LMFDB - Embedded newform 3800.2.d.k.3649.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(3649,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, 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("3800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.d (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:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5161984.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 4x^{3} + 25x^{2} - 20x + 8 $ x^6 - 4x^3 + 25x^2 - 20*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 760)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.5
Root			$0.432320 + 0.432320i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.k.3649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.76156i q^{3} +4.62620i q^{7} -0.103084 q^{9} +O(q^{10})$	$q+1.76156i q^{3} +4.62620i q^{7} -0.103084 q^{9} -5.52311 q^{11} +5.49084i q^{13} +6.62620i q^{17} +1.00000 q^{19} -8.14931 q^{21} +4.14931i q^{23} +5.10308i q^{27} +7.87859 q^{29} +1.25240 q^{31} -9.72928i q^{33} +0.387755i q^{37} -9.67243 q^{39} +6.77551 q^{41} -10.9817i q^{43} -1.72928i q^{47} -14.4017 q^{49} -11.6724 q^{51} +1.49084i q^{53} +1.76156i q^{57} -0.626198 q^{59} +15.0462 q^{61} -0.476886i q^{63} -5.22012i q^{67} -7.30925 q^{69} -11.0462 q^{71} -4.83237i q^{73} -25.5510i q^{77} +2.98168 q^{79} -9.29862 q^{81} -2.74760i q^{83} +13.8786i q^{87} -4.27072 q^{89} -25.4017 q^{91} +2.20617i q^{93} +13.7047i q^{97} +0.569343 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 8 q^{9}+O(q^{10}) $ 6 * q - 8 * q^9	$ 6 q - 8 q^{9} - 8 q^{11} + 6 q^{19} - 6 q^{21} - 6 q^{29} - 28 q^{31} + 10 q^{39} - 20 q^{41} - 8 q^{49} - 2 q^{51} + 14 q^{59} + 40 q^{61} - 66 q^{69} - 16 q^{71} - 28 q^{79} + 30 q^{81} - 36 q^{89} - 74 q^{91} - 72 q^{99}+O(q^{100}) $ 6 * q - 8 * q^9 - 8 * q^11 + 6 * q^19 - 6 * q^21 - 6 * q^29 - 28 * q^31 + 10 * q^39 - 20 * q^41 - 8 * q^49 - 2 * q^51 + 14 * q^59 + 40 * q^61 - 66 * q^69 - 16 * q^71 - 28 * q^79 + 30 * q^81 - 36 * q^89 - 74 * q^91 - 72 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$.

$n$	$401$	$951$	$1901$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.76156i	1.01704i	0.861052	+	0.508518i	$0.169806\pi$
$3$	1.76156i	1.01704i	−0.861052	+	0.508518i	$0.830194\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.62620i	1.74854i	0.485441	+	0.874269i	$0.338659\pi$
$7$	4.62620i	1.74854i	−0.485441	+	0.874269i	$0.661341\pi$
$8$	0	0
$9$	−0.103084	−0.0343612
$10$	0	0
$11$	−5.52311	−1.66528	−0.832641	−	0.553813i	$-0.813172\pi$
$11$	−5.52311	−1.66528	−0.832641	+	0.553813i	$0.813172\pi$
$12$	0	0
$13$	5.49084i	1.52288i	0.648233	+	0.761442i	$0.275508\pi$
$13$	5.49084i	1.52288i	−0.648233	+	0.761442i	$0.724492\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.62620i	1.60709i	0.595245	+	0.803545i	$0.297055\pi$
$17$	6.62620i	1.60709i	−0.595245	+	0.803545i	$0.702945\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−8.14931	−1.77833
$22$	0	0
$23$	4.14931i	0.865191i	0.901588	+	0.432596i	$0.142402\pi$
$23$	4.14931i	0.865191i	−0.901588	+	0.432596i	$0.857598\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.10308i	0.982089i
$28$	0	0
$29$	7.87859	1.46302	0.731509	−	0.681832i	$-0.238816\pi$
$29$	7.87859	1.46302	0.731509	+	0.681832i	$0.238816\pi$
$30$	0	0
$31$	1.25240	0.224937	0.112468	−	0.993655i	$-0.464124\pi$
$31$	1.25240	0.224937	0.112468	+	0.993655i	$0.464124\pi$
$32$	0	0
$33$	− 9.72928i	− 1.69365i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.387755i	0.0637466i	0.999492	+	0.0318733i	$0.0101473\pi$
$37$	0.387755i	0.0637466i	−0.999492	+	0.0318733i	$0.989853\pi$
$38$	0	0
$39$	−9.67243	−1.54883
$40$	0	0
$41$	6.77551	1.05816	0.529078	−	0.848573i	$-0.322538\pi$
$41$	6.77551	1.05816	0.529078	+	0.848573i	$0.322538\pi$
$42$	0	0
$43$	− 10.9817i	− 1.67469i	−0.546675	−	0.837345i	$-0.684107\pi$
$43$	− 10.9817i	− 1.67469i	0.546675	−	0.837345i	$-0.315893\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 1.72928i	− 0.252242i	−0.992015	−	0.126121i	$-0.959747\pi$
$47$	− 1.72928i	− 0.252242i	0.992015	−	0.126121i	$-0.0402527\pi$
$48$	0	0
$49$	−14.4017	−2.05739
$50$	0	0
$51$	−11.6724	−1.63447
$52$	0	0
$53$	1.49084i	0.204782i	0.994744	+	0.102391i	$0.0326494\pi$
$53$	1.49084i	0.204782i	−0.994744	+	0.102391i	$0.967351\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.76156i	0.233324i
$58$	0	0
$59$	−0.626198	−0.0815240	−0.0407620	−	0.999169i	$-0.512979\pi$
$59$	−0.626198	−0.0815240	−0.0407620	+	0.999169i	$0.512979\pi$
$60$	0	0
$61$	15.0462	1.92647	0.963236	−	0.268656i	$-0.0865796\pi$
$61$	15.0462	1.92647	0.963236	+	0.268656i	$0.0865796\pi$
$62$	0	0
$63$	− 0.476886i	− 0.0600819i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 5.22012i	− 0.637739i	−0.947799	−	0.318870i	$-0.896697\pi$
