LMFDB - Embedded newform 2025.2.b.a.649.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2025,2,Mod(649,2025)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2025.649"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2025 = 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2025.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-4,0,0,0,0,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$16.1697064093$
Analytic rank:	$1$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 405)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2025.649
Dual form			2025.2.b.a.649.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -2.00000 q^{4} -5.00000 q^{11} -4.00000i q^{13} -4.00000 q^{16} +4.00000i q^{17} +5.00000 q^{19} +10.0000i q^{22} +6.00000i q^{23} -8.00000 q^{26} -5.00000 q^{29} -9.00000 q^{31} +8.00000i q^{32} +8.00000 q^{34} -10.0000i q^{37} -10.0000i q^{38} -7.00000 q^{41} +2.00000i q^{43} +10.0000 q^{44} +12.0000 q^{46} -2.00000i q^{47} +7.00000 q^{49} +8.00000i q^{52} +8.00000i q^{53} +10.0000i q^{58} -1.00000 q^{59} -2.00000 q^{61} +18.0000i q^{62} +8.00000 q^{64} +6.00000i q^{67} -8.00000i q^{68} -1.00000 q^{71} +8.00000i q^{73} -20.0000 q^{74} -10.0000 q^{76} -12.0000 q^{79} +14.0000i q^{82} +6.00000i q^{83} +4.00000 q^{86} -9.00000 q^{89} -12.0000i q^{92} -4.00000 q^{94} +14.0000i q^{97} -14.0000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} - 10 q^{11} - 8 q^{16} + 10 q^{19} - 16 q^{26} - 10 q^{29} - 18 q^{31} + 16 q^{34} - 14 q^{41} + 20 q^{44} + 24 q^{46} + 14 q^{49} - 2 q^{59} - 4 q^{61} + 16 q^{64} - 2 q^{71} - 40 q^{74}+ \cdots - 8 q^{94}+O(q^{100}) $ 2 * q - 4 * q^4 - 10 * q^11 - 8 * q^16 + 10 * q^19 - 16 * q^26 - 10 * q^29 - 18 * q^31 + 16 * q^34 - 14 * q^41 + 20 * q^44 + 24 * q^46 + 14 * q^49 - 2 * q^59 - 4 * q^61 + 16 * q^64 - 2 * q^71 - 40 * q^74 - 20 * q^76 - 24 * q^79 + 8 * q^86 - 18 * q^89 - 8 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times$.

$n$	$326$	$1702$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.00000i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	− 2.00000i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	0	0
$15$	0	0
$16$	−4.00000	−1.00000
$17$	4.00000i	0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$	4.00000i	0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$	0	0
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	0	0
$21$	0	0
$22$	10.0000i	2.13201i
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	0	0
$25$	0	0
$26$	−8.00000	−1.56893
$27$	0	0
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	−9.00000	−1.61645	−0.808224	−	0.588875i	$-0.799571\pi$
$31$	−9.00000	−1.61645	−0.808224	+	0.588875i	$0.799571\pi$
$32$	8.00000i	1.41421i
$33$	0	0
$34$	8.00000	1.37199
$35$	0	0
$36$	0	0
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	− 10.0000i	− 1.62221i
$39$	0	0
$40$	0	0
$41$	−7.00000	−1.09322	−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000	−1.09322	−0.546608	+	0.837389i	$0.684081\pi$
$42$	0	0
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	10.0000	1.50756
$45$	0	0
$46$	12.0000	1.76930
$47$	− 2.00000i	− 0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$	− 2.00000i	− 0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	0	0
$52$	8.00000i	1.10940i
$53$	8.00000i	1.09888i	0.835532	+	0.549442i	$0.185160\pi$
$53$	8.00000i	1.09888i	−0.835532	+	0.549442i	$0.814840\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	10.0000i	1.31306i
$59$	−1.00000	−0.130189	−0.0650945	−	0.997879i	$-0.520735\pi$
$59$	−1.00000	−0.130189	−0.0650945	+	0.997879i	$0.520735\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	18.0000i	2.28600i
$63$	0	0
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	6.00000i	0.733017i	0.930415	+	0.366508i	$0.119447\pi$
$67$	6.00000i	0.733017i	−0.930415	+	0.366508i	$0.880553\pi$
$68$	− 8.00000i	− 0.970143i
$69$	0	0
$70$	0	0
$71$	−1.00000	−0.118678	−0.0593391	−	0.998238i	$-0.518899\pi$
$71$	−1.00000	−0.118678	−0.0593391	+	0.998238i	$0.518899\pi$
$72$	0	0
$73$	8.00000i	0.936329i	0.883641	+	0.468165i	$0.155085\pi$
$73$	8.00000i	0.936329i	−0.883641	+	0.468165i	$0.844915\pi$
$74$	−20.0000	−2.32495
$75$	0	0
$76$	−10.0000	−1.14708
$77$	0	0
