LMFDB - Embedded newform 2025.2.b.n.649.5

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:	$8$
Coefficient field:	8.0.34810603776.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 12x^{6} + 42x^{4} + 49x^{2} + 9 $ x^8 + 12x^6 + 42x^4 + 49*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 225)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.5
Root			$0.473255i$ of defining polynomial
Character	$\chi$	$=$	2025.649
Dual form			2025.2.b.n.649.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.473255i q^{2} +1.77603 q^{4} +2.56305i q^{7} +1.78702i q^{8} +O(q^{10})$	$q+0.473255i q^{2} +1.77603 q^{4} +2.56305i q^{7} +1.78702i q^{8} -6.16860 q^{11} -2.13230i q^{13} -1.21298 q^{14} +2.70634 q^{16} +3.16860i q^{17} -0.356267 q^{19} -2.91932i q^{22} +4.21298i q^{23} +1.00912 q^{26} +4.55206i q^{28} -1.68623 q^{29} -8.25840 q^{31} +4.85484i q^{32} -1.49956 q^{34} +3.63274i q^{37} -0.168605i q^{38} -2.73353 q^{41} +7.67817i q^{43} -10.9556 q^{44} -1.99381 q^{46} -11.4289i q^{47} +0.430757 q^{49} -3.78702i q^{52} +9.43507i q^{53} -4.58024 q^{56} -0.798017i q^{58} -10.2159 q^{59} -0.0109932 q^{61} -3.90833i q^{62} +3.11511 q^{64} +0.982817i q^{67} +5.62754i q^{68} -6.43507 q^{71} -6.61467i q^{73} -1.71921 q^{74} -0.632740 q^{76} -15.8105i q^{77} +9.47138 q^{79} -1.29366i q^{82} +10.4198i q^{83} -3.63373 q^{86} -11.0234i q^{88} +6.26940 q^{89} +5.46519 q^{91} +7.48237i q^{92} +5.40877 q^{94} +7.20679i q^{97} +0.203858i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4}+O(q^{10}) $ 8 * q - 8 * q^4	$ 8 q - 8 q^{4} - 2 q^{11} - 6 q^{14} + 8 q^{16} - 4 q^{19} - 20 q^{26} - 2 q^{29} - 8 q^{31} - 18 q^{34} - 10 q^{41} - 44 q^{44} + 6 q^{49} - 60 q^{56} - 34 q^{59} - 26 q^{61} - 38 q^{64} - 16 q^{71} - 80 q^{74} + 22 q^{76} + 14 q^{79} - 68 q^{86} + 18 q^{89} - 34 q^{91} - 6 q^{94}+O(q^{100}) $ 8 * q - 8 * q^4 - 2 * q^11 - 6 * q^14 + 8 * q^16 - 4 * q^19 - 20 * q^26 - 2 * q^29 - 8 * q^31 - 18 * q^34 - 10 * q^41 - 44 * q^44 + 6 * q^49 - 60 * q^56 - 34 * q^59 - 26 * q^61 - 38 * q^64 - 16 * q^71 - 80 * q^74 + 22 * q^76 + 14 * q^79 - 68 * q^86 + 18 * q^89 - 34 * q^91 - 6 * 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 $

$2$	0.473255i	0.334641i	0.985903	+	0.167321i	$0.0535115\pi$
$2$	0.473255i	0.334641i	−0.985903	+	0.167321i	$0.946488\pi$
$3$	0	0
$3$	0	0
$4$	1.77603	0.888015
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.56305i	0.968743i	0.874862	+	0.484372i	$0.160952\pi$
$7$	2.56305i	0.968743i	−0.874862	+	0.484372i	$0.839048\pi$
$8$	1.78702i	0.631808i
$9$	0	0
$10$	0	0
$11$	−6.16860	−1.85990	−0.929952	−	0.367681i	$-0.880152\pi$
$11$	−6.16860	−1.85990	−0.929952	+	0.367681i	$0.880152\pi$
$12$	0	0
$13$	− 2.13230i	− 0.591393i	−0.955282	−	0.295696i	$-0.904448\pi$
$13$	− 2.13230i	− 0.591393i	0.955282	−	0.295696i	$-0.0955517\pi$
$14$	−1.21298	−0.324182
$15$	0	0
$16$	2.70634	0.676586
$17$	3.16860i	0.768500i	0.923229	+	0.384250i	$0.125540\pi$
$17$	3.16860i	0.768500i	−0.923229	+	0.384250i	$0.874460\pi$
$18$	0	0
$19$	−0.356267	−0.0817332	−0.0408666	−	0.999165i	$-0.513012\pi$
$19$	−0.356267	−0.0817332	−0.0408666	+	0.999165i	$0.513012\pi$
$20$	0	0
$21$	0	0
$22$	− 2.91932i	− 0.622401i
$23$	4.21298i	0.878466i	0.898373	+	0.439233i	$0.144750\pi$
$23$	4.21298i	0.878466i	−0.898373	+	0.439233i	$0.855250\pi$
$24$	0	0
$25$	0	0
$26$	1.00912	0.197905
$27$	0	0
$28$	4.55206i	0.860259i
$29$	−1.68623	−0.313125	−0.156563	−	0.987668i	$-0.550041\pi$
$29$	−1.68623	−0.313125	−0.156563	+	0.987668i	$0.550041\pi$
$30$	0	0
$31$	−8.25840	−1.48325	−0.741627	−	0.670813i	$-0.765945\pi$
$31$	−8.25840	−1.48325	−0.741627	+	0.670813i	$0.765945\pi$
$32$	4.85484i	0.858222i
$33$	0	0
$34$	−1.49956	−0.257172
$35$	0	0
$36$	0	0
$37$	3.63274i	0.597219i	0.954375	+	0.298609i	$0.0965228\pi$
$37$	3.63274i	0.597219i	−0.954375	+	0.298609i	$0.903477\pi$
$38$	− 0.168605i	− 0.0273513i
$39$	0	0
$40$	0	0
$41$	−2.73353	−0.426906	−0.213453	−	0.976953i	$-0.568471\pi$
$41$	−2.73353	−0.426906	−0.213453	+	0.976953i	$0.568471\pi$
$42$	0	0
$43$	7.67817i	1.17091i	0.810705	+	0.585455i	$0.199084\pi$
$43$	7.67817i	1.17091i	−0.810705	+	0.585455i	$0.800916\pi$
$44$	−10.9556	−1.65162
$45$	0	0
$46$	−1.99381	−0.293971
$47$	− 11.4289i	− 1.66707i	−0.552464	−	0.833537i	$-0.686312\pi$
$47$	− 11.4289i	− 1.66707i	0.552464	−	0.833537i	$-0.313688\pi$
$48$	0	0
$49$	0.430757	0.0615367
$50$	0	0
$51$	0	0
$52$	− 3.78702i	− 0.525166i
$53$	9.43507i	1.29601i	0.761637	+	0.648003i	$0.224396\pi$
$53$	9.43507i	1.29601i	−0.761637	+	0.648003i	$0.775604\pi$
$54$	0	0
$55$	0	0
$56$	−4.58024	−0.612060
$57$	0	0
$58$	− 0.798017i	− 0.104785i
$59$	−10.2159	−1.33000	−0.664999	−	0.746844i	$-0.731568\pi$
$59$	−10.2159	−1.33000	−0.664999	+	0.746844i	$0.731568\pi$
$60$	0	0
$61$	−0.0109932	−0.00140753	−0.000703767	−	1.00000i	$-0.500224\pi$
$61$	−0.0109932	−0.00140753	−0.000703767		1.00000i	$0.500224\pi$
$62$	− 3.90833i	− 0.496358i
$63$	0	0
$64$	3.11511	0.389389
$65$	0	0
$66$	0	0
$67$	0.982817i	0.120070i	0.998196	+	0.0600351i	$0.0191213\pi$
$67$	0.982817i	0.120070i	−0.998196	+	0.0600351i	$0.980879\pi$
