LMFDB - Embedded newform 2025.2.b.f.649.2

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 = [2,0,0,4,0,0,0,0,0,0,6] 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:	$0$
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 405)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{4} +2.00000i q^{7} +3.00000 q^{11} +4.00000i q^{13} +4.00000 q^{16} +6.00000i q^{17} +1.00000 q^{19} -6.00000i q^{23} +4.00000i q^{28} -9.00000 q^{29} -1.00000 q^{31} +8.00000i q^{37} -3.00000 q^{41} +4.00000i q^{43} +6.00000 q^{44} -12.0000i q^{47} +3.00000 q^{49} +8.00000i q^{52} +6.00000i q^{53} +3.00000 q^{59} -10.0000 q^{61} +8.00000 q^{64} +14.0000i q^{67} +12.0000i q^{68} +3.00000 q^{71} -2.00000i q^{73} +2.00000 q^{76} +6.00000i q^{77} +16.0000 q^{79} -12.0000i q^{83} +15.0000 q^{89} -8.00000 q^{91} -12.0000i q^{92} -4.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{4} + 6 q^{11} + 8 q^{16} + 2 q^{19} - 18 q^{29} - 2 q^{31} - 6 q^{41} + 12 q^{44} + 6 q^{49} + 6 q^{59} - 20 q^{61} + 16 q^{64} + 6 q^{71} + 4 q^{76} + 32 q^{79} + 30 q^{89} - 16 q^{91}+O(q^{100}) $ 2 * q + 4 * q^4 + 6 * q^11 + 8 * q^16 + 2 * q^19 - 18 * q^29 - 2 * q^31 - 6 * q^41 + 12 * q^44 + 6 * q^49 + 6 * q^59 - 20 * q^61 + 16 * q^64 + 6 * q^71 + 4 * q^76 + 32 * q^79 + 30 * q^89 - 16 * q^91

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 $

$2$	0	0	1.00000			$0$
$2$	0	0	−1.00000			$\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	4.00000i	0.755929i
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.00000i	1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$	8.00000i	1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	0	0
$47$	− 12.0000i	− 1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$	− 12.0000i	− 1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	8.00000i	1.10940i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	0	0
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	14.0000i	1.71037i	0.518321	+	0.855186i	$0.326557\pi$
$67$	14.0000i	1.71037i	−0.518321	+	0.855186i	$0.673443\pi$
$68$	12.0000i	1.45521i
$69$	0	0
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	0	0
$75$	0	0
$76$	2.00000	0.229416
$77$	6.00000i	0.683763i
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	− 12.0000i	− 1.25109i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 4.00000i	− 0.406138i	−0.979164	−	0.203069i	$-0.934908\pi$
$97$	− 4.00000i	− 0.406138i	0.979164	−	0.203069i	$-0.0650917\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	0	0
$103$	− 14.0000i	− 1.37946i	−0.724066	−	0.689730i	$-0.757729\pi$
$103$	− 14.0000i	− 1.37946i	0.724066	−	0.689730i	$-0.242271\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.0000i	1.74013i	0.492941	+	0.870063i	$0.335922\pi$
$107$	18.0000i	1.74013i	−0.492941	+	0.870063i	$0.664078\pi$
$108$	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
$112$	8.00000i	0.755929i
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	0	0
$115$	0	0
$116$	−18.0000	−1.67126
$117$	0	0
$118$	0	0
$119$	−12.0000	−1.10004
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	0	0
$124$	−2.00000	−0.179605
$125$	0	0
$126$	0	0
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.00000	0.786334	0.393167	−	0.919467i	$-0.371379\pi$
$131$	9.00000	0.786334	0.393167	+	0.919467i	$0.371379\pi$
$132$	0	0
$133$	2.00000i	0.173422i
$134$	0	0
$135$	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
