LMFDB - Embedded newform 4020.2.g.b.1609.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1609,4020)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4020.1609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.g (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:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1609.13
Character	$\chi$	$=$	4020.1609
Dual form			4020.2.g.b.1609.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-2.22339 + 0.237781i) q^{5} -1.39060i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(-2.22339 + 0.237781i) q^{5} -1.39060i q^{7} -1.00000 q^{9} +1.82441 q^{11} -1.45733i q^{13} +(-0.237781 - 2.22339i) q^{15} +6.33480i q^{17} +0.666098 q^{19} +1.39060 q^{21} -3.19688i q^{23} +(4.88692 - 1.05736i) q^{25} -1.00000i q^{27} -2.76644 q^{29} -3.01977 q^{31} +1.82441i q^{33} +(0.330659 + 3.09185i) q^{35} -2.72957i q^{37} +1.45733 q^{39} -8.72619 q^{41} +0.406578i q^{43} +(2.22339 - 0.237781i) q^{45} -10.1676i q^{47} +5.06623 q^{49} -6.33480 q^{51} +1.57208i q^{53} +(-4.05638 + 0.433811i) q^{55} +0.666098i q^{57} -3.25825 q^{59} +6.68081 q^{61} +1.39060i q^{63} +(0.346526 + 3.24022i) q^{65} -1.00000i q^{67} +3.19688 q^{69} +3.41448 q^{71} +6.44766i q^{73} +(1.05736 + 4.88692i) q^{75} -2.53703i q^{77} +15.1051 q^{79} +1.00000 q^{81} -2.66126i q^{83} +(-1.50630 - 14.0847i) q^{85} -2.76644i q^{87} +15.2132 q^{89} -2.02657 q^{91} -3.01977i q^{93} +(-1.48099 + 0.158386i) q^{95} -13.0028i q^{97} -1.82441 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 4 q^{5} - 24 q^{9}+O(q^{10}) $ 24 * q - 4 * q^5 - 24 * q^9	$ 24 q - 4 q^{5} - 24 q^{9} - 8 q^{11} - 2 q^{15} + 16 q^{19} - 20 q^{21} + 10 q^{25} + 36 q^{29} - 2 q^{35} + 4 q^{39} - 24 q^{41} + 4 q^{45} - 4 q^{51} - 4 q^{55} + 24 q^{59} - 4 q^{61} - 20 q^{65} - 4 q^{69} + 20 q^{71} - 12 q^{75} - 28 q^{79} + 24 q^{81} - 16 q^{85} + 48 q^{89} - 20 q^{91} - 4 q^{95} + 8 q^{99}+O(q^{100}) $ 24 * q - 4 * q^5 - 24 * q^9 - 8 * q^11 - 2 * q^15 + 16 * q^19 - 20 * q^21 + 10 * q^25 + 36 * q^29 - 2 * q^35 + 4 * q^39 - 24 * q^41 + 4 * q^45 - 4 * q^51 - 4 * q^55 + 24 * q^59 - 4 * q^61 - 20 * q^65 - 4 * q^69 + 20 * q^71 - 12 * q^75 - 28 * q^79 + 24 * q^81 - 16 * q^85 + 48 * q^89 - 20 * q^91 - 4 * q^95 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times$.

$n$	$1141$	$2011$	$2681$	$3217$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−2.22339	+	0.237781i	−0.994330	+	0.106339i
$5$	−2.22339	+	0.237781i	−0.994330	+	0.106339i
$6$		0			0
$7$		−	1.39060i		−	0.525598i	−0.964851	−	0.262799i	$-0.915354\pi$
$7$		−	1.39060i		−	0.525598i	0.964851	−	0.262799i	$-0.0846456\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	1.82441			0.550081			0.275040	−	0.961433i	$-0.411309\pi$
$11$	1.82441			0.550081			0.275040	+	0.961433i	$0.411309\pi$
$12$		0			0
$13$		−	1.45733i		−	0.404191i	−0.979366	−	0.202096i	$-0.935225\pi$
$13$		−	1.45733i		−	0.404191i	0.979366	−	0.202096i	$-0.0647752\pi$
$14$		0			0
$15$	−0.237781	−	2.22339i	−0.0613949	−	0.574077i
$16$		0			0
$17$			6.33480i			1.53642i	0.640201	+	0.768208i	$0.278851\pi$
$17$			6.33480i			1.53642i	−0.640201	+	0.768208i	$0.721149\pi$
$18$		0			0
$19$	0.666098			0.152813			0.0764066	−	0.997077i	$-0.475655\pi$
$19$	0.666098			0.152813			0.0764066	+	0.997077i	$0.475655\pi$
$20$		0			0
$21$	1.39060			0.303454
$22$		0			0
$23$		−	3.19688i		−	0.666595i	−0.942822	−	0.333298i	$-0.891839\pi$
$23$		−	3.19688i		−	0.666595i	0.942822	−	0.333298i	$-0.108161\pi$
$24$		0			0
$25$	4.88692	−	1.05736i	0.977384	−	0.211472i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−2.76644			−0.513715			−0.256857	−	0.966449i	$-0.582687\pi$
$29$	−2.76644			−0.513715			−0.256857	+	0.966449i	$0.582687\pi$
$30$		0			0
$31$	−3.01977			−0.542367			−0.271184	−	0.962528i	$-0.587415\pi$
$31$	−3.01977			−0.542367			−0.271184	+	0.962528i	$0.587415\pi$
$32$		0			0
$33$			1.82441i			0.317589i
$34$		0			0
$35$	0.330659	+	3.09185i	0.0558916	+	0.522617i
$36$		0			0
$37$		−	2.72957i		−	0.448738i	−0.974504	−	0.224369i	$-0.927968\pi$
$37$		−	2.72957i		−	0.448738i	0.974504	−	0.224369i	$-0.0720321\pi$
$38$		0			0
$39$	1.45733			0.233360
$40$		0			0
$41$	−8.72619			−1.36280			−0.681401	−	0.731910i	$-0.738629\pi$
$41$	−8.72619			−1.36280			−0.681401	+	0.731910i	$0.738629\pi$
$42$		0			0
$43$			0.406578i			0.0620026i	0.999519	+	0.0310013i	$0.00986961\pi$
$43$			0.406578i			0.0620026i	−0.999519	+	0.0310013i	$0.990130\pi$
$44$		0			0
$45$	2.22339	−	0.237781i	0.331443	−	0.0354464i
$46$		0			0
$47$		−	10.1676i		−	1.48309i	−0.670902	−	0.741546i	$-0.734093\pi$
$47$		−	10.1676i		−	1.48309i	0.670902	−	0.741546i	$-0.265907\pi$
$48$		0			0
$49$	5.06623			0.723747
$50$		0			0
$51$	−6.33480			−0.887050
$52$		0			0
$53$			1.57208i			0.215941i	0.994154	+	0.107971i	$0.0344352\pi$
$53$			1.57208i			0.215941i	−0.994154	+	0.107971i	$0.965565\pi$
$54$		0			0
$55$	−4.05638	+	0.433811i	−0.546962	+	0.0584951i
$56$		0			0
$57$			0.666098i			0.0882268i
$58$		0			0
$59$	−3.25825			−0.424188			−0.212094	−	0.977249i	$-0.568028\pi$
$59$	−3.25825			−0.424188			−0.212094	+	0.977249i	$0.568028\pi$
$60$		0			0
$61$	6.68081			0.855390			0.427695	−	0.903923i	$-0.359326\pi$
$61$	6.68081			0.855390			0.427695	+	0.903923i	$0.359326\pi$
$62$		0			0
$63$			1.39060i			0.175199i
$64$		0			0
$65$	0.346526	+	3.24022i	0.0429813	+	0.401899i
$66$		0			0
$67$		−	1.00000i		−	0.122169i
$67$		−	1.00000i		−	0.122169i
$68$		0			0
$69$	3.19688			0.384859
$70$		0			0
$71$	3.41448			0.405224			0.202612	−	0.979259i	$-0.435057\pi$
