LMFDB - Embedded newform 8009.2.a.b.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8009,2,Mod(1,8009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8009, 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("8009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8009 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8009.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:	$63.9521869788$
Analytic rank:	$0$
Dimension:	$361$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.62631 q^{2} -2.69396 q^{3} +4.89749 q^{4} +2.82018 q^{5} +7.07518 q^{6} -2.28331 q^{7} -7.60969 q^{8} +4.25744 q^{9} +O(q^{10})$	\(q-2.62631 q^{2} -2.69396 q^{3} +4.89749 q^{4} +2.82018 q^{5} +7.07518 q^{6} -2.28331 q^{7} -7.60969 q^{8} +4.25744 q^{9} -7.40666 q^{10} +3.71723 q^{11} -13.1937 q^{12} -5.01783 q^{13} +5.99668 q^{14} -7.59746 q^{15} +10.1904 q^{16} +0.715753 q^{17} -11.1813 q^{18} +1.50761 q^{19} +13.8118 q^{20} +6.15116 q^{21} -9.76259 q^{22} +7.23201 q^{23} +20.5002 q^{24} +2.95342 q^{25} +13.1784 q^{26} -3.38751 q^{27} -11.1825 q^{28} +5.97645 q^{29} +19.9533 q^{30} -7.45476 q^{31} -11.5437 q^{32} -10.0141 q^{33} -1.87979 q^{34} -6.43935 q^{35} +20.8508 q^{36} -1.87418 q^{37} -3.95944 q^{38} +13.5179 q^{39} -21.4607 q^{40} +10.9862 q^{41} -16.1548 q^{42} +7.48211 q^{43} +18.2051 q^{44} +12.0068 q^{45} -18.9935 q^{46} +6.37182 q^{47} -27.4526 q^{48} -1.78648 q^{49} -7.75658 q^{50} -1.92821 q^{51} -24.5748 q^{52} -5.29194 q^{53} +8.89663 q^{54} +10.4833 q^{55} +17.3753 q^{56} -4.06144 q^{57} -15.6960 q^{58} +1.12602 q^{59} -37.2085 q^{60} +6.36632 q^{61} +19.5785 q^{62} -9.72107 q^{63} +9.93660 q^{64} -14.1512 q^{65} +26.3001 q^{66} -9.27657 q^{67} +3.50539 q^{68} -19.4828 q^{69} +16.9117 q^{70} +6.37486 q^{71} -32.3978 q^{72} -7.09344 q^{73} +4.92218 q^{74} -7.95640 q^{75} +7.38349 q^{76} -8.48760 q^{77} -35.5020 q^{78} -2.54233 q^{79} +28.7388 q^{80} -3.64651 q^{81} -28.8530 q^{82} -5.59239 q^{83} +30.1252 q^{84} +2.01855 q^{85} -19.6503 q^{86} -16.1003 q^{87} -28.2870 q^{88} +11.5329 q^{89} -31.5334 q^{90} +11.4573 q^{91} +35.4186 q^{92} +20.0829 q^{93} -16.7343 q^{94} +4.25173 q^{95} +31.0984 q^{96} +12.9211 q^{97} +4.69185 q^{98} +15.8259 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9}+O(q^{10}) $ 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9	\( 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9} + 65 q^{10} + 33 q^{11} + 52 q^{12} + 89 q^{13} + 32 q^{14} + 55 q^{15} + 512 q^{16} + 42 q^{17} + 34 q^{18} + 191 q^{19} + 48 q^{20} + 53 q^{21} + 61 q^{22} + 52 q^{23} + 139 q^{24} + 458 q^{25} + 57 q^{26} + 80 q^{27} + 194 q^{28} + 47 q^{29} + 32 q^{30} + 254 q^{31} + 55 q^{32} + 40 q^{33} + 122 q^{34} + 93 q^{35} + 519 q^{36} + 43 q^{37} + 25 q^{38} + 210 q^{39} + 184 q^{40} + 54 q^{41} + 48 q^{42} + 151 q^{43} + 56 q^{44} + 82 q^{45} + 101 q^{46} + 117 q^{47} + 77 q^{48} + 563 q^{49} + 38 q^{50} + 143 q^{51} + 241 q^{52} + 14 q^{53} + 164 q^{54} + 452 q^{55} + 52 q^{56} + 21 q^{57} + 55 q^{58} + 125 q^{59} + 39 q^{60} + 227 q^{61} + 58 q^{62} + 292 q^{63} + 710 q^{64} + 15 q^{65} + 105 q^{66} + 120 q^{67} + 125 q^{68} + 136 q^{69} + 88 q^{70} + 105 q^{71} + 78 q^{72} + 108 q^{73} + 41 q^{74} + 128 q^{75} + 461 q^{76} + 28 q^{77} + 13 q^{78} + 400 q^{79} + 59 q^{80} + 485 q^{81} + 175 q^{82} + 97 q^{83} + 76 q^{84} + 144 q^{85} - 14 q^{86} + 327 q^{87} + 145 q^{88} + 52 q^{89} + 60 q^{90} + 192 q^{91} + 11 q^{92} + 32 q^{93} + 366 q^{94} + 182 q^{95} + 275 q^{96} + 117 q^{97} + 42 q^{98} + 111 q^{99}+O(q^{100}) \) 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9 + 65 * q^10 + 33 * q^11 + 52 * q^12 + 89 * q^13 + 32 * q^14 + 55 * q^15 + 512 * q^16 + 42 * q^17 + 34 * q^18 + 191 * q^19 + 48 * q^20 + 53 * q^21 + 61 * q^22 + 52 * q^23 + 139 * q^24 + 458 * q^25 + 57 * q^26 + 80 * q^27 + 194 * q^28 + 47 * q^29 + 32 * q^30 + 254 * q^31 + 55 * q^32 + 40 * q^33 + 122 * q^34 + 93 * q^35 + 519 * q^36 + 43 * q^37 + 25 * q^38 + 210 * q^39 + 184 * q^40 + 54 * q^41 + 48 * q^42 + 151 * q^43 + 56 * q^44 + 82 * q^45 + 101 * q^46 + 117 * q^47 + 77 * q^48 + 563 * q^49 + 38 * q^50 + 143 * q^51 + 241 * q^52 + 14 * q^53 + 164 * q^54 + 452 * q^55 + 52 * q^56 + 21 * q^57 + 55 * q^58 + 125 * q^59 + 39 * q^60 + 227 * q^61 + 58 * q^62 + 292 * q^63 + 710 * q^64 + 15 * q^65 + 105 * q^66 + 120 * q^67 + 125 * q^68 + 136 * q^69 + 88 * q^70 + 105 * q^71 + 78 * q^72 + 108 * q^73 + 41 * q^74 + 128 * q^75 + 461 * q^76 + 28 * q^77 + 13 * q^78 + 400 * q^79 + 59 * q^80 + 485 * q^81 + 175 * q^82 + 97 * q^83 + 76 * q^84 + 144 * q^85 - 14 * q^86 + 327 * q^87 + 145 * q^88 + 52 * q^89 + 60 * q^90 + 192 * q^91 + 11 * q^92 + 32 * q^93 + 366 * q^94 + 182 * q^95 + 275 * q^96 + 117 * q^97 + 42 * q^98 + 111 * 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 $

$2$	−2.62631	−1.85708	−0.928540	−	0.371233i	$-0.878935\pi$
$2$	−2.62631	−1.85708	−0.928540	+	0.371233i	$0.878935\pi$
$3$	−2.69396	−1.55536	−0.777680	−	0.628660i	$-0.783604\pi$
$3$	−2.69396	−1.55536	−0.777680	+	0.628660i	$0.783604\pi$
$4$	4.89749	2.44874
$5$	2.82018	1.26122	0.630611	−	0.776099i	$-0.282804\pi$
$5$	2.82018	1.26122	0.630611	+	0.776099i	$0.282804\pi$
$6$	7.07518	2.88843
$7$	−2.28331	−0.863011	−0.431506	−	0.902110i	$-0.642017\pi$
$7$	−2.28331	−0.863011	−0.431506	+	0.902110i	$0.642017\pi$
$8$	−7.60969	−2.69043
$9$	4.25744	1.41915
$10$	−7.40666	−2.34219
$11$	3.71723	1.12079	0.560394	−	0.828226i	$-0.310650\pi$
$11$	3.71723	1.12079	0.560394	+	0.828226i	$0.310650\pi$
$12$	−13.1937	−3.80868
$13$	−5.01783	−1.39170	−0.695848	−	0.718189i	$-0.744971\pi$
$13$	−5.01783	−1.39170	−0.695848	+	0.718189i	$0.744971\pi$
$14$	5.99668	1.60268
$15$	−7.59746	−1.96166
$16$	10.1904	2.54760
$17$	0.715753	0.173596	0.0867978	−	0.996226i	$-0.472337\pi$
$17$	0.715753	0.173596	0.0867978	+	0.996226i	$0.472337\pi$
$18$	−11.1813	−2.63547
$19$	1.50761	0.345869	0.172935	−	0.984933i	$-0.444675\pi$
$19$	1.50761	0.345869	0.172935	+	0.984933i	$0.444675\pi$
$20$	13.8118	3.08841
$21$	6.15116	1.34229
$22$	−9.76259	−2.08139
$23$	7.23201	1.50798	0.753989	−	0.656887i	$-0.228127\pi$
$23$	7.23201	1.50798	0.753989	+	0.656887i	$0.228127\pi$
$24$	20.5002	4.18459