$139$	7.00000	0.593732	0.296866	−	0.954919i	$-0.404058\pi$
$139$	7.00000	0.593732	0.296866	+	0.954919i	$0.404058\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	12.0000i	1.00349i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	16.0000i	1.31519i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−1.00000	−0.0813788	−0.0406894	−	0.999172i	$-0.512955\pi$
$151$	−1.00000	−0.0813788	−0.0406894	+	0.999172i	$0.512955\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 16.0000i	− 1.27694i	−0.769647	−	0.638470i	$-0.779568\pi$
$157$	− 16.0000i	− 1.27694i	0.769647	−	0.638470i	$-0.220432\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	− 2.00000i	− 0.156652i	−0.996928	−	0.0783260i	$-0.975042\pi$
$163$	− 2.00000i	− 0.156652i	0.996928	−	0.0783260i	$-0.0249575\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	0	0
$167$	− 6.00000i	− 0.464294i	−0.972681	−	0.232147i	$-0.925425\pi$
$167$	− 6.00000i	− 0.464294i	0.972681	−	0.232147i	$-0.0745750\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	8.00000i	0.609994i
$173$	− 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
$176$	12.0000	0.904534
$177$	0	0
$178$	0	0
$179$	−9.00000	−0.672692	−0.336346	−	0.941739i	$-0.609191\pi$
$179$	−9.00000	−0.672692	−0.336346	+	0.941739i	$0.609191\pi$
$180$	0	0
$181$	11.0000	0.817624	0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000	0.817624	0.408812	+	0.912619i	$0.365943\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	18.0000i	1.31629i
$188$	− 24.0000i	− 1.75038i
$189$	0	0
$190$	0	0
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	0	0
$193$	4.00000i	0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$	4.00000i	0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	0	0
$195$	0	0
$196$	6.00000	0.428571
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 18.0000i	− 1.26335i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	16.0000i	1.10940i
$209$	3.00000	0.207514
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	12.0000i	0.824163i
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2.00000i	− 0.135769i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−24.0000	−1.61441
$222$	0	0
$223$	− 20.0000i	− 1.33930i	−0.742677	−	0.669650i	$-0.766444\pi$
$223$	− 20.0000i	− 1.33930i	0.742677	−	0.669650i	$-0.233556\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.0000i	1.19470i	0.801980	+	0.597351i	$0.203780\pi$
$227$	18.0000i	1.19470i	−0.801980	+	0.597351i	$0.796220\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.0000i	0.786146i	0.919507	+	0.393073i	$0.128588\pi$
$233$	12.0000i	0.786146i	−0.919507	+	0.393073i	$0.871412\pi$
$234$	0	0
$235$	0	0
$236$	6.00000	0.390567
$237$	0	0
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	0	0
$243$	0	0
$244$	−20.0000	−1.28037
$245$	0	0
$246$	0	0
$247$	4.00000i	0.254514i
$248$	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
$253$	− 18.0000i	− 1.13165i
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	− 6.00000i	− 0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$	− 6.00000i	− 0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.0000i	1.10993i	0.831875	+	0.554964i	$0.187268\pi$
$263$	18.0000i	1.10993i	−0.831875	+	0.554964i	$0.812732\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	28.0000i	1.71037i
$269$	−3.00000	−0.182913	−0.0914566	−	0.995809i	$-0.529152\pi$
