LMFDB - Embedded newform 6025.2.a.o.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6025,2,Mod(1,6025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6025, 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("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6025 = 5^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6025.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:	$48.1098672178$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.394755 q^{2} +2.83806 q^{3} -1.84417 q^{4} +1.12034 q^{6} -0.914717 q^{7} -1.51750 q^{8} +5.05456 q^{9} +O(q^{10})$	\(q+0.394755 q^{2} +2.83806 q^{3} -1.84417 q^{4} +1.12034 q^{6} -0.914717 q^{7} -1.51750 q^{8} +5.05456 q^{9} +1.45138 q^{11} -5.23385 q^{12} +4.92764 q^{13} -0.361089 q^{14} +3.08930 q^{16} +2.55248 q^{17} +1.99531 q^{18} +2.57620 q^{19} -2.59602 q^{21} +0.572938 q^{22} -1.67084 q^{23} -4.30676 q^{24} +1.94521 q^{26} +5.83095 q^{27} +1.68689 q^{28} -4.06036 q^{29} +1.90713 q^{31} +4.25452 q^{32} +4.11909 q^{33} +1.00760 q^{34} -9.32146 q^{36} -7.44516 q^{37} +1.01696 q^{38} +13.9849 q^{39} -8.87187 q^{41} -1.02479 q^{42} +10.2313 q^{43} -2.67659 q^{44} -0.659570 q^{46} -1.40332 q^{47} +8.76759 q^{48} -6.16329 q^{49} +7.24408 q^{51} -9.08740 q^{52} +9.75483 q^{53} +2.30179 q^{54} +1.38809 q^{56} +7.31139 q^{57} -1.60285 q^{58} +6.52367 q^{59} +11.1771 q^{61} +0.752848 q^{62} -4.62349 q^{63} -4.49910 q^{64} +1.62603 q^{66} -1.14970 q^{67} -4.70720 q^{68} -4.74192 q^{69} +2.51297 q^{71} -7.67031 q^{72} -8.07631 q^{73} -2.93901 q^{74} -4.75094 q^{76} -1.32760 q^{77} +5.52061 q^{78} +11.2886 q^{79} +1.38488 q^{81} -3.50221 q^{82} -0.985950 q^{83} +4.78750 q^{84} +4.03886 q^{86} -11.5235 q^{87} -2.20247 q^{88} +18.4736 q^{89} -4.50740 q^{91} +3.08130 q^{92} +5.41254 q^{93} -0.553968 q^{94} +12.0746 q^{96} -0.323709 q^{97} -2.43299 q^{98} +7.33607 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9}+O(q^{10}) $ 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9	$ 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9} + q^{11} + 14 q^{12} + 9 q^{13} - q^{14} + 43 q^{16} + 12 q^{17} + 42 q^{18} + 2 q^{21} + 5 q^{22} + 77 q^{23} - 2 q^{24} + 2 q^{26} + 38 q^{27} + 42 q^{28} + 2 q^{29} + q^{31} + 72 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 28 q^{37} + 23 q^{38} - 2 q^{39} - 2 q^{41} + 37 q^{42} + 31 q^{43} + 3 q^{44} + 14 q^{46} + 96 q^{47} + 13 q^{48} + 40 q^{49} - 10 q^{51} + 42 q^{52} + 54 q^{53} + 4 q^{54} - 15 q^{56} + 37 q^{57} + 27 q^{58} + q^{59} + 5 q^{61} + 39 q^{62} + 70 q^{63} + 65 q^{64} - 52 q^{66} + 34 q^{67} + 52 q^{68} + 21 q^{69} - 9 q^{71} + 70 q^{72} + 25 q^{73} + 22 q^{74} - 47 q^{76} + 54 q^{77} + 58 q^{78} + 13 q^{79} + 12 q^{81} - 5 q^{82} + 63 q^{83} + 95 q^{84} - 18 q^{86} + 47 q^{87} + 13 q^{88} + 19 q^{89} - 31 q^{91} + 137 q^{92} + 52 q^{93} + 120 q^{94} - 49 q^{96} + 36 q^{97} + 64 q^{98} + 16 q^{99}+O(q^{100}) $ 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9 + q^11 + 14 * q^12 + 9 * q^13 - q^14 + 43 * q^16 + 12 * q^17 + 42 * q^18 + 2 * q^21 + 5 * q^22 + 77 * q^23 - 2 * q^24 + 2 * q^26 + 38 * q^27 + 42 * q^28 + 2 * q^29 + q^31 + 72 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 28 * q^37 + 23 * q^38 - 2 * q^39 - 2 * q^41 + 37 * q^42 + 31 * q^43 + 3 * q^44 + 14 * q^46 + 96 * q^47 + 13 * q^48 + 40 * q^49 - 10 * q^51 + 42 * q^52 + 54 * q^53 + 4 * q^54 - 15 * q^56 + 37 * q^57 + 27 * q^58 + q^59 + 5 * q^61 + 39 * q^62 + 70 * q^63 + 65 * q^64 - 52 * q^66 + 34 * q^67 + 52 * q^68 + 21 * q^69 - 9 * q^71 + 70 * q^72 + 25 * q^73 + 22 * q^74 - 47 * q^76 + 54 * q^77 + 58 * q^78 + 13 * q^79 + 12 * q^81 - 5 * q^82 + 63 * q^83 + 95 * q^84 - 18 * q^86 + 47 * q^87 + 13 * q^88 + 19 * q^89 - 31 * q^91 + 137 * q^92 + 52 * q^93 + 120 * q^94 - 49 * q^96 + 36 * q^97 + 64 * q^98 + 16 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.394755	0.279134	0.139567	−	0.990213i	$-0.455429\pi$
$2$	0.394755	0.279134	0.139567	+	0.990213i	$0.455429\pi$
$3$	2.83806	1.63855	0.819276	−	0.573399i	$-0.194376\pi$
$3$	2.83806	1.63855	0.819276	+	0.573399i	$0.194376\pi$
$4$	−1.84417	−0.922084
$5$	0	0
$5$	0	0
$6$	1.12034	0.457375
$7$	−0.914717	−0.345731	−0.172865	−	0.984945i	$-0.555303\pi$
$7$	−0.914717	−0.345731	−0.172865	+	0.984945i	$0.555303\pi$
$8$	−1.51750	−0.536518
$9$	5.05456	1.68485
$10$	0	0
$11$	1.45138	0.437607	0.218803	−	0.975769i	$-0.429785\pi$
$11$	1.45138	0.437607	0.218803	+	0.975769i	$0.429785\pi$
$12$	−5.23385	−1.51088
$13$	4.92764	1.36668	0.683340	−	0.730100i	$-0.260527\pi$
$13$	4.92764	1.36668	0.683340	+	0.730100i	$0.260527\pi$
$14$	−0.361089	−0.0965050
$15$	0	0
$16$	3.08930	0.772324
$17$	2.55248	0.619067	0.309533	−	0.950889i	$-0.399827\pi$
$17$	2.55248	0.619067	0.309533	+	0.950889i	$0.399827\pi$
$18$	1.99531	0.470299
$19$	2.57620	0.591020	0.295510	−	0.955340i	$-0.404510\pi$
$19$	2.57620	0.591020	0.295510	+	0.955340i	$0.404510\pi$
$20$	0	0
$21$	−2.59602	−0.566498
$22$	0.572938	0.122151
$23$	−1.67084	−0.348393	−0.174197	−	0.984711i	$-0.555733\pi$
$23$	−1.67084	−0.348393	−0.174197	+	0.984711i	$0.555733\pi$
$24$	−4.30676	−0.879113
$25$	0	0
$26$	1.94521	0.381487
$27$	5.83095	1.12217
$28$	1.68689	0.318793
$29$	−4.06036	−0.753990	−0.376995	−	0.926215i	$-0.623043\pi$
$29$	−4.06036	−0.753990	−0.376995	+	0.926215i	$0.623043\pi$
$30$	0	0
$31$	1.90713	0.342530	0.171265	−	0.985225i	$-0.445215\pi$
$31$	1.90713	0.342530	0.171265	+	0.985225i	$0.445215\pi$
$32$	4.25452	0.752100
$33$	4.11909	0.717042
$34$	1.00760	0.172802
$35$	0	0
$36$	−9.32146	−1.55358
$37$	−7.44516	−1.22398	−0.611988	−	0.790867i	$-0.709630\pi$
$37$	−7.44516	−1.22398	−0.611988	+	0.790867i	$0.709630\pi$
