LMFDB - Embedded newform 8007.2.a.j.1.64

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, 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("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.9362168984$
Analytic rank:	$0$
Dimension:	$64$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.64
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.78814 q^{2} -1.00000 q^{3} +5.77371 q^{4} +3.79788 q^{5} -2.78814 q^{6} -2.58361 q^{7} +10.5216 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.78814 q^{2} -1.00000 q^{3} +5.77371 q^{4} +3.79788 q^{5} -2.78814 q^{6} -2.58361 q^{7} +10.5216 q^{8} +1.00000 q^{9} +10.5890 q^{10} +0.664575 q^{11} -5.77371 q^{12} +6.95532 q^{13} -7.20346 q^{14} -3.79788 q^{15} +17.7883 q^{16} +1.00000 q^{17} +2.78814 q^{18} -2.85900 q^{19} +21.9279 q^{20} +2.58361 q^{21} +1.85293 q^{22} -2.47830 q^{23} -10.5216 q^{24} +9.42389 q^{25} +19.3924 q^{26} -1.00000 q^{27} -14.9170 q^{28} -0.211880 q^{29} -10.5890 q^{30} -3.22564 q^{31} +28.5530 q^{32} -0.664575 q^{33} +2.78814 q^{34} -9.81224 q^{35} +5.77371 q^{36} -2.00843 q^{37} -7.97129 q^{38} -6.95532 q^{39} +39.9599 q^{40} +3.13177 q^{41} +7.20346 q^{42} -3.66634 q^{43} +3.83707 q^{44} +3.79788 q^{45} -6.90983 q^{46} -0.211833 q^{47} -17.7883 q^{48} -0.324967 q^{49} +26.2751 q^{50} -1.00000 q^{51} +40.1580 q^{52} -11.9231 q^{53} -2.78814 q^{54} +2.52398 q^{55} -27.1838 q^{56} +2.85900 q^{57} -0.590750 q^{58} -6.48686 q^{59} -21.9279 q^{60} -3.24254 q^{61} -8.99354 q^{62} -2.58361 q^{63} +44.0331 q^{64} +26.4155 q^{65} -1.85293 q^{66} +2.82490 q^{67} +5.77371 q^{68} +2.47830 q^{69} -27.3579 q^{70} -10.2034 q^{71} +10.5216 q^{72} -8.75240 q^{73} -5.59978 q^{74} -9.42389 q^{75} -16.5070 q^{76} -1.71700 q^{77} -19.3924 q^{78} +6.49333 q^{79} +67.5579 q^{80} +1.00000 q^{81} +8.73180 q^{82} +3.66616 q^{83} +14.9170 q^{84} +3.79788 q^{85} -10.2223 q^{86} +0.211880 q^{87} +6.99241 q^{88} -6.87978 q^{89} +10.5890 q^{90} -17.9698 q^{91} -14.3090 q^{92} +3.22564 q^{93} -0.590620 q^{94} -10.8581 q^{95} -28.5530 q^{96} -9.30948 q^{97} -0.906053 q^{98} +0.664575 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 64 q + 5 q^{2} - 64 q^{3} + 77 q^{4} - 3 q^{5} - 5 q^{6} + 5 q^{7} + 18 q^{8} + 64 q^{9}+O(q^{10}) $ 64 * q + 5 * q^2 - 64 * q^3 + 77 * q^4 - 3 * q^5 - 5 * q^6 + 5 * q^7 + 18 * q^8 + 64 * q^9	\( 64 q + 5 q^{2} - 64 q^{3} + 77 q^{4} - 3 q^{5} - 5 q^{6} + 5 q^{7} + 18 q^{8} + 64 q^{9} + 12 q^{10} - 7 q^{11} - 77 q^{12} + 24 q^{13} - 14 q^{14} + 3 q^{15} + 103 q^{16} + 64 q^{17} + 5 q^{18} + 26 q^{19} - 24 q^{20} - 5 q^{21} + 25 q^{22} + 20 q^{23} - 18 q^{24} + 141 q^{25} + 9 q^{26} - 64 q^{27} + 14 q^{28} + 5 q^{29} - 12 q^{30} + 11 q^{31} + 31 q^{32} + 7 q^{33} + 5 q^{34} - 3 q^{35} + 77 q^{36} + 50 q^{37} + 8 q^{38} - 24 q^{39} + 28 q^{40} - 9 q^{41} + 14 q^{42} + 59 q^{43} - 6 q^{44} - 3 q^{45} + 11 q^{47} - 103 q^{48} + 163 q^{49} + 20 q^{50} - 64 q^{51} + 65 q^{52} + 39 q^{53} - 5 q^{54} + 35 q^{55} - 34 q^{56} - 26 q^{57} - 27 q^{58} - 65 q^{59} + 24 q^{60} + 15 q^{61} + 18 q^{62} + 5 q^{63} + 152 q^{64} + 49 q^{65} - 25 q^{66} + 56 q^{67} + 77 q^{68} - 20 q^{69} + 28 q^{70} - 18 q^{71} + 18 q^{72} + 37 q^{73} - 76 q^{74} - 141 q^{75} + 30 q^{76} + 80 q^{77} - 9 q^{78} + 20 q^{79} - 144 q^{80} + 64 q^{81} + 27 q^{82} + 3 q^{83} - 14 q^{84} - 3 q^{85} + 12 q^{86} - 5 q^{87} + 108 q^{88} + 42 q^{89} + 12 q^{90} + 25 q^{91} + 18 q^{92} - 11 q^{93} + 60 q^{94} + 42 q^{95} - 31 q^{96} + 72 q^{97} + 18 q^{98} - 7 q^{99}+O(q^{100}) \) 64 * q + 5 * q^2 - 64 * q^3 + 77 * q^4 - 3 * q^5 - 5 * q^6 + 5 * q^7 + 18 * q^8 + 64 * q^9 + 12 * q^10 - 7 * q^11 - 77 * q^12 + 24 * q^13 - 14 * q^14 + 3 * q^15 + 103 * q^16 + 64 * q^17 + 5 * q^18 + 26 * q^19 - 24 * q^20 - 5 * q^21 + 25 * q^22 + 20 * q^23 - 18 * q^24 + 141 * q^25 + 9 * q^26 - 64 * q^27 + 14 * q^28 + 5 * q^29 - 12 * q^30 + 11 * q^31 + 31 * q^32 + 7 * q^33 + 5 * q^34 - 3 * q^35 + 77 * q^36 + 50 * q^37 + 8 * q^38 - 24 * q^39 + 28 * q^40 - 9 * q^41 + 14 * q^42 + 59 * q^43 - 6 * q^44 - 3 * q^45 + 11 * q^47 - 103 * q^48 + 163 * q^49 + 20 * q^50 - 64 * q^51 + 65 * q^52 + 39 * q^53 - 5 * q^54 + 35 * q^55 - 34 * q^56 - 26 * q^57 - 27 * q^58 - 65 * q^59 + 24 * q^60 + 15 * q^61 + 18 * q^62 + 5 * q^63 + 152 * q^64 + 49 * q^65 - 25 * q^66 + 56 * q^67 + 77 * q^68 - 20 * q^69 + 28 * q^70 - 18 * q^71 + 18 * q^72 + 37 * q^73 - 76 * q^74 - 141 * q^75 + 30 * q^76 + 80 * q^77 - 9 * q^78 + 20 * q^79 - 144 * q^80 + 64 * q^81 + 27 * q^82 + 3 * q^83 - 14 * q^84 - 3 * q^85 + 12 * q^86 - 5 * q^87 + 108 * q^88 + 42 * q^89 + 12 * q^90 + 25 * q^91 + 18 * q^92 - 11 * q^93 + 60 * q^94 + 42 * q^95 - 31 * q^96 + 72 * q^97 + 18 * q^98 - 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.78814	1.97151	0.985756	−	0.168185i	$-0.0537905\pi$
$2$	2.78814	1.97151	0.985756	+	0.168185i	$0.0537905\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.77371	2.88686
$5$	3.79788	1.69846	0.849232	−	0.528020i	$-0.177065\pi$
$5$	3.79788	1.69846	0.849232	+	0.528020i	$0.177065\pi$
$6$	−2.78814	−1.13825
$7$	−2.58361	−0.976512	−0.488256	−	0.872700i	$-0.662367\pi$
$7$	−2.58361	−0.976512	−0.488256	+	0.872700i	$0.662367\pi$
$8$	10.5216	3.71996
$9$	1.00000	0.333333
$10$	10.5890	3.34854
$11$	0.664575	0.200377	0.100189	−	0.994968i	$-0.468055\pi$
$11$	0.664575	0.200377	0.100189	+	0.994968i	$0.468055\pi$
$12$	−5.77371	−1.66673
$13$	6.95532	1.92906	0.964529	−	0.263978i	$-0.0850346\pi$
$13$	6.95532	1.92906	0.964529	+	0.263978i	$0.0850346\pi$
$14$	−7.20346	−1.92520
$15$	−3.79788	−0.980608
$16$	17.7883	4.44708
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	2.78814	0.657170
$19$	−2.85900	−0.655900	−0.327950	−	0.944695i	$-0.606358\pi$
$19$	−2.85900	−0.655900	−0.327950	+	0.944695i	$0.606358\pi$
$20$	21.9279	4.90322
$21$	2.58361	0.563790
$22$	1.85293	0.395045
$23$	−2.47830	−0.516761	−0.258380	−	0.966043i	$-0.583189\pi$
$23$	−2.47830	−0.516761	−0.258380	+	0.966043i	$0.583189\pi$
$24$	−10.5216	−2.14772
$25$	9.42389	1.88478
$26$	19.3924	3.80316
$27$	−1.00000	−0.192450
$28$	−14.9170	−2.81905
$29$	−0.211880	−0.0393451	−0.0196726	−	0.999806i	$-0.506262\pi$
$29$	−0.211880	−0.0393451	−0.0196726	+	0.999806i	$0.506262\pi$
