LMFDB - Embedded newform 8006.2.a.d.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8006,2,Mod(1,8006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8006, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8006 = 2 \cdot 4003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8006.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:	$63.9282318582$
Analytic rank:	$0$
Dimension:	$98$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	8006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.57565 q^{3} +1.00000 q^{4} -1.28503 q^{5} -2.57565 q^{6} -4.37657 q^{7} +1.00000 q^{8} +3.63399 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.57565 q^{3} +1.00000 q^{4} -1.28503 q^{5} -2.57565 q^{6} -4.37657 q^{7} +1.00000 q^{8} +3.63399 q^{9} -1.28503 q^{10} +2.79864 q^{11} -2.57565 q^{12} -4.04947 q^{13} -4.37657 q^{14} +3.30979 q^{15} +1.00000 q^{16} -6.99069 q^{17} +3.63399 q^{18} -3.85756 q^{19} -1.28503 q^{20} +11.2725 q^{21} +2.79864 q^{22} -0.0900235 q^{23} -2.57565 q^{24} -3.34871 q^{25} -4.04947 q^{26} -1.63295 q^{27} -4.37657 q^{28} -3.53839 q^{29} +3.30979 q^{30} -2.95804 q^{31} +1.00000 q^{32} -7.20832 q^{33} -6.99069 q^{34} +5.62401 q^{35} +3.63399 q^{36} -3.00296 q^{37} -3.85756 q^{38} +10.4300 q^{39} -1.28503 q^{40} +10.2308 q^{41} +11.2725 q^{42} -11.7691 q^{43} +2.79864 q^{44} -4.66978 q^{45} -0.0900235 q^{46} -10.7727 q^{47} -2.57565 q^{48} +12.1543 q^{49} -3.34871 q^{50} +18.0056 q^{51} -4.04947 q^{52} -2.11723 q^{53} -1.63295 q^{54} -3.59633 q^{55} -4.37657 q^{56} +9.93575 q^{57} -3.53839 q^{58} -5.46893 q^{59} +3.30979 q^{60} -11.7023 q^{61} -2.95804 q^{62} -15.9044 q^{63} +1.00000 q^{64} +5.20367 q^{65} -7.20832 q^{66} -1.84635 q^{67} -6.99069 q^{68} +0.231869 q^{69} +5.62401 q^{70} +10.2023 q^{71} +3.63399 q^{72} +1.50826 q^{73} -3.00296 q^{74} +8.62511 q^{75} -3.85756 q^{76} -12.2484 q^{77} +10.4300 q^{78} -2.96313 q^{79} -1.28503 q^{80} -6.69607 q^{81} +10.2308 q^{82} -14.7252 q^{83} +11.2725 q^{84} +8.98323 q^{85} -11.7691 q^{86} +9.11367 q^{87} +2.79864 q^{88} +2.21416 q^{89} -4.66978 q^{90} +17.7228 q^{91} -0.0900235 q^{92} +7.61888 q^{93} -10.7727 q^{94} +4.95707 q^{95} -2.57565 q^{96} -16.9619 q^{97} +12.1543 q^{98} +10.1702 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 98 q + 98 q^{2} + 16 q^{3} + 98 q^{4} + 4 q^{5} + 16 q^{6} + 29 q^{7} + 98 q^{8} + 130 q^{9}+O(q^{10}) $ 98 * q + 98 * q^2 + 16 * q^3 + 98 * q^4 + 4 * q^5 + 16 * q^6 + 29 * q^7 + 98 * q^8 + 130 * q^9	\( 98 q + 98 q^{2} + 16 q^{3} + 98 q^{4} + 4 q^{5} + 16 q^{6} + 29 q^{7} + 98 q^{8} + 130 q^{9} + 4 q^{10} + 51 q^{11} + 16 q^{12} + 31 q^{13} + 29 q^{14} + 57 q^{15} + 98 q^{16} + 35 q^{17} + 130 q^{18} + 77 q^{19} + 4 q^{20} + 46 q^{21} + 51 q^{22} + 73 q^{23} + 16 q^{24} + 150 q^{25} + 31 q^{26} + 52 q^{27} + 29 q^{28} + 20 q^{29} + 57 q^{30} + 59 q^{31} + 98 q^{32} + 27 q^{33} + 35 q^{34} + 48 q^{35} + 130 q^{36} + 41 q^{37} + 77 q^{38} + 64 q^{39} + 4 q^{40} + 29 q^{41} + 46 q^{42} + 94 q^{43} + 51 q^{44} - 3 q^{45} + 73 q^{46} + 58 q^{47} + 16 q^{48} + 149 q^{49} + 150 q^{50} + 58 q^{51} + 31 q^{52} - 11 q^{53} + 52 q^{54} + 56 q^{55} + 29 q^{56} + 64 q^{57} + 20 q^{58} + 45 q^{59} + 57 q^{60} + 73 q^{61} + 59 q^{62} + 53 q^{63} + 98 q^{64} + 39 q^{65} + 27 q^{66} + 133 q^{67} + 35 q^{68} + 13 q^{69} + 48 q^{70} + 67 q^{71} + 130 q^{72} + 42 q^{73} + 41 q^{74} + 36 q^{75} + 77 q^{76} - 25 q^{77} + 64 q^{78} + 154 q^{79} + 4 q^{80} + 198 q^{81} + 29 q^{82} + 69 q^{83} + 46 q^{84} + 81 q^{85} + 94 q^{86} + 25 q^{87} + 51 q^{88} + 32 q^{89} - 3 q^{90} + 95 q^{91} + 73 q^{92} - 23 q^{93} + 58 q^{94} + 50 q^{95} + 16 q^{96} + 76 q^{97} + 149 q^{98} + 149 q^{99}+O(q^{100}) \) 98 * q + 98 * q^2 + 16 * q^3 + 98 * q^4 + 4 * q^5 + 16 * q^6 + 29 * q^7 + 98 * q^8 + 130 * q^9 + 4 * q^10 + 51 * q^11 + 16 * q^12 + 31 * q^13 + 29 * q^14 + 57 * q^15 + 98 * q^16 + 35 * q^17 + 130 * q^18 + 77 * q^19 + 4 * q^20 + 46 * q^21 + 51 * q^22 + 73 * q^23 + 16 * q^24 + 150 * q^25 + 31 * q^26 + 52 * q^27 + 29 * q^28 + 20 * q^29 + 57 * q^30 + 59 * q^31 + 98 * q^32 + 27 * q^33 + 35 * q^34 + 48 * q^35 + 130 * q^36 + 41 * q^37 + 77 * q^38 + 64 * q^39 + 4 * q^40 + 29 * q^41 + 46 * q^42 + 94 * q^43 + 51 * q^44 - 3 * q^45 + 73 * q^46 + 58 * q^47 + 16 * q^48 + 149 * q^49 + 150 * q^50 + 58 * q^51 + 31 * q^52 - 11 * q^53 + 52 * q^54 + 56 * q^55 + 29 * q^56 + 64 * q^57 + 20 * q^58 + 45 * q^59 + 57 * q^60 + 73 * q^61 + 59 * q^62 + 53 * q^63 + 98 * q^64 + 39 * q^65 + 27 * q^66 + 133 * q^67 + 35 * q^68 + 13 * q^69 + 48 * q^70 + 67 * q^71 + 130 * q^72 + 42 * q^73 + 41 * q^74 + 36 * q^75 + 77 * q^76 - 25 * q^77 + 64 * q^78 + 154 * q^79 + 4 * q^80 + 198 * q^81 + 29 * q^82 + 69 * q^83 + 46 * q^84 + 81 * q^85 + 94 * q^86 + 25 * q^87 + 51 * q^88 + 32 * q^89 - 3 * q^90 + 95 * q^91 + 73 * q^92 - 23 * q^93 + 58 * q^94 + 50 * q^95 + 16 * q^96 + 76 * q^97 + 149 * q^98 + 149 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.57565	−1.48705	−0.743527	−	0.668706i	$-0.766849\pi$
$3$	−2.57565	−1.48705	−0.743527	+	0.668706i	$0.766849\pi$
$4$	1.00000	0.500000
$5$	−1.28503	−0.574682	−0.287341	−	0.957828i	$-0.592771\pi$
$5$	−1.28503	−0.574682	−0.287341	+	0.957828i	$0.592771\pi$
$6$	−2.57565	−1.05151
$7$	−4.37657	−1.65419	−0.827093	−	0.562065i	$-0.810007\pi$
$7$	−4.37657	−1.65419	−0.827093	+	0.562065i	$0.810007\pi$
$8$	1.00000	0.353553
$9$	3.63399	1.21133
$10$	−1.28503	−0.406361
$11$	2.79864	0.843821	0.421911	−	0.906637i	$-0.361360\pi$
$11$	2.79864	0.843821	0.421911	+	0.906637i	$0.361360\pi$
$12$	−2.57565	−0.743527
$13$	−4.04947	−1.12312	−0.561560	−	0.827436i	$-0.689799\pi$
$13$	−4.04947	−1.12312	−0.561560	+	0.827436i	$0.689799\pi$
$14$	−4.37657	−1.16969
$15$	3.30979	0.854583
$16$	1.00000	0.250000
$17$	−6.99069	−1.69549	−0.847746	−	0.530403i	$-0.822041\pi$
$17$	−6.99069	−1.69549	−0.847746	+	0.530403i	$0.822041\pi$
$18$	3.63399	0.856541
$19$	−3.85756	−0.884986	−0.442493	−	0.896772i	$-0.645906\pi$
$19$	−3.85756	−0.884986	−0.442493	+	0.896772i	$0.645906\pi$
$20$	−1.28503	−0.287341
$21$	11.2725	2.45987
$22$	2.79864	0.596672
$23$	−0.0900235	−0.0187712	−0.00938560	−	0.999956i	$-0.502988\pi$
$23$	−0.0900235	−0.0187712	−0.00938560	+	0.999956i	$0.502988\pi$
$24$	−2.57565	−0.525753
$25$	−3.34871	−0.669741