$25$	2.95342	0.590683
$26$	13.1784	2.58449
$27$	−3.38751	−0.651926
$28$	−11.1825	−2.11329
$29$	5.97645	1.10980	0.554899	−	0.831918i	$-0.312757\pi$
$29$	5.97645	1.10980	0.554899	+	0.831918i	$0.312757\pi$
$30$	19.9533	3.64295
$31$	−7.45476	−1.33891	−0.669457	−	0.742851i	$-0.733473\pi$
$31$	−7.45476	−1.33891	−0.669457	+	0.742851i	$0.733473\pi$
$32$	−11.5437	−2.04066
$33$	−10.0141	−1.74323
$34$	−1.87979	−0.322381
$35$	−6.43935	−1.08845
$36$	20.8508	3.47513
$37$	−1.87418	−0.308114	−0.154057	−	0.988062i	$-0.549234\pi$
$37$	−1.87418	−0.308114	−0.154057	+	0.988062i	$0.549234\pi$
$38$	−3.95944	−0.642306
$39$	13.5179	2.16459
$40$	−21.4607	−3.39323
$41$	10.9862	1.71575	0.857875	−	0.513858i	$-0.171784\pi$
$41$	10.9862	1.71575	0.857875	+	0.513858i	$0.171784\pi$
$42$	−16.1548	−2.49275
$43$	7.48211	1.14101	0.570506	−	0.821294i	$-0.306747\pi$
$43$	7.48211	1.14101	0.570506	+	0.821294i	$0.306747\pi$
$44$	18.2051	2.74452
$45$	12.0068	1.78986
$46$	−18.9935	−2.80043
$47$	6.37182	0.929425	0.464713	−	0.885462i	$-0.346158\pi$
$47$	6.37182	0.929425	0.464713	+	0.885462i	$0.346158\pi$
$48$	−27.4526	−3.96244
$49$	−1.78648	−0.255212
$50$	−7.75658	−1.09695
$51$	−1.92821	−0.270004
$52$	−24.5748	−3.40791
$53$	−5.29194	−0.726903	−0.363452	−	0.931613i	$-0.618402\pi$
$53$	−5.29194	−0.726903	−0.363452	+	0.931613i	$0.618402\pi$
$54$	8.89663	1.21068
$55$	10.4833	1.41356
$56$	17.3753	2.32187
$57$	−4.06144	−0.537951
$58$	−15.6960	−2.06098
$59$	1.12602	0.146595	0.0732977	−	0.997310i	$-0.476648\pi$
$59$	1.12602	0.146595	0.0732977	+	0.997310i	$0.476648\pi$
$60$	−37.2085	−4.80359
$61$	6.36632	0.815123	0.407562	−	0.913178i	$-0.366379\pi$
$61$	6.36632	0.815123	0.407562	+	0.913178i	$0.366379\pi$
$62$	19.5785	2.48647
$63$	−9.72107	−1.22474
$64$	9.93660	1.24208
$65$	−14.1512	−1.75524
$66$	26.3001	3.23731
$67$	−9.27657	−1.13331	−0.566657	−	0.823954i	$-0.691764\pi$
$67$	−9.27657	−1.13331	−0.566657	+	0.823954i	$0.691764\pi$
$68$	3.50539	0.425091
$69$	−19.4828	−2.34545
$70$	16.9117	2.02134
$71$	6.37486	0.756557	0.378278	−	0.925692i	$-0.376516\pi$
$71$	6.37486	0.756557	0.378278	+	0.925692i	$0.376516\pi$
$72$	−32.3978	−3.81812
$73$	−7.09344	−0.830224	−0.415112	−	0.909770i	$-0.636258\pi$
$73$	−7.09344	−0.830224	−0.415112	+	0.909770i	$0.636258\pi$
$74$	4.92218	0.572192
$75$	−7.95640	−0.918726
$76$	7.38349	0.846944
$77$	−8.48760	−0.967252
$78$	−35.5020	−4.01981
$79$	−2.54233	−0.286034	−0.143017	−	0.989720i	$-0.545680\pi$
$79$	−2.54233	−0.286034	−0.143017	+	0.989720i	$0.545680\pi$
$80$	28.7388	3.21309
$81$	−3.64651	−0.405168
$82$	−28.8530	−3.18629
$83$	−5.59239	−0.613844	−0.306922	−	0.951735i	$-0.599299\pi$
$83$	−5.59239	−0.613844	−0.306922	+	0.951735i	$0.599299\pi$
$84$	30.1252	3.28693
$85$	2.01855	0.218943
$86$	−19.6503	−2.11895
$87$	−16.1003	−1.72614
$88$	−28.2870	−3.01540
$89$	11.5329	1.22249	0.611244	−	0.791442i	$-0.290669\pi$
$89$	11.5329	1.22249	0.611244	+	0.791442i	$0.290669\pi$
$90$	−31.5334	−3.32391
$91$	11.4573	1.20105
$92$	35.4186	3.69265
$93$	20.0829	2.08250
$94$	−16.7343	−1.72602
$95$	4.25173	0.436218
$96$	31.0984	3.17397
$97$	12.9211	1.31194	0.655972	−	0.754786i	$-0.272259\pi$
$97$	12.9211	1.31194	0.655972	+	0.754786i	$0.272259\pi$
$98$	4.69185	0.473949
$99$	15.8259	1.59056
$100$	14.4643	1.44643
$101$	−2.72275	−0.270923	−0.135462	−	0.990783i	$-0.543252\pi$
$101$	−2.72275	−0.270923	−0.135462	+	0.990783i	$0.543252\pi$
$102$	5.06408	0.501418
$103$	19.4798	1.91940	0.959702	−	0.281019i	$-0.0906723\pi$
$103$	19.4798	1.91940	0.959702	+	0.281019i	$0.0906723\pi$
$104$	38.1841	3.74426
$105$	17.3474	1.69293
$106$	13.8982	1.34992
$107$	−11.7413	−1.13508	−0.567540	−	0.823346i	$-0.692105\pi$
$107$	−11.7413	−1.13508	−0.567540	+	0.823346i	$0.692105\pi$
$108$	−16.5903	−1.59640
$109$	−14.4319	−1.38233	−0.691164	−	0.722698i	$-0.742902\pi$
$109$	−14.4319	−1.38233	−0.691164	+	0.722698i	$0.742902\pi$
$110$	−27.5323	−2.62510
$111$	5.04898	0.479228
$112$	−23.2679	−2.19861
$113$	−14.7277	−1.38547	−0.692734	−	0.721194i	$-0.743594\pi$
$113$	−14.7277	−1.38547	−0.692734	+	0.721194i	$0.743594\pi$
$114$	10.6666	0.999018
$115$	20.3956	1.90190
$116$	29.2696	2.71761
$117$	−21.3631	−1.97502
$118$	−2.95727	−0.272239
$119$	−1.63429	−0.149815
$120$	57.8143	5.27770
$121$	2.81781	0.256165
$122$	−16.7199	−1.51375
$123$	−29.5963	−2.66861
$124$	−36.5096	−3.27866
$125$	−5.77173	−0.516240
$126$	25.5305	2.27444
$127$	16.0580	1.42492	0.712459	−	0.701714i	$-0.247582\pi$
$127$	16.0580	1.42492	0.712459	+	0.701714i	$0.247582\pi$
$128$	−3.00909	−0.265968
$129$	−20.1565	−1.77469
$130$	37.1654	3.25962
$131$	11.2584	0.983648	0.491824	−	0.870695i	$-0.336330\pi$
$131$	11.2584	0.983648	0.491824	+	0.870695i	$0.336330\pi$
$132$	−49.0439	−4.26872
$133$	−3.44234	−0.298489
$134$	24.3631	2.10465
$135$	−9.55338	−0.822224
$136$	−5.44666	−0.467047
$137$	16.1491	1.37971	0.689855	−	0.723947i	$-0.257674\pi$
$137$	16.1491	1.37971	0.689855	+	0.723947i	$0.257674\pi$
$138$	51.1677	4.35568
$139$	5.04422	0.427845	0.213923	−	0.976851i	$-0.431376\pi$
$139$	5.04422	0.427845	0.213923	+	0.976851i	$0.431376\pi$
$140$	−31.5366	−2.66533
$141$	−17.1654	−1.44559
$142$	−16.7423	−1.40499
$143$	−18.6524	−1.55980
$144$	43.3850	3.61542
$145$	16.8547	1.39970
$146$	18.6295	1.54179
$147$	4.81272	0.396947
$148$	−9.17879	−0.754492
$149$	11.7024	0.958694	0.479347	−	0.877625i	$-0.340873\pi$
$149$	11.7024	0.958694	0.479347	+	0.877625i	$0.340873\pi$
$150$	20.8959	1.70615
$151$	13.1084	1.06675	0.533375	−	0.845879i	$-0.320923\pi$
$151$	13.1084	1.06675	0.533375	+	0.845879i	$0.320923\pi$
$152$	−11.4724	−0.930537
$153$	3.04728	0.246358
$154$	22.2910	1.79626
$155$	−21.0238	−1.68867
$156$	66.2035	5.30052
$157$	5.19291	0.414439	0.207219	−	0.978294i	$-0.433559\pi$
$157$	5.19291	0.414439	0.207219	+	0.978294i	$0.433559\pi$
$158$	6.67694	0.531189
$159$	14.2563	1.13060
$160$	−32.5554	−2.57373
$161$	−16.5129	−1.30140
$162$	9.57685	0.752429
$163$	−13.4191	−1.05106	−0.525532	−	0.850774i	$-0.676134\pi$
$163$	−13.4191	−1.05106	−0.525532	+	0.850774i	$0.676134\pi$
$164$	53.8046	4.20143
$165$	−28.2415	−2.19860
$166$	14.6873	1.13996
$167$	3.60439	0.278916	0.139458	−	0.990228i	$-0.455464\pi$
