LMFDB - Embedded newform 9025.2.a.ck.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, 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("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.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:	$72.0649878242$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 22x^{14} + 190x^{12} - 820x^{10} + 1862x^{8} - 2154x^{6} + 1163x^{4} - 256x^{2} + 16 $ x^16 - 22x^14 + 190x^12 - 820x^10 + 1862x^8 - 2154x^6 + 1163x^4 - 256*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1805)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.805332$ of defining polynomial
Character	$\chi$	$=$	9025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.805332 q^{2} -1.95838 q^{3} -1.35144 q^{4} +1.57714 q^{6} +1.03713 q^{7} +2.69902 q^{8} +0.835236 q^{9} +O(q^{10})$	\(q-0.805332 q^{2} -1.95838 q^{3} -1.35144 q^{4} +1.57714 q^{6} +1.03713 q^{7} +2.69902 q^{8} +0.835236 q^{9} +1.80539 q^{11} +2.64663 q^{12} +2.36455 q^{13} -0.835236 q^{14} +0.529273 q^{16} -6.49986 q^{17} -0.672642 q^{18} -2.03110 q^{21} -1.45394 q^{22} +7.26351 q^{23} -5.28570 q^{24} -1.90425 q^{26} +4.23942 q^{27} -1.40162 q^{28} -2.22179 q^{29} +5.25331 q^{31} -5.82428 q^{32} -3.53564 q^{33} +5.23455 q^{34} -1.12877 q^{36} -6.67893 q^{37} -4.63069 q^{39} -4.43199 q^{41} +1.63571 q^{42} -6.26252 q^{43} -2.43988 q^{44} -5.84953 q^{46} +8.38928 q^{47} -1.03652 q^{48} -5.92436 q^{49} +12.7292 q^{51} -3.19556 q^{52} -0.0601745 q^{53} -3.41414 q^{54} +2.79924 q^{56} +1.78928 q^{58} +12.8100 q^{59} -13.4454 q^{61} -4.23066 q^{62} +0.866251 q^{63} +3.63194 q^{64} +2.84736 q^{66} +6.49953 q^{67} +8.78418 q^{68} -14.2247 q^{69} -13.5401 q^{71} +2.25432 q^{72} -8.23357 q^{73} +5.37876 q^{74} +1.87243 q^{77} +3.72924 q^{78} +6.11205 q^{79} -10.8081 q^{81} +3.56922 q^{82} -4.95287 q^{83} +2.74491 q^{84} +5.04341 q^{86} +4.35110 q^{87} +4.87280 q^{88} -13.6216 q^{89} +2.45236 q^{91} -9.81620 q^{92} -10.2880 q^{93} -6.75615 q^{94} +11.4061 q^{96} +18.4456 q^{97} +4.77107 q^{98} +1.50793 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9}+O(q^{10}) $ 16 * q + 12 * q^4 - 10 * q^6 + 6 * q^9	$ 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9} - 22 q^{11} - 6 q^{14} + 8 q^{16} + 20 q^{21} - 14 q^{24} - 16 q^{26} - 2 q^{29} - 16 q^{31} + 8 q^{34} + 18 q^{36} - 36 q^{39} - 26 q^{41} - 64 q^{44} - 2 q^{46} - 20 q^{49} + 38 q^{51} - 12 q^{54} - 6 q^{56} - 10 q^{59} - 30 q^{61} - 16 q^{64} + 4 q^{66} - 68 q^{69} + 20 q^{71} - 40 q^{74} - 12 q^{79} - 48 q^{81} + 2 q^{84} + 20 q^{86} + 86 q^{91} + 38 q^{94} + 22 q^{96} - 32 q^{99}+O(q^{100}) $ 16 * q + 12 * q^4 - 10 * q^6 + 6 * q^9 - 22 * q^11 - 6 * q^14 + 8 * q^16 + 20 * q^21 - 14 * q^24 - 16 * q^26 - 2 * q^29 - 16 * q^31 + 8 * q^34 + 18 * q^36 - 36 * q^39 - 26 * q^41 - 64 * q^44 - 2 * q^46 - 20 * q^49 + 38 * q^51 - 12 * q^54 - 6 * q^56 - 10 * q^59 - 30 * q^61 - 16 * q^64 + 4 * q^66 - 68 * q^69 + 20 * q^71 - 40 * q^74 - 12 * q^79 - 48 * q^81 + 2 * q^84 + 20 * q^86 + 86 * q^91 + 38 * q^94 + 22 * q^96 - 32 * q^99

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 $

$2$	−0.805332	−0.569456	−0.284728	−	0.958608i	$-0.591903\pi$
$2$	−0.805332	−0.569456	−0.284728	+	0.958608i	$0.591903\pi$
$3$	−1.95838	−1.13067	−0.565334	−	0.824862i	$-0.691253\pi$
$3$	−1.95838	−1.13067	−0.565334	+	0.824862i	$0.691253\pi$
$4$	−1.35144	−0.675720
$5$	0	0
$5$	0	0
$6$	1.57714	0.643866
$7$	1.03713	0.391999	0.196000	−	0.980604i	$-0.437205\pi$
$7$	1.03713	0.391999	0.196000	+	0.980604i	$0.437205\pi$
$8$	2.69902	0.954248
$9$	0.835236	0.278412
$10$	0	0
$11$	1.80539	0.544347	0.272173	−	0.962248i	$-0.412258\pi$
$11$	1.80539	0.544347	0.272173	+	0.962248i	$0.412258\pi$
$12$	2.64663	0.764016
$13$	2.36455	0.655810	0.327905	−	0.944711i	$-0.393657\pi$
$13$	2.36455	0.655810	0.327905	+	0.944711i	$0.393657\pi$
$14$	−0.835236	−0.223226
$15$	0	0
$16$	0.529273	0.132318
$17$	−6.49986	−1.57645	−0.788224	−	0.615388i	$-0.788999\pi$
$17$	−6.49986	−1.57645	−0.788224	+	0.615388i	$0.788999\pi$
$18$	−0.672642	−0.158543
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−2.03110	−0.443222
$22$	−1.45394	−0.309981
$23$	7.26351	1.51455	0.757273	−	0.653099i	$-0.226531\pi$
$23$	7.26351	1.51455	0.757273	+	0.653099i	$0.226531\pi$
$24$	−5.28570	−1.07894
$25$	0	0
$26$	−1.90425	−0.373454
$27$	4.23942	0.815877
$28$	−1.40162	−0.264882
$29$	−2.22179	−0.412576	−0.206288	−	0.978491i	$-0.566138\pi$
$29$	−2.22179	−0.412576	−0.206288	+	0.978491i	$0.566138\pi$
$30$	0	0
$31$	5.25331	0.943522	0.471761	−	0.881727i	$-0.343619\pi$
$31$	5.25331	0.943522	0.471761	+	0.881727i	$0.343619\pi$
$32$	−5.82428	−1.02960
$33$	−3.53564	−0.615476
$34$	5.23455	0.897717
$35$	0	0
$36$	−1.12877	−0.188129
$37$	−6.67893	−1.09801	−0.549005	−	0.835819i	$-0.684993\pi$
$37$	−6.67893	−1.09801	−0.549005	+	0.835819i	$0.684993\pi$
$38$	0	0
$39$	−4.63069	−0.741503
$40$	0	0
$41$	−4.43199	−0.692161	−0.346080	−	0.938205i	$-0.612488\pi$
$41$	−4.43199	−0.692161	−0.346080	+	0.938205i	$0.612488\pi$
$42$	1.63571	0.252395
$43$	−6.26252	−0.955025	−0.477513	−	0.878625i	$-0.658462\pi$
$43$	−6.26252	−0.955025	−0.477513	+	0.878625i	$0.658462\pi$
$44$	−2.43988	−0.367826
$45$	0	0
$46$	−5.84953	−0.862467
$47$	8.38928	1.22370	0.611851	−	0.790973i	$-0.290425\pi$
$47$	8.38928	1.22370	0.611851	+	0.790973i	$0.290425\pi$
$48$	−1.03652	−0.149608
$49$	−5.92436	−0.846336
$50$	0	0
$51$	12.7292	1.78244
$52$	−3.19556	−0.443144