$67$	− 5.22012i	− 0.637739i	0.947799	−	0.318870i	$-0.103303\pi$
$68$	0	0
$69$	−7.30925	−0.879930
$70$	0	0
$71$	−11.0462	−1.31095	−0.655473	−	0.755219i	$-0.727531\pi$
$71$	−11.0462	−1.31095	−0.655473	+	0.755219i	$0.727531\pi$
$72$	0	0
$73$	− 4.83237i	− 0.565586i	−0.959181	−	0.282793i	$-0.908739\pi$
$73$	− 4.83237i	− 0.565586i	0.959181	−	0.282793i	$-0.0912609\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 25.5510i	− 2.91181i
$78$	0	0
$79$	2.98168	0.335465	0.167732	−	0.985833i	$-0.446356\pi$
$79$	2.98168	0.335465	0.167732	+	0.985833i	$0.446356\pi$
$80$	0	0
$81$	−9.29862	−1.03318
$82$	0	0
$83$	− 2.74760i	− 0.301589i	−0.988565	−	0.150794i	$-0.951817\pi$
$83$	− 2.74760i	− 0.301589i	0.988565	−	0.150794i	$-0.0481831\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	13.8786i	1.48794i
$88$	0	0
$89$	−4.27072	−0.452695	−0.226348	−	0.974047i	$-0.572679\pi$
$89$	−4.27072	−0.452695	−0.226348	+	0.974047i	$0.572679\pi$
$90$	0	0
$91$	−25.4017	−2.66282
$92$	0	0
$93$	2.20617i	0.228769i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	13.7047i	1.39150i	0.718283	+	0.695751i	$0.244928\pi$
$97$	13.7047i	1.39150i	−0.718283	+	0.695751i	$0.755072\pi$
$98$	0	0
$99$	0.569343	0.0572211
$100$	0	0
$101$	7.04623	0.701126	0.350563	−	0.936539i	$-0.385990\pi$
$101$	7.04623	0.701126	0.350563	+	0.936539i	$0.385990\pi$
$102$	0	0
$103$	0.658473i	0.0648813i	0.999474	+	0.0324407i	$0.0103280\pi$
$103$	0.658473i	0.0648813i	−0.999474	+	0.0324407i	$0.989672\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.22012i	0.504648i	0.967643	+	0.252324i	$0.0811949\pi$
$107$	5.22012i	0.504648i	−0.967643	+	0.252324i	$0.918805\pi$
$108$	0	0
$109$	11.8140	1.13158	0.565790	−	0.824549i	$-0.308571\pi$
$109$	11.8140	1.13158	0.565790	+	0.824549i	$0.308571\pi$
$110$	0	0
$111$	−0.683053	−0.0648325
$112$	0	0
$113$	− 0.0891304i	− 0.00838468i	−0.999991	−	0.00419234i	$-0.998666\pi$
$113$	− 0.0891304i	− 0.00838468i	0.999991	−	0.00419234i	$-0.00133447\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 0.566016i	− 0.0523282i
$118$	0	0
$119$	−30.6541	−2.81006
$120$	0	0
$121$	19.5048	1.77316
$122$	0	0
$123$	11.9354i	1.07618i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 13.3694i	− 1.18635i	−0.805075	−	0.593173i	$-0.797875\pi$
$127$	− 13.3694i	− 1.18635i	0.805075	−	0.593173i	$-0.202125\pi$
$128$	0	0
$129$	19.3449	1.70322
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	4.62620i	0.401142i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.6262i	0.907857i	0.891038	+	0.453929i	$0.149978\pi$
$137$	10.6262i	0.907857i	−0.891038	+	0.453929i	$0.850022\pi$
$138$	0	0
$139$	12.5693	1.06612	0.533059	−	0.846078i	$-0.321042\pi$
$139$	12.5693	1.06612	0.533059	+	0.846078i	$0.321042\pi$
$140$	0	0
$141$	3.04623	0.256539
$142$	0	0
$143$	− 30.3265i	− 2.53603i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 25.3694i	− 2.09244i
$148$	0	0
$149$	−7.04623	−0.577250	−0.288625	−	0.957442i	$-0.593198\pi$
$149$	−7.04623	−0.577250	−0.288625	+	0.957442i	$0.593198\pi$
$150$	0	0
$151$	4.47689	0.364324	0.182162	−	0.983269i	$-0.441691\pi$
$151$	4.47689	0.364324	0.182162	+	0.983269i	$0.441691\pi$
$152$	0	0
$153$	− 0.683053i	− 0.0552216i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 8.27072i	− 0.660075i	−0.943968	−	0.330038i	$-0.892939\pi$
$157$	− 8.27072i	− 0.660075i	0.943968	−	0.330038i	$-0.107061\pi$
$158$	0	0
$159$	−2.62620	−0.208271
$160$	0	0
$161$	−19.1955	−1.51282
$162$	0	0
$163$	− 18.5693i	− 1.45446i	−0.686392	−	0.727232i	$-0.740807\pi$
$163$	− 18.5693i	− 1.45446i	0.686392	−	0.727232i	$-0.259193\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.3694i	1.65362i	0.562484	+	0.826808i	$0.309846\pi$
$167$	21.3694i	1.65362i	−0.562484	+	0.826808i	$0.690154\pi$
$168$	0	0
$169$	−17.1493	−1.31918
$170$	0	0
$171$	−0.103084	−0.00788301
$172$	0	0
$173$	− 12.3878i	− 0.941824i	−0.882180	−	0.470912i	$-0.843925\pi$
$173$	− 12.3878i	− 0.941824i	0.882180	−	0.470912i	$-0.156075\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1.10308i	− 0.0829128i
$178$	0	0
$179$	−12.7755	−0.954886	−0.477443	−	0.878663i	$-0.658436\pi$
$179$	−12.7755	−0.954886	−0.477443	+	0.878663i	$0.658436\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	26.5048i	1.95929i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 36.5972i	− 2.67626i
$188$	0	0
$189$	−23.6079	−1.71722
$190$	0	0
$191$	−19.1955	−1.38894	−0.694470	−	0.719521i	$-0.744361\pi$
$191$	−19.1955	−1.38894	−0.694470	+	0.719521i	$0.744361\pi$
$192$	0	0
$193$	− 13.7047i	− 0.986486i	−0.869892	−	0.493243i	$-0.835811\pi$
$193$	− 13.7047i	− 0.986486i	0.869892	−	0.493243i	$-0.164189\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	0	0
$199$	0.561647	0.0398141	0.0199071	−	0.999802i	$-0.493663\pi$
$199$	0.561647	0.0398141	0.0199071	+	0.999802i	$0.493663\pi$
