LMFDB - Embedded newform 8043.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.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:	$64.2236783457$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.70928 q^{2} +1.00000 q^{3} +5.34017 q^{4} +1.70928 q^{5} -2.70928 q^{6} -1.00000 q^{7} -9.04945 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.70928 q^{2} +1.00000 q^{3} +5.34017 q^{4} +1.70928 q^{5} -2.70928 q^{6} -1.00000 q^{7} -9.04945 q^{8} +1.00000 q^{9} -4.63090 q^{10} +2.63090 q^{11} +5.34017 q^{12} +2.29072 q^{13} +2.70928 q^{14} +1.70928 q^{15} +13.8371 q^{16} +0.340173 q^{17} -2.70928 q^{18} +8.68035 q^{19} +9.12783 q^{20} -1.00000 q^{21} -7.12783 q^{22} +8.49693 q^{23} -9.04945 q^{24} -2.07838 q^{25} -6.20620 q^{26} +1.00000 q^{27} -5.34017 q^{28} +9.60197 q^{29} -4.63090 q^{30} -10.3402 q^{31} -19.3896 q^{32} +2.63090 q^{33} -0.921622 q^{34} -1.70928 q^{35} +5.34017 q^{36} +10.0000 q^{37} -23.5174 q^{38} +2.29072 q^{39} -15.4680 q^{40} +4.44748 q^{41} +2.70928 q^{42} +7.41855 q^{43} +14.0494 q^{44} +1.70928 q^{45} -23.0205 q^{46} +5.26180 q^{47} +13.8371 q^{48} +1.00000 q^{49} +5.63090 q^{50} +0.340173 q^{51} +12.2329 q^{52} +8.34017 q^{53} -2.70928 q^{54} +4.49693 q^{55} +9.04945 q^{56} +8.68035 q^{57} -26.0144 q^{58} +12.6803 q^{59} +9.12783 q^{60} -11.8660 q^{61} +28.0144 q^{62} -1.00000 q^{63} +24.8576 q^{64} +3.91548 q^{65} -7.12783 q^{66} +4.09890 q^{67} +1.81658 q^{68} +8.49693 q^{69} +4.63090 q^{70} -11.0205 q^{71} -9.04945 q^{72} -8.83710 q^{73} -27.0928 q^{74} -2.07838 q^{75} +46.3545 q^{76} -2.63090 q^{77} -6.20620 q^{78} +2.04945 q^{79} +23.6514 q^{80} +1.00000 q^{81} -12.0494 q^{82} -2.83710 q^{83} -5.34017 q^{84} +0.581449 q^{85} -20.0989 q^{86} +9.60197 q^{87} -23.8082 q^{88} -10.3896 q^{89} -4.63090 q^{90} -2.29072 q^{91} +45.3751 q^{92} -10.3402 q^{93} -14.2557 q^{94} +14.8371 q^{95} -19.3896 q^{96} +1.02893 q^{97} -2.70928 q^{98} +2.63090 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 3 q^{3} + 5 q^{4} - 2 q^{5} - q^{6} - 3 q^{7} - 9 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - q^2 + 3 * q^3 + 5 * q^4 - 2 * q^5 - q^6 - 3 * q^7 - 9 * q^8 + 3 * q^9	\( 3 q - q^{2} + 3 q^{3} + 5 q^{4} - 2 q^{5} - q^{6} - 3 q^{7} - 9 q^{8} + 3 q^{9} - 10 q^{10} + 4 q^{11} + 5 q^{12} + 14 q^{13} + q^{14} - 2 q^{15} + 13 q^{16} - 10 q^{17} - q^{18} + 4 q^{19} + 6 q^{20} - 3 q^{21} + 8 q^{23} - 9 q^{24} - 3 q^{25} + 6 q^{26} + 3 q^{27} - 5 q^{28} + 10 q^{29} - 10 q^{30} - 20 q^{31} - 29 q^{32} + 4 q^{33} - 6 q^{34} + 2 q^{35} + 5 q^{36} + 30 q^{37} - 20 q^{38} + 14 q^{39} - 14 q^{40} + 14 q^{41} + q^{42} + 8 q^{43} + 24 q^{44} - 2 q^{45} - 36 q^{46} + 8 q^{47} + 13 q^{48} + 3 q^{49} + 13 q^{50} - 10 q^{51} + 14 q^{52} + 14 q^{53} - q^{54} - 4 q^{55} + 9 q^{56} + 4 q^{57} - 10 q^{58} + 16 q^{59} + 6 q^{60} - 22 q^{61} + 16 q^{62} - 3 q^{63} + 13 q^{64} - 20 q^{65} - 24 q^{67} + 10 q^{68} + 8 q^{69} + 10 q^{70} - 9 q^{72} + 2 q^{73} - 10 q^{74} - 3 q^{75} + 60 q^{76} - 4 q^{77} + 6 q^{78} - 12 q^{79} + 34 q^{80} + 3 q^{81} - 18 q^{82} + 20 q^{83} - 5 q^{84} + 16 q^{85} - 24 q^{86} + 10 q^{87} - 28 q^{88} - 2 q^{89} - 10 q^{90} - 14 q^{91} + 24 q^{92} - 20 q^{93} + 16 q^{95} - 29 q^{96} + 18 q^{97} - q^{98} + 4 q^{99}+O(q^{100}) \) 3 * q - q^2 + 3 * q^3 + 5 * q^4 - 2 * q^5 - q^6 - 3 * q^7 - 9 * q^8 + 3 * q^9 - 10 * q^10 + 4 * q^11 + 5 * q^12 + 14 * q^13 + q^14 - 2 * q^15 + 13 * q^16 - 10 * q^17 - q^18 + 4 * q^19 + 6 * q^20 - 3 * q^21 + 8 * q^23 - 9 * q^24 - 3 * q^25 + 6 * q^26 + 3 * q^27 - 5 * q^28 + 10 * q^29 - 10 * q^30 - 20 * q^31 - 29 * q^32 + 4 * q^33 - 6 * q^34 + 2 * q^35 + 5 * q^36 + 30 * q^37 - 20 * q^38 + 14 * q^39 - 14 * q^40 + 14 * q^41 + q^42 + 8 * q^43 + 24 * q^44 - 2 * q^45 - 36 * q^46 + 8 * q^47 + 13 * q^48 + 3 * q^49 + 13 * q^50 - 10 * q^51 + 14 * q^52 + 14 * q^53 - q^54 - 4 * q^55 + 9 * q^56 + 4 * q^57 - 10 * q^58 + 16 * q^59 + 6 * q^60 - 22 * q^61 + 16 * q^62 - 3 * q^63 + 13 * q^64 - 20 * q^65 - 24 * q^67 + 10 * q^68 + 8 * q^69 + 10 * q^70 - 9 * q^72 + 2 * q^73 - 10 * q^74 - 3 * q^75 + 60 * q^76 - 4 * q^77 + 6 * q^78 - 12 * q^79 + 34 * q^80 + 3 * q^81 - 18 * q^82 + 20 * q^83 - 5 * q^84 + 16 * q^85 - 24 * q^86 + 10 * q^87 - 28 * q^88 - 2 * q^89 - 10 * q^90 - 14 * q^91 + 24 * q^92 - 20 * q^93 + 16 * q^95 - 29 * q^96 + 18 * q^97 - q^98 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.70928	−1.91575	−0.957873	−	0.287190i	$-0.907279\pi$
$2$	−2.70928	−1.91575	−0.957873	+	0.287190i	$0.907279\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.34017	2.67009
$5$	1.70928	0.764411	0.382206	−	0.924077i	$-0.375165\pi$
$5$	1.70928	0.764411	0.382206	+	0.924077i	$0.375165\pi$
$6$	−2.70928	−1.10606
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−9.04945	−3.19946
$9$	1.00000	0.333333
$10$	−4.63090	−1.46442
$11$	2.63090	0.793245	0.396623	−	0.917982i	$-0.370182\pi$
$11$	2.63090	0.793245	0.396623	+	0.917982i	$0.370182\pi$
$12$	5.34017	1.54158
$13$	2.29072	0.635333	0.317666	−	0.948203i	$-0.397101\pi$
$13$	2.29072	0.635333	0.317666	+	0.948203i	$0.397101\pi$
$14$	2.70928	0.724084
$15$	1.70928	0.441333
$16$	13.8371	3.45928
$17$	0.340173	0.0825041	0.0412520	−	0.999149i	$-0.486865\pi$
$17$	0.340173	0.0825041	0.0412520	+	0.999149i	$0.486865\pi$
$18$	−2.70928	−0.638582
$19$	8.68035	1.99141	0.995704	−	0.0925938i	$-0.0295158\pi$
$19$	8.68035	1.99141	0.995704	+	0.0925938i	$0.0295158\pi$
$20$	9.12783	2.04104
$21$	−1.00000	−0.218218
$22$	−7.12783	−1.51966
$23$	8.49693	1.77173	0.885866	−	0.463941i	$-0.153565\pi$
$23$	8.49693	1.77173	0.885866	+	0.463941i	$0.153565\pi$
$24$	−9.04945	−1.84721
$25$	−2.07838	−0.415676
$26$	−6.20620	−1.21714
$27$	1.00000	0.192450
$28$	−5.34017	−1.00920
$29$	9.60197	1.78304	0.891520	−	0.452981i	$-0.149639\pi$
$29$	9.60197	1.78304	0.891520	+	0.452981i	$0.149639\pi$
$30$	−4.63090	−0.845482
$31$	−10.3402	−1.85715	−0.928575	−	0.371145i	$-0.878965\pi$
$31$	−10.3402	−1.85715	−0.928575	+	0.371145i	$0.878965\pi$
$32$	−19.3896	−3.42763
$33$	2.63090	0.457980
$34$	−0.921622	−0.158057
$35$	−1.70928	−0.288920
