LMFDB - Embedded newform 4225.2.a.bv.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4225,2,Mod(1,4225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4225, 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("4225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.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:	$33.7367948540$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 16x^{8} + 84x^{6} - 163x^{4} + 118x^{2} - 27 $ x^10 - 16x^8 + 84x^6 - 163x^4 + 118x^2 - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 325)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.887996$ of defining polynomial
Character	$\chi$	$=$	4225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.887996 q^{2} +3.06193 q^{3} -1.21146 q^{4} -2.71898 q^{6} +3.56304 q^{7} +2.85177 q^{8} +6.37543 q^{9} +O(q^{10})$	\(q-0.887996 q^{2} +3.06193 q^{3} -1.21146 q^{4} -2.71898 q^{6} +3.56304 q^{7} +2.85177 q^{8} +6.37543 q^{9} +0.942991 q^{11} -3.70942 q^{12} -3.16396 q^{14} -0.109431 q^{16} -0.959900 q^{17} -5.66135 q^{18} -3.33965 q^{19} +10.9098 q^{21} -0.837373 q^{22} +6.59999 q^{23} +8.73191 q^{24} +10.3353 q^{27} -4.31649 q^{28} +8.39726 q^{29} +4.13976 q^{31} -5.60636 q^{32} +2.88738 q^{33} +0.852387 q^{34} -7.72359 q^{36} -7.97574 q^{37} +2.96560 q^{38} -0.797392 q^{41} -9.68784 q^{42} +9.00874 q^{43} -1.14240 q^{44} -5.86076 q^{46} -10.5985 q^{47} -0.335071 q^{48} +5.69524 q^{49} -2.93915 q^{51} -7.44476 q^{53} -9.17773 q^{54} +10.1610 q^{56} -10.2258 q^{57} -7.45673 q^{58} -1.00360 q^{59} +4.52656 q^{61} -3.67609 q^{62} +22.7159 q^{63} +5.19729 q^{64} -2.56398 q^{66} +1.11972 q^{67} +1.16288 q^{68} +20.2087 q^{69} -1.85378 q^{71} +18.1812 q^{72} -4.69721 q^{73} +7.08243 q^{74} +4.04586 q^{76} +3.35991 q^{77} +5.64117 q^{79} +12.5198 q^{81} +0.708081 q^{82} -0.187778 q^{83} -13.2168 q^{84} -7.99972 q^{86} +25.7118 q^{87} +2.68919 q^{88} +9.48713 q^{89} -7.99564 q^{92} +12.6757 q^{93} +9.41144 q^{94} -17.1663 q^{96} +16.5439 q^{97} -5.05735 q^{98} +6.01197 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 6 q^{3} + 12 q^{4} + 16 q^{9}+O(q^{10}) $ 10 * q + 6 * q^3 + 12 * q^4 + 16 * q^9	$ 10 q + 6 q^{3} + 12 q^{4} + 16 q^{9} + 28 q^{12} - 8 q^{14} + 24 q^{16} + 16 q^{17} + 24 q^{22} + 26 q^{23} + 30 q^{27} - 14 q^{29} - 6 q^{36} + 62 q^{38} - 64 q^{42} + 8 q^{43} + 52 q^{48} - 2 q^{49} - 16 q^{51} + 24 q^{53} - 42 q^{56} - 26 q^{61} + 34 q^{62} + 34 q^{64} - 42 q^{66} + 26 q^{68} + 40 q^{69} + 52 q^{74} + 48 q^{77} + 4 q^{79} + 34 q^{81} - 2 q^{82} + 98 q^{87} - 12 q^{88} + 34 q^{92} - 10 q^{94}+O(q^{100}) $ 10 * q + 6 * q^3 + 12 * q^4 + 16 * q^9 + 28 * q^12 - 8 * q^14 + 24 * q^16 + 16 * q^17 + 24 * q^22 + 26 * q^23 + 30 * q^27 - 14 * q^29 - 6 * q^36 + 62 * q^38 - 64 * q^42 + 8 * q^43 + 52 * q^48 - 2 * q^49 - 16 * q^51 + 24 * q^53 - 42 * q^56 - 26 * q^61 + 34 * q^62 + 34 * q^64 - 42 * q^66 + 26 * q^68 + 40 * q^69 + 52 * q^74 + 48 * q^77 + 4 * q^79 + 34 * q^81 - 2 * q^82 + 98 * q^87 - 12 * q^88 + 34 * q^92 - 10 * q^94

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 $

$2$	−0.887996	−0.627908	−0.313954	−	0.949438i	$-0.601654\pi$
$2$	−0.887996	−0.627908	−0.313954	+	0.949438i	$0.601654\pi$
$3$	3.06193	1.76781	0.883904	−	0.467669i	$-0.154906\pi$
$3$	3.06193	1.76781	0.883904	+	0.467669i	$0.154906\pi$
$4$	−1.21146	−0.605732
$5$	0	0
$5$	0	0
$6$	−2.71898	−1.11002
$7$	3.56304	1.34670	0.673351	−	0.739323i	$-0.264854\pi$
$7$	3.56304	1.34670	0.673351	+	0.739323i	$0.264854\pi$
$8$	2.85177	1.00825
$9$	6.37543	2.12514
$10$	0	0
$11$	0.942991	0.284323	0.142161	−	0.989844i	$-0.454595\pi$
$11$	0.942991	0.284323	0.142161	+	0.989844i	$0.454595\pi$
$12$	−3.70942	−1.07082
$13$	0	0
$13$	0	0
$14$	−3.16396	−0.845605
$15$	0	0
$16$	−0.109431	−0.0273578
$17$	−0.959900	−0.232810	−0.116405	−	0.993202i	$-0.537137\pi$
$17$	−0.959900	−0.232810	−0.116405	+	0.993202i	$0.537137\pi$
$18$	−5.66135	−1.33439
$19$	−3.33965	−0.766169	−0.383084	−	0.923713i	$-0.625138\pi$
$19$	−3.33965	−0.766169	−0.383084	+	0.923713i	$0.625138\pi$
$20$	0	0
$21$	10.9098	2.38071
$22$	−0.837373	−0.178528
$23$	6.59999	1.37619	0.688096	−	0.725620i	$-0.258447\pi$
$23$	6.59999	1.37619	0.688096	+	0.725620i	$0.258447\pi$
$24$	8.73191	1.78239
$25$	0	0
$26$	0	0
$27$	10.3353	1.98903
$28$	−4.31649	−0.815740
$29$	8.39726	1.55933	0.779666	−	0.626196i	$-0.215389\pi$
$29$	8.39726	1.55933	0.779666	+	0.626196i	$0.215389\pi$
$30$	0	0
$31$	4.13976	0.743524	0.371762	−	0.928328i	$-0.378754\pi$
$31$	4.13976	0.743524	0.371762	+	0.928328i	$0.378754\pi$
$32$	−5.60636	−0.991074
$33$	2.88738	0.502627
$34$	0.852387	0.146183
$35$	0	0
$36$	−7.72359	−1.28727
$37$	−7.97574	−1.31120	−0.655602	−	0.755107i	$-0.727585\pi$
$37$	−7.97574	−1.31120	−0.655602	+	0.755107i	$0.727585\pi$
$38$	2.96560	0.481083
$39$	0	0
$40$	0	0
$41$	−0.797392	−0.124532	−0.0622659	−	0.998060i	$-0.519833\pi$
$41$	−0.797392	−0.124532	−0.0622659	+	0.998060i	$0.519833\pi$
$42$	−9.68784	−1.49487
$43$	9.00874	1.37382	0.686910	−	0.726743i	$-0.258967\pi$
$43$	9.00874	1.37382	0.686910	+	0.726743i	$0.258967\pi$
$44$	−1.14240	−0.172223
$45$	0	0
$46$	−5.86076	−0.864122
$47$	−10.5985	−1.54595	−0.772976	−	0.634435i	$-0.781233\pi$
$47$	−10.5985	−1.54595	−0.772976	+	0.634435i	$0.781233\pi$
$48$	−0.335071	−0.0483633
$49$	5.69524	0.813606
$50$	0	0
$51$	−2.93915	−0.411563
$52$	0	0
$53$	−7.44476	−1.02262	−0.511308	−	0.859397i	$-0.670839\pi$
