LMFDB - Embedded newform 1001.2.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1001,2,Mod(1,1001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1001, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1001 = 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1001.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:	$7.99302524233$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.3212905625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 5x^{6} + 16x^{5} + 5x^{4} - 22x^{3} + 4x^{2} + 4x - 1 $ x^8 - 3x^7 - 5x^6 + 16x^5 + 5x^4 - 22x^3 + 4x^2 + 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.49988$ of defining polynomial
Character	$\chi$	$=$	1001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.34808 q^{2} -0.833162 q^{3} -0.182668 q^{4} -0.0592996 q^{5} +1.12317 q^{6} -1.00000 q^{7} +2.94242 q^{8} -2.30584 q^{9} +O(q^{10})$	\(q-1.34808 q^{2} -0.833162 q^{3} -0.182668 q^{4} -0.0592996 q^{5} +1.12317 q^{6} -1.00000 q^{7} +2.94242 q^{8} -2.30584 q^{9} +0.0799408 q^{10} -1.00000 q^{11} +0.152192 q^{12} +1.00000 q^{13} +1.34808 q^{14} +0.0494061 q^{15} -3.60130 q^{16} +1.98413 q^{17} +3.10847 q^{18} -5.99360 q^{19} +0.0108322 q^{20} +0.833162 q^{21} +1.34808 q^{22} +2.15671 q^{23} -2.45151 q^{24} -4.99648 q^{25} -1.34808 q^{26} +4.42063 q^{27} +0.182668 q^{28} -5.38261 q^{29} -0.0666036 q^{30} +7.70525 q^{31} -1.02999 q^{32} +0.833162 q^{33} -2.67477 q^{34} +0.0592996 q^{35} +0.421204 q^{36} -0.226839 q^{37} +8.07988 q^{38} -0.833162 q^{39} -0.174484 q^{40} +2.81018 q^{41} -1.12317 q^{42} +6.22709 q^{43} +0.182668 q^{44} +0.136735 q^{45} -2.90742 q^{46} -8.26199 q^{47} +3.00046 q^{48} +1.00000 q^{49} +6.73568 q^{50} -1.65310 q^{51} -0.182668 q^{52} +8.54929 q^{53} -5.95938 q^{54} +0.0592996 q^{55} -2.94242 q^{56} +4.99364 q^{57} +7.25621 q^{58} +4.40803 q^{59} -0.00902494 q^{60} -3.46161 q^{61} -10.3873 q^{62} +2.30584 q^{63} +8.59111 q^{64} -0.0592996 q^{65} -1.12317 q^{66} -0.0542603 q^{67} -0.362438 q^{68} -1.79689 q^{69} -0.0799408 q^{70} +7.32990 q^{71} -6.78476 q^{72} +11.7642 q^{73} +0.305798 q^{74} +4.16288 q^{75} +1.09484 q^{76} +1.00000 q^{77} +1.12317 q^{78} +13.1908 q^{79} +0.213555 q^{80} +3.23443 q^{81} -3.78836 q^{82} +2.62861 q^{83} -0.152192 q^{84} -0.117658 q^{85} -8.39464 q^{86} +4.48458 q^{87} -2.94242 q^{88} +1.86537 q^{89} -0.184331 q^{90} -1.00000 q^{91} -0.393962 q^{92} -6.41972 q^{93} +11.1379 q^{94} +0.355418 q^{95} +0.858151 q^{96} +14.2434 q^{97} -1.34808 q^{98} +2.30584 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} - q^{3} + 12 q^{4} - 5 q^{5} + q^{6} - 8 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10}) $ 8 * q + 2 * q^2 - q^3 + 12 * q^4 - 5 * q^5 + q^6 - 8 * q^7 + 9 * q^8 + 3 * q^9	\( 8 q + 2 q^{2} - q^{3} + 12 q^{4} - 5 q^{5} + q^{6} - 8 q^{7} + 9 q^{8} + 3 q^{9} - q^{10} - 8 q^{11} + q^{12} + 8 q^{13} - 2 q^{14} + 13 q^{15} + 16 q^{16} - 5 q^{17} + 12 q^{18} + 31 q^{19} - 8 q^{20} + q^{21} - 2 q^{22} + q^{23} + 35 q^{24} + 11 q^{25} + 2 q^{26} - 13 q^{27} - 12 q^{28} - 7 q^{29} + 24 q^{30} + 20 q^{31} + 10 q^{32} + q^{33} + 7 q^{34} + 5 q^{35} + 23 q^{36} + 18 q^{37} + 10 q^{38} - q^{39} - 16 q^{40} + 2 q^{41} - q^{42} + 5 q^{43} - 12 q^{44} - 27 q^{45} + 28 q^{46} - 11 q^{47} + 16 q^{48} + 8 q^{49} + 49 q^{50} - q^{51} + 12 q^{52} - 2 q^{53} - 80 q^{54} + 5 q^{55} - 9 q^{56} + 10 q^{57} - 36 q^{58} - 13 q^{59} + 39 q^{60} + 10 q^{61} + 16 q^{62} - 3 q^{63} + 31 q^{64} - 5 q^{65} - q^{66} + 24 q^{67} - 46 q^{68} - 24 q^{69} + q^{70} + 30 q^{71} + 9 q^{72} - 16 q^{73} - 7 q^{74} - 32 q^{75} + 48 q^{76} + 8 q^{77} + q^{78} + 29 q^{79} - 40 q^{80} + 16 q^{81} - 30 q^{82} + 50 q^{83} - q^{84} + 41 q^{85} + 32 q^{86} + 33 q^{87} - 9 q^{88} - 9 q^{89} - 82 q^{90} - 8 q^{91} + 18 q^{92} - 13 q^{93} + 6 q^{94} - 7 q^{95} + 76 q^{96} - 18 q^{97} + 2 q^{98} - 3 q^{99}+O(q^{100}) \) 8 * q + 2 * q^2 - q^3 + 12 * q^4 - 5 * q^5 + q^6 - 8 * q^7 + 9 * q^8 + 3 * q^9 - q^10 - 8 * q^11 + q^12 + 8 * q^13 - 2 * q^14 + 13 * q^15 + 16 * q^16 - 5 * q^17 + 12 * q^18 + 31 * q^19 - 8 * q^20 + q^21 - 2 * q^22 + q^23 + 35 * q^24 + 11 * q^25 + 2 * q^26 - 13 * q^27 - 12 * q^28 - 7 * q^29 + 24 * q^30 + 20 * q^31 + 10 * q^32 + q^33 + 7 * q^34 + 5 * q^35 + 23 * q^36 + 18 * q^37 + 10 * q^38 - q^39 - 16 * q^40 + 2 * q^41 - q^42 + 5 * q^43 - 12 * q^44 - 27 * q^45 + 28 * q^46 - 11 * q^47 + 16 * q^48 + 8 * q^49 + 49 * q^50 - q^51 + 12 * q^52 - 2 * q^53 - 80 * q^54 + 5 * q^55 - 9 * q^56 + 10 * q^57 - 36 * q^58 - 13 * q^59 + 39 * q^60 + 10 * q^61 + 16 * q^62 - 3 * q^63 + 31 * q^64 - 5 * q^65 - q^66 + 24 * q^67 - 46 * q^68 - 24 * q^69 + q^70 + 30 * q^71 + 9 * q^72 - 16 * q^73 - 7 * q^74 - 32 * q^75 + 48 * q^76 + 8 * q^77 + q^78 + 29 * q^79 - 40 * q^80 + 16 * q^81 - 30 * q^82 + 50 * q^83 - q^84 + 41 * q^85 + 32 * q^86 + 33 * q^87 - 9 * q^88 - 9 * q^89 - 82 * q^90 - 8 * q^91 + 18 * q^92 - 13 * q^93 + 6 * q^94 - 7 * q^95 + 76 * q^96 - 18 * q^97 + 2 * 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 $

$2$	−1.34808	−0.953240	−0.476620	−	0.879110i	$-0.658138\pi$
$2$	−1.34808	−0.953240	−0.476620	+	0.879110i	$0.658138\pi$
$3$	−0.833162	−0.481026	−0.240513	−	0.970646i	$-0.577316\pi$
$3$	−0.833162	−0.481026	−0.240513	+	0.970646i	$0.577316\pi$
$4$	−0.182668	−0.0913342
$5$	−0.0592996	−0.0265196	−0.0132598	−	0.999912i	$-0.504221\pi$
$5$	−0.0592996	−0.0265196	−0.0132598	+	0.999912i	$0.504221\pi$
$6$	1.12317	0.458533
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.94242	1.04030
$9$	−2.30584	−0.768614
$10$	0.0799408	0.0252795
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0.152192	0.0439342
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.34808	0.360291
$15$	0.0494061	0.0127566
$16$	−3.60130	−0.900324
$17$	1.98413	0.481222	0.240611	−	0.970622i	$-0.422652\pi$
$17$	1.98413	0.481222	0.240611	+	0.970622i	$0.422652\pi$
$18$	3.10847	0.732673
$19$	−5.99360	−1.37503	−0.687513	−	0.726172i	$-0.741298\pi$
$19$	−5.99360	−1.37503	−0.687513	+	0.726172i	$0.741298\pi$
$20$	0.0108322	0.00242214
$21$	0.833162	0.181811
$22$	1.34808	0.287413
$23$	2.15671	0.449705	0.224852	−	0.974393i	$-0.427810\pi$
$23$	2.15671	0.449705	0.224852	+	0.974393i	$0.427810\pi$
$24$	−2.45151	−0.500413
$25$	−4.99648	−0.999297
$26$	−1.34808	−0.264381
$27$	4.42063	0.850750
$28$	0.182668	0.0345211
$29$	−5.38261	−0.999525	−0.499763	−	0.866162i	$-0.666579\pi$