$36$	5.34017	0.890029
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	−23.5174	−3.81503
$39$	2.29072	0.366810
$40$	−15.4680	−2.44571
$41$	4.44748	0.694580	0.347290	−	0.937758i	$-0.387102\pi$
$41$	4.44748	0.694580	0.347290	+	0.937758i	$0.387102\pi$
$42$	2.70928	0.418050
$43$	7.41855	1.13132	0.565659	−	0.824639i	$-0.308622\pi$
$43$	7.41855	1.13132	0.565659	+	0.824639i	$0.308622\pi$
$44$	14.0494	2.11803
$45$	1.70928	0.254804
$46$	−23.0205	−3.39419
$47$	5.26180	0.767512	0.383756	−	0.923435i	$-0.374630\pi$
$47$	5.26180	0.767512	0.383756	+	0.923435i	$0.374630\pi$
$48$	13.8371	1.99721
$49$	1.00000	0.142857
$50$	5.63090	0.796329
$51$	0.340173	0.0476337
$52$	12.2329	1.69639
$53$	8.34017	1.14561	0.572805	−	0.819691i	$-0.305855\pi$
$53$	8.34017	1.14561	0.572805	+	0.819691i	$0.305855\pi$
$54$	−2.70928	−0.368686
$55$	4.49693	0.606366
$56$	9.04945	1.20928
$57$	8.68035	1.14974
$58$	−26.0144	−3.41585
$59$	12.6803	1.65084	0.825420	−	0.564519i	$-0.190938\pi$
$59$	12.6803	1.65084	0.825420	+	0.564519i	$0.190938\pi$
$60$	9.12783	1.17840
$61$	−11.8660	−1.51929	−0.759645	−	0.650338i	$-0.774627\pi$
$61$	−11.8660	−1.51929	−0.759645	+	0.650338i	$0.774627\pi$
$62$	28.0144	3.55783
$63$	−1.00000	−0.125988
$64$	24.8576	3.10720
$65$	3.91548	0.485655
$66$	−7.12783	−0.877375
$67$	4.09890	0.500760	0.250380	−	0.968148i	$-0.419444\pi$
$67$	4.09890	0.500760	0.250380	+	0.968148i	$0.419444\pi$
$68$	1.81658	0.220293
$69$	8.49693	1.02291
$70$	4.63090	0.553498
$71$	−11.0205	−1.30789	−0.653947	−	0.756540i	$-0.726888\pi$
$71$	−11.0205	−1.30789	−0.653947	+	0.756540i	$0.726888\pi$
$72$	−9.04945	−1.06649
$73$	−8.83710	−1.03430	−0.517152	−	0.855893i	$-0.673008\pi$
$73$	−8.83710	−1.03430	−0.517152	+	0.855893i	$0.673008\pi$
$74$	−27.0928	−3.14947
$75$	−2.07838	−0.239990
$76$	46.3545	5.31723
$77$	−2.63090	−0.299819
$78$	−6.20620	−0.702714
$79$	2.04945	0.230581	0.115290	−	0.993332i	$-0.463220\pi$
$79$	2.04945	0.230581	0.115290	+	0.993332i	$0.463220\pi$
$80$	23.6514	2.64431
$81$	1.00000	0.111111
$82$	−12.0494	−1.33064
$83$	−2.83710	−0.311412	−0.155706	−	0.987803i	$-0.549765\pi$
$83$	−2.83710	−0.311412	−0.155706	+	0.987803i	$0.549765\pi$
$84$	−5.34017	−0.582661
$85$	0.581449	0.0630670
$86$	−20.0989	−2.16732
$87$	9.60197	1.02944
$88$	−23.8082	−2.53796
$89$	−10.3896	−1.10130	−0.550649	−	0.834737i	$-0.685620\pi$
$89$	−10.3896	−1.10130	−0.550649	+	0.834737i	$0.685620\pi$
$90$	−4.63090	−0.488139
$91$	−2.29072	−0.240133
$92$	45.3751	4.73068
$93$	−10.3402	−1.07223
$94$	−14.2557	−1.47036
$95$	14.8371	1.52225
$96$	−19.3896	−1.97894
$97$	1.02893	0.104472	0.0522360	−	0.998635i	$-0.483365\pi$
$97$	1.02893	0.104472	0.0522360	+	0.998635i	$0.483365\pi$
$98$	−2.70928	−0.273678
$99$	2.63090	0.264415
$100$	−11.0989	−1.10989
$101$	−11.1773	−1.11218	−0.556090	−	0.831122i	$-0.687699\pi$
$101$	−11.1773	−1.11218	−0.556090	+	0.831122i	$0.687699\pi$
$102$	−0.921622	−0.0912542
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−20.7298	−2.03272
$105$	−1.70928	−0.166808
$106$	−22.5958	−2.19470
$107$	7.31124	0.706805	0.353402	−	0.935471i	$-0.385025\pi$
$107$	7.31124	0.706805	0.353402	+	0.935471i	$0.385025\pi$
$108$	5.34017	0.513858
$109$	−11.3607	−1.08816	−0.544078	−	0.839034i	$-0.683121\pi$
$109$	−11.3607	−1.08816	−0.544078	+	0.839034i	$0.683121\pi$
$110$	−12.1834	−1.16164
$111$	10.0000	0.949158
$112$	−13.8371	−1.30748
$113$	−10.5958	−0.996771	−0.498386	−	0.866955i	$-0.666074\pi$
$113$	−10.5958	−0.996771	−0.498386	+	0.866955i	$0.666074\pi$
$114$	−23.5174	−2.20261
$115$	14.5236	1.35433
$116$	51.2762	4.76087
$117$	2.29072	0.211778
$118$	−34.3545	−3.16259
$119$	−0.340173	−0.0311836
$120$	−15.4680	−1.41203
$121$	−4.07838	−0.370762
$122$	32.1483	2.91057
$123$	4.44748	0.401016
$124$	−55.2183	−4.95875
$125$	−12.0989	−1.08216
$126$	2.70928	0.241361
$127$	−21.4680	−1.90498	−0.952488	−	0.304575i	$-0.901486\pi$
$127$	−21.4680	−1.90498	−0.952488	+	0.304575i	$0.901486\pi$
$128$	−28.5669	−2.52498
$129$	7.41855	0.653167
$130$	−10.6081	−0.930393
$131$	6.52359	0.569969	0.284984	−	0.958532i	$-0.408012\pi$
$131$	6.52359	0.569969	0.284984	+	0.958532i	$0.408012\pi$
$132$	14.0494	1.22285
$133$	−8.68035	−0.752681
$134$	−11.1050	−0.959329
$135$	1.70928	0.147111
$136$	−3.07838	−0.263969
$137$	−4.83710	−0.413261	−0.206631	−	0.978419i	$-0.566250\pi$
$137$	−4.83710	−0.413261	−0.206631	+	0.978419i	$0.566250\pi$
$138$	−23.0205	−1.95964
$139$	−0.496928	−0.0421489	−0.0210745	−	0.999778i	$-0.506709\pi$
$139$	−0.496928	−0.0421489	−0.0210745	+	0.999778i	$0.506709\pi$
$140$	−9.12783	−0.771442
$141$	5.26180	0.443123
$142$	29.8576	2.50560
$143$	6.02666	0.503975
$144$	13.8371	1.15309
$145$	16.4124	1.36298
$146$	23.9421	1.98147
$147$	1.00000	0.0824786
$148$	53.4017	4.38960
$149$	−5.23513	−0.428879	−0.214439	−	0.976737i	$-0.568792\pi$
$149$	−5.23513	−0.428879	−0.214439	+	0.976737i	$0.568792\pi$
$150$	5.63090	0.459761
$151$	−22.6309	−1.84168	−0.920838	−	0.389945i	$-0.872494\pi$
$151$	−22.6309	−1.84168	−0.920838	+	0.389945i	$0.872494\pi$
$152$	−78.5523	−6.37144
$153$	0.340173	0.0275014
$154$	7.12783	0.574377
$155$	−17.6742	−1.41963
$156$	12.2329	0.979413
$157$	22.1217	1.76550	0.882751	−	0.469841i	$-0.155689\pi$
$157$	22.1217	1.76550	0.882751	+	0.469841i	$0.155689\pi$
$158$	−5.55252	−0.441735
$159$	8.34017	0.661419
$160$	−33.1422	−2.62012
$161$	−8.49693	−0.669652
$162$	−2.70928	−0.212861
$163$	−8.57304	−0.671492	−0.335746	−	0.941953i	$-0.608988\pi$
$163$	−8.57304	−0.671492	−0.335746	+	0.941953i	$0.608988\pi$
$164$	23.7503	1.85459
$165$	4.49693	0.350085
$166$	7.68649	0.596587
$167$	−10.8371	−0.838600	−0.419300	−	0.907848i	$-0.637724\pi$
$167$	−10.8371	−0.838600	−0.419300	+	0.907848i	$0.637724\pi$
$168$	9.04945	0.698180
$169$	−7.75258	−0.596352
$170$	−1.57531	−0.120820
$171$	8.68035	0.663803
$172$	39.6163	3.02072
$173$	11.7587	0.893999	0.447000	−	0.894534i	$-0.352493\pi$
$173$	11.7587	0.893999	0.447000	+	0.894534i	$0.352493\pi$
$174$	−26.0144	−1.97214
$175$	2.07838	0.157111
$176$	36.4040	2.74405
$177$	12.6803	0.953113
$178$	28.1483	2.10981
