LMFDB - Embedded newform 6045.2.a.bh.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.2695680219$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 2 x^{16} - 25 x^{15} + 47 x^{14} + 252 x^{13} - 437 x^{12} - 1319 x^{11} + 2056 x^{10} + \cdots + 8 $ x^17 - 2x^16 - 25x^15 + 47x^14 + 252x^13 - 437x^12 - 1319x^11 + 2056x^10 + 3854x^9 - 5201x^8 - 6304x^7 + 6915x^6 + 5469x^5 - 4238x^4 - 2198x^3 + 760x^2 + 306x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$1.39113$ of defining polynomial
Character	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.39113 q^{2} -1.00000 q^{3} -0.0647466 q^{4} +1.00000 q^{5} -1.39113 q^{6} +1.38286 q^{7} -2.87234 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.39113 q^{2} -1.00000 q^{3} -0.0647466 q^{4} +1.00000 q^{5} -1.39113 q^{6} +1.38286 q^{7} -2.87234 q^{8} +1.00000 q^{9} +1.39113 q^{10} +5.69921 q^{11} +0.0647466 q^{12} +1.00000 q^{13} +1.92375 q^{14} -1.00000 q^{15} -3.86631 q^{16} -1.02783 q^{17} +1.39113 q^{18} -2.05308 q^{19} -0.0647466 q^{20} -1.38286 q^{21} +7.92836 q^{22} -0.355455 q^{23} +2.87234 q^{24} +1.00000 q^{25} +1.39113 q^{26} -1.00000 q^{27} -0.0895356 q^{28} +8.85962 q^{29} -1.39113 q^{30} +1.00000 q^{31} +0.366116 q^{32} -5.69921 q^{33} -1.42985 q^{34} +1.38286 q^{35} -0.0647466 q^{36} +9.22459 q^{37} -2.85611 q^{38} -1.00000 q^{39} -2.87234 q^{40} -8.53906 q^{41} -1.92375 q^{42} -11.2283 q^{43} -0.369004 q^{44} +1.00000 q^{45} -0.494485 q^{46} +12.1171 q^{47} +3.86631 q^{48} -5.08769 q^{49} +1.39113 q^{50} +1.02783 q^{51} -0.0647466 q^{52} -1.95405 q^{53} -1.39113 q^{54} +5.69921 q^{55} -3.97205 q^{56} +2.05308 q^{57} +12.3249 q^{58} +5.78963 q^{59} +0.0647466 q^{60} +1.90976 q^{61} +1.39113 q^{62} +1.38286 q^{63} +8.24195 q^{64} +1.00000 q^{65} -7.92836 q^{66} -5.58550 q^{67} +0.0665485 q^{68} +0.355455 q^{69} +1.92375 q^{70} -0.557844 q^{71} -2.87234 q^{72} +3.48698 q^{73} +12.8326 q^{74} -1.00000 q^{75} +0.132930 q^{76} +7.88122 q^{77} -1.39113 q^{78} +4.77841 q^{79} -3.86631 q^{80} +1.00000 q^{81} -11.8790 q^{82} +7.59662 q^{83} +0.0895356 q^{84} -1.02783 q^{85} -15.6200 q^{86} -8.85962 q^{87} -16.3701 q^{88} -7.46202 q^{89} +1.39113 q^{90} +1.38286 q^{91} +0.0230145 q^{92} -1.00000 q^{93} +16.8565 q^{94} -2.05308 q^{95} -0.366116 q^{96} -0.592459 q^{97} -7.07766 q^{98} +5.69921 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 2 q^{2} - 17 q^{3} + 20 q^{4} + 17 q^{5} - 2 q^{6} + 18 q^{7} + 9 q^{8} + 17 q^{9}+O(q^{10}) $ 17 * q + 2 * q^2 - 17 * q^3 + 20 * q^4 + 17 * q^5 - 2 * q^6 + 18 * q^7 + 9 * q^8 + 17 * q^9	\( 17 q + 2 q^{2} - 17 q^{3} + 20 q^{4} + 17 q^{5} - 2 q^{6} + 18 q^{7} + 9 q^{8} + 17 q^{9} + 2 q^{10} + 3 q^{11} - 20 q^{12} + 17 q^{13} + q^{14} - 17 q^{15} + 26 q^{16} + 2 q^{18} + 10 q^{19} + 20 q^{20} - 18 q^{21} + 5 q^{22} + 16 q^{23} - 9 q^{24} + 17 q^{25} + 2 q^{26} - 17 q^{27} + 36 q^{28} - 3 q^{29} - 2 q^{30} + 17 q^{31} + 20 q^{32} - 3 q^{33} + q^{34} + 18 q^{35} + 20 q^{36} + 14 q^{37} + 22 q^{38} - 17 q^{39} + 9 q^{40} - 6 q^{41} - q^{42} + 24 q^{43} - 15 q^{44} + 17 q^{45} + 6 q^{46} + 25 q^{47} - 26 q^{48} + 31 q^{49} + 2 q^{50} + 20 q^{52} - 15 q^{53} - 2 q^{54} + 3 q^{55} + 31 q^{56} - 10 q^{57} + 44 q^{58} + 16 q^{59} - 20 q^{60} - 5 q^{61} + 2 q^{62} + 18 q^{63} + 35 q^{64} + 17 q^{65} - 5 q^{66} + 50 q^{67} + 13 q^{68} - 16 q^{69} + q^{70} + 16 q^{71} + 9 q^{72} + 33 q^{73} + 2 q^{74} - 17 q^{75} + 9 q^{77} - 2 q^{78} - 10 q^{79} + 26 q^{80} + 17 q^{81} + 61 q^{82} + 27 q^{83} - 36 q^{84} - 12 q^{86} + 3 q^{87} + 23 q^{88} - 24 q^{89} + 2 q^{90} + 18 q^{91} - 21 q^{92} - 17 q^{93} + 6 q^{94} + 10 q^{95} - 20 q^{96} + 48 q^{97} + 14 q^{98} + 3 q^{99}+O(q^{100}) \) 17 * q + 2 * q^2 - 17 * q^3 + 20 * q^4 + 17 * q^5 - 2 * q^6 + 18 * q^7 + 9 * q^8 + 17 * q^9 + 2 * q^10 + 3 * q^11 - 20 * q^12 + 17 * q^13 + q^14 - 17 * q^15 + 26 * q^16 + 2 * q^18 + 10 * q^19 + 20 * q^20 - 18 * q^21 + 5 * q^22 + 16 * q^23 - 9 * q^24 + 17 * q^25 + 2 * q^26 - 17 * q^27 + 36 * q^28 - 3 * q^29 - 2 * q^30 + 17 * q^31 + 20 * q^32 - 3 * q^33 + q^34 + 18 * q^35 + 20 * q^36 + 14 * q^37 + 22 * q^38 - 17 * q^39 + 9 * q^40 - 6 * q^41 - q^42 + 24 * q^43 - 15 * q^44 + 17 * q^45 + 6 * q^46 + 25 * q^47 - 26 * q^48 + 31 * q^49 + 2 * q^50 + 20 * q^52 - 15 * q^53 - 2 * q^54 + 3 * q^55 + 31 * q^56 - 10 * q^57 + 44 * q^58 + 16 * q^59 - 20 * q^60 - 5 * q^61 + 2 * q^62 + 18 * q^63 + 35 * q^64 + 17 * q^65 - 5 * q^66 + 50 * q^67 + 13 * q^68 - 16 * q^69 + q^70 + 16 * q^71 + 9 * q^72 + 33 * q^73 + 2 * q^74 - 17 * q^75 + 9 * q^77 - 2 * q^78 - 10 * q^79 + 26 * q^80 + 17 * q^81 + 61 * q^82 + 27 * q^83 - 36 * q^84 - 12 * q^86 + 3 * q^87 + 23 * q^88 - 24 * q^89 + 2 * q^90 + 18 * q^91 - 21 * q^92 - 17 * q^93 + 6 * q^94 + 10 * q^95 - 20 * q^96 + 48 * q^97 + 14 * q^98 + 3 * 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 $

$2$	1.39113	0.983680	0.491840	−	0.870686i	$-0.336324\pi$
$2$	1.39113	0.983680	0.491840	+	0.870686i	$0.336324\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.0647466	−0.0323733
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.39113	−0.567928
$7$	1.38286	0.522673	0.261337	−	0.965248i	$-0.415837\pi$
$7$	1.38286	0.522673	0.261337	+	0.965248i	$0.415837\pi$
$8$	−2.87234	−1.01553
$9$	1.00000	0.333333
$10$	1.39113	0.439915
$11$	5.69921	1.71838	0.859188	−	0.511661i	$-0.170969\pi$
$11$	5.69921	1.71838	0.859188	+	0.511661i	$0.170969\pi$
$12$	0.0647466	0.0186907
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.92375	0.514143
$15$	−1.00000	−0.258199
$16$	−3.86631	−0.966579
$17$	−1.02783	−0.249285	−0.124643	−	0.992202i	$-0.539778\pi$
$17$	−1.02783	−0.249285	−0.124643	+	0.992202i	$0.539778\pi$
$18$	1.39113	0.327893
$19$	−2.05308	−0.471009	−0.235504	−	0.971873i	$-0.575674\pi$
$19$	−2.05308	−0.471009	−0.235504	+	0.971873i	$0.575674\pi$
$20$	−0.0647466	−0.0144778
$21$	−1.38286	−0.301765
$22$	7.92836	1.69033
$23$	−0.355455	−0.0741174	−0.0370587	−	0.999313i	$-0.511799\pi$
$23$	−0.355455	−0.0741174	−0.0370587	+	0.999313i	$0.511799\pi$
$24$	2.87234	0.586314
$25$	1.00000	0.200000
$26$	1.39113	0.272824
$27$	−1.00000	−0.192450
$28$	−0.0895356	−0.0169206
$29$	8.85962	1.64519	0.822595	−	0.568627i	$-0.192525\pi$
$29$	8.85962	1.64519	0.822595	+	0.568627i	$0.192525\pi$
$30$	−1.39113	−0.253985
