LMFDB - Embedded newform 4012.2.b.b.237.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(237,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4012.237");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.b (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:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$46$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			237.9
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.b.237.38

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.27623i q^{3} +2.55864i q^{5} -1.00518i q^{7} -2.18122 q^{9} +O(q^{10})$	$q-2.27623i q^{3} +2.55864i q^{5} -1.00518i q^{7} -2.18122 q^{9} +4.20954i q^{11} -6.65655 q^{13} +5.82405 q^{15} +(2.10815 - 3.54340i) q^{17} +1.96359 q^{19} -2.28803 q^{21} -7.06438i q^{23} -1.54664 q^{25} -1.86373i q^{27} +2.50313i q^{29} +7.19153i q^{31} +9.58187 q^{33} +2.57190 q^{35} +5.03511i q^{37} +15.1518i q^{39} -11.6991i q^{41} +2.35785 q^{43} -5.58095i q^{45} -9.91253 q^{47} +5.98961 q^{49} +(-8.06560 - 4.79863i) q^{51} +0.260490 q^{53} -10.7707 q^{55} -4.46958i q^{57} +1.00000 q^{59} -10.4088i q^{61} +2.19252i q^{63} -17.0317i q^{65} +3.87736 q^{67} -16.0801 q^{69} +5.64239i q^{71} -8.82479i q^{73} +3.52050i q^{75} +4.23135 q^{77} -13.6738i q^{79} -10.7859 q^{81} +2.15593 q^{83} +(9.06629 + 5.39399i) q^{85} +5.69770 q^{87} +4.88983 q^{89} +6.69104i q^{91} +16.3696 q^{93} +5.02412i q^{95} -12.2137i q^{97} -9.18192i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 54 q^{9}+O(q^{10}) $ 46 * q - 54 * q^9	$ 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) $ 46 * q - 54 * q^9 + 8 * q^13 - 10 * q^15 + q^17 - 20 * q^19 - 24 * q^21 - 54 * q^25 + 2 * q^33 + 26 * q^35 - 38 * q^43 + 6 * q^47 - 66 * q^49 + 26 * q^51 + 18 * q^53 - 20 * q^55 + 46 * q^59 + 48 * q^67 + 28 * q^69 + 22 * q^77 + 70 * q^81 - 52 * q^83 - 2 * q^85 + 44 * q^87 - 76 * q^89 - 26 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times$.

$n$	$2007$	$3129$	$3777$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		−	2.27623i		−	1.31418i	−0.753811	−	0.657091i	$-0.771787\pi$
$3$		−	2.27623i		−	1.31418i	0.753811	−	0.657091i	$-0.228213\pi$
$4$		0			0
$5$			2.55864i			1.14426i	0.820163	+	0.572129i	$0.193882\pi$
$5$			2.55864i			1.14426i	−0.820163	+	0.572129i	$0.806118\pi$
$6$		0			0
$7$		−	1.00518i		−	0.379923i	−0.981792	−	0.189962i	$-0.939164\pi$
$7$		−	1.00518i		−	0.379923i	0.981792	−	0.189962i	$-0.0608363\pi$
$8$		0			0
$9$	−2.18122			−0.727073
$10$		0			0
$11$			4.20954i			1.26922i	0.772831	+	0.634612i	$0.218840\pi$
$11$			4.20954i			1.26922i	−0.772831	+	0.634612i	$0.781160\pi$
$12$		0			0
$13$	−6.65655			−1.84619			−0.923097	−	0.384568i	$-0.874350\pi$
$13$	−6.65655			−1.84619			−0.923097	+	0.384568i	$0.874350\pi$
$14$		0			0
$15$	5.82405			1.50376
$16$		0			0
$17$	2.10815	−	3.54340i	0.511301	−	0.859402i
$17$	2.10815	−	3.54340i	0.511301	−	0.859402i
$18$		0			0
$19$	1.96359			0.450478			0.225239	−	0.974304i	$-0.427684\pi$
$19$	1.96359			0.450478			0.225239	+	0.974304i	$0.427684\pi$
$20$		0			0
$21$	−2.28803			−0.499288
$22$		0			0
$23$		−	7.06438i		−	1.47302i	−0.676424	−	0.736512i	$-0.736471\pi$
$23$		−	7.06438i		−	1.47302i	0.676424	−	0.736512i	$-0.263529\pi$
$24$		0			0
$25$	−1.54664			−0.309328
$26$		0			0
$27$		−	1.86373i		−	0.358675i
$28$		0			0
$29$			2.50313i			0.464820i	0.972618	+	0.232410i	$0.0746611\pi$
$29$			2.50313i			0.464820i	−0.972618	+	0.232410i	$0.925339\pi$
$30$		0			0
$31$			7.19153i			1.29164i	0.763491	+	0.645818i	$0.223484\pi$
$31$			7.19153i			1.29164i	−0.763491	+	0.645818i	$0.776516\pi$
$32$		0			0
$33$	9.58187			1.66799
$34$		0			0
$35$	2.57190			0.434730
$36$		0			0
$37$			5.03511i			0.827767i	0.910330	+	0.413884i	$0.135828\pi$
$37$			5.03511i			0.827767i	−0.910330	+	0.413884i	$0.864172\pi$
$38$		0			0
$39$			15.1518i			2.42623i
$40$		0			0
$41$		−	11.6991i		−	1.82709i	−0.406736	−	0.913546i	$-0.633333\pi$
$41$		−	11.6991i		−	1.82709i	0.406736	−	0.913546i	$-0.366667\pi$
$42$		0			0
$43$	2.35785			0.359569			0.179785	−	0.983706i	$-0.442460\pi$
$43$	2.35785			0.359569			0.179785	+	0.983706i	$0.442460\pi$
$44$		0			0
$45$		−	5.58095i		−	0.831960i
$46$		0			0
$47$	−9.91253			−1.44589			−0.722946	−	0.690905i	$-0.757212\pi$
$47$	−9.91253			−1.44589			−0.722946	+	0.690905i	$0.757212\pi$
$48$		0			0
$49$	5.98961			0.855658
$50$		0			0
$51$	−8.06560	−	4.79863i	−1.12941	−	0.671943i
$52$		0			0
$53$	0.260490			0.0357811			0.0178905	−	0.999840i	$-0.494305\pi$
$53$	0.260490			0.0357811			0.0178905	+	0.999840i	$0.494305\pi$
$54$		0			0
$55$	−10.7707			−1.45232
$56$		0			0
$57$		−	4.46958i		−	0.592010i
$58$		0			0
$59$	1.00000			0.130189
$59$	1.00000			0.130189
$60$		0			0
$61$		−	10.4088i		−	1.33271i	−0.745634	−	0.666355i	$-0.767853\pi$
$61$		−	10.4088i		−	1.33271i	0.745634	−	0.666355i	$-0.232147\pi$
$62$		0			0
$63$			2.19252i			0.276232i
$64$		0			0
$65$		−	17.0317i		−	2.11252i
$66$		0			0
$67$	3.87736			0.473695			0.236848	−	0.971547i	$-0.423886\pi$
$67$	3.87736			0.473695			0.236848	+	0.971547i	$0.423886\pi$
$68$		0			0
$69$	−16.0801			−1.93582
$70$		0			0
$71$			5.64239i			0.669629i	0.942284	+	0.334814i	$0.108674\pi$
$71$			5.64239i			0.669629i	−0.942284	+	0.334814i	$0.891326\pi$
$72$		0			0
