LMFDB - Embedded newform 8039.2.a.a.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8039,2,Mod(1,8039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8039, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8039 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8039.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:	$64.1917381849$
Analytic rank:	$1$
Dimension:	$279$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.51220 q^{2} -0.224247 q^{3} +4.31117 q^{4} +3.23846 q^{5} +0.563355 q^{6} -3.25585 q^{7} -5.80612 q^{8} -2.94971 q^{9} +O(q^{10})$	\(q-2.51220 q^{2} -0.224247 q^{3} +4.31117 q^{4} +3.23846 q^{5} +0.563355 q^{6} -3.25585 q^{7} -5.80612 q^{8} -2.94971 q^{9} -8.13568 q^{10} +0.115856 q^{11} -0.966767 q^{12} -4.71479 q^{13} +8.17936 q^{14} -0.726217 q^{15} +5.96381 q^{16} +3.05617 q^{17} +7.41028 q^{18} +0.801664 q^{19} +13.9616 q^{20} +0.730116 q^{21} -0.291054 q^{22} +2.75505 q^{23} +1.30201 q^{24} +5.48766 q^{25} +11.8445 q^{26} +1.33421 q^{27} -14.0365 q^{28} -6.15333 q^{29} +1.82440 q^{30} +3.10283 q^{31} -3.37008 q^{32} -0.0259804 q^{33} -7.67772 q^{34} -10.5440 q^{35} -12.7167 q^{36} +8.79097 q^{37} -2.01394 q^{38} +1.05728 q^{39} -18.8029 q^{40} -1.39140 q^{41} -1.83420 q^{42} -1.64550 q^{43} +0.499475 q^{44} -9.55254 q^{45} -6.92125 q^{46} -8.38187 q^{47} -1.33737 q^{48} +3.60057 q^{49} -13.7861 q^{50} -0.685338 q^{51} -20.3262 q^{52} -8.36596 q^{53} -3.35180 q^{54} +0.375196 q^{55} +18.9039 q^{56} -0.179771 q^{57} +15.4584 q^{58} +8.04061 q^{59} -3.13084 q^{60} +7.86856 q^{61} -7.79494 q^{62} +9.60383 q^{63} -3.46130 q^{64} -15.2687 q^{65} +0.0652681 q^{66} +12.0288 q^{67} +13.1757 q^{68} -0.617812 q^{69} +26.4886 q^{70} +8.81535 q^{71} +17.1264 q^{72} +11.7812 q^{73} -22.0847 q^{74} -1.23059 q^{75} +3.45610 q^{76} -0.377210 q^{77} -2.65610 q^{78} -6.71147 q^{79} +19.3136 q^{80} +8.54995 q^{81} +3.49548 q^{82} +4.61201 q^{83} +3.14765 q^{84} +9.89730 q^{85} +4.13382 q^{86} +1.37987 q^{87} -0.672674 q^{88} -6.70388 q^{89} +23.9979 q^{90} +15.3506 q^{91} +11.8775 q^{92} -0.695801 q^{93} +21.0570 q^{94} +2.59616 q^{95} +0.755731 q^{96} +9.35729 q^{97} -9.04538 q^{98} -0.341742 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 279 q - 13 q^{2} - 12 q^{3} + 227 q^{4} - 20 q^{5} - 40 q^{6} - 57 q^{7} - 39 q^{8} + 175 q^{9}+O(q^{10}) $ 279 * q - 13 * q^2 - 12 * q^3 + 227 * q^4 - 20 * q^5 - 40 * q^6 - 57 * q^7 - 39 * q^8 + 175 * q^9	\( 279 q - 13 q^{2} - 12 q^{3} + 227 q^{4} - 20 q^{5} - 40 q^{6} - 57 q^{7} - 39 q^{8} + 175 q^{9} - 42 q^{10} - 53 q^{11} - 36 q^{12} - 75 q^{13} - 31 q^{14} - 60 q^{15} + 127 q^{16} - 55 q^{17} - 57 q^{18} - 113 q^{19} - 43 q^{20} - 103 q^{21} - 73 q^{22} - 30 q^{23} - 106 q^{24} + 75 q^{25} - 42 q^{26} - 45 q^{27} - 146 q^{28} - 92 q^{29} - 76 q^{30} - 84 q^{31} - 71 q^{32} - 117 q^{33} - 106 q^{34} - 49 q^{35} + 67 q^{36} - 123 q^{37} - 21 q^{38} - 92 q^{39} - 97 q^{40} - 116 q^{41} - 19 q^{42} - 126 q^{43} - 131 q^{44} - 85 q^{45} - 183 q^{46} - 42 q^{47} - 47 q^{48} - 22 q^{49} - 64 q^{50} - 90 q^{51} - 158 q^{52} - 60 q^{53} - 117 q^{54} - 99 q^{55} - 65 q^{56} - 182 q^{57} - 93 q^{58} - 58 q^{59} - 141 q^{60} - 217 q^{61} - 16 q^{62} - 141 q^{63} - 47 q^{64} - 197 q^{65} - 53 q^{66} - 147 q^{67} - 90 q^{68} - 103 q^{69} - 118 q^{70} - 78 q^{71} - 135 q^{72} - 282 q^{73} - 98 q^{74} - 53 q^{75} - 296 q^{76} - 53 q^{77} - 27 q^{78} - 153 q^{79} - 52 q^{80} - 89 q^{81} - 81 q^{82} - 54 q^{83} - 164 q^{84} - 303 q^{85} - 82 q^{86} - 29 q^{87} - 203 q^{88} - 185 q^{89} - 56 q^{90} - 163 q^{91} - 66 q^{92} - 156 q^{93} - 134 q^{94} - 69 q^{95} - 189 q^{96} - 212 q^{97} - 13 q^{98} - 181 q^{99}+O(q^{100}) \) 279 * q - 13 * q^2 - 12 * q^3 + 227 * q^4 - 20 * q^5 - 40 * q^6 - 57 * q^7 - 39 * q^8 + 175 * q^9 - 42 * q^10 - 53 * q^11 - 36 * q^12 - 75 * q^13 - 31 * q^14 - 60 * q^15 + 127 * q^16 - 55 * q^17 - 57 * q^18 - 113 * q^19 - 43 * q^20 - 103 * q^21 - 73 * q^22 - 30 * q^23 - 106 * q^24 + 75 * q^25 - 42 * q^26 - 45 * q^27 - 146 * q^28 - 92 * q^29 - 76 * q^30 - 84 * q^31 - 71 * q^32 - 117 * q^33 - 106 * q^34 - 49 * q^35 + 67 * q^36 - 123 * q^37 - 21 * q^38 - 92 * q^39 - 97 * q^40 - 116 * q^41 - 19 * q^42 - 126 * q^43 - 131 * q^44 - 85 * q^45 - 183 * q^46 - 42 * q^47 - 47 * q^48 - 22 * q^49 - 64 * q^50 - 90 * q^51 - 158 * q^52 - 60 * q^53 - 117 * q^54 - 99 * q^55 - 65 * q^56 - 182 * q^57 - 93 * q^58 - 58 * q^59 - 141 * q^60 - 217 * q^61 - 16 * q^62 - 141 * q^63 - 47 * q^64 - 197 * q^65 - 53 * q^66 - 147 * q^67 - 90 * q^68 - 103 * q^69 - 118 * q^70 - 78 * q^71 - 135 * q^72 - 282 * q^73 - 98 * q^74 - 53 * q^75 - 296 * q^76 - 53 * q^77 - 27 * q^78 - 153 * q^79 - 52 * q^80 - 89 * q^81 - 81 * q^82 - 54 * q^83 - 164 * q^84 - 303 * q^85 - 82 * q^86 - 29 * q^87 - 203 * q^88 - 185 * q^89 - 56 * q^90 - 163 * q^91 - 66 * q^92 - 156 * q^93 - 134 * q^94 - 69 * q^95 - 189 * q^96 - 212 * q^97 - 13 * q^98 - 181 * 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 $

$2$	−2.51220	−1.77640	−0.888198	−	0.459461i	$-0.848043\pi$
$2$	−2.51220	−1.77640	−0.888198	+	0.459461i	$0.848043\pi$
$3$	−0.224247	−0.129469	−0.0647346	−	0.997903i	$-0.520620\pi$
$3$	−0.224247	−0.129469	−0.0647346	+	0.997903i	$0.520620\pi$
$4$	4.31117	2.15558
$5$	3.23846	1.44829	0.724143	−	0.689650i	$-0.242236\pi$
$5$	3.23846	1.44829	0.724143	+	0.689650i	$0.242236\pi$
$6$	0.563355	0.229989
$7$	−3.25585	−1.23060	−0.615298	−	0.788294i	$-0.710964\pi$
$7$	−3.25585	−1.23060	−0.615298	+	0.788294i	$0.710964\pi$
$8$	−5.80612	−2.05277
$9$	−2.94971	−0.983238
$10$	−8.13568	−2.57273
$11$	0.115856	0.0349319	0.0174660	−	0.999847i	$-0.494440\pi$
$11$	0.115856	0.0349319	0.0174660	+	0.999847i	$0.494440\pi$
$12$	−0.966767	−0.279082
$13$	−4.71479	−1.30765	−0.653823	−	0.756647i	$-0.726836\pi$
$13$	−4.71479	−1.30765	−0.653823	+	0.756647i	$0.726836\pi$
$14$	8.17936	2.18603
$15$	−0.726217	−0.187508
$16$	5.96381	1.49095
$17$	3.05617	0.741230	0.370615	−	0.928787i	$-0.379147\pi$
$17$	3.05617	0.741230	0.370615	+	0.928787i	$0.379147\pi$
$18$	7.41028	1.74662
$19$	0.801664	0.183914	0.0919571	−	0.995763i	$-0.470688\pi$
$19$	0.801664	0.183914	0.0919571	+	0.995763i	$0.470688\pi$
$20$	13.9616	3.12190
$21$	0.730116	0.159324
$22$	−0.291054	−0.0620529
$23$	2.75505	0.574468	0.287234	−	0.957860i	$-0.407264\pi$