$29$	−5.38261	−0.999525	−0.499763	+	0.866162i	$0.666579\pi$
$30$	−0.0666036	−0.0121601
$31$	7.70525	1.38390	0.691952	−	0.721944i	$-0.256751\pi$
$31$	7.70525	1.38390	0.691952	+	0.721944i	$0.256751\pi$
$32$	−1.02999	−0.182079
$33$	0.833162	0.145035
$34$	−2.67477	−0.458720
$35$	0.0592996	0.0100235
$36$	0.421204	0.0702007
$37$	−0.226839	−0.0372921	−0.0186461	−	0.999826i	$-0.505936\pi$
$37$	−0.226839	−0.0372921	−0.0186461	+	0.999826i	$0.505936\pi$
$38$	8.07988	1.31073
$39$	−0.833162	−0.133413
$40$	−0.174484	−0.0275884
$41$	2.81018	0.438876	0.219438	−	0.975626i	$-0.429578\pi$
$41$	2.81018	0.438876	0.219438	+	0.975626i	$0.429578\pi$
$42$	−1.12317	−0.173309
$43$	6.22709	0.949622	0.474811	−	0.880088i	$-0.342516\pi$
$43$	6.22709	0.949622	0.474811	+	0.880088i	$0.342516\pi$
$44$	0.182668	0.0275383
$45$	0.136735	0.0203833
$46$	−2.90742	−0.428676
$47$	−8.26199	−1.20514	−0.602568	−	0.798068i	$-0.705856\pi$
$47$	−8.26199	−1.20514	−0.602568	+	0.798068i	$0.705856\pi$
$48$	3.00046	0.433079
$49$	1.00000	0.142857
$50$	6.73568	0.952569
$51$	−1.65310	−0.231480
$52$	−0.182668	−0.0253316
$53$	8.54929	1.17434	0.587168	−	0.809465i	$-0.300243\pi$
$53$	8.54929	1.17434	0.587168	+	0.809465i	$0.300243\pi$
$54$	−5.95938	−0.810968
$55$	0.0592996	0.00799595
$56$	−2.94242	−0.393198
$57$	4.99364	0.661424
$58$	7.25621	0.952787
$59$	4.40803	0.573877	0.286938	−	0.957949i	$-0.407362\pi$
$59$	4.40803	0.573877	0.286938	+	0.957949i	$0.407362\pi$
$60$	−0.00902494	−0.00116512
$61$	−3.46161	−0.443214	−0.221607	−	0.975136i	$-0.571130\pi$
$61$	−3.46161	−0.443214	−0.221607	+	0.975136i	$0.571130\pi$
$62$	−10.3873	−1.31919
$63$	2.30584	0.290509
$64$	8.59111	1.07389
$65$	−0.0592996	−0.00735520
$66$	−1.12317	−0.138253
$67$	−0.0542603	−0.00662895	−0.00331447	−	0.999995i	$-0.501055\pi$
$67$	−0.0542603	−0.00662895	−0.00331447	+	0.999995i	$0.501055\pi$
$68$	−0.362438	−0.0439520
$69$	−1.79689	−0.216320
$70$	−0.0799408	−0.00955475
$71$	7.32990	0.869899	0.434949	−	0.900455i	$-0.356766\pi$
$71$	7.32990	0.869899	0.434949	+	0.900455i	$0.356766\pi$
$72$	−6.78476	−0.799591
$73$	11.7642	1.37689	0.688447	−	0.725287i	$-0.258293\pi$
$73$	11.7642	1.37689	0.688447	+	0.725287i	$0.258293\pi$
$74$	0.305798	0.0355483
$75$	4.16288	0.480688
$76$	1.09484	0.125587
$77$	1.00000	0.113961
$78$	1.12317	0.127174
$79$	13.1908	1.48408	0.742042	−	0.670353i	$-0.233857\pi$
$79$	13.1908	1.48408	0.742042	+	0.670353i	$0.233857\pi$
$80$	0.213555	0.0238762
$81$	3.23443	0.359381
$82$	−3.78836	−0.418354
$83$	2.62861	0.288528	0.144264	−	0.989539i	$-0.453919\pi$
$83$	2.62861	0.288528	0.144264	+	0.989539i	$0.453919\pi$
$84$	−0.152192	−0.0166056
$85$	−0.117658	−0.0127618
$86$	−8.39464	−0.905217
$87$	4.48458	0.480798
$88$	−2.94242	−0.313663
$89$	1.86537	0.197729	0.0988644	−	0.995101i	$-0.468479\pi$
$89$	1.86537	0.197729	0.0988644	+	0.995101i	$0.468479\pi$
$90$	−0.184331	−0.0194302
$91$	−1.00000	−0.104828
$92$	−0.393962	−0.0410734
$93$	−6.41972	−0.665694
$94$	11.1379	1.14878
$95$	0.355418	0.0364651
$96$	0.858151	0.0875846
$97$	14.2434	1.44620	0.723100	−	0.690743i	$-0.242717\pi$
$97$	14.2434	1.44620	0.723100	+	0.690743i	$0.242717\pi$
$98$	−1.34808	−0.136177
$99$	2.30584	0.231746
$100$	0.912700	0.0912700
$101$	−2.18900	−0.217813	−0.108907	−	0.994052i	$-0.534735\pi$
$101$	−2.18900	−0.217813	−0.108907	+	0.994052i	$0.534735\pi$
$102$	2.22852	0.220656
$103$	5.72540	0.564141	0.282070	−	0.959394i	$-0.408979\pi$
$103$	5.72540	0.564141	0.282070	+	0.959394i	$0.408979\pi$
$104$	2.94242	0.288528
$105$	−0.0494061	−0.00482155
$106$	−11.5252	−1.11942
$107$	11.5543	1.11700	0.558500	−	0.829505i	$-0.311377\pi$
$107$	11.5543	1.11700	0.558500	+	0.829505i	$0.311377\pi$
$108$	−0.807509	−0.0777026
$109$	−0.150039	−0.0143712	−0.00718559	−	0.999974i	$-0.502287\pi$
$109$	−0.150039	−0.0143712	−0.00718559	+	0.999974i	$0.502287\pi$
$110$	−0.0799408	−0.00762206
$111$	0.188994	0.0179385
$112$	3.60130	0.340290
$113$	−16.1797	−1.52205	−0.761027	−	0.648720i	$-0.775305\pi$
$113$	−16.1797	−1.52205	−0.761027	+	0.648720i	$0.775305\pi$
$114$	−6.73185	−0.630495
$115$	−0.127892	−0.0119260
$116$	0.983233	0.0912909
$117$	−2.30584	−0.213175
$118$	−5.94240	−0.547042
$119$	−1.98413	−0.181885
$120$	0.145374	0.0132707
$121$	1.00000	0.0909091
$122$	4.66654	0.422489
$123$	−2.34134	−0.211111
$124$	−1.40751	−0.126398
$125$	0.592787	0.0530205
$126$	−3.10847	−0.276924
$127$	16.4212	1.45715	0.728574	−	0.684967i	$-0.240183\pi$
$127$	16.4212	1.45715	0.728574	+	0.684967i	$0.240183\pi$
$128$	−9.52155	−0.841594
$129$	−5.18817	−0.456793
$130$	0.0799408	0.00701127
$131$	9.38775	0.820212	0.410106	−	0.912038i	$-0.365492\pi$
$131$	9.38775	0.820212	0.410106	+	0.912038i	$0.365492\pi$
$132$	−0.152192	−0.0132467
$133$	5.99360	0.519711
$134$	0.0731474	0.00631898
$135$	−0.262141	−0.0225615
$136$	5.83814	0.500617
$137$	−13.0380	−1.11391	−0.556954	−	0.830543i	$-0.688030\pi$
$137$	−13.0380	−1.11391	−0.556954	+	0.830543i	$0.688030\pi$
$138$	2.42235	0.206205
$139$	0.452387	0.0383710	0.0191855	−	0.999816i	$-0.493893\pi$
$139$	0.452387	0.0383710	0.0191855	+	0.999816i	$0.493893\pi$
$140$	−0.0108322	−0.000915485 0
$141$	6.88358	0.579702
$142$	−9.88132	−0.829222
$143$	−1.00000	−0.0836242
$144$	8.30401	0.692001
$145$	0.319186	0.0265070
$146$	−15.8591	−1.31251
$147$	−0.833162	−0.0687180
$148$	0.0414363	0.00340605
$149$	8.94722	0.732985	0.366492	−	0.930421i	$-0.380559\pi$
$149$	8.94722	0.732985	0.366492	+	0.930421i	$0.380559\pi$
$150$	−5.61191	−0.458211
$151$	7.90085	0.642962	0.321481	−	0.946916i	$-0.395819\pi$
$151$	7.90085	0.642962	0.321481	+	0.946916i	$0.395819\pi$
$152$	−17.6357	−1.43044
$153$	−4.57509	−0.369874
$154$	−1.34808	−0.108632
$155$	−0.456918	−0.0367005
$156$	0.152192	0.0121851
$157$	−9.61240	−0.767153	−0.383577	−	0.923509i	$-0.625308\pi$
$157$	−9.61240	−0.767153	−0.383577	+	0.923509i	$0.625308\pi$
$158$	−17.7824	−1.41469
$159$	−7.12295	−0.564886
$160$	0.0610781	0.00482865
$161$	−2.15671	−0.169972
$162$	−4.36028	−0.342576
$163$	5.55578	0.435162	0.217581	−	0.976042i	$-0.430183\pi$
$163$	5.55578	0.435162	0.217581	+	0.976042i	$0.430183\pi$
$164$	−0.513331	−0.0400844
$165$	−0.0494061	−0.00384626
$166$	−3.54359	−0.275036
$167$	−7.44331	−0.575981	−0.287990	−	0.957633i	$-0.592987\pi$
$167$	−7.44331	−0.575981	−0.287990	+	0.957633i	$0.592987\pi$
$168$	2.45151	0.189138
$169$	1.00000	0.0769231
$170$	0.158613	0.0121650
