LMFDB - Embedded newform 6025.2.a.n.1.6

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			1.6
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)

$f(q)$	$=$	$q-1.94004 q^{2} +1.97205 q^{3} +1.76376 q^{4} -3.82586 q^{6} -2.17744 q^{7} +0.458324 q^{8} +0.888995 q^{9} +O(q^{10})$	\(q-1.94004 q^{2} +1.97205 q^{3} +1.76376 q^{4} -3.82586 q^{6} -2.17744 q^{7} +0.458324 q^{8} +0.888995 q^{9} -4.36561 q^{11} +3.47822 q^{12} -1.66571 q^{13} +4.22432 q^{14} -4.41668 q^{16} +0.0716082 q^{17} -1.72469 q^{18} +1.98127 q^{19} -4.29403 q^{21} +8.46946 q^{22} -7.31877 q^{23} +0.903839 q^{24} +3.23154 q^{26} -4.16301 q^{27} -3.84047 q^{28} +4.53224 q^{29} -4.85339 q^{31} +7.65188 q^{32} -8.60922 q^{33} -0.138923 q^{34} +1.56797 q^{36} -0.626446 q^{37} -3.84375 q^{38} -3.28486 q^{39} +0.630715 q^{41} +8.33058 q^{42} +4.25577 q^{43} -7.69987 q^{44} +14.1987 q^{46} +4.25718 q^{47} -8.70992 q^{48} -2.25876 q^{49} +0.141215 q^{51} -2.93790 q^{52} +6.53836 q^{53} +8.07642 q^{54} -0.997972 q^{56} +3.90718 q^{57} -8.79272 q^{58} -6.42444 q^{59} +0.998477 q^{61} +9.41578 q^{62} -1.93573 q^{63} -6.01161 q^{64} +16.7022 q^{66} +4.31517 q^{67} +0.126299 q^{68} -14.4330 q^{69} +0.437661 q^{71} +0.407448 q^{72} +2.76071 q^{73} +1.21533 q^{74} +3.49448 q^{76} +9.50585 q^{77} +6.37276 q^{78} +0.0176016 q^{79} -10.8767 q^{81} -1.22361 q^{82} +7.97730 q^{83} -7.57361 q^{84} -8.25637 q^{86} +8.93781 q^{87} -2.00086 q^{88} -3.58851 q^{89} +3.62697 q^{91} -12.9085 q^{92} -9.57115 q^{93} -8.25910 q^{94} +15.0899 q^{96} +9.52612 q^{97} +4.38208 q^{98} -3.88100 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 9 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 20 q^{7} + 27 q^{8} + 38 q^{9}+O(q^{10}) $ 40 * q + 9 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 20 * q^7 + 27 * q^8 + 38 * q^9	$ 40 q + 9 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 20 q^{7} + 27 q^{8} + 38 q^{9} + q^{11} + 26 q^{12} + 11 q^{13} - q^{14} + 43 q^{16} + 20 q^{17} + 18 q^{18} + 2 q^{21} + 23 q^{22} + 79 q^{23} - 2 q^{24} + 2 q^{26} + 26 q^{27} + 30 q^{28} + 2 q^{29} + q^{31} + 68 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 16 q^{37} + 45 q^{38} - 2 q^{39} - 2 q^{41} + 19 q^{42} + 25 q^{43} + 3 q^{44} + 14 q^{46} + 88 q^{47} + 75 q^{48} + 40 q^{49} - 10 q^{51} + 18 q^{52} + 34 q^{53} + 4 q^{54} - 15 q^{56} + 51 q^{57} + 53 q^{58} + q^{59} + 9 q^{61} + 39 q^{62} + 110 q^{63} + 17 q^{64} + 26 q^{66} + 30 q^{67} + 44 q^{68} - 7 q^{69} + 5 q^{71} + 18 q^{72} + 23 q^{73} - 18 q^{74} + 43 q^{76} + 30 q^{77} + 46 q^{78} + 5 q^{79} + 44 q^{81} + 5 q^{82} + 65 q^{83} - 65 q^{84} + 40 q^{86} + 33 q^{87} + 71 q^{88} - 9 q^{89} + q^{91} + 117 q^{92} + 68 q^{93} - 72 q^{94} + 83 q^{96} - 8 q^{97} + 76 q^{98} - 40 q^{99}+O(q^{100}) $ 40 * q + 9 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 20 * q^7 + 27 * q^8 + 38 * q^9 + q^11 + 26 * q^12 + 11 * q^13 - q^14 + 43 * q^16 + 20 * q^17 + 18 * q^18 + 2 * q^21 + 23 * q^22 + 79 * q^23 - 2 * q^24 + 2 * q^26 + 26 * q^27 + 30 * q^28 + 2 * q^29 + q^31 + 68 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 16 * q^37 + 45 * q^38 - 2 * q^39 - 2 * q^41 + 19 * q^42 + 25 * q^43 + 3 * q^44 + 14 * q^46 + 88 * q^47 + 75 * q^48 + 40 * q^49 - 10 * q^51 + 18 * q^52 + 34 * q^53 + 4 * q^54 - 15 * q^56 + 51 * q^57 + 53 * q^58 + q^59 + 9 * q^61 + 39 * q^62 + 110 * q^63 + 17 * q^64 + 26 * q^66 + 30 * q^67 + 44 * q^68 - 7 * q^69 + 5 * q^71 + 18 * q^72 + 23 * q^73 - 18 * q^74 + 43 * q^76 + 30 * q^77 + 46 * q^78 + 5 * q^79 + 44 * q^81 + 5 * q^82 + 65 * q^83 - 65 * q^84 + 40 * q^86 + 33 * q^87 + 71 * q^88 - 9 * q^89 + q^91 + 117 * q^92 + 68 * q^93 - 72 * q^94 + 83 * q^96 - 8 * q^97 + 76 * q^98 - 40 * 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 $

$2$	−1.94004	−1.37182	−0.685908	−	0.727689i	$-0.740595\pi$
$2$	−1.94004	−1.37182	−0.685908	+	0.727689i	$0.740595\pi$
$3$	1.97205	1.13857	0.569283	−	0.822142i	$-0.307221\pi$
$3$	1.97205	1.13857	0.569283	+	0.822142i	$0.307221\pi$
$4$	1.76376	0.881878
$5$	0	0
$5$	0	0
$6$	−3.82586	−1.56190
$7$	−2.17744	−0.822995	−0.411497	−	0.911411i	$-0.634994\pi$
$7$	−2.17744	−0.822995	−0.411497	+	0.911411i	$0.634994\pi$
$8$	0.458324	0.162042
$9$	0.888995	0.296332
$10$	0	0
$11$	−4.36561	−1.31628	−0.658140	−	0.752895i	$-0.728657\pi$
$11$	−4.36561	−1.31628	−0.658140	+	0.752895i	$0.728657\pi$
$12$	3.47822	1.00408
$13$	−1.66571	−0.461984	−0.230992	−	0.972956i	$-0.574197\pi$
$13$	−1.66571	−0.461984	−0.230992	+	0.972956i	$0.574197\pi$
$14$	4.22432	1.12900
$15$	0	0
$16$	−4.41668	−1.10417
$17$	0.0716082	0.0173675	0.00868377	−	0.999962i	$-0.497236\pi$
$17$	0.0716082	0.0173675	0.00868377	+	0.999962i	$0.497236\pi$
$18$	−1.72469	−0.406512
$19$	1.98127	0.454536	0.227268	−	0.973832i	$-0.427021\pi$
$19$	1.98127	0.454536	0.227268	+	0.973832i	$0.427021\pi$
$20$	0	0
$21$	−4.29403	−0.937033
$22$	8.46946	1.80569
$23$	−7.31877	−1.52607	−0.763035	−	0.646357i	$-0.776292\pi$
$23$	−7.31877	−1.52607	−0.763035	+	0.646357i	$0.776292\pi$
$24$	0.903839	0.184495
$25$	0	0
$26$	3.23154	0.633757
$27$	−4.16301	−0.801173
$28$	−3.84047	−0.725781
$29$	4.53224	0.841615	0.420807	−	0.907150i	$-0.361747\pi$
$29$	4.53224	0.841615	0.420807	+	0.907150i	$0.361747\pi$
$30$	0	0
$31$	−4.85339	−0.871695	−0.435848	−	0.900020i	$-0.643551\pi$
$31$	−4.85339	−0.871695	−0.435848	+	0.900020i	$0.643551\pi$
$32$	7.65188	1.35267
$33$	−8.60922	−1.49867
$34$	−0.138923	−0.0238251
$35$	0	0
$36$	1.56797	0.261328
$37$	−0.626446	−0.102987	−0.0514936	−	0.998673i	$-0.516398\pi$
$37$	−0.626446	−0.102987	−0.0514936	+	0.998673i	$0.516398\pi$
$38$	−3.84375	−0.623539
$39$	−3.28486	−0.525999
$40$	0	0
$41$	0.630715	0.0985012	0.0492506	−	0.998786i	$-0.484317\pi$
