LMFDB - Embedded newform 2028.4.b.a.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2028,4,Mod(337,2028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2028, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2028.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2028 = 2^{2} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2028.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:	$119.655873492$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2028.337
Dual form			2028.4.b.a.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +6.00000i q^{5} -4.00000i q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +6.00000i q^{5} -4.00000i q^{7} +9.00000 q^{9} +36.0000i q^{11} -18.0000i q^{15} -66.0000 q^{17} -56.0000i q^{19} +12.0000i q^{21} -96.0000 q^{23} +89.0000 q^{25} -27.0000 q^{27} +222.000 q^{29} -260.000i q^{31} -108.000i q^{33} +24.0000 q^{35} -106.000i q^{37} +90.0000i q^{41} -44.0000 q^{43} +54.0000i q^{45} +168.000i q^{47} +327.000 q^{49} +198.000 q^{51} +30.0000 q^{53} -216.000 q^{55} +168.000i q^{57} +348.000i q^{59} -346.000 q^{61} -36.0000i q^{63} +256.000i q^{67} +288.000 q^{69} +168.000i q^{71} -814.000i q^{73} -267.000 q^{75} +144.000 q^{77} +200.000 q^{79} +81.0000 q^{81} -1236.00i q^{83} -396.000i q^{85} -666.000 q^{87} +318.000i q^{89} +780.000i q^{93} +336.000 q^{95} +502.000i q^{97} +324.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 18 * q^9	$ 2 q - 6 q^{3} + 18 q^{9} - 132 q^{17} - 192 q^{23} + 178 q^{25} - 54 q^{27} + 444 q^{29} + 48 q^{35} - 88 q^{43} + 654 q^{49} + 396 q^{51} + 60 q^{53} - 432 q^{55} - 692 q^{61} + 576 q^{69} - 534 q^{75} + 288 q^{77} + 400 q^{79} + 162 q^{81} - 1332 q^{87} + 672 q^{95}+O(q^{100}) $ 2 * q - 6 * q^3 + 18 * q^9 - 132 * q^17 - 192 * q^23 + 178 * q^25 - 54 * q^27 + 444 * q^29 + 48 * q^35 - 88 * q^43 + 654 * q^49 + 396 * q^51 + 60 * q^53 - 432 * q^55 - 692 * q^61 + 576 * q^69 - 534 * q^75 + 288 * q^77 + 400 * q^79 + 162 * q^81 - 1332 * q^87 + 672 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times$.

$n$	$677$	$1015$	$1861$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	6.00000i	0.536656i	0.963328	+	0.268328i	$0.0864711\pi$
$5$	6.00000i	0.536656i	−0.963328	+	0.268328i	$0.913529\pi$
$6$	0	0
$7$	− 4.00000i	− 0.215980i	−0.994152	−	0.107990i	$-0.965559\pi$
$7$	− 4.00000i	− 0.215980i	0.994152	−	0.107990i	$-0.0344414\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	36.0000i	0.986764i	0.869813	+	0.493382i	$0.164240\pi$
$11$	36.0000i	0.986764i	−0.869813	+	0.493382i	$0.835760\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 18.0000i	− 0.309839i
$16$	0	0
$17$	−66.0000	−0.941609	−0.470804	−	0.882238i	$-0.656036\pi$
$17$	−66.0000	−0.941609	−0.470804	+	0.882238i	$0.656036\pi$
$18$	0	0
$19$	− 56.0000i	− 0.676173i	−0.941115	−	0.338086i	$-0.890220\pi$
$19$	− 56.0000i	− 0.676173i	0.941115	−	0.338086i	$-0.109780\pi$
$20$	0	0
$21$	12.0000i	0.124696i
$22$	0	0
$23$	−96.0000	−0.870321	−0.435161	−	0.900353i	$-0.643308\pi$
$23$	−96.0000	−0.870321	−0.435161	+	0.900353i	$0.643308\pi$
$24$	0	0
$25$	89.0000	0.712000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	222.000	1.42153	0.710765	−	0.703430i	$-0.248349\pi$
$29$	222.000	1.42153	0.710765	+	0.703430i	$0.248349\pi$
$30$	0	0
$31$	− 260.000i	− 1.50637i	−0.657810	−	0.753184i	$-0.728517\pi$
$31$	− 260.000i	− 1.50637i	0.657810	−	0.753184i	$-0.271483\pi$
$32$	0	0
$33$	− 108.000i	− 0.569709i
$34$	0	0
$35$	24.0000	0.115907
$36$	0	0
$37$	− 106.000i	− 0.470981i	−0.971877	−	0.235490i	$-0.924330\pi$
$37$	− 106.000i	− 0.470981i	0.971877	−	0.235490i	$-0.0756696\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	90.0000i	0.342820i	0.985200	+	0.171410i	$0.0548323\pi$
$41$	90.0000i	0.342820i	−0.985200	+	0.171410i	$0.945168\pi$
$42$	0	0
$43$	−44.0000	−0.156045	−0.0780225	−	0.996952i	$-0.524861\pi$
$43$	−44.0000	−0.156045	−0.0780225	+	0.996952i	$0.524861\pi$
$44$	0	0
$45$	54.0000i	0.178885i
$46$	0	0
$47$	168.000i	0.521390i	0.965421	+	0.260695i	$0.0839517\pi$
$47$	168.000i	0.521390i	−0.965421	+	0.260695i	$0.916048\pi$
$48$	0	0
$49$	327.000	0.953353
$50$	0	0
$51$	198.000	0.543638
$52$	0	0
$53$	30.0000	0.0777513	0.0388756	−	0.999244i	$-0.487622\pi$
$53$	30.0000	0.0777513	0.0388756	+	0.999244i	$0.487622\pi$
$54$	0	0
$55$	−216.000	−0.529553
$56$	0	0
$57$	168.000i	0.390388i
$58$	0	0
$59$	348.000i	0.767894i	0.923355	+	0.383947i	$0.125435\pi$
$59$	348.000i	0.767894i	−0.923355	+	0.383947i	$0.874565\pi$
$60$	0	0
$61$	−346.000	−0.726242	−0.363121	−	0.931742i	$-0.618289\pi$
$61$	−346.000	−0.726242	−0.363121	+	0.931742i	$0.618289\pi$
$62$	0	0
$63$	− 36.0000i	− 0.0719932i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	256.000i	0.466797i	0.972381	+	0.233398i	$0.0749846\pi$
$67$	256.000i	0.466797i	−0.972381	+	0.233398i	$0.925015\pi$
$68$	0	0
$69$	288.000	0.502480
