LMFDB - Embedded newform 4032.2.b.n.3583.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(3583,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4032.3583");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.b (of order $2$, degree $1$, not 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:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2312.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x + 4 $ x^4 - x^3 - 2*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3583.1
Root			$1.28078 - 0.599676i$ of defining polynomial
Character	$\chi$	$=$	4032.3583
Dual form			4032.2.b.n.3583.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.33513i q^{5} +(-1.56155 - 2.13578i) q^{7} +O(q^{10})$	$q-3.33513i q^{5} +(-1.56155 - 2.13578i) q^{7} -0.936426i q^{11} +1.87285i q^{13} -5.20798i q^{17} -7.12311 q^{19} -0.936426i q^{23} -6.12311 q^{25} -2.00000 q^{29} +(-7.12311 + 5.20798i) q^{35} -1.12311 q^{37} -1.46228i q^{41} +9.06897i q^{43} -6.24621 q^{47} +(-2.12311 + 6.67026i) q^{49} +12.2462 q^{53} -3.12311 q^{55} +4.00000 q^{59} +4.79741i q^{61} +6.24621 q^{65} -10.9418i q^{67} -3.86098i q^{71} -6.67026i q^{73} +(-2.00000 + 1.46228i) q^{77} +2.39871i q^{79} -10.2462 q^{83} -17.3693 q^{85} +1.46228i q^{89} +(4.00000 - 2.92456i) q^{91} +23.7565i q^{95} +10.4160i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{7}+O(q^{10}) $ 4 * q + 2 * q^7	$ 4 q + 2 q^{7} - 12 q^{19} - 8 q^{25} - 8 q^{29} - 12 q^{35} + 12 q^{37} + 8 q^{47} + 8 q^{49} + 16 q^{53} + 4 q^{55} + 16 q^{59} - 8 q^{65} - 8 q^{77} - 8 q^{83} - 20 q^{85} + 16 q^{91}+O(q^{100}) $ 4 * q + 2 * q^7 - 12 * q^19 - 8 * q^25 - 8 * q^29 - 12 * q^35 + 12 * q^37 + 8 * q^47 + 8 * q^49 + 16 * q^53 + 4 * q^55 + 16 * q^59 - 8 * q^65 - 8 * q^77 - 8 * q^83 - 20 * q^85 + 16 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	3.33513i		−	1.49152i	−0.666217	−	0.745758i	$-0.732087\pi$
$5$		−	3.33513i		−	1.49152i	0.666217	−	0.745758i	$-0.267913\pi$
$6$		0			0
$7$	−1.56155	−	2.13578i	−0.590211	−	0.807249i
$7$	−1.56155	−	2.13578i	−0.590211	−	0.807249i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	0.936426i		−	0.282343i	−0.989985	−	0.141172i	$-0.954913\pi$
$11$		−	0.936426i		−	0.282343i	0.989985	−	0.141172i	$-0.0450869\pi$
$12$		0			0
$13$			1.87285i			0.519436i	0.965685	+	0.259718i	$0.0836296\pi$
$13$			1.87285i			0.519436i	−0.965685	+	0.259718i	$0.916370\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	5.20798i		−	1.26312i	−0.775326	−	0.631561i	$-0.782415\pi$
$17$		−	5.20798i		−	1.26312i	0.775326	−	0.631561i	$-0.217585\pi$
$18$		0			0
$19$	−7.12311			−1.63415			−0.817076	−	0.576530i	$-0.804407\pi$
$19$	−7.12311			−1.63415			−0.817076	+	0.576530i	$0.804407\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	0.936426i		−	0.195258i	−0.995223	−	0.0976292i	$-0.968874\pi$
$23$		−	0.936426i		−	0.195258i	0.995223	−	0.0976292i	$-0.0311259\pi$
$24$		0			0
$25$	−6.12311			−1.22462
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−7.12311	+	5.20798i	−1.20402	+	0.880310i
$36$		0			0
$37$	−1.12311			−0.184637			−0.0923187	−	0.995730i	$-0.529428\pi$
$37$	−1.12311			−0.184637			−0.0923187	+	0.995730i	$0.529428\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	1.46228i		−	0.228370i	−0.993460	−	0.114185i	$-0.963574\pi$
$41$		−	1.46228i		−	0.228370i	0.993460	−	0.114185i	$-0.0364256\pi$
$42$		0			0
$43$			9.06897i			1.38300i	0.722374	+	0.691502i	$0.243051\pi$
$43$			9.06897i			1.38300i	−0.722374	+	0.691502i	$0.756949\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.24621			−0.911104			−0.455552	−	0.890209i	$-0.650558\pi$
$47$	−6.24621			−0.911104			−0.455552	+	0.890209i	$0.650558\pi$
$48$		0			0
$49$	−2.12311	+	6.67026i	−0.303301	+	0.952895i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	12.2462			1.68215			0.841073	−	0.540921i	$-0.181924\pi$
$53$	12.2462			1.68215			0.841073	+	0.540921i	$0.181924\pi$
$54$		0			0
$55$	−3.12311			−0.421119
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.00000			0.520756			0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000			0.520756			0.260378	+	0.965507i	$0.416153\pi$
$60$		0			0
$61$			4.79741i			0.614246i	0.951670	+	0.307123i	$0.0993662\pi$
$61$			4.79741i			0.614246i	−0.951670	+	0.307123i	$0.900634\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.24621			0.774747
$66$		0			0
$67$		−	10.9418i		−	1.33676i	−0.743822	−	0.668378i	$-0.766989\pi$
$67$		−	10.9418i		−	1.33676i	0.743822	−	0.668378i	$-0.233011\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	3.86098i		−	0.458215i	−0.973401	−	0.229107i	$-0.926419\pi$
$71$		−	3.86098i		−	0.458215i	0.973401	−	0.229107i	$-0.0735807\pi$
$72$		0			0
$73$		−	6.67026i		−	0.780695i	−0.920668	−	0.390348i	$-0.872355\pi$
$73$		−	6.67026i		−	0.780695i	0.920668	−	0.390348i	$-0.127645\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.00000	+	1.46228i	−0.227921	+	0.166642i
$78$		0			0
$79$			2.39871i			0.269875i	0.990854	+	0.134938i	$0.0430834\pi$
$79$			2.39871i			0.269875i	−0.990854	+	0.134938i	$0.956917\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−10.2462			−1.12467			−0.562334	−	0.826910i	$-0.690096\pi$
