LMFDB - Embedded newform 4012.2.b.a.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:	$40$
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.a.237.32

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.18094i q^{3} -0.351694i q^{5} +3.15514i q^{7} -1.75651 q^{9} +O(q^{10})$	$q-2.18094i q^{3} -0.351694i q^{5} +3.15514i q^{7} -1.75651 q^{9} -3.05901i q^{11} -2.45298 q^{13} -0.767023 q^{15} +(0.112376 + 4.12157i) q^{17} +2.18282 q^{19} +6.88117 q^{21} -0.185893i q^{23} +4.87631 q^{25} -2.71199i q^{27} -4.16972i q^{29} +6.26324i q^{31} -6.67153 q^{33} +1.10964 q^{35} -11.6664i q^{37} +5.34980i q^{39} +6.35306i q^{41} -2.08817 q^{43} +0.617752i q^{45} +0.937671 q^{47} -2.95489 q^{49} +(8.98891 - 0.245087i) q^{51} +9.12001 q^{53} -1.07584 q^{55} -4.76059i q^{57} -1.00000 q^{59} +1.52355i q^{61} -5.54202i q^{63} +0.862697i q^{65} +8.22725 q^{67} -0.405422 q^{69} -11.2719i q^{71} +11.1539i q^{73} -10.6350i q^{75} +9.65161 q^{77} -14.0379i q^{79} -11.1842 q^{81} +3.09527 q^{83} +(1.44953 - 0.0395221i) q^{85} -9.09392 q^{87} +6.46956 q^{89} -7.73948i q^{91} +13.6598 q^{93} -0.767682i q^{95} -16.4606i q^{97} +5.37318i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	2.18094i		−	1.25917i	−0.776933	−	0.629584i	$-0.783225\pi$
$3$		−	2.18094i		−	1.25917i	0.776933	−	0.629584i	$-0.216775\pi$
$4$		0			0
$5$		−	0.351694i		−	0.157282i	−0.996903	−	0.0786411i	$-0.974942\pi$
$5$		−	0.351694i		−	0.157282i	0.996903	−	0.0786411i	$-0.0250581\pi$
$6$		0			0
$7$			3.15514i			1.19253i	0.802788	+	0.596265i	$0.203349\pi$
$7$			3.15514i			1.19253i	−0.802788	+	0.596265i	$0.796651\pi$
$8$		0			0
$9$	−1.75651			−0.585502
$10$		0			0
$11$		−	3.05901i		−	0.922328i	−0.887315	−	0.461164i	$-0.847432\pi$
$11$		−	3.05901i		−	0.922328i	0.887315	−	0.461164i	$-0.152568\pi$
$12$		0			0
$13$	−2.45298			−0.680334			−0.340167	−	0.940365i	$-0.610484\pi$
$13$	−2.45298			−0.680334			−0.340167	+	0.940365i	$0.610484\pi$
$14$		0			0
$15$	−0.767023			−0.198045
$16$		0			0
$17$	0.112376	+	4.12157i	0.0272553	+	0.999629i
$17$	0.112376	+	4.12157i	0.0272553	+	0.999629i
$18$		0			0
$19$	2.18282			0.500772			0.250386	−	0.968146i	$-0.419442\pi$
$19$	2.18282			0.500772			0.250386	+	0.968146i	$0.419442\pi$
$20$		0			0
$21$	6.88117			1.50159
$22$		0			0
$23$		−	0.185893i		−	0.0387614i	−0.999812	−	0.0193807i	$-0.993831\pi$
$23$		−	0.185893i		−	0.0387614i	0.999812	−	0.0193807i	$-0.00616945\pi$
$24$		0			0
$25$	4.87631			0.975262
$26$		0			0
$27$		−	2.71199i		−	0.521922i
$28$		0			0
$29$		−	4.16972i		−	0.774298i	−0.922017	−	0.387149i	$-0.873460\pi$
$29$		−	4.16972i		−	0.774298i	0.922017	−	0.387149i	$-0.126540\pi$
$30$		0			0
$31$			6.26324i			1.12491i	0.826827	+	0.562456i	$0.190143\pi$
$31$			6.26324i			1.12491i	−0.826827	+	0.562456i	$0.809857\pi$
$32$		0			0
$33$	−6.67153			−1.16136
$34$		0			0
$35$	1.10964			0.187564
$36$		0			0
$37$		−	11.6664i		−	1.91794i	−0.283513	−	0.958968i	$-0.591500\pi$
$37$		−	11.6664i		−	1.91794i	0.283513	−	0.958968i	$-0.408500\pi$
$38$		0			0
$39$			5.34980i			0.856654i
$40$		0			0
$41$			6.35306i			0.992181i	0.868271	+	0.496090i	$0.165232\pi$
$41$			6.35306i			0.992181i	−0.868271	+	0.496090i	$0.834768\pi$
$42$		0			0
$43$	−2.08817			−0.318443			−0.159221	−	0.987243i	$-0.550898\pi$
$43$	−2.08817			−0.318443			−0.159221	+	0.987243i	$0.550898\pi$
$44$		0			0
$45$			0.617752i			0.0920891i
$46$		0			0
$47$	0.937671			0.136773			0.0683867	−	0.997659i	$-0.478215\pi$
$47$	0.937671			0.136773			0.0683867	+	0.997659i	$0.478215\pi$
$48$		0			0
$49$	−2.95489			−0.422128
$50$		0			0
$51$	8.98891	−	0.245087i	1.25870	−	0.0343190i
$52$		0			0
$53$	9.12001			1.25273			0.626365	−	0.779530i	$-0.284542\pi$
$53$	9.12001			1.25273			0.626365	+	0.779530i	$0.284542\pi$
$54$		0			0
$55$	−1.07584			−0.145066
$56$		0			0
$57$		−	4.76059i		−	0.630556i
$58$		0			0
$59$	−1.00000			−0.130189
$59$	−1.00000			−0.130189
$60$		0			0
$61$			1.52355i			0.195070i	0.995232	+	0.0975352i	$0.0310959\pi$
$61$			1.52355i			0.195070i	−0.995232	+	0.0975352i	$0.968904\pi$
$62$		0			0
$63$		−	5.54202i		−	0.698229i
$64$		0			0
$65$			0.862697i			0.107004i
$66$		0			0
$67$	8.22725			1.00512			0.502559	−	0.864543i	$-0.332392\pi$
$67$	8.22725			1.00512			0.502559	+	0.864543i	$0.332392\pi$
$68$		0			0
$69$	−0.405422			−0.0488071
$70$		0			0
$71$		−	11.2719i		−	1.33773i	−0.743382	−	0.668867i	$-0.766780\pi$
$71$		−	11.2719i		−	1.33773i	0.743382	−	0.668867i	$-0.233220\pi$
$72$		0			0
$73$			11.1539i			1.30547i	0.757588	+	0.652733i	$0.226378\pi$
$73$			11.1539i			1.30547i	−0.757588	+	0.652733i	$0.773622\pi$
$74$		0			0
$75$		−	10.6350i		−	1.22802i
$76$		0			0
$77$	9.65161			1.09990
$78$		0			0
$79$		−	14.0379i		−	1.57939i	−0.613502	−	0.789693i	$-0.710240\pi$
$79$		−	14.0379i		−	1.57939i	0.613502	−	0.789693i	$-0.289760\pi$
$80$		0			0
$81$	−11.1842			−1.24269
$82$		0			0
$83$	3.09527			0.339750			0.169875	−	0.985466i	$-0.445664\pi$
$83$	3.09527			0.339750			0.169875	+	0.985466i	$0.445664\pi$
$84$		0			0
$85$	1.44953	−	0.0395221i	0.157224	−	0.00428677i
$86$		0			0
$87$	−9.09392			−0.974970
$88$		0			0
$89$	6.46956			0.685772			0.342886	−	0.939377i	$-0.388596\pi$
