LMFDB - Embedded newform 4031.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4031,2,Mod(1,4031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4031, 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("4031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4031 = 29 \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4031.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:	$32.1876970548$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	4031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{2} +2.41421 q^{3} +3.82843 q^{4} +2.00000 q^{5} +5.82843 q^{6} -3.82843 q^{7} +4.41421 q^{8} +2.82843 q^{9} +O(q^{10})$	\(q+2.41421 q^{2} +2.41421 q^{3} +3.82843 q^{4} +2.00000 q^{5} +5.82843 q^{6} -3.82843 q^{7} +4.41421 q^{8} +2.82843 q^{9} +4.82843 q^{10} +9.24264 q^{12} +5.00000 q^{13} -9.24264 q^{14} +4.82843 q^{15} +3.00000 q^{16} +4.41421 q^{17} +6.82843 q^{18} -5.24264 q^{19} +7.65685 q^{20} -9.24264 q^{21} +3.17157 q^{23} +10.6569 q^{24} -1.00000 q^{25} +12.0711 q^{26} -0.414214 q^{27} -14.6569 q^{28} +1.00000 q^{29} +11.6569 q^{30} +4.82843 q^{31} -1.58579 q^{32} +10.6569 q^{34} -7.65685 q^{35} +10.8284 q^{36} +8.48528 q^{37} -12.6569 q^{38} +12.0711 q^{39} +8.82843 q^{40} +2.00000 q^{41} -22.3137 q^{42} -2.41421 q^{43} +5.65685 q^{45} +7.65685 q^{46} +4.82843 q^{47} +7.24264 q^{48} +7.65685 q^{49} -2.41421 q^{50} +10.6569 q^{51} +19.1421 q^{52} +6.82843 q^{53} -1.00000 q^{54} -16.8995 q^{56} -12.6569 q^{57} +2.41421 q^{58} -11.6569 q^{59} +18.4853 q^{60} +10.8995 q^{61} +11.6569 q^{62} -10.8284 q^{63} -9.82843 q^{64} +10.0000 q^{65} -12.3137 q^{67} +16.8995 q^{68} +7.65685 q^{69} -18.4853 q^{70} -14.3137 q^{71} +12.4853 q^{72} -4.07107 q^{73} +20.4853 q^{74} -2.41421 q^{75} -20.0711 q^{76} +29.1421 q^{78} -2.48528 q^{79} +6.00000 q^{80} -9.48528 q^{81} +4.82843 q^{82} -7.00000 q^{83} -35.3848 q^{84} +8.82843 q^{85} -5.82843 q^{86} +2.41421 q^{87} -16.8284 q^{89} +13.6569 q^{90} -19.1421 q^{91} +12.1421 q^{92} +11.6569 q^{93} +11.6569 q^{94} -10.4853 q^{95} -3.82843 q^{96} +9.24264 q^{97} +18.4853 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 6 q^{6} - 2 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 4 * q^5 + 6 * q^6 - 2 * q^7 + 6 * q^8	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 6 q^{6} - 2 q^{7} + 6 q^{8} + 4 q^{10} + 10 q^{12} + 10 q^{13} - 10 q^{14} + 4 q^{15} + 6 q^{16} + 6 q^{17} + 8 q^{18} - 2 q^{19} + 4 q^{20} - 10 q^{21} + 12 q^{23} + 10 q^{24} - 2 q^{25} + 10 q^{26} + 2 q^{27} - 18 q^{28} + 2 q^{29} + 12 q^{30} + 4 q^{31} - 6 q^{32} + 10 q^{34} - 4 q^{35} + 16 q^{36} - 14 q^{38} + 10 q^{39} + 12 q^{40} + 4 q^{41} - 22 q^{42} - 2 q^{43} + 4 q^{46} + 4 q^{47} + 6 q^{48} + 4 q^{49} - 2 q^{50} + 10 q^{51} + 10 q^{52} + 8 q^{53} - 2 q^{54} - 14 q^{56} - 14 q^{57} + 2 q^{58} - 12 q^{59} + 20 q^{60} + 2 q^{61} + 12 q^{62} - 16 q^{63} - 14 q^{64} + 20 q^{65} - 2 q^{67} + 14 q^{68} + 4 q^{69} - 20 q^{70} - 6 q^{71} + 8 q^{72} + 6 q^{73} + 24 q^{74} - 2 q^{75} - 26 q^{76} + 30 q^{78} + 12 q^{79} + 12 q^{80} - 2 q^{81} + 4 q^{82} - 14 q^{83} - 34 q^{84} + 12 q^{85} - 6 q^{86} + 2 q^{87} - 28 q^{89} + 16 q^{90} - 10 q^{91} - 4 q^{92} + 12 q^{93} + 12 q^{94} - 4 q^{95} - 2 q^{96} + 10 q^{97} + 20 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 4 * q^5 + 6 * q^6 - 2 * q^7 + 6 * q^8 + 4 * q^10 + 10 * q^12 + 10 * q^13 - 10 * q^14 + 4 * q^15 + 6 * q^16 + 6 * q^17 + 8 * q^18 - 2 * q^19 + 4 * q^20 - 10 * q^21 + 12 * q^23 + 10 * q^24 - 2 * q^25 + 10 * q^26 + 2 * q^27 - 18 * q^28 + 2 * q^29 + 12 * q^30 + 4 * q^31 - 6 * q^32 + 10 * q^34 - 4 * q^35 + 16 * q^36 - 14 * q^38 + 10 * q^39 + 12 * q^40 + 4 * q^41 - 22 * q^42 - 2 * q^43 + 4 * q^46 + 4 * q^47 + 6 * q^48 + 4 * q^49 - 2 * q^50 + 10 * q^51 + 10 * q^52 + 8 * q^53 - 2 * q^54 - 14 * q^56 - 14 * q^57 + 2 * q^58 - 12 * q^59 + 20 * q^60 + 2 * q^61 + 12 * q^62 - 16 * q^63 - 14 * q^64 + 20 * q^65 - 2 * q^67 + 14 * q^68 + 4 * q^69 - 20 * q^70 - 6 * q^71 + 8 * q^72 + 6 * q^73 + 24 * q^74 - 2 * q^75 - 26 * q^76 + 30 * q^78 + 12 * q^79 + 12 * q^80 - 2 * q^81 + 4 * q^82 - 14 * q^83 - 34 * q^84 + 12 * q^85 - 6 * q^86 + 2 * q^87 - 28 * q^89 + 16 * q^90 - 10 * q^91 - 4 * q^92 + 12 * q^93 + 12 * q^94 - 4 * q^95 - 2 * q^96 + 10 * q^97 + 20 * q^98

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 $

$2$	2.41421	1.70711	0.853553	−	0.521005i	$-0.174443\pi$
$2$	2.41421	1.70711	0.853553	+	0.521005i	$0.174443\pi$
$3$	2.41421	1.39385	0.696923	−	0.717146i	$-0.254552\pi$
$3$	2.41421	1.39385	0.696923	+	0.717146i	$0.254552\pi$
$4$	3.82843	1.91421
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	5.82843	2.37945
$7$	−3.82843	−1.44701	−0.723505	−	0.690319i	$-0.757470\pi$
$7$	−3.82843	−1.44701	−0.723505	+	0.690319i	$0.757470\pi$
$8$	4.41421	1.56066
$9$	2.82843	0.942809
$10$	4.82843	1.52688
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	9.24264	2.66812
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	−9.24264	−2.47020
$15$	4.82843	1.24669
$16$	3.00000	0.750000
$17$	4.41421	1.07060	0.535302	−	0.844661i	$-0.320198\pi$
$17$	4.41421	1.07060	0.535302	+	0.844661i	$0.320198\pi$
$18$	6.82843	1.60948
$19$	−5.24264	−1.20274	−0.601372	−	0.798969i	$-0.705379\pi$
$19$	−5.24264	−1.20274	−0.601372	+	0.798969i	$0.705379\pi$
$20$	7.65685	1.71212
$21$	−9.24264	−2.01691
$22$	0	0
$23$	3.17157	0.661319	0.330659	−	0.943750i	$-0.392729\pi$
$23$	3.17157	0.661319	0.330659	+	0.943750i	$0.392729\pi$
$24$	10.6569	2.17532
$25$	−1.00000	−0.200000
$26$	12.0711	2.36733
$27$	−0.414214	−0.0797154
$28$	−14.6569	−2.76989
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	11.6569	2.12824
$31$	4.82843	0.867211	0.433606	−	0.901103i	$-0.357241\pi$
$31$	4.82843	0.867211	0.433606	+	0.901103i	$0.357241\pi$
$32$	−1.58579	−0.280330
$33$	0	0
$34$	10.6569	1.82764
$35$	−7.65685	−1.29424
$36$	10.8284	1.80474
