LMFDB - Embedded newform 322.2.c.a.321.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [322,2,Mod(321,322)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(322, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("322.321");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 322 = 2 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	322.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:	$2.57118294509$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.2
Root			$2.57794i$ of defining polynomial
Character	$\chi$	$=$	322.321
Dual form			322.2.c.a.321.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.41421i q^{3} +1.00000 q^{4} +1.41421i q^{6} +2.64575 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.41421i q^{3} +1.00000 q^{4} +1.41421i q^{6} +2.64575 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.74166i q^{11} -1.41421i q^{12} -4.24264i q^{13} -2.64575 q^{14} +1.00000 q^{16} -1.00000 q^{18} +5.29150 q^{19} -3.74166i q^{21} -3.74166i q^{22} +(-3.00000 - 3.74166i) q^{23} +1.41421i q^{24} -5.00000 q^{25} +4.24264i q^{26} -5.65685i q^{27} +2.64575 q^{28} +6.00000 q^{29} -8.48528i q^{31} -1.00000 q^{32} +5.29150 q^{33} +1.00000 q^{36} +11.2250i q^{37} -5.29150 q^{38} -6.00000 q^{39} +5.65685i q^{41} +3.74166i q^{42} -11.2250i q^{43} +3.74166i q^{44} +(3.00000 + 3.74166i) q^{46} -2.82843i q^{47} -1.41421i q^{48} +7.00000 q^{49} +5.00000 q^{50} -4.24264i q^{52} +3.74166i q^{53} +5.65685i q^{54} -2.64575 q^{56} -7.48331i q^{57} -6.00000 q^{58} +9.89949i q^{59} -10.5830 q^{61} +8.48528i q^{62} +2.64575 q^{63} +1.00000 q^{64} -5.29150 q^{66} +11.2250i q^{67} +(-5.29150 + 4.24264i) q^{69} +6.00000 q^{71} -1.00000 q^{72} +8.48528i q^{73} -11.2250i q^{74} +7.07107i q^{75} +5.29150 q^{76} +9.89949i q^{77} +6.00000 q^{78} -5.00000 q^{81} -5.65685i q^{82} -15.8745 q^{83} -3.74166i q^{84} +11.2250i q^{86} -8.48528i q^{87} -3.74166i q^{88} -15.8745 q^{89} -11.2250i q^{91} +(-3.00000 - 3.74166i) q^{92} -12.0000 q^{93} +2.82843i q^{94} +1.41421i q^{96} +5.29150 q^{97} -7.00000 q^{98} +3.74166i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 4 * q^9	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 4 q^{9} + 4 q^{16} - 4 q^{18} - 12 q^{23} - 20 q^{25} + 24 q^{29} - 4 q^{32} + 4 q^{36} - 24 q^{39} + 12 q^{46} + 28 q^{49} + 20 q^{50} - 24 q^{58} + 4 q^{64} + 24 q^{71} - 4 q^{72} + 24 q^{78} - 20 q^{81} - 12 q^{92} - 48 q^{93} - 28 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 4 * q^9 + 4 * q^16 - 4 * q^18 - 12 * q^23 - 20 * q^25 + 24 * q^29 - 4 * q^32 + 4 * q^36 - 24 * q^39 + 12 * q^46 + 28 * q^49 + 20 * q^50 - 24 * q^58 + 4 * q^64 + 24 * q^71 - 4 * q^72 + 24 * q^78 - 20 * q^81 - 12 * q^92 - 48 * q^93 - 28 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/322\mathbb{Z}\right)^\times$.

$n$	$185$	$281$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		−	1.41421i		−	0.816497i	−0.912871	−	0.408248i	$-0.866140\pi$
$3$		−	1.41421i		−	0.816497i	0.912871	−	0.408248i	$-0.133860\pi$
$4$	1.00000			0.500000
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$			1.41421i			0.577350i
$7$	2.64575			1.00000
$7$	2.64575			1.00000
$8$	−1.00000			−0.353553
$9$	1.00000			0.333333
$10$		0			0
$11$			3.74166i			1.12815i	0.825723	+	0.564076i	$0.190768\pi$
$11$			3.74166i			1.12815i	−0.825723	+	0.564076i	$0.809232\pi$
$12$		−	1.41421i		−	0.408248i
$13$		−	4.24264i		−	1.17670i	−0.808608	−	0.588348i	$-0.799778\pi$
$13$		−	4.24264i		−	1.17670i	0.808608	−	0.588348i	$-0.200222\pi$
$14$	−2.64575			−0.707107
$15$		0			0
$16$	1.00000			0.250000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−1.00000			−0.235702
$19$	5.29150			1.21395			0.606977	−	0.794719i	$-0.292382\pi$
$19$	5.29150			1.21395			0.606977	+	0.794719i	$0.292382\pi$
$20$		0			0
$21$		−	3.74166i		−	0.816497i
$22$		−	3.74166i		−	0.797724i
$23$	−3.00000	−	3.74166i	−0.625543	−	0.780189i
$23$	−3.00000	−	3.74166i	−0.625543	−	0.780189i
$24$			1.41421i			0.288675i
$25$	−5.00000			−1.00000
$26$			4.24264i			0.832050i
$27$		−	5.65685i		−	1.08866i
$28$	2.64575			0.500000
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$		−	8.48528i		−	1.52400i	−0.647576	−	0.762001i	$-0.724217\pi$
$31$		−	8.48528i		−	1.52400i	0.647576	−	0.762001i	$-0.275783\pi$
$32$	−1.00000			−0.176777
$33$	5.29150			0.921132
$34$		0			0
$35$		0			0
$36$	1.00000			0.166667
$37$			11.2250i			1.84537i	0.385550	+	0.922687i	$0.374012\pi$
$37$			11.2250i			1.84537i	−0.385550	+	0.922687i	$0.625988\pi$
$38$	−5.29150			−0.858395
$39$	−6.00000			−0.960769
$40$		0			0
$41$			5.65685i			0.883452i	0.897150	+	0.441726i	$0.145634\pi$
$41$			5.65685i			0.883452i	−0.897150	+	0.441726i	$0.854366\pi$
$42$			3.74166i			0.577350i
$43$		−	11.2250i		−	1.71179i	−0.517148	−	0.855896i	$-0.673006\pi$
$43$		−	11.2250i		−	1.71179i	0.517148	−	0.855896i	$-0.326994\pi$
$44$			3.74166i			0.564076i
$45$		0			0
$46$	3.00000	+	3.74166i	0.442326	+	0.551677i
$47$		−	2.82843i		−	0.412568i	−0.978492	−	0.206284i	$-0.933863\pi$
$47$		−	2.82843i		−	0.412568i	0.978492	−	0.206284i	$-0.0661372\pi$
$48$		−	1.41421i		−	0.204124i
$49$	7.00000			1.00000
$50$	5.00000			0.707107
$51$		0			0
$52$		−	4.24264i		−	0.588348i
$53$			3.74166i			0.513956i	0.966417	+	0.256978i	$0.0827268\pi$
$53$			3.74166i			0.513956i	−0.966417	+	0.256978i	$0.917273\pi$
$54$			5.65685i			0.769800i
$55$		0			0
$56$	−2.64575			−0.353553
$57$		−	7.48331i		−	0.991189i
$58$	−6.00000			−0.787839
$59$			9.89949i			1.28880i	0.764687	+	0.644402i	$0.222894\pi$
$59$			9.89949i			1.28880i	−0.764687	+	0.644402i	$0.777106\pi$
$60$		0			0
$61$	−10.5830			−1.35501			−0.677507	−	0.735516i	$-0.736940\pi$
$61$	−10.5830			−1.35501			−0.677507	+	0.735516i	$0.736940\pi$
$62$			8.48528i			1.07763i
$63$	2.64575			0.333333
$64$	1.00000			0.125000
$65$		0			0
$66$	−5.29150			−0.651339
$67$			11.2250i			1.37135i	0.727908	+	0.685674i	$0.240493\pi$