$269$	−3.00000	−0.182913	−0.0914566	+	0.995809i	$0.529152\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	24.0000i	1.45521i
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 16.0000i	− 0.961347i	−0.876900	−	0.480673i	$-0.840392\pi$
$277$	− 16.0000i	− 0.961347i	0.876900	−	0.480673i	$-0.159608\pi$
$278$	0	0
$279$	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
$283$	10.0000i	0.594438i	0.954809	+	0.297219i	$0.0960592\pi$
$283$	10.0000i	0.594438i	−0.954809	+	0.297219i	$0.903941\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	0	0
$287$	− 6.00000i	− 0.354169i
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	0	0
$292$	− 4.00000i	− 0.234082i
$293$	− 12.0000i	− 0.701047i	−0.936554	−	0.350524i	$-0.886004\pi$
$293$	− 12.0000i	− 0.701047i	0.936554	−	0.350524i	$-0.113996\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	24.0000	1.38796
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	0	0
$304$	4.00000	0.229416
$305$	0	0
$306$	0	0
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	12.0000i	0.683763i
$309$	0	0
$310$	0	0
$311$	−27.0000	−1.53103	−0.765515	−	0.643418i	$-0.777516\pi$
$311$	−27.0000	−1.53103	−0.765515	+	0.643418i	$0.777516\pi$
$312$	0	0
$313$	− 8.00000i	− 0.452187i	−0.974106	−	0.226093i	$-0.927405\pi$
$313$	− 8.00000i	− 0.452187i	0.974106	−	0.226093i	$-0.0725954\pi$
$314$	0	0
$315$	0	0
$316$	32.0000	1.80014
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	−27.0000	−1.51171
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.00000i	0.333849i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	−25.0000	−1.37412	−0.687062	−	0.726599i	$-0.741100\pi$
$331$	−25.0000	−1.37412	−0.687062	+	0.726599i	$0.741100\pi$
$332$	− 24.0000i	− 1.31717i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 4.00000i	− 0.217894i	−0.994048	−	0.108947i	$-0.965252\pi$
$337$	− 4.00000i	− 0.217894i	0.994048	−	0.108947i	$-0.0347479\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.00000	−0.162459
$342$	0	0
$343$	20.0000i	1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 18.0000i	− 0.966291i	−0.875540	−	0.483145i	$-0.839494\pi$
$347$	− 18.0000i	− 0.966291i	0.875540	−	0.483145i	$-0.160506\pi$
$348$	0	0
$349$	−5.00000	−0.267644	−0.133822	−	0.991005i	$-0.542725\pi$
$349$	−5.00000	−0.267644	−0.133822	+	0.991005i	$0.542725\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.0000i	0.638696i	0.947638	+	0.319348i	$0.103464\pi$
$353$	12.0000i	0.638696i	−0.947638	+	0.319348i	$0.896536\pi$
$354$	0	0
$355$	0	0
$356$	30.0000	1.59000
$357$	0	0
$358$	0	0
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	0	0
$364$	−16.0000	−0.838628
$365$	0	0
$366$	0	0
$367$	− 10.0000i	− 0.521996i	−0.965339	−	0.260998i	$-0.915948\pi$
$367$	− 10.0000i	− 0.521996i	0.965339	−	0.260998i	$-0.0840516\pi$
$368$	− 24.0000i	− 1.25109i
$369$	0	0
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	− 38.0000i	− 1.96757i	−0.179364	−	0.983783i	$-0.557404\pi$
$373$	− 38.0000i	− 1.96757i	0.179364	−	0.983783i	$-0.442596\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 36.0000i	− 1.85409i
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	− 8.00000i	− 0.406138i
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	36.0000	1.82060
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 22.0000i	− 1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$	− 22.0000i	− 1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	− 4.00000i	− 0.199254i