$26$	−4.04947	−0.794166
$27$	−1.63295	−0.314262
$28$	−4.37657	−0.827093
$29$	−3.53839	−0.657063	−0.328531	−	0.944493i	$-0.606554\pi$
$29$	−3.53839	−0.657063	−0.328531	+	0.944493i	$0.606554\pi$
$30$	3.30979	0.604281
$31$	−2.95804	−0.531279	−0.265640	−	0.964072i	$-0.585583\pi$
$31$	−2.95804	−0.531279	−0.265640	+	0.964072i	$0.585583\pi$
$32$	1.00000	0.176777
$33$	−7.20832	−1.25481
$34$	−6.99069	−1.19889
$35$	5.62401	0.950631
$36$	3.63399	0.605666
$37$	−3.00296	−0.493683	−0.246842	−	0.969056i	$-0.579393\pi$
$37$	−3.00296	−0.493683	−0.246842	+	0.969056i	$0.579393\pi$
$38$	−3.85756	−0.625779
$39$	10.4300	1.67014
$40$	−1.28503	−0.203181
$41$	10.2308	1.59778	0.798889	−	0.601479i	$-0.205422\pi$
$41$	10.2308	1.59778	0.798889	+	0.601479i	$0.205422\pi$
$42$	11.2725	1.73939
$43$	−11.7691	−1.79476	−0.897382	−	0.441255i	$-0.854534\pi$
$43$	−11.7691	−1.79476	−0.897382	+	0.441255i	$0.854534\pi$
$44$	2.79864	0.421911
$45$	−4.66978	−0.696130
$46$	−0.0900235	−0.0132732
$47$	−10.7727	−1.57136	−0.785679	−	0.618634i	$-0.787686\pi$
$47$	−10.7727	−1.57136	−0.785679	+	0.618634i	$0.787686\pi$
$48$	−2.57565	−0.371764
$49$	12.1543	1.73633
$50$	−3.34871	−0.473578
$51$	18.0056	2.52129
$52$	−4.04947	−0.561560
$53$	−2.11723	−0.290825	−0.145412	−	0.989371i	$-0.546451\pi$
$53$	−2.11723	−0.290825	−0.145412	+	0.989371i	$0.546451\pi$
$54$	−1.63295	−0.222216
$55$	−3.59633	−0.484928
$56$	−4.37657	−0.584843
$57$	9.93575	1.31602
$58$	−3.53839	−0.464614
$59$	−5.46893	−0.711995	−0.355997	−	0.934487i	$-0.615859\pi$
$59$	−5.46893	−0.711995	−0.355997	+	0.934487i	$0.615859\pi$
$60$	3.30979	0.427291
$61$	−11.7023	−1.49833	−0.749165	−	0.662383i	$-0.769545\pi$
$61$	−11.7023	−1.49833	−0.749165	+	0.662383i	$0.769545\pi$
$62$	−2.95804	−0.375671
$63$	−15.9044	−2.00377
$64$	1.00000	0.125000
$65$	5.20367	0.645436
$66$	−7.20832	−0.887283
$67$	−1.84635	−0.225568	−0.112784	−	0.993620i	$-0.535977\pi$
$67$	−1.84635	−0.225568	−0.112784	+	0.993620i	$0.535977\pi$
$68$	−6.99069	−0.847746
$69$	0.231869	0.0279138
$70$	5.62401	0.672197
$71$	10.2023	1.21079	0.605395	−	0.795925i	$-0.293015\pi$
$71$	10.2023	1.21079	0.605395	+	0.795925i	$0.293015\pi$
$72$	3.63399	0.428270
$73$	1.50826	0.176528	0.0882642	−	0.996097i	$-0.471868\pi$
$73$	1.50826	0.176528	0.0882642	+	0.996097i	$0.471868\pi$
$74$	−3.00296	−0.349087
$75$	8.62511	0.995942
$76$	−3.85756	−0.442493
$77$	−12.2484	−1.39584
$78$	10.4300	1.18097
$79$	−2.96313	−0.333378	−0.166689	−	0.986010i	$-0.553308\pi$
$79$	−2.96313	−0.333378	−0.166689	+	0.986010i	$0.553308\pi$
$80$	−1.28503	−0.143670
$81$	−6.69607	−0.744007
$82$	10.2308	1.12980
$83$	−14.7252	−1.61630	−0.808152	−	0.588975i	$-0.799532\pi$
$83$	−14.7252	−1.61630	−0.808152	+	0.588975i	$0.799532\pi$
$84$	11.2725	1.22993
$85$	8.98323	0.974368
$86$	−11.7691	−1.26909
$87$	9.11367	0.977088
$88$	2.79864	0.298336
$89$	2.21416	0.234701	0.117350	−	0.993091i	$-0.462560\pi$
$89$	2.21416	0.234701	0.117350	+	0.993091i	$0.462560\pi$
$90$	−4.66978	−0.492238
$91$	17.7228	1.85785
$92$	−0.0900235	−0.00938560
$93$	7.61888	0.790042
$94$	−10.7727	−1.11112
$95$	4.95707	0.508585
$96$	−2.57565	−0.262877
$97$	−16.9619	−1.72222	−0.861112	−	0.508416i	$-0.830231\pi$
$97$	−16.9619	−1.72222	−0.861112	+	0.508416i	$0.830231\pi$
$98$	12.1543	1.22777
$99$	10.1702	1.02215
$100$	−3.34871	−0.334871
$101$	−10.4472	−1.03954	−0.519769	−	0.854307i	$-0.673982\pi$
$101$	−10.4472	−1.03954	−0.519769	+	0.854307i	$0.673982\pi$
$102$	18.0056	1.78282
$103$	−5.08792	−0.501327	−0.250664	−	0.968074i	$-0.580649\pi$
$103$	−5.08792	−0.501327	−0.250664	+	0.968074i	$0.580649\pi$
$104$	−4.04947	−0.397083
$105$	−14.4855	−1.41364
$106$	−2.11723	−0.205644
$107$	11.5948	1.12091	0.560456	−	0.828184i	$-0.310626\pi$
$107$	11.5948	1.12091	0.560456	+	0.828184i	$0.310626\pi$
$108$	−1.63295	−0.157131
$109$	10.4671	1.00256	0.501282	−	0.865284i	$-0.332862\pi$
$109$	10.4671	1.00256	0.501282	+	0.865284i	$0.332862\pi$
$110$	−3.59633	−0.342896
$111$	7.73458	0.734134
$112$	−4.37657	−0.413547
$113$	−9.58702	−0.901871	−0.450935	−	0.892557i	$-0.648909\pi$
$113$	−9.58702	−0.901871	−0.450935	+	0.892557i	$0.648909\pi$
$114$	9.93575	0.930568
$115$	0.115683	0.0107875
$116$	−3.53839	−0.328531
$117$	−14.7157	−1.36047
$118$	−5.46893	−0.503456
$119$	30.5952	2.80466
$120$	3.30979	0.302141
$121$	−3.16763	−0.287966
$122$	−11.7023	−1.05948
$123$	−26.3509	−2.37598
$124$	−2.95804	−0.265640
$125$	10.7283	0.959569
$126$	−15.9044	−1.41688
$127$	5.41249	0.480280	0.240140	−	0.970738i	$-0.422807\pi$
$127$	5.41249	0.480280	0.240140	+	0.970738i	$0.422807\pi$
$128$	1.00000	0.0883883
$129$	30.3130	2.66891
$130$	5.20367	0.456392
$131$	−0.739590	−0.0646183	−0.0323091	−	0.999478i	$-0.510286\pi$
$131$	−0.739590	−0.0646183	−0.0323091	+	0.999478i	$0.510286\pi$
$132$	−7.20832	−0.627404
$133$	16.8829	1.46393
$134$	−1.84635	−0.159501
$135$	2.09839	0.180600
$136$	−6.99069	−0.599447
$137$	2.35097	0.200857	0.100428	−	0.994944i	$-0.467979\pi$
$137$	2.35097	0.200857	0.100428	+	0.994944i	$0.467979\pi$
$138$	0.231869	0.0197380
$139$	15.1434	1.28445	0.642223	−	0.766518i	$-0.278012\pi$
$139$	15.1434	1.28445	0.642223	+	0.766518i	$0.278012\pi$
$140$	5.62401	0.475315
$141$	27.7467	2.33670
$142$	10.2023	0.856158
$143$	−11.3330	−0.947712
$144$	3.63399	0.302833
$145$	4.54693	0.377602
$146$	1.50826	0.124824
$147$	−31.3054	−2.58202
$148$	−3.00296	−0.246842
$149$	6.71956	0.550488	0.275244	−	0.961374i	$-0.411241\pi$
$149$	6.71956	0.550488	0.275244	+	0.961374i	$0.411241\pi$
$150$	8.62511	0.704237
$151$	−15.1749	−1.23492	−0.617458	−	0.786604i	$-0.711837\pi$
$151$	−15.1749	−1.23492	−0.617458	+	0.786604i	$0.711837\pi$
$152$	−3.85756	−0.312890
$153$	−25.4041	−2.05380
$154$	−12.2484	−0.987006
$155$	3.80116	0.305317
$156$	10.4300	0.835070
$157$	−12.5813	−1.00410	−0.502048	−	0.864840i	$-0.667420\pi$
$157$	−12.5813	−1.00410	−0.502048	+	0.864840i	$0.667420\pi$
$158$	−2.96313	−0.235734
$159$	5.45326	0.432472
$160$	−1.28503	−0.101590
$161$	0.393994	0.0310511
$162$	−6.69607	−0.526093
$163$	−6.13624	−0.480627	−0.240314	−	0.970695i	$-0.577250\pi$
$163$	−6.13624	−0.480627	−0.240314	+	0.970695i	$0.577250\pi$
$164$	10.2308	0.798889
$165$	9.26289	0.721115
$166$	−14.7252	−1.14290
$167$	3.13446	0.242552	0.121276	−	0.992619i	$-0.461301\pi$
$167$	3.13446	0.242552	0.121276	+	0.992619i	$0.461301\pi$
