LMFDB - Embedded newform 4012.2.b.a.237.8

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			237.8
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.a.237.33

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.21522i q^{3} +2.69018i q^{5} -2.88510i q^{7} -1.90721 q^{9} +O(q^{10})$	$q-2.21522i q^{3} +2.69018i q^{5} -2.88510i q^{7} -1.90721 q^{9} -5.22259i q^{11} -3.34041 q^{13} +5.95933 q^{15} +(-3.60700 - 1.99739i) q^{17} +2.82638 q^{19} -6.39115 q^{21} -0.998743i q^{23} -2.23704 q^{25} -2.42078i q^{27} -3.66316i q^{29} +4.60738i q^{31} -11.5692 q^{33} +7.76144 q^{35} -3.04660i q^{37} +7.39976i q^{39} +6.46816i q^{41} -5.39877 q^{43} -5.13072i q^{45} +2.80764 q^{47} -1.32383 q^{49} +(-4.42465 + 7.99031i) q^{51} -6.70201 q^{53} +14.0497 q^{55} -6.26105i q^{57} -1.00000 q^{59} +9.33710i q^{61} +5.50249i q^{63} -8.98630i q^{65} +0.00861422 q^{67} -2.21244 q^{69} +2.93987i q^{71} -7.83587i q^{73} +4.95554i q^{75} -15.0677 q^{77} +5.00280i q^{79} -11.0842 q^{81} +6.57241 q^{83} +(5.37332 - 9.70346i) q^{85} -8.11472 q^{87} +10.6826 q^{89} +9.63744i q^{91} +10.2064 q^{93} +7.60345i q^{95} -3.82880i q^{97} +9.96056i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	2.21522i		−	1.27896i	−0.768808	−	0.639479i	$-0.779150\pi$
$3$		−	2.21522i		−	1.27896i	0.768808	−	0.639479i	$-0.220850\pi$
$4$		0			0
$5$			2.69018i			1.20308i	0.798842	+	0.601541i	$0.205447\pi$
$5$			2.69018i			1.20308i	−0.798842	+	0.601541i	$0.794553\pi$
$6$		0			0
$7$		−	2.88510i		−	1.09047i	−0.838284	−	0.545234i	$-0.816441\pi$
$7$		−	2.88510i		−	1.09047i	0.838284	−	0.545234i	$-0.183559\pi$
$8$		0			0
$9$	−1.90721			−0.635736
$10$		0			0
$11$		−	5.22259i		−	1.57467i	−0.616525	−	0.787335i	$-0.711460\pi$
$11$		−	5.22259i		−	1.57467i	0.616525	−	0.787335i	$-0.288540\pi$
$12$		0			0
$13$	−3.34041			−0.926464			−0.463232	−	0.886237i	$-0.653310\pi$
$13$	−3.34041			−0.926464			−0.463232	+	0.886237i	$0.653310\pi$
$14$		0			0
$15$	5.95933			1.53869
$16$		0			0
$17$	−3.60700	−	1.99739i	−0.874826	−	0.484437i
$17$	−3.60700	−	1.99739i	−0.874826	−	0.484437i
$18$		0			0
$19$	2.82638			0.648415			0.324208	−	0.945986i	$-0.394902\pi$
$19$	2.82638			0.648415			0.324208	+	0.945986i	$0.394902\pi$
$20$		0			0
$21$	−6.39115			−1.39466
$22$		0			0
$23$		−	0.998743i		−	0.208252i	−0.994564	−	0.104126i	$-0.966795\pi$
$23$		−	0.998743i		−	0.208252i	0.994564	−	0.104126i	$-0.0332046\pi$
$24$		0			0
$25$	−2.23704			−0.447409
$26$		0			0
$27$		−	2.42078i		−	0.465879i
$28$		0			0
$29$		−	3.66316i		−	0.680232i	−0.940383	−	0.340116i	$-0.889534\pi$
$29$		−	3.66316i		−	0.680232i	0.940383	−	0.340116i	$-0.110466\pi$
$30$		0			0
$31$			4.60738i			0.827510i	0.910388	+	0.413755i	$0.135783\pi$
$31$			4.60738i			0.827510i	−0.910388	+	0.413755i	$0.864217\pi$
$32$		0			0
$33$	−11.5692			−2.01394
$34$		0			0
$35$	7.76144			1.31192
$36$		0			0
$37$		−	3.04660i		−	0.500858i	−0.968135	−	0.250429i	$-0.919428\pi$
$37$		−	3.04660i		−	0.500858i	0.968135	−	0.250429i	$-0.0805717\pi$
$38$		0			0
$39$			7.39976i			1.18491i
$40$		0			0
$41$			6.46816i			1.01016i	0.863074	+	0.505078i	$0.168536\pi$
$41$			6.46816i			1.01016i	−0.863074	+	0.505078i	$0.831464\pi$
$42$		0			0
$43$	−5.39877			−0.823305			−0.411653	−	0.911341i	$-0.635048\pi$
$43$	−5.39877			−0.823305			−0.411653	+	0.911341i	$0.635048\pi$
$44$		0			0
$45$		−	5.13072i		−	0.764843i
$46$		0			0
$47$	2.80764			0.409536			0.204768	−	0.978811i	$-0.434356\pi$
$47$	2.80764			0.409536			0.204768	+	0.978811i	$0.434356\pi$
$48$		0			0
$49$	−1.32383			−0.189119
$50$		0			0
$51$	−4.42465	+	7.99031i	−0.619575	+	1.11887i
$52$		0			0
$53$	−6.70201			−0.920593			−0.460296	−	0.887765i	$-0.652257\pi$
$53$	−6.70201			−0.920593			−0.460296	+	0.887765i	$0.652257\pi$
$54$		0			0
$55$	14.0497			1.89446
$56$		0			0
$57$		−	6.26105i		−	0.829296i
$58$		0			0
$59$	−1.00000			−0.130189
$59$	−1.00000			−0.130189
$60$		0			0
$61$			9.33710i			1.19549i	0.801685	+	0.597747i	$0.203937\pi$
$61$			9.33710i			1.19549i	−0.801685	+	0.597747i	$0.796063\pi$
$62$		0			0
$63$			5.50249i			0.693249i
$64$		0			0
$65$		−	8.98630i		−	1.11461i
$66$		0			0
$67$	0.00861422			0.00105239			0.000526197	−	1.00000i	$-0.499833\pi$
$67$	0.00861422			0.00105239			0.000526197		1.00000i	$0.499833\pi$
$68$		0			0
$69$	−2.21244			−0.266346
$70$		0			0
$71$			2.93987i			0.348898i	0.984666	+	0.174449i	$0.0558145\pi$
$71$			2.93987i			0.348898i	−0.984666	+	0.174449i	$0.944186\pi$
$72$		0			0
$73$		−	7.83587i		−	0.917119i	−0.888664	−	0.458559i	$-0.848366\pi$
$73$		−	7.83587i		−	0.917119i	0.888664	−	0.458559i	$-0.151634\pi$
$74$		0			0
$75$			4.95554i			0.572217i
$76$		0			0
$77$	−15.0677			−1.71713
$78$		0			0
$79$			5.00280i			0.562859i	0.959582	+	0.281429i	$0.0908085\pi$
$79$			5.00280i			0.562859i	−0.959582	+	0.281429i	$0.909192\pi$
$80$		0			0
$81$	−11.0842			−1.23158
$82$		0			0
$83$	6.57241			0.721415			0.360708	−	0.932679i	$-0.382535\pi$
$83$	6.57241			0.721415			0.360708	+	0.932679i	$0.382535\pi$
$84$		0			0
$85$	5.37332	−	9.70346i	0.582818	−	1.05249i
$86$		0			0
$87$	−8.11472			−0.869989
$88$		0			0
$89$	10.6826			1.13236			0.566178	−	0.824283i	$-0.308421\pi$
