LMFDB - Embedded newform 6025.2.a.q.1.4

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:	$66$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q-2.60935 q^{2} +2.94245 q^{3} +4.80870 q^{4} -7.67788 q^{6} +2.28979 q^{7} -7.32887 q^{8} +5.65803 q^{9} +O(q^{10})$	\(q-2.60935 q^{2} +2.94245 q^{3} +4.80870 q^{4} -7.67788 q^{6} +2.28979 q^{7} -7.32887 q^{8} +5.65803 q^{9} +3.96558 q^{11} +14.1494 q^{12} +0.894019 q^{13} -5.97485 q^{14} +9.50617 q^{16} +2.17979 q^{17} -14.7638 q^{18} +1.92680 q^{19} +6.73759 q^{21} -10.3476 q^{22} -1.87669 q^{23} -21.5648 q^{24} -2.33281 q^{26} +7.82113 q^{27} +11.0109 q^{28} +8.71597 q^{29} -2.56862 q^{31} -10.1472 q^{32} +11.6685 q^{33} -5.68783 q^{34} +27.2078 q^{36} -5.21854 q^{37} -5.02768 q^{38} +2.63061 q^{39} -11.8551 q^{41} -17.5807 q^{42} -4.07586 q^{43} +19.0693 q^{44} +4.89694 q^{46} +11.8800 q^{47} +27.9715 q^{48} -1.75688 q^{49} +6.41393 q^{51} +4.29907 q^{52} +2.60774 q^{53} -20.4081 q^{54} -16.7815 q^{56} +5.66951 q^{57} -22.7430 q^{58} +14.2748 q^{59} +4.37731 q^{61} +6.70241 q^{62} +12.9557 q^{63} +7.46516 q^{64} -30.4473 q^{66} -6.37203 q^{67} +10.4820 q^{68} -5.52208 q^{69} +6.53115 q^{71} -41.4670 q^{72} +7.54616 q^{73} +13.6170 q^{74} +9.26538 q^{76} +9.08033 q^{77} -6.86417 q^{78} -1.39473 q^{79} +6.03922 q^{81} +30.9342 q^{82} -4.31151 q^{83} +32.3990 q^{84} +10.6353 q^{86} +25.6463 q^{87} -29.0632 q^{88} +17.1100 q^{89} +2.04711 q^{91} -9.02444 q^{92} -7.55803 q^{93} -30.9992 q^{94} -29.8576 q^{96} -17.5833 q^{97} +4.58432 q^{98} +22.4374 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9}+O(q^{10}) $ 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9	$ 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9} + 48 q^{11} + 30 q^{14} + 98 q^{16} + 12 q^{19} + 18 q^{21} + 42 q^{24} + 48 q^{26} + 56 q^{29} + 48 q^{31} + 8 q^{34} + 158 q^{36} + 84 q^{39} + 56 q^{41} + 144 q^{44} + 36 q^{46} + 98 q^{49} + 44 q^{51} + 86 q^{54} + 104 q^{56} + 108 q^{59} + 22 q^{61} + 136 q^{64} + 74 q^{66} + 20 q^{69} + 212 q^{71} + 84 q^{74} + 6 q^{76} + 66 q^{79} + 162 q^{81} - 52 q^{84} + 100 q^{86} + 54 q^{89} + 72 q^{91} - 96 q^{94} + 122 q^{96} + 112 q^{99}+O(q^{100}) $ 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9 + 48 * q^11 + 30 * q^14 + 98 * q^16 + 12 * q^19 + 18 * q^21 + 42 * q^24 + 48 * q^26 + 56 * q^29 + 48 * q^31 + 8 * q^34 + 158 * q^36 + 84 * q^39 + 56 * q^41 + 144 * q^44 + 36 * q^46 + 98 * q^49 + 44 * q^51 + 86 * q^54 + 104 * q^56 + 108 * q^59 + 22 * q^61 + 136 * q^64 + 74 * q^66 + 20 * q^69 + 212 * q^71 + 84 * q^74 + 6 * q^76 + 66 * q^79 + 162 * q^81 - 52 * q^84 + 100 * q^86 + 54 * q^89 + 72 * q^91 - 96 * q^94 + 122 * q^96 + 112 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.60935	−1.84509	−0.922544	−	0.385892i	$-0.873894\pi$
$2$	−2.60935	−1.84509	−0.922544	+	0.385892i	$0.873894\pi$
$3$	2.94245	1.69883	0.849413	−	0.527729i	$-0.176956\pi$
$3$	2.94245	1.69883	0.849413	+	0.527729i	$0.176956\pi$
$4$	4.80870	2.40435
$5$	0	0
$5$	0	0
$6$	−7.67788	−3.13448
$7$	2.28979	0.865458	0.432729	−	0.901524i	$-0.357551\pi$
$7$	2.28979	0.865458	0.432729	+	0.901524i	$0.357551\pi$
$8$	−7.32887	−2.59115
$9$	5.65803	1.88601
$10$	0	0
$11$	3.96558	1.19567	0.597834	−	0.801620i	$-0.296028\pi$
$11$	3.96558	1.19567	0.597834	+	0.801620i	$0.296028\pi$
$12$	14.1494	4.08457
$13$	0.894019	0.247956	0.123978	−	0.992285i	$-0.460435\pi$
$13$	0.894019	0.247956	0.123978	+	0.992285i	$0.460435\pi$
$14$	−5.97485	−1.59685
$15$	0	0
$16$	9.50617	2.37654
$17$	2.17979	0.528677	0.264338	−	0.964430i	$-0.414846\pi$
$17$	2.17979	0.528677	0.264338	+	0.964430i	$0.414846\pi$
$18$	−14.7638	−3.47985
$19$	1.92680	0.442037	0.221019	−	0.975270i	$-0.429062\pi$
$19$	1.92680	0.442037	0.221019	+	0.975270i	$0.429062\pi$
$20$	0	0
$21$	6.73759	1.47026
$22$	−10.3476	−2.20611
$23$	−1.87669	−0.391317	−0.195659	−	0.980672i	$-0.562684\pi$
$23$	−1.87669	−0.391317	−0.195659	+	0.980672i	$0.562684\pi$
$24$	−21.5648	−4.40191
$25$	0	0
$26$	−2.33281	−0.457501
$27$	7.82113	1.50518
$28$	11.0109	2.08086
$29$	8.71597	1.61852	0.809258	−	0.587454i	$-0.199870\pi$
$29$	8.71597	1.61852	0.809258	+	0.587454i	$0.199870\pi$
$30$	0	0
$31$	−2.56862	−0.461337	−0.230669	−	0.973032i	$-0.574091\pi$
$31$	−2.56862	−0.461337	−0.230669	+	0.973032i	$0.574091\pi$
$32$	−10.1472	−1.79378
$33$	11.6685	2.03123
$34$	−5.68783	−0.975455
$35$	0	0
$36$	27.2078	4.53463
$37$	−5.21854	−0.857923	−0.428961	−	0.903323i	$-0.641120\pi$
$37$	−5.21854	−0.857923	−0.428961	+	0.903323i	$0.641120\pi$
$38$	−5.02768	−0.815598
$39$	2.63061	0.421234
$40$	0	0
$41$	−11.8551	−1.85146	−0.925731	−	0.378183i	$-0.876549\pi$
$41$	−11.8551	−1.85146	−0.925731	+	0.378183i	$0.876549\pi$
$42$	−17.5807	−2.71276
$43$	−4.07586	−0.621563	−0.310781	−	0.950481i	$-0.600591\pi$
$43$	−4.07586	−0.621563	−0.310781	+	0.950481i	$0.600591\pi$
$44$	19.0693	2.87480
$45$	0	0
$46$	4.89694	0.722015
$47$	11.8800	1.73288	0.866441	−	0.499279i	$-0.166402\pi$
$47$	11.8800	1.73288	0.866441	+	0.499279i	$0.166402\pi$
$48$	27.9715	4.03733
$49$	−1.75688	−0.250983
$50$	0	0
$51$	6.41393	0.898130
$52$	4.29907	0.596173
$53$	2.60774	0.358201	0.179100	−	0.983831i	$-0.442681\pi$
$53$	2.60774	0.358201	0.179100	+	0.983831i	$0.442681\pi$
$54$	−20.4081	−2.77718
$55$	0	0
$56$	−16.7815	−2.24253
$57$	5.66951	0.750945
$58$	−22.7430	−2.98630
$59$	14.2748	1.85842	0.929211	−	0.369549i	$-0.120488\pi$
