LMFDB - Embedded newform 4012.2.b.a.237.20

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:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			237.20
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.a.237.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0489925i q^{3} -3.24065i q^{5} +0.683113i q^{7} +2.99760 q^{9} +O(q^{10})$	$q-0.0489925i q^{3} -3.24065i q^{5} +0.683113i q^{7} +2.99760 q^{9} -4.91743i q^{11} +3.27939 q^{13} -0.158767 q^{15} +(-3.23363 + 2.55806i) q^{17} -4.26440 q^{19} +0.0334674 q^{21} -5.93405i q^{23} -5.50183 q^{25} -0.293837i q^{27} -3.54850i q^{29} -1.31063i q^{31} -0.240917 q^{33} +2.21373 q^{35} -2.49309i q^{37} -0.160666i q^{39} +0.139098i q^{41} +0.999285 q^{43} -9.71418i q^{45} +5.28089 q^{47} +6.53336 q^{49} +(0.125325 + 0.158423i) q^{51} -7.01995 q^{53} -15.9357 q^{55} +0.208924i q^{57} -1.00000 q^{59} +2.89178i q^{61} +2.04770i q^{63} -10.6274i q^{65} -9.43849 q^{67} -0.290724 q^{69} -5.35052i q^{71} +8.00548i q^{73} +0.269548i q^{75} +3.35916 q^{77} -8.36353i q^{79} +8.97840 q^{81} +8.80513 q^{83} +(8.28977 + 10.4791i) q^{85} -0.173850 q^{87} -4.01361 q^{89} +2.24020i q^{91} -0.0642109 q^{93} +13.8194i q^{95} +10.6066i q^{97} -14.7405i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 36 q^{9}+O(q^{10}) $ 40 * q - 36 * q^9	$ 40 q - 36 q^{9} + 8 q^{13} + 20 q^{15} - 8 q^{17} + 20 q^{19} + 6 q^{21} - 24 q^{25} - 14 q^{33} - 52 q^{35} + 22 q^{43} - 10 q^{47} + 8 q^{49} - 6 q^{51} - 2 q^{53} - 12 q^{55} - 40 q^{59} - 24 q^{67} - 36 q^{69} - 38 q^{77} + 16 q^{81} + 32 q^{83} - 22 q^{85} - 18 q^{87} + 40 q^{89} + 22 q^{93}+O(q^{100}) $ 40 * q - 36 * q^9 + 8 * q^13 + 20 * q^15 - 8 * q^17 + 20 * q^19 + 6 * q^21 - 24 * q^25 - 14 * q^33 - 52 * q^35 + 22 * q^43 - 10 * q^47 + 8 * q^49 - 6 * q^51 - 2 * q^53 - 12 * q^55 - 40 * q^59 - 24 * q^67 - 36 * q^69 - 38 * q^77 + 16 * q^81 + 32 * q^83 - 22 * q^85 - 18 * q^87 + 40 * q^89 + 22 * 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
$3$		−	0.0489925i		−	0.0282858i	−0.999900	−	0.0141429i	$-0.995498\pi$
$3$		−	0.0489925i		−	0.0282858i	0.999900	−	0.0141429i	$-0.00450198\pi$
$4$		0			0
$5$		−	3.24065i		−	1.44926i	−0.689136	−	0.724632i	$-0.742010\pi$
$5$		−	3.24065i		−	1.44926i	0.689136	−	0.724632i	$-0.257990\pi$
$6$		0			0
$7$			0.683113i			0.258193i	0.991632	+	0.129096i	$0.0412076\pi$
$7$			0.683113i			0.258193i	−0.991632	+	0.129096i	$0.958792\pi$
$8$		0			0
$9$	2.99760			0.999200
$10$		0			0
$11$		−	4.91743i		−	1.48266i	−0.671140	−	0.741331i	$-0.734195\pi$
$11$		−	4.91743i		−	1.48266i	0.671140	−	0.741331i	$-0.265805\pi$
$12$		0			0
$13$	3.27939			0.909540			0.454770	−	0.890609i	$-0.349721\pi$
$13$	3.27939			0.909540			0.454770	+	0.890609i	$0.349721\pi$
$14$		0			0
$15$	−0.158767			−0.0409936
$16$		0			0
$17$	−3.23363	+	2.55806i	−0.784270	+	0.620420i
$17$	−3.23363	+	2.55806i	−0.784270	+	0.620420i
$18$		0			0
$19$	−4.26440			−0.978321			−0.489161	−	0.872194i	$-0.662697\pi$
$19$	−4.26440			−0.978321			−0.489161	+	0.872194i	$0.662697\pi$
$20$		0			0
$21$	0.0334674			0.00730319
$22$		0			0
$23$		−	5.93405i		−	1.23733i	−0.785653	−	0.618667i	$-0.787673\pi$
$23$		−	5.93405i		−	1.23733i	0.785653	−	0.618667i	$-0.212327\pi$
$24$		0			0
$25$	−5.50183			−1.10037
$26$		0			0
$27$		−	0.293837i		−	0.0565490i
$28$		0			0
$29$		−	3.54850i		−	0.658941i	−0.944166	−	0.329470i	$-0.893130\pi$
$29$		−	3.54850i		−	0.658941i	0.944166	−	0.329470i	$-0.106870\pi$
$30$		0			0
$31$		−	1.31063i		−	0.235396i	−0.993049	−	0.117698i	$-0.962449\pi$
$31$		−	1.31063i		−	0.235396i	0.993049	−	0.117698i	$-0.0375514\pi$
$32$		0			0
$33$	−0.240917			−0.0419383
$34$		0			0
$35$	2.21373			0.374189
$36$		0			0
$37$		−	2.49309i		−	0.409861i	−0.978777	−	0.204930i	$-0.934303\pi$
$37$		−	2.49309i		−	0.409861i	0.978777	−	0.204930i	$-0.0656968\pi$
$38$		0			0
$39$		−	0.160666i		−	0.0257271i
$40$		0			0
$41$			0.139098i			0.0217235i	0.999941	+	0.0108617i	$0.00345747\pi$
$41$			0.139098i			0.0217235i	−0.999941	+	0.0108617i	$0.996543\pi$
$42$		0			0
$43$	0.999285			0.152390			0.0761948	−	0.997093i	$-0.475723\pi$
$43$	0.999285			0.152390			0.0761948	+	0.997093i	$0.475723\pi$
$44$		0			0
$45$		−	9.71418i		−	1.44810i
$46$		0			0
$47$	5.28089			0.770297			0.385148	−	0.922855i	$-0.374150\pi$
$47$	5.28089			0.770297			0.385148	+	0.922855i	$0.374150\pi$
$48$		0			0
$49$	6.53336			0.933337
$50$		0			0
$51$	0.125325	+	0.158423i	0.0175491	+	0.0221837i
$52$		0			0
$53$	−7.01995			−0.964265			−0.482132	−	0.876098i	$-0.660138\pi$
$53$	−7.01995			−0.964265			−0.482132	+	0.876098i	$0.660138\pi$
$54$		0			0
$55$	−15.9357			−2.14877
$56$		0			0
$57$			0.208924i			0.0276726i
$58$		0			0
$59$	−1.00000			−0.130189
$59$	−1.00000			−0.130189
$60$		0			0
$61$			2.89178i			0.370254i	0.982715	+	0.185127i	$0.0592697\pi$
$61$			2.89178i			0.370254i	−0.982715	+	0.185127i	$0.940730\pi$
$62$		0			0
$63$			2.04770i			0.257986i
$64$		0			0
$65$		−	10.6274i		−	1.31816i
$66$		0			0
$67$	−9.43849			−1.15310			−0.576548	−	0.817064i	$-0.695600\pi$
$67$	−9.43849			−1.15310			−0.576548	+	0.817064i	$0.695600\pi$
$68$		0			0
$69$	−0.290724			−0.0349990
$70$		0			0
$71$		−	5.35052i		−	0.634990i	−0.948260	−	0.317495i	$-0.897158\pi$
$71$		−	5.35052i		−	0.634990i	0.948260	−	0.317495i	$-0.102842\pi$
$72$		0			0
$73$			8.00548i			0.936971i	0.883471	+	0.468485i	$0.155200\pi$