$68$	5.62754i	0.682439i
$69$	0	0
$70$	0	0
$71$	−6.43507	−0.763703	−0.381851	−	0.924224i	$-0.624713\pi$
$71$	−6.43507	−0.763703	−0.381851	+	0.924224i	$0.624713\pi$
$72$	0	0
$73$	− 6.61467i	− 0.774189i	−0.922040	−	0.387094i	$-0.873479\pi$
$73$	− 6.61467i	− 0.774189i	0.922040	−	0.387094i	$-0.126521\pi$
$74$	−1.71921	−0.199854
$75$	0	0
$76$	−0.632740	−0.0725803
$77$	− 15.8105i	− 1.80177i
$78$	0	0
$79$	9.47138	1.06561	0.532807	−	0.846237i	$-0.321137\pi$
$79$	9.47138	1.06561	0.532807	+	0.846237i	$0.321137\pi$
$80$	0	0
$81$	0	0
$82$	− 1.29366i	− 0.142860i
$83$	10.4198i	1.14372i	0.820352	+	0.571859i	$0.193777\pi$
$83$	10.4198i	1.14372i	−0.820352	+	0.571859i	$0.806223\pi$
$84$	0	0
$85$	0	0
$86$	−3.63373	−0.391835
$87$	0	0
$88$	− 11.0234i	− 1.17510i
$89$	6.26940	0.664555	0.332277	−	0.943182i	$-0.392183\pi$
$89$	6.26940	0.664555	0.332277	+	0.943182i	$0.392183\pi$
$90$	0	0
$91$	5.46519	0.572908
$92$	7.48237i	0.780091i
$93$	0	0
$94$	5.40877	0.557872
$95$	0	0
$96$	0	0
$97$	7.20679i	0.731738i	0.930666	+	0.365869i	$0.119228\pi$
$97$	7.20679i	0.731738i	−0.930666	+	0.365869i	$0.880772\pi$
$98$	0.203858i	0.0205927i
$99$	0	0
$100$	0	0
$101$	6.97094	0.693634	0.346817	−	0.937933i	$-0.387262\pi$
$101$	6.97094	0.693634	0.346817	+	0.937933i	$0.387262\pi$
$102$	0	0
$103$	6.11511i	0.602540i	0.953539	+	0.301270i	$0.0974106\pi$
$103$	6.11511i	0.602540i	−0.953539	+	0.301270i	$0.902589\pi$
$104$	3.81046	0.373647
$105$	0	0
$106$	−4.46519	−0.433698
$107$	− 14.5349i	− 1.40514i	−0.711615	−	0.702570i	$-0.752036\pi$
$107$	− 14.5349i	− 1.40514i	0.711615	−	0.702570i	$-0.247964\pi$
$108$	0	0
$109$	−1.90214	−0.182192	−0.0910958	−	0.995842i	$-0.529037\pi$
$109$	−1.90214	−0.182192	−0.0910958	+	0.995842i	$0.529037\pi$
$110$	0	0
$111$	0	0
$112$	6.93650i	0.655438i
$113$	6.57925i	0.618924i	0.950912	+	0.309462i	$0.100149\pi$
$113$	6.57925i	0.618924i	−0.950912	+	0.309462i	$0.899851\pi$
$114$	0	0
$115$	0	0
$116$	−2.99480	−0.278060
$117$	0	0
$118$	− 4.83472i	− 0.445072i
$119$	−8.12130	−0.744479
$120$	0	0
$121$	27.0517	2.45924
$122$	− 0.00520257i	0 0.000471019i
$123$	0	0
$124$	−14.6672	−1.31715
$125$	0	0
$126$	0	0
$127$	− 9.25840i	− 0.821550i	−0.911737	−	0.410775i	$-0.865258\pi$
$127$	− 9.25840i	− 0.821550i	0.911737	−	0.410775i	$-0.134742\pi$
$128$	11.1839i	0.988528i
$129$	0	0
$130$	0	0
$131$	0.269397	0.0235373	0.0117687	−	0.999931i	$-0.496254\pi$
$131$	0.269397	0.0235373	0.0117687	+	0.999931i	$0.496254\pi$
$132$	0	0
$133$	− 0.913130i	− 0.0791784i
$134$	−0.465123	−0.0401805
$135$	0	0
$136$	−5.66237	−0.485544
$137$	3.47618i	0.296990i	0.988913	+	0.148495i	$0.0474430\pi$
$137$	3.47618i	0.296990i	−0.988913	+	0.148495i	$0.952557\pi$
$138$	0	0
$139$	−14.7479	−1.25090	−0.625448	−	0.780266i	$-0.715084\pi$
$139$	−14.7479	−1.25090	−0.625448	+	0.780266i	$0.715084\pi$
$140$	0	0
$141$	0	0
$142$	− 3.04543i	− 0.255567i
$143$	13.1533i	1.09993i
$144$	0	0
$145$	0	0
$146$	3.13042	0.259076
$147$	0	0
$148$	6.45186i	0.530339i
$149$	10.1533	0.831790	0.415895	−	0.909413i	$-0.363468\pi$
$149$	10.1533	0.831790	0.415895	+	0.909413i	$0.363468\pi$
$150$	0	0
$151$	−10.3162	−0.839521	−0.419761	−	0.907635i	$-0.637886\pi$
$151$	−10.3162	−0.839521	−0.419761	+	0.907635i	$0.637886\pi$
$152$	− 0.636657i	− 0.0516397i
$153$	0	0
$154$	7.48237	0.602947
$155$	0	0
$156$	0	0
$157$	1.06261i	0.0848055i	0.999101	+	0.0424028i	$0.0135013\pi$
$157$	1.06261i	0.0848055i	−0.999101	+	0.0424028i	$0.986499\pi$
$158$	4.48237i	0.356598i
$159$	0	0
$160$	0	0
$161$	−10.7981	−0.851008
$162$	0	0
$163$	17.1386i	1.34240i	0.741278	+	0.671198i	$0.234220\pi$
$163$	17.1386i	1.34240i	−0.741278	+	0.671198i	$0.765780\pi$
$164$	−4.85484	−0.379099
$165$	0	0
$166$	−4.93120	−0.382735
$167$	− 4.37345i	− 0.338428i	−0.985579	−	0.169214i	$-0.945877\pi$
$167$	− 4.37345i	− 0.338428i	0.985579	−	0.169214i	$-0.0541229\pi$
$168$	0	0
$169$	8.45331	0.650255
$170$	0	0
$171$	0	0
$172$	13.6367i	1.03979i
$173$	14.6601i	1.11459i	0.830316	+	0.557293i	$0.188160\pi$
$173$	14.6601i	1.11459i	−0.830316	+	0.557293i	$0.811840\pi$
$174$	0	0
$175$	0	0
$176$	−16.6944	−1.25838
$177$	0	0
$178$	2.96702i	0.222388i
$179$	6.87014	0.513499	0.256749	−	0.966478i	$-0.417349\pi$
$179$	6.87014	0.513499	0.256749	+	0.966478i	$0.417349\pi$
$180$	0	0
$181$	−10.9709	−0.815463	−0.407732	−	0.913102i	$-0.633680\pi$
$181$	−10.9709	−0.815463	−0.407732	+	0.913102i	$0.633680\pi$
$182$	2.58643i	0.191719i
$183$	0	0
$184$	−7.52869	−0.555022
$185$	0	0
$186$	0	0
$187$	− 19.5459i	− 1.42934i
$188$	− 20.2980i	− 1.48039i
$189$	0	0
$190$	0	0
$191$	13.7325	0.993652	0.496826	−	0.867850i	$-0.334499\pi$
$191$	13.7325	0.993652	0.496826	+	0.867850i	$0.334499\pi$
$192$	0	0
$193$	0.482374i	0.0347220i	0.999849	+	0.0173610i	$0.00552646\pi$
$193$	0.482374i	0.0347220i	−0.999849	+	0.0173610i	$0.994474\pi$
$194$	−3.41064	−0.244870
$195$	0	0
$196$	0.765037	0.0546455
$197$	5.53488i	0.394344i	0.980369	+	0.197172i	$0.0631757\pi$
$197$	5.53488i	0.394344i	−0.980369	+	0.197172i	$0.936824\pi$
$198$	0	0
$199$	−17.4590	−1.23764	−0.618818	−	0.785534i	$-0.712388\pi$
$199$	−17.4590	−1.23764	−0.618818	+	0.785534i	$0.712388\pi$