$167$	3.60439	0.278916	0.139458	+	0.990228i	$0.455464\pi$
$168$	−46.8084	−3.61135
$169$	12.1786	0.936817
$170$	−5.30134	−0.406594
$171$	6.41855	0.490839
$172$	36.6436	2.79404
$173$	13.0745	0.994040	0.497020	−	0.867739i	$-0.334428\pi$
$173$	13.0745	0.994040	0.497020	+	0.867739i	$0.334428\pi$
$174$	42.2844	3.20557
$175$	−6.74357	−0.509766
$176$	37.8801	2.85532
$177$	−3.03346	−0.228009
$178$	−30.2890	−2.27026
$179$	15.3048	1.14394	0.571968	−	0.820276i	$-0.306180\pi$
$179$	15.3048	1.14394	0.571968	+	0.820276i	$0.306180\pi$
$180$	58.8029	4.38291
$181$	−6.25076	−0.464615	−0.232308	−	0.972642i	$-0.574628\pi$
$181$	−6.25076	−0.464615	−0.232308	+	0.972642i	$0.574628\pi$
$182$	−30.0903	−2.23044
$183$	−17.1506	−1.26781
$184$	−55.0333	−4.05711
$185$	−5.28553	−0.388600
$186$	−52.7437	−3.86736
$187$	2.66062	0.194564
$188$	31.2059	2.27592
$189$	7.73473	0.562619
$190$	−11.1663	−0.810091
$191$	−0.311663	−0.0225511	−0.0112756	−	0.999936i	$-0.503589\pi$
$191$	−0.311663	−0.0225511	−0.0112756	+	0.999936i	$0.503589\pi$
$192$	−26.7689	−1.93188
$193$	5.96010	0.429018	0.214509	−	0.976722i	$-0.431185\pi$
$193$	5.96010	0.429018	0.214509	+	0.976722i	$0.431185\pi$
$194$	−33.9349	−2.43638
$195$	38.1228	2.73003
$196$	−8.74928	−0.624949
$197$	16.6211	1.18420	0.592102	−	0.805863i	$-0.298298\pi$
$197$	16.6211	1.18420	0.592102	+	0.805863i	$0.298298\pi$
$198$	−41.5637	−2.95380
$199$	−12.5926	−0.892664	−0.446332	−	0.894867i	$-0.647270\pi$
$199$	−12.5926	−0.892664	−0.446332	+	0.894867i	$0.647270\pi$
$200$	−22.4746	−1.58919
$201$	24.9908	1.76271
$202$	7.15077	0.503126
$203$	−13.6461	−0.957768
$204$	−9.44340	−0.661170
$205$	30.9830	2.16394
$206$	−51.1600	−3.56449
$207$	30.7898	2.14004
$208$	−51.1337	−3.54548
$209$	5.60413	0.387646
$210$	−45.5596	−3.14391
$211$	−22.9257	−1.57827	−0.789136	−	0.614218i	$-0.789471\pi$
$211$	−22.9257	−1.57827	−0.789136	+	0.614218i	$0.789471\pi$
$212$	−25.9172	−1.78000
$213$	−17.1736	−1.17672
$214$	30.8364	2.10793
$215$	21.1009	1.43907
$216$	25.7779	1.75396
$217$	17.0215	1.15550
$218$	37.9026	2.56709
$219$	19.1095	1.29130
$220$	51.3416	3.46145
$221$	−3.59153	−0.241592
$222$	−13.2602	−0.889965
$223$	0.641576	0.0429631	0.0214815	−	0.999769i	$-0.493162\pi$
$223$	0.641576	0.0429631	0.0214815	+	0.999769i	$0.493162\pi$
$224$	26.3580	1.76112
$225$	12.5740	0.838267
$226$	38.6795	2.57292
$227$	−9.20069	−0.610671	−0.305336	−	0.952245i	$-0.598769\pi$
$227$	−9.20069	−0.610671	−0.305336	+	0.952245i	$0.598769\pi$
$228$	−19.8909	−1.31730
$229$	−15.0267	−0.992993	−0.496496	−	0.868039i	$-0.665380\pi$
$229$	−15.0267	−0.992993	−0.496496	+	0.868039i	$0.665380\pi$
$230$	−53.5650	−3.53197
$231$	22.8653	1.50443
$232$	−45.4789	−2.98584
$233$	−8.33862	−0.546281	−0.273140	−	0.961974i	$-0.588062\pi$
$233$	−8.33862	−0.546281	−0.273140	+	0.961974i	$0.588062\pi$
$234$	56.1061	3.66777
$235$	17.9697	1.17221
$236$	5.51467	0.358974
$237$	6.84894	0.444887
$238$	4.29214	0.278218
$239$	20.3570	1.31679	0.658393	−	0.752674i	$-0.271236\pi$
$239$	20.3570	1.31679	0.658393	+	0.752674i	$0.271236\pi$
$240$	−77.4212	−4.99752
$241$	−14.5362	−0.936358	−0.468179	−	0.883634i	$-0.655090\pi$
$241$	−14.5362	−0.936358	−0.468179	+	0.883634i	$0.655090\pi$
$242$	−7.40044	−0.475718
$243$	19.9861	1.28211
$244$	31.1789	1.99603
$245$	−5.03821	−0.321879
$246$	77.7291	4.95582
$247$	−7.56492	−0.481344
$248$	56.7284	3.60226
$249$	15.0657	0.954749
$250$	15.1583	0.958698
$251$	6.37221	0.402210	0.201105	−	0.979570i	$-0.435547\pi$
$251$	6.37221	0.402210	0.201105	+	0.979570i	$0.435547\pi$
$252$	−47.6088	−2.99907
$253$	26.8830	1.69012
$254$	−42.1732	−2.64618
$255$	−5.43791	−0.340535
$256$	−11.9704	−0.748152
$257$	−24.8573	−1.55055	−0.775277	−	0.631621i	$-0.782390\pi$
$257$	−24.8573	−1.55055	−0.775277	+	0.631621i	$0.782390\pi$
$258$	52.9373	3.29573
$259$	4.27935	0.265906
$260$	−69.3052	−4.29813
$261$	25.4444	1.57497
$262$	−29.5679	−1.82671
$263$	2.64695	0.163218	0.0816091	−	0.996664i	$-0.473994\pi$
$263$	2.64695	0.163218	0.0816091	+	0.996664i	$0.473994\pi$
$264$	76.2041	4.69004
$265$	−14.9242	−0.916787
$266$	9.04064	0.554317
$267$	−31.0693	−1.90141
$268$	−45.4319	−2.77519
$269$	−19.1795	−1.16939	−0.584696	−	0.811252i	$-0.698786\pi$
$269$	−19.1795	−1.16939	−0.584696	+	0.811252i	$0.698786\pi$
$270$	25.0901	1.52693
$271$	2.28694	0.138922	0.0694609	−	0.997585i	$-0.477872\pi$
$271$	2.28694	0.138922	0.0694609	+	0.997585i	$0.477872\pi$
$272$	7.29381	0.442252
$273$	−30.8655	−1.86806
$274$	−42.4125	−2.56223
$275$	10.9785	0.662031
$276$	−95.4166	−5.74340
$277$	−14.2115	−0.853887	−0.426943	−	0.904278i	$-0.640410\pi$
$277$	−14.2115	−0.853887	−0.426943	+	0.904278i	$0.640410\pi$
$278$	−13.2477	−0.794542
$279$	−31.7382	−1.90012
$280$	49.0015	2.92840
$281$	3.12789	0.186594	0.0932971	−	0.995638i	$-0.470259\pi$
$281$	3.12789	0.186594	0.0932971	+	0.995638i	$0.470259\pi$
$282$	45.0817	2.68458
$283$	5.56278	0.330673	0.165336	−	0.986237i	$-0.447129\pi$
$283$	5.56278	0.330673	0.165336	+	0.986237i	$0.447129\pi$
$284$	31.2208	1.85261
$285$	−11.4540	−0.678476
$286$	48.9870	2.89666
$287$	−25.0849	−1.48071
$288$	−49.1468	−2.89600
$289$	−16.4877	−0.969865
$290$	−44.2655	−2.59936
$291$	−34.8091	−2.04055
$292$	−34.7400	−2.03300
$293$	−25.6280	−1.49720	−0.748601	−	0.663021i	$-0.769274\pi$
$293$	−25.6280	−1.49720	−0.748601	+	0.663021i	$0.769274\pi$
$294$	−12.6397	−0.737162
$295$	3.17558	0.184889
$296$	14.2619	0.828959
$297$	−12.5921	−0.730670
$298$	−30.7340	−1.78037
$299$	−36.2890	−2.09865
$300$	−38.9664	−2.24972
$301$	−17.0840	−0.984706
$302$	−34.4268	−1.98104
$303$	7.33498	0.421384
$304$	15.3631	0.881136
$305$	17.9542	1.02805
$306$	−8.00308	−0.457506
$307$	10.7476	0.613398	0.306699	−	0.951807i	$-0.400775\pi$
$307$	10.7476	0.613398	0.306699	+	0.951807i	$0.400775\pi$
$308$	−41.5679	−2.36855
$309$	−52.4780	−2.98537
$310$	55.2149	3.13599
$311$	1.35490	0.0768295	0.0384148	−	0.999262i	$-0.487769\pi$
$311$	1.35490	0.0768295	0.0384148	+	0.999262i	$0.487769\pi$
$312$	−102.867	−5.82368
$313$	−4.45589	−0.251862	−0.125931	−	0.992039i	$-0.540192\pi$
$313$	−4.45589	−0.251862	−0.125931	+	0.992039i	$0.540192\pi$
$314$	−13.6382	−0.769646
$315$	−27.4152	−1.54467
$316$	−12.4510	−0.700425
$317$	−27.2291	−1.52934	−0.764668	−	0.644424i	$-0.777097\pi$