$30$	−10.5890	−1.93328
$31$	−3.22564	−0.579343	−0.289671	−	0.957126i	$-0.593546\pi$
$31$	−3.22564	−0.579343	−0.289671	+	0.957126i	$0.593546\pi$
$32$	28.5530	5.04751
$33$	−0.664575	−0.115688
$34$	2.78814	0.478162
$35$	−9.81224	−1.65857
$36$	5.77371	0.962285
$37$	−2.00843	−0.330184	−0.165092	−	0.986278i	$-0.552792\pi$
$37$	−2.00843	−0.330184	−0.165092	+	0.986278i	$0.552792\pi$
$38$	−7.97129	−1.29311
$39$	−6.95532	−1.11374
$40$	39.9599	6.31821
$41$	3.13177	0.489100	0.244550	−	0.969637i	$-0.421360\pi$
$41$	3.13177	0.489100	0.244550	+	0.969637i	$0.421360\pi$
$42$	7.20346	1.11152
$43$	−3.66634	−0.559111	−0.279556	−	0.960130i	$-0.590187\pi$
$43$	−3.66634	−0.559111	−0.279556	+	0.960130i	$0.590187\pi$
$44$	3.83707	0.578460
$45$	3.79788	0.566155
$46$	−6.90983	−1.01880
$47$	−0.211833	−0.0308990	−0.0154495	−	0.999881i	$-0.504918\pi$
$47$	−0.211833	−0.0308990	−0.0154495	+	0.999881i	$0.504918\pi$
$48$	−17.7883	−2.56752
$49$	−0.324967	−0.0464239
$50$	26.2751	3.71586
$51$	−1.00000	−0.140028
$52$	40.1580	5.56891
$53$	−11.9231	−1.63777	−0.818883	−	0.573960i	$-0.805406\pi$
$53$	−11.9231	−1.63777	−0.818883	+	0.573960i	$0.805406\pi$
$54$	−2.78814	−0.379417
$55$	2.52398	0.340333
$56$	−27.1838	−3.63258
$57$	2.85900	0.378684
$58$	−0.590750	−0.0775693
$59$	−6.48686	−0.844518	−0.422259	−	0.906475i	$-0.638763\pi$
$59$	−6.48686	−0.844518	−0.422259	+	0.906475i	$0.638763\pi$
$60$	−21.9279	−2.83088
$61$	−3.24254	−0.415164	−0.207582	−	0.978218i	$-0.566559\pi$
$61$	−3.24254	−0.415164	−0.207582	+	0.978218i	$0.566559\pi$
$62$	−8.99354	−1.14218
$63$	−2.58361	−0.325504
$64$	44.0331	5.50414
$65$	26.4155	3.27643
$66$	−1.85293	−0.228080
$67$	2.82490	0.345117	0.172558	−	0.984999i	$-0.444797\pi$
$67$	2.82490	0.345117	0.172558	+	0.984999i	$0.444797\pi$
$68$	5.77371	0.700165
$69$	2.47830	0.298352
$70$	−27.3579	−3.26989
$71$	−10.2034	−1.21092	−0.605458	−	0.795877i	$-0.707010\pi$
$71$	−10.2034	−1.21092	−0.605458	+	0.795877i	$0.707010\pi$
$72$	10.5216	1.23999
$73$	−8.75240	−1.02439	−0.512195	−	0.858869i	$-0.671168\pi$
$73$	−8.75240	−1.02439	−0.512195	+	0.858869i	$0.671168\pi$
$74$	−5.59978	−0.650961
$75$	−9.42389	−1.08818
$76$	−16.5070	−1.89349
$77$	−1.71700	−0.195671
$78$	−19.3924	−2.19575
$79$	6.49333	0.730556	0.365278	−	0.930898i	$-0.380974\pi$
$79$	6.49333	0.730556	0.365278	+	0.930898i	$0.380974\pi$
$80$	67.5579	7.55320
$81$	1.00000	0.111111
$82$	8.73180	0.964266
$83$	3.66616	0.402414	0.201207	−	0.979549i	$-0.435514\pi$
$83$	3.66616	0.402414	0.201207	+	0.979549i	$0.435514\pi$
$84$	14.9170	1.62758
$85$	3.79788	0.411938
$86$	−10.2223	−1.10229
$87$	0.211880	0.0227159
$88$	6.99241	0.745394
$89$	−6.87978	−0.729255	−0.364628	−	0.931153i	$-0.618804\pi$
$89$	−6.87978	−0.729255	−0.364628	+	0.931153i	$0.618804\pi$
$90$	10.5890	1.11618
$91$	−17.9698	−1.88375
$92$	−14.3090	−1.49181
$93$	3.22564	0.334484
$94$	−0.590620	−0.0609178
$95$	−10.8581	−1.11402
$96$	−28.5530	−2.91418
$97$	−9.30948	−0.945234	−0.472617	−	0.881268i	$-0.656691\pi$
$97$	−9.30948	−0.945234	−0.472617	+	0.881268i	$0.656691\pi$
$98$	−0.906053	−0.0915252
$99$	0.664575	0.0667923
$100$	54.4108	5.44108
$101$	3.63339	0.361536	0.180768	−	0.983526i	$-0.442142\pi$
$101$	3.63339	0.361536	0.180768	+	0.983526i	$0.442142\pi$
$102$	−2.78814	−0.276067
$103$	14.2788	1.40693	0.703464	−	0.710731i	$-0.251636\pi$
$103$	14.2788	1.40693	0.703464	+	0.710731i	$0.251636\pi$
$104$	73.1812	7.17601
$105$	9.81224	0.957576
$106$	−33.2433	−3.22887
$107$	−0.972444	−0.0940097	−0.0470049	−	0.998895i	$-0.514968\pi$
$107$	−0.972444	−0.0940097	−0.0470049	+	0.998895i	$0.514968\pi$
$108$	−5.77371	−0.555576
$109$	13.0230	1.24737	0.623687	−	0.781675i	$-0.285634\pi$
$109$	13.0230	1.24737	0.623687	+	0.781675i	$0.285634\pi$
$110$	7.03720	0.670970
$111$	2.00843	0.190632
$112$	−45.9581	−4.34263
$113$	−12.9886	−1.22186	−0.610932	−	0.791683i	$-0.709205\pi$
$113$	−12.9886	−1.22186	−0.610932	+	0.791683i	$0.709205\pi$
$114$	7.97129	0.746579
$115$	−9.41228	−0.877699
$116$	−1.22333	−0.113584
$117$	6.95532	0.643019
$118$	−18.0863	−1.66498
$119$	−2.58361	−0.236839
$120$	−39.9599	−3.64782
$121$	−10.5583	−0.959849
$122$	−9.04064	−0.818501
$123$	−3.13177	−0.282382
$124$	−18.6239	−1.67248
$125$	16.8014	1.50276
$126$	−7.20346	−0.641735
$127$	16.8160	1.49218	0.746090	−	0.665845i	$-0.231929\pi$
$127$	16.8160	1.49218	0.746090	+	0.665845i	$0.231929\pi$
$128$	65.6644	5.80397
$129$	3.66634	0.322803
$130$	73.6499	6.45953
$131$	14.6667	1.28144	0.640719	−	0.767776i	$-0.278636\pi$
$131$	14.6667	1.28144	0.640719	+	0.767776i	$0.278636\pi$
$132$	−3.83707	−0.333974
$133$	7.38654	0.640494
$134$	7.87621	0.680401
$135$	−3.79788	−0.326869
$136$	10.5216	0.902222
$137$	16.0525	1.37146	0.685731	−	0.727855i	$-0.259483\pi$
$137$	16.0525	1.37146	0.685731	+	0.727855i	$0.259483\pi$
$138$	6.90983	0.588204
$139$	−20.2048	−1.71375	−0.856873	−	0.515527i	$-0.827596\pi$
$139$	−20.2048	−1.71375	−0.856873	+	0.515527i	$0.827596\pi$
$140$	−56.6530	−4.78805
$141$	0.211833	0.0178396
$142$	−28.4484	−2.38733
$143$	4.62233	0.386539
$144$	17.7883	1.48236
$145$	−0.804694	−0.0668262
$146$	−24.4029	−2.01960
$147$	0.324967	0.0268028
$148$	−11.5961	−0.953193
$149$	−0.633669	−0.0519122	−0.0259561	−	0.999663i	$-0.508263\pi$
$149$	−0.633669	−0.0519122	−0.0259561	+	0.999663i	$0.508263\pi$
$150$	−26.2751	−2.14535
$151$	2.59962	0.211554	0.105777	−	0.994390i	$-0.466267\pi$
$151$	2.59962	0.211554	0.105777	+	0.994390i	$0.466267\pi$
$152$	−30.0813	−2.43992
$153$	1.00000	0.0808452
$154$	−4.78724	−0.385767
$155$	−12.2506	−0.983993
$156$	−40.1580	−3.21521
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	18.1043	1.44030
$159$	11.9231	0.945565
$160$	108.441	8.57301
$161$	6.40295	0.504623
$162$	2.78814	0.219057
$163$	−0.492799	−0.0385990	−0.0192995	−	0.999814i	$-0.506144\pi$
$163$	−0.492799	−0.0385990	−0.0192995	+	0.999814i	$0.506144\pi$
$164$	18.0819	1.41196
$165$	−2.52398	−0.196491
$166$	10.2218	0.793363
$167$	1.91638	0.148294	0.0741471	−	0.997247i	$-0.476377\pi$
$167$	1.91638	0.148294	0.0741471	+	0.997247i	$0.476377\pi$
$168$	27.1838	2.09727
$169$	35.3764	2.72126
$170$	10.5890	0.812140
$171$	−2.85900	−0.218633
$172$	−21.1684	−1.61407
$173$	−10.6222	−0.807591	−0.403796	−	0.914849i	$-0.632309\pi$