$67$			11.2250i			1.37135i	−0.727908	+	0.685674i	$0.759507\pi$
$68$		0			0
$69$	−5.29150	+	4.24264i	−0.637022	+	0.510754i
$70$		0			0
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$	−1.00000			−0.117851
$73$			8.48528i			0.993127i	0.868000	+	0.496564i	$0.165405\pi$
$73$			8.48528i			0.993127i	−0.868000	+	0.496564i	$0.834595\pi$
$74$		−	11.2250i		−	1.30488i
$75$			7.07107i			0.816497i
$76$	5.29150			0.606977
$77$			9.89949i			1.12815i
$78$	6.00000			0.679366
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−5.00000			−0.555556
$82$		−	5.65685i		−	0.624695i
$83$	−15.8745			−1.74245			−0.871227	−	0.490881i	$-0.836675\pi$
$83$	−15.8745			−1.74245			−0.871227	+	0.490881i	$0.836675\pi$
$84$		−	3.74166i		−	0.408248i
$85$		0			0
$86$			11.2250i			1.21042i
$87$		−	8.48528i		−	0.909718i
$88$		−	3.74166i		−	0.398862i
$89$	−15.8745			−1.68269			−0.841347	−	0.540495i	$-0.818237\pi$
$89$	−15.8745			−1.68269			−0.841347	+	0.540495i	$0.818237\pi$
$90$		0			0
$91$		−	11.2250i		−	1.17670i
$92$	−3.00000	−	3.74166i	−0.312772	−	0.390095i
$93$	−12.0000			−1.24434
$94$			2.82843i			0.291730i
$95$		0			0
$96$			1.41421i			0.144338i
$97$	5.29150			0.537271			0.268635	−	0.963242i	$-0.413427\pi$
$97$	5.29150			0.537271			0.268635	+	0.963242i	$0.413427\pi$
$98$	−7.00000			−0.707107
$99$			3.74166i			0.376051i
$100$	−5.00000			−0.500000
$101$			9.89949i			0.985037i	0.870302	+	0.492518i	$0.163924\pi$
$101$			9.89949i			0.985037i	−0.870302	+	0.492518i	$0.836076\pi$
$102$		0			0
$103$	5.29150			0.521387			0.260694	−	0.965422i	$-0.416049\pi$
$103$	5.29150			0.521387			0.260694	+	0.965422i	$0.416049\pi$
$104$			4.24264i			0.416025i
$105$		0			0
$106$		−	3.74166i		−	0.363422i
$107$			3.74166i			0.361720i	0.983509	+	0.180860i	$0.0578880\pi$
$107$			3.74166i			0.361720i	−0.983509	+	0.180860i	$0.942112\pi$
$108$		−	5.65685i		−	0.544331i
$109$			11.2250i			1.07516i	0.843214	+	0.537579i	$0.180661\pi$
$109$			11.2250i			1.07516i	−0.843214	+	0.537579i	$0.819339\pi$
$110$		0			0
$111$	15.8745			1.50674
$112$	2.64575			0.250000
$113$		−	7.48331i		−	0.703971i	−0.936005	−	0.351986i	$-0.885507\pi$
$113$		−	7.48331i		−	0.703971i	0.936005	−	0.351986i	$-0.114493\pi$
$114$			7.48331i			0.700877i
$115$		0			0
$116$	6.00000			0.557086
$117$		−	4.24264i		−	0.392232i
$118$		−	9.89949i		−	0.911322i
$119$		0			0
$120$		0			0
$121$	−3.00000			−0.272727
$122$	10.5830			0.958140
$123$	8.00000			0.721336
$124$		−	8.48528i		−	0.762001i
$125$		0			0
$126$	−2.64575			−0.235702
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000			−0.0883883
$129$	−15.8745			−1.39767
$130$		0			0
$131$		−	7.07107i		−	0.617802i	−0.951094	−	0.308901i	$-0.900039\pi$
$131$		−	7.07107i		−	0.617802i	0.951094	−	0.308901i	$-0.0999612\pi$
$132$	5.29150			0.460566
$133$	14.0000			1.21395
$134$		−	11.2250i		−	0.969690i
$135$		0			0
$136$		0			0
$137$		−	7.48331i		−	0.639343i	−0.947528	−	0.319671i	$-0.896427\pi$
$137$		−	7.48331i		−	0.639343i	0.947528	−	0.319671i	$-0.103573\pi$
$138$	5.29150	−	4.24264i	0.450443	−	0.361158i
$139$			4.24264i			0.359856i	0.983680	+	0.179928i	$0.0575865\pi$
$139$			4.24264i			0.359856i	−0.983680	+	0.179928i	$0.942414\pi$
$140$		0			0
$141$	−4.00000			−0.336861
$142$	−6.00000			−0.503509
$143$	15.8745			1.32749
$144$	1.00000			0.0833333
$145$		0			0
$146$		−	8.48528i		−	0.702247i
$147$		−	9.89949i		−	0.816497i
$148$			11.2250i			0.922687i
$149$			3.74166i			0.306529i	0.988185	+	0.153264i	$0.0489786\pi$
$149$			3.74166i			0.306529i	−0.988185	+	0.153264i	$0.951021\pi$
$150$		−	7.07107i		−	0.577350i
$151$	2.00000			0.162758			0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000			0.162758			0.0813788	+	0.996683i	$0.474068\pi$
$152$	−5.29150			−0.429198
$153$		0			0
$154$		−	9.89949i		−	0.797724i
$155$		0			0
$156$	−6.00000			−0.480384
$157$	21.1660			1.68923			0.844616	−	0.535373i	$-0.179829\pi$
$157$	21.1660			1.68923			0.844616	+	0.535373i	$0.179829\pi$
$158$		0			0
$159$	5.29150			0.419643
$160$		0			0
$161$	−7.93725	−	9.89949i	−0.625543	−	0.780189i
$162$	5.00000			0.392837
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$			5.65685i			0.441726i
$165$		0			0
$166$	15.8745			1.23210
$167$		−	19.7990i		−	1.53209i	−0.642786	−	0.766046i	$-0.722221\pi$
$167$		−	19.7990i		−	1.53209i	0.642786	−	0.766046i	$-0.277779\pi$
$168$			3.74166i			0.288675i
$169$	−5.00000			−0.384615
$170$		0			0
$171$	5.29150			0.404651
$172$		−	11.2250i		−	0.855896i
$173$			9.89949i			0.752645i	0.926489	+	0.376322i	$0.122811\pi$
$173$			9.89949i			0.752645i	−0.926489	+	0.376322i	$0.877189\pi$
$174$			8.48528i			0.643268i
$175$	−13.2288			−1.00000
$176$			3.74166i			0.282038i
$177$	14.0000			1.05230
$178$	15.8745			1.18984
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	−10.5830			−0.786629			−0.393314	−	0.919404i	$-0.628672\pi$
$181$	−10.5830			−0.786629			−0.393314	+	0.919404i	$0.628672\pi$
$182$			11.2250i			0.832050i
$183$			14.9666i			1.10637i
$184$	3.00000	+	3.74166i	0.221163	+	0.275839i
$185$		0			0
$186$	12.0000			0.879883
$187$		0			0
$188$		−	2.82843i		−	0.206284i
$189$		−	14.9666i		−	1.08866i
$190$		0			0
$191$			14.9666i			1.08295i	0.840718	+	0.541474i	$0.182133\pi$
$191$			14.9666i			1.08295i	−0.840718	+	0.541474i	$0.817867\pi$
$192$		−	1.41421i		−	0.102062i
$193$	−16.0000			−1.15171			−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000			−1.15171			−0.575853	+	0.817554i	$0.695330\pi$
$194$	−5.29150			−0.379908
$195$		0			0
$196$	7.00000			0.500000
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		−	3.74166i		−	0.265908i
$199$	−10.5830			−0.750209			−0.375105	−	0.926982i	$-0.622393\pi$
$199$	−10.5830			−0.750209			−0.375105	+	0.926982i	$0.622393\pi$
$200$	5.00000			0.353553
$201$	15.8745			1.11970
$202$		−	9.89949i		−	0.696526i
$203$	15.8745			1.11417
$204$		0			0
$205$		0			0
$206$	−5.29150			−0.368676
