LMFDB - Embedded newform 8005.2.a.e.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8005,2,Mod(1,8005)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8005, 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("8005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8005 = 5 \cdot 1601 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8005.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:	$63.9202468180$
Analytic rank:	$1$
Dimension:	$126$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	8005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.32335 q^{2} +0.155880 q^{3} +3.39795 q^{4} +1.00000 q^{5} -0.362164 q^{6} +0.705093 q^{7} -3.24794 q^{8} -2.97570 q^{9} +O(q^{10})$	\(q-2.32335 q^{2} +0.155880 q^{3} +3.39795 q^{4} +1.00000 q^{5} -0.362164 q^{6} +0.705093 q^{7} -3.24794 q^{8} -2.97570 q^{9} -2.32335 q^{10} +2.88914 q^{11} +0.529673 q^{12} +6.58008 q^{13} -1.63818 q^{14} +0.155880 q^{15} +0.750183 q^{16} -5.43683 q^{17} +6.91360 q^{18} +7.53937 q^{19} +3.39795 q^{20} +0.109910 q^{21} -6.71247 q^{22} -6.98618 q^{23} -0.506288 q^{24} +1.00000 q^{25} -15.2878 q^{26} -0.931493 q^{27} +2.39587 q^{28} +0.0855153 q^{29} -0.362164 q^{30} +7.75161 q^{31} +4.75293 q^{32} +0.450359 q^{33} +12.6317 q^{34} +0.705093 q^{35} -10.1113 q^{36} -6.77649 q^{37} -17.5166 q^{38} +1.02570 q^{39} -3.24794 q^{40} -10.5195 q^{41} -0.255359 q^{42} -1.20112 q^{43} +9.81715 q^{44} -2.97570 q^{45} +16.2313 q^{46} +4.61169 q^{47} +0.116939 q^{48} -6.50284 q^{49} -2.32335 q^{50} -0.847493 q^{51} +22.3588 q^{52} -13.2662 q^{53} +2.16418 q^{54} +2.88914 q^{55} -2.29010 q^{56} +1.17524 q^{57} -0.198682 q^{58} -14.5520 q^{59} +0.529673 q^{60} -6.82227 q^{61} -18.0097 q^{62} -2.09815 q^{63} -12.5431 q^{64} +6.58008 q^{65} -1.04634 q^{66} +10.3670 q^{67} -18.4741 q^{68} -1.08901 q^{69} -1.63818 q^{70} +10.1159 q^{71} +9.66489 q^{72} -7.19623 q^{73} +15.7442 q^{74} +0.155880 q^{75} +25.6184 q^{76} +2.03711 q^{77} -2.38307 q^{78} -5.08839 q^{79} +0.750183 q^{80} +8.78190 q^{81} +24.4406 q^{82} -7.78782 q^{83} +0.373469 q^{84} -5.43683 q^{85} +2.79063 q^{86} +0.0133301 q^{87} -9.38373 q^{88} -5.46104 q^{89} +6.91360 q^{90} +4.63957 q^{91} -23.7387 q^{92} +1.20832 q^{93} -10.7146 q^{94} +7.53937 q^{95} +0.740888 q^{96} +10.1880 q^{97} +15.1084 q^{98} -8.59721 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 126 q - 15 q^{2} - 46 q^{3} + 119 q^{4} + 126 q^{5} - 10 q^{6} - 60 q^{7} - 57 q^{8} + 112 q^{9}+O(q^{10}) $ 126 * q - 15 * q^2 - 46 * q^3 + 119 * q^4 + 126 * q^5 - 10 * q^6 - 60 * q^7 - 57 * q^8 + 112 * q^9	\( 126 q - 15 q^{2} - 46 q^{3} + 119 q^{4} + 126 q^{5} - 10 q^{6} - 60 q^{7} - 57 q^{8} + 112 q^{9} - 15 q^{10} - 35 q^{11} - 86 q^{12} - 36 q^{13} - 31 q^{14} - 46 q^{15} + 101 q^{16} - 62 q^{17} - 45 q^{18} - 29 q^{19} + 119 q^{20} - 16 q^{21} - 67 q^{22} - 107 q^{23} - 9 q^{24} + 126 q^{25} - 53 q^{26} - 181 q^{27} - 100 q^{28} - 39 q^{29} - 10 q^{30} - 29 q^{31} - 99 q^{32} - 72 q^{33} - 18 q^{34} - 60 q^{35} + 93 q^{36} - 72 q^{37} - 93 q^{38} - 8 q^{39} - 57 q^{40} - 28 q^{41} - 22 q^{42} - 103 q^{43} - 47 q^{44} + 112 q^{45} + q^{46} - 130 q^{47} - 134 q^{48} + 116 q^{49} - 15 q^{50} - 46 q^{51} - 117 q^{52} - 103 q^{53} + 11 q^{54} - 35 q^{55} - 84 q^{56} - 70 q^{57} - 77 q^{58} - 219 q^{59} - 86 q^{60} - 4 q^{61} - 77 q^{62} - 145 q^{63} + 51 q^{64} - 36 q^{65} + 16 q^{66} - 150 q^{67} - 130 q^{68} - 21 q^{69} - 31 q^{70} - 92 q^{71} - 115 q^{72} - 79 q^{73} - 57 q^{74} - 46 q^{75} - 38 q^{76} - 75 q^{77} - 23 q^{78} - 20 q^{79} + 101 q^{80} + 142 q^{81} - 63 q^{82} - 243 q^{83} - 2 q^{84} - 62 q^{85} - 14 q^{86} - 107 q^{87} - 121 q^{88} - 84 q^{89} - 45 q^{90} - 82 q^{91} - 228 q^{92} - 149 q^{93} - 29 q^{95} + 38 q^{96} - 85 q^{97} - 48 q^{98} - 33 q^{99}+O(q^{100}) \) 126 * q - 15 * q^2 - 46 * q^3 + 119 * q^4 + 126 * q^5 - 10 * q^6 - 60 * q^7 - 57 * q^8 + 112 * q^9 - 15 * q^10 - 35 * q^11 - 86 * q^12 - 36 * q^13 - 31 * q^14 - 46 * q^15 + 101 * q^16 - 62 * q^17 - 45 * q^18 - 29 * q^19 + 119 * q^20 - 16 * q^21 - 67 * q^22 - 107 * q^23 - 9 * q^24 + 126 * q^25 - 53 * q^26 - 181 * q^27 - 100 * q^28 - 39 * q^29 - 10 * q^30 - 29 * q^31 - 99 * q^32 - 72 * q^33 - 18 * q^34 - 60 * q^35 + 93 * q^36 - 72 * q^37 - 93 * q^38 - 8 * q^39 - 57 * q^40 - 28 * q^41 - 22 * q^42 - 103 * q^43 - 47 * q^44 + 112 * q^45 + q^46 - 130 * q^47 - 134 * q^48 + 116 * q^49 - 15 * q^50 - 46 * q^51 - 117 * q^52 - 103 * q^53 + 11 * q^54 - 35 * q^55 - 84 * q^56 - 70 * q^57 - 77 * q^58 - 219 * q^59 - 86 * q^60 - 4 * q^61 - 77 * q^62 - 145 * q^63 + 51 * q^64 - 36 * q^65 + 16 * q^66 - 150 * q^67 - 130 * q^68 - 21 * q^69 - 31 * q^70 - 92 * q^71 - 115 * q^72 - 79 * q^73 - 57 * q^74 - 46 * q^75 - 38 * q^76 - 75 * q^77 - 23 * q^78 - 20 * q^79 + 101 * q^80 + 142 * q^81 - 63 * q^82 - 243 * q^83 - 2 * q^84 - 62 * q^85 - 14 * q^86 - 107 * q^87 - 121 * q^88 - 84 * q^89 - 45 * q^90 - 82 * q^91 - 228 * q^92 - 149 * q^93 - 29 * q^95 + 38 * q^96 - 85 * q^97 - 48 * q^98 - 33 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.32335	−1.64286	−0.821428	−	0.570312i	$-0.806822\pi$
$2$	−2.32335	−1.64286	−0.821428	+	0.570312i	$0.806822\pi$
$3$	0.155880	0.0899974	0.0449987	−	0.998987i	$-0.485672\pi$
$3$	0.155880	0.0899974	0.0449987	+	0.998987i	$0.485672\pi$
$4$	3.39795	1.69898
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.362164	−0.147853
$7$	0.705093	0.266500	0.133250	−	0.991082i	$-0.457459\pi$
$7$	0.705093	0.266500	0.133250	+	0.991082i	$0.457459\pi$
$8$	−3.24794	−1.14832
$9$	−2.97570	−0.991900
$10$	−2.32335	−0.734708
$11$	2.88914	0.871107	0.435554	−	0.900163i	$-0.356553\pi$
$11$	2.88914	0.871107	0.435554	+	0.900163i	$0.356553\pi$
$12$	0.529673	0.152903
$13$	6.58008	1.82499	0.912493	−	0.409093i	$-0.134155\pi$
$13$	6.58008	1.82499	0.912493	+	0.409093i	$0.134155\pi$
$14$	−1.63818	−0.437821
$15$	0.155880	0.0402481
$16$	0.750183	0.187546
$17$	−5.43683	−1.31862	−0.659312	−	0.751869i	$-0.729152\pi$
$17$	−5.43683	−1.31862	−0.659312	+	0.751869i	$0.729152\pi$
$18$	6.91360	1.62955
$19$	7.53937	1.72965	0.864826	−	0.502072i	$-0.167429\pi$
$19$	7.53937	1.72965	0.864826	+	0.502072i	$0.167429\pi$
$20$	3.39795	0.759806
$21$	0.109910	0.0239843
$22$	−6.71247	−1.43110
$23$	−6.98618	−1.45672	−0.728359	−	0.685196i	$-0.759717\pi$
