LMFDB - Embedded newform 4012.2.b.b.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:	$46$
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.b.237.31

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.43806i q^{3} +2.66437i q^{5} +3.93282i q^{7} +0.931993 q^{9} +O(q^{10})$	$q-1.43806i q^{3} +2.66437i q^{5} +3.93282i q^{7} +0.931993 q^{9} -4.93836i q^{11} +5.59880 q^{13} +3.83152 q^{15} +(-2.22849 - 3.46898i) q^{17} -6.57098 q^{19} +5.65561 q^{21} -6.13081i q^{23} -2.09887 q^{25} -5.65443i q^{27} +1.64457i q^{29} -9.54938i q^{31} -7.10165 q^{33} -10.4785 q^{35} -6.71951i q^{37} -8.05138i q^{39} +9.66660i q^{41} +9.17449 q^{43} +2.48318i q^{45} -8.93156 q^{47} -8.46706 q^{49} +(-4.98859 + 3.20470i) q^{51} +5.74193 q^{53} +13.1576 q^{55} +9.44944i q^{57} +1.00000 q^{59} -11.2509i q^{61} +3.66536i q^{63} +14.9173i q^{65} -0.829919 q^{67} -8.81645 q^{69} +5.67113i q^{71} -5.67456i q^{73} +3.01829i q^{75} +19.4217 q^{77} -2.84110i q^{79} -5.33541 q^{81} +4.48971 q^{83} +(9.24265 - 5.93753i) q^{85} +2.36499 q^{87} +14.0734 q^{89} +22.0190i q^{91} -13.7326 q^{93} -17.5075i q^{95} -11.1433i q^{97} -4.60252i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 54 q^{9}+O(q^{10}) $ 46 * q - 54 * q^9	$ 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) $ 46 * q - 54 * q^9 + 8 * q^13 - 10 * q^15 + q^17 - 20 * q^19 - 24 * q^21 - 54 * q^25 + 2 * q^33 + 26 * q^35 - 38 * q^43 + 6 * q^47 - 66 * q^49 + 26 * q^51 + 18 * q^53 - 20 * q^55 + 46 * q^59 + 48 * q^67 + 28 * q^69 + 22 * q^77 + 70 * q^81 - 52 * q^83 - 2 * q^85 + 44 * q^87 - 76 * q^89 - 26 * q^93

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	1.43806i		−	0.830262i	−0.909762	−	0.415131i	$-0.863736\pi$
$3$		−	1.43806i		−	0.830262i	0.909762	−	0.415131i	$-0.136264\pi$
$4$		0			0
$5$			2.66437i			1.19154i	0.803154	+	0.595771i	$0.203154\pi$
$5$			2.66437i			1.19154i	−0.803154	+	0.595771i	$0.796846\pi$
$6$		0			0
$7$			3.93282i			1.48647i	0.669033	+	0.743233i	$0.266709\pi$
$7$			3.93282i			1.48647i	−0.669033	+	0.743233i	$0.733291\pi$
$8$		0			0
$9$	0.931993			0.310664
$10$		0			0
$11$		−	4.93836i		−	1.48897i	−0.667638	−	0.744486i	$-0.732694\pi$
$11$		−	4.93836i		−	1.48897i	0.667638	−	0.744486i	$-0.267306\pi$
$12$		0			0
$13$	5.59880			1.55283			0.776413	−	0.630224i	$-0.217037\pi$
$13$	5.59880			1.55283			0.776413	+	0.630224i	$0.217037\pi$
$14$		0			0
$15$	3.83152			0.989293
$16$		0			0
$17$	−2.22849	−	3.46898i	−0.540489	−	0.841351i
$17$	−2.22849	−	3.46898i	−0.540489	−	0.841351i
$18$		0			0
$19$	−6.57098			−1.50749			−0.753743	−	0.657169i	$-0.771754\pi$
$19$	−6.57098			−1.50749			−0.753743	+	0.657169i	$0.771754\pi$
$20$		0			0
$21$	5.65561			1.23416
$22$		0			0
$23$		−	6.13081i		−	1.27836i	−0.769056	−	0.639181i	$-0.779273\pi$
$23$		−	6.13081i		−	1.27836i	0.769056	−	0.639181i	$-0.220727\pi$
$24$		0			0
$25$	−2.09887			−0.419774
$26$		0			0
$27$		−	5.65443i		−	1.08820i
$28$		0			0
$29$			1.64457i			0.305389i	0.988273	+	0.152695i	$0.0487951\pi$
$29$			1.64457i			0.305389i	−0.988273	+	0.152695i	$0.951205\pi$
$30$		0			0
$31$		−	9.54938i		−	1.71512i	−0.514384	−	0.857560i	$-0.671980\pi$
$31$		−	9.54938i		−	1.71512i	0.514384	−	0.857560i	$-0.328020\pi$
$32$		0			0
$33$	−7.10165			−1.23624
$34$		0			0
$35$	−10.4785			−1.77119
$36$		0			0
$37$		−	6.71951i		−	1.10468i	−0.833619	−	0.552341i	$-0.813735\pi$
$37$		−	6.71951i		−	1.10468i	0.833619	−	0.552341i	$-0.186265\pi$
$38$		0			0
$39$		−	8.05138i		−	1.28925i
$40$		0			0
$41$			9.66660i			1.50967i	0.655915	+	0.754835i	$0.272283\pi$
$41$			9.66660i			1.50967i	−0.655915	+	0.754835i	$0.727717\pi$
$42$		0			0
$43$	9.17449			1.39910			0.699548	−	0.714585i	$-0.253385\pi$
$43$	9.17449			1.39910			0.699548	+	0.714585i	$0.253385\pi$
$44$		0			0
$45$			2.48318i			0.370170i
$46$		0			0
$47$	−8.93156			−1.30280			−0.651401	−	0.758734i	$-0.725818\pi$
$47$	−8.93156			−1.30280			−0.651401	+	0.758734i	$0.725818\pi$
$48$		0			0
$49$	−8.46706			−1.20958
$50$		0			0
$51$	−4.98859	+	3.20470i	−0.698542	+	0.448748i
$52$		0			0
$53$	5.74193			0.788715			0.394357	−	0.918957i	$-0.370967\pi$
$53$	5.74193			0.788715			0.394357	+	0.918957i	$0.370967\pi$
$54$		0			0
$55$	13.1576			1.77417
$56$		0			0
$57$			9.44944i			1.25161i
$58$		0			0
$59$	1.00000			0.130189
$59$	1.00000			0.130189
$60$		0			0
$61$		−	11.2509i		−	1.44054i	−0.693696	−	0.720268i	$-0.744019\pi$
$61$		−	11.2509i		−	1.44054i	0.693696	−	0.720268i	$-0.255981\pi$
$62$		0			0
$63$			3.66536i			0.461792i
$64$		0			0
$65$			14.9173i			1.85026i
$66$		0			0
$67$	−0.829919			−0.101391			−0.0506953	−	0.998714i	$-0.516144\pi$
$67$	−0.829919			−0.101391			−0.0506953	+	0.998714i	$0.516144\pi$
$68$		0			0
$69$	−8.81645			−1.06138
$70$		0			0
$71$			5.67113i			0.673039i	0.941677	+	0.336519i	$0.109250\pi$
$71$			5.67113i			0.673039i	−0.941677	+	0.336519i	$0.890750\pi$
$72$		0			0
$73$		−	5.67456i		−	0.664157i	−0.943252	−	0.332079i	$-0.892250\pi$
$73$		−	5.67456i		−	0.664157i	0.943252	−	0.332079i	$-0.107750\pi$
$74$		0			0
$75$			3.01829i			0.348523i
$76$		0			0
$77$	19.4217			2.21331
$78$		0			0
$79$		−	2.84110i		−	0.319649i	−0.987145	−	0.159824i	$-0.948907\pi$
$79$		−	2.84110i		−	0.319649i	0.987145	−	0.159824i	$-0.0510928\pi$
