LMFDB - Embedded newform 6025.2.a.n.1.8

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			1.8
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)

$f(q)$	$=$	$q-1.82970 q^{2} -1.30801 q^{3} +1.34779 q^{4} +2.39326 q^{6} +2.38020 q^{7} +1.19335 q^{8} -1.28911 q^{9} +O(q^{10})$	\(q-1.82970 q^{2} -1.30801 q^{3} +1.34779 q^{4} +2.39326 q^{6} +2.38020 q^{7} +1.19335 q^{8} -1.28911 q^{9} +4.04418 q^{11} -1.76292 q^{12} -0.674709 q^{13} -4.35503 q^{14} -4.87905 q^{16} -0.179609 q^{17} +2.35868 q^{18} -2.69583 q^{19} -3.11332 q^{21} -7.39962 q^{22} +0.408797 q^{23} -1.56092 q^{24} +1.23451 q^{26} +5.61020 q^{27} +3.20799 q^{28} -6.62106 q^{29} -0.446964 q^{31} +6.54046 q^{32} -5.28983 q^{33} +0.328630 q^{34} -1.73744 q^{36} +6.53749 q^{37} +4.93255 q^{38} +0.882526 q^{39} +5.81509 q^{41} +5.69643 q^{42} +1.03255 q^{43} +5.45069 q^{44} -0.747975 q^{46} -9.54749 q^{47} +6.38184 q^{48} -1.33467 q^{49} +0.234931 q^{51} -0.909363 q^{52} +3.34160 q^{53} -10.2650 q^{54} +2.84041 q^{56} +3.52618 q^{57} +12.1145 q^{58} -2.88182 q^{59} -6.56286 q^{61} +0.817809 q^{62} -3.06833 q^{63} -2.20896 q^{64} +9.67878 q^{66} +15.3500 q^{67} -0.242075 q^{68} -0.534711 q^{69} -5.25710 q^{71} -1.53836 q^{72} +6.90422 q^{73} -11.9616 q^{74} -3.63340 q^{76} +9.62594 q^{77} -1.61475 q^{78} +5.78996 q^{79} -3.47088 q^{81} -10.6398 q^{82} +17.8600 q^{83} -4.19609 q^{84} -1.88925 q^{86} +8.66041 q^{87} +4.82613 q^{88} +5.55025 q^{89} -1.60594 q^{91} +0.550971 q^{92} +0.584634 q^{93} +17.4690 q^{94} -8.55499 q^{96} +18.0075 q^{97} +2.44204 q^{98} -5.21338 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.82970	−1.29379	−0.646895	−	0.762579i	$-0.723933\pi$
$2$	−1.82970	−1.29379	−0.646895	+	0.762579i	$0.723933\pi$
$3$	−1.30801	−0.755180	−0.377590	−	0.925973i	$-0.623247\pi$
$3$	−1.30801	−0.755180	−0.377590	+	0.925973i	$0.623247\pi$
$4$	1.34779	0.673893
$5$	0	0
$5$	0	0
$6$	2.39326	0.977045
$7$	2.38020	0.899629	0.449815	−	0.893122i	$-0.351490\pi$
$7$	2.38020	0.899629	0.449815	+	0.893122i	$0.351490\pi$
$8$	1.19335	0.421914
$9$	−1.28911	−0.429703
$10$	0	0
$11$	4.04418	1.21937	0.609683	−	0.792645i	$-0.291297\pi$
$11$	4.04418	1.21937	0.609683	+	0.792645i	$0.291297\pi$
$12$	−1.76292	−0.508911
$13$	−0.674709	−0.187131	−0.0935653	−	0.995613i	$-0.529826\pi$
$13$	−0.674709	−0.187131	−0.0935653	+	0.995613i	$0.529826\pi$
$14$	−4.35503	−1.16393
$15$	0	0
$16$	−4.87905	−1.21976
$17$	−0.179609	−0.0435617	−0.0217808	−	0.999763i	$-0.506934\pi$
$17$	−0.179609	−0.0435617	−0.0217808	+	0.999763i	$0.506934\pi$
$18$	2.35868	0.555945
$19$	−2.69583	−0.618466	−0.309233	−	0.950986i	$-0.600072\pi$
$19$	−2.69583	−0.618466	−0.309233	+	0.950986i	$0.600072\pi$
$20$	0	0
$21$	−3.11332	−0.679382
$22$	−7.39962	−1.57760
$23$	0.408797	0.0852402	0.0426201	−	0.999091i	$-0.486429\pi$
$23$	0.408797	0.0852402	0.0426201	+	0.999091i	$0.486429\pi$
$24$	−1.56092	−0.318621
$25$	0	0
$26$	1.23451	0.242108
$27$	5.61020	1.07968
$28$	3.20799	0.606254
$29$	−6.62106	−1.22950	−0.614750	−	0.788722i	$-0.710743\pi$
$29$	−6.62106	−1.22950	−0.614750	+	0.788722i	$0.710743\pi$
$30$	0	0
$31$	−0.446964	−0.0802772	−0.0401386	−	0.999194i	$-0.512780\pi$
$31$	−0.446964	−0.0802772	−0.0401386	+	0.999194i	$0.512780\pi$
$32$	6.54046	1.15620
$33$	−5.28983	−0.920841
$34$	0.328630	0.0563596
$35$	0	0
$36$	−1.73744	−0.289574
$37$	6.53749	1.07476	0.537379	−	0.843341i	$-0.319415\pi$
$37$	6.53749	1.07476	0.537379	+	0.843341i	$0.319415\pi$
$38$	4.93255	0.800165
$39$	0.882526	0.141317
$40$	0	0
$41$	5.81509	0.908164	0.454082	−	0.890960i	$-0.349967\pi$
$41$	5.81509	0.908164	0.454082	+	0.890960i	$0.349967\pi$
$42$	5.69643	0.878978
$43$	1.03255	0.157463	0.0787313	−	0.996896i	$-0.474913\pi$
$43$	1.03255	0.157463	0.0787313	+	0.996896i	$0.474913\pi$
$44$	5.45069	0.821722
$45$	0	0
$46$	−0.747975	−0.110283
$47$	−9.54749	−1.39264	−0.696322	−	0.717730i	$-0.745181\pi$
$47$	−9.54749	−1.39264	−0.696322	+	0.717730i	$0.745181\pi$
$48$	6.38184	0.921140
$49$	−1.33467	−0.190667
$50$	0	0
$51$	0.234931	0.0328969
$52$	−0.909363	−0.126106
$53$	3.34160	0.459004	0.229502	−	0.973308i	$-0.426290\pi$
$53$	3.34160	0.459004	0.229502	+	0.973308i	$0.426290\pi$
$54$	−10.2650	−1.39688
$55$	0	0
$56$	2.84041	0.379566
$57$	3.52618	0.467053
$58$	12.1145	1.59071
$59$	−2.88182	−0.375181	−0.187590	−	0.982247i	$-0.560068\pi$
$59$	−2.88182	−0.375181	−0.187590	+	0.982247i	$0.560068\pi$
$60$	0	0
$61$	−6.56286	−0.840288	−0.420144	−	0.907457i	$-0.638020\pi$
$61$	−6.56286	−0.840288	−0.420144	+	0.907457i	$0.638020\pi$
$62$	0.817809	0.103862
$63$	−3.06833	−0.386573
$64$	−2.20896	−0.276120
$65$	0	0
$66$	9.67878	1.19137
$67$	15.3500	1.87530	0.937650	−	0.347580i	$-0.112996\pi$
$67$	15.3500	1.87530	0.937650	+	0.347580i	$0.112996\pi$
$68$	−0.242075	−0.0293559
$69$	−0.534711	−0.0643717
$70$	0	0
$71$	−5.25710	−0.623903	−0.311951	−	0.950098i	$-0.600983\pi$
$71$	−5.25710	−0.623903	−0.311951	+	0.950098i	$0.600983\pi$
$72$	−1.53836	−0.181298
$73$	6.90422	0.808078	0.404039	−	0.914742i	$-0.367606\pi$
$73$	6.90422	0.808078	0.404039	+	0.914742i	$0.367606\pi$
$74$	−11.9616	−1.39051
$75$	0	0
$76$	−3.63340	−0.416780
$77$	9.62594	1.09698
