LMFDB - Embedded newform 8007.2.a.g.1.20

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:	$56$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q-0.882919 q^{2} -1.00000 q^{3} -1.22045 q^{4} -0.454468 q^{5} +0.882919 q^{6} +1.65699 q^{7} +2.84340 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.882919 q^{2} -1.00000 q^{3} -1.22045 q^{4} -0.454468 q^{5} +0.882919 q^{6} +1.65699 q^{7} +2.84340 q^{8} +1.00000 q^{9} +0.401259 q^{10} +5.61736 q^{11} +1.22045 q^{12} -4.13881 q^{13} -1.46298 q^{14} +0.454468 q^{15} -0.0695876 q^{16} -1.00000 q^{17} -0.882919 q^{18} -6.42910 q^{19} +0.554657 q^{20} -1.65699 q^{21} -4.95968 q^{22} +2.54155 q^{23} -2.84340 q^{24} -4.79346 q^{25} +3.65424 q^{26} -1.00000 q^{27} -2.02227 q^{28} +7.34405 q^{29} -0.401259 q^{30} +0.941045 q^{31} -5.62536 q^{32} -5.61736 q^{33} +0.882919 q^{34} -0.753047 q^{35} -1.22045 q^{36} -4.50783 q^{37} +5.67637 q^{38} +4.13881 q^{39} -1.29224 q^{40} +8.13631 q^{41} +1.46298 q^{42} -0.294939 q^{43} -6.85573 q^{44} -0.454468 q^{45} -2.24398 q^{46} +6.62138 q^{47} +0.0695876 q^{48} -4.25440 q^{49} +4.23224 q^{50} +1.00000 q^{51} +5.05123 q^{52} +13.6603 q^{53} +0.882919 q^{54} -2.55291 q^{55} +4.71147 q^{56} +6.42910 q^{57} -6.48421 q^{58} -8.44326 q^{59} -0.554657 q^{60} +5.43598 q^{61} -0.830867 q^{62} +1.65699 q^{63} +5.10592 q^{64} +1.88096 q^{65} +4.95968 q^{66} +1.26198 q^{67} +1.22045 q^{68} -2.54155 q^{69} +0.664880 q^{70} +8.78828 q^{71} +2.84340 q^{72} +8.71576 q^{73} +3.98005 q^{74} +4.79346 q^{75} +7.84641 q^{76} +9.30788 q^{77} -3.65424 q^{78} -0.271962 q^{79} +0.0316254 q^{80} +1.00000 q^{81} -7.18370 q^{82} -4.61340 q^{83} +2.02227 q^{84} +0.454468 q^{85} +0.260407 q^{86} -7.34405 q^{87} +15.9724 q^{88} +1.64256 q^{89} +0.401259 q^{90} -6.85795 q^{91} -3.10184 q^{92} -0.941045 q^{93} -5.84615 q^{94} +2.92182 q^{95} +5.62536 q^{96} -14.5911 q^{97} +3.75629 q^{98} +5.61736 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + q^{2} - 56 q^{3} + 61 q^{4} + q^{5} - q^{6} + 19 q^{7} + 56 q^{9}+O(q^{10}) $ 56 * q + q^2 - 56 * q^3 + 61 * q^4 + q^5 - q^6 + 19 * q^7 + 56 * q^9	\( 56 q + q^{2} - 56 q^{3} + 61 q^{4} + q^{5} - q^{6} + 19 q^{7} + 56 q^{9} + 8 q^{10} - 7 q^{11} - 61 q^{12} + 8 q^{13} - 8 q^{14} - q^{15} + 71 q^{16} - 56 q^{17} + q^{18} - 2 q^{19} - 4 q^{20} - 19 q^{21} + 47 q^{22} + 16 q^{23} + 85 q^{25} - 11 q^{26} - 56 q^{27} + 52 q^{28} + 17 q^{29} - 8 q^{30} + 23 q^{31} + 11 q^{32} + 7 q^{33} - q^{34} - 41 q^{35} + 61 q^{36} + 58 q^{37} - 22 q^{38} - 8 q^{39} + 38 q^{40} - q^{41} + 8 q^{42} + 27 q^{43} + 2 q^{44} + q^{45} + 46 q^{46} + 5 q^{47} - 71 q^{48} + 59 q^{49} - 4 q^{50} + 56 q^{51} + 25 q^{52} + 15 q^{53} - q^{54} + 9 q^{55} - 36 q^{56} + 2 q^{57} + 89 q^{58} - 61 q^{59} + 4 q^{60} + 47 q^{61} + 8 q^{62} + 19 q^{63} + 88 q^{64} + 39 q^{65} - 47 q^{66} + 20 q^{67} - 61 q^{68} - 16 q^{69} + 36 q^{70} - 2 q^{71} + 93 q^{73} + 48 q^{74} - 85 q^{75} + 38 q^{76} + 26 q^{77} + 11 q^{78} + 72 q^{79} + 42 q^{80} + 56 q^{81} + 33 q^{82} - 11 q^{83} - 52 q^{84} - q^{85} - 4 q^{86} - 17 q^{87} + 130 q^{88} - 6 q^{89} + 8 q^{90} + 37 q^{91} + 132 q^{92} - 23 q^{93} - 32 q^{94} + 12 q^{95} - 11 q^{96} + 100 q^{97} + 42 q^{98} - 7 q^{99}+O(q^{100}) \) 56 * q + q^2 - 56 * q^3 + 61 * q^4 + q^5 - q^6 + 19 * q^7 + 56 * q^9 + 8 * q^10 - 7 * q^11 - 61 * q^12 + 8 * q^13 - 8 * q^14 - q^15 + 71 * q^16 - 56 * q^17 + q^18 - 2 * q^19 - 4 * q^20 - 19 * q^21 + 47 * q^22 + 16 * q^23 + 85 * q^25 - 11 * q^26 - 56 * q^27 + 52 * q^28 + 17 * q^29 - 8 * q^30 + 23 * q^31 + 11 * q^32 + 7 * q^33 - q^34 - 41 * q^35 + 61 * q^36 + 58 * q^37 - 22 * q^38 - 8 * q^39 + 38 * q^40 - q^41 + 8 * q^42 + 27 * q^43 + 2 * q^44 + q^45 + 46 * q^46 + 5 * q^47 - 71 * q^48 + 59 * q^49 - 4 * q^50 + 56 * q^51 + 25 * q^52 + 15 * q^53 - q^54 + 9 * q^55 - 36 * q^56 + 2 * q^57 + 89 * q^58 - 61 * q^59 + 4 * q^60 + 47 * q^61 + 8 * q^62 + 19 * q^63 + 88 * q^64 + 39 * q^65 - 47 * q^66 + 20 * q^67 - 61 * q^68 - 16 * q^69 + 36 * q^70 - 2 * q^71 + 93 * q^73 + 48 * q^74 - 85 * q^75 + 38 * q^76 + 26 * q^77 + 11 * q^78 + 72 * q^79 + 42 * q^80 + 56 * q^81 + 33 * q^82 - 11 * q^83 - 52 * q^84 - q^85 - 4 * q^86 - 17 * q^87 + 130 * q^88 - 6 * q^89 + 8 * q^90 + 37 * q^91 + 132 * q^92 - 23 * q^93 - 32 * q^94 + 12 * q^95 - 11 * q^96 + 100 * q^97 + 42 * 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 $

$2$	−0.882919	−0.624318	−0.312159	−	0.950030i	$-0.601052\pi$
$2$	−0.882919	−0.624318	−0.312159	+	0.950030i	$0.601052\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.22045	−0.610227
$5$	−0.454468	−0.203244	−0.101622	−	0.994823i	$-0.532403\pi$
$5$	−0.454468	−0.203244	−0.101622	+	0.994823i	$0.532403\pi$
$6$	0.882919	0.360450
$7$	1.65699	0.626281	0.313141	−	0.949707i	$-0.398619\pi$
$7$	1.65699	0.626281	0.313141	+	0.949707i	$0.398619\pi$
$8$	2.84340	1.00529
$9$	1.00000	0.333333
$10$	0.401259	0.126889
$11$	5.61736	1.69370	0.846849	−	0.531833i	$-0.178497\pi$
$11$	5.61736	1.69370	0.846849	+	0.531833i	$0.178497\pi$
$12$	1.22045	0.352314
$13$	−4.13881	−1.14790	−0.573950	−	0.818890i	$-0.694590\pi$
$13$	−4.13881	−1.14790	−0.573950	+	0.818890i	$0.694590\pi$
$14$	−1.46298	−0.390999
$15$	0.454468	0.117343
$16$	−0.0695876	−0.0173969
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−0.882919	−0.208106
$19$	−6.42910	−1.47494	−0.737468	−	0.675382i	$-0.763979\pi$
$19$	−6.42910	−1.47494	−0.737468	+	0.675382i	$0.763979\pi$
$20$	0.554657	0.124025
$21$	−1.65699	−0.361584
$22$	−4.95968	−1.05741
$23$	2.54155	0.529949	0.264975	−	0.964255i	$-0.414636\pi$
$23$	2.54155	0.529949	0.264975	+	0.964255i	$0.414636\pi$
$24$	−2.84340	−0.580407
$25$	−4.79346	−0.958692
$26$	3.65424	0.716655
$27$	−1.00000	−0.192450
$28$	−2.02227	−0.382174
$29$	7.34405	1.36376	0.681878	−	0.731466i	$-0.261163\pi$
$29$	7.34405	1.36376	0.681878	+	0.731466i	$0.261163\pi$
$30$	−0.401259	−0.0732595
$31$	0.941045	0.169017	0.0845084	−	0.996423i	$-0.473068\pi$
$31$	0.941045	0.169017	0.0845084	+	0.996423i	$0.473068\pi$
$32$	−5.62536	−0.994433
$33$	−5.61736	−0.977857
$34$	0.882919	0.151419
$35$	−0.753047	−0.127288
$36$	−1.22045	−0.203409
$37$	−4.50783	−0.741083	−0.370541	−	0.928816i	$-0.620828\pi$