$179$	−9.15449	−0.684239	−0.342119	−	0.939657i	$-0.611145\pi$
$179$	−9.15449	−0.684239	−0.342119	+	0.939657i	$0.611145\pi$
$180$	9.12783	0.680348
$181$	18.3896	1.36689	0.683445	−	0.730002i	$-0.260481\pi$
$181$	18.3896	1.36689	0.683445	+	0.730002i	$0.260481\pi$
$182$	6.20620	0.460034
$183$	−11.8660	−0.877162
$184$	−76.8925	−5.66859
$185$	17.0928	1.25668
$186$	28.0144	2.05411
$187$	0.894960	0.0654460
$188$	28.0989	2.04932
$189$	−1.00000	−0.0727393
$190$	−40.1978	−2.91625
$191$	24.8865	1.80073	0.900364	−	0.435138i	$-0.143300\pi$
$191$	24.8865	1.80073	0.900364	+	0.435138i	$0.143300\pi$
$192$	24.8576	1.79394
$193$	−11.3919	−0.820006	−0.410003	−	0.912084i	$-0.634472\pi$
$193$	−11.3919	−0.820006	−0.410003	+	0.912084i	$0.634472\pi$
$194$	−2.78765	−0.200142
$195$	3.91548	0.280393
$196$	5.34017	0.381441
$197$	1.50307	0.107089	0.0535447	−	0.998565i	$-0.482948\pi$
$197$	1.50307	0.107089	0.0535447	+	0.998565i	$0.482948\pi$
$198$	−7.12783	−0.506553
$199$	−1.44521	−0.102448	−0.0512242	−	0.998687i	$-0.516312\pi$
$199$	−1.44521	−0.102448	−0.0512242	+	0.998687i	$0.516312\pi$
$200$	18.8082	1.32994
$201$	4.09890	0.289114
$202$	30.2823	2.13066
$203$	−9.60197	−0.673926
$204$	1.81658	0.127186
$205$	7.60197	0.530944
$206$	0	0
$207$	8.49693	0.590577
$208$	31.6970	2.19779
$209$	22.8371	1.57968
$210$	4.63090	0.319562
$211$	−24.5730	−1.69168	−0.845839	−	0.533438i	$-0.820900\pi$
$211$	−24.5730	−1.69168	−0.845839	+	0.533438i	$0.820900\pi$
$212$	44.5380	3.05888
$213$	−11.0205	−0.755114
$214$	−19.8082	−1.35406
$215$	12.6803	0.864792
$216$	−9.04945	−0.615737
$217$	10.3402	0.701937
$218$	30.7792	2.08463
$219$	−8.83710	−0.597156
$220$	24.0144	1.61905
$221$	0.779243	0.0524175
$222$	−27.0928	−1.81835
$223$	9.17727	0.614556	0.307278	−	0.951620i	$-0.400582\pi$
$223$	9.17727	0.614556	0.307278	+	0.951620i	$0.400582\pi$
$224$	19.3896	1.29552
$225$	−2.07838	−0.138559
$226$	28.7070	1.90956
$227$	10.6537	0.707110	0.353555	−	0.935414i	$-0.384973\pi$
$227$	10.6537	0.707110	0.353555	+	0.935414i	$0.384973\pi$
$228$	46.3545	3.06990
$229$	−10.9939	−0.726495	−0.363247	−	0.931693i	$-0.618332\pi$
$229$	−10.9939	−0.726495	−0.363247	+	0.931693i	$0.618332\pi$
$230$	−39.3484	−2.59456
$231$	−2.63090	−0.173100
$232$	−86.8925	−5.70477
$233$	2.18342	0.143040	0.0715202	−	0.997439i	$-0.477215\pi$
$233$	2.18342	0.143040	0.0715202	+	0.997439i	$0.477215\pi$
$234$	−6.20620	−0.405712
$235$	8.99386	0.586695
$236$	67.7152	4.40789
$237$	2.04945	0.133126
$238$	0.921622	0.0597399
$239$	−5.15449	−0.333416	−0.166708	−	0.986006i	$-0.553314\pi$
$239$	−5.15449	−0.333416	−0.166708	+	0.986006i	$0.553314\pi$
$240$	23.6514	1.52669
$241$	12.1340	0.781618	0.390809	−	0.920472i	$-0.372195\pi$
$241$	12.1340	0.781618	0.390809	+	0.920472i	$0.372195\pi$
$242$	11.0494	0.710285
$243$	1.00000	0.0641500
$244$	−63.3667	−4.05663
$245$	1.70928	0.109202
$246$	−12.0494	−0.768245
$247$	19.8843	1.26521
$248$	93.5729	5.94188
$249$	−2.83710	−0.179794
$250$	32.7792	2.07314
$251$	−23.5174	−1.48441	−0.742204	−	0.670174i	$-0.766219\pi$
$251$	−23.5174	−1.48441	−0.742204	+	0.670174i	$0.766219\pi$
$252$	−5.34017	−0.336399
$253$	22.3545	1.40542
$254$	58.1627	3.64945
$255$	0.581449	0.0364118
$256$	27.6803	1.73002
$257$	5.12783	0.319865	0.159933	−	0.987128i	$-0.448872\pi$
$257$	5.12783	0.319865	0.159933	+	0.987128i	$0.448872\pi$
$258$	−20.0989	−1.25130
$259$	−10.0000	−0.621370
$260$	20.9093	1.29674
$261$	9.60197	0.594347
$262$	−17.6742	−1.09192
$263$	−0.496928	−0.0306419	−0.0153210	−	0.999883i	$-0.504877\pi$
$263$	−0.496928	−0.0306419	−0.0153210	+	0.999883i	$0.504877\pi$
$264$	−23.8082	−1.46529
$265$	14.2557	0.875718
$266$	23.5174	1.44195
$267$	−10.3896	−0.635834
$268$	21.8888	1.33707
$269$	−14.4885	−0.883381	−0.441690	−	0.897168i	$-0.645621\pi$
$269$	−14.4885	−0.883381	−0.441690	+	0.897168i	$0.645621\pi$
$270$	−4.63090	−0.281827
$271$	14.2413	0.865096	0.432548	−	0.901611i	$-0.357615\pi$
$271$	14.2413	0.865096	0.432548	+	0.901611i	$0.357615\pi$
$272$	4.70701	0.285404
$273$	−2.29072	−0.138641
$274$	13.1050	0.791704
$275$	−5.46800	−0.329733
$276$	45.3751	2.73126
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	1.34632	0.0807467
$279$	−10.3402	−0.619050
$280$	15.4680	0.924390
$281$	−2.49693	−0.148954	−0.0744771	−	0.997223i	$-0.523729\pi$
$281$	−2.49693	−0.148954	−0.0744771	+	0.997223i	$0.523729\pi$
$282$	−14.2557	−0.848912
$283$	16.2290	0.964713	0.482357	−	0.875975i	$-0.339781\pi$
$283$	16.2290	0.964713	0.482357	+	0.875975i	$0.339781\pi$
$284$	−58.8515	−3.49219
$285$	14.8371	0.878874
$286$	−16.3279	−0.965488
$287$	−4.44748	−0.262526
$288$	−19.3896	−1.14254
$289$	−16.8843	−0.993193
$290$	−44.4657	−2.61112
$291$	1.02893	0.0603169
$292$	−47.1917	−2.76168
$293$	18.8104	1.09892	0.549459	−	0.835521i	$-0.314834\pi$
$293$	18.8104	1.09892	0.549459	+	0.835521i	$0.314834\pi$
$294$	−2.70928	−0.158008
$295$	21.6742	1.26192
$296$	−90.4945	−5.25989
$297$	2.63090	0.152660
$298$	14.1834	0.821623
$299$	19.4641	1.12564
$300$	−11.0989	−0.640795
$301$	−7.41855	−0.427598
$302$	61.3133	3.52819
$303$	−11.1773	−0.642118
$304$	120.111	6.88883
$305$	−20.2823	−1.16136
$306$	−0.921622	−0.0526856
$307$	−1.34632	−0.0768383	−0.0384192	−	0.999262i	$-0.512232\pi$
$307$	−1.34632	−0.0768383	−0.0384192	+	0.999262i	$0.512232\pi$
$308$	−14.0494	−0.800542
$309$	0	0
$310$	47.8843	2.71964
$311$	−6.15676	−0.349118	−0.174559	−	0.984647i	$-0.555850\pi$
$311$	−6.15676	−0.349118	−0.174559	+	0.984647i	$0.555850\pi$
$312$	−20.7298	−1.17359
$313$	−34.8781	−1.97143	−0.985714	−	0.168425i	$-0.946132\pi$
$313$	−34.8781	−1.97143	−0.985714	+	0.168425i	$0.946132\pi$
$314$	−59.9337	−3.38226
$315$	−1.70928	−0.0963068
$316$	10.9444	0.615671
$317$	25.6020	1.43795	0.718975	−	0.695036i	$-0.244612\pi$
$317$	25.6020	1.43795	0.718975	+	0.695036i	$0.244612\pi$
$318$	−22.5958	−1.26711
$319$	25.2618	1.41439
$320$	42.4885	2.37518
$321$	7.31124	0.408074
$322$	23.0205	1.28288
$323$	2.95282	0.164299
$324$	5.34017	0.296676
$325$	−4.76099	−0.264092
$326$	23.2267	1.28641
$327$	−11.3607	−0.628248
$328$	−40.2472	−2.22228
$329$	−5.26180	−0.290092