$53$	−0.0601745	−0.00826560	−0.00413280	−	0.999991i	$-0.501316\pi$
$53$	−0.0601745	−0.00826560	−0.00413280	+	0.999991i	$0.501316\pi$
$54$	−3.41414	−0.464606
$55$	0	0
$56$	2.79924	0.374065
$57$	0	0
$58$	1.78928	0.234944
$59$	12.8100	1.66772	0.833860	−	0.551976i	$-0.186126\pi$
$59$	12.8100	1.66772	0.833860	+	0.551976i	$0.186126\pi$
$60$	0	0
$61$	−13.4454	−1.72151	−0.860753	−	0.509024i	$-0.830007\pi$
$61$	−13.4454	−1.72151	−0.860753	+	0.509024i	$0.830007\pi$
$62$	−4.23066	−0.537294
$63$	0.866251	0.109137
$64$	3.63194	0.453992
$65$	0	0
$66$	2.84736	0.350486
$67$	6.49953	0.794043	0.397022	−	0.917809i	$-0.370044\pi$
$67$	6.49953	0.794043	0.397022	+	0.917809i	$0.370044\pi$
$68$	8.78418	1.06524
$69$	−14.2247	−1.71245
$70$	0	0
$71$	−13.5401	−1.60692	−0.803459	−	0.595361i	$-0.797009\pi$
$71$	−13.5401	−1.60692	−0.803459	+	0.595361i	$0.797009\pi$
$72$	2.25432	0.265674
$73$	−8.23357	−0.963666	−0.481833	−	0.876263i	$-0.660029\pi$
$73$	−8.23357	−0.963666	−0.481833	+	0.876263i	$0.660029\pi$
$74$	5.37876	0.625268
$75$	0	0
$76$	0	0
$77$	1.87243	0.213384
$78$	3.72924	0.422253
$79$	6.11205	0.687659	0.343830	−	0.939032i	$-0.388276\pi$
$79$	6.11205	0.687659	0.343830	+	0.939032i	$0.388276\pi$
$80$	0	0
$81$	−10.8081	−1.20090
$82$	3.56922	0.394155
$83$	−4.95287	−0.543648	−0.271824	−	0.962347i	$-0.587627\pi$
$83$	−4.95287	−0.543648	−0.271824	+	0.962347i	$0.587627\pi$
$84$	2.74491	0.299494
$85$	0	0
$86$	5.04341	0.543845
$87$	4.35110	0.466487
$88$	4.87280	0.519442
$89$	−13.6216	−1.44389	−0.721945	−	0.691951i	$-0.756751\pi$
$89$	−13.6216	−1.44389	−0.721945	+	0.691951i	$0.756751\pi$
$90$	0	0
$91$	2.45236	0.257077
$92$	−9.81620	−1.02341
$93$	−10.2880	−1.06681
$94$	−6.75615	−0.696844
$95$	0	0
$96$	11.4061	1.16413
$97$	18.4456	1.87287	0.936435	−	0.350841i	$-0.114104\pi$
$97$	18.4456	1.87287	0.936435	+	0.350841i	$0.114104\pi$
$98$	4.77107	0.481951
$99$	1.50793	0.151553
$100$	0	0
$101$	−5.68380	−0.565559	−0.282780	−	0.959185i	$-0.591257\pi$
$101$	−5.68380	−0.565559	−0.282780	+	0.959185i	$0.591257\pi$
$102$	−10.2512	−1.01502
$103$	−6.57772	−0.648122	−0.324061	−	0.946036i	$-0.605048\pi$
$103$	−6.57772	−0.648122	−0.324061	+	0.946036i	$0.605048\pi$
$104$	6.38199	0.625805
$105$	0	0
$106$	0.0484604	0.00470689
$107$	12.0277	1.16276	0.581382	−	0.813631i	$-0.302512\pi$
$107$	12.0277	1.16276	0.581382	+	0.813631i	$0.302512\pi$
$108$	−5.72933	−0.551305
$109$	2.78067	0.266340	0.133170	−	0.991093i	$-0.457484\pi$
$109$	2.78067	0.266340	0.133170	+	0.991093i	$0.457484\pi$
$110$	0	0
$111$	13.0799	1.24149
$112$	0.548927	0.0518687
$113$	5.86007	0.551269	0.275635	−	0.961262i	$-0.411112\pi$
$113$	5.86007	0.551269	0.275635	+	0.961262i	$0.411112\pi$
$114$	0	0
$115$	0	0
$116$	3.00262	0.278786
$117$	1.97496	0.182585
$118$	−10.3163	−0.949692
$119$	−6.74122	−0.617967
$120$	0	0
$121$	−7.74055	−0.703686
$122$	10.8280	0.980321
$123$	8.67951	0.782605
$124$	−7.09953	−0.637557
$125$	0	0
$126$	−0.697620	−0.0621489
$127$	10.5530	0.936428	0.468214	−	0.883615i	$-0.344898\pi$
$127$	10.5530	0.936428	0.468214	+	0.883615i	$0.344898\pi$
$128$	8.72366	0.771069
$129$	12.2644	1.07982
$130$	0	0
$131$	−13.6545	−1.19300	−0.596501	−	0.802612i	$-0.703443\pi$
$131$	−13.6545	−1.19300	−0.596501	+	0.802612i	$0.703443\pi$
$132$	4.77821	0.415890
$133$	0	0
$134$	−5.23427	−0.452172
$135$	0	0
$136$	−17.5433	−1.50432
$137$	15.6619	1.33809	0.669043	−	0.743224i	$-0.266704\pi$
$137$	15.6619	1.33809	0.669043	+	0.743224i	$0.266704\pi$
$138$	11.4556	0.975164
$139$	−2.37339	−0.201308	−0.100654	−	0.994921i	$-0.532094\pi$
$139$	−2.37339	−0.201308	−0.100654	+	0.994921i	$0.532094\pi$
$140$	0	0
$141$	−16.4294	−1.38360
$142$	10.9043	0.915068
$143$	4.26895	0.356988
$144$	0.442068	0.0368390
$145$	0	0
$146$	6.63075	0.548765
$147$	11.6021	0.956926
$148$	9.02618	0.741947
$149$	3.19551	0.261786	0.130893	−	0.991396i	$-0.458216\pi$
$149$	3.19551	0.261786	0.130893	+	0.991396i	$0.458216\pi$
$150$	0	0
$151$	14.4241	1.17382	0.586908	−	0.809654i	$-0.300345\pi$
$151$	14.4241	1.17382	0.586908	+	0.809654i	$0.300345\pi$
$152$	0	0
$153$	−5.42892	−0.438902
$154$	−1.50793	−0.121513
$155$	0	0
$156$	6.25810	0.501049
$157$	−10.0688	−0.803574	−0.401787	−	0.915733i	$-0.631611\pi$
$157$	−10.0688	−0.803574	−0.401787	+	0.915733i	$0.631611\pi$
$158$	−4.92223	−0.391591
$159$	0.117844	0.00934565
$160$	0	0
$161$	7.53322	0.593701
$162$	8.70410	0.683859
$163$	4.33397	0.339462	0.169731	−	0.985490i	$-0.445710\pi$
$163$	4.33397	0.339462	0.169731	+	0.985490i	$0.445710\pi$
$164$	5.98958	0.467707
$165$	0	0
$166$	3.98870	0.309583
$167$	−6.45124	−0.499212	−0.249606	−	0.968348i	$-0.580301\pi$
$167$	−6.45124	−0.499212	−0.249606	+	0.968348i	$0.580301\pi$
$168$	−5.48197	−0.422943
$169$	−7.40888	−0.569914
$170$	0	0
$171$	0	0
$172$	8.46342	0.645330
$173$	6.89080	0.523898	0.261949	−	0.965082i	$-0.415635\pi$
$173$	6.89080	0.523898	0.261949	+	0.965082i	$0.415635\pi$
$174$	−3.50408	−0.265643
$175$	0	0
$176$	0.955547	0.0720271
$177$	−25.0868	−1.88564
$178$	10.9699	0.822231
$179$	−14.2092	−1.06205	−0.531023	−	0.847357i	$-0.678192\pi$
$179$	−14.2092	−1.06205	−0.531023	+	0.847357i	$0.678192\pi$
$180$	0	0
$181$	5.08595	0.378036	0.189018	−	0.981974i	$-0.439470\pi$
$181$	5.08595	0.378036	0.189018	+	0.981974i	$0.439470\pi$
$182$	−1.97496	−0.146394
$183$	26.3311	1.94645
$184$	19.6044	1.44525
$185$	0	0
$186$	8.28521	0.607501
$187$	−11.7348	−0.858135
$188$	−11.3376	−0.826880
$189$	4.39684	0.319823
