LMFDB - Embedded newform 7225.2.a.bg.1.3

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:	$0$
Dimension:	$6$
Coefficient field:	6.6.93924352.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 17x^{4} + 73x^{2} - 67 $ x^6 - 17x^4 + 73x^2 - 67
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 425)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.12261$ 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)

$f(q)$	$=$	$q+0.311108 q^{2} -1.12261 q^{3} -1.90321 q^{4} -0.349253 q^{6} -3.60843 q^{7} -1.21432 q^{8} -1.73975 q^{9} +O(q^{10})$	\(q+0.311108 q^{2} -1.12261 q^{3} -1.90321 q^{4} -0.349253 q^{6} -3.60843 q^{7} -1.21432 q^{8} -1.73975 q^{9} +5.74499 q^{11} +2.13656 q^{12} +2.59210 q^{13} -1.12261 q^{14} +3.42864 q^{16} -0.541249 q^{18} -4.28100 q^{19} +4.05086 q^{21} +1.78731 q^{22} -1.47186 q^{23} +1.36321 q^{24} +0.806424 q^{26} +5.32089 q^{27} +6.86760 q^{28} -5.50439 q^{29} -8.33946 q^{31} +3.49532 q^{32} -6.44938 q^{33} +3.31111 q^{36} +6.20290 q^{37} -1.33185 q^{38} -2.90992 q^{39} -3.95768 q^{41} +1.26025 q^{42} -9.76049 q^{43} -10.9339 q^{44} -0.457908 q^{46} +6.62222 q^{47} -3.84902 q^{48} +6.02074 q^{49} -4.93332 q^{52} -0.658781 q^{53} +1.65537 q^{54} +4.38178 q^{56} +4.80589 q^{57} -1.71246 q^{58} -5.95407 q^{59} -8.76357 q^{61} -2.59447 q^{62} +6.27775 q^{63} -5.76986 q^{64} -2.00645 q^{66} -0.428639 q^{67} +1.65233 q^{69} -1.36321 q^{71} +2.11261 q^{72} +3.25917 q^{73} +1.92977 q^{74} +8.14764 q^{76} -20.7304 q^{77} -0.905299 q^{78} -1.89597 q^{79} -0.754037 q^{81} -1.23127 q^{82} +16.6637 q^{83} -7.70964 q^{84} -3.03657 q^{86} +6.17929 q^{87} -6.97626 q^{88} -3.52543 q^{89} -9.35342 q^{91} +2.80127 q^{92} +9.36196 q^{93} +2.06022 q^{94} -3.92388 q^{96} -11.7073 q^{97} +1.87310 q^{98} -9.99483 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 2 q^{4} + 6 q^{8} + 16 q^{9}+O(q^{10}) $ 6 * q + 2 * q^2 + 2 * q^4 + 6 * q^8 + 16 * q^9	$ 6 q + 2 q^{2} + 2 q^{4} + 6 q^{8} + 16 q^{9} + 2 q^{13} - 6 q^{16} - 16 q^{18} - 12 q^{19} - 2 q^{21} - 22 q^{26} - 6 q^{32} + 28 q^{33} + 20 q^{36} + 32 q^{38} + 34 q^{42} + 8 q^{43} + 40 q^{47} - 4 q^{49} - 30 q^{52} + 10 q^{53} + 4 q^{59} - 22 q^{64} + 16 q^{66} + 24 q^{67} + 24 q^{69} + 52 q^{72} + 36 q^{76} - 44 q^{77} + 34 q^{81} + 20 q^{83} - 6 q^{84} - 4 q^{86} + 76 q^{87} - 8 q^{89} + 30 q^{93} + 40 q^{94} - 16 q^{98}+O(q^{100}) $ 6 * q + 2 * q^2 + 2 * q^4 + 6 * q^8 + 16 * q^9 + 2 * q^13 - 6 * q^16 - 16 * q^18 - 12 * q^19 - 2 * q^21 - 22 * q^26 - 6 * q^32 + 28 * q^33 + 20 * q^36 + 32 * q^38 + 34 * q^42 + 8 * q^43 + 40 * q^47 - 4 * q^49 - 30 * q^52 + 10 * q^53 + 4 * q^59 - 22 * q^64 + 16 * q^66 + 24 * q^67 + 24 * q^69 + 52 * q^72 + 36 * q^76 - 44 * q^77 + 34 * q^81 + 20 * q^83 - 6 * q^84 - 4 * q^86 + 76 * q^87 - 8 * q^89 + 30 * q^93 + 40 * q^94 - 16 * 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 $

$2$	0.311108	0.219986	0.109993	−	0.993932i	$-0.464917\pi$
$2$	0.311108	0.219986	0.109993	+	0.993932i	$0.464917\pi$
$3$	−1.12261	−0.648139	−0.324070	−	0.946033i	$-0.605051\pi$
$3$	−1.12261	−0.648139	−0.324070	+	0.946033i	$0.605051\pi$
$4$	−1.90321	−0.951606
$5$	0	0
$5$	0	0
$6$	−0.349253	−0.142582
$7$	−3.60843	−1.36386	−0.681929	−	0.731419i	$-0.738859\pi$
$7$	−3.60843	−1.36386	−0.681929	+	0.731419i	$0.738859\pi$
$8$	−1.21432	−0.429327
$9$	−1.73975	−0.579916
$10$	0	0
$11$	5.74499	1.73218	0.866090	−	0.499888i	$-0.166626\pi$
$11$	5.74499	1.73218	0.866090	+	0.499888i	$0.166626\pi$
$12$	2.13656	0.616773
$13$	2.59210	0.718920	0.359460	−	0.933160i	$-0.382961\pi$
$13$	2.59210	0.718920	0.359460	+	0.933160i	$0.382961\pi$
$14$	−1.12261	−0.300030
$15$	0	0
$16$	3.42864	0.857160
$17$	0	0
$17$	0	0
$18$	−0.541249	−0.127574
$19$	−4.28100	−0.982128	−0.491064	−	0.871124i	$-0.663392\pi$
$19$	−4.28100	−0.982128	−0.491064	+	0.871124i	$0.663392\pi$
$20$	0	0
$21$	4.05086	0.883969
$22$	1.78731	0.381056
$23$	−1.47186	−0.306905	−0.153452	−	0.988156i	$-0.549039\pi$
$23$	−1.47186	−0.306905	−0.153452	+	0.988156i	$0.549039\pi$
$24$	1.36321	0.278264
$25$	0	0
$26$	0.806424	0.158153
$27$	5.32089	1.02401
$28$	6.86760	1.29785
$29$	−5.50439	−1.02214	−0.511070	−	0.859539i	$-0.670751\pi$
$29$	−5.50439	−1.02214	−0.511070	+	0.859539i	$0.670751\pi$
$30$	0	0
$31$	−8.33946	−1.49781	−0.748906	−	0.662676i	$-0.769421\pi$
$31$	−8.33946	−1.49781	−0.748906	+	0.662676i	$0.769421\pi$
$32$	3.49532	0.617890
$33$	−6.44938	−1.12269
$34$	0	0
$35$	0	0
$36$	3.31111	0.551851
$37$	6.20290	1.01975	0.509875	−	0.860248i	$-0.329692\pi$
$37$	6.20290	1.01975	0.509875	+	0.860248i	$0.329692\pi$
$38$	−1.33185	−0.216055
$39$	−2.90992	−0.465960
$40$	0	0
$41$	−3.95768	−0.618086	−0.309043	−	0.951048i	$-0.600009\pi$
$41$	−3.95768	−0.618086	−0.309043	+	0.951048i	$0.600009\pi$
$42$	1.26025	0.194461
$43$	−9.76049	−1.48846	−0.744230	−	0.667923i	$-0.767184\pi$
$43$	−9.76049	−1.48846	−0.744230	+	0.667923i	$0.767184\pi$
$44$	−10.9339	−1.64835
$45$	0	0
$46$	−0.457908	−0.0675148
$47$	6.62222	0.965949	0.482975	−	0.875634i	$-0.339556\pi$
$47$	6.62222	0.965949	0.482975	+	0.875634i	$0.339556\pi$
$48$	−3.84902	−0.555559
$49$	6.02074	0.860106
$50$	0	0
$51$	0	0
$52$	−4.93332	−0.684129
$53$	−0.658781	−0.0904905	−0.0452452	−	0.998976i	$-0.514407\pi$
$53$	−0.658781	−0.0904905	−0.0452452	+	0.998976i	$0.514407\pi$
$54$	1.65537	0.225267
$55$	0	0
$56$	4.38178	0.585540
$57$	4.80589	0.636555
$58$	−1.71246	−0.224857