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	0	0
$82$	14.0000i	1.54604i
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	0	0
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	0	0
$91$	0	0
$92$	− 12.0000i	− 1.25109i
$93$	0	0
$94$	−4.00000	−0.412568
$95$	0	0
$96$	0	0
$97$	14.0000i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	14.0000i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	− 14.0000i	− 1.41421i
$99$	0	0
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	− 2.00000i	− 0.197066i	−0.995134	−	0.0985329i	$-0.968585\pi$
$103$	− 2.00000i	− 0.197066i	0.995134	−	0.0985329i	$-0.0314150\pi$
$104$	0	0
$105$	0	0
$106$	16.0000	1.55406
$107$	6.00000i	0.580042i	0.957020	+	0.290021i	$0.0936623\pi$
$107$	6.00000i	0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$108$	0	0
$109$	1.00000	0.0957826	0.0478913	−	0.998853i	$-0.484750\pi$
$109$	1.00000	0.0957826	0.0478913	+	0.998853i	$0.484750\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 16.0000i	− 1.50515i	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	− 16.0000i	− 1.50515i	0.658505	−	0.752577i	$-0.271189\pi$
$114$	0	0
$115$	0	0
$116$	10.0000	0.928477
$117$	0	0
$118$	2.00000i	0.184115i
$119$	0	0
$120$	0	0
$121$	14.0000	1.27273
$122$	4.00000i	0.362143i
$123$	0	0
$124$	18.0000	1.61645
$125$	0	0
$126$	0	0
$127$	16.0000i	1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$	16.0000i	1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	0	0
$133$	0	0
$134$	12.0000	1.03664
$135$	0	0
$136$	0	0
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	0	0
$139$	19.0000	1.61156	0.805779	−	0.592216i	$-0.201747\pi$
$139$	19.0000	1.61156	0.805779	+	0.592216i	$0.201747\pi$
$140$	0	0
$141$	0	0
$142$	2.00000i	0.167836i
$143$	20.0000i	1.67248i
$144$	0	0
$145$	0	0
$146$	16.0000	1.32417
$147$	0	0
$148$	20.0000i	1.64399i
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−5.00000	−0.406894	−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000	−0.406894	−0.203447	+	0.979086i	$0.565214\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	24.0000i	1.90934i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 8.00000i	− 0.626608i	−0.949653	−	0.313304i	$-0.898564\pi$
$163$	− 8.00000i	− 0.626608i	0.949653	−	0.313304i	$-0.101436\pi$
$164$	14.0000	1.09322
$165$	0	0
$166$	12.0000	0.931381
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	− 4.00000i	− 0.304997i
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	0	0
$176$	20.0000	1.50756
$177$	0	0
$178$	18.0000i	1.34916i
$179$	23.0000	1.71910	0.859550	−	0.511051i	$-0.170744\pi$
$179$	23.0000	1.71910	0.859550	+	0.511051i	$0.170744\pi$
$180$	0	0
$181$	−25.0000	−1.85824	−0.929118	−	0.369784i	$-0.879432\pi$
$181$	−25.0000	−1.85824	−0.929118	+	0.369784i	$0.879432\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 20.0000i	− 1.46254i
$188$	4.00000i	0.291730i
$189$	0	0
$190$	0	0
$191$	−1.00000	−0.0723575	−0.0361787	−	0.999345i	$-0.511519\pi$
$191$	−1.00000	−0.0723575	−0.0361787	+	0.999345i	$0.511519\pi$
$192$	0	0
$193$	− 26.0000i	− 1.87152i	−0.352636	−	0.935760i	$-0.614715\pi$
$193$	− 26.0000i	− 1.87152i	0.352636	−	0.935760i	$-0.385285\pi$
$194$	28.0000	2.01028
$195$	0	0
$196$	−14.0000	−1.00000
$197$	− 12.0000i	− 0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$	− 12.0000i	− 0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	0	0
$202$	− 6.00000i	− 0.422159i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−4.00000	−0.278693
$207$	0	0
$208$	16.0000i	1.10940i
$209$	−25.0000	−1.72929
$210$	0	0
$211$	11.0000	0.757271	0.378636	−	0.925546i	$-0.376393\pi$
$211$	11.0000	0.757271	0.378636	+	0.925546i	$0.376393\pi$
$212$	− 16.0000i	− 1.09888i
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 2.00000i	− 0.135457i
$219$	0	0
$220$	0	0
$221$	16.0000	1.07628
$222$	0	0
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	0	0
$225$	0	0
$226$	−32.0000	−2.12861
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	0	0
$235$	0	0
$236$	2.00000	0.130189
$237$	0	0
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	−11.0000	−0.708572	−0.354286	−	0.935137i	$-0.615276\pi$
$241$	−11.0000	−0.708572	−0.354286	+	0.935137i	$0.615276\pi$
$242$	− 28.0000i	− 1.79991i
$243$	0	0
