LMFDB - Embedded newform 7225.2.a.bx.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, 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("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7225 = 5^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7225.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:	$57.6919154604$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 425)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	7225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.555780 q^{2} -2.50339 q^{3} -1.69111 q^{4} -1.39134 q^{6} +4.03587 q^{7} -2.05144 q^{8} +3.26698 q^{9} +O(q^{10})$	\(q+0.555780 q^{2} -2.50339 q^{3} -1.69111 q^{4} -1.39134 q^{6} +4.03587 q^{7} -2.05144 q^{8} +3.26698 q^{9} +0.420560 q^{11} +4.23351 q^{12} +4.96035 q^{13} +2.24306 q^{14} +2.24207 q^{16} +1.81572 q^{18} -5.04055 q^{19} -10.1034 q^{21} +0.233739 q^{22} -8.05347 q^{23} +5.13557 q^{24} +2.75686 q^{26} -0.668350 q^{27} -6.82510 q^{28} -0.185558 q^{29} -3.41416 q^{31} +5.34898 q^{32} -1.05283 q^{33} -5.52481 q^{36} +3.81527 q^{37} -2.80144 q^{38} -12.4177 q^{39} +3.90854 q^{41} -5.61525 q^{42} -5.15014 q^{43} -0.711213 q^{44} -4.47596 q^{46} -6.56400 q^{47} -5.61277 q^{48} +9.28828 q^{49} -8.38849 q^{52} -0.871603 q^{53} -0.371456 q^{54} -8.27937 q^{56} +12.6185 q^{57} -0.103129 q^{58} -14.3445 q^{59} +3.41024 q^{61} -1.89752 q^{62} +13.1851 q^{63} -1.51128 q^{64} -0.585140 q^{66} +7.21621 q^{67} +20.1610 q^{69} -7.48836 q^{71} -6.70202 q^{72} +3.58559 q^{73} +2.12045 q^{74} +8.52413 q^{76} +1.69733 q^{77} -6.90151 q^{78} +14.0842 q^{79} -8.12779 q^{81} +2.17229 q^{82} -13.6689 q^{83} +17.0859 q^{84} -2.86234 q^{86} +0.464523 q^{87} -0.862755 q^{88} +15.1176 q^{89} +20.0193 q^{91} +13.6193 q^{92} +8.54699 q^{93} -3.64814 q^{94} -13.3906 q^{96} -10.1591 q^{97} +5.16224 q^{98} +1.37396 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 8 q^{2} + 24 q^{4} - 24 q^{8} + 24 q^{9}+O(q^{10}) $ 24 * q - 8 * q^2 + 24 * q^4 - 24 * q^8 + 24 * q^9	$ 24 q - 8 q^{2} + 24 q^{4} - 24 q^{8} + 24 q^{9} - 16 q^{13} + 24 q^{16} - 40 q^{18} - 16 q^{21} - 16 q^{26} - 56 q^{32} - 48 q^{33} + 24 q^{36} - 48 q^{38} - 32 q^{43} - 88 q^{47} + 16 q^{49} - 48 q^{52} - 48 q^{53} - 8 q^{59} + 72 q^{64} + 32 q^{66} - 40 q^{67} - 48 q^{69} - 120 q^{72} + 32 q^{76} - 120 q^{77} - 24 q^{81} - 104 q^{83} + 40 q^{84} - 16 q^{86} - 64 q^{87} + 16 q^{89} + 72 q^{93} + 112 q^{94} - 48 q^{98}+O(q^{100}) $ 24 * q - 8 * q^2 + 24 * q^4 - 24 * q^8 + 24 * q^9 - 16 * q^13 + 24 * q^16 - 40 * q^18 - 16 * q^21 - 16 * q^26 - 56 * q^32 - 48 * q^33 + 24 * q^36 - 48 * q^38 - 32 * q^43 - 88 * q^47 + 16 * q^49 - 48 * q^52 - 48 * q^53 - 8 * q^59 + 72 * q^64 + 32 * q^66 - 40 * q^67 - 48 * q^69 - 120 * q^72 + 32 * q^76 - 120 * q^77 - 24 * q^81 - 104 * q^83 + 40 * q^84 - 16 * q^86 - 64 * q^87 + 16 * q^89 + 72 * q^93 + 112 * q^94 - 48 * 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 $

$2$	0.555780	0.392996	0.196498	−	0.980504i	$-0.437043\pi$
$2$	0.555780	0.392996	0.196498	+	0.980504i	$0.437043\pi$
$3$	−2.50339	−1.44533	−0.722667	−	0.691196i	$-0.757084\pi$
$3$	−2.50339	−1.44533	−0.722667	+	0.691196i	$0.757084\pi$
$4$	−1.69111	−0.845554
$5$	0	0
$5$	0	0
$6$	−1.39134	−0.568010
$7$	4.03587	1.52542	0.762709	−	0.646742i	$-0.223869\pi$
$7$	4.03587	1.52542	0.762709	+	0.646742i	$0.223869\pi$
$8$	−2.05144	−0.725295
$9$	3.26698	1.08899
$10$	0	0
$11$	0.420560	0.126804	0.0634018	−	0.997988i	$-0.479805\pi$
$11$	0.420560	0.126804	0.0634018	+	0.997988i	$0.479805\pi$
$12$	4.23351	1.22211
$13$	4.96035	1.37575	0.687877	−	0.725828i	$-0.258543\pi$
$13$	4.96035	1.37575	0.687877	+	0.725828i	$0.258543\pi$
$14$	2.24306	0.599482
$15$	0	0
$16$	2.24207	0.560517
$17$	0	0
$17$	0	0
$18$	1.81572	0.427969
$19$	−5.04055	−1.15638	−0.578191	−	0.815901i	$-0.696241\pi$
$19$	−5.04055	−1.15638	−0.578191	+	0.815901i	$0.696241\pi$
$20$	0	0
$21$	−10.1034	−2.20474
$22$	0.233739	0.0498333
$23$	−8.05347	−1.67927	−0.839633	−	0.543155i	$-0.817230\pi$
$23$	−8.05347	−1.67927	−0.839633	+	0.543155i	$0.817230\pi$
$24$	5.13557	1.04829
$25$	0	0
$26$	2.75686	0.540665
$27$	−0.668350	−0.128624
$28$	−6.82510	−1.28982
$29$	−0.185558	−0.0344572	−0.0172286	−	0.999852i	$-0.505484\pi$
$29$	−0.185558	−0.0344572	−0.0172286	+	0.999852i	$0.505484\pi$
$30$	0	0
$31$	−3.41416	−0.613201	−0.306601	−	0.951838i	$-0.599192\pi$
$31$	−3.41416	−0.613201	−0.306601	+	0.951838i	$0.599192\pi$
$32$	5.34898	0.945575
$33$	−1.05283	−0.183274
$34$	0	0
$35$	0	0
$36$	−5.52481	−0.920802
$37$	3.81527	0.627227	0.313614	−	0.949551i	$-0.398460\pi$
$37$	3.81527	0.627227	0.313614	+	0.949551i	$0.398460\pi$
$38$	−2.80144	−0.454453
$39$	−12.4177	−1.98842
$40$	0	0
$41$	3.90854	0.610411	0.305206	−	0.952287i	$-0.401275\pi$
$41$	3.90854	0.610411	0.305206	+	0.952287i	$0.401275\pi$
$42$	−5.61525	−0.866453
$43$	−5.15014	−0.785389	−0.392694	−	0.919669i	$-0.628457\pi$
$43$	−5.15014	−0.785389	−0.392694	+	0.919669i	$0.628457\pi$
$44$	−0.711213	−0.107219
$45$	0	0
$46$	−4.47596	−0.659944
$47$	−6.56400	−0.957458	−0.478729	−	0.877963i	$-0.658902\pi$
$47$	−6.56400	−0.957458	−0.478729	+	0.877963i	$0.658902\pi$
$48$	−5.61277	−0.810134
$49$	9.28828	1.32690
$50$	0	0
$51$	0	0
$52$	−8.38849	−1.16327
$53$	−0.871603	−0.119724	−0.0598619	−	0.998207i	$-0.519066\pi$
$53$	−0.871603	−0.119724	−0.0598619	+	0.998207i	$0.519066\pi$
$54$	−0.371456	−0.0505487
$55$	0	0
$56$	−8.27937	−1.10638
$57$	12.6185	1.67136
