LMFDB - Embedded newform 6025.2.a.m.1.16

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:	$40$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
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)

$f(q)$	$=$	$q-0.863616 q^{2} -1.78313 q^{3} -1.25417 q^{4} +1.53994 q^{6} -4.46828 q^{7} +2.81035 q^{8} +0.179539 q^{9} +O(q^{10})$	\(q-0.863616 q^{2} -1.78313 q^{3} -1.25417 q^{4} +1.53994 q^{6} -4.46828 q^{7} +2.81035 q^{8} +0.179539 q^{9} -4.87897 q^{11} +2.23634 q^{12} -4.21291 q^{13} +3.85888 q^{14} +0.0812674 q^{16} +4.42667 q^{17} -0.155053 q^{18} -7.05265 q^{19} +7.96750 q^{21} +4.21356 q^{22} -6.18188 q^{23} -5.01121 q^{24} +3.63834 q^{26} +5.02924 q^{27} +5.60396 q^{28} +7.05304 q^{29} +4.29646 q^{31} -5.69089 q^{32} +8.69982 q^{33} -3.82294 q^{34} -0.225172 q^{36} +2.88399 q^{37} +6.09079 q^{38} +7.51215 q^{39} +8.91367 q^{41} -6.88087 q^{42} -1.06677 q^{43} +6.11904 q^{44} +5.33877 q^{46} -9.89044 q^{47} -0.144910 q^{48} +12.9655 q^{49} -7.89331 q^{51} +5.28369 q^{52} -10.8801 q^{53} -4.34333 q^{54} -12.5574 q^{56} +12.5758 q^{57} -6.09112 q^{58} +9.41453 q^{59} +4.32997 q^{61} -3.71049 q^{62} -0.802231 q^{63} +4.75221 q^{64} -7.51330 q^{66} +0.904559 q^{67} -5.55178 q^{68} +11.0231 q^{69} +13.2074 q^{71} +0.504569 q^{72} -10.6288 q^{73} -2.49066 q^{74} +8.84520 q^{76} +21.8006 q^{77} -6.48762 q^{78} -7.12919 q^{79} -9.50638 q^{81} -7.69799 q^{82} +12.4045 q^{83} -9.99258 q^{84} +0.921279 q^{86} -12.5765 q^{87} -13.7116 q^{88} +6.12202 q^{89} +18.8244 q^{91} +7.75310 q^{92} -7.66113 q^{93} +8.54155 q^{94} +10.1476 q^{96} -2.18721 q^{97} -11.1972 q^{98} -0.875966 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.863616	−0.610669	−0.305335	−	0.952245i	$-0.598768\pi$
$2$	−0.863616	−0.610669	−0.305335	+	0.952245i	$0.598768\pi$
$3$	−1.78313	−1.02949	−0.514744	−	0.857344i	$-0.672113\pi$
$3$	−1.78313	−1.02949	−0.514744	+	0.857344i	$0.672113\pi$
$4$	−1.25417	−0.627083
$5$	0	0
$5$	0	0
$6$	1.53994	0.628677
$7$	−4.46828	−1.68885	−0.844425	−	0.535674i	$-0.820058\pi$
$7$	−4.46828	−1.68885	−0.844425	+	0.535674i	$0.820058\pi$
$8$	2.81035	0.993609
$9$	0.179539	0.0598464
$10$	0	0
$11$	−4.87897	−1.47106	−0.735532	−	0.677490i	$-0.763068\pi$
$11$	−4.87897	−1.47106	−0.735532	+	0.677490i	$0.763068\pi$
$12$	2.23634	0.645575
$13$	−4.21291	−1.16845	−0.584225	−	0.811591i	$-0.698602\pi$
$13$	−4.21291	−1.16845	−0.584225	+	0.811591i	$0.698602\pi$
$14$	3.85888	1.03133
$15$	0	0
$16$	0.0812674	0.0203168
$17$	4.42667	1.07362	0.536812	−	0.843702i	$-0.319628\pi$
$17$	4.42667	1.07362	0.536812	+	0.843702i	$0.319628\pi$
$18$	−0.155053	−0.0365464
$19$	−7.05265	−1.61799	−0.808995	−	0.587816i	$-0.799988\pi$
$19$	−7.05265	−1.61799	−0.808995	+	0.587816i	$0.799988\pi$
$20$	0	0
$21$	7.96750	1.73865
$22$	4.21356	0.898333
$23$	−6.18188	−1.28901	−0.644505	−	0.764600i	$-0.722937\pi$
$23$	−6.18188	−1.28901	−0.644505	+	0.764600i	$0.722937\pi$
$24$	−5.01121	−1.02291
$25$	0	0
$26$	3.63834	0.713537
$27$	5.02924	0.967877
$28$	5.60396	1.05905
$29$	7.05304	1.30972	0.654859	−	0.755752i	$-0.272728\pi$
$29$	7.05304	1.30972	0.654859	+	0.755752i	$0.272728\pi$
$30$	0	0
$31$	4.29646	0.771667	0.385834	−	0.922568i	$-0.373914\pi$
$31$	4.29646	0.771667	0.385834	+	0.922568i	$0.373914\pi$
$32$	−5.69089	−1.00602
$33$	8.69982	1.51444
$34$	−3.82294	−0.655629
$35$	0	0
$36$	−0.225172	−0.0375287
$37$	2.88399	0.474125	0.237062	−	0.971494i	$-0.423815\pi$
$37$	2.88399	0.474125	0.237062	+	0.971494i	$0.423815\pi$
$38$	6.09079	0.988056
$39$	7.51215	1.20291
$40$	0	0
$41$	8.91367	1.39208	0.696040	−	0.718003i	$-0.254943\pi$
$41$	8.91367	1.39208	0.696040	+	0.718003i	$0.254943\pi$
$42$	−6.88087	−1.06174
$43$	−1.06677	−0.162681	−0.0813403	−	0.996686i	$-0.525920\pi$
$43$	−1.06677	−0.162681	−0.0813403	+	0.996686i	$0.525920\pi$
$44$	6.11904	0.922480
$45$	0	0
$46$	5.33877	0.787159
$47$	−9.89044	−1.44267	−0.721334	−	0.692587i	$-0.756471\pi$
$47$	−9.89044	−1.44267	−0.721334	+	0.692587i	$0.756471\pi$
$48$	−0.144910	−0.0209160
$49$	12.9655	1.85221
$50$	0	0
$51$	−7.89331	−1.10528
$52$	5.28369	0.732716
$53$	−10.8801	−1.49450	−0.747249	−	0.664544i	$-0.768626\pi$
$53$	−10.8801	−1.49450	−0.747249	+	0.664544i	$0.768626\pi$
$54$	−4.34333	−0.591053
$55$	0	0
$56$	−12.5574	−1.67806
$57$	12.5758	1.66570
$58$	−6.09112	−0.799804
$59$	9.41453	1.22567	0.612833	−	0.790212i	$-0.290030\pi$
$59$	9.41453	1.22567	0.612833	+	0.790212i	$0.290030\pi$
$60$	0	0
$61$	4.32997	0.554396	0.277198	−	0.960813i	$-0.410594\pi$
$61$	4.32997	0.554396	0.277198	+	0.960813i	$0.410594\pi$
$62$	−3.71049	−0.471233
$63$	−0.802231	−0.101072
$64$	4.75221	0.594026
$65$	0	0
$66$	−7.51330	−0.924824
$67$	0.904559	0.110509	0.0552547	−	0.998472i	$-0.482403\pi$
$67$	0.904559	0.110509	0.0552547	+	0.998472i	$0.482403\pi$
$68$	−5.55178	−0.673252
$69$	11.0231	1.32702
$70$	0	0
$71$	13.2074	1.56743	0.783714	−	0.621122i	$-0.213323\pi$
$71$	13.2074	1.56743	0.783714	+	0.621122i	$0.213323\pi$
$72$	0.504569	0.0594640
$73$	−10.6288	−1.24400	−0.622002	−	0.783015i	$-0.713681\pi$
$73$	−10.6288	−1.24400	−0.622002	+	0.783015i	$0.713681\pi$
$74$	−2.49066	−0.289533
$75$	0	0
$76$	8.84520	1.01461