$168$	11.2725	0.869694
$169$	3.39817	0.261398
$170$	8.98323	0.688982
$171$	−14.0184	−1.07201
$172$	−11.7691	−0.897382
$173$	−14.4959	−1.10211	−0.551053	−	0.834470i	$-0.685774\pi$
$173$	−14.4959	−1.10211	−0.551053	+	0.834470i	$0.685774\pi$
$174$	9.11367	0.690906
$175$	14.6558	1.10788
$176$	2.79864	0.210955
$177$	14.0861	1.05877
$178$	2.21416	0.165959
$179$	23.1012	1.72666	0.863332	−	0.504636i	$-0.168373\pi$
$179$	23.1012	1.72666	0.863332	+	0.504636i	$0.168373\pi$
$180$	−4.66978	−0.348065
$181$	−9.62394	−0.715342	−0.357671	−	0.933848i	$-0.616429\pi$
$181$	−9.62394	−0.715342	−0.357671	+	0.933848i	$0.616429\pi$
$182$	17.7228	1.31370
$183$	30.1412	2.22810
$184$	−0.0900235	−0.00663662
$185$	3.85888	0.283711
$186$	7.61888	0.558644
$187$	−19.5644	−1.43069
$188$	−10.7727	−0.785679
$189$	7.14672	0.519847
$190$	4.95707	0.359624
$191$	15.5587	1.12579	0.562894	−	0.826529i	$-0.309688\pi$
$191$	15.5587	1.12579	0.562894	+	0.826529i	$0.309688\pi$
$192$	−2.57565	−0.185882
$193$	−16.3862	−1.17951	−0.589754	−	0.807583i	$-0.700775\pi$
$193$	−16.3862	−1.17951	−0.589754	+	0.807583i	$0.700775\pi$
$194$	−16.9619	−1.21780
$195$	−13.4029	−0.959799
$196$	12.1543	0.868167
$197$	−15.8770	−1.13119	−0.565594	−	0.824684i	$-0.691353\pi$
$197$	−15.8770	−1.13119	−0.565594	+	0.824684i	$0.691353\pi$
$198$	10.1702	0.722767
$199$	−19.0565	−1.35088	−0.675439	−	0.737416i	$-0.736046\pi$
$199$	−19.0565	−1.35088	−0.675439	+	0.737416i	$0.736046\pi$
$200$	−3.34871	−0.236789
$201$	4.75557	0.335432
$202$	−10.4472	−0.735065
$203$	15.4860	1.08690
$204$	18.0056	1.26064
$205$	−13.1468	−0.918213
$206$	−5.08792	−0.354492
$207$	−0.327145	−0.0227381
$208$	−4.04947	−0.280780
$209$	−10.7959	−0.746770
$210$	−14.4855	−0.999594
$211$	−15.3391	−1.05599	−0.527994	−	0.849248i	$-0.677056\pi$
$211$	−15.3391	−1.05599	−0.527994	+	0.849248i	$0.677056\pi$
$212$	−2.11723	−0.145412
$213$	−26.2776	−1.80051
$214$	11.5948	0.792605
$215$	15.1236	1.03142
$216$	−1.63295	−0.111108
$217$	12.9461	0.878835
$218$	10.4671	0.708920
$219$	−3.88475	−0.262507
$220$	−3.59633	−0.242464
$221$	28.3086	1.90424
$222$	7.73458	0.519111
$223$	−3.67923	−0.246379	−0.123190	−	0.992383i	$-0.539312\pi$
$223$	−3.67923	−0.246379	−0.123190	+	0.992383i	$0.539312\pi$
$224$	−4.37657	−0.292422
$225$	−12.1692	−0.811278
$226$	−9.58702	−0.637719
$227$	23.5804	1.56509	0.782543	−	0.622597i	$-0.213922\pi$
$227$	23.5804	1.56509	0.782543	+	0.622597i	$0.213922\pi$
$228$	9.93575	0.658011
$229$	−4.58723	−0.303132	−0.151566	−	0.988447i	$-0.548432\pi$
$229$	−4.58723	−0.303132	−0.151566	+	0.988447i	$0.548432\pi$
$230$	0.115683	0.00762789
$231$	31.5477	2.07569
$232$	−3.53839	−0.232307
$233$	15.6764	1.02699	0.513497	−	0.858091i	$-0.328350\pi$
$233$	15.6764	1.02699	0.513497	+	0.858091i	$0.328350\pi$
$234$	−14.7157	−0.961998
$235$	13.8432	0.903031
$236$	−5.46893	−0.355997
$237$	7.63200	0.495751
$238$	30.5952	1.98319
$239$	0.343407	0.0222132	0.0111066	−	0.999938i	$-0.496465\pi$
$239$	0.343407	0.0222132	0.0111066	+	0.999938i	$0.496465\pi$
$240$	3.30979	0.213646
$241$	17.1416	1.10419	0.552095	−	0.833781i	$-0.313829\pi$
$241$	17.1416	1.10419	0.552095	+	0.833781i	$0.313829\pi$
$242$	−3.16763	−0.203623
$243$	22.1456	1.42064
$244$	−11.7023	−0.749165
$245$	−15.6186	−0.997839
$246$	−26.3509	−1.68007
$247$	15.6211	0.993945
$248$	−2.95804	−0.187836
$249$	37.9271	2.40353
$250$	10.7283	0.678518
$251$	9.42995	0.595213	0.297607	−	0.954689i	$-0.403812\pi$
$251$	9.42995	0.595213	0.297607	+	0.954689i	$0.403812\pi$
$252$	−15.9044	−1.00188
$253$	−0.251943	−0.0158395
$254$	5.41249	0.339610
$255$	−23.1377	−1.44894
$256$	1.00000	0.0625000
$257$	18.0960	1.12880	0.564398	−	0.825503i	$-0.309108\pi$
$257$	18.0960	1.12880	0.564398	+	0.825503i	$0.309108\pi$
$258$	30.3130	1.88721
$259$	13.1426	0.816644
$260$	5.20367	0.322718
$261$	−12.8585	−0.795921
$262$	−0.739590	−0.0456920
$263$	17.2296	1.06243	0.531213	−	0.847239i	$-0.321737\pi$
$263$	17.2296	1.06243	0.531213	+	0.847239i	$0.321737\pi$
$264$	−7.20832	−0.443642
$265$	2.72070	0.167132
$266$	16.8829	1.03516
$267$	−5.70292	−0.349013
$268$	−1.84635	−0.112784
$269$	11.8856	0.724678	0.362339	−	0.932046i	$-0.381978\pi$
$269$	11.8856	0.724678	0.362339	+	0.932046i	$0.381978\pi$
$270$	2.09839	0.127704
$271$	−2.17719	−0.132255	−0.0661275	−	0.997811i	$-0.521064\pi$
$271$	−2.17719	−0.132255	−0.0661275	+	0.997811i	$0.521064\pi$
$272$	−6.99069	−0.423873
$273$	−45.6477	−2.76272
$274$	2.35097	0.142027
$275$	−9.37181	−0.565142
$276$	0.231869	0.0139569
$277$	6.63119	0.398430	0.199215	−	0.979956i	$-0.436161\pi$
$277$	6.63119	0.398430	0.199215	+	0.979956i	$0.436161\pi$
$278$	15.1434	0.908241
$279$	−10.7495	−0.643556
$280$	5.62401	0.336099
$281$	−5.73809	−0.342306	−0.171153	−	0.985244i	$-0.554749\pi$
$281$	−5.73809	−0.342306	−0.171153	+	0.985244i	$0.554749\pi$
$282$	27.7467	1.65229
$283$	9.86107	0.586180	0.293090	−	0.956085i	$-0.405317\pi$
$283$	9.86107	0.586180	0.293090	+	0.956085i	$0.405317\pi$
$284$	10.2023	0.605395
$285$	−12.7677	−0.756294
$286$	−11.3330	−0.670134
$287$	−44.7756	−2.64302
$288$	3.63399	0.214135
$289$	31.8698	1.87469
$290$	4.54693	0.267005
$291$	43.6881	2.56104
$292$	1.50826	0.0882642
$293$	10.9284	0.638443	0.319221	−	0.947680i	$-0.396579\pi$
$293$	10.9284	0.638443	0.319221	+	0.947680i	$0.396579\pi$
$294$	−31.3054	−1.82577
$295$	7.02773	0.409170
$296$	−3.00296	−0.174543
$297$	−4.57004	−0.265180
$298$	6.71956	0.389254
$299$	0.364547	0.0210823
$300$	8.62511	0.497971
$301$	51.5081	2.96887
$302$	−15.1749	−0.873217
$303$	26.9085	1.54585
$304$	−3.85756	−0.221246
$305$	15.0378	0.861063
$306$	−25.4041	−1.45226
$307$	29.6438	1.69186	0.845930	−	0.533294i	$-0.179046\pi$
$307$	29.6438	1.69186	0.845930	+	0.533294i	$0.179046\pi$
$308$	−12.2484	−0.697919
$309$	13.1047	0.745501
$310$	3.80116	0.215891
$311$	−17.1217	−0.970883	−0.485441	−	0.874269i	$-0.661341\pi$
$311$	−17.1217	−0.970883	−0.485441	+	0.874269i	$0.661341\pi$
$312$	10.4300	0.590484
$313$	−8.37011	−0.473107	−0.236553	−	0.971619i	$-0.576018\pi$
$313$	−8.37011	−0.473107	−0.236553	+	0.971619i	$0.576018\pi$
$314$	−12.5813	−0.710003
$315$	20.4376	1.15153
$316$	−2.96313	−0.166689
$317$	15.3078	0.859770	0.429885	−	0.902884i	$-0.358554\pi$
$317$	15.3078	0.859770	0.429885	+	0.902884i	$0.358554\pi$