$190$	0	0
$191$	6.63072	0.479782	0.239891	−	0.970800i	$-0.422888\pi$
$191$	6.63072	0.479782	0.239891	+	0.970800i	$0.422888\pi$
$192$	−7.11269	−0.513314
$193$	13.7686	0.991088	0.495544	−	0.868583i	$-0.334969\pi$
$193$	13.7686	0.991088	0.495544	+	0.868583i	$0.334969\pi$
$194$	−14.8549	−1.06652
$195$	0	0
$196$	8.00641	0.571887
$197$	−0.729726	−0.0519908	−0.0259954	−	0.999662i	$-0.508276\pi$
$197$	−0.729726	−0.0519908	−0.0259954	+	0.999662i	$0.508276\pi$
$198$	−1.21438	−0.0863026
$199$	5.72434	0.405787	0.202894	−	0.979201i	$-0.434965\pi$
$199$	5.72434	0.405787	0.202894	+	0.979201i	$0.434965\pi$
$200$	0	0
$201$	−12.7285	−0.897800
$202$	4.57735	0.322061
$203$	−2.30429	−0.161729
$204$	−17.2027	−1.20443
$205$	0	0
$206$	5.29725	0.369077
$207$	6.06674	0.421668
$208$	1.25150	0.0867756
$209$	0	0
$210$	0	0
$211$	18.4265	1.26853	0.634267	−	0.773114i	$-0.281302\pi$
$211$	18.4265	1.26853	0.634267	+	0.773114i	$0.281302\pi$
$212$	0.0813222	0.00558523
$213$	26.5167	1.81689
$214$	−9.68631	−0.662143
$215$	0	0
$216$	11.4423	0.778549
$217$	5.44838	0.369860
$218$	−2.23937	−0.151669
$219$	16.1244	1.08959
$220$	0	0
$221$	−15.3693	−1.03385
$222$	−10.5336	−0.706971
$223$	−5.88051	−0.393788	−0.196894	−	0.980425i	$-0.563086\pi$
$223$	−5.88051	−0.393788	−0.196894	+	0.980425i	$0.563086\pi$
$224$	−6.04056	−0.403602
$225$	0	0
$226$	−4.71930	−0.313923
$227$	5.89316	0.391143	0.195571	−	0.980689i	$-0.437344\pi$
$227$	5.89316	0.391143	0.195571	+	0.980689i	$0.437344\pi$
$228$	0	0
$229$	16.2890	1.07641	0.538205	−	0.842814i	$-0.319103\pi$
$229$	16.2890	1.07641	0.538205	+	0.842814i	$0.319103\pi$
$230$	0	0
$231$	−3.66693	−0.241266
$232$	−5.99666	−0.393700
$233$	−22.9771	−1.50528	−0.752640	−	0.658432i	$-0.771220\pi$
$233$	−22.9771	−1.50528	−0.752640	+	0.658432i	$0.771220\pi$
$234$	−1.59050	−0.103974
$235$	0	0
$236$	−17.3120	−1.12691
$237$	−11.9697	−0.777515
$238$	5.42892	0.351905
$239$	−4.58060	−0.296294	−0.148147	−	0.988965i	$-0.547331\pi$
$239$	−4.58060	−0.296294	−0.148147	+	0.988965i	$0.547331\pi$
$240$	0	0
$241$	−8.67191	−0.558607	−0.279303	−	0.960203i	$-0.590104\pi$
$241$	−8.67191	−0.558607	−0.279303	+	0.960203i	$0.590104\pi$
$242$	6.23371	0.400718
$243$	8.44804	0.541942
$244$	18.1706	1.16326
$245$	0	0
$246$	−6.98988	−0.445659
$247$	0	0
$248$	14.1788	0.900354
$249$	9.69958	0.614686
$250$	0	0
$251$	2.98418	0.188360	0.0941800	−	0.995555i	$-0.469977\pi$
$251$	2.98418	0.188360	0.0941800	+	0.995555i	$0.469977\pi$
$252$	−1.17069	−0.0737464
$253$	13.1135	0.824438
$254$	−8.49867	−0.533254
$255$	0	0
$256$	−14.2893	−0.893082
$257$	−5.09001	−0.317506	−0.158753	−	0.987318i	$-0.550747\pi$
$257$	−5.09001	−0.317506	−0.158753	+	0.987318i	$0.550747\pi$
$258$	−9.87689	−0.614908
$259$	−6.92694	−0.430419
$260$	0	0
$261$	−1.85572	−0.114866
$262$	10.9964	0.679362
$263$	15.5925	0.961472	0.480736	−	0.876865i	$-0.340370\pi$
$263$	15.5925	0.961472	0.480736	+	0.876865i	$0.340370\pi$
$264$	−9.54277	−0.587317
$265$	0	0
$266$	0	0
$267$	26.6763	1.63256
$268$	−8.78372	−0.536551
$269$	−19.2926	−1.17629	−0.588145	−	0.808756i	$-0.700141\pi$
$269$	−19.2926	−1.17629	−0.588145	+	0.808756i	$0.700141\pi$
$270$	0	0
$271$	−27.4796	−1.66927	−0.834635	−	0.550804i	$-0.814321\pi$
$271$	−27.4796	−1.66927	−0.834635	+	0.550804i	$0.814321\pi$
$272$	−3.44020	−0.208593
$273$	−4.80264	−0.290669
$274$	−12.6130	−0.761981
$275$	0	0
$276$	19.2238	1.15714
$277$	16.9570	1.01885	0.509424	−	0.860516i	$-0.329858\pi$
$277$	16.9570	1.01885	0.509424	+	0.860516i	$0.329858\pi$
$278$	1.91137	0.114636
$279$	4.38775	0.262688
$280$	0	0
$281$	12.4942	0.745339	0.372670	−	0.927964i	$-0.378442\pi$
$281$	12.4942	0.745339	0.372670	+	0.927964i	$0.378442\pi$
$282$	13.2311	0.787900
$283$	30.8774	1.83547	0.917737	−	0.397189i	$-0.130014\pi$
$283$	30.8774	1.83547	0.917737	+	0.397189i	$0.130014\pi$
$284$	18.2987	1.08583
$285$	0	0
$286$	−3.43793	−0.203289
$287$	−4.59657	−0.271327
$288$	−4.86465	−0.286652
$289$	25.2482	1.48519
$290$	0	0
$291$	−36.1235	−2.11760
$292$	11.1272	0.651169
$293$	25.1823	1.47117	0.735583	−	0.677434i	$-0.236908\pi$
$293$	25.1823	1.47117	0.735583	+	0.677434i	$0.236908\pi$
$294$	−9.34355	−0.544927
$295$	0	0
$296$	−18.0266	−1.04777
$297$	7.65383	0.444120
$298$	−2.57344	−0.149076
$299$	17.1750	0.993254
$300$	0	0
$301$	−6.49507	−0.374369
$302$	−11.6162	−0.668436
$303$	11.1310	0.639460
$304$	0	0
$305$	0	0
$306$	4.37208	0.249935
$307$	6.07770	0.346873	0.173436	−	0.984845i	$-0.444513\pi$
$307$	6.07770	0.346873	0.173436	+	0.984845i	$0.444513\pi$
$308$	−2.53048	−0.144188
$309$	12.8817	0.732812
$310$	0	0
$311$	8.22815	0.466576	0.233288	−	0.972408i	$-0.425052\pi$
$311$	8.22815	0.466576	0.233288	+	0.972408i	$0.425052\pi$
$312$	−12.4983	−0.707578
$313$	−28.7995	−1.62784	−0.813922	−	0.580975i	$-0.802671\pi$
$313$	−28.7995	−1.62784	−0.813922	+	0.580975i	$0.802671\pi$
$314$	8.10868	0.457600
$315$	0	0
$316$	−8.26007	−0.464665
$317$	−13.0478	−0.732840	−0.366420	−	0.930450i	$-0.619417\pi$
$317$	−13.0478	−0.732840	−0.366420	+	0.930450i	$0.619417\pi$
$318$	−0.0949037	−0.00532193
$319$	−4.01120	−0.224584
$320$	0	0
$321$	−23.5548	−1.31470
$322$	−6.06674	−0.338086
$323$	0	0
$324$	14.6065	0.811472
$325$	0	0
$326$	−3.49028	−0.193309
$327$	−5.44561	−0.301143
$328$	−11.9620	−0.660493
$329$	8.70080	0.479690
$330$	0	0
$331$	−4.28808	−0.235694	−0.117847	−	0.993032i	$-0.537599\pi$
$331$	−4.28808	−0.235694	−0.117847	+	0.993032i	$0.537599\pi$
