LMFDB - Embedded newform 7225.2.a.bv.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$57.6919154604$
Analytic rank:	$0$
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			1.15
Root			$-2.50743$ 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)

$f(q)$	$=$	$q+2.50743 q^{2} -0.598298 q^{3} +4.28722 q^{4} -1.50019 q^{6} +3.52603 q^{7} +5.73504 q^{8} -2.64204 q^{9} +O(q^{10})$	\(q+2.50743 q^{2} -0.598298 q^{3} +4.28722 q^{4} -1.50019 q^{6} +3.52603 q^{7} +5.73504 q^{8} -2.64204 q^{9} +3.05962 q^{11} -2.56503 q^{12} +5.62679 q^{13} +8.84128 q^{14} +5.80580 q^{16} -6.62474 q^{18} -4.57297 q^{19} -2.10961 q^{21} +7.67178 q^{22} +0.455839 q^{23} -3.43126 q^{24} +14.1088 q^{26} +3.37562 q^{27} +15.1168 q^{28} +9.46268 q^{29} +4.24066 q^{31} +3.08756 q^{32} -1.83056 q^{33} -11.3270 q^{36} -9.85143 q^{37} -11.4664 q^{38} -3.36650 q^{39} -0.505315 q^{41} -5.28972 q^{42} +9.70924 q^{43} +13.1172 q^{44} +1.14299 q^{46} -4.72552 q^{47} -3.47360 q^{48} +5.43287 q^{49} +24.1233 q^{52} -12.0203 q^{53} +8.46414 q^{54} +20.2219 q^{56} +2.73600 q^{57} +23.7270 q^{58} -10.8652 q^{59} +13.8337 q^{61} +10.6332 q^{62} -9.31591 q^{63} -3.86974 q^{64} -4.59001 q^{66} -3.04488 q^{67} -0.272728 q^{69} +4.20889 q^{71} -15.1522 q^{72} -8.48279 q^{73} -24.7018 q^{74} -19.6053 q^{76} +10.7883 q^{77} -8.44126 q^{78} +3.66719 q^{79} +5.90649 q^{81} -1.26704 q^{82} -2.54818 q^{83} -9.04438 q^{84} +24.3453 q^{86} -5.66150 q^{87} +17.5470 q^{88} +4.72824 q^{89} +19.8402 q^{91} +1.95428 q^{92} -2.53718 q^{93} -11.8489 q^{94} -1.84728 q^{96} +3.05976 q^{97} +13.6226 q^{98} -8.08363 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.50743	1.77302	0.886511	−	0.462707i	$-0.153122\pi$
$2$	2.50743	1.77302	0.886511	+	0.462707i	$0.153122\pi$
$3$	−0.598298	−0.345427	−0.172714	−	0.984972i	$-0.555254\pi$
$3$	−0.598298	−0.345427	−0.172714	+	0.984972i	$0.555254\pi$
$4$	4.28722	2.14361
$5$	0	0
$5$	0	0
$6$	−1.50019	−0.612451
$7$	3.52603	1.33271	0.666357	−	0.745633i	$-0.267853\pi$
$7$	3.52603	1.33271	0.666357	+	0.745633i	$0.267853\pi$
$8$	5.73504	2.02764
$9$	−2.64204	−0.880680
$10$	0	0
$11$	3.05962	0.922509	0.461254	−	0.887268i	$-0.347400\pi$
$11$	3.05962	0.922509	0.461254	+	0.887268i	$0.347400\pi$
$12$	−2.56503	−0.740461
$13$	5.62679	1.56059	0.780296	−	0.625411i	$-0.215069\pi$
$13$	5.62679	1.56059	0.780296	+	0.625411i	$0.215069\pi$
$14$	8.84128	2.36293
$15$	0	0
$16$	5.80580	1.45145
$17$	0	0
$17$	0	0
$18$	−6.62474	−1.56147
$19$	−4.57297	−1.04911	−0.524556	−	0.851376i	$-0.675769\pi$
$19$	−4.57297	−1.04911	−0.524556	+	0.851376i	$0.675769\pi$
$20$	0	0
$21$	−2.10961	−0.460356
$22$	7.67178	1.63563
$23$	0.455839	0.0950490	0.0475245	−	0.998870i	$-0.484867\pi$
$23$	0.455839	0.0950490	0.0475245	+	0.998870i	$0.484867\pi$
$24$	−3.43126	−0.700404
$25$	0	0
$26$	14.1088	2.76696
$27$	3.37562	0.649638
$28$	15.1168	2.85682
$29$	9.46268	1.75718	0.878588	−	0.477581i	$-0.158486\pi$
$29$	9.46268	1.75718	0.878588	+	0.477581i	$0.158486\pi$
$30$	0	0
$31$	4.24066	0.761646	0.380823	−	0.924648i	$-0.375641\pi$
$31$	4.24066	0.761646	0.380823	+	0.924648i	$0.375641\pi$
$32$	3.08756	0.545809
$33$	−1.83056	−0.318660
$34$	0	0
$35$	0	0
$36$	−11.3270	−1.88783
$37$	−9.85143	−1.61957	−0.809783	−	0.586730i	$-0.800415\pi$
$37$	−9.85143	−1.61957	−0.809783	+	0.586730i	$0.800415\pi$
$38$	−11.4664	−1.86010
$39$	−3.36650	−0.539071
$40$	0	0
$41$	−0.505315	−0.0789169	−0.0394584	−	0.999221i	$-0.512563\pi$
$41$	−0.505315	−0.0789169	−0.0394584	+	0.999221i	$0.512563\pi$
$42$	−5.28972	−0.816221
$43$	9.70924	1.48065	0.740323	−	0.672252i	$-0.234673\pi$
$43$	9.70924	1.48065	0.740323	+	0.672252i	$0.234673\pi$
$44$	13.1172	1.97750
$45$	0	0
$46$	1.14299	0.168524
$47$	−4.72552	−0.689288	−0.344644	−	0.938734i	$-0.612000\pi$
$47$	−4.72552	−0.689288	−0.344644	+	0.938734i	$0.612000\pi$
$48$	−3.47360	−0.501370
$49$	5.43287	0.776125
$50$	0	0
$51$	0	0
$52$	24.1233	3.34530
$53$	−12.0203	−1.65112	−0.825559	−	0.564316i	$-0.809140\pi$
$53$	−12.0203	−1.65112	−0.825559	+	0.564316i	$0.809140\pi$
$54$	8.46414	1.15182
$55$	0	0
$56$	20.2219	2.70227
$57$	2.73600	0.362392
$58$	23.7270	3.11551
$59$	−10.8652	−1.41453	−0.707267	−	0.706946i	$-0.750072\pi$
$59$	−10.8652	−1.41453	−0.707267	+	0.706946i	$0.750072\pi$
$60$	0	0
$61$	13.8337	1.77122	0.885612	−	0.464425i	$-0.153739\pi$
$61$	13.8337	1.77122	0.885612	+	0.464425i	$0.153739\pi$
$62$	10.6332	1.35041
$63$	−9.31591	−1.17369
$64$	−3.86974	−0.483718
$65$	0	0
$66$	−4.59001	−0.564991
$67$	−3.04488	−0.371991	−0.185996	−	0.982551i	$-0.559551\pi$
$67$	−3.04488	−0.371991	−0.185996	+	0.982551i	$0.559551\pi$
$68$	0	0
$69$	−0.272728	−0.0328325
$70$	0	0
$71$	4.20889	0.499503	0.249752	−	0.968310i	$-0.419651\pi$
$71$	4.20889	0.499503	0.249752	+	0.968310i	$0.419651\pi$
$72$	−15.1522	−1.78571
$73$	−8.48279	−0.992836	−0.496418	−	0.868084i	$-0.665352\pi$
$73$	−8.48279	−0.992836	−0.496418	+	0.868084i	$0.665352\pi$
$74$	−24.7018	−2.87153
$75$	0	0
$76$	−19.6053	−2.24888
$77$	10.7883	1.22944
$78$	−8.44126	−0.955785
$79$	3.66719	0.412591	0.206296	−	0.978490i	$-0.433859\pi$