$404$	−18.0000	−0.895533
$405$	0	0
$406$	0	0
$407$	24.0000i	1.18964i
$408$	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
$412$	− 28.0000i	− 1.37946i
$413$	6.00000i	0.295241i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	5.00000	0.243685	0.121843	−	0.992549i	$-0.461120\pi$
$421$	5.00000	0.243685	0.121843	+	0.992549i	$0.461120\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 20.0000i	− 0.967868i
$428$	36.0000i	1.74013i
$429$	0	0
$430$	0	0
$431$	−15.0000	−0.722525	−0.361262	−	0.932464i	$-0.617654\pi$
$431$	−15.0000	−0.722525	−0.361262	+	0.932464i	$0.617654\pi$
$432$	0	0
$433$	− 26.0000i	− 1.24948i	−0.780833	−	0.624740i	$-0.785205\pi$
$433$	− 26.0000i	− 1.24948i	0.780833	−	0.624740i	$-0.214795\pi$
$434$	0	0
$435$	0	0
$436$	2.00000	0.0957826
$437$	− 6.00000i	− 0.287019i
$438$	0	0
$439$	1.00000	0.0477274	0.0238637	−	0.999715i	$-0.492403\pi$
$439$	1.00000	0.0477274	0.0238637	+	0.999715i	$0.492403\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	16.0000i	0.755929i
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	0	0
$451$	−9.00000	−0.423793
$452$	12.0000i	0.564433i
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 28.0000i	− 1.30978i	−0.755722	−	0.654892i	$-0.772714\pi$
$457$	− 28.0000i	− 1.30978i	0.755722	−	0.654892i	$-0.227286\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−21.0000	−0.978068	−0.489034	−	0.872265i	$-0.662651\pi$
$461$	−21.0000	−0.978068	−0.489034	+	0.872265i	$0.662651\pi$
$462$	0	0
$463$	− 26.0000i	− 1.20832i	−0.796862	−	0.604161i	$-0.793508\pi$
$463$	− 26.0000i	− 1.20832i	0.796862	−	0.604161i	$-0.206492\pi$
$464$	−36.0000	−1.67126
$465$	0	0
$466$	0	0
$467$	6.00000i	0.277647i	0.990317	+	0.138823i	$0.0443321\pi$
$467$	6.00000i	0.277647i	−0.990317	+	0.138823i	$0.955668\pi$
$468$	0	0
$469$	−28.0000	−1.29292
$470$	0	0
$471$	0	0
$472$	0	0
$473$	12.0000i	0.551761i
$474$	0	0
$475$	0	0
$476$	−24.0000	−1.10004
$477$	0	0
$478$	0	0
$479$	33.0000	1.50781	0.753904	−	0.656984i	$-0.228168\pi$
$479$	33.0000	1.50781	0.753904	+	0.656984i	$0.228168\pi$
$480$	0	0
$481$	−32.0000	−1.45907
$482$	0	0
$483$	0	0
$484$	−4.00000	−0.181818
$485$	0	0
$486$	0	0
$487$	− 16.0000i	− 0.725029i	−0.931978	−	0.362515i	$-0.881918\pi$
$487$	− 16.0000i	− 0.725029i	0.931978	−	0.362515i	$-0.118082\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9.00000	0.406164	0.203082	−	0.979162i	$-0.434904\pi$
$491$	9.00000	0.406164	0.203082	+	0.979162i	$0.434904\pi$
$492$	0	0
$493$	− 54.0000i	− 2.43204i
$494$	0	0
$495$	0	0
$496$	−4.00000	−0.179605
$497$	6.00000i	0.269137i
$498$	0	0
$499$	13.0000	0.581960	0.290980	−	0.956729i	$-0.406019\pi$
$499$	13.0000	0.581960	0.290980	+	0.956729i	$0.406019\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 12.0000i	− 0.535054i	−0.963550	−	0.267527i	$-0.913794\pi$
$503$	− 12.0000i	− 0.535054i	0.963550	−	0.267527i	$-0.0862064\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	4.00000i	0.177471i
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 36.0000i	− 1.58328i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	18.0000	0.786334
$525$	0	0
$526$	0	0
$527$	− 6.00000i	− 0.261364i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	4.00000i	0.173422i
$533$	− 12.0000i	− 0.519778i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	9.00000	0.387657
$540$	0	0
$541$	23.0000	0.988847	0.494424	−	0.869221i	$-0.335379\pi$
