LMFDB - Embedded newform 4012.2.b.a.237.16

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			237.16
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.a.237.25

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.622620i q^{3} +0.657734i q^{5} -2.30157i q^{7} +2.61234 q^{9} +O(q^{10})$	$q-0.622620i q^{3} +0.657734i q^{5} -2.30157i q^{7} +2.61234 q^{9} -4.95149i q^{11} +5.87745 q^{13} +0.409518 q^{15} +(2.82883 + 2.99962i) q^{17} +1.65864 q^{19} -1.43300 q^{21} +4.54507i q^{23} +4.56739 q^{25} -3.49436i q^{27} +3.32201i q^{29} -2.90571i q^{31} -3.08290 q^{33} +1.51382 q^{35} -0.0987401i q^{37} -3.65941i q^{39} +5.90940i q^{41} +9.96170 q^{43} +1.71823i q^{45} -8.82676 q^{47} +1.70279 q^{49} +(1.86762 - 1.76128i) q^{51} -10.2502 q^{53} +3.25677 q^{55} -1.03270i q^{57} -1.00000 q^{59} +11.0636i q^{61} -6.01248i q^{63} +3.86580i q^{65} +5.73061 q^{67} +2.82985 q^{69} -3.44370i q^{71} +0.343457i q^{73} -2.84374i q^{75} -11.3962 q^{77} -4.40969i q^{79} +5.66138 q^{81} -5.26518 q^{83} +(-1.97296 + 1.86062i) q^{85} +2.06835 q^{87} -10.6692 q^{89} -13.5273i q^{91} -1.80915 q^{93} +1.09094i q^{95} -5.04833i q^{97} -12.9350i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	0.622620i		−	0.359470i	−0.983715	−	0.179735i	$-0.942476\pi$
$3$		−	0.622620i		−	0.359470i	0.983715	−	0.179735i	$-0.0575240\pi$
$4$		0			0
$5$			0.657734i			0.294148i	0.989126	+	0.147074i	$0.0469855\pi$
$5$			0.657734i			0.294148i	−0.989126	+	0.147074i	$0.953015\pi$
$6$		0			0
$7$		−	2.30157i		−	0.869910i	−0.900452	−	0.434955i	$-0.856764\pi$
$7$		−	2.30157i		−	0.869910i	0.900452	−	0.434955i	$-0.143236\pi$
$8$		0			0
$9$	2.61234			0.870782
$10$		0			0
$11$		−	4.95149i		−	1.49293i	−0.665424	−	0.746465i	$-0.731749\pi$
$11$		−	4.95149i		−	1.49293i	0.665424	−	0.746465i	$-0.268251\pi$
$12$		0			0
$13$	5.87745			1.63011			0.815055	−	0.579383i	$-0.196707\pi$
$13$	5.87745			1.63011			0.815055	+	0.579383i	$0.196707\pi$
$14$		0			0
$15$	0.409518			0.105737
$16$		0			0
$17$	2.82883	+	2.99962i	0.686091	+	0.727516i
$17$	2.82883	+	2.99962i	0.686091	+	0.727516i
$18$		0			0
$19$	1.65864			0.380518			0.190259	−	0.981734i	$-0.439067\pi$
$19$	1.65864			0.380518			0.190259	+	0.981734i	$0.439067\pi$
$20$		0			0
$21$	−1.43300			−0.312706
$22$		0			0
$23$			4.54507i			0.947712i	0.880602	+	0.473856i	$0.157138\pi$
$23$			4.54507i			0.947712i	−0.880602	+	0.473856i	$0.842862\pi$
$24$		0			0
$25$	4.56739			0.913477
$26$		0			0
$27$		−	3.49436i		−	0.672489i
$28$		0			0
$29$			3.32201i			0.616881i	0.951244	+	0.308441i	$0.0998071\pi$
$29$			3.32201i			0.616881i	−0.951244	+	0.308441i	$0.900193\pi$
$30$		0			0
$31$		−	2.90571i		−	0.521882i	−0.965355	−	0.260941i	$-0.915967\pi$
$31$		−	2.90571i		−	0.521882i	0.965355	−	0.260941i	$-0.0840327\pi$
$32$		0			0
$33$	−3.08290			−0.536663
$34$		0			0
$35$	1.51382			0.255882
$36$		0			0
$37$		−	0.0987401i		−	0.0162328i	−0.999967	−	0.00811639i	$-0.997416\pi$
$37$		−	0.0987401i		−	0.0162328i	0.999967	−	0.00811639i	$-0.00258356\pi$
$38$		0			0
$39$		−	3.65941i		−	0.585975i
$40$		0			0
$41$			5.90940i			0.922893i	0.887168	+	0.461447i	$0.152669\pi$
$41$			5.90940i			0.922893i	−0.887168	+	0.461447i	$0.847331\pi$
$42$		0			0
$43$	9.96170			1.51915			0.759573	−	0.650423i	$-0.225408\pi$
$43$	9.96170			1.51915			0.759573	+	0.650423i	$0.225408\pi$
$44$		0			0
$45$			1.71823i			0.256138i
$46$		0			0
$47$	−8.82676			−1.28752			−0.643758	−	0.765230i	$-0.722625\pi$
$47$	−8.82676			−1.28752			−0.643758	+	0.765230i	$0.722625\pi$
$48$		0			0
$49$	1.70279			0.243256
$50$		0			0
$51$	1.86762	−	1.76128i	0.261520	−	0.246629i
$52$		0			0
$53$	−10.2502			−1.40797			−0.703987	−	0.710213i	$-0.748599\pi$
$53$	−10.2502			−1.40797			−0.703987	+	0.710213i	$0.748599\pi$
$54$		0			0
$55$	3.25677			0.439142
$56$		0			0
$57$		−	1.03270i		−	0.136785i
$58$		0			0
$59$	−1.00000			−0.130189
$59$	−1.00000			−0.130189
$60$		0			0
$61$			11.0636i			1.41655i	0.705934	+	0.708277i	$0.250527\pi$
$61$			11.0636i			1.41655i	−0.705934	+	0.708277i	$0.749473\pi$
$62$		0			0
$63$		−	6.01248i		−	0.757502i
$64$		0			0
$65$			3.86580i			0.479493i
$66$		0			0
$67$	5.73061			0.700106			0.350053	−	0.936730i	$-0.386164\pi$
$67$	5.73061			0.700106			0.350053	+	0.936730i	$0.386164\pi$
$68$		0			0
$69$	2.82985			0.340674
$70$		0			0
$71$		−	3.44370i		−	0.408692i	−0.978899	−	0.204346i	$-0.934493\pi$
$71$		−	3.44370i		−	0.408692i	0.978899	−	0.204346i	$-0.0655067\pi$
$72$		0			0
$73$			0.343457i			0.0401986i	0.999798	+	0.0200993i	$0.00639824\pi$
$73$			0.343457i			0.0401986i	−0.999798	+	0.0200993i	$0.993602\pi$
$74$		0			0
$75$		−	2.84374i		−	0.328367i
$76$		0			0
$77$	−11.3962			−1.29872
$78$		0			0
$79$		−	4.40969i		−	0.496128i	−0.968744	−	0.248064i	$-0.920206\pi$
$79$		−	4.40969i		−	0.496128i	0.968744	−	0.248064i	$-0.0797944\pi$
$80$		0			0
$81$	5.66138			0.629042
$82$		0			0
$83$	−5.26518			−0.577929			−0.288964	−	0.957340i	$-0.593311\pi$
$83$	−5.26518			−0.577929			−0.288964	+	0.957340i	$0.593311\pi$
$84$		0			0
$85$	−1.97296	+	1.86062i	−0.213997	+	0.201812i
$86$		0			0
$87$	2.06835			0.221750
$88$		0			0