$207$	−3.00000	−	3.74166i	−0.208514	−	0.260063i
$208$		−	4.24264i		−	0.294174i
$209$			19.7990i			1.36952i
$210$		0			0
$211$	−16.0000			−1.10149			−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000			−1.10149			−0.550743	+	0.834675i	$0.685655\pi$
$212$			3.74166i			0.256978i
$213$		−	8.48528i		−	0.581402i
$214$		−	3.74166i		−	0.255774i
$215$		0			0
$216$			5.65685i			0.384900i
$217$		−	22.4499i		−	1.52400i
$218$		−	11.2250i		−	0.760251i
$219$	12.0000			0.810885
$220$		0			0
$221$		0			0
$222$	−15.8745			−1.06543
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$	−2.64575			−0.176777
$225$	−5.00000			−0.333333
$226$			7.48331i			0.497783i
$227$	−15.8745			−1.05363			−0.526814	−	0.849981i	$-0.676614\pi$
$227$	−15.8745			−1.05363			−0.526814	+	0.849981i	$0.676614\pi$
$228$		−	7.48331i		−	0.495595i
$229$	−10.5830			−0.699345			−0.349672	−	0.936872i	$-0.613707\pi$
$229$	−10.5830			−0.699345			−0.349672	+	0.936872i	$0.613707\pi$
$230$		0			0
$231$	14.0000			0.921132
$232$	−6.00000			−0.393919
$233$	−18.0000			−1.17922			−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000			−1.17922			−0.589610	+	0.807688i	$0.700718\pi$
$234$			4.24264i			0.277350i
$235$		0			0
$236$			9.89949i			0.644402i
$237$		0			0
$238$		0			0
$239$	−6.00000			−0.388108			−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000			−0.388108			−0.194054	+	0.980991i	$0.562164\pi$
$240$		0			0
$241$	5.29150			0.340856			0.170428	−	0.985370i	$-0.445485\pi$
$241$	5.29150			0.340856			0.170428	+	0.985370i	$0.445485\pi$
$242$	3.00000			0.192847
$243$		−	9.89949i		−	0.635053i
$244$	−10.5830			−0.677507
$245$		0			0
$246$	−8.00000			−0.510061
$247$		−	22.4499i		−	1.42846i
$248$			8.48528i			0.538816i
$249$			22.4499i			1.42271i
$250$		0			0
$251$	15.8745			1.00199			0.500995	−	0.865450i	$-0.332967\pi$
$251$	15.8745			1.00199			0.500995	+	0.865450i	$0.332967\pi$
$252$	2.64575			0.166667
$253$	14.0000	−	11.2250i	0.880172	−	0.705708i
$254$	−8.00000			−0.501965
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	2.82843i		−	0.176432i	−0.996101	−	0.0882162i	$-0.971883\pi$
$257$		−	2.82843i		−	0.176432i	0.996101	−	0.0882162i	$-0.0281166\pi$
$258$	15.8745			0.988304
$259$			29.6985i			1.84537i
$260$		0			0
$261$	6.00000			0.371391
$262$			7.07107i			0.436852i
$263$		−	7.48331i		−	0.461441i	−0.973020	−	0.230720i	$-0.925892\pi$
$263$		−	7.48331i		−	0.461441i	0.973020	−	0.230720i	$-0.0741083\pi$
$264$	−5.29150			−0.325669
$265$		0			0
$266$	−14.0000			−0.858395
$267$			22.4499i			1.37391i
$268$			11.2250i			0.685674i
$269$			18.3848i			1.12094i	0.828175	+	0.560470i	$0.189379\pi$
$269$			18.3848i			1.12094i	−0.828175	+	0.560470i	$0.810621\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	−15.8745			−0.960769
$274$			7.48331i			0.452084i
$275$		−	18.7083i		−	1.12815i
$276$	−5.29150	+	4.24264i	−0.318511	+	0.255377i
$277$	2.00000			0.120168			0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000			0.120168			0.0600842	+	0.998193i	$0.480863\pi$
$278$		−	4.24264i		−	0.254457i
$279$		−	8.48528i		−	0.508001i
$280$		0			0
$281$			14.9666i			0.892834i	0.894825	+	0.446417i	$0.147300\pi$
$281$			14.9666i			0.892834i	−0.894825	+	0.446417i	$0.852700\pi$
$282$	4.00000			0.238197
$283$	−26.4575			−1.57274			−0.786368	−	0.617758i	$-0.788041\pi$
$283$	−26.4575			−1.57274			−0.786368	+	0.617758i	$0.788041\pi$
$284$	6.00000			0.356034
$285$		0			0
$286$	−15.8745			−0.938679
$287$			14.9666i			0.883452i
$288$	−1.00000			−0.0589256
$289$	−17.0000			−1.00000
$290$		0			0
$291$		−	7.48331i		−	0.438680i
$292$			8.48528i			0.496564i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$			9.89949i			0.577350i
$295$		0			0
$296$		−	11.2250i		−	0.652438i
$297$	21.1660			1.22818
$298$		−	3.74166i		−	0.216748i
$299$	−15.8745	+	12.7279i	−0.918046	+	0.736075i
$300$			7.07107i			0.408248i
$301$		−	29.6985i		−	1.71179i
$302$	−2.00000			−0.115087
$303$	14.0000			0.804279
$304$	5.29150			0.303488
$305$		0			0
$306$		0			0
$307$			29.6985i			1.69498i	0.530810	+	0.847491i	$0.321888\pi$
$307$			29.6985i			1.69498i	−0.530810	+	0.847491i	$0.678112\pi$
$308$			9.89949i			0.564076i
$309$		−	7.48331i		−	0.425711i
$310$		0			0
$311$		−	28.2843i		−	1.60385i	−0.597422	−	0.801927i	$-0.703808\pi$
$311$		−	28.2843i		−	1.60385i	0.597422	−	0.801927i	$-0.296192\pi$
$312$	6.00000			0.339683
$313$	−26.4575			−1.49547			−0.747734	−	0.663999i	$-0.768858\pi$
$313$	−26.4575			−1.49547			−0.747734	+	0.663999i	$0.768858\pi$
$314$	−21.1660			−1.19447
$315$		0			0
$316$		0			0
$317$	6.00000			0.336994			0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000			0.336994			0.168497	+	0.985702i	$0.446109\pi$
$318$	−5.29150			−0.296733
$319$			22.4499i			1.25696i
$320$		0			0
$321$	5.29150			0.295343
$322$	7.93725	+	9.89949i	0.442326	+	0.551677i
$323$		0			0
$324$	−5.00000			−0.277778
$325$			21.2132i			1.17670i
$326$	4.00000			0.221540
$327$	15.8745			0.877862
$328$		−	5.65685i		−	0.312348i
$329$		−	7.48331i		−	0.412568i
$330$		0			0
$331$	32.0000			1.75888			0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000			1.75888			0.879440	+	0.476011i	$0.157918\pi$
$332$	−15.8745			−0.871227
$333$			11.2250i			0.615125i
$334$			19.7990i			1.08335i
$335$		0			0
$336$		−	3.74166i		−	0.204124i
$337$			22.4499i			1.22293i	0.791273	+	0.611463i	$0.209419\pi$
$337$			22.4499i			1.22293i	−0.791273	+	0.611463i	$0.790581\pi$
$338$	5.00000			0.271964
$339$	−10.5830			−0.574790
$340$		0			0
$341$	31.7490			1.71931
$342$	−5.29150			−0.286132
$343$	18.5203			1.00000
$344$			11.2250i			0.605210i
$345$		0			0
$346$		−	9.89949i		−	0.532200i
$347$	24.0000			1.28839			0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000			1.28839			0.644194	+	0.764862i	$0.277193\pi$
$348$		−	8.48528i		−	0.454859i
$349$		−	4.24264i		−	0.227103i	−0.993532	−	0.113552i	$-0.963777\pi$
$349$		−	4.24264i		−	0.227103i	0.993532	−	0.113552i	$-0.0362227\pi$