$71$	3.41448			0.405224			0.202612	+	0.979259i	$0.435057\pi$
$72$		0			0
$73$			6.44766i			0.754641i	0.926083	+	0.377320i	$0.123154\pi$
$73$			6.44766i			0.754641i	−0.926083	+	0.377320i	$0.876846\pi$
$74$		0			0
$75$	1.05736	+	4.88692i	0.122094	+	0.564293i
$76$		0			0
$77$		−	2.53703i		−	0.289121i
$78$		0			0
$79$	15.1051			1.69945			0.849726	−	0.527225i	$-0.176768\pi$
$79$	15.1051			1.69945			0.849726	+	0.527225i	$0.176768\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	2.66126i		−	0.292111i	−0.989276	−	0.146056i	$-0.953342\pi$
$83$		−	2.66126i		−	0.292111i	0.989276	−	0.146056i	$-0.0466579\pi$
$84$		0			0
$85$	−1.50630	−	14.0847i	−0.163381	−	1.52770i
$86$		0			0
$87$		−	2.76644i		−	0.296593i
$88$		0			0
$89$	15.2132			1.61259			0.806296	−	0.591513i	$-0.201469\pi$
$89$	15.2132			1.61259			0.806296	+	0.591513i	$0.201469\pi$
$90$		0			0
$91$	−2.02657			−0.212442
$92$		0			0
$93$		−	3.01977i		−	0.313136i
$94$		0			0
$95$	−1.48099	+	0.158386i	−0.151947	+	0.0162500i
$96$		0			0
$97$		−	13.0028i		−	1.32024i	−0.751162	−	0.660118i	$-0.770506\pi$
$97$		−	13.0028i		−	1.32024i	0.751162	−	0.660118i	$-0.229494\pi$
$98$		0			0
$99$	−1.82441			−0.183360
$100$		0			0
$101$	7.53421			0.749682			0.374841	−	0.927089i	$-0.377697\pi$
$101$	7.53421			0.749682			0.374841	+	0.927089i	$0.377697\pi$
$102$		0			0
$103$		−	12.6775i		−	1.24915i	−0.780963	−	0.624577i	$-0.785271\pi$
$103$		−	12.6775i		−	1.24915i	0.780963	−	0.624577i	$-0.214729\pi$
$104$		0			0
$105$	−3.09185	+	0.330659i	−0.301733	+	0.0322690i
$106$		0			0
$107$			3.24648i			0.313849i	0.987611	+	0.156924i	$0.0501579\pi$
$107$			3.24648i			0.313849i	−0.987611	+	0.156924i	$0.949842\pi$
$108$		0			0
$109$	15.4343			1.47834			0.739170	−	0.673519i	$-0.235218\pi$
$109$	15.4343			1.47834			0.739170	+	0.673519i	$0.235218\pi$
$110$		0			0
$111$	2.72957			0.259079
$112$		0			0
$113$			14.6944i			1.38234i	0.722693	+	0.691169i	$0.242904\pi$
$113$			14.6944i			1.38234i	−0.722693	+	0.691169i	$0.757096\pi$
$114$		0			0
$115$	0.760159	+	7.10791i	0.0708851	+	0.662816i
$116$		0			0
$117$			1.45733i			0.134730i
$118$		0			0
$119$	8.80918			0.807536
$120$		0			0
$121$	−7.67152			−0.697411
$122$		0			0
$123$		−	8.72619i		−	0.786814i
$124$		0			0
$125$	−10.6141	+	3.51294i	−0.949354	+	0.314207i
$126$		0			0
$127$			16.9371i			1.50292i	0.659778	+	0.751460i	$0.270650\pi$
$127$			16.9371i			1.50292i	−0.659778	+	0.751460i	$0.729350\pi$
$128$		0			0
$129$	−0.406578			−0.0357972
$130$		0			0
$131$	19.1372			1.67203			0.836014	−	0.548709i	$-0.184880\pi$
$131$	19.1372			1.67203			0.836014	+	0.548709i	$0.184880\pi$
$132$		0			0
$133$		−	0.926276i		−	0.0803183i
$134$		0			0
$135$	0.237781	+	2.22339i	0.0204650	+	0.191359i
$136$		0			0
$137$			6.41529i			0.548095i	0.961716	+	0.274048i	$0.0883626\pi$
$137$			6.41529i			0.548095i	−0.961716	+	0.274048i	$0.911637\pi$
$138$		0			0
$139$	−2.35871			−0.200063			−0.100032	−	0.994984i	$-0.531894\pi$
$139$	−2.35871			−0.200063			−0.100032	+	0.994984i	$0.531894\pi$
$140$		0			0
$141$	10.1676			0.856263
$142$		0			0
$143$		−	2.65877i		−	0.222338i
$144$		0			0
$145$	6.15087	−	0.657808i	0.510802	−	0.0546280i
$146$		0			0
$147$			5.06623i			0.417856i
$148$		0			0
$149$	15.6986			1.28608			0.643038	−	0.765834i	$-0.277674\pi$
$149$	15.6986			1.28608			0.643038	+	0.765834i	$0.277674\pi$
$150$		0			0
$151$	4.06488			0.330795			0.165398	−	0.986227i	$-0.447109\pi$
$151$	4.06488			0.330795			0.165398	+	0.986227i	$0.447109\pi$
$152$		0			0
$153$		−	6.33480i		−	0.512139i
$154$		0			0
$155$	6.71413	−	0.718046i	0.539292	−	0.0576748i
$156$		0			0
$157$		−	20.1288i		−	1.60645i	−0.595675	−	0.803226i	$-0.703115\pi$
$157$		−	20.1288i		−	1.60645i	0.595675	−	0.803226i	$-0.296885\pi$
$158$		0			0
$159$	−1.57208			−0.124674
$160$		0			0
$161$	−4.44558			−0.350361
$162$		0			0
$163$		−	14.6538i		−	1.14777i	−0.818935	−	0.573887i	$-0.805435\pi$
$163$		−	14.6538i		−	1.14777i	0.818935	−	0.573887i	$-0.194565\pi$
$164$		0			0
$165$	−0.433811	−	4.05638i	−0.0337722	−	0.315789i
$166$		0			0
$167$			12.8759i			0.996368i	0.867071	+	0.498184i	$0.166000\pi$
$167$			12.8759i			0.996368i	−0.867071	+	0.498184i	$0.834000\pi$
$168$		0			0
$169$	10.8762			0.836630
$170$		0			0
$171$	−0.666098			−0.0509377
$172$		0			0
$173$		−	5.15310i		−	0.391783i	−0.980626	−	0.195891i	$-0.937240\pi$
$173$		−	5.15310i		−	0.391783i	0.980626	−	0.195891i	$-0.0627600\pi$
$174$		0			0
$175$	−1.47037	−	6.79575i	−0.111149	−	0.513711i
$176$		0			0
$177$		−	3.25825i		−	0.244905i
$178$		0			0
$179$	−3.94980			−0.295222			−0.147611	−	0.989046i	$-0.547158\pi$
$179$	−3.94980			−0.295222			−0.147611	+	0.989046i	$0.547158\pi$
$180$		0			0
$181$	−2.51516			−0.186951			−0.0934754	−	0.995622i	$-0.529798\pi$
$181$	−2.51516			−0.186951			−0.0934754	+	0.995622i	$0.529798\pi$
$182$		0			0
$183$			6.68081i			0.493859i
$184$		0			0
$185$	0.649040	+	6.06889i	0.0477184	+	0.446194i
$186$		0			0
$187$			11.5573i			0.845153i
$188$		0			0
$189$	−1.39060			−0.101151
$190$		0			0
$191$	6.23208			0.450937			0.225469	−	0.974250i	$-0.427609\pi$
$191$	6.23208			0.450937			0.225469	+	0.974250i	$0.427609\pi$
$192$		0			0
$193$			2.43899i			0.175562i	0.996140	+	0.0877811i	$0.0279776\pi$
$193$			2.43899i			0.175562i	−0.996140	+	0.0877811i	$0.972022\pi$
$194$		0			0
$195$	−3.24022	+	0.346526i	−0.232037	+	0.0248153i
$196$		0			0
$197$			19.2377i			1.37063i	0.728247	+	0.685314i	$0.240335\pi$
$197$			19.2377i			1.37063i	−0.728247	+	0.685314i	$0.759665\pi$
$198$		0			0
$199$	26.7905			1.89913			0.949564	−	0.313574i	$-0.101526\pi$