$79$	3.66719	0.412591	0.206296	+	0.978490i	$0.433859\pi$
$80$	0	0
$81$	5.90649	0.656277
$82$	−1.26704	−0.139921
$83$	−2.54818	−0.279699	−0.139849	−	0.990173i	$-0.544662\pi$
$83$	−2.54818	−0.279699	−0.139849	+	0.990173i	$0.544662\pi$
$84$	−9.04438	−0.986822
$85$	0	0
$86$	24.3453	2.62522
$87$	−5.66150	−0.606976
$88$	17.5470	1.87052
$89$	4.72824	0.501192	0.250596	−	0.968092i	$-0.419373\pi$
$89$	4.72824	0.501192	0.250596	+	0.968092i	$0.419373\pi$
$90$	0	0
$91$	19.8402	2.07982
$92$	1.95428	0.203748
$93$	−2.53718	−0.263093
$94$	−11.8489	−1.22212
$95$	0	0
$96$	−1.84728	−0.188537
$97$	3.05976	0.310672	0.155336	−	0.987862i	$-0.450354\pi$
$97$	3.05976	0.310672	0.155336	+	0.987862i	$0.450354\pi$
$98$	13.6226	1.37609
$99$	−8.08363	−0.812435
$100$	0	0
$101$	13.0465	1.29817	0.649086	−	0.760715i	$-0.275151\pi$
$101$	13.0465	1.29817	0.649086	+	0.760715i	$0.275151\pi$
$102$	0	0
$103$	8.23728	0.811643	0.405822	−	0.913952i	$-0.366985\pi$
$103$	8.23728	0.811643	0.405822	+	0.913952i	$0.366985\pi$
$104$	32.2699	3.16432
$105$	0	0
$106$	−30.1401	−2.92747
$107$	6.96309	0.673148	0.336574	−	0.941657i	$-0.390732\pi$
$107$	6.96309	0.673148	0.336574	+	0.941657i	$0.390732\pi$
$108$	14.4720	1.39257
$109$	−10.8324	−1.03756	−0.518779	−	0.854909i	$-0.673613\pi$
$109$	−10.8324	−1.03756	−0.518779	+	0.854909i	$0.673613\pi$
$110$	0	0
$111$	5.89409	0.559442
$112$	20.4714	1.93437
$113$	7.61490	0.716349	0.358175	−	0.933655i	$-0.383399\pi$
$113$	7.61490	0.716349	0.358175	+	0.933655i	$0.383399\pi$
$114$	6.86033	0.642529
$115$	0	0
$116$	40.5686	3.76670
$117$	−14.8662	−1.37438
$118$	−27.2439	−2.50800
$119$	0	0
$120$	0	0
$121$	−1.63875	−0.148977
$122$	34.6871	3.14042
$123$	0.302329	0.0272600
$124$	18.1806	1.63267
$125$	0	0
$126$	−23.3590	−2.08099
$127$	−7.44264	−0.660427	−0.330214	−	0.943906i	$-0.607121\pi$
$127$	−7.44264	−0.660427	−0.330214	+	0.943906i	$0.607121\pi$
$128$	−15.8782	−1.40345
$129$	−5.80902	−0.511455
$130$	0	0
$131$	1.70296	0.148789	0.0743943	−	0.997229i	$-0.476298\pi$
$131$	1.70296	0.148789	0.0743943	+	0.997229i	$0.476298\pi$
$132$	−7.84801	−0.683082
$133$	−16.1244	−1.39817
$134$	−7.63483	−0.659549
$135$	0	0
$136$	0	0
$137$	−0.390352	−0.0333500	−0.0166750	−	0.999861i	$-0.505308\pi$
$137$	−0.390352	−0.0333500	−0.0166750	+	0.999861i	$0.505308\pi$
$138$	−0.683846	−0.0582128
$139$	−18.6534	−1.58216	−0.791079	−	0.611714i	$-0.790480\pi$
$139$	−18.6534	−1.58216	−0.791079	+	0.611714i	$0.790480\pi$
$140$	0	0
$141$	2.82727	0.238099
$142$	10.5535	0.885630
$143$	17.2158	1.43966
$144$	−15.3392	−1.27826
$145$	0	0
$146$	−21.2700	−1.76032
$147$	−3.25048	−0.268095
$148$	−42.2352	−3.47172
$149$	−4.90578	−0.401897	−0.200948	−	0.979602i	$-0.564402\pi$
$149$	−4.90578	−0.401897	−0.200948	+	0.979602i	$0.564402\pi$
$150$	0	0
$151$	6.12643	0.498562	0.249281	−	0.968431i	$-0.419806\pi$
$151$	6.12643	0.498562	0.249281	+	0.968431i	$0.419806\pi$
$152$	−26.2262	−2.12722
$153$	0	0
$154$	27.0509	2.17982
$155$	0	0
$156$	−14.4329	−1.15556
$157$	4.45503	0.355550	0.177775	−	0.984071i	$-0.443110\pi$
$157$	4.45503	0.355550	0.177775	+	0.984071i	$0.443110\pi$
$158$	9.19523	0.731533
$159$	7.19173	0.570341
$160$	0	0
$161$	1.60730	0.126673
$162$	14.8101	1.16359
$163$	−7.75241	−0.607215	−0.303608	−	0.952797i	$-0.598191\pi$
$163$	−7.75241	−0.607215	−0.303608	+	0.952797i	$0.598191\pi$
$164$	−2.16639	−0.169167
$165$	0	0
$166$	−6.38939	−0.495913
$167$	24.7876	1.91812	0.959062	−	0.283197i	$-0.0913950\pi$
$167$	24.7876	1.91812	0.959062	+	0.283197i	$0.0913950\pi$
$168$	−12.0987	−0.933437
$169$	18.6608	1.43545
$170$	0	0
$171$	12.0820	0.923932
$172$	41.6256	3.17392
$173$	−0.830544	−0.0631451	−0.0315726	−	0.999501i	$-0.510052\pi$
$173$	−0.830544	−0.0631451	−0.0315726	+	0.999501i	$0.510052\pi$
$174$	−14.1958	−1.07618
$175$	0	0
$176$	17.7635	1.33898
$177$	6.50065	0.488619
$178$	11.8557	0.888626
$179$	11.7896	0.881195	0.440597	−	0.897705i	$-0.354767\pi$
$179$	11.7896	0.881195	0.440597	+	0.897705i	$0.354767\pi$
$180$	0	0
$181$	−1.23289	−0.0916397	−0.0458199	−	0.998950i	$-0.514590\pi$
$181$	−1.23289	−0.0916397	−0.0458199	+	0.998950i	$0.514590\pi$
$182$	49.7480	3.68757
$183$	−8.27668	−0.611830
$184$	2.61426	0.192726
$185$	0	0
$186$	−6.36181	−0.466470
$187$	0	0
$188$	−20.2593	−1.47756
$189$	11.9025	0.865782
$190$	0	0
$191$	−13.0772	−0.946235	−0.473118	−	0.880999i	$-0.656871\pi$
$191$	−13.0772	−0.946235	−0.473118	+	0.880999i	$0.656871\pi$
$192$	2.31526	0.167089
$193$	−0.307240	−0.0221156	−0.0110578	−	0.999939i	$-0.503520\pi$
$193$	−0.307240	−0.0221156	−0.0110578	+	0.999939i	$0.503520\pi$
$194$	7.67215	0.550828
$195$	0	0
$196$	23.2919	1.66371
$197$	14.1967	1.01147	0.505736	−	0.862688i	$-0.331221\pi$
$197$	14.1967	1.01147	0.505736	+	0.862688i	$0.331221\pi$
$198$	−20.2691	−1.44047
$199$	−15.7904	−1.11935	−0.559675	−	0.828712i	$-0.689074\pi$
$199$	−15.7904	−1.11935	−0.559675	+	0.828712i	$0.689074\pi$
$200$	0	0
$201$	1.82174	0.128496
$202$	32.7131	2.30169
$203$	33.3657	2.34181
$204$	0	0
$205$	0	0
$206$	20.6544	1.43906
$207$	−1.20435	−0.0837078