$59$	14.2748	1.85842	0.929211	+	0.369549i	$0.120488\pi$
$60$	0	0
$61$	4.37731	0.560457	0.280229	−	0.959933i	$-0.409590\pi$
$61$	4.37731	0.560457	0.280229	+	0.959933i	$0.409590\pi$
$62$	6.70241	0.851207
$63$	12.9557	1.63226
$64$	7.46516	0.933145
$65$	0	0
$66$	−30.4473	−3.74780
$67$	−6.37203	−0.778467	−0.389234	−	0.921139i	$-0.627260\pi$
$67$	−6.37203	−0.778467	−0.389234	+	0.921139i	$0.627260\pi$
$68$	10.4820	1.27112
$69$	−5.52208	−0.664780
$70$	0	0
$71$	6.53115	0.775104	0.387552	−	0.921848i	$-0.373321\pi$
$71$	6.53115	0.775104	0.387552	+	0.921848i	$0.373321\pi$
$72$	−41.4670	−4.88693
$73$	7.54616	0.883212	0.441606	−	0.897209i	$-0.354409\pi$
$73$	7.54616	0.883212	0.441606	+	0.897209i	$0.354409\pi$
$74$	13.6170	1.58294
$75$	0	0
$76$	9.26538	1.06281
$77$	9.08033	1.03480
$78$	−6.86417	−0.777215
$79$	−1.39473	−0.156919	−0.0784595	−	0.996917i	$-0.525000\pi$
$79$	−1.39473	−0.156919	−0.0784595	+	0.996917i	$0.525000\pi$
$80$	0	0
$81$	6.03922	0.671024
$82$	30.9342	3.41611
$83$	−4.31151	−0.473249	−0.236625	−	0.971601i	$-0.576041\pi$
$83$	−4.31151	−0.473249	−0.236625	+	0.971601i	$0.576041\pi$
$84$	32.3990	3.53502
$85$	0	0
$86$	10.6353	1.14684
$87$	25.6463	2.74958
$88$	−29.0632	−3.09815
$89$	17.1100	1.81366	0.906831	−	0.421495i	$-0.138495\pi$
$89$	17.1100	1.81366	0.906831	+	0.421495i	$0.138495\pi$
$90$	0	0
$91$	2.04711	0.214596
$92$	−9.02444	−0.940863
$93$	−7.55803	−0.783732
$94$	−30.9992	−3.19732
$95$	0	0
$96$	−29.8576	−3.04733
$97$	−17.5833	−1.78531	−0.892657	−	0.450737i	$-0.851161\pi$
$97$	−17.5833	−1.78531	−0.892657	+	0.450737i	$0.851161\pi$
$98$	4.58432	0.463086
$99$	22.4374	2.25504
$100$	0	0
$101$	4.83052	0.480655	0.240327	−	0.970692i	$-0.422745\pi$
$101$	4.83052	0.480655	0.240327	+	0.970692i	$0.422745\pi$
$102$	−16.7362	−1.65713
$103$	−17.1699	−1.69180	−0.845902	−	0.533338i	$-0.820937\pi$
$103$	−17.1699	−1.69180	−0.845902	+	0.533338i	$0.820937\pi$
$104$	−6.55214	−0.642491
$105$	0	0
$106$	−6.80450	−0.660912
$107$	−13.7849	−1.33264	−0.666319	−	0.745667i	$-0.732131\pi$
$107$	−13.7849	−1.33264	−0.666319	+	0.745667i	$0.732131\pi$
$108$	37.6094	3.61897
$109$	12.5248	1.19966	0.599829	−	0.800128i	$-0.295235\pi$
$109$	12.5248	1.19966	0.599829	+	0.800128i	$0.295235\pi$
$110$	0	0
$111$	−15.3553	−1.45746
$112$	21.7671	2.05680
$113$	−12.1095	−1.13917	−0.569585	−	0.821932i	$-0.692896\pi$
$113$	−12.1095	−1.13917	−0.569585	+	0.821932i	$0.692896\pi$
$114$	−14.7937	−1.38556
$115$	0	0
$116$	41.9125	3.89147
$117$	5.05839	0.467648
$118$	−37.2480	−3.42895
$119$	4.99125	0.457547
$120$	0	0
$121$	4.72583	0.429621
$122$	−11.4219	−1.03409
$123$	−34.8832	−3.14531
$124$	−12.3517	−1.10922
$125$	0	0
$126$	−33.8059	−3.01167
$127$	−3.92223	−0.348042	−0.174021	−	0.984742i	$-0.555676\pi$
$127$	−3.92223	−0.348042	−0.174021	+	0.984742i	$0.555676\pi$
$128$	0.815147	0.0720495
$129$	−11.9930	−1.05593
$130$	0	0
$131$	5.85377	0.511446	0.255723	−	0.966750i	$-0.417686\pi$
$131$	5.85377	0.511446	0.255723	+	0.966750i	$0.417686\pi$
$132$	56.1104	4.88379
$133$	4.41195	0.382565
$134$	16.6268	1.43634
$135$	0	0
$136$	−15.9754	−1.36988
$137$	12.1141	1.03498	0.517490	−	0.855689i	$-0.326867\pi$
$137$	12.1141	1.03498	0.517490	+	0.855689i	$0.326867\pi$
$138$	14.4090	1.22658
$139$	−1.20559	−0.102257	−0.0511285	−	0.998692i	$-0.516282\pi$
$139$	−1.20559	−0.102257	−0.0511285	+	0.998692i	$0.516282\pi$
$140$	0	0
$141$	34.9565	2.94387
$142$	−17.0420	−1.43014
$143$	3.54530	0.296473
$144$	53.7862	4.48218
$145$	0	0
$146$	−19.6906	−1.62960
$147$	−5.16954	−0.426377
$148$	−25.0944	−2.06275
$149$	−12.7350	−1.04329	−0.521644	−	0.853163i	$-0.674681\pi$
$149$	−12.7350	−1.04329	−0.521644	+	0.853163i	$0.674681\pi$
$150$	0	0
$151$	−22.6719	−1.84501	−0.922505	−	0.385985i	$-0.873862\pi$
$151$	−22.6719	−1.84501	−0.922505	+	0.385985i	$0.873862\pi$
$152$	−14.1212	−1.14538
$153$	12.3333	0.997090
$154$	−23.6937	−1.90930
$155$	0	0
$156$	12.6498	1.01279
$157$	1.36716	0.109111	0.0545555	−	0.998511i	$-0.482626\pi$
$157$	1.36716	0.109111	0.0545555	+	0.998511i	$0.482626\pi$
$158$	3.63933	0.289529
$159$	7.67316	0.608521
$160$	0	0
$161$	−4.29722	−0.338668
$162$	−15.7584	−1.23810
$163$	18.5743	1.45485	0.727426	−	0.686186i	$-0.240716\pi$
$163$	18.5743	1.45485	0.727426	+	0.686186i	$0.240716\pi$
$164$	−57.0078	−4.45156
$165$	0	0
$166$	11.2502	0.873187
$167$	−15.6123	−1.20812	−0.604060	−	0.796939i	$-0.706451\pi$
$167$	−15.6123	−1.20812	−0.604060	+	0.796939i	$0.706451\pi$
$168$	−49.3789	−3.80966
$169$	−12.2007	−0.938518
$170$	0	0
$171$	10.9019	0.833687
$172$	−19.5996	−1.49445
$173$	−0.362462	−0.0275575	−0.0137787	−	0.999905i	$-0.504386\pi$
$173$	−0.362462	−0.0275575	−0.0137787	+	0.999905i	$0.504386\pi$
$174$	−66.9202	−5.07321
$175$	0	0
$176$	37.6975	2.84155
$177$	42.0030	3.15714
$178$	−44.6461	−3.34636
$179$	14.5195	1.08524	0.542620	−	0.839978i	$-0.317432\pi$
$179$	14.5195	1.08524	0.542620	+	0.839978i	$0.317432\pi$
$180$	0	0
$181$	3.64948	0.271264	0.135632	−	0.990759i	$-0.456694\pi$
$181$	3.64948	0.271264	0.135632	+	0.990759i	$0.456694\pi$
$182$	−5.34163	−0.395948
$183$	12.8800	0.952119
$184$	13.7540	1.01396
$185$	0	0
$186$	19.7215	1.44605
$187$	8.64414	0.632122
$188$	57.1275	4.16645
$189$	17.9087	1.30267
$190$	0	0
$191$	−3.55783	−0.257436	−0.128718	−	0.991681i	$-0.541086\pi$
$191$	−3.55783	−0.257436	−0.128718	+	0.991681i	$0.541086\pi$
$192$	21.9659	1.58525