$38$	1.01696	0.164973
$39$	13.9849	2.23938
$40$	0	0
$41$	−8.87187	−1.38555	−0.692777	−	0.721152i	$-0.743613\pi$
$41$	−8.87187	−1.38555	−0.692777	+	0.721152i	$0.743613\pi$
$42$	−1.02479	−0.158129
$43$	10.2313	1.56026	0.780131	−	0.625616i	$-0.215152\pi$
$43$	10.2313	1.56026	0.780131	+	0.625616i	$0.215152\pi$
$44$	−2.67659	−0.403510
$45$	0	0
$46$	−0.659570	−0.0972483
$47$	−1.40332	−0.204696	−0.102348	−	0.994749i	$-0.532635\pi$
$47$	−1.40332	−0.204696	−0.102348	+	0.994749i	$0.532635\pi$
$48$	8.76759	1.26549
$49$	−6.16329	−0.880470
$50$	0	0
$51$	7.24408	1.01437
$52$	−9.08740	−1.26020
$53$	9.75483	1.33993	0.669964	−	0.742393i	$-0.266309\pi$
$53$	9.75483	1.33993	0.669964	+	0.742393i	$0.266309\pi$
$54$	2.30179	0.313234
$55$	0	0
$56$	1.38809	0.185491
$57$	7.31139	0.968417
$58$	−1.60285	−0.210464
$59$	6.52367	0.849309	0.424654	−	0.905356i	$-0.360396\pi$
$59$	6.52367	0.849309	0.424654	+	0.905356i	$0.360396\pi$
$60$	0	0
$61$	11.1771	1.43108	0.715538	−	0.698574i	$-0.246182\pi$
$61$	11.1771	1.43108	0.715538	+	0.698574i	$0.246182\pi$
$62$	0.752848	0.0956118
$63$	−4.62349	−0.582505
$64$	−4.49910	−0.562388
$65$	0	0
$66$	1.62603	0.200150
$67$	−1.14970	−0.140459	−0.0702294	−	0.997531i	$-0.522373\pi$
$67$	−1.14970	−0.140459	−0.0702294	+	0.997531i	$0.522373\pi$
$68$	−4.70720	−0.570832
$69$	−4.74192	−0.570861
$70$	0	0
$71$	2.51297	0.298235	0.149117	−	0.988819i	$-0.452357\pi$
$71$	2.51297	0.298235	0.149117	+	0.988819i	$0.452357\pi$
$72$	−7.67031	−0.903954
$73$	−8.07631	−0.945260	−0.472630	−	0.881261i	$-0.656695\pi$
$73$	−8.07631	−0.945260	−0.472630	+	0.881261i	$0.656695\pi$
$74$	−2.93901	−0.341653
$75$	0	0
$76$	−4.75094	−0.544970
$77$	−1.32760	−0.151294
$78$	5.52061	0.625086
$79$	11.2886	1.27007	0.635035	−	0.772484i	$-0.280986\pi$
$79$	11.2886	1.27007	0.635035	+	0.772484i	$0.280986\pi$
$80$	0	0
$81$	1.38488	0.153876
$82$	−3.50221	−0.386755
$83$	−0.985950	−0.108222	−0.0541110	−	0.998535i	$-0.517232\pi$
$83$	−0.985950	−0.108222	−0.0541110	+	0.998535i	$0.517232\pi$
$84$	4.78750	0.522359
$85$	0	0
$86$	4.03886	0.435522
$87$	−11.5235	−1.23545
$88$	−2.20247	−0.234784
$89$	18.4736	1.95820	0.979098	−	0.203390i	$-0.0651958\pi$
$89$	18.4736	1.95820	0.979098	+	0.203390i	$0.0651958\pi$
$90$	0	0
$91$	−4.50740	−0.472503
$92$	3.08130	0.321248
$93$	5.41254	0.561254
$94$	−0.553968	−0.0571374
$95$	0	0
$96$	12.0746	1.23235
$97$	−0.323709	−0.0328677	−0.0164338	−	0.999865i	$-0.505231\pi$
$97$	−0.323709	−0.0328677	−0.0164338	+	0.999865i	$0.505231\pi$
$98$	−2.43299	−0.245769
$99$	7.33607	0.737303
$100$	0	0
$101$	6.97479	0.694017	0.347009	−	0.937862i	$-0.387197\pi$
$101$	6.97479	0.694017	0.347009	+	0.937862i	$0.387197\pi$
$102$	2.85963	0.283146
$103$	9.44358	0.930503	0.465252	−	0.885178i	$-0.345964\pi$
$103$	9.44358	0.930503	0.465252	+	0.885178i	$0.345964\pi$
$104$	−7.47771	−0.733249
$105$	0	0
$106$	3.85076	0.374019
$107$	−5.08833	−0.491908	−0.245954	−	0.969282i	$-0.579101\pi$
$107$	−5.08833	−0.491908	−0.245954	+	0.969282i	$0.579101\pi$
$108$	−10.7533	−1.03473
$109$	6.10120	0.584389	0.292194	−	0.956359i	$-0.405615\pi$
$109$	6.10120	0.584389	0.292194	+	0.956359i	$0.405615\pi$
$110$	0	0
$111$	−21.1298	−2.00555
$112$	−2.82583	−0.267016
$113$	7.46973	0.702693	0.351346	−	0.936246i	$-0.385724\pi$
$113$	7.46973	0.702693	0.351346	+	0.936246i	$0.385724\pi$
$114$	2.88620	0.270318
$115$	0	0
$116$	7.48799	0.695243
$117$	24.9070	2.30266
$118$	2.57525	0.237071
$119$	−2.33480	−0.214030
$120$	0	0
$121$	−8.89350	−0.808500
$122$	4.41219	0.399461
$123$	−25.1789	−2.27030
$124$	−3.51707	−0.315842
$125$	0	0
$126$	−1.82514	−0.162597
$127$	−16.1450	−1.43264	−0.716319	−	0.697773i	$-0.754175\pi$
$127$	−16.1450	−1.43264	−0.716319	+	0.697773i	$0.754175\pi$
$128$	−10.2851	−0.909081
$129$	29.0371	2.55657
$130$	0	0
$131$	−5.56774	−0.486455	−0.243228	−	0.969969i	$-0.578206\pi$
$131$	−5.56774	−0.486455	−0.243228	+	0.969969i	$0.578206\pi$
$132$	−7.59630	−0.661173
$133$	−2.35649	−0.204334
$134$	−0.453851	−0.0392068
$135$	0	0
$136$	−3.87339	−0.332141
$137$	12.7857	1.09235	0.546177	−	0.837670i	$-0.316083\pi$
$137$	12.7857	1.09235	0.546177	+	0.837670i	$0.316083\pi$
$138$	−1.87190	−0.159346
$139$	−2.84560	−0.241360	−0.120680	−	0.992691i	$-0.538508\pi$
$139$	−2.84560	−0.241360	−0.120680	+	0.992691i	$0.538508\pi$
$140$	0	0
$141$	−3.98271	−0.335404
$142$	0.992007	0.0832474
$143$	7.15186	0.598069
$144$	15.6150	1.30125
$145$	0	0
$146$	−3.18816	−0.263854
$147$	−17.4918	−1.44270
$148$	13.7301	1.12861
$149$	20.5152	1.68067	0.840335	−	0.542067i	$-0.182358\pi$
$149$	20.5152	1.68067	0.840335	+	0.542067i	$0.182358\pi$
$150$	0	0
$151$	−0.0929076	−0.00756071	−0.00378035	−	0.999993i	$-0.501203\pi$
$151$	−0.0929076	−0.00756071	−0.00378035	+	0.999993i	$0.501203\pi$
$152$	−3.90938	−0.317093
$153$	12.9017	1.04304
$154$	−0.524076	−0.0422313
$155$	0	0
$156$	−25.7905	−2.06490
$157$	−5.17705	−0.413173	−0.206587	−	0.978428i	$-0.566236\pi$
$157$	−5.17705	−0.413173	−0.206587	+	0.978428i	$0.566236\pi$
$158$	4.45623	0.354519
$159$	27.6847	2.19554
$160$	0	0
$161$	1.52834	0.120450
$162$	0.546688	0.0429519
$163$	−9.15797	−0.717308	−0.358654	−	0.933471i	$-0.616764\pi$
$163$	−9.15797	−0.717308	−0.358654	+	0.933471i	$0.616764\pi$
$164$	16.3612	1.27760
$165$	0	0
$166$	−0.389208	−0.0302084
$167$	13.2872	1.02820	0.514098	−	0.857731i	$-0.328127\pi$
$167$	13.2872	1.02820	0.514098	+	0.857731i	$0.328127\pi$
$168$	3.93947	0.303936
$169$	11.2816	0.867816
$170$	0	0
$171$	13.0215	0.995781
$172$	−18.8683	−1.43869
$173$	−15.2825	−1.16191	−0.580955	−	0.813936i	$-0.697321\pi$
$173$	−15.2825	−1.16191	−0.580955	+	0.813936i	$0.697321\pi$
$174$	−4.54897	−0.344856
$175$	0	0