$83$	−10.2462			−1.12467			−0.562334	+	0.826910i	$0.690096\pi$
$84$		0			0
$85$	−17.3693			−1.88397
$86$		0			0
$87$		0			0
$88$		0			0
$89$			1.46228i			0.155001i	0.996992	+	0.0775006i	$0.0246940\pi$
$89$			1.46228i			0.155001i	−0.996992	+	0.0775006i	$0.975306\pi$
$90$		0			0
$91$	4.00000	−	2.92456i	0.419314	−	0.306577i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			23.7565i			2.43737i
$96$		0			0
$97$			10.4160i			1.05758i	0.848752	+	0.528791i	$0.177354\pi$
$97$			10.4160i			1.05758i	−0.848752	+	0.528791i	$0.822646\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	13.7511i		−	1.36829i	−0.729348	−	0.684143i	$-0.760176\pi$
$101$		−	13.7511i		−	1.36829i	0.729348	−	0.684143i	$-0.239824\pi$
$102$		0			0
$103$	−8.00000			−0.788263			−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000			−0.788263			−0.394132	+	0.919054i	$0.628955\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			9.47954i			0.916422i	0.888843	+	0.458211i	$0.151510\pi$
$107$			9.47954i			0.916422i	−0.888843	+	0.458211i	$0.848490\pi$
$108$		0			0
$109$	8.24621			0.789844			0.394922	−	0.918715i	$-0.370772\pi$
$109$	8.24621			0.789844			0.394922	+	0.918715i	$0.370772\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.24621			0.399450			0.199725	−	0.979852i	$-0.435995\pi$
$113$	4.24621			0.399450			0.199725	+	0.979852i	$0.435995\pi$
$114$		0			0
$115$	−3.12311			−0.291231
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−11.1231	+	8.13254i	−1.01965	+	0.745509i
$120$		0			0
$121$	10.1231			0.920282
$122$		0			0
$123$		0			0
$124$		0			0
$125$			3.74571i			0.335026i
$126$		0			0
$127$			9.89012i			0.877606i	0.898583	+	0.438803i	$0.144597\pi$
$127$			9.89012i			0.877606i	−0.898583	+	0.438803i	$0.855403\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.75379			0.502711			0.251355	−	0.967895i	$-0.419124\pi$
$131$	5.75379			0.502711			0.251355	+	0.967895i	$0.419124\pi$
$132$		0			0
$133$	11.1231	+	15.2134i	0.964496	+	1.31917i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.246211			−0.0210352			−0.0105176	−	0.999945i	$-0.503348\pi$
$137$	−0.246211			−0.0210352			−0.0105176	+	0.999945i	$0.503348\pi$
$138$		0			0
$139$	12.0000			1.01783			0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000			1.01783			0.508913	+	0.860818i	$0.330047\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.75379			0.146659
$144$		0			0
$145$			6.67026i			0.553935i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−10.0000			−0.819232			−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000			−0.819232			−0.409616	+	0.912258i	$0.634337\pi$
$150$		0			0
$151$			9.06897i			0.738022i	0.929425	+	0.369011i	$0.120304\pi$
$151$			9.06897i			0.738022i	−0.929425	+	0.369011i	$0.879696\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	21.8836i		−	1.74650i	−0.487268	−	0.873252i	$-0.662007\pi$
$157$		−	21.8836i		−	1.74650i	0.487268	−	0.873252i	$-0.337993\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.00000	+	1.46228i	−0.157622	+	0.115244i
$162$		0			0
$163$			15.7392i			1.23279i	0.787436	+	0.616396i	$0.211408\pi$
$163$			15.7392i			1.23279i	−0.787436	+	0.616396i	$0.788592\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−14.2462			−1.10240			−0.551202	−	0.834372i	$-0.685831\pi$
$167$	−14.2462			−1.10240			−0.551202	+	0.834372i	$0.685831\pi$
$168$		0			0
$169$	9.49242			0.730186
$170$		0			0
$171$		0			0
$172$		0			0
$173$			16.6757i			1.26783i	0.773404	+	0.633913i	$0.218552\pi$
$173$			16.6757i			1.26783i	−0.773404	+	0.633913i	$0.781448\pi$
$174$		0			0
$175$	9.56155	+	13.0776i	0.722785	+	0.988574i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			16.1498i			1.20709i	0.797328	+	0.603547i	$0.206246\pi$
$179$			16.1498i			1.20709i	−0.797328	+	0.603547i	$0.793754\pi$
$180$		0			0
$181$		−	1.87285i		−	0.139208i	−0.997575	−	0.0696040i	$-0.977826\pi$
$181$		−	1.87285i		−	0.139208i	0.997575	−	0.0696040i	$-0.0221736\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			3.74571i			0.275390i
$186$		0			0
$187$	−4.87689			−0.356634
$188$		0			0
$189$		0			0
$190$		0			0
$191$			2.80928i			0.203272i	0.994822	+	0.101636i	$0.0324078\pi$
$191$			2.80928i			0.203272i	−0.994822	+	0.101636i	$0.967592\pi$
$192$		0			0
$193$	−15.3693			−1.10631			−0.553154	−	0.833079i	$-0.686576\pi$
$193$	−15.3693			−1.10631			−0.553154	+	0.833079i	$0.686576\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−16.2462			−1.15749			−0.578747	−	0.815507i	$-0.696458\pi$
$197$	−16.2462			−1.15749			−0.578747	+	0.815507i	$0.696458\pi$
$198$		0			0
$199$	−3.12311			−0.221391			−0.110696	−	0.993854i	$-0.535308\pi$
$199$	−3.12311			−0.221391			−0.110696	+	0.993854i	$0.535308\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.12311	+	4.27156i	0.219199	+	0.299805i
$204$		0			0
$205$	−4.87689			−0.340617
$206$		0			0
$207$		0			0
$208$		0			0
$209$			6.67026i			0.461392i
$210$		0			0
$211$			12.8147i			0.882199i	0.897458	+	0.441099i	$0.145411\pi$
$211$			12.8147i			0.882199i	−0.897458	+	0.441099i	$0.854589\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	30.2462			2.06277