$77$	21.8006	2.48441
$78$	−6.48762	−0.734578
$79$	−7.12919	−0.802097	−0.401048	−	0.916057i	$-0.631354\pi$
$79$	−7.12919	−0.802097	−0.401048	+	0.916057i	$0.631354\pi$
$80$	0	0
$81$	−9.50638	−1.05626
$82$	−7.69799	−0.850101
$83$	12.4045	1.36157	0.680784	−	0.732484i	$-0.261639\pi$
$83$	12.4045	1.36157	0.680784	+	0.732484i	$0.261639\pi$
$84$	−9.99258	−1.09028
$85$	0	0
$86$	0.921279	0.0993440
$87$	−12.5765	−1.34834
$88$	−13.7116	−1.46166
$89$	6.12202	0.648933	0.324467	−	0.945897i	$-0.394815\pi$
$89$	6.12202	0.648933	0.324467	+	0.945897i	$0.394815\pi$
$90$	0	0
$91$	18.8244	1.97334
$92$	7.75310	0.808317
$93$	−7.66113	−0.794422
$94$	8.54155	0.880993
$95$	0	0
$96$	10.1476	1.03568
$97$	−2.18721	−0.222078	−0.111039	−	0.993816i	$-0.535418\pi$
$97$	−2.18721	−0.222078	−0.111039	+	0.993816i	$0.535418\pi$
$98$	−11.1972	−1.13109
$99$	−0.875966	−0.0880379
$100$	0	0
$101$	2.43263	0.242056	0.121028	−	0.992649i	$-0.461381\pi$
$101$	2.43263	0.242056	0.121028	+	0.992649i	$0.461381\pi$
$102$	6.81679	0.674963
$103$	−14.3979	−1.41867	−0.709335	−	0.704871i	$-0.751005\pi$
$103$	−14.3979	−1.41867	−0.709335	+	0.704871i	$0.751005\pi$
$104$	−11.8398	−1.16098
$105$	0	0
$106$	9.39624	0.912644
$107$	4.25527	0.411372	0.205686	−	0.978618i	$-0.434057\pi$
$107$	4.25527	0.411372	0.205686	+	0.978618i	$0.434057\pi$
$108$	−6.30750	−0.606940
$109$	−11.2743	−1.07988	−0.539940	−	0.841703i	$-0.681553\pi$
$109$	−11.2743	−1.07988	−0.539940	+	0.841703i	$0.681553\pi$
$110$	0	0
$111$	−5.14252	−0.488106
$112$	−0.363125	−0.0343121
$113$	8.46441	0.796265	0.398132	−	0.917328i	$-0.369658\pi$
$113$	8.46441	0.796265	0.398132	+	0.917328i	$0.369658\pi$
$114$	−10.8606	−1.01719
$115$	0	0
$116$	−8.84569	−0.821302
$117$	−0.756383	−0.0699276
$118$	−8.13054	−0.748477
$119$	−19.7796	−1.81319
$120$	0	0
$121$	12.8043	1.16403
$122$	−3.73943	−0.338552
$123$	−15.8942	−1.43313
$124$	−5.38848	−0.483900
$125$	0	0
$126$	0.692820	0.0617213
$127$	−3.13373	−0.278074	−0.139037	−	0.990287i	$-0.544401\pi$
$127$	−3.13373	−0.278074	−0.139037	+	0.990287i	$0.544401\pi$
$128$	7.27769	0.643263
$129$	1.90218	0.167478
$130$	0	0
$131$	−7.49908	−0.655197	−0.327599	−	0.944817i	$-0.606239\pi$
$131$	−7.49908	−0.655197	−0.327599	+	0.944817i	$0.606239\pi$
$132$	−10.9110	−0.949682
$133$	31.5132	2.73254
$134$	−0.781192	−0.0674847
$135$	0	0
$136$	12.4405	1.06676
$137$	10.5050	0.897506	0.448753	−	0.893656i	$-0.351868\pi$
$137$	10.5050	0.897506	0.448753	+	0.893656i	$0.351868\pi$
$138$	−9.51970	−0.810371
$139$	−15.3619	−1.30298	−0.651489	−	0.758658i	$-0.725855\pi$
$139$	−15.3619	−1.30298	−0.651489	+	0.758658i	$0.725855\pi$
$140$	0	0
$141$	17.6359	1.48521
$142$	−11.4061	−0.957180
$143$	20.5546	1.71887
$144$	0.0145907	0.00121589
$145$	0	0
$146$	9.17919	0.759675
$147$	−23.1191	−1.90683
$148$	−3.61700	−0.297316
$149$	14.2454	1.16703	0.583513	−	0.812103i	$-0.301678\pi$
$149$	14.2454	1.16703	0.583513	+	0.812103i	$0.301678\pi$
$150$	0	0
$151$	20.5372	1.67129	0.835647	−	0.549268i	$-0.185093\pi$
$151$	20.5372	1.67129	0.835647	+	0.549268i	$0.185093\pi$
$152$	−19.8204	−1.60765
$153$	0.794761	0.0642526
$154$	−18.8273	−1.51715
$155$	0	0
$156$	−9.42149	−0.754323
$157$	20.0226	1.59798	0.798988	−	0.601347i	$-0.205369\pi$
$157$	20.0226	1.59798	0.798988	+	0.601347i	$0.205369\pi$
$158$	6.15689	0.489816
$159$	19.4006	1.53857
$160$	0	0
$161$	27.6223	2.17695
$162$	8.20987	0.645028
$163$	−0.603296	−0.0472538	−0.0236269	−	0.999721i	$-0.507521\pi$
$163$	−0.603296	−0.0472538	−0.0236269	+	0.999721i	$0.507521\pi$
$164$	−11.1792	−0.872951
$165$	0	0
$166$	−10.7127	−0.831467
$167$	13.7518	1.06414	0.532072	−	0.846699i	$-0.321414\pi$
$167$	13.7518	1.06414	0.532072	+	0.846699i	$0.321414\pi$
$168$	22.3915	1.72754
$169$	4.74860	0.365277
$170$	0	0
$171$	−1.26623	−0.0968309
$172$	1.33791	0.102014
$173$	19.9322	1.51541	0.757707	−	0.652594i	$-0.226319\pi$
$173$	19.9322	1.51541	0.757707	+	0.652594i	$0.226319\pi$
$174$	10.8612	0.823389
$175$	0	0
$176$	−0.396501	−0.0298874
$177$	−16.7873	−1.26181
$178$	−5.28708	−0.396283
$179$	−3.96749	−0.296544	−0.148272	−	0.988947i	$-0.547371\pi$
$179$	−3.96749	−0.296544	−0.148272	+	0.988947i	$0.547371\pi$
$180$	0	0
$181$	−14.3092	−1.06360	−0.531799	−	0.846871i	$-0.678484\pi$
$181$	−14.3092	−1.06360	−0.531799	+	0.846871i	$0.678484\pi$
$182$	−16.2571	−1.20506
$183$	−7.72088	−0.570744
$184$	−17.3732	−1.28077
$185$	0	0
$186$	6.61628	0.485129
$187$	−21.5976	−1.57937
$188$	12.4043	0.904674
$189$	−22.4720	−1.63460
$190$	0	0
$191$	20.6397	1.49343	0.746717	−	0.665142i	$-0.231629\pi$
$191$	20.6397	1.49343	0.746717	+	0.665142i	$0.231629\pi$
$192$	−8.47379	−0.611543
$193$	−0.587792	−0.0423102	−0.0211551	−	0.999776i	$-0.506734\pi$
$193$	−0.587792	−0.0423102	−0.0211551	+	0.999776i	$0.506734\pi$
$194$	1.88891	0.135616
$195$	0	0
$196$	−16.2609	−1.16149
$197$	−17.3594	−1.23681	−0.618403	−	0.785861i	$-0.712220\pi$
$197$	−17.3594	−1.23681	−0.618403	+	0.785861i	$0.712220\pi$
$198$	0.756499	0.0537620
$199$	−5.81120	−0.411945	−0.205972	−	0.978558i	$-0.566036\pi$
$199$	−5.81120	−0.411945	−0.205972	+	0.978558i	$0.566036\pi$
$200$	0	0
$201$	−1.61294	−0.113768
$202$	−2.10086	−0.147816
$203$	−31.5149	−2.21192