$59$	−5.95407	−0.775154	−0.387577	−	0.921837i	$-0.626688\pi$
$59$	−5.95407	−0.775154	−0.387577	+	0.921837i	$0.626688\pi$
$60$	0	0
$61$	−8.76357	−1.12206	−0.561030	−	0.827796i	$-0.689595\pi$
$61$	−8.76357	−1.12206	−0.561030	+	0.827796i	$0.689595\pi$
$62$	−2.59447	−0.329498
$63$	6.27775	0.790922
$64$	−5.76986	−0.721232
$65$	0	0
$66$	−2.00645	−0.246977
$67$	−0.428639	−0.0523666	−0.0261833	−	0.999657i	$-0.508335\pi$
$67$	−0.428639	−0.0523666	−0.0261833	+	0.999657i	$0.508335\pi$
$68$	0	0
$69$	1.65233	0.198917
$70$	0	0
$71$	−1.36321	−0.161783	−0.0808915	−	0.996723i	$-0.525777\pi$
$71$	−1.36321	−0.161783	−0.0808915	+	0.996723i	$0.525777\pi$
$72$	2.11261	0.248973
$73$	3.25917	0.381457	0.190729	−	0.981643i	$-0.438915\pi$
$73$	3.25917	0.381457	0.190729	+	0.981643i	$0.438915\pi$
$74$	1.92977	0.224331
$75$	0	0
$76$	8.14764	0.934599
$77$	−20.7304	−2.36245
$78$	−0.905299	−0.102505
$79$	−1.89597	−0.213313	−0.106656	−	0.994296i	$-0.534015\pi$
$79$	−1.89597	−0.213313	−0.106656	+	0.994296i	$0.534015\pi$
$80$	0	0
$81$	−0.754037	−0.0837819
$82$	−1.23127	−0.135970
$83$	16.6637	1.82908	0.914540	−	0.404497i	$-0.132553\pi$
$83$	16.6637	1.82908	0.914540	+	0.404497i	$0.132553\pi$
$84$	−7.70964	−0.841190
$85$	0	0
$86$	−3.03657	−0.327441
$87$	6.17929	0.662489
$88$	−6.97626	−0.743671
$89$	−3.52543	−0.373695	−0.186847	−	0.982389i	$-0.559827\pi$
$89$	−3.52543	−0.373695	−0.186847	+	0.982389i	$0.559827\pi$
$90$	0	0
$91$	−9.35342	−0.980505
$92$	2.80127	0.292052
$93$	9.36196	0.970790
$94$	2.06022	0.212496
$95$	0	0
$96$	−3.92388	−0.400479
$97$	−11.7073	−1.18870	−0.594348	−	0.804208i	$-0.702590\pi$
$97$	−11.7073	−1.18870	−0.594348	+	0.804208i	$0.702590\pi$
$98$	1.87310	0.189212
$99$	−9.99483	−1.00452
$100$	0	0
$101$	3.96989	0.395019	0.197509	−	0.980301i	$-0.436715\pi$
$101$	3.96989	0.395019	0.197509	+	0.980301i	$0.436715\pi$
$102$	0	0
$103$	−4.23506	−0.417293	−0.208647	−	0.977991i	$-0.566906\pi$
$103$	−4.23506	−0.417293	−0.208647	+	0.977991i	$0.566906\pi$
$104$	−3.14764	−0.308652
$105$	0	0
$106$	−0.204952	−0.0199067
$107$	−11.8392	−1.14454	−0.572271	−	0.820065i	$-0.693937\pi$
$107$	−11.8392	−1.14454	−0.572271	+	0.820065i	$0.693937\pi$
$108$	−10.1268	−0.974449
$109$	−12.0227	−1.15157	−0.575785	−	0.817601i	$-0.695303\pi$
$109$	−12.0227	−1.15157	−0.575785	+	0.817601i	$0.695303\pi$
$110$	0	0
$111$	−6.96343	−0.660940
$112$	−12.3720	−1.16904
$113$	−10.7915	−1.01518	−0.507588	−	0.861600i	$-0.669463\pi$
$113$	−10.7915	−1.01518	−0.507588	+	0.861600i	$0.669463\pi$
$114$	1.49515	0.140034
$115$	0	0
$116$	10.4760	0.972675
$117$	−4.50961	−0.416913
$118$	−1.85236	−0.170523
$119$	0	0
$120$	0	0
$121$	22.0049	2.00045
$122$	−2.72641	−0.246838
$123$	4.44293	0.400605
$124$	15.8718	1.42533
$125$	0	0
$126$	1.95306	0.173992
$127$	17.1383	1.52078	0.760388	−	0.649469i	$-0.225009\pi$
$127$	17.1383	1.52078	0.760388	+	0.649469i	$0.225009\pi$
$128$	−8.78568	−0.776552
$129$	10.9572	0.964730
$130$	0	0
$131$	−8.89551	−0.777204	−0.388602	−	0.921406i	$-0.627042\pi$
$131$	−8.89551	−0.777204	−0.388602	+	0.921406i	$0.627042\pi$
$132$	12.2745	1.06836
$133$	15.4477	1.33948
$134$	−0.133353	−0.0115200
$135$	0	0
$136$	0	0
$137$	−14.0716	−1.20222	−0.601109	−	0.799167i	$-0.705274\pi$
$137$	−14.0716	−1.20222	−0.601109	+	0.799167i	$0.705274\pi$
$138$	0.514052	0.0437590
$139$	16.7876	1.42390	0.711952	−	0.702228i	$-0.247811\pi$
$139$	16.7876	1.42390	0.711952	+	0.702228i	$0.247811\pi$
$140$	0	0
$141$	−7.43416	−0.626070
$142$	−0.424104	−0.0355901
$143$	14.8916	1.24530
$144$	−5.96497	−0.497081
$145$	0	0
$146$	1.01395	0.0839155
$147$	−6.75895	−0.557468
$148$	−11.8054	−0.970400
$149$	−4.76494	−0.390359	−0.195179	−	0.980768i	$-0.562529\pi$
$149$	−4.76494	−0.390359	−0.195179	+	0.980768i	$0.562529\pi$
$150$	0	0
$151$	−2.99063	−0.243374	−0.121687	−	0.992569i	$-0.538830\pi$
$151$	−2.99063	−0.243374	−0.121687	+	0.992569i	$0.538830\pi$
$152$	5.19850	0.421654
$153$	0	0
$154$	−6.44938	−0.519706
$155$	0	0
$156$	5.53820	0.443411
$157$	13.9447	1.11291	0.556454	−	0.830878i	$-0.312162\pi$
$157$	13.9447	1.11291	0.556454	+	0.830878i	$0.312162\pi$
$158$	−0.589850	−0.0469260
$159$	0.739554	0.0586504
$160$	0	0
$161$	5.31111	0.418574
$162$	−0.234587	−0.0184309
$163$	8.65491	0.677905	0.338953	−	0.940803i	$-0.389927\pi$
$163$	8.65491	0.677905	0.338953	+	0.940803i	$0.389927\pi$
$164$	7.53230	0.588174
$165$	0	0
$166$	5.18421	0.402373
$167$	−0.773357	−0.0598442	−0.0299221	−	0.999552i	$-0.509526\pi$
$167$	−0.773357	−0.0598442	−0.0299221	+	0.999552i	$0.509526\pi$
$168$	−4.91903	−0.379512
$169$	−6.28100	−0.483154
$170$	0	0
$171$	7.44785	0.569551
$172$	18.5763	1.41643
$173$	2.41097	0.183302	0.0916511	−	0.995791i	$-0.470786\pi$
$173$	2.41097	0.183302	0.0916511	+	0.995791i	$0.470786\pi$
$174$	1.92242	0.145739
$175$	0	0
$176$	19.6975	1.48476
$177$	6.68409	0.502407
$178$	−1.09679	−0.0822077
$179$	18.4558	1.37945	0.689727	−	0.724070i	$-0.257731\pi$
$179$	18.4558	1.37945	0.689727	+	0.724070i	$0.257731\pi$
$180$	0	0
$181$	−20.6366	−1.53391	−0.766953	−	0.641703i	$-0.778228\pi$
$181$	−20.6366	−1.53391	−0.766953	+	0.641703i	$0.778228\pi$
$182$	−2.90992	−0.215698
$183$	9.83807	0.727251
$184$	1.78731	0.131762
$185$	0	0
$186$	2.91258	0.213561
$187$	0	0
$188$	−12.6035	−0.919203
$189$	−19.2000	−1.39660
$190$	0	0
$191$	22.8113	1.65057	0.825286	−	0.564716i	$-0.191014\pi$
$191$	22.8113	1.65057	0.825286	+	0.564716i	$0.191014\pi$
$192$	6.47730	0.467459