$317$	−27.2291	−1.52934	−0.764668	+	0.644424i	$0.777097\pi$
$318$	−37.4414	−2.09961
$319$	22.2158	1.24385
$320$	28.0230	1.56653
$321$	31.6308	1.76546
$322$	43.3680	2.41680
$323$	1.07908	0.0600413
$324$	−17.8587	−0.992152
$325$	−14.8197	−0.822052
$326$	35.2427	1.95191
$327$	38.8791	2.15002
$328$	−83.6013	−4.61611
$329$	−14.5489	−0.802104
$330$	74.1709	4.08298
$331$	−13.7245	−0.754364	−0.377182	−	0.926139i	$-0.623107\pi$
$331$	−13.7245	−0.754364	−0.377182	+	0.926139i	$0.623107\pi$
$332$	−27.3886	−1.50315
$333$	−7.97923	−0.437259
$334$	−9.46623	−0.517969
$335$	−26.1616	−1.42936
$336$	62.6828	3.41963
$337$	5.25318	0.286159	0.143079	−	0.989711i	$-0.454300\pi$
$337$	5.25318	0.286159	0.143079	+	0.989711i	$0.454300\pi$
$338$	−31.9848	−1.73974
$339$	39.6759	2.15490
$340$	9.88583	0.536135
$341$	−27.7111	−1.50064
$342$	−16.8571	−0.911527
$343$	20.0623	1.08326
$344$	−56.9366	−3.06981
$345$	−54.9449	−2.95813
$346$	−34.3378	−1.84601
$347$	14.1361	0.758866	0.379433	−	0.925219i	$-0.376119\pi$
$347$	14.1361	0.758866	0.379433	+	0.925219i	$0.376119\pi$
$348$	−78.8511	−4.22687
$349$	26.9533	1.44278	0.721388	−	0.692531i	$-0.243504\pi$
$349$	26.9533	1.44278	0.721388	+	0.692531i	$0.243504\pi$
$350$	17.7107	0.946676
$351$	16.9979	0.907282
$352$	−42.9108	−2.28715
$353$	27.9866	1.48958	0.744789	−	0.667300i	$-0.232550\pi$
$353$	27.9866	1.48958	0.744789	+	0.667300i	$0.232550\pi$
$354$	7.96679	0.423430
$355$	17.9783	0.954187
$356$	56.4824	2.99356
$357$	4.40271	0.233016
$358$	−40.1952	−2.12438
$359$	−10.4085	−0.549339	−0.274670	−	0.961539i	$-0.588568\pi$
$359$	−10.4085	−0.549339	−0.274670	+	0.961539i	$0.588568\pi$
$360$	−91.3677	−4.81550
$361$	−16.7271	−0.880375
$362$	16.4164	0.862827
$363$	−7.59109	−0.398429
$364$	56.1119	2.94106
$365$	−20.0048	−1.04710
$366$	45.0428	2.35442
$367$	−19.9828	−1.04309	−0.521546	−	0.853223i	$-0.674645\pi$
$367$	−19.9828	−1.04309	−0.521546	+	0.853223i	$0.674645\pi$
$368$	73.6970	3.84172
$369$	46.7730	2.43490
$370$	13.8814	0.721661
$371$	12.0831	0.627326
$372$	98.3555	5.09950
$373$	26.7085	1.38291	0.691456	−	0.722419i	$-0.256970\pi$
$373$	26.7085	1.38291	0.691456	+	0.722419i	$0.256970\pi$
$374$	−6.98760	−0.361320
$375$	15.5488	0.802939
$376$	−48.4875	−2.50055
$377$	−29.9888	−1.54450
$378$	−20.3138	−1.04483
$379$	30.4468	1.56394	0.781972	−	0.623313i	$-0.214214\pi$
$379$	30.4468	1.56394	0.781972	+	0.623313i	$0.214214\pi$
$380$	20.8228	1.06819
$381$	−43.2597	−2.21626
$382$	0.818522	0.0418792
$383$	−33.6301	−1.71842	−0.859210	−	0.511623i	$-0.829044\pi$
$383$	−33.6301	−1.71842	−0.859210	+	0.511623i	$0.829044\pi$
$384$	8.10637	0.413676
$385$	−23.9366	−1.21992
$386$	−15.6531	−0.796720
$387$	31.8547	1.61926
$388$	63.2811	3.21261
$389$	−35.7641	−1.81331	−0.906656	−	0.421870i	$-0.861374\pi$
$389$	−35.7641	−1.81331	−0.906656	+	0.421870i	$0.861374\pi$
$390$	−100.122	−5.06988
$391$	5.17633	0.261778
$392$	13.5946	0.686630
$393$	−30.3296	−1.52993
$394$	−43.6521	−2.19916
$395$	−7.16983	−0.360753
$396$	77.5071	3.89488
$397$	35.2928	1.77129	0.885647	−	0.464359i	$-0.153715\pi$
$397$	35.2928	1.77129	0.885647	+	0.464359i	$0.153715\pi$
$398$	33.0720	1.65775
$399$	9.27354	0.464258
$400$	30.0965	1.50483
$401$	−34.5278	−1.72424	−0.862119	−	0.506707i	$-0.830863\pi$
$401$	−34.5278	−1.72424	−0.862119	+	0.506707i	$0.830863\pi$
$402$	−65.6334	−3.27350
$403$	37.4067	1.86336
$404$	−13.3346	−0.663422
$405$	−10.2838	−0.511007
$406$	35.8388	1.77865
$407$	−6.96677	−0.345330
$408$	14.6731	0.726427
$409$	−20.3254	−1.00502	−0.502512	−	0.864570i	$-0.667591\pi$
$409$	−20.3254	−1.00502	−0.502512	+	0.864570i	$0.667591\pi$
$410$	−81.3708	−4.01862
$411$	−43.5051	−2.14595
$412$	95.4022	4.70013
$413$	−2.57106	−0.126513
$414$	−80.8636	−3.97423
$415$	−15.7715	−0.774195
$416$	57.9245	2.83998
$417$	−13.5889	−0.665454
$418$	−14.7182	−0.719889
$419$	3.60387	0.176060	0.0880302	−	0.996118i	$-0.471943\pi$
$419$	3.60387	0.176060	0.0880302	+	0.996118i	$0.471943\pi$
$420$	84.9586	4.14555
$421$	−13.0057	−0.633858	−0.316929	−	0.948449i	$-0.602652\pi$
$421$	−13.0057	−0.633858	−0.316929	+	0.948449i	$0.602652\pi$
$422$	60.2100	2.93098
$423$	27.1276	1.31899
$424$	40.2700	1.95568
$425$	2.11392	0.102540
$426$	45.1033	2.18526
$427$	−14.5363	−0.703460
$428$	−57.5031	−2.77952
$429$	50.2490	2.42604
$430$	−55.4175	−2.67247
$431$	16.3119	0.785717	0.392858	−	0.919599i	$-0.371486\pi$
$431$	16.3119	0.785717	0.392858	+	0.919599i	$0.371486\pi$
$432$	−34.5200	−1.66085
$433$	15.0702	0.724228	0.362114	−	0.932134i	$-0.382055\pi$
$433$	15.0702	0.724228	0.362114	+	0.932134i	$0.382055\pi$
$434$	−44.7038	−2.14585
$435$	−45.4058	−2.17704
$436$	−70.6801	−3.38496
$437$	10.9030	0.521563
$438$	−50.1873	−2.39804
$439$	40.2441	1.92075	0.960374	−	0.278716i	$-0.0899087\pi$
$439$	40.2441	1.92075	0.960374	+	0.278716i	$0.0899087\pi$
$440$	−79.7744	−3.80309
$441$	−7.60585	−0.362183
$442$	9.43245	0.448656
$443$	−6.65705	−0.316286	−0.158143	−	0.987416i	$-0.550551\pi$
$443$	−6.65705	−0.316286	−0.158143	+	0.987416i	$0.550551\pi$
$444$	24.7273	1.17351
$445$	32.5249	1.54183
$446$	−1.68497	−0.0797858
$447$	−31.5257	−1.49112
$448$	−22.6884	−1.07192
$449$	0.0185600	0.000875900 0	0.000437950	−	1.00000i	$-0.499861\pi$
$449$	0.0185600	0.000875900 0	0.000437950		1.00000i	$0.499861\pi$
$450$	−33.0232	−1.55673
$451$	40.8381	1.92299
$452$	−72.1288	−3.39265
$453$	−35.3137	−1.65918
$454$	24.1638	1.13406
$455$	32.3116	1.51479
$456$	30.9063	1.44732
$457$	−18.5419	−0.867352	−0.433676	−	0.901069i	$-0.642784\pi$
$457$	−18.5419	−0.867352	−0.433676	+	0.901069i	$0.642784\pi$
$458$	39.4647	1.84407
$459$	−2.42462	−0.113171
$460$	99.8870	4.65725
$461$	23.2839	1.08444	0.542219	−	0.840237i	$-0.317584\pi$
$461$	23.2839	1.08444	0.542219	+	0.840237i	$0.317584\pi$
$462$	−60.0513	−2.79384
$463$	15.0156	0.697833	0.348916	−	0.937154i	$-0.386550\pi$
$463$	15.0156	0.697833	0.348916	+	0.937154i	$0.386550\pi$
$464$	60.9024	2.82732
$465$	56.6373	2.62649
$466$	21.8998	1.01449
$467$	23.4063	1.08311	0.541557	−	0.840664i	$-0.317835\pi$
$467$	23.4063	1.08311	0.541557	+	0.840664i	$0.317835\pi$
$468$	−104.626	−4.83632
$469$	21.1813	0.978062
$470$	−47.1939	−2.17689
$471$	−13.9895	−0.644602