$37$	−4.50783	−0.741083	−0.370541	+	0.928816i	$0.620828\pi$
$38$	5.67637	0.920830
$39$	4.13881	0.662740
$40$	−1.29224	−0.204320
$41$	8.13631	1.27068	0.635339	−	0.772233i	$-0.280860\pi$
$41$	8.13631	1.27068	0.635339	+	0.772233i	$0.280860\pi$
$42$	1.46298	0.225743
$43$	−0.294939	−0.0449777	−0.0224889	−	0.999747i	$-0.507159\pi$
$43$	−0.294939	−0.0449777	−0.0224889	+	0.999747i	$0.507159\pi$
$44$	−6.85573	−1.03354
$45$	−0.454468	−0.0677481
$46$	−2.24398	−0.330857
$47$	6.62138	0.965828	0.482914	−	0.875668i	$-0.339578\pi$
$47$	6.62138	0.965828	0.482914	+	0.875668i	$0.339578\pi$
$48$	0.0695876	0.0100441
$49$	−4.25440	−0.607771
$50$	4.23224	0.598529
$51$	1.00000	0.140028
$52$	5.05123	0.700479
$53$	13.6603	1.87638	0.938191	−	0.346117i	$-0.112500\pi$
$53$	13.6603	1.87638	0.938191	+	0.346117i	$0.112500\pi$
$54$	0.882919	0.120150
$55$	−2.55291	−0.344235
$56$	4.71147	0.629597
$57$	6.42910	0.851555
$58$	−6.48421	−0.851418
$59$	−8.44326	−1.09922	−0.549609	−	0.835422i	$-0.685223\pi$
$59$	−8.44326	−1.09922	−0.549609	+	0.835422i	$0.685223\pi$
$60$	−0.554657	−0.0716060
$61$	5.43598	0.696006	0.348003	−	0.937493i	$-0.386860\pi$
$61$	5.43598	0.696006	0.348003	+	0.937493i	$0.386860\pi$
$62$	−0.830867	−0.105520
$63$	1.65699	0.208760
$64$	5.10592	0.638240
$65$	1.88096	0.233304
$66$	4.95968	0.610494
$67$	1.26198	0.154176	0.0770878	−	0.997024i	$-0.475438\pi$
$67$	1.26198	0.154176	0.0770878	+	0.997024i	$0.475438\pi$
$68$	1.22045	0.148002
$69$	−2.54155	−0.305966
$70$	0.664880	0.0794684
$71$	8.78828	1.04298	0.521488	−	0.853258i	$-0.325377\pi$
$71$	8.78828	1.04298	0.521488	+	0.853258i	$0.325377\pi$
$72$	2.84340	0.335098
$73$	8.71576	1.02010	0.510051	−	0.860144i	$-0.329626\pi$
$73$	8.71576	1.02010	0.510051	+	0.860144i	$0.329626\pi$
$74$	3.98005	0.462672
$75$	4.79346	0.553501
$76$	7.84641	0.900045
$77$	9.30788	1.06073
$78$	−3.65424	−0.413761
$79$	−0.271962	−0.0305981	−0.0152990	−	0.999883i	$-0.504870\pi$
$79$	−0.271962	−0.0305981	−0.0152990	+	0.999883i	$0.504870\pi$
$80$	0.0316254	0.00353583
$81$	1.00000	0.111111
$82$	−7.18370	−0.793307
$83$	−4.61340	−0.506386	−0.253193	−	0.967416i	$-0.581481\pi$
$83$	−4.61340	−0.506386	−0.253193	+	0.967416i	$0.581481\pi$
$84$	2.02227	0.220648
$85$	0.454468	0.0492940
$86$	0.260407	0.0280804
$87$	−7.34405	−0.787365
$88$	15.9724	1.70266
$89$	1.64256	0.174111	0.0870557	−	0.996203i	$-0.472254\pi$
$89$	1.64256	0.174111	0.0870557	+	0.996203i	$0.472254\pi$
$90$	0.401259	0.0422964
$91$	−6.85795	−0.718909
$92$	−3.10184	−0.323389
$93$	−0.941045	−0.0975819
$94$	−5.84615	−0.602984
$95$	2.92182	0.299773
$96$	5.62536	0.574136
$97$	−14.5911	−1.48150	−0.740749	−	0.671782i	$-0.765529\pi$
$97$	−14.5911	−1.48150	−0.740749	+	0.671782i	$0.765529\pi$
$98$	3.75629	0.379443
$99$	5.61736	0.564566
$100$	5.85019	0.585019
$101$	−18.9889	−1.88947	−0.944735	−	0.327836i	$-0.893681\pi$
$101$	−18.9889	−1.88947	−0.944735	+	0.327836i	$0.893681\pi$
$102$	−0.882919	−0.0874221
$103$	8.20097	0.808066	0.404033	−	0.914744i	$-0.367608\pi$
$103$	8.20097	0.808066	0.404033	+	0.914744i	$0.367608\pi$
$104$	−11.7683	−1.15398
$105$	0.753047	0.0734899
$106$	−12.0609	−1.17146
$107$	−3.24643	−0.313844	−0.156922	−	0.987611i	$-0.550157\pi$
$107$	−3.24643	−0.313844	−0.156922	+	0.987611i	$0.550157\pi$
$108$	1.22045	0.117438
$109$	−14.2296	−1.36295	−0.681474	−	0.731842i	$-0.738661\pi$
$109$	−14.2296	−1.36295	−0.681474	+	0.731842i	$0.738661\pi$
$110$	2.25402	0.214912
$111$	4.50783	0.427864
$112$	−0.115306	−0.0108954
$113$	−10.7777	−1.01388	−0.506941	−	0.861981i	$-0.669224\pi$
$113$	−10.7777	−1.01388	−0.506941	+	0.861981i	$0.669224\pi$
$114$	−5.67637	−0.531641
$115$	−1.15505	−0.107709
$116$	−8.96307	−0.832200
$117$	−4.13881	−0.382633
$118$	7.45471	0.686262
$119$	−1.65699	−0.151896
$120$	1.29224	0.117964
$121$	20.5547	1.86861
$122$	−4.79953	−0.434529
$123$	−8.13631	−0.733626
$124$	−1.14850	−0.103139
$125$	4.45082	0.398093
$126$	−1.46298	−0.130333
$127$	14.4501	1.28224	0.641119	−	0.767442i	$-0.278471\pi$
$127$	14.4501	1.28224	0.641119	+	0.767442i	$0.278471\pi$
$128$	6.74261	0.595968
$129$	0.294939	0.0259679
$130$	−1.66074	−0.145656
$131$	−2.22438	−0.194345	−0.0971727	−	0.995268i	$-0.530980\pi$
$131$	−2.22438	−0.194345	−0.0971727	+	0.995268i	$0.530980\pi$
$132$	6.85573	0.596714
$133$	−10.6529	−0.923725
$134$	−1.11423	−0.0962546
$135$	0.454468	0.0391144
$136$	−2.84340	−0.243820
$137$	3.06936	0.262233	0.131117	−	0.991367i	$-0.458144\pi$
$137$	3.06936	0.262233	0.131117	+	0.991367i	$0.458144\pi$
$138$	2.24398	0.191020
$139$	5.15575	0.437305	0.218653	−	0.975803i	$-0.429834\pi$
$139$	5.15575	0.437305	0.218653	+	0.975803i	$0.429834\pi$
$140$	0.919059	0.0776747
$141$	−6.62138	−0.557621
$142$	−7.75934	−0.651149
$143$	−23.2492	−1.94420
$144$	−0.0695876	−0.00579897
$145$	−3.33764	−0.277176
$146$	−7.69532	−0.636869
$147$	4.25440	0.350897
$148$	5.50160	0.452228
$149$	14.8430	1.21599	0.607995	−	0.793941i	$-0.291974\pi$
$149$	14.8430	1.21599	0.607995	+	0.793941i	$0.291974\pi$
$150$	−4.23224	−0.345561
$151$	−9.04785	−0.736303	−0.368152	−	0.929766i	$-0.620009\pi$
$151$	−9.04785	−0.736303	−0.368152	+	0.929766i	$0.620009\pi$
$152$	−18.2805	−1.48274
$153$	−1.00000	−0.0808452
$154$	−8.21811	−0.662234
$155$	−0.427675	−0.0343517
$156$	−5.05123	−0.404422
$157$	1.00000	0.0798087
$157$	1.00000	0.0798087
$158$	0.240120	0.0191029
$159$	−13.6603	−1.08333
$160$	2.55655	0.202113
$161$	4.21131	0.331897
$162$	−0.882919	−0.0693687
$163$	0.853127	0.0668220	0.0334110	−	0.999442i	$-0.489363\pi$
$163$	0.853127	0.0668220	0.0334110	+	0.999442i	$0.489363\pi$
$164$	−9.92998	−0.775401
$165$	2.55291	0.198744
$166$	4.07326	0.316146
$167$	10.6327	0.822781	0.411390	−	0.911459i	$-0.365043\pi$
$167$	10.6327	0.822781	0.411390	+	0.911459i	$0.365043\pi$
$168$	−4.71147	−0.363498
$169$	4.12977	0.317674
$170$	−0.401259	−0.0307752
$171$	−6.42910	−0.491645
$172$	0.359959	0.0274466
$173$	−24.4342	−1.85770	−0.928848	−	0.370461i	$-0.879200\pi$
$173$	−24.4342	−1.85770	−0.928848	+	0.370461i	$0.879200\pi$
$174$	6.48421	0.491566
$175$	−7.94269	−0.600411
$176$	−0.390899	−0.0294651
$177$	8.44326	0.634634
$178$	−1.45025	−0.108701
$179$	12.6432	0.944996	0.472498	−	0.881332i	$-0.343352\pi$
