LMFDB - Embedded newform 2025.2.b.l.649.6

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:	$6$
Coefficient field:	6.0.5089536.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 16x^2 - 24*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.6
Root			$1.66044 - 1.66044i$ of defining polynomial
Character	$\chi$	$=$	2025.649
Dual form			2025.2.b.l.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.51414i q^{2} -4.32088 q^{4} -0.514137i q^{7} -5.83502i q^{8} -3.32088 q^{11} +1.32088i q^{13} +1.29261 q^{14} +6.02827 q^{16} -3.32088i q^{17} +1.32088 q^{19} -8.34916i q^{22} -4.12763i q^{23} -3.32088 q^{26} +2.22153i q^{28} +1.38650 q^{29} +8.73566 q^{31} +3.48586i q^{32} +8.34916 q^{34} +0.292611i q^{37} +3.32088i q^{38} -11.3492 q^{41} -10.3492i q^{43} +14.3492 q^{44} +10.3774 q^{46} -4.86330i q^{47} +6.73566 q^{49} -5.70739i q^{52} +5.02827i q^{53} -3.00000 q^{56} +3.48586i q^{58} +5.02827 q^{59} +7.34916 q^{61} +21.9627i q^{62} +3.29261 q^{64} +9.44852i q^{67} +14.3492i q^{68} +8.99093 q^{71} -6.05655i q^{73} -0.735663 q^{74} -5.70739 q^{76} +1.70739i q^{77} +8.05655 q^{79} -28.5333i q^{82} -1.54241i q^{83} +26.0192 q^{86} +19.3774i q^{88} +3.00000 q^{89} +0.679116 q^{91} +17.8350i q^{92} +12.2270 q^{94} -12.2553i q^{97} +16.9344i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 10 q^{4} - 4 q^{11} + 18 q^{14} + 10 q^{16} - 8 q^{19} - 4 q^{26} + 14 q^{29} + 16 q^{31} + 8 q^{34} - 26 q^{41} + 44 q^{44} - 6 q^{46} + 4 q^{49} - 18 q^{56} + 4 q^{59} + 2 q^{61} + 30 q^{64} - 20 q^{71}+ \cdots + 62 q^{94}+O(q^{100}) $ 6 * q - 10 * q^4 - 4 * q^11 + 18 * q^14 + 10 * q^16 - 8 * q^19 - 4 * q^26 + 14 * q^29 + 16 * q^31 + 8 * q^34 - 26 * q^41 + 44 * q^44 - 6 * q^46 + 4 * q^49 - 18 * q^56 + 4 * q^59 + 2 * q^61 + 30 * q^64 - 20 * q^71 + 32 * q^74 - 24 * q^76 - 4 * q^79 + 56 * q^86 + 18 * q^89 + 20 * q^91 + 62 * q^94

Character values

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

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

Coefficient data

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

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

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

$2$	2.51414i	1.77776i	0.458137	+	0.888882i	$0.348517\pi$
$2$	2.51414i	1.77776i	−0.458137	+	0.888882i	$0.651483\pi$
$3$	0	0
$3$	0	0
$4$	−4.32088	−2.16044
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.514137i	− 0.194325i	−0.995269	−	0.0971627i	$-0.969023\pi$
$7$	− 0.514137i	− 0.194325i	0.995269	−	0.0971627i	$-0.0309767\pi$
$8$	− 5.83502i	− 2.06299i
$9$	0	0
$10$	0	0
$11$	−3.32088	−1.00128	−0.500642	−	0.865654i	$-0.666903\pi$
$11$	−3.32088	−1.00128	−0.500642	+	0.865654i	$0.666903\pi$
$12$	0	0
$13$	1.32088i	0.366347i	0.983081	+	0.183174i	$0.0586371\pi$
$13$	1.32088i	0.366347i	−0.983081	+	0.183174i	$0.941363\pi$
$14$	1.29261	0.345465
$15$	0	0
$16$	6.02827	1.50707
$17$	− 3.32088i	− 0.805433i	−0.915325	−	0.402716i	$-0.868066\pi$
$17$	− 3.32088i	− 0.805433i	0.915325	−	0.402716i	$-0.131934\pi$
$18$	0	0
$19$	1.32088	0.303032	0.151516	−	0.988455i	$-0.451585\pi$
$19$	1.32088	0.303032	0.151516	+	0.988455i	$0.451585\pi$
$20$	0	0
$21$	0	0
$22$	− 8.34916i	− 1.78005i
$23$	− 4.12763i	− 0.860671i	−0.902669	−	0.430335i	$-0.858395\pi$
$23$	− 4.12763i	− 0.860671i	0.902669	−	0.430335i	$-0.141605\pi$
$24$	0	0
$25$	0	0
$26$	−3.32088	−0.651279
$27$	0	0
$28$	2.22153i	0.419829i
$29$	1.38650	0.257467	0.128734	−	0.991679i	$-0.458909\pi$
$29$	1.38650	0.257467	0.128734	+	0.991679i	$0.458909\pi$
$30$	0	0
$31$	8.73566	1.56897	0.784486	−	0.620147i	$-0.212927\pi$
$31$	8.73566	1.56897	0.784486	+	0.620147i	$0.212927\pi$
$32$	3.48586i	0.616219i
$33$	0	0
$34$	8.34916	1.43187
$35$	0	0
$36$	0	0
$37$	0.292611i	0.0481049i	0.999711	+	0.0240524i	$0.00765687\pi$
$37$	0.292611i	0.0481049i	−0.999711	+	0.0240524i	$0.992343\pi$
$38$	3.32088i	0.538719i
$39$	0	0
$40$	0	0
$41$	−11.3492	−1.77244	−0.886220	−	0.463264i	$-0.846678\pi$
$41$	−11.3492	−1.77244	−0.886220	+	0.463264i	$0.846678\pi$
$42$	0	0
$43$	− 10.3492i	− 1.57823i	−0.614244	−	0.789116i	$-0.710539\pi$
$43$	− 10.3492i	− 1.57823i	0.614244	−	0.789116i	$-0.289461\pi$
$44$	14.3492	2.16322
$45$	0	0
$46$	10.3774	1.53007
$47$	− 4.86330i	− 0.709385i	−0.934983	−	0.354692i	$-0.884586\pi$
$47$	− 4.86330i	− 0.709385i	0.934983	−	0.354692i	$-0.115414\pi$
$48$	0	0
$49$	6.73566	0.962238
$50$	0	0
$51$	0	0
$52$	− 5.70739i	− 0.791472i
$53$	5.02827i	0.690687i	0.938476	+	0.345343i	$0.112238\pi$
$53$	5.02827i	0.690687i	−0.938476	+	0.345343i	$0.887762\pi$
$54$	0	0
$55$	0	0
$56$	−3.00000	−0.400892
$57$	0	0
$58$	3.48586i	0.457716i
$59$	5.02827	0.654625	0.327313	−	0.944916i	$-0.393857\pi$
$59$	5.02827	0.654625	0.327313	+	0.944916i	$0.393857\pi$
$60$	0	0
$61$	7.34916	0.940963	0.470482	−	0.882410i	$-0.344080\pi$
$61$	7.34916	0.940963	0.470482	+	0.882410i	$0.344080\pi$
$62$	21.9627i	2.78926i
$63$	0	0
$64$	3.29261	0.411576
$65$	0	0
$66$	0	0
$67$	9.44852i	1.15432i	0.816631	+	0.577160i	$0.195839\pi$
$67$	9.44852i	1.15432i	−0.816631	+	0.577160i	$0.804161\pi$