$23$	−6.98618	−1.45672	−0.728359	+	0.685196i	$0.759717\pi$
$24$	−0.506288	−0.103346
$25$	1.00000	0.200000
$26$	−15.2878	−2.99819
$27$	−0.931493	−0.179266
$28$	2.39587	0.452777
$29$	0.0855153	0.0158798	0.00793990	−	0.999968i	$-0.497473\pi$
$29$	0.0855153	0.0158798	0.00793990	+	0.999968i	$0.497473\pi$
$30$	−0.362164	−0.0661218
$31$	7.75161	1.39223	0.696116	−	0.717930i	$-0.254910\pi$
$31$	7.75161	1.39223	0.696116	+	0.717930i	$0.254910\pi$
$32$	4.75293	0.840208
$33$	0.450359	0.0783974
$34$	12.6317	2.16631
$35$	0.705093	0.119182
$36$	−10.1113	−1.68522
$37$	−6.77649	−1.11405	−0.557024	−	0.830496i	$-0.688057\pi$
$37$	−6.77649	−1.11405	−0.557024	+	0.830496i	$0.688057\pi$
$38$	−17.5166	−2.84157
$39$	1.02570	0.164244
$40$	−3.24794	−0.513544
$41$	−10.5195	−1.64288	−0.821438	−	0.570298i	$-0.806828\pi$
$41$	−10.5195	−1.64288	−0.821438	+	0.570298i	$0.806828\pi$
$42$	−0.255359	−0.0394028
$43$	−1.20112	−0.183170	−0.0915849	−	0.995797i	$-0.529193\pi$
$43$	−1.20112	−0.183170	−0.0915849	+	0.995797i	$0.529193\pi$
$44$	9.81715	1.47999
$45$	−2.97570	−0.443591
$46$	16.2313	2.39318
$47$	4.61169	0.672683	0.336342	−	0.941740i	$-0.390810\pi$
$47$	4.61169	0.672683	0.336342	+	0.941740i	$0.390810\pi$
$48$	0.116939	0.0168786
$49$	−6.50284	−0.928978
$50$	−2.32335	−0.328571
$51$	−0.847493	−0.118673
$52$	22.3588	3.10061
$53$	−13.2662	−1.82226	−0.911128	−	0.412123i	$-0.864787\pi$
$53$	−13.2662	−1.82226	−0.911128	+	0.412123i	$0.864787\pi$
$54$	2.16418	0.294508
$55$	2.88914	0.389571
$56$	−2.29010	−0.306027
$57$	1.17524	0.155664
$58$	−0.198682	−0.0260882
$59$	−14.5520	−1.89451	−0.947257	−	0.320474i	$-0.896158\pi$
$59$	−14.5520	−1.89451	−0.947257	+	0.320474i	$0.896158\pi$
$60$	0.529673	0.0683805
$61$	−6.82227	−0.873502	−0.436751	−	0.899582i	$-0.643871\pi$
$61$	−6.82227	−0.873502	−0.436751	+	0.899582i	$0.643871\pi$
$62$	−18.0097	−2.28724
$63$	−2.09815	−0.264342
$64$	−12.5431	−1.56789
$65$	6.58008	0.816158
$66$	−1.04634	−0.128796
$67$	10.3670	1.26653	0.633265	−	0.773935i	$-0.281714\pi$
$67$	10.3670	1.26653	0.633265	+	0.773935i	$0.281714\pi$
$68$	−18.4741	−2.24031
$69$	−1.08901	−0.131101
$70$	−1.63818	−0.195800
$71$	10.1159	1.20053	0.600266	−	0.799801i	$-0.295062\pi$
$71$	10.1159	1.20053	0.600266	+	0.799801i	$0.295062\pi$
$72$	9.66489	1.13902
$73$	−7.19623	−0.842255	−0.421128	−	0.907001i	$-0.638366\pi$
$73$	−7.19623	−0.842255	−0.421128	+	0.907001i	$0.638366\pi$
$74$	15.7442	1.83022
$75$	0.155880	0.0179995
$76$	25.6184	2.93864
$77$	2.03711	0.232150
$78$	−2.38307	−0.269829
$79$	−5.08839	−0.572489	−0.286244	−	0.958157i	$-0.592407\pi$
$79$	−5.08839	−0.572489	−0.286244	+	0.958157i	$0.592407\pi$
$80$	0.750183	0.0838730
$81$	8.78190	0.975767
$82$	24.4406	2.69901
$83$	−7.78782	−0.854824	−0.427412	−	0.904057i	$-0.640575\pi$
$83$	−7.78782	−0.854824	−0.427412	+	0.904057i	$0.640575\pi$
$84$	0.373469	0.0407488
$85$	−5.43683	−0.589707
$86$	2.79063	0.300922
$87$	0.0133301	0.00142914
$88$	−9.38373	−1.00031
$89$	−5.46104	−0.578869	−0.289434	−	0.957198i	$-0.593467\pi$
$89$	−5.46104	−0.578869	−0.289434	+	0.957198i	$0.593467\pi$
$90$	6.91360	0.728757
$91$	4.63957	0.486359
$92$	−23.7387	−2.47493
$93$	1.20832	0.125297
$94$	−10.7146	−1.10512
$95$	7.53937	0.773524
$96$	0.740888	0.0756165
$97$	10.1880	1.03444	0.517218	−	0.855854i	$-0.326968\pi$
$97$	10.1880	1.03444	0.517218	+	0.855854i	$0.326968\pi$
$98$	15.1084	1.52618
$99$	−8.59721	−0.864052
$100$	3.39795	0.339795
$101$	−9.32418	−0.927790	−0.463895	−	0.885890i	$-0.653549\pi$
$101$	−9.32418	−0.927790	−0.463895	+	0.885890i	$0.653549\pi$
$102$	1.96902	0.194962
$103$	−1.66907	−0.164459	−0.0822294	−	0.996613i	$-0.526204\pi$
$103$	−1.66907	−0.164459	−0.0822294	+	0.996613i	$0.526204\pi$
$104$	−21.3717	−2.09566
$105$	0.109910	0.0107261
$106$	30.8221	2.99371
$107$	0.967338	0.0935161	0.0467581	−	0.998906i	$-0.485111\pi$
$107$	0.967338	0.0935161	0.0467581	+	0.998906i	$0.485111\pi$
$108$	−3.16517	−0.304569
$109$	−13.6038	−1.30300	−0.651502	−	0.758647i	$-0.725861\pi$
$109$	−13.6038	−1.30300	−0.651502	+	0.758647i	$0.725861\pi$
$110$	−6.71247	−0.640009
$111$	−1.05632	−0.100261
$112$	0.528949	0.0499809
$113$	−15.3133	−1.44055	−0.720275	−	0.693689i	$-0.755984\pi$
$113$	−15.3133	−1.44055	−0.720275	+	0.693689i	$0.755984\pi$
$114$	−2.73049	−0.255734
$115$	−6.98618	−0.651464
$116$	0.290577	0.0269794
$117$	−19.5803	−1.81020
$118$	33.8095	3.11242
$119$	−3.83347	−0.351414
$120$	−0.506288	−0.0462176
$121$	−2.65289	−0.241172
$122$	15.8505	1.43504
$123$	−1.63979	−0.147854
$124$	26.3396	2.36537
$125$	1.00000	0.0894427
$126$	4.87473	0.434275
$127$	−3.56088	−0.315977	−0.157989	−	0.987441i	$-0.550501\pi$
$127$	−3.56088	−0.315977	−0.157989	+	0.987441i	$0.550501\pi$
$128$	19.6361	1.73560
$129$	−0.187231	−0.0164848
$130$	−15.2878	−1.34083
$131$	13.7140	1.19820	0.599099	−	0.800675i	$-0.295525\pi$
$131$	13.7140	1.19820	0.599099	+	0.800675i	$0.295525\pi$
$132$	1.53030	0.133195
$133$	5.31596	0.460952
$134$	−24.0861	−2.08073
$135$	−0.931493	−0.0801701
$136$	17.6585	1.51420
$137$	−7.34881	−0.627851	−0.313926	−	0.949448i	$-0.601644\pi$
$137$	−7.34881	−0.627851	−0.313926	+	0.949448i	$0.601644\pi$
$138$	2.53014	0.215380
$139$	−4.65458	−0.394797	−0.197398	−	0.980323i	$-0.563249\pi$
$139$	−4.65458	−0.394797	−0.197398	+	0.980323i	$0.563249\pi$
$140$	2.39587	0.202488
$141$	0.718870	0.0605398
$142$	−23.5027	−1.97230
$143$	19.0107	1.58976
$144$	−2.23232	−0.186027
$145$	0.0855153	0.00710166
$146$	16.7194	1.38370
$147$	−1.01366	−0.0836056
$148$	−23.0262	−1.89274
$149$	−3.49638	−0.286435	−0.143217	−	0.989691i	$-0.545745\pi$
$149$	−3.49638	−0.286435	−0.143217	+	0.989691i	$0.545745\pi$
$150$	−0.362164	−0.0295706
$151$	22.1096	1.79925	0.899626	−	0.436661i	$-0.143839\pi$
$151$	22.1096	1.79925	0.899626	+	0.436661i	$0.143839\pi$
$152$	−24.4874	−1.98619
$153$	16.1784	1.30794
$154$	−4.73292	−0.381389
$155$	7.75161	0.622625
$156$	3.48529	0.279047
$157$	−6.10536	−0.487261	−0.243631	−	0.969868i	$-0.578338\pi$
$157$	−6.10536	−0.487261	−0.243631	+	0.969868i	$0.578338\pi$
$158$	11.8221	0.940517
$159$	−2.06794	−0.163998
$160$	4.75293	0.375752
$161$	−4.92590	−0.388216
$162$	−20.4034	−1.60305
$163$	−16.8210	−1.31753	−0.658763	−	0.752351i	$-0.728920\pi$
$163$	−16.8210	−1.31753	−0.658763	+	0.752351i	$0.728920\pi$
$164$	−35.7449	−2.79121
$165$	0.450359	0.0350604
$166$	18.0938	1.40435
