LMFDB - Embedded newform 6025.2.a.j.1.11

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:	$1$
Dimension:	$25$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.782724 q^{2} +1.54504 q^{3} -1.38734 q^{4} -1.20934 q^{6} +2.90223 q^{7} +2.65136 q^{8} -0.612841 q^{9} +O(q^{10})$	\(q-0.782724 q^{2} +1.54504 q^{3} -1.38734 q^{4} -1.20934 q^{6} +2.90223 q^{7} +2.65136 q^{8} -0.612841 q^{9} -1.66779 q^{11} -2.14350 q^{12} -0.960719 q^{13} -2.27164 q^{14} +0.699406 q^{16} +0.171133 q^{17} +0.479686 q^{18} +3.15296 q^{19} +4.48407 q^{21} +1.30542 q^{22} -5.15649 q^{23} +4.09646 q^{24} +0.751978 q^{26} -5.58200 q^{27} -4.02639 q^{28} -5.41990 q^{29} +9.68125 q^{31} -5.85015 q^{32} -2.57681 q^{33} -0.133950 q^{34} +0.850221 q^{36} -0.699090 q^{37} -2.46790 q^{38} -1.48435 q^{39} -9.66061 q^{41} -3.50979 q^{42} +1.23181 q^{43} +2.31380 q^{44} +4.03611 q^{46} -2.13755 q^{47} +1.08061 q^{48} +1.42293 q^{49} +0.264408 q^{51} +1.33285 q^{52} -12.1850 q^{53} +4.36916 q^{54} +7.69484 q^{56} +4.87146 q^{57} +4.24229 q^{58} -8.87122 q^{59} +5.46623 q^{61} -7.57775 q^{62} -1.77861 q^{63} +3.18024 q^{64} +2.01693 q^{66} -3.38772 q^{67} -0.237420 q^{68} -7.96700 q^{69} +15.1252 q^{71} -1.62486 q^{72} +3.52794 q^{73} +0.547195 q^{74} -4.37424 q^{76} -4.84031 q^{77} +1.16184 q^{78} +11.0823 q^{79} -6.78590 q^{81} +7.56160 q^{82} -11.6227 q^{83} -6.22094 q^{84} -0.964167 q^{86} -8.37399 q^{87} -4.42191 q^{88} -13.8337 q^{89} -2.78823 q^{91} +7.15382 q^{92} +14.9579 q^{93} +1.67311 q^{94} -9.03874 q^{96} -2.32891 q^{97} -1.11376 q^{98} +1.02209 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9}+O(q^{10}) $ 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9	$ 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9} + 2 q^{11} - 20 q^{12} - 14 q^{13} - 5 q^{14} + 38 q^{16} - 7 q^{17} - 9 q^{18} + 30 q^{19} + q^{21} - q^{22} - 43 q^{23} - 6 q^{24} - 22 q^{26} - 42 q^{27} - 32 q^{28} - 4 q^{29} + 14 q^{31} - 26 q^{32} - 4 q^{33} + 7 q^{34} + 15 q^{36} - 16 q^{37} - 14 q^{38} - 21 q^{39} - q^{41} + 25 q^{42} - 35 q^{43} - 52 q^{44} - 27 q^{46} - 50 q^{47} - 26 q^{48} + 46 q^{49} - 7 q^{51} - 3 q^{52} - 4 q^{53} - 31 q^{54} - 51 q^{56} - 2 q^{58} + 6 q^{59} + 19 q^{61} - 28 q^{63} + 49 q^{64} - 27 q^{66} - 65 q^{67} + 25 q^{68} + 2 q^{69} - 34 q^{71} + 10 q^{72} - 8 q^{73} - 42 q^{74} + 71 q^{76} - q^{77} + 59 q^{78} - 12 q^{79} + 29 q^{81} - 11 q^{82} - 41 q^{83} - 10 q^{84} - 13 q^{86} - 40 q^{87} + 52 q^{88} - 24 q^{89} + 46 q^{91} - 85 q^{92} + 30 q^{93} + 14 q^{94} - 30 q^{96} - 9 q^{97} + 64 q^{98} + 2 q^{99}+O(q^{100}) $ 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9 + 2 * q^11 - 20 * q^12 - 14 * q^13 - 5 * q^14 + 38 * q^16 - 7 * q^17 - 9 * q^18 + 30 * q^19 + q^21 - q^22 - 43 * q^23 - 6 * q^24 - 22 * q^26 - 42 * q^27 - 32 * q^28 - 4 * q^29 + 14 * q^31 - 26 * q^32 - 4 * q^33 + 7 * q^34 + 15 * q^36 - 16 * q^37 - 14 * q^38 - 21 * q^39 - q^41 + 25 * q^42 - 35 * q^43 - 52 * q^44 - 27 * q^46 - 50 * q^47 - 26 * q^48 + 46 * q^49 - 7 * q^51 - 3 * q^52 - 4 * q^53 - 31 * q^54 - 51 * q^56 - 2 * q^58 + 6 * q^59 + 19 * q^61 - 28 * q^63 + 49 * q^64 - 27 * q^66 - 65 * q^67 + 25 * q^68 + 2 * q^69 - 34 * q^71 + 10 * q^72 - 8 * q^73 - 42 * q^74 + 71 * q^76 - q^77 + 59 * q^78 - 12 * q^79 + 29 * q^81 - 11 * q^82 - 41 * q^83 - 10 * q^84 - 13 * q^86 - 40 * q^87 + 52 * q^88 - 24 * q^89 + 46 * q^91 - 85 * q^92 + 30 * q^93 + 14 * q^94 - 30 * q^96 - 9 * q^97 + 64 * q^98 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.782724	−0.553470	−0.276735	−	0.960946i	$-0.589252\pi$
$2$	−0.782724	−0.553470	−0.276735	+	0.960946i	$0.589252\pi$
$3$	1.54504	0.892031	0.446016	−	0.895025i	$-0.352843\pi$
$3$	1.54504	0.892031	0.446016	+	0.895025i	$0.352843\pi$
$4$	−1.38734	−0.693671
$5$	0	0
$5$	0	0
$6$	−1.20934	−0.493712
$7$	2.90223	1.09694	0.548470	−	0.836171i	$-0.315211\pi$
$7$	2.90223	1.09694	0.548470	+	0.836171i	$0.315211\pi$
$8$	2.65136	0.937396
$9$	−0.612841	−0.204280
$10$	0	0
$11$	−1.66779	−0.502858	−0.251429	−	0.967876i	$-0.580900\pi$
$11$	−1.66779	−0.502858	−0.251429	+	0.967876i	$0.580900\pi$
$12$	−2.14350	−0.618777
$13$	−0.960719	−0.266456	−0.133228	−	0.991085i	$-0.542534\pi$
$13$	−0.960719	−0.266456	−0.133228	+	0.991085i	$0.542534\pi$
$14$	−2.27164	−0.607122
$15$	0	0
$16$	0.699406	0.174852
$17$	0.171133	0.0415058	0.0207529	−	0.999785i	$-0.493394\pi$
$17$	0.171133	0.0415058	0.0207529	+	0.999785i	$0.493394\pi$
$18$	0.479686	0.113063
$19$	3.15296	0.723339	0.361669	−	0.932306i	$-0.382207\pi$
$19$	3.15296	0.723339	0.361669	+	0.932306i	$0.382207\pi$
$20$	0	0
$21$	4.48407	0.978504
$22$	1.30542	0.278317
$23$	−5.15649	−1.07520	−0.537601	−	0.843199i	$-0.680669\pi$
$23$	−5.15649	−1.07520	−0.537601	+	0.843199i	$0.680669\pi$
$24$	4.09646	0.836186
$25$	0	0
$26$	0.751978	0.147475
$27$	−5.58200	−1.07426
$28$	−4.02639	−0.760915
$29$	−5.41990	−1.00645	−0.503225	−	0.864155i	$-0.667853\pi$
$29$	−5.41990	−1.00645	−0.503225	+	0.864155i	$0.667853\pi$
$30$	0	0
$31$	9.68125	1.73880	0.869402	−	0.494106i	$-0.164505\pi$
$31$	9.68125	1.73880	0.869402	+	0.494106i	$0.164505\pi$
$32$	−5.85015	−1.03417
$33$	−2.57681	−0.448565
$34$	−0.133950	−0.0229722
$35$	0	0
$36$	0.850221	0.141704
$37$	−0.699090	−0.114930	−0.0574649	−	0.998348i	$-0.518302\pi$
$37$	−0.699090	−0.114930	−0.0574649	+	0.998348i	$0.518302\pi$
$38$	−2.46790	−0.400346
$39$	−1.48435	−0.237687
$40$	0	0
$41$	−9.66061	−1.50873	−0.754367	−	0.656453i	$-0.772056\pi$
$41$	−9.66061	−1.50873	−0.754367	+	0.656453i	$0.772056\pi$
$42$	−3.50979	−0.541572
$43$	1.23181	0.187849	0.0939246	−	0.995579i	$-0.470059\pi$