$472$	−8.56866	−0.394405
$473$	27.8128	1.27883
$474$	−17.9874	−0.826190
$475$	4.45260	0.204299
$476$	−8.00390	−0.366858
$477$	−22.5301	−1.03158
$478$	−53.4638	−2.44538
$479$	−31.0709	−1.41967	−0.709833	−	0.704370i	$-0.751230\pi$
$479$	−31.0709	−1.41967	−0.709833	+	0.704370i	$0.751230\pi$
$480$	87.7032	4.00308
$481$	9.40433	0.428801
$482$	38.1765	1.73889
$483$	44.4852	2.02415
$484$	13.8002	0.627282
$485$	36.4399	1.65465
$486$	−52.4896	−2.38098
$487$	−36.1203	−1.63677	−0.818384	−	0.574671i	$-0.805130\pi$
$487$	−36.1203	−1.63677	−0.818384	+	0.574671i	$0.805130\pi$
$488$	−48.4457	−2.19303
$489$	36.1506	1.63479
$490$	13.2319	0.597755
$491$	43.2345	1.95114	0.975572	−	0.219681i	$-0.0705016\pi$
$491$	43.2345	1.95114	0.975572	+	0.219681i	$0.0705016\pi$
$492$	−144.948	−6.53475
$493$	4.27766	0.192656
$494$	19.8678	0.893895
$495$	44.6319	2.00605
$496$	−75.9670	−3.41102
$497$	−14.5558	−0.652917
$498$	−39.5671	−1.77305
$499$	−1.39233	−0.0623290	−0.0311645	−	0.999514i	$-0.509922\pi$
$499$	−1.39233	−0.0623290	−0.0311645	+	0.999514i	$0.509922\pi$
$500$	−28.2670	−1.26414
$501$	−9.71010	−0.433815
$502$	−16.7354	−0.746936
$503$	32.2625	1.43852	0.719258	−	0.694743i	$-0.244482\pi$
$503$	32.2625	1.43852	0.719258	+	0.694743i	$0.244482\pi$
$504$	73.9743	3.29508
$505$	−7.67863	−0.341695
$506$	−70.6031	−3.13869
$507$	−32.8088	−1.45709
$508$	78.6439	3.48926
$509$	8.13285	0.360482	0.180241	−	0.983622i	$-0.442312\pi$
$509$	8.13285	0.360482	0.180241	+	0.983622i	$0.442312\pi$
$510$	14.2816	0.632400
$511$	16.1965	0.716492
$512$	37.4562	1.65535
$513$	−5.10703	−0.225481
$514$	65.2828	2.87950
$515$	54.9366	2.42080
$516$	−98.7164	−4.34575
$517$	23.6855	1.04169
$518$	−11.2389	−0.493808
$519$	−35.2224	−1.54609
$520$	107.686	4.72235
$521$	−4.37462	−0.191656	−0.0958278	−	0.995398i	$-0.530550\pi$
$521$	−4.37462	−0.191656	−0.0958278	+	0.995398i	$0.530550\pi$
$522$	−66.8247	−2.92484
$523$	0.184742	0.00807820	0.00403910	−	0.999992i	$-0.498714\pi$
$523$	0.184742	0.00807820	0.00403910	+	0.999992i	$0.498714\pi$
$524$	55.1377	2.40870
$525$	18.1669	0.792870
$526$	−6.95171	−0.303109
$527$	−5.33577	−0.232430
$528$	−102.048	−4.44105
$529$	29.3019	1.27400
$530$	39.1956	1.70255
$531$	4.79397	0.208040
$532$	−16.8588	−0.730922
$533$	−55.1267	−2.38780
$534$	81.5975	3.53107
$535$	−33.1127	−1.43159
$536$	70.5918	3.04910
$537$	−41.2307	−1.77923
$538$	50.3712	2.17166
$539$	−6.64078	−0.286038
$540$	−46.7875	−2.01341
$541$	14.1251	0.607283	0.303642	−	0.952786i	$-0.401797\pi$
$541$	14.1251	0.607283	0.303642	+	0.952786i	$0.401797\pi$
$542$	−6.00621	−0.257989
$543$	16.8393	0.722644
$544$	−8.26247	−0.354250
$545$	−40.7006	−1.74342
$546$	81.0622	3.46914
$547$	13.4211	0.573846	0.286923	−	0.957954i	$-0.407368\pi$
$547$	13.4211	0.573846	0.286923	+	0.957954i	$0.407368\pi$
$548$	79.0900	3.37856
$549$	27.1042	1.15678
$550$	−28.8330	−1.22944
$551$	9.01014	0.383845
$552$	148.258	6.31027
$553$	5.80493	0.246851
$554$	37.3238	1.58574
$555$	14.2390	0.604414
$556$	24.7040	1.04768
$557$	−5.83410	−0.247199	−0.123599	−	0.992332i	$-0.539444\pi$
$557$	−5.83410	−0.247199	−0.123599	+	0.992332i	$0.539444\pi$
$558$	83.3543	3.52867
$559$	−37.5440	−1.58794
$560$	−65.6196	−2.77293
$561$	−7.16761	−0.302617
$562$	−8.21479	−0.346520
$563$	5.65730	0.238427	0.119213	−	0.992869i	$-0.461963\pi$
$563$	5.65730	0.238427	0.119213	+	0.992869i	$0.461963\pi$
$564$	−84.0676	−3.53988
$565$	−41.5348	−1.74738
$566$	−14.6096	−0.614085
$567$	8.32612	0.349664
$568$	−48.5107	−2.03546
$569$	−1.23054	−0.0515870	−0.0257935	−	0.999667i	$-0.508211\pi$
$569$	−1.23054	−0.0515870	−0.0257935	+	0.999667i	$0.508211\pi$
$570$	30.0817	1.25998
$571$	−24.3480	−1.01893	−0.509465	−	0.860491i	$-0.670157\pi$
$571$	−24.3480	−1.01893	−0.509465	+	0.860491i	$0.670157\pi$
$572$	−91.3501	−3.81954
$573$	0.839608	0.0350751
$574$	65.8805	2.74980
$575$	21.3591	0.890737
$576$	42.3045	1.76269
$577$	−44.0761	−1.83491	−0.917455	−	0.397839i	$-0.869760\pi$
$577$	−44.0761	−1.83491	−0.917455	+	0.397839i	$0.869760\pi$
$578$	43.3017	1.80112
$579$	−16.0563	−0.667277
$580$	82.5454	3.42751
$581$	12.7692	0.529754
$582$	91.4193	3.78945
$583$	−19.6714	−0.814704
$584$	53.9788	2.23366
$585$	−60.2479	−2.49094
$586$	67.3069	2.78042
$587$	18.2430	0.752971	0.376486	−	0.926422i	$-0.377132\pi$
$587$	18.2430	0.752971	0.376486	+	0.926422i	$0.377132\pi$
$588$	23.5703	0.972021
$589$	−11.2389	−0.463089
$590$	−8.34005	−0.343354
$591$	−44.7767	−1.84187
$592$	−19.0987	−0.784951
$593$	1.34559	0.0552567	0.0276284	−	0.999618i	$-0.491204\pi$
$593$	1.34559	0.0552567	0.0276284	+	0.999618i	$0.491204\pi$
$594$	33.0708	1.35691
$595$	−4.60899	−0.188950
$596$	57.3121	2.34760
$597$	33.9240	1.38842
$598$	95.3060	3.89735
$599$	24.7427	1.01096	0.505479	−	0.862839i	$-0.331316\pi$
$599$	24.7427	1.01096	0.505479	+	0.862839i	$0.331316\pi$
$600$	60.5457	2.47177
$601$	24.2590	0.989545	0.494772	−	0.869023i	$-0.335251\pi$
$601$	24.2590	0.989545	0.494772	+	0.869023i	$0.335251\pi$
$602$	44.8678	1.82868
$603$	−39.4945	−1.60834
$604$	64.1984	2.61220
$605$	7.94674	0.323081
$606$	−19.2639	−0.782543
$607$	−33.2239	−1.34852	−0.674259	−	0.738495i	$-0.735537\pi$
$607$	−33.2239	−1.34852	−0.674259	+	0.738495i	$0.735537\pi$
$608$	−17.4034	−0.705803
$609$	36.7621	1.48967
$610$	−47.1531	−1.90917
$611$	−31.9727	−1.29348
$612$	14.9240	0.603267
$613$	−12.4550	−0.503054	−0.251527	−	0.967850i	$-0.580933\pi$
$613$	−12.4550	−0.503054	−0.251527	+	0.967850i	$0.580933\pi$
$614$	−28.2265	−1.13913
$615$	−83.4670	−3.36571
$616$	64.5880	2.60232
$617$	21.8836	0.881001	0.440501	−	0.897752i	$-0.354801\pi$
$617$	21.8836	0.881001	0.440501	+	0.897752i	$0.354801\pi$
$618$	137.823	5.54406
$619$	20.7151	0.832612	0.416306	−	0.909225i	$-0.363325\pi$
$619$	20.7151	0.832612	0.416306	+	0.909225i	$0.363325\pi$
$620$	−102.964	−4.13512
$621$	−24.4985	−0.983089
$622$	−3.55839	−0.142678
$623$	−26.3333	−1.05502
$624$	137.752	5.51451
$625$	−31.0444	−1.24178
$626$	11.7025	0.467728
$627$	−15.0973	−0.602929
$628$	25.4322	1.01485
$629$	−1.34145	−0.0534872
$630$	72.0007	2.86857
$631$	24.2250	0.964384	0.482192	−	0.876066i	$-0.339841\pi$
$631$	24.2250	0.964384	0.482192	+	0.876066i	$0.339841\pi$