$193$	8.71075	0.627014	0.313507	−	0.949586i	$-0.398496\pi$
$193$	8.71075	0.627014	0.313507	+	0.949586i	$0.398496\pi$
$194$	45.8809	3.29406
$195$	0	0
$196$	−8.44831	−0.603451
$197$	−13.3359	−0.950147	−0.475073	−	0.879946i	$-0.657578\pi$
$197$	−13.3359	−0.950147	−0.475073	+	0.879946i	$0.657578\pi$
$198$	−58.5469	−4.16075
$199$	23.1339	1.63992	0.819959	−	0.572422i	$-0.193996\pi$
$199$	23.1339	1.63992	0.819959	+	0.572422i	$0.193996\pi$
$200$	0	0
$201$	−18.7494	−1.32248
$202$	−12.6045	−0.886850
$203$	19.9577	1.40076
$204$	30.8427	2.15942
$205$	0	0
$206$	44.8023	3.12153
$207$	−10.6184	−0.738028
$208$	8.49869	0.589278
$209$	7.64086	0.528530
$210$	0	0
$211$	21.7818	1.49952	0.749762	−	0.661708i	$-0.230168\pi$
$211$	21.7818	1.49952	0.749762	+	0.661708i	$0.230168\pi$
$212$	12.5398	0.861240
$213$	19.2176	1.31677
$214$	35.9696	2.45883
$215$	0	0
$216$	−57.3200	−3.90013
$217$	−5.88158	−0.399268
$218$	−32.6816	−2.21348
$219$	22.2042	1.50042
$220$	0	0
$221$	1.94877	0.131089
$222$	40.0673	2.68914
$223$	−0.447131	−0.0299421	−0.0149711	−	0.999888i	$-0.504766\pi$
$223$	−0.447131	−0.0299421	−0.0149711	+	0.999888i	$0.504766\pi$
$224$	−23.2348	−1.55244
$225$	0	0
$226$	31.5980	2.10187
$227$	13.1421	0.872274	0.436137	−	0.899880i	$-0.356346\pi$
$227$	13.1421	0.872274	0.436137	+	0.899880i	$0.356346\pi$
$228$	27.2629	1.80553
$229$	−6.59552	−0.435844	−0.217922	−	0.975966i	$-0.569928\pi$
$229$	−6.59552	−0.435844	−0.217922	+	0.975966i	$0.569928\pi$
$230$	0	0
$231$	26.7184	1.75794
$232$	−63.8782	−4.19381
$233$	6.23598	0.408533	0.204266	−	0.978915i	$-0.434519\pi$
$233$	6.23598	0.408533	0.204266	+	0.978915i	$0.434519\pi$
$234$	−13.1991	−0.862851
$235$	0	0
$236$	68.6432	4.46829
$237$	−4.10392	−0.266578
$238$	−13.0239	−0.844215
$239$	−11.1918	−0.723939	−0.361970	−	0.932190i	$-0.617896\pi$
$239$	−11.1918	−0.723939	−0.361970	+	0.932190i	$0.617896\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	−12.3313	−0.792687
$243$	−5.69327	−0.365224
$244$	21.0492	1.34753
$245$	0	0
$246$	91.0224	5.80337
$247$	1.72259	0.109606
$248$	18.8250	1.19539
$249$	−12.6864	−0.803969
$250$	0	0
$251$	−4.74944	−0.299782	−0.149891	−	0.988703i	$-0.547892\pi$
$251$	−4.74944	−0.299782	−0.149891	+	0.988703i	$0.547892\pi$
$252$	62.2999	3.92453
$253$	−7.44217	−0.467885
$254$	10.2345	0.642167
$255$	0	0
$256$	−17.0573	−1.06608
$257$	1.42368	0.0888068	0.0444034	−	0.999014i	$-0.485861\pi$
$257$	1.42368	0.0888068	0.0444034	+	0.999014i	$0.485861\pi$
$258$	31.2940	1.94828
$259$	−11.9493	−0.742496
$260$	0	0
$261$	49.3152	3.05254
$262$	−15.2745	−0.943663
$263$	−3.48067	−0.214627	−0.107314	−	0.994225i	$-0.534225\pi$
$263$	−3.48067	−0.214627	−0.107314	+	0.994225i	$0.534225\pi$
$264$	−85.5171	−5.26322
$265$	0	0
$266$	−11.5123	−0.705865
$267$	50.3455	3.08109
$268$	−30.6412	−1.87171
$269$	16.6456	1.01490	0.507450	−	0.861681i	$-0.330588\pi$
$269$	16.6456	1.01490	0.507450	+	0.861681i	$0.330588\pi$
$270$	0	0
$271$	11.7496	0.713738	0.356869	−	0.934154i	$-0.383844\pi$
$271$	11.7496	0.713738	0.356869	+	0.934154i	$0.383844\pi$
$272$	20.7215	1.25642
$273$	6.02353	0.364561
$274$	−31.6100	−1.90963
$275$	0	0
$276$	−26.5540	−1.59836
$277$	−16.2474	−0.976214	−0.488107	−	0.872784i	$-0.662312\pi$
$277$	−16.2474	−0.976214	−0.488107	+	0.872784i	$0.662312\pi$
$278$	3.14581	0.188673
$279$	−14.5333	−0.870087
$280$	0	0
$281$	−24.6882	−1.47277	−0.736387	−	0.676561i	$-0.763469\pi$
$281$	−24.6882	−1.47277	−0.736387	+	0.676561i	$0.763469\pi$
$282$	−91.2136	−5.43169
$283$	−27.4275	−1.63040	−0.815199	−	0.579181i	$-0.803372\pi$
$283$	−27.4275	−1.63040	−0.815199	+	0.579181i	$0.803372\pi$
$284$	31.4063	1.86362
$285$	0	0
$286$	−9.25093	−0.547019
$287$	−27.1457	−1.60236
$288$	−57.4130	−3.38309
$289$	−12.2485	−0.720501
$290$	0	0
$291$	−51.7380	−3.03294
$292$	36.2872	2.12355
$293$	−28.8112	−1.68317	−0.841584	−	0.540126i	$-0.818377\pi$
$293$	−28.8112	−1.68317	−0.841584	+	0.540126i	$0.818377\pi$
$294$	13.4891	0.786702
$295$	0	0
$296$	38.2460	2.22300
$297$	31.0153	1.79969
$298$	33.2299	1.92496
$299$	−1.67780	−0.0970295
$300$	0	0
$301$	−9.33284	−0.537936
$302$	59.1588	3.40420
$303$	14.2136	0.816549
$304$	18.3164	1.05052
$305$	0	0
$306$	−32.1819	−1.83972
$307$	10.4197	0.594686	0.297343	−	0.954771i	$-0.403900\pi$
$307$	10.4197	0.594686	0.297343	+	0.954771i	$0.403900\pi$
$308$	43.6645	2.48802
$309$	−50.5217	−2.87408
$310$	0	0
$311$	−12.3706	−0.701475	−0.350737	−	0.936474i	$-0.614069\pi$
$311$	−12.3706	−0.701475	−0.350737	+	0.936474i	$0.614069\pi$
$312$	−19.2794	−1.09148
$313$	4.88936	0.276363	0.138182	−	0.990407i	$-0.455874\pi$
$313$	4.88936	0.276363	0.138182	+	0.990407i	$0.455874\pi$
$314$	−3.56739	−0.201319
$315$	0	0
$316$	−6.70681	−0.377288
$317$	28.5741	1.60488	0.802442	−	0.596730i	$-0.203534\pi$
$317$	28.5741	1.60488	0.802442	+	0.596730i	$0.203534\pi$
$318$	−20.0219	−1.12277
$319$	34.5639	1.93521
$320$	0	0
$321$	−40.5615	−2.26392
$322$	11.2129	0.624873
$323$	4.20001	0.233695
$324$	29.0408	1.61338
$325$	0	0
$326$	−48.4668	−2.68433
$327$	36.8537	2.03801
$328$	86.8847	4.79741
$329$	27.2028	1.49974
$330$	0	0
$331$	15.9544	0.876932	0.438466	−	0.898748i	$-0.355522\pi$
$331$	15.9544	0.876932	0.438466	+	0.898748i	$0.355522\pi$
$332$	−20.7327	−1.13786
$333$	−29.5267	−1.61805
$334$	40.7380	2.22909
$335$	0	0
$336$	64.0486	3.49414
$337$	25.6316	1.39624	0.698122	−	0.715979i	$-0.254019\pi$
$337$	25.6316	1.39624	0.698122	+	0.715979i	$0.254019\pi$