$68$	14.3492i	1.74009i
$69$	0	0
$70$	0	0
$71$	8.99093	1.06703	0.533513	−	0.845792i	$-0.320871\pi$
$71$	8.99093	1.06703	0.533513	+	0.845792i	$0.320871\pi$
$72$	0	0
$73$	− 6.05655i	− 0.708865i	−0.935082	−	0.354433i	$-0.884674\pi$
$73$	− 6.05655i	− 0.708865i	0.935082	−	0.354433i	$-0.115326\pi$
$74$	−0.735663	−0.0855191
$75$	0	0
$76$	−5.70739	−0.654682
$77$	1.70739i	0.194575i
$78$	0	0
$79$	8.05655	0.906432	0.453216	−	0.891401i	$-0.350277\pi$
$79$	8.05655	0.906432	0.453216	+	0.891401i	$0.350277\pi$
$80$	0	0
$81$	0	0
$82$	− 28.5333i	− 3.15098i
$83$	− 1.54241i	− 0.169302i	−0.996411	−	0.0846508i	$-0.973023\pi$
$83$	− 1.54241i	− 0.169302i	0.996411	−	0.0846508i	$-0.0269775\pi$
$84$	0	0
$85$	0	0
$86$	26.0192	2.80572
$87$	0	0
$88$	19.3774i	2.06564i
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	0	0
$91$	0.679116	0.0711906
$92$	17.8350i	1.85943i
$93$	0	0
$94$	12.2270	1.26112
$95$	0	0
$96$	0	0
$97$	− 12.2553i	− 1.24433i	−0.782885	−	0.622167i	$-0.786253\pi$
$97$	− 12.2553i	− 1.24433i	0.782885	−	0.622167i	$-0.213747\pi$
$98$	16.9344i	1.71063i
$99$	0	0
$100$	0	0
$101$	11.6700	1.16121	0.580606	−	0.814184i	$-0.302816\pi$
$101$	11.6700	1.16121	0.580606	+	0.814184i	$0.302816\pi$
$102$	0	0
$103$	− 0.292611i	− 0.0288318i	−0.999896	−	0.0144159i	$-0.995411\pi$
$103$	− 0.292611i	− 0.0288318i	0.999896	−	0.0144159i	$-0.00458888\pi$
$104$	7.70739	0.755772
$105$	0	0
$106$	−12.6418	−1.22788
$107$	1.87237i	0.181009i	0.995896	+	0.0905043i	$0.0288479\pi$
$107$	1.87237i	0.181009i	−0.995896	+	0.0905043i	$0.971152\pi$
$108$	0	0
$109$	−5.54787	−0.531390	−0.265695	−	0.964057i	$-0.585601\pi$
$109$	−5.54787	−0.531390	−0.265695	+	0.964057i	$0.585601\pi$
$110$	0	0
$111$	0	0
$112$	− 3.09936i	− 0.292862i
$113$	− 7.80128i	− 0.733883i	−0.930244	−	0.366942i	$-0.880405\pi$
$113$	− 7.80128i	− 0.733883i	0.930244	−	0.366942i	$-0.119595\pi$
$114$	0	0
$115$	0	0
$116$	−5.99093	−0.556244
$117$	0	0
$118$	12.6418i	1.16377i
$119$	−1.70739	−0.156516
$120$	0	0
$121$	0.0282739	0.00257035
$122$	18.4768i	1.67281i
$123$	0	0
$124$	−37.7458	−3.38967
$125$	0	0
$126$	0	0
$127$	17.8916i	1.58762i	0.608166	+	0.793810i	$0.291906\pi$
$127$	17.8916i	1.58762i	−0.608166	+	0.793810i	$0.708094\pi$
$128$	15.2498i	1.34790i
$129$	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
$133$	− 0.679116i	− 0.0588868i
$134$	−23.7549	−2.05211
$135$	0	0
$136$	−19.3774	−1.66160
$137$	5.67004i	0.484424i	0.970223	+	0.242212i	$0.0778730\pi$
$137$	5.67004i	0.484424i	−0.970223	+	0.242212i	$0.922127\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	22.6044i	1.89692i
$143$	− 4.38650i	− 0.366818i
$144$	0	0
$145$	0	0
$146$	15.2270	1.26019
$147$	0	0
$148$	− 1.26434i	− 0.103928i
$149$	−17.6610	−1.44684	−0.723422	−	0.690407i	$-0.757432\pi$
$149$	−17.6610	−1.44684	−0.723422	+	0.690407i	$0.757432\pi$
$150$	0	0
$151$	1.26434	0.102890	0.0514451	−	0.998676i	$-0.483617\pi$
$151$	1.26434	0.102890	0.0514451	+	0.998676i	$0.483617\pi$
$152$	− 7.70739i	− 0.625152i
$153$	0	0
$154$	−4.29261	−0.345908
$155$	0	0
$156$	0	0
$157$	− 15.6700i	− 1.25061i	−0.780382	−	0.625303i	$-0.784975\pi$
$157$	− 15.6700i	− 1.25061i	0.780382	−	0.625303i	$-0.215025\pi$
$158$	20.2553i	1.61142i
$159$	0	0
$160$	0	0
$161$	−2.12217	−0.167250
$162$	0	0
$163$	− 15.7074i	− 1.23030i	−0.788411	−	0.615149i	$-0.789096\pi$
$163$	− 15.7074i	− 1.23030i	0.788411	−	0.615149i	$-0.210904\pi$
$164$	49.0384	3.82926
$165$	0	0
$166$	3.87783	0.300978
$167$	6.16498i	0.477060i	0.971135	+	0.238530i	$0.0766656\pi$
$167$	6.16498i	0.477060i	−0.971135	+	0.238530i	$0.923334\pi$
$168$	0	0
$169$	11.2553	0.865790
$170$	0	0
$171$	0	0
$172$	44.7175i	3.40968i
$173$	− 8.58522i	− 0.652722i	−0.945245	−	0.326361i	$-0.894177\pi$
$173$	− 8.58522i	− 0.652722i	0.945245	−	0.326361i	$-0.105823\pi$
$174$	0	0
$175$	0	0
$176$	−20.0192	−1.50900
$177$	0	0
$178$	7.54241i	0.565328i
$179$	1.06562	0.0796482	0.0398241	−	0.999207i	$-0.487320\pi$
$179$	1.06562	0.0796482	0.0398241	+	0.999207i	$0.487320\pi$
$180$	0	0
$181$	−12.6700	−0.941757	−0.470878	−	0.882198i	$-0.656063\pi$
$181$	−12.6700	−0.941757	−0.470878	+	0.882198i	$0.656063\pi$
$182$	1.70739i	0.126560i
$183$	0	0
$184$	−24.0848	−1.77556
$185$	0	0
$186$	0	0
$187$	11.0283i	0.806467i
$188$	21.0137i	1.53258i
$189$	0	0
$190$	0	0
$191$	−16.9344	−1.22533	−0.612664	−	0.790343i	$-0.709902\pi$
$191$	−16.9344	−1.22533	−0.612664	+	0.790343i	$0.709902\pi$
$192$	0	0
$193$	− 26.7175i	− 1.92317i	−0.274509	−	0.961585i	$-0.588515\pi$
$193$	− 26.7175i	− 1.92317i	0.274509	−	0.961585i	$-0.411485\pi$
$194$	30.8114	2.21213
$195$	0	0
$196$	−29.1040	−2.07886
$197$	− 14.2553i	− 1.01565i	−0.861462	−	0.507823i	$-0.830450\pi$
$197$	− 14.2553i	− 1.01565i	0.861462	−	0.507823i	$-0.169550\pi$
$198$	0	0
$199$	24.6610	1.74817	0.874085	−	0.485773i	$-0.161462\pi$
$199$	24.6610	1.74817	0.874085	+	0.485773i	$0.161462\pi$
$200$	0	0
$201$	0	0
$202$	29.3401i	2.06436i
$203$	− 0.712853i	− 0.0500325i
$204$	0	0
$205$	0	0
$206$	0.735663	0.0512561
$207$	0	0
$208$	7.96265i	0.552111i
$209$	−4.38650	−0.303421
$210$	0	0
