LMFDB - Embedded newform 7225.2.a.bo.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7225,2,Mod(1,7225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 7225 = 5^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,-3,3,21,0,9,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$57.6919154604$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 18 x^{10} + 55 x^{9} + 114 x^{8} - 354 x^{7} - 309 x^{6} + 936 x^{5} + 396 x^{4} + \cdots + 9 $ x^12 - 3x^11 - 18x^10 + 55x^9 + 114x^8 - 354x^7 - 309x^6 + 936x^5 + 396x^4 - 899x^3 - 297x^2 + 153*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1445)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.27524$ of defining polynomial
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)

$f(q)$	$=$	\(q-2.27524 q^{2} -1.70271 q^{3} +3.17670 q^{4} +3.87408 q^{6} -2.70724 q^{7} -2.67727 q^{8} -0.100765 q^{9} -2.17632 q^{11} -5.40901 q^{12} -5.37970 q^{13} +6.15962 q^{14} -0.261974 q^{16} +0.229265 q^{18} +8.53588 q^{19} +4.60966 q^{21} +4.95164 q^{22} +5.32570 q^{23} +4.55863 q^{24} +12.2401 q^{26} +5.27972 q^{27} -8.60011 q^{28} +1.31247 q^{29} +0.757667 q^{31} +5.95060 q^{32} +3.70565 q^{33} -0.320102 q^{36} +8.20380 q^{37} -19.4212 q^{38} +9.16010 q^{39} +0.0130696 q^{41} -10.4881 q^{42} +0.330596 q^{43} -6.91352 q^{44} -12.1172 q^{46} -11.3395 q^{47} +0.446067 q^{48} +0.329174 q^{49} -17.0897 q^{52} +8.04151 q^{53} -12.0126 q^{54} +7.24803 q^{56} -14.5342 q^{57} -2.98619 q^{58} -13.1546 q^{59} +7.20639 q^{61} -1.72387 q^{62} +0.272797 q^{63} -13.0151 q^{64} -8.43123 q^{66} -2.67143 q^{67} -9.06814 q^{69} -1.87106 q^{71} +0.269777 q^{72} -11.9780 q^{73} -18.6656 q^{74} +27.1160 q^{76} +5.89183 q^{77} -20.8414 q^{78} -17.0477 q^{79} -8.68755 q^{81} -0.0297365 q^{82} +3.35585 q^{83} +14.6435 q^{84} -0.752183 q^{86} -2.23477 q^{87} +5.82660 q^{88} -10.8909 q^{89} +14.5642 q^{91} +16.9182 q^{92} -1.29009 q^{93} +25.8001 q^{94} -10.1322 q^{96} +10.9119 q^{97} -0.748948 q^{98} +0.219298 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 3 q^{3} + 21 q^{4} + 9 q^{6} + 6 q^{7} - 12 q^{8} + 21 q^{9} + 6 q^{11} + 6 q^{12} - 9 q^{13} + 18 q^{14} + 39 q^{16} + 9 q^{18} + 27 q^{19} + 6 q^{21} + 15 q^{22} + 18 q^{23} + 36 q^{24}+ \cdots + 6 q^{99}+O(q^{100}) $ 12 * q - 3 * q^2 + 3 * q^3 + 21 * q^4 + 9 * q^6 + 6 * q^7 - 12 * q^8 + 21 * q^9 + 6 * q^11 + 6 * q^12 - 9 * q^13 + 18 * q^14 + 39 * q^16 + 9 * q^18 + 27 * q^19 + 6 * q^21 + 15 * q^22 + 18 * q^23 + 36 * q^24 + 6 * q^26 + 12 * q^27 + 39 * q^28 + 6 * q^31 - 12 * q^32 + 48 * q^36 + 21 * q^37 + 3 * q^38 + 9 * q^39 + 15 * q^41 - 30 * q^42 - 45 * q^43 + 42 * q^44 + 15 * q^46 - 3 * q^47 - 63 * q^48 + 12 * q^49 - 51 * q^52 + 12 * q^53 + 27 * q^54 - 3 * q^56 + 6 * q^57 - 51 * q^58 + 36 * q^59 - 3 * q^61 + 36 * q^62 + 72 * q^63 + 42 * q^64 - 39 * q^66 - 78 * q^67 - 66 * q^69 + 30 * q^71 + 12 * q^72 - 33 * q^73 - 9 * q^74 + 27 * q^76 + 39 * q^77 - 24 * q^78 + 3 * q^79 + 12 * q^81 - 51 * q^82 - 27 * q^83 + 27 * q^84 + 18 * q^86 - 24 * q^87 + 36 * q^88 + 6 * q^89 + 24 * q^91 - 3 * q^92 + 12 * q^93 + 21 * q^94 + 39 * q^96 - 6 * q^97 + 42 * q^98 + 6 * q^99

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 $

$2$	−2.27524	−1.60884	−0.804418	−	0.594064i	$-0.797522\pi$
$2$	−2.27524	−1.60884	−0.804418	+	0.594064i	$0.797522\pi$
$3$	−1.70271	−0.983062	−0.491531	−	0.870860i	$-0.663563\pi$
$3$	−1.70271	−0.983062	−0.491531	+	0.870860i	$0.663563\pi$
$4$	3.17670	1.58835
$5$	0	0
$5$	0	0
$6$	3.87408	1.58159
$7$	−2.70724	−1.02324	−0.511621	−	0.859211i	$-0.670955\pi$
$7$	−2.70724	−1.02324	−0.511621	+	0.859211i	$0.670955\pi$
$8$	−2.67727	−0.946559
$9$	−0.100765	−0.0335885
$10$	0	0
$11$	−2.17632	−0.656185	−0.328093	−	0.944646i	$-0.606406\pi$
$11$	−2.17632	−0.656185	−0.328093	+	0.944646i	$0.606406\pi$
$12$	−5.40901	−1.56145
$13$	−5.37970	−1.49206	−0.746031	−	0.665912i	$-0.768043\pi$
$13$	−5.37970	−1.49206	−0.746031	+	0.665912i	$0.768043\pi$
$14$	6.15962	1.64623
$15$	0	0
$16$	−0.261974	−0.0654935
$17$	0	0
$17$	0	0
$18$	0.229265	0.0540384
$19$	8.53588	1.95827	0.979133	−	0.203220i	$-0.0651406\pi$
$19$	8.53588	1.95827	0.979133	+	0.203220i	$0.0651406\pi$
$20$	0	0
$21$	4.60966	1.00591
$22$	4.95164	1.05569
$23$	5.32570	1.11049	0.555243	−	0.831688i	$-0.312625\pi$
$23$	5.32570	1.11049	0.555243	+	0.831688i	$0.312625\pi$
$24$	4.55863	0.930526
$25$	0	0
$26$	12.2401	2.40048
$27$	5.27972	1.01608
$28$	−8.60011	−1.62527
$29$	1.31247	0.243720	0.121860	−	0.992547i	$-0.461114\pi$
$29$	1.31247	0.243720	0.121860	+	0.992547i	$0.461114\pi$
$30$	0	0
$31$	0.757667	0.136081	0.0680405	−	0.997683i	$-0.478325\pi$
$31$	0.757667	0.136081	0.0680405	+	0.997683i	$0.478325\pi$
$32$	5.95060	1.05193
$33$	3.70565	0.645071
$34$	0	0
$35$	0	0
$36$	−0.320102	−0.0533503
$37$	8.20380	1.34870	0.674348	−	0.738414i	$-0.264425\pi$
$37$	8.20380	1.34870	0.674348	+	0.738414i	$0.264425\pi$
$38$	−19.4212	−3.15053
$39$	9.16010	1.46679
$40$	0	0
$41$	0.0130696	0.00204113	0.00102057	−	0.999999i	$-0.499675\pi$
$41$	0.0130696	0.00204113	0.00102057	+	0.999999i	$0.499675\pi$
$42$	−10.4881	−1.61834
$43$	0.330596	0.0504154	0.0252077	−	0.999682i	$-0.491975\pi$
$43$	0.330596	0.0504154	0.0252077	+	0.999682i	$0.491975\pi$
$44$	−6.91352	−1.04225
$45$	0	0
$46$	−12.1172	−1.78659
$47$	−11.3395	−1.65404	−0.827019	−	0.562175i	$-0.809965\pi$
$47$	−11.3395	−1.65404	−0.827019	+	0.562175i	$0.809965\pi$