$37$	8.48528	1.39497	0.697486	−	0.716599i	$-0.254302\pi$
$37$	8.48528	1.39497	0.697486	+	0.716599i	$0.254302\pi$
$38$	−12.6569	−2.05321
$39$	12.0711	1.93292
$40$	8.82843	1.39590
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−22.3137	−3.44308
$43$	−2.41421	−0.368164	−0.184082	−	0.982911i	$-0.558931\pi$
$43$	−2.41421	−0.368164	−0.184082	+	0.982911i	$0.558931\pi$
$44$	0	0
$45$	5.65685	0.843274
$46$	7.65685	1.12894
$47$	4.82843	0.704298	0.352149	−	0.935944i	$-0.385451\pi$
$47$	4.82843	0.704298	0.352149	+	0.935944i	$0.385451\pi$
$48$	7.24264	1.04539
$49$	7.65685	1.09384
$50$	−2.41421	−0.341421
$51$	10.6569	1.49226
$52$	19.1421	2.65454
$53$	6.82843	0.937957	0.468978	−	0.883210i	$-0.344622\pi$
$53$	6.82843	0.937957	0.468978	+	0.883210i	$0.344622\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−16.8995	−2.25829
$57$	−12.6569	−1.67644
$58$	2.41421	0.317002
$59$	−11.6569	−1.51759	−0.758797	−	0.651328i	$-0.774212\pi$
$59$	−11.6569	−1.51759	−0.758797	+	0.651328i	$0.774212\pi$
$60$	18.4853	2.38644
$61$	10.8995	1.39554	0.697769	−	0.716323i	$-0.254176\pi$
$61$	10.8995	1.39554	0.697769	+	0.716323i	$0.254176\pi$
$62$	11.6569	1.48042
$63$	−10.8284	−1.36425
$64$	−9.82843	−1.22855
$65$	10.0000	1.24035
$66$	0	0
$67$	−12.3137	−1.50436	−0.752179	−	0.658958i	$-0.770997\pi$
$67$	−12.3137	−1.50436	−0.752179	+	0.658958i	$0.770997\pi$
$68$	16.8995	2.04936
$69$	7.65685	0.921777
$70$	−18.4853	−2.20941
$71$	−14.3137	−1.69872	−0.849362	−	0.527810i	$-0.823013\pi$
$71$	−14.3137	−1.69872	−0.849362	+	0.527810i	$0.823013\pi$
$72$	12.4853	1.47140
$73$	−4.07107	−0.476482	−0.238241	−	0.971206i	$-0.576571\pi$
$73$	−4.07107	−0.476482	−0.238241	+	0.971206i	$0.576571\pi$
$74$	20.4853	2.38137
$75$	−2.41421	−0.278769
$76$	−20.0711	−2.30231
$77$	0	0
$78$	29.1421	3.29970
$79$	−2.48528	−0.279616	−0.139808	−	0.990179i	$-0.544649\pi$
$79$	−2.48528	−0.279616	−0.139808	+	0.990179i	$0.544649\pi$
$80$	6.00000	0.670820
$81$	−9.48528	−1.05392
$82$	4.82843	0.533211
$83$	−7.00000	−0.768350	−0.384175	−	0.923260i	$-0.625514\pi$
$83$	−7.00000	−0.768350	−0.384175	+	0.923260i	$0.625514\pi$
$84$	−35.3848	−3.86080
$85$	8.82843	0.957577
$86$	−5.82843	−0.628495
$87$	2.41421	0.258831
$88$	0	0
$89$	−16.8284	−1.78381	−0.891905	−	0.452223i	$-0.850631\pi$
$89$	−16.8284	−1.78381	−0.891905	+	0.452223i	$0.850631\pi$
$90$	13.6569	1.43956
$91$	−19.1421	−2.00664
$92$	12.1421	1.26591
$93$	11.6569	1.20876
$94$	11.6569	1.20231
$95$	−10.4853	−1.07577
$96$	−3.82843	−0.390737
$97$	9.24264	0.938448	0.469224	−	0.883079i	$-0.344534\pi$
$97$	9.24264	0.938448	0.469224	+	0.883079i	$0.344534\pi$
$98$	18.4853	1.86730
$99$	0	0
$100$	−3.82843	−0.382843
$101$	−18.1421	−1.80521	−0.902605	−	0.430470i	$-0.858348\pi$
$101$	−18.1421	−1.80521	−0.902605	+	0.430470i	$0.858348\pi$
$102$	25.7279	2.54744
$103$	−1.17157	−0.115439	−0.0577193	−	0.998333i	$-0.518383\pi$
$103$	−1.17157	−0.115439	−0.0577193	+	0.998333i	$0.518383\pi$
$104$	22.0711	2.16425
$105$	−18.4853	−1.80398
$106$	16.4853	1.60119
$107$	−2.51472	−0.243107	−0.121554	−	0.992585i	$-0.538788\pi$
$107$	−2.51472	−0.243107	−0.121554	+	0.992585i	$0.538788\pi$
$108$	−1.58579	−0.152592
$109$	−6.82843	−0.654045	−0.327022	−	0.945017i	$-0.606045\pi$
$109$	−6.82843	−0.654045	−0.327022	+	0.945017i	$0.606045\pi$
$110$	0	0
$111$	20.4853	1.94438
$112$	−11.4853	−1.08526
$113$	14.4853	1.36266	0.681330	−	0.731976i	$-0.261402\pi$
$113$	14.4853	1.36266	0.681330	+	0.731976i	$0.261402\pi$
$114$	−30.5563	−2.86186
$115$	6.34315	0.591501
$116$	3.82843	0.355461
$117$	14.1421	1.30744
$118$	−28.1421	−2.59069
$119$	−16.8995	−1.54917
$120$	21.3137	1.94567
$121$	−11.0000	−1.00000
$122$	26.3137	2.38233
$123$	4.82843	0.435365
$124$	18.4853	1.66003
$125$	−12.0000	−1.07331
$126$	−26.1421	−2.32893
$127$	−10.4853	−0.930418	−0.465209	−	0.885201i	$-0.654021\pi$
$127$	−10.4853	−0.930418	−0.465209	+	0.885201i	$0.654021\pi$
$128$	−20.5563	−1.81694
$129$	−5.82843	−0.513164
$130$	24.1421	2.11741
$131$	15.6569	1.36795	0.683973	−	0.729507i	$-0.260251\pi$
$131$	15.6569	1.36795	0.683973	+	0.729507i	$0.260251\pi$
$132$	0	0
$133$	20.0711	1.74038
$134$	−29.7279	−2.56810
$135$	−0.828427	−0.0712997
$136$	19.4853	1.67085
$137$	5.51472	0.471154	0.235577	−	0.971856i	$-0.424302\pi$
$137$	5.51472	0.471154	0.235577	+	0.971856i	$0.424302\pi$
$138$	18.4853	1.57357
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	−29.3137	−2.47746
$141$	11.6569	0.981684
$142$	−34.5563	−2.89990
$143$	0	0
$144$	8.48528	0.707107
$145$	2.00000	0.166091
$146$	−9.82843	−0.813406
$147$	18.4853	1.52464
$148$	32.4853	2.67027
$149$	4.82843	0.395560	0.197780	−	0.980246i	$-0.436627\pi$
$149$	4.82843	0.395560	0.197780	+	0.980246i	$0.436627\pi$
$150$	−5.82843	−0.475889
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	−23.1421	−1.87708
$153$	12.4853	1.00938
$154$	0	0
$155$	9.65685	0.775657
$156$	46.2132	3.70002
$157$	−24.2132	−1.93242	−0.966212	−	0.257749i	$-0.917019\pi$
$157$	−24.2132	−1.93242	−0.966212	+	0.257749i	$0.917019\pi$
$158$	−6.00000	−0.477334
$159$	16.4853	1.30737
$160$	−3.17157	−0.250735
$161$	−12.1421	−0.956934
$162$	−22.8995	−1.79915
$163$	−20.6274	−1.61566	−0.807832	−	0.589413i	$-0.799359\pi$
$163$	−20.6274	−1.61566	−0.807832	+	0.589413i	$0.799359\pi$
$164$	7.65685	0.597900
$165$	0	0
$166$	−16.8995	−1.31166
$167$	13.6569	1.05680	0.528400	−	0.848996i	$-0.322792\pi$
$167$	13.6569	1.05680	0.528400	+	0.848996i	$0.322792\pi$
$168$	−40.7990	−3.14771
$169$	12.0000	0.923077
$170$	21.3137	1.63469
$171$	−14.8284	−1.13396
$172$	−9.24264	−0.704745
$173$	9.82843	0.747241	0.373621	−	0.927582i	$-0.378116\pi$
$173$	9.82843	0.747241	0.373621	+	0.927582i	$0.378116\pi$
$174$	5.82843	0.441852
$175$	3.82843	0.289402
$176$	0	0
$177$	−28.1421	−2.11529
$178$	−40.6274	−3.04515
$179$	21.7990	1.62933	0.814667	−	0.579930i	$-0.196920\pi$
$179$	21.7990	1.62933	0.814667	+	0.579930i	$0.196920\pi$
