LMFDB - Embedded newform 6025.2.a.n.1.18

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			1.18
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)

$f(q)$	$=$	$q-0.230402 q^{2} +1.04564 q^{3} -1.94691 q^{4} -0.240918 q^{6} +1.53080 q^{7} +0.909377 q^{8} -1.90664 q^{9} +O(q^{10})$	\(q-0.230402 q^{2} +1.04564 q^{3} -1.94691 q^{4} -0.240918 q^{6} +1.53080 q^{7} +0.909377 q^{8} -1.90664 q^{9} +1.08670 q^{11} -2.03577 q^{12} +4.04835 q^{13} -0.352699 q^{14} +3.68431 q^{16} +5.09773 q^{17} +0.439293 q^{18} +2.30729 q^{19} +1.60067 q^{21} -0.250377 q^{22} -5.69274 q^{23} +0.950881 q^{24} -0.932749 q^{26} -5.13058 q^{27} -2.98034 q^{28} +1.38333 q^{29} +8.98284 q^{31} -2.66763 q^{32} +1.13629 q^{33} -1.17453 q^{34} +3.71206 q^{36} +1.25497 q^{37} -0.531604 q^{38} +4.23312 q^{39} +2.78663 q^{41} -0.368797 q^{42} -6.56905 q^{43} -2.11571 q^{44} +1.31162 q^{46} -3.61790 q^{47} +3.85246 q^{48} -4.65665 q^{49} +5.33039 q^{51} -7.88180 q^{52} +1.93686 q^{53} +1.18210 q^{54} +1.39207 q^{56} +2.41259 q^{57} -0.318723 q^{58} +3.87247 q^{59} -3.33009 q^{61} -2.06966 q^{62} -2.91868 q^{63} -6.75399 q^{64} -0.261804 q^{66} +9.93460 q^{67} -9.92485 q^{68} -5.95255 q^{69} +7.32783 q^{71} -1.73385 q^{72} +2.83121 q^{73} -0.289149 q^{74} -4.49209 q^{76} +1.66351 q^{77} -0.975320 q^{78} -5.12569 q^{79} +0.355172 q^{81} -0.642046 q^{82} +6.31069 q^{83} -3.11636 q^{84} +1.51352 q^{86} +1.44647 q^{87} +0.988217 q^{88} -7.31240 q^{89} +6.19722 q^{91} +11.0833 q^{92} +9.39281 q^{93} +0.833571 q^{94} -2.78938 q^{96} -4.89493 q^{97} +1.07290 q^{98} -2.07193 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.230402	−0.162919	−0.0814594	−	0.996677i	$-0.525958\pi$
$2$	−0.230402	−0.162919	−0.0814594	+	0.996677i	$0.525958\pi$
$3$	1.04564	0.603701	0.301850	−	0.953355i	$-0.402396\pi$
$3$	1.04564	0.603701	0.301850	+	0.953355i	$0.402396\pi$
$4$	−1.94691	−0.973457
$5$	0	0
$5$	0	0
$6$	−0.240918	−0.0983542
$7$	1.53080	0.578588	0.289294	−	0.957240i	$-0.406579\pi$
$7$	1.53080	0.578588	0.289294	+	0.957240i	$0.406579\pi$
$8$	0.909377	0.321513
$9$	−1.90664	−0.635545
$10$	0	0
$11$	1.08670	0.327651	0.163826	−	0.986489i	$-0.447617\pi$
$11$	1.08670	0.327651	0.163826	+	0.986489i	$0.447617\pi$
$12$	−2.03577	−0.587677
$13$	4.04835	1.12281	0.561406	−	0.827541i	$-0.310261\pi$
$13$	4.04835	1.12281	0.561406	+	0.827541i	$0.310261\pi$
$14$	−0.352699	−0.0942629
$15$	0	0
$16$	3.68431	0.921077
$17$	5.09773	1.23638	0.618191	−	0.786028i	$-0.287866\pi$
$17$	5.09773	1.23638	0.618191	+	0.786028i	$0.287866\pi$
$18$	0.439293	0.103542
$19$	2.30729	0.529328	0.264664	−	0.964341i	$-0.414739\pi$
$19$	2.30729	0.529328	0.264664	+	0.964341i	$0.414739\pi$
$20$	0	0
$21$	1.60067	0.349294
$22$	−0.250377	−0.0533806
$23$	−5.69274	−1.18702	−0.593509	−	0.804828i	$-0.702258\pi$
$23$	−5.69274	−1.18702	−0.593509	+	0.804828i	$0.702258\pi$
$24$	0.950881	0.194098
$25$	0	0
$26$	−0.932749	−0.182927
$27$	−5.13058	−0.987380
$28$	−2.98034	−0.563231
$29$	1.38333	0.256878	0.128439	−	0.991717i	$-0.459003\pi$
$29$	1.38333	0.256878	0.128439	+	0.991717i	$0.459003\pi$
$30$	0	0
$31$	8.98284	1.61336	0.806682	−	0.590985i	$-0.201261\pi$
$31$	8.98284	1.61336	0.806682	+	0.590985i	$0.201261\pi$
$32$	−2.66763	−0.471574
$33$	1.13629	0.197803
$34$	−1.17453	−0.201430
$35$	0	0
$36$	3.71206	0.618677
$37$	1.25497	0.206316	0.103158	−	0.994665i	$-0.467105\pi$
$37$	1.25497	0.206316	0.103158	+	0.994665i	$0.467105\pi$
$38$	−0.531604	−0.0862375
$39$	4.23312	0.677842
$40$	0	0
$41$	2.78663	0.435199	0.217599	−	0.976038i	$-0.430177\pi$
$41$	2.78663	0.435199	0.217599	+	0.976038i	$0.430177\pi$
$42$	−0.368797	−0.0569066
$43$	−6.56905	−1.00177	−0.500886	−	0.865513i	$-0.666992\pi$
$43$	−6.56905	−1.00177	−0.500886	+	0.865513i	$0.666992\pi$
$44$	−2.11571	−0.318955
$45$	0	0
$46$	1.31162	0.193387
$47$	−3.61790	−0.527725	−0.263862	−	0.964560i	$-0.584997\pi$
$47$	−3.61790	−0.527725	−0.263862	+	0.964560i	$0.584997\pi$
$48$	3.85246	0.556055
$49$	−4.65665	−0.665236
$50$	0	0
$51$	5.33039	0.746404
$52$	−7.88180	−1.09301
$53$	1.93686	0.266049	0.133024	−	0.991113i	$-0.457531\pi$
$53$	1.93686	0.266049	0.133024	+	0.991113i	$0.457531\pi$
$54$	1.18210	0.160863
$55$	0	0
$56$	1.39207	0.186024
$57$	2.41259	0.319556
$58$	−0.318723	−0.0418503
$59$	3.87247	0.504153	0.252076	−	0.967707i	$-0.418887\pi$
$59$	3.87247	0.504153	0.252076	+	0.967707i	$0.418887\pi$
$60$	0	0
$61$	−3.33009	−0.426374	−0.213187	−	0.977011i	$-0.568384\pi$
$61$	−3.33009	−0.426374	−0.213187	+	0.977011i	$0.568384\pi$
$62$	−2.06966	−0.262847
$63$	−2.91868	−0.367719
$64$	−6.75399	−0.844249
$65$	0	0
$66$	−0.261804	−0.0322259
$67$	9.93460	1.21370	0.606852	−	0.794815i	$-0.292432\pi$
$67$	9.93460	1.21370	0.606852	+	0.794815i	$0.292432\pi$
$68$	−9.92485	−1.20356
$69$	−5.95255	−0.716603
$70$	0	0
$71$	7.32783	0.869653	0.434827	−	0.900514i	$-0.356810\pi$
$71$	7.32783	0.869653	0.434827	+	0.900514i	$0.356810\pi$
$72$	−1.73385	−0.204336
$73$	2.83121	0.331369	0.165684	−	0.986179i	$-0.447017\pi$
$73$	2.83121	0.331369	0.165684	+	0.986179i	$0.447017\pi$
$74$	−0.289149	−0.0336128
$75$	0	0
$76$	−4.49209	−0.515278
$77$	1.66351	0.189575
$78$	−0.975320	−0.110433
$79$	−5.12569	−0.576685	−0.288343	−	0.957527i	$-0.593104\pi$
$79$	−5.12569	−0.576685	−0.288343	+	0.957527i	$0.593104\pi$
$80$	0	0
$81$	0.355172	0.0394636
$82$	−0.642046	−0.0709021
$83$	6.31069	0.692688	0.346344	−	0.938108i	$-0.387423\pi$
$83$	6.31069	0.692688	0.346344	+	0.938108i	$0.387423\pi$