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0.366116	0.0647208
$33$	−5.69921	−0.992104
$34$	−1.42985	−0.245217
$35$	1.38286	0.233747
$36$	−0.0647466	−0.0107911
$37$	9.22459	1.51651	0.758256	−	0.651956i	$-0.226051\pi$
$37$	9.22459	1.51651	0.758256	+	0.651956i	$0.226051\pi$
$38$	−2.85611	−0.463322
$39$	−1.00000	−0.160128
$40$	−2.87234	−0.454157
$41$	−8.53906	−1.33358	−0.666789	−	0.745247i	$-0.732332\pi$
$41$	−8.53906	−1.33358	−0.666789	+	0.745247i	$0.732332\pi$
$42$	−1.92375	−0.296841
$43$	−11.2283	−1.71230	−0.856148	−	0.516731i	$-0.827149\pi$
$43$	−11.2283	−1.71230	−0.856148	+	0.516731i	$0.827149\pi$
$44$	−0.369004	−0.0556295
$45$	1.00000	0.149071
$46$	−0.494485	−0.0729079
$47$	12.1171	1.76746	0.883731	−	0.467994i	$-0.155023\pi$
$47$	12.1171	1.76746	0.883731	+	0.467994i	$0.155023\pi$
$48$	3.86631	0.558054
$49$	−5.08769	−0.726813
$50$	1.39113	0.196736
$51$	1.02783	0.143925
$52$	−0.0647466	−0.00897873
$53$	−1.95405	−0.268409	−0.134204	−	0.990954i	$-0.542848\pi$
$53$	−1.95405	−0.268409	−0.134204	+	0.990954i	$0.542848\pi$
$54$	−1.39113	−0.189309
$55$	5.69921	0.768481
$56$	−3.97205	−0.530788
$57$	2.05308	0.271937
$58$	12.3249	1.61834
$59$	5.78963	0.753746	0.376873	−	0.926265i	$-0.376999\pi$
$59$	5.78963	0.753746	0.376873	+	0.926265i	$0.376999\pi$
$60$	0.0647466	0.00835875
$61$	1.90976	0.244519	0.122260	−	0.992498i	$-0.460986\pi$
$61$	1.90976	0.244519	0.122260	+	0.992498i	$0.460986\pi$
$62$	1.39113	0.176674
$63$	1.38286	0.174224
$64$	8.24195	1.03024
$65$	1.00000	0.124035
$66$	−7.92836	−0.975913
$67$	−5.58550	−0.682378	−0.341189	−	0.939995i	$-0.610830\pi$
$67$	−5.58550	−0.682378	−0.341189	+	0.939995i	$0.610830\pi$
$68$	0.0665485	0.00807019
$69$	0.355455	0.0427917
$70$	1.92375	0.229932
$71$	−0.557844	−0.0662039	−0.0331020	−	0.999452i	$-0.510539\pi$
$71$	−0.557844	−0.0662039	−0.0331020	+	0.999452i	$0.510539\pi$
$72$	−2.87234	−0.338508
$73$	3.48698	0.408120	0.204060	−	0.978958i	$-0.434586\pi$
$73$	3.48698	0.408120	0.204060	+	0.978958i	$0.434586\pi$
$74$	12.8326	1.49176
$75$	−1.00000	−0.115470
$76$	0.132930	0.0152481
$77$	7.88122	0.898149
$78$	−1.39113	−0.157515
$79$	4.77841	0.537613	0.268807	−	0.963194i	$-0.413371\pi$
$79$	4.77841	0.537613	0.268807	+	0.963194i	$0.413371\pi$
$80$	−3.86631	−0.432267
$81$	1.00000	0.111111
$82$	−11.8790	−1.31181
$83$	7.59662	0.833837	0.416919	−	0.908944i	$-0.363110\pi$
$83$	7.59662	0.833837	0.416919	+	0.908944i	$0.363110\pi$
$84$	0.0895356	0.00976914
$85$	−1.02783	−0.111484
$86$	−15.6200	−1.68435
$87$	−8.85962	−0.949851
$88$	−16.3701	−1.74505
$89$	−7.46202	−0.790973	−0.395486	−	0.918472i	$-0.629424\pi$
$89$	−7.46202	−0.790973	−0.395486	+	0.918472i	$0.629424\pi$
$90$	1.39113	0.146638
$91$	1.38286	0.144963
$92$	0.0230145	0.00239943
$93$	−1.00000	−0.103695
$94$	16.8565	1.73862
$95$	−2.05308	−0.210642
$96$	−0.366116	−0.0373666
$97$	−0.592459	−0.0601551	−0.0300775	−	0.999548i	$-0.509575\pi$
$97$	−0.592459	−0.0601551	−0.0300775	+	0.999548i	$0.509575\pi$
$98$	−7.07766	−0.714951
$99$	5.69921	0.572792
$100$	−0.0647466	−0.00647466
$101$	−7.85889	−0.781989	−0.390994	−	0.920393i	$-0.627869\pi$
$101$	−7.85889	−0.781989	−0.390994	+	0.920393i	$0.627869\pi$
$102$	1.42985	0.141576
$103$	0.801697	0.0789936	0.0394968	−	0.999220i	$-0.487425\pi$
$103$	0.801697	0.0789936	0.0394968	+	0.999220i	$0.487425\pi$
$104$	−2.87234	−0.281656
$105$	−1.38286	−0.134954
$106$	−2.71834	−0.264028
$107$	−4.32559	−0.418170	−0.209085	−	0.977897i	$-0.567049\pi$
$107$	−4.32559	−0.418170	−0.209085	+	0.977897i	$0.567049\pi$
$108$	0.0647466	0.00623024
$109$	−9.87105	−0.945475	−0.472738	−	0.881203i	$-0.656734\pi$
$109$	−9.87105	−0.945475	−0.472738	+	0.881203i	$0.656734\pi$
$110$	7.92836	0.755939
$111$	−9.22459	−0.875559
$112$	−5.34658	−0.505205
$113$	14.9684	1.40811	0.704053	−	0.710148i	$-0.251372\pi$
$113$	14.9684	1.40811	0.704053	+	0.710148i	$0.251372\pi$
$114$	2.85611	0.267499
$115$	−0.355455	−0.0331463
$116$	−0.573630	−0.0532602
$117$	1.00000	0.0924500
$118$	8.05415	0.741445
$119$	−1.42135	−0.130295
$120$	2.87234	0.262207
$121$	21.4809	1.95281
$122$	2.65673	0.240529
$123$	8.53906	0.769941
$124$	−0.0647466	−0.00581441
$125$	1.00000	0.0894427
$126$	1.92375	0.171381
$127$	2.53148	0.224633	0.112316	−	0.993672i	$-0.464173\pi$
$127$	2.53148	0.224633	0.112316	+	0.993672i	$0.464173\pi$
$128$	10.7334	0.948709
$129$	11.2283	0.988595
$130$	1.39113	0.122011
$131$	18.4764	1.61429	0.807144	−	0.590354i	$-0.201012\pi$
$131$	18.4764	1.61429	0.807144	+	0.590354i	$0.201012\pi$
$132$	0.369004	0.0321177
$133$	−2.83913	−0.246184
$134$	−7.77018	−0.671242
$135$	−1.00000	−0.0860663
$136$	2.95228	0.253156
$137$	15.1253	1.29224	0.646122	−	0.763234i	$-0.276390\pi$
$137$	15.1253	1.29224	0.646122	+	0.763234i	$0.276390\pi$
$138$	0.494485	0.0420934
$139$	3.14455	0.266717	0.133359	−	0.991068i	$-0.457424\pi$
$139$	3.14455	0.266717	0.133359	+	0.991068i	$0.457424\pi$
$140$	−0.0895356	−0.00756714
$141$	−12.1171	−1.02045
$142$	−0.776036	−0.0651235
$143$	5.69921	0.476592
$144$	−3.86631	−0.322193
$145$	8.85962	0.735752
$146$	4.85085	0.401460
$147$	5.08769	0.419626
$148$	−0.597260	−0.0490945
$149$	−12.2215	−1.00123	−0.500614	−	0.865671i	$-0.666892\pi$
$149$	−12.2215	−1.00123	−0.500614	+	0.865671i	$0.666892\pi$
$150$	−1.39113	−0.113586
$151$	−9.05394	−0.736799	−0.368399	−	0.929668i	$-0.620094\pi$
$151$	−9.05394	−0.736799	−0.368399	+	0.929668i	$0.620094\pi$
$152$	5.89714	0.478321
$153$	−1.02783	−0.0830951
$154$	10.9638	0.883491
$155$	1.00000	0.0803219
$156$	0.0647466	0.00518387
$157$	11.9013	0.949826	0.474913	−	0.880033i	$-0.342480\pi$
$157$	11.9013	0.949826	0.474913	+	0.880033i	$0.342480\pi$
$158$	6.64741	0.528839
$159$	1.95405	0.154966
$160$	0.366116	0.0289440
$161$	−0.491545	−0.0387392
$162$	1.39113	0.109298
$163$	6.92352	0.542292	0.271146	−	0.962538i	$-0.412597\pi$
$163$	6.92352	0.542292	0.271146	+	0.962538i	$0.412597\pi$
$164$	0.552875	0.0431723
$165$	−5.69921	−0.443683
$166$	10.5679	0.820229
$167$	13.6780	1.05843	0.529217	−	0.848486i	$-0.322486\pi$
$167$	13.6780	1.05843	0.529217	+	0.848486i	$0.322486\pi$
$168$	3.97205	0.306450
$169$	1.00000	0.0769231
$170$	−1.42985	−0.109664
$171$	−2.05308	−0.157003
$172$	0.726992	0.0554326
$173$	−4.79751	−0.364748	−0.182374	−	0.983229i	$-0.558378\pi$