$541$	23.0000	0.988847	0.494424	+	0.869221i	$0.335379\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.00000i	0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$	8.00000i	0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	− 24.0000i	− 1.02523i
$549$	0	0
$550$	0	0
$551$	−9.00000	−0.383413
$552$	0	0
$553$	32.0000i	1.36078i
$554$	0	0
$555$	0	0
$556$	14.0000	0.593732
$557$	− 24.0000i	− 1.01691i	−0.861088	−	0.508456i	$-0.830216\pi$
$557$	− 24.0000i	− 1.01691i	0.861088	−	0.508456i	$-0.169784\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−27.0000	−1.13190	−0.565949	−	0.824440i	$-0.691490\pi$
$569$	−27.0000	−1.13190	−0.565949	+	0.824440i	$0.691490\pi$
$570$	0	0
$571$	−19.0000	−0.795125	−0.397563	−	0.917575i	$-0.630144\pi$
$571$	−19.0000	−0.795125	−0.397563	+	0.917575i	$0.630144\pi$
$572$	24.0000i	1.00349i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 34.0000i	− 1.41544i	−0.706494	−	0.707719i	$-0.749724\pi$
$577$	− 34.0000i	− 1.41544i	0.706494	−	0.707719i	$-0.250276\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	24.0000	0.995688
$582$	0	0
$583$	18.0000i	0.745484i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	48.0000i	1.98117i	0.136892	+	0.990586i	$0.456289\pi$
$587$	48.0000i	1.98117i	−0.136892	+	0.990586i	$0.543711\pi$
$588$	0	0
$589$	−1.00000	−0.0412043
$590$	0	0
$591$	0	0
$592$	32.0000i	1.31519i
$593$	18.0000i	0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$	18.0000i	0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$	0	0
$595$	0	0
$596$	12.0000	0.491539
$597$	0	0
$598$	0	0
$599$	−21.0000	−0.858037	−0.429018	−	0.903296i	$-0.641140\pi$
$599$	−21.0000	−0.858037	−0.429018	+	0.903296i	$0.641140\pi$
$600$	0	0
$601$	−31.0000	−1.26452	−0.632258	−	0.774758i	$-0.717872\pi$
$601$	−31.0000	−1.26452	−0.632258	+	0.774758i	$0.717872\pi$
$602$	0	0
$603$	0	0
$604$	−2.00000	−0.0813788
$605$	0	0
$606$	0	0
$607$	20.0000i	0.811775i	0.913923	+	0.405887i	$0.133038\pi$
$607$	20.0000i	0.811775i	−0.913923	+	0.405887i	$0.866962\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	48.0000	1.94187
$612$	0	0
$613$	16.0000i	0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$	16.0000i	0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	0	0
$615$	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
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	30.0000i	1.20192i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	− 32.0000i	− 1.27694i
$629$	−48.0000	−1.91389
$630$	0	0
$631$	−25.0000	−0.995234	−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000	−0.995234	−0.497617	+	0.867397i	$0.665792\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	12.0000i	0.475457i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	0	0
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	24.0000	0.945732
$645$	0	0
$646$	0	0
$647$	− 6.00000i	− 0.235884i	−0.993020	−	0.117942i	$-0.962370\pi$
$647$	− 6.00000i	− 0.235884i	0.993020	−	0.117942i	$-0.0376297\pi$
$648$	0	0
$649$	9.00000	0.353281
$650$	0	0
$651$	0	0
$652$	− 4.00000i	− 0.156652i
$653$	− 24.0000i	− 0.939193i	−0.882881	−	0.469596i	$-0.844399\pi$
$653$	− 24.0000i	− 0.939193i	0.882881	−	0.469596i	$-0.155601\pi$
$654$	0	0
$655$	0	0
$656$	−12.0000	−0.468521
$657$	0	0
$658$	0	0
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	0	0
$661$	5.00000	0.194477	0.0972387	−	0.995261i	$-0.468999\pi$