$204$	9.89952	0.693105
$205$	0	0
$206$	12.4343	0.866338
$207$	−1.10989	−0.0771427
$208$	−0.342372	−0.0237392
$209$	34.4097	2.38017
$210$	0	0
$211$	16.0394	1.10420	0.552100	−	0.833778i	$-0.313827\pi$
$211$	16.0394	1.10420	0.552100	+	0.833778i	$0.313827\pi$
$212$	13.6455	0.937175
$213$	−23.5504	−1.61365
$214$	−3.67492	−0.251212
$215$	0	0
$216$	14.1339	0.961692
$217$	−19.1978	−1.30323
$218$	9.73666	0.659450
$219$	18.9525	1.28069
$220$	0	0
$221$	−18.6491	−1.25448
$222$	4.44116	0.298071
$223$	−20.5993	−1.37943	−0.689716	−	0.724080i	$-0.742265\pi$
$223$	−20.5993	−1.37943	−0.689716	+	0.724080i	$0.742265\pi$
$224$	25.4285	1.69901
$225$	0	0
$226$	−7.31000	−0.486254
$227$	5.17472	0.343459	0.171729	−	0.985144i	$-0.445065\pi$
$227$	5.17472	0.343459	0.171729	+	0.985144i	$0.445065\pi$
$228$	−15.7721	−1.04453
$229$	−16.7997	−1.11016	−0.555079	−	0.831798i	$-0.687312\pi$
$229$	−16.7997	−1.11016	−0.555079	+	0.831798i	$0.687312\pi$
$230$	0	0
$231$	−38.8732	−2.55767
$232$	19.8215	1.30135
$233$	−7.65140	−0.501260	−0.250630	−	0.968083i	$-0.580638\pi$
$233$	−7.65140	−0.501260	−0.250630	+	0.968083i	$0.580638\pi$
$234$	0.653225	0.0427026
$235$	0	0
$236$	−11.8074	−0.768595
$237$	12.7122	0.825749
$238$	17.0820	1.10726
$239$	−14.2392	−0.921058	−0.460529	−	0.887645i	$-0.652340\pi$
$239$	−14.2392	−0.921058	−0.460529	+	0.887645i	$0.652340\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−11.0580	−0.710837
$243$	1.86337	0.119535
$244$	−5.43050	−0.347652
$245$	0	0
$246$	13.7265	0.875169
$247$	29.7122	1.89054
$248$	12.0746	0.766736
$249$	−22.1187	−1.40172
$250$	0	0
$251$	13.5862	0.857552	0.428776	−	0.903411i	$-0.358945\pi$
$251$	13.5862	0.857552	0.428776	+	0.903411i	$0.358945\pi$
$252$	1.00613	0.0633804
$253$	30.1612	1.89622
$254$	2.70634	0.169811
$255$	0	0
$256$	−15.7895	−0.986847
$257$	−16.4354	−1.02521	−0.512605	−	0.858624i	$-0.671320\pi$
$257$	−16.4354	−1.02521	−0.512605	+	0.858624i	$0.671320\pi$
$258$	−1.64276	−0.102274
$259$	−12.8865	−0.800726
$260$	0	0
$261$	1.26630	0.0783819
$262$	6.47633	0.400109
$263$	−7.76798	−0.478994	−0.239497	−	0.970897i	$-0.576983\pi$
$263$	−7.76798	−0.478994	−0.239497	+	0.970897i	$0.576983\pi$
$264$	24.4495	1.50477
$265$	0	0
$266$	−27.2153	−1.66868
$267$	−10.9163	−0.668069
$268$	−1.13447	−0.0692986
$269$	9.89916	0.603562	0.301781	−	0.953377i	$-0.402419\pi$
$269$	9.89916	0.603562	0.301781	+	0.953377i	$0.402419\pi$
$270$	0	0
$271$	−25.2175	−1.53186	−0.765928	−	0.642927i	$-0.777720\pi$
$271$	−25.2175	−1.53186	−0.765928	+	0.642927i	$0.777720\pi$
$272$	0.359744	0.0218127
$273$	−33.5664	−2.03153
$274$	−9.07233	−0.548079
$275$	0	0
$276$	−13.8248	−0.832153
$277$	−0.823901	−0.0495034	−0.0247517	−	0.999694i	$-0.507880\pi$
$277$	−0.823901	−0.0495034	−0.0247517	+	0.999694i	$0.507880\pi$
$278$	13.2668	0.795688
$279$	0.771384	0.0461815
$280$	0	0
$281$	−4.87826	−0.291013	−0.145506	−	0.989357i	$-0.546481\pi$
$281$	−4.87826	−0.291013	−0.145506	+	0.989357i	$0.546481\pi$
$282$	−15.2307	−0.906972
$283$	−3.55456	−0.211296	−0.105648	−	0.994404i	$-0.533692\pi$
$283$	−3.55456	−0.211296	−0.105648	+	0.994404i	$0.533692\pi$
$284$	−16.5643	−0.982908
$285$	0	0
$286$	−17.7513	−1.04966
$287$	−39.8287	−2.35102
$288$	−1.02174	−0.0602065
$289$	2.59538	0.152669
$290$	0	0
$291$	3.90007	0.228626
$292$	13.3303	0.780095
$293$	26.1553	1.52801	0.764006	−	0.645209i	$-0.223230\pi$
$293$	26.1553	1.52801	0.764006	+	0.645209i	$0.223230\pi$
$294$	19.9661	1.16444
$295$	0	0
$296$	8.10503	0.471095
$297$	−24.5375	−1.42381
$298$	−12.3025	−0.712667
$299$	26.0437	1.50615
$300$	0	0
$301$	4.76662	0.274743
$302$	−17.7363	−1.02061
$303$	−4.33768	−0.249193
$304$	−0.573151	−0.0328725
$305$	0	0
$306$	−0.686368	−0.0392371
$307$	19.9773	1.14017	0.570083	−	0.821587i	$-0.306911\pi$
$307$	19.9773	1.14017	0.570083	+	0.821587i	$0.306911\pi$
$308$	−27.3416	−1.55793
$309$	25.6733	1.46050
$310$	0	0
$311$	12.8604	0.729247	0.364624	−	0.931155i	$-0.381198\pi$
$311$	12.8604	0.729247	0.364624	+	0.931155i	$0.381198\pi$
$312$	21.1118	1.19522
$313$	19.0290	1.07558	0.537792	−	0.843078i	$-0.319259\pi$
$313$	19.0290	1.07558	0.537792	+	0.843078i	$0.319259\pi$
$314$	−17.2918	−0.975835
$315$	0	0
$316$	8.94119	0.502981
$317$	14.7493	0.828401	0.414200	−	0.910186i	$-0.364061\pi$
$317$	14.7493	0.828401	0.414200	+	0.910186i	$0.364061\pi$
$318$	−16.7547	−0.939556
$319$	−34.4116	−1.92668
$320$	0	0
$321$	−7.58768	−0.423503
$322$	−23.8551	−1.32939
$323$	−31.2198	−1.73711
$324$	11.9226	0.662366
$325$	0	0
$326$	0.521017	0.0288564
$327$	20.1035	1.11172
$328$	25.0505	1.38318
$329$	44.1932	2.43645
$330$	0	0
$331$	−13.4560	−0.739609	−0.369804	−	0.929110i	$-0.620575\pi$
$331$	−13.4560	−0.739609	−0.369804	+	0.929110i	$0.620575\pi$
$332$	−15.5573	−0.853816
$333$	0.517790	0.0283747
$334$	−11.8763	−0.649840
$335$	0	0
$336$	0.647498	0.0353239
$337$	25.7142	1.40074	0.700370	−	0.713780i	$-0.253018\pi$
$337$	25.7142	1.40074	0.700370	+	0.713780i	$0.253018\pi$
$338$	−4.10097	−0.223063
$339$	−15.0931	−0.819745
$340$	0	0
$341$	−20.9623	−1.13517
$342$	1.09354	0.0591317
$343$	−26.6555	−1.43926
$344$	−2.99799	−0.161641
$345$	0	0
$346$	−17.2138	−0.925417