$180$	21.6569	1.61421
$181$	−12.1716	−0.904706	−0.452353	−	0.891839i	$-0.649415\pi$
$181$	−12.1716	−0.904706	−0.452353	+	0.891839i	$0.649415\pi$
$182$	−46.2132	−3.42555
$183$	26.3137	1.94517
$184$	14.0000	1.03209
$185$	16.9706	1.24770
$186$	28.1421	2.06348
$187$	0	0
$188$	18.4853	1.34818
$189$	1.58579	0.115349
$190$	−25.3137	−1.83645
$191$	−0.343146	−0.0248292	−0.0124146	−	0.999923i	$-0.503952\pi$
$191$	−0.343146	−0.0248292	−0.0124146	+	0.999923i	$0.503952\pi$
$192$	−23.7279	−1.71242
$193$	24.8284	1.78719	0.893595	−	0.448875i	$-0.148175\pi$
$193$	24.8284	1.78719	0.893595	+	0.448875i	$0.148175\pi$
$194$	22.3137	1.60203
$195$	24.1421	1.72885
$196$	29.3137	2.09384
$197$	−4.14214	−0.295115	−0.147557	−	0.989053i	$-0.547141\pi$
$197$	−4.14214	−0.295115	−0.147557	+	0.989053i	$0.547141\pi$
$198$	0	0
$199$	23.7990	1.68707	0.843533	−	0.537078i	$-0.180472\pi$
$199$	23.7990	1.68707	0.843533	+	0.537078i	$0.180472\pi$
$200$	−4.41421	−0.312132
$201$	−29.7279	−2.09685
$202$	−43.7990	−3.08169
$203$	−3.82843	−0.268703
$204$	40.7990	2.85650
$205$	4.00000	0.279372
$206$	−2.82843	−0.197066
$207$	8.97056	0.623497
$208$	15.0000	1.04006
$209$	0	0
$210$	−44.6274	−3.07958
$211$	−24.0711	−1.65712	−0.828560	−	0.559900i	$-0.810840\pi$
$211$	−24.0711	−1.65712	−0.828560	+	0.559900i	$0.810840\pi$
$212$	26.1421	1.79545
$213$	−34.5563	−2.36776
$214$	−6.07107	−0.415010
$215$	−4.82843	−0.329296
$216$	−1.82843	−0.124409
$217$	−18.4853	−1.25486
$218$	−16.4853	−1.11652
$219$	−9.82843	−0.664144
$220$	0	0
$221$	22.0711	1.48466
$222$	49.4558	3.31926
$223$	11.1716	0.748104	0.374052	−	0.927408i	$-0.377968\pi$
$223$	11.1716	0.748104	0.374052	+	0.927408i	$0.377968\pi$
$224$	6.07107	0.405640
$225$	−2.82843	−0.188562
$226$	34.9706	2.32621
$227$	20.6274	1.36909	0.684545	−	0.728971i	$-0.260001\pi$
$227$	20.6274	1.36909	0.684545	+	0.728971i	$0.260001\pi$
$228$	−48.4558	−3.20907
$229$	−3.51472	−0.232259	−0.116130	−	0.993234i	$-0.537049\pi$
$229$	−3.51472	−0.232259	−0.116130	+	0.993234i	$0.537049\pi$
$230$	15.3137	1.00976
$231$	0	0
$232$	4.41421	0.289807
$233$	−15.7990	−1.03503	−0.517513	−	0.855675i	$-0.673142\pi$
$233$	−15.7990	−1.03503	−0.517513	+	0.855675i	$0.673142\pi$
$234$	34.1421	2.23194
$235$	9.65685	0.629944
$236$	−44.6274	−2.90500
$237$	−6.00000	−0.389742
$238$	−40.7990	−2.64461
$239$	10.1716	0.657944	0.328972	−	0.944340i	$-0.393298\pi$
$239$	10.1716	0.657944	0.328972	+	0.944340i	$0.393298\pi$
$240$	14.4853	0.935021
$241$	12.9706	0.835507	0.417754	−	0.908560i	$-0.362817\pi$
$241$	12.9706	0.835507	0.417754	+	0.908560i	$0.362817\pi$
$242$	−26.5563	−1.70711
$243$	−21.6569	−1.38929
$244$	41.7279	2.67136
$245$	15.3137	0.978357
$246$	11.6569	0.743214
$247$	−26.2132	−1.66791
$248$	21.3137	1.35342
$249$	−16.8995	−1.07096
$250$	−28.9706	−1.83226
$251$	−4.97056	−0.313739	−0.156870	−	0.987619i	$-0.550140\pi$
$251$	−4.97056	−0.313739	−0.156870	+	0.987619i	$0.550140\pi$
$252$	−41.4558	−2.61147
$253$	0	0
$254$	−25.3137	−1.58832
$255$	21.3137	1.33472
$256$	−29.9706	−1.87316
$257$	6.68629	0.417079	0.208540	−	0.978014i	$-0.433129\pi$
$257$	6.68629	0.417079	0.208540	+	0.978014i	$0.433129\pi$
$258$	−14.0711	−0.876026
$259$	−32.4853	−2.01854
$260$	38.2843	2.37429
$261$	2.82843	0.175075
$262$	37.7990	2.33523
$263$	−18.4853	−1.13985	−0.569926	−	0.821696i	$-0.693028\pi$
$263$	−18.4853	−1.13985	−0.569926	+	0.821696i	$0.693028\pi$
$264$	0	0
$265$	13.6569	0.838934
$266$	48.4558	2.97102
$267$	−40.6274	−2.48636
$268$	−47.1421	−2.87966
$269$	17.7279	1.08089	0.540445	−	0.841379i	$-0.318256\pi$
$269$	17.7279	1.08089	0.540445	+	0.841379i	$0.318256\pi$
$270$	−2.00000	−0.121716
$271$	15.5858	0.946769	0.473385	−	0.880856i	$-0.343032\pi$
$271$	15.5858	0.946769	0.473385	+	0.880856i	$0.343032\pi$
$272$	13.2426	0.802953
$273$	−46.2132	−2.79695
$274$	13.3137	0.804311
$275$	0	0
$276$	29.3137	1.76448
$277$	26.8284	1.61196	0.805982	−	0.591940i	$-0.201638\pi$
$277$	26.8284	1.61196	0.805982	+	0.591940i	$0.201638\pi$
$278$	−2.41421	−0.144795
$279$	13.6569	0.817614
$280$	−33.7990	−2.01988
$281$	2.34315	0.139780	0.0698902	−	0.997555i	$-0.477735\pi$
$281$	2.34315	0.139780	0.0698902	+	0.997555i	$0.477735\pi$
$282$	28.1421	1.67584
$283$	−5.97056	−0.354913	−0.177457	−	0.984129i	$-0.556787\pi$
$283$	−5.97056	−0.354913	−0.177457	+	0.984129i	$0.556787\pi$
$284$	−54.7990	−3.25172
$285$	−25.3137	−1.49945
$286$	0	0
$287$	−7.65685	−0.451970
$288$	−4.48528	−0.264298
$289$	2.48528	0.146193
$290$	4.82843	0.283535
$291$	22.3137	1.30805
$292$	−15.5858	−0.912089
$293$	3.51472	0.205332	0.102666	−	0.994716i	$-0.467263\pi$
$293$	3.51472	0.205332	0.102666	+	0.994716i	$0.467263\pi$
$294$	44.6274	2.60272
$295$	−23.3137	−1.35738
$296$	37.4558	2.17708
$297$	0	0
$298$	11.6569	0.675263
$299$	15.8579	0.917084
$300$	−9.24264	−0.533624
$301$	9.24264	0.532737
$302$	−28.9706	−1.66707
$303$	−43.7990	−2.51619
$304$	−15.7279	−0.902058
$305$	21.7990	1.24821
$306$	30.1421	1.72311
$307$	4.68629	0.267461	0.133730	−	0.991018i	$-0.457304\pi$
$307$	4.68629	0.267461	0.133730	+	0.991018i	$0.457304\pi$
$308$	0	0
$309$	−2.82843	−0.160904
$310$	23.3137	1.32413
$311$	−7.65685	−0.434180	−0.217090	−	0.976152i	$-0.569657\pi$
$311$	−7.65685	−0.434180	−0.217090	+	0.976152i	$0.569657\pi$
$312$	53.2843	3.01663
$313$	0.857864	0.0484894	0.0242447	−	0.999706i	$-0.492282\pi$
$313$	0.857864	0.0484894	0.0242447	+	0.999706i	$0.492282\pi$
$314$	−58.4558	−3.29885
$315$	−21.6569	−1.22023
$316$	−9.51472	−0.535245
$317$	5.17157	0.290464	0.145232	−	0.989398i	$-0.453607\pi$
$317$	5.17157	0.290464	0.145232	+	0.989398i	$0.453607\pi$
$318$	39.7990	2.23182
$319$	0	0
$320$	−19.6569	−1.09885
$321$	−6.07107	−0.338854
$322$	−29.3137	−1.63359
$323$	−23.1421	−1.28766
$324$	−36.3137	−2.01743
$325$	−5.00000	−0.277350
$326$	−49.7990	−2.75811
$327$	−16.4853	−0.911638
$328$	8.82843	0.487468