$330$	−12.1834	−0.670675
$331$	−26.6225	−1.46330	−0.731652	−	0.681678i	$-0.761250\pi$
$331$	−26.6225	−1.46330	−0.731652	+	0.681678i	$0.761250\pi$
$332$	−15.1506	−0.831498
$333$	10.0000	0.547997
$334$	29.3607	1.60655
$335$	7.00614	0.382786
$336$	−13.8371	−0.754876
$337$	27.5753	1.50212	0.751061	−	0.660232i	$-0.229542\pi$
$337$	27.5753	1.50212	0.751061	+	0.660232i	$0.229542\pi$
$338$	21.0039	1.14246
$339$	−10.5958	−0.575486
$340$	3.10504	0.168394
$341$	−27.2039	−1.47318
$342$	−23.5174	−1.27168
$343$	−1.00000	−0.0539949
$344$	−67.1338	−3.61961
$345$	14.5236	0.781924
$346$	−31.8576	−1.71268
$347$	−28.3584	−1.52236	−0.761180	−	0.648541i	$-0.775380\pi$
$347$	−28.3584	−1.52236	−0.761180	+	0.648541i	$0.775380\pi$
$348$	51.2762	2.74869
$349$	−0.133969	−0.00717120	−0.00358560	−	0.999994i	$-0.501141\pi$
$349$	−0.133969	−0.00717120	−0.00358560	+	0.999994i	$0.501141\pi$
$350$	−5.63090	−0.300984
$351$	2.29072	0.122270
$352$	−51.0121	−2.71895
$353$	4.34017	0.231004	0.115502	−	0.993307i	$-0.463152\pi$
$353$	4.34017	0.231004	0.115502	+	0.993307i	$0.463152\pi$
$354$	−34.3545	−1.82592
$355$	−18.8371	−0.999770
$356$	−55.4824	−2.94056
$357$	−0.340173	−0.0180039
$358$	24.8020	1.31083
$359$	13.7359	0.724955	0.362478	−	0.931993i	$-0.381931\pi$
$359$	13.7359	0.724955	0.362478	+	0.931993i	$0.381931\pi$
$360$	−15.4680	−0.815235
$361$	56.3484	2.96571
$362$	−49.8225	−2.61861
$363$	−4.07838	−0.214059
$364$	−12.2329	−0.641176
$365$	−15.1050	−0.790634
$366$	32.1483	1.68042
$367$	24.0677	1.25632	0.628162	−	0.778083i	$-0.283808\pi$
$367$	24.0677	1.25632	0.628162	+	0.778083i	$0.283808\pi$
$368$	117.573	6.12891
$369$	4.44748	0.231527
$370$	−46.3090	−2.40749
$371$	−8.34017	−0.433000
$372$	−55.2183	−2.86294
$373$	−24.8371	−1.28602	−0.643008	−	0.765859i	$-0.722314\pi$
$373$	−24.8371	−1.28602	−0.643008	+	0.765859i	$0.722314\pi$
$374$	−2.42469	−0.125378
$375$	−12.0989	−0.624784
$376$	−47.6163	−2.45563
$377$	21.9955	1.13282
$378$	2.70928	0.139350
$379$	−21.2534	−1.09171	−0.545857	−	0.837879i	$-0.683796\pi$
$379$	−21.2534	−1.09171	−0.545857	+	0.837879i	$0.683796\pi$
$380$	79.2327	4.06455
$381$	−21.4680	−1.09984
$382$	−67.4245	−3.44974
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	−28.5669	−1.45780
$385$	−4.49693	−0.229185
$386$	30.8638	1.57092
$387$	7.41855	0.377106
$388$	5.49466	0.278949
$389$	35.8720	1.81878	0.909391	−	0.415942i	$-0.136548\pi$
$389$	35.8720	1.81878	0.909391	+	0.415942i	$0.136548\pi$
$390$	−10.6081	−0.537163
$391$	2.89043	0.146175
$392$	−9.04945	−0.457066
$393$	6.52359	0.329072
$394$	−4.07223	−0.205156
$395$	3.50307	0.176259
$396$	14.0494	0.706011
$397$	3.52973	0.177152	0.0885761	−	0.996069i	$-0.471768\pi$
$397$	3.52973	0.177152	0.0885761	+	0.996069i	$0.471768\pi$
$398$	3.91548	0.196265
$399$	−8.68035	−0.434561
$400$	−28.7587	−1.43794
$401$	1.71769	0.0857771	0.0428886	−	0.999080i	$-0.486344\pi$
$401$	1.71769	0.0857771	0.0428886	+	0.999080i	$0.486344\pi$
$402$	−11.1050	−0.553869
$403$	−23.6865	−1.17991
$404$	−59.6886	−2.96962
$405$	1.70928	0.0849346
$406$	26.0144	1.29107
$407$	26.3090	1.30409
$408$	−3.07838	−0.152402
$409$	13.4413	0.664631	0.332316	−	0.943168i	$-0.392170\pi$
$409$	13.4413	0.664631	0.332316	+	0.943168i	$0.392170\pi$
$410$	−20.5958	−1.01716
$411$	−4.83710	−0.238597
$412$	0	0
$413$	−12.6803	−0.623959
$414$	−23.0205	−1.13140
$415$	−4.84939	−0.238047
$416$	−44.4163	−2.17769
$417$	−0.496928	−0.0243347
$418$	−61.8720	−3.02626
$419$	−7.88428	−0.385172	−0.192586	−	0.981280i	$-0.561688\pi$
$419$	−7.88428	−0.385172	−0.192586	+	0.981280i	$0.561688\pi$
$420$	−9.12783	−0.445392
$421$	7.16290	0.349098	0.174549	−	0.984648i	$-0.444153\pi$
$421$	7.16290	0.349098	0.174549	+	0.984648i	$0.444153\pi$
$422$	66.5751	3.24083
$423$	5.26180	0.255837
$424$	−75.4740	−3.66534
$425$	−0.707008	−0.0342949
$426$	29.8576	1.44661
$427$	11.8660	0.574237
$428$	39.0433	1.88723
$429$	6.02666	0.290970
$430$	−34.3545	−1.65672
$431$	−19.9155	−0.959295	−0.479647	−	0.877461i	$-0.659235\pi$
$431$	−19.9155	−0.959295	−0.479647	+	0.877461i	$0.659235\pi$
$432$	13.8371	0.665738
$433$	−8.83710	−0.424684	−0.212342	−	0.977195i	$-0.568109\pi$
$433$	−8.83710	−0.424684	−0.212342	+	0.977195i	$0.568109\pi$
$434$	−28.0144	−1.34473
$435$	16.4124	0.786915
$436$	−60.6681	−2.90547
$437$	73.7563	3.52824
$438$	23.9421	1.14400
$439$	2.70701	0.129198	0.0645992	−	0.997911i	$-0.479423\pi$
$439$	2.70701	0.129198	0.0645992	+	0.997911i	$0.479423\pi$
$440$	−40.6947	−1.94004
$441$	1.00000	0.0476190
$442$	−2.11118	−0.100419
$443$	18.4619	0.877149	0.438575	−	0.898695i	$-0.355483\pi$
$443$	18.4619	0.877149	0.438575	+	0.898695i	$0.355483\pi$
$444$	53.4017	2.53433
$445$	−17.7587	−0.841844
$446$	−24.8638	−1.17733
$447$	−5.23513	−0.247613
$448$	−24.8576	−1.17441
$449$	4.55479	0.214954	0.107477	−	0.994208i	$-0.465723\pi$
$449$	4.55479	0.214954	0.107477	+	0.994208i	$0.465723\pi$
$450$	5.63090	0.265443
$451$	11.7009	0.550972
$452$	−56.5835	−2.66147
$453$	−22.6309	−1.06329
$454$	−28.8638	−1.35464
$455$	−3.91548	−0.183560
$456$	−78.5523	−3.67855
$457$	−7.36069	−0.344319	−0.172159	−	0.985069i	$-0.555074\pi$
$457$	−7.36069	−0.344319	−0.172159	+	0.985069i	$0.555074\pi$
$458$	29.7854	1.39178
$459$	0.340173	0.0158779
$460$	77.5585	3.61618
$461$	−33.6391	−1.56673	−0.783365	−	0.621562i	$-0.786499\pi$
$461$	−33.6391	−1.56673	−0.783365	+	0.621562i	$0.786499\pi$
$462$	7.12783	0.331616
$463$	8.88655	0.412993	0.206496	−	0.978447i	$-0.433794\pi$
$463$	8.88655	0.412993	0.206496	+	0.978447i	$0.433794\pi$
$464$	132.863	6.16803
$465$	−17.6742	−0.819622
$466$	−5.91548	−0.274029
$467$	28.8104	1.33319	0.666594	−	0.745421i	$-0.267751\pi$
$467$	28.8104	1.33319	0.666594	+	0.745421i	$0.267751\pi$
$468$	12.2329	0.565464
$469$	−4.09890	−0.189269
$470$	−24.3668	−1.12396
$471$	22.1217	1.01931
$472$	−114.750	−5.28180
$473$	19.5174	0.897413
$474$	−5.55252	−0.255036
$475$	−18.0410	−0.827780
$476$	−1.81658	−0.0832629
$477$	8.34017	0.381870
$478$	13.9649	0.638741
$479$	−2.52359	−0.115306	−0.0576529	−	0.998337i	$-0.518362\pi$
$479$	−2.52359	−0.115306	−0.0576529	+	0.998337i	$0.518362\pi$