$173$	−4.79751	−0.364748	−0.182374	+	0.983229i	$0.558378\pi$
$174$	−12.3249	−0.934350
$175$	1.38286	0.104535
$176$	−22.0349	−1.66094
$177$	−5.78963	−0.435175
$178$	−10.3807	−0.778064
$179$	−11.4389	−0.854980	−0.427490	−	0.904020i	$-0.640602\pi$
$179$	−11.4389	−0.854980	−0.427490	+	0.904020i	$0.640602\pi$
$180$	−0.0647466	−0.00482592
$181$	18.9693	1.40998	0.704990	−	0.709217i	$-0.250951\pi$
$181$	18.9693	1.40998	0.704990	+	0.709217i	$0.250951\pi$
$182$	1.92375	0.142598
$183$	−1.90976	−0.141173
$184$	1.02099	0.0752681
$185$	9.22459	0.678205
$186$	−1.39113	−0.102003
$187$	−5.85782	−0.428366
$188$	−0.784542	−0.0572186
$189$	−1.38286	−0.100588
$190$	−2.85611	−0.207204
$191$	−6.76037	−0.489163	−0.244582	−	0.969629i	$-0.578651\pi$
$191$	−6.76037	−0.489163	−0.244582	+	0.969629i	$0.578651\pi$
$192$	−8.24195	−0.594811
$193$	6.73173	0.484560	0.242280	−	0.970206i	$-0.422105\pi$
$193$	6.73173	0.484560	0.242280	+	0.970206i	$0.422105\pi$
$194$	−0.824189	−0.0591733
$195$	−1.00000	−0.0716115
$196$	0.329410	0.0235293
$197$	14.1930	1.01121	0.505604	−	0.862765i	$-0.331270\pi$
$197$	14.1930	1.01121	0.505604	+	0.862765i	$0.331270\pi$
$198$	7.92836	0.563444
$199$	10.1031	0.716188	0.358094	−	0.933686i	$-0.383427\pi$
$199$	10.1031	0.716188	0.358094	+	0.933686i	$0.383427\pi$
$200$	−2.87234	−0.203105
$201$	5.58550	0.393971
$202$	−10.9328	−0.769227
$203$	12.2516	0.859897
$204$	−0.0665485	−0.00465932
$205$	−8.53906	−0.596394
$206$	1.11527	0.0777044
$207$	−0.355455	−0.0247058
$208$	−3.86631	−0.268081
$209$	−11.7009	−0.809370
$210$	−1.92375	−0.132751
$211$	−23.4055	−1.61130	−0.805649	−	0.592393i	$-0.798183\pi$
$211$	−23.4055	−1.61130	−0.805649	+	0.592393i	$0.798183\pi$
$212$	0.126518	0.00868927
$213$	0.557844	0.0382229
$214$	−6.01747	−0.411346
$215$	−11.2283	−0.765762
$216$	2.87234	0.195438
$217$	1.38286	0.0938749
$218$	−13.7320	−0.930045
$219$	−3.48698	−0.235628
$220$	−0.369004	−0.0248782
$221$	−1.02783	−0.0691393
$222$	−12.8326	−0.861270
$223$	0.181212	0.0121349	0.00606743	−	0.999982i	$-0.498069\pi$
$223$	0.181212	0.0121349	0.00606743	+	0.999982i	$0.498069\pi$
$224$	0.506289	0.0338278
$225$	1.00000	0.0666667
$226$	20.8230	1.38513
$227$	19.6797	1.30619	0.653094	−	0.757277i	$-0.273471\pi$
$227$	19.6797	1.30619	0.653094	+	0.757277i	$0.273471\pi$
$228$	−0.132930	−0.00880350
$229$	9.44111	0.623886	0.311943	−	0.950101i	$-0.399020\pi$
$229$	9.44111	0.623886	0.311943	+	0.950101i	$0.399020\pi$
$230$	−0.494485	−0.0326054
$231$	−7.88122	−0.518546
$232$	−25.4478	−1.67073
$233$	14.9802	0.981385	0.490693	−	0.871333i	$-0.336744\pi$
$233$	14.9802	0.981385	0.490693	+	0.871333i	$0.336744\pi$
$234$	1.39113	0.0909413
$235$	12.1171	0.790433
$236$	−0.374859	−0.0244012
$237$	−4.77841	−0.310391
$238$	−1.97729	−0.128168
$239$	16.7441	1.08308	0.541542	−	0.840673i	$-0.317841\pi$
$239$	16.7441	1.08308	0.541542	+	0.840673i	$0.317841\pi$
$240$	3.86631	0.249570
$241$	−5.28252	−0.340277	−0.170139	−	0.985420i	$-0.554422\pi$
$241$	−5.28252	−0.340277	−0.170139	+	0.985420i	$0.554422\pi$
$242$	29.8829	1.92094
$243$	−1.00000	−0.0641500
$244$	−0.123650	−0.00791590
$245$	−5.08769	−0.325041
$246$	11.8790	0.757376
$247$	−2.05308	−0.130634
$248$	−2.87234	−0.182394
$249$	−7.59662	−0.481416
$250$	1.39113	0.0879830
$251$	11.9625	0.755065	0.377533	−	0.925996i	$-0.376773\pi$
$251$	11.9625	0.755065	0.377533	+	0.925996i	$0.376773\pi$
$252$	−0.0895356	−0.00564022
$253$	−2.02581	−0.127362
$254$	3.52163	0.220967
$255$	1.02783	0.0643652
$256$	−1.55227	−0.0970170
$257$	−13.7430	−0.857267	−0.428634	−	0.903478i	$-0.641005\pi$
$257$	−13.7430	−0.857267	−0.428634	+	0.903478i	$0.641005\pi$
$258$	15.6200	0.972461
$259$	12.7563	0.792641
$260$	−0.0647466	−0.00401541
$261$	8.85962	0.548397
$262$	25.7031	1.58794
$263$	8.72444	0.537972	0.268986	−	0.963144i	$-0.413312\pi$
$263$	8.72444	0.537972	0.268986	+	0.963144i	$0.413312\pi$
$264$	16.3701	1.00751
$265$	−1.95405	−0.120036
$266$	−3.94961	−0.242166
$267$	7.46202	0.456668
$268$	0.361642	0.0220908
$269$	3.17679	0.193692	0.0968460	−	0.995299i	$-0.469125\pi$
$269$	3.17679	0.193692	0.0968460	+	0.995299i	$0.469125\pi$
$270$	−1.39113	−0.0846617
$271$	9.62166	0.584474	0.292237	−	0.956346i	$-0.405600\pi$
$271$	9.62166	0.584474	0.292237	+	0.956346i	$0.405600\pi$
$272$	3.97391	0.240954
$273$	−1.38286	−0.0836947
$274$	21.0414	1.27116
$275$	5.69921	0.343675
$276$	−0.0230145	−0.00138531
$277$	29.6101	1.77910	0.889549	−	0.456839i	$-0.151019\pi$
$277$	29.6101	1.77910	0.889549	+	0.456839i	$0.151019\pi$
$278$	4.37449	0.262364
$279$	1.00000	0.0598684
$280$	−3.97205	−0.237375
$281$	−23.4937	−1.40152	−0.700760	−	0.713397i	$-0.747155\pi$
$281$	−23.4937	−1.40152	−0.700760	+	0.713397i	$0.747155\pi$
$282$	−16.8565	−1.00379
$283$	−22.6905	−1.34881	−0.674406	−	0.738361i	$-0.735600\pi$
$283$	−22.6905	−1.34881	−0.674406	+	0.738361i	$0.735600\pi$
$284$	0.0361185	0.00214324
$285$	2.05308	0.121614
$286$	7.92836	0.468814
$287$	−11.8084	−0.697025
$288$	0.366116	0.0215736
$289$	−15.9436	−0.937857
$290$	12.3249	0.723744
$291$	0.592459	0.0347305
$292$	−0.225770	−0.0132122
$293$	−0.841149	−0.0491405	−0.0245702	−	0.999698i	$-0.507822\pi$
$293$	−0.841149	−0.0491405	−0.0245702	+	0.999698i	$0.507822\pi$
$294$	7.07766	0.412777
$295$	5.78963	0.337085
$296$	−26.4961	−1.54006
$297$	−5.69921	−0.330701
$298$	−17.0018	−0.984888
$299$	−0.355455	−0.0205565
$300$	0.0647466	0.00373814
$301$	−15.5272	−0.894971
$302$	−12.5952	−0.724774
$303$	7.85889	0.451481
$304$	7.93785	0.455267
$305$	1.90976	0.109352
$306$	−1.42985	−0.0817390
$307$	29.2489	1.66933	0.834663	−	0.550761i	$-0.185663\pi$
$307$	29.2489	1.66933	0.834663	+	0.550761i	$0.185663\pi$
$308$	−0.510282	−0.0290760
$309$	−0.801697	−0.0456070
$310$	1.39113	0.0790111
$311$	27.0260	1.53251	0.766253	−	0.642539i	$-0.222119\pi$
$311$	27.0260	1.53251	0.766253	+	0.642539i	$0.222119\pi$
$312$	2.87234	0.162614
$313$	21.5023	1.21538	0.607691	−	0.794173i	$-0.292096\pi$
$313$	21.5023	1.21538	0.607691	+	0.794173i	$0.292096\pi$
$314$	16.5563	0.934325
$315$	1.38286	0.0779155
$316$	−0.309386	−0.0174043
$317$	−14.9069	−0.837257	−0.418629	−	0.908158i	$-0.637489\pi$
$317$	−14.9069	−0.837257	−0.418629	+	0.908158i	$0.637489\pi$
$318$	2.71834	0.152437
$319$	50.4928	2.82705
$320$	8.24195	0.460739
$321$	4.32559	0.241431