$211$	−5.37743	−0.370198	−0.185099	−	0.982720i	$-0.559261\pi$
$211$	−5.37743	−0.370198	−0.185099	+	0.982720i	$0.559261\pi$
$212$	− 21.7266i	− 1.49219i
$213$	0	0
$214$	−4.70739	−0.321791
$215$	0	0
$216$	0	0
$217$	− 4.49133i	− 0.304891i
$218$	− 13.9481i	− 0.944686i
$219$	0	0
$220$	0	0
$221$	4.38650	0.295068
$222$	0	0
$223$	− 8.66458i	− 0.580223i	−0.956993	−	0.290112i	$-0.906308\pi$
$223$	− 8.66458i	− 0.580223i	0.956993	−	0.290112i	$-0.0936924\pi$
$224$	1.79221	0.119747
$225$	0	0
$226$	19.6135	1.30467
$227$	3.32088i	0.220415i	0.993909	+	0.110207i	$0.0351515\pi$
$227$	3.32088i	0.220415i	−0.993909	+	0.110207i	$0.964848\pi$
$228$	0	0
$229$	25.3118	1.67265	0.836326	−	0.548233i	$-0.184699\pi$
$229$	25.3118	1.67265	0.836326	+	0.548233i	$0.184699\pi$
$230$	0	0
$231$	0	0
$232$	− 8.09029i	− 0.531153i
$233$	− 27.6327i	− 1.81028i	−0.425116	−	0.905139i	$-0.639767\pi$
$233$	− 27.6327i	− 1.81028i	0.425116	−	0.905139i	$-0.360233\pi$
$234$	0	0
$235$	0	0
$236$	−21.7266	−1.41428
$237$	0	0
$238$	− 4.29261i	− 0.278249i
$239$	4.19872	0.271592	0.135796	−	0.990737i	$-0.456641\pi$
$239$	4.19872	0.271592	0.135796	+	0.990737i	$0.456641\pi$
$240$	0	0
$241$	3.60442	0.232181	0.116091	−	0.993239i	$-0.462964\pi$
$241$	3.60442	0.232181	0.116091	+	0.993239i	$0.462964\pi$
$242$	0.0710844i	0.00456948i
$243$	0	0
$244$	−31.7549	−2.03290
$245$	0	0
$246$	0	0
$247$	1.74474i	0.111015i
$248$	− 50.9728i	− 3.23677i
$249$	0	0
$250$	0	0
$251$	−6.87783	−0.434125	−0.217062	−	0.976158i	$-0.569648\pi$
$251$	−6.87783	−0.434125	−0.217062	+	0.976158i	$0.569648\pi$
$252$	0	0
$253$	13.7074i	0.861776i
$254$	−44.9819	−2.82241
$255$	0	0
$256$	−31.7549	−1.98468
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	0	0
$259$	0.150442	0.00934801
$260$	0	0
$261$	0	0
$262$	− 15.0848i	− 0.931943i
$263$	− 6.23606i	− 0.384532i	−0.981343	−	0.192266i	$-0.938416\pi$
$263$	− 6.23606i	− 0.384532i	0.981343	−	0.192266i	$-0.0615837\pi$
$264$	0	0
$265$	0	0
$266$	1.70739	0.104687
$267$	0	0
$268$	− 40.8259i	− 2.49384i
$269$	−9.92345	−0.605044	−0.302522	−	0.953142i	$-0.597828\pi$
$269$	−9.92345	−0.605044	−0.302522	+	0.953142i	$0.597828\pi$
$270$	0	0
$271$	6.60442	0.401190	0.200595	−	0.979674i	$-0.435712\pi$
$271$	6.60442	0.401190	0.200595	+	0.979674i	$0.435712\pi$
$272$	− 20.0192i	− 1.21384i
$273$	0	0
$274$	−14.2553	−0.861192
$275$	0	0
$276$	0	0
$277$	22.6610i	1.36157i	0.732485	+	0.680783i	$0.238360\pi$
$277$	22.6610i	1.36157i	−0.732485	+	0.680783i	$0.761640\pi$
$278$	− 20.1131i	− 1.20630i
$279$	0	0
$280$	0	0
$281$	−15.5479	−0.927508	−0.463754	−	0.885964i	$-0.653498\pi$
$281$	−15.5479	−0.927508	−0.463754	+	0.885964i	$0.653498\pi$
$282$	0	0
$283$	− 0.645378i	− 0.0383637i	−0.999816	−	0.0191819i	$-0.993894\pi$
$283$	− 0.645378i	− 0.0383637i	0.999816	−	0.0191819i	$-0.00610615\pi$
$284$	−38.8488	−2.30525
$285$	0	0
$286$	11.0283	0.652116
$287$	5.83502i	0.344430i
$288$	0	0
$289$	5.97173	0.351278
$290$	0	0
$291$	0	0
$292$	26.1696i	1.53146i
$293$	1.37743i	0.0804704i	0.999190	+	0.0402352i	$0.0128107\pi$
$293$	1.37743i	0.0804704i	−0.999190	+	0.0402352i	$0.987189\pi$
$294$	0	0
$295$	0	0
$296$	1.70739	0.0992400
$297$	0	0
$298$	− 44.4021i	− 2.57214i
$299$	5.45213	0.315305
$300$	0	0
$301$	−5.32088	−0.306691
$302$	3.17872i	0.182915i
$303$	0	0
$304$	7.96265	0.456689
$305$	0	0
$306$	0	0
$307$	− 7.98546i	− 0.455754i	−0.973690	−	0.227877i	$-0.926822\pi$
$307$	− 7.98546i	− 0.455754i	0.973690	−	0.227877i	$-0.0731785\pi$
$308$	− 7.37743i	− 0.420368i
$309$	0	0
$310$	0	0
$311$	9.63270	0.546220	0.273110	−	0.961983i	$-0.411948\pi$
$311$	9.63270	0.546220	0.273110	+	0.961983i	$0.411948\pi$
$312$	0	0
$313$	24.5369i	1.38691i	0.720500	+	0.693455i	$0.243912\pi$
$313$	24.5369i	1.38691i	−0.720500	+	0.693455i	$0.756088\pi$
$314$	39.3966	2.22328
$315$	0	0
$316$	−34.8114	−1.95829
$317$	− 20.3492i	− 1.14292i	−0.820629	−	0.571461i	$-0.806377\pi$
$317$	− 20.3492i	− 1.14292i	0.820629	−	0.571461i	$-0.193623\pi$
$318$	0	0
$319$	−4.60442	−0.257798
$320$	0	0
$321$	0	0
$322$	− 5.33542i	− 0.297331i
$323$	− 4.38650i	− 0.244072i
$324$	0	0
$325$	0	0
$326$	39.4905	2.18718
$327$	0	0
$328$	66.2226i	3.65653i
$329$	−2.50040	−0.137851
$330$	0	0
$331$	16.4431	0.903792	0.451896	−	0.892071i	$-0.350748\pi$
$331$	16.4431	0.903792	0.451896	+	0.892071i	$0.350748\pi$
$332$	6.66458i	0.365766i
$333$	0	0
$334$	−15.4996	−0.848100
$335$	0	0
$336$	0	0
$337$	4.89703i	0.266758i	0.991065	+	0.133379i	$0.0425828\pi$
$337$	4.89703i	0.266758i	−0.991065	+	0.133379i	$0.957417\pi$
$338$	28.2973i	1.53917i
$339$	0	0
$340$	0	0
$341$	−29.0101	−1.57099
$342$	0	0
$343$	− 7.06201i	− 0.381313i
$344$	−60.3876	−3.25588
$345$	0	0
$346$	21.5844	1.16039
$347$	22.2745i	1.19576i	0.801587	+	0.597878i	$0.203989\pi$
$347$	22.2745i	1.19576i	−0.801587	+	0.597878i	$0.796011\pi$
$348$	0	0
$349$	2.94345	0.157559	0.0787797	−	0.996892i	$-0.474898\pi$
$349$	2.94345	0.157559	0.0787797	+	0.996892i	$0.474898\pi$
$350$	0	0
$351$	0	0
$352$	− 11.5761i	− 0.617011i
$353$	− 18.8296i	− 1.00220i	−0.865390	−	0.501098i	$-0.832930\pi$
