LMFDB - Embedded newform 2025.2.a.p.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,2,Mod(1,2025)]

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, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2025 = 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2025.a (trivial)

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:	yes
Analytic conductor:	$16.1697064093$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.11661.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 3 $ x^4 - 2x^3 - 4x^2 + 5*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 225)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.47325$ of defining polynomial
Character	$\chi$	$=$	2025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.473255 q^{2} -1.77603 q^{4} -2.56305 q^{7} -1.78702 q^{8} +O(q^{10})$	$q+0.473255 q^{2} -1.77603 q^{4} -2.56305 q^{7} -1.78702 q^{8} +6.16860 q^{11} -2.13230 q^{13} -1.21298 q^{14} +2.70634 q^{16} +3.16860 q^{17} +0.356267 q^{19} +2.91932 q^{22} -4.21298 q^{23} -1.00912 q^{26} +4.55206 q^{28} -1.68623 q^{29} -8.25840 q^{31} +4.85484 q^{32} +1.49956 q^{34} -3.63274 q^{37} +0.168605 q^{38} +2.73353 q^{41} +7.67817 q^{43} -10.9556 q^{44} -1.99381 q^{46} -11.4289 q^{47} -0.430757 q^{49} +3.78702 q^{52} -9.43507 q^{53} +4.58024 q^{56} -0.798017 q^{58} -10.2159 q^{59} -0.0109932 q^{61} -3.90833 q^{62} -3.11511 q^{64} -0.982817 q^{67} -5.62754 q^{68} +6.43507 q^{71} -6.61467 q^{73} -1.71921 q^{74} -0.632740 q^{76} -15.8105 q^{77} -9.47138 q^{79} +1.29366 q^{82} -10.4198 q^{83} +3.63373 q^{86} -11.0234 q^{88} +6.26940 q^{89} +5.46519 q^{91} +7.48237 q^{92} -5.40877 q^{94} -7.20679 q^{97} -0.203858 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 4 q^{4} - q^{7} - 9 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 4 * q^4 - q^7 - 9 * q^8	$ 4 q - 2 q^{2} + 4 q^{4} - q^{7} - 9 q^{8} + q^{11} + 2 q^{13} - 3 q^{14} + 4 q^{16} - 11 q^{17} + 2 q^{19} + 3 q^{22} - 15 q^{23} + 10 q^{26} - 4 q^{28} - q^{29} - 4 q^{31} - 10 q^{32} + 9 q^{34} - q^{37} - 23 q^{38} + 5 q^{41} - 10 q^{43} - 22 q^{44} - 20 q^{47} - 3 q^{49} + 17 q^{52} - 20 q^{53} + 30 q^{56} - 18 q^{58} - 17 q^{59} - 13 q^{61} + 6 q^{62} + 19 q^{64} + 17 q^{67} - 34 q^{68} + 8 q^{71} + 2 q^{73} - 40 q^{74} + 11 q^{76} - 12 q^{77} - 7 q^{79} + 12 q^{82} - 30 q^{83} + 34 q^{86} + 9 q^{88} + 9 q^{89} - 17 q^{91} + 12 q^{92} + 3 q^{94} - 19 q^{97} - 13 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 4 * q^4 - q^7 - 9 * q^8 + q^11 + 2 * q^13 - 3 * q^14 + 4 * q^16 - 11 * q^17 + 2 * q^19 + 3 * q^22 - 15 * q^23 + 10 * q^26 - 4 * q^28 - q^29 - 4 * q^31 - 10 * q^32 + 9 * q^34 - q^37 - 23 * q^38 + 5 * q^41 - 10 * q^43 - 22 * q^44 - 20 * q^47 - 3 * q^49 + 17 * q^52 - 20 * q^53 + 30 * q^56 - 18 * q^58 - 17 * q^59 - 13 * q^61 + 6 * q^62 + 19 * q^64 + 17 * q^67 - 34 * q^68 + 8 * q^71 + 2 * q^73 - 40 * q^74 + 11 * q^76 - 12 * q^77 - 7 * q^79 + 12 * q^82 - 30 * q^83 + 34 * q^86 + 9 * q^88 + 9 * q^89 - 17 * q^91 + 12 * q^92 + 3 * q^94 - 19 * q^97 - 13 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	0.473255	0.334641	0.167321	−	0.985903i	$-0.446488\pi$
$2$	0.473255	0.334641	0.167321	+	0.985903i	$0.446488\pi$
$3$	0	0
$3$	0	0
$4$	−1.77603	−0.888015
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.56305	−0.968743	−0.484372	−	0.874862i	$-0.660952\pi$
$7$	−2.56305	−0.968743	−0.484372	+	0.874862i	$0.660952\pi$
$8$	−1.78702	−0.631808
$9$	0	0
$10$	0	0
$11$	6.16860	1.85990	0.929952	−	0.367681i	$-0.119848\pi$
$11$	6.16860	1.85990	0.929952	+	0.367681i	$0.119848\pi$
$12$	0	0
$13$	−2.13230	−0.591393	−0.295696	−	0.955282i	$-0.595552\pi$
$13$	−2.13230	−0.591393	−0.295696	+	0.955282i	$0.595552\pi$
$14$	−1.21298	−0.324182
$15$	0	0
$16$	2.70634	0.676586
$17$	3.16860	0.768500	0.384250	−	0.923229i	$-0.374460\pi$
$17$	3.16860	0.768500	0.384250	+	0.923229i	$0.374460\pi$
$18$	0	0
$19$	0.356267	0.0817332	0.0408666	−	0.999165i	$-0.486988\pi$
$19$	0.356267	0.0817332	0.0408666	+	0.999165i	$0.486988\pi$
$20$	0	0
$21$	0	0
$22$	2.91932	0.622401
$23$	−4.21298	−0.878466	−0.439233	−	0.898373i	$-0.644750\pi$
$23$	−4.21298	−0.878466	−0.439233	+	0.898373i	$0.644750\pi$
$24$	0	0
$25$	0	0
$26$	−1.00912	−0.197905
$27$	0	0
$28$	4.55206	0.860259
$29$	−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	$0.765945\pi$
$32$	4.85484	0.858222
$33$	0	0
$34$	1.49956	0.257172
$35$	0	0
$36$	0	0
$37$	−3.63274	−0.597219	−0.298609	−	0.954375i	$-0.596523\pi$
$37$	−3.63274	−0.597219	−0.298609	+	0.954375i	$0.596523\pi$