$43$	1.23181	0.187849	0.0939246	+	0.995579i	$0.470059\pi$
$44$	2.31380	0.348818
$45$	0	0
$46$	4.03611	0.595092
$47$	−2.13755	−0.311794	−0.155897	−	0.987773i	$-0.549827\pi$
$47$	−2.13755	−0.311794	−0.155897	+	0.987773i	$0.549827\pi$
$48$	1.08061	0.155973
$49$	1.42293	0.203275
$50$	0	0
$51$	0.264408	0.0370245
$52$	1.33285	0.184833
$53$	−12.1850	−1.67373	−0.836866	−	0.547408i	$-0.815615\pi$
$53$	−12.1850	−1.67373	−0.836866	+	0.547408i	$0.815615\pi$
$54$	4.36916	0.594568
$55$	0	0
$56$	7.69484	1.02827
$57$	4.87146	0.645241
$58$	4.24229	0.557040
$59$	−8.87122	−1.15494	−0.577468	−	0.816414i	$-0.695959\pi$
$59$	−8.87122	−1.15494	−0.577468	+	0.816414i	$0.695959\pi$
$60$	0	0
$61$	5.46623	0.699878	0.349939	−	0.936772i	$-0.386202\pi$
$61$	5.46623	0.699878	0.349939	+	0.936772i	$0.386202\pi$
$62$	−7.57775	−0.962375
$63$	−1.77861	−0.224083
$64$	3.18024	0.397530
$65$	0	0
$66$	2.01693	0.248267
$67$	−3.38772	−0.413876	−0.206938	−	0.978354i	$-0.566350\pi$
$67$	−3.38772	−0.413876	−0.206938	+	0.978354i	$0.566350\pi$
$68$	−0.237420	−0.0287914
$69$	−7.96700	−0.959114
$70$	0	0
$71$	15.1252	1.79504	0.897518	−	0.440978i	$-0.145368\pi$
$71$	15.1252	1.79504	0.897518	+	0.440978i	$0.145368\pi$
$72$	−1.62486	−0.191492
$73$	3.52794	0.412914	0.206457	−	0.978456i	$-0.433807\pi$
$73$	3.52794	0.412914	0.206457	+	0.978456i	$0.433807\pi$
$74$	0.547195	0.0636101
$75$	0	0
$76$	−4.37424	−0.501759
$77$	−4.84031	−0.551605
$78$	1.16184	0.131552
$79$	11.0823	1.24686	0.623430	−	0.781880i	$-0.285739\pi$
$79$	11.0823	1.24686	0.623430	+	0.781880i	$0.285739\pi$
$80$	0	0
$81$	−6.78590	−0.753989
$82$	7.56160	0.835039
$83$	−11.6227	−1.27576	−0.637880	−	0.770136i	$-0.720188\pi$
$83$	−11.6227	−1.27576	−0.637880	+	0.770136i	$0.720188\pi$
$84$	−6.22094	−0.678760
$85$	0	0
$86$	−0.964167	−0.103969
$87$	−8.37399	−0.897786
$88$	−4.42191	−0.471377
$89$	−13.8337	−1.46637	−0.733187	−	0.680027i	$-0.761968\pi$
$89$	−13.8337	−1.46637	−0.733187	+	0.680027i	$0.761968\pi$
$90$	0	0
$91$	−2.78823	−0.292286
$92$	7.15382	0.745837
$93$	14.9579	1.55107
$94$	1.67311	0.172569
$95$	0	0
$96$	−9.03874	−0.922512
$97$	−2.32891	−0.236465	−0.118232	−	0.992986i	$-0.537723\pi$
$97$	−2.32891	−0.236465	−0.118232	+	0.992986i	$0.537723\pi$
$98$	−1.11376	−0.112507
$99$	1.02209	0.102724
$100$	0	0
$101$	17.2100	1.71246	0.856228	−	0.516597i	$-0.172802\pi$
$101$	17.2100	1.71246	0.856228	+	0.516597i	$0.172802\pi$
$102$	−0.206958	−0.0204919
$103$	12.3128	1.21322	0.606610	−	0.794999i	$-0.292529\pi$
$103$	12.3128	1.21322	0.606610	+	0.794999i	$0.292529\pi$
$104$	−2.54721	−0.249774
$105$	0	0
$106$	9.53746	0.926360
$107$	1.76866	0.170982	0.0854912	−	0.996339i	$-0.472754\pi$
$107$	1.76866	0.170982	0.0854912	+	0.996339i	$0.472754\pi$
$108$	7.74414	0.745180
$109$	−4.61122	−0.441675	−0.220838	−	0.975311i	$-0.570879\pi$
$109$	−4.61122	−0.441675	−0.220838	+	0.975311i	$0.570879\pi$
$110$	0	0
$111$	−1.08012	−0.102521
$112$	2.02984	0.191801
$113$	−9.59482	−0.902605	−0.451302	−	0.892371i	$-0.649040\pi$
$113$	−9.59482	−0.902605	−0.451302	+	0.892371i	$0.649040\pi$
$114$	−3.81301	−0.357121
$115$	0	0
$116$	7.51927	0.698146
$117$	0.588769	0.0544317
$118$	6.94372	0.639221
$119$	0.496667	0.0455294
$120$	0	0
$121$	−8.21847	−0.747134
$122$	−4.27855	−0.387361
$123$	−14.9261	−1.34584
$124$	−13.4312	−1.20616
$125$	0	0
$126$	1.39216	0.124023
$127$	−17.2240	−1.52838	−0.764191	−	0.644990i	$-0.776861\pi$
$127$	−17.2240	−1.52838	−0.764191	+	0.644990i	$0.776861\pi$
$128$	9.21105	0.814150
$129$	1.90320	0.167567
$130$	0	0
$131$	−9.31936	−0.814236	−0.407118	−	0.913375i	$-0.633466\pi$
$131$	−9.31936	−0.814236	−0.407118	+	0.913375i	$0.633466\pi$
$132$	3.57492	0.311157
$133$	9.15061	0.793459
$134$	2.65165	0.229068
$135$	0	0
$136$	0.453734	0.0389074
$137$	−2.20585	−0.188458	−0.0942291	−	0.995551i	$-0.530039\pi$
$137$	−2.20585	−0.188458	−0.0942291	+	0.995551i	$0.530039\pi$
$138$	6.23596	0.530840
$139$	−2.04596	−0.173536	−0.0867679	−	0.996229i	$-0.527654\pi$
$139$	−2.04596	−0.173536	−0.0867679	+	0.996229i	$0.527654\pi$
$140$	0	0
$141$	−3.30261	−0.278130
$142$	−11.8389	−0.993498
$143$	1.60228	0.133989
$144$	−0.428625	−0.0357188
$145$	0	0
$146$	−2.76140	−0.228535
$147$	2.19849	0.181328
$148$	0.969878	0.0797235
$149$	−19.8609	−1.62707	−0.813533	−	0.581519i	$-0.802459\pi$
$149$	−19.8609	−1.62707	−0.813533	+	0.581519i	$0.802459\pi$
$150$	0	0
$151$	9.22229	0.750499	0.375250	−	0.926924i	$-0.377557\pi$
$151$	9.22229	0.750499	0.375250	+	0.926924i	$0.377557\pi$
$152$	8.35962	0.678055
$153$	−0.104877	−0.00847883
$154$	3.78863	0.305296
$155$	0	0
$156$	2.05931	0.164876
$157$	−4.97110	−0.396737	−0.198368	−	0.980128i	$-0.563564\pi$
$157$	−4.97110	−0.396737	−0.198368	+	0.980128i	$0.563564\pi$
$158$	−8.67440	−0.690099
$159$	−18.8263	−1.49302
$160$	0	0
$161$	−14.9653	−1.17943
$162$	5.31149	0.417310
$163$	−4.20903	−0.329677	−0.164838	−	0.986321i	$-0.552710\pi$
$163$	−4.20903	−0.329677	−0.164838	+	0.986321i	$0.552710\pi$
$164$	13.4026	1.04657
$165$	0	0
$166$	9.09739	0.706094
$167$	−10.6989	−0.827909	−0.413954	−	0.910298i	$-0.635853\pi$
$167$	−10.6989	−0.827909	−0.413954	+	0.910298i	$0.635853\pi$
$168$	11.8889	0.917245
$169$	−12.0770	−0.929001
$170$	0	0
$171$	−1.93226	−0.147764
$172$	−1.70894	−0.130306
$173$	−9.08125	−0.690434	−0.345217	−	0.938523i	$-0.612195\pi$
$173$	−9.08125	−0.690434	−0.345217	+	0.938523i	$0.612195\pi$
$174$	6.55452	0.496897
$175$	0	0
$176$	−1.16646	−0.0879255
$177$	−13.7064	−1.03024
$178$	10.8280	0.811594
$179$	7.53959	0.563535	0.281768	−	0.959483i	$-0.409079\pi$
$179$	7.53959	0.563535	0.281768	+	0.959483i	$0.409079\pi$
$180$	0	0
$181$	−10.3885	−0.772174	−0.386087	−	0.922462i	$-0.626174\pi$