$179$	12.6432	0.944996	0.472498	+	0.881332i	$0.343352\pi$
$180$	0.554657	0.0413417
$181$	3.45216	0.256597	0.128299	−	0.991736i	$-0.459048\pi$
$181$	3.45216	0.256597	0.128299	+	0.991736i	$0.459048\pi$
$182$	6.05502	0.448828
$183$	−5.43598	−0.401839
$184$	7.22664	0.532755
$185$	2.04867	0.150621
$186$	0.830867	0.0609221
$187$	−5.61736	−0.410782
$188$	−8.08109	−0.589374
$189$	−1.65699	−0.120528
$190$	−2.57973	−0.187153
$191$	23.1914	1.67807	0.839036	−	0.544076i	$-0.183120\pi$
$191$	23.1914	1.67807	0.839036	+	0.544076i	$0.183120\pi$
$192$	−5.10592	−0.368488
$193$	19.8131	1.42618	0.713089	−	0.701074i	$-0.247296\pi$
$193$	19.8131	1.42618	0.713089	+	0.701074i	$0.247296\pi$
$194$	12.8827	0.924927
$195$	−1.88096	−0.134698
$196$	5.19230	0.370878
$197$	−11.1585	−0.795008	−0.397504	−	0.917601i	$-0.630123\pi$
$197$	−11.1585	−0.795008	−0.397504	+	0.917601i	$0.630123\pi$
$198$	−4.95968	−0.352469
$199$	−21.9836	−1.55838	−0.779188	−	0.626790i	$-0.784369\pi$
$199$	−21.9836	−1.55838	−0.779188	+	0.626790i	$0.784369\pi$
$200$	−13.6297	−0.963767
$201$	−1.26198	−0.0890133
$202$	16.7657	1.17963
$203$	12.1690	0.854095
$204$	−1.22045	−0.0854488
$205$	−3.69769	−0.258258
$206$	−7.24080	−0.504490
$207$	2.54155	0.176650
$208$	0.288010	0.0199699
$209$	−36.1146	−2.49810
$210$	−0.664880	−0.0458811
$211$	−0.594850	−0.0409512	−0.0204756	−	0.999790i	$-0.506518\pi$
$211$	−0.594850	−0.0409512	−0.0204756	+	0.999790i	$0.506518\pi$
$212$	−16.6717	−1.14502
$213$	−8.78828	−0.602163
$214$	2.86634	0.195939
$215$	0.134040	0.00914147
$216$	−2.84340	−0.193469
$217$	1.55930	0.105852
$218$	12.5636	0.850913
$219$	−8.71576	−0.588957
$220$	3.11571	0.210061
$221$	4.13881	0.278407
$222$	−3.98005	−0.267124
$223$	−12.9028	−0.864032	−0.432016	−	0.901866i	$-0.642198\pi$
$223$	−12.9028	−0.864032	−0.432016	+	0.901866i	$0.642198\pi$
$224$	−9.32114	−0.622795
$225$	−4.79346	−0.319564
$226$	9.51586	0.632985
$227$	−8.00181	−0.531099	−0.265549	−	0.964097i	$-0.585553\pi$
$227$	−8.00181	−0.531099	−0.265549	+	0.964097i	$0.585553\pi$
$228$	−7.84641	−0.519641
$229$	−15.8577	−1.04790	−0.523952	−	0.851748i	$-0.675543\pi$
$229$	−15.8577	−1.04790	−0.523952	+	0.851748i	$0.675543\pi$
$230$	1.01982	0.0672449
$231$	−9.30788	−0.612414
$232$	20.8821	1.37098
$233$	−4.50553	−0.295167	−0.147584	−	0.989050i	$-0.547150\pi$
$233$	−4.50553	−0.295167	−0.147584	+	0.989050i	$0.547150\pi$
$234$	3.65424	0.238885
$235$	−3.00921	−0.196299
$236$	10.3046	0.670772
$237$	0.271962	0.0176658
$238$	1.46298	0.0948312
$239$	8.00444	0.517764	0.258882	−	0.965909i	$-0.416646\pi$
$239$	8.00444	0.517764	0.258882	+	0.965909i	$0.416646\pi$
$240$	−0.0316254	−0.00204141
$241$	−22.7541	−1.46572	−0.732860	−	0.680379i	$-0.761815\pi$
$241$	−22.7541	−1.46572	−0.732860	+	0.680379i	$0.761815\pi$
$242$	−18.1482	−1.16661
$243$	−1.00000	−0.0641500
$244$	−6.63436	−0.424721
$245$	1.93349	0.123526
$246$	7.18370	0.458016
$247$	26.6088	1.69308
$248$	2.67577	0.169912
$249$	4.61340	0.292362
$250$	−3.92971	−0.248537
$251$	10.0596	0.634958	0.317479	−	0.948265i	$-0.397164\pi$
$251$	10.0596	0.634958	0.317479	+	0.948265i	$0.397164\pi$
$252$	−2.02227	−0.127391
$253$	14.2768	0.897574
$254$	−12.7583	−0.800524
$255$	−0.454468	−0.0284599
$256$	−16.1650	−1.01031
$257$	−1.81285	−0.113083	−0.0565414	−	0.998400i	$-0.518007\pi$
$257$	−1.81285	−0.113083	−0.0565414	+	0.998400i	$0.518007\pi$
$258$	−0.260407	−0.0162122
$259$	−7.46941	−0.464126
$260$	−2.29562	−0.142368
$261$	7.34405	0.454585
$262$	1.96395	0.121333
$263$	15.2117	0.937992	0.468996	−	0.883200i	$-0.344616\pi$
$263$	15.2117	0.937992	0.468996	+	0.883200i	$0.344616\pi$
$264$	−15.9724	−0.983034
$265$	−6.20816	−0.381364
$266$	9.40567	0.576699
$267$	−1.64256	−0.100523
$268$	−1.54019	−0.0940820
$269$	−15.2066	−0.927163	−0.463581	−	0.886054i	$-0.653436\pi$
$269$	−15.2066	−0.927163	−0.463581	+	0.886054i	$0.653436\pi$
$270$	−0.401259	−0.0244198
$271$	−2.53363	−0.153907	−0.0769536	−	0.997035i	$-0.524519\pi$
$271$	−2.53363	−0.153907	−0.0769536	+	0.997035i	$0.524519\pi$
$272$	0.0695876	0.00421937
$273$	6.85795	0.415062
$274$	−2.71000	−0.163717
$275$	−26.9266	−1.62373
$276$	3.10184	0.186709
$277$	21.9329	1.31782	0.658911	−	0.752221i	$-0.271017\pi$
$277$	21.9329	1.31782	0.658911	+	0.752221i	$0.271017\pi$
$278$	−4.55211	−0.273018
$279$	0.941045	0.0563389
$280$	−2.14122	−0.127962
$281$	−9.50537	−0.567043	−0.283521	−	0.958966i	$-0.591503\pi$
$281$	−9.50537	−0.567043	−0.283521	+	0.958966i	$0.591503\pi$
$282$	5.84615	0.348133
$283$	10.1319	0.602279	0.301139	−	0.953580i	$-0.402633\pi$
$283$	10.1319	0.602279	0.301139	+	0.953580i	$0.402633\pi$
$284$	−10.7257	−0.636452
$285$	−2.92182	−0.173074
$286$	20.5272	1.21380
$287$	13.4817	0.795802
$288$	−5.62536	−0.331478
$289$	1.00000	0.0588235
$290$	2.94687	0.173046
$291$	14.5911	0.855343
$292$	−10.6372	−0.622494
$293$	21.3813	1.24911	0.624555	−	0.780981i	$-0.285280\pi$
$293$	21.3813	1.24911	0.624555	+	0.780981i	$0.285280\pi$
$294$	−3.75629	−0.219071
$295$	3.83719	0.223410
$296$	−12.8176	−0.745006
$297$	−5.61736	−0.325952
$298$	−13.1052	−0.759165
$299$	−10.5190	−0.608329
$300$	−5.85019	−0.337761
$301$	−0.488709	−0.0281687
$302$	7.98852	0.459688
$303$	18.9889	1.09089
$304$	0.447386	0.0256593
$305$	−2.47048	−0.141459
$306$	0.882919	0.0504731
$307$	18.4808	1.05476	0.527378	−	0.849631i	$-0.323175\pi$
$307$	18.4808	1.05476	0.527378	+	0.849631i	$0.323175\pi$
$308$	−11.3598	−0.647287
$309$	−8.20097	−0.466537
$310$	0.377603	0.0214464
$311$	17.0791	0.968468	0.484234	−	0.874938i	$-0.339098\pi$
$311$	17.0791	0.968468	0.484234	+	0.874938i	$0.339098\pi$
$312$	11.7683	0.666249
$313$	9.88935	0.558979	0.279490	−	0.960149i	$-0.409835\pi$
$313$	9.88935	0.558979	0.279490	+	0.960149i	$0.409835\pi$
$314$	−0.882919	−0.0498260
$315$	−0.753047	−0.0424294
$316$	0.331917	0.0186718
$317$	7.96400	0.447303	0.223651	−	0.974669i	$-0.428202\pi$
$317$	7.96400	0.447303	0.223651	+	0.974669i	$0.428202\pi$
$318$	12.0609	0.676343
$319$	41.2542	2.30979
$320$	−2.32048	−0.129719
$321$	3.24643	0.181198
$322$	−3.71824	−0.207210
$323$	6.42910	0.357725
$324$	−1.22045	−0.0678030
$325$	19.8392	1.10048
$326$	−0.753242	−0.0417182
$327$	14.2296	0.786898
$328$	23.1348	1.27740