$38$	0.168605	0.0273513
$39$	0	0
$40$	0	0
$41$	2.73353	0.426906	0.213453	−	0.976953i	$-0.431529\pi$
$41$	2.73353	0.426906	0.213453	+	0.976953i	$0.431529\pi$
$42$	0	0
$43$	7.67817	1.17091	0.585455	−	0.810705i	$-0.300916\pi$
$43$	7.67817	1.17091	0.585455	+	0.810705i	$0.300916\pi$
$44$	−10.9556	−1.65162
$45$	0	0
$46$	−1.99381	−0.293971
$47$	−11.4289	−1.66707	−0.833537	−	0.552464i	$-0.813688\pi$
$47$	−11.4289	−1.66707	−0.833537	+	0.552464i	$0.813688\pi$
$48$	0	0
$49$	−0.430757	−0.0615367
$50$	0	0
$51$	0	0
$52$	3.78702	0.525166
$53$	−9.43507	−1.29601	−0.648003	−	0.761637i	$-0.724396\pi$
$53$	−9.43507	−1.29601	−0.648003	+	0.761637i	$0.724396\pi$
$54$	0	0
$55$	0	0
$56$	4.58024	0.612060
$57$	0	0
$58$	−0.798017	−0.104785
$59$	−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	$0.500224\pi$
$62$	−3.90833	−0.496358
$63$	0	0
$64$	−3.11511	−0.389389
$65$	0	0
$66$	0	0
$67$	−0.982817	−0.120070	−0.0600351	−	0.998196i	$-0.519121\pi$
$67$	−0.982817	−0.120070	−0.0600351	+	0.998196i	$0.519121\pi$
$68$	−5.62754	−0.682439
$69$	0	0
$70$	0	0
$71$	6.43507	0.763703	0.381851	−	0.924224i	$-0.375287\pi$
$71$	6.43507	0.763703	0.381851	+	0.924224i	$0.375287\pi$
$72$	0	0
$73$	−6.61467	−0.774189	−0.387094	−	0.922040i	$-0.626521\pi$
$73$	−6.61467	−0.774189	−0.387094	+	0.922040i	$0.626521\pi$
$74$	−1.71921	−0.199854
$75$	0	0
$76$	−0.632740	−0.0725803
$77$	−15.8105	−1.80177
$78$	0	0
$79$	−9.47138	−1.06561	−0.532807	−	0.846237i	$-0.678863\pi$
$79$	−9.47138	−1.06561	−0.532807	+	0.846237i	$0.678863\pi$
$80$	0	0
$81$	0	0
$82$	1.29366	0.142860
$83$	−10.4198	−1.14372	−0.571859	−	0.820352i	$-0.693777\pi$
$83$	−10.4198	−1.14372	−0.571859	+	0.820352i	$0.693777\pi$
$84$	0	0
$85$	0	0
$86$	3.63373	0.391835
$87$	0	0
$88$	−11.0234	−1.17510
$89$	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
$92$	7.48237	0.780091
$93$	0	0
$94$	−5.40877	−0.557872
$95$	0	0
$96$	0	0
$97$	−7.20679	−0.731738	−0.365869	−	0.930666i	$-0.619228\pi$
$97$	−7.20679	−0.731738	−0.365869	+	0.930666i	$0.619228\pi$
$98$	−0.203858	−0.0205927
$99$	0	0
$100$	0	0
$101$	−6.97094	−0.693634	−0.346817	−	0.937933i	$-0.612738\pi$
$101$	−6.97094	−0.693634	−0.346817	+	0.937933i	$0.612738\pi$
$102$	0	0
$103$	6.11511	0.602540	0.301270	−	0.953539i	$-0.402589\pi$
$103$	6.11511	0.602540	0.301270	+	0.953539i	$0.402589\pi$
$104$	3.81046	0.373647
$105$	0	0
$106$	−4.46519	−0.433698
$107$	−14.5349	−1.40514	−0.702570	−	0.711615i	$-0.747964\pi$
$107$	−14.5349	−1.40514	−0.702570	+	0.711615i	$0.747964\pi$
$108$	0	0
$109$	1.90214	0.182192	0.0910958	−	0.995842i	$-0.470963\pi$
$109$	1.90214	0.182192	0.0910958	+	0.995842i	$0.470963\pi$
$110$	0	0
$111$	0	0
$112$	−6.93650	−0.655438
$113$	−6.57925	−0.618924	−0.309462	−	0.950912i	$-0.600149\pi$
$113$	−6.57925	−0.618924	−0.309462	+	0.950912i	$0.600149\pi$
$114$	0	0
$115$	0	0
$116$	2.99480	0.278060
$117$	0	0
$118$	−4.83472	−0.445072
$119$	−8.12130	−0.744479
$120$	0	0
$121$	27.0517	2.45924
$122$	−0.00520257	−0.000471019 0
$123$	0	0
$124$	14.6672	1.31715
$125$	0	0
$126$	0	0
$127$	9.25840	0.821550	0.410775	−	0.911737i	$-0.365258\pi$
$127$	9.25840	0.821550	0.410775	+	0.911737i	$0.365258\pi$
$128$	−11.1839	−0.988528
$129$	0	0
$130$	0	0
$131$	−0.269397	−0.0235373	−0.0117687	−	0.999931i	$-0.503746\pi$
$131$	−0.269397	−0.0235373	−0.0117687	+	0.999931i	$0.503746\pi$
$132$	0	0
$133$	−0.913130	−0.0791784
$134$	−0.465123	−0.0401805
$135$	0	0
$136$	−5.66237	−0.485544
$137$	3.47618	0.296990	0.148495	−	0.988913i	$-0.452557\pi$
$137$	3.47618	0.296990	0.148495	+	0.988913i	$0.452557\pi$
$138$	0	0
$139$	14.7479	1.25090	0.625448	−	0.780266i	$-0.284916\pi$
$139$	14.7479	1.25090	0.625448	+	0.780266i	$0.284916\pi$
$140$	0	0
$141$	0	0
$142$	3.04543	0.255567
$143$	−13.1533	−1.09993
$144$	0	0
$145$	0	0
$146$	−3.13042	−0.259076
$147$	0	0
$148$	6.45186	0.530339
$149$	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	$0.637886\pi$
$152$	−0.636657	−0.0516397
$153$	0	0
$154$	−7.48237	−0.602947
$155$	0	0
$156$	0	0
$157$	−1.06261	−0.0848055	−0.0424028	−	0.999101i	$-0.513501\pi$
$157$	−1.06261	−0.0848055	−0.0424028	+	0.999101i	$0.513501\pi$
$158$	−4.48237	−0.356598
$159$	0	0
$160$	0	0
$161$	10.7981	0.851008
$162$	0	0
$163$	17.1386	1.34240	0.671198	−	0.741278i	$-0.265780\pi$
$163$	17.1386	1.34240	0.671198	+	0.741278i	$0.265780\pi$
$164$	−4.85484	−0.379099
$165$	0	0
$166$	−4.93120	−0.382735
$167$	−4.37345	−0.338428	−0.169214	−	0.985579i	$-0.554123\pi$
$167$	−4.37345	−0.338428	−0.169214	+	0.985579i	$0.554123\pi$