$53$	−7.44476	−1.02262	−0.511308	+	0.859397i	$0.670839\pi$
$54$	−9.17773	−1.24893
$55$	0	0
$56$	10.1610	1.35781
$57$	−10.2258	−1.35444
$58$	−7.45673	−0.979117
$59$	−1.00360	−0.130657	−0.0653287	−	0.997864i	$-0.520810\pi$
$59$	−1.00360	−0.130657	−0.0653287	+	0.997864i	$0.520810\pi$
$60$	0	0
$61$	4.52656	0.579567	0.289784	−	0.957092i	$-0.406417\pi$
$61$	4.52656	0.579567	0.289784	+	0.957092i	$0.406417\pi$
$62$	−3.67609	−0.466864
$63$	22.7159	2.86193
$64$	5.19729	0.649661
$65$	0	0
$66$	−2.56398	−0.315604
$67$	1.11972	0.136795	0.0683975	−	0.997658i	$-0.478211\pi$
$67$	1.11972	0.136795	0.0683975	+	0.997658i	$0.478211\pi$
$68$	1.16288	0.141020
$69$	20.2087	2.43284
$70$	0	0
$71$	−1.85378	−0.220003	−0.110002	−	0.993931i	$-0.535086\pi$
$71$	−1.85378	−0.220003	−0.110002	+	0.993931i	$0.535086\pi$
$72$	18.1812	2.14268
$73$	−4.69721	−0.549767	−0.274884	−	0.961478i	$-0.588639\pi$
$73$	−4.69721	−0.549767	−0.274884	+	0.961478i	$0.588639\pi$
$74$	7.08243	0.823315
$75$	0	0
$76$	4.04586	0.464092
$77$	3.35991	0.382898
$78$	0	0
$79$	5.64117	0.634681	0.317340	−	0.948312i	$-0.397210\pi$
$79$	5.64117	0.634681	0.317340	+	0.948312i	$0.397210\pi$
$80$	0	0
$81$	12.5198	1.39109
$82$	0.708081	0.0781945
$83$	−0.187778	−0.0206114	−0.0103057	−	0.999947i	$-0.503280\pi$
$83$	−0.187778	−0.0206114	−0.0103057	+	0.999947i	$0.503280\pi$
$84$	−13.2168	−1.44207
$85$	0	0
$86$	−7.99972	−0.862632
$87$	25.7118	2.75660
$88$	2.68919	0.286669
$89$	9.48713	1.00563	0.502817	−	0.864393i	$-0.332297\pi$
$89$	9.48713	1.00563	0.502817	+	0.864393i	$0.332297\pi$
$90$	0	0
$91$	0	0
$92$	−7.99564	−0.833603
$93$	12.6757	1.31441
$94$	9.41144	0.970715
$95$	0	0
$96$	−17.1663	−1.75203
$97$	16.5439	1.67977	0.839887	−	0.542761i	$-0.182621\pi$
$97$	16.5439	1.67977	0.839887	+	0.542761i	$0.182621\pi$
$98$	−5.05735	−0.510870
$99$	6.01197	0.604226
$100$	0	0
$101$	−5.50581	−0.547849	−0.273924	−	0.961751i	$-0.588322\pi$
$101$	−5.50581	−0.547849	−0.273924	+	0.961751i	$0.588322\pi$
$102$	2.60995	0.258424
$103$	−3.49795	−0.344664	−0.172332	−	0.985039i	$-0.555130\pi$
$103$	−3.49795	−0.344664	−0.172332	+	0.985039i	$0.555130\pi$
$104$	0	0
$105$	0	0
$106$	6.61092	0.642109
$107$	−5.74274	−0.555172	−0.277586	−	0.960701i	$-0.589534\pi$
$107$	−5.74274	−0.555172	−0.277586	+	0.960701i	$0.589534\pi$
$108$	−12.5209	−1.20482
$109$	−10.9903	−1.05268	−0.526339	−	0.850275i	$-0.676436\pi$
$109$	−10.9903	−1.05268	−0.526339	+	0.850275i	$0.676436\pi$
$110$	0	0
$111$	−24.4212	−2.31796
$112$	−0.389908	−0.0368428
$113$	13.7168	1.29037	0.645185	−	0.764027i	$-0.276780\pi$
$113$	13.7168	1.29037	0.645185	+	0.764027i	$0.276780\pi$
$114$	9.08046	0.850463
$115$	0	0
$116$	−10.1730	−0.944536
$117$	0	0
$118$	0.891191	0.0820408
$119$	−3.42016	−0.313526
$120$	0	0
$121$	−10.1108	−0.919161
$122$	−4.01957	−0.363915
$123$	−2.44156	−0.220148
$124$	−5.01517	−0.450376
$125$	0	0
$126$	−20.1716	−1.79703
$127$	−3.51347	−0.311770	−0.155885	−	0.987775i	$-0.549823\pi$
$127$	−3.51347	−0.311770	−0.155885	+	0.987775i	$0.549823\pi$
$128$	6.59755	0.583146
$129$	27.5841	2.42865
$130$	0	0
$131$	−18.0781	−1.57949	−0.789744	−	0.613437i	$-0.789787\pi$
$131$	−18.0781	−1.57949	−0.789744	+	0.613437i	$0.789787\pi$
$132$	−3.49795	−0.304457
$133$	−11.8993	−1.03180
$134$	−0.994303	−0.0858947
$135$	0	0
$136$	−2.73741	−0.234731
$137$	−0.780211	−0.0666579	−0.0333290	−	0.999444i	$-0.510611\pi$
$137$	−0.780211	−0.0666579	−0.0333290	+	0.999444i	$0.510611\pi$
$138$	−17.9453	−1.52760
$139$	0.278836	0.0236506	0.0118253	−	0.999930i	$-0.496236\pi$
$139$	0.278836	0.0236506	0.0118253	+	0.999930i	$0.496236\pi$
$140$	0	0
$141$	−32.4519	−2.73294
$142$	1.64615	0.138142
$143$	0	0
$144$	−0.697671	−0.0581392
$145$	0	0
$146$	4.17111	0.345203
$147$	17.4384	1.43830
$148$	9.66232	0.794238
$149$	−0.0102038	−0.000835928 0	−0.000417964	−	1.00000i	$-0.500133\pi$
$149$	−0.0102038	−0.000835928 0	−0.000417964		1.00000i	$0.500133\pi$
$150$	0	0
$151$	−0.408335	−0.0332298	−0.0166149	−	0.999862i	$-0.505289\pi$
$151$	−0.408335	−0.0332298	−0.0166149	+	0.999862i	$0.505289\pi$
$152$	−9.52391	−0.772491
$153$	−6.11977	−0.494754
$154$	−2.98359	−0.240425
$155$	0	0
$156$	0	0
$157$	0.950895	0.0758897	0.0379448	−	0.999280i	$-0.487919\pi$
$157$	0.950895	0.0758897	0.0379448	+	0.999280i	$0.487919\pi$
$158$	−5.00933	−0.398521
$159$	−22.7953	−1.80779
$160$	0	0
$161$	23.5160	1.85332
$162$	−11.1175	−0.873475
$163$	10.5013	0.822528	0.411264	−	0.911516i	$-0.365087\pi$
$163$	10.5013	0.822528	0.411264	+	0.911516i	$0.365087\pi$
$164$	0.966011	0.0754328
$165$	0	0
$166$	0.166746	0.0129420
$167$	7.97013	0.616747	0.308374	−	0.951265i	$-0.400215\pi$
$167$	7.97013	0.616747	0.308374	+	0.951265i	$0.400215\pi$
$168$	31.1121	2.40035
$169$	0	0
$170$	0	0
$171$	−21.2917	−1.62822
$172$	−10.9138	−0.832166
$173$	9.93290	0.755184	0.377592	−	0.925972i	$-0.376752\pi$
$173$	9.93290	0.755184	0.377592	+	0.925972i	$0.376752\pi$
$174$	−22.8320	−1.73089
$175$	0	0
$176$	−0.103193	−0.00777844
$177$	−3.07295	−0.230977
$178$	−8.42453	−0.631445
$179$	−10.6286	−0.794419	−0.397209	−	0.917728i	$-0.630021\pi$
$179$	−10.6286	−0.794419	−0.397209	+	0.917728i	$0.630021\pi$
$180$	0	0
$181$	−8.66166	−0.643816	−0.321908	−	0.946771i	$-0.604324\pi$
$181$	−8.66166	−0.643816	−0.321908	+	0.946771i	$0.604324\pi$
$182$	0	0
$183$	13.8600	1.02456
$184$	18.8216	1.38755
$185$	0	0
$186$	−11.2560	−0.825326
$187$	−0.905177	−0.0661931
$188$	12.8397	0.936432
$189$	36.8252	2.67864