$78$	−1.61475	−0.182835
$79$	5.78996	0.651422	0.325711	−	0.945469i	$-0.394396\pi$
$79$	5.78996	0.651422	0.325711	+	0.945469i	$0.394396\pi$
$80$	0	0
$81$	−3.47088	−0.385653
$82$	−10.6398	−1.17497
$83$	17.8600	1.96040	0.980198	−	0.198022i	$-0.0634518\pi$
$83$	17.8600	1.96040	0.980198	+	0.198022i	$0.0634518\pi$
$84$	−4.19609	−0.457831
$85$	0	0
$86$	−1.88925	−0.203723
$87$	8.66041	0.928493
$88$	4.82613	0.514468
$89$	5.55025	0.588326	0.294163	−	0.955755i	$-0.404959\pi$
$89$	5.55025	0.588326	0.294163	+	0.955755i	$0.404959\pi$
$90$	0	0
$91$	−1.60594	−0.168348
$92$	0.550971	0.0574427
$93$	0.584634	0.0606237
$94$	17.4690	1.80179
$95$	0	0
$96$	−8.55499	−0.873140
$97$	18.0075	1.82838	0.914192	−	0.405281i	$-0.132826\pi$
$97$	18.0075	1.82838	0.914192	+	0.405281i	$0.132826\pi$
$98$	2.44204	0.246683
$99$	−5.21338	−0.523965
$100$	0	0
$101$	2.80411	0.279020	0.139510	−	0.990221i	$-0.455447\pi$
$101$	2.80411	0.279020	0.139510	+	0.990221i	$0.455447\pi$
$102$	−0.429852	−0.0425617
$103$	−12.6590	−1.24732	−0.623662	−	0.781694i	$-0.714356\pi$
$103$	−12.6590	−1.24732	−0.623662	+	0.781694i	$0.714356\pi$
$104$	−0.805166	−0.0789530
$105$	0	0
$106$	−6.11412	−0.593855
$107$	−12.5404	−1.21232	−0.606161	−	0.795342i	$-0.707291\pi$
$107$	−12.5404	−1.21232	−0.606161	+	0.795342i	$0.707291\pi$
$108$	7.56135	0.727591
$109$	2.46607	0.236207	0.118103	−	0.993001i	$-0.462319\pi$
$109$	2.46607	0.236207	0.118103	+	0.993001i	$0.462319\pi$
$110$	0	0
$111$	−8.55111	−0.811636
$112$	−11.6131	−1.09733
$113$	−1.49154	−0.140312	−0.0701561	−	0.997536i	$-0.522350\pi$
$113$	−1.49154	−0.140312	−0.0701561	+	0.997536i	$0.522350\pi$
$114$	−6.45183	−0.604269
$115$	0	0
$116$	−8.92376	−0.828551
$117$	0.869773	0.0804105
$118$	5.27285	0.485405
$119$	−0.427505	−0.0391894
$120$	0	0
$121$	5.35538	0.486853
$122$	12.0080	1.08716
$123$	−7.60619	−0.685827
$124$	−0.602412	−0.0540982
$125$	0	0
$126$	5.61411	0.500145
$127$	4.20601	0.373223	0.186612	−	0.982434i	$-0.440249\pi$
$127$	4.20601	0.373223	0.186612	+	0.982434i	$0.440249\pi$
$128$	−9.03919	−0.798959
$129$	−1.35059	−0.118913
$130$	0	0
$131$	7.52300	0.657287	0.328644	−	0.944454i	$-0.393409\pi$
$131$	7.52300	0.657287	0.328644	+	0.944454i	$0.393409\pi$
$132$	−7.12956	−0.620548
$133$	−6.41661	−0.556390
$134$	−28.0858	−2.42625
$135$	0	0
$136$	−0.214337	−0.0183793
$137$	−15.1717	−1.29621	−0.648104	−	0.761552i	$-0.724438\pi$
$137$	−15.1717	−1.29621	−0.648104	+	0.761552i	$0.724438\pi$
$138$	0.978359	0.0832834
$139$	−17.1023	−1.45060	−0.725299	−	0.688434i	$-0.758298\pi$
$139$	−17.1023	−1.45060	−0.725299	+	0.688434i	$0.758298\pi$
$140$	0	0
$141$	12.4882	1.05170
$142$	9.61889	0.807199
$143$	−2.72864	−0.228181
$144$	6.28962	0.524135
$145$	0	0
$146$	−12.6326	−1.04548
$147$	1.74576	0.143988
$148$	8.81114	0.724271
$149$	−1.99087	−0.163098	−0.0815492	−	0.996669i	$-0.525987\pi$
$149$	−1.99087	−0.163098	−0.0815492	+	0.996669i	$0.525987\pi$
$150$	0	0
$151$	−2.63861	−0.214727	−0.107363	−	0.994220i	$-0.534241\pi$
$151$	−2.63861	−0.214727	−0.107363	+	0.994220i	$0.534241\pi$
$152$	−3.21708	−0.260940
$153$	0.231536	0.0187186
$154$	−17.6125	−1.41926
$155$	0	0
$156$	1.18946	0.0952327
$157$	8.56980	0.683945	0.341972	−	0.939710i	$-0.388905\pi$
$157$	8.56980	0.683945	0.341972	+	0.939710i	$0.388905\pi$
$158$	−10.5939	−0.842803
$159$	−4.37085	−0.346631
$160$	0	0
$161$	0.973018	0.0766845
$162$	6.35064	0.498954
$163$	−24.5931	−1.92628	−0.963140	−	0.269002i	$-0.913306\pi$
$163$	−24.5931	−1.92628	−0.963140	+	0.269002i	$0.913306\pi$
$164$	7.83749	0.612005
$165$	0	0
$166$	−32.6785	−2.53634
$167$	7.74846	0.599594	0.299797	−	0.954003i	$-0.403081\pi$
$167$	7.74846	0.599594	0.299797	+	0.954003i	$0.403081\pi$
$168$	−3.71529	−0.286641
$169$	−12.5448	−0.964982
$170$	0	0
$171$	3.47522	0.265757
$172$	1.39166	0.106113
$173$	−6.38241	−0.485245	−0.242623	−	0.970121i	$-0.578008\pi$
$173$	−6.38241	−0.485245	−0.242623	+	0.970121i	$0.578008\pi$
$174$	−15.8459	−1.20128
$175$	0	0
$176$	−19.7317	−1.48734
$177$	3.76945	0.283329
$178$	−10.1553	−0.761170
$179$	12.8601	0.961209	0.480605	−	0.876937i	$-0.340417\pi$
$179$	12.8601	0.961209	0.480605	+	0.876937i	$0.340417\pi$
$180$	0	0
$181$	2.85534	0.212236	0.106118	−	0.994354i	$-0.466158\pi$
$181$	2.85534	0.212236	0.106118	+	0.994354i	$0.466158\pi$
$182$	2.93838	0.217807
$183$	8.58429	0.634569
$184$	0.487840	0.0359640
$185$	0	0
$186$	−1.06970	−0.0784344
$187$	−0.726372	−0.0531176
$188$	−12.8680	−0.938493
$189$	13.3534	0.971315
$190$	0	0
$191$	20.2612	1.46605	0.733025	−	0.680202i	$-0.238108\pi$
$191$	20.2612	1.46605	0.733025	+	0.680202i	$0.238108\pi$
$192$	2.88934	0.208520
$193$	−2.24690	−0.161735	−0.0808676	−	0.996725i	$-0.525769\pi$
$193$	−2.24690	−0.161735	−0.0808676	+	0.996725i	$0.525769\pi$
$194$	−32.9482	−2.36555
$195$	0	0
$196$	−1.79885	−0.128489
$197$	23.6494	1.68495	0.842476	−	0.538733i	$-0.181097\pi$
$197$	23.6494	1.68495	0.842476	+	0.538733i	$0.181097\pi$
$198$	9.53891	0.677901
$199$	−14.2715	−1.01168	−0.505840	−	0.862628i	$-0.668817\pi$
$199$	−14.2715	−1.01168	−0.505840	+	0.862628i	$0.668817\pi$
$200$	0	0
$201$	−20.0780	−1.41619
$202$	−5.13067	−0.360993
$203$	−15.7594	−1.10609
$204$	0.316637	0.0221690