$200$	0	0
$201$	0	0
$202$	3.29903i	0.232119i
$203$	− 4.32190i	− 0.303338i
$204$	0	0
$205$	0	0
$206$	−2.89401	−0.201635
$207$	0	0
$208$	− 5.77073i	− 0.400128i
$209$	2.19767	0.152016
$210$	0	0
$211$	−1.63666	−0.112672	−0.0563360	−	0.998412i	$-0.517942\pi$
$211$	−1.63666	−0.112672	−0.0563360	+	0.998412i	$0.517942\pi$
$212$	16.7570i	1.15087i
$213$	0	0
$214$	6.87870	0.470218
$215$	0	0
$216$	0	0
$217$	− 21.1667i	− 1.43689i
$218$	− 0.900195i	− 0.0609689i
$219$	0	0
$220$	0	0
$221$	6.75641	0.454485
$222$	0	0
$223$	− 7.74785i	− 0.518835i	−0.965765	−	0.259417i	$-0.916469\pi$
$223$	− 7.74785i	− 0.518835i	0.965765	−	0.259417i	$-0.0835305\pi$
$224$	−12.4432	−0.831397
$225$	0	0
$226$	−3.11366	−0.207118
$227$	− 11.2603i	− 0.747371i	−0.927555	−	0.373685i	$-0.878094\pi$
$227$	− 11.2603i	− 0.747371i	0.927555	−	0.373685i	$-0.121906\pi$
$228$	0	0
$229$	10.4776	0.692377	0.346189	−	0.938165i	$-0.387476\pi$
$229$	10.4776	0.692377	0.346189	+	0.938165i	$0.387476\pi$
$230$	0	0
$231$	0	0
$232$	− 3.01333i	− 0.197835i
$233$	2.90214i	0.190125i	0.995471	+	0.0950627i	$0.0303051\pi$
$233$	2.90214i	0.190125i	−0.995471	+	0.0950627i	$0.969695\pi$
$234$	0	0
$235$	0	0
$236$	−18.1438	−1.18106
$237$	0	0
$238$	− 3.84344i	− 0.249133i
$239$	−16.3545	−1.05788	−0.528941	−	0.848659i	$-0.677411\pi$
$239$	−16.3545	−1.05788	−0.528941	+	0.848659i	$0.677411\pi$
$240$	0	0
$241$	17.5239	1.12881	0.564406	−	0.825497i	$-0.309105\pi$
$241$	17.5239	1.12881	0.564406	+	0.825497i	$0.309105\pi$
$242$	12.8023i	0.822965i
$243$	0	0
$244$	−0.0195242	−0.00124991
$245$	0	0
$246$	0	0
$247$	0.759666i	0.0483364i
$248$	− 14.7580i	− 0.937131i
$249$	0	0
$250$	0	0
$251$	8.46999	0.534621	0.267311	−	0.963610i	$-0.413865\pi$
$251$	8.46999	0.534621	0.267311	+	0.963610i	$0.413865\pi$
$252$	0	0
$253$	− 25.9882i	− 1.63386i
$254$	4.38158	0.274925
$255$	0	0
$256$	0.937390	0.0585869
$257$	− 2.87251i	− 0.179182i	−0.995979	−	0.0895910i	$-0.971444\pi$
$257$	− 2.87251i	− 0.179182i	0.995979	−	0.0895910i	$-0.0285560\pi$
$258$	0	0
$259$	−9.31091	−0.578552
$260$	0	0
$261$	0	0
$262$	0.127493i	0.00787656i
$263$	− 25.6239i	− 1.58003i	−0.613084	−	0.790017i	$-0.710071\pi$
$263$	− 25.6239i	− 1.58003i	0.613084	−	0.790017i	$-0.289929\pi$
$264$	0	0
$265$	0	0
$266$	0.432143	0.0264964
$267$	0	0
$268$	1.74551i	0.106624i
$269$	−0.337210	−0.0205600	−0.0102800	−	0.999947i	$-0.503272\pi$
$269$	−0.337210	−0.0205600	−0.0102800	+	0.999947i	$0.503272\pi$
$270$	0	0
$271$	21.5927	1.31166	0.655831	−	0.754908i	$-0.272318\pi$
$271$	21.5927	1.31166	0.655831	+	0.754908i	$0.272318\pi$
$272$	8.57533i	0.519956i
$273$	0	0
$274$	−1.64512	−0.0993853
$275$	0	0
$276$	0	0
$277$	24.1338i	1.45006i	0.688719	+	0.725028i	$0.258173\pi$
$277$	24.1338i	1.45006i	−0.688719	+	0.725028i	$0.741827\pi$
$278$	− 6.97949i	− 0.418602i
$279$	0	0
$280$	0	0
$281$	3.36726	0.200874	0.100437	−	0.994943i	$-0.467976\pi$
$281$	3.36726	0.200874	0.100437	+	0.994943i	$0.467976\pi$
$282$	0	0
$283$	− 21.8497i	− 1.29883i	−0.760434	−	0.649415i	$-0.775014\pi$
$283$	− 21.8497i	− 1.29883i	0.760434	−	0.649415i	$-0.224986\pi$
$284$	−11.4289	−0.678179
$285$	0	0
$286$	−6.22486	−0.368083
$287$	− 7.00619i	− 0.413562i
$288$	0	0
$289$	6.95994	0.409408
$290$	0	0
$291$	0	0
$292$	− 11.7479i	− 0.687491i
$293$	− 13.7540i	− 0.803520i	−0.915745	−	0.401760i	$-0.868399\pi$
$293$	− 13.7540i	− 0.803520i	0.915745	−	0.401760i	$-0.131601\pi$
$294$	0	0
$295$	0	0
$296$	−6.49179	−0.377328
$297$	0	0
$298$	4.80509i	0.278352i
$299$	8.98332	0.519519
$300$	0	0
$301$	−19.6796	−1.13431
$302$	− 4.88219i	− 0.280939i
$303$	0	0
$304$	−0.964180	−0.0552995
$305$	0	0
$306$	0	0
$307$	34.2183i	1.95294i	0.215644	+	0.976472i	$0.430815\pi$
$307$	34.2183i	1.95294i	−0.215644	+	0.976472i	$0.569185\pi$
$308$	− 28.0799i	− 1.60000i
$309$	0	0
$310$	0	0
$311$	23.0397	1.30646	0.653232	−	0.757158i	$-0.273413\pi$
$311$	23.0397	1.30646	0.653232	+	0.757158i	$0.273413\pi$
$312$	0	0
$313$	3.59130i	0.202992i	0.994836	+	0.101496i	$0.0323629\pi$
$313$	3.59130i	0.202992i	−0.994836	+	0.101496i	$0.967637\pi$
$314$	−0.502885	−0.0283794
$315$	0	0
$316$	16.8215	0.946281
$317$	13.1807i	0.740299i	0.928972	+	0.370150i	$0.120694\pi$
$317$	13.1807i	0.740299i	−0.928972	+	0.370150i	$0.879306\pi$
$318$	0	0
$319$	10.4017	0.582383
$320$	0	0
$321$	0	0
$322$	− 5.11024i	− 0.284783i
$323$	− 1.12887i	− 0.0628119i
$324$	0	0
$325$	0	0
$326$	−8.11090	−0.449221
$327$	0	0
$328$	− 4.88489i	− 0.269723i
$329$	29.2928	1.61497
$330$	0	0
$331$	1.18253	0.0649976	0.0324988	−	0.999472i	$-0.489653\pi$
$331$	1.18253	0.0649976	0.0324988	+	0.999472i	$0.489653\pi$
$332$	18.5058i	1.01564i
$333$	0	0
$334$	2.06975	0.113252
$335$	0	0
$336$	0	0
$337$	− 24.7995i	− 1.35091i	−0.737400	−	0.675457i	$-0.763947\pi$
$337$	− 24.7995i	− 1.35091i	0.737400	−	0.675457i	$-0.236053\pi$
$338$	4.00057i	0.217602i
$339$	0	0
$340$	0	0
$341$	50.9428	2.75871
$342$	0	0
$343$	19.0454i	1.02836i
$344$	−13.7211	−0.739790
$345$	0	0
$346$	−6.93796	−0.372987
$347$	22.1692i	1.19010i	0.803687	+	0.595052i	$0.202868\pi$
$347$	22.1692i	1.19010i	−0.803687	+	0.595052i	$0.797132\pi$