$208$	32.6680	2.26512
$209$	−13.9915	−0.967815
$210$	0	0
$211$	−6.85984	−0.472251	−0.236126	−	0.971723i	$-0.575878\pi$
$211$	−6.85984	−0.472251	−0.236126	+	0.971723i	$0.575878\pi$
$212$	−51.5337	−3.53935
$213$	−2.51817	−0.172542
$214$	17.4595	1.19351
$215$	0	0
$216$	19.3593	1.31724
$217$	14.9527	1.01506
$218$	−27.1616	−1.83961
$219$	5.07524	0.342953
$220$	0	0
$221$	0	0
$222$	14.7790	0.991904
$223$	13.4206	0.898711	0.449356	−	0.893353i	$-0.351654\pi$
$223$	13.4206	0.898711	0.449356	+	0.893353i	$0.351654\pi$
$224$	10.8868	0.727407
$225$	0	0
$226$	19.0938	1.27010
$227$	−3.47659	−0.230750	−0.115375	−	0.993322i	$-0.536807\pi$
$227$	−3.47659	−0.230750	−0.115375	+	0.993322i	$0.536807\pi$
$228$	11.7298	0.776826
$229$	−19.1819	−1.26758	−0.633789	−	0.773506i	$-0.718501\pi$
$229$	−19.1819	−1.26758	−0.633789	+	0.773506i	$0.718501\pi$
$230$	0	0
$231$	−6.45461	−0.424682
$232$	54.2689	3.56293
$233$	15.1991	0.995728	0.497864	−	0.867255i	$-0.334118\pi$
$233$	15.1991	0.995728	0.497864	+	0.867255i	$0.334118\pi$
$234$	−37.2760	−2.43681
$235$	0	0
$236$	−46.5817	−3.03221
$237$	−2.19407	−0.142520
$238$	0	0
$239$	−7.18197	−0.464563	−0.232281	−	0.972649i	$-0.574619\pi$
$239$	−7.18197	−0.464563	−0.232281	+	0.972649i	$0.574619\pi$
$240$	0	0
$241$	−14.0247	−0.903411	−0.451705	−	0.892167i	$-0.649184\pi$
$241$	−14.0247	−0.903411	−0.451705	+	0.892167i	$0.649184\pi$
$242$	−4.10906	−0.264140
$243$	−13.6607	−0.876334
$244$	59.3081	3.79681
$245$	0	0
$246$	0.758068	0.0483327
$247$	−25.7312	−1.63723
$248$	24.3204	1.54435
$249$	1.52457	0.0966157
$250$	0	0
$251$	−23.3367	−1.47300	−0.736499	−	0.676439i	$-0.763522\pi$
$251$	−23.3367	−1.47300	−0.736499	+	0.676439i	$0.763522\pi$
$252$	−39.9393	−2.51594
$253$	1.39469	0.0876836
$254$	−18.6619	−1.17095
$255$	0	0
$256$	−32.0741	−2.00463
$257$	−3.23211	−0.201613	−0.100807	−	0.994906i	$-0.532142\pi$
$257$	−3.23211	−0.201613	−0.100807	+	0.994906i	$0.532142\pi$
$258$	−14.5657	−0.906822
$259$	−34.7364	−2.15842
$260$	0	0
$261$	−25.0008	−1.54751
$262$	4.27007	0.263806
$263$	21.0607	1.29866	0.649331	−	0.760506i	$-0.275049\pi$
$263$	21.0607	1.29866	0.649331	+	0.760506i	$0.275049\pi$
$264$	−10.4983	−0.646129
$265$	0	0
$266$	−40.4309	−2.47898
$267$	−2.82890	−0.173126
$268$	−13.0541	−0.797403
$269$	−25.8825	−1.57809	−0.789043	−	0.614338i	$-0.789423\pi$
$269$	−25.8825	−1.57809	−0.789043	+	0.614338i	$0.789423\pi$
$270$	0	0
$271$	13.2896	0.807283	0.403642	−	0.914917i	$-0.367744\pi$
$271$	13.2896	0.807283	0.403642	+	0.914917i	$0.367744\pi$
$272$	0	0
$273$	−11.8704	−0.718427
$274$	−0.978782	−0.0591304
$275$	0	0
$276$	−1.16924	−0.0703801
$277$	10.2506	0.615898	0.307949	−	0.951403i	$-0.400357\pi$
$277$	10.2506	0.615898	0.307949	+	0.951403i	$0.400357\pi$
$278$	−46.7721	−2.80520
$279$	−11.2040	−0.670766
$280$	0	0
$281$	−9.95640	−0.593949	−0.296975	−	0.954885i	$-0.595978\pi$
$281$	−9.95640	−0.593949	−0.296975	+	0.954885i	$0.595978\pi$
$282$	7.08918	0.422155
$283$	14.6111	0.868538	0.434269	−	0.900783i	$-0.357007\pi$
$283$	14.6111	0.868538	0.434269	+	0.900783i	$0.357007\pi$
$284$	18.0444	1.07074
$285$	0	0
$286$	43.1675	2.55255
$287$	−1.78175	−0.105174
$288$	−8.15746	−0.480683
$289$	0	0
$290$	0	0
$291$	−1.83065	−0.107315
$292$	−36.3676	−2.12825
$293$	21.2483	1.24134	0.620669	−	0.784073i	$-0.286861\pi$
$293$	21.2483	1.24134	0.620669	+	0.784073i	$0.286861\pi$
$294$	−8.15035	−0.475338
$295$	0	0
$296$	−56.4984	−3.28390
$297$	10.3281	0.599297
$298$	−12.3009	−0.712572
$299$	2.56491	0.148333
$300$	0	0
$301$	34.2351	1.97328
$302$	15.3616	0.883961
$303$	−7.80567	−0.448424
$304$	−26.5498	−1.52273
$305$	0	0
$306$	0	0
$307$	−5.59078	−0.319082	−0.159541	−	0.987191i	$-0.551001\pi$
$307$	−5.59078	−0.319082	−0.159541	+	0.987191i	$0.551001\pi$
$308$	46.2518	2.63544
$309$	−4.92835	−0.280364
$310$	0	0
$311$	8.38479	0.475458	0.237729	−	0.971332i	$-0.423597\pi$
$311$	8.38479	0.475458	0.237729	+	0.971332i	$0.423597\pi$
$312$	−19.3070	−1.09304
$313$	−34.6879	−1.96068	−0.980339	−	0.197322i	$-0.936776\pi$
$313$	−34.6879	−1.96068	−0.980339	+	0.197322i	$0.936776\pi$
$314$	11.1707	0.630398
$315$	0	0
$316$	15.7220	0.884434
$317$	−11.6878	−0.656451	−0.328226	−	0.944599i	$-0.606451\pi$
$317$	−11.6878	−0.656451	−0.328226	+	0.944599i	$0.606451\pi$
$318$	18.0328	1.01123
$319$	28.9522	1.62101
$320$	0	0
$321$	−4.16600	−0.232524
$322$	4.03020	0.224594
$323$	0	0
$324$	25.3224	1.40680
$325$	0	0
$326$	−19.4386	−1.07661
$327$	6.48101	0.358401
$328$	−2.89800	−0.160015
$329$	−16.6623	−0.918623
$330$	0	0
$331$	−10.5772	−0.581375	−0.290688	−	0.956818i	$-0.593884\pi$
$331$	−10.5772	−0.581375	−0.290688	+	0.956818i	$0.593884\pi$
$332$	−10.9246	−0.599565
$333$	26.0279	1.42632
$334$	62.1533	3.40088
$335$	0	0
$336$	−12.2480	−0.668183
$337$	5.20193	0.283367	0.141684	−	0.989912i	$-0.454748\pi$
$337$	5.20193	0.283367	0.141684	+	0.989912i	$0.454748\pi$
$338$	46.7907	2.54508
$339$	−4.55598	−0.247447
$340$	0	0
$341$	12.9748	0.702625
$342$	30.2947	1.63815
$343$	−5.52573	−0.298361
$344$	55.6829	3.00222
$345$	0	0
$346$	−2.08253	−0.111958