$661$	5.00000	0.194477	0.0972387	+	0.995261i	$0.468999\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	54.0000i	2.09089i
$668$	− 12.0000i	− 0.464294i
$669$	0	0
$670$	0	0
$671$	−30.0000	−1.15814
$672$	0	0
$673$	34.0000i	1.31060i	0.755367	+	0.655302i	$0.227459\pi$
$673$	34.0000i	1.31060i	−0.755367	+	0.655302i	$0.772541\pi$
$674$	0	0
$675$	0	0
$676$	−6.00000	−0.230769
$677$	− 30.0000i	− 1.15299i	−0.817099	−	0.576497i	$-0.804419\pi$
$677$	− 30.0000i	− 1.15299i	0.817099	−	0.576497i	$-0.195581\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	16.0000i	0.609994i
$689$	−24.0000	−0.914327
$690$	0	0
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	− 12.0000i	− 0.456172i
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 18.0000i	− 0.681799i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−15.0000	−0.566542	−0.283271	−	0.959040i	$-0.591420\pi$
$701$	−15.0000	−0.566542	−0.283271	+	0.959040i	$0.591420\pi$
$702$	0	0
$703$	8.00000i	0.301726i
$704$	24.0000	0.904534
$705$	0	0
$706$	0	0
$707$	− 18.0000i	− 0.676960i
$708$	0	0
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.00000i	0.224702i
$714$	0	0
$715$	0	0
$716$	−18.0000	−0.672692
$717$	0	0
$718$	0	0
$719$	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
$721$	28.0000	1.04277
$722$	0	0
$723$	0	0
$724$	22.0000	0.817624
$725$	0	0
$726$	0	0
$727$	− 28.0000i	− 1.03846i	−0.854634	−	0.519231i	$-0.826218\pi$
$727$	− 28.0000i	− 1.03846i	0.854634	−	0.519231i	$-0.173782\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	− 14.0000i	− 0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$	− 14.0000i	− 0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	42.0000i	1.54709i
$738$	0	0
$739$	7.00000	0.257499	0.128750	−	0.991677i	$-0.458904\pi$
$739$	7.00000	0.257499	0.128750	+	0.991677i	$0.458904\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 18.0000i	− 0.660356i	−0.943919	−	0.330178i	$-0.892891\pi$
$743$	− 18.0000i	− 0.660356i	0.943919	−	0.330178i	$-0.107109\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	36.0000i	1.31629i
$749$	−36.0000	−1.31541
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	− 48.0000i	− 1.75038i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	20.0000i	0.726912i	0.931611	+	0.363456i	$0.118403\pi$
$757$	20.0000i	0.726912i	−0.931611	+	0.363456i	$0.881597\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−33.0000	−1.19625	−0.598125	−	0.801403i	$-0.704087\pi$
$761$	−33.0000	−1.19625	−0.598125	+	0.801403i	$0.704087\pi$
$762$	0	0
$763$	2.00000i	0.0724049i
$764$	30.0000	1.08536
$765$	0	0
$766$	0	0
$767$	12.0000i	0.433295i
$768$	0	0
$769$	−41.0000	−1.47850	−0.739249	−	0.673432i	$-0.764819\pi$
$769$	−41.0000	−1.47850	−0.739249	+	0.673432i	$0.764819\pi$
$770$	0	0
$771$	0	0
$772$	8.00000i	0.287926i
$773$	24.0000i	0.863220i	0.902060	+	0.431610i	$0.142054\pi$
$773$	24.0000i	0.863220i	−0.902060	+	0.431610i	$0.857946\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.00000	−0.107486
$780$	0	0
$781$	9.00000	0.322045
$782$	0	0
$783$	0	0
$784$	12.0000	0.428571
$785$	0	0
$786$	0	0
$787$	14.0000i	0.499046i	0.968369	+	0.249523i	$0.0802738\pi$
$787$	14.0000i	0.499046i	−0.968369	+	0.249523i	$0.919726\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	− 40.0000i	− 1.42044i
$794$	0	0
$795$	0	0
$796$	32.0000	1.13421
$797$	24.0000i	0.850124i	0.905164	+	0.425062i	$0.139748\pi$
$797$	24.0000i	0.850124i	−0.905164	+	0.425062i	$0.860252\pi$