$193$	19.7208	1.41953	0.709767	−	0.704437i	$-0.248800\pi$
$193$	19.7208	1.41953	0.709767	+	0.704437i	$0.248800\pi$
$194$	−3.64223	−0.261497
$195$	0	0
$196$	−11.4588	−0.818482
$197$	−12.8870	−0.918160	−0.459080	−	0.888395i	$-0.651821\pi$
$197$	−12.8870	−0.918160	−0.459080	+	0.888395i	$0.651821\pi$
$198$	−3.10947	−0.220980
$199$	−6.44350	−0.456767	−0.228384	−	0.973571i	$-0.573344\pi$
$199$	−6.44350	−0.456767	−0.228384	+	0.973571i	$0.573344\pi$
$200$	0	0
$201$	0.481195	0.0339409
$202$	1.23506	0.0868988
$203$	19.8622	1.39405
$204$	0	0
$205$	0	0
$206$	−1.31756	−0.0917988
$207$	2.56067	0.177979
$208$	8.88739	0.616230
$209$	−24.5943	−1.70122
$210$	0	0
$211$	11.3580	0.781920	0.390960	−	0.920408i	$-0.372143\pi$
$211$	11.3580	0.781920	0.390960	+	0.920408i	$0.372143\pi$
$212$	1.25380	0.0861113
$213$	1.53035	0.104858
$214$	−3.68328	−0.251784
$215$	0	0
$216$	−6.46126	−0.439633
$217$	30.0923	2.04280
$218$	−3.74037	−0.253330
$219$	−3.65878	−0.247237
$220$	0	0
$221$	0	0
$222$	−2.16638	−0.145398
$223$	0.888922	0.0595266	0.0297633	−	0.999557i	$-0.490525\pi$
$223$	0.888922	0.0595266	0.0297633	+	0.999557i	$0.490525\pi$
$224$	−12.6126	−0.842714
$225$	0	0
$226$	−3.35731	−0.223325
$227$	−12.9280	−0.858064	−0.429032	−	0.903289i	$-0.641145\pi$
$227$	−12.9280	−0.858064	−0.429032	+	0.903289i	$0.641145\pi$
$228$	−9.14662	−0.605750
$229$	−2.83161	−0.187118	−0.0935591	−	0.995614i	$-0.529824\pi$
$229$	−2.83161	−0.187118	−0.0935591	+	0.995614i	$0.529824\pi$
$230$	0	0
$231$	23.2721	1.53119
$232$	6.68409	0.438832
$233$	19.9381	1.30619	0.653094	−	0.757277i	$-0.273471\pi$
$233$	19.9381	1.30619	0.653094	+	0.757277i	$0.273471\pi$
$234$	−1.40297	−0.0917153
$235$	0	0
$236$	11.3319	0.737641
$237$	2.12843	0.138256
$238$	0	0
$239$	27.2859	1.76498	0.882490	−	0.470332i	$-0.155866\pi$
$239$	27.2859	1.76498	0.882490	+	0.470332i	$0.155866\pi$
$240$	0	0
$241$	3.57462	0.230262	0.115131	−	0.993350i	$-0.463271\pi$
$241$	3.57462	0.230262	0.115131	+	0.993350i	$0.463271\pi$
$242$	6.84590	0.440071
$243$	−15.1162	−0.969703
$244$	16.6789	1.06776
$245$	0	0
$246$	1.38223	0.0881278
$247$	−11.0968	−0.706072
$248$	10.1268	0.643051
$249$	−18.7068	−1.18550
$250$	0	0
$251$	17.6128	1.11171	0.555857	−	0.831278i	$-0.312390\pi$
$251$	17.6128	1.11171	0.555857	+	0.831278i	$0.312390\pi$
$252$	−11.9479	−0.752646
$253$	−8.45584	−0.531614
$254$	5.33185	0.334550
$255$	0	0
$256$	8.80642	0.550401
$257$	20.5462	1.28163	0.640817	−	0.767693i	$-0.278596\pi$
$257$	20.5462	1.28163	0.640817	+	0.767693i	$0.278596\pi$
$258$	3.40888	0.212227
$259$	−22.3827	−1.39079
$260$	0	0
$261$	9.57625	0.592755
$262$	−2.76746	−0.170974
$263$	8.38715	0.517174	0.258587	−	0.965988i	$-0.416743\pi$
$263$	8.38715	0.517174	0.258587	+	0.965988i	$0.416743\pi$
$264$	7.83161	0.482002
$265$	0	0
$266$	4.80589	0.294668
$267$	3.95768	0.242206
$268$	0.815792	0.0498324
$269$	12.4058	0.756395	0.378197	−	0.925725i	$-0.376544\pi$
$269$	12.4058	0.756395	0.378197	+	0.925725i	$0.376544\pi$
$270$	0	0
$271$	−7.13828	−0.433619	−0.216810	−	0.976214i	$-0.569565\pi$
$271$	−7.13828	−0.433619	−0.216810	+	0.976214i	$0.569565\pi$
$272$	0	0
$273$	10.5002	0.635503
$274$	−4.37778	−0.264472
$275$	0	0
$276$	−3.14473	−0.189290
$277$	20.4193	1.22688	0.613438	−	0.789743i	$-0.289786\pi$
$277$	20.4193	1.22688	0.613438	+	0.789743i	$0.289786\pi$
$278$	5.22275	0.313240
$279$	14.5086	0.868605
$280$	0	0
$281$	−29.1432	−1.73854	−0.869269	−	0.494340i	$-0.835410\pi$
$281$	−29.1432	−1.73854	−0.869269	+	0.494340i	$0.835410\pi$
$282$	−2.31283	−0.137727
$283$	9.16991	0.545095	0.272547	−	0.962142i	$-0.412134\pi$
$283$	9.16991	0.545095	0.272547	+	0.962142i	$0.412134\pi$
$284$	2.59447	0.153954
$285$	0	0
$286$	4.63290	0.273949
$287$	14.2810	0.842981
$288$	−6.08097	−0.358324
$289$	0	0
$290$	0	0
$291$	13.1427	0.770440
$292$	−6.20290	−0.362997
$293$	16.2351	0.948463	0.474231	−	0.880400i	$-0.342726\pi$
$293$	16.2351	0.948463	0.474231	+	0.880400i	$0.342726\pi$
$294$	−2.10276	−0.122636
$295$	0	0
$296$	−7.53230	−0.437806
$297$	30.5684	1.77376
$298$	−1.48241	−0.0858737
$299$	−3.81522	−0.220640
$300$	0	0
$301$	35.2200	2.03005
$302$	−0.930409	−0.0535390
$303$	−4.45664	−0.256027
$304$	−14.6780	−0.841841
$305$	0	0
$306$	0	0
$307$	−3.37778	−0.192780	−0.0963902	−	0.995344i	$-0.530730\pi$
$307$	−3.37778	−0.192780	−0.0963902	+	0.995344i	$0.530730\pi$
$308$	39.4543	2.24812
$309$	4.75432	0.270464
$310$	0	0
$311$	−7.95641	−0.451166	−0.225583	−	0.974224i	$-0.572429\pi$
$311$	−7.95641	−0.451166	−0.225583	+	0.974224i	$0.572429\pi$
$312$	3.53358	0.200049
$313$	−6.83380	−0.386269	−0.193135	−	0.981172i	$-0.561865\pi$
$313$	−6.83380	−0.386269	−0.193135	+	0.981172i	$0.561865\pi$
$314$	4.33830	0.244825
$315$	0	0
$316$	3.60843	0.202990
$317$	27.1550	1.52517	0.762587	−	0.646886i	$-0.223929\pi$
$317$	27.1550	1.52517	0.762587	+	0.646886i	$0.223929\pi$
$318$	0.230081	0.0129023
$319$	−31.6227	−1.77053
$320$	0	0
$321$	13.2908	0.741822
$322$	1.65233	0.0920806
$323$	0	0
$324$	1.43509	0.0797274
$325$	0	0
$326$	2.69261	0.149130
$327$	13.4968	0.746377
$328$	4.80589	0.265361
$329$	−23.8958	−1.31742
$330$	0	0
$331$	20.3783	1.12009	0.560045	−	0.828462i	$-0.310784\pi$
$331$	20.3783	1.12009	0.560045	+	0.828462i	$0.310784\pi$
$332$	−31.7146	−1.74056
$333$	−10.7915	−0.591369
$334$	−0.240597	−0.0131649
$335$	0	0
$336$	13.8889	0.757703
$337$	4.70775	0.256447	0.128224	−	0.991745i	$-0.459072\pi$