$73$			8.00548i			0.936971i	−0.883471	+	0.468485i	$0.844800\pi$
$74$		0			0
$75$			0.269548i			0.0311247i
$76$		0			0
$77$	3.35916			0.382812
$78$		0			0
$79$		−	8.36353i		−	0.940971i	−0.882408	−	0.470485i	$-0.844079\pi$
$79$		−	8.36353i		−	0.940971i	0.882408	−	0.470485i	$-0.155921\pi$
$80$		0			0
$81$	8.97840			0.997600
$82$		0			0
$83$	8.80513			0.966488			0.483244	−	0.875486i	$-0.339458\pi$
$83$	8.80513			0.966488			0.483244	+	0.875486i	$0.339458\pi$
$84$		0			0
$85$	8.28977	+	10.4791i	0.899152	+	1.13661i
$86$		0			0
$87$	−0.173850			−0.0186387
$88$		0			0
$89$	−4.01361			−0.425442			−0.212721	−	0.977113i	$-0.568233\pi$
$89$	−4.01361			−0.425442			−0.212721	+	0.977113i	$0.568233\pi$
$90$		0			0
$91$			2.24020i			0.234837i
$92$		0			0
$93$	−0.0642109			−0.00665836
$94$		0			0
$95$			13.8194i			1.41785i
$96$		0			0
$97$			10.6066i			1.07694i	0.842645	+	0.538470i	$0.180998\pi$
$97$			10.6066i			1.07694i	−0.842645	+	0.538470i	$0.819002\pi$
$98$		0			0
$99$		−	14.7405i		−	1.48147i
$100$		0			0
$101$	11.1491			1.10938			0.554689	−	0.832058i	$-0.312837\pi$
$101$	11.1491			1.10938			0.554689	+	0.832058i	$0.312837\pi$
$102$		0			0
$103$	−8.64979			−0.852289			−0.426145	−	0.904655i	$-0.640129\pi$
$103$	−8.64979			−0.852289			−0.426145	+	0.904655i	$0.640129\pi$
$104$		0			0
$105$		−	0.108456i		−	0.0105842i
$106$		0			0
$107$			9.43992i			0.912592i	0.889828	+	0.456296i	$0.150824\pi$
$107$			9.43992i			0.912592i	−0.889828	+	0.456296i	$0.849176\pi$
$108$		0			0
$109$			2.08441i			0.199650i	0.995005	+	0.0998252i	$0.0318284\pi$
$109$			2.08441i			0.199650i	−0.995005	+	0.0998252i	$0.968172\pi$
$110$		0			0
$111$	−0.122142			−0.0115932
$112$		0			0
$113$			14.3303i			1.34808i	0.738695	+	0.674040i	$0.235442\pi$
$113$			14.3303i			1.34808i	−0.738695	+	0.674040i	$0.764558\pi$
$114$		0			0
$115$	−19.2302			−1.79322
$116$		0			0
$117$	9.83031			0.908812
$118$		0			0
$119$	−1.74744	−	2.20893i	−0.160188	−	0.202493i
$120$		0			0
$121$	−13.1811			−1.19828
$122$		0			0
$123$	0.00681476			0.000614466
$124$		0			0
$125$			1.62625i			0.145456i
$126$		0			0
$127$	4.52999			0.401971			0.200986	−	0.979594i	$-0.435586\pi$
$127$	4.52999			0.401971			0.200986	+	0.979594i	$0.435586\pi$
$128$		0			0
$129$		−	0.0489574i		−	0.00431046i
$130$		0			0
$131$		−	5.88928i		−	0.514549i	−0.966338	−	0.257274i	$-0.917176\pi$
$131$		−	5.88928i		−	0.514549i	0.966338	−	0.257274i	$-0.0828244\pi$
$132$		0			0
$133$		−	2.91307i		−	0.252595i
$134$		0			0
$135$	−0.952224			−0.0819544
$136$		0			0
$137$	−13.3129			−1.13740			−0.568698	−	0.822546i	$-0.692553\pi$
$137$	−13.3129			−1.13740			−0.568698	+	0.822546i	$0.692553\pi$
$138$		0			0
$139$		−	13.1276i		−	1.11347i	−0.830691	−	0.556734i	$-0.812054\pi$
$139$		−	13.1276i		−	1.11347i	0.830691	−	0.556734i	$-0.187946\pi$
$140$		0			0
$141$		−	0.258724i		−	0.0217885i
$142$		0			0
$143$		−	16.1262i		−	1.34854i
$144$		0			0
$145$	−11.4995			−0.954979
$146$		0			0
$147$		−	0.320085i		−	0.0264002i
$148$		0			0
$149$	−10.5545			−0.864662			−0.432331	−	0.901715i	$-0.642309\pi$
$149$	−10.5545			−0.864662			−0.432331	+	0.901715i	$0.642309\pi$
$150$		0			0
$151$	−5.03524			−0.409762			−0.204881	−	0.978787i	$-0.565681\pi$
$151$	−5.03524			−0.409762			−0.204881	+	0.978787i	$0.565681\pi$
$152$		0			0
$153$	−9.69312	+	7.66803i	−0.783642	+	0.619923i
$154$		0			0
$155$	−4.24729			−0.341150
$156$		0			0
$157$	−17.7819			−1.41915			−0.709576	−	0.704629i	$-0.751114\pi$
$157$	−17.7819			−1.41915			−0.709576	+	0.704629i	$0.751114\pi$
$158$		0			0
$159$			0.343925i			0.0272750i
$160$		0			0
$161$	4.05363			0.319471
$162$		0			0
$163$		−	17.2590i		−	1.35183i	−0.736981	−	0.675913i	$-0.763749\pi$
$163$		−	17.2590i		−	1.35183i	0.736981	−	0.675913i	$-0.236251\pi$
$164$		0			0
$165$			0.780728i			0.0607796i
$166$		0			0
$167$		−	6.60513i		−	0.511120i	−0.966793	−	0.255560i	$-0.917740\pi$
$167$		−	6.60513i		−	0.511120i	0.966793	−	0.255560i	$-0.0822598\pi$
$168$		0			0
$169$	−2.24558			−0.172737
$170$		0			0
$171$	−12.7830			−0.977538
$172$		0			0
$173$		−	12.2516i		−	0.931473i	−0.884923	−	0.465736i	$-0.845789\pi$
$173$		−	12.2516i		−	0.931473i	0.884923	−	0.465736i	$-0.154211\pi$
$174$		0			0
$175$		−	3.75837i		−	0.284106i
$176$		0			0
$177$			0.0489925i			0.00368250i
$178$		0			0
$179$	13.4779			1.00739			0.503694	−	0.863882i	$-0.331974\pi$
$179$	13.4779			1.00739			0.503694	+	0.863882i	$0.331974\pi$
$180$		0			0
$181$			8.92689i			0.663531i	0.943362	+	0.331765i	$0.107644\pi$
$181$			8.92689i			0.663531i	−0.943362	+	0.331765i	$0.892356\pi$
$182$		0			0
$183$	0.141675			0.0104729
$184$		0			0
$185$	−8.07922			−0.593996
$186$		0			0
$187$	12.5791	+	15.9011i	0.919872	+	1.16281i
$188$		0			0
$189$	0.200724			0.0146005
$190$		0			0
$191$	−13.3040			−0.962647			−0.481324	−	0.876543i	$-0.659844\pi$
$191$	−13.3040			−0.962647			−0.481324	+	0.876543i	$0.659844\pi$
$192$		0			0
$193$			4.75883i			0.342548i	0.985223	+	0.171274i	$0.0547884\pi$
$193$			4.75883i			0.342548i	−0.985223	+	0.171274i	$0.945212\pi$
$194$		0			0
$195$	−0.520661			−0.0372853
$196$		0			0
$197$		−	11.4841i		−	0.818210i	−0.912487	−	0.409105i	$-0.865841\pi$
$197$		−	11.4841i		−	0.818210i	0.912487	−	0.409105i	$-0.134159\pi$
$198$		0			0
$199$		−	3.03750i		−	0.215322i	−0.994188	−	0.107661i	$-0.965664\pi$