$80$		0			0
$81$	−5.33541			−0.592823
$82$		0			0
$83$	4.48971			0.492810			0.246405	−	0.969167i	$-0.420751\pi$
$83$	4.48971			0.492810			0.246405	+	0.969167i	$0.420751\pi$
$84$		0			0
$85$	9.24265	−	5.93753i	1.00251	−	0.644016i
$86$		0			0
$87$	2.36499			0.253553
$88$		0			0
$89$	14.0734			1.49177			0.745887	−	0.666073i	$-0.232026\pi$
$89$	14.0734			1.49177			0.745887	+	0.666073i	$0.232026\pi$
$90$		0			0
$91$			22.0190i			2.30822i
$92$		0			0
$93$	−13.7326			−1.42400
$94$		0			0
$95$		−	17.5075i		−	1.79623i
$96$		0			0
$97$		−	11.1433i		−	1.13143i	−0.824601	−	0.565715i	$-0.808600\pi$
$97$		−	11.1433i		−	1.13143i	0.824601	−	0.565715i	$-0.191400\pi$
$98$		0			0
$99$		−	4.60252i		−	0.462571i
$100$		0			0
$101$	−3.64143			−0.362336			−0.181168	−	0.983452i	$-0.557988\pi$
$101$	−3.64143			−0.362336			−0.181168	+	0.983452i	$0.557988\pi$
$102$		0			0
$103$	4.72995			0.466056			0.233028	−	0.972470i	$-0.425137\pi$
$103$	4.72995			0.466056			0.233028	+	0.972470i	$0.425137\pi$
$104$		0			0
$105$			15.0687i			1.47055i
$106$		0			0
$107$		−	18.5925i		−	1.79740i	−0.438560	−	0.898702i	$-0.644511\pi$
$107$		−	18.5925i		−	1.79740i	0.438560	−	0.898702i	$-0.355489\pi$
$108$		0			0
$109$		−	0.790027i		−	0.0756708i	−0.999284	−	0.0378354i	$-0.987954\pi$
$109$		−	0.790027i		−	0.0756708i	0.999284	−	0.0378354i	$-0.0120463\pi$
$110$		0			0
$111$	−9.66304			−0.917175
$112$		0			0
$113$			11.5730i			1.08870i	0.838858	+	0.544350i	$0.183224\pi$
$113$			11.5730i			1.08870i	−0.838858	+	0.544350i	$0.816776\pi$
$114$		0			0
$115$	16.3347			1.52322
$116$		0			0
$117$	5.21804			0.482408
$118$		0			0
$119$	13.6429	−	8.76426i	1.25064	−	0.803418i
$120$		0			0
$121$	−13.3874			−1.21704
$122$		0			0
$123$	13.9011			1.25342
$124$		0			0
$125$			7.72968i			0.691364i
$126$		0			0
$127$	5.60393			0.497268			0.248634	−	0.968598i	$-0.420018\pi$
$127$	5.60393			0.497268			0.248634	+	0.968598i	$0.420018\pi$
$128$		0			0
$129$		−	13.1934i		−	1.16162i
$130$		0			0
$131$			20.2437i			1.76870i	0.466825	+	0.884350i	$0.345398\pi$
$131$			20.2437i			1.76870i	−0.466825	+	0.884350i	$0.654602\pi$
$132$		0			0
$133$		−	25.8425i		−	2.24083i
$134$		0			0
$135$	15.0655			1.29663
$136$		0			0
$137$	6.73196			0.575150			0.287575	−	0.957758i	$-0.407151\pi$
$137$	6.73196			0.575150			0.287575	+	0.957758i	$0.407151\pi$
$138$		0			0
$139$			0.142436i			0.0120812i	0.999982	+	0.00604061i	$0.00192280\pi$
$139$			0.142436i			0.0120812i	−0.999982	+	0.00604061i	$0.998077\pi$
$140$		0			0
$141$			12.8441i			1.08167i
$142$		0			0
$143$		−	27.6489i		−	2.31212i
$144$		0			0
$145$	−4.38175			−0.363884
$146$		0			0
$147$			12.1761i			1.00427i
$148$		0			0
$149$	19.5104			1.59836			0.799179	−	0.601093i	$-0.205268\pi$
$149$	19.5104			1.59836			0.799179	+	0.601093i	$0.205268\pi$
$150$		0			0
$151$	−19.0832			−1.55297			−0.776485	−	0.630135i	$-0.782999\pi$
$151$	−19.0832			−1.55297			−0.776485	+	0.630135i	$0.782999\pi$
$152$		0			0
$153$	−2.07694	−	3.23307i	−0.167911	−	0.261378i
$154$		0			0
$155$	25.4431			2.04364
$156$		0			0
$157$	−10.6262			−0.848063			−0.424032	−	0.905647i	$-0.639385\pi$
$157$	−10.6262			−0.848063			−0.424032	+	0.905647i	$0.639385\pi$
$158$		0			0
$159$		−	8.25722i		−	0.654840i
$160$		0			0
$161$	24.1114			1.90024
$162$		0			0
$163$			4.91575i			0.385031i	0.981294	+	0.192516i	$0.0616646\pi$
$163$			4.91575i			0.385031i	−0.981294	+	0.192516i	$0.938335\pi$
$164$		0			0
$165$		−	18.9214i		−	1.47303i
$166$		0			0
$167$			16.4227i			1.27083i	0.772171	+	0.635414i	$0.219171\pi$
$167$			16.4227i			1.27083i	−0.772171	+	0.635414i	$0.780829\pi$
$168$		0			0
$169$	18.3465			1.41127
$170$		0			0
$171$	−6.12411			−0.468322
$172$		0			0
$173$			17.5017i			1.33063i	0.746564	+	0.665313i	$0.231702\pi$
$173$			17.5017i			1.33063i	−0.746564	+	0.665313i	$0.768298\pi$
$174$		0			0
$175$		−	8.25447i		−	0.623980i
$176$		0			0
$177$		−	1.43806i		−	0.108091i
$178$		0			0
$179$	−3.84481			−0.287375			−0.143687	−	0.989623i	$-0.545896\pi$
$179$	−3.84481			−0.287375			−0.143687	+	0.989623i	$0.545896\pi$
$180$		0			0
$181$		−	4.67637i		−	0.347592i	−0.984782	−	0.173796i	$-0.944397\pi$
$181$		−	4.67637i		−	0.347592i	0.984782	−	0.173796i	$-0.0556033\pi$
$182$		0			0
$183$	−16.1795			−1.19602
$184$		0			0
$185$	17.9033			1.31627
$186$		0			0
$187$	−17.1311	+	11.0051i	−1.25275	+	0.804773i
$188$		0			0
$189$	22.2378			1.61756
$190$		0			0
$191$	5.08518			0.367951			0.183975	−	0.982931i	$-0.441103\pi$
$191$	5.08518			0.367951			0.183975	+	0.982931i	$0.441103\pi$
$192$		0			0
$193$		−	0.285281i		−	0.0205349i	−0.999947	−	0.0102675i	$-0.996732\pi$
$193$		−	0.285281i		−	0.0205349i	0.999947	−	0.0102675i	$-0.00326830\pi$
$194$		0			0
$195$	21.4519			1.53620
$196$		0			0
$197$		−	11.9224i		−	0.849434i	−0.905326	−	0.424717i	$-0.860374\pi$
$197$		−	11.9224i		−	0.849434i	0.905326	−	0.424717i	$-0.139626\pi$
$198$		0			0
$199$		−	12.3728i		−	0.877082i	−0.898711	−	0.438541i	$-0.855495\pi$
$199$		−	12.3728i		−	0.877082i	0.898711	−	0.438541i	$-0.144505\pi$
$200$		0			0
$201$			1.19347i			0.0841809i
$202$		0			0
$203$	−6.46780			−0.453950
$204$		0			0
$205$	−25.7554			−1.79884
$206$		0			0
$207$		−	5.71387i		−	0.397142i
$208$		0			0