$329$	−18.4853	−1.01913
$330$	0	0
$331$	−34.9706	−1.92216	−0.961078	−	0.276277i	$-0.910899\pi$
$331$	−34.9706	−1.92216	−0.961078	+	0.276277i	$0.910899\pi$
$332$	−26.7990	−1.47079
$333$	24.0000	1.31519
$334$	32.9706	1.80407
$335$	−24.6274	−1.34554
$336$	−27.7279	−1.51268
$337$	19.2426	1.04821	0.524107	−	0.851653i	$-0.324399\pi$
$337$	19.2426	1.04821	0.524107	+	0.851653i	$0.324399\pi$
$338$	28.9706	1.57579
$339$	34.9706	1.89934
$340$	33.7990	1.83301
$341$	0	0
$342$	−35.7990	−1.93579
$343$	−2.51472	−0.135782
$344$	−10.6569	−0.574579
$345$	15.3137	0.824462
$346$	23.7279	1.27562
$347$	20.9706	1.12576	0.562879	−	0.826539i	$-0.309694\pi$
$347$	20.9706	1.12576	0.562879	+	0.826539i	$0.309694\pi$
$348$	9.24264	0.495458
$349$	−27.9706	−1.49723	−0.748615	−	0.663005i	$-0.769281\pi$
$349$	−27.9706	−1.49723	−0.748615	+	0.663005i	$0.769281\pi$
$350$	9.24264	0.494040
$351$	−2.07107	−0.110545
$352$	0	0
$353$	35.4558	1.88712	0.943562	−	0.331196i	$-0.107452\pi$
$353$	35.4558	1.88712	0.943562	+	0.331196i	$0.107452\pi$
$354$	−67.9411	−3.61103
$355$	−28.6274	−1.51939
$356$	−64.4264	−3.41459
$357$	−40.7990	−2.15931
$358$	52.6274	2.78145
$359$	−8.48528	−0.447836	−0.223918	−	0.974608i	$-0.571885\pi$
$359$	−8.48528	−0.447836	−0.223918	+	0.974608i	$0.571885\pi$
$360$	24.9706	1.31606
$361$	8.48528	0.446594
$362$	−29.3848	−1.54443
$363$	−26.5563	−1.39385
$364$	−73.2843	−3.84114
$365$	−8.14214	−0.426179
$366$	63.5269	3.32060
$367$	18.1421	0.947012	0.473506	−	0.880791i	$-0.342988\pi$
$367$	18.1421	0.947012	0.473506	+	0.880791i	$0.342988\pi$
$368$	9.51472	0.495989
$369$	5.65685	0.294484
$370$	40.9706	2.12996
$371$	−26.1421	−1.35723
$372$	44.6274	2.31382
$373$	20.0000	1.03556	0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000	1.03556	0.517780	+	0.855514i	$0.326758\pi$
$374$	0	0
$375$	−28.9706	−1.49603
$376$	21.3137	1.09917
$377$	5.00000	0.257513
$378$	3.82843	0.196913
$379$	−8.27208	−0.424908	−0.212454	−	0.977171i	$-0.568146\pi$
$379$	−8.27208	−0.424908	−0.212454	+	0.977171i	$0.568146\pi$
$380$	−40.1421	−2.05925
$381$	−25.3137	−1.29686
$382$	−0.828427	−0.0423860
$383$	23.7990	1.21607	0.608036	−	0.793910i	$-0.291958\pi$
$383$	23.7990	1.21607	0.608036	+	0.793910i	$0.291958\pi$
$384$	−49.6274	−2.53254
$385$	0	0
$386$	59.9411	3.05092
$387$	−6.82843	−0.347108
$388$	35.3848	1.79639
$389$	−19.0416	−0.965449	−0.482724	−	0.875772i	$-0.660353\pi$
$389$	−19.0416	−0.965449	−0.482724	+	0.875772i	$0.660353\pi$
$390$	58.2843	2.95134
$391$	14.0000	0.708010
$392$	33.7990	1.70711
$393$	37.7990	1.90671
$394$	−10.0000	−0.503793
$395$	−4.97056	−0.250096
$396$	0	0
$397$	−4.00000	−0.200754	−0.100377	−	0.994949i	$-0.532005\pi$
$397$	−4.00000	−0.200754	−0.100377	+	0.994949i	$0.532005\pi$
$398$	57.4558	2.88000
$399$	48.4558	2.42583
$400$	−3.00000	−0.150000
$401$	22.9706	1.14710	0.573548	−	0.819172i	$-0.305567\pi$
$401$	22.9706	1.14710	0.573548	+	0.819172i	$0.305567\pi$
$402$	−71.7696	−3.57954
$403$	24.1421	1.20261
$404$	−69.4558	−3.45556
$405$	−18.9706	−0.942655
$406$	−9.24264	−0.458705
$407$	0	0
$408$	47.0416	2.32891
$409$	8.14214	0.402603	0.201301	−	0.979529i	$-0.435483\pi$
$409$	8.14214	0.402603	0.201301	+	0.979529i	$0.435483\pi$
$410$	9.65685	0.476918
$411$	13.3137	0.656717
$412$	−4.48528	−0.220974
$413$	44.6274	2.19597
$414$	21.6569	1.06438
$415$	−14.0000	−0.687233
$416$	−7.92893	−0.388748
$417$	−2.41421	−0.118225
$418$	0	0
$419$	−7.79899	−0.381006	−0.190503	−	0.981687i	$-0.561012\pi$
$419$	−7.79899	−0.381006	−0.190503	+	0.981687i	$0.561012\pi$
$420$	−70.7696	−3.45320
$421$	35.3137	1.72108	0.860542	−	0.509379i	$-0.170125\pi$
$421$	35.3137	1.72108	0.860542	+	0.509379i	$0.170125\pi$
$422$	−58.1127	−2.82888
$423$	13.6569	0.664019
$424$	30.1421	1.46383
$425$	−4.41421	−0.214121
$426$	−83.4264	−4.04202
$427$	−41.7279	−2.01936
$428$	−9.62742	−0.465359
$429$	0	0
$430$	−11.6569	−0.562143
$431$	10.8284	0.521587	0.260793	−	0.965395i	$-0.416016\pi$
$431$	10.8284	0.521587	0.260793	+	0.965395i	$0.416016\pi$
$432$	−1.24264	−0.0597866
$433$	4.34315	0.208718	0.104359	−	0.994540i	$-0.466721\pi$
$433$	4.34315	0.208718	0.104359	+	0.994540i	$0.466721\pi$
$434$	−44.6274	−2.14218
$435$	4.82843	0.231505
$436$	−26.1421	−1.25198
$437$	−16.6274	−0.795397
$438$	−23.7279	−1.13376
$439$	33.4558	1.59676	0.798380	−	0.602154i	$-0.205691\pi$
$439$	33.4558	1.59676	0.798380	+	0.602154i	$0.205691\pi$
$440$	0	0
$441$	21.6569	1.03128
$442$	53.2843	2.53447
$443$	−5.02944	−0.238956	−0.119478	−	0.992837i	$-0.538122\pi$
$443$	−5.02944	−0.238956	−0.119478	+	0.992837i	$0.538122\pi$
$444$	78.4264	3.72195
$445$	−33.6569	−1.59549
$446$	26.9706	1.27709
$447$	11.6569	0.551350
$448$	37.6274	1.77773
$449$	−27.5269	−1.29908	−0.649538	−	0.760329i	$-0.725038\pi$
$449$	−27.5269	−1.29908	−0.649538	+	0.760329i	$0.725038\pi$
$450$	−6.82843	−0.321895
$451$	0	0
$452$	55.4558	2.60842
$453$	−28.9706	−1.36116
$454$	49.7990	2.33718
$455$	−38.2843	−1.79479
$456$	−55.8701	−2.61636
$457$	−10.3431	−0.483832	−0.241916	−	0.970297i	$-0.577776\pi$
$457$	−10.3431	−0.483832	−0.241916	+	0.970297i	$0.577776\pi$
$458$	−8.48528	−0.396491
$459$	−1.82843	−0.0853437
$460$	24.2843	1.13226
$461$	−20.8284	−0.970077	−0.485038	−	0.874493i	$-0.661194\pi$
$461$	−20.8284	−0.970077	−0.485038	+	0.874493i	$0.661194\pi$
$462$	0	0
$463$	35.8284	1.66509	0.832544	−	0.553959i	$-0.186883\pi$
$463$	35.8284	1.66509	0.832544	+	0.553959i	$0.186883\pi$
$464$	3.00000	0.139272
$465$	23.3137	1.08115
$466$	−38.1421	−1.76690
$467$	−24.3431	−1.12647	−0.563233	−	0.826298i	$-0.690443\pi$
$467$	−24.3431	−1.12647	−0.563233	+	0.826298i	$0.690443\pi$
$468$	54.1421	2.50272
$469$	47.1421	2.17682
$470$	23.3137	1.07538
$471$	−58.4558	−2.69350
$472$	−51.4558	−2.36845
$473$	0	0
$474$	−14.4853	−0.665331
$475$	5.24264	0.240549
$476$	−64.6985	−2.96545
$477$	19.3137	0.884314
$478$	24.5563	1.12318