$353$	− 18.8296i	− 1.00220i	0.865390	−	0.501098i	$-0.167070\pi$
$354$	0	0
$355$	0	0
$356$	−12.9627	−0.687019
$357$	0	0
$358$	2.67912i	0.141596i
$359$	31.8770	1.68241	0.841203	−	0.540720i	$-0.181848\pi$
$359$	31.8770	1.68241	0.841203	+	0.540720i	$0.181848\pi$
$360$	0	0
$361$	−17.2553	−0.908172
$362$	− 31.8542i	− 1.67422i
$363$	0	0
$364$	−2.93438	−0.153803
$365$	0	0
$366$	0	0
$367$	− 18.3492i	− 0.957818i	−0.877864	−	0.478909i	$-0.841032\pi$
$367$	− 18.3492i	− 0.957818i	0.877864	−	0.478909i	$-0.158968\pi$
$368$	− 24.8825i	− 1.29709i
$369$	0	0
$370$	0	0
$371$	2.58522	0.134218
$372$	0	0
$373$	2.19872i	0.113845i	0.998379	+	0.0569226i	$0.0181288\pi$
$373$	2.19872i	0.113845i	−0.998379	+	0.0569226i	$0.981871\pi$
$374$	−27.7266	−1.43371
$375$	0	0
$376$	−28.3774	−1.46345
$377$	1.83141i	0.0943226i
$378$	0	0
$379$	−15.4713	−0.794709	−0.397354	−	0.917665i	$-0.630072\pi$
$379$	−15.4713	−0.794709	−0.397354	+	0.917665i	$0.630072\pi$
$380$	0	0
$381$	0	0
$382$	− 42.5753i	− 2.17834i
$383$	− 7.70739i	− 0.393829i	−0.980421	−	0.196915i	$-0.936908\pi$
$383$	− 7.70739i	− 0.393829i	0.980421	−	0.196915i	$-0.0630922\pi$
$384$	0	0
$385$	0	0
$386$	67.1715	3.41894
$387$	0	0
$388$	52.9536i	2.68831i
$389$	24.6327	1.24893	0.624464	−	0.781054i	$-0.285318\pi$
$389$	24.6327	1.24893	0.624464	+	0.781054i	$0.285318\pi$
$390$	0	0
$391$	−13.7074	−0.693212
$392$	− 39.3027i	− 1.98509i
$393$	0	0
$394$	35.8397	1.80558
$395$	0	0
$396$	0	0
$397$	− 6.77301i	− 0.339928i	−0.985450	−	0.169964i	$-0.945635\pi$
$397$	− 6.77301i	− 0.339928i	0.985450	−	0.169964i	$-0.0543651\pi$
$398$	62.0011i	3.10783i
$399$	0	0
$400$	0	0
$401$	−18.4996	−0.923826	−0.461913	−	0.886925i	$-0.652837\pi$
$401$	−18.4996	−0.923826	−0.461913	+	0.886925i	$0.652837\pi$
$402$	0	0
$403$	11.5388i	0.574789i
$404$	−50.4249	−2.50873
$405$	0	0
$406$	1.79221	0.0889459
$407$	− 0.971726i	− 0.0481667i
$408$	0	0
$409$	13.4148	0.663318	0.331659	−	0.943399i	$-0.392392\pi$
$409$	13.4148	0.663318	0.331659	+	0.943399i	$0.392392\pi$
$410$	0	0
$411$	0	0
$412$	1.26434i	0.0622894i
$413$	− 2.58522i	− 0.127210i
$414$	0	0
$415$	0	0
$416$	−4.60442	−0.225750
$417$	0	0
$418$	− 11.0283i	− 0.539411i
$419$	33.1150	1.61777	0.808886	−	0.587966i	$-0.200071\pi$
$419$	33.1150	1.61777	0.808886	+	0.587966i	$0.200071\pi$
$420$	0	0
$421$	−14.6983	−0.716352	−0.358176	−	0.933654i	$-0.616601\pi$
$421$	−14.6983	−0.716352	−0.358176	+	0.933654i	$0.616601\pi$
$422$	− 13.5196i	− 0.658124i
$423$	0	0
$424$	29.3401	1.42488
$425$	0	0
$426$	0	0
$427$	− 3.77847i	− 0.182853i
$428$	− 8.09029i	− 0.391059i
$429$	0	0
$430$	0	0
$431$	−32.7549	−1.57775	−0.788873	−	0.614556i	$-0.789335\pi$
$431$	−32.7549	−1.57775	−0.788873	+	0.614556i	$0.789335\pi$
$432$	0	0
$433$	11.8314i	0.568581i	0.958738	+	0.284291i	$0.0917581\pi$
$433$	11.8314i	0.568581i	−0.958738	+	0.284291i	$0.908242\pi$
$434$	11.2918	0.542024
$435$	0	0
$436$	23.9717	1.14804
$437$	− 5.45213i	− 0.260811i
$438$	0	0
$439$	−8.31181	−0.396701	−0.198351	−	0.980131i	$-0.563558\pi$
$439$	−8.31181	−0.396701	−0.198351	+	0.980131i	$0.563558\pi$
$440$	0	0
$441$	0	0
$442$	11.0283i	0.524561i
$443$	29.1751i	1.38615i	0.720865	+	0.693076i	$0.243745\pi$
$443$	29.1751i	1.38615i	−0.720865	+	0.693076i	$0.756255\pi$
$444$	0	0
$445$	0	0
$446$	21.7839	1.03150
$447$	0	0
$448$	− 1.69285i	− 0.0799798i
$449$	−18.9717	−0.895331	−0.447666	−	0.894201i	$-0.647744\pi$
$449$	−18.9717	−0.895331	−0.447666	+	0.894201i	$0.647744\pi$
$450$	0	0
$451$	37.6892	1.77472
$452$	33.7084i	1.58551i
$453$	0	0
$454$	−8.34916	−0.391845
$455$	0	0
$456$	0	0
$457$	− 23.2353i	− 1.08690i	−0.839442	−	0.543450i	$-0.817118\pi$
$457$	− 23.2353i	− 1.08690i	0.839442	−	0.543450i	$-0.182882\pi$
$458$	63.6374i	2.97358i
$459$	0	0
$460$	0	0
$461$	4.42571	0.206126	0.103063	−	0.994675i	$-0.467136\pi$
$461$	4.42571	0.206126	0.103063	+	0.994675i	$0.467136\pi$
$462$	0	0
$463$	19.5087i	0.906645i	0.891347	+	0.453322i	$0.149761\pi$
$463$	19.5087i	0.906645i	−0.891347	+	0.453322i	$0.850239\pi$
$464$	8.35823	0.388021
$465$	0	0
$466$	69.4724	3.21825
$467$	− 24.5935i	− 1.13805i	−0.822320	−	0.569026i	$-0.807321\pi$
$467$	− 24.5935i	− 1.13805i	0.822320	−	0.569026i	$-0.192679\pi$
$468$	0	0
$469$	4.85783	0.224314
$470$	0	0
$471$	0	0
$472$	− 29.3401i	− 1.35049i
$473$	34.3684i	1.58026i
$474$	0	0
$475$	0	0
$476$	7.37743	0.338144
$477$	0	0
$478$	10.5561i	0.482827i
$479$	−32.7549	−1.49661	−0.748304	−	0.663356i	$-0.769132\pi$
$479$	−32.7549	−1.49661	−0.748304	+	0.663356i	$0.769132\pi$
$480$	0	0
$481$	−0.386505	−0.0176231
$482$	9.06201i	0.412763i
$483$	0	0
$484$	−0.122168	−0.00555309
$485$	0	0
$486$	0	0
$487$	− 6.03735i	− 0.273578i	−0.990600	−	0.136789i	$-0.956322\pi$
$487$	− 6.03735i	− 0.273578i	0.990600	−	0.136789i	$-0.0436783\pi$
$488$	− 42.8825i	− 1.94120i
$489$	0	0
$490$	0	0
$491$	14.4431	0.651806	0.325903	−	0.945403i	$-0.394332\pi$
$491$	14.4431	0.651806	0.325903	+	0.945403i	$0.394332\pi$
$492$	0	0
$493$	− 4.60442i	− 0.207373i
$494$	−4.38650	−0.197358
$495$	0	0
$496$	52.6610	2.36455
$497$	− 4.62257i	− 0.207351i
$498$	0	0