$798$	0	0
$799$	72.0000	2.54718
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 6.00000i	− 0.211735i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	0	0
$811$	23.0000	0.807639	0.403820	−	0.914839i	$-0.367682\pi$
$811$	23.0000	0.807639	0.403820	+	0.914839i	$0.367682\pi$
$812$	− 36.0000i	− 1.26335i
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.00000i	0.139942i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	45.0000	1.57051	0.785255	−	0.619172i	$-0.212532\pi$
$821$	45.0000	1.57051	0.785255	+	0.619172i	$0.212532\pi$
$822$	0	0
$823$	22.0000i	0.766872i	0.923567	+	0.383436i	$0.125259\pi$
$823$	22.0000i	0.766872i	−0.923567	+	0.383436i	$0.874741\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	0	0
$829$	13.0000	0.451509	0.225754	−	0.974184i	$-0.427515\pi$
$829$	13.0000	0.451509	0.225754	+	0.974184i	$0.427515\pi$
$830$	0	0
$831$	0	0
$832$	32.0000i	1.10940i
$833$	18.0000i	0.623663i
$834$	0	0
$835$	0	0
$836$	6.00000	0.207514
$837$	0	0
$838$	0	0
$839$	45.0000	1.55357	0.776786	−	0.629764i	$-0.216849\pi$
$839$	45.0000	1.55357	0.776786	+	0.629764i	$0.216849\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	0	0
$844$	−26.0000	−0.894957
$845$	0	0
$846$	0	0
$847$	− 4.00000i	− 0.137442i
$848$	24.0000i	0.824163i
$849$	0	0
$850$	0	0
$851$	48.0000	1.64542
$852$	0	0
$853$	46.0000i	1.57501i	0.616308	+	0.787505i	$0.288628\pi$
$853$	46.0000i	1.57501i	−0.616308	+	0.787505i	$0.711372\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 48.0000i	− 1.63965i	−0.572615	−	0.819824i	$-0.694071\pi$
$857$	− 48.0000i	− 1.63965i	0.572615	−	0.819824i	$-0.305929\pi$
$858$	0	0
$859$	25.0000	0.852989	0.426494	−	0.904490i	$-0.359748\pi$
$859$	25.0000	0.852989	0.426494	+	0.904490i	$0.359748\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	30.0000i	1.02121i	0.859815	+	0.510606i	$0.170579\pi$
$863$	30.0000i	1.02121i	−0.859815	+	0.510606i	$0.829421\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	− 4.00000i	− 0.135769i
$869$	48.0000	1.62829
$870$	0	0
$871$	−56.0000	−1.89749
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	44.0000i	1.48577i	0.669417	+	0.742887i	$0.266544\pi$
$877$	44.0000i	1.48577i	−0.669417	+	0.742887i	$0.733456\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−39.0000	−1.31394	−0.656972	−	0.753915i	$-0.728163\pi$
$881$	−39.0000	−1.31394	−0.656972	+	0.753915i	$0.728163\pi$
$882$	0	0
$883$	46.0000i	1.54802i	0.633171	+	0.774012i	$0.281753\pi$
$883$	46.0000i	1.54802i	−0.633171	+	0.774012i	$0.718247\pi$
$884$	−48.0000	−1.61441
$885$	0	0
$886$	0	0
$887$	− 36.0000i	− 1.20876i	−0.796696	−	0.604381i	$-0.793421\pi$
$887$	− 36.0000i	− 1.20876i	0.796696	−	0.604381i	$-0.206579\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	0	0
$892$	− 40.0000i	− 1.33930i
$893$	− 12.0000i	− 0.401565i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.00000	0.300167
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	38.0000i	1.26177i	0.775877	+	0.630885i	$0.217308\pi$
$907$	38.0000i	1.26177i	−0.775877	+	0.630885i	$0.782692\pi$
$908$	36.0000i	1.19470i
$909$	0	0
$910$	0	0
$911$	9.00000	0.298183	0.149092	−	0.988823i	$-0.452365\pi$
$911$	9.00000	0.298183	0.149092	+	0.988823i	$0.452365\pi$
$912$	0	0
$913$	− 36.0000i	− 1.19143i
$914$	0	0
$915$	0	0
$916$	20.0000	0.660819
$917$	18.0000i	0.594412i
$918$	0	0
$919$	7.00000	0.230909	0.115454	−	0.993313i	$-0.463168\pi$