$209$			32.4499i			2.24461i
$210$		0			0
$211$		−	23.1918i		−	1.59659i	−0.602268	−	0.798294i	$-0.705736\pi$
$211$		−	23.1918i		−	1.59659i	0.602268	−	0.798294i	$-0.294264\pi$
$212$		0			0
$213$	8.15540			0.558799
$214$		0			0
$215$			24.4442i			1.66708i
$216$		0			0
$217$	37.5560			2.54947
$218$		0			0
$219$	−8.16034			−0.551425
$220$		0			0
$221$	−12.4769	−	19.4221i	−0.839285	−	1.30647i
$222$		0			0
$223$	−1.71690			−0.114972			−0.0574862	−	0.998346i	$-0.518309\pi$
$223$	−1.71690			−0.114972			−0.0574862	+	0.998346i	$0.518309\pi$
$224$		0			0
$225$	−1.95613			−0.130409
$226$		0			0
$227$			17.3219i			1.14970i	0.818260	+	0.574848i	$0.194939\pi$
$227$			17.3219i			1.14970i	−0.818260	+	0.574848i	$0.805061\pi$
$228$		0			0
$229$	27.1142			1.79175			0.895877	−	0.444302i	$-0.146548\pi$
$229$	27.1142			1.79175			0.895877	+	0.444302i	$0.146548\pi$
$230$		0			0
$231$		−	27.9295i		−	1.83762i
$232$		0			0
$233$		−	8.90548i		−	0.583418i	−0.956507	−	0.291709i	$-0.905776\pi$
$233$		−	8.90548i		−	0.583418i	0.956507	−	0.291709i	$-0.0942238\pi$
$234$		0			0
$235$		−	23.7970i		−	1.55234i
$236$		0			0
$237$	−4.08566			−0.265392
$238$		0			0
$239$	15.3486			0.992820			0.496410	−	0.868088i	$-0.334651\pi$
$239$	15.3486			0.992820			0.496410	+	0.868088i	$0.334651\pi$
$240$		0			0
$241$			9.83383i			0.633452i	0.948517	+	0.316726i	$0.102584\pi$
$241$			9.83383i			0.633452i	−0.948517	+	0.316726i	$0.897416\pi$
$242$		0			0
$243$		−	9.29067i		−	0.595997i
$244$		0			0
$245$		−	22.5594i		−	1.44127i
$246$		0			0
$247$	−36.7896			−2.34086
$248$		0			0
$249$		−	6.45646i		−	0.409161i
$250$		0			0
$251$	−16.2017			−1.02264			−0.511320	−	0.859391i	$-0.670843\pi$
$251$	−16.2017			−1.02264			−0.511320	+	0.859391i	$0.670843\pi$
$252$		0			0
$253$	−30.2762			−1.90345
$254$		0			0
$255$	−8.53850	−	13.2914i	−0.534702	−	0.832343i
$256$		0			0
$257$	−4.30176			−0.268336			−0.134168	−	0.990959i	$-0.542836\pi$
$257$	−4.30176			−0.268336			−0.134168	+	0.990959i	$0.542836\pi$
$258$		0			0
$259$	26.4266			1.64207
$260$		0			0
$261$			1.53273i			0.0948736i
$262$		0			0
$263$	−11.4217			−0.704295			−0.352148	−	0.935944i	$-0.614548\pi$
$263$	−11.4217			−0.704295			−0.352148	+	0.935944i	$0.614548\pi$
$264$		0			0
$265$			15.2986i			0.939787i
$266$		0			0
$267$		−	20.2383i		−	1.23856i
$268$		0			0
$269$			1.59411i			0.0971945i	0.998818	+	0.0485972i	$0.0154751\pi$
$269$			1.59411i			0.0971945i	−0.998818	+	0.0485972i	$0.984525\pi$
$270$		0			0
$271$	14.8878			0.904371			0.452185	−	0.891924i	$-0.350645\pi$
$271$	14.8878			0.904371			0.452185	+	0.891924i	$0.350645\pi$
$272$		0			0
$273$	31.6646			1.91643
$274$		0			0
$275$			10.3650i			0.625032i
$276$		0			0
$277$			3.21746i			0.193318i	0.995318	+	0.0966592i	$0.0308157\pi$
$277$			3.21746i			0.193318i	−0.995318	+	0.0966592i	$0.969184\pi$
$278$		0			0
$279$		−	8.89996i		−	0.532827i
$280$		0			0
$281$	−20.0117			−1.19380			−0.596898	−	0.802317i	$-0.703600\pi$
$281$	−20.0117			−1.19380			−0.596898	+	0.802317i	$0.703600\pi$
$282$		0			0
$283$		−	1.03613i		−	0.0615915i	−0.999526	−	0.0307958i	$-0.990196\pi$
$283$		−	1.03613i		−	0.0615915i	0.999526	−	0.0307958i	$-0.00980414\pi$
$284$		0			0
$285$	−25.1768			−1.49135
$286$		0			0
$287$	−38.0170			−2.24407
$288$		0			0
$289$	−7.06764	+	15.4612i	−0.415744	+	0.909482i
$290$		0			0
$291$	−16.0247			−0.939383
$292$		0			0
$293$	−19.0273			−1.11159			−0.555793	−	0.831321i	$-0.687585\pi$
$293$	−19.0273			−1.11159			−0.555793	+	0.831321i	$0.687585\pi$
$294$		0			0
$295$			2.66437i			0.155126i
$296$		0			0
$297$	−27.9236			−1.62029
$298$		0			0
$299$		−	34.3251i		−	1.98507i
$300$		0			0
$301$			36.0816i			2.07971i
$302$		0			0
$303$			5.23659i			0.300834i
$304$		0			0
$305$	29.9767			1.71646
$306$		0			0
$307$	12.5768			0.717794			0.358897	−	0.933377i	$-0.383153\pi$
$307$	12.5768			0.717794			0.358897	+	0.933377i	$0.383153\pi$
$308$		0			0
$309$		−	6.80194i		−	0.386949i
$310$		0			0
$311$		−	25.2938i		−	1.43428i	−0.696930	−	0.717139i	$-0.745451\pi$
$311$		−	25.2938i		−	1.43428i	0.696930	−	0.717139i	$-0.254549\pi$
$312$		0			0
$313$			15.6164i			0.882688i	0.897338	+	0.441344i	$0.145498\pi$
$313$			15.6164i			0.882688i	−0.897338	+	0.441344i	$0.854502\pi$
$314$		0			0
$315$	−9.76588			−0.550245
$316$		0			0
$317$			15.5970i			0.876017i	0.898971	+	0.438008i	$0.144316\pi$
$317$			15.5970i			0.876017i	−0.898971	+	0.438008i	$0.855684\pi$
$318$		0			0
$319$	8.12149			0.454716
$320$		0			0
$321$	−26.7371			−1.49232
$322$		0			0
$323$	14.6434	+	22.7946i	0.814780	+	1.26833i
$324$		0			0
$325$	−11.7511			−0.651836
$326$		0			0
$327$	−1.13610			−0.0628266
$328$		0			0
$329$		−	35.1262i		−	1.93657i
$330$		0			0
$331$	−13.6759			−0.751693			−0.375846	−	0.926682i	$-0.622648\pi$
$331$	−13.6759			−0.751693			−0.375846	+	0.926682i	$0.622648\pi$
$332$		0			0
$333$		−	6.26254i		−	0.343185i
$334$		0			0
$335$		−	2.21121i		−	0.120811i
$336$		0			0
$337$		−	22.7871i		−	1.24129i	−0.784091	−	0.620646i	$-0.786870\pi$
$337$		−	22.7871i		−	1.24129i	0.784091	−	0.620646i	$-0.213130\pi$
$338$		0			0
$339$	16.6427			0.903907
$340$		0			0
$341$	−47.1583			−2.55377
$342$		0			0
$343$		−	5.76967i		−	0.311533i
$344$		0			0
$345$		−	23.4903i		−	1.26467i