$89$	6.46956			0.685772			0.342886	+	0.939377i	$0.388596\pi$
$90$		0			0
$91$		−	7.73948i		−	0.811318i
$92$		0			0
$93$	13.6598			1.41645
$94$		0			0
$95$		−	0.767682i		−	0.0787625i
$96$		0			0
$97$		−	16.4606i		−	1.67132i	−0.549247	−	0.835660i	$-0.685085\pi$
$97$		−	16.4606i		−	1.67132i	0.549247	−	0.835660i	$-0.314915\pi$
$98$		0			0
$99$			5.37318i			0.540025i
$100$		0			0
$101$	14.7119			1.46389			0.731944	−	0.681364i	$-0.238613\pi$
$101$	14.7119			1.46389			0.731944	+	0.681364i	$0.238613\pi$
$102$		0			0
$103$	2.07455			0.204411			0.102206	−	0.994763i	$-0.467410\pi$
$103$	2.07455			0.204411			0.102206	+	0.994763i	$0.467410\pi$
$104$		0			0
$105$		−	2.42006i		−	0.236174i
$106$		0			0
$107$		−	2.19394i		−	0.212096i	−0.994361	−	0.106048i	$-0.966180\pi$
$107$		−	2.19394i		−	0.212096i	0.994361	−	0.106048i	$-0.0338198\pi$
$108$		0			0
$109$		−	19.6374i		−	1.88092i	−0.339908	−	0.940459i	$-0.610396\pi$
$109$		−	19.6374i		−	1.88092i	0.339908	−	0.940459i	$-0.389604\pi$
$110$		0			0
$111$	−25.4436			−2.41500
$112$		0			0
$113$		−	13.2005i		−	1.24180i	−0.783891	−	0.620899i	$-0.786768\pi$
$113$		−	13.2005i		−	1.24180i	0.783891	−	0.620899i	$-0.213232\pi$
$114$		0			0
$115$	−0.0653774			−0.00609648
$116$		0			0
$117$	4.30867			0.398337
$118$		0			0
$119$	−13.0041	+	0.354563i	−1.19209	+	0.0325028i
$120$		0			0
$121$	1.64243			0.149312
$122$		0			0
$123$	13.8556			1.24932
$124$		0			0
$125$		−	3.47344i		−	0.310674i
$126$		0			0
$127$	−4.50338			−0.399611			−0.199805	−	0.979836i	$-0.564031\pi$
$127$	−4.50338			−0.399611			−0.199805	+	0.979836i	$0.564031\pi$
$128$		0			0
$129$			4.55417i			0.400973i
$130$		0			0
$131$			5.08844i			0.444579i	0.974981	+	0.222290i	$0.0713531\pi$
$131$			5.08844i			0.444579i	−0.974981	+	0.222290i	$0.928647\pi$
$132$		0			0
$133$			6.88708i			0.597186i
$134$		0			0
$135$	−0.953788			−0.0820890
$136$		0			0
$137$	11.6396			0.994437			0.497218	−	0.867625i	$-0.334355\pi$
$137$	11.6396			0.994437			0.497218	+	0.867625i	$0.334355\pi$
$138$		0			0
$139$		−	9.21151i		−	0.781310i	−0.920537	−	0.390655i	$-0.872248\pi$
$139$		−	9.21151i		−	0.781310i	0.920537	−	0.390655i	$-0.127752\pi$
$140$		0			0
$141$		−	2.04501i		−	0.172221i
$142$		0			0
$143$			7.50369i			0.627490i
$144$		0			0
$145$	−1.46646			−0.121783
$146$		0			0
$147$			6.44445i			0.531529i
$148$		0			0
$149$	−1.31016			−0.107333			−0.0536663	−	0.998559i	$-0.517091\pi$
$149$	−1.31016			−0.107333			−0.0536663	+	0.998559i	$0.517091\pi$
$150$		0			0
$151$	1.62594			0.132317			0.0661587	−	0.997809i	$-0.478926\pi$
$151$	1.62594			0.132317			0.0661587	+	0.997809i	$0.478926\pi$
$152$		0			0
$153$	−0.197390	−	7.23957i	−0.0159580	−	0.585285i
$154$		0			0
$155$	2.20274			0.176929
$156$		0			0
$157$	−9.67197			−0.771907			−0.385954	−	0.922518i	$-0.626127\pi$
$157$	−9.67197			−0.771907			−0.385954	+	0.922518i	$0.626127\pi$
$158$		0			0
$159$		−	19.8902i		−	1.57740i
$160$		0			0
$161$	0.586518			0.0462241
$162$		0			0
$163$			19.2325i			1.50640i	0.657789	+	0.753202i	$0.271492\pi$
$163$			19.2325i			1.50640i	−0.657789	+	0.753202i	$0.728508\pi$
$164$		0			0
$165$			2.34634i			0.182662i
$166$		0			0
$167$			3.03663i			0.234982i	0.993074	+	0.117491i	$0.0374851\pi$
$167$			3.03663i			0.234982i	−0.993074	+	0.117491i	$0.962515\pi$
$168$		0			0
$169$	−6.98290			−0.537146
$170$		0			0
$171$	−3.83413			−0.293203
$172$		0			0
$173$			17.6581i			1.34252i	0.741223	+	0.671259i	$0.234246\pi$
$173$			17.6581i			1.34252i	−0.741223	+	0.671259i	$0.765754\pi$
$174$		0			0
$175$			15.3854i			1.16303i
$176$		0			0
$177$			2.18094i			0.163930i
$178$		0			0
$179$	5.54061			0.414125			0.207062	−	0.978328i	$-0.433610\pi$
$179$	5.54061			0.414125			0.207062	+	0.978328i	$0.433610\pi$
$180$		0			0
$181$			14.4867i			1.07679i	0.842693	+	0.538395i	$0.180969\pi$
$181$			14.4867i			1.07679i	−0.842693	+	0.538395i	$0.819031\pi$
$182$		0			0
$183$	3.32277			0.245626
$184$		0			0
$185$	−4.10298			−0.301657
$186$		0			0
$187$	12.6080	−	0.343761i	0.921985	−	0.0251383i
$188$		0			0
$189$	8.55669			0.622408
$190$		0			0
$191$	7.43091			0.537682			0.268841	−	0.963185i	$-0.413359\pi$
$191$	7.43091			0.537682			0.268841	+	0.963185i	$0.413359\pi$
$192$		0			0
$193$		−	2.08053i		−	0.149760i	−0.997193	−	0.0748799i	$-0.976143\pi$
$193$		−	2.08053i		−	0.149760i	0.997193	−	0.0748799i	$-0.0238573\pi$
$194$		0			0
$195$	1.88149			0.134736
$196$		0			0
$197$			3.44357i			0.245345i	0.992447	+	0.122672i	$0.0391464\pi$
$197$			3.44357i			0.245345i	−0.992447	+	0.122672i	$0.960854\pi$
$198$		0			0
$199$		−	10.3272i		−	0.732076i	−0.930600	−	0.366038i	$-0.880714\pi$
$199$		−	10.3272i		−	0.732076i	0.930600	−	0.366038i	$-0.119286\pi$
$200$		0			0
$201$		−	17.9431i		−	1.26561i
$202$		0			0
$203$	13.1560			0.923373
$204$		0			0
$205$	2.23433			0.156052
$206$		0			0
$207$			0.326523i			0.0226949i
$208$		0			0
$209$		−	6.67726i		−	0.461876i
$210$		0			0
$211$		−	23.1443i		−	1.59332i	−0.604429	−	0.796659i	$-0.706598\pi$
$211$		−	23.1443i		−	1.59332i	0.604429	−	0.796659i	$-0.293402\pi$
$212$		0			0
$213$	−24.5835			−1.68443
$214$		0			0