$190$	0	0
$191$	10.6087	0.767620	0.383810	−	0.923412i	$-0.374612\pi$
$191$	10.6087	0.767620	0.383810	+	0.923412i	$0.374612\pi$
$192$	15.9137	1.14848
$193$	−20.4437	−1.47157	−0.735784	−	0.677216i	$-0.763186\pi$
$193$	−20.4437	−1.47157	−0.735784	+	0.677216i	$0.763186\pi$
$194$	−14.6909	−1.05474
$195$	0	0
$196$	−6.89957	−0.492827
$197$	−11.4967	−0.819107	−0.409553	−	0.912286i	$-0.634315\pi$
$197$	−11.4967	−0.819107	−0.409553	+	0.912286i	$0.634315\pi$
$198$	−5.33861	−0.379398
$199$	−9.95544	−0.705723	−0.352861	−	0.935676i	$-0.614791\pi$
$199$	−9.95544	−0.705723	−0.352861	+	0.935676i	$0.614791\pi$
$200$	0	0
$201$	3.42849	0.241827
$202$	4.88914	0.343999
$203$	29.9198	2.09995
$204$	3.56067	0.249297
$205$	0	0
$206$	3.10617	0.216417
$207$	42.0777	2.92460
$208$	0	0
$209$	−3.14926	−0.217839
$210$	0	0
$211$	11.4570	0.788730	0.394365	−	0.918954i	$-0.370965\pi$
$211$	11.4570	0.788730	0.394365	+	0.918954i	$0.370965\pi$
$212$	9.01905	0.619431
$213$	−5.67615	−0.388923
$214$	5.09953	0.348597
$215$	0	0
$216$	29.4739	2.00545
$217$	14.7501	1.00130
$218$	9.75933	0.660985
$219$	−14.3825	−0.971883
$220$	0	0
$221$	0	0
$222$	21.6859	1.45546
$223$	4.92872	0.330051	0.165026	−	0.986289i	$-0.447229\pi$
$223$	4.92872	0.330051	0.165026	+	0.986289i	$0.447229\pi$
$224$	−19.9757	−1.33468
$225$	0	0
$226$	−12.1805	−0.810233
$227$	−24.2768	−1.61131	−0.805653	−	0.592388i	$-0.798185\pi$
$227$	−24.2768	−1.61131	−0.805653	+	0.592388i	$0.798185\pi$
$228$	12.3882	0.820426
$229$	−28.3643	−1.87437	−0.937183	−	0.348837i	$-0.886577\pi$
$229$	−28.3643	−1.87437	−0.937183	+	0.348837i	$0.886577\pi$
$230$	0	0
$231$	10.2878	0.676889
$232$	23.9470	1.57220
$233$	−12.6827	−0.830869	−0.415435	−	0.909623i	$-0.636371\pi$
$233$	−12.6827	−0.830869	−0.415435	+	0.909623i	$0.636371\pi$
$234$	0	0
$235$	0	0
$236$	1.21582	0.0791432
$237$	17.2729	1.12199
$238$	3.03709	0.196865
$239$	29.4374	1.90415	0.952073	−	0.305872i	$-0.0989480\pi$
$239$	29.4374	1.90415	0.952073	+	0.305872i	$0.0989480\pi$
$240$	0	0
$241$	12.0818	0.778255	0.389128	−	0.921184i	$-0.372777\pi$
$241$	12.0818	0.778255	0.389128	+	0.921184i	$0.372777\pi$
$242$	8.97832	0.577148
$243$	7.32875	0.470140
$244$	−5.48376	−0.351062
$245$	0	0
$246$	2.16810	0.138233
$247$	0	0
$248$	11.8056	0.749659
$249$	−0.574965	−0.0364369
$250$	0	0
$251$	3.92605	0.247810	0.123905	−	0.992294i	$-0.460458\pi$
$251$	3.92605	0.247810	0.123905	+	0.992294i	$0.460458\pi$
$252$	−27.5195	−1.73356
$253$	6.22373	0.391283
$254$	3.11994	0.195763
$255$	0	0
$256$	−16.2532	−1.01582
$257$	20.0600	1.25131	0.625654	−	0.780100i	$-0.284832\pi$
$257$	20.0600	1.25131	0.625654	+	0.780100i	$0.284832\pi$
$258$	−24.4946	−1.52497
$259$	−28.4179	−1.76580
$260$	0	0
$261$	53.5361	3.31380
$262$	16.0533	0.991773
$263$	15.9452	0.983223	0.491612	−	0.870815i	$-0.336408\pi$
$263$	15.9452	0.983223	0.491612	+	0.870815i	$0.336408\pi$
$264$	8.23412	0.506775
$265$	0	0
$266$	10.5665	0.647876
$267$	29.0489	1.77777
$268$	−1.35649	−0.0828611
$269$	20.8060	1.26856	0.634282	−	0.773101i	$-0.281296\pi$
$269$	20.8060	1.26856	0.634282	+	0.773101i	$0.281296\pi$
$270$	0	0
$271$	−8.70079	−0.528535	−0.264268	−	0.964449i	$-0.585130\pi$
$271$	−8.70079	−0.528535	−0.264268	+	0.964449i	$0.585130\pi$
$272$	0.105043	0.00636917
$273$	0	0
$274$	0.692824	0.0418550
$275$	0	0
$276$	−24.4821	−1.47365
$277$	−8.64034	−0.519148	−0.259574	−	0.965723i	$-0.583582\pi$
$277$	−8.64034	−0.519148	−0.259574	+	0.965723i	$0.583582\pi$
$278$	−0.247605	−0.0148504
$279$	26.3928	1.58009
$280$	0	0
$281$	−12.5197	−0.746862	−0.373431	−	0.927658i	$-0.621819\pi$
$281$	−12.5197	−0.746862	−0.373431	+	0.927658i	$0.621819\pi$
$282$	28.8172	1.71604
$283$	20.3296	1.20847	0.604236	−	0.796806i	$-0.293479\pi$
$283$	20.3296	1.20847	0.604236	+	0.796806i	$0.293479\pi$
$284$	2.24579	0.133263
$285$	0	0
$286$	0	0
$287$	−2.84114	−0.167707
$288$	−35.7429	−2.10617
$289$	−16.0786	−0.945800
$290$	0	0
$291$	50.6562	2.96952
$292$	5.69050	0.333011
$293$	25.8016	1.50734	0.753672	−	0.657250i	$-0.228281\pi$
$293$	25.8016	1.50734	0.753672	+	0.657250i	$0.228281\pi$
$294$	−15.4853	−0.903119
$295$	0	0
$296$	−22.7450	−1.32202
$297$	9.74612	0.565527
$298$	0.00906094	0.000524886 0
$299$	0	0
$300$	0	0
$301$	32.0985	1.85012
$302$	0.362600	0.0208653
$303$	−16.8584	−0.968491
$304$	0.365462	0.0209607
$305$	0	0
$306$	5.43433	0.310660
$307$	−5.73478	−0.327301	−0.163650	−	0.986518i	$-0.552327\pi$
$307$	−5.73478	−0.327301	−0.163650	+	0.986518i	$0.552327\pi$
$308$	−4.07041	−0.231933
$309$	−10.7105	−0.609299
$310$	0	0
$311$	16.1145	0.913771	0.456886	−	0.889525i	$-0.348965\pi$
$311$	16.1145	0.913771	0.456886	+	0.889525i	$0.348965\pi$
$312$	0	0
$313$	20.6515	1.16729	0.583646	−	0.812008i	$-0.301626\pi$
$313$	20.6515	1.16729	0.583646	+	0.812008i	$0.301626\pi$
$314$	−0.844391	−0.0476517
$315$	0	0
$316$	−6.83406	−0.384446
$317$	19.2638	1.08196	0.540981	−	0.841035i	$-0.318053\pi$
$317$	19.2638	1.08196	0.540981	+	0.841035i	$0.318053\pi$
$318$	20.2422	1.13512
$319$	7.91854	0.443353
$320$	0	0
$321$	−17.5839	−0.981436
$322$	−20.8821	−1.16371
$323$	3.20573	0.178372
$324$	−15.1673	−0.842625
$325$	0	0
$326$	−9.32514	−0.516472
$327$	−33.6515	−1.86093
$328$	−2.27398	−0.125559
$329$	−37.7629	−2.08194
$330$	0	0
$331$	19.8891	1.09320	0.546602	−	0.837392i	$-0.315921\pi$
$331$	19.8891	1.09320	0.546602	+	0.837392i	$0.315921\pi$
$332$	0.227487	0.0124849
$333$	−50.8488	−2.78649
$334$	−7.07744	−0.387261