$346$		0			0
$347$			26.7914i			1.43824i	0.694887	+	0.719119i	$0.255454\pi$
$347$			26.7914i			1.43824i	−0.694887	+	0.719119i	$0.744546\pi$
$348$		0			0
$349$	−5.34385			−0.286050			−0.143025	−	0.989719i	$-0.545683\pi$
$349$	−5.34385			−0.286050			−0.143025	+	0.989719i	$0.545683\pi$
$350$		0			0
$351$		−	31.6580i		−	1.68978i
$352$		0			0
$353$	−10.5447			−0.561236			−0.280618	−	0.959820i	$-0.590539\pi$
$353$	−10.5447			−0.561236			−0.280618	+	0.959820i	$0.590539\pi$
$354$		0			0
$355$	−15.1100			−0.801954
$356$		0			0
$357$	−12.6035	−	19.6192i	−0.667048	−	1.03836i
$358$		0			0
$359$	−4.03950			−0.213197			−0.106598	−	0.994302i	$-0.533996\pi$
$359$	−4.03950			−0.213197			−0.106598	+	0.994302i	$0.533996\pi$
$360$		0			0
$361$	24.1778			1.27251
$362$		0			0
$363$			19.2519i			1.01046i
$364$		0			0
$365$	15.1191			0.791372
$366$		0			0
$367$		−	34.8079i		−	1.81696i	−0.417933	−	0.908478i	$-0.637245\pi$
$367$		−	34.8079i		−	1.81696i	0.417933	−	0.908478i	$-0.362755\pi$
$368$		0			0
$369$			9.00921i			0.469001i
$370$		0			0
$371$			22.5820i			1.17240i
$372$		0			0
$373$	16.4544			0.851978			0.425989	−	0.904728i	$-0.359926\pi$
$373$	16.4544			0.851978			0.425989	+	0.904728i	$0.359926\pi$
$374$		0			0
$375$	11.1157			0.574014
$376$		0			0
$377$			9.20762i			0.474216i
$378$		0			0
$379$			24.3735i			1.25198i	0.779831	+	0.625990i	$0.215305\pi$
$379$			24.3735i			1.25198i	−0.779831	+	0.625990i	$0.784695\pi$
$380$		0			0
$381$		−	8.05876i		−	0.412863i
$382$		0			0
$383$	−29.3363			−1.49901			−0.749507	−	0.661997i	$-0.769709\pi$
$383$	−29.3363			−1.49901			−0.749507	+	0.661997i	$0.769709\pi$
$384$		0			0
$385$			51.7466i			2.63725i
$386$		0			0
$387$	8.55057			0.434650
$388$		0			0
$389$	−7.76624			−0.393764			−0.196882	−	0.980427i	$-0.563082\pi$
$389$	−7.76624			−0.393764			−0.196882	+	0.980427i	$0.563082\pi$
$390$		0			0
$391$	−21.2677	+	13.6625i	−1.07555	+	0.690940i
$392$		0			0
$393$	29.1116			1.46848
$394$		0			0
$395$	7.56974			0.380875
$396$		0			0
$397$		−	38.7854i		−	1.94658i	−0.229575	−	0.973291i	$-0.573734\pi$
$397$		−	38.7854i		−	1.94658i	0.229575	−	0.973291i	$-0.426266\pi$
$398$		0			0
$399$	−37.1629			−1.86047
$400$		0			0
$401$			3.17880i			0.158742i	0.996845	+	0.0793708i	$0.0252911\pi$
$401$			3.17880i			0.158742i	−0.996845	+	0.0793708i	$0.974709\pi$
$402$		0			0
$403$		−	53.4650i		−	2.66328i
$404$		0			0
$405$		−	14.2155i		−	0.706374i
$406$		0			0
$407$	−33.1834			−1.64484
$408$		0			0
$409$	−0.994863			−0.0491928			−0.0245964	−	0.999697i	$-0.507830\pi$
$409$	−0.994863			−0.0491928			−0.0245964	+	0.999697i	$0.507830\pi$
$410$		0			0
$411$		−	9.68093i		−	0.477525i
$412$		0			0
$413$			3.93282i			0.193521i
$414$		0			0
$415$			11.9623i			0.587204i
$416$		0			0
$417$	0.204830			0.0100306
$418$		0			0
$419$			1.88003i			0.0918454i	0.998945	+	0.0459227i	$0.0146228\pi$
$419$			1.88003i			0.0918454i	−0.998945	+	0.0459227i	$0.985377\pi$
$420$		0			0
$421$	25.5582			1.24563			0.622816	−	0.782368i	$-0.285988\pi$
$421$	25.5582			1.24563			0.622816	+	0.782368i	$0.285988\pi$
$422$		0			0
$423$	−8.32415			−0.404734
$424$		0			0
$425$	4.67732	+	7.28094i	0.226883	+	0.353177i
$426$		0			0
$427$	44.2479			2.14131
$428$		0			0
$429$	−39.7607			−1.91966
$430$		0			0
$431$		−	23.6517i		−	1.13926i	−0.821901	−	0.569630i	$-0.807087\pi$
$431$		−	23.6517i		−	1.13926i	0.821901	−	0.569630i	$-0.192913\pi$
$432$		0			0
$433$	−23.1584			−1.11292			−0.556462	−	0.830873i	$-0.687841\pi$
$433$	−23.1584			−1.11292			−0.556462	+	0.830873i	$0.687841\pi$
$434$		0			0
$435$			6.30120i			0.302119i
$436$		0			0
$437$			40.2854i			1.92711i
$438$		0			0
$439$		−	10.0893i		−	0.481535i	−0.970583	−	0.240768i	$-0.922601\pi$
$439$		−	10.0893i		−	0.481535i	0.970583	−	0.240768i	$-0.0773992\pi$
$440$		0			0
$441$	−7.89124			−0.375773
$442$		0			0
$443$	33.6961			1.60095			0.800475	−	0.599366i	$-0.204580\pi$
$443$	33.6961			1.60095			0.800475	+	0.599366i	$0.204580\pi$
$444$		0			0
$445$			37.4967i			1.77751i
$446$		0			0
$447$		−	28.0571i		−	1.32706i
$448$		0			0
$449$		−	18.4832i		−	0.872278i	−0.899879	−	0.436139i	$-0.856346\pi$
$449$		−	18.4832i		−	0.872278i	0.899879	−	0.436139i	$-0.143654\pi$
$450$		0			0
$451$	47.7372			2.24786
$452$		0			0
$453$			27.4428i			1.28937i
$454$		0			0
$455$	−58.6669			−2.75035
$456$		0			0
$457$	14.2318			0.665735			0.332868	−	0.942974i	$-0.391984\pi$
$457$	14.2318			0.665735			0.332868	+	0.942974i	$0.391984\pi$
$458$		0			0
$459$	−19.6151	+	12.6009i	−0.915554	+	0.588157i
$460$		0			0
$461$	−11.4183			−0.531804			−0.265902	−	0.964000i	$-0.585670\pi$
$461$	−11.4183			−0.531804			−0.265902	+	0.964000i	$0.585670\pi$
$462$		0			0
$463$	35.3216			1.64154			0.820768	−	0.571262i	$-0.193546\pi$
$463$	35.3216			1.64154			0.820768	+	0.571262i	$0.193546\pi$
$464$		0			0
$465$		−	36.5886i		−	1.69676i
$466$		0			0
$467$	3.77136			0.174518			0.0872589	−	0.996186i	$-0.472189\pi$
$467$	3.77136			0.174518			0.0872589	+	0.996186i	$0.472189\pi$
$468$		0			0
$469$		−	3.26392i		−	0.150714i
$470$		0			0
$471$			15.2811i			0.704115i
$472$		0			0
$473$		−	45.3070i		−	2.08322i
$474$		0			0
$475$	13.7916			0.632804
$476$		0			0
$477$	5.35144			0.245026
$478$		0			0