$48$	0.446067	0.0643842
$49$	0.329174	0.0470248
$50$	0	0
$51$	0	0
$52$	−17.0897	−2.36992
$53$	8.04151	1.10459	0.552293	−	0.833650i	$-0.313753\pi$
$53$	8.04151	1.10459	0.552293	+	0.833650i	$0.313753\pi$
$54$	−12.0126	−1.63471
$55$	0	0
$56$	7.24803	0.968559
$57$	−14.5342	−1.92510
$58$	−2.98619	−0.392106
$59$	−13.1546	−1.71259	−0.856293	−	0.516490i	$-0.827238\pi$
$59$	−13.1546	−1.71259	−0.856293	+	0.516490i	$0.827238\pi$
$60$	0	0
$61$	7.20639	0.922684	0.461342	−	0.887222i	$-0.347368\pi$
$61$	7.20639	0.922684	0.461342	+	0.887222i	$0.347368\pi$
$62$	−1.72387	−0.218932
$63$	0.272797	0.0343692
$64$	−13.0151	−1.62688
$65$	0	0
$66$	−8.43123	−1.03781
$67$	−2.67143	−0.326367	−0.163184	−	0.986596i	$-0.552176\pi$
$67$	−2.67143	−0.326367	−0.163184	+	0.986596i	$0.552176\pi$
$68$	0	0
$69$	−9.06814	−1.09168
$70$	0	0
$71$	−1.87106	−0.222054	−0.111027	−	0.993817i	$-0.535414\pi$
$71$	−1.87106	−0.222054	−0.111027	+	0.993817i	$0.535414\pi$
$72$	0.269777	0.0317935
$73$	−11.9780	−1.40192	−0.700962	−	0.713199i	$-0.747246\pi$
$73$	−11.9780	−1.40192	−0.700962	+	0.713199i	$0.747246\pi$
$74$	−18.6656	−2.16983
$75$	0	0
$76$	27.1160	3.11041
$77$	5.89183	0.671437
$78$	−20.8414	−2.35982
$79$	−17.0477	−1.91802	−0.959009	−	0.283375i	$-0.908546\pi$
$79$	−17.0477	−1.91802	−0.959009	+	0.283375i	$0.908546\pi$
$80$	0	0
$81$	−8.68755	−0.965283
$82$	−0.0297365	−0.00328385
$83$	3.35585	0.368352	0.184176	−	0.982893i	$-0.441038\pi$
$83$	3.35585	0.368352	0.184176	+	0.982893i	$0.441038\pi$
$84$	14.6435	1.59774
$85$	0	0
$86$	−0.752183	−0.0811100
$87$	−2.23477	−0.239592
$88$	5.82660	0.621118
$89$	−10.8909	−1.15444	−0.577218	−	0.816590i	$-0.695862\pi$
$89$	−10.8909	−1.15444	−0.577218	+	0.816590i	$0.695862\pi$
$90$	0	0
$91$	14.5642	1.52674
$92$	16.9182	1.76384
$93$	−1.29009	−0.133776
$94$	25.8001	2.66107
$95$	0	0
$96$	−10.1322	−1.03411
$97$	10.9119	1.10793	0.553965	−	0.832540i	$-0.313114\pi$
$97$	10.9119	1.10793	0.553965	+	0.832540i	$0.313114\pi$
$98$	−0.748948	−0.0756552
$99$	0.219298	0.0220403
$100$	0	0
$101$	−3.95599	−0.393636	−0.196818	−	0.980440i	$-0.563061\pi$
$101$	−3.95599	−0.393636	−0.196818	+	0.980440i	$0.563061\pi$
$102$	0	0
$103$	−7.16779	−0.706263	−0.353132	−	0.935574i	$-0.614883\pi$
$103$	−7.16779	−0.706263	−0.353132	+	0.935574i	$0.614883\pi$
$104$	14.4029	1.41232
$105$	0	0
$106$	−18.2963	−1.77710
$107$	3.70041	0.357732	0.178866	−	0.983873i	$-0.442757\pi$
$107$	3.70041	0.357732	0.178866	+	0.983873i	$0.442757\pi$
$108$	16.7721	1.61389
$109$	−2.73869	−0.262319	−0.131160	−	0.991361i	$-0.541870\pi$
$109$	−2.73869	−0.262319	−0.131160	+	0.991361i	$0.541870\pi$
$110$	0	0
$111$	−13.9687	−1.32585
$112$	0.709227	0.0670157
$113$	1.51487	0.142507	0.0712536	−	0.997458i	$-0.477300\pi$
$113$	1.51487	0.142507	0.0712536	+	0.997458i	$0.477300\pi$
$114$	33.0687	3.09716
$115$	0	0
$116$	4.16934	0.387114
$117$	0.542088	0.0501161
$118$	29.9299	2.75527
$119$	0	0
$120$	0	0
$121$	−6.26363	−0.569421
$122$	−16.3963	−1.48445
$123$	−0.0222538	−0.00200656
$124$	2.40688	0.216144
$125$	0	0
$126$	−0.620677	−0.0552943
$127$	−4.15113	−0.368353	−0.184176	−	0.982893i	$-0.558962\pi$
$127$	−4.15113	−0.368353	−0.184176	+	0.982893i	$0.558962\pi$
$128$	17.7112	1.56546
$129$	−0.562910	−0.0495614
$130$	0	0
$131$	12.3436	1.07847	0.539234	−	0.842156i	$-0.318714\pi$
$131$	12.3436	1.07847	0.539234	+	0.842156i	$0.318714\pi$
$132$	11.7717	1.02460
$133$	−23.1087	−2.00378
$134$	6.07814	0.525071
$135$	0	0
$136$	0	0
$137$	−13.1700	−1.12519	−0.562594	−	0.826733i	$-0.690197\pi$
$137$	−13.1700	−1.12519	−0.562594	+	0.826733i	$0.690197\pi$
$138$	20.6322	1.75633
$139$	−0.706325	−0.0599097	−0.0299548	−	0.999551i	$-0.509536\pi$
$139$	−0.706325	−0.0599097	−0.0299548	+	0.999551i	$0.509536\pi$
$140$	0	0
$141$	19.3079	1.62602
$142$	4.25710	0.357248
$143$	11.7080	0.979069
$144$	0.0263979	0.00219983
$145$	0	0
$146$	27.2529	2.25546
$147$	−0.560489	−0.0462283
$148$	26.0610	2.14220
$149$	7.41327	0.607319	0.303659	−	0.952781i	$-0.401792\pi$
$149$	7.41327	0.607319	0.303659	+	0.952781i	$0.401792\pi$
$150$	0	0
$151$	−2.19280	−0.178448	−0.0892238	−	0.996012i	$-0.528439\pi$
$151$	−2.19280	−0.178448	−0.0892238	+	0.996012i	$0.528439\pi$
$152$	−22.8529	−1.85361
$153$	0	0
$154$	−13.4053	−1.08023
$155$	0	0
$156$	29.0989	2.32978
$157$	−5.33731	−0.425963	−0.212982	−	0.977056i	$-0.568317\pi$
$157$	−5.33731	−0.425963	−0.212982	+	0.977056i	$0.568317\pi$
$158$	38.7876	3.08578
$159$	−13.6924	−1.08588
$160$	0	0
$161$	−14.4180	−1.13630
$162$	19.7662	1.55298
$163$	−7.78483	−0.609755	−0.304878	−	0.952392i	$-0.598616\pi$
$163$	−7.78483	−0.609755	−0.304878	+	0.952392i	$0.598616\pi$
$164$	0.0415183	0.00324204
$165$	0	0
$166$	−7.63534	−0.592617
$167$	−4.42165	−0.342157	−0.171079	−	0.985257i	$-0.554725\pi$
$167$	−4.42165	−0.342157	−0.171079	+	0.985257i	$0.554725\pi$
$168$	−12.3413	−0.952154
$169$	15.9412	1.22625
$170$	0	0
$171$	−0.860123	−0.0657752
$172$	1.05020	0.0800773
$173$	−14.6188	−1.11145	−0.555724	−	0.831367i	$-0.687559\pi$
$173$	−14.6188	−1.11145	−0.555724	+	0.831367i	$0.687559\pi$
$174$	5.08463	0.385465
$175$	0	0
$176$	0.570139	0.0429759
$177$	22.3986	1.68358
$178$	24.7794	1.85730
$179$	−5.42912	−0.405791	−0.202896	−	0.979200i	$-0.565035\pi$
$179$	−5.42912	−0.405791	−0.202896	+	0.979200i	$0.565035\pi$
$180$	0	0
$181$	18.3483	1.36382	0.681909	−	0.731437i	$-0.261150\pi$
$181$	18.3483	1.36382	0.681909	+	0.731437i	$0.261150\pi$