$89$	−10.6692			−1.13094			−0.565469	−	0.824770i	$-0.691305\pi$
$89$	−10.6692			−1.13094			−0.565469	+	0.824770i	$0.691305\pi$
$90$		0			0
$91$		−	13.5273i		−	1.41805i
$92$		0			0
$93$	−1.80915			−0.187601
$94$		0			0
$95$			1.09094i			0.111929i
$96$		0			0
$97$		−	5.04833i		−	0.512580i	−0.966600	−	0.256290i	$-0.917500\pi$
$97$		−	5.04833i		−	0.512580i	0.966600	−	0.256290i	$-0.0825002\pi$
$98$		0			0
$99$		−	12.9350i		−	1.30002i
$100$		0			0
$101$	13.4799			1.34130			0.670650	−	0.741774i	$-0.266015\pi$
$101$	13.4799			1.34130			0.670650	+	0.741774i	$0.266015\pi$
$102$		0			0
$103$	−7.09290			−0.698885			−0.349442	−	0.936958i	$-0.613629\pi$
$103$	−7.09290			−0.698885			−0.349442	+	0.936958i	$0.613629\pi$
$104$		0			0
$105$		−	0.942533i		−	0.0919818i
$106$		0			0
$107$			9.24189i			0.893448i	0.894672	+	0.446724i	$0.147409\pi$
$107$			9.24189i			0.893448i	−0.894672	+	0.446724i	$0.852591\pi$
$108$		0			0
$109$			0.424874i			0.0406956i	0.999793	+	0.0203478i	$0.00647735\pi$
$109$			0.424874i			0.0406956i	−0.999793	+	0.0203478i	$0.993523\pi$
$110$		0			0
$111$	−0.0614775			−0.00583519
$112$		0			0
$113$		−	13.6249i		−	1.28173i	−0.767655	−	0.640863i	$-0.778576\pi$
$113$		−	13.6249i		−	1.28173i	0.767655	−	0.640863i	$-0.221424\pi$
$114$		0			0
$115$	−2.98945			−0.278767
$116$		0			0
$117$	15.3539			1.41947
$118$		0			0
$119$	6.90383	−	6.51073i	0.632873	−	0.596838i
$120$		0			0
$121$	−13.5173			−1.22884
$122$		0			0
$123$	3.67931			0.331752
$124$		0			0
$125$			6.29280i			0.562845i
$126$		0			0
$127$	−6.96093			−0.617683			−0.308842	−	0.951114i	$-0.599941\pi$
$127$	−6.96093			−0.617683			−0.308842	+	0.951114i	$0.599941\pi$
$128$		0			0
$129$		−	6.20235i		−	0.546086i
$130$		0			0
$131$		−	15.0160i		−	1.31196i	−0.754780	−	0.655978i	$-0.772256\pi$
$131$		−	15.0160i		−	1.31196i	0.754780	−	0.655978i	$-0.227744\pi$
$132$		0			0
$133$		−	3.81747i		−	0.331017i
$134$		0			0
$135$	2.29836			0.197811
$136$		0			0
$137$	16.0544			1.37162			0.685810	−	0.727781i	$-0.259448\pi$
$137$	16.0544			1.37162			0.685810	+	0.727781i	$0.259448\pi$
$138$		0			0
$139$			19.3853i			1.64424i	0.569315	+	0.822119i	$0.307208\pi$
$139$			19.3853i			1.64424i	−0.569315	+	0.822119i	$0.692792\pi$
$140$		0			0
$141$			5.49571i			0.462823i
$142$		0			0
$143$		−	29.1021i		−	2.43364i
$144$		0			0
$145$	−2.18500			−0.181454
$146$		0			0
$147$		−	1.06019i		−	0.0874432i
$148$		0			0
$149$	3.35559			0.274901			0.137451	−	0.990509i	$-0.456109\pi$
$149$	3.35559			0.274901			0.137451	+	0.990509i	$0.456109\pi$
$150$		0			0
$151$	−2.57901			−0.209877			−0.104938	−	0.994479i	$-0.533465\pi$
$151$	−2.57901			−0.209877			−0.104938	+	0.994479i	$0.533465\pi$
$152$		0			0
$153$	7.38987	+	7.83605i	0.597436	+	0.633507i
$154$		0			0
$155$	1.91119			0.153510
$156$		0			0
$157$	2.01736			0.161002			0.0805012	−	0.996755i	$-0.474348\pi$
$157$	2.01736			0.161002			0.0805012	+	0.996755i	$0.474348\pi$
$158$		0			0
$159$			6.38197i			0.506124i
$160$		0			0
$161$	10.4608			0.824424
$162$		0			0
$163$		−	4.10397i		−	0.321448i	−0.986999	−	0.160724i	$-0.948617\pi$
$163$		−	4.10397i		−	0.321448i	0.986999	−	0.160724i	$-0.0513828\pi$
$164$		0			0
$165$		−	2.02773i		−	0.157858i
$166$		0			0
$167$			7.61789i			0.589490i	0.955576	+	0.294745i	$0.0952347\pi$
$167$			7.61789i			0.589490i	−0.955576	+	0.294745i	$0.904765\pi$
$168$		0			0
$169$	21.5444			1.65726
$170$		0			0
$171$	4.33294			0.331348
$172$		0			0
$173$		−	10.7140i		−	0.814573i	−0.913300	−	0.407286i	$-0.866475\pi$
$173$		−	10.7140i		−	0.814573i	0.913300	−	0.407286i	$-0.133525\pi$
$174$		0			0
$175$		−	10.5121i		−	0.794643i
$176$		0			0
$177$			0.622620i			0.0467990i
$178$		0			0
$179$	16.3304			1.22059			0.610296	−	0.792173i	$-0.291050\pi$
$179$	16.3304			1.22059			0.610296	+	0.792173i	$0.291050\pi$
$180$		0			0
$181$		−	8.83246i		−	0.656511i	−0.944589	−	0.328256i	$-0.893539\pi$
$181$		−	8.83246i		−	0.656511i	0.944589	−	0.328256i	$-0.106461\pi$
$182$		0			0
$183$	6.88844			0.509208
$184$		0			0
$185$	0.0649448			0.00477483
$186$		0			0
$187$	14.8526	−	14.0069i	1.08613	−	1.02429i
$188$		0			0
$189$	−8.04249			−0.585005
$190$		0			0
$191$	−10.9744			−0.794078			−0.397039	−	0.917802i	$-0.629962\pi$
$191$	−10.9744			−0.794078			−0.397039	+	0.917802i	$0.629962\pi$
$192$		0			0
$193$			8.83448i			0.635920i	0.948104	+	0.317960i	$0.102998\pi$
$193$			8.83448i			0.635920i	−0.948104	+	0.317960i	$0.897002\pi$
$194$		0			0
$195$	2.40692			0.172363
$196$		0			0
$197$			7.09515i			0.505508i	0.967531	+	0.252754i	$0.0813364\pi$
$197$			7.09515i			0.505508i	−0.967531	+	0.252754i	$0.918664\pi$
$198$		0			0
$199$		−	11.9518i		−	0.847242i	−0.905840	−	0.423621i	$-0.860759\pi$
$199$		−	11.9518i		−	0.847242i	0.905840	−	0.423621i	$-0.139241\pi$
$200$		0			0
$201$		−	3.56799i		−	0.251667i
$202$		0			0
$203$	7.64582			0.536631
$204$		0			0
$205$	−3.88681			−0.271467
$206$		0			0
$207$			11.8733i			0.825250i
$208$		0			0
$209$		−	8.21274i		−	0.568087i
$210$		0			0
$211$			15.1405i			1.04232i	0.853460	+	0.521158i	$0.174500\pi$
$211$			15.1405i			1.04232i	−0.853460	+	0.521158i	$0.825500\pi$
$212$		0			0
$213$	−2.14411			−0.146912