$167$	−19.6983	−1.52430	−0.762150	−	0.647400i	$-0.775856\pi$
$167$	−19.6983	−1.52430	−0.762150	+	0.647400i	$0.775856\pi$
$168$	−0.356980	−0.0275416
$169$	30.2974	2.33057
$170$	12.6317	0.968804
$171$	−22.4349	−1.71564
$172$	−4.08137	−0.311201
$173$	3.26818	0.248475	0.124238	−	0.992253i	$-0.460352\pi$
$173$	3.26818	0.248475	0.124238	+	0.992253i	$0.460352\pi$
$174$	−0.0309706	−0.00234787
$175$	0.705093	0.0533000
$176$	2.16738	0.163372
$177$	−2.26837	−0.170501
$178$	12.6879	0.950998
$179$	15.7815	1.17956	0.589781	−	0.807563i	$-0.299214\pi$
$179$	15.7815	1.17956	0.589781	+	0.807563i	$0.299214\pi$
$180$	−10.1113	−0.753652
$181$	−3.07431	−0.228512	−0.114256	−	0.993451i	$-0.536448\pi$
$181$	−3.07431	−0.228512	−0.114256	+	0.993451i	$0.536448\pi$
$182$	−10.7793	−0.799017
$183$	−1.06346	−0.0786129
$184$	22.6906	1.67278
$185$	−6.77649	−0.498217
$186$	−2.80735	−0.205845
$187$	−15.7077	−1.14866
$188$	15.6703	1.14287
$189$	−0.656789	−0.0477744
$190$	−17.5166	−1.27079
$191$	27.1126	1.96180	0.980899	−	0.194519i	$-0.0623147\pi$
$191$	27.1126	1.96180	0.980899	+	0.194519i	$0.0623147\pi$
$192$	−1.95522	−0.141106
$193$	−12.1535	−0.874826	−0.437413	−	0.899261i	$-0.644105\pi$
$193$	−12.1535	−0.874826	−0.437413	+	0.899261i	$0.644105\pi$
$194$	−23.6703	−1.69943
$195$	1.02570	0.0734521
$196$	−22.0964	−1.57831
$197$	11.3826	0.810977	0.405489	−	0.914100i	$-0.367101\pi$
$197$	11.3826	0.810977	0.405489	+	0.914100i	$0.367101\pi$
$198$	19.9743	1.41951
$199$	−11.3158	−0.802156	−0.401078	−	0.916044i	$-0.631364\pi$
$199$	−11.3158	−0.802156	−0.401078	+	0.916044i	$0.631364\pi$
$200$	−3.24794	−0.229664
$201$	1.61601	0.113984
$202$	21.6633	1.52423
$203$	0.0602962	0.00423197
$204$	−2.87974	−0.201622
$205$	−10.5195	−0.734716
$206$	3.87784	0.270182
$207$	20.7888	1.44492
$208$	4.93626	0.342268
$209$	21.7823	1.50671
$210$	−0.255359	−0.0176215
$211$	−2.62882	−0.180976	−0.0904878	−	0.995898i	$-0.528843\pi$
$211$	−2.62882	−0.180976	−0.0904878	+	0.995898i	$0.528843\pi$
$212$	−45.0780	−3.09597
$213$	1.57686	0.108045
$214$	−2.24747	−0.153634
$215$	−1.20112	−0.0819160
$216$	3.02543	0.205854
$217$	5.46561	0.371030
$218$	31.6063	2.14065
$219$	−1.12175	−0.0758008
$220$	9.81715	0.661872
$221$	−35.7748	−2.40647
$222$	2.45420	0.164715
$223$	−23.0872	−1.54603	−0.773017	−	0.634386i	$-0.781253\pi$
$223$	−23.0872	−1.54603	−0.773017	+	0.634386i	$0.781253\pi$
$224$	3.35126	0.223916
$225$	−2.97570	−0.198380
$226$	35.5780	2.36662
$227$	10.1560	0.674079	0.337040	−	0.941490i	$-0.390574\pi$
$227$	10.1560	0.674079	0.337040	+	0.941490i	$0.390574\pi$
$228$	3.99341	0.264470
$229$	−23.6365	−1.56195	−0.780973	−	0.624564i	$-0.785277\pi$
$229$	−23.6365	−1.56195	−0.780973	+	0.624564i	$0.785277\pi$
$230$	16.2313	1.07026
$231$	0.317545	0.0208929
$232$	−0.277748	−0.0182351
$233$	−15.0362	−0.985054	−0.492527	−	0.870297i	$-0.663927\pi$
$233$	−15.0362	−0.985054	−0.492527	+	0.870297i	$0.663927\pi$
$234$	45.4920	2.97390
$235$	4.61169	0.300833
$236$	−49.4472	−3.21874
$237$	−0.793179	−0.0515225
$238$	8.90649	0.577322
$239$	1.83357	0.118604	0.0593019	−	0.998240i	$-0.481113\pi$
$239$	1.83357	0.118604	0.0593019	+	0.998240i	$0.481113\pi$
$240$	0.116939	0.00754835
$241$	14.3088	0.921713	0.460857	−	0.887475i	$-0.347542\pi$
$241$	14.3088	0.921713	0.460857	+	0.887475i	$0.347542\pi$
$242$	6.16359	0.396211
$243$	4.16340	0.267082
$244$	−23.1818	−1.48406
$245$	−6.50284	−0.415451
$246$	3.80980	0.242904
$247$	49.6097	3.15659
$248$	−25.1767	−1.59873
$249$	−1.21397	−0.0769320
$250$	−2.32335	−0.146942
$251$	−18.5652	−1.17182	−0.585912	−	0.810375i	$-0.699263\pi$
$251$	−18.5652	−1.17182	−0.585912	+	0.810375i	$0.699263\pi$
$252$	−7.12940	−0.449110
$253$	−20.1840	−1.26896
$254$	8.27317	0.519105
$255$	−0.847493	−0.0530721
$256$	−20.5354	−1.28346
$257$	−3.80939	−0.237623	−0.118812	−	0.992917i	$-0.537908\pi$
$257$	−3.80939	−0.237623	−0.118812	+	0.992917i	$0.537908\pi$
$258$	0.435004	0.0270822
$259$	−4.77805	−0.296894
$260$	22.3588	1.38663
$261$	−0.254468	−0.0157512
$262$	−31.8624	−1.96847
$263$	−10.0928	−0.622346	−0.311173	−	0.950353i	$-0.600722\pi$
$263$	−10.0928	−0.622346	−0.311173	+	0.950353i	$0.600722\pi$
$264$	−1.46274	−0.0900252
$265$	−13.2662	−0.814938
$266$	−12.3508	−0.757278
$267$	−0.851266	−0.0520967
$268$	35.2266	2.15180
$269$	22.0435	1.34402	0.672009	−	0.740543i	$-0.265432\pi$
$269$	22.0435	1.34402	0.672009	+	0.740543i	$0.265432\pi$
$270$	2.16418	0.131708
$271$	−10.7429	−0.652582	−0.326291	−	0.945269i	$-0.605799\pi$
$271$	−10.7429	−0.652582	−0.326291	+	0.945269i	$0.605799\pi$
$272$	−4.07861	−0.247302
$273$	0.723216	0.0437710
$274$	17.0739	1.03147
$275$	2.88914	0.174221
$276$	−3.70039	−0.222737
$277$	−12.1469	−0.729835	−0.364918	−	0.931040i	$-0.618903\pi$
$277$	−12.1469	−0.729835	−0.364918	+	0.931040i	$0.618903\pi$
$278$	10.8142	0.648594
$279$	−23.0665	−1.38095
$280$	−2.29010	−0.136859
$281$	−15.5081	−0.925135	−0.462567	−	0.886584i	$-0.653072\pi$
$281$	−15.5081	−0.925135	−0.462567	+	0.886584i	$0.653072\pi$
$282$	−1.67019	−0.0994581
$283$	2.01709	0.119904	0.0599518	−	0.998201i	$-0.480905\pi$
$283$	2.01709	0.119904	0.0599518	+	0.998201i	$0.480905\pi$
$284$	34.3732	2.03968
$285$	1.17524	0.0696151
$286$	−44.1686	−2.61174
$287$	−7.41725	−0.437826
$288$	−14.1433	−0.833403
$289$	12.5591	0.738771
$290$	−0.198682	−0.0116670
$291$	1.58811	0.0930965
$292$	−24.4525	−1.43097
$293$	18.0535	1.05470	0.527349	−	0.849648i	$-0.323186\pi$
$293$	18.0535	1.05470	0.527349	+	0.849648i	$0.323186\pi$
$294$	2.35510	0.137352
$295$	−14.5520	−0.847253
$296$	22.0096	1.27928
$297$	−2.69121	−0.156160
$298$	8.12332	0.470571
$299$	−45.9696	−2.65849
$300$	0.529673	0.0305807
$301$	−0.846904	−0.0488148
$302$	−51.3683	−2.95591
$303$	−1.45345	−0.0834987
$304$	5.65591	0.324389
$305$	−6.82227	−0.390642
$306$	−37.5880	−2.14876
$307$	17.7406	1.01251	0.506254	−	0.862384i	$-0.331030\pi$
$307$	17.7406	1.01251	0.506254	+	0.862384i	$0.331030\pi$
$308$	6.92200	0.394418
$309$	−0.260175	−0.0148009
$310$	−18.0097	−1.02288
$311$	18.8684	1.06993	0.534964	−	0.844875i	$-0.320325\pi$
$311$	18.8684	1.06993	0.534964	+	0.844875i	$0.320325\pi$
$312$	−3.33142	−0.188604
$313$	−16.5208	−0.933813	−0.466907	−	0.884307i	$-0.654632\pi$
$313$	−16.5208	−0.933813	−0.466907	+	0.884307i	$0.654632\pi$
$314$	14.1849	0.800500
$315$	−2.09815	−0.118217
$316$	−17.2901	−0.972645