$338$	31.8360	1.73165
$339$	−35.6318	−1.93525
$340$	0	0
$341$	−10.1861	−0.551606
$342$	−28.4468	−1.53823
$343$	−20.0514	−1.08267
$344$	29.8714	1.61056
$345$	0	0
$346$	0.945790	0.0508460
$347$	−35.8668	−1.92543	−0.962715	−	0.270517i	$-0.912806\pi$
$347$	−35.8668	−1.92543	−0.962715	+	0.270517i	$0.912806\pi$
$348$	123.325	6.61094
$349$	13.7904	0.738182	0.369091	−	0.929393i	$-0.379669\pi$
$349$	13.7904	0.738182	0.369091	+	0.929393i	$0.379669\pi$
$350$	0	0
$351$	6.99224	0.373218
$352$	−40.2394	−2.14477
$353$	−3.95192	−0.210340	−0.105170	−	0.994454i	$-0.533539\pi$
$353$	−3.95192	−0.210340	−0.105170	+	0.994454i	$0.533539\pi$
$354$	−109.600	−5.82519
$355$	0	0
$356$	82.2770	4.36067
$357$	14.6865	0.777294
$358$	−37.8865	−2.00236
$359$	−4.76057	−0.251253	−0.125627	−	0.992078i	$-0.540094\pi$
$359$	−4.76057	−0.251253	−0.125627	+	0.992078i	$0.540094\pi$
$360$	0	0
$361$	−15.2875	−0.804603
$362$	−9.52277	−0.500506
$363$	13.9055	0.729851
$364$	9.84394	0.515963
$365$	0	0
$366$	−33.6085	−1.75674
$367$	6.88934	0.359621	0.179810	−	0.983701i	$-0.442452\pi$
$367$	6.88934	0.359621	0.179810	+	0.983701i	$0.442452\pi$
$368$	−17.8401	−0.929982
$369$	−67.0767	−3.49188
$370$	0	0
$371$	5.97117	0.310008
$372$	−36.3443	−1.88436
$373$	−29.1897	−1.51139	−0.755694	−	0.654925i	$-0.772700\pi$
$373$	−29.1897	−1.51139	−0.755694	+	0.654925i	$0.772700\pi$
$374$	−22.5556	−1.16632
$375$	0	0
$376$	−87.0673	−4.49015
$377$	7.79224	0.401321
$378$	−46.7301	−2.40354
$379$	−17.5075	−0.899300	−0.449650	−	0.893205i	$-0.648451\pi$
$379$	−17.5075	−0.899300	−0.449650	+	0.893205i	$0.648451\pi$
$380$	0	0
$381$	−11.5410	−0.591262
$382$	9.28362	0.474991
$383$	7.09088	0.362327	0.181164	−	0.983453i	$-0.442014\pi$
$383$	7.09088	0.362327	0.181164	+	0.983453i	$0.442014\pi$
$384$	2.39853	0.122400
$385$	0	0
$386$	−22.7294	−1.15690
$387$	−23.0613	−1.17227
$388$	−84.5527	−4.29251
$389$	−12.1197	−0.614491	−0.307245	−	0.951630i	$-0.599407\pi$
$389$	−12.1197	−0.614491	−0.307245	+	0.951630i	$0.599407\pi$
$390$	0	0
$391$	−4.09080	−0.206880
$392$	12.8760	0.650334
$393$	17.2244	0.868859
$394$	34.7981	1.75310
$395$	0	0
$396$	107.895	5.42190
$397$	24.1075	1.20992	0.604960	−	0.796256i	$-0.293189\pi$
$397$	24.1075	1.20992	0.604960	+	0.796256i	$0.293189\pi$
$398$	−60.3644	−3.02579
$399$	12.9820	0.649911
$400$	0	0
$401$	−10.9279	−0.545714	−0.272857	−	0.962055i	$-0.587969\pi$
$401$	−10.9279	−0.545714	−0.272857	+	0.962055i	$0.587969\pi$
$402$	48.9237	2.44009
$403$	−2.29639	−0.114391
$404$	23.2285	1.15566
$405$	0	0
$406$	−52.0766	−2.58452
$407$	−20.6945	−1.02579
$408$	−47.0069	−2.32719
$409$	31.4687	1.55603	0.778015	−	0.628246i	$-0.216227\pi$
$409$	31.4687	1.55603	0.778015	+	0.628246i	$0.216227\pi$
$410$	0	0
$411$	35.6452	1.75825
$412$	−82.5650	−4.06769
$413$	32.6863	1.60839
$414$	27.7070	1.36173
$415$	0	0
$416$	−9.07176	−0.444780
$417$	−3.54740	−0.173717
$418$	−19.9377	−0.975183
$419$	30.0203	1.46659	0.733293	−	0.679913i	$-0.237982\pi$
$419$	30.0203	1.46659	0.733293	+	0.679913i	$0.237982\pi$
$420$	0	0
$421$	−19.4645	−0.948641	−0.474321	−	0.880352i	$-0.657306\pi$
$421$	−19.4645	−0.948641	−0.474321	+	0.880352i	$0.657306\pi$
$422$	−56.8364	−2.76675
$423$	67.2177	3.26823
$424$	−19.1118	−0.928150
$425$	0	0
$426$	−50.1454	−2.42955
$427$	10.0231	0.485052
$428$	−66.2875	−3.20413
$429$	10.4319	0.503656
$430$	0	0
$431$	23.2925	1.12196	0.560981	−	0.827829i	$-0.310424\pi$
$431$	23.2925	1.12196	0.560981	+	0.827829i	$0.310424\pi$
$432$	74.3490	3.57712
$433$	−7.34512	−0.352984	−0.176492	−	0.984302i	$-0.556475\pi$
$433$	−7.34512	−0.352984	−0.176492	+	0.984302i	$0.556475\pi$
$434$	15.3471	0.736684
$435$	0	0
$436$	60.2280	2.88440
$437$	−3.61600	−0.172977
$438$	−57.9386	−2.76841
$439$	−18.6982	−0.892415	−0.446207	−	0.894930i	$-0.647226\pi$
$439$	−18.6982	−0.892415	−0.446207	+	0.894930i	$0.647226\pi$
$440$	0	0
$441$	−9.94049	−0.473357
$442$	−5.08503	−0.241870
$443$	32.9274	1.56443	0.782214	−	0.623009i	$-0.214090\pi$
$443$	32.9274	1.56443	0.782214	+	0.623009i	$0.214090\pi$
$444$	−73.8390	−3.50425
$445$	0	0
$446$	1.16672	0.0552458
$447$	−37.4720	−1.77236
$448$	17.0936	0.807597
$449$	−8.36090	−0.394575	−0.197288	−	0.980346i	$-0.563213\pi$
$449$	−8.36090	−0.394575	−0.197288	+	0.980346i	$0.563213\pi$
$450$	0	0
$451$	−47.0125	−2.21373
$452$	−58.2311	−2.73896
$453$	−66.7109	−3.13435
$454$	−34.2924	−1.60942
$455$	0	0
$456$	−41.5511	−1.94581
$457$	−37.6692	−1.76209	−0.881046	−	0.473030i	$-0.843160\pi$
$457$	−37.6692	−1.76209	−0.881046	+	0.473030i	$0.843160\pi$
$458$	17.2100	0.804170
$459$	17.0484	0.795753
$460$	0	0
$461$	−25.1740	−1.17247	−0.586234	−	0.810141i	$-0.699390\pi$
$461$	−25.1740	−1.17247	−0.586234	+	0.810141i	$0.699390\pi$
$462$	−69.7177	−3.24356
$463$	18.2153	0.846536	0.423268	−	0.906004i	$-0.360883\pi$
$463$	18.2153	0.846536	0.423268	+	0.906004i	$0.360883\pi$
$464$	82.8555	3.84647
$465$	0	0
$466$	−16.2718	−0.753778
$467$	−7.56476	−0.350055	−0.175028	−	0.984564i	$-0.556002\pi$
$467$	−7.56476	−0.350055	−0.175028	+	0.984564i	$0.556002\pi$
$468$	24.3242	1.12439
$469$	−14.5906	−0.673730
$470$	0	0
$471$	4.02279	0.185360
$472$	−104.618	−4.81544
$473$	−16.1631	−0.743182
$474$	10.7085	0.491860
$475$	0	0
$476$	24.0014	1.10010
$477$	14.7547	0.675570
$478$	29.2034	1.33573
$479$	16.7143	0.763696	0.381848	−	0.924225i	$-0.375288\pi$
$479$	16.7143	0.763696	0.381848	+	0.924225i	$0.375288\pi$