$41$	0.630715	0.0985012	0.0492506	+	0.998786i	$0.484317\pi$
$42$	8.33058	1.28544
$43$	4.25577	0.648999	0.324500	−	0.945886i	$-0.394804\pi$
$43$	4.25577	0.648999	0.324500	+	0.945886i	$0.394804\pi$
$44$	−7.69987	−1.16080
$45$	0	0
$46$	14.1987	2.09349
$47$	4.25718	0.620973	0.310487	−	0.950578i	$-0.399508\pi$
$47$	4.25718	0.620973	0.310487	+	0.950578i	$0.399508\pi$
$48$	−8.70992	−1.25717
$49$	−2.25876	−0.322680
$50$	0	0
$51$	0.141215	0.0197741
$52$	−2.93790	−0.407413
$53$	6.53836	0.898113	0.449056	−	0.893503i	$-0.351760\pi$
$53$	6.53836	0.898113	0.449056	+	0.893503i	$0.351760\pi$
$54$	8.07642	1.09906
$55$	0	0
$56$	−0.997972	−0.133360
$57$	3.90718	0.517519
$58$	−8.79272	−1.15454
$59$	−6.42444	−0.836391	−0.418196	−	0.908357i	$-0.637337\pi$
$59$	−6.42444	−0.836391	−0.418196	+	0.908357i	$0.637337\pi$
$60$	0	0
$61$	0.998477	0.127842	0.0639209	−	0.997955i	$-0.479639\pi$
$61$	0.998477	0.127842	0.0639209	+	0.997955i	$0.479639\pi$
$62$	9.41578	1.19581
$63$	−1.93573	−0.243879
$64$	−6.01161	−0.751451
$65$	0	0
$66$	16.7022	2.05590
$67$	4.31517	0.527182	0.263591	−	0.964635i	$-0.415093\pi$
$67$	4.31517	0.527182	0.263591	+	0.964635i	$0.415093\pi$
$68$	0.126299	0.0153160
$69$	−14.4330	−1.73753
$70$	0	0
$71$	0.437661	0.0519408	0.0259704	−	0.999663i	$-0.491732\pi$
$71$	0.437661	0.0519408	0.0259704	+	0.999663i	$0.491732\pi$
$72$	0.407448	0.0480182
$73$	2.76071	0.323116	0.161558	−	0.986863i	$-0.448348\pi$
$73$	2.76071	0.323116	0.161558	+	0.986863i	$0.448348\pi$
$74$	1.21533	0.141279
$75$	0	0
$76$	3.49448	0.400845
$77$	9.50585	1.08329
$78$	6.37276	0.721573
$79$	0.0176016	0.00198034	0.000990169	−	1.00000i	$-0.499685\pi$
$79$	0.0176016	0.00198034	0.000990169		1.00000i	$0.499685\pi$
$80$	0	0
$81$	−10.8767	−1.20852
$82$	−1.22361	−0.135125
$83$	7.97730	0.875622	0.437811	−	0.899067i	$-0.355754\pi$
$83$	7.97730	0.875622	0.437811	+	0.899067i	$0.355754\pi$
$84$	−7.57361	−0.826349
$85$	0	0
$86$	−8.25637	−0.890307
$87$	8.93781	0.958234
$88$	−2.00086	−0.213293
$89$	−3.58851	−0.380381	−0.190190	−	0.981747i	$-0.560911\pi$
$89$	−3.58851	−0.380381	−0.190190	+	0.981747i	$0.560911\pi$
$90$	0	0
$91$	3.62697	0.380210
$92$	−12.9085	−1.34581
$93$	−9.57115	−0.992482
$94$	−8.25910	−0.851861
$95$	0	0
$96$	15.0899	1.54011
$97$	9.52612	0.967231	0.483615	−	0.875281i	$-0.339323\pi$
$97$	9.52612	0.967231	0.483615	+	0.875281i	$0.339323\pi$
$98$	4.38208	0.442657
$99$	−3.88100	−0.390056
$100$	0	0
$101$	−1.00023	−0.0995266	−0.0497633	−	0.998761i	$-0.515847\pi$
$101$	−1.00023	−0.0995266	−0.0497633	+	0.998761i	$0.515847\pi$
$102$	−0.273963	−0.0271264
$103$	0.964154	0.0950009	0.0475004	−	0.998871i	$-0.484874\pi$
$103$	0.964154	0.0950009	0.0475004	+	0.998871i	$0.484874\pi$
$104$	−0.763433	−0.0748608
$105$	0	0
$106$	−12.6847	−1.23204
$107$	4.63700	0.448276	0.224138	−	0.974557i	$-0.428043\pi$
$107$	4.63700	0.448276	0.224138	+	0.974557i	$0.428043\pi$
$108$	−7.34254	−0.706536
$109$	4.31709	0.413502	0.206751	−	0.978394i	$-0.433711\pi$
$109$	4.31709	0.413502	0.206751	+	0.978394i	$0.433711\pi$
$110$	0	0
$111$	−1.23539	−0.117258
$112$	9.61705	0.908725
$113$	−5.40857	−0.508795	−0.254398	−	0.967100i	$-0.581877\pi$
$113$	−5.40857	−0.508795	−0.254398	+	0.967100i	$0.581877\pi$
$114$	−7.58009	−0.709940
$115$	0	0
$116$	7.99375	0.742202
$117$	−1.48080	−0.136900
$118$	12.4637	1.14737
$119$	−0.155922	−0.0142934
$120$	0	0
$121$	8.05855	0.732595
$122$	−1.93708	−0.175375
$123$	1.24380	0.112150
$124$	−8.56020	−0.768729
$125$	0	0
$126$	3.75540	0.334557
$127$	17.1513	1.52193	0.760966	−	0.648791i	$-0.224725\pi$
$127$	17.1513	1.52193	0.760966	+	0.648791i	$0.224725\pi$
$128$	−3.64101	−0.321823
$129$	8.39261	0.738928
$130$	0	0
$131$	0.558934	0.0488343	0.0244171	−	0.999702i	$-0.492227\pi$
$131$	0.558934	0.0488343	0.0244171	+	0.999702i	$0.492227\pi$
$132$	−15.1846	−1.32165
$133$	−4.31411	−0.374080
$134$	−8.37160	−0.723196
$135$	0	0
$136$	0.0328197	0.00281427
$137$	12.5365	1.07107	0.535534	−	0.844514i	$-0.320110\pi$
$137$	12.5365	1.07107	0.535534	+	0.844514i	$0.320110\pi$
$138$	28.0006	2.38357
$139$	−1.88569	−0.159942	−0.0799711	−	0.996797i	$-0.525483\pi$
$139$	−1.88569	−0.159942	−0.0799711	+	0.996797i	$0.525483\pi$
$140$	0	0
$141$	8.39539	0.707019
$142$	−0.849079	−0.0712532
$143$	7.27182	0.608100
$144$	−3.92640	−0.327200
$145$	0	0
$146$	−5.35588	−0.443256
$147$	−4.45439	−0.367392
$148$	−1.10490	−0.0908221
$149$	−2.89515	−0.237180	−0.118590	−	0.992943i	$-0.537837\pi$
$149$	−2.89515	−0.237180	−0.118590	+	0.992943i	$0.537837\pi$
$150$	0	0
$151$	−2.01768	−0.164196	−0.0820982	−	0.996624i	$-0.526162\pi$
$151$	−2.01768	−0.164196	−0.0820982	+	0.996624i	$0.526162\pi$
$152$	0.908066	0.0736539
$153$	0.0636593	0.00514655
$154$	−18.4417	−1.48608
$155$	0	0
$156$	−5.79369	−0.463867
$157$	−2.77781	−0.221693	−0.110847	−	0.993838i	$-0.535356\pi$
$157$	−2.77781	−0.221693	−0.110847	+	0.993838i	$0.535356\pi$
$158$	−0.0341479	−0.00271666
$159$	12.8940	1.02256
$160$	0	0
$161$	15.9362	1.25595
$162$	21.1012	1.65787
$163$	11.8971	0.931850	0.465925	−	0.884824i	$-0.345722\pi$
$163$	11.8971	0.931850	0.465925	+	0.884824i	$0.345722\pi$
$164$	1.11243	0.0868660
$165$	0	0
$166$	−15.4763	−1.20119
$167$	15.1860	1.17513	0.587565	−	0.809177i	$-0.300087\pi$
$167$	15.1860	1.17513	0.587565	+	0.809177i	$0.300087\pi$
$168$	−1.96805	−0.151839
$169$	−10.2254	−0.786571
$170$	0	0
$171$	1.76134	0.134693
$172$	7.50614	0.572338
$173$	5.95193	0.452517	0.226259	−	0.974067i	$-0.427351\pi$
$173$	5.95193	0.452517	0.226259	+	0.974067i	$0.427351\pi$
$174$	−17.3397	−1.31452
$175$	0	0
$176$	19.2815	1.45340
$177$	−12.6693	−0.952286
$178$	6.96184	0.521812
$179$	18.4317	1.37765	0.688824	−	0.724928i	$-0.258127\pi$