$89$	10.6826			1.13236			0.566178	+	0.824283i	$0.308421\pi$
$90$		0			0
$91$			9.63744i			1.01028i
$92$		0			0
$93$	10.2064			1.05835
$94$		0			0
$95$			7.60345i			0.780097i
$96$		0			0
$97$		−	3.82880i		−	0.388756i	−0.980927	−	0.194378i	$-0.937731\pi$
$97$		−	3.82880i		−	0.388756i	0.980927	−	0.194378i	$-0.0622688\pi$
$98$		0			0
$99$			9.96056i			1.00107i
$100$		0			0
$101$	−13.8291			−1.37604			−0.688021	−	0.725691i	$-0.741520\pi$
$101$	−13.8291			−1.37604			−0.688021	+	0.725691i	$0.741520\pi$
$102$		0			0
$103$	−14.6151			−1.44007			−0.720033	−	0.693940i	$-0.755873\pi$
$103$	−14.6151			−1.44007			−0.720033	+	0.693940i	$0.755873\pi$
$104$		0			0
$105$		−	17.1933i		−	1.67789i
$106$		0			0
$107$		−	7.91276i		−	0.764955i	−0.923965	−	0.382478i	$-0.875071\pi$
$107$		−	7.91276i		−	0.764955i	0.923965	−	0.382478i	$-0.124929\pi$
$108$		0			0
$109$			10.8402i			1.03830i	0.854683	+	0.519150i	$0.173751\pi$
$109$			10.8402i			1.03830i	−0.854683	+	0.519150i	$0.826249\pi$
$110$		0			0
$111$	−6.74890			−0.640577
$112$		0			0
$113$		−	5.22025i		−	0.491080i	−0.969386	−	0.245540i	$-0.921035\pi$
$113$		−	5.22025i		−	0.491080i	0.969386	−	0.245540i	$-0.0789653\pi$
$114$		0			0
$115$	2.68679			0.250545
$116$		0			0
$117$	6.37086			0.588986
$118$		0			0
$119$	−5.76267	+	10.4066i	−0.528263	+	0.953969i
$120$		0			0
$121$	−16.2754			−1.47959
$122$		0			0
$123$	14.3284			1.29195
$124$		0			0
$125$			7.43284i			0.664813i
$126$		0			0
$127$	−21.6540			−1.92148			−0.960742	−	0.277444i	$-0.910513\pi$
$127$	−21.6540			−1.92148			−0.960742	+	0.277444i	$0.910513\pi$
$128$		0			0
$129$			11.9595i			1.05297i
$130$		0			0
$131$		−	0.303276i		−	0.0264973i	−0.999912	−	0.0132487i	$-0.995783\pi$
$131$		−	0.303276i		−	0.0264973i	0.999912	−	0.0132487i	$-0.00421730\pi$
$132$		0			0
$133$		−	8.15439i		−	0.707076i
$134$		0			0
$135$	6.51232			0.560491
$136$		0			0
$137$	−13.6459			−1.16585			−0.582925	−	0.812526i	$-0.698092\pi$
$137$	−13.6459			−1.16585			−0.582925	+	0.812526i	$0.698092\pi$
$138$		0			0
$139$		−	3.53156i		−	0.299543i	−0.988721	−	0.149772i	$-0.952146\pi$
$139$		−	3.53156i		−	0.299543i	0.988721	−	0.149772i	$-0.0478538\pi$
$140$		0			0
$141$		−	6.21954i		−	0.523780i
$142$		0			0
$143$			17.4456i			1.45888i
$144$		0			0
$145$	9.85455			0.818376
$146$		0			0
$147$			2.93258i			0.241875i
$148$		0			0
$149$	5.29465			0.433755			0.216877	−	0.976199i	$-0.430413\pi$
$149$	5.29465			0.433755			0.216877	+	0.976199i	$0.430413\pi$
$150$		0			0
$151$	−3.88389			−0.316066			−0.158033	−	0.987434i	$-0.550515\pi$
$151$	−3.88389			−0.316066			−0.158033	+	0.987434i	$0.550515\pi$
$152$		0			0
$153$	6.87930	+	3.80943i	0.556158	+	0.307974i
$154$		0			0
$155$	−12.3947			−0.995563
$156$		0			0
$157$	−10.5404			−0.841217			−0.420609	−	0.907242i	$-0.638183\pi$
$157$	−10.5404			−0.841217			−0.420609	+	0.907242i	$0.638183\pi$
$158$		0			0
$159$			14.8464i			1.17740i
$160$		0			0
$161$	−2.88148			−0.227092
$162$		0			0
$163$		−	19.6196i		−	1.53673i	−0.640012	−	0.768365i	$-0.721071\pi$
$163$		−	19.6196i		−	1.53673i	0.640012	−	0.768365i	$-0.278929\pi$
$164$		0			0
$165$		−	31.1232i		−	2.42293i
$166$		0			0
$167$			22.0637i			1.70734i	0.520816	+	0.853669i	$0.325628\pi$
$167$			22.0637i			1.70734i	−0.520816	+	0.853669i	$0.674372\pi$
$168$		0			0
$169$	−1.84164			−0.141664
$170$		0			0
$171$	−5.39048			−0.412221
$172$		0			0
$173$		−	19.6719i		−	1.49563i	−0.663909	−	0.747813i	$-0.731104\pi$
$173$		−	19.6719i		−	1.49563i	0.663909	−	0.747813i	$-0.268896\pi$
$174$		0			0
$175$			6.45410i			0.487884i
$176$		0			0
$177$			2.21522i			0.166506i
$178$		0			0
$179$	−2.35425			−0.175965			−0.0879823	−	0.996122i	$-0.528042\pi$
$179$	−2.35425			−0.175965			−0.0879823	+	0.996122i	$0.528042\pi$
$180$		0			0
$181$		−	13.3467i		−	0.992056i	−0.868307	−	0.496028i	$-0.834791\pi$
$181$		−	13.3467i		−	0.992056i	0.868307	−	0.496028i	$-0.165209\pi$
$182$		0			0
$183$	20.6838			1.52899
$184$		0			0
$185$	8.19589			0.602574
$186$		0			0
$187$	−10.4315	+	18.8379i	−0.762829	+	1.37756i
$188$		0			0
$189$	−6.98420			−0.508026
$190$		0			0
$191$	−3.14581			−0.227623			−0.113811	−	0.993502i	$-0.536306\pi$
$191$	−3.14581			−0.227623			−0.113811	+	0.993502i	$0.536306\pi$
$192$		0			0
$193$			17.8210i			1.28278i	0.767215	+	0.641390i	$0.221642\pi$
$193$			17.8210i			1.28278i	−0.767215	+	0.641390i	$0.778358\pi$
$194$		0			0
$195$	−19.9066			−1.42554
$196$		0			0
$197$		−	4.56869i		−	0.325506i	−0.986667	−	0.162753i	$-0.947963\pi$
$197$		−	4.56869i		−	0.325506i	0.986667	−	0.162753i	$-0.0520373\pi$
$198$		0			0
$199$		−	7.98657i		−	0.566153i	−0.959097	−	0.283077i	$-0.908645\pi$
$199$		−	7.98657i		−	0.566153i	0.959097	−	0.283077i	$-0.0913550\pi$
$200$		0			0
$201$		−	0.0190824i		−	0.00134597i
$202$		0			0
$203$	−10.5686			−0.741771
$204$		0			0
$205$	−17.4005			−1.21530
$206$		0			0
$207$			1.90481i			0.132393i
$208$		0			0
$209$		−	14.7610i		−	1.02104i
$210$		0			0
$211$		−	6.01864i		−	0.414340i	−0.978305	−	0.207170i	$-0.933575\pi$
$211$		−	6.01864i		−	0.414340i	0.978305	−	0.207170i	$-0.0664253\pi$
$212$		0			0
$213$	6.51246			0.446226
