LMFDB - Embedded newform 2025.2.b.c.649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,2,Mod(649,2025)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 45)
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.c.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{4} -3.00000i q^{7} +3.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} +1.00000 q^{4} -3.00000i q^{7} +3.00000i q^{8} -2.00000 q^{11} +2.00000i q^{13} +3.00000 q^{14} -1.00000 q^{16} +4.00000i q^{17} +8.00000 q^{19} -2.00000i q^{22} -3.00000i q^{23} -2.00000 q^{26} -3.00000i q^{28} +1.00000 q^{29} +5.00000i q^{32} -4.00000 q^{34} -4.00000i q^{37} +8.00000i q^{38} +5.00000 q^{41} +8.00000i q^{43} -2.00000 q^{44} +3.00000 q^{46} +7.00000i q^{47} -2.00000 q^{49} +2.00000i q^{52} +2.00000i q^{53} +9.00000 q^{56} +1.00000i q^{58} +14.0000 q^{59} +7.00000 q^{61} -7.00000 q^{64} -3.00000i q^{67} +4.00000i q^{68} +2.00000 q^{71} -4.00000i q^{73} +4.00000 q^{74} +8.00000 q^{76} +6.00000i q^{77} +6.00000 q^{79} +5.00000i q^{82} -9.00000i q^{83} -8.00000 q^{86} -6.00000i q^{88} +15.0000 q^{89} +6.00000 q^{91} -3.00000i q^{92} -7.00000 q^{94} +2.00000i q^{97} -2.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4}+O(q^{10}) $ 2 * q + 2 * q^4	$ 2 q + 2 q^{4} - 4 q^{11} + 6 q^{14} - 2 q^{16} + 16 q^{19} - 4 q^{26} + 2 q^{29} - 8 q^{34} + 10 q^{41} - 4 q^{44} + 6 q^{46} - 4 q^{49} + 18 q^{56} + 28 q^{59} + 14 q^{61} - 14 q^{64} + 4 q^{71} + 8 q^{74} + 16 q^{76} + 12 q^{79} - 16 q^{86} + 30 q^{89} + 12 q^{91} - 14 q^{94}+O(q^{100}) $ 2 * q + 2 * q^4 - 4 * q^11 + 6 * q^14 - 2 * q^16 + 16 * q^19 - 4 * q^26 + 2 * q^29 - 8 * q^34 + 10 * q^41 - 4 * q^44 + 6 * q^46 - 4 * q^49 + 18 * q^56 + 28 * q^59 + 14 * q^61 - 14 * q^64 + 4 * q^71 + 8 * q^74 + 16 * q^76 + 12 * q^79 - 16 * q^86 + 30 * q^89 + 12 * q^91 - 14 * q^94

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000i	0.707107i	0.935414	+	0.353553i	$0.115027\pi$
$2$	1.00000i	0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.00000i	− 1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$	− 3.00000i	− 1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	3.00000i	1.06066i
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	3.00000	0.801784
$15$	0	0
$16$	−1.00000	−0.250000
$17$	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
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	0	0
$22$	− 2.00000i	− 0.426401i
$23$	− 3.00000i	− 0.625543i	−0.949828	−	0.312772i	$-0.898743\pi$
$23$	− 3.00000i	− 0.625543i	0.949828	−	0.312772i	$-0.101257\pi$
$24$	0	0
$25$	0	0
$26$	−2.00000	−0.392232
$27$	0	0
$28$	− 3.00000i	− 0.566947i
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	5.00000i	0.883883i
$33$	0	0
$34$	−4.00000	−0.685994
$35$	0	0
$36$	0	0
$37$	− 4.00000i	− 0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$	− 4.00000i	− 0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	8.00000i	1.29777i
$39$	0	0
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	3.00000	0.442326
$47$	7.00000i	1.02105i	0.859861	+	0.510527i	$0.170550\pi$
$47$	7.00000i	1.02105i	−0.859861	+	0.510527i	$0.829450\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	0	0
$52$	2.00000i	0.277350i
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	0	0
$56$	9.00000	1.20268
$57$	0	0
$58$	1.00000i	0.131306i
$59$	14.0000	1.82264	0.911322	−	0.411693i	$-0.135063\pi$
$59$	14.0000	1.82264	0.911322	+	0.411693i	$0.135063\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	0	0
$63$	0	0
$64$	−7.00000	−0.875000
$65$	0	0
$66$	0	0
$67$	− 3.00000i	− 0.366508i	−0.983066	−	0.183254i	$-0.941337\pi$
$67$	− 3.00000i	− 0.366508i	0.983066	−	0.183254i	$-0.0586631\pi$
$68$	4.00000i	0.485071i
$69$	0	0
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	8.00000	0.917663
$77$	6.00000i	0.683763i
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	0	0
$81$	0	0
$82$	5.00000i	0.552158i
$83$	− 9.00000i	− 0.987878i	−0.869496	−	0.493939i	$-0.835557\pi$
$83$	− 9.00000i	− 0.987878i	0.869496	−	0.493939i	$-0.164443\pi$
$84$	0	0
$85$	0	0
$86$	−8.00000	−0.862662
$87$	0	0
$88$	− 6.00000i	− 0.639602i
$89$	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
$91$	6.00000	0.628971
$92$	− 3.00000i	− 0.312772i