$23$	2.75505	0.574468	0.287234	+	0.957860i	$0.407264\pi$
$24$	1.30201	0.265771
$25$	5.48766	1.09753
$26$	11.8445	2.32290
$27$	1.33421	0.256768
$28$	−14.0365	−2.65265
$29$	−6.15333	−1.14265	−0.571323	−	0.820726i	$-0.693569\pi$
$29$	−6.15333	−1.14265	−0.571323	+	0.820726i	$0.693569\pi$
$30$	1.82440	0.333089
$31$	3.10283	0.557285	0.278643	−	0.960395i	$-0.410116\pi$
$31$	3.10283	0.557285	0.278643	+	0.960395i	$0.410116\pi$
$32$	−3.37008	−0.595752
$33$	−0.0259804	−0.00452261
$34$	−7.67772	−1.31672
$35$	−10.5440	−1.78226
$36$	−12.7167	−2.11945
$37$	8.79097	1.44523	0.722613	−	0.691252i	$-0.242941\pi$
$37$	8.79097	1.44523	0.722613	+	0.691252i	$0.242941\pi$
$38$	−2.01394	−0.326705
$39$	1.05728	0.169300
$40$	−18.8029	−2.97300
$41$	−1.39140	−0.217300	−0.108650	−	0.994080i	$-0.534653\pi$
$41$	−1.39140	−0.217300	−0.108650	+	0.994080i	$0.534653\pi$
$42$	−1.83420	−0.283023
$43$	−1.64550	−0.250936	−0.125468	−	0.992098i	$-0.540043\pi$
$43$	−1.64550	−0.250936	−0.125468	+	0.992098i	$0.540043\pi$
$44$	0.499475	0.0752987
$45$	−9.55254	−1.42401
$46$	−6.92125	−1.02048
$47$	−8.38187	−1.22262	−0.611311	−	0.791391i	$-0.709358\pi$
$47$	−8.38187	−1.22262	−0.611311	+	0.791391i	$0.709358\pi$
$48$	−1.33737	−0.193033
$49$	3.60057	0.514368
$50$	−13.7861	−1.94965
$51$	−0.685338	−0.0959665
$52$	−20.3262	−2.81874
$53$	−8.36596	−1.14915	−0.574577	−	0.818451i	$-0.694833\pi$
$53$	−8.36596	−1.14915	−0.574577	+	0.818451i	$0.694833\pi$
$54$	−3.35180	−0.456122
$55$	0.375196	0.0505914
$56$	18.9039	2.52613
$57$	−0.179771	−0.0238112
$58$	15.4584	2.02979
$59$	8.04061	1.04680	0.523399	−	0.852088i	$-0.324664\pi$
$59$	8.04061	1.04680	0.523399	+	0.852088i	$0.324664\pi$
$60$	−3.13084	−0.404190
$61$	7.86856	1.00747	0.503733	−	0.863860i	$-0.331960\pi$
$61$	7.86856	1.00747	0.503733	+	0.863860i	$0.331960\pi$
$62$	−7.79494	−0.989959
$63$	9.60383	1.20997
$64$	−3.46130	−0.432663
$65$	−15.2687	−1.89385
$66$	0.0652681	0.00803394
$67$	12.0288	1.46955	0.734777	−	0.678308i	$-0.237287\pi$
$67$	12.0288	1.46955	0.734777	+	0.678308i	$0.237287\pi$
$68$	13.1757	1.59778
$69$	−0.617812	−0.0743759
$70$	26.4886	3.16599
$71$	8.81535	1.04619	0.523095	−	0.852275i	$-0.324777\pi$
$71$	8.81535	1.04619	0.523095	+	0.852275i	$0.324777\pi$
$72$	17.1264	2.01836
$73$	11.7812	1.37888	0.689440	−	0.724342i	$-0.257856\pi$
$73$	11.7812	1.37888	0.689440	+	0.724342i	$0.257856\pi$
$74$	−22.0847	−2.56730
$75$	−1.23059	−0.142096
$76$	3.45610	0.396442
$77$	−0.377210	−0.0429871
$78$	−2.65610	−0.300744
$79$	−6.71147	−0.755099	−0.377550	−	0.925989i	$-0.623233\pi$
$79$	−6.71147	−0.755099	−0.377550	+	0.925989i	$0.623233\pi$
$80$	19.3136	2.15933
$81$	8.54995	0.949994
$82$	3.49548	0.386012
$83$	4.61201	0.506234	0.253117	−	0.967436i	$-0.418544\pi$
$83$	4.61201	0.506234	0.253117	+	0.967436i	$0.418544\pi$
$84$	3.14765	0.343437
$85$	9.89730	1.07351
$86$	4.13382	0.445761
$87$	1.37987	0.147937
$88$	−0.672674	−0.0717073
$89$	−6.70388	−0.710610	−0.355305	−	0.934750i	$-0.615623\pi$
$89$	−6.70388	−0.710610	−0.355305	+	0.934750i	$0.615623\pi$
$90$	23.9979	2.52960
$91$	15.3506	1.60919
$92$	11.8775	1.23831
$93$	−0.695801	−0.0721513
$94$	21.0570	2.17186
$95$	2.59616	0.266360
$96$	0.755731	0.0771315
$97$	9.35729	0.950089	0.475044	−	0.879962i	$-0.342432\pi$
$97$	9.35729	0.950089	0.475044	+	0.879962i	$0.342432\pi$
$98$	−9.04538	−0.913721
$99$	−0.341742	−0.0343464
$100$	23.6582	2.36582
$101$	−13.1477	−1.30825	−0.654123	−	0.756389i	$-0.726962\pi$
$101$	−13.1477	−1.30825	−0.654123	+	0.756389i	$0.726962\pi$
$102$	1.72171	0.170474
$103$	5.61329	0.553094	0.276547	−	0.961000i	$-0.410810\pi$
$103$	5.61329	0.553094	0.276547	+	0.961000i	$0.410810\pi$
$104$	27.3746	2.68430
$105$	2.36445	0.230747
$106$	21.0170	2.04135
$107$	0.694376	0.0671279	0.0335639	−	0.999437i	$-0.489314\pi$
$107$	0.694376	0.0671279	0.0335639	+	0.999437i	$0.489314\pi$
$108$	5.75199	0.553485
$109$	−6.85837	−0.656913	−0.328457	−	0.944519i	$-0.606528\pi$
$109$	−6.85837	−0.656913	−0.328457	+	0.944519i	$0.606528\pi$
$110$	−0.942568	−0.0898704
$111$	−1.97135	−0.187112
$112$	−19.4173	−1.83476
$113$	17.7861	1.67318	0.836590	−	0.547830i	$-0.184546\pi$
$113$	17.7861	1.67318	0.836590	+	0.547830i	$0.184546\pi$
$114$	0.451621	0.0422982
$115$	8.92213	0.831993
$116$	−26.5280	−2.46307
$117$	13.9073	1.28573
$118$	−20.1996	−1.85953
$119$	−9.95044	−0.912155
$120$	4.21650	0.384912
$121$	−10.9866	−0.998780
$122$	−19.7674	−1.78966
$123$	0.312018	0.0281337
$124$	13.3768	1.20127
$125$	1.57926	0.141253
$126$	−24.1268	−2.14938
$127$	−13.8479	−1.22880	−0.614401	−	0.788994i	$-0.710602\pi$
$127$	−13.8479	−1.22880	−0.614401	+	0.788994i	$0.710602\pi$
$128$	15.4357	1.36433
$129$	0.368998	0.0324884
$130$	38.3580	3.36422
$131$	−13.2733	−1.15970	−0.579848	−	0.814725i	$-0.696888\pi$
$131$	−13.2733	−1.15970	−0.579848	+	0.814725i	$0.696888\pi$
$132$	−0.112006	−0.00974886
$133$	−2.61010	−0.226324
$134$	−30.2189	−2.61051
$135$	4.32078	0.371874
$136$	−17.7445	−1.52158
$137$	−1.01381	−0.0866159	−0.0433080	−	0.999062i	$-0.513790\pi$
$137$	−1.01381	−0.0866159	−0.0433080	+	0.999062i	$0.513790\pi$
$138$	1.55207	0.132121
$139$	−20.6942	−1.75526	−0.877628	−	0.479342i	$-0.840875\pi$
$139$	−20.6942	−1.75526	−0.877628	+	0.479342i	$0.840875\pi$
$140$	−45.4568	−3.84180
$141$	1.87961	0.158292
$142$	−22.1459	−1.85845
$143$	−0.546237	−0.0456786
$144$	−17.5915	−1.46596
$145$	−19.9274	−1.65488
$146$	−29.5967	−2.44944
$147$	−0.807419	−0.0665948
$148$	37.8993	3.11531
$149$	18.4221	1.50920	0.754600	−	0.656185i	$-0.227831\pi$
$149$	18.4221	1.50920	0.754600	+	0.656185i	$0.227831\pi$
$150$	3.09150	0.252420
$151$	−6.36507	−0.517982	−0.258991	−	0.965880i	$-0.583390\pi$
$151$	−6.36507	−0.517982	−0.258991	+	0.965880i	$0.583390\pi$
$152$	−4.65455	−0.377534
$153$	−9.01483	−0.728806
$154$	0.947629	0.0763621
$155$	10.0484	0.807108
$156$	4.55810	0.364940
$157$	−9.29533	−0.741848	−0.370924	−	0.928663i	$-0.620959\pi$
$157$	−9.29533	−0.741848	−0.370924	+	0.928663i	$0.620959\pi$
$158$	16.8606	1.34136
$159$	1.87604	0.148780
$160$	−10.9139	−0.862819
$161$	−8.97004	−0.706938
$162$	−21.4792	−1.68757
$163$	2.08463	0.163281	0.0816403	−	0.996662i	$-0.473984\pi$
$163$	2.08463	0.163281	0.0816403	+	0.996662i	$0.473984\pi$
$164$	−5.99856	−0.468409
$165$	−0.0841366	−0.00655003
$166$	−11.5863	−0.899273
$167$	5.26428	0.407362	0.203681	−	0.979037i	$-0.434709\pi$