$214$		0			0
$215$			6.55215i			0.446853i
$216$		0			0
$217$	−6.68769			−0.453990
$218$		0			0
$219$	0.213843			0.0144502
$220$		0			0
$221$	16.6263	+	17.6301i	1.11840	+	1.18593i
$222$		0			0
$223$	15.2134			1.01876			0.509382	−	0.860541i	$-0.329874\pi$
$223$	15.2134			1.01876			0.509382	+	0.860541i	$0.329874\pi$
$224$		0			0
$225$	11.9316			0.795439
$226$		0			0
$227$			25.4086i			1.68642i	0.537580	+	0.843212i	$0.319338\pi$
$227$			25.4086i			1.68642i	−0.537580	+	0.843212i	$0.680662\pi$
$228$		0			0
$229$	−2.15250			−0.142241			−0.0711205	−	0.997468i	$-0.522657\pi$
$229$	−2.15250			−0.142241			−0.0711205	+	0.997468i	$0.522657\pi$
$230$		0			0
$231$			7.09549i			0.466849i
$232$		0			0
$233$		−	18.9689i		−	1.24269i	−0.783536	−	0.621346i	$-0.786586\pi$
$233$		−	18.9689i		−	1.24269i	0.783536	−	0.621346i	$-0.213414\pi$
$234$		0			0
$235$		−	5.80566i		−	0.378720i
$236$		0			0
$237$	−2.74556			−0.178343
$238$		0			0
$239$	−13.1307			−0.849352			−0.424676	−	0.905345i	$-0.639612\pi$
$239$	−13.1307			−0.849352			−0.424676	+	0.905345i	$0.639612\pi$
$240$		0			0
$241$		−	14.0923i		−	0.907764i	−0.891062	−	0.453882i	$-0.850039\pi$
$241$		−	14.0923i		−	0.907764i	0.891062	−	0.453882i	$-0.149961\pi$
$242$		0			0
$243$		−	14.0080i		−	0.898611i
$244$		0			0
$245$			1.11999i			0.0715533i
$246$		0			0
$247$	9.74857			0.620287
$248$		0			0
$249$			3.27821i			0.207748i
$250$		0			0
$251$	−16.6461			−1.05069			−0.525345	−	0.850889i	$-0.676064\pi$
$251$	−16.6461			−1.05069			−0.525345	+	0.850889i	$0.676064\pi$
$252$		0			0
$253$	22.5049			1.41487
$254$		0			0
$255$	1.15846	+	1.22840i	0.0725453	+	0.0769254i
$256$		0			0
$257$	−23.8213			−1.48593			−0.742966	−	0.669329i	$-0.766582\pi$
$257$	−23.8213			−1.48593			−0.742966	+	0.669329i	$0.766582\pi$
$258$		0			0
$259$	−0.227257			−0.0141211
$260$		0			0
$261$			8.67823i			0.537169i
$262$		0			0
$263$	−8.34578			−0.514623			−0.257311	−	0.966329i	$-0.582837\pi$
$263$	−8.34578			−0.514623			−0.257311	+	0.966329i	$0.582837\pi$
$264$		0			0
$265$		−	6.74191i		−	0.414152i
$266$		0			0
$267$			6.64288i			0.406538i
$268$		0			0
$269$		−	3.59767i		−	0.219354i	−0.993967	−	0.109677i	$-0.965018\pi$
$269$		−	3.59767i		−	0.219354i	0.993967	−	0.109677i	$-0.0349817\pi$
$270$		0			0
$271$	9.07502			0.551268			0.275634	−	0.961263i	$-0.411112\pi$
$271$	9.07502			0.551268			0.275634	+	0.961263i	$0.411112\pi$
$272$		0			0
$273$	−8.42238			−0.509746
$274$		0			0
$275$		−	22.6154i		−	1.36376i
$276$		0			0
$277$		−	14.8060i		−	0.889605i	−0.895629	−	0.444802i	$-0.853274\pi$
$277$		−	14.8060i		−	0.889605i	0.895629	−	0.444802i	$-0.146726\pi$
$278$		0			0
$279$		−	7.59073i		−	0.454445i
$280$		0			0
$281$	21.1798			1.26348			0.631740	−	0.775180i	$-0.282341\pi$
$281$	21.1798			1.26348			0.631740	+	0.775180i	$0.282341\pi$
$282$		0			0
$283$		−	12.6269i		−	0.750593i	−0.926905	−	0.375297i	$-0.877541\pi$
$283$		−	12.6269i		−	0.750593i	0.926905	−	0.375297i	$-0.122459\pi$
$284$		0			0
$285$	0.679243			0.0402349
$286$		0			0
$287$	13.6009			0.802834
$288$		0			0
$289$	−0.995482	+	16.9708i	−0.0585578	+	0.998284i
$290$		0			0
$291$	−3.14319			−0.184257
$292$		0			0
$293$	−9.79452			−0.572202			−0.286101	−	0.958199i	$-0.592359\pi$
$293$	−9.79452			−0.572202			−0.286101	+	0.958199i	$0.592359\pi$
$294$		0			0
$295$		−	0.657734i		−	0.0382948i
$296$		0			0
$297$	−17.3023			−1.00398
$298$		0			0
$299$			26.7134i			1.54488i
$300$		0			0
$301$		−	22.9275i		−	1.32152i
$302$		0			0
$303$		−	8.39284i		−	0.482156i
$304$		0			0
$305$	−7.27694			−0.416676
$306$		0			0
$307$	−12.3215			−0.703223			−0.351611	−	0.936146i	$-0.614366\pi$
$307$	−12.3215			−0.703223			−0.351611	+	0.936146i	$0.614366\pi$
$308$		0			0
$309$			4.41618i			0.251228i
$310$		0			0
$311$		−	22.3577i		−	1.26779i	−0.773420	−	0.633893i	$-0.781456\pi$
$311$		−	22.3577i		−	1.26779i	0.773420	−	0.633893i	$-0.218544\pi$
$312$		0			0
$313$			16.0638i			0.907980i	0.891007	+	0.453990i	$0.150000\pi$
$313$			16.0638i			0.907980i	−0.891007	+	0.453990i	$0.850000\pi$
$314$		0			0
$315$	3.95462			0.222817
$316$		0			0
$317$		−	21.6318i		−	1.21496i	−0.794333	−	0.607482i	$-0.792180\pi$
$317$		−	21.6318i		−	1.21496i	0.794333	−	0.607482i	$-0.207820\pi$
$318$		0			0
$319$	16.4489			0.920961
$320$		0			0
$321$	5.75418			0.321167
$322$		0			0
$323$	4.69201	+	4.97530i	0.261070	+	0.276833i
$324$		0			0
$325$	26.8446			1.48907
$326$		0			0
$327$	0.264535			0.0146288
$328$		0			0
$329$			20.3154i			1.12002i
$330$		0			0
$331$	−23.2360			−1.27717			−0.638584	−	0.769552i	$-0.720480\pi$
$331$	−23.2360			−1.27717			−0.638584	+	0.769552i	$0.720480\pi$
$332$		0			0
$333$		−	0.257943i		−	0.0141352i
$334$		0			0
$335$			3.76922i			0.205934i
$336$		0			0
$337$		−	24.4529i		−	1.33204i	−0.745936	−	0.666018i	$-0.767998\pi$
$337$		−	24.4529i		−	1.33204i	0.745936	−	0.666018i	$-0.232002\pi$
$338$		0			0
$339$	−8.48316			−0.460742
$340$		0			0
$341$	−14.3876			−0.779133
$342$		0			0
$343$		−	20.0301i		−	1.08152i
$344$		0			0
$345$			1.86129i			0.100208i
$346$		0			0
$347$			28.5027i			1.53010i	0.643969	+	0.765051i	$0.277287\pi$
$347$			28.5027i			1.53010i	−0.643969	+	0.765051i	$0.722713\pi$
