LMFDB - Embedded newform 8011.2.a.a.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8011,2,Mod(1,8011)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8011, 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("8011.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8011.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:	$63.9681570592$
Analytic rank:	$1$
Dimension:	$309$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	8011.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.58476 q^{2} -1.81476 q^{3} +4.68100 q^{4} +2.17232 q^{5} +4.69073 q^{6} -4.47910 q^{7} -6.92974 q^{8} +0.293363 q^{9} +O(q^{10})$	\(q-2.58476 q^{2} -1.81476 q^{3} +4.68100 q^{4} +2.17232 q^{5} +4.69073 q^{6} -4.47910 q^{7} -6.92974 q^{8} +0.293363 q^{9} -5.61492 q^{10} -2.18976 q^{11} -8.49490 q^{12} -4.47644 q^{13} +11.5774 q^{14} -3.94224 q^{15} +8.54973 q^{16} +6.20907 q^{17} -0.758275 q^{18} -7.73483 q^{19} +10.1686 q^{20} +8.12850 q^{21} +5.66001 q^{22} +3.08268 q^{23} +12.5758 q^{24} -0.281042 q^{25} +11.5705 q^{26} +4.91190 q^{27} -20.9666 q^{28} -8.85800 q^{29} +10.1897 q^{30} +3.24145 q^{31} -8.23955 q^{32} +3.97390 q^{33} -16.0490 q^{34} -9.73002 q^{35} +1.37323 q^{36} -11.0326 q^{37} +19.9927 q^{38} +8.12368 q^{39} -15.0536 q^{40} +0.364432 q^{41} -21.0102 q^{42} +3.43325 q^{43} -10.2503 q^{44} +0.637278 q^{45} -7.96798 q^{46} +11.0272 q^{47} -15.5157 q^{48} +13.0623 q^{49} +0.726427 q^{50} -11.2680 q^{51} -20.9542 q^{52} +3.54326 q^{53} -12.6961 q^{54} -4.75686 q^{55} +31.0390 q^{56} +14.0369 q^{57} +22.8958 q^{58} +8.97244 q^{59} -18.4536 q^{60} -6.76540 q^{61} -8.37838 q^{62} -1.31400 q^{63} +4.19782 q^{64} -9.72425 q^{65} -10.2716 q^{66} +12.6464 q^{67} +29.0646 q^{68} -5.59432 q^{69} +25.1498 q^{70} +1.11934 q^{71} -2.03293 q^{72} -1.54385 q^{73} +28.5166 q^{74} +0.510025 q^{75} -36.2067 q^{76} +9.80816 q^{77} -20.9978 q^{78} +6.30245 q^{79} +18.5727 q^{80} -9.79403 q^{81} -0.941970 q^{82} +13.6371 q^{83} +38.0495 q^{84} +13.4881 q^{85} -8.87414 q^{86} +16.0752 q^{87} +15.1745 q^{88} +14.4727 q^{89} -1.64721 q^{90} +20.0504 q^{91} +14.4300 q^{92} -5.88246 q^{93} -28.5027 q^{94} -16.8025 q^{95} +14.9528 q^{96} -2.46905 q^{97} -33.7630 q^{98} -0.642396 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 309 q - 33 q^{2} - 15 q^{3} + 273 q^{4} - 74 q^{5} - 32 q^{6} - 19 q^{7} - 93 q^{8} + 214 q^{9}+O(q^{10}) $ 309 * q - 33 * q^2 - 15 * q^3 + 273 * q^4 - 74 * q^5 - 32 * q^6 - 19 * q^7 - 93 * q^8 + 214 * q^9	\( 309 q - 33 q^{2} - 15 q^{3} + 273 q^{4} - 74 q^{5} - 32 q^{6} - 19 q^{7} - 93 q^{8} + 214 q^{9} - 23 q^{10} - 72 q^{11} - 42 q^{12} - 57 q^{13} - 77 q^{14} - 44 q^{15} + 205 q^{16} - 86 q^{17} - 82 q^{18} - 58 q^{19} - 134 q^{20} - 123 q^{21} - 31 q^{22} - 94 q^{23} - 84 q^{24} + 225 q^{25} - 92 q^{26} - 48 q^{27} - 36 q^{28} - 345 q^{29} - 85 q^{30} - 36 q^{31} - 199 q^{32} - 56 q^{33} - 28 q^{34} - 168 q^{35} + 65 q^{36} - 79 q^{37} - 66 q^{38} - 145 q^{39} - 54 q^{40} - 176 q^{41} - 48 q^{42} - 58 q^{43} - 194 q^{44} - 192 q^{45} - 44 q^{46} - 82 q^{47} - 81 q^{48} + 186 q^{49} - 206 q^{50} - 145 q^{51} - 86 q^{52} - 223 q^{53} - 117 q^{54} - 58 q^{55} - 216 q^{56} - 124 q^{57} - 151 q^{59} - 91 q^{60} - 184 q^{61} - 124 q^{62} - 78 q^{63} + 101 q^{64} - 194 q^{65} - 112 q^{66} - 53 q^{67} - 182 q^{68} - 243 q^{69} - 193 q^{71} - 208 q^{72} - 69 q^{73} - 236 q^{74} - 62 q^{75} - 142 q^{76} - 324 q^{77} - 20 q^{78} - 91 q^{79} - 223 q^{80} - 27 q^{81} + 2 q^{82} - 117 q^{83} - 157 q^{84} - 171 q^{85} - 203 q^{86} - 69 q^{87} - 36 q^{88} - 172 q^{89} - 10 q^{90} - 84 q^{91} - 226 q^{92} - 220 q^{93} - 96 q^{94} - 166 q^{95} - 118 q^{96} - 12 q^{97} - 116 q^{98} - 154 q^{99}+O(q^{100}) \) 309 * q - 33 * q^2 - 15 * q^3 + 273 * q^4 - 74 * q^5 - 32 * q^6 - 19 * q^7 - 93 * q^8 + 214 * q^9 - 23 * q^10 - 72 * q^11 - 42 * q^12 - 57 * q^13 - 77 * q^14 - 44 * q^15 + 205 * q^16 - 86 * q^17 - 82 * q^18 - 58 * q^19 - 134 * q^20 - 123 * q^21 - 31 * q^22 - 94 * q^23 - 84 * q^24 + 225 * q^25 - 92 * q^26 - 48 * q^27 - 36 * q^28 - 345 * q^29 - 85 * q^30 - 36 * q^31 - 199 * q^32 - 56 * q^33 - 28 * q^34 - 168 * q^35 + 65 * q^36 - 79 * q^37 - 66 * q^38 - 145 * q^39 - 54 * q^40 - 176 * q^41 - 48 * q^42 - 58 * q^43 - 194 * q^44 - 192 * q^45 - 44 * q^46 - 82 * q^47 - 81 * q^48 + 186 * q^49 - 206 * q^50 - 145 * q^51 - 86 * q^52 - 223 * q^53 - 117 * q^54 - 58 * q^55 - 216 * q^56 - 124 * q^57 - 151 * q^59 - 91 * q^60 - 184 * q^61 - 124 * q^62 - 78 * q^63 + 101 * q^64 - 194 * q^65 - 112 * q^66 - 53 * q^67 - 182 * q^68 - 243 * q^69 - 193 * q^71 - 208 * q^72 - 69 * q^73 - 236 * q^74 - 62 * q^75 - 142 * q^76 - 324 * q^77 - 20 * q^78 - 91 * q^79 - 223 * q^80 - 27 * q^81 + 2 * q^82 - 117 * q^83 - 157 * q^84 - 171 * q^85 - 203 * q^86 - 69 * q^87 - 36 * q^88 - 172 * q^89 - 10 * q^90 - 84 * q^91 - 226 * q^92 - 220 * q^93 - 96 * q^94 - 166 * q^95 - 118 * q^96 - 12 * q^97 - 116 * q^98 - 154 * 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 $

$2$	−2.58476	−1.82770	−0.913851	−	0.406049i	$-0.866906\pi$
$2$	−2.58476	−1.82770	−0.913851	+	0.406049i	$0.866906\pi$
$3$	−1.81476	−1.04775	−0.523877	−	0.851794i	$-0.675515\pi$
$3$	−1.81476	−1.04775	−0.523877	+	0.851794i	$0.675515\pi$
$4$	4.68100	2.34050
$5$	2.17232	0.971489	0.485745	−	0.874101i	$-0.338548\pi$
$5$	2.17232	0.971489	0.485745	+	0.874101i	$0.338548\pi$
$6$	4.69073	1.91498
$7$	−4.47910	−1.69294	−0.846470	−	0.532436i	$-0.821277\pi$
$7$	−4.47910	−1.69294	−0.846470	+	0.532436i	$0.821277\pi$
$8$	−6.92974	−2.45003
$9$	0.293363	0.0977878
$10$	−5.61492	−1.77559
$11$	−2.18976	−0.660238	−0.330119	−	0.943939i	$-0.607089\pi$
$11$	−2.18976	−0.660238	−0.330119	+	0.943939i	$0.607089\pi$
$12$	−8.49490	−2.45227
$13$	−4.47644	−1.24154	−0.620771	−	0.783992i	$-0.713180\pi$
$13$	−4.47644	−1.24154	−0.620771	+	0.783992i	$0.713180\pi$
$14$	11.5774	3.09419
$15$	−3.94224	−1.01788
$16$	8.54973	2.13743
$17$	6.20907	1.50592	0.752961	−	0.658066i	$-0.228625\pi$
$17$	6.20907	1.50592	0.752961	+	0.658066i	$0.228625\pi$
$18$	−0.758275	−0.178727
$19$	−7.73483	−1.77449	−0.887246	−	0.461297i	$-0.847384\pi$
$19$	−7.73483	−1.77449	−0.887246	+	0.461297i	$0.847384\pi$
$20$	10.1686	2.27377
$21$	8.12850	1.77378
$22$	5.66001	1.20672