$176$	4.48374	0.337974
$177$	18.5145	1.39164
$178$	7.29253	0.546598
$179$	−17.5071	−1.30854	−0.654272	−	0.756259i	$-0.727025\pi$
$179$	−17.5071	−1.30854	−0.654272	+	0.756259i	$0.727025\pi$
$180$	0	0
$181$	20.6553	1.53530	0.767649	−	0.640871i	$-0.221427\pi$
$181$	20.6553	1.53530	0.767649	+	0.640871i	$0.221427\pi$
$182$	−1.77931	−0.131892
$183$	31.7211	2.34489
$184$	2.53550	0.186919
$185$	0	0
$186$	2.13662	0.156665
$187$	3.70461	0.270908
$188$	2.58796	0.188747
$189$	−5.33367	−0.387967
$190$	0	0
$191$	11.1306	0.805384	0.402692	−	0.915335i	$-0.368075\pi$
$191$	11.1306	0.805384	0.402692	+	0.915335i	$0.368075\pi$
$192$	−12.7687	−0.921502
$193$	20.4977	1.47546	0.737730	−	0.675096i	$-0.235898\pi$
$193$	20.4977	1.47546	0.737730	+	0.675096i	$0.235898\pi$
$194$	−0.127786	−0.00917448
$195$	0	0
$196$	11.3662	0.811868
$197$	11.9194	0.849219	0.424610	−	0.905376i	$-0.360411\pi$
$197$	11.9194	0.849219	0.424610	+	0.905376i	$0.360411\pi$
$198$	2.89595	0.205806
$199$	−18.5474	−1.31479	−0.657395	−	0.753546i	$-0.728342\pi$
$199$	−18.5474	−1.31479	−0.657395	+	0.753546i	$0.728342\pi$
$200$	0	0
$201$	−3.26293	−0.230149
$202$	2.75333	0.193724
$203$	3.71408	0.260678
$204$	−13.3593	−0.935338
$205$	0	0
$206$	3.72789	0.259735
$207$	−8.44533	−0.586991
$208$	15.2229	1.05552
$209$	3.73903	0.258634
$210$	0	0
$211$	15.7666	1.08542	0.542708	−	0.839921i	$-0.317399\pi$
$211$	15.7666	1.08542	0.542708	+	0.839921i	$0.317399\pi$
$212$	−17.9895	−1.23553
$213$	7.13196	0.488674
$214$	−2.00864	−0.137308
$215$	0	0
$216$	−8.84848	−0.602063
$217$	−1.74448	−0.118423
$218$	2.40848	0.163123
$219$	−22.9210	−1.54886
$220$	0	0
$221$	12.5777	0.846067
$222$	−8.34107	−0.559816
$223$	5.04042	0.337531	0.168766	−	0.985656i	$-0.446022\pi$
$223$	5.04042	0.337531	0.168766	+	0.985656i	$0.446022\pi$
$224$	−3.89168	−0.260024
$225$	0	0
$226$	2.94871	0.196145
$227$	1.10003	0.0730116	0.0365058	−	0.999333i	$-0.488377\pi$
$227$	1.10003	0.0730116	0.0365058	+	0.999333i	$0.488377\pi$
$228$	−13.4834	−0.892962
$229$	−11.7371	−0.775612	−0.387806	−	0.921741i	$-0.626767\pi$
$229$	−11.7371	−0.775612	−0.387806	+	0.921741i	$0.626767\pi$
$230$	0	0
$231$	−3.76780	−0.247903
$232$	6.16161	0.404530
$233$	8.10335	0.530868	0.265434	−	0.964129i	$-0.414485\pi$
$233$	8.10335	0.530868	0.265434	+	0.964129i	$0.414485\pi$
$234$	9.83216	0.642749
$235$	0	0
$236$	−12.0307	−0.783134
$237$	32.0377	2.08107
$238$	−0.921671	−0.0597431
$239$	11.7706	0.761377	0.380688	−	0.924703i	$-0.375687\pi$
$239$	11.7706	0.761377	0.380688	+	0.924703i	$0.375687\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	−3.51075	−0.225680
$243$	−13.5625	−0.870033
$244$	−20.6124	−1.31957
$245$	0	0
$246$	−9.93947	−0.633717
$247$	12.6946	0.807735
$248$	−2.89407	−0.183774
$249$	−2.79818	−0.177327
$250$	0	0
$251$	7.33863	0.463210	0.231605	−	0.972810i	$-0.425602\pi$
$251$	7.33863	0.463210	0.231605	+	0.972810i	$0.425602\pi$
$252$	8.52650	0.537119
$253$	−2.42501	−0.152459
$254$	−6.37332	−0.399898
$255$	0	0
$256$	4.93812	0.308633
$257$	−27.1494	−1.69354	−0.846768	−	0.531962i	$-0.821455\pi$
$257$	−27.1494	−1.69354	−0.846768	+	0.531962i	$0.821455\pi$
$258$	11.4625	0.713625
$259$	6.81021	0.423166
$260$	0	0
$261$	−20.5233	−1.27036
$262$	−2.19789	−0.135786
$263$	1.66609	0.102735	0.0513677	−	0.998680i	$-0.483642\pi$
$263$	1.66609	0.102735	0.0513677	+	0.998680i	$0.483642\pi$
$264$	−6.25073	−0.384706
$265$	0	0
$266$	−0.930235	−0.0570364
$267$	52.4290	3.20861
$268$	2.12025	0.129515
$269$	8.63127	0.526258	0.263129	−	0.964761i	$-0.415246\pi$
$269$	8.63127	0.526258	0.263129	+	0.964761i	$0.415246\pi$
$270$	0	0
$271$	−20.7232	−1.25885	−0.629423	−	0.777063i	$-0.716709\pi$
$271$	−20.7232	−1.25885	−0.629423	+	0.777063i	$0.716709\pi$
$272$	7.88536	0.478120
$273$	−12.7922	−0.774221
$274$	5.04720	0.304913
$275$	0	0
$276$	8.74491	0.526382
$277$	−18.6452	−1.12028	−0.560142	−	0.828397i	$-0.689253\pi$
$277$	−18.6452	−1.12028	−0.560142	+	0.828397i	$0.689253\pi$
$278$	−1.12331	−0.0673718
$279$	9.63969	0.577113
$280$	0	0
$281$	−15.7875	−0.941803	−0.470901	−	0.882186i	$-0.656071\pi$
$281$	−15.7875	−0.941803	−0.470901	+	0.882186i	$0.656071\pi$
$282$	−1.57219	−0.0936226
$283$	−12.1471	−0.722073	−0.361036	−	0.932552i	$-0.617577\pi$
$283$	−12.1471	−0.722073	−0.361036	+	0.932552i	$0.617577\pi$
$284$	−4.63435	−0.274998
$285$	0	0
$286$	2.82323	0.166941
$287$	8.11525	0.479028
$288$	21.5047	1.26718
$289$	−10.4849	−0.616756
$290$	0	0
$291$	−0.918704	−0.0538554
$292$	14.8941	0.871610
$293$	15.4914	0.905017	0.452508	−	0.891760i	$-0.350529\pi$
$293$	15.4914	0.905017	0.452508	+	0.891760i	$0.350529\pi$
$294$	−6.90495	−0.402705
$295$	0	0
$296$	11.2980	0.656686
$297$	8.46291	0.491068
$298$	8.09846	0.469131
$299$	−8.23327	−0.476142
$300$	0	0
$301$	−9.35877	−0.539431
$302$	−0.0366757	−0.00211045
$303$	19.7948	1.13718
$304$	7.95863	0.456459
$305$	0	0
$306$	5.09298	0.291147
$307$	24.8884	1.42045	0.710227	−	0.703973i	$-0.248592\pi$
$307$	24.8884	1.42045	0.710227	+	0.703973i	$0.248592\pi$
$308$	2.44832	0.139506
$309$	26.8014	1.52468
$310$	0	0
$311$	3.96273	0.224706	0.112353	−	0.993668i	$-0.464161\pi$
$311$	3.96273	0.224706	0.112353	+	0.993668i	$0.464161\pi$
$312$	−21.2221	−1.20147
$313$	5.68639	0.321414	0.160707	−	0.987002i	$-0.448623\pi$
$313$	5.68639	0.321414	0.160707	+	0.987002i	$0.448623\pi$
$314$	−2.04366	−0.115331
$315$	0	0
$316$	−20.8181	−1.17111
$317$	−6.12565	−0.344051	−0.172025	−	0.985093i	$-0.555031\pi$
$317$	−6.12565	−0.344051	−0.172025	+	0.985093i	$0.555031\pi$
$318$	10.9287	0.612850
$319$	−5.89312	−0.329951
$320$	0	0
$321$	−14.4410	−0.806017
$322$	0.603320	0.0336217
$323$	6.57568	0.365881
$324$	−2.55395	−0.141886