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	9.75379			0.656111
$222$		0			0
$223$	27.1231			1.81630			0.908149	−	0.418648i	$-0.137496\pi$
$223$	27.1231			1.81630			0.908149	+	0.418648i	$0.137496\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−16.4924			−1.09464			−0.547320	−	0.836923i	$-0.684352\pi$
$227$	−16.4924			−1.09464			−0.547320	+	0.836923i	$0.684352\pi$
$228$		0			0
$229$			5.61856i			0.371285i	0.982617	+	0.185642i	$0.0594366\pi$
$229$			5.61856i			0.371285i	−0.982617	+	0.185642i	$0.940563\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−22.4924			−1.47353			−0.736764	−	0.676150i	$-0.763647\pi$
$233$	−22.4924			−1.47353			−0.736764	+	0.676150i	$0.763647\pi$
$234$		0			0
$235$			20.8319i			1.35893i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			16.1498i			1.04464i	0.852748	+	0.522322i	$0.174934\pi$
$239$			16.1498i			1.04464i	−0.852748	+	0.522322i	$0.825066\pi$
$240$		0			0
$241$		−	23.7565i		−	1.53029i	−0.643858	−	0.765145i	$-0.722667\pi$
$241$		−	23.7565i		−	1.53029i	0.643858	−	0.765145i	$-0.277333\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	22.2462	+	7.08084i	1.42126	+	0.452378i
$246$		0			0
$247$		−	13.3405i		−	0.848837i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5.75379			−0.363176			−0.181588	−	0.983375i	$-0.558124\pi$
$251$	−5.75379			−0.363176			−0.181588	+	0.983375i	$0.558124\pi$
$252$		0			0
$253$	−0.876894			−0.0551299
$254$		0			0
$255$		0			0
$256$		0			0
$257$			2.28343i			0.142436i	0.997461	+	0.0712181i	$0.0226886\pi$
$257$			2.28343i			0.142436i	−0.997461	+	0.0712181i	$0.977311\pi$
$258$		0			0
$259$	1.75379	+	2.39871i	0.108975	+	0.149048i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	24.6929i		−	1.52263i	−0.648382	−	0.761315i	$-0.724554\pi$
$263$		−	24.6929i		−	1.52263i	0.648382	−	0.761315i	$-0.275446\pi$
$264$		0			0
$265$		−	40.8427i		−	2.50895i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			10.8265i			0.660106i	0.943962	+	0.330053i	$0.107067\pi$
$269$			10.8265i			0.660106i	−0.943962	+	0.330053i	$0.892933\pi$
$270$		0			0
$271$	−28.4924			−1.73079			−0.865396	−	0.501089i	$-0.832933\pi$
$271$	−28.4924			−1.73079			−0.865396	+	0.501089i	$0.832933\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			5.73384i			0.345763i
$276$		0			0
$277$	5.12311			0.307818			0.153909	−	0.988085i	$-0.450814\pi$
$277$	5.12311			0.307818			0.153909	+	0.988085i	$0.450814\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−16.2462			−0.969168			−0.484584	−	0.874745i	$-0.661029\pi$
$281$	−16.2462			−0.969168			−0.484584	+	0.874745i	$0.661029\pi$
$282$		0			0
$283$	−8.87689			−0.527677			−0.263838	−	0.964567i	$-0.584989\pi$
$283$	−8.87689			−0.527677			−0.263838	+	0.964567i	$0.584989\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.12311	+	2.28343i	−0.184351	+	0.134786i
$288$		0			0
$289$	−10.1231			−0.595477
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	13.7511i		−	0.803348i	−0.915783	−	0.401674i	$-0.868429\pi$
$293$		−	13.7511i		−	0.803348i	0.915783	−	0.401674i	$-0.131571\pi$
$294$		0			0
$295$		−	13.3405i		−	0.776716i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.75379			0.101424
$300$		0			0
$301$	19.3693	−	14.1617i	1.11643	−	0.816265i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	16.0000			0.916157
$306$		0			0
$307$	−19.6155			−1.11952			−0.559759	−	0.828656i	$-0.689106\pi$
$307$	−19.6155			−1.11952			−0.559759	+	0.828656i	$0.689106\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−8.00000			−0.453638			−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000			−0.453638			−0.226819	+	0.973937i	$0.572833\pi$
$312$		0			0
$313$			22.9354i			1.29638i	0.761478	+	0.648191i	$0.224474\pi$
$313$			22.9354i			1.29638i	−0.761478	+	0.648191i	$0.775526\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−14.4924			−0.813976			−0.406988	−	0.913434i	$-0.633421\pi$
$317$	−14.4924			−0.813976			−0.406988	+	0.913434i	$0.633421\pi$
$318$		0			0
$319$			1.87285i			0.104860i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			37.0970i			2.06413i
$324$		0			0
$325$		−	11.4677i		−	0.636112i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	9.75379	+	13.3405i	0.537744	+	0.735487i
$330$		0			0
$331$		−	17.6121i		−	0.968048i	−0.875055	−	0.484024i	$-0.839175\pi$
$331$		−	17.6121i		−	0.968048i	0.875055	−	0.484024i	$-0.160825\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−36.4924			−1.99379
$336$		0			0
$337$	8.24621			0.449200			0.224600	−	0.974451i	$-0.427892\pi$
$337$	8.24621			0.449200			0.224600	+	0.974451i	$0.427892\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	17.5616	−	5.88148i	0.948235	−	0.317570i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	20.1261i		−	1.08042i	−0.841529	−	0.540212i	$-0.818344\pi$
$347$		−	20.1261i		−	1.08042i	0.841529	−	0.540212i	$-0.181656\pi$
$348$		0			0
$349$		−	21.8836i		−	1.17140i	−0.810526	−	0.585702i	$-0.800819\pi$
$349$		−	21.8836i		−	1.17140i	0.810526	−	0.585702i	$-0.199181\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	28.9645i		−	1.54162i	−0.637063	−	0.770812i	$-0.719851\pi$