$23$	3.08268	0.642782	0.321391	−	0.946947i	$-0.395850\pi$
$23$	3.08268	0.642782	0.321391	+	0.946947i	$0.395850\pi$
$24$	12.5758	2.56703
$25$	−0.281042	−0.0562084
$26$	11.5705	2.26917
$27$	4.91190	0.945296
$28$	−20.9666	−3.96232
$29$	−8.85800	−1.64489	−0.822444	−	0.568846i	$-0.807390\pi$
$29$	−8.85800	−1.64489	−0.822444	+	0.568846i	$0.807390\pi$
$30$	10.1897	1.86039
$31$	3.24145	0.582182	0.291091	−	0.956695i	$-0.405982\pi$
$31$	3.24145	0.582182	0.291091	+	0.956695i	$0.405982\pi$
$32$	−8.23955	−1.45656
$33$	3.97390	0.691767
$34$	−16.0490	−2.75238
$35$	−9.73002	−1.64467
$36$	1.37323	0.228872
$37$	−11.0326	−1.81375	−0.906873	−	0.421403i	$-0.861538\pi$
$37$	−11.0326	−1.81375	−0.906873	+	0.421403i	$0.861538\pi$
$38$	19.9927	3.24324
$39$	8.12368	1.30083
$40$	−15.0536	−2.38018
$41$	0.364432	0.0569147	0.0284573	−	0.999595i	$-0.490941\pi$
$41$	0.364432	0.0569147	0.0284573	+	0.999595i	$0.490941\pi$
$42$	−21.0102	−3.24195
$43$	3.43325	0.523566	0.261783	−	0.965127i	$-0.415690\pi$
$43$	3.43325	0.523566	0.261783	+	0.965127i	$0.415690\pi$
$44$	−10.2503	−1.54529
$45$	0.637278	0.0949998
$46$	−7.96798	−1.17481
$47$	11.0272	1.60848	0.804241	−	0.594304i	$-0.202572\pi$
$47$	11.0272	1.60848	0.804241	+	0.594304i	$0.202572\pi$
$48$	−15.5157	−2.23950
$49$	13.0623	1.86605
$50$	0.726427	0.102732
$51$	−11.2680	−1.57783
$52$	−20.9542	−2.90583
$53$	3.54326	0.486704	0.243352	−	0.969938i	$-0.421753\pi$
$53$	3.54326	0.486704	0.243352	+	0.969938i	$0.421753\pi$
$54$	−12.6961	−1.72772
$55$	−4.75686	−0.641414
$56$	31.0390	4.14776
$57$	14.0369	1.85923
$58$	22.8958	3.00637
$59$	8.97244	1.16811	0.584056	−	0.811713i	$-0.301465\pi$
$59$	8.97244	1.16811	0.584056	+	0.811713i	$0.301465\pi$
$60$	−18.4536	−2.38235
$61$	−6.76540	−0.866221	−0.433111	−	0.901341i	$-0.642584\pi$
$61$	−6.76540	−0.866221	−0.433111	+	0.901341i	$0.642584\pi$
$62$	−8.37838	−1.06405
$63$	−1.31400	−0.165549
$64$	4.19782	0.524728
$65$	−9.72425	−1.20614
$66$	−10.2716	−1.26434
$67$	12.6464	1.54501	0.772503	−	0.635011i	$-0.219005\pi$
$67$	12.6464	1.54501	0.772503	+	0.635011i	$0.219005\pi$
$68$	29.0646	3.52461
$69$	−5.59432	−0.673477
$70$	25.1498	3.00597
$71$	1.11934	0.132841	0.0664204	−	0.997792i	$-0.478842\pi$
$71$	1.11934	0.132841	0.0664204	+	0.997792i	$0.478842\pi$
$72$	−2.03293	−0.239583
$73$	−1.54385	−0.180694	−0.0903470	−	0.995910i	$-0.528798\pi$
$73$	−1.54385	−0.180694	−0.0903470	+	0.995910i	$0.528798\pi$
$74$	28.5166	3.31499
$75$	0.510025	0.0588926
$76$	−36.2067	−4.15319
$77$	9.80816	1.11774
$78$	−20.9978	−2.37753
$79$	6.30245	0.709081	0.354541	−	0.935041i	$-0.384637\pi$
$79$	6.30245	0.709081	0.354541	+	0.935041i	$0.384637\pi$
$80$	18.5727	2.07649
$81$	−9.79403	−1.08823
$82$	−0.941970	−0.104023
$83$	13.6371	1.49687	0.748435	−	0.663208i	$-0.230806\pi$
$83$	13.6371	1.49687	0.748435	+	0.663208i	$0.230806\pi$
$84$	38.0495	4.15154
$85$	13.4881	1.46299
$86$	−8.87414	−0.956923
$87$	16.0752	1.72344
$88$	15.1745	1.61760
$89$	14.4727	1.53411	0.767053	−	0.641583i	$-0.221722\pi$
$89$	14.4727	1.53411	0.767053	+	0.641583i	$0.221722\pi$
$90$	−1.64721	−0.173631
$91$	20.0504	2.10186
$92$	14.4300	1.50443
$93$	−5.88246	−0.609983
$94$	−28.5027	−2.93983
$95$	−16.8025	−1.72390
$96$	14.9528	1.52612
$97$	−2.46905	−0.250694	−0.125347	−	0.992113i	$-0.540004\pi$
$97$	−2.46905	−0.250694	−0.125347	+	0.992113i	$0.540004\pi$
$98$	−33.7630	−3.41058
$99$	−0.642396	−0.0645632
$100$	−1.31556	−0.131556
$101$	0.0247435	0.00246207	0.00123104	−	0.999999i	$-0.499608\pi$
$101$	0.0247435	0.00246207	0.00123104	+	0.999999i	$0.499608\pi$
$102$	29.1251	2.88381
$103$	1.74957	0.172390	0.0861950	−	0.996278i	$-0.472529\pi$
$103$	1.74957	0.172390	0.0861950	+	0.996278i	$0.472529\pi$
$104$	31.0206	3.04182
$105$	17.6577	1.72321
$106$	−9.15848	−0.889550
$107$	−8.80542	−0.851252	−0.425626	−	0.904899i	$-0.639946\pi$
$107$	−8.80542	−0.851252	−0.425626	+	0.904899i	$0.639946\pi$
$108$	22.9926	2.21246
$109$	−7.50392	−0.718745	−0.359373	−	0.933194i	$-0.617009\pi$
$109$	−7.50392	−0.718745	−0.359373	+	0.933194i	$0.617009\pi$
$110$	12.2953	1.17231
$111$	20.0215	1.90036
$112$	−38.2951	−3.61855
$113$	16.7080	1.57176	0.785879	−	0.618380i	$-0.212211\pi$
$113$	16.7080	1.57176	0.785879	+	0.618380i	$0.212211\pi$
$114$	−36.2820	−3.39812
$115$	6.69655	0.624456
$116$	−41.4642	−3.84986
$117$	−1.31322	−0.121408
$118$	−23.1916	−2.13496
$119$	−27.8111	−2.54943
$120$	27.3187	2.49384
$121$	−6.20494	−0.564086
$122$	17.4870	1.58320
$123$	−0.661357	−0.0596326
$124$	15.1732	1.36259
$125$	−11.4721	−1.02610
$126$	3.39639	0.302574
$127$	2.75655	0.244604	0.122302	−	0.992493i	$-0.460972\pi$
$127$	2.75655	0.244604	0.122302	+	0.992493i	$0.460972\pi$
$128$	5.62874	0.497515
$129$	−6.23054	−0.548568
$130$	25.1349	2.20447
$131$	−12.1043	−1.05755	−0.528777	−	0.848761i	$-0.677349\pi$
$131$	−12.1043	−1.05755	−0.528777	+	0.848761i	$0.677349\pi$
$132$	18.6018	1.61908
$133$	34.6451	3.00411
$134$	−32.6880	−2.82381
$135$	10.6702	0.918345
$136$	−43.0273	−3.68956
$137$	18.0724	1.54403	0.772014	−	0.635605i	$-0.219249\pi$
$137$	18.0724	1.54403	0.772014	+	0.635605i	$0.219249\pi$
$138$	14.4600	1.23092
$139$	16.5507	1.40381	0.701905	−	0.712270i	$-0.252333\pi$
$139$	16.5507	1.40381	0.701905	+	0.712270i	$0.252333\pi$
$140$	−45.5462	−3.84936
$141$	−20.0117	−1.68529
$142$	−2.89322	−0.242794
$143$	9.80234	0.819713
$144$	2.50818	0.209015
$145$	−19.2424	−1.59799
$146$	3.99049	0.330255
$147$	−23.7050	−1.95516
$148$	−51.6435	−4.24507
$149$	6.04823	0.495490	0.247745	−	0.968825i	$-0.420310\pi$
$149$	6.04823	0.495490	0.247745	+	0.968825i	$0.420310\pi$
$150$	−1.31829	−0.107638
$151$	10.2183	0.831550	0.415775	−	0.909468i	$-0.363510\pi$
$151$	10.2183	0.831550	0.415775	+	0.909468i	$0.363510\pi$
$152$	53.6003	4.34756
$153$	1.82152	0.147261
$154$	−25.3518	−2.04290
$155$	7.04145	0.565583
$156$	38.0269	3.04459
$157$	−22.2297	−1.77412	−0.887060	−	0.461654i	$-0.847256\pi$
$157$	−22.2297	−1.77412	−0.887060	+	0.461654i	$0.847256\pi$
$158$	−16.2903	−1.29599
$159$	−6.43017	−0.509946
$160$	−17.8989	−1.41503
$161$	−13.8076	−1.08819
$162$	25.3152	1.98895
$163$	−0.0523957	−0.00410395	−0.00205197	−	0.999998i	$-0.500653\pi$
$163$	−0.0523957	−0.00410395	−0.00205197	+	0.999998i	$0.500653\pi$
$164$	1.70590	0.133209
$165$	8.63256	0.672044
$166$	−35.2487	−2.73583