$171$	13.8203	1.05686
$172$	−1.13749	−0.0867330
$173$	−19.4274	−1.47704	−0.738518	−	0.674234i	$-0.764474\pi$
$173$	−19.4274	−1.47704	−0.738518	+	0.674234i	$0.764474\pi$
$174$	−6.04560	−0.458316
$175$	4.99648	0.377699
$176$	3.60130	0.271458
$177$	−3.67261	−0.276050
$178$	−2.51467	−0.188483
$179$	1.15988	0.0866934	0.0433467	−	0.999060i	$-0.486198\pi$
$179$	1.15988	0.0866934	0.0433467	+	0.999060i	$0.486198\pi$
$180$	−0.0249772	−0.00186169
$181$	3.06560	0.227864	0.113932	−	0.993489i	$-0.463655\pi$
$181$	3.06560	0.227864	0.113932	+	0.993489i	$0.463655\pi$
$182$	1.34808	0.0999267
$183$	2.88408	0.213197
$184$	6.34594	0.467829
$185$	0.0134515	0.000988971 0
$186$	8.65432	0.634566
$187$	−1.98413	−0.145094
$188$	1.50921	0.110070
$189$	−4.42063	−0.321553
$190$	−0.479133	−0.0347600
$191$	−7.60762	−0.550468	−0.275234	−	0.961377i	$-0.588755\pi$
$191$	−7.60762	−0.550468	−0.275234	+	0.961377i	$0.588755\pi$
$192$	−7.15778	−0.516569
$193$	19.1893	1.38128	0.690639	−	0.723199i	$-0.257329\pi$
$193$	19.1893	1.38128	0.690639	+	0.723199i	$0.257329\pi$
$194$	−19.2013	−1.37858
$195$	0.0494061	0.00353805
$196$	−0.182668	−0.0130477
$197$	1.84114	0.131176	0.0655879	−	0.997847i	$-0.479108\pi$
$197$	1.84114	0.131176	0.0655879	+	0.997847i	$0.479108\pi$
$198$	−3.10847	−0.220909
$199$	−22.5155	−1.59608	−0.798040	−	0.602604i	$-0.794130\pi$
$199$	−22.5155	−1.59608	−0.798040	+	0.602604i	$0.794130\pi$
$200$	−14.7018	−1.03957
$201$	0.0452076	0.00318870
$202$	2.95095	0.207628
$203$	5.38261	0.377785
$204$	0.301969	0.0211421
$205$	−0.166642	−0.0116388
$206$	−7.71833	−0.537761
$207$	−4.97302	−0.345649
$208$	−3.60130	−0.249705
$209$	5.99360	0.414586
$210$	0.0666036	0.00459609
$211$	15.0339	1.03498	0.517490	−	0.855689i	$-0.326867\pi$
$211$	15.0339	1.03498	0.517490	+	0.855689i	$0.326867\pi$
$212$	−1.56169	−0.107257
$213$	−6.10699	−0.418444
$214$	−15.5762	−1.06477
$215$	−0.369263	−0.0251836
$216$	13.0073	0.885037
$217$	−7.70525	−0.523066
$218$	0.202266	0.0136992
$219$	−9.80147	−0.662322
$220$	−0.0108322	−0.000730304 0
$221$	1.98413	0.133467
$222$	−0.254779	−0.0170997
$223$	−10.5549	−0.706811	−0.353406	−	0.935470i	$-0.614976\pi$
$223$	−10.5549	−0.706811	−0.353406	+	0.935470i	$0.614976\pi$
$224$	1.02999	0.0688193
$225$	11.5211	0.768073
$226$	21.8115	1.45088
$227$	0.737401	0.0489430	0.0244715	−	0.999701i	$-0.492210\pi$
$227$	0.737401	0.0489430	0.0244715	+	0.999701i	$0.492210\pi$
$228$	−0.912181	−0.0604106
$229$	−6.21684	−0.410821	−0.205410	−	0.978676i	$-0.565853\pi$
$229$	−6.21684	−0.410821	−0.205410	+	0.978676i	$0.565853\pi$
$230$	0.172409	0.0113683
$231$	−0.833162	−0.0548180
$232$	−15.8379	−1.03981
$233$	−22.3045	−1.46122	−0.730609	−	0.682796i	$-0.760764\pi$
$233$	−22.3045	−1.46122	−0.730609	+	0.682796i	$0.760764\pi$
$234$	3.10847	0.203207
$235$	0.489932	0.0319597
$236$	−0.805209	−0.0524146
$237$	−10.9901	−0.713884
$238$	2.67477	0.173380
$239$	17.0929	1.10565	0.552823	−	0.833299i	$-0.313551\pi$
$239$	17.0929	1.10565	0.552823	+	0.833299i	$0.313551\pi$
$240$	−0.177926	−0.0114851
$241$	20.6857	1.33248	0.666242	−	0.745736i	$-0.267902\pi$
$241$	20.6857	1.33248	0.666242	+	0.745736i	$0.267902\pi$
$242$	−1.34808	−0.0866581
$243$	−15.9567	−1.02362
$244$	0.632327	0.0404806
$245$	−0.0592996	−0.00378851
$246$	3.15632	0.201239
$247$	−5.99360	−0.381364
$248$	22.6721	1.43968
$249$	−2.19006	−0.138790
$250$	−0.799127	−0.0505412
$251$	0.418412	0.0264099	0.0132050	−	0.999913i	$-0.495797\pi$
$251$	0.418412	0.0264099	0.0132050	+	0.999913i	$0.495797\pi$
$252$	−0.421204	−0.0265334
$253$	−2.15671	−0.135591
$254$	−22.1372	−1.38901
$255$	0.0980281	0.00613876
$256$	−4.34636	−0.271647
$257$	−15.4487	−0.963666	−0.481833	−	0.876263i	$-0.660029\pi$
$257$	−15.4487	−0.963666	−0.481833	+	0.876263i	$0.660029\pi$
$258$	6.99409	0.435433
$259$	0.226839	0.0140951
$260$	0.0108322	0.000671782 0
$261$	12.4114	0.768249
$262$	−12.6555	−0.781858
$263$	−16.5449	−1.02020	−0.510100	−	0.860115i	$-0.670392\pi$
$263$	−16.5449	−1.02020	−0.510100	+	0.860115i	$0.670392\pi$
$264$	2.45151	0.150880
$265$	−0.506969	−0.0311429
$266$	−8.07988	−0.495409
$267$	−1.55415	−0.0951127
$268$	0.00991164	0.000605450 0
$269$	5.97084	0.364049	0.182024	−	0.983294i	$-0.441735\pi$
$269$	5.97084	0.364049	0.182024	+	0.983294i	$0.441735\pi$
$270$	0.353388	0.0215065
$271$	13.1913	0.801315	0.400658	−	0.916228i	$-0.368782\pi$
$271$	13.1913	0.801315	0.400658	+	0.916228i	$0.368782\pi$
$272$	−7.14543	−0.433256
$273$	0.833162	0.0504253
$274$	17.5763	1.06182
$275$	4.99648	0.301299
$276$	0.328235	0.0197574
$277$	−27.1564	−1.63167	−0.815834	−	0.578286i	$-0.803722\pi$
$277$	−27.1564	−1.63167	−0.815834	+	0.578286i	$0.803722\pi$
$278$	−0.609856	−0.0365767
$279$	−17.7671	−1.06369
$280$	0.174484	0.0104274
$281$	13.8174	0.824277	0.412138	−	0.911121i	$-0.364782\pi$
$281$	13.8174	0.824277	0.412138	+	0.911121i	$0.364782\pi$
$282$	−9.27964	−0.552595
$283$	−0.428160	−0.0254515	−0.0127257	−	0.999919i	$-0.504051\pi$
$283$	−0.428160	−0.0254515	−0.0127257	+	0.999919i	$0.504051\pi$
$284$	−1.33894	−0.0794515
$285$	−0.296121	−0.0175407
$286$	1.34808	0.0797139
$287$	−2.81018	−0.165880
$288$	2.37500	0.139948
$289$	−13.0632	−0.768426
$290$	−0.430290	−0.0252675
$291$	−11.8671	−0.695660
$292$	−2.14895	−0.125758
$293$	−1.19924	−0.0700605	−0.0350303	−	0.999386i	$-0.511153\pi$
$293$	−1.19924	−0.0700605	−0.0350303	+	0.999386i	$0.511153\pi$
$294$	1.12317	0.0655048
$295$	−0.261394	−0.0152190
$296$	−0.667456	−0.0387951
$297$	−4.42063	−0.256511
$298$	−12.0616	−0.698710
$299$	2.15671	0.124726
$300$	−0.760427	−0.0439033
$301$	−6.22709	−0.358923
$302$	−10.6510	−0.612897
$303$	1.82379	0.104774
$304$	21.5847	1.23797
$305$	0.205272	0.0117538
$306$	6.16760	0.352578
$307$	−20.7659	−1.18517	−0.592586	−	0.805507i	$-0.701893\pi$
$307$	−20.7659	−1.18517	−0.592586	+	0.805507i	$0.701893\pi$
$308$	−0.182668	−0.0104085
$309$	−4.77019	−0.271367
$310$	0.615964	0.0349844
$311$	−27.9332	−1.58394	−0.791972	−	0.610557i	$-0.790946\pi$
$311$	−27.9332	−1.58394	−0.791972	+	0.610557i	$0.790946\pi$
$312$	−2.45151	−0.138790
$313$	29.1267	1.64634	0.823170	−	0.567795i	$-0.192203\pi$
$313$	29.1267	1.64634	0.823170	+	0.567795i	$0.192203\pi$
$314$	12.9583	0.731281
$315$	−0.136735	−0.00770416
$316$	−2.40955	−0.135548
$317$	10.1125	0.567974	0.283987	−	0.958828i	$-0.408343\pi$
$317$	10.1125	0.567974	0.283987	+	0.958828i	$0.408343\pi$
$318$	9.60233	0.538472