$181$	−10.3885	−0.772174	−0.386087	+	0.922462i	$0.626174\pi$
$182$	2.18241	0.161771
$183$	8.44556	0.624313
$184$	−13.6717	−1.00789
$185$	0	0
$186$	−11.7079	−0.858468
$187$	−0.285414	−0.0208715
$188$	2.96552	0.216283
$189$	−16.2002	−1.17839
$190$	0	0
$191$	17.7008	1.28078	0.640391	−	0.768049i	$-0.278772\pi$
$191$	17.7008	1.28078	0.640391	+	0.768049i	$0.278772\pi$
$192$	4.91361	0.354609
$193$	−10.6107	−0.763775	−0.381887	−	0.924209i	$-0.624726\pi$
$193$	−10.6107	−0.763775	−0.381887	+	0.924209i	$0.624726\pi$
$194$	1.82289	0.130876
$195$	0	0
$196$	−1.97409	−0.141006
$197$	21.7613	1.55043	0.775213	−	0.631700i	$-0.217643\pi$
$197$	21.7613	1.55043	0.775213	+	0.631700i	$0.217643\pi$
$198$	−0.800016	−0.0568546
$199$	7.97231	0.565142	0.282571	−	0.959246i	$-0.408813\pi$
$199$	7.97231	0.565142	0.282571	+	0.959246i	$0.408813\pi$
$200$	0	0
$201$	−5.23417	−0.369190
$202$	−13.4707	−0.947793
$203$	−15.7298	−1.10402
$204$	−0.366824	−0.0256828
$205$	0	0
$206$	−9.63756	−0.671481
$207$	3.16011	0.219643
$208$	−0.671933	−0.0465902
$209$	−5.25848	−0.363737
$210$	0	0
$211$	−0.774749	−0.0533359	−0.0266680	−	0.999644i	$-0.508490\pi$
$211$	−0.774749	−0.0533359	−0.0266680	+	0.999644i	$0.508490\pi$
$212$	16.9047	1.16102
$213$	23.3692	1.60123
$214$	−1.38437	−0.0946336
$215$	0	0
$216$	−14.7999	−1.00700
$217$	28.0972	1.90736
$218$	3.60932	0.244454
$219$	5.45082	0.368332
$220$	0	0
$221$	−0.164411	−0.0110595
$222$	0.845439	0.0567422
$223$	10.0760	0.674736	0.337368	−	0.941373i	$-0.390463\pi$
$223$	10.0760	0.674736	0.337368	+	0.941373i	$0.390463\pi$
$224$	−16.9785	−1.13442
$225$	0	0
$226$	7.51010	0.499564
$227$	18.0345	1.19699	0.598494	−	0.801127i	$-0.295766\pi$
$227$	18.0345	1.19699	0.598494	+	0.801127i	$0.295766\pi$
$228$	−6.75839	−0.447585
$229$	12.3223	0.814282	0.407141	−	0.913365i	$-0.366526\pi$
$229$	12.3223	0.814282	0.407141	+	0.913365i	$0.366526\pi$
$230$	0	0
$231$	−7.47849	−0.492048
$232$	−14.3701	−0.943443
$233$	5.31403	0.348133	0.174067	−	0.984734i	$-0.444309\pi$
$233$	5.31403	0.348133	0.174067	+	0.984734i	$0.444309\pi$
$234$	−0.460843	−0.0301263
$235$	0	0
$236$	12.3074	0.801145
$237$	17.1227	1.11224
$238$	−0.388753	−0.0251991
$239$	−20.1067	−1.30059	−0.650297	−	0.759680i	$-0.725355\pi$
$239$	−20.1067	−1.30059	−0.650297	+	0.759680i	$0.725355\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	6.43280	0.413516
$243$	6.26148	0.401674
$244$	−7.58353	−0.485486
$245$	0	0
$246$	11.6830	0.744880
$247$	−3.02911	−0.192738
$248$	25.6684	1.62995
$249$	−17.9576	−1.13802
$250$	0	0
$251$	28.7710	1.81601	0.908003	−	0.418963i	$-0.137606\pi$
$251$	28.7710	1.81601	0.908003	+	0.418963i	$0.137606\pi$
$252$	2.46754	0.155440
$253$	8.59995	0.540674
$254$	13.4816	0.845913
$255$	0	0
$256$	−13.5702	−0.848137
$257$	−20.7242	−1.29274	−0.646370	−	0.763024i	$-0.723714\pi$
$257$	−20.7242	−1.29274	−0.646370	+	0.763024i	$0.723714\pi$
$258$	−1.48968	−0.0927434
$259$	−2.02892	−0.126071
$260$	0	0
$261$	3.32154	0.205598
$262$	7.29449	0.450655
$263$	−11.9937	−0.739565	−0.369783	−	0.929118i	$-0.620568\pi$
$263$	−11.9937	−0.739565	−0.369783	+	0.929118i	$0.620568\pi$
$264$	−6.83204	−0.420483
$265$	0	0
$266$	−7.16240	−0.439155
$267$	−21.3737	−1.30805
$268$	4.69993	0.287094
$269$	−0.764462	−0.0466101	−0.0233050	−	0.999728i	$-0.507419\pi$
$269$	−0.764462	−0.0466101	−0.0233050	+	0.999728i	$0.507419\pi$
$270$	0	0
$271$	−10.4235	−0.633185	−0.316592	−	0.948562i	$-0.602539\pi$
$271$	−10.4235	−0.633185	−0.316592	+	0.948562i	$0.602539\pi$
$272$	0.119691	0.00725736
$273$	−4.30793	−0.260728
$274$	1.72657	0.104306
$275$	0	0
$276$	11.0530	0.665310
$277$	−4.97370	−0.298841	−0.149420	−	0.988774i	$-0.547741\pi$
$277$	−4.97370	−0.298841	−0.149420	+	0.988774i	$0.547741\pi$
$278$	1.60142	0.0960468
$279$	−5.93307	−0.355204
$280$	0	0
$281$	15.9449	0.951192	0.475596	−	0.879664i	$-0.342232\pi$
$281$	15.9449	0.951192	0.475596	+	0.879664i	$0.342232\pi$
$282$	2.58503	0.153937
$283$	−2.86498	−0.170305	−0.0851526	−	0.996368i	$-0.527138\pi$
$283$	−2.86498	−0.170305	−0.0851526	+	0.996368i	$0.527138\pi$
$284$	−20.9839	−1.24517
$285$	0	0
$286$	−1.25414	−0.0741590
$287$	−28.0373	−1.65499
$288$	3.58522	0.211261
$289$	−16.9707	−0.998277
$290$	0	0
$291$	−3.59826	−0.210934
$292$	−4.89446	−0.286427
$293$	3.94035	0.230198	0.115099	−	0.993354i	$-0.463282\pi$
$293$	3.94035	0.230198	0.115099	+	0.993354i	$0.463282\pi$
$294$	−1.72081	−0.100360
$295$	0	0
$296$	−1.85354	−0.107735
$297$	9.30960	0.540198
$298$	15.5456	0.900531
$299$	4.95394	0.286494
$300$	0	0
$301$	3.57499	0.206059
$302$	−7.21851	−0.415378
$303$	26.5902	1.52756
$304$	2.20520	0.126477
$305$	0	0
$306$	0.0820900	0.00469278
$307$	−1.26971	−0.0724662	−0.0362331	−	0.999343i	$-0.511536\pi$
$307$	−1.26971	−0.0724662	−0.0362331	+	0.999343i	$0.511536\pi$
$308$	6.71517	0.382632
$309$	19.0239	1.08223
$310$	0	0
$311$	3.96632	0.224909	0.112455	−	0.993657i	$-0.464129\pi$
$311$	3.96632	0.224909	0.112455	+	0.993657i	$0.464129\pi$
$312$	−3.93555	−0.222806
$313$	−26.8368	−1.51691	−0.758453	−	0.651728i	$-0.774045\pi$
$313$	−26.8368	−1.51691	−0.758453	+	0.651728i	$0.774045\pi$
$314$	3.89100	0.219582
$315$	0	0
$316$	−15.3750	−0.864911
$317$	−32.6776	−1.83536	−0.917679	−	0.397324i	$-0.869939\pi$
$317$	−32.6776	−1.83536	−0.917679	+	0.397324i	$0.869939\pi$
$318$	14.7358	0.826342
$319$	9.03927	0.506102
$320$	0	0
$321$	2.73265	0.152522
$322$	11.7137	0.652779
$323$	0.539576	0.0300228
$324$	9.41437	0.523021
$325$	0	0
$326$	3.29451	0.182466
$327$	−7.12454	−0.393988
$328$	−25.6137	−1.41428
$329$	−6.20367	−0.342019
$330$	0	0
$331$	0.537543	0.0295460	0.0147730	−	0.999891i	$-0.495297\pi$