$70$	0	0
$71$	168.000i	0.280816i	0.990094	+	0.140408i	$0.0448414\pi$
$71$	168.000i	0.280816i	−0.990094	+	0.140408i	$0.955159\pi$
$72$	0	0
$73$	− 814.000i	− 1.30509i	−0.757750	−	0.652544i	$-0.773702\pi$
$73$	− 814.000i	− 1.30509i	0.757750	−	0.652544i	$-0.226298\pi$
$74$	0	0
$75$	−267.000	−0.411073
$76$	0	0
$77$	144.000	0.213121
$78$	0	0
$79$	200.000	0.284832	0.142416	−	0.989807i	$-0.454513\pi$
$79$	200.000	0.284832	0.142416	+	0.989807i	$0.454513\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 1236.00i	− 1.63456i	−0.576240	−	0.817281i	$-0.695480\pi$
$83$	− 1236.00i	− 1.63456i	0.576240	−	0.817281i	$-0.304520\pi$
$84$	0	0
$85$	− 396.000i	− 0.505320i
$86$	0	0
$87$	−666.000	−0.820721
$88$	0	0
$89$	318.000i	0.378741i	0.981906	+	0.189370i	$0.0606447\pi$
$89$	318.000i	0.378741i	−0.981906	+	0.189370i	$0.939355\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	780.000i	0.869701i
$94$	0	0
$95$	336.000	0.362872
$96$	0	0
$97$	502.000i	0.525468i	0.964868	+	0.262734i	$0.0846241\pi$
$97$	502.000i	0.525468i	−0.964868	+	0.262734i	$0.915376\pi$
$98$	0	0
$99$	324.000i	0.328921i
$100$	0	0
$101$	−1062.00	−1.04627	−0.523133	−	0.852251i	$-0.675237\pi$
$101$	−1062.00	−1.04627	−0.523133	+	0.852251i	$0.675237\pi$
$102$	0	0
$103$	64.0000	0.0612243	0.0306122	−	0.999531i	$-0.490254\pi$
$103$	64.0000	0.0612243	0.0306122	+	0.999531i	$0.490254\pi$
$104$	0	0
$105$	−72.0000	−0.0669189
$106$	0	0
$107$	−444.000	−0.401150	−0.200575	−	0.979678i	$-0.564281\pi$
$107$	−444.000	−0.401150	−0.200575	+	0.979678i	$0.564281\pi$
$108$	0	0
$109$	− 1382.00i	− 1.21442i	−0.794542	−	0.607209i	$-0.792289\pi$
$109$	− 1382.00i	− 1.21442i	0.794542	−	0.607209i	$-0.207711\pi$
$110$	0	0
$111$	318.000i	0.271921i
$112$	0	0
$113$	−870.000	−0.724272	−0.362136	−	0.932125i	$-0.617952\pi$
$113$	−870.000	−0.724272	−0.362136	+	0.932125i	$0.617952\pi$
$114$	0	0
$115$	− 576.000i	− 0.467063i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	264.000i	0.203368i
$120$	0	0
$121$	35.0000	0.0262960
$122$	0	0
$123$	− 270.000i	− 0.197927i
$124$	0	0
$125$	1284.00i	0.918756i
$126$	0	0
$127$	−464.000	−0.324200	−0.162100	−	0.986774i	$-0.551827\pi$
$127$	−464.000	−0.324200	−0.162100	+	0.986774i	$0.551827\pi$
$128$	0	0
$129$	132.000	0.0900927
$130$	0	0
$131$	1548.00	1.03244	0.516219	−	0.856457i	$-0.327339\pi$
$131$	1548.00	1.03244	0.516219	+	0.856457i	$0.327339\pi$
$132$	0	0
$133$	−224.000	−0.146040
$134$	0	0
$135$	− 162.000i	− 0.103280i
$136$	0	0
$137$	294.000i	0.183344i	0.995789	+	0.0916720i	$0.0292211\pi$
$137$	294.000i	0.183344i	−0.995789	+	0.0916720i	$0.970779\pi$
$138$	0	0
$139$	2564.00	1.56457	0.782286	−	0.622919i	$-0.214053\pi$
$139$	2564.00	1.56457	0.782286	+	0.622919i	$0.214053\pi$
$140$	0	0
$141$	− 504.000i	− 0.301025i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1332.00i	0.762873i
$146$	0	0
$147$	−981.000	−0.550418
$148$	0	0
$149$	− 114.000i	− 0.0626795i	−0.999509	−	0.0313397i	$-0.990023\pi$
$149$	− 114.000i	− 0.0626795i	0.999509	−	0.0313397i	$-0.00997738\pi$
$150$	0	0
$151$	2036.00i	1.09727i	0.836063	+	0.548634i	$0.184852\pi$
$151$	2036.00i	1.09727i	−0.836063	+	0.548634i	$0.815148\pi$
$152$	0	0
$153$	−594.000	−0.313870
$154$	0	0
$155$	1560.00	0.808401
$156$	0	0
$157$	2870.00	1.45892	0.729462	−	0.684022i	$-0.239771\pi$
$157$	2870.00	1.45892	0.729462	+	0.684022i	$0.239771\pi$
$158$	0	0
$159$	−90.0000	−0.0448897
$160$	0	0
$161$	384.000i	0.187972i
$162$	0	0
$163$	1472.00i	0.707337i	0.935371	+	0.353669i	$0.115066\pi$
$163$	1472.00i	0.707337i	−0.935371	+	0.353669i	$0.884934\pi$
$164$	0	0
$165$	648.000	0.305738
$166$	0	0
$167$	− 240.000i	− 0.111208i	−0.998453	−	0.0556041i	$-0.982292\pi$
$167$	− 240.000i	− 0.111208i	0.998453	−	0.0556041i	$-0.0177085\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 504.000i	− 0.225391i
$172$	0	0
$173$	306.000	0.134478	0.0672392	−	0.997737i	$-0.478581\pi$
$173$	306.000	0.134478	0.0672392	+	0.997737i	$0.478581\pi$
$174$	0	0
$175$	− 356.000i	− 0.153778i
$176$	0	0
$177$	− 1044.00i	− 0.443344i
$178$	0	0
$179$	2052.00	0.856836	0.428418	−	0.903581i	$-0.359071\pi$
$179$	2052.00	0.856836	0.428418	+	0.903581i	$0.359071\pi$
$180$	0	0
$181$	4498.00	1.84715	0.923574	−	0.383421i	$-0.125254\pi$
$181$	4498.00	1.84715	0.923574	+	0.383421i	$0.125254\pi$
$182$	0	0
$183$	1038.00	0.419296
$184$	0	0
$185$	636.000	0.252755
$186$	0	0
$187$	− 2376.00i	− 0.929146i
$188$	0	0
$189$	108.000i	0.0415653i
$190$	0	0
$191$	−4056.00	−1.53655	−0.768277	−	0.640117i	$-0.778886\pi$
$191$	−4056.00	−1.53655	−0.768277	+	0.640117i	$0.778886\pi$
$192$	0	0
$193$	− 2062.00i	− 0.769047i	−0.923115	−	0.384523i	$-0.874366\pi$
$193$	− 2062.00i	− 0.769047i	0.923115	−	0.384523i	$-0.125634\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4374.00i	1.58190i	0.611880	+	0.790951i	$0.290414\pi$
$197$	4374.00i	1.58190i	−0.611880	+	0.790951i	$0.709586\pi$