$499$	−20.9717	−0.938823	−0.469412	−	0.882979i	$-0.655534\pi$
$499$	−20.9717	−0.938823	−0.469412	+	0.882979i	$0.655534\pi$
$500$	0	0
$501$	0	0
$502$	− 17.2918i	− 0.771771i
$503$	5.31728i	0.237086i	0.992949	+	0.118543i	$0.0378223\pi$
$503$	5.31728i	0.237086i	−0.992949	+	0.118543i	$0.962178\pi$
$504$	0	0
$505$	0	0
$506$	−34.4623	−1.53203
$507$	0	0
$508$	− 77.3074i	− 3.42996i
$509$	−18.2270	−0.807897	−0.403949	−	0.914782i	$-0.632363\pi$
$509$	−18.2270	−0.807897	−0.403949	+	0.914782i	$0.632363\pi$
$510$	0	0
$511$	−3.11389	−0.137751
$512$	− 49.3365i	− 2.18038i
$513$	0	0
$514$	45.2545	1.99609
$515$	0	0
$516$	0	0
$517$	16.1504i	0.710296i
$518$	0.378232i	0.0166185i
$519$	0	0
$520$	0	0
$521$	40.1232	1.75783	0.878915	−	0.476978i	$-0.158268\pi$
$521$	40.1232	1.75783	0.878915	+	0.476978i	$0.158268\pi$
$522$	0	0
$523$	− 18.9873i	− 0.830257i	−0.909763	−	0.415129i	$-0.863737\pi$
$523$	− 18.9873i	− 0.830257i	0.909763	−	0.415129i	$-0.136263\pi$
$524$	25.9253	1.13255
$525$	0	0
$526$	15.6783	0.683607
$527$	− 29.0101i	− 1.26370i
$528$	0	0
$529$	5.96265	0.259246
$530$	0	0
$531$	0	0
$532$	2.93438i	0.127221i
$533$	− 14.9909i	− 0.649329i
$534$	0	0
$535$	0	0
$536$	55.1323	2.38135
$537$	0	0
$538$	− 24.9489i	− 1.07562i
$539$	−22.3684	−0.963473
$540$	0	0
$541$	16.5279	0.710589	0.355294	−	0.934754i	$-0.384381\pi$
$541$	16.5279	0.710589	0.355294	+	0.934754i	$0.384381\pi$
$542$	16.6044i	0.713221i
$543$	0	0
$544$	11.5761	0.496323
$545$	0	0
$546$	0	0
$547$	17.6737i	0.755671i	0.925873	+	0.377835i	$0.123331\pi$
$547$	17.6737i	0.755671i	−0.925873	+	0.377835i	$0.876669\pi$
$548$	− 24.4996i	− 1.04657i
$549$	0	0
$550$	0	0
$551$	1.83141	0.0780208
$552$	0	0
$553$	− 4.14217i	− 0.176143i
$554$	−56.9728	−2.42054
$555$	0	0
$556$	34.5671	1.46597
$557$	17.3401i	0.734723i	0.930078	+	0.367362i	$0.119739\pi$
$557$	17.3401i	0.734723i	−0.930078	+	0.367362i	$0.880261\pi$
$558$	0	0
$559$	13.6700	0.578181
$560$	0	0
$561$	0	0
$562$	− 39.0895i	− 1.64889i
$563$	− 12.9945i	− 0.547654i	−0.961779	−	0.273827i	$-0.911710\pi$
$563$	− 12.9945i	− 0.547654i	0.961779	−	0.273827i	$-0.0882896\pi$
$564$	0	0
$565$	0	0
$566$	1.62257	0.0682016
$567$	0	0
$568$	− 52.4623i	− 2.20127i
$569$	16.6802	0.699269	0.349635	−	0.936886i	$-0.386306\pi$
$569$	16.6802	0.699269	0.349635	+	0.936886i	$0.386306\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	18.9536i	0.792489i
$573$	0	0
$574$	−14.6700	−0.612316
$575$	0	0
$576$	0	0
$577$	− 23.5953i	− 0.982287i	−0.871079	−	0.491144i	$-0.836579\pi$
$577$	− 23.5953i	− 0.982287i	0.871079	−	0.491144i	$-0.163421\pi$
$578$	15.0137i	0.624489i
$579$	0	0
$580$	0	0
$581$	−0.793010	−0.0328996
$582$	0	0
$583$	− 16.6983i	− 0.691574i
$584$	−35.3401	−1.46238
$585$	0	0
$586$	−3.46305	−0.143057
$587$	28.1276i	1.16095i	0.814277	+	0.580476i	$0.197133\pi$
$587$	28.1276i	1.16095i	−0.814277	+	0.580476i	$0.802867\pi$
$588$	0	0
$589$	11.5388	0.475448
$590$	0	0
$591$	0	0
$592$	1.76394i	0.0724974i
$593$	9.17872i	0.376925i	0.982080	+	0.188462i	$0.0603503\pi$
$593$	9.17872i	0.376925i	−0.982080	+	0.188462i	$0.939650\pi$
$594$	0	0
$595$	0	0
$596$	76.3110	3.12582
$597$	0	0
$598$	13.7074i	0.560537i
$599$	31.4713	1.28588	0.642942	−	0.765915i	$-0.277714\pi$
$599$	31.4713	1.28588	0.642942	+	0.765915i	$0.277714\pi$
$600$	0	0
$601$	−29.2654	−1.19376	−0.596880	−	0.802330i	$-0.703593\pi$
$601$	−29.2654	−1.19376	−0.596880	+	0.802330i	$0.703593\pi$
$602$	− 13.3774i	− 0.545223i
$603$	0	0
$604$	−5.46305	−0.222288
$605$	0	0
$606$	0	0
$607$	− 44.2034i	− 1.79416i	−0.441868	−	0.897080i	$-0.645684\pi$
$607$	− 44.2034i	− 1.79416i	0.441868	−	0.897080i	$-0.354316\pi$
$608$	4.60442i	0.186734i
$609$	0	0
$610$	0	0
$611$	6.42385	0.259881
$612$	0	0
$613$	35.1715i	1.42056i	0.703918	+	0.710282i	$0.251432\pi$
$613$	35.1715i	1.42056i	−0.703918	+	0.710282i	$0.748568\pi$
$614$	20.0765	0.810224
$615$	0	0
$616$	9.96265	0.401407
$617$	− 7.42571i	− 0.298948i	−0.988766	−	0.149474i	$-0.952242\pi$
$617$	− 7.42571i	− 0.298948i	0.988766	−	0.149474i	$-0.0477580\pi$
$618$	0	0
$619$	−8.54787	−0.343568	−0.171784	−	0.985135i	$-0.554953\pi$
$619$	−8.54787	−0.343568	−0.171784	+	0.985135i	$0.554953\pi$
$620$	0	0
$621$	0	0
$622$	24.2179i	0.971050i
$623$	− 1.54241i	− 0.0617954i
$624$	0	0
$625$	0	0
$626$	−61.6892	−2.46560
$627$	0	0
$628$	67.7084i	2.70186i
$629$	0.971726	0.0387453
$630$	0	0
$631$	−2.36836	−0.0942829	−0.0471415	−	0.998888i	$-0.515011\pi$
$631$	−2.36836	−0.0942829	−0.0471415	+	0.998888i	$0.515011\pi$
$632$	− 47.0101i	− 1.86996i
$633$	0	0
$634$	51.1606	2.03185
$635$	0	0
$636$	0	0
$637$	8.89703i	0.352513i
$638$	− 11.5761i	− 0.458304i
$639$	0	0
$640$	0	0
$641$	−0.133096	−0.00525698	−0.00262849	−	0.999997i	$-0.500837\pi$
$641$	−0.133096	−0.00525698	−0.00262849	+	0.999997i	$0.500837\pi$
$642$	0	0
$643$	22.6464i	0.893088i	0.894762	+	0.446544i	$0.147345\pi$
$643$	22.6464i	0.893088i	−0.894762	+	0.446544i	$0.852655\pi$
$644$	9.16964	0.361335
$645$	0	0
$646$	11.0283	0.433902
$647$	46.3912i	1.82383i	0.410385	+	0.911913i	$0.365394\pi$
$647$	46.3912i	1.82383i	−0.410385	+	0.911913i	$0.634606\pi$