$337$	4.70775	0.256447	0.128224	+	0.991745i	$0.459072\pi$
$338$	−1.95407	−0.106287
$339$	12.1146	0.657976
$340$	0	0
$341$	−47.9101	−2.59448
$342$	2.31708	0.125294
$343$	3.53358	0.190795
$344$	11.8524	0.639036
$345$	0	0
$346$	0.750070	0.0403240
$347$	−35.3520	−1.89779	−0.948896	−	0.315588i	$-0.897798\pi$
$347$	−35.3520	−1.89779	−0.948896	+	0.315588i	$0.897798\pi$
$348$	−11.7605	−0.630428
$349$	25.0558	1.34121	0.670603	−	0.741817i	$-0.266036\pi$
$349$	25.0558	1.34121	0.670603	+	0.741817i	$0.266036\pi$
$350$	0	0
$351$	13.7923	0.736178
$352$	20.0806	1.07030
$353$	−13.6953	−0.728930	−0.364465	−	0.931217i	$-0.618748\pi$
$353$	−13.6953	−0.728930	−0.364465	+	0.931217i	$0.618748\pi$
$354$	2.07947	0.110523
$355$	0	0
$356$	6.70964	0.355610
$357$	0	0
$358$	5.74176	0.303461
$359$	−2.53480	−0.133781	−0.0668907	−	0.997760i	$-0.521308\pi$
$359$	−2.53480	−0.133781	−0.0668907	+	0.997760i	$0.521308\pi$
$360$	0	0
$361$	−0.673071	−0.0354248
$362$	−6.42021	−0.337439
$363$	−24.7029	−1.29657
$364$	17.8015	0.933054
$365$	0	0
$366$	3.06070	0.159985
$367$	36.3426	1.89707	0.948535	−	0.316673i	$-0.102566\pi$
$367$	36.3426	1.89707	0.948535	+	0.316673i	$0.102566\pi$
$368$	−5.04649	−0.263066
$369$	6.88536	0.358438
$370$	0	0
$371$	2.37716	0.123416
$372$	−17.8178	−0.923810
$373$	12.7714	0.661278	0.330639	−	0.943757i	$-0.392736\pi$
$373$	12.7714	0.661278	0.330639	+	0.943757i	$0.392736\pi$
$374$	0	0
$375$	0	0
$376$	−8.04149	−0.414708
$377$	−14.2680	−0.734837
$378$	−5.97328	−0.307232
$379$	22.1495	1.13774	0.568872	−	0.822426i	$-0.307380\pi$
$379$	22.1495	1.13774	0.568872	+	0.822426i	$0.307380\pi$
$380$	0	0
$381$	−19.2396	−0.985674
$382$	7.09679	0.363103
$383$	−1.95407	−0.0998482	−0.0499241	−	0.998753i	$-0.515898\pi$
$383$	−1.95407	−0.0998482	−0.0499241	+	0.998753i	$0.515898\pi$
$384$	9.86289	0.503314
$385$	0	0
$386$	6.13529	0.312278
$387$	16.9808	0.863182
$388$	22.2815	1.13117
$389$	7.91258	0.401184	0.200592	−	0.979675i	$-0.435713\pi$
$389$	7.91258	0.401184	0.200592	+	0.979675i	$0.435713\pi$
$390$	0	0
$391$	0	0
$392$	−7.31111	−0.369267
$393$	9.98619	0.503736
$394$	−4.00924	−0.201983
$395$	0	0
$396$	19.0223	0.955906
$397$	9.24476	0.463981	0.231991	−	0.972718i	$-0.425476\pi$
$397$	9.24476	0.463981	0.231991	+	0.972718i	$0.425476\pi$
$398$	−2.00462	−0.100483
$399$	−17.3417	−0.868171
$400$	0	0
$401$	−21.4848	−1.07290	−0.536450	−	0.843932i	$-0.680235\pi$
$401$	−21.4848	−1.07290	−0.536450	+	0.843932i	$0.680235\pi$
$402$	0.149703	0.00746653
$403$	−21.6168	−1.07681
$404$	−7.55554	−0.375902
$405$	0	0
$406$	6.17929	0.306673
$407$	35.6356	1.76639
$408$	0	0
$409$	18.1669	0.898293	0.449147	−	0.893458i	$-0.351728\pi$
$409$	18.1669	0.898293	0.449147	+	0.893458i	$0.351728\pi$
$410$	0	0
$411$	15.7969	0.779204
$412$	8.06022	0.397099
$413$	21.4848	1.05720
$414$	0.796644	0.0391529
$415$	0	0
$416$	9.06022	0.444214
$417$	−18.8459	−0.922888
$418$	−7.65147	−0.374246
$419$	14.3589	0.701476	0.350738	−	0.936474i	$-0.385931\pi$
$419$	14.3589	0.701476	0.350738	+	0.936474i	$0.385931\pi$
$420$	0	0
$421$	3.96989	0.193481	0.0967403	−	0.995310i	$-0.469158\pi$
$421$	3.96989	0.193481	0.0967403	+	0.995310i	$0.469158\pi$
$422$	3.53358	0.172012
$423$	−11.5210	−0.560169
$424$	0.799970	0.0388500
$425$	0	0
$426$	0.476104	0.0230673
$427$	31.6227	1.53033
$428$	22.5326	1.08915
$429$	−16.7175	−0.807127
$430$	0	0
$431$	−32.9337	−1.58636	−0.793181	−	0.608985i	$-0.791577\pi$
$431$	−32.9337	−1.58636	−0.793181	+	0.608985i	$0.791577\pi$
$432$	18.2434	0.877736
$433$	38.2306	1.83725	0.918623	−	0.395135i	$-0.129302\pi$
$433$	38.2306	1.83725	0.918623	+	0.395135i	$0.129302\pi$
$434$	9.36196	0.449389
$435$	0	0
$436$	22.8818	1.09584
$437$	6.30104	0.301420
$438$	−1.13828	−0.0543889
$439$	0.664702	0.0317245	0.0158622	−	0.999874i	$-0.494951\pi$
$439$	0.664702	0.0317245	0.0158622	+	0.999874i	$0.494951\pi$
$440$	0	0
$441$	−10.4746	−0.498789
$442$	0	0
$443$	27.5669	1.30974	0.654872	−	0.755740i	$-0.272723\pi$
$443$	27.5669	1.30974	0.654872	+	0.755740i	$0.272723\pi$
$444$	13.2529	0.628954
$445$	0	0
$446$	0.276551	0.0130950
$447$	5.34916	0.253007
$448$	20.8201	0.983658
$449$	−17.4756	−0.824723	−0.412362	−	0.911020i	$-0.635296\pi$
$449$	−17.4756	−0.824723	−0.412362	+	0.911020i	$0.635296\pi$
$450$	0	0
$451$	−22.7368	−1.07064
$452$	20.5385	0.966048
$453$	3.35731	0.157740
$454$	−4.02201	−0.188762
$455$	0	0
$456$	−5.83589	−0.273290
$457$	−13.0350	−0.609753	−0.304877	−	0.952392i	$-0.598615\pi$
$457$	−13.0350	−0.609753	−0.304877	+	0.952392i	$0.598615\pi$
$458$	−0.880937	−0.0411635
$459$	0	0
$460$	0	0
$461$	10.9491	0.509953	0.254976	−	0.966947i	$-0.417932\pi$
$461$	10.9491	0.509953	0.254976	+	0.966947i	$0.417932\pi$
$462$	7.24014	0.336842
$463$	−17.2400	−0.801210	−0.400605	−	0.916251i	$-0.631200\pi$
$463$	−17.2400	−0.801210	−0.400605	+	0.916251i	$0.631200\pi$
$464$	−18.8726	−0.876138
$465$	0	0
$466$	6.20290	0.287344
$467$	−0.0414872	−0.00191980	−0.000959899	−	1.00000i	$-0.500306\pi$
$467$	−0.0414872	−0.00191980	−0.000959899		1.00000i	$0.500306\pi$
$468$	8.58274	0.396737
$469$	1.54671	0.0714206
$470$	0	0
$471$	−15.6545	−0.721319
$472$	7.23014	0.332794
$473$	−56.0739	−2.57828
$474$	0.662171	0.0304145
$475$	0	0
$476$	0	0
$477$	1.14611	0.0524769
$478$	8.48886	0.388272
$479$	−35.4606	−1.62024	−0.810118	−	0.586266i	$-0.800597\pi$
$479$	−35.4606	−1.62024	−0.810118	+	0.586266i	$0.800597\pi$
$480$	0	0
$481$	16.0786	0.733119
$482$	1.11209	0.0506545