$58$	−0.103129	−0.0135415
$59$	−14.3445	−1.86749	−0.933746	−	0.357937i	$-0.883480\pi$
$59$	−14.3445	−1.86749	−0.933746	+	0.357937i	$0.883480\pi$
$60$	0	0
$61$	3.41024	0.436637	0.218318	−	0.975878i	$-0.429943\pi$
$61$	3.41024	0.436637	0.218318	+	0.975878i	$0.429943\pi$
$62$	−1.89752	−0.240985
$63$	13.1851	1.66117
$64$	−1.51128	−0.188910
$65$	0	0
$66$	−0.585140	−0.0720257
$67$	7.21621	0.881601	0.440800	−	0.897605i	$-0.354695\pi$
$67$	7.21621	0.881601	0.440800	+	0.897605i	$0.354695\pi$
$68$	0	0
$69$	20.1610	2.42710
$70$	0	0
$71$	−7.48836	−0.888705	−0.444352	−	0.895852i	$-0.646566\pi$
$71$	−7.48836	−0.888705	−0.444352	+	0.895852i	$0.646566\pi$
$72$	−6.70202	−0.789841
$73$	3.58559	0.419662	0.209831	−	0.977738i	$-0.432709\pi$
$73$	3.58559	0.419662	0.209831	+	0.977738i	$0.432709\pi$
$74$	2.12045	0.246498
$75$	0	0
$76$	8.52413	0.977784
$77$	1.69733	0.193428
$78$	−6.90151	−0.781442
$79$	14.0842	1.58459	0.792296	−	0.610137i	$-0.208886\pi$
$79$	14.0842	1.58459	0.792296	+	0.610137i	$0.208886\pi$
$80$	0	0
$81$	−8.12779	−0.903088
$82$	2.17229	0.239889
$83$	−13.6689	−1.50035	−0.750177	−	0.661237i	$-0.770032\pi$
$83$	−13.6689	−1.50035	−0.750177	+	0.661237i	$0.770032\pi$
$84$	17.0859	1.86423
$85$	0	0
$86$	−2.86234	−0.308654
$87$	0.464523	0.0498021
$88$	−0.862755	−0.0919700
$89$	15.1176	1.60246	0.801231	−	0.598355i	$-0.204179\pi$
$89$	15.1176	1.60246	0.801231	+	0.598355i	$0.204179\pi$
$90$	0	0
$91$	20.0193	2.09860
$92$	13.6193	1.41991
$93$	8.54699	0.886281
$94$	−3.64814	−0.376277
$95$	0	0
$96$	−13.3906	−1.36667
$97$	−10.1591	−1.03150	−0.515751	−	0.856738i	$-0.672487\pi$
$97$	−10.1591	−1.03150	−0.515751	+	0.856738i	$0.672487\pi$
$98$	5.16224	0.521465
$99$	1.37396	0.138088
$100$	0	0
$101$	−0.340331	−0.0338642	−0.0169321	−	0.999857i	$-0.505390\pi$
$101$	−0.340331	−0.0338642	−0.0169321	+	0.999857i	$0.505390\pi$
$102$	0	0
$103$	0.688204	0.0678107	0.0339054	−	0.999425i	$-0.489206\pi$
$103$	0.688204	0.0678107	0.0339054	+	0.999425i	$0.489206\pi$
$104$	−10.1759	−0.997827
$105$	0	0
$106$	−0.484419	−0.0470510
$107$	9.68440	0.936226	0.468113	−	0.883669i	$-0.344934\pi$
$107$	9.68440	0.936226	0.468113	+	0.883669i	$0.344934\pi$
$108$	1.13025	0.108759
$109$	10.4494	1.00087	0.500436	−	0.865774i	$-0.333173\pi$
$109$	10.4494	1.00087	0.500436	+	0.865774i	$0.333173\pi$
$110$	0	0
$111$	−9.55113	−0.906553
$112$	9.04870	0.855022
$113$	9.01565	0.848121	0.424061	−	0.905634i	$-0.360604\pi$
$113$	9.01565	0.848121	0.424061	+	0.905634i	$0.360604\pi$
$114$	7.01310	0.656837
$115$	0	0
$116$	0.313798	0.0291354
$117$	16.2054	1.49819
$118$	−7.97237	−0.733916
$119$	0	0
$120$	0	0
$121$	−10.8231	−0.983921
$122$	1.89534	0.171596
$123$	−9.78461	−0.882248
$124$	5.77372	0.518495
$125$	0	0
$126$	7.32802	0.652832
$127$	−13.6042	−1.20718	−0.603590	−	0.797295i	$-0.706264\pi$
$127$	−13.6042	−1.20718	−0.603590	+	0.797295i	$0.706264\pi$
$128$	−11.5379	−1.01982
$129$	12.8928	1.13515
$130$	0	0
$131$	13.5224	1.18145	0.590727	−	0.806871i	$-0.298841\pi$
$131$	13.5224	1.18145	0.590727	+	0.806871i	$0.298841\pi$
$132$	1.78044	0.154968
$133$	−20.3430	−1.76397
$134$	4.01063	0.346465
$135$	0	0
$136$	0	0
$137$	8.57682	0.732768	0.366384	−	0.930464i	$-0.380596\pi$
$137$	8.57682	0.732768	0.366384	+	0.930464i	$0.380596\pi$
$138$	11.2051	0.953840
$139$	13.1182	1.11267	0.556336	−	0.830958i	$-0.312207\pi$
$139$	13.1182	1.11267	0.556336	+	0.830958i	$0.312207\pi$
$140$	0	0
$141$	16.4323	1.38385
$142$	−4.16188	−0.349257
$143$	2.08612	0.174450
$144$	7.32478	0.610398
$145$	0	0
$146$	1.99280	0.164925
$147$	−23.2522	−1.91781
$148$	−6.45204	−0.530355
$149$	−8.25712	−0.676450	−0.338225	−	0.941065i	$-0.609826\pi$
$149$	−8.25712	−0.676450	−0.338225	+	0.941065i	$0.609826\pi$
$150$	0	0
$151$	6.34217	0.516118	0.258059	−	0.966129i	$-0.416917\pi$
$151$	6.34217	0.516118	0.258059	+	0.966129i	$0.416917\pi$
$152$	10.3404	0.838718
$153$	0	0
$154$	0.943340	0.0760165
$155$	0	0
$156$	20.9997	1.68132
$157$	−21.7433	−1.73531	−0.867653	−	0.497169i	$-0.834373\pi$
$157$	−21.7433	−1.73531	−0.867653	+	0.497169i	$0.834373\pi$
$158$	7.82769	0.622738
$159$	2.18196	0.173041
$160$	0	0
$161$	−32.5028	−2.56158
$162$	−4.51726	−0.354910
$163$	−13.5699	−1.06288	−0.531440	−	0.847096i	$-0.678349\pi$
$163$	−13.5699	−1.06288	−0.531440	+	0.847096i	$0.678349\pi$
$164$	−6.60976	−0.516136
$165$	0	0
$166$	−7.59688	−0.589633
$167$	−15.9981	−1.23797	−0.618986	−	0.785402i	$-0.712456\pi$
$167$	−15.9981	−1.23797	−0.618986	+	0.785402i	$0.712456\pi$
$168$	20.7265	1.59909
$169$	11.6051	0.892697
$170$	0	0
$171$	−16.4674	−1.25929
$172$	8.70944	0.664089
$173$	−2.51999	−0.191591	−0.0957955	−	0.995401i	$-0.530539\pi$
$173$	−2.51999	−0.191591	−0.0957955	+	0.995401i	$0.530539\pi$
$174$	0.258173	0.0195720
$175$	0	0
$176$	0.942923	0.0710755
$177$	35.9099	2.69915
$178$	8.40206	0.629761
$179$	7.45781	0.557423	0.278711	−	0.960375i	$-0.410093\pi$
$179$	7.45781	0.557423	0.278711	+	0.960375i	$0.410093\pi$
$180$	0	0
$181$	16.3032	1.21181	0.605904	−	0.795538i	$-0.292812\pi$
$181$	16.3032	1.21181	0.605904	+	0.795538i	$0.292812\pi$
$182$	11.1263	0.824740
$183$	−8.53718	−0.631086
$184$	16.5212	1.21796
$185$	0	0
$186$	4.75024	0.348305
$187$	0	0
$188$	11.1004	0.809583
$189$	−2.69738	−0.196205
$190$	0	0
$191$	4.90447	0.354875	0.177437	−	0.984132i	$-0.443219\pi$
$191$	4.90447	0.354875	0.177437	+	0.984132i	$0.443219\pi$
$192$	3.78332	0.273038
$193$	6.29053	0.452802	0.226401	−	0.974034i	$-0.427304\pi$