$479$	25.3137	1.15661	0.578306	−	0.815820i	$-0.303714\pi$
$479$	25.3137	1.15661	0.578306	+	0.815820i	$0.303714\pi$
$480$	−7.65685	−0.349486
$481$	42.4264	1.93448
$482$	31.3137	1.42630
$483$	−29.3137	−1.33382
$484$	−42.1127	−1.91421
$485$	18.4853	0.839373
$486$	−52.2843	−2.37166
$487$	−9.85786	−0.446702	−0.223351	−	0.974738i	$-0.571700\pi$
$487$	−9.85786	−0.446702	−0.223351	+	0.974738i	$0.571700\pi$
$488$	48.1127	2.17796
$489$	−49.7990	−2.25199
$490$	36.9706	1.67016
$491$	22.2843	1.00568	0.502838	−	0.864381i	$-0.332289\pi$
$491$	22.2843	1.00568	0.502838	+	0.864381i	$0.332289\pi$
$492$	18.4853	0.833381
$493$	4.41421	0.198806
$494$	−63.2843	−2.84729
$495$	0	0
$496$	14.4853	0.650408
$497$	54.7990	2.45807
$498$	−40.7990	−1.82825
$499$	−34.6274	−1.55014	−0.775068	−	0.631878i	$-0.782284\pi$
$499$	−34.6274	−1.55014	−0.775068	+	0.631878i	$0.782284\pi$
$500$	−45.9411	−2.05455
$501$	32.9706	1.47302
$502$	−12.0000	−0.535586
$503$	7.31371	0.326102	0.163051	−	0.986618i	$-0.447866\pi$
$503$	7.31371	0.326102	0.163051	+	0.986618i	$0.447866\pi$
$504$	−47.7990	−2.12914
$505$	−36.2843	−1.61463
$506$	0	0
$507$	28.9706	1.28663
$508$	−40.1421	−1.78102
$509$	−12.4853	−0.553400	−0.276700	−	0.960956i	$-0.589241\pi$
$509$	−12.4853	−0.553400	−0.276700	+	0.960956i	$0.589241\pi$
$510$	51.4558	2.27850
$511$	15.5858	0.689475
$512$	−31.2426	−1.38074
$513$	2.17157	0.0958773
$514$	16.1421	0.711999
$515$	−2.34315	−0.103251
$516$	−22.3137	−0.982306
$517$	0	0
$518$	−78.4264	−3.44586
$519$	23.7279	1.04154
$520$	44.1421	1.93576
$521$	2.82843	0.123916	0.0619578	−	0.998079i	$-0.480266\pi$
$521$	2.82843	0.123916	0.0619578	+	0.998079i	$0.480266\pi$
$522$	6.82843	0.298872
$523$	27.4853	1.20185	0.600924	−	0.799306i	$-0.294800\pi$
$523$	27.4853	1.20185	0.600924	+	0.799306i	$0.294800\pi$
$524$	59.9411	2.61854
$525$	9.24264	0.403382
$526$	−44.6274	−1.94585
$527$	21.3137	0.928440
$528$	0	0
$529$	−12.9411	−0.562658
$530$	32.9706	1.43215
$531$	−32.9706	−1.43080
$532$	76.8406	3.33146
$533$	10.0000	0.433148
$534$	−98.0833	−4.24448
$535$	−5.02944	−0.217442
$536$	−54.3553	−2.34779
$537$	52.6274	2.27104
$538$	42.7990	1.84520
$539$	0	0
$540$	−3.17157	−0.136483
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	37.6274	1.61624
$543$	−29.3848	−1.26102
$544$	−7.00000	−0.300123
$545$	−13.6569	−0.584995
$546$	−111.569	−4.77469
$547$	−18.1421	−0.775702	−0.387851	−	0.921722i	$-0.626782\pi$
$547$	−18.1421	−0.775702	−0.387851	+	0.921722i	$0.626782\pi$
$548$	21.1127	0.901890
$549$	30.8284	1.31573
$550$	0	0
$551$	−5.24264	−0.223344
$552$	33.7990	1.43858
$553$	9.51472	0.404607
$554$	64.7696	2.75179
$555$	40.9706	1.73910
$556$	−3.82843	−0.162361
$557$	0.343146	0.0145396	0.00726978	−	0.999974i	$-0.497686\pi$
$557$	0.343146	0.0145396	0.00726978	+	0.999974i	$0.497686\pi$
$558$	32.9706	1.39576
$559$	−12.0711	−0.510552
$560$	−22.9706	−0.970683
$561$	0	0
$562$	5.65685	0.238620
$563$	45.1127	1.90127	0.950637	−	0.310306i	$-0.100431\pi$
$563$	45.1127	1.90127	0.950637	+	0.310306i	$0.100431\pi$
$564$	44.6274	1.87915
$565$	28.9706	1.21880
$566$	−14.4142	−0.605875
$567$	36.3137	1.52503
$568$	−63.1838	−2.65113
$569$	27.7990	1.16539	0.582697	−	0.812689i	$-0.301997\pi$
$569$	27.7990	1.16539	0.582697	+	0.812689i	$0.301997\pi$
$570$	−61.1127	−2.55973
$571$	24.6274	1.03063	0.515313	−	0.857002i	$-0.327676\pi$
$571$	24.6274	1.03063	0.515313	+	0.857002i	$0.327676\pi$
$572$	0	0
$573$	−0.828427	−0.0346080
$574$	−18.4853	−0.771561
$575$	−3.17157	−0.132264
$576$	−27.7990	−1.15829
$577$	−16.7574	−0.697618	−0.348809	−	0.937194i	$-0.613414\pi$
$577$	−16.7574	−0.697618	−0.348809	+	0.937194i	$0.613414\pi$
$578$	6.00000	0.249567
$579$	59.9411	2.49107
$580$	7.65685	0.317934
$581$	26.7990	1.11181
$582$	53.8701	2.23299
$583$	0	0
$584$	−17.9706	−0.743627
$585$	28.2843	1.16941
$586$	8.48528	0.350524
$587$	−8.65685	−0.357307	−0.178653	−	0.983912i	$-0.557174\pi$
$587$	−8.65685	−0.357307	−0.178653	+	0.983912i	$0.557174\pi$
$588$	70.7696	2.91849
$589$	−25.3137	−1.04303
$590$	−56.2843	−2.31719
$591$	−10.0000	−0.411345
$592$	25.4558	1.04623
$593$	36.9411	1.51699	0.758495	−	0.651679i	$-0.225935\pi$
$593$	36.9411	1.51699	0.758495	+	0.651679i	$0.225935\pi$
$594$	0	0
$595$	−33.7990	−1.38562
$596$	18.4853	0.757187
$597$	57.4558	2.35151
$598$	38.2843	1.56556
$599$	22.2843	0.910511	0.455255	−	0.890361i	$-0.349548\pi$
$599$	22.2843	0.910511	0.455255	+	0.890361i	$0.349548\pi$
$600$	−10.6569	−0.435064
$601$	27.5147	1.12235	0.561174	−	0.827698i	$-0.310350\pi$
$601$	27.5147	1.12235	0.561174	+	0.827698i	$0.310350\pi$
$602$	22.3137	0.909439
$603$	−34.8284	−1.41832
$604$	−45.9411	−1.86932
$605$	−22.0000	−0.894427
$606$	−105.740	−4.29540
$607$	−30.8284	−1.25129	−0.625644	−	0.780109i	$-0.715164\pi$
$607$	−30.8284	−1.25129	−0.625644	+	0.780109i	$0.715164\pi$
$608$	8.31371	0.337165
$609$	−9.24264	−0.374531
$610$	52.6274	2.13082
$611$	24.1421	0.976686
$612$	47.7990	1.93216
$613$	14.7990	0.597726	0.298863	−	0.954296i	$-0.403393\pi$
$613$	14.7990	0.597726	0.298863	+	0.954296i	$0.403393\pi$
$614$	11.3137	0.456584
$615$	9.65685	0.389402
$616$	0	0
$617$	19.3848	0.780402	0.390201	−	0.920730i	$-0.372406\pi$
$617$	19.3848	0.780402	0.390201	+	0.920730i	$0.372406\pi$
$618$	−6.82843	−0.274680
$619$	−9.65685	−0.388142	−0.194071	−	0.980988i	$-0.562169\pi$
$619$	−9.65685	−0.388142	−0.194071	+	0.980988i	$0.562169\pi$
$620$	36.9706	1.48477
$621$	−1.31371	−0.0527173
$622$	−18.4853	−0.741192
$623$	64.4264	2.58119
$624$	36.2132	1.44969
$625$	−19.0000	−0.760000
$626$	2.07107	0.0827765
$627$	0	0
$628$	−92.6985	−3.69907
$629$	37.4558	1.49346
$630$	−52.2843	−2.08306
$631$	6.48528	0.258175	0.129087	−	0.991633i	$-0.458795\pi$
$631$	6.48528	0.258175	0.129087	+	0.991633i	$0.458795\pi$
$632$	−10.9706	−0.436386
$633$	−58.1127	−2.30977
$634$	12.4853	0.495854
$635$	−20.9706	−0.832191
$636$	63.1127	2.50258
$637$	38.2843	1.51688