$317$	−8.23083	−0.462290	−0.231145	−	0.972919i	$-0.574247\pi$
$317$	−8.23083	−0.462290	−0.231145	+	0.972919i	$0.574247\pi$
$318$	4.80455	0.269426
$319$	0.247065	0.0138330
$320$	−12.5431	−0.701180
$321$	0.150789	0.00841621
$322$	11.4446	0.637782
$323$	−40.9903	−2.28076
$324$	29.8405	1.65781
$325$	6.58008	0.364997
$326$	39.0812	2.16451
$327$	−2.12056	−0.117267
$328$	34.1668	1.88654
$329$	3.25167	0.179270
$330$	−1.04634	−0.0575992
$331$	27.5004	1.51156	0.755778	−	0.654828i	$-0.227259\pi$
$331$	27.5004	1.51156	0.755778	+	0.654828i	$0.227259\pi$
$332$	−26.4627	−1.45233
$333$	20.1648	1.10502
$334$	45.7660	2.50421
$335$	10.3670	0.566409
$336$	0.0824525	0.00449815
$337$	−28.6473	−1.56052	−0.780259	−	0.625457i	$-0.784913\pi$
$337$	−28.6473	−1.56052	−0.780259	+	0.625457i	$0.784913\pi$
$338$	−70.3915	−3.82879
$339$	−2.38703	−0.129646
$340$	−18.4741	−1.00190
$341$	22.3955	1.21278
$342$	52.1242	2.81855
$343$	−9.52076	−0.514073
$344$	3.90118	0.210337
$345$	−1.08901	−0.0586301
$346$	−7.59312	−0.408209
$347$	23.6859	1.27153	0.635764	−	0.771883i	$-0.280685\pi$
$347$	23.6859	1.27153	0.635764	+	0.771883i	$0.280685\pi$
$348$	0.0452952	0.00242808
$349$	−1.17865	−0.0630914	−0.0315457	−	0.999502i	$-0.510043\pi$
$349$	−1.17865	−0.0630914	−0.0315457	+	0.999502i	$0.510043\pi$
$350$	−1.63818	−0.0875643
$351$	−6.12929	−0.327158
$352$	13.7319	0.731911
$353$	−18.7200	−0.996365	−0.498182	−	0.867072i	$-0.665999\pi$
$353$	−18.7200	−0.996365	−0.498182	+	0.867072i	$0.665999\pi$
$354$	5.27022	0.280109
$355$	10.1159	0.536894
$356$	−18.5563	−0.983484
$357$	−0.597561	−0.0316263
$358$	−36.6659	−1.93785
$359$	−25.4066	−1.34091	−0.670455	−	0.741950i	$-0.733901\pi$
$359$	−25.4066	−1.34091	−0.670455	+	0.741950i	$0.733901\pi$
$360$	9.66489	0.509384
$361$	37.8422	1.99169
$362$	7.14269	0.375412
$363$	−0.413533	−0.0217048
$364$	15.7650	0.826312
$365$	−7.19623	−0.376668
$366$	2.47078	0.129150
$367$	4.60634	0.240449	0.120224	−	0.992747i	$-0.461639\pi$
$367$	4.60634	0.240449	0.120224	+	0.992747i	$0.461639\pi$
$368$	−5.24091	−0.273201
$369$	31.3030	1.62957
$370$	15.7442	0.818500
$371$	−9.35392	−0.485631
$372$	4.10582	0.212877
$373$	−12.2792	−0.635795	−0.317897	−	0.948125i	$-0.602977\pi$
$373$	−12.2792	−0.635795	−0.317897	+	0.948125i	$0.602977\pi$
$374$	36.4946	1.88709
$375$	0.155880	0.00804961
$376$	−14.9785	−0.772455
$377$	0.562697	0.0289804
$378$	1.52595	0.0784864
$379$	31.8519	1.63612	0.818062	−	0.575131i	$-0.195049\pi$
$379$	31.8519	1.63612	0.818062	+	0.575131i	$0.195049\pi$
$380$	25.6184	1.31420
$381$	−0.555070	−0.0284371
$382$	−62.9920	−3.22295
$383$	29.4722	1.50596	0.752980	−	0.658044i	$-0.228616\pi$
$383$	29.4722	1.50596	0.752980	+	0.658044i	$0.228616\pi$
$384$	3.06088	0.156200
$385$	2.03711	0.103821
$386$	28.2367	1.43721
$387$	3.57419	0.181686
$388$	34.6184	1.75748
$389$	25.2569	1.28058	0.640288	−	0.768135i	$-0.278815\pi$
$389$	25.2569	1.28058	0.640288	+	0.768135i	$0.278815\pi$
$390$	−2.38307	−0.120671
$391$	37.9826	1.92086
$392$	21.1208	1.06676
$393$	2.13774	0.107835
$394$	−26.4458	−1.33232
$395$	−5.08839	−0.256025
$396$	−29.2129	−1.46800
$397$	25.3560	1.27258	0.636291	−	0.771449i	$-0.280468\pi$
$397$	25.3560	1.27258	0.636291	+	0.771449i	$0.280468\pi$
$398$	26.2906	1.31783
$399$	0.828652	0.0414845
$400$	0.750183	0.0375091
$401$	23.4405	1.17056	0.585282	−	0.810830i	$-0.300984\pi$
$401$	23.4405	1.17056	0.585282	+	0.810830i	$0.300984\pi$
$402$	−3.75455	−0.187260
$403$	51.0062	2.54080
$404$	−31.6831	−1.57629
$405$	8.78190	0.436376
$406$	−0.140089	−0.00695251
$407$	−19.5782	−0.970455
$408$	2.75260	0.136274
$409$	−0.490470	−0.0242522	−0.0121261	−	0.999926i	$-0.503860\pi$
$409$	−0.490470	−0.0242522	−0.0121261	+	0.999926i	$0.503860\pi$
$410$	24.4406	1.20703
$411$	−1.14553	−0.0565050
$412$	−5.67144	−0.279412
$413$	−10.2605	−0.504888
$414$	−48.2996	−2.37380
$415$	−7.78782	−0.382289
$416$	31.2747	1.53337
$417$	−0.725557	−0.0355307
$418$	−50.6079	−2.47531
$419$	33.5803	1.64051	0.820253	−	0.572001i	$-0.193833\pi$
$419$	33.5803	1.64051	0.820253	+	0.572001i	$0.193833\pi$
$420$	0.373469	0.0182234
$421$	18.7674	0.914666	0.457333	−	0.889295i	$-0.348805\pi$
$421$	18.7674	0.914666	0.457333	+	0.889295i	$0.348805\pi$
$422$	6.10767	0.297317
$423$	−13.7230	−0.667235
$424$	43.0879	2.09253
$425$	−5.43683	−0.263725
$426$	−3.66360	−0.177502
$427$	−4.81034	−0.232788
$428$	3.28697	0.158882
$429$	2.96340	0.143074
$430$	2.79063	0.134576
$431$	−18.7560	−0.903443	−0.451722	−	0.892159i	$-0.649190\pi$
$431$	−18.7560	−0.903443	−0.451722	+	0.892159i	$0.649190\pi$
$432$	−0.698790	−0.0336205
$433$	13.6463	0.655798	0.327899	−	0.944713i	$-0.393659\pi$
$433$	13.6463	0.655798	0.327899	+	0.944713i	$0.393659\pi$
$434$	−12.6985	−0.609548
$435$	0.0133301	0.000639131 0
$436$	−46.2250	−2.21377
$437$	−52.6714	−2.51961
$438$	2.60621	0.124530
$439$	7.84379	0.374364	0.187182	−	0.982325i	$-0.440065\pi$
$439$	7.84379	0.374364	0.187182	+	0.982325i	$0.440065\pi$
$440$	−9.38373	−0.447352
$441$	19.3505	0.921453
$442$	83.1173	3.95349
$443$	−2.86048	−0.135905	−0.0679526	−	0.997689i	$-0.521647\pi$
$443$	−2.86048	−0.135905	−0.0679526	+	0.997689i	$0.521647\pi$
$444$	−3.58933	−0.170342
$445$	−5.46104	−0.258878
$446$	53.6396	2.53991
$447$	−0.545016	−0.0257784
$448$	−8.84405	−0.417842
$449$	−8.89937	−0.419987	−0.209994	−	0.977703i	$-0.567344\pi$
$449$	−8.89937	−0.419987	−0.209994	+	0.977703i	$0.567344\pi$
$450$	6.91360	0.325910
$451$	−30.3924	−1.43112
$452$	−52.0337	−2.44746
$453$	3.44644	0.161928
$454$	−23.5960	−1.10742
$455$	4.63957	0.217506
$456$	−3.81710	−0.178752
$457$	6.78806	0.317532	0.158766	−	0.987316i	$-0.449248\pi$
$457$	6.78806	0.317532	0.158766	+	0.987316i	$0.449248\pi$
$458$	54.9159	2.56605
$459$	5.06437	0.236384
$460$	−23.7387	−1.10682
$461$	24.9904	1.16392	0.581960	−	0.813217i	$-0.302286\pi$
$461$	24.9904	1.16392	0.581960	+	0.813217i	$0.302286\pi$
$462$	−0.737767	−0.0343240
$463$	4.91500	0.228419	0.114210	−	0.993457i	$-0.463566\pi$
$463$	4.91500	0.228419	0.114210	+	0.993457i	$0.463566\pi$
$464$	0.0641521	0.00297819
$465$	1.20832	0.0560346
$466$	34.9344	1.61830
$467$	−15.9191	−0.736649	−0.368324	−	0.929697i	$-0.620068\pi$
$467$	−15.9191	−0.736649	−0.368324	+	0.929697i	$0.620068\pi$
$468$	−66.5331	−3.07549
$469$	7.30969	0.337530
$470$	−10.7146	−0.494226
$471$	−0.951705	−0.0438522
$472$	47.2641	2.17551