$84$	−3.11636	−0.340023
$85$	0	0
$86$	1.51352	0.163207
$87$	1.44647	0.155078
$88$	0.988217	0.105344
$89$	−7.31240	−0.775113	−0.387556	−	0.921846i	$-0.626681\pi$
$89$	−7.31240	−0.775113	−0.387556	+	0.921846i	$0.626681\pi$
$90$	0	0
$91$	6.19722	0.649645
$92$	11.0833	1.15551
$93$	9.39281	0.973989
$94$	0.833571	0.0859763
$95$	0	0
$96$	−2.78938	−0.284690
$97$	−4.89493	−0.497005	−0.248502	−	0.968631i	$-0.579938\pi$
$97$	−4.89493	−0.497005	−0.248502	+	0.968631i	$0.579938\pi$
$98$	1.07290	0.108379
$99$	−2.07193	−0.208237
$100$	0	0
$101$	−2.13547	−0.212487	−0.106244	−	0.994340i	$-0.533882\pi$
$101$	−2.13547	−0.212487	−0.106244	+	0.994340i	$0.533882\pi$
$102$	−1.22813	−0.121603
$103$	−12.0399	−1.18633	−0.593164	−	0.805082i	$-0.702122\pi$
$103$	−12.0399	−1.18633	−0.593164	+	0.805082i	$0.702122\pi$
$104$	3.68148	0.360999
$105$	0	0
$106$	−0.446257	−0.0433443
$107$	−4.51984	−0.436950	−0.218475	−	0.975843i	$-0.570108\pi$
$107$	−4.51984	−0.436950	−0.218475	+	0.975843i	$0.570108\pi$
$108$	9.98880	0.961172
$109$	4.59602	0.440219	0.220110	−	0.975475i	$-0.429359\pi$
$109$	4.59602	0.440219	0.220110	+	0.975475i	$0.429359\pi$
$110$	0	0
$111$	1.31225	0.124553
$112$	5.63994	0.532924
$113$	9.49538	0.893250	0.446625	−	0.894721i	$-0.352626\pi$
$113$	9.49538	0.893250	0.446625	+	0.894721i	$0.352626\pi$
$114$	−0.555866	−0.0520616
$115$	0	0
$116$	−2.69323	−0.250060
$117$	−7.71874	−0.713598
$118$	−0.892225	−0.0821360
$119$	7.80361	0.715356
$120$	0	0
$121$	−9.81909	−0.892645
$122$	0.767259	0.0694644
$123$	2.91381	0.262730
$124$	−17.4888	−1.57054
$125$	0	0
$126$	0.672470	0.0599083
$127$	−7.44543	−0.660675	−0.330338	−	0.943863i	$-0.607163\pi$
$127$	−7.44543	−0.660675	−0.330338	+	0.943863i	$0.607163\pi$
$128$	6.89138	0.609118
$129$	−6.86887	−0.604770
$130$	0	0
$131$	1.20150	0.104976	0.0524879	−	0.998622i	$-0.483285\pi$
$131$	1.20150	0.104976	0.0524879	+	0.998622i	$0.483285\pi$
$132$	−2.21227	−0.192553
$133$	3.53200	0.306263
$134$	−2.28895	−0.197735
$135$	0	0
$136$	4.63576	0.397513
$137$	17.5079	1.49580	0.747901	−	0.663810i	$-0.231062\pi$
$137$	17.5079	1.49580	0.747901	+	0.663810i	$0.231062\pi$
$138$	1.37148	0.116748
$139$	−3.26785	−0.277175	−0.138588	−	0.990350i	$-0.544256\pi$
$139$	−3.26785	−0.277175	−0.138588	+	0.990350i	$0.544256\pi$
$140$	0	0
$141$	−3.78302	−0.318588
$142$	−1.68835	−0.141683
$143$	4.39933	0.367891
$144$	−7.02464	−0.585386
$145$	0	0
$146$	−0.652317	−0.0539862
$147$	−4.86918	−0.401603
$148$	−2.44333	−0.200840
$149$	16.7874	1.37528	0.687640	−	0.726052i	$-0.258647\pi$
$149$	16.7874	1.37528	0.687640	+	0.726052i	$0.258647\pi$
$150$	0	0
$151$	16.4256	1.33669	0.668347	−	0.743850i	$-0.267002\pi$
$151$	16.4256	1.33669	0.668347	+	0.743850i	$0.267002\pi$
$152$	2.09819	0.170186
$153$	−9.71952	−0.785777
$154$	−0.383277	−0.0308853
$155$	0	0
$156$	−8.24153	−0.659850
$157$	−0.346594	−0.0276612	−0.0138306	−	0.999904i	$-0.504403\pi$
$157$	−0.346594	−0.0276612	−0.0138306	+	0.999904i	$0.504403\pi$
$158$	1.18097	0.0939528
$159$	2.02526	0.160614
$160$	0	0
$161$	−8.71444	−0.686794
$162$	−0.0818324	−0.00642936
$163$	11.3264	0.887153	0.443577	−	0.896236i	$-0.353709\pi$
$163$	11.3264	0.887153	0.443577	+	0.896236i	$0.353709\pi$
$164$	−5.42534	−0.423648
$165$	0	0
$166$	−1.45400	−0.112852
$167$	12.5729	0.972920	0.486460	−	0.873703i	$-0.338288\pi$
$167$	12.5729	0.972920	0.486460	+	0.873703i	$0.338288\pi$
$168$	1.45561	0.112303
$169$	3.38917	0.260705
$170$	0	0
$171$	−4.39916	−0.336412
$172$	12.7894	0.975182
$173$	6.67331	0.507362	0.253681	−	0.967288i	$-0.418359\pi$
$173$	6.67331	0.507362	0.253681	+	0.967288i	$0.418359\pi$
$174$	−0.333269	−0.0252651
$175$	0	0
$176$	4.00372	0.301792
$177$	4.04921	0.304357
$178$	1.68479	0.126280
$179$	8.99772	0.672521	0.336261	−	0.941769i	$-0.390838\pi$
$179$	8.99772	0.672521	0.336261	+	0.941769i	$0.390838\pi$
$180$	0	0
$181$	7.72994	0.574562	0.287281	−	0.957846i	$-0.407249\pi$
$181$	7.72994	0.574562	0.287281	+	0.957846i	$0.407249\pi$
$182$	−1.42785	−0.105839
$183$	−3.48207	−0.257402
$184$	−5.17684	−0.381642
$185$	0	0
$186$	−2.16412	−0.158681
$187$	5.53969	0.405102
$188$	7.04374	0.513718
$189$	−7.85389	−0.571286
$190$	0	0
$191$	−0.321076	−0.0232322	−0.0116161	−	0.999933i	$-0.503698\pi$
$191$	−0.321076	−0.0232322	−0.0116161	+	0.999933i	$0.503698\pi$
$192$	−7.06224	−0.509673
$193$	−2.26293	−0.162889	−0.0814445	−	0.996678i	$-0.525953\pi$
$193$	−2.26293	−0.162889	−0.0814445	+	0.996678i	$0.525953\pi$
$194$	1.12780	0.0809714
$195$	0	0
$196$	9.06610	0.647579
$197$	11.3106	0.805850	0.402925	−	0.915233i	$-0.367994\pi$
$197$	11.3106	0.805850	0.402925	+	0.915233i	$0.367994\pi$
$198$	0.477378	0.0339258
$199$	6.90978	0.489821	0.244911	−	0.969546i	$-0.421241\pi$
$199$	6.90978	0.489821	0.244911	+	0.969546i	$0.421241\pi$
$200$	0	0
$201$	10.3880	0.732714
$202$	0.492017	0.0346182
$203$	2.11761	0.148627
$204$	−10.3778	−0.726593
$205$	0	0
$206$	2.77402	0.193275
$207$	10.8540	0.754404
$208$	14.9154	1.03420
$209$	2.50732	0.173435
$210$	0	0
$211$	21.8441	1.50381	0.751906	−	0.659270i	$-0.229135\pi$
$211$	21.8441	1.50381	0.751906	+	0.659270i	$0.229135\pi$
$212$	−3.77091	−0.258987
$213$	7.66227	0.525010
$214$	1.04138	0.0711873
$215$	0	0
$216$	−4.66563	−0.317456
$217$	13.7509	0.933474
$218$	−1.05893	−0.0717200
$219$	2.96043	0.200047
$220$	0	0
$221$	20.6374	1.38822
$222$	−0.302345	−0.0202921
$223$	6.84006	0.458044	0.229022	−	0.973421i	$-0.426447\pi$