$244$	4.00000	0.256074
$245$	0	0
$246$	0	0
$247$	− 20.0000i	− 1.27257i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	− 30.0000i	− 1.88608i
$254$	32.0000	2.00786
$255$	0	0
$256$	16.0000	1.00000
$257$	6.00000i	0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$	6.00000i	0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	30.0000i	1.85341i
$263$	10.0000i	0.616626i	0.951285	+	0.308313i	$0.0997645\pi$
$263$	10.0000i	0.616626i	−0.951285	+	0.308313i	$0.900236\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	− 12.0000i	− 0.733017i
$269$	−31.0000	−1.89010	−0.945052	−	0.326921i	$-0.893989\pi$
$269$	−31.0000	−1.89010	−0.945052	+	0.326921i	$0.893989\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	− 16.0000i	− 0.970143i
$273$	0	0
$274$	−24.0000	−1.44989
$275$	0	0
$276$	0	0
$277$	− 18.0000i	− 1.08152i	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	− 18.0000i	− 1.08152i	0.841178	−	0.540758i	$-0.181862\pi$
$278$	− 38.0000i	− 2.27909i
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	− 6.00000i	− 0.356663i	−0.983970	−	0.178331i	$-0.942930\pi$
$283$	− 6.00000i	− 0.356663i	0.983970	−	0.178331i	$-0.0570699\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	40.0000	2.36525
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	− 16.0000i	− 0.936329i
$293$	18.0000i	1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$	18.0000i	1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	4.00000i	0.231714i
$299$	24.0000	1.38796
$300$	0	0
$301$	0	0
$302$	10.0000i	0.575435i
$303$	0	0
$304$	−20.0000	−1.14708
$305$	0	0
$306$	0	0
$307$	10.0000i	0.570730i	0.958419	+	0.285365i	$0.0921148\pi$
$307$	10.0000i	0.570730i	−0.958419	+	0.285365i	$0.907885\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.00000	0.510343	0.255172	−	0.966896i	$-0.417868\pi$
$311$	9.00000	0.510343	0.255172	+	0.966896i	$0.417868\pi$
$312$	0	0
$313$	4.00000i	0.226093i	0.993590	+	0.113047i	$0.0360610\pi$
$313$	4.00000i	0.226093i	−0.993590	+	0.113047i	$0.963939\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	24.0000	1.35011
$317$	2.00000i	0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$	2.00000i	0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$	0	0
$319$	25.0000	1.39973
$320$	0	0
$321$	0	0
$322$	0	0
$323$	20.0000i	1.11283i
$324$	0	0
$325$	0	0
$326$	−16.0000	−0.886158
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−21.0000	−1.15426	−0.577132	−	0.816651i	$-0.695828\pi$
$331$	−21.0000	−1.15426	−0.577132	+	0.816651i	$0.695828\pi$
$332$	− 12.0000i	− 0.658586i
$333$	0	0
$334$	−24.0000	−1.31322
$335$	0	0
$336$	0	0
$337$	− 8.00000i	− 0.435788i	−0.975972	−	0.217894i	$-0.930081\pi$
$337$	− 8.00000i	− 0.435788i	0.975972	−	0.217894i	$-0.0699187\pi$
$338$	6.00000i	0.326357i
$339$	0	0
$340$	0	0
$341$	45.0000	2.43689
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−12.0000	−0.645124
$347$	− 20.0000i	− 1.07366i	−0.843692	−	0.536828i	$-0.819622\pi$
$347$	− 20.0000i	− 1.07366i	0.843692	−	0.536828i	$-0.180378\pi$
$348$	0	0
$349$	−13.0000	−0.695874	−0.347937	−	0.937518i	$-0.613118\pi$
$349$	−13.0000	−0.695874	−0.347937	+	0.937518i	$0.613118\pi$
$350$	0	0
$351$	0	0
$352$	− 40.0000i	− 2.13201i
$353$	− 18.0000i	− 0.958043i	−0.877803	−	0.479022i	$-0.840992\pi$
$353$	− 18.0000i	− 0.958043i	0.877803	−	0.479022i	$-0.159008\pi$
$354$	0	0
$355$	0	0
$356$	18.0000	0.953998
$357$	0	0
$358$	− 46.0000i	− 2.43118i
$359$	−27.0000	−1.42501	−0.712503	−	0.701669i	$-0.752438\pi$
$359$	−27.0000	−1.42501	−0.712503	+	0.701669i	$0.752438\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	50.0000i	2.62794i
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	18.0000i	0.939592i	0.882775	+	0.469796i	$0.155673\pi$
$367$	18.0000i	0.939592i	−0.882775	+	0.469796i	$0.844327\pi$
$368$	− 24.0000i	− 1.25109i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 16.0000i	− 0.828449i	−0.910175	−	0.414224i	$-0.864053\pi$
$373$	− 16.0000i	− 0.828449i	0.910175	−	0.414224i	$-0.135947\pi$
$374$	−40.0000	−2.06835
$375$	0	0
$376$	0	0
$377$	20.0000i	1.03005i
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	0	0
$382$	2.00000i	0.102329i
$383$	− 36.0000i	− 1.83951i	−0.392488	−	0.919757i	$-0.628386\pi$
$383$	− 36.0000i	− 1.83951i	0.392488	−	0.919757i	$-0.371614\pi$