$200$	0	0
$201$	9.19554	0.648603
$202$	0	0
$203$	36.4479i	2.55814i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 0.427726i	− 0.0297290i
$208$	0	0
$209$	−5.52311	−0.382042
$210$	0	0
$211$	15.6724	1.07893	0.539467	−	0.842007i	$-0.318626\pi$
$211$	15.6724	1.07893	0.539467	+	0.842007i	$0.318626\pi$
$212$	0	0
$213$	− 19.4586i	− 1.33328i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5.79383i	0.393311i
$218$	0	0
$219$	8.51249	0.575221
$220$	0	0
$221$	−36.3834	−2.44741
$222$	0	0
$223$	12.0525i	0.807094i	0.914959	+	0.403547i	$0.132223\pi$
$223$	12.0525i	0.807094i	−0.914959	+	0.403547i	$0.867777\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3.07850i	− 0.204327i	−0.994768	−	0.102164i	$-0.967423\pi$
$227$	− 3.07850i	− 0.204327i	0.994768	−	0.102164i	$-0.0325766\pi$
$228$	0	0
$229$	−17.7938	−1.17585	−0.587925	−	0.808916i	$-0.700055\pi$
$229$	−17.7938	−1.17585	−0.587925	+	0.808916i	$0.700055\pi$
$230$	0	0
$231$	45.0096	2.96141
$232$	0	0
$233$	8.50479i	0.557167i	0.960412	+	0.278584i	$0.0898650\pi$
$233$	8.50479i	0.557167i	−0.960412	+	0.278584i	$0.910135\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.25240i	0.341180i
$238$	0	0
$239$	23.6079	1.52707	0.763533	−	0.645768i	$-0.223463\pi$
$239$	23.6079	1.52707	0.763533	+	0.645768i	$0.223463\pi$
$240$	0	0
$241$	9.11078	0.586877	0.293438	−	0.955978i	$-0.405200\pi$
$241$	9.11078	0.586877	0.293438	+	0.955978i	$0.405200\pi$
$242$	0	0
$243$	− 1.07081i	− 0.0686924i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.49084i	0.349374i
$248$	0	0
$249$	4.84006	0.306726
$250$	0	0
$251$	13.5510	0.855333	0.427666	−	0.903937i	$-0.359336\pi$
$251$	13.5510	0.855333	0.427666	+	0.903937i	$0.359336\pi$
$252$	0	0
$253$	− 22.9171i	− 1.44079i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 10.6585i	− 0.664857i	−0.943128	−	0.332429i	$-0.892132\pi$
$257$	− 10.6585i	− 0.664857i	0.943128	−	0.332429i	$-0.107868\pi$
$258$	0	0
$259$	−1.79383	−0.111463
$260$	0	0
$261$	−0.812155	−0.0502711
$262$	0	0
$263$	− 17.9634i	− 1.10767i	−0.832627	−	0.553834i	$-0.813164\pi$
$263$	− 17.9634i	− 1.10767i	0.832627	−	0.553834i	$-0.186836\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 7.52311i	− 0.460407i
$268$	0	0
$269$	−15.5510	−0.948162	−0.474081	−	0.880481i	$-0.657220\pi$
$269$	−15.5510	−0.948162	−0.474081	+	0.880481i	$0.657220\pi$
$270$	0	0
$271$	8.29093	0.503638	0.251819	−	0.967774i	$-0.418971\pi$
$271$	8.29093	0.503638	0.251819	+	0.967774i	$0.418971\pi$
$272$	0	0
$273$	− 44.7466i	− 2.70819i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	28.5693i	1.71657i	0.513177	+	0.858283i	$0.328468\pi$
$277$	28.5693i	1.71657i	−0.513177	+	0.858283i	$0.671532\pi$
$278$	0	0
$279$	−0.129102	−0.00772911
$280$	0	0
$281$	−19.3169	−1.15235	−0.576176	−	0.817325i	$-0.695456\pi$
$281$	−19.3169	−1.15235	−0.576176	+	0.817325i	$0.695456\pi$
$282$	0	0
$283$	16.2986i	0.968853i	0.874832	+	0.484426i	$0.160972\pi$
$283$	16.2986i	0.968853i	−0.874832	+	0.484426i	$0.839028\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	31.3449i	1.85023i
$288$	0	0
$289$	−26.9065	−1.58274
$290$	0	0
$291$	−24.1416	−1.41521
$292$	0	0
$293$	− 26.3955i	− 1.54204i	−0.636812	−	0.771019i	$-0.719747\pi$
$293$	− 26.3955i	− 1.54204i	0.636812	−	0.771019i	$-0.280253\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 28.1849i	− 1.63545i
$298$	0	0
$299$	−22.7832	−1.31759
$300$	0	0
$301$	50.8034	2.92826
$302$	0	0
$303$	12.4123i	0.713070i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	30.2095i	1.72415i	0.506783	+	0.862073i	$0.330834\pi$
$307$	30.2095i	1.72415i	−0.506783	+	0.862073i	$0.669166\pi$
$308$	0	0
$309$	−1.15994	−0.0659866
$310$	0	0
$311$	−4.12141	−0.233703	−0.116852	−	0.993149i	$-0.537280\pi$
$311$	−4.12141	−0.233703	−0.116852	+	0.993149i	$0.537280\pi$
$312$	0	0
$313$	− 15.1031i	− 0.853677i	−0.904328	−	0.426838i	$-0.859627\pi$
$313$	− 15.1031i	− 0.853677i	0.904328	−	0.426838i	$-0.140373\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.3126i	0.747709i	0.927487	+	0.373854i	$0.121964\pi$
$317$	13.3126i	0.747709i	−0.927487	+	0.373854i	$0.878036\pi$
$318$	0	0
$319$	−43.5144	−2.43634
$320$	0	0
$321$	−9.19554	−0.513245
$322$	0	0
$323$	6.62620i	0.368692i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	20.8111i	1.15086i
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	16.9248	0.930272	0.465136	−	0.885239i	$-0.346005\pi$
$331$	16.9248	0.930272	0.465136	+	0.885239i	$0.346005\pi$
$332$	0	0
$333$	− 0.0399712i	− 0.00219041i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 1.16327i	− 0.0633671i	−0.999498	−	0.0316836i	$-0.989913\pi$
$337$	− 1.16327i	− 0.0633671i	0.999498	−	0.0316836i	$-0.0100869\pi$
$338$	0	0
$339$	0.157008	0.00852752
$340$	0	0
$341$	−6.91713	−0.374583
$342$	0	0