$223$	6.84006	0.458044	0.229022	+	0.973421i	$0.426447\pi$
$224$	−4.08360	−0.272847
$225$	0	0
$226$	−2.18775	−0.145527
$227$	−14.1436	−0.938746	−0.469373	−	0.883000i	$-0.655520\pi$
$227$	−14.1436	−0.938746	−0.469373	+	0.883000i	$0.655520\pi$
$228$	−4.69711	−0.311074
$229$	−24.3595	−1.60972	−0.804861	−	0.593464i	$-0.797760\pi$
$229$	−24.3595	−1.60972	−0.804861	+	0.593464i	$0.797760\pi$
$230$	0	0
$231$	1.73944	0.114447
$232$	1.25797	0.0825899
$233$	−7.29164	−0.477692	−0.238846	−	0.971058i	$-0.576769\pi$
$233$	−7.29164	−0.477692	−0.238846	+	0.971058i	$0.576769\pi$
$234$	1.77841	0.116258
$235$	0	0
$236$	−7.53937	−0.490771
$237$	−5.35963	−0.348145
$238$	−1.79797	−0.116545
$239$	−4.94493	−0.319861	−0.159931	−	0.987128i	$-0.551127\pi$
$239$	−4.94493	−0.319861	−0.159931	+	0.987128i	$0.551127\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	2.26234	0.145429
$243$	15.7631	1.01120
$244$	6.48340	0.415057
$245$	0	0
$246$	−0.671349	−0.0428036
$247$	9.34072	0.594336
$248$	8.16878	0.518718
$249$	6.59871	0.418176
$250$	0	0
$251$	8.77699	0.553999	0.276999	−	0.960870i	$-0.410660\pi$
$251$	8.77699	0.553999	0.276999	+	0.960870i	$0.410660\pi$
$252$	5.68242	0.357959
$253$	−6.18627	−0.388928
$254$	1.71544	0.107636
$255$	0	0
$256$	11.9202	0.745012
$257$	14.1648	0.883575	0.441787	−	0.897120i	$-0.354345\pi$
$257$	14.1648	0.883575	0.441787	+	0.897120i	$0.354345\pi$
$258$	1.58260	0.0985284
$259$	1.92111	0.119372
$260$	0	0
$261$	−2.63751	−0.163258
$262$	−0.276829	−0.0171025
$263$	16.1691	0.997029	0.498515	−	0.866881i	$-0.333879\pi$
$263$	16.1691	0.997029	0.498515	+	0.866881i	$0.333879\pi$
$264$	1.03332	0.0635964
$265$	0	0
$266$	−0.813779	−0.0498960
$267$	−7.64614	−0.467936
$268$	−19.3418	−1.18149
$269$	5.34121	0.325659	0.162830	−	0.986654i	$-0.447938\pi$
$269$	5.34121	0.325659	0.162830	+	0.986654i	$0.447938\pi$
$270$	0	0
$271$	−9.65289	−0.586371	−0.293186	−	0.956056i	$-0.594715\pi$
$271$	−9.65289	−0.586371	−0.293186	+	0.956056i	$0.594715\pi$
$272$	18.7816	1.13880
$273$	6.48006	0.392191
$274$	−4.03386	−0.243694
$275$	0	0
$276$	11.5891	0.697583
$277$	1.56795	0.0942091	0.0471045	−	0.998890i	$-0.485001\pi$
$277$	1.56795	0.0942091	0.0471045	+	0.998890i	$0.485001\pi$
$278$	0.752919	0.0451571
$279$	−17.1270	−1.02537
$280$	0	0
$281$	−23.3023	−1.39010	−0.695050	−	0.718962i	$-0.744618\pi$
$281$	−23.3023	−1.39010	−0.695050	+	0.718962i	$0.744618\pi$
$282$	0.871616	0.0519040
$283$	17.2315	1.02430	0.512152	−	0.858895i	$-0.328848\pi$
$283$	17.2315	1.02430	0.512152	+	0.858895i	$0.328848\pi$
$284$	−14.2667	−0.846570
$285$	0	0
$286$	−1.01361	−0.0599363
$287$	4.26578	0.251801
$288$	5.08619	0.299707
$289$	8.98687	0.528640
$290$	0	0
$291$	−5.11834	−0.300042
$292$	−5.51213	−0.322573
$293$	−10.5402	−0.615762	−0.307881	−	0.951425i	$-0.599620\pi$
$293$	−10.5402	−0.615762	−0.307881	+	0.951425i	$0.599620\pi$
$294$	1.12187	0.0654287
$295$	0	0
$296$	1.14124	0.0663335
$297$	−5.57538	−0.323516
$298$	−3.86786	−0.224059
$299$	−23.0462	−1.33280
$300$	0	0
$301$	−10.0559	−0.579613
$302$	−3.78448	−0.217773
$303$	−2.23293	−0.128279
$304$	8.50076	0.487552
$305$	0	0
$306$	2.23940	0.128018
$307$	25.8670	1.47631	0.738154	−	0.674633i	$-0.235698\pi$
$307$	25.8670	1.47631	0.738154	+	0.674633i	$0.235698\pi$
$308$	−3.23872	−0.184543
$309$	−12.5894	−0.716187
$310$	0	0
$311$	12.2169	0.692758	0.346379	−	0.938095i	$-0.387411\pi$
$311$	12.2169	0.692758	0.346379	+	0.938095i	$0.387411\pi$
$312$	3.84950	0.217935
$313$	26.1973	1.48076	0.740380	−	0.672189i	$-0.234646\pi$
$313$	26.1973	1.48076	0.740380	+	0.672189i	$0.234646\pi$
$314$	0.0798559	0.00450653
$315$	0	0
$316$	9.97928	0.561378
$317$	6.90950	0.388076	0.194038	−	0.980994i	$-0.437841\pi$
$317$	6.90950	0.388076	0.194038	+	0.980994i	$0.437841\pi$
$318$	−0.466624	−0.0261670
$319$	1.50326	0.0841666
$320$	0	0
$321$	−4.72613	−0.263787
$322$	2.00782	0.111892
$323$	11.7619	0.654452
$324$	−0.691490	−0.0384161
$325$	0	0
$326$	−2.60963	−0.144534
$327$	4.80579	0.265761
$328$	2.53410	0.139922
$329$	−5.53828	−0.305335
$330$	0	0
$331$	−4.03471	−0.221768	−0.110884	−	0.993833i	$-0.535368\pi$
$331$	−4.03471	−0.221768	−0.110884	+	0.993833i	$0.535368\pi$
$332$	−12.2864	−0.674303
$333$	−2.39278	−0.131124
$334$	−2.89682	−0.158507
$335$	0	0
$336$	5.89735	0.321727
$337$	5.92948	0.322999	0.161500	−	0.986873i	$-0.448367\pi$
$337$	5.92948	0.322999	0.161500	+	0.986873i	$0.448367\pi$
$338$	−0.780871	−0.0424738
$339$	9.92875	0.539256
$340$	0	0
$341$	9.76161	0.528621
$342$	1.01358	0.0548079
$343$	−17.8440	−0.963486
$344$	−5.97375	−0.322083
$345$	0	0
$346$	−1.53754	−0.0826589
$347$	−1.88122	−0.100989	−0.0504945	−	0.998724i	$-0.516080\pi$
$347$	−1.88122	−0.100989	−0.0504945	+	0.998724i	$0.516080\pi$
$348$	−2.81615	−0.150962
$349$	−21.3569	−1.14321	−0.571604	−	0.820530i	$-0.693678\pi$
$349$	−21.3569	−1.14321	−0.571604	+	0.820530i	$0.693678\pi$
$350$	0	0
$351$	−20.7704	−1.10864
$352$	−2.89890	−0.154512
$353$	−0.297877	−0.0158544	−0.00792720	−	0.999969i	$-0.502523\pi$
$353$	−0.297877	−0.0158544	−0.00792720	+	0.999969i	$0.502523\pi$
$354$	−0.932946	−0.0495855
$355$	0	0
$356$	14.2366	0.754539
$357$	8.15977	0.431861
$358$	−2.07309	−0.109566
$359$	10.0336	0.529554	0.264777	−	0.964310i	$-0.414702\pi$
$359$	10.0336	0.529554	0.264777	+	0.964310i	$0.414702\pi$
$360$	0	0
$361$	−13.6764	−0.719812
$362$	−1.78099	−0.0936069
$363$	−10.2672	−0.538890
$364$	−12.0655	−0.632402
$365$	0	0
$366$	0.802277	0.0419357
