LMFDB - Embedded newform 7225.2.a.bv.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:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 21 x^{13} - 2 x^{12} + 171 x^{11} + 30 x^{10} - 678 x^{9} - 153 x^{8} + 1350 x^{7} + 301 x^{6} + \cdots + 3 $ x^15 - 21x^13 - 2x^12 + 171x^11 + 30x^10 - 678x^9 - 153x^8 + 1350x^7 + 301x^6 - 1284x^5 - 183x^4 + 555x^3 + 18x^2 - 81*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.26761$ 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)

$f(q)$	$=$	$q-2.26761 q^{2} +3.43264 q^{3} +3.14206 q^{4} -7.78389 q^{6} +3.42036 q^{7} -2.58974 q^{8} +8.78301 q^{9} +O(q^{10})$	\(q-2.26761 q^{2} +3.43264 q^{3} +3.14206 q^{4} -7.78389 q^{6} +3.42036 q^{7} -2.58974 q^{8} +8.78301 q^{9} +2.42502 q^{11} +10.7855 q^{12} -1.02476 q^{13} -7.75603 q^{14} -0.411591 q^{16} -19.9164 q^{18} -2.06691 q^{19} +11.7408 q^{21} -5.49899 q^{22} +0.137743 q^{23} -8.88965 q^{24} +2.32376 q^{26} +19.8510 q^{27} +10.7470 q^{28} -5.43934 q^{29} +5.31479 q^{31} +6.11281 q^{32} +8.32421 q^{33} +27.5967 q^{36} -8.82117 q^{37} +4.68694 q^{38} -3.51764 q^{39} +7.18050 q^{41} -26.6237 q^{42} +1.19483 q^{43} +7.61954 q^{44} -0.312347 q^{46} +6.70288 q^{47} -1.41284 q^{48} +4.69883 q^{49} -3.21986 q^{52} -3.25677 q^{53} -45.0143 q^{54} -8.85784 q^{56} -7.09494 q^{57} +12.3343 q^{58} -5.09719 q^{59} -0.158376 q^{61} -12.0519 q^{62} +30.0410 q^{63} -13.0383 q^{64} -18.8761 q^{66} -9.49940 q^{67} +0.472821 q^{69} -0.926355 q^{71} -22.7457 q^{72} -4.72883 q^{73} +20.0030 q^{74} -6.49434 q^{76} +8.29442 q^{77} +7.97663 q^{78} +13.7016 q^{79} +41.7922 q^{81} -16.2826 q^{82} +3.72273 q^{83} +36.8904 q^{84} -2.70940 q^{86} -18.6713 q^{87} -6.28017 q^{88} -11.1399 q^{89} -3.50505 q^{91} +0.432795 q^{92} +18.2437 q^{93} -15.1995 q^{94} +20.9831 q^{96} +9.77007 q^{97} -10.6551 q^{98} +21.2990 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 9 q^{3} + 12 q^{4} - 9 q^{6} + 12 q^{7} - 6 q^{8} + 12 q^{9}+O(q^{10}) $ 15 * q + 9 * q^3 + 12 * q^4 - 9 * q^6 + 12 * q^7 - 6 * q^8 + 12 * q^9	$ 15 q + 9 q^{3} + 12 q^{4} - 9 q^{6} + 12 q^{7} - 6 q^{8} + 12 q^{9} - 6 q^{11} + 24 q^{12} + 6 q^{16} - 12 q^{18} + 6 q^{19} + 30 q^{21} + 12 q^{22} + 36 q^{23} - 18 q^{24} + 36 q^{26} + 36 q^{27} + 24 q^{28} + 18 q^{29} - 12 q^{32} - 12 q^{33} - 9 q^{36} + 12 q^{37} + 6 q^{38} - 9 q^{39} + 18 q^{41} - 36 q^{42} + 3 q^{43} + 12 q^{44} - 21 q^{46} + 3 q^{47} - 12 q^{48} + 15 q^{49} + 27 q^{52} - 21 q^{54} + 6 q^{56} + 39 q^{57} + 18 q^{58} - 12 q^{59} + 15 q^{61} + 54 q^{62} + 60 q^{63} - 36 q^{64} + 18 q^{66} + 24 q^{67} + 42 q^{69} - 6 q^{71} - 66 q^{72} - 9 q^{73} + 36 q^{74} - 18 q^{76} - 30 q^{77} + 30 q^{78} + 9 q^{79} + 51 q^{81} - 36 q^{82} - 15 q^{83} + 9 q^{84} - 36 q^{86} + 51 q^{87} + 30 q^{88} - 24 q^{89} - 27 q^{91} + 15 q^{92} + 42 q^{93} - 57 q^{94} - 42 q^{96} + 48 q^{97} - 12 q^{99}+O(q^{100}) $ 15 * q + 9 * q^3 + 12 * q^4 - 9 * q^6 + 12 * q^7 - 6 * q^8 + 12 * q^9 - 6 * q^11 + 24 * q^12 + 6 * q^16 - 12 * q^18 + 6 * q^19 + 30 * q^21 + 12 * q^22 + 36 * q^23 - 18 * q^24 + 36 * q^26 + 36 * q^27 + 24 * q^28 + 18 * q^29 - 12 * q^32 - 12 * q^33 - 9 * q^36 + 12 * q^37 + 6 * q^38 - 9 * q^39 + 18 * q^41 - 36 * q^42 + 3 * q^43 + 12 * q^44 - 21 * q^46 + 3 * q^47 - 12 * q^48 + 15 * q^49 + 27 * q^52 - 21 * q^54 + 6 * q^56 + 39 * q^57 + 18 * q^58 - 12 * q^59 + 15 * q^61 + 54 * q^62 + 60 * q^63 - 36 * q^64 + 18 * q^66 + 24 * q^67 + 42 * q^69 - 6 * q^71 - 66 * q^72 - 9 * q^73 + 36 * q^74 - 18 * q^76 - 30 * q^77 + 30 * q^78 + 9 * q^79 + 51 * q^81 - 36 * q^82 - 15 * q^83 + 9 * q^84 - 36 * q^86 + 51 * q^87 + 30 * q^88 - 24 * q^89 - 27 * q^91 + 15 * q^92 + 42 * q^93 - 57 * q^94 - 42 * q^96 + 48 * q^97 - 12 * 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 $

$2$	−2.26761	−1.60344	−0.801721	−	0.597698i	$-0.796082\pi$
$2$	−2.26761	−1.60344	−0.801721	+	0.597698i	$0.796082\pi$
$3$	3.43264	1.98183	0.990917	−	0.134471i	$-0.0429336\pi$
$3$	3.43264	1.98183	0.990917	+	0.134471i	$0.0429336\pi$
$4$	3.14206	1.57103
$5$	0	0
$5$	0	0
$6$	−7.78389	−3.17776
$7$	3.42036	1.29277	0.646386	−	0.763010i	$-0.276279\pi$
$7$	3.42036	1.29277	0.646386	+	0.763010i	$0.276279\pi$
$8$	−2.58974	−0.915612
$9$	8.78301	2.92767
$10$	0	0
$11$	2.42502	0.731170	0.365585	−	0.930778i	$-0.380869\pi$
$11$	2.42502	0.731170	0.365585	+	0.930778i	$0.380869\pi$
$12$	10.7855	3.11352
$13$	−1.02476	−0.284218	−0.142109	−	0.989851i	$-0.545388\pi$
$13$	−1.02476	−0.284218	−0.142109	+	0.989851i	$0.545388\pi$
$14$	−7.75603	−2.07289
$15$	0	0
$16$	−0.411591	−0.102898
$17$	0	0
$17$	0	0
$18$	−19.9164	−4.69435
$19$	−2.06691	−0.474181	−0.237090	−	0.971488i	$-0.576194\pi$
$19$	−2.06691	−0.474181	−0.237090	+	0.971488i	$0.576194\pi$
$20$	0	0
$21$	11.7408	2.56206
$22$	−5.49899	−1.17239
$23$	0.137743	0.0287213	0.0143607	−	0.999897i	$-0.495429\pi$
$23$	0.137743	0.0287213	0.0143607	+	0.999897i	$0.495429\pi$
$24$	−8.88965	−1.81459
$25$	0	0
$26$	2.32376	0.455727
$27$	19.8510	3.82032
$28$	10.7470	2.03098
$29$	−5.43934	−1.01006	−0.505030	−	0.863102i	$-0.668518\pi$
$29$	−5.43934	−1.01006	−0.505030	+	0.863102i	$0.668518\pi$
$30$	0	0
$31$	5.31479	0.954564	0.477282	−	0.878750i	$-0.341622\pi$
$31$	5.31479	0.954564	0.477282	+	0.878750i	$0.341622\pi$
$32$	6.11281	1.08060
$33$	8.32421	1.44906
$34$	0	0
$35$	0	0
$36$	27.5967	4.59945
$37$	−8.82117	−1.45019	−0.725096	−	0.688648i	$-0.758205\pi$
$37$	−8.82117	−1.45019	−0.725096	+	0.688648i	$0.758205\pi$
$38$	4.68694	0.760322
$39$	−3.51764	−0.563273
$40$	0	0