$205$	0	0
$206$	23.1620	1.61377
$207$	−0.526984	−0.0366279
$208$	3.29193	0.228255
$209$	−10.9024	−0.754136
$210$	0	0
$211$	6.35709	0.437640	0.218820	−	0.975765i	$-0.429779\pi$
$211$	6.35709	0.437640	0.218820	+	0.975765i	$0.429779\pi$
$212$	4.50376	0.309320
$213$	6.87634	0.471159
$214$	22.9450	1.56849
$215$	0	0
$216$	6.69495	0.455534
$217$	−1.06386	−0.0722197
$218$	−4.51216	−0.305602
$219$	−9.03079	−0.610244
$220$	0	0
$221$	0.121184	0.00815172
$222$	15.6459	1.05009
$223$	15.8001	1.05805	0.529025	−	0.848606i	$-0.322558\pi$
$223$	15.8001	1.05805	0.529025	+	0.848606i	$0.322558\pi$
$224$	15.5676	1.04015
$225$	0	0
$226$	2.72906	0.181535
$227$	−0.435612	−0.0289126	−0.0144563	−	0.999896i	$-0.504602\pi$
$227$	−0.435612	−0.0289126	−0.0144563	+	0.999896i	$0.504602\pi$
$228$	4.75253	0.314744
$229$	0.196525	0.0129867	0.00649336	−	0.999979i	$-0.497933\pi$
$229$	0.196525	0.0129867	0.00649336	+	0.999979i	$0.497933\pi$
$230$	0	0
$231$	−12.5908	−0.828416
$232$	−7.90126	−0.518743
$233$	20.3507	1.33322	0.666609	−	0.745408i	$-0.267745\pi$
$233$	20.3507	1.33322	0.666609	+	0.745408i	$0.267745\pi$
$234$	−1.59142	−0.104034
$235$	0	0
$236$	−3.88408	−0.252832
$237$	−7.57333	−0.491941
$238$	0.782205	0.0507028
$239$	−22.3983	−1.44882	−0.724411	−	0.689368i	$-0.757888\pi$
$239$	−22.3983	−1.44882	−0.724411	+	0.689368i	$0.757888\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−9.79872	−0.629885
$243$	−12.2907	−0.788446
$244$	−8.84533	−0.566264
$245$	0	0
$246$	13.9170	0.887317
$247$	1.81890	0.115734
$248$	−0.533386	−0.0338701
$249$	−23.3611	−1.48045
$250$	0	0
$251$	−12.9735	−0.818879	−0.409440	−	0.912337i	$-0.634276\pi$
$251$	−12.9735	−0.818879	−0.409440	+	0.912337i	$0.634276\pi$
$252$	−4.13545	−0.260509
$253$	1.65325	0.103939
$254$	−7.69572	−0.482873
$255$	0	0
$256$	20.9569	1.30981
$257$	−3.11640	−0.194396	−0.0971978	−	0.995265i	$-0.530988\pi$
$257$	−3.11640	−0.194396	−0.0971978	+	0.995265i	$0.530988\pi$
$258$	2.47116	0.153848
$259$	15.5605	0.966883
$260$	0	0
$261$	8.53526	0.528319
$262$	−13.7648	−0.850392
$263$	7.90511	0.487450	0.243725	−	0.969844i	$-0.421631\pi$
$263$	7.90511	0.487450	0.243725	+	0.969844i	$0.421631\pi$
$264$	−6.31264	−0.388516
$265$	0	0
$266$	11.7404	0.719852
$267$	−7.25979	−0.444292
$268$	20.6885	1.26375
$269$	22.3334	1.36169	0.680844	−	0.732428i	$-0.261613\pi$
$269$	22.3334	1.36169	0.680844	+	0.732428i	$0.261613\pi$
$270$	0	0
$271$	−14.5349	−0.882932	−0.441466	−	0.897278i	$-0.645541\pi$
$271$	−14.5349	−0.882932	−0.441466	+	0.897278i	$0.645541\pi$
$272$	0.876322	0.0531348
$273$	2.10058	0.127133
$274$	27.7596	1.67702
$275$	0	0
$276$	−0.720676	−0.0433796
$277$	−22.5163	−1.35287	−0.676437	−	0.736501i	$-0.736477\pi$
$277$	−22.5163	−1.35287	−0.676437	+	0.736501i	$0.736477\pi$
$278$	31.2920	1.87677
$279$	0.576185	0.0344953
$280$	0	0
$281$	14.6336	0.872965	0.436482	−	0.899713i	$-0.356224\pi$
$281$	14.6336	0.872965	0.436482	+	0.899713i	$0.356224\pi$
$282$	−22.8496	−1.36068
$283$	−14.0490	−0.835129	−0.417564	−	0.908647i	$-0.637116\pi$
$283$	−14.0490	−0.835129	−0.417564	+	0.908647i	$0.637116\pi$
$284$	−7.08544	−0.420444
$285$	0	0
$286$	4.99258	0.295218
$287$	13.8410	0.817011
$288$	−8.43136	−0.496823
$289$	−16.9677	−0.998102
$290$	0	0
$291$	−23.5540	−1.38076
$292$	9.30541	0.544558
$293$	−5.19656	−0.303586	−0.151793	−	0.988412i	$-0.548505\pi$
$293$	−5.19656	−0.303586	−0.151793	+	0.988412i	$0.548505\pi$
$294$	−3.19421	−0.186290
$295$	0	0
$296$	7.80154	0.453455
$297$	22.6886	1.31653
$298$	3.64269	0.211015
$299$	−0.275819	−0.0159510
$300$	0	0
$301$	2.45767	0.141658
$302$	4.82785	0.277811
$303$	−3.66781	−0.210710
$304$	13.1531	0.754381
$305$	0	0
$306$	−0.423640	−0.0242179
$307$	−18.2308	−1.04049	−0.520244	−	0.854018i	$-0.674159\pi$
$307$	−18.2308	−1.04049	−0.520244	+	0.854018i	$0.674159\pi$
$308$	12.9737	0.739245
$309$	16.5580	0.941954
$310$	0	0
$311$	26.0257	1.47578	0.737892	−	0.674919i	$-0.235821\pi$
$311$	26.0257	1.47578	0.737892	+	0.674919i	$0.235821\pi$
$312$	1.05317	0.0596238
$313$	21.6565	1.22410	0.612048	−	0.790821i	$-0.290346\pi$
$313$	21.6565	1.22410	0.612048	+	0.790821i	$0.290346\pi$
$314$	−15.6801	−0.884881
$315$	0	0
$316$	7.80363	0.438988
$317$	−4.89194	−0.274759	−0.137379	−	0.990519i	$-0.543868\pi$
$317$	−4.89194	−0.274759	−0.137379	+	0.990519i	$0.543868\pi$
$318$	7.99733	0.448468
$319$	−26.7767	−1.49921
$320$	0	0
$321$	16.4029	0.915522
$322$	−1.78033	−0.0992137
$323$	0.484197	0.0269414
$324$	−4.67800	−0.259889
$325$	0	0
$326$	44.9979	2.49220
$327$	−3.22564	−0.178379
$328$	6.93945	0.383167
$329$	−22.7249	−1.25286
$330$	0	0
$331$	8.70325	0.478374	0.239187	−	0.970974i	$-0.423119\pi$
$331$	8.70325	0.478374	0.239187	+	0.970974i	$0.423119\pi$
$332$	24.0715	1.32110
$333$	−8.42754	−0.461826
$334$	−14.1773	−0.775749
$335$	0	0
$336$	15.1900	0.828684
$337$	−24.0513	−1.31016	−0.655078	−	0.755561i	$-0.727364\pi$
$337$	−24.0513	−1.31016	−0.655078	+	0.755561i	$0.727364\pi$
$338$	22.9531	1.24848
$339$	1.95095	0.105961
$340$	0	0
$341$	−1.80760	−0.0978872
$342$	−6.35859	−0.343833
$343$	−19.8381	−1.07116
$344$	1.23220	0.0664357
$345$	0	0
$346$	11.6779	0.627806
$347$	21.8919	1.17522	0.587609	−	0.809145i	$-0.300069\pi$