$367$	−16.6581	−0.869545	−0.434772	−	0.900540i	$-0.643171\pi$
$367$	−16.6581	−0.869545	−0.434772	+	0.900540i	$0.643171\pi$
$368$	−20.9738	−1.09333
$369$	−5.31309	−0.276589
$370$	0	0
$371$	2.96495	0.153932
$372$	−18.2870	−0.948137
$373$	11.3025	0.585222	0.292611	−	0.956232i	$-0.405476\pi$
$373$	11.3025	0.585222	0.292611	+	0.956232i	$0.405476\pi$
$374$	−1.27635	−0.0659987
$375$	0	0
$376$	−3.29004	−0.169671
$377$	5.60022	0.288426
$378$	1.80955	0.0930733
$379$	−21.4003	−1.09926	−0.549629	−	0.835409i	$-0.685231\pi$
$379$	−21.4003	−1.09926	−0.549629	+	0.835409i	$0.685231\pi$
$380$	0	0
$381$	−7.78524	−0.398850
$382$	0.0739765	0.00378497
$383$	1.12371	0.0574188	0.0287094	−	0.999588i	$-0.490860\pi$
$383$	1.12371	0.0574188	0.0287094	+	0.999588i	$0.490860\pi$
$384$	7.20591	0.367725
$385$	0	0
$386$	0.521383	0.0265377
$387$	12.5248	0.636671
$388$	9.53001	0.483813
$389$	26.3859	1.33782	0.668908	−	0.743345i	$-0.266762\pi$
$389$	26.3859	1.33782	0.668908	+	0.743345i	$0.266762\pi$
$390$	0	0
$391$	−29.0200	−1.46761
$392$	−4.23465	−0.213882
$393$	1.25634	0.0633740
$394$	−2.60599	−0.131288
$395$	0	0
$396$	4.03388	0.202710
$397$	−10.7901	−0.541539	−0.270769	−	0.962644i	$-0.587278\pi$
$397$	−10.7901	−0.541539	−0.270769	+	0.962644i	$0.587278\pi$
$398$	−1.59203	−0.0798011
$399$	3.69320	0.184891
$400$	0	0
$401$	−16.1119	−0.804590	−0.402295	−	0.915510i	$-0.631787\pi$
$401$	−16.1119	−0.804590	−0.402295	+	0.915510i	$0.631787\pi$
$402$	−2.39342	−0.119373
$403$	36.3657	1.81150
$404$	4.15758	0.206847
$405$	0	0
$406$	−0.487901	−0.0242141
$407$	1.36378	0.0675998
$408$	4.84734	0.239979
$409$	−4.52397	−0.223696	−0.111848	−	0.993725i	$-0.535677\pi$
$409$	−4.52397	−0.223696	−0.111848	+	0.993725i	$0.535677\pi$
$410$	0	0
$411$	18.3070	0.903017
$412$	23.4407	1.15484
$413$	5.92798	0.291697
$414$	−2.50078	−0.122907
$415$	0	0
$416$	−10.7995	−0.529489
$417$	−3.41699	−0.167331
$418$	−0.577692	−0.0282558
$419$	23.3500	1.14072	0.570360	−	0.821395i	$-0.306804\pi$
$419$	23.3500	1.14072	0.570360	+	0.821395i	$0.306804\pi$
$420$	0	0
$421$	−20.7759	−1.01256	−0.506278	−	0.862371i	$-0.668979\pi$
$421$	−20.7759	−1.01256	−0.506278	+	0.862371i	$0.668979\pi$
$422$	−5.03293	−0.244999
$423$	6.89802	0.335393
$424$	1.76134	0.0855382
$425$	0	0
$426$	−1.76540	−0.0855340
$427$	−5.09770	−0.246695
$428$	8.79975	0.425352
$429$	4.60012	0.222096
$430$	0	0
$431$	−13.7490	−0.662264	−0.331132	−	0.943584i	$-0.607431\pi$
$431$	−13.7490	−0.662264	−0.331132	+	0.943584i	$0.607431\pi$
$432$	−18.9026	−0.909453
$433$	34.0815	1.63785	0.818926	−	0.573899i	$-0.194570\pi$
$433$	34.0815	1.63785	0.818926	+	0.573899i	$0.194570\pi$
$434$	−3.16824	−0.152080
$435$	0	0
$436$	−8.94806	−0.428535
$437$	−13.1348	−0.628322
$438$	−0.682089	−0.0325915
$439$	24.6124	1.17468	0.587342	−	0.809339i	$-0.300174\pi$
$439$	24.6124	1.17468	0.587342	+	0.809339i	$0.300174\pi$
$440$	0	0
$441$	8.87854	0.422788
$442$	−4.75490	−0.226168
$443$	4.80798	0.228434	0.114217	−	0.993456i	$-0.463564\pi$
$443$	4.80798	0.228434	0.114217	+	0.993456i	$0.463564\pi$
$444$	−2.55484	−0.121247
$445$	0	0
$446$	−1.57596	−0.0746240
$447$	17.5536	0.830257
$448$	−10.3390	−0.488472
$449$	−6.92367	−0.326748	−0.163374	−	0.986564i	$-0.552238\pi$
$449$	−6.92367	−0.326748	−0.163374	+	0.986564i	$0.552238\pi$
$450$	0	0
$451$	3.02822	0.142593
$452$	−18.4867	−0.869541
$453$	17.1752	0.806963
$454$	3.25872	0.152939
$455$	0	0
$456$	2.19396	0.102741
$457$	−18.4552	−0.863299	−0.431650	−	0.902041i	$-0.642068\pi$
$457$	−18.4552	−0.863299	−0.431650	+	0.902041i	$0.642068\pi$
$458$	5.61248	0.262254
$459$	−26.1543	−1.22078
$460$	0	0
$461$	−3.34766	−0.155916	−0.0779579	−	0.996957i	$-0.524840\pi$
$461$	−3.34766	−0.155916	−0.0779579	+	0.996957i	$0.524840\pi$
$462$	−0.400770	−0.0186455
$463$	17.9370	0.833605	0.416802	−	0.908997i	$-0.363151\pi$
$463$	17.9370	0.833605	0.416802	+	0.908997i	$0.363151\pi$
$464$	5.09662	0.236605
$465$	0	0
$466$	1.68001	0.0778249
$467$	−31.3201	−1.44932	−0.724660	−	0.689106i	$-0.758003\pi$
$467$	−31.3201	−1.44932	−0.724660	+	0.689106i	$0.758003\pi$
$468$	15.0277	0.694657
$469$	15.2079	0.702235
$470$	0	0
$471$	−0.362413	−0.0166991
$472$	3.52154	0.162092
$473$	−7.13857	−0.328232
$474$	1.23487	0.0567194
$475$	0	0
$476$	−15.1930	−0.696368
$477$	−3.69289	−0.169086
$478$	1.13932	0.0521114
$479$	23.7157	1.08360	0.541799	−	0.840508i	$-0.317743\pi$
$479$	23.7157	1.08360	0.541799	+	0.840508i	$0.317743\pi$
$480$	0	0
$481$	5.08058	0.231654
$482$	−0.230402	−0.0104945
$483$	−9.11217	−0.414618
$484$	19.1169	0.868952
$485$	0	0
$486$	−3.63185	−0.164744
$487$	7.54423	0.341862	0.170931	−	0.985283i	$-0.445323\pi$
$487$	7.54423	0.341862	0.170931	+	0.985283i	$0.445323\pi$
$488$	−3.02831	−0.137085
$489$	11.8434	0.535575
$490$	0	0
$491$	10.7212	0.483841	0.241921	−	0.970296i	$-0.422223\pi$
$491$	10.7212	0.483841	0.241921	+	0.970296i	$0.422223\pi$
$492$	−5.67295	−0.255756
$493$	7.05186	0.317600
$494$	−2.15212	−0.0968285
$495$	0	0
$496$	33.0955	1.48603
$497$	11.2174	0.503171
$498$	−1.52036	−0.0681288
$499$	0.0761574	0.00340928	0.00170464	−	0.999999i	$-0.499457\pi$
$499$	0.0761574	0.00340928	0.00170464	+	0.999999i	$0.499457\pi$
$500$	0	0
$501$	13.1467	0.587353
$502$	−2.02224	−0.0902569
$503$	10.2920	0.458899	0.229450	−	0.973321i	$-0.426307\pi$
$503$	10.2920	0.458899	0.229450	+	0.973321i	$0.426307\pi$
$504$	−2.65418	−0.118227
$505$	0	0
$506$	1.42533	0.0633636
$507$	3.54385	0.157388
$508$	14.4956	0.643139