$332$	6.69351	0.367354
$333$	−5.57849	−0.305699
$334$	5.19539	0.284279
$335$	0	0
$336$	−1.07500	−0.0586463
$337$	−5.26898	−0.287020	−0.143510	−	0.989649i	$-0.545839\pi$
$337$	−5.26898	−0.287020	−0.143510	+	0.989649i	$0.545839\pi$
$338$	5.96661	0.324541
$339$	−11.4762	−0.623303
$340$	0	0
$341$	9.48429	0.513603
$342$	0	0
$343$	−13.4043	−0.723763
$344$	−16.9027	−0.911331
$345$	0	0
$346$	−5.54938	−0.298337
$347$	−9.78768	−0.525430	−0.262715	−	0.964873i	$-0.584618\pi$
$347$	−9.78768	−0.525430	−0.262715	+	0.964873i	$0.584618\pi$
$348$	−5.88025	−0.315214
$349$	13.6788	0.732211	0.366106	−	0.930573i	$-0.380691\pi$
$349$	13.6788	0.732211	0.366106	+	0.930573i	$0.380691\pi$
$350$	0	0
$351$	10.0243	0.535060
$352$	−10.5151	−0.560458
$353$	16.9726	0.903363	0.451681	−	0.892179i	$-0.350824\pi$
$353$	16.9726	0.903363	0.451681	+	0.892179i	$0.350824\pi$
$354$	20.2032	1.07379
$355$	0	0
$356$	18.4088	0.975665
$357$	13.2018	0.698716
$358$	11.4431	0.604789
$359$	−10.2829	−0.542713	−0.271356	−	0.962479i	$-0.587472\pi$
$359$	−10.2829	−0.542713	−0.271356	+	0.962479i	$0.587472\pi$
$360$	0	0
$361$	0	0
$362$	−4.09588	−0.215275
$363$	15.1589	0.795636
$364$	−3.31422	−0.173712
$365$	0	0
$366$	−21.2053	−1.10842
$367$	36.2545	1.89247	0.946235	−	0.323480i	$-0.104853\pi$
$367$	36.2545	1.89247	0.946235	+	0.323480i	$0.104853\pi$
$368$	3.84438	0.200402
$369$	−3.70176	−0.192706
$370$	0	0
$371$	−0.0624089	−0.00324011
$372$	13.9036	0.720866
$373$	−20.9496	−1.08473	−0.542365	−	0.840143i	$-0.682471\pi$
$373$	−20.9496	−1.08473	−0.542365	+	0.840143i	$0.682471\pi$
$374$	9.45042	0.488670
$375$	0	0
$376$	22.6428	1.16772
$377$	−5.25354	−0.270571
$378$	−3.54092	−0.182125
$379$	−12.7015	−0.652434	−0.326217	−	0.945295i	$-0.605774\pi$
$379$	−12.7015	−0.652434	−0.326217	+	0.945295i	$0.605774\pi$
$380$	0	0
$381$	−20.6668	−1.05879
$382$	−5.33993	−0.273214
$383$	−8.28673	−0.423432	−0.211716	−	0.977331i	$-0.567905\pi$
$383$	−8.28673	−0.423432	−0.211716	+	0.977331i	$0.567905\pi$
$384$	−17.0842	−0.871824
$385$	0	0
$386$	−11.0883	−0.564381
$387$	−5.23068	−0.265891
$388$	−24.9282	−1.26554
$389$	−15.0072	−0.760898	−0.380449	−	0.924802i	$-0.624231\pi$
$389$	−15.0072	−0.760898	−0.380449	+	0.924802i	$0.624231\pi$
$390$	0	0
$391$	−47.2118	−2.38760
$392$	−15.9900	−0.807615
$393$	26.7407	1.34889
$394$	0.587671	0.0296064
$395$	0	0
$396$	−2.03788	−0.102407
$397$	9.68332	0.485992	0.242996	−	0.970027i	$-0.421870\pi$
$397$	9.68332	0.485992	0.242996	+	0.970027i	$0.421870\pi$
$398$	−4.60999	−0.231078
$399$	0	0
$400$	0	0
$401$	17.3572	0.866775	0.433387	−	0.901208i	$-0.357318\pi$
$401$	17.3572	0.866775	0.433387	+	0.901208i	$0.357318\pi$
$402$	10.2507	0.511257
$403$	12.4217	0.618771
$404$	7.68132	0.382160
$405$	0	0
$406$	1.85572	0.0920977
$407$	−12.0581	−0.597698
$408$	34.3563	1.70089
$409$	15.3330	0.758167	0.379083	−	0.925363i	$-0.376239\pi$
$409$	15.3330	0.758167	0.379083	+	0.925363i	$0.376239\pi$
$410$	0	0
$411$	−30.6719	−1.51293
$412$	8.88940	0.437949
$413$	13.2857	0.653745
$414$	−4.88574	−0.240121
$415$	0	0
$416$	−13.7718	−0.675220
$417$	4.64799	0.227613
$418$	0	0
$419$	1.03847	0.0507326	0.0253663	−	0.999678i	$-0.491925\pi$
$419$	1.03847	0.0507326	0.0253663	+	0.999678i	$0.491925\pi$
$420$	0	0
$421$	−31.9823	−1.55872	−0.779360	−	0.626577i	$-0.784456\pi$
$421$	−31.9823	−1.55872	−0.779360	+	0.626577i	$0.784456\pi$
$422$	−14.8395	−0.722374
$423$	7.00703	0.340693
$424$	−0.162412	−0.00788743
$425$	0	0
$426$	−21.3547	−1.03464
$427$	−13.9447	−0.674829
$428$	−16.2548	−0.785704
$429$	−8.36022	−0.403635
$430$	0	0
$431$	−9.65878	−0.465247	−0.232624	−	0.972567i	$-0.574731\pi$
$431$	−9.65878	−0.465247	−0.232624	+	0.972567i	$0.574731\pi$
$432$	2.24381	0.107955
$433$	−22.4423	−1.07851	−0.539254	−	0.842143i	$-0.681294\pi$
$433$	−22.4423	−1.07851	−0.539254	+	0.842143i	$0.681294\pi$
$434$	−4.38775	−0.210619
$435$	0	0
$436$	−3.75792	−0.179972
$437$	0	0
$438$	−12.9855	−0.620472
$439$	−20.8661	−0.995884	−0.497942	−	0.867210i	$-0.665911\pi$
$439$	−20.8661	−0.995884	−0.497942	+	0.867210i	$0.665911\pi$
$440$	0	0
$441$	−4.94824	−0.235630
$442$	12.3774	0.588732
$443$	−0.0662795	−0.00314903	−0.00157452	−	0.999999i	$-0.500501\pi$
$443$	−0.0662795	−0.00314903	−0.00157452	+	0.999999i	$0.500501\pi$
$444$	−17.6767	−0.838897
$445$	0	0
$446$	4.73576	0.224245
$447$	−6.25800	−0.295993
$448$	3.76680	0.177965
$449$	0.575177	0.0271443	0.0135721	−	0.999908i	$-0.495680\pi$
$449$	0.575177	0.0271443	0.0135721	+	0.999908i	$0.495680\pi$
$450$	0	0
$451$	−8.00150	−0.376776
$452$	−7.91954	−0.372504
$453$	−28.2478	−1.32720
$454$	−4.74595	−0.222738
$455$	0	0
$456$	0	0
$457$	−41.2016	−1.92733	−0.963665	−	0.267113i	$-0.913930\pi$
$457$	−41.2016	−1.92733	−0.963665	+	0.267113i	$0.913930\pi$
$458$	−13.1181	−0.612967
$459$	−27.5557	−1.28619
$460$	0	0
$461$	−32.5041	−1.51387	−0.756933	−	0.653492i	$-0.773303\pi$
$461$	−32.5041	−1.51387	−0.756933	+	0.653492i	$0.773303\pi$
$462$	2.95310	0.137390
$463$	−6.59889	−0.306676	−0.153338	−	0.988174i	$-0.549002\pi$
$463$	−6.59889	−0.306676	−0.153338	+	0.988174i	$0.549002\pi$
$464$	−1.17593	−0.0545913
$465$	0	0
$466$	18.5042	0.857190
$467$	−37.3500	−1.72835	−0.864176	−	0.503190i	$-0.832160\pi$
$467$	−37.3500	−1.72835	−0.864176	+	0.503190i	$0.832160\pi$
$468$	−2.66904	−0.123377
$469$	6.74087	0.311265
$470$	0	0
$471$	19.7184	0.908576
$472$	34.5745	1.59142
$473$	−11.3063	−0.519865
$474$	9.63957	0.442760
$475$	0	0
$476$	9.11036	0.417573