$214$		0			0
$215$		−	14.5236i		−	0.990504i
$216$		0			0
$217$	13.2928			0.902372
$218$		0			0
$219$	−17.3582			−1.17296
$220$		0			0
$221$	12.0489	+	6.67209i	0.810495	+	0.448814i
$222$		0			0
$223$	−1.87703			−0.125695			−0.0628477	−	0.998023i	$-0.520018\pi$
$223$	−1.87703			−0.125695			−0.0628477	+	0.998023i	$0.520018\pi$
$224$		0			0
$225$	4.26650			0.284434
$226$		0			0
$227$		−	0.466679i		−	0.0309745i	−0.999880	−	0.0154873i	$-0.995070\pi$
$227$		−	0.466679i		−	0.0309745i	0.999880	−	0.0154873i	$-0.00492995\pi$
$228$		0			0
$229$	−0.396612			−0.0262088			−0.0131044	−	0.999914i	$-0.504171\pi$
$229$	−0.396612			−0.0262088			−0.0131044	+	0.999914i	$0.504171\pi$
$230$		0			0
$231$			33.3783i			2.19613i
$232$		0			0
$233$			1.99160i			0.130474i	0.997870	+	0.0652372i	$0.0207804\pi$
$233$			1.99160i			0.130474i	−0.997870	+	0.0652372i	$0.979220\pi$
$234$		0			0
$235$			7.55303i			0.492706i
$236$		0			0
$237$	11.0823			0.719873
$238$		0			0
$239$	3.10469			0.200826			0.100413	−	0.994946i	$-0.467984\pi$
$239$	3.10469			0.200826			0.100413	+	0.994946i	$0.467984\pi$
$240$		0			0
$241$		−	15.7425i		−	1.01406i	−0.861928	−	0.507031i	$-0.830743\pi$
$241$		−	15.7425i		−	1.01406i	0.861928	−	0.507031i	$-0.169257\pi$
$242$		0			0
$243$			17.2916i			1.10926i
$244$		0			0
$245$		−	3.56134i		−	0.227525i
$246$		0			0
$247$	−9.44127			−0.600733
$248$		0			0
$249$		−	14.5593i		−	0.922661i
$250$		0			0
$251$	18.8882			1.19221			0.596107	−	0.802905i	$-0.296713\pi$
$251$	18.8882			1.19221			0.596107	+	0.802905i	$0.296713\pi$
$252$		0			0
$253$	−5.21603			−0.327929
$254$		0			0
$255$	−21.4953	−	11.9031i	−1.34609	−	0.745400i
$256$		0			0
$257$	0.163008			0.0101682			0.00508408	−	0.999987i	$-0.498382\pi$
$257$	0.163008			0.0101682			0.00508408	+	0.999987i	$0.498382\pi$
$258$		0			0
$259$	−8.78977			−0.546170
$260$		0			0
$261$			6.98641i			0.432448i
$262$		0			0
$263$	6.32728			0.390157			0.195078	−	0.980788i	$-0.437504\pi$
$263$	6.32728			0.390157			0.195078	+	0.980788i	$0.437504\pi$
$264$		0			0
$265$		−	18.0296i		−	1.10755i
$266$		0			0
$267$		−	23.6644i		−	1.44824i
$268$		0			0
$269$			0.163676i			0.00997947i	0.999988	+	0.00498974i	$0.00158829\pi$
$269$			0.163676i			0.00997947i	−0.999988	+	0.00498974i	$0.998412\pi$
$270$		0			0
$271$	2.86326			0.173931			0.0869654	−	0.996211i	$-0.472283\pi$
$271$	2.86326			0.173931			0.0869654	+	0.996211i	$0.472283\pi$
$272$		0			0
$273$	21.3491			1.29210
$274$		0			0
$275$			11.6832i			0.704521i
$276$		0			0
$277$			14.2188i			0.854327i	0.904174	+	0.427164i	$0.140487\pi$
$277$			14.2188i			0.854327i	−0.904174	+	0.427164i	$0.859513\pi$
$278$		0			0
$279$		−	8.78722i		−	0.526077i
$280$		0			0
$281$	2.27875			0.135939			0.0679695	−	0.997687i	$-0.478348\pi$
$281$	2.27875			0.135939			0.0679695	+	0.997687i	$0.478348\pi$
$282$		0			0
$283$			0.470581i			0.0279732i	0.999902	+	0.0139866i	$0.00445221\pi$
$283$			0.470581i			0.0279732i	−0.999902	+	0.0139866i	$0.995548\pi$
$284$		0			0
$285$	16.8433			0.997712
$286$		0			0
$287$	18.6613			1.10154
$288$		0			0
$289$	9.02091	+	14.4091i	0.530641	+	0.847596i
$290$		0			0
$291$	−8.48164			−0.497202
$292$		0			0
$293$	29.4349			1.71961			0.859804	−	0.510624i	$-0.170586\pi$
$293$	29.4349			1.71961			0.859804	+	0.510624i	$0.170586\pi$
$294$		0			0
$295$		−	2.69018i		−	0.156628i
$296$		0			0
$297$	−12.6427			−0.733606
$298$		0			0
$299$			3.33621i			0.192938i
$300$		0			0
$301$			15.5760i			0.897787i
$302$		0			0
$303$			30.6344i			1.75990i
$304$		0			0
$305$	−25.1184			−1.43828
$306$		0			0
$307$	−15.4379			−0.881090			−0.440545	−	0.897730i	$-0.645215\pi$
$307$	−15.4379			−0.881090			−0.440545	+	0.897730i	$0.645215\pi$
$308$		0			0
$309$			32.3756i			1.84178i
$310$		0			0
$311$		−	0.273856i		−	0.0155289i	−0.999970	−	0.00776446i	$-0.997528\pi$
$311$		−	0.273856i		−	0.0155289i	0.999970	−	0.00776446i	$-0.00247153\pi$
$312$		0			0
$313$			16.0372i			0.906475i	0.891390	+	0.453238i	$0.149731\pi$
$313$			16.0372i			0.906475i	−0.891390	+	0.453238i	$0.850269\pi$
$314$		0			0
$315$	−14.8027			−0.834036
$316$		0			0
$317$			27.9683i			1.57085i	0.618954	+	0.785427i	$0.287557\pi$
$317$			27.9683i			1.57085i	−0.618954	+	0.785427i	$0.712443\pi$
$318$		0			0
$319$	−19.1312			−1.07114
$320$		0			0
$321$	−17.5285			−0.978346
$322$		0			0
$323$	−10.1947	−	5.64536i	−0.567251	−	0.314116i
$324$		0			0
$325$	7.47265			0.414508
$326$		0			0
$327$	24.0134			1.32794
$328$		0			0
$329$		−	8.10033i		−	0.446585i
$330$		0			0
$331$	11.9835			0.658672			0.329336	−	0.944213i	$-0.393175\pi$
$331$	11.9835			0.658672			0.329336	+	0.944213i	$0.393175\pi$
$332$		0			0
$333$			5.81050i			0.318413i
$334$		0			0
$335$			0.0231738i			0.00126612i
$336$		0			0
$337$		−	6.40932i		−	0.349138i	−0.984645	−	0.174569i	$-0.944147\pi$
$337$		−	6.40932i		−	0.349138i	0.984645	−	0.174569i	$-0.0558532\pi$
$338$		0			0
$339$	−11.5640			−0.628071
$340$		0			0
$341$	24.0624			1.30305
$342$		0			0
$343$		−	16.3763i		−	0.884239i
$344$		0			0
$345$		−	5.95184i		−	0.320436i
$346$		0			0
$347$			29.2837i			1.57203i	0.618206	+	0.786016i	$0.287860\pi$
$347$			29.2837i			1.57203i	−0.618206	+	0.786016i	$0.712140\pi$