$347$	−18.3551	−0.985352	−0.492676	−	0.870213i	$-0.663981\pi$
$347$	−18.3551	−0.985352	−0.492676	+	0.870213i	$0.663981\pi$
$348$	−24.2721	−1.30112
$349$	12.3788	0.662625	0.331312	−	0.943521i	$-0.392509\pi$
$349$	12.3788	0.662625	0.331312	+	0.943521i	$0.392509\pi$
$350$	0	0
$351$	18.9939	1.01382
$352$	9.44675	0.503514
$353$	−14.1068	−0.750828	−0.375414	−	0.926857i	$-0.622499\pi$
$353$	−14.1068	−0.750828	−0.375414	+	0.926857i	$0.622499\pi$
$354$	16.2999	0.866332
$355$	0	0
$356$	20.2710	1.07436
$357$	0	0
$358$	29.5616	1.56238
$359$	−26.9401	−1.42185	−0.710923	−	0.703270i	$-0.751722\pi$
$359$	−26.9401	−1.42185	−0.710923	+	0.703270i	$0.751722\pi$
$360$	0	0
$361$	1.91207	0.100635
$362$	−3.09138	−0.162479
$363$	0.980461	0.0514609
$364$	85.0594	4.45832
$365$	0	0
$366$	−20.7532	−1.08479
$367$	−10.0103	−0.522533	−0.261266	−	0.965267i	$-0.584140\pi$
$367$	−10.0103	−0.522533	−0.261266	+	0.965267i	$0.584140\pi$
$368$	2.64651	0.137959
$369$	1.33506	0.0695005
$370$	0	0
$371$	−42.3840	−2.20047
$372$	−10.8774	−0.563969
$373$	−2.35960	−0.122175	−0.0610876	−	0.998132i	$-0.519457\pi$
$373$	−2.35960	−0.122175	−0.0610876	+	0.998132i	$0.519457\pi$
$374$	0	0
$375$	0	0
$376$	−27.1011	−1.39763
$377$	53.2445	2.74223
$378$	29.8448	1.53505
$379$	22.2045	1.14057	0.570283	−	0.821448i	$-0.306833\pi$
$379$	22.2045	1.14057	0.570283	+	0.821448i	$0.306833\pi$
$380$	0	0
$381$	4.45291	0.228130
$382$	−32.7903	−1.67770
$383$	−14.0370	−0.717259	−0.358630	−	0.933480i	$-0.616756\pi$
$383$	−14.0370	−0.717259	−0.358630	+	0.933480i	$0.616756\pi$
$384$	9.49992	0.484791
$385$	0	0
$386$	−0.770383	−0.0392115
$387$	−25.6522	−1.30397
$388$	13.1179	0.665959
$389$	25.3269	1.28412	0.642061	−	0.766653i	$-0.278079\pi$
$389$	25.3269	1.28412	0.642061	+	0.766653i	$0.278079\pi$
$390$	0	0
$391$	0	0
$392$	31.1578	1.57370
$393$	−1.01888	−0.0513957
$394$	35.5972	1.79336
$395$	0	0
$396$	−34.6563	−1.74154
$397$	12.5106	0.627889	0.313945	−	0.949441i	$-0.398349\pi$
$397$	12.5106	0.627889	0.313945	+	0.949441i	$0.398349\pi$
$398$	−39.5933	−1.98463
$399$	9.64721	0.482964
$400$	0	0
$401$	9.15876	0.457367	0.228683	−	0.973501i	$-0.426558\pi$
$401$	9.15876	0.457367	0.228683	+	0.973501i	$0.426558\pi$
$402$	4.56790	0.227826
$403$	23.8613	1.18862
$404$	55.9331	2.78277
$405$	0	0
$406$	83.6622	4.15208
$407$	−30.1416	−1.49406
$408$	0	0
$409$	30.1228	1.48948	0.744739	−	0.667356i	$-0.232574\pi$
$409$	30.1228	1.48948	0.744739	+	0.667356i	$0.232574\pi$
$410$	0	0
$411$	0.233547	0.0115200
$412$	35.3150	1.73985
$413$	−38.3112	−1.88517
$414$	−3.01981	−0.148416
$415$	0	0
$416$	17.3731	0.851785
$417$	11.1603	0.546521
$418$	−35.0828	−1.71596
$419$	37.9818	1.85553	0.927765	−	0.373165i	$-0.121727\pi$
$419$	37.9818	1.85553	0.927765	+	0.373165i	$0.121727\pi$
$420$	0	0
$421$	19.0464	0.928263	0.464131	−	0.885766i	$-0.346367\pi$
$421$	19.0464	0.928263	0.464131	+	0.885766i	$0.346367\pi$
$422$	−17.2006	−0.837312
$423$	12.4850	0.607042
$424$	−68.9370	−3.34788
$425$	0	0
$426$	−6.31414	−0.305921
$427$	48.7780	2.36054
$428$	29.8523	1.44297
$429$	−10.3002	−0.497298
$430$	0	0
$431$	−16.1877	−0.779732	−0.389866	−	0.920872i	$-0.627479\pi$
$431$	−16.1877	−0.779732	−0.389866	+	0.920872i	$0.627479\pi$
$432$	19.5982	0.942917
$433$	−3.06400	−0.147246	−0.0736232	−	0.997286i	$-0.523456\pi$
$433$	−3.06400	−0.147246	−0.0736232	+	0.997286i	$0.523456\pi$
$434$	37.4929	1.79972
$435$	0	0
$436$	−46.4409	−2.22412
$437$	−2.08454	−0.0997170
$438$	12.7258	0.608063
$439$	−5.66931	−0.270581	−0.135291	−	0.990806i	$-0.543197\pi$
$439$	−5.66931	−0.270581	−0.135291	+	0.990806i	$0.543197\pi$
$440$	0	0
$441$	−14.3539	−0.683518
$442$	0	0
$443$	3.21373	0.152689	0.0763444	−	0.997082i	$-0.475675\pi$
$443$	3.21373	0.152689	0.0763444	+	0.997082i	$0.475675\pi$
$444$	25.2693	1.19923
$445$	0	0
$446$	33.6513	1.59344
$447$	2.93512	0.138826
$448$	−13.6448	−0.644658
$449$	−13.4157	−0.633127	−0.316564	−	0.948571i	$-0.602529\pi$
$449$	−13.4157	−0.633127	−0.316564	+	0.948571i	$0.602529\pi$
$450$	0	0
$451$	−1.54607	−0.0728015
$452$	32.6467	1.53557
$453$	−3.66543	−0.172217
$454$	−8.71732	−0.409124
$455$	0	0
$456$	15.6911	0.734802
$457$	−20.3295	−0.950974	−0.475487	−	0.879723i	$-0.657728\pi$
$457$	−20.3295	−0.950974	−0.475487	+	0.879723i	$0.657728\pi$
$458$	−48.0974	−2.24744
$459$	0	0
$460$	0	0
$461$	−0.770520	−0.0358867	−0.0179434	−	0.999839i	$-0.505712\pi$
$461$	−0.770520	−0.0358867	−0.0179434	+	0.999839i	$0.505712\pi$
$462$	−16.1845	−0.752971
$463$	−33.1892	−1.54243	−0.771216	−	0.636573i	$-0.780351\pi$
$463$	−33.1892	−1.54243	−0.771216	+	0.636573i	$0.780351\pi$
$464$	54.9384	2.55045
$465$	0	0
$466$	38.1108	1.76545
$467$	27.5913	1.27677	0.638385	−	0.769717i	$-0.279603\pi$
$467$	27.5913	1.27677	0.638385	+	0.769717i	$0.279603\pi$
$468$	−63.7347	−2.94614
$469$	−10.7363	−0.495758
$470$	0	0
$471$	−2.66543	−0.122817
$472$	−62.3126	−2.86817
$473$	29.7065	1.36591
$474$	−5.50149	−0.252692
$475$	0	0
$476$	0	0
$477$	31.7581	1.45411
$478$	−18.0083	−0.823681
$479$	26.3785	1.20526	0.602632	−	0.798019i	$-0.294119\pi$
$479$	26.3785	1.20526	0.602632	+	0.798019i	$0.294119\pi$