$480$	−33.1422	−1.51273
$481$	22.9072	1.04448
$482$	−32.8743	−1.49738
$483$	−8.49693	−0.386624
$484$	−21.7792	−0.989966
$485$	1.75872	0.0798595
$486$	−2.70928	−0.122895
$487$	7.52586	0.341029	0.170515	−	0.985355i	$-0.445457\pi$
$487$	7.52586	0.341029	0.170515	+	0.985355i	$0.445457\pi$
$488$	107.381	4.86091
$489$	−8.57304	−0.387686
$490$	−4.63090	−0.209203
$491$	−12.7649	−0.576070	−0.288035	−	0.957620i	$-0.593002\pi$
$491$	−12.7649	−0.576070	−0.288035	+	0.957620i	$0.593002\pi$
$492$	23.7503	1.07075
$493$	3.26633	0.147108
$494$	−53.8720	−2.42382
$495$	4.49693	0.202122
$496$	−143.078	−6.42439
$497$	11.0205	0.494338
$498$	7.68649	0.344440
$499$	−2.25565	−0.100977	−0.0504884	−	0.998725i	$-0.516078\pi$
$499$	−2.25565	−0.100977	−0.0504884	+	0.998725i	$0.516078\pi$
$500$	−64.6102	−2.88946
$501$	−10.8371	−0.484166
$502$	63.7152	2.84375
$503$	−42.0866	−1.87655	−0.938275	−	0.345891i	$-0.887577\pi$
$503$	−42.0866	−1.87655	−0.938275	+	0.345891i	$0.887577\pi$
$504$	9.04945	0.403094
$505$	−19.1050	−0.850163
$506$	−60.5646	−2.69243
$507$	−7.75258	−0.344304
$508$	−114.643	−5.08645
$509$	−13.0205	−0.577124	−0.288562	−	0.957461i	$-0.593177\pi$
$509$	−13.0205	−0.577124	−0.288562	+	0.957461i	$0.593177\pi$
$510$	−1.57531	−0.0697557
$511$	8.83710	0.390930
$512$	−17.8599	−0.789303
$513$	8.68035	0.383247
$514$	−13.8927	−0.612780
$515$	0	0
$516$	39.6163	1.74401
$517$	13.8432	0.608825
$518$	27.0928	1.19039
$519$	11.7587	0.516151
$520$	−35.4329	−1.55384
$521$	−7.13170	−0.312446	−0.156223	−	0.987722i	$-0.549932\pi$
$521$	−7.13170	−0.312446	−0.156223	+	0.987722i	$0.549932\pi$
$522$	−26.0144	−1.13862
$523$	−40.3279	−1.76342	−0.881708	−	0.471796i	$-0.843606\pi$
$523$	−40.3279	−1.76342	−0.881708	+	0.471796i	$0.843606\pi$
$524$	34.8371	1.52187
$525$	2.07838	0.0907078
$526$	1.34632	0.0587022
$527$	−3.51745	−0.153222
$528$	36.4040	1.58428
$529$	49.1978	2.13903
$530$	−38.6225	−1.67765
$531$	12.6803	0.550280
$532$	−46.3545	−2.00972
$533$	10.1880	0.441289
$534$	28.1483	1.21810
$535$	12.4969	0.540289
$536$	−37.0928	−1.60216
$537$	−9.15449	−0.395045
$538$	39.2534	1.69233
$539$	2.63090	0.113321
$540$	9.12783	0.392799
$541$	−11.6286	−0.499954	−0.249977	−	0.968252i	$-0.580423\pi$
$541$	−11.6286	−0.499954	−0.249977	+	0.968252i	$0.580423\pi$
$542$	−38.5835	−1.65730
$543$	18.3896	0.789174
$544$	−6.59583	−0.282794
$545$	−19.4186	−0.831799
$546$	6.20620	0.265601
$547$	14.9444	0.638977	0.319488	−	0.947590i	$-0.396489\pi$
$547$	14.9444	0.638977	0.319488	+	0.947590i	$0.396489\pi$
$548$	−25.8310	−1.10344
$549$	−11.8660	−0.506430
$550$	14.8143	0.631685
$551$	83.3484	3.55076
$552$	−76.8925	−3.27276
$553$	−2.04945	−0.0871514
$554$	48.7670	2.07191
$555$	17.0928	0.725547
$556$	−2.65368	−0.112541
$557$	−3.36069	−0.142397	−0.0711985	−	0.997462i	$-0.522682\pi$
$557$	−3.36069	−0.142397	−0.0711985	+	0.997462i	$0.522682\pi$
$558$	28.0144	1.18594
$559$	16.9939	0.718764
$560$	−23.6514	−0.999455
$561$	0.894960	0.0377853
$562$	6.76487	0.285359
$563$	11.3197	0.477067	0.238533	−	0.971134i	$-0.423333\pi$
$563$	11.3197	0.477067	0.238533	+	0.971134i	$0.423333\pi$
$564$	28.0989	1.18318
$565$	−18.1112	−0.761943
$566$	−43.9688	−1.84815
$567$	−1.00000	−0.0419961
$568$	99.7296	4.18456
$569$	7.04718	0.295433	0.147717	−	0.989030i	$-0.452808\pi$
$569$	7.04718	0.295433	0.147717	+	0.989030i	$0.452808\pi$
$570$	−40.1978	−1.68370
$571$	−19.3112	−0.808150	−0.404075	−	0.914726i	$-0.632407\pi$
$571$	−19.3112	−0.808150	−0.404075	+	0.914726i	$0.632407\pi$
$572$	32.1834	1.34566
$573$	24.8865	1.03965
$574$	12.0494	0.502934
$575$	−17.6598	−0.736466
$576$	24.8576	1.03573
$577$	−37.7093	−1.56986	−0.784929	−	0.619586i	$-0.787301\pi$
$577$	−37.7093	−1.56986	−0.784929	+	0.619586i	$0.787301\pi$
$578$	45.7442	1.90271
$579$	−11.3919	−0.473431
$580$	87.6451	3.63926
$581$	2.83710	0.117703
$582$	−2.78765	−0.115552
$583$	21.9421	0.908751
$584$	79.9709	3.30922
$585$	3.91548	0.161885
$586$	−50.9627	−2.10525
$587$	0.863763	0.0356513	0.0178257	−	0.999841i	$-0.494326\pi$
$587$	0.863763	0.0356513	0.0178257	+	0.999841i	$0.494326\pi$
$588$	5.34017	0.220225
$589$	−89.7563	−3.69834
$590$	−58.7214	−2.41752
$591$	1.50307	0.0618281
$592$	138.371	5.68701
$593$	−31.5981	−1.29758	−0.648789	−	0.760968i	$-0.724724\pi$
$593$	−31.5981	−1.29758	−0.648789	+	0.760968i	$0.724724\pi$
$594$	−7.12783	−0.292458
$595$	−0.581449	−0.0238371
$596$	−27.9565	−1.14514
$597$	−1.44521	−0.0591486
$598$	−52.7337	−2.15644
$599$	−12.7115	−0.519380	−0.259690	−	0.965692i	$-0.583620\pi$
$599$	−12.7115	−0.519380	−0.259690	+	0.965692i	$0.583620\pi$
$600$	18.8082	0.767840
$601$	4.66209	0.190171	0.0950854	−	0.995469i	$-0.469688\pi$
$601$	4.66209	0.190171	0.0950854	+	0.995469i	$0.469688\pi$
$602$	20.0989	0.819170
$603$	4.09890	0.166920
$604$	−120.853	−4.91744
$605$	−6.97107	−0.283414
$606$	30.2823	1.23013
$607$	−14.0267	−0.569325	−0.284662	−	0.958628i	$-0.591881\pi$
$607$	−14.0267	−0.569325	−0.284662	+	0.958628i	$0.591881\pi$
$608$	−168.309	−6.82582
$609$	−9.60197	−0.389091
$610$	54.9504	2.22488
$611$	12.0533	0.487625
$612$	1.81658	0.0734310
$613$	14.6270	0.590780	0.295390	−	0.955377i	$-0.404550\pi$
$613$	14.6270	0.590780	0.295390	+	0.955377i	$0.404550\pi$
$614$	3.64754	0.147203
$615$	7.60197	0.306541
$616$	23.8082	0.959259
$617$	−16.8227	−0.677257	−0.338629	−	0.940920i	$-0.609963\pi$
$617$	−16.8227	−0.677257	−0.338629	+	0.940920i	$0.609963\pi$
$618$	0	0
$619$	19.4329	0.781075	0.390538	−	0.920587i	$-0.372289\pi$
$619$	19.4329	0.781075	0.390538	+	0.920587i	$0.372289\pi$
$620$	−94.3833	−3.79052
$621$	8.49693	0.340970
$622$	16.6803	0.668821
$623$	10.3896	0.416251
$624$	31.6970	1.26890
$625$	−10.2885	−0.411538
$626$	94.4945	3.77676
$627$	22.8371	0.912026
$628$	118.134	4.71404
$629$	3.40173	0.135636
$630$	4.63090	0.184499
$631$	−31.8843	−1.26929	−0.634647	−	0.772802i	$-0.718854\pi$
$631$	−31.8843	−1.26929	−0.634647	+	0.772802i	$0.718854\pi$
$632$	−18.5464	−0.737735
$633$	−24.5730	−0.976691
$634$	−69.3628	−2.75475
$635$	−36.6947	−1.45619
$636$	44.5380	1.76605
$637$	2.29072	0.0907618