$322$	−0.683805	−0.0381070
$323$	2.11022	0.117416
$324$	−0.0647466	−0.00359703
$325$	1.00000	0.0554700
$326$	9.63155	0.533442
$327$	9.87105	0.545870
$328$	24.5271	1.35428
$329$	16.7563	0.923805
$330$	−7.92836	−0.436442
$331$	−14.1190	−0.776048	−0.388024	−	0.921649i	$-0.626842\pi$
$331$	−14.1190	−0.776048	−0.388024	+	0.921649i	$0.626842\pi$
$332$	−0.491855	−0.0269941
$333$	9.22459	0.505504
$334$	19.0279	1.04116
$335$	−5.58550	−0.305169
$336$	5.34658	0.291680
$337$	−12.3038	−0.670228	−0.335114	−	0.942178i	$-0.608775\pi$
$337$	−12.3038	−0.670228	−0.335114	+	0.942178i	$0.608775\pi$
$338$	1.39113	0.0756677
$339$	−14.9684	−0.812970
$340$	0.0665485	0.00360910
$341$	5.69921	0.308629
$342$	−2.85611	−0.154441
$343$	−16.7156	−0.902559
$344$	32.2514	1.73888
$345$	0.355455	0.0191370
$346$	−6.67398	−0.358795
$347$	−35.3059	−1.89532	−0.947660	−	0.319282i	$-0.896558\pi$
$347$	−35.3059	−1.89532	−0.947660	+	0.319282i	$0.896558\pi$
$348$	0.573630	0.0307498
$349$	34.1896	1.83013	0.915065	−	0.403307i	$-0.132139\pi$
$349$	34.1896	1.83013	0.915065	+	0.403307i	$0.132139\pi$
$350$	1.92375	0.102829
$351$	−1.00000	−0.0533761
$352$	2.08657	0.111215
$353$	27.9285	1.48648	0.743242	−	0.669023i	$-0.233287\pi$
$353$	27.9285	1.48648	0.743242	+	0.669023i	$0.233287\pi$
$354$	−8.05415	−0.428073
$355$	−0.557844	−0.0296073
$356$	0.483140	0.0256064
$357$	1.42135	0.0752257
$358$	−15.9130	−0.841027
$359$	2.15475	0.113724	0.0568618	−	0.998382i	$-0.481891\pi$
$359$	2.15475	0.113724	0.0568618	+	0.998382i	$0.481891\pi$
$360$	−2.87234	−0.151386
$361$	−14.7849	−0.778151
$362$	26.3889	1.38697
$363$	−21.4809	−1.12746
$364$	−0.0895356	−0.00469294
$365$	3.48698	0.182517
$366$	−2.65673	−0.138869
$367$	13.0921	0.683403	0.341702	−	0.939808i	$-0.388997\pi$
$367$	13.0921	0.683403	0.341702	+	0.939808i	$0.388997\pi$
$368$	1.37430	0.0716403
$369$	−8.53906	−0.444526
$370$	12.8326	0.667137
$371$	−2.70218	−0.140290
$372$	0.0647466	0.00335695
$373$	31.2426	1.61768	0.808840	−	0.588029i	$-0.200096\pi$
$373$	31.2426	1.61768	0.808840	+	0.588029i	$0.200096\pi$
$374$	−8.14900	−0.421375
$375$	−1.00000	−0.0516398
$376$	−34.8045	−1.79490
$377$	8.85962	0.456294
$378$	−1.92375	−0.0989469
$379$	10.5649	0.542682	0.271341	−	0.962483i	$-0.412533\pi$
$379$	10.5649	0.542682	0.271341	+	0.962483i	$0.412533\pi$
$380$	0.132930	0.00681916
$381$	−2.53148	−0.129692
$382$	−9.40458	−0.481180
$383$	−11.1161	−0.568004	−0.284002	−	0.958824i	$-0.591662\pi$
$383$	−11.1161	−0.568004	−0.284002	+	0.958824i	$0.591662\pi$
$384$	−10.7334	−0.547737
$385$	7.88122	0.401664
$386$	9.36473	0.476652
$387$	−11.2283	−0.570765
$388$	0.0383597	0.00194742
$389$	−17.6292	−0.893835	−0.446917	−	0.894575i	$-0.647478\pi$
$389$	−17.6292	−0.893835	−0.446917	+	0.894575i	$0.647478\pi$
$390$	−1.39113	−0.0704428
$391$	0.365347	0.0184764
$392$	14.6136	0.738097
$393$	−18.4764	−0.932010
$394$	19.7443	0.994706
$395$	4.77841	0.240428
$396$	−0.369004	−0.0185432
$397$	−32.9536	−1.65389	−0.826947	−	0.562280i	$-0.809924\pi$
$397$	−32.9536	−1.65389	−0.826947	+	0.562280i	$0.809924\pi$
$398$	14.0547	0.704500
$399$	2.83913	0.142134
$400$	−3.86631	−0.193316
$401$	−10.4405	−0.521376	−0.260688	−	0.965423i	$-0.583949\pi$
$401$	−10.4405	−0.521376	−0.260688	+	0.965423i	$0.583949\pi$
$402$	7.77018	0.387541
$403$	1.00000	0.0498135
$404$	0.508836	0.0253155
$405$	1.00000	0.0496904
$406$	17.0437	0.845864
$407$	52.5728	2.60594
$408$	−2.95228	−0.146159
$409$	9.93105	0.491059	0.245529	−	0.969389i	$-0.421038\pi$
$409$	9.93105	0.491059	0.245529	+	0.969389i	$0.421038\pi$
$410$	−11.8790	−0.586661
$411$	−15.1253	−0.746078
$412$	−0.0519072	−0.00255728
$413$	8.00627	0.393963
$414$	−0.494485	−0.0243026
$415$	7.59662	0.372903
$416$	0.366116	0.0179503
$417$	−3.14455	−0.153989
$418$	−16.2776	−0.796161
$419$	−1.61769	−0.0790294	−0.0395147	−	0.999219i	$-0.512581\pi$
$419$	−1.61769	−0.0790294	−0.0395147	+	0.999219i	$0.512581\pi$
$420$	0.0895356	0.00436889
$421$	0.786448	0.0383291	0.0191646	−	0.999816i	$-0.493899\pi$
$421$	0.786448	0.0383291	0.0191646	+	0.999816i	$0.493899\pi$
$422$	−32.5601	−1.58500
$423$	12.1171	0.589154
$424$	5.61268	0.272576
$425$	−1.02783	−0.0498571
$426$	0.776036	0.0375991
$427$	2.64093	0.127804
$428$	0.280067	0.0135375
$429$	−5.69921	−0.275160
$430$	−15.6200	−0.753265
$431$	−4.07049	−0.196069	−0.0980344	−	0.995183i	$-0.531256\pi$
$431$	−4.07049	−0.196069	−0.0980344	+	0.995183i	$0.531256\pi$
$432$	3.86631	0.186018
$433$	−3.54812	−0.170512	−0.0852560	−	0.996359i	$-0.527171\pi$
$433$	−3.54812	−0.170512	−0.0852560	+	0.996359i	$0.527171\pi$
$434$	1.92375	0.0923429
$435$	−8.85962	−0.424786
$436$	0.639117	0.0306081
$437$	0.729777	0.0349100
$438$	−4.85085	−0.231783
$439$	−10.3172	−0.492414	−0.246207	−	0.969217i	$-0.579184\pi$
$439$	−10.3172	−0.492414	−0.246207	+	0.969217i	$0.579184\pi$
$440$	−16.3701	−0.780412
$441$	−5.08769	−0.242271
$442$	−1.42985	−0.0680110
$443$	5.30435	0.252018	0.126009	−	0.992029i	$-0.459783\pi$
$443$	5.30435	0.252018	0.126009	+	0.992029i	$0.459783\pi$
$444$	0.597260	0.0283447
$445$	−7.46202	−0.353734
$446$	0.252090	0.0119368
$447$	12.2215	0.578059
$448$	11.3975	0.538481
$449$	11.0852	0.523142	0.261571	−	0.965184i	$-0.415759\pi$
$449$	11.0852	0.523142	0.261571	+	0.965184i	$0.415759\pi$
$450$	1.39113	0.0655787
$451$	−48.6659	−2.29159
$452$	−0.969150	−0.0455850
$453$	9.05394	0.425391
$454$	27.3771	1.28487
$455$	1.38286	0.0648296
$456$	−5.89714	−0.276159
$457$	−30.9546	−1.44799	−0.723997	−	0.689803i	$-0.757697\pi$
$457$	−30.9546	−1.44799	−0.723997	+	0.689803i	$0.757697\pi$
$458$	13.1338	0.613704
$459$	1.02783	0.0479750
$460$	0.0230145	0.00107306
$461$	−37.7496	−1.75817	−0.879087	−	0.476662i	$-0.841847\pi$
$461$	−37.7496	−1.75817	−0.879087	+	0.476662i	$0.841847\pi$
$462$	−10.9638	−0.510084
$463$	−29.6344	−1.37723	−0.688614	−	0.725128i	$-0.741781\pi$
$463$	−29.6344	−1.37723	−0.688614	+	0.725128i	$0.741781\pi$
$464$	−34.2541	−1.59021
$465$	−1.00000	−0.0463739
$466$	20.8395	0.965369
$467$	18.7133	0.865947	0.432974	−	0.901407i	$-0.357464\pi$
$467$	18.7133	0.865947	0.432974	+	0.901407i	$0.357464\pi$
$468$	−0.0647466	−0.00299291
$469$	−7.72399	−0.356661
$470$	16.8565	0.777534
$471$	−11.9013	−0.548382
$472$	−16.6298	−0.765448
$473$	−63.9923	−2.94237
$474$	−6.64741	−0.305326
$475$	−2.05308	−0.0942018