$384$	0	0
$385$	0	0
$386$	−52.0000	−2.64673
$387$	0	0
$388$	− 28.0000i	− 1.42148i
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	0	0
$393$	0	0
$394$	−24.0000	−1.20910
$395$	0	0
$396$	0	0
$397$	− 38.0000i	− 1.90717i	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	− 38.0000i	− 1.90717i	0.301131	−	0.953583i	$-0.402636\pi$
$398$	32.0000i	1.60402i
$399$	0	0
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	36.0000i	1.79329i
$404$	−6.00000	−0.298511
$405$	0	0
$406$	0	0
$407$	50.0000i	2.47841i
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	0	0
$412$	4.00000i	0.197066i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	32.0000	1.56893
$417$	0	0
$418$	50.0000i	2.44558i
$419$	16.0000	0.781651	0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000	0.781651	0.390826	+	0.920465i	$0.372190\pi$
$420$	0	0
$421$	13.0000	0.633581	0.316791	−	0.948495i	$-0.397395\pi$
$421$	13.0000	0.633581	0.316791	+	0.948495i	$0.397395\pi$
$422$	− 22.0000i	− 1.07094i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 12.0000i	− 0.580042i
$429$	0	0
$430$	0	0
$431$	−3.00000	−0.144505	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000	−0.144505	−0.0722525	+	0.997386i	$0.523019\pi$
$432$	0	0
$433$	26.0000i	1.24948i	0.780833	+	0.624740i	$0.214795\pi$
$433$	26.0000i	1.24948i	−0.780833	+	0.624740i	$0.785205\pi$
$434$	0	0
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	30.0000i	1.43509i
$438$	0	0
$439$	29.0000	1.38409	0.692047	−	0.721852i	$-0.256709\pi$
$439$	29.0000	1.38409	0.692047	+	0.721852i	$0.256709\pi$
$440$	0	0
$441$	0	0
$442$	− 32.0000i	− 1.52208i
$443$	6.00000i	0.285069i	0.989790	+	0.142534i	$0.0455251\pi$
$443$	6.00000i	0.285069i	−0.989790	+	0.142534i	$0.954475\pi$
$444$	0	0
$445$	0	0
$446$	−16.0000	−0.757622
$447$	0	0
$448$	0	0
$449$	17.0000	0.802280	0.401140	−	0.916017i	$-0.368614\pi$
$449$	17.0000	0.802280	0.401140	+	0.916017i	$0.368614\pi$
$450$	0	0
$451$	35.0000	1.64809
$452$	32.0000i	1.50515i
$453$	0	0
$454$	−8.00000	−0.375459
$455$	0	0
$456$	0	0
$457$	− 38.0000i	− 1.77757i	−0.458329	−	0.888783i	$-0.651552\pi$
$457$	− 38.0000i	− 1.77757i	0.458329	−	0.888783i	$-0.348448\pi$
$458$	12.0000i	0.560723i
$459$	0	0
$460$	0	0
$461$	15.0000	0.698620	0.349310	−	0.937007i	$-0.386416\pi$
$461$	15.0000	0.698620	0.349310	+	0.937007i	$0.386416\pi$
$462$	0	0
$463$	6.00000i	0.278844i	0.990233	+	0.139422i	$0.0445244\pi$
$463$	6.00000i	0.278844i	−0.990233	+	0.139422i	$0.955476\pi$
$464$	20.0000	0.928477
$465$	0	0
$466$	12.0000	0.555889
$467$	− 4.00000i	− 0.185098i	−0.995708	−	0.0925490i	$-0.970499\pi$
$467$	− 4.00000i	− 0.185098i	0.995708	−	0.0925490i	$-0.0295015\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 10.0000i	− 0.459800i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	32.0000i	1.46365i
$479$	−15.0000	−0.685367	−0.342684	−	0.939451i	$-0.611336\pi$
$479$	−15.0000	−0.685367	−0.342684	+	0.939451i	$0.611336\pi$
$480$	0	0
$481$	−40.0000	−1.82384
$482$	22.0000i	1.00207i
$483$	0	0
$484$	−28.0000	−1.27273
$485$	0	0
$486$	0	0
$487$	8.00000i	0.362515i	0.983436	+	0.181257i	$0.0580167\pi$
$487$	8.00000i	0.362515i	−0.983436	+	0.181257i	$0.941983\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−43.0000	−1.94056	−0.970281	−	0.241979i	$-0.922203\pi$
$491$	−43.0000	−1.94056	−0.970281	+	0.241979i	$0.922203\pi$
$492$	0	0
$493$	− 20.0000i	− 0.900755i
$494$	−40.0000	−1.79969
$495$	0	0
$496$	36.0000	1.61645
$497$	0	0
$498$	0	0
$499$	−7.00000	−0.313363	−0.156682	−	0.987649i	$-0.550080\pi$
$499$	−7.00000	−0.313363	−0.156682	+	0.987649i	$0.550080\pi$
$500$	0	0
$501$	0	0
$502$	− 24.0000i	− 1.07117i
$503$	− 20.0000i	− 0.891756i	−0.895094	−	0.445878i	$-0.852892\pi$
$503$	− 20.0000i	− 0.891756i	0.895094	−	0.445878i	$-0.147108\pi$
$504$	0	0
$505$	0	0
$506$	−60.0000	−2.66733
$507$	0	0
$508$	− 32.0000i	− 1.41977i
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	0	0
$512$	− 32.0000i	− 1.41421i
$513$	0	0
$514$	12.0000	0.529297
$515$	0	0
$516$	0	0
$517$	10.0000i	0.439799i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	− 16.0000i	− 0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$	− 16.0000i	− 0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	30.0000	1.31056