$648$	0	0
$649$	−16.6983	−0.655466
$650$	0	0
$651$	0	0
$652$	67.8698i	2.65799i
$653$	− 36.4057i	− 1.42467i	−0.701842	−	0.712333i	$-0.747639\pi$
$653$	− 36.4057i	− 1.42467i	0.701842	−	0.712333i	$-0.252361\pi$
$654$	0	0
$655$	0	0
$656$	−68.4158	−2.67119
$657$	0	0
$658$	− 6.28635i	− 0.245067i
$659$	−19.1414	−0.745642	−0.372821	−	0.927903i	$-0.621609\pi$
$659$	−19.1414	−0.745642	−0.372821	+	0.927903i	$0.621609\pi$
$660$	0	0
$661$	39.9072	1.55221	0.776104	−	0.630605i	$-0.217193\pi$
$661$	39.9072	1.55221	0.776104	+	0.630605i	$0.217193\pi$
$662$	41.3401i	1.60673i
$663$	0	0
$664$	−9.00000	−0.349268
$665$	0	0
$666$	0	0
$667$	− 5.72298i	− 0.221595i
$668$	− 26.6382i	− 1.03066i
$669$	0	0
$670$	0	0
$671$	−24.4057	−0.942172
$672$	0	0
$673$	− 23.6508i	− 0.911673i	−0.890064	−	0.455836i	$-0.849340\pi$
$673$	− 23.6508i	− 0.911673i	0.890064	−	0.455836i	$-0.150660\pi$
$674$	−12.3118	−0.474233
$675$	0	0
$676$	−48.6327	−1.87049
$677$	− 14.8031i	− 0.568931i	−0.958686	−	0.284465i	$-0.908184\pi$
$677$	− 14.8031i	− 0.568931i	0.958686	−	0.284465i	$-0.0918161\pi$
$678$	0	0
$679$	−6.30088	−0.241806
$680$	0	0
$681$	0	0
$682$	− 72.9354i	− 2.79284i
$683$	4.95252i	0.189503i	0.995501	+	0.0947515i	$0.0302057\pi$
$683$	4.95252i	0.189503i	−0.995501	+	0.0947515i	$0.969794\pi$
$684$	0	0
$685$	0	0
$686$	17.7549	0.677884
$687$	0	0
$688$	− 62.3876i	− 2.37850i
$689$	−6.64177	−0.253031
$690$	0	0
$691$	−19.2088	−0.730739	−0.365369	−	0.930863i	$-0.619057\pi$
$691$	−19.2088	−0.730739	−0.365369	+	0.930863i	$0.619057\pi$
$692$	37.0957i	1.41017i
$693$	0	0
$694$	−56.0011	−2.12577
$695$	0	0
$696$	0	0
$697$	37.6892i	1.42758i
$698$	7.40024i	0.280103i
$699$	0	0
$700$	0	0
$701$	−29.3492	−1.10850	−0.554251	−	0.832349i	$-0.686995\pi$
$701$	−29.3492	−1.10850	−0.554251	+	0.832349i	$0.686995\pi$
$702$	0	0
$703$	0.386505i	0.0145773i
$704$	−10.9344	−0.412105
$705$	0	0
$706$	47.3401	1.78167
$707$	− 6.00000i	− 0.225653i
$708$	0	0
$709$	−38.7266	−1.45441	−0.727204	−	0.686422i	$-0.759180\pi$
$709$	−38.7266	−1.45441	−0.727204	+	0.686422i	$0.759180\pi$
$710$	0	0
$711$	0	0
$712$	− 17.5051i	− 0.656030i
$713$	− 36.0576i	− 1.35037i
$714$	0	0
$715$	0	0
$716$	−4.60442	−0.172075
$717$	0	0
$718$	80.1432i	2.99092i
$719$	15.0848	0.562569	0.281284	−	0.959624i	$-0.409240\pi$
$719$	15.0848	0.562569	0.281284	+	0.959624i	$0.409240\pi$
$720$	0	0
$721$	−0.150442	−0.00560275
$722$	− 43.3821i	− 1.61451i
$723$	0	0
$724$	54.7458	2.03461
$725$	0	0
$726$	0	0
$727$	12.3455i	0.457871i	0.973442	+	0.228936i	$0.0735245\pi$
$727$	12.3455i	0.457871i	−0.973442	+	0.228936i	$0.926475\pi$
$728$	− 3.96265i	− 0.146866i
$729$	0	0
$730$	0	0
$731$	−34.3684	−1.27116
$732$	0	0
$733$	22.0000i	0.812589i	0.913742	+	0.406294i	$0.133179\pi$
$733$	22.0000i	0.812589i	−0.913742	+	0.406294i	$0.866821\pi$
$734$	46.1323	1.70277
$735$	0	0
$736$	14.3884	0.530362
$737$	− 31.3774i	− 1.15580i
$738$	0	0
$739$	−29.7266	−1.09351	−0.546755	−	0.837293i	$-0.684137\pi$
$739$	−29.7266	−1.09351	−0.546755	+	0.837293i	$0.684137\pi$
$740$	0	0
$741$	0	0
$742$	6.49960i	0.238608i
$743$	− 48.3648i	− 1.77433i	−0.461452	−	0.887165i	$-0.652671\pi$
$743$	− 48.3648i	− 1.77433i	0.461452	−	0.887165i	$-0.347329\pi$
$744$	0	0
$745$	0	0
$746$	−5.52787	−0.202390
$747$	0	0
$748$	− 47.6519i	− 1.74233i
$749$	0.962653	0.0351746
$750$	0	0
$751$	−31.8205	−1.16115	−0.580573	−	0.814208i	$-0.697171\pi$
$751$	−31.8205	−1.16115	−0.580573	+	0.814208i	$0.697171\pi$
$752$	− 29.3173i	− 1.06909i
$753$	0	0
$754$	−4.60442	−0.167683
$755$	0	0
$756$	0	0
$757$	4.94531i	0.179740i	0.995953	+	0.0898701i	$0.0286452\pi$
$757$	4.94531i	0.179740i	−0.995953	+	0.0898701i	$0.971355\pi$
$758$	− 38.8970i	− 1.41280i
$759$	0	0
$760$	0	0
$761$	35.4249	1.28415	0.642076	−	0.766641i	$-0.278073\pi$
$761$	35.4249	1.28415	0.642076	+	0.766641i	$0.278073\pi$
$762$	0	0
$763$	2.85237i	0.103263i
$764$	73.1715	2.64725
$765$	0	0
$766$	19.3774	0.700135
$767$	6.64177i	0.239820i
$768$	0	0
$769$	−49.4249	−1.78231	−0.891154	−	0.453701i	$-0.850103\pi$
$769$	−49.4249	−1.78231	−0.891154	+	0.453701i	$0.850103\pi$
$770$	0	0
$771$	0	0
$772$	115.443i	4.15490i
$773$	− 12.6599i	− 0.455345i	−0.973738	−	0.227673i	$-0.926888\pi$
$773$	− 12.6599i	− 0.455345i	0.973738	−	0.227673i	$-0.0731116\pi$
$774$	0	0
$775$	0	0
$776$	−71.5097	−2.56705
$777$	0	0
$778$	61.9300i	2.22030i
$779$	−14.9909	−0.537106
$780$	0	0
$781$	−29.8578	−1.06840
$782$	− 34.4623i	− 1.23237i
$783$	0	0
$784$	40.6044	1.45016
$785$	0	0
$786$	0	0
$787$	30.9344i	1.10269i	0.834277	+	0.551346i	$0.185885\pi$
$787$	30.9344i	1.10269i	−0.834277	+	0.551346i	$0.814115\pi$
$788$	61.5953i	2.19424i
$789$	0	0
$790$	0	0
$791$	−4.01093	−0.142612
$792$	0	0
$793$	9.70739i	0.344720i
$794$	17.0283	0.604311
$795$	0	0
$796$	−106.557	−3.77682
$797$	− 30.5935i	− 1.08368i	−0.840483	−	0.541839i	$-0.817728\pi$
$797$	− 30.5935i	− 1.08368i	0.840483	−	0.541839i	$-0.182272\pi$
$798$	0	0
$799$	−16.1504	−0.571362
$800$	0	0
$801$	0	0
$802$	− 46.5105i	− 1.64234i
$803$	20.1131i	0.709776i
$804$	0	0
$805$	0	0
$806$	−29.0101	−1.02184
$807$	0	0