$477$	−0.0502599	−0.00230124
$478$	3.68890	0.168726
$479$	−12.7086	−0.580672	−0.290336	−	0.956925i	$-0.593767\pi$
$479$	−12.7086	−0.580672	−0.290336	+	0.956925i	$0.593767\pi$
$480$	0	0
$481$	−15.7927	−0.720085
$482$	6.98376	0.318102
$483$	−14.7529	−0.671279
$484$	10.4609	0.475495
$485$	0	0
$486$	−6.80347	−0.308612
$487$	−21.6054	−0.979033	−0.489517	−	0.871994i	$-0.662827\pi$
$487$	−21.6054	−0.979033	−0.489517	+	0.871994i	$0.662827\pi$
$488$	−36.2894	−1.64274
$489$	−8.48753	−0.383820
$490$	0	0
$491$	34.0900	1.53846	0.769229	−	0.638973i	$-0.220641\pi$
$491$	34.0900	1.53846	0.769229	+	0.638973i	$0.220641\pi$
$492$	−11.7298	−0.528822
$493$	14.4413	0.650404
$494$	0	0
$495$	0	0
$496$	2.78043	0.124845
$497$	−14.0429	−0.629911
$498$	−7.81138	−0.350036
$499$	−11.1398	−0.498684	−0.249342	−	0.968415i	$-0.580214\pi$
$499$	−11.1398	−0.498684	−0.249342	+	0.968415i	$0.580214\pi$
$500$	0	0
$501$	12.6339	0.564443
$502$	−2.40326	−0.107263
$503$	−13.2039	−0.588735	−0.294367	−	0.955692i	$-0.595109\pi$
$503$	−13.2039	−0.588735	−0.294367	+	0.955692i	$0.595109\pi$
$504$	2.33803	0.104144
$505$	0	0
$506$	−10.5607	−0.469481
$507$	14.5094	0.644384
$508$	−14.2618	−0.632763
$509$	−9.19539	−0.407579	−0.203789	−	0.979015i	$-0.565326\pi$
$509$	−9.19539	−0.407579	−0.203789	+	0.979015i	$0.565326\pi$
$510$	0	0
$511$	−8.53931	−0.377757
$512$	−5.93968	−0.262499
$513$	0	0
$514$	4.09915	0.180806
$515$	0	0
$516$	−16.5746	−0.729655
$517$	15.1460	0.666118
$518$	5.57849	0.245105
$519$	−13.4948	−0.592355
$520$	0	0
$521$	−21.4325	−0.938973	−0.469486	−	0.882940i	$-0.655561\pi$
$521$	−21.4325	−0.938973	−0.469486	+	0.882940i	$0.655561\pi$
$522$	1.49447	0.0654111
$523$	−9.04219	−0.395387	−0.197694	−	0.980264i	$-0.563345\pi$
$523$	−9.04219	−0.395387	−0.197694	+	0.980264i	$0.563345\pi$
$524$	18.4533	0.806136
$525$	0	0
$526$	−12.5571	−0.547516
$527$	−34.1458	−1.48741
$528$	−1.87132	−0.0814388
$529$	29.7585	1.29385
$530$	0	0
$531$	10.6994	0.464313
$532$	0	0
$533$	−10.4797	−0.453926
$534$	−21.4832	−0.929671
$535$	0	0
$536$	17.5424	0.757715
$537$	27.8270	1.20082
$538$	15.5369	0.669845
$539$	−10.6958	−0.460701
$540$	0	0
$541$	−23.9095	−1.02795	−0.513976	−	0.857805i	$-0.671828\pi$
$541$	−23.9095	−1.02795	−0.513976	+	0.857805i	$0.671828\pi$
$542$	22.1302	0.950575
$543$	−9.96020	−0.427433
$544$	37.8570	1.62311
$545$	0	0
$546$	3.86772	0.165523
$547$	−38.3031	−1.63772	−0.818861	−	0.573992i	$-0.805394\pi$
$547$	−38.3031	−1.63772	−0.818861	+	0.573992i	$0.805394\pi$
$548$	−21.1661	−0.904172
$549$	−11.2301	−0.479288
$550$	0	0
$551$	0	0
$552$	−38.3927	−1.63410
$553$	6.33901	0.269562
$554$	−13.6560	−0.580189
$555$	0	0
$556$	3.20750	0.136028
$557$	10.7640	0.456084	0.228042	−	0.973651i	$-0.426768\pi$
$557$	10.7640	0.456084	0.228042	+	0.973651i	$0.426768\pi$
$558$	−3.53360	−0.149589
$559$	−14.8081	−0.626315
$560$	0	0
$561$	22.9812	0.970266
$562$	−10.0619	−0.424438
$563$	−12.0769	−0.508981	−0.254490	−	0.967075i	$-0.581908\pi$
$563$	−12.0769	−0.508981	−0.254490	+	0.967075i	$0.581908\pi$
$564$	22.2033	0.934928
$565$	0	0
$566$	−24.8666	−1.04522
$567$	−11.2094	−0.470752
$568$	−36.5451	−1.53340
$569$	41.3125	1.73191	0.865954	−	0.500123i	$-0.166712\pi$
$569$	41.3125	1.73191	0.865954	+	0.500123i	$0.166712\pi$
$570$	0	0
$571$	0.646103	0.0270385	0.0135193	−	0.999909i	$-0.495697\pi$
$571$	0.646103	0.0270385	0.0135193	+	0.999909i	$0.495697\pi$
$572$	−5.76924	−0.241224
$573$	−12.9854	−0.542474
$574$	3.70176	0.154509
$575$	0	0
$576$	3.03352	0.126397
$577$	−9.70295	−0.403939	−0.201969	−	0.979392i	$-0.564734\pi$
$577$	−9.70295	−0.403939	−0.201969	+	0.979392i	$0.564734\pi$
$578$	−20.3332	−0.845749
$579$	−26.9642	−1.12059
$580$	0	0
$581$	−5.13678	−0.213110
$582$	29.0914	1.20588
$583$	−0.108639	−0.00449935
$584$	−22.2226	−0.919577
$585$	0	0
$586$	−20.2801	−0.837764
$587$	19.2787	0.795716	0.397858	−	0.917447i	$-0.369754\pi$
$587$	19.2787	0.795716	0.397858	+	0.917447i	$0.369754\pi$
$588$	−15.6796	−0.646615
$589$	0	0
$590$	0	0
$591$	1.42908	0.0587844
$592$	−3.53498	−0.145287
$593$	−35.2446	−1.44732	−0.723660	−	0.690156i	$-0.757542\pi$
$593$	−35.2446	−1.44732	−0.723660	+	0.690156i	$0.757542\pi$
$594$	−6.16387	−0.252907
$595$	0	0
$596$	−4.31854	−0.176894
$597$	−11.2104	−0.458811
$598$	−13.8315	−0.565614
$599$	0.181646	0.00742186	0.00371093	−	0.999993i	$-0.498819\pi$
$599$	0.181646	0.00742186	0.00371093	+	0.999993i	$0.498819\pi$
$600$	0	0
$601$	29.0186	1.18369	0.591847	−	0.806050i	$-0.298399\pi$
$601$	29.0186	1.18369	0.591847	+	0.806050i	$0.298399\pi$
$602$	5.23068	0.213187
$603$	5.42864	0.221071
$604$	−19.4933	−0.793172
$605$	0	0
$606$	−8.96417	−0.364144
$607$	31.8804	1.29398	0.646992	−	0.762497i	$-0.276027\pi$
$607$	31.8804	1.29398	0.646992	+	0.762497i	$0.276027\pi$
$608$	0	0
$609$	4.51267	0.182862
$610$	0	0
$611$	19.8369	0.802515
$612$	7.33686	0.296575
$613$	26.1763	1.05725	0.528626	−	0.848855i	$-0.322707\pi$
$613$	26.1763	1.05725	0.528626	+	0.848855i	$0.322707\pi$
$614$	−4.89457	−0.197529
$615$	0	0
$616$	5.05374	0.203621
$617$	−30.4273	−1.22496	−0.612479	−	0.790487i	$-0.709828\pi$
$617$	−30.4273	−1.22496	−0.612479	+	0.790487i	$0.709828\pi$
$618$	−10.3740	−0.417304
$619$	−25.1588	−1.01122	−0.505609	−	0.862763i	$-0.668732\pi$
$619$	−25.1588	−1.01122	−0.505609	+	0.862763i	$0.668732\pi$
$620$	0	0
$621$	30.7931	1.23568
$622$	−6.62639	−0.265694
$623$	−14.1274	−0.566004
$624$	−2.45090	−0.0981145
$625$	0	0
$626$	23.1931	0.926984