$318$	5.45326	0.305804
$319$	−9.90268	−0.554443
$320$	−1.28503	−0.0718352
$321$	−29.8642	−1.66686
$322$	0.393994	0.0219564
$323$	26.9670	1.50049
$324$	−6.69607	−0.372004
$325$	13.5605	0.752199
$326$	−6.13624	−0.339855
$327$	−26.9596	−1.49087
$328$	10.2308	0.564900
$329$	47.1474	2.59932
$330$	9.26289	0.509905
$331$	−6.50630	−0.357619	−0.178809	−	0.983884i	$-0.557225\pi$
$331$	−6.50630	−0.357619	−0.178809	+	0.983884i	$0.557225\pi$
$332$	−14.7252	−0.808152
$333$	−10.9127	−0.598014
$334$	3.13446	0.171510
$335$	2.37261	0.129630
$336$	11.2725	0.614966
$337$	−13.5354	−0.737320	−0.368660	−	0.929564i	$-0.620183\pi$
$337$	−13.5354	−0.737320	−0.368660	+	0.929564i	$0.620183\pi$
$338$	3.39817	0.184836
$339$	24.6928	1.34113
$340$	8.98323	0.487184
$341$	−8.27848	−0.448305
$342$	−14.0184	−0.758026
$343$	−22.5583	−1.21803
$344$	−11.7691	−0.634545
$345$	−0.297959	−0.0160415
$346$	−14.4959	−0.779307
$347$	−8.96665	−0.481355	−0.240677	−	0.970605i	$-0.577370\pi$
$347$	−8.96665	−0.481355	−0.240677	+	0.970605i	$0.577370\pi$
$348$	9.11367	0.488544
$349$	−17.8050	−0.953079	−0.476540	−	0.879153i	$-0.658109\pi$
$349$	−17.8050	−0.953079	−0.476540	+	0.879153i	$0.658109\pi$
$350$	14.6558	0.783387
$351$	6.61258	0.352953
$352$	2.79864	0.149168
$353$	−1.41826	−0.0754863	−0.0377432	−	0.999287i	$-0.512017\pi$
$353$	−1.41826	−0.0754863	−0.0377432	+	0.999287i	$0.512017\pi$
$354$	14.0861	0.748667
$355$	−13.1102	−0.695819
$356$	2.21416	0.117350
$357$	−78.8027	−4.17068
$358$	23.1012	1.22094
$359$	19.0225	1.00397	0.501984	−	0.864877i	$-0.332604\pi$
$359$	19.0225	1.00397	0.501984	+	0.864877i	$0.332604\pi$
$360$	−4.66978	−0.246119
$361$	−4.11921	−0.216800
$362$	−9.62394	−0.505823
$363$	8.15871	0.428221
$364$	17.7228	0.928925
$365$	−1.93815	−0.101448
$366$	30.1412	1.57550
$367$	−4.57170	−0.238641	−0.119320	−	0.992856i	$-0.538072\pi$
$367$	−4.57170	−0.238641	−0.119320	+	0.992856i	$0.538072\pi$
$368$	−0.0900235	−0.00469280
$369$	37.1786	1.93544
$370$	3.85888	0.200614
$371$	9.26622	0.481078
$372$	7.61888	0.395021
$373$	1.09076	0.0564773	0.0282387	−	0.999601i	$-0.491010\pi$
$373$	1.09076	0.0564773	0.0282387	+	0.999601i	$0.491010\pi$
$374$	−19.5644	−1.01165
$375$	−27.6324	−1.42693
$376$	−10.7727	−0.555559
$377$	14.3286	0.737960
$378$	7.14672	0.367587
$379$	−14.9442	−0.767634	−0.383817	−	0.923409i	$-0.625391\pi$
$379$	−14.9442	−0.767634	−0.383817	+	0.923409i	$0.625391\pi$
$380$	4.95707	0.254292
$381$	−13.9407	−0.714203
$382$	15.5587	0.796052
$383$	12.8645	0.657345	0.328672	−	0.944444i	$-0.393399\pi$
$383$	12.8645	0.657345	0.328672	+	0.944444i	$0.393399\pi$
$384$	−2.57565	−0.131438
$385$	15.7396	0.802162
$386$	−16.3862	−0.834038
$387$	−42.7687	−2.17405
$388$	−16.9619	−0.861112
$389$	−6.32601	−0.320742	−0.160371	−	0.987057i	$-0.551269\pi$
$389$	−6.32601	−0.320742	−0.160371	+	0.987057i	$0.551269\pi$
$390$	−13.4029	−0.678680
$391$	0.629326	0.0318264
$392$	12.1543	0.613887
$393$	1.90493	0.0960909
$394$	−15.8770	−0.799871
$395$	3.80770	0.191586
$396$	10.1702	0.511074
$397$	17.6785	0.887257	0.443629	−	0.896211i	$-0.353691\pi$
$397$	17.6785	0.887257	0.443629	+	0.896211i	$0.353691\pi$
$398$	−19.0565	−0.955215
$399$	−43.4845	−2.17695
$400$	−3.34871	−0.167435
$401$	−35.8395	−1.78974	−0.894871	−	0.446326i	$-0.852732\pi$
$401$	−35.8395	−1.78974	−0.894871	+	0.446326i	$0.852732\pi$
$402$	4.75557	0.237186
$403$	11.9785	0.596690
$404$	−10.4472	−0.519769
$405$	8.60463	0.427567
$406$	15.4860	0.768558
$407$	−8.40419	−0.416580
$408$	18.0056	0.891410
$409$	15.1948	0.751333	0.375666	−	0.926755i	$-0.377414\pi$
$409$	15.1948	0.751333	0.375666	+	0.926755i	$0.377414\pi$
$410$	−13.1468	−0.649275
$411$	−6.05528	−0.298685
$412$	−5.08792	−0.250664
$413$	23.9352	1.17777
$414$	−0.327145	−0.0160783
$415$	18.9223	0.928860
$416$	−4.04947	−0.198541
$417$	−39.0042	−1.91004
$418$	−10.7959	−0.528046
$419$	−0.384723	−0.0187949	−0.00939747	−	0.999956i	$-0.502991\pi$
$419$	−0.384723	−0.0187949	−0.00939747	+	0.999956i	$0.502991\pi$
$420$	−14.4855	−0.706820
$421$	−29.9341	−1.45890	−0.729449	−	0.684035i	$-0.760223\pi$
$421$	−29.9341	−1.45890	−0.729449	+	0.684035i	$0.760223\pi$
$422$	−15.3391	−0.746697
$423$	−39.1479	−1.90344
$424$	−2.11723	−0.102822
$425$	23.4098	1.13554
$426$	−26.2776	−1.27315
$427$	51.2161	2.47852
$428$	11.5948	0.560456
$429$	29.1899	1.40930
$430$	15.1236	0.729323
$431$	−1.98331	−0.0955329	−0.0477664	−	0.998859i	$-0.515210\pi$
$431$	−1.98331	−0.0955329	−0.0477664	+	0.998859i	$0.515210\pi$
$432$	−1.63295	−0.0785654
$433$	4.82868	0.232051	0.116026	−	0.993246i	$-0.462985\pi$
$433$	4.82868	0.232051	0.116026	+	0.993246i	$0.462985\pi$
$434$	12.9461	0.621430
$435$	−11.7113	−0.561515
$436$	10.4671	0.501282
$437$	0.347271	0.0166122
$438$	−3.88475	−0.185621
$439$	−21.3549	−1.01921	−0.509607	−	0.860407i	$-0.670209\pi$
$439$	−21.3549	−1.01921	−0.509607	+	0.860407i	$0.670209\pi$
$440$	−3.59633	−0.171448
$441$	44.1688	2.10328
$442$	28.3086	1.34650
$443$	23.7991	1.13073	0.565365	−	0.824841i	$-0.308735\pi$
$443$	23.7991	1.13073	0.565365	+	0.824841i	$0.308735\pi$
$444$	7.73458	0.367067
$445$	−2.84526	−0.134878
$446$	−3.67923	−0.174216
$447$	−17.3073	−0.818605
$448$	−4.37657	−0.206773
$449$	−11.2221	−0.529606	−0.264803	−	0.964303i	$-0.585307\pi$
$449$	−11.2221	−0.529606	−0.264803	+	0.964303i	$0.585307\pi$
$450$	−12.1692	−0.573660
$451$	28.6322	1.34824
$452$	−9.58702	−0.450935
$453$	39.0853	1.83639
$454$	23.5804	1.10668
$455$	−22.7742	−1.06767
$456$	9.93575	0.465284
$457$	19.2730	0.901551	0.450775	−	0.892637i	$-0.351148\pi$
$457$	19.2730	0.901551	0.450775	+	0.892637i	$0.351148\pi$
$458$	−4.58723	−0.214347
$459$	11.4155	0.532828
$460$	0.115683	0.00539373
$461$	−27.1866	−1.26621	−0.633103	−	0.774067i	$-0.718219\pi$
$461$	−27.1866	−1.26621	−0.633103	+	0.774067i	$0.718219\pi$
$462$	31.5477	1.46773
$463$	39.7020	1.84511	0.922553	−	0.385870i	$-0.126099\pi$
$463$	39.7020	1.84511	0.922553	+	0.385870i	$0.126099\pi$
$464$	−3.53839	−0.164266
$465$	−9.79047	−0.454022
$466$	15.6764	0.726195
$467$	33.7438	1.56148	0.780738	−	0.624859i	$-0.214844\pi$
$467$	33.7438	1.56148	0.780738	+	0.624859i	$0.214844\pi$
$468$	−14.7157	−0.680235
$469$	8.08069	0.373132
$470$	13.8432	0.638539
$471$	32.4051	1.49315
$472$	−5.46893	−0.251728
$473$	−32.9373	−1.51446
$474$	7.63200	0.350549