$632$	19.3463	0.769556
$633$	61.7611	2.45478
$634$	71.5119	2.84010
$635$	45.2865	1.79714
$636$	69.8200	2.76854
$637$	8.96427	0.355177
$638$	−58.3456	−2.30992
$639$	27.1406	1.07367
$640$	−8.48616	−0.335445
$641$	25.1221	0.992265	0.496133	−	0.868247i	$-0.334753\pi$
$641$	25.1221	0.992265	0.496133	+	0.868247i	$0.334753\pi$
$642$	−83.0721	−3.27859
$643$	−4.59528	−0.181220	−0.0906100	−	0.995886i	$-0.528882\pi$
$643$	−4.59528	−0.181220	−0.0906100	+	0.995886i	$0.528882\pi$
$644$	−80.8718	−3.18680
$645$	−56.8451	−2.23827
$646$	−2.83398	−0.111502
$647$	22.0852	0.868260	0.434130	−	0.900850i	$-0.357056\pi$
$647$	22.0852	0.868260	0.434130	+	0.900850i	$0.357056\pi$
$648$	27.7488	1.09008
$649$	4.18568	0.164302
$650$	38.9212	1.52661
$651$	−45.8554	−1.79722
$652$	−65.7198	−2.57379
$653$	−35.3601	−1.38375	−0.691874	−	0.722019i	$-0.743215\pi$
$653$	−35.3601	−1.38375	−0.691874	+	0.722019i	$0.743215\pi$
$654$	−102.108	−3.99275
$655$	31.7506	1.24060
$656$	111.953	4.37105
$657$	−30.1999	−1.17821
$658$	38.2097	1.48957
$659$	−3.95220	−0.153956	−0.0769779	−	0.997033i	$-0.524527\pi$
$659$	−3.95220	−0.153956	−0.0769779	+	0.997033i	$0.524527\pi$
$660$	−138.313	−5.38381
$661$	−40.7295	−1.58419	−0.792096	−	0.610396i	$-0.791010\pi$
$661$	−40.7295	−1.58419	−0.792096	+	0.610396i	$0.791010\pi$
$662$	36.0446	1.40091
$663$	9.67545	0.375763
$664$	42.5563	1.65151
$665$	−9.70802	−0.376461
$666$	20.9559	0.812024
$667$	43.2217	1.67355
$668$	17.6524	0.682994
$669$	−1.72838	−0.0668231
$670$	68.7084	2.65444
$671$	23.6651	0.913580
$672$	−71.0074	−2.73917
$673$	−12.5276	−0.482905	−0.241452	−	0.970413i	$-0.577624\pi$
$673$	−12.5276	−0.482905	−0.241452	+	0.970413i	$0.577624\pi$
$674$	−13.7965	−0.531420
$675$	−10.0047	−0.385082
$676$	59.6447	2.29403
$677$	−7.25259	−0.278740	−0.139370	−	0.990240i	$-0.544508\pi$
$677$	−7.25259	−0.278740	−0.139370	+	0.990240i	$0.544508\pi$
$678$	−104.201	−4.00182
$679$	−29.5030	−1.13222
$680$	−15.3606	−0.589050
$681$	24.7863	0.949814
$682$	72.7778	2.78681
$683$	33.6901	1.28912	0.644558	−	0.764556i	$-0.277042\pi$
$683$	33.6901	1.28912	0.644558	+	0.764556i	$0.277042\pi$
$684$	31.4348	1.20194
$685$	45.5434	1.74012
$686$	−52.6897	−2.01170
$687$	40.4814	1.54446
$688$	76.2457	2.90684
$689$	26.5540	1.01163
$690$	144.302	5.49349
$691$	−30.5536	−1.16231	−0.581157	−	0.813792i	$-0.697400\pi$
$691$	−30.5536	−1.16231	−0.581157	+	0.813792i	$0.697400\pi$
$692$	64.0324	2.43415
$693$	−36.1355	−1.37267
$694$	−37.1258	−1.40927
$695$	14.2256	0.539608
$696$	122.518	4.64405
$697$	7.86338	0.297847
$698$	−70.7876	−2.67935
$699$	22.4639	0.849664
$700$	−33.0266	−1.24829
$701$	35.6930	1.34811	0.674053	−	0.738683i	$-0.264552\pi$
$701$	35.6930	1.34811	0.674053	+	0.738683i	$0.264552\pi$
$702$	−44.6418	−1.68490
$703$	−2.82553	−0.106567
$704$	36.9367	1.39210
$705$	−48.4097	−1.82321
$706$	−73.5015	−2.76626
$707$	6.21688	0.233810
$708$	−14.8563	−0.558335
$709$	−22.4811	−0.844296	−0.422148	−	0.906527i	$-0.638724\pi$
$709$	−22.4811	−0.844296	−0.422148	+	0.906527i	$0.638724\pi$
$710$	−47.2164	−1.77200
$711$	−10.8238	−0.405925
$712$	−87.7620	−3.28902
$713$	−53.9129	−2.01905
$714$	−11.5629	−0.432730
$715$	−52.6032	−1.96725
$716$	74.9552	2.80121
$717$	−54.8411	−2.04808
$718$	27.3359	1.02017
$719$	−16.2063	−0.604395	−0.302197	−	0.953245i	$-0.597720\pi$
$719$	−16.2063	−0.604395	−0.302197	+	0.953245i	$0.597720\pi$
$720$	122.354	4.55985
$721$	−44.4785	−1.65647
$722$	43.9305	1.63493
$723$	39.1600	1.45637
$724$	−30.6130	−1.13772
$725$	17.6509	0.655539
$726$	19.9365	0.739914
$727$	8.88350	0.329471	0.164735	−	0.986338i	$-0.447323\pi$
$727$	8.88350	0.329471	0.164735	+	0.986338i	$0.447323\pi$
$728$	−87.1863	−3.23134
$729$	−42.9023	−1.58897
$730$	52.5386	1.94454
$731$	5.35535	0.198075
$732$	−83.9950	−3.10454
$733$	−29.7146	−1.09753	−0.548767	−	0.835975i	$-0.684903\pi$
$733$	−29.7146	−1.09753	−0.548767	+	0.835975i	$0.684903\pi$
$734$	52.4809	1.93710
$735$	13.5727	0.500638
$736$	−83.4844	−3.07728
$737$	−34.4832	−1.27020
$738$	−122.840	−4.52181
$739$	−7.49138	−0.275575	−0.137787	−	0.990462i	$-0.543999\pi$
$739$	−7.49138	−0.275575	−0.137787	+	0.990462i	$0.543999\pi$
$740$	−25.8858	−0.951582
$741$	20.3796	0.748664
$742$	−31.7340	−1.16499
$743$	−25.0543	−0.919154	−0.459577	−	0.888138i	$-0.651999\pi$
$743$	−25.0543	−0.919154	−0.459577	+	0.888138i	$0.651999\pi$
$744$	−152.824	−5.60281
$745$	33.0028	1.20913
$746$	−70.1446	−2.56818
$747$	−23.8093	−0.871136
$748$	13.0304	0.476437
$749$	26.8092	0.979586
$750$	−40.8360	−1.49112
$751$	−22.8018	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$751$	−22.8018	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$752$	64.9314	2.36780
$753$	−17.1665	−0.625582
$754$	78.7598	2.86826
$755$	36.9682	1.34541
$756$	37.8808	1.37771
$757$	29.2597	1.06346	0.531731	−	0.846913i	$-0.321542\pi$
$757$	29.2597	1.06346	0.531731	+	0.846913i	$0.321542\pi$
$758$	−79.9625	−2.90437
$759$	−72.4219	−2.62875
$760$	−32.3543	−1.17361
$761$	−1.22985	−0.0445821	−0.0222910	−	0.999752i	$-0.507096\pi$
$761$	−1.22985	−0.0445821	−0.0222910	+	0.999752i	$0.507096\pi$
$762$	113.613	4.11577
$763$	32.9526	1.19296
$764$	−1.52636	−0.0552219
$765$	8.59387	0.310712
$766$	88.3231	3.19124
$767$	−5.65018	−0.204016
$768$	32.2479	1.16365
$769$	50.9888	1.83870	0.919352	−	0.393437i	$-0.128714\pi$
$769$	50.9888	1.83870	0.919352	+	0.393437i	$0.128714\pi$
$770$	62.8648	2.26549
$771$	66.9646	2.41167
$772$	29.1895	1.05055
$773$	−20.9688	−0.754197	−0.377098	−	0.926173i	$-0.623078\pi$
$773$	−20.9688	−0.754197	−0.377098	+	0.926173i	$0.623078\pi$
$774$	−83.6601	−3.00710
$775$	−22.0170	−0.790875
$776$	−98.3259	−3.52969
$777$	−11.5284	−0.413579
$778$	93.9275	3.36746
$779$	16.5628	0.593425
$780$	186.706	6.68514
$781$	23.6968	0.847940
$782$	−13.5946	−0.486143
$783$	−20.2452	−0.723506
$784$	−18.2050	−0.650178
$785$	14.6449	0.522700
$786$	79.6549	2.84120
$787$	21.2395	0.757106	0.378553	−	0.925580i	$-0.376422\pi$
$787$	21.2395	0.757106	0.378553	+	0.925580i	$0.376422\pi$
$788$	81.4016	2.89981
$789$	−7.13080	−0.253863
$790$	18.8302	0.669947
$791$	33.6280	1.19567
$792$	−120.430	−4.27930
$793$	−31.9451	−1.13440
$794$	−92.6897	−3.28943
$795$	40.2053	1.42594