$179$	18.4317	1.37765	0.688824	+	0.724928i	$0.258127\pi$
$180$	0	0
$181$	−21.5047	−1.59843	−0.799214	−	0.601046i	$-0.794751\pi$
$181$	−21.5047	−1.59843	−0.799214	+	0.601046i	$0.794751\pi$
$182$	−7.03647	−0.521578
$183$	1.96905	0.145556
$184$	−3.35437	−0.247287
$185$	0	0
$186$	18.5684	1.36150
$187$	−0.312613	−0.0228606
$188$	7.50862	0.547623
$189$	9.06471	0.659361
$190$	0	0
$191$	7.91002	0.572349	0.286174	−	0.958178i	$-0.407616\pi$
$191$	7.91002	0.572349	0.286174	+	0.958178i	$0.407616\pi$
$192$	−11.8552	−0.855576
$193$	−12.1860	−0.877168	−0.438584	−	0.898690i	$-0.644520\pi$
$193$	−12.1860	−0.877168	−0.438584	+	0.898690i	$0.644520\pi$
$194$	−18.4811	−1.32686
$195$	0	0
$196$	−3.98390	−0.284564
$197$	4.05749	0.289084	0.144542	−	0.989499i	$-0.453829\pi$
$197$	4.05749	0.289084	0.144542	+	0.989499i	$0.453829\pi$
$198$	7.52931	0.535084
$199$	8.99892	0.637917	0.318958	−	0.947769i	$-0.396667\pi$
$199$	8.99892	0.637917	0.318958	+	0.947769i	$0.396667\pi$
$200$	0	0
$201$	8.50975	0.600231
$202$	1.94049	0.136532
$203$	−9.86867	−0.692645
$204$	0.249069	0.0174383
$205$	0	0
$206$	−1.87050	−0.130324
$207$	−6.50635	−0.452223
$208$	7.35689	0.510108
$209$	−8.64947	−0.598297
$210$	0	0
$211$	13.8447	0.953109	0.476554	−	0.879145i	$-0.341885\pi$
$211$	13.8447	0.953109	0.476554	+	0.879145i	$0.341885\pi$
$212$	11.5321	0.792026
$213$	0.863090	0.0591380
$214$	−8.99597	−0.614952
$215$	0	0
$216$	−1.90801	−0.129824
$217$	10.5680	0.717400
$218$	−8.37533	−0.567249
$219$	5.44426	0.367889
$220$	0	0
$221$	−0.119278	−0.00802352
$222$	2.39670	0.160856
$223$	15.7872	1.05719	0.528594	−	0.848875i	$-0.322719\pi$
$223$	15.7872	1.05719	0.528594	+	0.848875i	$0.322719\pi$
$224$	−16.6615	−1.11324
$225$	0	0
$226$	10.4928	0.697973
$227$	2.47562	0.164313	0.0821564	−	0.996619i	$-0.473819\pi$
$227$	2.47562	0.164313	0.0821564	+	0.996619i	$0.473819\pi$
$228$	6.89131	0.456388
$229$	3.35633	0.221793	0.110896	−	0.993832i	$-0.464628\pi$
$229$	3.35633	0.221793	0.110896	+	0.993832i	$0.464628\pi$
$230$	0	0
$231$	18.7460	1.23340
$232$	2.07723	0.136377
$233$	−11.1945	−0.733377	−0.366689	−	0.930344i	$-0.619509\pi$
$233$	−11.1945	−0.733377	−0.366689	+	0.930344i	$0.619509\pi$
$234$	2.87282	0.187802
$235$	0	0
$236$	−11.3311	−0.737595
$237$	0.0347114	0.00225475
$238$	0.302496	0.0196079
$239$	0.0382977	0.00247727	0.00123864	−	0.999999i	$-0.499606\pi$
$239$	0.0382977	0.00247727	0.00123864	+	0.999999i	$0.499606\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−15.6339	−1.00499
$243$	−8.96034	−0.574806
$244$	1.76107	0.112741
$245$	0	0
$246$	−2.41303	−0.153849
$247$	−3.30022	−0.209988
$248$	−2.22443	−0.141251
$249$	15.7317	0.996953
$250$	0	0
$251$	−2.43490	−0.153689	−0.0768446	−	0.997043i	$-0.524485\pi$
$251$	−2.43490	−0.153689	−0.0768446	+	0.997043i	$0.524485\pi$
$252$	−3.41416	−0.215072
$253$	31.9509	2.00874
$254$	−33.2742	−2.08781
$255$	0	0
$256$	19.0869	1.19293
$257$	16.8596	1.05167	0.525837	−	0.850585i	$-0.323752\pi$
$257$	16.8596	1.05167	0.525837	+	0.850585i	$0.323752\pi$
$258$	−16.2820	−1.01367
$259$	1.36405	0.0847579
$260$	0	0
$261$	4.02913	0.249397
$262$	−1.08435	−0.0669916
$263$	27.7473	1.71097	0.855486	−	0.517826i	$-0.173259\pi$
$263$	27.7473	1.71097	0.855486	+	0.517826i	$0.173259\pi$
$264$	−3.94581	−0.242848
$265$	0	0
$266$	8.36954	0.513169
$267$	−7.07672	−0.433089
$268$	7.61090	0.464910
$269$	27.4956	1.67643	0.838217	−	0.545337i	$-0.183598\pi$
$269$	27.4956	1.67643	0.838217	+	0.545337i	$0.183598\pi$
$270$	0	0
$271$	−4.89862	−0.297570	−0.148785	−	0.988870i	$-0.547536\pi$
$271$	−4.89862	−0.297570	−0.148785	+	0.988870i	$0.547536\pi$
$272$	−0.316270	−0.0191767
$273$	7.15259	0.432894
$274$	−24.3214	−1.46931
$275$	0	0
$276$	−25.4563	−1.53229
$277$	−26.3854	−1.58535	−0.792673	−	0.609648i	$-0.791311\pi$
$277$	−26.3854	−1.58535	−0.792673	+	0.609648i	$0.791311\pi$
$278$	3.65831	0.219411
$279$	−4.31464	−0.258311
$280$	0	0
$281$	−9.46824	−0.564828	−0.282414	−	0.959293i	$-0.591135\pi$
$281$	−9.46824	−0.564828	−0.282414	+	0.959293i	$0.591135\pi$
$282$	−16.2874	−0.969900
$283$	4.80963	0.285903	0.142951	−	0.989730i	$-0.454341\pi$
$283$	4.80963	0.285903	0.142951	+	0.989730i	$0.454341\pi$
$284$	0.771927	0.0458054
$285$	0	0
$286$	−14.1076	−0.834202
$287$	−1.37334	−0.0810659
$288$	6.80249	0.400840
$289$	−16.9949	−0.999698
$290$	0	0
$291$	18.7860	1.10126
$292$	4.86921	0.284949
$293$	−5.18759	−0.303062	−0.151531	−	0.988452i	$-0.548420\pi$
$293$	−5.18759	−0.303062	−0.151531	+	0.988452i	$0.548420\pi$
$294$	8.64170	0.503994
$295$	0	0
$296$	−0.287115	−0.0166882
$297$	18.1741	1.05457
$298$	5.61670	0.325367
$299$	12.1909	0.705020
$300$	0	0
$301$	−9.26668	−0.534123
$302$	3.91438	0.225247
$303$	−1.97251	−0.113318
$304$	−8.75065	−0.501884
$305$	0	0
$306$	−0.123502	−0.00706012
$307$	−4.84985	−0.276796	−0.138398	−	0.990377i	$-0.544195\pi$
$307$	−4.84985	−0.276796	−0.138398	+	0.990377i	$0.544195\pi$
$308$	16.7660	0.955331
$309$	1.90136	0.108165
$310$	0	0
$311$	−4.61908	−0.261924	−0.130962	−	0.991387i	$-0.541807\pi$
$311$	−4.61908	−0.261924	−0.130962	+	0.991387i	$0.541807\pi$
$312$	−1.50553	−0.0852339
$313$	−6.58860	−0.372410	−0.186205	−	0.982511i	$-0.559619\pi$
$313$	−6.58860	−0.372410	−0.186205	+	0.982511i	$0.559619\pi$
$314$	5.38907	0.304123
$315$	0	0
$316$	0.0310450	0.00174642
$317$	14.1896	0.796967	0.398483	−	0.917176i	$-0.369537\pi$
$317$	14.1896	0.796967	0.398483	+	0.917176i	$0.369537\pi$
$318$	−25.0149	−1.40276
$319$	−19.7860	−1.10780
$320$	0	0
$321$	9.14442	0.510392
$322$	−30.9168	−1.72293
$323$	0.141876	0.00789417
$324$	−19.1838	−1.06577
$325$	0	0
$326$	−23.0808	−1.27833
$327$	8.51353	0.470800
$328$	0.289072	0.0159613
$329$	−9.26975	−0.511058
$330$	0	0