$73$		−	8.82479i		−	1.03286i	−0.856328	−	0.516432i	$-0.827260\pi$
$73$		−	8.82479i		−	1.03286i	0.856328	−	0.516432i	$-0.172740\pi$
$74$		0			0
$75$			3.52050i			0.406513i
$76$		0			0
$77$	4.23135			0.482207
$78$		0			0
$79$		−	13.6738i		−	1.53842i	−0.638994	−	0.769212i	$-0.720649\pi$
$79$		−	13.6738i		−	1.53842i	0.638994	−	0.769212i	$-0.279351\pi$
$80$		0			0
$81$	−10.7859			−1.19844
$82$		0			0
$83$	2.15593			0.236644			0.118322	−	0.992975i	$-0.462248\pi$
$83$	2.15593			0.236644			0.118322	+	0.992975i	$0.462248\pi$
$84$		0			0
$85$	9.06629	+	5.39399i	0.983378	+	0.585061i
$86$		0			0
$87$	5.69770			0.610857
$88$		0			0
$89$	4.88983			0.518321			0.259161	−	0.965834i	$-0.416554\pi$
$89$	4.88983			0.518321			0.259161	+	0.965834i	$0.416554\pi$
$90$		0			0
$91$			6.69104i			0.701412i
$92$		0			0
$93$	16.3696			1.69744
$94$		0			0
$95$			5.02412i			0.515464i
$96$		0			0
$97$		−	12.2137i		−	1.24012i	−0.784556	−	0.620058i	$-0.787109\pi$
$97$		−	12.2137i		−	1.24012i	0.784556	−	0.620058i	$-0.212891\pi$
$98$		0			0
$99$		−	9.18192i		−	0.922818i
$100$		0			0
$101$	4.57122			0.454853			0.227427	−	0.973795i	$-0.426969\pi$
$101$	4.57122			0.454853			0.227427	+	0.973795i	$0.426969\pi$
$102$		0			0
$103$	6.37882			0.628524			0.314262	−	0.949336i	$-0.398243\pi$
$103$	6.37882			0.628524			0.314262	+	0.949336i	$0.398243\pi$
$104$		0			0
$105$		−	5.85423i		−	0.571315i
$106$		0			0
$107$		−	13.2575i		−	1.28165i	−0.767685	−	0.640827i	$-0.778592\pi$
$107$		−	13.2575i		−	1.28165i	0.767685	−	0.640827i	$-0.221408\pi$
$108$		0			0
$109$		−	17.6707i		−	1.69255i	−0.532748	−	0.846274i	$-0.678841\pi$
$109$		−	17.6707i		−	1.69255i	0.532748	−	0.846274i	$-0.321159\pi$
$110$		0			0
$111$	11.4611			1.08784
$112$		0			0
$113$		−	11.8665i		−	1.11630i	−0.829739	−	0.558152i	$-0.811511\pi$
$113$		−	11.8665i		−	1.11630i	0.829739	−	0.558152i	$-0.188489\pi$
$114$		0			0
$115$	18.0752			1.68552
$116$		0			0
$117$	14.5194			1.34232
$118$		0			0
$119$	−3.56177	−	2.11907i	−0.326507	−	0.194255i
$120$		0			0
$121$	−6.72020			−0.610927
$122$		0			0
$123$	−26.6298			−2.40113
$124$		0			0
$125$			8.83591i			0.790308i
$126$		0			0
$127$	−12.7130			−1.12810			−0.564048	−	0.825742i	$-0.690757\pi$
$127$	−12.7130			−1.12810			−0.564048	+	0.825742i	$0.690757\pi$
$128$		0			0
$129$		−	5.36701i		−	0.472539i
$130$		0			0
$131$			0.992715i			0.0867339i	0.999059	+	0.0433670i	$0.0138085\pi$
$131$			0.992715i			0.0867339i	−0.999059	+	0.0433670i	$0.986192\pi$
$132$		0			0
$133$		−	1.97377i		−	0.171147i
$134$		0			0
$135$	4.76862			0.410417
$136$		0			0
$137$	9.80553			0.837743			0.418871	−	0.908046i	$-0.362426\pi$
$137$	9.80553			0.837743			0.418871	+	0.908046i	$0.362426\pi$
$138$		0			0
$139$			14.2279i			1.20680i	0.797440	+	0.603398i	$0.206187\pi$
$139$			14.2279i			1.20680i	−0.797440	+	0.603398i	$0.793813\pi$
$140$		0			0
$141$			22.5632i			1.90016i
$142$		0			0
$143$		−	28.0210i		−	2.34323i
$144$		0			0
$145$	−6.40461			−0.531874
$146$		0			0
$147$		−	13.6337i		−	1.12449i
$148$		0			0
$149$	−19.7986			−1.62197			−0.810983	−	0.585070i	$-0.801067\pi$
$149$	−19.7986			−1.62197			−0.810983	+	0.585070i	$0.801067\pi$
$150$		0			0
$151$	−18.6733			−1.51961			−0.759804	−	0.650152i	$-0.774705\pi$
$151$	−18.6733			−1.51961			−0.759804	+	0.650152i	$0.774705\pi$
$152$		0			0
$153$	−4.59833	+	7.72894i	−0.371753	+	0.624848i
$154$		0			0
$155$	−18.4005			−1.47797
$156$		0			0
$157$	−0.00844362			−0.000673874			−0.000336937	−	1.00000i	$-0.500107\pi$
$157$	−0.00844362			−0.000673874			−0.000336937		1.00000i	$0.500107\pi$
$158$		0			0
$159$		−	0.592936i		−	0.0470229i
$160$		0			0
$161$	−7.10099			−0.559636
$162$		0			0
$163$			2.81580i			0.220550i	0.993901	+	0.110275i	$0.0351732\pi$
$163$			2.81580i			0.220550i	−0.993901	+	0.110275i	$0.964827\pi$
$164$		0			0
$165$			24.5166i			1.90861i
$166$		0			0
$167$		−	15.8605i		−	1.22732i	−0.789569	−	0.613661i	$-0.789696\pi$
$167$		−	15.8605i		−	1.22732i	0.789569	−	0.613661i	$-0.210304\pi$
$168$		0			0
$169$	31.3096			2.40843
$170$		0			0
$171$	−4.28302			−0.327531
$172$		0			0
$173$		−	15.4777i		−	1.17675i	−0.808589	−	0.588374i	$-0.799768\pi$
$173$		−	15.4777i		−	1.17675i	0.808589	−	0.588374i	$-0.200232\pi$
$174$		0			0
$175$			1.55465i			0.117521i
$176$		0			0
$177$		−	2.27623i		−	0.171092i
$178$		0			0
$179$	9.06073			0.677231			0.338616	−	0.940925i	$-0.390041\pi$
$179$	9.06073			0.677231			0.338616	+	0.940925i	$0.390041\pi$
$180$		0			0
$181$		−	2.64116i		−	0.196316i	−0.995171	−	0.0981579i	$-0.968705\pi$
$181$		−	2.64116i		−	0.196316i	0.995171	−	0.0981579i	$-0.0312950\pi$
$182$		0			0
$183$	−23.6928			−1.75142
$184$		0			0
$185$	−12.8830			−0.947180
$186$		0			0
$187$	14.9161	+	8.87433i	1.09077	+	0.648955i
$188$		0			0
$189$	−1.87339			−0.136269
$190$		0			0
$191$	−13.8146			−0.999592			−0.499796	−	0.866143i	$-0.666592\pi$
$191$	−13.8146			−0.999592			−0.499796	+	0.866143i	$0.666592\pi$
$192$		0			0
$193$		−	5.45032i		−	0.392322i	−0.980572	−	0.196161i	$-0.937152\pi$
$193$		−	5.45032i		−	0.392322i	0.980572	−	0.196161i	$-0.0628476\pi$
$194$		0			0
$195$	−38.7681			−2.77624
$196$		0			0
$197$			22.1005i			1.57459i	0.616574	+	0.787297i	$0.288520\pi$
$197$			22.1005i			1.57459i	−0.616574	+	0.787297i	$0.711480\pi$
$198$		0			0