$168$	0	0
$169$	−8.45331	−0.650255
$170$	0	0
$171$	0	0
$172$	−13.6367	−1.03979
$173$	−14.6601	−1.11459	−0.557293	−	0.830316i	$-0.688160\pi$
$173$	−14.6601	−1.11459	−0.557293	+	0.830316i	$0.688160\pi$
$174$	0	0
$175$	0	0
$176$	16.6944	1.25838
$177$	0	0
$178$	2.96702	0.222388
$179$	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	$0.633680\pi$
$182$	2.58643	0.191719
$183$	0	0
$184$	7.52869	0.555022
$185$	0	0
$186$	0	0
$187$	19.5459	1.42934
$188$	20.2980	1.48039
$189$	0	0
$190$	0	0
$191$	−13.7325	−0.993652	−0.496826	−	0.867850i	$-0.665501\pi$
$191$	−13.7325	−0.993652	−0.496826	+	0.867850i	$0.665501\pi$
$192$	0	0
$193$	0.482374	0.0347220	0.0173610	−	0.999849i	$-0.494474\pi$
$193$	0.482374	0.0347220	0.0173610	+	0.999849i	$0.494474\pi$
$194$	−3.41064	−0.244870
$195$	0	0
$196$	0.765037	0.0546455
$197$	5.53488	0.394344	0.197172	−	0.980369i	$-0.436824\pi$
$197$	5.53488	0.394344	0.197172	+	0.980369i	$0.436824\pi$
$198$	0	0
$199$	17.4590	1.23764	0.618818	−	0.785534i	$-0.287612\pi$
$199$	17.4590	1.23764	0.618818	+	0.785534i	$0.287612\pi$
$200$	0	0
$201$	0	0
$202$	−3.29903	−0.232119
$203$	4.32190	0.303338
$204$	0	0
$205$	0	0
$206$	2.89401	0.201635
$207$	0	0
$208$	−5.77073	−0.400128
$209$	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	$0.517942\pi$
$212$	16.7570	1.15087
$213$	0	0
$214$	−6.87870	−0.470218
$215$	0	0
$216$	0	0
$217$	21.1667	1.43689
$218$	0.900195	0.0609689
$219$	0	0
$220$	0	0
$221$	−6.75641	−0.454485
$222$	0	0
$223$	−7.74785	−0.518835	−0.259417	−	0.965765i	$-0.583531\pi$
$223$	−7.74785	−0.518835	−0.259417	+	0.965765i	$0.583531\pi$
$224$	−12.4432	−0.831397
$225$	0	0
$226$	−3.11366	−0.207118
$227$	−11.2603	−0.747371	−0.373685	−	0.927555i	$-0.621906\pi$
$227$	−11.2603	−0.747371	−0.373685	+	0.927555i	$0.621906\pi$
$228$	0	0
$229$	−10.4776	−0.692377	−0.346189	−	0.938165i	$-0.612524\pi$
$229$	−10.4776	−0.692377	−0.346189	+	0.938165i	$0.612524\pi$
$230$	0	0
$231$	0	0
$232$	3.01333	0.197835
$233$	−2.90214	−0.190125	−0.0950627	−	0.995471i	$-0.530305\pi$
$233$	−2.90214	−0.190125	−0.0950627	+	0.995471i	$0.530305\pi$
$234$	0	0
$235$	0	0
$236$	18.1438	1.18106
$237$	0	0
$238$	−3.84344	−0.249133
$239$	−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	$0.309105\pi$
$242$	12.8023	0.822965
$243$	0	0
$244$	0.0195242	0.00124991
$245$	0	0
$246$	0	0
$247$	−0.759666	−0.0483364
$248$	14.7580	0.937131
$249$	0	0
$250$	0	0
$251$	−8.46999	−0.534621	−0.267311	−	0.963610i	$-0.586135\pi$
$251$	−8.46999	−0.534621	−0.267311	+	0.963610i	$0.586135\pi$
$252$	0	0
$253$	−25.9882	−1.63386
$254$	4.38158	0.274925
$255$	0	0
$256$	0.937390	0.0585869
$257$	−2.87251	−0.179182	−0.0895910	−	0.995979i	$-0.528556\pi$
$257$	−2.87251	−0.179182	−0.0895910	+	0.995979i	$0.528556\pi$
$258$	0	0
$259$	9.31091	0.578552
$260$	0	0
$261$	0	0
$262$	−0.127493	−0.00787656
$263$	25.6239	1.58003	0.790017	−	0.613084i	$-0.210071\pi$
$263$	25.6239	1.58003	0.790017	+	0.613084i	$0.210071\pi$
$264$	0	0
$265$	0	0
$266$	−0.432143	−0.0264964
$267$	0	0
$268$	1.74551	0.106624
$269$	−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	$0.272318\pi$
$272$	8.57533	0.519956
$273$	0	0
$274$	1.64512	0.0993853
$275$	0	0
$276$	0	0
$277$	−24.1338	−1.45006	−0.725028	−	0.688719i	$-0.758173\pi$
$277$	−24.1338	−1.45006	−0.725028	+	0.688719i	$0.758173\pi$
$278$	6.97949	0.418602
$279$	0	0
$280$	0	0
$281$	−3.36726	−0.200874	−0.100437	−	0.994943i	$-0.532024\pi$
$281$	−3.36726	−0.200874	−0.100437	+	0.994943i	$0.532024\pi$
$282$	0	0
$283$	−21.8497	−1.29883	−0.649415	−	0.760434i	$-0.724986\pi$
$283$	−21.8497	−1.29883	−0.649415	+	0.760434i	$0.724986\pi$
$284$	−11.4289	−0.678179
$285$	0	0
$286$	−6.22486	−0.368083
$287$	−7.00619	−0.413562
$288$	0	0
$289$	−6.95994	−0.409408
$290$	0	0
$291$	0	0
$292$	11.7479	0.687491
$293$	13.7540	0.803520	0.401760	−	0.915745i	$-0.368399\pi$
$293$	13.7540	0.803520	0.401760	+	0.915745i	$0.368399\pi$
$294$	0	0
$295$	0	0
$296$	6.49179	0.377328
$297$	0	0
$298$	4.80509	0.278352
$299$	8.98332	0.519519
$300$	0	0
$301$	−19.6796	−1.13431
$302$	−4.88219	−0.280939
$303$	0	0
$304$	0.964180	0.0552995
$305$	0	0
$306$	0	0
$307$	−34.2183	−1.95294	−0.976472	−	0.215644i	$-0.930815\pi$
$307$	−34.2183	−1.95294	−0.976472	+	0.215644i	$0.930815\pi$
$308$	28.0799	1.60000
$309$	0	0
$310$	0	0
$311$	−23.0397	−1.30646	−0.653232	−	0.757158i	$-0.726587\pi$
$311$	−23.0397	−1.30646	−0.653232	+	0.757158i	$0.726587\pi$
$312$	0	0
$313$	3.59130	0.202992	0.101496	−	0.994836i	$-0.467637\pi$
$313$	3.59130	0.202992	0.101496	+	0.994836i	$0.467637\pi$