$331$	−17.3430	−0.953259	−0.476630	−	0.879104i	$-0.658142\pi$
$331$	−17.3430	−0.953259	−0.476630	+	0.879104i	$0.658142\pi$
$332$	14.0700	0.772192
$333$	−0.556908	−0.0305184
$334$	−29.4615	−1.61206
$335$	0	0
$336$	18.9653	1.03464
$337$	9.86979	0.537642	0.268821	−	0.963190i	$-0.413366\pi$
$337$	9.86979	0.537642	0.268821	+	0.963190i	$0.413366\pi$
$338$	19.8377	1.07903
$339$	−10.6660	−0.579297
$340$	0	0
$341$	21.1880	1.14740
$342$	−3.41708	−0.184774
$343$	20.1604	1.08856
$344$	1.95052	0.105165
$345$	0	0
$346$	−11.5470	−0.620770
$347$	17.6668	0.948402	0.474201	−	0.880417i	$-0.342737\pi$
$347$	17.6668	0.948402	0.474201	+	0.880417i	$0.342737\pi$
$348$	15.7641	0.845045
$349$	29.9032	1.60068	0.800342	−	0.599544i	$-0.204651\pi$
$349$	29.9032	1.60068	0.800342	+	0.599544i	$0.204651\pi$
$350$	0	0
$351$	6.93436	0.370129
$352$	−33.4051	−1.78050
$353$	−2.93118	−0.156011	−0.0780056	−	0.996953i	$-0.524855\pi$
$353$	−2.93118	−0.156011	−0.0780056	+	0.996953i	$0.524855\pi$
$354$	24.5790	1.30636
$355$	0	0
$356$	−6.32925	−0.335449
$357$	−0.307487	−0.0162740
$358$	−35.7582	−1.88988
$359$	−31.0407	−1.63826	−0.819132	−	0.573605i	$-0.805545\pi$
$359$	−31.0407	−1.63826	−0.819132	+	0.573605i	$0.805545\pi$
$360$	0	0
$361$	−15.0745	−0.793397
$362$	41.7199	2.19275
$363$	15.8919	0.834108
$364$	6.39709	0.335299
$365$	0	0
$366$	−3.82003	−0.199676
$367$	−11.2202	−0.585688	−0.292844	−	0.956160i	$-0.594602\pi$
$367$	−11.2202	−0.585688	−0.292844	+	0.956160i	$0.594602\pi$
$368$	32.3247	1.68504
$369$	0.560703	0.0291890
$370$	0	0
$371$	−14.2369	−0.739142
$372$	−16.8812	−0.875248
$373$	−30.2587	−1.56674	−0.783368	−	0.621558i	$-0.786500\pi$
$373$	−30.2587	−1.56674	−0.783368	+	0.621558i	$0.786500\pi$
$374$	0.606483	0.0313605
$375$	0	0
$376$	1.95117	0.100624
$377$	−7.54937	−0.388812
$378$	−17.5859	−0.904521
$379$	13.8675	0.712326	0.356163	−	0.934424i	$-0.384085\pi$
$379$	13.8675	0.712326	0.356163	+	0.934424i	$0.384085\pi$
$380$	0	0
$381$	33.8233	1.73282
$382$	−15.3457	−0.785157
$383$	14.6295	0.747530	0.373765	−	0.927523i	$-0.378067\pi$
$383$	14.6295	0.747530	0.373765	+	0.927523i	$0.378067\pi$
$384$	−7.18027	−0.366417
$385$	0	0
$386$	23.6414	1.20331
$387$	3.78336	0.192319
$388$	16.8017	0.852979
$389$	22.9557	1.16390	0.581950	−	0.813224i	$-0.302290\pi$
$389$	22.9557	1.16390	0.581950	+	0.813224i	$0.302290\pi$
$390$	0	0
$391$	−0.524084	−0.0265041
$392$	−1.03524	−0.0522877
$393$	1.10225	0.0556010
$394$	−7.87170	−0.396571
$395$	0	0
$396$	−6.84514	−0.343981
$397$	−16.0811	−0.807085	−0.403543	−	0.914961i	$-0.632221\pi$
$397$	−16.0811	−0.807085	−0.403543	+	0.914961i	$0.632221\pi$
$398$	−17.4583	−0.875104
$399$	−8.50765	−0.425915
$400$	0	0
$401$	−3.29235	−0.164412	−0.0822062	−	0.996615i	$-0.526197\pi$
$401$	−3.29235	−0.164412	−0.0822062	+	0.996615i	$0.526197\pi$
$402$	−16.5092	−0.823406
$403$	8.08433	0.402709
$404$	−1.76416	−0.0877703
$405$	0	0
$406$	19.1456	0.950181
$407$	2.73482	0.135560
$408$	0.0647223	0.00320423
$409$	−29.4516	−1.45629	−0.728143	−	0.685425i	$-0.759617\pi$
$409$	−29.4516	−1.45629	−0.728143	+	0.685425i	$0.759617\pi$
$410$	0	0
$411$	24.7227	1.21948
$412$	1.70053	0.0837792
$413$	13.9888	0.688345
$414$	12.6226	0.620366
$415$	0	0
$416$	−12.7458	−0.624914
$417$	−3.71868	−0.182105
$418$	16.7803	0.820752
$419$	−27.1211	−1.32495	−0.662476	−	0.749083i	$-0.730494\pi$
$419$	−27.1211	−1.32495	−0.662476	+	0.749083i	$0.730494\pi$
$420$	0	0
$421$	11.4245	0.556796	0.278398	−	0.960466i	$-0.410197\pi$
$421$	11.4245	0.556796	0.278398	+	0.960466i	$0.410197\pi$
$422$	−26.8593	−1.30749
$423$	3.78461	0.184014
$424$	2.99669	0.145532
$425$	0	0
$426$	−1.67443	−0.0811264
$427$	−2.17412	−0.105213
$428$	8.17854	0.395325
$429$	14.3404	0.692362
$430$	0	0
$431$	0.278711	0.0134251	0.00671253	−	0.999977i	$-0.497863\pi$
$431$	0.278711	0.0134251	0.00671253	+	0.999977i	$0.497863\pi$
$432$	18.3867	0.884630
$433$	−19.6133	−0.942554	−0.471277	−	0.881985i	$-0.656207\pi$
$433$	−19.6133	−0.942554	−0.471277	+	0.881985i	$0.656207\pi$
$434$	−20.5023	−0.984141
$435$	0	0
$436$	7.61429	0.364658
$437$	−14.5005	−0.693653
$438$	−10.5621	−0.504676
$439$	34.8411	1.66288	0.831438	−	0.555618i	$-0.187518\pi$
$439$	34.8411	1.66288	0.831438	+	0.555618i	$0.187518\pi$
$440$	0	0
$441$	−2.00803	−0.0956203
$442$	0.231405	0.0110068
$443$	15.8277	0.751997	0.375999	−	0.926620i	$-0.377300\pi$
$443$	15.8277	0.751997	0.375999	+	0.926620i	$0.377300\pi$
$444$	−2.17892	−0.103407
$445$	0	0
$446$	−30.6278	−1.45027
$447$	−5.70939	−0.270045
$448$	13.0899	0.618440
$449$	−28.6529	−1.35222	−0.676108	−	0.736803i	$-0.736335\pi$
$449$	−28.6529	−1.35222	−0.676108	+	0.736803i	$0.736335\pi$
$450$	0	0
$451$	−2.75346	−0.129655
$452$	−9.53939	−0.448695
$453$	−3.97897	−0.186948
$454$	−4.80281	−0.225407
$455$	0	0
$456$	1.79075	0.0838598
$457$	1.10016	0.0514634	0.0257317	−	0.999669i	$-0.491808\pi$
$457$	1.10016	0.0514634	0.0257317	+	0.999669i	$0.491808\pi$
$458$	−6.51142	−0.304259
$459$	−0.298106	−0.0139144
$460$	0	0
$461$	−15.2721	−0.711293	−0.355647	−	0.934621i	$-0.615739\pi$
$461$	−15.2721	−0.711293	−0.355647	+	0.934621i	$0.615739\pi$
$462$	−36.3681	−1.69200
$463$	12.1961	0.566799	0.283400	−	0.959002i	$-0.408538\pi$
$463$	12.1961	0.566799	0.283400	+	0.959002i	$0.408538\pi$
$464$	−20.0174	−0.929286
$465$	0	0
$466$	21.7178	1.00606
$467$	14.2364	0.658780	0.329390	−	0.944194i	$-0.393157\pi$
$467$	14.2364	0.658780	0.329390	+	0.944194i	$0.393157\pi$
$468$	−2.61178	−0.120729
$469$	−9.39602	−0.433868
$470$	0	0
$471$	−5.47799	−0.252413
$472$	−2.94448	−0.135530
$473$	−18.5790	−0.854265
$474$	−0.0673414	−0.00309309
$475$	0	0
$476$	−0.275009	−0.0126050
$477$	5.81257	0.266139
$478$	−0.0742990	−0.00339836
$479$	5.72591	0.261624	0.130812	−	0.991407i	$-0.458242\pi$