$167$	5.26428	0.407362	0.203681	+	0.979037i	$0.434709\pi$
$168$	−4.23914	−0.327057
$169$	9.22921	0.709939
$170$	−24.8640	−1.90698
$171$	−2.36468	−0.180831
$172$	−7.09400	−0.540913
$173$	−0.416548	−0.0316695	−0.0158348	−	0.999875i	$-0.505041\pi$
$173$	−0.416548	−0.0316695	−0.0158348	+	0.999875i	$0.505041\pi$
$174$	−3.46651	−0.262795
$175$	−17.8670	−1.35062
$176$	0.690944	0.0520819
$177$	−1.80308	−0.135528
$178$	16.8415	1.26232
$179$	−25.5694	−1.91114	−0.955572	−	0.294759i	$-0.904761\pi$
$179$	−25.5694	−1.91114	−0.955572	+	0.294759i	$0.904761\pi$
$180$	−41.1826	−3.06957
$181$	20.7952	1.54569	0.772847	−	0.634592i	$-0.218832\pi$
$181$	20.7952	1.54569	0.772847	+	0.634592i	$0.218832\pi$
$182$	−38.5639	−2.85855
$183$	−1.76450	−0.130436
$184$	−15.9961	−1.17925
$185$	28.4693	2.09310
$186$	1.74799	0.128169
$187$	0.354076	0.0258926
$188$	−36.1356	−2.63546
$189$	−4.34398	−0.315978
$190$	−6.52208	−0.473161
$191$	−1.03012	−0.0745366	−0.0372683	−	0.999305i	$-0.511866\pi$
$191$	−1.03012	−0.0745366	−0.0372683	+	0.999305i	$0.511866\pi$
$192$	0.776187	0.0560165
$193$	6.65049	0.478713	0.239356	−	0.970932i	$-0.423064\pi$
$193$	6.65049	0.478713	0.239356	+	0.970932i	$0.423064\pi$
$194$	−23.5074	−1.68773
$195$	3.42396	0.245195
$196$	15.5227	1.10876
$197$	18.9370	1.34921	0.674603	−	0.738181i	$-0.264315\pi$
$197$	18.9370	1.34921	0.674603	+	0.738181i	$0.264315\pi$
$198$	0.858526	0.0610128
$199$	7.76328	0.550324	0.275162	−	0.961398i	$-0.411269\pi$
$199$	7.76328	0.550324	0.275162	+	0.961398i	$0.411269\pi$
$200$	−31.8620	−2.25298
$201$	−2.69743	−0.190262
$202$	33.0297	2.32396
$203$	20.0343	1.40614
$204$	−2.95460	−0.206864
$205$	−4.50601	−0.314713
$206$	−14.1017	−0.982514
$207$	−8.12661	−0.564838
$208$	−28.1181	−1.94964
$209$	0.0928776	0.00642448
$210$	−5.93999	−0.409898
$211$	−0.655600	−0.0451333	−0.0225667	−	0.999745i	$-0.507184\pi$
$211$	−0.655600	−0.0451333	−0.0225667	+	0.999745i	$0.507184\pi$
$212$	−36.0671	−2.47710
$213$	−1.97682	−0.135449
$214$	−1.74441	−0.119246
$215$	−5.32888	−0.363427
$216$	−7.74656	−0.527087
$217$	−10.1024	−0.685793
$218$	17.2296	1.16694
$219$	−2.64189	−0.178523
$220$	1.61753	0.109054
$221$	−14.4092	−0.969267
$222$	4.95243	0.332386
$223$	−7.11420	−0.476402	−0.238201	−	0.971216i	$-0.576558\pi$
$223$	−7.11420	−0.476402	−0.238201	+	0.971216i	$0.576558\pi$
$224$	10.9725	0.733130
$225$	−16.1870	−1.07913
$226$	−44.6824	−2.97223
$227$	−6.18866	−0.410756	−0.205378	−	0.978683i	$-0.565842\pi$
$227$	−6.18866	−0.410756	−0.205378	+	0.978683i	$0.565842\pi$
$228$	−0.775022	−0.0513271
$229$	−23.3825	−1.54516	−0.772579	−	0.634919i	$-0.781033\pi$
$229$	−23.3825	−1.54516	−0.772579	+	0.634919i	$0.781033\pi$
$230$	−22.4142	−1.47795
$231$	0.0845884	0.00556551
$232$	35.7270	2.34559
$233$	−17.8692	−1.17065	−0.585324	−	0.810799i	$-0.699033\pi$
$233$	−17.8692	−1.17065	−0.585324	+	0.810799i	$0.699033\pi$
$234$	−34.9379	−2.28396
$235$	−27.1444	−1.77071
$236$	34.6644	2.25646
$237$	1.50503	0.0977621
$238$	24.9975	1.62035
$239$	4.48759	0.290278	0.145139	−	0.989411i	$-0.453637\pi$
$239$	4.48759	0.290278	0.145139	+	0.989411i	$0.453637\pi$
$240$	−4.33102	−0.279566
$241$	−20.9811	−1.35151	−0.675757	−	0.737124i	$-0.736183\pi$
$241$	−20.9811	−1.35151	−0.675757	+	0.737124i	$0.736183\pi$
$242$	27.6005	1.77423
$243$	−5.91992	−0.379763
$244$	33.9226	2.17167
$245$	11.6603	0.744952
$246$	−0.783852	−0.0499766
$247$	−3.77967	−0.240495
$248$	−18.0154	−1.14398
$249$	−1.03423	−0.0655418
$250$	−3.96741	−0.250921
$251$	−13.9960	−0.883417	−0.441709	−	0.897159i	$-0.645627\pi$
$251$	−13.9960	−0.883417	−0.441709	+	0.897159i	$0.645627\pi$
$252$	41.4037	2.60819
$253$	0.319189	0.0200673
$254$	34.7887	2.18284
$255$	−2.21944	−0.138987
$256$	−31.8549	−1.99093
$257$	−25.6984	−1.60302	−0.801511	−	0.597980i	$-0.795970\pi$
$257$	−25.6984	−1.60302	−0.801511	+	0.597980i	$0.795970\pi$
$258$	−0.926997	−0.0577123
$259$	−28.6221	−1.77849
$260$	−65.8258	−4.08234
$261$	18.1506	1.12349
$262$	33.3453	2.06008
$263$	−4.42472	−0.272840	−0.136420	−	0.990651i	$-0.543560\pi$
$263$	−4.42472	−0.272840	−0.136420	+	0.990651i	$0.543560\pi$
$264$	0.150845	0.00928389
$265$	−27.0929	−1.66430
$266$	6.55710	0.402041
$267$	1.50333	0.0920021
$268$	51.8583	3.16775
$269$	4.48356	0.273367	0.136684	−	0.990615i	$-0.456356\pi$
$269$	4.48356	0.273367	0.136684	+	0.990615i	$0.456356\pi$
$270$	−10.8547	−0.660595
$271$	20.9183	1.27069	0.635347	−	0.772227i	$-0.280857\pi$
$271$	20.9183	1.27069	0.635347	+	0.772227i	$0.280857\pi$
$272$	18.2264	1.10514
$273$	−3.44234	−0.208340
$274$	2.54691	0.153864
$275$	0.635778	0.0383389
$276$	−2.66349	−0.160323
$277$	−20.4897	−1.23111	−0.615554	−	0.788095i	$-0.711068\pi$
$277$	−20.4897	−1.23111	−0.615554	+	0.788095i	$0.711068\pi$
$278$	51.9879	3.11803
$279$	−9.15246	−0.547944
$280$	61.2195	3.65856
$281$	−17.3283	−1.03372	−0.516860	−	0.856070i	$-0.672899\pi$
$281$	−17.3283	−1.03372	−0.516860	+	0.856070i	$0.672899\pi$
$282$	−4.72197	−0.281189
$283$	3.40999	0.202703	0.101352	−	0.994851i	$-0.467683\pi$
$283$	3.40999	0.202703	0.101352	+	0.994851i	$0.467683\pi$
$284$	38.0044	2.25515
$285$	−0.582182	−0.0344855
$286$	1.37226	0.0811433
$287$	4.53020	0.267409
$288$	9.94077	0.585766
$289$	−7.65982	−0.450578
$290$	50.0616	2.93972
$291$	−2.09835	−0.123007
$292$	50.7905	2.97229
$293$	−17.7134	−1.03483	−0.517413	−	0.855736i	$-0.673105\pi$
$293$	−17.7134	−1.03483	−0.517413	+	0.855736i	$0.673105\pi$
$294$	2.02840	0.118299
$295$	26.0392	1.51606
$296$	−51.0414	−2.96672
$297$	0.154576	0.00896941
$298$	−46.2802	−2.68094
$299$	−12.9895	−0.751201
$300$	−5.30528	−0.306301
$301$	5.35749	0.308801
$302$	15.9903	0.920141
$303$	2.94834	0.169377
$304$	4.78097	0.274208
$305$	25.4820	1.45910
$306$	22.6471	1.29465
$307$	12.7845	0.729650	0.364825	−	0.931076i	$-0.381129\pi$
$307$	12.7845	0.729650	0.364825	+	0.931076i	$0.381129\pi$
$308$	−1.62622	−0.0926623
$309$	−1.25876	−0.0716086
$310$	−25.2437	−1.43374
$311$	25.7128	1.45804	0.729019	−	0.684494i	$-0.239977\pi$
$311$	25.7128	1.45804	0.729019	+	0.684494i	$0.239977\pi$
$312$	−6.13868	−0.347534
$313$	−3.02039	−0.170722	−0.0853612	−	0.996350i	$-0.527204\pi$
$313$	−3.02039	−0.170722	−0.0853612	+	0.996350i	$0.527204\pi$
$314$	23.3518	1.31782
$315$	31.1017	1.75238
$316$	−28.9343	−1.62768