$525$	0	0
$526$	20.0000	0.872041
$527$	− 36.0000i	− 1.56818i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	28.0000i	1.21281i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	62.0000i	2.67301i
$539$	−35.0000	−1.50756
$540$	0	0
$541$	3.00000	0.128980	0.0644900	−	0.997918i	$-0.479458\pi$
$541$	3.00000	0.128980	0.0644900	+	0.997918i	$0.479458\pi$
$542$	16.0000i	0.687259i
$543$	0	0
$544$	−32.0000	−1.37199
$545$	0	0
$546$	0	0
$547$	28.0000i	1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$	28.0000i	1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$	24.0000i	1.02523i
$549$	0	0
$550$	0	0
$551$	−25.0000	−1.06504
$552$	0	0
$553$	0	0
$554$	−36.0000	−1.52949
$555$	0	0
$556$	−38.0000	−1.61156
$557$	24.0000i	1.01691i	0.861088	+	0.508456i	$0.169784\pi$
$557$	24.0000i	1.01691i	−0.861088	+	0.508456i	$0.830216\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	12.0000i	0.506189i
$563$	− 18.0000i	− 0.758610i	−0.925272	−	0.379305i	$-0.876163\pi$
$563$	− 18.0000i	− 0.758610i	0.925272	−	0.379305i	$-0.123837\pi$
$564$	0	0
$565$	0	0
$566$	−12.0000	−0.504398
$567$	0	0
$568$	0	0
$569$	−3.00000	−0.125767	−0.0628833	−	0.998021i	$-0.520030\pi$
$569$	−3.00000	−0.125767	−0.0628833	+	0.998021i	$0.520030\pi$
$570$	0	0
$571$	5.00000	0.209243	0.104622	−	0.994512i	$-0.466637\pi$
$571$	5.00000	0.209243	0.104622	+	0.994512i	$0.466637\pi$
$572$	− 40.0000i	− 1.67248i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 16.0000i	− 0.666089i	−0.942911	−	0.333044i	$-0.891924\pi$
$577$	− 16.0000i	− 0.666089i	0.942911	−	0.333044i	$-0.108076\pi$
$578$	− 2.00000i	− 0.0831890i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 40.0000i	− 1.65663i
$584$	0	0
$585$	0	0
$586$	36.0000	1.48715
$587$	− 18.0000i	− 0.742940i	−0.928445	−	0.371470i	$-0.878854\pi$
$587$	− 18.0000i	− 0.742940i	0.928445	−	0.371470i	$-0.121146\pi$
$588$	0	0
$589$	−45.0000	−1.85419
$590$	0	0
$591$	0	0
$592$	40.0000i	1.64399i
$593$	− 14.0000i	− 0.574911i	−0.957794	−	0.287456i	$-0.907191\pi$
$593$	− 14.0000i	− 0.574911i	0.957794	−	0.287456i	$-0.0928094\pi$
$594$	0	0
$595$	0	0
$596$	4.00000	0.163846
$597$	0	0
$598$	− 48.0000i	− 1.96287i
$599$	−17.0000	−0.694601	−0.347301	−	0.937754i	$-0.612902\pi$
$599$	−17.0000	−0.694601	−0.347301	+	0.937754i	$0.612902\pi$
$600$	0	0
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	0	0
$603$	0	0
$604$	10.0000	0.406894
$605$	0	0
$606$	0	0
$607$	34.0000i	1.38002i	0.723801	+	0.690009i	$0.242393\pi$
$607$	34.0000i	1.38002i	−0.723801	+	0.690009i	$0.757607\pi$
$608$	40.0000i	1.62221i
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	− 4.00000i	− 0.161558i	−0.996732	−	0.0807792i	$-0.974259\pi$
$613$	− 4.00000i	− 0.161558i	0.996732	−	0.0807792i	$-0.0257409\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	0	0
$617$	24.0000i	0.966204i	0.875564	+	0.483102i	$0.160490\pi$
$617$	24.0000i	0.966204i	−0.875564	+	0.483102i	$0.839510\pi$
$618$	0	0
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	0	0
$622$	− 18.0000i	− 0.721734i
$623$	0	0
$624$	0	0
$625$	0	0
$626$	8.00000	0.319744
$627$	0	0
$628$	− 4.00000i	− 0.159617i
$629$	40.0000	1.59490
$630$	0	0
$631$	−17.0000	−0.676759	−0.338380	−	0.941010i	$-0.609879\pi$
$631$	−17.0000	−0.676759	−0.338380	+	0.941010i	$0.609879\pi$
$632$	0	0
$633$	0	0
$634$	4.00000	0.158860
$635$	0	0
$636$	0	0
$637$	− 28.0000i	− 1.10940i
$638$	− 50.0000i	− 1.97952i
$639$	0	0
$640$	0	0
$641$	−3.00000	−0.118493	−0.0592464	−	0.998243i	$-0.518870\pi$
$641$	−3.00000	−0.118493	−0.0592464	+	0.998243i	$0.518870\pi$
$642$	0	0
$643$	− 6.00000i	− 0.236617i	−0.992977	−	0.118308i	$-0.962253\pi$
$643$	− 6.00000i	− 0.236617i	0.992977	−	0.118308i	$-0.0377472\pi$
$644$	0	0
$645$	0	0
$646$	40.0000	1.57378
$647$	10.0000i	0.393141i	0.980490	+	0.196570i	$0.0629804\pi$
$647$	10.0000i	0.393141i	−0.980490	+	0.196570i	$0.937020\pi$
$648$	0	0
$649$	5.00000	0.196267
$650$	0	0
$651$	0	0
$652$	16.0000i	0.626608i
$653$	− 8.00000i	− 0.313064i	−0.987673	−	0.156532i	$-0.949969\pi$
$653$	− 8.00000i	− 0.313064i	0.987673	−	0.156532i	$-0.0500315\pi$
$654$	0	0
$655$	0	0
$656$	28.0000	1.09322
$657$	0	0
$658$	0	0
$659$	44.0000	1.71400	0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000	1.71400	0.856998	+	0.515319i	$0.172327\pi$