$199$		−	3.03750i		−	0.215322i	0.994188	−	0.107661i	$-0.0343362\pi$
$200$		0			0
$201$			0.462415i			0.0326162i
$202$		0			0
$203$	2.42403			0.170134
$204$		0			0
$205$	0.450769			0.0314831
$206$		0			0
$207$		−	17.7879i		−	1.23634i
$208$		0			0
$209$			20.9699i			1.45052i
$210$		0			0
$211$		−	6.40923i		−	0.441230i	−0.975361	−	0.220615i	$-0.929194\pi$
$211$		−	6.40923i		−	0.441230i	0.975361	−	0.220615i	$-0.0708064\pi$
$212$		0			0
$213$	−0.262135			−0.0179612
$214$		0			0
$215$		−	3.23833i		−	0.220853i
$216$		0			0
$217$	0.895307			0.0607774
$218$		0			0
$219$	0.392208			0.0265030
$220$		0			0
$221$	−10.6043	+	8.38887i	−0.713325	+	0.564297i
$222$		0			0
$223$	−12.4122			−0.831180			−0.415590	−	0.909552i	$-0.636425\pi$
$223$	−12.4122			−0.831180			−0.415590	+	0.909552i	$0.636425\pi$
$224$		0			0
$225$	−16.4923			−1.09948
$226$		0			0
$227$		−	15.2620i		−	1.01297i	−0.862248	−	0.506487i	$-0.830944\pi$
$227$		−	15.2620i		−	1.01297i	0.862248	−	0.506487i	$-0.169056\pi$
$228$		0			0
$229$	−3.71441			−0.245455			−0.122728	−	0.992440i	$-0.539164\pi$
$229$	−3.71441			−0.245455			−0.122728	+	0.992440i	$0.539164\pi$
$230$		0			0
$231$		−	0.164574i		−	0.0108281i
$232$		0			0
$233$		−	22.8089i		−	1.49426i	−0.664677	−	0.747131i	$-0.731431\pi$
$233$		−	22.8089i		−	1.49426i	0.664677	−	0.747131i	$-0.268569\pi$
$234$		0			0
$235$		−	17.1135i		−	1.11636i
$236$		0			0
$237$	−0.409750			−0.0266161
$238$		0			0
$239$	8.40980			0.543984			0.271992	−	0.962299i	$-0.412318\pi$
$239$	8.40980			0.543984			0.271992	+	0.962299i	$0.412318\pi$
$240$		0			0
$241$			18.9297i			1.21937i	0.792645	+	0.609684i	$0.208704\pi$
$241$			18.9297i			1.21937i	−0.792645	+	0.609684i	$0.791296\pi$
$242$		0			0
$243$		−	1.32139i		−	0.0847669i
$244$		0			0
$245$		−	21.1723i		−	1.35265i
$246$		0			0
$247$	−13.9847			−0.889822
$248$		0			0
$249$		−	0.431385i		−	0.0273379i
$250$		0			0
$251$	14.4236			0.910409			0.455204	−	0.890387i	$-0.349566\pi$
$251$	14.4236			0.910409			0.455204	+	0.890387i	$0.349566\pi$
$252$		0			0
$253$	−29.1803			−1.83455
$254$		0			0
$255$	0.513395	−	0.406136i	0.0321500	−	0.0254332i
$256$		0			0
$257$	25.9257			1.61720			0.808602	−	0.588356i	$-0.200225\pi$
$257$	25.9257			1.61720			0.808602	+	0.588356i	$0.200225\pi$
$258$		0			0
$259$	1.70306			0.105823
$260$		0			0
$261$		−	10.6370i		−	0.658413i
$262$		0			0
$263$	18.1357			1.11829			0.559146	−	0.829069i	$-0.311129\pi$
$263$	18.1357			1.11829			0.559146	+	0.829069i	$0.311129\pi$
$264$		0			0
$265$			22.7492i			1.39747i
$266$		0			0
$267$			0.196637i			0.0120340i
$268$		0			0
$269$		−	10.5280i		−	0.641901i	−0.947096	−	0.320951i	$-0.895998\pi$
$269$		−	10.5280i		−	0.641901i	0.947096	−	0.320951i	$-0.104002\pi$
$270$		0			0
$271$	−26.0398			−1.58180			−0.790902	−	0.611942i	$-0.790388\pi$
$271$	−26.0398			−1.58180			−0.790902	+	0.611942i	$0.790388\pi$
$272$		0			0
$273$	0.109753			0.00664254
$274$		0			0
$275$			27.0549i			1.63147i
$276$		0			0
$277$			0.840550i			0.0505037i	0.999681	+	0.0252519i	$0.00803877\pi$
$277$			0.840550i			0.0505037i	−0.999681	+	0.0252519i	$0.991961\pi$
$278$		0			0
$279$		−	3.92874i		−	0.235207i
$280$		0			0
$281$	−15.1241			−0.902226			−0.451113	−	0.892467i	$-0.648973\pi$
$281$	−15.1241			−0.902226			−0.451113	+	0.892467i	$0.648973\pi$
$282$		0			0
$283$		−	1.03449i		−	0.0614942i	−0.999527	−	0.0307471i	$-0.990211\pi$
$283$		−	1.03449i		−	0.0614942i	0.999527	−	0.0307471i	$-0.00978865\pi$
$284$		0			0
$285$	0.677049			0.0401049
$286$		0			0
$287$	−0.0950198			−0.00560884
$288$		0			0
$289$	3.91269	−	16.5436i	0.230158	−	0.973153i
$290$		0			0
$291$	0.519645			0.0304621
$292$		0			0
$293$	12.5141			0.731084			0.365542	−	0.930795i	$-0.380884\pi$
$293$	12.5141			0.731084			0.365542	+	0.930795i	$0.380884\pi$
$294$		0			0
$295$			3.24065i			0.188678i
$296$		0			0
$297$	−1.44492			−0.0838430
$298$		0			0
$299$		−	19.4601i		−	1.12541i
$300$		0			0
$301$			0.682625i			0.0393458i
$302$		0			0
$303$		−	0.546222i		−	0.0313797i
$304$		0			0
$305$	9.37124			0.536596
$306$		0			0
$307$	12.3533			0.705042			0.352521	−	0.935804i	$-0.385324\pi$
$307$	12.3533			0.705042			0.352521	+	0.935804i	$0.385324\pi$
$308$		0			0
$309$			0.423774i			0.0241077i
$310$		0			0
$311$			24.8843i			1.41106i	0.708679	+	0.705531i	$0.249291\pi$
$311$			24.8843i			1.41106i	−0.708679	+	0.705531i	$0.750709\pi$
$312$		0			0
$313$			3.82500i			0.216202i	0.994140	+	0.108101i	$0.0344770\pi$
$313$			3.82500i			0.216202i	−0.994140	+	0.108101i	$0.965523\pi$
$314$		0			0
$315$	6.63589			0.373890
$316$		0			0
$317$			19.5509i			1.09809i	0.835793	+	0.549045i	$0.185009\pi$
$317$			19.5509i			1.09809i	−0.835793	+	0.549045i	$0.814991\pi$
$318$		0			0
$319$	−17.4495			−0.976985
$320$		0			0
$321$	0.462485			0.0258134
$322$		0			0
$323$	13.7895	−	10.9086i	0.767268	−	0.606970i
$324$		0			0
$325$	−18.0427			−1.00083
$326$		0			0
$327$	0.102120			0.00564727
$328$		0			0
$329$			3.60745i			0.198885i
$330$		0			0
$331$	7.31472			0.402053			0.201027	−	0.979586i	$-0.435572\pi$
$331$	7.31472			0.402053			0.201027	+	0.979586i	$0.435572\pi$
$332$		0			0
$333$		−	7.47327i		−	0.409533i
$334$		0			0
$335$			30.5869i			1.67114i
$336$		0			0
$337$			3.52206i			0.191859i	0.995388	+	0.0959293i	$0.0305823\pi$
$337$			3.52206i			0.191859i	−0.995388	+	0.0959293i	$0.969418\pi$
$338$		0			0
$339$	0.702076			0.0381315