$347$	5.26197	0.282477	0.141239	−	0.989976i	$-0.454892\pi$
$347$	5.26197	0.282477	0.141239	+	0.989976i	$0.454892\pi$
$348$	15.7730	0.845521
$349$	−8.32889	−0.445835	−0.222918	−	0.974837i	$-0.571558\pi$
$349$	−8.32889	−0.445835	−0.222918	+	0.974837i	$0.571558\pi$
$350$	0	0
$351$	−21.1877	−1.13092
$352$	27.7657	1.47991
$353$	0.822306	0.0437669	0.0218835	−	0.999761i	$-0.493034\pi$
$353$	0.822306	0.0437669	0.0218835	+	0.999761i	$0.493034\pi$
$354$	14.4978	0.770548
$355$	0	0
$356$	−7.67804	−0.406935
$357$	35.2695	1.86666
$358$	3.42639	0.181090
$359$	−20.5407	−1.08410	−0.542049	−	0.840347i	$-0.682351\pi$
$359$	−20.5407	−1.08410	−0.542049	+	0.840347i	$0.682351\pi$
$360$	0	0
$361$	30.7399	1.61789
$362$	12.3577	0.649506
$363$	−22.8317	−1.19835
$364$	−23.6090	−1.23745
$365$	0	0
$366$	6.66788	0.348536
$367$	−36.2301	−1.89120	−0.945599	−	0.325335i	$-0.894523\pi$
$367$	−36.2301	−1.89120	−0.945599	+	0.325335i	$0.894523\pi$
$368$	−0.502385	−0.0261886
$369$	1.60035	0.0833111
$370$	0	0
$371$	48.6154	2.52398
$372$	9.60834	0.498169
$373$	−19.7597	−1.02312	−0.511559	−	0.859248i	$-0.670932\pi$
$373$	−19.7597	−1.02312	−0.511559	+	0.859248i	$0.670932\pi$
$374$	18.6520	0.964472
$375$	0	0
$376$	−27.7956	−1.43345
$377$	−29.7138	−1.53034
$378$	19.4072	0.998199
$379$	17.0402	0.875298	0.437649	−	0.899146i	$-0.355811\pi$
$379$	17.0402	0.875298	0.437649	+	0.899146i	$0.355811\pi$
$380$	0	0
$381$	5.58784	0.286274
$382$	−17.8248	−0.911994
$383$	3.83347	0.195881	0.0979406	−	0.995192i	$-0.468774\pi$
$383$	3.83347	0.195881	0.0979406	+	0.995192i	$0.468774\pi$
$384$	−12.9770	−0.662232
$385$	0	0
$386$	0.507627	0.0258375
$387$	−0.191527	−0.00973586
$388$	2.74313	0.139261
$389$	−0.0851908	−0.00431934	−0.00215967	−	0.999998i	$-0.500687\pi$
$389$	−0.0851908	−0.00431934	−0.00215967	+	0.999998i	$0.500687\pi$
$390$	0	0
$391$	−27.3651	−1.38391
$392$	36.4376	1.84038
$393$	13.3718	0.674518
$394$	14.9919	0.755279
$395$	0	0
$396$	1.09861	0.0552071
$397$	−19.5084	−0.979100	−0.489550	−	0.871975i	$-0.662839\pi$
$397$	−19.5084	−0.979100	−0.489550	+	0.871975i	$0.662839\pi$
$398$	5.01864	0.251562
$399$	−56.1920	−2.81312
$400$	0	0
$401$	26.3750	1.31710	0.658552	−	0.752535i	$-0.271169\pi$
$401$	26.3750	1.31710	0.658552	+	0.752535i	$0.271169\pi$
$402$	1.39296	0.0694747
$403$	−18.1006	−0.901655
$404$	−3.05092	−0.151789
$405$	0	0
$406$	27.2168	1.35075
$407$	−14.0709	−0.697468
$408$	−22.1830	−1.09822
$409$	14.0823	0.696327	0.348163	−	0.937434i	$-0.386805\pi$
$409$	14.0823	0.696327	0.348163	+	0.937434i	$0.386805\pi$
$410$	0	0
$411$	−18.7318	−0.923972
$412$	18.0574	0.889625
$413$	−42.0667	−2.06997
$414$	0.958519	0.0471086
$415$	0	0
$416$	23.9752	1.17548
$417$	27.3922	1.34140
$418$	−29.7168	−1.45349
$419$	11.1513	0.544775	0.272387	−	0.962188i	$-0.412187\pi$
$419$	11.1513	0.544775	0.272387	+	0.962188i	$0.412187\pi$
$420$	0	0
$421$	−0.491996	−0.0239784	−0.0119892	−	0.999928i	$-0.503816\pi$
$421$	−0.491996	−0.0239784	−0.0119892	+	0.999928i	$0.503816\pi$
$422$	−13.8519	−0.674301
$423$	−1.77572	−0.0863386
$424$	−30.5769	−1.48495
$425$	0	0
$426$	20.3385	0.985405
$427$	−19.3475	−0.936291
$428$	−5.33682	−0.257965
$429$	−36.6515	−1.76955
$430$	0	0
$431$	21.4135	1.03145	0.515726	−	0.856754i	$-0.327522\pi$
$431$	21.4135	1.03145	0.515726	+	0.856754i	$0.327522\pi$
$432$	0.408713	0.0196642
$433$	−22.0993	−1.06202	−0.531012	−	0.847364i	$-0.678188\pi$
$433$	−22.0993	−1.06202	−0.531012	+	0.847364i	$0.678188\pi$
$434$	16.5795	0.795842
$435$	0	0
$436$	14.1398	0.677175
$437$	43.5986	2.08561
$438$	−16.3677	−0.782077
$439$	−26.8126	−1.27970	−0.639848	−	0.768501i	$-0.721003\pi$
$439$	−26.8126	−1.27970	−0.639848	+	0.768501i	$0.721003\pi$
$440$	0	0
$441$	2.32782	0.110848
$442$	16.1057	0.766070
$443$	−37.7477	−1.79345	−0.896723	−	0.442592i	$-0.854059\pi$
$443$	−37.7477	−1.79345	−0.896723	+	0.442592i	$0.854059\pi$
$444$	6.44957	0.306083
$445$	0	0
$446$	17.7899	0.842377
$447$	−25.4013	−1.20144
$448$	−21.2342	−1.00322
$449$	−11.1114	−0.524378	−0.262189	−	0.965017i	$-0.584444\pi$
$449$	−11.1114	−0.524378	−0.262189	+	0.965017i	$0.584444\pi$
$450$	0	0
$451$	−43.4895	−2.04784
$452$	−10.6158	−0.499324
$453$	−36.6204	−1.72058
$454$	−4.46898	−0.209739
$455$	0	0
$456$	35.3423	1.65506
$457$	−14.3862	−0.672960	−0.336480	−	0.941691i	$-0.609236\pi$
$457$	−14.3862	−0.672960	−0.336480	+	0.941691i	$0.609236\pi$
$458$	14.5085	0.677939
$459$	22.2628	1.03914
$460$	0	0
$461$	−19.3620	−0.901779	−0.450890	−	0.892580i	$-0.648893\pi$
$461$	−19.3620	−0.901779	−0.450890	+	0.892580i	$0.648893\pi$
$462$	33.5715	1.56189
$463$	7.39222	0.343545	0.171773	−	0.985137i	$-0.445051\pi$
$463$	7.39222	0.343545	0.171773	+	0.985137i	$0.445051\pi$
$464$	0.573182	0.0266093
$465$	0	0
$466$	6.60788	0.306104
$467$	12.3281	0.570475	0.285238	−	0.958457i	$-0.407927\pi$
$467$	12.3281	0.570475	0.285238	+	0.958457i	$0.407927\pi$
$468$	0.948630	0.0438504
$469$	−4.04182	−0.186634
$470$	0	0
$471$	−35.7028	−1.64510
$472$	26.4581	1.21783
$473$	5.20473	0.239314
$474$	−10.9785	−0.504259
$475$	0	0
$476$	24.8069	1.13702
$477$	−1.95341	−0.0894404
$478$	12.2972	0.562461
$479$	20.8189	0.951242	0.475621	−	0.879650i	$-0.342223\pi$