$325$	0	0
$326$	−3.61515	−0.200225
$327$	17.3155	0.957552
$328$	13.4631	0.743375
$329$	1.28364	0.0707695
$330$	0	0
$331$	−20.4475	−1.12390	−0.561949	−	0.827172i	$-0.689948\pi$
$331$	−20.4475	−1.12390	−0.561949	+	0.827172i	$0.689948\pi$
$332$	1.81826	0.0997899
$333$	−37.6320	−2.06222
$334$	5.24519	0.287004
$335$	0	0
$336$	−8.01987	−0.437520
$337$	26.6948	1.45416	0.727079	−	0.686554i	$-0.240878\pi$
$337$	26.6948	1.45416	0.727079	+	0.686554i	$0.240878\pi$
$338$	4.45347	0.242237
$339$	21.1995	1.15140
$340$	0	0
$341$	2.76796	0.149894
$342$	5.14031	0.277956
$343$	12.0407	0.650136
$344$	−15.5261	−0.837110
$345$	0	0
$346$	−6.03285	−0.324328
$347$	24.4990	1.31518	0.657588	−	0.753378i	$-0.271577\pi$
$347$	24.4990	1.31518	0.657588	+	0.753378i	$0.271577\pi$
$348$	21.2513	1.13919
$349$	5.05782	0.270739	0.135369	−	0.990795i	$-0.456778\pi$
$349$	5.05782	0.270739	0.135369	+	0.990795i	$0.456778\pi$
$350$	0	0
$351$	28.7328	1.53364
$352$	6.17492	0.329124
$353$	−2.80153	−0.149111	−0.0745553	−	0.997217i	$-0.523754\pi$
$353$	−2.80153	−0.149111	−0.0745553	+	0.997217i	$0.523754\pi$
$354$	7.30869	0.388453
$355$	0	0
$356$	−34.0684	−1.80562
$357$	−6.62628	−0.350700
$358$	−6.91102	−0.365259
$359$	−9.63713	−0.508629	−0.254314	−	0.967122i	$-0.581850\pi$
$359$	−9.63713	−0.508629	−0.254314	+	0.967122i	$0.581850\pi$
$360$	0	0
$361$	−12.3632	−0.650696
$362$	8.15378	0.428553
$363$	−25.2403	−1.32477
$364$	8.31240	0.435688
$365$	0	0
$366$	12.5221	0.654538
$367$	3.97610	0.207551	0.103775	−	0.994601i	$-0.466908\pi$
$367$	3.97610	0.207551	0.103775	+	0.994601i	$0.466908\pi$
$368$	−5.16171	−0.269073
$369$	−44.8434	−2.33445
$370$	0	0
$371$	−8.92291	−0.463254
$372$	−9.98163	−0.517524
$373$	18.3385	0.949532	0.474766	−	0.880112i	$-0.342533\pi$
$373$	18.3385	0.949532	0.474766	+	0.880112i	$0.342533\pi$
$374$	1.46241	0.0756195
$375$	0	0
$376$	2.12955	0.109823
$377$	−20.0080	−1.03046
$378$	−2.10549	−0.108295
$379$	8.05914	0.413970	0.206985	−	0.978344i	$-0.433635\pi$
$379$	8.05914	0.413970	0.206985	+	0.978344i	$0.433635\pi$
$380$	0	0
$381$	−45.8205	−2.34745
$382$	4.39387	0.224810
$383$	−22.0010	−1.12420	−0.562100	−	0.827069i	$-0.690007\pi$
$383$	−22.0010	−1.12420	−0.562100	+	0.827069i	$0.690007\pi$
$384$	−29.1896	−1.48958
$385$	0	0
$386$	8.09157	0.411850
$387$	51.7148	2.62881
$388$	0.596974	0.0303068
$389$	16.1284	0.817743	0.408871	−	0.912592i	$-0.365922\pi$
$389$	16.1284	0.817743	0.408871	+	0.912592i	$0.365922\pi$
$390$	0	0
$391$	−4.26477	−0.215679
$392$	9.35282	0.472388
$393$	−15.8015	−0.797082
$394$	4.70522	0.237046
$395$	0	0
$396$	−13.5290	−0.679856
$397$	1.94734	0.0977340	0.0488670	−	0.998805i	$-0.484439\pi$
$397$	1.94734	0.0977340	0.0488670	+	0.998805i	$0.484439\pi$
$398$	−7.32167	−0.367002
$399$	−6.68785	−0.334811
$400$	0	0
$401$	5.35156	0.267244	0.133622	−	0.991032i	$-0.457339\pi$
$401$	5.35156	0.267244	0.133622	+	0.991032i	$0.457339\pi$
$402$	−1.28805	−0.0642423
$403$	9.39764	0.468130
$404$	−12.8627	−0.639943
$405$	0	0
$406$	1.46615	0.0727639
$407$	−10.8057	−0.535620
$408$	−10.9929	−0.544230
$409$	−18.2185	−0.900849	−0.450425	−	0.892814i	$-0.648727\pi$
$409$	−18.2185	−0.900849	−0.450425	+	0.892814i	$0.648727\pi$
$410$	0	0
$411$	36.2865	1.78988
$412$	−17.4155	−0.858002
$413$	−5.96731	−0.293632
$414$	−3.33383	−0.163849
$415$	0	0
$416$	20.9647	1.02788
$417$	−8.07596	−0.395482
$418$	1.47600	0.0721935
$419$	−23.6777	−1.15673	−0.578365	−	0.815778i	$-0.696309\pi$
$419$	−23.6777	−1.15673	−0.578365	+	0.815778i	$0.696309\pi$
$420$	0	0
$421$	−17.0983	−0.833322	−0.416661	−	0.909062i	$-0.636800\pi$
$421$	−17.0983	−0.833322	−0.416661	+	0.909062i	$0.636800\pi$
$422$	6.22393	0.302976
$423$	−7.09317	−0.344882
$424$	−14.8030	−0.718896
$425$	0	0
$426$	2.81537	0.136405
$427$	−10.2238	−0.494767
$428$	9.38375	0.453581
$429$	20.2974	0.979967
$430$	0	0
$431$	−10.9883	−0.529289	−0.264644	−	0.964346i	$-0.585255\pi$
$431$	−10.9883	−0.529289	−0.264644	+	0.964346i	$0.585255\pi$
$432$	18.0135	0.866676
$433$	−26.0020	−1.24957	−0.624787	−	0.780795i	$-0.714814\pi$
$433$	−26.0020	−1.24957	−0.624787	+	0.780795i	$0.714814\pi$
$434$	−0.688643	−0.0330559
$435$	0	0
$436$	−11.2516	−0.538856
$437$	−4.30440	−0.205907
$438$	−9.04817	−0.432338
$439$	−9.57265	−0.456878	−0.228439	−	0.973558i	$-0.573362\pi$
$439$	−9.57265	−0.456878	−0.228439	+	0.973558i	$0.573362\pi$
$440$	0	0
$441$	−31.1527	−1.48346
$442$	4.96510	0.236166
$443$	23.2988	1.10696	0.553479	−	0.832863i	$-0.313300\pi$
$443$	23.2988	1.10696	0.553479	+	0.832863i	$0.313300\pi$
$444$	38.9669	1.84929
$445$	0	0
$446$	1.98973	0.0942163
$447$	58.2232	2.75387
$448$	4.11541	0.194435
$449$	−31.0942	−1.46743	−0.733713	−	0.679460i	$-0.762214\pi$
$449$	−31.0942	−1.46743	−0.733713	+	0.679460i	$0.762214\pi$
$450$	0	0
$451$	−12.8764	−0.606328
$452$	−13.7754	−0.647942
$453$	−0.263677	−0.0123886
$454$	0.434242	0.0203800
$455$	0	0
$456$	−11.0950	−0.519573
$457$	−20.1997	−0.944903	−0.472452	−	0.881357i	$-0.656631\pi$
$457$	−20.1997	−0.944903	−0.472452	+	0.881357i	$0.656631\pi$
$458$	−4.63329	−0.216499
$459$	14.8834	0.694696
$460$	0	0
$461$	24.7858	1.15439	0.577194	−	0.816607i	$-0.304148\pi$
$461$	24.7858	1.15439	0.577194	+	0.816607i	$0.304148\pi$
$462$	−1.48736	−0.0691981
$463$	17.0202	0.790994	0.395497	−	0.918467i	$-0.370572\pi$
$463$	17.0202	0.790994	0.395497	+	0.918467i	$0.370572\pi$
$464$	−12.5437	−0.582325
$465$	0	0
$466$	3.19883	0.148183
$467$	37.5394	1.73712	0.868559	−	0.495586i	$-0.165047\pi$
$467$	37.5394	1.73712	0.868559	+	0.495586i	$0.165047\pi$
$468$	−45.9328	−2.12324
$469$	1.05165	0.0485609
$470$	0	0
$471$	−14.6927	−0.677006
$472$	−9.89968	−0.455670
$473$	14.8495	0.682782
