LMFDB - Embedded newform 8018.2.a.f.1.21

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8018,2,Mod(1,8018)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8018, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8018.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8018 = 2 \cdot 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8018.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:	$64.0240523407$
Analytic rank:	$1$
Dimension:	$34$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.21
Character	$\chi$	$=$	8018.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +0.744353 q^{3} +1.00000 q^{4} +3.33272 q^{5} -0.744353 q^{6} +0.241764 q^{7} -1.00000 q^{8} -2.44594 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.744353 q^{3} +1.00000 q^{4} +3.33272 q^{5} -0.744353 q^{6} +0.241764 q^{7} -1.00000 q^{8} -2.44594 q^{9} -3.33272 q^{10} -2.45952 q^{11} +0.744353 q^{12} +5.12319 q^{13} -0.241764 q^{14} +2.48072 q^{15} +1.00000 q^{16} -1.97083 q^{17} +2.44594 q^{18} -1.00000 q^{19} +3.33272 q^{20} +0.179958 q^{21} +2.45952 q^{22} -8.65364 q^{23} -0.744353 q^{24} +6.10702 q^{25} -5.12319 q^{26} -4.05370 q^{27} +0.241764 q^{28} +0.510085 q^{29} -2.48072 q^{30} +2.52120 q^{31} -1.00000 q^{32} -1.83075 q^{33} +1.97083 q^{34} +0.805733 q^{35} -2.44594 q^{36} -1.19235 q^{37} +1.00000 q^{38} +3.81347 q^{39} -3.33272 q^{40} -2.23170 q^{41} -0.179958 q^{42} -6.78049 q^{43} -2.45952 q^{44} -8.15163 q^{45} +8.65364 q^{46} -1.86664 q^{47} +0.744353 q^{48} -6.94155 q^{49} -6.10702 q^{50} -1.46699 q^{51} +5.12319 q^{52} -10.7402 q^{53} +4.05370 q^{54} -8.19689 q^{55} -0.241764 q^{56} -0.744353 q^{57} -0.510085 q^{58} +6.19008 q^{59} +2.48072 q^{60} +1.12684 q^{61} -2.52120 q^{62} -0.591341 q^{63} +1.00000 q^{64} +17.0742 q^{65} +1.83075 q^{66} -8.01859 q^{67} -1.97083 q^{68} -6.44136 q^{69} -0.805733 q^{70} +10.2160 q^{71} +2.44594 q^{72} -14.5320 q^{73} +1.19235 q^{74} +4.54578 q^{75} -1.00000 q^{76} -0.594624 q^{77} -3.81347 q^{78} -12.1135 q^{79} +3.33272 q^{80} +4.32043 q^{81} +2.23170 q^{82} -2.48354 q^{83} +0.179958 q^{84} -6.56822 q^{85} +6.78049 q^{86} +0.379683 q^{87} +2.45952 q^{88} +12.7259 q^{89} +8.15163 q^{90} +1.23861 q^{91} -8.65364 q^{92} +1.87666 q^{93} +1.86664 q^{94} -3.33272 q^{95} -0.744353 q^{96} -1.80708 q^{97} +6.94155 q^{98} +6.01583 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q - 34 q^{2} - 10 q^{3} + 34 q^{4} + 7 q^{5} + 10 q^{6} - 6 q^{7} - 34 q^{8} + 38 q^{9}+O(q^{10}) $ 34 * q - 34 * q^2 - 10 * q^3 + 34 * q^4 + 7 * q^5 + 10 * q^6 - 6 * q^7 - 34 * q^8 + 38 * q^9	\( 34 q - 34 q^{2} - 10 q^{3} + 34 q^{4} + 7 q^{5} + 10 q^{6} - 6 q^{7} - 34 q^{8} + 38 q^{9} - 7 q^{10} + 4 q^{11} - 10 q^{12} - 5 q^{13} + 6 q^{14} - 17 q^{15} + 34 q^{16} + 8 q^{17} - 38 q^{18} - 34 q^{19} + 7 q^{20} - 4 q^{22} - 24 q^{23} + 10 q^{24} + 5 q^{25} + 5 q^{26} - 31 q^{27} - 6 q^{28} + 24 q^{29} + 17 q^{30} - 28 q^{31} - 34 q^{32} - 16 q^{33} - 8 q^{34} + 2 q^{35} + 38 q^{36} - 36 q^{37} + 34 q^{38} - 9 q^{39} - 7 q^{40} + 9 q^{41} - 24 q^{43} + 4 q^{44} + 22 q^{45} + 24 q^{46} - 29 q^{47} - 10 q^{48} + 22 q^{49} - 5 q^{50} - 21 q^{51} - 5 q^{52} - 9 q^{53} + 31 q^{54} - 28 q^{55} + 6 q^{56} + 10 q^{57} - 24 q^{58} - 28 q^{59} - 17 q^{60} + 23 q^{61} + 28 q^{62} - 27 q^{63} + 34 q^{64} - 6 q^{65} + 16 q^{66} - 51 q^{67} + 8 q^{68} - 17 q^{69} - 2 q^{70} - 31 q^{71} - 38 q^{72} - 26 q^{73} + 36 q^{74} - 109 q^{75} - 34 q^{76} - 28 q^{77} + 9 q^{78} - 36 q^{79} + 7 q^{80} + 6 q^{81} - 9 q^{82} - 10 q^{83} - 32 q^{85} + 24 q^{86} - 8 q^{87} - 4 q^{88} - 34 q^{89} - 22 q^{90} - 41 q^{91} - 24 q^{92} - 33 q^{93} + 29 q^{94} - 7 q^{95} + 10 q^{96} - 56 q^{97} - 22 q^{98} - 28 q^{99}+O(q^{100}) \) 34 * q - 34 * q^2 - 10 * q^3 + 34 * q^4 + 7 * q^5 + 10 * q^6 - 6 * q^7 - 34 * q^8 + 38 * q^9 - 7 * q^10 + 4 * q^11 - 10 * q^12 - 5 * q^13 + 6 * q^14 - 17 * q^15 + 34 * q^16 + 8 * q^17 - 38 * q^18 - 34 * q^19 + 7 * q^20 - 4 * q^22 - 24 * q^23 + 10 * q^24 + 5 * q^25 + 5 * q^26 - 31 * q^27 - 6 * q^28 + 24 * q^29 + 17 * q^30 - 28 * q^31 - 34 * q^32 - 16 * q^33 - 8 * q^34 + 2 * q^35 + 38 * q^36 - 36 * q^37 + 34 * q^38 - 9 * q^39 - 7 * q^40 + 9 * q^41 - 24 * q^43 + 4 * q^44 + 22 * q^45 + 24 * q^46 - 29 * q^47 - 10 * q^48 + 22 * q^49 - 5 * q^50 - 21 * q^51 - 5 * q^52 - 9 * q^53 + 31 * q^54 - 28 * q^55 + 6 * q^56 + 10 * q^57 - 24 * q^58 - 28 * q^59 - 17 * q^60 + 23 * q^61 + 28 * q^62 - 27 * q^63 + 34 * q^64 - 6 * q^65 + 16 * q^66 - 51 * q^67 + 8 * q^68 - 17 * q^69 - 2 * q^70 - 31 * q^71 - 38 * q^72 - 26 * q^73 + 36 * q^74 - 109 * q^75 - 34 * q^76 - 28 * q^77 + 9 * q^78 - 36 * q^79 + 7 * q^80 + 6 * q^81 - 9 * q^82 - 10 * q^83 - 32 * q^85 + 24 * q^86 - 8 * q^87 - 4 * q^88 - 34 * q^89 - 22 * q^90 - 41 * q^91 - 24 * q^92 - 33 * q^93 + 29 * q^94 - 7 * q^95 + 10 * q^96 - 56 * q^97 - 22 * q^98 - 28 * 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
$3$	0.744353	0.429753	0.214876	−	0.976641i	$-0.431065\pi$
$3$	0.744353	0.429753	0.214876	+	0.976641i	$0.431065\pi$
$4$	1.00000	0.500000
$5$	3.33272	1.49044	0.745219	−	0.666820i	$-0.232345\pi$
$5$	3.33272	1.49044	0.745219	+	0.666820i	$0.232345\pi$
$6$	−0.744353	−0.303881
$7$	0.241764	0.0913784	0.0456892	−	0.998956i	$-0.485452\pi$
$7$	0.241764	0.0913784	0.0456892	+	0.998956i	$0.485452\pi$
$8$	−1.00000	−0.353553
$9$	−2.44594	−0.815313
$10$	−3.33272	−1.05390
$11$	−2.45952	−0.741573	−0.370786	−	0.928718i	$-0.620912\pi$
$11$	−2.45952	−0.741573	−0.370786	+	0.928718i	$0.620912\pi$
$12$	0.744353	0.214876
$13$	5.12319	1.42092	0.710459	−	0.703739i	$-0.248487\pi$
$13$	5.12319	1.42092	0.710459	+	0.703739i	$0.248487\pi$
$14$	−0.241764	−0.0646143
$15$	2.48072	0.640519
$16$	1.00000	0.250000
$17$	−1.97083	−0.477996	−0.238998	−	0.971020i	$-0.576819\pi$
$17$	−1.97083	−0.477996	−0.238998	+	0.971020i	$0.576819\pi$
$18$	2.44594	0.576513
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	3.33272	0.745219
$21$	0.179958	0.0392701
$22$	2.45952	0.524371
$23$	−8.65364	−1.80441	−0.902204	−	0.431310i	$-0.858052\pi$
$23$	−8.65364	−1.80441	−0.902204	+	0.431310i	$0.858052\pi$
$24$	−0.744353	−0.151940
$25$	6.10702	1.22140
$26$	−5.12319	−1.00474
$27$	−4.05370	−0.780135
$28$	0.241764	0.0456892
$29$	0.510085	0.0947204	0.0473602	−	0.998878i	$-0.484919\pi$