$473$	−3.47021	−0.159561
$474$	1.84283	0.0846440
$475$	7.53937	0.345930
$476$	−13.0260	−0.597044
$477$	39.4763	1.80750
$478$	−4.26003	−0.194849
$479$	30.9296	1.41321	0.706606	−	0.707607i	$-0.250225\pi$
$479$	30.9296	1.41321	0.706606	+	0.707607i	$0.250225\pi$
$480$	0.740888	0.0338167
$481$	−44.5898	−2.03312
$482$	−33.2444	−1.51424
$483$	−0.767850	−0.0349384
$484$	−9.01440	−0.409746
$485$	10.1880	0.462614
$486$	−9.67304	−0.438778
$487$	−34.7018	−1.57249	−0.786245	−	0.617915i	$-0.787978\pi$
$487$	−34.7018	−1.57249	−0.786245	+	0.617915i	$0.787978\pi$
$488$	22.1583	1.00306
$489$	−2.62207	−0.118574
$490$	15.1084	0.682527
$491$	−34.1466	−1.54102	−0.770508	−	0.637431i	$-0.779997\pi$
$491$	−34.1466	−1.54102	−0.770508	+	0.637431i	$0.779997\pi$
$492$	−5.57192	−0.251201
$493$	−0.464932	−0.0209395
$494$	−115.261	−5.18582
$495$	−8.59721	−0.386416
$496$	5.81513	0.261107
$497$	7.13262	0.319942
$498$	2.82047	0.126388
$499$	13.3604	0.598094	0.299047	−	0.954238i	$-0.403331\pi$
$499$	13.3604	0.598094	0.299047	+	0.954238i	$0.403331\pi$
$500$	3.39795	0.151961
$501$	−3.07057	−0.137183
$502$	43.1334	1.92514
$503$	−22.7693	−1.01523	−0.507616	−	0.861583i	$-0.669473\pi$
$503$	−22.7693	−1.01523	−0.507616	+	0.861583i	$0.669473\pi$
$504$	6.81464	0.303548
$505$	−9.32418	−0.414920
$506$	46.8945	2.08472
$507$	4.72276	0.209745
$508$	−12.0997	−0.536838
$509$	−13.0231	−0.577237	−0.288619	−	0.957444i	$-0.593196\pi$
$509$	−13.0231	−0.577237	−0.288619	+	0.957444i	$0.593196\pi$
$510$	1.96902	0.0871898
$511$	−5.07401	−0.224461
$512$	8.43866	0.372940
$513$	−7.02287	−0.310067
$514$	8.85055	0.390381
$515$	−1.66907	−0.0735482
$516$	−0.636204	−0.0280073
$517$	13.3238	0.585979
$518$	11.1011	0.487754
$519$	0.509444	0.0223621
$520$	−21.3717	−0.937210
$521$	38.6872	1.69491	0.847457	−	0.530863i	$-0.178132\pi$
$521$	38.6872	1.69491	0.847457	+	0.530863i	$0.178132\pi$
$522$	0.591218	0.0258769
$523$	1.03586	0.0452952	0.0226476	−	0.999744i	$-0.492790\pi$
$523$	1.03586	0.0452952	0.0226476	+	0.999744i	$0.492790\pi$
$524$	46.5996	2.03571
$525$	0.109910	0.00479686
$526$	23.4490	1.02242
$527$	−42.1442	−1.83583
$528$	0.337851	0.0147031
$529$	25.8066	1.12203
$530$	30.8221	1.33883
$531$	43.3025	1.87917
$532$	18.0634	0.783147
$533$	−69.2194	−2.99822
$534$	1.97779	0.0855873
$535$	0.967338	0.0418217
$536$	−33.6713	−1.45438
$537$	2.46002	0.106158
$538$	−51.2148	−2.20803
$539$	−18.7876	−0.809239
$540$	−3.16517	−0.136207
$541$	−16.4241	−0.706126	−0.353063	−	0.935600i	$-0.614860\pi$
$541$	−16.4241	−0.706126	−0.353063	+	0.935600i	$0.614860\pi$
$542$	24.9594	1.07210
$543$	−0.479223	−0.0205654
$544$	−25.8409	−1.10792
$545$	−13.6038	−0.582721
$546$	−1.68028	−0.0719095
$547$	40.6938	1.73994	0.869971	−	0.493103i	$-0.164137\pi$
$547$	40.6938	1.73994	0.869971	+	0.493103i	$0.164137\pi$
$548$	−24.9709	−1.06671
$549$	20.3010	0.866427
$550$	−6.71247	−0.286221
$551$	0.644732	0.0274665
$552$	3.53702	0.150546
$553$	−3.58779	−0.152568
$554$	28.2214	1.19901
$555$	−1.05632	−0.0448383
$556$	−15.8161	−0.670750
$557$	12.4810	0.528837	0.264418	−	0.964408i	$-0.414820\pi$
$557$	12.4810	0.528837	0.264418	+	0.964408i	$0.414820\pi$
$558$	53.5915	2.26871
$559$	−7.90349	−0.334282
$560$	0.528949	0.0223522
$561$	−2.44852	−0.103377
$562$	36.0307	1.51986
$563$	36.8522	1.55313	0.776567	−	0.630035i	$-0.216959\pi$
$563$	36.8522	1.55313	0.776567	+	0.630035i	$0.216959\pi$
$564$	2.44269	0.102856
$565$	−15.3133	−0.644233
$566$	−4.68641	−0.196984
$567$	6.19206	0.260042
$568$	−32.8557	−1.37859
$569$	8.47256	0.355188	0.177594	−	0.984104i	$-0.443169\pi$
$569$	8.47256	0.355188	0.177594	+	0.984104i	$0.443169\pi$
$570$	−2.73049	−0.114368
$571$	−2.29800	−0.0961683	−0.0480842	−	0.998843i	$-0.515312\pi$
$571$	−2.29800	−0.0961683	−0.0480842	+	0.998843i	$0.515312\pi$
$572$	64.5976	2.70096
$573$	4.22631	0.176557
$574$	17.2329	0.719286
$575$	−6.98618	−0.291344
$576$	37.3245	1.55519
$577$	−39.9324	−1.66241	−0.831203	−	0.555969i	$-0.812347\pi$
$577$	−39.9324	−1.66241	−0.831203	+	0.555969i	$0.812347\pi$
$578$	−29.1792	−1.21369
$579$	−1.89448	−0.0787320
$580$	0.290577	0.0120656
$581$	−5.49114	−0.227811
$582$	−3.68973	−0.152944
$583$	−38.3279	−1.58738
$584$	23.3729	0.967177
$585$	−19.5803	−0.809548
$586$	−41.9447	−1.73272
$587$	−36.8210	−1.51976	−0.759882	−	0.650061i	$-0.774743\pi$
$587$	−36.8210	−1.51976	−0.759882	+	0.650061i	$0.774743\pi$
$588$	−3.44438	−0.142044
$589$	58.4423	2.40807
$590$	33.8095	1.39191
$591$	1.77432	0.0729858
$592$	−5.08361	−0.208935
$593$	−43.6189	−1.79122	−0.895608	−	0.444845i	$-0.853259\pi$
$593$	−43.6189	−1.79122	−0.895608	+	0.444845i	$0.853259\pi$
$594$	6.25262	0.256548
$595$	−3.83347	−0.157157
$596$	−11.8805	−0.486646
$597$	−1.76391	−0.0721920
$598$	106.803	4.36752
$599$	5.69938	0.232870	0.116435	−	0.993198i	$-0.462853\pi$
$599$	5.69938	0.232870	0.116435	+	0.993198i	$0.462853\pi$
$600$	−0.506288	−0.0206691
$601$	−19.4616	−0.793855	−0.396928	−	0.917850i	$-0.629924\pi$
$601$	−19.4616	−0.793855	−0.396928	+	0.917850i	$0.629924\pi$
$602$	1.96766	0.0801956
$603$	−30.8491	−1.25627
$604$	75.1273	3.05689
$605$	−2.65289	−0.107855
$606$	3.37688	0.137176
$607$	−1.35580	−0.0550303	−0.0275152	−	0.999621i	$-0.508759\pi$
$607$	−1.35580	−0.0550303	−0.0275152	+	0.999621i	$0.508759\pi$
$608$	35.8342	1.45327
$609$	0.00939898	0.000380866 0
$610$	15.8505	0.641769
$611$	30.3452	1.22764
$612$	54.9734	2.22217
$613$	32.5544	1.31486	0.657429	−	0.753516i	$-0.271644\pi$
$613$	32.5544	1.31486	0.657429	+	0.753516i	$0.271644\pi$
$614$	−41.2176	−1.66341
$615$	−1.63979	−0.0661225
$616$	−6.61640	−0.266582
$617$	−7.11978	−0.286632	−0.143316	−	0.989677i	$-0.545776\pi$
$617$	−7.11978	−0.286632	−0.143316	+	0.989677i	$0.545776\pi$
$618$	0.604478	0.0243157
$619$	−34.8500	−1.40074	−0.700370	−	0.713780i	$-0.746982\pi$
$619$	−34.8500	−1.40074	−0.700370	+	0.713780i	$0.746982\pi$
$620$	26.3396	1.05782
$621$	6.50757	0.261140
$622$	−43.8378	−1.75774
$623$	−3.85054	−0.154269
$624$	0.769465	0.0308032
$625$	1.00000	0.0400000
$626$	38.3837	1.53412
$627$	3.39542	0.135600
$628$	−20.7457	−0.827845
$629$	36.8426	1.46901
$630$	4.87473	0.194214
$631$	−18.8618	−0.750875	−0.375438	−	0.926848i	$-0.622508\pi$
$631$	−18.8618	−0.750875	−0.375438	+	0.926848i	$0.622508\pi$
$632$	16.5268	0.657399
$633$	−0.409781	−0.0162873
$634$	19.1231	0.759475
$635$	−3.56088	−0.141309
