LMFDB - Embedded newform 2025.2.b.g.649.3

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 = [4,0,0,-8,0,0,0,0,0,0,-16] 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:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 405)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.3
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2025.649
Dual form			2025.2.b.g.649.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051i q^{2} +1.46410 q^{4} -4.73205i q^{7} +2.53590i q^{8} -5.73205 q^{11} -1.46410i q^{13} +3.46410 q^{14} +1.07180 q^{16} -2.73205i q^{17} -4.46410 q^{19} -4.19615i q^{22} +3.46410i q^{23} +1.07180 q^{26} -6.92820i q^{28} -3.19615 q^{29} -3.00000 q^{31} +5.85641i q^{32} +2.00000 q^{34} -2.73205i q^{37} -3.26795i q^{38} -7.19615 q^{41} -0.196152i q^{43} -8.39230 q^{44} -2.53590 q^{46} -8.73205i q^{47} -15.3923 q^{49} -2.14359i q^{52} -6.73205i q^{53} +12.0000 q^{56} -2.33975i q^{58} +8.26795 q^{59} +4.00000 q^{61} -2.19615i q^{62} -2.14359 q^{64} +3.46410i q^{67} -4.00000i q^{68} -3.73205 q^{71} +7.66025i q^{73} +2.00000 q^{74} -6.53590 q^{76} +27.1244i q^{77} -15.4641 q^{79} -5.26795i q^{82} -2.19615i q^{83} +0.143594 q^{86} -14.5359i q^{88} -5.19615 q^{89} -6.92820 q^{91} +5.07180i q^{92} +6.39230 q^{94} -9.66025i q^{97} -11.2679i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} - 16 q^{11} + 32 q^{16} - 4 q^{19} + 32 q^{26} + 8 q^{29} - 12 q^{31} + 8 q^{34} - 8 q^{41} + 8 q^{44} - 24 q^{46} - 20 q^{49} + 48 q^{56} + 40 q^{59} + 16 q^{61} - 64 q^{64} - 8 q^{71} + 8 q^{74}+ \cdots - 16 q^{94}+O(q^{100}) $ 4 * q - 8 * q^4 - 16 * q^11 + 32 * q^16 - 4 * q^19 + 32 * q^26 + 8 * q^29 - 12 * q^31 + 8 * q^34 - 8 * q^41 + 8 * q^44 - 24 * q^46 - 20 * q^49 + 48 * q^56 + 40 * q^59 + 16 * q^61 - 64 * q^64 - 8 * q^71 + 8 * q^74 - 40 * q^76 - 48 * q^79 + 56 * q^86 - 16 * 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 $

$2$	0.732051i	0.517638i	0.965926	+	0.258819i	$0.0833333\pi$
$2$	0.732051i	0.517638i	−0.965926	+	0.258819i	$0.916667\pi$
$3$	0	0
$3$	0	0
$4$	1.46410	0.732051
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.73205i	− 1.78855i	−0.447521	−	0.894274i	$-0.647693\pi$
$7$	− 4.73205i	− 1.78855i	0.447521	−	0.894274i	$-0.352307\pi$
$8$	2.53590i	0.896575i
$9$	0	0
$10$	0	0
$11$	−5.73205	−1.72828	−0.864139	−	0.503253i	$-0.832136\pi$
$11$	−5.73205	−1.72828	−0.864139	+	0.503253i	$0.832136\pi$
$12$	0	0
$13$	− 1.46410i	− 0.406069i	−0.979172	−	0.203034i	$-0.934920\pi$
$13$	− 1.46410i	− 0.406069i	0.979172	−	0.203034i	$-0.0650803\pi$
$14$	3.46410	0.925820
$15$	0	0
$16$	1.07180	0.267949
$17$	− 2.73205i	− 0.662620i	−0.943522	−	0.331310i	$-0.892509\pi$
$17$	− 2.73205i	− 0.662620i	0.943522	−	0.331310i	$-0.107491\pi$
$18$	0	0
$19$	−4.46410	−1.02414	−0.512068	−	0.858945i	$-0.671120\pi$
$19$	−4.46410	−1.02414	−0.512068	+	0.858945i	$0.671120\pi$
$20$	0	0
$21$	0	0
$22$	− 4.19615i	− 0.894623i
$23$	3.46410i	0.722315i	0.932505	+	0.361158i	$0.117618\pi$
$23$	3.46410i	0.722315i	−0.932505	+	0.361158i	$0.882382\pi$
$24$	0	0
$25$	0	0
$26$	1.07180	0.210197
$27$	0	0
$28$	− 6.92820i	− 1.30931i
$29$	−3.19615	−0.593511	−0.296755	−	0.954954i	$-0.595905\pi$
$29$	−3.19615	−0.593511	−0.296755	+	0.954954i	$0.595905\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	5.85641i	1.03528i
$33$	0	0
$34$	2.00000	0.342997
$35$	0	0
$36$	0	0
$37$	− 2.73205i	− 0.449146i	−0.974457	−	0.224573i	$-0.927901\pi$
$37$	− 2.73205i	− 0.449146i	0.974457	−	0.224573i	$-0.0720988\pi$
$38$	− 3.26795i	− 0.530131i
$39$	0	0
$40$	0	0
$41$	−7.19615	−1.12385	−0.561925	−	0.827188i	$-0.689939\pi$
$41$	−7.19615	−1.12385	−0.561925	+	0.827188i	$0.689939\pi$
$42$	0	0
$43$	− 0.196152i	− 0.0299130i	−0.999888	−	0.0149565i	$-0.995239\pi$
$43$	− 0.196152i	− 0.0299130i	0.999888	−	0.0149565i	$-0.00476097\pi$
$44$	−8.39230	−1.26519
$45$	0	0
$46$	−2.53590	−0.373898
$47$	− 8.73205i	− 1.27370i	−0.770988	−	0.636850i	$-0.780237\pi$
$47$	− 8.73205i	− 1.27370i	0.770988	−	0.636850i	$-0.219763\pi$
$48$	0	0
$49$	−15.3923	−2.19890
$50$	0	0
$51$	0	0
$52$	− 2.14359i	− 0.297263i
$53$	− 6.73205i	− 0.924718i	−0.886693	−	0.462359i	$-0.847003\pi$
$53$	− 6.73205i	− 0.924718i	0.886693	−	0.462359i	$-0.152997\pi$
$54$	0	0
$55$	0	0
$56$	12.0000	1.60357
$57$	0	0
$58$	− 2.33975i	− 0.307224i
$59$	8.26795	1.07640	0.538198	−	0.842819i	$-0.319105\pi$
$59$	8.26795	1.07640	0.538198	+	0.842819i	$0.319105\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	− 2.19615i	− 0.278912i
$63$	0	0
$64$	−2.14359	−0.267949
$65$	0	0
$66$	0	0
$67$	3.46410i	0.423207i	0.977356	+	0.211604i	$0.0678686\pi$
$67$	3.46410i	0.423207i	−0.977356	+	0.211604i	$0.932131\pi$
$68$	− 4.00000i	− 0.485071i
$69$	0	0
$70$	0	0
$71$	−3.73205	−0.442913	−0.221456	−	0.975170i	$-0.571081\pi$
$71$	−3.73205	−0.442913	−0.221456	+	0.975170i	$0.571081\pi$
$72$	0	0
$73$	7.66025i	0.896565i	0.893892	+	0.448282i	$0.147964\pi$
