LMFDB - Embedded newform 1002.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1002,2,Mod(1,1002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1002, 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("1002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1002 = 2 \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1002.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:	$8.00101028253$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.2777.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + x + 2 $ x^4 - x^3 - 4*x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.36234$ of defining polynomial
Character	$\chi$	$=$	1002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.87806 q^{5} -1.00000 q^{6} -3.28324 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.87806 q^{5} -1.00000 q^{6} -3.28324 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.87806 q^{10} -1.00000 q^{12} +4.43662 q^{13} -3.28324 q^{14} +1.87806 q^{15} +1.00000 q^{16} +6.72468 q^{17} +1.00000 q^{18} +1.69324 q^{19} -1.87806 q^{20} +3.28324 q^{21} -1.69324 q^{23} -1.00000 q^{24} -1.47289 q^{25} +4.43662 q^{26} -1.00000 q^{27} -3.28324 q^{28} +9.75612 q^{29} +1.87806 q^{30} +1.59000 q^{31} +1.00000 q^{32} +6.72468 q^{34} +6.16612 q^{35} +1.00000 q^{36} +4.53985 q^{37} +1.69324 q^{38} -4.43662 q^{39} -1.87806 q^{40} +3.33821 q^{41} +3.28324 q^{42} +7.16130 q^{43} -1.87806 q^{45} -1.69324 q^{46} +4.47289 q^{47} -1.00000 q^{48} +3.77965 q^{49} -1.47289 q^{50} -6.72468 q^{51} +4.43662 q^{52} -0.184825 q^{53} -1.00000 q^{54} -3.28324 q^{56} -1.69324 q^{57} +9.75612 q^{58} -13.4760 q^{59} +1.87806 q^{60} -5.75612 q^{61} +1.59000 q^{62} -3.28324 q^{63} +1.00000 q^{64} -8.33224 q^{65} -8.45418 q^{67} +6.72468 q^{68} +1.69324 q^{69} +6.16612 q^{70} +11.1426 q^{71} +1.00000 q^{72} -10.6294 q^{73} +4.53985 q^{74} +1.47289 q^{75} +1.69324 q^{76} -4.43662 q^{78} -10.7876 q^{79} -1.87806 q^{80} +1.00000 q^{81} +3.33821 q^{82} +14.6656 q^{83} +3.28324 q^{84} -12.6294 q^{85} +7.16130 q^{86} -9.75612 q^{87} +8.16612 q^{89} -1.87806 q^{90} -14.5665 q^{91} -1.69324 q^{92} -1.59000 q^{93} +4.47289 q^{94} -3.18000 q^{95} -1.00000 q^{96} +10.2290 q^{97} +3.77965 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 5 q^{5} - 4 q^{6} + q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 5 * q^5 - 4 * q^6 + q^7 + 4 * q^8 + 4 * q^9	$ 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 5 q^{5} - 4 q^{6} + q^{7} + 4 q^{8} + 4 q^{9} + 5 q^{10} - 4 q^{12} + 8 q^{13} + q^{14} - 5 q^{15} + 4 q^{16} + 10 q^{17} + 4 q^{18} - 2 q^{19} + 5 q^{20} - q^{21} + 2 q^{23} - 4 q^{24} + 5 q^{25} + 8 q^{26} - 4 