$329$	10.9715	0.604880
$330$	−2.25402	−0.124080
$331$	10.5485	0.579796	0.289898	−	0.957058i	$-0.406379\pi$
$331$	10.5485	0.579796	0.289898	+	0.957058i	$0.406379\pi$
$332$	5.63044	0.309010
$333$	−4.50783	−0.247028
$334$	−9.38779	−0.513677
$335$	−0.573530	−0.0313353
$336$	0.115306	0.00629044
$337$	22.5200	1.22674	0.613370	−	0.789796i	$-0.289813\pi$
$337$	22.5200	1.22674	0.613370	+	0.789796i	$0.289813\pi$
$338$	−3.64625	−0.198330
$339$	10.7777	0.585365
$340$	−0.554657	−0.0300805
$341$	5.28619	0.286263
$342$	5.67637	0.306943
$343$	−18.6484	−1.00692
$344$	−0.838628	−0.0452158
$345$	1.15505	0.0621860
$346$	21.5734	1.15979
$347$	15.5414	0.834304	0.417152	−	0.908837i	$-0.363028\pi$
$347$	15.5414	0.834304	0.417152	+	0.908837i	$0.363028\pi$
$348$	8.96307	0.480471
$349$	4.45413	0.238424	0.119212	−	0.992869i	$-0.461963\pi$
$349$	4.45413	0.238424	0.119212	+	0.992869i	$0.461963\pi$
$350$	7.01276	0.374848
$351$	4.13881	0.220913
$352$	−31.5997	−1.68427
$353$	19.6236	1.04446	0.522230	−	0.852805i	$-0.325100\pi$
$353$	19.6236	1.04446	0.522230	+	0.852805i	$0.325100\pi$
$354$	−7.45471	−0.396214
$355$	−3.99399	−0.211979
$356$	−2.00467	−0.106247
$357$	1.65699	0.0876969
$358$	−11.1629	−0.589978
$359$	−28.1727	−1.48690	−0.743450	−	0.668792i	$-0.766812\pi$
$359$	−28.1727	−1.48690	−0.743450	+	0.668792i	$0.766812\pi$
$360$	−1.29224	−0.0681068
$361$	22.3333	1.17544
$362$	−3.04798	−0.160198
$363$	−20.5547	−1.07884
$364$	8.36981	0.438697
$365$	−3.96104	−0.207330
$366$	4.79953	0.250876
$367$	7.53847	0.393505	0.196753	−	0.980453i	$-0.436960\pi$
$367$	7.53847	0.393505	0.196753	+	0.980453i	$0.436960\pi$
$368$	−0.176860	−0.00921948
$369$	8.13631	0.423559
$370$	−1.80881	−0.0940354
$371$	22.6349	1.17514
$372$	1.14850	0.0595470
$373$	23.3606	1.20957	0.604784	−	0.796390i	$-0.293260\pi$
$373$	23.3606	1.20957	0.604784	+	0.796390i	$0.293260\pi$
$374$	4.95968	0.256459
$375$	−4.45082	−0.229839
$376$	18.8273	0.970941
$377$	−30.3956	−1.56546
$378$	1.46298	0.0752478
$379$	−31.9755	−1.64247	−0.821237	−	0.570587i	$-0.806716\pi$
$379$	−31.9755	−1.64247	−0.821237	+	0.570587i	$0.806716\pi$
$380$	−3.56595	−0.182929
$381$	−14.4501	−0.740300
$382$	−20.4761	−1.04765
$383$	27.7047	1.41564	0.707822	−	0.706391i	$-0.249678\pi$
$383$	27.7047	1.41564	0.707822	+	0.706391i	$0.249678\pi$
$384$	−6.74261	−0.344082
$385$	−4.23014	−0.215588
$386$	−17.4934	−0.890389
$387$	−0.294939	−0.0149926
$388$	17.8077	0.904050
$389$	−34.5498	−1.75175	−0.875873	−	0.482541i	$-0.839714\pi$
$389$	−34.5498	−1.75175	−0.875873	+	0.482541i	$0.839714\pi$
$390$	1.66074	0.0840946
$391$	−2.54155	−0.128532
$392$	−12.0970	−0.610989
$393$	2.22438	0.112205
$394$	9.85203	0.496338
$395$	0.123598	0.00621889
$396$	−6.85573	−0.344513
$397$	−4.49732	−0.225714	−0.112857	−	0.993611i	$-0.536000\pi$
$397$	−4.49732	−0.225714	−0.112857	+	0.993611i	$0.536000\pi$
$398$	19.4098	0.972923
$399$	10.6529	0.533313
$400$	0.333566	0.0166783
$401$	11.5963	0.579093	0.289547	−	0.957164i	$-0.406495\pi$
$401$	11.5963	0.579093	0.289547	+	0.957164i	$0.406495\pi$
$402$	1.11423	0.0555726
$403$	−3.89481	−0.194014
$404$	23.1751	1.15300
$405$	−0.454468	−0.0225827
$406$	−10.7442	−0.533227
$407$	−25.3221	−1.25517
$408$	2.84340	0.140769
$409$	−8.16928	−0.403945	−0.201972	−	0.979391i	$-0.564735\pi$
$409$	−8.16928	−0.403945	−0.201972	+	0.979391i	$0.564735\pi$
$410$	3.26477	0.161235
$411$	−3.06936	−0.151401
$412$	−10.0089	−0.493103
$413$	−13.9903	−0.688420
$414$	−2.24398	−0.110286
$415$	2.09664	0.102920
$416$	23.2823	1.14151
$417$	−5.15575	−0.252478
$418$	31.8862	1.55961
$419$	21.6718	1.05874	0.529368	−	0.848392i	$-0.322429\pi$
$419$	21.6718	1.05874	0.529368	+	0.848392i	$0.322429\pi$
$420$	−0.919059	−0.0448455
$421$	6.40359	0.312092	0.156046	−	0.987750i	$-0.450125\pi$
$421$	6.40359	0.312092	0.156046	+	0.987750i	$0.450125\pi$
$422$	0.525205	0.0255666
$423$	6.62138	0.321943
$424$	38.8416	1.88632
$425$	4.79346	0.232517
$426$	7.75934	0.375941
$427$	9.00734	0.435896
$428$	3.96212	0.191516
$429$	23.2492	1.12248
$430$	−0.118347	−0.00570719
$431$	18.5436	0.893213	0.446607	−	0.894730i	$-0.352632\pi$
$431$	18.5436	0.893213	0.446607	+	0.894730i	$0.352632\pi$
$432$	0.0695876	0.00334804
$433$	8.82278	0.423996	0.211998	−	0.977270i	$-0.432003\pi$
$433$	8.82278	0.423996	0.211998	+	0.977270i	$0.432003\pi$
$434$	−1.37673	−0.0660854
$435$	3.33764	0.160028
$436$	17.3665	0.831707
$437$	−16.3399	−0.781641
$438$	7.69532	0.367696
$439$	26.2810	1.25432	0.627161	−	0.778890i	$-0.284217\pi$
$439$	26.2810	1.25432	0.627161	+	0.778890i	$0.284217\pi$
$440$	−7.25895	−0.346057
$441$	−4.25440	−0.202590
$442$	−3.65424	−0.173814
$443$	−14.1845	−0.673925	−0.336963	−	0.941518i	$-0.609400\pi$
$443$	−14.1845	−0.673925	−0.336963	+	0.941518i	$0.609400\pi$
$444$	−5.50160	−0.261094
$445$	−0.746493	−0.0353872
$446$	11.3921	0.539431
$447$	−14.8430	−0.702052
$448$	8.46043	0.399718
$449$	−17.9149	−0.845455	−0.422728	−	0.906257i	$-0.638927\pi$
$449$	−17.9149	−0.845455	−0.422728	+	0.906257i	$0.638927\pi$
$450$	4.23224	0.199510
$451$	45.7046	2.15214
$452$	13.1537	0.618698
$453$	9.04785	0.425105
$454$	7.06496	0.331575
$455$	3.11672	0.146114
$456$	18.2805	0.856063
$457$	−1.68610	−0.0788724	−0.0394362	−	0.999222i	$-0.512556\pi$
$457$	−1.68610	−0.0788724	−0.0394362	+	0.999222i	$0.512556\pi$
$458$	14.0010	0.654225
$459$	1.00000	0.0466760
$460$	1.40969	0.0657270
$461$	−4.31844	−0.201130	−0.100565	−	0.994931i	$-0.532065\pi$
$461$	−4.31844	−0.201130	−0.100565	+	0.994931i	$0.532065\pi$
$462$	8.21811	0.382341
$463$	18.0316	0.838001	0.419001	−	0.907986i	$-0.362381\pi$
$463$	18.0316	0.838001	0.419001	+	0.907986i	$0.362381\pi$
$464$	−0.511055	−0.0237251
$465$	0.427675	0.0198330
$466$	3.97802	0.184278
$467$	13.4587	0.622794	0.311397	−	0.950280i	$-0.399203\pi$
$467$	13.4587	0.622794	0.311397	+	0.950280i	$0.399203\pi$
$468$	5.05123	0.233493
$469$	2.09108	0.0965573
$470$	2.65689	0.122553
$471$	−1.00000	−0.0460776
$472$	−24.0076	−1.10504
$473$	−1.65678	−0.0761787
$474$	−0.240120	−0.0110291
$475$	30.8176	1.41401
$476$	2.02227	0.0926907
$477$	13.6603	0.625461
$478$	−7.06728	−0.323250
$479$	40.7000	1.85963	0.929815	−	0.368027i	$-0.119967\pi$