$808$	− 68.0950i	− 2.39557i
$809$	2.89703	0.101854	0.0509271	−	0.998702i	$-0.483782\pi$
$809$	2.89703	0.101854	0.0509271	+	0.998702i	$0.483782\pi$
$810$	0	0
$811$	−14.8861	−0.522722	−0.261361	−	0.965241i	$-0.584171\pi$
$811$	−14.8861	−0.522722	−0.261361	+	0.965241i	$0.584171\pi$
$812$	3.08016i	0.108092i
$813$	0	0
$814$	2.44305	0.0856289
$815$	0	0
$816$	0	0
$817$	− 13.6700i	− 0.478254i
$818$	33.7266i	1.17922i
$819$	0	0
$820$	0	0
$821$	−8.95173	−0.312417	−0.156209	−	0.987724i	$-0.549927\pi$
$821$	−8.95173	−0.312417	−0.156209	+	0.987724i	$0.549927\pi$
$822$	0	0
$823$	− 2.99454i	− 0.104383i	−0.998637	−	0.0521915i	$-0.983379\pi$
$823$	− 2.99454i	− 0.104383i	0.998637	−	0.0521915i	$-0.0166206\pi$
$824$	−1.70739	−0.0594797
$825$	0	0
$826$	6.49960	0.226150
$827$	31.9663i	1.11158i	0.831324	+	0.555788i	$0.187583\pi$
$827$	31.9663i	1.11158i	−0.831324	+	0.555788i	$0.812417\pi$
$828$	0	0
$829$	−22.7458	−0.789994	−0.394997	−	0.918682i	$-0.629254\pi$
$829$	−22.7458	−0.789994	−0.394997	+	0.918682i	$0.629254\pi$
$830$	0	0
$831$	0	0
$832$	4.34916i	0.150780i
$833$	− 22.3684i	− 0.775018i
$834$	0	0
$835$	0	0
$836$	18.9536	0.655523
$837$	0	0
$838$	83.2555i	2.87601i
$839$	23.2643	0.803174	0.401587	−	0.915821i	$-0.368459\pi$
$839$	23.2643	0.803174	0.401587	+	0.915821i	$0.368459\pi$
$840$	0	0
$841$	−27.0776	−0.933710
$842$	− 36.9536i	− 1.27350i
$843$	0	0
$844$	23.2353	0.799791
$845$	0	0
$846$	0	0
$847$	− 0.0145366i	0 0.000499485i
$848$	30.3118i	1.04091i
$849$	0	0
$850$	0	0
$851$	1.20779	0.0414025
$852$	0	0
$853$	− 10.9909i	− 0.376322i	−0.982138	−	0.188161i	$-0.939747\pi$
$853$	− 10.9909i	− 0.376322i	0.982138	−	0.188161i	$-0.0602527\pi$
$854$	9.49960	0.325070
$855$	0	0
$856$	10.9253	0.373419
$857$	− 16.1504i	− 0.551689i	−0.961202	−	0.275844i	$-0.911043\pi$
$857$	− 16.1504i	− 0.551689i	0.961202	−	0.275844i	$-0.0889574\pi$
$858$	0	0
$859$	−28.5188	−0.973049	−0.486524	−	0.873667i	$-0.661736\pi$
$859$	−28.5188	−0.973049	−0.486524	+	0.873667i	$0.661736\pi$
$860$	0	0
$861$	0	0
$862$	− 82.3502i	− 2.80486i
$863$	− 12.2890i	− 0.418322i	−0.977881	−	0.209161i	$-0.932927\pi$
$863$	− 12.2890i	− 0.418322i	0.977881	−	0.209161i	$-0.0670734\pi$
$864$	0	0
$865$	0	0
$866$	−29.7458	−1.01080
$867$	0	0
$868$	19.4065i	0.658700i
$869$	−26.7549	−0.907597
$870$	0	0
$871$	−12.4804	−0.422882
$872$	32.3720i	1.09625i
$873$	0	0
$874$	13.7074	0.463659
$875$	0	0
$876$	0	0
$877$	− 39.7002i	− 1.34058i	−0.742099	−	0.670290i	$-0.766170\pi$
$877$	− 39.7002i	− 1.34058i	0.742099	−	0.670290i	$-0.233830\pi$
$878$	− 20.8970i	− 0.705241i
$879$	0	0
$880$	0	0
$881$	32.1040	1.08161	0.540806	−	0.841147i	$-0.318119\pi$
$881$	32.1040	1.08161	0.540806	+	0.841147i	$0.318119\pi$
$882$	0	0
$883$	− 13.5051i	− 0.454482i	−0.973839	−	0.227241i	$-0.927030\pi$
$883$	− 13.5051i	− 0.454482i	0.973839	−	0.227241i	$-0.0729704\pi$
$884$	−18.9536	−0.637478
$885$	0	0
$886$	−73.3502	−2.46425
$887$	35.1222i	1.17929i	0.807664	+	0.589643i	$0.200732\pi$
$887$	35.1222i	1.17929i	−0.807664	+	0.589643i	$0.799268\pi$
$888$	0	0
$889$	9.19872	0.308515
$890$	0	0
$891$	0	0
$892$	37.4386i	1.25354i
$893$	− 6.42385i	− 0.214966i
$894$	0	0
$895$	0	0
$896$	7.84049	0.261932
$897$	0	0
$898$	− 47.6975i	− 1.59169i
$899$	12.1120	0.403959
$900$	0	0
$901$	16.6983	0.556302
$902$	94.7559i	3.15503i
$903$	0	0
$904$	−45.5207	−1.51399
$905$	0	0
$906$	0	0
$907$	15.1186i	0.502004i	0.967987	+	0.251002i	$0.0807600\pi$
$907$	15.1186i	0.502004i	−0.967987	+	0.251002i	$0.919240\pi$
$908$	− 14.3492i	− 0.476194i
$909$	0	0
$910$	0	0
$911$	52.5561	1.74126	0.870631	−	0.491936i	$-0.163711\pi$
$911$	52.5561	1.74126	0.870631	+	0.491936i	$0.163711\pi$
$912$	0	0
$913$	5.12217i	0.169519i
$914$	58.4166	1.93225
$915$	0	0
$916$	−109.369	−3.61367
$917$	3.08482i	0.101870i
$918$	0	0
$919$	54.5489	1.79940	0.899702	−	0.436505i	$-0.143784\pi$
$919$	54.5489	1.79940	0.899702	+	0.436505i	$0.143784\pi$
$920$	0	0
$921$	0	0
$922$	11.1268i	0.366443i
$923$	11.8760i	0.390903i
$924$	0	0
$925$	0	0
$926$	−49.0475	−1.61180
$927$	0	0
$928$	4.83317i	0.158656i
$929$	−20.3793	−0.668623	−0.334311	−	0.942463i	$-0.608504\pi$
$929$	−20.3793	−0.668623	−0.334311	+	0.942463i	$0.608504\pi$
$930$	0	0
$931$	8.89703	0.291588
$932$	119.398i	3.91100i
$933$	0	0
$934$	61.8314	2.02319
$935$	0	0
$936$	0	0
$937$	49.1979i	1.60723i	0.595152	+	0.803613i	$0.297092\pi$
$937$	49.1979i	1.60723i	−0.595152	+	0.803613i	$0.702908\pi$
$938$	12.2133i	0.398777i
$939$	0	0
$940$	0	0
$941$	−23.2371	−0.757508	−0.378754	−	0.925497i	$-0.623647\pi$
$941$	−23.2371	−0.757508	−0.378754	+	0.925497i	$0.623647\pi$
$942$	0	0
$943$	46.8452i	1.52549i
$944$	30.3118	0.986565
$945$	0	0
$946$	−86.4068	−2.80933
$947$	37.1642i	1.20767i	0.797108	+	0.603837i	$0.206362\pi$
$947$	37.1642i	1.20767i	−0.797108	+	0.603837i	$0.793638\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	0	0
$951$	0	0
$952$	9.96265i	0.322891i
$953$	23.5761i	0.763706i	0.924223	+	0.381853i	$0.124714\pi$
$953$	23.5761i	0.763706i	−0.924223	+	0.381853i	$0.875286\pi$
$954$	0	0
$955$	0	0
$956$	−18.1422	−0.586760