$199$		−	11.8021i		−	0.836629i	−0.908302	−	0.418315i	$-0.862621\pi$
$199$		−	11.8021i		−	0.836629i	0.908302	−	0.418315i	$-0.137379\pi$
$200$		0			0
$201$		−	8.82576i		−	0.622521i
$202$		0			0
$203$	2.51610			0.176596
$204$		0			0
$205$	29.9338			2.09066
$206$		0			0
$207$			15.4090i			1.07100i
$208$		0			0
$209$			8.26580i			0.571758i
$210$		0			0
$211$		−	28.7529i		−	1.97943i	−0.143060	−	0.989714i	$-0.545694\pi$
$211$		−	28.7529i		−	1.97943i	0.143060	−	0.989714i	$-0.454306\pi$
$212$		0			0
$213$	12.8434			0.880014
$214$		0			0
$215$			6.03289i			0.411440i
$216$		0			0
$217$	7.22879			0.490722
$218$		0			0
$219$	−20.0872			−1.35737
$220$		0			0
$221$	−14.0330	+	23.5868i	−0.943961	+	1.58662i
$222$		0			0
$223$	19.4175			1.30029			0.650147	−	0.759808i	$-0.274707\pi$
$223$	19.4175			1.30029			0.650147	+	0.759808i	$0.274707\pi$
$224$		0			0
$225$	3.37356			0.224904
$226$		0			0
$227$			21.4500i			1.42368i	0.702339	+	0.711842i	$0.252139\pi$
$227$			21.4500i			1.42368i	−0.702339	+	0.711842i	$0.747861\pi$
$228$		0			0
$229$	10.6539			0.704032			0.352016	−	0.935994i	$-0.385496\pi$
$229$	10.6539			0.704032			0.352016	+	0.935994i	$0.385496\pi$
$230$		0			0
$231$		−	9.63153i		−	0.633708i
$232$		0			0
$233$			24.7340i			1.62038i	0.586166	+	0.810191i	$0.300636\pi$
$233$			24.7340i			1.62038i	−0.586166	+	0.810191i	$0.699364\pi$
$234$		0			0
$235$		−	25.3626i		−	1.65447i
$236$		0			0
$237$	−31.1247			−2.02177
$238$		0			0
$239$	−23.6967			−1.53282			−0.766408	−	0.642355i	$-0.777958\pi$
$239$	−23.6967			−1.53282			−0.766408	+	0.642355i	$0.777958\pi$
$240$		0			0
$241$			6.04958i			0.389688i	0.980834	+	0.194844i	$0.0624200\pi$
$241$			6.04958i			0.389688i	−0.980834	+	0.194844i	$0.937580\pi$
$242$		0			0
$243$			18.9601i			1.21629i
$244$		0			0
$245$			15.3253i			0.979094i
$246$		0			0
$247$	−13.0707			−0.831670
$248$		0			0
$249$		−	4.90739i		−	0.310994i
$250$		0			0
$251$	−4.14423			−0.261581			−0.130791	−	0.991410i	$-0.541752\pi$
$251$	−4.14423			−0.261581			−0.130791	+	0.991410i	$0.541752\pi$
$252$		0			0
$253$	29.7378			1.86960
$254$		0			0
$255$	12.2780	−	20.6370i	0.768876	−	1.29234i
$256$		0			0
$257$	29.4226			1.83533			0.917664	−	0.397357i	$-0.130072\pi$
$257$	29.4226			1.83533			0.917664	+	0.397357i	$0.130072\pi$
$258$		0			0
$259$	5.06120			0.314488
$260$		0			0
$261$		−	5.45988i		−	0.337958i
$262$		0			0
$263$	−7.77247			−0.479271			−0.239636	−	0.970863i	$-0.577028\pi$
$263$	−7.77247			−0.479271			−0.239636	+	0.970863i	$0.577028\pi$
$264$		0			0
$265$			0.666501i			0.0409428i
$266$		0			0
$267$		−	11.1304i		−	0.681168i
$268$		0			0
$269$			3.48426i			0.212439i	0.994343	+	0.106220i	$0.0338747\pi$
$269$			3.48426i			0.212439i	−0.994343	+	0.106220i	$0.966125\pi$
$270$		0			0
$271$	32.3356			1.96425			0.982124	−	0.188233i	$-0.0602760\pi$
$271$	32.3356			1.96425			0.982124	+	0.188233i	$0.0602760\pi$
$272$		0			0
$273$	15.2303			0.921782
$274$		0			0
$275$		−	6.51063i		−	0.392606i
$276$		0			0
$277$		−	28.1946i		−	1.69405i	−0.531553	−	0.847025i	$-0.678391\pi$
$277$		−	28.1946i		−	1.69405i	0.531553	−	0.847025i	$-0.321609\pi$
$278$		0			0
$279$		−	15.6863i		−	0.939114i
$280$		0			0
$281$	−12.6074			−0.752092			−0.376046	−	0.926601i	$-0.622717\pi$
$281$	−12.6074			−0.752092			−0.376046	+	0.926601i	$0.622717\pi$
$282$		0			0
$283$			31.0152i			1.84366i	0.387594	+	0.921830i	$0.373306\pi$
$283$			31.0152i			1.84366i	−0.387594	+	0.921830i	$0.626694\pi$
$284$		0			0
$285$	11.4360			0.677413
$286$		0			0
$287$	−11.7597			−0.694154
$288$		0			0
$289$	−8.11142	−	14.9400i	−0.477142	−	0.878826i
$290$		0			0
$291$	−27.8013			−1.62974
$292$		0			0
$293$	−3.10723			−0.181527			−0.0907633	−	0.995872i	$-0.528931\pi$
$293$	−3.10723			−0.181527			−0.0907633	+	0.995872i	$0.528931\pi$
$294$		0			0
$295$			2.55864i			0.148970i
$296$		0			0
$297$	7.84545			0.455239
$298$		0			0
$299$			47.0243i			2.71949i
$300$		0			0
$301$		−	2.37007i		−	0.136609i
$302$		0			0
$303$		−	10.4051i		−	0.597760i
$304$		0			0
$305$	26.6324			1.52497
$306$		0			0
$307$	−28.4347			−1.62285			−0.811426	−	0.584455i	$-0.801308\pi$
$307$	−28.4347			−1.62285			−0.811426	+	0.584455i	$0.801308\pi$
$308$		0			0
$309$		−	14.5197i		−	0.825995i
$310$		0			0
$311$		−	33.6607i		−	1.90873i	−0.298651	−	0.954363i	$-0.596537\pi$
$311$		−	33.6607i		−	1.90873i	0.298651	−	0.954363i	$-0.403463\pi$
$312$		0			0
$313$			12.3042i			0.695474i	0.937592	+	0.347737i	$0.113050\pi$
$313$			12.3042i			0.695474i	−0.937592	+	0.347737i	$0.886950\pi$
$314$		0			0
$315$	−5.60988			−0.316081
$316$		0			0
$317$		−	30.4950i		−	1.71277i	−0.516339	−	0.856384i	$-0.672706\pi$
$317$		−	30.4950i		−	1.71277i	0.516339	−	0.856384i	$-0.327294\pi$
$318$		0			0
$319$	−10.5370			−0.589960
$320$		0			0
$321$	−30.1772			−1.68433
$322$		0			0
$323$	4.13954	−	6.95779i	0.230330	−	0.387142i
$324$		0			0
$325$	10.2953			0.571079
$326$		0			0
$327$	−40.2226			−2.22431
$328$		0			0
$329$			9.96390i			0.549328i
$330$		0			0
$331$	−2.55529			−0.140451			−0.0702256	−	0.997531i	$-0.522372\pi$
$331$	−2.55529			−0.140451			−0.0702256	+	0.997531i	$0.522372\pi$
$332$		0			0
$333$		−	10.9827i		−	0.601847i
$334$		0			0
$335$			9.92077i			0.542030i
$336$		0			0
$337$		−	10.8364i		−	0.590297i	−0.955451	−	0.295149i	$-0.904631\pi$