$193$	6.29053	0.452802	0.226401	+	0.974034i	$0.427304\pi$
$194$	−5.64624	−0.405376
$195$	0	0
$196$	−15.7075	−1.12196
$197$	−22.8122	−1.62530	−0.812650	−	0.582752i	$-0.801976\pi$
$197$	−22.8122	−1.62530	−0.812650	+	0.582752i	$0.801976\pi$
$198$	0.763619	0.0542680
$199$	−23.4002	−1.65880	−0.829398	−	0.558658i	$-0.811317\pi$
$199$	−23.4002	−1.65880	−0.829398	+	0.558658i	$0.811317\pi$
$200$	0	0
$201$	−18.0650	−1.27421
$202$	−0.189149	−0.0133085
$203$	−0.748887	−0.0525615
$204$	0	0
$205$	0	0
$206$	0.382490	0.0266493
$207$	−26.3105	−1.82871
$208$	11.1214	0.771133
$209$	−2.11986	−0.146633
$210$	0	0
$211$	−1.80689	−0.124392	−0.0621959	−	0.998064i	$-0.519810\pi$
$211$	−1.80689	−0.124392	−0.0621959	+	0.998064i	$0.519810\pi$
$212$	1.47398	0.101233
$213$	18.7463	1.28448
$214$	5.38239	0.367933
$215$	0	0
$216$	1.37108	0.0932904
$217$	−13.7791	−0.935388
$218$	5.80757	0.393338
$219$	−8.97614	−0.606552
$220$	0	0
$221$	0	0
$222$	−5.30832	−0.356271
$223$	8.89829	0.595874	0.297937	−	0.954586i	$-0.403702\pi$
$223$	8.89829	0.595874	0.297937	+	0.954586i	$0.403702\pi$
$224$	21.5878	1.44240
$225$	0	0
$226$	5.01072	0.333308
$227$	16.6329	1.10397	0.551983	−	0.833855i	$-0.313871\pi$
$227$	16.6329	1.10397	0.551983	+	0.833855i	$0.313871\pi$
$228$	−21.3392	−1.41323
$229$	1.37325	0.0907468	0.0453734	−	0.998970i	$-0.485552\pi$
$229$	1.37325	0.0907468	0.0453734	+	0.998970i	$0.485552\pi$
$230$	0	0
$231$	−4.24908	−0.279569
$232$	0.380661	0.0249916
$233$	1.66320	0.108960	0.0544800	−	0.998515i	$-0.482650\pi$
$233$	1.66320	0.108960	0.0544800	+	0.998515i	$0.482650\pi$
$234$	9.00661	0.588780
$235$	0	0
$236$	24.2581	1.57907
$237$	−35.2582	−2.29027
$238$	0	0
$239$	−18.0943	−1.17042	−0.585212	−	0.810880i	$-0.698989\pi$
$239$	−18.0943	−1.17042	−0.585212	+	0.810880i	$0.698989\pi$
$240$	0	0
$241$	7.01381	0.451799	0.225900	−	0.974151i	$-0.427468\pi$
$241$	7.01381	0.451799	0.225900	+	0.974151i	$0.427468\pi$
$242$	−6.01528	−0.386677
$243$	22.3521	1.43389
$244$	−5.76709	−0.369200
$245$	0	0
$246$	−5.43809	−0.346720
$247$	−25.0029	−1.59090
$248$	7.00396	0.444752
$249$	34.2186	2.16851
$250$	0	0
$251$	−5.17763	−0.326809	−0.163405	−	0.986559i	$-0.552248\pi$
$251$	−5.17763	−0.326809	−0.163405	+	0.986559i	$0.552248\pi$
$252$	−22.2975	−1.40461
$253$	−3.38697	−0.212937
$254$	−7.56096	−0.474417
$255$	0	0
$256$	−3.38998	−0.211874
$257$	−2.70058	−0.168457	−0.0842287	−	0.996446i	$-0.526843\pi$
$257$	−2.70058	−0.168457	−0.0842287	+	0.996446i	$0.526843\pi$
$258$	7.16557	0.446109
$259$	15.3980	0.956783
$260$	0	0
$261$	−0.606212	−0.0375236
$262$	7.51545	0.464306
$263$	−24.3885	−1.50386	−0.751931	−	0.659242i	$-0.770877\pi$
$263$	−24.3885	−1.50386	−0.751931	+	0.659242i	$0.770877\pi$
$264$	2.15981	0.132927
$265$	0	0
$266$	−11.3063	−0.693231
$267$	−37.8453	−2.31610
$268$	−12.2034	−0.745442
$269$	−7.65414	−0.466681	−0.233340	−	0.972395i	$-0.574966\pi$
$269$	−7.65414	−0.466681	−0.233340	+	0.972395i	$0.574966\pi$
$270$	0	0
$271$	0.0146097	0.000887478 0	0.000443739	−	1.00000i	$-0.499859\pi$
$271$	0.0146097	0.000887478 0	0.000443739		1.00000i	$0.499859\pi$
$272$	0	0
$273$	−50.1163	−3.03318
$274$	4.76683	0.287974
$275$	0	0
$276$	−34.0945	−2.05225
$277$	−2.43015	−0.146014	−0.0730068	−	0.997331i	$-0.523259\pi$
$277$	−2.43015	−0.146014	−0.0730068	+	0.997331i	$0.523259\pi$
$278$	7.29083	0.437275
$279$	−11.1540	−0.667772
$280$	0	0
$281$	7.50385	0.447642	0.223821	−	0.974630i	$-0.428147\pi$
$281$	7.50385	0.447642	0.223821	+	0.974630i	$0.428147\pi$
$282$	9.13273	0.543846
$283$	−8.59847	−0.511126	−0.255563	−	0.966792i	$-0.582261\pi$
$283$	−8.59847	−0.511126	−0.255563	+	0.966792i	$0.582261\pi$
$284$	12.6636	0.751448
$285$	0	0
$286$	1.15943	0.0685583
$287$	15.7744	0.931131
$288$	17.4750	1.02972
$289$	0	0
$290$	0	0
$291$	25.4323	1.49087
$292$	−6.06362	−0.354847
$293$	−20.6490	−1.20633	−0.603164	−	0.797617i	$-0.706094\pi$
$293$	−20.6490	−1.20633	−0.603164	+	0.797617i	$0.706094\pi$
$294$	−12.9231	−0.753691
$295$	0	0
$296$	−7.82682	−0.454925
$297$	−0.281081	−0.0163100
$298$	−4.58914	−0.265842
$299$	−39.9480	−2.31025
$300$	0	0
$301$	−20.7853	−1.19805
$302$	3.52485	0.202832
$303$	0.851981	0.0489450
$304$	−11.3013	−0.648172
$305$	0	0
$306$	0	0
$307$	−29.4692	−1.68190	−0.840949	−	0.541115i	$-0.818002\pi$
$307$	−29.4692	−1.68190	−0.840949	+	0.541115i	$0.818002\pi$
$308$	−2.87036	−0.163554
$309$	−1.72284	−0.0980092
$310$	0	0
$311$	4.29068	0.243302	0.121651	−	0.992573i	$-0.461181\pi$
$311$	4.29068	0.243302	0.121651	+	0.992573i	$0.461181\pi$
$312$	25.4742	1.44219
$313$	10.5224	0.594761	0.297381	−	0.954759i	$-0.403887\pi$
$313$	10.5224	0.594761	0.297381	+	0.954759i	$0.403887\pi$
$314$	−12.0845	−0.681968
$315$	0	0
$316$	−23.8178	−1.33986
$317$	−15.2781	−0.858105	−0.429052	−	0.903280i	$-0.641152\pi$
$317$	−15.2781	−0.858105	−0.429052	+	0.903280i	$0.641152\pi$
$318$	1.21269	0.0680044
$319$	−0.0780381	−0.00436929
$320$	0	0
$321$	−24.2439	−1.35316
$322$	−18.0644	−1.00669
$323$	0	0
$324$	13.7450	0.763610
$325$	0	0
$326$	−7.54190	−0.417707
$327$	−26.1590	−1.44660
$328$	−8.01815	−0.442728
$329$	−26.4915	−1.46052
$330$	0	0
$331$	−12.7193	−0.699114	−0.349557	−	0.936915i	$-0.613668\pi$
$331$	−12.7193	−0.699114	−0.349557	+	0.936915i	$0.613668\pi$
$332$	23.1156	1.26863
$333$	12.4644	0.683046
$334$	−8.89144	−0.486518
$335$	0	0
$336$	−22.6525	−1.23579
$337$	1.89743	0.103360	0.0516798	−	0.998664i	$-0.483542\pi$
$337$	1.89743	0.103360	0.0516798	+	0.998664i	$0.483542\pi$