$93$	0	0
$94$	−7.00000	−0.721995
$95$	0	0
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	− 2.00000i	− 0.202031i
$99$	0	0
$100$	0	0
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	0	0
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−2.00000	−0.194257
$107$	3.00000i	0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$	3.00000i	0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$	0	0
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	0	0
$111$	0	0
$112$	3.00000i	0.283473i
$113$	8.00000i	0.752577i	0.926503	+	0.376288i	$0.122800\pi$
$113$	8.00000i	0.752577i	−0.926503	+	0.376288i	$0.877200\pi$
$114$	0	0
$115$	0	0
$116$	1.00000	0.0928477
$117$	0	0
$118$	14.0000i	1.28880i
$119$	12.0000	1.10004
$120$	0	0
$121$	−7.00000	−0.636364
$122$	7.00000i	0.633750i
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 5.00000i	− 0.443678i	−0.975083	−	0.221839i	$-0.928794\pi$
$127$	− 5.00000i	− 0.443678i	0.975083	−	0.221839i	$-0.0712060\pi$
$128$	3.00000i	0.265165i
$129$	0	0
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	− 24.0000i	− 2.08106i
$134$	3.00000	0.259161
$135$	0	0
$136$	−12.0000	−1.02899
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	0	0
$142$	2.00000i	0.167836i
$143$	− 4.00000i	− 0.334497i
$144$	0	0
$145$	0	0
$146$	4.00000	0.331042
$147$	0	0
$148$	− 4.00000i	− 0.328798i
$149$	−17.0000	−1.39269	−0.696347	−	0.717705i	$-0.745193\pi$
$149$	−17.0000	−1.39269	−0.696347	+	0.717705i	$0.745193\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	24.0000i	1.94666i
$153$	0	0
$154$	−6.00000	−0.483494
$155$	0	0
$156$	0	0
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	6.00000i	0.477334i
$159$	0	0
$160$	0	0
$161$	−9.00000	−0.709299
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	5.00000	0.390434
$165$	0	0
$166$	9.00000	0.698535
$167$	− 9.00000i	− 0.696441i	−0.937413	−	0.348220i	$-0.886786\pi$
$167$	− 9.00000i	− 0.696441i	0.937413	−	0.348220i	$-0.113214\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	8.00000i	0.609994i
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	2.00000	0.150756
$177$	0	0
$178$	15.0000i	1.12430i
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	6.00000i	0.444750i
$183$	0	0
$184$	9.00000	0.663489
$185$	0	0
$186$	0	0
$187$	− 8.00000i	− 0.585018i
$188$	7.00000i	0.510527i
$189$	0	0
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−2.00000	−0.142857
$197$	− 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
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	0	0
$202$	− 18.0000i	− 1.26648i
$203$	− 3.00000i	− 0.210559i
$204$	0	0
$205$	0	0
$206$	8.00000	0.557386
$207$	0	0
$208$	− 2.00000i	− 0.138675i
$209$	−16.0000	−1.10674
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	2.00000i	0.137361i
$213$	0	0
$214$	−3.00000	−0.205076
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 5.00000i	− 0.338643i
$219$	0	0
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	19.0000i	1.27233i	0.771551	+	0.636167i	$0.219481\pi$
$223$	19.0000i	1.27233i	−0.771551	+	0.636167i	$0.780519\pi$
$224$	15.0000	1.00223
$225$	0	0
$226$	−8.00000	−0.532152
$227$	− 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
$229$	−15.0000	−0.991228	−0.495614	−	0.868543i	$-0.665057\pi$
$229$	−15.0000	−0.991228	−0.495614	+	0.868543i	$0.665057\pi$
$230$	0	0
$231$	0	0
$232$	3.00000i	0.196960i
$233$	− 24.0000i	− 1.57229i	−0.618041	−	0.786146i	$-0.712073\pi$
$233$	− 24.0000i	− 1.57229i	0.618041	−	0.786146i	$-0.287927\pi$
$234$	0	0
$235$	0	0
$236$	14.0000	0.911322
$237$	0	0
$238$	12.0000i	0.777844i
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−11.0000	−0.708572	−0.354286	−	0.935137i	$-0.615276\pi$
$241$	−11.0000	−0.708572	−0.354286	+	0.935137i	$0.615276\pi$
$242$	− 7.00000i	− 0.449977i
$243$	0	0
$244$	7.00000	0.448129
$245$	0	0
$246$	0	0
$247$	16.0000i	1.01806i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	6.00000i	0.377217i
$254$	5.00000	0.313728
$255$	0	0
$256$	−17.0000	−1.06250
$257$	− 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
$259$	−12.0000	−0.745644
$260$	0	0
$261$	0	0
$262$	− 6.00000i	− 0.370681i
$263$	16.0000i	0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$	16.0000i	0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$	0	0