$41$	7.18050	1.12141	0.560703	−	0.828017i	$-0.310531\pi$
$41$	7.18050	1.12141	0.560703	+	0.828017i	$0.310531\pi$
$42$	−26.6237	−4.10812
$43$	1.19483	0.182209	0.0911047	−	0.995841i	$-0.470960\pi$
$43$	1.19483	0.182209	0.0911047	+	0.995841i	$0.470960\pi$
$44$	7.61954	1.14869
$45$	0	0
$46$	−0.312347	−0.0460530
$47$	6.70288	0.977715	0.488858	−	0.872364i	$-0.337414\pi$
$47$	6.70288	0.977715	0.488858	+	0.872364i	$0.337414\pi$
$48$	−1.41284	−0.203926
$49$	4.69883	0.671262
$50$	0	0
$51$	0	0
$52$	−3.21986	−0.446514
$53$	−3.25677	−0.447352	−0.223676	−	0.974664i	$-0.571806\pi$
$53$	−3.25677	−0.447352	−0.223676	+	0.974664i	$0.571806\pi$
$54$	−45.0143	−6.12567
$55$	0	0
$56$	−8.85784	−1.18368
$57$	−7.09494	−0.939748
$58$	12.3343	1.61957
$59$	−5.09719	−0.663597	−0.331799	−	0.943350i	$-0.607655\pi$
$59$	−5.09719	−0.663597	−0.331799	+	0.943350i	$0.607655\pi$
$60$	0	0
$61$	−0.158376	−0.0202779	−0.0101390	−	0.999949i	$-0.503227\pi$
$61$	−0.158376	−0.0202779	−0.0101390	+	0.999949i	$0.503227\pi$
$62$	−12.0519	−1.53059
$63$	30.0410	3.78481
$64$	−13.0383	−1.62979
$65$	0	0
$66$	−18.8761	−2.32348
$67$	−9.49940	−1.16054	−0.580268	−	0.814425i	$-0.697052\pi$
$67$	−9.49940	−1.16054	−0.580268	+	0.814425i	$0.697052\pi$
$68$	0	0
$69$	0.472821	0.0569209
$70$	0	0
$71$	−0.926355	−0.109938	−0.0549691	−	0.998488i	$-0.517506\pi$
$71$	−0.926355	−0.109938	−0.0549691	+	0.998488i	$0.517506\pi$
$72$	−22.7457	−2.68061
$73$	−4.72883	−0.553467	−0.276734	−	0.960947i	$-0.589252\pi$
$73$	−4.72883	−0.553467	−0.276734	+	0.960947i	$0.589252\pi$
$74$	20.0030	2.32530
$75$	0	0
$76$	−6.49434	−0.744952
$77$	8.29442	0.945237
$78$	7.97663	0.903175
$79$	13.7016	1.54156	0.770778	−	0.637104i	$-0.219868\pi$
$79$	13.7016	1.54156	0.770778	+	0.637104i	$0.219868\pi$
$80$	0	0
$81$	41.7922	4.64358
$82$	−16.2826	−1.79811
$83$	3.72273	0.408623	0.204312	−	0.978906i	$-0.434504\pi$
$83$	3.72273	0.408623	0.204312	+	0.978906i	$0.434504\pi$
$84$	36.8904	4.02507
$85$	0	0
$86$	−2.70940	−0.292162
$87$	−18.6713	−2.00177
$88$	−6.28017	−0.669468
$89$	−11.1399	−1.18082	−0.590412	−	0.807102i	$-0.701035\pi$
$89$	−11.1399	−1.18082	−0.590412	+	0.807102i	$0.701035\pi$
$90$	0	0
$91$	−3.50505	−0.367429
$92$	0.432795	0.0451220
$93$	18.2437	1.89179
$94$	−15.1995	−1.56771
$95$	0	0
$96$	20.9831	2.14158
$97$	9.77007	0.992000	0.496000	−	0.868322i	$-0.334802\pi$
$97$	9.77007	0.992000	0.496000	+	0.868322i	$0.334802\pi$
$98$	−10.6551	−1.07633
$99$	21.2990	2.14063
$100$	0	0
$101$	14.2577	1.41869	0.709345	−	0.704861i	$-0.248991\pi$
$101$	14.2577	1.41869	0.709345	+	0.704861i	$0.248991\pi$
$102$	0	0
$103$	10.4146	1.02618	0.513090	−	0.858335i	$-0.328501\pi$
$103$	10.4146	1.02618	0.513090	+	0.858335i	$0.328501\pi$
$104$	2.65387	0.260233
$105$	0	0
$106$	7.38509	0.717303
$107$	7.92321	0.765966	0.382983	−	0.923755i	$-0.374897\pi$
$107$	7.92321	0.765966	0.382983	+	0.923755i	$0.374897\pi$
$108$	62.3729	6.00184
$109$	−3.24221	−0.310548	−0.155274	−	0.987871i	$-0.549626\pi$
$109$	−3.24221	−0.310548	−0.155274	+	0.987871i	$0.549626\pi$
$110$	0	0
$111$	−30.2799	−2.87404
$112$	−1.40779	−0.133023
$113$	15.0504	1.41582	0.707911	−	0.706301i	$-0.249638\pi$
$113$	15.0504	1.41582	0.707911	+	0.706301i	$0.249638\pi$
$114$	16.0886	1.50683
$115$	0	0
$116$	−17.0907	−1.58683
$117$	−9.00049	−0.832096
$118$	11.5584	1.06404
$119$	0	0
$120$	0	0
$121$	−5.11929	−0.465390
$122$	0.359135	0.0325145
$123$	24.6481	2.22244
$124$	16.6994	1.49965
$125$	0	0
$126$	−68.1213	−6.06873
$127$	10.6659	0.946446	0.473223	−	0.880943i	$-0.343091\pi$
$127$	10.6659	0.946446	0.473223	+	0.880943i	$0.343091\pi$
$128$	17.3401	1.53267
$129$	4.10141	0.361109
$130$	0	0
$131$	−16.2383	−1.41875	−0.709375	−	0.704832i	$-0.751022\pi$
$131$	−16.2383	−1.41875	−0.709375	+	0.704832i	$0.751022\pi$
$132$	26.1551	2.27651
$133$	−7.06956	−0.613008
$134$	21.5409	1.86085
$135$	0	0
$136$	0	0
$137$	13.7574	1.17537	0.587685	−	0.809090i	$-0.300039\pi$
$137$	13.7574	1.17537	0.587685	+	0.809090i	$0.300039\pi$
$138$	−1.07217	−0.0912695
$139$	−7.97632	−0.676542	−0.338271	−	0.941049i	$-0.609842\pi$
$139$	−7.97632	−0.676542	−0.338271	+	0.941049i	$0.609842\pi$
$140$	0	0
$141$	23.0086	1.93767
$142$	2.10061	0.176279
$143$	−2.48506	−0.207812
$144$	−3.61500	−0.301250
$145$	0	0
$146$	10.7231	0.887453
$147$	16.1294	1.33033
$148$	−27.7166	−2.27829
$149$	6.52895	0.534872	0.267436	−	0.963576i	$-0.413824\pi$
$149$	6.52895	0.534872	0.267436	+	0.963576i	$0.413824\pi$
$150$	0	0
$151$	20.0371	1.63059	0.815297	−	0.579043i	$-0.196574\pi$
$151$	20.0371	1.63059	0.815297	+	0.579043i	$0.196574\pi$
$152$	5.35275	0.434166
$153$	0	0
$154$	−18.8085	−1.51563
$155$	0	0
$156$	−11.0526	−0.884917
$157$	−8.59665	−0.686088	−0.343044	−	0.939319i	$-0.611458\pi$
$157$	−8.59665	−0.686088	−0.343044	+	0.939319i	$0.611458\pi$
$158$	−31.0700	−2.47180
$159$	−11.1793	−0.886577
$160$	0	0
$161$	0.471129	0.0371302
$162$	−94.7685	−7.44572
$163$	8.02629	0.628668	0.314334	−	0.949312i	$-0.398219\pi$
$163$	8.02629	0.628668	0.314334	+	0.949312i	$0.398219\pi$
$164$	22.5615	1.76176
$165$	0	0
$166$	−8.44171	−0.655204
$167$	10.2559	0.793625	0.396813	−	0.917900i	$-0.370116\pi$
$167$	10.2559	0.793625	0.396813	+	0.917900i	$0.370116\pi$
$168$	−30.4058	−2.34585
$169$	−11.9499	−0.919220
$170$	0	0
$171$	−18.1537	−1.38825
$172$	3.75421	0.286256
$173$	10.5979	0.805746	0.402873	−	0.915256i	$-0.368012\pi$
$173$	10.5979	0.805746	0.402873	+	0.915256i	$0.368012\pi$
$174$	42.3392	3.20972
$175$	0	0
$176$	−0.998114	−0.0752357
$177$	−17.4968	−1.31514
$178$	25.2609	1.89338
$179$	−11.6583	−0.871384	−0.435692	−	0.900096i	$-0.643496\pi$