$29$	0.510085	0.0947204	0.0473602	+	0.998878i	$0.484919\pi$
$30$	−2.48072	−0.452916
$31$	2.52120	0.452821	0.226411	−	0.974032i	$-0.427301\pi$
$31$	2.52120	0.452821	0.226411	+	0.974032i	$0.427301\pi$
$32$	−1.00000	−0.176777
$33$	−1.83075	−0.318693
$34$	1.97083	0.337994
$35$	0.805733	0.136194
$36$	−2.44594	−0.407656
$37$	−1.19235	−0.196021	−0.0980104	−	0.995185i	$-0.531248\pi$
$37$	−1.19235	−0.196021	−0.0980104	+	0.995185i	$0.531248\pi$
$38$	1.00000	0.162221
$39$	3.81347	0.610643
$40$	−3.33272	−0.526949
$41$	−2.23170	−0.348533	−0.174266	−	0.984699i	$-0.555755\pi$
$41$	−2.23170	−0.348533	−0.174266	+	0.984699i	$0.555755\pi$
$42$	−0.179958	−0.0277681
$43$	−6.78049	−1.03401	−0.517007	−	0.855981i	$-0.672954\pi$
$43$	−6.78049	−1.03401	−0.517007	+	0.855981i	$0.672954\pi$
$44$	−2.45952	−0.370786
$45$	−8.15163	−1.21517
$46$	8.65364	1.27591
$47$	−1.86664	−0.272278	−0.136139	−	0.990690i	$-0.543469\pi$
$47$	−1.86664	−0.272278	−0.136139	+	0.990690i	$0.543469\pi$
$48$	0.744353	0.107438
$49$	−6.94155	−0.991650
$50$	−6.10702	−0.863663
$51$	−1.46699	−0.205420
$52$	5.12319	0.710459
$53$	−10.7402	−1.47528	−0.737639	−	0.675195i	$-0.764059\pi$
$53$	−10.7402	−1.47528	−0.737639	+	0.675195i	$0.764059\pi$
$54$	4.05370	0.551639
$55$	−8.19689	−1.10527
$56$	−0.241764	−0.0323071
$57$	−0.744353	−0.0985920
$58$	−0.510085	−0.0669774
$59$	6.19008	0.805879	0.402940	−	0.915227i	$-0.367988\pi$
$59$	6.19008	0.805879	0.402940	+	0.915227i	$0.367988\pi$
$60$	2.48072	0.320260
$61$	1.12684	0.144277	0.0721387	−	0.997395i	$-0.477018\pi$
$61$	1.12684	0.144277	0.0721387	+	0.997395i	$0.477018\pi$
$62$	−2.52120	−0.320193
$63$	−0.591341	−0.0745020
$64$	1.00000	0.125000
$65$	17.0742	2.11779
$66$	1.83075	0.225350
$67$	−8.01859	−0.979627	−0.489813	−	0.871827i	$-0.662935\pi$
$67$	−8.01859	−0.979627	−0.489813	+	0.871827i	$0.662935\pi$
$68$	−1.97083	−0.238998
$69$	−6.44136	−0.775449
$70$	−0.805733	−0.0963036
$71$	10.2160	1.21242	0.606209	−	0.795305i	$-0.292690\pi$
$71$	10.2160	1.21242	0.606209	+	0.795305i	$0.292690\pi$
$72$	2.44594	0.288257
$73$	−14.5320	−1.70084	−0.850420	−	0.526105i	$-0.823652\pi$
$73$	−14.5320	−1.70084	−0.850420	+	0.526105i	$0.823652\pi$
$74$	1.19235	0.138608
$75$	4.54578	0.524902
$76$	−1.00000	−0.114708
$77$	−0.594624	−0.0677637
$78$	−3.81347	−0.431790
$79$	−12.1135	−1.36288	−0.681439	−	0.731875i	$-0.738645\pi$
$79$	−12.1135	−1.36288	−0.681439	+	0.731875i	$0.738645\pi$
$80$	3.33272	0.372609
$81$	4.32043	0.480048
$82$	2.23170	0.246450
$83$	−2.48354	−0.272603	−0.136302	−	0.990667i	$-0.543522\pi$
$83$	−2.48354	−0.272603	−0.136302	+	0.990667i	$0.543522\pi$
$84$	0.179958	0.0196350
$85$	−6.56822	−0.712424
$86$	6.78049	0.731159
$87$	0.379683	0.0407063
$88$	2.45952	0.262186
$89$	12.7259	1.34894	0.674470	−	0.738302i	$-0.264372\pi$
$89$	12.7259	1.34894	0.674470	+	0.738302i	$0.264372\pi$
$90$	8.15163	0.859257
$91$	1.23861	0.129841
$92$	−8.65364	−0.902204
$93$	1.87666	0.194601
$94$	1.86664	0.192529
$95$	−3.33272	−0.341930
$96$	−0.744353	−0.0759702
$97$	−1.80708	−0.183481	−0.0917404	−	0.995783i	$-0.529243\pi$
$97$	−1.80708	−0.183481	−0.0917404	+	0.995783i	$0.529243\pi$
$98$	6.94155	0.701202
$99$	6.01583	0.604614
$100$	6.10702	0.610702
$101$	9.38449	0.933791	0.466896	−	0.884312i	$-0.345372\pi$
$101$	9.38449	0.933791	0.466896	+	0.884312i	$0.345372\pi$
$102$	1.46699	0.145254
$103$	6.74955	0.665053	0.332526	−	0.943094i	$-0.392099\pi$
$103$	6.74955	0.665053	0.332526	+	0.943094i	$0.392099\pi$
$104$	−5.12319	−0.502370
$105$	0.599750	0.0585296
$106$	10.7402	1.04318
$107$	4.38034	0.423463	0.211732	−	0.977328i	$-0.432090\pi$
$107$	4.38034	0.423463	0.211732	+	0.977328i	$0.432090\pi$
$108$	−4.05370	−0.390068
$109$	−4.60985	−0.441544	−0.220772	−	0.975325i	$-0.570858\pi$
$109$	−4.60985	−0.441544	−0.220772	+	0.975325i	$0.570858\pi$
$110$	8.19689	0.781542
$111$	−0.887528	−0.0842404
$112$	0.241764	0.0228446
$113$	3.55423	0.334354	0.167177	−	0.985927i	$-0.446535\pi$
$113$	3.55423	0.334354	0.167177	+	0.985927i	$0.446535\pi$
$114$	0.744353	0.0697151
$115$	−28.8401	−2.68936
$116$	0.510085	0.0473602
$117$	−12.5310	−1.15849
$118$	−6.19008	−0.569843
$119$	−0.476476	−0.0436785
$120$	−2.48072	−0.226458
$121$	−4.95077	−0.450070
$122$	−1.12684	−0.102020
$123$	−1.66117	−0.149783
$124$	2.52120	0.226411
$125$	3.68940	0.329990
$126$	0.591341	0.0526808
$127$	−15.8407	−1.40563	−0.702816	−	0.711372i	$-0.748074\pi$
$127$	−15.8407	−1.40563	−0.702816	+	0.711372i	$0.748074\pi$
$128$	−1.00000	−0.0883883
$129$	−5.04708	−0.444370
$130$	−17.0742	−1.49750
$131$	0.345532	0.0301893	0.0150946	−	0.999886i	$-0.495195\pi$
$131$	0.345532	0.0301893	0.0150946	+	0.999886i	$0.495195\pi$
$132$	−1.83075	−0.159346
$133$	−0.241764	−0.0209636
$134$	8.01859	0.692701
$135$	−13.5099	−1.16274
$136$	1.97083	0.168997
$137$	−7.33702	−0.626844	−0.313422	−	0.949614i	$-0.601475\pi$
$137$	−7.33702	−0.626844	−0.313422	+	0.949614i	$0.601475\pi$
$138$	6.44136	0.548325
$139$	1.67042	0.141683	0.0708415	−	0.997488i	$-0.477432\pi$
$139$	1.67042	0.141683	0.0708415	+	0.997488i	$0.477432\pi$
$140$	0.805733	0.0680969
$141$	−1.38944	−0.117012
$142$	−10.2160	−0.857309
$143$	−12.6006	−1.05371
$144$	−2.44594	−0.203828
$145$	1.69997	0.141175
$146$	14.5320	1.20267
$147$	−5.16696	−0.426164
$148$	−1.19235	−0.0980104
$149$	5.52826	0.452893	0.226446	−	0.974024i	$-0.427289\pi$
$149$	5.52826	0.452893	0.226446	+	0.974024i	$0.427289\pi$
$150$	−4.54578	−0.371162
$151$	−9.06202	−0.737456	−0.368728	−	0.929537i	$-0.620207\pi$
$151$	−9.06202	−0.737456	−0.368728	+	0.929537i	$0.620207\pi$
$152$	1.00000	0.0811107
$153$	4.82053	0.389716
$154$	0.594624	0.0479162
$155$	8.40246	0.674902
$156$	3.81347	0.305322
$157$	−11.3414	−0.905139	−0.452570	−	0.891729i	$-0.649493\pi$
$157$	−11.3414	−0.905139	−0.452570	+	0.891729i	$0.649493\pi$
$158$	12.1135	0.963700
$159$	−7.99449	−0.634004
$160$	−3.33272	−0.263475
$161$	−2.09214	−0.164884
$162$	−4.32043	−0.339445
$163$	−11.7485	−0.920216	−0.460108	−	0.887863i	$-0.652189\pi$
$163$	−11.7485	−0.920216	−0.460108	+	0.887863i	$0.652189\pi$
$164$	−2.23170	−0.174266
$165$	−6.10138	−0.474992
$166$	2.48354	0.192760
$167$	3.59930	0.278522	0.139261	−	0.990256i	$-0.455527\pi$
$167$	3.59930	0.278522	0.139261	+	0.990256i	$0.455527\pi$
$168$	−0.179958	−0.0138841
$169$	13.2471	1.01901