$636$	−7.02676	−0.278629
$637$	−42.7892	−1.69537
$638$	−0.574019	−0.0227256
$639$	−30.1018	−1.19081
$640$	19.6361	0.776186
$641$	45.0948	1.78114	0.890568	−	0.454850i	$-0.150307\pi$
$641$	45.0948	1.78114	0.890568	+	0.454850i	$0.150307\pi$
$642$	−0.350335	−0.0138266
$643$	−28.9478	−1.14159	−0.570795	−	0.821092i	$-0.693365\pi$
$643$	−28.9478	−1.14159	−0.570795	+	0.821092i	$0.693365\pi$
$644$	−16.7380	−0.659569
$645$	−0.187231	−0.00737223
$646$	95.2348	3.74696
$647$	−0.952161	−0.0374333	−0.0187167	−	0.999825i	$-0.505958\pi$
$647$	−0.952161	−0.0374333	−0.0187167	+	0.999825i	$0.505958\pi$
$648$	−28.5231	−1.12049
$649$	−42.0428	−1.65033
$650$	−15.2878	−0.599638
$651$	0.851979	0.0333917
$652$	−57.1571	−2.23845
$653$	34.3099	1.34265	0.671326	−	0.741163i	$-0.265725\pi$
$653$	34.3099	1.34265	0.671326	+	0.741163i	$0.265725\pi$
$654$	4.92679	0.192653
$655$	13.7140	0.535851
$656$	−7.89157	−0.308114
$657$	21.4138	0.835433
$658$	−7.55476	−0.294515
$659$	−44.5268	−1.73452	−0.867258	−	0.497859i	$-0.834120\pi$
$659$	−44.5268	−1.73452	−0.867258	+	0.497859i	$0.834120\pi$
$660$	1.53030	0.0595668
$661$	23.1928	0.902094	0.451047	−	0.892500i	$-0.351051\pi$
$661$	23.1928	0.902094	0.451047	+	0.892500i	$0.351051\pi$
$662$	−63.8929	−2.48327
$663$	−5.57657	−0.216576
$664$	25.2943	0.981611
$665$	5.31596	0.206144
$666$	−46.8499	−1.81540
$667$	−0.597425	−0.0231324
$668$	−66.9339	−2.58975
$669$	−3.59883	−0.139139
$670$	−24.0861	−0.930529
$671$	−19.7105	−0.760914
$672$	0.522395	0.0201518
$673$	−19.0252	−0.733367	−0.366683	−	0.930346i	$-0.619507\pi$
$673$	−19.0252	−0.733367	−0.366683	+	0.930346i	$0.619507\pi$
$674$	66.5577	2.56371
$675$	−0.931493	−0.0358532
$676$	102.949	3.95959
$677$	0.972426	0.0373734	0.0186867	−	0.999825i	$-0.494051\pi$
$677$	0.972426	0.0373734	0.0186867	+	0.999825i	$0.494051\pi$
$678$	5.54591	0.212989
$679$	7.18349	0.275677
$680$	17.6585	0.677171
$681$	1.58312	0.0606654
$682$	−52.0325	−1.99243
$683$	19.6679	0.752572	0.376286	−	0.926504i	$-0.377201\pi$
$683$	19.6679	0.752572	0.376286	+	0.926504i	$0.377201\pi$
$684$	−76.2329	−2.91484
$685$	−7.34881	−0.280784
$686$	22.1201	0.844548
$687$	−3.68447	−0.140571
$688$	−0.901063	−0.0343527
$689$	−87.2928	−3.32559
$690$	2.53014	0.0963208
$691$	−14.8602	−0.565310	−0.282655	−	0.959222i	$-0.591215\pi$
$691$	−14.8602	−0.565310	−0.282655	+	0.959222i	$0.591215\pi$
$692$	11.1051	0.422153
$693$	−6.06183	−0.230270
$694$	−55.0307	−2.08894
$695$	−4.65458	−0.176558
$696$	−0.0432954	−0.00164111
$697$	57.1929	2.16634
$698$	2.73841	0.103650
$699$	−2.34384	−0.0886523
$700$	2.39587	0.0905555
$701$	−18.1748	−0.686454	−0.343227	−	0.939253i	$-0.611520\pi$
$701$	−18.1748	−0.686454	−0.343227	+	0.939253i	$0.611520\pi$
$702$	14.2405	0.537473
$703$	−51.0905	−1.92691
$704$	−36.2387	−1.36580
$705$	0.718870	0.0270742
$706$	43.4931	1.63688
$707$	−6.57441	−0.247256
$708$	−7.70783	−0.289678
$709$	−4.35273	−0.163470	−0.0817352	−	0.996654i	$-0.526046\pi$
$709$	−4.35273	−0.163470	−0.0817352	+	0.996654i	$0.526046\pi$
$710$	−23.5027	−0.882040
$711$	15.1415	0.567852
$712$	17.7371	0.664726
$713$	−54.1541	−2.02809
$714$	1.38834	0.0519575
$715$	19.0107	0.710961
$716$	53.6247	2.00405
$717$	0.285817	0.0106740
$718$	59.0285	2.20292
$719$	14.5545	0.542791	0.271395	−	0.962468i	$-0.412515\pi$
$719$	14.5545	0.542791	0.271395	+	0.962468i	$0.412515\pi$
$720$	−2.23232	−0.0831937
$721$	−1.17685	−0.0438283
$722$	−87.9206	−3.27207
$723$	2.23046	0.0829518
$724$	−10.4464	−0.388236
$725$	0.0855153	0.00317596
$726$	0.960782	0.0356579
$727$	−0.311761	−0.0115626	−0.00578129	−	0.999983i	$-0.501840\pi$
$727$	−0.311761	−0.0115626	−0.00578129	+	0.999983i	$0.501840\pi$
$728$	−15.0690	−0.558495
$729$	−25.6967	−0.951730
$730$	16.7194	0.618811
$731$	6.53031	0.241532
$732$	−3.61357	−0.133562
$733$	−36.5562	−1.35023	−0.675117	−	0.737710i	$-0.735907\pi$
$733$	−36.5562	−1.35023	−0.675117	+	0.737710i	$0.735907\pi$
$734$	−10.7021	−0.395023
$735$	−1.01366	−0.0373895
$736$	−33.2048	−1.22395
$737$	29.9517	1.10328
$738$	−72.7278	−2.67715
$739$	50.1534	1.84492	0.922462	−	0.386088i	$-0.126174\pi$
$739$	50.1534	1.84492	0.922462	+	0.386088i	$0.126174\pi$
$740$	−23.0262	−0.846460
$741$	7.73316	0.284085
$742$	21.7324	0.797823
$743$	22.8461	0.838142	0.419071	−	0.907953i	$-0.362356\pi$
$743$	22.8461	0.838142	0.419071	+	0.907953i	$0.362356\pi$
$744$	−3.92455	−0.143881
$745$	−3.49638	−0.128098
$746$	28.5290	1.04452
$747$	23.1742	0.847901
$748$	−53.3742	−1.95155
$749$	0.682063	0.0249221
$750$	−0.362164	−0.0132244
$751$	−36.1534	−1.31926	−0.659628	−	0.751592i	$-0.729286\pi$
$751$	−36.1534	−1.31926	−0.659628	+	0.751592i	$0.729286\pi$
$752$	3.45961	0.126159
$753$	−2.89394	−0.105461
$754$	−1.30734	−0.0476106
$755$	22.1096	0.804650
$756$	−2.23174	−0.0811675
$757$	−25.0267	−0.909612	−0.454806	−	0.890590i	$-0.650291\pi$
$757$	−25.0267	−0.909612	−0.454806	+	0.890590i	$0.650291\pi$
$758$	−74.0032	−2.68792
$759$	−3.14628	−0.114203
$760$	−24.4874	−0.888252
$761$	−15.7440	−0.570721	−0.285360	−	0.958420i	$-0.592113\pi$
$761$	−15.7440	−0.570721	−0.285360	+	0.958420i	$0.592113\pi$
$762$	1.28962	0.0467181
$763$	−9.59192	−0.347251
$764$	92.1273	3.33305
$765$	16.1784	0.584930
$766$	−68.4742	−2.47408
$767$	−95.7536	−3.45746
$768$	−3.20106	−0.115508
$769$	−10.4718	−0.377624	−0.188812	−	0.982013i	$-0.560464\pi$
$769$	−10.4718	−0.377624	−0.188812	+	0.982013i	$0.560464\pi$
$770$	−4.73292	−0.170563
$771$	−0.593808	−0.0213855
$772$	−41.2969	−1.48631
$773$	−24.8209	−0.892745	−0.446372	−	0.894847i	$-0.647284\pi$
$773$	−24.8209	−0.892745	−0.446372	+	0.894847i	$0.647284\pi$
$774$	−8.30409	−0.298484
$775$	7.75161	0.278446
$776$	−33.0900	−1.18786
$777$	−0.744803	−0.0267197
$778$	−58.6806	−2.10380
$779$	−79.3107	−2.84160
$780$	3.48529	0.124793
$781$	29.2261	1.04579
$782$	−88.2469	−3.15570
$783$	−0.0796569	−0.00284670
$784$	−4.87832	−0.174226
$785$	−6.10536	−0.217910
$786$	−4.96672	−0.177157
$787$	29.5786	1.05436	0.527181	−	0.849753i	$-0.323249\pi$
$787$	29.5786	1.05436	0.527181	+	0.849753i	$0.323249\pi$
$788$	38.6776	1.37783
$789$	−1.57326	−0.0560095
$790$	11.8221	0.420612
$791$	−10.7973	−0.383907
$792$	27.9232	0.992207
$793$	−44.8911	−1.59413
$794$	−58.9109	−2.09067
$795$	−2.06794	−0.0733423
$796$	−38.4506	−1.36285
$797$	−43.9541	−1.55693	−0.778467	−	0.627686i	$-0.784002\pi$