$73$	7.66025i	0.896565i	−0.893892	+	0.448282i	$0.852036\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−6.53590	−0.749719
$77$	27.1244i	3.09111i
$78$	0	0
$79$	−15.4641	−1.73985	−0.869924	−	0.493186i	$-0.835832\pi$
$79$	−15.4641	−1.73985	−0.869924	+	0.493186i	$0.835832\pi$
$80$	0	0
$81$	0	0
$82$	− 5.26795i	− 0.581748i
$83$	− 2.19615i	− 0.241059i	−0.992710	−	0.120530i	$-0.961541\pi$
$83$	− 2.19615i	− 0.241059i	0.992710	−	0.120530i	$-0.0384592\pi$
$84$	0	0
$85$	0	0
$86$	0.143594	0.0154841
$87$	0	0
$88$	− 14.5359i	− 1.54953i
$89$	−5.19615	−0.550791	−0.275396	−	0.961331i	$-0.588809\pi$
$89$	−5.19615	−0.550791	−0.275396	+	0.961331i	$0.588809\pi$
$90$	0	0
$91$	−6.92820	−0.726273
$92$	5.07180i	0.528771i
$93$	0	0
$94$	6.39230	0.659316
$95$	0	0
$96$	0	0
$97$	− 9.66025i	− 0.980850i	−0.871483	−	0.490425i	$-0.836842\pi$
$97$	− 9.66025i	− 0.980850i	0.871483	−	0.490425i	$-0.163158\pi$
$98$	− 11.2679i	− 1.13823i
$99$	0	0
$100$	0	0
$101$	−2.66025	−0.264705	−0.132353	−	0.991203i	$-0.542253\pi$
$101$	−2.66025	−0.264705	−0.132353	+	0.991203i	$0.542253\pi$
$102$	0	0
$103$	0.535898i	0.0528036i	0.999651	+	0.0264018i	$0.00840494\pi$
$103$	0.535898i	0.0528036i	−0.999651	+	0.0264018i	$0.991595\pi$
$104$	3.71281	0.364071
$105$	0	0
$106$	4.92820	0.478669
$107$	− 8.53590i	− 0.825196i	−0.910913	−	0.412598i	$-0.864621\pi$
$107$	− 8.53590i	− 0.825196i	0.910913	−	0.412598i	$-0.135379\pi$
$108$	0	0
$109$	6.07180	0.581573	0.290786	−	0.956788i	$-0.406083\pi$
$109$	6.07180	0.581573	0.290786	+	0.956788i	$0.406083\pi$
$110$	0	0
$111$	0	0
$112$	− 5.07180i	− 0.479240i
$113$	19.1244i	1.79907i	0.436851	+	0.899534i	$0.356094\pi$
$113$	19.1244i	1.79907i	−0.436851	+	0.899534i	$0.643906\pi$
$114$	0	0
$115$	0	0
$116$	−4.67949	−0.434480
$117$	0	0
$118$	6.05256i	0.557183i
$119$	−12.9282	−1.18513
$120$	0	0
$121$	21.8564	1.98695
$122$	2.92820i	0.265107i
$123$	0	0
$124$	−4.39230	−0.394441
$125$	0	0
$126$	0	0
$127$	− 14.5885i	− 1.29452i	−0.762271	−	0.647258i	$-0.775916\pi$
$127$	− 14.5885i	− 1.29452i	0.762271	−	0.647258i	$-0.224084\pi$
$128$	10.1436i	0.896575i
$129$	0	0
$130$	0	0
$131$	15.5885	1.36197	0.680985	−	0.732297i	$-0.261552\pi$
$131$	15.5885	1.36197	0.680985	+	0.732297i	$0.261552\pi$
$132$	0	0
$133$	21.1244i	1.83171i
$134$	−2.53590	−0.219068
$135$	0	0
$136$	6.92820	0.594089
$137$	2.53590i	0.216656i	0.994115	+	0.108328i	$0.0345497\pi$
$137$	2.53590i	0.216656i	−0.994115	+	0.108328i	$0.965450\pi$
$138$	0	0
$139$	−0.607695	−0.0515440	−0.0257720	−	0.999668i	$-0.508204\pi$
$139$	−0.607695	−0.0515440	−0.0257720	+	0.999668i	$0.508204\pi$
$140$	0	0
$141$	0	0
$142$	− 2.73205i	− 0.229269i
$143$	8.39230i	0.701800i
$144$	0	0
$145$	0	0
$146$	−5.60770	−0.464096
$147$	0	0
$148$	− 4.00000i	− 0.328798i
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	0	0
$151$	5.39230	0.438820	0.219410	−	0.975633i	$-0.429587\pi$
$151$	5.39230	0.438820	0.219410	+	0.975633i	$0.429587\pi$
$152$	− 11.3205i	− 0.918214i
$153$	0	0
$154$	−19.8564	−1.60007
$155$	0	0
$156$	0	0
$157$	− 19.1244i	− 1.52629i	−0.646227	−	0.763145i	$-0.723654\pi$
$157$	− 19.1244i	− 1.52629i	0.646227	−	0.763145i	$-0.276346\pi$
$158$	− 11.3205i	− 0.900611i
$159$	0	0
$160$	0	0
$161$	16.3923	1.29189
$162$	0	0
$163$	− 12.7321i	− 0.997251i	−0.866817	−	0.498626i	$-0.833838\pi$
$163$	− 12.7321i	− 0.997251i	0.866817	−	0.498626i	$-0.166162\pi$
$164$	−10.5359	−0.822715
$165$	0	0
$166$	1.60770	0.124781
$167$	− 17.6603i	− 1.36659i	−0.730142	−	0.683296i	$-0.760546\pi$
$167$	− 17.6603i	− 1.36659i	0.730142	−	0.683296i	$-0.239454\pi$
$168$	0	0
$169$	10.8564	0.835108
$170$	0	0
$171$	0	0
$172$	− 0.287187i	− 0.0218978i
$173$	− 8.53590i	− 0.648972i	−0.945890	−	0.324486i	$-0.894809\pi$
$173$	− 8.53590i	− 0.648972i	0.945890	−	0.324486i	$-0.105191\pi$
$174$	0	0
$175$	0	0
$176$	−6.14359	−0.463091
$177$	0	0
$178$	− 3.80385i	− 0.285110i
$179$	−8.12436	−0.607243	−0.303621	−	0.952793i	$-0.598196\pi$
$179$	−8.12436	−0.607243	−0.303621	+	0.952793i	$0.598196\pi$
$180$	0	0
$181$	26.4641	1.96706	0.983531	−	0.180742i	$-0.0578498\pi$
$181$	26.4641	1.96706	0.983531	+	0.180742i	$0.0578498\pi$
$182$	− 5.07180i	− 0.375947i
$183$	0	0
$184$	−8.78461	−0.647610
$185$	0	0
$186$	0	0
$187$	15.6603i	1.14519i
$188$	− 12.7846i	− 0.932413i
$189$	0	0
$190$	0	0
$191$	−8.12436	−0.587858	−0.293929	−	0.955827i	$-0.594963\pi$
$191$	−8.12436	−0.587858	−0.293929	+	0.955827i	$0.594963\pi$
$192$	0	0
$193$	5.26795i	0.379195i	0.981862	+	0.189598i	$0.0607184\pi$
$193$	5.26795i	0.379195i	−0.981862	+	0.189598i	$0.939282\pi$
$194$	7.07180	0.507725
$195$	0	0
$196$	−22.5359	−1.60971
$197$	− 13.8564i	− 0.987228i	−0.869681	−	0.493614i	$-0.835676\pi$
$197$	− 13.8564i	− 0.987228i	0.869681	−	0.493614i	$-0.164324\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	0	0
$201$	0	0
$202$	− 1.94744i	− 0.137021i
$203$	15.1244i	1.06152i
$204$	0	0
$205$	0	0
$206$	−0.392305	−0.0273332
$207$	0	0
$208$	− 1.56922i	− 0.108806i