$319$	5.38261	0.301368
$320$	−0.509449	−0.0284791
$321$	−9.62663	−0.537306
$322$	2.90742	0.162024
$323$	−11.8921	−0.661693
$324$	−0.590828	−0.0328238
$325$	−4.99648	−0.277155
$326$	−7.48966	−0.414814
$327$	0.125007	0.00691291
$328$	8.26873	0.456564
$329$	8.26199	0.455498
$330$	0.0666036	0.00366641
$331$	−5.23273	−0.287617	−0.143808	−	0.989606i	$-0.545935\pi$
$331$	−5.23273	−0.287617	−0.143808	+	0.989606i	$0.545935\pi$
$332$	−0.480165	−0.0263525
$333$	0.523055	0.0286632
$334$	10.0342	0.549048
$335$	0.00321761	0.000175797 0
$336$	−3.00046	−0.163689
$337$	8.12780	0.442749	0.221375	−	0.975189i	$-0.428946\pi$
$337$	8.12780	0.442749	0.221375	+	0.975189i	$0.428946\pi$
$338$	−1.34808	−0.0733261
$339$	13.4803	0.732148
$340$	0.0214924	0.00116559
$341$	−7.70525	−0.417263
$342$	−18.6309	−1.00744
$343$	−1.00000	−0.0539949
$344$	18.3227	0.987894
$345$	0.106555	0.00573671
$346$	26.1897	1.40797
$347$	24.3455	1.30693	0.653467	−	0.756955i	$-0.273314\pi$
$347$	24.3455	1.30693	0.653467	+	0.756955i	$0.273314\pi$
$348$	−0.819192	−0.0439133
$349$	−0.485080	−0.0259657	−0.0129829	−	0.999916i	$-0.504133\pi$
$349$	−0.485080	−0.0259657	−0.0129829	+	0.999916i	$0.504133\pi$
$350$	−6.73568	−0.360037
$351$	4.42063	0.235956
$352$	1.02999	0.0548988
$353$	−0.803786	−0.0427812	−0.0213906	−	0.999771i	$-0.506809\pi$
$353$	−0.803786	−0.0427812	−0.0213906	+	0.999771i	$0.506809\pi$
$354$	4.95098	0.263142
$355$	−0.434660	−0.0230693
$356$	−0.340744	−0.0180594
$357$	1.65310	0.0874914
$358$	−1.56361	−0.0826396
$359$	12.3257	0.650527	0.325263	−	0.945623i	$-0.394547\pi$
$359$	12.3257	0.650527	0.325263	+	0.945623i	$0.394547\pi$
$360$	0.402333	0.0212048
$361$	16.9232	0.890697
$362$	−4.13268	−0.217209
$363$	−0.833162	−0.0437297
$364$	0.182668	0.00957443
$365$	−0.697611	−0.0365146
$366$	−3.88799	−0.203228
$367$	29.8215	1.55667	0.778334	−	0.627850i	$-0.216065\pi$
$367$	29.8215	1.55667	0.778334	+	0.627850i	$0.216065\pi$
$368$	−7.76694	−0.404880
$369$	−6.47983	−0.337326
$370$	−0.0181337	−0.000942726 0
$371$	−8.54929	−0.443857
$372$	1.17268	0.0608007
$373$	−2.75021	−0.142401	−0.0712003	−	0.997462i	$-0.522683\pi$
$373$	−2.75021	−0.142401	−0.0712003	+	0.997462i	$0.522683\pi$
$374$	2.67477	0.138309
$375$	−0.493888	−0.0255042
$376$	−24.3103	−1.25371
$377$	−5.38261	−0.277218
$378$	5.95938	0.306517
$379$	37.0367	1.90245	0.951224	−	0.308500i	$-0.0998270\pi$
$379$	37.0367	1.90245	0.951224	+	0.308500i	$0.0998270\pi$
$380$	−0.0649236	−0.00333051
$381$	−13.6815	−0.700927
$382$	10.2557	0.524728
$383$	15.7279	0.803660	0.401830	−	0.915714i	$-0.368374\pi$
$383$	15.7279	0.803660	0.401830	+	0.915714i	$0.368374\pi$
$384$	7.93300	0.404829
$385$	−0.0592996	−0.00302219
$386$	−25.8688	−1.31669
$387$	−14.3587	−0.729892
$388$	−2.60182	−0.132088
$389$	−14.2937	−0.724721	−0.362360	−	0.932038i	$-0.618029\pi$
$389$	−14.2937	−0.724721	−0.362360	+	0.932038i	$0.618029\pi$
$390$	−0.0666036	−0.00337261
$391$	4.27918	0.216408
$392$	2.94242	0.148615
$393$	−7.82152	−0.394543
$394$	−2.48201	−0.125042
$395$	−0.782210	−0.0393573
$396$	−0.421204	−0.0211663
$397$	−7.33362	−0.368064	−0.184032	−	0.982920i	$-0.558915\pi$
$397$	−7.33362	−0.368064	−0.184032	+	0.982920i	$0.558915\pi$
$398$	30.3528	1.52145
$399$	−4.99364	−0.249995
$400$	17.9938	0.899691
$401$	6.44045	0.321621	0.160810	−	0.986985i	$-0.448589\pi$
$401$	6.44045	0.321621	0.160810	+	0.986985i	$0.448589\pi$
$402$	−0.0609437	−0.00303959
$403$	7.70525	0.383826
$404$	0.399861	0.0198938
$405$	−0.191800	−0.00953062
$406$	−7.25621	−0.360120
$407$	0.226839	0.0112440
$408$	−4.86412	−0.240810
$409$	16.3829	0.810081	0.405040	−	0.914299i	$-0.367257\pi$
$409$	16.3829	0.810081	0.405040	+	0.914299i	$0.367257\pi$
$410$	0.224648	0.0110946
$411$	10.8627	0.535819
$412$	−1.04585	−0.0515254
$413$	−4.40803	−0.216905
$414$	6.70406	0.329486
$415$	−0.155876	−0.00765164
$416$	−1.02999	−0.0504995
$417$	−0.376912	−0.0184574
$418$	−8.07988	−0.395200
$419$	−18.2207	−0.890139	−0.445070	−	0.895496i	$-0.646821\pi$
$419$	−18.2207	−0.890139	−0.445070	+	0.895496i	$0.646821\pi$
$420$	0.00902494	0.000440372 0
$421$	34.5414	1.68344	0.841722	−	0.539911i	$-0.181542\pi$
$421$	34.5414	1.68344	0.841722	+	0.539911i	$0.181542\pi$
$422$	−20.2670	−0.986584
$423$	19.0508	0.926283
$424$	25.1556	1.22166
$425$	−9.91367	−0.480883
$426$	8.23274	0.398878
$427$	3.46161	0.167519
$428$	−2.11061	−0.102020
$429$	0.833162	0.0402254
$430$	0.497798	0.0240060
$431$	25.5547	1.23093	0.615464	−	0.788165i	$-0.288969\pi$
$431$	25.5547	1.23093	0.615464	+	0.788165i	$0.288969\pi$
$432$	−15.9200	−0.765950
$433$	−25.3711	−1.21926	−0.609628	−	0.792688i	$-0.708681\pi$
$433$	−25.3711	−1.21926	−0.609628	+	0.792688i	$0.708681\pi$
$434$	10.3873	0.498608
$435$	−0.265934	−0.0127506
$436$	0.0274075	0.00131258
$437$	−12.9264	−0.618356
$438$	13.2132	0.631352
$439$	23.0831	1.10169	0.550847	−	0.834606i	$-0.314305\pi$
$439$	23.0831	1.10169	0.550847	+	0.834606i	$0.314305\pi$
$440$	0.174484	0.00831821
$441$	−2.30584	−0.109802
$442$	−2.67477	−0.127226
$443$	12.9511	0.615324	0.307662	−	0.951496i	$-0.400453\pi$
$443$	12.9511	0.615324	0.307662	+	0.951496i	$0.400453\pi$
$444$	−0.0345232	−0.00163840
$445$	−0.110616	−0.00524368
$446$	14.2290	0.673761
$447$	−7.45448	−0.352585
$448$	−8.59111	−0.405892
$449$	−3.90370	−0.184227	−0.0921136	−	0.995749i	$-0.529362\pi$
$449$	−3.90370	−0.184227	−0.0921136	+	0.995749i	$0.529362\pi$
$450$	−15.5314	−0.732158
$451$	−2.81018	−0.132326
$452$	2.95551	0.139016
$453$	−6.58269	−0.309282
$454$	−0.994079	−0.0466544
$455$	0.0592996	0.00278001
$456$	14.6934	0.688081
$457$	−1.42795	−0.0667967	−0.0333983	−	0.999442i	$-0.510633\pi$
$457$	−1.42795	−0.0667967	−0.0333983	+	0.999442i	$0.510633\pi$
$458$	8.38083	0.391611
$459$	8.77109	0.409399
$460$	0.0233618	0.00108925
$461$	11.1857	0.520972	0.260486	−	0.965478i	$-0.416117\pi$
$461$	11.1857	0.520972	0.260486	+	0.965478i	$0.416117\pi$
$462$	1.12317	0.0522547
$463$	27.9442	1.29868	0.649338	−	0.760500i	$-0.275046\pi$
$463$	27.9442	1.29868	0.649338	+	0.760500i	$0.275046\pi$
$464$	19.3844	0.899896
$465$	0.380687	0.0176539
$466$	30.0684	1.39289
$467$	−36.1032	−1.67066	−0.835329	−	0.549750i	$-0.814723\pi$
$467$	−36.1032	−1.67066	−0.835329	+	0.549750i	$0.814723\pi$
$468$	0.421204	0.0194702
$469$	0.0542603	0.00250551
$470$	−0.660470	−0.0304652
$471$	8.00869	0.369021
$472$	12.9703	0.597006
$473$	−6.22709	−0.286322