$348$	0	0
$349$	−14.9185	−0.798569	−0.399285	−	0.916827i	$-0.630741\pi$
$349$	−14.9185	−0.798569	−0.399285	+	0.916827i	$0.630741\pi$
$350$	0	0
$351$	0	0
$352$	− 29.9476i	− 1.59621i
$353$	− 16.9145i	− 0.900269i	−0.892961	−	0.450134i	$-0.851376\pi$
$353$	− 16.9145i	− 0.900269i	0.892961	−	0.450134i	$-0.148624\pi$
$354$	0	0
$355$	0	0
$356$	11.1346	0.590135
$357$	0	0
$358$	3.25133i	0.171838i
$359$	−0.636657	−0.0336015	−0.0168007	−	0.999859i	$-0.505348\pi$
$359$	−0.636657	−0.0336015	−0.0168007	+	0.999859i	$0.505348\pi$
$360$	0	0
$361$	−18.8731	−0.993320
$362$	− 5.19205i	− 0.272888i
$363$	0	0
$364$	9.70634	0.508751
$365$	0	0
$366$	0	0
$367$	20.0979i	1.04910i	0.851379	+	0.524552i	$0.175767\pi$
$367$	20.0979i	1.04910i	−0.851379	+	0.524552i	$0.824233\pi$
$368$	11.4018i	0.594358i
$369$	0	0
$370$	0	0
$371$	−24.1826	−1.25550
$372$	0	0
$373$	19.6429i	1.01707i	0.861041	+	0.508536i	$0.169813\pi$
$373$	19.6429i	1.01707i	−0.861041	+	0.508536i	$0.830187\pi$
$374$	9.25017	0.478315
$375$	0	0
$376$	20.4237	1.05327
$377$	3.59555i	0.185180i
$378$	0	0
$379$	7.94219	0.407963	0.203982	−	0.978975i	$-0.434612\pi$
$379$	7.94219	0.407963	0.203982	+	0.978975i	$0.434612\pi$
$380$	0	0
$381$	0	0
$382$	6.49899i	0.332517i
$383$	31.1888i	1.59367i	0.604195	+	0.796836i	$0.293495\pi$
$383$	31.1888i	1.59367i	−0.604195	+	0.796836i	$0.706505\pi$
$384$	0	0
$385$	0	0
$386$	−0.228285	−0.0116194
$387$	0	0
$388$	12.7995i	0.649795i
$389$	31.4494	1.59455	0.797274	−	0.603618i	$-0.206275\pi$
$389$	31.4494	1.59455	0.797274	+	0.603618i	$0.206275\pi$
$390$	0	0
$391$	−13.3493	−0.675101
$392$	0.769772i	0.0388794i
$393$	0	0
$394$	−2.61941	−0.131964
$395$	0	0
$396$	0	0
$397$	17.7174i	0.889211i	0.895726	+	0.444606i	$0.146656\pi$
$397$	17.7174i	0.889211i	−0.895726	+	0.444606i	$0.853344\pi$
$398$	− 8.26255i	− 0.414164i
$399$	0	0
$400$	0	0
$401$	−7.14247	−0.356678	−0.178339	−	0.983969i	$-0.557072\pi$
$401$	−7.14247	−0.356678	−0.178339	+	0.983969i	$0.557072\pi$
$402$	0	0
$403$	17.6094i	0.877185i
$404$	12.3806	0.615958
$405$	0	0
$406$	2.04536	0.101509
$407$	− 22.4089i	− 1.11077i
$408$	0	0
$409$	−24.7518	−1.22390	−0.611948	−	0.790898i	$-0.709614\pi$
$409$	−24.7518	−1.22390	−0.611948	+	0.790898i	$0.709614\pi$
$410$	0	0
$411$	0	0
$412$	10.8606i	0.535065i
$413$	− 26.1839i	− 1.28843i
$414$	0	0
$415$	0	0
$416$	10.3520	0.507546
$417$	0	0
$418$	1.04006i	0.0508708i
$419$	10.6591	0.520732	0.260366	−	0.965510i	$-0.416157\pi$
$419$	10.6591	0.520732	0.260366	+	0.965510i	$0.416157\pi$
$420$	0	0
$421$	−8.17861	−0.398601	−0.199301	−	0.979938i	$-0.563867\pi$
$421$	−8.17861	−0.398601	−0.199301	+	0.979938i	$0.563867\pi$
$422$	− 0.774555i	− 0.0377048i
$423$	0	0
$424$	−16.8607	−0.818828
$425$	0	0
$426$	0	0
$427$	− 0.0281761i	− 0.00136354i
$428$	− 25.8144i	− 1.24779i
$429$	0	0
$430$	0	0
$431$	1.67248	0.0805604	0.0402802	−	0.999188i	$-0.487175\pi$
$431$	1.67248	0.0805604	0.0402802	+	0.999188i	$0.487175\pi$
$432$	0	0
$433$	9.95994i	0.478644i	0.970940	+	0.239322i	$0.0769252\pi$
$433$	9.95994i	0.478644i	−0.970940	+	0.239322i	$0.923075\pi$
$434$	10.0173	0.480843
$435$	0	0
$436$	−3.37825	−0.161789
$437$	− 1.50094i	− 0.0717998i
$438$	0	0
$439$	−12.8158	−0.611663	−0.305832	−	0.952086i	$-0.598934\pi$
$439$	−12.8158	−0.611663	−0.305832	+	0.952086i	$0.598934\pi$
$440$	0	0
$441$	0	0
$442$	3.19750i	0.152090i
$443$	7.75404i	0.368406i	0.982888	+	0.184203i	$0.0589703\pi$
$443$	7.75404i	0.368406i	−0.982888	+	0.184203i	$0.941030\pi$
$444$	0	0
$445$	0	0
$446$	3.66671	0.173624
$447$	0	0
$448$	7.98420i	0.377218i
$449$	33.3401	1.57342	0.786709	−	0.617324i	$-0.211783\pi$
$449$	33.3401	1.57342	0.786709	+	0.617324i	$0.211783\pi$
$450$	0	0
$451$	16.8621	0.794004
$452$	11.6849i	0.549614i
$453$	0	0
$454$	5.32898	0.250101
$455$	0	0
$456$	0	0
$457$	− 38.2192i	− 1.78782i	−0.448246	−	0.893910i	$-0.647951\pi$
$457$	− 38.2192i	− 1.78782i	0.448246	−	0.893910i	$-0.352049\pi$
$458$	4.95856i	0.231698i
$459$	0	0
$460$	0	0
$461$	31.3033	1.45794	0.728971	−	0.684545i	$-0.239999\pi$
$461$	31.3033	1.45794	0.728971	+	0.684545i	$0.239999\pi$
$462$	0	0
$463$	− 12.0852i	− 0.561645i	−0.959760	−	0.280823i	$-0.909393\pi$
$463$	− 12.0852i	− 0.561645i	0.959760	−	0.280823i	$-0.0906073\pi$
$464$	−4.56352	−0.211856
$465$	0	0
$466$	−1.37345	−0.0636238
$467$	− 7.60466i	− 0.351902i	−0.984399	−	0.175951i	$-0.943700\pi$
$467$	− 7.60466i	− 0.351902i	0.984399	−	0.175951i	$-0.0563000\pi$
$468$	0	0
$469$	−2.51901	−0.116317
$470$	0	0
$471$	0	0
$472$	− 18.2561i	− 0.840303i
$473$	− 47.3636i	− 2.17778i
$474$	0	0
$475$	0	0
$476$	−14.4237	−0.661108
$477$	0	0
$478$	− 7.73982i	− 0.354011i
$479$	32.4833	1.48420	0.742101	−	0.670289i	$-0.233830\pi$
$479$	32.4833	1.48420	0.742101	+	0.670289i	$0.233830\pi$
$480$	0	0
$481$	7.74608	0.353191
$482$	8.29326i	0.377748i
$483$	0	0
$484$	48.0446	2.18385
$485$	0	0
$486$	0	0
$487$	− 4.46121i	− 0.202157i	−0.994878	−	0.101078i	$-0.967771\pi$
$487$	− 4.46121i	− 0.202157i	0.994878	−	0.101078i	$-0.0322293\pi$
$488$	− 0.0196451i	0 0.000889291i
$489$	0	0
$490$	0	0
$491$	32.8420	1.48214	0.741070	−	0.671428i	$-0.234319\pi$