$475$	12.9178	0.592711
$476$	30.5952	1.40233
$477$	−7.69402	−0.352285
$478$	0.343407	0.0157071
$479$	33.0219	1.50881	0.754405	−	0.656409i	$-0.227925\pi$
$479$	33.0219	1.50881	0.754405	+	0.656409i	$0.227925\pi$
$480$	3.30979	0.151070
$481$	12.1604	0.554465
$482$	17.1416	0.780780
$483$	−1.01479	−0.0461746
$484$	−3.16763	−0.143983
$485$	21.7965	0.989730
$486$	22.1456	1.00455
$487$	−33.7043	−1.52729	−0.763643	−	0.645638i	$-0.776591\pi$
$487$	−33.7043	−1.52729	−0.763643	+	0.645638i	$0.776591\pi$
$488$	−11.7023	−0.529740
$489$	15.8048	0.714719
$490$	−15.6186	−0.705579
$491$	−19.0580	−0.860074	−0.430037	−	0.902811i	$-0.641499\pi$
$491$	−19.0580	−0.860074	−0.430037	+	0.902811i	$0.641499\pi$
$492$	−26.3509	−1.18799
$493$	24.7358	1.11404
$494$	15.6211	0.702825
$495$	−13.0690	−0.587409
$496$	−2.95804	−0.132820
$497$	−44.6511	−2.00287
$498$	37.9271	1.69955
$499$	−17.4190	−0.779781	−0.389890	−	0.920861i	$-0.627487\pi$
$499$	−17.4190	−0.779781	−0.389890	+	0.920861i	$0.627487\pi$
$500$	10.7283	0.479785
$501$	−8.07329	−0.360688
$502$	9.42995	0.420879
$503$	31.8138	1.41851	0.709253	−	0.704954i	$-0.249032\pi$
$503$	31.8138	1.41851	0.709253	+	0.704954i	$0.249032\pi$
$504$	−15.9044	−0.708439
$505$	13.4250	0.597404
$506$	−0.251943	−0.0112002
$507$	−8.75252	−0.388713
$508$	5.41249	0.240140
$509$	−18.6891	−0.828381	−0.414190	−	0.910190i	$-0.635935\pi$
$509$	−18.6891	−0.828381	−0.414190	+	0.910190i	$0.635935\pi$
$510$	−23.1377	−1.02455
$511$	−6.60100	−0.292011
$512$	1.00000	0.0441942
$513$	6.29921	0.278117
$514$	18.0960	0.798179
$515$	6.53811	0.288104
$516$	30.3130	1.33446
$517$	−30.1489	−1.32595
$518$	13.1426	0.577454
$519$	37.3365	1.63889
$520$	5.20367	0.228196
$521$	−17.7991	−0.779792	−0.389896	−	0.920859i	$-0.627489\pi$
$521$	−17.7991	−0.779792	−0.389896	+	0.920859i	$0.627489\pi$
$522$	−12.8585	−0.562801
$523$	23.8678	1.04366	0.521832	−	0.853048i	$-0.325249\pi$
$523$	23.8678	1.04366	0.521832	+	0.853048i	$0.325249\pi$
$524$	−0.739590	−0.0323091
$525$	−37.7484	−1.64747
$526$	17.2296	0.751248
$527$	20.6787	0.900780
$528$	−7.20832	−0.313702
$529$	−22.9919	−0.999648
$530$	2.72070	0.118180
$531$	−19.8741	−0.862462
$532$	16.8829	0.731966
$533$	−41.4291	−1.79450
$534$	−5.70292	−0.246789
$535$	−14.8996	−0.644168
$536$	−1.84635	−0.0797503
$537$	−59.5007	−2.56765
$538$	11.8856	0.512425
$539$	34.0156	1.46515
$540$	2.09839	0.0903002
$541$	−20.8481	−0.896328	−0.448164	−	0.893951i	$-0.647922\pi$
$541$	−20.8481	−0.896328	−0.448164	+	0.893951i	$0.647922\pi$
$542$	−2.17719	−0.0935184
$543$	24.7880	1.06375
$544$	−6.99069	−0.299723
$545$	−13.4505	−0.576155
$546$	−45.6477	−1.95354
$547$	−30.2981	−1.29545	−0.647727	−	0.761873i	$-0.724280\pi$
$547$	−30.2981	−1.29545	−0.647727	+	0.761873i	$0.724280\pi$
$548$	2.35097	0.100428
$549$	−42.5262	−1.81498
$550$	−9.37181	−0.399615
$551$	13.6496	0.581491
$552$	0.231869	0.00986902
$553$	12.9683	0.551470
$554$	6.63119	0.281732
$555$	−9.93914	−0.421893
$556$	15.1434	0.642223
$557$	9.74905	0.413080	0.206540	−	0.978438i	$-0.433780\pi$
$557$	9.74905	0.413080	0.206540	+	0.978438i	$0.433780\pi$
$558$	−10.7495	−0.455062
$559$	47.6584	2.01573
$560$	5.62401	0.237658
$561$	50.3912	2.12752
$562$	−5.73809	−0.242047
$563$	−22.1943	−0.935378	−0.467689	−	0.883893i	$-0.654913\pi$
$563$	−22.1943	−0.935378	−0.467689	+	0.883893i	$0.654913\pi$
$564$	27.7467	1.16835
$565$	12.3196	0.518288
$566$	9.86107	0.414492
$567$	29.3058	1.23073
$568$	10.2023	0.428079
$569$	29.7468	1.24705	0.623525	−	0.781803i	$-0.285700\pi$
$569$	29.7468	1.24705	0.623525	+	0.781803i	$0.285700\pi$
$570$	−12.7677	−0.534780
$571$	10.5976	0.443496	0.221748	−	0.975104i	$-0.428824\pi$
$571$	10.5976	0.443496	0.221748	+	0.975104i	$0.428824\pi$
$572$	−11.3330	−0.473856
$573$	−40.0738	−1.67411
$574$	−44.7756	−1.86890
$575$	0.301462	0.0125718
$576$	3.63399	0.151416
$577$	−6.13168	−0.255265	−0.127633	−	0.991822i	$-0.540738\pi$
$577$	−6.13168	−0.255265	−0.127633	+	0.991822i	$0.540738\pi$
$578$	31.8698	1.32561
$579$	42.2053	1.75399
$580$	4.54693	0.188801
$581$	64.4459	2.67367
$582$	43.6881	1.81093
$583$	−5.92537	−0.245404
$584$	1.50826	0.0624122
$585$	18.9101	0.781837
$586$	10.9284	0.451447
$587$	−15.1292	−0.624447	−0.312224	−	0.950009i	$-0.601074\pi$
$587$	−15.1292	−0.624447	−0.312224	+	0.950009i	$0.601074\pi$
$588$	−31.3054	−1.29101
$589$	11.4108	0.470175
$590$	7.02773	0.289327
$591$	40.8936	1.68214
$592$	−3.00296	−0.123421
$593$	−2.64161	−0.108478	−0.0542390	−	0.998528i	$-0.517273\pi$
$593$	−2.64161	−0.108478	−0.0542390	+	0.998528i	$0.517273\pi$
$594$	−4.57004	−0.187511
$595$	−39.3157	−1.61179
$596$	6.71956	0.275244
$597$	49.0829	2.00883
$598$	0.364547	0.0149074
$599$	−29.7769	−1.21665	−0.608326	−	0.793687i	$-0.708159\pi$
$599$	−29.7769	−1.21665	−0.608326	+	0.793687i	$0.708159\pi$
$600$	8.62511	0.352119
$601$	−34.7015	−1.41551	−0.707753	−	0.706460i	$-0.750291\pi$
$601$	−34.7015	−1.41551	−0.707753	+	0.706460i	$0.750291\pi$
$602$	51.5081	2.09931
$603$	−6.70964	−0.273238
$604$	−15.1749	−0.617458
$605$	4.07049	0.165489
$606$	26.9085	1.09308
$607$	−27.8602	−1.13081	−0.565406	−	0.824813i	$-0.691280\pi$
$607$	−27.8602	−1.13081	−0.565406	+	0.824813i	$0.691280\pi$
$608$	−3.85756	−0.156445
$609$	−39.8866	−1.61629
$610$	15.0378	0.608864
$611$	43.6236	1.76482
$612$	−25.4041	−1.02690
$613$	−21.4667	−0.867032	−0.433516	−	0.901146i	$-0.642727\pi$
$613$	−21.4667	−0.867032	−0.433516	+	0.901146i	$0.642727\pi$
$614$	29.6438	1.19633
$615$	33.8616	1.36543
$616$	−12.2484	−0.493503
$617$	8.26228	0.332627	0.166313	−	0.986073i	$-0.446814\pi$
$617$	8.26228	0.332627	0.166313	+	0.986073i	$0.446814\pi$
$618$	13.1047	0.527149
$619$	−23.0828	−0.927776	−0.463888	−	0.885894i	$-0.653546\pi$
$619$	−23.0828	−0.927776	−0.463888	+	0.885894i	$0.653546\pi$
$620$	3.80116	0.152658
$621$	0.147004	0.00589907
$622$	−17.1217	−0.686518
$623$	−9.69043	−0.388239
$624$	10.4300	0.417535
$625$	2.95735	0.118294
$626$	−8.37011	−0.334537
$627$	27.8066	1.11049
$628$	−12.5813	−0.502048
$629$	20.9927	0.837035
$630$	20.4376	0.814254
$631$	43.8417	1.74531	0.872655	−	0.488337i	$-0.162396\pi$
$631$	43.8417	1.74531	0.872655	+	0.488337i	$0.162396\pi$
$632$	−2.96313	−0.117867
$633$	39.5083	1.57031
$634$	15.3078	0.607949
$635$	−6.95519	−0.276008
$636$	5.45326	0.216236