$199$	26.7905			1.89913			0.949564	+	0.313574i	$0.101526\pi$
$200$		0			0
$201$	1.00000			0.0705346
$202$		0			0
$203$			3.84701i			0.270007i
$204$		0			0
$205$	19.4017	−	2.07493i	1.35507	−	0.144919i
$206$		0			0
$207$			3.19688i			0.222198i
$208$		0			0
$209$	1.21524			0.0840597
$210$		0			0
$211$	−11.1067			−0.764618			−0.382309	−	0.924035i	$-0.624871\pi$
$211$	−11.1067			−0.764618			−0.382309	+	0.924035i	$0.624871\pi$
$212$		0			0
$213$			3.41448i			0.233956i
$214$		0			0
$215$	−0.0966768	−	0.903982i	−0.00659330	−	0.0616511i
$216$		0			0
$217$			4.19930i			0.285067i
$218$		0			0
$219$	−6.44766			−0.435692
$220$		0			0
$221$	9.23191			0.621005
$222$		0			0
$223$		−	13.8549i		−	0.927790i	−0.885890	−	0.463895i	$-0.846452\pi$
$223$		−	13.8549i		−	0.927790i	0.885890	−	0.463895i	$-0.153548\pi$
$224$		0			0
$225$	−4.88692	+	1.05736i	−0.325795	+	0.0704908i
$226$		0			0
$227$			23.3208i			1.54786i	0.633272	+	0.773929i	$0.281712\pi$
$227$			23.3208i			1.54786i	−0.633272	+	0.773929i	$0.718288\pi$
$228$		0			0
$229$	−1.58939			−0.105030			−0.0525150	−	0.998620i	$-0.516724\pi$
$229$	−1.58939			−0.105030			−0.0525150	+	0.998620i	$0.516724\pi$
$230$		0			0
$231$	2.53703			0.166924
$232$		0			0
$233$		−	3.82548i		−	0.250616i	−0.992118	−	0.125308i	$-0.960008\pi$
$233$		−	3.82548i		−	0.250616i	0.992118	−	0.125308i	$-0.0399919\pi$
$234$		0			0
$235$	2.41766	+	22.6065i	0.157711	+	1.47468i
$236$		0			0
$237$			15.1051i			0.981179i
$238$		0			0
$239$	12.5263			0.810260			0.405130	−	0.914259i	$-0.367226\pi$
$239$	12.5263			0.810260			0.405130	+	0.914259i	$0.367226\pi$
$240$		0			0
$241$	−6.62063			−0.426472			−0.213236	−	0.977001i	$-0.568400\pi$
$241$	−6.62063			−0.426472			−0.213236	+	0.977001i	$0.568400\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−11.2642	+	1.20466i	−0.719643	+	0.0769626i
$246$		0			0
$247$		−	0.970725i		−	0.0617658i
$248$		0			0
$249$	2.66126			0.168651
$250$		0			0
$251$	7.34156			0.463395			0.231698	−	0.972788i	$-0.425572\pi$
$251$	7.34156			0.463395			0.231698	+	0.972788i	$0.425572\pi$
$252$		0			0
$253$		−	5.83243i		−	0.366681i
$254$		0			0
$255$	14.0847	−	1.50630i	0.882020	−	0.0943281i
$256$		0			0
$257$		−	3.59398i		−	0.224186i	−0.993698	−	0.112093i	$-0.964244\pi$
$257$		−	3.59398i		−	0.224186i	0.993698	−	0.112093i	$-0.0357555\pi$
$258$		0			0
$259$	−3.79574			−0.235856
$260$		0			0
$261$	2.76644			0.171238
$262$		0			0
$263$			19.1102i			1.17838i	0.807993	+	0.589191i	$0.200554\pi$
$263$			19.1102i			1.17838i	−0.807993	+	0.589191i	$0.799446\pi$
$264$		0			0
$265$	−0.373810	−	3.49534i	−0.0229630	−	0.214717i
$266$		0			0
$267$			15.2132i			0.931030i
$268$		0			0
$269$	10.6321			0.648248			0.324124	−	0.946015i	$-0.394931\pi$
$269$	10.6321			0.648248			0.324124	+	0.946015i	$0.394931\pi$
$270$		0			0
$271$	5.42042			0.329267			0.164634	−	0.986355i	$-0.447356\pi$
$271$	5.42042			0.329267			0.164634	+	0.986355i	$0.447356\pi$
$272$		0			0
$273$		−	2.02657i		−	0.122653i
$274$		0			0
$275$	8.91576	−	1.92906i	0.537640	−	0.116327i
$276$		0			0
$277$		−	7.04365i		−	0.423212i	−0.977355	−	0.211606i	$-0.932131\pi$
$277$		−	7.04365i		−	0.423212i	0.977355	−	0.211606i	$-0.0678694\pi$
$278$		0			0
$279$	3.01977			0.180789
$280$		0			0
$281$	17.4752			1.04248			0.521240	−	0.853410i	$-0.325469\pi$
$281$	17.4752			1.04248			0.521240	+	0.853410i	$0.325469\pi$
$282$		0			0
$283$			1.16212i			0.0690808i	0.999403	+	0.0345404i	$0.0109967\pi$
$283$			1.16212i			0.0690808i	−0.999403	+	0.0345404i	$0.989003\pi$
$284$		0			0
$285$	−0.158386	−	1.48099i	−0.00938195	−	0.0877265i
$286$		0			0
$287$			12.1346i			0.716285i
$288$		0			0
$289$	−23.1297			−1.36057
$290$		0			0
$291$	13.0028			0.762239
$292$		0			0
$293$		−	3.17912i		−	0.185726i	−0.995679	−	0.0928630i	$-0.970398\pi$
$293$		−	3.17912i		−	0.185726i	0.995679	−	0.0928630i	$-0.0296019\pi$
$294$		0			0
$295$	7.24436	−	0.774751i	0.421783	−	0.0451077i
$296$		0			0
$297$		−	1.82441i		−	0.105863i
$298$		0			0
$299$	−4.65891			−0.269432
$300$		0			0
$301$	0.565388			0.0325884
$302$		0			0
$303$			7.53421i			0.432829i
$304$		0			0
$305$	−14.8540	+	1.58857i	−0.850540	+	0.0909614i
$306$		0			0
$307$		−	14.5327i		−	0.829424i	−0.909953	−	0.414712i	$-0.863882\pi$
$307$		−	14.5327i		−	0.829424i	0.909953	−	0.414712i	$-0.136118\pi$
$308$		0			0
$309$	12.6775			0.721200
$310$		0			0
$311$	−24.8841			−1.41105			−0.705523	−	0.708687i	$-0.749288\pi$
$311$	−24.8841			−1.41105			−0.705523	+	0.708687i	$0.749288\pi$
$312$		0			0
$313$		−	32.3113i		−	1.82634i	−0.407578	−	0.913170i	$-0.633627\pi$
$313$		−	32.3113i		−	1.82634i	0.407578	−	0.913170i	$-0.366373\pi$
$314$		0			0
$315$	−0.330659	−	3.09185i	−0.0186305	−	0.174206i
$316$		0			0
$317$		−	3.80173i		−	0.213527i	−0.994284	−	0.106763i	$-0.965951\pi$
$317$		−	3.80173i		−	0.213527i	0.994284	−	0.106763i	$-0.0340487\pi$
$318$		0			0
$319$	−5.04712			−0.282585
$320$		0			0
$321$	−3.24648			−0.181201
$322$		0			0
$323$			4.21960i			0.234785i
$324$		0			0
$325$	−1.54093	−	7.12186i	−0.0854752	−	0.395050i
$326$		0			0
$327$			15.4343i			0.853520i
$328$		0			0
$329$	−14.1390			−0.779510
$330$		0			0
$331$	8.89401			0.488859			0.244429	−	0.969667i	$-0.421399\pi$
$331$	8.89401			0.488859			0.244429	+	0.969667i	$0.421399\pi$
$332$		0			0
$333$			2.72957i			0.149579i
$334$		0			0
$335$	0.237781	+	2.22339i	0.0129914	+	0.121477i
$336$		0			0
$337$			23.8171i			1.29740i	0.761045	+	0.648699i	$0.224687\pi$
$337$			23.8171i			1.29740i	−0.761045	+	0.648699i	$0.775313\pi$
$338$		0			0
$339$	−14.6944			−0.798093