$350$	13.2288			0.707107
$351$	−24.0000			−1.28103
$352$		−	3.74166i		−	0.199431i
$353$		−	28.2843i		−	1.50542i	−0.658352	−	0.752710i	$-0.728746\pi$
$353$		−	28.2843i		−	1.50542i	0.658352	−	0.752710i	$-0.271254\pi$
$354$	−14.0000			−0.744092
$355$		0			0
$356$	−15.8745			−0.841347
$357$		0			0
$358$	12.0000			0.634220
$359$		−	29.9333i		−	1.57982i	−0.613225	−	0.789908i	$-0.710128\pi$
$359$		−	29.9333i		−	1.57982i	0.613225	−	0.789908i	$-0.289872\pi$
$360$		0			0
$361$	9.00000			0.473684
$362$	10.5830			0.556230
$363$			4.24264i			0.222681i
$364$		−	11.2250i		−	0.588348i
$365$		0			0
$366$		−	14.9666i		−	0.782318i
$367$	5.29150			0.276214			0.138107	−	0.990417i	$-0.455898\pi$
$367$	5.29150			0.276214			0.138107	+	0.990417i	$0.455898\pi$
$368$	−3.00000	−	3.74166i	−0.156386	−	0.195047i
$369$			5.65685i			0.294484i
$370$		0			0
$371$			9.89949i			0.513956i
$372$	−12.0000			−0.622171
$373$			11.2250i			0.581207i	0.956844	+	0.290604i	$0.0938561\pi$
$373$			11.2250i			0.581207i	−0.956844	+	0.290604i	$0.906144\pi$
$374$		0			0
$375$		0			0
$376$			2.82843i			0.145865i
$377$		−	25.4558i		−	1.31104i
$378$			14.9666i			0.769800i
$379$			11.2250i			0.576588i	0.957542	+	0.288294i	$0.0930881\pi$
$379$			11.2250i			0.576588i	−0.957542	+	0.288294i	$0.906912\pi$
$380$		0			0
$381$		−	11.3137i		−	0.579619i
$382$		−	14.9666i		−	0.765759i
$383$	15.8745			0.811149			0.405575	−	0.914062i	$-0.367071\pi$
$383$	15.8745			0.811149			0.405575	+	0.914062i	$0.367071\pi$
$384$			1.41421i			0.0721688i
$385$		0			0
$386$	16.0000			0.814379
$387$		−	11.2250i		−	0.570597i
$388$	5.29150			0.268635
$389$		−	18.7083i		−	0.948548i	−0.880377	−	0.474274i	$-0.842711\pi$
$389$		−	18.7083i		−	0.948548i	0.880377	−	0.474274i	$-0.157289\pi$
$390$		0			0
$391$		0			0
$392$	−7.00000			−0.353553
$393$	−10.0000			−0.504433
$394$	−6.00000			−0.302276
$395$		0			0
$396$			3.74166i			0.188025i
$397$		−	12.7279i		−	0.638796i	−0.947621	−	0.319398i	$-0.896519\pi$
$397$		−	12.7279i		−	0.638796i	0.947621	−	0.319398i	$-0.103481\pi$
$398$	10.5830			0.530478
$399$		−	19.7990i		−	0.991189i
$400$	−5.00000			−0.250000
$401$			14.9666i			0.747398i	0.927550	+	0.373699i	$0.121911\pi$
$401$			14.9666i			0.747398i	−0.927550	+	0.373699i	$0.878089\pi$
$402$	−15.8745			−0.791748
$403$	−36.0000			−1.79329
$404$			9.89949i			0.492518i
$405$		0			0
$406$	−15.8745			−0.787839
$407$	−42.0000			−2.08186
$408$		0			0
$409$			8.48528i			0.419570i	0.977748	+	0.209785i	$0.0672764\pi$
$409$			8.48528i			0.419570i	−0.977748	+	0.209785i	$0.932724\pi$
$410$		0			0
$411$	−10.5830			−0.522021
$412$	5.29150			0.260694
$413$			26.1916i			1.28880i
$414$	3.00000	+	3.74166i	0.147442	+	0.183892i
$415$		0			0
$416$			4.24264i			0.208013i
$417$	6.00000			0.293821
$418$		−	19.7990i		−	0.968400i
$419$	15.8745			0.775520			0.387760	−	0.921760i	$-0.373249\pi$
$419$	15.8745			0.775520			0.387760	+	0.921760i	$0.373249\pi$
$420$		0			0
$421$		−	11.2250i		−	0.547072i	−0.961862	−	0.273536i	$-0.911807\pi$
$421$		−	11.2250i		−	0.547072i	0.961862	−	0.273536i	$-0.0881932\pi$
$422$	16.0000			0.778868
$423$		−	2.82843i		−	0.137523i
$424$		−	3.74166i		−	0.181711i
$425$		0			0
$426$			8.48528i			0.411113i
$427$	−28.0000			−1.35501
$428$			3.74166i			0.180860i
$429$		−	22.4499i		−	1.08389i
$430$		0			0
$431$		−	29.9333i		−	1.44183i	−0.693021	−	0.720917i	$-0.743721\pi$
$431$		−	29.9333i		−	1.44183i	0.693021	−	0.720917i	$-0.256279\pi$
$432$		−	5.65685i		−	0.272166i
$433$	21.1660			1.01717			0.508587	−	0.861011i	$-0.330168\pi$
$433$	21.1660			1.01717			0.508587	+	0.861011i	$0.330168\pi$
$434$			22.4499i			1.07763i
$435$		0			0
$436$			11.2250i			0.537579i
$437$	−15.8745	−	19.7990i	−0.759381	−	0.947114i
$438$	−12.0000			−0.573382
$439$			16.9706i			0.809961i	0.914325	+	0.404980i	$0.132722\pi$
$439$			16.9706i			0.809961i	−0.914325	+	0.404980i	$0.867278\pi$
$440$		0			0
$441$	7.00000			0.333333
$442$		0			0
$443$	24.0000			1.14027			0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000			1.14027			0.570137	+	0.821549i	$0.306890\pi$
$444$	15.8745			0.753371
$445$		0			0
$446$		0			0
$447$	5.29150			0.250279
$448$	2.64575			0.125000
$449$	12.0000			0.566315			0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000			0.566315			0.283158	+	0.959073i	$0.408618\pi$
$450$	5.00000			0.235702
$451$	−21.1660			−0.996669
$452$		−	7.48331i		−	0.351986i
$453$		−	2.82843i		−	0.132891i
$454$	15.8745			0.745028
$455$		0			0
$456$			7.48331i			0.350438i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	10.5830			0.494511
$459$		0			0
$460$		0			0
$461$		−	15.5563i		−	0.724531i	−0.932075	−	0.362266i	$-0.882003\pi$
$461$		−	15.5563i		−	0.724531i	0.932075	−	0.362266i	$-0.117997\pi$
$462$	−14.0000			−0.651339
$463$	26.0000			1.20832			0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000			1.20832			0.604161	+	0.796862i	$0.293508\pi$
$464$	6.00000			0.278543
$465$		0			0
$466$	18.0000			0.833834
$467$	−15.8745			−0.734585			−0.367292	−	0.930106i	$-0.619715\pi$
$467$	−15.8745			−0.734585			−0.367292	+	0.930106i	$0.619715\pi$
$468$		−	4.24264i		−	0.196116i
$469$			29.6985i			1.37135i
$470$		0			0
$471$		−	29.9333i		−	1.37925i
$472$		−	9.89949i		−	0.455661i
$473$	42.0000			1.93116
$474$		0			0
$475$	−26.4575			−1.21395
$476$		0			0
$477$			3.74166i			0.171319i
$478$	6.00000			0.274434
$479$	15.8745			0.725325			0.362662	−	0.931921i	$-0.381868\pi$
$479$	15.8745			0.725325			0.362662	+	0.931921i	$0.381868\pi$
$480$		0			0
$481$	47.6235			2.17145
$482$	−5.29150			−0.241021
$483$	−14.0000	+	11.2250i	−0.637022	+	0.510754i
$484$	−3.00000			−0.136364
$485$		0			0
$486$			9.89949i			0.449050i
$487$	−16.0000			−0.725029			−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000			−0.725029			−0.362515	+	0.931978i	$0.618082\pi$
$488$	10.5830			0.479070
$489$			5.65685i			0.255812i
$490$		0			0