$348$		0			0
$349$	−21.0353			−1.12599			−0.562996	−	0.826460i	$-0.690351\pi$
$349$	−21.0353			−1.12599			−0.562996	+	0.826460i	$0.690351\pi$
$350$		0			0
$351$			8.08640i			0.431620i
$352$		0			0
$353$	−16.5395			−0.880308			−0.440154	−	0.897922i	$-0.645076\pi$
$353$	−16.5395			−0.880308			−0.440154	+	0.897922i	$0.645076\pi$
$354$		0			0
$355$	−7.90876			−0.419753
$356$		0			0
$357$	23.0529	+	12.7656i	1.22009	+	0.675626i
$358$		0			0
$359$	20.6116			1.08784			0.543918	−	0.839138i	$-0.316940\pi$
$359$	20.6116			1.08784			0.543918	+	0.839138i	$0.316940\pi$
$360$		0			0
$361$	−11.0116			−0.579558
$362$		0			0
$363$			36.0537i			1.89233i
$364$		0			0
$365$	21.0799			1.10337
$366$		0			0
$367$		−	8.13955i		−	0.424881i	−0.977174	−	0.212440i	$-0.931859\pi$
$367$		−	8.13955i		−	0.424881i	0.977174	−	0.212440i	$-0.0681412\pi$
$368$		0			0
$369$		−	12.3361i		−	0.642192i
$370$		0			0
$371$			19.3360i			1.00388i
$372$		0			0
$373$	−29.5907			−1.53215			−0.766073	−	0.642753i	$-0.777792\pi$
$373$	−29.5907			−1.53215			−0.766073	+	0.642753i	$0.777792\pi$
$374$		0			0
$375$	16.4654			0.850269
$376$		0			0
$377$			12.2365i			0.630211i
$378$		0			0
$379$			0.769837i			0.0395439i	0.999805	+	0.0197719i	$0.00629401\pi$
$379$			0.769837i			0.0395439i	−0.999805	+	0.0197719i	$0.993706\pi$
$380$		0			0
$381$			47.9685i			2.45750i
$382$		0			0
$383$	23.1646			1.18365			0.591827	−	0.806065i	$-0.298407\pi$
$383$	23.1646			1.18365			0.591827	+	0.806065i	$0.298407\pi$
$384$		0			0
$385$		−	40.5348i		−	2.06585i
$386$		0			0
$387$	10.2966			0.523404
$388$		0			0
$389$	−10.2814			−0.521286			−0.260643	−	0.965435i	$-0.583935\pi$
$389$	−10.2814			−0.521286			−0.260643	+	0.965435i	$0.583935\pi$
$390$		0			0
$391$	−1.99487	+	3.60247i	−0.100885	+	0.182185i
$392$		0			0
$393$	−0.671823			−0.0338890
$394$		0			0
$395$	−13.4584			−0.677166
$396$		0			0
$397$		−	12.0500i		−	0.604774i	−0.953185	−	0.302387i	$-0.902216\pi$
$397$		−	12.0500i		−	0.604774i	0.953185	−	0.302387i	$-0.0977835\pi$
$398$		0			0
$399$	−18.0638			−0.904320
$400$		0			0
$401$		−	0.140633i		−	0.00702288i	−0.999994	−	0.00351144i	$-0.998882\pi$
$401$		−	0.140633i		−	0.00702288i	0.999994	−	0.00351144i	$-0.00111773\pi$
$402$		0			0
$403$		−	15.3906i		−	0.766658i
$404$		0			0
$405$		−	29.8184i		−	1.48169i
$406$		0			0
$407$	−15.9112			−0.788687
$408$		0			0
$409$	4.48881			0.221957			0.110979	−	0.993823i	$-0.464601\pi$
$409$	4.48881			0.221957			0.110979	+	0.993823i	$0.464601\pi$
$410$		0			0
$411$			30.2288i			1.49108i
$412$		0			0
$413$			2.88510i			0.141967i
$414$		0			0
$415$			17.6809i			0.867923i
$416$		0			0
$417$	−7.82319			−0.383103
$418$		0			0
$419$		−	19.8978i		−	0.972070i	−0.873939	−	0.486035i	$-0.838443\pi$
$419$		−	19.8978i		−	0.972070i	0.873939	−	0.486035i	$-0.161557\pi$
$420$		0			0
$421$	19.3428			0.942712			0.471356	−	0.881943i	$-0.343765\pi$
$421$	19.3428			0.942712			0.471356	+	0.881943i	$0.343765\pi$
$422$		0			0
$423$	−5.35474			−0.260357
$424$		0			0
$425$	8.06901	+	4.46824i	0.391405	+	0.216741i
$426$		0			0
$427$	26.9385			1.30365
$428$		0			0
$429$	38.6459			1.86584
$430$		0			0
$431$			6.21944i			0.299580i	0.988718	+	0.149790i	$0.0478597\pi$
$431$			6.21944i			0.299580i	−0.988718	+	0.149790i	$0.952140\pi$
$432$		0			0
$433$	−16.0179			−0.769773			−0.384887	−	0.922964i	$-0.625759\pi$
$433$	−16.0179			−0.769773			−0.384887	+	0.922964i	$0.625759\pi$
$434$		0			0
$435$		−	21.8300i		−	1.04667i
$436$		0			0
$437$		−	2.82282i		−	0.135034i
$438$		0			0
$439$		−	23.2431i		−	1.10933i	−0.832073	−	0.554666i	$-0.812846\pi$
$439$		−	23.2431i		−	1.10933i	0.832073	−	0.554666i	$-0.187154\pi$
$440$		0			0
$441$	2.52482			0.120229
$442$		0			0
$443$	−8.36059			−0.397223			−0.198612	−	0.980078i	$-0.563643\pi$
$443$	−8.36059			−0.397223			−0.198612	+	0.980078i	$0.563643\pi$
$444$		0			0
$445$			28.7381i			1.36232i
$446$		0			0
$447$		−	11.7288i		−	0.554754i
$448$		0			0
$449$			10.8470i			0.511900i	0.966690	+	0.255950i	$0.0823883\pi$
$449$			10.8470i			0.511900i	−0.966690	+	0.255950i	$0.917612\pi$
$450$		0			0
$451$	33.7805			1.59066
$452$		0			0
$453$			8.60368i			0.404236i
$454$		0			0
$455$	−25.9264			−1.21545
$456$		0			0
$457$	−18.0272			−0.843275			−0.421638	−	0.906764i	$-0.638545\pi$
$457$	−18.0272			−0.843275			−0.421638	+	0.906764i	$0.638545\pi$
$458$		0			0
$459$	−4.83523	+	8.73175i	−0.225689	+	0.407563i
$460$		0			0
$461$	9.11099			0.424341			0.212171	−	0.977233i	$-0.431947\pi$
$461$	9.11099			0.424341			0.212171	+	0.977233i	$0.431947\pi$
$462$		0			0
$463$	−20.4302			−0.949471			−0.474735	−	0.880129i	$-0.657456\pi$
$463$	−20.4302			−0.949471			−0.474735	+	0.880129i	$0.657456\pi$
$464$		0			0
$465$			27.4569i			1.27328i
$466$		0			0
$467$	0.978580			0.0452833			0.0226416	−	0.999744i	$-0.492792\pi$
$467$	0.978580			0.0452833			0.0226416	+	0.999744i	$0.492792\pi$
$468$		0			0
$469$		−	0.0248529i		−	0.00114760i
$470$		0			0
$471$			23.3494i			1.07588i
$472$		0			0
$473$			28.1956i			1.29643i
$474$		0			0
$475$	−6.32272			−0.290106
$476$		0			0
$477$	12.7821			0.585253
$478$		0			0