$509$	28.5944	1.26742	0.633712	−	0.773569i	$-0.281530\pi$
$509$	28.5944	1.26742	0.633712	+	0.773569i	$0.281530\pi$
$510$	0	0
$511$	4.33402	0.191726
$512$	−16.5292	−0.730495
$513$	−11.8377	−0.522648
$514$	−3.26359	−0.143951
$515$	0	0
$516$	13.3731	0.588718
$517$	−3.93156	−0.172910
$518$	−0.442629	−0.0194480
$519$	6.97788	0.306295
$520$	0	0
$521$	25.9647	1.13753	0.568766	−	0.822499i	$-0.307421\pi$
$521$	25.9647	1.13753	0.568766	+	0.822499i	$0.307421\pi$
$522$	0.607688	0.0265978
$523$	22.0503	0.964192	0.482096	−	0.876118i	$-0.339876\pi$
$523$	22.0503	0.964192	0.482096	+	0.876118i	$0.339876\pi$
$524$	−2.33923	−0.102190
$525$	0	0
$526$	−3.72539	−0.162435
$527$	45.7921	1.99473
$528$	4.18645	0.182192
$529$	9.40724	0.409010
$530$	0	0
$531$	−7.38339	−0.320412
$532$	−6.87650	−0.298134
$533$	11.2813	0.488646
$534$	1.76169	0.0762356
$535$	0	0
$536$	9.03430	0.390222
$537$	9.40838	0.406001
$538$	−1.23063	−0.0530561
$539$	−5.06037	−0.217965
$540$	0	0
$541$	−12.0290	−0.517169	−0.258584	−	0.965989i	$-0.583256\pi$
$541$	−12.0290	−0.517169	−0.258584	+	0.965989i	$0.583256\pi$
$542$	2.22404	0.0955309
$543$	8.08273	0.346863
$544$	−13.5988	−0.583046
$545$	0	0
$546$	−1.49302	−0.0638953
$547$	18.8127	0.804373	0.402187	−	0.915558i	$-0.368250\pi$
$547$	18.8127	0.804373	0.402187	+	0.915558i	$0.368250\pi$
$548$	−34.0864	−1.45610
$549$	6.34927	0.270980
$550$	0	0
$551$	3.19175	0.135973
$552$	−5.41312	−0.230398
$553$	−7.84640	−0.333663
$554$	−0.361259	−0.0153484
$555$	0	0
$556$	6.36222	0.269818
$557$	21.3322	0.903875	0.451938	−	0.892050i	$-0.350733\pi$
$557$	21.3322	0.903875	0.451938	+	0.892050i	$0.350733\pi$
$558$	3.94610	0.167052
$559$	−26.5939	−1.12480
$560$	0	0
$561$	5.79252	0.244560
$562$	5.36890	0.226473
$563$	2.58915	0.109120	0.0545598	−	0.998511i	$-0.482624\pi$
$563$	2.58915	0.109120	0.0545598	+	0.998511i	$0.482624\pi$
$564$	7.36522	0.310132
$565$	0	0
$566$	−3.97017	−0.166878
$567$	0.543698	0.0228332
$568$	6.66376	0.279605
$569$	−6.51247	−0.273017	−0.136508	−	0.990639i	$-0.543588\pi$
$569$	−6.51247	−0.273017	−0.136508	+	0.990639i	$0.543588\pi$
$570$	0	0
$571$	−28.0880	−1.17545	−0.587723	−	0.809062i	$-0.699975\pi$
$571$	−28.0880	−1.17545	−0.587723	+	0.809062i	$0.699975\pi$
$572$	−8.56512	−0.358126
$573$	−0.335730	−0.0140253
$574$	−0.982843	−0.0410231
$575$	0	0
$576$	12.8774	0.536558
$577$	−27.1681	−1.13102	−0.565512	−	0.824740i	$-0.691321\pi$
$577$	−27.1681	−1.13102	−0.565512	+	0.824740i	$0.691321\pi$
$578$	−2.07059	−0.0861253
$579$	−2.36621	−0.0983362
$580$	0	0
$581$	9.66041	0.400781
$582$	1.17927	0.0488825
$583$	2.10478	0.0871711
$584$	2.57464	0.106539
$585$	0	0
$586$	2.42847	0.100319
$587$	9.38467	0.387347	0.193673	−	0.981066i	$-0.437960\pi$
$587$	9.38467	0.387347	0.193673	+	0.981066i	$0.437960\pi$
$588$	9.47988	0.390944
$589$	20.7260	0.853999
$590$	0	0
$591$	11.8269	0.486492
$592$	4.62371	0.190033
$593$	−16.4860	−0.676999	−0.338500	−	0.940967i	$-0.609919\pi$
$593$	−16.4860	−0.676999	−0.338500	+	0.940967i	$0.609919\pi$
$594$	1.28458	0.0527069
$595$	0	0
$596$	−32.6837	−1.33878
$597$	7.22514	0.295705
$598$	5.30989	0.217138
$599$	−44.1444	−1.80369	−0.901846	−	0.432057i	$-0.857788\pi$
$599$	−44.1444	−1.80369	−0.901846	+	0.432057i	$0.857788\pi$
$600$	0	0
$601$	44.1270	1.79998	0.899989	−	0.435913i	$-0.143575\pi$
$601$	44.1270	1.79998	0.899989	+	0.435913i	$0.143575\pi$
$602$	2.31690	0.0944298
$603$	−18.9417	−0.771365
$604$	−31.9792	−1.30121
$605$	0	0
$606$	0.514472	0.0208990
$607$	−26.3335	−1.06884	−0.534422	−	0.845218i	$-0.679471\pi$
$607$	−26.3335	−1.06884	−0.534422	+	0.845218i	$0.679471\pi$
$608$	−6.15498	−0.249617
$609$	2.21425	0.0897261
$610$	0	0
$611$	−14.6465	−0.592536
$612$	18.9231	0.764920
$613$	4.62380	0.186753	0.0933767	−	0.995631i	$-0.470234\pi$
$613$	4.62380	0.186753	0.0933767	+	0.995631i	$0.470234\pi$
$614$	−5.95981	−0.240518
$615$	0	0
$616$	1.51276	0.0609509
$617$	15.1645	0.610499	0.305249	−	0.952272i	$-0.401260\pi$
$617$	15.1645	0.610499	0.305249	+	0.952272i	$0.401260\pi$
$618$	2.90063	0.116680
$619$	5.73669	0.230577	0.115289	−	0.993332i	$-0.463221\pi$
$619$	5.73669	0.230577	0.115289	+	0.993332i	$0.463221\pi$
$620$	0	0
$621$	29.2070	1.17204
$622$	−2.81480	−0.112863
$623$	−11.1938	−0.448471
$624$	15.5961	0.624345
$625$	0	0
$626$	−6.03591	−0.241244
$627$	2.62176	0.104703
$628$	0.674789	0.0269270
$629$	6.39752	0.255086
$630$	0	0
$631$	30.1959	1.20208	0.601040	−	0.799219i	$-0.294753\pi$
$631$	30.1959	1.20208	0.601040	+	0.799219i	$0.294753\pi$
$632$	−4.66118	−0.185412
$633$	22.8411	0.907853
$634$	−1.59196	−0.0632249
$635$	0	0
$636$	−3.94301	−0.156351
$637$	−18.8518	−0.746934
$638$	−0.346355	−0.0137123
$639$	−13.9715	−0.552704
$640$	0	0
$641$	−26.2515	−1.03687	−0.518436	−	0.855117i	$-0.673485\pi$
$641$	−26.2515	−1.03687	−0.518436	+	0.855117i	$0.673485\pi$
$642$	1.08891	0.0429758
$643$	0.312363	0.0123184	0.00615920	−	0.999981i	$-0.498039\pi$
$643$	0.312363	0.0123184	0.00615920	+	0.999981i	$0.498039\pi$
$644$	16.9663	0.668565
$645$	0	0
$646$	−2.70997	−0.106622
$647$	−10.7566	−0.422887	−0.211444	−	0.977390i	$-0.567816\pi$
$647$	−10.7566	−0.422887	−0.211444	+	0.977390i	$0.567816\pi$
$648$	0.322985	0.0126881
$649$	4.20820	0.165186
$650$	0	0
$651$	14.3785	0.563539
$652$	−22.0516	−0.863606
$653$	−17.7309	−0.693863	−0.346932	−	0.937890i	$-0.612776\pi$
$653$	−17.7309	−0.693863	−0.346932	+	0.937890i	$0.612776\pi$
$654$	−1.10726	−0.0432974
$655$	0	0