$660$	0	0
$661$	25.0000	0.972387	0.486194	−	0.873851i	$-0.338385\pi$
$661$	25.0000	0.972387	0.486194	+	0.873851i	$0.338385\pi$
$662$	42.0000i	1.63238i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 30.0000i	− 1.16160i
$668$	24.0000i	0.928588i
$669$	0	0
$670$	0	0
$671$	10.0000	0.386046
$672$	0	0
$673$	− 42.0000i	− 1.61898i	−0.587133	−	0.809491i	$-0.699743\pi$
$673$	− 42.0000i	− 1.61898i	0.587133	−	0.809491i	$-0.300257\pi$
$674$	−16.0000	−0.616297
$675$	0	0
$676$	6.00000	0.230769
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	− 90.0000i	− 3.44628i
$683$	48.0000i	1.83667i	0.395805	+	0.918334i	$0.370466\pi$
$683$	48.0000i	1.83667i	−0.395805	+	0.918334i	$0.629534\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	− 8.00000i	− 0.304997i
$689$	32.0000	1.21910
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	12.0000i	0.456172i
$693$	0	0
$694$	−40.0000	−1.51838
$695$	0	0
$696$	0	0
$697$	− 28.0000i	− 1.06058i
$698$	26.0000i	0.984115i
$699$	0	0
$700$	0	0
$701$	−19.0000	−0.717620	−0.358810	−	0.933411i	$-0.616817\pi$
$701$	−19.0000	−0.717620	−0.358810	+	0.933411i	$0.616817\pi$
$702$	0	0
$703$	− 50.0000i	− 1.88579i
$704$	−40.0000	−1.50756
$705$	0	0
$706$	−36.0000	−1.35488
$707$	0	0
$708$	0	0
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 54.0000i	− 2.02232i
$714$	0	0
$715$	0	0
$716$	−46.0000	−1.71910
$717$	0	0
$718$	54.0000i	2.01526i
$719$	27.0000	1.00693	0.503465	−	0.864016i	$-0.332058\pi$
$719$	27.0000	1.00693	0.503465	+	0.864016i	$0.332058\pi$
$720$	0	0
$721$	0	0
$722$	− 12.0000i	− 0.446594i
$723$	0	0
$724$	50.0000	1.85824
$725$	0	0
$726$	0	0
$727$	44.0000i	1.63187i	0.578144	+	0.815935i	$0.303777\pi$
$727$	44.0000i	1.63187i	−0.578144	+	0.815935i	$0.696223\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	46.0000i	1.69905i	0.527549	+	0.849524i	$0.323111\pi$
$733$	46.0000i	1.69905i	−0.527549	+	0.849524i	$0.676889\pi$
$734$	36.0000	1.32878
$735$	0	0
$736$	−48.0000	−1.76930
$737$	− 30.0000i	− 1.10506i
$738$	0	0
$739$	35.0000	1.28750	0.643748	−	0.765238i	$-0.277379\pi$
$739$	35.0000	1.28750	0.643748	+	0.765238i	$0.277379\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 4.00000i	− 0.146746i	−0.997305	−	0.0733729i	$-0.976624\pi$
$743$	− 4.00000i	− 0.146746i	0.997305	−	0.0733729i	$-0.0233763\pi$
$744$	0	0
$745$	0	0
$746$	−32.0000	−1.17160
$747$	0	0
$748$	40.0000i	1.46254i
$749$	0	0
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	8.00000i	0.291730i
$753$	0	0
$754$	40.0000	1.45671
$755$	0	0
$756$	0	0
$757$	28.0000i	1.01768i	0.860862	+	0.508839i	$0.169925\pi$
$757$	28.0000i	1.01768i	−0.860862	+	0.508839i	$0.830075\pi$
$758$	8.00000i	0.290573i
$759$	0	0
$760$	0	0
$761$	27.0000	0.978749	0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000	0.978749	0.489375	+	0.872074i	$0.337225\pi$
$762$	0	0
$763$	0	0
$764$	2.00000	0.0723575
$765$	0	0
$766$	−72.0000	−2.60147
$767$	4.00000i	0.144432i
$768$	0	0
$769$	−5.00000	−0.180305	−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000	−0.180305	−0.0901523	+	0.995928i	$0.528735\pi$
$770$	0	0
$771$	0	0
$772$	52.0000i	1.87152i
$773$	18.0000i	0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$	18.0000i	0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	− 12.0000i	− 0.430221i
$779$	−35.0000	−1.25401
$780$	0	0
$781$	5.00000	0.178914
$782$	48.0000i	1.71648i
$783$	0	0
$784$	−28.0000	−1.00000
$785$	0	0
$786$	0	0
$787$	4.00000i	0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$	4.00000i	0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$	24.0000i	0.854965i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	8.00000i	0.284088i
$794$	−76.0000	−2.69714
$795$	0	0
$796$	32.0000	1.13421
$797$	− 4.00000i	− 0.141687i	−0.997487	−	0.0708436i	$-0.977431\pi$
$797$	− 4.00000i	− 0.141687i	0.997487	−	0.0708436i	$-0.0225691\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	0	0
$802$	60.0000i	2.11867i
$803$	− 40.0000i	− 1.41157i
$804$	0	0
$805$	0	0
$806$	72.0000	2.53609
$807$	0	0
$808$	0	0
$809$	7.00000	0.246107	0.123053	−	0.992400i	$-0.460731\pi$
$809$	7.00000	0.246107	0.123053	+	0.992400i	$0.460731\pi$
$810$	0	0
$811$	−21.0000	−0.737410	−0.368705	−	0.929547i	$-0.620199\pi$