$638$	−68.4412	−2.70961
$639$	−11.0205	−0.435965
$640$	−48.8287	−1.93012
$641$	0.470266	0.0185744	0.00928720	−	0.999957i	$-0.497044\pi$
$641$	0.470266	0.0185744	0.00928720	+	0.999957i	$0.497044\pi$
$642$	−19.8082	−0.781766
$643$	−0.299135	−0.0117967	−0.00589837	−	0.999983i	$-0.501878\pi$
$643$	−0.299135	−0.0117967	−0.00589837	+	0.999983i	$0.501878\pi$
$644$	−45.3751	−1.78803
$645$	12.6803	0.499288
$646$	−8.00000	−0.314756
$647$	24.6803	0.970285	0.485142	−	0.874435i	$-0.338768\pi$
$647$	24.6803	0.970285	0.485142	+	0.874435i	$0.338768\pi$
$648$	−9.04945	−0.355496
$649$	33.3607	1.30952
$650$	12.8988	0.505934
$651$	10.3402	0.405263
$652$	−45.7815	−1.79294
$653$	5.18181	0.202780	0.101390	−	0.994847i	$-0.467671\pi$
$653$	5.18181	0.202780	0.101390	+	0.994847i	$0.467671\pi$
$654$	30.7792	1.20356
$655$	11.1506	0.435690
$656$	61.5402	2.40274
$657$	−8.83710	−0.344768
$658$	14.2557	0.555743
$659$	−25.8576	−1.00727	−0.503635	−	0.863917i	$-0.668004\pi$
$659$	−25.8576	−1.00727	−0.503635	+	0.863917i	$0.668004\pi$
$660$	24.0144	0.934758
$661$	−1.10504	−0.0429811	−0.0214905	−	0.999769i	$-0.506841\pi$
$661$	−1.10504	−0.0429811	−0.0214905	+	0.999769i	$0.506841\pi$
$662$	72.1276	2.80332
$663$	0.779243	0.0302633
$664$	25.6742	0.996352
$665$	−14.8371	−0.575358
$666$	−27.0928	−1.04982
$667$	81.5872	3.15907
$668$	−57.8720	−2.23913
$669$	9.17727	0.354814
$670$	−18.9816	−0.733322
$671$	−31.2183	−1.20517
$672$	19.3896	0.747971
$673$	−24.0579	−0.927362	−0.463681	−	0.886002i	$-0.653472\pi$
$673$	−24.0579	−0.927362	−0.463681	+	0.886002i	$0.653472\pi$
$674$	−74.7091	−2.87769
$675$	−2.07838	−0.0799968
$676$	−41.4001	−1.59231
$677$	15.9733	0.613905	0.306953	−	0.951725i	$-0.400691\pi$
$677$	15.9733	0.613905	0.306953	+	0.951725i	$0.400691\pi$
$678$	28.7070	1.10249
$679$	−1.02893	−0.0394867
$680$	−5.26180	−0.201781
$681$	10.6537	0.408250
$682$	73.7030	2.82223
$683$	12.2290	0.467929	0.233965	−	0.972245i	$-0.424830\pi$
$683$	12.2290	0.467929	0.233965	+	0.972245i	$0.424830\pi$
$684$	46.3545	1.77241
$685$	−8.26794	−0.315902
$686$	2.70928	0.103441
$687$	−10.9939	−0.419442
$688$	102.651	3.91354
$689$	19.1050	0.727844
$690$	−39.3484	−1.49797
$691$	20.7649	0.789933	0.394966	−	0.918696i	$-0.370756\pi$
$691$	20.7649	0.789933	0.394966	+	0.918696i	$0.370756\pi$
$692$	62.7936	2.38706
$693$	−2.63090	−0.0999395
$694$	76.8308	2.91646
$695$	−0.849388	−0.0322191
$696$	−86.8925	−3.29365
$697$	1.51291	0.0573056
$698$	0.362959	0.0137382
$699$	2.18342	0.0825844
$700$	11.0989	0.419499
$701$	−13.5486	−0.511725	−0.255863	−	0.966713i	$-0.582359\pi$
$701$	−13.5486	−0.511725	−0.255863	+	0.966713i	$0.582359\pi$
$702$	−6.20620	−0.234238
$703$	86.8035	3.27385
$704$	65.3979	2.46477
$705$	8.99386	0.338728
$706$	−11.7587	−0.442545
$707$	11.1773	0.420365
$708$	67.7152	2.54489
$709$	−32.8371	−1.23322	−0.616612	−	0.787267i	$-0.711495\pi$
$709$	−32.8371	−1.23322	−0.616612	+	0.787267i	$0.711495\pi$
$710$	51.0349	1.91531
$711$	2.04945	0.0768603
$712$	94.0203	3.52356
$713$	−87.8597	−3.29037
$714$	0.921622	0.0344908
$715$	10.3012	0.385244
$716$	−48.8865	−1.82698
$717$	−5.15449	−0.192498
$718$	−37.2144	−1.38883
$719$	−3.31965	−0.123802	−0.0619011	−	0.998082i	$-0.519716\pi$
$719$	−3.31965	−0.123802	−0.0619011	+	0.998082i	$0.519716\pi$
$720$	23.6514	0.881436
$721$	0	0
$722$	−152.663	−5.68154
$723$	12.1340	0.451267
$724$	98.2038	3.64971
$725$	−19.9565	−0.741166
$726$	11.0494	0.410083
$727$	−17.9733	−0.666594	−0.333297	−	0.942822i	$-0.608161\pi$
$727$	−17.9733	−0.666594	−0.333297	+	0.942822i	$0.608161\pi$
$728$	20.7298	0.768297
$729$	1.00000	0.0370370
$730$	40.9237	1.51465
$731$	2.52359	0.0933384
$732$	−63.3667	−2.34210
$733$	−32.4079	−1.19701	−0.598506	−	0.801118i	$-0.704239\pi$
$733$	−32.4079	−1.19701	−0.598506	+	0.801118i	$0.704239\pi$
$734$	−65.2060	−2.40680
$735$	1.70928	0.0630476
$736$	−164.752	−6.07285
$737$	10.7838	0.397226
$738$	−12.0494	−0.443546
$739$	26.7831	0.985233	0.492616	−	0.870247i	$-0.336041\pi$
$739$	26.7831	0.985233	0.492616	+	0.870247i	$0.336041\pi$
$740$	91.2783	3.35546
$741$	19.8843	0.730467
$742$	22.5958	0.829519
$743$	−21.7359	−0.797414	−0.398707	−	0.917078i	$-0.630541\pi$
$743$	−21.7359	−0.797414	−0.398707	+	0.917078i	$0.630541\pi$
$744$	93.5729	3.43055
$745$	−8.94828	−0.327840
$746$	67.2905	2.46368
$747$	−2.83710	−0.103804
$748$	4.77924	0.174746
$749$	−7.31124	−0.267147
$750$	32.7792	1.19693
$751$	−15.4719	−0.564577	−0.282288	−	0.959330i	$-0.591093\pi$
$751$	−15.4719	−0.564577	−0.282288	+	0.959330i	$0.591093\pi$
$752$	72.8080	2.65503
$753$	−23.5174	−0.857023
$754$	−59.5918	−2.17020
$755$	−38.6824	−1.40780
$756$	−5.34017	−0.194220
$757$	42.0821	1.52950	0.764750	−	0.644328i	$-0.222863\pi$
$757$	42.0821	1.52950	0.764750	+	0.644328i	$0.222863\pi$
$758$	57.5813	2.09145
$759$	22.3545	0.811419
$760$	−134.268	−4.87040
$761$	−15.5441	−0.563474	−0.281737	−	0.959492i	$-0.590910\pi$
$761$	−15.5441	−0.563474	−0.281737	+	0.959492i	$0.590910\pi$
$762$	58.1627	2.10701
$763$	11.3607	0.411285
$764$	132.898	4.80810
$765$	0.581449	0.0210223
$766$	2.70928	0.0978901
$767$	29.0472	1.04883
$768$	27.6803	0.998828
$769$	6.15222	0.221855	0.110927	−	0.993829i	$-0.464618\pi$
$769$	6.15222	0.221855	0.110927	+	0.993829i	$0.464618\pi$
$770$	12.1834	0.439060
$771$	5.12783	0.184674
$772$	−60.8347	−2.18949
$773$	31.0784	1.11781	0.558906	−	0.829231i	$-0.311221\pi$
$773$	31.0784	1.11781	0.558906	+	0.829231i	$0.311221\pi$
$774$	−20.0989	−0.722440
$775$	21.4908	0.771972
$776$	−9.31124	−0.334254
$777$	−10.0000	−0.358748
$778$	−97.1871	−3.48433
$779$	38.6057	1.38319
$780$	20.9093	0.748674
$781$	−28.9939	−1.03748
$782$	−7.83096	−0.280034
$783$	9.60197	0.343146
$784$	13.8371	0.494182
$785$	37.8120	1.34957
$786$	−17.6742	−0.630418
$787$	−51.8453	−1.84809	−0.924043	−	0.382288i	$-0.875136\pi$
$787$	−51.8453	−1.84809	−0.924043	+	0.382288i	$0.875136\pi$
$788$	8.02666	0.285938
$789$	−0.496928	−0.0176911
$790$	−9.49079	−0.337667
$791$	10.5958	0.376744
$792$	−23.8082	−0.845987
$793$	−27.1818	−0.965254
$794$	−9.56302	−0.339379
$795$	14.2557	0.505596