$797$	−43.9541	−1.55693	−0.778467	+	0.627686i	$0.784002\pi$
$798$	−1.92525	−0.0681531
$799$	−25.0729	−0.887017
$800$	4.75293	0.168042
$801$	16.2504	0.574180
$802$	−54.4605	−1.92307
$803$	−20.7909	−0.733695
$804$	5.49112	0.193657
$805$	−4.92590	−0.173615
$806$	−118.505	−4.17417
$807$	3.43615	0.120958
$808$	30.2843	1.06540
$809$	25.8117	0.907490	0.453745	−	0.891132i	$-0.350088\pi$
$809$	25.8117	0.907490	0.453745	+	0.891132i	$0.350088\pi$
$810$	−20.4034	−0.716904
$811$	−5.98989	−0.210334	−0.105167	−	0.994455i	$-0.533538\pi$
$811$	−5.98989	−0.210334	−0.105167	+	0.994455i	$0.533538\pi$
$812$	0.204884	0.00719001
$813$	−1.67460	−0.0587307
$814$	45.4870	1.59432
$815$	−16.8210	−0.589215
$816$	−0.635775	−0.0222566
$817$	−9.05573	−0.316820
$818$	1.13953	0.0398428
$819$	−13.8060	−0.482419
$820$	−35.7449	−1.24827
$821$	−56.2935	−1.96466	−0.982329	−	0.187162i	$-0.940071\pi$
$821$	−56.2935	−1.96466	−0.982329	+	0.187162i	$0.940071\pi$
$822$	2.66147	0.0928296
$823$	−19.3047	−0.672921	−0.336461	−	0.941698i	$-0.609230\pi$
$823$	−19.3047	−0.672921	−0.336461	+	0.941698i	$0.609230\pi$
$824$	5.42105	0.188851
$825$	0.450359	0.0156795
$826$	23.8388	0.829459
$827$	−13.3532	−0.464335	−0.232168	−	0.972676i	$-0.574582\pi$
$827$	−13.3532	−0.464335	−0.232168	+	0.972676i	$0.574582\pi$
$828$	70.6393	2.45488
$829$	−8.57654	−0.297876	−0.148938	−	0.988847i	$-0.547585\pi$
$829$	−8.57654	−0.297876	−0.148938	+	0.988847i	$0.547585\pi$
$830$	18.0938	0.628046
$831$	−1.89346	−0.0656833
$832$	−82.5345	−2.86137
$833$	35.3548	1.22497
$834$	1.68572	0.0583718
$835$	−19.6983	−0.681688
$836$	74.0152	2.55987
$837$	−7.22057	−0.249579
$838$	−78.0188	−2.69511
$839$	2.65681	0.0917233	0.0458617	−	0.998948i	$-0.485397\pi$
$839$	2.65681	0.0917233	0.0458617	+	0.998948i	$0.485397\pi$
$840$	−0.356980	−0.0123170
$841$	−28.9927	−0.999748
$842$	−43.6032	−1.50267
$843$	−2.41740	−0.0832597
$844$	−8.93262	−0.307473
$845$	30.2974	1.04226
$846$	31.8833	1.09617
$847$	−1.87054	−0.0642723
$848$	−9.95209	−0.341756
$849$	0.314424	0.0107910
$850$	12.6317	0.433262
$851$	47.3417	1.62285
$852$	5.35810	0.183565
$853$	−9.21134	−0.315390	−0.157695	−	0.987488i	$-0.550406\pi$
$853$	−9.21134	−0.315390	−0.157695	+	0.987488i	$0.550406\pi$
$854$	11.1761	0.382438
$855$	−22.4349	−0.767258
$856$	−3.14185	−0.107386
$857$	−37.3058	−1.27434	−0.637170	−	0.770723i	$-0.719895\pi$
$857$	−37.3058	−1.27434	−0.637170	+	0.770723i	$0.719895\pi$
$858$	−6.88500	−0.235050
$859$	−22.4131	−0.764726	−0.382363	−	0.924012i	$-0.624890\pi$
$859$	−22.4131	−0.764726	−0.382363	+	0.924012i	$0.624890\pi$
$860$	−4.08137	−0.139173
$861$	−1.15620	−0.0394032
$862$	43.5767	1.48423
$863$	−23.4721	−0.799001	−0.399500	−	0.916733i	$-0.630816\pi$
$863$	−23.4721	−0.799001	−0.399500	+	0.916733i	$0.630816\pi$
$864$	−4.42732	−0.150621
$865$	3.26818	0.111121
$866$	−31.7051	−1.07738
$867$	1.95771	0.0664874
$868$	18.5719	0.630371
$869$	−14.7011	−0.498699
$870$	−0.0309706	−0.00105000
$871$	68.2156	2.31140
$872$	44.1841	1.49626
$873$	−30.3165	−1.02606
$874$	122.374	4.13936
$875$	0.705093	0.0238365
$876$	−3.81165	−0.128784
$877$	32.3378	1.09197	0.545985	−	0.837795i	$-0.316155\pi$
$877$	32.3378	1.09197	0.545985	+	0.837795i	$0.316155\pi$
$878$	−18.2239	−0.615026
$879$	2.81419	0.0949202
$880$	2.16738	0.0730624
$881$	22.5794	0.760718	0.380359	−	0.924839i	$-0.375800\pi$
$881$	22.5794	0.760718	0.380359	+	0.924839i	$0.375800\pi$
$882$	−44.9580	−1.51382
$883$	−27.8363	−0.936767	−0.468383	−	0.883525i	$-0.655163\pi$
$883$	−27.8363	−0.936767	−0.468383	+	0.883525i	$0.655163\pi$
$884$	−121.561	−4.08854
$885$	−2.26837	−0.0762505
$886$	6.64588	0.223273
$887$	−22.3411	−0.750142	−0.375071	−	0.926996i	$-0.622382\pi$
$887$	−22.3411	−0.750142	−0.375071	+	0.926996i	$0.622382\pi$
$888$	3.43086	0.115132
$889$	−2.51075	−0.0842079
$890$	12.6879	0.425299
$891$	25.3721	0.849998
$892$	−78.4492	−2.62667
$893$	34.7692	1.16351
$894$	1.26626	0.0423502
$895$	15.7815	0.527516
$896$	13.8453	0.462539
$897$	−7.16574	−0.239257
$898$	20.6764	0.689979
$899$	0.662882	0.0221083
$900$	−10.1113	−0.337043
$901$	72.1262	2.40287
$902$	70.6121	2.35113
$903$	−0.132016	−0.00439320
$904$	49.7365	1.65421
$905$	−3.07431	−0.102193
$906$	−8.00729	−0.266024
$907$	27.7553	0.921600	0.460800	−	0.887504i	$-0.347563\pi$
$907$	27.7553	0.921600	0.460800	+	0.887504i	$0.347563\pi$
$908$	34.5097	1.14525
$909$	27.7460	0.920276
$910$	−10.7793	−0.357331
$911$	18.8121	0.623271	0.311636	−	0.950202i	$-0.399123\pi$
$911$	18.8121	0.623271	0.311636	+	0.950202i	$0.399123\pi$
$912$	0.881643	0.0291941
$913$	−22.5001	−0.744644
$914$	−15.7710	−0.521660
$915$	−1.06346	−0.0351568
$916$	−80.3159	−2.65371
$917$	9.66965	0.319320
$918$	−11.7663	−0.388346
$919$	−42.3315	−1.39639	−0.698193	−	0.715909i	$-0.746012\pi$
$919$	−42.3315	−1.39639	−0.698193	+	0.715909i	$0.746012\pi$
$920$	22.6906	0.748089
$921$	2.76540	0.0911232
$922$	−58.0615	−1.91215
$923$	66.5631	2.19095
$924$	1.07900	0.0354966
$925$	−6.77649	−0.222810
$926$	−11.4193	−0.375260
$927$	4.96667	0.163127
$928$	0.406449	0.0133423
$929$	−21.0475	−0.690545	−0.345273	−	0.938502i	$-0.612214\pi$
$929$	−21.0475	−0.690545	−0.345273	+	0.938502i	$0.612214\pi$
$930$	−2.80735	−0.0920568
$931$	−49.0274	−1.60681
$932$	−51.0923	−1.67358
$933$	2.94120	0.0962907
$934$	36.9857	1.21021
$935$	−15.7077	−0.513698
$936$	63.5957	2.07869
$937$	57.6499	1.88334	0.941670	−	0.336537i	$-0.109256\pi$
$937$	57.6499	1.88334	0.941670	+	0.336537i	$0.109256\pi$
$938$	−16.9830	−0.554514
$939$	−2.57527	−0.0840408
$940$	15.6703	0.511109
$941$	−38.9407	−1.26943	−0.634715	−	0.772747i	$-0.718882\pi$
$941$	−38.9407	−1.26943	−0.634715	+	0.772747i	$0.718882\pi$
$942$	2.21114	0.0720429
$943$	73.4913	2.39321
$944$	−10.9167	−0.355308
$945$	−0.656789	−0.0213653
$946$	8.06252	0.262135
$947$	−16.7373	−0.543890	−0.271945	−	0.962313i	$-0.587667\pi$
$947$	−16.7373	−0.543890	−0.271945	+	0.962313i	$0.587667\pi$
$948$	−2.69518	−0.0875355
$949$	−47.3518	−1.53710
$950$	−17.5166	−0.568314
$951$	−1.28302	−0.0416049
$952$	12.4509	0.403535
$953$	−22.4367	−0.726796	−0.363398	−	0.931634i	$-0.618383\pi$
$953$	−22.4367	−0.726796	−0.363398	+	0.931634i	$0.618383\pi$
$954$	−91.7173	−2.96946
$955$	27.1126	0.877342
$956$	6.23039	0.201505
$957$	0.0385126	0.00124493
$958$	−71.8604	−2.32170
$959$	−5.18160	−0.167322
$960$	−1.95522	−0.0631044
$961$	29.0875	0.938308