q^{27} + q^{28} + 14 q^{29} - 5 q^{30} + q^{31} + 4 q^{32} + 10 q^{34} + 5 q^{35} + 4 q^{36} + 5 q^{37} - 2 q^{38} - 8 q^{39} + 5 q^{40} + 14 q^{41} - q^{42} + 2 q^{43} + 5 q^{45} + 2 q^{46} + 7 q^{47} - 4 q^{48} + 13 q^{49} + 5 q^{50} - 10 q^{51} + 8 q^{52} + 3 q^{53} - 4 q^{54} + q^{56} + 2 q^{57} + 14 q^{58} - 5 q^{59} - 5 q^{60} + 2 q^{61} + q^{62} + q^{63} + 4 q^{64} + 6 q^{65} - 7 q^{67} + 10 q^{68} - 2 q^{69} + 5 q^{70} + 2 q^{71} + 4 q^{72} + 2 q^{73} + 5 q^{74} - 5 q^{75} - 2 q^{76} - 8 q^{78} - 10 q^{79} + 5 q^{80} + 4 q^{81} + 14 q^{82} + 13 q^{83} - q^{84} - 6 q^{85} + 2 q^{86} - 14 q^{87} + 13 q^{89} + 5 q^{90} - 30 q^{91} + 2 q^{92} - q^{93} + 7 q^{94} - 2 q^{95} - 4 q^{96} + 5 q^{97} + 13 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 5 * q^5 - 4 * q^6 + q^7 + 4 * q^8 + 4 * q^9 + 5 * q^10 - 4 * q^12 + 8 * q^13 + q^14 - 5 * q^15 + 4 * q^16 + 10 * q^17 + 4 * q^18 - 2 * q^19 + 5 * q^20 - q^21 + 2 * q^23 - 4 * q^24 + 5 * q^25 + 8 * q^26 - 4 * q^27 + q^28 + 14 * q^29 - 5 * q^30 + q^31 + 4 * q^32 + 10 * q^34 + 5 * q^35 + 4 * q^36 + 5 * q^37 - 2 * q^38 - 8 * q^39 + 5 * q^40 + 14 * q^41 - q^42 + 2 * q^43 + 5 * q^45 + 2 * q^46 + 7 * q^47 - 4 * q^48 + 13 * q^49 + 5 * q^50 - 10 * q^51 + 8 * q^52 + 3 * q^53 - 4 * q^54 + q^56 + 2 * q^57 + 14 * q^58 - 5 * q^59 - 5 * q^60 + 2 * q^61 + q^62 + q^63 + 4 * q^64 + 6 * q^65 - 7 * q^67 + 10 * q^68 - 2 * q^69 + 5 * q^70 + 2 * q^71 + 4 * q^72 + 2 * q^73 + 5 * q^74 - 5 * q^75 - 2 * q^76 - 8 * q^78 - 10 * q^79 + 5 * q^80 + 4 * q^81 + 14 * q^82 + 13 * q^83 - q^84 - 6 * q^85 + 2 * q^86 - 14 * q^87 + 13 * q^89 + 5 * q^90 - 30 * q^91 + 2 * q^92 - q^93 + 7 * q^94 - 2 * q^95 - 4 * q^96 + 5 * q^97 + 13 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.87806	−0.839895	−0.419947	−	0.907548i	$-0.637951\pi$
$5$	−1.87806	−0.839895	−0.419947	+	0.907548i	$0.637951\pi$
$6$	−1.00000	−0.408248
$7$	−3.28324	−1.24095	−0.620474	−	0.784227i	$-0.713060\pi$
$7$	−3.28324	−1.24095	−0.620474	+	0.784227i	$0.713060\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.87806	−0.593895
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−1.00000	−0.288675
$13$	4.43662	1.23050	0.615248	−	0.788333i	$-0.289056\pi$
$13$	4.43662	1.23050	0.615248	+	0.788333i	$0.289056\pi$
$14$	−3.28324	−0.877482
$15$	1.87806	0.484913
$16$	1.00000	0.250000
$17$	6.72468	1.63097	0.815487	−	0.578775i	$-0.196469\pi$
$17$	6.72468	1.63097	0.815487	+	0.578775i	$0.196469\pi$
$18$	1.00000	0.235702
$19$	1.69324	0.388455	0.194228	−	0.980957i	$-0.437780\pi$
$19$	1.69324	0.388455	0.194228	+	0.980957i	$0.437780\pi$
$20$	−1.87806	−0.419947
$21$	3.28324	0.716461
$22$	0	0