$474$	14.8156	0.680502
$475$	29.9469	1.37406
$476$	0.362438	0.0166123
$477$	−19.7133	−0.902610
$478$	−23.0426	−1.05394
$479$	−15.0516	−0.687725	−0.343862	−	0.939020i	$-0.611735\pi$
$479$	−15.0516	−0.687725	−0.343862	+	0.939020i	$0.611735\pi$
$480$	−0.0508880	−0.00232271
$481$	−0.226839	−0.0103430
$482$	−27.8861	−1.27018
$483$	1.79689	0.0817612
$484$	−0.182668	−0.00830311
$485$	−0.844628	−0.0383526
$486$	21.5109	0.975756
$487$	25.8145	1.16977	0.584883	−	0.811118i	$-0.301140\pi$
$487$	25.8145	1.16977	0.584883	+	0.811118i	$0.301140\pi$
$488$	−10.1855	−0.461077
$489$	−4.62886	−0.209324
$490$	0.0799408	0.00361136
$491$	−13.8044	−0.622984	−0.311492	−	0.950249i	$-0.600829\pi$
$491$	−13.8044	−0.622984	−0.311492	+	0.950249i	$0.600829\pi$
$492$	0.427688	0.0192817
$493$	−10.6798	−0.480993
$494$	8.07988	0.363531
$495$	−0.136735	−0.00614580
$496$	−27.7489	−1.24596
$497$	−7.32990	−0.328791
$498$	2.95239	0.132300
$499$	25.1225	1.12464	0.562318	−	0.826921i	$-0.309910\pi$
$499$	25.1225	1.12464	0.562318	+	0.826921i	$0.309910\pi$
$500$	−0.108284	−0.00484259
$501$	6.20148	0.277062
$502$	−0.564055	−0.0251750
$503$	27.0112	1.20437	0.602185	−	0.798357i	$-0.294297\pi$
$503$	27.0112	1.20437	0.602185	+	0.798357i	$0.294297\pi$
$504$	6.78476	0.302217
$505$	0.129807	0.00577632
$506$	2.90742	0.129251
$507$	−0.833162	−0.0370020
$508$	−2.99964	−0.133088
$509$	1.29493	0.0573968	0.0286984	−	0.999588i	$-0.490864\pi$
$509$	1.29493	0.0573968	0.0286984	+	0.999588i	$0.490864\pi$
$510$	−0.132150	−0.00585171
$511$	−11.7642	−0.520417
$512$	24.9024	1.10054
$513$	−26.4955	−1.16980
$514$	20.8262	0.918605
$515$	−0.339514	−0.0149608
$516$	0.947715	0.0417208
$517$	8.26199	0.363362
$518$	−0.305798	−0.0134360
$519$	16.1861	0.710493
$520$	−0.174484	−0.00765164
$521$	−21.3929	−0.937238	−0.468619	−	0.883400i	$-0.655248\pi$
$521$	−21.3929	−0.937238	−0.468619	+	0.883400i	$0.655248\pi$
$522$	−16.7317	−0.732325
$523$	−3.92514	−0.171634	−0.0858172	−	0.996311i	$-0.527350\pi$
$523$	−3.92514	−0.171634	−0.0858172	+	0.996311i	$0.527350\pi$
$524$	−1.71485	−0.0749134
$525$	−4.16288	−0.181683
$526$	22.3039	0.972495
$527$	15.2882	0.665965
$528$	−3.00046	−0.130578
$529$	−18.3486	−0.797766
$530$	0.683437	0.0296866
$531$	−10.1642	−0.441090
$532$	−1.09484	−0.0474674
$533$	2.81018	0.121722
$534$	2.09513	0.0906652
$535$	−0.685167	−0.0296223
$536$	−0.159657	−0.00689611
$537$	−0.966367	−0.0417018
$538$	−8.04920	−0.347026
$539$	−1.00000	−0.0430730
$540$	0.0478849	0.00206064
$541$	5.65356	0.243066	0.121533	−	0.992587i	$-0.461219\pi$
$541$	5.65356	0.243066	0.121533	+	0.992587i	$0.461219\pi$
$542$	−17.7830	−0.763845
$543$	−2.55414	−0.109609
$544$	−2.04364	−0.0876203
$545$	0.00889728	0.000381117 0
$546$	−1.12317	−0.0480674
$547$	32.3114	1.38153	0.690767	−	0.723077i	$-0.257273\pi$
$547$	32.3114	1.38153	0.690767	+	0.723077i	$0.257273\pi$
$548$	2.38162	0.101738
$549$	7.98192	0.340660
$550$	−6.73568	−0.287210
$551$	32.2612	1.37437
$552$	−5.28720	−0.225038
$553$	−13.1908	−0.560931
$554$	36.6091	1.55537
$555$	−0.0112072	−0.000475721 0
$556$	−0.0826369	−0.00350458
$557$	−30.2716	−1.28265	−0.641324	−	0.767270i	$-0.721614\pi$
$557$	−30.2716	−1.28265	−0.641324	+	0.767270i	$0.721614\pi$
$558$	23.9515	1.01395
$559$	6.22709	0.263378
$560$	−0.213555	−0.00902435
$561$	1.65310	0.0697940
$562$	−18.6270	−0.785733
$563$	27.3504	1.15268	0.576341	−	0.817209i	$-0.304480\pi$
$563$	27.3504	1.15268	0.576341	+	0.817209i	$0.304480\pi$
$564$	−1.25741	−0.0529466
$565$	0.959447	0.0403642
$566$	0.577196	0.0242613
$567$	−3.23443	−0.135833
$568$	21.5676	0.904958
$569$	21.2213	0.889644	0.444822	−	0.895619i	$-0.353267\pi$
$569$	21.2213	0.889644	0.444822	+	0.895619i	$0.353267\pi$
$570$	0.399196	0.0167205
$571$	34.7001	1.45215	0.726077	−	0.687614i	$-0.241342\pi$
$571$	34.7001	1.45215	0.726077	+	0.687614i	$0.241342\pi$
$572$	0.182668	0.00763775
$573$	6.33838	0.264790
$574$	3.78836	0.158123
$575$	−10.7760	−0.449388
$576$	−19.8097	−0.825405
$577$	−34.1668	−1.42238	−0.711191	−	0.702999i	$-0.751844\pi$
$577$	−34.1668	−1.42238	−0.711191	+	0.702999i	$0.751844\pi$
$578$	17.6103	0.732494
$579$	−15.9878	−0.664431
$580$	−0.0583053	−0.00242099
$581$	−2.62861	−0.109053
$582$	15.9978	0.663131
$583$	−8.54929	−0.354076
$584$	34.6152	1.43239
$585$	0.136735	0.00565331
$586$	1.61668	0.0667845
$587$	−23.8506	−0.984419	−0.492209	−	0.870477i	$-0.663811\pi$
$587$	−23.8506	−0.984419	−0.492209	+	0.870477i	$0.663811\pi$
$588$	0.152192	0.00627631
$589$	−46.1822	−1.90290
$590$	0.352382	0.0145073
$591$	−1.53397	−0.0630990
$592$	0.816914	0.0335750
$593$	−1.85436	−0.0761492	−0.0380746	−	0.999275i	$-0.512122\pi$
$593$	−1.85436	−0.0761492	−0.0380746	+	0.999275i	$0.512122\pi$
$594$	5.95938	0.244516
$595$	0.117658	0.00482351
$596$	−1.63437	−0.0669466
$597$	18.7590	0.767757
$598$	−2.90742	−0.118893
$599$	17.3889	0.710490	0.355245	−	0.934773i	$-0.384397\pi$
$599$	17.3889	0.710490	0.355245	+	0.934773i	$0.384397\pi$
$600$	12.2489	0.500061
$601$	−4.60277	−0.187751	−0.0938755	−	0.995584i	$-0.529926\pi$
$601$	−4.60277	−0.187751	−0.0938755	+	0.995584i	$0.529926\pi$
$602$	8.39464	0.342140
$603$	0.125116	0.00509510
$604$	−1.44324	−0.0587245
$605$	−0.0592996	−0.00241087
$606$	−2.45862	−0.0998747
$607$	−4.08415	−0.165770	−0.0828852	−	0.996559i	$-0.526413\pi$
$607$	−4.08415	−0.165770	−0.0828852	+	0.996559i	$0.526413\pi$
$608$	6.17336	0.250363
$609$	−4.48458	−0.181725
$610$	−0.276724	−0.0112042
$611$	−8.26199	−0.334244
$612$	0.835724	0.0337821
$613$	5.10883	0.206344	0.103172	−	0.994664i	$-0.467101\pi$
$613$	5.10883	0.206344	0.103172	+	0.994664i	$0.467101\pi$
$614$	27.9942	1.12975
$615$	0.138840	0.00559858
$616$	2.94242	0.118554
$617$	8.45596	0.340424	0.170212	−	0.985407i	$-0.445555\pi$
$617$	8.45596	0.340424	0.170212	+	0.985407i	$0.445555\pi$
$618$	6.43062	0.258677
$619$	−39.3825	−1.58292	−0.791458	−	0.611224i	$-0.790678\pi$
$619$	−39.3825	−1.58292	−0.791458	+	0.611224i	$0.790678\pi$
$620$	0.0834645	0.00335201
$621$	9.53399	0.382586
$622$	37.6563	1.50988
$623$	−1.86537	−0.0747344
$624$	3.00046	0.120115
$625$	24.9473	0.997891
$626$	−39.2653	−1.56936
$627$	−4.99364	−0.199427
$628$	1.75588	0.0700674
$629$	−0.450078	−0.0179458
$630$	0.184331	0.00734391
$631$	0.998138	0.0397352	0.0198676	−	0.999803i	$-0.493676\pi$
$631$	0.998138	0.0397352	0.0198676	+	0.999803i	$0.493676\pi$
$632$	38.8130	1.54390
$633$	−12.5257	−0.497852