$167$	−1.20017	−0.0928720	−0.0464360	−	0.998921i	$-0.514786\pi$
$167$	−1.20017	−0.0928720	−0.0464360	+	0.998921i	$0.514786\pi$
$168$	−56.3284	−4.34583
$169$	7.03853	0.541425
$170$	−34.8635	−2.67390
$171$	−2.26912	−0.173524
$172$	16.0710	1.22541
$173$	−20.1102	−1.52895	−0.764476	−	0.644653i	$-0.777002\pi$
$173$	−20.1102	−1.52895	−0.764476	+	0.644653i	$0.777002\pi$
$174$	−41.5505	−3.14993
$175$	1.25882	0.0951575
$176$	−18.7219	−1.41122
$177$	−16.2829	−1.22389
$178$	−37.4086	−2.80389
$179$	−13.3877	−1.00064	−0.500322	−	0.865839i	$-0.666785\pi$
$179$	−13.3877	−1.00064	−0.500322	+	0.865839i	$0.666785\pi$
$180$	2.98310	0.222347
$181$	5.50448	0.409145	0.204572	−	0.978851i	$-0.434420\pi$
$181$	5.50448	0.409145	0.204572	+	0.978851i	$0.434420\pi$
$182$	−51.8256	−3.84157
$183$	12.2776	0.907586
$184$	−21.3621	−1.57484
$185$	−23.9663	−1.76204
$186$	15.2048	1.11487
$187$	−13.5964	−0.994267
$188$	51.6182	3.76465
$189$	−22.0009	−1.60033
$190$	43.4304	3.15078
$191$	0.459224	0.0332283	0.0166141	−	0.999862i	$-0.494711\pi$
$191$	0.459224	0.0332283	0.0166141	+	0.999862i	$0.494711\pi$
$192$	−7.61805	−0.549785
$193$	−1.33642	−0.0961973	−0.0480986	−	0.998843i	$-0.515316\pi$
$193$	−1.33642	−0.0961973	−0.0480986	+	0.998843i	$0.515316\pi$
$194$	6.38189	0.458193
$195$	17.6472	1.26374
$196$	61.1447	4.36748
$197$	−6.15116	−0.438252	−0.219126	−	0.975697i	$-0.570321\pi$
$197$	−6.15116	−0.438252	−0.219126	+	0.975697i	$0.570321\pi$
$198$	1.66044	0.118002
$199$	5.90957	0.418919	0.209459	−	0.977817i	$-0.432830\pi$
$199$	5.90957	0.418919	0.209459	+	0.977817i	$0.432830\pi$
$200$	1.94755	0.137712
$201$	−22.9502	−1.61879
$202$	−0.0639561	−0.00449993
$203$	39.6758	2.78470
$204$	−52.7454	−3.69292
$205$	0.791661	0.0552920
$206$	−4.52222	−0.315078
$207$	0.904344	0.0628563
$208$	−38.2724	−2.65371
$209$	16.9374	1.17159
$210$	−45.6409	−3.14952
$211$	−10.4459	−0.719122	−0.359561	−	0.933122i	$-0.617074\pi$
$211$	−10.4459	−0.719122	−0.359561	+	0.933122i	$0.617074\pi$
$212$	16.5860	1.13913
$213$	−2.03133	−0.139185
$214$	22.7599	1.55584
$215$	7.45811	0.508639
$216$	−34.0382	−2.31601
$217$	−14.5188	−0.985599
$218$	19.3959	1.31365
$219$	2.80172	0.189323
$220$	−22.2668	−1.50123
$221$	−27.7946	−1.86966
$222$	−51.7509	−3.47329
$223$	6.57368	0.440206	0.220103	−	0.975477i	$-0.429361\pi$
$223$	6.57368	0.440206	0.220103	+	0.975477i	$0.429361\pi$
$224$	36.9058	2.46587
$225$	−0.0824475	−0.00549650
$226$	−43.1862	−2.87271
$227$	7.04736	0.467750	0.233875	−	0.972267i	$-0.424859\pi$
$227$	7.04736	0.467750	0.233875	+	0.972267i	$0.424859\pi$
$228$	65.7066	4.35152
$229$	−21.5599	−1.42472	−0.712359	−	0.701815i	$-0.752373\pi$
$229$	−21.5599	−1.42472	−0.712359	+	0.701815i	$0.752373\pi$
$230$	−17.3090	−1.14132
$231$	−17.7995	−1.17112
$232$	61.3836	4.03003
$233$	−24.5995	−1.61157	−0.805783	−	0.592211i	$-0.798255\pi$
$233$	−24.5995	−1.61157	−0.805783	+	0.592211i	$0.798255\pi$
$234$	3.39437	0.221897
$235$	23.9545	1.56262
$236$	42.0000	2.73397
$237$	−11.4375	−0.742943
$238$	71.8850	4.65961
$239$	−26.5061	−1.71454	−0.857270	−	0.514868i	$-0.827841\pi$
$239$	−26.5061	−1.71454	−0.857270	+	0.514868i	$0.827841\pi$
$240$	−33.7051	−2.17565
$241$	10.8578	0.699412	0.349706	−	0.936859i	$-0.386281\pi$
$241$	10.8578	0.699412	0.349706	+	0.936859i	$0.386281\pi$
$242$	16.0383	1.03098
$243$	3.03813	0.194896
$244$	−31.6688	−2.02739
$245$	28.3755	1.81284
$246$	1.70945	0.108991
$247$	34.6245	2.20310
$248$	−22.4624	−1.42636
$249$	−24.7482	−1.56835
$250$	29.6526	1.87540
$251$	−15.5714	−0.982861	−0.491430	−	0.870917i	$-0.663526\pi$
$251$	−15.5714	−0.982861	−0.491430	+	0.870917i	$0.663526\pi$
$252$	−6.15085	−0.387467
$253$	−6.75033	−0.424389
$254$	−7.12502	−0.447064
$255$	−24.4776	−1.53285
$256$	−22.9446	−1.43404
$257$	29.0273	1.81067	0.905336	−	0.424695i	$-0.139619\pi$
$257$	29.0273	1.81067	0.905336	+	0.424695i	$0.139619\pi$
$258$	16.1045	1.00262
$259$	49.4161	3.07056
$260$	−45.5192	−2.82298
$261$	−2.59861	−0.160850
$262$	31.2866	1.93289
$263$	5.22956	0.322468	0.161234	−	0.986916i	$-0.448453\pi$
$263$	5.22956	0.322468	0.161234	+	0.986916i	$0.448453\pi$
$264$	−27.5381	−1.69485
$265$	7.69708	0.472828
$266$	−89.5492	−5.49062
$267$	−26.2646	−1.60737
$268$	59.1978	3.61608
$269$	22.3990	1.36569	0.682845	−	0.730563i	$-0.260742\pi$
$269$	22.3990	1.36569	0.682845	+	0.730563i	$0.260742\pi$
$270$	−27.5799	−1.67846
$271$	−17.0593	−1.03628	−0.518140	−	0.855296i	$-0.673375\pi$
$271$	−17.0593	−1.03628	−0.518140	+	0.855296i	$0.673375\pi$
$272$	53.0859	3.21881
$273$	−36.3868	−2.20223
$274$	−46.7128	−2.82203
$275$	0.615415	0.0371109
$276$	−26.1870	−1.57627
$277$	8.60530	0.517042	0.258521	−	0.966006i	$-0.416765\pi$
$277$	8.60530	0.517042	0.258521	+	0.966006i	$0.416765\pi$
$278$	−42.7796	−2.56575
$279$	0.950923	0.0569303
$280$	67.4265	4.02950
$281$	30.1185	1.79672	0.898361	−	0.439258i	$-0.144759\pi$
$281$	30.1185	1.79672	0.898361	+	0.439258i	$0.144759\pi$
$282$	51.7256	3.08021
$283$	1.89885	0.112875	0.0564374	−	0.998406i	$-0.482026\pi$
$283$	1.89885	0.112875	0.0564374	+	0.998406i	$0.482026\pi$
$284$	5.23961	0.310914
$285$	30.4925	1.80622
$286$	−25.3367	−1.49819
$287$	−1.63233	−0.0963532
$288$	−2.41718	−0.142434
$289$	21.5526	1.26780
$290$	49.7370	2.92065
$291$	4.48073	0.262665
$292$	−7.22676	−0.422914
$293$	27.4289	1.60241	0.801206	−	0.598389i	$-0.204192\pi$
$293$	27.4289	1.60241	0.801206	+	0.598389i	$0.204192\pi$
$294$	61.2719	3.57345
$295$	19.4910	1.13481
$296$	76.4530	4.44374
$297$	−10.7559	−0.624121
$298$	−15.6332	−0.905609
$299$	−13.7994	−0.798041
$300$	2.38742	0.137838
$301$	−15.3779	−0.886366
$302$	−26.4118	−1.51983
$303$	−0.0449036	−0.00257964
$304$	−66.1307	−3.79286
$305$	−14.6966	−0.841525
$306$	−4.70818	−0.269149
$307$	0.0418578	0.00238895	0.00119447	−	0.999999i	$-0.499620\pi$
$307$	0.0418578	0.00238895	0.00119447	+	0.999999i	$0.499620\pi$
$308$	45.9120	2.61608
$309$	−3.17505	−0.180622
$310$	−18.2005	−1.03372
$311$	−1.58404	−0.0898226	−0.0449113	−	0.998991i	$-0.514301\pi$
$311$	−1.58404	−0.0898226	−0.0449113	+	0.998991i	$0.514301\pi$
$312$	−56.2950	−3.18708
$313$	−30.5923	−1.72918	−0.864589	−	0.502480i	$-0.832421\pi$
$313$	−30.5923	−1.72918	−0.864589	+	0.502480i	$0.832421\pi$
$314$	57.4584	3.24256
$315$	−2.85443	−0.160829
$316$	29.5018	1.65960