$337$		−	10.8364i		−	0.590297i	0.955451	−	0.295149i	$-0.0953692\pi$
$338$		0			0
$339$	−27.0108			−1.46703
$340$		0			0
$341$	−30.2730			−1.63937
$342$		0			0
$343$		−	13.0569i		−	0.705008i
$344$		0			0
$345$		−	41.1433i		−	2.21508i
$346$		0			0
$347$			0.510915i			0.0274273i	0.999906	+	0.0137137i	$0.00436533\pi$
$347$			0.510915i			0.0274273i	−0.999906	+	0.0137137i	$0.995635\pi$
$348$		0			0
$349$	−17.3955			−0.931157			−0.465579	−	0.885007i	$-0.654154\pi$
$349$	−17.3955			−0.931157			−0.465579	+	0.885007i	$0.654154\pi$
$350$		0			0
$351$			12.4060i			0.662184i
$352$		0			0
$353$	29.8410			1.58827			0.794137	−	0.607739i	$-0.207923\pi$
$353$	29.8410			1.58827			0.794137	+	0.607739i	$0.207923\pi$
$354$		0			0
$355$	−14.4369			−0.766229
$356$		0			0
$357$	−4.82350	+	8.10740i	−0.255287	+	0.429089i
$358$		0			0
$359$	−29.4112			−1.55226			−0.776131	−	0.630571i	$-0.782821\pi$
$359$	−29.4112			−1.55226			−0.776131	+	0.630571i	$0.782821\pi$
$360$		0			0
$361$	−15.1443			−0.797069
$362$		0			0
$363$			15.2967i			0.802870i
$364$		0			0
$365$	22.5794			1.18186
$366$		0			0
$367$		−	16.6869i		−	0.871049i	−0.900177	−	0.435525i	$-0.856563\pi$
$367$		−	16.6869i		−	0.871049i	0.900177	−	0.435525i	$-0.143437\pi$
$368$		0			0
$369$			25.5183i			1.32843i
$370$		0			0
$371$		−	0.261840i		−	0.0135941i
$372$		0			0
$373$	8.46715			0.438413			0.219206	−	0.975679i	$-0.429653\pi$
$373$	8.46715			0.438413			0.219206	+	0.975679i	$0.429653\pi$
$374$		0			0
$375$	20.1126			1.03861
$376$		0			0
$377$		−	16.6622i		−	0.858147i
$378$		0			0
$379$			25.6790i			1.31904i	0.751686	+	0.659521i	$0.229241\pi$
$379$			25.6790i			1.31904i	−0.751686	+	0.659521i	$0.770759\pi$
$380$		0			0
$381$			28.9377i			1.48252i
$382$		0			0
$383$	18.0159			0.920569			0.460284	−	0.887771i	$-0.347747\pi$
$383$	18.0159			0.920569			0.460284	+	0.887771i	$0.347747\pi$
$384$		0			0
$385$			10.8265i			0.551770i
$386$		0			0
$387$	−5.14299			−0.261433
$388$		0			0
$389$	−16.5210			−0.837649			−0.418825	−	0.908067i	$-0.637558\pi$
$389$	−16.5210			−0.837649			−0.418825	+	0.908067i	$0.637558\pi$
$390$		0			0
$391$	−25.0319	−	14.8928i	−1.26592	−	0.753159i
$392$		0			0
$393$	2.25965			0.113984
$394$		0			0
$395$	34.9864			1.76035
$396$		0			0
$397$			7.10777i			0.356729i	0.983965	+	0.178364i	$0.0570806\pi$
$397$			7.10777i			0.356729i	−0.983965	+	0.178364i	$0.942919\pi$
$398$		0			0
$399$	−4.49274			−0.224918
$400$		0			0
$401$			23.7556i			1.18630i	0.805093	+	0.593148i	$0.202115\pi$
$401$			23.7556i			1.18630i	−0.805093	+	0.593148i	$0.797885\pi$
$402$		0			0
$403$		−	47.8707i		−	2.38461i
$404$		0			0
$405$		−	27.5973i		−	1.37132i
$406$		0			0
$407$	−21.1955			−1.05062
$408$		0			0
$409$	−35.3537			−1.74813			−0.874063	−	0.485813i	$-0.838524\pi$
$409$	−35.3537			−1.74813			−0.874063	+	0.485813i	$0.838524\pi$
$410$		0			0
$411$		−	22.3196i		−	1.10095i
$412$		0			0
$413$		−	1.00518i		−	0.0494618i
$414$		0			0
$415$			5.51625i			0.270782i
$416$		0			0
$417$	32.3860			1.58595
$418$		0			0
$419$			1.19866i			0.0585585i	0.999571	+	0.0292792i	$0.00932120\pi$
$419$			1.19866i			0.0585585i	−0.999571	+	0.0292792i	$0.990679\pi$
$420$		0			0
$421$	10.8949			0.530983			0.265492	−	0.964113i	$-0.414466\pi$
$421$	10.8949			0.530983			0.265492	+	0.964113i	$0.414466\pi$
$422$		0			0
$423$	21.6214			1.05127
$424$		0			0
$425$	−3.26054	+	5.48036i	−0.158160	+	0.265837i
$426$		0			0
$427$	−10.4627			−0.506328
$428$		0			0
$429$	−63.7822			−3.07943
$430$		0			0
$431$			0.595134i			0.0286666i	0.999897	+	0.0143333i	$0.00456259\pi$
$431$			0.595134i			0.0286666i	−0.999897	+	0.0143333i	$0.995437\pi$
$432$		0			0
$433$	20.6870			0.994154			0.497077	−	0.867706i	$-0.334407\pi$
$433$	20.6870			0.994154			0.497077	+	0.867706i	$0.334407\pi$
$434$		0			0
$435$			14.5784i			0.698979i
$436$		0			0
$437$		−	13.8715i		−	0.663566i
$438$		0			0
$439$		−	8.10626i		−	0.386891i	−0.981111	−	0.193445i	$-0.938034\pi$
$439$		−	8.10626i		−	0.386891i	0.981111	−	0.193445i	$-0.0619662\pi$
$440$		0			0
$441$	−13.0647			−0.622126
$442$		0			0
$443$	6.07515			0.288639			0.144320	−	0.989531i	$-0.453901\pi$
$443$	6.07515			0.288639			0.144320	+	0.989531i	$0.453901\pi$
$444$		0			0
$445$			12.5113i			0.593094i
$446$		0			0
$447$			45.0662i			2.13156i
$448$		0			0
$449$			23.9914i			1.13222i	0.824329	+	0.566111i	$0.191553\pi$
$449$			23.9914i			1.13222i	−0.824329	+	0.566111i	$0.808447\pi$
$450$		0			0
$451$	49.2478			2.31899
$452$		0			0
$453$			42.5046i			1.99704i
$454$		0			0
$455$	−17.1200			−0.802596
$456$		0			0
$457$	3.45563			0.161647			0.0808237	−	0.996728i	$-0.474245\pi$
$457$	3.45563			0.161647			0.0808237	+	0.996728i	$0.474245\pi$
$458$		0			0
$459$	−6.60396	−	3.92903i	−0.308246	−	0.183391i
$460$		0			0
$461$	41.5005			1.93287			0.966436	−	0.256906i	$-0.0827032\pi$
$461$	41.5005			1.93287			0.966436	+	0.256906i	$0.0827032\pi$
$462$		0			0
$463$	0.955366			0.0443996			0.0221998	−	0.999754i	$-0.492933\pi$
$463$	0.955366			0.0443996			0.0221998	+	0.999754i	$0.492933\pi$
$464$		0			0
$465$			41.8838i			1.94232i
$466$		0			0
$467$	−28.3197			−1.31048			−0.655240	−	0.755421i	$-0.727432\pi$
$467$	−28.3197			−1.31048			−0.655240	+	0.755421i	$0.727432\pi$
$468$		0			0
$469$		−	3.89746i		−	0.179968i
$470$		0			0
