−7.94410e8			−1.02352			−0.511762	+	0.859127i	$0.671007\pi$
$920$			1.10572e8i			0.141998i
$921$			5.90879e8i			0.756344i
$922$			7.62002e8i			0.972218i
$923$			7.13409e8i			0.907263i
$924$	−3.11036e8			−0.394270
$925$		−	5.17270e8i		−	0.653570i
$926$		−	2.43510e9i		−	3.06679i
$927$	−1.48668e8			−0.186628
$928$		−	1.66216e8i		−	0.207984i
$929$		−	1.95534e8i		−	0.243880i	−0.992537	−	0.121940i	$-0.961088\pi$
$929$		−	1.95534e8i		−	0.243880i	0.992537	−	0.121940i	$-0.0389115\pi$
$930$	9.44535e8	−	5.25014e8i	1.17427	−	0.652712i
$931$	2.93612e8			0.363853
$932$	−6.35974e8			−0.785582
$933$		−	1.15399e9i		−	1.42088i
$934$	1.12916e9			1.38585
$935$	1.45641e8			0.178176
$936$			2.01860e8i			0.246163i
$937$	6.58304e7			0.0800217			0.0400108	−	0.999199i	$-0.487261\pi$
$937$	6.58304e7			0.0800217			0.0400108	+	0.999199i	$0.487261\pi$
$938$	−2.00056e8			−0.242406
$939$	−5.61477e8			−0.678165
$940$	1.34554e9			1.61999
$941$			1.66149e8i			0.199402i	0.995017	+	0.0997008i	$0.0317885\pi$
$941$			1.66149e8i			0.199402i	−0.995017	+	0.0997008i	$0.968211\pi$
$942$		−	2.72296e9i		−	3.25753i
$943$			2.53251e8i			0.302006i
$944$	−1.08614e8			−0.129113
$945$			2.56245e8i			0.303641i
$946$	−3.66366e8			−0.432755
$947$			1.32111e9i			1.55557i	0.628532	+	0.777784i	$0.283656\pi$
$947$			1.32111e9i			1.55557i	−0.628532	+	0.777784i	$0.716344\pi$
$948$	−2.79742e8			−0.328347
$949$	2.01726e9			2.36027
$950$	−2.52614e8			−0.294637
$951$			1.11844e8i			0.130039i
$952$			1.15053e8i			0.133349i
$953$		−	1.12882e9i		−	1.30421i	−0.758130	−	0.652103i	$-0.773887\pi$
$953$		−	1.12882e9i		−	1.30421i	0.758130	−	0.652103i	$-0.226113\pi$
$954$		−	4.64731e8i		−	0.535249i
$955$	5.75267e8			0.660480
$956$		−	1.60239e9i		−	1.83398i
$957$			8.36132e7i			0.0953979i
$958$	−1.81426e9			−2.06349
$959$			2.24575e8i			0.254628i
$960$		−	1.21317e9i		−	1.37122i
$961$	−4.68538e8	+	7.53747e8i	−0.527928	+	0.849289i
$962$	3.33027e9			3.74072
$963$	−2.11188e8			−0.236478
$964$		−	6.11939e8i		−	0.683088i
$965$	3.17837e7			0.0353690
$966$	−2.11930e8			−0.235105
$967$		−	1.00049e9i		−	1.10645i	−0.833032	−	0.553225i	$-0.813397\pi$
$967$		−	1.00049e9i		−	1.10645i	0.833032	−	0.553225i	$-0.186603\pi$
$968$	−4.22046e8			−0.465301
$969$	−2.09431e8			−0.230181
$970$	4.09871e8			0.449088
$971$	1.18447e9			1.29379			0.646896	−	0.762578i	$-0.276067\pi$
$971$	1.18447e9			1.29379			0.646896	+	0.762578i	$0.276067\pi$
$972$		−	6.50638e8i		−	0.708502i
$973$		−	3.32667e8i		−	0.361136i
$974$		−	1.10071e9i		−	1.19124i
$975$	6.27473e8			0.676989
$976$		−	6.56464e8i		−	0.706093i
$977$	−9.22899e8			−0.989624			−0.494812	−	0.869000i	$-0.664763\pi$
$977$	−9.22899e8			−0.989624			−0.494812	+	0.869000i	$0.664763\pi$
$978$			6.10764e8i			0.652915i
$979$	3.15737e8			0.336494
$980$	−7.97927e8			−0.847784
$981$	−3.70989e8			−0.392965
$982$			9.27655e8i			0.979607i
$983$			2.25206e8i			0.237094i	0.992948	+	0.118547i	$0.0378236\pi$
$983$			2.25206e8i			0.237094i	−0.992948	+	0.118547i	$0.962176\pi$
$984$			7.26438e8i			0.762453i
$985$			2.93138e8i			0.306735i
$986$	1.05564e8			0.110125
$987$			7.55600e8i			0.785851i
$988$		−	9.52761e8i		−	0.987900i
$989$	−1.46238e8			−0.151172
$990$			1.56994e8i			0.161799i
$991$			1.20205e9i			1.23510i	0.786532	+	0.617549i	$0.211874\pi$
$991$			1.20205e9i			1.23510i	−0.786532	+	0.617549i	$0.788126\pi$
$992$	6.10382e8	+	1.09812e9i	0.625268	+	1.12490i
$993$	1.41481e9			1.44494
$994$	−4.38314e8			−0.446299
$995$			1.06692e8i			0.108309i
$996$	−1.04196e9			−1.05457
$997$	−4.96926e8			−0.501425			−0.250713	−	0.968062i	$-0.580665\pi$
$997$	−4.96926e8			−0.501425			−0.250713	+	0.968062i	$0.580665\pi$