$480$	0	0
$481$	−55.4320	−2.52748
$482$	−35.1660	−1.60177
$483$	−0.961645	−0.0437564
$484$	−7.02568	−0.319349
$485$	0	0
$486$	−34.2533	−1.55376
$487$	−37.7174	−1.70914	−0.854569	−	0.519338i	$-0.826179\pi$
$487$	−37.7174	−1.70914	−0.854569	+	0.519338i	$0.826179\pi$
$488$	79.3369	3.59141
$489$	4.63825	0.209749
$490$	0	0
$491$	−10.4059	−0.469613	−0.234806	−	0.972042i	$-0.575446\pi$
$491$	−10.4059	−0.469613	−0.234806	+	0.972042i	$0.575446\pi$
$492$	1.29615	0.0584349
$493$	0	0
$494$	−64.5191	−2.90285
$495$	0	0
$496$	24.6204	1.10549
$497$	14.8407	0.665695
$498$	3.82276	0.171302
$499$	−26.5160	−1.18702	−0.593509	−	0.804827i	$-0.702258\pi$
$499$	−26.5160	−1.18702	−0.593509	+	0.804827i	$0.702258\pi$
$500$	0	0
$501$	−14.8304	−0.662572
$502$	−58.5151	−2.61166
$503$	25.4913	1.13660	0.568301	−	0.822821i	$-0.307601\pi$
$503$	25.4913	1.13660	0.568301	+	0.822821i	$0.307601\pi$
$504$	−53.4271	−2.37983
$505$	0	0
$506$	3.49710	0.155465
$507$	−11.1647	−0.495842
$508$	−31.9082	−1.41570
$509$	19.9285	0.883315	0.441657	−	0.897184i	$-0.354391\pi$
$509$	19.9285	0.883315	0.441657	+	0.897184i	$0.354391\pi$
$510$	0	0
$511$	−29.9106	−1.32317
$512$	−48.6673	−2.15081
$513$	−15.4366	−0.681543
$514$	−8.10429	−0.357465
$515$	0	0
$516$	−24.9045	−1.09636
$517$	−14.4583	−0.635874
$518$	−87.0993	−3.82692
$519$	0.496913	0.0218121
$520$	0	0
$521$	−13.1306	−0.575262	−0.287631	−	0.957741i	$-0.592868\pi$
$521$	−13.1306	−0.575262	−0.287631	+	0.957741i	$0.592868\pi$
$522$	−62.6877	−2.74377
$523$	−6.15211	−0.269013	−0.134506	−	0.990913i	$-0.542945\pi$
$523$	−6.15211	−0.269013	−0.134506	+	0.990913i	$0.542945\pi$
$524$	7.30098	0.318945
$525$	0	0
$526$	52.8084	2.30256
$527$	0	0
$528$	−10.6279	−0.462519
$529$	−22.7922	−0.990966
$530$	0	0
$531$	28.7064	1.24575
$532$	−69.1289	−2.99712
$533$	−2.84330	−0.123157
$534$	−7.09326	−0.306956
$535$	0	0
$536$	−17.4625	−0.754266
$537$	−7.05368	−0.304389
$538$	−64.8987	−2.79798
$539$	16.6225	0.715982
$540$	0	0
$541$	18.7146	0.804605	0.402302	−	0.915507i	$-0.368210\pi$
$541$	18.7146	0.804605	0.402302	+	0.915507i	$0.368210\pi$
$542$	33.3227	1.43133
$543$	0.737633	0.0316549
$544$	0	0
$545$	0	0
$546$	−29.7641	−1.27379
$547$	−10.4089	−0.445052	−0.222526	−	0.974927i	$-0.571430\pi$
$547$	−10.4089	−0.445052	−0.222526	+	0.974927i	$0.571430\pi$
$548$	−1.67353	−0.0714895
$549$	−36.5492	−1.55988
$550$	0	0
$551$	−43.2726	−1.84347
$552$	−1.56410	−0.0665727
$553$	12.9306	0.549866
$554$	25.7027	1.09200
$555$	0	0
$556$	−79.9711	−3.39153
$557$	33.2151	1.40737	0.703685	−	0.710512i	$-0.251536\pi$
$557$	33.2151	1.40737	0.703685	+	0.710512i	$0.251536\pi$
$558$	−28.0933	−1.18928
$559$	54.6319	2.31068
$560$	0	0
$561$	0	0
$562$	−24.9650	−1.05309
$563$	−21.7736	−0.917649	−0.458824	−	0.888527i	$-0.651729\pi$
$563$	−21.7736	−0.917649	−0.458824	+	0.888527i	$0.651729\pi$
$564$	12.1211	0.510391
$565$	0	0
$566$	36.6363	1.53994
$567$	20.8265	0.874629
$568$	24.1382	1.01281
$569$	9.98336	0.418524	0.209262	−	0.977860i	$-0.432894\pi$
$569$	9.98336	0.418524	0.209262	+	0.977860i	$0.432894\pi$
$570$	0	0
$571$	−9.57244	−0.400594	−0.200297	−	0.979735i	$-0.564191\pi$
$571$	−9.57244	−0.400594	−0.200297	+	0.979735i	$0.564191\pi$
$572$	73.8080	3.08607
$573$	7.82408	0.326856
$574$	−4.46763	−0.186475
$575$	0	0
$576$	10.2240	0.426001
$577$	11.3210	0.471301	0.235650	−	0.971838i	$-0.424278\pi$
$577$	11.3210	0.471301	0.235650	+	0.971838i	$0.424278\pi$
$578$	0	0
$579$	0.183821	0.00763933
$580$	0	0
$581$	−8.98495	−0.372759
$582$	−4.59023	−0.190271
$583$	−36.7775	−1.52317
$584$	−48.6492	−2.01312
$585$	0	0
$586$	53.2786	2.20092
$587$	−3.34528	−0.138074	−0.0690372	−	0.997614i	$-0.521993\pi$
$587$	−3.34528	−0.138074	−0.0690372	+	0.997614i	$0.521993\pi$
$588$	−13.9355	−0.574690
$589$	−19.3924	−0.799051
$590$	0	0
$591$	−8.49384	−0.349390
$592$	−57.1954	−2.35072
$593$	38.7935	1.59306	0.796529	−	0.604601i	$-0.206667\pi$
$593$	38.7935	1.59306	0.796529	+	0.604601i	$0.206667\pi$
$594$	25.8970	1.06257
$595$	0	0
$596$	−21.0321	−0.861510
$597$	9.44735	0.386654
$598$	6.43134	0.262997
$599$	−43.4267	−1.77437	−0.887184	−	0.461416i	$-0.847342\pi$
$599$	−43.4267	−1.77437	−0.887184	+	0.461416i	$0.847342\pi$
$600$	0	0
$601$	14.4771	0.590533	0.295266	−	0.955415i	$-0.404592\pi$
$601$	14.4771	0.590533	0.295266	+	0.955415i	$0.404592\pi$
$602$	85.8421	3.49866
$603$	8.04469	0.327605
$604$	26.2653	1.06872
$605$	0	0
$606$	−19.5722	−0.795066
$607$	−10.8496	−0.440371	−0.220185	−	0.975458i	$-0.570666\pi$
$607$	−10.8496	−0.440371	−0.220185	+	0.975458i	$0.570666\pi$
$608$	−14.1193	−0.572615
$609$	−19.9626	−0.808926
$610$	0	0
$611$	−26.5895	−1.07570
$612$	0	0
$613$	−10.5871	−0.427608	−0.213804	−	0.976877i	$-0.568585\pi$
$613$	−10.5871	−0.427608	−0.213804	+	0.976877i	$0.568585\pi$
$614$	−14.0185	−0.565740
$615$	0	0
$616$	61.8713	2.49287
$617$	−14.1871	−0.571152	−0.285576	−	0.958356i	$-0.592185\pi$
$617$	−14.1871	−0.571152	−0.285576	+	0.958356i	$0.592185\pi$
$618$	−12.3575	−0.497091
$619$	40.6923	1.63556	0.817780	−	0.575531i	$-0.195205\pi$