$634$	−13.6325	−0.541416
$635$	−0.973771	−0.0386429
$636$	1.30114	0.0515935
$637$	1.00000	0.0396214
$638$	−7.25621	−0.287276
$639$	−16.9016	−0.668616
$640$	0.564624	0.0223187
$641$	−22.1088	−0.873246	−0.436623	−	0.899645i	$-0.643826\pi$
$641$	−22.1088	−0.873246	−0.436623	+	0.899645i	$0.643826\pi$
$642$	12.9775	0.512181
$643$	25.5702	1.00839	0.504196	−	0.863590i	$-0.331789\pi$
$643$	25.5702	1.00839	0.504196	+	0.863590i	$0.331789\pi$
$644$	0.393962	0.0155243
$645$	0.307656	0.0121140
$646$	16.0315	0.630752
$647$	−33.0265	−1.29840	−0.649202	−	0.760616i	$-0.724897\pi$
$647$	−33.0265	−1.29840	−0.649202	+	0.760616i	$0.724897\pi$
$648$	9.51705	0.373865
$649$	−4.40803	−0.173030
$650$	6.73568	0.264195
$651$	6.41972	0.251609
$652$	−1.01487	−0.0397452
$653$	35.5221	1.39009	0.695044	−	0.718967i	$-0.255385\pi$
$653$	35.5221	1.39009	0.695044	+	0.718967i	$0.255385\pi$
$654$	−0.168520	−0.00658966
$655$	−0.556689	−0.0217517
$656$	−10.1203	−0.395131
$657$	−27.1263	−1.05830
$658$	−11.1379	−0.434199
$659$	7.41704	0.288927	0.144463	−	0.989510i	$-0.453854\pi$
$659$	7.41704	0.288927	0.144463	+	0.989510i	$0.453854\pi$
$660$	0.00902494	0.000351295 0
$661$	−14.9789	−0.582613	−0.291307	−	0.956630i	$-0.594090\pi$
$661$	−14.9789	−0.582613	−0.291307	+	0.956630i	$0.594090\pi$
$662$	7.05416	0.274168
$663$	−1.65310	−0.0642011
$664$	7.73449	0.300157
$665$	−0.355418	−0.0137825
$666$	−0.705122	−0.0273229
$667$	−11.6087	−0.449491
$668$	1.35966	0.0526068
$669$	8.79398	0.339995
$670$	−0.00433761	−0.000167576 0
$671$	3.46161	0.133634
$672$	−0.858151	−0.0331039
$673$	−34.4641	−1.32849	−0.664247	−	0.747513i	$-0.731248\pi$
$673$	−34.4641	−1.32849	−0.664247	+	0.747513i	$0.731248\pi$
$674$	−10.9570	−0.422046
$675$	−22.0876	−0.850151
$676$	−0.182668	−0.00702571
$677$	47.6028	1.82952	0.914762	−	0.403993i	$-0.132378\pi$
$677$	47.6028	1.82952	0.914762	+	0.403993i	$0.132378\pi$
$678$	−18.1726	−0.697913
$679$	−14.2434	−0.546612
$680$	−0.346199	−0.0132761
$681$	−0.614374	−0.0235429
$682$	10.3873	0.397751
$683$	11.3237	0.433290	0.216645	−	0.976250i	$-0.430489\pi$
$683$	11.3237	0.433290	0.216645	+	0.976250i	$0.430489\pi$
$684$	−2.52453	−0.0965279
$685$	0.773145	0.0295403
$686$	1.34808	0.0514701
$687$	5.17964	0.197616
$688$	−22.4256	−0.854967
$689$	8.54929	0.325702
$690$	−0.143645	−0.00546845
$691$	−29.1312	−1.10820	−0.554102	−	0.832449i	$-0.686938\pi$
$691$	−29.1312	−1.10820	−0.554102	+	0.832449i	$0.686938\pi$
$692$	3.54877	0.134904
$693$	−2.30584	−0.0875917
$694$	−32.8198	−1.24582
$695$	−0.0268264	−0.00101758
$696$	13.1955	0.500175
$697$	5.57576	0.211197
$698$	0.653929	0.0247516
$699$	18.5833	0.702884
$700$	−0.912700	−0.0344968
$701$	−42.8548	−1.61860	−0.809301	−	0.587394i	$-0.800154\pi$
$701$	−42.8548	−1.61860	−0.809301	+	0.587394i	$0.800154\pi$
$702$	−5.95938	−0.224922
$703$	1.35958	0.0512776
$704$	−8.59111	−0.323790
$705$	−0.408193	−0.0153734
$706$	1.08357	0.0407808
$707$	2.18900	0.0823257
$708$	0.670869	0.0252128
$709$	14.0772	0.528679	0.264340	−	0.964430i	$-0.414846\pi$
$709$	14.0772	0.528679	0.264340	+	0.964430i	$0.414846\pi$
$710$	0.585958	0.0219906
$711$	−30.4160	−1.14069
$712$	5.48870	0.205698
$713$	16.6180	0.622348
$714$	−2.22852	−0.0834002
$715$	0.0592996	0.00221768
$716$	−0.211873	−0.00791808
$717$	−14.2411	−0.531844
$718$	−16.6161	−0.620108
$719$	1.04951	0.0391402	0.0195701	−	0.999808i	$-0.493770\pi$
$719$	1.04951	0.0391402	0.0195701	+	0.999808i	$0.493770\pi$
$720$	−0.492424	−0.0183516
$721$	−5.72540	−0.213225
$722$	−22.8140	−0.849048
$723$	−17.2345	−0.640960
$724$	−0.559988	−0.0208118
$725$	26.8941	0.998822
$726$	1.12317	0.0416848
$727$	−25.9396	−0.962048	−0.481024	−	0.876707i	$-0.659735\pi$
$727$	−25.9396	−0.962048	−0.481024	+	0.876707i	$0.659735\pi$
$728$	−2.94242	−0.109053
$729$	3.59122	0.133008
$730$	0.940438	0.0348072
$731$	12.3553	0.456979
$732$	−0.526831	−0.0194722
$733$	−20.8505	−0.770129	−0.385065	−	0.922890i	$-0.625821\pi$
$733$	−20.8505	−0.770129	−0.385065	+	0.922890i	$0.625821\pi$
$734$	−40.2019	−1.48388
$735$	0.0494061	0.00182237
$736$	−2.22139	−0.0818816
$737$	0.0542603	0.00199870
$738$	8.73536	0.321553
$739$	22.7254	0.835967	0.417984	−	0.908455i	$-0.362737\pi$
$739$	22.7254	0.835967	0.417984	+	0.908455i	$0.362737\pi$
$740$	−0.00245716	−9.03269e−5 0
$741$	4.99364	0.183446
$742$	11.5252	0.423102
$743$	−44.9794	−1.65014	−0.825068	−	0.565034i	$-0.808863\pi$
$743$	−44.9794	−1.65014	−0.825068	+	0.565034i	$0.808863\pi$
$744$	−18.8895	−0.692523
$745$	−0.530566	−0.0194384
$746$	3.70752	0.135742
$747$	−6.06117	−0.221767
$748$	0.362438	0.0132520
$749$	−11.5543	−0.422186
$750$	0.665802	0.0243117
$751$	−13.8275	−0.504574	−0.252287	−	0.967652i	$-0.581183\pi$
$751$	−13.8275	−0.504574	−0.252287	+	0.967652i	$0.581183\pi$
$752$	29.7539	1.08501
$753$	−0.348605	−0.0127039
$754$	7.25621	0.264256
$755$	−0.468517	−0.0170511
$756$	0.807509	0.0293688
$757$	−25.9369	−0.942692	−0.471346	−	0.881948i	$-0.656232\pi$
$757$	−25.9369	−0.942692	−0.471346	+	0.881948i	$0.656232\pi$
$758$	−49.9286	−1.81349
$759$	1.79689	0.0652228
$760$	1.04579	0.0379348
$761$	47.7229	1.72996	0.864978	−	0.501811i	$-0.167333\pi$
$761$	47.7229	1.72996	0.864978	+	0.501811i	$0.167333\pi$
$762$	18.4439	0.668151
$763$	0.150039	0.00543179
$764$	1.38967	0.0502766
$765$	0.271301	0.00980889
$766$	−21.2026	−0.766080
$767$	4.40803	0.159165
$768$	3.62122	0.130670
$769$	5.31525	0.191673	0.0958364	−	0.995397i	$-0.469447\pi$
$769$	5.31525	0.191673	0.0958364	+	0.995397i	$0.469447\pi$
$770$	0.0799408	0.00288087
$771$	12.8713	0.463549
$772$	−3.50529	−0.126158
$773$	12.1158	0.435777	0.217888	−	0.975974i	$-0.430083\pi$
$773$	12.1158	0.435777	0.217888	+	0.975974i	$0.430083\pi$
$774$	19.3567	0.695762
$775$	−38.4991	−1.38293
$776$	41.9101	1.50449
$777$	−0.188994	−0.00678011
$778$	19.2692	0.690833
$779$	−16.8431	−0.603467
$780$	−0.00902494	−0.000323145 0
$781$	−7.32990	−0.262284
$782$	−5.76870	−0.206288
$783$	−23.7945	−0.850346
$784$	−3.60130	−0.128618
$785$	0.570011	0.0203446
$786$	10.5441	0.376094
$787$	21.8154	0.777634	0.388817	−	0.921315i	$-0.372884\pi$
$787$	21.8154	0.777634	0.388817	+	0.921315i	$0.372884\pi$
$788$	−0.336318	−0.0119808
$789$	13.7845	0.490743
$790$	1.05449	0.0375169
$791$	16.1797	0.575282
$792$	6.78476	0.241086
$793$	−3.46161	−0.122925
$794$	9.88634	0.350853
$795$	0.422388	0.0149805
$796$	4.11287	0.145777