$182$	−33.1369	−2.45627
$183$	−12.2704	−0.907056
$184$	−14.2583	−1.05114
$185$	0	0
$186$	2.93526	0.215224
$187$	0	0
$188$	−36.0222	−2.62719
$189$	−14.2935	−1.03970
$190$	0	0
$191$	4.91654	0.355748	0.177874	−	0.984053i	$-0.443078\pi$
$191$	4.91654	0.355748	0.177874	+	0.984053i	$0.443078\pi$
$192$	22.1609	1.59933
$193$	−17.5487	−1.26318	−0.631591	−	0.775302i	$-0.717598\pi$
$193$	−17.5487	−1.26318	−0.631591	+	0.775302i	$0.717598\pi$
$194$	−24.8270	−1.78248
$195$	0	0
$196$	1.04569	0.0746919
$197$	21.7620	1.55048	0.775239	−	0.631668i	$-0.217629\pi$
$197$	21.7620	1.55048	0.775239	+	0.631668i	$0.217629\pi$
$198$	−0.498955	−0.0354592
$199$	−0.241280	−0.0171039	−0.00855196	−	0.999963i	$-0.502722\pi$
$199$	−0.241280	−0.0171039	−0.00855196	+	0.999963i	$0.502722\pi$
$200$	0	0
$201$	4.54869	0.320840
$202$	9.00082	0.633295
$203$	−3.55319	−0.249385
$204$	0	0
$205$	0	0
$206$	16.3084	1.13626
$207$	−0.536647	−0.0372995
$208$	1.40934	0.0977203
$209$	−18.5768	−1.28499
$210$	0	0
$211$	−9.41485	−0.648145	−0.324072	−	0.946032i	$-0.605052\pi$
$211$	−9.41485	−0.648145	−0.324072	+	0.946032i	$0.605052\pi$
$212$	25.5455	1.75447
$213$	3.18588	0.218293
$214$	−8.41931	−0.575532
$215$	0	0
$216$	−14.1352	−0.961781
$217$	−2.05119	−0.139244
$218$	6.23118	0.422029
$219$	20.3952	1.37818
$220$	0	0
$221$	0	0
$222$	31.7821	2.13308
$223$	14.8338	0.993345	0.496672	−	0.867938i	$-0.334555\pi$
$223$	14.8338	0.993345	0.496672	+	0.867938i	$0.334555\pi$
$224$	−16.1097	−1.07638
$225$	0	0
$226$	−3.44669	−0.229271
$227$	9.83158	0.652545	0.326272	−	0.945276i	$-0.394207\pi$
$227$	9.83158	0.652545	0.326272	+	0.945276i	$0.394207\pi$
$228$	−46.1707	−3.05773
$229$	−12.4315	−0.821500	−0.410750	−	0.911748i	$-0.634733\pi$
$229$	−12.4315	−0.821500	−0.410750	+	0.911748i	$0.634733\pi$
$230$	0	0
$231$	−10.0321	−0.660064
$232$	−3.51385	−0.230696
$233$	12.5812	0.824224	0.412112	−	0.911133i	$-0.364791\pi$
$233$	12.5812	0.824224	0.412112	+	0.911133i	$0.364791\pi$
$234$	−1.23338	−0.0806285
$235$	0	0
$236$	−41.7883	−2.72019
$237$	29.0274	1.88553
$238$	0	0
$239$	−15.7966	−1.02179	−0.510897	−	0.859642i	$-0.670687\pi$
$239$	−15.7966	−1.02179	−0.510897	+	0.859642i	$0.670687\pi$
$240$	0	0
$241$	8.45284	0.544496	0.272248	−	0.962227i	$-0.412233\pi$
$241$	8.45284	0.544496	0.272248	+	0.962227i	$0.412233\pi$
$242$	14.2512	0.916104
$243$	−1.04674	−0.0671482
$244$	22.8926	1.46555
$245$	0	0
$246$	0.0506328	0.00322823
$247$	−45.9205	−2.92185
$248$	−2.02848	−0.128809
$249$	−5.71404	−0.362113
$250$	0	0
$251$	14.7900	0.933540	0.466770	−	0.884379i	$-0.345418\pi$
$251$	14.7900	0.933540	0.466770	+	0.884379i	$0.345418\pi$
$252$	0.866594	0.0545903
$253$	−11.5904	−0.728684
$254$	9.44480	0.592619
$255$	0	0
$256$	−14.2669	−0.891684
$257$	−13.8681	−0.865070	−0.432535	−	0.901617i	$-0.642381\pi$
$257$	−13.8681	−0.865070	−0.432535	+	0.901617i	$0.642381\pi$
$258$	1.28075	0.0797362
$259$	−22.2097	−1.38004
$260$	0	0
$261$	−0.132252	−0.00818620
$262$	−28.0847	−1.73508
$263$	−0.382638	−0.0235945	−0.0117972	−	0.999930i	$-0.503755\pi$
$263$	−0.382638	−0.0235945	−0.0117972	+	0.999930i	$0.503755\pi$
$264$	−9.92104	−0.610598
$265$	0	0
$266$	52.5778	3.22375
$267$	18.5441	1.13488
$268$	−8.48634	−0.518386
$269$	21.8946	1.33494	0.667468	−	0.744638i	$-0.267378\pi$
$269$	21.8946	1.33494	0.667468	+	0.744638i	$0.267378\pi$
$270$	0	0
$271$	16.9450	1.02933	0.514667	−	0.857390i	$-0.327916\pi$
$271$	16.9450	1.02933	0.514667	+	0.857390i	$0.327916\pi$
$272$	0	0
$273$	−24.7986	−1.50088
$274$	29.9648	1.81024
$275$	0	0
$276$	−28.8068	−1.73396
$277$	−24.4581	−1.46954	−0.734771	−	0.678315i	$-0.762710\pi$
$277$	−24.4581	−1.46954	−0.734771	+	0.678315i	$0.762710\pi$
$278$	1.60706	0.0963848
$279$	−0.0763467	−0.00457076
$280$	0	0
$281$	−13.2014	−0.787529	−0.393765	−	0.919211i	$-0.628827\pi$
$281$	−13.2014	−0.787529	−0.393765	+	0.919211i	$0.628827\pi$
$282$	−43.9301	−2.61600
$283$	13.7732	0.818730	0.409365	−	0.912371i	$-0.365750\pi$
$283$	13.7732	0.818730	0.409365	+	0.912371i	$0.365750\pi$
$284$	−5.94379	−0.352699
$285$	0	0
$286$	−26.6384	−1.57516
$287$	−0.0353827	−0.00208857
$288$	−0.599615	−0.0353326
$289$	0	0
$290$	0	0
$291$	−18.5798	−1.08916
$292$	−38.0507	−2.22675
$293$	−21.1370	−1.23483	−0.617417	−	0.786636i	$-0.711821\pi$
$293$	−21.1370	−1.23483	−0.617417	+	0.786636i	$0.711821\pi$
$294$	1.27524	0.0743737
$295$	0	0
$296$	−21.9638	−1.27662
$297$	−11.4904	−0.666738
$298$	−16.8669	−0.977076
$299$	−28.6507	−1.65691
$300$	0	0
$301$	−0.895003	−0.0515871
$302$	4.98914	0.287093
$303$	6.73592	0.386969
$304$	−2.23618	−0.128254
$305$	0	0
$306$	0	0
$307$	23.2068	1.32448	0.662242	−	0.749290i	$-0.269605\pi$
$307$	23.2068	1.32448	0.662242	+	0.749290i	$0.269605\pi$
$308$	18.7166	1.06648
$309$	12.2047	0.694301
$310$	0	0
$311$	19.0929	1.08266	0.541330	−	0.840810i	$-0.317921\pi$
$311$	19.0929	1.08266	0.541330	+	0.840810i	$0.317921\pi$
$312$	−24.5241	−1.38840
$313$	−31.9061	−1.80344	−0.901720	−	0.432320i	$-0.857695\pi$
$313$	−31.9061	−1.80344	−0.901720	+	0.432320i	$0.857695\pi$
$314$	12.1436	0.685305
$315$	0	0
$316$	−54.1555	−3.04649
$317$	−7.35179	−0.412917	−0.206459	−	0.978455i	$-0.566194\pi$
$317$	−7.35179	−0.412917	−0.206459	+	0.978455i	$0.566194\pi$
$318$	31.1534	1.74700
$319$	−2.85637	−0.159926
$320$	0	0
$321$	−6.30074	−0.351673
$322$	32.8043	1.82811
$323$	0	0
$324$	−27.5977	−1.53321
$325$	0	0
$326$	17.7123	0.980995
$327$	4.66321	0.257876