$619$	40.6923	1.63556	0.817780	+	0.575531i	$0.195205\pi$
$620$	0	0
$621$	1.53874	0.0617475
$622$	21.0243	0.842997
$623$	16.6719	0.667946
$624$	−19.5452	−0.782434
$625$	0	0
$626$	−86.9776	−3.47633
$627$	8.37110	0.334310
$628$	19.0997	0.762160
$629$	0	0
$630$	0	0
$631$	−40.3897	−1.60789	−0.803944	−	0.594705i	$-0.797269\pi$
$631$	−40.3897	−1.60789	−0.803944	+	0.594705i	$0.797269\pi$
$632$	21.0315	0.836588
$633$	4.10423	0.163128
$634$	−29.3063	−1.16390
$635$	0	0
$636$	30.8325	1.22259
$637$	30.5697	1.21121
$638$	72.5956	2.87409
$639$	−11.1201	−0.439902
$640$	0	0
$641$	−23.1261	−0.913426	−0.456713	−	0.889614i	$-0.650973\pi$
$641$	−23.1261	−0.913426	−0.456713	+	0.889614i	$0.650973\pi$
$642$	−10.4460	−0.412270
$643$	−3.81570	−0.150477	−0.0752383	−	0.997166i	$-0.523972\pi$
$643$	−3.81570	−0.150477	−0.0752383	+	0.997166i	$0.523972\pi$
$644$	6.89085	0.271538
$645$	0	0
$646$	0	0
$647$	−43.6126	−1.71459	−0.857294	−	0.514827i	$-0.827856\pi$
$647$	−43.6126	−1.71459	−0.857294	+	0.514827i	$0.827856\pi$
$648$	33.8740	1.33070
$649$	−33.2435	−1.30492
$650$	0	0
$651$	−8.94617	−0.350628
$652$	−33.2362	−1.30163
$653$	−24.7307	−0.967785	−0.483893	−	0.875127i	$-0.660777\pi$
$653$	−24.7307	−0.967785	−0.483893	+	0.875127i	$0.660777\pi$
$654$	16.2507	0.635453
$655$	0	0
$656$	−2.93375	−0.114544
$657$	22.4119	0.874371
$658$	−41.7796	−1.62874
$659$	−18.2763	−0.711942	−0.355971	−	0.934497i	$-0.615850\pi$
$659$	−18.2763	−0.711942	−0.355971	+	0.934497i	$0.615850\pi$
$660$	0	0
$661$	−24.7823	−0.963921	−0.481960	−	0.876193i	$-0.660075\pi$
$661$	−24.7823	−0.963921	−0.481960	+	0.876193i	$0.660075\pi$
$662$	−26.5216	−1.03079
$663$	0	0
$664$	−14.6139	−0.567130
$665$	0	0
$666$	65.2632	2.52890
$667$	4.31346	0.167018
$668$	106.270	4.11171
$669$	−8.02953	−0.310440
$670$	0	0
$671$	42.3258	1.63397
$672$	−6.51357	−0.251266
$673$	−10.5928	−0.408323	−0.204162	−	0.978937i	$-0.565447\pi$
$673$	−10.5928	−0.408323	−0.204162	+	0.978937i	$0.565447\pi$
$674$	13.0435	0.502416
$675$	0	0
$676$	80.0029	3.07703
$677$	8.56246	0.329082	0.164541	−	0.986370i	$-0.447386\pi$
$677$	8.56246	0.329082	0.164541	+	0.986370i	$0.447386\pi$
$678$	−11.4238	−0.438728
$679$	10.7888	0.414037
$680$	0	0
$681$	2.08004	0.0797072
$682$	32.5334	1.24577
$683$	−42.3447	−1.62028	−0.810138	−	0.586239i	$-0.800608\pi$
$683$	−42.3447	−1.62028	−0.810138	+	0.586239i	$0.800608\pi$
$684$	51.7980	1.98055
$685$	0	0
$686$	−13.8554	−0.529001
$687$	11.4765	0.437856
$688$	56.3699	2.14908
$689$	−67.6358	−2.57672
$690$	0	0
$691$	43.4629	1.65341	0.826703	−	0.562638i	$-0.190214\pi$
$691$	43.4629	1.65341	0.826703	+	0.562638i	$0.190214\pi$
$692$	−3.56072	−0.135358
$693$	−28.5031	−1.08274
$694$	−46.0241	−1.74705
$695$	0	0
$696$	−32.4689	−1.23073
$697$	0	0
$698$	31.0391	1.17485
$699$	−9.09361	−0.343952
$700$	0	0
$701$	−18.3541	−0.693224	−0.346612	−	0.938009i	$-0.612668\pi$
$701$	−18.3541	−0.693224	−0.346612	+	0.938009i	$0.612668\pi$
$702$	47.6260	1.79753
$703$	45.0503	1.69911
$704$	−11.8399	−0.446234
$705$	0	0
$706$	−35.3718	−1.33124
$707$	46.0022	1.73009
$708$	27.8697	1.04741
$709$	−33.0240	−1.24024	−0.620121	−	0.784506i	$-0.712917\pi$
$709$	−33.0240	−1.24024	−0.620121	+	0.784506i	$0.712917\pi$
$710$	0	0
$711$	−9.68886	−0.363361
$712$	27.1167	1.01624
$713$	1.93306	0.0723937
$714$	0	0
$715$	0	0
$716$	50.5445	1.88894
$717$	4.29696	0.160473
$718$	−67.5505	−2.52096
$719$	−26.1373	−0.974758	−0.487379	−	0.873191i	$-0.662047\pi$
$719$	−26.1373	−0.974758	−0.487379	+	0.873191i	$0.662047\pi$
$720$	0	0
$721$	29.0449	1.08169
$722$	4.79438	0.178428
$723$	8.39095	0.312063
$724$	−5.28565	−0.196440
$725$	0	0
$726$	2.45844	0.0912413
$727$	−17.3108	−0.642024	−0.321012	−	0.947075i	$-0.604023\pi$
$727$	−17.3108	−0.642024	−0.321012	+	0.947075i	$0.604023\pi$
$728$	113.785	4.21714
$729$	−9.54631	−0.353567
$730$	0	0
$731$	0	0
$732$	−35.4839	−1.31152
$733$	7.71364	0.284910	0.142455	−	0.989801i	$-0.454500\pi$
$733$	7.71364	0.284910	0.142455	+	0.989801i	$0.454500\pi$
$734$	−25.1001	−0.926462
$735$	0	0
$736$	1.40743	0.0518786
$737$	−9.31616	−0.343165
$738$	3.34758	0.123226
$739$	−3.73009	−0.137213	−0.0686067	−	0.997644i	$-0.521855\pi$
$739$	−3.73009	−0.137213	−0.0686067	+	0.997644i	$0.521855\pi$
$740$	0	0
$741$	15.3949	0.565546
$742$	−106.275	−3.90148
$743$	−4.74617	−0.174120	−0.0870601	−	0.996203i	$-0.527747\pi$
$743$	−4.74617	−0.174120	−0.0870601	+	0.996203i	$0.527747\pi$
$744$	−14.5508	−0.533459
$745$	0	0
$746$	−5.91653	−0.216619
$747$	6.73239	0.246325
$748$	0	0
$749$	24.5521	0.897113
$750$	0	0
$751$	−31.6567	−1.15517	−0.577584	−	0.816331i	$-0.696004\pi$
$751$	−31.6567	−1.15517	−0.577584	+	0.816331i	$0.696004\pi$
$752$	−27.4354	−1.00047
$753$	13.9623	0.508814
$754$	133.507	4.86204
$755$	0	0
$756$	51.0287	1.85590
$757$	4.00212	0.145459	0.0727297	−	0.997352i	$-0.476829\pi$
$757$	4.00212	0.145459	0.0727297	+	0.997352i	$0.476829\pi$
$758$	55.6762	2.02225
$759$	−0.834442	−0.0302883
$760$	0	0
$761$	−23.0522	−0.835641	−0.417820	−	0.908530i	$-0.637206\pi$
$761$	−23.0522	−0.835641	−0.417820	+	0.908530i	$0.637206\pi$