$215$			0.734395i			0.0500853i
$216$		0			0
$217$	−19.7614			−1.34149
$218$		0			0
$219$	24.3260			1.64380
$220$		0			0
$221$	−0.275657	−	10.1101i	−0.0185427	−	0.680081i
$222$		0			0
$223$	−6.84912			−0.458651			−0.229325	−	0.973350i	$-0.573652\pi$
$223$	−6.84912			−0.458651			−0.229325	+	0.973350i	$0.573652\pi$
$224$		0			0
$225$	−8.56527			−0.571018
$226$		0			0
$227$		−	16.7446i		−	1.11138i	−0.831391	−	0.555688i	$-0.812455\pi$
$227$		−	16.7446i		−	1.11138i	0.831391	−	0.555688i	$-0.187545\pi$
$228$		0			0
$229$	−1.33345			−0.0881167			−0.0440584	−	0.999029i	$-0.514029\pi$
$229$	−1.33345			−0.0881167			−0.0440584	+	0.999029i	$0.514029\pi$
$230$		0			0
$231$		−	21.0496i		−	1.38496i
$232$		0			0
$233$			22.6162i			1.48164i	0.671706	+	0.740818i	$0.265562\pi$
$233$			22.6162i			1.48164i	−0.671706	+	0.740818i	$0.734438\pi$
$234$		0			0
$235$		−	0.329773i		−	0.0215120i
$236$		0			0
$237$	−30.6158			−1.98871
$238$		0			0
$239$	−13.8140			−0.893552			−0.446776	−	0.894646i	$-0.647428\pi$
$239$	−13.8140			−0.893552			−0.446776	+	0.894646i	$0.647428\pi$
$240$		0			0
$241$		−	7.61998i		−	0.490846i	−0.969416	−	0.245423i	$-0.921073\pi$
$241$		−	7.61998i		−	0.490846i	0.969416	−	0.245423i	$-0.0789268\pi$
$242$		0			0
$243$			16.2561i			1.04283i
$244$		0			0
$245$			1.03922i			0.0663932i
$246$		0			0
$247$	−5.35440			−0.340692
$248$		0			0
$249$		−	6.75061i		−	0.427803i
$250$		0			0
$251$	−21.0204			−1.32680			−0.663399	−	0.748266i	$-0.730887\pi$
$251$	−21.0204			−1.32680			−0.663399	+	0.748266i	$0.730887\pi$
$252$		0			0
$253$	−0.568650			−0.0357507
$254$		0			0
$255$	−0.0861954	−	3.16134i	−0.00539776	−	0.197971i
$256$		0			0
$257$	16.4299			1.02487			0.512433	−	0.858727i	$-0.328744\pi$
$257$	16.4299			1.02487			0.512433	+	0.858727i	$0.328744\pi$
$258$		0			0
$259$	36.8090			2.28720
$260$		0			0
$261$			7.32414i			0.453353i
$262$		0			0
$263$	8.45848			0.521572			0.260786	−	0.965397i	$-0.416018\pi$
$263$	8.45848			0.521572			0.260786	+	0.965397i	$0.416018\pi$
$264$		0			0
$265$		−	3.20745i		−	0.197032i
$266$		0			0
$267$		−	14.1097i		−	0.863502i
$268$		0			0
$269$		−	8.15538i		−	0.497242i	−0.968601	−	0.248621i	$-0.920023\pi$
$269$		−	8.15538i		−	0.497242i	0.968601	−	0.248621i	$-0.0799774\pi$
$270$		0			0
$271$	−25.2413			−1.53330			−0.766650	−	0.642065i	$-0.778078\pi$
$271$	−25.2413			−1.53330			−0.766650	+	0.642065i	$0.778078\pi$
$272$		0			0
$273$	−16.8794			−1.02159
$274$		0			0
$275$		−	14.9167i		−	0.899511i
$276$		0			0
$277$			12.9161i			0.776056i	0.921648	+	0.388028i	$0.126844\pi$
$277$			12.9161i			0.776056i	−0.921648	+	0.388028i	$0.873156\pi$
$278$		0			0
$279$		−	11.0014i		−	0.658638i
$280$		0			0
$281$	29.9182			1.78477			0.892386	−	0.451272i	$-0.149030\pi$
$281$	29.9182			1.78477			0.892386	+	0.451272i	$0.149030\pi$
$282$		0			0
$283$		−	22.0086i		−	1.30827i	−0.756376	−	0.654137i	$-0.773032\pi$
$283$		−	22.0086i		−	1.30827i	0.756376	−	0.654137i	$-0.226968\pi$
$284$		0			0
$285$	−1.67427			−0.0991752
$286$		0			0
$287$	−20.0448			−1.18321
$288$		0			0
$289$	−16.9747	+	0.926336i	−0.998514	+	0.0544904i
$290$		0			0
$291$	−35.8996			−2.10447
$292$		0			0
$293$	−1.32978			−0.0776867			−0.0388433	−	0.999245i	$-0.512367\pi$
$293$	−1.32978			−0.0776867			−0.0388433	+	0.999245i	$0.512367\pi$
$294$		0			0
$295$			0.351694i			0.0204764i
$296$		0			0
$297$	−8.29601			−0.481383
$298$		0			0
$299$			0.455992i			0.0263707i
$300$		0			0
$301$		−	6.58846i		−	0.379752i
$302$		0			0
$303$		−	32.0858i		−	1.84328i
$304$		0			0
$305$	0.535822			0.0306811
$306$		0			0
$307$	−13.3192			−0.760167			−0.380084	−	0.924952i	$-0.624105\pi$
$307$	−13.3192			−0.760167			−0.380084	+	0.924952i	$0.624105\pi$
$308$		0			0
$309$		−	4.52446i		−	0.257388i
$310$		0			0
$311$			9.02736i			0.511895i	0.966691	+	0.255947i	$0.0823874\pi$
$311$			9.02736i			0.511895i	−0.966691	+	0.255947i	$0.917613\pi$
$312$		0			0
$313$			15.6629i			0.885321i	0.896689	+	0.442661i	$0.145965\pi$
$313$			15.6629i			0.885321i	−0.896689	+	0.442661i	$0.854035\pi$
$314$		0			0
$315$	−1.94909			−0.109819
$316$		0			0
$317$		−	27.2230i		−	1.52900i	−0.644625	−	0.764499i	$-0.722986\pi$
$317$		−	27.2230i		−	1.52900i	0.644625	−	0.764499i	$-0.277014\pi$
$318$		0			0
$319$	−12.7552			−0.714156
$320$		0			0
$321$	−4.78486			−0.267065
$322$		0			0
$323$	0.245297	+	8.99663i	0.0136487	+	0.500586i
$324$		0			0
$325$	−11.9615			−0.663504
$326$		0			0
$327$	−42.8279			−2.36839
$328$		0			0
$329$			2.95848i			0.163106i
$330$		0			0
$331$	3.69474			0.203081			0.101541	−	0.994831i	$-0.467623\pi$
$331$	3.69474			0.203081			0.101541	+	0.994831i	$0.467623\pi$
$332$		0			0
$333$			20.4920i			1.12296i
$334$		0			0
$335$		−	2.89347i		−	0.158087i
$336$		0			0
$337$		−	3.09585i		−	0.168642i	−0.996439	−	0.0843208i	$-0.973128\pi$
$337$		−	3.09585i		−	0.168642i	0.996439	−	0.0843208i	$-0.0268720\pi$
$338$		0			0
$339$	−28.7895			−1.56363
$340$		0			0
$341$	19.1594			1.03754
$342$		0			0
$343$			12.7629i			0.689130i
$344$		0			0
$345$			0.142584i			0.00767648i
$346$		0			0
$347$		−	26.6898i		−	1.43278i	−0.697699	−	0.716391i	$-0.745793\pi$