$338$	6.44986	0.350826
$339$	−22.5697	−1.22582
$340$	0	0
$341$	−1.43586	−0.0777561
$342$	−9.15224	−0.494896
$343$	9.23521	0.498654
$344$	10.5652	0.569638
$345$	0	0
$346$	−1.40056	−0.0752944
$347$	−25.5714	−1.37275	−0.686373	−	0.727250i	$-0.740798\pi$
$347$	−25.5714	−1.37275	−0.686373	+	0.727250i	$0.740798\pi$
$348$	−0.785560	−0.0421104
$349$	15.3328	0.820746	0.410373	−	0.911918i	$-0.365399\pi$
$349$	15.3328	0.820746	0.410373	+	0.911918i	$0.365399\pi$
$350$	0	0
$351$	−3.31525	−0.176955
$352$	2.24957	0.119902
$353$	−20.0627	−1.06783	−0.533915	−	0.845538i	$-0.679280\pi$
$353$	−20.0627	−1.06783	−0.533915	+	0.845538i	$0.679280\pi$
$354$	19.9580	1.06075
$355$	0	0
$356$	−25.5655	−1.35497
$357$	0	0
$358$	4.14490	0.219065
$359$	−16.7709	−0.885137	−0.442568	−	0.896735i	$-0.645933\pi$
$359$	−16.7709	−0.885137	−0.442568	+	0.896735i	$0.645933\pi$
$360$	0	0
$361$	6.40719	0.337221
$362$	9.06099	0.476235
$363$	27.0945	1.42210
$364$	−33.8549	−1.77448
$365$	0	0
$366$	−4.74479	−0.248014
$367$	9.66088	0.504294	0.252147	−	0.967689i	$-0.418863\pi$
$367$	9.66088	0.504294	0.252147	+	0.967689i	$0.418863\pi$
$368$	−18.0564	−0.941256
$369$	12.7691	0.664733
$370$	0	0
$371$	−3.51768	−0.182629
$372$	−14.4539	−0.749399
$373$	−8.84764	−0.458113	−0.229057	−	0.973413i	$-0.573564\pi$
$373$	−8.84764	−0.458113	−0.229057	+	0.973413i	$0.573564\pi$
$374$	0	0
$375$	0	0
$376$	13.4657	0.694440
$377$	−0.920430	−0.0474046
$378$	−1.49915	−0.0771078
$379$	−9.05019	−0.464877	−0.232438	−	0.972611i	$-0.574670\pi$
$379$	−9.05019	−0.464877	−0.232438	+	0.972611i	$0.574670\pi$
$380$	0	0
$381$	34.0568	1.74478
$382$	2.72580	0.139464
$383$	−16.4906	−0.842631	−0.421315	−	0.906914i	$-0.638431\pi$
$383$	−16.4906	−0.842631	−0.421315	+	0.906914i	$0.638431\pi$
$384$	28.8839	1.47398
$385$	0	0
$386$	3.49615	0.177949
$387$	−16.8254	−0.855282
$388$	17.1802	0.872192
$389$	−26.3924	−1.33815	−0.669074	−	0.743196i	$-0.733309\pi$
$389$	−26.3924	−1.33815	−0.669074	+	0.743196i	$0.733309\pi$
$390$	0	0
$391$	0	0
$392$	−19.0544	−0.962392
$393$	−33.8518	−1.70760
$394$	−12.6785	−0.638736
$395$	0	0
$396$	−2.32352	−0.116761
$397$	−0.897712	−0.0450549	−0.0225274	−	0.999746i	$-0.507171\pi$
$397$	−0.897712	−0.0450549	−0.0225274	+	0.999746i	$0.507171\pi$
$398$	−13.0054	−0.651900
$399$	50.9266	2.54952
$400$	0	0
$401$	−15.2264	−0.760371	−0.380186	−	0.924910i	$-0.624140\pi$
$401$	−15.2264	−0.760371	−0.380186	+	0.924910i	$0.624140\pi$
$402$	−10.0402	−0.500758
$403$	−16.9354	−0.843614
$404$	0.575536	0.0286340
$405$	0	0
$406$	−0.416216	−0.0206565
$407$	1.60455	0.0795346
$408$	0	0
$409$	5.69707	0.281702	0.140851	−	0.990031i	$-0.455016\pi$
$409$	5.69707	0.281702	0.140851	+	0.990031i	$0.455016\pi$
$410$	0	0
$411$	−21.4712	−1.05909
$412$	−1.16383	−0.0573377
$413$	−57.8925	−2.84870
$414$	−14.6229	−0.718674
$415$	0	0
$416$	26.5328	1.30088
$417$	−32.8400	−1.60818
$418$	−1.17817	−0.0576263
$419$	−2.42262	−0.118353	−0.0591764	−	0.998248i	$-0.518847\pi$
$419$	−2.42262	−0.118353	−0.0591764	+	0.998248i	$0.518847\pi$
$420$	0	0
$421$	35.5336	1.73180	0.865901	−	0.500215i	$-0.166746\pi$
$421$	35.5336	1.73180	0.865901	+	0.500215i	$0.166746\pi$
$422$	−1.00424	−0.0488854
$423$	−21.4445	−1.04266
$424$	1.78804	0.0868351
$425$	0	0
$426$	10.4188	0.504793
$427$	13.7633	0.666053
$428$	−16.3774	−0.791630
$429$	−5.22239	−0.252139
$430$	0	0
$431$	11.1438	0.536778	0.268389	−	0.963311i	$-0.413509\pi$
$431$	11.1438	0.536778	0.268389	+	0.963311i	$0.413509\pi$
$432$	−1.49849	−0.0720959
$433$	−24.2605	−1.16589	−0.582944	−	0.812513i	$-0.698099\pi$
$433$	−24.2605	−1.16589	−0.582944	+	0.812513i	$0.698099\pi$
$434$	−7.65816	−0.367603
$435$	0	0
$436$	−17.6711	−0.846292
$437$	40.5940	1.94187
$438$	−4.98876	−0.238372
$439$	18.9055	0.902312	0.451156	−	0.892445i	$-0.351012\pi$
$439$	18.9055	0.902312	0.451156	+	0.892445i	$0.351012\pi$
$440$	0	0
$441$	30.3446	1.44498
$442$	0	0
$443$	8.80318	0.418252	0.209126	−	0.977889i	$-0.432938\pi$
$443$	8.80318	0.418252	0.209126	+	0.977889i	$0.432938\pi$
$444$	16.1520	0.766540
$445$	0	0
$446$	4.94549	0.234176
$447$	20.6708	0.977697
$448$	−6.09932	−0.288166
$449$	−37.8204	−1.78486	−0.892428	−	0.451190i	$-0.851000\pi$
$449$	−37.8204	−1.78486	−0.892428	+	0.451190i	$0.851000\pi$
$450$	0	0
$451$	1.64377	0.0774023
$452$	−15.2464	−0.717133
$453$	−15.8769	−0.745964
$454$	9.24425	0.433854
$455$	0	0
$456$	−25.8861	−1.21223
$457$	−1.10481	−0.0516807	−0.0258404	−	0.999666i	$-0.508226\pi$
$457$	−1.10481	−0.0516807	−0.0258404	+	0.999666i	$0.508226\pi$
$458$	0.763223	0.0356631
$459$	0	0
$460$	0	0
$461$	5.67143	0.264145	0.132072	−	0.991240i	$-0.457837\pi$
$461$	5.67143	0.264145	0.132072	+	0.991240i	$0.457837\pi$
$462$	−2.36155	−0.109869
$463$	−8.91271	−0.414209	−0.207104	−	0.978319i	$-0.566404\pi$
$463$	−8.91271	−0.414209	−0.207104	+	0.978319i	$0.566404\pi$
$464$	−0.416032	−0.0193138
$465$	0	0
$466$	0.924374	0.0428208
$467$	10.4339	0.482822	0.241411	−	0.970423i	$-0.422390\pi$
$467$	10.4339	0.482822	0.241411	+	0.970423i	$0.422390\pi$
$468$	−27.4050	−1.26680
$469$	29.1237	1.34481
$470$	0	0
$471$	54.4321	2.50810
$472$	29.4269	1.35448
$473$	−2.16594	−0.0995901
$474$	−19.5958	−0.900064
$475$	0	0
$476$	0	0
$477$	−2.84751	−0.130378
$478$	−10.0565	−0.459972
$479$	−33.0306	−1.50921	−0.754603	−	0.656181i	$-0.772171\pi$
$479$	−33.0306	−1.50921	−0.754603	+	0.656181i	$0.772171\pi$
$480$	0	0
$481$	18.9251	0.862910
$482$	3.89814	0.177555
$483$	81.3673	3.70234