$173$	−10.6222	−0.807591	−0.403796	+	0.914849i	$0.632309\pi$
$174$	0.590750	0.0447847
$175$	−24.3477	−1.84051
$176$	11.8217	0.891093
$177$	6.48686	0.487582
$178$	−19.1818	−1.43773
$179$	−19.7374	−1.47524	−0.737620	−	0.675216i	$-0.764050\pi$
$179$	−19.7374	−1.47524	−0.737620	+	0.675216i	$0.764050\pi$
$180$	21.9279	1.63441
$181$	17.8259	1.32499	0.662495	−	0.749067i	$-0.269498\pi$
$181$	17.8259	1.32499	0.662495	+	0.749067i	$0.269498\pi$
$182$	−50.1023	−3.71383
$183$	3.24254	0.239695
$184$	−26.0757	−1.92233
$185$	−7.62777	−0.560805
$186$	8.99354	0.659438
$187$	0.664575	0.0485986
$188$	−1.22306	−0.0892011
$189$	2.58361	0.187930
$190$	−30.2740	−2.19631
$191$	−24.7571	−1.79136	−0.895681	−	0.444697i	$-0.853311\pi$
$191$	−24.7571	−1.79136	−0.895681	+	0.444697i	$0.853311\pi$
$192$	−44.0331	−3.17782
$193$	−5.67569	−0.408545	−0.204272	−	0.978914i	$-0.565483\pi$
$193$	−5.67569	−0.408545	−0.204272	+	0.978914i	$0.565483\pi$
$194$	−25.9561	−1.86354
$195$	−26.4155	−1.89165
$196$	−1.87627	−0.134019
$197$	−17.0004	−1.21123	−0.605614	−	0.795759i	$-0.707072\pi$
$197$	−17.0004	−1.21123	−0.605614	+	0.795759i	$0.707072\pi$
$198$	1.85293	0.131682
$199$	8.99560	0.637681	0.318841	−	0.947808i	$-0.396707\pi$
$199$	8.99560	0.637681	0.318841	+	0.947808i	$0.396707\pi$
$200$	99.1547	7.01130
$201$	−2.82490	−0.199253
$202$	10.1304	0.712772
$203$	0.547415	0.0384210
$204$	−5.77371	−0.404241
$205$	11.8941	0.830719
$206$	39.8111	2.77377
$207$	−2.47830	−0.172254
$208$	123.723	8.57867
$209$	−1.90002	−0.131427
$210$	27.3579	1.88787
$211$	−16.2103	−1.11597	−0.557983	−	0.829852i	$-0.688425\pi$
$211$	−16.2103	−1.11597	−0.557983	+	0.829852i	$0.688425\pi$
$212$	−68.8407	−4.72800
$213$	10.2034	0.699123
$214$	−2.71131	−0.185341
$215$	−13.9243	−0.949630
$216$	−10.5216	−0.715906
$217$	8.33380	0.565735
$218$	36.3098	2.45921
$219$	8.75240	0.591432
$220$	14.5727	0.982492
$221$	6.95532	0.467865
$222$	5.59978	0.375832
$223$	−5.15546	−0.345235	−0.172618	−	0.984989i	$-0.555222\pi$
$223$	−5.15546	−0.345235	−0.172618	+	0.984989i	$0.555222\pi$
$224$	−73.7699	−4.92896
$225$	9.42389	0.628260
$226$	−36.2140	−2.40892
$227$	9.82219	0.651921	0.325961	−	0.945383i	$-0.394312\pi$
$227$	9.82219	0.651921	0.325961	+	0.945383i	$0.394312\pi$
$228$	16.5070	1.09321
$229$	20.1511	1.33162	0.665812	−	0.746119i	$-0.268085\pi$
$229$	20.1511	1.33162	0.665812	+	0.746119i	$0.268085\pi$
$230$	−26.2427	−1.73039
$231$	1.71700	0.112970
$232$	−2.22932	−0.146362
$233$	20.2633	1.32749	0.663747	−	0.747957i	$-0.268965\pi$
$233$	20.2633	1.32749	0.663747	+	0.747957i	$0.268965\pi$
$234$	19.3924	1.26772
$235$	−0.804517	−0.0524809
$236$	−37.4533	−2.43800
$237$	−6.49333	−0.421787
$238$	−7.20346	−0.466931
$239$	20.6863	1.33808	0.669042	−	0.743224i	$-0.266705\pi$
$239$	20.6863	1.33808	0.669042	+	0.743224i	$0.266705\pi$
$240$	−67.5579	−4.36084
$241$	−1.89118	−0.121822	−0.0609109	−	0.998143i	$-0.519401\pi$
$241$	−1.89118	−0.121822	−0.0609109	+	0.998143i	$0.519401\pi$
$242$	−29.4381	−1.89235
$243$	−1.00000	−0.0641500
$244$	−18.7215	−1.19852
$245$	−1.23419	−0.0788493
$246$	−8.73180	−0.556719
$247$	−19.8853	−1.26527
$248$	−33.9390	−2.15513
$249$	−3.66616	−0.232334
$250$	46.8447	2.96272
$251$	3.68319	0.232481	0.116240	−	0.993221i	$-0.462916\pi$
$251$	3.68319	0.232481	0.116240	+	0.993221i	$0.462916\pi$
$252$	−14.9170	−0.939683
$253$	−1.64702	−0.103547
$254$	46.8854	2.94185
$255$	−3.79788	−0.237832
$256$	95.0151	5.93844
$257$	−20.3107	−1.26695	−0.633475	−	0.773763i	$-0.718372\pi$
$257$	−20.3107	−1.26695	−0.633475	+	0.773763i	$0.718372\pi$
$258$	10.2223	0.636410
$259$	5.18899	0.322428
$260$	152.515	9.45859
$261$	−0.211880	−0.0131150
$262$	40.8928	2.52637
$263$	25.4289	1.56802	0.784008	−	0.620751i	$-0.213172\pi$
$263$	25.4289	1.56802	0.784008	+	0.620751i	$0.213172\pi$
$264$	−6.99241	−0.430353
$265$	−45.2826	−2.78169
$266$	20.5947	1.26274
$267$	6.87978	0.421036
$268$	16.3102	0.996302
$269$	−8.37766	−0.510795	−0.255398	−	0.966836i	$-0.582206\pi$
$269$	−8.37766	−0.510795	−0.255398	+	0.966836i	$0.582206\pi$
$270$	−10.5890	−0.644427
$271$	27.4330	1.66644	0.833219	−	0.552943i	$-0.186495\pi$
$271$	27.4330	1.66644	0.833219	+	0.552943i	$0.186495\pi$
$272$	17.7883	1.07858
$273$	17.9698	1.08758
$274$	44.7567	2.70385
$275$	6.26289	0.377666
$276$	14.3090	0.861299
$277$	8.72498	0.524233	0.262117	−	0.965036i	$-0.415579\pi$
$277$	8.72498	0.524233	0.262117	+	0.965036i	$0.415579\pi$
$278$	−56.3337	−3.37867
$279$	−3.22564	−0.193114
$280$	−103.241	−6.16981
$281$	27.4762	1.63909	0.819545	−	0.573015i	$-0.194226\pi$
$281$	27.4762	1.63909	0.819545	+	0.573015i	$0.194226\pi$
$282$	0.590620	0.0351709
$283$	−2.70295	−0.160674	−0.0803368	−	0.996768i	$-0.525600\pi$
$283$	−2.70295	−0.160674	−0.0803368	+	0.996768i	$0.525600\pi$
$284$	−58.9113	−3.49574
$285$	10.8581	0.643181
$286$	12.8877	0.762065
$287$	−8.09126	−0.477612
$288$	28.5530	1.68250
$289$	1.00000	0.0588235
$290$	−2.24360	−0.131749
$291$	9.30948	0.545731
$292$	−50.5338	−2.95727
$293$	14.9056	0.870793	0.435397	−	0.900239i	$-0.356608\pi$
$293$	14.9056	0.870793	0.435397	+	0.900239i	$0.356608\pi$
$294$	0.906053	0.0528421
$295$	−24.6363	−1.43438
$296$	−21.1319	−1.22827
$297$	−0.664575	−0.0385626
$298$	−1.76676	−0.102345
$299$	−17.2373	−0.996861
$300$	−54.4108	−3.14141
$301$	9.47238	0.545979
$302$	7.24810	0.417081
$303$	−3.63339	−0.208733
$304$	−50.8568	−2.91684
$305$	−12.3148	−0.705141
$306$	2.78814	0.159387
$307$	−32.1155	−1.83293	−0.916464	−	0.400118i	$-0.868969\pi$
$307$	−32.1155	−1.83293	−0.916464	+	0.400118i	$0.868969\pi$
$308$	−9.91348	−0.564873
$309$	−14.2788	−0.812290
$310$	−34.1564	−1.93995
$311$	−20.1670	−1.14356	−0.571782	−	0.820406i	$-0.693748\pi$
$311$	−20.1670	−1.14356	−0.571782	+	0.820406i	$0.693748\pi$
$312$	−73.1812	−4.14307
$313$	23.3757	1.32128	0.660638	−	0.750705i	$-0.270286\pi$
$313$	23.3757	1.32128	0.660638	+	0.750705i	$0.270286\pi$
$314$	−2.78814	−0.157344
$315$	−9.81224	−0.552857
$316$	37.4906	2.10901
$317$	20.5557	1.15452	0.577262	−	0.816559i	$-0.304121\pi$
$317$	20.5557	1.15452	0.577262	+	0.816559i	$0.304121\pi$
$318$	33.2433	1.86419
$319$	−0.140810	−0.00788385
$320$	167.233	9.34859
$321$	0.972444	0.0542765
$322$	17.8523	0.994870