$317$	−20.9842	−1.17859	−0.589295	−	0.807918i	$-0.700595\pi$
$317$	−20.9842	−1.17859	−0.589295	+	0.807918i	$0.700595\pi$
$318$	16.6205	0.932030
$319$	19.3969	1.08602
$320$	9.11900	0.509767
$321$	15.9797	0.891903
$322$	35.6894	1.98889
$323$	−48.0261	−2.67224
$324$	−45.8458	−2.54699
$325$	1.25807	0.0697851
$326$	0.135430	0.00750079
$327$	13.6178	0.753068
$328$	−2.52542	−0.139443
$329$	−49.3919	−2.72306
$330$	−22.3131	−1.22830
$331$	8.82134	0.484864	0.242432	−	0.970168i	$-0.422055\pi$
$331$	8.82134	0.484864	0.242432	+	0.970168i	$0.422055\pi$
$332$	63.8354	3.50342
$333$	−3.23656	−0.177362
$334$	3.10216	0.169742
$335$	27.4720	1.50096
$336$	69.4965	3.79135
$337$	34.5454	1.88181	0.940905	−	0.338670i	$-0.109977\pi$
$337$	34.5454	1.88181	0.940905	+	0.338670i	$0.109977\pi$
$338$	−18.1929	−0.989565
$339$	−30.3211	−1.64681
$340$	63.1376	3.42412
$341$	−7.09800	−0.384378
$342$	5.86513	0.317150
$343$	−27.1538	−1.46617
$344$	−23.7915	−1.28275
$345$	−12.1526	−0.654276
$346$	51.9801	2.79447
$347$	24.2506	1.30184	0.650921	−	0.759146i	$-0.274383\pi$
$347$	24.2506	1.30184	0.650921	+	0.759146i	$0.274383\pi$
$348$	75.2478	4.03370
$349$	−22.3961	−1.19884	−0.599418	−	0.800436i	$-0.704601\pi$
$349$	−22.3961	−1.19884	−0.599418	+	0.800436i	$0.704601\pi$
$350$	−3.25374	−0.173920
$351$	−21.9878	−1.17362
$352$	18.0427	0.961677
$353$	4.02392	0.214171	0.107086	−	0.994250i	$-0.465848\pi$
$353$	4.02392	0.214171	0.107086	+	0.994250i	$0.465848\pi$
$354$	42.0873	2.23692
$355$	2.43155	0.129053
$356$	67.7468	3.59057
$357$	50.4705	2.67118
$358$	34.6040	1.82888
$359$	16.4485	0.868117	0.434059	−	0.900885i	$-0.357081\pi$
$359$	16.4485	0.868117	0.434059	+	0.900885i	$0.357081\pi$
$360$	−4.41617	−0.232753
$361$	40.8276	2.14882
$362$	−14.2278	−0.747795
$363$	11.2605	0.591023
$364$	93.8560	4.91939
$365$	−3.35373	−0.175542
$366$	−31.7347	−1.65880
$367$	8.07393	0.421456	0.210728	−	0.977545i	$-0.432417\pi$
$367$	8.07393	0.421456	0.210728	+	0.977545i	$0.432417\pi$
$368$	26.3561	1.37390
$369$	0.106911	0.00556556
$370$	61.9471	3.22048
$371$	−15.8706	−0.823961
$372$	−27.5358	−1.42766
$373$	−1.73236	−0.0896982	−0.0448491	−	0.998994i	$-0.514281\pi$
$373$	−1.73236	−0.0896982	−0.0448491	+	0.998994i	$0.514281\pi$
$374$	35.1434	1.81722
$375$	20.8191	1.07510
$376$	−76.4156	−3.94083
$377$	39.6523	2.04220
$378$	56.8671	2.92493
$379$	15.8671	0.815038	0.407519	−	0.913197i	$-0.366394\pi$
$379$	15.8671	0.815038	0.407519	+	0.913197i	$0.366394\pi$
$380$	−78.6524	−4.03478
$381$	−5.00248	−0.256285
$382$	−1.18698	−0.0607314
$383$	−9.39082	−0.479849	−0.239924	−	0.970792i	$-0.577123\pi$
$383$	−9.39082	−0.479849	−0.239924	+	0.970792i	$0.577123\pi$
$384$	−10.2148	−0.521273
$385$	21.3064	1.08588
$386$	3.45432	0.175820
$387$	1.00719	0.0511984
$388$	−11.5576	−0.586748
$389$	−27.5792	−1.39832	−0.699160	−	0.714965i	$-0.746442\pi$
$389$	−27.5792	−1.39832	−0.699160	+	0.714965i	$0.746442\pi$
$390$	−45.6138	−2.30975
$391$	19.1406	0.967979
$392$	−90.5185	−4.57188
$393$	21.9664	1.10806
$394$	15.8993	0.800994
$395$	13.6909	0.688865
$396$	−3.00705	−0.151110
$397$	9.91414	0.497577	0.248788	−	0.968558i	$-0.419968\pi$
$397$	9.91414	0.497577	0.248788	+	0.968558i	$0.419968\pi$
$398$	−15.2748	−0.765659
$399$	−62.8726	−3.14757
$400$	−2.40283	−0.120142
$401$	−22.6488	−1.13103	−0.565513	−	0.824739i	$-0.691322\pi$
$401$	−22.6488	−1.13103	−0.565513	+	0.824739i	$0.691322\pi$
$402$	59.3209	2.95866
$403$	−14.5102	−0.722803
$404$	0.115824	0.00576247
$405$	−21.2757	−1.05720
$406$	−102.553	−5.08960
$407$	24.1587	1.19750
$408$	78.0842	3.86575
$409$	−0.269714	−0.0133365	−0.00666825	−	0.999978i	$-0.502123\pi$
$409$	−0.269714	−0.0133365	−0.00666825	+	0.999978i	$0.502123\pi$
$410$	−2.04626	−0.101057
$411$	−32.7971	−1.61776
$412$	8.18972	0.403479
$413$	−40.1885	−1.97754
$414$	−2.33752	−0.114883
$415$	29.6242	1.45419
$416$	36.8839	1.80838
$417$	−30.0356	−1.47085
$418$	−43.7792	−2.14131
$419$	−8.20748	−0.400962	−0.200481	−	0.979698i	$-0.564250\pi$
$419$	−8.20748	−0.400962	−0.200481	+	0.979698i	$0.564250\pi$
$420$	82.6555	4.03318
$421$	−30.3914	−1.48119	−0.740593	−	0.671953i	$-0.765456\pi$
$421$	−30.3914	−1.48119	−0.740593	+	0.671953i	$0.765456\pi$
$422$	27.0000	1.31434
$423$	3.23498	0.157290
$424$	−24.5539	−1.19244
$425$	−1.74501	−0.0846454
$426$	5.25051	0.254388
$427$	30.3029	1.46646
$428$	−41.2181	−1.99235
$429$	−17.7889	−0.858857
$430$	−19.2774	−0.929641
$431$	5.68296	0.273738	0.136869	−	0.990589i	$-0.456296\pi$
$431$	5.68296	0.273738	0.136869	+	0.990589i	$0.456296\pi$
$432$	41.9955	2.02051
$433$	15.1116	0.726215	0.363108	−	0.931747i	$-0.381716\pi$
$433$	15.1116	0.726215	0.363108	+	0.931747i	$0.381716\pi$
$434$	37.5276	1.80138
$435$	34.9203	1.67430
$436$	−35.1258	−1.68222
$437$	−23.8440	−1.14061
$438$	−7.24179	−0.346026
$439$	−26.4749	−1.26358	−0.631790	−	0.775140i	$-0.717679\pi$
$439$	−26.4749	−1.26358	−0.631790	+	0.775140i	$0.717679\pi$
$440$	32.9638	1.57149
$441$	3.83201	0.182477
$442$	71.8423	3.41719
$443$	−11.5267	−0.547650	−0.273825	−	0.961780i	$-0.588289\pi$
$443$	−11.5267	−0.547650	−0.273825	+	0.961780i	$0.588289\pi$
$444$	93.7207	4.44779
$445$	31.4394	1.49037
$446$	−16.9914	−0.804566
$447$	−10.9761	−0.519152
$448$	−18.8025	−0.888333
$449$	38.7332	1.82793	0.913966	−	0.405791i	$-0.133004\pi$
$449$	38.7332	1.82793	0.913966	+	0.405791i	$0.133004\pi$
$450$	0.213107	0.0100460
$451$	−0.798019	−0.0375772
$452$	78.2102	3.67870
$453$	−18.5437	−0.871259
$454$	−18.2157	−0.854907
$455$	43.5559	2.04193
$456$	−97.2719	−4.55517
$457$	12.0146	0.562019	0.281009	−	0.959705i	$-0.409331\pi$
$457$	12.0146	0.562019	0.281009	+	0.959705i	$0.409331\pi$
$458$	55.7272	2.60396
$459$	30.4984	1.42354
$460$	31.3465	1.46154
$461$	−25.6141	−1.19297	−0.596484	−	0.802625i	$-0.703436\pi$
$461$	−25.6141	−1.19297	−0.596484	+	0.802625i	$0.703436\pi$
$462$	46.0074	2.14046
$463$	5.05715	0.235026	0.117513	−	0.993071i	$-0.462508\pi$
$463$	5.05715	0.235026	0.117513	+	0.993071i	$0.462508\pi$
$464$	−75.7335	−3.51584
$465$	−12.7786	−0.592592
$466$	63.5838	2.94546
$467$	−33.4184	−1.54642	−0.773209	−	0.634151i	$-0.781350\pi$
$467$	−33.4184	−1.54642	−0.773209	+	0.634151i	$0.781350\pi$
$468$	−6.14720	−0.284154
$469$	−56.6446	−2.61560
$470$	−61.9168	−2.85601
$471$	40.3415	1.85884
$472$	−62.1767	−2.86191