$957$	0	0
$958$	− 82.3502i	− 2.66061i
$959$	2.91518	0.0941360
$960$	0	0
$961$	45.3118	1.46167
$962$	− 0.971726i	− 0.0313297i
$963$	0	0
$964$	−15.5743	−0.501614
$965$	0	0
$966$	0	0
$967$	8.38290i	0.269576i	0.990874	+	0.134788i	$0.0430353\pi$
$967$	8.38290i	0.269576i	−0.990874	+	0.134788i	$0.956965\pi$
$968$	− 0.164979i	− 0.00530261i
$969$	0	0
$970$	0	0
$971$	13.2078	0.423858	0.211929	−	0.977285i	$-0.432025\pi$
$971$	13.2078	0.423858	0.211929	+	0.977285i	$0.432025\pi$
$972$	0	0
$973$	4.11310i	0.131860i
$974$	15.1787	0.486357
$975$	0	0
$976$	44.3027	1.41810
$977$	− 14.3310i	− 0.458490i	−0.973369	−	0.229245i	$-0.926374\pi$
$977$	− 14.3310i	− 0.458490i	0.973369	−	0.229245i	$-0.0736257\pi$
$978$	0	0
$979$	−9.96265	−0.318408
$980$	0	0
$981$	0	0
$982$	36.3118i	1.15876i
$983$	32.3082i	1.03047i	0.857048	+	0.515236i	$0.172296\pi$
$983$	32.3082i	1.03047i	−0.857048	+	0.515236i	$0.827704\pi$
$984$	0	0
$985$	0	0
$986$	11.5761	0.368660
$987$	0	0
$988$	− 7.53880i	− 0.239841i
$989$	−42.7175	−1.35834
$990$	0	0
$991$	−39.6700	−1.26016	−0.630080	−	0.776530i	$-0.716978\pi$
$991$	−39.6700	−1.26016	−0.630080	+	0.776530i	$0.716978\pi$
$992$	30.4513i	0.966831i
$993$	0	0
$994$	11.6218	0.368620
$995$	0	0
$996$	0	0
$997$	38.6874i	1.22524i	0.790377	+	0.612621i	$0.209885\pi$
$997$	38.6874i	1.22524i	−0.790377	+	0.612621i	$0.790115\pi$
$998$	− 52.7258i	− 1.66901i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.b.l.649.6		6
3.2	odd	2		2025.2.b.m.649.1		6
5.2	odd	4		2025.2.a.n.1.1		3
5.3	odd	4		405.2.a.j.1.3		3
5.4	even	2	inner	2025.2.b.l.649.1		6
9.2	odd	6		675.2.k.b.199.1		12
9.4	even	3		225.2.k.b.124.1		12
9.5	odd	6		675.2.k.b.424.6		12
9.7	even	3		225.2.k.b.49.6		12
15.2	even	4		2025.2.a.o.1.3		3
15.8	even	4		405.2.a.i.1.1		3
15.14	odd	2		2025.2.b.m.649.6		6
20.3	even	4		6480.2.a.bv.1.3		3
45.2	even	12		675.2.e.b.226.1		6
45.4	even	6		225.2.k.b.124.6		12
45.7	odd	12		225.2.e.b.76.3		6
45.13	odd	12		45.2.e.b.16.1	✓	6
45.14	odd	6		675.2.k.b.424.1		12
45.22	odd	12		225.2.e.b.151.3		6
45.23	even	12		135.2.e.b.46.3		6
45.29	odd	6		675.2.k.b.199.6		12
45.32	even	12		675.2.e.b.451.1		6
45.34	even	6		225.2.k.b.49.1		12
45.38	even	12		135.2.e.b.91.3		6
45.43	odd	12		45.2.e.b.31.1	yes	6
60.23	odd	4		6480.2.a.bs.1.3		3
180.23	odd	12		2160.2.q.k.721.1		6
180.43	even	12		720.2.q.i.481.1		6
180.83	odd	12		2160.2.q.k.1441.1		6
180.103	even	12		720.2.q.i.241.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.e.b.16.1	✓	6	45.13	odd	12
45.2.e.b.31.1	yes	6	45.43	odd	12
135.2.e.b.46.3		6	45.23	even	12
135.2.e.b.91.3		6	45.38	even	12
225.2.e.b.76.3		6	45.7	odd	12
225.2.e.b.151.3		6	45.22	odd	12
225.2.k.b.49.1		12	45.34	even	6
225.2.k.b.49.6		12	9.7	even	3
225.2.k.b.124.1		12	9.4	even	3
225.2.k.b.124.6		12	45.4	even	6
405.2.a.i.1.1		3	15.8	even	4
405.2.a.j.1.3		3	5.3	odd	4
675.2.e.b.226.1		6	45.2	even	12
675.2.e.b.451.1		6	45.32	even	12
675.2.k.b.199.1		12	9.2	odd	6
675.2.k.b.199.6		12	45.29	odd	6
675.2.k.b.424.1		12	45.14	odd	6
675.2.k.b.424.6		12	9.5	odd	6
720.2.q.i.241.1		6	180.103	even	12
720.2.q.i.481.1		6	180.43	even	12
2025.2.a.n.1.1		3	5.2	odd	4
2025.2.a.o.1.3		3	15.2	even	4
2025.2.b.l.649.1		6	5.4	even	2	inner
2025.2.b.l.649.6		6	1.1	even	1	trivial
2025.2.b.m.649.1		6	3.2	odd	2
2025.2.b.m.649.6		6	15.14	odd	2
2160.2.q.k.721.1		6	180.23	odd	12
2160.2.q.k.1441.1		6	180.83	odd	12
6480.2.a.bs.1.3		3	60.23	odd	4
6480.2.a.bv.1.3		3	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.51414i q^{2} -4.32088 q^{4} -0.514137i q^{7} -5.83502i q^{8} -3.32088 q^{11} +1.32088i q^{13} +1.29261 q^{14} +6.02827 q^{16} -3.32088i q^{17} +1.32088 q^{19} -8.34916i q^{22} -4.12763i q^{23} -3.32088 q^{26} +2.22153i q^{28} +1.38650 q^{29} +8.73566 q^{31} +3.48586i q^{32} +8.34916 q^{34} +0.292611i q^{37} +3.32088i q^{38} -11.3492 q^{41} -10.3492i q^{43} +14.3492 q^{44} +10.3774 q^{46} -4.86330i q^{47} +6.73566 q^{49} -5.70739i q^{52} +5.02827i q^{53} -3.00000 q^{56} +3.48586i q^{58} +5.02827 q^{59} +7.34916 q^{61} +21.9627i q^{62} +3.29261 q^{64} +9.44852i q^{67} +14.3492i q^{68} +8.99093 q^{71} -6.05655i q^{73} -0.735663 q^{74} -5.70739 q^{76} +1.70739i q^{77} +8.05655 q^{79} -28.5333i q^{82} -1.54241i q^{83} +26.0192 q^{86} +19.3774i q^{88} +3.00000 q^{89} +0.679116 q^{91} +17.8350i q^{92} +12.2270 q^{94} -12.2553i q^{97} +16.9344i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 10 q^{4} - 4 q^{11} + 18 q^{14} + 10 q^{16} - 8 q^{19} - 4 q^{26} + 14 q^{29} + 16 q^{31} + 8 q^{34} - 26 q^{41} + 44 q^{44} - 6 q^{46} + 4 q^{49} - 18 q^{56} + 4 q^{59} + 2 q^{61} + 30 q^{64} - 20 q^{71}+ \cdots + 62 q^{94}+O(q^{100}) \) 6 * q - 10 * q^4 - 4 * q^11 + 18 * q^14 + 10 * q^16 - 8 * q^19 - 4 * q^26 + 14 * q^29 + 16 * q^31 + 8 * q^34 - 26 * q^41 + 44 * q^44 - 6 * q^46 + 4 * q^49 - 18 * q^56 + 4 * q^59 + 2 * q^61 + 30 * q^64 - 20 * q^71 + 32 * q^74 - 24 * q^76 - 4 * q^79 + 56 * q^86 + 18 * q^89 + 20 * q^91 + 62 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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