$170$	6.56822	0.503759
$171$	2.44594	0.187046
$172$	−6.78049	−0.517007
$173$	16.9599	1.28944	0.644720	−	0.764419i	$-0.276974\pi$
$173$	16.9599	1.28944	0.644720	+	0.764419i	$0.276974\pi$
$174$	−0.379683	−0.0287837
$175$	1.47646	0.111610
$176$	−2.45952	−0.185393
$177$	4.60760	0.346329
$178$	−12.7259	−0.953845
$179$	1.04088	0.0777994	0.0388997	−	0.999243i	$-0.487615\pi$
$179$	1.04088	0.0777994	0.0388997	+	0.999243i	$0.487615\pi$
$180$	−8.15163	−0.607586
$181$	−0.881957	−0.0655554	−0.0327777	−	0.999463i	$-0.510435\pi$
$181$	−0.881957	−0.0655554	−0.0327777	+	0.999463i	$0.510435\pi$
$182$	−1.23861	−0.0918116
$183$	0.838769	0.0620036
$184$	8.65364	0.637955
$185$	−3.97376	−0.292157
$186$	−1.87666	−0.137604
$187$	4.84729	0.354469
$188$	−1.86664	−0.136139
$189$	−0.980041	−0.0712875
$190$	3.33272	0.241781
$191$	6.52637	0.472232	0.236116	−	0.971725i	$-0.424126\pi$
$191$	6.52637	0.472232	0.236116	+	0.971725i	$0.424126\pi$
$192$	0.744353	0.0537191
$193$	8.04858	0.579349	0.289675	−	0.957125i	$-0.406453\pi$
$193$	8.04858	0.579349	0.289675	+	0.957125i	$0.406453\pi$
$194$	1.80708	0.129741
$195$	12.7092	0.910126
$196$	−6.94155	−0.495825
$197$	13.2469	0.943805	0.471903	−	0.881651i	$-0.343567\pi$
$197$	13.2469	0.943805	0.471903	+	0.881651i	$0.343567\pi$
$198$	−6.01583	−0.427526
$199$	−17.1696	−1.21712	−0.608562	−	0.793507i	$-0.708253\pi$
$199$	−17.1696	−1.21712	−0.608562	+	0.793507i	$0.708253\pi$
$200$	−6.10702	−0.431832
$201$	−5.96866	−0.420997
$202$	−9.38449	−0.660290
$203$	0.123320	0.00865540
$204$	−1.46699	−0.102710
$205$	−7.43763	−0.519467
$206$	−6.74955	−0.470263
$207$	21.1663	1.47116
$208$	5.12319	0.355230
$209$	2.45952	0.170128
$210$	−0.599750	−0.0413867
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	−10.7402	−0.737639
$213$	7.60432	0.521040
$214$	−4.38034	−0.299434
$215$	−22.5975	−1.54113
$216$	4.05370	0.275819
$217$	0.609537	0.0413781
$218$	4.60985	0.312218
$219$	−10.8169	−0.730940
$220$	−8.19689	−0.552634
$221$	−10.0969	−0.679193
$222$	0.887528	0.0595670
$223$	−19.3741	−1.29739	−0.648693	−	0.761050i	$-0.724684\pi$
$223$	−19.3741	−1.29739	−0.648693	+	0.761050i	$0.724684\pi$
$224$	−0.241764	−0.0161536
$225$	−14.9374	−0.995827
$226$	−3.55423	−0.236424
$227$	−9.76508	−0.648131	−0.324066	−	0.946035i	$-0.605050\pi$
$227$	−9.76508	−0.648131	−0.324066	+	0.946035i	$0.605050\pi$
$228$	−0.744353	−0.0492960
$229$	7.52711	0.497405	0.248703	−	0.968580i	$-0.419996\pi$
$229$	7.52711	0.497405	0.248703	+	0.968580i	$0.419996\pi$
$230$	28.8401	1.90166
$231$	−0.442610	−0.0291216
$232$	−0.510085	−0.0334887
$233$	−15.4176	−1.01004	−0.505020	−	0.863107i	$-0.668515\pi$
$233$	−15.4176	−1.01004	−0.505020	+	0.863107i	$0.668515\pi$
$234$	12.5310	0.819178
$235$	−6.22099	−0.405813
$236$	6.19008	0.402940
$237$	−9.01673	−0.585700
$238$	0.476476	0.0308854
$239$	−24.7937	−1.60377	−0.801887	−	0.597476i	$-0.796170\pi$
$239$	−24.7937	−1.60377	−0.801887	+	0.597476i	$0.796170\pi$
$240$	2.48072	0.160130
$241$	4.12130	0.265476	0.132738	−	0.991151i	$-0.457623\pi$
$241$	4.12130	0.265476	0.132738	+	0.991151i	$0.457623\pi$
$242$	4.95077	0.318248
$243$	15.3770	0.986437
$244$	1.12684	0.0721387
$245$	−23.1342	−1.47799
$246$	1.66117	0.105912
$247$	−5.12319	−0.325981
$248$	−2.52120	−0.160097
$249$	−1.84863	−0.117152
$250$	−3.68940	−0.233338
$251$	−9.35417	−0.590430	−0.295215	−	0.955431i	$-0.595391\pi$
$251$	−9.35417	−0.590430	−0.295215	+	0.955431i	$0.595391\pi$
$252$	−0.591341	−0.0372510
$253$	21.2838	1.33810
$254$	15.8407	0.993931
$255$	−4.88908	−0.306166
$256$	1.00000	0.0625000
$257$	−5.50307	−0.343272	−0.171636	−	0.985160i	$-0.554905\pi$
$257$	−5.50307	−0.343272	−0.171636	+	0.985160i	$0.554905\pi$
$258$	5.04708	0.314217
$259$	−0.288267	−0.0179121
$260$	17.0742	1.05890
$261$	−1.24764	−0.0772267
$262$	−0.345532	−0.0213470
$263$	−28.0509	−1.72969	−0.864847	−	0.502036i	$-0.832585\pi$
$263$	−28.0509	−1.72969	−0.864847	+	0.502036i	$0.832585\pi$
$264$	1.83075	0.112675
$265$	−35.7940	−2.19881
$266$	0.241764	0.0148235
$267$	9.47254	0.579710
$268$	−8.01859	−0.489813
$269$	24.8061	1.51245	0.756227	−	0.654309i	$-0.227040\pi$
$269$	24.8061	1.51245	0.756227	+	0.654309i	$0.227040\pi$
$270$	13.5099	0.822183
$271$	−4.57671	−0.278015	−0.139008	−	0.990291i	$-0.544391\pi$
$271$	−4.57671	−0.278015	−0.139008	+	0.990291i	$0.544391\pi$
$272$	−1.97083	−0.119499
$273$	0.921960	0.0557996
$274$	7.33702	0.443246
$275$	−15.0203	−0.905760
$276$	−6.44136	−0.387724
$277$	11.1915	0.672432	0.336216	−	0.941785i	$-0.390853\pi$
$277$	11.1915	0.672432	0.336216	+	0.941785i	$0.390853\pi$
$278$	−1.67042	−0.100185
$279$	−6.16671	−0.369191
$280$	−0.805733	−0.0481518
$281$	−13.4588	−0.802882	−0.401441	−	0.915885i	$-0.631491\pi$
$281$	−13.4588	−0.802882	−0.401441	+	0.915885i	$0.631491\pi$
$282$	1.38944	0.0827399
$283$	19.5100	1.15975	0.579875	−	0.814706i	$-0.303101\pi$
$283$	19.5100	1.15975	0.579875	+	0.814706i	$0.303101\pi$
$284$	10.2160	0.606209
$285$	−2.48072	−0.146945
$286$	12.6006	0.745088
$287$	−0.539546	−0.0318484
$288$	2.44594	0.144128
$289$	−13.1158	−0.771520
$290$	−1.69997	−0.0998257
$291$	−1.34510	−0.0788513
$292$	−14.5320	−0.850420
$293$	24.4438	1.42802	0.714012	−	0.700134i	$-0.246876\pi$
$293$	24.4438	1.42802	0.714012	+	0.700134i	$0.246876\pi$
$294$	5.16696	0.301344
$295$	20.6298	1.20111
$296$	1.19235	0.0693038
$297$	9.97015	0.578527
$298$	−5.52826	−0.320243
$299$	−44.3343	−2.56392
$300$	4.54578	0.262451
$301$	−1.63928	−0.0944866
$302$	9.06202	0.521460
$303$	6.98537	0.401299
$304$	−1.00000	−0.0573539
$305$	3.75545	0.215037
$306$	−4.82053	−0.275571
$307$	−8.78387	−0.501322	−0.250661	−	0.968075i	$-0.580648\pi$
$307$	−8.78387	−0.501322	−0.250661	+	0.968075i	$0.580648\pi$
$308$	−0.594624	−0.0338819
$309$	5.02405	0.285808
$310$	−8.40246	−0.477228
$311$	7.96037	0.451391	0.225696	−	0.974198i	$-0.427535\pi$
$311$	7.96037	0.451391	0.225696	+	0.974198i	$0.427535\pi$
$312$	−3.81347	−0.215895
$313$	23.8940	1.35057	0.675285	−	0.737557i	$-0.264020\pi$
$313$	23.8940	1.35057	0.675285	+	0.737557i	$0.264020\pi$
$314$	11.3414	0.640030
$315$	−1.97077	−0.111041
$316$	−12.1135	−0.681439
$317$	26.1551	1.46901	0.734507	−	0.678601i	$-0.237413\pi$
$317$	26.1551	1.46901	0.734507	+	0.678601i	$0.237413\pi$
$318$	7.99449	0.448309
$319$	−1.25456	−0.0702420