$317$	−6.52940	−0.366728	−0.183364	−	0.983045i	$-0.558699\pi$
$317$	−6.52940	−0.366728	−0.183364	+	0.983045i	$0.558699\pi$
$318$	−4.71300	−0.264292
$319$	−0.712901	−0.0399148
$320$	−11.2093	−0.626619
$321$	−0.155712	−0.00869099
$322$	22.5346	1.25580
$323$	2.45002	0.136323
$324$	36.8602	2.04779
$325$	−25.8731	−1.43518
$326$	−5.23701	−0.290051
$327$	1.53797	0.0850500
$328$	8.07864	0.446068
$329$	27.2901	1.50455
$330$	0.211368	0.0116354
$331$	−34.8718	−1.91673	−0.958365	−	0.285547i	$-0.907825\pi$
$331$	−34.8718	−1.91673	−0.958365	+	0.285547i	$0.907825\pi$
$332$	19.8832	1.09123
$333$	−25.9308	−1.42100
$334$	−13.2249	−0.723636
$335$	38.9549	2.12834
$336$	4.35428	0.237545
$337$	1.65313	0.0900516	0.0450258	−	0.998986i	$-0.485663\pi$
$337$	1.65313	0.0900516	0.0450258	+	0.998986i	$0.485663\pi$
$338$	−23.1857	−1.26113
$339$	−3.98849	−0.216625
$340$	42.6689	2.31405
$341$	0.359482	0.0194670
$342$	5.94055	0.321228
$343$	11.0680	0.597617
$344$	9.55394	0.515114
$345$	−2.00076	−0.107717
$346$	1.04645	0.0562576
$347$	13.5993	0.730050	0.365025	−	0.930998i	$-0.381060\pi$
$347$	13.5993	0.730050	0.365025	+	0.930998i	$0.381060\pi$
$348$	5.94884	0.318891
$349$	−4.31850	−0.231164	−0.115582	−	0.993298i	$-0.536873\pi$
$349$	−4.31850	−0.231164	−0.115582	+	0.993298i	$0.536873\pi$
$350$	44.8855	2.39923
$351$	−6.29050	−0.335762
$352$	−0.390444	−0.0208108
$353$	31.7613	1.69048	0.845241	−	0.534386i	$-0.179457\pi$
$353$	31.7613	1.69048	0.845241	+	0.534386i	$0.179457\pi$
$354$	4.52971	0.240752
$355$	28.5482	1.51518
$356$	−28.9015	−1.53178
$357$	2.23136	0.118096
$358$	64.2354	3.39495
$359$	−24.3434	−1.28479	−0.642397	−	0.766372i	$-0.722060\pi$
$359$	−24.3434	−1.28479	−0.642397	+	0.766372i	$0.722060\pi$
$360$	55.4632	2.92317
$361$	−18.3573	−0.966176
$362$	−52.2418	−2.74577
$363$	2.46371	0.129311
$364$	66.1792	3.46873
$365$	38.1529	1.99701
$366$	4.43279	0.231705
$367$	−22.1517	−1.15631	−0.578156	−	0.815926i	$-0.696227\pi$
$367$	−22.1517	−1.15631	−0.578156	+	0.815926i	$0.696227\pi$
$368$	16.4306	0.856505
$369$	4.10424	0.213658
$370$	−71.5206	−3.71818
$371$	27.2383	1.41414
$372$	−2.99972	−0.155528
$373$	−27.2088	−1.40882	−0.704408	−	0.709795i	$-0.748788\pi$
$373$	−27.2088	−1.40882	−0.704408	+	0.709795i	$0.748788\pi$
$374$	−0.889511	−0.0459955
$375$	−0.354144	−0.0182879
$376$	48.6661	2.50976
$377$	29.0117	1.49418
$378$	10.9130	0.561302
$379$	6.57250	0.337607	0.168803	−	0.985650i	$-0.446010\pi$
$379$	6.57250	0.337607	0.168803	+	0.985650i	$0.446010\pi$
$380$	11.1925	0.574162
$381$	3.10535	0.159092
$382$	2.58786	0.132407
$383$	−26.4784	−1.35298	−0.676491	−	0.736451i	$-0.736500\pi$
$383$	−26.4784	−1.35298	−0.676491	+	0.736451i	$0.736500\pi$
$384$	−3.46140	−0.176639
$385$	−1.22158	−0.0622576
$386$	−16.7074	−0.850384
$387$	4.85374	0.246729
$388$	40.3408	2.04800
$389$	−24.4114	−1.23771	−0.618855	−	0.785506i	$-0.712403\pi$
$389$	−24.4114	−1.23771	−0.618855	+	0.785506i	$0.712403\pi$
$390$	−8.60168	−0.435563
$391$	8.41990	0.425813
$392$	−20.9054	−1.05588
$393$	2.97651	0.150145
$394$	−47.5736	−2.39672
$395$	−21.7349	−1.09360
$396$	−1.47331	−0.0740365
$397$	13.0485	0.654885	0.327443	−	0.944871i	$-0.393813\pi$
$397$	13.0485	0.654885	0.327443	+	0.944871i	$0.393813\pi$
$398$	−19.5029	−0.977593
$399$	0.585307	0.0293020
$400$	32.7274	1.63637
$401$	−13.6637	−0.682332	−0.341166	−	0.940003i	$-0.610822\pi$
$401$	−13.6637	−0.682332	−0.341166	+	0.940003i	$0.610822\pi$
$402$	6.77649	0.337981
$403$	−14.6292	−0.728732
$404$	−56.6819	−2.82003
$405$	27.6887	1.37586
$406$	−50.3303	−2.49785
$407$	1.01849	0.0504846
$408$	3.97915	0.196997
$409$	−22.6725	−1.12108	−0.560542	−	0.828126i	$-0.689407\pi$
$409$	−22.6725	−1.12108	−0.560542	+	0.828126i	$0.689407\pi$
$410$	11.3200	0.559055
$411$	0.227345	0.0112141
$412$	24.1998	1.19224
$413$	−26.1790	−1.28819
$414$	20.4157	1.00338
$415$	14.9358	0.733172
$416$	15.8892	0.779033
$417$	4.64061	0.227252
$418$	−0.233327	−0.0114124
$419$	25.5450	1.24796	0.623979	−	0.781441i	$-0.285515\pi$
$419$	25.5450	1.24796	0.623979	+	0.781441i	$0.285515\pi$
$420$	10.1936	0.497395
$421$	23.0381	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$421$	23.0381	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$422$	1.64700	0.0801747
$423$	24.7241	1.20213
$424$	48.5738	2.35895
$425$	16.7712	0.813523
$426$	4.96617	0.240611
$427$	−25.6189	−1.23978
$428$	2.99357	0.144700
$429$	0.122492	0.00591397
$430$	13.3872	0.645589
$431$	−4.92354	−0.237158	−0.118579	−	0.992945i	$-0.537834\pi$
$431$	−4.92354	−0.237158	−0.118579	+	0.992945i	$0.537834\pi$
$432$	7.95696	0.382829
$433$	−18.4552	−0.886902	−0.443451	−	0.896299i	$-0.646246\pi$
$433$	−18.4552	−0.886902	−0.443451	+	0.896299i	$0.646246\pi$
$434$	25.3792	1.21824
$435$	4.46865	0.214256
$436$	−29.5676	−1.41603
$437$	2.20862	0.105653
$438$	6.63697	0.317127
$439$	1.49382	0.0712960	0.0356480	−	0.999364i	$-0.488650\pi$
$439$	1.49382	0.0712960	0.0356480	+	0.999364i	$0.488650\pi$
$440$	−2.17843	−0.103853
$441$	−10.6207	−0.505746
$442$	36.1988	1.72180
$443$	7.57917	0.360097	0.180049	−	0.983658i	$-0.442374\pi$
$443$	7.57917	0.360097	0.180049	+	0.983658i	$0.442374\pi$
$444$	−8.49882	−0.403336
$445$	−21.7103	−1.02917
$446$	17.8723	0.846278
$447$	−4.13111	−0.195395
$448$	11.2695	0.532433
$449$	12.3061	0.580759	0.290379	−	0.956912i	$-0.406218\pi$
$449$	12.3061	0.580759	0.290379	+	0.956912i	$0.406218\pi$
$450$	40.6651	1.91697
$451$	−0.161202	−0.00759072
$452$	76.6790	3.60668
$453$	1.42735	0.0670627
$454$	15.5472	0.729665
$455$	49.7125	2.33056
$456$	1.04377	0.0488790
$457$	−11.4019	−0.533358	−0.266679	−	0.963785i	$-0.585926\pi$
$457$	−11.4019	−0.533358	−0.266679	+	0.963785i	$0.585926\pi$
$458$	58.7415	2.74481
$459$	4.07756	0.190324
$460$	38.4648	1.79343
$461$	−36.6815	−1.70843	−0.854213	−	0.519923i	$-0.825961\pi$
$461$	−36.6815	−1.70843	−0.854213	+	0.519923i	$0.825961\pi$
$462$	−0.212503	−0.00988654
$463$	14.2211	0.660910	0.330455	−	0.943822i	$-0.392798\pi$
$463$	14.2211	0.660910	0.330455	+	0.943822i	$0.392798\pi$
$464$	−36.6973	−1.70363
$465$	−2.25333	−0.104496
$466$	44.8910	2.07954
$467$	6.25867	0.289617	0.144808	−	0.989460i	$-0.453743\pi$
$467$	6.25867	0.289617	0.144808	+	0.989460i	$0.453743\pi$
$468$	59.9565	2.77149
$469$	−39.1641	−1.80843
$470$	68.1923	3.14547
$471$	2.08445	0.0960464
$472$	−46.6847	−2.14884
$473$	−0.190641	−0.00876567