$209$	25.5885	1.76999
$210$	0	0
$211$	−8.85641	−0.609700	−0.304850	−	0.952400i	$-0.598606\pi$
$211$	−8.85641	−0.609700	−0.304850	+	0.952400i	$0.598606\pi$
$212$	− 9.85641i	− 0.676941i
$213$	0	0
$214$	6.24871	0.427153
$215$	0	0
$216$	0	0
$217$	14.1962i	0.963698i
$218$	4.44486i	0.301044i
$219$	0	0
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	− 16.7846i	− 1.12398i	−0.827144	−	0.561990i	$-0.810036\pi$
$223$	− 16.7846i	− 1.12398i	0.827144	−	0.561990i	$-0.189964\pi$
$224$	27.7128	1.85164
$225$	0	0
$226$	−14.0000	−0.931266
$227$	− 18.0526i	− 1.19819i	−0.800678	−	0.599095i	$-0.795527\pi$
$227$	− 18.0526i	− 1.19819i	0.800678	−	0.599095i	$-0.204473\pi$
$228$	0	0
$229$	12.0000	0.792982	0.396491	−	0.918039i	$-0.370228\pi$
$229$	12.0000	0.792982	0.396491	+	0.918039i	$0.370228\pi$
$230$	0	0
$231$	0	0
$232$	− 8.10512i	− 0.532127i
$233$	28.0526i	1.83778i	0.394509	+	0.918892i	$0.370915\pi$
$233$	28.0526i	1.83778i	−0.394509	+	0.918892i	$0.629085\pi$
$234$	0	0
$235$	0	0
$236$	12.1051	0.787976
$237$	0	0
$238$	− 9.46410i	− 0.613467i
$239$	0.535898	0.0346644	0.0173322	−	0.999850i	$-0.494483\pi$
$239$	0.535898	0.0346644	0.0173322	+	0.999850i	$0.494483\pi$
$240$	0	0
$241$	−16.3205	−1.05130	−0.525648	−	0.850702i	$-0.676177\pi$
$241$	−16.3205	−1.05130	−0.525648	+	0.850702i	$0.676177\pi$
$242$	16.0000i	1.02852i
$243$	0	0
$244$	5.85641	0.374918
$245$	0	0
$246$	0	0
$247$	6.53590i	0.415869i
$248$	− 7.60770i	− 0.483089i
$249$	0	0
$250$	0	0
$251$	−10.3923	−0.655956	−0.327978	−	0.944685i	$-0.606367\pi$
$251$	−10.3923	−0.655956	−0.327978	+	0.944685i	$0.606367\pi$
$252$	0	0
$253$	− 19.8564i	− 1.24836i
$254$	10.6795	0.670091
$255$	0	0
$256$	−11.7128	−0.732051
$257$	10.3923i	0.648254i	0.946014	+	0.324127i	$0.105071\pi$
$257$	10.3923i	0.648254i	−0.946014	+	0.324127i	$0.894929\pi$
$258$	0	0
$259$	−12.9282	−0.803319
$260$	0	0
$261$	0	0
$262$	11.4115i	0.705007i
$263$	− 21.3205i	− 1.31468i	−0.753595	−	0.657339i	$-0.771682\pi$
$263$	− 21.3205i	− 1.31468i	0.753595	−	0.657339i	$-0.228318\pi$
$264$	0	0
$265$	0	0
$266$	−15.4641	−0.948165
$267$	0	0
$268$	5.07180i	0.309809i
$269$	−10.6603	−0.649967	−0.324984	−	0.945720i	$-0.605359\pi$
$269$	−10.6603	−0.649967	−0.324984	+	0.945720i	$0.605359\pi$
$270$	0	0
$271$	−2.92820	−0.177876	−0.0889378	−	0.996037i	$-0.528347\pi$
$271$	−2.92820	−0.177876	−0.0889378	+	0.996037i	$0.528347\pi$
$272$	− 2.92820i	− 0.177548i
$273$	0	0
$274$	−1.85641	−0.112150
$275$	0	0
$276$	0	0
$277$	3.80385i	0.228551i	0.993449	+	0.114276i	$0.0364547\pi$
$277$	3.80385i	0.228551i	−0.993449	+	0.114276i	$0.963545\pi$
$278$	− 0.444864i	− 0.0266812i
$279$	0	0
$280$	0	0
$281$	15.4641	0.922511	0.461255	−	0.887267i	$-0.347399\pi$
$281$	15.4641	0.922511	0.461255	+	0.887267i	$0.347399\pi$
$282$	0	0
$283$	29.3205i	1.74292i	0.490464	+	0.871462i	$0.336827\pi$
$283$	29.3205i	1.74292i	−0.490464	+	0.871462i	$0.663173\pi$
$284$	−5.46410	−0.324235
$285$	0	0
$286$	−6.14359	−0.363278
$287$	34.0526i	2.01006i
$288$	0	0
$289$	9.53590	0.560935
$290$	0	0
$291$	0	0
$292$	11.2154i	0.656331i
$293$	− 28.7321i	− 1.67854i	−0.543712	−	0.839272i	$-0.682981\pi$
$293$	− 28.7321i	− 1.67854i	0.543712	−	0.839272i	$-0.317019\pi$
$294$	0	0
$295$	0	0
$296$	6.92820	0.402694
$297$	0	0
$298$	5.85641i	0.339253i
$299$	5.07180	0.293310
$300$	0	0
$301$	−0.928203	−0.0535007
$302$	3.94744i	0.227150i
$303$	0	0
$304$	−4.78461	−0.274416
$305$	0	0
$306$	0	0
$307$	14.0526i	0.802022i	0.916073	+	0.401011i	$0.131341\pi$
$307$	14.0526i	0.802022i	−0.916073	+	0.401011i	$0.868659\pi$
$308$	39.7128i	2.26285i
$309$	0	0
$310$	0	0
$311$	19.7321	1.11890	0.559451	−	0.828863i	$-0.311012\pi$
$311$	19.7321	1.11890	0.559451	+	0.828863i	$0.311012\pi$
$312$	0	0
$313$	9.07180i	0.512768i	0.966575	+	0.256384i	$0.0825312\pi$
$313$	9.07180i	0.512768i	−0.966575	+	0.256384i	$0.917469\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	−22.6410	−1.27366
$317$	6.19615i	0.348011i	0.984745	+	0.174005i	$0.0556710\pi$
$317$	6.19615i	0.348011i	−0.984745	+	0.174005i	$0.944329\pi$
$318$	0	0
$319$	18.3205	1.02575
$320$	0	0
$321$	0	0
$322$	12.0000i	0.668734i
$323$	12.1962i	0.678612i
$324$	0	0
$325$	0	0
$326$	9.32051	0.516215
$327$	0	0
$328$	− 18.2487i	− 1.00762i
$329$	−41.3205	−2.27807
$330$	0	0
$331$	−0.464102	−0.0255093	−0.0127547	−	0.999919i	$-0.504060\pi$
$331$	−0.464102	−0.0255093	−0.0127547	+	0.999919i	$0.504060\pi$
$332$	− 3.21539i	− 0.176467i
$333$	0	0
$334$	12.9282	0.707400
$335$	0	0
$336$	0	0
$337$	27.3205i	1.48824i	0.668044	+	0.744121i	$0.267132\pi$
$337$	27.3205i	1.48824i	−0.668044	+	0.744121i	$0.732868\pi$
$338$	7.94744i	0.432284i
$339$	0	0
$340$	0	0
$341$	17.1962	0.931224
$342$	0	0
$343$	39.7128i	2.14429i
$344$	0.497423	0.0268192
$345$	0	0
$346$	6.24871	0.335933
$347$	− 28.5885i	− 1.53471i	−0.641223	−	0.767354i	$-0.721573\pi$
$347$	− 28.5885i	− 1.53471i	0.641223	−	0.767354i	$-0.278427\pi$
$348$	0	0
$349$	18.8564	1.00936	0.504680	−	0.863306i	$-0.331610\pi$