$796$	−7.71769	−0.273546
$797$	47.1605	1.67051	0.835254	−	0.549864i	$-0.185320\pi$
$797$	47.1605	1.67051	0.835254	+	0.549864i	$0.185320\pi$
$798$	23.5174	0.832509
$799$	1.78992	0.0633228
$800$	40.2990	1.42478
$801$	−10.3896	−0.367099
$802$	−4.65368	−0.164327
$803$	−23.2495	−0.820457
$804$	21.8888	0.771959
$805$	−14.5236	−0.511889
$806$	64.1732	2.26041
$807$	−14.4885	−0.510020
$808$	101.148	3.55838
$809$	36.4391	1.28113	0.640565	−	0.767904i	$-0.278700\pi$
$809$	36.4391	1.28113	0.640565	+	0.767904i	$0.278700\pi$
$810$	−4.63090	−0.162713
$811$	51.0493	1.79258	0.896291	−	0.443466i	$-0.146251\pi$
$811$	51.0493	1.79258	0.896291	+	0.443466i	$0.146251\pi$
$812$	−51.2762	−1.79944
$813$	14.2413	0.499463
$814$	−71.2783	−2.49830
$815$	−14.6537	−0.513296
$816$	4.70701	0.164778
$817$	64.3956	2.25292
$818$	−36.4163	−1.27327
$819$	−2.29072	−0.0800444
$820$	40.5958	1.41767
$821$	−42.2823	−1.47566	−0.737831	−	0.674985i	$-0.764150\pi$
$821$	−42.2823	−1.47566	−0.737831	+	0.674985i	$0.764150\pi$
$822$	13.1050	0.457091
$823$	20.9939	0.731800	0.365900	−	0.930654i	$-0.380761\pi$
$823$	20.9939	0.731800	0.365900	+	0.930654i	$0.380761\pi$
$824$	0	0
$825$	−5.46800	−0.190371
$826$	34.3545	1.19535
$827$	11.0433	0.384013	0.192007	−	0.981394i	$-0.438500\pi$
$827$	11.0433	0.384013	0.192007	+	0.981394i	$0.438500\pi$
$828$	45.3751	1.57689
$829$	25.6332	0.890277	0.445138	−	0.895462i	$-0.353155\pi$
$829$	25.6332	0.890277	0.445138	+	0.895462i	$0.353155\pi$
$830$	13.1383	0.456038
$831$	−18.0000	−0.624413
$832$	56.9420	1.97411
$833$	0.340173	0.0117863
$834$	1.34632	0.0466191
$835$	−18.5236	−0.641035
$836$	121.954	4.21787
$837$	−10.3402	−0.357409
$838$	21.3607	0.737893
$839$	37.3074	1.28799	0.643997	−	0.765028i	$-0.277275\pi$
$839$	37.3074	1.28799	0.643997	+	0.765028i	$0.277275\pi$
$840$	15.4680	0.533697
$841$	63.1978	2.17923
$842$	−19.4063	−0.668784
$843$	−2.49693	−0.0859988
$844$	−131.224	−4.51693
$845$	−13.2513	−0.455858
$846$	−14.2557	−0.490119
$847$	4.07838	0.140135
$848$	115.404	3.96298
$849$	16.2290	0.556978
$850$	1.91548	0.0657004
$851$	84.9693	2.91271
$852$	−58.8515	−2.01622
$853$	8.89043	0.304402	0.152201	−	0.988350i	$-0.451364\pi$
$853$	8.89043	0.304402	0.152201	+	0.988350i	$0.451364\pi$
$854$	−32.1483	−1.10009
$855$	14.8371	0.507418
$856$	−66.1627	−2.26140
$857$	−20.6986	−0.707051	−0.353525	−	0.935425i	$-0.615017\pi$
$857$	−20.6986	−0.707051	−0.353525	+	0.935425i	$0.615017\pi$
$858$	−16.3279	−0.557425
$859$	3.18956	0.108826	0.0544132	−	0.998519i	$-0.482671\pi$
$859$	3.18956	0.108826	0.0544132	+	0.998519i	$0.482671\pi$
$860$	67.7152	2.30907
$861$	−4.44748	−0.151570
$862$	53.9565	1.83777
$863$	16.5197	0.562338	0.281169	−	0.959658i	$-0.409278\pi$
$863$	16.5197	0.562338	0.281169	+	0.959658i	$0.409278\pi$
$864$	−19.3896	−0.659648
$865$	20.0989	0.683383
$866$	23.9421	0.813587
$867$	−16.8843	−0.573420
$868$	55.2183	1.87423
$869$	5.39189	0.182907
$870$	−44.4657	−1.50753
$871$	9.38944	0.318149
$872$	102.808	3.48152
$873$	1.02893	0.0348240
$874$	−199.826	−6.75922
$875$	12.0989	0.409017
$876$	−47.1917	−1.59446
$877$	−16.4703	−0.556161	−0.278081	−	0.960558i	$-0.589698\pi$
$877$	−16.4703	−0.556161	−0.278081	+	0.960558i	$0.589698\pi$
$878$	−7.33403	−0.247512
$879$	18.8104	0.634460
$880$	62.2245	2.09759
$881$	−31.1689	−1.05011	−0.525053	−	0.851070i	$-0.675954\pi$
$881$	−31.1689	−1.05011	−0.525053	+	0.851070i	$0.675954\pi$
$882$	−2.70928	−0.0912260
$883$	9.05559	0.304745	0.152372	−	0.988323i	$-0.451309\pi$
$883$	9.05559	0.304745	0.152372	+	0.988323i	$0.451309\pi$
$884$	4.16129	0.139959
$885$	21.6742	0.728570
$886$	−50.0183	−1.68040
$887$	−41.7275	−1.40107	−0.700537	−	0.713616i	$-0.747056\pi$
$887$	−41.7275	−1.40107	−0.700537	+	0.713616i	$0.747056\pi$
$888$	−90.4945	−3.03680
$889$	21.4680	0.720014
$890$	48.1133	1.61276
$891$	2.63090	0.0881384
$892$	49.0082	1.64092
$893$	45.6742	1.52843
$894$	14.1834	0.474364
$895$	−15.6475	−0.523040
$896$	28.5669	0.954353
$897$	19.4641	0.649888
$898$	−12.3402	−0.411797
$899$	−99.2860	−3.31137
$900$	−11.0989	−0.369963
$901$	2.83710	0.0945176
$902$	−31.7009	−1.05552
$903$	−7.41855	−0.246874
$904$	95.8864	3.18913
$905$	31.4329	1.04487
$906$	61.3133	2.03700
$907$	12.3584	0.410355	0.205177	−	0.978725i	$-0.434223\pi$
$907$	12.3584	0.410355	0.205177	+	0.978725i	$0.434223\pi$
$908$	56.8925	1.88804
$909$	−11.1773	−0.370727
$910$	10.6081	0.351655
$911$	22.4163	0.742685	0.371342	−	0.928496i	$-0.378898\pi$
$911$	22.4163	0.742685	0.371342	+	0.928496i	$0.378898\pi$
$912$	120.111	3.97727
$913$	−7.46412	−0.247026
$914$	19.9421	0.659627
$915$	−20.2823	−0.670513
$916$	−58.7091	−1.93980
$917$	−6.52359	−0.215428
$918$	−0.921622	−0.0304181
$919$	−35.5630	−1.17312	−0.586558	−	0.809907i	$-0.699517\pi$
$919$	−35.5630	−1.17312	−0.586558	+	0.809907i	$0.699517\pi$
$920$	−131.430	−4.33313
$921$	−1.34632	−0.0443626
$922$	91.1377	3.00146
$923$	−25.2450	−0.830948
$924$	−14.0494	−0.462193
$925$	−20.7838	−0.683366
$926$	−24.0761	−0.791190
$927$	0	0
$928$	−186.179	−6.11161
$929$	32.7975	1.07605	0.538025	−	0.842929i	$-0.319170\pi$
$929$	32.7975	1.07605	0.538025	+	0.842929i	$0.319170\pi$
$930$	47.8843	1.57019
$931$	8.68035	0.284487
$932$	11.6598	0.381930
$933$	−6.15676	−0.201563
$934$	−78.0554	−2.55405
$935$	1.52973	0.0500276
$936$	−20.7298	−0.677575
$937$	44.6225	1.45775	0.728877	−	0.684645i	$-0.240043\pi$
$937$	44.6225	1.45775	0.728877	+	0.684645i	$0.240043\pi$
$938$	11.1050	0.362592
$939$	−34.8781	−1.13821
$940$	48.0288	1.56653
$941$	6.33564	0.206536	0.103268	−	0.994654i	$-0.467070\pi$
$941$	6.33564	0.206536	0.103268	+	0.994654i	$0.467070\pi$
$942$	−59.9337	−1.95275
$943$	37.7899	1.23061
$944$	175.459	5.71071
$945$	−1.70928	−0.0556027
$946$	−52.8781	−1.71922
$947$	−17.6826	−0.574608	−0.287304	−	0.957839i	$-0.592759\pi$
$947$	−17.6826	−0.574608	−0.287304	+	0.957839i	$0.592759\pi$
$948$	10.9444	0.355458
$949$	−20.2434	−0.657127
$950$	48.8781	1.58582
$951$	25.6020	0.830201
$952$	3.07838	0.0997708
$953$	−31.7054	−1.02704	−0.513519	−	0.858078i	$-0.671659\pi$