$479$	40.7000	1.85963	0.929815	+	0.368027i	$0.119967\pi$
$480$	−2.55655	−0.116690
$481$	18.6571	0.850689
$482$	20.0900	0.915076
$483$	−4.21131	−0.191621
$484$	−25.0861	−1.14028
$485$	6.63118	0.301106
$486$	0.882919	0.0400500
$487$	−43.1888	−1.95707	−0.978536	−	0.206076i	$-0.933931\pi$
$487$	−43.1888	−1.95707	−0.978536	+	0.206076i	$0.933931\pi$
$488$	15.4567	0.699691
$489$	−0.853127	−0.0385797
$490$	−1.70712	−0.0771197
$491$	−25.3229	−1.14280	−0.571402	−	0.820670i	$-0.693600\pi$
$491$	−25.3229	−1.14280	−0.571402	+	0.820670i	$0.693600\pi$
$492$	9.92998	0.447678
$493$	−7.34405	−0.330759
$494$	−23.4935	−1.05702
$495$	−2.55291	−0.114745
$496$	−0.0654851	−0.00294037
$497$	14.5620	0.653197
$498$	−4.07326	−0.182527
$499$	19.7273	0.883116	0.441558	−	0.897233i	$-0.354426\pi$
$499$	19.7273	0.883116	0.441558	+	0.897233i	$0.354426\pi$
$500$	−5.43201	−0.242927
$501$	−10.6327	−0.475033
$502$	−8.88185	−0.396416
$503$	42.8258	1.90951	0.954754	−	0.297397i	$-0.0961184\pi$
$503$	42.8258	1.90951	0.954754	+	0.297397i	$0.0961184\pi$
$504$	4.71147	0.209866
$505$	8.62987	0.384024
$506$	−12.6053	−0.560372
$507$	−4.12977	−0.183409
$508$	−17.6356	−0.782455
$509$	32.8214	1.45478	0.727391	−	0.686223i	$-0.240733\pi$
$509$	32.8214	1.45478	0.727391	+	0.686223i	$0.240733\pi$
$510$	0.401259	0.0177680
$511$	14.4419	0.638872
$512$	0.787187	0.0347891
$513$	6.42910	0.283852
$514$	1.60060	0.0705996
$515$	−3.72708	−0.164235
$516$	−0.359959	−0.0158463
$517$	37.1947	1.63582
$518$	6.59489	0.289763
$519$	24.4342	1.07254
$520$	5.34832	0.234539
$521$	9.54168	0.418028	0.209014	−	0.977913i	$-0.432975\pi$
$521$	9.54168	0.418028	0.209014	+	0.977913i	$0.432975\pi$
$522$	−6.48421	−0.283806
$523$	3.11070	0.136021	0.0680107	−	0.997685i	$-0.478335\pi$
$523$	3.11070	0.136021	0.0680107	+	0.997685i	$0.478335\pi$
$524$	2.71476	0.118595
$525$	7.94269	0.346647
$526$	−13.4307	−0.585605
$527$	−0.941045	−0.0409926
$528$	0.390899	0.0170117
$529$	−16.5405	−0.719154
$530$	5.48131	0.238093
$531$	−8.44326	−0.366406
$532$	13.0014	0.563682
$533$	−33.6747	−1.45861
$534$	1.45025	0.0627585
$535$	1.47540	0.0637871
$536$	3.58832	0.154992
$537$	−12.6432	−0.545594
$538$	13.4262	0.578845
$539$	−23.8985	−1.02938
$540$	−0.554657	−0.0238687
$541$	16.7200	0.718851	0.359425	−	0.933174i	$-0.382973\pi$
$541$	16.7200	0.718851	0.359425	+	0.933174i	$0.382973\pi$
$542$	2.23699	0.0960871
$543$	−3.45216	−0.148146
$544$	5.62536	0.241185
$545$	6.46690	0.277012
$546$	−6.05502	−0.259131
$547$	10.1137	0.432432	0.216216	−	0.976346i	$-0.430628\pi$
$547$	10.1137	0.432432	0.216216	+	0.976346i	$0.430628\pi$
$548$	−3.74602	−0.160022
$549$	5.43598	0.232002
$550$	23.7740	1.01373
$551$	−47.2156	−2.01145
$552$	−7.22664	−0.307586
$553$	−0.450637	−0.0191630
$554$	−19.3650	−0.822740
$555$	−2.04867	−0.0869610
$556$	−6.29235	−0.266855
$557$	0.0787677	0.00333750	0.00166875	−	0.999999i	$-0.499469\pi$
$557$	0.0787677	0.00333750	0.00166875	+	0.999999i	$0.499469\pi$
$558$	−0.830867	−0.0351734
$559$	1.22070	0.0516299
$560$	0.0524028	0.00221442
$561$	5.61736	0.237165
$562$	8.39248	0.354015
$563$	−12.4593	−0.525098	−0.262549	−	0.964919i	$-0.584563\pi$
$563$	−12.4593	−0.525098	−0.262549	+	0.964919i	$0.584563\pi$
$564$	8.08109	0.340275
$565$	4.89813	0.206066
$566$	−8.94565	−0.376014
$567$	1.65699	0.0695868
$568$	24.9886	1.04850
$569$	7.11694	0.298358	0.149179	−	0.988810i	$-0.452337\pi$
$569$	7.11694	0.298358	0.149179	+	0.988810i	$0.452337\pi$
$570$	2.57973	0.108053
$571$	−25.8068	−1.07998	−0.539990	−	0.841671i	$-0.681572\pi$
$571$	−25.8068	−1.07998	−0.539990	+	0.841671i	$0.681572\pi$
$572$	28.3746	1.18640
$573$	−23.1914	−0.968835
$574$	−11.9033	−0.496834
$575$	−12.1828	−0.508058
$576$	5.10592	0.212747
$577$	3.67049	0.152805	0.0764023	−	0.997077i	$-0.475657\pi$
$577$	3.67049	0.152805	0.0764023	+	0.997077i	$0.475657\pi$
$578$	−0.882919	−0.0367246
$579$	−19.8131	−0.823404
$580$	4.07343	0.169140
$581$	−7.64433	−0.317140
$582$	−12.8827	−0.534007
$583$	76.7347	3.17803
$584$	24.7824	1.02550
$585$	1.88096	0.0777681
$586$	−18.8780	−0.779843
$587$	9.85729	0.406854	0.203427	−	0.979090i	$-0.434792\pi$
$587$	9.85729	0.406854	0.203427	+	0.979090i	$0.434792\pi$
$588$	−5.19230	−0.214127
$589$	−6.05007	−0.249289
$590$	−3.38793	−0.139479
$591$	11.1585	0.458998
$592$	0.313689	0.0128926
$593$	−14.1827	−0.582416	−0.291208	−	0.956660i	$-0.594057\pi$
$593$	−14.1827	−0.582416	−0.291208	+	0.956660i	$0.594057\pi$
$594$	4.95968	0.203498
$595$	0.753047	0.0308719
$596$	−18.1152	−0.742029
$597$	21.9836	0.899729
$598$	9.28742	0.379791
$599$	25.3151	1.03435	0.517173	−	0.855881i	$-0.326984\pi$
$599$	25.3151	1.03435	0.517173	+	0.855881i	$0.326984\pi$
$600$	13.6297	0.556431
$601$	−4.61297	−0.188167	−0.0940834	−	0.995564i	$-0.529992\pi$
$601$	−4.61297	−0.188167	−0.0940834	+	0.995564i	$0.529992\pi$
$602$	0.431490	0.0175862
$603$	1.26198	0.0513918
$604$	11.0425	0.449312
$605$	−9.34148	−0.379785
$606$	−16.7657	−0.681060
$607$	−17.8461	−0.724349	−0.362174	−	0.932110i	$-0.617966\pi$
$607$	−17.8461	−0.724349	−0.362174	+	0.932110i	$0.617966\pi$
$608$	36.1660	1.46672
$609$	−12.1690	−0.493112
$610$	2.18124	0.0883157
$611$	−27.4047	−1.10867
$612$	1.22045	0.0493339
$613$	−3.79018	−0.153084	−0.0765420	−	0.997066i	$-0.524388\pi$
$613$	−3.79018	−0.153084	−0.0765420	+	0.997066i	$0.524388\pi$
$614$	−16.3171	−0.658504
$615$	3.69769	0.149105
$616$	26.4660	1.06635
$617$	−12.9411	−0.520987	−0.260494	−	0.965476i	$-0.583885\pi$
$617$	−12.9411	−0.520987	−0.260494	+	0.965476i	$0.583885\pi$
$618$	7.24080	0.291268
$619$	26.9559	1.08345	0.541725	−	0.840555i	$-0.317771\pi$
$619$	26.9559	1.08345	0.541725	+	0.840555i	$0.317771\pi$
$620$	0.521958	0.0209623
$621$	−2.54155	−0.101989
$622$	−15.0795	−0.604632
$623$	2.72170	0.109043
$624$	−0.288010	−0.0115296
$625$	21.9445	0.877781
$626$	−8.73150	−0.348981
$627$	36.1146	1.44228
$628$	−1.22045	−0.0487014
$629$	4.50783	0.179739
$630$	0.664880	0.0264895
$631$	−5.72609	−0.227952	−0.113976	−	0.993483i	$-0.536359\pi$
$631$	−5.72609	−0.227952	−0.113976	+	0.993483i	$0.536359\pi$
$632$	−0.773296	−0.0307601
$633$	0.594850	0.0236432
$634$	−7.03157	−0.279259
$635$	−6.56711	−0.260608
$636$	16.6717	0.661077
$637$	17.6082	0.697661