$480$	0	0
$481$	−4.66547	−0.212727
$482$	2.60935	0.118853
$483$	−12.6444	−0.575339
$484$	22.7251	1.03296
$485$	0	0
$486$	14.8557	0.673869
$487$	−4.37958	−0.198458	−0.0992289	−	0.995065i	$-0.531638\pi$
$487$	−4.37958	−0.198458	−0.0992289	+	0.995065i	$0.531638\pi$
$488$	−32.0807	−1.45223
$489$	54.6540	2.47154
$490$	0	0
$491$	−20.8725	−0.941964	−0.470982	−	0.882143i	$-0.656100\pi$
$491$	−20.8725	−0.941964	−0.470982	+	0.882143i	$0.656100\pi$
$492$	−167.743	−7.56242
$493$	18.9990	0.855672
$494$	−4.49484	−0.202232
$495$	0	0
$496$	−24.4177	−1.09639
$497$	14.9549	0.670820
$498$	33.1033	1.48339
$499$	26.5033	1.18645	0.593225	−	0.805037i	$-0.297855\pi$
$499$	26.5033	1.18645	0.593225	+	0.805037i	$0.297855\pi$
$500$	0	0
$501$	−45.9386	−2.05239
$502$	12.3929	0.553124
$503$	4.77911	0.213090	0.106545	−	0.994308i	$-0.466021\pi$
$503$	4.77911	0.213090	0.106545	+	0.994308i	$0.466021\pi$
$504$	−94.9504	−4.22943
$505$	0	0
$506$	19.4192	0.863289
$507$	−35.9001	−1.59438
$508$	−18.8608	−0.836813
$509$	−22.4912	−0.996905	−0.498452	−	0.866917i	$-0.666098\pi$
$509$	−22.4912	−0.996905	−0.498452	+	0.866917i	$0.666098\pi$
$510$	0	0
$511$	17.2791	0.764382
$512$	42.8782	1.89497
$513$	15.0697	0.665345
$514$	−3.71488	−0.163856
$515$	0	0
$516$	−57.6708	−2.53882
$517$	47.1113	2.07195
$518$	31.1800	1.36997
$519$	−1.06653	−0.0468154
$520$	0	0
$521$	−39.5254	−1.73164	−0.865820	−	0.500356i	$-0.833203\pi$
$521$	−39.5254	−1.73164	−0.865820	+	0.500356i	$0.833203\pi$
$522$	−128.681	−5.63220
$523$	2.35446	0.102953	0.0514766	−	0.998674i	$-0.483607\pi$
$523$	2.35446	0.102953	0.0514766	+	0.998674i	$0.483607\pi$
$524$	28.1490	1.22970
$525$	0	0
$526$	9.08227	0.396006
$527$	−5.59905	−0.243898
$528$	110.923	4.82731
$529$	−19.4780	−0.846871
$530$	0	0
$531$	80.7673	3.50500
$532$	21.2157	0.919818
$533$	−10.5987	−0.459081
$534$	−131.369	−5.68489
$535$	0	0
$536$	46.6997	2.01712
$537$	42.7230	1.84364
$538$	−43.4341	−1.87258
$539$	−6.96706	−0.300092
$540$	0	0
$541$	36.5474	1.57129	0.785647	−	0.618675i	$-0.212330\pi$
$541$	36.5474	1.57129	0.785647	+	0.618675i	$0.212330\pi$
$542$	−30.6588	−1.31691
$543$	10.7384	0.460830
$544$	−22.1187	−0.948332
$545$	0	0
$546$	−15.7175	−0.672646
$547$	−33.9370	−1.45104	−0.725520	−	0.688201i	$-0.758401\pi$
$547$	−33.9370	−1.45104	−0.725520	+	0.688201i	$0.758401\pi$
$548$	58.2532	2.48845
$549$	24.7670	1.05703
$550$	0	0
$551$	16.7939	0.715444
$552$	40.4706	1.72254
$553$	−3.19362	−0.135807
$554$	42.3952	1.80120
$555$	0	0
$556$	−5.79733	−0.245862
$557$	14.5375	0.615972	0.307986	−	0.951391i	$-0.400345\pi$
$557$	14.5375	0.615972	0.307986	+	0.951391i	$0.400345\pi$
$558$	37.9225	1.60539
$559$	−3.64389	−0.154120
$560$	0	0
$561$	25.4350	1.07387
$562$	64.4200	2.71740
$563$	10.6923	0.450628	0.225314	−	0.974286i	$-0.427659\pi$
$563$	10.6923	0.450628	0.225314	+	0.974286i	$0.427659\pi$
$564$	168.095	7.07808
$565$	0	0
$566$	71.5680	3.00823
$567$	13.8285	0.580743
$568$	−47.8659	−2.00841
$569$	3.81786	0.160053	0.0800265	−	0.996793i	$-0.474500\pi$
$569$	3.81786	0.160053	0.0800265	+	0.996793i	$0.474500\pi$
$570$	0	0
$571$	−23.6649	−0.990347	−0.495173	−	0.868794i	$-0.664895\pi$
$571$	−23.6649	−0.990347	−0.495173	+	0.868794i	$0.664895\pi$
$572$	17.0483	0.712825
$573$	−10.4688	−0.437338
$574$	70.8326	2.95650
$575$	0	0
$576$	42.2381	1.75992
$577$	10.1337	0.421872	0.210936	−	0.977500i	$-0.432349\pi$
$577$	10.1337	0.421872	0.210936	+	0.977500i	$0.432349\pi$
$578$	31.9606	1.32939
$579$	25.6310	1.06519
$580$	0	0
$581$	−9.87243	−0.409577
$582$	135.003	5.59603
$583$	10.3412	0.428289
$584$	−55.3048	−2.28853
$585$	0	0
$586$	75.1784	3.10559
$587$	−1.79171	−0.0739518	−0.0369759	−	0.999316i	$-0.511772\pi$
$587$	−1.79171	−0.0739518	−0.0369759	+	0.999316i	$0.511772\pi$
$588$	−24.8588	−1.02516
$589$	−4.94920	−0.203928
$590$	0	0
$591$	−39.2404	−1.61413
$592$	−49.6083	−2.03889
$593$	26.3703	1.08290	0.541448	−	0.840734i	$-0.317876\pi$
$593$	26.3703	1.08290	0.541448	+	0.840734i	$0.317876\pi$
$594$	−80.9298	−3.32059
$595$	0	0
$596$	−61.2385	−2.50843
$597$	68.0704	2.78594
$598$	4.37796	0.179028
$599$	−29.5793	−1.20858	−0.604288	−	0.796766i	$-0.706543\pi$
$599$	−29.5793	−1.20858	−0.604288	+	0.796766i	$0.706543\pi$
$600$	0	0
$601$	−30.6450	−1.25003	−0.625017	−	0.780611i	$-0.714908\pi$
$601$	−30.6450	−1.25003	−0.625017	+	0.780611i	$0.714908\pi$
$602$	24.3526	0.992539
$603$	−36.0531	−1.46820
$604$	−109.022	−4.43605
$605$	0	0
$606$	−37.0882	−1.50660
$607$	−23.8653	−0.968662	−0.484331	−	0.874885i	$-0.660937\pi$
$607$	−23.8653	−0.968662	−0.484331	+	0.874885i	$0.660937\pi$
$608$	−19.5515	−0.792919
$609$	58.7246	2.37964
$610$	0	0
$611$	10.6210	0.429679
$612$	59.3072	2.39735
$613$	39.3153	1.58793	0.793964	−	0.607965i	$-0.208014\pi$
$613$	39.3153	1.58793	0.793964	+	0.607965i	$0.208014\pi$
$614$	−27.1887	−1.09725
$615$	0	0
$616$	−66.5485	−2.68132
$617$	9.56356	0.385015	0.192507	−	0.981296i	$-0.438338\pi$
$617$	9.56356	0.385015	0.192507	+	0.981296i	$0.438338\pi$
$618$	131.829	5.30293
$619$	−24.2783	−0.975828	−0.487914	−	0.872892i	$-0.662242\pi$
$619$	−24.2783	−0.975828	−0.487914	+	0.872892i	$0.662242\pi$
$620$	0	0
$621$	−14.6779	−0.589002
$622$	32.2793	1.29428
$623$	39.1783	1.56965
$624$	25.0070	1.00108
$625$	0	0
$626$	−12.7580	−0.509914
$627$	22.4829	0.897880
$628$	6.57424	0.262341
$629$	−11.3753	−0.453564
$630$	0	0
$631$	0.222716	0.00886617	0.00443308	−	0.999990i	$-0.498589\pi$