$349$	18.8564	1.00936	0.504680	+	0.863306i	$0.331610\pi$
$350$	0	0
$351$	0	0
$352$	− 33.5692i	− 1.78925i
$353$	− 25.5167i	− 1.35811i	−0.734085	−	0.679057i	$-0.762389\pi$
$353$	− 25.5167i	− 1.35811i	0.734085	−	0.679057i	$-0.237611\pi$
$354$	0	0
$355$	0	0
$356$	−7.60770	−0.403207
$357$	0	0
$358$	− 5.94744i	− 0.314332i
$359$	6.12436	0.323231	0.161616	−	0.986854i	$-0.448330\pi$
$359$	6.12436	0.323231	0.161616	+	0.986854i	$0.448330\pi$
$360$	0	0
$361$	0.928203	0.0488528
$362$	19.3731i	1.01823i
$363$	0	0
$364$	−10.1436	−0.531669
$365$	0	0
$366$	0	0
$367$	31.1769i	1.62742i	0.581270	+	0.813711i	$0.302556\pi$
$367$	31.1769i	1.62742i	−0.581270	+	0.813711i	$0.697444\pi$
$368$	3.71281i	0.193544i
$369$	0	0
$370$	0	0
$371$	−31.8564	−1.65390
$372$	0	0
$373$	− 20.0526i	− 1.03828i	−0.854689	−	0.519141i	$-0.826252\pi$
$373$	− 20.0526i	− 1.03828i	0.854689	−	0.519141i	$-0.173748\pi$
$374$	−11.4641	−0.592795
$375$	0	0
$376$	22.1436	1.14197
$377$	4.67949i	0.241006i
$378$	0	0
$379$	−2.39230	−0.122884	−0.0614422	−	0.998111i	$-0.519570\pi$
$379$	−2.39230	−0.122884	−0.0614422	+	0.998111i	$0.519570\pi$
$380$	0	0
$381$	0	0
$382$	− 5.94744i	− 0.304298i
$383$	− 2.53590i	− 0.129578i	−0.997899	−	0.0647892i	$-0.979363\pi$
$383$	− 2.53590i	− 0.129578i	0.997899	−	0.0647892i	$-0.0206375\pi$
$384$	0	0
$385$	0	0
$386$	−3.85641	−0.196286
$387$	0	0
$388$	− 14.1436i	− 0.718032i
$389$	−27.4641	−1.39249	−0.696243	−	0.717807i	$-0.745146\pi$
$389$	−27.4641	−1.39249	−0.696243	+	0.717807i	$0.745146\pi$
$390$	0	0
$391$	9.46410	0.478620
$392$	− 39.0333i	− 1.97148i
$393$	0	0
$394$	10.1436	0.511027
$395$	0	0
$396$	0	0
$397$	14.3923i	0.722329i	0.932502	+	0.361165i	$0.117621\pi$
$397$	14.3923i	0.722329i	−0.932502	+	0.361165i	$0.882379\pi$
$398$	1.46410i	0.0733888i
$399$	0	0
$400$	0	0
$401$	24.9282	1.24486	0.622428	−	0.782677i	$-0.286147\pi$
$401$	24.9282	1.24486	0.622428	+	0.782677i	$0.286147\pi$
$402$	0	0
$403$	4.39230i	0.218796i
$404$	−3.89488	−0.193778
$405$	0	0
$406$	−11.0718	−0.549484
$407$	15.6603i	0.776250i
$408$	0	0
$409$	17.8564	0.882942	0.441471	−	0.897275i	$-0.354457\pi$
$409$	17.8564	0.882942	0.441471	+	0.897275i	$0.354457\pi$
$410$	0	0
$411$	0	0
$412$	0.784610i	0.0386549i
$413$	− 39.1244i	− 1.92518i
$414$	0	0
$415$	0	0
$416$	8.57437	0.420393
$417$	0	0
$418$	18.7321i	0.916215i
$419$	−20.3923	−0.996229	−0.498115	−	0.867111i	$-0.665974\pi$
$419$	−20.3923	−0.996229	−0.498115	+	0.867111i	$0.665974\pi$
$420$	0	0
$421$	33.7846	1.64656	0.823281	−	0.567635i	$-0.192141\pi$
$421$	33.7846	1.64656	0.823281	+	0.567635i	$0.192141\pi$
$422$	− 6.48334i	− 0.315604i
$423$	0	0
$424$	17.0718	0.829080
$425$	0	0
$426$	0	0
$427$	− 18.9282i	− 0.916000i
$428$	− 12.4974i	− 0.604086i
$429$	0	0
$430$	0	0
$431$	−21.3397	−1.02790	−0.513950	−	0.857820i	$-0.671818\pi$
$431$	−21.3397	−1.02790	−0.513950	+	0.857820i	$0.671818\pi$
$432$	0	0
$433$	35.4641i	1.70430i	0.523301	+	0.852148i	$0.324700\pi$
$433$	35.4641i	1.70430i	−0.523301	+	0.852148i	$0.675300\pi$
$434$	−10.3923	−0.498847
$435$	0	0
$436$	8.88973	0.425741
$437$	− 15.4641i	− 0.739748i
$438$	0	0
$439$	−5.39230	−0.257361	−0.128680	−	0.991686i	$-0.541074\pi$
$439$	−5.39230	−0.257361	−0.128680	+	0.991686i	$0.541074\pi$
$440$	0	0
$441$	0	0
$442$	− 2.92820i	− 0.139280i
$443$	0.339746i	0.0161418i	0.999967	+	0.00807091i	$0.00256908\pi$
$443$	0.339746i	0.0161418i	−0.999967	+	0.00807091i	$0.997431\pi$
$444$	0	0
$445$	0	0
$446$	12.2872	0.581815
$447$	0	0
$448$	10.1436i	0.479240i
$449$	−8.12436	−0.383412	−0.191706	−	0.981452i	$-0.561402\pi$
$449$	−8.12436	−0.383412	−0.191706	+	0.981452i	$0.561402\pi$
$450$	0	0
$451$	41.2487	1.94233
$452$	28.0000i	1.31701i
$453$	0	0
$454$	13.2154	0.620229
$455$	0	0
$456$	0	0
$457$	2.73205i	0.127800i	0.997956	+	0.0639000i	$0.0203539\pi$
$457$	2.73205i	0.127800i	−0.997956	+	0.0639000i	$0.979646\pi$
$458$	8.78461i	0.410478i
$459$	0	0
$460$	0	0
$461$	−37.0526	−1.72571	−0.862855	−	0.505452i	$-0.831326\pi$
$461$	−37.0526	−1.72571	−0.862855	+	0.505452i	$0.831326\pi$
$462$	0	0
$463$	10.3923i	0.482971i	0.970404	+	0.241486i	$0.0776347\pi$
$463$	10.3923i	0.482971i	−0.970404	+	0.241486i	$0.922365\pi$
$464$	−3.42563	−0.159031
$465$	0	0
$466$	−20.5359	−0.951307
$467$	37.3731i	1.72942i	0.502272	+	0.864710i	$0.332498\pi$
$467$	37.3731i	1.72942i	−0.502272	+	0.864710i	$0.667502\pi$
$468$	0	0
$469$	16.3923	0.756926
$470$	0	0
$471$	0	0
$472$	20.9667i	0.965070i
$473$	1.12436i	0.0516979i
$474$	0	0
$475$	0	0
$476$	−18.9282	−0.867573
$477$	0	0
$478$	0.392305i	0.0179436i
$479$	11.8756	0.542612	0.271306	−	0.962493i	$-0.412544\pi$
$479$	11.8756	0.542612	0.271306	+	0.962493i	$0.412544\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	− 11.9474i	− 0.544191i
$483$	0	0
$484$	32.0000	1.45455
$485$	0	0
$486$	0	0
$487$	24.3923i	1.10532i	0.833407	+	0.552660i	$0.186387\pi$
$487$	24.3923i	1.10532i	−0.833407	+	0.552660i	$0.813613\pi$