$473$	−7.51800	−0.345678
$474$	29.5631	1.35788
$475$	2.17381	0.0997413
$476$	−130.183	−5.96695
$477$	1.03946	0.0475937
$478$	68.5121	3.13367
$479$	−11.5816	−0.529175	−0.264587	−	0.964362i	$-0.585236\pi$
$479$	−11.5816	−0.529175	−0.264587	+	0.964362i	$0.585236\pi$
$480$	32.4823	1.48261
$481$	49.3867	2.25184
$482$	−28.0648	−1.27832
$483$	25.0575	1.14016
$484$	−29.0453	−1.32024
$485$	−5.36355	−0.243546
$486$	−7.85284	−0.356212
$487$	26.2985	1.19170	0.595850	−	0.803096i	$-0.296815\pi$
$487$	26.2985	1.19170	0.595850	+	0.803096i	$0.296815\pi$
$488$	46.8825	2.12227
$489$	0.0950857	0.00429992
$490$	−73.3439	−3.31334
$491$	12.2771	0.554057	0.277029	−	0.960862i	$-0.410650\pi$
$491$	12.2771	0.554057	0.277029	+	0.960862i	$0.410650\pi$
$492$	−3.09581	−0.139570
$493$	−54.9999	−2.47707
$494$	−89.4961	−4.02662
$495$	−1.39549	−0.0627225
$496$	27.7135	1.24437
$497$	−5.01362	−0.224892
$498$	63.9681	2.86648
$499$	21.0717	0.943298	0.471649	−	0.881786i	$-0.343659\pi$
$499$	21.0717	0.943298	0.471649	+	0.881786i	$0.343659\pi$
$500$	−53.7008	−2.40157
$501$	2.17802	0.0973070
$502$	40.2485	1.79638
$503$	−33.1787	−1.47937	−0.739684	−	0.672955i	$-0.765025\pi$
$503$	−33.1787	−1.47937	−0.739684	+	0.672955i	$0.765025\pi$
$504$	9.10570	0.405600
$505$	0.0537507	0.00239188
$506$	17.4480	0.775658
$507$	−12.7733	−0.567280
$508$	12.9034	0.572496
$509$	−22.9257	−1.01616	−0.508081	−	0.861309i	$-0.669645\pi$
$509$	−22.9257	−1.01616	−0.508081	+	0.861309i	$0.669645\pi$
$510$	63.2689	2.80159
$511$	6.91506	0.305904
$512$	48.0488	2.12348
$513$	−37.9927	−1.67742
$514$	−75.0287	−3.30937
$515$	3.80062	0.167475
$516$	−29.1651	−1.28392
$517$	−24.1469	−1.06198
$518$	−127.729	−5.61208
$519$	36.4953	1.60196
$520$	67.3865	2.95509
$521$	18.7562	0.821725	0.410862	−	0.911697i	$-0.365228\pi$
$521$	18.7562	0.821725	0.410862	+	0.911697i	$0.365228\pi$
$522$	6.71680	0.293986
$523$	−9.12664	−0.399080	−0.199540	−	0.979890i	$-0.563945\pi$
$523$	−9.12664	−0.399080	−0.199540	+	0.979890i	$0.563945\pi$
$524$	−56.6600	−2.47520
$525$	−2.28445	−0.0997016
$526$	−13.5172	−0.589377
$527$	20.1264	0.876720
$528$	33.9758	1.47861
$529$	−13.4971	−0.586831
$530$	−19.8951	−0.864189
$531$	2.63219	0.114227
$532$	162.173	7.03111
$533$	−1.63136	−0.0706619
$534$	67.8877	2.93779
$535$	−19.1282	−0.826982
$536$	−87.6364	−3.78531
$537$	24.2955	1.04843
$538$	−57.8961	−2.49608
$539$	−28.6034	−1.23204
$540$	49.9472	2.14939
$541$	−1.02263	−0.0439662	−0.0219831	−	0.999758i	$-0.506998\pi$
$541$	−1.02263	−0.0439662	−0.0219831	+	0.999758i	$0.506998\pi$
$542$	44.0943	1.89401
$543$	−9.98932	−0.428683
$544$	−51.1600	−2.19347
$545$	−16.3009	−0.698253
$546$	94.0511	4.02502
$547$	7.98807	0.341545	0.170773	−	0.985310i	$-0.445374\pi$
$547$	7.98807	0.341545	0.170773	+	0.985310i	$0.445374\pi$
$548$	84.5968	3.61380
$549$	−1.98472	−0.0847059
$550$	−1.59070	−0.0678278
$551$	68.5151	2.91884
$552$	38.7672	1.65004
$553$	−28.2293	−1.20043
$554$	−22.2427	−0.945000
$555$	43.4931	1.84618
$556$	77.4737	3.28562
$557$	13.1179	0.555825	0.277912	−	0.960606i	$-0.410357\pi$
$557$	13.1179	0.555825	0.277912	+	0.960606i	$0.410357\pi$
$558$	−2.45791	−0.104052
$559$	−15.3688	−0.650029
$560$	−83.1891	−3.51538
$561$	24.6742	1.04175
$562$	−77.8493	−3.28387
$563$	24.7088	1.04135	0.520676	−	0.853754i	$-0.325680\pi$
$563$	24.7088	1.04135	0.520676	+	0.853754i	$0.325680\pi$
$564$	−93.6749	−3.94442
$565$	36.2951	1.52695
$566$	−4.90807	−0.206302
$567$	43.8684	1.84230
$568$	−7.75671	−0.325464
$569$	5.32794	0.223359	0.111679	−	0.993744i	$-0.464377\pi$
$569$	5.32794	0.223359	0.111679	+	0.993744i	$0.464377\pi$
$570$	−78.8160	−3.30124
$571$	−31.0452	−1.29920	−0.649601	−	0.760275i	$-0.725064\pi$
$571$	−31.0452	−1.29920	−0.649601	+	0.760275i	$0.725064\pi$
$572$	45.8847	1.91854
$573$	−0.833382	−0.0348150
$574$	4.21918	0.176105
$575$	−0.866361	−0.0361298
$576$	1.23149	0.0513120
$577$	−13.0999	−0.545358	−0.272679	−	0.962105i	$-0.587910\pi$
$577$	−13.0999	−0.545358	−0.272679	+	0.962105i	$0.587910\pi$
$578$	−55.7083	−2.31716
$579$	2.42528	0.100791
$580$	−90.0735	−3.74010
$581$	−61.0821	−2.53411
$582$	−11.5816	−0.480074
$583$	−7.75889	−0.321341
$584$	10.6985	0.442706
$585$	−2.85274	−0.117946
$586$	−70.8971	−2.92873
$587$	−4.83274	−0.199469	−0.0997343	−	0.995014i	$-0.531799\pi$
$587$	−4.83274	−0.199469	−0.0997343	+	0.995014i	$0.531799\pi$
$588$	−110.963	−4.57604
$589$	−25.0721	−1.03308
$590$	−50.3796	−2.07409
$591$	11.1629	0.459180
$592$	−94.3257	−3.87676
$593$	5.52926	0.227059	0.113530	−	0.993535i	$-0.463784\pi$
$593$	5.52926	0.227059	0.113530	+	0.993535i	$0.463784\pi$
$594$	27.8014	1.14071
$595$	−60.4144	−2.47675
$596$	28.3117	1.15969
$597$	−10.7245	−0.438924
$598$	35.6682	1.45858
$599$	−7.20388	−0.294343	−0.147171	−	0.989111i	$-0.547017\pi$
$599$	−7.20388	−0.294343	−0.147171	+	0.989111i	$0.547017\pi$
$600$	−3.53434	−0.144289
$601$	−14.9011	−0.607830	−0.303915	−	0.952699i	$-0.598294\pi$
$601$	−14.9011	−0.607830	−0.303915	+	0.952699i	$0.598294\pi$
$602$	39.7482	1.62001
$603$	3.71000	0.151083
$604$	47.8316	1.94624
$605$	−13.4791	−0.548003
$606$	0.116065	0.00471482
$607$	34.4216	1.39713	0.698565	−	0.715546i	$-0.253822\pi$
$607$	34.4216	1.39713	0.698565	+	0.715546i	$0.253822\pi$
$608$	63.7315	2.58466
$609$	−72.0022	−2.91768
$610$	37.9872	1.53806
$611$	−49.3626	−1.99700
$612$	8.52651	0.344664
$613$	−15.7292	−0.635297	−0.317648	−	0.948209i	$-0.602893\pi$
$613$	−15.7292	−0.635297	−0.317648	+	0.948209i	$0.602893\pi$
$614$	−0.108192	−0.00436629
$615$	−1.43668	−0.0579324
$616$	−67.9680	−2.73851
$617$	−0.632171	−0.0254502	−0.0127251	−	0.999919i	$-0.504051\pi$
$617$	−0.632171	−0.0254502	−0.0127251	+	0.999919i	$0.504051\pi$
$618$	8.20675	0.330124
$619$	3.54839	0.142622	0.0713110	−	0.997454i	$-0.477282\pi$
$619$	3.54839	0.142622	0.0713110	+	0.997454i	$0.477282\pi$
$620$	32.9610	1.32375
$621$	15.1418	0.607620
$622$	4.09436	0.164169
$623$	−64.8248	−2.59715
$624$	69.4553	2.78044
$625$	−23.5158	−0.940632
$626$	79.0738	3.16042
$627$	−30.7374	−1.22753
$628$	−104.057	−4.15232
$629$	−68.5022	−2.73136
$630$	7.37803	0.293948
$631$	20.4059	0.812346	0.406173	−	0.913796i	$-0.366863\pi$
$631$	20.4059	0.812346	0.406173	+	0.913796i	$0.366863\pi$
$632$	−43.6743	−1.73727
$633$	18.9567	0.753463
$634$	54.2392	2.15411