$347$		−	26.6898i		−	1.43278i	0.697699	−	0.716391i	$-0.254207\pi$
$348$		0			0
$349$	4.04582			0.216568			0.108284	−	0.994120i	$-0.465464\pi$
$349$	4.04582			0.216568			0.108284	+	0.994120i	$0.465464\pi$
$350$		0			0
$351$			6.65244i			0.355081i
$352$		0			0
$353$	24.0259			1.27877			0.639384	−	0.768888i	$-0.279189\pi$
$353$	24.0259			1.27877			0.639384	+	0.768888i	$0.279189\pi$
$354$		0			0
$355$	−3.96427			−0.210402
$356$		0			0
$357$	0.773282	+	28.3613i	0.0409264	+	1.50104i
$358$		0			0
$359$	17.3718			0.916850			0.458425	−	0.888733i	$-0.348414\pi$
$359$	17.3718			0.916850			0.458425	+	0.888733i	$0.348414\pi$
$360$		0			0
$361$	−14.2353			−0.749227
$362$		0			0
$363$		−	3.58205i		−	0.188009i
$364$		0			0
$365$	3.92276			0.205326
$366$		0			0
$367$			2.70486i			0.141193i	0.997505	+	0.0705963i	$0.0224902\pi$
$367$			2.70486i			0.141193i	−0.997505	+	0.0705963i	$0.977510\pi$
$368$		0			0
$369$		−	11.1592i		−	0.580924i
$370$		0			0
$371$			28.7749i			1.49392i
$372$		0			0
$373$	4.13114			0.213903			0.106951	−	0.994264i	$-0.465891\pi$
$373$	4.13114			0.213903			0.106951	+	0.994264i	$0.465891\pi$
$374$		0			0
$375$	−7.57536			−0.391190
$376$		0			0
$377$			10.2282i			0.526781i
$378$		0			0
$379$		−	14.2370i		−	0.731305i	−0.930751	−	0.365652i	$-0.880846\pi$
$379$		−	14.2370i		−	0.731305i	0.930751	−	0.365652i	$-0.119154\pi$
$380$		0			0
$381$			9.82162i			0.503177i
$382$		0			0
$383$	−17.7101			−0.904946			−0.452473	−	0.891778i	$-0.649458\pi$
$383$	−17.7101			−0.904946			−0.452473	+	0.891778i	$0.649458\pi$
$384$		0			0
$385$		−	3.39441i		−	0.172995i
$386$		0			0
$387$	3.66788			0.186449
$388$		0			0
$389$	−15.8273			−0.802474			−0.401237	−	0.915974i	$-0.631420\pi$
$389$	−15.8273			−0.802474			−0.401237	+	0.915974i	$0.631420\pi$
$390$		0			0
$391$	0.766172	−	0.0208900i	0.0387470	−	0.00105645i
$392$		0			0
$393$	11.0976			0.559800
$394$		0			0
$395$	−4.93704			−0.248409
$396$		0			0
$397$			21.8360i			1.09592i	0.836505	+	0.547959i	$0.184595\pi$
$397$			21.8360i			1.09592i	−0.836505	+	0.547959i	$0.815405\pi$
$398$		0			0
$399$	15.0203			0.751957
$400$		0			0
$401$			6.79022i			0.339088i	0.985523	+	0.169544i	$0.0542294\pi$
$401$			6.79022i			0.339088i	−0.985523	+	0.169544i	$0.945771\pi$
$402$		0			0
$403$		−	15.3636i		−	0.765315i
$404$		0			0
$405$			3.93341i			0.195453i
$406$		0			0
$407$	−35.6875			−1.76897
$408$		0			0
$409$	36.3216			1.79599			0.897993	−	0.440010i	$-0.145025\pi$
$409$	36.3216			1.79599			0.897993	+	0.440010i	$0.145025\pi$
$410$		0			0
$411$		−	25.3853i		−	1.25216i
$412$		0			0
$413$		−	3.15514i		−	0.155254i
$414$		0			0
$415$		−	1.08859i		−	0.0534367i
$416$		0			0
$417$	−20.0898			−0.983800
$418$		0			0
$419$		−	9.58585i		−	0.468299i	−0.972201	−	0.234150i	$-0.924769\pi$
$419$		−	9.58585i		−	0.468299i	0.972201	−	0.234150i	$-0.0752306\pi$
$420$		0			0
$421$	−14.0521			−0.684856			−0.342428	−	0.939544i	$-0.611249\pi$
$421$	−14.0521			−0.684856			−0.342428	+	0.939544i	$0.611249\pi$
$422$		0			0
$423$	−1.64703			−0.0800811
$424$		0			0
$425$	0.547983	+	20.0981i	0.0265811	+	0.974900i
$426$		0			0
$427$	−4.80701			−0.232627
$428$		0			0
$429$	16.3651			0.790115
$430$		0			0
$431$		−	12.4509i		−	0.599739i	−0.953980	−	0.299869i	$-0.903057\pi$
$431$		−	12.4509i		−	0.599739i	0.953980	−	0.299869i	$-0.0969431\pi$
$432$		0			0
$433$	9.66470			0.464456			0.232228	−	0.972661i	$-0.425398\pi$
$433$	9.66470			0.464456			0.232228	+	0.972661i	$0.425398\pi$
$434$		0			0
$435$			3.19827i			0.153345i
$436$		0			0
$437$		−	0.405770i		−	0.0194106i
$438$		0			0
$439$		−	25.2281i		−	1.20407i	−0.798470	−	0.602035i	$-0.794357\pi$
$439$		−	25.2281i		−	1.20407i	0.798470	−	0.602035i	$-0.205643\pi$
$440$		0			0
$441$	5.19029			0.247157
$442$		0			0
$443$	−3.40622			−0.161834			−0.0809172	−	0.996721i	$-0.525785\pi$
$443$	−3.40622			−0.161834			−0.0809172	+	0.996721i	$0.525785\pi$
$444$		0			0
$445$		−	2.27530i		−	0.107860i
$446$		0			0
$447$			2.85739i			0.135150i
$448$		0			0
$449$			27.1662i			1.28205i	0.767519	+	0.641026i	$0.221491\pi$
$449$			27.1662i			1.28205i	−0.767519	+	0.641026i	$0.778509\pi$
$450$		0			0
$451$	19.4341			0.915116
$452$		0			0
$453$		−	3.54609i		−	0.166610i
$454$		0			0
$455$	−2.72193			−0.127606
$456$		0			0
$457$	−25.8143			−1.20754			−0.603772	−	0.797157i	$-0.706336\pi$
$457$	−25.8143			−1.20754			−0.603772	+	0.797157i	$0.706336\pi$
$458$		0			0
$459$	11.1777	−	0.304764i	0.521728	−	0.0142251i
$460$		0			0
$461$	9.32367			0.434247			0.217123	−	0.976144i	$-0.430333\pi$
$461$	9.32367			0.434247			0.217123	+	0.976144i	$0.430333\pi$
$462$		0			0
$463$	27.7473			1.28953			0.644764	−	0.764382i	$-0.276956\pi$
$463$	27.7473			1.28953			0.644764	+	0.764382i	$0.276956\pi$
$464$		0			0
$465$		−	4.80405i		−	0.222783i
$466$		0			0
$467$	9.93640			0.459802			0.229901	−	0.973214i	$-0.426160\pi$
$467$	9.93640			0.459802			0.229901	+	0.973214i	$0.426160\pi$
$468$		0			0
$469$			25.9581i			1.19863i
$470$		0			0
$471$			21.0940i			0.971960i
$472$		0			0
$473$			6.38774i			0.293708i
$474$		0			0
$475$	10.6441			0.488384
$476$		0			0
$477$	−16.0194			−0.733476
$478$		0			0