$328$	−0.0349910	−0.00193205
$329$	30.6988	1.69248
$330$	0	0
$331$	16.3546	0.898930	0.449465	−	0.893298i	$-0.351615\pi$
$331$	16.3546	0.898930	0.449465	+	0.893298i	$0.351615\pi$
$332$	10.6605	0.585072
$333$	−0.826660	−0.0453007
$334$	10.0603	0.550475
$335$	0	0
$336$	−1.20761	−0.0658806
$337$	2.73232	0.148839	0.0744196	−	0.997227i	$-0.476290\pi$
$337$	2.73232	0.148839	0.0744196	+	0.997227i	$0.476290\pi$
$338$	−36.2700	−1.97283
$339$	−2.57939	−0.140093
$340$	0	0
$341$	−1.64893	−0.0892944
$342$	1.95698	0.105821
$343$	18.0596	0.975125
$344$	−0.885095	−0.0477211
$345$	0	0
$346$	33.2613	1.78814
$347$	−18.9609	−1.01788	−0.508938	−	0.860803i	$-0.669962\pi$
$347$	−18.9609	−1.01788	−0.508938	+	0.860803i	$0.669962\pi$
$348$	−7.09919	−0.380557
$349$	−32.3382	−1.73102	−0.865511	−	0.500890i	$-0.833006\pi$
$349$	−32.3382	−1.73102	−0.865511	+	0.500890i	$0.833006\pi$
$350$	0	0
$351$	−28.4033	−1.51606
$352$	−12.9504	−0.690259
$353$	13.0916	0.696796	0.348398	−	0.937347i	$-0.386726\pi$
$353$	13.0916	0.696796	0.348398	+	0.937347i	$0.386726\pi$
$354$	−50.9620	−2.70860
$355$	0	0
$356$	−34.5972	−1.83365
$357$	0	0
$358$	12.3525	0.652851
$359$	8.64264	0.456141	0.228071	−	0.973645i	$-0.426758\pi$
$359$	8.64264	0.456141	0.228071	+	0.973645i	$0.426758\pi$
$360$	0	0
$361$	53.8613	2.83481
$362$	−41.7467	−2.19416
$363$	10.6652	0.559776
$364$	46.2660	2.42500
$365$	0	0
$366$	27.9181	1.45930
$367$	−6.61035	−0.345057	−0.172529	−	0.985005i	$-0.555194\pi$
$367$	−6.61035	−0.345057	−0.172529	+	0.985005i	$0.555194\pi$
$368$	−1.39519	−0.0727295
$369$	−0.00131697	−6.85586e−5 0
$370$	0	0
$371$	−21.7703	−1.13026
$372$	−4.09823	−0.212483
$373$	7.14349	0.369876	0.184938	−	0.982750i	$-0.440792\pi$
$373$	7.14349	0.369876	0.184938	+	0.982750i	$0.440792\pi$
$374$	0	0
$375$	0	0
$376$	30.3589	1.56564
$377$	−7.06073	−0.363646
$378$	32.5211	1.67270
$379$	−7.19225	−0.369441	−0.184720	−	0.982791i	$-0.559138\pi$
$379$	−7.19225	−0.369441	−0.184720	+	0.982791i	$0.559138\pi$
$380$	0	0
$381$	7.06818	0.362114
$382$	−11.1863	−0.572340
$383$	12.2903	0.628006	0.314003	−	0.949422i	$-0.398330\pi$
$383$	12.2903	0.628006	0.314003	+	0.949422i	$0.398330\pi$
$384$	−30.1570	−1.53895
$385$	0	0
$386$	39.9274	2.03225
$387$	−0.0333126	−0.00169338
$388$	34.6637	1.75978
$389$	10.6413	0.539536	0.269768	−	0.962925i	$-0.413053\pi$
$389$	10.6413	0.539536	0.269768	+	0.962925i	$0.413053\pi$
$390$	0	0
$391$	0	0
$392$	−0.881288	−0.0445117
$393$	−21.0177	−1.06020
$394$	−49.5137	−2.49446
$395$	0	0
$396$	0.696644	0.0350077
$397$	24.0554	1.20731	0.603654	−	0.797247i	$-0.293711\pi$
$397$	24.0554	1.20731	0.603654	+	0.797247i	$0.293711\pi$
$398$	0.548970	0.0275174
$399$	39.3476	1.96984
$400$	0	0
$401$	22.9193	1.14453	0.572267	−	0.820067i	$-0.306064\pi$
$401$	22.9193	1.14453	0.572267	+	0.820067i	$0.306064\pi$
$402$	−10.3493	−0.516178
$403$	−4.07603	−0.203041
$404$	−12.5670	−0.625232
$405$	0	0
$406$	8.08435	0.401220
$407$	−17.8541	−0.884995
$408$	0	0
$409$	−7.50913	−0.371303	−0.185651	−	0.982616i	$-0.559439\pi$
$409$	−7.50913	−0.371303	−0.185651	+	0.982616i	$0.559439\pi$
$410$	0	0
$411$	22.4247	1.10613
$412$	−22.7699	−1.12179
$413$	35.6128	1.75239
$414$	1.22100	0.0600088
$415$	0	0
$416$	−32.0124	−1.56954
$417$	1.20267	0.0588949
$418$	42.2667	2.06733
$419$	13.3993	0.654598	0.327299	−	0.944921i	$-0.393862\pi$
$419$	13.3993	0.654598	0.327299	+	0.944921i	$0.393862\pi$
$420$	0	0
$421$	−20.5078	−0.999491	−0.499745	−	0.866172i	$-0.666573\pi$
$421$	−20.5078	−0.999491	−0.499745	+	0.866172i	$0.666573\pi$
$422$	21.4210	1.04276
$423$	1.14263	0.0555566
$424$	−21.5293	−1.04556
$425$	0	0
$426$	−7.24862	−0.351197
$427$	−19.5095	−0.944130
$428$	11.7551	0.568204
$429$	−19.9353	−0.962486
$430$	0	0
$431$	8.98646	0.432863	0.216431	−	0.976298i	$-0.430558\pi$
$431$	8.98646	0.432863	0.216431	+	0.976298i	$0.430558\pi$
$432$	−1.38315	−0.0665467
$433$	−26.3738	−1.26744	−0.633721	−	0.773562i	$-0.718473\pi$
$433$	−26.3738	−1.26744	−0.633721	+	0.773562i	$0.718473\pi$
$434$	4.66694	0.224020
$435$	0	0
$436$	−8.70001	−0.416655
$437$	45.4596	2.17463
$438$	−46.4039	−2.21726
$439$	−29.3558	−1.40108	−0.700538	−	0.713615i	$-0.747057\pi$
$439$	−29.3558	−1.40108	−0.700538	+	0.713615i	$0.747057\pi$
$440$	0	0
$441$	−0.0331693	−0.00157949
$442$	0	0
$443$	23.4483	1.11406	0.557030	−	0.830492i	$-0.311941\pi$
$443$	23.4483	1.11406	0.557030	+	0.830492i	$0.311941\pi$
$444$	−44.3744	−2.10592
$445$	0	0
$446$	−33.7504	−1.59813
$447$	−12.6227	−0.597032
$448$	35.2350	1.66470
$449$	−11.8222	−0.557926	−0.278963	−	0.960302i	$-0.589991\pi$
$449$	−11.8222	−0.557926	−0.278963	+	0.960302i	$0.589991\pi$
$450$	0	0
$451$	−0.0284437	−0.00133936
$452$	4.81230	0.226351
$453$	3.73371	0.175425
$454$	−22.3692	−1.04984
$455$	0	0
$456$	38.9119	1.82222
$457$	3.31478	0.155059	0.0775293	−	0.996990i	$-0.475297\pi$
$457$	3.31478	0.155059	0.0775293	+	0.996990i	$0.475297\pi$
$458$	28.2847	1.32166
$459$	0	0
$460$	0	0
$461$	36.2559	1.68860	0.844302	−	0.535868i	$-0.180016\pi$
$461$	36.2559	1.68860	0.844302	+	0.535868i	$0.180016\pi$
$462$	22.8254	1.06193
$463$	−10.9581	−0.509268	−0.254634	−	0.967038i	$-0.581955\pi$
$463$	−10.9581	−0.509268	−0.254634	+	0.967038i	$0.581955\pi$
$464$	−0.343834	−0.0159621
$465$	0	0
$466$	−28.6253	−1.32604
$467$	34.3919	1.59147	0.795734	−	0.605647i	$-0.207086\pi$
$467$	34.3919	1.59147	0.795734	+	0.605647i	$0.207086\pi$
$468$	1.72205	0.0796019