$638$	−36.4241	−1.44204
$639$	8.78828	0.347659
$640$	−3.06430	−0.121127
$641$	−33.1206	−1.30818	−0.654092	−	0.756415i	$-0.726949\pi$
$641$	−33.1206	−1.30818	−0.654092	+	0.756415i	$0.726949\pi$
$642$	−2.86634	−0.113125
$643$	−16.1992	−0.638834	−0.319417	−	0.947614i	$-0.603487\pi$
$643$	−16.1992	−0.638834	−0.319417	+	0.947614i	$0.603487\pi$
$644$	−5.13970	−0.202533
$645$	−0.134040	−0.00527783
$646$	−5.67637	−0.223334
$647$	34.4458	1.35420	0.677102	−	0.735889i	$-0.263235\pi$
$647$	34.4458	1.35420	0.677102	+	0.735889i	$0.263235\pi$
$648$	2.84340	0.111699
$649$	−47.4288	−1.86174
$650$	−17.5164	−0.687051
$651$	−1.55930	−0.0611137
$652$	−1.04120	−0.0407766
$653$	47.5394	1.86036	0.930181	−	0.367102i	$-0.119650\pi$
$653$	47.5394	1.86036	0.930181	+	0.367102i	$0.119650\pi$
$654$	−12.5636	−0.491275
$655$	1.01091	0.0394996
$656$	−0.566186	−0.0221059
$657$	8.71576	0.340034
$658$	−9.68698	−0.377638
$659$	−13.1491	−0.512217	−0.256108	−	0.966648i	$-0.582440\pi$
$659$	−13.1491	−0.512217	−0.256108	+	0.966648i	$0.582440\pi$
$660$	−3.11571	−0.121279
$661$	−33.8213	−1.31550	−0.657749	−	0.753237i	$-0.728491\pi$
$661$	−33.8213	−1.31550	−0.657749	+	0.753237i	$0.728491\pi$
$662$	−9.31344	−0.361977
$663$	−4.13881	−0.160738
$664$	−13.1177	−0.509067
$665$	4.84141	0.187742
$666$	3.98005	0.154224
$667$	18.6653	0.722722
$668$	−12.9767	−0.502083
$669$	12.9028	0.498849
$670$	0.506381	0.0195632
$671$	30.5359	1.17882
$672$	9.32114	0.359571
$673$	−34.0650	−1.31311	−0.656555	−	0.754278i	$-0.727987\pi$
$673$	−34.0650	−1.31311	−0.656555	+	0.754278i	$0.727987\pi$
$674$	−19.8833	−0.765876
$675$	4.79346	0.184500
$676$	−5.04019	−0.193853
$677$	21.1646	0.813422	0.406711	−	0.913557i	$-0.366676\pi$
$677$	21.1646	0.813422	0.406711	+	0.913557i	$0.366676\pi$
$678$	−9.51586	−0.365454
$679$	−24.1772	−0.927835
$680$	1.29224	0.0495550
$681$	8.00181	0.306630
$682$	−4.66728	−0.178719
$683$	38.4172	1.46999	0.734996	−	0.678072i	$-0.237184\pi$
$683$	38.4172	1.46999	0.734996	+	0.678072i	$0.237184\pi$
$684$	7.84641	0.300015
$685$	−1.39493	−0.0532975
$686$	16.4650	0.628637
$687$	15.8577	0.605007
$688$	0.0205241	0.000782473 0
$689$	−56.5373	−2.15390
$690$	−1.01982	−0.0388238
$691$	−9.08859	−0.345746	−0.172873	−	0.984944i	$-0.555305\pi$
$691$	−9.08859	−0.345746	−0.172873	+	0.984944i	$0.555305\pi$
$692$	29.8208	1.13362
$693$	9.30788	0.353577
$694$	−13.7218	−0.520871
$695$	−2.34313	−0.0888799
$696$	−20.8821	−0.791533
$697$	−8.13631	−0.308185
$698$	−3.93264	−0.148853
$699$	4.50553	0.170415
$700$	9.69368	0.366387
$701$	40.0111	1.51120	0.755600	−	0.655034i	$-0.227346\pi$
$701$	40.0111	1.51120	0.755600	+	0.655034i	$0.227346\pi$
$702$	−3.65424	−0.137920
$703$	28.9813	1.09305
$704$	28.6818	1.08099
$705$	3.00921	0.113333
$706$	−17.3261	−0.652076
$707$	−31.4644	−1.18334
$708$	−10.3046	−0.387271
$709$	−2.48349	−0.0932693	−0.0466347	−	0.998912i	$-0.514850\pi$
$709$	−2.48349	−0.0932693	−0.0466347	+	0.998912i	$0.514850\pi$
$710$	3.52638	0.132343
$711$	−0.271962	−0.0101994
$712$	4.67047	0.175033
$713$	2.39171	0.0895703
$714$	−1.46298	−0.0547508
$715$	10.5660	0.395147
$716$	−15.4304	−0.576662
$717$	−8.00444	−0.298931
$718$	24.8742	0.928298
$719$	−18.7140	−0.697914	−0.348957	−	0.937139i	$-0.613464\pi$
$719$	−18.7140	−0.697914	−0.348957	+	0.937139i	$0.613464\pi$
$720$	0.0316254	0.00117861
$721$	13.5889	0.506077
$722$	−19.7185	−0.733846
$723$	22.7541	0.846234
$724$	−4.21320	−0.156582
$725$	−35.2034	−1.30742
$726$	18.1482	0.673542
$727$	−9.72055	−0.360515	−0.180258	−	0.983619i	$-0.557693\pi$
$727$	−9.72055	−0.360515	−0.180258	+	0.983619i	$0.557693\pi$
$728$	−19.4999	−0.722714
$729$	1.00000	0.0370370
$730$	3.49728	0.129440
$731$	0.294939	0.0109087
$732$	6.63436	0.245213
$733$	−15.9400	−0.588756	−0.294378	−	0.955689i	$-0.595112\pi$
$733$	−15.9400	−0.588756	−0.294378	+	0.955689i	$0.595112\pi$
$734$	−6.65587	−0.245672
$735$	−1.93349	−0.0713179
$736$	−14.2971	−0.526999
$737$	7.08900	0.261127
$738$	−7.18370	−0.264436
$739$	7.24955	0.266679	0.133340	−	0.991070i	$-0.457430\pi$
$739$	7.24955	0.266679	0.133340	+	0.991070i	$0.457430\pi$
$740$	−2.50030	−0.0919129
$741$	−26.6088	−0.977500
$742$	−19.9848	−0.733664
$743$	−13.1591	−0.482761	−0.241380	−	0.970431i	$-0.577600\pi$
$743$	−13.1591	−0.482761	−0.241380	+	0.970431i	$0.577600\pi$
$744$	−2.67577	−0.0980985
$745$	−6.74569	−0.247143
$746$	−20.6256	−0.755155
$747$	−4.61340	−0.168795
$748$	6.85573	0.250670
$749$	−5.37929	−0.196555
$750$	3.92971	0.143493
$751$	13.3104	0.485703	0.242852	−	0.970063i	$-0.421917\pi$
$751$	13.3104	0.485703	0.242852	+	0.970063i	$0.421917\pi$
$752$	−0.460767	−0.0168024
$753$	−10.0596	−0.366593
$754$	26.8369	0.977343
$755$	4.11196	0.149650
$756$	2.02227	0.0735493
$757$	18.7770	0.682463	0.341232	−	0.939979i	$-0.389156\pi$
$757$	18.7770	0.682463	0.341232	+	0.939979i	$0.389156\pi$
$758$	28.2318	1.02543
$759$	−14.2768	−0.518215
$760$	8.30791	0.301360
$761$	9.97208	0.361488	0.180744	−	0.983530i	$-0.442149\pi$
$761$	9.97208	0.361488	0.180744	+	0.983530i	$0.442149\pi$
$762$	12.7583	0.462183
$763$	−23.5782	−0.853589
$764$	−28.3040	−1.02400
$765$	0.454468	0.0164313
$766$	−24.4610	−0.883813
$767$	34.9451	1.26179
$768$	16.1650	0.583305
$769$	41.3951	1.49274	0.746372	−	0.665529i	$-0.231794\pi$
$769$	41.3951	1.49274	0.746372	+	0.665529i	$0.231794\pi$
$770$	3.73487	0.134595
$771$	1.81285	0.0652884
$772$	−24.1809	−0.870291
$773$	17.3714	0.624806	0.312403	−	0.949950i	$-0.398866\pi$
$773$	17.3714	0.624806	0.312403	+	0.949950i	$0.398866\pi$
$774$	0.260407	0.00936014
$775$	−4.51086	−0.162035
$776$	−41.4883	−1.48934
$777$	7.46941	0.267964
$778$	30.5047	1.09365
$779$	−52.3091	−1.87417
$780$	2.29562	0.0821965
$781$	49.3669	1.76649
$782$	2.24398	0.0802446
$783$	−7.34405	−0.262455
$784$	0.296054	0.0105733
$785$	−0.454468	−0.0162207
$786$	−1.96395	−0.0700518
$787$	−11.6575	−0.415546	−0.207773	−	0.978177i	$-0.566621\pi$
$787$	−11.6575	−0.415546	−0.207773	+	0.978177i	$0.566621\pi$
$788$	13.6184	0.485135
$789$	−15.2117	−0.541550
$790$	−0.109127	−0.00388257
$791$	−17.8585	−0.634976
$792$	15.9724	0.567555
$793$	−22.4985	−0.798945
$794$	3.97077	0.140917
$795$	6.20816	0.220181
$796$	26.8300	0.950963