$179$	−11.6583	−0.871384	−0.435692	+	0.900096i	$0.643496\pi$
$180$	0	0
$181$	−11.5402	−0.857779	−0.428890	−	0.903357i	$-0.641095\pi$
$181$	−11.5402	−0.857779	−0.428890	+	0.903357i	$0.641095\pi$
$182$	7.94809	0.589151
$183$	−0.543647	−0.0401875
$184$	−0.356718	−0.0262976
$185$	0	0
$186$	−41.3697	−3.03337
$187$	0	0
$188$	21.0608	1.53602
$189$	67.8974	4.93881
$190$	0	0
$191$	1.94231	0.140540	0.0702702	−	0.997528i	$-0.477614\pi$
$191$	1.94231	0.140540	0.0702702	+	0.997528i	$0.477614\pi$
$192$	−44.7557	−3.22997
$193$	−24.7207	−1.77943	−0.889717	−	0.456513i	$-0.849098\pi$
$193$	−24.7207	−1.77943	−0.889717	+	0.456513i	$0.849098\pi$
$194$	−22.1547	−1.59062
$195$	0	0
$196$	14.7640	1.05457
$197$	16.6252	1.18450	0.592249	−	0.805755i	$-0.298240\pi$
$197$	16.6252	1.18450	0.592249	+	0.805755i	$0.298240\pi$
$198$	−48.2977	−3.43237
$199$	−23.7206	−1.68151	−0.840753	−	0.541419i	$-0.817887\pi$
$199$	−23.7206	−1.68151	−0.840753	+	0.541419i	$0.817887\pi$
$200$	0	0
$201$	−32.6080	−2.29999
$202$	−32.3308	−2.27479
$203$	−18.6045	−1.30578
$204$	0	0
$205$	0	0
$206$	−23.6162	−1.64542
$207$	1.20980	0.0840866
$208$	0.421782	0.0292453
$209$	−5.01228	−0.346707
$210$	0	0
$211$	−5.10588	−0.351503	−0.175752	−	0.984435i	$-0.556236\pi$
$211$	−5.10588	−0.351503	−0.175752	+	0.984435i	$0.556236\pi$
$212$	−10.2330	−0.702802
$213$	−3.17984	−0.217879
$214$	−17.9668	−1.22818
$215$	0	0
$216$	−51.4089	−3.49793
$217$	18.1785	1.23403
$218$	7.35208	0.497945
$219$	−16.2324	−1.09688
$220$	0	0
$221$	0	0
$222$	68.6630	4.60836
$223$	−0.490240	−0.0328289	−0.0164144	−	0.999865i	$-0.505225\pi$
$223$	−0.490240	−0.0328289	−0.0164144	+	0.999865i	$0.505225\pi$
$224$	20.9080	1.39697
$225$	0	0
$226$	−34.1284	−2.27019
$227$	−7.18453	−0.476854	−0.238427	−	0.971160i	$-0.576632\pi$
$227$	−7.18453	−0.476854	−0.238427	+	0.971160i	$0.576632\pi$
$228$	−22.2927	−1.47637
$229$	16.0572	1.06109	0.530546	−	0.847656i	$-0.321987\pi$
$229$	16.0572	1.06109	0.530546	+	0.847656i	$0.321987\pi$
$230$	0	0
$231$	28.4718	1.87330
$232$	14.0865	0.924822
$233$	4.16731	0.273010	0.136505	−	0.990639i	$-0.456413\pi$
$233$	4.16731	0.273010	0.136505	+	0.990639i	$0.456413\pi$
$234$	20.4096	1.33422
$235$	0	0
$236$	−16.0157	−1.04253
$237$	47.0328	3.05511
$238$	0	0
$239$	−15.3960	−0.995886	−0.497943	−	0.867210i	$-0.665911\pi$
$239$	−15.3960	−0.995886	−0.497943	+	0.867210i	$0.665911\pi$
$240$	0	0
$241$	−24.9443	−1.60680	−0.803401	−	0.595438i	$-0.796979\pi$
$241$	−24.9443	−1.60680	−0.803401	+	0.595438i	$0.796979\pi$
$242$	11.6086	0.746226
$243$	83.9047	5.38249
$244$	−0.497626	−0.0318572
$245$	0	0
$246$	−55.8922	−3.56356
$247$	2.11809	0.134771
$248$	−13.7639	−0.874010
$249$	12.7788	0.809823
$250$	0	0
$251$	−21.6937	−1.36930	−0.684649	−	0.728873i	$-0.740044\pi$
$251$	−21.6937	−1.36930	−0.684649	+	0.728873i	$0.740044\pi$
$252$	94.3906	5.94605
$253$	0.334028	0.0210002
$254$	−24.1861	−1.51757
$255$	0	0
$256$	−13.2441	−0.827757
$257$	23.1571	1.44450	0.722249	−	0.691633i	$-0.243108\pi$
$257$	23.1571	1.44450	0.722249	+	0.691633i	$0.243108\pi$
$258$	−9.30040	−0.579018
$259$	−30.1715	−1.87477
$260$	0	0
$261$	−47.7737	−2.95712
$262$	36.8222	2.27488
$263$	20.9731	1.29326	0.646629	−	0.762804i	$-0.276178\pi$
$263$	20.9731	1.29326	0.646629	+	0.762804i	$0.276178\pi$
$264$	−21.5575	−1.32678
$265$	0	0
$266$	16.0310	0.982924
$267$	−38.2392	−2.34020
$268$	−29.8477	−1.82324
$269$	7.00248	0.426949	0.213474	−	0.976949i	$-0.431522\pi$
$269$	7.00248	0.426949	0.213474	+	0.976949i	$0.431522\pi$
$270$	0	0
$271$	2.09634	0.127344	0.0636719	−	0.997971i	$-0.479719\pi$
$271$	2.09634	0.127344	0.0636719	+	0.997971i	$0.479719\pi$
$272$	0	0
$273$	−12.0316	−0.728184
$274$	−31.1963	−1.88464
$275$	0	0
$276$	1.48563	0.0894244
$277$	−9.92108	−0.596100	−0.298050	−	0.954550i	$-0.596336\pi$
$277$	−9.92108	−0.596100	−0.298050	+	0.954550i	$0.596336\pi$
$278$	18.0872	1.08480
$279$	46.6798	2.79465
$280$	0	0
$281$	24.6826	1.47244	0.736221	−	0.676742i	$-0.236609\pi$
$281$	24.6826	1.47244	0.736221	+	0.676742i	$0.236609\pi$
$282$	−52.1745	−3.10694
$283$	−4.24664	−0.252437	−0.126218	−	0.992002i	$-0.540284\pi$
$283$	−4.24664	−0.252437	−0.126218	+	0.992002i	$0.540284\pi$
$284$	−2.91066	−0.172716
$285$	0	0
$286$	5.63516	0.333214
$287$	24.5599	1.44972
$288$	53.6889	3.16365
$289$	0	0
$290$	0	0
$291$	33.5371	1.96598
$292$	−14.8582	−0.869513
$293$	15.8192	0.924169	0.462085	−	0.886836i	$-0.347102\pi$
$293$	15.8192	0.924169	0.462085	+	0.886836i	$0.347102\pi$
$294$	−36.5752	−2.13311
$295$	0	0
$296$	22.8446	1.32781
$297$	48.1390	2.79331
$298$	−14.8051	−0.857637
$299$	−0.141153	−0.00816311
$300$	0	0
$301$	4.08673	0.235555
$302$	−45.4363	−2.61456
$303$	48.9414	2.81161
$304$	0.850719	0.0487921
$305$	0	0
$306$	0	0
$307$	14.7934	0.844306	0.422153	−	0.906525i	$-0.361275\pi$
$307$	14.7934	0.844306	0.422153	+	0.906525i	$0.361275\pi$
$308$	26.0615	1.48499
$309$	35.7495	2.03372
$310$	0	0
$311$	−24.5242	−1.39064	−0.695319	−	0.718702i	$-0.744737\pi$
$311$	−24.5242	−1.39064	−0.695319	+	0.718702i	$0.744737\pi$
$312$	9.10977	0.515739
$313$	18.6529	1.05432	0.527161	−	0.849765i	$-0.323256\pi$
$313$	18.6529	1.05432	0.527161	+	0.849765i	$0.323256\pi$
$314$	19.4939	1.10010
$315$	0	0
$316$	43.0513	2.42183
$317$	−29.8708	−1.67771	−0.838855	−	0.544355i	$-0.816775\pi$
$317$	−29.8708	−1.67771	−0.838855	+	0.544355i	$0.816775\pi$
$318$	25.3503	1.42158
$319$	−13.1905	−0.738525
$320$	0	0
$321$	27.1975	1.51802
$322$	−1.06834	−0.0595361
$323$	0	0
$324$	131.314	7.29520
$325$	0	0
$326$	−18.2005	−1.00803