$23$	−1.69324	−0.353064	−0.176532	−	0.984295i	$-0.556488\pi$
$23$	−1.69324	−0.353064	−0.176532	+	0.984295i	$0.556488\pi$
$24$	−1.00000	−0.204124
$25$	−1.47289	−0.294577
$26$	4.43662	0.870093
$27$	−1.00000	−0.192450
$28$	−3.28324	−0.620474
$29$	9.75612	1.81167	0.905833	−	0.423634i	$-0.139246\pi$
$29$	9.75612	1.81167	0.905833	+	0.423634i	$0.139246\pi$
$30$	1.87806	0.342886
$31$	1.59000	0.285573	0.142786	−	0.989754i	$-0.454394\pi$
$31$	1.59000	0.285573	0.142786	+	0.989754i	$0.454394\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.72468	1.15327
$35$	6.16612	1.04226
$36$	1.00000	0.166667
$37$	4.53985	0.746348	0.373174	−	0.927761i	$-0.378270\pi$
$37$	4.53985	0.746348	0.373174	+	0.927761i	$0.378270\pi$
$38$	1.69324	0.274679
$39$	−4.43662	−0.710428
$40$	−1.87806	−0.296948
$41$	3.33821	0.521340	0.260670	−	0.965428i	$-0.416056\pi$
$41$	3.33821	0.521340	0.260670	+	0.965428i	$0.416056\pi$
$42$	3.28324	0.506615
$43$	7.16130	1.09209	0.546044	−	0.837757i	$-0.316133\pi$
$43$	7.16130	1.09209	0.546044	+	0.837757i	$0.316133\pi$
$44$	0	0
$45$	−1.87806	−0.279965
$46$	−1.69324	−0.249654
$47$	4.47289	0.652437	0.326219	−	0.945294i	$-0.394225\pi$
$47$	4.47289	0.652437	0.326219	+	0.945294i	$0.394225\pi$
$48$	−1.00000	−0.144338
$49$	3.77965	0.539950
$50$	−1.47289	−0.208297
$51$	−6.72468	−0.941644
$52$	4.43662	0.615248
$53$	−0.184825	−0.0253877	−0.0126938	−	0.999919i	$-0.504041\pi$
$53$	−0.184825	−0.0253877	−0.0126938	+	0.999919i	$0.504041\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−3.28324	−0.438741
$57$	−1.69324	−0.224275
$58$	9.75612	1.28104
$59$	−13.4760	−1.75442	−0.877212	−	0.480104i	$-0.840599\pi$
$59$	−13.4760	−1.75442	−0.877212	+	0.480104i	$0.840599\pi$
$60$	1.87806	0.242457
$61$	−5.75612	−0.736996	−0.368498	−	0.929629i	$-0.620128\pi$
$61$	−5.75612	−0.736996	−0.368498	+	0.929629i	$0.620128\pi$
$62$	1.59000	0.201930
$63$	−3.28324	−0.413649
$64$	1.00000	0.125000
$65$	−8.33224	−1.03349
$66$	0	0
$67$	−8.45418	−1.03284	−0.516421	−	0.856335i	$-0.672736\pi$
$67$	−8.45418	−1.03284	−0.516421	+	0.856335i	$0.672736\pi$
$68$	6.72468	0.815487
$69$	1.69324	0.203842
$70$	6.16612	0.736993
$71$	11.1426	1.32238	0.661191	−	0.750217i	$-0.270051\pi$
$71$	11.1426	1.32238	0.661191	+	0.750217i	$0.270051\pi$
$72$	1.00000	0.117851
$73$	−10.6294	−1.24407	−0.622036	−	0.782988i	$-0.713694\pi$
$73$	−10.6294	−1.24407	−0.622036	+	0.782988i	$0.713694\pi$
$74$	4.53985	0.527747
$75$	1.47289	0.170074
$76$	1.69324	0.194228
$77$	0	0
$78$	−4.43662	−0.502348
$79$	−10.7876	−1.21370	−0.606848	−	0.794818i	$-0.707566\pi$
$79$	−10.7876	−1.21370	−0.606848	+	0.794818i	$0.707566\pi$
$80$	−1.87806	−0.209974
$81$	1.00000	0.111111