$340$		0			0
$341$	−6.44492			−0.349012
$342$		0			0
$343$			9.24482i			0.499173i
$344$		0			0
$345$			0.942134i			0.0507228i
$346$		0			0
$347$			18.6356i			1.00041i	0.865907	+	0.500206i	$0.166742\pi$
$347$			18.6356i			1.00041i	−0.865907	+	0.500206i	$0.833258\pi$
$348$		0			0
$349$	10.5154			0.562875			0.281438	−	0.959579i	$-0.409189\pi$
$349$	10.5154			0.562875			0.281438	+	0.959579i	$0.409189\pi$
$350$		0			0
$351$		−	0.963608i		−	0.0514336i
$352$		0			0
$353$	−23.2741			−1.23876			−0.619379	−	0.785092i	$-0.712615\pi$
$353$	−23.2741			−1.23876			−0.619379	+	0.785092i	$0.712615\pi$
$354$		0			0
$355$	−17.3392			−0.920268
$356$		0			0
$357$	−0.108221	+	0.0856115i	−0.00572767	+	0.00453104i
$358$		0			0
$359$	−1.81607			−0.0958486			−0.0479243	−	0.998851i	$-0.515261\pi$
$359$	−1.81607			−0.0958486			−0.0479243	+	0.998851i	$0.515261\pi$
$360$		0			0
$361$	−0.814869			−0.0428878
$362$		0			0
$363$			0.645775i			0.0338944i
$364$		0			0
$365$	25.9430			1.35792
$366$		0			0
$367$			12.4339i			0.649045i	0.945878	+	0.324523i	$0.105204\pi$
$367$			12.4339i			0.649045i	−0.945878	+	0.324523i	$0.894796\pi$
$368$		0			0
$369$			0.416961i			0.0217061i
$370$		0			0
$371$		−	4.79543i		−	0.248966i
$372$		0			0
$373$	12.6015			0.652481			0.326240	−	0.945287i	$-0.394218\pi$
$373$	12.6015			0.652481			0.326240	+	0.945287i	$0.394218\pi$
$374$		0			0
$375$	0.0796738			0.00411434
$376$		0			0
$377$		−	11.6369i		−	0.599333i
$378$		0			0
$379$		−	5.51376i		−	0.283223i	−0.989922	−	0.141611i	$-0.954772\pi$
$379$		−	5.51376i		−	0.283223i	0.989922	−	0.141611i	$-0.0452284\pi$
$380$		0			0
$381$		−	0.221935i		−	0.0113701i
$382$		0			0
$383$	−16.8083			−0.858863			−0.429431	−	0.903100i	$-0.641286\pi$
$383$	−16.8083			−0.858863			−0.429431	+	0.903100i	$0.641286\pi$
$384$		0			0
$385$		−	10.8859i		−	0.554796i
$386$		0			0
$387$	2.99546			0.152268
$388$		0			0
$389$	26.6932			1.35340			0.676700	−	0.736259i	$-0.263409\pi$
$389$	26.6932			1.35340			0.676700	+	0.736259i	$0.263409\pi$
$390$		0			0
$391$	15.1796	+	19.1885i	0.767667	+	0.970404i
$392$		0			0
$393$	−0.288530			−0.0145544
$394$		0			0
$395$	−27.1033			−1.36371
$396$		0			0
$397$			29.0352i			1.45723i	0.684921	+	0.728617i	$0.259837\pi$
$397$			29.0352i			1.45723i	−0.684921	+	0.728617i	$0.740163\pi$
$398$		0			0
$399$	−0.142718			−0.00714486
$400$		0			0
$401$			9.79832i			0.489305i	0.969611	+	0.244652i	$0.0786738\pi$
$401$			9.79832i			0.489305i	−0.969611	+	0.244652i	$0.921326\pi$
$402$		0			0
$403$		−	4.29806i		−	0.214102i
$404$		0			0
$405$		−	29.0959i		−	1.44579i
$406$		0			0
$407$	−12.2596			−0.607684
$408$		0			0
$409$	2.07930			0.102815			0.0514075	−	0.998678i	$-0.483629\pi$
$409$	2.07930			0.102815			0.0514075	+	0.998678i	$0.483629\pi$
$410$		0			0
$411$			0.652231i			0.0321722i
$412$		0			0
$413$		−	0.683113i		−	0.0336138i
$414$		0			0
$415$		−	28.5344i		−	1.40070i
$416$		0			0
$417$	−0.643152			−0.0314953
$418$		0			0
$419$		−	35.1374i		−	1.71657i	−0.513170	−	0.858287i	$-0.671529\pi$
$419$		−	35.1374i		−	1.71657i	0.513170	−	0.858287i	$-0.328471\pi$
$420$		0			0
$421$	16.5910			0.808596			0.404298	−	0.914627i	$-0.367516\pi$
$421$	16.5910			0.808596			0.404298	+	0.914627i	$0.367516\pi$
$422$		0			0
$423$	15.8300			0.769680
$424$		0			0
$425$	17.7909	−	14.0740i	0.862983	−	0.682688i
$426$		0			0
$427$	−1.97541			−0.0955969
$428$		0			0
$429$	−0.790062			−0.0381445
$430$		0			0
$431$		−	31.9602i		−	1.53947i	−0.638365	−	0.769734i	$-0.720389\pi$
$431$		−	31.9602i		−	1.53947i	0.638365	−	0.769734i	$-0.279611\pi$
$432$		0			0
$433$	−25.4504			−1.22307			−0.611534	−	0.791218i	$-0.709447\pi$
$433$	−25.4504			−1.22307			−0.611534	+	0.791218i	$0.709447\pi$
$434$		0			0
$435$			0.563387i			0.0270123i
$436$		0			0
$437$			25.3052i			1.21051i
$438$		0			0
$439$		−	37.4810i		−	1.78887i	−0.447197	−	0.894435i	$-0.647578\pi$
$439$		−	37.4810i		−	1.78887i	0.447197	−	0.894435i	$-0.352422\pi$
$440$		0			0
$441$	19.5844			0.932590
$442$		0			0
$443$	−7.82528			−0.371790			−0.185895	−	0.982570i	$-0.559519\pi$
$443$	−7.82528			−0.371790			−0.185895	+	0.982570i	$0.559519\pi$
$444$		0			0
$445$			13.0067i			0.616578i
$446$		0			0
$447$			0.517093i			0.0244577i
$448$		0			0
$449$		−	0.746331i		−	0.0352215i	−0.999845	−	0.0176108i	$-0.994394\pi$
$449$		−	0.746331i		−	0.0352215i	0.999845	−	0.0176108i	$-0.00560597\pi$
$450$		0			0
$451$	0.684005			0.0322086
$452$		0			0
$453$			0.246689i			0.0115905i
$454$		0			0
$455$	7.25970			0.340340
$456$		0			0
$457$	4.71390			0.220507			0.110253	−	0.993903i	$-0.464834\pi$
$457$	4.71390			0.220507			0.110253	+	0.993903i	$0.464834\pi$
$458$		0			0
$459$	0.751652	+	0.950160i	0.0350841	+	0.0443497i
$460$		0			0
$461$	21.9077			1.02034			0.510171	−	0.860073i	$-0.329582\pi$
$461$	21.9077			1.02034			0.510171	+	0.860073i	$0.329582\pi$
$462$		0			0
$463$	23.9411			1.11264			0.556319	−	0.830969i	$-0.312213\pi$
$463$	23.9411			1.11264			0.556319	+	0.830969i	$0.312213\pi$
$464$		0			0
$465$			0.208085i			0.00964971i
$466$		0			0
$467$	−27.0609			−1.25223			−0.626115	−	0.779731i	$-0.715356\pi$
$467$	−27.0609			−1.25223			−0.626115	+	0.779731i	$0.715356\pi$
$468$		0			0
$469$		−	6.44756i		−	0.297721i
$470$		0			0
$471$			0.871180i			0.0401419i
$472$		0			0
$473$		−	4.91391i		−	0.225942i
$474$		0			0
$475$	23.4620			1.07651