$353$		−	28.9645i		−	1.54162i	0.637063	−	0.770812i	$-0.280149\pi$
$354$		0			0
$355$	−12.8769			−0.683435
$356$		0			0
$357$		0			0
$358$		0			0
$359$			22.8201i			1.20440i	0.798346	+	0.602199i	$0.205708\pi$
$359$			22.8201i			1.20440i	−0.798346	+	0.602199i	$0.794292\pi$
$360$		0			0
$361$	31.7386			1.67045
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−22.2462			−1.16442
$366$		0			0
$367$	−33.3693			−1.74186			−0.870932	−	0.491403i	$-0.836484\pi$
$367$	−33.3693			−1.74186			−0.870932	+	0.491403i	$0.836484\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−19.1231	−	26.1552i	−0.992822	−	1.35791i
$372$		0			0
$373$	10.0000			0.517780			0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000			0.517780			0.258890	+	0.965907i	$0.416643\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	3.74571i		−	0.192914i
$378$		0			0
$379$		−	25.1035i		−	1.28948i	−0.764402	−	0.644740i	$-0.776966\pi$
$379$		−	25.1035i		−	1.28948i	0.764402	−	0.644740i	$-0.223034\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9.75379			0.498395			0.249198	−	0.968453i	$-0.419833\pi$
$383$	9.75379			0.498395			0.249198	+	0.968453i	$0.419833\pi$
$384$		0			0
$385$	4.87689	+	6.67026i	0.248550	+	0.339948i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−16.2462			−0.823716			−0.411858	−	0.911248i	$-0.635120\pi$
$389$	−16.2462			−0.823716			−0.411858	+	0.911248i	$0.635120\pi$
$390$		0			0
$391$	−4.87689			−0.246635
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8.00000			0.402524
$396$		0			0
$397$			18.1379i			0.910317i	0.890410	+	0.455159i	$0.150417\pi$
$397$			18.1379i			0.910317i	−0.890410	+	0.455159i	$0.849583\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−8.24621			−0.411796			−0.205898	−	0.978573i	$-0.566012\pi$
$401$	−8.24621			−0.411796			−0.205898	+	0.978573i	$0.566012\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$			1.05171i			0.0521311i
$408$		0			0
$409$			0.821147i			0.0406031i	0.999794	+	0.0203016i	$0.00646263\pi$
$409$			0.821147i			0.0406031i	−0.999794	+	0.0203016i	$0.993537\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6.24621	−	8.54312i	−0.307356	−	0.420379i
$414$		0			0
$415$			34.1725i			1.67746i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−16.4924			−0.805708			−0.402854	−	0.915264i	$-0.631982\pi$
$419$	−16.4924			−0.805708			−0.402854	+	0.915264i	$0.631982\pi$
$420$		0			0
$421$	−10.8769			−0.530107			−0.265054	−	0.964234i	$-0.585390\pi$
$421$	−10.8769			−0.530107			−0.265054	+	0.964234i	$0.585390\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			31.8890i			1.54685i
$426$		0			0
$427$	10.2462	−	7.49141i	0.495849	−	0.362535i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	4.68213i		−	0.225530i	−0.993622	−	0.112765i	$-0.964029\pi$
$431$		−	4.68213i		−	0.225530i	0.993622	−	0.112765i	$-0.0359708\pi$
$432$		0			0
$433$			13.3405i			0.641105i	0.947231	+	0.320552i	$0.103869\pi$
$433$			13.3405i			0.641105i	−0.947231	+	0.320552i	$0.896131\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			6.67026i			0.319082i
$438$		0			0
$439$	−6.63068			−0.316465			−0.158233	−	0.987402i	$-0.550580\pi$
$439$	−6.63068			−0.316465			−0.158233	+	0.987402i	$0.550580\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	18.8438i		−	0.895296i	−0.894210	−	0.447648i	$-0.852262\pi$
$443$		−	18.8438i		−	0.895296i	0.894210	−	0.447648i	$-0.147738\pi$
$444$		0			0
$445$	4.87689			0.231187
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−11.7538			−0.554696			−0.277348	−	0.960770i	$-0.589455\pi$
$449$	−11.7538			−0.554696			−0.277348	+	0.960770i	$0.589455\pi$
$450$		0			0
$451$	−1.36932			−0.0644786
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−9.75379	−	13.3405i	−0.457265	−	0.625414i
$456$		0			0
$457$	0.246211			0.0115173			0.00575864	−	0.999983i	$-0.498167\pi$
$457$	0.246211			0.0115173			0.00575864	+	0.999983i	$0.498167\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			6.25969i			0.291543i	0.989318	+	0.145771i	$0.0465664\pi$
$461$			6.25969i			0.291543i	−0.989318	+	0.145771i	$0.953434\pi$
$462$		0			0
$463$		−	39.2652i		−	1.82481i	−0.409292	−	0.912404i	$-0.634224\pi$
$463$		−	39.2652i		−	1.82481i	0.409292	−	0.912404i	$-0.365776\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	34.2462			1.58473			0.792363	−	0.610050i	$-0.208851\pi$
$467$	34.2462			1.58473			0.792363	+	0.610050i	$0.208851\pi$
$468$		0			0
$469$	−23.3693	+	17.0862i	−1.07909	+	0.788969i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	8.49242			0.390482
$474$		0			0
$475$	43.6155			2.00122
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−12.4924			−0.570793			−0.285397	−	0.958409i	$-0.592125\pi$
$479$	−12.4924			−0.570793			−0.285397	+	0.958409i	$0.592125\pi$
$480$		0			0
$481$		−	2.10341i		−	0.0959073i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	34.7386			1.57740
$486$		0			0
$487$			1.57756i			0.0714860i	0.999361	+	0.0357430i	$0.0113798\pi$
$487$			1.57756i			0.0714860i	−0.999361	+	0.0357430i	$0.988620\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	11.3524i		−	0.512326i	−0.966634	−	0.256163i	$-0.917542\pi$