$627$	0	0
$628$	13.6073	0.542991
$629$	43.4121	1.73096
$630$	0	0
$631$	−22.4203	−0.892539	−0.446269	−	0.894899i	$-0.647248\pi$
$631$	−22.4203	−0.892539	−0.446269	+	0.894899i	$0.647248\pi$
$632$	16.4965	0.656198
$633$	−36.0861	−1.43429
$634$	10.5078	0.417320
$635$	0	0
$636$	−0.159259	−0.00631505
$637$	−14.0085	−0.555035
$638$	3.23035	0.127891
$639$	−11.3092	−0.447385
$640$	0	0
$641$	0.599952	0.0236967	0.0118483	−	0.999930i	$-0.496228\pi$
$641$	0.599952	0.0236967	0.0118483	+	0.999930i	$0.496228\pi$
$642$	18.9694	0.748664
$643$	28.0144	1.10478	0.552390	−	0.833586i	$-0.313716\pi$
$643$	28.0144	1.10478	0.552390	+	0.833586i	$0.313716\pi$
$644$	−10.1807	−0.401176
$645$	0	0
$646$	0	0
$647$	−14.5244	−0.571013	−0.285506	−	0.958377i	$-0.592162\pi$
$647$	−14.5244	−0.571013	−0.285506	+	0.958377i	$0.592162\pi$
$648$	−29.1713	−1.14596
$649$	23.1271	0.907818
$650$	0	0
$651$	−10.6700	−0.418189
$652$	−5.85710	−0.229382
$653$	22.9792	0.899247	0.449623	−	0.893218i	$-0.351558\pi$
$653$	22.9792	0.899247	0.449623	+	0.893218i	$0.351558\pi$
$654$	4.38552	0.171487
$655$	0	0
$656$	−2.34573	−0.0915856
$657$	−6.87697	−0.268296
$658$	−7.00703	−0.273162
$659$	−29.7896	−1.16044	−0.580219	−	0.814460i	$-0.697033\pi$
$659$	−29.7896	−1.16044	−0.580219	+	0.814460i	$0.697033\pi$
$660$	0	0
$661$	25.1870	0.979660	0.489830	−	0.871818i	$-0.337059\pi$
$661$	25.1870	0.979660	0.489830	+	0.871818i	$0.337059\pi$
$662$	3.45333	0.134217
$663$	30.0988	1.16894
$664$	−13.3679	−0.518775
$665$	0	0
$666$	4.49253	0.174082
$667$	−16.1380	−0.624865
$668$	8.71846	0.337327
$669$	11.5163	0.445244
$670$	0	0
$671$	−24.2742	−0.937096
$672$	11.8297	0.456340
$673$	20.9432	0.807302	0.403651	−	0.914913i	$-0.367741\pi$
$673$	20.9432	0.807302	0.403651	+	0.914913i	$0.367741\pi$
$674$	4.24328	0.163445
$675$	0	0
$676$	10.0127	0.385102
$677$	−33.8768	−1.30199	−0.650995	−	0.759082i	$-0.725648\pi$
$677$	−33.8768	−1.30199	−0.650995	+	0.759082i	$0.725648\pi$
$678$	9.24217	0.354943
$679$	19.1306	0.734164
$680$	0	0
$681$	−11.5410	−0.442253
$682$	−7.63800	−0.292474
$683$	−30.3942	−1.16300	−0.581501	−	0.813546i	$-0.697534\pi$
$683$	−30.3942	−1.16300	−0.581501	+	0.813546i	$0.697534\pi$
$684$	0	0
$685$	0	0
$686$	10.7949	0.412151
$687$	−31.9000	−1.21706
$688$	−3.31458	−0.126367
$689$	−0.142286	−0.00542066
$690$	0	0
$691$	−9.00959	−0.342741	−0.171371	−	0.985207i	$-0.554820\pi$
$691$	−9.00959	−0.342741	−0.171371	+	0.985207i	$0.554820\pi$
$692$	−9.31251	−0.354009
$693$	1.56393	0.0594086
$694$	7.88233	0.299209
$695$	0	0
$696$	11.7437	0.445144
$697$	28.8073	1.09116
$698$	−11.0160	−0.416962
$699$	44.9978	1.70197
$700$	0	0
$701$	1.50058	0.0566762	0.0283381	−	0.999598i	$-0.490978\pi$
$701$	1.50058	0.0566762	0.0283381	+	0.999598i	$0.490978\pi$
$702$	−8.07292	−0.304693
$703$	0	0
$704$	6.55708	0.247129
$705$	0	0
$706$	−13.6686	−0.514425
$707$	−5.89486	−0.221699
$708$	33.9033	1.27416
$709$	−40.6894	−1.52812	−0.764061	−	0.645144i	$-0.776798\pi$
$709$	−40.6894	−1.52812	−0.764061	+	0.645144i	$0.776798\pi$
$710$	0	0
$711$	5.10500	0.191453
$712$	−36.7651	−1.37783
$713$	38.1574	1.42901
$714$	−10.6319	−0.397888
$715$	0	0
$716$	19.2029	0.717647
$717$	8.97053	0.335011
$718$	8.28117	0.309051
$719$	−18.5007	−0.689959	−0.344980	−	0.938610i	$-0.612114\pi$
$719$	−18.5007	−0.689959	−0.344980	+	0.938610i	$0.612114\pi$
$720$	0	0
$721$	−6.82197	−0.254064
$722$	0	0
$723$	16.9829	0.631599
$724$	−6.87336	−0.255446
$725$	0	0
$726$	−12.2080	−0.453080
$727$	−0.175782	−0.00651939	−0.00325970	−	0.999995i	$-0.501038\pi$
$727$	−0.175782	−0.00651939	−0.00325970	+	0.999995i	$0.501038\pi$
$728$	6.61897	0.245315
$729$	15.8798	0.588142
$730$	0	0
$731$	40.7055	1.50555
$732$	−35.5849	−1.31526
$733$	−31.7823	−1.17390	−0.586952	−	0.809621i	$-0.699672\pi$
$733$	−31.7823	−1.17390	−0.586952	+	0.809621i	$0.699672\pi$
$734$	−29.1969	−1.07768
$735$	0	0
$736$	−42.3047	−1.55937
$737$	11.7342	0.432235
$738$	2.98115	0.109738
$739$	1.60789	0.0591473	0.0295737	−	0.999563i	$-0.490585\pi$
$739$	1.60789	0.0591473	0.0295737	+	0.999563i	$0.490585\pi$
$740$	0	0
$741$	0	0
$742$	0.0502599	0.00184510
$743$	13.7363	0.503935	0.251968	−	0.967736i	$-0.418922\pi$
$743$	13.7363	0.503935	0.251968	+	0.967736i	$0.418922\pi$
$744$	−27.7674	−1.01800
$745$	0	0
$746$	16.8714	0.617706
$747$	−4.13682	−0.151358
$748$	15.8589	0.579859
$749$	12.4744	0.455803
$750$	0	0
$751$	−44.1497	−1.61104	−0.805522	−	0.592565i	$-0.798115\pi$
$751$	−44.1497	−1.61104	−0.805522	+	0.592565i	$0.798115\pi$
$752$	4.44022	0.161918
$753$	−5.84415	−0.212973
$754$	4.23084	0.154078
$755$	0	0
$756$	−5.94207	−0.216111
$757$	25.5242	0.927693	0.463846	−	0.885916i	$-0.346469\pi$
$757$	25.5242	0.927693	0.463846	+	0.885916i	$0.346469\pi$
$758$	10.2289	0.371532
$759$	−25.6812	−0.932167
$760$	0	0
$761$	−1.51764	−0.0550143	−0.0275072	−	0.999622i	$-0.508757\pi$
$761$	−1.51764	−0.0550143	−0.0275072	+	0.999622i	$0.508757\pi$
$762$	16.6436	0.602934
$763$	2.88393	0.104405
$764$	−8.96102	−0.324198
$765$	0	0
$766$	6.67357	0.241126
$767$	30.2899	1.09371
$768$	27.9838	1.00978
$769$	34.9179	1.25917	0.629586	−	0.776931i	$-0.283224\pi$
$769$	34.9179	1.25917	0.629586	+	0.776931i	$0.283224\pi$
$770$	0	0
$771$	9.96815	0.358994
$772$	−18.6075	−0.669698
$773$	44.5250	1.60145	0.800726	−	0.599031i	$-0.204447\pi$
$773$	44.5250	1.60145	0.800726	+	0.599031i	$0.204447\pi$
$774$	4.21244	0.151413
$775$	0	0
$776$	49.7852	1.78718
$777$	13.5656	0.486661