$265$	0	0
$266$	24.0000	1.47153
$267$	0	0
$268$	− 3.00000i	− 0.183254i
$269$	−25.0000	−1.52428	−0.762138	−	0.647414i	$-0.775850\pi$
$269$	−25.0000	−1.52428	−0.762138	+	0.647414i	$0.775850\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	− 4.00000i	− 0.242536i
$273$	0	0
$274$	−12.0000	−0.724947
$275$	0	0
$276$	0	0
$277$	12.0000i	0.721010i	0.932757	+	0.360505i	$0.117396\pi$
$277$	12.0000i	0.721010i	−0.932757	+	0.360505i	$0.882604\pi$
$278$	16.0000i	0.959616i
$279$	0	0
$280$	0	0
$281$	−15.0000	−0.894825	−0.447412	−	0.894328i	$-0.647654\pi$
$281$	−15.0000	−0.894825	−0.447412	+	0.894328i	$0.647654\pi$
$282$	0	0
$283$	− 21.0000i	− 1.24832i	−0.781296	−	0.624160i	$-0.785441\pi$
$283$	− 21.0000i	− 1.24832i	0.781296	−	0.624160i	$-0.214559\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	4.00000	0.236525
$287$	− 15.0000i	− 0.885422i
$288$	0	0
$289$	1.00000	0.0588235
$290$	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
$296$	12.0000	0.697486
$297$	0	0
$298$	− 17.0000i	− 0.984784i
$299$	6.00000	0.346989
$300$	0	0
$301$	24.0000	1.38334
$302$	− 2.00000i	− 0.115087i
$303$	0	0
$304$	−8.00000	−0.458831
$305$	0	0
$306$	0	0
$307$	7.00000i	0.399511i	0.979846	+	0.199756i	$0.0640148\pi$
$307$	7.00000i	0.399511i	−0.979846	+	0.199756i	$0.935985\pi$
$308$	6.00000i	0.341882i
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	− 14.0000i	− 0.791327i	−0.918396	−	0.395663i	$-0.870515\pi$
$313$	− 14.0000i	− 0.791327i	0.918396	−	0.395663i	$-0.129485\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	6.00000	0.337526
$317$	− 34.0000i	− 1.90963i	−0.297200	−	0.954815i	$-0.596053\pi$
$317$	− 34.0000i	− 1.90963i	0.297200	−	0.954815i	$-0.403947\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	0	0
$321$	0	0
$322$	− 9.00000i	− 0.501550i
$323$	32.0000i	1.78053i
$324$	0	0
$325$	0	0
$326$	−4.00000	−0.221540
$327$	0	0
$328$	15.0000i	0.828236i
$329$	21.0000	1.15777
$330$	0	0
$331$	−6.00000	−0.329790	−0.164895	−	0.986311i	$-0.552728\pi$
$331$	−6.00000	−0.329790	−0.164895	+	0.986311i	$0.552728\pi$
$332$	− 9.00000i	− 0.493939i
$333$	0	0
$334$	9.00000	0.492458
$335$	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	$-0.0699187\pi$
$338$	9.00000i	0.489535i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 15.0000i	− 0.809924i
$344$	−24.0000	−1.29399
$345$	0	0
$346$	0	0
$347$	4.00000i	0.214731i	0.994220	+	0.107366i	$0.0342415\pi$
$347$	4.00000i	0.214731i	−0.994220	+	0.107366i	$0.965758\pi$
$348$	0	0
$349$	5.00000	0.267644	0.133822	−	0.991005i	$-0.457275\pi$
$349$	5.00000	0.267644	0.133822	+	0.991005i	$0.457275\pi$
$350$	0	0
$351$	0	0
$352$	− 10.0000i	− 0.533002i
$353$	− 24.0000i	− 1.27739i	−0.769460	−	0.638696i	$-0.779474\pi$
$353$	− 24.0000i	− 1.27739i	0.769460	−	0.638696i	$-0.220526\pi$
$354$	0	0
$355$	0	0
$356$	15.0000	0.794998
$357$	0	0
$358$	2.00000i	0.105703i
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	− 7.00000i	− 0.367912i
$363$	0	0
$364$	6.00000	0.314485
$365$	0	0
$366$	0	0
$367$	24.0000i	1.25279i	0.779506	+	0.626395i	$0.215470\pi$
$367$	24.0000i	1.25279i	−0.779506	+	0.626395i	$0.784530\pi$
$368$	3.00000i	0.156386i
$369$	0	0
$370$	0	0
$371$	6.00000	0.311504
$372$	0	0
$373$	− 10.0000i	− 0.517780i	−0.965907	−	0.258890i	$-0.916643\pi$
$373$	− 10.0000i	− 0.517780i	0.965907	−	0.258890i	$-0.0833568\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	−21.0000	−1.08299
$377$	2.00000i	0.103005i
$378$	0	0
$379$	26.0000	1.33553	0.667765	−	0.744372i	$-0.267251\pi$
$379$	26.0000	1.33553	0.667765	+	0.744372i	$0.267251\pi$
$380$	0	0
$381$	0	0
$382$	8.00000i	0.409316i
$383$	− 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
$386$	−10.0000	−0.508987
$387$	0	0
$388$	2.00000i	0.101535i
$389$	−33.0000	−1.67317	−0.836583	−	0.547840i	$-0.815450\pi$
$389$	−33.0000	−1.67317	−0.836583	+	0.547840i	$0.815450\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	− 6.00000i	− 0.303046i
$393$	0	0
$394$	12.0000	0.604551
$395$	0	0
$396$	0	0
$397$	34.0000i	1.70641i	0.521575	+	0.853206i	$0.325345\pi$
$397$	34.0000i	1.70641i	−0.521575	+	0.853206i	$0.674655\pi$
$398$	− 4.00000i	− 0.200502i
$399$	0	0