$471$			0.0192196i			0.000885593i
$472$		0			0
$473$			9.92547i			0.456373i
$474$		0			0
$475$	−3.03696			−0.139345
$476$		0			0
$477$	−0.568186			−0.0260155
$478$		0			0
$479$			18.0796i			0.826079i	0.910713	+	0.413039i	$0.135533\pi$
$479$			18.0796i			0.826079i	−0.910713	+	0.413039i	$0.864467\pi$
$480$		0			0
$481$		−	33.5164i		−	1.52822i
$482$		0			0
$483$			16.1635i			0.735463i
$484$		0			0
$485$	31.2505			1.41901
$486$		0			0
$487$			20.7402i			0.939828i	0.882712	+	0.469914i	$0.155715\pi$
$487$			20.7402i			0.939828i	−0.882712	+	0.469914i	$0.844285\pi$
$488$		0			0
$489$	6.40940			0.289843
$490$		0			0
$491$	9.46942			0.427349			0.213674	−	0.976905i	$-0.431457\pi$
$491$	9.46942			0.427349			0.213674	+	0.976905i	$0.431457\pi$
$492$		0			0
$493$	8.86960	+	5.27697i	0.399467	+	0.237663i
$494$		0			0
$495$	23.4932			1.05594
$496$		0			0
$497$	5.67163			0.254408
$498$		0			0
$499$		−	30.8318i		−	1.38022i	−0.723705	−	0.690110i	$-0.757562\pi$
$499$		−	30.8318i		−	1.38022i	0.723705	−	0.690110i	$-0.242438\pi$
$500$		0			0
$501$	−36.1021			−1.61292
$502$		0			0
$503$		−	21.0282i		−	0.937603i	−0.883303	−	0.468802i	$-0.844686\pi$
$503$		−	21.0282i		−	0.937603i	0.883303	−	0.468802i	$-0.155314\pi$
$504$		0			0
$505$			11.6961i			0.520470i
$506$		0			0
$507$		−	71.2678i		−	3.16511i
$508$		0			0
$509$	−7.87582			−0.349090			−0.174545	−	0.984649i	$-0.555845\pi$
$509$	−7.87582			−0.349090			−0.174545	+	0.984649i	$0.555845\pi$
$510$		0			0
$511$	−8.87052			−0.392409
$512$		0			0
$513$		−	3.65961i		−	0.161576i
$514$		0			0
$515$			16.3211i			0.719194i
$516$		0			0
$517$		−	41.7272i		−	1.83516i
$518$		0			0
$519$	−35.2308			−1.54646
$520$		0			0
$521$		−	4.28579i		−	0.187764i	−0.995583	−	0.0938820i	$-0.970072\pi$
$521$		−	4.28579i		−	0.187764i	0.995583	−	0.0938820i	$-0.0299276\pi$
$522$		0			0
$523$	9.03467			0.395058			0.197529	−	0.980297i	$-0.436708\pi$
$523$	9.03467			0.395058			0.197529	+	0.980297i	$0.436708\pi$
$524$		0			0
$525$	3.53875			0.154444
$526$		0			0
$527$	25.4825	+	15.1608i	1.11003	+	0.660415i
$528$		0			0
$529$	−26.9054			−1.16980
$530$		0			0
$531$	−2.18122			−0.0946569
$532$		0			0
$533$			77.8755i			3.37316i
$534$		0			0
$535$	33.9212			1.46654
$536$		0			0
$537$		−	20.6243i		−	0.890005i
$538$		0			0
$539$			25.2135i			1.08602i
$540$		0			0
$541$			27.7720i			1.19401i	0.802237	+	0.597006i	$0.203643\pi$
$541$			27.7720i			1.19401i	−0.802237	+	0.597006i	$0.796357\pi$
$542$		0			0
$543$	−6.01189			−0.257995
$544$		0			0
$545$	45.2130			1.93671
$546$		0			0
$547$			12.2621i			0.524289i	0.965029	+	0.262144i	$0.0844297\pi$
$547$			12.2621i			0.524289i	−0.965029	+	0.262144i	$0.915570\pi$
$548$		0			0
$549$			22.7039i			0.968978i
$550$		0			0
$551$			4.91512i			0.209391i
$552$		0			0
$553$	−13.7447			−0.584483
$554$		0			0
$555$			29.3247i			1.24477i
$556$		0			0
$557$	−7.41588			−0.314221			−0.157111	−	0.987581i	$-0.550218\pi$
$557$	−7.41588			−0.314221			−0.157111	+	0.987581i	$0.550218\pi$
$558$		0			0
$559$	−15.6951			−0.663834
$560$		0			0
$561$	20.2000	−	33.9524i	0.852845	−	1.43347i
$562$		0			0
$563$	−2.04178			−0.0860509			−0.0430255	−	0.999074i	$-0.513700\pi$
$563$	−2.04178			−0.0860509			−0.0430255	+	0.999074i	$0.513700\pi$
$564$		0			0
$565$	30.3620			1.27734
$566$		0			0
$567$			10.8418i			0.455314i
$568$		0			0
$569$	−29.6366			−1.24243			−0.621216	−	0.783639i	$-0.713361\pi$
$569$	−29.6366			−1.24243			−0.621216	+	0.783639i	$0.713361\pi$
$570$		0			0
$571$			29.3519i			1.22834i	0.789174	+	0.614169i	$0.210509\pi$
$571$			29.3519i			1.22834i	−0.789174	+	0.614169i	$0.789491\pi$
$572$		0			0
$573$			31.4453i			1.31364i
$574$		0			0
$575$			10.9260i			0.455647i
$576$		0			0
$577$	−3.38873			−0.141075			−0.0705374	−	0.997509i	$-0.522471\pi$
$577$	−3.38873			−0.141075			−0.0705374	+	0.997509i	$0.522471\pi$
$578$		0			0
$579$	−12.4062			−0.515583
$580$		0			0
$581$		−	2.16710i		−	0.0899066i
$582$		0			0
$583$			1.09654i			0.0454142i
$584$		0			0
$585$			37.1499i			1.53596i
$586$		0			0
$587$	−30.9740			−1.27844			−0.639218	−	0.769026i	$-0.720742\pi$
$587$	−30.9740			−1.27844			−0.639218	+	0.769026i	$0.720742\pi$
$588$		0			0
$589$			14.1212i			0.581854i
$590$		0			0
$591$	50.3057			2.06930
$592$		0			0
$593$	22.5290			0.925155			0.462577	−	0.886579i	$-0.346925\pi$
$593$	22.5290			0.925155			0.462577	+	0.886579i	$0.346925\pi$
$594$		0			0
$595$	5.42195	−	9.11328i	0.222278	−	0.373608i
$596$		0			0
$597$	−26.8643			−1.09948
$598$		0			0
$599$	36.2890			1.48273			0.741364	−	0.671103i	$-0.234179\pi$
$599$	36.2890			1.48273			0.741364	+	0.671103i	$0.234179\pi$
$600$		0			0
$601$			5.37020i			0.219055i	0.993984	+	0.109527i	$0.0349338\pi$
$601$			5.37020i			0.219055i	−0.993984	+	0.109527i	$0.965066\pi$
$602$		0			0
$603$	−8.45738			−0.344411
$604$		0			0
$605$		−	17.1946i		−	0.699059i
$606$		0			0
$607$			29.1625i			1.18367i	0.806059	+	0.591835i	$0.201596\pi$
$607$			29.1625i			1.18367i	−0.806059	+	0.591835i	$0.798404\pi$
$608$		0			0
$609$		−	5.72722i		−	0.232079i
$610$		0			0
$611$	65.9832			2.66940
$612$		0			0
$613$	−29.7291			−1.20075			−0.600374	−	0.799719i	$-0.704982\pi$
$613$	−29.7291			−1.20075			−0.600374	+	0.799719i	$0.704982\pi$
$614$		0			0
$615$		−	68.1361i		−	2.74751i