$82$	3.33821	0.368643
$83$	14.6656	1.60976	0.804881	−	0.593436i	$-0.202229\pi$
$83$	14.6656	1.60976	0.804881	+	0.593436i	$0.202229\pi$
$84$	3.28324	0.358231
$85$	−12.6294	−1.36985
$86$	7.16130	0.772223
$87$	−9.75612	−1.04597
$88$	0	0
$89$	8.16612	0.865607	0.432804	−	0.901488i	$-0.357524\pi$
$89$	8.16612	0.865607	0.432804	+	0.901488i	$0.357524\pi$
$90$	−1.87806	−0.197965
$91$	−14.5665	−1.52698
$92$	−1.69324	−0.176532
$93$	−1.59000	−0.164875
$94$	4.47289	0.461343
$95$	−3.18000	−0.326261
$96$	−1.00000	−0.102062
$97$	10.2290	1.03860	0.519299	−	0.854592i	$-0.326193\pi$
$97$	10.2290	1.03860	0.519299	+	0.854592i	$0.326193\pi$
$98$	3.77965	0.381802
$99$	0	0
$100$	−1.47289	−0.147289
$101$	9.56165	0.951420	0.475710	−	0.879602i	$-0.342191\pi$
$101$	9.56165	0.951420	0.475710	+	0.879602i	$0.342191\pi$
$102$	−6.72468	−0.665843
$103$	18.8034	1.85275	0.926377	−	0.376597i	$-0.122906\pi$
$103$	18.8034	1.85275	0.926377	+	0.376597i	$0.122906\pi$
$104$	4.43662	0.435046
$105$	−6.16612	−0.601752
$106$	−0.184825	−0.0179518
$107$	−4.07253	−0.393707	−0.196853	−	0.980433i	$-0.563072\pi$
$107$	−4.07253	−0.393707	−0.196853	+	0.980433i	$0.563072\pi$
$108$	−1.00000	−0.0962250
$109$	−1.18409	−0.113415	−0.0567074	−	0.998391i	$-0.518060\pi$
$109$	−1.18409	−0.113415	−0.0567074	+	0.998391i	$0.518060\pi$
$110$	0	0
$111$	−4.53985	−0.430904
$112$	−3.28324	−0.310237
$113$	−19.9302	−1.87487	−0.937436	−	0.348158i	$-0.886807\pi$
$113$	−19.9302	−1.87487	−0.937436	+	0.348158i	$0.886807\pi$
$114$	−1.69324	−0.158586
$115$	3.18000	0.296537
$116$	9.75612	0.905833
$117$	4.43662	0.410166
$118$	−13.4760	−1.24056
$119$	−22.0787	−2.02395
$120$	1.87806	0.171443
$121$	−11.0000	−1.00000
$122$	−5.75612	−0.521135
$123$	−3.33821	−0.300996
$124$	1.59000	0.142786
$125$	12.1565	1.08731
$126$	−3.28324	−0.292494
$127$	6.42583	0.570201	0.285100	−	0.958498i	$-0.407973\pi$
$127$	6.42583	0.570201	0.285100	+	0.958498i	$0.407973\pi$
$128$	1.00000	0.0883883
$129$	−7.16130	−0.630517
$130$	−8.33224	−0.730786
$131$	1.24697	0.108948	0.0544742	−	0.998515i	$-0.482652\pi$
$131$	1.24697	0.108948	0.0544742	+	0.998515i	$0.482652\pi$
$132$	0	0
$133$	−5.55930	−0.482052
$134$	−8.45418	−0.730330
$135$	1.87806	0.161638
$136$	6.72468	0.576637
$137$	21.0020	1.79432	0.897159	−	0.441708i	$-0.145627\pi$
$137$	21.0020	1.79432	0.897159	+	0.441708i	$0.145627\pi$
$138$	1.69324	0.144138
$139$	−4.62553	−0.392332	−0.196166	−	0.980571i	$-0.562849\pi$
$139$	−4.62553	−0.392332	−0.196166	+	0.980571i	$0.562849\pi$
$140$	6.16612	0.521132
$141$	−4.47289	−0.376685
$142$	11.1426	0.935066
$143$	0	0
$144$	1.00000	0.0833333
$145$	−18.3226	−1.52161
$146$	−10.6294	−0.879692
$147$	−3.77965	−0.311740