$476$		0			0
$477$	−21.0430			−0.963493
$478$		0			0
$479$			22.9594i			1.04904i	0.851398	+	0.524521i	$0.175755\pi$
$479$			22.9594i			1.04904i	−0.851398	+	0.524521i	$0.824245\pi$
$480$		0			0
$481$		−	8.17581i		−	0.372785i
$482$		0			0
$483$		−	0.198597i		−	0.00903648i
$484$		0			0
$485$	34.3724			1.56077
$486$		0			0
$487$		−	2.31526i		−	0.104914i	−0.998623	−	0.0524572i	$-0.983295\pi$
$487$		−	2.31526i		−	0.104914i	0.998623	−	0.0524572i	$-0.0167053\pi$
$488$		0			0
$489$	−0.845559			−0.0382375
$490$		0			0
$491$	17.1787			0.775262			0.387631	−	0.921815i	$-0.373293\pi$
$491$	17.1787			0.775262			0.387631	+	0.921815i	$0.373293\pi$
$492$		0			0
$493$	9.07727	+	11.4745i	0.408820	+	0.516787i
$494$		0			0
$495$	−47.7688			−2.14705
$496$		0			0
$497$	3.65501			0.163950
$498$		0			0
$499$		−	12.0648i		−	0.540094i	−0.962847	−	0.270047i	$-0.912961\pi$
$499$		−	12.0648i		−	0.540094i	0.962847	−	0.270047i	$-0.0870392\pi$
$500$		0			0
$501$	−0.323601			−0.0144574
$502$		0			0
$503$			1.96124i			0.0874472i	0.999044	+	0.0437236i	$0.0139221\pi$
$503$			1.96124i			0.0874472i	−0.999044	+	0.0437236i	$0.986078\pi$
$504$		0			0
$505$		−	36.1304i		−	1.60778i
$506$		0			0
$507$			0.110016i			0.00488599i
$508$		0			0
$509$	24.0763			1.06716			0.533582	−	0.845748i	$-0.320846\pi$
$509$	24.0763			1.06716			0.533582	+	0.845748i	$0.320846\pi$
$510$		0			0
$511$	−5.46865			−0.241919
$512$		0			0
$513$			1.25304i			0.0553231i
$514$		0			0
$515$			28.0310i			1.23519i
$516$		0			0
$517$		−	25.9684i		−	1.14209i
$518$		0			0
$519$	−0.600237			−0.0263475
$520$		0			0
$521$		−	27.6641i		−	1.21198i	−0.795470	−	0.605992i	$-0.792776\pi$
$521$		−	27.6641i		−	1.21198i	0.795470	−	0.605992i	$-0.207224\pi$
$522$		0			0
$523$	11.3168			0.494848			0.247424	−	0.968907i	$-0.420416\pi$
$523$	11.3168			0.494848			0.247424	+	0.968907i	$0.420416\pi$
$524$		0			0
$525$	−0.184132			−0.00803617
$526$		0			0
$527$	3.35266	+	4.23808i	0.146044	+	0.184614i
$528$		0			0
$529$	−12.2129			−0.530997
$530$		0			0
$531$	−2.99760			−0.130085
$532$		0			0
$533$			0.456158i			0.0197584i
$534$		0			0
$535$	30.5915			1.32259
$536$		0			0
$537$		−	0.660317i		−	0.0284948i
$538$		0			0
$539$		−	32.1273i		−	1.38382i
$540$		0			0
$541$		−	10.1776i		−	0.437569i	−0.975773	−	0.218784i	$-0.929791\pi$
$541$		−	10.1776i		−	0.437569i	0.975773	−	0.218784i	$-0.0702091\pi$
$542$		0			0
$543$	0.437350			0.0187685
$544$		0			0
$545$	6.75485			0.289346
$546$		0			0
$547$		−	4.74932i		−	0.203066i	−0.994832	−	0.101533i	$-0.967625\pi$
$547$		−	4.74932i		−	0.203066i	0.994832	−	0.101533i	$-0.0323748\pi$
$548$		0			0
$549$			8.66839i			0.369958i
$550$		0			0
$551$			15.1322i			0.644655i
$552$		0			0
$553$	5.71324			0.242952
$554$		0			0
$555$			0.395821i			0.0168017i
$556$		0			0
$557$	19.2709			0.816534			0.408267	−	0.912863i	$-0.366133\pi$
$557$	19.2709			0.816534			0.408267	+	0.912863i	$0.366133\pi$
$558$		0			0
$559$	3.27705			0.138604
$560$		0			0
$561$	0.779036	−	0.616279i	0.0328909	−	0.0260193i
$562$		0			0
$563$	−43.9832			−1.85367			−0.926835	−	0.375469i	$-0.877482\pi$
$563$	−43.9832			−1.85367			−0.926835	+	0.375469i	$0.877482\pi$
$564$		0			0
$565$	46.4395			1.95372
$566$		0			0
$567$			6.13327i			0.257573i
$568$		0			0
$569$	11.4443			0.479771			0.239885	−	0.970801i	$-0.422890\pi$
$569$	11.4443			0.479771			0.239885	+	0.970801i	$0.422890\pi$
$570$		0			0
$571$		−	6.84839i		−	0.286596i	−0.989680	−	0.143298i	$-0.954229\pi$
$571$		−	6.84839i		−	0.286596i	0.989680	−	0.143298i	$-0.0457708\pi$
$572$		0			0
$573$			0.651798i			0.0272292i
$574$		0			0
$575$			32.6481i			1.36152i
$576$		0			0
$577$	−12.9026			−0.537143			−0.268572	−	0.963260i	$-0.586552\pi$
$577$	−12.9026			−0.537143			−0.268572	+	0.963260i	$0.586552\pi$
$578$		0			0
$579$	0.233147			0.00968926
$580$		0			0
$581$			6.01490i			0.249540i
$582$		0			0
$583$			34.5201i			1.42968i
$584$		0			0
$585$		−	31.8566i		−	1.31711i
$586$		0			0
$587$	30.2829			1.24991			0.624954	−	0.780661i	$-0.285118\pi$
$587$	30.2829			1.24991			0.624954	+	0.780661i	$0.285118\pi$
$588$		0			0
$589$			5.58904i			0.230293i
$590$		0			0
$591$	−0.562636			−0.0231437
$592$		0			0
$593$	−5.66107			−0.232472			−0.116236	−	0.993222i	$-0.537083\pi$
$593$	−5.66107			−0.232472			−0.116236	+	0.993222i	$0.537083\pi$
$594$		0			0
$595$	−7.15839	+	5.66285i	−0.293465	+	0.232154i
$596$		0			0
$597$	−0.148814			−0.00609057
$598$		0			0
$599$	−15.4922			−0.632994			−0.316497	−	0.948593i	$-0.602507\pi$
$599$	−15.4922			−0.632994			−0.316497	+	0.948593i	$0.602507\pi$
$600$		0			0
$601$			28.0315i			1.14343i	0.820453	+	0.571715i	$0.193722\pi$
$601$			28.0315i			1.14343i	−0.820453	+	0.571715i	$0.806278\pi$
$602$		0			0
$603$	−28.2928			−1.15217
$604$		0			0
$605$			42.7154i			1.73663i
$606$		0			0
$607$		−	2.40745i		−	0.0977152i	−0.998806	−	0.0488576i	$-0.984442\pi$
$607$		−	2.40745i		−	0.0977152i	0.998806	−	0.0488576i	$-0.0155581\pi$
$608$		0			0
$609$		−	0.118759i		−	0.00481237i
$610$		0			0
$611$	17.3181			0.700616
$612$		0			0
$613$	22.7129			0.917365			0.458683	−	0.888600i	$-0.348321\pi$
$613$	22.7129			0.917365			0.458683	+	0.888600i	$0.348321\pi$
$614$		0			0
$615$		−	0.0220843i		−	0.000890523i
$616$		0			0
$617$			16.3841i			0.659600i	0.944051	+	0.329800i	$0.106981\pi$
$617$			16.3841i			0.659600i	−0.944051	+	0.329800i	$0.893019\pi$