$347$	21.8919	1.17522	0.587609	+	0.809145i	$0.300069\pi$
$348$	11.6724	0.625705
$349$	11.5027	0.615725	0.307862	−	0.951431i	$-0.400386\pi$
$349$	11.5027	0.615725	0.307862	+	0.951431i	$0.400386\pi$
$350$	0	0
$351$	−3.78525	−0.202042
$352$	26.4508	1.40983
$353$	3.28049	0.174603	0.0873015	−	0.996182i	$-0.472176\pi$
$353$	3.28049	0.174603	0.0873015	+	0.996182i	$0.472176\pi$
$354$	−6.89695	−0.366569
$355$	0	0
$356$	7.48055	0.396469
$357$	0.559182	0.0295950
$358$	−23.5301	−1.24360
$359$	10.1801	0.537286	0.268643	−	0.963240i	$-0.413425\pi$
$359$	10.1801	0.537286	0.268643	+	0.963240i	$0.413425\pi$
$360$	0	0
$361$	−11.7325	−0.617500
$362$	−5.22441	−0.274589
$363$	−7.00490	−0.367662
$364$	−2.16446	−0.113449
$365$	0	0
$366$	−15.7066	−0.820999
$367$	20.0515	1.04668	0.523340	−	0.852124i	$-0.324686\pi$
$367$	20.0515	1.04668	0.523340	+	0.852124i	$0.324686\pi$
$368$	−1.99454	−0.103973
$369$	−7.49628	−0.390241
$370$	0	0
$371$	7.95367	0.412934
$372$	0.787962	0.0408539
$373$	6.42080	0.332456	0.166228	−	0.986087i	$-0.446841\pi$
$373$	6.42080	0.332456	0.166228	+	0.986087i	$0.446841\pi$
$374$	1.32904	0.0687230
$375$	0	0
$376$	−11.3935	−0.587576
$377$	4.46728	0.230077
$378$	−24.4326	−1.25668
$379$	13.0776	0.671750	0.335875	−	0.941907i	$-0.390968\pi$
$379$	13.0776	0.671750	0.335875	+	0.941907i	$0.390968\pi$
$380$	0	0
$381$	−5.50151	−0.281851
$382$	−37.0719	−1.89676
$383$	−10.7717	−0.550406	−0.275203	−	0.961386i	$-0.588745\pi$
$383$	−10.7717	−0.550406	−0.275203	+	0.961386i	$0.588745\pi$
$384$	11.8234	0.603358
$385$	0	0
$386$	4.11114	0.209251
$387$	−1.33107	−0.0676621
$388$	24.2703	1.23214
$389$	10.3707	0.525816	0.262908	−	0.964821i	$-0.415318\pi$
$389$	10.3707	0.525816	0.262908	+	0.964821i	$0.415318\pi$
$390$	0	0
$391$	−0.0734238	−0.00371320
$392$	−1.59273	−0.0804451
$393$	−9.84016	−0.496370
$394$	−43.2713	−2.17998
$395$	0	0
$396$	−7.02653	−0.353096
$397$	35.3263	1.77298	0.886488	−	0.462752i	$-0.153138\pi$
$397$	35.3263	1.77298	0.886488	+	0.462752i	$0.153138\pi$
$398$	26.1125	1.30890
$399$	8.39299	0.420175
$400$	0	0
$401$	38.5979	1.92749	0.963743	−	0.266832i	$-0.0859770\pi$
$401$	38.5979	1.92749	0.963743	+	0.266832i	$0.0859770\pi$
$402$	36.7366	1.83225
$403$	0.301571	0.0150223
$404$	3.77934	0.188029
$405$	0	0
$406$	28.8349	1.43105
$407$	26.4388	1.31052
$408$	0.280356	0.0138797
$409$	9.00233	0.445137	0.222568	−	0.974917i	$-0.428556\pi$
$409$	9.00233	0.445137	0.222568	+	0.974917i	$0.428556\pi$
$410$	0	0
$411$	19.8448	0.978870
$412$	−17.0616	−0.840562
$413$	−6.85929	−0.337524
$414$	0.964221	0.0473889
$415$	0	0
$416$	−4.41291	−0.216360
$417$	22.3700	1.09546
$418$	19.9481	0.975694
$419$	1.06081	0.0518239	0.0259120	−	0.999664i	$-0.491751\pi$
$419$	1.06081	0.0518239	0.0259120	+	0.999664i	$0.491751\pi$
$420$	0	0
$421$	−6.85413	−0.334050	−0.167025	−	0.985953i	$-0.553416\pi$
$421$	−6.85413	−0.334050	−0.167025	+	0.985953i	$0.553416\pi$
$422$	−11.6315	−0.566215
$423$	12.3077	0.598423
$424$	3.98771	0.193660
$425$	0	0
$426$	−12.5816	−0.609581
$427$	−15.6209	−0.755948
$428$	−16.9017	−0.816976
$429$	3.56909	0.172317
$430$	0	0
$431$	6.01999	0.289973	0.144986	−	0.989434i	$-0.453686\pi$
$431$	6.01999	0.289973	0.144986	+	0.989434i	$0.453686\pi$
$432$	−27.3724	−1.31696
$433$	4.85105	0.233126	0.116563	−	0.993183i	$-0.462812\pi$
$433$	4.85105	0.233126	0.116563	+	0.993183i	$0.462812\pi$
$434$	1.94654	0.0934371
$435$	0	0
$436$	3.32373	0.159178
$437$	−1.10205	−0.0527182
$438$	16.5236	0.789528
$439$	−10.8745	−0.519013	−0.259507	−	0.965741i	$-0.583560\pi$
$439$	−10.8745	−0.519013	−0.259507	+	0.965741i	$0.583560\pi$
$440$	0	0
$441$	1.72053	0.0819301
$442$	−0.221730	−0.0105466
$443$	−6.95807	−0.330588	−0.165294	−	0.986244i	$-0.552857\pi$
$443$	−6.95807	−0.330588	−0.165294	+	0.986244i	$0.552857\pi$
$444$	−11.5251	−0.546956
$445$	0	0
$446$	−28.9093	−1.36889
$447$	2.60408	0.123169
$448$	−5.25776	−0.248406
$449$	11.8216	0.557895	0.278947	−	0.960306i	$-0.410015\pi$
$449$	11.8216	0.557895	0.278947	+	0.960306i	$0.410015\pi$
$450$	0	0
$451$	23.5172	1.10738
$452$	−2.01028	−0.0945554
$453$	3.45132	0.162157
$454$	0.797037	0.0374068
$455$	0	0
$456$	4.20797	0.197056
$457$	−33.0812	−1.54747	−0.773737	−	0.633507i	$-0.781615\pi$
$457$	−33.0812	−1.54747	−0.773737	+	0.633507i	$0.781615\pi$
$458$	−0.359580	−0.0168021
$459$	−1.00764	−0.0470328
$460$	0	0
$461$	28.4181	1.32356	0.661781	−	0.749697i	$-0.269801\pi$
$461$	28.4181	1.32356	0.661781	+	0.749697i	$0.269801\pi$
$462$	23.0374	1.07180
$463$	15.4999	0.720342	0.360171	−	0.932886i	$-0.382718\pi$
$463$	15.4999	0.720342	0.360171	+	0.932886i	$0.382718\pi$
$464$	32.3044	1.49970
$465$	0	0
$466$	−37.2355	−1.72490
$467$	−6.97094	−0.322577	−0.161288	−	0.986907i	$-0.551565\pi$
$467$	−6.97094	−0.322577	−0.161288	+	0.986907i	$0.551565\pi$
$468$	1.17227	0.0541881
$469$	36.5360	1.68708
$470$	0	0
$471$	−11.2094	−0.516502
$472$	−3.43903	−0.158294
$473$	4.17582	0.192004
$474$	13.8569	0.636468
$475$	0	0
$476$	−0.576186	−0.0264094
$477$	−4.30769	−0.197235
$478$	40.9820	1.87447
$479$	28.1264	1.28513	0.642565	−	0.766231i	$-0.277870\pi$
$479$	28.1264	1.28513	0.642565	+	0.766231i	$0.277870\pi$