$491$	32.8420	1.48214	0.741070	+	0.671428i	$0.234319\pi$
$492$	0	0
$493$	− 5.34300i	− 0.240637i
$494$	−0.359515	−0.0161754
$495$	0	0
$496$	−22.3501	−1.00355
$497$	− 16.4934i	− 0.739832i
$498$	0	0
$499$	34.2021	1.53109	0.765547	−	0.643380i	$-0.222468\pi$
$499$	34.2021	1.53109	0.765547	+	0.643380i	$0.222468\pi$
$500$	0	0
$501$	0	0
$502$	4.00846i	0.178906i
$503$	− 22.1773i	− 0.988837i	−0.869224	−	0.494419i	$-0.835381\pi$
$503$	− 22.1773i	− 0.988837i	0.869224	−	0.494419i	$-0.164619\pi$
$504$	0	0
$505$	0	0
$506$	12.2990	0.546758
$507$	0	0
$508$	− 16.4432i	− 0.729549i
$509$	21.5632	0.955773	0.477887	−	0.878422i	$-0.341403\pi$
$509$	21.5632	0.955773	0.477887	+	0.878422i	$0.341403\pi$
$510$	0	0
$511$	16.9538	0.749990
$512$	22.8115i	1.00813i
$513$	0	0
$514$	1.35943	0.0599617
$515$	0	0
$516$	0	0
$517$	70.5003i	3.10060i
$518$	− 4.40643i	− 0.193607i
$519$	0	0
$520$	0	0
$521$	20.2626	0.887718	0.443859	−	0.896097i	$-0.353609\pi$
$521$	20.2626	0.887718	0.443859	+	0.896097i	$0.353609\pi$
$522$	0	0
$523$	− 31.8114i	− 1.39101i	−0.718520	−	0.695507i	$-0.755180\pi$
$523$	− 31.8114i	− 1.39101i	0.718520	−	0.695507i	$-0.244820\pi$
$524$	0.478457	0.0209015
$525$	0	0
$526$	12.1266	0.528745
$527$	− 26.1676i	− 1.13988i
$528$	0	0
$529$	5.25083	0.228297
$530$	0	0
$531$	0	0
$532$	− 1.62175i	− 0.0703117i
$533$	5.82870i	0.252469i
$534$	0	0
$535$	0	0
$536$	−1.75632	−0.0758613
$537$	0	0
$538$	− 0.159586i	− 0.00688024i
$539$	−2.65717	−0.114452
$540$	0	0
$541$	−15.1315	−0.650553	−0.325277	−	0.945619i	$-0.605457\pi$
$541$	−15.1315	−0.650553	−0.325277	+	0.945619i	$0.605457\pi$
$542$	10.2188i	0.438937i
$543$	0	0
$544$	−15.3831	−0.659543
$545$	0	0
$546$	0	0
$547$	− 4.08744i	− 0.174766i	−0.996175	−	0.0873831i	$-0.972150\pi$
$547$	− 4.08744i	− 0.174766i	0.996175	−	0.0873831i	$-0.0278504\pi$
$548$	6.17381i	0.263732i
$549$	0	0
$550$	0	0
$551$	0.600748	0.0255927
$552$	0	0
$553$	24.2757i	1.03231i
$554$	−11.4214	−0.485249
$555$	0	0
$556$	−26.1926	−1.11082
$557$	− 13.1425i	− 0.556864i	−0.960456	−	0.278432i	$-0.910185\pi$
$557$	− 13.1425i	− 0.556864i	0.960456	−	0.278432i	$-0.0898147\pi$
$558$	0	0
$559$	16.3721	0.692467
$560$	0	0
$561$	0	0
$562$	1.59357i	0.0672207i
$563$	24.5221i	1.03348i	0.856141	+	0.516742i	$0.172855\pi$
$563$	24.5221i	1.03348i	−0.856141	+	0.516742i	$0.827145\pi$
$564$	0	0
$565$	0	0
$566$	10.3405	0.434642
$567$	0	0
$568$	− 11.4996i	− 0.482514i
$569$	−22.7299	−0.952885	−0.476442	−	0.879206i	$-0.658074\pi$
$569$	−22.7299	−0.952885	−0.476442	+	0.879206i	$0.658074\pi$
$570$	0	0
$571$	−0.494186	−0.0206810	−0.0103405	−	0.999947i	$-0.503292\pi$
$571$	−0.494186	−0.0206810	−0.0103405	+	0.999947i	$0.503292\pi$
$572$	23.3607i	0.976758i
$573$	0	0
$574$	3.31571	0.138395
$575$	0	0
$576$	0	0
$577$	− 9.41187i	− 0.391821i	−0.980622	−	0.195911i	$-0.937234\pi$
$577$	− 9.41187i	− 0.391821i	0.980622	−	0.195911i	$-0.0627662\pi$
$578$	3.29382i	0.137005i
$579$	0	0
$580$	0	0
$581$	−26.7064	−1.10797
$582$	0	0
$583$	− 58.2012i	− 2.41045i
$584$	11.8206	0.489139
$585$	0	0
$586$	6.50916	0.268891
$587$	9.97321i	0.411638i	0.978590	+	0.205819i	$0.0659859\pi$
$587$	9.97321i	0.411638i	−0.978590	+	0.205819i	$0.934014\pi$
$588$	0	0
$589$	2.94219	0.121231
$590$	0	0
$591$	0	0
$592$	9.83144i	0.404070i
$593$	− 38.3421i	− 1.57452i	−0.616621	−	0.787260i	$-0.711499\pi$
$593$	− 38.3421i	− 1.57452i	0.616621	−	0.787260i	$-0.288501\pi$
$594$	0	0
$595$	0	0
$596$	18.0326	0.738642
$597$	0	0
$598$	4.25140i	0.173852i
$599$	10.1533	0.414852	0.207426	−	0.978251i	$-0.433491\pi$
$599$	10.1533	0.414852	0.207426	+	0.978251i	$0.433491\pi$
$600$	0	0
$601$	−21.2743	−0.867796	−0.433898	−	0.900962i	$-0.642862\pi$
$601$	−21.2743	−0.867796	−0.433898	+	0.900962i	$0.642862\pi$
$602$	− 9.31344i	− 0.379587i
$603$	0	0
$604$	−18.3219	−0.745508
$605$	0	0
$606$	0	0
$607$	− 37.7355i	− 1.53164i	−0.643056	−	0.765819i	$-0.722334\pi$
$607$	− 37.7355i	− 1.53164i	0.643056	−	0.765819i	$-0.277666\pi$
$608$	− 1.72962i	− 0.0701452i
$609$	0	0
$610$	0	0
$611$	−24.3698	−0.985895
$612$	0	0
$613$	32.2633i	1.30310i	0.758605	+	0.651551i	$0.225881\pi$
$613$	32.2633i	1.30310i	−0.758605	+	0.651551i	$0.774119\pi$
$614$	−16.1940	−0.653536
$615$	0	0
$616$	28.2537	1.13837
$617$	− 26.0178i	− 1.04744i	−0.851891	−	0.523719i	$-0.824544\pi$
$617$	− 26.0178i	− 1.04744i	0.851891	−	0.523719i	$-0.175456\pi$
$618$	0	0
$619$	11.8815	0.477559	0.238780	−	0.971074i	$-0.423253\pi$
$619$	11.8815	0.477559	0.238780	+	0.971074i	$0.423253\pi$
$620$	0	0
$621$	0	0
$622$	10.9037i	0.437197i
$623$	16.0688i	0.643783i
$624$	0	0
$625$	0	0
$626$	−1.69960	−0.0679296
$627$	0	0
$628$	1.88723i	0.0753086i
$629$	−11.5107	−0.458962
$630$	0	0
$631$	−13.2726	−0.528372	−0.264186	−	0.964472i	$-0.585103\pi$
$631$	−13.2726	−0.528372	−0.264186	+	0.964472i	$0.585103\pi$
$632$	16.9256i	0.673263i
$633$	0	0
$634$	−6.23780	−0.247735
$635$	0	0
$636$	0	0
$637$	− 0.918501i	− 0.0363923i
$638$	4.92265i	0.194890i
$639$	0	0
$640$	0	0
$641$	44.8149	1.77008	0.885042	−	0.465511i	$-0.154129\pi$
$641$	44.8149	1.77008	0.885042	+	0.465511i	$0.154129\pi$
$642$	0	0