$474$	−3.78094	−0.173664
$475$	4.39925	0.201852
$476$	−42.8980	−1.96623
$477$	24.6772	1.12989
$478$	−11.2737	−0.515649
$479$	22.1797	1.01342	0.506708	−	0.862118i	$-0.330862\pi$
$479$	22.1797	1.01342	0.506708	+	0.862118i	$0.330862\pi$
$480$	2.44741	0.111708
$481$	−41.4476	−1.88985
$482$	52.7089	2.40083
$483$	2.01151	0.0915267
$484$	−47.3649	−2.15295
$485$	30.3033	1.37600
$486$	14.8720	0.674610
$487$	−32.0218	−1.45105	−0.725523	−	0.688197i	$-0.758402\pi$
$487$	−32.0218	−1.45105	−0.725523	+	0.688197i	$0.758402\pi$
$488$	−45.6857	−2.06810
$489$	−0.467472	−0.0211398
$490$	−29.2931	−1.32333
$491$	29.1486	1.31546	0.657730	−	0.753254i	$-0.271517\pi$
$491$	29.1486	1.31546	0.657730	+	0.753254i	$0.271517\pi$
$492$	1.34516	0.0606445
$493$	−18.8056	−0.846963
$494$	9.49531	0.427214
$495$	−1.10672	−0.0497434
$496$	18.5047	0.830886
$497$	−28.7015	−1.28744
$498$	2.59820	0.116428
$499$	14.0558	0.629224	0.314612	−	0.949220i	$-0.398126\pi$
$499$	14.0558	0.629224	0.314612	+	0.949220i	$0.398126\pi$
$500$	6.80843	0.304482
$501$	−1.18050	−0.0527408
$502$	35.1607	1.56930
$503$	22.7129	1.01272	0.506360	−	0.862322i	$-0.330991\pi$
$503$	22.7129	1.01272	0.506360	+	0.862322i	$0.330991\pi$
$504$	−55.7610	−2.48379
$505$	−42.5784	−1.89471
$506$	−0.801869	−0.0356474
$507$	−2.06963	−0.0919153
$508$	−59.7005	−2.64878
$509$	14.3692	0.636905	0.318452	−	0.947939i	$-0.396837\pi$
$509$	14.3692	0.636905	0.318452	+	0.947939i	$0.396837\pi$
$510$	5.57569	0.246896
$511$	−38.3577	−1.69685
$512$	49.1547	2.17235
$513$	1.06958	0.0472233
$514$	64.5596	2.84760
$515$	18.1784	0.801038
$516$	1.59081	0.0700315
$517$	−0.971091	−0.0427085
$518$	71.9046	3.15930
$519$	0.0934096	0.00410023
$520$	88.6517	3.88763
$521$	−7.42580	−0.325330	−0.162665	−	0.986681i	$-0.552009\pi$
$521$	−7.42580	−0.325330	−0.162665	+	0.986681i	$0.552009\pi$
$522$	−45.5979	−1.99577
$523$	−6.27621	−0.274440	−0.137220	−	0.990541i	$-0.543817\pi$
$523$	−6.27621	−0.274440	−0.137220	+	0.990541i	$0.543817\pi$
$524$	−57.2235	−2.49982
$525$	4.00662	0.174863
$526$	11.1158	0.484672
$527$	9.48278	0.413077
$528$	−0.154942	−0.00674300
$529$	−15.4097	−0.669987
$530$	68.0628	2.95646
$531$	−23.7175	−1.02925
$532$	−11.2526	−0.487861
$533$	6.56016	0.284152
$534$	−3.77666	−0.163432
$535$	2.24871	0.0972204
$536$	−69.8408	−3.01666
$537$	5.73386	0.247434
$538$	−11.2636	−0.485609
$539$	0.417149	0.0179679
$540$	18.6276	0.801604
$541$	13.6623	0.587389	0.293694	−	0.955899i	$-0.405115\pi$
$541$	13.6623	0.587389	0.293694	+	0.955899i	$0.405115\pi$
$542$	−52.5509	−2.25726
$543$	−4.66327	−0.200120
$544$	−10.2995	−0.441589
$545$	−22.2106	−0.951398
$546$	8.64786	0.370094
$547$	−1.24983	−0.0534388	−0.0267194	−	0.999643i	$-0.508506\pi$
$547$	−1.24983	−0.0534388	−0.0267194	+	0.999643i	$0.508506\pi$
$548$	−4.37072	−0.186708
$549$	−23.2100	−0.990578
$550$	−1.59720	−0.0681050
$551$	−4.93290	−0.210149
$552$	3.58709	0.152677
$553$	21.8516	0.929223
$554$	51.4743	2.18693
$555$	−6.38415	−0.270992
$556$	−89.2160	−3.78360
$557$	43.9064	1.86037	0.930187	−	0.367085i	$-0.119644\pi$
$557$	43.9064	1.86037	0.930187	+	0.367085i	$0.119644\pi$
$558$	22.9929	0.973365
$559$	7.75816	0.328135
$560$	−62.8822	−2.65726
$561$	−0.0794006	−0.00335229
$562$	43.5322	1.83629
$563$	4.37913	0.184559	0.0922793	−	0.995733i	$-0.470585\pi$
$563$	4.37913	0.184559	0.0922793	+	0.995733i	$0.470585\pi$
$564$	8.10332	0.341211
$565$	57.5998	2.42324
$566$	−8.56660	−0.360081
$567$	−27.8374	−1.16906
$568$	−51.1829	−2.14759
$569$	11.6907	0.490099	0.245050	−	0.969511i	$-0.421196\pi$
$569$	11.6907	0.490099	0.245050	+	0.969511i	$0.421196\pi$
$570$	1.46256	0.0612598
$571$	−29.4930	−1.23424	−0.617122	−	0.786868i	$-0.711701\pi$
$571$	−29.4930	−1.23424	−0.617122	+	0.786868i	$0.711701\pi$
$572$	−2.35492	−0.0984640
$573$	0.231001	0.00965020
$574$	−11.3808	−0.475024
$575$	15.1188	0.630496
$576$	10.2098	0.425410
$577$	1.70783	0.0710977	0.0355489	−	0.999368i	$-0.488682\pi$
$577$	1.70783	0.0710977	0.0355489	+	0.999368i	$0.488682\pi$
$578$	19.2430	0.800404
$579$	−1.49135	−0.0619786
$580$	−85.9101	−3.56722
$581$	−15.0160	−0.622970
$582$	5.27147	0.218510
$583$	−0.969248	−0.0401421
$584$	−68.4028	−2.83053
$585$	45.0382	1.86210
$586$	44.4995	1.83826
$587$	−20.4260	−0.843069	−0.421535	−	0.906812i	$-0.638508\pi$
$587$	−20.4260	−0.843069	−0.421535	+	0.906812i	$0.638508\pi$
$588$	−3.48092	−0.143551
$589$	2.48743	0.102493
$590$	−65.4158	−2.69313
$591$	−4.24657	−0.174681
$592$	52.4277	2.15477
$593$	−40.2729	−1.65381	−0.826904	−	0.562343i	$-0.809900\pi$
$593$	−40.2729	−1.65381	−0.826904	+	0.562343i	$0.809900\pi$
$594$	−0.388326	−0.0159332
$595$	−32.2242	−1.32106
$596$	79.4209	3.25321
$597$	−1.74089	−0.0712500
$598$	32.6322	1.33443
$599$	19.9531	0.815262	0.407631	−	0.913147i	$-0.366355\pi$
$599$	19.9531	0.815262	0.407631	+	0.913147i	$0.366355\pi$
$600$	7.14496	0.291692
$601$	−13.4006	−0.546624	−0.273312	−	0.961925i	$-0.588119\pi$
$601$	−13.4006	−0.546624	−0.273312	+	0.961925i	$0.588119\pi$
$602$	−13.4591	−0.548552
$603$	−35.4816	−1.44492
$604$	−27.4409	−1.11655
$605$	−35.5796	−1.44652
$606$	−7.40682	−0.300881
$607$	−18.4380	−0.748374	−0.374187	−	0.927353i	$-0.622078\pi$
$607$	−18.4380	−0.748374	−0.374187	+	0.927353i	$0.622078\pi$
$608$	−2.70167	−0.109567
$609$	−4.49265	−0.182051
$610$	−64.0161	−2.59193
$611$	39.5187	1.59876
$612$	−38.8644	−1.57100
$613$	−21.7063	−0.876708	−0.438354	−	0.898802i	$-0.644438\pi$
$613$	−21.7063	−0.876708	−0.438354	+	0.898802i	$0.644438\pi$
$614$	−32.1172	−1.29615
$615$	1.01046	0.0407456
$616$	2.19013	0.0882427
$617$	−15.5444	−0.625794	−0.312897	−	0.949787i	$-0.601299\pi$
$617$	−15.5444	−0.625794	−0.312897	+	0.949787i	$0.601299\pi$
$618$	3.16227	0.127205
$619$	−26.6991	−1.07313	−0.536563	−	0.843860i	$-0.680278\pi$
$619$	−26.6991	−1.07313	−0.536563	+	0.843860i	$0.680278\pi$
$620$	43.3204	1.73979
$621$	3.67581	0.147505
$622$	−64.5957	−2.59005
$623$	21.8268	0.874474
$624$	6.30541	0.252418
$625$	−22.3239	−0.892957
$626$	7.58783	0.303271
$627$	−0.0208275	−0.000831772 0
$628$	−40.0737	−1.59911
$629$	26.8667	1.07125
$630$	−78.1337	−3.11292
$631$	44.4941	1.77128	0.885641	−	0.464371i	$-0.153720\pi$
$631$	44.4941	1.77128	0.885641	+	0.464371i	$0.153720\pi$
$632$	38.9676	1.55005
$633$	0.147016	0.00584338
$634$	16.4032	0.651454
$635$	−44.8459	−1.77966