$488$	10.1436i	0.459179i
$489$	0	0
$490$	0	0
$491$	−13.8756	−0.626199	−0.313100	−	0.949720i	$-0.601367\pi$
$491$	−13.8756	−0.626199	−0.313100	+	0.949720i	$0.601367\pi$
$492$	0	0
$493$	8.73205i	0.393272i
$494$	−4.78461	−0.215270
$495$	0	0
$496$	−3.21539	−0.144375
$497$	17.6603i	0.792171i
$498$	0	0
$499$	−24.3205	−1.08874	−0.544368	−	0.838847i	$-0.683230\pi$
$499$	−24.3205	−1.08874	−0.544368	+	0.838847i	$0.683230\pi$
$500$	0	0
$501$	0	0
$502$	− 7.60770i	− 0.339548i
$503$	7.32051i	0.326405i	0.986593	+	0.163203i	$0.0521824\pi$
$503$	7.32051i	0.326405i	−0.986593	+	0.163203i	$0.947818\pi$
$504$	0	0
$505$	0	0
$506$	14.5359	0.646200
$507$	0	0
$508$	− 21.3590i	− 0.947652i
$509$	6.78461	0.300723	0.150361	−	0.988631i	$-0.451956\pi$
$509$	6.78461	0.300723	0.150361	+	0.988631i	$0.451956\pi$
$510$	0	0
$511$	36.2487	1.60355
$512$	11.7128i	0.517638i
$513$	0	0
$514$	−7.60770	−0.335561
$515$	0	0
$516$	0	0
$517$	50.0526i	2.20131i
$518$	− 9.46410i	− 0.415829i
$519$	0	0
$520$	0	0
$521$	19.4641	0.852738	0.426369	−	0.904549i	$-0.359793\pi$
$521$	19.4641	0.852738	0.426369	+	0.904549i	$0.359793\pi$
$522$	0	0
$523$	− 22.2487i	− 0.972868i	−0.873717	−	0.486434i	$-0.838297\pi$
$523$	− 22.2487i	− 0.972868i	0.873717	−	0.486434i	$-0.161703\pi$
$524$	22.8231	0.997031
$525$	0	0
$526$	15.6077	0.680528
$527$	8.19615i	0.357030i
$528$	0	0
$529$	11.0000	0.478261
$530$	0	0
$531$	0	0
$532$	30.9282i	1.34091i
$533$	10.5359i	0.456360i
$534$	0	0
$535$	0	0
$536$	−8.78461	−0.379437
$537$	0	0
$538$	− 7.80385i	− 0.336448i
$539$	88.2295	3.80031
$540$	0	0
$541$	−24.4641	−1.05179	−0.525897	−	0.850548i	$-0.676270\pi$
$541$	−24.4641	−1.05179	−0.525897	+	0.850548i	$0.676270\pi$
$542$	− 2.14359i	− 0.0920752i
$543$	0	0
$544$	16.0000	0.685994
$545$	0	0
$546$	0	0
$547$	− 33.8564i	− 1.44760i	−0.690012	−	0.723798i	$-0.742395\pi$
$547$	− 33.8564i	− 1.44760i	0.690012	−	0.723798i	$-0.257605\pi$
$548$	3.71281i	0.158604i
$549$	0	0
$550$	0	0
$551$	14.2679	0.607835
$552$	0	0
$553$	73.1769i	3.11180i
$554$	−2.78461	−0.118307
$555$	0	0
$556$	−0.889727	−0.0377328
$557$	− 2.53590i	− 0.107449i	−0.998556	−	0.0537247i	$-0.982891\pi$
$557$	− 2.53590i	− 0.107449i	0.998556	−	0.0537247i	$-0.0171094\pi$
$558$	0	0
$559$	−0.287187	−0.0121467
$560$	0	0
$561$	0	0
$562$	11.3205i	0.477527i
$563$	16.7321i	0.705172i	0.935779	+	0.352586i	$0.114698\pi$
$563$	16.7321i	0.705172i	−0.935779	+	0.352586i	$0.885302\pi$
$564$	0	0
$565$	0	0
$566$	−21.4641	−0.902203
$567$	0	0
$568$	− 9.46410i	− 0.397105i
$569$	32.9090	1.37962	0.689808	−	0.723993i	$-0.257695\pi$
$569$	32.9090	1.37962	0.689808	+	0.723993i	$0.257695\pi$
$570$	0	0
$571$	−39.7846	−1.66493	−0.832467	−	0.554075i	$-0.813072\pi$
$571$	−39.7846	−1.66493	−0.832467	+	0.554075i	$0.813072\pi$
$572$	12.2872i	0.513753i
$573$	0	0
$574$	−24.9282	−1.04048
$575$	0	0
$576$	0	0
$577$	15.2679i	0.635613i	0.948156	+	0.317807i	$0.102946\pi$
$577$	15.2679i	0.635613i	−0.948156	+	0.317807i	$0.897054\pi$
$578$	6.98076i	0.290361i
$579$	0	0
$580$	0	0
$581$	−10.3923	−0.431145
$582$	0	0
$583$	38.5885i	1.59817i
$584$	−19.4256	−0.803838
$585$	0	0
$586$	21.0333	0.868878
$587$	− 11.6603i	− 0.481270i	−0.970616	−	0.240635i	$-0.922644\pi$
$587$	− 11.6603i	− 0.481270i	0.970616	−	0.240635i	$-0.0773557\pi$
$588$	0	0
$589$	13.3923	0.551820
$590$	0	0
$591$	0	0
$592$	− 2.92820i	− 0.120348i
$593$	0.143594i	0.00589668i	0.999996	+	0.00294834i	$0.000938487\pi$
$593$	0.143594i	0.00589668i	−0.999996	+	0.00294834i	$0.999062\pi$
$594$	0	0
$595$	0	0
$596$	11.7128	0.479776
$597$	0	0
$598$	3.71281i	0.151828i
$599$	−27.1962	−1.11120	−0.555602	−	0.831448i	$-0.687512\pi$
$599$	−27.1962	−1.11120	−0.555602	+	0.831448i	$0.687512\pi$
$600$	0	0
$601$	−31.2487	−1.27466	−0.637331	−	0.770590i	$-0.719961\pi$
$601$	−31.2487	−1.27466	−0.637331	+	0.770590i	$0.719961\pi$
$602$	− 0.679492i	− 0.0276940i
$603$	0	0
$604$	7.89488	0.321238
$605$	0	0
$606$	0	0
$607$	− 16.1962i	− 0.657382i	−0.944438	−	0.328691i	$-0.893393\pi$
$607$	− 16.1962i	− 0.657382i	0.944438	−	0.328691i	$-0.106607\pi$
$608$	− 26.1436i	− 1.06026i
$609$	0	0
$610$	0	0
$611$	−12.7846	−0.517210
$612$	0	0
$613$	− 1.46410i	− 0.0591345i	−0.999563	−	0.0295673i	$-0.990587\pi$
$613$	− 1.46410i	− 0.0591345i	0.999563	−	0.0295673i	$-0.00941292\pi$
$614$	−10.2872	−0.415157
$615$	0	0
$616$	−68.7846	−2.77141
$617$	− 6.92820i	− 0.278919i	−0.990228	−	0.139459i	$-0.955464\pi$
$617$	− 6.92820i	− 0.278919i	0.990228	−	0.139459i	$-0.0445365\pi$
$618$	0	0
$619$	11.8564	0.476549	0.238275	−	0.971198i	$-0.423418\pi$
$619$	11.8564	0.476549	0.238275	+	0.971198i	$0.423418\pi$
$620$	0	0
$621$	0	0
$622$	14.4449i	0.579186i
$623$	24.5885i	0.985116i
$624$	0	0
$625$	0	0
$626$	−6.64102	−0.265428
$627$	0	0
$628$	− 28.0000i	− 1.11732i
$629$	−7.46410	−0.297613
$630$	0	0
$631$	−32.7128	−1.30228	−0.651138	−	0.758959i	$-0.725708\pi$
$631$	−32.7128	−1.30228	−0.651138	+	0.758959i	$0.725708\pi$
$632$	− 39.2154i	− 1.55990i