$491$		−	11.3524i		−	0.512326i	0.966634	−	0.256163i	$-0.0824585\pi$
$492$		0			0
$493$			10.4160i			0.469112i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−8.24621	+	6.02913i	−0.369893	+	0.270444i
$498$		0			0
$499$		−	13.8664i		−	0.620744i	−0.950615	−	0.310372i	$-0.899546\pi$
$499$		−	13.8664i		−	0.620744i	0.950615	−	0.310372i	$-0.100454\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	26.7386			1.19222			0.596108	−	0.802904i	$-0.296713\pi$
$503$	26.7386			1.19222			0.596108	+	0.802904i	$0.296713\pi$
$504$		0			0
$505$	−45.8617			−2.04082
$506$		0			0
$507$		0			0
$508$		0			0
$509$			3.33513i			0.147827i	0.997265	+	0.0739136i	$0.0235489\pi$
$509$			3.33513i			0.147827i	−0.997265	+	0.0739136i	$0.976451\pi$
$510$		0			0
$511$	−14.2462	+	10.4160i	−0.630215	+	0.460775i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			26.6811i			1.17571i
$516$		0			0
$517$			5.84912i			0.257244i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			29.7856i			1.30493i	0.757818	+	0.652466i	$0.226266\pi$
$521$			29.7856i			1.30493i	−0.757818	+	0.652466i	$0.773734\pi$
$522$		0			0
$523$	−32.4924			−1.42079			−0.710397	−	0.703801i	$-0.751485\pi$
$523$	−32.4924			−1.42079			−0.710397	+	0.703801i	$0.751485\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	22.1231			0.961874
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.73863			0.118623
$534$		0			0
$535$	31.6155			1.36686
$536$		0			0
$537$		0			0
$538$		0			0
$539$	6.24621	+	1.98813i	0.269043	+	0.0856349i
$540$		0			0
$541$	−2.87689			−0.123687			−0.0618437	−	0.998086i	$-0.519698\pi$
$541$	−2.87689			−0.123687			−0.0618437	+	0.998086i	$0.519698\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	27.5022i		−	1.17806i
$546$		0			0
$547$		−	16.7909i		−	0.717929i	−0.933351	−	0.358964i	$-0.883130\pi$
$547$		−	16.7909i		−	0.717929i	0.933351	−	0.358964i	$-0.116870\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	14.2462			0.606909
$552$		0			0
$553$	5.12311	−	3.74571i	0.217857	−	0.159284i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	14.0000			0.593199			0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000			0.593199			0.296600	+	0.955002i	$0.404147\pi$
$558$		0			0
$559$	−16.9848			−0.718382
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.24621			0.0946665			0.0473333	−	0.998879i	$-0.484928\pi$
$563$	2.24621			0.0946665			0.0473333	+	0.998879i	$0.484928\pi$
$564$		0			0
$565$		−	14.1617i		−	0.595786i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	30.9848			1.29895			0.649476	−	0.760382i	$-0.274988\pi$
$569$	30.9848			1.29895			0.649476	+	0.760382i	$0.274988\pi$
$570$		0			0
$571$		−	8.83841i		−	0.369876i	−0.982750	−	0.184938i	$-0.940792\pi$
$571$		−	8.83841i		−	0.369876i	0.982750	−	0.184938i	$-0.0592084\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			5.73384i			0.239118i
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	16.0000	+	21.8836i	0.663792	+	0.907887i
$582$		0			0
$583$		−	11.4677i		−	0.474943i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−21.7538			−0.897875			−0.448937	−	0.893563i	$-0.648197\pi$
$587$	−21.7538			−0.897875			−0.448937	+	0.893563i	$0.648197\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	26.0399i		−	1.06933i	−0.845064	−	0.534666i	$-0.820438\pi$
$593$		−	26.0399i		−	1.06933i	0.845064	−	0.534666i	$-0.179562\pi$
$594$		0			0
$595$	27.1231	+	37.0970i	1.11194	+	1.52083i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	17.2015i		−	0.702835i	−0.936219	−	0.351417i	$-0.885700\pi$
$599$		−	17.2015i		−	0.702835i	0.936219	−	0.351417i	$-0.114300\pi$
$600$		0			0
$601$			17.0862i			0.696962i	0.937316	+	0.348481i	$0.113302\pi$
$601$			17.0862i			0.696962i	−0.937316	+	0.348481i	$0.886698\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	33.7619i		−	1.37262i
$606$		0			0
$607$	7.61553			0.309105			0.154552	−	0.987985i	$-0.450606\pi$
$607$	7.61553			0.309105			0.154552	+	0.987985i	$0.450606\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	11.6982i		−	0.473260i
$612$		0			0
$613$	−8.73863			−0.352950			−0.176475	−	0.984305i	$-0.556469\pi$
$613$	−8.73863			−0.352950			−0.176475	+	0.984305i	$0.556469\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−32.2462			−1.29818			−0.649092	−	0.760710i	$-0.724851\pi$
$617$	−32.2462			−1.29818			−0.649092	+	0.760710i	$0.724851\pi$
$618$		0			0
$619$	−20.0000			−0.803868			−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000			−0.803868			−0.401934	+	0.915669i	$0.631662\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	3.12311	−	2.28343i	0.125125	−	0.0914835i
$624$		0			0
$625$	−18.1231			−0.724924
$626$		0			0
$627$		0			0
$628$		0			0
$629$			5.84912i			0.233220i
$630$		0			0
$631$			40.3169i			1.60499i	0.596659	+	0.802495i	$0.296494\pi$
$631$			40.3169i			1.60499i	−0.596659	+	0.802495i	$0.703506\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	32.9848			1.30896
$636$		0			0
$637$	−12.4924	−	3.97626i	−0.494968	−	0.157545i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	42.4924			1.67835			0.839175	−	0.543862i	$-0.183038\pi$
$641$	42.4924			1.67835			0.839175	+	0.543862i	$0.183038\pi$