$327$	−11.1293	−0.615454
$328$	−18.5956	−1.02677
$329$	22.9262	1.26396
$330$	0	0
$331$	20.5330	1.12860	0.564298	−	0.825571i	$-0.309147\pi$
$331$	20.5330	1.12860	0.564298	+	0.825571i	$0.309147\pi$
$332$	11.6970	0.641958
$333$	−77.4764	−4.24568
$334$	−23.2564	−1.27253
$335$	0	0
$336$	−4.83242	−0.263630
$337$	−16.2183	−0.883468	−0.441734	−	0.897146i	$-0.645637\pi$
$337$	−16.2183	−0.883468	−0.441734	+	0.897146i	$0.645637\pi$
$338$	27.0976	1.47392
$339$	51.6626	2.80593
$340$	0	0
$341$	12.8884	0.697949
$342$	41.1654	2.22597
$343$	−7.87081	−0.424984
$344$	−3.09429	−0.166833
$345$	0	0
$346$	−24.0320	−1.29197
$347$	−15.8091	−0.848674	−0.424337	−	0.905504i	$-0.639493\pi$
$347$	−15.8091	−0.848674	−0.424337	+	0.905504i	$0.639493\pi$
$348$	−58.6662	−3.14484
$349$	−24.7012	−1.32222	−0.661111	−	0.750288i	$-0.729915\pi$
$349$	−24.7012	−1.32222	−0.661111	+	0.750288i	$0.729915\pi$
$350$	0	0
$351$	−20.3425	−1.08580
$352$	14.8237	0.790104
$353$	3.41463	0.181742	0.0908712	−	0.995863i	$-0.471035\pi$
$353$	3.41463	0.181742	0.0908712	+	0.995863i	$0.471035\pi$
$354$	39.6759	2.10875
$355$	0	0
$356$	−35.0021	−1.85511
$357$	0	0
$358$	26.4366	1.39722
$359$	0.736037	0.0388466	0.0194233	−	0.999811i	$-0.493817\pi$
$359$	0.736037	0.0388466	0.0194233	+	0.999811i	$0.493817\pi$
$360$	0	0
$361$	−14.7279	−0.775152
$362$	26.1688	1.37540
$363$	−17.5727	−0.922326
$364$	−11.0131	−0.577241
$365$	0	0
$366$	1.23278	0.0644384
$367$	5.20818	0.271865	0.135932	−	0.990718i	$-0.456597\pi$
$367$	5.20818	0.271865	0.135932	+	0.990718i	$0.456597\pi$
$368$	−0.0566936	−0.00295536
$369$	63.0664	3.28311
$370$	0	0
$371$	−11.1393	−0.578324
$372$	57.3229	2.97205
$373$	−3.30245	−0.170995	−0.0854973	−	0.996338i	$-0.527248\pi$
$373$	−3.30245	−0.170995	−0.0854973	+	0.996338i	$0.527248\pi$
$374$	0	0
$375$	0	0
$376$	−17.3587	−0.895208
$377$	5.57402	0.287077
$378$	−153.965	−7.91910
$379$	−28.2481	−1.45101	−0.725504	−	0.688218i	$-0.758393\pi$
$379$	−28.2481	−1.45101	−0.725504	+	0.688218i	$0.758393\pi$
$380$	0	0
$381$	36.6122	1.87570
$382$	−4.40440	−0.225349
$383$	−20.4634	−1.04563	−0.522816	−	0.852445i	$-0.675119\pi$
$383$	−20.4634	−1.04563	−0.522816	+	0.852445i	$0.675119\pi$
$384$	59.5224	3.03749
$385$	0	0
$386$	56.0569	2.85322
$387$	10.4942	0.533449
$388$	30.6981	1.55846
$389$	7.61199	0.385943	0.192972	−	0.981204i	$-0.438187\pi$
$389$	7.61199	0.385943	0.192972	+	0.981204i	$0.438187\pi$
$390$	0	0
$391$	0	0
$392$	−12.1688	−0.614615
$393$	−55.7403	−2.81173
$394$	−37.6996	−1.89928
$395$	0	0
$396$	66.9225	3.36298
$397$	−29.3656	−1.47382	−0.736908	−	0.675993i	$-0.763715\pi$
$397$	−29.3656	−1.47382	−0.736908	+	0.675993i	$0.763715\pi$
$398$	53.7890	2.69620
$399$	−24.2672	−1.21488
$400$	0	0
$401$	−6.89608	−0.344374	−0.172187	−	0.985064i	$-0.555083\pi$
$401$	−6.89608	−0.344374	−0.172187	+	0.985064i	$0.555083\pi$
$402$	73.9423	3.68791
$403$	−5.44639	−0.271304
$404$	44.7984	2.22880
$405$	0	0
$406$	42.1877	2.09374
$407$	−21.3915	−1.06034
$408$	0	0
$409$	−8.18646	−0.404794	−0.202397	−	0.979304i	$-0.564873\pi$
$409$	−8.18646	−0.404794	−0.202397	+	0.979304i	$0.564873\pi$
$410$	0	0
$411$	47.2240	2.32939
$412$	32.7232	1.61216
$413$	−17.4342	−0.857881
$414$	−2.74334	−0.134828
$415$	0	0
$416$	−6.26417	−0.307126
$417$	−27.3798	−1.34080
$418$	11.3659	0.555925
$419$	−12.3625	−0.603949	−0.301974	−	0.953316i	$-0.597646\pi$
$419$	−12.3625	−0.603949	−0.301974	+	0.953316i	$0.597646\pi$
$420$	0	0
$421$	−1.21592	−0.0592602	−0.0296301	−	0.999561i	$-0.509433\pi$
$421$	−1.21592	−0.0592602	−0.0296301	+	0.999561i	$0.509433\pi$
$422$	11.5782	0.563616
$423$	58.8714	2.86243
$424$	8.43419	0.409601
$425$	0	0
$426$	7.21064	0.349357
$427$	−0.541702	−0.0262148
$428$	24.8952	1.20335
$429$	−8.53033	−0.411848
$430$	0	0
$431$	−6.89719	−0.332226	−0.166113	−	0.986107i	$-0.553122\pi$
$431$	−6.89719	−0.332226	−0.166113	+	0.986107i	$0.553122\pi$
$432$	−8.17048	−0.393102
$433$	−37.6103	−1.80744	−0.903718	−	0.428129i	$-0.859173\pi$
$433$	−37.6103	−1.80744	−0.903718	+	0.428129i	$0.859173\pi$
$434$	−41.2217	−1.97870
$435$	0	0
$436$	−10.1872	−0.487879
$437$	−0.284701	−0.0136191
$438$	36.8087	1.75879
$439$	5.40953	0.258183	0.129091	−	0.991633i	$-0.458794\pi$
$439$	5.40953	0.258183	0.129091	+	0.991633i	$0.458794\pi$
$440$	0	0
$441$	41.2699	1.96523
$442$	0	0
$443$	1.90937	0.0907169	0.0453584	−	0.998971i	$-0.485557\pi$
$443$	1.90937	0.0907169	0.0453584	+	0.998971i	$0.485557\pi$
$444$	−95.1412	−4.51520
$445$	0	0
$446$	1.11167	0.0526393
$447$	22.4115	1.06003
$448$	−44.5956	−2.10694
$449$	−14.7434	−0.695783	−0.347891	−	0.937535i	$-0.613102\pi$
$449$	−14.7434	−0.695783	−0.347891	+	0.937535i	$0.613102\pi$
$450$	0	0
$451$	17.4128	0.819939
$452$	47.2892	2.22430
$453$	68.7800	3.23157
$454$	16.2917	0.764608
$455$	0	0
$456$	18.3741	0.860445
$457$	−13.1498	−0.615123	−0.307562	−	0.951528i	$-0.599513\pi$
$457$	−13.1498	−0.615123	−0.307562	+	0.951528i	$0.599513\pi$
$458$	−36.4115	−1.70140
$459$	0	0
$460$	0	0
$461$	−14.5314	−0.676796	−0.338398	−	0.941003i	$-0.609885\pi$
$461$	−14.5314	−0.676796	−0.338398	+	0.941003i	$0.609885\pi$
$462$	−64.5629	−3.00374
$463$	15.8296	0.735665	0.367832	−	0.929892i	$-0.380100\pi$
$463$	15.8296	0.735665	0.367832	+	0.929892i	$0.380100\pi$
$464$	2.23878	0.103933
$465$	0	0
$466$	−9.44984	−0.437755
$467$	−38.0410	−1.76033	−0.880164	−	0.474669i	$-0.842568\pi$
$467$	−38.0410	−1.76033	−0.880164	+	0.474669i	$0.842568\pi$
$468$	−28.2801	−1.30725
$469$	−32.4913	−1.50031
$470$	0	0
$471$	−29.5092	−1.35971
$472$	13.2004	0.607597