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	0	0
$404$	−18.0000	−0.895533
$405$	0	0
$406$	3.00000	0.148888
$407$	8.00000i	0.396545i
$408$	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
$412$	− 8.00000i	− 0.394132i
$413$	− 42.0000i	− 2.06668i
$414$	0	0
$415$	0	0
$416$	−10.0000	−0.490290
$417$	0	0
$418$	− 16.0000i	− 0.782586i
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	0	0
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	− 22.0000i	− 1.07094i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	− 21.0000i	− 1.01626i
$428$	3.00000i	0.145010i
$429$	0	0
$430$	0	0
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	0	0
$433$	− 28.0000i	− 1.34559i	−0.739827	−	0.672797i	$-0.765093\pi$
$433$	− 28.0000i	− 1.34559i	0.739827	−	0.672797i	$-0.234907\pi$
$434$	0	0
$435$	0	0
$436$	−5.00000	−0.239457
$437$	− 24.0000i	− 1.14808i
$438$	0	0
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	0	0
$442$	− 8.00000i	− 0.380521i
$443$	− 15.0000i	− 0.712672i	−0.934358	−	0.356336i	$-0.884026\pi$
$443$	− 15.0000i	− 0.712672i	0.934358	−	0.356336i	$-0.115974\pi$
$444$	0	0
$445$	0	0
$446$	−19.0000	−0.899676
$447$	0	0
$448$	21.0000i	0.992157i
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	0	0
$451$	−10.0000	−0.470882
$452$	8.00000i	0.376288i
$453$	0	0
$454$	4.00000	0.187729
$455$	0	0
$456$	0	0
$457$	− 20.0000i	− 0.935561i	−0.883845	−	0.467780i	$-0.845054\pi$
$457$	− 20.0000i	− 0.935561i	0.883845	−	0.467780i	$-0.154946\pi$
$458$	− 15.0000i	− 0.700904i
$459$	0	0
$460$	0	0
$461$	9.00000	0.419172	0.209586	−	0.977790i	$-0.432788\pi$
$461$	9.00000	0.419172	0.209586	+	0.977790i	$0.432788\pi$
$462$	0	0
$463$	36.0000i	1.67306i	0.547920	+	0.836531i	$0.315420\pi$
$463$	36.0000i	1.67306i	−0.547920	+	0.836531i	$0.684580\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	24.0000	1.11178
$467$	20.0000i	0.925490i	0.886492	+	0.462745i	$0.153135\pi$
$467$	20.0000i	0.925490i	−0.886492	+	0.462745i	$0.846865\pi$
$468$	0	0
$469$	−9.00000	−0.415581
$470$	0	0
$471$	0	0
$472$	42.0000i	1.93321i
$473$	− 16.0000i	− 0.735681i
$474$	0	0
$475$	0	0
$476$	12.0000	0.550019
$477$	0	0
$478$	8.00000i	0.365911i
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	− 11.0000i	− 0.501036i
$483$	0	0
$484$	−7.00000	−0.318182
$485$	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	$-0.118082\pi$
$488$	21.0000i	0.950625i
$489$	0	0
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	4.00000i	0.180151i
$494$	−16.0000	−0.719874
$495$	0	0
$496$	0	0
$497$	− 6.00000i	− 0.269137i
$498$	0	0
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7.00000i	0.312115i	0.987748	+	0.156057i	$0.0498784\pi$
$503$	7.00000i	0.312115i	−0.987748	+	0.156057i	$0.950122\pi$
$504$	0	0
$505$	0	0
$506$	−6.00000	−0.266733
$507$	0	0
$508$	− 5.00000i	− 0.221839i
$509$	−43.0000	−1.90594	−0.952971	−	0.303062i	$-0.901991\pi$
$509$	−43.0000	−1.90594	−0.952971	+	0.303062i	$0.901991\pi$
$510$	0	0
$511$	−12.0000	−0.530849
$512$	− 11.0000i	− 0.486136i
$513$	0	0
$514$	6.00000	0.264649
$515$	0	0
$516$	0	0
$517$	− 14.0000i	− 0.615719i
$518$	− 12.0000i	− 0.527250i
$519$	0	0
$520$	0	0
$521$	11.0000	0.481919	0.240959	−	0.970535i	$-0.422538\pi$
$521$	11.0000	0.481919	0.240959	+	0.970535i	$0.422538\pi$
$522$	0	0
$523$	29.0000i	1.26808i	0.773300	+	0.634041i	$0.218605\pi$
$523$	29.0000i	1.26808i	−0.773300	+	0.634041i	$0.781395\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	−16.0000	−0.697633
$527$	0	0
$528$	0	0
$529$	14.0000	0.608696
$530$	0	0
$531$	0	0
$532$	− 24.0000i	− 1.04053i
$533$	10.0000i	0.433148i
$534$	0	0
$535$	0	0
$536$	9.00000	0.388741
$537$	0	0
$538$	− 25.0000i	− 1.07783i
$539$	4.00000	0.172292
$540$	0	0
$541$	−39.0000	−1.67674	−0.838370	−	0.545101i	$-0.816491\pi$
$541$	−39.0000	−1.67674	−0.838370	+	0.545101i	$0.816491\pi$
$542$	− 8.00000i	− 0.343629i
$543$	0	0
$544$	−20.0000	−0.857493
$545$	0	0
$546$	0	0
$547$	− 29.0000i	− 1.23995i	−0.784621	−	0.619975i	$-0.787143\pi$
$547$	− 29.0000i	− 1.23995i	0.784621	−	0.619975i	$-0.212857\pi$
$548$	12.0000i	0.512615i
$549$	0	0
$550$	0	0
$551$	8.00000	0.340811
$552$	0	0
$553$	− 18.0000i	− 0.765438i
$554$	−12.0000	−0.509831
$555$	0	0
$556$	16.0000	0.678551