$778$	12.0858	0.433297
$779$	0	0
$780$	0	0
$781$	−24.4453	−0.874720
$782$	38.0212	1.35963
$783$	−9.41910	−0.336611
$784$	−3.13560	−0.111986
$785$	0	0
$786$	−21.5351	−0.768133
$787$	−23.3866	−0.833643	−0.416822	−	0.908988i	$-0.636856\pi$
$787$	−23.3866	−0.833643	−0.416822	+	0.908988i	$0.636856\pi$
$788$	0.986181	0.0351312
$789$	−30.5359	−1.08711
$790$	0	0
$791$	6.07768	0.216097
$792$	4.06994	0.144619
$793$	−31.7924	−1.12898
$794$	−7.79829	−0.276751
$795$	0	0
$796$	−7.73610	−0.274199
$797$	−19.6099	−0.694618	−0.347309	−	0.937751i	$-0.612905\pi$
$797$	−19.6099	−0.694618	−0.347309	+	0.937751i	$0.612905\pi$
$798$	0	0
$799$	−54.5292	−1.92910
$800$	0	0
$801$	−11.3773	−0.401996
$802$	−13.9783	−0.493590
$803$	−14.8648	−0.524569
$804$	17.2018	0.606662
$805$	0	0
$806$	−10.0036	−0.352362
$807$	37.7821	1.32999
$808$	−15.3407	−0.539684
$809$	−28.1670	−0.990298	−0.495149	−	0.868808i	$-0.664887\pi$
$809$	−28.1670	−0.990298	−0.495149	+	0.868808i	$0.664887\pi$
$810$	0	0
$811$	3.22068	0.113093	0.0565467	−	0.998400i	$-0.481991\pi$
$811$	3.22068	0.113093	0.0565467	+	0.998400i	$0.481991\pi$
$812$	3.11411	0.109284
$813$	53.8155	1.88739
$814$	9.71078	0.340363
$815$	0	0
$816$	6.73721	0.235850
$817$	0	0
$818$	−12.3481	−0.431742
$819$	2.04830	0.0715733
$820$	0	0
$821$	−8.81189	−0.307537	−0.153768	−	0.988107i	$-0.549141\pi$
$821$	−8.81189	−0.307537	−0.153768	+	0.988107i	$0.549141\pi$
$822$	24.7010	0.861548
$823$	12.2640	0.427497	0.213748	−	0.976889i	$-0.431433\pi$
$823$	12.2640	0.427497	0.213748	+	0.976889i	$0.431433\pi$
$824$	−17.7534	−0.618470
$825$	0	0
$826$	−10.6994	−0.372279
$827$	−45.8102	−1.59298	−0.796489	−	0.604653i	$-0.793312\pi$
$827$	−45.8102	−1.59298	−0.796489	+	0.604653i	$0.793312\pi$
$828$	−8.19884	−0.284930
$829$	−46.7917	−1.62514	−0.812572	−	0.582861i	$-0.801933\pi$
$829$	−46.7917	−1.62514	−0.812572	+	0.582861i	$0.801933\pi$
$830$	0	0
$831$	−33.2082	−1.15198
$832$	8.58791	0.297732
$833$	38.5075	1.33421
$834$	−3.74318	−0.129616
$835$	0	0
$836$	0	0
$837$	22.2710	0.769798
$838$	−0.836314	−0.0288900
$839$	40.7277	1.40608	0.703038	−	0.711152i	$-0.251826\pi$
$839$	40.7277	1.40608	0.703038	+	0.711152i	$0.251826\pi$
$840$	0	0
$841$	−24.0637	−0.829781
$842$	25.7563	0.887622
$843$	−24.4683	−0.842732
$844$	−24.9024	−0.857174
$845$	0	0
$846$	−5.64298	−0.194010
$847$	−8.02798	−0.275845
$848$	−0.0318487	−0.00109369
$849$	−60.4696	−2.07531
$850$	0	0
$851$	−48.5125	−1.66299
$852$	−35.8357	−1.22771
$853$	28.2038	0.965678	0.482839	−	0.875709i	$-0.339606\pi$
$853$	28.2038	0.965678	0.482839	+	0.875709i	$0.339606\pi$
$854$	11.2301	0.384285
$855$	0	0
$856$	32.4631	1.10957
$857$	−8.80423	−0.300747	−0.150373	−	0.988629i	$-0.548048\pi$
$857$	−8.80423	−0.300747	−0.150373	+	0.988629i	$0.548048\pi$
$858$	6.73275	0.229852
$859$	19.9249	0.679830	0.339915	−	0.940456i	$-0.389602\pi$
$859$	19.9249	0.679830	0.339915	+	0.940456i	$0.389602\pi$
$860$	0	0
$861$	9.00180	0.306781
$862$	7.77852	0.264938
$863$	−1.20780	−0.0411141	−0.0205571	−	0.999789i	$-0.506544\pi$
$863$	−1.20780	−0.0411141	−0.0205571	+	0.999789i	$0.506544\pi$
$864$	−24.6916	−0.840025
$865$	0	0
$866$	18.0735	0.614162
$867$	−49.4455	−1.67926
$868$	−7.36316	−0.249922
$869$	11.0347	0.374325
$870$	0	0
$871$	15.3685	0.520741
$872$	7.50510	0.254155
$873$	15.4065	0.521430
$874$	0	0
$875$	0	0
$876$	−21.7912	−0.736256
$877$	−8.95984	−0.302552	−0.151276	−	0.988492i	$-0.548338\pi$
$877$	−8.95984	−0.302552	−0.151276	+	0.988492i	$0.548338\pi$
$878$	16.8041	0.567112
$879$	−49.3164	−1.66340
$880$	0	0
$881$	−3.13906	−0.105758	−0.0528789	−	0.998601i	$-0.516840\pi$
$881$	−3.13906	−0.105758	−0.0528789	+	0.998601i	$0.516840\pi$
$882$	3.98497	0.134181
$883$	−12.3080	−0.414197	−0.207099	−	0.978320i	$-0.566402\pi$
$883$	−12.3080	−0.414197	−0.207099	+	0.978320i	$0.566402\pi$
$884$	20.7707	0.698593
$885$	0	0
$886$	0.0533770	0.00179323
$887$	−29.3370	−0.985041	−0.492521	−	0.870301i	$-0.663924\pi$
$887$	−29.3370	−0.985041	−0.492521	+	0.870301i	$0.663924\pi$
$888$	35.3028	1.18469
$889$	10.9449	0.367079
$890$	0	0
$891$	−19.5129	−0.653706
$892$	7.94716	0.266091
$893$	0	0
$894$	5.03977	0.168555
$895$	0	0
$896$	9.04759	0.302259
$897$	−33.6350	−1.12304
$898$	−0.463209	−0.0154575
$899$	−11.6717	−0.389274
$900$	0	0
$901$	0.391126	0.0130303
$902$	6.44386	0.214557
$903$	12.7198	0.423288
$904$	15.8165	0.526048
$905$	0	0
$906$	22.7489	0.755780
$907$	−1.85256	−0.0615131	−0.0307566	−	0.999527i	$-0.509792\pi$
$907$	−1.85256	−0.0615131	−0.0307566	+	0.999527i	$0.509792\pi$
$908$	−7.96426	−0.264303
$909$	−4.74732	−0.157459
$910$	0	0
$911$	33.0581	1.09526	0.547632	−	0.836720i	$-0.315530\pi$
$911$	33.0581	1.09526	0.547632	+	0.836720i	$0.315530\pi$
$912$	0	0
$913$	−8.94188	−0.295933
$914$	33.1810	1.09753
$915$	0	0
$916$	−22.0137	−0.727352
$917$	−14.1616	−0.467656
$918$	22.1914	0.732427
$919$	50.8506	1.67741	0.838703	−	0.544590i	$-0.183315\pi$
$919$	50.8506	1.67741	0.838703	+	0.544590i	$0.183315\pi$
$920$	0	0
$921$	−11.9024	−0.392198
$922$	26.1766	0.862080
$923$	−32.0164	−1.05383
$924$	4.95564	0.163029
$925$	0	0
$926$	5.31429	0.174638
$927$	−5.49395	−0.180445
$928$	12.9403	0.424787
$929$	8.01818	0.263068	0.131534	−	0.991312i	$-0.458010\pi$
$929$	8.01818	0.263068	0.131534	+	0.991312i	$0.458010\pi$
$930$	0	0
$931$	0	0
$932$	31.0522	1.01715
$933$	−16.1138	−0.527543
$934$	30.0791	0.984220
$935$	0	0
$936$	5.33047	0.174232
$937$	−52.9527	−1.72989	−0.864944	−	0.501869i	$-0.832646\pi$