$320$	3.33272	0.186305
$321$	3.26052	0.181984
$322$	2.09214	0.116591
$323$	1.97083	0.109660
$324$	4.32043	0.240024
$325$	31.2875	1.73552
$326$	11.7485	0.650691
$327$	−3.43136	−0.189754
$328$	2.23170	0.123225
$329$	−0.451287	−0.0248803
$330$	6.10138	0.335870
$331$	−21.1601	−1.16306	−0.581531	−	0.813524i	$-0.697546\pi$
$331$	−21.1601	−1.16306	−0.581531	+	0.813524i	$0.697546\pi$
$332$	−2.48354	−0.136302
$333$	2.91641	0.159818
$334$	−3.59930	−0.196945
$335$	−26.7237	−1.46007
$336$	0.179958	0.00981752
$337$	−0.905939	−0.0493497	−0.0246748	−	0.999696i	$-0.507855\pi$
$337$	−0.905939	−0.0493497	−0.0246748	+	0.999696i	$0.507855\pi$
$338$	−13.2471	−0.720548
$339$	2.64560	0.143689
$340$	−6.56822	−0.356212
$341$	−6.20094	−0.335800
$342$	−2.44594	−0.132261
$343$	−3.37057	−0.181994
$344$	6.78049	0.365579
$345$	−21.4673	−1.15576
$346$	−16.9599	−0.911771
$347$	−3.75311	−0.201478	−0.100739	−	0.994913i	$-0.532121\pi$
$347$	−3.75311	−0.201478	−0.100739	+	0.994913i	$0.532121\pi$
$348$	0.379683	0.0203532
$349$	29.0410	1.55453	0.777265	−	0.629173i	$-0.216606\pi$
$349$	29.0410	1.55453	0.777265	+	0.629173i	$0.216606\pi$
$350$	−1.47646	−0.0789202
$351$	−20.7679	−1.10851
$352$	2.45952	0.131093
$353$	−2.35144	−0.125155	−0.0625773	−	0.998040i	$-0.519932\pi$
$353$	−2.35144	−0.125155	−0.0625773	+	0.998040i	$0.519932\pi$
$354$	−4.60760	−0.244891
$355$	34.0471	1.80703
$356$	12.7259	0.674470
$357$	−0.354667	−0.0187710
$358$	−1.04088	−0.0550125
$359$	20.0239	1.05682	0.528411	−	0.848989i	$-0.322788\pi$
$359$	20.0239	1.05682	0.528411	+	0.848989i	$0.322788\pi$
$360$	8.15163	0.429629
$361$	1.00000	0.0526316
$362$	0.881957	0.0463546
$363$	−3.68512	−0.193419
$364$	1.23861	0.0649206
$365$	−48.4310	−2.53499
$366$	−0.838769	−0.0438432
$367$	29.2296	1.52577	0.762886	−	0.646533i	$-0.223782\pi$
$367$	29.2296	1.52577	0.762886	+	0.646533i	$0.223782\pi$
$368$	−8.65364	−0.451102
$369$	5.45860	0.284163
$370$	3.97376	0.206586
$371$	−2.59660	−0.134809
$372$	1.87666	0.0973005
$373$	−19.1164	−0.989811	−0.494906	−	0.868947i	$-0.664797\pi$
$373$	−19.1164	−0.989811	−0.494906	+	0.868947i	$0.664797\pi$
$374$	−4.84729	−0.250647
$375$	2.74621	0.141814
$376$	1.86664	0.0962646
$377$	2.61326	0.134590
$378$	0.980041	0.0504079
$379$	−31.3021	−1.60788	−0.803941	−	0.594709i	$-0.797267\pi$
$379$	−31.3021	−1.60788	−0.803941	+	0.594709i	$0.797267\pi$
$380$	−3.33272	−0.170965
$381$	−11.7910	−0.604073
$382$	−6.52637	−0.333918
$383$	35.0634	1.79166	0.895829	−	0.444399i	$-0.146583\pi$
$383$	35.0634	1.79166	0.895829	+	0.444399i	$0.146583\pi$
$384$	−0.744353	−0.0379851
$385$	−1.98172	−0.100998
$386$	−8.04858	−0.409662
$387$	16.5847	0.843045
$388$	−1.80708	−0.0917404
$389$	13.0595	0.662143	0.331072	−	0.943606i	$-0.392590\pi$
$389$	13.0595	0.662143	0.331072	+	0.943606i	$0.392590\pi$
$390$	−12.7092	−0.643556
$391$	17.0548	0.862500
$392$	6.94155	0.350601
$393$	0.257198	0.0129739
$394$	−13.2469	−0.667371
$395$	−40.3710	−2.03128
$396$	6.01583	0.302307
$397$	−1.05800	−0.0530993	−0.0265496	−	0.999647i	$-0.508452\pi$
$397$	−1.05800	−0.0530993	−0.0265496	+	0.999647i	$0.508452\pi$
$398$	17.1696	0.860636
$399$	−0.179958	−0.00900918
$400$	6.10702	0.305351
$401$	27.0454	1.35058	0.675291	−	0.737552i	$-0.264018\pi$
$401$	27.0454	1.35058	0.675291	+	0.737552i	$0.264018\pi$
$402$	5.96866	0.297690
$403$	12.9166	0.643422
$404$	9.38449	0.466896
$405$	14.3988	0.715481
$406$	−0.123320	−0.00612029
$407$	2.93260	0.145364
$408$	1.46699	0.0726270
$409$	−7.90994	−0.391121	−0.195561	−	0.980692i	$-0.562653\pi$
$409$	−7.90994	−0.391121	−0.195561	+	0.980692i	$0.562653\pi$
$410$	7.43763	0.367318
$411$	−5.46134	−0.269388
$412$	6.74955	0.332526
$413$	1.49654	0.0736399
$414$	−21.1663	−1.04027
$415$	−8.27693	−0.406298
$416$	−5.12319	−0.251185
$417$	1.24338	0.0608886
$418$	−2.45952	−0.120299
$419$	−40.0838	−1.95822	−0.979112	−	0.203321i	$-0.934826\pi$
$419$	−40.0838	−1.95822	−0.979112	+	0.203321i	$0.934826\pi$
$420$	0.599750	0.0292648
$421$	−22.6055	−1.10172	−0.550862	−	0.834596i	$-0.685701\pi$
$421$	−22.6055	−1.10172	−0.550862	+	0.834596i	$0.685701\pi$
$422$	1.00000	0.0486792
$423$	4.56569	0.221991
$424$	10.7402	0.521589
$425$	−12.0359	−0.583827
$426$	−7.60432	−0.368431
$427$	0.272431	0.0131838
$428$	4.38034	0.211732
$429$	−9.37929	−0.452836
$430$	22.5975	1.08975
$431$	10.9068	0.525364	0.262682	−	0.964883i	$-0.415393\pi$
$431$	10.9068	0.525364	0.262682	+	0.964883i	$0.415393\pi$
$432$	−4.05370	−0.195034
$433$	−24.7796	−1.19083	−0.595415	−	0.803418i	$-0.703012\pi$
$433$	−24.7796	−1.19083	−0.595415	+	0.803418i	$0.703012\pi$
$434$	−0.609537	−0.0292587
$435$	1.26538	0.0606702
$436$	−4.60985	−0.220772
$437$	8.65364	0.413960
$438$	10.8169	0.516853
$439$	32.4133	1.54700	0.773501	−	0.633795i	$-0.218504\pi$
$439$	32.4133	1.54700	0.773501	+	0.633795i	$0.218504\pi$
$440$	8.19689	0.390771
$441$	16.9786	0.808505
$442$	10.0969	0.480262
$443$	7.56302	0.359330	0.179665	−	0.983728i	$-0.442499\pi$
$443$	7.56302	0.359330	0.179665	+	0.983728i	$0.442499\pi$
$444$	−0.887528	−0.0421202
$445$	42.4118	2.01051
$446$	19.3741	0.917390
$447$	4.11498	0.194632
$448$	0.241764	0.0114223
$449$	−32.3950	−1.52882	−0.764408	−	0.644733i	$-0.776969\pi$
$449$	−32.3950	−1.52882	−0.764408	+	0.644733i	$0.776969\pi$
$450$	14.9374	0.704156
$451$	5.48890	0.258462
$452$	3.55423	0.167177
$453$	−6.74534	−0.316924
$454$	9.76508	0.458298
$455$	4.12793	0.193520
$456$	0.744353	0.0348575
$457$	16.5296	0.773220	0.386610	−	0.922243i	$-0.373646\pi$
$457$	16.5296	0.773220	0.386610	+	0.922243i	$0.373646\pi$
$458$	−7.52711	−0.351719
$459$	7.98915	0.372902
$460$	−28.8401	−1.34468
$461$	−16.1635	−0.752807	−0.376404	−	0.926456i	$-0.622839\pi$
$461$	−16.1635	−0.752807	−0.376404	+	0.926456i	$0.622839\pi$
$462$	0.442610	0.0205921
$463$	8.83227	0.410470	0.205235	−	0.978713i	$-0.434204\pi$
$463$	8.83227	0.410470	0.205235	+	0.978713i	$0.434204\pi$
$464$	0.510085	0.0236801
$465$	6.25440	0.290041
$466$	15.4176	0.714207
$467$	−0.989841	−0.0458044	−0.0229022	−	0.999738i	$-0.507291\pi$
$467$	−0.989841	−0.0458044	−0.0229022	+	0.999738i	$0.507291\pi$
$468$	−12.5310	−0.579246
$469$	−1.93861	−0.0895167
$470$	6.22099	0.286953
$471$	−8.44198	−0.388986
$472$	−6.19008	−0.284921
$473$	16.6767	0.766797
$474$	9.01673	0.414152
$475$	−6.10702	−0.280209