$314$	−0.502885	−0.0283794
$315$	0	0
$316$	16.8215	0.946281
$317$	13.1807	0.740299	0.370150	−	0.928972i	$-0.379306\pi$
$317$	13.1807	0.740299	0.370150	+	0.928972i	$0.379306\pi$
$318$	0	0
$319$	−10.4017	−0.582383
$320$	0	0
$321$	0	0
$322$	5.11024	0.284783
$323$	1.12887	0.0628119
$324$	0	0
$325$	0	0
$326$	8.11090	0.449221
$327$	0	0
$328$	−4.88489	−0.269723
$329$	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	$0.489653\pi$
$332$	18.5058	1.01564
$333$	0	0
$334$	−2.06975	−0.113252
$335$	0	0
$336$	0	0
$337$	24.7995	1.35091	0.675457	−	0.737400i	$-0.263947\pi$
$337$	24.7995	1.35091	0.675457	+	0.737400i	$0.263947\pi$
$338$	−4.00057	−0.217602
$339$	0	0
$340$	0	0
$341$	−50.9428	−2.75871
$342$	0	0
$343$	19.0454	1.02836
$344$	−13.7211	−0.739790
$345$	0	0
$346$	−6.93796	−0.372987
$347$	22.1692	1.19010	0.595052	−	0.803687i	$-0.297132\pi$
$347$	22.1692	1.19010	0.595052	+	0.803687i	$0.297132\pi$
$348$	0	0
$349$	14.9185	0.798569	0.399285	−	0.916827i	$-0.369259\pi$
$349$	14.9185	0.798569	0.399285	+	0.916827i	$0.369259\pi$
$350$	0	0
$351$	0	0
$352$	29.9476	1.59621
$353$	16.9145	0.900269	0.450134	−	0.892961i	$-0.351376\pi$
$353$	16.9145	0.900269	0.450134	+	0.892961i	$0.351376\pi$
$354$	0	0
$355$	0	0
$356$	−11.1346	−0.590135
$357$	0	0
$358$	3.25133	0.171838
$359$	−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
$362$	−5.19205	−0.272888
$363$	0	0
$364$	−9.70634	−0.508751
$365$	0	0
$366$	0	0
$367$	−20.0979	−1.04910	−0.524552	−	0.851379i	$-0.675767\pi$
$367$	−20.0979	−1.04910	−0.524552	+	0.851379i	$0.675767\pi$
$368$	−11.4018	−0.594358
$369$	0	0
$370$	0	0
$371$	24.1826	1.25550
$372$	0	0
$373$	19.6429	1.01707	0.508536	−	0.861041i	$-0.330187\pi$
$373$	19.6429	1.01707	0.508536	+	0.861041i	$0.330187\pi$
$374$	9.25017	0.478315
$375$	0	0
$376$	20.4237	1.05327
$377$	3.59555	0.185180
$378$	0	0
$379$	−7.94219	−0.407963	−0.203982	−	0.978975i	$-0.565388\pi$
$379$	−7.94219	−0.407963	−0.203982	+	0.978975i	$0.565388\pi$
$380$	0	0
$381$	0	0
$382$	−6.49899	−0.332517
$383$	−31.1888	−1.59367	−0.796836	−	0.604195i	$-0.793495\pi$
$383$	−31.1888	−1.59367	−0.796836	+	0.604195i	$0.793495\pi$
$384$	0	0
$385$	0	0
$386$	0.228285	0.0116194
$387$	0	0
$388$	12.7995	0.649795
$389$	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
$392$	0.769772	0.0388794
$393$	0	0
$394$	2.61941	0.131964
$395$	0	0
$396$	0	0
$397$	−17.7174	−0.889211	−0.444606	−	0.895726i	$-0.646656\pi$
$397$	−17.7174	−0.889211	−0.444606	+	0.895726i	$0.646656\pi$
$398$	8.26255	0.414164
$399$	0	0
$400$	0	0
$401$	7.14247	0.356678	0.178339	−	0.983969i	$-0.442928\pi$
$401$	7.14247	0.356678	0.178339	+	0.983969i	$0.442928\pi$
$402$	0	0
$403$	17.6094	0.877185
$404$	12.3806	0.615958
$405$	0	0
$406$	2.04536	0.101509
$407$	−22.4089	−1.11077
$408$	0	0
$409$	24.7518	1.22390	0.611948	−	0.790898i	$-0.290386\pi$
$409$	24.7518	1.22390	0.611948	+	0.790898i	$0.290386\pi$
$410$	0	0
$411$	0	0
$412$	−10.8606	−0.535065
$413$	26.1839	1.28843
$414$	0	0
$415$	0	0
$416$	−10.3520	−0.507546
$417$	0	0
$418$	1.04006	0.0508708
$419$	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	$0.563867\pi$
$422$	−0.774555	−0.0377048
$423$	0	0
$424$	16.8607	0.818828
$425$	0	0
$426$	0	0
$427$	0.0281761	0.00136354
$428$	25.8144	1.24779
$429$	0	0
$430$	0	0
$431$	−1.67248	−0.0805604	−0.0402802	−	0.999188i	$-0.512825\pi$
$431$	−1.67248	−0.0805604	−0.0402802	+	0.999188i	$0.512825\pi$
$432$	0	0
$433$	9.95994	0.478644	0.239322	−	0.970940i	$-0.423075\pi$
$433$	9.95994	0.478644	0.239322	+	0.970940i	$0.423075\pi$
$434$	10.0173	0.480843
$435$	0	0
$436$	−3.37825	−0.161789
$437$	−1.50094	−0.0717998
$438$	0	0
$439$	12.8158	0.611663	0.305832	−	0.952086i	$-0.401066\pi$
$439$	12.8158	0.611663	0.305832	+	0.952086i	$0.401066\pi$
$440$	0	0
$441$	0	0
$442$	−3.19750	−0.152090
$443$	−7.75404	−0.368406	−0.184203	−	0.982888i	$-0.558970\pi$
$443$	−7.75404	−0.368406	−0.184203	+	0.982888i	$0.558970\pi$
$444$	0	0
$445$	0	0
$446$	−3.66671	−0.173624
$447$	0	0
$448$	7.98420	0.377218
$449$	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
$452$	11.6849	0.549614
$453$	0	0
$454$	−5.32898	−0.250101
$455$	0	0
$456$	0	0
$457$	38.2192	1.78782	0.893910	−	0.448246i	$-0.147951\pi$
$457$	38.2192	1.78782	0.893910	+	0.448246i	$0.147951\pi$
$458$	−4.95856	−0.231698
$459$	0	0
$460$	0	0
$461$	−31.3033	−1.45794	−0.728971	−	0.684545i	$-0.760001\pi$
$461$	−31.3033	−1.45794	−0.728971	+	0.684545i	$0.760001\pi$
$462$	0	0
$463$	−12.0852	−0.561645	−0.280823	−	0.959760i	$-0.590607\pi$