$483$	−5.96230	−0.271294
$484$	−41.8800	−1.90364
$485$	0	0
$486$	−4.70276	−0.213321
$487$	−24.6691	−1.11787	−0.558933	−	0.829213i	$-0.688789\pi$
$487$	−24.6691	−1.11787	−0.558933	+	0.829213i	$0.688789\pi$
$488$	10.6418	0.481730
$489$	−9.71609	−0.439377
$490$	0	0
$491$	−8.58073	−0.387243	−0.193621	−	0.981076i	$-0.562023\pi$
$491$	−8.58073	−0.387243	−0.193621	+	0.981076i	$0.562023\pi$
$492$	−8.45584	−0.381219
$493$	0	0
$494$	−3.45230	−0.155326
$495$	0	0
$496$	−28.5930	−1.28386
$497$	4.91903	0.220649
$498$	−5.81984	−0.260793
$499$	−8.48917	−0.380027	−0.190014	−	0.981781i	$-0.560853\pi$
$499$	−8.48917	−0.380027	−0.190014	+	0.981781i	$0.560853\pi$
$500$	0	0
$501$	0.868178	0.0387873
$502$	5.47949	0.244562
$503$	−24.1879	−1.07849	−0.539244	−	0.842150i	$-0.681290\pi$
$503$	−24.1879	−1.07849	−0.539244	+	0.842150i	$0.681290\pi$
$504$	−7.62320	−0.339564
$505$	0	0
$506$	−2.63068	−0.116948
$507$	7.05111	0.313151
$508$	−32.6178	−1.44718
$509$	−25.8671	−1.14654	−0.573270	−	0.819366i	$-0.694325\pi$
$509$	−25.8671	−1.14654	−0.573270	+	0.819366i	$0.694325\pi$
$510$	0	0
$511$	−11.7605	−0.520253
$512$	20.3111	0.897633
$513$	−22.7787	−1.00570
$514$	6.39207	0.281942
$515$	0	0
$516$	−20.8539	−0.918042
$517$	38.0446	1.67320
$518$	−6.96343	−0.305956
$519$	−2.70657	−0.118805
$520$	0	0
$521$	−11.4900	−0.503385	−0.251693	−	0.967807i	$-0.580987\pi$
$521$	−11.4900	−0.503385	−0.251693	+	0.967807i	$0.580987\pi$
$522$	2.97925	0.130398
$523$	33.0509	1.44521	0.722606	−	0.691260i	$-0.242944\pi$
$523$	33.0509	1.44521	0.722606	+	0.691260i	$0.242944\pi$
$524$	16.9300	0.739592
$525$	0	0
$526$	2.60931	0.113771
$527$	0	0
$528$	−22.1126	−0.962328
$529$	−20.8336	−0.905810
$530$	0	0
$531$	10.3586	0.449524
$532$	−29.4002	−1.27466
$533$	−10.2587	−0.444354
$534$	1.23127	0.0532820
$535$	0	0
$536$	0.520505	0.0224824
$537$	−20.7187	−0.894078
$538$	3.85954	0.166397
$539$	34.5891	1.48986
$540$	0	0
$541$	−19.7884	−0.850770	−0.425385	−	0.905013i	$-0.639861\pi$
$541$	−19.7884	−0.850770	−0.425385	+	0.905013i	$0.639861\pi$
$542$	−2.22077	−0.0953904
$543$	23.1669	0.994185
$544$	0	0
$545$	0	0
$546$	3.26671	0.139802
$547$	15.4815	0.661940	0.330970	−	0.943641i	$-0.392624\pi$
$547$	15.4815	0.661940	0.330970	+	0.943641i	$0.392624\pi$
$548$	26.7812	1.14404
$549$	15.2464	0.650700
$550$	0	0
$551$	23.5643	1.00387
$552$	−2.00645	−0.0854003
$553$	6.84146	0.290928
$554$	6.35260	0.269896
$555$	0	0
$556$	−31.9503	−1.35500
$557$	−15.3477	−0.650302	−0.325151	−	0.945662i	$-0.605415\pi$
$557$	−15.3477	−0.650302	−0.325151	+	0.945662i	$0.605415\pi$
$558$	4.51373	0.191081
$559$	−25.3002	−1.07008
$560$	0	0
$561$	0	0
$562$	−9.06668	−0.382455
$563$	−14.3827	−0.606159	−0.303079	−	0.952965i	$-0.598015\pi$
$563$	−14.3827	−0.606159	−0.303079	+	0.952965i	$0.598015\pi$
$564$	14.1488	0.595772
$565$	0	0
$566$	2.85283	0.119913
$567$	2.72089	0.114267
$568$	1.65537	0.0694578
$569$	12.1432	0.509069	0.254535	−	0.967064i	$-0.418078\pi$
$569$	12.1432	0.509069	0.254535	+	0.967064i	$0.418078\pi$
$570$	0	0
$571$	−2.88663	−0.120802	−0.0604009	−	0.998174i	$-0.519238\pi$
$571$	−2.88663	−0.120802	−0.0604009	+	0.998174i	$0.519238\pi$
$572$	−28.3419	−1.18503
$573$	−25.6082	−1.06980
$574$	4.44293	0.185444
$575$	0	0
$576$	10.0381	0.418254
$577$	−3.10970	−0.129458	−0.0647291	−	0.997903i	$-0.520618\pi$
$577$	−3.10970	−0.129458	−0.0647291	+	0.997903i	$0.520618\pi$
$578$	0	0
$579$	−22.1388	−0.920055
$580$	0	0
$581$	−60.1298	−2.49460
$582$	4.08880	0.169486
$583$	−3.78469	−0.156746
$584$	−3.95768	−0.163770
$585$	0	0
$586$	5.05086	0.208649
$587$	−10.2034	−0.421140	−0.210570	−	0.977579i	$-0.567532\pi$
$587$	−10.2034	−0.421140	−0.210570	+	0.977579i	$0.567532\pi$
$588$	12.8637	0.530490
$589$	35.7012	1.47104
$590$	0	0
$591$	14.4671	0.595095
$592$	21.2675	0.874089
$593$	16.2301	0.666492	0.333246	−	0.942840i	$-0.391856\pi$
$593$	16.2301	0.666492	0.333246	+	0.942840i	$0.391856\pi$
$594$	9.51008	0.390203
$595$	0	0
$596$	9.06868	0.371468
$597$	7.23353	0.296049
$598$	−1.18694	−0.0485378
$599$	−8.43309	−0.344567	−0.172283	−	0.985047i	$-0.555114\pi$
$599$	−8.43309	−0.344567	−0.172283	+	0.985047i	$0.555114\pi$
$600$	0	0
$601$	20.2535	0.826160	0.413080	−	0.910695i	$-0.364453\pi$
$601$	20.2535	0.826160	0.413080	+	0.910695i	$0.364453\pi$
$602$	10.9572	0.446583
$603$	0.745724	0.0303682
$604$	5.69181	0.231596
$605$	0	0
$606$	−1.38649	−0.0563225
$607$	−22.7732	−0.924334	−0.462167	−	0.886793i	$-0.652928\pi$
$607$	−22.7732	−0.924334	−0.462167	+	0.886793i	$0.652928\pi$
$608$	−14.9634	−0.606847
$609$	−22.2975	−0.903540
$610$	0	0
$611$	17.1655	0.694441
$612$	0	0
$613$	−8.40636	−0.339530	−0.169765	−	0.985485i	$-0.554301\pi$
$613$	−8.40636	−0.339530	−0.169765	+	0.985485i	$0.554301\pi$
$614$	−1.05086	−0.0424091
$615$	0	0
$616$	25.1733	1.01426
$617$	−14.8168	−0.596500	−0.298250	−	0.954488i	$-0.596403\pi$
$617$	−14.8168	−0.596500	−0.298250	+	0.954488i	$0.596403\pi$
$618$	1.47911	0.0594984
$619$	31.6221	1.27100	0.635500	−	0.772101i	$-0.280794\pi$
$619$	31.6221	1.27100	0.635500	+	0.772101i	$0.280794\pi$
$620$	0	0
$621$	−7.83161	−0.314272
$622$	−2.47530	−0.0992505
$623$	12.7212	0.509666
$624$	−9.97707	−0.399403
$625$	0	0
$626$	−2.12605	−0.0849740
$627$	27.6098	1.10263
$628$	−26.5397	−1.05905
$629$	0	0
$630$	0	0
$631$	17.3733	0.691622	0.345811	−	0.938304i	$-0.387604\pi$
$631$	17.3733	0.691622	0.345811	+	0.938304i	$0.387604\pi$
$632$	2.30231	0.0915810