$343$	− 34.2418i	− 1.84888i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 3.70138i	− 0.198700i	−0.995053	−	0.0993501i	$-0.968324\pi$
$347$	− 3.70138i	− 0.198700i	0.995053	−	0.0993501i	$-0.0316764\pi$
$348$	0	0
$349$	2.54144	0.136040	0.0680200	−	0.997684i	$-0.478332\pi$
$349$	2.54144	0.136040	0.0680200	+	0.997684i	$0.478332\pi$
$350$	0	0
$351$	−28.0202	−1.49561
$352$	0	0
$353$	− 21.3738i	− 1.13761i	−0.822471	−	0.568806i	$-0.807405\pi$
$353$	− 21.3738i	− 1.13761i	0.822471	−	0.568806i	$-0.192595\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 53.9990i	− 2.85793i
$358$	0	0
$359$	4.42003	0.233280	0.116640	−	0.993174i	$-0.462788\pi$
$359$	4.42003	0.233280	0.116640	+	0.993174i	$0.462788\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	34.3588i	1.80337i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 3.28030i	− 0.171230i	−0.996328	−	0.0856152i	$-0.972714\pi$
$367$	− 3.28030i	− 0.171230i	0.996328	−	0.0856152i	$-0.0272856\pi$
$368$	0	0
$369$	−0.698445	−0.0363596
$370$	0	0
$371$	−6.89692	−0.358070
$372$	0	0
$373$	− 14.8078i	− 0.766718i	−0.923599	−	0.383359i	$-0.874767\pi$
$373$	− 14.8078i	− 0.766718i	0.923599	−	0.383359i	$-0.125233\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	43.2601i	2.22801i
$378$	0	0
$379$	19.7370	1.01382	0.506910	−	0.861999i	$-0.330788\pi$
$379$	19.7370	1.01382	0.506910	+	0.861999i	$0.330788\pi$
$380$	0	0
$381$	23.5510	1.20656
$382$	0	0
$383$	− 10.2095i	− 0.521681i	−0.965382	−	0.260840i	$-0.916000\pi$
$383$	− 10.2095i	− 0.521681i	0.965382	−	0.260840i	$-0.0839996\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.13203i	0.0575444i
$388$	0	0
$389$	−10.7110	−0.543067	−0.271534	−	0.962429i	$-0.587531\pi$
$389$	−10.7110	−0.543067	−0.271534	+	0.962429i	$0.587531\pi$
$390$	0	0
$391$	−27.4942	−1.39044
$392$	0	0
$393$	− 7.04623i	− 0.355435i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 20.5693i	− 1.03235i	−0.856484	−	0.516173i	$-0.827356\pi$
$397$	− 20.5693i	− 1.03235i	0.856484	−	0.516173i	$-0.172644\pi$
$398$	0	0
$399$	−8.14931	−0.407976
$400$	0	0
$401$	−23.9634	−1.19667	−0.598336	−	0.801245i	$-0.704171\pi$
$401$	−23.9634	−1.19667	−0.598336	+	0.801245i	$0.704171\pi$
$402$	0	0
$403$	6.87671i	0.342553i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 2.14162i	− 0.106156i
$408$	0	0
$409$	4.50479	0.222748	0.111374	−	0.993779i	$-0.464475\pi$
$409$	4.50479	0.222748	0.111374	+	0.993779i	$0.464475\pi$
$410$	0	0
$411$	−18.7187	−0.923323
$412$	0	0
$413$	− 2.89692i	− 0.142548i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	22.1416i	1.08428i
$418$	0	0
$419$	31.7572	1.55144	0.775720	−	0.631077i	$-0.217387\pi$
$419$	31.7572	1.55144	0.775720	+	0.631077i	$0.217387\pi$
$420$	0	0
$421$	−21.4296	−1.04442	−0.522208	−	0.852818i	$-0.674891\pi$
$421$	−21.4296	−1.04442	−0.522208	+	0.852818i	$0.674891\pi$
$422$	0	0
$423$	0.178261i	0.00866734i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	69.6068i	3.36851i
$428$	0	0
$429$	53.4219	2.57923
$430$	0	0
$431$	−10.6831	−0.514585	−0.257292	−	0.966334i	$-0.582830\pi$
$431$	−10.6831	−0.514585	−0.257292	+	0.966334i	$0.582830\pi$
$432$	0	0
$433$	− 24.6864i	− 1.18635i	−0.805073	−	0.593176i	$-0.797874\pi$
$433$	− 24.6864i	− 1.18635i	0.805073	−	0.593176i	$-0.202126\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.14931i	0.198489i
$438$	0	0
$439$	14.9817	0.715036	0.357518	−	0.933906i	$-0.383623\pi$
$439$	14.9817	0.715036	0.357518	+	0.933906i	$0.383623\pi$
$440$	0	0
$441$	1.48458	0.0706944
$442$	0	0
$443$	− 4.54144i	− 0.215770i	−0.994163	−	0.107885i	$-0.965592\pi$
$443$	− 4.54144i	− 0.215770i	0.994163	−	0.107885i	$-0.0344079\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 12.4123i	− 0.587083i
$448$	0	0
$449$	16.0925	0.759450	0.379725	−	0.925099i	$-0.376019\pi$
$449$	16.0925	0.759450	0.379725	+	0.925099i	$0.376019\pi$
$450$	0	0
$451$	−37.4219	−1.76213
$452$	0	0
$453$	7.88629i	0.370530i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.65410i	0.217710i	0.994058	+	0.108855i	$0.0347184\pi$
$457$	4.65410i	0.217710i	−0.994058	+	0.108855i	$0.965282\pi$
$458$	0	0
$459$	−33.8140	−1.57830
$460$	0	0
$461$	19.0096	0.885365	0.442682	−	0.896679i	$-0.354027\pi$
$461$	19.0096	0.885365	0.442682	+	0.896679i	$0.354027\pi$
$462$	0	0
$463$	− 10.6339i	− 0.494199i	−0.968990	−	0.247099i	$-0.920523\pi$
$463$	− 10.6339i	− 0.494199i	0.968990	−	0.247099i	$-0.0794774\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 32.8805i	− 1.52153i	−0.649029	−	0.760764i	$-0.724825\pi$
$467$	− 32.8805i	− 1.52153i	0.649029	−	0.760764i	$-0.275175\pi$
$468$	0	0
$469$	24.1493	1.11511
$470$	0	0
$471$	14.5693	0.671320
$472$	0	0
$473$	60.6531i	2.78883i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 0.153681i	− 0.00703658i
$478$	0	0