$474$	12.6470	0.580898
$475$	0	0
$476$	4.30576	0.197354
$477$	49.3063	2.25758
$478$	4.64650	0.212526
$479$	−35.6396	−1.62841	−0.814206	−	0.580576i	$-0.802828\pi$
$479$	−35.6396	−1.62841	−0.814206	+	0.580576i	$0.802828\pi$
$480$	0	0
$481$	−36.6870	−1.67278
$482$	−0.394755	−0.0179806
$483$	4.33752	0.197364
$484$	16.4011	0.745505
$485$	0	0
$486$	−5.35385	−0.242856
$487$	−3.24681	−0.147127	−0.0735636	−	0.997291i	$-0.523437\pi$
$487$	−3.24681	−0.147127	−0.0735636	+	0.997291i	$0.523437\pi$
$488$	−16.9612	−0.767798
$489$	−25.9908	−1.17535
$490$	0	0
$491$	−40.8177	−1.84208	−0.921038	−	0.389474i	$-0.872657\pi$
$491$	−40.8177	−1.84208	−0.921038	+	0.389474i	$0.872657\pi$
$492$	46.4341	2.09341
$493$	−10.3640	−0.466771
$494$	5.01123	0.225466
$495$	0	0
$496$	5.89169	0.264545
$497$	−2.29866	−0.103109
$498$	−1.10459	−0.0494981
$499$	6.69403	0.299666	0.149833	−	0.988711i	$-0.452126\pi$
$499$	6.69403	0.299666	0.149833	+	0.988711i	$0.452126\pi$
$500$	0	0
$501$	37.7099	1.68475
$502$	2.89696	0.129298
$503$	−18.7565	−0.836313	−0.418156	−	0.908375i	$-0.637324\pi$
$503$	−18.7565	−0.836313	−0.418156	+	0.908375i	$0.637324\pi$
$504$	7.01616	0.312525
$505$	0	0
$506$	−0.957285	−0.0425565
$507$	32.0178	1.42196
$508$	29.7741	1.32101
$509$	−16.1174	−0.714390	−0.357195	−	0.934030i	$-0.616267\pi$
$509$	−16.1174	−0.714390	−0.357195	+	0.934030i	$0.616267\pi$
$510$	0	0
$511$	7.38754	0.326805
$512$	22.5195	0.995231
$513$	15.0217	0.663223
$514$	−10.7174	−0.472723
$515$	0	0
$516$	−53.5493	−2.35738
$517$	−2.03675	−0.0895762
$518$	2.68836	0.118120
$519$	−43.3727	−1.90385
$520$	0	0
$521$	14.5796	0.638744	0.319372	−	0.947629i	$-0.396528\pi$
$521$	14.5796	0.638744	0.319372	+	0.947629i	$0.396528\pi$
$522$	−8.10168	−0.354601
$523$	−24.1026	−1.05393	−0.526967	−	0.849886i	$-0.676671\pi$
$523$	−24.1026	−1.05393	−0.526967	+	0.849886i	$0.676671\pi$
$524$	10.2678	0.448553
$525$	0	0
$526$	0.657696	0.0286769
$527$	4.86791	0.212049
$528$	12.7251	0.553789
$529$	−20.2083	−0.878622
$530$	0	0
$531$	32.9742	1.43096
$532$	4.34577	0.188413
$533$	−43.7174	−1.89361
$534$	20.6966	0.895630
$535$	0	0
$536$	1.74468	0.0753587
$537$	−49.6862	−2.14412
$538$	3.40723	0.146896
$539$	−8.94527	−0.385300
$540$	0	0
$541$	38.6411	1.66131	0.830656	−	0.556786i	$-0.187966\pi$
$541$	38.6411	1.66131	0.830656	+	0.556786i	$0.187966\pi$
$542$	−8.18059	−0.351386
$543$	58.6209	2.51566
$544$	10.8596	0.465600
$545$	0	0
$546$	−5.04979	−0.216111
$547$	−15.7613	−0.673906	−0.336953	−	0.941521i	$-0.609396\pi$
$547$	−15.7613	−0.673906	−0.336953	+	0.941521i	$0.609396\pi$
$548$	−23.5790	−1.00724
$549$	56.4951	2.41115
$550$	0	0
$551$	−10.4603	−0.445623
$552$	7.19588	0.306277
$553$	−10.3259	−0.439102
$554$	−7.36029	−0.312709
$555$	0	0
$556$	5.24776	0.222555
$557$	−17.5493	−0.743588	−0.371794	−	0.928315i	$-0.621257\pi$
$557$	−17.5493	−0.743588	−0.371794	+	0.928315i	$0.621257\pi$
$558$	3.80531	0.161092
$559$	50.4163	2.13238
$560$	0	0
$561$	10.5139	0.443897
$562$	−6.23218	−0.262889
$563$	19.0895	0.804528	0.402264	−	0.915524i	$-0.368223\pi$
$563$	19.0895	0.804528	0.402264	+	0.915524i	$0.368223\pi$
$564$	7.34478	0.309271
$565$	0	0
$566$	−4.79514	−0.201555
$567$	−1.26677	−0.0531995
$568$	−3.81344	−0.160009
$569$	−23.0261	−0.965303	−0.482651	−	0.875813i	$-0.660326\pi$
$569$	−23.0261	−0.965303	−0.482651	+	0.875813i	$0.660326\pi$
$570$	0	0
$571$	−23.8202	−0.996845	−0.498422	−	0.866934i	$-0.666087\pi$
$571$	−23.8202	−0.996845	−0.498422	+	0.866934i	$0.666087\pi$
$572$	−13.1892	−0.551470
$573$	31.5894	1.31966
$574$	3.20353	0.133713
$575$	0	0
$576$	−22.7410	−0.947540
$577$	21.0817	0.877642	0.438821	−	0.898574i	$-0.355396\pi$
$577$	21.0817	0.877642	0.438821	+	0.898574i	$0.355396\pi$
$578$	−4.13894	−0.172157
$579$	58.1737	2.41762
$580$	0	0
$581$	0.901865	0.0374157
$582$	−0.362663	−0.0150329
$583$	14.1579	0.586362
$584$	12.2558	0.507149
$585$	0	0
$586$	6.11530	0.252621
$587$	−11.5202	−0.475490	−0.237745	−	0.971328i	$-0.576408\pi$
$587$	−11.5202	−0.475490	−0.237745	+	0.971328i	$0.576408\pi$
$588$	32.2578	1.33029
$589$	4.91314	0.202442
$590$	0	0
$591$	33.8278	1.39149
$592$	−23.0003	−0.945306
$593$	−33.0333	−1.35651	−0.678257	−	0.734825i	$-0.737264\pi$
$593$	−33.0333	−1.35651	−0.678257	+	0.734825i	$0.737264\pi$
$594$	3.34077	0.137074
$595$	0	0
$596$	−37.8335	−1.54972
$597$	−52.6385	−2.15435
$598$	−3.25012	−0.132907
$599$	−16.3195	−0.666798	−0.333399	−	0.942786i	$-0.608196\pi$
$599$	−16.3195	−0.666798	−0.333399	+	0.942786i	$0.608196\pi$
$600$	0	0
$601$	−29.3590	−1.19758	−0.598789	−	0.800907i	$-0.704351\pi$
$601$	−29.3590	−1.19758	−0.598789	+	0.800907i	$0.704351\pi$
$602$	−3.69442	−0.150573
$603$	−5.81125	−0.236652
$604$	0.171337	0.00697161
$605$	0	0
$606$	7.81410	0.317426
$607$	−20.4777	−0.831162	−0.415581	−	0.909556i	$-0.636422\pi$
$607$	−20.4777	−0.831162	−0.415581	+	0.909556i	$0.636422\pi$
$608$	10.9605	0.444506
$609$	10.5408	0.427134
$610$	0	0
$611$	−6.91506	−0.279754
$612$	−23.7928	−0.961768
$613$	−36.7934	−1.48607	−0.743036	−	0.669251i	$-0.766615\pi$
$613$	−36.7934	−1.48607	−0.743036	+	0.669251i	$0.766615\pi$
$614$	9.82480	0.396497
$615$	0	0
$616$	2.01464	0.0811721
$617$	−21.7664	−0.876282	−0.438141	−	0.898906i	$-0.644363\pi$
$617$	−21.7664	−0.876282	−0.438141	+	0.898906i	$0.644363\pi$
$618$	10.5800	0.425589
$619$	32.2247	1.29522	0.647611	−	0.761972i	$-0.275768\pi$
$619$	32.2247	1.29522	0.647611	+	0.761972i	$0.275768\pi$
$620$	0	0
$621$	−9.74256	−0.390955
$622$	1.56430	0.0627229
$623$	−16.8981	−0.677008
$624$	43.2035	1.72953
$625$	0	0
$626$	2.24473	0.0897174
$627$	10.6116	0.423786
$628$	9.54735	0.380981
$629$	−19.0036	−0.757723
$630$	0	0