$148$	4.53985	0.373174
$149$	5.87806	0.481550	0.240775	−	0.970581i	$-0.422598\pi$
$149$	5.87806	0.481550	0.240775	+	0.970581i	$0.422598\pi$
$150$	1.47289	0.120261
$151$	7.66080	0.623427	0.311714	−	0.950176i	$-0.399097\pi$
$151$	7.66080	0.623427	0.311714	+	0.950176i	$0.399097\pi$
$152$	1.69324	0.137340
$153$	6.72468	0.543658
$154$	0	0
$155$	−2.98612	−0.239851
$156$	−4.43662	−0.355214
$157$	1.38647	0.110653	0.0553263	−	0.998468i	$-0.482380\pi$
$157$	1.38647	0.110653	0.0553263	+	0.998468i	$0.482380\pi$
$158$	−10.7876	−0.858213
$159$	0.184825	0.0146576
$160$	−1.87806	−0.148474
$161$	5.55930	0.438134
$162$	1.00000	0.0785674
$163$	−9.20165	−0.720729	−0.360364	−	0.932812i	$-0.617348\pi$
$163$	−9.20165	−0.720729	−0.360364	+	0.932812i	$0.617348\pi$
$164$	3.33821	0.260670
$165$	0	0
$166$	14.6656	1.13827
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	3.28324	0.253307
$169$	6.68359	0.514122
$170$	−12.6294	−0.968628
$171$	1.69324	0.129485
$172$	7.16130	0.546044
$173$	5.75612	0.437630	0.218815	−	0.975766i	$-0.429781\pi$
$173$	5.75612	0.437630	0.218815	+	0.975766i	$0.429781\pi$
$174$	−9.75612	−0.739610
$175$	4.83583	0.365555
$176$	0	0
$177$	13.4760	1.01292
$178$	8.16612	0.612077
$179$	−7.38647	−0.552091	−0.276045	−	0.961145i	$-0.589024\pi$
$179$	−7.38647	−0.552091	−0.276045	+	0.961145i	$0.589024\pi$
$180$	−1.87806	−0.139982
$181$	−16.1968	−1.20390	−0.601950	−	0.798534i	$-0.705609\pi$
$181$	−16.1968	−1.20390	−0.601950	+	0.798534i	$0.705609\pi$
$182$	−14.5665	−1.07974
$183$	5.75612	0.425505
$184$	−1.69324	−0.124827
$185$	−8.52613	−0.626853
$186$	−1.59000	−0.116585
$187$	0	0
$188$	4.47289	0.326219
$189$	3.28324	0.238820
$190$	−3.18000	−0.230702
$191$	−7.07677	−0.512057	−0.256028	−	0.966669i	$-0.582414\pi$
$191$	−7.07677	−0.512057	−0.256028	+	0.966669i	$0.582414\pi$
$192$	−1.00000	−0.0721688
$193$	−10.8732	−0.782673	−0.391336	−	0.920248i	$-0.627987\pi$
$193$	−10.8732	−0.782673	−0.391336	+	0.920248i	$0.627987\pi$
$194$	10.2290	0.734400
$195$	8.33224	0.596684
$196$	3.77965	0.269975
$197$	−13.1613	−0.937704	−0.468852	−	0.883277i	$-0.655332\pi$
$197$	−13.1613	−0.937704	−0.468852	+	0.883277i	$0.655332\pi$
$198$	0	0
$199$	4.87324	0.345455	0.172727	−	0.984970i	$-0.444742\pi$
$199$	4.87324	0.345455	0.172727	+	0.984970i	$0.444742\pi$
$200$	−1.47289	−0.104149
$201$	8.45418	0.596312
$202$	9.56165	0.672756
$203$	−32.0317	−2.24818
$204$	−6.72468	−0.470822
$205$	−6.26936	−0.437871
$206$	18.8034	1.31009
$207$	−1.69324	−0.117688
$208$	4.43662	0.307624
$209$	0	0
$210$	−6.16612	−0.425503
$211$	−14.3068	−0.984918	−0.492459	−	0.870336i	$-0.663902\pi$
$211$	−14.3068	−0.984918	−0.492459	+	0.870336i	$0.663902\pi$