$340$		0			0
$341$	−5.50931			−0.298346
$342$		0			0
$343$		−	16.7793i		−	0.905997i
$344$		0			0
$345$	−7.10791	+	0.760159i	−0.382677	+	0.0409256i
$346$		0			0
$347$		−	9.85261i		−	0.528916i	−0.964397	−	0.264458i	$-0.914807\pi$
$347$		−	9.85261i		−	0.528916i	0.964397	−	0.264458i	$-0.0851930\pi$
$348$		0			0
$349$	−4.79292			−0.256559			−0.128280	−	0.991738i	$-0.540945\pi$
$349$	−4.79292			−0.256559			−0.128280	+	0.991738i	$0.540945\pi$
$350$		0			0
$351$	−1.45733			−0.0777866
$352$		0			0
$353$		−	2.27830i		−	0.121262i	−0.998160	−	0.0606309i	$-0.980689\pi$
$353$		−	2.27830i		−	0.121262i	0.998160	−	0.0606309i	$-0.0193113\pi$
$354$		0			0
$355$	−7.59172	+	0.811900i	−0.402926	+	0.0430912i
$356$		0			0
$357$			8.80918i			0.466231i
$358$		0			0
$359$	25.9354			1.36882			0.684408	−	0.729099i	$-0.260061\pi$
$359$	25.9354			1.36882			0.684408	+	0.729099i	$0.260061\pi$
$360$		0			0
$361$	−18.5563			−0.976648
$362$		0			0
$363$		−	7.67152i		−	0.402650i
$364$		0			0
$365$	−1.53313	−	14.3356i	−0.0802478	−	0.750362i
$366$		0			0
$367$		−	5.93742i		−	0.309931i	−0.987920	−	0.154966i	$-0.950473\pi$
$367$		−	5.93742i		−	0.309931i	0.987920	−	0.154966i	$-0.0495266\pi$
$368$		0			0
$369$	8.72619			0.454267
$370$		0			0
$371$	2.18613			0.113498
$372$		0			0
$373$		−	0.268148i		−	0.0138842i	−0.999976	−	0.00694208i	$-0.997790\pi$
$373$		−	0.268148i		−	0.0138842i	0.999976	−	0.00694208i	$-0.00220975\pi$
$374$		0			0
$375$	−3.51294	−	10.6141i	−0.181408	−	0.548110i
$376$		0			0
$377$			4.03162i			0.207639i
$378$		0			0
$379$	34.4569			1.76993			0.884966	−	0.465655i	$-0.154181\pi$
$379$	34.4569			1.76993			0.884966	+	0.465655i	$0.154181\pi$
$380$		0			0
$381$	−16.9371			−0.867711
$382$		0			0
$383$		−	24.7915i		−	1.26679i	−0.773829	−	0.633394i	$-0.781661\pi$
$383$		−	24.7915i		−	1.26679i	0.773829	−	0.633394i	$-0.218339\pi$
$384$		0			0
$385$	0.603258	+	5.64080i	0.0307449	+	0.287482i
$386$		0			0
$387$		−	0.406578i		−	0.0206675i
$388$		0			0
$389$	−28.4119			−1.44054			−0.720270	−	0.693694i	$-0.755982\pi$
$389$	−28.4119			−1.44054			−0.720270	+	0.693694i	$0.755982\pi$
$390$		0			0
$391$	20.2516			1.02417
$392$		0			0
$393$			19.1372i			0.965345i
$394$		0			0
$395$	−33.5844	+	3.59170i	−1.68982	+	0.180718i
$396$		0			0
$397$		−	26.9086i		−	1.35050i	−0.737587	−	0.675252i	$-0.764035\pi$
$397$		−	26.9086i		−	1.35050i	0.737587	−	0.675252i	$-0.235965\pi$
$398$		0			0
$399$	0.926276			0.0463718
$400$		0			0
$401$	4.36395			0.217925			0.108963	−	0.994046i	$-0.465247\pi$
$401$	4.36395			0.217925			0.108963	+	0.994046i	$0.465247\pi$
$402$		0			0
$403$			4.40081i			0.219220i
$404$		0			0
$405$	−2.22339	+	0.237781i	−0.110481	+	0.0118155i
$406$		0			0
$407$		−	4.97985i		−	0.246842i
$408$		0			0
$409$	−0.144249			−0.00713267			−0.00356633	−	0.999994i	$-0.501135\pi$
$409$	−0.144249			−0.00713267			−0.00356633	+	0.999994i	$0.501135\pi$
$410$		0			0
$411$	−6.41529			−0.316443
$412$		0			0
$413$			4.53092i			0.222952i
$414$		0			0
$415$	0.632799	+	5.91702i	0.0310629	+	0.290455i
$416$		0			0
$417$		−	2.35871i		−	0.115507i
$418$		0			0
$419$	5.70744			0.278827			0.139413	−	0.990234i	$-0.455478\pi$
$419$	5.70744			0.278827			0.139413	+	0.990234i	$0.455478\pi$
$420$		0			0
$421$	−36.1779			−1.76321			−0.881603	−	0.471993i	$-0.843535\pi$
$421$	−36.1779			−1.76321			−0.881603	+	0.471993i	$0.843535\pi$
$422$		0			0
$423$			10.1676i			0.494364i
$424$		0			0
$425$	6.69818	+	30.9577i	0.324909	+	1.50167i
$426$		0			0
$427$		−	9.29034i		−	0.449591i
$428$		0			0
$429$	2.65877			0.128367
$430$		0			0
$431$	12.4901			0.601628			0.300814	−	0.953683i	$-0.402742\pi$
$431$	12.4901			0.601628			0.300814	+	0.953683i	$0.402742\pi$
$432$		0			0
$433$			25.4916i			1.22505i	0.790452	+	0.612525i	$0.209846\pi$
$433$			25.4916i			1.22505i	−0.790452	+	0.612525i	$0.790154\pi$
$434$		0			0
$435$	0.657808	+	6.15087i	0.0315395	+	0.294912i
$436$		0			0
$437$		−	2.12943i		−	0.101865i
$438$		0			0
$439$	18.5000			0.882957			0.441479	−	0.897272i	$-0.354454\pi$
$439$	18.5000			0.882957			0.441479	+	0.897272i	$0.354454\pi$
$440$		0			0
$441$	−5.06623			−0.241249
$442$		0			0
$443$			2.98174i			0.141667i	0.997488	+	0.0708334i	$0.0225659\pi$
$443$			2.98174i			0.141667i	−0.997488	+	0.0708334i	$0.977434\pi$
$444$		0			0
$445$	−33.8248	+	3.61741i	−1.60345	+	0.171481i
$446$		0			0
$447$			15.6986i			0.742516i
$448$		0			0
$449$	2.16132			0.101999			0.0509995	−	0.998699i	$-0.483759\pi$
$449$	2.16132			0.101999			0.0509995	+	0.998699i	$0.483759\pi$
$450$		0			0
$451$	−15.9202			−0.749651
$452$		0			0
$453$			4.06488i			0.190985i
$454$		0			0
$455$	4.50585	−	0.481880i	0.211237	−	0.0225909i
$456$		0			0
$457$		−	19.3780i		−	0.906464i	−0.891393	−	0.453232i	$-0.850271\pi$
$457$		−	19.3780i		−	0.906464i	0.891393	−	0.453232i	$-0.149729\pi$
$458$		0			0
$459$	6.33480			0.295683
$460$		0			0
$461$	−29.3444			−1.36670			−0.683352	−	0.730090i	$-0.739478\pi$
$461$	−29.3444			−1.36670			−0.683352	+	0.730090i	$0.739478\pi$
$462$		0			0
$463$			14.4629i			0.672149i	0.941835	+	0.336074i	$0.109099\pi$
$463$			14.4629i			0.672149i	−0.941835	+	0.336074i	$0.890901\pi$
$464$		0			0
$465$	0.718046	+	6.71413i	0.0332986	+	0.311360i
$466$		0			0
$467$		−	38.4318i		−	1.77841i	−0.457507	−	0.889206i	$-0.651257\pi$
$467$		−	38.4318i		−	1.77841i	0.457507	−	0.889206i	$-0.348743\pi$
$468$		0			0
$469$	−1.39060			−0.0642120
$470$		0			0
$471$	20.1288			0.927485
$472$		0			0
$473$			0.741766i			0.0341065i
$474$		0			0
$475$	3.25517	−	0.704306i	0.149357	−	0.0323158i