$469$	7.23222	0.333953
$470$	0	0
$471$	9.08791	0.418749
$472$	35.2185	1.62106
$473$	−0.719482	−0.0330818
$474$	−66.0442	−3.03351
$475$	0	0
$476$	0	0
$477$	−0.810307	−0.0371014
$478$	35.9409	1.64390
$479$	27.5154	1.25721	0.628606	−	0.777724i	$-0.283626\pi$
$479$	27.5154	1.25721	0.628606	+	0.777724i	$0.283626\pi$
$480$	0	0
$481$	−44.1340	−2.01234
$482$	−19.2322	−0.876004
$483$	24.5497	1.11705
$484$	−19.8977	−0.904440
$485$	0	0
$486$	2.38158	0.108030
$487$	−10.6229	−0.481368	−0.240684	−	0.970604i	$-0.577372\pi$
$487$	−10.6229	−0.481368	−0.240684	+	0.970604i	$0.577372\pi$
$488$	−19.2935	−0.873375
$489$	13.2553	0.599427
$490$	0	0
$491$	36.2103	1.63415	0.817074	−	0.576533i	$-0.195595\pi$
$491$	36.2103	1.63415	0.817074	+	0.576533i	$0.195595\pi$
$492$	−0.0706938	−0.00318712
$493$	0	0
$494$	104.480	4.70078
$495$	0	0
$496$	−0.198489	−0.00891242
$497$	5.06541	0.227215
$498$	13.0008	0.582580
$499$	−20.8740	−0.934447	−0.467224	−	0.884139i	$-0.654746\pi$
$499$	−20.8740	−0.934447	−0.467224	+	0.884139i	$0.654746\pi$
$500$	0	0
$501$	7.52880	0.336362
$502$	−33.6509	−1.50191
$503$	−7.63988	−0.340645	−0.170323	−	0.985388i	$-0.554481\pi$
$503$	−7.63988	−0.340645	−0.170323	+	0.985388i	$0.554481\pi$
$504$	−0.730351	−0.0325324
$505$	0	0
$506$	26.3710	1.17233
$507$	−27.1433	−1.20548
$508$	−13.1869	−0.585074
$509$	−20.9805	−0.929945	−0.464972	−	0.885325i	$-0.653936\pi$
$509$	−20.9805	−0.929945	−0.464972	+	0.885325i	$0.653936\pi$
$510$	0	0
$511$	32.4275	1.43451
$512$	−2.96165	−0.130888
$513$	45.0671	1.98976
$514$	31.5533	1.39175
$515$	0	0
$516$	−1.78820	−0.0787209
$517$	24.6784	1.08536
$518$	50.5323	2.22026
$519$	24.8917	1.09262
$520$	0	0
$521$	−6.11774	−0.268023	−0.134012	−	0.990980i	$-0.542786\pi$
$521$	−6.11774	−0.268023	−0.134012	+	0.990980i	$0.542786\pi$
$522$	0.300905	0.0131703
$523$	5.40916	0.236526	0.118263	−	0.992982i	$-0.462267\pi$
$523$	5.40916	0.236526	0.118263	+	0.992982i	$0.462267\pi$
$524$	39.2120	1.71298
$525$	0	0
$526$	0.870591	0.0379596
$527$	0	0
$528$	−0.970784	−0.0422480
$529$	5.36307	0.233177
$530$	0	0
$531$	1.32553	0.0575232
$532$	−73.4095	−3.18271
$533$	−0.0703107	−0.00304550
$534$	−42.1923	−1.82584
$535$	0	0
$536$	7.15215	0.308926
$537$	9.24423	0.398918
$538$	−49.8154	−2.14769
$539$	−0.716388	−0.0308570
$540$	0	0
$541$	−20.4944	−0.881124	−0.440562	−	0.897722i	$-0.645221\pi$
$541$	−20.4944	−0.881124	−0.440562	+	0.897722i	$0.645221\pi$
$542$	−38.5538	−1.65603
$543$	−31.2419	−1.34072
$544$	0	0
$545$	0	0
$546$	56.4227	2.41467
$547$	−10.1545	−0.434175	−0.217088	−	0.976152i	$-0.569656\pi$
$547$	−10.1545	−0.434175	−0.217088	+	0.976152i	$0.569656\pi$
$548$	−41.8371	−1.78719
$549$	−0.726156	−0.0309916
$550$	0	0
$551$	11.2031	0.477270
$552$	24.2779	1.03334
$553$	46.1523	1.96260
$554$	55.6479	2.36425
$555$	0	0
$556$	−2.24378	−0.0951575
$557$	−30.1117	−1.27587	−0.637937	−	0.770088i	$-0.720212\pi$
$557$	−30.1117	−1.27587	−0.637937	+	0.770088i	$0.720212\pi$
$558$	0.173707	0.00735360
$559$	−1.77851	−0.0752228
$560$	0	0
$561$	0	0
$562$	30.0363	1.26700
$563$	−12.8482	−0.541486	−0.270743	−	0.962652i	$-0.587269\pi$
$563$	−12.8482	−0.541486	−0.270743	+	0.962652i	$0.587269\pi$
$564$	61.3355	2.58269
$565$	0	0
$566$	−31.3372	−1.31720
$567$	23.5193	0.987719
$568$	5.00933	0.210187
$569$	19.6602	0.824196	0.412098	−	0.911139i	$-0.364796\pi$
$569$	19.6602	0.824196	0.412098	+	0.911139i	$0.364796\pi$
$570$	0	0
$571$	−20.2253	−0.846403	−0.423202	−	0.906036i	$-0.639094\pi$
$571$	−20.2253	−0.846403	−0.423202	+	0.906036i	$0.639094\pi$
$572$	37.1927	1.55510
$573$	−8.37146	−0.349723
$574$	0.0805040	0.00336017
$575$	0	0
$576$	1.31147	0.0546446
$577$	−34.5597	−1.43874	−0.719369	−	0.694628i	$-0.755569\pi$
$577$	−34.5597	−1.43874	−0.719369	+	0.694628i	$0.755569\pi$
$578$	0	0
$579$	29.8804	1.24179
$580$	0	0
$581$	−9.08509	−0.376913
$582$	42.2734	1.75229
$583$	−17.5009	−0.724814
$584$	32.0685	1.32700
$585$	0	0
$586$	48.0916	1.98665
$587$	−23.7081	−0.978540	−0.489270	−	0.872132i	$-0.662737\pi$
$587$	−23.7081	−0.978540	−0.489270	+	0.872132i	$0.662737\pi$
$588$	−1.78050	−0.0734268
$589$	6.46736	0.266483
$590$	0	0
$591$	−37.0545	−1.52422
$592$	−2.14918	−0.0883308
$593$	11.5371	0.473770	0.236885	−	0.971538i	$-0.423873\pi$
$593$	11.5371	0.473770	0.236885	+	0.971538i	$0.423873\pi$
$594$	26.1433	1.07267
$595$	0	0
$596$	23.5497	0.964635
$597$	0.410831	0.0168142
$598$	65.1871	2.66570
$599$	−9.83844	−0.401988	−0.200994	−	0.979593i	$-0.564417\pi$
$599$	−9.83844	−0.401988	−0.200994	+	0.979593i	$0.564417\pi$
$600$	0	0
$601$	10.1246	0.412991	0.206496	−	0.978448i	$-0.433794\pi$
$601$	10.1246	0.412991	0.206496	+	0.978448i	$0.433794\pi$
$602$	2.03634	0.0829952
$603$	0.269188	0.0109622
$604$	−6.96587	−0.283437
$605$	0	0
$606$	−15.3258	−0.622569
$607$	0.707519	0.0287173	0.0143587	−	0.999897i	$-0.495429\pi$
$607$	0.707519	0.0287173	0.0143587	+	0.999897i	$0.495429\pi$
$608$	50.7936	2.05995
$609$	6.05007	0.245161
$610$	0	0
$611$	61.0032	2.46792
$612$	0	0
$613$	28.1775	1.13808	0.569039	−	0.822310i	$-0.307315\pi$
$613$	28.1775	1.13808	0.569039	+	0.822310i	$0.307315\pi$
$614$	−52.8010	−2.13088
$615$	0	0
$616$	−15.7740	−0.635554
$617$	39.8676	1.60501	0.802505	−	0.596645i	$-0.203500\pi$
$617$	39.8676	1.60501	0.802505	+	0.596645i	$0.203500\pi$
$618$	−27.7686	−1.11702
$619$	37.4591	1.50561	0.752804	−	0.658244i	$-0.228701\pi$
$619$	37.4591	1.50561	0.752804	+	0.658244i	$0.228701\pi$