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	− 15.0000i	− 0.632737i
$563$	21.0000i	0.885044i	0.896758	+	0.442522i	$0.145916\pi$
$563$	21.0000i	0.885044i	−0.896758	+	0.442522i	$0.854084\pi$
$564$	0	0
$565$	0	0
$566$	21.0000	0.882696
$567$	0	0
$568$	6.00000i	0.251754i
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	− 4.00000i	− 0.167248i
$573$	0	0
$574$	15.0000	0.626088
$575$	0	0
$576$	0	0
$577$	− 10.0000i	− 0.416305i	−0.978096	−	0.208153i	$-0.933255\pi$
$577$	− 10.0000i	− 0.416305i	0.978096	−	0.208153i	$-0.0667451\pi$
$578$	1.00000i	0.0415945i
$579$	0	0
$580$	0	0
$581$	−27.0000	−1.12015
$582$	0	0
$583$	− 4.00000i	− 0.165663i
$584$	12.0000	0.496564
$585$	0	0
$586$	12.0000	0.495715
$587$	− 33.0000i	− 1.36206i	−0.732257	−	0.681028i	$-0.761533\pi$
$587$	− 33.0000i	− 1.36206i	0.732257	−	0.681028i	$-0.238467\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	4.00000i	0.164399i
$593$	− 20.0000i	− 0.821302i	−0.911793	−	0.410651i	$-0.865302\pi$
$593$	− 20.0000i	− 0.821302i	0.911793	−	0.410651i	$-0.134698\pi$
$594$	0	0
$595$	0	0
$596$	−17.0000	−0.696347
$597$	0	0
$598$	6.00000i	0.245358i
$599$	10.0000	0.408589	0.204294	−	0.978909i	$-0.434510\pi$
$599$	10.0000	0.408589	0.204294	+	0.978909i	$0.434510\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	24.0000i	0.978167i
$603$	0	0
$604$	−2.00000	−0.0813788
$605$	0	0
$606$	0	0
$607$	− 41.0000i	− 1.66414i	−0.554672	−	0.832069i	$-0.687156\pi$
$607$	− 41.0000i	− 1.66414i	0.554672	−	0.832069i	$-0.312844\pi$
$608$	40.0000i	1.62221i
$609$	0	0
$610$	0	0
$611$	−14.0000	−0.566379
$612$	0	0
$613$	44.0000i	1.77714i	0.458738	+	0.888572i	$0.348302\pi$
$613$	44.0000i	1.77714i	−0.458738	+	0.888572i	$0.651698\pi$
$614$	−7.00000	−0.282497
$615$	0	0
$616$	−18.0000	−0.725241
$617$	− 36.0000i	− 1.44931i	−0.689114	−	0.724653i	$-0.742000\pi$
$617$	− 36.0000i	− 1.44931i	0.689114	−	0.724653i	$-0.258000\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 45.0000i	− 1.80289i
$624$	0	0
$625$	0	0
$626$	14.0000	0.559553
$627$	0	0
$628$	14.0000i	0.558661i
$629$	16.0000	0.637962
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	18.0000i	0.716002i
$633$	0	0
$634$	34.0000	1.35031
$635$	0	0
$636$	0	0
$637$	− 4.00000i	− 0.158486i
$638$	− 2.00000i	− 0.0791808i
$639$	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
$643$	9.00000i	0.354925i	0.984128	+	0.177463i	$0.0567889\pi$
$643$	9.00000i	0.354925i	−0.984128	+	0.177463i	$0.943211\pi$
$644$	−9.00000	−0.354650
$645$	0	0
$646$	−32.0000	−1.25902
$647$	− 17.0000i	− 0.668339i	−0.942513	−	0.334169i	$-0.891544\pi$
$647$	− 17.0000i	− 0.668339i	0.942513	−	0.334169i	$-0.108456\pi$
$648$	0	0
$649$	−28.0000	−1.09910
$650$	0	0
$651$	0	0
$652$	4.00000i	0.156652i
$653$	4.00000i	0.156532i	0.996933	+	0.0782660i	$0.0249384\pi$
$653$	4.00000i	0.156532i	−0.996933	+	0.0782660i	$0.975062\pi$
$654$	0	0
$655$	0	0
$656$	−5.00000	−0.195217
$657$	0	0
$658$	21.0000i	0.818665i
$659$	8.00000	0.311636	0.155818	−	0.987786i	$-0.450199\pi$
$659$	8.00000	0.311636	0.155818	+	0.987786i	$0.450199\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	− 6.00000i	− 0.233197i
$663$	0	0
$664$	27.0000	1.04780
$665$	0	0
$666$	0	0
$667$	− 3.00000i	− 0.116160i
$668$	− 9.00000i	− 0.348220i
$669$	0	0
$670$	0	0
$671$	−14.0000	−0.540464
$672$	0	0
$673$	− 6.00000i	− 0.231283i	−0.993291	−	0.115642i	$-0.963108\pi$
$673$	− 6.00000i	− 0.231283i	0.993291	−	0.115642i	$-0.0368924\pi$
$674$	8.00000	0.308148
$675$	0	0
$676$	9.00000	0.346154
$677$	− 42.0000i	− 1.61419i	−0.590421	−	0.807096i	$-0.701038\pi$
$677$	− 42.0000i	− 1.61419i	0.590421	−	0.807096i	$-0.298962\pi$
$678$	0	0
$679$	6.00000	0.230259
$680$	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
$686$	15.0000	0.572703
$687$	0	0
$688$	− 8.00000i	− 0.304997i
$689$	−4.00000	−0.152388
$690$	0	0
$691$	−14.0000	−0.532585	−0.266293	−	0.963892i	$-0.585799\pi$
$691$	−14.0000	−0.532585	−0.266293	+	0.963892i	$0.585799\pi$
$692$	0	0
$693$	0	0
$694$	−4.00000	−0.151838
$695$	0	0
$696$	0	0
$697$	20.0000i	0.757554i
$698$	5.00000i	0.189253i
$699$	0	0
$700$	0	0
$701$	23.0000	0.868698	0.434349	−	0.900745i	$-0.356978\pi$
$701$	23.0000	0.868698	0.434349	+	0.900745i	$0.356978\pi$
$702$	0	0
$703$	− 32.0000i	− 1.20690i