$348$		0			0
$349$	−35.3333			−1.89135			−0.945673	−	0.325120i	$-0.894595\pi$
$349$	−35.3333			−1.89135			−0.945673	+	0.325120i	$0.894595\pi$
$350$		0			0
$351$		−	20.5379i		−	1.09623i
$352$		0			0
$353$	−12.3735			−0.658574			−0.329287	−	0.944230i	$-0.606808\pi$
$353$	−12.3735			−0.658574			−0.329287	+	0.944230i	$0.606808\pi$
$354$		0			0
$355$	2.26504			0.120216
$356$		0			0
$357$	−4.05371	−	4.29846i	−0.214545	−	0.227499i
$358$		0			0
$359$	−4.34832			−0.229495			−0.114748	−	0.993395i	$-0.536606\pi$
$359$	−4.34832			−0.229495			−0.114748	+	0.993395i	$0.536606\pi$
$360$		0			0
$361$	−16.2489			−0.855206
$362$		0			0
$363$			8.41611i			0.441731i
$364$		0			0
$365$	−0.225903			−0.0118243
$366$		0			0
$367$		−	9.34265i		−	0.487682i	−0.969815	−	0.243841i	$-0.921592\pi$
$367$		−	9.34265i		−	0.487682i	0.969815	−	0.243841i	$-0.0784075\pi$
$368$		0			0
$369$			15.4374i			0.803638i
$370$		0			0
$371$			23.5915i			1.22481i
$372$		0			0
$373$	−11.1148			−0.575502			−0.287751	−	0.957705i	$-0.592908\pi$
$373$	−11.1148			−0.575502			−0.287751	+	0.957705i	$0.592908\pi$
$374$		0			0
$375$	3.91802			0.202326
$376$		0			0
$377$			19.5249i			1.00558i
$378$		0			0
$379$			20.7143i			1.06402i	0.846737	+	0.532011i	$0.178563\pi$
$379$			20.7143i			1.06402i	−0.846737	+	0.532011i	$0.821437\pi$
$380$		0			0
$381$			4.33401i			0.222038i
$382$		0			0
$383$	9.71098			0.496208			0.248104	−	0.968733i	$-0.420193\pi$
$383$	9.71098			0.496208			0.248104	+	0.968733i	$0.420193\pi$
$384$		0			0
$385$		−	7.49566i		−	0.382014i
$386$		0			0
$387$	26.0234			1.32284
$388$		0			0
$389$	23.3374			1.18325			0.591627	−	0.806212i	$-0.298486\pi$
$389$	23.3374			1.18325			0.591627	+	0.806212i	$0.298486\pi$
$390$		0			0
$391$	−13.6335	+	12.8572i	−0.689475	+	0.650217i
$392$		0			0
$393$	−9.34927			−0.471608
$394$		0			0
$395$	2.90040			0.145935
$396$		0			0
$397$		−	26.3261i		−	1.32127i	−0.750707	−	0.660636i	$-0.770287\pi$
$397$		−	26.3261i		−	1.32127i	0.750707	−	0.660636i	$-0.229713\pi$
$398$		0			0
$399$	−2.37683			−0.118990
$400$		0			0
$401$		−	29.6366i		−	1.47998i	−0.672618	−	0.739990i	$-0.734830\pi$
$401$		−	29.6366i		−	1.47998i	0.672618	−	0.739990i	$-0.265170\pi$
$402$		0			0
$403$		−	17.0782i		−	0.850725i
$404$		0			0
$405$			3.72368i			0.185031i
$406$		0			0
$407$	−0.488911			−0.0242344
$408$		0			0
$409$	17.5656			0.868562			0.434281	−	0.900777i	$-0.357002\pi$
$409$	17.5656			0.868562			0.434281	+	0.900777i	$0.357002\pi$
$410$		0			0
$411$		−	9.99579i		−	0.493056i
$412$		0			0
$413$			2.30157i			0.113253i
$414$		0			0
$415$		−	3.46309i		−	0.169996i
$416$		0			0
$417$	12.0697			0.591054
$418$		0			0
$419$		−	12.1170i		−	0.591956i	−0.955195	−	0.295978i	$-0.904354\pi$
$419$		−	12.1170i		−	0.591956i	0.955195	−	0.295978i	$-0.0956455\pi$
$420$		0			0
$421$	−27.0793			−1.31976			−0.659882	−	0.751369i	$-0.729394\pi$
$421$	−27.0793			−1.31976			−0.659882	+	0.751369i	$0.729394\pi$
$422$		0			0
$423$	−23.0585			−1.12114
$424$		0			0
$425$	12.9203	+	13.7004i	0.626729	+	0.664569i
$426$		0			0
$427$	25.4637			1.23228
$428$		0			0
$429$	−18.1196			−0.874820
$430$		0			0
$431$		−	13.2262i		−	0.637085i	−0.947909	−	0.318542i	$-0.896807\pi$
$431$		−	13.2262i		−	0.637085i	0.947909	−	0.318542i	$-0.103193\pi$
$432$		0			0
$433$	6.32303			0.303865			0.151933	−	0.988391i	$-0.451450\pi$
$433$	6.32303			0.303865			0.151933	+	0.988391i	$0.451450\pi$
$434$		0			0
$435$			1.36042i			0.0652273i
$436$		0			0
$437$			7.53863i			0.360622i
$438$		0			0
$439$			29.6986i			1.41744i	0.705490	+	0.708720i	$0.250727\pi$
$439$			29.6986i			1.41744i	−0.705490	+	0.708720i	$0.749273\pi$
$440$		0			0
$441$	4.44828			0.211823
$442$		0			0
$443$	−27.6019			−1.31141			−0.655704	−	0.755018i	$-0.727628\pi$
$443$	−27.6019			−1.31141			−0.655704	+	0.755018i	$0.727628\pi$
$444$		0			0
$445$		−	7.01753i		−	0.332663i
$446$		0			0
$447$		−	2.08926i		−	0.0988186i
$448$		0			0
$449$			22.3044i			1.05261i	0.850297	+	0.526304i	$0.176423\pi$
$449$			22.3044i			1.05261i	−0.850297	+	0.526304i	$0.823577\pi$
$450$		0			0
$451$	29.2603			1.37782
$452$		0			0
$453$			1.60574i			0.0754443i
$454$		0			0
$455$	8.89739			0.417116
$456$		0			0
$457$	−13.5256			−0.632703			−0.316351	−	0.948642i	$-0.602458\pi$
$457$	−13.5256			−0.632703			−0.316351	+	0.948642i	$0.602458\pi$
$458$		0			0
$459$	10.4818	−	9.88493i	0.489246	−	0.461389i
$460$		0			0
$461$	−15.6525			−0.729008			−0.364504	−	0.931202i	$-0.618761\pi$
$461$	−15.6525			−0.729008			−0.364504	+	0.931202i	$0.618761\pi$
$462$		0			0
$463$	20.0063			0.929770			0.464885	−	0.885371i	$-0.346096\pi$
$463$	20.0063			0.929770			0.464885	+	0.885371i	$0.346096\pi$
$464$		0			0
$465$		−	1.18994i		−	0.0551823i
$466$		0			0
$467$	18.9503			0.876917			0.438458	−	0.898751i	$-0.355525\pi$
$467$	18.9503			0.876917			0.438458	+	0.898751i	$0.355525\pi$
$468$		0			0
$469$		−	13.1894i		−	0.609029i
$470$		0			0
$471$		−	1.25604i		−	0.0578755i
$472$		0			0
$473$		−	49.3253i		−	2.26798i
$474$		0			0
$475$	7.57565			0.347595
$476$		0			0
$477$	−26.7771			−1.22604
$478$		0			0