$476$		0			0
$477$		−	1.57208i		−	0.0719804i
$478$		0			0
$479$	21.3424			0.975159			0.487579	−	0.873079i	$-0.337880\pi$
$479$	21.3424			0.975159			0.487579	+	0.873079i	$0.337880\pi$
$480$		0			0
$481$	−3.97788			−0.181376
$482$		0			0
$483$		−	4.44558i		−	0.202281i
$484$		0			0
$485$	3.09183	+	28.9103i	0.140393	+	1.31275i
$486$		0			0
$487$			26.7184i			1.21073i	0.795949	+	0.605363i	$0.206972\pi$
$487$			26.7184i			1.21073i	−0.795949	+	0.605363i	$0.793028\pi$
$488$		0			0
$489$	14.6538			0.662668
$490$		0			0
$491$	−2.03516			−0.0918453			−0.0459227	−	0.998945i	$-0.514623\pi$
$491$	−2.03516			−0.0918453			−0.0459227	+	0.998945i	$0.514623\pi$
$492$		0			0
$493$		−	17.5248i		−	0.789279i
$494$		0			0
$495$	4.05638	−	0.433811i	0.182321	−	0.0194984i
$496$		0			0
$497$		−	4.74818i		−	0.212985i
$498$		0			0
$499$	1.31221			0.0587426			0.0293713	−	0.999569i	$-0.490649\pi$
$499$	1.31221			0.0587426			0.0293713	+	0.999569i	$0.490649\pi$
$500$		0			0
$501$	−12.8759			−0.575253
$502$		0			0
$503$		−	13.3315i		−	0.594423i	−0.954812	−	0.297212i	$-0.903943\pi$
$503$		−	13.3315i		−	0.594423i	0.954812	−	0.297212i	$-0.0960566\pi$
$504$		0			0
$505$	−16.7515	+	1.79149i	−0.745431	+	0.0797205i
$506$		0			0
$507$			10.8762i			0.483028i
$508$		0			0
$509$	−5.73525			−0.254211			−0.127105	−	0.991889i	$-0.540569\pi$
$509$	−5.73525			−0.254211			−0.127105	+	0.991889i	$0.540569\pi$
$510$		0			0
$511$	8.96611			0.396638
$512$		0			0
$513$		−	0.666098i		−	0.0294089i
$514$		0			0
$515$	3.01448	+	28.1871i	0.132834	+	1.24207i
$516$		0			0
$517$		−	18.5498i		−	0.815820i
$518$		0			0
$519$	5.15310			0.226196
$520$		0			0
$521$	−26.0379			−1.14074			−0.570370	−	0.821388i	$-0.693200\pi$
$521$	−26.0379			−1.14074			−0.570370	+	0.821388i	$0.693200\pi$
$522$		0			0
$523$			1.04151i			0.0455423i	0.999741	+	0.0227711i	$0.00724890\pi$
$523$			1.04151i			0.0455423i	−0.999741	+	0.0227711i	$0.992751\pi$
$524$		0			0
$525$	6.79575	−	1.47037i	0.296591	−	0.0641721i
$526$		0			0
$527$		−	19.1297i		−	0.833302i
$528$		0			0
$529$	12.7800			0.555650
$530$		0			0
$531$	3.25825			0.141396
$532$		0			0
$533$			12.7170i			0.550832i
$534$		0			0
$535$	−0.771952	−	7.21818i	−0.0333744	−	0.312069i
$536$		0			0
$537$		−	3.94980i		−	0.170446i
$538$		0			0
$539$	9.24289			0.398119
$540$		0			0
$541$	25.3064			1.08801			0.544003	−	0.839083i	$-0.316908\pi$
$541$	25.3064			1.08801			0.544003	+	0.839083i	$0.316908\pi$
$542$		0			0
$543$		−	2.51516i		−	0.107936i
$544$		0			0
$545$	−34.3165	+	3.67000i	−1.46996	+	0.157205i
$546$		0			0
$547$			12.1676i			0.520248i	0.965575	+	0.260124i	$0.0837634\pi$
$547$			12.1676i			0.520248i	−0.965575	+	0.260124i	$0.916237\pi$
$548$		0			0
$549$	−6.68081			−0.285130
$550$		0			0
$551$	−1.84272			−0.0785024
$552$		0			0
$553$		−	21.0051i		−	0.893228i
$554$		0			0
$555$	−6.06889	+	0.649040i	−0.257610	+	0.0275502i
$556$		0			0
$557$		−	23.1359i		−	0.980301i	−0.871638	−	0.490151i	$-0.836942\pi$
$557$		−	23.1359i		−	0.980301i	0.871638	−	0.490151i	$-0.163058\pi$
$558$		0			0
$559$	0.592520			0.0250609
$560$		0			0
$561$	−11.5573			−0.487949
$562$		0			0
$563$			11.2956i			0.476052i	0.971259	+	0.238026i	$0.0765003\pi$
$563$			11.2956i			0.476052i	−0.971259	+	0.238026i	$0.923500\pi$
$564$		0			0
$565$	−3.49407	−	32.6715i	−0.146996	−	1.37450i
$566$		0			0
$567$		−	1.39060i		−	0.0583997i
$568$		0			0
$569$	22.0712			0.925272			0.462636	−	0.886548i	$-0.346904\pi$
$569$	22.0712			0.925272			0.462636	+	0.886548i	$0.346904\pi$
$570$		0			0
$571$	21.8508			0.914426			0.457213	−	0.889357i	$-0.348848\pi$
$571$	21.8508			0.914426			0.457213	+	0.889357i	$0.348848\pi$
$572$		0			0
$573$			6.23208i			0.260349i
$574$		0			0
$575$	−3.38026	−	15.6229i	−0.140966	−	0.651520i
$576$		0			0
$577$			9.53816i			0.397079i	0.980093	+	0.198539i	$0.0636198\pi$
$577$			9.53816i			0.397079i	−0.980093	+	0.198539i	$0.936380\pi$
$578$		0			0
$579$	−2.43899			−0.101361
$580$		0			0
$581$	−3.70075			−0.153533
$582$		0			0
$583$			2.86811i			0.118785i
$584$		0			0
$585$	−0.346526	−	3.24022i	−0.0143271	−	0.133966i
$586$		0			0
$587$		−	43.6613i		−	1.80210i	−0.433719	−	0.901048i	$-0.642799\pi$
$587$		−	43.6613i		−	1.80210i	0.433719	−	0.901048i	$-0.357201\pi$
$588$		0			0
$589$	−2.01146			−0.0828809
$590$		0			0
$591$	−19.2377			−0.791333
$592$		0			0
$593$			12.7622i			0.524079i	0.965057	+	0.262039i	$0.0843950\pi$
$593$			12.7622i			0.524079i	−0.965057	+	0.262039i	$0.915605\pi$
$594$		0			0
$595$	−19.5862	+	2.09466i	−0.802958	+	0.0858727i
$596$		0			0
$597$			26.7905i			1.09646i
$598$		0			0
$599$	−39.0679			−1.59627			−0.798135	−	0.602479i	$-0.794180\pi$
$599$	−39.0679			−1.59627			−0.798135	+	0.602479i	$0.794180\pi$
$600$		0			0
$601$	−8.39897			−0.342601			−0.171301	−	0.985219i	$-0.554797\pi$
$601$	−8.39897			−0.342601			−0.171301	+	0.985219i	$0.554797\pi$
$602$		0			0
$603$			1.00000i			0.0407231i
$604$		0			0
$605$	17.0568	−	1.82415i	0.693457	−	0.0741620i
$606$		0			0
$607$			2.01435i			0.0817599i	0.999164	+	0.0408799i	$0.0130161\pi$
$607$			2.01435i			0.0817599i	−0.999164	+	0.0408799i	$0.986984\pi$
$608$		0			0
$609$	−3.84701			−0.155889
$610$		0			0
$611$	−14.8175			−0.599452
$612$		0			0
$613$		−	1.49351i		−	0.0603221i	−0.999545	−	0.0301611i	$-0.990398\pi$
$613$		−	1.49351i		−	0.0603221i	0.999545	−	0.0301611i	$-0.00960202\pi$
$614$		0			0
$615$	2.07493	+	19.4017i	0.0836691	+	0.782353i
$616$		0			0
$617$		−	41.4277i		−	1.66781i	−0.551904	−	0.833907i	$-0.686099\pi$
$617$		−	41.4277i		−	1.66781i	0.551904	−	0.833907i	$-0.313901\pi$