$962$	103.598	3.34013
$963$	−2.87851	−0.0927587
$964$	48.6208	1.56597
$965$	−12.1535	−0.391234
$966$	1.78398	0.0573987
$967$	35.2490	1.13353	0.566766	−	0.823879i	$-0.308194\pi$
$967$	35.2490	1.13353	0.566766	+	0.823879i	$0.308194\pi$
$968$	8.61642	0.276942
$969$	−6.38957	−0.205263
$970$	−23.6703	−0.760008
$971$	43.5115	1.39635	0.698176	−	0.715926i	$-0.253995\pi$
$971$	43.5115	1.39635	0.698176	+	0.715926i	$0.253995\pi$
$972$	14.1470	0.453767
$973$	−3.28191	−0.105213
$974$	80.6245	2.58337
$975$	1.02570	0.0328488
$976$	−5.11795	−0.163822
$977$	20.4838	0.655335	0.327667	−	0.944793i	$-0.393737\pi$
$977$	20.4838	0.655335	0.327667	+	0.944793i	$0.393737\pi$
$978$	6.09197	0.194800
$979$	−15.7777	−0.504257
$980$	−22.0964	−0.705842
$981$	40.4807	1.29245
$982$	79.3345	2.53167
$983$	−13.8586	−0.442021	−0.221011	−	0.975271i	$-0.570936\pi$
$983$	−13.8586	−0.442021	−0.221011	+	0.975271i	$0.570936\pi$
$984$	5.32592	0.169784
$985$	11.3826	0.362680
$986$	1.08020	0.0344006
$987$	0.506870	0.0161338
$988$	168.571	5.36297
$989$	8.39127	0.266827
$990$	19.9743	0.634826
$991$	7.61471	0.241889	0.120945	−	0.992659i	$-0.461408\pi$
$991$	7.61471	0.241889	0.120945	+	0.992659i	$0.461408\pi$
$992$	36.8429	1.16976
$993$	4.28676	0.136036
$994$	−16.5716	−0.525618
$995$	−11.3158	−0.358735
$996$	−4.12500	−0.130706
$997$	−39.8119	−1.26085	−0.630427	−	0.776248i	$-0.717120\pi$
$997$	−39.8119	−1.26085	−0.630427	+	0.776248i	$0.717120\pi$
$998$	−31.0409	−0.982583
$999$	6.31225	0.199711

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8005.2.a.e.1.16	✓	126

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8005.2.a.e.1.16	✓	126	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:	\( N \)	\(=\)	\( 8005 = 5 \cdot 1601 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8005.a (trivial)

\(f(q)\)	\(=\)	\(q-2.32335 q^{2} +0.155880 q^{3} +3.39795 q^{4} +1.00000 q^{5} -0.362164 q^{6} +0.705093 q^{7} -3.24794 q^{8} -2.97570 q^{9} +O(q^{10})\)	\(q-2.32335 q^{2} +0.155880 q^{3} +3.39795 q^{4} +1.00000 q^{5} -0.362164 q^{6} +0.705093 q^{7} -3.24794 q^{8} -2.97570 q^{9} -2.32335 q^{10} +2.88914 q^{11} +0.529673 q^{12} +6.58008 q^{13} -1.63818 q^{14} +0.155880 q^{15} +0.750183 q^{16} -5.43683 q^{17} +6.91360 q^{18} +7.53937 q^{19} +3.39795 q^{20} +0.109910 q^{21} -6.71247 q^{22} -6.98618 q^{23} -0.506288 q^{24} +1.00000 q^{25} -15.2878 q^{26} -0.931493 q^{27} +2.39587 q^{28} +0.0855153 q^{29} -0.362164 q^{30} +7.75161 q^{31} +4.75293 q^{32} +0.450359 q^{33} +12.6317 q^{34} +0.705093 q^{35} -10.1113 q^{36} -6.77649 q^{37} -17.5166 q^{38} +1.02570 q^{39} -3.24794 q^{40} -10.5195 q^{41} -0.255359 q^{42} -1.20112 q^{43} +9.81715 q^{44} -2.97570 q^{45} +16.2313 q^{46} +4.61169 q^{47} +0.116939 q^{48} -6.50284 q^{49} -2.32335 q^{50} -0.847493 q^{51} +22.3588 q^{52} -13.2662 q^{53} +2.16418 q^{54} +2.88914 q^{55} -2.29010 q^{56} +1.17524 q^{57} -0.198682 q^{58} -14.5520 q^{59} +0.529673 q^{60} -6.82227 q^{61} -18.0097 q^{62} -2.09815 q^{63} -12.5431 q^{64} +6.58008 q^{65} -1.04634 q^{66} +10.3670 q^{67} -18.4741 q^{68} -1.08901 q^{69} -1.63818 q^{70} +10.1159 q^{71} +9.66489 q^{72} -7.19623 q^{73} +15.7442 q^{74} +0.155880 q^{75} +25.6184 q^{76} +2.03711 q^{77} -2.38307 q^{78} -5.08839 q^{79} +0.750183 q^{80} +8.78190 q^{81} +24.4406 q^{82} -7.78782 q^{83} +0.373469 q^{84} -5.43683 q^{85} +2.79063 q^{86} +0.0133301 q^{87} -9.38373 q^{88} -5.46104 q^{89} +6.91360 q^{90} +4.63957 q^{91} -23.7387 q^{92} +1.20832 q^{93} -10.7146 q^{94} +7.53937 q^{95} +0.740888 q^{96} +10.1880 q^{97} +15.1084 q^{98} -8.59721 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 15 q^{2} - 46 q^{3} + 119 q^{4} + 126 q^{5} - 10 q^{6} - 60 q^{7} - 57 q^{8} + 112 q^{9}+O(q^{10}) \) 126 * q - 15 * q^2 - 46 * q^3 + 119 * q^4 + 126 * q^5 - 10 * q^6 - 60 * q^7 - 57 * q^8 + 112 * q^9	\( 126 q - 15 q^{2} - 46 q^{3} + 119 q^{4} + 126 q^{5} - 10 q^{6} - 60 q^{7} - 57 q^{8} + 112 q^{9} - 15 q^{10} - 35 q^{11} - 86 q^{12} - 36 q^{13} - 31 q^{14} - 46 q^{15} + 101 q^{16} - 62 q^{17} - 45 q^{18} - 29 q^{19} + 119 q^{20} - 16 q^{21} - 67 q^{22} - 107 q^{23} - 9 q^{24} + 126 q^{25} - 53 q^{26} - 181 q^{27} - 100 q^{28} - 39 q^{29} - 10 q^{30} - 29 q^{31} - 99 q^{32} - 72 q^{33} - 18 q^{34} - 60 q^{35} + 93 q^{36} - 72 q^{37} - 93 q^{38} - 8 q^{39} - 57 q^{40} - 28 q^{41} - 22 q^{42} - 103 q^{43} - 47 q^{44} + 112 q^{45} + q^{46} - 130 q^{47} - 134 q^{48} + 116 q^{49} - 15 q^{50} - 46 q^{51} - 117 q^{52} - 103 q^{53} + 11 q^{54} - 35 q^{55} - 84 q^{56} - 70 q^{57} - 77 q^{58} - 219 q^{59} - 86 q^{60} - 4 q^{61} - 77 q^{62} - 145 q^{63} + 51 q^{64} - 36 q^{65} + 16 q^{66} - 150 q^{67} - 130 q^{68} - 21 q^{69} - 31 q^{70} - 92 q^{71} - 115 q^{72} - 79 q^{73} - 57 q^{74} - 46 q^{75} - 38 q^{76} - 75 q^{77} - 23 q^{78} - 20 q^{79} + 101 q^{80} + 142 q^{81} - 63 q^{82} - 243 q^{83} - 2 q^{84} - 62 q^{85} - 14 q^{86} - 107 q^{87} - 121 q^{88} - 84 q^{89} - 45 q^{90} - 82 q^{91} - 228 q^{92} - 149 q^{93} - 29 q^{95} + 38 q^{96} - 85 q^{97} - 48 q^{98} - 33 q^{99}+O(q^{100}) \) 126 * q - 15 * q^2 - 46 * q^3 + 119 * q^4 + 126 * q^5 - 10 * q^6 - 60 * q^7 - 57 * q^8 + 112 * q^9 - 15 * q^10 - 35 * q^11 - 86 * q^12 - 36 * q^13 - 31 * q^14 - 46 * q^15 + 101 * q^16 - 62 * q^17 - 45 * q^18 - 29 * q^19 + 119 * q^20 - 16 * q^21 - 67 * q^22 - 107 * q^23 - 9 * q^24 + 126 * q^25 - 53 * q^26 - 181 * q^27 - 100 * q^28 - 39 * q^29 - 10 * q^30 - 29 * q^31 - 99 * q^32 - 72 * q^33 - 18 * q^34 - 60 * q^35 + 93 * q^36 - 72 * q^37 - 93 * q^38 - 8 * q^39 - 57 * q^40 - 28 * q^41 - 22 * q^42 - 103 * q^43 - 47 * q^44 + 112 * q^45 + q^46 - 130 * q^47 - 134 * q^48 + 116 * q^49 - 15 * q^50 - 46 * q^51 - 117 * q^52 - 103 * q^53 + 11 * q^54 - 35 * q^55 - 84 * q^56 - 70 * q^57 - 77 * q^58 - 219 * q^59 - 86 * q^60 - 4 * q^61 - 77 * q^62 - 145 * q^63 + 51 * q^64 - 36 * q^65 + 16 * q^66 - 150 * q^67 - 130 * q^68 - 21 * q^69 - 31 * q^70 - 92 * q^71 - 115 * q^72 - 79 * q^73 - 57 * q^74 - 46 * q^75 - 38 * q^76 - 75 * q^77 - 23 * q^78 - 20 * q^79 + 101 * q^80 + 142 * q^81 - 63 * q^82 - 243 * q^83 - 2 * q^84 - 62 * q^85 - 14 * q^86 - 107 * q^87 - 121 * q^88 - 84 * q^89 - 45 * q^90 - 82 * q^91 - 228 * q^92 - 149 * q^93 - 29 * q^95 + 38 * q^96 - 85 * q^97 - 48 * q^98 - 33 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more