$479$		−	27.4761i		−	1.25542i	−0.778449	−	0.627708i	$-0.783993\pi$
$479$		−	27.4761i		−	1.25542i	0.778449	−	0.627708i	$-0.216007\pi$
$480$		0			0
$481$		−	0.580340i		−	0.0264612i
$482$		0			0
$483$		−	6.51308i		−	0.296355i
$484$		0			0
$485$	3.32046			0.150774
$486$		0			0
$487$			34.6896i			1.57194i	0.618268	+	0.785968i	$0.287835\pi$
$487$			34.6896i			1.57194i	−0.618268	+	0.785968i	$0.712165\pi$
$488$		0			0
$489$	−2.55521			−0.115551
$490$		0			0
$491$	−9.30137			−0.419765			−0.209883	−	0.977727i	$-0.567308\pi$
$491$	−9.30137			−0.419765			−0.209883	+	0.977727i	$0.567308\pi$
$492$		0			0
$493$	−9.96477	+	9.39738i	−0.448791	+	0.423237i
$494$		0			0
$495$	8.50779			0.382397
$496$		0			0
$497$	−7.92590			−0.355525
$498$		0			0
$499$		−	8.15466i		−	0.365053i	−0.983201	−	0.182526i	$-0.941573\pi$
$499$		−	8.15466i		−	0.365053i	0.983201	−	0.182526i	$-0.0584275\pi$
$500$		0			0
$501$	4.74305			0.211904
$502$		0			0
$503$			6.83090i			0.304575i	0.988336	+	0.152287i	$0.0486639\pi$
$503$			6.83090i			0.304575i	−0.988336	+	0.152287i	$0.951336\pi$
$504$		0			0
$505$			8.86619i			0.394540i
$506$		0			0
$507$		−	13.4140i		−	0.595735i
$508$		0			0
$509$	4.24635			0.188216			0.0941081	−	0.995562i	$-0.470000\pi$
$509$	4.24635			0.188216			0.0941081	+	0.995562i	$0.470000\pi$
$510$		0			0
$511$	0.790489			0.0349692
$512$		0			0
$513$		−	5.79588i		−	0.255894i
$514$		0			0
$515$		−	4.66525i		−	0.205575i
$516$		0			0
$517$			43.7056i			1.92217i
$518$		0			0
$519$	−6.67077			−0.292814
$520$		0			0
$521$		−	19.1698i		−	0.839846i	−0.907560	−	0.419923i	$-0.862057\pi$
$521$		−	19.1698i		−	0.839846i	0.907560	−	0.419923i	$-0.137943\pi$
$522$		0			0
$523$	−4.58455			−0.200468			−0.100234	−	0.994964i	$-0.531959\pi$
$523$	−4.58455			−0.200468			−0.100234	+	0.994964i	$0.531959\pi$
$524$		0			0
$525$	−6.54506			−0.285650
$526$		0			0
$527$	8.71605	−	8.21976i	0.379677	−	0.358058i
$528$		0			0
$529$	2.34236			0.101842
$530$		0			0
$531$	−2.61234			−0.113366
$532$		0			0
$533$			34.7322i			1.50442i
$534$		0			0
$535$	−6.07871			−0.262806
$536$		0			0
$537$		−	10.1676i		−	0.438766i
$538$		0			0
$539$		−	8.43137i		−	0.363165i
$540$		0			0
$541$			32.7884i			1.40968i	0.709364	+	0.704842i	$0.248982\pi$
$541$			32.7884i			1.40968i	−0.709364	+	0.704842i	$0.751018\pi$
$542$		0			0
$543$	−5.49926			−0.235996
$544$		0			0
$545$	−0.279454			−0.0119705
$546$		0			0
$547$		−	2.89032i		−	0.123581i	−0.998089	−	0.0617905i	$-0.980319\pi$
$547$		−	2.89032i		−	0.123581i	0.998089	−	0.0617905i	$-0.0196811\pi$
$548$		0			0
$549$			28.9021i			1.23351i
$550$		0			0
$551$			5.51001i			0.234735i
$552$		0			0
$553$	−10.1492			−0.431587
$554$		0			0
$555$		−	0.0404359i		−	0.00171641i
$556$		0			0
$557$	−28.8022			−1.22039			−0.610195	−	0.792251i	$-0.708909\pi$
$557$	−28.8022			−1.22039			−0.610195	+	0.792251i	$0.708909\pi$
$558$		0			0
$559$	58.5494			2.47637
$560$		0			0
$561$	−8.72098	−	9.24752i	−0.368200	−	0.390431i
$562$		0			0
$563$	−31.6437			−1.33362			−0.666811	−	0.745227i	$-0.732341\pi$
$563$	−31.6437			−1.33362			−0.666811	+	0.745227i	$0.732341\pi$
$564$		0			0
$565$	8.96159			0.377017
$566$		0			0
$567$		−	13.0300i		−	0.547210i
$568$		0			0
$569$	16.6688			0.698793			0.349397	−	0.936975i	$-0.386387\pi$
$569$	16.6688			0.698793			0.349397	+	0.936975i	$0.386387\pi$
$570$		0			0
$571$			3.44631i			0.144223i	0.997397	+	0.0721117i	$0.0229738\pi$
$571$			3.44631i			0.144223i	−0.997397	+	0.0721117i	$0.977026\pi$
$572$		0			0
$573$			6.83286i			0.285447i
$574$		0			0
$575$			20.7591i			0.865713i
$576$		0			0
$577$	−0.125018			−0.00520459			−0.00260229	−	0.999997i	$-0.500828\pi$
$577$	−0.125018			−0.00520459			−0.00260229	+	0.999997i	$0.500828\pi$
$578$		0			0
$579$	5.50052			0.228594
$580$		0			0
$581$			12.1182i			0.502746i
$582$		0			0
$583$			50.7538i			2.10201i
$584$		0			0
$585$			10.0988i			0.417534i
$586$		0			0
$587$	−8.81207			−0.363713			−0.181857	−	0.983325i	$-0.558211\pi$
$587$	−8.81207			−0.363713			−0.181857	+	0.983325i	$0.558211\pi$
$588$		0			0
$589$		−	4.81953i		−	0.198585i
$590$		0			0
$591$	4.41758			0.181715
$592$		0			0
$593$	0.711914			0.0292348			0.0146174	−	0.999893i	$-0.495347\pi$
$593$	0.711914			0.0292348			0.0146174	+	0.999893i	$0.495347\pi$
$594$		0			0
$595$	4.28233	+	4.54089i	0.175558	+	0.186158i
$596$		0			0
$597$	−7.44143			−0.304558
$598$		0			0
$599$	−19.0642			−0.778941			−0.389470	−	0.921039i	$-0.627342\pi$
$599$	−19.0642			−0.778941			−0.389470	+	0.921039i	$0.627342\pi$
$600$		0			0
$601$			45.0789i			1.83881i	0.393314	+	0.919404i	$0.371328\pi$
$601$			45.0789i			1.83881i	−0.393314	+	0.919404i	$0.628672\pi$
$602$		0			0
$603$	14.9703			0.609639
$604$		0			0
$605$		−	8.89077i		−	0.361461i
$606$		0			0
$607$		−	3.25816i		−	0.132245i	−0.997812	−	0.0661223i	$-0.978937\pi$
$607$		−	3.25816i		−	0.132245i	0.997812	−	0.0661223i	$-0.0210628\pi$
$608$		0			0
$609$		−	4.76044i		−	0.192903i
$610$		0			0
$611$	−51.8788			−2.09879
$612$		0			0
$613$	−5.91658			−0.238968			−0.119484	−	0.992836i	$-0.538124\pi$
$613$	−5.91658			−0.238968			−0.119484	+	0.992836i	$0.538124\pi$
$614$		0			0
$615$			2.42001i			0.0975841i