$642$		0			0
$643$	11.6155			0.458072			0.229036	−	0.973418i	$-0.426443\pi$
$643$	11.6155			0.458072			0.229036	+	0.973418i	$0.426443\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	32.9848			1.29677			0.648384	−	0.761313i	$-0.275445\pi$
$647$	32.9848			1.29677			0.648384	+	0.761313i	$0.275445\pi$
$648$		0			0
$649$		−	3.74571i		−	0.147032i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	16.7386			0.655033			0.327517	−	0.944845i	$-0.393788\pi$
$653$	16.7386			0.655033			0.327517	+	0.944845i	$0.393788\pi$
$654$		0			0
$655$		−	19.1896i		−	0.749801i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	26.7963i		−	1.04384i	−0.852995	−	0.521919i	$-0.825217\pi$
$659$		−	26.7963i		−	1.04384i	0.852995	−	0.521919i	$-0.174783\pi$
$660$		0			0
$661$			8.54312i			0.332289i	0.986101	+	0.166144i	$0.0531318\pi$
$661$			8.54312i			0.332289i	−0.986101	+	0.166144i	$0.946868\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	50.7386	−	37.0970i	1.96756	−	1.43856i
$666$		0			0
$667$			1.87285i			0.0725171i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4.49242			0.173428
$672$		0			0
$673$	−27.8617			−1.07399			−0.536996	−	0.843585i	$-0.680441\pi$
$673$	−27.8617			−1.07399			−0.536996	+	0.843585i	$0.680441\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	9.18425i		−	0.352979i	−0.984302	−	0.176490i	$-0.943526\pi$
$677$		−	9.18425i		−	0.352979i	0.984302	−	0.176490i	$-0.0564742\pi$
$678$		0			0
$679$	22.2462	−	16.2651i	0.853731	−	0.624197i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	32.1843i		−	1.23150i	−0.787942	−	0.615750i	$-0.788853\pi$
$683$		−	32.1843i		−	1.23150i	0.787942	−	0.615750i	$-0.211147\pi$
$684$		0			0
$685$			0.821147i			0.0313744i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			22.9354i			0.873767i
$690$		0			0
$691$	−12.0000			−0.456502			−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000			−0.456502			−0.228251	+	0.973602i	$0.573301\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	40.0216i		−	1.51811i
$696$		0			0
$697$	−7.61553			−0.288459
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−34.0000			−1.28416			−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000			−1.28416			−0.642081	+	0.766637i	$0.721929\pi$
$702$		0			0
$703$	8.00000			0.301726
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−29.3693	+	21.4731i	−1.10455	+	0.807578i
$708$		0			0
$709$	−6.00000			−0.225335			−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000			−0.225335			−0.112667	+	0.993633i	$0.535939\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		−	5.84912i		−	0.218745i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−28.4924			−1.06259			−0.531294	−	0.847187i	$-0.678294\pi$
$719$	−28.4924			−1.06259			−0.531294	+	0.847187i	$0.678294\pi$
$720$		0			0
$721$	12.4924	+	17.0862i	0.465242	+	0.636325i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	12.2462			0.454813
$726$		0			0
$727$	32.9848			1.22334			0.611670	−	0.791113i	$-0.290498\pi$
$727$	32.9848			1.22334			0.611670	+	0.791113i	$0.290498\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	47.2311			1.74690
$732$		0			0
$733$			36.0453i			1.33136i	0.746235	+	0.665682i	$0.231859\pi$
$733$			36.0453i			1.33136i	−0.746235	+	0.665682i	$0.768141\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−10.2462			−0.377424
$738$		0			0
$739$			36.5712i			1.34529i	0.739964	+	0.672646i	$0.234842\pi$
$739$			36.5712i			1.34529i	−0.739964	+	0.672646i	$0.765158\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	35.1089i		−	1.28802i	−0.765017	−	0.644010i	$-0.777269\pi$
$743$		−	35.1089i		−	1.28802i	0.765017	−	0.644010i	$-0.222731\pi$
$744$		0			0
$745$			33.3513i			1.22190i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	20.2462	−	14.8028i	0.739780	−	0.540883i
$750$		0			0
$751$		−	28.8492i		−	1.05272i	−0.850261	−	0.526361i	$-0.823556\pi$
$751$		−	28.8492i		−	1.05272i	0.850261	−	0.526361i	$-0.176444\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	30.2462			1.10077
$756$		0			0
$757$	34.9848			1.27155			0.635773	−	0.771876i	$-0.280681\pi$
$757$	34.9848			1.27155			0.635773	+	0.771876i	$0.280681\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	29.7856i		−	1.07973i	−0.841752	−	0.539864i	$-0.818476\pi$
$761$		−	29.7856i		−	1.07973i	0.841752	−	0.539864i	$-0.181524\pi$
$762$		0			0
$763$	−12.8769	−	17.6121i	−0.466175	−	0.637600i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			7.49141i			0.270499i
$768$		0			0
$769$			32.5302i			1.17307i	0.809925	+	0.586534i	$0.199508\pi$
$769$			32.5302i			1.17307i	−0.809925	+	0.586534i	$0.800492\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	3.33513i		−	0.119956i	−0.998200	−	0.0599782i	$-0.980897\pi$
$773$		−	3.33513i		−	0.119956i	0.998200	−	0.0599782i	$-0.0191031\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$			10.4160i			0.373191i
$780$		0			0
$781$	−3.61553			−0.129374
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−72.9848			−2.60494
$786$		0			0
$787$	44.9848			1.60354			0.801768	−	0.597635i	$-0.203893\pi$