$323$	−2.85900	−0.159079
$324$	5.77371	0.320762
$325$	65.5462	3.63585
$326$	−1.37399	−0.0760983
$327$	−13.0230	−0.720171
$328$	32.9513	1.81943
$329$	0.547294	0.0301733
$330$	−7.03720	−0.387385
$331$	−24.0978	−1.32454	−0.662268	−	0.749267i	$-0.730406\pi$
$331$	−24.0978	−1.32454	−0.662268	+	0.749267i	$0.730406\pi$
$332$	21.1674	1.16171
$333$	−2.00843	−0.110061
$334$	5.34314	0.292364
$335$	10.7286	0.586168
$336$	45.9581	2.50722
$337$	−25.2138	−1.37348	−0.686740	−	0.726903i	$-0.740959\pi$
$337$	−25.2138	−1.37348	−0.686740	+	0.726903i	$0.740959\pi$
$338$	98.6343	5.36500
$339$	12.9886	0.705443
$340$	21.9279	1.18921
$341$	−2.14368	−0.116087
$342$	−7.97129	−0.431038
$343$	18.9248	1.02185
$344$	−38.5758	−2.07987
$345$	9.41228	0.506740
$346$	−29.6162	−1.59217
$347$	−6.77782	−0.363853	−0.181926	−	0.983312i	$-0.558233\pi$
$347$	−6.77782	−0.363853	−0.181926	+	0.983312i	$0.558233\pi$
$348$	1.22333	0.0655775
$349$	9.53644	0.510474	0.255237	−	0.966879i	$-0.417847\pi$
$349$	9.53644	0.510474	0.255237	+	0.966879i	$0.417847\pi$
$350$	−67.8846	−3.62859
$351$	−6.95532	−0.371247
$352$	18.9756	1.01141
$353$	5.54174	0.294957	0.147478	−	0.989065i	$-0.452884\pi$
$353$	5.54174	0.294957	0.147478	+	0.989065i	$0.452884\pi$
$354$	18.0863	0.961274
$355$	−38.7512	−2.05670
$356$	−39.7219	−2.10525
$357$	2.58361	0.136739
$358$	−55.0305	−2.90845
$359$	−21.4444	−1.13179	−0.565897	−	0.824476i	$-0.691470\pi$
$359$	−21.4444	−1.13179	−0.565897	+	0.824476i	$0.691470\pi$
$360$	39.9599	2.10607
$361$	−10.8261	−0.569796
$362$	49.7011	2.61223
$363$	10.5583	0.554169
$364$	−103.753	−5.43811
$365$	−33.2406	−1.73989
$366$	9.04064	0.472562
$367$	−29.7080	−1.55075	−0.775374	−	0.631503i	$-0.782438\pi$
$367$	−29.7080	−1.55075	−0.775374	+	0.631503i	$0.782438\pi$
$368$	−44.0847	−2.29808
$369$	3.13177	0.163033
$370$	−21.2673	−1.10563
$371$	30.8047	1.59930
$372$	18.6239	0.965606
$373$	−36.5481	−1.89239	−0.946195	−	0.323598i	$-0.895108\pi$
$373$	−36.5481	−1.89239	−0.946195	+	0.323598i	$0.895108\pi$
$374$	1.85293	0.0958126
$375$	−16.8014	−0.867622
$376$	−2.22883	−0.114943
$377$	−1.47369	−0.0758990
$378$	7.20346	0.370506
$379$	15.0134	0.771188	0.385594	−	0.922668i	$-0.373996\pi$
$379$	15.0134	0.771188	0.385594	+	0.922668i	$0.373996\pi$
$380$	−62.6918	−3.21602
$381$	−16.8160	−0.861510
$382$	−69.0262	−3.53169
$383$	−7.31968	−0.374018	−0.187009	−	0.982358i	$-0.559879\pi$
$383$	−7.31968	−0.374018	−0.187009	+	0.982358i	$0.559879\pi$
$384$	−65.6644	−3.35092
$385$	−6.52097	−0.332339
$386$	−15.8246	−0.805451
$387$	−3.66634	−0.186370
$388$	−53.7502	−2.72876
$389$	36.4469	1.84793	0.923966	−	0.382474i	$-0.124928\pi$
$389$	36.4469	1.84793	0.923966	+	0.382474i	$0.124928\pi$
$390$	−73.6499	−3.72941
$391$	−2.47830	−0.125333
$392$	−3.41918	−0.172695
$393$	−14.6667	−0.739838
$394$	−47.3994	−2.38795
$395$	24.6609	1.24082
$396$	3.83707	0.192820
$397$	−7.42903	−0.372852	−0.186426	−	0.982469i	$-0.559691\pi$
$397$	−7.42903	−0.372852	−0.186426	+	0.982469i	$0.559691\pi$
$398$	25.0810	1.25720
$399$	−7.38654	−0.369789
$400$	167.635	8.38176
$401$	26.0628	1.30152	0.650758	−	0.759285i	$-0.274451\pi$
$401$	26.0628	1.30152	0.650758	+	0.759285i	$0.274451\pi$
$402$	−7.87621	−0.392830
$403$	−22.4354	−1.11759
$404$	20.9782	1.04370
$405$	3.79788	0.188718
$406$	1.52627	0.0757474
$407$	−1.33475	−0.0661612
$408$	−10.5216	−0.520898
$409$	39.3817	1.94730	0.973649	−	0.228052i	$-0.0732355\pi$
$409$	39.3817	1.94730	0.973649	+	0.228052i	$0.0732355\pi$
$410$	33.1623	1.63777
$411$	−16.0525	−0.791813
$412$	82.4414	4.06160
$413$	16.7595	0.824682
$414$	−6.90983	−0.339600
$415$	13.9236	0.683485
$416$	198.595	9.73694
$417$	20.2048	0.989432
$418$	−5.29752	−0.259110
$419$	−29.9177	−1.46157	−0.730787	−	0.682606i	$-0.760847\pi$
$419$	−29.9177	−1.46157	−0.730787	+	0.682606i	$0.760847\pi$
$420$	56.6530	2.76438
$421$	−14.1202	−0.688175	−0.344087	−	0.938938i	$-0.611812\pi$
$421$	−14.1202	−0.688175	−0.344087	+	0.938938i	$0.611812\pi$
$422$	−45.1967	−2.20014
$423$	−0.211833	−0.0102997
$424$	−125.451	−6.09242
$425$	9.42389	0.457126
$426$	28.4484	1.37833
$427$	8.37744	0.405413
$428$	−5.61461	−0.271393
$429$	−4.62233	−0.223168
$430$	−38.8229	−1.87221
$431$	27.5496	1.32702	0.663509	−	0.748169i	$-0.269067\pi$
$431$	27.5496	1.32702	0.663509	+	0.748169i	$0.269067\pi$
$432$	−17.7883	−0.855841
$433$	−28.4351	−1.36651	−0.683253	−	0.730182i	$-0.739435\pi$
$433$	−28.4351	−1.36651	−0.683253	+	0.730182i	$0.739435\pi$
$434$	23.2358	1.11535
$435$	0.804694	0.0385821
$436$	75.1908	3.60099
$437$	7.08545	0.338943
$438$	24.4029	1.16602
$439$	−34.3971	−1.64168	−0.820842	−	0.571156i	$-0.806495\pi$
$439$	−34.3971	−1.64168	−0.820842	+	0.571156i	$0.806495\pi$
$440$	26.5564	1.26602
$441$	−0.324967	−0.0154746
$442$	19.3924	0.922401
$443$	−30.8581	−1.46611	−0.733055	−	0.680169i	$-0.761906\pi$
$443$	−30.8581	−1.46611	−0.733055	+	0.680169i	$0.761906\pi$
$444$	11.5961	0.550326
$445$	−26.1286	−1.23861
$446$	−14.3741	−0.680635
$447$	0.633669	0.0299715
$448$	−113.764	−5.37486
$449$	13.8459	0.653430	0.326715	−	0.945123i	$-0.394058\pi$
$449$	13.8459	0.653430	0.326715	+	0.945123i	$0.394058\pi$
$450$	26.2751	1.23862
$451$	2.08130	0.0980044
$452$	−74.9923	−3.52734
$453$	−2.59962	−0.122141
$454$	27.3856	1.28527
$455$	−68.2472	−3.19948
$456$	30.0813	1.40869
$457$	19.4608	0.910336	0.455168	−	0.890406i	$-0.349579\pi$
$457$	19.4608	0.910336	0.455168	+	0.890406i	$0.349579\pi$
$458$	56.1841	2.62531
$459$	−1.00000	−0.0466760
$460$	−54.3438	−2.53379
$461$	−9.80361	−0.456600	−0.228300	−	0.973591i	$-0.573317\pi$
$461$	−9.80361	−0.456600	−0.228300	+	0.973591i	$0.573317\pi$
$462$	4.78724	0.222723
$463$	−9.88041	−0.459181	−0.229591	−	0.973287i	$-0.573739\pi$
$463$	−9.88041	−0.459181	−0.229591	+	0.973287i	$0.573739\pi$
$464$	−3.76899	−0.174971
$465$	12.2506	0.568108
$466$	56.4969	2.61717
$467$	−11.8097	−0.546490	−0.273245	−	0.961945i	$-0.588097\pi$
$467$	−11.8097	−0.546490	−0.273245	+	0.961945i	$0.588097\pi$
$468$	40.1580	1.85630
$469$	−7.29844	−0.337011
$470$	−2.24310	−0.103467
$471$	1.00000	0.0460776
$472$	−68.2523	−3.14157
$473$	−2.43656	−0.112033
$474$	−18.1043	−0.831558
$475$	−26.9429	−1.23623
$476$	−14.9170	−0.683720
$477$	−11.9231	−0.545922
$478$	57.6762	2.63805