$643$	14.9255i	0.588605i	0.955712	+	0.294302i	$0.0950873\pi$
$643$	14.9255i	0.588605i	−0.955712	+	0.294302i	$0.904913\pi$
$644$	−19.1777	−0.755708
$645$	0	0
$646$	0.534242	0.0210195
$647$	− 41.2684i	− 1.62243i	−0.584749	−	0.811214i	$-0.698807\pi$
$647$	− 41.2684i	− 1.62243i	0.584749	−	0.811214i	$-0.301193\pi$
$648$	0	0
$649$	63.0179	2.47367
$650$	0	0
$651$	0	0
$652$	30.4386i	1.19207i
$653$	− 27.6252i	− 1.08106i	−0.841326	−	0.540528i	$-0.818224\pi$
$653$	− 27.6252i	− 1.08106i	0.841326	−	0.540528i	$-0.181776\pi$
$654$	0	0
$655$	0	0
$656$	−7.39788	−0.288839
$657$	0	0
$658$	13.8630i	0.540435i
$659$	−40.0225	−1.55905	−0.779527	−	0.626369i	$-0.784540\pi$
$659$	−40.0225	−1.55905	−0.779527	+	0.626369i	$0.784540\pi$
$660$	0	0
$661$	24.9929	0.972112	0.486056	−	0.873928i	$-0.338435\pi$
$661$	24.9929	0.972112	0.486056	+	0.873928i	$0.338435\pi$
$662$	0.559636i	0.0217509i
$663$	0	0
$664$	−18.6204	−0.722610
$665$	0	0
$666$	0	0
$667$	− 7.10405i	− 0.275070i
$668$	− 7.76738i	− 0.300529i
$669$	0	0
$670$	0	0
$671$	0.0678126	0.00261788
$672$	0	0
$673$	40.8048i	1.57291i	0.617649	+	0.786454i	$0.288085\pi$
$673$	40.8048i	1.57291i	−0.617649	+	0.786454i	$0.711915\pi$
$674$	11.7365	0.452072
$675$	0	0
$676$	15.0133	0.577436
$677$	41.1894i	1.58304i	0.611146	+	0.791518i	$0.290709\pi$
$677$	41.1894i	1.58304i	−0.611146	+	0.791518i	$0.709291\pi$
$678$	0	0
$679$	−18.4714	−0.708867
$680$	0	0
$681$	0	0
$682$	24.1089i	0.923178i
$683$	− 1.33820i	− 0.0512047i	−0.999672	−	0.0256023i	$-0.991850\pi$
$683$	− 1.33820i	− 0.0512047i	0.999672	−	0.0256023i	$-0.00815037\pi$
$684$	0	0
$685$	0	0
$686$	−9.01333	−0.344131
$687$	0	0
$688$	20.7798i	0.792221i
$689$	20.1184	0.766449
$690$	0	0
$691$	−25.2813	−0.961748	−0.480874	−	0.876790i	$-0.659680\pi$
$691$	−25.2813	−0.961748	−0.480874	+	0.876790i	$0.659680\pi$
$692$	26.0368i	0.989770i
$693$	0	0
$694$	−10.4917	−0.398258
$695$	0	0
$696$	0	0
$697$	− 8.66148i	− 0.328077i
$698$	− 7.06025i	− 0.267234i
$699$	0	0
$700$	0	0
$701$	−18.2064	−0.687645	−0.343822	−	0.939035i	$-0.611722\pi$
$701$	−18.2064	−0.687645	−0.343822	+	0.939035i	$0.611722\pi$
$702$	0	0
$703$	− 1.29422i	− 0.0488126i
$704$	−19.2159	−0.724227
$705$	0	0
$706$	8.00487	0.301267
$707$	17.8669i	0.671953i
$708$	0	0
$709$	−41.8206	−1.57061	−0.785304	−	0.619111i	$-0.787493\pi$
$709$	−41.8206	−1.57061	−0.785304	+	0.619111i	$0.787493\pi$
$710$	0	0
$711$	0	0
$712$	11.2036i	0.419871i
$713$	− 34.7925i	− 1.30299i
$714$	0	0
$715$	0	0
$716$	12.2016	0.455995
$717$	0	0
$718$	− 0.301301i	− 0.0112444i
$719$	−48.9786	−1.82660	−0.913298	−	0.407293i	$-0.866473\pi$
$719$	−48.9786	−1.82660	−0.913298	+	0.407293i	$0.866473\pi$
$720$	0	0
$721$	−15.6734	−0.583707
$722$	− 8.93177i	− 0.332406i
$723$	0	0
$724$	−19.4847	−0.724144
$725$	0	0
$726$	0	0
$727$	− 43.8009i	− 1.62449i	−0.583319	−	0.812243i	$-0.698246\pi$
$727$	− 43.8009i	− 1.62449i	0.583319	−	0.812243i	$-0.301754\pi$
$728$	9.76642i	0.361968i
$729$	0	0
$730$	0	0
$731$	−24.3291	−0.899843
$732$	0	0
$733$	6.78664i	0.250670i	0.992114	+	0.125335i	$0.0400006\pi$
$733$	6.78664i	0.250670i	−0.992114	+	0.125335i	$0.959999\pi$
$734$	−9.51144	−0.351074
$735$	0	0
$736$	−20.4533	−0.753919
$737$	− 6.06261i	− 0.223319i
$738$	0	0
$739$	−28.7245	−1.05665	−0.528324	−	0.849043i	$-0.677179\pi$
$739$	−28.7245	−1.05665	−0.528324	+	0.849043i	$0.677179\pi$
$740$	0	0
$741$	0	0
$742$	− 11.4445i	− 0.420142i
$743$	31.4523i	1.15387i	0.816789	+	0.576937i	$0.195752\pi$
$743$	31.4523i	1.15387i	−0.816789	+	0.576937i	$0.804248\pi$
$744$	0	0
$745$	0	0
$746$	−9.29610	−0.340354
$747$	0	0
$748$	− 34.7141i	− 1.26927i
$749$	37.2537	1.36122
$750$	0	0
$751$	10.9532	0.399687	0.199844	−	0.979828i	$-0.435957\pi$
$751$	10.9532	0.399687	0.199844	+	0.979828i	$0.435957\pi$
$752$	− 30.9305i	− 1.12792i
$753$	0	0
$754$	−1.70161	−0.0619689
$755$	0	0
$756$	0	0
$757$	− 45.7942i	− 1.66442i	−0.554461	−	0.832210i	$-0.687075\pi$
$757$	− 45.7942i	− 1.66442i	0.554461	−	0.832210i	$-0.312925\pi$
$758$	3.75868i	0.136521i
$759$	0	0
$760$	0	0
$761$	−33.9138	−1.22937	−0.614687	−	0.788771i	$-0.710717\pi$
$761$	−33.9138	−1.22937	−0.614687	+	0.788771i	$0.710717\pi$
$762$	0	0
$763$	− 4.87528i	− 0.176497i
$764$	24.3894	0.882378
$765$	0	0
$766$	−14.7602	−0.533309
$767$	21.7833i	0.786551i
$768$	0	0
$769$	−7.15972	−0.258186	−0.129093	−	0.991632i	$-0.541207\pi$
$769$	−7.15972	−0.258186	−0.129093	+	0.991632i	$0.541207\pi$
$770$	0	0
$771$	0	0
$772$	0.856710i	0.0308337i
$773$	14.5998i	0.525117i	0.964916	+	0.262558i	$0.0845663\pi$
$773$	14.5998i	0.525117i	−0.964916	+	0.262558i	$0.915434\pi$
$774$	0	0
$775$	0	0
$776$	−12.8787	−0.462318
$777$	0	0
$778$	14.8836i	0.533602i
$779$	0.973866	0.0348924
$780$	0	0
$781$	39.6954	1.42041
$782$	− 6.31760i	− 0.225917i
$783$	0	0
$784$	1.16578	0.0416348
$785$	0	0
$786$	0	0
$787$	18.4728i	0.658483i	0.944246	+	0.329242i	$0.106793\pi$
$787$	18.4728i	0.658483i	−0.944246	+	0.329242i	$0.893207\pi$
$788$	9.83011i	0.350183i
$789$	0	0
$790$	0	0
$791$	−16.8630	−0.599578
$792$	0	0
$793$	0.0234407i	0 0.000832405i
$794$	−8.38484	−0.297567
$795$	0	0
$796$	−31.0077	−1.09904
$797$	41.0374i	1.45362i	0.686838	+	0.726810i	$0.258998\pi$