$479$			0.917745i			0.0419329i	0.999780	+	0.0209664i	$0.00667431\pi$
$479$			0.917745i			0.0419329i	−0.999780	+	0.0209664i	$0.993326\pi$
$480$		0			0
$481$			10.1769i			0.464027i
$482$		0			0
$483$			6.38311i			0.290442i
$484$		0			0
$485$	10.3001			0.467705
$486$		0			0
$487$		−	30.6257i		−	1.38778i	−0.720080	−	0.693891i	$-0.755895\pi$
$487$		−	30.6257i		−	1.38778i	0.720080	−	0.693891i	$-0.244105\pi$
$488$		0			0
$489$	−43.4619			−1.96541
$490$		0			0
$491$	4.44655			0.200670			0.100335	−	0.994954i	$-0.468009\pi$
$491$	4.44655			0.200670			0.100335	+	0.994954i	$0.468009\pi$
$492$		0			0
$493$	−7.31675	+	13.2130i	−0.329530	+	0.595085i
$494$		0			0
$495$	−26.7956			−1.20437
$496$		0			0
$497$	8.48183			0.380462
$498$		0			0
$499$		−	24.0215i		−	1.07535i	−0.843152	−	0.537675i	$-0.819303\pi$
$499$		−	24.0215i		−	1.07535i	0.843152	−	0.537675i	$-0.180697\pi$
$500$		0			0
$501$	48.8759			2.18361
$502$		0			0
$503$			36.1919i			1.61372i	0.590745	+	0.806859i	$0.298834\pi$
$503$			36.1919i			1.61372i	−0.590745	+	0.806859i	$0.701166\pi$
$504$		0			0
$505$		−	37.2026i		−	1.65549i
$506$		0			0
$507$			4.07963i			0.181183i
$508$		0			0
$509$	28.9495			1.28316			0.641582	−	0.767054i	$-0.278278\pi$
$509$	28.9495			1.28316			0.641582	+	0.767054i	$0.278278\pi$
$510$		0			0
$511$	−22.6073			−1.00009
$512$		0			0
$513$		−	6.84203i		−	0.302083i
$514$		0			0
$515$		−	39.3171i		−	1.73252i
$516$		0			0
$517$		−	14.6631i		−	0.644884i
$518$		0			0
$519$	−43.5776			−1.91284
$520$		0			0
$521$		−	6.41117i		−	0.280878i	−0.990089	−	0.140439i	$-0.955149\pi$
$521$		−	6.41117i		−	0.280878i	0.990089	−	0.140439i	$-0.0448515\pi$
$522$		0			0
$523$	43.7666			1.91378			0.956889	−	0.290452i	$-0.0938058\pi$
$523$	43.7666			1.91378			0.956889	+	0.290452i	$0.0938058\pi$
$524$		0			0
$525$	14.2973			0.623984
$526$		0			0
$527$	9.20271	−	16.6188i	0.400876	−	0.723927i
$528$		0			0
$529$	22.0025			0.956631
$530$		0			0
$531$	1.90721			0.0827657
$532$		0			0
$533$		−	21.6063i		−	0.935873i
$534$		0			0
$535$	21.2867			0.920305
$536$		0			0
$537$			5.21518i			0.225051i
$538$		0			0
$539$			6.91382i			0.297800i
$540$		0			0
$541$		−	9.20225i		−	0.395636i	−0.980239	−	0.197818i	$-0.936615\pi$
$541$		−	9.20225i		−	0.395636i	0.980239	−	0.197818i	$-0.0633855\pi$
$542$		0			0
$543$	−29.5660			−1.26880
$544$		0			0
$545$	−29.1620			−1.24916
$546$		0			0
$547$		−	42.9729i		−	1.83739i	−0.394970	−	0.918694i	$-0.629245\pi$
$547$		−	42.9729i		−	1.83739i	0.394970	−	0.918694i	$-0.370755\pi$
$548$		0			0
$549$		−	17.8078i		−	0.760018i
$550$		0			0
$551$		−	10.3535i		−	0.441073i
$552$		0			0
$553$	14.4336			0.613779
$554$		0			0
$555$		−	18.1557i		−	0.770667i
$556$		0			0
$557$	−16.4046			−0.695083			−0.347542	−	0.937665i	$-0.612983\pi$
$557$	−16.4046			−0.695083			−0.347542	+	0.937665i	$0.612983\pi$
$558$		0			0
$559$	18.0341			0.762763
$560$		0			0
$561$	41.7301	+	23.1081i	1.76185	+	0.975626i
$562$		0			0
$563$	22.6447			0.954362			0.477181	−	0.878805i	$-0.341659\pi$
$563$	22.6447			0.954362			0.477181	+	0.878805i	$0.341659\pi$
$564$		0			0
$565$	14.0434			0.590810
$566$		0			0
$567$			31.9790i			1.34299i
$568$		0			0
$569$	−2.12845			−0.0892292			−0.0446146	−	0.999004i	$-0.514206\pi$
$569$	−2.12845			−0.0892292			−0.0446146	+	0.999004i	$0.514206\pi$
$570$		0			0
$571$			2.83309i			0.118561i	0.998241	+	0.0592806i	$0.0188807\pi$
$571$			2.83309i			0.118561i	−0.998241	+	0.0592806i	$0.981119\pi$
$572$		0			0
$573$			6.96867i			0.291120i
$574$		0			0
$575$			2.23423i			0.0931739i
$576$		0			0
$577$	18.3846			0.765360			0.382680	−	0.923881i	$-0.375001\pi$
$577$	18.3846			0.765360			0.382680	+	0.923881i	$0.375001\pi$
$578$		0			0
$579$	39.4774			1.64062
$580$		0			0
$581$		−	18.9621i		−	0.786680i
$582$		0			0
$583$			35.0019i			1.44963i
$584$		0			0
$585$			17.1387i			0.708599i
$586$		0			0
$587$	36.1761			1.49315			0.746574	−	0.665302i	$-0.231697\pi$
$587$	36.1761			1.49315			0.746574	+	0.665302i	$0.231697\pi$
$588$		0			0
$589$			13.0222i			0.536570i
$590$		0			0
$591$	−10.1207			−0.416308
$592$		0			0
$593$	17.5358			0.720107			0.360054	−	0.932932i	$-0.382758\pi$
$593$	17.5358			0.720107			0.360054	+	0.932932i	$0.382758\pi$
$594$		0			0
$595$	−27.9955	−	15.5026i	−1.14770	−	0.635544i
$596$		0			0
$597$	−17.6920			−0.724086
$598$		0			0
$599$	−13.6140			−0.556254			−0.278127	−	0.960544i	$-0.589714\pi$
$599$	−13.6140			−0.556254			−0.278127	+	0.960544i	$0.589714\pi$
$600$		0			0
$601$		−	29.7568i		−	1.21381i	−0.794776	−	0.606903i	$-0.792412\pi$
$601$		−	29.7568i		−	1.21381i	0.794776	−	0.606903i	$-0.207588\pi$
$602$		0			0
$603$	−0.0164291			−0.000669044
$604$		0			0
$605$		−	43.7838i		−	1.78006i
$606$		0			0
$607$		−	44.1956i		−	1.79385i	−0.442187	−	0.896923i	$-0.645797\pi$
$607$		−	44.1956i		−	1.79385i	0.442187	−	0.896923i	$-0.354203\pi$
$608$		0			0
$609$			23.4118i			0.948694i
$610$		0			0
$611$	−9.37867			−0.379420
$612$		0			0
$613$	−18.3523			−0.741243			−0.370622	−	0.928784i	$-0.620855\pi$
$613$	−18.3523			−0.741243			−0.370622	+	0.928784i	$0.620855\pi$
$614$		0			0
$615$			38.5459i			1.55432i
$616$		0			0