$479$	5.72591	0.261624	0.130812	+	0.991407i	$0.458242\pi$
$480$	0	0
$481$	1.04348	0.0475784
$482$	−1.94004	−0.0883664
$483$	31.4270	1.42998
$484$	14.2133	0.646059
$485$	0	0
$486$	17.3834	0.788528
$487$	−3.38045	−0.153183	−0.0765914	−	0.997063i	$-0.524404\pi$
$487$	−3.38045	−0.153183	−0.0765914	+	0.997063i	$0.524404\pi$
$488$	0.457626	0.0207157
$489$	23.4616	1.06097
$490$	0	0
$491$	−3.30094	−0.148970	−0.0744848	−	0.997222i	$-0.523731\pi$
$491$	−3.30094	−0.148970	−0.0744848	+	0.997222i	$0.523731\pi$
$492$	2.19377	0.0989026
$493$	0.324545	0.0146168
$494$	6.40256	0.288065
$495$	0	0
$496$	21.4359	0.962499
$497$	−0.952980	−0.0427470
$498$	−30.5200	−1.36764
$499$	41.5620	1.86057	0.930285	−	0.366836i	$-0.119559\pi$
$499$	41.5620	1.86057	0.930285	+	0.366836i	$0.119559\pi$
$500$	0	0
$501$	29.9477	1.33796
$502$	4.72380	0.210833
$503$	−10.2117	−0.455317	−0.227659	−	0.973741i	$-0.573107\pi$
$503$	−10.2117	−0.455317	−0.227659	+	0.973741i	$0.573107\pi$
$504$	−0.887192	−0.0395187
$505$	0	0
$506$	−61.9860	−2.75562
$507$	−20.1651	−0.895563
$508$	30.2507	1.34216
$509$	−26.3198	−1.16660	−0.583302	−	0.812255i	$-0.698240\pi$
$509$	−26.3198	−1.16660	−0.583302	+	0.812255i	$0.698240\pi$
$510$	0	0
$511$	−6.01127	−0.265923
$512$	−29.7474	−1.31466
$513$	−8.24808	−0.364161
$514$	−32.7084	−1.44270
$515$	0	0
$516$	14.8025	0.651644
$517$	−18.5852	−0.817375
$518$	−2.64631	−0.116272
$519$	11.7375	0.515220
$520$	0	0
$521$	−31.7751	−1.39209	−0.696047	−	0.717996i	$-0.745059\pi$
$521$	−31.7751	−1.39209	−0.696047	+	0.717996i	$0.745059\pi$
$522$	−7.81668	−0.342127
$523$	−8.95404	−0.391533	−0.195766	−	0.980651i	$-0.562719\pi$
$523$	−8.95404	−0.391533	−0.195766	+	0.980651i	$0.562719\pi$
$524$	0.985823	0.0430659
$525$	0	0
$526$	−53.8309	−2.34714
$527$	−0.347543	−0.0151392
$528$	38.0241	1.65479
$529$	30.5644	1.32889
$530$	0	0
$531$	−5.71130	−0.247849
$532$	−7.60903	−0.329893
$533$	−1.05059	−0.0455059
$534$	13.7291	0.594118
$535$	0	0
$536$	1.97775	0.0854256
$537$	36.3483	1.56854
$538$	−53.3425	−2.29976
$539$	9.86086	0.424737
$540$	0	0
$541$	−21.9916	−0.945494	−0.472747	−	0.881198i	$-0.656738\pi$
$541$	−21.9916	−0.945494	−0.472747	+	0.881198i	$0.656738\pi$
$542$	9.50353	0.408211
$543$	−42.4083	−1.81992
$544$	0.547938	0.0234926
$545$	0	0
$546$	−13.8763	−0.593851
$547$	0.398248	0.0170279	0.00851393	−	0.999964i	$-0.497290\pi$
$547$	0.398248	0.0170279	0.00851393	+	0.999964i	$0.497290\pi$
$548$	22.1114	0.944550
$549$	0.887641	0.0378836
$550$	0	0
$551$	8.97960	0.382544
$552$	−6.61500	−0.281553
$553$	−0.0383265	−0.00162981
$554$	51.1887	2.17480
$555$	0	0
$556$	−3.32590	−0.141049
$557$	36.5203	1.54741	0.773706	−	0.633544i	$-0.218401\pi$
$557$	36.5203	1.54741	0.773706	+	0.633544i	$0.218401\pi$
$558$	8.37058	0.354355
$559$	−7.08887	−0.299827
$560$	0	0
$561$	−0.616490	−0.0260282
$562$	18.3688	0.774840
$563$	−8.44871	−0.356071	−0.178035	−	0.984024i	$-0.556974\pi$
$563$	−8.44871	−0.356071	−0.178035	+	0.984024i	$0.556974\pi$
$564$	14.8074	0.623504
$565$	0	0
$566$	−9.33087	−0.392206
$567$	23.6833	0.994605
$568$	0.200590	0.00841659
$569$	36.1838	1.51691	0.758453	−	0.651728i	$-0.225956\pi$
$569$	36.1838	1.51691	0.758453	+	0.651728i	$0.225956\pi$
$570$	0	0
$571$	−12.1968	−0.510421	−0.255211	−	0.966886i	$-0.582145\pi$
$571$	−12.1968	−0.510421	−0.255211	+	0.966886i	$0.582145\pi$
$572$	12.8257	0.536270
$573$	15.5990	0.651657
$574$	2.66434	0.111208
$575$	0	0
$576$	−5.34429	−0.222679
$577$	−16.5429	−0.688691	−0.344346	−	0.938843i	$-0.611899\pi$
$577$	−16.5429	−0.688691	−0.344346	+	0.938843i	$0.611899\pi$
$578$	32.9707	1.37140
$579$	−24.0315	−0.998714
$580$	0	0
$581$	−17.3701	−0.720632
$582$	−36.4456	−1.51072
$583$	−28.5439	−1.18217
$584$	1.26530	0.0523584
$585$	0	0
$586$	10.0641	0.415746
$587$	10.4589	0.431683	0.215842	−	0.976428i	$-0.430750\pi$
$587$	10.4589	0.431683	0.215842	+	0.976428i	$0.430750\pi$
$588$	−7.85646	−0.323995
$589$	−9.61591	−0.396217
$590$	0	0
$591$	8.00160	0.329142
$592$	2.76681	0.113715
$593$	−32.9825	−1.35443	−0.677215	−	0.735785i	$-0.736813\pi$
$593$	−32.9825	−1.35443	−0.677215	+	0.735785i	$0.736813\pi$
$594$	−35.2585	−1.44667
$595$	0	0
$596$	−5.10633	−0.209163
$597$	17.7464	0.726310
$598$	−23.6509	−0.967157
$599$	37.7237	1.54135	0.770674	−	0.637230i	$-0.219920\pi$
$599$	37.7237	1.54135	0.770674	+	0.637230i	$0.219920\pi$
$600$	0	0
$601$	−18.2571	−0.744724	−0.372362	−	0.928088i	$-0.621452\pi$
$601$	−18.2571	−0.744724	−0.372362	+	0.928088i	$0.621452\pi$
$602$	17.9777	0.732718
$603$	3.83616	0.156221
$604$	−3.55869	−0.144801
$605$	0	0
$606$	3.82674	0.155451
$607$	−23.2159	−0.942304	−0.471152	−	0.882052i	$-0.656162\pi$
$607$	−23.2159	−0.942304	−0.471152	+	0.882052i	$0.656162\pi$
$608$	15.1605	0.614839
$609$	−19.4615	−0.788621
$610$	0	0
$611$	−7.09121	−0.286880
$612$	0.112279	0.00453863
$613$	−3.36525	−0.135921	−0.0679606	−	0.997688i	$-0.521649\pi$
$613$	−3.36525	−0.135921	−0.0679606	+	0.997688i	$0.521649\pi$
$614$	9.40891	0.379713
$615$	0	0
$616$	4.35676	0.175539
$617$	20.9066	0.841669	0.420835	−	0.907137i	$-0.361737\pi$
$617$	20.9066	0.841669	0.420835	+	0.907137i	$0.361737\pi$
$618$	−3.68872	−0.148382
$619$	36.7733	1.47805	0.739023	−	0.673681i	$-0.235288\pi$
$619$	36.7733	1.47805	0.739023	+	0.673681i	$0.235288\pi$
$620$	0	0
$621$	30.4682	1.22265
$622$	8.96121	0.359312
$623$	7.81375	0.313051
$624$	14.5082	0.580792
$625$	0	0
$626$	12.7821	0.510877
$627$	−17.0572	−0.681200
$628$	−4.89938	−0.195507
$629$	−0.0448587	−0.00178863
$630$	0	0
$631$	−16.4111	−0.653316	−0.326658	−	0.945143i	$-0.605922\pi$
$631$	−16.4111	−0.653316	−0.326658	+	0.945143i	$0.605922\pi$
$632$	0.00806725	0.000320898 0
$633$	27.3025	1.08518