$620$	0	0
$621$	28.1182	1.12834
$622$	−43.4409	−1.74182
$623$	29.4844	1.18127
$624$	−2.39971	−0.0960651
$625$	0	0
$626$	72.5940	2.90144
$627$	31.6310	1.26322
$628$	−16.9550	−0.676579
$629$	0	0
$630$	0	0
$631$	−10.5005	−0.418020	−0.209010	−	0.977913i	$-0.567024\pi$
$631$	−10.5005	−0.418020	−0.209010	+	0.977913i	$0.567024\pi$
$632$	45.6414	1.81552
$633$	16.0308	0.637167
$634$	16.7271	0.664316
$635$	0	0
$636$	−43.4966	−1.72475
$637$	−1.77086	−0.0701639
$638$	6.49891	0.257294
$639$	0.188538	0.00745845
$640$	0	0
$641$	44.9746	1.77639	0.888195	−	0.459466i	$-0.151959\pi$
$641$	44.9746	1.77639	0.888195	+	0.459466i	$0.151959\pi$
$642$	14.3357	0.565784
$643$	−3.17835	−0.125342	−0.0626710	−	0.998034i	$-0.519962\pi$
$643$	−3.17835	−0.125342	−0.0626710	+	0.998034i	$0.519962\pi$
$644$	−45.8016	−1.80484
$645$	0	0
$646$	0	0
$647$	−3.28494	−0.129144	−0.0645721	−	0.997913i	$-0.520568\pi$
$647$	−3.28494	−0.129144	−0.0645721	+	0.997913i	$0.520568\pi$
$648$	23.2589	0.913697
$649$	28.6287	1.12377
$650$	0	0
$651$	3.49259	0.136885
$652$	−24.7301	−0.968505
$653$	33.9120	1.32708	0.663540	−	0.748140i	$-0.269053\pi$
$653$	33.9120	1.32708	0.663540	+	0.748140i	$0.269053\pi$
$654$	−10.6099	−0.414880
$655$	0	0
$656$	−0.00342390	−0.000133681 0
$657$	1.20697	0.0470885
$658$	−69.8471	−2.72292
$659$	27.5756	1.07419	0.537097	−	0.843521i	$-0.319521\pi$
$659$	27.5756	1.07419	0.537097	+	0.843521i	$0.319521\pi$
$660$	0	0
$661$	−19.4561	−0.756756	−0.378378	−	0.925651i	$-0.623518\pi$
$661$	−19.4561	−0.756756	−0.378378	+	0.925651i	$0.623518\pi$
$662$	−37.2106	−1.44623
$663$	0	0
$664$	−8.98451	−0.348667
$665$	0	0
$666$	1.88085	0.0728813
$667$	6.98985	0.270648
$668$	−14.0463	−0.543466
$669$	−25.2577	−0.976520
$670$	0	0
$671$	−15.6834	−0.605452
$672$	27.4302	1.05814
$673$	2.96580	0.114323	0.0571616	−	0.998365i	$-0.481795\pi$
$673$	2.96580	0.114323	0.0571616	+	0.998365i	$0.481795\pi$
$674$	−6.21668	−0.239458
$675$	0	0
$676$	50.6404	1.94771
$677$	−4.50634	−0.173193	−0.0865964	−	0.996243i	$-0.527599\pi$
$677$	−4.50634	−0.173193	−0.0865964	+	0.996243i	$0.527599\pi$
$678$	5.86873	0.225387
$679$	−29.5411	−1.13368
$680$	0	0
$681$	−16.7404	−0.641492
$682$	3.75170	0.143660
$683$	−48.5495	−1.85769	−0.928847	−	0.370464i	$-0.879199\pi$
$683$	−48.5495	−1.85769	−0.928847	+	0.370464i	$0.879199\pi$
$684$	−2.73235	−0.104474
$685$	0	0
$686$	−41.0898	−1.56881
$687$	21.1674	0.807585
$688$	−0.0866074	−0.00330188
$689$	−43.2609	−1.64811
$690$	0	0
$691$	−1.63196	−0.0620827	−0.0310413	−	0.999518i	$-0.509882\pi$
$691$	−1.63196	−0.0620827	−0.0310413	+	0.999518i	$0.509882\pi$
$692$	−46.4396	−1.76537
$693$	−0.593693	−0.0225525
$694$	43.1406	1.63759
$695$	0	0
$696$	5.98309	0.226788
$697$	0	0
$698$	73.5769	2.78493
$699$	−21.4222	−0.810264
$700$	0	0
$701$	−40.4449	−1.52758	−0.763792	−	0.645463i	$-0.776664\pi$
$701$	−40.4449	−1.52758	−0.763792	+	0.645463i	$0.776664\pi$
$702$	64.6242	2.43908
$703$	70.0267	2.64111
$704$	28.3250	1.06754
$705$	0	0
$706$	−29.7865	−1.12103
$707$	10.7098	0.402785
$708$	71.1535	2.67411
$709$	−47.3049	−1.77657	−0.888286	−	0.459291i	$-0.848103\pi$
$709$	−47.3049	−1.77657	−0.888286	+	0.459291i	$0.848103\pi$
$710$	0	0
$711$	1.71782	0.0644234
$712$	29.1580	1.09274
$713$	4.03511	0.151116
$714$	0	0
$715$	0	0
$716$	−17.2467	−0.644539
$717$	26.8970	1.00449
$718$	−19.6640	−0.733856
$719$	8.17117	0.304733	0.152367	−	0.988324i	$-0.451311\pi$
$719$	8.17117	0.304733	0.152367	+	0.988324i	$0.451311\pi$
$720$	0	0
$721$	19.4050	0.722679
$722$	−122.547	−4.56074
$723$	−14.3928	−0.535273
$724$	58.2871	2.16622
$725$	0	0
$726$	−24.2658	−0.900587
$727$	18.8237	0.698133	0.349067	−	0.937098i	$-0.386499\pi$
$727$	18.8237	0.698133	0.349067	+	0.937098i	$0.386499\pi$
$728$	−38.9923	−1.44515
$729$	27.8449	1.03129
$730$	0	0
$731$	0	0
$732$	−38.9795	−1.44072
$733$	−25.9444	−0.958278	−0.479139	−	0.877739i	$-0.659051\pi$
$733$	−25.9444	−0.958278	−0.479139	+	0.877739i	$0.659051\pi$
$734$	15.0401	0.555140
$735$	0	0
$736$	31.6911	1.16815
$737$	5.81389	0.214158
$738$	0.00299641	0.000110299 0
$739$	18.9248	0.696162	0.348081	−	0.937465i	$-0.386833\pi$
$739$	18.9248	0.696162	0.348081	+	0.937465i	$0.386833\pi$
$740$	0	0
$741$	78.1895	2.87236
$742$	49.5327	1.81840
$743$	0.0747338	0.00274172	0.00137086	−	0.999999i	$-0.499564\pi$
$743$	0.0747338	0.00274172	0.00137086	+	0.999999i	$0.499564\pi$
$744$	3.45392	0.126627
$745$	0	0
$746$	−16.2531	−0.595069
$747$	−0.338153	−0.0123724
$748$	0	0
$749$	−10.0179	−0.366047
$750$	0	0
$751$	−31.3032	−1.14227	−0.571135	−	0.820856i	$-0.693497\pi$
$751$	−31.3032	−1.14227	−0.571135	+	0.820856i	$0.693497\pi$
$752$	2.97066	0.108329
$753$	−25.1832	−0.917728
$754$	16.0648	0.585046
$755$	0	0
$756$	−45.4061	−1.65140
$757$	10.9290	0.397223	0.198612	−	0.980078i	$-0.436357\pi$
$757$	10.9290	0.397223	0.198612	+	0.980078i	$0.436357\pi$
$758$	16.3641	0.594369
$759$	19.7352	0.716342
$760$	0	0
$761$	−34.7125	−1.25833	−0.629164	−	0.777272i	$-0.716603\pi$
$761$	−34.7125	−1.25833	−0.629164	+	0.777272i	$0.716603\pi$
$762$	−16.0818	−0.582582
$763$	7.41431	0.268416
$764$	15.6184	0.565053
$765$	0	0
$766$	−27.9634	−1.01036
$767$	70.7680	2.55528
$768$	24.2925	0.876581
$769$	28.5727	1.03036	0.515178	−	0.857083i	$-0.327726\pi$
$769$	28.5727	1.03036	0.515178	+	0.857083i	$0.327726\pi$
$770$	0	0
$771$	23.6135	0.850418
$772$	−55.7469	−2.00638