$616$		0			0
$617$			0.474131i			0.0190878i	0.999954	+	0.00954391i	$0.00303797\pi$
$617$			0.474131i			0.0190878i	−0.999954	+	0.00954391i	$0.996962\pi$
$618$		0			0
$619$		−	41.5114i		−	1.66848i	−0.551399	−	0.834242i	$-0.685906\pi$
$619$		−	41.5114i		−	1.66848i	0.551399	−	0.834242i	$-0.314094\pi$
$620$		0			0
$621$	−13.1661			−0.528338
$622$		0			0
$623$		−	4.91517i		−	0.196922i
$624$		0			0
$625$	−30.3411			−1.21364
$626$		0			0
$627$	18.8149			0.751393
$628$		0			0
$629$	17.8414	+	10.6148i	0.711384	+	0.423238i
$630$		0			0
$631$	−3.42025			−0.136158			−0.0680791	−	0.997680i	$-0.521687\pi$
$631$	−3.42025			−0.136158			−0.0680791	+	0.997680i	$0.521687\pi$
$632$		0			0
$633$	−65.4481			−2.60133
$634$		0			0
$635$		−	32.5280i		−	1.29083i
$636$		0			0
$637$	−39.8701			−1.57971
$638$		0			0
$639$		−	12.3073i		−	0.486869i
$640$		0			0
$641$		−	1.25740i		−	0.0496643i	−0.999692	−	0.0248322i	$-0.992095\pi$
$641$		−	1.25740i		−	0.0496643i	0.999692	−	0.0248322i	$-0.00790514\pi$
$642$		0			0
$643$		−	43.0990i		−	1.69966i	−0.527059	−	0.849829i	$-0.676705\pi$
$643$		−	43.0990i		−	1.69966i	0.527059	−	0.849829i	$-0.323295\pi$
$644$		0			0
$645$	13.7323			0.540707
$646$		0			0
$647$	6.97098			0.274058			0.137029	−	0.990567i	$-0.456245\pi$
$647$	6.97098			0.274058			0.137029	+	0.990567i	$0.456245\pi$
$648$		0			0
$649$			4.20954i			0.165239i
$650$		0			0
$651$		−	16.4544i		−	0.644898i
$652$		0			0
$653$		−	29.9941i		−	1.17376i	−0.809674	−	0.586880i	$-0.800356\pi$
$653$		−	29.9941i		−	1.17376i	0.809674	−	0.586880i	$-0.199644\pi$
$654$		0			0
$655$	−2.54000			−0.0992460
$656$		0			0
$657$			19.2488i			0.750967i
$658$		0			0
$659$	34.2812			1.33541			0.667703	−	0.744428i	$-0.267278\pi$
$659$	34.2812			1.33541			0.667703	+	0.744428i	$0.267278\pi$
$660$		0			0
$661$	31.6709			1.23186			0.615928	−	0.787803i	$-0.288781\pi$
$661$	31.6709			1.23186			0.615928	+	0.787803i	$0.288781\pi$
$662$		0			0
$663$	53.6890	+	31.9423i	2.08511	+	1.24054i
$664$		0			0
$665$	5.05015			0.195837
$666$		0			0
$667$	17.6831			0.684691
$668$		0			0
$669$		−	44.1988i		−	1.70882i
$670$		0			0
$671$	43.8162			1.69151
$672$		0			0
$673$		−	29.9400i		−	1.15410i	−0.816708	−	0.577051i	$-0.804203\pi$
$673$		−	29.9400i		−	1.15410i	0.816708	−	0.577051i	$-0.195797\pi$
$674$		0			0
$675$			2.88252i			0.110948i
$676$		0			0
$677$		−	3.31186i		−	0.127285i	−0.997973	−	0.0636426i	$-0.979728\pi$
$677$		−	3.31186i		−	0.127285i	0.997973	−	0.0636426i	$-0.0202718\pi$
$678$		0			0
$679$	−12.2770			−0.471149
$680$		0			0
$681$	48.8250			1.87098
$682$		0			0
$683$		−	4.80915i		−	0.184017i	−0.995758	−	0.0920085i	$-0.970671\pi$
$683$		−	4.80915i		−	0.184017i	0.995758	−	0.0920085i	$-0.0293287\pi$
$684$		0			0
$685$			25.0888i			0.958594i
$686$		0			0
$687$		−	24.2508i		−	0.925226i
$688$		0			0
$689$	−1.73397			−0.0660588
$690$		0			0
$691$			14.1889i			0.539772i	0.962892	+	0.269886i	$0.0869861\pi$
$691$			14.1889i			0.539772i	−0.962892	+	0.269886i	$0.913014\pi$
$692$		0			0
$693$	−9.22951			−0.350600
$694$		0			0
$695$	−36.4041			−1.38089
$696$		0			0
$697$	−41.4546	−	24.6634i	−1.57021	−	0.934194i
$698$		0			0
$699$	56.3004			2.12948
$700$		0			0
$701$	−2.33659			−0.0882517			−0.0441258	−	0.999026i	$-0.514050\pi$
$701$	−2.33659			−0.0882517			−0.0441258	+	0.999026i	$0.514050\pi$
$702$		0			0
$703$			9.88689i			0.372891i
$704$		0			0
$705$	−57.7311			−2.17428
$706$		0			0
$707$		−	4.59491i		−	0.172809i
$708$		0			0
$709$		−	5.65401i		−	0.212341i	−0.994348	−	0.106170i	$-0.966141\pi$
$709$		−	5.65401i		−	0.212341i	0.994348	−	0.106170i	$-0.0338589\pi$
$710$		0			0
$711$			29.8256i			1.11855i
$712$		0			0
$713$	50.8036			1.90261
$714$		0			0
$715$	71.6956			2.68126
$716$		0			0
$717$			53.9392i			2.01440i
$718$		0			0
$719$			27.2950i			1.01793i	0.860787	+	0.508966i	$0.169972\pi$
$719$			27.2950i			1.01793i	−0.860787	+	0.508966i	$0.830028\pi$
$720$		0			0
$721$		−	6.41188i		−	0.238791i
$722$		0			0
$723$	13.7702			0.512120
$724$		0			0
$725$		−	3.87144i		−	0.143782i
$726$		0			0
$727$	7.29246			0.270462			0.135231	−	0.990814i	$-0.456822\pi$
$727$	7.29246			0.270462			0.135231	+	0.990814i	$0.456822\pi$
$728$		0			0
$729$	10.7997			0.399987
$730$		0			0
$731$	4.97070	−	8.35482i	0.183848	−	0.309014i
$732$		0			0
$733$	44.4584			1.64211			0.821055	−	0.570850i	$-0.193386\pi$
$733$	44.4584			1.64211			0.821055	+	0.570850i	$0.193386\pi$
$734$		0			0
$735$	34.8838			1.28671
$736$		0			0
$737$			16.3219i			0.601225i
$738$		0			0
$739$	39.3561			1.44774			0.723868	−	0.689939i	$-0.242362\pi$
$739$	39.3561			1.44774			0.723868	+	0.689939i	$0.242362\pi$
$740$		0			0
$741$			29.7520i			1.09297i
$742$		0			0
$743$			35.7207i			1.31047i	0.755426	+	0.655234i	$0.227430\pi$
$743$			35.7207i			1.31047i	−0.755426	+	0.655234i	$0.772570\pi$
$744$		0			0
$745$		−	50.6575i		−	1.85595i
$746$		0			0
$747$	−4.70256			−0.172058
$748$		0			0
$749$	−13.3262			−0.486930
$750$		0			0
$751$			37.7059i			1.37591i	0.725754	+	0.687954i	$0.241491\pi$
$751$			37.7059i			1.37591i	−0.725754	+	0.687954i	$0.758509\pi$
$752$		0			0
$753$			9.43321i			0.343765i
$754$		0			0
$755$		−	47.7782i		−	1.73883i
$756$		0			0
$757$	3.21403			0.116816			0.0584080	−	0.998293i	$-0.481398\pi$