$633$	0	0
$634$	−4.53590	−0.180144
$635$	0	0
$636$	0	0
$637$	22.5359i	0.892905i
$638$	13.4115i	0.530968i
$639$	0	0
$640$	0	0
$641$	9.33975	0.368898	0.184449	−	0.982842i	$-0.440950\pi$
$641$	9.33975	0.368898	0.184449	+	0.982842i	$0.440950\pi$
$642$	0	0
$643$	− 41.6603i	− 1.64292i	−0.570266	−	0.821460i	$-0.693160\pi$
$643$	− 41.6603i	− 1.64292i	0.570266	−	0.821460i	$-0.306840\pi$
$644$	24.0000	0.945732
$645$	0	0
$646$	−8.92820	−0.351275
$647$	11.4641i	0.450700i	0.974278	+	0.225350i	$0.0723526\pi$
$647$	11.4641i	0.450700i	−0.974278	+	0.225350i	$0.927647\pi$
$648$	0	0
$649$	−47.3923	−1.86031
$650$	0	0
$651$	0	0
$652$	− 18.6410i	− 0.730039i
$653$	17.4641i	0.683423i	0.939805	+	0.341712i	$0.111007\pi$
$653$	17.4641i	0.683423i	−0.939805	+	0.341712i	$0.888993\pi$
$654$	0	0
$655$	0	0
$656$	−7.71281	−0.301135
$657$	0	0
$658$	− 30.2487i	− 1.17922i
$659$	1.46410	0.0570333	0.0285167	−	0.999593i	$-0.490922\pi$
$659$	1.46410	0.0570333	0.0285167	+	0.999593i	$0.490922\pi$
$660$	0	0
$661$	18.3205	0.712585	0.356293	−	0.934374i	$-0.384041\pi$
$661$	18.3205	0.712585	0.356293	+	0.934374i	$0.384041\pi$
$662$	− 0.339746i	− 0.0132046i
$663$	0	0
$664$	5.56922	0.216128
$665$	0	0
$666$	0	0
$667$	− 11.0718i	− 0.428702i
$668$	− 25.8564i	− 1.00041i
$669$	0	0
$670$	0	0
$671$	−22.9282	−0.885133
$672$	0	0
$673$	10.3923i	0.400594i	0.979735	+	0.200297i	$0.0641907\pi$
$673$	10.3923i	0.400594i	−0.979735	+	0.200297i	$0.935809\pi$
$674$	−20.0000	−0.770371
$675$	0	0
$676$	15.8949	0.611342
$677$	− 18.0000i	− 0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$	− 18.0000i	− 0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$	0	0
$679$	−45.7128	−1.75430
$680$	0	0
$681$	0	0
$682$	12.5885i	0.482037i
$683$	19.6077i	0.750268i	0.926971	+	0.375134i	$0.122403\pi$
$683$	19.6077i	0.750268i	−0.926971	+	0.375134i	$0.877597\pi$
$684$	0	0
$685$	0	0
$686$	−29.0718	−1.10997
$687$	0	0
$688$	− 0.210236i	− 0.00801515i
$689$	−9.85641	−0.375499
$690$	0	0
$691$	−17.7128	−0.673827	−0.336914	−	0.941536i	$-0.609383\pi$
$691$	−17.7128	−0.673827	−0.336914	+	0.941536i	$0.609383\pi$
$692$	− 12.4974i	− 0.475081i
$693$	0	0
$694$	20.9282	0.794424
$695$	0	0
$696$	0	0
$697$	19.6603i	0.744685i
$698$	13.8038i	0.522483i
$699$	0	0
$700$	0	0
$701$	−20.8038	−0.785750	−0.392875	−	0.919592i	$-0.628520\pi$
$701$	−20.8038	−0.785750	−0.392875	+	0.919592i	$0.628520\pi$
$702$	0	0
$703$	12.1962i	0.459987i
$704$	12.2872	0.463091
$705$	0	0
$706$	18.6795	0.703012
$707$	12.5885i	0.473438i
$708$	0	0
$709$	22.5359	0.846353	0.423177	−	0.906047i	$-0.360915\pi$
$709$	22.5359	0.846353	0.423177	+	0.906047i	$0.360915\pi$
$710$	0	0
$711$	0	0
$712$	− 13.1769i	− 0.493826i
$713$	− 10.3923i	− 0.389195i
$714$	0	0
$715$	0	0
$716$	−11.8949	−0.444533
$717$	0	0
$718$	4.48334i	0.167317i
$719$	−8.41154	−0.313698	−0.156849	−	0.987623i	$-0.550134\pi$
$719$	−8.41154	−0.313698	−0.156849	+	0.987623i	$0.550134\pi$
$720$	0	0
$721$	2.53590	0.0944418
$722$	0.679492i	0.0252881i
$723$	0	0
$724$	38.7461	1.43999
$725$	0	0
$726$	0	0
$727$	− 8.39230i	− 0.311253i	−0.987816	−	0.155627i	$-0.950260\pi$
$727$	− 8.39230i	− 0.311253i	0.987816	−	0.155627i	$-0.0497397\pi$
$728$	− 17.5692i	− 0.651159i
$729$	0	0
$730$	0	0
$731$	−0.535898	−0.0198209
$732$	0	0
$733$	− 34.7846i	− 1.28480i	−0.766370	−	0.642399i	$-0.777939\pi$
$733$	− 34.7846i	− 1.28480i	0.766370	−	0.642399i	$-0.222061\pi$
$734$	−22.8231	−0.842415
$735$	0	0
$736$	−20.2872	−0.747796
$737$	− 19.8564i	− 0.731420i
$738$	0	0
$739$	−22.4641	−0.826355	−0.413178	−	0.910650i	$-0.635581\pi$
$739$	−22.4641	−0.826355	−0.413178	+	0.910650i	$0.635581\pi$
$740$	0	0
$741$	0	0
$742$	− 23.3205i	− 0.856123i
$743$	− 19.9090i	− 0.730389i	−0.930931	−	0.365195i	$-0.881002\pi$
$743$	− 19.9090i	− 0.730389i	0.930931	−	0.365195i	$-0.118998\pi$
$744$	0	0
$745$	0	0
$746$	14.6795	0.537454
$747$	0	0
$748$	22.9282i	0.838338i
$749$	−40.3923	−1.47590
$750$	0	0
$751$	48.7846	1.78018	0.890088	−	0.455789i	$-0.150643\pi$
$751$	48.7846	1.78018	0.890088	+	0.455789i	$0.150643\pi$
$752$	− 9.35898i	− 0.341287i
$753$	0	0
$754$	−3.42563	−0.124754
$755$	0	0
$756$	0	0
$757$	− 9.17691i	− 0.333541i	−0.985996	−	0.166770i	$-0.946666\pi$
$757$	− 9.17691i	− 0.333541i	0.985996	−	0.166770i	$-0.0533338\pi$
$758$	− 1.75129i	− 0.0636097i
$759$	0	0
$760$	0	0
$761$	23.4449	0.849876	0.424938	−	0.905223i	$-0.360296\pi$
$761$	23.4449	0.849876	0.424938	+	0.905223i	$0.360296\pi$
$762$	0	0
$763$	− 28.7321i	− 1.04017i
$764$	−11.8949	−0.430342
$765$	0	0
$766$	1.85641	0.0670747
$767$	− 12.1051i	− 0.437090i
$768$	0	0
$769$	9.53590	0.343873	0.171937	−	0.985108i	$-0.444998\pi$
$769$	9.53590	0.343873	0.171937	+	0.985108i	$0.444998\pi$
$770$	0	0
$771$	0	0
$772$	7.71281i	0.277590i
$773$	− 1.51666i	− 0.0545505i	−0.999628	−	0.0272752i	$-0.991317\pi$
$773$	− 1.51666i	− 0.0545505i	0.999628	−	0.0272752i	$-0.00868306\pi$
$774$	0	0
$775$	0	0
$776$	24.4974	0.879406