$634$	−27.5284	−1.09329
$635$	0	0
$636$	22.7419	0.901773
$637$	3.76243	0.149073
$638$	38.3856	1.51970
$639$	0.389078	0.0153917
$640$	0	0
$641$	21.8184	0.861775	0.430887	−	0.902406i	$-0.358201\pi$
$641$	21.8184	0.861775	0.430887	+	0.902406i	$0.358201\pi$
$642$	−17.7405	−0.700163
$643$	45.6763	1.80130	0.900649	−	0.434547i	$-0.143091\pi$
$643$	45.6763	1.80130	0.900649	+	0.434547i	$0.143091\pi$
$644$	28.1075	1.10759
$645$	0	0
$646$	−0.275244	−0.0108293
$647$	−6.32643	−0.248717	−0.124359	−	0.992237i	$-0.539687\pi$
$647$	−6.32643	−0.248717	−0.124359	+	0.992237i	$0.539687\pi$
$648$	−4.98504	−0.195831
$649$	28.0466	1.10093
$650$	0	0
$651$	20.8406	0.816807
$652$	20.9835	0.821778
$653$	43.8009	1.71406	0.857030	−	0.515266i	$-0.172307\pi$
$653$	43.8009	1.71406	0.857030	+	0.515266i	$0.172307\pi$
$654$	−16.5166	−0.645850
$655$	0	0
$656$	−2.78567	−0.108762
$657$	2.45425	0.0957496
$658$	17.9837	0.701077
$659$	−12.4255	−0.484027	−0.242014	−	0.970273i	$-0.577808\pi$
$659$	−12.4255	−0.484027	−0.242014	+	0.970273i	$0.577808\pi$
$660$	0	0
$661$	12.5408	0.487783	0.243891	−	0.969803i	$-0.421576\pi$
$661$	12.5408	0.487783	0.243891	+	0.969803i	$0.421576\pi$
$662$	33.6462	1.30770
$663$	−0.235223	−0.00913531
$664$	3.65619	0.141888
$665$	0	0
$666$	1.08042	0.0418655
$667$	−33.1704	−1.28436
$668$	26.7844	1.03632
$669$	31.1332	1.20368
$670$	0	0
$671$	−4.35896	−0.168276
$672$	−32.8574	−1.26750
$673$	−18.5386	−0.714612	−0.357306	−	0.933987i	$-0.616305\pi$
$673$	−18.5386	−0.714612	−0.357306	+	0.933987i	$0.616305\pi$
$674$	−19.1478	−0.737545
$675$	0	0
$676$	−18.0351	−0.693659
$677$	−4.90600	−0.188553	−0.0942765	−	0.995546i	$-0.530054\pi$
$677$	−4.90600	−0.188553	−0.0942765	+	0.995546i	$0.530054\pi$
$678$	20.6924	0.794689
$679$	−20.7425	−0.796026
$680$	0	0
$681$	4.88206	0.187081
$682$	−41.1056	−1.57402
$683$	15.4975	0.592997	0.296498	−	0.955033i	$-0.404181\pi$
$683$	15.4975	0.592997	0.296498	+	0.955033i	$0.404181\pi$
$684$	3.10658	0.118783
$685$	0	0
$686$	−39.1120	−1.49330
$687$	6.61887	0.252525
$688$	−18.7964	−0.716605
$689$	−10.8910	−0.414913
$690$	0	0
$691$	21.7098	0.825879	0.412939	−	0.910759i	$-0.364502\pi$
$691$	21.7098	0.825879	0.412939	+	0.910759i	$0.364502\pi$
$692$	10.4978	0.399065
$693$	8.45065	0.321014
$694$	−34.2742	−1.30103
$695$	0	0
$696$	4.09641	0.155274
$697$	0.0451644	0.00171072
$698$	−58.0135	−2.19584
$699$	−22.0762	−0.834998
$700$	0	0
$701$	40.7810	1.54028	0.770139	−	0.637876i	$-0.220187\pi$
$701$	40.7810	1.54028	0.770139	+	0.637876i	$0.220187\pi$
$702$	−13.4529	−0.507748
$703$	−1.24116	−0.0468113
$704$	26.2443	0.989120
$705$	0	0
$706$	5.68661	0.214019
$707$	2.17794	0.0819098
$708$	−22.3456	−0.839800
$709$	43.3369	1.62755	0.813776	−	0.581179i	$-0.197408\pi$
$709$	43.3369	1.62755	0.813776	+	0.581179i	$0.197408\pi$
$710$	0	0
$711$	0.0156478	0.000586837 0
$712$	−1.64470	−0.0616377
$713$	35.5209	1.33027
$714$	0.596538	0.0223249
$715$	0	0
$716$	32.5090	1.21492
$717$	0.0755251	0.00282054
$718$	60.2202	2.24740
$719$	14.6186	0.545183	0.272591	−	0.962130i	$-0.412119\pi$
$719$	14.6186	0.545183	0.272591	+	0.962130i	$0.412119\pi$
$720$	0	0
$721$	−2.09939	−0.0781852
$722$	29.2452	1.08839
$723$	1.97205	0.0733415
$724$	−37.9290	−1.40962
$725$	0	0
$726$	−30.8309	−1.14424
$727$	−41.8049	−1.55046	−0.775230	−	0.631679i	$-0.782366\pi$
$727$	−41.8049	−1.55046	−0.775230	+	0.631679i	$0.782366\pi$
$728$	1.66233	0.0616100
$729$	14.9598	0.554065
$730$	0	0
$731$	0.304748	0.0112715
$732$	3.47292	0.128363
$733$	−49.6357	−1.83333	−0.916667	−	0.399651i	$-0.869131\pi$
$733$	−49.6357	−1.83333	−0.916667	+	0.399651i	$0.869131\pi$
$734$	21.7676	0.803456
$735$	0	0
$736$	−56.0024	−2.06428
$737$	−18.8383	−0.693919
$738$	−1.08779	−0.0400419
$739$	42.6329	1.56828	0.784138	−	0.620586i	$-0.213105\pi$
$739$	42.6329	1.56828	0.784138	+	0.620586i	$0.213105\pi$
$740$	0	0
$741$	−6.50821	−0.239085
$742$	27.6201	1.01397
$743$	45.9469	1.68563	0.842813	−	0.538206i	$-0.180898\pi$
$743$	45.9469	1.68563	0.842813	+	0.538206i	$0.180898\pi$
$744$	−4.38669	−0.160824
$745$	0	0
$746$	58.7031	2.14927
$747$	7.09178	0.259475
$748$	−0.551374	−0.0201602
$749$	−10.0968	−0.368929
$750$	0	0
$751$	0.779037	0.0284275	0.0142137	−	0.999899i	$-0.495475\pi$
$751$	0.779037	0.0284275	0.0142137	+	0.999899i	$0.495475\pi$
$752$	−18.8026	−0.685660
$753$	−4.80175	−0.174985
$754$	14.6461	0.533379
$755$	0	0
$756$	15.9879	0.581475
$757$	26.6711	0.969377	0.484689	−	0.874687i	$-0.338933\pi$
$757$	26.6711	0.969377	0.484689	+	0.874687i	$0.338933\pi$
$758$	−26.9035	−0.977179
$759$	63.0089	2.28708
$760$	0	0
$761$	−18.0268	−0.653471	−0.326736	−	0.945116i	$-0.605949\pi$
$761$	−18.0268	−0.653471	−0.326736	+	0.945116i	$0.605949\pi$
$762$	−65.6186	−2.37711
$763$	−9.40020	−0.340310
$764$	13.9513	0.504742
$765$	0	0
$766$	−28.3817	−1.02547
$767$	10.7012	0.386399
$768$	37.6404	1.35823
$769$	47.4868	1.71242	0.856210	−	0.516628i	$-0.172813\pi$
$769$	47.4868	1.71242	0.856210	+	0.516628i	$0.172813\pi$
$770$	0	0
$771$	33.2481	1.19740
$772$	−21.4931	−0.773555
$773$	−13.5834	−0.488560	−0.244280	−	0.969705i	$-0.578552\pi$
$773$	−13.5834	−0.488560	−0.244280	+	0.969705i	$0.578552\pi$
$774$	−7.33987	−0.263826
$775$	0	0
$776$	4.36605	0.156732
$777$	2.68998	0.0965024
$778$	−44.5350	−1.59666
$779$	1.24962	0.0447723
$780$	0	0
$781$	−1.91066	−0.0683686
$782$	1.01674	0.0363587
$783$	−18.8678	−0.674279
$784$	9.97621	0.356293
$785$	0	0
$786$	−2.13840	−0.0762744
$787$	−7.63137	−0.272029	−0.136015	−	0.990707i	$-0.543429\pi$
$787$	−7.63137	−0.272029	−0.136015	+	0.990707i	$0.543429\pi$
$788$	7.15643	0.254937
$789$	54.7192	1.94805
$790$	0	0
$791$	11.7768	0.418736
$792$	−1.77876	−0.0632054
$793$	−1.66317	−0.0590609
$794$	31.1979	1.10717