$491$	−36.0000			−1.62466			−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000			−1.62466			−0.812329	+	0.583200i	$0.801800\pi$
$492$	8.00000			0.360668
$493$		0			0
$494$			22.4499i			1.01007i
$495$		0			0
$496$		−	8.48528i		−	0.381000i
$497$	15.8745			0.712069
$498$		−	22.4499i		−	1.00601i
$499$	32.0000			1.43252			0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000			1.43252			0.716258	+	0.697835i	$0.245853\pi$
$500$		0			0
$501$	−28.0000			−1.25095
$502$	−15.8745			−0.708514
$503$	−15.8745			−0.707809			−0.353905	−	0.935282i	$-0.615146\pi$
$503$	−15.8745			−0.707809			−0.353905	+	0.935282i	$0.615146\pi$
$504$	−2.64575			−0.117851
$505$		0			0
$506$	−14.0000	+	11.2250i	−0.622376	+	0.499011i
$507$			7.07107i			0.314037i
$508$	8.00000			0.354943
$509$		−	32.5269i		−	1.44173i	−0.693075	−	0.720865i	$-0.743745\pi$
$509$		−	32.5269i		−	1.44173i	0.693075	−	0.720865i	$-0.256255\pi$
$510$		0			0
$511$			22.4499i			0.993127i
$512$	−1.00000			−0.0441942
$513$		−	29.9333i		−	1.32159i
$514$			2.82843i			0.124757i
$515$		0			0
$516$	−15.8745			−0.698836
$517$	10.5830			0.465440
$518$		−	29.6985i		−	1.30488i
$519$	14.0000			0.614532
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$	−6.00000			−0.262613
$523$	−26.4575			−1.15691			−0.578453	−	0.815716i	$-0.696343\pi$
$523$	−26.4575			−1.15691			−0.578453	+	0.815716i	$0.696343\pi$
$524$		−	7.07107i		−	0.308901i
$525$			18.7083i			0.816497i
$526$			7.48331i			0.326288i
$527$		0			0
$528$	5.29150			0.230283
$529$	−5.00000	+	22.4499i	−0.217391	+	0.976085i
$530$		0			0
$531$			9.89949i			0.429601i
$532$	14.0000			0.606977
$533$	24.0000			1.03956
$534$		−	22.4499i		−	0.971504i
$535$		0			0
$536$		−	11.2250i		−	0.484845i
$537$			16.9706i			0.732334i
$538$		−	18.3848i		−	0.792624i
$539$			26.1916i			1.12815i
$540$		0			0
$541$	−10.0000			−0.429934			−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000			−0.429934			−0.214967	+	0.976621i	$0.568964\pi$
$542$		0			0
$543$			14.9666i			0.642280i
$544$		0			0
$545$		0			0
$546$	15.8745			0.679366
$547$	8.00000			0.342055			0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000			0.342055			0.171028	+	0.985266i	$0.445291\pi$
$548$		−	7.48331i		−	0.319671i
$549$	−10.5830			−0.451672
$550$			18.7083i			0.797724i
$551$	31.7490			1.35255
$552$	5.29150	−	4.24264i	0.225221	−	0.180579i
$553$		0			0
$554$	−2.00000			−0.0849719
$555$		0			0
$556$			4.24264i			0.179928i
$557$			3.74166i			0.158539i	0.996853	+	0.0792696i	$0.0252588\pi$
$557$			3.74166i			0.158539i	−0.996853	+	0.0792696i	$0.974741\pi$
$558$			8.48528i			0.359211i
$559$	−47.6235			−2.01426
$560$		0			0
$561$		0			0
$562$		−	14.9666i		−	0.631329i
$563$	15.8745			0.669031			0.334515	−	0.942390i	$-0.391427\pi$
$563$	15.8745			0.669031			0.334515	+	0.942390i	$0.391427\pi$
$564$	−4.00000			−0.168430
$565$		0			0
$566$	26.4575			1.11209
$567$	−13.2288			−0.555556
$568$	−6.00000			−0.251754
$569$			14.9666i			0.627434i	0.949517	+	0.313717i	$0.101574\pi$
$569$			14.9666i			0.627434i	−0.949517	+	0.313717i	$0.898426\pi$
$570$		0			0
$571$		−	33.6749i		−	1.40925i	−0.709579	−	0.704626i	$-0.751115\pi$
$571$		−	33.6749i		−	1.40925i	0.709579	−	0.704626i	$-0.248885\pi$
$572$	15.8745			0.663747
$573$	21.1660			0.884223
$574$		−	14.9666i		−	0.624695i
$575$	15.0000	+	18.7083i	0.625543	+	0.780189i
$576$	1.00000			0.0416667
$577$		−	8.48528i		−	0.353247i	−0.984278	−	0.176623i	$-0.943483\pi$
$577$		−	8.48528i		−	0.353247i	0.984278	−	0.176623i	$-0.0565175\pi$
$578$	17.0000			0.707107
$579$			22.6274i			0.940363i
$580$		0			0
$581$	−42.0000			−1.74245
$582$			7.48331i			0.310193i
$583$	−14.0000			−0.579821
$584$		−	8.48528i		−	0.351123i
$585$		0			0
$586$		0			0
$587$			35.3553i			1.45927i	0.683836	+	0.729636i	$0.260310\pi$
$587$			35.3553i			1.45927i	−0.683836	+	0.729636i	$0.739690\pi$
$588$		−	9.89949i		−	0.408248i
$589$		−	44.8999i		−	1.85007i
$590$		0			0
$591$		−	8.48528i		−	0.349038i
$592$			11.2250i			0.461344i
$593$		−	19.7990i		−	0.813047i	−0.913640	−	0.406524i	$-0.866741\pi$
$593$		−	19.7990i		−	0.813047i	0.913640	−	0.406524i	$-0.133259\pi$
$594$	−21.1660			−0.868452
$595$		0			0
$596$			3.74166i			0.153264i
$597$			14.9666i			0.612543i
$598$	15.8745	−	12.7279i	0.649157	−	0.520483i
$599$	24.0000			0.980613			0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000			0.980613			0.490307	+	0.871550i	$0.336885\pi$
$600$		−	7.07107i		−	0.288675i
$601$		−	25.4558i		−	1.03837i	−0.854663	−	0.519183i	$-0.826236\pi$
$601$		−	25.4558i		−	1.03837i	0.854663	−	0.519183i	$-0.173764\pi$
$602$			29.6985i			1.21042i
$603$			11.2250i			0.457116i
$604$	2.00000			0.0813788
$605$		0			0
$606$	−14.0000			−0.568711
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$	−5.29150			−0.214599
$609$		−	22.4499i		−	0.909718i
$610$		0			0
$611$	−12.0000			−0.485468
$612$		0			0
$613$			11.2250i			0.453372i	0.973968	+	0.226686i	$0.0727892\pi$
$613$			11.2250i			0.453372i	−0.973968	+	0.226686i	$0.927211\pi$
$614$		−	29.6985i		−	1.19853i
$615$		0			0
$616$		−	9.89949i		−	0.398862i
$617$		−	7.48331i		−	0.301267i	−0.988590	−	0.150633i	$-0.951869\pi$
$617$		−	7.48331i		−	0.301267i	0.988590	−	0.150633i	$-0.0481313\pi$
$618$			7.48331i			0.301023i
$619$	5.29150			0.212683			0.106342	−	0.994330i	$-0.466086\pi$
$619$	5.29150			0.212683			0.106342	+	0.994330i	$0.466086\pi$
$620$		0			0
$621$	−21.1660	+	16.9706i	−0.849363	+	0.681005i
$622$			28.2843i			1.13410i
$623$	−42.0000			−1.68269
$624$	−6.00000			−0.240192
$625$	25.0000			1.00000
$626$	26.4575			1.05745
$627$	28.0000			1.11821
$628$	21.1660			0.844616
$629$		0			0
$630$		0			0
$631$			22.4499i			0.893718i	0.894604	+	0.446859i	$0.147457\pi$
$631$			22.4499i			0.893718i	−0.894604	+	0.446859i	$0.852543\pi$
$632$		0			0
$633$			22.6274i			0.899359i
$634$	−6.00000			−0.238290
$635$		0			0
$636$	5.29150			0.209822
$637$		−	29.6985i		−	1.17670i