$331$	0.537543	0.0295460	0.0147730	+	0.999891i	$0.495297\pi$
$332$	16.1247	0.884958
$333$	0.428431	0.0234779
$334$	8.37432	0.458222
$335$	0	0
$336$	3.13618	0.171093
$337$	−20.8846	−1.13765	−0.568827	−	0.822457i	$-0.692603\pi$
$337$	−20.8846	−1.13765	−0.568827	+	0.822457i	$0.692603\pi$
$338$	9.45297	0.514174
$339$	−14.8244	−0.805151
$340$	0	0
$341$	−16.1463	−0.874371
$342$	1.51243	0.0817829
$343$	−16.1859	−0.873958
$344$	3.26597	0.176089
$345$	0	0
$346$	7.10811	0.382134
$347$	14.1621	0.760263	0.380132	−	0.924932i	$-0.375879\pi$
$347$	14.1621	0.760263	0.380132	+	0.924932i	$0.375879\pi$
$348$	11.6176	0.622768
$349$	19.5927	1.04878	0.524388	−	0.851480i	$-0.324294\pi$
$349$	19.5927	1.04878	0.524388	+	0.851480i	$0.324294\pi$
$350$	0	0
$351$	5.36273	0.286241
$352$	9.75683	0.520041
$353$	−23.0084	−1.22461	−0.612307	−	0.790620i	$-0.709758\pi$
$353$	−23.0084	−1.22461	−0.612307	+	0.790620i	$0.709758\pi$
$354$	10.7284	0.570205
$355$	0	0
$356$	19.1922	1.01718
$357$	0.767372	0.0406136
$358$	−5.90142	−0.311900
$359$	−20.4440	−1.07899	−0.539496	−	0.841988i	$-0.681385\pi$
$359$	−20.4440	−1.07899	−0.539496	+	0.841988i	$0.681385\pi$
$360$	0	0
$361$	−9.05884	−0.476781
$362$	8.13136	0.427375
$363$	−12.6979	−0.666467
$364$	3.86823	0.202750
$365$	0	0
$366$	−6.61054	−0.345538
$367$	−9.22692	−0.481641	−0.240821	−	0.970570i	$-0.577417\pi$
$367$	−9.22692	−0.481641	−0.240821	+	0.970570i	$0.577417\pi$
$368$	−3.60648	−0.188001
$369$	5.92042	0.308205
$370$	0	0
$371$	−35.3635	−1.83598
$372$	−20.7518	−1.07593
$373$	−11.3543	−0.587902	−0.293951	−	0.955820i	$-0.594970\pi$
$373$	−11.3543	−0.587902	−0.293951	+	0.955820i	$0.594970\pi$
$374$	0.223400	0.0115518
$375$	0	0
$376$	−5.66741	−0.292274
$377$	5.20701	0.268174
$378$	12.6803	0.652205
$379$	25.6584	1.31798	0.658992	−	0.752150i	$-0.270983\pi$
$379$	25.6584	1.31798	0.658992	+	0.752150i	$0.270983\pi$
$380$	0	0
$381$	−26.6118	−1.36336
$382$	−13.8548	−0.708874
$383$	18.2415	0.932100	0.466050	−	0.884759i	$-0.345677\pi$
$383$	18.2415	0.932100	0.466050	+	0.884759i	$0.345677\pi$
$384$	14.2315	0.726247
$385$	0	0
$386$	8.30525	0.422726
$387$	−0.754904	−0.0383739
$388$	3.23099	0.164029
$389$	33.7036	1.70884	0.854420	−	0.519583i	$-0.173913\pi$
$389$	33.7036	1.70884	0.854420	+	0.519583i	$0.173913\pi$
$390$	0	0
$391$	−0.882445	−0.0446272
$392$	3.77269	0.190549
$393$	−14.3988	−0.726324
$394$	−17.0331	−0.858113
$395$	0	0
$396$	−1.41799	−0.0712568
$397$	−14.9316	−0.749394	−0.374697	−	0.927147i	$-0.622253\pi$
$397$	−14.9316	−0.749394	−0.374697	+	0.927147i	$0.622253\pi$
$398$	−6.24012	−0.312789
$399$	14.1381	0.707790
$400$	0	0
$401$	−15.2288	−0.760491	−0.380245	−	0.924886i	$-0.624160\pi$
$401$	−15.2288	−0.760491	−0.380245	+	0.924886i	$0.624160\pi$
$402$	4.09691	0.204336
$403$	−9.30096	−0.463314
$404$	−23.8761	−1.18788
$405$	0	0
$406$	12.3121	0.611039
$407$	1.16594	0.0577933
$408$	0.701039	0.0347066
$409$	33.8086	1.67173	0.835863	−	0.548939i	$-0.184968\pi$
$409$	33.8086	1.67173	0.835863	+	0.548939i	$0.184968\pi$
$410$	0	0
$411$	−3.40813	−0.168111
$412$	−17.0821	−0.841577
$413$	−25.7463	−1.26689
$414$	−2.47349	−0.121566
$415$	0	0
$416$	5.62035	0.275561
$417$	−3.16109	−0.154799
$418$	4.11594	0.201317
$419$	−33.2024	−1.62204	−0.811021	−	0.585018i	$-0.801088\pi$
$419$	−33.2024	−1.62204	−0.811021	+	0.585018i	$0.801088\pi$
$420$	0	0
$421$	−39.7993	−1.93970	−0.969849	−	0.243708i	$-0.921636\pi$
$421$	−39.7993	−1.93970	−0.969849	+	0.243708i	$0.921636\pi$
$422$	0.606415	0.0295198
$423$	1.30998	0.0636935
$424$	−32.3066	−1.56895
$425$	0	0
$426$	−18.2916	−0.886231
$427$	15.8642	0.767724
$428$	−2.45373	−0.118606
$429$	2.47559	0.119523
$430$	0	0
$431$	−19.1835	−0.924036	−0.462018	−	0.886871i	$-0.652874\pi$
$431$	−19.1835	−0.924036	−0.462018	+	0.886871i	$0.652874\pi$
$432$	−3.90408	−0.187835
$433$	−22.1750	−1.06566	−0.532831	−	0.846221i	$-0.678872\pi$
$433$	−22.1750	−1.06566	−0.532831	+	0.846221i	$0.678872\pi$
$434$	−21.9923	−1.05567
$435$	0	0
$436$	6.39735	0.306377
$437$	−16.2582	−0.777735
$438$	−4.26649	−0.203861
$439$	−3.71907	−0.177502	−0.0887508	−	0.996054i	$-0.528287\pi$
$439$	−3.71907	−0.177502	−0.0887508	+	0.996054i	$0.528287\pi$
$440$	0	0
$441$	−0.872029	−0.0415252
$442$	0.128688	0.00612108
$443$	10.3055	0.489627	0.244813	−	0.969570i	$-0.421273\pi$
$443$	10.3055	0.489627	0.244813	+	0.969570i	$0.421273\pi$
$444$	1.49850	0.0711158
$445$	0	0
$446$	−7.88669	−0.373446
$447$	−30.6859	−1.45139
$448$	9.22979	0.436067
$449$	15.1992	0.717296	0.358648	−	0.933473i	$-0.383238\pi$
$449$	15.1992	0.717296	0.358648	+	0.933473i	$0.383238\pi$
$450$	0	0
$451$	16.1119	0.758679
$452$	13.3113	0.626111
$453$	14.2488	0.669469
$454$	−14.1160	−0.662497
$455$	0	0
$456$	12.9160	0.604846
$457$	15.6966	0.734256	0.367128	−	0.930170i	$-0.380341\pi$
$457$	15.6966	0.734256	0.367128	+	0.930170i	$0.380341\pi$
$458$	−9.64498	−0.450680
$459$	−0.955264	−0.0445879
$460$	0	0
$461$	−2.24117	−0.104382	−0.0521908	−	0.998637i	$-0.516620\pi$
$461$	−2.24117	−0.104382	−0.0521908	+	0.998637i	$0.516620\pi$
$462$	5.85359	0.272334
$463$	−21.1905	−0.984805	−0.492403	−	0.870368i	$-0.663881\pi$
$463$	−21.1905	−0.984805	−0.492403	+	0.870368i	$0.663881\pi$
$464$	−3.79071	−0.175979
$465$	0	0
$466$	−4.15942	−0.192681
$467$	−9.83046	−0.454899	−0.227450	−	0.973790i	$-0.573039\pi$
$467$	−9.83046	−0.454899	−0.227450	+	0.973790i	$0.573039\pi$
$468$	−0.816824	−0.0377577
$469$	−9.83194	−0.453997
$470$	0	0
$471$	−7.68056	−0.353902
$472$	−23.5208	−1.08263
$473$	−2.05440	−0.0944615
$474$	−13.4023	−0.615589
$475$	0	0
$476$	−0.689047	−0.0315824
$477$	7.46744	0.341911