$795$	0	0
$796$	15.8719	0.562564
$797$	−8.53983	−0.302496	−0.151248	−	0.988496i	$-0.548329\pi$
$797$	−8.53983	−0.302496	−0.151248	+	0.988496i	$0.548329\pi$
$798$	16.5052	0.584277
$799$	0.304849	0.0107848
$800$	0	0
$801$	−3.19016	−0.112719
$802$	6.38730	0.225543
$803$	−12.0522	−0.425312
$804$	15.0091	0.529330
$805$	0	0
$806$	−15.6839	−0.552443
$807$	54.2227	1.90873
$808$	−0.458429	−0.0161275
$809$	16.9918	0.597401	0.298701	−	0.954347i	$-0.403447\pi$
$809$	16.9918	0.597401	0.298701	+	0.954347i	$0.403447\pi$
$810$	0	0
$811$	17.8223	0.625824	0.312912	−	0.949782i	$-0.398695\pi$
$811$	17.8223	0.625824	0.312912	+	0.949782i	$0.398695\pi$
$812$	−17.4059	−0.610828
$813$	−9.66035	−0.338803
$814$	−5.30566	−0.185963
$815$	0	0
$816$	−0.623702	−0.0218339
$817$	8.43185	0.294993
$818$	57.1372	1.99776
$819$	3.22436	0.112668
$820$	0	0
$821$	3.07237	0.107226	0.0536132	−	0.998562i	$-0.482926\pi$
$821$	3.07237	0.107226	0.0536132	+	0.998562i	$0.482926\pi$
$822$	−47.9630	−1.67290
$823$	−4.41018	−0.153729	−0.0768645	−	0.997042i	$-0.524491\pi$
$823$	−4.41018	−0.153729	−0.0768645	+	0.997042i	$0.524491\pi$
$824$	0.441895	0.0153941
$825$	0	0
$826$	−27.1389	−0.944283
$827$	28.3646	0.986335	0.493167	−	0.869934i	$-0.335839\pi$
$827$	28.3646	0.986335	0.493167	+	0.869934i	$0.335839\pi$
$828$	−11.4756	−0.398805
$829$	4.30568	0.149542	0.0747712	−	0.997201i	$-0.476177\pi$
$829$	4.30568	0.149542	0.0747712	+	0.997201i	$0.476177\pi$
$830$	0	0
$831$	−52.0334	−1.80502
$832$	10.0136	0.347158
$833$	−0.161746	−0.00560416
$834$	7.21439	0.249814
$835$	0	0
$836$	−15.2556	−0.527624
$837$	20.2047	0.698378
$838$	52.6160	1.81759
$839$	21.1295	0.729471	0.364736	−	0.931111i	$-0.381159\pi$
$839$	21.1295	0.729471	0.364736	+	0.931111i	$0.381159\pi$
$840$	0	0
$841$	−8.45884	−0.291684
$842$	−22.1640	−0.763821
$843$	−18.6719	−0.643094
$844$	24.4187	0.840526
$845$	0	0
$846$	−7.34230	−0.252433
$847$	−17.5470	−0.602922
$848$	−28.8778	−0.991669
$849$	9.48484	0.325519
$850$	0	0
$851$	4.58482	0.157166
$852$	1.52228	0.0521525
$853$	−3.80199	−0.130178	−0.0650889	−	0.997879i	$-0.520733\pi$
$853$	−3.80199	−0.130178	−0.0650889	+	0.997879i	$0.520733\pi$
$854$	4.21788	0.144333
$855$	0	0
$856$	2.12525	0.0726395
$857$	−7.91925	−0.270516	−0.135258	−	0.990810i	$-0.543186\pi$
$857$	−7.91925	−0.270516	−0.135258	+	0.990810i	$0.543186\pi$
$858$	−27.8210	−0.949793
$859$	−16.5085	−0.563262	−0.281631	−	0.959523i	$-0.590875\pi$
$859$	−16.5085	−0.563262	−0.281631	+	0.959523i	$0.590875\pi$
$860$	0	0
$861$	−2.70831	−0.0922989
$862$	−0.540711	−0.0184167
$863$	35.3271	1.20255	0.601274	−	0.799043i	$-0.294660\pi$
$863$	35.3271	1.20255	0.601274	+	0.799043i	$0.294660\pi$
$864$	−31.8549	−1.08373
$865$	0	0
$866$	38.0505	1.29301
$867$	−33.5148	−1.13822
$868$	18.6393	0.632659
$869$	−0.0768419	−0.00260668
$870$	0	0
$871$	−7.18781	−0.243549
$872$	1.97863	0.0670047
$873$	8.46867	0.286621
$874$	28.1316	0.951564
$875$	0	0
$876$	9.60235	0.324433
$877$	15.4911	0.523098	0.261549	−	0.965190i	$-0.415767\pi$
$877$	15.4911	0.523098	0.261549	+	0.965190i	$0.415767\pi$
$878$	−67.5931	−2.28116
$879$	−10.2302	−0.345057
$880$	0	0
$881$	28.1913	0.949791	0.474895	−	0.880042i	$-0.342486\pi$
$881$	28.1913	0.949791	0.474895	+	0.880042i	$0.342486\pi$
$882$	3.89565	0.131173
$883$	35.7208	1.20210	0.601050	−	0.799212i	$-0.294749\pi$
$883$	35.7208	1.20210	0.601050	+	0.799212i	$0.294749\pi$
$884$	−0.210378	−0.00707576
$885$	0	0
$886$	−30.7064	−1.03160
$887$	31.5038	1.05780	0.528898	−	0.848686i	$-0.322606\pi$
$887$	31.5038	1.05780	0.528898	+	0.848686i	$0.322606\pi$
$888$	−0.566207	−0.0190007
$889$	−37.3459	−1.25254
$890$	0	0
$891$	47.4833	1.59075
$892$	27.8447	0.932311
$893$	8.43464	0.282255
$894$	11.0764	0.370451
$895$	0	0
$896$	7.92808	0.264859
$897$	24.0412	0.802711
$898$	55.5879	1.85499
$899$	−21.9967	−0.733632
$900$	0	0
$901$	0.468200	0.0155980
$902$	5.34182	0.177863
$903$	−18.2744	−0.608134
$904$	−2.47888	−0.0824462
$905$	0	0
$906$	7.71936	0.256459
$907$	−26.2739	−0.872410	−0.436205	−	0.899847i	$-0.643678\pi$
$907$	−26.2739	−0.872410	−0.436205	+	0.899847i	$0.643678\pi$
$908$	4.36639	0.144904
$909$	−0.889199	−0.0294929
$910$	0	0
$911$	31.3870	1.03990	0.519949	−	0.854198i	$-0.325951\pi$
$911$	31.3870	1.03990	0.519949	+	0.854198i	$0.325951\pi$
$912$	−17.2568	−0.571428
$913$	−34.8258	−1.15256
$914$	−2.13436	−0.0705983
$915$	0	0
$916$	5.91975	0.195594
$917$	−1.21704	−0.0401903
$918$	0.578338	0.0190880
$919$	−20.4383	−0.674198	−0.337099	−	0.941469i	$-0.609446\pi$
$919$	−20.4383	−0.674198	−0.337099	+	0.941469i	$0.609446\pi$
$920$	0	0
$921$	−9.56417	−0.315150
$922$	29.6285	0.975763
$923$	−0.729014	−0.0239958
$924$	33.0634	1.08771
$925$	0	0
$926$	−23.6609	−0.777544
$927$	0.857128	0.0281518
$928$	34.6801	1.13843
$929$	50.3370	1.65150	0.825751	−	0.564034i	$-0.190751\pi$
$929$	50.3370	1.65150	0.825751	+	0.564034i	$0.190751\pi$
$930$	0	0
$931$	−4.47522	−0.146670
$932$	−19.7444	−0.646749
$933$	−9.10908	−0.298218
$934$	−27.6191	−0.903725
$935$	0	0
$936$	−0.678688	−0.0221836
$937$	−26.1648	−0.854766	−0.427383	−	0.904071i	$-0.640564\pi$
$937$	−26.1648	−0.854766	−0.427383	+	0.904071i	$0.640564\pi$
$938$	18.2287	0.595187
$939$	−12.9931	−0.424013
$940$	0	0
$941$	60.8568	1.98388	0.991938	−	0.126724i	$-0.0404462\pi$
$941$	60.8568	1.98388	0.991938	+	0.126724i	$0.0404462\pi$
$942$	10.6275	0.346264
$943$	−4.61606	−0.150320
$944$	28.3747	0.923517
$945$	0	0
$946$	36.0441	1.17189
$947$	0.532319	0.0172981	0.00864903	−	0.999963i	$-0.497247\pi$
$947$	0.532319	0.0172981	0.00864903	+	0.999963i	$0.497247\pi$
$948$	0.0612224	0.00198841
$949$	−4.59853	−0.149274
$950$	0	0
$951$	27.9826	0.907399
$952$	−0.0714630	−0.00231613