$463$	−12.0852	−0.561645	−0.280823	+	0.959760i	$0.590607\pi$
$464$	−4.56352	−0.211856
$465$	0	0
$466$	−1.37345	−0.0636238
$467$	−7.60466	−0.351902	−0.175951	−	0.984399i	$-0.556300\pi$
$467$	−7.60466	−0.351902	−0.175951	+	0.984399i	$0.556300\pi$
$468$	0	0
$469$	2.51901	0.116317
$470$	0	0
$471$	0	0
$472$	18.2561	0.840303
$473$	47.3636	2.17778
$474$	0	0
$475$	0	0
$476$	14.4237	0.661108
$477$	0	0
$478$	−7.73982	−0.354011
$479$	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
$482$	8.29326	0.377748
$483$	0	0
$484$	−48.0446	−2.18385
$485$	0	0
$486$	0	0
$487$	4.46121	0.202157	0.101078	−	0.994878i	$-0.467771\pi$
$487$	4.46121	0.202157	0.101078	+	0.994878i	$0.467771\pi$
$488$	0.0196451	0.000889291 0
$489$	0	0
$490$	0	0
$491$	−32.8420	−1.48214	−0.741070	−	0.671428i	$-0.765681\pi$
$491$	−32.8420	−1.48214	−0.741070	+	0.671428i	$0.765681\pi$
$492$	0	0
$493$	−5.34300	−0.240637
$494$	−0.359515	−0.0161754
$495$	0	0
$496$	−22.3501	−1.00355
$497$	−16.4934	−0.739832
$498$	0	0
$499$	−34.2021	−1.53109	−0.765547	−	0.643380i	$-0.777532\pi$
$499$	−34.2021	−1.53109	−0.765547	+	0.643380i	$0.777532\pi$
$500$	0	0
$501$	0	0
$502$	−4.00846	−0.178906
$503$	22.1773	0.988837	0.494419	−	0.869224i	$-0.335381\pi$
$503$	22.1773	0.988837	0.494419	+	0.869224i	$0.335381\pi$
$504$	0	0
$505$	0	0
$506$	−12.2990	−0.546758
$507$	0	0
$508$	−16.4432	−0.729549
$509$	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
$512$	22.8115	1.00813
$513$	0	0
$514$	−1.35943	−0.0599617
$515$	0	0
$516$	0	0
$517$	−70.5003	−3.10060
$518$	4.40643	0.193607
$519$	0	0
$520$	0	0
$521$	−20.2626	−0.887718	−0.443859	−	0.896097i	$-0.646391\pi$
$521$	−20.2626	−0.887718	−0.443859	+	0.896097i	$0.646391\pi$
$522$	0	0
$523$	−31.8114	−1.39101	−0.695507	−	0.718520i	$-0.744820\pi$
$523$	−31.8114	−1.39101	−0.695507	+	0.718520i	$0.744820\pi$
$524$	0.478457	0.0209015
$525$	0	0
$526$	12.1266	0.528745
$527$	−26.1676	−1.13988
$528$	0	0
$529$	−5.25083	−0.228297
$530$	0	0
$531$	0	0
$532$	1.62175	0.0703117
$533$	−5.82870	−0.252469
$534$	0	0
$535$	0	0
$536$	1.75632	0.0758613
$537$	0	0
$538$	−0.159586	−0.00688024
$539$	−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	$0.605457\pi$
$542$	10.2188	0.438937
$543$	0	0
$544$	15.3831	0.659543
$545$	0	0
$546$	0	0
$547$	4.08744	0.174766	0.0873831	−	0.996175i	$-0.472150\pi$
$547$	4.08744	0.174766	0.0873831	+	0.996175i	$0.472150\pi$
$548$	−6.17381	−0.263732
$549$	0	0
$550$	0	0
$551$	−0.600748	−0.0255927
$552$	0	0
$553$	24.2757	1.03231
$554$	−11.4214	−0.485249
$555$	0	0
$556$	−26.1926	−1.11082
$557$	−13.1425	−0.556864	−0.278432	−	0.960456i	$-0.589815\pi$
$557$	−13.1425	−0.556864	−0.278432	+	0.960456i	$0.589815\pi$
$558$	0	0
$559$	−16.3721	−0.692467
$560$	0	0
$561$	0	0
$562$	−1.59357	−0.0672207
$563$	−24.5221	−1.03348	−0.516742	−	0.856141i	$-0.672855\pi$
$563$	−24.5221	−1.03348	−0.516742	+	0.856141i	$0.672855\pi$
$564$	0	0
$565$	0	0
$566$	−10.3405	−0.434642
$567$	0	0
$568$	−11.4996	−0.482514
$569$	−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	$0.503292\pi$
$572$	23.3607	0.976758
$573$	0	0
$574$	−3.31571	−0.138395
$575$	0	0
$576$	0	0
$577$	9.41187	0.391821	0.195911	−	0.980622i	$-0.437234\pi$
$577$	9.41187	0.391821	0.195911	+	0.980622i	$0.437234\pi$
$578$	−3.29382	−0.137005
$579$	0	0
$580$	0	0
$581$	26.7064	1.10797
$582$	0	0
$583$	−58.2012	−2.41045
$584$	11.8206	0.489139
$585$	0	0
$586$	6.50916	0.268891
$587$	9.97321	0.411638	0.205819	−	0.978590i	$-0.434014\pi$
$587$	9.97321	0.411638	0.205819	+	0.978590i	$0.434014\pi$
$588$	0	0
$589$	−2.94219	−0.121231
$590$	0	0
$591$	0	0
$592$	−9.83144	−0.404070
$593$	38.3421	1.57452	0.787260	−	0.616621i	$-0.211499\pi$
$593$	38.3421	1.57452	0.787260	+	0.616621i	$0.211499\pi$
$594$	0	0
$595$	0	0
$596$	−18.0326	−0.738642
$597$	0	0
$598$	4.25140	0.173852
$599$	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	$0.642862\pi$
$602$	−9.31344	−0.379587
$603$	0	0
$604$	18.3219	0.745508
$605$	0	0
$606$	0	0
$607$	37.7355	1.53164	0.765819	−	0.643056i	$-0.222334\pi$
$607$	37.7355	1.53164	0.765819	+	0.643056i	$0.222334\pi$
$608$	1.72962	0.0701452
$609$	0	0
$610$	0	0
$611$	24.3698	0.985895
$612$	0	0
$613$	32.2633	1.30310	0.651551	−	0.758605i	$-0.274119\pi$
$613$	32.2633	1.30310	0.651551	+	0.758605i	$0.274119\pi$
$614$	−16.1940	−0.653536
$615$	0	0
$616$	28.2537	1.13837
$617$	−26.0178	−1.04744	−0.523719	−	0.851891i	$-0.675456\pi$
$617$	−26.0178	−1.04744	−0.523719	+	0.851891i	$0.675456\pi$
$618$	0	0
$619$	−11.8815	−0.477559	−0.238780	−	0.971074i	$-0.576747\pi$