$479$			27.0604i			1.23642i	0.786013	+	0.618210i	$0.212142\pi$
$479$			27.0604i			1.23642i	−0.786013	+	0.618210i	$0.787858\pi$
$480$		0			0
$481$			28.6173i			1.30484i
$482$		0			0
$483$		−	1.27916i		−	0.0582039i
$484$		0			0
$485$	−5.78909			−0.262869
$486$		0			0
$487$		−	41.3115i		−	1.87200i	−0.351999	−	0.936000i	$-0.614498\pi$
$487$		−	41.3115i		−	1.87200i	0.351999	−	0.936000i	$-0.385502\pi$
$488$		0			0
$489$	41.9449			1.89682
$490$		0			0
$491$	10.8285			0.488684			0.244342	−	0.969689i	$-0.421428\pi$
$491$	10.8285			0.488684			0.244342	+	0.969689i	$0.421428\pi$
$492$		0			0
$493$	17.1858	−	0.468578i	0.774010	−	0.0211037i
$494$		0			0
$495$	1.88971			0.0849363
$496$		0			0
$497$	35.5645			1.59529
$498$		0			0
$499$		−	0.00603515i		−	0.000270170i	−1.00000		0.000135085i	$-0.999957\pi$
$499$		−	0.00603515i		−	0.000270170i	1.00000		0.000135085i	$-4.29990e-5\pi$
$500$		0			0
$501$	6.62272			0.295881
$502$		0			0
$503$		−	4.60375i		−	0.205271i	−0.994719	−	0.102635i	$-0.967272\pi$
$503$		−	4.60375i		−	0.205271i	0.994719	−	0.102635i	$-0.0327275\pi$
$504$		0			0
$505$		−	5.17408i		−	0.230244i
$506$		0			0
$507$			15.2293i			0.676357i
$508$		0			0
$509$	−25.2421			−1.11883			−0.559417	−	0.828886i	$-0.688975\pi$
$509$	−25.2421			−1.11883			−0.559417	+	0.828886i	$0.688975\pi$
$510$		0			0
$511$	−35.1921			−1.55681
$512$		0			0
$513$		−	5.91977i		−	0.261364i
$514$		0			0
$515$		−	0.729605i		−	0.0321502i
$516$		0			0
$517$		−	2.86835i		−	0.126150i
$518$		0			0
$519$	38.5112			1.69045
$520$		0			0
$521$			15.3418i			0.672135i	0.941838	+	0.336067i	$0.109097\pi$
$521$			15.3418i			0.672135i	−0.941838	+	0.336067i	$0.890903\pi$
$522$		0			0
$523$	20.0003			0.874554			0.437277	−	0.899327i	$-0.355943\pi$
$523$	20.0003			0.874554			0.437277	+	0.899327i	$0.355943\pi$
$524$		0			0
$525$	33.5547			1.46445
$526$		0			0
$527$	−25.8144	+	0.703841i	−1.12449	+	0.0306598i
$528$		0			0
$529$	22.9654			0.998498
$530$		0			0
$531$	1.75651			0.0762259
$532$		0			0
$533$		−	15.5839i		−	0.675014i
$534$		0			0
$535$	−0.771596			−0.0333590
$536$		0			0
$537$		−	12.0838i		−	0.521452i
$538$		0			0
$539$			9.03906i			0.389340i
$540$		0			0
$541$			31.9114i			1.37198i	0.727611	+	0.685990i	$0.240631\pi$
$541$			31.9114i			1.37198i	−0.727611	+	0.685990i	$0.759369\pi$
$542$		0			0
$543$	31.5947			1.35586
$544$		0			0
$545$	−6.90633			−0.295835
$546$		0			0
$547$			15.6886i			0.670797i	0.942076	+	0.335398i	$0.108871\pi$
$547$			15.6886i			0.670797i	−0.942076	+	0.335398i	$0.891129\pi$
$548$		0			0
$549$		−	2.67612i		−	0.114214i
$550$		0			0
$551$		−	9.10173i		−	0.387747i
$552$		0			0
$553$	44.2915			1.88347
$554$		0			0
$555$			8.94837i			0.379837i
$556$		0			0
$557$	29.9233			1.26789			0.633945	−	0.773378i	$-0.281434\pi$
$557$	29.9233			1.26789			0.633945	+	0.773378i	$0.281434\pi$
$558$		0			0
$559$	5.12223			0.216647
$560$		0			0
$561$	−0.749723	−	27.4972i	−0.0316533	−	1.16093i
$562$		0			0
$563$	−37.1922			−1.56746			−0.783732	−	0.621099i	$-0.786686\pi$
$563$	−37.1922			−1.56746			−0.783732	+	0.621099i	$0.786686\pi$
$564$		0			0
$565$	−4.64253			−0.195313
$566$		0			0
$567$		−	35.2877i		−	1.48194i
$568$		0			0
$569$	−2.45025			−0.102720			−0.0513600	−	0.998680i	$-0.516356\pi$
$569$	−2.45025			−0.102720			−0.0513600	+	0.998680i	$0.516356\pi$
$570$		0			0
$571$		−	5.67107i		−	0.237327i	−0.992935	−	0.118663i	$-0.962139\pi$
$571$		−	5.67107i		−	0.237327i	0.992935	−	0.118663i	$-0.0378610\pi$
$572$		0			0
$573$		−	16.2064i		−	0.677031i
$574$		0			0
$575$		−	0.906473i		−	0.0378025i
$576$		0			0
$577$	27.6082			1.14934			0.574672	−	0.818383i	$-0.305129\pi$
$577$	27.6082			1.14934			0.574672	+	0.818383i	$0.305129\pi$
$578$		0			0
$579$	−4.53751			−0.188573
$580$		0			0
$581$			9.76602i			0.405163i
$582$		0			0
$583$		−	27.8982i		−	1.15543i
$584$		0			0
$585$		−	1.51533i		−	0.0626513i
$586$		0			0
$587$	28.6893			1.18413			0.592067	−	0.805888i	$-0.298312\pi$
$587$	28.6893			1.18413			0.592067	+	0.805888i	$0.298312\pi$
$588$		0			0
$589$			13.6715i			0.563324i
$590$		0			0
$591$	7.51023			0.308930
$592$		0			0
$593$	28.9561			1.18908			0.594542	−	0.804064i	$-0.297333\pi$
$593$	28.9561			1.18908			0.594542	+	0.804064i	$0.297333\pi$
$594$		0			0
$595$	0.124698	+	4.57347i	0.00511211	+	0.187494i
$596$		0			0
$597$	−22.5230			−0.921806
$598$		0			0
$599$	11.7362			0.479527			0.239764	−	0.970831i	$-0.422930\pi$
$599$	11.7362			0.479527			0.239764	+	0.970831i	$0.422930\pi$
$600$		0			0
$601$		−	41.0974i		−	1.67640i	−0.545363	−	0.838200i	$-0.683608\pi$
$601$		−	41.0974i		−	1.67640i	0.545363	−	0.838200i	$-0.316392\pi$
$602$		0			0
$603$	−14.4512			−0.588499
$604$		0			0
$605$		−	0.577633i		−	0.0234841i
$606$		0			0
$607$			36.2430i			1.47106i	0.677493	+	0.735530i	$0.263067\pi$
$607$			36.2430i			1.47106i	−0.677493	+	0.735530i	$0.736933\pi$
$608$		0			0
$609$		−	28.6926i		−	1.16268i
$610$		0			0
$611$	−2.30009			−0.0930515
$612$		0			0
$613$	−0.690148			−0.0278748			−0.0139374	−	0.999903i	$-0.504437\pi$
$613$	−0.690148			−0.0278748			−0.0139374	+	0.999903i	$0.504437\pi$
$614$		0			0
$615$		−	4.87294i		−	0.196496i
$616$		0			0