$479$	−12.4402	−0.568409	−0.284205	−	0.958764i	$-0.591729\pi$
$479$	−12.4402	−0.568409	−0.284205	+	0.958764i	$0.591729\pi$
$480$	0	0
$481$	−2.12910	−0.0970787
$482$	0	0
$483$	− 33.8140i	− 1.53859i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 40.9571i	− 1.85594i	−0.372651	−	0.927972i	$-0.621551\pi$
$487$	− 40.9571i	− 1.85594i	0.372651	−	0.927972i	$-0.378449\pi$
$488$	0	0
$489$	32.7110	1.47924
$490$	0	0
$491$	43.3449	1.95613	0.978063	−	0.208310i	$-0.0667962\pi$
$491$	43.3449	1.95613	0.978063	+	0.208310i	$0.0667962\pi$
$492$	0	0
$493$	52.2051i	2.35120i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 51.1020i	− 2.29224i
$498$	0	0
$499$	−25.4094	−1.13748	−0.568741	−	0.822517i	$-0.692569\pi$
$499$	−25.4094	−1.13748	−0.568741	+	0.822517i	$0.692569\pi$
$500$	0	0
$501$	−37.6435	−1.68179
$502$	0	0
$503$	34.1127i	1.52101i	0.649333	+	0.760504i	$0.275048\pi$
$503$	34.1127i	1.52101i	−0.649333	+	0.760504i	$0.724952\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 30.2095i	− 1.34165i
$508$	0	0
$509$	22.3632	0.991230	0.495615	−	0.868542i	$-0.334943\pi$
$509$	22.3632	0.991230	0.495615	+	0.868542i	$0.334943\pi$
$510$	0	0
$511$	22.3555	0.988948
$512$	0	0
$513$	5.10308i	0.225307i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	9.55102i	0.420053i
$518$	0	0
$519$	21.8217	0.957868
$520$	0	0
$521$	2.29862	0.100705	0.0503523	−	0.998732i	$-0.483966\pi$
$521$	2.29862	0.100705	0.0503523	+	0.998732i	$0.483966\pi$
$522$	0	0
$523$	27.3771i	1.19712i	0.801079	+	0.598559i	$0.204260\pi$
$523$	27.3771i	1.19712i	−0.801079	+	0.598559i	$0.795740\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.29862i	0.361494i
$528$	0	0
$529$	5.78321	0.251444
$530$	0	0
$531$	0.0645508	0.00280127
$532$	0	0
$533$	37.2032i	1.61145i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 22.5048i	− 0.971153i
$538$	0	0
$539$	79.5423	3.42613
$540$	0	0
$541$	15.4586	0.664616	0.332308	−	0.943171i	$-0.392173\pi$
$541$	15.4586	0.664616	0.332308	+	0.943171i	$0.392173\pi$
$542$	0	0
$543$	38.7543i	1.66310i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	13.9754i	0.597546i	0.954324	+	0.298773i	$0.0965773\pi$
$547$	13.9754i	0.597546i	−0.954324	+	0.298773i	$0.903423\pi$
$548$	0	0
$549$	−1.55102	−0.0661960
$550$	0	0
$551$	7.87859	0.335639
$552$	0	0
$553$	13.7938i	0.586573i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 30.6618i	− 1.29918i	−0.760284	−	0.649591i	$-0.774940\pi$
$557$	− 30.6618i	− 1.29918i	0.760284	−	0.649591i	$-0.225060\pi$
$558$	0	0
$559$	60.2986	2.55036
$560$	0	0
$561$	64.4681	2.72185
$562$	0	0
$563$	− 2.02458i	− 0.0853259i	−0.999090	−	0.0426629i	$-0.986416\pi$
$563$	− 2.02458i	− 0.0853259i	0.999090	−	0.0426629i	$-0.0135842\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 43.0173i	− 1.80656i
$568$	0	0
$569$	28.2620	1.18480	0.592402	−	0.805643i	$-0.298180\pi$
$569$	28.2620	1.18480	0.592402	+	0.805643i	$0.298180\pi$
$570$	0	0
$571$	10.6464	0.445538	0.222769	−	0.974871i	$-0.428490\pi$
$571$	10.6464	0.445538	0.222769	+	0.974871i	$0.428490\pi$
$572$	0	0
$573$	− 33.8140i	− 1.41260i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	9.30925i	0.387549i	0.981046	+	0.193775i	$0.0620730\pi$
$577$	9.30925i	0.387549i	−0.981046	+	0.193775i	$0.937927\pi$
$578$	0	0
$579$	24.1416	1.00329
$580$	0	0
$581$	12.7110	0.527339
$582$	0	0
$583$	− 8.23407i	− 0.341021i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	42.3911i	1.74967i	0.484424	+	0.874834i	$0.339029\pi$
$587$	42.3911i	1.74967i	−0.484424	+	0.874834i	$0.660971\pi$
$588$	0	0
$589$	1.25240	0.0516041
$590$	0	0
$591$	−3.52311	−0.144922
$592$	0	0
$593$	28.5048i	1.17055i	0.810834	+	0.585276i	$0.199014\pi$
$593$	28.5048i	1.17055i	−0.810834	+	0.585276i	$0.800986\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.989374i	0.0404924i
$598$	0	0
$599$	−25.6156	−1.04662	−0.523312	−	0.852141i	$-0.675304\pi$
$599$	−25.6156	−1.04662	−0.523312	+	0.852141i	$0.675304\pi$
$600$	0	0
$601$	−29.2803	−1.19437	−0.597184	−	0.802104i	$-0.703714\pi$
$601$	−29.2803	−1.19437	−0.597184	+	0.802104i	$0.703714\pi$
$602$	0	0
$603$	0.538109i	0.0219135i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	20.0525i	0.813905i	0.913449	+	0.406953i	$0.133409\pi$
$607$	20.0525i	0.813905i	−0.913449	+	0.406953i	$0.866591\pi$
$608$	0	0
$609$	−64.2051	−2.60172
$610$	0	0
$611$	9.49521	0.384135
$612$	0	0
$613$	− 7.89881i	− 0.319030i	−0.987196	−	0.159515i	$-0.949007\pi$
$613$	− 7.89881i	− 0.319030i	0.987196	−	0.159515i	$-0.0509930\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	41.4094i	1.66708i	0.552459	+	0.833540i	$0.313689\pi$
$617$	41.4094i	1.66708i	−0.552459	+	0.833540i	$0.686311\pi$
$618$	0	0
$619$	−26.9450	−1.08301	−0.541506	−	0.840697i	$-0.682146\pi$
$619$	−26.9450	−1.08301	−0.541506	+	0.840697i	$0.682146\pi$
$620$	0	0
$621$	−21.1743	−0.849695