$636$	8.08794	0.320708
$637$	−16.9759	−0.672611
$638$	1.79095	0.0709045
$639$	−26.0027	−1.02865
$640$	49.9878	1.97594
$641$	−23.4816	−0.927467	−0.463734	−	0.885975i	$-0.653491\pi$
$641$	−23.4816	−0.927467	−0.463734	+	0.885975i	$0.653491\pi$
$642$	0.391180	0.0154386
$643$	29.5514	1.16539	0.582697	−	0.812690i	$-0.301998\pi$
$643$	29.5514	1.16539	0.582697	+	0.812690i	$0.301998\pi$
$644$	−38.6713	−1.52386
$645$	1.19499	0.0470525
$646$	−6.15495	−0.242163
$647$	20.9033	0.821794	0.410897	−	0.911682i	$-0.365216\pi$
$647$	20.9033	0.821794	0.410897	+	0.911682i	$0.365216\pi$
$648$	−49.6420	−1.95012
$649$	0.931553	0.0365667
$650$	64.9985	2.54945
$651$	2.26543	0.0887891
$652$	8.98717	0.351965
$653$	21.6239	0.846210	0.423105	−	0.906081i	$-0.360940\pi$
$653$	21.6239	0.846210	0.423105	+	0.906081i	$0.360940\pi$
$654$	−3.86370	−0.151082
$655$	−42.9852	−1.67957
$656$	−8.29806	−0.323985
$657$	−34.7511	−1.35577
$658$	−68.5584	−2.67268
$659$	28.7354	1.11937	0.559685	−	0.828705i	$-0.310922\pi$
$659$	28.7354	1.11937	0.559685	+	0.828705i	$0.310922\pi$
$660$	−0.362727	−0.0141191
$661$	10.6771	0.415289	0.207645	−	0.978204i	$-0.433420\pi$
$661$	10.6771	0.415289	0.207645	+	0.978204i	$0.433420\pi$
$662$	87.6051	3.40487
$663$	3.23122	0.125490
$664$	−26.7779	−1.03918
$665$	−8.45271	−0.327782
$666$	65.1436	2.52426
$667$	−16.9527	−0.656413
$668$	22.6952	0.878103
$669$	1.59534	0.0616794
$670$	−97.8627	−3.78077
$671$	0.911620	0.0351927
$672$	−2.46055	−0.0949177
$673$	−11.0691	−0.426682	−0.213341	−	0.976978i	$-0.568434\pi$
$673$	−11.0691	−0.426682	−0.213341	+	0.976978i	$0.568434\pi$
$674$	−4.15299	−0.159967
$675$	7.32167	0.281811
$676$	39.7887	1.53033
$677$	9.85313	0.378687	0.189343	−	0.981911i	$-0.439364\pi$
$677$	9.85313	0.378687	0.189343	+	0.981911i	$0.439364\pi$
$678$	10.0199	0.384812
$679$	−30.4660	−1.16918
$680$	−57.4649	−2.20368
$681$	1.38779	0.0531802
$682$	−0.903092	−0.0345812
$683$	−19.4626	−0.744716	−0.372358	−	0.928089i	$-0.621451\pi$
$683$	−19.4626	−0.744716	−0.372358	+	0.928089i	$0.621451\pi$
$684$	−10.1945	−0.389797
$685$	−3.28320	−0.125445
$686$	−27.8051	−1.06160
$687$	5.24345	0.200050
$688$	−9.81343	−0.374133
$689$	39.4437	1.50269
$690$	5.02632	0.191349
$691$	31.2096	1.18727	0.593634	−	0.804735i	$-0.297693\pi$
$691$	31.2096	1.18727	0.593634	+	0.804735i	$0.297693\pi$
$692$	−1.79581	−0.0682663
$693$	1.11266	0.0422665
$694$	−34.1643	−1.29686
$695$	−67.0173	−2.54211
$696$	−8.01167	−0.303682
$697$	−4.25236	−0.161070
$698$	10.8490	0.410639
$699$	4.00711	0.151563
$700$	−77.0276	−2.91137
$701$	22.7666	0.859883	0.429941	−	0.902857i	$-0.358534\pi$
$701$	22.7666	0.859883	0.429941	+	0.902857i	$0.358534\pi$
$702$	15.8030	0.596446
$703$	7.04740	0.265798
$704$	−0.401013	−0.0151137
$705$	6.08706	0.229252
$706$	−79.7907	−3.00296
$707$	42.8070	1.60992
$708$	−7.77339	−0.292142
$709$	−24.7417	−0.929194	−0.464597	−	0.885522i	$-0.653801\pi$
$709$	−24.7417	−0.929194	−0.464597	+	0.885522i	$0.653801\pi$
$710$	−71.7188	−2.69156
$711$	19.7969	0.742442
$712$	38.9235	1.45872
$713$	8.54846	0.320142
$714$	−5.60563	−0.209785
$715$	−1.76897	−0.0661557
$716$	−110.234	−4.11963
$717$	−1.00633	−0.0375821
$718$	61.1555	2.28230
$719$	−47.8049	−1.78282	−0.891411	−	0.453196i	$-0.850284\pi$
$719$	−47.8049	−1.78282	−0.891411	+	0.453196i	$0.850284\pi$
$720$	−56.9696	−2.12313
$721$	−18.2760	−0.680635
$722$	46.1174	1.71631
$723$	4.70496	0.174980
$724$	89.6515	3.33187
$725$	−33.7674	−1.25409
$726$	−6.18934	−0.229708
$727$	−2.98613	−0.110749	−0.0553747	−	0.998466i	$-0.517635\pi$
$727$	−2.98613	−0.110749	−0.0553747	+	0.998466i	$0.517635\pi$
$728$	−89.1277	−3.30329
$729$	−24.3223	−0.900827
$730$	−95.8478	−3.54749
$731$	−5.02891	−0.186001
$732$	−7.60706	−0.281165
$733$	33.8426	1.25000	0.625002	−	0.780623i	$-0.285098\pi$
$733$	33.8426	1.25000	0.625002	+	0.780623i	$0.285098\pi$
$734$	55.6497	2.05407
$735$	−2.61480	−0.0964483
$736$	−9.28474	−0.342240
$737$	1.39361	0.0513344
$738$	−10.3107	−0.379541
$739$	9.39870	0.345737	0.172868	−	0.984945i	$-0.444696\pi$
$739$	9.39870	0.345737	0.172868	+	0.984945i	$0.444696\pi$
$740$	122.736	4.51185
$741$	0.847581	0.0311367
$742$	−68.4283	−2.51208
$743$	8.11083	0.297557	0.148779	−	0.988871i	$-0.452466\pi$
$743$	8.11083	0.297557	0.148779	+	0.988871i	$0.452466\pi$
$744$	4.03990	0.148110
$745$	59.6594	2.18575
$746$	68.3540	2.50262
$747$	−13.6041	−0.497749
$748$	1.52648	0.0558136
$749$	−2.26079	−0.0826073
$750$	0.889681	0.0324865
$751$	4.25647	0.155321	0.0776604	−	0.996980i	$-0.475255\pi$
$751$	4.25647	0.155321	0.0776604	+	0.996980i	$0.475255\pi$
$752$	−49.9879	−1.82287
$753$	3.13855	0.114375
$754$	−72.8832	−2.65425
$755$	−20.6131	−0.750186
$756$	−18.7276	−0.681117
$757$	−36.0728	−1.31109	−0.655545	−	0.755156i	$-0.727561\pi$
$757$	−36.0728	−1.31109	−0.655545	+	0.755156i	$0.727561\pi$
$758$	−16.5115	−0.599723
$759$	−0.0715773	−0.00259809
$760$	−15.0736	−0.546777
$761$	−25.1953	−0.913328	−0.456664	−	0.889639i	$-0.650956\pi$
$761$	−25.1953	−0.913328	−0.456664	+	0.889639i	$0.650956\pi$
$762$	−7.80127	−0.282610
$763$	22.3299	0.808395
$764$	−4.44100	−0.160670
$765$	−29.1942	−1.05552
$766$	66.5191	2.40343
$767$	−37.9097	−1.36884
$768$	7.14337	0.257764
$769$	−44.8144	−1.61605	−0.808024	−	0.589149i	$-0.799463\pi$
$769$	−44.8144	−1.61605	−0.808024	+	0.589149i	$0.799463\pi$
$770$	3.06886	0.110594
$771$	5.76280	0.207542
$772$	28.6714	1.03191
$773$	8.54208	0.307237	0.153619	−	0.988130i	$-0.450907\pi$
$773$	8.54208	0.307237	0.153619	+	0.988130i	$0.450907\pi$
$774$	−12.1936	−0.438289
$775$	17.0273	0.611638
$776$	−54.3295	−1.95032
$777$	6.41843	0.230260
$778$	61.3265	2.19866
$779$	−1.11544	−0.0399646
$780$	14.7612	0.528537
$781$	1.02131	0.0365454
$782$	−21.1525	−0.756412
$783$	−8.20982	−0.293395
$784$	21.4732	0.766899
$785$	−30.1026	−1.07441
$786$	−7.47759	−0.266717
$787$	−20.1616	−0.718682	−0.359341	−	0.933206i	$-0.616998\pi$
$787$	−20.1616	−0.718682	−0.359341	+	0.933206i	$0.616998\pi$
$788$	81.6406	2.90833
$789$	0.992232	0.0353244
$790$	54.6024	1.94267
$791$	−57.9091	−2.05901
$792$	1.98420	0.0705053
$793$	−37.0986	−1.31741
$794$	−32.7805	−1.16334
$795$	6.07550	0.215476
$796$	33.4688	1.18627
$797$	27.8776	0.987477	0.493738	−	0.869611i	$-0.335630\pi$
$797$	27.8776	0.987477	0.493738	+	0.869611i	$0.335630\pi$
$798$	−1.47041	−0.0520520
$799$	−25.6164	−0.906244