$998$		−	2.26469e9i		−	2.27833i
$999$	−1.34136e9			−1.34539

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	31.7.b.c.30.2	yes	12
3.2	odd	2		279.7.d.f.154.11		12
4.3	odd	2		496.7.e.c.433.4		12
31.30	odd	2	inner	31.7.b.c.30.1	✓	12
93.92	even	2		279.7.d.f.154.12		12
124.123	even	2		496.7.e.c.433.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.7.b.c.30.1	✓	12	31.30	odd	2	inner
31.7.b.c.30.2	yes	12	1.1	even	1	trivial
279.7.d.f.154.11		12	3.2	odd	2
279.7.d.f.154.12		12	93.92	even	2
496.7.e.c.433.4		12	4.3	odd	2
496.7.e.c.433.9		12	124.123	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 31 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	31.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-12.4307 q^{2} +30.2629i q^{3} +90.5217 q^{4} +96.4253 q^{5} -376.188i q^{6} +161.968 q^{7} -329.682 q^{8} -186.843 q^{9} -1198.63 q^{10} +701.001i q^{11} +2739.45i q^{12} +3277.00i q^{13} -2013.37 q^{14} +2918.11i q^{15} -1695.21 q^{16} -2154.64i q^{17} +2322.59 q^{18} -3211.85 q^{19} +8728.58 q^{20} +4901.61i q^{21} -8713.91i q^{22} -3478.24i q^{23} -9977.14i q^{24} -6327.15 q^{25} -40735.3i q^{26} +16407.2i q^{27} +14661.6 q^{28} -3941.36i q^{29} -36274.1i q^{30} +(14473.5 + 26038.8i) q^{31} +42172.3 q^{32} -21214.3 q^{33} +26783.7i q^{34} +15617.8 q^{35} -16913.4 q^{36} +81754.0i q^{37} +39925.4 q^{38} -99171.5 q^{39} -31789.7 q^{40} -72810.2 q^{41} -60930.4i q^{42} -42043.8i q^{43} +63455.8i q^{44} -18016.4 q^{45} +43236.8i q^{46} +154153. q^{47} -51302.1i q^{48} -91415.4 q^{49} +78650.8 q^{50} +65205.7 q^{51} +296640. i q^{52} -200092. i q^{53} -203953. i q^{54} +67594.2i q^{55} -53397.9 q^{56} -97199.8i q^{57} +48993.7i q^{58} +64070.9 q^{59} +264152. i q^{60} +387246. i q^{61} +(-179916. - 323680. i) q^{62} -30262.6 q^{63} -415737. q^{64} +315986. i q^{65} +263708. q^{66} +99363.7 q^{67} -195042. i q^{68} +105262. q^{69} -194140. q^{70} +217702. q^{71} +61598.9 q^{72} -615580. i q^{73} -1.01626e6i q^{74} -191478. i q^{75} -290742. q^{76} +113540. i q^{77} +1.23277e6 q^{78} +102116. i q^{79} -163462. q^{80} -632739. q^{81} +905080. q^{82} +380355. i q^{83} +443702. i q^{84} -207762. i q^{85} +522633. i q^{86} +119277. q^{87} -231108. i q^{88} -450409. i q^{89} +223956. q^{90} +530768. i q^{91} -314856. i q^{92} +(-788011. + 438011. i) q^{93} -1.91623e6 q^{94} -309703. q^{95} +1.27626e6i q^{96} -341949. q^{97} +1.13636e6 q^{98} -130977. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{2} + 122 q^{4} - 146 q^{5} + 6 q^{7} + 1142 q^{8} - 5668 q^{9} - 3004 q^{10} - 1312 q^{14} - 1102 q^{16} - 16382 q^{18} + 10430 q^{19} - 8052 q^{20} + 29618 q^{25} + 136504 q^{28} + 90076 q^{31}+ \cdots + 408810 q^{98}+O(q^{100}) \) 12 * q - 2 * q^2 + 122 * q^4 - 146 * q^5 + 6 * q^7 + 1142 * q^8 - 5668 * q^9 - 3004 * q^10 - 1312 * q^14 - 1102 * q^16 - 16382 * q^18 + 10430 * q^19 - 8052 * q^20 + 29618 * q^25 + 136504 * q^28 + 90076 * q^31 + 38406 * q^32 - 166248 * q^33 - 198898 * q^35 + 26982 * q^36 + 448196 * q^38 + 208224 * q^39 - 354932 * q^40 - 116690 * q^41 - 363818 * q^45 - 221120 * q^47 - 580182 * q^49 - 60318 * q^50 + 179784 * q^51 + 750496 * q^56 + 799510 * q^59 - 73070 * q^62 - 469874 * q^63 - 322286 * q^64 - 511884 * q^66 - 13744 * q^67 + 475128 * q^69 - 71312 * q^70 + 640678 * q^71 - 118870 * q^72 + 2044196 * q^76 + 1373868 * q^78 + 1356020 * q^80 - 678724 * q^81 - 488056 * q^82 - 1051200 * q^87 - 2598532 * q^90 - 2260440 * q^93 - 5045848 * q^94 + 3828206 * q^95 + 3708966 * q^97 + 408810 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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