$335$	0	0
$336$	−1.19387	−0.0651310
$337$	−14.2197	−0.774599	−0.387299	−	0.921954i	$-0.626592\pi$
$337$	−14.2197	−0.774599	−0.387299	+	0.921954i	$0.626592\pi$
$338$	0	0
$339$	42.0000	2.28112
$340$	0	0
$341$	3.90376	0.211401
$342$	18.9069	1.02237
$343$	−4.64891	−0.251017
$344$	25.6908	1.38516
$345$	0	0
$346$	−8.82037	−0.474186
$347$	5.48270	0.294327	0.147163	−	0.989112i	$-0.452986\pi$
$347$	5.48270	0.294327	0.147163	+	0.989112i	$0.452986\pi$
$348$	−31.1489	−1.66976
$349$	5.37900	0.287931	0.143966	−	0.989583i	$-0.454014\pi$
$349$	5.37900	0.287931	0.143966	+	0.989583i	$0.454014\pi$
$350$	0	0
$351$	0	0
$352$	−5.28675	−0.281785
$353$	−0.238408	−0.0126892	−0.00634460	−	0.999980i	$-0.502020\pi$
$353$	−0.238408	−0.0126892	−0.00634460	+	0.999980i	$0.502020\pi$
$354$	2.72877	0.145032
$355$	0	0
$356$	−11.4933	−0.609144
$357$	−10.4723	−0.554253
$358$	9.43815	0.498822
$359$	0.536191	0.0282991	0.0141495	−	0.999900i	$-0.495496\pi$
$359$	0.536191	0.0282991	0.0141495	+	0.999900i	$0.495496\pi$
$360$	0	0
$361$	−7.84673	−0.412986
$362$	7.69152	0.404257
$363$	−30.9585	−1.62490
$364$	0	0
$365$	0	0
$366$	−12.3077	−0.643331
$367$	10.0441	0.524299	0.262149	−	0.965027i	$-0.415569\pi$
$367$	10.0441	0.524299	0.262149	+	0.965027i	$0.415569\pi$
$368$	−0.722244	−0.0376496
$369$	−5.08372	−0.264648
$370$	0	0
$371$	−26.5260	−1.37716
$372$	−15.3561	−0.796177
$373$	4.46872	0.231382	0.115691	−	0.993285i	$-0.463092\pi$
$373$	4.46872	0.231382	0.115691	+	0.993285i	$0.463092\pi$
$374$	0.803794	0.0415632
$375$	0	0
$376$	−30.2245	−1.55871
$377$	0	0
$378$	−32.7006	−1.68194
$379$	−26.0561	−1.33841	−0.669205	−	0.743078i	$-0.733365\pi$
$379$	−26.0561	−1.33841	−0.669205	+	0.743078i	$0.733365\pi$
$380$	0	0
$381$	−10.7580	−0.551149
$382$	−9.42050	−0.481995
$383$	−31.8585	−1.62789	−0.813947	−	0.580939i	$-0.802686\pi$
$383$	−31.8585	−1.62789	−0.813947	+	0.580939i	$0.802686\pi$
$384$	20.2012	1.03089
$385$	0	0
$386$	18.1539	0.924009
$387$	57.4345	2.91956
$388$	−20.0423	−1.01749
$389$	11.6351	0.589923	0.294962	−	0.955509i	$-0.404693\pi$
$389$	11.6351	0.589923	0.294962	+	0.955509i	$0.404693\pi$
$390$	0	0
$391$	−6.33533	−0.320391
$392$	16.2415	0.820319
$393$	−55.3538	−2.79223
$394$	10.2090	0.514324
$395$	0	0
$396$	−7.28328	−0.365999
$397$	−8.07621	−0.405334	−0.202667	−	0.979248i	$-0.564961\pi$
$397$	−8.07621	−0.405334	−0.202667	+	0.979248i	$0.564961\pi$
$398$	8.84039	0.443129
$399$	−36.4349	−1.82402
$400$	0	0
$401$	31.2240	1.55925	0.779625	−	0.626246i	$-0.215409\pi$
$401$	31.2240	1.55925	0.779625	+	0.626246i	$0.215409\pi$
$402$	−3.04449	−0.151845
$403$	0	0
$404$	6.67009	0.331849
$405$	0	0
$406$	−26.5686	−1.31858
$407$	−7.52106	−0.372805
$408$	−8.38176	−0.414959
$409$	−32.1963	−1.59201	−0.796003	−	0.605293i	$-0.793056\pi$
$409$	−32.1963	−1.59201	−0.796003	+	0.605293i	$0.793056\pi$
$410$	0	0
$411$	−2.38895	−0.117838
$412$	4.23764	0.208774
$413$	−3.57586	−0.175956
$414$	−37.3649	−1.83638
$415$	0	0
$416$	0	0
$417$	0.853777	0.0418096
$418$	2.79653	0.136783
$419$	−21.8280	−1.06637	−0.533185	−	0.845999i	$-0.679005\pi$
$419$	−21.8280	−1.06637	−0.533185	+	0.845999i	$0.679005\pi$
$420$	0	0
$421$	4.55712	0.222100	0.111050	−	0.993815i	$-0.464579\pi$
$421$	4.55712	0.222100	0.111050	+	0.993815i	$0.464579\pi$
$422$	−10.1737	−0.495250
$423$	−67.5700	−3.28537
$424$	−21.2307	−1.03105
$425$	0	0
$426$	5.04040	0.244208
$427$	16.1283	0.780504
$428$	6.95712	0.336285
$429$	0	0
$430$	0	0
$431$	29.7238	1.43174	0.715871	−	0.698232i	$-0.246030\pi$
$431$	29.7238	1.43174	0.715871	+	0.698232i	$0.246030\pi$
$432$	−1.13101	−0.0544156
$433$	−7.78222	−0.373990	−0.186995	−	0.982361i	$-0.559875\pi$
$433$	−7.78222	−0.373990	−0.186995	+	0.982361i	$0.559875\pi$
$434$	−13.0981	−0.628727
$435$	0	0
$436$	13.3143	0.637641
$437$	−22.0417	−1.05440
$438$	12.7716	0.610253
$439$	4.19882	0.200399	0.100199	−	0.994967i	$-0.468052\pi$
$439$	4.19882	0.200399	0.100199	+	0.994967i	$0.468052\pi$
$440$	0	0
$441$	36.3096	1.72903
$442$	0	0
$443$	−21.5287	−1.02286	−0.511430	−	0.859325i	$-0.670884\pi$
$443$	−21.5287	−1.02286	−0.511430	+	0.859325i	$0.670884\pi$
$444$	29.5854	1.40406
$445$	0	0
$446$	−4.37668	−0.207242
$447$	−0.0312433	−0.00147776
$448$	18.5181	0.874899
$449$	24.0900	1.13688	0.568439	−	0.822726i	$-0.307548\pi$
$449$	24.0900	1.13688	0.568439	+	0.822726i	$0.307548\pi$
$450$	0	0
$451$	−0.751934	−0.0354072
$452$	−16.6174	−0.781618
$453$	−1.25029	−0.0587439
$454$	21.5577	1.01175
$455$	0	0
$456$	−29.1615	−1.36561
$457$	32.8242	1.53545	0.767726	−	0.640779i	$-0.221388\pi$
$457$	32.8242	1.53545	0.767726	+	0.640779i	$0.221388\pi$
$458$	25.1874	1.17693
$459$	−9.92088	−0.463067
$460$	0	0
$461$	24.2274	1.12838	0.564191	−	0.825644i	$-0.309188\pi$
$461$	24.2274	1.12838	0.564191	+	0.825644i	$0.309188\pi$
$462$	−9.13555	−0.425024
$463$	−3.28092	−0.152477	−0.0762385	−	0.997090i	$-0.524291\pi$
$463$	−3.28092	−0.152477	−0.0762385	+	0.997090i	$0.524291\pi$
$464$	−0.918922	−0.0426599
$465$	0	0
$466$	11.2622	0.521709
$467$	−7.88097	−0.364688	−0.182344	−	0.983235i	$-0.558368\pi$
$467$	−7.88097	−0.364688	−0.182344	+	0.983235i	$0.558368\pi$
$468$	0	0
$469$	3.98959	0.184222
$470$	0	0
$471$	2.91158	0.134158
$472$	−2.86203	−0.131735
$473$	8.49516	0.390608
$474$	−15.3382	−0.704509
$475$	0	0
$476$	4.14340	0.189912
$477$	−47.4635	−2.17320
$478$	−26.1403	−1.19563
$479$	−33.2549	−1.51945	−0.759727	−	0.650243i	$-0.774667\pi$