$473$	2.89748	0.133226
$474$	−106.652	−4.89869
$475$	0	0
$476$	0	0
$477$	−28.6042	−1.30970
$478$	34.9122	1.59685
$479$	26.1491	1.19478	0.597391	−	0.801950i	$-0.296204\pi$
$479$	26.1491	1.19478	0.597391	+	0.801950i	$0.296204\pi$
$480$	0	0
$481$	9.03960	0.412170
$482$	56.5639	2.57642
$483$	1.61722	0.0735859
$484$	−16.0851	−0.731141
$485$	0	0
$486$	−190.263	−8.63051
$487$	−19.1129	−0.866086	−0.433043	−	0.901373i	$-0.642560\pi$
$487$	−19.1129	−0.866086	−0.433043	+	0.901373i	$0.642560\pi$
$488$	0.410152	0.0185667
$489$	27.5514	1.24592
$490$	0	0
$491$	12.3853	0.558940	0.279470	−	0.960154i	$-0.409841\pi$
$491$	12.3853	0.558940	0.279470	+	0.960154i	$0.409841\pi$
$492$	77.4456	3.49152
$493$	0	0
$494$	−4.80299	−0.216097
$495$	0	0
$496$	−2.18752	−0.0982224
$497$	−3.16846	−0.142125
$498$	−28.9773	−1.29851
$499$	−38.5417	−1.72536	−0.862681	−	0.505748i	$-0.831217\pi$
$499$	−38.5417	−1.72536	−0.862681	+	0.505748i	$0.831217\pi$
$500$	0	0
$501$	35.2048	1.57283
$502$	49.1930	2.19559
$503$	30.0204	1.33854	0.669272	−	0.743018i	$-0.266606\pi$
$503$	30.0204	1.33854	0.669272	+	0.743018i	$0.266606\pi$
$504$	−77.7985	−3.46542
$505$	0	0
$506$	−0.757446	−0.0336726
$507$	−41.0196	−1.82174
$508$	33.5129	1.48689
$509$	−17.1828	−0.761616	−0.380808	−	0.924654i	$-0.624354\pi$
$509$	−17.1828	−0.761616	−0.380808	+	0.924654i	$0.624354\pi$
$510$	0	0
$511$	−16.1743	−0.715508
$512$	−4.64780	−0.205406
$513$	−41.0301	−1.81152
$514$	−52.5112	−2.31617
$515$	0	0
$516$	12.8869	0.567312
$517$	16.2546	0.714876
$518$	68.4173	3.00608
$519$	36.3789	1.59686
$520$	0	0
$521$	39.5173	1.73128	0.865642	−	0.500663i	$-0.166911\pi$
$521$	39.5173	1.73128	0.865642	+	0.500663i	$0.166911\pi$
$522$	108.332	4.74157
$523$	−21.3500	−0.933572	−0.466786	−	0.884370i	$-0.654588\pi$
$523$	−21.3500	−0.933572	−0.466786	+	0.884370i	$0.654588\pi$
$524$	−51.0217	−2.22890
$525$	0	0
$526$	−47.5589	−2.07367
$527$	0	0
$528$	−3.42617	−0.149105
$529$	−22.9810	−0.999175
$530$	0	0
$531$	−44.7686	−1.94279
$532$	−22.2130	−0.963054
$533$	−7.35830	−0.318723
$534$	86.7115	3.75238
$535$	0	0
$536$	24.6010	1.06260
$537$	−40.0188	−1.72694
$538$	−15.8789	−0.684588
$539$	11.3948	0.490807
$540$	0	0
$541$	18.9258	0.813684	0.406842	−	0.913498i	$-0.366630\pi$
$541$	18.9258	0.813684	0.406842	+	0.913498i	$0.366630\pi$
$542$	−4.75369	−0.204188
$543$	−39.6135	−1.69998
$544$	0	0
$545$	0	0
$546$	27.2829	1.16760
$547$	−16.3672	−0.699810	−0.349905	−	0.936785i	$-0.613786\pi$
$547$	−16.3672	−0.699810	−0.349905	+	0.936785i	$0.613786\pi$
$548$	43.2264	1.84654
$549$	−1.39102	−0.0593671
$550$	0	0
$551$	11.2426	0.478951
$552$	−1.22448	−0.0521175
$553$	46.8645	1.99288
$554$	22.4972	0.955813
$555$	0	0
$556$	−25.0620	−1.06287
$557$	−36.2444	−1.53572	−0.767862	−	0.640615i	$-0.778679\pi$
$557$	−36.2444	−1.53572	−0.767862	+	0.640615i	$0.778679\pi$
$558$	−105.852	−4.48106
$559$	−1.22441	−0.0517871
$560$	0	0
$561$	0	0
$562$	−55.9705	−2.36097
$563$	4.23956	0.178676	0.0893382	−	0.996001i	$-0.471525\pi$
$563$	4.23956	0.178676	0.0893382	+	0.996001i	$0.471525\pi$
$564$	72.2942	3.04414
$565$	0	0
$566$	9.62974	0.404768
$567$	142.944	6.00310
$568$	2.39902	0.100661
$569$	−33.6529	−1.41080	−0.705401	−	0.708808i	$-0.749233\pi$
$569$	−33.6529	−1.41080	−0.705401	+	0.708808i	$0.749233\pi$
$570$	0	0
$571$	12.7549	0.533775	0.266888	−	0.963728i	$-0.414005\pi$
$571$	12.7549	0.533775	0.266888	+	0.963728i	$0.414005\pi$
$572$	−7.80821	−0.326478
$573$	6.66724	0.278528
$574$	−55.6922	−2.32455
$575$	0	0
$576$	−114.515	−4.77148
$577$	24.8192	1.03324	0.516618	−	0.856216i	$-0.327191\pi$
$577$	24.8192	1.03324	0.516618	+	0.856216i	$0.327191\pi$
$578$	0	0
$579$	−84.8572	−3.52654
$580$	0	0
$581$	12.7331	0.528257
$582$	−76.0491	−3.15234
$583$	−7.89772	−0.327090
$584$	12.2464	0.506761
$585$	0	0
$586$	−35.8718	−1.48185
$587$	−19.6733	−0.812006	−0.406003	−	0.913872i	$-0.633078\pi$
$587$	−19.6733	−0.812006	−0.406003	+	0.913872i	$0.633078\pi$
$588$	50.6795	2.08999
$589$	−10.9852	−0.452636
$590$	0	0
$591$	57.0684	2.34748
$592$	3.63071	0.149221
$593$	0.0626815	0.00257402	0.00128701	−	0.999999i	$-0.499590\pi$
$593$	0.0626815	0.00257402	0.00128701	+	0.999999i	$0.499590\pi$
$594$	−109.160	−4.47891
$595$	0	0
$596$	20.5143	0.840299
$597$	−81.4241	−3.33247
$598$	0.320081	0.0130891
$599$	5.94198	0.242783	0.121391	−	0.992605i	$-0.461264\pi$
$599$	5.94198	0.242783	0.121391	+	0.992605i	$0.461264\pi$
$600$	0	0
$601$	24.5776	1.00254	0.501271	−	0.865290i	$-0.332866\pi$
$601$	24.5776	1.00254	0.501271	+	0.865290i	$0.332866\pi$
$602$	−9.26712	−0.377700
$603$	−83.4334	−3.39767
$604$	62.9576	2.56171
$605$	0	0
$606$	−110.980	−4.50826
$607$	14.4393	0.586075	0.293037	−	0.956101i	$-0.405334\pi$
$607$	14.4393	0.586075	0.293037	+	0.956101i	$0.405334\pi$
$608$	−12.6346	−0.512401
$609$	−63.8624	−2.58784
$610$	0	0
$611$	−6.86885	−0.277884
$612$	0	0
$613$	38.2629	1.54542	0.772711	−	0.634758i	$-0.218900\pi$
$613$	38.2629	1.54542	0.772711	+	0.634758i	$0.218900\pi$
$614$	−33.5457	−1.35380
$615$	0	0
$616$	−21.4804	−0.865470
$617$	20.9468	0.843286	0.421643	−	0.906762i	$-0.361454\pi$
$617$	20.9468	0.843286	0.421643	+	0.906762i	$0.361454\pi$
$618$	−81.0660	−3.26095
$619$	−2.67715	−0.107604	−0.0538018	−	0.998552i	$-0.517134\pi$
$619$	−2.67715	−0.107604	−0.0538018	+	0.998552i	$0.517134\pi$
$620$	0	0
$621$	2.73433	0.109725
$622$	55.6112	2.22981
$623$	−38.1023	−1.52654
$624$	1.44783	0.0579594
$625$	0	0
$626$	−42.2974	−1.69055
$627$	−17.2054	−0.687116
$628$	−27.0112	−1.07786
$629$	0	0
$630$	0	0
$631$	30.4073	1.21049	0.605247	−	0.796038i	$-0.293074\pi$