$479$			16.0809i			0.734755i	0.930072	+	0.367377i	$0.119744\pi$
$479$			16.0809i			0.734755i	−0.930072	+	0.367377i	$0.880256\pi$
$480$		0			0
$481$		−	37.6212i		−	1.71538i
$482$		0			0
$483$		−	34.6735i		−	1.57770i
$484$		0			0
$485$	29.6898			1.34815
$486$		0			0
$487$		−	27.0721i		−	1.22675i	−0.789790	−	0.613377i	$-0.789811\pi$
$487$		−	27.0721i		−	1.22675i	0.789790	−	0.613377i	$-0.210189\pi$
$488$		0			0
$489$	7.06913			0.319677
$490$		0			0
$491$	20.3412			0.917985			0.458993	−	0.888440i	$-0.348210\pi$
$491$	20.3412			0.917985			0.458993	+	0.888440i	$0.348210\pi$
$492$		0			0
$493$	5.70498	−	3.66491i	0.256940	−	0.165059i
$494$		0			0
$495$	12.2628			0.551173
$496$		0			0
$497$	−22.3035			−1.00045
$498$		0			0
$499$		−	13.1752i		−	0.589803i	−0.955528	−	0.294901i	$-0.904713\pi$
$499$		−	13.1752i		−	0.589803i	0.955528	−	0.294901i	$-0.0952868\pi$
$500$		0			0
$501$	23.6168			1.05512
$502$		0			0
$503$		−	3.05368i		−	0.136157i	−0.997680	−	0.0680785i	$-0.978313\pi$
$503$		−	3.05368i		−	0.136157i	0.997680	−	0.0680785i	$-0.0216868\pi$
$504$		0			0
$505$		−	9.70213i		−	0.431739i
$506$		0			0
$507$		−	26.3833i		−	1.17172i
$508$		0			0
$509$	−0.477036			−0.0211443			−0.0105721	−	0.999944i	$-0.503365\pi$
$509$	−0.477036			−0.0211443			−0.0105721	+	0.999944i	$0.503365\pi$
$510$		0			0
$511$	22.3170			0.987247
$512$		0			0
$513$			37.1551i			1.64044i
$514$		0			0
$515$			12.6023i			0.555326i
$516$		0			0
$517$			44.1073i			1.93984i
$518$		0			0
$519$	25.1684			1.10477
$520$		0			0
$521$		−	7.29393i		−	0.319553i	−0.987153	−	0.159776i	$-0.948923\pi$
$521$		−	7.29393i		−	0.319553i	0.987153	−	0.159776i	$-0.0510773\pi$
$522$		0			0
$523$	15.0795			0.659382			0.329691	−	0.944089i	$-0.393055\pi$
$523$	15.0795			0.659382			0.329691	+	0.944089i	$0.393055\pi$
$524$		0			0
$525$	−11.8704			−0.518067
$526$		0			0
$527$	−33.1266	+	21.2807i	−1.44302	+	0.927003i
$528$		0			0
$529$	−14.5868			−0.634209
$530$		0			0
$531$	0.931993			0.0404451
$532$		0			0
$533$			54.1213i			2.34426i
$534$		0			0
$535$	49.5373			2.14168
$536$		0			0
$537$			5.52906i			0.238596i
$538$		0			0
$539$			41.8134i			1.80103i
$540$		0			0
$541$		−	9.43840i		−	0.405789i	−0.979201	−	0.202894i	$-0.934965\pi$
$541$		−	9.43840i		−	0.405789i	0.979201	−	0.202894i	$-0.0650348\pi$
$542$		0			0
$543$	−6.72489			−0.288593
$544$		0			0
$545$	2.10492			0.0901650
$546$		0			0
$547$			37.4448i			1.60102i	0.599318	+	0.800511i	$0.295438\pi$
$547$			37.4448i			1.60102i	−0.599318	+	0.800511i	$0.704562\pi$
$548$		0			0
$549$		−	10.4858i		−	0.447523i
$550$		0			0
$551$		−	10.8064i		−	0.460370i
$552$		0			0
$553$	11.1735			0.475147
$554$		0			0
$555$		−	25.7459i		−	1.09285i
$556$		0			0
$557$	18.3004			0.775415			0.387707	−	0.921783i	$-0.373267\pi$
$557$	18.3004			0.775415			0.387707	+	0.921783i	$0.373267\pi$
$558$		0			0
$559$	51.3661			2.17255
$560$		0			0
$561$	15.8260	+	24.6355i	0.668173	+	1.04011i
$562$		0			0
$563$	14.9555			0.630298			0.315149	−	0.949042i	$-0.397946\pi$
$563$	14.9555			0.630298			0.315149	+	0.949042i	$0.397946\pi$
$564$		0			0
$565$	−30.8349			−1.29723
$566$		0			0
$567$		−	20.9832i		−	0.881211i
$568$		0			0
$569$	25.0101			1.04848			0.524239	−	0.851571i	$-0.324350\pi$
$569$	25.0101			1.04848			0.524239	+	0.851571i	$0.324350\pi$
$570$		0			0
$571$			34.4005i			1.43961i	0.694174	+	0.719807i	$0.255770\pi$
$571$			34.4005i			1.43961i	−0.694174	+	0.719807i	$0.744230\pi$
$572$		0			0
$573$		−	7.31278i		−	0.305496i
$574$		0			0
$575$			12.8678i			0.536623i
$576$		0			0
$577$	−36.2824			−1.51045			−0.755227	−	0.655463i	$-0.772474\pi$
$577$	−36.2824			−1.51045			−0.755227	+	0.655463i	$0.772474\pi$
$578$		0			0
$579$	−0.410250			−0.0170494
$580$		0			0
$581$			17.6572i			0.732545i
$582$		0			0
$583$		−	28.3557i		−	1.17437i
$584$		0			0
$585$			13.9028i			0.574810i
$586$		0			0
$587$	−17.4887			−0.721836			−0.360918	−	0.932598i	$-0.617537\pi$
$587$	−17.4887			−0.721836			−0.360918	+	0.932598i	$0.617537\pi$
$588$		0			0
$589$			62.7488i			2.58552i
$590$		0			0
$591$	−17.1450			−0.705253
$592$		0			0
$593$	37.2797			1.53090			0.765448	−	0.643498i	$-0.222518\pi$
$593$	37.2797			1.53090			0.765448	+	0.643498i	$0.222518\pi$
$594$		0			0
$595$	23.3512	+	36.3496i	0.957307	+	1.49019i
$596$		0			0
$597$	−17.7927			−0.728208
$598$		0			0
$599$	−9.61792			−0.392978			−0.196489	−	0.980506i	$-0.562954\pi$
$599$	−9.61792			−0.392978			−0.196489	+	0.980506i	$0.562954\pi$
$600$		0			0
$601$			38.0506i			1.55212i	0.630661	+	0.776059i	$0.282784\pi$
$601$			38.0506i			1.55212i	−0.630661	+	0.776059i	$0.717216\pi$
$602$		0			0
$603$	−0.773479			−0.0314985
$604$		0			0
$605$		−	35.6691i		−	1.45015i
$606$		0			0
$607$			25.9507i			1.05331i	0.850080	+	0.526654i	$0.176554\pi$
$607$			25.9507i			1.05331i	−0.850080	+	0.526654i	$0.823446\pi$
$608$		0			0
$609$			9.30106i			0.376898i
$610$		0			0
$611$	−50.0060			−2.02302
$612$		0			0
$613$	−36.9504			−1.49241			−0.746206	−	0.665715i	$-0.768127\pi$
$613$	−36.9504			−1.49241			−0.746206	+	0.665715i	$0.768127\pi$
$614$		0			0
$615$			37.0377i			1.49351i
$616$		0			0
$617$			25.6827i			1.03395i	0.856001	+	0.516974i	$0.172942\pi$
$617$			25.6827i			1.03395i	−0.856001	+	0.516974i	$0.827058\pi$