$484$	18.3031	0.831959
$485$	0	0
$486$	12.4228	0.563512
$487$	−25.3479	−1.14862	−0.574312	−	0.818636i	$-0.694730\pi$
$487$	−25.3479	−1.14862	−0.574312	+	0.818636i	$0.694730\pi$
$488$	−6.99592	−0.316690
$489$	33.9709	1.53622
$490$	0	0
$491$	26.0273	1.17460	0.587298	−	0.809371i	$-0.300192\pi$
$491$	26.0273	1.17460	0.587298	+	0.809371i	$0.300192\pi$
$492$	16.5468	0.745989
$493$	0	0
$494$	−13.8961	−0.625216
$495$	0	0
$496$	−7.65478	−0.343710
$497$	−30.2221	−1.35565
$498$	19.0180	0.852216
$499$	18.2209	0.815679	0.407840	−	0.913054i	$-0.366282\pi$
$499$	18.2209	0.815679	0.407840	+	0.913054i	$0.366282\pi$
$500$	0	0
$501$	40.0496	1.78929
$502$	−2.87762	−0.128435
$503$	20.0595	0.894407	0.447204	−	0.894432i	$-0.352420\pi$
$503$	20.0595	0.894407	0.447204	+	0.894432i	$0.352420\pi$
$504$	−27.0485	−1.20484
$505$	0	0
$506$	−1.88241	−0.0836833
$507$	−29.0520	−1.29025
$508$	23.0062	1.02074
$509$	−8.75911	−0.388241	−0.194120	−	0.980978i	$-0.562185\pi$
$509$	−8.75911	−0.388241	−0.194120	+	0.980978i	$0.562185\pi$
$510$	0	0
$511$	14.4710	0.640159
$512$	21.1917	0.936551
$513$	3.36886	0.148739
$514$	−1.50093	−0.0662030
$515$	0	0
$516$	−21.8032	−0.959831
$517$	−2.76056	−0.121409
$518$	8.55788	0.376011
$519$	6.30852	0.276913
$520$	0	0
$521$	−20.5856	−0.901870	−0.450935	−	0.892557i	$-0.648909\pi$
$521$	−20.5856	−0.901870	−0.450935	+	0.892557i	$0.648909\pi$
$522$	−0.336921	−0.0147466
$523$	36.8267	1.61032	0.805159	−	0.593059i	$-0.202080\pi$
$523$	36.8267	1.61032	0.805159	+	0.593059i	$0.202080\pi$
$524$	−22.8678	−0.998984
$525$	0	0
$526$	−13.5547	−0.591011
$527$	0	0
$528$	−2.36051	−0.102728
$529$	41.8584	1.81993
$530$	0	0
$531$	−46.8631	−2.03368
$532$	34.4023	1.49153
$533$	19.3877	0.839775
$534$	−21.0337	−0.910215
$535$	0	0
$536$	−14.8037	−0.639421
$537$	−18.6698	−0.805663
$538$	−4.25401	−0.183404
$539$	3.90628	0.168255
$540$	0	0
$541$	−39.3467	−1.69165	−0.845823	−	0.533464i	$-0.820890\pi$
$541$	−39.3467	−1.69165	−0.845823	+	0.533464i	$0.820890\pi$
$542$	0.00811979	0.000348775 0
$543$	−40.8133	−1.75147
$544$	0	0
$545$	0	0
$546$	−27.8536	−1.19203
$547$	16.4348	0.702702	0.351351	−	0.936244i	$-0.385722\pi$
$547$	16.4348	0.702702	0.351351	+	0.936244i	$0.385722\pi$
$548$	−14.5043	−0.619595
$549$	11.1412	0.475494
$550$	0	0
$551$	0.935313	0.0398457
$552$	−41.3592	−1.76036
$553$	56.8419	2.41716
$554$	−1.35063	−0.0573827
$555$	0	0
$556$	−22.1843	−0.940824
$557$	−8.41654	−0.356620	−0.178310	−	0.983974i	$-0.557063\pi$
$557$	−8.41654	−0.356620	−0.178310	+	0.983974i	$0.557063\pi$
$558$	−6.19916	−0.262431
$559$	−25.5465	−1.08050
$560$	0	0
$561$	0	0
$562$	4.17049	0.175921
$563$	−12.6973	−0.535126	−0.267563	−	0.963540i	$-0.586218\pi$
$563$	−12.6973	−0.535126	−0.267563	+	0.963540i	$0.586218\pi$
$564$	−27.7888	−1.17012
$565$	0	0
$566$	−4.77886	−0.200870
$567$	−32.8027	−1.37759
$568$	15.3619	0.644573
$569$	16.8526	0.706499	0.353249	−	0.935529i	$-0.385077\pi$
$569$	16.8526	0.706499	0.353249	+	0.935529i	$0.385077\pi$
$570$	0	0
$571$	4.03344	0.168794	0.0843972	−	0.996432i	$-0.473104\pi$
$571$	4.03344	0.168794	0.0843972	+	0.996432i	$0.473104\pi$
$572$	−3.52786	−0.147507
$573$	−12.2778	−0.512913
$574$	8.76707	0.365931
$575$	0	0
$576$	−4.93731	−0.205721
$577$	12.3496	0.514119	0.257060	−	0.966396i	$-0.417246\pi$
$577$	12.3496	0.514119	0.257060	+	0.966396i	$0.417246\pi$
$578$	0	0
$579$	−15.7477	−0.654451
$580$	0	0
$581$	−55.1659	−2.28867
$582$	14.1347	0.585904
$583$	−0.366561	−0.0151814
$584$	−7.35564	−0.304378
$585$	0	0
$586$	−11.4763	−0.474082
$587$	36.6844	1.51413	0.757064	−	0.653341i	$-0.226633\pi$
$587$	36.6844	1.51413	0.757064	+	0.653341i	$0.226633\pi$
$588$	39.3220	1.62161
$589$	17.2093	0.709095
$590$	0	0
$591$	57.1078	2.34910
$592$	8.55410	0.351571
$593$	17.1445	0.704041	0.352021	−	0.935992i	$-0.385495\pi$
$593$	17.1445	0.704041	0.352021	+	0.935992i	$0.385495\pi$
$594$	−0.156219	−0.00640976
$595$	0	0
$596$	13.9637	0.571975
$597$	58.5799	2.39752
$598$	−22.2023	−0.907920
$599$	19.1061	0.780654	0.390327	−	0.920676i	$-0.372362\pi$
$599$	19.1061	0.780654	0.390327	+	0.920676i	$0.372362\pi$
$600$	0	0
$601$	3.60770	0.147161	0.0735807	−	0.997289i	$-0.476557\pi$
$601$	3.60770	0.147161	0.0735807	+	0.997289i	$0.476557\pi$
$602$	−11.5521	−0.470827
$603$	23.5752	0.960057
$604$	−10.7253	−0.436406
$605$	0	0
$606$	0.473514	0.0192352
$607$	24.0681	0.976893	0.488447	−	0.872594i	$-0.337564\pi$
$607$	24.0681	0.976893	0.488447	+	0.872594i	$0.337564\pi$
$608$	−26.9618	−1.09345
$609$	1.87476	0.0759690
$610$	0	0
$611$	−32.5597	−1.31723
$612$	0	0
$613$	−1.37144	−0.0553919	−0.0276959	−	0.999616i	$-0.508817\pi$
$613$	−1.37144	−0.0553919	−0.0276959	+	0.999616i	$0.508817\pi$
$614$	−16.3784	−0.660978
$615$	0	0
$616$	−3.48197	−0.140293
$617$	−9.37017	−0.377229	−0.188614	−	0.982051i	$-0.560400\pi$
$617$	−9.37017	−0.377229	−0.188614	+	0.982051i	$0.560400\pi$
$618$	−0.957522	−0.0385172
$619$	−26.7267	−1.07424	−0.537118	−	0.843507i	$-0.680487\pi$
$619$	−26.7267	−1.07424	−0.537118	+	0.843507i	$0.680487\pi$
$620$	0	0
$621$	5.38254	0.215994
$622$	2.38467	0.0956168
$623$	61.0127	2.44442
$624$	−27.8413	−1.11454
$625$	0	0
$626$	5.84814	0.233739
$627$	5.30683	0.211934
$628$	36.7703	1.46730
$629$	0	0
$630$	0	0
$631$	−0.0544273	−0.00216672	−0.00108336	−	0.999999i	$-0.500345\pi$
$631$	−0.0544273	−0.00216672	−0.00108336	+	0.999999i	$0.500345\pi$
$632$	−28.8929	−1.14930
$633$	4.52337	0.179788
$634$	−8.49127	−0.337231
$635$	0	0
$636$	−3.68994	−0.146316