$704$	14.0000	0.527645
$705$	0	0
$706$	24.0000	0.903252
$707$	54.0000i	2.03088i
$708$	0	0
$709$	41.0000	1.53979	0.769894	−	0.638172i	$-0.220309\pi$
$709$	41.0000	1.53979	0.769894	+	0.638172i	$0.220309\pi$
$710$	0	0
$711$	0	0
$712$	45.0000i	1.68645i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	2.00000	0.0747435
$717$	0	0
$718$	− 24.0000i	− 0.895672i
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	45.0000i	1.67473i
$723$	0	0
$724$	−7.00000	−0.260153
$725$	0	0
$726$	0	0
$727$	23.0000i	0.853023i	0.904482	+	0.426511i	$0.140258\pi$
$727$	23.0000i	0.853023i	−0.904482	+	0.426511i	$0.859742\pi$
$728$	18.0000i	0.667124i
$729$	0	0
$730$	0	0
$731$	−32.0000	−1.18356
$732$	0	0
$733$	34.0000i	1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$733$	34.0000i	1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$734$	−24.0000	−0.885856
$735$	0	0
$736$	15.0000	0.552907
$737$	6.00000i	0.221013i
$738$	0	0
$739$	2.00000	0.0735712	0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000	0.0735712	0.0367856	+	0.999323i	$0.488288\pi$
$740$	0	0
$741$	0	0
$742$	6.00000i	0.220267i
$743$	29.0000i	1.06391i	0.846774	+	0.531953i	$0.178542\pi$
$743$	29.0000i	1.06391i	−0.846774	+	0.531953i	$0.821458\pi$
$744$	0	0
$745$	0	0
$746$	10.0000	0.366126
$747$	0	0
$748$	− 8.00000i	− 0.292509i
$749$	9.00000	0.328853
$750$	0	0
$751$	10.0000	0.364905	0.182453	−	0.983215i	$-0.441596\pi$
$751$	10.0000	0.364905	0.182453	+	0.983215i	$0.441596\pi$
$752$	− 7.00000i	− 0.255264i
$753$	0	0
$754$	−2.00000	−0.0728357
$755$	0	0
$756$	0	0
$757$	− 26.0000i	− 0.944986i	−0.881334	−	0.472493i	$-0.843354\pi$
$757$	− 26.0000i	− 0.944986i	0.881334	−	0.472493i	$-0.156646\pi$
$758$	26.0000i	0.944363i
$759$	0	0
$760$	0	0
$761$	15.0000	0.543750	0.271875	−	0.962333i	$-0.412356\pi$
$761$	15.0000	0.543750	0.271875	+	0.962333i	$0.412356\pi$
$762$	0	0
$763$	15.0000i	0.543036i
$764$	8.00000	0.289430
$765$	0	0
$766$	36.0000	1.30073
$767$	28.0000i	1.01102i
$768$	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
$772$	10.0000i	0.359908i
$773$	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
$776$	−6.00000	−0.215387
$777$	0	0
$778$	− 33.0000i	− 1.18311i
$779$	40.0000	1.43315
$780$	0	0
$781$	−4.00000	−0.143131
$782$	12.0000i	0.429119i
$783$	0	0
$784$	2.00000	0.0714286
$785$	0	0
$786$	0	0
$787$	28.0000i	0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$	28.0000i	0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$	− 12.0000i	− 0.427482i
$789$	0	0
$790$	0	0
$791$	24.0000	0.853342
$792$	0	0
$793$	14.0000i	0.497155i
$794$	−34.0000	−1.20661
$795$	0	0
$796$	−4.00000	−0.141776
$797$	26.0000i	0.920967i	0.887668	+	0.460484i	$0.152324\pi$
$797$	26.0000i	0.920967i	−0.887668	+	0.460484i	$0.847676\pi$
$798$	0	0
$799$	−28.0000	−0.990569
$800$	0	0
$801$	0	0
$802$	− 18.0000i	− 0.635602i
$803$	8.00000i	0.282314i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	− 54.0000i	− 1.89971i
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	−42.0000	−1.47482	−0.737410	−	0.675446i	$-0.763951\pi$
$811$	−42.0000	−1.47482	−0.737410	+	0.675446i	$0.763951\pi$
$812$	− 3.00000i	− 0.105279i
$813$	0	0
$814$	−8.00000	−0.280400
$815$	0	0
$816$	0	0
$817$	64.0000i	2.23908i
$818$	− 14.0000i	− 0.489499i
$819$	0	0
$820$	0	0
$821$	15.0000	0.523504	0.261752	−	0.965135i	$-0.415700\pi$
$821$	15.0000	0.523504	0.261752	+	0.965135i	$0.415700\pi$
$822$	0	0
$823$	− 53.0000i	− 1.84746i	−0.383040	−	0.923732i	$-0.625123\pi$
$823$	− 53.0000i	− 1.84746i	0.383040	−	0.923732i	$-0.374877\pi$
$824$	24.0000	0.836080
$825$	0	0
$826$	42.0000	1.46137
$827$	37.0000i	1.28662i	0.765607	+	0.643308i	$0.222439\pi$
$827$	37.0000i	1.28662i	−0.765607	+	0.643308i	$0.777561\pi$
$828$	0	0
$829$	3.00000	0.104194	0.0520972	−	0.998642i	$-0.483409\pi$
$829$	3.00000	0.104194	0.0520972	+	0.998642i	$0.483409\pi$
$830$	0	0
$831$	0	0
$832$	− 14.0000i	− 0.485363i
$833$	− 8.00000i	− 0.277184i
$834$	0	0
$835$	0	0
$836$	−16.0000	−0.553372
$837$	0	0
$838$	− 26.0000i	− 0.898155i
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	34.0000i	1.17172i
$843$	0	0
$844$	−22.0000	−0.757271
$845$	0	0
$846$	0	0
$847$	21.0000i	0.721569i
$848$	− 2.00000i	− 0.0686803i
$849$	0	0
$850$	0	0