$618$		0			0
$619$			3.17128i			0.127464i	0.997967	+	0.0637322i	$0.0203004\pi$
$619$			3.17128i			0.127464i	−0.997967	+	0.0637322i	$0.979700\pi$
$620$		0			0
$621$	−34.6662			−1.39111
$622$		0			0
$623$			55.3480i			2.21747i
$624$		0			0
$625$	−31.0891			−1.24356
$626$		0			0
$627$	46.6648			1.86361
$628$		0			0
$629$	−23.3099	+	14.9744i	−0.929425	+	0.597068i
$630$		0			0
$631$	−10.2218			−0.406925			−0.203463	−	0.979083i	$-0.565220\pi$
$631$	−10.2218			−0.406925			−0.203463	+	0.979083i	$0.565220\pi$
$632$		0			0
$633$	−33.3511			−1.32559
$634$		0			0
$635$			14.9309i			0.592516i
$636$		0			0
$637$	−47.4053			−1.87827
$638$		0			0
$639$			5.28545i			0.209089i
$640$		0			0
$641$			20.8179i			0.822256i	0.911578	+	0.411128i	$0.134865\pi$
$641$			20.8179i			0.822256i	−0.911578	+	0.411128i	$0.865135\pi$
$642$		0			0
$643$			26.7086i			1.05329i	0.850087	+	0.526643i	$0.176549\pi$
$643$			26.7086i			1.05329i	−0.850087	+	0.526643i	$0.823451\pi$
$644$		0			0
$645$	35.1522			1.38412
$646$		0			0
$647$	42.0526			1.65326			0.826630	−	0.562746i	$-0.190255\pi$
$647$	42.0526			1.65326			0.826630	+	0.562746i	$0.190255\pi$
$648$		0			0
$649$		−	4.93836i		−	0.193848i
$650$		0			0
$651$		−	54.0076i		−	2.11673i
$652$		0			0
$653$			1.46051i			0.0571543i	0.999592	+	0.0285772i	$0.00909763\pi$
$653$			1.46051i			0.0571543i	−0.999592	+	0.0285772i	$0.990902\pi$
$654$		0			0
$655$	−53.9367			−2.10748
$656$		0			0
$657$		−	5.28865i		−	0.206330i
$658$		0			0
$659$	35.2978			1.37501			0.687504	−	0.726181i	$-0.258707\pi$
$659$	35.2978			1.37501			0.687504	+	0.726181i	$0.258707\pi$
$660$		0			0
$661$	30.3176			1.17922			0.589608	−	0.807689i	$-0.299282\pi$
$661$	30.3176			1.17922			0.589608	+	0.807689i	$0.299282\pi$
$662$		0			0
$663$	−27.9301	+	17.9425i	−1.08471	+	0.696827i
$664$		0			0
$665$	68.8539			2.67004
$666$		0			0
$667$	10.0826			0.390398
$668$		0			0
$669$			2.46901i			0.0954573i
$670$		0			0
$671$	−55.5612			−2.14492
$672$		0			0
$673$		−	12.7565i		−	0.491727i	−0.969305	−	0.245863i	$-0.920929\pi$
$673$		−	12.7565i		−	0.491727i	0.969305	−	0.245863i	$-0.0790714\pi$
$674$		0			0
$675$			11.8679i			0.456796i
$676$		0			0
$677$		−	26.4740i		−	1.01748i	−0.860921	−	0.508739i	$-0.830112\pi$
$677$		−	26.4740i		−	1.01748i	0.860921	−	0.508739i	$-0.169888\pi$
$678$		0			0
$679$	43.8245			1.68183
$680$		0			0
$681$	24.9099			0.954550
$682$		0			0
$683$		−	4.49043i		−	0.171821i	−0.996303	−	0.0859107i	$-0.972620\pi$
$683$		−	4.49043i		−	0.171821i	0.996303	−	0.0859107i	$-0.0273800\pi$
$684$		0			0
$685$			17.9364i			0.685316i
$686$		0			0
$687$		−	38.9917i		−	1.48763i
$688$		0			0
$689$	32.1479			1.22474
$690$		0			0
$691$		−	0.816424i		−	0.0310582i	−0.999879	−	0.0155291i	$-0.995057\pi$
$691$		−	0.816424i		−	0.0310582i	0.999879	−	0.0155291i	$-0.00494327\pi$
$692$		0			0
$693$	18.1009			0.687596
$694$		0			0
$695$	−0.379501			−0.0143953
$696$		0			0
$697$	33.5333	−	21.5420i	1.27016	−	0.815960i
$698$		0			0
$699$	−12.8066			−0.484390
$700$		0			0
$701$	−12.5635			−0.474518			−0.237259	−	0.971446i	$-0.576249\pi$
$701$	−12.5635			−0.474518			−0.237259	+	0.971446i	$0.576249\pi$
$702$		0			0
$703$			44.1538i			1.66529i
$704$		0			0
$705$	−34.2214			−1.28885
$706$		0			0
$707$		−	14.3211i		−	0.538600i
$708$		0			0
$709$			44.5609i			1.67352i	0.547571	+	0.836759i	$0.315552\pi$
$709$			44.5609i			1.67352i	−0.547571	+	0.836759i	$0.684448\pi$
$710$		0			0
$711$		−	2.64788i		−	0.0993034i
$712$		0			0
$713$	−58.5454			−2.19254
$714$		0			0
$715$	73.6669			2.75498
$716$		0			0
$717$		−	22.0722i		−	0.824301i
$718$		0			0
$719$		−	3.38195i		−	0.126126i	−0.998010	−	0.0630628i	$-0.979913\pi$
$719$		−	3.38195i		−	0.126126i	0.998010	−	0.0630628i	$-0.0200868\pi$
$720$		0			0
$721$			18.6020i			0.692776i
$722$		0			0
$723$	14.1416			0.525932
$724$		0			0
$725$		−	3.45174i		−	0.128194i
$726$		0			0
$727$	−27.2106			−1.00919			−0.504593	−	0.863357i	$-0.668358\pi$
$727$	−27.2106			−1.00919			−0.504593	+	0.863357i	$0.668358\pi$
$728$		0			0
$729$	−29.3667			−1.08766
$730$		0			0
$731$	−20.4453	−	31.8261i	−0.756196	−	1.17713i
$732$		0			0
$733$	−2.20268			−0.0813578			−0.0406789	−	0.999172i	$-0.512952\pi$
$733$	−2.20268			−0.0813578			−0.0406789	+	0.999172i	$0.512952\pi$
$734$		0			0
$735$	−32.4417			−1.19663
$736$		0			0
$737$			4.09844i			0.150968i
$738$		0			0
$739$	−12.9774			−0.477381			−0.238690	−	0.971096i	$-0.576718\pi$
$739$	−12.9774			−0.477381			−0.238690	+	0.971096i	$0.576718\pi$
$740$		0			0
$741$			52.9055i			1.94353i
$742$		0			0
$743$		−	20.5515i		−	0.753962i	−0.926221	−	0.376981i	$-0.876962\pi$
$743$		−	20.5515i		−	0.753962i	0.926221	−	0.376981i	$-0.123038\pi$
$744$		0			0
$745$			51.9830i			1.90451i
$746$		0			0
$747$	4.18438			0.153099
$748$		0			0
$749$	73.1209			2.67178
$750$		0			0
$751$		−	45.4481i		−	1.65843i	−0.558933	−	0.829213i	$-0.688789\pi$
$751$		−	45.4481i		−	1.65843i	0.558933	−	0.829213i	$-0.311211\pi$
$752$		0			0
$753$			23.2989i			0.849059i
$754$		0			0
$755$		−	50.8448i		−	1.85043i
$756$		0			0
$757$	21.3682			0.776640			0.388320	−	0.921525i	$-0.373056\pi$
$757$	21.3682			0.776640			0.388320	+	0.921525i	$0.373056\pi$
$758$		0			0
$759$			43.5388i			1.58036i