$631$	0.222716	0.00886617	0.00443308	+	0.999990i	$0.498589\pi$
$632$	10.2218	0.406600
$633$	64.0920	2.54743
$634$	−74.5599	−2.96115
$635$	0	0
$636$	36.8979	1.46310
$637$	−1.57069	−0.0622328
$638$	−90.1892	−3.57062
$639$	36.9534	1.46185
$640$	0	0
$641$	−17.4626	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$641$	−17.4626	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$642$	105.839	4.17713
$643$	17.7047	0.698206	0.349103	−	0.937084i	$-0.386486\pi$
$643$	17.7047	0.698206	0.349103	+	0.937084i	$0.386486\pi$
$644$	−20.6640	−0.814277
$645$	0	0
$646$	−10.9593	−0.431188
$647$	26.5931	1.04548	0.522742	−	0.852491i	$-0.324909\pi$
$647$	26.5931	1.04548	0.522742	+	0.852491i	$0.324909\pi$
$648$	−44.2606	−1.73872
$649$	56.6079	2.22206
$650$	0	0
$651$	−17.3063	−0.678286
$652$	89.3182	3.49797
$653$	41.9890	1.64316	0.821578	−	0.570096i	$-0.193094\pi$
$653$	41.9890	1.64316	0.821578	+	0.570096i	$0.193094\pi$
$654$	−96.1640	−3.76031
$655$	0	0
$656$	−112.697	−4.40008
$657$	42.6964	1.66575
$658$	−70.9815	−2.76715
$659$	−1.99515	−0.0777199	−0.0388599	−	0.999245i	$-0.512373\pi$
$659$	−1.99515	−0.0777199	−0.0388599	+	0.999245i	$0.512373\pi$
$660$	0	0
$661$	40.2608	1.56596	0.782982	−	0.622044i	$-0.213697\pi$
$661$	40.2608	1.56596	0.782982	+	0.622044i	$0.213697\pi$
$662$	−41.6305	−1.61802
$663$	5.73418	0.222697
$664$	31.5985	1.22626
$665$	0	0
$666$	77.0453	2.98545
$667$	−16.3572	−0.633353
$668$	−75.0750	−2.90474
$669$	−1.31566	−0.0508664
$670$	0	0
$671$	17.3586	0.670120
$672$	−68.3674	−2.63733
$673$	0.560012	0.0215869	0.0107934	−	0.999942i	$-0.496564\pi$
$673$	0.560012	0.0215869	0.0107934	+	0.999942i	$0.496564\pi$
$674$	−66.8819	−2.57619
$675$	0	0
$676$	−58.6696	−2.25652
$677$	36.7862	1.41381	0.706905	−	0.707308i	$-0.250091\pi$
$677$	36.7862	1.41381	0.706905	+	0.707308i	$0.250091\pi$
$678$	92.9757	3.57071
$679$	−40.2620	−1.54511
$680$	0	0
$681$	38.6701	1.48184
$682$	26.5790	1.01776
$683$	25.8141	0.987749	0.493874	−	0.869533i	$-0.335580\pi$
$683$	25.8141	0.987749	0.493874	+	0.869533i	$0.335580\pi$
$684$	52.4238	2.00447
$685$	0	0
$686$	52.3210	1.99763
$687$	−19.4070	−0.740423
$688$	−38.7458	−1.47717
$689$	2.33137	0.0888181
$690$	0	0
$691$	19.4917	0.741498	0.370749	−	0.928733i	$-0.379101\pi$
$691$	19.4917	0.741498	0.370749	+	0.928733i	$0.379101\pi$
$692$	−1.74297	−0.0662578
$693$	51.3768	1.95164
$694$	93.5890	3.55259
$695$	0	0
$696$	−187.959	−7.12455
$697$	−25.8417	−0.978825
$698$	−35.9839	−1.36201
$699$	18.3491	0.694026
$700$	0	0
$701$	6.75563	0.255157	0.127578	−	0.991828i	$-0.459280\pi$
$701$	6.75563	0.255157	0.127578	+	0.991828i	$0.459280\pi$
$702$	−18.2452	−0.688620
$703$	−10.0551	−0.379234
$704$	29.6037	1.11573
$705$	0	0
$706$	10.3119	0.388095
$707$	11.0609	0.415986
$708$	201.980	7.59086
$709$	30.2355	1.13552	0.567759	−	0.823195i	$-0.307811\pi$
$709$	30.2355	1.13552	0.567759	+	0.823195i	$0.307811\pi$
$710$	0	0
$711$	−7.89140	−0.295951
$712$	−125.397	−4.69946
$713$	4.82050	0.180529
$714$	−38.3223	−1.43417
$715$	0	0
$716$	69.8200	2.60930
$717$	−32.9314	−1.22985
$718$	12.4220	0.463584
$719$	0.540522	0.0201581	0.0100790	−	0.999949i	$-0.496792\pi$
$719$	0.540522	0.0201581	0.0100790	+	0.999949i	$0.496792\pi$
$720$	0	0
$721$	−39.3155	−1.46418
$722$	39.8903	1.48456
$723$	−2.94245	−0.109431
$724$	17.5493	0.652213
$725$	0	0
$726$	−36.2843	−1.34664
$727$	−44.5818	−1.65345	−0.826723	−	0.562609i	$-0.809798\pi$
$727$	−44.5818	−1.65345	−0.826723	+	0.562609i	$0.809798\pi$
$728$	−15.0030	−0.556048
$729$	−34.8698	−1.29148
$730$	0	0
$731$	−8.88452	−0.328606
$732$	61.9362	2.28923
$733$	−1.22238	−0.0451497	−0.0225749	−	0.999745i	$-0.507186\pi$
$733$	−1.22238	−0.0451497	−0.0225749	+	0.999745i	$0.507186\pi$
$734$	−17.9767	−0.663532
$735$	0	0
$736$	19.0431	0.701938
$737$	−25.2688	−0.930788
$738$	175.027	6.44282
$739$	−1.13737	−0.0418389	−0.0209194	−	0.999781i	$-0.506659\pi$
$739$	−1.13737	−0.0418389	−0.0209194	+	0.999781i	$0.506659\pi$
$740$	0	0
$741$	5.06865	0.186201
$742$	−15.5809	−0.571991
$743$	−13.3543	−0.489921	−0.244960	−	0.969533i	$-0.578775\pi$
$743$	−13.3543	−0.489921	−0.244960	+	0.969533i	$0.578775\pi$
$744$	55.3918	2.03076
$745$	0	0
$746$	76.1662	2.78864
$747$	−24.3946	−0.892553
$748$	41.5670	1.51984
$749$	−31.5645	−1.15334
$750$	0	0
$751$	−18.4435	−0.673012	−0.336506	−	0.941681i	$-0.609245\pi$
$751$	−18.4435	−0.673012	−0.336506	+	0.941681i	$0.609245\pi$
$752$	112.934	4.11827
$753$	−13.9750	−0.509278
$754$	−20.3327	−0.740472
$755$	0	0
$756$	86.1176	3.13207
$757$	51.4141	1.86868	0.934339	−	0.356384i	$-0.115991\pi$
$757$	51.4141	1.86868	0.934339	+	0.356384i	$0.115991\pi$
$758$	45.6832	1.65929
$759$	−21.8982	−0.794856
$760$	0	0
$761$	−4.20935	−0.152589	−0.0762943	−	0.997085i	$-0.524309\pi$
$761$	−4.20935	−0.152589	−0.0762943	+	0.997085i	$0.524309\pi$
$762$	30.1144	1.09093
$763$	28.6791	1.03825
$764$	−17.1085	−0.618965
$765$	0	0
$766$	−18.5026	−0.668526
$767$	12.7620	0.460807
$768$	−50.1904	−1.81109
$769$	15.9062	0.573593	0.286797	−	0.957992i	$-0.407410\pi$
$769$	15.9062	0.573593	0.286797	+	0.957992i	$0.407410\pi$
$770$	0	0
$771$	4.18912	0.150867
$772$	41.8874	1.50756
$773$	29.4017	1.05751	0.528753	−	0.848776i	$-0.322660\pi$
$773$	29.4017	1.05751	0.528753	+	0.848776i	$0.322660\pi$
$774$	60.1750	2.16295
$775$	0	0
$776$	128.866	4.62601
$777$	−35.1604	−1.26137
$778$	31.6244	1.13379