$619$	−11.8815	−0.477559	−0.238780	+	0.971074i	$0.576747\pi$
$620$	0	0
$621$	0	0
$622$	−10.9037	−0.437197
$623$	−16.0688	−0.643783
$624$	0	0
$625$	0	0
$626$	1.69960	0.0679296
$627$	0	0
$628$	1.88723	0.0753086
$629$	−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	$0.585103\pi$
$632$	16.9256	0.673263
$633$	0	0
$634$	6.23780	0.247735
$635$	0	0
$636$	0	0
$637$	0.918501	0.0363923
$638$	−4.92265	−0.194890
$639$	0	0
$640$	0	0
$641$	−44.8149	−1.77008	−0.885042	−	0.465511i	$-0.845871\pi$
$641$	−44.8149	−1.77008	−0.885042	+	0.465511i	$0.845871\pi$
$642$	0	0
$643$	14.9255	0.588605	0.294302	−	0.955712i	$-0.404913\pi$
$643$	14.9255	0.588605	0.294302	+	0.955712i	$0.404913\pi$
$644$	−19.1777	−0.755708
$645$	0	0
$646$	0.534242	0.0210195
$647$	−41.2684	−1.62243	−0.811214	−	0.584749i	$-0.801193\pi$
$647$	−41.2684	−1.62243	−0.811214	+	0.584749i	$0.801193\pi$
$648$	0	0
$649$	−63.0179	−2.47367
$650$	0	0
$651$	0	0
$652$	−30.4386	−1.19207
$653$	27.6252	1.08106	0.540528	−	0.841326i	$-0.318224\pi$
$653$	27.6252	1.08106	0.540528	+	0.841326i	$0.318224\pi$
$654$	0	0
$655$	0	0
$656$	7.39788	0.288839
$657$	0	0
$658$	13.8630	0.540435
$659$	−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	$0.338435\pi$
$662$	0.559636	0.0217509
$663$	0	0
$664$	18.6204	0.722610
$665$	0	0
$666$	0	0
$667$	7.10405	0.275070
$668$	7.76738	0.300529
$669$	0	0
$670$	0	0
$671$	−0.0678126	−0.00261788
$672$	0	0
$673$	40.8048	1.57291	0.786454	−	0.617649i	$-0.211915\pi$
$673$	40.8048	1.57291	0.786454	+	0.617649i	$0.211915\pi$
$674$	11.7365	0.452072
$675$	0	0
$676$	15.0133	0.577436
$677$	41.1894	1.58304	0.791518	−	0.611146i	$-0.209291\pi$
$677$	41.1894	1.58304	0.791518	+	0.611146i	$0.209291\pi$
$678$	0	0
$679$	18.4714	0.708867
$680$	0	0
$681$	0	0
$682$	−24.1089	−0.923178
$683$	1.33820	0.0512047	0.0256023	−	0.999672i	$-0.491850\pi$
$683$	1.33820	0.0512047	0.0256023	+	0.999672i	$0.491850\pi$
$684$	0	0
$685$	0	0
$686$	9.01333	0.344131
$687$	0	0
$688$	20.7798	0.792221
$689$	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	$0.659680\pi$
$692$	26.0368	0.989770
$693$	0	0
$694$	10.4917	0.398258
$695$	0	0
$696$	0	0
$697$	8.66148	0.328077
$698$	7.06025	0.267234
$699$	0	0
$700$	0	0
$701$	18.2064	0.687645	0.343822	−	0.939035i	$-0.388278\pi$
$701$	18.2064	0.687645	0.343822	+	0.939035i	$0.388278\pi$
$702$	0	0
$703$	−1.29422	−0.0488126
$704$	−19.2159	−0.724227
$705$	0	0
$706$	8.00487	0.301267
$707$	17.8669	0.671953
$708$	0	0
$709$	41.8206	1.57061	0.785304	−	0.619111i	$-0.212507\pi$
$709$	41.8206	1.57061	0.785304	+	0.619111i	$0.212507\pi$
$710$	0	0
$711$	0	0
$712$	−11.2036	−0.419871
$713$	34.7925	1.30299
$714$	0	0
$715$	0	0
$716$	−12.2016	−0.455995
$717$	0	0
$718$	−0.301301	−0.0112444
$719$	−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
$722$	−8.93177	−0.332406
$723$	0	0
$724$	19.4847	0.724144
$725$	0	0
$726$	0	0
$727$	43.8009	1.62449	0.812243	−	0.583319i	$-0.198246\pi$
$727$	43.8009	1.62449	0.812243	+	0.583319i	$0.198246\pi$
$728$	−9.76642	−0.361968
$729$	0	0
$730$	0	0
$731$	24.3291	0.899843
$732$	0	0
$733$	6.78664	0.250670	0.125335	−	0.992114i	$-0.459999\pi$
$733$	6.78664	0.250670	0.125335	+	0.992114i	$0.459999\pi$
$734$	−9.51144	−0.351074
$735$	0	0
$736$	−20.4533	−0.753919
$737$	−6.06261	−0.223319
$738$	0	0
$739$	28.7245	1.05665	0.528324	−	0.849043i	$-0.322821\pi$
$739$	28.7245	1.05665	0.528324	+	0.849043i	$0.322821\pi$
$740$	0	0
$741$	0	0
$742$	11.4445	0.420142
$743$	−31.4523	−1.15387	−0.576937	−	0.816789i	$-0.695752\pi$
$743$	−31.4523	−1.15387	−0.576937	+	0.816789i	$0.695752\pi$
$744$	0	0
$745$	0	0
$746$	9.29610	0.340354
$747$	0	0
$748$	−34.7141	−1.26927
$749$	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	$0.435957\pi$
$752$	−30.9305	−1.12792
$753$	0	0
$754$	1.70161	0.0619689
$755$	0	0
$756$	0	0
$757$	45.7942	1.66442	0.832210	−	0.554461i	$-0.187075\pi$
$757$	45.7942	1.66442	0.832210	+	0.554461i	$0.187075\pi$
$758$	−3.75868	−0.136521
$759$	0	0
$760$	0	0
$761$	33.9138	1.22937	0.614687	−	0.788771i	$-0.289283\pi$
$761$	33.9138	1.22937	0.614687	+	0.788771i	$0.289283\pi$
$762$	0	0
$763$	−4.87528	−0.176497
$764$	24.3894	0.882378
$765$	0	0
$766$	−14.7602	−0.533309
$767$	21.7833	0.786551
$768$	0	0
$769$	7.15972	0.258186	0.129093	−	0.991632i	$-0.458793\pi$
$769$	7.15972	0.258186	0.129093	+	0.991632i	$0.458793\pi$
$770$	0	0
$771$	0	0
$772$	−0.856710	−0.0308337
$773$	−14.5998	−0.525117	−0.262558	−	0.964916i	$-0.584566\pi$
$773$	−14.5998	−0.525117	−0.262558	+	0.964916i	$0.584566\pi$