$953$	−13.6973	−0.443700	−0.221850	−	0.975081i	$-0.571209\pi$
$953$	−13.6973	−0.443700	−0.221850	+	0.975081i	$0.571209\pi$
$954$	−11.2766	−0.365094
$955$	0	0
$956$	0.0675477	0.00218465
$957$	−39.0190	−1.26130
$958$	−11.1085	−0.358899
$959$	−27.2975	−0.881483
$960$	0	0
$961$	−7.44457	−0.240147
$962$	−2.02438	−0.0652688
$963$	4.12227	0.132838
$964$	1.76376	0.0568067
$965$	0	0
$966$	−60.9696	−1.96167
$967$	18.8849	0.607296	0.303648	−	0.952784i	$-0.401795\pi$
$967$	18.8849	0.607296	0.303648	+	0.952784i	$0.401795\pi$
$968$	3.69342	0.118711
$969$	0.279786	0.00898803
$970$	0	0
$971$	−5.21718	−0.167427	−0.0837136	−	0.996490i	$-0.526678\pi$
$971$	−5.21718	−0.167427	−0.0837136	+	0.996490i	$0.526678\pi$
$972$	−15.8038	−0.506908
$973$	4.10598	0.131632
$974$	6.55821	0.210139
$975$	0	0
$976$	−4.40995	−0.141159
$977$	52.5890	1.68247	0.841236	−	0.540668i	$-0.181829\pi$
$977$	52.5890	1.68247	0.841236	+	0.540668i	$0.181829\pi$
$978$	−45.5165	−1.45546
$979$	15.6660	0.500688
$980$	0	0
$981$	3.83787	0.122534
$982$	6.40396	0.204359
$983$	−22.7553	−0.725782	−0.362891	−	0.931832i	$-0.618210\pi$
$983$	−22.7553	−0.725782	−0.362891	+	0.931832i	$0.618210\pi$
$984$	0.570065	0.0181730
$985$	0	0
$986$	−0.629631	−0.0200515
$987$	−18.2804	−0.581873
$988$	−5.82078	−0.185184
$989$	−31.1470	−0.990418
$990$	0	0
$991$	−19.3263	−0.613921	−0.306961	−	0.951722i	$-0.599312\pi$
$991$	−19.3263	−0.613921	−0.306961	+	0.951722i	$0.599312\pi$
$992$	−37.1376	−1.17912
$993$	−34.2014	−1.08535
$994$	1.84882	0.0586410
$995$	0	0
$996$	27.7468	0.879191
$997$	−2.47505	−0.0783856	−0.0391928	−	0.999232i	$-0.512479\pi$
$997$	−2.47505	−0.0783856	−0.0391928	+	0.999232i	$0.512479\pi$
$998$	−80.6320	−2.55236
$999$	2.60791	0.0825105

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.n.1.6	yes	40
5.4	even	2		6025.2.a.m.1.35	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.m.1.35	✓	40	5.4	even	2
6025.2.a.n.1.6	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)

\(f(q)\)	\(=\)	\(q-1.94004 q^{2} +1.97205 q^{3} +1.76376 q^{4} -3.82586 q^{6} -2.17744 q^{7} +0.458324 q^{8} +0.888995 q^{9} +O(q^{10})\)	\(q-1.94004 q^{2} +1.97205 q^{3} +1.76376 q^{4} -3.82586 q^{6} -2.17744 q^{7} +0.458324 q^{8} +0.888995 q^{9} -4.36561 q^{11} +3.47822 q^{12} -1.66571 q^{13} +4.22432 q^{14} -4.41668 q^{16} +0.0716082 q^{17} -1.72469 q^{18} +1.98127 q^{19} -4.29403 q^{21} +8.46946 q^{22} -7.31877 q^{23} +0.903839 q^{24} +3.23154 q^{26} -4.16301 q^{27} -3.84047 q^{28} +4.53224 q^{29} -4.85339 q^{31} +7.65188 q^{32} -8.60922 q^{33} -0.138923 q^{34} +1.56797 q^{36} -0.626446 q^{37} -3.84375 q^{38} -3.28486 q^{39} +0.630715 q^{41} +8.33058 q^{42} +4.25577 q^{43} -7.69987 q^{44} +14.1987 q^{46} +4.25718 q^{47} -8.70992 q^{48} -2.25876 q^{49} +0.141215 q^{51} -2.93790 q^{52} +6.53836 q^{53} +8.07642 q^{54} -0.997972 q^{56} +3.90718 q^{57} -8.79272 q^{58} -6.42444 q^{59} +0.998477 q^{61} +9.41578 q^{62} -1.93573 q^{63} -6.01161 q^{64} +16.7022 q^{66} +4.31517 q^{67} +0.126299 q^{68} -14.4330 q^{69} +0.437661 q^{71} +0.407448 q^{72} +2.76071 q^{73} +1.21533 q^{74} +3.49448 q^{76} +9.50585 q^{77} +6.37276 q^{78} +0.0176016 q^{79} -10.8767 q^{81} -1.22361 q^{82} +7.97730 q^{83} -7.57361 q^{84} -8.25637 q^{86} +8.93781 q^{87} -2.00086 q^{88} -3.58851 q^{89} +3.62697 q^{91} -12.9085 q^{92} -9.57115 q^{93} -8.25910 q^{94} +15.0899 q^{96} +9.52612 q^{97} +4.38208 q^{98} -3.88100 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 9 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 20 q^{7} + 27 q^{8} + 38 q^{9}+O(q^{10}) \) 40 * q + 9 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 20 * q^7 + 27 * q^8 + 38 * q^9	\( 40 q + 9 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 20 q^{7} + 27 q^{8} + 38 q^{9} + q^{11} + 26 q^{12} + 11 q^{13} - q^{14} + 43 q^{16} + 20 q^{17} + 18 q^{18} + 2 q^{21} + 23 q^{22} + 79 q^{23} - 2 q^{24} + 2 q^{26} + 26 q^{27} + 30 q^{28} + 2 q^{29} + q^{31} + 68 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 16 q^{37} + 45 q^{38} - 2 q^{39} - 2 q^{41} + 19 q^{42} + 25 q^{43} + 3 q^{44} + 14 q^{46} + 88 q^{47} + 75 q^{48} + 40 q^{49} - 10 q^{51} + 18 q^{52} + 34 q^{53} + 4 q^{54} - 15 q^{56} + 51 q^{57} + 53 q^{58} + q^{59} + 9 q^{61} + 39 q^{62} + 110 q^{63} + 17 q^{64} + 26 q^{66} + 30 q^{67} + 44 q^{68} - 7 q^{69} + 5 q^{71} + 18 q^{72} + 23 q^{73} - 18 q^{74} + 43 q^{76} + 30 q^{77} + 46 q^{78} + 5 q^{79} + 44 q^{81} + 5 q^{82} + 65 q^{83} - 65 q^{84} + 40 q^{86} + 33 q^{87} + 71 q^{88} - 9 q^{89} + q^{91} + 117 q^{92} + 68 q^{93} - 72 q^{94} + 83 q^{96} - 8 q^{97} + 76 q^{98} - 40 q^{99}+O(q^{100}) \) 40 * q + 9 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 20 * q^7 + 27 * q^8 + 38 * q^9 + q^11 + 26 * q^12 + 11 * q^13 - q^14 + 43 * q^16 + 20 * q^17 + 18 * q^18 + 2 * q^21 + 23 * q^22 + 79 * q^23 - 2 * q^24 + 2 * q^26 + 26 * q^27 + 30 * q^28 + 2 * q^29 + q^31 + 68 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 16 * q^37 + 45 * q^38 - 2 * q^39 - 2 * q^41 + 19 * q^42 + 25 * q^43 + 3 * q^44 + 14 * q^46 + 88 * q^47 + 75 * q^48 + 40 * q^49 - 10 * q^51 + 18 * q^52 + 34 * q^53 + 4 * q^54 - 15 * q^56 + 51 * q^57 + 53 * q^58 + q^59 + 9 * q^61 + 39 * q^62 + 110 * q^63 + 17 * q^64 + 26 * q^66 + 30 * q^67 + 44 * q^68 - 7 * q^69 + 5 * q^71 + 18 * q^72 + 23 * q^73 - 18 * q^74 + 43 * q^76 + 30 * q^77 + 46 * q^78 + 5 * q^79 + 44 * q^81 + 5 * q^82 + 65 * q^83 - 65 * q^84 + 40 * q^86 + 33 * q^87 + 71 * q^88 - 9 * q^89 + q^91 + 117 * q^92 + 68 * q^93 - 72 * q^94 + 83 * q^96 - 8 * q^97 + 76 * q^98 - 40 * q^99

Introduction

L-functions

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Database

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