$631$	−28.4198	−1.13138	−0.565688	−	0.824620i	$-0.691389\pi$
$631$	−28.4198	−1.13138	−0.565688	+	0.824620i	$0.691389\pi$
$632$	−17.1305	−0.681415
$633$	44.7465	1.77851
$634$	−2.41813	−0.0960361
$635$	0	0
$636$	−51.0553	−2.02448
$637$	−30.3705	−1.20332
$638$	−2.32634	−0.0921005
$639$	12.7020	0.502482
$640$	0	0
$641$	3.03782	0.119987	0.0599933	−	0.998199i	$-0.480892\pi$
$641$	3.03782	0.119987	0.0599933	+	0.998199i	$0.480892\pi$
$642$	−5.70064	−0.224986
$643$	−13.6989	−0.540231	−0.270115	−	0.962828i	$-0.587062\pi$
$643$	−13.6989	−0.540231	−0.270115	+	0.962828i	$0.587062\pi$
$644$	−2.81852	−0.111065
$645$	0	0
$646$	2.59578	0.102130
$647$	8.63280	0.339390	0.169695	−	0.985497i	$-0.445722\pi$
$647$	8.63280	0.339390	0.169695	+	0.985497i	$0.445722\pi$
$648$	−2.10156	−0.0825571
$649$	9.46830	0.371663
$650$	0	0
$651$	−4.95094	−0.194043
$652$	16.8888	0.661418
$653$	18.7137	0.732324	0.366162	−	0.930551i	$-0.380672\pi$
$653$	18.7137	0.732324	0.366162	+	0.930551i	$0.380672\pi$
$654$	6.83539	0.267285
$655$	0	0
$656$	−27.4078	−1.07010
$657$	−40.8222	−1.59262
$658$	0.506724	0.0197542
$659$	−40.5919	−1.58124	−0.790618	−	0.612309i	$-0.790241\pi$
$659$	−40.5919	−1.58124	−0.790618	+	0.612309i	$0.790241\pi$
$660$	0	0
$661$	24.0673	0.936109	0.468054	−	0.883700i	$-0.344955\pi$
$661$	24.0673	0.936109	0.468054	+	0.883700i	$0.344955\pi$
$662$	−8.07176	−0.313718
$663$	35.6962	1.38632
$664$	1.49618	0.0580631
$665$	0	0
$666$	−14.8554	−0.575635
$667$	6.78420	0.262685
$668$	−24.5039	−0.948084
$669$	14.3050	0.553062
$670$	0	0
$671$	16.2221	0.626249
$672$	−11.0448	−0.426063
$673$	−17.2653	−0.665530	−0.332765	−	0.943010i	$-0.607982\pi$
$673$	−17.2653	−0.665530	−0.332765	+	0.943010i	$0.607982\pi$
$674$	10.5379	0.405904
$675$	0	0
$676$	−20.8052	−0.800200
$677$	3.05418	0.117382	0.0586908	−	0.998276i	$-0.481307\pi$
$677$	3.05418	0.117382	0.0586908	+	0.998276i	$0.481307\pi$
$678$	8.36860	0.321394
$679$	0.296102	0.0113634
$680$	0	0
$681$	3.12195	0.119633
$682$	1.09267	0.0418404
$683$	−41.1144	−1.57320	−0.786599	−	0.617464i	$-0.788160\pi$
$683$	−41.1144	−1.57320	−0.786599	+	0.617464i	$0.788160\pi$
$684$	−24.0139	−0.918194
$685$	0	0
$686$	4.75312	0.181475
$687$	−33.3106	−1.27088
$688$	31.6076	1.20503
$689$	48.0682	1.83125
$690$	0	0
$691$	−28.1743	−1.07180	−0.535901	−	0.844281i	$-0.680028\pi$
$691$	−28.1743	−1.07180	−0.535901	+	0.844281i	$0.680028\pi$
$692$	28.1836	1.07138
$693$	−6.71043	−0.254908
$694$	9.67110	0.367110
$695$	0	0
$696$	17.4870	0.662843
$697$	−22.6453	−0.857750
$698$	1.99660	0.0755723
$699$	22.9977	0.869855
$700$	0	0
$701$	10.7417	0.405710	0.202855	−	0.979209i	$-0.434978\pi$
$701$	10.7417	0.405710	0.202855	+	0.979209i	$0.434978\pi$
$702$	11.3424	0.428091
$703$	−19.1802	−0.723394
$704$	−6.52990	−0.246105
$705$	0	0
$706$	−1.10592	−0.0416218
$707$	−6.37996	−0.239943
$708$	−34.1439	−1.28321
$709$	1.52953	0.0574429	0.0287214	−	0.999587i	$-0.490856\pi$
$709$	1.52953	0.0574429	0.0287214	+	0.999587i	$0.490856\pi$
$710$	0	0
$711$	57.0590	2.13988
$712$	−28.0337	−1.05061
$713$	−3.18650	−0.119335
$714$	−2.61575	−0.0978921
$715$	0	0
$716$	32.2861	1.20659
$717$	33.4056	1.24756
$718$	−3.80430	−0.141975
$719$	−11.7860	−0.439542	−0.219771	−	0.975551i	$-0.570531\pi$
$719$	−11.7860	−0.439542	−0.219771	+	0.975551i	$0.570531\pi$
$720$	0	0
$721$	−8.63820	−0.321703
$722$	−4.88044	−0.181631
$723$	−2.83806	−0.105548
$724$	−38.0919	−1.41567
$725$	0	0
$726$	−9.96370	−0.369788
$727$	16.4848	0.611389	0.305694	−	0.952130i	$-0.401111\pi$
$727$	16.4848	0.611389	0.305694	+	0.952130i	$0.401111\pi$
$728$	6.83999	0.253507
$729$	−42.6457	−1.57947
$730$	0	0
$731$	26.1152	0.965907
$732$	−58.4991	−2.16219
$733$	3.25393	0.120187	0.0600934	−	0.998193i	$-0.480860\pi$
$733$	3.25393	0.120187	0.0600934	+	0.998193i	$0.480860\pi$
$734$	1.56958	0.0579343
$735$	0	0
$736$	−7.10860	−0.262027
$737$	−1.66866	−0.0614657
$738$	−17.7021	−0.651624
$739$	−4.47473	−0.164606	−0.0823028	−	0.996607i	$-0.526227\pi$
$739$	−4.47473	−0.164606	−0.0823028	+	0.996607i	$0.526227\pi$
$740$	0	0
$741$	36.0279	1.32352
$742$	−3.52236	−0.129310
$743$	29.1746	1.07031	0.535157	−	0.844753i	$-0.320253\pi$
$743$	29.1746	1.07031	0.535157	+	0.844753i	$0.320253\pi$
$744$	−8.21354	−0.301123
$745$	0	0
$746$	7.23921	0.265046
$747$	−4.98354	−0.182338
$748$	−6.83193	−0.249800
$749$	4.65439	0.170068
$750$	0	0
$751$	14.1087	0.514835	0.257418	−	0.966300i	$-0.417128\pi$
$751$	14.1087	0.514835	0.257418	+	0.966300i	$0.417128\pi$
$752$	−4.33528	−0.158091
$753$	20.8274	0.758994
$754$	−7.89825	−0.287637
$755$	0	0
$756$	9.83619	0.357739
$757$	−39.4960	−1.43551	−0.717754	−	0.696297i	$-0.754830\pi$
$757$	−39.4960	−1.43551	−0.717754	+	0.696297i	$0.754830\pi$
$758$	3.18138	0.115553
$759$	−6.88232	−0.249812
$760$	0	0
$761$	−12.5926	−0.456481	−0.228241	−	0.973605i	$-0.573297\pi$
$761$	−12.5926	−0.456481	−0.228241	+	0.973605i	$0.573297\pi$
$762$	−18.0878	−0.655253
$763$	−5.58087	−0.202041
$764$	−20.5268	−0.742632
$765$	0	0
$766$	−8.68501	−0.313802
$767$	32.1463	1.16073
$768$	14.0147	0.505711
$769$	7.46452	0.269178	0.134589	−	0.990902i	$-0.457029\pi$
$769$	7.46452	0.269178	0.134589	+	0.990902i	$0.457029\pi$
$770$	0	0
$771$	−77.0516	−2.77495
$772$	−37.8013	−1.36050
$773$	41.9524	1.50892	0.754461	−	0.656345i	$-0.227898\pi$
$773$	41.9524	1.50892	0.754461	+	0.656345i	$0.227898\pi$
$774$	20.4147	0.733790
$775$	0	0
$776$	0.491230	0.0176341
$777$	19.3278	0.693380
$778$	6.36676	0.228259
$779$	−22.8557	−0.818890
$780$	0	0
$781$	3.64727	0.130510
$782$	−1.68354	−0.0602032
$783$	−23.6758	−0.846103
$784$	−19.0402	−0.680008
$785$	0	0
$786$	−6.23773	−0.222493
$787$	7.34094	0.261676	0.130838	−	0.991404i	$-0.458233\pi$