$774$	0	0
$775$	0	0
$776$	12.8787	0.462318
$777$	0	0
$778$	14.8836	0.533602
$779$	0.973866	0.0348924
$780$	0	0
$781$	39.6954	1.42041
$782$	−6.31760	−0.225917
$783$	0	0
$784$	−1.16578	−0.0416348
$785$	0	0
$786$	0	0
$787$	−18.4728	−0.658483	−0.329242	−	0.944246i	$-0.606793\pi$
$787$	−18.4728	−0.658483	−0.329242	+	0.944246i	$0.606793\pi$
$788$	−9.83011	−0.350183
$789$	0	0
$790$	0	0
$791$	16.8630	0.599578
$792$	0	0
$793$	0.0234407	0.000832405 0
$794$	−8.38484	−0.297567
$795$	0	0
$796$	−31.0077	−1.09904
$797$	41.0374	1.45362	0.726810	−	0.686838i	$-0.241002\pi$
$797$	41.0374	1.45362	0.726810	+	0.686838i	$0.241002\pi$
$798$	0	0
$799$	−36.2136	−1.28115
$800$	0	0
$801$	0	0
$802$	3.38021	0.119359
$803$	−40.8033	−1.43992
$804$	0	0
$805$	0	0
$806$	8.33371	0.293543
$807$	0	0
$808$	12.4572	0.438244
$809$	−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	$0.265859\pi$
$812$	−7.67583	−0.269369
$813$	0	0
$814$	−10.6051	−0.371710
$815$	0	0
$816$	0	0
$817$	2.73547	0.0957021
$818$	11.7139	0.409566
$819$	0	0
$820$	0	0
$821$	0.668560	0.0233329	0.0116665	−	0.999932i	$-0.496286\pi$
$821$	0.668560	0.0233329	0.0116665	+	0.999932i	$0.496286\pi$
$822$	0	0
$823$	1.42033	0.0495096	0.0247548	−	0.999694i	$-0.492119\pi$
$823$	1.42033	0.0495096	0.0247548	+	0.999694i	$0.492119\pi$
$824$	−10.9279	−0.380690
$825$	0	0
$826$	12.3917	0.431161
$827$	49.8169	1.73230	0.866152	−	0.499782i	$-0.166586\pi$
$827$	49.8169	1.73230	0.866152	+	0.499782i	$0.166586\pi$
$828$	0	0
$829$	36.4150	1.26475	0.632373	−	0.774664i	$-0.282081\pi$
$829$	36.4150	1.26475	0.632373	+	0.774664i	$0.282081\pi$
$830$	0	0
$831$	0	0
$832$	6.64235	0.230282
$833$	−1.36490	−0.0472909
$834$	0	0
$835$	0	0
$836$	−3.90312	−0.134992
$837$	0	0
$838$	5.04447	0.174258
$839$	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
$842$	−3.87056	−0.133388
$843$	0	0
$844$	2.90675	0.100055
$845$	0	0
$846$	0	0
$847$	−69.3349	−2.38238
$848$	−25.5345	−0.876860
$849$	0	0
$850$	0	0
$851$	15.3046	0.524637
$852$	0	0
$853$	−26.9084	−0.921326	−0.460663	−	0.887575i	$-0.652388\pi$
$853$	−26.9084	−0.921326	−0.460663	+	0.887575i	$0.652388\pi$
$854$	0.0133345	0.000456296 0
$855$	0	0
$856$	25.9742	0.887779
$857$	−48.8408	−1.66837	−0.834184	−	0.551486i	$-0.814061\pi$
$857$	−48.8408	−1.66837	−0.834184	+	0.551486i	$0.814061\pi$
$858$	0	0
$859$	−41.4094	−1.41287	−0.706435	−	0.707778i	$-0.749698\pi$
$859$	−41.4094	−1.41287	−0.706435	+	0.707778i	$0.749698\pi$
$860$	0	0
$861$	0	0
$862$	−0.791507	−0.0269588
$863$	−50.8101	−1.72960	−0.864799	−	0.502119i	$-0.832554\pi$
$863$	−50.8101	−1.72960	−0.864799	+	0.502119i	$0.832554\pi$
$864$	0	0
$865$	0	0
$866$	4.71359	0.160174
$867$	0	0
$868$	−37.5928	−1.27598
$869$	−58.4252	−1.98194
$870$	0	0
$871$	2.09566	0.0710087
$872$	−3.39916	−0.115110
$873$	0	0
$874$	−0.710328	−0.0240272
$875$	0	0
$876$	0	0
$877$	0.409589	0.0138308	0.00691542	−	0.999976i	$-0.497799\pi$
$877$	0.409589	0.0138308	0.00691542	+	0.999976i	$0.497799\pi$
$878$	6.06512	0.204688
$879$	0	0
$880$	0	0
$881$	5.32851	0.179522	0.0897610	−	0.995963i	$-0.471390\pi$
$881$	5.32851	0.179522	0.0897610	+	0.995963i	$0.471390\pi$
$882$	0	0
$883$	14.2064	0.478083	0.239042	−	0.971009i	$-0.423167\pi$
$883$	14.2064	0.478083	0.239042	+	0.971009i	$0.423167\pi$
$884$	11.9996	0.403590
$885$	0	0
$886$	−3.66964	−0.123284
$887$	−7.23193	−0.242825	−0.121412	−	0.992602i	$-0.538742\pi$
$887$	−7.23193	−0.242825	−0.121412	+	0.992602i	$0.538742\pi$
$888$	0	0
$889$	−23.7298	−0.795871
$890$	0	0
$891$	0	0
$892$	13.7604	0.460733
$893$	−4.07173	−0.136255
$894$	0	0
$895$	0	0
$896$	28.6650	0.957629
$897$	0	0
$898$	15.7784	0.526531
$899$	13.9256	0.464444
$900$	0	0
$901$	−29.8960	−0.995981
$902$	7.98006	0.265707
$903$	0	0
$904$	11.7573	0.391041
$905$	0	0
$906$	0	0
$907$	−38.8101	−1.28867	−0.644335	−	0.764743i	$-0.722866\pi$
$907$	−38.8101	−1.28867	−0.644335	+	0.764743i	$0.722866\pi$
$908$	19.9986	0.663677
$909$	0	0
$910$	0	0
$911$	40.2781	1.33447	0.667236	−	0.744846i	$-0.267477\pi$
$911$	40.2781	1.33447	0.667236	+	0.744846i	$0.267477\pi$
$912$	0	0
$913$	−64.2754	−2.12721
$914$	18.0874	0.598279
$915$	0	0
$916$	18.6085	0.614842
$917$	0.690479	0.0228016
$918$	0	0
$919$	−13.0468	−0.430375	−0.215187	−	0.976573i	$-0.569036\pi$
$919$	−13.0468	−0.430375	−0.215187	+	0.976573i	$0.569036\pi$
$920$	0	0
$921$	0	0
$922$	−14.8144	−0.487888
$923$	−13.7215	−0.451648
$924$	0	0
$925$	0	0
$926$	−5.71936	−0.187950
$927$	0	0
$928$	−8.18638	−0.268731
$929$	−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