$478$	15.7380	0.719839
$479$	2.94988	0.134784	0.0673918	−	0.997727i	$-0.478532\pi$
$479$	2.94988	0.134784	0.0673918	+	0.997727i	$0.478532\pi$
$480$	0	0
$481$	0.671629	0.0306237
$482$	0.782724	0.0356521
$483$	−23.1220	−1.05209
$484$	11.4018	0.518265
$485$	0	0
$486$	−4.90101	−0.222314
$487$	−20.4577	−0.927029	−0.463514	−	0.886089i	$-0.653412\pi$
$487$	−20.4577	−0.927029	−0.463514	+	0.886089i	$0.653412\pi$
$488$	14.4929	0.656063
$489$	−6.50314	−0.294082
$490$	0	0
$491$	−12.8271	−0.578881	−0.289440	−	0.957196i	$-0.593469\pi$
$491$	−12.8271	−0.578881	−0.289440	+	0.957196i	$0.593469\pi$
$492$	20.7076	0.933570
$493$	−0.927524	−0.0417736
$494$	2.37096	0.106674
$495$	0	0
$496$	6.77112	0.304032
$497$	43.8969	1.96905
$498$	14.0559	0.629858
$499$	40.9396	1.83271	0.916355	−	0.400366i	$-0.131117\pi$
$499$	40.9396	1.83271	0.916355	+	0.400366i	$0.131117\pi$
$500$	0	0
$501$	−16.5303	−0.738521
$502$	−22.5197	−1.00510
$503$	−6.00073	−0.267559	−0.133780	−	0.991011i	$-0.542711\pi$
$503$	−6.00073	−0.267559	−0.133780	+	0.991011i	$0.542711\pi$
$504$	−4.71571	−0.210055
$505$	0	0
$506$	−6.73139	−0.299247
$507$	−18.6595	−0.828698
$508$	23.8956	1.06019
$509$	23.2519	1.03062	0.515311	−	0.857003i	$-0.327676\pi$
$509$	23.2519	1.03062	0.515311	+	0.857003i	$0.327676\pi$
$510$	0	0
$511$	10.2389	0.452941
$512$	−7.80038	−0.344731
$513$	−17.5998	−0.777051
$514$	16.2213	0.715492
$515$	0	0
$516$	−2.64039	−0.116237
$517$	3.56499	0.156788
$518$	1.58808	0.0697764
$519$	−14.0309	−0.615889
$520$	0	0
$521$	−25.4599	−1.11542	−0.557708	−	0.830037i	$-0.688319\pi$
$521$	−25.4599	−1.11542	−0.557708	+	0.830037i	$0.688319\pi$
$522$	−2.59985	−0.113792
$523$	26.6451	1.16511	0.582554	−	0.812792i	$-0.302054\pi$
$523$	26.6451	1.16511	0.582554	+	0.812792i	$0.302054\pi$
$524$	12.9291	0.564813
$525$	0	0
$526$	9.38778	0.409327
$527$	1.65678	0.0721705
$528$	−1.80224	−0.0784323
$529$	3.58937	0.156059
$530$	0	0
$531$	5.43665	0.235931
$532$	−12.6950	−0.550400
$533$	9.28114	0.402011
$534$	16.7297	0.723967
$535$	0	0
$536$	−8.98205	−0.387965
$537$	11.6490	0.502691
$538$	0.598363	0.0257973
$539$	−2.37315	−0.102219
$540$	0	0
$541$	−5.65440	−0.243101	−0.121551	−	0.992585i	$-0.538787\pi$
$541$	−5.65440	−0.243101	−0.121551	+	0.992585i	$0.538787\pi$
$542$	8.15875	0.350449
$543$	−16.0507	−0.688803
$544$	−1.00115	−0.0429241
$545$	0	0
$546$	3.37192	0.144305
$547$	−44.4599	−1.90097	−0.950485	−	0.310770i	$-0.899413\pi$
$547$	−44.4599	−1.90097	−0.950485	+	0.310770i	$0.899413\pi$
$548$	3.06027	0.130728
$549$	−3.34993	−0.142972
$550$	0	0
$551$	−17.0887	−0.728005
$552$	−21.1233	−0.899069
$553$	32.1634	1.36773
$554$	3.89304	0.165399
$555$	0	0
$556$	2.83844	0.120377
$557$	34.5486	1.46387	0.731935	−	0.681375i	$-0.238618\pi$
$557$	34.5486	1.46387	0.731935	+	0.681375i	$0.238618\pi$
$558$	4.64396	0.196594
$559$	−1.18342	−0.0500535
$560$	0	0
$561$	−0.440977	−0.0186181
$562$	−12.4804	−0.526456
$563$	−3.98100	−0.167779	−0.0838895	−	0.996475i	$-0.526734\pi$
$563$	−3.98100	−0.167779	−0.0838895	+	0.996475i	$0.526734\pi$
$564$	4.58186	0.192931
$565$	0	0
$566$	2.24249	0.0942587
$567$	−19.6942	−0.827080
$568$	40.1024	1.68266
$569$	−43.8910	−1.84001	−0.920003	−	0.391912i	$-0.871814\pi$
$569$	−43.8910	−1.84001	−0.920003	+	0.391912i	$0.871814\pi$
$570$	0	0
$571$	−10.0439	−0.420323	−0.210161	−	0.977667i	$-0.567399\pi$
$571$	−10.0439	−0.420323	−0.210161	+	0.977667i	$0.567399\pi$
$572$	−2.22291	−0.0929446
$573$	27.3484	1.14250
$574$	21.9455	0.915987
$575$	0	0
$576$	−1.94898	−0.0812077
$577$	−32.2089	−1.34087	−0.670437	−	0.741967i	$-0.733893\pi$
$577$	−32.2089	−1.34087	−0.670437	+	0.741967i	$0.733893\pi$
$578$	13.2834	0.552516
$579$	−16.3940	−0.681311
$580$	0	0
$581$	−33.7318	−1.39943
$582$	2.81645	0.116746
$583$	20.3220	0.841650
$584$	9.35382	0.387064
$585$	0	0
$586$	−3.08421	−0.127407
$587$	−13.7236	−0.566435	−0.283218	−	0.959056i	$-0.591402\pi$
$587$	−13.7236	−0.566435	−0.283218	+	0.959056i	$0.591402\pi$
$588$	−3.05005	−0.125782
$589$	30.5246	1.25774
$590$	0	0
$591$	33.6221	1.38303
$592$	−0.488948	−0.0200956
$593$	−19.0496	−0.782271	−0.391136	−	0.920333i	$-0.627918\pi$
$593$	−19.0496	−0.782271	−0.391136	+	0.920333i	$0.627918\pi$
$594$	−7.28685	−0.298983
$595$	0	0
$596$	27.5538	1.12865
$597$	12.3176	0.504124
$598$	−3.87757	−0.158565
$599$	−2.77926	−0.113557	−0.0567787	−	0.998387i	$-0.518083\pi$
$599$	−2.77926	−0.113557	−0.0567787	+	0.998387i	$0.518083\pi$
$600$	0	0
$601$	−4.19810	−0.171244	−0.0856221	−	0.996328i	$-0.527288\pi$
$601$	−4.19810	−0.171244	−0.0856221	+	0.996328i	$0.527288\pi$
$602$	−2.79823	−0.114047
$603$	2.07613	0.0845468
$604$	−12.7945	−0.520600
$605$	0	0
$606$	−20.8128	−0.845461
$607$	14.1349	0.573719	0.286860	−	0.957973i	$-0.407389\pi$
$607$	14.1349	0.573719	0.286860	+	0.957973i	$0.407389\pi$
$608$	−18.4453	−0.748056
$609$	−24.3032	−0.984816
$610$	0	0
$611$	2.05359	0.0830793
$612$	0.145501	0.00588152
$613$	−37.7846	−1.52611	−0.763053	−	0.646336i	$-0.776300\pi$
$613$	−37.7846	−1.52611	−0.763053	+	0.646336i	$0.776300\pi$
$614$	0.993833	0.0401078
$615$	0	0
$616$	−12.8334	−0.517072
$617$	34.3834	1.38422	0.692112	−	0.721790i	$-0.256680\pi$
$617$	34.3834	1.38422	0.692112	+	0.721790i	$0.256680\pi$
$618$	−14.8905	−0.598982
$619$	−25.3877	−1.02042	−0.510208	−	0.860051i	$-0.670432\pi$
$619$	−25.3877	−1.02042	−0.510208	+	0.860051i	$0.670432\pi$
$620$	0	0
$621$	28.7835	1.15504
$622$	−3.10453	−0.124480
$623$	−40.1487	−1.60852
$624$	−1.03817	−0.0415599
$625$	0	0
$626$	21.0058	0.839561
$627$	−8.12458	−0.324464
$628$	6.89662	0.275205
$629$	−0.119637	−0.00477025
$630$	0	0
$631$	18.2834	0.727849	0.363925	−	0.931428i	$-0.381437\pi$