$638$		−	22.4499i		−	0.888802i
$639$	6.00000			0.237356
$640$		0			0
$641$		−	29.9333i		−	1.18229i	−0.806564	−	0.591146i	$-0.798676\pi$
$641$		−	29.9333i		−	1.18229i	0.806564	−	0.591146i	$-0.201324\pi$
$642$	−5.29150			−0.208839
$643$	5.29150			0.208676			0.104338	−	0.994542i	$-0.466728\pi$
$643$	5.29150			0.208676			0.104338	+	0.994542i	$0.466728\pi$
$644$	−7.93725	−	9.89949i	−0.312772	−	0.390095i
$645$		0			0
$646$		0			0
$647$			31.1127i			1.22317i	0.791180	+	0.611583i	$0.209467\pi$
$647$			31.1127i			1.22317i	−0.791180	+	0.611583i	$0.790533\pi$
$648$	5.00000			0.196419
$649$	−37.0405			−1.45397
$650$		−	21.2132i		−	0.832050i
$651$	−31.7490			−1.24434
$652$	−4.00000			−0.156652
$653$	18.0000			0.704394			0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000			0.704394			0.352197	+	0.935926i	$0.385435\pi$
$654$	−15.8745			−0.620742
$655$		0			0
$656$			5.65685i			0.220863i
$657$			8.48528i			0.331042i
$658$			7.48331i			0.291730i
$659$		−	18.7083i		−	0.728771i	−0.931248	−	0.364386i	$-0.881279\pi$
$659$		−	18.7083i		−	0.728771i	0.931248	−	0.364386i	$-0.118721\pi$
$660$		0			0
$661$	21.1660			0.823262			0.411631	−	0.911351i	$-0.364959\pi$
$661$	21.1660			0.823262			0.411631	+	0.911351i	$0.364959\pi$
$662$	−32.0000			−1.24372
$663$		0			0
$664$	15.8745			0.616050
$665$		0			0
$666$		−	11.2250i		−	0.434959i
$667$	−18.0000	−	22.4499i	−0.696963	−	0.869265i
$668$		−	19.7990i		−	0.766046i
$669$		0			0
$670$		0			0
$671$		−	39.5980i		−	1.52866i
$672$			3.74166i			0.144338i
$673$	20.0000			0.770943			0.385472	−	0.922720i	$-0.374039\pi$
$673$	20.0000			0.770943			0.385472	+	0.922720i	$0.374039\pi$
$674$		−	22.4499i		−	0.864740i
$675$			28.2843i			1.08866i
$676$	−5.00000			−0.192308
$677$	−31.7490			−1.22021			−0.610107	−	0.792319i	$-0.708874\pi$
$677$	−31.7490			−1.22021			−0.610107	+	0.792319i	$0.708874\pi$
$678$	10.5830			0.406438
$679$	14.0000			0.537271
$680$		0			0
$681$			22.4499i			0.860284i
$682$	−31.7490			−1.21573
$683$	12.0000			0.459167			0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000			0.459167			0.229584	+	0.973289i	$0.426264\pi$
$684$	5.29150			0.202326
$685$		0			0
$686$	−18.5203			−0.707107
$687$			14.9666i			0.571013i
$688$		−	11.2250i		−	0.427948i
$689$	15.8745			0.604771
$690$		0			0
$691$			29.6985i			1.12978i	0.825165	+	0.564892i	$0.191082\pi$
$691$			29.6985i			1.12978i	−0.825165	+	0.564892i	$0.808918\pi$
$692$			9.89949i			0.376322i
$693$			9.89949i			0.376051i
$694$	−24.0000			−0.911028
$695$		0			0
$696$			8.48528i			0.321634i
$697$		0			0
$698$			4.24264i			0.160586i
$699$			25.4558i			0.962828i
$700$	−13.2288			−0.500000
$701$			26.1916i			0.989243i	0.869108	+	0.494622i	$0.164693\pi$
$701$			26.1916i			0.989243i	−0.869108	+	0.494622i	$0.835307\pi$
$702$	24.0000			0.905822
$703$			59.3970i			2.24020i
$704$			3.74166i			0.141019i
$705$		0			0
$706$			28.2843i			1.06449i
$707$			26.1916i			0.985037i
$708$	14.0000			0.526152
$709$		−	11.2250i		−	0.421563i	−0.977533	−	0.210781i	$-0.932399\pi$
$709$		−	11.2250i		−	0.421563i	0.977533	−	0.210781i	$-0.0676008\pi$
$710$		0			0
$711$		0			0
$712$	15.8745			0.594922
$713$	−31.7490	+	25.4558i	−1.18901	+	0.953329i
$714$		0			0
$715$		0			0
$716$	−12.0000			−0.448461
$717$			8.48528i			0.316889i
$718$			29.9333i			1.11710i
$719$		−	19.7990i		−	0.738378i	−0.929354	−	0.369189i	$-0.879636\pi$
$719$		−	19.7990i		−	0.738378i	0.929354	−	0.369189i	$-0.120364\pi$
$720$		0			0
$721$	14.0000			0.521387
$722$	−9.00000			−0.334945
$723$		−	7.48331i		−	0.278307i
$724$	−10.5830			−0.393314
$725$	−30.0000			−1.11417
$726$		−	4.24264i		−	0.157459i
$727$	−10.5830			−0.392502			−0.196251	−	0.980554i	$-0.562877\pi$
$727$	−10.5830			−0.392502			−0.196251	+	0.980554i	$0.562877\pi$
$728$			11.2250i			0.416025i
$729$	−29.0000			−1.07407
$730$		0			0
$731$		0			0
$732$			14.9666i			0.553183i
$733$	21.1660			0.781784			0.390892	−	0.920436i	$-0.372167\pi$
$733$	21.1660			0.781784			0.390892	+	0.920436i	$0.372167\pi$
$734$	−5.29150			−0.195313
$735$		0			0
$736$	3.00000	+	3.74166i	0.110581	+	0.137919i
$737$	−42.0000			−1.54709
$738$		−	5.65685i		−	0.208232i
$739$	−16.0000			−0.588570			−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000			−0.588570			−0.294285	+	0.955718i	$0.595081\pi$
$740$		0			0
$741$	−31.7490			−1.16633
$742$		−	9.89949i		−	0.363422i
$743$		−	7.48331i		−	0.274536i	−0.990534	−	0.137268i	$-0.956168\pi$
$743$		−	7.48331i		−	0.274536i	0.990534	−	0.137268i	$-0.0438322\pi$
$744$	12.0000			0.439941
$745$		0			0
$746$		−	11.2250i		−	0.410975i
$747$	−15.8745			−0.580818
$748$		0			0
$749$			9.89949i			0.361720i
$750$		0			0
$751$		−	22.4499i		−	0.819210i	−0.912263	−	0.409605i	$-0.865667\pi$
$751$		−	22.4499i		−	0.819210i	0.912263	−	0.409605i	$-0.134333\pi$
$752$		−	2.82843i		−	0.103142i
$753$		−	22.4499i		−	0.818121i
$754$			25.4558i			0.927047i
$755$		0			0
$756$		−	14.9666i		−	0.544331i
$757$		−	33.6749i		−	1.22394i	−0.790883	−	0.611968i	$-0.790378\pi$
$757$		−	33.6749i		−	1.22394i	0.790883	−	0.611968i	$-0.209622\pi$
$758$		−	11.2250i		−	0.407709i
$759$	−15.8745	−	19.7990i	−0.576208	−	0.718658i
$760$		0			0
$761$		−	2.82843i		−	0.102530i	−0.998685	−	0.0512652i	$-0.983675\pi$
$761$		−	2.82843i		−	0.102530i	0.998685	−	0.0512652i	$-0.0163254\pi$
$762$			11.3137i			0.409852i
$763$			29.6985i			1.07516i
$764$			14.9666i			0.541474i
$765$		0			0
$766$	−15.8745			−0.573569
$767$	42.0000			1.51653
$768$		−	1.41421i		−	0.0510310i
$769$	21.1660			0.763266			0.381633	−	0.924314i	$-0.375362\pi$
$769$	21.1660			0.763266			0.381633	+	0.924314i	$0.375362\pi$
$770$		0			0
$771$	−4.00000			−0.144056
$772$	−16.0000			−0.575853
$773$	31.7490			1.14193			0.570966	−	0.820973i	$-0.306569\pi$
$773$	31.7490			1.14193			0.570966	+	0.820973i	$0.306569\pi$
$774$			11.2250i			0.403473i
$775$			42.4264i			1.52400i
$776$	−5.29150			−0.189954