$617$		−	21.6122i		−	0.870074i	−0.900413	−	0.435037i	$-0.856735\pi$
$617$		−	21.6122i		−	0.870074i	0.900413	−	0.435037i	$-0.143265\pi$
$618$		0			0
$619$			0.907391i			0.0364711i	0.999834	+	0.0182356i	$0.00580488\pi$
$619$			0.907391i			0.0364711i	−0.999834	+	0.0182356i	$0.994195\pi$
$620$		0			0
$621$	−2.41774			−0.0970204
$622$		0			0
$623$		−	30.8205i		−	1.23480i
$624$		0			0
$625$	−31.1809			−1.24723
$626$		0			0
$627$	−32.6989			−1.30587
$628$		0			0
$629$	−6.08524	+	10.9891i	−0.242634	+	0.438164i
$630$		0			0
$631$	5.47031			0.217769			0.108885	−	0.994054i	$-0.465272\pi$
$631$	5.47031			0.217769			0.108885	+	0.994054i	$0.465272\pi$
$632$		0			0
$633$	−13.3326			−0.529924
$634$		0			0
$635$		−	58.2531i		−	2.31170i
$636$		0			0
$637$	4.42214			0.175212
$638$		0			0
$639$		−	5.60694i		−	0.221807i
$640$		0			0
$641$		−	30.4843i		−	1.20406i	−0.798474	−	0.602030i	$-0.794359\pi$
$641$		−	30.4843i		−	1.20406i	0.798474	−	0.602030i	$-0.205641\pi$
$642$		0			0
$643$			2.79033i			0.110040i	0.998485	+	0.0550200i	$0.0175223\pi$
$643$			2.79033i			0.110040i	−0.998485	+	0.0550200i	$0.982478\pi$
$644$		0			0
$645$	−32.1731			−1.26681
$646$		0			0
$647$	33.9424			1.33441			0.667206	−	0.744873i	$-0.267490\pi$
$647$	33.9424			1.33441			0.667206	+	0.744873i	$0.267490\pi$
$648$		0			0
$649$			5.22259i			0.205005i
$650$		0			0
$651$		−	29.4464i		−	1.15410i
$652$		0			0
$653$			16.8474i			0.659289i	0.944105	+	0.329644i	$0.106929\pi$
$653$			16.8474i			0.659289i	−0.944105	+	0.329644i	$0.893071\pi$
$654$		0			0
$655$	0.815865			0.0318785
$656$		0			0
$657$			14.9446i			0.583045i
$658$		0			0
$659$	−21.1179			−0.822638			−0.411319	−	0.911491i	$-0.634932\pi$
$659$	−21.1179			−0.822638			−0.411319	+	0.911491i	$0.634932\pi$
$660$		0			0
$661$	15.2160			0.591835			0.295917	−	0.955214i	$-0.404375\pi$
$661$	15.2160			0.591835			0.295917	+	0.955214i	$0.404375\pi$
$662$		0			0
$663$	14.7802	−	26.6909i	0.574014	−	1.03659i
$664$		0			0
$665$	21.9367			0.850670
$666$		0			0
$667$	−3.65856			−0.141660
$668$		0			0
$669$			4.15804i			0.160759i
$670$		0			0
$671$	48.7639			1.88251
$672$		0			0
$673$		−	37.0247i		−	1.42720i	−0.700556	−	0.713598i	$-0.747064\pi$
$673$		−	37.0247i		−	1.42720i	0.700556	−	0.713598i	$-0.252936\pi$
$674$		0			0
$675$			5.41539i			0.208438i
$676$		0			0
$677$			17.6609i			0.678762i	0.940649	+	0.339381i	$0.110218\pi$
$677$			17.6609i			0.678762i	−0.940649	+	0.339381i	$0.889782\pi$
$678$		0			0
$679$	−11.0465			−0.423925
$680$		0			0
$681$	−1.03380			−0.0396152
$682$		0			0
$683$		−	40.1233i		−	1.53528i	−0.640884	−	0.767638i	$-0.721432\pi$
$683$		−	40.1233i		−	1.53528i	0.640884	−	0.767638i	$-0.278568\pi$
$684$		0			0
$685$		−	36.7100i		−	1.40262i
$686$		0			0
$687$			0.878583i			0.0335200i
$688$		0			0
$689$	22.3875			0.852896
$690$		0			0
$691$		−	46.8916i		−	1.78384i	−0.452191	−	0.891921i	$-0.649358\pi$
$691$		−	46.8916i		−	1.78384i	0.452191	−	0.891921i	$-0.350642\pi$
$692$		0			0
$693$	28.7373			1.09164
$694$		0			0
$695$	9.50052			0.360375
$696$		0			0
$697$	12.9194	−	23.3306i	0.489357	−	0.883711i
$698$		0			0
$699$	4.41184			0.166871
$700$		0			0
$701$	−19.0688			−0.720218			−0.360109	−	0.932910i	$-0.617261\pi$
$701$	−19.0688			−0.720218			−0.360109	+	0.932910i	$0.617261\pi$
$702$		0			0
$703$		−	8.61085i		−	0.324764i
$704$		0			0
$705$	16.7316			0.630150
$706$		0			0
$707$			39.8983i			1.50053i
$708$		0			0
$709$		−	5.61950i		−	0.211045i	−0.994417	−	0.105522i	$-0.966349\pi$
$709$		−	5.61950i		−	0.211045i	0.994417	−	0.105522i	$-0.0336514\pi$
$710$		0			0
$711$		−	9.54137i		−	0.357829i
$712$		0			0
$713$	4.60159			0.172331
$714$		0			0
$715$	−46.9318			−1.75515
$716$		0			0
$717$		−	6.87758i		−	0.256848i
$718$		0			0
$719$			19.9178i			0.742809i	0.928471	+	0.371405i	$0.121124\pi$
$719$			19.9178i			0.742809i	−0.928471	+	0.371405i	$0.878876\pi$
$720$		0			0
$721$			42.1660i			1.57034i
$722$		0			0
$723$	−34.8731			−1.29694
$724$		0			0
$725$			8.19465i			0.304342i
$726$		0			0
$727$	−41.2044			−1.52819			−0.764093	−	0.645107i	$-0.776813\pi$
$727$	−41.2044			−1.52819			−0.764093	+	0.645107i	$0.776813\pi$
$728$		0			0
$729$	5.05214			0.187116
$730$		0			0
$731$	19.4734	+	10.7834i	0.720249	+	0.398840i
$732$		0			0
$733$	−8.24801			−0.304647			−0.152324	−	0.988331i	$-0.548676\pi$
$733$	−8.24801			−0.304647			−0.152324	+	0.988331i	$0.548676\pi$
$734$		0			0
$735$	−7.88915			−0.290996
$736$		0			0
$737$		−	0.0449885i		−	0.00165717i
$738$		0			0
$739$	1.66998			0.0614312			0.0307156	−	0.999528i	$-0.490221\pi$
$739$	1.66998			0.0614312			0.0307156	+	0.999528i	$0.490221\pi$
$740$		0			0
$741$			20.9145i			0.768313i
$742$		0			0
$743$		−	34.7175i		−	1.27366i	−0.771004	−	0.636830i	$-0.780245\pi$
$743$		−	34.7175i		−	1.27366i	0.771004	−	0.636830i	$-0.219755\pi$
$744$		0			0
$745$			14.2435i			0.521843i
$746$		0			0
$747$	−12.5349			−0.458629
$748$		0			0
$749$	−22.8291			−0.834159
$750$		0			0
$751$		−	10.9905i		−	0.401049i	−0.979689	−	0.200524i	$-0.935735\pi$
$751$		−	10.9905i		−	0.401049i	0.979689	−	0.200524i	$-0.0642646\pi$
$752$		0			0
$753$		−	41.8416i		−	1.52479i
$754$		0			0
$755$		−	10.4483i		−	0.380254i