$480$	0	0
$481$	−4.41090	−0.201120
$482$	−1.82970	−0.0833403
$483$	−1.27272	−0.0579107
$484$	7.21791	0.328087
$485$	0	0
$486$	22.4882	1.02008
$487$	39.0225	1.76828	0.884138	−	0.467226i	$-0.154747\pi$
$487$	39.0225	1.76828	0.884138	+	0.467226i	$0.154747\pi$
$488$	−7.83181	−0.354529
$489$	32.1680	1.45469
$490$	0	0
$491$	38.6915	1.74612	0.873061	−	0.487611i	$-0.162131\pi$
$491$	38.6915	1.74612	0.873061	+	0.487611i	$0.162131\pi$
$492$	−10.2515	−0.462174
$493$	1.18920	0.0535590
$494$	−3.32803	−0.149735
$495$	0	0
$496$	2.18076	0.0979190
$497$	−12.5129	−0.561281
$498$	42.7438	1.91539
$499$	4.92933	0.220667	0.110333	−	0.993895i	$-0.464808\pi$
$499$	4.92933	0.220667	0.110333	+	0.993895i	$0.464808\pi$
$500$	0	0
$501$	−10.1351	−0.452802
$502$	23.7375	1.05946
$503$	−6.37441	−0.284221	−0.142110	−	0.989851i	$-0.545389\pi$
$503$	−6.37441	−0.284221	−0.142110	+	0.989851i	$0.545389\pi$
$504$	−3.66160	−0.163101
$505$	0	0
$506$	−3.02494	−0.134475
$507$	16.4087	0.728735
$508$	5.66880	0.251512
$509$	0.122008	0.00540789	0.00270394	−	0.999996i	$-0.499139\pi$
$509$	0.122008	0.00540789	0.00270394	+	0.999996i	$0.499139\pi$
$510$	0	0
$511$	16.4334	0.726971
$512$	−20.2663	−0.895655
$513$	−15.1242	−0.667748
$514$	5.70206	0.251507
$515$	0	0
$516$	−1.82030	−0.0801344
$517$	−38.6117	−1.69814
$518$	−28.4710	−1.25094
$519$	8.34826	0.366448
$520$	0	0
$521$	−12.6353	−0.553561	−0.276780	−	0.960933i	$-0.589267\pi$
$521$	−12.6353	−0.553561	−0.276780	+	0.960933i	$0.589267\pi$
$522$	−15.6169	−0.683534
$523$	9.15035	0.400117	0.200058	−	0.979784i	$-0.435887\pi$
$523$	9.15035	0.400117	0.200058	+	0.979784i	$0.435887\pi$
$524$	10.1394	0.442941
$525$	0	0
$526$	−14.4639	−0.630658
$527$	0.0802790	0.00349701
$528$	25.8093	1.12321
$529$	−22.8329	−0.992734
$530$	0	0
$531$	3.71498	0.161216
$532$	−8.64821	−0.374947
$533$	−3.92349	−0.169945
$534$	13.2832	0.574821
$535$	0	0
$536$	18.3180	0.791216
$537$	−16.8212	−0.725886
$538$	−40.8632	−1.76174
$539$	−5.39764	−0.232493
$540$	0	0
$541$	22.7731	0.979091	0.489546	−	0.871978i	$-0.337163\pi$
$541$	22.7731	0.979091	0.489546	+	0.871978i	$0.337163\pi$
$542$	26.5944	1.14233
$543$	−3.73482	−0.160276
$544$	−1.17473	−0.0503660
$545$	0	0
$546$	−3.84343	−0.164484
$547$	−31.4075	−1.34289	−0.671445	−	0.741054i	$-0.734326\pi$
$547$	−31.4075	−1.34289	−0.671445	+	0.741054i	$0.734326\pi$
$548$	−20.4482	−0.873505
$549$	8.46024	0.361074
$550$	0	0
$551$	17.8492	0.760404
$552$	−0.638100	−0.0271593
$553$	13.7812	0.586038
$554$	41.1980	1.75033
$555$	0	0
$556$	−23.0502	−0.977547
$557$	9.52366	0.403530	0.201765	−	0.979434i	$-0.435332\pi$
$557$	9.52366	0.403530	0.201765	+	0.979434i	$0.435332\pi$
$558$	−1.05424	−0.0446297
$559$	−0.696671	−0.0294660
$560$	0	0
$561$	0.950103	0.0401134
$562$	−26.7750	−1.12943
$563$	−17.1676	−0.723527	−0.361764	−	0.932270i	$-0.617825\pi$
$563$	−17.1676	−0.723527	−0.361764	+	0.932270i	$0.617825\pi$
$564$	16.8314	0.708731
$565$	0	0
$566$	25.7055	1.08048
$567$	−8.26136	−0.346945
$568$	−6.27358	−0.263233
$569$	2.07125	0.0868313	0.0434156	−	0.999057i	$-0.486176\pi$
$569$	2.07125	0.0868313	0.0434156	+	0.999057i	$0.486176\pi$
$570$	0	0
$571$	−17.4411	−0.729888	−0.364944	−	0.931029i	$-0.618912\pi$
$571$	−17.4411	−0.729888	−0.364944	+	0.931029i	$0.618912\pi$
$572$	−3.67763	−0.153769
$573$	−26.5019	−1.10713
$574$	−25.3249	−1.05704
$575$	0	0
$576$	2.84759	0.118650
$577$	−24.5310	−1.02124	−0.510620	−	0.859806i	$-0.670584\pi$
$577$	−24.5310	−1.02124	−0.510620	+	0.859806i	$0.670584\pi$
$578$	31.0458	1.29134
$579$	2.93897	0.122139
$580$	0	0
$581$	42.5104	1.76363
$582$	43.0966	1.78641
$583$	13.5140	0.559694
$584$	8.23917	0.340939
$585$	0	0
$586$	9.50813	0.392777
$587$	3.73304	0.154079	0.0770396	−	0.997028i	$-0.475453\pi$
$587$	3.73304	0.154079	0.0770396	+	0.997028i	$0.475453\pi$
$588$	2.35291	0.0970324
$589$	1.20494	0.0496487
$590$	0	0
$591$	−30.9337	−1.27244
$592$	−31.8967	−1.31095
$593$	−13.2439	−0.543863	−0.271931	−	0.962317i	$-0.587662\pi$
$593$	−13.2439	−0.543863	−0.271931	+	0.962317i	$0.587662\pi$
$594$	−41.5133	−1.70331
$595$	0	0
$596$	−2.68327	−0.109911
$597$	18.6673	0.764000
$598$	0.504665	0.0206373
$599$	3.90478	0.159545	0.0797724	−	0.996813i	$-0.474581\pi$
$599$	3.90478	0.159545	0.0797724	+	0.996813i	$0.474581\pi$
$600$	0	0
$601$	−17.8815	−0.729401	−0.364701	−	0.931125i	$-0.618829\pi$
$601$	−17.8815	−0.729401	−0.364701	+	0.931125i	$0.618829\pi$
$602$	−4.49679	−0.183276
$603$	−19.7878	−0.805822
$604$	−3.55628	−0.144703
$605$	0	0
$606$	6.71097	0.272615
$607$	12.7914	0.519189	0.259594	−	0.965718i	$-0.416411\pi$
$607$	12.7914	0.519189	0.259594	+	0.965718i	$0.416411\pi$
$608$	−17.6320	−0.715071
$609$	20.6135	0.835300
$610$	0	0
$611$	6.44177	0.260606
$612$	0.312061	0.0126143
$613$	44.9620	1.81600	0.907998	−	0.418974i	$-0.137610\pi$
$613$	44.9620	1.81600	0.907998	+	0.418974i	$0.137610\pi$
$614$	33.3569	1.34617
$615$	0	0
$616$	11.4871	0.462830
$617$	26.2124	1.05527	0.527637	−	0.849470i	$-0.323078\pi$
$617$	26.2124	1.05527	0.527637	+	0.849470i	$0.323078\pi$
$618$	−30.2962	−1.21869
$619$	−42.7956	−1.72010	−0.860050	−	0.510210i	$-0.829568\pi$
$619$	−42.7956	−1.72010	−0.860050	+	0.510210i	$0.829568\pi$