$631$	30.4073	1.21049	0.605247	+	0.796038i	$0.293074\pi$
$632$	−35.4837	−1.41147
$633$	−17.5267	−0.696622
$634$	67.7353	2.69011
$635$	0	0
$636$	−35.1260	−1.39284
$637$	−4.81518	−0.190785
$638$	29.9109	1.18418
$639$	−8.13619	−0.321863
$640$	0	0
$641$	−20.2069	−0.798124	−0.399062	−	0.916924i	$-0.630664\pi$
$641$	−20.2069	−0.798124	−0.399062	+	0.916924i	$0.630664\pi$
$642$	−61.6734	−2.43405
$643$	6.06073	0.239012	0.119506	−	0.992833i	$-0.461869\pi$
$643$	6.06073	0.239012	0.119506	+	0.992833i	$0.461869\pi$
$644$	1.48031	0.0583325
$645$	0	0
$646$	0	0
$647$	−3.03490	−0.119314	−0.0596572	−	0.998219i	$-0.519001\pi$
$647$	−3.03490	−0.119314	−0.0596572	+	0.998219i	$0.519001\pi$
$648$	−108.231	−4.25172
$649$	−12.3608	−0.485203
$650$	0	0
$651$	62.4001	2.44565
$652$	25.2191	0.987655
$653$	19.3596	0.757599	0.378799	−	0.925479i	$-0.376337\pi$
$653$	19.3596	0.757599	0.378799	+	0.925479i	$0.376337\pi$
$654$	25.2370	0.986846
$655$	0	0
$656$	−2.95543	−0.115390
$657$	−41.5333	−1.62037
$658$	−51.9878	−2.02669
$659$	18.1972	0.708864	0.354432	−	0.935082i	$-0.384674\pi$
$659$	18.1972	0.708864	0.354432	+	0.935082i	$0.384674\pi$
$660$	0	0
$661$	−2.85266	−0.110956	−0.0554779	−	0.998460i	$-0.517668\pi$
$661$	−2.85266	−0.110956	−0.0554779	+	0.998460i	$0.517668\pi$
$662$	−46.5609	−1.80964
$663$	0	0
$664$	−9.64091	−0.374140
$665$	0	0
$666$	175.686	6.80771
$667$	−0.749229	−0.0290102
$668$	32.2246	1.24681
$669$	−1.68282	−0.0650615
$670$	0	0
$671$	−0.384064	−0.0148266
$672$	71.7696	2.76857
$673$	−9.43727	−0.363780	−0.181890	−	0.983319i	$-0.558221\pi$
$673$	−9.43727	−0.363780	−0.181890	+	0.983319i	$0.558221\pi$
$674$	36.7768	1.41659
$675$	0	0
$676$	−37.5472	−1.44412
$677$	15.4108	0.592284	0.296142	−	0.955144i	$-0.404300\pi$
$677$	15.4108	0.592284	0.296142	+	0.955144i	$0.404300\pi$
$678$	−117.151	−4.49914
$679$	33.4171	1.28243
$680$	0	0
$681$	−24.6619	−0.945046
$682$	−29.2260	−1.11912
$683$	19.8675	0.760210	0.380105	−	0.924943i	$-0.375888\pi$
$683$	19.8675	0.760210	0.380105	+	0.924943i	$0.375888\pi$
$684$	−57.0398	−2.18097
$685$	0	0
$686$	17.8479	0.681437
$687$	55.1186	2.10291
$688$	−0.491780	−0.0187489
$689$	3.33741	0.127145
$690$	0	0
$691$	−26.3727	−1.00326	−0.501632	−	0.865081i	$-0.667267\pi$
$691$	−26.3727	−1.00326	−0.501632	+	0.865081i	$0.667267\pi$
$692$	33.2993	1.26585
$693$	72.8500	2.76734
$694$	35.8488	1.36080
$695$	0	0
$696$	48.3538	1.83284
$697$	0	0
$698$	56.0126	2.12011
$699$	14.3049	0.541060
$700$	0	0
$701$	28.7217	1.08480	0.542402	−	0.840119i	$-0.317515\pi$
$701$	28.7217	1.08480	0.542402	+	0.840119i	$0.317515\pi$
$702$	46.1289	1.74102
$703$	18.2325	0.687653
$704$	−31.6181	−1.19165
$705$	0	0
$706$	−7.74305	−0.291414
$707$	48.7663	1.83404
$708$	−54.9760	−2.06612
$709$	−23.6034	−0.886443	−0.443222	−	0.896412i	$-0.646165\pi$
$709$	−23.6034	−0.886443	−0.443222	+	0.896412i	$0.646165\pi$
$710$	0	0
$711$	120.342	4.51316
$712$	28.8494	1.08118
$713$	0.732073	0.0274163
$714$	0	0
$715$	0	0
$716$	−36.6311	−1.36897
$717$	−52.8490	−1.97368
$718$	−1.66905	−0.0622882
$719$	28.7333	1.07157	0.535786	−	0.844354i	$-0.320015\pi$
$719$	28.7333	1.07157	0.535786	+	0.844354i	$0.320015\pi$
$720$	0	0
$721$	35.6216	1.32662
$722$	33.3971	1.24291
$723$	−85.6247	−3.18442
$724$	−36.2601	−1.34760
$725$	0	0
$726$	39.8480	1.47890
$727$	30.9774	1.14889	0.574445	−	0.818543i	$-0.305218\pi$
$727$	30.9774	1.14889	0.574445	+	0.818543i	$0.305218\pi$
$728$	9.07717	0.336422
$729$	162.638	6.02362
$730$	0	0
$731$	0	0
$732$	−1.70817	−0.0631358
$733$	28.7630	1.06239	0.531193	−	0.847251i	$-0.321744\pi$
$733$	28.7630	1.06239	0.531193	+	0.847251i	$0.321744\pi$
$734$	−11.8101	−0.435920
$735$	0	0
$736$	0.841995	0.0310363
$737$	−23.0362	−0.848550
$738$	−143.010	−5.26427
$739$	−51.3664	−1.88954	−0.944772	−	0.327729i	$-0.893717\pi$
$739$	−51.3664	−1.88954	−0.944772	+	0.327729i	$0.893717\pi$
$740$	0	0
$741$	7.27062	0.267093
$742$	25.2596	0.927310
$743$	10.2981	0.377800	0.188900	−	0.981996i	$-0.439508\pi$
$743$	10.2981	0.377800	0.188900	+	0.981996i	$0.439508\pi$
$744$	−47.2466	−1.73214
$745$	0	0
$746$	7.48868	0.274180
$747$	32.6968	1.19631
$748$	0	0
$749$	27.1002	0.990220
$750$	0	0
$751$	−43.0781	−1.57194	−0.785972	−	0.618262i	$-0.787837\pi$
$751$	−43.0781	−1.57194	−0.785972	+	0.618262i	$0.787837\pi$
$752$	−2.75884	−0.100605
$753$	−74.4668	−2.71372
$754$	−12.6397	−0.460311
$755$	0	0
$756$	213.338	7.75901
$757$	−4.41027	−0.160294	−0.0801469	−	0.996783i	$-0.525539\pi$
$757$	−4.41027	−0.160294	−0.0801469	+	0.996783i	$0.525539\pi$
$758$	64.0557	2.32661
$759$	1.14660	0.0416189
$760$	0	0
$761$	−11.3291	−0.410679	−0.205339	−	0.978691i	$-0.565830\pi$
$761$	−11.3291	−0.410679	−0.205339	+	0.978691i	$0.565830\pi$
$762$	−83.0222	−3.00758
$763$	−11.0895	−0.401468
$764$	6.10284	0.220793
$765$	0	0
$766$	46.4031	1.67661
$767$	5.22340	0.188606
$768$	−45.4622	−1.64048
$769$	−37.9671	−1.36913	−0.684565	−	0.728952i	$-0.740008\pi$
$769$	−37.9671	−1.36913	−0.684565	+	0.728952i	$0.740008\pi$
$770$	0	0
$771$	79.4899	2.86276
$772$	−77.6738	−2.79554
$773$	−31.1395	−1.12001	−0.560004	−	0.828490i	$-0.689201\pi$
$773$	−31.1395	−1.12001	−0.560004	+	0.828490i	$0.689201\pi$
$774$	−23.7967	−0.855355
$775$	0	0
$776$	−25.3019	−0.908287
$777$	−103.568	−3.71548
$778$	−17.2610	−0.618838
$779$	−14.8414	−0.531749
$780$	0	0
$781$	−2.24643	−0.0803835
$782$	0	0
$783$	−107.976	−3.85875
$784$	−1.93400	−0.0690713
$785$	0	0
$786$	126.397	4.50844
$787$	−41.3688	−1.47464	−0.737319	−	0.675544i	$-0.763909\pi$