$476$	0.0920274	0.00421807
$477$	−1.95405	−0.0894696
$478$	23.2933	1.06541
$479$	−15.1840	−0.693775	−0.346888	−	0.937907i	$-0.612761\pi$
$479$	−15.1840	−0.693775	−0.346888	+	0.937907i	$0.612761\pi$
$480$	−0.366116	−0.0167108
$481$	9.22459	0.420605
$482$	−7.34870	−0.334724
$483$	0.491545	0.0223661
$484$	−1.39082	−0.0632190
$485$	−0.592459	−0.0269022
$486$	−1.39113	−0.0631031
$487$	36.5329	1.65547	0.827733	−	0.561123i	$-0.189630\pi$
$487$	36.5329	1.65547	0.827733	+	0.561123i	$0.189630\pi$
$488$	−5.48547	−0.248316
$489$	−6.92352	−0.313093
$490$	−7.07766	−0.319736
$491$	14.5711	0.657583	0.328792	−	0.944402i	$-0.393359\pi$
$491$	14.5711	0.657583	0.328792	+	0.944402i	$0.393359\pi$
$492$	−0.552875	−0.0249255
$493$	−9.10619	−0.410122
$494$	−2.85611	−0.128502
$495$	5.69921	0.256160
$496$	−3.86631	−0.173603
$497$	−0.771422	−0.0346030
$498$	−10.5679	−0.473560
$499$	11.3538	0.508265	0.254133	−	0.967169i	$-0.418210\pi$
$499$	11.3538	0.508265	0.254133	+	0.967169i	$0.418210\pi$
$500$	−0.0647466	−0.00289555
$501$	−13.6780	−0.611088
$502$	16.6414	0.742743
$503$	0.488796	0.0217944	0.0108972	−	0.999941i	$-0.496531\pi$
$503$	0.488796	0.0217944	0.0108972	+	0.999941i	$0.496531\pi$
$504$	−3.97205	−0.176929
$505$	−7.85889	−0.349716
$506$	−2.81817	−0.125283
$507$	−1.00000	−0.0444116
$508$	−0.163905	−0.00727210
$509$	30.1355	1.33573	0.667867	−	0.744281i	$-0.267208\pi$
$509$	30.1355	1.33573	0.667867	+	0.744281i	$0.267208\pi$
$510$	1.42985	0.0633148
$511$	4.82202	0.213313
$512$	−23.6263	−1.04414
$513$	2.05308	0.0906457
$514$	−19.1184	−0.843277
$515$	0.801697	0.0353270
$516$	−0.726992	−0.0320041
$517$	69.0579	3.03716
$518$	17.7458	0.779705
$519$	4.79751	0.210587
$520$	−2.87234	−0.125960
$521$	−37.2900	−1.63371	−0.816853	−	0.576846i	$-0.804283\pi$
$521$	−37.2900	−1.63371	−0.816853	+	0.576846i	$0.804283\pi$
$522$	12.3249	0.539447
$523$	−39.4490	−1.72498	−0.862492	−	0.506070i	$-0.831098\pi$
$523$	−39.4490	−1.72498	−0.862492	+	0.506070i	$0.831098\pi$
$524$	−1.19628	−0.0522598
$525$	−1.38286	−0.0603531
$526$	12.1369	0.529192
$527$	−1.02783	−0.0447730
$528$	22.0349	0.958947
$529$	−22.8737	−0.994507
$530$	−2.71834	−0.118077
$531$	5.78963	0.251249
$532$	0.183824	0.00796978
$533$	−8.53906	−0.369868
$534$	10.3807	0.449216
$535$	−4.32559	−0.187011
$536$	16.0435	0.692972
$537$	11.4389	0.493623
$538$	4.41933	0.190531
$539$	−28.9958	−1.24894
$540$	0.0647466	0.00278625
$541$	−26.2359	−1.12797	−0.563984	−	0.825785i	$-0.690732\pi$
$541$	−26.2359	−1.12797	−0.563984	+	0.825785i	$0.690732\pi$
$542$	13.3850	0.574935
$543$	−18.9693	−0.814053
$544$	−0.376305	−0.0161340
$545$	−9.87105	−0.422829
$546$	−1.92375	−0.0823288
$547$	−23.7750	−1.01654	−0.508272	−	0.861197i	$-0.669716\pi$
$547$	−23.7750	−1.01654	−0.508272	+	0.861197i	$0.669716\pi$
$548$	−0.979314	−0.0418342
$549$	1.90976	0.0815065
$550$	7.92836	0.338066
$551$	−18.1895	−0.774899
$552$	−1.02099	−0.0434561
$553$	6.60789	0.280996
$554$	41.1916	1.75006
$555$	−9.22459	−0.391562
$556$	−0.203599	−0.00863451
$557$	9.12354	0.386577	0.193288	−	0.981142i	$-0.438085\pi$
$557$	9.12354	0.386577	0.193288	+	0.981142i	$0.438085\pi$
$558$	1.39113	0.0588914
$559$	−11.2283	−0.474905
$560$	−5.34658	−0.225934
$561$	5.85782	0.247317
$562$	−32.6829	−1.37865
$563$	1.60749	0.0677477	0.0338738	−	0.999426i	$-0.489216\pi$
$563$	1.60749	0.0677477	0.0338738	+	0.999426i	$0.489216\pi$
$564$	0.784542	0.0330352
$565$	14.9684	0.629724
$566$	−31.5656	−1.32680
$567$	1.38286	0.0580748
$568$	1.60232	0.0672318
$569$	37.3411	1.56542	0.782711	−	0.622386i	$-0.213836\pi$
$569$	37.3411	1.56542	0.782711	+	0.622386i	$0.213836\pi$
$570$	2.85611	0.119629
$571$	−9.45636	−0.395736	−0.197868	−	0.980229i	$-0.563402\pi$
$571$	−9.45636	−0.395736	−0.197868	+	0.980229i	$0.563402\pi$
$572$	−0.369004	−0.0154288
$573$	6.76037	0.282419
$574$	−16.4270	−0.685650
$575$	−0.355455	−0.0148235
$576$	8.24195	0.343414
$577$	39.0598	1.62608	0.813040	−	0.582207i	$-0.197811\pi$
$577$	39.0598	1.62608	0.813040	+	0.582207i	$0.197811\pi$
$578$	−22.1796	−0.922551
$579$	−6.73173	−0.279761
$580$	−0.573630	−0.0238187
$581$	10.5051	0.435824
$582$	0.824189	0.0341637
$583$	−11.1365	−0.461227
$584$	−10.0158	−0.414456
$585$	1.00000	0.0413449
$586$	−1.17015	−0.0483385
$587$	−38.9456	−1.60746	−0.803729	−	0.594996i	$-0.797154\pi$
$587$	−38.9456	−1.60746	−0.803729	+	0.594996i	$0.797154\pi$
$588$	−0.329410	−0.0135847
$589$	−2.05308	−0.0845957
$590$	8.05415	0.331584
$591$	−14.1930	−0.583822
$592$	−35.6652	−1.46583
$593$	−18.8270	−0.773131	−0.386565	−	0.922262i	$-0.626339\pi$
$593$	−18.8270	−0.773131	−0.386565	+	0.922262i	$0.626339\pi$
$594$	−7.92836	−0.325304
$595$	−1.42135	−0.0582696
$596$	0.791303	0.0324130
$597$	−10.1031	−0.413491
$598$	−0.494485	−0.0202210
$599$	18.7150	0.764675	0.382337	−	0.924023i	$-0.375119\pi$
$599$	18.7150	0.764675	0.382337	+	0.924023i	$0.375119\pi$
$600$	2.87234	0.117263
$601$	−30.6386	−1.24978	−0.624888	−	0.780714i	$-0.714855\pi$
$601$	−30.6386	−1.24978	−0.624888	+	0.780714i	$0.714855\pi$
$602$	−21.6004	−0.880365
$603$	−5.58550	−0.227459
$604$	0.586211	0.0238526
$605$	21.4809	0.873325
$606$	10.9328	0.444113
$607$	41.3499	1.67834	0.839170	−	0.543869i	$-0.183041\pi$
$607$	41.3499	1.67834	0.839170	+	0.543869i	$0.183041\pi$
$608$	−0.751666	−0.0304841
$609$	−12.2516	−0.496462
$610$	2.65673	0.107568
$611$	12.1171	0.490206
$612$	0.0665485	0.00269006
$613$	−32.5125	−1.31317	−0.656583	−	0.754253i	$-0.727999\pi$
$613$	−32.5125	−1.31317	−0.656583	+	0.754253i	$0.727999\pi$
$614$	40.6892	1.64208
$615$	8.53906	0.344328
$616$	−22.6375	−0.912093
$617$	−25.0088	−1.00682	−0.503409	−	0.864049i	$-0.667921\pi$
$617$	−25.0088	−1.00682	−0.503409	+	0.864049i	$0.667921\pi$
$618$	−1.11527	−0.0448627
$619$	−32.0070	−1.28647	−0.643236	−	0.765668i	$-0.722408\pi$
$619$	−32.0070	−1.28647	−0.643236	+	0.765668i	$0.722408\pi$
$620$	−0.0647466	−0.00260028
$621$	0.355455	0.0142639
$622$	37.5968	1.50750
$623$	−10.3190	−0.413420
$624$	3.86631	0.154776
$625$	1.00000	0.0400000
$626$	29.9126	1.19555
$627$	11.7009	0.467290
$628$	−0.770567	−0.0307490
$629$	−9.48131	−0.378045
$630$	1.92375	0.0766440
$631$	35.2395	1.40286	0.701432	−	0.712737i	$-0.252545\pi$
$631$	35.2395	1.40286	0.701432	+	0.712737i	$0.252545\pi$
$632$	−13.7252	−0.545960