$616$		0			0
$617$		−	26.0364i		−	1.04818i	−0.851662	−	0.524092i	$-0.824405\pi$
$617$		−	26.0364i		−	1.04818i	0.851662	−	0.524092i	$-0.175595\pi$
$618$		0			0
$619$			1.53120i			0.0615441i	0.999526	+	0.0307721i	$0.00979660\pi$
$619$			1.53120i			0.0615441i	−0.999526	+	0.0307721i	$0.990203\pi$
$620$		0			0
$621$	15.8821			0.637326
$622$		0			0
$623$			24.5560i			0.983814i
$624$		0			0
$625$	18.6979			0.747918
$626$		0			0
$627$	−5.11341			−0.204210
$628$		0			0
$629$	0.296183	−	0.279319i	0.0118096	−	0.0111372i
$630$		0			0
$631$	−9.09233			−0.361960			−0.180980	−	0.983487i	$-0.557927\pi$
$631$	−9.09233			−0.361960			−0.180980	+	0.983487i	$0.557927\pi$
$632$		0			0
$633$	9.42678			0.374681
$634$		0			0
$635$		−	4.57844i		−	0.181690i
$636$		0			0
$637$	10.0081			0.396535
$638$		0			0
$639$		−	8.99613i		−	0.355881i
$640$		0			0
$641$			24.9863i			0.986898i	0.869775	+	0.493449i	$0.164264\pi$
$641$			24.9863i			0.986898i	−0.869775	+	0.493449i	$0.835736\pi$
$642$		0			0
$643$		−	31.9116i		−	1.25847i	−0.777214	−	0.629236i	$-0.783368\pi$
$643$		−	31.9116i		−	1.25847i	0.777214	−	0.629236i	$-0.216632\pi$
$644$		0			0
$645$	4.07950			0.160630
$646$		0			0
$647$	−45.4646			−1.78740			−0.893700	−	0.448666i	$-0.851899\pi$
$647$	−45.4646			−1.78740			−0.893700	+	0.448666i	$0.851899\pi$
$648$		0			0
$649$			4.95149i			0.194363i
$650$		0			0
$651$			4.16389i			0.163196i
$652$		0			0
$653$			15.0004i			0.587010i	0.955958	+	0.293505i	$0.0948217\pi$
$653$			15.0004i			0.587010i	−0.955958	+	0.293505i	$0.905178\pi$
$654$		0			0
$655$	9.87655			0.385909
$656$		0			0
$657$			0.897228i			0.0350042i
$658$		0			0
$659$	17.7916			0.693064			0.346532	−	0.938038i	$-0.387359\pi$
$659$	17.7916			0.693064			0.346532	+	0.938038i	$0.387359\pi$
$660$		0			0
$661$	−23.6194			−0.918687			−0.459344	−	0.888259i	$-0.651915\pi$
$661$	−23.6194			−0.918687			−0.459344	+	0.888259i	$0.651915\pi$
$662$		0			0
$663$	10.9769	−	10.3518i	0.426306	−	0.402032i
$664$		0			0
$665$	2.51088			0.0973678
$666$		0			0
$667$	−15.0987			−0.584626
$668$		0			0
$669$		−	9.47215i		−	0.366215i
$670$		0			0
$671$	54.7816			2.11482
$672$		0			0
$673$		−	5.45680i		−	0.210344i	−0.994454	−	0.105172i	$-0.966461\pi$
$673$		−	5.45680i		−	0.210344i	0.994454	−	0.105172i	$-0.0335394\pi$
$674$		0			0
$675$		−	15.9601i		−	0.614303i
$676$		0			0
$677$			36.9448i			1.41990i	0.704251	+	0.709951i	$0.251283\pi$
$677$			36.9448i			1.41990i	−0.704251	+	0.709951i	$0.748717\pi$
$678$		0			0
$679$	−11.6191			−0.445898
$680$		0			0
$681$	15.8199			0.606218
$682$		0			0
$683$			0.562151i			0.0215101i	0.999942	+	0.0107550i	$0.00342350\pi$
$683$			0.562151i			0.0215101i	−0.999942	+	0.0107550i	$0.996576\pi$
$684$		0			0
$685$			10.5595i			0.403459i
$686$		0			0
$687$			1.34019i			0.0511313i
$688$		0			0
$689$	−60.2450			−2.29515
$690$		0			0
$691$		−	24.5464i		−	0.933788i	−0.884313	−	0.466894i	$-0.845373\pi$
$691$		−	24.5464i		−	0.933788i	0.884313	−	0.466894i	$-0.154627\pi$
$692$		0			0
$693$	−29.7708			−1.13090
$694$		0			0
$695$	−12.7504			−0.483649
$696$		0			0
$697$	−17.7260	+	16.7167i	−0.671419	+	0.633189i
$698$		0			0
$699$	−11.8104			−0.446710
$700$		0			0
$701$	−29.7605			−1.12404			−0.562018	−	0.827125i	$-0.689975\pi$
$701$	−29.7605			−1.12404			−0.562018	+	0.827125i	$0.689975\pi$
$702$		0			0
$703$		−	0.163774i		−	0.00617687i
$704$		0			0
$705$	−3.61472			−0.136138
$706$		0			0
$707$		−	31.0249i		−	1.16681i
$708$		0			0
$709$		−	11.0676i		−	0.415651i	−0.978166	−	0.207826i	$-0.933361\pi$
$709$		−	11.0676i		−	0.415651i	0.978166	−	0.207826i	$-0.0666387\pi$
$710$		0			0
$711$		−	11.5196i		−	0.432020i
$712$		0			0
$713$	13.2067			0.494594
$714$		0			0
$715$	19.1415			0.715850
$716$		0			0
$717$			8.17541i			0.305316i
$718$		0			0
$719$		−	34.7480i		−	1.29588i	−0.761691	−	0.647940i	$-0.775631\pi$
$719$		−	34.7480i		−	1.29588i	0.761691	−	0.647940i	$-0.224369\pi$
$720$		0			0
$721$			16.3248i			0.607967i
$722$		0			0
$723$	−8.77413			−0.326314
$724$		0			0
$725$			15.1729i			0.563507i
$726$		0			0
$727$	31.4537			1.16655			0.583277	−	0.812273i	$-0.301770\pi$
$727$	31.4537			1.16655			0.583277	+	0.812273i	$0.301770\pi$
$728$		0			0
$729$	8.26252			0.306019
$730$		0			0
$731$	28.1799	+	29.8814i	1.04227	+	1.10520i
$732$		0			0
$733$	−38.7153			−1.42998			−0.714992	−	0.699133i	$-0.753569\pi$
$733$	−38.7153			−1.42998			−0.714992	+	0.699133i	$0.753569\pi$
$734$		0			0
$735$	0.697325			0.0257212
$736$		0			0
$737$		−	28.3751i		−	1.04521i
$738$		0			0
$739$	45.8338			1.68602			0.843012	−	0.537894i	$-0.180780\pi$
$739$	45.8338			1.68602			0.843012	+	0.537894i	$0.180780\pi$
$740$		0			0
$741$		−	6.06965i		−	0.222974i
$742$		0			0
$743$			39.9475i			1.46553i	0.680481	+	0.732765i	$0.261771\pi$
$743$			39.9475i			1.46553i	−0.680481	+	0.732765i	$0.738229\pi$
$744$		0			0
$745$			2.20709i			0.0808615i
$746$		0			0
$747$	−13.7545			−0.503250
$748$		0			0
$749$	21.2708			0.777219
$750$		0			0
$751$			35.6803i			1.30199i	0.759081	+	0.650996i	$0.225649\pi$
$751$			35.6803i			1.30199i	−0.759081	+	0.650996i	$0.774351\pi$
$752$		0			0
$753$			10.3642i			0.377691i