$637$	−49.2186	−1.95011
$638$	−9.90268	−0.392051
$639$	37.0751	1.46667
$640$	−1.28503	−0.0507952
$641$	−25.9658	−1.02559	−0.512793	−	0.858512i	$-0.671389\pi$
$641$	−25.9658	−1.02559	−0.512793	+	0.858512i	$0.671389\pi$
$642$	−29.8642	−1.17865
$643$	33.7829	1.33227	0.666134	−	0.745832i	$-0.267948\pi$
$643$	33.7829	1.33227	0.666134	+	0.745832i	$0.267948\pi$
$644$	0.393994	0.0155255
$645$	−38.9530	−1.53377
$646$	26.9670	1.06100
$647$	27.5988	1.08502	0.542510	−	0.840049i	$-0.317474\pi$
$647$	27.5988	1.08502	0.542510	+	0.840049i	$0.317474\pi$
$648$	−6.69607	−0.263046
$649$	−15.3056	−0.600796
$650$	13.5605	0.531885
$651$	−33.3446	−1.30688
$652$	−6.13624	−0.240314
$653$	−30.5250	−1.19454	−0.597268	−	0.802042i	$-0.703747\pi$
$653$	−30.5250	−1.19454	−0.597268	+	0.802042i	$0.703747\pi$
$654$	−26.9596	−1.05420
$655$	0.950393	0.0371349
$656$	10.2308	0.399444
$657$	5.48101	0.213834
$658$	47.1474	1.83800
$659$	8.96645	0.349283	0.174642	−	0.984632i	$-0.444123\pi$
$659$	8.96645	0.349283	0.174642	+	0.984632i	$0.444123\pi$
$660$	9.26289	0.360558
$661$	−15.2039	−0.591363	−0.295682	−	0.955287i	$-0.595547\pi$
$661$	−15.2039	−0.591363	−0.295682	+	0.955287i	$0.595547\pi$
$662$	−6.50630	−0.252875
$663$	−72.9131	−2.83171
$664$	−14.7252	−0.571449
$665$	−21.6950	−0.841295
$666$	−10.9127	−0.422860
$667$	0.318538	0.0123339
$668$	3.13446	0.121276
$669$	9.47642	0.366379
$670$	2.37261	0.0916621
$671$	−32.7506	−1.26432
$672$	11.2725	0.434847
$673$	38.4105	1.48062	0.740309	−	0.672267i	$-0.234679\pi$
$673$	38.4105	1.48062	0.740309	+	0.672267i	$0.234679\pi$
$674$	−13.5354	−0.521364
$675$	5.46827	0.210474
$676$	3.39817	0.130699
$677$	−34.1890	−1.31399	−0.656995	−	0.753895i	$-0.728173\pi$
$677$	−34.1890	−1.31399	−0.656995	+	0.753895i	$0.728173\pi$
$678$	24.6928	0.948323
$679$	74.2350	2.84888
$680$	8.98323	0.344491
$681$	−60.7349	−2.32737
$682$	−8.27848	−0.316999
$683$	−30.7487	−1.17657	−0.588283	−	0.808655i	$-0.700196\pi$
$683$	−30.7487	−1.17657	−0.588283	+	0.808655i	$0.700196\pi$
$684$	−14.0184	−0.536006
$685$	−3.02106	−0.115429
$686$	−22.5583	−0.861279
$687$	11.8151	0.450774
$688$	−11.7691	−0.448691
$689$	8.57367	0.326631
$690$	−0.297959	−0.0113431
$691$	−20.2765	−0.771355	−0.385678	−	0.922634i	$-0.626032\pi$
$691$	−20.2765	−0.771355	−0.385678	+	0.922634i	$0.626032\pi$
$692$	−14.4959	−0.551053
$693$	−44.5107	−1.69082
$694$	−8.96665	−0.340369
$695$	−19.4597	−0.738148
$696$	9.11367	0.345453
$697$	−71.5201	−2.70902
$698$	−17.8050	−0.673929
$699$	−40.3770	−1.52720
$700$	14.6558	0.553938
$701$	4.40574	0.166403	0.0832013	−	0.996533i	$-0.473486\pi$
$701$	4.40574	0.166403	0.0832013	+	0.996533i	$0.473486\pi$
$702$	6.61258	0.249576
$703$	11.5841	0.436902
$704$	2.79864	0.105478
$705$	−35.6553	−1.34286
$706$	−1.41826	−0.0533769
$707$	45.7230	1.71959
$708$	14.0861	0.529387
$709$	−2.75341	−0.103406	−0.0517032	−	0.998662i	$-0.516465\pi$
$709$	−2.75341	−0.103406	−0.0517032	+	0.998662i	$0.516465\pi$
$710$	−13.1102	−0.492018
$711$	−10.7680	−0.403831
$712$	2.21416	0.0829793
$713$	0.266293	0.00997275
$714$	−78.8027	−2.94912
$715$	14.5632	0.544633
$716$	23.1012	0.863332
$717$	−0.884498	−0.0330322
$718$	19.0225	0.709913
$719$	3.78501	0.141157	0.0705785	−	0.997506i	$-0.477515\pi$
$719$	3.78501	0.141157	0.0705785	+	0.997506i	$0.477515\pi$
$720$	−4.66978	−0.174032
$721$	22.2676	0.829289
$722$	−4.11921	−0.153301
$723$	−44.1509	−1.64199
$724$	−9.62394	−0.357671
$725$	11.8490	0.440062
$726$	8.15871	0.302798
$727$	−17.2021	−0.637990	−0.318995	−	0.947756i	$-0.603345\pi$
$727$	−17.2021	−0.637990	−0.318995	+	0.947756i	$0.603345\pi$
$728$	17.7228	0.656849
$729$	−36.9512	−1.36856
$730$	−1.93815	−0.0717343
$731$	82.2738	3.04301
$732$	30.1412	1.11405
$733$	−44.8733	−1.65743	−0.828716	−	0.559669i	$-0.810928\pi$
$733$	−44.8733	−1.65743	−0.828716	+	0.559669i	$0.810928\pi$
$734$	−4.57170	−0.168744
$735$	40.2282	1.48384
$736$	−0.0900235	−0.00331831
$737$	−5.16728	−0.190339
$738$	37.1786	1.36856
$739$	1.30462	0.0479914	0.0239957	−	0.999712i	$-0.492361\pi$
$739$	1.30462	0.0479914	0.0239957	+	0.999712i	$0.492361\pi$
$740$	3.85888	0.141855
$741$	−40.2345	−1.47805
$742$	9.26622	0.340174
$743$	13.6014	0.498989	0.249494	−	0.968376i	$-0.419736\pi$
$743$	13.6014	0.498989	0.249494	+	0.968376i	$0.419736\pi$
$744$	7.61888	0.279322
$745$	−8.63482	−0.316355
$746$	1.09076	0.0399355
$747$	−53.5114	−1.95788
$748$	−19.5644	−0.715346
$749$	−50.7455	−1.85420
$750$	−27.6324	−1.00899
$751$	46.6592	1.70262	0.851309	−	0.524665i	$-0.175809\pi$
$751$	46.6592	1.70262	0.851309	+	0.524665i	$0.175809\pi$
$752$	−10.7727	−0.392840
$753$	−24.2883	−0.885114
$754$	14.3286	0.521817
$755$	19.5002	0.709683
$756$	7.14672	0.259924
$757$	32.2831	1.17335	0.586674	−	0.809823i	$-0.300437\pi$
$757$	32.2831	1.17335	0.586674	+	0.809823i	$0.300437\pi$
$758$	−14.9442	−0.542799
$759$	0.648919	0.0235543
$760$	4.95707	0.179812
$761$	−31.0659	−1.12614	−0.563069	−	0.826410i	$-0.690379\pi$
$761$	−31.0659	−1.12614	−0.563069	+	0.826410i	$0.690379\pi$
$762$	−13.9407	−0.505018
$763$	−45.8099	−1.65843
$764$	15.5587	0.562894
$765$	32.6450	1.18028
$766$	12.8645	0.464813
$767$	22.1463	0.799655
$768$	−2.57565	−0.0929409
$769$	−15.7726	−0.568774	−0.284387	−	0.958710i	$-0.591790\pi$
$769$	−15.7726	−0.568774	−0.284387	+	0.958710i	$0.591790\pi$
$770$	15.7396	0.567214
$771$	−46.6090	−1.67858
$772$	−16.3862	−0.589754
$773$	−45.0344	−1.61978	−0.809888	−	0.586584i	$-0.800472\pi$
$773$	−45.0344	−1.61978	−0.809888	+	0.586584i	$0.800472\pi$
$774$	−42.7687	−1.53729
$775$	9.90560	0.355820
$776$	−16.9619	−0.608898
$777$	−33.8509	−1.21439
$778$	−6.32601	−0.226798
$779$	−39.4658	−1.41401
$780$	−13.4029	−0.479899
$781$	28.5525	1.02169
$782$	0.629326	0.0225047
$783$	5.77802	0.206490
$784$	12.1543	0.434083
$785$	16.1673	0.577036
$786$	1.90493	0.0679465
$787$	3.21426	0.114576	0.0572881	−	0.998358i	$-0.481755\pi$
$787$	3.21426	0.114576	0.0572881	+	0.998358i	$0.481755\pi$
$788$	−15.8770	−0.565594
$789$	−44.3776	−1.57988
$790$	3.80770	0.135472
$791$	41.9582	1.49186
$792$	10.1702	0.361384
$793$	47.3882	1.68280
$794$	17.6785	0.627386
$795$	−7.00759	−0.248534
$796$	−19.0565	−0.675439
$797$	27.0987	0.959885	0.479943	−	0.877300i	$-0.340657\pi$
$797$	27.0987	0.959885	0.479943	+	0.877300i	$0.340657\pi$
$798$	−43.4845	−1.53933