$479$	−33.2549	−1.51945	−0.759727	+	0.650243i	$0.774667\pi$
$480$	0	0
$481$	0	0
$482$	−10.7286	−0.488673
$483$	72.0044	3.27631
$484$	12.2488	0.556765
$485$	0	0
$486$	−6.50790	−0.295204
$487$	−3.18231	−0.144204	−0.0721022	−	0.997397i	$-0.522971\pi$
$487$	−3.18231	−0.144204	−0.0721022	+	0.997397i	$0.522971\pi$
$488$	12.9087	0.584349
$489$	32.1544	1.45407
$490$	0	0
$491$	32.5839	1.47049	0.735245	−	0.677802i	$-0.237067\pi$
$491$	32.5839	1.47049	0.735245	+	0.677802i	$0.237067\pi$
$492$	2.95786	0.133351
$493$	−8.06053	−0.363028
$494$	0	0
$495$	0	0
$496$	−0.453019	−0.0203412
$497$	−6.60509	−0.296279
$498$	0.510566	0.0228790
$499$	−9.77164	−0.437439	−0.218719	−	0.975788i	$-0.570188\pi$
$499$	−9.77164	−0.437439	−0.218719	+	0.975788i	$0.570188\pi$
$500$	0	0
$501$	24.4040	1.09029
$502$	−3.48632	−0.155602
$503$	−18.5746	−0.828201	−0.414100	−	0.910231i	$-0.635904\pi$
$503$	−18.5746	−0.828201	−0.414100	+	0.910231i	$0.635904\pi$
$504$	64.7804	2.88555
$505$	0	0
$506$	−5.52665	−0.245689
$507$	0	0
$508$	4.25644	0.188849
$509$	−1.96638	−0.0871584	−0.0435792	−	0.999050i	$-0.513876\pi$
$509$	−1.96638	−0.0871584	−0.0435792	+	0.999050i	$0.513876\pi$
$510$	0	0
$511$	−16.7364	−0.740373
$512$	1.23765	0.0546971
$513$	−34.5164	−1.52394
$514$	−17.8132	−0.785707
$515$	0	0
$516$	−33.4172	−1.47111
$517$	−9.99430	−0.439549
$518$	25.2350	1.10876
$519$	30.4139	1.33502
$520$	0	0
$521$	−21.0065	−0.920312	−0.460156	−	0.887838i	$-0.652206\pi$
$521$	−21.0065	−0.920312	−0.460156	+	0.887838i	$0.652206\pi$
$522$	−47.5399	−2.08076
$523$	20.2897	0.887206	0.443603	−	0.896223i	$-0.353700\pi$
$523$	20.2897	0.887206	0.443603	+	0.896223i	$0.353700\pi$
$524$	21.9009	0.956746
$525$	0	0
$526$	−14.1593	−0.617374
$527$	−3.97376	−0.173100
$528$	−0.315969	−0.0137508
$529$	20.5598	0.893905
$530$	0	0
$531$	−6.39836	−0.277665
$532$	14.4156	0.624994
$533$	0	0
$534$	−25.7953	−1.11627
$535$	0	0
$536$	3.19317	0.137924
$537$	−32.5440	−1.40438
$538$	−18.4757	−0.796542
$539$	5.37056	0.231326
$540$	0	0
$541$	−42.0546	−1.80807	−0.904035	−	0.427458i	$-0.859409\pi$
$541$	−42.0546	−1.80807	−0.904035	+	0.427458i	$0.859409\pi$
$542$	7.72627	0.331872
$543$	−26.5214	−1.13814
$544$	5.38154	0.230732
$545$	0	0
$546$	0	0
$547$	−31.0572	−1.32791	−0.663956	−	0.747772i	$-0.731124\pi$
$547$	−31.0572	−1.32791	−0.663956	+	0.747772i	$0.731124\pi$
$548$	0.945197	0.0403768
$549$	28.8588	1.23166
$550$	0	0
$551$	−28.0439	−1.19471
$552$	57.6305	2.45292
$553$	20.0997	0.854726
$554$	7.67259	0.325977
$555$	0	0
$556$	−0.337799	−0.0143259
$557$	−3.28840	−0.139334	−0.0696669	−	0.997570i	$-0.522194\pi$
$557$	−3.28840	−0.139334	−0.0696669	+	0.997570i	$0.522194\pi$
$558$	−23.4367	−0.992153
$559$	0	0
$560$	0	0
$561$	−2.77159	−0.117017
$562$	11.1174	0.468961
$563$	36.5000	1.53829	0.769147	−	0.639072i	$-0.220682\pi$
$563$	36.5000	1.53829	0.769147	+	0.639072i	$0.220682\pi$
$564$	39.3143	1.65543
$565$	0	0
$566$	−18.0526	−0.758809
$567$	44.6085	1.87338
$568$	−5.28655	−0.221819
$569$	−12.4563	−0.522194	−0.261097	−	0.965313i	$-0.584084\pi$
$569$	−12.4563	−0.522194	−0.261097	+	0.965313i	$0.584084\pi$
$570$	0	0
$571$	−12.8215	−0.536565	−0.268282	−	0.963340i	$-0.586456\pi$
$571$	−12.8215	−0.536565	−0.268282	+	0.963340i	$0.586456\pi$
$572$	0	0
$573$	32.4832	1.35700
$574$	2.52292	0.105305
$575$	0	0
$576$	33.1349	1.38062
$577$	22.7641	0.947681	0.473841	−	0.880611i	$-0.342867\pi$
$577$	22.7641	0.947681	0.473841	+	0.880611i	$0.342867\pi$
$578$	14.2777	0.593875
$579$	−62.5972	−2.60145
$580$	0	0
$581$	−0.669061	−0.0277573
$582$	−44.9825	−1.86458
$583$	−7.02034	−0.290753
$584$	−13.3954	−0.554304
$585$	0	0
$586$	−22.9117	−0.946474
$587$	18.5958	0.767533	0.383766	−	0.923430i	$-0.374627\pi$
$587$	18.5958	0.767533	0.383766	+	0.923430i	$0.374627\pi$
$588$	−21.1260	−0.871223
$589$	−13.8254	−0.569664
$590$	0	0
$591$	−35.2021	−1.44802
$592$	0.872795	0.0358717
$593$	36.4829	1.49817	0.749086	−	0.662473i	$-0.230493\pi$
$593$	36.4829	1.49817	0.749086	+	0.662473i	$0.230493\pi$
$594$	−8.65452	−0.355099
$595$	0	0
$596$	0.0123615	0.000506348 0
$597$	−30.4829	−1.24758
$598$	0	0
$599$	−46.6153	−1.90465	−0.952324	−	0.305089i	$-0.901314\pi$
$599$	−46.6153	−1.90465	−0.952324	+	0.305089i	$0.901314\pi$
$600$	0	0
$601$	−2.75628	−0.112431	−0.0562155	−	0.998419i	$-0.517903\pi$
$601$	−2.75628	−0.112431	−0.0562155	+	0.998419i	$0.517903\pi$
$602$	−28.5033	−1.16171
$603$	7.13867	0.290709
$604$	0.494683	0.0201283
$605$	0	0
$606$	14.9702	0.608123
$607$	−7.97853	−0.323839	−0.161919	−	0.986804i	$-0.551768\pi$
$607$	−7.97853	−0.323839	−0.161919	+	0.986804i	$0.551768\pi$
$608$	18.7233	0.759329
$609$	91.6122	3.71232
$610$	0	0
$611$	0	0
$612$	7.41388	0.299688
$613$	−28.3639	−1.14561	−0.572804	−	0.819693i	$-0.694144\pi$
$613$	−28.3639	−1.14561	−0.572804	+	0.819693i	$0.694144\pi$
$614$	5.09246	0.205515
$615$	0	0
$616$	9.58169	0.386057
$617$	28.8176	1.16015	0.580075	−	0.814563i	$-0.303023\pi$
$617$	28.8176	1.16015	0.580075	+	0.814563i	$0.303023\pi$
$618$	9.51088	0.382584
$619$	11.8346	0.475671	0.237836	−	0.971305i	$-0.423562\pi$
$619$	11.8346	0.475671	0.237836	+	0.971305i	$0.423562\pi$
$620$	0	0
$621$	68.2130	2.73729
$622$	−14.3096	−0.573764
$623$	33.8030	1.35429
$624$	0	0
$625$	0	0
$626$	−18.3385	−0.732952
$627$	−9.64283	−0.385097
$628$	−1.15197	−0.0459688
$629$	7.65592	0.305261
$630$	0	0
$631$	−22.0122	−0.876290	−0.438145	−	0.898904i	$-0.644364\pi$
$631$	−22.0122	−0.876290	−0.438145	+	0.898904i	$0.644364\pi$