$656$	10.2668	0.400852
$657$	−5.39810	−0.210600
$658$	1.27603	0.0497449
$659$	−47.8260	−1.86304	−0.931518	−	0.363696i	$-0.881515\pi$
$659$	−47.8260	−1.86304	−0.931518	+	0.363696i	$0.881515\pi$
$660$	0	0
$661$	−27.6104	−1.07392	−0.536961	−	0.843607i	$-0.680428\pi$
$661$	−27.6104	−1.07392	−0.536961	+	0.843607i	$0.680428\pi$
$662$	0.929605	0.0361301
$663$	21.5793	0.838071
$664$	5.73880	0.222709
$665$	0	0
$666$	0.551301	0.0213625
$667$	−7.87495	−0.304919
$668$	−24.4784	−0.947097
$669$	7.15224	0.276522
$670$	0	0
$671$	−3.61879	−0.139702
$672$	−4.26998	−0.164718
$673$	−7.90820	−0.304839	−0.152419	−	0.988316i	$-0.548706\pi$
$673$	−7.90820	−0.304839	−0.152419	+	0.988316i	$0.548706\pi$
$674$	−1.36616	−0.0526226
$675$	0	0
$676$	−6.59842	−0.253786
$677$	5.57451	0.214246	0.107123	−	0.994246i	$-0.465836\pi$
$677$	5.57451	0.214246	0.107123	+	0.994246i	$0.465836\pi$
$678$	−2.28760	−0.0878549
$679$	−7.49316	−0.287561
$680$	0	0
$681$	−14.7892	−0.566722
$682$	−2.24910	−0.0861223
$683$	−28.3828	−1.08604	−0.543018	−	0.839721i	$-0.682719\pi$
$683$	−28.3828	−1.08604	−0.543018	+	0.839721i	$0.682719\pi$
$684$	8.56479	0.327483
$685$	0	0
$686$	4.11129	0.156970
$687$	−25.4713	−0.971790
$688$	−24.2024	−0.922708
$689$	7.84111	0.298722
$690$	0	0
$691$	0.606217	0.0230616	0.0115308	−	0.999934i	$-0.496330\pi$
$691$	0.606217	0.0230616	0.0115308	+	0.999934i	$0.496330\pi$
$692$	−12.9924	−0.493896
$693$	−3.17172	−0.120484
$694$	0.433436	0.0164530
$695$	0	0
$696$	1.31539	0.0498596
$697$	14.2055	0.538072
$698$	4.92067	0.186250
$699$	−7.62444	−0.288383
$700$	0	0
$701$	−21.9873	−0.830448	−0.415224	−	0.909719i	$-0.636297\pi$
$701$	−21.9873	−0.830448	−0.415224	+	0.909719i	$0.636297\pi$
$702$	4.78554	0.180619
$703$	2.89559	0.109209
$704$	−7.33953	−0.276619
$705$	0	0
$706$	0.0686315	0.00258298
$707$	−3.26898	−0.122943
$708$	−7.88347	−0.296279
$709$	−41.1374	−1.54495	−0.772474	−	0.635046i	$-0.780981\pi$
$709$	−41.1374	−1.54495	−0.772474	+	0.635046i	$0.780981\pi$
$710$	0	0
$711$	9.77283	0.366510
$712$	−6.64973	−0.249209
$713$	−51.1369	−1.91509
$714$	−1.88003	−0.0703582
$715$	0	0
$716$	−17.5178	−0.654671
$717$	−5.17062	−0.193100
$718$	−2.31177	−0.0862744
$719$	5.26010	0.196169	0.0980844	−	0.995178i	$-0.468728\pi$
$719$	5.26010	0.196169	0.0980844	+	0.995178i	$0.468728\pi$
$720$	0	0
$721$	−18.4307	−0.686395
$722$	3.15108	0.117271
$723$	1.04564	0.0388878
$724$	−15.0495	−0.559311
$725$	0	0
$726$	2.36559	0.0877954
$727$	8.87043	0.328986	0.164493	−	0.986378i	$-0.447401\pi$
$727$	8.87043	0.328986	0.164493	+	0.986378i	$0.447401\pi$
$728$	5.63561	0.208870
$729$	15.4170	0.571001
$730$	0	0
$731$	−33.4873	−1.23857
$732$	6.77930	0.250570
$733$	42.9508	1.58643	0.793213	−	0.608945i	$-0.208407\pi$
$733$	42.9508	1.58643	0.793213	+	0.608945i	$0.208407\pi$
$734$	3.83805	0.141665
$735$	0	0
$736$	15.1861	0.559767
$737$	10.7959	0.397672
$738$	1.22415	0.0450615
$739$	24.9819	0.918974	0.459487	−	0.888185i	$-0.348033\pi$
$739$	24.9819	0.918974	0.459487	+	0.888185i	$0.348033\pi$
$740$	0	0
$741$	9.76703	0.358801
$742$	−0.683130	−0.0250785
$743$	−35.4538	−1.30067	−0.650336	−	0.759647i	$-0.725372\pi$
$743$	−35.4538	−1.30067	−0.650336	+	0.759647i	$0.725372\pi$
$744$	8.54161	0.313151
$745$	0	0
$746$	−2.60412	−0.0953436
$747$	−12.0322	−0.440235
$748$	−10.7853	−0.394350
$749$	−6.91898	−0.252814
$750$	0	0
$751$	−26.3011	−0.959740	−0.479870	−	0.877340i	$-0.659316\pi$
$751$	−26.3011	−0.959740	−0.479870	+	0.877340i	$0.659316\pi$
$752$	−13.3295	−0.486075
$753$	9.17758	0.334450
$754$	−1.29030	−0.0469900
$755$	0	0
$756$	15.2908	0.556123
$757$	−1.71032	−0.0621627	−0.0310813	−	0.999517i	$-0.509895\pi$
$757$	−1.71032	−0.0621627	−0.0310813	+	0.999517i	$0.509895\pi$
$758$	4.93066	0.179090
$759$	−6.46862	−0.234796
$760$	0	0
$761$	−12.2248	−0.443148	−0.221574	−	0.975144i	$-0.571119\pi$
$761$	−12.2248	−0.443148	−0.221574	+	0.975144i	$0.571119\pi$
$762$	1.79374	0.0649802
$763$	7.03559	0.254705
$764$	0.625107	0.0226156
$765$	0	0
$766$	−0.258905	−0.00935460
$767$	15.6771	0.566068
$768$	12.4642	0.449764
$769$	38.6314	1.39308	0.696542	−	0.717516i	$-0.254721\pi$
$769$	38.6314	1.39308	0.696542	+	0.717516i	$0.254721\pi$
$770$	0	0
$771$	14.8113	0.533415
$772$	4.40573	0.158566
$773$	−17.0301	−0.612532	−0.306266	−	0.951946i	$-0.599080\pi$
$773$	−17.0301	−0.612532	−0.306266	+	0.951946i	$0.599080\pi$
$774$	−2.88574	−0.103726
$775$	0	0
$776$	−4.45134	−0.159794
$777$	2.00879	0.0720651
$778$	−6.07935	−0.217955
$779$	6.42956	0.230363
$780$	0	0
$781$	7.96312	0.284943
$782$	6.68628	0.239101
$783$	−7.09730	−0.253637
$784$	−17.1565	−0.612733
$785$	0	0
$786$	−0.289463	−0.0103248
$787$	−17.9144	−0.638580	−0.319290	−	0.947657i	$-0.603444\pi$
$787$	−17.9144	−0.638580	−0.319290	+	0.947657i	$0.603444\pi$
$788$	−22.0209	−0.784461
$789$	16.9071	0.601907
$790$	0	0
$791$	14.5355	0.516824
$792$	−1.88417	−0.0669511
$793$	−13.4814	−0.478738
$794$	2.48606	0.0882268
$795$	0	0
$796$	−13.4528	−0.476820
$797$	39.0586	1.38353	0.691764	−	0.722124i	$-0.256834\pi$
$797$	39.0586	1.38353	0.691764	+	0.722124i	$0.256834\pi$
$798$	−0.850920	−0.0301222
$799$	−18.4431	−0.652469
$800$	0	0
$801$	13.9421	0.492620
$802$	3.71221	0.131083
$803$	3.07667	0.108573
$804$	−20.2246	−0.713266
$805$	0	0
$806$	−8.37873	−0.295128
$807$	5.58499	0.196601
$808$	−1.94195	−0.0683175
$809$	39.4098	1.38558	0.692788	−	0.721142i	$-0.256382\pi$
$809$	39.4098	1.38558	0.692788	+	0.721142i	$0.256382\pi$
$810$	0	0