$620$	0	0
$621$	2.29343	0.0920324
$622$	−47.6192	−1.90935
$623$	13.2107	0.529275
$624$	−4.30588	−0.172373
$625$	0	0
$626$	−39.6247	−1.58372
$627$	14.2605	0.569509
$628$	11.5503	0.460906
$629$	−1.17420	−0.0468182
$630$	0	0
$631$	15.9593	0.635329	0.317664	−	0.948203i	$-0.397102\pi$
$631$	15.9593	0.635329	0.317664	+	0.948203i	$0.397102\pi$
$632$	6.90947	0.274844
$633$	−8.31515	−0.330497
$634$	8.95076	0.355480
$635$	0	0
$636$	−5.89097	−0.233592
$637$	0.900512	0.0356796
$638$	48.9933	1.93966
$639$	6.77697	0.268093
$640$	0	0
$641$	5.13101	0.202663	0.101331	−	0.994853i	$-0.467690\pi$
$641$	5.13101	0.202663	0.101331	+	0.994853i	$0.467690\pi$
$642$	−30.0124	−1.18449
$643$	2.22845	0.0878816	0.0439408	−	0.999034i	$-0.486009\pi$
$643$	2.22845	0.0878816	0.0439408	+	0.999034i	$0.486009\pi$
$644$	1.31142	0.0516772
$645$	0	0
$646$	−0.885932	−0.0348565
$647$	7.67318	0.301664	0.150832	−	0.988559i	$-0.451805\pi$
$647$	7.67318	0.301664	0.150832	+	0.988559i	$0.451805\pi$
$648$	−4.14198	−0.162712
$649$	−11.6546	−0.457483
$650$	0	0
$651$	1.39154	0.0545389
$652$	−33.1462	−1.29811
$653$	40.9865	1.60393	0.801963	−	0.597374i	$-0.203789\pi$
$653$	40.9865	1.60393	0.801963	+	0.597374i	$0.203789\pi$
$654$	5.90195	0.230784
$655$	0	0
$656$	−28.3721	−1.10774
$657$	−8.90029	−0.347233
$658$	41.5796	1.62094
$659$	23.2195	0.904503	0.452252	−	0.891890i	$-0.350621\pi$
$659$	23.2195	0.904503	0.452252	+	0.891890i	$0.350621\pi$
$660$	0	0
$661$	−13.2639	−0.515907	−0.257954	−	0.966157i	$-0.583048\pi$
$661$	−13.2639	−0.515907	−0.257954	+	0.966157i	$0.583048\pi$
$662$	−15.9243	−0.618916
$663$	−0.158510	−0.00615602
$664$	21.3134	0.827118
$665$	0	0
$666$	15.4198	0.597506
$667$	−2.70667	−0.104803
$668$	10.4433	0.404062
$669$	−20.6666	−0.799018
$670$	0	0
$671$	−26.5414	−1.02462
$672$	−20.3626	−0.785503
$673$	46.9378	1.80932	0.904660	−	0.426135i	$-0.140125\pi$
$673$	46.9378	1.80932	0.904660	+	0.426135i	$0.140125\pi$
$674$	44.0065	1.69507
$675$	0	0
$676$	−16.9077	−0.650295
$677$	17.1387	0.658694	0.329347	−	0.944209i	$-0.393171\pi$
$677$	17.1387	0.658694	0.329347	+	0.944209i	$0.393171\pi$
$678$	−3.56964	−0.137091
$679$	42.8614	1.64487
$680$	0	0
$681$	0.569785	0.0218342
$682$	3.30736	0.126646
$683$	26.9244	1.03023	0.515117	−	0.857120i	$-0.327749\pi$
$683$	26.9244	1.03023	0.515117	+	0.857120i	$0.327749\pi$
$684$	4.68385	0.179091
$685$	0	0
$686$	36.2978	1.38585
$687$	−0.257056	−0.00980731
$688$	−5.03786	−0.192067
$689$	−2.25461	−0.0858937
$690$	0	0
$691$	17.7690	0.675964	0.337982	−	0.941153i	$-0.390256\pi$
$691$	17.7690	0.675964	0.337982	+	0.941153i	$0.390256\pi$
$692$	−8.60212	−0.327003
$693$	−12.4089	−0.471374
$694$	−40.0555	−1.52049
$695$	0	0
$696$	10.3349	0.391745
$697$	−1.04444	−0.0395611
$698$	−21.0464	−0.796618
$699$	−26.6189	−1.00682
$700$	0	0
$701$	1.31982	0.0498489	0.0249244	−	0.999689i	$-0.492065\pi$
$701$	1.31982	0.0498489	0.0249244	+	0.999689i	$0.492065\pi$
$702$	6.92586	0.261400
$703$	−17.6240	−0.664701
$704$	−8.93343	−0.336691
$705$	0	0
$706$	−6.00230	−0.225900
$707$	6.67434	0.251014
$708$	5.08041	0.190934
$709$	−2.57683	−0.0967749	−0.0483874	−	0.998829i	$-0.515408\pi$
$709$	−2.57683	−0.0967749	−0.0483874	+	0.998829i	$0.515408\pi$
$710$	0	0
$711$	−7.46389	−0.279918
$712$	6.62342	0.248223
$713$	−0.182718	−0.00684284
$714$	−1.02313	−0.0382898
$715$	0	0
$716$	17.3327	0.647752
$717$	29.2972	1.09412
$718$	−18.6265	−0.695136
$719$	40.5061	1.51062	0.755312	−	0.655366i	$-0.227485\pi$
$719$	40.5061	1.51062	0.755312	+	0.655366i	$0.227485\pi$
$720$	0	0
$721$	−30.1308	−1.12213
$722$	21.4669	0.798915
$723$	−1.30801	−0.0486454
$724$	3.84839	0.143024
$725$	0	0
$726$	12.8168	0.475677
$727$	50.5991	1.87662	0.938308	−	0.345801i	$-0.112393\pi$
$727$	50.5991	1.87662	0.938308	+	0.345801i	$0.112393\pi$
$728$	−1.91645	−0.0710285
$729$	26.4889	0.981072
$730$	0	0
$731$	−0.185456	−0.00685933
$732$	11.5698	0.427632
$733$	−4.94419	−0.182618	−0.0913089	−	0.995823i	$-0.529105\pi$
$733$	−4.94419	−0.182618	−0.0913089	+	0.995823i	$0.529105\pi$
$734$	−36.6882	−1.35418
$735$	0	0
$736$	2.67372	0.0985547
$737$	62.0781	2.28668
$738$	13.7159	0.504889
$739$	26.8692	0.988399	0.494199	−	0.869349i	$-0.335461\pi$
$739$	26.8692	0.988399	0.494199	+	0.869349i	$0.335461\pi$
$740$	0	0
$741$	−2.37914	−0.0874000
$742$	−14.5528	−0.534250
$743$	20.6026	0.755836	0.377918	−	0.925839i	$-0.376640\pi$
$743$	20.6026	0.755836	0.377918	+	0.925839i	$0.376640\pi$
$744$	0.697675	0.0255780
$745$	0	0
$746$	−11.7481	−0.430129
$747$	−23.0235	−0.842387
$748$	−0.978994	−0.0357956
$749$	−29.8485	−1.09064
$750$	0	0
$751$	−22.5216	−0.821824	−0.410912	−	0.911675i	$-0.634790\pi$
$751$	−22.5216	−0.821824	−0.410912	+	0.911675i	$0.634790\pi$
$752$	46.5826	1.69869
$753$	16.9695	0.618402
$754$	−8.17377	−0.297671
$755$	0	0
$756$	17.9975	0.654562
$757$	−1.78846	−0.0650025	−0.0325013	−	0.999472i	$-0.510347\pi$
$757$	−1.78846	−0.0650025	−0.0325013	+	0.999472i	$0.510347\pi$
$758$	−23.9280	−0.869103
$759$	−2.16247	−0.0784926
$760$	0	0
$761$	−43.2071	−1.56626	−0.783128	−	0.621860i	$-0.786377\pi$
$761$	−43.2071	−1.56626	−0.783128	+	0.621860i	$0.786377\pi$
$762$	10.0661	0.364656