$198$	0	0
$199$	2536.00	0.903378	0.451689	−	0.892175i	$-0.350822\pi$
$199$	2536.00	0.903378	0.451689	+	0.892175i	$0.350822\pi$
$200$	0	0
$201$	− 768.000i	− 0.269505i
$202$	0	0
$203$	− 888.000i	− 0.307022i
$204$	0	0
$205$	−540.000	−0.183977
$206$	0	0
$207$	−864.000	−0.290107
$208$	0	0
$209$	2016.00	0.667223
$210$	0	0
$211$	−4444.00	−1.44994	−0.724971	−	0.688780i	$-0.758147\pi$
$211$	−4444.00	−1.44994	−0.724971	+	0.688780i	$0.758147\pi$
$212$	0	0
$213$	− 504.000i	− 0.162129i
$214$	0	0
$215$	− 264.000i	− 0.0837426i
$216$	0	0
$217$	−1040.00	−0.325345
$218$	0	0
$219$	2442.00i	0.753493i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	2716.00i	0.815591i	0.913073	+	0.407796i	$0.133702\pi$
$223$	2716.00i	0.815591i	−0.913073	+	0.407796i	$0.866298\pi$
$224$	0	0
$225$	801.000	0.237333
$226$	0	0
$227$	4692.00i	1.37189i	0.727653	+	0.685945i	$0.240611\pi$
$227$	4692.00i	1.37189i	−0.727653	+	0.685945i	$0.759389\pi$
$228$	0	0
$229$	6446.00i	1.86010i	0.367429	+	0.930052i	$0.380238\pi$
$229$	6446.00i	1.86010i	−0.367429	+	0.930052i	$0.619762\pi$
$230$	0	0
$231$	−432.000	−0.123046
$232$	0	0
$233$	3102.00	0.872184	0.436092	−	0.899902i	$-0.356362\pi$
$233$	3102.00	0.872184	0.436092	+	0.899902i	$0.356362\pi$
$234$	0	0
$235$	−1008.00	−0.279807
$236$	0	0
$237$	−600.000	−0.164448
$238$	0	0
$239$	− 816.000i	− 0.220848i	−0.993885	−	0.110424i	$-0.964779\pi$
$239$	− 816.000i	− 0.220848i	0.993885	−	0.110424i	$-0.0352209\pi$
$240$	0	0
$241$	3818.00i	1.02049i	0.860028	+	0.510247i	$0.170446\pi$
$241$	3818.00i	1.02049i	−0.860028	+	0.510247i	$0.829554\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	1962.00i	0.511623i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	3708.00i	0.943715i
$250$	0	0
$251$	6612.00	1.66273	0.831366	−	0.555725i	$-0.187559\pi$
$251$	6612.00	1.66273	0.831366	+	0.555725i	$0.187559\pi$
$252$	0	0
$253$	− 3456.00i	− 0.858802i
$254$	0	0
$255$	1188.00i	0.291747i
$256$	0	0
$257$	4806.00	1.16650	0.583249	−	0.812293i	$-0.301781\pi$
$257$	4806.00	1.16650	0.583249	+	0.812293i	$0.301781\pi$
$258$	0	0
$259$	−424.000	−0.101722
$260$	0	0
$261$	1998.00	0.473843
$262$	0	0
$263$	−4584.00	−1.07476	−0.537379	−	0.843341i	$-0.680586\pi$
$263$	−4584.00	−1.07476	−0.537379	+	0.843341i	$0.680586\pi$
$264$	0	0
$265$	180.000i	0.0417257i
$266$	0	0
$267$	− 954.000i	− 0.218666i
$268$	0	0
$269$	7134.00	1.61698	0.808490	−	0.588510i	$-0.200285\pi$
$269$	7134.00	1.61698	0.808490	+	0.588510i	$0.200285\pi$
$270$	0	0
$271$	3140.00i	0.703843i	0.936030	+	0.351921i	$0.114472\pi$
$271$	3140.00i	0.703843i	−0.936030	+	0.351921i	$0.885528\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3204.00i	0.702576i
$276$	0	0
$277$	4786.00	1.03813	0.519067	−	0.854734i	$-0.326280\pi$
$277$	4786.00	1.03813	0.519067	+	0.854734i	$0.326280\pi$
$278$	0	0
$279$	− 2340.00i	− 0.502122i
$280$	0	0
$281$	3798.00i	0.806298i	0.915134	+	0.403149i	$0.132084\pi$
$281$	3798.00i	0.806298i	−0.915134	+	0.403149i	$0.867916\pi$
$282$	0	0
$283$	−3572.00	−0.750295	−0.375147	−	0.926965i	$-0.622408\pi$
$283$	−3572.00	−0.750295	−0.375147	+	0.926965i	$0.622408\pi$
$284$	0	0
$285$	−1008.00	−0.209504
$286$	0	0
$287$	360.000	0.0740423
$288$	0	0
$289$	−557.000	−0.113373
$290$	0	0
$291$	− 1506.00i	− 0.303379i
$292$	0	0
$293$	7122.00i	1.42004i	0.704182	+	0.710020i	$0.251314\pi$
$293$	7122.00i	1.42004i	−0.704182	+	0.710020i	$0.748686\pi$
$294$	0	0
$295$	−2088.00	−0.412095
$296$	0	0
$297$	− 972.000i	− 0.189903i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	176.000i	0.0337026i
$302$	0	0
$303$	3186.00	0.604062
$304$	0	0
$305$	− 2076.00i	− 0.389742i
$306$	0	0
$307$	− 6856.00i	− 1.27457i	−0.770629	−	0.637284i	$-0.780058\pi$
$307$	− 6856.00i	− 1.27457i	0.770629	−	0.637284i	$-0.219942\pi$
$308$	0	0
$309$	−192.000	−0.0353479
$310$	0	0
$311$	8832.00	1.61034	0.805172	−	0.593042i	$-0.202073\pi$
$311$	8832.00	1.61034	0.805172	+	0.593042i	$0.202073\pi$
$312$	0	0
$313$	3626.00	0.654804	0.327402	−	0.944885i	$-0.393827\pi$
$313$	3626.00	0.654804	0.327402	+	0.944885i	$0.393827\pi$
$314$	0	0
$315$	216.000	0.0386356
$316$	0	0
$317$	− 10146.0i	− 1.79765i	−0.438304	−	0.898827i	$-0.644421\pi$
$317$	− 10146.0i	− 1.79765i	0.438304	−	0.898827i	$-0.355579\pi$
$318$	0	0
$319$	7992.00i	1.40272i
$320$	0	0
$321$	1332.00	0.231604
$322$	0	0
$323$	3696.00i	0.636690i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4146.00i	0.701145i
$328$	0	0
$329$	672.000	0.112610
$330$	0	0
$331$	− 6536.00i	− 1.08535i	−0.839943	−	0.542675i	$-0.817411\pi$
$331$	− 6536.00i	− 1.08535i	0.839943	−	0.542675i	$-0.182589\pi$
$332$	0	0
$333$	− 954.000i	− 0.156994i
$334$	0	0
$335$	−1536.00	−0.250509
$336$	0	0
$337$	6094.00	0.985048	0.492524	−	0.870299i	$-0.336074\pi$
$337$	6094.00	0.985048	0.492524	+	0.870299i	$0.336074\pi$
$338$	0	0
$339$	2610.00	0.418159
$340$	0	0
$341$	9360.00	1.48643
$342$	0	0