$777$	0	0
$778$	− 20.1051i	− 0.720803i
$779$	32.1244	1.15097
$780$	0	0
$781$	21.3923	0.765477
$782$	6.92820i	0.247752i
$783$	0	0
$784$	−16.4974	−0.589194
$785$	0	0
$786$	0	0
$787$	− 48.0526i	− 1.71289i	−0.516239	−	0.856444i	$-0.672668\pi$
$787$	− 48.0526i	− 1.71289i	0.516239	−	0.856444i	$-0.327332\pi$
$788$	− 20.2872i	− 0.722701i
$789$	0	0
$790$	0	0
$791$	90.4974	3.21772
$792$	0	0
$793$	− 5.85641i	− 0.207967i
$794$	−10.5359	−0.373905
$795$	0	0
$796$	2.92820	0.103787
$797$	− 15.6077i	− 0.552853i	−0.961035	−	0.276426i	$-0.910850\pi$
$797$	− 15.6077i	− 0.552853i	0.961035	−	0.276426i	$-0.0891502\pi$
$798$	0	0
$799$	−23.8564	−0.843979
$800$	0	0
$801$	0	0
$802$	18.2487i	0.644384i
$803$	− 43.9090i	− 1.54951i
$804$	0	0
$805$	0	0
$806$	−3.21539	−0.113257
$807$	0	0
$808$	− 6.74613i	− 0.237328i
$809$	−45.4449	−1.59776	−0.798878	−	0.601493i	$-0.794573\pi$
$809$	−45.4449	−1.59776	−0.798878	+	0.601493i	$0.794573\pi$
$810$	0	0
$811$	−18.4641	−0.648362	−0.324181	−	0.945995i	$-0.605089\pi$
$811$	−18.4641	−0.648362	−0.324181	+	0.945995i	$0.605089\pi$
$812$	22.1436i	0.777088i
$813$	0	0
$814$	−11.4641	−0.401817
$815$	0	0
$816$	0	0
$817$	0.875644i	0.0306349i
$818$	13.0718i	0.457045i
$819$	0	0
$820$	0	0
$821$	−37.7321	−1.31686	−0.658429	−	0.752643i	$-0.728779\pi$
$821$	−37.7321	−1.31686	−0.658429	+	0.752643i	$0.728779\pi$
$822$	0	0
$823$	− 3.85641i	− 0.134426i	−0.997739	−	0.0672129i	$-0.978589\pi$
$823$	− 3.85641i	− 0.134426i	0.997739	−	0.0672129i	$-0.0214107\pi$
$824$	−1.35898	−0.0473424
$825$	0	0
$826$	28.6410	0.996548
$827$	− 11.6077i	− 0.403639i	−0.979423	−	0.201820i	$-0.935315\pi$
$827$	− 11.6077i	− 0.403639i	0.979423	−	0.201820i	$-0.0646855\pi$
$828$	0	0
$829$	23.7846	0.826074	0.413037	−	0.910714i	$-0.364468\pi$
$829$	23.7846	0.826074	0.413037	+	0.910714i	$0.364468\pi$
$830$	0	0
$831$	0	0
$832$	3.13844i	0.108806i
$833$	42.0526i	1.45703i
$834$	0	0
$835$	0	0
$836$	37.4641	1.29572
$837$	0	0
$838$	− 14.9282i	− 0.515686i
$839$	−33.1962	−1.14606	−0.573029	−	0.819535i	$-0.694232\pi$
$839$	−33.1962	−1.14606	−0.573029	+	0.819535i	$0.694232\pi$
$840$	0	0
$841$	−18.7846	−0.647745
$842$	24.7321i	0.852323i
$843$	0	0
$844$	−12.9667	−0.446331
$845$	0	0
$846$	0	0
$847$	− 103.426i	− 3.55375i
$848$	− 7.21539i	− 0.247778i
$849$	0	0
$850$	0	0
$851$	9.46410	0.324425
$852$	0	0
$853$	− 10.4833i	− 0.358943i	−0.983763	−	0.179471i	$-0.942561\pi$
$853$	− 10.4833i	− 0.358943i	0.983763	−	0.179471i	$-0.0574387\pi$
$854$	13.8564	0.474156
$855$	0	0
$856$	21.6462	0.739851
$857$	44.4974i	1.52000i	0.649921	+	0.760002i	$0.274802\pi$
$857$	44.4974i	1.52000i	−0.649921	+	0.760002i	$0.725198\pi$
$858$	0	0
$859$	11.0000	0.375315	0.187658	−	0.982235i	$-0.439910\pi$
$859$	11.0000	0.375315	0.187658	+	0.982235i	$0.439910\pi$
$860$	0	0
$861$	0	0
$862$	− 15.6218i	− 0.532080i
$863$	23.1244i	0.787162i	0.919290	+	0.393581i	$0.128764\pi$
$863$	23.1244i	0.787162i	−0.919290	+	0.393581i	$0.871236\pi$
$864$	0	0
$865$	0	0
$866$	−25.9615	−0.882209
$867$	0	0
$868$	20.7846i	0.705476i
$869$	88.6410	3.00694
$870$	0	0
$871$	5.07180	0.171851
$872$	15.3975i	0.521424i
$873$	0	0
$874$	11.3205	0.382922
$875$	0	0
$876$	0	0
$877$	33.4641i	1.13000i	0.825090	+	0.565001i	$0.191124\pi$
$877$	33.4641i	1.13000i	−0.825090	+	0.565001i	$0.808876\pi$
$878$	− 3.94744i	− 0.133220i
$879$	0	0
$880$	0	0
$881$	−47.0526	−1.58524	−0.792620	−	0.609715i	$-0.791284\pi$
$881$	−47.0526	−1.58524	−0.792620	+	0.609715i	$0.791284\pi$
$882$	0	0
$883$	30.1962i	1.01618i	0.861304	+	0.508091i	$0.169649\pi$
$883$	30.1962i	1.01618i	−0.861304	+	0.508091i	$0.830351\pi$
$884$	−5.85641	−0.196972
$885$	0	0
$886$	−0.248711	−0.00835562
$887$	33.8038i	1.13502i	0.823366	+	0.567511i	$0.192094\pi$
$887$	33.8038i	1.13502i	−0.823366	+	0.567511i	$0.807906\pi$
$888$	0	0
$889$	−69.0333	−2.31530
$890$	0	0
$891$	0	0
$892$	− 24.5744i	− 0.822811i
$893$	38.9808i	1.30444i
$894$	0	0
$895$	0	0
$896$	48.0000	1.60357
$897$	0	0
$898$	− 5.94744i	− 0.198469i
$899$	9.58846	0.319793
$900$	0	0
$901$	−18.3923	−0.612737
$902$	30.1962i	1.00542i
$903$	0	0
$904$	−48.4974	−1.61300
$905$	0	0
$906$	0	0
$907$	− 30.0000i	− 0.996134i	−0.867139	−	0.498067i	$-0.834043\pi$
$907$	− 30.0000i	− 0.996134i	0.867139	−	0.498067i	$-0.165957\pi$
$908$	− 26.4308i	− 0.877136i
$909$	0	0
$910$	0	0
$911$	23.5885	0.781520	0.390760	−	0.920493i	$-0.372212\pi$
$911$	23.5885	0.781520	0.390760	+	0.920493i	$0.372212\pi$
$912$	0	0
$913$	12.5885i	0.416617i
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	17.5692	0.580503
$917$	− 73.7654i	− 2.43595i
$918$	0	0
$919$	56.9615	1.87899	0.939494	−	0.342566i	$-0.111296\pi$
$919$	56.9615	1.87899	0.939494	+	0.342566i	$0.111296\pi$
$920$	0	0
$921$	0	0
$922$	− 27.1244i	− 0.893293i
$923$	5.46410i	0.179853i
$924$	0	0
$925$	0	0
$926$	−7.60770	−0.250004
$927$	0	0
$928$	− 18.7180i	− 0.614447i
$929$	44.3731	1.45583	0.727917	−	0.685666i	$-0.240489\pi$