$618$		0			0
$619$	−9.21773			−0.370492			−0.185246	−	0.982692i	$-0.559308\pi$
$619$	−9.21773			−0.370492			−0.185246	+	0.982692i	$0.559308\pi$
$620$		0			0
$621$	−3.19688			−0.128286
$622$		0			0
$623$		−	21.1554i		−	0.847574i
$624$		0			0
$625$	22.7640	−	10.3345i	0.910559	−	0.413379i
$626$		0			0
$627$			1.21524i			0.0485319i
$628$		0			0
$629$	17.2913			0.689448
$630$		0			0
$631$	24.2213			0.964235			0.482117	−	0.876107i	$-0.339868\pi$
$631$	24.2213			0.964235			0.482117	+	0.876107i	$0.339868\pi$
$632$		0			0
$633$		−	11.1067i		−	0.441452i
$634$		0			0
$635$	−4.02732	−	37.6577i	−0.159819	−	1.49440i
$636$		0			0
$637$		−	7.38318i		−	0.292532i
$638$		0			0
$639$	−3.41448			−0.135075
$640$		0			0
$641$	32.4655			1.28231			0.641155	−	0.767411i	$-0.278455\pi$
$641$	32.4655			1.28231			0.641155	+	0.767411i	$0.278455\pi$
$642$		0			0
$643$		−	25.7206i		−	1.01432i	−0.861851	−	0.507161i	$-0.830695\pi$
$643$		−	25.7206i		−	1.01432i	0.861851	−	0.507161i	$-0.169305\pi$
$644$		0			0
$645$	0.903982	−	0.0966768i	0.0355943	−	0.00380664i
$646$		0			0
$647$			17.3169i			0.680796i	0.940281	+	0.340398i	$0.110562\pi$
$647$			17.3169i			0.680796i	−0.940281	+	0.340398i	$0.889438\pi$
$648$		0			0
$649$	−5.94439			−0.233338
$650$		0			0
$651$	−4.19930			−0.164583
$652$		0			0
$653$			25.1296i			0.983396i	0.870766	+	0.491698i	$0.163624\pi$
$653$			25.1296i			0.983396i	−0.870766	+	0.491698i	$0.836376\pi$
$654$		0			0
$655$	−42.5495	+	4.55048i	−1.66255	+	0.177802i
$656$		0			0
$657$		−	6.44766i		−	0.251547i
$658$		0			0
$659$	9.48428			0.369455			0.184728	−	0.982790i	$-0.440860\pi$
$659$	9.48428			0.369455			0.184728	+	0.982790i	$0.440860\pi$
$660$		0			0
$661$	13.3968			0.521076			0.260538	−	0.965464i	$-0.416100\pi$
$661$	13.3968			0.521076			0.260538	+	0.965464i	$0.416100\pi$
$662$		0			0
$663$			9.23191i			0.358538i
$664$		0			0
$665$	0.220251	+	2.05947i	0.00854097	+	0.0798629i
$666$		0			0
$667$			8.84397i			0.342440i
$668$		0			0
$669$	13.8549			0.535660
$670$		0			0
$671$	12.1885			0.470534
$672$		0			0
$673$		−	8.93996i		−	0.344610i	−0.985044	−	0.172305i	$-0.944879\pi$
$673$		−	8.93996i		−	0.344610i	0.985044	−	0.172305i	$-0.0551215\pi$
$674$		0			0
$675$	−1.05736	−	4.88692i	−0.0406979	−	0.188098i
$676$		0			0
$677$		−	0.229227i		−	0.00880989i	−0.999990	−	0.00440495i	$-0.998598\pi$
$677$		−	0.229227i		−	0.00880989i	0.999990	−	0.00440495i	$-0.00140214\pi$
$678$		0			0
$679$	−18.0817			−0.693913
$680$		0			0
$681$	−23.3208			−0.893657
$682$		0			0
$683$			7.80200i			0.298535i	0.988797	+	0.149268i	$0.0476916\pi$
$683$			7.80200i			0.298535i	−0.988797	+	0.149268i	$0.952308\pi$
$684$		0			0
$685$	−1.52544	−	14.2637i	−0.0582839	−	0.544987i
$686$		0			0
$687$		−	1.58939i		−	0.0606391i
$688$		0			0
$689$	2.29104			0.0872815
$690$		0			0
$691$	−20.4808			−0.779127			−0.389564	−	0.921000i	$-0.627374\pi$
$691$	−20.4808			−0.779127			−0.389564	+	0.921000i	$0.627374\pi$
$692$		0			0
$693$			2.53703i			0.0963737i
$694$		0			0
$695$	5.24433	−	0.560857i	0.198929	−	0.0212745i
$696$		0			0
$697$		−	55.2787i		−	2.09383i
$698$		0			0
$699$	3.82548			0.144693
$700$		0			0
$701$	−18.2710			−0.690086			−0.345043	−	0.938587i	$-0.612136\pi$
$701$	−18.2710			−0.690086			−0.345043	+	0.938587i	$0.612136\pi$
$702$		0			0
$703$		−	1.81816i		−	0.0685731i
$704$		0			0
$705$	−22.6065	+	2.41766i	−0.851408	+	0.0910543i
$706$		0			0
$707$		−	10.4771i		−	0.394031i
$708$		0			0
$709$	−26.4900			−0.994853			−0.497427	−	0.867506i	$-0.665722\pi$
$709$	−26.4900			−0.994853			−0.497427	+	0.867506i	$0.665722\pi$
$710$		0			0
$711$	−15.1051			−0.566484
$712$		0			0
$713$			9.65385i			0.361540i
$714$		0			0
$715$	0.632207	+	5.91149i	0.0236432	+	0.221077i
$716$		0			0
$717$			12.5263i			0.467804i
$718$		0			0
$719$	36.3250			1.35469			0.677347	−	0.735664i	$-0.263130\pi$
$719$	36.3250			1.35469			0.677347	+	0.735664i	$0.263130\pi$
$720$		0			0
$721$	−17.6294			−0.656553
$722$		0			0
$723$		−	6.62063i		−	0.246224i
$724$		0			0
$725$	−13.5194	+	2.92513i	−0.502097	+	0.108636i
$726$		0			0
$727$			18.3394i			0.680171i	0.940394	+	0.340086i	$0.110456\pi$
$727$			18.3394i			0.680171i	−0.940394	+	0.340086i	$0.889544\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−2.57559			−0.0952618
$732$		0			0
$733$		−	3.98303i		−	0.147117i	−0.997291	−	0.0735583i	$-0.976565\pi$
$733$		−	3.98303i		−	0.147117i	0.997291	−	0.0735583i	$-0.0234355\pi$
$734$		0			0
$735$	−1.20466	−	11.2642i	−0.0444344	−	0.415486i
$736$		0			0
$737$		−	1.82441i		−	0.0672031i
$738$		0			0
$739$	−1.74243			−0.0640962			−0.0320481	−	0.999486i	$-0.510203\pi$
$739$	−1.74243			−0.0640962			−0.0320481	+	0.999486i	$0.510203\pi$
$740$		0			0
$741$	0.970725			0.0356605
$742$		0			0
$743$			8.57132i			0.314451i	0.987563	+	0.157226i	$0.0502550\pi$
$743$			8.57132i			0.314451i	−0.987563	+	0.157226i	$0.949745\pi$
$744$		0			0
$745$	−34.9040	+	3.73283i	−1.27878	+	0.136760i
$746$		0			0
$747$			2.66126i			0.0973705i
$748$		0			0
$749$	4.51455			0.164958
$750$		0			0
$751$	17.3695			0.633821			0.316910	−	0.948456i	$-0.397355\pi$
$751$	17.3695			0.633821			0.316910	+	0.948456i	$0.397355\pi$
$752$		0			0
$753$			7.34156i			0.267541i
$754$		0			0
$755$	−9.03781	+	0.966553i	−0.328920	+	0.0351765i
$756$		0			0
$757$		−	26.3124i		−	0.956339i	−0.878268	−	0.478170i	$-0.841300\pi$
$757$		−	26.3124i		−	0.956339i	0.878268	−	0.478170i	$-0.158700\pi$
$758$		0			0
$759$	5.83243			0.211704
$760$		0			0
$761$	4.81399			0.174507			0.0872535	−	0.996186i	$-0.472191\pi$