$618$		0			0
$619$		−	28.6438i		−	1.15129i	−0.817700	−	0.575645i	$-0.804751\pi$
$619$		−	28.6438i		−	1.15129i	0.817700	−	0.575645i	$-0.195249\pi$
$620$		0			0
$621$	−1.74364			−0.0699700
$622$		0			0
$623$		−	2.74175i		−	0.109846i
$624$		0			0
$625$	−22.2390			−0.889561
$626$		0			0
$627$	1.02737			0.0410291
$628$		0			0
$629$	6.37745	+	8.06171i	0.254286	+	0.321441i
$630$		0			0
$631$	−21.7481			−0.865776			−0.432888	−	0.901448i	$-0.642505\pi$
$631$	−21.7481			−0.865776			−0.432888	+	0.901448i	$0.642505\pi$
$632$		0			0
$633$	−0.314004			−0.0124805
$634$		0			0
$635$		−	14.6801i		−	0.582562i
$636$		0			0
$637$	21.4254			0.848907
$638$		0			0
$639$		−	16.0387i		−	0.634482i
$640$		0			0
$641$		−	41.3051i		−	1.63145i	−0.578437	−	0.815727i	$-0.696337\pi$
$641$		−	41.3051i		−	1.63145i	0.578437	−	0.815727i	$-0.303663\pi$
$642$		0			0
$643$			13.9674i			0.550821i	0.961327	+	0.275410i	$0.0888138\pi$
$643$			13.9674i			0.550821i	−0.961327	+	0.275410i	$0.911186\pi$
$644$		0			0
$645$	−0.158654			−0.00624699
$646$		0			0
$647$	26.0490			1.02409			0.512047	−	0.858958i	$-0.328887\pi$
$647$	26.0490			1.02409			0.512047	+	0.858958i	$0.328887\pi$
$648$		0			0
$649$			4.91743i			0.193026i
$650$		0			0
$651$		−	0.0438633i		−	0.00171914i
$652$		0			0
$653$		−	3.47444i		−	0.135965i	−0.997687	−	0.0679826i	$-0.978344\pi$
$653$		−	3.47444i		−	0.135965i	0.997687	−	0.0679826i	$-0.0216562\pi$
$654$		0			0
$655$	−19.0851			−0.745717
$656$		0			0
$657$			23.9972i			0.936221i
$658$		0			0
$659$	−1.56191			−0.0608435			−0.0304218	−	0.999537i	$-0.509685\pi$
$659$	−1.56191			−0.0608435			−0.0304218	+	0.999537i	$0.509685\pi$
$660$		0			0
$661$	7.75015			0.301446			0.150723	−	0.988576i	$-0.451840\pi$
$661$	7.75015			0.301446			0.150723	+	0.988576i	$0.451840\pi$
$662$		0			0
$663$	0.410992	+	0.519533i	0.0159616	+	0.0201770i
$664$		0			0
$665$	−9.44025			−0.366077
$666$		0			0
$667$	−21.0570			−0.815330
$668$		0			0
$669$			0.608102i			0.0235106i
$670$		0			0
$671$	14.2201			0.548961
$672$		0			0
$673$		−	5.82518i		−	0.224544i	−0.993678	−	0.112272i	$-0.964187\pi$
$673$		−	5.82518i		−	0.224544i	0.993678	−	0.112272i	$-0.0358128\pi$
$674$		0			0
$675$			1.61664i			0.0622245i
$676$		0			0
$677$		−	33.2788i		−	1.27901i	−0.768788	−	0.639504i	$-0.779140\pi$
$677$		−	33.2788i		−	1.27901i	0.768788	−	0.639504i	$-0.220860\pi$
$678$		0			0
$679$	−7.24553			−0.278058
$680$		0			0
$681$	−0.747722			−0.0286528
$682$		0			0
$683$			35.1884i			1.34645i	0.739439	+	0.673223i	$0.235091\pi$
$683$			35.1884i			1.34645i	−0.739439	+	0.673223i	$0.764909\pi$
$684$		0			0
$685$			43.1424i			1.64839i
$686$		0			0
$687$			0.181978i			0.00694290i
$688$		0			0
$689$	−23.0212			−0.877038
$690$		0			0
$691$			48.0275i			1.82705i	0.406781	+	0.913526i	$0.366651\pi$
$691$			48.0275i			1.82705i	−0.406781	+	0.913526i	$0.633349\pi$
$692$		0			0
$693$	10.0694			0.382506
$694$		0			0
$695$	−42.5419			−1.61371
$696$		0			0
$697$	−0.355821	−	0.449792i	−0.0134777	−	0.0170371i
$698$		0			0
$699$	−1.11746			−0.0422664
$700$		0			0
$701$	44.4239			1.67787			0.838934	−	0.544233i	$-0.183179\pi$
$701$	44.4239			1.67787			0.838934	+	0.544233i	$0.183179\pi$
$702$		0			0
$703$			10.6315i			0.400975i
$704$		0			0
$705$	−0.838433			−0.0315772
$706$		0			0
$707$			7.61611i			0.286433i
$708$		0			0
$709$			41.4581i			1.55699i	0.627650	+	0.778496i	$0.284017\pi$
$709$			41.4581i			1.55699i	−0.627650	+	0.778496i	$0.715983\pi$
$710$		0			0
$711$		−	25.0705i		−	0.940218i
$712$		0			0
$713$	−7.77733			−0.291263
$714$		0			0
$715$	−52.2594			−1.95439
$716$		0			0
$717$		−	0.412016i		−	0.0153870i
$718$		0			0
$719$		−	18.9666i		−	0.707334i	−0.935371	−	0.353667i	$-0.884935\pi$
$719$		−	18.9666i		−	0.707334i	0.935371	−	0.353667i	$-0.115065\pi$
$720$		0			0
$721$		−	5.90879i		−	0.220055i
$722$		0			0
$723$	0.927411			0.0344908
$724$		0			0
$725$			19.5233i			0.725075i
$726$		0			0
$727$	16.5341			0.613214			0.306607	−	0.951836i	$-0.400806\pi$
$727$	16.5341			0.613214			0.306607	+	0.951836i	$0.400806\pi$
$728$		0			0
$729$	26.8705			0.995203
$730$		0			0
$731$	−3.23132	+	2.55623i	−0.119515	+	0.0945455i
$732$		0			0
$733$	42.1724			1.55767			0.778837	−	0.627227i	$-0.215810\pi$
$733$	42.1724			1.55767			0.778837	+	0.627227i	$0.215810\pi$
$734$		0			0
$735$	−1.03728			−0.0382608
$736$		0			0
$737$			46.4131i			1.70965i
$738$		0			0
$739$	−26.6908			−0.981836			−0.490918	−	0.871206i	$-0.663339\pi$
$739$	−26.6908			−0.981836			−0.490918	+	0.871206i	$0.663339\pi$
$740$		0			0
$741$			0.685143i			0.0251693i
$742$		0			0
$743$			46.7733i			1.71595i	0.513695	+	0.857973i	$0.328276\pi$
$743$			46.7733i			1.71595i	−0.513695	+	0.857973i	$0.671724\pi$
$744$		0			0
$745$			34.2036i			1.25312i
$746$		0			0
$747$	26.3943			0.965715
$748$		0			0
$749$	−6.44854			−0.235624
$750$		0			0
$751$			25.0389i			0.913684i	0.889548	+	0.456842i	$0.151019\pi$
$751$			25.0389i			0.913684i	−0.889548	+	0.456842i	$0.848981\pi$
$752$		0			0
$753$		−	0.706647i		−	0.0257516i
$754$		0			0
$755$			16.3175i			0.593854i
$756$		0			0
$757$	11.1156			0.404003			0.202002	−	0.979385i	$-0.435255\pi$
$757$	11.1156			0.404003			0.202002	+	0.979385i	$0.435255\pi$
$758$		0			0
$759$			1.42961i			0.0518917i
$760$		0			0
$761$	−45.8045			−1.66041			−0.830206	−	0.557456i	$-0.811777\pi$