$633$	−12.7506	−0.506793
$634$	8.44812	0.335518
$635$	0	0
$636$	−1.40753	−0.0558121
$637$	15.6064	0.618348
$638$	−9.83807	−0.389493
$639$	2.37164	0.0938205
$640$	0	0
$641$	7.26842	0.287085	0.143543	−	0.989644i	$-0.454151\pi$
$641$	7.26842	0.287085	0.143543	+	0.989644i	$0.454151\pi$
$642$	4.13488	0.163191
$643$	−4.40507	−0.173719	−0.0868595	−	0.996221i	$-0.527683\pi$
$643$	−4.40507	−0.173719	−0.0868595	+	0.996221i	$0.527683\pi$
$644$	−10.1082	−0.398317
$645$	0	0
$646$	0	0
$647$	0.198022	0.00778504	0.00389252	−	0.999992i	$-0.498761\pi$
$647$	0.198022	0.00778504	0.00389252	+	0.999992i	$0.498761\pi$
$648$	0.915643	0.0359698
$649$	−34.2061	−1.34271
$650$	0	0
$651$	−33.7820	−1.32402
$652$	−16.4721	−0.645098
$653$	9.92723	0.388482	0.194241	−	0.980954i	$-0.437776\pi$
$653$	9.92723	0.388482	0.194241	+	0.980954i	$0.437776\pi$
$654$	4.19897	0.164193
$655$	0	0
$656$	−13.5695	−0.529798
$657$	−5.67014	−0.221213
$658$	−7.43416	−0.289814
$659$	3.00492	0.117055	0.0585276	−	0.998286i	$-0.481359\pi$
$659$	3.00492	0.117055	0.0585276	+	0.998286i	$0.481359\pi$
$660$	0	0
$661$	−23.6923	−0.921523	−0.460762	−	0.887524i	$-0.652424\pi$
$661$	−23.6923	−0.921523	−0.460762	+	0.887524i	$0.652424\pi$
$662$	6.33984	0.246405
$663$	0	0
$664$	−20.2351	−0.785273
$665$	0	0
$666$	−3.35731	−0.130093
$667$	8.10171	0.313699
$668$	1.47186	0.0569481
$669$	−0.997912	−0.0385815
$670$	0	0
$671$	−50.3466	−1.94361
$672$	14.1590	0.546196
$673$	−30.1968	−1.16400	−0.582001	−	0.813188i	$-0.697730\pi$
$673$	−30.1968	−1.16400	−0.582001	+	0.813188i	$0.697730\pi$
$674$	1.46462	0.0564150
$675$	0	0
$676$	11.9541	0.459772
$677$	47.9413	1.84253	0.921266	−	0.388933i	$-0.127156\pi$
$677$	47.9413	1.84253	0.921266	+	0.388933i	$0.127156\pi$
$678$	3.76895	0.144746
$679$	42.2449	1.62121
$680$	0	0
$681$	14.5131	0.556145
$682$	−14.9052	−0.570750
$683$	−1.41477	−0.0541347	−0.0270674	−	0.999634i	$-0.508617\pi$
$683$	−1.41477	−0.0541347	−0.0270674	+	0.999634i	$0.508617\pi$
$684$	−14.1748	−0.541989
$685$	0	0
$686$	1.09932	0.0419723
$687$	3.17880	0.121279
$688$	−33.4652	−1.27585
$689$	−1.70763	−0.0650554
$690$	0	0
$691$	−1.36321	−0.0518588	−0.0259294	−	0.999664i	$-0.508255\pi$
$691$	−1.36321	−0.0518588	−0.0259294	+	0.999664i	$0.508255\pi$
$692$	−4.58858	−0.174432
$693$	36.0656	1.37002
$694$	−10.9983	−0.417489
$695$	0	0
$696$	−7.50363	−0.284424
$697$	0	0
$698$	7.79505	0.295047
$699$	−22.3827	−0.846592
$700$	0	0
$701$	−40.1990	−1.51829	−0.759147	−	0.650919i	$-0.774384\pi$
$701$	−40.1990	−1.51829	−0.759147	+	0.650919i	$0.774384\pi$
$702$	4.29089	0.161949
$703$	−26.5546	−1.00153
$704$	−33.1478	−1.24930
$705$	0	0
$706$	−4.26073	−0.160355
$707$	−14.3251	−0.538749
$708$	−12.7212	−0.478094
$709$	19.4729	0.731322	0.365661	−	0.930748i	$-0.380843\pi$
$709$	19.4729	0.731322	0.365661	+	0.930748i	$0.380843\pi$
$710$	0	0
$711$	3.29850	0.123704
$712$	4.28100	0.160437
$713$	12.2745	0.459685
$714$	0	0
$715$	0	0
$716$	−35.1254	−1.31270
$717$	−30.6314	−1.14395
$718$	−0.788595	−0.0294301
$719$	21.9765	0.819586	0.409793	−	0.912179i	$-0.365601\pi$
$719$	21.9765	0.819586	0.409793	+	0.912179i	$0.365601\pi$
$720$	0	0
$721$	15.2819	0.569128
$722$	−0.209398	−0.00779297
$723$	−4.01291	−0.149242
$724$	39.2758	1.45967
$725$	0	0
$726$	−7.68528	−0.285227
$727$	36.7007	1.36116	0.680578	−	0.732676i	$-0.261729\pi$
$727$	36.7007	1.36116	0.680578	+	0.732676i	$0.261729\pi$
$728$	11.3580	0.420957
$729$	19.2317	0.712284
$730$	0	0
$731$	0	0
$732$	−18.7239	−0.692056
$733$	−24.8666	−0.918471	−0.459235	−	0.888315i	$-0.651877\pi$
$733$	−24.8666	−0.918471	−0.459235	+	0.888315i	$0.651877\pi$
$734$	11.3065	0.417330
$735$	0	0
$736$	−5.14462	−0.189633
$737$	−2.46253	−0.0907085
$738$	2.14209	0.0788514
$739$	32.7368	1.20424	0.602122	−	0.798404i	$-0.294322\pi$
$739$	32.7368	1.20424	0.602122	+	0.798404i	$0.294322\pi$
$740$	0	0
$741$	12.4574	0.457633
$742$	0.739554	0.0271499
$743$	−21.3762	−0.784215	−0.392108	−	0.919919i	$-0.628254\pi$
$743$	−21.3762	−0.784215	−0.392108	+	0.919919i	$0.628254\pi$
$744$	−11.3684	−0.416786
$745$	0	0
$746$	3.97328	0.145472
$747$	−28.9906	−1.06071
$748$	0	0
$749$	42.7210	1.56099
$750$	0	0
$751$	44.5557	1.62586	0.812930	−	0.582362i	$-0.197871\pi$
$751$	44.5557	1.62586	0.812930	+	0.582362i	$0.197871\pi$
$752$	22.7052	0.827973
$753$	−19.7724	−0.720545
$754$	−4.43887	−0.161654
$755$	0	0
$756$	36.5417	1.32901
$757$	−12.1793	−0.442664	−0.221332	−	0.975199i	$-0.571040\pi$
$757$	−12.1793	−0.442664	−0.221332	+	0.975199i	$0.571040\pi$
$758$	6.89089	0.250288
$759$	9.49260	0.344560
$760$	0	0
$761$	7.18712	0.260533	0.130266	−	0.991479i	$-0.458417\pi$
$761$	7.18712	0.260533	0.130266	+	0.991479i	$0.458417\pi$
$762$	−5.98559	−0.216835
$763$	43.3832	1.57058
$764$	−43.4148	−1.57069
$765$	0	0
$766$	−0.607926	−0.0219652
$767$	−15.4336	−0.557274
$768$	−9.88618	−0.356737
$769$	14.9748	0.540005	0.270003	−	0.962860i	$-0.412975\pi$
$769$	14.9748	0.540005	0.270003	+	0.962860i	$0.412975\pi$
$770$	0	0
$771$	−23.0653	−0.830678
$772$	−37.5328	−1.35084
$773$	26.2938	0.945721	0.472860	−	0.881137i	$-0.343222\pi$
$773$	26.2938	0.945721	0.472860	+	0.881137i	$0.343222\pi$
$774$	5.28286	0.189888
$775$	0	0
$776$	14.2164	0.510339
$777$	25.1270	0.901428
$778$	2.46167	0.0882550
$779$	16.9428	0.607039
$780$	0	0
$781$	−7.83161	−0.280237
$782$	0	0
$783$	−29.2883	−1.04668
$784$	20.6430	0.737249
$785$	0	0
$786$	3.10678	0.110815
$787$	9.35342	0.333413	0.166707	−	0.986007i	$-0.446687\pi$