$638$	0	0
$639$	−40.4853	−1.60157
$640$	−41.1127	−1.62512
$641$	6.82843	0.269707	0.134853	−	0.990866i	$-0.456944\pi$
$641$	6.82843	0.269707	0.134853	+	0.990866i	$0.456944\pi$
$642$	−14.6569	−0.578460
$643$	24.3431	0.960000	0.480000	−	0.877269i	$-0.340637\pi$
$643$	24.3431	0.960000	0.480000	+	0.877269i	$0.340637\pi$
$644$	−46.4853	−1.83178
$645$	−11.6569	−0.458988
$646$	−55.8701	−2.19818
$647$	−32.6569	−1.28387	−0.641937	−	0.766758i	$-0.721869\pi$
$647$	−32.6569	−1.28387	−0.641937	+	0.766758i	$0.721869\pi$
$648$	−41.8701	−1.64481
$649$	0	0
$650$	−12.0711	−0.473466
$651$	−44.6274	−1.74909
$652$	−78.9706	−3.09273
$653$	2.27208	0.0889133	0.0444566	−	0.999011i	$-0.485844\pi$
$653$	2.27208	0.0889133	0.0444566	+	0.999011i	$0.485844\pi$
$654$	−39.7990	−1.55626
$655$	31.3137	1.22353
$656$	6.00000	0.234261
$657$	−11.5147	−0.449232
$658$	−44.6274	−1.73976
$659$	−7.58579	−0.295500	−0.147750	−	0.989025i	$-0.547203\pi$
$659$	−7.58579	−0.295500	−0.147750	+	0.989025i	$0.547203\pi$
$660$	0	0
$661$	−45.3137	−1.76250	−0.881249	−	0.472651i	$-0.843297\pi$
$661$	−45.3137	−1.76250	−0.881249	+	0.472651i	$0.843297\pi$
$662$	−84.4264	−3.28133
$663$	53.2843	2.06939
$664$	−30.8995	−1.19913
$665$	40.1421	1.55665
$666$	57.9411	2.24517
$667$	3.17157	0.122804
$668$	52.2843	2.02294
$669$	26.9706	1.04274
$670$	−59.4558	−2.29698
$671$	0	0
$672$	14.6569	0.565400
$673$	14.5147	0.559501	0.279751	−	0.960073i	$-0.409748\pi$
$673$	14.5147	0.559501	0.279751	+	0.960073i	$0.409748\pi$
$674$	46.4558	1.78941
$675$	0.414214	0.0159431
$676$	45.9411	1.76697
$677$	21.9411	0.843266	0.421633	−	0.906767i	$-0.361457\pi$
$677$	21.9411	0.843266	0.421633	+	0.906767i	$0.361457\pi$
$678$	84.4264	3.24238
$679$	−35.3848	−1.35794
$680$	38.9706	1.49445
$681$	49.7990	1.90830
$682$	0	0
$683$	13.1421	0.502870	0.251435	−	0.967874i	$-0.419098\pi$
$683$	13.1421	0.502870	0.251435	+	0.967874i	$0.419098\pi$
$684$	−56.7696	−2.17064
$685$	11.0294	0.421413
$686$	−6.07107	−0.231794
$687$	−8.48528	−0.323734
$688$	−7.24264	−0.276123
$689$	34.1421	1.30071
$690$	36.9706	1.40745
$691$	1.51472	0.0576226	0.0288113	−	0.999585i	$-0.490828\pi$
$691$	1.51472	0.0576226	0.0288113	+	0.999585i	$0.490828\pi$
$692$	37.6274	1.43038
$693$	0	0
$694$	50.6274	1.92179
$695$	−2.00000	−0.0758643
$696$	10.6569	0.403947
$697$	8.82843	0.334401
$698$	−67.5269	−2.55593
$699$	−38.1421	−1.44267
$700$	14.6569	0.553977
$701$	−29.6863	−1.12124	−0.560618	−	0.828075i	$-0.689436\pi$
$701$	−29.6863	−1.12124	−0.560618	+	0.828075i	$0.689436\pi$
$702$	−5.00000	−0.188713
$703$	−44.4853	−1.67779
$704$	0	0
$705$	23.3137	0.878045
$706$	85.5980	3.22152
$707$	69.4558	2.61216
$708$	−107.740	−4.04912
$709$	−26.8284	−1.00756	−0.503781	−	0.863831i	$-0.668058\pi$
$709$	−26.8284	−1.00756	−0.503781	+	0.863831i	$0.668058\pi$
$710$	−69.1127	−2.59375
$711$	−7.02944	−0.263624
$712$	−74.2843	−2.78392
$713$	15.3137	0.573503
$714$	−98.4975	−3.68618
$715$	0	0
$716$	83.4558	3.11889
$717$	24.5563	0.917074
$718$	−20.4853	−0.764504
$719$	9.68629	0.361238	0.180619	−	0.983553i	$-0.442190\pi$
$719$	9.68629	0.361238	0.180619	+	0.983553i	$0.442190\pi$
$720$	16.9706	0.632456
$721$	4.48528	0.167041
$722$	20.4853	0.762383
$723$	31.3137	1.16457
$724$	−46.5980	−1.73180
$725$	−1.00000	−0.0371391
$726$	−64.1127	−2.37945
$727$	−25.8701	−0.959467	−0.479734	−	0.877414i	$-0.659267\pi$
$727$	−25.8701	−0.959467	−0.479734	+	0.877414i	$0.659267\pi$
$728$	−84.4975	−3.13168
$729$	−23.8284	−0.882534
$730$	−19.6569	−0.727533
$731$	−10.6569	−0.394158
$732$	100.740	3.72346
$733$	−26.9706	−0.996180	−0.498090	−	0.867125i	$-0.665965\pi$
$733$	−26.9706	−0.996180	−0.498090	+	0.867125i	$0.665965\pi$
$734$	43.7990	1.61665
$735$	36.9706	1.36368
$736$	−5.02944	−0.185388
$737$	0	0
$738$	13.6569	0.502716
$739$	−49.1127	−1.80664	−0.903320	−	0.428968i	$-0.858877\pi$
$739$	−49.1127	−1.80664	−0.903320	+	0.428968i	$0.858877\pi$
$740$	64.9706	2.38837
$741$	−63.2843	−2.32481
$742$	−63.1127	−2.31694
$743$	18.4142	0.675552	0.337776	−	0.941227i	$-0.390325\pi$
$743$	18.4142	0.675552	0.337776	+	0.941227i	$0.390325\pi$
$744$	51.4558	1.88646
$745$	9.65685	0.353800
$746$	48.2843	1.76781
$747$	−19.7990	−0.724407
$748$	0	0
$749$	9.62742	0.351778
$750$	−69.9411	−2.55389
$751$	−10.0711	−0.367498	−0.183749	−	0.982973i	$-0.558823\pi$
$751$	−10.0711	−0.367498	−0.183749	+	0.982973i	$0.558823\pi$
$752$	14.4853	0.528224
$753$	−12.0000	−0.437304
$754$	12.0711	0.439602
$755$	−24.0000	−0.873449
$756$	6.07107	0.220803
$757$	−40.7696	−1.48179	−0.740897	−	0.671618i	$-0.765599\pi$
$757$	−40.7696	−1.48179	−0.740897	+	0.671618i	$0.765599\pi$
$758$	−19.9706	−0.725364
$759$	0	0
$760$	−46.2843	−1.67891
$761$	−30.5147	−1.10616	−0.553079	−	0.833129i	$-0.686547\pi$
$761$	−30.5147	−1.10616	−0.553079	+	0.833129i	$0.686547\pi$
$762$	−61.1127	−2.21388
$763$	26.1421	0.946409
$764$	−1.31371	−0.0475283
$765$	24.9706	0.902813
$766$	57.4558	2.07596
$767$	−58.2843	−2.10452
$768$	−72.3553	−2.61090
$769$	−21.1716	−0.763466	−0.381733	−	0.924273i	$-0.624673\pi$
$769$	−21.1716	−0.763466	−0.381733	+	0.924273i	$0.624673\pi$
$770$	0	0
$771$	16.1421	0.581345
$772$	95.0538	3.42106
$773$	42.8284	1.54043	0.770216	−	0.637783i	$-0.220149\pi$
$773$	42.8284	1.54043	0.770216	+	0.637783i	$0.220149\pi$
$774$	−16.4853	−0.592551
$775$	−4.82843	−0.173442
$776$	40.7990	1.46460
$777$	−78.4264	−2.81353
$778$	−45.9706	−1.64812
$779$	−10.4853	−0.375674
$780$	92.4264	3.30940
$781$	0	0
$782$	33.7990	1.20865
$783$	−0.414214	−0.0148028
$784$	22.9706	0.820377
$785$	−48.4264	−1.72841
$786$	91.2548	3.25495
$787$	0.343146	0.0122318	0.00611591	−	0.999981i	$-0.498053\pi$
$787$	0.343146	0.0122318	0.00611591	+	0.999981i	$0.498053\pi$
$788$	−15.8579	−0.564913
$789$	−44.6274	−1.58878
$790$	−12.0000	−0.426941
$791$	−55.4558	−1.97178
$792$	0	0
$793$	54.4975	1.93526
$794$	−9.65685	−0.342709
$795$	32.9706	1.16935