$796$	−61.6720	−2.18591
$797$	−26.3520	−0.933437	−0.466718	−	0.884406i	$-0.654564\pi$
$797$	−26.3520	−0.933437	−0.466718	+	0.884406i	$0.654564\pi$
$798$	−24.3552	−0.862163
$799$	4.56065	0.161344
$800$	−34.0935	−1.20539
$801$	49.1008	1.73489
$802$	90.6806	3.20204
$803$	−26.3679	−0.930505
$804$	122.392	4.31643
$805$	−46.5694	−1.64136
$806$	−98.2415	−3.46041
$807$	51.6688	1.81883
$808$	20.7192	0.728901
$809$	8.01497	0.281791	0.140896	−	0.990024i	$-0.455002\pi$
$809$	8.01497	0.281791	0.140896	+	0.990024i	$0.455002\pi$
$810$	27.0085	0.948980
$811$	−5.59792	−0.196569	−0.0982847	−	0.995158i	$-0.531336\pi$
$811$	−5.59792	−0.196569	−0.0982847	+	0.995158i	$0.531336\pi$
$812$	−66.8316	−2.34533
$813$	−6.16094	−0.216074
$814$	18.2969	0.641305
$815$	−37.8443	−1.32563
$816$	−19.6493	−0.687862
$817$	11.2801	0.394641
$818$	53.3807	1.86641
$819$	48.7787	1.70447
$820$	151.739	5.29894
$821$	−25.2052	−0.879666	−0.439833	−	0.898080i	$-0.644962\pi$
$821$	−25.2052	−0.879666	−0.439833	+	0.898080i	$0.644962\pi$
$822$	114.258	3.98520
$823$	−37.5897	−1.31029	−0.655147	−	0.755501i	$-0.727393\pi$
$823$	−37.5897	−1.31029	−0.655147	+	0.755501i	$0.727393\pi$
$824$	−148.235	−5.16403
$825$	−29.5758	−1.02970
$826$	6.75238	0.234945
$827$	41.4545	1.44151	0.720757	−	0.693188i	$-0.243794\pi$
$827$	41.4545	1.44151	0.720757	+	0.693188i	$0.243794\pi$
$828$	150.793	5.24041
$829$	2.68862	0.0933798	0.0466899	−	0.998909i	$-0.485133\pi$
$829$	2.68862	0.0933798	0.0466899	+	0.998909i	$0.485133\pi$
$830$	41.4209	1.43774
$831$	38.2853	1.32810
$832$	−49.8602	−1.72859
$833$	−1.27868	−0.0443037
$834$	35.6887	1.23580
$835$	10.1650	0.351775
$836$	27.4461	0.949245
$837$	25.2530	0.872873
$838$	−9.46486	−0.326958
$839$	−34.6585	−1.19654	−0.598271	−	0.801293i	$-0.704146\pi$
$839$	−34.6585	−1.19654	−0.598271	+	0.801293i	$0.704146\pi$
$840$	−132.008	−4.55472
$841$	6.71790	0.231652
$842$	34.1569	1.17713
$843$	−8.42642	−0.290221
$844$	−112.278	−3.86478
$845$	34.3459	1.18154
$846$	−71.2455	−2.44947
$847$	−6.43395	−0.221073
$848$	−53.9270	−1.85186
$849$	−14.9859	−0.514315
$850$	−5.55179	−0.190425
$851$	−13.5541	−0.464629
$852$	−84.1077	−2.88148
$853$	15.0820	0.516397	0.258199	−	0.966092i	$-0.416871\pi$
$853$	15.0820	0.516397	0.258199	+	0.966092i	$0.416871\pi$
$854$	38.1767	1.30638
$855$	18.1015	0.619058
$856$	89.3480	3.05385
$857$	38.2461	1.30646	0.653231	−	0.757159i	$-0.273413\pi$
$857$	38.2461	1.30646	0.653231	+	0.757159i	$0.273413\pi$
$858$	−131.969	−4.50536
$859$	−30.5218	−1.04139	−0.520695	−	0.853743i	$-0.674327\pi$
$859$	−30.5218	−1.04139	−0.520695	+	0.853743i	$0.674327\pi$
$860$	103.341	3.52391
$861$	67.5777	2.30304
$862$	−42.8401	−1.45914
$863$	20.3606	0.693084	0.346542	−	0.938035i	$-0.387356\pi$
$863$	20.3606	0.693084	0.346542	+	0.938035i	$0.387356\pi$
$864$	39.1045	1.33036
$865$	36.8726	1.25371
$866$	−39.5790	−1.34495
$867$	44.4173	1.50849
$868$	83.3628	2.82952
$869$	−9.45043	−0.320584
$870$	119.250	4.04294
$871$	46.5483	1.57723
$872$	109.822	3.71906
$873$	55.0110	1.86184
$874$	−28.6347	−0.968583
$875$	13.1787	0.445520
$876$	93.5883	3.16206
$877$	−12.2353	−0.413158	−0.206579	−	0.978430i	$-0.566233\pi$
$877$	−12.2353	−0.413158	−0.206579	+	0.978430i	$0.566233\pi$
$878$	−105.693	−3.56698
$879$	69.0408	2.32869
$880$	106.829	3.60119
$881$	3.02030	0.101756	0.0508782	−	0.998705i	$-0.483798\pi$
$881$	3.02030	0.101756	0.0508782	+	0.998705i	$0.483798\pi$
$882$	19.9753	0.672603
$883$	13.1588	0.442828	0.221414	−	0.975180i	$-0.428933\pi$
$883$	13.1588	0.442828	0.221414	+	0.975180i	$0.428933\pi$
$884$	−17.5895	−0.591597
$885$	−8.55490	−0.287570
$886$	17.4834	0.587368
$887$	−0.231319	−0.00776694	−0.00388347	−	0.999992i	$-0.501236\pi$
$887$	−0.231319	−0.00776694	−0.00388347	+	0.999992i	$0.501236\pi$
$888$	−38.4212	−1.28933
$889$	−36.6654	−1.22972
$890$	−85.4205	−2.86330
$891$	−13.5549	−0.454107
$892$	3.14211	0.105206
$893$	9.60620	0.321459
$894$	82.7962	2.76912
$895$	43.1624	1.44276
$896$	6.87068	0.229533
$897$	97.7612	3.26415
$898$	−0.0487442	−0.00162662
$899$	−44.5530	−1.48592
$900$	61.5810	2.05270
$901$	−3.78772	−0.126187
$902$	−107.253	−3.57115
$903$	46.0237	1.53157
$904$	112.073	3.72750
$905$	−17.6283	−0.585984
$906$	92.7445	3.08123
$907$	29.8560	0.991351	0.495676	−	0.868508i	$-0.334921\pi$
$907$	29.8560	0.991351	0.495676	+	0.868508i	$0.334921\pi$
$908$	−45.0602	−1.49538
$909$	−11.5919	−0.384480
$910$	−84.8601	−2.81309
$911$	13.9017	0.460585	0.230292	−	0.973121i	$-0.426032\pi$
$911$	13.9017	0.460585	0.230292	+	0.973121i	$0.426032\pi$
$912$	−41.3877	−1.37048
$913$	−20.7882	−0.687989
$914$	48.6966	1.61074
$915$	−48.3679	−1.59899
$916$	−73.5931	−2.43158
$917$	−25.7064	−0.848899
$918$	6.36779	0.210168
$919$	−34.6040	−1.14148	−0.570740	−	0.821131i	$-0.693344\pi$
$919$	−34.6040	−1.14148	−0.570740	+	0.821131i	$0.693344\pi$
$920$	−155.204	−5.11692
$921$	−28.9537	−0.954056
$922$	−61.1506	−2.01389
$923$	−31.9880	−1.05290
$924$	111.982	3.68395
$925$	−5.53524	−0.181998
$926$	−39.4355	−1.29593
$927$	82.9343	2.72392
$928$	−68.9905	−2.26473
$929$	6.56997	0.215554	0.107777	−	0.994175i	$-0.465627\pi$
$929$	6.56997	0.215554	0.107777	+	0.994175i	$0.465627\pi$
$930$	−148.747	−4.87760
$931$	−2.69332	−0.0882699
$932$	−40.8383	−1.33770
$933$	−3.65006	−0.119498
$934$	−61.4721	−2.01143
$935$	7.50343	0.245388
$936$	162.567	5.31366
$937$	33.5636	1.09647	0.548237	−	0.836323i	$-0.315299\pi$
$937$	33.5636	1.09647	0.548237	+	0.836323i	$0.315299\pi$
$938$	−55.6286	−1.81634
$939$	12.0040	0.391736
$940$	88.0062	2.87045
$941$	24.8479	0.810017	0.405009	−	0.914313i	$-0.367269\pi$
$941$	24.8479	0.810017	0.405009	+	0.914313i	$0.367269\pi$
$942$	36.7407	1.19708
$943$	79.4520	2.58731
$944$	11.4746	0.373466
$945$	21.8133	0.709588
$946$	−73.0448	−2.37489
$947$	56.5643	1.83809	0.919047	−	0.394148i	$-0.128960\pi$
$947$	56.5643	1.83809	0.919047	+	0.394148i	$0.128960\pi$
$948$	33.5426	1.08941
$949$	35.5937	1.15542
$950$	−11.6939	−0.379400
$951$	73.3541	2.37867
$952$	12.4364	0.403067
$953$	27.0760	0.877077	0.438539	−	0.898712i	$-0.355496\pi$
$953$	27.0760	0.877077	0.438539	+	0.898712i	$0.355496\pi$
$954$	59.1710	1.91573
$955$	−0.878945	−0.0284420
$956$	99.6982	3.22447
$957$	−59.8487	−1.93463