$797$	−41.8532	−1.48252	−0.741258	−	0.671220i	$-0.765771\pi$
$797$	−41.8532	−1.48252	−0.741258	+	0.671220i	$0.765771\pi$
$798$	−9.40567	−0.332957
$799$	−6.62138	−0.234248
$800$	26.9649	0.953354
$801$	1.64256	0.0580371
$802$	−10.2386	−0.361538
$803$	48.9596	1.72775
$804$	1.54019	0.0543183
$805$	−1.91391	−0.0674563
$806$	3.43880	0.121127
$807$	15.2066	0.535298
$808$	−53.9931	−1.89947
$809$	1.02463	0.0360240	0.0180120	−	0.999838i	$-0.494266\pi$
$809$	1.02463	0.0360240	0.0180120	+	0.999838i	$0.494266\pi$
$810$	0.401259	0.0140988
$811$	−49.8392	−1.75009	−0.875045	−	0.484041i	$-0.839169\pi$
$811$	−49.8392	−1.75009	−0.875045	+	0.484041i	$0.839169\pi$
$812$	−14.8517	−0.521192
$813$	2.53363	0.0888583
$814$	22.3574	0.783626
$815$	−0.387719	−0.0135812
$816$	−0.0695876	−0.00243605
$817$	1.89619	0.0663392
$818$	7.21281	0.252190
$819$	−6.85795	−0.239636
$820$	4.51286	0.157596
$821$	57.2556	1.99824	0.999118	−	0.0419985i	$-0.0133725\pi$
$821$	57.2556	1.99824	0.999118	+	0.0419985i	$0.0133725\pi$
$822$	2.71000	0.0945222
$823$	9.62745	0.335592	0.167796	−	0.985822i	$-0.446335\pi$
$823$	9.62745	0.335592	0.167796	+	0.985822i	$0.446335\pi$
$824$	23.3187	0.812344
$825$	26.9266	0.937464
$826$	12.3524	0.429793
$827$	1.44341	0.0501922	0.0250961	−	0.999685i	$-0.492011\pi$
$827$	1.44341	0.0501922	0.0250961	+	0.999685i	$0.492011\pi$
$828$	−3.10184	−0.107796
$829$	−22.6344	−0.786124	−0.393062	−	0.919512i	$-0.628584\pi$
$829$	−22.6344	−0.786124	−0.393062	+	0.919512i	$0.628584\pi$
$830$	−1.85117	−0.0642550
$831$	−21.9329	−0.760845
$832$	−21.1324	−0.732635
$833$	4.25440	0.147406
$834$	4.55211	0.157627
$835$	−4.83221	−0.167226
$836$	44.0761	1.52440
$837$	−0.941045	−0.0325273
$838$	−19.1344	−0.660988
$839$	−53.8467	−1.85899	−0.929497	−	0.368830i	$-0.879758\pi$
$839$	−53.8467	−1.85899	−0.929497	+	0.368830i	$0.879758\pi$
$840$	2.14122	0.0738789
$841$	24.9351	0.859830
$842$	−5.65386	−0.194845
$843$	9.50537	0.327382
$844$	0.725987	0.0249895
$845$	−1.87685	−0.0645656
$846$	−5.84615	−0.200995
$847$	34.0589	1.17028
$848$	−0.950586	−0.0326433
$849$	−10.1319	−0.347726
$850$	−4.23224	−0.145165
$851$	−11.4569	−0.392736
$852$	10.7257	0.367456
$853$	−20.0750	−0.687354	−0.343677	−	0.939088i	$-0.611673\pi$
$853$	−20.0750	−0.687354	−0.343677	+	0.939088i	$0.611673\pi$
$854$	−7.95275	−0.272138
$855$	2.92182	0.0999242
$856$	−9.23090	−0.315506
$857$	1.27413	0.0435236	0.0217618	−	0.999763i	$-0.493072\pi$
$857$	1.27413	0.0435236	0.0217618	+	0.999763i	$0.493072\pi$
$858$	−20.5272	−0.700786
$859$	50.2212	1.71352	0.856762	−	0.515711i	$-0.172472\pi$
$859$	50.2212	1.71352	0.856762	+	0.515711i	$0.172472\pi$
$860$	−0.163590	−0.00557837
$861$	−13.4817	−0.459456
$862$	−16.3725	−0.557650
$863$	12.2403	0.416664	0.208332	−	0.978058i	$-0.433197\pi$
$863$	12.2403	0.416664	0.208332	+	0.978058i	$0.433197\pi$
$864$	5.62536	0.191379
$865$	11.1046	0.377566
$866$	−7.78980	−0.264708
$867$	−1.00000	−0.0339618
$868$	−1.90305	−0.0645937
$869$	−1.52771	−0.0518239
$870$	−2.94687	−0.0999081
$871$	−5.22310	−0.176978
$872$	−40.4604	−1.37016
$873$	−14.5911	−0.493833
$874$	14.4268	0.487993
$875$	7.37494	0.249318
$876$	10.6372	0.359397
$877$	37.4394	1.26424	0.632120	−	0.774870i	$-0.282185\pi$
$877$	37.4394	1.26424	0.632120	+	0.774870i	$0.282185\pi$
$878$	−23.2040	−0.783096
$879$	−21.3813	−0.721174
$880$	0.177651	0.00598862
$881$	20.5678	0.692946	0.346473	−	0.938060i	$-0.387379\pi$
$881$	20.5678	0.692946	0.346473	+	0.938060i	$0.387379\pi$
$882$	3.75629	0.126481
$883$	25.1256	0.845545	0.422773	−	0.906236i	$-0.361057\pi$
$883$	25.1256	0.845545	0.422773	+	0.906236i	$0.361057\pi$
$884$	−5.05123	−0.169891
$885$	−3.83719	−0.128986
$886$	12.5238	0.420744
$887$	−10.0628	−0.337875	−0.168938	−	0.985627i	$-0.554034\pi$
$887$	−10.0628	−0.337875	−0.168938	+	0.985627i	$0.554034\pi$
$888$	12.8176	0.430129
$889$	23.9436	0.803042
$890$	0.659093	0.0220929
$891$	5.61736	0.188189
$892$	15.7472	0.527255
$893$	−42.5695	−1.42453
$894$	13.1052	0.438304
$895$	−5.74593	−0.192065
$896$	11.1724	0.373244
$897$	10.5190	0.351219
$898$	15.8174	0.527833
$899$	6.91108	0.230498
$900$	5.85019	0.195006
$901$	−13.6603	−0.455090
$902$	−40.3535	−1.34362
$903$	0.488709	0.0162632
$904$	−30.6454	−1.01925
$905$	−1.56890	−0.0521519
$906$	−7.98852	−0.265401
$907$	8.48932	0.281883	0.140942	−	0.990018i	$-0.454987\pi$
$907$	8.48932	0.281883	0.140942	+	0.990018i	$0.454987\pi$
$908$	9.76584	0.324091
$909$	−18.9889	−0.629823
$910$	−2.75181	−0.0912217
$911$	59.5650	1.97348	0.986739	−	0.162316i	$-0.0518965\pi$
$911$	59.5650	1.97348	0.986739	+	0.162316i	$0.0518965\pi$
$912$	−0.447386	−0.0148144
$913$	−25.9151	−0.857666
$914$	1.48869	0.0492415
$915$	2.47048	0.0816716
$916$	19.3535	0.639458
$917$	−3.68577	−0.121715
$918$	−0.882919	−0.0291407
$919$	32.3416	1.06685	0.533425	−	0.845847i	$-0.320905\pi$
$919$	32.3416	1.06685	0.533425	+	0.845847i	$0.320905\pi$
$920$	−3.28428	−0.108279
$921$	−18.4808	−0.608964
$922$	3.81283	0.125569
$923$	−36.3730	−1.19723
$924$	11.3598	0.373711
$925$	21.6081	0.710470
$926$	−15.9205	−0.523179
$927$	8.20097	0.269355
$928$	−41.3129	−1.35616
$929$	41.4597	1.36025	0.680125	−	0.733096i	$-0.261926\pi$
$929$	41.4597	1.36025	0.680125	+	0.733096i	$0.261926\pi$
$930$	−0.377603	−0.0123821
$931$	27.3520	0.896424
$932$	5.49879	0.180119
$933$	−17.0791	−0.559145
$934$	−11.8829	−0.388822
$935$	2.55291	0.0834892
$936$	−11.7683	−0.384659
$937$	20.2731	0.662293	0.331147	−	0.943579i	$-0.392565\pi$
$937$	20.2731	0.662293	0.331147	+	0.943579i	$0.392565\pi$
$938$	−1.84626	−0.0602825
$939$	−9.88935	−0.322727
$940$	3.67260	0.119787
$941$	−1.67009	−0.0544432	−0.0272216	−	0.999629i	$-0.508666\pi$
$941$	−1.67009	−0.0544432	−0.0272216	+	0.999629i	$0.508666\pi$
$942$	0.882919	0.0287671
$943$	20.6788	0.673395
$944$	0.587546	0.0191230
$945$	0.753047	0.0244966
$946$	1.46280	0.0475597
$947$	−44.8209	−1.45648	−0.728242	−	0.685320i	$-0.759662\pi$
$947$	−44.8209	−1.45648	−0.728242	+	0.685320i	$0.759662\pi$
$948$	−0.331917	−0.0107801
$949$	−36.0729	−1.17098
$950$	−27.2095	−0.882792
$951$	−7.96400	−0.258250
$952$	−4.71147	−0.152700
$953$	50.1480	1.62445	0.812226	−	0.583342i	$-0.198255\pi$
$953$	50.1480	1.62445	0.812226	+	0.583342i	$0.198255\pi$