$937$	−52.9527	−1.72989	−0.864944	+	0.501869i	$0.832646\pi$
$938$	−5.42864	−0.177251
$939$	56.4002	1.84055
$940$	0	0
$941$	29.3673	0.957347	0.478674	−	0.877993i	$-0.341118\pi$
$941$	29.3673	0.957347	0.478674	+	0.877993i	$0.341118\pi$
$942$	−15.8799	−0.517394
$943$	−32.1918	−1.04831
$944$	6.77999	0.220670
$945$	0	0
$946$	9.10534	0.296040
$947$	0.230323	0.00748449	0.00374225	−	0.999993i	$-0.498809\pi$
$947$	0.230323	0.00748449	0.00374225	+	0.999993i	$0.498809\pi$
$948$	16.1763	0.525382
$949$	−19.4687	−0.631981
$950$	0	0
$951$	25.5526	0.828599
$952$	−18.1947	−0.589694
$953$	−9.62121	−0.311662	−0.155831	−	0.987784i	$-0.549805\pi$
$953$	−9.62121	−0.311662	−0.155831	+	0.987784i	$0.549805\pi$
$954$	0.0404759	0.00131046
$955$	0	0
$956$	6.19040	0.200212
$957$	7.85545	0.253931
$958$	10.2347	0.330667
$959$	16.2435	0.524529
$960$	0	0
$961$	−3.40276	−0.109767
$962$	12.7184	0.410056
$963$	10.0460	0.323728
$964$	11.7196	0.377462
$965$	0	0
$966$	11.8810	0.382264
$967$	21.1871	0.681330	0.340665	−	0.940185i	$-0.389348\pi$
$967$	21.1871	0.681330	0.340665	+	0.940185i	$0.389348\pi$
$968$	−20.8919	−0.671492
$969$	0	0
$970$	0	0
$971$	18.2019	0.584126	0.292063	−	0.956399i	$-0.405658\pi$
$971$	18.2019	0.584126	0.292063	+	0.956399i	$0.405658\pi$
$972$	−11.4170	−0.366201
$973$	−2.46152	−0.0789128
$974$	17.3995	0.557516
$975$	0	0
$976$	−7.11628	−0.227787
$977$	−12.8125	−0.409909	−0.204955	−	0.978771i	$-0.565705\pi$
$977$	−12.8125	−0.409909	−0.204955	+	0.978771i	$0.565705\pi$
$978$	6.83528	0.218568
$979$	−24.5924	−0.785977
$980$	0	0
$981$	2.32252	0.0741524
$982$	−27.4537	−0.876083
$983$	−41.3211	−1.31794	−0.658970	−	0.752169i	$-0.729007\pi$
$983$	−41.3211	−1.31794	−0.658970	+	0.752169i	$0.729007\pi$
$984$	23.4262	0.746799
$985$	0	0
$986$	−11.6301	−0.370376
$987$	−17.0394	−0.542371
$988$	0	0
$989$	−45.4879	−1.44643
$990$	0	0
$991$	−28.2455	−0.897249	−0.448624	−	0.893720i	$-0.648086\pi$
$991$	−28.2455	−0.897249	−0.448624	+	0.893720i	$0.648086\pi$
$992$	−30.5968	−0.971448
$993$	8.39767	0.266492
$994$	11.3092	0.358706
$995$	0	0
$996$	−13.1084	−0.415356
$997$	−34.1678	−1.08211	−0.541053	−	0.840988i	$-0.681974\pi$
$997$	−34.1678	−1.08211	−0.541053	+	0.840988i	$0.681974\pi$
$998$	8.97120	0.283979
$999$	−28.3148	−0.895841

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.ck.1.6		16
5.2	odd	4		1805.2.b.j.1084.6	yes	16
5.3	odd	4		1805.2.b.j.1084.11	yes	16
5.4	even	2	inner	9025.2.a.ck.1.11		16
19.18	odd	2		9025.2.a.cl.1.11		16
95.18	even	4		1805.2.b.i.1084.6	✓	16
95.37	even	4		1805.2.b.i.1084.11	yes	16
95.94	odd	2		9025.2.a.cl.1.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.b.i.1084.6	✓	16	95.18	even	4
1805.2.b.i.1084.11	yes	16	95.37	even	4
1805.2.b.j.1084.6	yes	16	5.2	odd	4
1805.2.b.j.1084.11	yes	16	5.3	odd	4
9025.2.a.ck.1.6		16	1.1	even	1	trivial
9025.2.a.ck.1.11		16	5.4	even	2	inner
9025.2.a.cl.1.6		16	95.94	odd	2
9025.2.a.cl.1.11		16	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q-0.805332 q^{2} -1.95838 q^{3} -1.35144 q^{4} +1.57714 q^{6} +1.03713 q^{7} +2.69902 q^{8} +0.835236 q^{9} +O(q^{10})\)	\(q-0.805332 q^{2} -1.95838 q^{3} -1.35144 q^{4} +1.57714 q^{6} +1.03713 q^{7} +2.69902 q^{8} +0.835236 q^{9} +1.80539 q^{11} +2.64663 q^{12} +2.36455 q^{13} -0.835236 q^{14} +0.529273 q^{16} -6.49986 q^{17} -0.672642 q^{18} -2.03110 q^{21} -1.45394 q^{22} +7.26351 q^{23} -5.28570 q^{24} -1.90425 q^{26} +4.23942 q^{27} -1.40162 q^{28} -2.22179 q^{29} +5.25331 q^{31} -5.82428 q^{32} -3.53564 q^{33} +5.23455 q^{34} -1.12877 q^{36} -6.67893 q^{37} -4.63069 q^{39} -4.43199 q^{41} +1.63571 q^{42} -6.26252 q^{43} -2.43988 q^{44} -5.84953 q^{46} +8.38928 q^{47} -1.03652 q^{48} -5.92436 q^{49} +12.7292 q^{51} -3.19556 q^{52} -0.0601745 q^{53} -3.41414 q^{54} +2.79924 q^{56} +1.78928 q^{58} +12.8100 q^{59} -13.4454 q^{61} -4.23066 q^{62} +0.866251 q^{63} +3.63194 q^{64} +2.84736 q^{66} +6.49953 q^{67} +8.78418 q^{68} -14.2247 q^{69} -13.5401 q^{71} +2.25432 q^{72} -8.23357 q^{73} +5.37876 q^{74} +1.87243 q^{77} +3.72924 q^{78} +6.11205 q^{79} -10.8081 q^{81} +3.56922 q^{82} -4.95287 q^{83} +2.74491 q^{84} +5.04341 q^{86} +4.35110 q^{87} +4.87280 q^{88} -13.6216 q^{89} +2.45236 q^{91} -9.81620 q^{92} -10.2880 q^{93} -6.75615 q^{94} +11.4061 q^{96} +18.4456 q^{97} +4.77107 q^{98} +1.50793 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9}+O(q^{10}) \) 16 * q + 12 * q^4 - 10 * q^6 + 6 * q^9	\( 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9} - 22 q^{11} - 6 q^{14} + 8 q^{16} + 20 q^{21} - 14 q^{24} - 16 q^{26} - 2 q^{29} - 16 q^{31} + 8 q^{34} + 18 q^{36} - 36 q^{39} - 26 q^{41} - 64 q^{44} - 2 q^{46} - 20 q^{49} + 38 q^{51} - 12 q^{54} - 6 q^{56} - 10 q^{59} - 30 q^{61} - 16 q^{64} + 4 q^{66} - 68 q^{69} + 20 q^{71} - 40 q^{74} - 12 q^{79} - 48 q^{81} + 2 q^{84} + 20 q^{86} + 86 q^{91} + 38 q^{94} + 22 q^{96} - 32 q^{99}+O(q^{100}) \) 16 * q + 12 * q^4 - 10 * q^6 + 6 * q^9 - 22 * q^11 - 6 * q^14 + 8 * q^16 + 20 * q^21 - 14 * q^24 - 16 * q^26 - 2 * q^29 - 16 * q^31 + 8 * q^34 + 18 * q^36 - 36 * q^39 - 26 * q^41 - 64 * q^44 - 2 * q^46 - 20 * q^49 + 38 * q^51 - 12 * q^54 - 6 * q^56 - 10 * q^59 - 30 * q^61 - 16 * q^64 + 4 * q^66 - 68 * q^69 + 20 * q^71 - 40 * q^74 - 12 * q^79 - 48 * q^81 + 2 * q^84 + 20 * q^86 + 86 * q^91 + 38 * q^94 + 22 * q^96 - 32 * q^99

Introduction

L-functions

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