$811$	−21.0000	−0.737410	−0.368705	+	0.929547i	$0.620199\pi$
$812$	0	0
$813$	0	0
$814$	100.000	3.50500
$815$	0	0
$816$	0	0
$817$	10.0000i	0.349856i
$818$	28.0000i	0.978997i
$819$	0	0
$820$	0	0
$821$	−3.00000	−0.104701	−0.0523504	−	0.998629i	$-0.516671\pi$
$821$	−3.00000	−0.104701	−0.0523504	+	0.998629i	$0.516671\pi$
$822$	0	0
$823$	46.0000i	1.60346i	0.597687	+	0.801730i	$0.296087\pi$
$823$	46.0000i	1.60346i	−0.597687	+	0.801730i	$0.703913\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 44.0000i	− 1.53003i	−0.644013	−	0.765015i	$-0.722732\pi$
$827$	− 44.0000i	− 1.53003i	0.644013	−	0.765015i	$-0.277268\pi$
$828$	0	0
$829$	−27.0000	−0.937749	−0.468874	−	0.883265i	$-0.655340\pi$
$829$	−27.0000	−0.937749	−0.468874	+	0.883265i	$0.655340\pi$
$830$	0	0
$831$	0	0
$832$	− 32.0000i	− 1.10940i
$833$	28.0000i	0.970143i
$834$	0	0
$835$	0	0
$836$	50.0000	1.72929
$837$	0	0
$838$	− 32.0000i	− 1.10542i
$839$	13.0000	0.448810	0.224405	−	0.974496i	$-0.427956\pi$
$839$	13.0000	0.448810	0.224405	+	0.974496i	$0.427956\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	− 26.0000i	− 0.896019i
$843$	0	0
$844$	−22.0000	−0.757271
$845$	0	0
$846$	0	0
$847$	0	0
$848$	− 32.0000i	− 1.09888i
$849$	0	0
$850$	0	0
$851$	60.0000	2.05677
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 32.0000i	− 1.09310i	−0.837427	−	0.546550i	$-0.815941\pi$
$857$	− 32.0000i	− 1.09310i	0.837427	−	0.546550i	$-0.184059\pi$
$858$	0	0
$859$	−55.0000	−1.87658	−0.938288	−	0.345855i	$-0.887589\pi$
$859$	−55.0000	−1.87658	−0.938288	+	0.345855i	$0.887589\pi$
$860$	0	0
$861$	0	0
$862$	6.00000i	0.204361i
$863$	− 44.0000i	− 1.49778i	−0.662696	−	0.748889i	$-0.730588\pi$
$863$	− 44.0000i	− 1.49778i	0.662696	−	0.748889i	$-0.269412\pi$
$864$	0	0
$865$	0	0
$866$	52.0000	1.76703
$867$	0	0
$868$	0	0
$869$	60.0000	2.03536
$870$	0	0
$871$	24.0000	0.813209
$872$	0	0
$873$	0	0
$874$	60.0000	2.02953
$875$	0	0
$876$	0	0
$877$	− 24.0000i	− 0.810422i	−0.914223	−	0.405211i	$-0.867198\pi$
$877$	− 24.0000i	− 0.810422i	0.914223	−	0.405211i	$-0.132802\pi$
$878$	− 58.0000i	− 1.95741i
$879$	0	0
$880$	0	0
$881$	25.0000	0.842271	0.421136	−	0.906998i	$-0.361632\pi$
$881$	25.0000	0.842271	0.421136	+	0.906998i	$0.361632\pi$
$882$	0	0
$883$	52.0000i	1.74994i	0.484178	+	0.874970i	$0.339119\pi$
$883$	52.0000i	1.74994i	−0.484178	+	0.874970i	$0.660881\pi$
$884$	−32.0000	−1.07628
$885$	0	0
$886$	12.0000	0.403148
$887$	− 6.00000i	− 0.201460i	−0.994914	−	0.100730i	$-0.967882\pi$
$887$	− 6.00000i	− 0.201460i	0.994914	−	0.100730i	$-0.0321179\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	16.0000i	0.535720i
$893$	− 10.0000i	− 0.334637i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	− 34.0000i	− 1.13459i
$899$	45.0000	1.50083
$900$	0	0
$901$	−32.0000	−1.06607
$902$	− 70.0000i	− 2.33075i
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 18.0000i	− 0.597680i	−0.954303	−	0.298840i	$-0.903400\pi$
$907$	− 18.0000i	− 0.597680i	0.954303	−	0.298840i	$-0.0965997\pi$
$908$	8.00000i	0.265489i
$909$	0	0
$910$	0	0
$911$	49.0000	1.62344	0.811721	−	0.584045i	$-0.198531\pi$
$911$	49.0000	1.62344	0.811721	+	0.584045i	$0.198531\pi$
$912$	0	0
$913$	− 30.0000i	− 0.992855i
$914$	−76.0000	−2.51386
$915$	0	0
$916$	12.0000	0.396491
$917$	0	0
$918$	0	0
$919$	11.0000	0.362857	0.181428	−	0.983404i	$-0.441928\pi$
$919$	11.0000	0.362857	0.181428	+	0.983404i	$0.441928\pi$
$920$	0	0
$921$	0	0
$922$	− 30.0000i	− 0.987997i
$923$	4.00000i	0.131662i
$924$	0	0
$925$	0	0
$926$	12.0000	0.394344
$927$	0	0
$928$	− 40.0000i	− 1.31306i
$929$	1.00000	0.0328089	0.0164045	−	0.999865i	$-0.494778\pi$
$929$	1.00000	0.0328089	0.0164045	+	0.999865i	$0.494778\pi$
$930$	0	0
$931$	35.0000	1.14708
$932$	− 12.0000i	− 0.393073i
$933$	0	0
$934$	−8.00000	−0.261768
$935$	0	0
$936$	0	0
$937$	42.0000i	1.37208i	0.727564	+	0.686040i	$0.240653\pi$
$937$	42.0000i	1.37208i	−0.727564	+	0.686040i	$0.759347\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−46.0000	−1.49956	−0.749779	−	0.661689i	$-0.769840\pi$
$941$	−46.0000	−1.49956	−0.749779	+	0.661689i	$0.769840\pi$
$942$	0	0
$943$	− 42.0000i	− 1.36771i
$944$	4.00000	0.130189
$945$	0	0
$946$	−20.0000	−0.650256
$947$	− 18.0000i	− 0.584921i	−0.956278	−	0.292461i	$-0.905526\pi$