$954$	−12.0609	−0.390487
$955$	−10.5398	−0.341059
$956$	−9.76904	−0.315953
$957$	−41.2542	−1.33356
$958$	−35.9348	−1.16100
$959$	5.08589	0.164232
$960$	2.32048	0.0748931
$961$	−30.1144	−0.971433
$962$	−16.4727	−0.531101
$963$	−3.24643	−0.104615
$964$	27.7703	0.894421
$965$	−9.00442	−0.289863
$966$	3.71824	0.119633
$967$	21.2583	0.683622	0.341811	−	0.939769i	$-0.388960\pi$
$967$	21.2583	0.683622	0.341811	+	0.939769i	$0.388960\pi$
$968$	58.4454	1.87851
$969$	−6.42910	−0.206532
$970$	−5.85480	−0.187986
$971$	−49.4596	−1.58724	−0.793618	−	0.608417i	$-0.791805\pi$
$971$	−49.4596	−1.58724	−0.793618	+	0.608417i	$0.791805\pi$
$972$	1.22045	0.0391461
$973$	8.54301	0.273876
$974$	38.1322	1.22184
$975$	−19.8392	−0.635364
$976$	−0.378277	−0.0121084
$977$	29.9623	0.958578	0.479289	−	0.877657i	$-0.340895\pi$
$977$	29.9623	0.958578	0.479289	+	0.877657i	$0.340895\pi$
$978$	0.753242	0.0240860
$979$	9.22687	0.294892
$980$	−2.35973	−0.0753790
$981$	−14.2296	−0.454316
$982$	22.3581	0.713474
$983$	−8.04897	−0.256722	−0.128361	−	0.991728i	$-0.540972\pi$
$983$	−8.04897	−0.256722	−0.128361	+	0.991728i	$0.540972\pi$
$984$	−23.1348	−0.737510
$985$	5.07117	0.161581
$986$	6.48421	0.206499
$987$	−10.9715	−0.349228
$988$	−32.4748	−1.03316
$989$	−0.749600	−0.0238359
$990$	2.25402	0.0716373
$991$	20.6321	0.655399	0.327699	−	0.944782i	$-0.393727\pi$
$991$	20.6321	0.655399	0.327699	+	0.944782i	$0.393727\pi$
$992$	−5.29372	−0.168076
$993$	−10.5485	−0.334745
$994$	−12.8571	−0.407803
$995$	9.99086	0.316731
$996$	−5.63044	−0.178407
$997$	18.1044	0.573371	0.286686	−	0.958025i	$-0.407447\pi$
$997$	18.1044	0.573371	0.286686	+	0.958025i	$0.407447\pi$
$998$	−17.4176	−0.551345
$999$	4.50783	0.142621

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.g.1.20	✓	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.g.1.20	✓	56	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)

\(f(q)\)	\(=\)	\(q-0.882919 q^{2} -1.00000 q^{3} -1.22045 q^{4} -0.454468 q^{5} +0.882919 q^{6} +1.65699 q^{7} +2.84340 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.882919 q^{2} -1.00000 q^{3} -1.22045 q^{4} -0.454468 q^{5} +0.882919 q^{6} +1.65699 q^{7} +2.84340 q^{8} +1.00000 q^{9} +0.401259 q^{10} +5.61736 q^{11} +1.22045 q^{12} -4.13881 q^{13} -1.46298 q^{14} +0.454468 q^{15} -0.0695876 q^{16} -1.00000 q^{17} -0.882919 q^{18} -6.42910 q^{19} +0.554657 q^{20} -1.65699 q^{21} -4.95968 q^{22} +2.54155 q^{23} -2.84340 q^{24} -4.79346 q^{25} +3.65424 q^{26} -1.00000 q^{27} -2.02227 q^{28} +7.34405 q^{29} -0.401259 q^{30} +0.941045 q^{31} -5.62536 q^{32} -5.61736 q^{33} +0.882919 q^{34} -0.753047 q^{35} -1.22045 q^{36} -4.50783 q^{37} +5.67637 q^{38} +4.13881 q^{39} -1.29224 q^{40} +8.13631 q^{41} +1.46298 q^{42} -0.294939 q^{43} -6.85573 q^{44} -0.454468 q^{45} -2.24398 q^{46} +6.62138 q^{47} +0.0695876 q^{48} -4.25440 q^{49} +4.23224 q^{50} +1.00000 q^{51} +5.05123 q^{52} +13.6603 q^{53} +0.882919 q^{54} -2.55291 q^{55} +4.71147 q^{56} +6.42910 q^{57} -6.48421 q^{58} -8.44326 q^{59} -0.554657 q^{60} +5.43598 q^{61} -0.830867 q^{62} +1.65699 q^{63} +5.10592 q^{64} +1.88096 q^{65} +4.95968 q^{66} +1.26198 q^{67} +1.22045 q^{68} -2.54155 q^{69} +0.664880 q^{70} +8.78828 q^{71} +2.84340 q^{72} +8.71576 q^{73} +3.98005 q^{74} +4.79346 q^{75} +7.84641 q^{76} +9.30788 q^{77} -3.65424 q^{78} -0.271962 q^{79} +0.0316254 q^{80} +1.00000 q^{81} -7.18370 q^{82} -4.61340 q^{83} +2.02227 q^{84} +0.454468 q^{85} +0.260407 q^{86} -7.34405 q^{87} +15.9724 q^{88} +1.64256 q^{89} +0.401259 q^{90} -6.85795 q^{91} -3.10184 q^{92} -0.941045 q^{93} -5.84615 q^{94} +2.92182 q^{95} +5.62536 q^{96} -14.5911 q^{97} +3.75629 q^{98} +5.61736 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + q^{2} - 56 q^{3} + 61 q^{4} + q^{5} - q^{6} + 19 q^{7} + 56 q^{9}+O(q^{10}) \) 56 * q + q^2 - 56 * q^3 + 61 * q^4 + q^5 - q^6 + 19 * q^7 + 56 * q^9	\( 56 q + q^{2} - 56 q^{3} + 61 q^{4} + q^{5} - q^{6} + 19 q^{7} + 56 q^{9} + 8 q^{10} - 7 q^{11} - 61 q^{12} + 8 q^{13} - 8 q^{14} - q^{15} + 71 q^{16} - 56 q^{17} + q^{18} - 2 q^{19} - 4 q^{20} - 19 q^{21} + 47 q^{22} + 16 q^{23} + 85 q^{25} - 11 q^{26} - 56 q^{27} + 52 q^{28} + 17 q^{29} - 8 q^{30} + 23 q^{31} + 11 q^{32} + 7 q^{33} - q^{34} - 41 q^{35} + 61 q^{36} + 58 q^{37} - 22 q^{38} - 8 q^{39} + 38 q^{40} - q^{41} + 8 q^{42} + 27 q^{43} + 2 q^{44} + q^{45} + 46 q^{46} + 5 q^{47} - 71 q^{48} + 59 q^{49} - 4 q^{50} + 56 q^{51} + 25 q^{52} + 15 q^{53} - q^{54} + 9 q^{55} - 36 q^{56} + 2 q^{57} + 89 q^{58} - 61 q^{59} + 4 q^{60} + 47 q^{61} + 8 q^{62} + 19 q^{63} + 88 q^{64} + 39 q^{65} - 47 q^{66} + 20 q^{67} - 61 q^{68} - 16 q^{69} + 36 q^{70} - 2 q^{71} + 93 q^{73} + 48 q^{74} - 85 q^{75} + 38 q^{76} + 26 q^{77} + 11 q^{78} + 72 q^{79} + 42 q^{80} + 56 q^{81} + 33 q^{82} - 11 q^{83} - 52 q^{84} - q^{85} - 4 q^{86} - 17 q^{87} + 130 q^{88} - 6 q^{89} + 8 q^{90} + 37 q^{91} + 132 q^{92} - 23 q^{93} - 32 q^{94} + 12 q^{95} - 11 q^{96} + 100 q^{97} + 42 q^{98} - 7 q^{99}+O(q^{100}) \) 56 * q + q^2 - 56 * q^3 + 61 * q^4 + q^5 - q^6 + 19 * q^7 + 56 * q^9 + 8 * q^10 - 7 * q^11 - 61 * q^12 + 8 * q^13 - 8 * q^14 - q^15 + 71 * q^16 - 56 * q^17 + q^18 - 2 * q^19 - 4 * q^20 - 19 * q^21 + 47 * q^22 + 16 * q^23 + 85 * q^25 - 11 * q^26 - 56 * q^27 + 52 * q^28 + 17 * q^29 - 8 * q^30 + 23 * q^31 + 11 * q^32 + 7 * q^33 - q^34 - 41 * q^35 + 61 * q^36 + 58 * q^37 - 22 * q^38 - 8 * q^39 + 38 * q^40 - q^41 + 8 * q^42 + 27 * q^43 + 2 * q^44 + q^45 + 46 * q^46 + 5 * q^47 - 71 * q^48 + 59 * q^49 - 4 * q^50 + 56 * q^51 + 25 * q^52 + 15 * q^53 - q^54 + 9 * q^55 - 36 * q^56 + 2 * q^57 + 89 * q^58 - 61 * q^59 + 4 * q^60 + 47 * q^61 + 8 * q^62 + 19 * q^63 + 88 * q^64 + 39 * q^65 - 47 * q^66 + 20 * q^67 - 61 * q^68 - 16 * q^69 + 36 * q^70 - 2 * q^71 + 93 * q^73 + 48 * q^74 - 85 * q^75 + 38 * q^76 + 26 * q^77 + 11 * q^78 + 72 * q^79 + 42 * q^80 + 56 * q^81 + 33 * q^82 - 11 * q^83 - 52 * q^84 - q^85 - 4 * q^86 - 17 * q^87 + 130 * q^88 - 6 * q^89 + 8 * q^90 + 37 * q^91 + 132 * q^92 - 23 * q^93 - 32 * q^94 + 12 * q^95 - 11 * q^96 + 100 * q^97 + 42 * q^98 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more