$796$	91.1127	3.22940
$797$	−31.0416	−1.09955	−0.549775	−	0.835312i	$-0.685287\pi$
$797$	−31.0416	−1.09955	−0.549775	+	0.835312i	$0.685287\pi$
$798$	116.983	4.14114
$799$	21.3137	0.754025
$800$	1.58579	0.0560660
$801$	−47.5980	−1.68179
$802$	55.4558	1.95821
$803$	0	0
$804$	−113.811	−4.01381
$805$	−24.2843	−0.855908
$806$	58.2843	2.05298
$807$	42.7990	1.50660
$808$	−80.0833	−2.81732
$809$	11.3848	0.400267	0.200134	−	0.979769i	$-0.435862\pi$
$809$	11.3848	0.400267	0.200134	+	0.979769i	$0.435862\pi$
$810$	−45.7990	−1.60921
$811$	−14.3431	−0.503656	−0.251828	−	0.967772i	$-0.581032\pi$
$811$	−14.3431	−0.503656	−0.251828	+	0.967772i	$0.581032\pi$
$812$	−14.6569	−0.514355
$813$	37.6274	1.31965
$814$	0	0
$815$	−41.2548	−1.44509
$816$	31.9706	1.11919
$817$	12.6569	0.442807
$818$	19.6569	0.687286
$819$	−54.1421	−1.89188
$820$	15.3137	0.534778
$821$	−54.4264	−1.89949	−0.949747	−	0.313018i	$-0.898660\pi$
$821$	−54.4264	−1.89949	−0.949747	+	0.313018i	$0.898660\pi$
$822$	32.1421	1.12109
$823$	0.414214	0.0144386	0.00721929	−	0.999974i	$-0.497702\pi$
$823$	0.414214	0.0144386	0.00721929	+	0.999974i	$0.497702\pi$
$824$	−5.17157	−0.180160
$825$	0	0
$826$	107.740	3.74876
$827$	−33.2426	−1.15596	−0.577980	−	0.816051i	$-0.696159\pi$
$827$	−33.2426	−1.15596	−0.577980	+	0.816051i	$0.696159\pi$
$828$	34.3431	1.19351
$829$	−32.7696	−1.13813	−0.569067	−	0.822291i	$-0.692695\pi$
$829$	−32.7696	−1.13813	−0.569067	+	0.822291i	$0.692695\pi$
$830$	−33.7990	−1.17318
$831$	64.7696	2.24683
$832$	−49.1421	−1.70370
$833$	33.7990	1.17107
$834$	−5.82843	−0.201822
$835$	27.3137	0.945230
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	−18.8284	−0.650417
$839$	22.0000	0.759524	0.379762	−	0.925084i	$-0.376006\pi$
$839$	22.0000	0.759524	0.379762	+	0.925084i	$0.376006\pi$
$840$	−81.5980	−2.81540
$841$	1.00000	0.0344828
$842$	85.2548	2.93808
$843$	5.65685	0.194832
$844$	−92.1543	−3.17208
$845$	24.0000	0.825625
$846$	32.9706	1.13355
$847$	42.1127	1.44701
$848$	20.4853	0.703467
$849$	−14.4142	−0.494695
$850$	−10.6569	−0.365527
$851$	26.9117	0.922521
$852$	−132.296	−4.53240
$853$	31.1005	1.06486	0.532431	−	0.846474i	$-0.321279\pi$
$853$	31.1005	1.06486	0.532431	+	0.846474i	$0.321279\pi$
$854$	−100.740	−3.44726
$855$	−29.6569	−1.01424
$856$	−11.1005	−0.379407
$857$	19.9411	0.681176	0.340588	−	0.940213i	$-0.389374\pi$
$857$	19.9411	0.681176	0.340588	+	0.940213i	$0.389374\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	−18.4853	−0.630343
$861$	−18.4853	−0.629977
$862$	26.1421	0.890405
$863$	15.0294	0.511608	0.255804	−	0.966729i	$-0.417660\pi$
$863$	15.0294	0.511608	0.255804	+	0.966729i	$0.417660\pi$
$864$	0.656854	0.0223466
$865$	19.6569	0.668353
$866$	10.4853	0.356304
$867$	6.00000	0.203771
$868$	−70.7696	−2.40208
$869$	0	0
$870$	11.6569	0.395204
$871$	−61.5685	−2.08617
$872$	−30.1421	−1.02074
$873$	26.1421	0.884777
$874$	−40.1421	−1.35783
$875$	45.9411	1.55309
$876$	−37.6274	−1.27131
$877$	−29.3137	−0.989854	−0.494927	−	0.868935i	$-0.664805\pi$
$877$	−29.3137	−0.989854	−0.494927	+	0.868935i	$0.664805\pi$
$878$	80.7696	2.72584
$879$	8.48528	0.286201
$880$	0	0
$881$	−22.6274	−0.762337	−0.381169	−	0.924506i	$-0.624478\pi$
$881$	−22.6274	−0.762337	−0.381169	+	0.924506i	$0.624478\pi$
$882$	52.2843	1.76050
$883$	49.4264	1.66333	0.831666	−	0.555277i	$-0.187388\pi$
$883$	49.4264	1.66333	0.831666	+	0.555277i	$0.187388\pi$
$884$	84.4975	2.84196
$885$	−56.2843	−1.89198
$886$	−12.1421	−0.407923
$887$	35.5858	1.19485	0.597427	−	0.801923i	$-0.296190\pi$
$887$	35.5858	1.19485	0.597427	+	0.801923i	$0.296190\pi$
$888$	90.4264	3.03451
$889$	40.1421	1.34632
$890$	−81.2548	−2.72367
$891$	0	0
$892$	42.7696	1.43203
$893$	−25.3137	−0.847091
$894$	28.1421	0.941214
$895$	43.5980	1.45732
$896$	78.6985	2.62913
$897$	38.2843	1.27827
$898$	−66.4558	−2.21766
$899$	4.82843	0.161037
$900$	−10.8284	−0.360948
$901$	30.1421	1.00418
$902$	0	0
$903$	22.3137	0.742554
$904$	63.9411	2.12665
$905$	−24.3431	−0.809194
$906$	−69.9411	−2.32364
$907$	−1.31371	−0.0436210	−0.0218105	−	0.999762i	$-0.506943\pi$
$907$	−1.31371	−0.0436210	−0.0218105	+	0.999762i	$0.506943\pi$
$908$	78.9706	2.62073
$909$	−51.3137	−1.70197
$910$	−92.4264	−3.06391
$911$	−24.8284	−0.822602	−0.411301	−	0.911499i	$-0.634926\pi$
$911$	−24.8284	−0.822602	−0.411301	+	0.911499i	$0.634926\pi$
$912$	−37.9706	−1.25733
$913$	0	0
$914$	−24.9706	−0.825953
$915$	52.6274	1.73981
$916$	−13.4558	−0.444594
$917$	−59.9411	−1.97943
$918$	−4.41421	−0.145691
$919$	2.34315	0.0772932	0.0386466	−	0.999253i	$-0.487695\pi$
$919$	2.34315	0.0772932	0.0386466	+	0.999253i	$0.487695\pi$
$920$	28.0000	0.923133
$921$	11.3137	0.372799
$922$	−50.2843	−1.65602
$923$	−71.5685	−2.35571
$924$	0	0
$925$	−8.48528	−0.278994
$926$	86.4975	2.84248
$927$	−3.31371	−0.108836
$928$	−1.58579	−0.0520560
$929$	6.97056	0.228697	0.114348	−	0.993441i	$-0.463522\pi$
$929$	6.97056	0.228697	0.114348	+	0.993441i	$0.463522\pi$
$930$	56.2843	1.84563
$931$	−40.1421	−1.31561
$932$	−60.4853	−1.98126
$933$	−18.4853	−0.605181
$934$	−58.7696	−1.92300
$935$	0	0
$936$	62.4264	2.04047
$937$	19.1716	0.626308	0.313154	−	0.949702i	$-0.398614\pi$
$937$	19.1716	0.626308	0.313154	+	0.949702i	$0.398614\pi$
$938$	113.811	3.71607
$939$	2.07107	0.0675867
$940$	36.9706	1.20585
$941$	−50.9706	−1.66159	−0.830796	−	0.556576i	$-0.812115\pi$
$941$	−50.9706	−1.66159	−0.830796	+	0.556576i	$0.812115\pi$
$942$	−141.125	−4.59810
$943$	6.34315	0.206561
$944$	−34.9706	−1.13819
$945$	3.17157	0.103171
$946$	0	0
$947$	−30.8284	−1.00179	−0.500895	−	0.865508i	$-0.666996\pi$
$947$	−30.8284	−1.00179	−0.500895	+	0.865508i	$0.666996\pi$
$948$	−22.9706	−0.746049
$949$	−20.3553	−0.660762
$950$	12.6569	0.410643
$951$	12.4853	0.404863
$952$	−74.5980	−2.41773
$953$	−11.6569	−0.377603	−0.188801	−	0.982015i	$-0.560460\pi$
$953$	−11.6569	−0.377603	−0.188801	+	0.982015i	$0.560460\pi$