$787$	−41.3688	−1.47464	−0.737319	+	0.675544i	$0.763909\pi$
$788$	52.2374	1.86088
$789$	71.9932	2.56303
$790$	0	0
$791$	51.4777	1.83034
$792$	−55.1588	−1.95998
$793$	0.162297	0.00576335
$794$	66.5897	2.36318
$795$	0	0
$796$	−74.5314	−2.64169
$797$	29.7948	1.05538	0.527692	−	0.849436i	$-0.323057\pi$
$797$	29.7948	1.05538	0.527692	+	0.849436i	$0.323057\pi$
$798$	55.0286	1.94799
$799$	0	0
$800$	0	0
$801$	−97.8416	−3.45706
$802$	15.6376	0.552184
$803$	−11.4675	−0.404679
$804$	−102.456	−3.61335
$805$	0	0
$806$	12.3503	0.435020
$807$	24.0370	0.846142
$808$	−36.9236	−1.29897
$809$	55.7952	1.96165	0.980827	−	0.194882i	$-0.0624323\pi$
$809$	55.7952	1.96165	0.980827	+	0.194882i	$0.0624323\pi$
$810$	0	0
$811$	21.0043	0.737560	0.368780	−	0.929517i	$-0.379776\pi$
$811$	21.0043	0.737560	0.368780	+	0.929517i	$0.379776\pi$
$812$	−58.4563	−2.05141
$813$	7.19599	0.252374
$814$	48.5076	1.70019
$815$	0	0
$816$	0	0
$817$	−2.46960	−0.0864002
$818$	18.5637	0.649064
$819$	−30.7849	−1.07571
$820$	0	0
$821$	−46.0106	−1.60578	−0.802890	−	0.596127i	$-0.796705\pi$
$821$	−46.0106	−1.60578	−0.802890	+	0.596127i	$0.796705\pi$
$822$	−107.086	−3.73504
$823$	−19.2464	−0.670887	−0.335444	−	0.942060i	$-0.608886\pi$
$823$	−19.2464	−0.670887	−0.335444	+	0.942060i	$0.608886\pi$
$824$	−26.9711	−0.939582
$825$	0	0
$826$	39.5340	1.37556
$827$	−19.7642	−0.687269	−0.343635	−	0.939103i	$-0.611658\pi$
$827$	−19.7642	−0.687269	−0.343635	+	0.939103i	$0.611658\pi$
$828$	3.80125	0.132102
$829$	11.7163	0.406923	0.203461	−	0.979083i	$-0.434781\pi$
$829$	11.7163	0.406923	0.203461	+	0.979083i	$0.434781\pi$
$830$	0	0
$831$	−34.0555	−1.18137
$832$	13.3611	0.463214
$833$	0	0
$834$	62.0868	2.14989
$835$	0	0
$836$	−15.7489	−0.544687
$837$	105.504	3.64674
$838$	28.0334	0.968397
$839$	−17.6619	−0.609756	−0.304878	−	0.952391i	$-0.598616\pi$
$839$	−17.6619	−0.609756	−0.304878	+	0.952391i	$0.598616\pi$
$840$	0	0
$841$	0.586373	0.0202197
$842$	2.75723	0.0950203
$843$	84.7265	2.91814
$844$	−16.0430	−0.552222
$845$	0	0
$846$	−133.498	−4.58974
$847$	−17.5098	−0.601644
$848$	1.34046	0.0460315
$849$	−14.5772	−0.500288
$850$	0	0
$851$	−1.21505	−0.0416514
$852$	−9.99125	−0.342294
$853$	−27.4151	−0.938676	−0.469338	−	0.883019i	$-0.655507\pi$
$853$	−27.4151	−0.938676	−0.469338	+	0.883019i	$0.655507\pi$
$854$	1.22837	0.0420339
$855$	0	0
$856$	−20.5191	−0.701327
$857$	45.3787	1.55011	0.775054	−	0.631896i	$-0.217723\pi$
$857$	45.3787	1.55011	0.775054	+	0.631896i	$0.217723\pi$
$858$	19.3435	0.660375
$859$	−0.0923067	−0.00314946	−0.00157473	−	0.999999i	$-0.500501\pi$
$859$	−0.0923067	−0.00314946	−0.00157473	+	0.999999i	$0.500501\pi$
$860$	0	0
$861$	84.3052	2.87311
$862$	15.6401	0.532705
$863$	−35.9426	−1.22350	−0.611751	−	0.791051i	$-0.709534\pi$
$863$	−35.9426	−1.22350	−0.611751	+	0.791051i	$0.709534\pi$
$864$	121.345	4.12825
$865$	0	0
$866$	85.2855	2.89812
$867$	0	0
$868$	57.1178	1.93870
$869$	33.2267	1.12714
$870$	0	0
$871$	9.73462	0.329845
$872$	8.39649	0.284341
$873$	85.8106	2.90425
$874$	0.645592	0.0218375
$875$	0	0
$876$	−51.0030	−1.72323
$877$	4.49035	0.151628	0.0758142	−	0.997122i	$-0.475844\pi$
$877$	4.49035	0.151628	0.0758142	+	0.997122i	$0.475844\pi$
$878$	−12.2667	−0.413981
$879$	54.3017	1.83155
$880$	0	0
$881$	−3.06388	−0.103225	−0.0516123	−	0.998667i	$-0.516436\pi$
$881$	−3.06388	−0.103225	−0.0516123	+	0.998667i	$0.516436\pi$
$882$	−93.5841	−3.15114
$883$	12.9210	0.434825	0.217413	−	0.976080i	$-0.430238\pi$
$883$	12.9210	0.434825	0.217413	+	0.976080i	$0.430238\pi$
$884$	0	0
$885$	0	0
$886$	−4.32971	−0.145459
$887$	43.5319	1.46166	0.730828	−	0.682561i	$-0.239134\pi$
$887$	43.5319	1.46166	0.730828	+	0.682561i	$0.239134\pi$
$888$	78.4171	2.63151
$889$	36.4812	1.22354
$890$	0	0
$891$	101.347	3.39525
$892$	−1.54036	−0.0515751
$893$	−13.8542	−0.463614
$894$	−50.8206	−1.69969
$895$	0	0
$896$	59.3095	1.98139
$897$	−0.484529	−0.0161779
$898$	33.4322	1.11565
$899$	−28.9089	−0.964166
$900$	0	0
$901$	0	0
$902$	−39.4855	−1.31472
$903$	14.0283	0.466832
$904$	−38.9766	−1.29634
$905$	0	0
$906$	−155.966	−5.18163
$907$	12.6116	0.418760	0.209380	−	0.977834i	$-0.432855\pi$
$907$	12.6116	0.418760	0.209380	+	0.977834i	$0.432855\pi$
$908$	−22.5742	−0.749151
$909$	125.225	4.15346
$910$	0	0
$911$	35.2743	1.16869	0.584345	−	0.811505i	$-0.301351\pi$
$911$	35.2743	1.16869	0.584345	+	0.811505i	$0.301351\pi$
$912$	2.92021	0.0966979
$913$	9.02769	0.298773
$914$	29.8187	0.986315
$915$	0	0
$916$	50.4527	1.66700
$917$	−55.5408	−1.83412
$918$	0	0
$919$	24.4423	0.806278	0.403139	−	0.915139i	$-0.367919\pi$
$919$	24.4423	0.806278	0.403139	+	0.915139i	$0.367919\pi$
$920$	0	0
$921$	50.7805	1.67327
$922$	32.9516	1.08520
$923$	0.949293	0.0312464
$924$	89.4599	2.94301
$925$	0	0
$926$	−35.8954	−1.17960
$927$	91.4714	3.00432
$928$	−33.2496	−1.09147
$929$	−12.5213	−0.410811	−0.205405	−	0.978677i	$-0.565851\pi$
$929$	−12.5213	−0.410811	−0.205405	+	0.978677i	$0.565851\pi$
$930$	0	0
$931$	−9.71205	−0.318300
$932$	13.0939	0.428906
$933$	−84.1826	−2.75601
$934$	86.2622	2.82259
$935$	0	0
$936$	23.3089	0.761876
$937$	−54.0150	−1.76459	−0.882296	−	0.470695i	$-0.844003\pi$
$937$	−54.0150	−1.76459	−0.882296	+	0.470695i	$0.844003\pi$
$938$	73.6777	2.40566
$939$	64.0285	2.08949
$940$	0	0
$941$	9.12106	0.297338	0.148669	−	0.988887i	$-0.452501\pi$
$941$	9.12106	0.297338	0.148669	+	0.988887i	$0.452501\pi$
$942$	66.9154	2.18022
$943$	0.989062	0.0322083
$944$	2.09795	0.0682826
$945$	0	0
$946$	−6.57035	−0.213620
$947$	−26.1164	−0.848668	−0.424334	−	0.905506i	$-0.639492\pi$