$760$		0			0
$761$	13.0180			0.471902			0.235951	−	0.971765i	$-0.424180\pi$
$761$	13.0180			0.471902			0.235951	+	0.971765i	$0.424180\pi$
$762$		0			0
$763$	3.10703			0.112482
$764$		0			0
$765$	8.61409	−	5.53374i	0.311443	−	0.200073i
$766$		0			0
$767$	5.59880			0.202161
$768$		0			0
$769$	2.63006			0.0948425			0.0474212	−	0.998875i	$-0.484900\pi$
$769$	2.63006			0.0948425			0.0474212	+	0.998875i	$0.484900\pi$
$770$		0			0
$771$			6.18618i			0.222790i
$772$		0			0
$773$	−29.6444			−1.06624			−0.533118	−	0.846041i	$-0.678980\pi$
$773$	−29.6444			−1.06624			−0.533118	+	0.846041i	$0.678980\pi$
$774$		0			0
$775$			20.0429i			0.719963i
$776$		0			0
$777$		−	38.0030i		−	1.36335i
$778$		0			0
$779$		−	63.5191i		−	2.27581i
$780$		0			0
$781$	28.0061			1.00214
$782$		0			0
$783$	9.29911			0.332323
$784$		0			0
$785$		−	28.3121i		−	1.01050i
$786$		0			0
$787$			32.9315i			1.17388i	0.809631	+	0.586940i	$0.199667\pi$
$787$			32.9315i			1.17388i	−0.809631	+	0.586940i	$0.800333\pi$
$788$		0			0
$789$			16.4251i			0.584750i
$790$		0			0
$791$	−45.5147			−1.61831
$792$		0			0
$793$		−	62.9917i		−	2.23690i
$794$		0			0
$795$	22.0003			0.780270
$796$		0			0
$797$	−38.4945			−1.36355			−0.681773	−	0.731564i	$-0.738791\pi$
$797$	−38.4945			−1.36355			−0.681773	+	0.731564i	$0.738791\pi$
$798$		0			0
$799$	19.9039	+	30.9834i	0.704150	+	1.09611i
$800$		0			0
$801$	13.1163			0.463441
$802$		0			0
$803$	−28.0230			−0.988912
$804$		0			0
$805$			64.2416i			2.26422i
$806$		0			0
$807$	2.29242			0.0806969
$808$		0			0
$809$		−	51.1518i		−	1.79840i	−0.437537	−	0.899200i	$-0.644149\pi$
$809$		−	51.1518i		−	1.79840i	0.437537	−	0.899200i	$-0.355851\pi$
$810$		0			0
$811$		−	26.9215i		−	0.945341i	−0.881239	−	0.472671i	$-0.843290\pi$
$811$		−	26.9215i		−	0.945341i	0.881239	−	0.472671i	$-0.156710\pi$
$812$		0			0
$813$		−	21.4095i		−	0.750865i
$814$		0			0
$815$	−13.0974			−0.458781
$816$		0			0
$817$	−60.2854			−2.10912
$818$		0			0
$819$			20.5216i			0.717083i
$820$		0			0
$821$			14.8185i			0.517170i	0.965988	+	0.258585i	$0.0832563\pi$
$821$			14.8185i			0.517170i	−0.965988	+	0.258585i	$0.916744\pi$
$822$		0			0
$823$			21.7596i			0.758493i	0.925296	+	0.379247i	$0.123817\pi$
$823$			21.7596i			0.758493i	−0.925296	+	0.379247i	$0.876183\pi$
$824$		0			0
$825$	14.9054			0.518941
$826$		0			0
$827$			35.9365i			1.24963i	0.780771	+	0.624817i	$0.214827\pi$
$827$			35.9365i			1.24963i	−0.780771	+	0.624817i	$0.785173\pi$
$828$		0			0
$829$	14.3853			0.499622			0.249811	−	0.968295i	$-0.419632\pi$
$829$	14.3853			0.499622			0.249811	+	0.968295i	$0.419632\pi$
$830$		0			0
$831$	4.62689			0.160505
$832$		0			0
$833$	18.8688	+	29.3721i	0.653764	+	1.01768i
$834$		0			0
$835$	−43.7562			−1.51425
$836$		0			0
$837$	−53.9963			−1.86639
$838$		0			0
$839$			30.9984i			1.07018i	0.844794	+	0.535091i	$0.179723\pi$
$839$			30.9984i			1.07018i	−0.844794	+	0.535091i	$0.820277\pi$
$840$		0			0
$841$	26.2954			0.906737
$842$		0			0
$843$			28.7779i			0.991164i
$844$		0			0
$845$			48.8819i			1.68159i
$846$		0			0
$847$		−	52.6503i		−	1.80909i
$848$		0			0
$849$	−1.49001			−0.0511371
$850$		0			0
$851$	−41.1961			−1.41218
$852$		0			0
$853$		−	52.3410i		−	1.79212i	−0.443932	−	0.896060i	$-0.646417\pi$
$853$		−	52.3410i		−	1.79212i	0.443932	−	0.896060i	$-0.353583\pi$
$854$		0			0
$855$		−	16.3169i		−	0.558026i
$856$		0			0
$857$			0.316408i			0.0108083i	0.999985	+	0.00540414i	$0.00172020\pi$
$857$			0.316408i			0.0108083i	−0.999985	+	0.00540414i	$0.998280\pi$
$858$		0			0
$859$	−20.6211			−0.703584			−0.351792	−	0.936078i	$-0.614428\pi$
$859$	−20.6211			−0.703584			−0.351792	+	0.936078i	$0.614428\pi$
$860$		0			0
$861$			54.6706i			1.86317i
$862$		0			0
$863$	41.2299			1.40348			0.701740	−	0.712433i	$-0.252407\pi$
$863$	41.2299			1.40348			0.701740	+	0.712433i	$0.252407\pi$
$864$		0			0
$865$	−46.6309			−1.58550
$866$		0			0
$867$	22.2341	+	10.1637i	0.755109	+	0.345176i
$868$		0			0
$869$	−14.0304			−0.475948
$870$		0			0
$871$	−4.64654			−0.157442
$872$		0			0
$873$		−	10.3855i		−	0.351495i
$874$		0			0
$875$	−30.3994			−1.02769
$876$		0			0
$877$		−	1.04684i		−	0.0353492i	−0.999844	−	0.0176746i	$-0.994374\pi$
$877$		−	1.04684i		−	0.0353492i	0.999844	−	0.0176746i	$-0.00562629\pi$
$878$		0			0
$879$			27.3623i			0.922908i
$880$		0			0
$881$		−	29.3598i		−	0.989156i	−0.869133	−	0.494578i	$-0.835323\pi$
$881$		−	29.3598i		−	0.989156i	0.869133	−	0.494578i	$-0.164677\pi$
$882$		0			0
$883$	−4.26706			−0.143598			−0.0717990	−	0.997419i	$-0.522874\pi$
$883$	−4.26706			−0.143598			−0.0717990	+	0.997419i	$0.522874\pi$
$884$		0			0
$885$	3.83152			0.128795
$886$		0			0
$887$			16.3118i			0.547696i	0.961773	+	0.273848i	$0.0882965\pi$
$887$			16.3118i			0.547696i	−0.961773	+	0.273848i	$0.911703\pi$
$888$		0			0
$889$			22.0392i			0.739172i
$890$		0			0
$891$			26.3482i			0.882697i
$892$		0			0
$893$	58.6891			1.96396
$894$		0			0
$895$		−	10.2440i		−	0.342419i
$896$		0			0
$897$	−49.3615			−1.64813
$898$		0			0
$899$	15.7046			0.523779
$900$		0			0
$901$	−12.7958	−	19.9186i	−0.426292	−	0.663586i
$902$		0			0
$903$	51.8874			1.72670
$904$		0			0
$905$	12.4596			0.414171
$906$		0			0