$343$	− 2680.00i	− 0.421885i
$344$	0	0
$345$	1728.00i	0.269659i
$346$	0	0
$347$	−2724.00	−0.421418	−0.210709	−	0.977549i	$-0.567577\pi$
$347$	−2724.00	−0.421418	−0.210709	+	0.977549i	$0.567577\pi$
$348$	0	0
$349$	− 1522.00i	− 0.233441i	−0.993165	−	0.116720i	$-0.962762\pi$
$349$	− 1522.00i	− 0.233441i	0.993165	−	0.116720i	$-0.0372381\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1362.00i	0.205360i	0.994714	+	0.102680i	$0.0327417\pi$
$353$	1362.00i	0.205360i	−0.994714	+	0.102680i	$0.967258\pi$
$354$	0	0
$355$	−1008.00	−0.150702
$356$	0	0
$357$	− 792.000i	− 0.117415i
$358$	0	0
$359$	8880.00i	1.30548i	0.757581	+	0.652742i	$0.226381\pi$
$359$	8880.00i	1.30548i	−0.757581	+	0.652742i	$0.773619\pi$
$360$	0	0
$361$	3723.00	0.542790
$362$	0	0
$363$	−105.000	−0.0151820
$364$	0	0
$365$	4884.00	0.700384
$366$	0	0
$367$	−3712.00	−0.527970	−0.263985	−	0.964527i	$-0.585037\pi$
$367$	−3712.00	−0.527970	−0.263985	+	0.964527i	$0.585037\pi$
$368$	0	0
$369$	810.000i	0.114273i
$370$	0	0
$371$	− 120.000i	− 0.0167927i
$372$	0	0
$373$	5726.00	0.794855	0.397428	−	0.917634i	$-0.369903\pi$
$373$	5726.00	0.794855	0.397428	+	0.917634i	$0.369903\pi$
$374$	0	0
$375$	− 3852.00i	− 0.530444i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	13168.0i	1.78468i	0.451361	+	0.892341i	$0.350939\pi$
$379$	13168.0i	1.78468i	−0.451361	+	0.892341i	$0.649061\pi$
$380$	0	0
$381$	1392.00	0.187177
$382$	0	0
$383$	− 4872.00i	− 0.649994i	−0.945715	−	0.324997i	$-0.894637\pi$
$383$	− 4872.00i	− 0.649994i	0.945715	−	0.324997i	$-0.105363\pi$
$384$	0	0
$385$	864.000i	0.114373i
$386$	0	0
$387$	−396.000	−0.0520150
$388$	0	0
$389$	1266.00	0.165010	0.0825048	−	0.996591i	$-0.473708\pi$
$389$	1266.00	0.165010	0.0825048	+	0.996591i	$0.473708\pi$
$390$	0	0
$391$	6336.00	0.819502
$392$	0	0
$393$	−4644.00	−0.596078
$394$	0	0
$395$	1200.00i	0.152857i
$396$	0	0
$397$	− 4882.00i	− 0.617180i	−0.951195	−	0.308590i	$-0.900143\pi$
$397$	− 4882.00i	− 0.617180i	0.951195	−	0.308590i	$-0.0998571\pi$
$398$	0	0
$399$	672.000	0.0843160
$400$	0	0
$401$	− 90.0000i	− 0.0112079i	−0.999984	−	0.00560397i	$-0.998216\pi$
$401$	− 90.0000i	− 0.0112079i	0.999984	−	0.00560397i	$-0.00178381\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	486.000i	0.0596285i
$406$	0	0
$407$	3816.00	0.464747
$408$	0	0
$409$	− 2354.00i	− 0.284591i	−0.989824	−	0.142296i	$-0.954552\pi$
$409$	− 2354.00i	− 0.284591i	0.989824	−	0.142296i	$-0.0454484\pi$
$410$	0	0
$411$	− 882.000i	− 0.105854i
$412$	0	0
$413$	1392.00	0.165849
$414$	0	0
$415$	7416.00	0.877198
$416$	0	0
$417$	−7692.00	−0.903307
$418$	0	0
$419$	7020.00	0.818495	0.409248	−	0.912423i	$-0.365791\pi$
$419$	7020.00	0.818495	0.409248	+	0.912423i	$0.365791\pi$
$420$	0	0
$421$	− 302.000i	− 0.0349610i	−0.999847	−	0.0174805i	$-0.994436\pi$
$421$	− 302.000i	− 0.0349610i	0.999847	−	0.0174805i	$-0.00556450\pi$
$422$	0	0
$423$	1512.00i	0.173797i
$424$	0	0
$425$	−5874.00	−0.670426
$426$	0	0
$427$	1384.00i	0.156854i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 9816.00i	− 1.09703i	−0.836141	−	0.548515i	$-0.815193\pi$
$431$	− 9816.00i	− 1.09703i	0.836141	−	0.548515i	$-0.184807\pi$
$432$	0	0
$433$	14782.0	1.64059	0.820297	−	0.571937i	$-0.193808\pi$
$433$	14782.0	1.64059	0.820297	+	0.571937i	$0.193808\pi$
$434$	0	0
$435$	− 3996.00i	− 0.440445i
$436$	0	0
$437$	5376.00i	0.588487i
$438$	0	0
$439$	−3584.00	−0.389647	−0.194823	−	0.980838i	$-0.562413\pi$
$439$	−3584.00	−0.389647	−0.194823	+	0.980838i	$0.562413\pi$
$440$	0	0
$441$	2943.00	0.317784
$442$	0	0
$443$	180.000	0.0193049	0.00965244	−	0.999953i	$-0.496927\pi$
$443$	180.000	0.0193049	0.00965244	+	0.999953i	$0.496927\pi$
$444$	0	0
$445$	−1908.00	−0.203254
$446$	0	0
$447$	342.000i	0.0361880i
$448$	0	0
$449$	− 3450.00i	− 0.362618i	−0.983426	−	0.181309i	$-0.941967\pi$
$449$	− 3450.00i	− 0.362618i	0.983426	−	0.181309i	$-0.0580334\pi$
$450$	0	0
$451$	−3240.00	−0.338283
$452$	0	0
$453$	− 6108.00i	− 0.633507i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16654.0i	1.70469i	0.522984	+	0.852343i	$0.324819\pi$
$457$	16654.0i	1.70469i	−0.522984	+	0.852343i	$0.675181\pi$
$458$	0	0
$459$	1782.00	0.181213
$460$	0	0
$461$	14046.0i	1.41906i	0.704674	+	0.709531i	$0.251093\pi$
$461$	14046.0i	1.41906i	−0.704674	+	0.709531i	$0.748907\pi$
$462$	0	0
$463$	− 4588.00i	− 0.460524i	−0.973129	−	0.230262i	$-0.926042\pi$
$463$	− 4588.00i	− 0.460524i	0.973129	−	0.230262i	$-0.0739583\pi$
$464$	0	0
$465$	−4680.00	−0.466731
$466$	0	0
$467$	15372.0	1.52319	0.761597	−	0.648051i	$-0.224416\pi$
$467$	15372.0	1.52319	0.761597	+	0.648051i	$0.224416\pi$
$468$	0	0
$469$	1024.00	0.100819
$470$	0	0
$471$	−8610.00	−0.842310
$472$	0	0
$473$	− 1584.00i	− 0.153980i
$474$	0	0
$475$	− 4984.00i	− 0.481435i
$476$	0	0
$477$	270.000	0.0259171
$478$	0	0
$479$	12864.0i	1.22708i	0.789664	+	0.613540i	$0.210255\pi$