$476$	−0.476476	−0.0218393
$477$	26.2698	1.20281
$478$	24.7937	1.13404
$479$	33.1586	1.51505	0.757527	−	0.652804i	$-0.226407\pi$
$479$	33.1586	1.51505	0.757527	+	0.652804i	$0.226407\pi$
$480$	−2.48072	−0.113229
$481$	−6.10863	−0.278529
$482$	−4.12130	−0.187720
$483$	−1.55729	−0.0708593
$484$	−4.95077	−0.225035
$485$	−6.02248	−0.273467
$486$	−15.3770	−0.697516
$487$	7.49646	0.339697	0.169848	−	0.985470i	$-0.445672\pi$
$487$	7.49646	0.339697	0.169848	+	0.985470i	$0.445672\pi$
$488$	−1.12684	−0.0510098
$489$	−8.74505	−0.395465
$490$	23.1342	1.04510
$491$	26.8333	1.21097	0.605485	−	0.795857i	$-0.292979\pi$
$491$	26.8333	1.21097	0.605485	+	0.795857i	$0.292979\pi$
$492$	−1.66117	−0.0748914
$493$	−1.00529	−0.0452760
$494$	5.12319	0.230503
$495$	20.0491	0.901139
$496$	2.52120	0.113205
$497$	2.46987	0.110789
$498$	1.84863	0.0828390
$499$	12.9066	0.577778	0.288889	−	0.957363i	$-0.406714\pi$
$499$	12.9066	0.577778	0.288889	+	0.957363i	$0.406714\pi$
$500$	3.68940	0.164995
$501$	2.67915	0.119695
$502$	9.35417	0.417497
$503$	32.3270	1.44139	0.720694	−	0.693253i	$-0.243823\pi$
$503$	32.3270	1.44139	0.720694	+	0.693253i	$0.243823\pi$
$504$	0.591341	0.0263404
$505$	31.2759	1.39176
$506$	−21.2838	−0.946179
$507$	9.86053	0.437921
$508$	−15.8407	−0.702816
$509$	4.09275	0.181408	0.0907039	−	0.995878i	$-0.471088\pi$
$509$	4.09275	0.181408	0.0907039	+	0.995878i	$0.471088\pi$
$510$	4.88908	0.216492
$511$	−3.51332	−0.155420
$512$	−1.00000	−0.0441942
$513$	4.05370	0.178975
$514$	5.50307	0.242730
$515$	22.4943	0.991219
$516$	−5.04708	−0.222185
$517$	4.59104	0.201914
$518$	0.288267	0.0126657
$519$	12.6242	0.554140
$520$	−17.0742	−0.748752
$521$	9.40381	0.411988	0.205994	−	0.978553i	$-0.433957\pi$
$521$	9.40381	0.411988	0.205994	+	0.978553i	$0.433957\pi$
$522$	1.24764	0.0546075
$523$	−37.7724	−1.65167	−0.825835	−	0.563912i	$-0.809296\pi$
$523$	−37.7724	−1.65167	−0.825835	+	0.563912i	$0.809296\pi$
$524$	0.345532	0.0150946
$525$	1.09901	0.0479647
$526$	28.0509	1.22308
$527$	−4.96886	−0.216447
$528$	−1.83075	−0.0796732
$529$	51.8854	2.25589
$530$	35.7940	1.55479
$531$	−15.1405	−0.657044
$532$	−0.241764	−0.0104818
$533$	−11.4334	−0.495237
$534$	−9.47254	−0.409917
$535$	14.5984	0.631146
$536$	8.01859	0.346350
$537$	0.774786	0.0334345
$538$	−24.8061	−1.06947
$539$	17.0729	0.735381
$540$	−13.5099	−0.581371
$541$	−21.1877	−0.910931	−0.455465	−	0.890254i	$-0.650527\pi$
$541$	−21.1877	−0.910931	−0.455465	+	0.890254i	$0.650527\pi$
$542$	4.57671	0.196586
$543$	−0.656488	−0.0281726
$544$	1.97083	0.0844986
$545$	−15.3633	−0.658093
$546$	−0.921960	−0.0394563
$547$	35.7847	1.53005	0.765023	−	0.644003i	$-0.222728\pi$
$547$	35.7847	1.53005	0.765023	+	0.644003i	$0.222728\pi$
$548$	−7.33702	−0.313422
$549$	−2.75619	−0.117631
$550$	15.0203	0.640469
$551$	−0.510085	−0.0217303
$552$	6.44136	0.274163
$553$	−2.92862	−0.124538
$554$	−11.1915	−0.475481
$555$	−2.95788	−0.125555
$556$	1.67042	0.0708415
$557$	−11.0737	−0.469209	−0.234605	−	0.972091i	$-0.575380\pi$
$557$	−11.0737	−0.469209	−0.234605	+	0.972091i	$0.575380\pi$
$558$	6.16671	0.261057
$559$	−34.7377	−1.46925
$560$	0.805733	0.0340484
$561$	3.60810	0.152334
$562$	13.4588	0.567723
$563$	−26.6894	−1.12483	−0.562413	−	0.826857i	$-0.690127\pi$
$563$	−26.6894	−1.12483	−0.562413	+	0.826857i	$0.690127\pi$
$564$	−1.38944	−0.0585060
$565$	11.8453	0.498334
$566$	−19.5100	−0.820067
$567$	1.04453	0.0438660
$568$	−10.2160	−0.428654
$569$	6.33830	0.265715	0.132858	−	0.991135i	$-0.457585\pi$
$569$	6.33830	0.265715	0.132858	+	0.991135i	$0.457585\pi$
$570$	2.48072	0.103906
$571$	−30.0702	−1.25840	−0.629199	−	0.777244i	$-0.716617\pi$
$571$	−30.0702	−1.25840	−0.629199	+	0.777244i	$0.716617\pi$
$572$	−12.6006	−0.526857
$573$	4.85793	0.202943
$574$	0.539546	0.0225202
$575$	−52.8480	−2.20391
$576$	−2.44594	−0.101914
$577$	10.5755	0.440263	0.220132	−	0.975470i	$-0.429351\pi$
$577$	10.5755	0.440263	0.220132	+	0.975470i	$0.429351\pi$
$578$	13.1158	0.545547
$579$	5.99098	0.248977
$580$	1.69997	0.0705874
$581$	−0.600431	−0.0249101
$582$	1.34510	0.0557563
$583$	26.4157	1.09403
$584$	14.5320	0.601337
$585$	−41.7624	−1.72666
$586$	−24.4438	−1.00977
$587$	−41.0145	−1.69285	−0.846424	−	0.532509i	$-0.821249\pi$
$587$	−41.0145	−1.69285	−0.846424	+	0.532509i	$0.821249\pi$
$588$	−5.16696	−0.213082
$589$	−2.52120	−0.103884
$590$	−20.6298	−0.849315
$591$	9.86040	0.405603
$592$	−1.19235	−0.0490052
$593$	−2.60367	−0.106920	−0.0534599	−	0.998570i	$-0.517025\pi$
$593$	−2.60367	−0.106920	−0.0534599	+	0.998570i	$0.517025\pi$
$594$	−9.97015	−0.409080
$595$	−1.58796	−0.0651001
$596$	5.52826	0.226446
$597$	−12.7803	−0.523062
$598$	44.3343	1.81296
$599$	−24.7010	−1.00926	−0.504629	−	0.863337i	$-0.668371\pi$
$599$	−24.7010	−1.00926	−0.504629	+	0.863337i	$0.668371\pi$
$600$	−4.54578	−0.185581
$601$	−39.3108	−1.60352	−0.801760	−	0.597646i	$-0.796103\pi$
$601$	−39.3108	−1.60352	−0.801760	+	0.597646i	$0.796103\pi$
$602$	1.63928	0.0668121
$603$	19.6130	0.798702
$604$	−9.06202	−0.368728
$605$	−16.4995	−0.670801
$606$	−6.98537	−0.283761
$607$	4.98753	0.202438	0.101219	−	0.994864i	$-0.467726\pi$
$607$	4.98753	0.202438	0.101219	+	0.994864i	$0.467726\pi$
$608$	1.00000	0.0405554
$609$	0.0917939	0.00371968
$610$	−3.75545	−0.152054
$611$	−9.56316	−0.386884
$612$	4.82053	0.194858
$613$	−1.14404	−0.0462073	−0.0231037	−	0.999733i	$-0.507355\pi$
$613$	−1.14404	−0.0462073	−0.0231037	+	0.999733i	$0.507355\pi$
$614$	8.78387	0.354488
$615$	−5.53622	−0.223242
$616$	0.594624	0.0239581
$617$	−14.3365	−0.577166	−0.288583	−	0.957455i	$-0.593184\pi$
$617$	−14.3365	−0.577166	−0.288583	+	0.957455i	$0.593184\pi$
$618$	−5.02405	−0.202097
$619$	29.3734	1.18062	0.590309	−	0.807177i	$-0.299006\pi$
$619$	29.3734	1.18062	0.590309	+	0.807177i	$0.299006\pi$
$620$	8.40246	0.337451
$621$	35.0793	1.40768
$622$	−7.96037	−0.319182
$623$	3.07666	0.123264
$624$	3.81347	0.152661
$625$	−18.2394	−0.729575
$626$	−23.8940	−0.954998
$627$	1.83075	0.0731131
$628$	−11.3414	−0.452570
$629$	2.34991	0.0936972
$630$	1.97077	0.0785175
$631$	25.7381	1.02462	0.512309	−	0.858801i	$-0.328790\pi$
$631$	25.7381	1.02462	0.512309	+	0.858801i	$0.328790\pi$
$632$	12.1135	0.481850
$633$	−0.744353	−0.0295854
$634$	−26.1551	−1.03875
$635$	−52.7925	−2.09501
$636$	−7.99449	−0.317002