$622$	0	0
$623$	− 19.7572i	− 0.791555i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 9.72928i	− 0.388550i
$628$	0	0
$629$	−2.56934	−0.102446
$630$	0	0
$631$	−15.4865	−0.616507	−0.308253	−	0.951304i	$-0.599744\pi$
$631$	−15.4865	−0.616507	−0.308253	+	0.951304i	$0.599744\pi$
$632$	0	0
$633$	27.6079i	1.09731i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 79.0775i	− 3.13316i
$638$	0	0
$639$	1.13869	0.0450457
$640$	0	0
$641$	9.93545	0.392427	0.196213	−	0.980561i	$-0.437135\pi$
$641$	9.93545	0.392427	0.196213	+	0.980561i	$0.437135\pi$
$642$	0	0
$643$	5.25240i	0.207134i	0.994622	+	0.103567i	$0.0330257\pi$
$643$	5.25240i	0.207134i	−0.994622	+	0.103567i	$0.966974\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.64452i	− 0.0646528i	−0.999477	−	0.0323264i	$-0.989708\pi$
$647$	− 1.64452i	− 0.0646528i	0.999477	−	0.0323264i	$-0.0102916\pi$
$648$	0	0
$649$	3.45856	0.135760
$650$	0	0
$651$	−10.2062	−0.400011
$652$	0	0
$653$	− 30.9450i	− 1.21097i	−0.795856	−	0.605486i	$-0.792979\pi$
$653$	− 30.9450i	− 1.21097i	0.795856	−	0.605486i	$-0.207021\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0.498138i	0.0194342i
$658$	0	0
$659$	−17.3372	−0.675360	−0.337680	−	0.941261i	$-0.609642\pi$
$659$	−17.3372	−0.675360	−0.337680	+	0.941261i	$0.609642\pi$
$660$	0	0
$661$	8.83237	0.343539	0.171770	−	0.985137i	$-0.445052\pi$
$661$	8.83237	0.343539	0.171770	+	0.985137i	$0.445052\pi$
$662$	0	0
$663$	− 64.0914i	− 2.48910i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	32.6907i	1.26579i
$668$	0	0
$669$	−21.2311	−0.820843
$670$	0	0
$671$	−83.1020	−3.20812
$672$	0	0
$673$	29.6402i	1.14254i	0.820761	+	0.571272i	$0.193550\pi$
$673$	29.6402i	1.14254i	−0.820761	+	0.571272i	$0.806450\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 23.2847i	− 0.894903i	−0.894308	−	0.447451i	$-0.852332\pi$
$677$	− 23.2847i	− 0.894903i	0.894308	−	0.447451i	$-0.147668\pi$
$678$	0	0
$679$	−63.4007	−2.43309
$680$	0	0
$681$	5.42296	0.207808
$682$	0	0
$683$	27.3415i	1.04619i	0.852273	+	0.523097i	$0.175224\pi$
$683$	27.3415i	1.04619i	−0.852273	+	0.523097i	$0.824776\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 31.3449i	− 1.19588i
$688$	0	0
$689$	−8.18596	−0.311860
$690$	0	0
$691$	38.8313	1.47721	0.738607	−	0.674137i	$-0.235484\pi$
$691$	38.8313	1.47721	0.738607	+	0.674137i	$0.235484\pi$
$692$	0	0
$693$	2.63389i	0.100053i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	44.8959i	1.70055i
$698$	0	0
$699$	−14.9817	−0.566659
$700$	0	0
$701$	−19.5144	−0.737048	−0.368524	−	0.929618i	$-0.620137\pi$
$701$	−19.5144	−0.737048	−0.368524	+	0.929618i	$0.620137\pi$
$702$	0	0
$703$	0.387755i	0.0146245i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	32.5972i	1.22595i
$708$	0	0
$709$	11.1387	0.418322	0.209161	−	0.977881i	$-0.432927\pi$
$709$	11.1387	0.418322	0.209161	+	0.977881i	$0.432927\pi$
$710$	0	0
$711$	−0.307362	−0.0115270
$712$	0	0
$713$	5.19658i	0.194614i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	41.5866i	1.55308i
$718$	0	0
$719$	16.1214	0.601227	0.300613	−	0.953746i	$-0.402809\pi$
$719$	16.1214	0.601227	0.300613	+	0.953746i	$0.402809\pi$
$720$	0	0
$721$	−3.04623	−0.113447
$722$	0	0
$723$	16.0492i	0.596875i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	8.21386i	0.304635i	0.988332	+	0.152318i	$0.0486737\pi$
$727$	8.21386i	0.304635i	−0.988332	+	0.152318i	$0.951326\pi$
$728$	0	0
$729$	−26.0096	−0.963318
$730$	0	0
$731$	72.7668	2.69138
$732$	0	0
$733$	28.6743i	1.05911i	0.848276	+	0.529555i	$0.177641\pi$
$733$	28.6743i	1.05911i	−0.848276	+	0.529555i	$0.822359\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	28.8313i	1.06202i
$738$	0	0
$739$	11.7014	0.430442	0.215221	−	0.976565i	$-0.430953\pi$
$739$	11.7014	0.430442	0.215221	+	0.976565i	$0.430953\pi$
$740$	0	0
$741$	−9.67243	−0.355325
$742$	0	0
$743$	1.91087i	0.0701030i	0.999386	+	0.0350515i	$0.0111595\pi$
$743$	1.91087i	0.0701030i	−0.999386	+	0.0350515i	$0.988840\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.283233i	0.0103630i
$748$	0	0
$749$	−24.1493	−0.882397
$750$	0	0
$751$	9.55976	0.348841	0.174420	−	0.984671i	$-0.444195\pi$
$751$	9.55976	0.348841	0.174420	+	0.984671i	$0.444195\pi$
$752$	0	0
$753$	23.8709i	0.869904i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	33.6435i	1.22279i	0.791324	+	0.611397i	$0.209392\pi$
$757$	33.6435i	1.22279i	−0.791324	+	0.611397i	$0.790608\pi$
$758$	0	0
$759$	40.3698	1.46533
$760$	0	0
$761$	−22.4113	−0.812409	−0.406204	−	0.913782i	$-0.633148\pi$
$761$	−22.4113	−0.812409	−0.406204	+	0.913782i	$0.633148\pi$
$762$	0	0
$763$	54.6541i	1.97861i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 3.43835i	− 0.124152i
$768$	0	0
$769$	−4.98937	−0.179921	−0.0899607	−	0.995945i	$-0.528674\pi$