$756$		0			0
$757$	38.6826			1.40594			0.702971	−	0.711218i	$-0.251856\pi$
$757$	38.6826			1.40594			0.702971	+	0.711218i	$0.251856\pi$
$758$		0			0
$759$			11.5547i			0.419407i
$760$		0			0
$761$	1.39363			0.0505189			0.0252595	−	0.999681i	$-0.491959\pi$
$761$	1.39363			0.0505189			0.0252595	+	0.999681i	$0.491959\pi$
$762$		0			0
$763$	31.2750			1.13223
$764$		0			0
$765$	−10.2480	+	18.5065i	−0.370518	+	0.669104i
$766$		0			0
$767$	3.34041			0.120615
$768$		0			0
$769$	−51.2276			−1.84731			−0.923657	−	0.383220i	$-0.874815\pi$
$769$	−51.2276			−1.84731			−0.923657	+	0.383220i	$0.874815\pi$
$770$		0			0
$771$		−	0.361099i		−	0.0130047i
$772$		0			0
$773$	31.7532			1.14208			0.571042	−	0.820921i	$-0.306539\pi$
$773$	31.7532			1.14208			0.571042	+	0.820921i	$0.306539\pi$
$774$		0			0
$775$		−	10.3069i		−	0.370235i
$776$		0			0
$777$			19.4713i			0.698528i
$778$		0			0
$779$			18.2814i			0.655001i
$780$		0			0
$781$	15.3537			0.549400
$782$		0			0
$783$	−8.86771			−0.316906
$784$		0			0
$785$		−	28.3556i		−	1.01205i
$786$		0			0
$787$			35.9002i			1.27970i	0.768499	+	0.639851i	$0.221004\pi$
$787$			35.9002i			1.27970i	−0.768499	+	0.639851i	$0.778996\pi$
$788$		0			0
$789$		−	14.0163i		−	0.498994i
$790$		0			0
$791$	−15.0610			−0.535507
$792$		0			0
$793$		−	31.1898i		−	1.10758i
$794$		0			0
$795$	−39.9395			−1.41651
$796$		0			0
$797$	−12.5143			−0.443279			−0.221639	−	0.975129i	$-0.571141\pi$
$797$	−12.5143			−0.443279			−0.221639	+	0.975129i	$0.571141\pi$
$798$		0			0
$799$	−10.1271	−	5.60793i	−0.358273	−	0.198394i
$800$		0			0
$801$	−20.3740			−0.719879
$802$		0			0
$803$	−40.9235			−1.44416
$804$		0			0
$805$		−	7.75168i		−	0.273211i
$806$		0			0
$807$	0.362578			0.0127633
$808$		0			0
$809$			4.49685i			0.158101i	0.996871	+	0.0790503i	$0.0251888\pi$
$809$			4.49685i			0.158101i	−0.996871	+	0.0790503i	$0.974811\pi$
$810$		0			0
$811$			17.6215i			0.618775i	0.950936	+	0.309387i	$0.100124\pi$
$811$			17.6215i			0.618775i	−0.950936	+	0.309387i	$0.899876\pi$
$812$		0			0
$813$		−	6.34276i		−	0.222450i
$814$		0			0
$815$	52.7803			1.84881
$816$		0			0
$817$	−15.2590			−0.533844
$818$		0			0
$819$		−	18.3806i		−	0.642270i
$820$		0			0
$821$		−	45.4965i		−	1.58784i	−0.608022	−	0.793920i	$-0.708037\pi$
$821$		−	45.4965i		−	1.58784i	0.608022	−	0.793920i	$-0.291963\pi$
$822$		0			0
$823$			34.8121i			1.21347i	0.794903	+	0.606736i	$0.207522\pi$
$823$			34.8121i			1.21347i	−0.794903	+	0.606736i	$0.792478\pi$
$824$		0			0
$825$	25.8808			0.901053
$826$		0			0
$827$		−	27.3617i		−	0.951460i	−0.879591	−	0.475730i	$-0.842184\pi$
$827$		−	27.3617i		−	0.951460i	0.879591	−	0.475730i	$-0.157816\pi$
$828$		0			0
$829$	−10.4543			−0.363091			−0.181546	−	0.983383i	$-0.558110\pi$
$829$	−10.4543			−0.363091			−0.181546	+	0.983383i	$0.558110\pi$
$830$		0			0
$831$	31.4979			1.09265
$832$		0			0
$833$	4.77506	+	2.64420i	0.165446	+	0.0916161i
$834$		0			0
$835$	−59.3551			−2.05407
$836$		0			0
$837$	11.1534			0.385520
$838$		0			0
$839$		−	42.5304i		−	1.46831i	−0.678980	−	0.734157i	$-0.737578\pi$
$839$		−	42.5304i		−	1.46831i	0.678980	−	0.734157i	$-0.262422\pi$
$840$		0			0
$841$	15.5812			0.537284
$842$		0			0
$843$		−	5.04794i		−	0.173860i
$844$		0			0
$845$		−	4.95432i		−	0.170434i
$846$		0			0
$847$			46.9564i			1.61344i
$848$		0			0
$849$	1.04244			0.0357765
$850$		0			0
$851$	−3.04277			−0.104305
$852$		0			0
$853$		−	14.4058i		−	0.493246i	−0.969111	−	0.246623i	$-0.920679\pi$
$853$		−	14.4058i		−	0.493246i	0.969111	−	0.246623i	$-0.0793210\pi$
$854$		0			0
$855$		−	14.5013i		−	0.495936i
$856$		0			0
$857$		−	38.5986i		−	1.31850i	−0.751923	−	0.659251i	$-0.770873\pi$
$857$		−	38.5986i		−	1.31850i	0.751923	−	0.659251i	$-0.229127\pi$
$858$		0			0
$859$	−24.1478			−0.823913			−0.411957	−	0.911203i	$-0.635154\pi$
$859$	−24.1478			−0.823913			−0.411957	+	0.911203i	$0.635154\pi$
$860$		0			0
$861$		−	41.3389i		−	1.40883i
$862$		0			0
$863$	−11.3792			−0.387354			−0.193677	−	0.981065i	$-0.562041\pi$
$863$	−11.3792			−0.387354			−0.193677	+	0.981065i	$0.562041\pi$
$864$		0			0
$865$	52.9208			1.79936
$866$		0			0
$867$	31.9194	−	19.9833i	1.08404	−	0.678669i
$868$		0			0
$869$	26.1276			0.886317
$870$		0			0
$871$	−0.0287751			−0.000975005
$872$		0			0
$873$			7.30231i			0.247146i
$874$		0			0
$875$	21.4445			0.724957
$876$		0			0
$877$			45.7414i			1.54458i	0.635273	+	0.772288i	$0.280888\pi$
$877$			45.7414i			1.54458i	−0.635273	+	0.772288i	$0.719112\pi$
$878$		0			0
$879$		−	65.2049i		−	2.19931i
$880$		0			0
$881$		−	34.9173i		−	1.17639i	−0.808718	−	0.588196i	$-0.799838\pi$
$881$		−	34.9173i		−	1.17639i	0.808718	−	0.588196i	$-0.200162\pi$
$882$		0			0
$883$	−13.6779			−0.460298			−0.230149	−	0.973155i	$-0.573921\pi$
$883$	−13.6779			−0.460298			−0.230149	+	0.973155i	$0.573921\pi$
$884$		0			0
$885$	−5.95933			−0.200321
$886$		0			0
$887$		−	9.73184i		−	0.326763i	−0.986563	−	0.163382i	$-0.947760\pi$
$887$		−	9.73184i		−	0.326763i	0.986563	−	0.163382i	$-0.0522402\pi$
$888$		0			0
$889$			62.4741i			2.09531i
$890$		0			0
$891$			57.8881i			1.93933i
$892$		0			0
$893$	7.93544			0.265549