$761$	−45.8045			−1.66041			−0.830206	+	0.557456i	$0.811777\pi$
$762$		0			0
$763$	−1.42389			−0.0515482
$764$		0			0
$765$	24.8494	+	31.4120i	0.898433	+	1.13570i
$766$		0			0
$767$	−3.27939			−0.118412
$768$		0			0
$769$	46.3819			1.67258			0.836288	−	0.548291i	$-0.184721\pi$
$769$	46.3819			1.67258			0.836288	+	0.548291i	$0.184721\pi$
$770$		0			0
$771$		−	1.27017i		−	0.0457439i
$772$		0			0
$773$	20.8664			0.750514			0.375257	−	0.926921i	$-0.377554\pi$
$773$	20.8664			0.750514			0.375257	+	0.926921i	$0.377554\pi$
$774$		0			0
$775$			7.21085i			0.259021i
$776$		0			0
$777$		−	0.0834371i		−	0.00299329i
$778$		0			0
$779$		−	0.593170i		−	0.0212525i
$780$		0			0
$781$	−26.3108			−0.941475
$782$		0			0
$783$	−1.04268			−0.0372624
$784$		0			0
$785$			57.6250i			2.05673i
$786$		0			0
$787$			10.4463i			0.372372i	0.982515	+	0.186186i	$0.0596127\pi$
$787$			10.4463i			0.372372i	−0.982515	+	0.186186i	$0.940387\pi$
$788$		0			0
$789$		−	0.888510i		−	0.0316318i
$790$		0			0
$791$	−9.78921			−0.348064
$792$		0			0
$793$			9.48327i			0.336761i
$794$		0			0
$795$	1.11454			0.0395287
$796$		0			0
$797$	53.1902			1.88409			0.942047	−	0.335480i	$-0.108898\pi$
$797$	53.1902			1.88409			0.942047	+	0.335480i	$0.108898\pi$
$798$		0			0
$799$	−17.0764	+	13.5088i	−0.604121	+	0.477907i
$800$		0			0
$801$	−12.0312			−0.425102
$802$		0			0
$803$	39.3664			1.38921
$804$		0			0
$805$		−	13.1364i		−	0.462997i
$806$		0			0
$807$	−0.515791			−0.0181567
$808$		0			0
$809$		−	46.4060i		−	1.63155i	−0.578371	−	0.815774i	$-0.696311\pi$
$809$		−	46.4060i		−	1.63155i	0.578371	−	0.815774i	$-0.303689\pi$
$810$		0			0
$811$			21.8965i			0.768891i	0.923148	+	0.384445i	$0.125607\pi$
$811$			21.8965i			0.768891i	−0.923148	+	0.384445i	$0.874393\pi$
$812$		0			0
$813$			1.27575i			0.0447426i
$814$		0			0
$815$	−55.9303			−1.95915
$816$		0			0
$817$	−4.26135			−0.149086
$818$		0			0
$819$			6.71522i			0.234649i
$820$		0			0
$821$			6.46533i			0.225642i	0.993615	+	0.112821i	$0.0359886\pi$
$821$			6.46533i			0.225642i	−0.993615	+	0.112821i	$0.964011\pi$
$822$		0			0
$823$			7.19594i			0.250835i	0.992104	+	0.125417i	$0.0400270\pi$
$823$			7.19594i			0.250835i	−0.992104	+	0.125417i	$0.959973\pi$
$824$		0			0
$825$	1.32548			0.0461474
$826$		0			0
$827$		−	27.0317i		−	0.939986i	−0.882670	−	0.469993i	$-0.844257\pi$
$827$		−	27.0317i		−	0.939986i	0.882670	−	0.469993i	$-0.155743\pi$
$828$		0			0
$829$	−2.76860			−0.0961576			−0.0480788	−	0.998844i	$-0.515310\pi$
$829$	−2.76860			−0.0961576			−0.0480788	+	0.998844i	$0.515310\pi$
$830$		0			0
$831$	0.0411806			0.00142854
$832$		0			0
$833$	−21.1264	+	16.7127i	−0.731988	+	0.579061i
$834$		0			0
$835$	−21.4049			−0.740748
$836$		0			0
$837$	−0.385111			−0.0133114
$838$		0			0
$839$			18.0721i			0.623918i	0.950096	+	0.311959i	$0.100985\pi$
$839$			18.0721i			0.623918i	−0.950096	+	0.311959i	$0.899015\pi$
$840$		0			0
$841$	16.4081			0.565797
$842$		0			0
$843$			0.740965i			0.0255202i
$844$		0			0
$845$			7.27713i			0.250341i
$846$		0			0
$847$		−	9.00420i		−	0.309388i
$848$		0			0
$849$	−0.0506823			−0.00173941
$850$		0			0
$851$	−14.7941			−0.507135
$852$		0			0
$853$		−	45.1680i		−	1.54652i	−0.634087	−	0.773261i	$-0.718624\pi$
$853$		−	45.1680i		−	1.54652i	0.634087	−	0.773261i	$-0.281376\pi$
$854$		0			0
$855$			41.4252i			1.41671i
$856$		0			0
$857$			19.4363i			0.663930i	0.943292	+	0.331965i	$0.107712\pi$
$857$			19.4363i			0.663930i	−0.943292	+	0.331965i	$0.892288\pi$
$858$		0			0
$859$	27.7316			0.946191			0.473095	−	0.881011i	$-0.343137\pi$
$859$	27.7316			0.946191			0.473095	+	0.881011i	$0.343137\pi$
$860$		0			0
$861$			0.00465525i			0.000158651i
$862$		0			0
$863$	35.5727			1.21091			0.605454	−	0.795880i	$-0.292991\pi$
$863$	35.5727			1.21091			0.605454	+	0.795880i	$0.292991\pi$
$864$		0			0
$865$	−39.7032			−1.34995
$866$		0			0
$867$	−0.810512	−	0.191692i	−0.0275264	−	0.00651022i
$868$		0			0
$869$	−41.1271			−1.39514
$870$		0			0
$871$	−30.9525			−1.04879
$872$		0			0
$873$			31.7944i			1.07608i
$874$		0			0
$875$	−1.11091			−0.0375556
$876$		0			0
$877$		−	43.6139i		−	1.47274i	−0.676581	−	0.736368i	$-0.736539\pi$
$877$		−	43.6139i		−	1.47274i	0.676581	−	0.736368i	$-0.263461\pi$
$878$		0			0
$879$		−	0.613099i		−	0.0206793i
$880$		0			0
$881$		−	20.2762i		−	0.683122i	−0.939860	−	0.341561i	$-0.889044\pi$
$881$		−	20.2762i		−	0.683122i	0.939860	−	0.341561i	$-0.110956\pi$
$882$		0			0
$883$	−17.5944			−0.592098			−0.296049	−	0.955173i	$-0.595669\pi$
$883$	−17.5944			−0.592098			−0.296049	+	0.955173i	$0.595669\pi$
$884$		0			0
$885$	0.158767			0.00533691
$886$		0			0
$887$		−	8.86467i		−	0.297646i	−0.988864	−	0.148823i	$-0.952451\pi$
$887$		−	8.86467i		−	0.297646i	0.988864	−	0.148823i	$-0.0475485\pi$
$888$		0			0
$889$			3.09449i			0.103786i
$890$		0			0
$891$		−	44.1507i		−	1.47910i
$892$		0			0
$893$	−22.5198			−0.753598
$894$		0			0
$895$		−	43.6773i		−	1.45997i
$896$		0			0
$897$	−0.953397			−0.0318330
$898$		0			0
$899$	−4.65077			−0.155112
$900$		0			0
$901$	22.6999	−	17.9574i	0.756244	−	0.598249i
$902$		0			0
$903$	0.0334435			0.00111293
$904$		0			0
$905$	28.9290			0.961631
$906$		0			0
$907$		−	44.3935i		−	1.47406i	−0.675860	−	0.737030i	$-0.736228\pi$
$907$		−	44.3935i		−	1.47406i	0.675860	−	0.737030i	$-0.263772\pi$
$908$		0			0