$633$	23.4055	0.930283
$634$	−20.7375	−0.823593
$635$	2.53148	0.100459
$636$	−0.126518	−0.00501675
$637$	−5.08769	−0.201582
$638$	70.2423	2.78092
$639$	−0.557844	−0.0220680
$640$	10.7334	0.424276
$641$	−29.9891	−1.18450	−0.592249	−	0.805755i	$-0.701760\pi$
$641$	−29.9891	−1.18450	−0.592249	+	0.805755i	$0.701760\pi$
$642$	6.01747	0.237491
$643$	−36.2006	−1.42761	−0.713806	−	0.700343i	$-0.753030\pi$
$643$	−36.2006	−1.42761	−0.713806	+	0.700343i	$0.753030\pi$
$644$	0.0318259	0.00125412
$645$	11.2283	0.442113
$646$	2.93560	0.115499
$647$	−22.1275	−0.869923	−0.434961	−	0.900449i	$-0.643238\pi$
$647$	−22.1275	−0.869923	−0.434961	+	0.900449i	$0.643238\pi$
$648$	−2.87234	−0.112836
$649$	32.9963	1.29522
$650$	1.39113	0.0545648
$651$	−1.38286	−0.0541987
$652$	−0.448274	−0.0175558
$653$	24.7516	0.968606	0.484303	−	0.874900i	$-0.339073\pi$
$653$	24.7516	0.968606	0.484303	+	0.874900i	$0.339073\pi$
$654$	13.7320	0.536962
$655$	18.4764	0.721932
$656$	33.0147	1.28901
$657$	3.48698	0.136040
$658$	23.3103	0.908729
$659$	−32.8594	−1.28002	−0.640010	−	0.768367i	$-0.721070\pi$
$659$	−32.8594	−1.28002	−0.640010	+	0.768367i	$0.721070\pi$
$660$	0.369004	0.0143635
$661$	−34.1399	−1.32789	−0.663943	−	0.747783i	$-0.731118\pi$
$661$	−34.1399	−1.32789	−0.663943	+	0.747783i	$0.731118\pi$
$662$	−19.6414	−0.763383
$663$	1.02783	0.0399176
$664$	−21.8201	−0.846783
$665$	−2.83913	−0.110097
$666$	12.8326	0.497255
$667$	−3.14919	−0.121937
$668$	−0.885603	−0.0342650
$669$	−0.181212	−0.00700607
$670$	−7.77018	−0.300188
$671$	10.8841	0.420176
$672$	−0.506289	−0.0195305
$673$	−43.7272	−1.68556	−0.842780	−	0.538258i	$-0.819083\pi$
$673$	−43.7272	−1.68556	−0.842780	+	0.538258i	$0.819083\pi$
$674$	−17.1162	−0.659290
$675$	−1.00000	−0.0384900
$676$	−0.0647466	−0.00249025
$677$	11.6657	0.448351	0.224175	−	0.974549i	$-0.428031\pi$
$677$	11.6657	0.448351	0.224175	+	0.974549i	$0.428031\pi$
$678$	−20.8230	−0.799702
$679$	−0.819289	−0.0314414
$680$	2.95228	0.113215
$681$	−19.6797	−0.754128
$682$	7.92836	0.303593
$683$	−18.6252	−0.712672	−0.356336	−	0.934358i	$-0.615974\pi$
$683$	−18.6252	−0.712672	−0.356336	+	0.934358i	$0.615974\pi$
$684$	0.132930	0.00508270
$685$	15.1253	0.577909
$686$	−23.2537	−0.887829
$687$	−9.44111	−0.360201
$688$	43.4120	1.65507
$689$	−1.95405	−0.0744432
$690$	0.494485	0.0188247
$691$	−1.56695	−0.0596096	−0.0298048	−	0.999556i	$-0.509489\pi$
$691$	−1.56695	−0.0596096	−0.0298048	+	0.999556i	$0.509489\pi$
$692$	0.310622	0.0118081
$693$	7.88122	0.299383
$694$	−49.1152	−1.86439
$695$	3.14455	0.119279
$696$	25.4478	0.964598
$697$	8.77670	0.332441
$698$	47.5624	1.80026
$699$	−14.9802	−0.566603
$700$	−0.0895356	−0.00338413
$701$	31.0338	1.17213	0.586066	−	0.810263i	$-0.300676\pi$
$701$	31.0338	1.17213	0.586066	+	0.810263i	$0.300676\pi$
$702$	−1.39113	−0.0525050
$703$	−18.9388	−0.714291
$704$	46.9726	1.77034
$705$	−12.1171	−0.456357
$706$	38.8523	1.46222
$707$	−10.8678	−0.408724
$708$	0.374859	0.0140881
$709$	−40.4519	−1.51920	−0.759601	−	0.650390i	$-0.774606\pi$
$709$	−40.4519	−1.51920	−0.759601	+	0.650390i	$0.774606\pi$
$710$	−0.776036	−0.0291241
$711$	4.77841	0.179204
$712$	21.4335	0.803253
$713$	−0.355455	−0.0133119
$714$	1.97729	0.0739981
$715$	5.69921	0.213138
$716$	0.740627	0.0276785
$717$	−16.7441	−0.625319
$718$	2.99755	0.111868
$719$	8.27802	0.308718	0.154359	−	0.988015i	$-0.450669\pi$
$719$	8.27802	0.308718	0.154359	+	0.988015i	$0.450669\pi$
$720$	−3.86631	−0.144089
$721$	1.10864	0.0412878
$722$	−20.5677	−0.765451
$723$	5.28252	0.196459
$724$	−1.22820	−0.0456457
$725$	8.85962	0.329038
$726$	−29.8829	−1.10906
$727$	5.52282	0.204830	0.102415	−	0.994742i	$-0.467343\pi$
$727$	5.52282	0.204830	0.102415	+	0.994742i	$0.467343\pi$
$728$	−3.97205	−0.147214
$729$	1.00000	0.0370370
$730$	4.85085	0.179538
$731$	11.5408	0.426850
$732$	0.123650	0.00457024
$733$	27.6952	1.02295	0.511473	−	0.859299i	$-0.329100\pi$
$733$	27.6952	1.02295	0.511473	+	0.859299i	$0.329100\pi$
$734$	18.2129	0.672250
$735$	5.08769	0.187662
$736$	−0.130138	−0.00479694
$737$	−31.8329	−1.17258
$738$	−11.8790	−0.437271
$739$	−47.2191	−1.73698	−0.868491	−	0.495706i	$-0.834909\pi$
$739$	−47.2191	−1.73698	−0.868491	+	0.495706i	$0.834909\pi$
$740$	−0.597260	−0.0219557
$741$	2.05308	0.0754218
$742$	−3.75909	−0.138001
$743$	17.0883	0.626910	0.313455	−	0.949603i	$-0.398514\pi$
$743$	17.0883	0.626910	0.313455	+	0.949603i	$0.398514\pi$
$744$	2.87234	0.105305
$745$	−12.2215	−0.447763
$746$	43.4626	1.59128
$747$	7.59662	0.277946
$748$	0.379273	0.0138676
$749$	−5.98169	−0.218566
$750$	−1.39113	−0.0507970
$751$	43.8714	1.60089	0.800446	−	0.599405i	$-0.204596\pi$
$751$	43.8714	1.60089	0.800446	+	0.599405i	$0.204596\pi$
$752$	−46.8486	−1.70839
$753$	−11.9625	−0.435937
$754$	12.3249	0.448847
$755$	−9.05394	−0.329506
$756$	0.0895356	0.00325638
$757$	−44.1352	−1.60412	−0.802061	−	0.597243i	$-0.796263\pi$
$757$	−44.1352	−1.60412	−0.802061	+	0.597243i	$0.796263\pi$
$758$	14.6972	0.533826
$759$	2.02581	0.0735322
$760$	5.89714	0.213912
$761$	−38.5820	−1.39860	−0.699299	−	0.714829i	$-0.746504\pi$
$761$	−38.5820	−1.39860	−0.699299	+	0.714829i	$0.746504\pi$
$762$	−3.52163	−0.127575
$763$	−13.6503	−0.494174
$764$	0.437711	0.0158358
$765$	−1.02783	−0.0371613
$766$	−15.4639	−0.558734
$767$	5.78963	0.209051
$768$	1.55227	0.0560128
$769$	26.4293	0.953065	0.476533	−	0.879157i	$-0.341893\pi$
$769$	26.4293	0.953065	0.476533	+	0.879157i	$0.341893\pi$
$770$	10.9638	0.395109
$771$	13.7430	0.494944
$772$	−0.435856	−0.0156868
$773$	−32.1596	−1.15670	−0.578350	−	0.815789i	$-0.696303\pi$
$773$	−32.1596	−1.15670	−0.578350	+	0.815789i	$0.696303\pi$
$774$	−15.6200	−0.561451
$775$	1.00000	0.0359211
$776$	1.70174	0.0610890
$777$	−12.7563	−0.457631
$778$	−24.5246	−0.879248
$779$	17.5314	0.628127
$780$	0.0647466	0.00231830
$781$	−3.17927	−0.113763
$782$	0.508247	0.0181749
$783$	−8.85962	−0.316617
$784$	19.6706	0.702522
$785$	11.9013	0.424775
$786$	−25.7031	−0.916800
$787$	−20.1650	−0.718806	−0.359403	−	0.933183i	$-0.617020\pi$
$787$	−20.1650	−0.718806	−0.359403	+	0.933183i	$0.617020\pi$
$788$	−0.918947	−0.0327361
$789$	−8.72444	−0.310598
$790$	6.64741	0.236504
$791$	20.6992	0.735979
$792$	−16.3701	−0.581684
$793$	1.90976	0.0678175
$794$	−45.8429	−1.62690
$795$	1.95405	0.0693028