$769$	−4.98937	−0.179921	−0.0899607	+	0.995945i	$0.528674\pi$
$770$	0	0
$771$	18.7755	0.676183
$772$	0	0
$773$	16.6508i	0.598887i	0.954114	+	0.299443i	$0.0968010\pi$
$773$	16.6508i	0.598887i	−0.954114	+	0.299443i	$0.903199\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 3.15994i	− 0.113362i
$778$	0	0
$779$	6.77551	0.242758
$780$	0	0
$781$	61.0096	2.18309
$782$	0	0
$783$	40.2051i	1.43681i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	43.8453i	1.56292i	0.623958	+	0.781458i	$0.285524\pi$
$787$	43.8453i	1.56292i	−0.623958	+	0.781458i	$0.714476\pi$
$788$	0	0
$789$	31.6435	1.12654
$790$	0	0
$791$	0.412335	0.0146609
$792$	0	0
$793$	82.6164i	2.93380i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 49.0052i	− 1.73585i	−0.496692	−	0.867927i	$-0.665452\pi$
$797$	− 49.0052i	− 1.73585i	0.496692	−	0.867927i	$-0.334548\pi$
$798$	0	0
$799$	11.4586	0.405375
$800$	0	0
$801$	0.440241	0.0155552
$802$	0	0
$803$	26.6897i	0.941859i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 27.3940i	− 0.964315i
$808$	0	0
$809$	−9.19554	−0.323298	−0.161649	−	0.986848i	$-0.551681\pi$
$809$	−9.19554	−0.323298	−0.161649	+	0.986848i	$0.551681\pi$
$810$	0	0
$811$	−18.9527	−0.665520	−0.332760	−	0.943011i	$-0.607980\pi$
$811$	−18.9527	−0.665520	−0.332760	+	0.943011i	$0.607980\pi$
$812$	0	0
$813$	14.6049i	0.512218i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 10.9817i	− 0.384200i
$818$	0	0
$819$	2.61850	0.0914979
$820$	0	0
$821$	−45.7572	−1.59694	−0.798468	−	0.602037i	$-0.794356\pi$
$821$	−45.7572	−1.59694	−0.798468	+	0.602037i	$0.794356\pi$
$822$	0	0
$823$	− 11.3738i	− 0.396466i	−0.980155	−	0.198233i	$-0.936480\pi$
$823$	− 11.3738i	− 0.396466i	0.980155	−	0.198233i	$-0.0635202\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11.8386i	0.411669i	0.978587	+	0.205835i	$0.0659909\pi$
$827$	11.8386i	0.411669i	−0.978587	+	0.205835i	$0.934009\pi$
$828$	0	0
$829$	26.9894	0.937380	0.468690	−	0.883363i	$-0.344726\pi$
$829$	26.9894	0.937380	0.468690	+	0.883363i	$0.344726\pi$
$830$	0	0
$831$	−50.3265	−1.74581
$832$	0	0
$833$	− 95.4286i	− 3.30640i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6.39108i	0.220908i
$838$	0	0
$839$	−2.81215	−0.0970864	−0.0485432	−	0.998821i	$-0.515458\pi$
$839$	−2.81215	−0.0970864	−0.0485432	+	0.998821i	$0.515458\pi$
$840$	0	0
$841$	33.0722	1.14042
$842$	0	0
$843$	− 34.0279i	− 1.17198i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	90.2330i	3.10044i
$848$	0	0
$849$	−28.7110	−0.985358
$850$	0	0
$851$	−1.60892	−0.0551530
$852$	0	0
$853$	− 39.6068i	− 1.35611i	−0.735010	−	0.678056i	$-0.762823\pi$
$853$	− 39.6068i	− 1.35611i	0.735010	−	0.678056i	$-0.237177\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 17.8742i	− 0.610572i	−0.952261	−	0.305286i	$-0.901248\pi$
$857$	− 17.8742i	− 0.610572i	0.952261	−	0.305286i	$-0.0987520\pi$
$858$	0	0
$859$	−4.12910	−0.140883	−0.0704416	−	0.997516i	$-0.522441\pi$
$859$	−4.12910	−0.140883	−0.0704416	+	0.997516i	$0.522441\pi$
$860$	0	0
$861$	−55.2158	−1.88175
$862$	0	0
$863$	14.7509i	0.502128i	0.967971	+	0.251064i	$0.0807804\pi$
$863$	14.7509i	0.502128i	−0.967971	+	0.251064i	$0.919220\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 47.3973i	− 1.60970i
$868$	0	0
$869$	−16.4681	−0.558644
$870$	0	0
$871$	28.6628	0.971203
$872$	0	0
$873$	− 1.41273i	− 0.0478137i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 16.8357i	− 0.568501i	−0.958750	−	0.284250i	$-0.908255\pi$
$877$	− 16.8357i	− 0.568501i	0.958750	−	0.284250i	$-0.0917447\pi$
$878$	0	0
$879$	46.4971	1.56831
$880$	0	0
$881$	45.4373	1.53082	0.765411	−	0.643542i	$-0.222536\pi$
$881$	45.4373	1.53082	0.765411	+	0.643542i	$0.222536\pi$
$882$	0	0
$883$	38.0837i	1.28162i	0.767700	+	0.640810i	$0.221401\pi$
$883$	38.0837i	1.28162i	−0.767700	+	0.640810i	$0.778599\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 21.3049i	− 0.715348i	−0.933847	−	0.357674i	$-0.883570\pi$
$887$	− 21.3049i	− 0.715348i	0.933847	−	0.357674i	$-0.116430\pi$
$888$	0	0
$889$	61.8496	2.07437
$890$	0	0
$891$	51.3574	1.72054
$892$	0	0
$893$	− 1.72928i	− 0.0578682i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 40.1339i	− 1.34003i
$898$	0	0
$899$	9.86712	0.329087
$900$	0	0
$901$	−9.87859	−0.329104
$902$	0	0
$903$	89.4931i	2.97814i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 12.4600i	− 0.413728i	−0.978370	−	0.206864i	$-0.933674\pi$
$907$	− 12.4600i	− 0.413728i	0.978370	−	0.206864i	$-0.0663257\pi$
$908$	0	0
$909$	−0.726351	−0.0240916
$910$	0	0
$911$	35.2803	1.16889	0.584444	−	0.811434i	$-0.301313\pi$
$911$	35.2803	1.16889	0.584444	+	0.811434i	$0.301313\pi$
$912$	0	0
$913$	15.1753i	0.502230i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 18.5048i	− 0.611082i
$918$	0	0
$919$	0.392124	0.0129350	0.00646749	−	0.999979i	$-0.497941\pi$