$763$	5.86973	0.212498
$764$	27.3078	0.987961
$765$	0	0
$766$	19.7088	0.712110
$767$	1.94439	0.0702078
$768$	−27.4118	−0.989140
$769$	13.7423	0.495562	0.247781	−	0.968816i	$-0.420299\pi$
$769$	13.7423	0.495562	0.247781	+	0.968816i	$0.420299\pi$
$770$	0	0
$771$	4.07628	0.146804
$772$	−3.02834	−0.108992
$773$	−11.0021	−0.395716	−0.197858	−	0.980231i	$-0.563399\pi$
$773$	−11.0021	−0.395716	−0.197858	+	0.980231i	$0.563399\pi$
$774$	2.43545	0.0875405
$775$	0	0
$776$	21.4893	0.771421
$777$	−20.3533	−0.730171
$778$	−18.9753	−0.680296
$779$	−15.6765	−0.561669
$780$	0	0
$781$	−21.2606	−0.760766
$782$	0.134343	0.00480411
$783$	−37.1454	−1.32747
$784$	6.51191	0.232568
$785$	0	0
$786$	18.0045	0.642199
$787$	14.1147	0.503133	0.251567	−	0.967840i	$-0.419054\pi$
$787$	14.1147	0.503133	0.251567	+	0.967840i	$0.419054\pi$
$788$	31.8744	1.13548
$789$	−10.3400	−0.368112
$790$	0	0
$791$	−3.55016	−0.126229
$792$	−6.22141	−0.221068
$793$	4.42802	0.157244
$794$	−64.6363	−2.29386
$795$	0	0
$796$	−19.2349	−0.681763
$797$	−13.8969	−0.492255	−0.246128	−	0.969237i	$-0.579158\pi$
$797$	−13.8969	−0.492255	−0.246128	+	0.969237i	$0.579158\pi$
$798$	−15.3566	−0.543618
$799$	1.71482	0.0606659
$800$	0	0
$801$	−7.15488	−0.252805
$802$	−70.6224	−2.49376
$803$	27.9219	0.985342
$804$	−27.0608	−0.954360
$805$	0	0
$806$	−0.551783	−0.0194357
$807$	−29.2123	−1.02832
$808$	3.34630	0.117722
$809$	43.9876	1.54652	0.773261	−	0.634088i	$-0.218624\pi$
$809$	43.9876	1.54652	0.773261	+	0.634088i	$0.218624\pi$
$810$	0	0
$811$	−33.3435	−1.17085	−0.585425	−	0.810727i	$-0.699072\pi$
$811$	−33.3435	−1.17085	−0.585425	+	0.810727i	$0.699072\pi$
$812$	−21.2403	−0.745389
$813$	19.0118	0.666773
$814$	−48.3749	−1.69554
$815$	0	0
$816$	−1.14624	−0.0401264
$817$	−2.78358	−0.0973852
$818$	−16.4715	−0.575914
$819$	2.07023	0.0723396
$820$	0	0
$821$	−48.5742	−1.69525	−0.847625	−	0.530595i	$-0.821969\pi$
$821$	−48.5742	−1.69525	−0.847625	+	0.530595i	$0.821969\pi$
$822$	−36.3099	−1.26645
$823$	−20.6620	−0.720232	−0.360116	−	0.932908i	$-0.617263\pi$
$823$	−20.6620	−0.720232	−0.360116	+	0.932908i	$0.617263\pi$
$824$	−15.1066	−0.526263
$825$	0	0
$826$	12.5504	0.436685
$827$	28.5377	0.992351	0.496176	−	0.868222i	$-0.334737\pi$
$827$	28.5377	0.992351	0.496176	+	0.868222i	$0.334737\pi$
$828$	−0.710262	−0.0246833
$829$	−16.9509	−0.588729	−0.294364	−	0.955693i	$-0.595108\pi$
$829$	−16.9509	−0.588729	−0.294364	+	0.955693i	$0.595108\pi$
$830$	0	0
$831$	29.4516	1.02166
$832$	1.49040	0.0516705
$833$	0.239719	0.00830577
$834$	−40.9302	−1.41730
$835$	0	0
$836$	−14.6941	−0.508207
$837$	−2.50756	−0.0866739
$838$	−1.94096	−0.0670493
$839$	−31.5272	−1.08844	−0.544220	−	0.838943i	$-0.683174\pi$
$839$	−31.5272	−1.08844	−0.544220	+	0.838943i	$0.683174\pi$
$840$	0	0
$841$	14.8384	0.511668
$842$	12.5410	0.432190
$843$	−19.1408	−0.659246
$844$	8.56800	0.294923
$845$	0	0
$846$	−22.5194	−0.774234
$847$	12.7469	0.437987
$848$	−16.3038	−0.559876
$849$	18.3763	0.630673
$850$	0	0
$851$	2.67251	0.0916125
$852$	9.26783	0.317511
$853$	20.9945	0.718839	0.359419	−	0.933176i	$-0.382975\pi$
$853$	20.9945	0.718839	0.359419	+	0.933176i	$0.382975\pi$
$854$	28.5815	0.978038
$855$	0	0
$856$	−14.9651	−0.511496
$857$	−19.7902	−0.676022	−0.338011	−	0.941142i	$-0.609754\pi$
$857$	−19.7902	−0.676022	−0.338011	+	0.941142i	$0.609754\pi$
$858$	−6.53035	−0.222943
$859$	55.7379	1.90175	0.950876	−	0.309572i	$-0.100186\pi$
$859$	55.7379	1.90175	0.950876	+	0.309572i	$0.100186\pi$
$860$	0	0
$861$	−18.1042	−0.616991
$862$	−11.0147	−0.375164
$863$	26.8047	0.912442	0.456221	−	0.889866i	$-0.349203\pi$
$863$	26.8047	0.912442	0.456221	+	0.889866i	$0.349203\pi$
$864$	36.6933	1.24833
$865$	0	0
$866$	−8.87594	−0.301617
$867$	22.1940	0.753747
$868$	−1.43386	−0.0486683
$869$	23.4156	0.794321
$870$	0	0
$871$	−10.3568	−0.350926
$872$	2.94289	0.0996589
$873$	−23.2136	−0.785662
$874$	2.01641	0.0682062
$875$	0	0
$876$	−12.1716	−0.411239
$877$	26.9711	0.910750	0.455375	−	0.890300i	$-0.349505\pi$
$877$	26.9711	0.910750	0.455375	+	0.890300i	$0.349505\pi$
$878$	19.8971	0.671494
$879$	6.79716	0.229262
$880$	0	0
$881$	12.7603	0.429907	0.214953	−	0.976624i	$-0.431040\pi$
$881$	12.7603	0.429907	0.214953	+	0.976624i	$0.431040\pi$
$882$	−3.14805	−0.106000
$883$	28.3020	0.952437	0.476218	−	0.879327i	$-0.342007\pi$
$883$	28.3020	0.952437	0.476218	+	0.879327i	$0.342007\pi$
$884$	0.163330	0.00549338
$885$	0	0
$886$	12.7312	0.427712
$887$	37.4927	1.25888	0.629440	−	0.777049i	$-0.283284\pi$
$887$	37.4927	1.25888	0.629440	+	0.777049i	$0.283284\pi$
$888$	−10.2045	−0.342441
$889$	10.0111	0.335763
$890$	0	0
$891$	−14.0368	−0.470252
$892$	21.2951	0.713012
$893$	25.7384	0.861303
$894$	−4.76467	−0.159355
$895$	0	0
$896$	−21.5151	−0.718767
$897$	0.360774	0.0120459
$898$	−21.6299	−0.721799
$899$	2.95938	0.0987007
$900$	0	0
$901$	−0.600183	−0.0199950
$902$	−43.0294	−1.43272
$903$	−3.21466	−0.106977
$904$	−1.77993	−0.0591997
$905$	0	0
$906$	−6.31487	−0.209798
$907$	18.1253	0.601842	0.300921	−	0.953649i	$-0.402706\pi$
$907$	18.1253	0.601842	0.300921	+	0.953649i	$0.402706\pi$
$908$	−0.587112	−0.0194840
$909$	−3.61481	−0.119896
$910$	0	0