$800$	−18.4938	−0.653856
$801$	19.7745	0.698698
$802$	34.3260	1.21209
$803$	1.36492	0.0481670
$804$	−11.6291	−0.410126
$805$	−29.0491	−1.02385
$806$	36.7515	1.29452
$807$	−1.00543	−0.0353927
$808$	76.3371	2.68553
$809$	−36.6479	−1.28847	−0.644236	−	0.764827i	$-0.722825\pi$
$809$	−36.6479	−1.28847	−0.644236	+	0.764827i	$0.722825\pi$
$810$	−69.5597	−2.44408
$811$	0.390142	0.0136998	0.00684988	−	0.999977i	$-0.497820\pi$
$811$	0.390142	0.0136998	0.00684988	+	0.999977i	$0.497820\pi$
$812$	86.3714	3.03104
$813$	−4.69086	−0.164516
$814$	−2.55865	−0.0896806
$815$	6.75099	0.236477
$816$	−4.08723	−0.143082
$817$	−1.31913	−0.0461507
$818$	56.9580	1.99149
$819$	−45.2800	−1.58221
$820$	−19.4261	−0.678390
$821$	41.7924	1.45857	0.729283	−	0.684212i	$-0.239854\pi$
$821$	41.7924	1.45857	0.729283	+	0.684212i	$0.239854\pi$
$822$	−0.571137	−0.0199207
$823$	−12.6082	−0.439495	−0.219748	−	0.975557i	$-0.570523\pi$
$823$	−12.6082	−0.439495	−0.219748	+	0.975557i	$0.570523\pi$
$824$	−32.5914	−1.13538
$825$	−0.142572	−0.00496370
$826$	65.7670	2.28833
$827$	−34.2541	−1.19113	−0.595566	−	0.803306i	$-0.703072\pi$
$827$	−34.2541	−1.19113	−0.595566	+	0.803306i	$0.703072\pi$
$828$	−35.0351	−1.21756
$829$	−23.7474	−0.824782	−0.412391	−	0.911007i	$-0.635306\pi$
$829$	−23.7474	−0.824782	−0.412391	+	0.911007i	$0.635306\pi$
$830$	−37.5219	−1.30240
$831$	4.59476	0.159391
$832$	16.3193	0.565770
$833$	11.0040	0.381265
$834$	−11.6582	−0.403689
$835$	17.0482	0.589977
$836$	0.400411	0.0138485
$837$	4.13982	0.143093
$838$	−64.1743	−2.21687
$839$	−51.7346	−1.78608	−0.893038	−	0.449981i	$-0.851431\pi$
$839$	−51.7346	−1.78608	−0.893038	+	0.449981i	$0.851431\pi$
$840$	−13.7283	−0.473671
$841$	8.86351	0.305638
$842$	−57.8765	−1.99456
$843$	3.88582	0.133835
$844$	−2.82640	−0.0972886
$845$	29.8885	1.02819
$846$	−62.1120	−2.13546
$847$	35.7707	1.22909
$848$	−49.8931	−1.71333
$849$	−0.764682	−0.0262438
$850$	−42.1327	−1.44514
$851$	24.2196	0.830236
$852$	−8.52238	−0.291972
$853$	−53.9680	−1.84783	−0.923915	−	0.382598i	$-0.875029\pi$
$853$	−53.9680	−1.84783	−0.923915	+	0.382598i	$0.875029\pi$
$854$	64.3598	2.20235
$855$	−7.65793	−0.261896
$856$	−4.03163	−0.137798
$857$	50.9194	1.73937	0.869686	−	0.493605i	$-0.164321\pi$
$857$	50.9194	1.73937	0.869686	+	0.493605i	$0.164321\pi$
$858$	−0.307725	−0.0105056
$859$	−10.7180	−0.365693	−0.182847	−	0.983141i	$-0.558531\pi$
$859$	−10.7180	−0.365693	−0.182847	+	0.983141i	$0.558531\pi$
$860$	−22.9737	−0.783396
$861$	−1.01588	−0.0346212
$862$	12.3689	0.421287
$863$	10.3625	0.352745	0.176372	−	0.984324i	$-0.443564\pi$
$863$	10.3625	0.352745	0.176372	+	0.984324i	$0.443564\pi$
$864$	−4.49638	−0.152970
$865$	−1.34897	−0.0458665
$866$	46.3633	1.57549
$867$	1.71769	0.0583359
$868$	−43.5530	−1.47828
$869$	−0.777565	−0.0263771
$870$	−11.2262	−0.380603
$871$	−56.7133	−1.92166
$872$	39.8205	1.34849
$873$	−27.6013	−0.934163
$874$	−5.54851	−0.187681
$875$	−5.14182	−0.173825
$876$	−11.3896	−0.384820
$877$	−37.9982	−1.28311	−0.641555	−	0.767077i	$-0.721710\pi$
$877$	−37.9982	−1.28311	−0.641555	+	0.767077i	$0.721710\pi$
$878$	−3.75277	−0.126650
$879$	3.97217	0.133978
$880$	2.23760	0.0754294
$881$	−25.0514	−0.844002	−0.422001	−	0.906595i	$-0.638672\pi$
$881$	−25.0514	−0.844002	−0.422001	+	0.906595i	$0.638672\pi$
$882$	26.6813	0.898405
$883$	38.7198	1.30302	0.651512	−	0.758638i	$-0.274135\pi$
$883$	38.7198	1.30302	0.651512	+	0.758638i	$0.274135\pi$
$884$	−62.1204	−2.08934
$885$	−5.83922	−0.196283
$886$	−19.0404	−0.639676
$887$	31.3550	1.05280	0.526400	−	0.850237i	$-0.323542\pi$
$887$	31.3550	1.05280	0.526400	+	0.850237i	$0.323542\pi$
$888$	11.4459	0.384099
$889$	45.0867	1.51216
$890$	54.5406	1.82821
$891$	0.990564	0.0331851
$892$	−30.6705	−1.02692
$893$	−6.71944	−0.224858
$894$	10.3782	0.347099
$895$	−82.8055	−2.76788
$896$	−50.2562	−1.67894
$897$	2.91285	0.0972573
$898$	−30.9153	−1.03166
$899$	−19.0928	−0.636779
$900$	−69.7849	−2.32616
$901$	−25.5678	−0.851787
$902$	0.404973	0.0134841
$903$	−1.20140	−0.0399802
$904$	−103.268	−3.43466
$905$	67.3445	2.23861
$906$	−3.58579	−0.119130
$907$	19.2610	0.639552	0.319776	−	0.947493i	$-0.396392\pi$
$907$	19.2610	0.639552	0.319776	+	0.947493i	$0.396392\pi$
$908$	−26.6803	−0.885418
$909$	38.7819	1.28632
$910$	−124.888	−4.14000
$911$	−28.4273	−0.941839	−0.470920	−	0.882176i	$-0.656078\pi$
$911$	−28.4273	−0.941839	−0.470920	+	0.882176i	$0.656078\pi$
$912$	−1.07212	−0.0355014
$913$	0.534330	0.0176837
$914$	28.6439	0.947455
$915$	−5.71428	−0.188908
$916$	−100.806	−3.33071
$917$	43.2160	1.42712
$918$	−10.2437	−0.338091
$919$	26.7192	0.881384	0.440692	−	0.897658i	$-0.354733\pi$
$919$	26.7192	0.881384	0.440692	+	0.897658i	$0.354733\pi$
$920$	−51.8029	−1.70789
$921$	−2.86689	−0.0944671
$922$	92.1513	3.03484
$923$	−41.5625	−1.36805
$924$	0.364674	0.0119969
$925$	48.2418	1.58618
$926$	−35.7263	−1.17404
$927$	−16.5576	−0.543823
$928$	20.7372	0.680733
$929$	−31.6729	−1.03915	−0.519577	−	0.854424i	$-0.673910\pi$
$929$	−31.6729	−1.03915	−0.519577	+	0.854424i	$0.673910\pi$
$930$	5.66082	0.185626
$931$	2.88645	0.0945996
$932$	−77.0370	−2.52343
$933$	−5.76602	−0.188771
$934$	−15.7231	−0.514474
$935$	1.14666	0.0374999
$936$	−80.7472	−2.63931
$937$	−24.2732	−0.792971	−0.396486	−	0.918041i	$-0.629770\pi$
$937$	−24.2732	−0.792971	−0.396486	+	0.918041i	$0.629770\pi$
$938$	98.3881	3.21249
$939$	0.677313	0.0221033
$940$	−117.024	−3.81690
$941$	36.5880	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$941$	36.5880	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$942$	−5.23657	−0.170617
$943$	−3.83338	−0.124832
$944$	47.9527	1.56073
$945$	−14.0678	−0.457626
$946$	0.478928	0.0155713
$947$	−4.33095	−0.140737	−0.0703685	−	0.997521i	$-0.522418\pi$
$947$	−4.33095	−0.140737	−0.0703685	+	0.997521i	$0.522418\pi$
$948$	6.48843	0.210734
$949$	−55.5457	−1.80309
$950$	−11.0518	−0.358568
$951$	1.46420	0.0474799
$952$	57.7734	1.87245
$953$	−48.6270	−1.57518	−0.787591	−	0.616198i	$-0.788672\pi$
$953$	−48.6270	−1.57518	−0.787591	+	0.616198i	$0.788672\pi$
$954$	−61.9941	−2.00713
$955$	−3.33600	−0.107950
$956$	19.3467	0.625718
$957$	0.159866	0.00516774
$958$	−55.7199	−1.80023
$959$	3.30083	0.106589
$960$	2.51365	0.0811279
$961$	−21.3724	−0.689433
$962$	104.125	3.35711
$963$	−2.04821	−0.0660027