$958$	81.6018	2.63643
$959$	−36.8734	−1.19071
$960$	−75.4930	−2.43653
$961$	24.5735	0.792692
$962$	−24.6987	−0.796317
$963$	−49.9881	−1.61084
$964$	−71.1908	−2.29290
$965$	16.8086	0.541087
$966$	−116.832	−3.75900
$967$	16.5028	0.530696	0.265348	−	0.964153i	$-0.414513\pi$
$967$	16.5028	0.530696	0.265348	+	0.964153i	$0.414513\pi$
$968$	−21.4427	−0.689194
$969$	−2.90699	−0.0933860
$970$	−95.7025	−3.07282
$971$	39.2012	1.25803	0.629013	−	0.777395i	$-0.283459\pi$
$971$	39.2012	1.25803	0.629013	+	0.777395i	$0.283459\pi$
$972$	97.8816	3.13955
$973$	−11.5175	−0.369235
$974$	94.8631	3.03961
$975$	39.9239	1.27859
$976$	64.8753	2.07661
$977$	10.8178	0.346093	0.173046	−	0.984914i	$-0.444639\pi$
$977$	10.8178	0.346093	0.173046	+	0.984914i	$0.444639\pi$
$978$	−94.9425	−3.03593
$979$	42.8706	1.37015
$980$	−24.6746	−0.788200
$981$	−61.4431	−1.96173
$982$	−113.547	−3.62343
$983$	−4.33148	−0.138153	−0.0690763	−	0.997611i	$-0.522005\pi$
$983$	−4.33148	−0.138153	−0.0690763	+	0.997611i	$0.522005\pi$
$984$	225.219	7.17972
$985$	46.8745	1.49355
$986$	−11.2344	−0.357778
$987$	39.1941	1.24756
$988$	−37.0491	−1.17869
$989$	54.1107	1.72062
$990$	−117.217	−3.72540
$991$	36.9435	1.17355	0.586775	−	0.809750i	$-0.300397\pi$
$991$	36.9435	1.17355	0.586775	+	0.809750i	$0.300397\pi$
$992$	86.0558	2.73228
$993$	36.9732	1.17331
$994$	38.2280	1.21252
$995$	−35.5133	−1.12585
$996$	73.7840	2.33794
$997$	−51.4313	−1.62884	−0.814422	−	0.580273i	$-0.802946\pi$
$997$	−51.4313	−1.62884	−0.814422	+	0.580273i	$0.802946\pi$
$998$	3.65667	0.115750
$999$	6.34880	0.200867

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8009.2.a.b.1.20	✓	361

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8009.2.a.b.1.20	✓	361	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:	\( N \)	\(=\)	\( 8009 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.62631 q^{2} -2.69396 q^{3} +4.89749 q^{4} +2.82018 q^{5} +7.07518 q^{6} -2.28331 q^{7} -7.60969 q^{8} +4.25744 q^{9} +O(q^{10})\)	\(q-2.62631 q^{2} -2.69396 q^{3} +4.89749 q^{4} +2.82018 q^{5} +7.07518 q^{6} -2.28331 q^{7} -7.60969 q^{8} +4.25744 q^{9} -7.40666 q^{10} +3.71723 q^{11} -13.1937 q^{12} -5.01783 q^{13} +5.99668 q^{14} -7.59746 q^{15} +10.1904 q^{16} +0.715753 q^{17} -11.1813 q^{18} +1.50761 q^{19} +13.8118 q^{20} +6.15116 q^{21} -9.76259 q^{22} +7.23201 q^{23} +20.5002 q^{24} +2.95342 q^{25} +13.1784 q^{26} -3.38751 q^{27} -11.1825 q^{28} +5.97645 q^{29} +19.9533 q^{30} -7.45476 q^{31} -11.5437 q^{32} -10.0141 q^{33} -1.87979 q^{34} -6.43935 q^{35} +20.8508 q^{36} -1.87418 q^{37} -3.95944 q^{38} +13.5179 q^{39} -21.4607 q^{40} +10.9862 q^{41} -16.1548 q^{42} +7.48211 q^{43} +18.2051 q^{44} +12.0068 q^{45} -18.9935 q^{46} +6.37182 q^{47} -27.4526 q^{48} -1.78648 q^{49} -7.75658 q^{50} -1.92821 q^{51} -24.5748 q^{52} -5.29194 q^{53} +8.89663 q^{54} +10.4833 q^{55} +17.3753 q^{56} -4.06144 q^{57} -15.6960 q^{58} +1.12602 q^{59} -37.2085 q^{60} +6.36632 q^{61} +19.5785 q^{62} -9.72107 q^{63} +9.93660 q^{64} -14.1512 q^{65} +26.3001 q^{66} -9.27657 q^{67} +3.50539 q^{68} -19.4828 q^{69} +16.9117 q^{70} +6.37486 q^{71} -32.3978 q^{72} -7.09344 q^{73} +4.92218 q^{74} -7.95640 q^{75} +7.38349 q^{76} -8.48760 q^{77} -35.5020 q^{78} -2.54233 q^{79} +28.7388 q^{80} -3.64651 q^{81} -28.8530 q^{82} -5.59239 q^{83} +30.1252 q^{84} +2.01855 q^{85} -19.6503 q^{86} -16.1003 q^{87} -28.2870 q^{88} +11.5329 q^{89} -31.5334 q^{90} +11.4573 q^{91} +35.4186 q^{92} +20.0829 q^{93} -16.7343 q^{94} +4.25173 q^{95} +31.0984 q^{96} +12.9211 q^{97} +4.69185 q^{98} +15.8259 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9}+O(q^{10}) \) 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9	\( 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9} + 65 q^{10} + 33 q^{11} + 52 q^{12} + 89 q^{13} + 32 q^{14} + 55 q^{15} + 512 q^{16} + 42 q^{17} + 34 q^{18} + 191 q^{19} + 48 q^{20} + 53 q^{21} + 61 q^{22} + 52 q^{23} + 139 q^{24} + 458 q^{25} + 57 q^{26} + 80 q^{27} + 194 q^{28} + 47 q^{29} + 32 q^{30} + 254 q^{31} + 55 q^{32} + 40 q^{33} + 122 q^{34} + 93 q^{35} + 519 q^{36} + 43 q^{37} + 25 q^{38} + 210 q^{39} + 184 q^{40} + 54 q^{41} + 48 q^{42} + 151 q^{43} + 56 q^{44} + 82 q^{45} + 101 q^{46} + 117 q^{47} + 77 q^{48} + 563 q^{49} + 38 q^{50} + 143 q^{51} + 241 q^{52} + 14 q^{53} + 164 q^{54} + 452 q^{55} + 52 q^{56} + 21 q^{57} + 55 q^{58} + 125 q^{59} + 39 q^{60} + 227 q^{61} + 58 q^{62} + 292 q^{63} + 710 q^{64} + 15 q^{65} + 105 q^{66} + 120 q^{67} + 125 q^{68} + 136 q^{69} + 88 q^{70} + 105 q^{71} + 78 q^{72} + 108 q^{73} + 41 q^{74} + 128 q^{75} + 461 q^{76} + 28 q^{77} + 13 q^{78} + 400 q^{79} + 59 q^{80} + 485 q^{81} + 175 q^{82} + 97 q^{83} + 76 q^{84} + 144 q^{85} - 14 q^{86} + 327 q^{87} + 145 q^{88} + 52 q^{89} + 60 q^{90} + 192 q^{91} + 11 q^{92} + 32 q^{93} + 366 q^{94} + 182 q^{95} + 275 q^{96} + 117 q^{97} + 42 q^{98} + 111 q^{99}+O(q^{100}) \) 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9 + 65 * q^10 + 33 * q^11 + 52 * q^12 + 89 * q^13 + 32 * q^14 + 55 * q^15 + 512 * q^16 + 42 * q^17 + 34 * q^18 + 191 * q^19 + 48 * q^20 + 53 * q^21 + 61 * q^22 + 52 * q^23 + 139 * q^24 + 458 * q^25 + 57 * q^26 + 80 * q^27 + 194 * q^28 + 47 * q^29 + 32 * q^30 + 254 * q^31 + 55 * q^32 + 40 * q^33 + 122 * q^34 + 93 * q^35 + 519 * q^36 + 43 * q^37 + 25 * q^38 + 210 * q^39 + 184 * q^40 + 54 * q^41 + 48 * q^42 + 151 * q^43 + 56 * q^44 + 82 * q^45 + 101 * q^46 + 117 * q^47 + 77 * q^48 + 563 * q^49 + 38 * q^50 + 143 * q^51 + 241 * q^52 + 14 * q^53 + 164 * q^54 + 452 * q^55 + 52 * q^56 + 21 * q^57 + 55 * q^58 + 125 * q^59 + 39 * q^60 + 227 * q^61 + 58 * q^62 + 292 * q^63 + 710 * q^64 + 15 * q^65 + 105 * q^66 + 120 * q^67 + 125 * q^68 + 136 * q^69 + 88 * q^70 + 105 * q^71 + 78 * q^72 + 108 * q^73 + 41 * q^74 + 128 * q^75 + 461 * q^76 + 28 * q^77 + 13 * q^78 + 400 * q^79 + 59 * q^80 + 485 * q^81 + 175 * q^82 + 97 * q^83 + 76 * q^84 + 144 * q^85 - 14 * q^86 + 327 * q^87 + 145 * q^88 + 52 * q^89 + 60 * q^90 + 192 * q^91 + 11 * q^92 + 32 * q^93 + 366 * q^94 + 182 * q^95 + 275 * q^96 + 117 * q^97 + 42 * q^98 + 111 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more