$796$	−0.654140	−0.0231854
$797$	11.5442	0.408918	0.204459	−	0.978875i	$-0.434456\pi$
$797$	11.5442	0.408918	0.204459	+	0.978875i	$0.434456\pi$
$798$	3.94961	0.139815
$799$	−12.4543	−0.440603
$800$	0.366116	0.0129442
$801$	−7.46202	−0.263658
$802$	−14.5242	−0.512867
$803$	19.8730	0.701303
$804$	−0.361642	−0.0127541
$805$	−0.491545	−0.0173247
$806$	1.39113	0.0490006
$807$	−3.17679	−0.111828
$808$	22.5734	0.794129
$809$	20.1762	0.709358	0.354679	−	0.934988i	$-0.384590\pi$
$809$	20.1762	0.709358	0.354679	+	0.934988i	$0.384590\pi$
$810$	1.39113	0.0488795
$811$	20.2322	0.710450	0.355225	−	0.934781i	$-0.384404\pi$
$811$	20.2322	0.710450	0.355225	+	0.934781i	$0.384404\pi$
$812$	−0.793252	−0.0278377
$813$	−9.62166	−0.337446
$814$	73.1358	2.56341
$815$	6.92352	0.242520
$816$	−3.97391	−0.139115
$817$	23.0525	0.806507
$818$	13.8154	0.483045
$819$	1.38286	0.0483212
$820$	0.552875	0.0193072
$821$	−18.7635	−0.654851	−0.327425	−	0.944877i	$-0.606181\pi$
$821$	−18.7635	−0.654851	−0.327425	+	0.944877i	$0.606181\pi$
$822$	−21.0414	−0.733902
$823$	8.65470	0.301684	0.150842	−	0.988558i	$-0.451802\pi$
$823$	8.65470	0.301684	0.150842	+	0.988558i	$0.451802\pi$
$824$	−2.30275	−0.0802200
$825$	−5.69921	−0.198421
$826$	11.1378	0.387533
$827$	−5.33576	−0.185543	−0.0927713	−	0.995687i	$-0.529573\pi$
$827$	−5.33576	−0.185543	−0.0927713	+	0.995687i	$0.529573\pi$
$828$	0.0230145	0.000799808 0
$829$	27.4816	0.954475	0.477238	−	0.878774i	$-0.341638\pi$
$829$	27.4816	0.954475	0.477238	+	0.878774i	$0.341638\pi$
$830$	10.5679	0.366818
$831$	−29.6101	−1.02716
$832$	8.24195	0.285738
$833$	5.22928	0.181184
$834$	−4.37449	−0.151476
$835$	13.6780	0.473346
$836$	0.757595	0.0262020
$837$	−1.00000	−0.0345651
$838$	−2.25043	−0.0777397
$839$	52.7965	1.82274	0.911369	−	0.411589i	$-0.135026\pi$
$839$	52.7965	1.82274	0.911369	+	0.411589i	$0.135026\pi$
$840$	3.97205	0.137049
$841$	49.4929	1.70665
$842$	1.09405	0.0377036
$843$	23.4937	0.809168
$844$	1.51542	0.0521630
$845$	1.00000	0.0344010
$846$	16.8565	0.579539
$847$	29.7052	1.02068
$848$	7.55495	0.259438
$849$	22.6905	0.778737
$850$	−1.42985	−0.0490434
$851$	−3.27892	−0.112400
$852$	−0.0361185	−0.00123740
$853$	−14.8287	−0.507726	−0.253863	−	0.967240i	$-0.581701\pi$
$853$	−14.8287	−0.507726	−0.253863	+	0.967240i	$0.581701\pi$
$854$	3.67389	0.125718
$855$	−2.05308	−0.0702139
$856$	12.4245	0.424662
$857$	−46.6148	−1.59233	−0.796165	−	0.605079i	$-0.793142\pi$
$857$	−46.6148	−1.59233	−0.796165	+	0.605079i	$0.793142\pi$
$858$	−7.92836	−0.270670
$859$	33.4203	1.14029	0.570143	−	0.821546i	$-0.306888\pi$
$859$	33.4203	1.14029	0.570143	+	0.821546i	$0.306888\pi$
$860$	0.726992	0.0247902
$861$	11.8084	0.402428
$862$	−5.66260	−0.192869
$863$	−21.1905	−0.721334	−0.360667	−	0.932695i	$-0.617451\pi$
$863$	−21.1905	−0.721334	−0.360667	+	0.932695i	$0.617451\pi$
$864$	−0.366116	−0.0124555
$865$	−4.79751	−0.163120
$866$	−4.93591	−0.167729
$867$	15.9436	0.541472
$868$	−0.0895356	−0.00303904
$869$	27.2331	0.923821
$870$	−12.3249	−0.417854
$871$	−5.58550	−0.189258
$872$	28.3530	0.960154
$873$	−0.592459	−0.0200517
$874$	1.01522	0.0343403
$875$	1.38286	0.0467493
$876$	0.225770	0.00762806
$877$	51.4875	1.73861	0.869305	−	0.494276i	$-0.164567\pi$
$877$	51.4875	1.73861	0.869305	+	0.494276i	$0.164567\pi$
$878$	−14.3526	−0.484378
$879$	0.841149	0.0283713
$880$	−22.0349	−0.742797
$881$	14.0107	0.472034	0.236017	−	0.971749i	$-0.424158\pi$
$881$	14.0107	0.472034	0.236017	+	0.971749i	$0.424158\pi$
$882$	−7.07766	−0.238317
$883$	9.35606	0.314856	0.157428	−	0.987530i	$-0.449680\pi$
$883$	9.35606	0.314856	0.157428	+	0.987530i	$0.449680\pi$
$884$	0.0665485	0.00223827
$885$	−5.78963	−0.194616
$886$	7.37907	0.247905
$887$	27.4835	0.922806	0.461403	−	0.887191i	$-0.347346\pi$
$887$	27.4835	0.922806	0.461403	+	0.887191i	$0.347346\pi$
$888$	26.4961	0.889152
$889$	3.50069	0.117410
$890$	−10.3807	−0.347961
$891$	5.69921	0.190931
$892$	−0.0117329	−0.000392845 0
$893$	−24.8774	−0.832491
$894$	17.0018	0.568625
$895$	−11.4389	−0.382359
$896$	14.8428	0.495865
$897$	0.355455	0.0118683
$898$	15.4210	0.514605
$899$	8.85962	0.295485
$900$	−0.0647466	−0.00215822
$901$	2.00843	0.0669104
$902$	−67.7007	−2.25419
$903$	15.5272	0.516712
$904$	−42.9942	−1.42997
$905$	18.9693	0.630562
$906$	12.5952	0.418449
$907$	−21.2866	−0.706809	−0.353404	−	0.935471i	$-0.614976\pi$
$907$	−21.2866	−0.706809	−0.353404	+	0.935471i	$0.614976\pi$
$908$	−1.27419	−0.0422856
$909$	−7.85889	−0.260663
$910$	1.92375	0.0637716
$911$	−19.4531	−0.644511	−0.322256	−	0.946653i	$-0.604441\pi$
$911$	−19.4531	−0.644511	−0.322256	+	0.946653i	$0.604441\pi$
$912$	−7.93785	−0.262849
$913$	43.2947	1.43285
$914$	−43.0620	−1.42436
$915$	−1.90976	−0.0631346
$916$	−0.611279	−0.0201972
$917$	25.5503	0.843745
$918$	1.42985	0.0471921
$919$	−45.0388	−1.48569	−0.742846	−	0.669462i	$-0.766525\pi$
$919$	−45.0388	−1.48569	−0.742846	+	0.669462i	$0.766525\pi$
$920$	1.02099	0.0336609
$921$	−29.2489	−0.963786
$922$	−52.5147	−1.72948
$923$	−0.557844	−0.0183617
$924$	0.510282	0.0167870
$925$	9.22459	0.303303
$926$	−41.2255	−1.35475
$927$	0.801697	0.0263312
$928$	3.24365	0.106478
$929$	−29.0453	−0.952944	−0.476472	−	0.879190i	$-0.658085\pi$
$929$	−29.0453	−0.952944	−0.476472	+	0.879190i	$0.658085\pi$
$930$	−1.39113	−0.0456171
$931$	10.4454	0.342335
$932$	−0.969916	−0.0317707
$933$	−27.0260	−0.884793
$934$	26.0327	0.851815
$935$	−5.85782	−0.191571
$936$	−2.87234	−0.0938853
$937$	−27.7486	−0.906508	−0.453254	−	0.891381i	$-0.649737\pi$
$937$	−27.7486	−0.906508	−0.453254	+	0.891381i	$0.649737\pi$
$938$	−10.7451	−0.350840
$939$	−21.5023	−0.701702
$940$	−0.784542	−0.0255889
$941$	−29.9596	−0.976655	−0.488328	−	0.872660i	$-0.662393\pi$
$941$	−29.9596	−0.976655	−0.488328	+	0.872660i	$0.662393\pi$
$942$	−16.5563	−0.539433
$943$	3.03525	0.0988414
$944$	−22.3845	−0.728554
$945$	−1.38286	−0.0449845
$946$	−89.0218	−2.89435
$947$	−29.1840	−0.948351	−0.474175	−	0.880430i	$-0.657254\pi$
$947$	−29.1840	−0.948351	−0.474175	+	0.880430i	$0.657254\pi$
$948$	0.309386	0.0100484
$949$	3.48698	0.113192
$950$	−2.85611	−0.0926644
$951$	14.9069	0.483391
$952$	4.08259	0.132318
$953$	−43.3885	−1.40549	−0.702746	−	0.711441i	$-0.748043\pi$
$953$	−43.3885	−1.40549	−0.702746	+	0.711441i	$0.748043\pi$