$851$	−12.0000	−0.411355
$852$	0	0
$853$	54.0000i	1.84892i	0.381273	+	0.924462i	$0.375486\pi$
$853$	54.0000i	1.84892i	−0.381273	+	0.924462i	$0.624514\pi$
$854$	21.0000	0.718605
$855$	0	0
$856$	−9.00000	−0.307614
$857$	10.0000i	0.341593i	0.985306	+	0.170797i	$0.0546341\pi$
$857$	10.0000i	0.341593i	−0.985306	+	0.170797i	$0.945366\pi$
$858$	0	0
$859$	−22.0000	−0.750630	−0.375315	−	0.926897i	$-0.622466\pi$
$859$	−22.0000	−0.750630	−0.375315	+	0.926897i	$0.622466\pi$
$860$	0	0
$861$	0	0
$862$	− 30.0000i	− 1.02180i
$863$	− 17.0000i	− 0.578687i	−0.957225	−	0.289343i	$-0.906563\pi$
$863$	− 17.0000i	− 0.578687i	0.957225	−	0.289343i	$-0.0934369\pi$
$864$	0	0
$865$	0	0
$866$	28.0000	0.951479
$867$	0	0
$868$	0	0
$869$	−12.0000	−0.407072
$870$	0	0
$871$	6.00000	0.203302
$872$	− 15.0000i	− 0.507964i
$873$	0	0
$874$	24.0000	0.811812
$875$	0	0
$876$	0	0
$877$	− 18.0000i	− 0.607817i	−0.952701	−	0.303908i	$-0.901708\pi$
$877$	− 18.0000i	− 0.607817i	0.952701	−	0.303908i	$-0.0982917\pi$
$878$	− 28.0000i	− 0.944954i
$879$	0	0
$880$	0	0
$881$	−35.0000	−1.17918	−0.589590	−	0.807703i	$-0.700711\pi$
$881$	−35.0000	−1.17918	−0.589590	+	0.807703i	$0.700711\pi$
$882$	0	0
$883$	− 23.0000i	− 0.774012i	−0.922077	−	0.387006i	$-0.873509\pi$
$883$	− 23.0000i	− 0.774012i	0.922077	−	0.387006i	$-0.126491\pi$
$884$	−8.00000	−0.269069
$885$	0	0
$886$	15.0000	0.503935
$887$	36.0000i	1.20876i	0.796696	+	0.604381i	$0.206579\pi$
$887$	36.0000i	1.20876i	−0.796696	+	0.604381i	$0.793421\pi$
$888$	0	0
$889$	−15.0000	−0.503084
$890$	0	0
$891$	0	0
$892$	19.0000i	0.636167i
$893$	56.0000i	1.87397i
$894$	0	0
$895$	0	0
$896$	9.00000	0.300669
$897$	0	0
$898$	26.0000i	0.867631i
$899$	0	0
$900$	0	0
$901$	−8.00000	−0.266519
$902$	− 10.0000i	− 0.332964i
$903$	0	0
$904$	−24.0000	−0.798228
$905$	0	0
$906$	0	0
$907$	− 51.0000i	− 1.69343i	−0.532049	−	0.846714i	$-0.678578\pi$
$907$	− 51.0000i	− 1.69343i	0.532049	−	0.846714i	$-0.321422\pi$
$908$	− 4.00000i	− 0.132745i
$909$	0	0
$910$	0	0
$911$	−50.0000	−1.65657	−0.828287	−	0.560304i	$-0.810684\pi$
$911$	−50.0000	−1.65657	−0.828287	+	0.560304i	$0.810684\pi$
$912$	0	0
$913$	18.0000i	0.595713i
$914$	20.0000	0.661541
$915$	0	0
$916$	−15.0000	−0.495614
$917$	18.0000i	0.594412i
$918$	0	0
$919$	−10.0000	−0.329870	−0.164935	−	0.986304i	$-0.552741\pi$
$919$	−10.0000	−0.329870	−0.164935	+	0.986304i	$0.552741\pi$
$920$	0	0
$921$	0	0
$922$	9.00000i	0.296399i
$923$	4.00000i	0.131662i
$924$	0	0
$925$	0	0
$926$	−36.0000	−1.18303
$927$	0	0
$928$	5.00000i	0.164133i
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	−16.0000	−0.524379
$932$	− 24.0000i	− 0.786146i
$933$	0	0
$934$	−20.0000	−0.654420
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	− 9.00000i	− 0.293860i
$939$	0	0
$940$	0	0
$941$	−7.00000	−0.228193	−0.114097	−	0.993470i	$-0.536397\pi$
$941$	−7.00000	−0.228193	−0.114097	+	0.993470i	$0.536397\pi$
$942$	0	0
$943$	− 15.0000i	− 0.488467i
$944$	−14.0000	−0.455661
$945$	0	0
$946$	16.0000	0.520205
$947$	57.0000i	1.85225i	0.377215	+	0.926126i	$0.376882\pi$
$947$	57.0000i	1.85225i	−0.377215	+	0.926126i	$0.623118\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	0	0
$951$	0	0
$952$	36.0000i	1.16677i
$953$	26.0000i	0.842223i	0.907009	+	0.421111i	$0.138360\pi$
$953$	26.0000i	0.842223i	−0.907009	+	0.421111i	$0.861640\pi$
$954$	0	0
$955$	0	0
$956$	8.00000	0.258738
$957$	0	0
$958$	18.0000i	0.581554i
$959$	36.0000	1.16250
$960$	0	0
$961$	−31.0000	−1.00000
$962$	8.00000i	0.257930i
$963$	0	0
$964$	−11.0000	−0.354286
$965$	0	0
$966$	0	0
$967$	41.0000i	1.31847i	0.751936	+	0.659236i	$0.229120\pi$
$967$	41.0000i	1.31847i	−0.751936	+	0.659236i	$0.770880\pi$
$968$	− 21.0000i	− 0.674966i
$969$	0	0
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	0	0
$973$	− 48.0000i	− 1.53881i
$974$	16.0000	0.512673
$975$	0	0
$976$	−7.00000	−0.224065
$977$	− 38.0000i	− 1.21573i	−0.794041	−	0.607864i	$-0.792027\pi$
$977$	− 38.0000i	− 1.21573i	0.794041	−	0.607864i	$-0.207973\pi$
$978$	0	0
$979$	−30.0000	−0.958804
$980$	0	0
$981$	0	0
$982$	20.0000i	0.638226i
$983$	− 3.00000i	− 0.0956851i	−0.998855	−	0.0478426i	$-0.984765\pi$
$983$	− 3.00000i	− 0.0956851i	0.998855	−	0.0478426i	$-0.0152346\pi$
$984$	0	0
$985$	0	0