$479$	−12.4148	−0.567245	−0.283623	−	0.958936i	$-0.591536\pi$
$479$	−12.4148	−0.567245	−0.283623	+	0.958936i	$0.591536\pi$
$480$	−108.441	−4.94963
$481$	−13.9693	−0.636943
$482$	−5.27288	−0.240173
$483$	−6.40295	−0.291344
$484$	−60.9608	−2.77095
$485$	−35.3563	−1.60545
$486$	−2.78814	−0.126472
$487$	−0.267048	−0.0121011	−0.00605056	−	0.999982i	$-0.501926\pi$
$487$	−0.267048	−0.0121011	−0.00605056	+	0.999982i	$0.501926\pi$
$488$	−34.1168	−1.54439
$489$	0.492799	0.0222851
$490$	−3.44108	−0.155452
$491$	17.3601	0.783450	0.391725	−	0.920082i	$-0.371878\pi$
$491$	17.3601	0.783450	0.391725	+	0.920082i	$0.371878\pi$
$492$	−18.0819	−0.815196
$493$	−0.211880	−0.00954259
$494$	−55.4428	−2.49449
$495$	2.52398	0.113444
$496$	−57.3788	−2.57638
$497$	26.3615	1.18247
$498$	−10.2218	−0.458048
$499$	−2.28421	−0.102255	−0.0511277	−	0.998692i	$-0.516282\pi$
$499$	−2.28421	−0.102255	−0.0511277	+	0.998692i	$0.516282\pi$
$500$	97.0066	4.33827
$501$	−1.91638	−0.0856177
$502$	10.2692	0.458338
$503$	42.9445	1.91480	0.957401	−	0.288761i	$-0.0932434\pi$
$503$	42.9445	1.91480	0.957401	+	0.288761i	$0.0932434\pi$
$504$	−27.1838	−1.21086
$505$	13.7992	0.614056
$506$	−4.59211	−0.204144
$507$	−35.3764	−1.57112
$508$	97.0908	4.30771
$509$	−27.8559	−1.23469	−0.617344	−	0.786693i	$-0.711792\pi$
$509$	−27.8559	−1.23469	−0.617344	+	0.786693i	$0.711792\pi$
$510$	−10.5890	−0.468889
$511$	22.6128	1.00033
$512$	133.586	5.90374
$513$	2.85900	0.126228
$514$	−56.6291	−2.49780
$515$	54.2290	2.38961
$516$	21.1684	0.931886
$517$	−0.140779	−0.00619146
$518$	14.4676	0.635671
$519$	10.6222	0.466263
$520$	277.934	12.1882
$521$	−30.1141	−1.31932	−0.659661	−	0.751564i	$-0.729300\pi$
$521$	−30.1141	−1.31932	−0.659661	+	0.751564i	$0.729300\pi$
$522$	−0.590750	−0.0258564
$523$	−40.7775	−1.78308	−0.891538	−	0.452946i	$-0.850373\pi$
$523$	−40.7775	−1.78308	−0.891538	+	0.452946i	$0.850373\pi$
$524$	84.6814	3.69933
$525$	24.3477	1.06262
$526$	70.8994	3.09136
$527$	−3.22564	−0.140511
$528$	−11.8217	−0.514473
$529$	−16.8580	−0.732958
$530$	−126.254	−5.48413
$531$	−6.48686	−0.281506
$532$	42.6477	1.84901
$533$	21.7824	0.943502
$534$	19.1818	0.830077
$535$	−3.69323	−0.159672
$536$	29.7226	1.28382
$537$	19.7374	0.851730
$538$	−23.3581	−1.00704
$539$	−0.215965	−0.00930228
$540$	−21.9279	−0.943625
$541$	−31.4843	−1.35362	−0.676809	−	0.736159i	$-0.736638\pi$
$541$	−31.4843	−1.35362	−0.676809	+	0.736159i	$0.736638\pi$
$542$	76.4870	3.28540
$543$	−17.8259	−0.764983
$544$	28.5530	1.22420
$545$	49.4596	2.11862
$546$	50.1023	2.14418
$547$	−5.37848	−0.229967	−0.114984	−	0.993367i	$-0.536682\pi$
$547$	−5.37848	−0.229967	−0.114984	+	0.993367i	$0.536682\pi$
$548$	92.6827	3.95921
$549$	−3.24254	−0.138388
$550$	17.4618	0.744573
$551$	0.605765	0.0258064
$552$	26.0757	1.10986
$553$	−16.7762	−0.713397
$554$	24.3265	1.03353
$555$	7.62777	0.323781
$556$	−116.657	−4.94734
$557$	4.40229	0.186531	0.0932655	−	0.995641i	$-0.470269\pi$
$557$	4.40229	0.186531	0.0932655	+	0.995641i	$0.470269\pi$
$558$	−8.99354	−0.380727
$559$	−25.5005	−1.07856
$560$	−174.543	−7.37580
$561$	−0.664575	−0.0280584
$562$	76.6073	3.23148
$563$	−22.9664	−0.967918	−0.483959	−	0.875091i	$-0.660802\pi$
$563$	−22.9664	−0.967918	−0.483959	+	0.875091i	$0.660802\pi$
$564$	1.22306	0.0515003
$565$	−49.3291	−2.07529
$566$	−7.53619	−0.316770
$567$	−2.58361	−0.108501
$568$	−107.356	−4.50456
$569$	20.1583	0.845079	0.422539	−	0.906345i	$-0.361139\pi$
$569$	20.1583	0.845079	0.422539	+	0.906345i	$0.361139\pi$
$570$	30.2740	1.26804
$571$	32.2056	1.34776	0.673881	−	0.738840i	$-0.264626\pi$
$571$	32.2056	1.34776	0.673881	+	0.738840i	$0.264626\pi$
$572$	26.6880	1.11588
$573$	24.7571	1.03424
$574$	−22.5596	−0.941618
$575$	−23.3552	−0.973980
$576$	44.0331	1.83471
$577$	9.88984	0.411720	0.205860	−	0.978582i	$-0.434001\pi$
$577$	9.88984	0.411720	0.205860	+	0.978582i	$0.434001\pi$
$578$	2.78814	0.115971
$579$	5.67569	0.235873
$580$	−4.64607	−0.192918
$581$	−9.47193	−0.392962
$582$	25.9561	1.07592
$583$	−7.92381	−0.328171
$584$	−92.0895	−3.81069
$585$	26.4155	1.09214
$586$	41.5588	1.71678
$587$	35.7825	1.47690	0.738452	−	0.674307i	$-0.235557\pi$
$587$	35.7825	1.47690	0.738452	+	0.674307i	$0.235557\pi$
$588$	1.87627	0.0773759
$589$	9.22212	0.379991
$590$	−68.6895	−2.82790
$591$	17.0004	0.699303
$592$	−35.7266	−1.46835
$593$	17.8512	0.733061	0.366531	−	0.930406i	$-0.380545\pi$
$593$	17.8512	0.733061	0.366531	+	0.930406i	$0.380545\pi$
$594$	−1.85293	−0.0760265
$595$	−9.81224	−0.402262
$596$	−3.65862	−0.149863
$597$	−8.99560	−0.368165
$598$	−48.0601	−1.96532
$599$	12.0607	0.492786	0.246393	−	0.969170i	$-0.420755\pi$
$599$	12.0607	0.492786	0.246393	+	0.969170i	$0.420755\pi$
$600$	−99.1547	−4.04797
$601$	32.7876	1.33743	0.668717	−	0.743517i	$-0.266844\pi$
$601$	32.7876	1.33743	0.668717	+	0.743517i	$0.266844\pi$
$602$	26.4103	1.07640
$603$	2.82490	0.115039
$604$	15.0095	0.610726
$605$	−40.0993	−1.63027
$606$	−10.1304	−0.411519
$607$	−31.6270	−1.28370	−0.641851	−	0.766830i	$-0.721833\pi$
$607$	−31.6270	−1.28370	−0.641851	+	0.766830i	$0.721833\pi$
$608$	−81.6331	−3.31066
$609$	−0.547415	−0.0221824
$610$	−34.3353	−1.39019
$611$	−1.47337	−0.0596060
$612$	5.77371	0.233388
$613$	0.314089	0.0126860	0.00634298	−	0.999980i	$-0.497981\pi$
$613$	0.314089	0.0126860	0.00634298	+	0.999980i	$0.497981\pi$
$614$	−89.5424	−3.61364
$615$	−11.8941	−0.479616
$616$	−18.0657	−0.727886
$617$	32.5235	1.30935	0.654674	−	0.755912i	$-0.272806\pi$
$617$	32.5235	1.30935	0.654674	+	0.755912i	$0.272806\pi$
$618$	−39.8111	−1.60144
$619$	−16.0069	−0.643371	−0.321686	−	0.946847i	$-0.604249\pi$
$619$	−16.0069	−0.643371	−0.321686	+	0.946847i	$0.604249\pi$
$620$	−70.7315	−2.84064
$621$	2.47830	0.0994506
$622$	−56.2282	−2.25455
$623$	17.7747	0.712127
$624$	−123.723	−4.95290
$625$	16.6903	0.667613
$626$	65.1748	2.60491
$627$	1.90002	0.0758795
$628$	−5.77371	−0.230396
$629$	−2.00843	−0.0800813
$630$	−27.3579	−1.08996
$631$	13.8172	0.550054	0.275027	−	0.961437i	$-0.411313\pi$
$631$	13.8172	0.550054	0.275027	+	0.961437i	$0.411313\pi$
$632$	68.3204	2.71764
$633$	16.2103	0.644303
$634$	57.3121	2.27616
$635$	63.8652	2.53441
$636$	68.8407	2.72971
$637$	−2.26025	−0.0895543