$754$		0			0
$755$		−	1.69630i		−	0.0617348i
$756$		0			0
$757$	16.6653			0.605711			0.302856	−	0.953036i	$-0.402060\pi$
$757$	16.6653			0.605711			0.302856	+	0.953036i	$0.402060\pi$
$758$		0			0
$759$		−	14.0120i		−	0.508602i
$760$		0			0
$761$	23.8880			0.865940			0.432970	−	0.901408i	$-0.357466\pi$
$761$	23.8880			0.865940			0.432970	+	0.901408i	$0.357466\pi$
$762$		0			0
$763$	0.977877			0.0354015
$764$		0			0
$765$	−5.15404	+	4.86057i	−0.186345	+	0.175734i
$766$		0			0
$767$	−5.87745			−0.212222
$768$		0			0
$769$	−14.8308			−0.534811			−0.267405	−	0.963584i	$-0.586166\pi$
$769$	−14.8308			−0.534811			−0.267405	+	0.963584i	$0.586166\pi$
$770$		0			0
$771$			14.8316i			0.534147i
$772$		0			0
$773$	41.8106			1.50382			0.751912	−	0.659263i	$-0.229132\pi$
$773$	41.8106			1.50382			0.751912	+	0.659263i	$0.229132\pi$
$774$		0			0
$775$		−	13.2715i		−	0.476727i
$776$		0			0
$777$			0.141495i			0.00507609i
$778$		0			0
$779$			9.80157i			0.351178i
$780$		0			0
$781$	−17.0514			−0.610148
$782$		0			0
$783$	11.6083			0.414846
$784$		0			0
$785$			1.32688i			0.0473585i
$786$		0			0
$787$			13.4131i			0.478127i	0.971004	+	0.239063i	$0.0768404\pi$
$787$			13.4131i			0.478127i	−0.971004	+	0.239063i	$0.923160\pi$
$788$		0			0
$789$			5.19624i			0.184991i
$790$		0			0
$791$	−31.3587			−1.11499
$792$		0			0
$793$			65.0260i			2.30914i
$794$		0			0
$795$	−4.19764			−0.148875
$796$		0			0
$797$	−11.3450			−0.401862			−0.200931	−	0.979605i	$-0.564397\pi$
$797$	−11.3450			−0.401862			−0.200931	+	0.979605i	$0.564397\pi$
$798$		0			0
$799$	−24.9694	−	26.4770i	−0.883353	−	0.936687i
$800$		0			0
$801$	−27.8717			−0.984800
$802$		0			0
$803$	1.70062			0.0600137
$804$		0			0
$805$			6.88041i			0.242503i
$806$		0			0
$807$	−2.23998			−0.0788511
$808$		0			0
$809$			36.5945i			1.28660i	0.765616	+	0.643298i	$0.222434\pi$
$809$			36.5945i			1.28660i	−0.765616	+	0.643298i	$0.777566\pi$
$810$		0			0
$811$		−	54.1462i		−	1.90133i	−0.310218	−	0.950665i	$-0.600402\pi$
$811$		−	54.1462i		−	1.90133i	0.310218	−	0.950665i	$-0.399598\pi$
$812$		0			0
$813$		−	5.65028i		−	0.198164i
$814$		0			0
$815$	2.69932			0.0945531
$816$		0			0
$817$	16.5229			0.578062
$818$		0			0
$819$		−	35.3381i		−	1.23481i
$820$		0			0
$821$		−	41.1270i		−	1.43534i	−0.696383	−	0.717671i	$-0.745208\pi$
$821$		−	41.1270i		−	1.43534i	0.696383	−	0.717671i	$-0.254792\pi$
$822$		0			0
$823$		−	36.2394i		−	1.26323i	−0.775283	−	0.631614i	$-0.782393\pi$
$823$		−	36.2394i		−	1.26323i	0.775283	−	0.631614i	$-0.217607\pi$
$824$		0			0
$825$	−14.0808			−0.490230
$826$		0			0
$827$			20.7157i			0.720354i	0.932884	+	0.360177i	$0.117284\pi$
$827$			20.7157i			0.720354i	−0.932884	+	0.360177i	$0.882716\pi$
$828$		0			0
$829$	−9.94825			−0.345517			−0.172759	−	0.984964i	$-0.555268\pi$
$829$	−9.94825			−0.345517			−0.172759	+	0.984964i	$0.555268\pi$
$830$		0			0
$831$	−9.21849			−0.319786
$832$		0			0
$833$	4.81691	+	5.10774i	0.166896	+	0.176973i
$834$		0			0
$835$	−5.01055			−0.173397
$836$		0			0
$837$	−10.1536			−0.350960
$838$		0			0
$839$			18.7456i			0.647169i	0.946199	+	0.323584i	$0.104888\pi$
$839$			18.7456i			0.647169i	−0.946199	+	0.323584i	$0.895112\pi$
$840$		0			0
$841$	17.9643			0.619458
$842$		0			0
$843$		−	13.1869i		−	0.454183i
$844$		0			0
$845$			14.1705i			0.487479i
$846$		0			0
$847$			31.1109i			1.06898i
$848$		0			0
$849$	−7.86177			−0.269815
$850$		0			0
$851$	0.448781			0.0153840
$852$		0			0
$853$			8.72532i			0.298749i	0.988781	+	0.149375i	$0.0477261\pi$
$853$			8.72532i			0.298749i	−0.988781	+	0.149375i	$0.952274\pi$
$854$		0			0
$855$			2.84992i			0.0974653i
$856$		0			0
$857$			46.4202i			1.58569i	0.609426	+	0.792843i	$0.291400\pi$
$857$			46.4202i			1.58569i	−0.609426	+	0.792843i	$0.708600\pi$
$858$		0			0
$859$	−55.2285			−1.88437			−0.942186	−	0.335091i	$-0.891233\pi$
$859$	−55.2285			−1.88437			−0.942186	+	0.335091i	$0.891233\pi$
$860$		0			0
$861$		−	8.46817i		−	0.288594i
$862$		0			0
$863$	23.9573			0.815515			0.407757	−	0.913090i	$-0.366311\pi$
$863$	23.9573			0.815515			0.407757	+	0.913090i	$0.366311\pi$
$864$		0			0
$865$	7.04699			0.239605
$866$		0			0
$867$	10.5664	+	0.619807i	0.358853	+	0.0210497i
$868$		0			0
$869$	−21.8345			−0.740685
$870$		0			0
$871$	33.6814			1.14125
$872$		0			0
$873$		−	13.1880i		−	0.446345i
$874$		0			0
$875$	14.4833			0.489624
$876$		0			0
$877$			21.4779i			0.725257i	0.931934	+	0.362629i	$0.118121\pi$
$877$			21.4779i			0.725257i	−0.931934	+	0.362629i	$0.881879\pi$
$878$		0			0
$879$			6.09826i			0.205689i
$880$		0			0
$881$			52.4116i			1.76579i	0.469568	+	0.882896i	$0.344410\pi$
$881$			52.4116i			1.76579i	−0.469568	+	0.882896i	$0.655590\pi$
$882$		0			0
$883$	−39.3039			−1.32268			−0.661340	−	0.750086i	$-0.730012\pi$
$883$	−39.3039			−1.32268			−0.661340	+	0.750086i	$0.730012\pi$
$884$		0			0
$885$	−0.409518			−0.0137658
$886$		0			0
$887$			12.1989i			0.409598i	0.978804	+	0.204799i	$0.0656541\pi$
$887$			12.1989i			0.409598i	−0.978804	+	0.204799i	$0.934346\pi$
$888$		0			0
$889$			16.0211i			0.537329i
$890$		0			0
$891$		−	28.0323i		−	0.939117i
$892$		0			0