$479$	20.8189	0.951242	0.475621	+	0.879650i	$0.342223\pi$
$480$	0	0
$481$	−12.1500	−0.553992
$482$	−0.863616	−0.0393367
$483$	−49.2541	−2.24114
$484$	−16.0588	−0.729943
$485$	0	0
$486$	−1.60924	−0.0729965
$487$	0.385407	0.0174645	0.00873223	−	0.999962i	$-0.497220\pi$
$487$	0.385407	0.0174645	0.00873223	+	0.999962i	$0.497220\pi$
$488$	12.1687	0.550853
$489$	1.07575	0.0486473
$490$	0	0
$491$	−35.8812	−1.61930	−0.809648	−	0.586915i	$-0.800342\pi$
$491$	−35.8812	−1.61930	−0.809648	+	0.586915i	$0.800342\pi$
$492$	19.9340	0.898693
$493$	31.2215	1.40614
$494$	−25.6599	−1.15450
$495$	0	0
$496$	0.349162	0.0156778
$497$	−59.0142	−2.64715
$498$	19.1021	0.855986
$499$	40.4917	1.81266	0.906329	−	0.422572i	$-0.138873\pi$
$499$	40.4917	1.81266	0.906329	+	0.422572i	$0.138873\pi$
$500$	0	0
$501$	−24.5211	−1.09552
$502$	−11.7332	−0.523680
$503$	−14.3322	−0.639042	−0.319521	−	0.947579i	$-0.603522\pi$
$503$	−14.3322	−0.639042	−0.319521	+	0.947579i	$0.603522\pi$
$504$	−2.25455	−0.100426
$505$	0	0
$506$	−26.0477	−1.15796
$507$	−8.46736	−0.376049
$508$	3.93022	0.174375
$509$	−22.2491	−0.986172	−0.493086	−	0.869981i	$-0.664131\pi$
$509$	−22.2491	−0.986172	−0.493086	+	0.869981i	$0.664131\pi$
$510$	0	0
$511$	47.4923	2.10094
$512$	−0.919264	−0.0406261
$513$	−35.4695	−1.56602
$514$	14.1939	0.626064
$515$	0	0
$516$	−2.38565	−0.105023
$517$	48.2551	2.12226
$518$	11.1290	0.488979
$519$	−35.5416	−1.56010
$520$	0	0
$521$	−31.9023	−1.39766	−0.698832	−	0.715286i	$-0.746297\pi$
$521$	−31.9023	−1.39766	−0.698832	+	0.715286i	$0.746297\pi$
$522$	−1.09360	−0.0478654
$523$	17.6485	0.771717	0.385858	−	0.922558i	$-0.373905\pi$
$523$	17.6485	0.771717	0.385858	+	0.922558i	$0.373905\pi$
$524$	9.40509	0.410863
$525$	0	0
$526$	6.70856	0.292507
$527$	19.0190	0.828481
$528$	0.707011	0.0307687
$529$	15.2156	0.661548
$530$	0	0
$531$	1.69028	0.0733518
$532$	−39.5228	−1.71353
$533$	−37.5525	−1.62658
$534$	9.42753	0.407969
$535$	0	0
$536$	2.54213	0.109803
$537$	7.07453	0.305289
$538$	−8.54907	−0.368577
$539$	−63.2583	−2.72473
$540$	0	0
$541$	7.92842	0.340869	0.170435	−	0.985369i	$-0.445483\pi$
$541$	7.92842	0.340869	0.170435	+	0.985369i	$0.445483\pi$
$542$	21.7783	0.935457
$543$	25.5152	1.09496
$544$	−25.1917	−1.08008
$545$	0	0
$546$	28.9885	1.24059
$547$	3.48255	0.148903	0.0744515	−	0.997225i	$-0.476279\pi$
$547$	3.48255	0.148903	0.0744515	+	0.997225i	$0.476279\pi$
$548$	−13.1751	−0.562811
$549$	0.777400	0.0331786
$550$	0	0
$551$	−49.7427	−2.11911
$552$	30.9787	1.31854
$553$	31.8552	1.35462
$554$	0.711535	0.0302302
$555$	0	0
$556$	19.2664	0.817076
$557$	21.6663	0.918029	0.459014	−	0.888429i	$-0.348203\pi$
$557$	21.6663	0.918029	0.459014	+	0.888429i	$0.348203\pi$
$558$	−0.666180	−0.0282016
$559$	4.49420	0.190084
$560$	0	0
$561$	38.5112	1.62594
$562$	4.21295	0.177712
$563$	−0.669139	−0.0282008	−0.0141004	−	0.999901i	$-0.504488\pi$
$563$	−0.669139	−0.0282008	−0.0141004	+	0.999901i	$0.504488\pi$
$564$	−22.1184	−0.931351
$565$	0	0
$566$	3.06977	0.129032
$567$	42.4772	1.78387
$568$	37.1174	1.55741
$569$	18.2582	0.765424	0.382712	−	0.923868i	$-0.374990\pi$
$569$	18.2582	0.765424	0.382712	+	0.923868i	$0.374990\pi$
$570$	0	0
$571$	−3.88771	−0.162696	−0.0813479	−	0.996686i	$-0.525922\pi$
$571$	−3.88771	−0.162696	−0.0813479	+	0.996686i	$0.525922\pi$
$572$	−25.7790	−1.07787
$573$	−36.8031	−1.53747
$574$	34.3968	1.43569
$575$	0	0
$576$	0.853208	0.0355503
$577$	−14.9733	−0.623347	−0.311673	−	0.950189i	$-0.600889\pi$
$577$	−14.9733	−0.623347	−0.311673	+	0.950189i	$0.600889\pi$
$578$	−2.24141	−0.0932304
$579$	1.04811	0.0435578
$580$	0	0
$581$	−55.4266	−2.29948
$582$	−3.36817	−0.139615
$583$	53.0837	2.19850
$584$	−29.8706	−1.23605
$585$	0	0
$586$	−22.5882	−0.933109
$587$	20.0331	0.826855	0.413428	−	0.910537i	$-0.364331\pi$
$587$	20.0331	0.826855	0.413428	+	0.910537i	$0.364331\pi$
$588$	28.9952	1.19574
$589$	−30.3015	−1.24855
$590$	0	0
$591$	30.9540	1.27328
$592$	0.234374	0.00963273
$593$	−11.4761	−0.471267	−0.235633	−	0.971842i	$-0.575716\pi$
$593$	−11.4761	−0.471267	−0.235633	+	0.971842i	$0.575716\pi$
$594$	21.1910	0.869476
$595$	0	0
$596$	−17.8661	−0.731823
$597$	10.3621	0.424092
$598$	−22.4918	−0.919756
$599$	38.1191	1.55751	0.778753	−	0.627331i	$-0.215853\pi$
$599$	38.1191	1.55751	0.778753	+	0.627331i	$0.215853\pi$
$600$	0	0
$601$	−44.3051	−1.80724	−0.903622	−	0.428331i	$-0.859102\pi$
$601$	−44.3051	−1.80724	−0.903622	+	0.428331i	$0.859102\pi$
$602$	−4.11653	−0.167777
$603$	0.162404	0.00661360
$604$	−25.7571	−1.04804
$605$	0	0
$606$	3.74609	0.152175
$607$	−27.6129	−1.12077	−0.560386	−	0.828231i	$-0.689347\pi$
$607$	−27.6129	−1.12077	−0.560386	+	0.828231i	$0.689347\pi$
$608$	40.1359	1.62772
$609$	56.1951	2.27714
$610$	0	0
$611$	41.6675	1.68569
$612$	−0.996762	−0.0402917
$613$	12.8791	0.520182	0.260091	−	0.965584i	$-0.416248\pi$
$613$	12.8791	0.520182	0.260091	+	0.965584i	$0.416248\pi$
$614$	−17.2528	−0.696265
$615$	0	0
$616$	61.2673	2.46853
$617$	−33.7453	−1.35853	−0.679267	−	0.733891i	$-0.737702\pi$
$617$	−33.7453	−1.35853	−0.679267	+	0.733891i	$0.737702\pi$
$618$	−22.1719	−0.891885
$619$	32.8600	1.32075	0.660377	−	0.750934i	$-0.270396\pi$