$779$	−22.8424	−0.818415
$780$	0	0
$781$	25.8998	0.926767
$782$	10.6743	0.381712
$783$	68.1687	2.43615
$784$	−16.7012	−0.596472
$785$	0	0
$786$	−44.9446	−1.60312
$787$	−45.1897	−1.61084	−0.805420	−	0.592705i	$-0.798060\pi$
$787$	−45.1897	−1.61084	−0.805420	+	0.592705i	$0.798060\pi$
$788$	−64.1285	−2.28448
$789$	−10.2417	−0.364614
$790$	0	0
$791$	−27.7283	−0.985904
$792$	−164.441	−5.84314
$793$	3.91340	0.138969
$794$	−62.9048	−2.23241
$795$	0	0
$796$	111.244	3.94294
$797$	28.9432	1.02522	0.512611	−	0.858621i	$-0.328678\pi$
$797$	28.9432	1.02522	0.512611	+	0.858621i	$0.328678\pi$
$798$	−33.8744	−1.19914
$799$	25.8960	0.916135
$800$	0	0
$801$	96.8092	3.42058
$802$	28.5147	1.00689
$803$	29.9249	1.05603
$804$	−90.1602	−3.17970
$805$	0	0
$806$	5.99208	0.211062
$807$	48.9789	1.72414
$808$	−35.4022	−1.24545
$809$	−4.25320	−0.149534	−0.0747672	−	0.997201i	$-0.523821\pi$
$809$	−4.25320	−0.149534	−0.0747672	+	0.997201i	$0.523821\pi$
$810$	0	0
$811$	−48.0195	−1.68619	−0.843096	−	0.537763i	$-0.819270\pi$
$811$	−48.0195	−1.68619	−0.843096	+	0.537763i	$0.819270\pi$
$812$	95.9705	3.36791
$813$	34.5727	1.21252
$814$	53.9993	1.89267
$815$	0	0
$816$	60.9719	2.13444
$817$	−7.85335	−0.274754
$818$	−82.1129	−2.87101
$819$	11.5826	0.404729
$820$	0	0
$821$	−13.6434	−0.476159	−0.238079	−	0.971246i	$-0.576518\pi$
$821$	−13.6434	−0.476159	−0.238079	+	0.971246i	$0.576518\pi$
$822$	−93.0108	−3.24413
$823$	28.7887	1.00351	0.501755	−	0.865010i	$-0.332688\pi$
$823$	28.7887	1.00351	0.501755	+	0.865010i	$0.332688\pi$
$824$	125.836	4.38371
$825$	0	0
$826$	−85.2898	−2.96761
$827$	1.06464	0.0370211	0.0185106	−	0.999829i	$-0.494108\pi$
$827$	1.06464	0.0370211	0.0185106	+	0.999829i	$0.494108\pi$
$828$	−51.0606	−1.77448
$829$	21.6767	0.752864	0.376432	−	0.926444i	$-0.377151\pi$
$829$	21.6767	0.752864	0.376432	+	0.926444i	$0.377151\pi$
$830$	0	0
$831$	−47.8073	−1.65842
$832$	6.67399	0.231379
$833$	−3.82964	−0.132689
$834$	9.25641	0.320523
$835$	0	0
$836$	36.7426	1.27077
$837$	−20.0895	−0.694394
$838$	−78.3333	−2.70598
$839$	−18.4837	−0.638127	−0.319063	−	0.947733i	$-0.603368\pi$
$839$	−18.4837	−0.638127	−0.319063	+	0.947733i	$0.603368\pi$
$840$	0	0
$841$	46.9681	1.61959
$842$	50.7896	1.75033
$843$	−72.6438	−2.50199
$844$	104.742	3.60538
$845$	0	0
$846$	−175.394	−6.03018
$847$	10.8211	0.371818
$848$	24.7896	0.851279
$849$	−80.7042	−2.76976
$850$	0	0
$851$	9.79359	0.335720
$852$	92.4116	3.16597
$853$	3.69419	0.126487	0.0632434	−	0.997998i	$-0.479856\pi$
$853$	3.69419	0.126487	0.0632434	+	0.997998i	$0.479856\pi$
$854$	−26.1538	−0.894963
$855$	0	0
$856$	101.028	3.45306
$857$	−12.7715	−0.436266	−0.218133	−	0.975919i	$-0.569997\pi$
$857$	−12.7715	−0.436266	−0.218133	+	0.975919i	$0.569997\pi$
$858$	−27.2204	−0.929290
$859$	−40.9231	−1.39628	−0.698140	−	0.715961i	$-0.745989\pi$
$859$	−40.9231	−1.39628	−0.698140	+	0.715961i	$0.745989\pi$
$860$	0	0
$861$	−79.8750	−2.72213
$862$	−60.7783	−2.07012
$863$	20.5851	0.700726	0.350363	−	0.936614i	$-0.386058\pi$
$863$	20.5851	0.700726	0.350363	+	0.936614i	$0.386058\pi$
$864$	−79.3623	−2.69996
$865$	0	0
$866$	19.1660	0.651287
$867$	−36.0407	−1.22401
$868$	−28.2827	−0.959979
$869$	−5.53090	−0.187623
$870$	0	0
$871$	−5.69671	−0.193026
$872$	−91.7926	−3.10849
$873$	−99.4868	−3.36712
$874$	9.43541	0.319157
$875$	0	0
$876$	106.773	3.60754
$877$	5.66899	0.191428	0.0957141	−	0.995409i	$-0.469487\pi$
$877$	5.66899	0.191428	0.0957141	+	0.995409i	$0.469487\pi$
$878$	48.7900	1.64658
$879$	−84.7756	−2.85941
$880$	0	0
$881$	−6.97987	−0.235158	−0.117579	−	0.993064i	$-0.537513\pi$
$881$	−6.97987	−0.235158	−0.117579	+	0.993064i	$0.537513\pi$
$882$	25.9382	0.873385
$883$	−4.33326	−0.145826	−0.0729130	−	0.997338i	$-0.523230\pi$
$883$	−4.33326	−0.145826	−0.0729130	+	0.997338i	$0.523230\pi$
$884$	9.37106	0.315183
$885$	0	0
$886$	−85.9191	−2.88651
$887$	22.6088	0.759129	0.379565	−	0.925165i	$-0.376074\pi$
$887$	22.6088	0.759129	0.379565	+	0.925165i	$0.376074\pi$
$888$	112.537	3.77650
$889$	−8.98106	−0.301215
$890$	0	0
$891$	23.9490	0.802322
$892$	−2.15012	−0.0719912
$893$	22.8904	0.765999
$894$	97.7775	3.27017
$895$	0	0
$896$	1.86651	0.0623558
$897$	−4.93684	−0.164836
$898$	21.8165	0.728026
$899$	−22.3880	−0.746681
$900$	0	0
$901$	5.68433	0.189373
$902$	122.672	4.08453
$903$	−27.4615	−0.913860
$904$	88.7492	2.95176
$905$	0	0
$906$	174.072	5.78315
$907$	−24.5600	−0.815500	−0.407750	−	0.913094i	$-0.633687\pi$
$907$	−24.5600	−0.815500	−0.407750	+	0.913094i	$0.633687\pi$
$908$	63.1966	2.09725
$909$	27.3312	0.906520
$910$	0	0
$911$	41.9000	1.38821	0.694105	−	0.719874i	$-0.255800\pi$
$911$	41.9000	1.38821	0.694105	+	0.719874i	$0.255800\pi$
$912$	53.8953	1.78465
$913$	−17.0976	−0.565849
$914$	98.2922	3.25122
$915$	0	0
$916$	−31.7158	−1.04792
$917$	13.4039	0.442635
$918$	−44.4853	−1.46823
$919$	−20.2671	−0.668550	−0.334275	−	0.942476i	$-0.608491\pi$
$919$	−20.2671	−0.668550	−0.334275	+	0.942476i	$0.608491\pi$
$920$	0	0
$921$	30.6596	1.01027
$922$	65.6877	2.16331
$923$	5.83897	0.192192
$924$	128.481	4.22671
$925$	0	0
$926$	−47.5300	−1.56193
$927$	−97.1480	−3.19076
$928$	−88.4424	−2.90326
$929$	13.0876	0.429390	0.214695	−	0.976681i	$-0.431124\pi$
$929$	13.0876	0.429390	0.214695	+	0.976681i	$0.431124\pi$
$930$	0	0
$931$	−3.38515	−0.110944
$932$	29.9869	0.982254
$933$	−36.4000	−1.19168
$934$	19.7391	0.645883
$935$	0	0
$936$	−37.0722	−1.21174