$932$	5.15428	0.168834
$933$	0	0
$934$	−3.59894	−0.117761
$935$	0	0
$936$	0	0
$937$	16.9141	0.552559	0.276280	−	0.961077i	$-0.410898\pi$
$937$	16.9141	0.552559	0.276280	+	0.961077i	$0.410898\pi$
$938$	1.19213	0.0389246
$939$	0	0
$940$	0	0
$941$	−57.2093	−1.86497	−0.932485	−	0.361209i	$-0.882364\pi$
$941$	−57.2093	−1.86497	−0.932485	+	0.361209i	$0.882364\pi$
$942$	0	0
$943$	−11.5163	−0.375023
$944$	−27.6478	−0.899858
$945$	0	0
$946$	22.4150	0.728775
$947$	−38.8746	−1.26325	−0.631627	−	0.775272i	$-0.717613\pi$
$947$	−38.8746	−1.26325	−0.631627	+	0.775272i	$0.717613\pi$
$948$	0	0
$949$	14.1044	0.457850
$950$	0	0
$951$	0	0
$952$	14.5130	0.470368
$953$	−54.4516	−1.76386	−0.881930	−	0.471381i	$-0.843756\pi$
$953$	−54.4516	−1.76386	−0.881930	+	0.471381i	$0.843756\pi$
$954$	0	0
$955$	0	0
$956$	29.0460	0.939415
$957$	0	0
$958$	15.3729	0.496675
$959$	−8.90965	−0.287707
$960$	0	0
$961$	37.2012	1.20004
$962$	3.66587	0.118192
$963$	0	0
$964$	−31.1229	−1.00240
$965$	0	0
$966$	0	0
$967$	19.5701	0.629333	0.314666	−	0.949202i	$-0.398107\pi$
$967$	19.5701	0.629333	0.314666	+	0.949202i	$0.398107\pi$
$968$	−48.3420	−1.55377
$969$	0	0
$970$	0	0
$971$	−6.31009	−0.202500	−0.101250	−	0.994861i	$-0.532284\pi$
$971$	−6.31009	−0.202500	−0.101250	+	0.994861i	$0.532284\pi$
$972$	0	0
$973$	−37.7995	−1.21180
$974$	2.11129	0.0676500
$975$	0	0
$976$	−0.0297513	−0.000952317 0
$977$	7.41911	0.237358	0.118679	−	0.992933i	$-0.462134\pi$
$977$	7.41911	0.237358	0.118679	+	0.992933i	$0.462134\pi$
$978$	0	0
$979$	38.6734	1.23601
$980$	0	0
$981$	0	0
$982$	−15.5426	−0.495986
$983$	−10.9827	−0.350295	−0.175148	−	0.984542i	$-0.556040\pi$
$983$	−10.9827	−0.350295	−0.175148	+	0.984542i	$0.556040\pi$
$984$	0	0
$985$	0	0
$986$	−2.52860	−0.0805270
$987$	0	0
$988$	1.34919	0.0429234
$989$	−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	$0.610242\pi$
$992$	−40.0932	−1.27296
$993$	0	0
$994$	−7.80559	−0.247578
$995$	0	0
$996$	0	0
$997$	30.5348	0.967047	0.483524	−	0.875331i	$-0.339357\pi$
$997$	30.5348	0.967047	0.483524	+	0.875331i	$0.339357\pi$
$998$	−16.1863	−0.512368
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+0.473255 q^{2} -1.77603 q^{4} -2.56305 q^{7} -1.78702 q^{8} +O(q^{10})\)	\(q+0.473255 q^{2} -1.77603 q^{4} -2.56305 q^{7} -1.78702 q^{8} +6.16860 q^{11} -2.13230 q^{13} -1.21298 q^{14} +2.70634 q^{16} +3.16860 q^{17} +0.356267 q^{19} +2.91932 q^{22} -4.21298 q^{23} -1.00912 q^{26} +4.55206 q^{28} -1.68623 q^{29} -8.25840 q^{31} +4.85484 q^{32} +1.49956 q^{34} -3.63274 q^{37} +0.168605 q^{38} +2.73353 q^{41} +7.67817 q^{43} -10.9556 q^{44} -1.99381 q^{46} -11.4289 q^{47} -0.430757 q^{49} +3.78702 q^{52} -9.43507 q^{53} +4.58024 q^{56} -0.798017 q^{58} -10.2159 q^{59} -0.0109932 q^{61} -3.90833 q^{62} -3.11511 q^{64} -0.982817 q^{67} -5.62754 q^{68} +6.43507 q^{71} -6.61467 q^{73} -1.71921 q^{74} -0.632740 q^{76} -15.8105 q^{77} -9.47138 q^{79} +1.29366 q^{82} -10.4198 q^{83} +3.63373 q^{86} -11.0234 q^{88} +6.26940 q^{89} +5.46519 q^{91} +7.48237 q^{92} -5.40877 q^{94} -7.20679 q^{97} -0.203858 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 4 q^{4} - q^{7} - 9 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 4 * q^4 - q^7 - 9 * q^8	\( 4 q - 2 q^{2} + 4 q^{4} - q^{7} - 9 q^{8} + q^{11} + 2 q^{13} - 3 q^{14} + 4 q^{16} - 11 q^{17} + 2 q^{19} + 3 q^{22} - 15 q^{23} + 10 q^{26} - 4 q^{28} - q^{29} - 4 q^{31} - 10 q^{32} + 9 q^{34} - q^{37} - 23 q^{38} + 5 q^{41} - 10 q^{43} - 22 q^{44} - 20 q^{47} - 3 q^{49} + 17 q^{52} - 20 q^{53} + 30 q^{56} - 18 q^{58} - 17 q^{59} - 13 q^{61} + 6 q^{62} + 19 q^{64} + 17 q^{67} - 34 q^{68} + 8 q^{71} + 2 q^{73} - 40 q^{74} + 11 q^{76} - 12 q^{77} - 7 q^{79} + 12 q^{82} - 30 q^{83} + 34 q^{86} + 9 q^{88} + 9 q^{89} - 17 q^{91} + 12 q^{92} + 3 q^{94} - 19 q^{97} - 13 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 4 * q^4 - q^7 - 9 * q^8 + q^11 + 2 * q^13 - 3 * q^14 + 4 * q^16 - 11 * q^17 + 2 * q^19 + 3 * q^22 - 15 * q^23 + 10 * q^26 - 4 * q^28 - q^29 - 4 * q^31 - 10 * q^32 + 9 * q^34 - q^37 - 23 * q^38 + 5 * q^41 - 10 * q^43 - 22 * q^44 - 20 * q^47 - 3 * q^49 + 17 * q^52 - 20 * q^53 + 30 * q^56 - 18 * q^58 - 17 * q^59 - 13 * q^61 + 6 * q^62 + 19 * q^64 + 17 * q^67 - 34 * q^68 + 8 * q^71 + 2 * q^73 - 40 * q^74 + 11 * q^76 - 12 * q^77 - 7 * q^79 + 12 * q^82 - 30 * q^83 + 34 * q^86 + 9 * q^88 + 9 * q^89 - 17 * q^91 + 12 * q^92 + 3 * q^94 - 19 * q^97 - 13 * q^98

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