$787$	9.35342	0.333413	0.166707	+	0.986007i	$0.446687\pi$
$788$	24.5267	0.873727
$789$	−9.41550	−0.335201
$790$	0	0
$791$	38.9403	1.38456
$792$	12.1369	0.431267
$793$	−22.7161	−0.806672
$794$	2.87612	0.102070
$795$	0	0
$796$	12.2633	0.434663
$797$	29.4800	1.04423	0.522117	−	0.852874i	$-0.325142\pi$
$797$	29.4800	1.04423	0.522117	+	0.852874i	$0.325142\pi$
$798$	−5.39514	−0.190986
$799$	0	0
$800$	0	0
$801$	6.13335	0.216711
$802$	−6.68409	−0.236024
$803$	18.7239	0.660753
$804$	−0.915816	−0.0322983
$805$	0	0
$806$	−6.72514	−0.236883
$807$	−13.9269	−0.490249
$808$	−4.82071	−0.169592
$809$	−35.1219	−1.23482	−0.617410	−	0.786642i	$-0.711818\pi$
$809$	−35.1219	−1.23482	−0.617410	+	0.786642i	$0.711818\pi$
$810$	0	0
$811$	17.5682	0.616902	0.308451	−	0.951240i	$-0.400189\pi$
$811$	17.5682	0.616902	0.308451	+	0.951240i	$0.400189\pi$
$812$	−37.8020	−1.32659
$813$	8.01350	0.281046
$814$	11.0865	0.388582
$815$	0	0
$816$	0	0
$817$	41.7846	1.46186
$818$	5.65185	0.197612
$819$	16.2726	0.568610
$820$	0	0
$821$	13.1043	0.457343	0.228672	−	0.973504i	$-0.426562\pi$
$821$	13.1043	0.457343	0.228672	+	0.973504i	$0.426562\pi$
$822$	4.91454	0.171414
$823$	−23.7638	−0.828355	−0.414178	−	0.910196i	$-0.635931\pi$
$823$	−23.7638	−0.828355	−0.414178	+	0.910196i	$0.635931\pi$
$824$	5.14272	0.179155
$825$	0	0
$826$	6.68409	0.232569
$827$	28.9423	1.00642	0.503211	−	0.864164i	$-0.332152\pi$
$827$	28.9423	1.00642	0.503211	+	0.864164i	$0.332152\pi$
$828$	−4.87350	−0.169366
$829$	48.3131	1.67798	0.838992	−	0.544144i	$-0.183145\pi$
$829$	48.3131	1.67798	0.838992	+	0.544144i	$0.183145\pi$
$830$	0	0
$831$	−22.9229	−0.795187
$832$	−14.9561	−0.518509
$833$	0	0
$834$	−5.86311	−0.203023
$835$	0	0
$836$	46.8081	1.61889
$837$	−44.3733	−1.53377
$838$	4.46715	0.154315
$839$	36.8238	1.27130	0.635650	−	0.771978i	$-0.280732\pi$
$839$	36.8238	1.27130	0.635650	+	0.771978i	$0.280732\pi$
$840$	0	0
$841$	1.29835	0.0447707
$842$	1.23506	0.0425631
$843$	32.7164	1.12681
$844$	−21.6168	−0.744079
$845$	0	0
$846$	−3.58427	−0.123230
$847$	−79.4031	−2.72832
$848$	−2.25872	−0.0775648
$849$	−10.2942	−0.353297
$850$	0	0
$851$	−9.12981	−0.312966
$852$	−2.91258	−0.0997833
$853$	29.8138	1.02080	0.510402	−	0.859936i	$-0.329497\pi$
$853$	29.8138	1.02080	0.510402	+	0.859936i	$0.329497\pi$
$854$	9.83807	0.336652
$855$	0	0
$856$	14.3766	0.491383
$857$	1.71246	0.0584965	0.0292483	−	0.999572i	$-0.490689\pi$
$857$	1.71246	0.0584965	0.0292483	+	0.999572i	$0.490689\pi$
$858$	−5.20094	−0.177557
$859$	−22.5749	−0.770246	−0.385123	−	0.922865i	$-0.625841\pi$
$859$	−22.5749	−0.770246	−0.385123	+	0.922865i	$0.625841\pi$
$860$	0	0
$861$	−16.0320	−0.546369
$862$	−10.2459	−0.348978
$863$	−54.2578	−1.84696	−0.923479	−	0.383650i	$-0.874667\pi$
$863$	−54.2578	−1.84696	−0.923479	+	0.383650i	$0.874667\pi$
$864$	18.5982	0.632723
$865$	0	0
$866$	11.8938	0.404169
$867$	0	0
$868$	−57.2721	−1.94394
$869$	−10.8923	−0.369496
$870$	0	0
$871$	−1.11108	−0.0376474
$872$	14.5995	0.494400
$873$	20.3677	0.689343
$874$	1.96030	0.0663082
$875$	0	0
$876$	6.96343	0.235273
$877$	14.8989	0.503099	0.251549	−	0.967844i	$-0.419060\pi$
$877$	14.8989	0.503099	0.251549	+	0.967844i	$0.419060\pi$
$878$	0.206794	0.00697896
$879$	−18.2256	−0.614736
$880$	0	0
$881$	22.3330	0.752419	0.376209	−	0.926535i	$-0.377227\pi$
$881$	22.3330	0.752419	0.376209	+	0.926535i	$0.377227\pi$
$882$	−3.25872	−0.109727
$883$	−25.3319	−0.852485	−0.426242	−	0.904609i	$-0.640163\pi$
$883$	−25.3319	−0.852485	−0.426242	+	0.904609i	$0.640163\pi$
$884$	0	0
$885$	0	0
$886$	8.57628	0.288126
$887$	−6.35813	−0.213485	−0.106743	−	0.994287i	$-0.534042\pi$
$887$	−6.35813	−0.213485	−0.106743	+	0.994287i	$0.534042\pi$
$888$	8.45584	0.283759
$889$	−61.8422	−2.07412
$890$	0	0
$891$	−4.33194	−0.145125
$892$	−1.69181	−0.0566459
$893$	−28.3497	−0.948686
$894$	1.66417	0.0556581
$895$	0	0
$896$	31.7025	1.05911
$897$	4.28300	0.143005
$898$	−5.43679	−0.181428
$899$	45.9037	1.53097
$900$	0	0
$901$	0	0
$902$	−7.07361	−0.235525
$903$	−39.5383	−1.31575
$904$	13.1043	0.435843
$905$	0	0
$906$	1.04449	0.0347007
$907$	13.0295	0.432636	0.216318	−	0.976323i	$-0.430595\pi$
$907$	13.0295	0.432636	0.216318	+	0.976323i	$0.430595\pi$
$908$	24.6048	0.816539
$909$	−6.90660	−0.229078
$910$	0	0
$911$	20.7685	0.688093	0.344046	−	0.938953i	$-0.388202\pi$
$911$	20.7685	0.688093	0.344046	+	0.938953i	$0.388202\pi$
$912$	16.4777	0.545630
$913$	95.7328	3.16829
$914$	−4.05530	−0.134137
$915$	0	0
$916$	5.38916	0.178063
$917$	32.0988	1.06000
$918$	0	0
$919$	−5.10970	−0.168553	−0.0842766	−	0.996442i	$-0.526858\pi$
$919$	−5.10970	−0.168553	−0.0842766	+	0.996442i	$0.526858\pi$
$920$	0	0
$921$	3.79193	0.124948
$922$	3.40636	0.112183
$923$	−3.53358	−0.116309
$924$	−44.2918	−1.45709
$925$	0	0
$926$	−5.36349	−0.176255
$927$	7.36794	0.241995
$928$	−19.2396	−0.631571
$929$	−39.6073	−1.29947	−0.649737	−	0.760159i	$-0.725121\pi$
$929$	−39.6073	−1.29947	−0.649737	+	0.760159i	$0.725121\pi$
$930$	0	0
$931$	−25.7748	−0.844734
$932$	−37.9464	−1.24298
$933$	8.93194	0.292419
$934$	−0.0129070	−0.000422330 0
$935$	0	0
$936$	5.47610	0.178992
$937$	−39.5575	−1.29229	−0.646144	−	0.763215i	$-0.723620\pi$
$937$	−39.5575	−1.29229	−0.646144	+	0.763215i	$0.723620\pi$
$938$	0.481195	0.0157116
$939$	7.67169	0.250356
$940$	0	0
$941$	53.2283	1.73519	0.867597	−	0.497268i	$-0.165663\pi$