$799$	75.3085	2.66422
$800$	−3.34871	−0.118395
$801$	8.04626	0.284300
$802$	−35.8395	−1.26554
$803$	4.22107	0.148958
$804$	4.75557	0.167716
$805$	−0.506293	−0.0178445
$806$	11.9785	0.421924
$807$	−30.6132	−1.07764
$808$	−10.4472	−0.367532
$809$	−14.3449	−0.504338	−0.252169	−	0.967683i	$-0.581144\pi$
$809$	−14.3449	−0.504338	−0.252169	+	0.967683i	$0.581144\pi$
$810$	8.60463	0.302336
$811$	−5.95889	−0.209245	−0.104622	−	0.994512i	$-0.533363\pi$
$811$	−5.95889	−0.209245	−0.104622	+	0.994512i	$0.533363\pi$
$812$	15.4860	0.543452
$813$	5.60769	0.196670
$814$	−8.40419	−0.294567
$815$	7.88523	0.276208
$816$	18.0056	0.630322
$817$	45.3999	1.58834
$818$	15.1948	0.531272
$819$	64.4044	2.25047
$820$	−13.1468	−0.459107
$821$	14.1233	0.492907	0.246454	−	0.969155i	$-0.420735\pi$
$821$	14.1233	0.492907	0.246454	+	0.969155i	$0.420735\pi$
$822$	−6.05528	−0.211202
$823$	−25.9742	−0.905405	−0.452703	−	0.891662i	$-0.649540\pi$
$823$	−25.9742	−0.905405	−0.452703	+	0.891662i	$0.649540\pi$
$824$	−5.08792	−0.177246
$825$	24.1385	0.840396
$826$	23.9352	0.832811
$827$	5.24976	0.182552	0.0912760	−	0.995826i	$-0.470905\pi$
$827$	5.24976	0.182552	0.0912760	+	0.995826i	$0.470905\pi$
$828$	−0.327145	−0.0113691
$829$	4.09297	0.142155	0.0710774	−	0.997471i	$-0.477356\pi$
$829$	4.09297	0.142155	0.0710774	+	0.997471i	$0.477356\pi$
$830$	18.9223	0.656803
$831$	−17.0797	−0.592487
$832$	−4.04947	−0.140390
$833$	−84.9672	−2.94394
$834$	−39.0042	−1.35060
$835$	−4.02787	−0.139390
$836$	−10.7959	−0.373385
$837$	4.83033	0.166961
$838$	−0.384723	−0.0132900
$839$	−46.3660	−1.60073	−0.800366	−	0.599512i	$-0.795361\pi$
$839$	−46.3660	−1.60073	−0.800366	+	0.599512i	$0.795361\pi$
$840$	−14.4855	−0.499797
$841$	−16.4798	−0.568268
$842$	−29.9341	−1.03160
$843$	14.7793	0.509028
$844$	−15.3391	−0.527994
$845$	−4.36674	−0.150221
$846$	−39.1479	−1.34593
$847$	13.8633	0.476350
$848$	−2.11723	−0.0727061
$849$	−25.3987	−0.871681
$850$	23.4098	0.802948
$851$	0.270337	0.00926702
$852$	−26.2776	−0.900256
$853$	−33.8199	−1.15797	−0.578985	−	0.815338i	$-0.696551\pi$
$853$	−33.8199	−1.15797	−0.578985	+	0.815338i	$0.696551\pi$
$854$	51.2161	1.75258
$855$	18.0140	0.616065
$856$	11.5948	0.396302
$857$	17.3548	0.592828	0.296414	−	0.955060i	$-0.404209\pi$
$857$	17.3548	0.592828	0.296414	+	0.955060i	$0.404209\pi$
$858$	29.1899	0.996525
$859$	−23.3241	−0.795807	−0.397904	−	0.917427i	$-0.630262\pi$
$859$	−23.3241	−0.795807	−0.397904	+	0.917427i	$0.630262\pi$
$860$	15.1236	0.515709
$861$	115.327	3.93032
$862$	−1.98331	−0.0675520
$863$	13.1657	0.448166	0.224083	−	0.974570i	$-0.428061\pi$
$863$	13.1657	0.448166	0.224083	+	0.974570i	$0.428061\pi$
$864$	−1.63295	−0.0555541
$865$	18.6277	0.633360
$866$	4.82868	0.164085
$867$	−82.0855	−2.78777
$868$	12.9461	0.439418
$869$	−8.29273	−0.281311
$870$	−11.7113	−0.397051
$871$	7.47675	0.253340
$872$	10.4671	0.354460
$873$	−61.6396	−2.08618
$874$	0.347271	0.0117466
$875$	−46.9532	−1.58731
$876$	−3.88475	−0.131254
$877$	−6.00057	−0.202625	−0.101312	−	0.994855i	$-0.532304\pi$
$877$	−6.00057	−0.202625	−0.101312	+	0.994855i	$0.532304\pi$
$878$	−21.3549	−0.720693
$879$	−28.1477	−0.949400
$880$	−3.59633	−0.121232
$881$	−45.5963	−1.53618	−0.768088	−	0.640344i	$-0.778792\pi$
$881$	−45.5963	−1.53618	−0.768088	+	0.640344i	$0.778792\pi$
$882$	44.1688	1.48724
$883$	−19.7319	−0.664031	−0.332016	−	0.943274i	$-0.607729\pi$
$883$	−19.7319	−0.664031	−0.332016	+	0.943274i	$0.607729\pi$
$884$	28.3086	0.952120
$885$	−18.1010	−0.608458
$886$	23.7991	0.799546
$887$	−40.1089	−1.34673	−0.673363	−	0.739312i	$-0.735151\pi$
$887$	−40.1089	−1.34673	−0.673363	+	0.739312i	$0.735151\pi$
$888$	7.73458	0.259555
$889$	−23.6881	−0.794474
$890$	−2.84526	−0.0953733
$891$	−18.7399	−0.627809
$892$	−3.67923	−0.123190
$893$	41.5563	1.39063
$894$	−17.3073	−0.578841
$895$	−29.6857	−0.992283
$896$	−4.37657	−0.146211
$897$	−0.938947	−0.0313505
$898$	−11.2221	−0.374488
$899$	10.4667	0.349084
$900$	−12.1692	−0.405639
$901$	14.8009	0.493091
$902$	28.6322	0.953348
$903$	−132.667	−4.41488
$904$	−9.58702	−0.318859
$905$	12.3670	0.411094
$906$	39.0853	1.29852
$907$	−53.3282	−1.77073	−0.885367	−	0.464893i	$-0.846093\pi$
$907$	−53.3282	−1.77073	−0.885367	+	0.464893i	$0.846093\pi$
$908$	23.5804	0.782543
$909$	−37.9652	−1.25923
$910$	−22.7742	−0.754958
$911$	−15.8567	−0.525357	−0.262679	−	0.964883i	$-0.584606\pi$
$911$	−15.8567	−0.525357	−0.262679	+	0.964883i	$0.584606\pi$
$912$	9.93575	0.329006
$913$	−41.2106	−1.36387
$914$	19.2730	0.637493
$915$	−38.7322	−1.28045
$916$	−4.58723	−0.151566
$917$	3.23686	0.106891
$918$	11.4155	0.376766
$919$	−20.1207	−0.663722	−0.331861	−	0.943328i	$-0.607676\pi$
$919$	−20.1207	−0.663722	−0.331861	+	0.943328i	$0.607676\pi$
$920$	0.115683	0.00381394
$921$	−76.3521	−2.51589
$922$	−27.1866	−0.895344
$923$	−41.3139	−1.35986
$924$	31.5477	1.03784
$925$	10.0560	0.330640
$926$	39.7020	1.30469
$927$	−18.4895	−0.607273
$928$	−3.53839	−0.116153
$929$	−5.72351	−0.187782	−0.0938912	−	0.995582i	$-0.529931\pi$
$929$	−5.72351	−0.187782	−0.0938912	+	0.995582i	$0.529931\pi$
$930$	−9.79047	−0.321042
$931$	−46.8861	−1.53663
$932$	15.6764	0.513497
$933$	44.0996	1.44376
$934$	33.7438	1.10413
$935$	25.1408	0.822192
$936$	−14.7157	−0.480999
$937$	43.4414	1.41917	0.709584	−	0.704621i	$-0.248883\pi$
$937$	43.4414	1.41917	0.709584	+	0.704621i	$0.248883\pi$
$938$	8.08069	0.263844
$939$	21.5585	0.703536
$940$	13.8432	0.451515
$941$	−14.9809	−0.488363	−0.244182	−	0.969730i	$-0.578519\pi$
$941$	−14.9809	−0.488363	−0.244182	+	0.969730i	$0.578519\pi$
$942$	32.4051	1.05581
$943$	−0.921010	−0.0299922
$944$	−5.46893	−0.177999
$945$	−9.18373	−0.298747
$946$	−32.9373	−1.07088
$947$	21.4435	0.696821	0.348410	−	0.937342i	$-0.386722\pi$
$947$	21.4435	0.696821	0.348410	+	0.937342i	$0.386722\pi$
$948$	7.63200	0.247876
$949$	−6.10764	−0.198263
$950$	12.9178	0.419110
$951$	−39.4275	−1.27852
$952$	30.5952	0.991597
$953$	40.7067	1.31862	0.659309	−	0.751872i	$-0.270849\pi$
$953$	40.7067	1.31862	0.659309	+	0.751872i	$0.270849\pi$
$954$	−7.69402	−0.249103
$955$	−19.9933	−0.646969
$956$	0.343407	0.0111066
$957$	25.5059	0.824488
$958$	33.0219	1.06689
$959$	−10.2892	−0.332254
$960$	3.30979	0.106823
$961$	−22.2500	−0.717742
$962$	12.1604	0.392066