$919$	7.00000	0.230909	0.115454	+	0.993313i	$0.463168\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	12.0000i	0.394985i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−39.0000	−1.27955	−0.639774	−	0.768563i	$-0.720972\pi$
$929$	−39.0000	−1.27955	−0.639774	+	0.768563i	$0.720972\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	24.0000i	0.786146i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	14.0000i	0.457360i	0.973502	+	0.228680i	$0.0734410\pi$
$937$	14.0000i	0.457360i	−0.973502	+	0.228680i	$0.926559\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	18.0000i	0.586161i
$944$	12.0000	0.390567
$945$	0	0
$946$	0	0
$947$	− 30.0000i	− 0.974869i	−0.873160	−	0.487435i	$-0.837933\pi$
$947$	− 30.0000i	− 0.974869i	0.873160	−	0.487435i	$-0.162067\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 48.0000i	− 1.55487i	−0.628962	−	0.777436i	$-0.716520\pi$
$953$	− 48.0000i	− 1.55487i	0.628962	−	0.777436i	$-0.283480\pi$
$954$	0	0
$955$	0	0
$956$	−48.0000	−1.55243
$957$	0	0
$958$	0	0
$959$	24.0000	0.775000
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	0	0
$964$	34.0000	1.09507
$965$	0	0
$966$	0	0
$967$	50.0000i	1.60789i	0.594703	+	0.803946i	$0.297270\pi$
$967$	50.0000i	1.60789i	−0.594703	+	0.803946i	$0.702730\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.00000	0.0962746	0.0481373	−	0.998841i	$-0.484672\pi$
$971$	3.00000	0.0962746	0.0481373	+	0.998841i	$0.484672\pi$
$972$	0	0
$973$	14.0000i	0.448819i
$974$	0	0
$975$	0	0
$976$	−40.0000	−1.28037
$977$	48.0000i	1.53566i	0.640656	+	0.767828i	$0.278662\pi$
$977$	48.0000i	1.53566i	−0.640656	+	0.767828i	$0.721338\pi$
$978$	0	0
$979$	45.0000	1.43821
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 6.00000i	− 0.191370i	−0.995412	−	0.0956851i	$-0.969496\pi$
$983$	− 6.00000i	− 0.191370i	0.995412	−	0.0956851i	$-0.0305042\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	8.00000i	0.254514i
$989$	24.0000	0.763156
$990$	0	0
$991$	47.0000	1.49300	0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000	1.49300	0.746502	+	0.665383i	$0.231732\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 10.0000i	− 0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	− 10.0000i	− 0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$	0	0
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+2.00000 q^{4} +2.00000i q^{7} +3.00000 q^{11} +4.00000i q^{13} +4.00000 q^{16} +6.00000i q^{17} +1.00000 q^{19} -6.00000i q^{23} +4.00000i q^{28} -9.00000 q^{29} -1.00000 q^{31} +8.00000i q^{37} -3.00000 q^{41} +4.00000i q^{43} +6.00000 q^{44} -12.0000i q^{47} +3.00000 q^{49} +8.00000i q^{52} +6.00000i q^{53} +3.00000 q^{59} -10.0000 q^{61} +8.00000 q^{64} +14.0000i q^{67} +12.0000i q^{68} +3.00000 q^{71} -2.00000i q^{73} +2.00000 q^{76} +6.00000i q^{77} +16.0000 q^{79} -12.0000i q^{83} +15.0000 q^{89} -8.00000 q^{91} -12.0000i q^{92} -4.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{4} + 6 q^{11} + 8 q^{16} + 2 q^{19} - 18 q^{29} - 2 q^{31} - 6 q^{41} + 12 q^{44} + 6 q^{49} + 6 q^{59} - 20 q^{61} + 16 q^{64} + 6 q^{71} + 4 q^{76} + 32 q^{79} + 30 q^{89} - 16 q^{91}+O(q^{100}) \) 2 * q + 4 * q^4 + 6 * q^11 + 8 * q^16 + 2 * q^19 - 18 * q^29 - 2 * q^31 - 6 * q^41 + 12 * q^44 + 6 * q^49 + 6 * q^59 - 20 * q^61 + 16 * q^64 + 6 * q^71 + 4 * q^76 + 32 * q^79 + 30 * q^89 - 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more