$937$	22.6923	0.741325	0.370662	−	0.928768i	$-0.379131\pi$
$937$	22.6923	0.741325	0.370662	+	0.928768i	$0.379131\pi$
$938$	38.0719	1.24309
$939$	14.3867	0.469493
$940$	0	0
$941$	−34.9122	−1.13811	−0.569053	−	0.822301i	$-0.692690\pi$
$941$	−34.9122	−1.13811	−0.569053	+	0.822301i	$0.692690\pi$
$942$	−10.4969	−0.342006
$943$	22.2484	0.724509
$944$	135.699	4.41662
$945$	0	0
$946$	42.1753	1.37124
$947$	0.398371	0.0129453	0.00647266	−	0.999979i	$-0.497940\pi$
$947$	0.398371	0.0129453	0.00647266	+	0.999979i	$0.497940\pi$
$948$	−19.7345	−0.640946
$949$	6.74641	0.218998
$950$	0	0
$951$	84.0781	2.72642
$952$	−36.5802	−1.18557
$953$	−27.1036	−0.877972	−0.438986	−	0.898494i	$-0.644662\pi$
$953$	−27.1036	−0.877972	−0.438986	+	0.898494i	$0.644662\pi$
$954$	−38.5001	−1.24649
$955$	0	0
$956$	−53.8181	−1.74060
$957$	101.703	3.28758
$958$	−43.6134	−1.40909
$959$	27.7387	0.895731
$960$	0	0
$961$	−24.4022	−0.787168
$962$	12.1738	0.392500
$963$	−77.9955	−2.51337
$964$	−4.80870	−0.154878
$965$	0	0
$966$	32.9936	1.06155
$967$	45.7065	1.46982	0.734912	−	0.678163i	$-0.237224\pi$
$967$	45.7065	1.46982	0.734912	+	0.678163i	$0.237224\pi$
$968$	−34.6349	−1.11321
$969$	12.3583	0.397007
$970$	0	0
$971$	−49.0764	−1.57494	−0.787468	−	0.616355i	$-0.788609\pi$
$971$	−49.0764	−1.57494	−0.787468	+	0.616355i	$0.788609\pi$
$972$	−27.3772	−0.878125
$973$	−2.76055	−0.0884992
$974$	11.4278	0.366172
$975$	0	0
$976$	41.6114	1.33195
$977$	−21.0980	−0.674984	−0.337492	−	0.941328i	$-0.609579\pi$
$977$	−21.0980	−0.674984	−0.337492	+	0.941328i	$0.609579\pi$
$978$	−142.611	−4.56021
$979$	67.8513	2.16854
$980$	0	0
$981$	70.8657	2.26257
$982$	54.4637	1.73801
$983$	−29.3787	−0.937034	−0.468517	−	0.883455i	$-0.655211\pi$
$983$	−29.3787	−0.937034	−0.468517	+	0.883455i	$0.655211\pi$
$984$	255.654	8.14996
$985$	0	0
$986$	−49.5750	−1.57879
$987$	80.0428	2.54779
$988$	8.28342	0.263531
$989$	7.64913	0.243228
$990$	0	0
$991$	19.1180	0.607303	0.303652	−	0.952783i	$-0.401794\pi$
$991$	19.1180	0.607303	0.303652	+	0.952783i	$0.401794\pi$
$992$	26.0642	0.827539
$993$	46.9450	1.48975
$994$	−39.0226	−1.23772
$995$	0	0
$996$	−61.0051	−1.93302
$997$	16.9400	0.536494	0.268247	−	0.963350i	$-0.413556\pi$
$997$	16.9400	0.536494	0.268247	+	0.963350i	$0.413556\pi$
$998$	−69.1562	−2.18910
$999$	−40.8149	−1.29133

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.q.1.4		66
5.2	odd	4		1205.2.b.d.724.4	✓	66
5.3	odd	4		1205.2.b.d.724.63	yes	66
5.4	even	2	inner	6025.2.a.q.1.63		66

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.d.724.4	✓	66	5.2	odd	4
1205.2.b.d.724.63	yes	66	5.3	odd	4
6025.2.a.q.1.4		66	1.1	even	1	trivial
6025.2.a.q.1.63		66	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.60935 q^{2} +2.94245 q^{3} +4.80870 q^{4} -7.67788 q^{6} +2.28979 q^{7} -7.32887 q^{8} +5.65803 q^{9} +O(q^{10})\)	\(q-2.60935 q^{2} +2.94245 q^{3} +4.80870 q^{4} -7.67788 q^{6} +2.28979 q^{7} -7.32887 q^{8} +5.65803 q^{9} +3.96558 q^{11} +14.1494 q^{12} +0.894019 q^{13} -5.97485 q^{14} +9.50617 q^{16} +2.17979 q^{17} -14.7638 q^{18} +1.92680 q^{19} +6.73759 q^{21} -10.3476 q^{22} -1.87669 q^{23} -21.5648 q^{24} -2.33281 q^{26} +7.82113 q^{27} +11.0109 q^{28} +8.71597 q^{29} -2.56862 q^{31} -10.1472 q^{32} +11.6685 q^{33} -5.68783 q^{34} +27.2078 q^{36} -5.21854 q^{37} -5.02768 q^{38} +2.63061 q^{39} -11.8551 q^{41} -17.5807 q^{42} -4.07586 q^{43} +19.0693 q^{44} +4.89694 q^{46} +11.8800 q^{47} +27.9715 q^{48} -1.75688 q^{49} +6.41393 q^{51} +4.29907 q^{52} +2.60774 q^{53} -20.4081 q^{54} -16.7815 q^{56} +5.66951 q^{57} -22.7430 q^{58} +14.2748 q^{59} +4.37731 q^{61} +6.70241 q^{62} +12.9557 q^{63} +7.46516 q^{64} -30.4473 q^{66} -6.37203 q^{67} +10.4820 q^{68} -5.52208 q^{69} +6.53115 q^{71} -41.4670 q^{72} +7.54616 q^{73} +13.6170 q^{74} +9.26538 q^{76} +9.08033 q^{77} -6.86417 q^{78} -1.39473 q^{79} +6.03922 q^{81} +30.9342 q^{82} -4.31151 q^{83} +32.3990 q^{84} +10.6353 q^{86} +25.6463 q^{87} -29.0632 q^{88} +17.1100 q^{89} +2.04711 q^{91} -9.02444 q^{92} -7.55803 q^{93} -30.9992 q^{94} -29.8576 q^{96} -17.5833 q^{97} +4.58432 q^{98} +22.4374 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9}+O(q^{10}) \) 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9	\( 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9} + 48 q^{11} + 30 q^{14} + 98 q^{16} + 12 q^{19} + 18 q^{21} + 42 q^{24} + 48 q^{26} + 56 q^{29} + 48 q^{31} + 8 q^{34} + 158 q^{36} + 84 q^{39} + 56 q^{41} + 144 q^{44} + 36 q^{46} + 98 q^{49} + 44 q^{51} + 86 q^{54} + 104 q^{56} + 108 q^{59} + 22 q^{61} + 136 q^{64} + 74 q^{66} + 20 q^{69} + 212 q^{71} + 84 q^{74} + 6 q^{76} + 66 q^{79} + 162 q^{81} - 52 q^{84} + 100 q^{86} + 54 q^{89} + 72 q^{91} - 96 q^{94} + 122 q^{96} + 112 q^{99}+O(q^{100}) \) 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9 + 48 * q^11 + 30 * q^14 + 98 * q^16 + 12 * q^19 + 18 * q^21 + 42 * q^24 + 48 * q^26 + 56 * q^29 + 48 * q^31 + 8 * q^34 + 158 * q^36 + 84 * q^39 + 56 * q^41 + 144 * q^44 + 36 * q^46 + 98 * q^49 + 44 * q^51 + 86 * q^54 + 104 * q^56 + 108 * q^59 + 22 * q^61 + 136 * q^64 + 74 * q^66 + 20 * q^69 + 212 * q^71 + 84 * q^74 + 6 * q^76 + 66 * q^79 + 162 * q^81 - 52 * q^84 + 100 * q^86 + 54 * q^89 + 72 * q^91 - 96 * q^94 + 122 * q^96 + 112 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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