$773$	−31.3773	−1.12856	−0.564281	−	0.825583i	$-0.690847\pi$
$773$	−31.3773	−1.12856	−0.564281	+	0.825583i	$0.690847\pi$
$774$	0.0757941	0.00272436
$775$	0	0
$776$	−29.2140	−1.04872
$777$	37.8167	1.35667
$778$	−24.2115	−0.868025
$779$	0.111561	0.00399708
$780$	0	0
$781$	4.07202	0.145708
$782$	0	0
$783$	6.92950	0.247640
$784$	−0.0862349	−0.00307982
$785$	0	0
$786$	47.8201	1.70569
$787$	19.5079	0.695380	0.347690	−	0.937610i	$-0.386966\pi$
$787$	19.5079	0.695380	0.347690	+	0.937610i	$0.386966\pi$
$788$	69.1314	2.46270
$789$	0.651523	0.0231948
$790$	0	0
$791$	−4.10113	−0.145819
$792$	−0.587121	−0.0208624
$793$	−38.7683	−1.37670
$794$	−54.7318	−1.94236
$795$	0	0
$796$	−0.766476	−0.0271670
$797$	16.1734	0.572892	0.286446	−	0.958096i	$-0.407526\pi$
$797$	16.1734	0.572892	0.286446	+	0.958096i	$0.407526\pi$
$798$	−89.5250	−3.16915
$799$	0	0
$800$	0	0
$801$	1.09743	0.0387757
$802$	−52.1468	−1.84137
$803$	26.0681	0.919922
$804$	14.4498	0.509606
$805$	0	0
$806$	9.27392	0.326660
$807$	−37.2802	−1.31233
$808$	10.5913	0.372600
$809$	37.5852	1.32143	0.660713	−	0.750639i	$-0.270254\pi$
$809$	37.5852	1.32143	0.660713	+	0.750639i	$0.270254\pi$
$810$	0	0
$811$	34.8967	1.22539	0.612695	−	0.790320i	$-0.290085\pi$
$811$	34.8967	1.22539	0.612695	+	0.790320i	$0.290085\pi$
$812$	−11.2874	−0.396111
$813$	−28.8524	−1.01190
$814$	40.6223	1.42381
$815$	0	0
$816$	0	0
$817$	2.82193	0.0987267
$818$	17.0850	0.597365
$819$	−1.46757	−0.0512809
$820$	0	0
$821$	30.8990	1.07838	0.539191	−	0.842184i	$-0.318730\pi$
$821$	30.8990	1.07838	0.539191	+	0.842184i	$0.318730\pi$
$822$	−51.0215	−1.77958
$823$	−42.4684	−1.48035	−0.740177	−	0.672412i	$-0.765258\pi$
$823$	−42.4684	−1.48035	−0.740177	+	0.672412i	$0.765258\pi$
$824$	19.1901	0.668520
$825$	0	0
$826$	−81.0275	−2.81931
$827$	−20.4596	−0.711449	−0.355724	−	0.934591i	$-0.615766\pi$
$827$	−20.4596	−0.711449	−0.355724	+	0.934591i	$0.615766\pi$
$828$	−1.70477	−0.0592447
$829$	−38.1368	−1.32455	−0.662273	−	0.749263i	$-0.730408\pi$
$829$	−38.1368	−1.32455	−0.662273	+	0.749263i	$0.730408\pi$
$830$	0	0
$831$	41.6451	1.44465
$832$	70.0172	2.42741
$833$	0	0
$834$	−2.73636	−0.0947522
$835$	0	0
$836$	−59.0130	−2.04101
$837$	4.00027	0.138270
$838$	−30.4866	−1.05314
$839$	37.7486	1.30323	0.651613	−	0.758552i	$-0.274093\pi$
$839$	37.7486	1.30323	0.651613	+	0.758552i	$0.274093\pi$
$840$	0	0
$841$	−27.2774	−0.940600
$842$	46.6602	1.60802
$843$	22.4782	0.774190
$844$	−29.9082	−1.02948
$845$	0	0
$846$	−2.59976	−0.0893814
$847$	16.9572	0.582655
$848$	−2.10667	−0.0723432
$849$	−23.4518	−0.804863
$850$	0	0
$851$	43.6910	1.49771
$852$	10.1206	0.346725
$853$	19.5859	0.670608	0.335304	−	0.942110i	$-0.391161\pi$
$853$	19.5859	0.670608	0.335304	+	0.942110i	$0.391161\pi$
$854$	44.3887	1.51895
$855$	0	0
$856$	−9.90701	−0.338615
$857$	14.2754	0.487639	0.243820	−	0.969821i	$-0.421599\pi$
$857$	14.2754	0.487639	0.243820	+	0.969821i	$0.421599\pi$
$858$	45.3575	1.54848
$859$	50.2428	1.71426	0.857130	−	0.515099i	$-0.172245\pi$
$859$	50.2428	1.71426	0.857130	+	0.515099i	$0.172245\pi$
$860$	0	0
$861$	0.0602466	0.00205320
$862$	−20.4463	−0.696405
$863$	−13.1930	−0.449095	−0.224548	−	0.974463i	$-0.572090\pi$
$863$	−13.1930	−0.449095	−0.224548	+	0.974463i	$0.572090\pi$
$864$	31.4175	1.06884
$865$	0	0
$866$	60.0065	2.03910
$867$	0	0
$868$	−6.51602	−0.221168
$869$	37.1013	1.25858
$870$	0	0
$871$	14.3715	0.486960
$872$	7.33223	0.248301
$873$	−1.09954	−0.0372137
$874$	−103.431	−3.49861
$875$	0	0
$876$	64.7894	2.18903
$877$	−6.22918	−0.210345	−0.105172	−	0.994454i	$-0.533539\pi$
$877$	−6.22918	−0.210345	−0.105172	+	0.994454i	$0.533539\pi$
$878$	66.7914	2.25410
$879$	35.9902	1.21392
$880$	0	0
$881$	−30.3408	−1.02221	−0.511104	−	0.859519i	$-0.670763\pi$
$881$	−30.3408	−1.02221	−0.511104	+	0.859519i	$0.670763\pi$
$882$	0.0754681	0.00254114
$883$	−27.0762	−0.911185	−0.455593	−	0.890188i	$-0.650573\pi$
$883$	−27.0762	−0.911185	−0.455593	+	0.890188i	$0.650573\pi$
$884$	0	0
$885$	0	0
$886$	−53.3504	−1.79234
$887$	28.1506	0.945205	0.472603	−	0.881276i	$-0.343315\pi$
$887$	28.1506	0.945205	0.472603	+	0.881276i	$0.343315\pi$
$888$	37.3981	1.25500
$889$	11.2381	0.376914
$890$	0	0
$891$	18.9069	0.633405
$892$	47.1225	1.57778
$893$	−96.7927	−3.23905
$894$	28.7196	0.960526
$895$	0	0
$896$	−47.9484	−1.60185
$897$	48.7839	1.62885
$898$	26.8984	0.897611
$899$	0.994419	0.0331657
$900$	0	0
$901$	0	0
$902$	0.0647162	0.00215481
$903$	1.52393	0.0507134
$904$	−4.05573	−0.134891
$905$	0	0
$906$	−8.49507	−0.282230
$907$	37.7225	1.25256	0.626278	−	0.779600i	$-0.284577\pi$
$907$	37.7225	1.25256	0.626278	+	0.779600i	$0.284577\pi$
$908$	31.2320	1.03647
$909$	0.398628	0.0132216
$910$	0	0
$911$	17.9517	0.594765	0.297383	−	0.954758i	$-0.403886\pi$
$911$	17.9517	0.594765	0.297383	+	0.954758i	$0.403886\pi$
$912$	3.80757	0.126081
$913$	−7.30340	−0.241707
$914$	−7.54190	−0.249464
$915$	0	0
$916$	−39.4913	−1.30483
$917$	−33.4172	−1.10353
$918$	0	0
$919$	−25.0436	−0.826112	−0.413056	−	0.910706i	$-0.635539\pi$
$919$	−25.0436	−0.826112	−0.413056	+	0.910706i	$0.635539\pi$
$920$	0	0
$921$	−39.5146	−1.30205
$922$	−82.4906	−2.71668
$923$	10.0657	0.331318
$924$	−31.8690	−1.04841
$925$	0	0
$926$	24.9324	0.819328
$927$	0.722266	0.0237223
$928$	7.81001	0.256376
$929$	21.1573	0.694148	0.347074	−	0.937838i	$-0.387175\pi$
$929$	21.1573	0.694148	0.347074	+	0.937838i	$0.387175\pi$
$930$	0	0
$931$	2.80979	0.0920871