$479$	12864.0i	1.22708i	−0.789664	+	0.613540i	$0.789745\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 1152.00i	− 0.108525i
$484$	0	0
$485$	−3012.00	−0.281996
$486$	0	0
$487$	10276.0i	0.956160i	0.878316	+	0.478080i	$0.158667\pi$
$487$	10276.0i	0.956160i	−0.878316	+	0.478080i	$0.841333\pi$
$488$	0	0
$489$	− 4416.00i	− 0.408381i
$490$	0	0
$491$	11220.0	1.03127	0.515633	−	0.856810i	$-0.327557\pi$
$491$	11220.0	1.03127	0.515633	+	0.856810i	$0.327557\pi$
$492$	0	0
$493$	−14652.0	−1.33853
$494$	0	0
$495$	−1944.00	−0.176518
$496$	0	0
$497$	672.000	0.0606505
$498$	0	0
$499$	− 17264.0i	− 1.54878i	−0.632707	−	0.774392i	$-0.718056\pi$
$499$	− 17264.0i	− 1.54878i	0.632707	−	0.774392i	$-0.281944\pi$
$500$	0	0
$501$	720.000i	0.0642060i
$502$	0	0
$503$	−1896.00	−0.168069	−0.0840343	−	0.996463i	$-0.526781\pi$
$503$	−1896.00	−0.168069	−0.0840343	+	0.996463i	$0.526781\pi$
$504$	0	0
$505$	− 6372.00i	− 0.561486i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 5010.00i	− 0.436276i	−0.975918	−	0.218138i	$-0.930002\pi$
$509$	− 5010.00i	− 0.436276i	0.975918	−	0.218138i	$-0.0699982\pi$
$510$	0	0
$511$	−3256.00	−0.281873
$512$	0	0
$513$	1512.00i	0.130129i
$514$	0	0
$515$	384.000i	0.0328564i
$516$	0	0
$517$	−6048.00	−0.514489
$518$	0	0
$519$	−918.000	−0.0776411
$520$	0	0
$521$	8610.00	0.724013	0.362007	−	0.932176i	$-0.382092\pi$
$521$	8610.00	0.724013	0.362007	+	0.932176i	$0.382092\pi$
$522$	0	0
$523$	−5308.00	−0.443791	−0.221895	−	0.975070i	$-0.571224\pi$
$523$	−5308.00	−0.443791	−0.221895	+	0.975070i	$0.571224\pi$
$524$	0	0
$525$	1068.00i	0.0887835i
$526$	0	0
$527$	17160.0i	1.41841i
$528$	0	0
$529$	−2951.00	−0.242541
$530$	0	0
$531$	3132.00i	0.255965i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 2664.00i	− 0.215280i
$536$	0	0
$537$	−6156.00	−0.494695
$538$	0	0
$539$	11772.0i	0.940735i
$540$	0	0
$541$	6182.00i	0.491285i	0.969361	+	0.245642i	$0.0789989\pi$
$541$	6182.00i	0.491285i	−0.969361	+	0.245642i	$0.921001\pi$
$542$	0	0
$543$	−13494.0	−1.06645
$544$	0	0
$545$	8292.00	0.651725
$546$	0	0
$547$	1292.00	0.100991	0.0504954	−	0.998724i	$-0.483920\pi$
$547$	1292.00	0.100991	0.0504954	+	0.998724i	$0.483920\pi$
$548$	0	0
$549$	−3114.00	−0.242081
$550$	0	0
$551$	− 12432.0i	− 0.961200i
$552$	0	0
$553$	− 800.000i	− 0.0615180i
$554$	0	0
$555$	−1908.00	−0.145928
$556$	0	0
$557$	− 12774.0i	− 0.971727i	−0.874035	−	0.485863i	$-0.838505\pi$
$557$	− 12774.0i	− 0.971727i	0.874035	−	0.485863i	$-0.161495\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	7128.00i	0.536443i
$562$	0	0
$563$	−16908.0	−1.26570	−0.632848	−	0.774276i	$-0.718114\pi$
$563$	−16908.0	−1.26570	−0.632848	+	0.774276i	$0.718114\pi$
$564$	0	0
$565$	− 5220.00i	− 0.388685i
$566$	0	0
$567$	− 324.000i	− 0.0239977i
$568$	0	0
$569$	11214.0	0.826213	0.413107	−	0.910683i	$-0.364444\pi$
$569$	11214.0	0.826213	0.413107	+	0.910683i	$0.364444\pi$
$570$	0	0
$571$	−25220.0	−1.84838	−0.924189	−	0.381935i	$-0.875258\pi$
$571$	−25220.0	−1.84838	−0.924189	+	0.381935i	$0.875258\pi$
$572$	0	0
$573$	12168.0	0.887130
$574$	0	0
$575$	−8544.00	−0.619669
$576$	0	0
$577$	17710.0i	1.27778i	0.769300	+	0.638888i	$0.220605\pi$
$577$	17710.0i	1.27778i	−0.769300	+	0.638888i	$0.779395\pi$
$578$	0	0
$579$	6186.00i	0.444009i
$580$	0	0
$581$	−4944.00	−0.353032
$582$	0	0
$583$	1080.00i	0.0767222i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 20028.0i	− 1.40825i	−0.710075	−	0.704126i	$-0.751339\pi$
$587$	− 20028.0i	− 1.40825i	0.710075	−	0.704126i	$-0.248661\pi$
$588$	0	0
$589$	−14560.0	−1.01856
$590$	0	0
$591$	− 13122.0i	− 0.913311i
$592$	0	0
$593$	19926.0i	1.37987i	0.723871	+	0.689935i	$0.242361\pi$
$593$	19926.0i	1.37987i	−0.723871	+	0.689935i	$0.757639\pi$
$594$	0	0
$595$	−1584.00	−0.109139
$596$	0	0
$597$	−7608.00	−0.521566
$598$	0	0
$599$	−1704.00	−0.116233	−0.0581165	−	0.998310i	$-0.518509\pi$
$599$	−1704.00	−0.116233	−0.0581165	+	0.998310i	$0.518509\pi$
$600$	0	0
$601$	11018.0	0.747810	0.373905	−	0.927467i	$-0.378019\pi$
$601$	11018.0	0.747810	0.373905	+	0.927467i	$0.378019\pi$
$602$	0	0
$603$	2304.00i	0.155599i
$604$	0	0
$605$	210.000i	0.0141119i
$606$	0	0
$607$	−448.000	−0.0299568	−0.0149784	−	0.999888i	$-0.504768\pi$
$607$	−448.000	−0.0299568	−0.0149784	+	0.999888i	$0.504768\pi$
$608$	0	0
$609$	2664.00i	0.177259i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	12586.0i	0.829272i	0.909987	+	0.414636i	$0.136091\pi$
$613$	12586.0i	0.829272i	−0.909987	+	0.414636i	$0.863909\pi$
$614$	0	0
$615$	1620.00	0.106219
$616$	0	0
$617$	29610.0i	1.93202i	0.258513	+	0.966008i	$0.416768\pi$
$617$	29610.0i	1.93202i	−0.258513	+	0.966008i	$0.583232\pi$
$618$	0	0
$619$	− 7120.00i	− 0.462321i	−0.972916	−	0.231161i	$-0.925748\pi$
$619$	− 7120.00i	− 0.462321i	0.972916	−	0.231161i	$-0.0742523\pi$
$620$	0	0
$621$	2592.00	0.167493
$622$	0	0
$623$	1272.00	0.0818003
$624$	0	0