$761$	4.81399			0.174507			0.0872535	+	0.996186i	$0.472191\pi$
$762$		0			0
$763$		−	21.4630i		−	0.777012i
$764$		0			0
$765$	1.50630	+	14.0847i	0.0544603	+	0.509235i
$766$		0			0
$767$			4.74835i			0.171453i
$768$		0			0
$769$	−38.7444			−1.39716			−0.698579	−	0.715533i	$-0.746184\pi$
$769$	−38.7444			−1.39716			−0.698579	+	0.715533i	$0.746184\pi$
$770$		0			0
$771$	3.59398			0.129434
$772$		0			0
$773$		−	52.9095i		−	1.90302i	−0.307616	−	0.951511i	$-0.599531\pi$
$773$		−	52.9095i		−	1.90302i	0.307616	−	0.951511i	$-0.400469\pi$
$774$		0			0
$775$	−14.7574	+	3.19299i	−0.530101	+	0.114696i
$776$		0			0
$777$		−	3.79574i		−	0.136171i
$778$		0			0
$779$	−5.81249			−0.208254
$780$		0			0
$781$	6.22942			0.222906
$782$		0			0
$783$			2.76644i			0.0988644i
$784$		0			0
$785$	4.78625	+	44.7541i	0.170829	+	1.59734i
$786$		0			0
$787$		−	23.6246i		−	0.842127i	−0.907031	−	0.421064i	$-0.861657\pi$
$787$		−	23.6246i		−	0.842127i	0.907031	−	0.421064i	$-0.138343\pi$
$788$		0			0
$789$	−19.1102			−0.680340
$790$		0			0
$791$	20.4341			0.726553
$792$		0			0
$793$		−	9.73615i		−	0.345741i
$794$		0			0
$795$	3.49534	−	0.373810i	0.123967	−	0.0132577i
$796$		0			0
$797$		−	37.3841i		−	1.32421i	−0.749410	−	0.662106i	$-0.769663\pi$
$797$		−	37.3841i		−	1.32421i	0.749410	−	0.662106i	$-0.230337\pi$
$798$		0			0
$799$	64.4095			2.27865
$800$		0			0
$801$	−15.2132			−0.537530
$802$		0			0
$803$			11.7632i			0.415114i
$804$		0			0
$805$	9.88426	−	1.05708i	0.348374	−	0.0372571i
$806$		0			0
$807$			10.6321i			0.374266i
$808$		0			0
$809$	7.68015			0.270020			0.135010	−	0.990844i	$-0.456893\pi$
$809$	7.68015			0.270020			0.135010	+	0.990844i	$0.456893\pi$
$810$		0			0
$811$	23.1303			0.812215			0.406107	−	0.913825i	$-0.366886\pi$
$811$	23.1303			0.812215			0.406107	+	0.913825i	$0.366886\pi$
$812$		0			0
$813$			5.42042i			0.190103i
$814$		0			0
$815$	3.48440	+	32.5811i	0.122053	+	1.14127i
$816$		0			0
$817$			0.270821i			0.00947482i
$818$		0			0
$819$	2.02657			0.0708140
$820$		0			0
$821$	−48.9539			−1.70850			−0.854251	−	0.519860i	$-0.825984\pi$
$821$	−48.9539			−1.70850			−0.854251	+	0.519860i	$0.825984\pi$
$822$		0			0
$823$		−	18.6832i		−	0.651255i	−0.945498	−	0.325627i	$-0.894425\pi$
$823$		−	18.6832i		−	0.651255i	0.945498	−	0.325627i	$-0.105575\pi$
$824$		0			0
$825$	1.92906	+	8.91576i	0.0671613	+	0.310407i
$826$		0			0
$827$			49.0107i			1.70427i	0.523321	+	0.852135i	$0.324693\pi$
$827$			49.0107i			1.70427i	−0.523321	+	0.852135i	$0.675307\pi$
$828$		0			0
$829$	−30.3303			−1.05342			−0.526708	−	0.850046i	$-0.676574\pi$
$829$	−30.3303			−1.05342			−0.526708	+	0.850046i	$0.676574\pi$
$830$		0			0
$831$	7.04365			0.244342
$832$		0			0
$833$			32.0936i			1.11198i
$834$		0			0
$835$	−3.06165	−	28.6282i	−0.105953	−	0.990719i
$836$		0			0
$837$			3.01977i			0.104379i
$838$		0			0
$839$	18.8402			0.650437			0.325218	−	0.945639i	$-0.394562\pi$
$839$	18.8402			0.650437			0.325218	+	0.945639i	$0.394562\pi$
$840$		0			0
$841$	−21.3468			−0.736097
$842$		0			0
$843$			17.4752i			0.601877i
$844$		0			0
$845$	−24.1820	+	2.58615i	−0.831886	+	0.0889664i
$846$		0			0
$847$			10.6680i			0.366558i
$848$		0			0
$849$	−1.16212			−0.0398838
$850$		0			0
$851$	−8.72609			−0.299127
$852$		0			0
$853$		−	3.72313i		−	0.127478i	−0.997967	−	0.0637389i	$-0.979698\pi$
$853$		−	3.72313i		−	0.127478i	0.997967	−	0.0637389i	$-0.0203025\pi$
$854$		0			0
$855$	1.48099	−	0.158386i	0.0506489	−	0.00541667i
$856$		0			0
$857$			34.9828i			1.19499i	0.801872	+	0.597495i	$0.203837\pi$
$857$			34.9828i			1.19499i	−0.801872	+	0.597495i	$0.796163\pi$
$858$		0			0
$859$	23.5460			0.803381			0.401690	−	0.915776i	$-0.368423\pi$
$859$	23.5460			0.803381			0.401690	+	0.915776i	$0.368423\pi$
$860$		0			0
$861$	−12.1346			−0.413548
$862$		0			0
$863$			46.6988i			1.58965i	0.606842	+	0.794823i	$0.292436\pi$
$863$			46.6988i			1.58965i	−0.606842	+	0.794823i	$0.707564\pi$
$864$		0			0
$865$	1.22531	+	11.4573i	0.0416618	+	0.389561i
$866$		0			0
$867$		−	23.1297i		−	0.785527i
$868$		0			0
$869$	27.5578			0.934836
$870$		0			0
$871$	−1.45733			−0.0493798
$872$		0			0
$873$			13.0028i			0.440079i
$874$		0			0
$875$	4.88510	+	14.7600i	0.165147	+	0.498978i
$876$		0			0
$877$			10.7300i			0.362325i	0.983453	+	0.181162i	$0.0579860\pi$
$877$			10.7300i			0.362325i	−0.983453	+	0.181162i	$0.942014\pi$
$878$		0			0
$879$	3.17912			0.107229
$880$		0			0
$881$	−9.47980			−0.319383			−0.159691	−	0.987167i	$-0.551050\pi$
$881$	−9.47980			−0.319383			−0.159691	+	0.987167i	$0.551050\pi$
$882$		0			0
$883$			28.6783i			0.965103i	0.875868	+	0.482552i	$0.160290\pi$
$883$			28.6783i			0.965103i	−0.875868	+	0.482552i	$0.839710\pi$
$884$		0			0
$885$	0.774751	+	7.24436i	0.0260430	+	0.243516i
$886$		0			0
$887$			20.3975i			0.684881i	0.939539	+	0.342441i	$0.111254\pi$
$887$			20.3975i			0.684881i	−0.939539	+	0.342441i	$0.888746\pi$
$888$		0			0
$889$	23.5527			0.789931
$890$		0			0
$891$	1.82441			0.0611201
$892$		0			0
$893$		−	6.77259i		−	0.226636i
$894$		0			0
$895$	8.78194	−	0.939189i	0.293548	−	0.0313936i
$896$		0			0
$897$		−	4.65891i		−	0.155557i
$898$		0			0
$899$	8.35402			0.278622
$900$		0			0
$901$	−9.95879			−0.331775
$902$		0			0
$903$			0.565388i			0.0188149i
$904$		0			0
$905$	5.59219	−	0.598059i	0.185891	−	0.0198802i
$906$		0			0
$907$			8.66809i			0.287819i	0.989591	+	0.143910i	$0.0459674\pi$
$907$			8.66809i			0.287819i	−0.989591	+	0.143910i	$0.954033\pi$
$908$		0			0
$909$	−7.53421			−0.249894
$910$		0			0
$911$	−38.2448			−1.26711			−0.633553	−	0.773700i	$-0.718404\pi$