$637$	46.0731	1.82548
$638$	−0.0433720	−0.00171711
$639$	−24.4643	−0.967793
$640$	0	0
$641$	−35.0538	−1.38454	−0.692272	−	0.721637i	$-0.743390\pi$
$641$	−35.0538	−1.38454	−0.692272	+	0.721637i	$0.743390\pi$
$642$	−13.4742	−0.531786
$643$	31.7691	1.25285	0.626425	−	0.779481i	$-0.284517\pi$
$643$	31.7691	1.25285	0.626425	+	0.779481i	$0.284517\pi$
$644$	54.9658	2.16596
$645$	0	0
$646$	0	0
$647$	−47.2437	−1.85734	−0.928671	−	0.370904i	$-0.879048\pi$
$647$	−47.2437	−1.85734	−0.928671	+	0.370904i	$0.879048\pi$
$648$	16.6737	0.655005
$649$	−6.03271	−0.236805
$650$	0	0
$651$	34.4946	1.35195
$652$	22.9483	0.898723
$653$	−18.2590	−0.714530	−0.357265	−	0.934003i	$-0.616291\pi$
$653$	−18.2590	−0.714530	−0.357265	+	0.934003i	$0.616291\pi$
$654$	−14.5386	−0.568506
$655$	0	0
$656$	8.76320	0.342146
$657$	11.7140	0.457008
$658$	−14.7234	−0.573979
$659$	31.0240	1.20852	0.604261	−	0.796786i	$-0.293468\pi$
$659$	31.0240	1.20852	0.604261	+	0.796786i	$0.293468\pi$
$660$	0	0
$661$	−26.5079	−1.03104	−0.515519	−	0.856878i	$-0.672401\pi$
$661$	−26.5079	−1.03104	−0.515519	+	0.856878i	$0.672401\pi$
$662$	−7.06911	−0.274749
$663$	0	0
$664$	28.0409	1.08820
$665$	0	0
$666$	6.92747	0.268434
$667$	1.49438	0.0578627
$668$	27.0546	1.04677
$669$	−22.2759	−0.861237
$670$	0	0
$671$	1.43421	0.0553671
$672$	−54.0428	−2.08475
$673$	−42.1664	−1.62539	−0.812697	−	0.582687i	$-0.802001\pi$
$673$	−42.1664	−1.62539	−0.812697	+	0.582687i	$0.802001\pi$
$674$	1.05455	0.0406199
$675$	0	0
$676$	−19.6254	−0.754824
$677$	−9.02111	−0.346710	−0.173355	−	0.984859i	$-0.555461\pi$
$677$	−9.02111	−0.346710	−0.173355	+	0.984859i	$0.555461\pi$
$678$	−12.5438	−0.481741
$679$	−41.0009	−1.57347
$680$	0	0
$681$	−41.6388	−1.59560
$682$	−0.798022	−0.0305578
$683$	32.1144	1.22883	0.614413	−	0.788985i	$-0.289393\pi$
$683$	32.1144	1.22883	0.614413	+	0.788985i	$0.289393\pi$
$684$	27.8481	1.06480
$685$	0	0
$686$	5.13274	0.195969
$687$	−3.43778	−0.131159
$688$	−11.5470	−0.440223
$689$	−4.32345	−0.164711
$690$	0	0
$691$	−8.79278	−0.334493	−0.167247	−	0.985915i	$-0.553488\pi$
$691$	−8.79278	−0.334493	−0.167247	+	0.985915i	$0.553488\pi$
$692$	4.26157	0.162001
$693$	5.54513	0.210642
$694$	−14.2121	−0.539483
$695$	0	0
$696$	−0.952944	−0.0361212
$697$	0	0
$698$	8.52166	0.322549
$699$	−4.16365	−0.157484
$700$	0	0
$701$	19.0206	0.718399	0.359199	−	0.933261i	$-0.383050\pi$
$701$	19.0206	0.718399	0.359199	+	0.933261i	$0.383050\pi$
$702$	−1.84255	−0.0695425
$703$	−19.2311	−0.725314
$704$	−0.635583	−0.0239544
$705$	0	0
$706$	−11.1504	−0.419652
$707$	−1.37353	−0.0516570
$708$	−60.7275	−2.28228
$709$	−21.9528	−0.824454	−0.412227	−	0.911081i	$-0.635249\pi$
$709$	−21.9528	−0.824454	−0.412227	+	0.911081i	$0.635249\pi$
$710$	0	0
$711$	46.0126	1.72561
$712$	−31.0129	−1.16226
$713$	27.4959	1.02973
$714$	0	0
$715$	0	0
$716$	−12.6120	−0.471331
$717$	45.2972	1.69166
$718$	−9.32095	−0.347855
$719$	36.9733	1.37887	0.689436	−	0.724346i	$-0.257858\pi$
$719$	36.9733	1.37887	0.689436	+	0.724346i	$0.257858\pi$
$720$	0	0
$721$	2.77750	0.103440
$722$	3.56099	0.132526
$723$	−17.5583	−0.653001
$724$	−27.5705	−1.02465
$725$	0	0
$726$	15.0586	0.558877
$727$	−44.1513	−1.63748	−0.818741	−	0.574163i	$-0.805328\pi$
$727$	−44.1513	−1.63748	−0.818741	+	0.574163i	$0.805328\pi$
$728$	−41.0686	−1.52210
$729$	−31.5727	−1.16936
$730$	0	0
$731$	0	0
$732$	14.4373	0.533618
$733$	5.30089	0.195793	0.0978965	−	0.995197i	$-0.468789\pi$
$733$	5.30089	0.195793	0.0978965	+	0.995197i	$0.468789\pi$
$734$	5.36932	0.198185
$735$	0	0
$736$	−43.0779	−1.58787
$737$	3.03485	0.111790
$738$	7.09681	0.261237
$739$	2.64136	0.0971642	0.0485821	−	0.998819i	$-0.484530\pi$
$739$	2.64136	0.0971642	0.0485821	+	0.998819i	$0.484530\pi$
$740$	0	0
$741$	62.5921	2.29938
$742$	−1.95506	−0.0717723
$743$	11.7618	0.431497	0.215748	−	0.976449i	$-0.430781\pi$
$743$	11.7618	0.431497	0.215748	+	0.976449i	$0.430781\pi$
$744$	−17.5337	−0.642815
$745$	0	0
$746$	−4.91734	−0.180037
$747$	−44.6559	−1.63387
$748$	0	0
$749$	39.0850	1.42814
$750$	0	0
$751$	15.3736	0.560990	0.280495	−	0.959855i	$-0.409501\pi$
$751$	15.3736	0.560990	0.280495	+	0.959855i	$0.409501\pi$
$752$	−14.7169	−0.536671
$753$	12.9617	0.472349
$754$	−0.511556	−0.0186298
$755$	0	0
$756$	4.56156	0.165902
$757$	6.04560	0.219731	0.109866	−	0.993946i	$-0.464958\pi$
$757$	6.04560	0.219731	0.109866	+	0.993946i	$0.464958\pi$
$758$	−5.02991	−0.182695
$759$	8.47891	0.307765
$760$	0	0
$761$	−13.0506	−0.473084	−0.236542	−	0.971621i	$-0.576014\pi$
$761$	−13.0506	−0.473084	−0.236542	+	0.971621i	$0.576014\pi$
$762$	18.9281	0.685691
$763$	42.1725	1.52675
$764$	−8.29399	−0.300066
$765$	0	0
$766$	−9.16515	−0.331150
$767$	−71.1536	−2.56921
$768$	8.48645	0.306228
$769$	−33.3051	−1.20101	−0.600507	−	0.799620i	$-0.705034\pi$
$769$	−33.3051	−1.20101	−0.600507	+	0.799620i	$0.705034\pi$
$770$	0	0
$771$	6.76061	0.243477
$772$	−10.6380	−0.382869
$773$	10.2351	0.368131	0.184065	−	0.982914i	$-0.441074\pi$
$773$	10.2351	0.368131	0.184065	+	0.982914i	$0.441074\pi$
$774$	−9.35121	−0.336122
$775$	0	0
$776$	20.8409	0.748144
$777$	−38.5472	−1.38287
$778$	−14.6684	−0.525886
$779$	−19.7012	−0.705869
$780$	0	0
$781$	−3.14930	−0.112691
$782$	0	0
$783$	0.124017	0.00443202
$784$	20.8249	0.743748
$785$	0	0
$786$	−18.8141	−0.671078
$787$	2.89479	0.103188	0.0515941	−	0.998668i	$-0.483570\pi$
$787$	2.89479	0.103188	0.0515941	+	0.998668i	$0.483570\pi$
$788$	38.5779	1.37428