$637$	−35.5629	−1.40905
$638$	1.25456	0.0496686
$639$	−24.9877	−0.988500
$640$	−3.33272	−0.131737
$641$	40.8435	1.61322	0.806610	−	0.591084i	$-0.201300\pi$
$641$	40.8435	1.61322	0.806610	+	0.591084i	$0.201300\pi$
$642$	−3.26052	−0.128682
$643$	−24.0529	−0.948553	−0.474277	−	0.880376i	$-0.657290\pi$
$643$	−24.0529	−0.948553	−0.474277	+	0.880376i	$0.657290\pi$
$644$	−2.09214	−0.0824419
$645$	−16.8205	−0.662306
$646$	−1.97083	−0.0775412
$647$	38.7765	1.52446	0.762231	−	0.647305i	$-0.224104\pi$
$647$	38.7765	1.52446	0.762231	+	0.647305i	$0.224104\pi$
$648$	−4.32043	−0.169722
$649$	−15.2246	−0.597618
$650$	−31.2875	−1.22720
$651$	0.453711	0.0177823
$652$	−11.7485	−0.460108
$653$	−5.00095	−0.195702	−0.0978511	−	0.995201i	$-0.531197\pi$
$653$	−5.00095	−0.195702	−0.0978511	+	0.995201i	$0.531197\pi$
$654$	3.43136	0.134177
$655$	1.15156	0.0449952
$656$	−2.23170	−0.0871332
$657$	35.5443	1.38672
$658$	0.451287	0.0175930
$659$	−4.53825	−0.176785	−0.0883926	−	0.996086i	$-0.528173\pi$
$659$	−4.53825	−0.176785	−0.0883926	+	0.996086i	$0.528173\pi$
$660$	−6.10138	−0.237496
$661$	10.3541	0.402726	0.201363	−	0.979517i	$-0.435463\pi$
$661$	10.3541	0.402726	0.201363	+	0.979517i	$0.435463\pi$
$662$	21.1601	0.822410
$663$	−7.51569	−0.291885
$664$	2.48354	0.0963799
$665$	−0.805733	−0.0312450
$666$	−2.91641	−0.113009
$667$	−4.41409	−0.170914
$668$	3.59930	0.139261
$669$	−14.4212	−0.557555
$670$	26.7237	1.03243
$671$	−2.77149	−0.106992
$672$	−0.179958	−0.00694204
$673$	4.57825	0.176478	0.0882392	−	0.996099i	$-0.471876\pi$
$673$	4.57825	0.176478	0.0882392	+	0.996099i	$0.471876\pi$
$674$	0.905939	0.0348955
$675$	−24.7560	−0.952861
$676$	13.2471	0.509504
$677$	3.89842	0.149829	0.0749143	−	0.997190i	$-0.476132\pi$
$677$	3.89842	0.149829	0.0749143	+	0.997190i	$0.476132\pi$
$678$	−2.64560	−0.101604
$679$	−0.436887	−0.0167662
$680$	6.56822	0.251880
$681$	−7.26867	−0.278536
$682$	6.20094	0.237446
$683$	−45.3644	−1.73582	−0.867910	−	0.496722i	$-0.834537\pi$
$683$	−45.3644	−1.73582	−0.867910	+	0.496722i	$0.834537\pi$
$684$	2.44594	0.0935228
$685$	−24.4522	−0.934272
$686$	3.37057	0.128689
$687$	5.60283	0.213761
$688$	−6.78049	−0.258504
$689$	−55.0240	−2.09625
$690$	21.4673	0.817244
$691$	6.91342	0.262999	0.131499	−	0.991316i	$-0.458021\pi$
$691$	6.91342	0.262999	0.131499	+	0.991316i	$0.458021\pi$
$692$	16.9599	0.644720
$693$	1.45441	0.0552486
$694$	3.75311	0.142466
$695$	5.56704	0.211170
$696$	−0.379683	−0.0143919
$697$	4.39830	0.166597
$698$	−29.0410	−1.09922
$699$	−11.4761	−0.434067
$700$	1.47646	0.0558050
$701$	−14.9829	−0.565898	−0.282949	−	0.959135i	$-0.591313\pi$
$701$	−14.9829	−0.565898	−0.282949	+	0.959135i	$0.591313\pi$
$702$	20.7679	0.783834
$703$	1.19235	0.0449702
$704$	−2.45952	−0.0926966
$705$	−4.63061	−0.174399
$706$	2.35144	0.0884977
$707$	2.26884	0.0853283
$708$	4.60760	0.173164
$709$	−5.75134	−0.215996	−0.107998	−	0.994151i	$-0.534444\pi$
$709$	−5.75134	−0.215996	−0.107998	+	0.994151i	$0.534444\pi$
$710$	−34.0471	−1.27777
$711$	29.6289	1.11117
$712$	−12.7259	−0.476922
$713$	−21.8176	−0.817074
$714$	0.354667	0.0132731
$715$	−41.9942	−1.57050
$716$	1.04088	0.0388997
$717$	−18.4553	−0.689226
$718$	−20.0239	−0.747285
$719$	6.79398	0.253373	0.126686	−	0.991943i	$-0.459566\pi$
$719$	6.79398	0.253373	0.126686	+	0.991943i	$0.459566\pi$
$720$	−8.15163	−0.303793
$721$	1.63180	0.0607714
$722$	−1.00000	−0.0372161
$723$	3.06770	0.114089
$724$	−0.881957	−0.0327777
$725$	3.11510	0.115692
$726$	3.68512	0.136768
$727$	−40.4653	−1.50077	−0.750387	−	0.660999i	$-0.770133\pi$
$727$	−40.4653	−1.50077	−0.750387	+	0.660999i	$0.770133\pi$
$728$	−1.23861	−0.0459058
$729$	−1.51535	−0.0561240
$730$	48.4310	1.79251
$731$	13.3632	0.494255
$732$	0.838769	0.0310018
$733$	−9.62793	−0.355616	−0.177808	−	0.984065i	$-0.556901\pi$
$733$	−9.62793	−0.355616	−0.177808	+	0.984065i	$0.556901\pi$
$734$	−29.2296	−1.07888
$735$	−17.2200	−0.635171
$736$	8.65364	0.318977
$737$	19.7219	0.726464
$738$	−5.45860	−0.200934
$739$	−19.0808	−0.701898	−0.350949	−	0.936395i	$-0.614141\pi$
$739$	−19.0808	−0.701898	−0.350949	+	0.936395i	$0.614141\pi$
$740$	−3.97376	−0.146078
$741$	−3.81347	−0.140091
$742$	2.59660	0.0953240
$743$	−24.6466	−0.904195	−0.452098	−	0.891968i	$-0.649324\pi$
$743$	−24.6466	−0.904195	−0.452098	+	0.891968i	$0.649324\pi$
$744$	−1.87666	−0.0688019
$745$	18.4241	0.675008
$746$	19.1164	0.699902
$747$	6.07457	0.222257
$748$	4.84729	0.177234
$749$	1.05901	0.0386954
$750$	−2.74621	−0.100278
$751$	4.46452	0.162913	0.0814563	−	0.996677i	$-0.474043\pi$
$751$	4.46452	0.162913	0.0814563	+	0.996677i	$0.474043\pi$
$752$	−1.86664	−0.0680694
$753$	−6.96280	−0.253739
$754$	−2.61326	−0.0951694
$755$	−30.2012	−1.09913
$756$	−0.980041	−0.0356437
$757$	45.2251	1.64373	0.821867	−	0.569680i	$-0.192933\pi$
$757$	45.2251	1.64373	0.821867	+	0.569680i	$0.192933\pi$
$758$	31.3021	1.13694
$759$	15.8426	0.575052
$760$	3.33272	0.120890
$761$	−13.5058	−0.489585	−0.244793	−	0.969575i	$-0.578720\pi$
$761$	−13.5058	−0.489585	−0.244793	+	0.969575i	$0.578720\pi$
$762$	11.7910	0.427144
$763$	−1.11450	−0.0403475
$764$	6.52637	0.236116
$765$	16.0655	0.580848
$766$	−35.0634	−1.26689
$767$	31.7130	1.14509
$768$	0.744353	0.0268595
$769$	−17.5404	−0.632524	−0.316262	−	0.948672i	$-0.602428\pi$
$769$	−17.5404	−0.632524	−0.316262	+	0.948672i	$0.602428\pi$
$770$	1.98172	0.0714161
$771$	−4.09622	−0.147522
$772$	8.04858	0.289675
$773$	−41.9178	−1.50768	−0.753840	−	0.657058i	$-0.771801\pi$
$773$	−41.9178	−1.50768	−0.753840	+	0.657058i	$0.771801\pi$
$774$	−16.5847	−0.596123
$775$	15.3970	0.553078
$776$	1.80708	0.0648703
$777$	−0.214573	−0.00769775
$778$	−13.0595	−0.468206
$779$	2.23170	0.0799589
$780$	12.7092	0.455063
$781$	−25.1265	−0.899096
$782$	−17.0548	−0.609880
$783$	−2.06773	−0.0738947
$784$	−6.94155	−0.247912
$785$	−37.7976	−1.34905
$786$	−0.257198	−0.00917394
$787$	−7.87240	−0.280621	−0.140310	−	0.990108i	$-0.544810\pi$
$787$	−7.87240	−0.280621	−0.140310	+	0.990108i	$0.544810\pi$
$788$	13.2469	0.471903
$789$	−20.8798	−0.743340
$790$	40.3710	1.43633
$791$	0.859287	0.0305527
$792$	−6.01583	−0.213763
$793$	5.77303	0.205006
$794$	1.05800	0.0375469
$795$	−26.6434	−0.944944
$796$	−17.1696	−0.608562
$797$	9.16988	0.324814	0.162407	−	0.986724i	$-0.448074\pi$