$811$	−34.5217	−1.21222	−0.606111	−	0.795380i	$-0.707271\pi$
$811$	−34.5217	−1.21222	−0.606111	+	0.795380i	$0.707271\pi$
$812$	−4.12280	−0.144682
$813$	−10.0934	−0.353993
$814$	−0.314217	−0.0110133
$815$	0	0
$816$	19.6388	0.687496
$817$	−15.1567	−0.530266
$818$	1.04233	0.0364442
$819$	−11.8158	−0.412879
$820$	0	0
$821$	28.7720	1.00415	0.502075	−	0.864824i	$-0.332570\pi$
$821$	28.7720	1.00415	0.502075	+	0.864824i	$0.332570\pi$
$822$	−4.21797	−0.147118
$823$	−2.85341	−0.0994636	−0.0497318	−	0.998763i	$-0.515837\pi$
$823$	−2.85341	−0.0994636	−0.0497318	+	0.998763i	$0.515837\pi$
$824$	−10.9488	−0.381420
$825$	0	0
$826$	−1.36582	−0.0475229
$827$	43.5715	1.51513	0.757565	−	0.652759i	$-0.226389\pi$
$827$	43.5715	1.51513	0.757565	+	0.652759i	$0.226389\pi$
$828$	−21.1318	−0.734380
$829$	28.1396	0.977330	0.488665	−	0.872471i	$-0.337484\pi$
$829$	28.1396	0.977330	0.488665	+	0.872471i	$0.337484\pi$
$830$	0	0
$831$	1.63951	0.0568741
$832$	−27.3425	−0.947932
$833$	−23.7384	−0.822485
$834$	0.787282	0.0272614
$835$	0	0
$836$	−4.88154	−0.168832
$837$	−46.0871	−1.59300
$838$	−5.37988	−0.185845
$839$	−34.8055	−1.20162	−0.600810	−	0.799392i	$-0.705155\pi$
$839$	−34.8055	−1.20162	−0.600810	+	0.799392i	$0.705155\pi$
$840$	0	0
$841$	−27.0864	−0.934013
$842$	4.78681	0.164964
$843$	−24.3658	−0.839204
$844$	−42.5287	−1.46390
$845$	0	0
$846$	−1.58932	−0.0546419
$847$	−15.0311	−0.516473
$848$	7.13600	0.245051
$849$	18.0179	0.618373
$850$	0	0
$851$	−7.14424	−0.244901
$852$	−14.9178	−0.511075
$853$	−34.8717	−1.19398	−0.596992	−	0.802247i	$-0.703638\pi$
$853$	−34.8717	−1.19398	−0.596992	+	0.802247i	$0.703638\pi$
$854$	1.17452	0.0401913
$855$	0	0
$856$	−4.11024	−0.140485
$857$	28.0916	0.959592	0.479796	−	0.877380i	$-0.340711\pi$
$857$	28.0916	0.959592	0.479796	+	0.877380i	$0.340711\pi$
$858$	−1.05988	−0.0361836
$859$	−14.0686	−0.480014	−0.240007	−	0.970771i	$-0.577150\pi$
$859$	−14.0686	−0.480014	−0.240007	+	0.970771i	$0.577150\pi$
$860$	0	0
$861$	4.46047	0.152012
$862$	3.16779	0.107895
$863$	55.2635	1.88119	0.940596	−	0.339529i	$-0.110268\pi$
$863$	55.2635	1.88119	0.940596	+	0.339529i	$0.110268\pi$
$864$	13.6865	0.465623
$865$	0	0
$866$	−7.85245	−0.266837
$867$	9.39704	0.319140
$868$	−26.7719	−0.908697
$869$	−5.57007	−0.188952
$870$	0	0
$871$	40.2188	1.36276
$872$	4.17952	0.141536
$873$	9.33285	0.315869
$874$	3.02628	0.102365
$875$	0	0
$876$	−5.76371	−0.194738
$877$	−27.8286	−0.939703	−0.469852	−	0.882745i	$-0.655693\pi$
$877$	−27.8286	−0.939703	−0.469852	+	0.882745i	$0.655693\pi$
$878$	−5.67074	−0.191378
$879$	−11.0212	−0.371736
$880$	0	0
$881$	26.0455	0.877496	0.438748	−	0.898610i	$-0.355422\pi$
$881$	26.0455	0.877496	0.438748	+	0.898610i	$0.355422\pi$
$882$	−2.04563	−0.0688801
$883$	−13.6534	−0.459472	−0.229736	−	0.973253i	$-0.573786\pi$
$883$	−13.6534	−0.459472	−0.229736	+	0.973253i	$0.573786\pi$
$884$	−40.1793	−1.35138
$885$	0	0
$886$	−1.10777	−0.0372162
$887$	32.9764	1.10724	0.553620	−	0.832770i	$-0.313246\pi$
$887$	32.9764	1.10724	0.553620	+	0.832770i	$0.313246\pi$
$888$	1.19333	0.0400456
$889$	−11.3975	−0.382259
$890$	0	0
$891$	0.385964	0.0129303
$892$	−13.3170	−0.445887
$893$	−8.34754	−0.279340
$894$	−4.04439	−0.135264
$895$	0	0
$896$	10.5493	0.352428
$897$	−24.0980	−0.804610
$898$	1.59523	0.0532334
$899$	12.4263	0.414439
$900$	0	0
$901$	9.87361	0.328938
$902$	−0.697709	−0.0232312
$903$	−10.5149	−0.349913
$904$	8.63488	0.287192
$905$	0	0
$906$	−3.95721	−0.131469
$907$	−13.8518	−0.459940	−0.229970	−	0.973198i	$-0.573863\pi$
$907$	−13.8518	−0.459940	−0.229970	+	0.973198i	$0.573863\pi$
$908$	27.5365	0.913830
$909$	4.07157	0.135045
$910$	0	0
$911$	−39.7840	−1.31810	−0.659052	−	0.752097i	$-0.729042\pi$
$911$	−39.7840	−1.31810	−0.659052	+	0.752097i	$0.729042\pi$
$912$	8.88873	0.294335
$913$	6.85780	0.226960
$914$	4.25212	0.140648
$915$	0	0
$916$	47.4259	1.56700
$917$	1.83926	0.0607378
$918$	6.02600	0.198888
$919$	−6.64094	−0.219064	−0.109532	−	0.993983i	$-0.534935\pi$
$919$	−6.64094	−0.219064	−0.109532	+	0.993983i	$0.534935\pi$
$920$	0	0
$921$	27.0476	0.891248
$922$	0.771306	0.0254016
$923$	29.6656	0.976457
$924$	−3.38654	−0.111409
$925$	0	0
$926$	−4.13273	−0.135810
$927$	22.9557	0.753966
$928$	−3.69022	−0.121137
$929$	−8.75083	−0.287105	−0.143553	−	0.989643i	$-0.545853\pi$
$929$	−8.75083	−0.287105	−0.143553	+	0.989643i	$0.545853\pi$
$930$	0	0
$931$	−10.7442	−0.352128
$932$	14.1962	0.465012
$933$	12.7745	0.418218
$934$	7.21621	0.236122
$935$	0	0
$936$	−7.01924	−0.229431
$937$	−1.37031	−0.0447661	−0.0223831	−	0.999749i	$-0.507125\pi$
$937$	−1.37031	−0.0447661	−0.0223831	+	0.999749i	$0.507125\pi$
$938$	−3.50393	−0.114407
$939$	27.3930	0.893935
$940$	0	0
$941$	6.48786	0.211498	0.105749	−	0.994393i	$-0.466276\pi$
$941$	6.48786	0.211498	0.105749	+	0.994393i	$0.466276\pi$
$942$	0.0835006	0.00272060
$943$	−15.8636	−0.516589
$944$	14.2674	0.464363
$945$	0	0
$946$	1.64474	0.0534751
$947$	−45.5135	−1.47899	−0.739495	−	0.673162i	$-0.764936\pi$
$947$	−45.5135	−1.47899	−0.739495	+	0.673162i	$0.764936\pi$
$948$	10.4347	0.338904
$949$	11.4618	0.372064
$950$	0	0
$951$	7.22485	0.234282
$952$	7.09642	0.229996
$953$	−16.5368	−0.535679	−0.267839	−	0.963464i	$-0.586310\pi$
$953$	−16.5368	−0.535679	−0.267839	+	0.963464i	$0.586310\pi$
$954$	0.850850	0.0275473
$955$	0	0
$956$	9.62736	0.311371
$957$	1.57187	0.0508114
$958$	−5.46415	−0.176539
$959$	26.8011	0.865453
$960$	0	0