$789$	61.0541	2.17358
$790$	0	0
$791$	36.3860	1.29374
$792$	−2.81860	−0.100155
$793$	16.9160	0.600705
$794$	−0.498930	−0.0177064
$795$	0	0
$796$	39.5723	1.40260
$797$	−17.3882	−0.615923	−0.307961	−	0.951399i	$-0.599647\pi$
$797$	−17.3882	−0.615923	−0.307961	+	0.951399i	$0.599647\pi$
$798$	28.3040	1.00195
$799$	0	0
$800$	0	0
$801$	49.3889	1.74507
$802$	−8.46254	−0.298823
$803$	1.50796	0.0532146
$804$	30.5499	1.07741
$805$	0	0
$806$	−9.41237	−0.331537
$807$	19.1613	0.674510
$808$	0.698169	0.0245615
$809$	−37.8663	−1.33131	−0.665654	−	0.746261i	$-0.731847\pi$
$809$	−37.8663	−1.33131	−0.665654	+	0.746261i	$0.731847\pi$
$810$	0	0
$811$	−4.46329	−0.156727	−0.0783636	−	0.996925i	$-0.524970\pi$
$811$	−4.46329	−0.156727	−0.0783636	+	0.996925i	$0.524970\pi$
$812$	1.26645	0.0444436
$813$	−0.0365739	−0.00128270
$814$	0.891777	0.0312568
$815$	0	0
$816$	0	0
$817$	25.9596	0.908210
$818$	3.16632	0.110708
$819$	65.4028	2.28536
$820$	0	0
$821$	3.36602	0.117475	0.0587375	−	0.998273i	$-0.481293\pi$
$821$	3.36602	0.117475	0.0587375	+	0.998273i	$0.481293\pi$
$822$	−11.9332	−0.416219
$823$	−12.1803	−0.424579	−0.212289	−	0.977207i	$-0.568092\pi$
$823$	−12.1803	−0.424579	−0.212289	+	0.977207i	$0.568092\pi$
$824$	−1.41181	−0.0491828
$825$	0	0
$826$	−32.1755	−1.11953
$827$	18.4858	0.642816	0.321408	−	0.946941i	$-0.395844\pi$
$827$	18.4858	0.642816	0.321408	+	0.946941i	$0.395844\pi$
$828$	44.4939	1.54627
$829$	−27.8002	−0.965540	−0.482770	−	0.875747i	$-0.660369\pi$
$829$	−27.8002	−0.965540	−0.482770	+	0.875747i	$0.660369\pi$
$830$	0	0
$831$	6.08362	0.211038
$832$	−7.49646	−0.259893
$833$	0	0
$834$	−18.2518	−0.632009
$835$	0	0
$836$	3.58491	0.123987
$837$	2.28186	0.0788724
$838$	−1.34644	−0.0465121
$839$	−39.9107	−1.37787	−0.688935	−	0.724823i	$-0.741922\pi$
$839$	−39.9107	−1.37787	−0.688935	+	0.724823i	$0.741922\pi$
$840$	0	0
$841$	−28.9656	−0.998813
$842$	19.7489	0.680591
$843$	−18.7851	−0.646993
$844$	3.05565	0.105180
$845$	0	0
$846$	−11.9184	−0.409763
$847$	−43.6808	−1.50089
$848$	−1.95419	−0.0671072
$849$	21.5254	0.738749
$850$	0	0
$851$	−30.7262	−1.05328
$852$	−31.7020	−1.08609
$853$	−41.6170	−1.42494	−0.712469	−	0.701703i	$-0.752423\pi$
$853$	−41.6170	−1.42494	−0.712469	+	0.701703i	$0.752423\pi$
$854$	7.64937	0.261756
$855$	0	0
$856$	−19.8670	−0.679040
$857$	−10.6131	−0.362538	−0.181269	−	0.983434i	$-0.558020\pi$
$857$	−10.6131	−0.362538	−0.181269	+	0.983434i	$0.558020\pi$
$858$	−2.90250	−0.0990897
$859$	45.3309	1.54667	0.773335	−	0.633998i	$-0.218587\pi$
$859$	45.3309	1.54667	0.773335	+	0.633998i	$0.218587\pi$
$860$	0	0
$861$	−39.4894	−1.34580
$862$	6.19350	0.210952
$863$	−8.98488	−0.305849	−0.152925	−	0.988238i	$-0.548869\pi$
$863$	−8.98488	−0.305849	−0.152925	+	0.988238i	$0.548869\pi$
$864$	−3.57499	−0.121624
$865$	0	0
$866$	−13.4835	−0.458189
$867$	0	0
$868$	23.3020	0.790921
$869$	5.92323	0.200932
$870$	0	0
$871$	35.7949	1.21287
$872$	−21.4364	−0.725927
$873$	−33.1896	−1.12330
$874$	22.5613	0.763148
$875$	0	0
$876$	15.1796	0.512872
$877$	11.7160	0.395622	0.197811	−	0.980240i	$-0.436617\pi$
$877$	11.7160	0.395622	0.197811	+	0.980240i	$0.436617\pi$
$878$	10.5073	0.354605
$879$	51.6926	1.74355
$880$	0	0
$881$	4.31806	0.145479	0.0727396	−	0.997351i	$-0.476826\pi$
$881$	4.31806	0.145479	0.0727396	+	0.997351i	$0.476826\pi$
$882$	16.8649	0.567871
$883$	−29.8534	−1.00465	−0.502323	−	0.864680i	$-0.667521\pi$
$883$	−29.8534	−1.00465	−0.502323	+	0.864680i	$0.667521\pi$
$884$	0	0
$885$	0	0
$886$	4.89263	0.164371
$887$	11.4344	0.383928	0.191964	−	0.981402i	$-0.438514\pi$
$887$	11.4344	0.383928	0.191964	+	0.981402i	$0.438514\pi$
$888$	19.5936	0.657518
$889$	−54.9050	−1.84145
$890$	0	0
$891$	−3.41822	−0.114515
$892$	−15.0480	−0.503844
$893$	33.0862	1.10719
$894$	11.4884	0.384231
$895$	0	0
$896$	−46.5655	−1.55564
$897$	100.006	3.33909
$898$	−21.0198	−0.701441
$899$	0.633523	0.0211292
$900$	0	0
$901$	0	0
$902$	0.913577	0.0304188
$903$	52.0338	1.73158
$904$	−18.4951	−0.615138
$905$	0	0
$906$	−8.82408	−0.293161
$907$	−34.2774	−1.13816	−0.569081	−	0.822282i	$-0.692701\pi$
$907$	−34.2774	−1.13816	−0.569081	+	0.822282i	$0.692701\pi$
$908$	−28.1281	−0.933463
$909$	−1.11185	−0.0368778
$910$	0	0
$911$	−24.2471	−0.803342	−0.401671	−	0.915784i	$-0.631570\pi$
$911$	−24.2471	−0.803342	−0.401671	+	0.915784i	$0.631570\pi$
$912$	28.2915	0.936825
$913$	−5.74858	−0.190250
$914$	−0.614030	−0.0203103
$915$	0	0
$916$	−2.32231	−0.0767313
$917$	54.5745	1.80221
$918$	0	0
$919$	−42.9958	−1.41830	−0.709151	−	0.705057i	$-0.750921\pi$
$919$	−42.9958	−1.41830	−0.709151	+	0.705057i	$0.750921\pi$
$920$	0	0
$921$	73.7731	2.43091
$922$	3.15207	0.103808
$923$	−37.1449	−1.22264
$924$	7.18565	0.236391
$925$	0	0
$926$	−4.95350	−0.162782
$927$	2.24835	0.0738454
$928$	−0.992544	−0.0325819
$929$	29.4737	0.967001	0.483500	−	0.875344i	$-0.339365\pi$
$929$	29.4737	0.967001	0.483500	+	0.875344i	$0.339365\pi$
$930$	0	0
$931$	−46.8181	−1.53440
$932$	−2.81265	−0.0921316
$933$	−10.7413	−0.351653
$934$	5.79894	0.189747
$935$	0	0
$936$	−33.2444	−1.08663
$937$	−22.9716	−0.750450	−0.375225	−	0.926934i	$-0.622435\pi$
$937$	−22.9716	−0.750450	−0.375225	+	0.926934i	$0.622435\pi$
$938$	16.1864	0.528504
$939$	−26.3417	−0.859629
$940$	0	0
$941$	6.91682	0.225482	0.112741	−	0.993624i	$-0.464037\pi$
$941$	6.91682	0.225482	0.112741	+	0.993624i	$0.464037\pi$
$942$	30.2523	0.985672
$943$	−31.4773	−1.02504
$944$	−32.1613	−1.04676
$945$	0	0