$777$	42.0000			1.50674
$778$			18.7083i			0.670725i
$779$			29.9333i			1.07247i
$780$		0			0
$781$			22.4499i			0.803322i
$782$		0			0
$783$		−	33.9411i		−	1.21296i
$784$	7.00000			0.250000
$785$		0			0
$786$	10.0000			0.356688
$787$	−26.4575			−0.943108			−0.471554	−	0.881837i	$-0.656307\pi$
$787$	−26.4575			−0.943108			−0.471554	+	0.881837i	$0.656307\pi$
$788$	6.00000			0.213741
$789$	−10.5830			−0.376765
$790$		0			0
$791$		−	19.7990i		−	0.703971i
$792$		−	3.74166i		−	0.132954i
$793$			44.8999i			1.59444i
$794$			12.7279i			0.451697i
$795$		0			0
$796$	−10.5830			−0.375105
$797$	31.7490			1.12461			0.562304	−	0.826931i	$-0.309915\pi$
$797$	31.7490			1.12461			0.562304	+	0.826931i	$0.309915\pi$
$798$			19.7990i			0.700877i
$799$		0			0
$800$	5.00000			0.176777
$801$	−15.8745			−0.560898
$802$		−	14.9666i		−	0.528490i
$803$	−31.7490			−1.12040
$804$	15.8745			0.559851
$805$		0			0
$806$	36.0000			1.26805
$807$	26.0000			0.915243
$808$		−	9.89949i		−	0.348263i
$809$	−24.0000			−0.843795			−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000			−0.843795			−0.421898	+	0.906644i	$0.638636\pi$
$810$		0			0
$811$		−	4.24264i		−	0.148979i	−0.997222	−	0.0744896i	$-0.976267\pi$
$811$		−	4.24264i		−	0.148979i	0.997222	−	0.0744896i	$-0.0237328\pi$
$812$	15.8745			0.557086
$813$		0			0
$814$	42.0000			1.47210
$815$		0			0
$816$		0			0
$817$		−	59.3970i		−	2.07804i
$818$		−	8.48528i		−	0.296681i
$819$		−	11.2250i		−	0.392232i
$820$		0			0
$821$	6.00000			0.209401			0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000			0.209401			0.104701	+	0.994504i	$0.466612\pi$
$822$	10.5830			0.369125
$823$	−16.0000			−0.557725			−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000			−0.557725			−0.278862	+	0.960331i	$0.589957\pi$
$824$	−5.29150			−0.184338
$825$	−26.4575			−0.921132
$826$		−	26.1916i		−	0.911322i
$827$		−	18.7083i		−	0.650551i	−0.945619	−	0.325275i	$-0.894543\pi$
$827$		−	18.7083i		−	0.650551i	0.945619	−	0.325275i	$-0.105457\pi$
$828$	−3.00000	−	3.74166i	−0.104257	−	0.130032i
$829$		−	29.6985i		−	1.03147i	−0.856748	−	0.515736i	$-0.827519\pi$
$829$		−	29.6985i		−	1.03147i	0.856748	−	0.515736i	$-0.172481\pi$
$830$		0			0
$831$		−	2.82843i		−	0.0981170i
$832$		−	4.24264i		−	0.147087i
$833$		0			0
$834$	−6.00000			−0.207763
$835$		0			0
$836$			19.7990i			0.684762i
$837$	−48.0000			−1.65912
$838$	−15.8745			−0.548376
$839$	47.6235			1.64415			0.822073	−	0.569382i	$-0.192817\pi$
$839$	47.6235			1.64415			0.822073	+	0.569382i	$0.192817\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$			11.2250i			0.386838i
$843$	21.1660			0.728996
$844$	−16.0000			−0.550743
$845$		0			0
$846$			2.82843i			0.0972433i
$847$	−7.93725			−0.272727
$848$			3.74166i			0.128489i
$849$			37.4166i			1.28413i
$850$		0			0
$851$	42.0000	−	33.6749i	1.43974	−	1.15436i
$852$		−	8.48528i		−	0.290701i
$853$			29.6985i			1.01686i	0.861104	+	0.508428i	$0.169773\pi$
$853$			29.6985i			1.01686i	−0.861104	+	0.508428i	$0.830227\pi$
$854$	28.0000			0.958140
$855$		0			0
$856$		−	3.74166i		−	0.127887i
$857$			48.0833i			1.64249i	0.570574	+	0.821246i	$0.306721\pi$
$857$			48.0833i			1.64249i	−0.570574	+	0.821246i	$0.693279\pi$
$858$			22.4499i			0.766428i
$859$		−	46.6690i		−	1.59233i	−0.605081	−	0.796164i	$-0.706859\pi$
$859$		−	46.6690i		−	1.59233i	0.605081	−	0.796164i	$-0.293141\pi$
$860$		0			0
$861$	21.1660			0.721336
$862$			29.9333i			1.01953i
$863$	24.0000			0.816970			0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000			0.816970			0.408485	+	0.912765i	$0.366057\pi$
$864$			5.65685i			0.192450i
$865$		0			0
$866$	−21.1660			−0.719250
$867$			24.0416i			0.816497i
$868$		−	22.4499i		−	0.762001i
$869$		0			0
$870$		0			0
$871$	47.6235			1.61366
$872$		−	11.2250i		−	0.380126i
$873$	5.29150			0.179090
$874$	15.8745	+	19.7990i	0.536963	+	0.669711i
$875$		0			0
$876$	12.0000			0.405442
$877$	−34.0000			−1.14810			−0.574049	−	0.818821i	$-0.694628\pi$
$877$	−34.0000			−1.14810			−0.574049	+	0.818821i	$0.694628\pi$
$878$		−	16.9706i		−	0.572729i
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	−7.00000			−0.235702
$883$	−16.0000			−0.538443			−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000			−0.538443			−0.269221	+	0.963078i	$0.586766\pi$
$884$		0			0
$885$		0			0
$886$	−24.0000			−0.806296
$887$		−	36.7696i		−	1.23460i	−0.786728	−	0.617300i	$-0.788226\pi$
$887$		−	36.7696i		−	1.23460i	0.786728	−	0.617300i	$-0.211774\pi$
$888$	−15.8745			−0.532714
$889$	21.1660			0.709885
$890$		0			0
$891$		−	18.7083i		−	0.626751i
$892$		0			0
$893$		−	14.9666i		−	0.500839i
$894$	−5.29150			−0.176974
$895$		0			0
$896$	−2.64575			−0.0883883
$897$	18.0000	+	22.4499i	0.601003	+	0.749582i
$898$	−12.0000			−0.400445
$899$		−	50.9117i		−	1.69800i
$900$	−5.00000			−0.166667
$901$		0			0
$902$	21.1660			0.704751
$903$	−42.0000			−1.39767
$904$			7.48331i			0.248891i
$905$		0			0
$906$			2.82843i			0.0939682i
$907$			33.6749i			1.11816i	0.829115	+	0.559079i	$0.188845\pi$
$907$			33.6749i			1.11816i	−0.829115	+	0.559079i	$0.811155\pi$
$908$	−15.8745			−0.526814
$909$			9.89949i			0.328346i
$910$		0			0
$911$			14.9666i			0.495867i	0.968777	+	0.247933i	$0.0797514\pi$
$911$			14.9666i			0.495867i	−0.968777	+	0.247933i	$0.920249\pi$
$912$		−	7.48331i		−	0.247797i
$913$		−	59.3970i		−	1.96575i
$914$		0			0
$915$		0			0
$916$	−10.5830			−0.349672
$917$		−	18.7083i		−	0.617802i
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	42.0000			1.38395
$922$			15.5563i			0.512321i
$923$		−	25.4558i		−	0.837889i
$924$	14.0000			0.460566
$925$		−	56.1249i		−	1.84537i
$926$	−26.0000			−0.854413
$927$	5.29150			0.173796
$928$	−6.00000			−0.196960
$929$			22.6274i			0.742381i	0.928557	+	0.371191i	$0.121050\pi$
$929$			22.6274i			0.742381i	−0.928557	+	0.371191i	$0.878950\pi$
$930$		0			0
$931$	37.0405			1.21395
$932$	−18.0000			−0.589610