$617$			43.2992i			1.74316i	0.490251	+	0.871581i	$0.336905\pi$
$617$			43.2992i			1.74316i	−0.490251	+	0.871581i	$0.663095\pi$
$618$		0			0
$619$			22.2663i			0.894957i	0.894295	+	0.447479i	$0.147678\pi$
$619$			22.2663i			0.894957i	−0.894295	+	0.447479i	$0.852322\pi$
$620$		0			0
$621$	−0.504140			−0.0202304
$622$		0			0
$623$			20.4124i			0.817804i
$624$		0			0
$625$	23.1600			0.926399
$626$		0			0
$627$	−14.5627			−0.581579
$628$		0			0
$629$	48.0837	−	1.31102i	1.91722	−	0.0522739i
$630$		0			0
$631$	−43.8827			−1.74694			−0.873471	−	0.486876i	$-0.838136\pi$
$631$	−43.8827			−1.74694			−0.873471	+	0.486876i	$0.838136\pi$
$632$		0			0
$633$	−50.4764			−2.00625
$634$		0			0
$635$			1.58381i			0.0628517i
$636$		0			0
$637$	7.24829			0.287188
$638$		0			0
$639$			19.7992i			0.783246i
$640$		0			0
$641$			33.8096i			1.33540i	0.744431	+	0.667699i	$0.232721\pi$
$641$			33.8096i			1.33540i	−0.744431	+	0.667699i	$0.767279\pi$
$642$		0			0
$643$			24.7563i			0.976293i	0.872762	+	0.488146i	$0.162327\pi$
$643$			24.7563i			0.976293i	−0.872762	+	0.488146i	$0.837673\pi$
$644$		0			0
$645$	1.60167			0.0630658
$646$		0			0
$647$	−35.8215			−1.40829			−0.704144	−	0.710057i	$-0.748669\pi$
$647$	−35.8215			−1.40829			−0.704144	+	0.710057i	$0.748669\pi$
$648$		0			0
$649$			3.05901i			0.120077i
$650$		0			0
$651$			43.0985i			1.68916i
$652$		0			0
$653$			34.2925i			1.34197i	0.741472	+	0.670984i	$0.234128\pi$
$653$			34.2925i			1.34197i	−0.741472	+	0.670984i	$0.765872\pi$
$654$		0			0
$655$	1.78957			0.0699244
$656$		0			0
$657$		−	19.5919i		−	0.764353i
$658$		0			0
$659$	34.9055			1.35973			0.679863	−	0.733339i	$-0.262039\pi$
$659$	34.9055			1.35973			0.679863	+	0.733339i	$0.262039\pi$
$660$		0			0
$661$	20.8883			0.812461			0.406231	−	0.913771i	$-0.366843\pi$
$661$	20.8883			0.812461			0.406231	+	0.913771i	$0.366843\pi$
$662$		0			0
$663$	−22.0496	+	0.601192i	−0.856336	+	0.0233484i
$664$		0			0
$665$	2.42214			0.0939267
$666$		0			0
$667$	−0.775122			−0.0300129
$668$		0			0
$669$			14.9375i			0.577518i
$670$		0			0
$671$	4.66056			0.179919
$672$		0			0
$673$		−	4.81349i		−	0.185546i	−0.995687	−	0.0927732i	$-0.970427\pi$
$673$		−	4.81349i		−	0.185546i	0.995687	−	0.0927732i	$-0.0295732\pi$
$674$		0			0
$675$		−	13.2245i		−	0.509011i
$676$		0			0
$677$			38.9649i			1.49754i	0.662828	+	0.748772i	$0.269356\pi$
$677$			38.9649i			1.49754i	−0.662828	+	0.748772i	$0.730644\pi$
$678$		0			0
$679$	51.9354			1.99310
$680$		0			0
$681$	−36.5189			−1.39941
$682$		0			0
$683$		−	11.6215i		−	0.444684i	−0.974969	−	0.222342i	$-0.928630\pi$
$683$		−	11.6215i		−	0.444684i	0.974969	−	0.222342i	$-0.0713702\pi$
$684$		0			0
$685$		−	4.09357i		−	0.156407i
$686$		0			0
$687$			2.90817i			0.110954i
$688$		0			0
$689$	−22.3712			−0.852274
$690$		0			0
$691$		−	43.6018i		−	1.65869i	−0.558736	−	0.829346i	$-0.688713\pi$
$691$		−	43.6018i		−	1.65869i	0.558736	−	0.829346i	$-0.311287\pi$
$692$		0			0
$693$	−16.9531			−0.643996
$694$		0			0
$695$	−3.23963			−0.122886
$696$		0			0
$697$	−26.1846	+	0.713934i	−0.991812	+	0.0270422i
$698$		0			0
$699$	49.3246			1.86563
$700$		0			0
$701$	9.24982			0.349361			0.174680	−	0.984625i	$-0.444111\pi$
$701$	9.24982			0.349361			0.174680	+	0.984625i	$0.444111\pi$
$702$		0			0
$703$		−	25.4655i		−	0.960449i
$704$		0			0
$705$	−0.719216			−0.0270872
$706$		0			0
$707$			46.4181i			1.74573i
$708$		0			0
$709$		−	19.1933i		−	0.720821i	−0.932794	−	0.360411i	$-0.882637\pi$
$709$		−	19.1933i		−	0.720821i	0.932794	−	0.360411i	$-0.117363\pi$
$710$		0			0
$711$			24.6577i			0.924734i
$712$		0			0
$713$	1.16429			0.0436032
$714$		0			0
$715$	2.63900			0.0986931
$716$		0			0
$717$			30.1275i			1.12513i
$718$		0			0
$719$			25.7806i			0.961455i	0.876870	+	0.480727i	$0.159627\pi$
$719$			25.7806i			0.961455i	−0.876870	+	0.480727i	$0.840373\pi$
$720$		0			0
$721$			6.54548i			0.243766i
$722$		0			0
$723$	−16.6187			−0.618057
$724$		0			0
$725$		−	20.3329i		−	0.755143i
$726$		0			0
$727$	−49.4495			−1.83398			−0.916991	−	0.398908i	$-0.869389\pi$
$727$	−49.4495			−1.83398			−0.916991	+	0.398908i	$0.869389\pi$
$728$		0			0
$729$	1.90107			0.0704101
$730$		0			0
$731$	−0.234661	−	8.60654i	−0.00867925	−	0.318324i
$732$		0			0
$733$	−15.3618			−0.567400			−0.283700	−	0.958913i	$-0.591562\pi$
$733$	−15.3618			−0.567400			−0.283700	+	0.958913i	$0.591562\pi$
$734$		0			0
$735$	2.26647			0.0836001
$736$		0			0
$737$		−	25.1673i		−	0.927048i
$738$		0			0
$739$	14.6919			0.540451			0.270226	−	0.962797i	$-0.412902\pi$
$739$	14.6919			0.540451			0.270226	+	0.962797i	$0.412902\pi$
$740$		0			0
$741$			11.6776i			0.428988i
$742$		0			0
$743$			30.9398i			1.13507i	0.823349	+	0.567535i	$0.192103\pi$
$743$			30.9398i			1.13507i	−0.823349	+	0.567535i	$0.807897\pi$
$744$		0			0
$745$			0.460775i			0.0168815i
$746$		0			0
$747$	−5.43687			−0.198925
$748$		0			0
$749$	6.92219			0.252931
$750$		0			0
$751$		−	0.350527i		−	0.0127909i	−0.999980	−	0.00639545i	$-0.997964\pi$
$751$		−	0.350527i		−	0.0127909i	0.999980	−	0.00639545i	$-0.00203575\pi$
$752$		0			0
$753$			45.8443i			1.67066i
$754$		0			0
$755$		−	0.571834i		−	0.0208112i