$909$	33.4206			1.10849
$910$		0			0
$911$			12.8474i			0.425652i	0.977090	+	0.212826i	$0.0682668\pi$
$911$			12.8474i			0.425652i	−0.977090	+	0.212826i	$0.931733\pi$
$912$		0			0
$913$		−	43.2986i		−	1.43297i
$914$		0			0
$915$		−	0.459120i		−	0.0151780i
$916$		0			0
$917$	4.02305			0.132853
$918$		0			0
$919$	35.5235			1.17181			0.585905	−	0.810379i	$-0.300739\pi$
$919$	35.5235			1.17181			0.585905	+	0.810379i	$0.300739\pi$
$920$		0			0
$921$		−	0.605220i		−	0.0199427i
$922$		0			0
$923$		−	17.5465i		−	0.577549i
$924$		0			0
$925$			13.7165i			0.450996i
$926$		0			0
$927$	−25.9286			−0.851607
$928$		0			0
$929$			39.6585i			1.30115i	0.759440	+	0.650577i	$0.225473\pi$
$929$			39.6585i			1.30115i	−0.759440	+	0.650577i	$0.774527\pi$
$930$		0			0
$931$	−27.8609			−0.913103
$932$		0			0
$933$	1.21914			0.0399130
$934$		0			0
$935$	51.5301	−	40.7644i	1.68521	−	1.33314i
$936$		0			0
$937$	32.0906			1.04836			0.524178	−	0.851609i	$-0.324373\pi$
$937$	32.0906			1.04836			0.524178	+	0.851609i	$0.324373\pi$
$938$		0			0
$939$	0.187396			0.00611545
$940$		0			0
$941$		−	9.89361i		−	0.322522i	−0.986912	−	0.161261i	$-0.948444\pi$
$941$		−	9.89361i		−	0.322522i	0.986912	−	0.161261i	$-0.0515562\pi$
$942$		0			0
$943$	0.825415			0.0268792
$944$		0			0
$945$		−	0.650477i		−	0.0211600i
$946$		0			0
$947$			39.9372i			1.29778i	0.760880	+	0.648892i	$0.224767\pi$
$947$			39.9372i			1.29778i	−0.760880	+	0.648892i	$0.775233\pi$
$948$		0			0
$949$			26.2531i			0.852213i
$950$		0			0
$951$	0.957849			0.0310604
$952$		0			0
$953$	−36.0717			−1.16848			−0.584238	−	0.811583i	$-0.698606\pi$
$953$	−36.0717			−1.16848			−0.584238	+	0.811583i	$0.698606\pi$
$954$		0			0
$955$			43.1138i			1.39513i
$956$		0			0
$957$			0.854895i			0.0276348i
$958$		0			0
$959$		−	9.09421i		−	0.293667i
$960$		0			0
$961$	29.2823			0.944589
$962$		0			0
$963$			28.2971i			0.911861i
$964$		0			0
$965$	15.4217			0.496443
$966$		0			0
$967$	−4.38259			−0.140935			−0.0704673	−	0.997514i	$-0.522449\pi$
$967$	−4.38259			−0.140935			−0.0704673	+	0.997514i	$0.522449\pi$
$968$		0			0
$969$	−0.534438	−	0.675581i	−0.0171686	−	0.0217028i
$970$		0			0
$971$	25.4680			0.817306			0.408653	−	0.912690i	$-0.365999\pi$
$971$	25.4680			0.817306			0.408653	+	0.912690i	$0.365999\pi$
$972$		0			0
$973$	8.96763			0.287489
$974$		0			0
$975$			0.883954i			0.0283092i
$976$		0			0
$977$	45.7553			1.46384			0.731921	−	0.681389i	$-0.238624\pi$
$977$	45.7553			1.46384			0.731921	+	0.681389i	$0.238624\pi$
$978$		0			0
$979$			19.7367i			0.630787i
$980$		0			0
$981$			6.24823i			0.199491i
$982$		0			0
$983$			30.9359i			0.986702i	0.869830	+	0.493351i	$0.164228\pi$
$983$			30.9359i			0.986702i	−0.869830	+	0.493351i	$0.835772\pi$
$984$		0			0
$985$	−37.2161			−1.18580
$986$		0			0
$987$	0.176738			0.00562562
$988$		0			0
$989$		−	5.92980i		−	0.188557i
$990$		0			0
$991$			32.7702i			1.04098i	0.853867	+	0.520491i	$0.174251\pi$
$991$			32.7702i			1.04098i	−0.853867	+	0.520491i	$0.825749\pi$
$992$		0			0
$993$		−	0.358366i		−	0.0113724i
$994$		0			0
$995$	−9.84347			−0.312059
$996$		0			0
$997$			45.8764i			1.45292i	0.687208	+	0.726461i	$0.258836\pi$
$997$			45.8764i			1.45292i	−0.687208	+	0.726461i	$0.741164\pi$
$998$		0			0
$999$	−0.732561			−0.0231772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.a.237.20	✓	40
17.16	even	2	inner	4012.2.b.a.237.21	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.a.237.20	✓	40	1.1	even	1	trivial
4012.2.b.a.237.21	yes	40	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)

\(f(q)\)	\(=\)	\(q-0.0489925i q^{3} -3.24065i q^{5} +0.683113i q^{7} +2.99760 q^{9} +O(q^{10})\)	\(q-0.0489925i q^{3} -3.24065i q^{5} +0.683113i q^{7} +2.99760 q^{9} -4.91743i q^{11} +3.27939 q^{13} -0.158767 q^{15} +(-3.23363 + 2.55806i) q^{17} -4.26440 q^{19} +0.0334674 q^{21} -5.93405i q^{23} -5.50183 q^{25} -0.293837i q^{27} -3.54850i q^{29} -1.31063i q^{31} -0.240917 q^{33} +2.21373 q^{35} -2.49309i q^{37} -0.160666i q^{39} +0.139098i q^{41} +0.999285 q^{43} -9.71418i q^{45} +5.28089 q^{47} +6.53336 q^{49} +(0.125325 + 0.158423i) q^{51} -7.01995 q^{53} -15.9357 q^{55} +0.208924i q^{57} -1.00000 q^{59} +2.89178i q^{61} +2.04770i q^{63} -10.6274i q^{65} -9.43849 q^{67} -0.290724 q^{69} -5.35052i q^{71} +8.00548i q^{73} +0.269548i q^{75} +3.35916 q^{77} -8.36353i q^{79} +8.97840 q^{81} +8.80513 q^{83} +(8.28977 + 10.4791i) q^{85} -0.173850 q^{87} -4.01361 q^{89} +2.24020i q^{91} -0.0642109 q^{93} +13.8194i q^{95} +10.6066i q^{97} -14.7405i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 36 q^{9}+O(q^{10}) \) 40 * q - 36 * q^9	\( 40 q - 36 q^{9} + 8 q^{13} + 20 q^{15} - 8 q^{17} + 20 q^{19} + 6 q^{21} - 24 q^{25} - 14 q^{33} - 52 q^{35} + 22 q^{43} - 10 q^{47} + 8 q^{49} - 6 q^{51} - 2 q^{53} - 12 q^{55} - 40 q^{59} - 24 q^{67} - 36 q^{69} - 38 q^{77} + 16 q^{81} + 32 q^{83} - 22 q^{85} - 18 q^{87} + 40 q^{89} + 22 q^{93}+O(q^{100}) \) 40 * q - 36 * q^9 + 8 * q^13 + 20 * q^15 - 8 * q^17 + 20 * q^19 + 6 * q^21 - 24 * q^25 - 14 * q^33 - 52 * q^35 + 22 * q^43 - 10 * q^47 + 8 * q^49 - 6 * q^51 - 2 * q^53 - 12 * q^55 - 40 * q^59 - 24 * q^67 - 36 * q^69 - 38 * q^77 + 16 * q^81 + 32 * q^83 - 22 * q^85 - 18 * q^87 + 40 * q^89 + 22 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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