$762$	11.1654	0.404479
$763$	−38.1954	−1.38277
$764$	−56.0649	−2.02836
$765$	0	0
$766$	−35.1969	−1.27172
$767$	−61.1365	−2.20751
$768$	19.1899	0.692455
$769$	−29.1001	−1.04938	−0.524688	−	0.851295i	$-0.675818\pi$
$769$	−29.1001	−1.04938	−0.524688	+	0.851295i	$0.675818\pi$
$770$	0	0
$771$	1.93376	0.0696427
$772$	−1.31720	−0.0474072
$773$	2.49408	0.0897059	0.0448529	−	0.998994i	$-0.485718\pi$
$773$	2.49408	0.0897059	0.0448529	+	0.998994i	$0.485718\pi$
$774$	−64.3212	−2.31198
$775$	0	0
$776$	17.5479	0.629932
$777$	20.7827	0.745576
$778$	63.5054	2.27678
$779$	2.31079	0.0827926
$780$	0	0
$781$	12.8776	0.460796
$782$	0	0
$783$	31.9424	1.14153
$784$	31.5422	1.12651
$785$	0	0
$786$	−2.55477	−0.0911257
$787$	44.4841	1.58569	0.792844	−	0.609425i	$-0.208600\pi$
$787$	44.4841	1.58569	0.792844	+	0.609425i	$0.208600\pi$
$788$	60.8643	2.16820
$789$	−12.6006	−0.448593
$790$	0	0
$791$	26.8503	0.954688
$792$	−46.3600	−1.64733
$793$	77.8394	2.76416
$794$	31.3695	1.11326
$795$	0	0
$796$	−67.6968	−2.39945
$797$	−43.4377	−1.53864	−0.769322	−	0.638861i	$-0.779406\pi$
$797$	−43.4377	−1.53864	−0.769322	+	0.638861i	$0.779406\pi$
$798$	24.1897	0.856307
$799$	0	0
$800$	0	0
$801$	−12.4922	−0.441390
$802$	22.9650	0.810921
$803$	−25.9541	−0.915900
$804$	7.81021	0.275445
$805$	0	0
$806$	59.8307	2.10745
$807$	15.4855	0.545114
$808$	74.8221	2.63223
$809$	−18.4878	−0.649995	−0.324998	−	0.945715i	$-0.605363\pi$
$809$	−18.4878	−0.649995	−0.324998	+	0.945715i	$0.605363\pi$
$810$	0	0
$811$	−23.5730	−0.827758	−0.413879	−	0.910332i	$-0.635826\pi$
$811$	−23.5730	−0.827758	−0.413879	+	0.910332i	$0.635826\pi$
$812$	143.046	5.01993
$813$	−7.95111	−0.278858
$814$	−75.5780	−2.64901
$815$	0	0
$816$	0	0
$817$	−44.4001	−1.55336
$818$	75.5310	2.64088
$819$	−52.4187	−1.83166
$820$	0	0
$821$	−52.8639	−1.84496	−0.922481	−	0.386042i	$-0.873842\pi$
$821$	−52.8639	−1.84496	−0.922481	+	0.386042i	$0.873842\pi$
$822$	0.585603	0.0204253
$823$	43.2073	1.50611	0.753057	−	0.657956i	$-0.228579\pi$
$823$	43.2073	1.50611	0.753057	+	0.657956i	$0.228579\pi$
$824$	47.2412	1.64572
$825$	0	0
$826$	−96.0626	−3.34245
$827$	41.5689	1.44549	0.722745	−	0.691114i	$-0.242880\pi$
$827$	41.5689	1.44549	0.722745	+	0.691114i	$0.242880\pi$
$828$	−5.16329	−0.179437
$829$	−38.5545	−1.33905	−0.669526	−	0.742789i	$-0.733503\pi$
$829$	−38.5545	−1.33905	−0.669526	+	0.742789i	$0.733503\pi$
$830$	0	0
$831$	−6.13291	−0.212748
$832$	−21.7742	−0.754886
$833$	0	0
$834$	27.9836	0.968994
$835$	0	0
$836$	−59.9848	−2.07462
$837$	14.3149	0.494794
$838$	95.2367	3.28990
$839$	−40.9756	−1.41463	−0.707317	−	0.706896i	$-0.750095\pi$
$839$	−40.9756	−1.41463	−0.707317	+	0.706896i	$0.750095\pi$
$840$	0	0
$841$	60.5423	2.08766
$842$	47.7575	1.64583
$843$	5.95689	0.205166
$844$	−29.4096	−1.01232
$845$	0	0
$846$	31.3053	1.07630
$847$	−5.77828	−0.198544
$848$	−69.7875	−2.39651
$849$	−8.74177	−0.300017
$850$	0	0
$851$	−4.49067	−0.153938
$852$	−10.7959	−0.369863
$853$	22.9121	0.784494	0.392247	−	0.919860i	$-0.371698\pi$
$853$	22.9121	0.784494	0.392247	+	0.919860i	$0.371698\pi$
$854$	122.308	4.18528
$855$	0	0
$856$	39.9336	1.36490
$857$	19.4301	0.663719	0.331860	−	0.943329i	$-0.392324\pi$
$857$	19.4301	0.663719	0.331860	+	0.943329i	$0.392324\pi$
$858$	−25.8270	−0.881720
$859$	18.3698	0.626768	0.313384	−	0.949626i	$-0.398537\pi$
$859$	18.3698	0.626768	0.313384	+	0.949626i	$0.398537\pi$
$860$	0	0
$861$	1.06602	0.0363298
$862$	−40.5895	−1.38248
$863$	−23.1327	−0.787447	−0.393724	−	0.919229i	$-0.628813\pi$
$863$	−23.1327	−0.787447	−0.393724	+	0.919229i	$0.628813\pi$
$864$	10.4224	0.354578
$865$	0	0
$866$	−7.68277	−0.261071
$867$	0	0
$868$	64.1055	2.17588
$869$	11.2202	0.380619
$870$	0	0
$871$	−17.1329	−0.580526
$872$	−62.1244	−2.10380
$873$	−8.08401	−0.273602
$874$	−5.22684	−0.176801
$875$	0	0
$876$	21.7586	0.735156
$877$	−52.7767	−1.78214	−0.891071	−	0.453864i	$-0.850045\pi$
$877$	−52.7767	−1.78214	−0.891071	+	0.453864i	$0.850045\pi$
$878$	−14.2154	−0.479747
$879$	−12.7128	−0.428792
$880$	0	0
$881$	−6.90383	−0.232596	−0.116298	−	0.993214i	$-0.537103\pi$
$881$	−6.90383	−0.232596	−0.116298	+	0.993214i	$0.537103\pi$
$882$	−35.9914	−1.21189
$883$	−3.78214	−0.127279	−0.0636395	−	0.997973i	$-0.520271\pi$
$883$	−3.78214	−0.127279	−0.0636395	+	0.997973i	$0.520271\pi$
$884$	0	0
$885$	0	0
$886$	8.05820	0.270721
$887$	18.1504	0.609431	0.304716	−	0.952443i	$-0.401439\pi$
$887$	18.1504	0.609431	0.304716	+	0.952443i	$0.401439\pi$
$888$	33.8029	1.13435
$889$	−26.2430	−0.880160
$890$	0	0
$891$	18.0716	0.605421
$892$	57.5371	1.92649
$893$	21.6097	0.723140
$894$	7.35960	0.246142
$895$	0	0
$896$	−55.9871	−1.87040
$897$	−1.53458	−0.0512382
$898$	−33.6390	−1.12255
$899$	40.1280	1.33834
$900$	0	0
$901$	0	0
$902$	−3.87666	−0.129079
$903$	−20.4828	−0.681623
$904$	43.6718	1.45250
$905$	0	0
$906$	−9.19081	−0.305344
$907$	18.2255	0.605167	0.302584	−	0.953123i	$-0.402151\pi$
$907$	18.2255	0.605167	0.302584	+	0.953123i	$0.402151\pi$
$908$	−14.9049	−0.494637
$909$	−34.4693	−1.14327
$910$	0	0