$919$	0.392124	0.0129350	0.00646749	+	0.999979i	$0.497941\pi$
$920$	0	0
$921$	−53.2158	−1.75352
$922$	0	0
$923$	− 60.6531i	− 1.99642i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 0.0678779i	− 0.00222940i
$928$	0	0
$929$	13.4942	0.442729	0.221365	−	0.975191i	$-0.428949\pi$
$929$	13.4942	0.442729	0.221365	+	0.975191i	$0.428949\pi$
$930$	0	0
$931$	−14.4017	−0.471997
$932$	0	0
$933$	− 7.26009i	− 0.237685i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	53.4296i	1.74547i	0.488195	+	0.872735i	$0.337656\pi$
$937$	53.4296i	1.74547i	−0.488195	+	0.872735i	$0.662344\pi$
$938$	0	0
$939$	26.6049	0.868220
$940$	0	0
$941$	−0.0568550	−0.00185342	−0.000926710	−	1.00000i	$-0.500295\pi$
$941$	−0.0568550	−0.00185342	−0.000926710		1.00000i	$0.500295\pi$
$942$	0	0
$943$	28.1137i	0.915508i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 2.03664i	− 0.0661820i	−0.999452	−	0.0330910i	$-0.989465\pi$
$947$	− 2.03664i	− 0.0661820i	0.999452	−	0.0330910i	$-0.0105351\pi$
$948$	0	0
$949$	26.5337	0.861322
$950$	0	0
$951$	−23.4509	−0.760446
$952$	0	0
$953$	17.5352i	0.568020i	0.958821	+	0.284010i	$0.0916649\pi$
$953$	17.5352i	0.568020i	−0.958821	+	0.284010i	$0.908335\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 76.6531i	− 2.47784i
$958$	0	0
$959$	−49.1589	−1.58742
$960$	0	0
$961$	−29.4315	−0.949403
$962$	0	0
$963$	− 0.538109i	− 0.0173403i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	14.9904i	0.482059i	0.970518	+	0.241030i	$0.0774851\pi$
$967$	14.9904i	0.482059i	−0.970518	+	0.241030i	$0.922515\pi$
$968$	0	0
$969$	−11.6724	−0.374972
$970$	0	0
$971$	−20.8805	−0.670087	−0.335043	−	0.942203i	$-0.608751\pi$
$971$	−20.8805	−0.670087	−0.335043	+	0.942203i	$0.608751\pi$
$972$	0	0
$973$	58.1483i	1.86415i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.0246i	0.768614i	0.923205	+	0.384307i	$0.125560\pi$
$977$	24.0246i	0.768614i	−0.923205	+	0.384307i	$0.874440\pi$
$978$	0	0
$979$	23.5877	0.753865
$980$	0	0
$981$	−1.21784	−0.0388825
$982$	0	0
$983$	− 30.9205i	− 0.986209i	−0.869970	−	0.493105i	$-0.835862\pi$
$983$	− 30.9205i	− 0.986209i	0.869970	−	0.493105i	$-0.164138\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	14.0925i	0.448568i
$988$	0	0
$989$	45.5664	1.44893
$990$	0	0
$991$	2.56934	0.0816179	0.0408089	−	0.999167i	$-0.487007\pi$
$991$	2.56934	0.0816179	0.0408089	+	0.999167i	$0.487007\pi$
$992$	0	0
$993$	29.8140i	0.946120i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 35.1878i	− 1.11441i	−0.830375	−	0.557205i	$-0.811874\pi$
$997$	− 35.1878i	− 1.11441i	0.830375	−	0.557205i	$-0.188126\pi$
$998$	0	0
$999$	−1.97875	−0.0626048

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.k.3649.5		6
5.2	odd	4		760.2.a.h.1.3	✓	3
5.3	odd	4		3800.2.a.y.1.1		3
5.4	even	2	inner	3800.2.d.k.3649.2		6
15.2	even	4		6840.2.a.bj.1.1		3
20.3	even	4		7600.2.a.bo.1.3		3
20.7	even	4		1520.2.a.r.1.1		3
40.27	even	4		6080.2.a.bs.1.3		3
40.37	odd	4		6080.2.a.bw.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.h.1.3	✓	3	5.2	odd	4
1520.2.a.r.1.1		3	20.7	even	4
3800.2.a.y.1.1		3	5.3	odd	4
3800.2.d.k.3649.2		6	5.4	even	2	inner
3800.2.d.k.3649.5		6	1.1	even	1	trivial
6080.2.a.bs.1.3		3	40.27	even	4
6080.2.a.bw.1.1		3	40.37	odd	4
6840.2.a.bj.1.1		3	15.2	even	4
7600.2.a.bo.1.3		3	20.3	even	4

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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.76156i q^{3} +4.62620i q^{7} -0.103084 q^{9} +O(q^{10})\)	\(q+1.76156i q^{3} +4.62620i q^{7} -0.103084 q^{9} -5.52311 q^{11} +5.49084i q^{13} +6.62620i q^{17} +1.00000 q^{19} -8.14931 q^{21} +4.14931i q^{23} +5.10308i q^{27} +7.87859 q^{29} +1.25240 q^{31} -9.72928i q^{33} +0.387755i q^{37} -9.67243 q^{39} +6.77551 q^{41} -10.9817i q^{43} -1.72928i q^{47} -14.4017 q^{49} -11.6724 q^{51} +1.49084i q^{53} +1.76156i q^{57} -0.626198 q^{59} +15.0462 q^{61} -0.476886i q^{63} -5.22012i q^{67} -7.30925 q^{69} -11.0462 q^{71} -4.83237i q^{73} -25.5510i q^{77} +2.98168 q^{79} -9.29862 q^{81} -2.74760i q^{83} +13.8786i q^{87} -4.27072 q^{89} -25.4017 q^{91} +2.20617i q^{93} +13.7047i q^{97} +0.569343 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{9}+O(q^{10}) \) 6 * q - 8 * q^9	\( 6 q - 8 q^{9} - 8 q^{11} + 6 q^{19} - 6 q^{21} - 6 q^{29} - 28 q^{31} + 10 q^{39} - 20 q^{41} - 8 q^{49} - 2 q^{51} + 14 q^{59} + 40 q^{61} - 66 q^{69} - 16 q^{71} - 28 q^{79} + 30 q^{81} - 36 q^{89} - 74 q^{91} - 72 q^{99}+O(q^{100}) \) 6 * q - 8 * q^9 - 8 * q^11 + 6 * q^19 - 6 * q^21 - 6 * q^29 - 28 * q^31 + 10 * q^39 - 20 * q^41 - 8 * q^49 - 2 * q^51 + 14 * q^59 + 40 * q^61 - 66 * q^69 - 16 * q^71 - 28 * q^79 + 30 * q^81 - 36 * q^89 - 74 * q^91 - 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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