$797$	−12.9601	−0.459071	−0.229535	−	0.973300i	$-0.573721\pi$
$797$	−12.9601	−0.459071	−0.229535	+	0.973300i	$0.573721\pi$
$798$	6.73185	0.238305
$799$	−16.3929	−0.579937
$800$	5.14634	0.181951
$801$	−4.30124	−0.151977
$802$	−8.68227	−0.306582
$803$	−11.7642	−0.415149
$804$	−0.00825800	−0.000291237 0
$805$	0.127892	0.00450759
$806$	−10.3873	−0.365878
$807$	−4.97468	−0.175117
$808$	−6.44095	−0.226592
$809$	−6.59273	−0.231788	−0.115894	−	0.993262i	$-0.536973\pi$
$809$	−6.59273	−0.231788	−0.115894	+	0.993262i	$0.536973\pi$
$810$	0.258563	0.00908497
$811$	−27.5152	−0.966191	−0.483095	−	0.875568i	$-0.660488\pi$
$811$	−27.5152	−0.966191	−0.483095	+	0.875568i	$0.660488\pi$
$812$	−0.983233	−0.0345047
$813$	−10.9905	−0.385454
$814$	−0.305798	−0.0107182
$815$	−0.329455	−0.0115403
$816$	5.95330	0.208407
$817$	−37.3227	−1.30575
$818$	−22.0855	−0.772201
$819$	2.30584	0.0805726
$820$	0.0304403	0.00106302
$821$	46.0908	1.60858	0.804290	−	0.594238i	$-0.202546\pi$
$821$	46.0908	1.60858	0.804290	+	0.594238i	$0.202546\pi$
$822$	−14.6439	−0.510764
$823$	37.5798	1.30995	0.654974	−	0.755651i	$-0.272680\pi$
$823$	37.5798	1.30995	0.654974	+	0.755651i	$0.272680\pi$
$824$	16.8465	0.586877
$825$	−4.16288	−0.144933
$826$	5.94240	0.206763
$827$	−40.1987	−1.39785	−0.698923	−	0.715197i	$-0.746337\pi$
$827$	−40.1987	−1.39785	−0.698923	+	0.715197i	$0.746337\pi$
$828$	0.908415	0.0315696
$829$	−33.9940	−1.18066	−0.590331	−	0.807162i	$-0.701003\pi$
$829$	−33.9940	−1.18066	−0.590331	+	0.807162i	$0.701003\pi$
$830$	0.210134	0.00729384
$831$	22.6257	0.784876
$832$	8.59111	0.297843
$833$	1.98413	0.0687460
$834$	0.508109	0.0175944
$835$	0.441385	0.0152748
$836$	−1.09484	−0.0378659
$837$	34.0620	1.17736
$838$	24.5630	0.848516
$839$	7.32502	0.252888	0.126444	−	0.991974i	$-0.459644\pi$
$839$	7.32502	0.252888	0.126444	+	0.991974i	$0.459644\pi$
$840$	−0.145374	−0.00501587
$841$	−0.0275347	−0.000949473 0
$842$	−46.5647	−1.60473
$843$	−11.5121	−0.396499
$844$	−2.74623	−0.0945291
$845$	−0.0592996	−0.00203997
$846$	−25.6821	−0.882970
$847$	−1.00000	−0.0343604
$848$	−30.7885	−1.05728
$849$	0.356727	0.0122428
$850$	13.3645	0.458397
$851$	−0.489225	−0.0167704
$852$	1.11555	0.0382183
$853$	−50.3712	−1.72468	−0.862339	−	0.506331i	$-0.831001\pi$
$853$	−50.3712	−1.72468	−0.862339	+	0.506331i	$0.831001\pi$
$854$	−4.66654	−0.159686
$855$	−0.819537	−0.0280276
$856$	33.9977	1.16202
$857$	16.5549	0.565504	0.282752	−	0.959193i	$-0.408753\pi$
$857$	16.5549	0.565504	0.282752	+	0.959193i	$0.408753\pi$
$858$	−1.12317	−0.0383445
$859$	−27.8396	−0.949875	−0.474938	−	0.880019i	$-0.657529\pi$
$859$	−27.8396	−0.949875	−0.474938	+	0.880019i	$0.657529\pi$
$860$	0.0674528	0.00230012
$861$	2.34134	0.0797925
$862$	−34.4499	−1.17337
$863$	35.7355	1.21645	0.608226	−	0.793764i	$-0.291881\pi$
$863$	35.7355	1.21645	0.608226	+	0.793764i	$0.291881\pi$
$864$	−4.55321	−0.154903
$865$	1.15203	0.0391703
$866$	34.2024	1.16224
$867$	10.8838	0.369633
$868$	1.40751	0.0477739
$869$	−13.1908	−0.447468
$870$	0.358501	0.0121543
$871$	−0.0542603	−0.00183854
$872$	−0.441479	−0.0149504
$873$	−32.8431	−1.11157
$874$	17.4259	0.589441
$875$	−0.592787	−0.0200399
$876$	1.79042	0.0604927
$877$	30.0425	1.01446	0.507232	−	0.861810i	$-0.330669\pi$
$877$	30.0425	1.01446	0.507232	+	0.861810i	$0.330669\pi$
$878$	−31.1179	−1.05018
$879$	0.999164	0.0337009
$880$	−0.213555	−0.00719894
$881$	−52.2659	−1.76088	−0.880441	−	0.474156i	$-0.842753\pi$
$881$	−52.2659	−1.76088	−0.880441	+	0.474156i	$0.842753\pi$
$882$	3.10847	0.104668
$883$	−1.09232	−0.0367595	−0.0183798	−	0.999831i	$-0.505851\pi$
$883$	−1.09232	−0.0367595	−0.0183798	+	0.999831i	$0.505851\pi$
$884$	−0.362438	−0.0121901
$885$	0.217784	0.00732072
$886$	−17.4591	−0.586551
$887$	54.4452	1.82809	0.914046	−	0.405612i	$-0.132941\pi$
$887$	54.4452	1.82809	0.914046	+	0.405612i	$0.132941\pi$
$888$	0.556099	0.0186615
$889$	−16.4212	−0.550750
$890$	0.149119	0.00499848
$891$	−3.23443	−0.108357
$892$	1.92806	0.0645561
$893$	49.5191	1.65709
$894$	10.0493	0.336098
$895$	−0.0687803	−0.00229907
$896$	9.52155	0.318093
$897$	−1.79689	−0.0599963
$898$	5.26252	0.175613
$899$	−41.4743	−1.38325
$900$	−2.10454	−0.0701514
$901$	16.9629	0.565116
$902$	3.78836	0.126139
$903$	5.18817	0.172652
$904$	−47.6074	−1.58340
$905$	−0.181788	−0.00604285
$906$	8.87402	0.294820
$907$	4.81333	0.159824	0.0799119	−	0.996802i	$-0.474536\pi$
$907$	4.81333	0.159824	0.0799119	+	0.996802i	$0.474536\pi$
$908$	−0.134700	−0.00447017
$909$	5.04748	0.167414
$910$	−0.0799408	−0.00265001
$911$	−8.36851	−0.277261	−0.138631	−	0.990344i	$-0.544270\pi$
$911$	−8.36851	−0.277261	−0.138631	+	0.990344i	$0.544270\pi$
$912$	−17.9836	−0.595496
$913$	−2.62861	−0.0869945
$914$	1.92500	0.0636732
$915$	−0.171025	−0.00565390
$916$	1.13562	0.0375220
$917$	−9.38775	−0.310011
$918$	−11.8242	−0.390256
$919$	−18.0266	−0.594642	−0.297321	−	0.954778i	$-0.596093\pi$
$919$	−18.0266	−0.594642	−0.297321	+	0.954778i	$0.596093\pi$
$920$	−0.376312	−0.0124066
$921$	17.3014	0.570099
$922$	−15.0793	−0.496611
$923$	7.32990	0.241266
$924$	0.152192	0.00500676
$925$	1.13340	0.0372659
$926$	−37.6711	−1.23795
$927$	−13.2019	−0.433606
$928$	5.54405	0.181992
$929$	12.1322	0.398046	0.199023	−	0.979995i	$-0.436223\pi$
$929$	12.1322	0.398046	0.199023	+	0.979995i	$0.436223\pi$
$930$	−0.513198	−0.0168284
$931$	−5.99360	−0.196432
$932$	4.07433	0.133459
$933$	23.2729	0.761919
$934$	48.6702	1.59254
$935$	0.117658	0.00384783
$936$	−6.78476	−0.221767
$937$	−50.6508	−1.65469	−0.827345	−	0.561695i	$-0.810150\pi$
$937$	−50.6508	−1.65469	−0.827345	+	0.561695i	$0.810150\pi$
$938$	−0.0731474	−0.00238835
$939$	−24.2673	−0.791933
$940$	−0.0894952	−0.00291901
$941$	−44.6840	−1.45666	−0.728329	−	0.685228i	$-0.759703\pi$
$941$	−44.6840	−1.45666	−0.728329	+	0.685228i	$0.759703\pi$
$942$	−10.7964	−0.351765
$943$	6.06074	0.197365
$944$	−15.8746	−0.516675
$945$	0.262141	0.00852745
$946$	8.39464	0.272933
$947$	−47.8664	−1.55545	−0.777724	−	0.628605i	$-0.783626\pi$
$947$	−47.8664	−1.55545	−0.777724	+	0.628605i	$0.783626\pi$
$948$	2.00754	0.0652020
$949$	11.7642	0.381882
$950$	−40.3710	−1.30981
$951$	−8.42535	−0.273211
$952$	−5.83814	−0.189215
$953$	17.2364	0.558342	0.279171	−	0.960241i	$-0.409940\pi$
$953$	17.2364	0.558342	0.279171	+	0.960241i	$0.409940\pi$
$954$	26.5752	0.860404
$955$	0.451129	0.0145982
$956$	−3.12233	−0.100983