$911$	−38.2448			−1.26711			−0.633553	+	0.773700i	$0.718404\pi$
$912$		0			0
$913$		−	4.85524i		−	0.160685i
$914$		0			0
$915$	−1.58857	−	14.8540i	−0.0525166	−	0.491059i
$916$		0			0
$917$		−	26.6122i		−	0.878814i
$918$		0			0
$919$	−29.5263			−0.973984			−0.486992	−	0.873406i	$-0.661906\pi$
$919$	−29.5263			−0.973984			−0.486992	+	0.873406i	$0.661906\pi$
$920$		0			0
$921$	14.5327			0.478868
$922$		0			0
$923$		−	4.97603i		−	0.163788i
$924$		0			0
$925$	−2.88614	−	13.3392i	−0.0948956	−	0.438589i
$926$		0			0
$927$			12.6775i			0.416385i
$928$		0			0
$929$	−34.8196			−1.14239			−0.571197	−	0.820813i	$-0.693521\pi$
$929$	−34.8196			−1.14239			−0.571197	+	0.820813i	$0.693521\pi$
$930$		0			0
$931$	3.37460			0.110598
$932$		0			0
$933$		−	24.8841i		−	0.814668i
$934$		0			0
$935$	−2.74811	−	25.6964i	−0.0898728	−	0.840361i
$936$		0			0
$937$			60.6511i			1.98138i	0.136124	+	0.990692i	$0.456536\pi$
$937$			60.6511i			1.98138i	−0.136124	+	0.990692i	$0.543464\pi$
$938$		0			0
$939$	32.3113			1.05444
$940$		0			0
$941$	−13.0899			−0.426719			−0.213359	−	0.976974i	$-0.568441\pi$
$941$	−13.0899			−0.426719			−0.213359	+	0.976974i	$0.568441\pi$
$942$		0			0
$943$			27.8966i			0.908438i
$944$		0			0
$945$	3.09185	−	0.330659i	0.100578	−	0.0107563i
$946$		0			0
$947$		−	34.6325i		−	1.12541i	−0.826659	−	0.562703i	$-0.809762\pi$
$947$		−	34.6325i		−	1.12541i	0.826659	−	0.562703i	$-0.190238\pi$
$948$		0			0
$949$	9.39637			0.305019
$950$		0			0
$951$	3.80173			0.123280
$952$		0			0
$953$			16.2682i			0.526977i	0.964662	+	0.263489i	$0.0848732\pi$
$953$			16.2682i			0.526977i	−0.964662	+	0.263489i	$0.915127\pi$
$954$		0			0
$955$	−13.8563	+	1.48187i	−0.448380	+	0.0479523i
$956$		0			0
$957$		−	5.04712i		−	0.163150i
$958$		0			0
$959$	8.92110			0.288077
$960$		0			0
$961$	−21.8810			−0.705838
$962$		0			0
$963$		−	3.24648i		−	0.104616i
$964$		0			0
$965$	−0.579946	−	5.42282i	−0.0186691	−	0.174567i
$966$		0			0
$967$			19.7927i			0.636489i	0.948009	+	0.318245i	$0.103093\pi$
$967$			19.7927i			0.636489i	−0.948009	+	0.318245i	$0.896907\pi$
$968$		0			0
$969$	−4.21960			−0.135553
$970$		0			0
$971$	−53.0842			−1.70355			−0.851776	−	0.523906i	$-0.824474\pi$
$971$	−53.0842			−1.70355			−0.851776	+	0.523906i	$0.824474\pi$
$972$		0			0
$973$			3.28002i			0.105153i
$974$		0			0
$975$	7.12186	−	1.54093i	0.228082	−	0.0493491i
$976$		0			0
$977$			9.63017i			0.308096i	0.988063	+	0.154048i	$0.0492311\pi$
$977$			9.63017i			0.308096i	−0.988063	+	0.154048i	$0.950769\pi$
$978$		0			0
$979$	27.7551			0.887056
$980$		0			0
$981$	−15.4343			−0.492780
$982$		0			0
$983$		−	37.8897i		−	1.20849i	−0.796798	−	0.604246i	$-0.793474\pi$
$983$		−	37.8897i		−	1.20849i	0.796798	−	0.604246i	$-0.206526\pi$
$984$		0			0
$985$	−4.57436	−	42.7729i	−0.145751	−	1.36286i
$986$		0			0
$987$		−	14.1390i		−	0.450050i
$988$		0			0
$989$	1.29978			0.0413307
$990$		0			0
$991$	−32.5911			−1.03529			−0.517645	−	0.855596i	$-0.673191\pi$
$991$	−32.5911			−1.03529			−0.517645	+	0.855596i	$0.673191\pi$
$992$		0			0
$993$			8.89401i			0.282243i
$994$		0			0
$995$	−59.5657	+	6.37028i	−1.88836	+	0.201951i
$996$		0			0
$997$			3.38853i			0.107316i	0.998559	+	0.0536579i	$0.0170880\pi$
$997$			3.38853i			0.107316i	−0.998559	+	0.0536579i	$0.982912\pi$
$998$		0			0
$999$	−2.72957			−0.0863597

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.g.b.1609.13	yes	24
5.4	even	2	inner	4020.2.g.b.1609.1	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.g.b.1609.1	✓	24	5.4	even	2	inner
4020.2.g.b.1609.13	yes	24	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-2.22339 + 0.237781i) q^{5} -1.39060i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(-2.22339 + 0.237781i) q^{5} -1.39060i q^{7} -1.00000 q^{9} +1.82441 q^{11} -1.45733i q^{13} +(-0.237781 - 2.22339i) q^{15} +6.33480i q^{17} +0.666098 q^{19} +1.39060 q^{21} -3.19688i q^{23} +(4.88692 - 1.05736i) q^{25} -1.00000i q^{27} -2.76644 q^{29} -3.01977 q^{31} +1.82441i q^{33} +(0.330659 + 3.09185i) q^{35} -2.72957i q^{37} +1.45733 q^{39} -8.72619 q^{41} +0.406578i q^{43} +(2.22339 - 0.237781i) q^{45} -10.1676i q^{47} +5.06623 q^{49} -6.33480 q^{51} +1.57208i q^{53} +(-4.05638 + 0.433811i) q^{55} +0.666098i q^{57} -3.25825 q^{59} +6.68081 q^{61} +1.39060i q^{63} +(0.346526 + 3.24022i) q^{65} -1.00000i q^{67} +3.19688 q^{69} +3.41448 q^{71} +6.44766i q^{73} +(1.05736 + 4.88692i) q^{75} -2.53703i q^{77} +15.1051 q^{79} +1.00000 q^{81} -2.66126i q^{83} +(-1.50630 - 14.0847i) q^{85} -2.76644i q^{87} +15.2132 q^{89} -2.02657 q^{91} -3.01977i q^{93} +(-1.48099 + 0.158386i) q^{95} -13.0028i q^{97} -1.82441 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 4 q^{5} - 24 q^{9}+O(q^{10}) \) 24 * q - 4 * q^5 - 24 * q^9	\( 24 q - 4 q^{5} - 24 q^{9} - 8 q^{11} - 2 q^{15} + 16 q^{19} - 20 q^{21} + 10 q^{25} + 36 q^{29} - 2 q^{35} + 4 q^{39} - 24 q^{41} + 4 q^{45} - 4 q^{51} - 4 q^{55} + 24 q^{59} - 4 q^{61} - 20 q^{65} - 4 q^{69} + 20 q^{71} - 12 q^{75} - 28 q^{79} + 24 q^{81} - 16 q^{85} + 48 q^{89} - 20 q^{91} - 4 q^{95} + 8 q^{99}+O(q^{100}) \) 24 * q - 4 * q^5 - 24 * q^9 - 8 * q^11 - 2 * q^15 + 16 * q^19 - 20 * q^21 + 10 * q^25 + 36 * q^29 - 2 * q^35 + 4 * q^39 - 24 * q^41 + 4 * q^45 - 4 * q^51 - 4 * q^55 + 24 * q^59 - 4 * q^61 - 20 * q^65 - 4 * q^69 + 20 * q^71 - 12 * q^75 - 28 * q^79 + 24 * q^81 - 16 * q^85 + 48 * q^89 - 20 * q^91 - 4 * q^95 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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