$638$	−0.392598	−0.0155431
$639$	−10.2034	−0.403639
$640$	249.386	9.85783
$641$	36.0796	1.42506	0.712528	−	0.701643i	$-0.247550\pi$
$641$	36.0796	1.42506	0.712528	+	0.701643i	$0.247550\pi$
$642$	2.71131	0.107007
$643$	−3.27662	−0.129217	−0.0646087	−	0.997911i	$-0.520580\pi$
$643$	−3.27662	−0.129217	−0.0646087	+	0.997911i	$0.520580\pi$
$644$	36.9688	1.45677
$645$	13.9243	0.548269
$646$	−7.97129	−0.313626
$647$	−15.6273	−0.614372	−0.307186	−	0.951650i	$-0.599387\pi$
$647$	−15.6273	−0.614372	−0.307186	+	0.951650i	$0.599387\pi$
$648$	10.5216	0.413329
$649$	−4.31101	−0.169222
$650$	182.752	7.16811
$651$	−8.33380	−0.326627
$652$	−2.84528	−0.111430
$653$	31.0651	1.21567	0.607835	−	0.794063i	$-0.292038\pi$
$653$	31.0651	1.21567	0.607835	+	0.794063i	$0.292038\pi$
$654$	−36.3098	−1.41983
$655$	55.7025	2.17648
$656$	55.7089	2.17507
$657$	−8.75240	−0.341464
$658$	1.52593	0.0594870
$659$	−30.8692	−1.20249	−0.601247	−	0.799063i	$-0.705329\pi$
$659$	−30.8692	−1.20249	−0.601247	+	0.799063i	$0.705329\pi$
$660$	−14.5727	−0.567242
$661$	−7.34279	−0.285601	−0.142801	−	0.989751i	$-0.545611\pi$
$661$	−7.34279	−0.285601	−0.142801	+	0.989751i	$0.545611\pi$
$662$	−67.1880	−2.61134
$663$	−6.95532	−0.270122
$664$	38.5740	1.49696
$665$	28.0532	1.08786
$666$	−5.59978	−0.216987
$667$	0.525101	0.0203320
$668$	11.0647	0.428104
$669$	5.15546	0.199322
$670$	29.9129	1.15564
$671$	−2.15491	−0.0831894
$672$	73.7699	2.84573
$673$	−1.40979	−0.0543435	−0.0271717	−	0.999631i	$-0.508650\pi$
$673$	−1.40979	−0.0543435	−0.0271717	+	0.999631i	$0.508650\pi$
$674$	−70.2994	−2.70783
$675$	−9.42389	−0.362726
$676$	204.253	7.85589
$677$	10.8985	0.418863	0.209432	−	0.977823i	$-0.432839\pi$
$677$	10.8985	0.418863	0.209432	+	0.977823i	$0.432839\pi$
$678$	36.2140	1.39079
$679$	24.0520	0.923033
$680$	39.9599	1.53239
$681$	−9.82219	−0.376387
$682$	−5.97688	−0.228867
$683$	22.3826	0.856448	0.428224	−	0.903673i	$-0.359139\pi$
$683$	22.3826	0.856448	0.428224	+	0.903673i	$0.359139\pi$
$684$	−16.5070	−0.631163
$685$	60.9656	2.32938
$686$	52.7651	2.01458
$687$	−20.1511	−0.768814
$688$	−65.2180	−2.48641
$689$	−82.9291	−3.15935
$690$	26.2427	0.999043
$691$	30.3984	1.15641	0.578205	−	0.815891i	$-0.303753\pi$
$691$	30.3984	1.15641	0.578205	+	0.815891i	$0.303753\pi$
$692$	−61.3295	−2.33140
$693$	−1.71700	−0.0652235
$694$	−18.8975	−0.717339
$695$	−76.7353	−2.91074
$696$	2.22932	0.0845022
$697$	3.13177	0.118624
$698$	26.5889	1.00640
$699$	−20.2633	−0.766429
$700$	−140.576	−5.31329
$701$	−37.8608	−1.42998	−0.714991	−	0.699133i	$-0.753569\pi$
$701$	−37.8608	−1.42998	−0.714991	+	0.699133i	$0.753569\pi$
$702$	−19.3924	−0.731918
$703$	5.74210	0.216567
$704$	29.2633	1.10290
$705$	0.804517	0.0302999
$706$	15.4511	0.581511
$707$	−9.38726	−0.353044
$708$	37.4533	1.40758
$709$	7.03515	0.264210	0.132105	−	0.991236i	$-0.457826\pi$
$709$	7.03515	0.264210	0.132105	+	0.991236i	$0.457826\pi$
$710$	−108.044	−4.05480
$711$	6.49333	0.243519
$712$	−72.3865	−2.71280
$713$	7.99410	0.299382
$714$	7.20346	0.269583
$715$	17.5551	0.656522
$716$	−113.958	−4.25881
$717$	−20.6863	−0.772543
$718$	−59.7901	−2.23135
$719$	21.4748	0.800875	0.400437	−	0.916324i	$-0.368858\pi$
$719$	21.4748	0.800875	0.400437	+	0.916324i	$0.368858\pi$
$720$	67.5579	2.51773
$721$	−36.8907	−1.37388
$722$	−30.1847	−1.12336
$723$	1.89118	0.0703338
$724$	102.922	3.82505
$725$	−1.99673	−0.0741568
$726$	29.4381	1.09255
$727$	27.0178	1.00204	0.501018	−	0.865437i	$-0.332959\pi$
$727$	27.0178	1.00204	0.501018	+	0.865437i	$0.332959\pi$
$728$	−189.072	−7.00746
$729$	1.00000	0.0370370
$730$	−92.6793	−3.43021
$731$	−3.66634	−0.135604
$732$	18.7215	0.691965
$733$	44.5102	1.64402	0.822011	−	0.569472i	$-0.192852\pi$
$733$	44.5102	1.64402	0.822011	+	0.569472i	$0.192852\pi$
$734$	−82.8301	−3.05732
$735$	1.23419	0.0455236
$736$	−70.7629	−2.60836
$737$	1.87736	0.0691534
$738$	8.73180	0.321422
$739$	52.2911	1.92356	0.961780	−	0.273824i	$-0.0882887\pi$
$739$	52.2911	1.92356	0.961780	+	0.273824i	$0.0882887\pi$
$740$	−44.0406	−1.61896
$741$	19.8853	0.730503
$742$	85.8877	3.15304
$743$	−21.9228	−0.804269	−0.402135	−	0.915581i	$-0.631732\pi$
$743$	−21.9228	−0.804269	−0.402135	+	0.915581i	$0.631732\pi$
$744$	33.9390	1.24426
$745$	−2.40660	−0.0881710
$746$	−101.901	−3.73087
$747$	3.66616	0.134138
$748$	3.83707	0.140297
$749$	2.51241	0.0918016
$750$	−46.8447	−1.71053
$751$	6.23347	0.227463	0.113731	−	0.993512i	$-0.463720\pi$
$751$	6.23347	0.227463	0.113731	+	0.993512i	$0.463720\pi$
$752$	−3.76816	−0.137410
$753$	−3.68319	−0.134223
$754$	−4.10885	−0.149636
$755$	9.87305	0.359317
$756$	14.9170	0.542526
$757$	−41.5934	−1.51174	−0.755870	−	0.654722i	$-0.772786\pi$
$757$	−41.5934	−1.51174	−0.755870	+	0.654722i	$0.772786\pi$
$758$	41.8595	1.52041
$759$	1.64702	0.0597829
$760$	−114.245	−4.14411
$761$	−42.8581	−1.55361	−0.776803	−	0.629743i	$-0.783160\pi$
$761$	−42.8581	−1.55361	−0.776803	+	0.629743i	$0.783160\pi$
$762$	−46.8854	−1.69848
$763$	−33.6462	−1.21808
$764$	−142.940	−5.17140
$765$	3.79788	0.137313
$766$	−20.4083	−0.737381
$767$	−45.1182	−1.62912
$768$	−95.0151	−3.42856
$769$	30.1074	1.08570	0.542850	−	0.839829i	$-0.317345\pi$
$769$	30.1074	1.08570	0.542850	+	0.839829i	$0.317345\pi$
$770$	−18.1814	−0.655211
$771$	20.3107	0.731474
$772$	−32.7698	−1.17941
$773$	11.0782	0.398455	0.199228	−	0.979953i	$-0.436157\pi$
$773$	11.0782	0.398455	0.199228	+	0.979953i	$0.436157\pi$
$774$	−10.2223	−0.367431
$775$	−30.3981	−1.09193
$776$	−97.9509	−3.51623
$777$	−5.18899	−0.186154
$778$	101.619	3.64322
$779$	−8.95373	−0.320801
$780$	−152.515	−5.46092
$781$	−6.78090	−0.242640
$782$	−6.90983	−0.247095
$783$	0.211880	0.00757197
$784$	−5.78062	−0.206451
$785$	−3.79788	−0.135552
$786$	−40.8928	−1.45860
$787$	20.4996	0.730733	0.365367	−	0.930864i	$-0.380944\pi$
$787$	20.4996	0.730733	0.365367	+	0.930864i	$0.380944\pi$
$788$	−98.1553	−3.49664
$789$	−25.4289	−0.905295
$790$	68.7579	2.44630
$791$	33.5574	1.19316
$792$	6.99241	0.248465
$793$	−22.5529	−0.800876
$794$	−20.7132	−0.735083
$795$	45.2826	1.60601
$796$	51.9380	1.84089
$797$	24.2092	0.857534	0.428767	−	0.903415i	$-0.358948\pi$
$797$	24.2092	0.857534	0.428767	+	0.903415i	$0.358948\pi$