$635$	5.98810	0.237630
$636$	−30.0996	−1.19353
$637$	−58.4728	−2.31677
$638$	−50.1364	−1.98492
$639$	0.328373	0.0129902
$640$	12.2274	0.483330
$641$	14.0579	0.555252	0.277626	−	0.960689i	$-0.410452\pi$
$641$	14.0579	0.555252	0.277626	+	0.960689i	$0.410452\pi$
$642$	−41.3038	−1.63013
$643$	9.67431	0.381518	0.190759	−	0.981637i	$-0.438905\pi$
$643$	9.67431	0.381518	0.190759	+	0.981637i	$0.438905\pi$
$644$	−64.6334	−2.54691
$645$	−13.5347	−0.532928
$646$	124.136	4.88407
$647$	−29.4287	−1.15696	−0.578482	−	0.815695i	$-0.696355\pi$
$647$	−29.4287	−1.15696	−0.578482	+	0.815695i	$0.696355\pi$
$648$	67.8701	2.66619
$649$	−19.6475	−0.771232
$650$	−3.25181	−0.127546
$651$	26.3481	1.03266
$652$	−0.245264	−0.00960528
$653$	16.4997	0.645682	0.322841	−	0.946453i	$-0.395362\pi$
$653$	16.4997	0.645682	0.322841	+	0.946453i	$0.395362\pi$
$654$	−35.1989	−1.37638
$655$	−26.2943	−1.02740
$656$	3.11580	0.121651
$657$	−0.452909	−0.0176697
$658$	127.666	4.97695
$659$	23.4551	0.913681	0.456841	−	0.889549i	$-0.348981\pi$
$659$	23.4551	0.913681	0.456841	+	0.889549i	$0.348981\pi$
$660$	40.4090	1.57292
$661$	−20.2330	−0.786974	−0.393487	−	0.919330i	$-0.628731\pi$
$661$	−20.2330	−0.786974	−0.393487	+	0.919330i	$0.628731\pi$
$662$	−22.8011	−0.886188
$663$	50.4405	1.95895
$664$	−94.5018	−3.66738
$665$	75.2600	2.91846
$666$	8.36574	0.324166
$667$	−27.3063	−1.05731
$668$	−5.61799	−0.217367
$669$	−11.9297	−0.461227
$670$	−71.0086	−2.74330
$671$	14.8146	0.571912
$672$	−66.9752	−2.58363
$673$	16.1750	0.623500	0.311750	−	0.950164i	$-0.399085\pi$
$673$	16.1750	0.623500	0.311750	+	0.950164i	$0.399085\pi$
$674$	−89.2918	−3.43939
$675$	−1.38045	−0.0531336
$676$	32.9473	1.26721
$677$	−18.0833	−0.694998	−0.347499	−	0.937680i	$-0.612969\pi$
$677$	−18.0833	−0.694998	−0.347499	+	0.937680i	$0.612969\pi$
$678$	78.3728	3.00989
$679$	11.0591	0.424409
$680$	−93.4688	−3.58436
$681$	−12.7893	−0.490086
$682$	18.3467	0.702530
$683$	24.1333	0.923434	0.461717	−	0.887027i	$-0.347234\pi$
$683$	24.1333	0.923434	0.461717	+	0.887027i	$0.347234\pi$
$684$	−10.6217	−0.406132
$685$	39.2589	1.50001
$686$	70.1860	2.67972
$687$	39.1261	1.49275
$688$	29.3534	1.11909
$689$	−15.8612	−0.604263
$690$	31.4117	1.19582
$691$	36.4749	1.38757	0.693785	−	0.720183i	$-0.255942\pi$
$691$	36.4749	1.38757	0.693785	+	0.720183i	$0.255942\pi$
$692$	−94.1358	−3.57851
$693$	2.87736	0.109302
$694$	−62.6821	−2.37938
$695$	35.9533	1.36379
$696$	−111.397	−4.22248
$697$	2.26278	0.0857090
$698$	57.8886	2.19111
$699$	44.6422	1.68852
$700$	5.89251	0.222716
$701$	13.1373	0.496188	0.248094	−	0.968736i	$-0.420196\pi$
$701$	13.1373	0.496188	0.248094	+	0.968736i	$0.420196\pi$
$702$	56.8334	2.14504
$703$	85.3352	3.21848
$704$	−9.19223	−0.346445
$705$	−43.4718	−1.63724
$706$	−10.4009	−0.391442
$707$	−0.110829	−0.00416814
$708$	−76.2200	−2.86452
$709$	−6.05540	−0.227415	−0.113708	−	0.993514i	$-0.536273\pi$
$709$	−6.05540	−0.227415	−0.113708	+	0.993514i	$0.536273\pi$
$710$	−6.28499	−0.235871
$711$	1.84891	0.0693395
$712$	−100.292	−3.75861
$713$	9.99234	0.374216
$714$	−130.454	−4.88212
$715$	21.2938	0.796342
$716$	−62.6678	−2.34201
$717$	48.1023	1.79641
$718$	−42.5154	−1.58666
$719$	0.486388	0.0181392	0.00906961	−	0.999959i	$-0.497113\pi$
$719$	0.486388	0.0181392	0.00906961	+	0.999959i	$0.497113\pi$
$720$	5.44856	0.203056
$721$	−7.83649	−0.291846
$722$	−105.530	−3.92740
$723$	−19.7043	−0.732812
$724$	25.7664	0.957602
$725$	2.48947	0.0924566
$726$	−29.1057	−1.08021
$727$	26.5832	0.985917	0.492959	−	0.870053i	$-0.335915\pi$
$727$	26.5832	0.985917	0.492959	+	0.870053i	$0.335915\pi$
$728$	−138.944	−5.14961
$729$	23.8686	0.884022
$730$	8.66860	0.320839
$731$	21.3173	0.788449
$732$	57.4714	2.12420
$733$	−44.6287	−1.64840	−0.824200	−	0.566299i	$-0.808375\pi$
$733$	−44.6287	−1.64840	−0.824200	+	0.566299i	$0.808375\pi$
$734$	−20.8692	−0.770296
$735$	−51.4948	−1.89941
$736$	−25.3999	−0.936252
$737$	−27.6926	−1.02007
$738$	−0.276339	−0.0101722
$739$	−30.7861	−1.13248	−0.566242	−	0.824239i	$-0.691603\pi$
$739$	−30.7861	−1.13248	−0.566242	+	0.824239i	$0.691603\pi$
$740$	−112.186	−4.12404
$741$	−62.8353	−2.30831
$742$	41.0217	1.50596
$743$	50.9887	1.87059	0.935297	−	0.353863i	$-0.115132\pi$
$743$	50.9887	1.87059	0.935297	+	0.353863i	$0.115132\pi$
$744$	40.7639	1.49448
$745$	13.1387	0.481364
$746$	4.47774	0.163942
$747$	4.00064	0.146376
$748$	−63.6447	−2.32708
$749$	39.4403	1.44112
$750$	−53.8125	−1.96495
$751$	9.39105	0.342684	0.171342	−	0.985212i	$-0.445190\pi$
$751$	9.39105	0.342684	0.171342	+	0.985212i	$0.445190\pi$
$752$	94.2796	3.43802
$753$	28.2585	1.02980
$754$	−102.492	−3.73253
$755$	22.1973	0.807842
$756$	−102.986	−3.74557
$757$	−17.6787	−0.642542	−0.321271	−	0.946987i	$-0.604110\pi$
$757$	−17.6787	−0.642542	−0.321271	+	0.946987i	$0.604110\pi$
$758$	−41.0127	−1.48965
$759$	12.2502	0.444655
$760$	116.437	4.22361
$761$	−36.4348	−1.32076	−0.660380	−	0.750931i	$-0.729605\pi$
$761$	−36.4348	−1.32076	−0.660380	+	0.750931i	$0.729605\pi$
$762$	12.9302	0.468413
$763$	33.6108	1.21679
$764$	2.14962	0.0777707
$765$	3.95691	0.143062
$766$	24.2730	0.877021
$767$	−40.1646	−1.45026
$768$	41.6390	1.50252
$769$	−20.2360	−0.729728	−0.364864	−	0.931061i	$-0.618885\pi$
$769$	−20.2360	−0.729728	−0.364864	+	0.931061i	$0.618885\pi$
$770$	−55.0721	−1.98466
$771$	−52.6777	−1.89714
$772$	−6.25576	−0.225150
$773$	48.3697	1.73974	0.869868	−	0.493285i	$-0.164204\pi$
$773$	48.3697	1.73974	0.869868	+	0.493285i	$0.164204\pi$
$774$	−2.60335	−0.0935754
$775$	−0.910984	−0.0327235
$776$	17.1098	0.614207
$777$	−89.6784	−3.21720
$778$	71.2856	2.55571
$779$	−2.81882	−0.100995
$780$	82.6065	2.95779
$781$	−2.45108	−0.0877066
$782$	−49.4738	−1.76918
$783$	−43.5096	−1.55491
$784$	111.679	3.98855
$785$	−48.2898	−1.72354
$786$	−56.7778	−2.02520
$787$	16.5168	0.588759	0.294380	−	0.955689i	$-0.404887\pi$
$787$	16.5168	0.588759	0.294380	+	0.955689i	$0.404887\pi$
$788$	−28.7936	−1.02573
$789$	−9.49041	−0.337867
$790$	−35.3878	−1.25904
$791$	−74.8368	−2.66089
$792$	4.45164	0.158182
$793$	30.2849	1.07545
$794$	−25.6257	−0.909422
$795$	−13.9684	−0.495407
$796$	27.6627	0.980478
$797$	−53.9622	−1.91144	−0.955720	−	0.294279i	$-0.904920\pi$
$797$	−53.9622	−1.91144	−0.955720	+	0.294279i	$0.904920\pi$