$631$	18.2834	0.727849	0.363925	+	0.931428i	$0.381437\pi$
$632$	29.3832	1.16880
$633$	−1.19702	−0.0475773
$634$	25.5776	1.01581
$635$	0	0
$636$	26.1185	1.03567
$637$	−1.36703	−0.0541639
$638$	−7.07525	−0.280112
$639$	−9.26937	−0.366691
$640$	0	0
$641$	−17.0455	−0.673257	−0.336629	−	0.941637i	$-0.609287\pi$
$641$	−17.0455	−0.673257	−0.336629	+	0.941637i	$0.609287\pi$
$642$	−2.13891	−0.0844161
$643$	22.3404	0.881021	0.440510	−	0.897748i	$-0.354798\pi$
$643$	22.3404	0.881021	0.440510	+	0.897748i	$0.354798\pi$
$644$	20.7620	0.818138
$645$	0	0
$646$	−0.422339	−0.0166167
$647$	−16.7185	−0.657270	−0.328635	−	0.944457i	$-0.606589\pi$
$647$	−16.7185	−0.657270	−0.328635	+	0.944457i	$0.606589\pi$
$648$	−17.9918	−0.706786
$649$	14.7954	0.580768
$650$	0	0
$651$	43.4114	1.70143
$652$	5.83937	0.228687
$653$	30.6924	1.20108	0.600542	−	0.799593i	$-0.294951\pi$
$653$	30.6924	1.20108	0.600542	+	0.799593i	$0.294951\pi$
$654$	5.57655	0.218060
$655$	0	0
$656$	−6.75669	−0.263805
$657$	−2.16207	−0.0843503
$658$	4.85576	0.189297
$659$	30.0362	1.17004	0.585021	−	0.811018i	$-0.301086\pi$
$659$	30.0362	1.17004	0.585021	+	0.811018i	$0.301086\pi$
$660$	0	0
$661$	0.866875	0.0337175	0.0168588	−	0.999858i	$-0.494633\pi$
$661$	0.866875	0.0337175	0.0168588	+	0.999858i	$0.494633\pi$
$662$	−0.420748	−0.0163528
$663$	−0.254022	−0.00986539
$664$	−30.8160	−1.19589
$665$	0	0
$666$	−0.335344	−0.0129943
$667$	27.9477	1.08214
$668$	14.8431	0.574297
$669$	15.5678	0.601885
$670$	0	0
$671$	−9.11652	−0.351940
$672$	−26.2325	−1.01194
$673$	7.41264	0.285736	0.142868	−	0.989742i	$-0.454367\pi$
$673$	7.41264	0.285736	0.142868	+	0.989742i	$0.454367\pi$
$674$	16.3469	0.629657
$675$	0	0
$676$	16.7550	0.644422
$677$	−37.8243	−1.45371	−0.726853	−	0.686793i	$-0.759018\pi$
$677$	−37.8243	−1.45371	−0.726853	+	0.686793i	$0.759018\pi$
$678$	11.6034	0.445627
$679$	−6.75902	−0.259387
$680$	0	0
$681$	27.8640	1.06775
$682$	12.6381	0.483938
$683$	45.5147	1.74157	0.870786	−	0.491662i	$-0.163610\pi$
$683$	45.5147	1.74157	0.870786	+	0.491662i	$0.163610\pi$
$684$	2.68071	0.102500
$685$	0	0
$686$	12.6691	0.483709
$687$	19.0385	0.726365
$688$	0.861535	0.0328457
$689$	11.7063	0.445975
$690$	0	0
$691$	11.7354	0.446436	0.223218	−	0.974769i	$-0.428344\pi$
$691$	11.7354	0.446436	0.223218	+	0.974769i	$0.428344\pi$
$692$	12.5988	0.478935
$693$	2.96634	0.112682
$694$	−11.0850	−0.420783
$695$	0	0
$696$	−22.2024	−0.841580
$697$	−1.65325	−0.0626213
$698$	−15.3357	−0.580465
$699$	8.21040	0.310546
$700$	0	0
$701$	16.2368	0.613255	0.306628	−	0.951830i	$-0.400799\pi$
$701$	16.2368	0.613255	0.306628	+	0.951830i	$0.400799\pi$
$702$	−4.19754	−0.158426
$703$	−2.20420	−0.0831331
$704$	−5.30398	−0.199901
$705$	0	0
$706$	18.0092	0.677787
$707$	49.9473	1.87846
$708$	19.0155	0.714647
$709$	35.5273	1.33426	0.667128	−	0.744943i	$-0.267523\pi$
$709$	35.5273	1.33426	0.667128	+	0.744943i	$0.267523\pi$
$710$	0	0
$711$	−6.79171	−0.254709
$712$	−36.6782	−1.37457
$713$	−49.9212	−1.86956
$714$	−0.600640	−0.0224784
$715$	0	0
$716$	−10.4600	−0.390908
$717$	−31.0657	−1.16017
$718$	16.0020	0.597189
$719$	52.0962	1.94286	0.971430	−	0.237325i	$-0.0762706\pi$
$719$	52.0962	1.94286	0.971430	+	0.237325i	$0.0762706\pi$
$720$	0	0
$721$	35.7347	1.33083
$722$	7.09057	0.263884
$723$	−1.54504	−0.0574608
$724$	14.4125	0.535635
$725$	0	0
$726$	9.93895	0.368869
$727$	21.2446	0.787920	0.393960	−	0.919128i	$-0.371105\pi$
$727$	21.2446	0.787920	0.393960	+	0.919128i	$0.371105\pi$
$728$	−7.39258	−0.273987
$729$	30.0320	1.11229
$730$	0	0
$731$	0.210803	0.00779684
$732$	−11.7169	−0.433068
$733$	−10.2512	−0.378636	−0.189318	−	0.981916i	$-0.560628\pi$
$733$	−10.2512	−0.378636	−0.189318	+	0.981916i	$0.560628\pi$
$734$	7.22213	0.266574
$735$	0	0
$736$	30.1662	1.11194
$737$	5.65001	0.208121
$738$	−4.63406	−0.170582
$739$	21.9869	0.808801	0.404400	−	0.914582i	$-0.367480\pi$
$739$	21.9869	0.808801	0.404400	+	0.914582i	$0.367480\pi$
$740$	0	0
$741$	−4.68011	−0.171928
$742$	27.6799	1.01616
$743$	8.36247	0.306789	0.153395	−	0.988165i	$-0.450979\pi$
$743$	8.36247	0.306789	0.153395	+	0.988165i	$0.450979\pi$
$744$	39.6588	1.45396
$745$	0	0
$746$	8.88726	0.325386
$747$	7.12289	0.260613
$748$	0.395967	0.0144780
$749$	5.13304	0.187557
$750$	0	0
$751$	−23.1974	−0.846485	−0.423242	−	0.906016i	$-0.639108\pi$
$751$	−23.1974	−0.846485	−0.423242	+	0.906016i	$0.639108\pi$
$752$	−1.49502	−0.0545177
$753$	44.4524	1.61993
$754$	−4.07565	−0.148426
$755$	0	0
$756$	22.4753	0.817418
$757$	0.542993	0.0197354	0.00986771	−	0.999951i	$-0.496859\pi$
$757$	0.542993	0.0197354	0.00986771	+	0.999951i	$0.496859\pi$
$758$	−20.0834	−0.729464
$759$	13.2873	0.482298
$760$	0	0
$761$	−4.37488	−0.158589	−0.0792947	−	0.996851i	$-0.525267\pi$
$761$	−4.37488	−0.158589	−0.0792947	+	0.996851i	$0.525267\pi$
$762$	20.8297	0.754580
$763$	−13.3828	−0.484491
$764$	−24.5570	−0.888442
$765$	0	0
$766$	−14.2781	−0.515889
$767$	8.52276	0.307739
$768$	−20.9665	−0.756565
$769$	−31.7834	−1.14614	−0.573069	−	0.819507i	$-0.694247\pi$
$769$	−31.7834	−1.14614	−0.573069	+	0.819507i	$0.694247\pi$
$770$	0	0
$771$	−32.0198	−1.15316
$772$	14.7207	0.529809
$773$	−5.09447	−0.183235	−0.0916176	−	0.995794i	$-0.529204\pi$
$773$	−5.09447	−0.183235	−0.0916176	+	0.995794i	$0.529204\pi$
$774$	0.590882	0.0212388
$775$	0	0
$776$	−6.17476	−0.221661
$777$	−3.13477	−0.112459
$778$	−26.3806	−0.945791
$779$	−30.4595	−1.09133
$780$	0	0
$781$	−25.2257	−0.902648
$782$	0.690711	0.0246998
$783$	30.2539	1.08119
$784$	0.995204	0.0355430
$785$	0	0
$786$	11.2703	0.401998
$787$	−25.6615	−0.914735	−0.457367	−	0.889278i	$-0.651208\pi$