$797$	9.16988	0.324814	0.162407	+	0.986724i	$0.448074\pi$
$798$	0.179958	0.00637045
$799$	3.67883	0.130148
$800$	−6.10702	−0.215916
$801$	−31.1267	−1.09981
$802$	−27.0454	−0.955005
$803$	35.7417	1.26130
$804$	−5.96866	−0.210499
$805$	−6.97252	−0.245749
$806$	−12.9166	−0.454968
$807$	18.4645	0.649981
$808$	−9.38449	−0.330145
$809$	−7.05668	−0.248100	−0.124050	−	0.992276i	$-0.539588\pi$
$809$	−7.05668	−0.248100	−0.124050	+	0.992276i	$0.539588\pi$
$810$	−14.3988	−0.505922
$811$	23.1592	0.813231	0.406615	−	0.913599i	$-0.366709\pi$
$811$	23.1592	0.813231	0.406615	+	0.913599i	$0.366709\pi$
$812$	0.123320	0.00432770
$813$	−3.40669	−0.119478
$814$	−2.93260	−0.102788
$815$	−39.1545	−1.37152
$816$	−1.46699	−0.0513550
$817$	6.78049	0.237219
$818$	7.90994	0.276565
$819$	−3.02955	−0.105861
$820$	−7.43763	−0.259733
$821$	−32.8920	−1.14794	−0.573969	−	0.818877i	$-0.694597\pi$
$821$	−32.8920	−1.14794	−0.573969	+	0.818877i	$0.694597\pi$
$822$	5.46134	0.190486
$823$	−44.6254	−1.55554	−0.777772	−	0.628547i	$-0.783650\pi$
$823$	−44.6254	−1.55554	−0.777772	+	0.628547i	$0.783650\pi$
$824$	−6.74955	−0.235132
$825$	−11.1804	−0.389253
$826$	−1.49654	−0.0520713
$827$	−29.1894	−1.01502	−0.507508	−	0.861647i	$-0.669433\pi$
$827$	−29.1894	−1.01502	−0.507508	+	0.861647i	$0.669433\pi$
$828$	21.1663	0.735578
$829$	−27.2867	−0.947705	−0.473852	−	0.880604i	$-0.657137\pi$
$829$	−27.2867	−0.947705	−0.473852	+	0.880604i	$0.657137\pi$
$830$	8.27693	0.287296
$831$	8.33043	0.288979
$832$	5.12319	0.177615
$833$	13.6806	0.474005
$834$	−1.24338	−0.0430548
$835$	11.9954	0.415120
$836$	2.45952	0.0850642
$837$	−10.2202	−0.353262
$838$	40.0838	1.38467
$839$	6.47523	0.223550	0.111775	−	0.993734i	$-0.464346\pi$
$839$	6.47523	0.223550	0.111775	+	0.993734i	$0.464346\pi$
$840$	−0.599750	−0.0206933
$841$	−28.7398	−0.991028
$842$	22.6055	0.779037
$843$	−10.0181	−0.345041
$844$	−1.00000	−0.0344214
$845$	44.1489	1.51877
$846$	−4.56569	−0.156972
$847$	−1.19692	−0.0411267
$848$	−10.7402	−0.368819
$849$	14.5223	0.498405
$850$	12.0359	0.412828
$851$	10.3181	0.353701
$852$	7.60432	0.260520
$853$	−53.3703	−1.82736	−0.913682	−	0.406429i	$-0.866774\pi$
$853$	−53.3703	−1.82736	−0.913682	+	0.406429i	$0.866774\pi$
$854$	−0.272431	−0.00932238
$855$	8.15163	0.278780
$856$	−4.38034	−0.149717
$857$	7.45818	0.254766	0.127383	−	0.991854i	$-0.459342\pi$
$857$	7.45818	0.254766	0.127383	+	0.991854i	$0.459342\pi$
$858$	9.37929	0.320204
$859$	9.80449	0.334525	0.167262	−	0.985912i	$-0.446507\pi$
$859$	9.80449	0.334525	0.167262	+	0.985912i	$0.446507\pi$
$860$	−22.5975	−0.770567
$861$	−0.401612	−0.0136869
$862$	−10.9068	−0.371488
$863$	−11.9705	−0.407482	−0.203741	−	0.979025i	$-0.565310\pi$
$863$	−11.9705	−0.407482	−0.203741	+	0.979025i	$0.565310\pi$
$864$	4.05370	0.137910
$865$	56.5227	1.92183
$866$	24.7796	0.842044
$867$	−9.76281	−0.331563
$868$	0.609537	0.0206890
$869$	29.7934	1.01067
$870$	−1.26538	−0.0429003
$871$	−41.0808	−1.39197
$872$	4.60985	0.156109
$873$	4.42000	0.149594
$874$	−8.65364	−0.292714
$875$	0.891965	0.0301539
$876$	−10.8169	−0.365470
$877$	−22.6913	−0.766229	−0.383115	−	0.923701i	$-0.625149\pi$
$877$	−22.6913	−0.766229	−0.383115	+	0.923701i	$0.625149\pi$
$878$	−32.4133	−1.09390
$879$	18.1948	0.613697
$880$	−8.19689	−0.276317
$881$	12.0726	0.406735	0.203367	−	0.979102i	$-0.434811\pi$
$881$	12.0726	0.406735	0.203367	+	0.979102i	$0.434811\pi$
$882$	−16.9786	−0.571699
$883$	−22.5696	−0.759527	−0.379764	−	0.925084i	$-0.623995\pi$
$883$	−22.5696	−0.759527	−0.379764	+	0.925084i	$0.623995\pi$
$884$	−10.0969	−0.339597
$885$	15.3558	0.516181
$886$	−7.56302	−0.254084
$887$	−0.650037	−0.0218261	−0.0109130	−	0.999940i	$-0.503474\pi$
$887$	−0.650037	−0.0218261	−0.0109130	+	0.999940i	$0.503474\pi$
$888$	0.887528	0.0297835
$889$	−3.82971	−0.128444
$890$	−42.4118	−1.42165
$891$	−10.6262	−0.355990
$892$	−19.3741	−0.648693
$893$	1.86664	0.0624647
$894$	−4.11498	−0.137625
$895$	3.46898	0.115955
$896$	−0.241764	−0.00807678
$897$	−33.0003	−1.10185
$898$	32.3950	1.08104
$899$	1.28603	0.0428914
$900$	−14.9374	−0.497913
$901$	21.1671	0.705177
$902$	−5.48890	−0.182761
$903$	−1.22020	−0.0406058
$904$	−3.55423	−0.118212
$905$	−2.93932	−0.0977062
$906$	6.74534	0.224099
$907$	−1.87585	−0.0622865	−0.0311433	−	0.999515i	$-0.509915\pi$
$907$	−1.87585	−0.0622865	−0.0311433	+	0.999515i	$0.509915\pi$
$908$	−9.76508	−0.324066
$909$	−22.9539	−0.761332
$910$	−4.12793	−0.136839
$911$	13.8414	0.458586	0.229293	−	0.973358i	$-0.426359\pi$
$911$	13.8414	0.458586	0.229293	+	0.973358i	$0.426359\pi$
$912$	−0.744353	−0.0246480
$913$	6.10830	0.202155
$914$	−16.5296	−0.546749
$915$	2.79538	0.0924125
$916$	7.52711	0.248703
$917$	0.0835373	0.00275865
$918$	−7.98915	−0.263681
$919$	−52.4220	−1.72924	−0.864621	−	0.502424i	$-0.832441\pi$
$919$	−52.4220	−1.72924	−0.864621	+	0.502424i	$0.832441\pi$
$920$	28.8401	0.950832
$921$	−6.53830	−0.215444
$922$	16.1635	0.532315
$923$	52.3386	1.72275
$924$	−0.442610	−0.0145608
$925$	−7.28169	−0.239421
$926$	−8.83227	−0.290246
$927$	−16.5090	−0.542226
$928$	−0.510085	−0.0167444
$929$	−12.2532	−0.402016	−0.201008	−	0.979590i	$-0.564422\pi$
$929$	−12.2532	−0.402016	−0.201008	+	0.979590i	$0.564422\pi$
$930$	−6.25440	−0.205090
$931$	6.94155	0.227500
$932$	−15.4176	−0.505020
$933$	5.92533	0.193986
$934$	0.989841	0.0323886
$935$	16.1547	0.528314
$936$	12.5310	0.409589
$937$	35.3671	1.15539	0.577696	−	0.816252i	$-0.303952\pi$
$937$	35.3671	1.15539	0.577696	+	0.816252i	$0.303952\pi$
$938$	1.93861	0.0632979
$939$	17.7856	0.580411
$940$	−6.22099	−0.202906
$941$	33.5416	1.09343	0.546713	−	0.837320i	$-0.315879\pi$
$941$	33.5416	1.09343	0.546713	+	0.837320i	$0.315879\pi$
$942$	8.44198	0.275054
$943$	19.3123	0.628895
$944$	6.19008	0.201470
$945$	−3.26620	−0.106250
$946$	−16.6767	−0.542207
$947$	−24.3038	−0.789769	−0.394884	−	0.918731i	$-0.629215\pi$
$947$	−24.3038	−0.789769	−0.394884	+	0.918731i	$0.629215\pi$
$948$	−9.01673	−0.292850
$949$	−74.4501	−2.41675
$950$	6.10702	0.198138
$951$	19.4686	0.631313
$952$	0.476476	0.0154427
$953$	−19.3590	−0.627101	−0.313550	−	0.949572i	$-0.601518\pi$
$953$	−19.3590	−0.627101	−0.313550	+	0.949572i	$0.601518\pi$
$954$	−26.2698	−0.850517
$955$	21.7506	0.703832
$956$	−24.7937	−0.801887
$957$	−0.933838	−0.0301867