$954$	46.6274	1.50962
$955$	−0.686292	−0.0222079
$956$	38.9411	1.25945
$957$	0	0
$958$	61.1127	1.97446
$959$	−21.1127	−0.681765
$960$	−47.4558	−1.53163
$961$	−7.68629	−0.247945
$962$	102.426	3.30236
$963$	−7.11270	−0.229204
$964$	49.6569	1.59934
$965$	49.6569	1.59851
$966$	−70.7696	−2.27697
$967$	−22.2843	−0.716614	−0.358307	−	0.933604i	$-0.616646\pi$
$967$	−22.2843	−0.716614	−0.358307	+	0.933604i	$0.616646\pi$
$968$	−48.5563	−1.56066
$969$	−55.8701	−1.79480
$970$	44.6274	1.43290
$971$	39.5980	1.27076	0.635380	−	0.772200i	$-0.280844\pi$
$971$	39.5980	1.27076	0.635380	+	0.772200i	$0.280844\pi$
$972$	−82.9117	−2.65939
$973$	3.82843	0.122734
$974$	−23.7990	−0.762569
$975$	−12.0711	−0.386584
$976$	32.6985	1.04665
$977$	47.9706	1.53471	0.767357	−	0.641220i	$-0.221571\pi$
$977$	47.9706	1.53471	0.767357	+	0.641220i	$0.221571\pi$
$978$	−120.225	−3.84438
$979$	0	0
$980$	58.6274	1.87278
$981$	−19.3137	−0.616639
$982$	53.7990	1.71679
$983$	−5.78680	−0.184570	−0.0922851	−	0.995733i	$-0.529417\pi$
$983$	−5.78680	−0.184570	−0.0922851	+	0.995733i	$0.529417\pi$
$984$	21.3137	0.679456
$985$	−8.28427	−0.263959
$986$	10.6569	0.339383
$987$	−44.6274	−1.42051
$988$	−100.355	−3.19273
$989$	−7.65685	−0.243474
$990$	0	0
$991$	−14.9706	−0.475556	−0.237778	−	0.971320i	$-0.576419\pi$
$991$	−14.9706	−0.475556	−0.237778	+	0.971320i	$0.576419\pi$
$992$	−7.65685	−0.243105
$993$	−84.4264	−2.67919
$994$	132.296	4.19619
$995$	47.5980	1.50896
$996$	−64.6985	−2.05005
$997$	−17.7990	−0.563700	−0.281850	−	0.959459i	$-0.590948\pi$
$997$	−17.7990	−0.563700	−0.281850	+	0.959459i	$0.590948\pi$
$998$	−83.5980	−2.64625
$999$	−3.51472	−0.111201

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.a.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.a.1.2	✓	2	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:	\( N \)	\(=\)	\( 4031 = 29 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4031.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{2} +2.41421 q^{3} +3.82843 q^{4} +2.00000 q^{5} +5.82843 q^{6} -3.82843 q^{7} +4.41421 q^{8} +2.82843 q^{9} +O(q^{10})\)	\(q+2.41421 q^{2} +2.41421 q^{3} +3.82843 q^{4} +2.00000 q^{5} +5.82843 q^{6} -3.82843 q^{7} +4.41421 q^{8} +2.82843 q^{9} +4.82843 q^{10} +9.24264 q^{12} +5.00000 q^{13} -9.24264 q^{14} +4.82843 q^{15} +3.00000 q^{16} +4.41421 q^{17} +6.82843 q^{18} -5.24264 q^{19} +7.65685 q^{20} -9.24264 q^{21} +3.17157 q^{23} +10.6569 q^{24} -1.00000 q^{25} +12.0711 q^{26} -0.414214 q^{27} -14.6569 q^{28} +1.00000 q^{29} +11.6569 q^{30} +4.82843 q^{31} -1.58579 q^{32} +10.6569 q^{34} -7.65685 q^{35} +10.8284 q^{36} +8.48528 q^{37} -12.6569 q^{38} +12.0711 q^{39} +8.82843 q^{40} +2.00000 q^{41} -22.3137 q^{42} -2.41421 q^{43} +5.65685 q^{45} +7.65685 q^{46} +4.82843 q^{47} +7.24264 q^{48} +7.65685 q^{49} -2.41421 q^{50} +10.6569 q^{51} +19.1421 q^{52} +6.82843 q^{53} -1.00000 q^{54} -16.8995 q^{56} -12.6569 q^{57} +2.41421 q^{58} -11.6569 q^{59} +18.4853 q^{60} +10.8995 q^{61} +11.6569 q^{62} -10.8284 q^{63} -9.82843 q^{64} +10.0000 q^{65} -12.3137 q^{67} +16.8995 q^{68} +7.65685 q^{69} -18.4853 q^{70} -14.3137 q^{71} +12.4853 q^{72} -4.07107 q^{73} +20.4853 q^{74} -2.41421 q^{75} -20.0711 q^{76} +29.1421 q^{78} -2.48528 q^{79} +6.00000 q^{80} -9.48528 q^{81} +4.82843 q^{82} -7.00000 q^{83} -35.3848 q^{84} +8.82843 q^{85} -5.82843 q^{86} +2.41421 q^{87} -16.8284 q^{89} +13.6569 q^{90} -19.1421 q^{91} +12.1421 q^{92} +11.6569 q^{93} +11.6569 q^{94} -10.4853 q^{95} -3.82843 q^{96} +9.24264 q^{97} +18.4853 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 6 q^{6} - 2 q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 4 * q^5 + 6 * q^6 - 2 * q^7 + 6 * q^8	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 6 q^{6} - 2 q^{7} + 6 q^{8} + 4 q^{10} + 10 q^{12} + 10 q^{13} - 10 q^{14} + 4 q^{15} + 6 q^{16} + 6 q^{17} + 8 q^{18} - 2 q^{19} + 4 q^{20} - 10 q^{21} + 12 q^{23} + 10 q^{24} - 2 q^{25} + 10 q^{26} + 2 q^{27} - 18 q^{28} + 2 q^{29} + 12 q^{30} + 4 q^{31} - 6 q^{32} + 10 q^{34} - 4 q^{35} + 16 q^{36} - 14 q^{38} + 10 q^{39} + 12 q^{40} + 4 q^{41} - 22 q^{42} - 2 q^{43} + 4 q^{46} + 4 q^{47} + 6 q^{48} + 4 q^{49} - 2 q^{50} + 10 q^{51} + 10 q^{52} + 8 q^{53} - 2 q^{54} - 14 q^{56} - 14 q^{57} + 2 q^{58} - 12 q^{59} + 20 q^{60} + 2 q^{61} + 12 q^{62} - 16 q^{63} - 14 q^{64} + 20 q^{65} - 2 q^{67} + 14 q^{68} + 4 q^{69} - 20 q^{70} - 6 q^{71} + 8 q^{72} + 6 q^{73} + 24 q^{74} - 2 q^{75} - 26 q^{76} + 30 q^{78} + 12 q^{79} + 12 q^{80} - 2 q^{81} + 4 q^{82} - 14 q^{83} - 34 q^{84} + 12 q^{85} - 6 q^{86} + 2 q^{87} - 28 q^{89} + 16 q^{90} - 10 q^{91} - 4 q^{92} + 12 q^{93} + 12 q^{94} - 4 q^{95} - 2 q^{96} + 10 q^{97} + 20 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 4 * q^5 + 6 * q^6 - 2 * q^7 + 6 * q^8 + 4 * q^10 + 10 * q^12 + 10 * q^13 - 10 * q^14 + 4 * q^15 + 6 * q^16 + 6 * q^17 + 8 * q^18 - 2 * q^19 + 4 * q^20 - 10 * q^21 + 12 * q^23 + 10 * q^24 - 2 * q^25 + 10 * q^26 + 2 * q^27 - 18 * q^28 + 2 * q^29 + 12 * q^30 + 4 * q^31 - 6 * q^32 + 10 * q^34 - 4 * q^35 + 16 * q^36 - 14 * q^38 + 10 * q^39 + 12 * q^40 + 4 * q^41 - 22 * q^42 - 2 * q^43 + 4 * q^46 + 4 * q^47 + 6 * q^48 + 4 * q^49 - 2 * q^50 + 10 * q^51 + 10 * q^52 + 8 * q^53 - 2 * q^54 - 14 * q^56 - 14 * q^57 + 2 * q^58 - 12 * q^59 + 20 * q^60 + 2 * q^61 + 12 * q^62 - 16 * q^63 - 14 * q^64 + 20 * q^65 - 2 * q^67 + 14 * q^68 + 4 * q^69 - 20 * q^70 - 6 * q^71 + 8 * q^72 + 6 * q^73 + 24 * q^74 - 2 * q^75 - 26 * q^76 + 30 * q^78 + 12 * q^79 + 12 * q^80 - 2 * q^81 + 4 * q^82 - 14 * q^83 - 34 * q^84 + 12 * q^85 - 6 * q^86 + 2 * q^87 - 28 * q^89 + 16 * q^90 - 10 * q^91 - 4 * q^92 + 12 * q^93 + 12 * q^94 - 4 * q^95 - 2 * q^96 + 10 * q^97 + 20 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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