$787$	7.34094	0.261676	0.130838	+	0.991404i	$0.458233\pi$
$788$	−21.9813	−0.783052
$789$	4.72845	0.168337
$790$	0	0
$791$	−6.83269	−0.242942
$792$	−11.1325	−0.395577
$793$	55.0765	1.95582
$794$	0.768720	0.0272808
$795$	0	0
$796$	34.2045	1.21235
$797$	−17.9997	−0.637584	−0.318792	−	0.947825i	$-0.603277\pi$
$797$	−17.9997	−0.637584	−0.318792	+	0.947825i	$0.603277\pi$
$798$	−2.64006	−0.0934571
$799$	−3.58195	−0.126720
$800$	0	0
$801$	93.3758	3.29927
$802$	2.11255	0.0745968
$803$	−11.7218	−0.413652
$804$	6.01739	0.212217
$805$	0	0
$806$	3.70976	0.130671
$807$	24.4960	0.862301
$808$	−10.5843	−0.372353
$809$	−16.8968	−0.594059	−0.297030	−	0.954868i	$-0.595996\pi$
$809$	−16.8968	−0.594059	−0.297030	+	0.954868i	$0.595996\pi$
$810$	0	0
$811$	31.2792	1.09836	0.549181	−	0.835703i	$-0.314940\pi$
$811$	31.2792	1.09836	0.549181	+	0.835703i	$0.314940\pi$
$812$	−6.84940	−0.240367
$813$	−58.8136	−2.06268
$814$	−4.26561	−0.149510
$815$	0	0
$816$	22.3791	0.783425
$817$	26.3579	0.922146
$818$	−7.19185	−0.251457
$819$	−22.7829	−0.796099
$820$	0	0
$821$	−4.27439	−0.149177	−0.0745887	−	0.997214i	$-0.523764\pi$
$821$	−4.27439	−0.149177	−0.0745887	+	0.997214i	$0.523764\pi$
$822$	14.3242	0.499615
$823$	−25.5442	−0.890415	−0.445207	−	0.895428i	$-0.646870\pi$
$823$	−25.5442	−0.890415	−0.445207	+	0.895428i	$0.646870\pi$
$824$	−14.3307	−0.499232
$825$	0	0
$826$	−2.35562	−0.0819626
$827$	−5.38049	−0.187098	−0.0935490	−	0.995615i	$-0.529821\pi$
$827$	−5.38049	−0.187098	−0.0935490	+	0.995615i	$0.529821\pi$
$828$	15.5746	0.541256
$829$	−40.8463	−1.41865	−0.709325	−	0.704881i	$-0.751000\pi$
$829$	−40.8463	−1.41865	−0.709325	+	0.704881i	$0.751000\pi$
$830$	0	0
$831$	−52.9162	−1.83564
$832$	−22.1699	−0.768605
$833$	−15.7317	−0.545070
$834$	−3.18802	−0.110392
$835$	0	0
$836$	−6.89541	−0.238483
$837$	11.1204	0.384376
$838$	−9.34687	−0.322882
$839$	−8.01547	−0.276725	−0.138362	−	0.990382i	$-0.544184\pi$
$839$	−8.01547	−0.276725	−0.138362	+	0.990382i	$0.544184\pi$
$840$	0	0
$841$	−12.5135	−0.431499
$842$	−6.74964	−0.232608
$843$	−44.8058	−1.54319
$844$	−29.0763	−1.00085
$845$	0	0
$846$	−2.80006	−0.0962681
$847$	8.13504	0.279523
$848$	30.1356	1.03486
$849$	−34.4743	−1.18315
$850$	0	0
$851$	12.4396	0.426425
$852$	−13.1525	−0.450598
$853$	13.9527	0.477732	0.238866	−	0.971053i	$-0.423224\pi$
$853$	13.9527	0.477732	0.238866	+	0.971053i	$0.423224\pi$
$854$	−4.03591	−0.138106
$855$	0	0
$856$	7.72156	0.263918
$857$	10.6846	0.364978	0.182489	−	0.983208i	$-0.441585\pi$
$857$	10.6846	0.364978	0.182489	+	0.983208i	$0.441585\pi$
$858$	8.01248	0.273542
$859$	−35.6966	−1.21795	−0.608976	−	0.793189i	$-0.708419\pi$
$859$	−35.6966	−1.21795	−0.608976	+	0.793189i	$0.708419\pi$
$860$	0	0
$861$	23.0315	0.784913
$862$	−4.33769	−0.147742
$863$	40.7480	1.38708	0.693538	−	0.720420i	$-0.256051\pi$
$863$	40.7480	1.38708	0.693538	+	0.720420i	$0.256051\pi$
$864$	24.8079	0.843981
$865$	0	0
$866$	−10.2644	−0.348798
$867$	−29.7566	−1.01059
$868$	3.21712	0.109196
$869$	16.3841	0.555791
$870$	0	0
$871$	−5.66533	−0.191962
$872$	−9.25859	−0.313535
$873$	−1.63621	−0.0553772
$874$	−1.69918	−0.0574757
$875$	0	0
$876$	42.2702	1.42818
$877$	−33.8708	−1.14374	−0.571868	−	0.820346i	$-0.693781\pi$
$877$	−33.8708	−1.14374	−0.571868	+	0.820346i	$0.693781\pi$
$878$	−3.77885	−0.127530
$879$	43.9654	1.48292
$880$	0	0
$881$	−28.2949	−0.953278	−0.476639	−	0.879099i	$-0.658145\pi$
$881$	−28.2949	−0.953278	−0.476639	+	0.879099i	$0.658145\pi$
$882$	−12.2977	−0.414084
$883$	12.2965	0.413809	0.206904	−	0.978361i	$-0.433661\pi$
$883$	12.2965	0.413809	0.206904	+	0.978361i	$0.433661\pi$
$884$	−23.1954	−0.780145
$885$	0	0
$886$	9.19729	0.308989
$887$	−26.8689	−0.902171	−0.451085	−	0.892481i	$-0.648963\pi$
$887$	−26.8689	−0.902171	−0.451085	+	0.892481i	$0.648963\pi$
$888$	32.0645	1.07601
$889$	14.7681	0.495307
$890$	0	0
$891$	2.00999	0.0673370
$892$	−9.29538	−0.311232
$893$	−3.61523	−0.120979
$894$	22.9839	0.768696
$895$	0	0
$896$	9.40794	0.314297
$897$	−23.3665	−0.780184
$898$	−12.2746	−0.409608
$899$	−7.74363	−0.258265
$900$	0	0
$901$	24.8990	0.829505
$902$	−5.08303	−0.169246
$903$	−26.5607	−0.883885
$904$	−11.3353	−0.377008
$905$	0	0
$906$	−0.104088	−0.00345808
$907$	20.7501	0.688997	0.344498	−	0.938787i	$-0.388049\pi$
$907$	20.7501	0.688997	0.344498	+	0.938787i	$0.388049\pi$
$908$	−2.02864	−0.0673229
$909$	35.2545	1.16932
$910$	0	0
$911$	9.82491	0.325514	0.162757	−	0.986666i	$-0.447961\pi$
$911$	9.82491	0.325514	0.162757	+	0.986666i	$0.447961\pi$
$912$	22.5870	0.747932
$913$	−1.43099	−0.0473587
$914$	−7.97393	−0.263754
$915$	0	0
$916$	21.6453	0.715179
$917$	5.09290	0.168183
$918$	5.87528	0.193913
$919$	37.0086	1.22080	0.610401	−	0.792092i	$-0.291008\pi$
$919$	37.0086	1.22080	0.610401	+	0.792092i	$0.291008\pi$
$920$	0	0
$921$	70.6346	2.32749
$922$	9.78429	0.322229
$923$	12.3830	0.407592
$924$	6.94846	0.228588
$925$	0	0
$926$	6.71878	0.220793
$927$	47.7331	1.56776
$928$	−17.2749	−0.567076
$929$	−26.1323	−0.857373	−0.428687	−	0.903453i	$-0.641024\pi$
$929$	−26.1323	−0.857373	−0.428687	+	0.903453i	$0.641024\pi$
$930$	0	0
$931$	−15.8778	−0.520375
$932$	−14.9439	−0.489505
$933$	11.2464	0.368192
$934$	14.8189	0.484888
$935$	0	0
$936$	−37.7965	−1.23542
$937$	−53.7559	−1.75613	−0.878064	−	0.478543i	$-0.841165\pi$
$937$	−53.7559	−1.75613	−0.878064	+	0.478543i	$0.841165\pi$
$938$	0.415145	0.0135550
$939$	16.1383	0.526653
$940$	0	0
$941$	16.6675	0.543345	0.271673	−	0.962390i	$-0.412423\pi$
$941$	16.6675	0.543345	0.271673	+	0.962390i	$0.412423\pi$
$942$	−5.80003	−0.188975
$943$	14.8234	0.482718
$944$	20.1535	0.655942
$945$	0	0