$958$	−33.1586	−1.07131
$959$	−1.77383	−0.0572800
$960$	2.48072	0.0800649
$961$	−24.6435	−0.794953
$962$	6.10863	0.196950
$963$	−10.7140	−0.345255
$964$	4.12130	0.132738
$965$	26.8236	0.863484
$966$	1.55729	0.0501051
$967$	−29.4808	−0.948038	−0.474019	−	0.880515i	$-0.657197\pi$
$967$	−29.4808	−0.948038	−0.474019	+	0.880515i	$0.657197\pi$
$968$	4.95077	0.159124
$969$	1.46699	0.0471266
$970$	6.02248	0.193370
$971$	14.5776	0.467817	0.233909	−	0.972259i	$-0.424848\pi$
$971$	14.5776	0.467817	0.233909	+	0.972259i	$0.424848\pi$
$972$	15.3770	0.493218
$973$	0.403848	0.0129468
$974$	−7.49646	−0.240202
$975$	23.2889	0.745842
$976$	1.12684	0.0360694
$977$	36.3543	1.16308	0.581538	−	0.813519i	$-0.302451\pi$
$977$	36.3543	1.16308	0.581538	+	0.813519i	$0.302451\pi$
$978$	8.74505	0.279636
$979$	−31.2995	−1.00034
$980$	−23.1342	−0.738996
$981$	11.2754	0.359996
$982$	−26.8333	−0.856285
$983$	−29.3324	−0.935559	−0.467780	−	0.883845i	$-0.654946\pi$
$983$	−29.3324	−0.935559	−0.467780	+	0.883845i	$0.654946\pi$
$984$	1.66117	0.0529562
$985$	44.1483	1.40668
$986$	1.00529	0.0320149
$987$	−0.335917	−0.0106924
$988$	−5.12319	−0.162990
$989$	58.6759	1.86578
$990$	−20.0491	−0.637201
$991$	10.7261	0.340726	0.170363	−	0.985381i	$-0.445506\pi$
$991$	10.7261	0.340726	0.170363	+	0.985381i	$0.445506\pi$
$992$	−2.52120	−0.0800483
$993$	−15.7506	−0.499829
$994$	−2.46987	−0.0783395
$995$	−57.2216	−1.81405
$996$	−1.84863	−0.0585760
$997$	56.2754	1.78226	0.891130	−	0.453748i	$-0.149914\pi$
$997$	56.2754	1.78226	0.891130	+	0.453748i	$0.149914\pi$
$998$	−12.9066	−0.408551
$999$	4.83342	0.152923

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8018.2.a.f.1.21	✓	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8018.2.a.f.1.21	✓	34	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:	\( N \)	\(=\)	\( 8018 = 2 \cdot 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8018.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.744353 q^{3} +1.00000 q^{4} +3.33272 q^{5} -0.744353 q^{6} +0.241764 q^{7} -1.00000 q^{8} -2.44594 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.744353 q^{3} +1.00000 q^{4} +3.33272 q^{5} -0.744353 q^{6} +0.241764 q^{7} -1.00000 q^{8} -2.44594 q^{9} -3.33272 q^{10} -2.45952 q^{11} +0.744353 q^{12} +5.12319 q^{13} -0.241764 q^{14} +2.48072 q^{15} +1.00000 q^{16} -1.97083 q^{17} +2.44594 q^{18} -1.00000 q^{19} +3.33272 q^{20} +0.179958 q^{21} +2.45952 q^{22} -8.65364 q^{23} -0.744353 q^{24} +6.10702 q^{25} -5.12319 q^{26} -4.05370 q^{27} +0.241764 q^{28} +0.510085 q^{29} -2.48072 q^{30} +2.52120 q^{31} -1.00000 q^{32} -1.83075 q^{33} +1.97083 q^{34} +0.805733 q^{35} -2.44594 q^{36} -1.19235 q^{37} +1.00000 q^{38} +3.81347 q^{39} -3.33272 q^{40} -2.23170 q^{41} -0.179958 q^{42} -6.78049 q^{43} -2.45952 q^{44} -8.15163 q^{45} +8.65364 q^{46} -1.86664 q^{47} +0.744353 q^{48} -6.94155 q^{49} -6.10702 q^{50} -1.46699 q^{51} +5.12319 q^{52} -10.7402 q^{53} +4.05370 q^{54} -8.19689 q^{55} -0.241764 q^{56} -0.744353 q^{57} -0.510085 q^{58} +6.19008 q^{59} +2.48072 q^{60} +1.12684 q^{61} -2.52120 q^{62} -0.591341 q^{63} +1.00000 q^{64} +17.0742 q^{65} +1.83075 q^{66} -8.01859 q^{67} -1.97083 q^{68} -6.44136 q^{69} -0.805733 q^{70} +10.2160 q^{71} +2.44594 q^{72} -14.5320 q^{73} +1.19235 q^{74} +4.54578 q^{75} -1.00000 q^{76} -0.594624 q^{77} -3.81347 q^{78} -12.1135 q^{79} +3.33272 q^{80} +4.32043 q^{81} +2.23170 q^{82} -2.48354 q^{83} +0.179958 q^{84} -6.56822 q^{85} +6.78049 q^{86} +0.379683 q^{87} +2.45952 q^{88} +12.7259 q^{89} +8.15163 q^{90} +1.23861 q^{91} -8.65364 q^{92} +1.87666 q^{93} +1.86664 q^{94} -3.33272 q^{95} -0.744353 q^{96} -1.80708 q^{97} +6.94155 q^{98} +6.01583 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q - 34 q^{2} - 10 q^{3} + 34 q^{4} + 7 q^{5} + 10 q^{6} - 6 q^{7} - 34 q^{8} + 38 q^{9}+O(q^{10}) \) 34 * q - 34 * q^2 - 10 * q^3 + 34 * q^4 + 7 * q^5 + 10 * q^6 - 6 * q^7 - 34 * q^8 + 38 * q^9	\( 34 q - 34 q^{2} - 10 q^{3} + 34 q^{4} + 7 q^{5} + 10 q^{6} - 6 q^{7} - 34 q^{8} + 38 q^{9} - 7 q^{10} + 4 q^{11} - 10 q^{12} - 5 q^{13} + 6 q^{14} - 17 q^{15} + 34 q^{16} + 8 q^{17} - 38 q^{18} - 34 q^{19} + 7 q^{20} - 4 q^{22} - 24 q^{23} + 10 q^{24} + 5 q^{25} + 5 q^{26} - 31 q^{27} - 6 q^{28} + 24 q^{29} + 17 q^{30} - 28 q^{31} - 34 q^{32} - 16 q^{33} - 8 q^{34} + 2 q^{35} + 38 q^{36} - 36 q^{37} + 34 q^{38} - 9 q^{39} - 7 q^{40} + 9 q^{41} - 24 q^{43} + 4 q^{44} + 22 q^{45} + 24 q^{46} - 29 q^{47} - 10 q^{48} + 22 q^{49} - 5 q^{50} - 21 q^{51} - 5 q^{52} - 9 q^{53} + 31 q^{54} - 28 q^{55} + 6 q^{56} + 10 q^{57} - 24 q^{58} - 28 q^{59} - 17 q^{60} + 23 q^{61} + 28 q^{62} - 27 q^{63} + 34 q^{64} - 6 q^{65} + 16 q^{66} - 51 q^{67} + 8 q^{68} - 17 q^{69} - 2 q^{70} - 31 q^{71} - 38 q^{72} - 26 q^{73} + 36 q^{74} - 109 q^{75} - 34 q^{76} - 28 q^{77} + 9 q^{78} - 36 q^{79} + 7 q^{80} + 6 q^{81} - 9 q^{82} - 10 q^{83} - 32 q^{85} + 24 q^{86} - 8 q^{87} - 4 q^{88} - 34 q^{89} - 22 q^{90} - 41 q^{91} - 24 q^{92} - 33 q^{93} + 29 q^{94} - 7 q^{95} + 10 q^{96} - 56 q^{97} - 22 q^{98} - 28 q^{99}+O(q^{100}) \) 34 * q - 34 * q^2 - 10 * q^3 + 34 * q^4 + 7 * q^5 + 10 * q^6 - 6 * q^7 - 34 * q^8 + 38 * q^9 - 7 * q^10 + 4 * q^11 - 10 * q^12 - 5 * q^13 + 6 * q^14 - 17 * q^15 + 34 * q^16 + 8 * q^17 - 38 * q^18 - 34 * q^19 + 7 * q^20 - 4 * q^22 - 24 * q^23 + 10 * q^24 + 5 * q^25 + 5 * q^26 - 31 * q^27 - 6 * q^28 + 24 * q^29 + 17 * q^30 - 28 * q^31 - 34 * q^32 - 16 * q^33 - 8 * q^34 + 2 * q^35 + 38 * q^36 - 36 * q^37 + 34 * q^38 - 9 * q^39 - 7 * q^40 + 9 * q^41 - 24 * q^43 + 4 * q^44 + 22 * q^45 + 24 * q^46 - 29 * q^47 - 10 * q^48 + 22 * q^49 - 5 * q^50 - 21 * q^51 - 5 * q^52 - 9 * q^53 + 31 * q^54 - 28 * q^55 + 6 * q^56 + 10 * q^57 - 24 * q^58 - 28 * q^59 - 17 * q^60 + 23 * q^61 + 28 * q^62 - 27 * q^63 + 34 * q^64 - 6 * q^65 + 16 * q^66 - 51 * q^67 + 8 * q^68 - 17 * q^69 - 2 * q^70 - 31 * q^71 - 38 * q^72 - 26 * q^73 + 36 * q^74 - 109 * q^75 - 34 * q^76 - 28 * q^77 + 9 * q^78 - 36 * q^79 + 7 * q^80 + 6 * q^81 - 9 * q^82 - 10 * q^83 - 32 * q^85 + 24 * q^86 - 8 * q^87 - 4 * q^88 - 34 * q^89 - 22 * q^90 - 41 * q^91 - 24 * q^92 - 33 * q^93 + 29 * q^94 - 7 * q^95 + 10 * q^96 - 56 * q^97 - 22 * q^98 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more