$632$	16.0873	0.639918
$633$	35.0804	1.39432
$634$	−17.1062	−0.679373
$635$	0	0
$636$	27.6157	1.09503
$637$	0	0
$638$	−7.03163	−0.278385
$639$	−11.8186	−0.467538
$640$	0	0
$641$	−2.36286	−0.0933272	−0.0466636	−	0.998911i	$-0.514859\pi$
$641$	−2.36286	−0.0933272	−0.0466636	+	0.998911i	$0.514859\pi$
$642$	15.6144	0.616252
$643$	−28.6095	−1.12825	−0.564125	−	0.825689i	$-0.690786\pi$
$643$	−28.6095	−1.12825	−0.564125	+	0.825689i	$0.690786\pi$
$644$	−28.4888	−1.12261
$645$	0	0
$646$	−2.84668	−0.112001
$647$	31.0046	1.21892	0.609458	−	0.792818i	$-0.291387\pi$
$647$	31.0046	1.21892	0.609458	+	0.792818i	$0.291387\pi$
$648$	35.7035	1.40257
$649$	−0.946384	−0.0371488
$650$	0	0
$651$	45.1639	1.77011
$652$	−12.7220	−0.498231
$653$	19.7419	0.772559	0.386279	−	0.922382i	$-0.373760\pi$
$653$	19.7419	0.772559	0.386279	+	0.922382i	$0.373760\pi$
$654$	29.8824	1.16849
$655$	0	0
$656$	0.0872596	0.00340691
$657$	−29.9467	−1.16833
$658$	33.5333	1.30726
$659$	−34.3831	−1.33937	−0.669687	−	0.742643i	$-0.733572\pi$
$659$	−34.3831	−1.33937	−0.669687	+	0.742643i	$0.733572\pi$
$660$	0	0
$661$	2.91906	0.113538	0.0567692	−	0.998387i	$-0.481920\pi$
$661$	2.91906	0.113538	0.0567692	+	0.998387i	$0.481920\pi$
$662$	−17.6615	−0.686432
$663$	0	0
$664$	−0.535500	−0.0207814
$665$	0	0
$666$	45.1535	1.74966
$667$	55.4218	2.14594
$668$	−9.65552	−0.373583
$669$	15.0914	0.583467
$670$	0	0
$671$	4.26851	0.164784
$672$	−61.1641	−2.35946
$673$	36.1865	1.39489	0.697444	−	0.716640i	$-0.254321\pi$
$673$	36.1865	1.39489	0.697444	+	0.716640i	$0.254321\pi$
$674$	12.6271	0.486377
$675$	0	0
$676$	0	0
$677$	−17.8383	−0.685580	−0.342790	−	0.939412i	$-0.611372\pi$
$677$	−17.8383	−0.685580	−0.342790	+	0.939412i	$0.611372\pi$
$678$	−37.2958	−1.43234
$679$	58.9464	2.26216
$680$	0	0
$681$	−74.3338	−2.84848
$682$	−3.46653	−0.132740
$683$	−47.9902	−1.83629	−0.918147	−	0.396240i	$-0.870315\pi$
$683$	−47.9902	−1.83629	−0.918147	+	0.396240i	$0.870315\pi$
$684$	25.7941	0.986262
$685$	0	0
$686$	4.12821	0.157616
$687$	−86.8496	−3.31352
$688$	−0.985837	−0.0375847
$689$	0	0
$690$	0	0
$691$	−30.7768	−1.17081	−0.585403	−	0.810742i	$-0.699064\pi$
$691$	−30.7768	−1.17081	−0.585403	+	0.810742i	$0.699064\pi$
$692$	−12.0333	−0.457439
$693$	21.4209	0.813712
$694$	−4.86861	−0.184810
$695$	0	0
$696$	73.3241	2.77934
$697$	0.765417	0.0289922
$698$	−4.77653	−0.180794
$699$	−38.8335	−1.46882
$700$	0	0
$701$	−6.69487	−0.252862	−0.126431	−	0.991975i	$-0.540352\pi$
$701$	−6.69487	−0.252862	−0.126431	+	0.991975i	$0.540352\pi$
$702$	0	0
$703$	26.6362	1.00460
$704$	4.90100	0.184713
$705$	0	0
$706$	0.211706	0.00796765
$707$	−19.6174	−0.737789
$708$	3.72276	0.139910
$709$	−14.5935	−0.548071	−0.274036	−	0.961720i	$-0.588359\pi$
$709$	−14.5935	−0.548071	−0.274036	+	0.961720i	$0.588359\pi$
$710$	0	0
$711$	35.9648	1.34879
$712$	27.0551	1.01393
$713$	27.3224	1.02323
$714$	9.29936	0.348020
$715$	0	0
$716$	12.8762	0.481204
$717$	90.1352	3.36616
$718$	−0.476136	−0.0177692
$719$	−15.0560	−0.561493	−0.280746	−	0.959782i	$-0.590582\pi$
$719$	−15.0560	−0.561493	−0.280746	+	0.959782i	$0.590582\pi$
$720$	0	0
$721$	−12.4633	−0.464159
$722$	6.96787	0.259317
$723$	36.9936	1.37581
$724$	10.4933	0.389980
$725$	0	0
$726$	27.4910	1.02029
$727$	38.0451	1.41102	0.705508	−	0.708702i	$-0.250719\pi$
$727$	38.0451	1.41102	0.705508	+	0.708702i	$0.250719\pi$
$728$	0	0
$729$	−15.1192	−0.559971
$730$	0	0
$731$	−8.64749	−0.319839
$732$	−16.7909	−0.620610
$733$	−1.60488	−0.0592776	−0.0296388	−	0.999561i	$-0.509436\pi$
$733$	−1.60488	−0.0592776	−0.0296388	+	0.999561i	$0.509436\pi$
$734$	−8.91914	−0.329211
$735$	0	0
$736$	−37.0019	−1.36391
$737$	1.05588	0.0388939
$738$	4.51432	0.166174
$739$	37.6687	1.38567	0.692833	−	0.721098i	$-0.256362\pi$
$739$	37.6687	1.38567	0.692833	+	0.721098i	$0.256362\pi$
$740$	0	0
$741$	0	0
$742$	23.5549	0.864729
$743$	−45.6903	−1.67621	−0.838106	−	0.545507i	$-0.816337\pi$
$743$	−45.6903	−1.67621	−0.838106	+	0.545507i	$0.816337\pi$
$744$	36.1481	1.32525
$745$	0	0
$746$	−3.96821	−0.145286
$747$	−1.19717	−0.0438021
$748$	1.09659	0.0400953
$749$	−20.4616	−0.747651
$750$	0	0
$751$	−38.3301	−1.39868	−0.699342	−	0.714787i	$-0.746524\pi$
$751$	−38.3301	−1.39868	−0.699342	+	0.714787i	$0.746524\pi$
$752$	1.15981	0.0422938
$753$	12.0213	0.438081
$754$	0	0
$755$	0	0
$756$	−44.6123	−1.62253
$757$	−36.1056	−1.31228	−0.656141	−	0.754639i	$-0.727812\pi$
$757$	−36.1056	−1.31228	−0.656141	+	0.754639i	$0.727812\pi$
$758$	23.1377	0.840398
$759$	19.0566	0.691712
$760$	0	0
$761$	11.4601	0.415428	0.207714	−	0.978190i	$-0.433398\pi$
$761$	11.4601	0.415428	0.207714	+	0.978190i	$0.433398\pi$
$762$	9.55306	0.346071
$763$	−39.1588	−1.41764
$764$	−12.8521	−0.464972
$765$	0	0
$766$	28.2902	1.02217
$767$	0	0
$768$	−49.7661	−1.79578
$769$	−19.8697	−0.716521	−0.358260	−	0.933622i	$-0.616630\pi$
$769$	−19.8697	−0.716521	−0.358260	+	0.933622i	$0.616630\pi$
$770$	0	0
$771$	61.4224	2.21207
$772$	24.7668	0.891375
$773$	−46.3506	−1.66712	−0.833558	−	0.552432i	$-0.813700\pi$
$773$	−46.3506	−1.66712	−0.833558	+	0.552432i	$0.813700\pi$
$774$	−51.0016	−1.83322
$775$	0	0
$776$	47.1792	1.69364
$777$	−87.0136	−3.12160
$778$	−10.3319	−0.370417
$779$	2.66301	0.0954123
$780$	0	0
$781$	−1.74810	−0.0625519
$782$	5.62575	0.201176
$783$	86.7884	3.10156
$784$	−0.623237	−0.0222585
$785$	0	0
$786$	49.1540	1.75326
$787$	−35.5663	−1.26780	−0.633901	−	0.773414i	$-0.718547\pi$