$619$	32.8600	1.32075	0.660377	+	0.750934i	$0.270396\pi$
$620$	0	0
$621$	−31.0901	−1.24760
$622$	−11.1065	−0.445329
$623$	−27.3549	−1.09595
$624$	0.610493	0.0244393
$625$	0	0
$626$	−16.4338	−0.656825
$627$	−61.3568	−2.45035
$628$	−25.1117	−1.00206
$629$	12.7665	0.509032
$630$	0	0
$631$	−36.2885	−1.44462	−0.722311	−	0.691568i	$-0.756920\pi$
$631$	−36.2885	−1.44462	−0.722311	+	0.691568i	$0.756920\pi$
$632$	−20.0355	−0.796971
$633$	−28.6003	−1.13676
$634$	−12.7377	−0.505879
$635$	0	0
$636$	−24.3316	−0.964811
$637$	−54.6225	−2.16422
$638$	29.7184	1.17656
$639$	2.37124	0.0938050
$640$	0	0
$641$	1.36425	0.0538846	0.0269423	−	0.999637i	$-0.491423\pi$
$641$	1.36425	0.0538846	0.0269423	+	0.999637i	$0.491423\pi$
$642$	6.55285	0.258620
$643$	24.2573	0.956615	0.478307	−	0.878192i	$-0.341250\pi$
$643$	24.2573	0.956615	0.478307	+	0.878192i	$0.341250\pi$
$644$	−34.6430	−1.36513
$645$	0	0
$646$	26.9619	1.06080
$647$	−36.2190	−1.42391	−0.711957	−	0.702223i	$-0.752191\pi$
$647$	−36.2190	−1.42391	−0.711957	+	0.702223i	$0.752191\pi$
$648$	−26.7163	−1.04951
$649$	−45.9332	−1.80303
$650$	0	0
$651$	34.2321	1.34166
$652$	0.756634	0.0296321
$653$	−44.2805	−1.73283	−0.866414	−	0.499326i	$-0.833581\pi$
$653$	−44.2805	−1.73283	−0.866414	+	0.499326i	$0.833581\pi$
$654$	−17.3617	−0.678896
$655$	0	0
$656$	0.724391	0.0282827
$657$	−1.90828	−0.0744493
$658$	−38.1660	−1.48787
$659$	−6.69953	−0.260977	−0.130488	−	0.991450i	$-0.541655\pi$
$659$	−6.69953	−0.260977	−0.130488	+	0.991450i	$0.541655\pi$
$660$	0	0
$661$	7.94369	0.308974	0.154487	−	0.987995i	$-0.450628\pi$
$661$	7.94369	0.308974	0.154487	+	0.987995i	$0.450628\pi$
$662$	11.6208	0.451656
$663$	33.2538	1.29147
$664$	34.8609	1.35287
$665$	0	0
$666$	−0.447172	−0.0173275
$667$	−43.6010	−1.68824
$668$	−17.2470	−0.667307
$669$	36.7312	1.42011
$670$	0	0
$671$	−21.1258	−0.815552
$672$	−45.3422	−1.74911
$673$	30.5587	1.17795	0.588976	−	0.808151i	$-0.299531\pi$
$673$	30.5587	1.17795	0.588976	+	0.808151i	$0.299531\pi$
$674$	−22.2072	−0.855389
$675$	0	0
$676$	−5.95554	−0.229059
$677$	−30.0759	−1.15591	−0.577955	−	0.816069i	$-0.696149\pi$
$677$	−30.0759	−1.15591	−0.577955	+	0.816069i	$0.696149\pi$
$678$	13.0347	0.500593
$679$	9.77307	0.375056
$680$	0	0
$681$	−9.22719	−0.353587
$682$	18.1034	0.693214
$683$	35.2537	1.34895	0.674473	−	0.738299i	$-0.264371\pi$
$683$	35.2537	1.34895	0.674473	+	0.738299i	$0.264371\pi$
$684$	1.58806	0.0607211
$685$	0	0
$686$	23.0201	0.878913
$687$	29.9561	1.14289
$688$	−0.0866935	−0.00330516
$689$	45.8369	1.74625
$690$	0	0
$691$	19.0594	0.725053	0.362526	−	0.931974i	$-0.381914\pi$
$691$	19.0594	0.725053	0.362526	+	0.931974i	$0.381914\pi$
$692$	−24.9983	−0.950291
$693$	3.91406	0.148683
$694$	−4.54432	−0.172500
$695$	0	0
$696$	−35.3443	−1.33972
$697$	39.4578	1.49457
$698$	7.19296	0.272258
$699$	13.6434	0.516041
$700$	0	0
$701$	9.84151	0.371709	0.185854	−	0.982577i	$-0.440495\pi$
$701$	9.84151	0.371709	0.185854	+	0.982577i	$0.440495\pi$
$702$	18.2981	0.690616
$703$	−20.3398	−0.767129
$704$	−23.1859	−0.873850
$705$	0	0
$706$	−0.710157	−0.0267271
$707$	−10.8697	−0.408795
$708$	21.0541	0.791260
$709$	25.9322	0.973905	0.486952	−	0.873429i	$-0.338109\pi$
$709$	25.9322	0.973905	0.486952	+	0.873429i	$0.338109\pi$
$710$	0	0
$711$	−1.27997	−0.0480026
$712$	17.2050	0.644786
$713$	−26.5602	−0.994687
$714$	−30.4593	−1.13991
$715$	0	0
$716$	4.97589	0.185958
$717$	25.3903	0.948218
$718$	17.7393	0.662025
$719$	34.6219	1.29118	0.645590	−	0.763684i	$-0.276612\pi$
$719$	34.6219	1.29118	0.645590	+	0.763684i	$0.276612\pi$
$720$	0	0
$721$	64.3340	2.39592
$722$	−26.5475	−0.987996
$723$	−1.78313	−0.0663152
$724$	17.9462	0.666964
$725$	0	0
$726$	19.7179	0.731798
$727$	29.4963	1.09396	0.546979	−	0.837146i	$-0.315778\pi$
$727$	29.4963	1.09396	0.546979	+	0.837146i	$0.315778\pi$
$728$	52.9033	1.96073
$729$	25.1965	0.933205
$730$	0	0
$731$	−4.72223	−0.174658
$732$	9.68327	0.357904
$733$	41.6264	1.53751	0.768753	−	0.639546i	$-0.220878\pi$
$733$	41.6264	1.53751	0.768753	+	0.639546i	$0.220878\pi$
$734$	31.2889	1.15490
$735$	0	0
$736$	35.1804	1.29677
$737$	−4.41331	−0.162566
$738$	−1.38209	−0.0508755
$739$	−1.16792	−0.0429625	−0.0214813	−	0.999769i	$-0.506838\pi$
$739$	−1.16792	−0.0429625	−0.0214813	+	0.999769i	$0.506838\pi$
$740$	0	0
$741$	−52.9806	−1.94629
$742$	−41.9850	−1.54132
$743$	32.1005	1.17765	0.588827	−	0.808259i	$-0.299590\pi$
$743$	32.1005	1.17765	0.588827	+	0.808259i	$0.299590\pi$
$744$	−21.5305	−0.789346
$745$	0	0
$746$	17.0648	0.624787
$747$	2.22709	0.0814850
$748$	27.0869	0.990397
$749$	−19.0137	−0.694746
$750$	0	0
$751$	−19.0338	−0.694554	−0.347277	−	0.937763i	$-0.612894\pi$
$751$	−19.0338	−0.694554	−0.347277	+	0.937763i	$0.612894\pi$
$752$	−0.803770	−0.0293105
$753$	−24.2259	−0.882839
$754$	25.6613	0.934531
$755$	0	0
$756$	28.1837	1.02503
$757$	11.0948	0.403249	0.201624	−	0.979463i	$-0.435378\pi$
$757$	11.0948	0.403249	0.201624	+	0.979463i	$0.435378\pi$
$758$	−14.7162	−0.534517
$759$	−53.7812	−1.95213
$760$	0	0
$761$	−9.09041	−0.329527	−0.164764	−	0.986333i	$-0.552686\pi$
$761$	−9.09041	−0.329527	−0.164764	+	0.986333i	$0.552686\pi$