$212$	−0.184825	−0.0126938
$213$	−11.1426	−0.763478
$214$	−4.07253	−0.278393
$215$	−13.4494	−0.917239
$216$	−1.00000	−0.0680414
$217$	−5.22035	−0.354381
$218$	−1.18409	−0.0801964
$219$	10.6294	0.718266
$220$	0	0
$221$	29.8348	2.00691
$222$	−4.53985	−0.304695
$223$	3.28324	0.219862	0.109931	−	0.993939i	$-0.464937\pi$
$223$	3.28324	0.219862	0.109931	+	0.993939i	$0.464937\pi$
$224$	−3.28324	−0.219371
$225$	−1.47289	−0.0981924
$226$	−19.9302	−1.32573
$227$	−16.3963	−1.08826	−0.544129	−	0.839001i	$-0.683140\pi$
$227$	−16.3963	−1.08826	−0.544129	+	0.839001i	$0.683140\pi$
$228$	−1.69324	−0.112137
$229$	−6.50359	−0.429769	−0.214885	−	0.976639i	$-0.568938\pi$
$229$	−6.50359	−0.429769	−0.214885	+	0.976639i	$0.568938\pi$
$230$	3.18000	0.209683
$231$	0	0
$232$	9.75612	0.640521
$233$	17.1119	1.12104	0.560519	−	0.828142i	$-0.310602\pi$
$233$	17.1119	1.12104	0.560519	+	0.828142i	$0.310602\pi$
$234$	4.43662	0.290031
$235$	−8.40035	−0.547979
$236$	−13.4760	−0.877212
$237$	10.7876	0.700728
$238$	−22.0787	−1.43115
$239$	−4.87324	−0.315224	−0.157612	−	0.987501i	$-0.550379\pi$
$239$	−4.87324	−0.315224	−0.157612	+	0.987501i	$0.550379\pi$
$240$	1.87806	0.121228
$241$	3.18965	0.205463	0.102732	−	0.994709i	$-0.467242\pi$
$241$	3.18965	0.205463	0.102732	+	0.994709i	$0.467242\pi$
$242$	−11.0000	−0.707107
$243$	−1.00000	−0.0641500
$244$	−5.75612	−0.368498
$245$	−7.09841	−0.453501
$246$	−3.33821	−0.212836
$247$	7.51225	0.477993
$248$	1.59000	0.100965
$249$	−14.6656	−0.929396
$250$	12.1565	0.768843
$251$	3.31394	0.209174	0.104587	−	0.994516i	$-0.466648\pi$
$251$	3.31394	0.209174	0.104587	+	0.994516i	$0.466648\pi$
$252$	−3.28324	−0.206825
$253$	0	0
$254$	6.42583	0.403193
$255$	12.6294	0.790881
$256$	1.00000	0.0625000
$257$	0.905670	0.0564942	0.0282471	−	0.999601i	$-0.491007\pi$
$257$	0.905670	0.0564942	0.0282471	+	0.999601i	$0.491007\pi$
$258$	−7.16130	−0.445843
$259$	−14.9054	−0.926178
$260$	−8.33224	−0.516744
$261$	9.75612	0.603889
$262$	1.24697	0.0770382
$263$	27.0926	1.67060	0.835301	−	0.549793i	$-0.185294\pi$
$263$	27.0926	1.67060	0.835301	+	0.549793i	$0.185294\pi$
$264$	0	0
$265$	0.347113	0.0213230
$266$	−5.55930	−0.340862
$267$	−8.16612	−0.499759
$268$	−8.45418	−0.516421
$269$	25.4628	1.55250	0.776248	−	0.630427i	$-0.217120\pi$
$269$	25.4628	1.55250	0.776248	+	0.630427i	$0.217120\pi$
$270$	1.87806	0.114295
$271$	−20.2743	−1.23158	−0.615789	−	0.787911i	$-0.711163\pi$
$271$	−20.2743	−1.23158	−0.615789	+	0.787911i	$0.711163\pi$
$272$	6.72468	0.407744
$273$	14.5665	0.881603
$274$	21.0020	1.26877
$275$	0	0
$276$	1.69324	0.101921
$277$	−31.3205	−1.88186	−0.940932	−	0.338595i	$-0.890048\pi$