$941$	53.2283	1.73519	0.867597	+	0.497268i	$0.165663\pi$
$942$	−4.87022	−0.158680
$943$	5.82516	0.189693
$944$	−20.4143	−0.664430
$945$	0	0
$946$	−17.4450	−0.567187
$947$	−0.773357	−0.0251307	−0.0125654	−	0.999921i	$-0.504000\pi$
$947$	−0.773357	−0.0251307	−0.0125654	+	0.999921i	$0.504000\pi$
$948$	−4.05086	−0.131566
$949$	8.44812	0.274238
$950$	0	0
$951$	−30.4844	−0.988525
$952$	0	0
$953$	36.8731	1.19444	0.597218	−	0.802079i	$-0.296273\pi$
$953$	36.8731	1.19444	0.597218	+	0.802079i	$0.296273\pi$
$954$	0.356564	0.0115442
$955$	0	0
$956$	−51.9309	−1.67956
$957$	35.4999	1.14755
$958$	−11.0321	−0.356430
$959$	50.7763	1.63965
$960$	0	0
$961$	38.5466	1.24344
$962$	5.00217	0.161276
$963$	20.5973	0.663738
$964$	−6.80327	−0.219118
$965$	0	0
$966$	−1.85492	−0.0596810
$967$	−17.8306	−0.573392	−0.286696	−	0.958022i	$-0.592557\pi$
$967$	−17.8306	−0.573392	−0.286696	+	0.958022i	$0.592557\pi$
$968$	−26.7210	−0.858846
$969$	0	0
$970$	0	0
$971$	20.3783	0.653970	0.326985	−	0.945030i	$-0.393967\pi$
$971$	20.3783	0.653970	0.326985	+	0.945030i	$0.393967\pi$
$972$	28.7693	0.922775
$973$	−60.5768	−1.94200
$974$	−7.67476	−0.245915
$975$	0	0
$976$	−30.0471	−0.961785
$977$	8.64587	0.276606	0.138303	−	0.990390i	$-0.455835\pi$
$977$	8.64587	0.276606	0.138303	+	0.990390i	$0.455835\pi$
$978$	−3.02275	−0.0966569
$979$	−20.2535	−0.647306
$980$	0	0
$981$	20.9165	0.667813
$982$	−2.66953	−0.0851882
$983$	61.1615	1.95075	0.975374	−	0.220558i	$-0.0707877\pi$
$983$	61.1615	1.95075	0.975374	+	0.220558i	$0.0707877\pi$
$984$	−5.39514	−0.171991
$985$	0	0
$986$	0	0
$987$	26.8256	0.853869
$988$	21.1195	0.671902
$989$	14.3661	0.456815
$990$	0	0
$991$	27.0979	0.860792	0.430396	−	0.902640i	$-0.358374\pi$
$991$	27.0979	0.860792	0.430396	+	0.902640i	$0.358374\pi$
$992$	−29.1491	−0.925484
$993$	−22.8768	−0.725974
$994$	1.53035	0.0485397
$995$	0	0
$996$	35.6031	1.12813
$997$	48.3759	1.53208	0.766040	−	0.642793i	$-0.222225\pi$
$997$	48.3759	1.53208	0.766040	+	0.642793i	$0.222225\pi$
$998$	−2.64105	−0.0836009
$999$	33.0049	1.04423

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bg.1.3		6
5.4	even	2		7225.2.a.ba.1.4		6
17.4	even	4		425.2.d.a.101.4	yes	6
17.13	even	4		425.2.d.a.101.3	✓	6
17.16	even	2	inner	7225.2.a.bg.1.4		6
85.4	even	4		425.2.d.b.101.3	yes	6
85.13	odd	4		425.2.c.c.424.8		12
85.38	odd	4		425.2.c.c.424.7		12
85.47	odd	4		425.2.c.c.424.5		12
85.64	even	4		425.2.d.b.101.4	yes	6
85.72	odd	4		425.2.c.c.424.6		12
85.84	even	2		7225.2.a.ba.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.c.c.424.5		12	85.47	odd	4
425.2.c.c.424.6		12	85.72	odd	4
425.2.c.c.424.7		12	85.38	odd	4
425.2.c.c.424.8		12	85.13	odd	4
425.2.d.a.101.3	✓	6	17.13	even	4
425.2.d.a.101.4	yes	6	17.4	even	4
425.2.d.b.101.3	yes	6	85.4	even	4
425.2.d.b.101.4	yes	6	85.64	even	4
7225.2.a.ba.1.3		6	85.84	even	2
7225.2.a.ba.1.4		6	5.4	even	2
7225.2.a.bg.1.3		6	1.1	even	1	trivial
7225.2.a.bg.1.4		6	17.16	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.311108 q^{2} -1.12261 q^{3} -1.90321 q^{4} -0.349253 q^{6} -3.60843 q^{7} -1.21432 q^{8} -1.73975 q^{9} +O(q^{10})\)	\(q+0.311108 q^{2} -1.12261 q^{3} -1.90321 q^{4} -0.349253 q^{6} -3.60843 q^{7} -1.21432 q^{8} -1.73975 q^{9} +5.74499 q^{11} +2.13656 q^{12} +2.59210 q^{13} -1.12261 q^{14} +3.42864 q^{16} -0.541249 q^{18} -4.28100 q^{19} +4.05086 q^{21} +1.78731 q^{22} -1.47186 q^{23} +1.36321 q^{24} +0.806424 q^{26} +5.32089 q^{27} +6.86760 q^{28} -5.50439 q^{29} -8.33946 q^{31} +3.49532 q^{32} -6.44938 q^{33} +3.31111 q^{36} +6.20290 q^{37} -1.33185 q^{38} -2.90992 q^{39} -3.95768 q^{41} +1.26025 q^{42} -9.76049 q^{43} -10.9339 q^{44} -0.457908 q^{46} +6.62222 q^{47} -3.84902 q^{48} +6.02074 q^{49} -4.93332 q^{52} -0.658781 q^{53} +1.65537 q^{54} +4.38178 q^{56} +4.80589 q^{57} -1.71246 q^{58} -5.95407 q^{59} -8.76357 q^{61} -2.59447 q^{62} +6.27775 q^{63} -5.76986 q^{64} -2.00645 q^{66} -0.428639 q^{67} +1.65233 q^{69} -1.36321 q^{71} +2.11261 q^{72} +3.25917 q^{73} +1.92977 q^{74} +8.14764 q^{76} -20.7304 q^{77} -0.905299 q^{78} -1.89597 q^{79} -0.754037 q^{81} -1.23127 q^{82} +16.6637 q^{83} -7.70964 q^{84} -3.03657 q^{86} +6.17929 q^{87} -6.97626 q^{88} -3.52543 q^{89} -9.35342 q^{91} +2.80127 q^{92} +9.36196 q^{93} +2.06022 q^{94} -3.92388 q^{96} -11.7073 q^{97} +1.87310 q^{98} -9.99483 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 2 q^{4} + 6 q^{8} + 16 q^{9}+O(q^{10}) \) 6 * q + 2 * q^2 + 2 * q^4 + 6 * q^8 + 16 * q^9	\( 6 q + 2 q^{2} + 2 q^{4} + 6 q^{8} + 16 q^{9} + 2 q^{13} - 6 q^{16} - 16 q^{18} - 12 q^{19} - 2 q^{21} - 22 q^{26} - 6 q^{32} + 28 q^{33} + 20 q^{36} + 32 q^{38} + 34 q^{42} + 8 q^{43} + 40 q^{47} - 4 q^{49} - 30 q^{52} + 10 q^{53} + 4 q^{59} - 22 q^{64} + 16 q^{66} + 24 q^{67} + 24 q^{69} + 52 q^{72} + 36 q^{76} - 44 q^{77} + 34 q^{81} + 20 q^{83} - 6 q^{84} - 4 q^{86} + 76 q^{87} - 8 q^{89} + 30 q^{93} + 40 q^{94} - 16 q^{98}+O(q^{100}) \) 6 * q + 2 * q^2 + 2 * q^4 + 6 * q^8 + 16 * q^9 + 2 * q^13 - 6 * q^16 - 16 * q^18 - 12 * q^19 - 2 * q^21 - 22 * q^26 - 6 * q^32 + 28 * q^33 + 20 * q^36 + 32 * q^38 + 34 * q^42 + 8 * q^43 + 40 * q^47 - 4 * q^49 - 30 * q^52 + 10 * q^53 + 4 * q^59 - 22 * q^64 + 16 * q^66 + 24 * q^67 + 24 * q^69 + 52 * q^72 + 36 * q^76 - 44 * q^77 + 34 * q^81 + 20 * q^83 - 6 * q^84 - 4 * q^86 + 76 * q^87 - 8 * q^89 + 30 * q^93 + 40 * q^94 - 16 * q^98

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