$625$	3421.00	0.218944
$626$	0	0
$627$	−6048.00	−0.385221
$628$	0	0
$629$	6996.00i	0.443480i
$630$	0	0
$631$	− 15580.0i	− 0.982932i	−0.870897	−	0.491466i	$-0.836461\pi$
$631$	− 15580.0i	− 0.982932i	0.870897	−	0.491466i	$-0.163539\pi$
$632$	0	0
$633$	13332.0	0.837124
$634$	0	0
$635$	− 2784.00i	− 0.173984i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1512.00i	0.0936053i
$640$	0	0
$641$	19806.0	1.22042	0.610211	−	0.792239i	$-0.291085\pi$
$641$	19806.0	1.22042	0.610211	+	0.792239i	$0.291085\pi$
$642$	0	0
$643$	− 24032.0i	− 1.47392i	−0.675937	−	0.736959i	$-0.736261\pi$
$643$	− 24032.0i	− 1.47392i	0.675937	−	0.736959i	$-0.263739\pi$
$644$	0	0
$645$	792.000i	0.0483488i
$646$	0	0
$647$	2808.00	0.170624	0.0853121	−	0.996354i	$-0.472811\pi$
$647$	2808.00	0.170624	0.0853121	+	0.996354i	$0.472811\pi$
$648$	0	0
$649$	−12528.0	−0.757730
$650$	0	0
$651$	3120.00	0.187838
$652$	0	0
$653$	23886.0	1.43144	0.715721	−	0.698386i	$-0.246098\pi$
$653$	23886.0	1.43144	0.715721	+	0.698386i	$0.246098\pi$
$654$	0	0
$655$	9288.00i	0.554064i
$656$	0	0
$657$	− 7326.00i	− 0.435030i
$658$	0	0
$659$	−3948.00	−0.233372	−0.116686	−	0.993169i	$-0.537227\pi$
$659$	−3948.00	−0.233372	−0.116686	+	0.993169i	$0.537227\pi$
$660$	0	0
$661$	5750.00i	0.338350i	0.985586	+	0.169175i	$0.0541102\pi$
$661$	5750.00i	0.338350i	−0.985586	+	0.169175i	$0.945890\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1344.00i	− 0.0783731i
$666$	0	0
$667$	−21312.0	−1.23719
$668$	0	0
$669$	− 8148.00i	− 0.470882i
$670$	0	0
$671$	− 12456.0i	− 0.716630i
$672$	0	0
$673$	−28082.0	−1.60844	−0.804221	−	0.594330i	$-0.797417\pi$
$673$	−28082.0	−1.60844	−0.804221	+	0.594330i	$0.797417\pi$
$674$	0	0
$675$	−2403.00	−0.137024
$676$	0	0
$677$	−27954.0	−1.58694	−0.793471	−	0.608608i	$-0.791728\pi$
$677$	−27954.0	−1.58694	−0.793471	+	0.608608i	$0.791728\pi$
$678$	0	0
$679$	2008.00	0.113490
$680$	0	0
$681$	− 14076.0i	− 0.792061i
$682$	0	0
$683$	− 28428.0i	− 1.59263i	−0.604881	−	0.796316i	$-0.706779\pi$
$683$	− 28428.0i	− 1.59263i	0.604881	−	0.796316i	$-0.293221\pi$
$684$	0	0
$685$	−1764.00	−0.0983927
$686$	0	0
$687$	− 19338.0i	− 1.07393i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 21680.0i	− 1.19355i	−0.802407	−	0.596777i	$-0.796448\pi$
$691$	− 21680.0i	− 1.19355i	0.802407	−	0.596777i	$-0.203552\pi$
$692$	0	0
$693$	1296.00	0.0710404
$694$	0	0
$695$	15384.0i	0.839638i
$696$	0	0
$697$	− 5940.00i	− 0.322803i
$698$	0	0
$699$	−9306.00	−0.503555
$700$	0	0
$701$	7482.00	0.403126	0.201563	−	0.979476i	$-0.435398\pi$
$701$	7482.00	0.403126	0.201563	+	0.979476i	$0.435398\pi$
$702$	0	0
$703$	−5936.00	−0.318464
$704$	0	0
$705$	3024.00	0.161547
$706$	0	0
$707$	4248.00i	0.225972i
$708$	0	0
$709$	2270.00i	0.120242i	0.998191	+	0.0601210i	$0.0191487\pi$
$709$	2270.00i	0.120242i	−0.998191	+	0.0601210i	$0.980851\pi$
$710$	0	0
$711$	1800.00	0.0949441
$712$	0	0
$713$	24960.0i	1.31102i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2448.00i	0.127507i
$718$	0	0
$719$	−36024.0	−1.86852	−0.934262	−	0.356588i	$-0.883940\pi$
$719$	−36024.0	−1.86852	−0.934262	+	0.356588i	$0.883940\pi$
$720$	0	0
$721$	− 256.000i	− 0.0132232i
$722$	0	0
$723$	− 11454.0i	− 0.589182i
$724$	0	0
$725$	19758.0	1.01213
$726$	0	0
$727$	21544.0	1.09907	0.549534	−	0.835471i	$-0.314805\pi$
$727$	21544.0	1.09907	0.549534	+	0.835471i	$0.314805\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	2904.00	0.146933
$732$	0	0
$733$	1018.00i	0.0512970i	0.999671	+	0.0256485i	$0.00816506\pi$
$733$	1018.00i	0.0512970i	−0.999671	+	0.0256485i	$0.991835\pi$
$734$	0	0
$735$	− 5886.00i	− 0.295386i
$736$	0	0
$737$	−9216.00	−0.460618
$738$	0	0
$739$	− 24568.0i	− 1.22293i	−0.791270	−	0.611467i	$-0.790580\pi$
$739$	− 24568.0i	− 1.22293i	0.791270	−	0.611467i	$-0.209420\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 16968.0i	− 0.837814i	−0.908029	−	0.418907i	$-0.862413\pi$
$743$	− 16968.0i	− 0.837814i	0.908029	−	0.418907i	$-0.137587\pi$
$744$	0	0
$745$	684.000	0.0336373
$746$	0	0
$747$	− 11124.0i	− 0.544854i
$748$	0	0
$749$	1776.00i	0.0866404i
$750$	0	0
$751$	−3224.00	−0.156652	−0.0783259	−	0.996928i	$-0.524957\pi$
$751$	−3224.00	−0.156652	−0.0783259	+	0.996928i	$0.524957\pi$
$752$	0	0
$753$	−19836.0	−0.959979
$754$	0	0
$755$	−12216.0	−0.588855
$756$	0	0
$757$	−31570.0	−1.51576	−0.757881	−	0.652393i	$-0.773765\pi$
$757$	−31570.0	−1.51576	−0.757881	+	0.652393i	$0.773765\pi$
$758$	0	0
$759$	10368.0i	0.495829i
$760$	0	0
$761$	− 34890.0i	− 1.66197i	−0.556293	−	0.830987i	$-0.687777\pi$
$761$	− 34890.0i	− 1.66197i	0.556293	−	0.830987i	$-0.312223\pi$
$762$	0	0
$763$	−5528.00	−0.262290
$764$	0	0
$765$	− 3564.00i	− 0.168440i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 11522.0i	− 0.540304i	−0.962818	−	0.270152i	$-0.912926\pi$
$769$	− 11522.0i	− 0.540304i	0.962818	−	0.270152i	$-0.0870740\pi$
$770$	0	0