$787$	−35.5663	−1.26780	−0.633901	+	0.773414i	$0.718547\pi$
$788$	13.9278	0.496159
$789$	48.8231	1.73815
$790$	0	0
$791$	48.8735	1.73774
$792$	17.1447	0.609212
$793$	0	0
$794$	7.17165	0.254512
$795$	0	0
$796$	12.0607	0.427478
$797$	−15.7100	−0.556478	−0.278239	−	0.960512i	$-0.589751\pi$
$797$	−15.7100	−0.556478	−0.278239	+	0.960512i	$0.589751\pi$
$798$	32.3540	1.14532
$799$	10.1735	0.359913
$800$	0	0
$801$	60.4845	2.13711
$802$	−27.7268	−0.979066
$803$	−4.42943	−0.156311
$804$	−4.15349	−0.146482
$805$	0	0
$806$	0	0
$807$	63.7066	2.24258
$808$	−15.7013	−0.552369
$809$	−49.8734	−1.75346	−0.876728	−	0.480987i	$-0.840278\pi$
$809$	−49.8734	−1.75346	−0.876728	+	0.480987i	$0.840278\pi$
$810$	0	0
$811$	38.3395	1.34628	0.673141	−	0.739514i	$-0.264945\pi$
$811$	38.3395	1.34628	0.673141	+	0.739514i	$0.264945\pi$
$812$	−36.2467	−1.27201
$813$	−26.6412	−0.934349
$814$	6.67867	0.234087
$815$	0	0
$816$	0.321635	0.0112595
$817$	−30.0860	−1.05258
$818$	28.5902	0.999633
$819$	0	0
$820$	0	0
$821$	32.1738	1.12287	0.561436	−	0.827520i	$-0.310249\pi$
$821$	32.1738	1.12287	0.561436	+	0.827520i	$0.310249\pi$
$822$	2.12138	0.0739916
$823$	−11.2002	−0.390413	−0.195207	−	0.980762i	$-0.562538\pi$
$823$	−11.2002	−0.390413	−0.195207	+	0.980762i	$0.562538\pi$
$824$	−9.97535	−0.347508
$825$	0	0
$826$	3.17535	0.110484
$827$	−8.21374	−0.285620	−0.142810	−	0.989750i	$-0.545614\pi$
$827$	−8.21374	−0.285620	−0.142810	+	0.989750i	$0.545614\pi$
$828$	−50.9756	−1.77153
$829$	35.0948	1.21889	0.609447	−	0.792827i	$-0.291391\pi$
$829$	35.0948	1.21889	0.609447	+	0.792827i	$0.291391\pi$
$830$	0	0
$831$	−26.4561	−0.917753
$832$	0	0
$833$	−5.46686	−0.189416
$834$	−0.758150	−0.0262526
$835$	0	0
$836$	3.81521	0.131952
$837$	42.7858	1.47889
$838$	19.3832	0.669582
$839$	−17.7011	−0.611108	−0.305554	−	0.952175i	$-0.598842\pi$
$839$	−17.7011	−0.611108	−0.305554	+	0.952175i	$0.598842\pi$
$840$	0	0
$841$	41.5140	1.43152
$842$	−4.04670	−0.139459
$843$	−38.3345	−1.32031
$844$	−13.8797	−0.477759
$845$	0	0
$846$	60.0019	2.06291
$847$	−36.0250	−1.23784
$848$	0.814689	0.0279765
$849$	62.2479	2.13634
$850$	0	0
$851$	−52.6398	−1.80447
$852$	6.87644	0.235583
$853$	−20.4558	−0.700394	−0.350197	−	0.936676i	$-0.613885\pi$
$853$	−20.4558	−0.700394	−0.350197	+	0.936676i	$0.613885\pi$
$854$	−14.3219	−0.490085
$855$	0	0
$856$	−16.3770	−0.559753
$857$	−36.3625	−1.24212	−0.621060	−	0.783763i	$-0.713298\pi$
$857$	−36.3625	−1.24212	−0.621060	+	0.783763i	$0.713298\pi$
$858$	0	0
$859$	−51.0474	−1.74172	−0.870858	−	0.491534i	$-0.836436\pi$
$859$	−51.0474	−1.74172	−0.870858	+	0.491534i	$0.836436\pi$
$860$	0	0
$861$	−8.69937	−0.296474
$862$	−26.3946	−0.899003
$863$	24.9638	0.849776	0.424888	−	0.905246i	$-0.360313\pi$
$863$	24.9638	0.849776	0.424888	+	0.905246i	$0.360313\pi$
$864$	−57.9435	−1.97128
$865$	0	0
$866$	6.91058	0.234831
$867$	−49.2316	−1.67199
$868$	−17.8692	−0.606522
$869$	5.31957	0.180454
$870$	0	0
$871$	0	0
$872$	−31.3417	−1.06137
$873$	105.474	3.56976
$874$	19.5729	0.662063
$875$	0	0
$876$	17.4239	0.588700
$877$	35.9738	1.21475	0.607374	−	0.794416i	$-0.292223\pi$
$877$	35.9738	1.21475	0.607374	+	0.794416i	$0.292223\pi$
$878$	−3.72854	−0.125832
$879$	79.0027	2.66469
$880$	0	0
$881$	−7.77696	−0.262013	−0.131006	−	0.991382i	$-0.541821\pi$
$881$	−7.77696	−0.262013	−0.131006	+	0.991382i	$0.541821\pi$
$882$	−32.2428	−1.08567
$883$	−38.0635	−1.28094	−0.640470	−	0.767983i	$-0.721260\pi$
$883$	−38.0635	−1.28094	−0.640470	+	0.767983i	$0.721260\pi$
$884$	0	0
$885$	0	0
$886$	19.1174	0.642262
$887$	22.3640	0.750910	0.375455	−	0.926841i	$-0.377486\pi$
$887$	22.3640	0.750910	0.375455	+	0.926841i	$0.377486\pi$
$888$	−69.6435	−2.33708
$889$	−12.5186	−0.419861
$890$	0	0
$891$	11.8060	0.395518
$892$	−5.97096	−0.199922
$893$	35.3953	1.18446
$894$	0.0277440	0.000927897 0
$895$	0	0
$896$	23.5073	0.785324
$897$	0	0
$898$	−21.3918	−0.713855
$899$	34.7627	1.15940
$900$	0	0
$901$	7.14622	0.238075
$902$	0.667714	0.0222325
$903$	98.2833	3.27066
$904$	39.1172	1.30102
$905$	0	0
$906$	1.11026	0.0368858
$907$	−39.4230	−1.30902	−0.654510	−	0.756053i	$-0.727125\pi$
$907$	−39.4230	−1.30902	−0.654510	+	0.756053i	$0.727125\pi$
$908$	29.4104	0.976019
$909$	−35.1019	−1.16426
$910$	0	0
$911$	−32.8525	−1.08845	−0.544227	−	0.838938i	$-0.683177\pi$
$911$	−32.8525	−1.08845	−0.544227	+	0.838938i	$0.683177\pi$
$912$	1.11902	0.0370544
$913$	−0.177073	−0.00586027
$914$	−29.1478	−0.964122
$915$	0	0
$916$	34.3623	1.13536
$917$	−64.4128	−2.12710
$918$	8.80970	0.290764
$919$	8.54325	0.281816	0.140908	−	0.990023i	$-0.454998\pi$
$919$	8.54325	0.281816	0.140908	+	0.990023i	$0.454998\pi$
$920$	0	0
$921$	−17.5595	−0.578605
$922$	−21.5138	−0.708520
$923$	0	0
$924$	−12.4633	−0.410013
$925$	0	0
$926$	2.91344	0.0957416
$927$	−22.3010	−0.732459
$928$	−47.0780	−1.54541
$929$	−0.365640	−0.0119962	−0.00599812	−	0.999982i	$-0.501909\pi$
$929$	−0.365640	−0.0119962	−0.00599812	+	0.999982i	$0.501909\pi$
$930$	0	0
$931$	−19.0201	−0.623359
$932$	15.3646	0.503284
$933$	49.3416	1.61537
$934$	6.99827	0.228990
$935$	0	0
$936$	0	0
$937$	−9.19666	−0.300442	−0.150221	−	0.988652i	$-0.547999\pi$
$937$	−9.19666	−0.300442	−0.150221	+	0.988652i	$0.547999\pi$
$938$	−3.54274	−0.115675
$939$	63.2335	2.06355
$940$	0	0
$941$	3.66316	0.119416	0.0597078	−	0.998216i	$-0.480983\pi$
$941$	3.66316	0.119416	0.0597078	+	0.998216i	$0.480983\pi$