$277$	−31.3205	−1.88186	−0.940932	+	0.338595i	$0.890048\pi$
$278$	−4.62553	−0.277421
$279$	1.59000	0.0951909
$280$	6.16612	0.368496
$281$	−11.6784	−0.696673	−0.348336	−	0.937370i	$-0.613253\pi$
$281$	−11.6784	−0.696673	−0.348336	+	0.937370i	$0.613253\pi$
$282$	−4.47289	−0.266356
$283$	−23.2842	−1.38410	−0.692051	−	0.721849i	$-0.743293\pi$
$283$	−23.2842	−1.38410	−0.692051	+	0.721849i	$0.743293\pi$
$284$	11.1426	0.661191
$285$	3.18000	0.188367
$286$	0	0
$287$	−10.9601	−0.646956
$288$	1.00000	0.0589256
$289$	28.2213	1.66008
$290$	−18.3226	−1.07594
$291$	−10.2290	−0.599635
$292$	−10.6294	−0.622036
$293$	10.2439	0.598454	0.299227	−	0.954182i	$-0.403271\pi$
$293$	10.2439	0.598454	0.299227	+	0.954182i	$0.403271\pi$
$294$	−3.77965	−0.220434
$295$	25.3087	1.47353
$296$	4.53985	0.263874
$297$	0	0
$298$	5.87806	0.340507
$299$	−7.51225	−0.434444
$300$	1.47289	0.0850371
$301$	−23.5122	−1.35522
$302$	7.66080	0.440830
$303$	−9.56165	−0.549303
$304$	1.69324	0.0971138
$305$	10.8104	0.618999
$306$	6.72468	0.384424
$307$	12.6414	0.721481	0.360740	−	0.932666i	$-0.382524\pi$
$307$	12.6414	0.721481	0.360740	+	0.932666i	$0.382524\pi$
$308$	0	0
$309$	−18.8034	−1.06969
$310$	−2.98612	−0.169600
$311$	34.2352	1.94130	0.970650	−	0.240497i	$-0.0773103\pi$
$311$	34.2352	1.94130	0.970650	+	0.240497i	$0.0773103\pi$
$312$	−4.43662	−0.251174
$313$	32.9713	1.86365	0.931823	−	0.362914i	$-0.118218\pi$
$313$	32.9713	1.86365	0.931823	+	0.362914i	$0.118218\pi$
$314$	1.38647	0.0782432
$315$	6.16612	0.347422
$316$	−10.7876	−0.606848
$317$	7.50260	0.421388	0.210694	−	0.977552i	$-0.432428\pi$
$317$	7.50260	0.421388	0.210694	+	0.977552i	$0.432428\pi$
$318$	0.184825	0.0103645
$319$	0	0
$320$	−1.87806	−0.104987
$321$	4.07253	0.227307
$322$	5.55930	0.309808
$323$	11.3865	0.633560
$324$	1.00000	0.0555556
$325$	−6.53463	−0.362476
$326$	−9.20165	−0.509632
$327$	1.18409	0.0654801
$328$	3.33821	0.184322
$329$	−14.6855	−0.809640
$330$	0	0
$331$	−8.35095	−0.459010	−0.229505	−	0.973308i	$-0.573711\pi$
$331$	−8.35095	−0.459010	−0.229505	+	0.973308i	$0.573711\pi$
$332$	14.6656	0.804881
$333$	4.53985	0.248783
$334$	−1.00000	−0.0547176
$335$	15.8775	0.867479
$336$	3.28324	0.179115
$337$	−3.42682	−0.186671	−0.0933354	−	0.995635i	$-0.529753\pi$
$337$	−3.42682	−0.186671	−0.0933354	+	0.995635i	$0.529753\pi$
$338$	6.68359	0.363539
$339$	19.9302	1.08246
$340$	−12.6294	−0.684923
$341$	0	0
$342$	1.69324	0.0915597
$343$	10.5732	0.570898
$344$	7.16130	0.386111
$345$	−3.18000	−0.171206
$346$	5.75612	0.309451
$347$	−24.3747	−1.30850	−0.654251	−	0.756277i	$-0.727016\pi$
$347$	−24.3747	−1.30850	−0.654251	+	0.756277i	$0.727016\pi$