$893$	−14.6404			−0.489923
$894$		0			0
$895$			10.7411i			0.359034i
$896$		0			0
$897$	16.6323			0.555336
$898$		0			0
$899$	9.65280			0.321939
$900$		0			0
$901$	−28.9960	−	30.7467i	−0.965998	−	1.02432i
$902$		0			0
$903$	−14.2751			−0.475046
$904$		0			0
$905$	5.80941			0.193111
$906$		0			0
$907$			14.1687i			0.470463i	0.971939	+	0.235231i	$0.0755847\pi$
$907$			14.1687i			0.470463i	−0.971939	+	0.235231i	$0.924415\pi$
$908$		0			0
$909$	35.2141			1.16798
$910$		0			0
$911$			26.0637i			0.863530i	0.901986	+	0.431765i	$0.142109\pi$
$911$			26.0637i			0.863530i	−0.901986	+	0.431765i	$0.857891\pi$
$912$		0			0
$913$			26.0705i			0.862808i
$914$		0			0
$915$			4.53077i			0.149782i
$916$		0			0
$917$	−34.5604			−1.14128
$918$		0			0
$919$	2.85489			0.0941740			0.0470870	−	0.998891i	$-0.485006\pi$
$919$	2.85489			0.0941740			0.0470870	+	0.998891i	$0.485006\pi$
$920$		0			0
$921$			7.67158i			0.252787i
$922$		0			0
$923$		−	20.2401i		−	0.666213i
$924$		0			0
$925$		−	0.450984i		−	0.0148283i
$926$		0			0
$927$	−18.5291			−0.608576
$928$		0			0
$929$		−	44.2608i		−	1.45215i	−0.687615	−	0.726076i	$-0.741342\pi$
$929$		−	44.2608i		−	1.45215i	0.687615	−	0.726076i	$-0.258658\pi$
$930$		0			0
$931$	2.82432			0.0925634
$932$		0			0
$933$	−13.9203			−0.455731
$934$		0			0
$935$	9.21282	+	9.76907i	0.301292	+	0.319483i
$936$		0			0
$937$	−18.7968			−0.614064			−0.307032	−	0.951699i	$-0.599336\pi$
$937$	−18.7968			−0.614064			−0.307032	+	0.951699i	$0.599336\pi$
$938$		0			0
$939$	10.0016			0.326391
$940$		0			0
$941$		−	39.1558i		−	1.27644i	−0.769852	−	0.638222i	$-0.779670\pi$
$941$		−	39.1558i		−	1.27644i	0.769852	−	0.638222i	$-0.220330\pi$
$942$		0			0
$943$	−26.8586			−0.874637
$944$		0			0
$945$		−	5.28982i		−	0.172078i
$946$		0			0
$947$			53.9734i			1.75390i	0.480583	+	0.876949i	$0.340425\pi$
$947$			53.9734i			1.75390i	−0.480583	+	0.876949i	$0.659575\pi$
$948$		0			0
$949$			2.01865i			0.0655281i
$950$		0			0
$951$	−13.4684			−0.436743
$952$		0			0
$953$	−15.0512			−0.487558			−0.243779	−	0.969831i	$-0.578387\pi$
$953$	−15.0512			−0.487558			−0.243779	+	0.969831i	$0.578387\pi$
$954$		0			0
$955$		−	7.21822i		−	0.233576i
$956$		0			0
$957$		−	10.2414i		−	0.331057i
$958$		0			0
$959$		−	36.9503i		−	1.19319i
$960$		0			0
$961$	22.5568			0.727640
$962$		0			0
$963$			24.1430i			0.777998i
$964$		0			0
$965$	−5.81074			−0.187054
$966$		0			0
$967$	50.7473			1.63192			0.815961	−	0.578106i	$-0.196208\pi$
$967$	50.7473			1.63192			0.815961	+	0.578106i	$0.196208\pi$
$968$		0			0
$969$	3.09772	−	2.92133i	0.0995130	−	0.0938468i
$970$		0			0
$971$	19.8599			0.637334			0.318667	−	0.947867i	$-0.396765\pi$
$971$	19.8599			0.637334			0.318667	+	0.947867i	$0.396765\pi$
$972$		0			0
$973$	44.6165			1.43034
$974$		0			0
$975$		−	16.7140i		−	0.535275i
$976$		0			0
$977$	5.54495			0.177399			0.0886994	−	0.996058i	$-0.471729\pi$
$977$	5.54495			0.177399			0.0886994	+	0.996058i	$0.471729\pi$
$978$		0			0
$979$			52.8287i			1.68841i
$980$		0			0
$981$			1.10992i			0.0354370i
$982$		0			0
$983$			34.3514i			1.09564i	0.836596	+	0.547820i	$0.184542\pi$
$983$			34.3514i			1.09564i	−0.836596	+	0.547820i	$0.815458\pi$
$984$		0			0
$985$	−4.66672			−0.148694
$986$		0			0
$987$	12.6487			0.402614
$988$		0			0
$989$			45.2766i			1.43971i
$990$		0			0
$991$			20.5093i			0.651499i	0.945456	+	0.325750i	$0.105617\pi$
$991$			20.5093i			0.651499i	−0.945456	+	0.325750i	$0.894383\pi$
$992$		0			0
$993$			14.4672i			0.459103i
$994$		0			0
$995$	7.86112			0.249214
$996$		0			0
$997$			28.5019i			0.902663i	0.892356	+	0.451332i	$0.149051\pi$
$997$			28.5019i			0.902663i	−0.892356	+	0.451332i	$0.850949\pi$
$998$		0			0
$999$	−0.345033			−0.0109164

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.a.237.16	✓	40
17.16	even	2	inner	4012.2.b.a.237.25	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.a.237.16	✓	40	1.1	even	1	trivial
4012.2.b.a.237.25	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)

\(f(q)\)	\(=\)	\(q-0.622620i q^{3} +0.657734i q^{5} -2.30157i q^{7} +2.61234 q^{9} +O(q^{10})\)	\(q-0.622620i q^{3} +0.657734i q^{5} -2.30157i q^{7} +2.61234 q^{9} -4.95149i q^{11} +5.87745 q^{13} +0.409518 q^{15} +(2.82883 + 2.99962i) q^{17} +1.65864 q^{19} -1.43300 q^{21} +4.54507i q^{23} +4.56739 q^{25} -3.49436i q^{27} +3.32201i q^{29} -2.90571i q^{31} -3.08290 q^{33} +1.51382 q^{35} -0.0987401i q^{37} -3.65941i q^{39} +5.90940i q^{41} +9.96170 q^{43} +1.71823i q^{45} -8.82676 q^{47} +1.70279 q^{49} +(1.86762 - 1.76128i) q^{51} -10.2502 q^{53} +3.25677 q^{55} -1.03270i q^{57} -1.00000 q^{59} +11.0636i q^{61} -6.01248i q^{63} +3.86580i q^{65} +5.73061 q^{67} +2.82985 q^{69} -3.44370i q^{71} +0.343457i q^{73} -2.84374i q^{75} -11.3962 q^{77} -4.40969i q^{79} +5.66138 q^{81} -5.26518 q^{83} +(-1.97296 + 1.86062i) q^{85} +2.06835 q^{87} -10.6692 q^{89} -13.5273i q^{91} -1.80915 q^{93} +1.09094i q^{95} -5.04833i q^{97} -12.9350i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 36 q^{9}+O(q^{10}) \) 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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