$756$		0			0
$757$	−17.2573			−0.627226			−0.313613	−	0.949551i	$-0.601539\pi$
$757$	−17.2573			−0.627226			−0.313613	+	0.949551i	$0.601539\pi$
$758$		0			0
$759$			1.24019i			0.0450161i
$760$		0			0
$761$	−12.0319			−0.436154			−0.218077	−	0.975932i	$-0.569978\pi$
$761$	−12.0319			−0.436154			−0.218077	+	0.975932i	$0.569978\pi$
$762$		0			0
$763$	61.9586			2.24305
$764$		0			0
$765$	−2.54611	+	0.0694208i	−0.0920548	+	0.00250991i
$766$		0			0
$767$	2.45298			0.0885719
$768$		0			0
$769$	13.8329			0.498829			0.249414	−	0.968397i	$-0.419762\pi$
$769$	13.8329			0.498829			0.249414	+	0.968397i	$0.419762\pi$
$770$		0			0
$771$		−	35.8326i		−	1.29048i
$772$		0			0
$773$	−5.38303			−0.193614			−0.0968070	−	0.995303i	$-0.530863\pi$
$773$	−5.38303			−0.193614			−0.0968070	+	0.995303i	$0.530863\pi$
$774$		0			0
$775$			30.5415i			1.09708i
$776$		0			0
$777$		−	80.2782i		−	2.87996i
$778$		0			0
$779$			13.8676i			0.496857i
$780$		0			0
$781$	−34.4810			−1.23383
$782$		0			0
$783$	−11.3082			−0.404123
$784$		0			0
$785$			3.40157i			0.121407i
$786$		0			0
$787$			9.27251i			0.330529i	0.986249	+	0.165265i	$0.0528478\pi$
$787$			9.27251i			0.330529i	−0.986249	+	0.165265i	$0.947152\pi$
$788$		0			0
$789$		−	18.4475i		−	0.656747i
$790$		0			0
$791$	41.6494			1.48088
$792$		0			0
$793$		−	3.73723i		−	0.132713i
$794$		0			0
$795$	−6.99526			−0.248096
$796$		0			0
$797$	3.89199			0.137862			0.0689308	−	0.997621i	$-0.478041\pi$
$797$	3.89199			0.137862			0.0689308	+	0.997621i	$0.478041\pi$
$798$		0			0
$799$	0.105372	+	3.86468i	0.00372780	+	0.136723i
$800$		0			0
$801$	−11.3638			−0.401521
$802$		0			0
$803$	34.1199			1.20407
$804$		0			0
$805$		−	0.206275i		−	0.00727023i
$806$		0			0
$807$	−17.7864			−0.626111
$808$		0			0
$809$			28.2684i			0.993865i	0.867789	+	0.496933i	$0.165540\pi$
$809$			28.2684i			0.993865i	−0.867789	+	0.496933i	$0.834460\pi$
$810$		0			0
$811$		−	20.7336i		−	0.728055i	−0.931388	−	0.364028i	$-0.881401\pi$
$811$		−	20.7336i		−	0.728055i	0.931388	−	0.364028i	$-0.118599\pi$
$812$		0			0
$813$			55.0498i			1.93068i
$814$		0			0
$815$	6.76394			0.236931
$816$		0			0
$817$	−4.55808			−0.159467
$818$		0			0
$819$			13.5945i			0.475029i
$820$		0			0
$821$			37.0710i			1.29379i	0.762580	+	0.646894i	$0.223932\pi$
$821$			37.0710i			1.29379i	−0.762580	+	0.646894i	$0.776068\pi$
$822$		0			0
$823$			0.142181i			0.00495611i	0.999997	+	0.00247805i	$0.000788790\pi$
$823$			0.142181i			0.00495611i	−0.999997	+	0.00247805i	$0.999211\pi$
$824$		0			0
$825$	−32.5325			−1.13264
$826$		0			0
$827$			44.7020i			1.55444i	0.629229	+	0.777220i	$0.283371\pi$
$827$			44.7020i			1.55444i	−0.629229	+	0.777220i	$0.716629\pi$
$828$		0			0
$829$	4.85095			0.168480			0.0842402	−	0.996445i	$-0.473154\pi$
$829$	4.85095			0.168480			0.0842402	+	0.996445i	$0.473154\pi$
$830$		0			0
$831$	28.1694			0.977184
$832$		0			0
$833$	−0.332061	−	12.1788i	−0.0115052	−	0.421971i
$834$		0			0
$835$	1.06796			0.0369584
$836$		0			0
$837$	16.9858			0.587116
$838$		0			0
$839$		−	46.6218i		−	1.60956i	−0.593572	−	0.804781i	$-0.702283\pi$
$839$		−	46.6218i		−	1.60956i	0.593572	−	0.804781i	$-0.297717\pi$
$840$		0			0
$841$	11.6134			0.400463
$842$		0			0
$843$		−	65.2499i		−	2.24733i
$844$		0			0
$845$			2.45584i			0.0844835i
$846$		0			0
$847$			5.18210i			0.178059i
$848$		0			0
$849$	−47.9994			−1.64734
$850$		0			0
$851$	−2.16870			−0.0743419
$852$		0			0
$853$			27.7643i			0.950633i	0.879815	+	0.475317i	$0.157666\pi$
$853$			27.7643i			0.950633i	−0.879815	+	0.475317i	$0.842334\pi$
$854$		0			0
$855$			1.34844i			0.0461156i
$856$		0			0
$857$			2.44453i			0.0835034i	0.999128	+	0.0417517i	$0.0132939\pi$
$857$			2.44453i			0.0835034i	−0.999128	+	0.0417517i	$0.986706\pi$
$858$		0			0
$859$	16.5762			0.565571			0.282785	−	0.959183i	$-0.408742\pi$
$859$	16.5762			0.565571			0.282785	+	0.959183i	$0.408742\pi$
$860$		0			0
$861$			43.7165i			1.48985i
$862$		0			0
$863$	−30.2605			−1.03008			−0.515039	−	0.857166i	$-0.672223\pi$
$863$	−30.2605			−1.03008			−0.515039	+	0.857166i	$0.672223\pi$
$864$		0			0
$865$	6.21023			0.211154
$866$		0			0
$867$	2.02028	+	37.0209i	0.0686125	+	1.25730i
$868$		0			0
$869$	−42.9421			−1.45671
$870$		0			0
$871$	−20.1813			−0.683816
$872$		0			0
$873$			28.9131i			0.978561i
$874$		0			0
$875$	10.9592			0.370488
$876$		0			0
$877$		−	54.3322i		−	1.83467i	−0.398117	−	0.917335i	$-0.630336\pi$
$877$		−	54.3322i		−	1.83467i	0.398117	−	0.917335i	$-0.369664\pi$
$878$		0			0
$879$			2.90018i			0.0978205i
$880$		0			0
$881$			34.6733i			1.16817i	0.811692	+	0.584086i	$0.198547\pi$
$881$			34.6733i			1.16817i	−0.811692	+	0.584086i	$0.801453\pi$
$882$		0			0
$883$	19.3794			0.652167			0.326084	−	0.945341i	$-0.394271\pi$
$883$	19.3794			0.652167			0.326084	+	0.945341i	$0.394271\pi$
$884$		0			0
$885$	0.767023			0.0257832
$886$		0			0
$887$		−	34.3442i		−	1.15317i	−0.817039	−	0.576583i	$-0.804386\pi$
$887$		−	34.3442i		−	1.15317i	0.817039	−	0.576583i	$-0.195614\pi$
$888$		0			0
$889$		−	14.2088i		−	0.476548i
$890$		0			0
$891$			34.2126i			1.14617i
$892$		0			0
$893$	2.04676			0.0684923