$946$	5.86192	0.190587
$947$	−16.3180	−0.530265	−0.265133	−	0.964212i	$-0.585416\pi$
$947$	−16.3180	−0.530265	−0.265133	+	0.964212i	$0.585416\pi$
$948$	−59.0830	−1.91893
$949$	−39.7971	−1.29187
$950$	0	0
$951$	−17.3849	−0.563745
$952$	3.54306	0.114831
$953$	21.4661	0.695354	0.347677	−	0.937614i	$-0.386971\pi$
$953$	21.4661	0.695354	0.347677	+	0.937614i	$0.386971\pi$
$954$	19.4639	0.630167
$955$	0	0
$956$	−21.7070	−0.702054
$957$	−16.7250	−0.540642
$958$	−14.0689	−0.454545
$959$	−11.6953	−0.377660
$960$	0	0
$961$	−27.3629	−0.882673
$962$	−14.4824	−0.466930
$963$	−25.7193	−0.828792
$964$	1.84417	0.0593967
$965$	0	0
$966$	1.71226	0.0550909
$967$	9.77699	0.314407	0.157203	−	0.987566i	$-0.449752\pi$
$967$	9.77699	0.314407	0.157203	+	0.987566i	$0.449752\pi$
$968$	13.4959	0.433775
$969$	18.6622	0.599515
$970$	0	0
$971$	4.26655	0.136920	0.0684601	−	0.997654i	$-0.478191\pi$
$971$	4.26655	0.136920	0.0684601	+	0.997654i	$0.478191\pi$
$972$	25.0115	0.802244
$973$	2.60292	0.0834457
$974$	−1.28169	−0.0410681
$975$	0	0
$976$	34.5293	1.10525
$977$	29.1863	0.933751	0.466876	−	0.884323i	$-0.345380\pi$
$977$	29.1863	0.933751	0.466876	+	0.884323i	$0.345380\pi$
$978$	−10.2600	−0.328079
$979$	26.8121	0.856920
$980$	0	0
$981$	30.8389	0.984609
$982$	−16.1130	−0.514185
$983$	21.2509	0.677797	0.338899	−	0.940823i	$-0.389946\pi$
$983$	21.2509	0.677797	0.338899	+	0.940823i	$0.389946\pi$
$984$	38.2090	1.21806
$985$	0	0
$986$	−4.09123	−0.130291
$987$	3.64305	0.115960
$988$	−23.4109	−0.744800
$989$	−17.0949	−0.543585
$990$	0	0
$991$	7.91679	0.251485	0.125743	−	0.992063i	$-0.459869\pi$
$991$	7.91679	0.251485	0.125743	+	0.992063i	$0.459869\pi$
$992$	8.11392	0.257617
$993$	−58.0312	−1.84157
$994$	−0.907406	−0.0287812
$995$	0	0
$996$	5.16032	0.163511
$997$	20.3074	0.643141	0.321570	−	0.946886i	$-0.395789\pi$
$997$	20.3074	0.643141	0.321570	+	0.946886i	$0.395789\pi$
$998$	2.64250	0.0836469
$999$	−43.4123	−1.37351

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.o.1.20	yes	40
5.4	even	2		6025.2.a.l.1.21	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.l.1.21	✓	40	5.4	even	2
6025.2.a.o.1.20	yes	40	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q+0.394755 q^{2} +2.83806 q^{3} -1.84417 q^{4} +1.12034 q^{6} -0.914717 q^{7} -1.51750 q^{8} +5.05456 q^{9} +O(q^{10})\)	\(q+0.394755 q^{2} +2.83806 q^{3} -1.84417 q^{4} +1.12034 q^{6} -0.914717 q^{7} -1.51750 q^{8} +5.05456 q^{9} +1.45138 q^{11} -5.23385 q^{12} +4.92764 q^{13} -0.361089 q^{14} +3.08930 q^{16} +2.55248 q^{17} +1.99531 q^{18} +2.57620 q^{19} -2.59602 q^{21} +0.572938 q^{22} -1.67084 q^{23} -4.30676 q^{24} +1.94521 q^{26} +5.83095 q^{27} +1.68689 q^{28} -4.06036 q^{29} +1.90713 q^{31} +4.25452 q^{32} +4.11909 q^{33} +1.00760 q^{34} -9.32146 q^{36} -7.44516 q^{37} +1.01696 q^{38} +13.9849 q^{39} -8.87187 q^{41} -1.02479 q^{42} +10.2313 q^{43} -2.67659 q^{44} -0.659570 q^{46} -1.40332 q^{47} +8.76759 q^{48} -6.16329 q^{49} +7.24408 q^{51} -9.08740 q^{52} +9.75483 q^{53} +2.30179 q^{54} +1.38809 q^{56} +7.31139 q^{57} -1.60285 q^{58} +6.52367 q^{59} +11.1771 q^{61} +0.752848 q^{62} -4.62349 q^{63} -4.49910 q^{64} +1.62603 q^{66} -1.14970 q^{67} -4.70720 q^{68} -4.74192 q^{69} +2.51297 q^{71} -7.67031 q^{72} -8.07631 q^{73} -2.93901 q^{74} -4.75094 q^{76} -1.32760 q^{77} +5.52061 q^{78} +11.2886 q^{79} +1.38488 q^{81} -3.50221 q^{82} -0.985950 q^{83} +4.78750 q^{84} +4.03886 q^{86} -11.5235 q^{87} -2.20247 q^{88} +18.4736 q^{89} -4.50740 q^{91} +3.08130 q^{92} +5.41254 q^{93} -0.553968 q^{94} +12.0746 q^{96} -0.323709 q^{97} -2.43299 q^{98} +7.33607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9}+O(q^{10}) \) 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9	\( 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9} + q^{11} + 14 q^{12} + 9 q^{13} - q^{14} + 43 q^{16} + 12 q^{17} + 42 q^{18} + 2 q^{21} + 5 q^{22} + 77 q^{23} - 2 q^{24} + 2 q^{26} + 38 q^{27} + 42 q^{28} + 2 q^{29} + q^{31} + 72 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 28 q^{37} + 23 q^{38} - 2 q^{39} - 2 q^{41} + 37 q^{42} + 31 q^{43} + 3 q^{44} + 14 q^{46} + 96 q^{47} + 13 q^{48} + 40 q^{49} - 10 q^{51} + 42 q^{52} + 54 q^{53} + 4 q^{54} - 15 q^{56} + 37 q^{57} + 27 q^{58} + q^{59} + 5 q^{61} + 39 q^{62} + 70 q^{63} + 65 q^{64} - 52 q^{66} + 34 q^{67} + 52 q^{68} + 21 q^{69} - 9 q^{71} + 70 q^{72} + 25 q^{73} + 22 q^{74} - 47 q^{76} + 54 q^{77} + 58 q^{78} + 13 q^{79} + 12 q^{81} - 5 q^{82} + 63 q^{83} + 95 q^{84} - 18 q^{86} + 47 q^{87} + 13 q^{88} + 19 q^{89} - 31 q^{91} + 137 q^{92} + 52 q^{93} + 120 q^{94} - 49 q^{96} + 36 q^{97} + 64 q^{98} + 16 q^{99}+O(q^{100}) \) 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9 + q^11 + 14 * q^12 + 9 * q^13 - q^14 + 43 * q^16 + 12 * q^17 + 42 * q^18 + 2 * q^21 + 5 * q^22 + 77 * q^23 - 2 * q^24 + 2 * q^26 + 38 * q^27 + 42 * q^28 + 2 * q^29 + q^31 + 72 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 28 * q^37 + 23 * q^38 - 2 * q^39 - 2 * q^41 + 37 * q^42 + 31 * q^43 + 3 * q^44 + 14 * q^46 + 96 * q^47 + 13 * q^48 + 40 * q^49 - 10 * q^51 + 42 * q^52 + 54 * q^53 + 4 * q^54 - 15 * q^56 + 37 * q^57 + 27 * q^58 + q^59 + 5 * q^61 + 39 * q^62 + 70 * q^63 + 65 * q^64 - 52 * q^66 + 34 * q^67 + 52 * q^68 + 21 * q^69 - 9 * q^71 + 70 * q^72 + 25 * q^73 + 22 * q^74 - 47 * q^76 + 54 * q^77 + 58 * q^78 + 13 * q^79 + 12 * q^81 - 5 * q^82 + 63 * q^83 + 95 * q^84 - 18 * q^86 + 47 * q^87 + 13 * q^88 + 19 * q^89 - 31 * q^91 + 137 * q^92 + 52 * q^93 + 120 * q^94 - 49 * q^96 + 36 * q^97 + 64 * q^98 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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