$911$	26.9725	0.893638	0.446819	−	0.894624i	$-0.352557\pi$
$911$	26.9725	0.893638	0.446819	+	0.894624i	$0.352557\pi$
$912$	15.8847	0.525994
$913$	−7.79645	−0.258025
$914$	−50.9749	−1.68610
$915$	0	0
$916$	−82.2371	−2.71719
$917$	6.00470	0.198293
$918$	0	0
$919$	−1.07053	−0.0353134	−0.0176567	−	0.999844i	$-0.505621\pi$
$919$	−1.07053	−0.0353134	−0.0176567	+	0.999844i	$0.505621\pi$
$920$	0	0
$921$	3.34495	0.110220
$922$	−1.93203	−0.0636280
$923$	23.6825	0.779520
$924$	−27.6723	−0.910352
$925$	0	0
$926$	−83.2196	−2.73477
$927$	−21.7632	−0.714798
$928$	29.2166	0.959082
$929$	−10.3780	−0.340492	−0.170246	−	0.985402i	$-0.554456\pi$
$929$	−10.3780	−0.340492	−0.170246	+	0.985402i	$0.554456\pi$
$930$	0	0
$931$	−24.8444	−0.814242
$932$	65.1620	2.13445
$933$	−5.01660	−0.164236
$934$	69.1832	2.26374
$935$	0	0
$936$	−85.2584	−2.78676
$937$	−7.66960	−0.250555	−0.125277	−	0.992122i	$-0.539982\pi$
$937$	−7.66960	−0.250555	−0.125277	+	0.992122i	$0.539982\pi$
$938$	−26.9206	−0.878989
$939$	20.7537	0.677272
$940$	0	0
$941$	46.9716	1.53123	0.765615	−	0.643300i	$-0.222435\pi$
$941$	46.9716	1.53123	0.765615	+	0.643300i	$0.222435\pi$
$942$	−6.68339	−0.217757
$943$	−0.230342	−0.00750097
$944$	−63.0814	−2.05313
$945$	0	0
$946$	74.4872	2.42179
$947$	15.6994	0.510163	0.255081	−	0.966920i	$-0.417898\pi$
$947$	15.6994	0.510163	0.255081	+	0.966920i	$0.417898\pi$
$948$	−9.40646	−0.305508
$949$	−47.7309	−1.54941
$950$	0	0
$951$	6.99278	0.226756
$952$	0	0
$953$	17.7725	0.575708	0.287854	−	0.957674i	$-0.407058\pi$
$953$	17.7725	0.575708	0.287854	+	0.957674i	$0.407058\pi$
$954$	79.6314	2.57816
$955$	0	0
$956$	−30.7907	−0.995841
$957$	−17.3220	−0.559941
$958$	66.1423	2.13696
$959$	−1.37639	−0.0444461
$960$	0	0
$961$	−13.0168	−0.419896
$962$	−138.992	−4.48128
$963$	−18.3968	−0.592828
$964$	−60.1270	−1.93656
$965$	0	0
$966$	−2.41126	−0.0775810
$967$	−11.9909	−0.385600	−0.192800	−	0.981238i	$-0.561757\pi$
$967$	−11.9909	−0.385600	−0.192800	+	0.981238i	$0.561757\pi$
$968$	−9.39831	−0.302073
$969$	0	0
$970$	0	0
$971$	19.5354	0.626920	0.313460	−	0.949601i	$-0.398512\pi$
$971$	19.5354	0.626920	0.313460	+	0.949601i	$0.398512\pi$
$972$	−58.5664	−1.87852
$973$	−65.7723	−2.10856
$974$	−94.5738	−3.03034
$975$	0	0
$976$	80.3157	2.57084
$977$	−37.0958	−1.18680	−0.593400	−	0.804907i	$-0.702215\pi$
$977$	−37.0958	−1.18680	−0.593400	+	0.804907i	$0.702215\pi$
$978$	11.6301	0.371889
$979$	14.4666	0.462354
$980$	0	0
$981$	28.6197	0.913756
$982$	−26.0921	−0.832634
$983$	−2.44368	−0.0779412	−0.0389706	−	0.999240i	$-0.512408\pi$
$983$	−2.44368	−0.0779412	−0.0389706	+	0.999240i	$0.512408\pi$
$984$	1.73387	0.0552737
$985$	0	0
$986$	0	0
$987$	9.96902	0.317318
$988$	−110.315	−3.50959
$989$	4.42585	0.140734
$990$	0	0
$991$	41.1240	1.30635	0.653174	−	0.757208i	$-0.273437\pi$
$991$	41.1240	1.30635	0.653174	+	0.757208i	$0.273437\pi$
$992$	13.0933	0.415713
$993$	6.32831	0.200823
$994$	37.2120	1.18029
$995$	0	0
$996$	6.53616	0.207106
$997$	22.7241	0.719679	0.359839	−	0.933014i	$-0.382832\pi$
$997$	22.7241	0.719679	0.359839	+	0.933014i	$0.382832\pi$
$998$	−66.4870	−2.10461
$999$	−33.2547	−1.05213

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bv.1.15	yes	15
5.4	even	2		7225.2.a.bt.1.1	✓	15
17.16	even	2		7225.2.a.bu.1.15	yes	15
85.84	even	2		7225.2.a.bw.1.1	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7225.2.a.bt.1.1	✓	15	5.4	even	2
7225.2.a.bu.1.15	yes	15	17.16	even	2
7225.2.a.bv.1.15	yes	15	1.1	even	1	trivial
7225.2.a.bw.1.1	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)

\(f(q)\)	\(=\)	\(q+2.50743 q^{2} -0.598298 q^{3} +4.28722 q^{4} -1.50019 q^{6} +3.52603 q^{7} +5.73504 q^{8} -2.64204 q^{9} +O(q^{10})\)	\(q+2.50743 q^{2} -0.598298 q^{3} +4.28722 q^{4} -1.50019 q^{6} +3.52603 q^{7} +5.73504 q^{8} -2.64204 q^{9} +3.05962 q^{11} -2.56503 q^{12} +5.62679 q^{13} +8.84128 q^{14} +5.80580 q^{16} -6.62474 q^{18} -4.57297 q^{19} -2.10961 q^{21} +7.67178 q^{22} +0.455839 q^{23} -3.43126 q^{24} +14.1088 q^{26} +3.37562 q^{27} +15.1168 q^{28} +9.46268 q^{29} +4.24066 q^{31} +3.08756 q^{32} -1.83056 q^{33} -11.3270 q^{36} -9.85143 q^{37} -11.4664 q^{38} -3.36650 q^{39} -0.505315 q^{41} -5.28972 q^{42} +9.70924 q^{43} +13.1172 q^{44} +1.14299 q^{46} -4.72552 q^{47} -3.47360 q^{48} +5.43287 q^{49} +24.1233 q^{52} -12.0203 q^{53} +8.46414 q^{54} +20.2219 q^{56} +2.73600 q^{57} +23.7270 q^{58} -10.8652 q^{59} +13.8337 q^{61} +10.6332 q^{62} -9.31591 q^{63} -3.86974 q^{64} -4.59001 q^{66} -3.04488 q^{67} -0.272728 q^{69} +4.20889 q^{71} -15.1522 q^{72} -8.48279 q^{73} -24.7018 q^{74} -19.6053 q^{76} +10.7883 q^{77} -8.44126 q^{78} +3.66719 q^{79} +5.90649 q^{81} -1.26704 q^{82} -2.54818 q^{83} -9.04438 q^{84} +24.3453 q^{86} -5.66150 q^{87} +17.5470 q^{88} +4.72824 q^{89} +19.8402 q^{91} +1.95428 q^{92} -2.53718 q^{93} -11.8489 q^{94} -1.84728 q^{96} +3.05976 q^{97} +13.6226 q^{98} -8.08363 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 9 q^{3} + 12 q^{4} - 9 q^{6} + 12 q^{7} - 6 q^{8} + 12 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more