$911$	−13.6975	−0.453817	−0.226909	−	0.973916i	$-0.572862\pi$
$911$	−13.6975	−0.453817	−0.226909	+	0.973916i	$0.572862\pi$
$912$	−17.2044	−0.569694
$913$	72.2292	2.39044
$914$	60.5286	2.00211
$915$	0	0
$916$	0.264873	0.00875166
$917$	17.9062	0.591315
$918$	1.84368	0.0608506
$919$	−17.7519	−0.585581	−0.292791	−	0.956177i	$-0.594584\pi$
$919$	−17.7519	−0.585581	−0.292791	+	0.956177i	$0.594584\pi$
$920$	0	0
$921$	23.8461	0.785756
$922$	−51.9964	−1.71241
$923$	3.54701	0.116751
$924$	−16.9697	−0.558263
$925$	0	0
$926$	−28.3601	−0.931972
$927$	16.3188	0.535978
$928$	−43.3047	−1.42155
$929$	34.6402	1.13651	0.568254	−	0.822853i	$-0.307619\pi$
$929$	34.6402	1.13651	0.568254	+	0.822853i	$0.307619\pi$
$930$	0	0
$931$	3.59804	0.117921
$932$	27.4284	0.898446
$933$	−34.0419	−1.11448
$934$	12.7547	0.417346
$935$	0	0
$936$	1.03795	0.0339263
$937$	−33.3169	−1.08842	−0.544208	−	0.838950i	$-0.683170\pi$
$937$	−33.3169	−1.08842	−0.544208	+	0.838950i	$0.683170\pi$
$938$	−66.8498	−2.18272
$939$	−28.3269	−0.924412
$940$	0	0
$941$	−35.5187	−1.15788	−0.578938	−	0.815372i	$-0.696533\pi$
$941$	−35.5187	−1.15788	−0.578938	+	0.815372i	$0.696533\pi$
$942$	20.5098	0.668245
$943$	2.37719	0.0774120
$944$	14.0605	0.457631
$945$	0	0
$946$	−7.64048	−0.248413
$947$	−35.8730	−1.16571	−0.582857	−	0.812574i	$-0.698065\pi$
$947$	−35.8730	−1.16571	−0.582857	+	0.812574i	$0.698065\pi$
$948$	−10.2072	−0.331515
$949$	−4.65834	−0.151216
$950$	0	0
$951$	6.39871	0.207492
$952$	−0.510165	−0.0165345
$953$	36.1237	1.17016	0.585081	−	0.810975i	$-0.301063\pi$
$953$	36.1237	1.17016	0.585081	+	0.810975i	$0.301063\pi$
$954$	7.88176	0.255181
$955$	0	0
$956$	−30.1881	−0.976351
$957$	35.0242	1.13217
$958$	−51.4628	−1.66269
$959$	−36.1117	−1.16611
$960$	0	0
$961$	−30.8002	−0.993556
$962$	8.07061	0.260207
$963$	16.1659	0.520938
$964$	1.34779	0.0434093
$965$	0	0
$966$	2.32869	0.0749242
$967$	−7.85468	−0.252589	−0.126295	−	0.991993i	$-0.540308\pi$
$967$	−7.85468	−0.252589	−0.126295	+	0.991993i	$0.540308\pi$
$968$	6.39086	0.205410
$969$	−0.633334	−0.0203456
$970$	0	0
$971$	−16.3966	−0.526192	−0.263096	−	0.964770i	$-0.584744\pi$
$971$	−16.3966	−0.526192	−0.263096	+	0.964770i	$0.584744\pi$
$972$	−16.5652	−0.531328
$973$	−40.7068	−1.30500
$974$	−71.3992	−2.28778
$975$	0	0
$976$	32.0205	1.02495
$977$	−7.82651	−0.250392	−0.125196	−	0.992132i	$-0.539956\pi$
$977$	−7.82651	−0.250392	−0.125196	+	0.992132i	$0.539956\pi$
$978$	−58.8577	−1.88206
$979$	22.4462	0.717384
$980$	0	0
$981$	−3.17903	−0.101499
$982$	−70.7936	−2.25912
$983$	−4.20613	−0.134155	−0.0670774	−	0.997748i	$-0.521367\pi$
$983$	−4.20613	−0.134155	−0.0670774	+	0.997748i	$0.521367\pi$
$984$	−9.07688	−0.289360
$985$	0	0
$986$	−2.17588	−0.0692941
$987$	29.7244	0.946138
$988$	2.45149	0.0779922
$989$	0.422104	0.0134221
$990$	0	0
$991$	2.98329	0.0947672	0.0473836	−	0.998877i	$-0.484912\pi$
$991$	2.98329	0.0947672	0.0473836	+	0.998877i	$0.484912\pi$
$992$	−2.92335	−0.0928165
$993$	−11.3839	−0.361259
$994$	22.8948	0.726180
$995$	0	0
$996$	−31.4858	−0.997666
$997$	3.01518	0.0954917	0.0477458	−	0.998860i	$-0.484796\pi$
$997$	3.01518	0.0954917	0.0477458	+	0.998860i	$0.484796\pi$
$998$	−9.01916	−0.285497
$999$	36.6766	1.16040

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.n.1.8	yes	40
5.4	even	2		6025.2.a.m.1.33	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.m.1.33	✓	40	5.4	even	2
6025.2.a.n.1.8	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)

\(f(q)\)	\(=\)	\(q-1.82970 q^{2} -1.30801 q^{3} +1.34779 q^{4} +2.39326 q^{6} +2.38020 q^{7} +1.19335 q^{8} -1.28911 q^{9} +O(q^{10})\)	\(q-1.82970 q^{2} -1.30801 q^{3} +1.34779 q^{4} +2.39326 q^{6} +2.38020 q^{7} +1.19335 q^{8} -1.28911 q^{9} +4.04418 q^{11} -1.76292 q^{12} -0.674709 q^{13} -4.35503 q^{14} -4.87905 q^{16} -0.179609 q^{17} +2.35868 q^{18} -2.69583 q^{19} -3.11332 q^{21} -7.39962 q^{22} +0.408797 q^{23} -1.56092 q^{24} +1.23451 q^{26} +5.61020 q^{27} +3.20799 q^{28} -6.62106 q^{29} -0.446964 q^{31} +6.54046 q^{32} -5.28983 q^{33} +0.328630 q^{34} -1.73744 q^{36} +6.53749 q^{37} +4.93255 q^{38} +0.882526 q^{39} +5.81509 q^{41} +5.69643 q^{42} +1.03255 q^{43} +5.45069 q^{44} -0.747975 q^{46} -9.54749 q^{47} +6.38184 q^{48} -1.33467 q^{49} +0.234931 q^{51} -0.909363 q^{52} +3.34160 q^{53} -10.2650 q^{54} +2.84041 q^{56} +3.52618 q^{57} +12.1145 q^{58} -2.88182 q^{59} -6.56286 q^{61} +0.817809 q^{62} -3.06833 q^{63} -2.20896 q^{64} +9.67878 q^{66} +15.3500 q^{67} -0.242075 q^{68} -0.534711 q^{69} -5.25710 q^{71} -1.53836 q^{72} +6.90422 q^{73} -11.9616 q^{74} -3.63340 q^{76} +9.62594 q^{77} -1.61475 q^{78} +5.78996 q^{79} -3.47088 q^{81} -10.6398 q^{82} +17.8600 q^{83} -4.19609 q^{84} -1.88925 q^{86} +8.66041 q^{87} +4.82613 q^{88} +5.55025 q^{89} -1.60594 q^{91} +0.550971 q^{92} +0.584634 q^{93} +17.4690 q^{94} -8.55499 q^{96} +18.0075 q^{97} +2.44204 q^{98} -5.21338 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 9 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 20 q^{7} + 27 q^{8} + 38 q^{9}+O(q^{10}) \) 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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