$787$	44.9848			1.60354			0.801768	+	0.597635i	$0.203893\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−6.63068	−	9.06897i	−0.235760	−	0.322455i
$792$		0			0
$793$	−8.98485			−0.319061
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	39.6110i		−	1.40309i	−0.712623	−	0.701547i	$-0.752493\pi$
$797$		−	39.6110i		−	1.40309i	0.712623	−	0.701547i	$-0.247507\pi$
$798$		0			0
$799$			32.5302i			1.15083i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.24621			−0.220424
$804$		0			0
$805$	4.87689	+	6.67026i	0.171888	+	0.235096i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−22.4924			−0.790791			−0.395396	−	0.918511i	$-0.629393\pi$
$809$	−22.4924			−0.790791			−0.395396	+	0.918511i	$0.629393\pi$
$810$		0			0
$811$	−0.492423			−0.0172913			−0.00864565	−	0.999963i	$-0.502752\pi$
$811$	−0.492423			−0.0172913			−0.00864565	+	0.999963i	$0.502752\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	52.4924			1.83873
$816$		0			0
$817$		−	64.5992i		−	2.26004i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−57.2311			−1.99738			−0.998689	−	0.0511922i	$-0.983698\pi$
$821$	−57.2311			−1.99738			−0.998689	+	0.0511922i	$0.983698\pi$
$822$		0			0
$823$		−	13.0452i		−	0.454728i	−0.973810	−	0.227364i	$-0.926989\pi$
$823$		−	13.0452i		−	0.454728i	0.973810	−	0.227364i	$-0.0730108\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	55.9408i		−	1.94525i	−0.232373	−	0.972627i	$-0.574649\pi$
$827$		−	55.9408i		−	1.94525i	0.232373	−	0.972627i	$-0.425351\pi$
$828$		0			0
$829$			21.0625i			0.731531i	0.930707	+	0.365765i	$0.119193\pi$
$829$			21.0625i			0.731531i	−0.930707	+	0.365765i	$0.880807\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	34.7386	+	11.0571i	1.20362	+	0.383106i
$834$		0			0
$835$			47.5130i			1.64426i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	42.7386			1.47550			0.737751	−	0.675073i	$-0.235888\pi$
$839$	42.7386			1.47550			0.737751	+	0.675073i	$0.235888\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	31.6585i		−	1.08908i
$846$		0			0
$847$	−15.8078	−	21.6207i	−0.543161	−	0.742897i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			1.05171i			0.0360520i
$852$		0			0
$853$			50.2070i			1.71905i	0.511090	+	0.859527i	$0.329242\pi$
$853$			50.2070i			1.71905i	−0.511090	+	0.859527i	$0.670758\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	56.4667i		−	1.92887i	−0.264329	−	0.964433i	$-0.585150\pi$
$857$		−	56.4667i		−	1.92887i	0.264329	−	0.964433i	$-0.414850\pi$
$858$		0			0
$859$	25.8617			0.882391			0.441196	−	0.897411i	$-0.354555\pi$
$859$	25.8617			0.882391			0.441196	+	0.897411i	$0.354555\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			8.65840i			0.294735i	0.989082	+	0.147368i	$0.0470800\pi$
$863$			8.65840i			0.294735i	−0.989082	+	0.147368i	$0.952920\pi$
$864$		0			0
$865$	55.6155			1.89098
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.24621			0.0761975
$870$		0			0
$871$	20.4924			0.694359
$872$		0			0
$873$		0			0
$874$		0			0
$875$	8.00000	−	5.84912i	0.270449	−	0.197736i
$876$		0			0
$877$	−36.2462			−1.22395			−0.611974	−	0.790878i	$-0.709624\pi$
$877$	−36.2462			−1.22395			−0.611974	+	0.790878i	$0.709624\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			46.8719i			1.57915i	0.613652	+	0.789577i	$0.289700\pi$
$881$			46.8719i			1.57915i	−0.613652	+	0.789577i	$0.710300\pi$
$882$		0			0
$883$			18.6638i			0.628087i	0.949409	+	0.314043i	$0.101684\pi$
$883$			18.6638i			0.628087i	−0.949409	+	0.314043i	$0.898316\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	26.7386			0.897795			0.448898	−	0.893583i	$-0.351817\pi$
$887$	26.7386			0.897795			0.448898	+	0.893583i	$0.351817\pi$
$888$		0			0
$889$	21.1231	−	15.4439i	0.708446	−	0.517973i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	44.4924			1.48888
$894$		0			0
$895$	53.8617			1.80040
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		−	63.7781i		−	2.12476i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−6.24621			−0.207631
$906$		0			0
$907$		−	38.4440i		−	1.27651i	−0.769824	−	0.638256i	$-0.779656\pi$
$907$		−	38.4440i		−	1.27651i	0.769824	−	0.638256i	$-0.220344\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			5.73384i			0.189971i	0.995479	+	0.0949853i	$0.0302804\pi$
$911$			5.73384i			0.189971i	−0.995479	+	0.0949853i	$0.969720\pi$
$912$		0			0
$913$			9.59482i			0.317542i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−8.98485	−	12.2888i	−0.296706	−	0.405813i
$918$		0			0
$919$			9.06897i			0.299158i	0.988750	+	0.149579i	$0.0477918\pi$
$919$			9.06897i			0.299158i	−0.988750	+	0.149579i	$0.952208\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	7.23106			0.238013
$924$		0			0
$925$	6.87689			0.226111
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	3.56569i		−	0.116987i	−0.998288	−	0.0584933i	$-0.981370\pi$
$929$		−	3.56569i		−	0.116987i	0.998288	−	0.0584933i	$-0.0186296\pi$
$930$		0			0
$931$	15.1231	−	47.5130i	0.495640	−	1.55718i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			16.2651i			0.531925i
$936$		0			0
$937$		−	28.7845i		−	0.940348i	−0.882574	−	0.470174i	$-0.844191\pi$