$947$	− 18.0000i	− 0.584921i	0.956278	−	0.292461i	$-0.0944741\pi$
$948$	0	0
$949$	32.0000	1.03876
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 16.0000i	− 0.518291i	−0.965838	−	0.259145i	$-0.916559\pi$
$953$	− 16.0000i	− 0.518291i	0.965838	−	0.259145i	$-0.0834409\pi$
$954$	0	0
$955$	0	0
$956$	32.0000	1.03495
$957$	0	0
$958$	30.0000i	0.969256i
$959$	0	0
$960$	0	0
$961$	50.0000	1.61290
$962$	80.0000i	2.57930i
$963$	0	0
$964$	22.0000	0.708572
$965$	0	0
$966$	0	0
$967$	− 40.0000i	− 1.28631i	−0.765735	−	0.643157i	$-0.777624\pi$
$967$	− 40.0000i	− 1.28631i	0.765735	−	0.643157i	$-0.222376\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	27.0000	0.866471	0.433236	−	0.901281i	$-0.357372\pi$
$971$	27.0000	0.866471	0.433236	+	0.901281i	$0.357372\pi$
$972$	0	0
$973$	0	0
$974$	16.0000	0.512673
$975$	0	0
$976$	8.00000	0.256074
$977$	28.0000i	0.895799i	0.894084	+	0.447900i	$0.147828\pi$
$977$	28.0000i	0.895799i	−0.894084	+	0.447900i	$0.852172\pi$
$978$	0	0
$979$	45.0000	1.43821
$980$	0	0
$981$	0	0
$982$	86.0000i	2.74437i
$983$	48.0000i	1.53096i	0.643458	+	0.765481i	$0.277499\pi$
$983$	48.0000i	1.53096i	−0.643458	+	0.765481i	$0.722501\pi$
$984$	0	0
$985$	0	0
$986$	−40.0000	−1.27386
$987$	0	0
$988$	40.0000i	1.27257i
$989$	−12.0000	−0.381578
$990$	0	0
$991$	−1.00000	−0.0317660	−0.0158830	−	0.999874i	$-0.505056\pi$
$991$	−1.00000	−0.0317660	−0.0158830	+	0.999874i	$0.505056\pi$
$992$	− 72.0000i	− 2.28600i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	48.0000i	1.52018i	0.649821	+	0.760088i	$0.274844\pi$
$997$	48.0000i	1.52018i	−0.649821	+	0.760088i	$0.725156\pi$
$998$	14.0000i	0.443162i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.b.a.649.1		2
3.2	odd	2		2025.2.b.b.649.2		2
5.2	odd	4		2025.2.a.f.1.1		1
5.3	odd	4		405.2.a.a.1.1	✓	1
5.4	even	2	inner	2025.2.b.a.649.2		2
15.2	even	4		2025.2.a.a.1.1		1
15.8	even	4		405.2.a.f.1.1	yes	1
15.14	odd	2		2025.2.b.b.649.1		2
20.3	even	4		6480.2.a.f.1.1		1
45.13	odd	12		405.2.e.g.136.1		2
45.23	even	12		405.2.e.a.136.1		2
45.38	even	12		405.2.e.a.271.1		2
45.43	odd	12		405.2.e.g.271.1		2
60.23	odd	4		6480.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.2.a.a.1.1	✓	1	5.3	odd	4
405.2.a.f.1.1	yes	1	15.8	even	4
405.2.e.a.136.1		2	45.23	even	12
405.2.e.a.271.1		2	45.38	even	12
405.2.e.g.136.1		2	45.13	odd	12
405.2.e.g.271.1		2	45.43	odd	12
2025.2.a.a.1.1		1	15.2	even	4
2025.2.a.f.1.1		1	5.2	odd	4
2025.2.b.a.649.1		2	1.1	even	1	trivial
2025.2.b.a.649.2		2	5.4	even	2	inner
2025.2.b.b.649.1		2	15.14	odd	2
2025.2.b.b.649.2		2	3.2	odd	2
6480.2.a.f.1.1		1	20.3	even	4
6480.2.a.r.1.1		1	60.23	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2025 = 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2025.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -2.00000 q^{4} -5.00000 q^{11} -4.00000i q^{13} -4.00000 q^{16} +4.00000i q^{17} +5.00000 q^{19} +10.0000i q^{22} +6.00000i q^{23} -8.00000 q^{26} -5.00000 q^{29} -9.00000 q^{31} +8.00000i q^{32} +8.00000 q^{34} -10.0000i q^{37} -10.0000i q^{38} -7.00000 q^{41} +2.00000i q^{43} +10.0000 q^{44} +12.0000 q^{46} -2.00000i q^{47} +7.00000 q^{49} +8.00000i q^{52} +8.00000i q^{53} +10.0000i q^{58} -1.00000 q^{59} -2.00000 q^{61} +18.0000i q^{62} +8.00000 q^{64} +6.00000i q^{67} -8.00000i q^{68} -1.00000 q^{71} +8.00000i q^{73} -20.0000 q^{74} -10.0000 q^{76} -12.0000 q^{79} +14.0000i q^{82} +6.00000i q^{83} +4.00000 q^{86} -9.00000 q^{89} -12.0000i q^{92} -4.00000 q^{94} +14.0000i q^{97} -14.0000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} - 10 q^{11} - 8 q^{16} + 10 q^{19} - 16 q^{26} - 10 q^{29} - 18 q^{31} + 16 q^{34} - 14 q^{41} + 20 q^{44} + 24 q^{46} + 14 q^{49} - 2 q^{59} - 4 q^{61} + 16 q^{64} - 2 q^{71} - 40 q^{74}+ \cdots - 8 q^{94}+O(q^{100}) \) 2 * q - 4 * q^4 - 10 * q^11 - 8 * q^16 + 10 * q^19 - 16 * q^26 - 10 * q^29 - 18 * q^31 + 16 * q^34 - 14 * q^41 + 20 * q^44 + 24 * q^46 + 14 * q^49 - 2 * q^59 - 4 * q^61 + 16 * q^64 - 2 * q^71 - 40 * q^74 - 20 * q^76 - 24 * q^79 + 8 * q^86 - 18 * q^89 - 8 * q^94

Introduction

L-functions

Modular forms

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Database

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