$894$		0			0
$895$		−	1.94860i		−	0.0651345i
$896$		0			0
$897$	0.994491			0.0332051
$898$		0			0
$899$	26.1160			0.871016
$900$		0			0
$901$	1.02487	+	37.5888i	0.0341435	+	1.25226i
$902$		0			0
$903$	−14.3690			−0.478172
$904$		0			0
$905$	5.09489			0.169360
$906$		0			0
$907$		−	26.5477i		−	0.881501i	−0.897630	−	0.440750i	$-0.854712\pi$
$907$		−	26.5477i		−	0.881501i	0.897630	−	0.440750i	$-0.145288\pi$
$908$		0			0
$909$	−25.8416			−0.857110
$910$		0			0
$911$			21.5343i			0.713463i	0.934207	+	0.356731i	$0.116109\pi$
$911$			21.5343i			0.713463i	−0.934207	+	0.356731i	$0.883891\pi$
$912$		0			0
$913$		−	9.46849i		−	0.313361i
$914$		0			0
$915$		−	1.16860i		−	0.0386326i
$916$		0			0
$917$	−16.0547			−0.530174
$918$		0			0
$919$	−42.7888			−1.41147			−0.705736	−	0.708474i	$-0.749384\pi$
$919$	−42.7888			−1.41147			−0.705736	+	0.708474i	$0.749384\pi$
$920$		0			0
$921$			29.0484i			0.957178i
$922$		0			0
$923$			27.6498i			0.910105i
$924$		0			0
$925$		−	56.8888i		−	1.87049i
$926$		0			0
$927$	−3.64395			−0.119683
$928$		0			0
$929$			30.0099i			0.984592i	0.870428	+	0.492296i	$0.163842\pi$
$929$			30.0099i			0.984592i	−0.870428	+	0.492296i	$0.836158\pi$
$930$		0			0
$931$	−6.44999			−0.211390
$932$		0			0
$933$	19.6881			0.644561
$934$		0			0
$935$	−0.120899	−	4.43414i	−0.00395381	−	0.145012i
$936$		0			0
$937$	7.14190			0.233316			0.116658	−	0.993172i	$-0.462782\pi$
$937$	7.14190			0.233316			0.116658	+	0.993172i	$0.462782\pi$
$938$		0			0
$939$	34.1599			1.11477
$940$		0			0
$941$			29.0643i			0.947468i	0.880668	+	0.473734i	$0.157094\pi$
$941$			29.0643i			0.947468i	−0.880668	+	0.473734i	$0.842906\pi$
$942$		0			0
$943$	1.18099			0.0384583
$944$		0			0
$945$		−	3.00933i		−	0.0978936i
$946$		0			0
$947$		−	51.4993i		−	1.67350i	−0.547583	−	0.836751i	$-0.684452\pi$
$947$		−	51.4993i		−	1.67350i	0.547583	−	0.836751i	$-0.315548\pi$
$948$		0			0
$949$		−	27.3603i		−	0.888152i
$950$		0			0
$951$	−59.3719			−1.92526
$952$		0			0
$953$	10.8763			0.352318			0.176159	−	0.984362i	$-0.443633\pi$
$953$	10.8763			0.352318			0.176159	+	0.984362i	$0.443633\pi$
$954$		0			0
$955$		−	2.61340i		−	0.0845678i
$956$		0			0
$957$			27.8184i			0.899242i
$958$		0			0
$959$			36.7245i			1.18590i
$960$		0			0
$961$	−8.22823			−0.265427
$962$		0			0
$963$			3.85367i			0.124183i
$964$		0			0
$965$	−0.731709			−0.0235545
$966$		0			0
$967$	34.8948			1.12214			0.561070	−	0.827768i	$-0.310390\pi$
$967$	34.8948			1.12214			0.561070	+	0.827768i	$0.310390\pi$
$968$		0			0
$969$	19.6211	−	0.534979i	0.630322	−	0.0171860i
$970$		0			0
$971$	7.32177			0.234967			0.117483	−	0.993075i	$-0.462517\pi$
$971$	7.32177			0.234967			0.117483	+	0.993075i	$0.462517\pi$
$972$		0			0
$973$	29.0636			0.931736
$974$		0			0
$975$			26.0873i			0.835462i
$976$		0			0
$977$	−39.6906			−1.26982			−0.634908	−	0.772588i	$-0.718962\pi$
$977$	−39.6906			−1.26982			−0.634908	+	0.772588i	$0.718962\pi$
$978$		0			0
$979$		−	19.7905i		−	0.632506i
$980$		0			0
$981$			34.4931i			1.10128i
$982$		0			0
$983$			24.5172i			0.781976i	0.920396	+	0.390988i	$0.127867\pi$
$983$			24.5172i			0.781976i	−0.920396	+	0.390988i	$0.872133\pi$
$984$		0			0
$985$	1.21108			0.0385883
$986$		0			0
$987$	6.45228			0.205378
$988$		0			0
$989$			0.388176i			0.0123433i
$990$		0			0
$991$		−	8.43615i		−	0.267983i	−0.990982	−	0.133992i	$-0.957220\pi$
$991$		−	8.43615i		−	0.267983i	0.990982	−	0.133992i	$-0.0427795\pi$
$992$		0			0
$993$		−	8.05801i		−	0.255713i
$994$		0			0
$995$	−3.63201			−0.115143
$996$		0			0
$997$		−	9.40806i		−	0.297956i	−0.988840	−	0.148978i	$-0.952402\pi$
$997$		−	9.40806i		−	0.297956i	0.988840	−	0.148978i	$-0.0475984\pi$
$998$		0			0
$999$	−31.6390			−1.00101

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.a.237.9	✓	40
17.16	even	2	inner	4012.2.b.a.237.32	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.a.237.9	✓	40	1.1	even	1	trivial
4012.2.b.a.237.32	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)

\(f(q)\)	\(=\)	\(q-2.18094i q^{3} -0.351694i q^{5} +3.15514i q^{7} -1.75651 q^{9} +O(q^{10})\)	\(q-2.18094i q^{3} -0.351694i q^{5} +3.15514i q^{7} -1.75651 q^{9} -3.05901i q^{11} -2.45298 q^{13} -0.767023 q^{15} +(0.112376 + 4.12157i) q^{17} +2.18282 q^{19} +6.88117 q^{21} -0.185893i q^{23} +4.87631 q^{25} -2.71199i q^{27} -4.16972i q^{29} +6.26324i q^{31} -6.67153 q^{33} +1.10964 q^{35} -11.6664i q^{37} +5.34980i q^{39} +6.35306i q^{41} -2.08817 q^{43} +0.617752i q^{45} +0.937671 q^{47} -2.95489 q^{49} +(8.98891 - 0.245087i) q^{51} +9.12001 q^{53} -1.07584 q^{55} -4.76059i q^{57} -1.00000 q^{59} +1.52355i q^{61} -5.54202i q^{63} +0.862697i q^{65} +8.22725 q^{67} -0.405422 q^{69} -11.2719i q^{71} +11.1539i q^{73} -10.6350i q^{75} +9.65161 q^{77} -14.0379i q^{79} -11.1842 q^{81} +3.09527 q^{83} +(1.44953 - 0.0395221i) q^{85} -9.09392 q^{87} +6.46956 q^{89} -7.73948i q^{91} +13.6598 q^{93} -0.767682i q^{95} -16.4606i q^{97} +5.37318i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 36 q^{9}+O(q^{10}) \) 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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