$963$	42.1355	1.35780
$964$	17.1416	0.552095
$965$	21.0568	0.677841
$966$	−1.01479	−0.0326504
$967$	49.9460	1.60615	0.803077	−	0.595875i	$-0.203195\pi$
$967$	49.9460	1.60615	0.803077	+	0.595875i	$0.203195\pi$
$968$	−3.16763	−0.101811
$969$	−69.4577	−2.23130
$970$	21.7965	0.699845
$971$	23.8436	0.765177	0.382589	−	0.923919i	$-0.375033\pi$
$971$	23.8436	0.765177	0.382589	+	0.923919i	$0.375033\pi$
$972$	22.1456	0.710321
$973$	−66.2761	−2.12471
$974$	−33.7043	−1.07995
$975$	−34.9271	−1.11856
$976$	−11.7023	−0.374583
$977$	5.01379	0.160405	0.0802027	−	0.996779i	$-0.474443\pi$
$977$	5.01379	0.160405	0.0802027	+	0.996779i	$0.474443\pi$
$978$	15.8048	0.505382
$979$	6.19664	0.198045
$980$	−15.6186	−0.498919
$981$	38.0373	1.21444
$982$	−19.0580	−0.608164
$983$	−11.5736	−0.369139	−0.184570	−	0.982819i	$-0.559089\pi$
$983$	−11.5736	−0.369139	−0.184570	+	0.982819i	$0.559089\pi$
$984$	−26.3509	−0.840037
$985$	20.4024	0.650073
$986$	24.7358	0.787748
$987$	−121.435	−3.86533
$988$	15.6211	0.496972
$989$	1.05949	0.0336899
$990$	−13.0690	−0.415361
$991$	4.80346	0.152587	0.0762934	−	0.997085i	$-0.475691\pi$
$991$	4.80346	0.152587	0.0762934	+	0.997085i	$0.475691\pi$
$992$	−2.95804	−0.0939178
$993$	16.7580	0.531798
$994$	−44.6511	−1.41625
$995$	24.4881	0.776325
$996$	37.9271	1.20177
$997$	6.02009	0.190658	0.0953290	−	0.995446i	$-0.469610\pi$
$997$	6.02009	0.190658	0.0953290	+	0.995446i	$0.469610\pi$
$998$	−17.4190	−0.551388
$999$	4.90368	0.155146

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8006.2.a.d.1.13	✓	98

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8006.2.a.d.1.13	✓	98	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:	\( N \)	\(=\)	\( 8006 = 2 \cdot 4003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8006.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.57565 q^{3} +1.00000 q^{4} -1.28503 q^{5} -2.57565 q^{6} -4.37657 q^{7} +1.00000 q^{8} +3.63399 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.57565 q^{3} +1.00000 q^{4} -1.28503 q^{5} -2.57565 q^{6} -4.37657 q^{7} +1.00000 q^{8} +3.63399 q^{9} -1.28503 q^{10} +2.79864 q^{11} -2.57565 q^{12} -4.04947 q^{13} -4.37657 q^{14} +3.30979 q^{15} +1.00000 q^{16} -6.99069 q^{17} +3.63399 q^{18} -3.85756 q^{19} -1.28503 q^{20} +11.2725 q^{21} +2.79864 q^{22} -0.0900235 q^{23} -2.57565 q^{24} -3.34871 q^{25} -4.04947 q^{26} -1.63295 q^{27} -4.37657 q^{28} -3.53839 q^{29} +3.30979 q^{30} -2.95804 q^{31} +1.00000 q^{32} -7.20832 q^{33} -6.99069 q^{34} +5.62401 q^{35} +3.63399 q^{36} -3.00296 q^{37} -3.85756 q^{38} +10.4300 q^{39} -1.28503 q^{40} +10.2308 q^{41} +11.2725 q^{42} -11.7691 q^{43} +2.79864 q^{44} -4.66978 q^{45} -0.0900235 q^{46} -10.7727 q^{47} -2.57565 q^{48} +12.1543 q^{49} -3.34871 q^{50} +18.0056 q^{51} -4.04947 q^{52} -2.11723 q^{53} -1.63295 q^{54} -3.59633 q^{55} -4.37657 q^{56} +9.93575 q^{57} -3.53839 q^{58} -5.46893 q^{59} +3.30979 q^{60} -11.7023 q^{61} -2.95804 q^{62} -15.9044 q^{63} +1.00000 q^{64} +5.20367 q^{65} -7.20832 q^{66} -1.84635 q^{67} -6.99069 q^{68} +0.231869 q^{69} +5.62401 q^{70} +10.2023 q^{71} +3.63399 q^{72} +1.50826 q^{73} -3.00296 q^{74} +8.62511 q^{75} -3.85756 q^{76} -12.2484 q^{77} +10.4300 q^{78} -2.96313 q^{79} -1.28503 q^{80} -6.69607 q^{81} +10.2308 q^{82} -14.7252 q^{83} +11.2725 q^{84} +8.98323 q^{85} -11.7691 q^{86} +9.11367 q^{87} +2.79864 q^{88} +2.21416 q^{89} -4.66978 q^{90} +17.7228 q^{91} -0.0900235 q^{92} +7.61888 q^{93} -10.7727 q^{94} +4.95707 q^{95} -2.57565 q^{96} -16.9619 q^{97} +12.1543 q^{98} +10.1702 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 98 q + 98 q^{2} + 16 q^{3} + 98 q^{4} + 4 q^{5} + 16 q^{6} + 29 q^{7} + 98 q^{8} + 130 q^{9}+O(q^{10}) \) 98 * q + 98 * q^2 + 16 * q^3 + 98 * q^4 + 4 * q^5 + 16 * q^6 + 29 * q^7 + 98 * q^8 + 130 * q^9	\( 98 q + 98 q^{2} + 16 q^{3} + 98 q^{4} + 4 q^{5} + 16 q^{6} + 29 q^{7} + 98 q^{8} + 130 q^{9} + 4 q^{10} + 51 q^{11} + 16 q^{12} + 31 q^{13} + 29 q^{14} + 57 q^{15} + 98 q^{16} + 35 q^{17} + 130 q^{18} + 77 q^{19} + 4 q^{20} + 46 q^{21} + 51 q^{22} + 73 q^{23} + 16 q^{24} + 150 q^{25} + 31 q^{26} + 52 q^{27} + 29 q^{28} + 20 q^{29} + 57 q^{30} + 59 q^{31} + 98 q^{32} + 27 q^{33} + 35 q^{34} + 48 q^{35} + 130 q^{36} + 41 q^{37} + 77 q^{38} + 64 q^{39} + 4 q^{40} + 29 q^{41} + 46 q^{42} + 94 q^{43} + 51 q^{44} - 3 q^{45} + 73 q^{46} + 58 q^{47} + 16 q^{48} + 149 q^{49} + 150 q^{50} + 58 q^{51} + 31 q^{52} - 11 q^{53} + 52 q^{54} + 56 q^{55} + 29 q^{56} + 64 q^{57} + 20 q^{58} + 45 q^{59} + 57 q^{60} + 73 q^{61} + 59 q^{62} + 53 q^{63} + 98 q^{64} + 39 q^{65} + 27 q^{66} + 133 q^{67} + 35 q^{68} + 13 q^{69} + 48 q^{70} + 67 q^{71} + 130 q^{72} + 42 q^{73} + 41 q^{74} + 36 q^{75} + 77 q^{76} - 25 q^{77} + 64 q^{78} + 154 q^{79} + 4 q^{80} + 198 q^{81} + 29 q^{82} + 69 q^{83} + 46 q^{84} + 81 q^{85} + 94 q^{86} + 25 q^{87} + 51 q^{88} + 32 q^{89} - 3 q^{90} + 95 q^{91} + 73 q^{92} - 23 q^{93} + 58 q^{94} + 50 q^{95} + 16 q^{96} + 76 q^{97} + 149 q^{98} + 149 q^{99}+O(q^{100}) \) 98 * q + 98 * q^2 + 16 * q^3 + 98 * q^4 + 4 * q^5 + 16 * q^6 + 29 * q^7 + 98 * q^8 + 130 * q^9 + 4 * q^10 + 51 * q^11 + 16 * q^12 + 31 * q^13 + 29 * q^14 + 57 * q^15 + 98 * q^16 + 35 * q^17 + 130 * q^18 + 77 * q^19 + 4 * q^20 + 46 * q^21 + 51 * q^22 + 73 * q^23 + 16 * q^24 + 150 * q^25 + 31 * q^26 + 52 * q^27 + 29 * q^28 + 20 * q^29 + 57 * q^30 + 59 * q^31 + 98 * q^32 + 27 * q^33 + 35 * q^34 + 48 * q^35 + 130 * q^36 + 41 * q^37 + 77 * q^38 + 64 * q^39 + 4 * q^40 + 29 * q^41 + 46 * q^42 + 94 * q^43 + 51 * q^44 - 3 * q^45 + 73 * q^46 + 58 * q^47 + 16 * q^48 + 149 * q^49 + 150 * q^50 + 58 * q^51 + 31 * q^52 - 11 * q^53 + 52 * q^54 + 56 * q^55 + 29 * q^56 + 64 * q^57 + 20 * q^58 + 45 * q^59 + 57 * q^60 + 73 * q^61 + 59 * q^62 + 53 * q^63 + 98 * q^64 + 39 * q^65 + 27 * q^66 + 133 * q^67 + 35 * q^68 + 13 * q^69 + 48 * q^70 + 67 * q^71 + 130 * q^72 + 42 * q^73 + 41 * q^74 + 36 * q^75 + 77 * q^76 - 25 * q^77 + 64 * q^78 + 154 * q^79 + 4 * q^80 + 198 * q^81 + 29 * q^82 + 69 * q^83 + 46 * q^84 + 81 * q^85 + 94 * q^86 + 25 * q^87 + 51 * q^88 + 32 * q^89 - 3 * q^90 + 95 * q^91 + 73 * q^92 - 23 * q^93 + 58 * q^94 + 50 * q^95 + 16 * q^96 + 76 * q^97 + 149 * q^98 + 149 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more