$798$	162.511	5.75281
$799$	68.4686	2.42225
$800$	2.31566	0.0818710
$801$	4.24577	0.150017
$802$	58.5417	2.06718
$803$	3.38067	0.119301
$804$	−107.430	−3.78876
$805$	−29.9945	−1.05717
$806$	37.5053	1.32107
$807$	−40.6488	−1.43091
$808$	−0.171466	−0.00603215
$809$	41.0414	1.44294	0.721469	−	0.692447i	$-0.243467\pi$
$809$	41.0414	1.44294	0.721469	+	0.692447i	$0.243467\pi$
$810$	54.9927	1.93225
$811$	0.880947	0.0309342	0.0154671	−	0.999880i	$-0.495076\pi$
$811$	0.880947	0.0309342	0.0154671	+	0.999880i	$0.495076\pi$
$812$	185.722	6.51758
$813$	30.9586	1.08577
$814$	−62.4446	−2.18868
$815$	−0.113820	−0.00398694
$816$	−96.3384	−3.37252
$817$	−26.5556	−0.929063
$818$	0.697147	0.0243752
$819$	5.88206	0.205536
$820$	3.70576	0.129411
$821$	6.50719	0.227102	0.113551	−	0.993532i	$-0.463777\pi$
$821$	6.50719	0.227102	0.113551	+	0.993532i	$0.463777\pi$
$822$	84.7727	2.95679
$823$	−46.4247	−1.61826	−0.809131	−	0.587628i	$-0.800062\pi$
$823$	−46.4247	−1.61826	−0.809131	+	0.587628i	$0.800062\pi$
$824$	−12.1240	−0.422361
$825$	−1.11683	−0.0388831
$826$	103.878	3.61436
$827$	−33.9089	−1.17913	−0.589564	−	0.807722i	$-0.700700\pi$
$827$	−33.9089	−1.17913	−0.589564	+	0.807722i	$0.700700\pi$
$828$	4.23323	0.147115
$829$	29.1322	1.01180	0.505902	−	0.862591i	$-0.331160\pi$
$829$	29.1322	1.01180	0.505902	+	0.862591i	$0.331160\pi$
$830$	−76.5714	−2.65783
$831$	−15.6166	−0.541733
$832$	−18.7913	−0.651471
$833$	81.1050	2.81012
$834$	77.6348	2.68827
$835$	−2.60715	−0.0902242
$836$	79.2841	2.74210
$837$	15.9217	0.550334
$838$	21.2144	0.732839
$839$	21.8667	0.754922	0.377461	−	0.926025i	$-0.376797\pi$
$839$	21.8667	0.754922	0.377461	+	0.926025i	$0.376797\pi$
$840$	−122.363	−4.22193
$841$	49.4641	1.70566
$842$	78.5546	2.70717
$843$	−54.6580	−1.88252
$844$	−48.8970	−1.68310
$845$	15.2899	0.525989
$846$	−8.36164	−0.287479
$847$	27.7926	0.954963
$848$	30.2939	1.04030
$849$	−3.44596	−0.118265
$850$	4.51044	0.154707
$851$	−34.0099	−1.16584
$852$	−9.50865	−0.325761
$853$	26.8259	0.918503	0.459251	−	0.888306i	$-0.348118\pi$
$853$	26.8259	0.918503	0.459251	+	0.888306i	$0.348118\pi$
$854$	−78.3258	−2.68025
$855$	−4.92924	−0.168576
$856$	61.0193	2.08560
$857$	43.5575	1.48790	0.743948	−	0.668237i	$-0.232951\pi$
$857$	43.5575	1.48790	0.743948	+	0.668237i	$0.232951\pi$
$858$	45.9801	1.56974
$859$	34.0573	1.16202	0.581010	−	0.813896i	$-0.302658\pi$
$859$	34.0573	1.16202	0.581010	+	0.813896i	$0.302658\pi$
$860$	34.9114	1.19047
$861$	2.96228	0.100954
$862$	−14.6891	−0.500313
$863$	−42.7112	−1.45391	−0.726953	−	0.686687i	$-0.759064\pi$
$863$	−42.7112	−1.45391	−0.726953	+	0.686687i	$0.759064\pi$
$864$	−40.4719	−1.37688
$865$	−43.6857	−1.48536
$866$	−39.0598	−1.32731
$867$	−39.1128	−1.32834
$868$	−67.9623	−2.30679
$869$	−13.8009	−0.468163
$870$	−90.2608	−3.06013
$871$	−56.6109	−1.91819
$872$	52.0002	1.76095
$873$	−0.724328	−0.0245148
$874$	61.6310	2.08470
$875$	51.3846	1.73712
$876$	13.1149	0.443110
$877$	−18.5526	−0.626477	−0.313239	−	0.949674i	$-0.601414\pi$
$877$	−18.5526	−0.626477	−0.313239	+	0.949674i	$0.601414\pi$
$878$	68.4314	2.30945
$879$	−49.7769	−1.67893
$880$	−40.6699	−1.37098
$881$	−44.9825	−1.51550	−0.757749	−	0.652546i	$-0.773701\pi$
$881$	−44.9825	−1.51550	−0.757749	+	0.652546i	$0.773701\pi$
$882$	−9.90484	−0.333513
$883$	−27.2174	−0.915937	−0.457969	−	0.888968i	$-0.651423\pi$
$883$	−27.2174	−0.915937	−0.457969	+	0.888968i	$0.651423\pi$
$884$	−130.106	−4.37594
$885$	−35.3715	−1.18900
$886$	29.7938	1.00094
$887$	38.9487	1.30777	0.653885	−	0.756594i	$-0.273138\pi$
$887$	38.9487	1.30777	0.653885	+	0.756594i	$0.273138\pi$
$888$	−138.744	−4.65594
$889$	−12.3469	−0.414100
$890$	−81.2633	−2.72395
$891$	21.4466	0.718488
$892$	30.7714	1.03030
$893$	−85.2934	−2.85424
$894$	28.3706	0.948855
$895$	−29.0823	−0.972115
$896$	−25.2117	−0.842263
$897$	25.0427	0.836150
$898$	−100.116	−3.34092
$899$	−28.7127	−0.957624
$900$	−0.385936	−0.0128645
$901$	22.0004	0.732938
$902$	2.06269	0.0686800
$903$	27.9072	0.928693
$904$	−115.782	−3.85086
$905$	11.9575	0.397480
$906$	47.9311	1.59240
$907$	−51.0169	−1.69399	−0.846994	−	0.531602i	$-0.821590\pi$
$907$	−51.0169	−1.69399	−0.846994	+	0.531602i	$0.821590\pi$
$908$	32.9887	1.09477
$909$	0.00725884	0.000240761 0
$910$	−112.582	−3.73204
$911$	−3.56713	−0.118184	−0.0590921	−	0.998253i	$-0.518821\pi$
$911$	−3.56713	−0.118184	−0.0590921	+	0.998253i	$0.518821\pi$
$912$	120.012	3.97398
$913$	−29.8621	−0.988290
$914$	−31.0549	−1.02720
$915$	26.6708	0.881711
$916$	−100.922	−3.33455
$917$	54.2162	1.79038
$918$	−78.8310	−2.60181
$919$	17.4908	0.576967	0.288484	−	0.957485i	$-0.406849\pi$
$919$	17.4908	0.576967	0.288484	+	0.957485i	$0.406849\pi$
$920$	−46.4053	−1.52994
$921$	−0.0759619	−0.00250303
$922$	66.2063	2.18039
$923$	−5.01065	−0.164927
$924$	−83.3193	−2.74100
$925$	3.10062	0.101948
$926$	−13.0715	−0.429557
$927$	0.513259	0.0168576
$928$	72.9859	2.39588
$929$	−19.8701	−0.651917	−0.325959	−	0.945384i	$-0.605687\pi$
$929$	−19.8701	−0.651917	−0.325959	+	0.945384i	$0.605687\pi$
$930$	33.0296	1.08308
$931$	−101.035	−3.31128
$932$	−115.150	−3.77187
$933$	2.87465	0.0941119
$934$	86.3786	2.82639
$935$	−29.5357	−0.965919
$936$	9.10030	0.297453
$937$	−7.96577	−0.260230	−0.130115	−	0.991499i	$-0.541535\pi$
$937$	−7.96577	−0.260230	−0.130115	+	0.991499i	$0.541535\pi$
$938$	146.413	4.78054
$939$	55.5177	1.81175
$940$	112.131	3.65732
$941$	15.0220	0.489703	0.244851	−	0.969561i	$-0.421261\pi$
$941$	15.0220	0.489703	0.244851	+	0.969561i	$0.421261\pi$
$942$	−104.273	−3.39741
$943$	1.12342	0.0365837
$944$	76.7120	2.49676
$945$	−47.7929	−1.55470
$946$	19.4323	0.631797
$947$	22.1786	0.720707	0.360353	−	0.932816i	$-0.382656\pi$
$947$	22.1786	0.720707	0.360353	+	0.932816i	$0.382656\pi$
$948$	−53.5387	−1.73886
$949$	6.91096	0.224339
$950$	−5.61879	−0.182298
$951$	38.0814	1.23487
$952$	192.723	6.24620
$953$	−46.6018	−1.50958	−0.754791	−	0.655966i	$-0.772261\pi$
$953$	−46.6018	−1.50958	−0.754791	+	0.655966i	$0.772261\pi$
$954$	−2.68676	−0.0869872
$955$	0.997579	0.0322809
$956$	−124.075	−4.01288
$957$	−35.2008	−1.13788
$958$	29.9356	0.967174
$959$	−80.9480	−2.61395
$960$	−16.5488	−0.534111
$961$	−20.4930	−0.661065
$962$	−127.653	−4.11570
$963$	−2.58319	−0.0832421
$964$	50.8253	1.63697