$771$	−14418.0	−0.673478
$772$	0	0
$773$	28158.0i	1.31018i	0.755549	+	0.655092i	$0.227370\pi$
$773$	28158.0i	1.31018i	−0.755549	+	0.655092i	$0.772630\pi$
$774$	0	0
$775$	− 23140.0i	− 1.07253i
$776$	0	0
$777$	1272.00	0.0587294
$778$	0	0
$779$	5040.00	0.231806
$780$	0	0
$781$	−6048.00	−0.277099
$782$	0	0
$783$	−5994.00	−0.273574
$784$	0	0
$785$	17220.0i	0.782940i
$786$	0	0
$787$	14504.0i	0.656940i	0.944514	+	0.328470i	$0.106533\pi$
$787$	14504.0i	0.656940i	−0.944514	+	0.328470i	$0.893467\pi$
$788$	0	0
$789$	13752.0	0.620512
$790$	0	0
$791$	3480.00i	0.156428i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 540.000i	− 0.0240903i
$796$	0	0
$797$	18090.0	0.803991	0.401995	−	0.915642i	$-0.368317\pi$
$797$	18090.0	0.803991	0.401995	+	0.915642i	$0.368317\pi$
$798$	0	0
$799$	− 11088.0i	− 0.490945i
$800$	0	0
$801$	2862.00i	0.126247i
$802$	0	0
$803$	29304.0	1.28782
$804$	0	0
$805$	−2304.00	−0.100876
$806$	0	0
$807$	−21402.0	−0.933564
$808$	0	0
$809$	36402.0	1.58199	0.790993	−	0.611826i	$-0.209565\pi$
$809$	36402.0	1.58199	0.790993	+	0.611826i	$0.209565\pi$
$810$	0	0
$811$	32368.0i	1.40147i	0.713420	+	0.700736i	$0.247145\pi$
$811$	32368.0i	1.40147i	−0.713420	+	0.700736i	$0.752855\pi$
$812$	0	0
$813$	− 9420.00i	− 0.406364i
$814$	0	0
$815$	−8832.00	−0.379597
$816$	0	0
$817$	2464.00i	0.105513i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 35778.0i	− 1.52090i	−0.649395	−	0.760451i	$-0.724978\pi$
$821$	− 35778.0i	− 1.52090i	0.649395	−	0.760451i	$-0.275022\pi$
$822$	0	0
$823$	10240.0	0.433711	0.216855	−	0.976204i	$-0.430420\pi$
$823$	10240.0	0.433711	0.216855	+	0.976204i	$0.430420\pi$
$824$	0	0
$825$	− 9612.00i	− 0.405633i
$826$	0	0
$827$	− 16284.0i	− 0.684704i	−0.939572	−	0.342352i	$-0.888776\pi$
$827$	− 16284.0i	− 0.684704i	0.939572	−	0.342352i	$-0.111224\pi$
$828$	0	0
$829$	−14150.0	−0.592822	−0.296411	−	0.955060i	$-0.595790\pi$
$829$	−14150.0	−0.592822	−0.296411	+	0.955060i	$0.595790\pi$
$830$	0	0
$831$	−14358.0	−0.599366
$832$	0	0
$833$	−21582.0	−0.897685
$834$	0	0
$835$	1440.00	0.0596805
$836$	0	0
$837$	7020.00i	0.289900i
$838$	0	0
$839$	39576.0i	1.62850i	0.580511	+	0.814252i	$0.302853\pi$
$839$	39576.0i	1.62850i	−0.580511	+	0.814252i	$0.697147\pi$
$840$	0	0
$841$	24895.0	1.02075
$842$	0	0
$843$	− 11394.0i	− 0.465516i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 140.000i	− 0.00567941i
$848$	0	0
$849$	10716.0	0.433183
$850$	0	0
$851$	10176.0i	0.409905i
$852$	0	0
$853$	− 6922.00i	− 0.277848i	−0.990303	−	0.138924i	$-0.955636\pi$
$853$	− 6922.00i	− 0.277848i	0.990303	−	0.138924i	$-0.0443645\pi$
$854$	0	0
$855$	3024.00	0.120957
$856$	0	0
$857$	−48162.0	−1.91970	−0.959850	−	0.280514i	$-0.909495\pi$
$857$	−48162.0	−1.91970	−0.959850	+	0.280514i	$0.909495\pi$
$858$	0	0
$859$	−27652.0	−1.09834	−0.549170	−	0.835711i	$-0.685056\pi$
$859$	−27652.0	−1.09834	−0.549170	+	0.835711i	$0.685056\pi$
$860$	0	0
$861$	−1080.00	−0.0427483
$862$	0	0
$863$	− 648.000i	− 0.0255599i	−0.999918	−	0.0127799i	$-0.995932\pi$
$863$	− 648.000i	− 0.0255599i	0.999918	−	0.0127799i	$-0.00406809\pi$
$864$	0	0
$865$	1836.00i	0.0721686i
$866$	0	0
$867$	1671.00	0.0654558
$868$	0	0
$869$	7200.00i	0.281062i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	4518.00i	0.175156i
$874$	0	0
$875$	5136.00	0.198433
$876$	0	0
$877$	− 7166.00i	− 0.275916i	−0.990438	−	0.137958i	$-0.955946\pi$
$877$	− 7166.00i	− 0.275916i	0.990438	−	0.137958i	$-0.0440540\pi$
$878$	0	0
$879$	− 21366.0i	− 0.819860i
$880$	0	0
$881$	37062.0	1.41731	0.708655	−	0.705555i	$-0.249302\pi$
$881$	37062.0	1.41731	0.708655	+	0.705555i	$0.249302\pi$
$882$	0	0
$883$	−24716.0	−0.941970	−0.470985	−	0.882141i	$-0.656101\pi$
$883$	−24716.0	−0.941970	−0.470985	+	0.882141i	$0.656101\pi$
$884$	0	0
$885$	6264.00	0.237923
$886$	0	0
$887$	48672.0	1.84244	0.921221	−	0.389040i	$-0.127193\pi$
$887$	48672.0	1.84244	0.921221	+	0.389040i	$0.127193\pi$
$888$	0	0
$889$	1856.00i	0.0700205i
$890$	0	0
$891$	2916.00i	0.109640i
$892$	0	0
$893$	9408.00	0.352550
$894$	0	0
$895$	12312.0i	0.459827i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 57720.0i	− 2.14135i
$900$	0	0
$901$	−1980.00	−0.0732113
$902$	0	0
$903$	− 528.000i	− 0.0194582i
$904$	0	0
$905$	26988.0i	0.991283i
$906$	0	0
$907$	9484.00	0.347201	0.173600	−	0.984816i	$-0.444460\pi$
$907$	9484.00	0.347201	0.173600	+	0.984816i	$0.444460\pi$
$908$	0	0
$909$	−9558.00	−0.348756
$910$	0	0
$911$	12792.0	0.465223	0.232611	−	0.972570i	$-0.425273\pi$
$911$	12792.0	0.465223	0.232611	+	0.972570i	$0.425273\pi$
$912$	0	0
$913$	44496.0	1.61293
$914$	0	0
$915$	6228.00i	0.225018i
$916$	0	0
$917$	− 6192.00i	− 0.222986i
$918$	0	0
$919$	−18592.0	−0.667349	−0.333674	−	0.942688i	$-0.608289\pi$
$919$	−18592.0	−0.667349	−0.333674	+	0.942688i	$0.608289\pi$
$920$	0	0
$921$	20568.0i	0.735873i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 9434.00i	− 0.335338i