$757$	3.21403			0.116816			0.0584080	+	0.998293i	$0.481398\pi$
$758$		0			0
$759$		−	67.6900i		−	2.45699i
$760$		0			0
$761$	−20.2104			−0.732627			−0.366313	−	0.930492i	$-0.619380\pi$
$761$	−20.2104			−0.732627			−0.366313	+	0.930492i	$0.619380\pi$
$762$		0			0
$763$	−17.7623			−0.643038
$764$		0			0
$765$	−19.7756	−	11.7655i	−0.714987	−	0.425382i
$766$		0			0
$767$	−6.65655			−0.240354
$768$		0			0
$769$	8.75506			0.315716			0.157858	−	0.987462i	$-0.449541\pi$
$769$	8.75506			0.315716			0.157858	+	0.987462i	$0.449541\pi$
$770$		0			0
$771$		−	66.9725i		−	2.41195i
$772$		0			0
$773$	48.8172			1.75583			0.877916	−	0.478814i	$-0.158933\pi$
$773$	48.8172			1.75583			0.877916	+	0.478814i	$0.158933\pi$
$774$		0			0
$775$		−	11.1227i		−	0.399539i
$776$		0			0
$777$		−	11.5205i		−	0.413294i
$778$		0			0
$779$		−	22.9722i		−	0.823065i
$780$		0			0
$781$	−23.7519			−0.849909
$782$		0			0
$783$	4.66517			0.166719
$784$		0			0
$785$		−	0.0216042i		−	0.000771086i
$786$		0			0
$787$			20.5500i			0.732527i	0.930511	+	0.366264i	$0.119363\pi$
$787$			20.5500i			0.732527i	−0.930511	+	0.366264i	$0.880637\pi$
$788$		0			0
$789$			17.6919i			0.629849i
$790$		0			0
$791$	−11.9280			−0.424110
$792$		0			0
$793$			69.2867i			2.46044i
$794$		0			0
$795$	1.51711			0.0538063
$796$		0			0
$797$	1.49022			0.0527864			0.0263932	−	0.999652i	$-0.491598\pi$
$797$	1.49022			0.0527864			0.0263932	+	0.999652i	$0.491598\pi$
$798$		0			0
$799$	−20.8971	+	35.1241i	−0.739286	+	1.24260i
$800$		0			0
$801$	−10.6658			−0.376858
$802$		0			0
$803$	37.1483			1.31093
$804$		0			0
$805$		−	18.1689i		−	0.640368i
$806$		0			0
$807$	7.93098			0.279184
$808$		0			0
$809$			15.0847i			0.530351i	0.964200	+	0.265175i	$0.0854298\pi$
$809$			15.0847i			0.530351i	−0.964200	+	0.265175i	$0.914570\pi$
$810$		0			0
$811$			15.2886i			0.536857i	0.963300	+	0.268429i	$0.0865043\pi$
$811$			15.2886i			0.536857i	−0.963300	+	0.268429i	$0.913496\pi$
$812$		0			0
$813$		−	73.6033i		−	2.58138i
$814$		0			0
$815$	−7.20461			−0.252367
$816$		0			0
$817$	4.62985			0.161978
$818$		0			0
$819$		−	14.5946i		−	0.509978i
$820$		0			0
$821$			24.2068i			0.844823i	0.906404	+	0.422412i	$0.138816\pi$
$821$			24.2068i			0.844823i	−0.906404	+	0.422412i	$0.861184\pi$
$822$		0			0
$823$		−	35.7573i		−	1.24642i	−0.782054	−	0.623211i	$-0.785828\pi$
$823$		−	35.7573i		−	1.24642i	0.782054	−	0.623211i	$-0.214172\pi$
$824$		0			0
$825$	−14.8197			−0.515955
$826$		0			0
$827$			7.32093i			0.254574i	0.991866	+	0.127287i	$0.0406269\pi$
$827$			7.32093i			0.254574i	−0.991866	+	0.127287i	$0.959373\pi$
$828$		0			0
$829$	6.26753			0.217680			0.108840	−	0.994059i	$-0.465286\pi$
$829$	6.26753			0.217680			0.108840	+	0.994059i	$0.465286\pi$
$830$		0			0
$831$	−64.1774			−2.22629
$832$		0			0
$833$	12.6270	−	21.2236i	0.437499	−	0.735354i
$834$		0			0
$835$	40.5813			1.40437
$836$		0			0
$837$	13.4031			0.463278
$838$		0			0
$839$		−	4.79735i		−	0.165623i	−0.996565	−	0.0828115i	$-0.973610\pi$
$839$		−	4.79735i		−	0.165623i	0.996565	−	0.0828115i	$-0.0263899\pi$
$840$		0			0
$841$	22.7343			0.783943
$842$		0			0
$843$			28.6972i			0.988385i
$844$		0			0
$845$			80.1100i			2.75587i
$846$		0			0
$847$			6.75503i			0.232105i
$848$		0			0
$849$	70.5976			2.42290
$850$		0			0
$851$	35.5699			1.21932
$852$		0			0
$853$			10.1214i			0.346552i	0.984873	+	0.173276i	$0.0554352\pi$
$853$			10.1214i			0.346552i	−0.984873	+	0.173276i	$0.944565\pi$
$854$		0			0
$855$		−	10.9587i		−	0.374780i
$856$		0			0
$857$			0.759286i			0.0259367i	0.999916	+	0.0129684i	$0.00412807\pi$
$857$			0.759286i			0.0259367i	−0.999916	+	0.0129684i	$0.995872\pi$
$858$		0			0
$859$	28.7082			0.979512			0.489756	−	0.871859i	$-0.337086\pi$
$859$	28.7082			0.979512			0.489756	+	0.871859i	$0.337086\pi$
$860$		0			0
$861$			26.7678i			0.912245i
$862$		0			0
$863$	32.5354			1.10752			0.553759	−	0.832677i	$-0.313193\pi$
$863$	32.5354			1.10752			0.553759	+	0.832677i	$0.313193\pi$
$864$		0			0
$865$	39.6018			1.34650
$866$		0			0
$867$	−34.0070	+	18.4634i	−1.15494	+	0.627051i
$868$		0			0
$869$	57.5604			1.95260
$870$		0			0
$871$	−25.8098			−0.874533
$872$		0			0
$873$			26.6408i			0.901655i
$874$		0			0
$875$	8.88170			0.300256
$876$		0			0
$877$		−	44.0504i		−	1.48748i	−0.668472	−	0.743738i	$-0.733051\pi$
$877$		−	44.0504i		−	1.48748i	0.668472	−	0.743738i	$-0.266949\pi$
$878$		0			0
$879$			7.07278i			0.238559i
$880$		0			0
$881$		−	8.31272i		−	0.280063i	−0.990147	−	0.140031i	$-0.955280\pi$
$881$		−	8.31272i		−	0.280063i	0.990147	−	0.140031i	$-0.0447203\pi$
$882$		0			0
$883$	15.5567			0.523526			0.261763	−	0.965132i	$-0.415696\pi$
$883$	15.5567			0.523526			0.261763	+	0.965132i	$0.415696\pi$
$884$		0			0
$885$	5.82405			0.195773
$886$		0			0
$887$			21.0174i			0.705694i	0.935681	+	0.352847i	$0.114786\pi$
$887$			21.0174i			0.705694i	−0.935681	+	0.352847i	$0.885214\pi$
$888$		0			0
$889$			12.7789i			0.428590i
$890$		0			0
$891$		−	45.4038i		−	1.52109i
$892$		0			0
$893$	−19.4642			−0.651343
$894$		0			0
$895$			23.1832i			0.774927i
$896$		0			0
$897$	107.038			3.57390
$898$		0			0
$899$	−18.0013			−0.600378
$900$		0			0
$901$	0.549152	−	0.923022i	0.0182949	−	0.0307503i
$902$		0			0
$903$	−5.39482			−0.179529
$904$		0			0
$905$	6.75778			0.224636
$906$		0			0