$929$	44.3731	1.45583	0.727917	+	0.685666i	$0.240489\pi$
$930$	0	0
$931$	68.7128	2.25197
$932$	41.0718i	1.34535i
$933$	0	0
$934$	−27.3590	−0.895213
$935$	0	0
$936$	0	0
$937$	− 4.14359i	− 0.135365i	−0.997707	−	0.0676827i	$-0.978439\pi$
$937$	− 4.14359i	− 0.135365i	0.997707	−	0.0676827i	$-0.0215605\pi$
$938$	12.0000i	0.391814i
$939$	0	0
$940$	0	0
$941$	35.1769	1.14673	0.573367	−	0.819298i	$-0.305637\pi$
$941$	35.1769	1.14673	0.573367	+	0.819298i	$0.305637\pi$
$942$	0	0
$943$	− 24.9282i	− 0.811774i
$944$	8.86156	0.288419
$945$	0	0
$946$	−0.823085	−0.0267608
$947$	− 57.7128i	− 1.87541i	−0.347427	−	0.937707i	$-0.612944\pi$
$947$	− 57.7128i	− 1.87541i	0.347427	−	0.937707i	$-0.387056\pi$
$948$	0	0
$949$	11.2154	0.364067
$950$	0	0
$951$	0	0
$952$	− 32.7846i	− 1.06256i
$953$	− 36.3923i	− 1.17886i	−0.807819	−	0.589431i	$-0.799352\pi$
$953$	− 36.3923i	− 1.17886i	0.807819	−	0.589431i	$-0.200648\pi$
$954$	0	0
$955$	0	0
$956$	0.784610	0.0253761
$957$	0	0
$958$	8.69358i	0.280877i
$959$	12.0000	0.387500
$960$	0	0
$961$	−22.0000	−0.709677
$962$	− 2.92820i	− 0.0944091i
$963$	0	0
$964$	−23.8949	−0.769602
$965$	0	0
$966$	0	0
$967$	− 25.8038i	− 0.829796i	−0.909868	−	0.414898i	$-0.863817\pi$
$967$	− 25.8038i	− 0.829796i	0.909868	−	0.414898i	$-0.136183\pi$
$968$	55.4256i	1.78145i
$969$	0	0
$970$	0	0
$971$	17.4449	0.559832	0.279916	−	0.960024i	$-0.409693\pi$
$971$	17.4449	0.559832	0.279916	+	0.960024i	$0.409693\pi$
$972$	0	0
$973$	2.87564i	0.0921889i
$974$	−17.8564	−0.572156
$975$	0	0
$976$	4.28719	0.137230
$977$	5.46410i	0.174812i	0.996173	+	0.0874060i	$0.0278578\pi$
$977$	5.46410i	0.174812i	−0.996173	+	0.0874060i	$0.972142\pi$
$978$	0	0
$979$	29.7846	0.951920
$980$	0	0
$981$	0	0
$982$	− 10.1577i	− 0.324144i
$983$	− 48.5885i	− 1.54973i	−0.632126	−	0.774866i	$-0.717818\pi$
$983$	− 48.5885i	− 1.54973i	0.632126	−	0.774866i	$-0.282182\pi$
$984$	0	0
$985$	0	0
$986$	−6.39230	−0.203572
$987$	0	0
$988$	9.56922i	0.304437i
$989$	0.679492	0.0216066
$990$	0	0
$991$	30.8564	0.980186	0.490093	−	0.871670i	$-0.336963\pi$
$991$	30.8564	0.980186	0.490093	+	0.871670i	$0.336963\pi$
$992$	− 17.5692i	− 0.557823i
$993$	0	0
$994$	−12.9282	−0.410058
$995$	0	0
$996$	0	0
$997$	25.5167i	0.808121i	0.914732	+	0.404060i	$0.132401\pi$
$997$	25.5167i	0.808121i	−0.914732	+	0.404060i	$0.867599\pi$
$998$	− 17.8038i	− 0.563571i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.b.g.649.3		4
3.2	odd	2		2025.2.b.h.649.2		4
5.2	odd	4		2025.2.a.m.1.1		2
5.3	odd	4		405.2.a.g.1.2	✓	2
5.4	even	2	inner	2025.2.b.g.649.2		4
15.2	even	4		2025.2.a.g.1.2		2
15.8	even	4		405.2.a.h.1.1	yes	2
15.14	odd	2		2025.2.b.h.649.3		4
20.3	even	4		6480.2.a.br.1.2		2
45.13	odd	12		405.2.e.l.136.1		4
45.23	even	12		405.2.e.i.136.2		4
45.38	even	12		405.2.e.i.271.2		4
45.43	odd	12		405.2.e.l.271.1		4
60.23	odd	4		6480.2.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.2.a.g.1.2	✓	2	5.3	odd	4
405.2.a.h.1.1	yes	2	15.8	even	4
405.2.e.i.136.2		4	45.23	even	12
405.2.e.i.271.2		4	45.38	even	12
405.2.e.l.136.1		4	45.13	odd	12
405.2.e.l.271.1		4	45.43	odd	12
2025.2.a.g.1.2		2	15.2	even	4
2025.2.a.m.1.1		2	5.2	odd	4
2025.2.b.g.649.2		4	5.4	even	2	inner
2025.2.b.g.649.3		4	1.1	even	1	trivial
2025.2.b.h.649.2		4	3.2	odd	2
2025.2.b.h.649.3		4	15.14	odd	2
6480.2.a.bi.1.2		2	60.23	odd	4
6480.2.a.br.1.2		2	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)

\(f(q)\)	\(=\)	\(q+0.732051i q^{2} +1.46410 q^{4} -4.73205i q^{7} +2.53590i q^{8} -5.73205 q^{11} -1.46410i q^{13} +3.46410 q^{14} +1.07180 q^{16} -2.73205i q^{17} -4.46410 q^{19} -4.19615i q^{22} +3.46410i q^{23} +1.07180 q^{26} -6.92820i q^{28} -3.19615 q^{29} -3.00000 q^{31} +5.85641i q^{32} +2.00000 q^{34} -2.73205i q^{37} -3.26795i q^{38} -7.19615 q^{41} -0.196152i q^{43} -8.39230 q^{44} -2.53590 q^{46} -8.73205i q^{47} -15.3923 q^{49} -2.14359i q^{52} -6.73205i q^{53} +12.0000 q^{56} -2.33975i q^{58} +8.26795 q^{59} +4.00000 q^{61} -2.19615i q^{62} -2.14359 q^{64} +3.46410i q^{67} -4.00000i q^{68} -3.73205 q^{71} +7.66025i q^{73} +2.00000 q^{74} -6.53590 q^{76} +27.1244i q^{77} -15.4641 q^{79} -5.26795i q^{82} -2.19615i q^{83} +0.143594 q^{86} -14.5359i q^{88} -5.19615 q^{89} -6.92820 q^{91} +5.07180i q^{92} +6.39230 q^{94} -9.66025i q^{97} -11.2679i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 16 q^{11} + 32 q^{16} - 4 q^{19} + 32 q^{26} + 8 q^{29} - 12 q^{31} + 8 q^{34} - 8 q^{41} + 8 q^{44} - 24 q^{46} - 20 q^{49} + 48 q^{56} + 40 q^{59} + 16 q^{61} - 64 q^{64} - 8 q^{71} + 8 q^{74}+ \cdots - 16 q^{94}+O(q^{100}) \) 4 * q - 8 * q^4 - 16 * q^11 + 32 * q^16 - 4 * q^19 + 32 * q^26 + 8 * q^29 - 12 * q^31 + 8 * q^34 - 8 * q^41 + 8 * q^44 - 24 * q^46 - 20 * q^49 + 48 * q^56 + 40 * q^59 + 16 * q^61 - 64 * q^64 - 8 * q^71 + 8 * q^74 - 40 * q^76 - 48 * q^79 + 56 * q^86 - 16 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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