$942$	−2.58547	−0.0842391
$943$	−5.26278	−0.171380
$944$	0.109825	0.00357450
$945$	0	0
$946$	−7.54367	−0.245266
$947$	30.6437	0.995787	0.497893	−	0.867238i	$-0.334107\pi$
$947$	30.6437	0.995787	0.497893	+	0.867238i	$0.334107\pi$
$948$	−20.9254	−0.679627
$949$	0	0
$950$	0	0
$951$	58.9844	1.91270
$952$	−9.75350	−0.316113
$953$	5.94775	0.192667	0.0963333	−	0.995349i	$-0.469289\pi$
$953$	5.94775	0.192667	0.0963333	+	0.995349i	$0.469289\pi$
$954$	42.1474	1.36457
$955$	0	0
$956$	−35.6623	−1.15340
$957$	24.2460	0.783763
$958$	29.5302	0.954077
$959$	−2.77992	−0.0897683
$960$	0	0
$961$	−13.8623	−0.447173
$962$	0	0
$963$	−36.6124	−1.17982
$964$	−14.6366	−0.471414
$965$	0	0
$966$	−63.9396	−2.05722
$967$	−1.71181	−0.0550482	−0.0275241	−	0.999621i	$-0.508762\pi$
$967$	−1.71181	−0.0550482	−0.0275241	+	0.999621i	$0.508762\pi$
$968$	−28.8335	−0.926745
$969$	9.81573	0.315327
$970$	0	0
$971$	−14.2668	−0.457844	−0.228922	−	0.973445i	$-0.573520\pi$
$971$	−14.2668	−0.457844	−0.228922	+	0.973445i	$0.573520\pi$
$972$	−8.87851	−0.284778
$973$	0.993503	0.0318502
$974$	2.82588	0.0905471
$975$	0	0
$976$	−0.495347	−0.0158557
$977$	−47.5178	−1.52023	−0.760115	−	0.649788i	$-0.774858\pi$
$977$	−47.5178	−1.52023	−0.760115	+	0.649788i	$0.774858\pi$
$978$	−28.5530	−0.913023
$979$	8.94628	0.285924
$980$	0	0
$981$	−70.0678	−2.23709
$982$	−28.9344	−0.923332
$983$	−35.3161	−1.12641	−0.563204	−	0.826318i	$-0.690432\pi$
$983$	−35.3161	−1.12641	−0.563204	+	0.826318i	$0.690432\pi$
$984$	−6.96276	−0.221965
$985$	0	0
$986$	7.15772	0.227948
$987$	−115.627	−3.68046
$988$	0	0
$989$	59.4575	1.89064
$990$	0	0
$991$	−61.3288	−1.94817	−0.974087	−	0.226173i	$-0.927378\pi$
$991$	−61.3288	−1.94817	−0.974087	+	0.226173i	$0.927378\pi$
$992$	−23.2090	−0.736887
$993$	60.8991	1.93258
$994$	5.86529	0.186036
$995$	0	0
$996$	0.696548	0.0220710
$997$	20.6596	0.654295	0.327148	−	0.944973i	$-0.393913\pi$
$997$	20.6596	0.654295	0.327148	+	0.944973i	$0.393913\pi$
$998$	8.67717	0.274671
$999$	−82.4319	−2.60803

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.bv.1.4		10
5.4	even	2		4225.2.a.bu.1.7		10
13.6	odd	12		325.2.n.e.101.4	✓	10
13.11	odd	12		325.2.n.e.251.4	yes	10
13.12	even	2	inner	4225.2.a.bv.1.7		10
65.19	odd	12		325.2.n.f.101.2	yes	10
65.24	odd	12		325.2.n.f.251.2	yes	10
65.32	even	12		325.2.m.d.49.7		20
65.37	even	12		325.2.m.d.199.4		20
65.58	even	12		325.2.m.d.49.4		20
65.63	even	12		325.2.m.d.199.7		20
65.64	even	2		4225.2.a.bu.1.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.2.m.d.49.4		20	65.58	even	12
325.2.m.d.49.7		20	65.32	even	12
325.2.m.d.199.4		20	65.37	even	12
325.2.m.d.199.7		20	65.63	even	12
325.2.n.e.101.4	✓	10	13.6	odd	12
325.2.n.e.251.4	yes	10	13.11	odd	12
325.2.n.f.101.2	yes	10	65.19	odd	12
325.2.n.f.251.2	yes	10	65.24	odd	12
4225.2.a.bu.1.4		10	65.64	even	2
4225.2.a.bu.1.7		10	5.4	even	2
4225.2.a.bv.1.4		10	1.1	even	1	trivial
4225.2.a.bv.1.7		10	13.12	even	2	inner

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:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q-0.887996 q^{2} +3.06193 q^{3} -1.21146 q^{4} -2.71898 q^{6} +3.56304 q^{7} +2.85177 q^{8} +6.37543 q^{9} +O(q^{10})\)	\(q-0.887996 q^{2} +3.06193 q^{3} -1.21146 q^{4} -2.71898 q^{6} +3.56304 q^{7} +2.85177 q^{8} +6.37543 q^{9} +0.942991 q^{11} -3.70942 q^{12} -3.16396 q^{14} -0.109431 q^{16} -0.959900 q^{17} -5.66135 q^{18} -3.33965 q^{19} +10.9098 q^{21} -0.837373 q^{22} +6.59999 q^{23} +8.73191 q^{24} +10.3353 q^{27} -4.31649 q^{28} +8.39726 q^{29} +4.13976 q^{31} -5.60636 q^{32} +2.88738 q^{33} +0.852387 q^{34} -7.72359 q^{36} -7.97574 q^{37} +2.96560 q^{38} -0.797392 q^{41} -9.68784 q^{42} +9.00874 q^{43} -1.14240 q^{44} -5.86076 q^{46} -10.5985 q^{47} -0.335071 q^{48} +5.69524 q^{49} -2.93915 q^{51} -7.44476 q^{53} -9.17773 q^{54} +10.1610 q^{56} -10.2258 q^{57} -7.45673 q^{58} -1.00360 q^{59} +4.52656 q^{61} -3.67609 q^{62} +22.7159 q^{63} +5.19729 q^{64} -2.56398 q^{66} +1.11972 q^{67} +1.16288 q^{68} +20.2087 q^{69} -1.85378 q^{71} +18.1812 q^{72} -4.69721 q^{73} +7.08243 q^{74} +4.04586 q^{76} +3.35991 q^{77} +5.64117 q^{79} +12.5198 q^{81} +0.708081 q^{82} -0.187778 q^{83} -13.2168 q^{84} -7.99972 q^{86} +25.7118 q^{87} +2.68919 q^{88} +9.48713 q^{89} -7.99564 q^{92} +12.6757 q^{93} +9.41144 q^{94} -17.1663 q^{96} +16.5439 q^{97} -5.05735 q^{98} +6.01197 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 6 q^{3} + 12 q^{4} + 16 q^{9}+O(q^{10}) \) 10 * q + 6 * q^3 + 12 * q^4 + 16 * q^9	\( 10 q + 6 q^{3} + 12 q^{4} + 16 q^{9} + 28 q^{12} - 8 q^{14} + 24 q^{16} + 16 q^{17} + 24 q^{22} + 26 q^{23} + 30 q^{27} - 14 q^{29} - 6 q^{36} + 62 q^{38} - 64 q^{42} + 8 q^{43} + 52 q^{48} - 2 q^{49} - 16 q^{51} + 24 q^{53} - 42 q^{56} - 26 q^{61} + 34 q^{62} + 34 q^{64} - 42 q^{66} + 26 q^{68} + 40 q^{69} + 52 q^{74} + 48 q^{77} + 4 q^{79} + 34 q^{81} - 2 q^{82} + 98 q^{87} - 12 q^{88} + 34 q^{92} - 10 q^{94}+O(q^{100}) \) 10 * q + 6 * q^3 + 12 * q^4 + 16 * q^9 + 28 * q^12 - 8 * q^14 + 24 * q^16 + 16 * q^17 + 24 * q^22 + 26 * q^23 + 30 * q^27 - 14 * q^29 - 6 * q^36 + 62 * q^38 - 64 * q^42 + 8 * q^43 + 52 * q^48 - 2 * q^49 - 16 * q^51 + 24 * q^53 - 42 * q^56 - 26 * q^61 + 34 * q^62 + 34 * q^64 - 42 * q^66 + 26 * q^68 + 40 * q^69 + 52 * q^74 + 48 * q^77 + 4 * q^79 + 34 * q^81 - 2 * q^82 + 98 * q^87 - 12 * q^88 + 34 * q^92 - 10 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more