$907$		−	48.9651i		−	1.62586i	−0.582363	−	0.812929i	$-0.697872\pi$
$907$		−	48.9651i		−	1.62586i	0.582363	−	0.812929i	$-0.302128\pi$
$908$		0			0
$909$	−9.97083			−0.330711
$910$		0			0
$911$			5.64946i			0.187175i	0.995611	+	0.0935875i	$0.0298335\pi$
$911$			5.64946i			0.187175i	−0.995611	+	0.0935875i	$0.970167\pi$
$912$		0			0
$913$			9.07547i			0.300354i
$914$		0			0
$915$		−	60.6214i		−	2.00408i
$916$		0			0
$917$	0.997860			0.0329522
$918$		0			0
$919$	−19.7510			−0.651527			−0.325763	−	0.945451i	$-0.605621\pi$
$919$	−19.7510			−0.651527			−0.325763	+	0.945451i	$0.605621\pi$
$920$		0			0
$921$			64.7238i			2.13272i
$922$		0			0
$923$		−	37.5588i		−	1.23626i
$924$		0			0
$925$		−	7.78750i		−	0.256051i
$926$		0			0
$927$	−13.9136			−0.456983
$928$		0			0
$929$			13.6660i			0.448367i	0.974547	+	0.224183i	$0.0719715\pi$
$929$			13.6660i			0.448367i	−0.974547	+	0.224183i	$0.928029\pi$
$930$		0			0
$931$	11.7611			0.385456
$932$		0			0
$933$	−76.6195			−2.50841
$934$		0			0
$935$	−22.7062	+	38.1649i	−0.742573	+	1.24813i
$936$		0			0
$937$	51.7283			1.68989			0.844945	−	0.534853i	$-0.179633\pi$
$937$	51.7283			1.68989			0.844945	+	0.534853i	$0.179633\pi$
$938$		0			0
$939$	28.0072			0.913979
$940$		0			0
$941$		−	21.7528i		−	0.709120i	−0.935033	−	0.354560i	$-0.884631\pi$
$941$		−	21.7528i		−	0.709120i	0.935033	−	0.354560i	$-0.115369\pi$
$942$		0			0
$943$	−82.6468			−2.69135
$944$		0			0
$945$		−	4.79333i		−	0.155927i
$946$		0			0
$947$			12.9877i			0.422044i	0.977481	+	0.211022i	$0.0676791\pi$
$947$			12.9877i			0.422044i	−0.977481	+	0.211022i	$0.932321\pi$
$948$		0			0
$949$			58.7426i			1.90686i
$950$		0			0
$951$	−69.4135			−2.25089
$952$		0			0
$953$	47.8930			1.55141			0.775704	−	0.631097i	$-0.217395\pi$
$953$	47.8930			1.55141			0.775704	+	0.631097i	$0.217395\pi$
$954$		0			0
$955$		−	35.3467i		−	1.14379i
$956$		0			0
$957$			23.9847i			0.775314i
$958$		0			0
$959$		−	9.85634i		−	0.318278i
$960$		0			0
$961$	−20.7180			−0.668324
$962$		0			0
$963$			28.9176i			0.931856i
$964$		0			0
$965$	13.9454			0.448918
$966$		0			0
$967$	7.36657			0.236893			0.118447	−	0.992960i	$-0.462209\pi$
$967$	7.36657			0.236893			0.118447	+	0.992960i	$0.462209\pi$
$968$		0			0
$969$	−15.8375	−	9.42254i	−0.508775	−	0.302696i
$970$		0			0
$971$	−16.0629			−0.515484			−0.257742	−	0.966214i	$-0.582979\pi$
$971$	−16.0629			−0.515484			−0.257742	+	0.966214i	$0.582979\pi$
$972$		0			0
$973$	14.3016			0.458490
$974$		0			0
$975$		−	23.4344i		−	0.750501i
$976$		0			0
$977$	11.6732			0.373459			0.186730	−	0.982411i	$-0.440211\pi$
$977$	11.6732			0.373459			0.186730	+	0.982411i	$0.440211\pi$
$978$		0			0
$979$			20.5839i			0.657866i
$980$		0			0
$981$			38.5437i			1.23061i
$982$		0			0
$983$			30.6125i			0.976386i	0.872736	+	0.488193i	$0.162344\pi$
$983$			30.6125i			0.976386i	−0.872736	+	0.488193i	$0.837656\pi$
$984$		0			0
$985$	−56.5472			−1.80174
$986$		0			0
$987$	22.6801			0.721916
$988$		0			0
$989$		−	16.6568i		−	0.529654i
$990$		0			0
$991$			12.3755i			0.393122i	0.980492	+	0.196561i	$0.0629773\pi$
$991$			12.3755i			0.393122i	−0.980492	+	0.196561i	$0.937023\pi$
$992$		0			0
$993$			5.81642i			0.184578i
$994$		0			0
$995$	30.1973			0.957320
$996$		0			0
$997$		−	8.64552i		−	0.273806i	−0.990584	−	0.136903i	$-0.956285\pi$
$997$		−	8.64552i		−	0.273806i	0.990584	−	0.136903i	$-0.0437149\pi$
$998$		0			0
$999$	9.38410			0.296900

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.b.237.9	✓	46
17.16	even	2	inner	4012.2.b.b.237.38	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.b.237.9	✓	46	1.1	even	1	trivial
4012.2.b.b.237.38	yes	46	17.16	even	2	inner

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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.27623i q^{3} +2.55864i q^{5} -1.00518i q^{7} -2.18122 q^{9} +O(q^{10})\)	\(q-2.27623i q^{3} +2.55864i q^{5} -1.00518i q^{7} -2.18122 q^{9} +4.20954i q^{11} -6.65655 q^{13} +5.82405 q^{15} +(2.10815 - 3.54340i) q^{17} +1.96359 q^{19} -2.28803 q^{21} -7.06438i q^{23} -1.54664 q^{25} -1.86373i q^{27} +2.50313i q^{29} +7.19153i q^{31} +9.58187 q^{33} +2.57190 q^{35} +5.03511i q^{37} +15.1518i q^{39} -11.6991i q^{41} +2.35785 q^{43} -5.58095i q^{45} -9.91253 q^{47} +5.98961 q^{49} +(-8.06560 - 4.79863i) q^{51} +0.260490 q^{53} -10.7707 q^{55} -4.46958i q^{57} +1.00000 q^{59} -10.4088i q^{61} +2.19252i q^{63} -17.0317i q^{65} +3.87736 q^{67} -16.0801 q^{69} +5.64239i q^{71} -8.82479i q^{73} +3.52050i q^{75} +4.23135 q^{77} -13.6738i q^{79} -10.7859 q^{81} +2.15593 q^{83} +(9.06629 + 5.39399i) q^{85} +5.69770 q^{87} +4.88983 q^{89} +6.69104i q^{91} +16.3696 q^{93} +5.02412i q^{95} -12.2137i q^{97} -9.18192i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 54 q^{9}+O(q^{10}) \) 46 * q - 54 * q^9	\( 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) \) 46 * q - 54 * q^9 + 8 * q^13 - 10 * q^15 + q^17 - 20 * q^19 - 24 * q^21 - 54 * q^25 + 2 * q^33 + 26 * q^35 - 38 * q^43 + 6 * q^47 - 66 * q^49 + 26 * q^51 + 18 * q^53 - 20 * q^55 + 46 * q^59 + 48 * q^67 + 28 * q^69 + 22 * q^77 + 70 * q^81 - 52 * q^83 - 2 * q^85 + 44 * q^87 - 76 * q^89 - 26 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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