$964$	−90.4532	−2.91330
$965$	21.5374	0.693313
$966$	−5.05331	−0.162588
$967$	38.9720	1.25325	0.626627	−	0.779319i	$-0.284435\pi$
$967$	38.9720	1.25325	0.626627	+	0.779319i	$0.284435\pi$
$968$	63.7893	2.05027
$969$	−0.549410	−0.0176496
$970$	−76.1279	−2.44432
$971$	10.2663	0.329460	0.164730	−	0.986339i	$-0.447325\pi$
$971$	10.2663	0.329460	0.164730	+	0.986339i	$0.447325\pi$
$972$	−25.5218	−0.818611
$973$	67.3771	2.16001
$974$	80.4453	2.57763
$975$	5.80198	0.185812
$976$	46.9266	1.50208
$977$	−53.0127	−1.69603	−0.848014	−	0.529974i	$-0.822202\pi$
$977$	−53.0127	−1.69603	−0.848014	+	0.529974i	$0.822202\pi$
$978$	1.17438	0.0375527
$979$	−0.776685	−0.0248230
$980$	50.2696	1.60580
$981$	20.2302	0.645902
$982$	−73.2273	−2.33678
$983$	−43.7228	−1.39454	−0.697270	−	0.716808i	$-0.745602\pi$
$983$	−43.7228	−1.39454	−0.697270	+	0.716808i	$0.745602\pi$
$984$	−1.81161	−0.0577521
$985$	61.3268	1.95404
$986$	47.2436	1.50454
$987$	−6.11974	−0.194793
$988$	−16.2948	−0.518406
$989$	−4.53342	−0.144154
$990$	2.78031	0.0883639
$991$	43.4167	1.37918	0.689588	−	0.724202i	$-0.257792\pi$
$991$	43.4167	1.37918	0.689588	+	0.724202i	$0.257792\pi$
$992$	−10.4568	−0.332004
$993$	7.81991	0.248157
$994$	72.1039	2.28700
$995$	25.1411	0.797026
$996$	−4.45874	−0.141281
$997$	−26.7199	−0.846227	−0.423113	−	0.906077i	$-0.639063\pi$
$997$	−26.7199	−0.846227	−0.423113	+	0.906077i	$0.639063\pi$
$998$	−35.3110	−1.11775
$999$	11.7290	0.371088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8039.2.a.a.1.19	✓	279

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8039.2.a.a.1.19	✓	279	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:	\( N \)	\(=\)	\( 8039 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8039.a (trivial)

\(f(q)\)	\(=\)	\(q-2.51220 q^{2} -0.224247 q^{3} +4.31117 q^{4} +3.23846 q^{5} +0.563355 q^{6} -3.25585 q^{7} -5.80612 q^{8} -2.94971 q^{9} +O(q^{10})\)	\(q-2.51220 q^{2} -0.224247 q^{3} +4.31117 q^{4} +3.23846 q^{5} +0.563355 q^{6} -3.25585 q^{7} -5.80612 q^{8} -2.94971 q^{9} -8.13568 q^{10} +0.115856 q^{11} -0.966767 q^{12} -4.71479 q^{13} +8.17936 q^{14} -0.726217 q^{15} +5.96381 q^{16} +3.05617 q^{17} +7.41028 q^{18} +0.801664 q^{19} +13.9616 q^{20} +0.730116 q^{21} -0.291054 q^{22} +2.75505 q^{23} +1.30201 q^{24} +5.48766 q^{25} +11.8445 q^{26} +1.33421 q^{27} -14.0365 q^{28} -6.15333 q^{29} +1.82440 q^{30} +3.10283 q^{31} -3.37008 q^{32} -0.0259804 q^{33} -7.67772 q^{34} -10.5440 q^{35} -12.7167 q^{36} +8.79097 q^{37} -2.01394 q^{38} +1.05728 q^{39} -18.8029 q^{40} -1.39140 q^{41} -1.83420 q^{42} -1.64550 q^{43} +0.499475 q^{44} -9.55254 q^{45} -6.92125 q^{46} -8.38187 q^{47} -1.33737 q^{48} +3.60057 q^{49} -13.7861 q^{50} -0.685338 q^{51} -20.3262 q^{52} -8.36596 q^{53} -3.35180 q^{54} +0.375196 q^{55} +18.9039 q^{56} -0.179771 q^{57} +15.4584 q^{58} +8.04061 q^{59} -3.13084 q^{60} +7.86856 q^{61} -7.79494 q^{62} +9.60383 q^{63} -3.46130 q^{64} -15.2687 q^{65} +0.0652681 q^{66} +12.0288 q^{67} +13.1757 q^{68} -0.617812 q^{69} +26.4886 q^{70} +8.81535 q^{71} +17.1264 q^{72} +11.7812 q^{73} -22.0847 q^{74} -1.23059 q^{75} +3.45610 q^{76} -0.377210 q^{77} -2.65610 q^{78} -6.71147 q^{79} +19.3136 q^{80} +8.54995 q^{81} +3.49548 q^{82} +4.61201 q^{83} +3.14765 q^{84} +9.89730 q^{85} +4.13382 q^{86} +1.37987 q^{87} -0.672674 q^{88} -6.70388 q^{89} +23.9979 q^{90} +15.3506 q^{91} +11.8775 q^{92} -0.695801 q^{93} +21.0570 q^{94} +2.59616 q^{95} +0.755731 q^{96} +9.35729 q^{97} -9.04538 q^{98} -0.341742 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 279 q - 13 q^{2} - 12 q^{3} + 227 q^{4} - 20 q^{5} - 40 q^{6} - 57 q^{7} - 39 q^{8} + 175 q^{9}+O(q^{10}) \) 279 * q - 13 * q^2 - 12 * q^3 + 227 * q^4 - 20 * q^5 - 40 * q^6 - 57 * q^7 - 39 * q^8 + 175 * q^9	\( 279 q - 13 q^{2} - 12 q^{3} + 227 q^{4} - 20 q^{5} - 40 q^{6} - 57 q^{7} - 39 q^{8} + 175 q^{9} - 42 q^{10} - 53 q^{11} - 36 q^{12} - 75 q^{13} - 31 q^{14} - 60 q^{15} + 127 q^{16} - 55 q^{17} - 57 q^{18} - 113 q^{19} - 43 q^{20} - 103 q^{21} - 73 q^{22} - 30 q^{23} - 106 q^{24} + 75 q^{25} - 42 q^{26} - 45 q^{27} - 146 q^{28} - 92 q^{29} - 76 q^{30} - 84 q^{31} - 71 q^{32} - 117 q^{33} - 106 q^{34} - 49 q^{35} + 67 q^{36} - 123 q^{37} - 21 q^{38} - 92 q^{39} - 97 q^{40} - 116 q^{41} - 19 q^{42} - 126 q^{43} - 131 q^{44} - 85 q^{45} - 183 q^{46} - 42 q^{47} - 47 q^{48} - 22 q^{49} - 64 q^{50} - 90 q^{51} - 158 q^{52} - 60 q^{53} - 117 q^{54} - 99 q^{55} - 65 q^{56} - 182 q^{57} - 93 q^{58} - 58 q^{59} - 141 q^{60} - 217 q^{61} - 16 q^{62} - 141 q^{63} - 47 q^{64} - 197 q^{65} - 53 q^{66} - 147 q^{67} - 90 q^{68} - 103 q^{69} - 118 q^{70} - 78 q^{71} - 135 q^{72} - 282 q^{73} - 98 q^{74} - 53 q^{75} - 296 q^{76} - 53 q^{77} - 27 q^{78} - 153 q^{79} - 52 q^{80} - 89 q^{81} - 81 q^{82} - 54 q^{83} - 164 q^{84} - 303 q^{85} - 82 q^{86} - 29 q^{87} - 203 q^{88} - 185 q^{89} - 56 q^{90} - 163 q^{91} - 66 q^{92} - 156 q^{93} - 134 q^{94} - 69 q^{95} - 189 q^{96} - 212 q^{97} - 13 q^{98} - 181 q^{99}+O(q^{100}) \) 279 * q - 13 * q^2 - 12 * q^3 + 227 * q^4 - 20 * q^5 - 40 * q^6 - 57 * q^7 - 39 * q^8 + 175 * q^9 - 42 * q^10 - 53 * q^11 - 36 * q^12 - 75 * q^13 - 31 * q^14 - 60 * q^15 + 127 * q^16 - 55 * q^17 - 57 * q^18 - 113 * q^19 - 43 * q^20 - 103 * q^21 - 73 * q^22 - 30 * q^23 - 106 * q^24 + 75 * q^25 - 42 * q^26 - 45 * q^27 - 146 * q^28 - 92 * q^29 - 76 * q^30 - 84 * q^31 - 71 * q^32 - 117 * q^33 - 106 * q^34 - 49 * q^35 + 67 * q^36 - 123 * q^37 - 21 * q^38 - 92 * q^39 - 97 * q^40 - 116 * q^41 - 19 * q^42 - 126 * q^43 - 131 * q^44 - 85 * q^45 - 183 * q^46 - 42 * q^47 - 47 * q^48 - 22 * q^49 - 64 * q^50 - 90 * q^51 - 158 * q^52 - 60 * q^53 - 117 * q^54 - 99 * q^55 - 65 * q^56 - 182 * q^57 - 93 * q^58 - 58 * q^59 - 141 * q^60 - 217 * q^61 - 16 * q^62 - 141 * q^63 - 47 * q^64 - 197 * q^65 - 53 * q^66 - 147 * q^67 - 90 * q^68 - 103 * q^69 - 118 * q^70 - 78 * q^71 - 135 * q^72 - 282 * q^73 - 98 * q^74 - 53 * q^75 - 296 * q^76 - 53 * q^77 - 27 * q^78 - 153 * q^79 - 52 * q^80 - 89 * q^81 - 81 * q^82 - 54 * q^83 - 164 * q^84 - 303 * q^85 - 82 * q^86 - 29 * q^87 - 203 * q^88 - 185 * q^89 - 56 * q^90 - 163 * q^91 - 66 * q^92 - 156 * q^93 - 134 * q^94 - 69 * q^95 - 189 * q^96 - 212 * q^97 - 13 * q^98 - 181 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more