$946$	−1.20379	−0.0391385
$947$	39.4991	1.28355	0.641774	−	0.766894i	$-0.278199\pi$
$947$	39.4991	1.28355	0.641774	+	0.766894i	$0.278199\pi$
$948$	59.6254	1.93654
$949$	17.7858	0.577351
$950$	0	0
$951$	38.2471	1.24025
$952$	0	0
$953$	30.8702	0.999983	0.499992	−	0.866030i	$-0.333336\pi$
$953$	30.8702	0.999983	0.499992	+	0.866030i	$0.333336\pi$
$954$	−1.58259	−0.0512381
$955$	0	0
$956$	30.5995	0.989658
$957$	0.195360	0.00631509
$958$	−18.3577	−0.593112
$959$	34.6150	1.11778
$960$	0	0
$961$	−19.3435	−0.623984
$962$	10.5182	0.339120
$963$	31.6387	1.01954
$964$	−11.8611	−0.382021
$965$	0	0
$966$	45.2223	1.45500
$967$	−12.1454	−0.390569	−0.195285	−	0.980747i	$-0.562563\pi$
$967$	−12.1454	−0.390569	−0.195285	+	0.980747i	$0.562563\pi$
$968$	22.2030	0.713633
$969$	0	0
$970$	0	0
$971$	59.3265	1.90388	0.951939	−	0.306287i	$-0.0990867\pi$
$971$	59.3265	1.90388	0.951939	+	0.306287i	$0.0990867\pi$
$972$	−37.7998	−1.21243
$973$	52.9434	1.69729
$974$	−14.0879	−0.451404
$975$	0	0
$976$	7.64599	0.244742
$977$	2.28110	0.0729788	0.0364894	−	0.999334i	$-0.488382\pi$
$977$	2.28110	0.0729788	0.0364894	+	0.999334i	$0.488382\pi$
$978$	18.8804	0.603727
$979$	6.35786	0.203198
$980$	0	0
$981$	34.1380	1.08994
$982$	14.4654	0.461611
$983$	0.541983	0.0172866	0.00864328	−	0.999963i	$-0.497249\pi$
$983$	0.541983	0.0172866	0.00864328	+	0.999963i	$0.497249\pi$
$984$	20.0726	0.639890
$985$	0	0
$986$	0	0
$987$	66.3186	2.11094
$988$	42.2826	1.34519
$989$	41.4765	1.31888
$990$	0	0
$991$	−12.9204	−0.410430	−0.205215	−	0.978717i	$-0.565789\pi$
$991$	−12.9204	−0.410430	−0.205215	+	0.978717i	$0.565789\pi$
$992$	−18.2623	−0.579828
$993$	31.8413	1.01045
$994$	−16.7968	−0.532763
$995$	0	0
$996$	−57.8673	−1.83360
$997$	12.3411	0.390847	0.195424	−	0.980719i	$-0.437392\pi$
$997$	12.3411	0.390847	0.195424	+	0.980719i	$0.437392\pi$
$998$	10.1268	0.320558
$999$	−2.54994	−0.0806765

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bx.1.15		24
5.4	even	2		7225.2.a.cb.1.10		24
17.3	odd	16		425.2.m.c.26.4	✓	24
17.6	odd	16		425.2.m.c.376.4	yes	24
17.16	even	2	inner	7225.2.a.bx.1.16		24
85.3	even	16		425.2.n.e.349.3		24
85.23	even	16		425.2.n.d.274.4		24
85.37	even	16		425.2.n.d.349.4		24
85.54	odd	16		425.2.m.d.26.3	yes	24
85.57	even	16		425.2.n.e.274.3		24
85.74	odd	16		425.2.m.d.376.3	yes	24
85.84	even	2		7225.2.a.cb.1.9		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.m.c.26.4	✓	24	17.3	odd	16
425.2.m.c.376.4	yes	24	17.6	odd	16
425.2.m.d.26.3	yes	24	85.54	odd	16
425.2.m.d.376.3	yes	24	85.74	odd	16
425.2.n.d.274.4		24	85.23	even	16
425.2.n.d.349.4		24	85.37	even	16
425.2.n.e.274.3		24	85.57	even	16
425.2.n.e.349.3		24	85.3	even	16
7225.2.a.bx.1.15		24	1.1	even	1	trivial
7225.2.a.bx.1.16		24	17.16	even	2	inner
7225.2.a.cb.1.9		24	85.84	even	2
7225.2.a.cb.1.10		24	5.4	even	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:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7225.a (trivial)

\(f(q)\)	\(=\)	\(q+0.555780 q^{2} -2.50339 q^{3} -1.69111 q^{4} -1.39134 q^{6} +4.03587 q^{7} -2.05144 q^{8} +3.26698 q^{9} +O(q^{10})\)	\(q+0.555780 q^{2} -2.50339 q^{3} -1.69111 q^{4} -1.39134 q^{6} +4.03587 q^{7} -2.05144 q^{8} +3.26698 q^{9} +0.420560 q^{11} +4.23351 q^{12} +4.96035 q^{13} +2.24306 q^{14} +2.24207 q^{16} +1.81572 q^{18} -5.04055 q^{19} -10.1034 q^{21} +0.233739 q^{22} -8.05347 q^{23} +5.13557 q^{24} +2.75686 q^{26} -0.668350 q^{27} -6.82510 q^{28} -0.185558 q^{29} -3.41416 q^{31} +5.34898 q^{32} -1.05283 q^{33} -5.52481 q^{36} +3.81527 q^{37} -2.80144 q^{38} -12.4177 q^{39} +3.90854 q^{41} -5.61525 q^{42} -5.15014 q^{43} -0.711213 q^{44} -4.47596 q^{46} -6.56400 q^{47} -5.61277 q^{48} +9.28828 q^{49} -8.38849 q^{52} -0.871603 q^{53} -0.371456 q^{54} -8.27937 q^{56} +12.6185 q^{57} -0.103129 q^{58} -14.3445 q^{59} +3.41024 q^{61} -1.89752 q^{62} +13.1851 q^{63} -1.51128 q^{64} -0.585140 q^{66} +7.21621 q^{67} +20.1610 q^{69} -7.48836 q^{71} -6.70202 q^{72} +3.58559 q^{73} +2.12045 q^{74} +8.52413 q^{76} +1.69733 q^{77} -6.90151 q^{78} +14.0842 q^{79} -8.12779 q^{81} +2.17229 q^{82} -13.6689 q^{83} +17.0859 q^{84} -2.86234 q^{86} +0.464523 q^{87} -0.862755 q^{88} +15.1176 q^{89} +20.0193 q^{91} +13.6193 q^{92} +8.54699 q^{93} -3.64814 q^{94} -13.3906 q^{96} -10.1591 q^{97} +5.16224 q^{98} +1.37396 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 8 q^{2} + 24 q^{4} - 24 q^{8} + 24 q^{9}+O(q^{10}) \) 24 * q - 8 * q^2 + 24 * q^4 - 24 * q^8 + 24 * q^9	\( 24 q - 8 q^{2} + 24 q^{4} - 24 q^{8} + 24 q^{9} - 16 q^{13} + 24 q^{16} - 40 q^{18} - 16 q^{21} - 16 q^{26} - 56 q^{32} - 48 q^{33} + 24 q^{36} - 48 q^{38} - 32 q^{43} - 88 q^{47} + 16 q^{49} - 48 q^{52} - 48 q^{53} - 8 q^{59} + 72 q^{64} + 32 q^{66} - 40 q^{67} - 48 q^{69} - 120 q^{72} + 32 q^{76} - 120 q^{77} - 24 q^{81} - 104 q^{83} + 40 q^{84} - 16 q^{86} - 64 q^{87} + 16 q^{89} + 72 q^{93} + 112 q^{94} - 48 q^{98}+O(q^{100}) \) 24 * q - 8 * q^2 + 24 * q^4 - 24 * q^8 + 24 * q^9 - 16 * q^13 + 24 * q^16 - 40 * q^18 - 16 * q^21 - 16 * q^26 - 56 * q^32 - 48 * q^33 + 24 * q^36 - 48 * q^38 - 32 * q^43 - 88 * q^47 + 16 * q^49 - 48 * q^52 - 48 * q^53 - 8 * q^59 + 72 * q^64 + 32 * q^66 - 40 * q^67 - 48 * q^69 - 120 * q^72 + 32 * q^76 - 120 * q^77 - 24 * q^81 - 104 * q^83 + 40 * q^84 - 16 * q^86 - 64 * q^87 + 16 * q^89 + 72 * q^93 + 112 * q^94 - 48 * q^98

Introduction

L-functions

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