$953$	−31.7054	−1.02704	−0.513519	+	0.858078i	$0.671659\pi$
$954$	−22.5958	−0.731567
$955$	42.5380	1.37650
$956$	−27.5259	−0.890250
$957$	25.2618	0.816598
$958$	6.83710	0.220897
$959$	4.83710	0.156198
$960$	42.4885	1.37131
$961$	75.9192	2.44901
$962$	−62.0620	−2.00096
$963$	7.31124	0.235602
$964$	64.7975	2.08699
$965$	−19.4719	−0.626822
$966$	23.0205	0.740673
$967$	42.1933	1.35684	0.678422	−	0.734673i	$-0.262664\pi$
$967$	42.1933	1.35684	0.678422	+	0.734673i	$0.262664\pi$
$968$	36.9071	1.18624
$969$	2.95282	0.0948582
$970$	−4.76487	−0.152991
$971$	20.9795	0.673264	0.336632	−	0.941636i	$-0.390712\pi$
$971$	20.9795	0.673264	0.336632	+	0.941636i	$0.390712\pi$
$972$	5.34017	0.171286
$973$	0.496928	0.0159308
$974$	−20.3896	−0.653326
$975$	−4.76099	−0.152474
$976$	−164.191	−5.25564
$977$	7.12395	0.227915	0.113958	−	0.993486i	$-0.463647\pi$
$977$	7.12395	0.227915	0.113958	+	0.993486i	$0.463647\pi$
$978$	23.2267	0.742709
$979$	−27.3340	−0.873599
$980$	9.12783	0.291578
$981$	−11.3607	−0.362719
$982$	34.5835	1.10361
$983$	−21.8576	−0.697150	−0.348575	−	0.937281i	$-0.613334\pi$
$983$	−21.8576	−0.697150	−0.348575	+	0.937281i	$0.613334\pi$
$984$	−40.2472	−1.28304
$985$	2.56916	0.0818603
$986$	−8.84939	−0.281822
$987$	−5.26180	−0.167485
$988$	106.186	3.37821
$989$	63.0349	2.00439
$990$	−12.1834	−0.387214
$991$	−13.7731	−0.437517	−0.218758	−	0.975779i	$-0.570201\pi$
$991$	−13.7731	−0.437517	−0.218758	+	0.975779i	$0.570201\pi$
$992$	200.492	6.36563
$993$	−26.6225	−0.844839
$994$	−29.8576	−0.947026
$995$	−2.47027	−0.0783127
$996$	−15.1506	−0.480066
$997$	−10.1217	−0.320557	−0.160278	−	0.987072i	$-0.551239\pi$
$997$	−10.1217	−0.320557	−0.160278	+	0.987072i	$0.551239\pi$
$998$	6.11118	0.193446
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.m.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.m.1.1	✓	3	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q-2.70928 q^{2} +1.00000 q^{3} +5.34017 q^{4} +1.70928 q^{5} -2.70928 q^{6} -1.00000 q^{7} -9.04945 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.70928 q^{2} +1.00000 q^{3} +5.34017 q^{4} +1.70928 q^{5} -2.70928 q^{6} -1.00000 q^{7} -9.04945 q^{8} +1.00000 q^{9} -4.63090 q^{10} +2.63090 q^{11} +5.34017 q^{12} +2.29072 q^{13} +2.70928 q^{14} +1.70928 q^{15} +13.8371 q^{16} +0.340173 q^{17} -2.70928 q^{18} +8.68035 q^{19} +9.12783 q^{20} -1.00000 q^{21} -7.12783 q^{22} +8.49693 q^{23} -9.04945 q^{24} -2.07838 q^{25} -6.20620 q^{26} +1.00000 q^{27} -5.34017 q^{28} +9.60197 q^{29} -4.63090 q^{30} -10.3402 q^{31} -19.3896 q^{32} +2.63090 q^{33} -0.921622 q^{34} -1.70928 q^{35} +5.34017 q^{36} +10.0000 q^{37} -23.5174 q^{38} +2.29072 q^{39} -15.4680 q^{40} +4.44748 q^{41} +2.70928 q^{42} +7.41855 q^{43} +14.0494 q^{44} +1.70928 q^{45} -23.0205 q^{46} +5.26180 q^{47} +13.8371 q^{48} +1.00000 q^{49} +5.63090 q^{50} +0.340173 q^{51} +12.2329 q^{52} +8.34017 q^{53} -2.70928 q^{54} +4.49693 q^{55} +9.04945 q^{56} +8.68035 q^{57} -26.0144 q^{58} +12.6803 q^{59} +9.12783 q^{60} -11.8660 q^{61} +28.0144 q^{62} -1.00000 q^{63} +24.8576 q^{64} +3.91548 q^{65} -7.12783 q^{66} +4.09890 q^{67} +1.81658 q^{68} +8.49693 q^{69} +4.63090 q^{70} -11.0205 q^{71} -9.04945 q^{72} -8.83710 q^{73} -27.0928 q^{74} -2.07838 q^{75} +46.3545 q^{76} -2.63090 q^{77} -6.20620 q^{78} +2.04945 q^{79} +23.6514 q^{80} +1.00000 q^{81} -12.0494 q^{82} -2.83710 q^{83} -5.34017 q^{84} +0.581449 q^{85} -20.0989 q^{86} +9.60197 q^{87} -23.8082 q^{88} -10.3896 q^{89} -4.63090 q^{90} -2.29072 q^{91} +45.3751 q^{92} -10.3402 q^{93} -14.2557 q^{94} +14.8371 q^{95} -19.3896 q^{96} +1.02893 q^{97} -2.70928 q^{98} +2.63090 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 3 q^{3} + 5 q^{4} - 2 q^{5} - q^{6} - 3 q^{7} - 9 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - q^2 + 3 * q^3 + 5 * q^4 - 2 * q^5 - q^6 - 3 * q^7 - 9 * q^8 + 3 * q^9	\( 3 q - q^{2} + 3 q^{3} + 5 q^{4} - 2 q^{5} - q^{6} - 3 q^{7} - 9 q^{8} + 3 q^{9} - 10 q^{10} + 4 q^{11} + 5 q^{12} + 14 q^{13} + q^{14} - 2 q^{15} + 13 q^{16} - 10 q^{17} - q^{18} + 4 q^{19} + 6 q^{20} - 3 q^{21} + 8 q^{23} - 9 q^{24} - 3 q^{25} + 6 q^{26} + 3 q^{27} - 5 q^{28} + 10 q^{29} - 10 q^{30} - 20 q^{31} - 29 q^{32} + 4 q^{33} - 6 q^{34} + 2 q^{35} + 5 q^{36} + 30 q^{37} - 20 q^{38} + 14 q^{39} - 14 q^{40} + 14 q^{41} + q^{42} + 8 q^{43} + 24 q^{44} - 2 q^{45} - 36 q^{46} + 8 q^{47} + 13 q^{48} + 3 q^{49} + 13 q^{50} - 10 q^{51} + 14 q^{52} + 14 q^{53} - q^{54} - 4 q^{55} + 9 q^{56} + 4 q^{57} - 10 q^{58} + 16 q^{59} + 6 q^{60} - 22 q^{61} + 16 q^{62} - 3 q^{63} + 13 q^{64} - 20 q^{65} - 24 q^{67} + 10 q^{68} + 8 q^{69} + 10 q^{70} - 9 q^{72} + 2 q^{73} - 10 q^{74} - 3 q^{75} + 60 q^{76} - 4 q^{77} + 6 q^{78} - 12 q^{79} + 34 q^{80} + 3 q^{81} - 18 q^{82} + 20 q^{83} - 5 q^{84} + 16 q^{85} - 24 q^{86} + 10 q^{87} - 28 q^{88} - 2 q^{89} - 10 q^{90} - 14 q^{91} + 24 q^{92} - 20 q^{93} + 16 q^{95} - 29 q^{96} + 18 q^{97} - q^{98} + 4 q^{99}+O(q^{100}) \) 3 * q - q^2 + 3 * q^3 + 5 * q^4 - 2 * q^5 - q^6 - 3 * q^7 - 9 * q^8 + 3 * q^9 - 10 * q^10 + 4 * q^11 + 5 * q^12 + 14 * q^13 + q^14 - 2 * q^15 + 13 * q^16 - 10 * q^17 - q^18 + 4 * q^19 + 6 * q^20 - 3 * q^21 + 8 * q^23 - 9 * q^24 - 3 * q^25 + 6 * q^26 + 3 * q^27 - 5 * q^28 + 10 * q^29 - 10 * q^30 - 20 * q^31 - 29 * q^32 + 4 * q^33 - 6 * q^34 + 2 * q^35 + 5 * q^36 + 30 * q^37 - 20 * q^38 + 14 * q^39 - 14 * q^40 + 14 * q^41 + q^42 + 8 * q^43 + 24 * q^44 - 2 * q^45 - 36 * q^46 + 8 * q^47 + 13 * q^48 + 3 * q^49 + 13 * q^50 - 10 * q^51 + 14 * q^52 + 14 * q^53 - q^54 - 4 * q^55 + 9 * q^56 + 4 * q^57 - 10 * q^58 + 16 * q^59 + 6 * q^60 - 22 * q^61 + 16 * q^62 - 3 * q^63 + 13 * q^64 - 20 * q^65 - 24 * q^67 + 10 * q^68 + 8 * q^69 + 10 * q^70 - 9 * q^72 + 2 * q^73 - 10 * q^74 - 3 * q^75 + 60 * q^76 - 4 * q^77 + 6 * q^78 - 12 * q^79 + 34 * q^80 + 3 * q^81 - 18 * q^82 + 20 * q^83 - 5 * q^84 + 16 * q^85 - 24 * q^86 + 10 * q^87 - 28 * q^88 - 2 * q^89 - 10 * q^90 - 14 * q^91 + 24 * q^92 - 20 * q^93 + 16 * q^95 - 29 * q^96 + 18 * q^97 - q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more