$798$	−20.5947	−0.729044
$799$	−0.211833	−0.00749412
$800$	269.081	9.51344
$801$	−6.87978	−0.243085
$802$	72.6668	2.56595
$803$	−5.81663	−0.205264
$804$	−16.3102	−0.575215
$805$	24.3176	0.857084
$806$	−62.5529	−2.20333
$807$	8.37766	0.294908
$808$	38.2292	1.34490
$809$	−49.6525	−1.74569	−0.872844	−	0.488000i	$-0.837727\pi$
$809$	−49.6525	−1.74569	−0.872844	+	0.488000i	$0.837727\pi$
$810$	10.5890	0.372060
$811$	36.0917	1.26735	0.633675	−	0.773599i	$-0.281546\pi$
$811$	36.0917	1.26735	0.633675	+	0.773599i	$0.281546\pi$
$812$	3.16061	0.110916
$813$	−27.4330	−0.962118
$814$	−3.72147	−0.130438
$815$	−1.87159	−0.0655590
$816$	−17.7883	−0.622716
$817$	10.4821	0.366721
$818$	109.802	3.83912
$819$	−17.9698	−0.627916
$820$	68.6730	2.39817
$821$	−1.30555	−0.0455640	−0.0227820	−	0.999740i	$-0.507252\pi$
$821$	−1.30555	−0.0455640	−0.0227820	+	0.999740i	$0.507252\pi$
$822$	−44.7567	−1.56107
$823$	−11.9659	−0.417106	−0.208553	−	0.978011i	$-0.566875\pi$
$823$	−11.9659	−0.417106	−0.208553	+	0.978011i	$0.566875\pi$
$824$	150.236	5.23371
$825$	−6.26289	−0.218046
$826$	46.7278	1.62587
$827$	−11.4925	−0.399632	−0.199816	−	0.979833i	$-0.564034\pi$
$827$	−11.4925	−0.399632	−0.199816	+	0.979833i	$0.564034\pi$
$828$	−14.3090	−0.497271
$829$	−19.2017	−0.666904	−0.333452	−	0.942767i	$-0.608214\pi$
$829$	−19.2017	−0.666904	−0.333452	+	0.942767i	$0.608214\pi$
$830$	38.8210	1.34750
$831$	−8.72498	−0.302666
$832$	306.264	10.6178
$833$	−0.324967	−0.0112594
$834$	56.3337	1.95068
$835$	7.27820	0.251872
$836$	−10.9702	−0.379411
$837$	3.22564	0.111495
$838$	−83.4145	−2.88151
$839$	−9.47175	−0.327001	−0.163501	−	0.986543i	$-0.552279\pi$
$839$	−9.47175	−0.327001	−0.163501	+	0.986543i	$0.552279\pi$
$840$	103.241	3.56214
$841$	−28.9551	−0.998452
$842$	−39.3690	−1.35674
$843$	−27.4762	−0.946329
$844$	−93.5938	−3.22163
$845$	134.355	4.62197
$846$	−0.590620	−0.0203059
$847$	27.2786	0.937304
$848$	−212.092	−7.28328
$849$	2.70295	0.0927649
$850$	26.2751	0.901229
$851$	4.97748	0.170626
$852$	58.9113	2.01827
$853$	1.43719	0.0492086	0.0246043	−	0.999697i	$-0.492167\pi$
$853$	1.43719	0.0492086	0.0246043	+	0.999697i	$0.492167\pi$
$854$	23.3575	0.799276
$855$	−10.8581	−0.371341
$856$	−10.2317	−0.349712
$857$	−23.8314	−0.814063	−0.407032	−	0.913414i	$-0.633436\pi$
$857$	−23.8314	−0.814063	−0.407032	+	0.913414i	$0.633436\pi$
$858$	−12.8877	−0.439979
$859$	1.57339	0.0536835	0.0268417	−	0.999640i	$-0.491455\pi$
$859$	1.57339	0.0536835	0.0268417	+	0.999640i	$0.491455\pi$
$860$	−80.3950	−2.74145
$861$	8.09126	0.275750
$862$	76.8121	2.61623
$863$	3.73957	0.127297	0.0636483	−	0.997972i	$-0.479726\pi$
$863$	3.73957	0.127297	0.0636483	+	0.997972i	$0.479726\pi$
$864$	−28.5530	−0.971394
$865$	−40.3418	−1.37166
$866$	−79.2811	−2.69408
$867$	−1.00000	−0.0339618
$868$	48.1170	1.63320
$869$	4.31531	0.146387
$870$	2.24360	0.0760651
$871$	19.6481	0.665750
$872$	137.023	4.64017
$873$	−9.30948	−0.315078
$874$	19.7552	0.668230
$875$	−43.4083	−1.46747
$876$	50.5338	1.70738
$877$	3.77336	0.127417	0.0637086	−	0.997969i	$-0.479707\pi$
$877$	3.77336	0.127417	0.0637086	+	0.997969i	$0.479707\pi$
$878$	−95.9038	−3.23660
$879$	−14.9056	−0.502753
$880$	44.8973	1.51349
$881$	−22.2606	−0.749980	−0.374990	−	0.927029i	$-0.622354\pi$
$881$	−22.2606	−0.749980	−0.374990	+	0.927029i	$0.622354\pi$
$882$	−0.906053	−0.0305084
$883$	−0.539858	−0.0181677	−0.00908383	−	0.999959i	$-0.502892\pi$
$883$	−0.539858	−0.0181677	−0.00908383	+	0.999959i	$0.502892\pi$
$884$	40.1580	1.35066
$885$	24.6363	0.828141
$886$	−86.0365	−2.89045
$887$	8.28447	0.278165	0.139083	−	0.990281i	$-0.455585\pi$
$887$	8.28447	0.278165	0.139083	+	0.990281i	$0.455585\pi$
$888$	21.1319	0.709142
$889$	−43.4460	−1.45713
$890$	−72.8501	−2.44194
$891$	0.664575	0.0222641
$892$	−29.7661	−0.996644
$893$	0.605631	0.0202667
$894$	1.76676	0.0590892
$895$	−74.9602	−2.50564
$896$	−169.651	−5.66765
$897$	17.2373	0.575538
$898$	38.6044	1.28825
$899$	0.683449	0.0227943
$900$	54.4108	1.81369
$901$	−11.9231	−0.397217
$902$	5.80294	0.193217
$903$	−9.47238	−0.315221
$904$	−136.661	−4.54528
$905$	67.7007	2.25045
$906$	−7.24810	−0.240802
$907$	45.6384	1.51540	0.757699	−	0.652604i	$-0.226324\pi$
$907$	45.6384	1.51540	0.757699	+	0.652604i	$0.226324\pi$
$908$	56.7105	1.88200
$909$	3.63339	0.120512
$910$	−190.283	−6.30781
$911$	−0.378322	−0.0125344	−0.00626719	−	0.999980i	$-0.501995\pi$
$911$	−0.378322	−0.0125344	−0.00626719	+	0.999980i	$0.501995\pi$
$912$	50.8568	1.68404
$913$	2.43644	0.0806344
$914$	54.2593	1.79474
$915$	12.3148	0.407114
$916$	116.347	3.84421
$917$	−37.8931	−1.25134
$918$	−2.78814	−0.0920223
$919$	40.2507	1.32775	0.663873	−	0.747845i	$-0.268911\pi$
$919$	40.2507	1.32775	0.663873	+	0.747845i	$0.268911\pi$
$920$	−99.0325	−3.26500
$921$	32.1155	1.05824
$922$	−27.3338	−0.900191
$923$	−70.9676	−2.33593
$924$	9.91348	0.326129
$925$	−18.9272	−0.622323
$926$	−27.5479	−0.905281
$927$	14.2788	0.468976
$928$	−6.04981	−0.198595
$929$	−11.1844	−0.366947	−0.183474	−	0.983025i	$-0.558734\pi$
$929$	−11.1844	−0.366947	−0.183474	+	0.983025i	$0.558734\pi$
$930$	34.1564	1.12003
$931$	0.929081	0.0304494
$932$	116.995	3.83228
$933$	20.1670	0.660236
$934$	−32.9272	−1.07741
$935$	2.52398	0.0825429
$936$	73.1812	2.39200
$937$	−22.3123	−0.728910	−0.364455	−	0.931221i	$-0.618745\pi$
$937$	−22.3123	−0.728910	−0.364455	+	0.931221i	$0.618745\pi$
$938$	−20.3490	−0.664420
$939$	−23.3757	−0.762839
$940$	−4.64505	−0.151505
$941$	5.02399	0.163777	0.0818887	−	0.996641i	$-0.473905\pi$
$941$	5.02399	0.163777	0.0818887	+	0.996641i	$0.473905\pi$
$942$	2.78814	0.0908424
$943$	−7.76145	−0.252748
$944$	−115.390	−3.75564
$945$	9.81224	0.319192
$946$	−6.79346	−0.220874
$947$	−5.83865	−0.189731	−0.0948654	−	0.995490i	$-0.530242\pi$
$947$	−5.83865	−0.189731	−0.0948654	+	0.995490i	$0.530242\pi$
$948$	−37.4906	−1.21764
$949$	−60.8757	−1.97611
$950$	−75.1206	−2.43723
$951$	−20.5557	−0.666564
$952$	−27.1838	−0.881031
$953$	35.7087	1.15672	0.578359	−	0.815783i	$-0.303693\pi$
$953$	35.7087	1.15672	0.578359	+	0.815783i	$0.303693\pi$
$954$	−33.2433	−1.07629
$955$	−94.0245	−3.04256
$956$	119.437	3.86286
$957$	0.140810	0.00455175
$958$	−34.6141	−1.11833
$959$	−41.4735	−1.33925