$933$	−40.0000			−1.30954
$934$	15.8745			0.519430
$935$		0			0
$936$			4.24264i			0.138675i
$937$	5.29150			0.172866			0.0864329	−	0.996258i	$-0.472453\pi$
$937$	5.29150			0.172866			0.0864329	+	0.996258i	$0.472453\pi$
$938$		−	29.6985i		−	0.969690i
$939$			37.4166i			1.22104i
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$			29.9333i			0.975278i
$943$	21.1660	−	16.9706i	0.689260	−	0.552638i
$944$			9.89949i			0.322201i
$945$		0			0
$946$	−42.0000			−1.36554
$947$	−24.0000			−0.779895			−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000			−0.779895			−0.389948	+	0.920837i	$0.627507\pi$
$948$		0			0
$949$	36.0000			1.16861
$950$	26.4575			0.858395
$951$		−	8.48528i		−	0.275154i
$952$		0			0
$953$		−	52.3832i		−	1.69686i	−0.529309	−	0.848429i	$-0.677549\pi$
$953$		−	52.3832i		−	1.69686i	0.529309	−	0.848429i	$-0.322451\pi$
$954$		−	3.74166i		−	0.121141i
$955$		0			0
$956$	−6.00000			−0.194054
$957$	31.7490			1.02630
$958$	−15.8745			−0.512882
$959$		−	19.7990i		−	0.639343i
$960$		0			0
$961$	−41.0000			−1.32258
$962$	−47.6235			−1.53544
$963$			3.74166i			0.120573i
$964$	5.29150			0.170428
$965$		0			0
$966$	14.0000	−	11.2250i	0.450443	−	0.361158i
$967$	−22.0000			−0.707472			−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000			−0.707472			−0.353736	+	0.935345i	$0.615089\pi$
$968$	3.00000			0.0964237
$969$		0			0
$970$		0			0
$971$	15.8745			0.509437			0.254719	−	0.967015i	$-0.418017\pi$
$971$	15.8745			0.509437			0.254719	+	0.967015i	$0.418017\pi$
$972$		−	9.89949i		−	0.317526i
$973$			11.2250i			0.359856i
$974$	16.0000			0.512673
$975$	30.0000			0.960769
$976$	−10.5830			−0.338754
$977$		−	52.3832i		−	1.67589i	−0.545757	−	0.837944i	$-0.683758\pi$
$977$		−	52.3832i		−	1.67589i	0.545757	−	0.837944i	$-0.316242\pi$
$978$		−	5.65685i		−	0.180886i
$979$		−	59.3970i		−	1.89834i
$980$		0			0
$981$			11.2250i			0.358386i
$982$	36.0000			1.14881
$983$	−31.7490			−1.01264			−0.506318	−	0.862347i	$-0.668994\pi$
$983$	−31.7490			−1.01264			−0.506318	+	0.862347i	$0.668994\pi$
$984$	−8.00000			−0.255031
$985$		0			0
$986$		0			0
$987$	−10.5830			−0.336861
$988$		−	22.4499i		−	0.714228i
$989$	−42.0000	+	33.6749i	−1.33552	+	1.07080i
$990$		0			0
$991$	32.0000			1.01651			0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000			1.01651			0.508257	+	0.861206i	$0.330290\pi$
$992$			8.48528i			0.269408i
$993$		−	45.2548i		−	1.43612i
$994$	−15.8745			−0.503509
$995$		0			0
$996$			22.4499i			0.711354i
$997$		−	29.6985i		−	0.940560i	−0.882517	−	0.470280i	$-0.844153\pi$
$997$		−	29.6985i		−	0.940560i	0.882517	−	0.470280i	$-0.155847\pi$
$998$	−32.0000			−1.01294
$999$	63.4980			2.00899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.c.a.321.2	yes	4
3.2	odd	2		2898.2.g.f.2575.3		4
4.3	odd	2		2576.2.f.d.321.3		4
7.6	odd	2	inner	322.2.c.a.321.3	yes	4
21.20	even	2		2898.2.g.f.2575.1		4
23.22	odd	2	inner	322.2.c.a.321.1	✓	4
28.27	even	2		2576.2.f.d.321.2		4
69.68	even	2		2898.2.g.f.2575.2		4
92.91	even	2		2576.2.f.d.321.4		4
161.160	even	2	inner	322.2.c.a.321.4	yes	4
483.482	odd	2		2898.2.g.f.2575.4		4
644.643	odd	2		2576.2.f.d.321.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.c.a.321.1	✓	4	23.22	odd	2	inner
322.2.c.a.321.2	yes	4	1.1	even	1	trivial
322.2.c.a.321.3	yes	4	7.6	odd	2	inner
322.2.c.a.321.4	yes	4	161.160	even	2	inner
2576.2.f.d.321.1		4	644.643	odd	2
2576.2.f.d.321.2		4	28.27	even	2
2576.2.f.d.321.3		4	4.3	odd	2
2576.2.f.d.321.4		4	92.91	even	2
2898.2.g.f.2575.1		4	21.20	even	2
2898.2.g.f.2575.2		4	69.68	even	2
2898.2.g.f.2575.3		4	3.2	odd	2
2898.2.g.f.2575.4		4	483.482	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 322 = 2 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	322.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.41421i q^{3} +1.00000 q^{4} +1.41421i q^{6} +2.64575 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.41421i q^{3} +1.00000 q^{4} +1.41421i q^{6} +2.64575 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.74166i q^{11} -1.41421i q^{12} -4.24264i q^{13} -2.64575 q^{14} +1.00000 q^{16} -1.00000 q^{18} +5.29150 q^{19} -3.74166i q^{21} -3.74166i q^{22} +(-3.00000 - 3.74166i) q^{23} +1.41421i q^{24} -5.00000 q^{25} +4.24264i q^{26} -5.65685i q^{27} +2.64575 q^{28} +6.00000 q^{29} -8.48528i q^{31} -1.00000 q^{32} +5.29150 q^{33} +1.00000 q^{36} +11.2250i q^{37} -5.29150 q^{38} -6.00000 q^{39} +5.65685i q^{41} +3.74166i q^{42} -11.2250i q^{43} +3.74166i q^{44} +(3.00000 + 3.74166i) q^{46} -2.82843i q^{47} -1.41421i q^{48} +7.00000 q^{49} +5.00000 q^{50} -4.24264i q^{52} +3.74166i q^{53} +5.65685i q^{54} -2.64575 q^{56} -7.48331i q^{57} -6.00000 q^{58} +9.89949i q^{59} -10.5830 q^{61} +8.48528i q^{62} +2.64575 q^{63} +1.00000 q^{64} -5.29150 q^{66} +11.2250i q^{67} +(-5.29150 + 4.24264i) q^{69} +6.00000 q^{71} -1.00000 q^{72} +8.48528i q^{73} -11.2250i q^{74} +7.07107i q^{75} +5.29150 q^{76} +9.89949i q^{77} +6.00000 q^{78} -5.00000 q^{81} -5.65685i q^{82} -15.8745 q^{83} -3.74166i q^{84} +11.2250i q^{86} -8.48528i q^{87} -3.74166i q^{88} -15.8745 q^{89} -11.2250i q^{91} +(-3.00000 - 3.74166i) q^{92} -12.0000 q^{93} +2.82843i q^{94} +1.41421i q^{96} +5.29150 q^{97} -7.00000 q^{98} +3.74166i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 4 q^{9} + 4 q^{16} - 4 q^{18} - 12 q^{23} - 20 q^{25} + 24 q^{29} - 4 q^{32} + 4 q^{36} - 24 q^{39} + 12 q^{46} + 28 q^{49} + 20 q^{50} - 24 q^{58} + 4 q^{64} + 24 q^{71} - 4 q^{72} + 24 q^{78} - 20 q^{81} - 12 q^{92} - 48 q^{93} - 28 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 4 * q^9 + 4 * q^16 - 4 * q^18 - 12 * q^23 - 20 * q^25 + 24 * q^29 - 4 * q^32 + 4 * q^36 - 24 * q^39 + 12 * q^46 + 28 * q^49 + 20 * q^50 - 24 * q^58 + 4 * q^64 + 24 * q^71 - 4 * q^72 + 24 * q^78 - 20 * q^81 - 12 * q^92 - 48 * q^93 - 28 * q^98

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