$926$	0	0
$927$	576.000	0.0204081
$928$	0	0
$929$	15378.0i	0.543096i	0.962425	+	0.271548i	$0.0875355\pi$
$929$	15378.0i	0.543096i	−0.962425	+	0.271548i	$0.912464\pi$
$930$	0	0
$931$	− 18312.0i	− 0.644631i
$932$	0	0
$933$	−26496.0	−0.929732
$934$	0	0
$935$	14256.0	0.498632
$936$	0	0
$937$	−37078.0	−1.29273	−0.646364	−	0.763030i	$-0.723711\pi$
$937$	−37078.0	−1.29273	−0.646364	+	0.763030i	$0.723711\pi$
$938$	0	0
$939$	−10878.0	−0.378051
$940$	0	0
$941$	− 10842.0i	− 0.375599i	−0.982207	−	0.187800i	$-0.939864\pi$
$941$	− 10842.0i	− 0.375599i	0.982207	−	0.187800i	$-0.0601356\pi$
$942$	0	0
$943$	− 8640.00i	− 0.298364i
$944$	0	0
$945$	−648.000	−0.0223063
$946$	0	0
$947$	41508.0i	1.42432i	0.702018	+	0.712159i	$0.252282\pi$
$947$	41508.0i	1.42432i	−0.702018	+	0.712159i	$0.747718\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	30438.0i	1.03788i
$952$	0	0
$953$	−38706.0	−1.31565	−0.657823	−	0.753173i	$-0.728522\pi$
$953$	−38706.0	−1.31565	−0.657823	+	0.753173i	$0.728522\pi$
$954$	0	0
$955$	− 24336.0i	− 0.824602i
$956$	0	0
$957$	− 23976.0i	− 0.809858i
$958$	0	0
$959$	1176.00	0.0395986
$960$	0	0
$961$	−37809.0	−1.26914
$962$	0	0
$963$	−3996.00	−0.133717
$964$	0	0
$965$	12372.0	0.412714
$966$	0	0
$967$	24388.0i	0.811029i	0.914089	+	0.405515i	$0.132908\pi$
$967$	24388.0i	0.811029i	−0.914089	+	0.405515i	$0.867092\pi$
$968$	0	0
$969$	− 11088.0i	− 0.367593i
$970$	0	0
$971$	−14100.0	−0.466005	−0.233002	−	0.972476i	$-0.574855\pi$
$971$	−14100.0	−0.466005	−0.233002	+	0.972476i	$0.574855\pi$
$972$	0	0
$973$	− 10256.0i	− 0.337916i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 44838.0i	− 1.46826i	−0.679006	−	0.734132i	$-0.737589\pi$
$977$	− 44838.0i	− 1.46826i	0.679006	−	0.734132i	$-0.262411\pi$
$978$	0	0
$979$	−11448.0	−0.373728
$980$	0	0
$981$	− 12438.0i	− 0.404806i
$982$	0	0
$983$	13176.0i	0.427517i	0.976887	+	0.213758i	$0.0685706\pi$
$983$	13176.0i	0.427517i	−0.976887	+	0.213758i	$0.931429\pi$
$984$	0	0
$985$	−26244.0	−0.848937
$986$	0	0
$987$	−2016.00	−0.0650152
$988$	0	0
$989$	4224.00	0.135809
$990$	0	0
$991$	−43648.0	−1.39912	−0.699558	−	0.714576i	$-0.746620\pi$
$991$	−43648.0	−1.39912	−0.699558	+	0.714576i	$0.746620\pi$
$992$	0	0
$993$	19608.0i	0.626627i
$994$	0	0
$995$	15216.0i	0.484804i
$996$	0	0
$997$	62750.0	1.99329	0.996646	−	0.0818317i	$-0.0260770\pi$
$997$	62750.0	1.99329	0.996646	+	0.0818317i	$0.0260770\pi$
$998$	0	0
$999$	2862.00i	0.0906403i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2028.4.b.a.337.2		2
13.5	odd	4		2028.4.a.a.1.1		1
13.8	odd	4		156.4.a.a.1.1	✓	1
13.12	even	2	inner	2028.4.b.a.337.1		2
39.8	even	4		468.4.a.b.1.1		1
52.47	even	4		624.4.a.h.1.1		1
104.21	odd	4		2496.4.a.n.1.1		1
104.99	even	4		2496.4.a.e.1.1		1
156.47	odd	4		1872.4.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.4.a.a.1.1	✓	1	13.8	odd	4
468.4.a.b.1.1		1	39.8	even	4
624.4.a.h.1.1		1	52.47	even	4
1872.4.a.j.1.1		1	156.47	odd	4
2028.4.a.a.1.1		1	13.5	odd	4
2028.4.b.a.337.1		2	13.12	even	2	inner
2028.4.b.a.337.2		2	1.1	even	1	trivial
2496.4.a.e.1.1		1	104.99	even	4
2496.4.a.n.1.1		1	104.21	odd	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:	\( N \)	\(=\)	\( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2028.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +6.00000i q^{5} -4.00000i q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +6.00000i q^{5} -4.00000i q^{7} +9.00000 q^{9} +36.0000i q^{11} -18.0000i q^{15} -66.0000 q^{17} -56.0000i q^{19} +12.0000i q^{21} -96.0000 q^{23} +89.0000 q^{25} -27.0000 q^{27} +222.000 q^{29} -260.000i q^{31} -108.000i q^{33} +24.0000 q^{35} -106.000i q^{37} +90.0000i q^{41} -44.0000 q^{43} +54.0000i q^{45} +168.000i q^{47} +327.000 q^{49} +198.000 q^{51} +30.0000 q^{53} -216.000 q^{55} +168.000i q^{57} +348.000i q^{59} -346.000 q^{61} -36.0000i q^{63} +256.000i q^{67} +288.000 q^{69} +168.000i q^{71} -814.000i q^{73} -267.000 q^{75} +144.000 q^{77} +200.000 q^{79} +81.0000 q^{81} -1236.00i q^{83} -396.000i q^{85} -666.000 q^{87} +318.000i q^{89} +780.000i q^{93} +336.000 q^{95} +502.000i q^{97} +324.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 18 * q^9	\( 2 q - 6 q^{3} + 18 q^{9} - 132 q^{17} - 192 q^{23} + 178 q^{25} - 54 q^{27} + 444 q^{29} + 48 q^{35} - 88 q^{43} + 654 q^{49} + 396 q^{51} + 60 q^{53} - 432 q^{55} - 692 q^{61} + 576 q^{69} - 534 q^{75} + 288 q^{77} + 400 q^{79} + 162 q^{81} - 1332 q^{87} + 672 q^{95}+O(q^{100}) \) 2 * q - 6 * q^3 + 18 * q^9 - 132 * q^17 - 192 * q^23 + 178 * q^25 - 54 * q^27 + 444 * q^29 + 48 * q^35 - 88 * q^43 + 654 * q^49 + 396 * q^51 + 60 * q^53 - 432 * q^55 - 692 * q^61 + 576 * q^69 - 534 * q^75 + 288 * q^77 + 400 * q^79 + 162 * q^81 - 1332 * q^87 + 672 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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