$937$		−	28.7845i		−	0.940348i	0.882574	−	0.470174i	$-0.155809\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			53.7727i			1.75294i	0.481457	+	0.876470i	$0.340108\pi$
$941$			53.7727i			1.75294i	−0.481457	+	0.876470i	$0.659892\pi$
$942$		0			0
$943$	−1.36932			−0.0445911
$944$		0			0
$945$		0			0
$946$		0			0
$947$			48.6800i			1.58189i	0.611889	+	0.790943i	$0.290410\pi$
$947$			48.6800i			1.58189i	−0.611889	+	0.790943i	$0.709590\pi$
$948$		0			0
$949$	12.4924			0.405521
$950$		0			0
$951$		0			0
$952$		0			0
$953$	21.2311			0.687741			0.343871	−	0.939017i	$-0.388262\pi$
$953$	21.2311			0.687741			0.343871	+	0.939017i	$0.388262\pi$
$954$		0			0
$955$	9.36932			0.303184
$956$		0			0
$957$		0			0
$958$		0			0
$959$	0.384472	+	0.525853i	0.0124152	+	0.0169807i
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$			51.2587i			1.65008i
$966$		0			0
$967$		−	8.83841i		−	0.284224i	−0.989851	−	0.142112i	$-0.954611\pi$
$967$		−	8.83841i		−	0.284224i	0.989851	−	0.142112i	$-0.0453893\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−12.0000			−0.385098			−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000			−0.385098			−0.192549	+	0.981287i	$0.561675\pi$
$972$		0			0
$973$	−18.7386	−	25.6294i	−0.600733	−	0.821639i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	11.2614			0.360283			0.180142	−	0.983641i	$-0.442344\pi$
$977$	11.2614			0.360283			0.180142	+	0.983641i	$0.442344\pi$
$978$		0			0
$979$	1.36932			0.0437636
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−5.26137			−0.167812			−0.0839058	−	0.996474i	$-0.526739\pi$
$983$	−5.26137			−0.167812			−0.0839058	+	0.996474i	$0.526739\pi$
$984$		0			0
$985$			54.1833i			1.72642i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	8.49242			0.270043
$990$		0			0
$991$		−	0.525853i		−	0.0167043i	−0.999965	−	0.00835213i	$-0.997341\pi$
$991$		−	0.525853i		−	0.0167043i	0.999965	−	0.00835213i	$-0.00265860\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			10.4160i			0.330208i
$996$		0			0
$997$		−	19.7802i		−	0.626446i	−0.949680	−	0.313223i	$-0.898591\pi$
$997$		−	19.7802i		−	0.626446i	0.949680	−	0.313223i	$-0.101409\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.b.n.3583.1		4
3.2	odd	2		1344.2.b.f.895.4		4
4.3	odd	2		4032.2.b.j.3583.1		4
7.6	odd	2		4032.2.b.j.3583.4		4
8.3	odd	2		252.2.b.d.55.3		4
8.5	even	2		252.2.b.e.55.4		4
12.11	even	2		1344.2.b.e.895.4		4
21.20	even	2		1344.2.b.e.895.1		4
24.5	odd	2		84.2.b.a.55.1	✓	4
24.11	even	2		84.2.b.b.55.2	yes	4
28.27	even	2	inner	4032.2.b.n.3583.4		4
56.13	odd	2		252.2.b.d.55.4		4
56.27	even	2		252.2.b.e.55.3		4
84.83	odd	2		1344.2.b.f.895.1		4
168.5	even	6		588.2.o.a.31.2		8
168.11	even	6		588.2.o.a.19.2		8
168.53	odd	6		588.2.o.c.19.4		8
168.59	odd	6		588.2.o.c.19.2		8
168.83	odd	2		84.2.b.a.55.2	yes	4
168.101	even	6		588.2.o.a.19.4		8
168.107	even	6		588.2.o.a.31.4		8
168.125	even	2		84.2.b.b.55.1	yes	4
168.131	odd	6		588.2.o.c.31.4		8
168.149	odd	6		588.2.o.c.31.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.2.b.a.55.1	✓	4	24.5	odd	2
84.2.b.a.55.2	yes	4	168.83	odd	2
84.2.b.b.55.1	yes	4	168.125	even	2
84.2.b.b.55.2	yes	4	24.11	even	2
252.2.b.d.55.3		4	8.3	odd	2
252.2.b.d.55.4		4	56.13	odd	2
252.2.b.e.55.3		4	56.27	even	2
252.2.b.e.55.4		4	8.5	even	2
588.2.o.a.19.2		8	168.11	even	6
588.2.o.a.19.4		8	168.101	even	6
588.2.o.a.31.2		8	168.5	even	6
588.2.o.a.31.4		8	168.107	even	6
588.2.o.c.19.2		8	168.59	odd	6
588.2.o.c.19.4		8	168.53	odd	6
588.2.o.c.31.2		8	168.149	odd	6
588.2.o.c.31.4		8	168.131	odd	6
1344.2.b.e.895.1		4	21.20	even	2
1344.2.b.e.895.4		4	12.11	even	2
1344.2.b.f.895.1		4	84.83	odd	2
1344.2.b.f.895.4		4	3.2	odd	2
4032.2.b.j.3583.1		4	4.3	odd	2
4032.2.b.j.3583.4		4	7.6	odd	2
4032.2.b.n.3583.1		4	1.1	even	1	trivial
4032.2.b.n.3583.4		4	28.27	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:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.33513i q^{5} +(-1.56155 - 2.13578i) q^{7} +O(q^{10})\)	\(q-3.33513i q^{5} +(-1.56155 - 2.13578i) q^{7} -0.936426i q^{11} +1.87285i q^{13} -5.20798i q^{17} -7.12311 q^{19} -0.936426i q^{23} -6.12311 q^{25} -2.00000 q^{29} +(-7.12311 + 5.20798i) q^{35} -1.12311 q^{37} -1.46228i q^{41} +9.06897i q^{43} -6.24621 q^{47} +(-2.12311 + 6.67026i) q^{49} +12.2462 q^{53} -3.12311 q^{55} +4.00000 q^{59} +4.79741i q^{61} +6.24621 q^{65} -10.9418i q^{67} -3.86098i q^{71} -6.67026i q^{73} +(-2.00000 + 1.46228i) q^{77} +2.39871i q^{79} -10.2462 q^{83} -17.3693 q^{85} +1.46228i q^{89} +(4.00000 - 2.92456i) q^{91} +23.7565i q^{95} +10.4160i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{7}+O(q^{10}) \) 4 * q + 2 * q^7	\( 4 q + 2 q^{7} - 12 q^{19} - 8 q^{25} - 8 q^{29} - 12 q^{35} + 12 q^{37} + 8 q^{47} + 8 q^{49} + 16 q^{53} + 4 q^{55} + 16 q^{59} - 8 q^{65} - 8 q^{77} - 8 q^{83} - 20 q^{85} + 16 q^{91}+O(q^{100}) \) 4 * q + 2 * q^7 - 12 * q^19 - 8 * q^25 - 8 * q^29 - 12 * q^35 + 12 * q^37 + 8 * q^47 + 8 * q^49 + 16 * q^53 + 4 * q^55 + 16 * q^59 - 8 * q^65 - 8 * q^77 - 8 * q^83 - 20 * q^85 + 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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