$986$	−4.00000	−0.127386
$987$	0	0
$988$	16.0000i	0.509028i
$989$	24.0000	0.763156
$990$	0	0
$991$	26.0000	0.825917	0.412959	−	0.910750i	$-0.364495\pi$
$991$	26.0000	0.825917	0.412959	+	0.910750i	$0.364495\pi$
$992$	0	0
$993$	0	0
$994$	6.00000	0.190308
$995$	0	0
$996$	0	0
$997$	− 18.0000i	− 0.570066i	−0.958518	−	0.285033i	$-0.907995\pi$
$997$	− 18.0000i	− 0.570066i	0.958518	−	0.285033i	$-0.0920045\pi$
$998$	32.0000i	1.01294i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.b.c.649.2		2
3.2	odd	2		2025.2.b.d.649.1		2
5.2	odd	4		2025.2.a.b.1.1		1
5.3	odd	4		405.2.a.e.1.1		1
5.4	even	2	inner	2025.2.b.c.649.1		2
9.2	odd	6		675.2.k.a.199.1		4
9.4	even	3		225.2.k.a.124.1		4
9.5	odd	6		675.2.k.a.424.2		4
9.7	even	3		225.2.k.a.49.2		4
15.2	even	4		2025.2.a.e.1.1		1
15.8	even	4		405.2.a.b.1.1		1
15.14	odd	2		2025.2.b.d.649.2		2
20.3	even	4		6480.2.a.k.1.1		1
45.2	even	12		675.2.e.a.226.1		2
45.4	even	6		225.2.k.a.124.2		4
45.7	odd	12		225.2.e.a.76.1		2
45.13	odd	12		45.2.e.a.16.1	✓	2
45.14	odd	6		675.2.k.a.424.1		4
45.22	odd	12		225.2.e.a.151.1		2
45.23	even	12		135.2.e.a.46.1		2
45.29	odd	6		675.2.k.a.199.2		4
45.32	even	12		675.2.e.a.451.1		2
45.34	even	6		225.2.k.a.49.1		4
45.38	even	12		135.2.e.a.91.1		2
45.43	odd	12		45.2.e.a.31.1	yes	2
60.23	odd	4		6480.2.a.x.1.1		1
180.23	odd	12		2160.2.q.a.721.1		2
180.43	even	12		720.2.q.d.481.1		2
180.83	odd	12		2160.2.q.a.1441.1		2
180.103	even	12		720.2.q.d.241.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.e.a.16.1	✓	2	45.13	odd	12
45.2.e.a.31.1	yes	2	45.43	odd	12
135.2.e.a.46.1		2	45.23	even	12
135.2.e.a.91.1		2	45.38	even	12
225.2.e.a.76.1		2	45.7	odd	12
225.2.e.a.151.1		2	45.22	odd	12
225.2.k.a.49.1		4	45.34	even	6
225.2.k.a.49.2		4	9.7	even	3
225.2.k.a.124.1		4	9.4	even	3
225.2.k.a.124.2		4	45.4	even	6
405.2.a.b.1.1		1	15.8	even	4
405.2.a.e.1.1		1	5.3	odd	4
675.2.e.a.226.1		2	45.2	even	12
675.2.e.a.451.1		2	45.32	even	12
675.2.k.a.199.1		4	9.2	odd	6
675.2.k.a.199.2		4	45.29	odd	6
675.2.k.a.424.1		4	45.14	odd	6
675.2.k.a.424.2		4	9.5	odd	6
720.2.q.d.241.1		2	180.103	even	12
720.2.q.d.481.1		2	180.43	even	12
2025.2.a.b.1.1		1	5.2	odd	4
2025.2.a.e.1.1		1	15.2	even	4
2025.2.b.c.649.1		2	5.4	even	2	inner
2025.2.b.c.649.2		2	1.1	even	1	trivial
2025.2.b.d.649.1		2	3.2	odd	2
2025.2.b.d.649.2		2	15.14	odd	2
2160.2.q.a.721.1		2	180.23	odd	12
2160.2.q.a.1441.1		2	180.83	odd	12
6480.2.a.k.1.1		1	20.3	even	4
6480.2.a.x.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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{4} -3.00000i q^{7} +3.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{4} -3.00000i q^{7} +3.00000i q^{8} -2.00000 q^{11} +2.00000i q^{13} +3.00000 q^{14} -1.00000 q^{16} +4.00000i q^{17} +8.00000 q^{19} -2.00000i q^{22} -3.00000i q^{23} -2.00000 q^{26} -3.00000i q^{28} +1.00000 q^{29} +5.00000i q^{32} -4.00000 q^{34} -4.00000i q^{37} +8.00000i q^{38} +5.00000 q^{41} +8.00000i q^{43} -2.00000 q^{44} +3.00000 q^{46} +7.00000i q^{47} -2.00000 q^{49} +2.00000i q^{52} +2.00000i q^{53} +9.00000 q^{56} +1.00000i q^{58} +14.0000 q^{59} +7.00000 q^{61} -7.00000 q^{64} -3.00000i q^{67} +4.00000i q^{68} +2.00000 q^{71} -4.00000i q^{73} +4.00000 q^{74} +8.00000 q^{76} +6.00000i q^{77} +6.00000 q^{79} +5.00000i q^{82} -9.00000i q^{83} -8.00000 q^{86} -6.00000i q^{88} +15.0000 q^{89} +6.00000 q^{91} -3.00000i q^{92} -7.00000 q^{94} +2.00000i q^{97} -2.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4}+O(q^{10}) \) 2 * q + 2 * q^4	\( 2 q + 2 q^{4} - 4 q^{11} + 6 q^{14} - 2 q^{16} + 16 q^{19} - 4 q^{26} + 2 q^{29} - 8 q^{34} + 10 q^{41} - 4 q^{44} + 6 q^{46} - 4 q^{49} + 18 q^{56} + 28 q^{59} + 14 q^{61} - 14 q^{64} + 4 q^{71} + 8 q^{74} + 16 q^{76} + 12 q^{79} - 16 q^{86} + 30 q^{89} + 12 q^{91} - 14 q^{94}+O(q^{100}) \) 2 * q + 2 * q^4 - 4 * q^11 + 6 * q^14 - 2 * q^16 + 16 * q^19 - 4 * q^26 + 2 * q^29 - 8 * q^34 + 10 * q^41 - 4 * q^44 + 6 * q^46 - 4 * q^49 + 18 * q^56 + 28 * q^59 + 14 * q^61 - 14 * q^64 + 4 * q^71 + 8 * q^74 + 16 * q^76 + 12 * q^79 - 16 * q^86 + 30 * q^89 + 12 * q^91 - 14 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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