$947$	−26.1164	−0.848668	−0.424334	+	0.905506i	$0.639492\pi$
$948$	147.780	4.79966
$949$	4.84592	0.157305
$950$	0	0
$951$	−102.536	−3.32494
$952$	0	0
$953$	−5.63840	−0.182646	−0.0913228	−	0.995821i	$-0.529110\pi$
$953$	−5.63840	−0.182646	−0.0913228	+	0.995821i	$0.529110\pi$
$954$	64.8633	2.10003
$955$	0	0
$956$	−48.3752	−1.56457
$957$	−45.2782	−1.46364
$958$	−59.2959	−1.91576
$959$	47.0551	1.51949
$960$	0	0
$961$	−2.75304	−0.0888079
$962$	−20.4983	−0.660891
$963$	69.5896	2.24249
$964$	−78.3764	−2.52433
$965$	0	0
$966$	−3.66721	−0.117991
$967$	−17.5965	−0.565866	−0.282933	−	0.959140i	$-0.591307\pi$
$967$	−17.5965	−0.565866	−0.282933	+	0.959140i	$0.591307\pi$
$968$	13.2576	0.426117
$969$	0	0
$970$	0	0
$971$	−11.2359	−0.360578	−0.180289	−	0.983614i	$-0.557703\pi$
$971$	−11.2359	−0.360578	−0.180289	+	0.983614i	$0.557703\pi$
$972$	263.633	8.45604
$973$	−27.2818	−0.874616
$974$	43.3405	1.38872
$975$	0	0
$976$	0.0651860	0.00208655
$977$	12.4815	0.399319	0.199660	−	0.979865i	$-0.436016\pi$
$977$	12.4815	0.399319	0.199660	+	0.979865i	$0.436016\pi$
$978$	−62.4758	−1.99775
$979$	−27.0144	−0.863384
$980$	0	0
$981$	−28.4764	−0.909181
$982$	−28.0850	−0.896228
$983$	0.723270	0.0230687	0.0115344	−	0.999933i	$-0.496328\pi$
$983$	0.723270	0.0230687	0.0115344	+	0.999933i	$0.496328\pi$
$984$	−63.8321	−2.03489
$985$	0	0
$986$	0	0
$987$	78.6975	2.50497
$988$	6.65515	0.211728
$989$	0.164579	0.00523330
$990$	0	0
$991$	36.2760	1.15234	0.576172	−	0.817329i	$-0.304546\pi$
$991$	36.2760	1.15234	0.576172	+	0.817329i	$0.304546\pi$
$992$	32.4883	1.03150
$993$	70.4824	2.23669
$994$	7.18484	0.227889
$995$	0	0
$996$	40.1517	1.27226
$997$	13.6801	0.433254	0.216627	−	0.976254i	$-0.430495\pi$
$997$	13.6801	0.433254	0.216627	+	0.976254i	$0.430495\pi$
$998$	87.3975	2.76652
$999$	−175.109	−5.54020

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bv.1.3	yes	15
5.4	even	2		7225.2.a.bt.1.13	✓	15
17.16	even	2		7225.2.a.bu.1.3	yes	15
85.84	even	2		7225.2.a.bw.1.13	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7225.2.a.bt.1.13	✓	15	5.4	even	2
7225.2.a.bu.1.3	yes	15	17.16	even	2
7225.2.a.bv.1.3	yes	15	1.1	even	1	trivial
7225.2.a.bw.1.13	yes	15	85.84	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.26761 q^{2} +3.43264 q^{3} +3.14206 q^{4} -7.78389 q^{6} +3.42036 q^{7} -2.58974 q^{8} +8.78301 q^{9} +O(q^{10})\)	\(q-2.26761 q^{2} +3.43264 q^{3} +3.14206 q^{4} -7.78389 q^{6} +3.42036 q^{7} -2.58974 q^{8} +8.78301 q^{9} +2.42502 q^{11} +10.7855 q^{12} -1.02476 q^{13} -7.75603 q^{14} -0.411591 q^{16} -19.9164 q^{18} -2.06691 q^{19} +11.7408 q^{21} -5.49899 q^{22} +0.137743 q^{23} -8.88965 q^{24} +2.32376 q^{26} +19.8510 q^{27} +10.7470 q^{28} -5.43934 q^{29} +5.31479 q^{31} +6.11281 q^{32} +8.32421 q^{33} +27.5967 q^{36} -8.82117 q^{37} +4.68694 q^{38} -3.51764 q^{39} +7.18050 q^{41} -26.6237 q^{42} +1.19483 q^{43} +7.61954 q^{44} -0.312347 q^{46} +6.70288 q^{47} -1.41284 q^{48} +4.69883 q^{49} -3.21986 q^{52} -3.25677 q^{53} -45.0143 q^{54} -8.85784 q^{56} -7.09494 q^{57} +12.3343 q^{58} -5.09719 q^{59} -0.158376 q^{61} -12.0519 q^{62} +30.0410 q^{63} -13.0383 q^{64} -18.8761 q^{66} -9.49940 q^{67} +0.472821 q^{69} -0.926355 q^{71} -22.7457 q^{72} -4.72883 q^{73} +20.0030 q^{74} -6.49434 q^{76} +8.29442 q^{77} +7.97663 q^{78} +13.7016 q^{79} +41.7922 q^{81} -16.2826 q^{82} +3.72273 q^{83} +36.8904 q^{84} -2.70940 q^{86} -18.6713 q^{87} -6.28017 q^{88} -11.1399 q^{89} -3.50505 q^{91} +0.432795 q^{92} +18.2437 q^{93} -15.1995 q^{94} +20.9831 q^{96} +9.77007 q^{97} -10.6551 q^{98} +21.2990 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 9 q^{3} + 12 q^{4} - 9 q^{6} + 12 q^{7} - 6 q^{8} + 12 q^{9}+O(q^{10}) \) 15 * q + 9 * q^3 + 12 * q^4 - 9 * q^6 + 12 * q^7 - 6 * q^8 + 12 * q^9	\( 15 q + 9 q^{3} + 12 q^{4} - 9 q^{6} + 12 q^{7} - 6 q^{8} + 12 q^{9} - 6 q^{11} + 24 q^{12} + 6 q^{16} - 12 q^{18} + 6 q^{19} + 30 q^{21} + 12 q^{22} + 36 q^{23} - 18 q^{24} + 36 q^{26} + 36 q^{27} + 24 q^{28} + 18 q^{29} - 12 q^{32} - 12 q^{33} - 9 q^{36} + 12 q^{37} + 6 q^{38} - 9 q^{39} + 18 q^{41} - 36 q^{42} + 3 q^{43} + 12 q^{44} - 21 q^{46} + 3 q^{47} - 12 q^{48} + 15 q^{49} + 27 q^{52} - 21 q^{54} + 6 q^{56} + 39 q^{57} + 18 q^{58} - 12 q^{59} + 15 q^{61} + 54 q^{62} + 60 q^{63} - 36 q^{64} + 18 q^{66} + 24 q^{67} + 42 q^{69} - 6 q^{71} - 66 q^{72} - 9 q^{73} + 36 q^{74} - 18 q^{76} - 30 q^{77} + 30 q^{78} + 9 q^{79} + 51 q^{81} - 36 q^{82} - 15 q^{83} + 9 q^{84} - 36 q^{86} + 51 q^{87} + 30 q^{88} - 24 q^{89} - 27 q^{91} + 15 q^{92} + 42 q^{93} - 57 q^{94} - 42 q^{96} + 48 q^{97} - 12 q^{99}+O(q^{100}) \) 15 * q + 9 * q^3 + 12 * q^4 - 9 * q^6 + 12 * q^7 - 6 * q^8 + 12 * q^9 - 6 * q^11 + 24 * q^12 + 6 * q^16 - 12 * q^18 + 6 * q^19 + 30 * q^21 + 12 * q^22 + 36 * q^23 - 18 * q^24 + 36 * q^26 + 36 * q^27 + 24 * q^28 + 18 * q^29 - 12 * q^32 - 12 * q^33 - 9 * q^36 + 12 * q^37 + 6 * q^38 - 9 * q^39 + 18 * q^41 - 36 * q^42 + 3 * q^43 + 12 * q^44 - 21 * q^46 + 3 * q^47 - 12 * q^48 + 15 * q^49 + 27 * q^52 - 21 * q^54 + 6 * q^56 + 39 * q^57 + 18 * q^58 - 12 * q^59 + 15 * q^61 + 54 * q^62 + 60 * q^63 - 36 * q^64 + 18 * q^66 + 24 * q^67 + 42 * q^69 - 6 * q^71 - 66 * q^72 - 9 * q^73 + 36 * q^74 - 18 * q^76 - 30 * q^77 + 30 * q^78 + 9 * q^79 + 51 * q^81 - 36 * q^82 - 15 * q^83 + 9 * q^84 - 36 * q^86 + 51 * q^87 + 30 * q^88 - 24 * q^89 - 27 * q^91 + 15 * q^92 + 42 * q^93 - 57 * q^94 - 42 * q^96 + 48 * q^97 - 12 * q^99

Introduction

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