$762$	−4.82575	−0.174818
$763$	50.3766	1.82376
$764$	−25.8856	−0.936508
$765$	0	0
$766$	−3.31065	−0.119619
$767$	−39.6625	−1.43213
$768$	28.1548	1.01595
$769$	−40.6241	−1.46494	−0.732471	−	0.680798i	$-0.761633\pi$
$769$	−40.6241	−1.46494	−0.732471	+	0.680798i	$0.761633\pi$
$770$	0	0
$771$	29.3063	1.05544
$772$	0.737189	0.0265320
$773$	−18.5441	−0.666986	−0.333493	−	0.942753i	$-0.608227\pi$
$773$	−18.5441	−0.666986	−0.333493	+	0.942753i	$0.608227\pi$
$774$	0.165406	0.00594539
$775$	0	0
$776$	−6.14683	−0.220658
$777$	22.9782	0.824338
$778$	0.0735721	0.00263769
$779$	−62.8650	−2.25237
$780$	0	0
$781$	−64.4384	−2.30579
$782$	23.6330	0.845113
$783$	35.4714	1.26765
$784$	1.05367	0.0376312
$785$	0	0
$786$	−11.5481	−0.411907
$787$	34.4337	1.22743	0.613715	−	0.789528i	$-0.289674\pi$
$787$	34.4337	1.22743	0.613715	+	0.789528i	$0.289674\pi$
$788$	21.7716	0.775580
$789$	13.8513	0.493119
$790$	0	0
$791$	−37.8213	−1.34477
$792$	−2.46177	−0.0874753
$793$	−18.2418	−0.647784
$794$	16.8478	0.597906
$795$	0	0
$796$	7.28821	0.258324
$797$	−34.7177	−1.22976	−0.614882	−	0.788619i	$-0.710796\pi$
$797$	−34.7177	−1.22976	−0.614882	+	0.788619i	$0.710796\pi$
$798$	48.5284	1.71789
$799$	−43.7817	−1.54888
$800$	0	0
$801$	1.09914	0.0388363
$802$	−22.7779	−0.804315
$803$	51.8575	1.83001
$804$	2.02290	0.0713421
$805$	0	0
$806$	15.6320	0.550613
$807$	−17.6514	−0.621360
$808$	6.83654	0.240509
$809$	−40.7764	−1.43362	−0.716812	−	0.697267i	$-0.754399\pi$
$809$	−40.7764	−1.43362	−0.716812	+	0.697267i	$0.754399\pi$
$810$	0	0
$811$	−11.5291	−0.404842	−0.202421	−	0.979299i	$-0.564881\pi$
$811$	−11.5291	−0.404842	−0.202421	+	0.979299i	$0.564881\pi$
$812$	39.5250	1.38706
$813$	44.9660	1.57703
$814$	12.1519	0.425922
$815$	0	0
$816$	−0.641468	−0.0224559
$817$	7.52355	0.263216
$818$	−12.1617	−0.425225
$819$	3.37973	0.118097
$820$	0	0
$821$	14.9443	0.521561	0.260780	−	0.965398i	$-0.416020\pi$
$821$	14.9443	0.521561	0.260780	+	0.965398i	$0.416020\pi$
$822$	16.1771	0.564241
$823$	55.9373	1.94985	0.974926	−	0.222531i	$-0.0714320\pi$
$823$	55.9373	1.94985	0.974926	+	0.222531i	$0.0714320\pi$
$824$	−40.4633	−1.40960
$825$	0	0
$826$	36.3295	1.26407
$827$	12.2473	0.425879	0.212939	−	0.977065i	$-0.431696\pi$
$827$	12.2473	0.425879	0.212939	+	0.977065i	$0.431696\pi$
$828$	1.39199	0.0483749
$829$	−9.12091	−0.316782	−0.158391	−	0.987376i	$-0.550631\pi$
$829$	−9.12091	−0.316782	−0.158391	+	0.987376i	$0.550631\pi$
$830$	0	0
$831$	1.46912	0.0509632
$832$	−20.0206	−0.694090
$833$	57.3940	1.98858
$834$	−23.6563	−0.819152
$835$	0	0
$836$	−43.1555	−1.49256
$837$	21.6079	0.746879
$838$	−9.63042	−0.332677
$839$	2.17487	0.0750849	0.0375424	−	0.999295i	$-0.488047\pi$
$839$	2.17487	0.0750849	0.0375424	+	0.999295i	$0.488047\pi$
$840$	0	0
$841$	20.7454	0.715359
$842$	0.424895	0.0146429
$843$	8.69855	0.299594
$844$	−20.1161	−0.692426
$845$	0	0
$846$	1.53354	0.0527243
$847$	−57.2133	−1.96587
$848$	−0.884198	−0.0303635
$849$	6.33822	0.217527
$850$	0	0
$851$	−17.8285	−0.611152
$852$	29.5362	1.01189
$853$	15.2442	0.521953	0.260976	−	0.965345i	$-0.415956\pi$
$853$	15.2442	0.521953	0.260976	+	0.965345i	$0.415956\pi$
$854$	16.7088	0.571764
$855$	0	0
$856$	11.9588	0.408744
$857$	36.0848	1.23263	0.616317	−	0.787498i	$-0.288624\pi$
$857$	36.0848	1.23263	0.616317	+	0.787498i	$0.288624\pi$
$858$	31.6529	1.08061
$859$	42.2315	1.44092	0.720460	−	0.693497i	$-0.243931\pi$
$859$	42.2315	1.44092	0.720460	+	0.693497i	$0.243931\pi$
$860$	0	0
$861$	71.0197	2.42034
$862$	−18.4930	−0.629876
$863$	−13.0172	−0.443111	−0.221556	−	0.975148i	$-0.571113\pi$
$863$	−13.0172	−0.443111	−0.221556	+	0.975148i	$0.571113\pi$
$864$	−28.6208	−0.973700
$865$	0	0
$866$	19.0853	0.648545
$867$	−4.62789	−0.157171
$868$	24.0772	0.817234
$869$	34.7831	1.17994
$870$	0	0
$871$	−3.81082	−0.129125
$872$	−31.6847	−1.07298
$873$	−0.392690	−0.0132906
$874$	−37.6525	−1.27361
$875$	0	0
$876$	−23.7695	−0.803098
$877$	−33.8522	−1.14311	−0.571554	−	0.820564i	$-0.693659\pi$
$877$	−33.8522	−1.14311	−0.571554	+	0.820564i	$0.693659\pi$
$878$	23.1558	0.781471
$879$	−46.6383	−1.57307
$880$	0	0
$881$	21.1314	0.711935	0.355967	−	0.934498i	$-0.384151\pi$
$881$	21.1314	0.711935	0.355967	+	0.934498i	$0.384151\pi$
$882$	−2.01034	−0.0676917
$883$	25.3429	0.852858	0.426429	−	0.904521i	$-0.359771\pi$
$883$	25.3429	0.852858	0.426429	+	0.904521i	$0.359771\pi$
$884$	23.3891	0.786662
$885$	0	0
$886$	32.5995	1.09520
$887$	23.7697	0.798109	0.399054	−	0.916927i	$-0.369338\pi$
$887$	23.7697	0.798109	0.399054	+	0.916927i	$0.369338\pi$
$888$	−14.4523	−0.484987
$889$	14.0024	0.469625
$890$	0	0
$891$	46.3813	1.55383
$892$	25.8350	0.865019
$893$	69.7539	2.33422
$894$	21.9370	0.733683
$895$	0	0
$896$	−32.5187	−1.08637
$897$	−46.4392	−1.55056
$898$	9.59595	0.320221
$899$	30.3031	1.01067
$900$	0	0
$901$	−48.1626	−1.60453
$902$	37.5582	1.25055
$903$	−8.49948	−0.282845
$904$	23.7880	0.791176
$905$	0	0
$906$	31.6260	1.05070
$907$	3.00923	0.0999198	0.0499599	−	0.998751i	$-0.484091\pi$
$907$	3.00923	0.0999198	0.0499599	+	0.998751i	$0.484091\pi$
$908$	−6.48997	−0.215377
$909$	0.436752	0.0144862
$910$	0	0