$797$	41.0374i	1.45362i	−0.686838	+	0.726810i	$0.741002\pi$
$798$	0	0
$799$	36.2136	1.28115
$800$	0	0
$801$	0	0
$802$	− 3.38021i	− 0.119359i
$803$	40.8033i	1.43992i
$804$	0	0
$805$	0	0
$806$	−8.33371	−0.293543
$807$	0	0
$808$	12.4572i	0.438244i
$809$	−7.19375	−0.252919	−0.126459	−	0.991972i	$-0.540361\pi$
$809$	−7.19375	−0.252919	−0.126459	+	0.991972i	$0.540361\pi$
$810$	0	0
$811$	38.2183	1.34203	0.671014	−	0.741445i	$-0.265859\pi$
$811$	38.2183	1.34203	0.671014	+	0.741445i	$0.265859\pi$
$812$	− 7.67583i	− 0.269369i
$813$	0	0
$814$	10.6051	0.371710
$815$	0	0
$816$	0	0
$817$	− 2.73547i	− 0.0957021i
$818$	− 11.7139i	− 0.409566i
$819$	0	0
$820$	0	0
$821$	−0.668560	−0.0233329	−0.0116665	−	0.999932i	$-0.503714\pi$
$821$	−0.668560	−0.0233329	−0.0116665	+	0.999932i	$0.503714\pi$
$822$	0	0
$823$	1.42033i	0.0495096i	0.999694	+	0.0247548i	$0.00788051\pi$
$823$	1.42033i	0.0495096i	−0.999694	+	0.0247548i	$0.992119\pi$
$824$	−10.9279	−0.380690
$825$	0	0
$826$	12.3917	0.431161
$827$	49.8169i	1.73230i	0.499782	+	0.866152i	$0.333414\pi$
$827$	49.8169i	1.73230i	−0.499782	+	0.866152i	$0.666586\pi$
$828$	0	0
$829$	−36.4150	−1.26475	−0.632373	−	0.774664i	$-0.717919\pi$
$829$	−36.4150	−1.26475	−0.632373	+	0.774664i	$0.717919\pi$
$830$	0	0
$831$	0	0
$832$	− 6.64235i	− 0.230282i
$833$	1.36490i	0.0472909i
$834$	0	0
$835$	0	0
$836$	3.90312	0.134992
$837$	0	0
$838$	5.04447i	0.174258i
$839$	20.0890	0.693549	0.346774	−	0.937949i	$-0.387277\pi$
$839$	20.0890	0.693549	0.346774	+	0.937949i	$0.387277\pi$
$840$	0	0
$841$	−26.1566	−0.901953
$842$	− 3.87056i	− 0.133388i
$843$	0	0
$844$	−2.90675	−0.100055
$845$	0	0
$846$	0	0
$847$	69.3349i	2.38238i
$848$	25.5345i	0.876860i
$849$	0	0
$850$	0	0
$851$	−15.3046	−0.524637
$852$	0	0
$853$	− 26.9084i	− 0.921326i	−0.887575	−	0.460663i	$-0.847612\pi$
$853$	− 26.9084i	− 0.921326i	0.887575	−	0.460663i	$-0.152388\pi$
$854$	0.0133345	0.000456296 0
$855$	0	0
$856$	25.9742	0.887779
$857$	− 48.8408i	− 1.66837i	−0.551486	−	0.834184i	$-0.685939\pi$
$857$	− 48.8408i	− 1.66837i	0.551486	−	0.834184i	$-0.314061\pi$
$858$	0	0
$859$	41.4094	1.41287	0.706435	−	0.707778i	$-0.250302\pi$
$859$	41.4094	1.41287	0.706435	+	0.707778i	$0.250302\pi$
$860$	0	0
$861$	0	0
$862$	0.791507i	0.0269588i
$863$	50.8101i	1.72960i	0.502119	+	0.864799i	$0.332554\pi$
$863$	50.8101i	1.72960i	−0.502119	+	0.864799i	$0.667446\pi$
$864$	0	0
$865$	0	0
$866$	−4.71359	−0.160174
$867$	0	0
$868$	− 37.5928i	− 1.27598i
$869$	−58.4252	−1.98194
$870$	0	0
$871$	2.09566	0.0710087
$872$	− 3.39916i	− 0.115110i
$873$	0	0
$874$	0.710328	0.0240272
$875$	0	0
$876$	0	0
$877$	− 0.409589i	− 0.0138308i	−0.999976	−	0.00691542i	$-0.997799\pi$
$877$	− 0.409589i	− 0.0138308i	0.999976	−	0.00691542i	$-0.00220127\pi$
$878$	− 6.06512i	− 0.204688i
$879$	0	0
$880$	0	0
$881$	−5.32851	−0.179522	−0.0897610	−	0.995963i	$-0.528610\pi$
$881$	−5.32851	−0.179522	−0.0897610	+	0.995963i	$0.528610\pi$
$882$	0	0
$883$	14.2064i	0.478083i	0.971009	+	0.239042i	$0.0768333\pi$
$883$	14.2064i	0.478083i	−0.971009	+	0.239042i	$0.923167\pi$
$884$	11.9996	0.403590
$885$	0	0
$886$	−3.66964	−0.123284
$887$	− 7.23193i	− 0.242825i	−0.992602	−	0.121412i	$-0.961258\pi$
$887$	− 7.23193i	− 0.242825i	0.992602	−	0.121412i	$-0.0387423\pi$
$888$	0	0
$889$	23.7298	0.795871
$890$	0	0
$891$	0	0
$892$	− 13.7604i	− 0.460733i
$893$	4.07173i	0.136255i
$894$	0	0
$895$	0	0
$896$	−28.6650	−0.957629
$897$	0	0
$898$	15.7784i	0.526531i
$899$	13.9256	0.464444
$900$	0	0
$901$	−29.8960	−0.995981
$902$	7.98006i	0.265707i
$903$	0	0
$904$	−11.7573	−0.391041
$905$	0	0
$906$	0	0
$907$	38.8101i	1.28867i	0.764743	+	0.644335i	$0.222866\pi$
$907$	38.8101i	1.28867i	−0.764743	+	0.644335i	$0.777134\pi$
$908$	− 19.9986i	− 0.663677i
$909$	0	0
$910$	0	0
$911$	−40.2781	−1.33447	−0.667236	−	0.744846i	$-0.732523\pi$
$911$	−40.2781	−1.33447	−0.667236	+	0.744846i	$0.732523\pi$
$912$	0	0
$913$	− 64.2754i	− 2.12721i
$914$	18.0874	0.598279
$915$	0	0
$916$	18.6085	0.614842
$917$	0.690479i	0.0228016i
$918$	0	0
$919$	13.0468	0.430375	0.215187	−	0.976573i	$-0.430964\pi$
$919$	13.0468	0.430375	0.215187	+	0.976573i	$0.430964\pi$
$920$	0	0
$921$	0	0
$922$	14.8144i	0.487888i
$923$	13.7215i	0.451648i
$924$	0	0
$925$	0	0
$926$	5.71936	0.187950
$927$	0	0
$928$	− 8.18638i	− 0.268731i
$929$	−0.293825	−0.00964008	−0.00482004	−	0.999988i	$-0.501534\pi$
$929$	−0.293825	−0.00964008	−0.00482004	+	0.999988i	$0.501534\pi$
$930$	0	0
$931$	−0.153464	−0.00502959
$932$	5.15428i	0.168834i
$933$	0	0
$934$	3.59894	0.117761
$935$	0	0
$936$	0	0
$937$	− 16.9141i	− 0.552559i	−0.961077	−	0.276280i	$-0.910898\pi$
$937$	− 16.9141i	− 0.552559i	0.961077	−	0.276280i	$-0.0891016\pi$
$938$	− 1.19213i	− 0.0389246i
$939$	0	0
$940$	0	0
$941$	57.2093	1.86497	0.932485	−	0.361209i	$-0.117636\pi$
$941$	57.2093	1.86497	0.932485	+	0.361209i	$0.117636\pi$
$942$	0	0
$943$	− 11.5163i	− 0.375023i
$944$	−27.6478	−0.899858
$945$	0	0
$946$	22.4150	0.728775
$947$	− 38.8746i	− 1.26325i	−0.775272	−	0.631627i	$-0.782387\pi$
$947$	− 38.8746i	− 1.26325i	0.775272	−	0.631627i	$-0.217613\pi$