$787$	−25.6615	−0.914735	−0.457367	+	0.889278i	$0.651208\pi$
$788$	−30.1903	−1.07549
$789$	−18.5308	−0.659715
$790$	0	0
$791$	−27.8463	−0.990102
$792$	2.70993	0.0962931
$793$	−5.25151	−0.186487
$794$	11.6873	0.414766
$795$	0	0
$796$	−11.0603	−0.392023
$797$	−12.1111	−0.428997	−0.214499	−	0.976724i	$-0.568812\pi$
$797$	−12.1111	−0.428997	−0.214499	+	0.976724i	$0.568812\pi$
$798$	−11.0662	−0.391740
$799$	−0.365806	−0.0129413
$800$	0	0
$801$	8.47789	0.299552
$802$	11.9200	0.420908
$803$	−5.88387	−0.207637
$804$	7.26159	0.256097
$805$	0	0
$806$	7.28009	0.256430
$807$	−1.18113	−0.0415776
$808$	45.6298	1.60525
$809$	−32.7662	−1.15200	−0.576000	−	0.817450i	$-0.695387\pi$
$809$	−32.7662	−1.15200	−0.576000	+	0.817450i	$0.695387\pi$
$810$	0	0
$811$	24.7651	0.869620	0.434810	−	0.900522i	$-0.356816\pi$
$811$	24.7651	0.869620	0.434810	+	0.900522i	$0.356816\pi$
$812$	21.8226	0.765824
$813$	−16.1048	−0.564821
$814$	−0.912607	−0.0319868
$815$	0	0
$816$	0.184928	0.00647379
$817$	3.88385	0.135879
$818$	−26.4628	−0.925249
$819$	1.70874	0.0597082
$820$	0	0
$821$	−35.5435	−1.24048	−0.620238	−	0.784414i	$-0.712964\pi$
$821$	−35.5435	−1.24048	−0.620238	+	0.784414i	$0.712964\pi$
$822$	2.66763	0.0930441
$823$	−29.5066	−1.02854	−0.514268	−	0.857629i	$-0.671936\pi$
$823$	−29.5066	−1.02854	−0.514268	+	0.857629i	$0.671936\pi$
$824$	32.6457	1.13727
$825$	0	0
$826$	20.1523	0.701187
$827$	29.5722	1.02833	0.514163	−	0.857693i	$-0.328103\pi$
$827$	29.5722	1.02833	0.514163	+	0.857693i	$0.328103\pi$
$828$	−4.38416	−0.152360
$829$	−21.6365	−0.751466	−0.375733	−	0.926728i	$-0.622609\pi$
$829$	−21.6365	−0.751466	−0.375733	+	0.926728i	$0.622609\pi$
$830$	0	0
$831$	−7.68458	−0.266575
$832$	−3.05532	−0.105924
$833$	0.243510	0.00843712
$834$	2.47426	0.0856767
$835$	0	0
$836$	7.29532	0.252314
$837$	−54.0407	−1.86792
$838$	25.9883	0.897750
$839$	19.6053	0.676848	0.338424	−	0.940994i	$-0.390106\pi$
$839$	19.6053	0.676848	0.338424	+	0.940994i	$0.390106\pi$
$840$	0	0
$841$	0.375361	0.0129435
$842$	31.1518	1.07356
$843$	24.6355	0.848493
$844$	1.07484	0.0369976
$845$	0	0
$846$	−1.02535	−0.0352524
$847$	−23.8519	−0.819560
$848$	−8.52223	−0.292655
$849$	−4.42651	−0.151918
$850$	0	0
$851$	3.60485	0.123573
$852$	−32.4210	−1.11073
$853$	35.2053	1.20541	0.602704	−	0.797965i	$-0.294090\pi$
$853$	35.2053	1.20541	0.602704	+	0.797965i	$0.294090\pi$
$854$	−12.4173	−0.424912
$855$	0	0
$856$	4.68934	0.160278
$857$	13.2046	0.451059	0.225529	−	0.974236i	$-0.427589\pi$
$857$	13.2046	0.451059	0.225529	+	0.974236i	$0.427589\pi$
$858$	−1.93770	−0.0661522
$859$	30.2808	1.03317	0.516584	−	0.856237i	$-0.327203\pi$
$859$	30.2808	1.03317	0.516584	+	0.856237i	$0.327203\pi$
$860$	0	0
$861$	−43.3189	−1.47630
$862$	15.0154	0.511426
$863$	10.2413	0.348618	0.174309	−	0.984691i	$-0.444231\pi$
$863$	10.2413	0.348618	0.174309	+	0.984691i	$0.444231\pi$
$864$	32.6555	1.11096
$865$	0	0
$866$	17.3569	0.589812
$867$	−26.2205	−0.890494
$868$	−38.9804	−1.32308
$869$	−18.4830	−0.626993
$870$	0	0
$871$	3.25465	0.110280
$872$	−12.2260	−0.414024
$873$	1.42725	0.0483051
$874$	12.7257	0.430453
$875$	0	0
$876$	−7.56215	−0.255501
$877$	14.0827	0.475540	0.237770	−	0.971321i	$-0.423584\pi$
$877$	14.0827	0.475540	0.237770	+	0.971321i	$0.423584\pi$
$878$	2.91101	0.0982417
$879$	6.08801	0.205344
$880$	0	0
$881$	15.6951	0.528781	0.264391	−	0.964416i	$-0.414829\pi$
$881$	15.6951	0.528781	0.264391	+	0.964416i	$0.414829\pi$
$882$	0.682558	0.0229829
$883$	−22.0436	−0.741827	−0.370913	−	0.928667i	$-0.620955\pi$
$883$	−22.0436	−0.741827	−0.370913	+	0.928667i	$0.620955\pi$
$884$	0.228094	0.00767163
$885$	0	0
$886$	−8.06633	−0.270994
$887$	42.9750	1.44296	0.721480	−	0.692435i	$-0.243462\pi$
$887$	42.9750	1.44296	0.721480	+	0.692435i	$0.243462\pi$
$888$	−2.86379	−0.0961026
$889$	−49.9879	−1.67654
$890$	0	0
$891$	11.3175	0.379149
$892$	−13.9788	−0.468045
$893$	−6.73962	−0.225533
$894$	24.0186	0.803302
$895$	0	0
$896$	26.7326	0.893073
$897$	7.65405	0.255561
$898$	−11.8968	−0.397002
$899$	−52.4714	−1.75002
$900$	0	0
$901$	−2.08525	−0.0694697
$902$	−12.6112	−0.419906
$903$	5.52352	0.183811
$904$	−25.4393	−0.846097
$905$	0	0
$906$	−11.1529	−0.370531
$907$	−32.2906	−1.07219	−0.536096	−	0.844157i	$-0.680102\pi$
$907$	−32.2906	−1.07219	−0.536096	+	0.844157i	$0.680102\pi$
$908$	−25.0200	−0.830317
$909$	−10.5470	−0.349822
$910$	0	0
$911$	23.3877	0.774870	0.387435	−	0.921897i	$-0.373361\pi$
$911$	23.3877	0.774870	0.387435	+	0.921897i	$0.373361\pi$
$912$	3.40713	0.112821
$913$	19.3843	0.641526
$914$	−12.2861	−0.406388
$915$	0	0
$916$	−17.0953	−0.564844
$917$	−27.0469	−0.893168
$918$	0.747708	0.0246780
$919$	9.04929	0.298508	0.149254	−	0.988799i	$-0.452313\pi$
$919$	9.04929	0.298508	0.149254	+	0.988799i	$0.452313\pi$
$920$	0	0
$921$	−1.96176	−0.0646421
$922$	1.75422	0.0577721
$923$	−14.5311	−0.478297
$924$	10.3752	0.341320
$925$	0	0
$926$	16.5863	0.545060
$927$	−7.54582	−0.247837
$928$	31.7073	1.04084
$929$	−42.2077	−1.38479	−0.692395	−	0.721518i	$-0.743445\pi$
$929$	−42.2077	−1.38479	−0.692395	+	0.721518i	$0.743445\pi$
$930$	0	0
$931$	4.48644	0.147037
$932$	−7.37238	−0.241490
$933$	6.12813	0.200626
$934$	7.69454	0.251773
$935$	0	0
$936$	1.56103	0.0510240
$937$	27.5398	0.899687	0.449843	−	0.893107i	$-0.351480\pi$
$937$	27.5398	0.899687	0.449843	+	0.893107i	$0.351480\pi$
$938$	7.69569	0.251273
$939$	−41.4640	−1.35313
$940$	0	0
$941$	−20.8723	−0.680416	−0.340208	−	0.940350i	$-0.610497\pi$
$941$	−20.8723	−0.680416	−0.340208	+	0.940350i	$0.610497\pi$
$942$	6.01176	0.195874
$943$	49.8148	1.62219
$944$	−6.20459	−0.201942
$945$	0	0