$911$	41.3377	1.36958	0.684790	−	0.728741i	$-0.259894\pi$
$911$	41.3377	1.36958	0.684790	+	0.728741i	$0.259894\pi$
$912$	1.02200	0.0338418
$913$	−60.5210	−2.00295
$914$	12.4242	0.410956
$915$	0	0
$916$	21.0697	0.696162
$917$	33.5080	1.10653
$918$	−19.2265	−0.634569
$919$	44.1334	1.45583	0.727913	−	0.685670i	$-0.240491\pi$
$919$	44.1334	1.45583	0.727913	+	0.685670i	$0.240491\pi$
$920$	0	0
$921$	−35.6221	−1.17379
$922$	16.7214	0.550689
$923$	−55.6415	−1.83146
$924$	48.7535	1.60387
$925$	0	0
$926$	−6.38404	−0.209793
$927$	−2.58499	−0.0849024
$928$	−40.1381	−1.31760
$929$	−2.77420	−0.0910184	−0.0455092	−	0.998964i	$-0.514491\pi$
$929$	−2.77420	−0.0910184	−0.0455092	+	0.998964i	$0.514491\pi$
$930$	0	0
$931$	−91.4412	−2.99686
$932$	9.59614	0.314332
$933$	−22.9317	−0.750752
$934$	−10.6467	−0.348372
$935$	0	0
$936$	−2.12570	−0.0694807
$937$	10.9483	0.357666	0.178833	−	0.983879i	$-0.442768\pi$
$937$	10.9483	0.357666	0.178833	+	0.983879i	$0.442768\pi$
$938$	3.49058	0.113972
$939$	−33.9311	−1.10730
$940$	0	0
$941$	−17.7147	−0.577483	−0.288742	−	0.957407i	$-0.593237\pi$
$941$	−17.7147	−0.577483	−0.288742	+	0.957407i	$0.593237\pi$
$942$	30.8335	1.00461
$943$	−55.1032	−1.79441
$944$	0.765094	0.0249017
$945$	0	0
$946$	−4.49489	−0.146141
$947$	26.0877	0.847737	0.423869	−	0.905724i	$-0.360672\pi$
$947$	26.0877	0.847737	0.423869	+	0.905724i	$0.360672\pi$
$948$	−15.9433	−0.517814
$949$	44.7781	1.45356
$950$	0	0
$951$	−26.2998	−0.852829
$952$	−55.5876	−1.80160
$953$	4.09691	0.132712	0.0663560	−	0.997796i	$-0.478863\pi$
$953$	4.09691	0.132712	0.0663560	+	0.997796i	$0.478863\pi$
$954$	1.68700	0.0546185
$955$	0	0
$956$	17.8583	0.577580
$957$	61.3602	1.98349
$958$	−17.9796	−0.580894
$959$	−46.9394	−1.51575
$960$	0	0
$961$	−12.5404	−0.404530
$962$	10.4929	0.338306
$963$	0.763988	0.0246192
$964$	−1.25417	−0.0403940
$965$	0	0
$966$	42.5367	1.36859
$967$	40.0920	1.28927	0.644636	−	0.764490i	$-0.277009\pi$
$967$	40.0920	1.28927	0.644636	+	0.764490i	$0.277009\pi$
$968$	35.9847	1.15659
$969$	55.6688	1.78834
$970$	0	0
$971$	18.0956	0.580716	0.290358	−	0.956918i	$-0.406226\pi$
$971$	18.0956	0.580716	0.290358	+	0.956918i	$0.406226\pi$
$972$	−2.33698	−0.0749585
$973$	68.6412	2.20053
$974$	−0.332844	−0.0106650
$975$	0	0
$976$	0.351885	0.0112636
$977$	30.3451	0.970826	0.485413	−	0.874285i	$-0.338669\pi$
$977$	30.3451	0.970826	0.485413	+	0.874285i	$0.338669\pi$
$978$	−0.929039	−0.0297074
$979$	−29.8692	−0.954622
$980$	0	0
$981$	−2.02418	−0.0646270
$982$	30.9876	0.988854
$983$	−2.21982	−0.0708013	−0.0354007	−	0.999373i	$-0.511271\pi$
$983$	−2.21982	−0.0708013	−0.0354007	+	0.999373i	$0.511271\pi$
$984$	−44.6683	−1.42397
$985$	0	0
$986$	−26.9634	−0.858689
$987$	−78.8021	−2.50830
$988$	−37.2640	−1.18553
$989$	6.59463	0.209697
$990$	0	0
$991$	−56.1646	−1.78413	−0.892063	−	0.451910i	$-0.850743\pi$
$991$	−56.1646	−1.78413	−0.892063	+	0.451910i	$0.850743\pi$
$992$	−24.4507	−0.776310
$993$	23.9938	0.761419
$994$	50.9657	1.61653
$995$	0	0
$996$	27.7406	0.878994
$997$	32.9517	1.04359	0.521796	−	0.853071i	$-0.325262\pi$
$997$	32.9517	1.04359	0.521796	+	0.853071i	$0.325262\pi$
$998$	−34.9693	−1.10693
$999$	14.5043	0.458895

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.m.1.16	✓	40
5.4	even	2		6025.2.a.n.1.25	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.m.1.16	✓	40	1.1	even	1	trivial
6025.2.a.n.1.25	yes	40	5.4	even	2

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)

\(f(q)\)	\(=\)	\(q-0.863616 q^{2} -1.78313 q^{3} -1.25417 q^{4} +1.53994 q^{6} -4.46828 q^{7} +2.81035 q^{8} +0.179539 q^{9} +O(q^{10})\)	\(q-0.863616 q^{2} -1.78313 q^{3} -1.25417 q^{4} +1.53994 q^{6} -4.46828 q^{7} +2.81035 q^{8} +0.179539 q^{9} -4.87897 q^{11} +2.23634 q^{12} -4.21291 q^{13} +3.85888 q^{14} +0.0812674 q^{16} +4.42667 q^{17} -0.155053 q^{18} -7.05265 q^{19} +7.96750 q^{21} +4.21356 q^{22} -6.18188 q^{23} -5.01121 q^{24} +3.63834 q^{26} +5.02924 q^{27} +5.60396 q^{28} +7.05304 q^{29} +4.29646 q^{31} -5.69089 q^{32} +8.69982 q^{33} -3.82294 q^{34} -0.225172 q^{36} +2.88399 q^{37} +6.09079 q^{38} +7.51215 q^{39} +8.91367 q^{41} -6.88087 q^{42} -1.06677 q^{43} +6.11904 q^{44} +5.33877 q^{46} -9.89044 q^{47} -0.144910 q^{48} +12.9655 q^{49} -7.89331 q^{51} +5.28369 q^{52} -10.8801 q^{53} -4.34333 q^{54} -12.5574 q^{56} +12.5758 q^{57} -6.09112 q^{58} +9.41453 q^{59} +4.32997 q^{61} -3.71049 q^{62} -0.802231 q^{63} +4.75221 q^{64} -7.51330 q^{66} +0.904559 q^{67} -5.55178 q^{68} +11.0231 q^{69} +13.2074 q^{71} +0.504569 q^{72} -10.6288 q^{73} -2.49066 q^{74} +8.84520 q^{76} +21.8006 q^{77} -6.48762 q^{78} -7.12919 q^{79} -9.50638 q^{81} -7.69799 q^{82} +12.4045 q^{83} -9.99258 q^{84} +0.921279 q^{86} -12.5765 q^{87} -13.7116 q^{88} +6.12202 q^{89} +18.8244 q^{91} +7.75310 q^{92} -7.66113 q^{93} +8.54155 q^{94} +10.1476 q^{96} -2.18721 q^{97} -11.1972 q^{98} -0.875966 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 9 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 20 q^{7} - 27 q^{8} + 38 q^{9}+O(q^{10}) \) 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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