$932$	39.9668	1.30916
$933$	−32.5098	−1.06432
$934$	−78.2497	−2.56041
$935$	0	0
$936$	−1.45132	−0.0474378
$937$	6.56252	0.214388	0.107194	−	0.994238i	$-0.465813\pi$
$937$	6.56252	0.214388	0.107194	+	0.994238i	$0.465813\pi$
$938$	−16.4550	−0.537275
$939$	54.3270	1.77289
$940$	0	0
$941$	18.5109	0.603439	0.301719	−	0.953397i	$-0.402439\pi$
$941$	18.5109	0.603439	0.301719	+	0.953397i	$0.402439\pi$
$942$	−20.6771	−0.673698
$943$	0.0696049	0.00226665
$944$	3.44617	0.112163
$945$	0	0
$946$	1.63699	0.0532232
$947$	−30.7922	−1.00061	−0.500307	−	0.865848i	$-0.666779\pi$
$947$	−30.7922	−1.00061	−0.500307	+	0.865848i	$0.666779\pi$
$948$	92.2113	2.99488
$949$	64.4383	2.09176
$950$	0	0
$951$	12.5180	0.405924
$952$	0	0
$953$	12.7261	0.412240	0.206120	−	0.978527i	$-0.433916\pi$
$953$	12.7261	0.412240	0.206120	+	0.978527i	$0.433916\pi$
$954$	1.84364	0.0596900
$955$	0	0
$956$	−50.1809	−1.62297
$957$	4.86358	0.157217
$958$	−62.6041	−2.02265
$959$	35.6544	1.15134
$960$	0	0
$961$	−30.4259	−0.981482
$962$	100.415	3.23752
$963$	−0.372874	−0.0120157
$964$	26.8522	0.864850
$965$	0	0
$966$	−55.8563	−1.79715
$967$	−6.17919	−0.198709	−0.0993547	−	0.995052i	$-0.531678\pi$
$967$	−6.17919	−0.198709	−0.0993547	+	0.995052i	$0.531678\pi$
$968$	16.7694	0.538990
$969$	0	0
$970$	0	0
$971$	8.79368	0.282202	0.141101	−	0.989995i	$-0.454936\pi$
$971$	8.79368	0.282202	0.141101	+	0.989995i	$0.454936\pi$
$972$	−3.32517	−0.106655
$973$	1.91219	0.0613021
$974$	24.1695	0.774442
$975$	0	0
$976$	−1.88789	−0.0604298
$977$	14.5316	0.464906	0.232453	−	0.972608i	$-0.425325\pi$
$977$	14.5316	0.464906	0.232453	+	0.972608i	$0.425325\pi$
$978$	−30.1590	−0.964380
$979$	23.7021	0.757524
$980$	0	0
$981$	0.275966	0.00881091
$982$	−82.3870	−2.62907
$983$	22.7184	0.724603	0.362302	−	0.932061i	$-0.381991\pi$
$983$	22.7184	0.724603	0.362302	+	0.932061i	$0.381991\pi$
$984$	0.0595796	0.00189933
$985$	0	0
$986$	0	0
$987$	−52.2713	−1.66381
$988$	−145.876	−4.64093
$989$	1.76065	0.0559855
$990$	0	0
$991$	−3.88859	−0.123525	−0.0617626	−	0.998091i	$-0.519672\pi$
$991$	−3.88859	−0.123525	−0.0617626	+	0.998091i	$0.519672\pi$
$992$	4.50857	0.143147
$993$	−27.8472	−0.883705
$994$	−11.5250	−0.365551
$995$	0	0
$996$	−18.1518	−0.575162
$997$	−31.1914	−0.987842	−0.493921	−	0.869507i	$-0.664437\pi$
$997$	−31.1914	−0.987842	−0.493921	+	0.869507i	$0.664437\pi$
$998$	47.4932	1.50337
$999$	43.3137	1.37039

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bo.1.3		12
5.4	even	2		1445.2.a.r.1.10	✓	12
17.16	even	2		7225.2.a.bn.1.3		12
85.4	even	4		1445.2.d.i.866.5		24
85.64	even	4		1445.2.d.i.866.6		24
85.84	even	2		1445.2.a.s.1.10	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1445.2.a.r.1.10	✓	12	5.4	even	2
1445.2.a.s.1.10	yes	12	85.84	even	2
1445.2.d.i.866.5		24	85.4	even	4
1445.2.d.i.866.6		24	85.64	even	4
7225.2.a.bn.1.3		12	17.16	even	2
7225.2.a.bo.1.3		12	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7225.a (trivial)

\(f(q)\)	\(=\)	\(q-2.27524 q^{2} -1.70271 q^{3} +3.17670 q^{4} +3.87408 q^{6} -2.70724 q^{7} -2.67727 q^{8} -0.100765 q^{9} -2.17632 q^{11} -5.40901 q^{12} -5.37970 q^{13} +6.15962 q^{14} -0.261974 q^{16} +0.229265 q^{18} +8.53588 q^{19} +4.60966 q^{21} +4.95164 q^{22} +5.32570 q^{23} +4.55863 q^{24} +12.2401 q^{26} +5.27972 q^{27} -8.60011 q^{28} +1.31247 q^{29} +0.757667 q^{31} +5.95060 q^{32} +3.70565 q^{33} -0.320102 q^{36} +8.20380 q^{37} -19.4212 q^{38} +9.16010 q^{39} +0.0130696 q^{41} -10.4881 q^{42} +0.330596 q^{43} -6.91352 q^{44} -12.1172 q^{46} -11.3395 q^{47} +0.446067 q^{48} +0.329174 q^{49} -17.0897 q^{52} +8.04151 q^{53} -12.0126 q^{54} +7.24803 q^{56} -14.5342 q^{57} -2.98619 q^{58} -13.1546 q^{59} +7.20639 q^{61} -1.72387 q^{62} +0.272797 q^{63} -13.0151 q^{64} -8.43123 q^{66} -2.67143 q^{67} -9.06814 q^{69} -1.87106 q^{71} +0.269777 q^{72} -11.9780 q^{73} -18.6656 q^{74} +27.1160 q^{76} +5.89183 q^{77} -20.8414 q^{78} -17.0477 q^{79} -8.68755 q^{81} -0.0297365 q^{82} +3.35585 q^{83} +14.6435 q^{84} -0.752183 q^{86} -2.23477 q^{87} +5.82660 q^{88} -10.8909 q^{89} +14.5642 q^{91} +16.9182 q^{92} -1.29009 q^{93} +25.8001 q^{94} -10.1322 q^{96} +10.9119 q^{97} -0.748948 q^{98} +0.219298 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 3 q^{3} + 21 q^{4} + 9 q^{6} + 6 q^{7} - 12 q^{8} + 21 q^{9} + 6 q^{11} + 6 q^{12} - 9 q^{13} + 18 q^{14} + 39 q^{16} + 9 q^{18} + 27 q^{19} + 6 q^{21} + 15 q^{22} + 18 q^{23} + 36 q^{24}+ \cdots + 6 q^{99}+O(q^{100}) \) 12 * q - 3 * q^2 + 3 * q^3 + 21 * q^4 + 9 * q^6 + 6 * q^7 - 12 * q^8 + 21 * q^9 + 6 * q^11 + 6 * q^12 - 9 * q^13 + 18 * q^14 + 39 * q^16 + 9 * q^18 + 27 * q^19 + 6 * q^21 + 15 * q^22 + 18 * q^23 + 36 * q^24 + 6 * q^26 + 12 * q^27 + 39 * q^28 + 6 * q^31 - 12 * q^32 + 48 * q^36 + 21 * q^37 + 3 * q^38 + 9 * q^39 + 15 * q^41 - 30 * q^42 - 45 * q^43 + 42 * q^44 + 15 * q^46 - 3 * q^47 - 63 * q^48 + 12 * q^49 - 51 * q^52 + 12 * q^53 + 27 * q^54 - 3 * q^56 + 6 * q^57 - 51 * q^58 + 36 * q^59 - 3 * q^61 + 36 * q^62 + 72 * q^63 + 42 * q^64 - 39 * q^66 - 78 * q^67 - 66 * q^69 + 30 * q^71 + 12 * q^72 - 33 * q^73 - 9 * q^74 + 27 * q^76 + 39 * q^77 - 24 * q^78 + 3 * q^79 + 12 * q^81 - 51 * q^82 - 27 * q^83 + 27 * q^84 + 18 * q^86 - 24 * q^87 + 36 * q^88 + 6 * q^89 + 24 * q^91 - 3 * q^92 + 12 * q^93 + 21 * q^94 + 39 * q^96 - 6 * q^97 + 42 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more