$954$	−2.71834	−0.0880095
$955$	−6.76037	−0.218760
$956$	−1.08412	−0.0350630
$957$	−50.4928	−1.63220
$958$	−21.1230	−0.682453
$959$	20.9163	0.675422
$960$	−8.24195	−0.266008
$961$	1.00000	0.0322581
$962$	12.8326	0.413741
$963$	−4.32559	−0.139390
$964$	0.342025	0.0110159
$965$	6.73173	0.216702
$966$	0.683805	0.0220011
$967$	39.1337	1.25845	0.629227	−	0.777221i	$-0.283372\pi$
$967$	39.1337	1.25845	0.629227	+	0.777221i	$0.283372\pi$
$968$	−61.7006	−1.98313
$969$	−2.11022	−0.0677900
$970$	−0.824189	−0.0264631
$971$	−18.7151	−0.600597	−0.300299	−	0.953845i	$-0.597086\pi$
$971$	−18.7151	−0.600597	−0.300299	+	0.953845i	$0.597086\pi$
$972$	0.0647466	0.00207675
$973$	4.34848	0.139406
$974$	50.8222	1.62845
$975$	−1.00000	−0.0320256
$976$	−7.38372	−0.236347
$977$	3.13677	0.100354	0.0501771	−	0.998740i	$-0.484021\pi$
$977$	3.13677	0.100354	0.0501771	+	0.998740i	$0.484021\pi$
$978$	−9.63155	−0.307983
$979$	−42.5276	−1.35919
$980$	0.329410	0.0105226
$981$	−9.87105	−0.315158
$982$	20.2703	0.646852
$983$	−41.8392	−1.33446	−0.667232	−	0.744850i	$-0.732521\pi$
$983$	−41.8392	−1.33446	−0.667232	+	0.744850i	$0.732521\pi$
$984$	−24.5271	−0.781895
$985$	14.1930	0.452226
$986$	−12.6679	−0.403429
$987$	−16.7563	−0.533359
$988$	0.132930	0.00422906
$989$	3.99114	0.126911
$990$	7.92836	0.251980
$991$	38.5024	1.22307	0.611535	−	0.791217i	$-0.290552\pi$
$991$	38.5024	1.22307	0.611535	+	0.791217i	$0.290552\pi$
$992$	0.366116	0.0116242
$993$	14.1190	0.448052
$994$	−1.07315	−0.0340383
$995$	10.1031	0.320289
$996$	0.491855	0.0155850
$997$	35.6184	1.12805	0.564023	−	0.825759i	$-0.309253\pi$
$997$	35.6184	1.12805	0.564023	+	0.825759i	$0.309253\pi$
$998$	15.7946	0.499971
$999$	−9.22459	−0.291853

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.bh.1.12	✓	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.bh.1.12	✓	17	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:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

\(f(q)\)	\(=\)	\(q+1.39113 q^{2} -1.00000 q^{3} -0.0647466 q^{4} +1.00000 q^{5} -1.39113 q^{6} +1.38286 q^{7} -2.87234 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.39113 q^{2} -1.00000 q^{3} -0.0647466 q^{4} +1.00000 q^{5} -1.39113 q^{6} +1.38286 q^{7} -2.87234 q^{8} +1.00000 q^{9} +1.39113 q^{10} +5.69921 q^{11} +0.0647466 q^{12} +1.00000 q^{13} +1.92375 q^{14} -1.00000 q^{15} -3.86631 q^{16} -1.02783 q^{17} +1.39113 q^{18} -2.05308 q^{19} -0.0647466 q^{20} -1.38286 q^{21} +7.92836 q^{22} -0.355455 q^{23} +2.87234 q^{24} +1.00000 q^{25} +1.39113 q^{26} -1.00000 q^{27} -0.0895356 q^{28} +8.85962 q^{29} -1.39113 q^{30} +1.00000 q^{31} +0.366116 q^{32} -5.69921 q^{33} -1.42985 q^{34} +1.38286 q^{35} -0.0647466 q^{36} +9.22459 q^{37} -2.85611 q^{38} -1.00000 q^{39} -2.87234 q^{40} -8.53906 q^{41} -1.92375 q^{42} -11.2283 q^{43} -0.369004 q^{44} +1.00000 q^{45} -0.494485 q^{46} +12.1171 q^{47} +3.86631 q^{48} -5.08769 q^{49} +1.39113 q^{50} +1.02783 q^{51} -0.0647466 q^{52} -1.95405 q^{53} -1.39113 q^{54} +5.69921 q^{55} -3.97205 q^{56} +2.05308 q^{57} +12.3249 q^{58} +5.78963 q^{59} +0.0647466 q^{60} +1.90976 q^{61} +1.39113 q^{62} +1.38286 q^{63} +8.24195 q^{64} +1.00000 q^{65} -7.92836 q^{66} -5.58550 q^{67} +0.0665485 q^{68} +0.355455 q^{69} +1.92375 q^{70} -0.557844 q^{71} -2.87234 q^{72} +3.48698 q^{73} +12.8326 q^{74} -1.00000 q^{75} +0.132930 q^{76} +7.88122 q^{77} -1.39113 q^{78} +4.77841 q^{79} -3.86631 q^{80} +1.00000 q^{81} -11.8790 q^{82} +7.59662 q^{83} +0.0895356 q^{84} -1.02783 q^{85} -15.6200 q^{86} -8.85962 q^{87} -16.3701 q^{88} -7.46202 q^{89} +1.39113 q^{90} +1.38286 q^{91} +0.0230145 q^{92} -1.00000 q^{93} +16.8565 q^{94} -2.05308 q^{95} -0.366116 q^{96} -0.592459 q^{97} -7.07766 q^{98} +5.69921 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 2 q^{2} - 17 q^{3} + 20 q^{4} + 17 q^{5} - 2 q^{6} + 18 q^{7} + 9 q^{8} + 17 q^{9}+O(q^{10}) \) 17 * q + 2 * q^2 - 17 * q^3 + 20 * q^4 + 17 * q^5 - 2 * q^6 + 18 * q^7 + 9 * q^8 + 17 * q^9	\( 17 q + 2 q^{2} - 17 q^{3} + 20 q^{4} + 17 q^{5} - 2 q^{6} + 18 q^{7} + 9 q^{8} + 17 q^{9} + 2 q^{10} + 3 q^{11} - 20 q^{12} + 17 q^{13} + q^{14} - 17 q^{15} + 26 q^{16} + 2 q^{18} + 10 q^{19} + 20 q^{20} - 18 q^{21} + 5 q^{22} + 16 q^{23} - 9 q^{24} + 17 q^{25} + 2 q^{26} - 17 q^{27} + 36 q^{28} - 3 q^{29} - 2 q^{30} + 17 q^{31} + 20 q^{32} - 3 q^{33} + q^{34} + 18 q^{35} + 20 q^{36} + 14 q^{37} + 22 q^{38} - 17 q^{39} + 9 q^{40} - 6 q^{41} - q^{42} + 24 q^{43} - 15 q^{44} + 17 q^{45} + 6 q^{46} + 25 q^{47} - 26 q^{48} + 31 q^{49} + 2 q^{50} + 20 q^{52} - 15 q^{53} - 2 q^{54} + 3 q^{55} + 31 q^{56} - 10 q^{57} + 44 q^{58} + 16 q^{59} - 20 q^{60} - 5 q^{61} + 2 q^{62} + 18 q^{63} + 35 q^{64} + 17 q^{65} - 5 q^{66} + 50 q^{67} + 13 q^{68} - 16 q^{69} + q^{70} + 16 q^{71} + 9 q^{72} + 33 q^{73} + 2 q^{74} - 17 q^{75} + 9 q^{77} - 2 q^{78} - 10 q^{79} + 26 q^{80} + 17 q^{81} + 61 q^{82} + 27 q^{83} - 36 q^{84} - 12 q^{86} + 3 q^{87} + 23 q^{88} - 24 q^{89} + 2 q^{90} + 18 q^{91} - 21 q^{92} - 17 q^{93} + 6 q^{94} + 10 q^{95} - 20 q^{96} + 48 q^{97} + 14 q^{98} + 3 q^{99}+O(q^{100}) \) 17 * q + 2 * q^2 - 17 * q^3 + 20 * q^4 + 17 * q^5 - 2 * q^6 + 18 * q^7 + 9 * q^8 + 17 * q^9 + 2 * q^10 + 3 * q^11 - 20 * q^12 + 17 * q^13 + q^14 - 17 * q^15 + 26 * q^16 + 2 * q^18 + 10 * q^19 + 20 * q^20 - 18 * q^21 + 5 * q^22 + 16 * q^23 - 9 * q^24 + 17 * q^25 + 2 * q^26 - 17 * q^27 + 36 * q^28 - 3 * q^29 - 2 * q^30 + 17 * q^31 + 20 * q^32 - 3 * q^33 + q^34 + 18 * q^35 + 20 * q^36 + 14 * q^37 + 22 * q^38 - 17 * q^39 + 9 * q^40 - 6 * q^41 - q^42 + 24 * q^43 - 15 * q^44 + 17 * q^45 + 6 * q^46 + 25 * q^47 - 26 * q^48 + 31 * q^49 + 2 * q^50 + 20 * q^52 - 15 * q^53 - 2 * q^54 + 3 * q^55 + 31 * q^56 - 10 * q^57 + 44 * q^58 + 16 * q^59 - 20 * q^60 - 5 * q^61 + 2 * q^62 + 18 * q^63 + 35 * q^64 + 17 * q^65 - 5 * q^66 + 50 * q^67 + 13 * q^68 - 16 * q^69 + q^70 + 16 * q^71 + 9 * q^72 + 33 * q^73 + 2 * q^74 - 17 * q^75 + 9 * q^77 - 2 * q^78 - 10 * q^79 + 26 * q^80 + 17 * q^81 + 61 * q^82 + 27 * q^83 - 36 * q^84 - 12 * q^86 + 3 * q^87 + 23 * q^88 - 24 * q^89 + 2 * q^90 + 18 * q^91 - 21 * q^92 - 17 * q^93 + 6 * q^94 + 10 * q^95 - 20 * q^96 + 48 * q^97 + 14 * q^98 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more