$961$	49.6913	1.60295
$962$	−1.17058	−0.0377409
$963$	8.61770	0.277701
$964$	−1.94691	−0.0627059
$965$	0	0
$966$	2.09946	0.0675491
$967$	−35.3262	−1.13601	−0.568007	−	0.823024i	$-0.692285\pi$
$967$	−35.3262	−1.13601	−0.568007	+	0.823024i	$0.692285\pi$
$968$	−8.92926	−0.286997
$969$	12.2988	0.395093
$970$	0	0
$971$	27.9626	0.897363	0.448681	−	0.893692i	$-0.351894\pi$
$971$	27.9626	0.897363	0.448681	+	0.893692i	$0.351894\pi$
$972$	−30.6894	−0.984364
$973$	−5.00242	−0.160370
$974$	−1.73821	−0.0556957
$975$	0	0
$976$	−12.2691	−0.392723
$977$	36.7829	1.17679	0.588395	−	0.808574i	$-0.299760\pi$
$977$	36.7829	1.17679	0.588395	+	0.808574i	$0.299760\pi$
$978$	−2.72873	−0.0872553
$979$	−7.94636	−0.253967
$980$	0	0
$981$	−8.76294	−0.279779
$982$	−2.47019	−0.0788268
$983$	−40.4060	−1.28875	−0.644375	−	0.764709i	$-0.722882\pi$
$983$	−40.4060	−1.28875	−0.644375	+	0.764709i	$0.722882\pi$
$984$	2.64976	0.0844712
$985$	0	0
$986$	−1.62476	−0.0517430
$987$	−5.79105	−0.184331
$988$	−18.1856	−0.578560
$989$	37.3959	1.18912
$990$	0	0
$991$	−1.63298	−0.0518734	−0.0259367	−	0.999664i	$-0.508257\pi$
$991$	−1.63298	−0.0518734	−0.0259367	+	0.999664i	$0.508257\pi$
$992$	−23.9628	−0.760821
$993$	−4.21885	−0.133881
$994$	−2.58452	−0.0819760
$995$	0	0
$996$	−12.8471	−0.407077
$997$	5.44754	0.172525	0.0862626	−	0.996272i	$-0.472508\pi$
$997$	5.44754	0.172525	0.0862626	+	0.996272i	$0.472508\pi$
$998$	−0.0175468	−0.000555435 0
$999$	−6.43874	−0.203713

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.n.1.18	yes	40
5.4	even	2		6025.2.a.m.1.23	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.m.1.23	✓	40	5.4	even	2
6025.2.a.n.1.18	yes	40	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.230402 q^{2} +1.04564 q^{3} -1.94691 q^{4} -0.240918 q^{6} +1.53080 q^{7} +0.909377 q^{8} -1.90664 q^{9} +O(q^{10})\)	\(q-0.230402 q^{2} +1.04564 q^{3} -1.94691 q^{4} -0.240918 q^{6} +1.53080 q^{7} +0.909377 q^{8} -1.90664 q^{9} +1.08670 q^{11} -2.03577 q^{12} +4.04835 q^{13} -0.352699 q^{14} +3.68431 q^{16} +5.09773 q^{17} +0.439293 q^{18} +2.30729 q^{19} +1.60067 q^{21} -0.250377 q^{22} -5.69274 q^{23} +0.950881 q^{24} -0.932749 q^{26} -5.13058 q^{27} -2.98034 q^{28} +1.38333 q^{29} +8.98284 q^{31} -2.66763 q^{32} +1.13629 q^{33} -1.17453 q^{34} +3.71206 q^{36} +1.25497 q^{37} -0.531604 q^{38} +4.23312 q^{39} +2.78663 q^{41} -0.368797 q^{42} -6.56905 q^{43} -2.11571 q^{44} +1.31162 q^{46} -3.61790 q^{47} +3.85246 q^{48} -4.65665 q^{49} +5.33039 q^{51} -7.88180 q^{52} +1.93686 q^{53} +1.18210 q^{54} +1.39207 q^{56} +2.41259 q^{57} -0.318723 q^{58} +3.87247 q^{59} -3.33009 q^{61} -2.06966 q^{62} -2.91868 q^{63} -6.75399 q^{64} -0.261804 q^{66} +9.93460 q^{67} -9.92485 q^{68} -5.95255 q^{69} +7.32783 q^{71} -1.73385 q^{72} +2.83121 q^{73} -0.289149 q^{74} -4.49209 q^{76} +1.66351 q^{77} -0.975320 q^{78} -5.12569 q^{79} +0.355172 q^{81} -0.642046 q^{82} +6.31069 q^{83} -3.11636 q^{84} +1.51352 q^{86} +1.44647 q^{87} +0.988217 q^{88} -7.31240 q^{89} +6.19722 q^{91} +11.0833 q^{92} +9.39281 q^{93} +0.833571 q^{94} -2.78938 q^{96} -4.89493 q^{97} +1.07290 q^{98} -2.07193 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 9 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 20 q^{7} + 27 q^{8} + 38 q^{9}+O(q^{10}) \) 40 * q + 9 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 20 * q^7 + 27 * q^8 + 38 * q^9	\( 40 q + 9 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 20 q^{7} + 27 q^{8} + 38 q^{9} + q^{11} + 26 q^{12} + 11 q^{13} - q^{14} + 43 q^{16} + 20 q^{17} + 18 q^{18} + 2 q^{21} + 23 q^{22} + 79 q^{23} - 2 q^{24} + 2 q^{26} + 26 q^{27} + 30 q^{28} + 2 q^{29} + q^{31} + 68 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 16 q^{37} + 45 q^{38} - 2 q^{39} - 2 q^{41} + 19 q^{42} + 25 q^{43} + 3 q^{44} + 14 q^{46} + 88 q^{47} + 75 q^{48} + 40 q^{49} - 10 q^{51} + 18 q^{52} + 34 q^{53} + 4 q^{54} - 15 q^{56} + 51 q^{57} + 53 q^{58} + q^{59} + 9 q^{61} + 39 q^{62} + 110 q^{63} + 17 q^{64} + 26 q^{66} + 30 q^{67} + 44 q^{68} - 7 q^{69} + 5 q^{71} + 18 q^{72} + 23 q^{73} - 18 q^{74} + 43 q^{76} + 30 q^{77} + 46 q^{78} + 5 q^{79} + 44 q^{81} + 5 q^{82} + 65 q^{83} - 65 q^{84} + 40 q^{86} + 33 q^{87} + 71 q^{88} - 9 q^{89} + q^{91} + 117 q^{92} + 68 q^{93} - 72 q^{94} + 83 q^{96} - 8 q^{97} + 76 q^{98} - 40 q^{99}+O(q^{100}) \) 40 * q + 9 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 20 * q^7 + 27 * q^8 + 38 * q^9 + q^11 + 26 * q^12 + 11 * q^13 - q^14 + 43 * q^16 + 20 * q^17 + 18 * q^18 + 2 * q^21 + 23 * q^22 + 79 * q^23 - 2 * q^24 + 2 * q^26 + 26 * q^27 + 30 * q^28 + 2 * q^29 + q^31 + 68 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 16 * q^37 + 45 * q^38 - 2 * q^39 - 2 * q^41 + 19 * q^42 + 25 * q^43 + 3 * q^44 + 14 * q^46 + 88 * q^47 + 75 * q^48 + 40 * q^49 - 10 * q^51 + 18 * q^52 + 34 * q^53 + 4 * q^54 - 15 * q^56 + 51 * q^57 + 53 * q^58 + q^59 + 9 * q^61 + 39 * q^62 + 110 * q^63 + 17 * q^64 + 26 * q^66 + 30 * q^67 + 44 * q^68 - 7 * q^69 + 5 * q^71 + 18 * q^72 + 23 * q^73 - 18 * q^74 + 43 * q^76 + 30 * q^77 + 46 * q^78 + 5 * q^79 + 44 * q^81 + 5 * q^82 + 65 * q^83 - 65 * q^84 + 40 * q^86 + 33 * q^87 + 71 * q^88 - 9 * q^89 + q^91 + 117 * q^92 + 68 * q^93 - 72 * q^94 + 83 * q^96 - 8 * q^97 + 76 * q^98 - 40 * q^99

Introduction

L-functions

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Database

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