$965$	−2.90312	−0.0934546
$966$	−64.7678	−2.08387
$967$	−20.7255	−0.666488	−0.333244	−	0.942841i	$-0.608143\pi$
$967$	−20.7255	−0.666488	−0.333244	+	0.942841i	$0.608143\pi$
$968$	42.9986	1.38203
$969$	87.1560	2.79985
$970$	13.8635	0.445130
$971$	−54.3482	−1.74412	−0.872059	−	0.489401i	$-0.837215\pi$
$971$	−54.3482	−1.74412	−0.872059	+	0.489401i	$0.837215\pi$
$972$	14.2215	0.456154
$973$	−74.1322	−2.37657
$974$	−67.9754	−2.17807
$975$	−2.28310	−0.0731176
$976$	−57.8424	−1.85149
$977$	−49.8551	−1.59501	−0.797503	−	0.603315i	$-0.793846\pi$
$977$	−49.8551	−1.59501	−0.797503	+	0.603315i	$0.793846\pi$
$978$	−0.245774	−0.00785898
$979$	−31.6918	−1.01288
$980$	132.826	4.24296
$981$	−2.20138	−0.0702845
$982$	−31.7334	−1.01265
$983$	−53.7286	−1.71368	−0.856838	−	0.515586i	$-0.827574\pi$
$983$	−53.7286	−1.71368	−0.856838	+	0.515586i	$0.827574\pi$
$984$	4.58303	0.146102
$985$	−13.3623	−0.425757
$986$	142.162	4.52735
$987$	89.6346	2.85310
$988$	162.077	5.15636
$989$	10.5836	0.336539
$990$	3.60700	0.114638
$991$	−10.1073	−0.321070	−0.160535	−	0.987030i	$-0.551322\pi$
$991$	−10.1073	−0.321070	−0.160535	+	0.987030i	$0.551322\pi$
$992$	−26.7081	−0.847983
$993$	−16.0086	−0.508019
$994$	12.9590	0.411035
$995$	12.8375	0.406975
$996$	−115.846	−3.67072
$997$	18.5368	0.587066	0.293533	−	0.955949i	$-0.405169\pi$
$997$	18.5368	0.587066	0.293533	+	0.955949i	$0.405169\pi$
$998$	−54.4653	−1.72407
$999$	−54.1910	−1.71453

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8011.2.a.a.1.17	✓	309

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8011.2.a.a.1.17	✓	309	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:	\( N \)	\(=\)	\( 8011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8011.a (trivial)

\(f(q)\)	\(=\)	\(q-2.58476 q^{2} -1.81476 q^{3} +4.68100 q^{4} +2.17232 q^{5} +4.69073 q^{6} -4.47910 q^{7} -6.92974 q^{8} +0.293363 q^{9} +O(q^{10})\)	\(q-2.58476 q^{2} -1.81476 q^{3} +4.68100 q^{4} +2.17232 q^{5} +4.69073 q^{6} -4.47910 q^{7} -6.92974 q^{8} +0.293363 q^{9} -5.61492 q^{10} -2.18976 q^{11} -8.49490 q^{12} -4.47644 q^{13} +11.5774 q^{14} -3.94224 q^{15} +8.54973 q^{16} +6.20907 q^{17} -0.758275 q^{18} -7.73483 q^{19} +10.1686 q^{20} +8.12850 q^{21} +5.66001 q^{22} +3.08268 q^{23} +12.5758 q^{24} -0.281042 q^{25} +11.5705 q^{26} +4.91190 q^{27} -20.9666 q^{28} -8.85800 q^{29} +10.1897 q^{30} +3.24145 q^{31} -8.23955 q^{32} +3.97390 q^{33} -16.0490 q^{34} -9.73002 q^{35} +1.37323 q^{36} -11.0326 q^{37} +19.9927 q^{38} +8.12368 q^{39} -15.0536 q^{40} +0.364432 q^{41} -21.0102 q^{42} +3.43325 q^{43} -10.2503 q^{44} +0.637278 q^{45} -7.96798 q^{46} +11.0272 q^{47} -15.5157 q^{48} +13.0623 q^{49} +0.726427 q^{50} -11.2680 q^{51} -20.9542 q^{52} +3.54326 q^{53} -12.6961 q^{54} -4.75686 q^{55} +31.0390 q^{56} +14.0369 q^{57} +22.8958 q^{58} +8.97244 q^{59} -18.4536 q^{60} -6.76540 q^{61} -8.37838 q^{62} -1.31400 q^{63} +4.19782 q^{64} -9.72425 q^{65} -10.2716 q^{66} +12.6464 q^{67} +29.0646 q^{68} -5.59432 q^{69} +25.1498 q^{70} +1.11934 q^{71} -2.03293 q^{72} -1.54385 q^{73} +28.5166 q^{74} +0.510025 q^{75} -36.2067 q^{76} +9.80816 q^{77} -20.9978 q^{78} +6.30245 q^{79} +18.5727 q^{80} -9.79403 q^{81} -0.941970 q^{82} +13.6371 q^{83} +38.0495 q^{84} +13.4881 q^{85} -8.87414 q^{86} +16.0752 q^{87} +15.1745 q^{88} +14.4727 q^{89} -1.64721 q^{90} +20.0504 q^{91} +14.4300 q^{92} -5.88246 q^{93} -28.5027 q^{94} -16.8025 q^{95} +14.9528 q^{96} -2.46905 q^{97} -33.7630 q^{98} -0.642396 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 309 q - 33 q^{2} - 15 q^{3} + 273 q^{4} - 74 q^{5} - 32 q^{6} - 19 q^{7} - 93 q^{8} + 214 q^{9}+O(q^{10}) \) 309 * q - 33 * q^2 - 15 * q^3 + 273 * q^4 - 74 * q^5 - 32 * q^6 - 19 * q^7 - 93 * q^8 + 214 * q^9	\( 309 q - 33 q^{2} - 15 q^{3} + 273 q^{4} - 74 q^{5} - 32 q^{6} - 19 q^{7} - 93 q^{8} + 214 q^{9} - 23 q^{10} - 72 q^{11} - 42 q^{12} - 57 q^{13} - 77 q^{14} - 44 q^{15} + 205 q^{16} - 86 q^{17} - 82 q^{18} - 58 q^{19} - 134 q^{20} - 123 q^{21} - 31 q^{22} - 94 q^{23} - 84 q^{24} + 225 q^{25} - 92 q^{26} - 48 q^{27} - 36 q^{28} - 345 q^{29} - 85 q^{30} - 36 q^{31} - 199 q^{32} - 56 q^{33} - 28 q^{34} - 168 q^{35} + 65 q^{36} - 79 q^{37} - 66 q^{38} - 145 q^{39} - 54 q^{40} - 176 q^{41} - 48 q^{42} - 58 q^{43} - 194 q^{44} - 192 q^{45} - 44 q^{46} - 82 q^{47} - 81 q^{48} + 186 q^{49} - 206 q^{50} - 145 q^{51} - 86 q^{52} - 223 q^{53} - 117 q^{54} - 58 q^{55} - 216 q^{56} - 124 q^{57} - 151 q^{59} - 91 q^{60} - 184 q^{61} - 124 q^{62} - 78 q^{63} + 101 q^{64} - 194 q^{65} - 112 q^{66} - 53 q^{67} - 182 q^{68} - 243 q^{69} - 193 q^{71} - 208 q^{72} - 69 q^{73} - 236 q^{74} - 62 q^{75} - 142 q^{76} - 324 q^{77} - 20 q^{78} - 91 q^{79} - 223 q^{80} - 27 q^{81} + 2 q^{82} - 117 q^{83} - 157 q^{84} - 171 q^{85} - 203 q^{86} - 69 q^{87} - 36 q^{88} - 172 q^{89} - 10 q^{90} - 84 q^{91} - 226 q^{92} - 220 q^{93} - 96 q^{94} - 166 q^{95} - 118 q^{96} - 12 q^{97} - 116 q^{98} - 154 q^{99}+O(q^{100}) \) 309 * q - 33 * q^2 - 15 * q^3 + 273 * q^4 - 74 * q^5 - 32 * q^6 - 19 * q^7 - 93 * q^8 + 214 * q^9 - 23 * q^10 - 72 * q^11 - 42 * q^12 - 57 * q^13 - 77 * q^14 - 44 * q^15 + 205 * q^16 - 86 * q^17 - 82 * q^18 - 58 * q^19 - 134 * q^20 - 123 * q^21 - 31 * q^22 - 94 * q^23 - 84 * q^24 + 225 * q^25 - 92 * q^26 - 48 * q^27 - 36 * q^28 - 345 * q^29 - 85 * q^30 - 36 * q^31 - 199 * q^32 - 56 * q^33 - 28 * q^34 - 168 * q^35 + 65 * q^36 - 79 * q^37 - 66 * q^38 - 145 * q^39 - 54 * q^40 - 176 * q^41 - 48 * q^42 - 58 * q^43 - 194 * q^44 - 192 * q^45 - 44 * q^46 - 82 * q^47 - 81 * q^48 + 186 * q^49 - 206 * q^50 - 145 * q^51 - 86 * q^52 - 223 * q^53 - 117 * q^54 - 58 * q^55 - 216 * q^56 - 124 * q^57 - 151 * q^59 - 91 * q^60 - 184 * q^61 - 124 * q^62 - 78 * q^63 + 101 * q^64 - 194 * q^65 - 112 * q^66 - 53 * q^67 - 182 * q^68 - 243 * q^69 - 193 * q^71 - 208 * q^72 - 69 * q^73 - 236 * q^74 - 62 * q^75 - 142 * q^76 - 324 * q^77 - 20 * q^78 - 91 * q^79 - 223 * q^80 - 27 * q^81 + 2 * q^82 - 117 * q^83 - 157 * q^84 - 171 * q^85 - 203 * q^86 - 69 * q^87 - 36 * q^88 - 172 * q^89 - 10 * q^90 - 84 * q^91 - 226 * q^92 - 220 * q^93 - 96 * q^94 - 166 * q^95 - 118 * q^96 - 12 * q^97 - 116 * q^98 - 154 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more