$957$	−4.48458	−0.144966
$958$	20.2908	0.655566
$959$	13.0380	0.421018
$960$	0.424453	0.0136992
$961$	28.3709	0.915189
$962$	0.305798	0.00985933
$963$	−26.6424	−0.858541
$964$	−3.77863	−0.121701
$965$	−1.13792	−0.0366309
$966$	−2.42235	−0.0779380
$967$	−9.14174	−0.293979	−0.146989	−	0.989138i	$-0.546958\pi$
$967$	−9.14174	−0.293979	−0.146989	+	0.989138i	$0.546958\pi$
$968$	2.94242	0.0945730
$969$	9.90802	0.318292
$970$	1.13863	0.0365592
$971$	19.5248	0.626581	0.313290	−	0.949657i	$-0.398569\pi$
$971$	19.5248	0.626581	0.313290	+	0.949657i	$0.398569\pi$
$972$	2.91478	0.0934917
$973$	−0.452387	−0.0145029
$974$	−34.8001	−1.11507
$975$	4.16288	0.133319
$976$	12.4663	0.399036
$977$	−5.22495	−0.167161	−0.0835804	−	0.996501i	$-0.526636\pi$
$977$	−5.22495	−0.167161	−0.0835804	+	0.996501i	$0.526636\pi$
$978$	6.24010	0.199536
$979$	−1.86537	−0.0596175
$980$	0.0108322	0.000346021 0
$981$	0.345967	0.0110459
$982$	18.6095	0.593853
$983$	−9.92859	−0.316673	−0.158336	−	0.987385i	$-0.550613\pi$
$983$	−9.92859	−0.316673	−0.158336	+	0.987385i	$0.550613\pi$
$984$	−6.88920	−0.219620
$985$	−0.109179	−0.00347873
$986$	14.3973	0.458502
$987$	−6.88358	−0.219107
$988$	1.09484	0.0348316
$989$	13.4300	0.427049
$990$	0.184331	0.00585842
$991$	27.9472	0.887771	0.443886	−	0.896083i	$-0.353600\pi$
$991$	27.9472	0.887771	0.443886	+	0.896083i	$0.353600\pi$
$992$	−7.93635	−0.251979
$993$	4.35971	0.138351
$994$	9.88132	0.313416
$995$	1.33516	0.0423274
$996$	0.400055	0.0126762
$997$	−38.4531	−1.21782	−0.608911	−	0.793238i	$-0.708394\pi$
$997$	−38.4531	−1.21782	−0.608911	+	0.793238i	$0.708394\pi$
$998$	−33.8672	−1.07205
$999$	−1.00277	−0.0317263

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1001.2.a.m.1.3	✓	8
3.2	odd	2		9009.2.a.bm.1.6		8
7.6	odd	2		7007.2.a.t.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1001.2.a.m.1.3	✓	8	1.1	even	1	trivial
7007.2.a.t.1.3		8	7.6	odd	2
9009.2.a.bm.1.6		8	3.2	odd	2

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:	\( N \)	\(=\)	\( 1001 = 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1001.a (trivial)

\(f(q)\)	\(=\)	\(q-1.34808 q^{2} -0.833162 q^{3} -0.182668 q^{4} -0.0592996 q^{5} +1.12317 q^{6} -1.00000 q^{7} +2.94242 q^{8} -2.30584 q^{9} +O(q^{10})\)	\(q-1.34808 q^{2} -0.833162 q^{3} -0.182668 q^{4} -0.0592996 q^{5} +1.12317 q^{6} -1.00000 q^{7} +2.94242 q^{8} -2.30584 q^{9} +0.0799408 q^{10} -1.00000 q^{11} +0.152192 q^{12} +1.00000 q^{13} +1.34808 q^{14} +0.0494061 q^{15} -3.60130 q^{16} +1.98413 q^{17} +3.10847 q^{18} -5.99360 q^{19} +0.0108322 q^{20} +0.833162 q^{21} +1.34808 q^{22} +2.15671 q^{23} -2.45151 q^{24} -4.99648 q^{25} -1.34808 q^{26} +4.42063 q^{27} +0.182668 q^{28} -5.38261 q^{29} -0.0666036 q^{30} +7.70525 q^{31} -1.02999 q^{32} +0.833162 q^{33} -2.67477 q^{34} +0.0592996 q^{35} +0.421204 q^{36} -0.226839 q^{37} +8.07988 q^{38} -0.833162 q^{39} -0.174484 q^{40} +2.81018 q^{41} -1.12317 q^{42} +6.22709 q^{43} +0.182668 q^{44} +0.136735 q^{45} -2.90742 q^{46} -8.26199 q^{47} +3.00046 q^{48} +1.00000 q^{49} +6.73568 q^{50} -1.65310 q^{51} -0.182668 q^{52} +8.54929 q^{53} -5.95938 q^{54} +0.0592996 q^{55} -2.94242 q^{56} +4.99364 q^{57} +7.25621 q^{58} +4.40803 q^{59} -0.00902494 q^{60} -3.46161 q^{61} -10.3873 q^{62} +2.30584 q^{63} +8.59111 q^{64} -0.0592996 q^{65} -1.12317 q^{66} -0.0542603 q^{67} -0.362438 q^{68} -1.79689 q^{69} -0.0799408 q^{70} +7.32990 q^{71} -6.78476 q^{72} +11.7642 q^{73} +0.305798 q^{74} +4.16288 q^{75} +1.09484 q^{76} +1.00000 q^{77} +1.12317 q^{78} +13.1908 q^{79} +0.213555 q^{80} +3.23443 q^{81} -3.78836 q^{82} +2.62861 q^{83} -0.152192 q^{84} -0.117658 q^{85} -8.39464 q^{86} +4.48458 q^{87} -2.94242 q^{88} +1.86537 q^{89} -0.184331 q^{90} -1.00000 q^{91} -0.393962 q^{92} -6.41972 q^{93} +11.1379 q^{94} +0.355418 q^{95} +0.858151 q^{96} +14.2434 q^{97} -1.34808 q^{98} +2.30584 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} - q^{3} + 12 q^{4} - 5 q^{5} + q^{6} - 8 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) 8 * q + 2 * q^2 - q^3 + 12 * q^4 - 5 * q^5 + q^6 - 8 * q^7 + 9 * q^8 + 3 * q^9	\( 8 q + 2 q^{2} - q^{3} + 12 q^{4} - 5 q^{5} + q^{6} - 8 q^{7} + 9 q^{8} + 3 q^{9} - q^{10} - 8 q^{11} + q^{12} + 8 q^{13} - 2 q^{14} + 13 q^{15} + 16 q^{16} - 5 q^{17} + 12 q^{18} + 31 q^{19} - 8 q^{20} + q^{21} - 2 q^{22} + q^{23} + 35 q^{24} + 11 q^{25} + 2 q^{26} - 13 q^{27} - 12 q^{28} - 7 q^{29} + 24 q^{30} + 20 q^{31} + 10 q^{32} + q^{33} + 7 q^{34} + 5 q^{35} + 23 q^{36} + 18 q^{37} + 10 q^{38} - q^{39} - 16 q^{40} + 2 q^{41} - q^{42} + 5 q^{43} - 12 q^{44} - 27 q^{45} + 28 q^{46} - 11 q^{47} + 16 q^{48} + 8 q^{49} + 49 q^{50} - q^{51} + 12 q^{52} - 2 q^{53} - 80 q^{54} + 5 q^{55} - 9 q^{56} + 10 q^{57} - 36 q^{58} - 13 q^{59} + 39 q^{60} + 10 q^{61} + 16 q^{62} - 3 q^{63} + 31 q^{64} - 5 q^{65} - q^{66} + 24 q^{67} - 46 q^{68} - 24 q^{69} + q^{70} + 30 q^{71} + 9 q^{72} - 16 q^{73} - 7 q^{74} - 32 q^{75} + 48 q^{76} + 8 q^{77} + q^{78} + 29 q^{79} - 40 q^{80} + 16 q^{81} - 30 q^{82} + 50 q^{83} - q^{84} + 41 q^{85} + 32 q^{86} + 33 q^{87} - 9 q^{88} - 9 q^{89} - 82 q^{90} - 8 q^{91} + 18 q^{92} - 13 q^{93} + 6 q^{94} - 7 q^{95} + 76 q^{96} - 18 q^{97} + 2 q^{98} - 3 q^{99}+O(q^{100}) \) 8 * q + 2 * q^2 - q^3 + 12 * q^4 - 5 * q^5 + q^6 - 8 * q^7 + 9 * q^8 + 3 * q^9 - q^10 - 8 * q^11 + q^12 + 8 * q^13 - 2 * q^14 + 13 * q^15 + 16 * q^16 - 5 * q^17 + 12 * q^18 + 31 * q^19 - 8 * q^20 + q^21 - 2 * q^22 + q^23 + 35 * q^24 + 11 * q^25 + 2 * q^26 - 13 * q^27 - 12 * q^28 - 7 * q^29 + 24 * q^30 + 20 * q^31 + 10 * q^32 + q^33 + 7 * q^34 + 5 * q^35 + 23 * q^36 + 18 * q^37 + 10 * q^38 - q^39 - 16 * q^40 + 2 * q^41 - q^42 + 5 * q^43 - 12 * q^44 - 27 * q^45 + 28 * q^46 - 11 * q^47 + 16 * q^48 + 8 * q^49 + 49 * q^50 - q^51 + 12 * q^52 - 2 * q^53 - 80 * q^54 + 5 * q^55 - 9 * q^56 + 10 * q^57 - 36 * q^58 - 13 * q^59 + 39 * q^60 + 10 * q^61 + 16 * q^62 - 3 * q^63 + 31 * q^64 - 5 * q^65 - q^66 + 24 * q^67 - 46 * q^68 - 24 * q^69 + q^70 + 30 * q^71 + 9 * q^72 - 16 * q^73 - 7 * q^74 - 32 * q^75 + 48 * q^76 + 8 * q^77 + q^78 + 29 * q^79 - 40 * q^80 + 16 * q^81 - 30 * q^82 + 50 * q^83 - q^84 + 41 * q^85 + 32 * q^86 + 33 * q^87 - 9 * q^88 - 9 * q^89 - 82 * q^90 - 8 * q^91 + 18 * q^92 - 13 * q^93 + 6 * q^94 - 7 * q^95 + 76 * q^96 - 18 * q^97 + 2 * q^98 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more