$894$		0			0
$895$		−	6.33333i		−	0.211700i
$896$		0			0
$897$	7.39046			0.246760
$898$		0			0
$899$	16.8776			0.562899
$900$		0			0
$901$	24.1742	+	13.3865i	0.805358	+	0.445969i
$902$		0			0
$903$	34.5044			1.14823
$904$		0			0
$905$	35.9051			1.19353
$906$		0			0
$907$		−	13.4670i		−	0.447164i	−0.974685	−	0.223582i	$-0.928225\pi$
$907$		−	13.4670i		−	0.447164i	0.974685	−	0.223582i	$-0.0717751\pi$
$908$		0			0
$909$	26.3749			0.874799
$910$		0			0
$911$		−	42.1610i		−	1.39686i	−0.715681	−	0.698428i	$-0.753883\pi$
$911$		−	42.1610i		−	1.39686i	0.715681	−	0.698428i	$-0.246117\pi$
$912$		0			0
$913$		−	34.3250i		−	1.13599i
$914$		0			0
$915$			55.6429i			1.83950i
$916$		0			0
$917$	−0.874982			−0.0288945
$918$		0			0
$919$	−29.5626			−0.975179			−0.487589	−	0.873073i	$-0.662124\pi$
$919$	−29.5626			−0.975179			−0.487589	+	0.873073i	$0.662124\pi$
$920$		0			0
$921$			34.1985i			1.12688i
$922$		0			0
$923$		−	9.82038i		−	0.323242i
$924$		0			0
$925$			6.81538i			0.224088i
$926$		0			0
$927$	27.8740			0.915501
$928$		0			0
$929$		−	51.9452i		−	1.70427i	−0.523324	−	0.852134i	$-0.675308\pi$
$929$		−	51.9452i		−	1.70427i	0.523324	−	0.852134i	$-0.324692\pi$
$930$		0			0
$931$	−3.74164			−0.122627
$932$		0			0
$933$	−0.606651			−0.0198609
$934$		0			0
$935$	−50.6772	−	28.0626i	−1.65732	−	0.917746i
$936$		0			0
$937$	−31.8778			−1.04140			−0.520701	−	0.853739i	$-0.674329\pi$
$937$	−31.8778			−1.04140			−0.520701	+	0.853739i	$0.674329\pi$
$938$		0			0
$939$	35.5259			1.15934
$940$		0			0
$941$		−	45.3828i		−	1.47944i	−0.672916	−	0.739719i	$-0.734959\pi$
$941$		−	45.3828i		−	1.47944i	0.672916	−	0.739719i	$-0.265041\pi$
$942$		0			0
$943$	6.46002			0.210367
$944$		0			0
$945$		−	18.7887i		−	0.611197i
$946$		0			0
$947$		−	2.85118i		−	0.0926508i	−0.998926	−	0.0463254i	$-0.985249\pi$
$947$		−	2.85118i		−	0.0926508i	0.998926	−	0.0463254i	$-0.0147511\pi$
$948$		0			0
$949$			26.1750i			0.849678i
$950$		0			0
$951$	61.9559			2.00906
$952$		0			0
$953$	47.1308			1.52672			0.763359	−	0.645975i	$-0.223549\pi$
$953$	47.1308			1.52672			0.763359	+	0.645975i	$0.223549\pi$
$954$		0			0
$955$		−	8.46278i		−	0.273849i
$956$		0			0
$957$			42.3798i			1.36995i
$958$		0			0
$959$			39.3699i			1.27132i
$960$		0			0
$961$	9.77206			0.315228
$962$		0			0
$963$			15.0913i			0.486309i
$964$		0			0
$965$	−47.9415			−1.54329
$966$		0			0
$967$	6.64785			0.213780			0.106890	−	0.994271i	$-0.465911\pi$
$967$	6.64785			0.213780			0.106890	+	0.994271i	$0.465911\pi$
$968$		0			0
$969$	−12.5057	+	22.5836i	−0.401742	+	0.725490i
$970$		0			0
$971$	−17.5953			−0.564661			−0.282330	−	0.959317i	$-0.591107\pi$
$971$	−17.5953			−0.564661			−0.282330	+	0.959317i	$0.591107\pi$
$972$		0			0
$973$	−10.1889			−0.326642
$974$		0			0
$975$		−	16.5536i		−	0.530139i
$976$		0			0
$977$	−12.8418			−0.410845			−0.205422	−	0.978673i	$-0.565857\pi$
$977$	−12.8418			−0.410845			−0.205422	+	0.978673i	$0.565857\pi$
$978$		0			0
$979$		−	55.7910i		−	1.78309i
$980$		0			0
$981$		−	20.6745i		−	0.660084i
$982$		0			0
$983$			36.8329i			1.17479i	0.809302	+	0.587393i	$0.199846\pi$
$983$			36.8329i			1.17479i	−0.809302	+	0.587393i	$0.800154\pi$
$984$		0			0
$985$	12.2906			0.391610
$986$		0			0
$987$	−17.9440			−0.571164
$988$		0			0
$989$			5.39199i			0.171455i
$990$		0			0
$991$			57.7375i			1.83409i	0.398780	+	0.917047i	$0.369434\pi$
$991$			57.7375i			1.83409i	−0.398780	+	0.917047i	$0.630566\pi$
$992$		0			0
$993$		−	26.5461i		−	0.842415i
$994$		0			0
$995$	21.4853			0.681129
$996$		0			0
$997$		−	17.4041i		−	0.551192i	−0.961274	−	0.275596i	$-0.911125\pi$
$997$		−	17.4041i		−	0.551192i	0.961274	−	0.275596i	$-0.0888752\pi$
$998$		0			0
$999$	−7.37515			−0.233339

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.a.237.8	✓	40
17.16	even	2	inner	4012.2.b.a.237.33	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.a.237.8	✓	40	1.1	even	1	trivial
4012.2.b.a.237.33	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)

\(f(q)\)	\(=\)	\(q-2.21522i q^{3} +2.69018i q^{5} -2.88510i q^{7} -1.90721 q^{9} +O(q^{10})\)	\(q-2.21522i q^{3} +2.69018i q^{5} -2.88510i q^{7} -1.90721 q^{9} -5.22259i q^{11} -3.34041 q^{13} +5.95933 q^{15} +(-3.60700 - 1.99739i) q^{17} +2.82638 q^{19} -6.39115 q^{21} -0.998743i q^{23} -2.23704 q^{25} -2.42078i q^{27} -3.66316i q^{29} +4.60738i q^{31} -11.5692 q^{33} +7.76144 q^{35} -3.04660i q^{37} +7.39976i q^{39} +6.46816i q^{41} -5.39877 q^{43} -5.13072i q^{45} +2.80764 q^{47} -1.32383 q^{49} +(-4.42465 + 7.99031i) q^{51} -6.70201 q^{53} +14.0497 q^{55} -6.26105i q^{57} -1.00000 q^{59} +9.33710i q^{61} +5.50249i q^{63} -8.98630i q^{65} +0.00861422 q^{67} -2.21244 q^{69} +2.93987i q^{71} -7.83587i q^{73} +4.95554i q^{75} -15.0677 q^{77} +5.00280i q^{79} -11.0842 q^{81} +6.57241 q^{83} +(5.37332 - 9.70346i) q^{85} -8.11472 q^{87} +10.6826 q^{89} +9.63744i q^{91} +10.2064 q^{93} +7.60345i q^{95} -3.82880i q^{97} +9.96056i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 36 q^{9}+O(q^{10}) \) 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

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