$960$	−167.233	−5.39741
$961$	−20.5952	−0.664362
$962$	−38.9482	−1.25574
$963$	−0.972444	−0.0313366
$964$	−10.9191	−0.351682
$965$	−21.5556	−0.693899
$966$	−17.8523	−0.574388
$967$	−4.48427	−0.144204	−0.0721022	−	0.997397i	$-0.522971\pi$
$967$	−4.48427	−0.144204	−0.0721022	+	0.997397i	$0.522971\pi$
$968$	−111.091	−3.57060
$969$	2.85900	0.0918443
$970$	−98.5782	−3.16515
$971$	−4.30204	−0.138059	−0.0690295	−	0.997615i	$-0.521990\pi$
$971$	−4.30204	−0.138059	−0.0690295	+	0.997615i	$0.521990\pi$
$972$	−5.77371	−0.185192
$973$	52.2012	1.67349
$974$	−0.744568	−0.0238575
$975$	−65.5462	−2.09916
$976$	−57.6793	−1.84627
$977$	21.1506	0.676667	0.338333	−	0.941026i	$-0.390137\pi$
$977$	21.1506	0.676667	0.338333	+	0.941026i	$0.390137\pi$
$978$	1.37399	0.0439354
$979$	−4.57213	−0.146126
$980$	−7.12584	−0.227626
$981$	13.0230	0.415791
$982$	48.4023	1.54458
$983$	−48.9858	−1.56241	−0.781203	−	0.624278i	$-0.785393\pi$
$983$	−48.9858	−1.56241	−0.781203	+	0.624278i	$0.785393\pi$
$984$	−32.9513	−1.05045
$985$	−64.5654	−2.05723
$986$	−0.590750	−0.0188133
$987$	−0.547294	−0.0174206
$988$	−114.812	−3.65265
$989$	9.08627	0.288927
$990$	7.03720	0.223657
$991$	−23.1010	−0.733828	−0.366914	−	0.930255i	$-0.619586\pi$
$991$	−23.1010	−0.733828	−0.366914	+	0.930255i	$0.619586\pi$
$992$	−92.1019	−2.92424
$993$	24.0978	0.764721
$994$	73.4995	2.33126
$995$	34.1642	1.08308
$996$	−21.1674	−0.670714
$997$	−51.8978	−1.64362	−0.821810	−	0.569761i	$-0.807036\pi$
$997$	−51.8978	−1.64362	−0.821810	+	0.569761i	$0.807036\pi$
$998$	−6.36870	−0.201598
$999$	2.00843	0.0635439

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.j.1.64	✓	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.j.1.64	✓	64	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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q+2.78814 q^{2} -1.00000 q^{3} +5.77371 q^{4} +3.79788 q^{5} -2.78814 q^{6} -2.58361 q^{7} +10.5216 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.78814 q^{2} -1.00000 q^{3} +5.77371 q^{4} +3.79788 q^{5} -2.78814 q^{6} -2.58361 q^{7} +10.5216 q^{8} +1.00000 q^{9} +10.5890 q^{10} +0.664575 q^{11} -5.77371 q^{12} +6.95532 q^{13} -7.20346 q^{14} -3.79788 q^{15} +17.7883 q^{16} +1.00000 q^{17} +2.78814 q^{18} -2.85900 q^{19} +21.9279 q^{20} +2.58361 q^{21} +1.85293 q^{22} -2.47830 q^{23} -10.5216 q^{24} +9.42389 q^{25} +19.3924 q^{26} -1.00000 q^{27} -14.9170 q^{28} -0.211880 q^{29} -10.5890 q^{30} -3.22564 q^{31} +28.5530 q^{32} -0.664575 q^{33} +2.78814 q^{34} -9.81224 q^{35} +5.77371 q^{36} -2.00843 q^{37} -7.97129 q^{38} -6.95532 q^{39} +39.9599 q^{40} +3.13177 q^{41} +7.20346 q^{42} -3.66634 q^{43} +3.83707 q^{44} +3.79788 q^{45} -6.90983 q^{46} -0.211833 q^{47} -17.7883 q^{48} -0.324967 q^{49} +26.2751 q^{50} -1.00000 q^{51} +40.1580 q^{52} -11.9231 q^{53} -2.78814 q^{54} +2.52398 q^{55} -27.1838 q^{56} +2.85900 q^{57} -0.590750 q^{58} -6.48686 q^{59} -21.9279 q^{60} -3.24254 q^{61} -8.99354 q^{62} -2.58361 q^{63} +44.0331 q^{64} +26.4155 q^{65} -1.85293 q^{66} +2.82490 q^{67} +5.77371 q^{68} +2.47830 q^{69} -27.3579 q^{70} -10.2034 q^{71} +10.5216 q^{72} -8.75240 q^{73} -5.59978 q^{74} -9.42389 q^{75} -16.5070 q^{76} -1.71700 q^{77} -19.3924 q^{78} +6.49333 q^{79} +67.5579 q^{80} +1.00000 q^{81} +8.73180 q^{82} +3.66616 q^{83} +14.9170 q^{84} +3.79788 q^{85} -10.2223 q^{86} +0.211880 q^{87} +6.99241 q^{88} -6.87978 q^{89} +10.5890 q^{90} -17.9698 q^{91} -14.3090 q^{92} +3.22564 q^{93} -0.590620 q^{94} -10.8581 q^{95} -28.5530 q^{96} -9.30948 q^{97} -0.906053 q^{98} +0.664575 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 5 q^{2} - 64 q^{3} + 77 q^{4} - 3 q^{5} - 5 q^{6} + 5 q^{7} + 18 q^{8} + 64 q^{9}+O(q^{10}) \) 64 * q + 5 * q^2 - 64 * q^3 + 77 * q^4 - 3 * q^5 - 5 * q^6 + 5 * q^7 + 18 * q^8 + 64 * q^9	\( 64 q + 5 q^{2} - 64 q^{3} + 77 q^{4} - 3 q^{5} - 5 q^{6} + 5 q^{7} + 18 q^{8} + 64 q^{9} + 12 q^{10} - 7 q^{11} - 77 q^{12} + 24 q^{13} - 14 q^{14} + 3 q^{15} + 103 q^{16} + 64 q^{17} + 5 q^{18} + 26 q^{19} - 24 q^{20} - 5 q^{21} + 25 q^{22} + 20 q^{23} - 18 q^{24} + 141 q^{25} + 9 q^{26} - 64 q^{27} + 14 q^{28} + 5 q^{29} - 12 q^{30} + 11 q^{31} + 31 q^{32} + 7 q^{33} + 5 q^{34} - 3 q^{35} + 77 q^{36} + 50 q^{37} + 8 q^{38} - 24 q^{39} + 28 q^{40} - 9 q^{41} + 14 q^{42} + 59 q^{43} - 6 q^{44} - 3 q^{45} + 11 q^{47} - 103 q^{48} + 163 q^{49} + 20 q^{50} - 64 q^{51} + 65 q^{52} + 39 q^{53} - 5 q^{54} + 35 q^{55} - 34 q^{56} - 26 q^{57} - 27 q^{58} - 65 q^{59} + 24 q^{60} + 15 q^{61} + 18 q^{62} + 5 q^{63} + 152 q^{64} + 49 q^{65} - 25 q^{66} + 56 q^{67} + 77 q^{68} - 20 q^{69} + 28 q^{70} - 18 q^{71} + 18 q^{72} + 37 q^{73} - 76 q^{74} - 141 q^{75} + 30 q^{76} + 80 q^{77} - 9 q^{78} + 20 q^{79} - 144 q^{80} + 64 q^{81} + 27 q^{82} + 3 q^{83} - 14 q^{84} - 3 q^{85} + 12 q^{86} - 5 q^{87} + 108 q^{88} + 42 q^{89} + 12 q^{90} + 25 q^{91} + 18 q^{92} - 11 q^{93} + 60 q^{94} + 42 q^{95} - 31 q^{96} + 72 q^{97} + 18 q^{98} - 7 q^{99}+O(q^{100}) \) 64 * q + 5 * q^2 - 64 * q^3 + 77 * q^4 - 3 * q^5 - 5 * q^6 + 5 * q^7 + 18 * q^8 + 64 * q^9 + 12 * q^10 - 7 * q^11 - 77 * q^12 + 24 * q^13 - 14 * q^14 + 3 * q^15 + 103 * q^16 + 64 * q^17 + 5 * q^18 + 26 * q^19 - 24 * q^20 - 5 * q^21 + 25 * q^22 + 20 * q^23 - 18 * q^24 + 141 * q^25 + 9 * q^26 - 64 * q^27 + 14 * q^28 + 5 * q^29 - 12 * q^30 + 11 * q^31 + 31 * q^32 + 7 * q^33 + 5 * q^34 - 3 * q^35 + 77 * q^36 + 50 * q^37 + 8 * q^38 - 24 * q^39 + 28 * q^40 - 9 * q^41 + 14 * q^42 + 59 * q^43 - 6 * q^44 - 3 * q^45 + 11 * q^47 - 103 * q^48 + 163 * q^49 + 20 * q^50 - 64 * q^51 + 65 * q^52 + 39 * q^53 - 5 * q^54 + 35 * q^55 - 34 * q^56 - 26 * q^57 - 27 * q^58 - 65 * q^59 + 24 * q^60 + 15 * q^61 + 18 * q^62 + 5 * q^63 + 152 * q^64 + 49 * q^65 - 25 * q^66 + 56 * q^67 + 77 * q^68 - 20 * q^69 + 28 * q^70 - 18 * q^71 + 18 * q^72 + 37 * q^73 - 76 * q^74 - 141 * q^75 + 30 * q^76 + 80 * q^77 - 9 * q^78 + 20 * q^79 - 144 * q^80 + 64 * q^81 + 27 * q^82 + 3 * q^83 - 14 * q^84 - 3 * q^85 + 12 * q^86 - 5 * q^87 + 108 * q^88 + 42 * q^89 + 12 * q^90 + 25 * q^91 + 18 * q^92 - 11 * q^93 + 60 * q^94 + 42 * q^95 - 31 * q^96 + 72 * q^97 + 18 * q^98 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more