$946$	1.60803	0.0522816
$947$	28.4729	0.925245	0.462623	−	0.886555i	$-0.346909\pi$
$947$	28.4729	0.925245	0.462623	+	0.886555i	$0.346909\pi$
$948$	−23.7550	−0.771527
$949$	−3.38936	−0.110023
$950$	0	0
$951$	−50.4883	−1.63720
$952$	1.31684	0.0426790
$953$	−16.4316	−0.532271	−0.266135	−	0.963936i	$-0.585747\pi$
$953$	−16.4316	−0.532271	−0.266135	+	0.963936i	$0.585747\pi$
$954$	−5.84495	−0.189237
$955$	0	0
$956$	27.8949	0.902185
$957$	13.9661	0.451459
$958$	−2.30894	−0.0745986
$959$	−6.40187	−0.206727
$960$	0	0
$961$	62.7266	2.02344
$962$	−0.525701	−0.0169493
$963$	−1.08391	−0.0349284
$964$	1.38734	0.0446833
$965$	0	0
$966$	18.0982	0.582299
$967$	41.5931	1.33754	0.668772	−	0.743468i	$-0.266820\pi$
$967$	41.5931	1.33754	0.668772	+	0.743468i	$0.266820\pi$
$968$	−21.7901	−0.700360
$969$	0.833668	0.0267813
$970$	0	0
$971$	−8.79027	−0.282093	−0.141047	−	0.990003i	$-0.545047\pi$
$971$	−8.79027	−0.282093	−0.141047	+	0.990003i	$0.545047\pi$
$972$	−8.68682	−0.278630
$973$	−5.93783	−0.190358
$974$	16.0128	0.513082
$975$	0	0
$976$	3.82311	0.122375
$977$	41.5165	1.32823	0.664115	−	0.747631i	$-0.268809\pi$
$977$	41.5165	1.32823	0.664115	+	0.747631i	$0.268809\pi$
$978$	5.09016	0.162765
$979$	23.0718	0.737378
$980$	0	0
$981$	2.82595	0.0902256
$982$	10.0401	0.320393
$983$	37.6652	1.20133	0.600666	−	0.799500i	$-0.294902\pi$
$983$	37.6652	1.20133	0.600666	+	0.799500i	$0.294902\pi$
$984$	−39.5743	−1.26158
$985$	0	0
$986$	0.725996	0.0231204
$987$	−9.58493	−0.305092
$988$	4.20241	0.133697
$989$	−6.35181	−0.201976
$990$	0	0
$991$	24.7152	0.785104	0.392552	−	0.919730i	$-0.371592\pi$
$991$	24.7152	0.785104	0.392552	+	0.919730i	$0.371592\pi$
$992$	−56.6368	−1.79822
$993$	0.830527	0.0263560
$994$	−34.3592	−1.08981
$995$	0	0
$996$	24.9134	0.789410
$997$	−13.6219	−0.431411	−0.215706	−	0.976458i	$-0.569205\pi$
$997$	−13.6219	−0.431411	−0.215706	+	0.976458i	$0.569205\pi$
$998$	−32.0444	−1.01435
$999$	3.90232	0.123464

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.j.1.11		25
5.4	even	2		1205.2.a.e.1.15	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.a.e.1.15	✓	25	5.4	even	2
6025.2.a.j.1.11		25	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.782724 q^{2} +1.54504 q^{3} -1.38734 q^{4} -1.20934 q^{6} +2.90223 q^{7} +2.65136 q^{8} -0.612841 q^{9} +O(q^{10})\)	\(q-0.782724 q^{2} +1.54504 q^{3} -1.38734 q^{4} -1.20934 q^{6} +2.90223 q^{7} +2.65136 q^{8} -0.612841 q^{9} -1.66779 q^{11} -2.14350 q^{12} -0.960719 q^{13} -2.27164 q^{14} +0.699406 q^{16} +0.171133 q^{17} +0.479686 q^{18} +3.15296 q^{19} +4.48407 q^{21} +1.30542 q^{22} -5.15649 q^{23} +4.09646 q^{24} +0.751978 q^{26} -5.58200 q^{27} -4.02639 q^{28} -5.41990 q^{29} +9.68125 q^{31} -5.85015 q^{32} -2.57681 q^{33} -0.133950 q^{34} +0.850221 q^{36} -0.699090 q^{37} -2.46790 q^{38} -1.48435 q^{39} -9.66061 q^{41} -3.50979 q^{42} +1.23181 q^{43} +2.31380 q^{44} +4.03611 q^{46} -2.13755 q^{47} +1.08061 q^{48} +1.42293 q^{49} +0.264408 q^{51} +1.33285 q^{52} -12.1850 q^{53} +4.36916 q^{54} +7.69484 q^{56} +4.87146 q^{57} +4.24229 q^{58} -8.87122 q^{59} +5.46623 q^{61} -7.57775 q^{62} -1.77861 q^{63} +3.18024 q^{64} +2.01693 q^{66} -3.38772 q^{67} -0.237420 q^{68} -7.96700 q^{69} +15.1252 q^{71} -1.62486 q^{72} +3.52794 q^{73} +0.547195 q^{74} -4.37424 q^{76} -4.84031 q^{77} +1.16184 q^{78} +11.0823 q^{79} -6.78590 q^{81} +7.56160 q^{82} -11.6227 q^{83} -6.22094 q^{84} -0.964167 q^{86} -8.37399 q^{87} -4.42191 q^{88} -13.8337 q^{89} -2.78823 q^{91} +7.15382 q^{92} +14.9579 q^{93} +1.67311 q^{94} -9.03874 q^{96} -2.32891 q^{97} -1.11376 q^{98} +1.02209 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9}+O(q^{10}) \) 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9	\( 25 q - 6 q^{2} - 15 q^{3} + 32 q^{4} - q^{6} - 19 q^{7} - 15 q^{8} + 32 q^{9} + 2 q^{11} - 20 q^{12} - 14 q^{13} - 5 q^{14} + 38 q^{16} - 7 q^{17} - 9 q^{18} + 30 q^{19} + q^{21} - q^{22} - 43 q^{23} - 6 q^{24} - 22 q^{26} - 42 q^{27} - 32 q^{28} - 4 q^{29} + 14 q^{31} - 26 q^{32} - 4 q^{33} + 7 q^{34} + 15 q^{36} - 16 q^{37} - 14 q^{38} - 21 q^{39} - q^{41} + 25 q^{42} - 35 q^{43} - 52 q^{44} - 27 q^{46} - 50 q^{47} - 26 q^{48} + 46 q^{49} - 7 q^{51} - 3 q^{52} - 4 q^{53} - 31 q^{54} - 51 q^{56} - 2 q^{58} + 6 q^{59} + 19 q^{61} - 28 q^{63} + 49 q^{64} - 27 q^{66} - 65 q^{67} + 25 q^{68} + 2 q^{69} - 34 q^{71} + 10 q^{72} - 8 q^{73} - 42 q^{74} + 71 q^{76} - q^{77} + 59 q^{78} - 12 q^{79} + 29 q^{81} - 11 q^{82} - 41 q^{83} - 10 q^{84} - 13 q^{86} - 40 q^{87} + 52 q^{88} - 24 q^{89} + 46 q^{91} - 85 q^{92} + 30 q^{93} + 14 q^{94} - 30 q^{96} - 9 q^{97} + 64 q^{98} + 2 q^{99}+O(q^{100}) \) 25 * q - 6 * q^2 - 15 * q^3 + 32 * q^4 - q^6 - 19 * q^7 - 15 * q^8 + 32 * q^9 + 2 * q^11 - 20 * q^12 - 14 * q^13 - 5 * q^14 + 38 * q^16 - 7 * q^17 - 9 * q^18 + 30 * q^19 + q^21 - q^22 - 43 * q^23 - 6 * q^24 - 22 * q^26 - 42 * q^27 - 32 * q^28 - 4 * q^29 + 14 * q^31 - 26 * q^32 - 4 * q^33 + 7 * q^34 + 15 * q^36 - 16 * q^37 - 14 * q^38 - 21 * q^39 - q^41 + 25 * q^42 - 35 * q^43 - 52 * q^44 - 27 * q^46 - 50 * q^47 - 26 * q^48 + 46 * q^49 - 7 * q^51 - 3 * q^52 - 4 * q^53 - 31 * q^54 - 51 * q^56 - 2 * q^58 + 6 * q^59 + 19 * q^61 - 28 * q^63 + 49 * q^64 - 27 * q^66 - 65 * q^67 + 25 * q^68 + 2 * q^69 - 34 * q^71 + 10 * q^72 - 8 * q^73 - 42 * q^74 + 71 * q^76 - q^77 + 59 * q^78 - 12 * q^79 + 29 * q^81 - 11 * q^82 - 41 * q^83 - 10 * q^84 - 13 * q^86 - 40 * q^87 + 52 * q^88 - 24 * q^89 + 46 * q^91 - 85 * q^92 + 30 * q^93 + 14 * q^94 - 30 * q^96 - 9 * q^97 + 64 * q^98 + 2 * q^99

Introduction

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