$907$		−	10.9667i		−	0.364145i	−0.983285	−	0.182072i	$-0.941719\pi$
$907$		−	10.9667i		−	0.364145i	0.983285	−	0.182072i	$-0.0582805\pi$
$908$		0			0
$909$	−3.39379			−0.112565
$910$		0			0
$911$		−	2.12531i		−	0.0704148i	−0.999380	−	0.0352074i	$-0.988791\pi$
$911$		−	2.12531i		−	0.0704148i	0.999380	−	0.0352074i	$-0.0112092\pi$
$912$		0			0
$913$		−	22.1718i		−	0.733780i
$914$		0			0
$915$		−	43.1081i		−	1.42511i
$916$		0			0
$917$	−79.6148			−2.62911
$918$		0			0
$919$	37.3877			1.23331			0.616653	−	0.787235i	$-0.288488\pi$
$919$	37.3877			1.23331			0.616653	+	0.787235i	$0.288488\pi$
$920$		0			0
$921$		−	18.0861i		−	0.595957i
$922$		0			0
$923$			31.7515i			1.04511i
$924$		0			0
$925$			14.1034i			0.463716i
$926$		0			0
$927$	4.40828			0.144787
$928$		0			0
$929$			4.99158i			0.163768i	0.996642	+	0.0818842i	$0.0260938\pi$
$929$			4.99158i			0.163768i	−0.996642	+	0.0818842i	$0.973906\pi$
$930$		0			0
$931$	55.6369			1.82342
$932$		0			0
$933$	−36.3739			−1.19083
$934$		0			0
$935$	−29.3217	−	45.6435i	−0.958922	−	1.49270i
$936$		0			0
$937$	13.2993			0.434470			0.217235	−	0.976119i	$-0.430296\pi$
$937$	13.2993			0.434470			0.217235	+	0.976119i	$0.430296\pi$
$938$		0			0
$939$	22.4572			0.732863
$940$		0			0
$941$			34.1840i			1.11437i	0.830390	+	0.557183i	$0.188118\pi$
$941$			34.1840i			1.11437i	−0.830390	+	0.557183i	$0.811882\pi$
$942$		0			0
$943$	59.2641			1.92990
$944$		0			0
$945$			59.2498i			1.92740i
$946$		0			0
$947$		−	28.5123i		−	0.926524i	−0.886221	−	0.463262i	$-0.846679\pi$
$947$		−	28.5123i		−	0.926524i	0.886221	−	0.463262i	$-0.153321\pi$
$948$		0			0
$949$		−	31.7707i		−	1.03132i
$950$		0			0
$951$	22.4294			0.727324
$952$		0			0
$953$	−50.1117			−1.62328			−0.811639	−	0.584159i	$-0.801424\pi$
$953$	−50.1117			−1.62328			−0.811639	+	0.584159i	$0.801424\pi$
$954$		0			0
$955$			13.5488i			0.438429i
$956$		0			0
$957$		−	11.6792i		−	0.377534i
$958$		0			0
$959$			26.4756i			0.854940i
$960$		0			0
$961$	−60.1907			−1.94164
$962$		0			0
$963$		−	17.3281i		−	0.558390i
$964$		0			0
$965$	0.760093			0.0244683
$966$		0			0
$967$	−41.3654			−1.33022			−0.665110	−	0.746745i	$-0.731616\pi$
$967$	−41.3654			−1.33022			−0.665110	+	0.746745i	$0.731616\pi$
$968$		0			0
$969$	32.7799	−	21.0580i	1.05304	−	0.676481i
$970$		0			0
$971$	−2.23803			−0.0718219			−0.0359109	−	0.999355i	$-0.511433\pi$
$971$	−2.23803			−0.0718219			−0.0359109	+	0.999355i	$0.511433\pi$
$972$		0			0
$973$	−0.560173			−0.0179583
$974$		0			0
$975$			16.8988i			0.541195i
$976$		0			0
$977$	−4.31764			−0.138134			−0.0690668	−	0.997612i	$-0.522002\pi$
$977$	−4.31764			−0.138134			−0.0690668	+	0.997612i	$0.522002\pi$
$978$		0			0
$979$		−	69.4994i		−	2.22121i
$980$		0			0
$981$		−	0.736300i		−	0.0235082i
$982$		0			0
$983$		−	21.2921i		−	0.679113i	−0.940586	−	0.339557i	$-0.889723\pi$
$983$		−	21.2921i		−	0.679113i	0.940586	−	0.339557i	$-0.110277\pi$
$984$		0			0
$985$	31.7656			1.01214
$986$		0			0
$987$	−50.5135			−1.60786
$988$		0			0
$989$		−	56.2471i		−	1.78855i
$990$		0			0
$991$		−	53.9694i		−	1.71439i	−0.514989	−	0.857197i	$-0.672204\pi$
$991$		−	53.9694i		−	1.71439i	0.514989	−	0.857197i	$-0.327796\pi$
$992$		0			0
$993$			19.6667i			0.624102i
$994$		0			0
$995$	32.9656			1.04508
$996$		0			0
$997$			12.5273i			0.396744i	0.980127	+	0.198372i	$0.0635654\pi$
$997$			12.5273i			0.396744i	−0.980127	+	0.198372i	$0.936435\pi$
$998$		0			0
$999$	−37.9950			−1.20211

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.b.237.16	✓	46
17.16	even	2	inner	4012.2.b.b.237.31	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.b.237.16	✓	46	1.1	even	1	trivial
4012.2.b.b.237.31	yes	46	17.16	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.43806i q^{3} +2.66437i q^{5} +3.93282i q^{7} +0.931993 q^{9} +O(q^{10})\)	\(q-1.43806i q^{3} +2.66437i q^{5} +3.93282i q^{7} +0.931993 q^{9} -4.93836i q^{11} +5.59880 q^{13} +3.83152 q^{15} +(-2.22849 - 3.46898i) q^{17} -6.57098 q^{19} +5.65561 q^{21} -6.13081i q^{23} -2.09887 q^{25} -5.65443i q^{27} +1.64457i q^{29} -9.54938i q^{31} -7.10165 q^{33} -10.4785 q^{35} -6.71951i q^{37} -8.05138i q^{39} +9.66660i q^{41} +9.17449 q^{43} +2.48318i q^{45} -8.93156 q^{47} -8.46706 q^{49} +(-4.98859 + 3.20470i) q^{51} +5.74193 q^{53} +13.1576 q^{55} +9.44944i q^{57} +1.00000 q^{59} -11.2509i q^{61} +3.66536i q^{63} +14.9173i q^{65} -0.829919 q^{67} -8.81645 q^{69} +5.67113i q^{71} -5.67456i q^{73} +3.01829i q^{75} +19.4217 q^{77} -2.84110i q^{79} -5.33541 q^{81} +4.48971 q^{83} +(9.24265 - 5.93753i) q^{85} +2.36499 q^{87} +14.0734 q^{89} +22.0190i q^{91} -13.7326 q^{93} -17.5075i q^{95} -11.1433i q^{97} -4.60252i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 54 q^{9}+O(q^{10}) \) 46 * q - 54 * q^9	\( 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) \) 46 * q - 54 * q^9 + 8 * q^13 - 10 * q^15 + q^17 - 20 * q^19 - 24 * q^21 - 54 * q^25 + 2 * q^33 + 26 * q^35 - 38 * q^43 + 6 * q^47 - 66 * q^49 + 26 * q^51 + 18 * q^53 - 20 * q^55 + 46 * q^59 + 48 * q^67 + 28 * q^69 + 22 * q^77 + 70 * q^81 - 52 * q^83 - 2 * q^85 + 44 * q^87 - 76 * q^89 - 26 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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