$948$	0	0
$949$	−14.1044	−0.457850
$950$	0	0
$951$	0	0
$952$	− 14.5130i	− 0.470368i
$953$	54.4516i	1.76386i	0.471381	+	0.881930i	$0.343756\pi$
$953$	54.4516i	1.76386i	−0.471381	+	0.881930i	$0.656244\pi$
$954$	0	0
$955$	0	0
$956$	−29.0460	−0.939415
$957$	0	0
$958$	15.3729i	0.496675i
$959$	−8.90965	−0.287707
$960$	0	0
$961$	37.2012	1.20004
$962$	3.66587i	0.118192i
$963$	0	0
$964$	31.1229	1.00240
$965$	0	0
$966$	0	0
$967$	− 19.5701i	− 0.629333i	−0.949202	−	0.314666i	$-0.898107\pi$
$967$	− 19.5701i	− 0.629333i	0.949202	−	0.314666i	$-0.101893\pi$
$968$	48.3420i	1.55377i
$969$	0	0
$970$	0	0
$971$	6.31009	0.202500	0.101250	−	0.994861i	$-0.467716\pi$
$971$	6.31009	0.202500	0.101250	+	0.994861i	$0.467716\pi$
$972$	0	0
$973$	− 37.7995i	− 1.21180i
$974$	2.11129	0.0676500
$975$	0	0
$976$	−0.0297513	−0.000952317 0
$977$	7.41911i	0.237358i	0.992933	+	0.118679i	$0.0378660\pi$
$977$	7.41911i	0.237358i	−0.992933	+	0.118679i	$0.962134\pi$
$978$	0	0
$979$	−38.6734	−1.23601
$980$	0	0
$981$	0	0
$982$	15.5426i	0.495986i
$983$	10.9827i	0.350295i	0.984542	+	0.175148i	$0.0560403\pi$
$983$	10.9827i	0.350295i	−0.984542	+	0.175148i	$0.943960\pi$
$984$	0	0
$985$	0	0
$986$	2.52860	0.0805270
$987$	0	0
$988$	1.34919i	0.0429234i
$989$	−32.3479	−1.02860
$990$	0	0
$991$	−21.3721	−0.678908	−0.339454	−	0.940623i	$-0.610242\pi$
$991$	−21.3721	−0.678908	−0.339454	+	0.940623i	$0.610242\pi$
$992$	− 40.0932i	− 1.27296i
$993$	0	0
$994$	7.80559	0.247578
$995$	0	0
$996$	0	0
$997$	− 30.5348i	− 0.967047i	−0.875331	−	0.483524i	$-0.839357\pi$
$997$	− 30.5348i	− 0.967047i	0.875331	−	0.483524i	$-0.160643\pi$
$998$	16.1863i	0.512368i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.b.n.649.5		8
3.2	odd	2		2025.2.b.o.649.4		8
5.2	odd	4		2025.2.a.y.1.2		4
5.3	odd	4		2025.2.a.q.1.3		4
5.4	even	2	inner	2025.2.b.n.649.4		8
9.2	odd	6		675.2.k.c.199.4		16
9.4	even	3		225.2.k.c.124.4		16
9.5	odd	6		675.2.k.c.424.5		16
9.7	even	3		225.2.k.c.49.5		16
15.2	even	4		2025.2.a.p.1.3		4
15.8	even	4		2025.2.a.z.1.2		4
15.14	odd	2		2025.2.b.o.649.5		8
45.2	even	12		675.2.e.e.226.2		8
45.4	even	6		225.2.k.c.124.5		16
45.7	odd	12		225.2.e.c.76.3	✓	8
45.13	odd	12		225.2.e.e.151.2	yes	8
45.14	odd	6		675.2.k.c.424.4		16
45.22	odd	12		225.2.e.c.151.3	yes	8
45.23	even	12		675.2.e.c.451.3		8
45.29	odd	6		675.2.k.c.199.5		16
45.32	even	12		675.2.e.e.451.2		8
45.34	even	6		225.2.k.c.49.4		16
45.38	even	12		675.2.e.c.226.3		8
45.43	odd	12		225.2.e.e.76.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.e.c.76.3	✓	8	45.7	odd	12
225.2.e.c.151.3	yes	8	45.22	odd	12
225.2.e.e.76.2	yes	8	45.43	odd	12
225.2.e.e.151.2	yes	8	45.13	odd	12
225.2.k.c.49.4		16	45.34	even	6
225.2.k.c.49.5		16	9.7	even	3
225.2.k.c.124.4		16	9.4	even	3
225.2.k.c.124.5		16	45.4	even	6
675.2.e.c.226.3		8	45.38	even	12
675.2.e.c.451.3		8	45.23	even	12
675.2.e.e.226.2		8	45.2	even	12
675.2.e.e.451.2		8	45.32	even	12
675.2.k.c.199.4		16	9.2	odd	6
675.2.k.c.199.5		16	45.29	odd	6
675.2.k.c.424.4		16	45.14	odd	6
675.2.k.c.424.5		16	9.5	odd	6
2025.2.a.p.1.3		4	15.2	even	4
2025.2.a.q.1.3		4	5.3	odd	4
2025.2.a.y.1.2		4	5.2	odd	4
2025.2.a.z.1.2		4	15.8	even	4
2025.2.b.n.649.4		8	5.4	even	2	inner
2025.2.b.n.649.5		8	1.1	even	1	trivial
2025.2.b.o.649.4		8	3.2	odd	2
2025.2.b.o.649.5		8	15.14	odd	2

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)

\(f(q)\)	\(=\)	\(q+0.473255i q^{2} +1.77603 q^{4} +2.56305i q^{7} +1.78702i q^{8} +O(q^{10})\)	\(q+0.473255i q^{2} +1.77603 q^{4} +2.56305i q^{7} +1.78702i q^{8} -6.16860 q^{11} -2.13230i q^{13} -1.21298 q^{14} +2.70634 q^{16} +3.16860i q^{17} -0.356267 q^{19} -2.91932i q^{22} +4.21298i q^{23} +1.00912 q^{26} +4.55206i q^{28} -1.68623 q^{29} -8.25840 q^{31} +4.85484i q^{32} -1.49956 q^{34} +3.63274i q^{37} -0.168605i q^{38} -2.73353 q^{41} +7.67817i q^{43} -10.9556 q^{44} -1.99381 q^{46} -11.4289i q^{47} +0.430757 q^{49} -3.78702i q^{52} +9.43507i q^{53} -4.58024 q^{56} -0.798017i q^{58} -10.2159 q^{59} -0.0109932 q^{61} -3.90833i q^{62} +3.11511 q^{64} +0.982817i q^{67} +5.62754i q^{68} -6.43507 q^{71} -6.61467i q^{73} -1.71921 q^{74} -0.632740 q^{76} -15.8105i q^{77} +9.47138 q^{79} -1.29366i q^{82} +10.4198i q^{83} -3.63373 q^{86} -11.0234i q^{88} +6.26940 q^{89} +5.46519 q^{91} +7.48237i q^{92} +5.40877 q^{94} +7.20679i q^{97} +0.203858i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 8 * q - 8 * q^4	\( 8 q - 8 q^{4} - 2 q^{11} - 6 q^{14} + 8 q^{16} - 4 q^{19} - 20 q^{26} - 2 q^{29} - 8 q^{31} - 18 q^{34} - 10 q^{41} - 44 q^{44} + 6 q^{49} - 60 q^{56} - 34 q^{59} - 26 q^{61} - 38 q^{64} - 16 q^{71} - 80 q^{74} + 22 q^{76} + 14 q^{79} - 68 q^{86} + 18 q^{89} - 34 q^{91} - 6 q^{94}+O(q^{100}) \) 8 * q - 8 * q^4 - 2 * q^11 - 6 * q^14 + 8 * q^16 - 4 * q^19 - 20 * q^26 - 2 * q^29 - 8 * q^31 - 18 * q^34 - 10 * q^41 - 44 * q^44 + 6 * q^49 - 60 * q^56 - 34 * q^59 - 26 * q^61 - 38 * q^64 - 16 * q^71 - 80 * q^74 + 22 * q^76 + 14 * q^79 - 68 * q^86 + 18 * q^89 - 34 * q^91 - 6 * q^94

Introduction

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