$348$	−9.75612	−0.522983
$349$	−16.0521	−0.859249	−0.429625	−	0.903008i	$-0.641354\pi$
$349$	−16.0521	−0.859249	−0.429625	+	0.903008i	$0.641354\pi$
$350$	4.83583	0.258486
$351$	−4.43662	−0.236809
$352$	0	0
$353$	−3.67837	−0.195780	−0.0978899	−	0.995197i	$-0.531209\pi$
$353$	−3.67837	−0.195780	−0.0978899	+	0.995197i	$0.531209\pi$
$354$	13.4760	0.716240
$355$	−20.9265	−1.11066
$356$	8.16612	0.432804
$357$	22.0787	1.16853
$358$	−7.38647	−0.390387
$359$	−14.9361	−0.788299	−0.394149	−	0.919046i	$-0.628961\pi$
$359$	−14.9361	−0.788299	−0.394149	+	0.919046i	$0.628961\pi$
$360$	−1.87806	−0.0989825
$361$	−16.1330	−0.849103
$362$	−16.1968	−0.851286
$363$	11.0000	0.577350
$364$	−14.5665	−0.763491
$365$	19.9626	1.04489
$366$	5.75612	0.300877
$367$	−6.32260	−0.330037	−0.165018	−	0.986290i	$-0.552768\pi$
$367$	−6.32260	−0.330037	−0.165018	+	0.986290i	$0.552768\pi$
$368$	−1.69324	−0.0882661
$369$	3.33821	0.173780
$370$	−8.52613	−0.443252
$371$	0.606824	0.0315047
$372$	−1.59000	−0.0824377
$373$	33.2383	1.72101	0.860507	−	0.509439i	$-0.170147\pi$
$373$	33.2383	1.72101	0.860507	+	0.509439i	$0.170147\pi$
$374$	0	0
$375$	−12.1565	−0.627758
$376$	4.47289	0.230671
$377$	43.2842	2.22925
$378$	3.28324	0.168872
$379$	8.24771	0.423656	0.211828	−	0.977307i	$-0.432058\pi$
$379$	8.24771	0.423656	0.211828	+	0.977307i	$0.432058\pi$
$380$	−3.18000	−0.163131
$381$	−6.42583	−0.329205
$382$	−7.07677	−0.362079
$383$	19.0701	0.974435	0.487217	−	0.873281i	$-0.338012\pi$
$383$	19.0701	0.974435	0.487217	+	0.873281i	$0.338012\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−10.8732	−0.553433
$387$	7.16130	0.364029
$388$	10.2290	0.519299
$389$	11.0610	0.560815	0.280408	−	0.959881i	$-0.409530\pi$
$389$	11.0610	0.560815	0.280408	+	0.959881i	$0.409530\pi$
$390$	8.33224	0.421920
$391$	−11.3865	−0.575839
$392$	3.77965	0.190901
$393$	−1.24697	−0.0629014
$394$	−13.1613	−0.663057
$395$	20.2597	1.01938
$396$	0	0
$397$	26.0158	1.30570	0.652849	−	0.757488i	$-0.273574\pi$
$397$	26.0158	1.30570	0.652849	+	0.757488i	$0.273574\pi$
$398$	4.87324	0.244273
$399$	5.55930	0.278313
$400$	−1.47289	−0.0736443
$401$	22.7405	1.13561	0.567804	−	0.823164i	$-0.307793\pi$
$401$	22.7405	1.13561	0.567804	+	0.823164i	$0.307793\pi$
$402$	8.45418	0.421656
$403$	7.05423	0.351396
$404$	9.56165	0.475710
$405$	−1.87806	−0.0933216
$406$	−32.0317	−1.58971
$407$	0	0
$408$	−6.72468	−0.332921
$409$	−28.1513	−1.39199	−0.695995	−	0.718047i	$-0.745036\pi$
$409$	−28.1513	−1.39199	−0.695995	+	0.718047i	$0.745036\pi$
$410$	−6.26936	−0.309621
$411$	−21.0020	−1.03595
$412$	18.8034	0.926377
$413$	44.2448	2.17715
$414$	−1.69324	−0.0832180
$415$	−27.5429	−1.35203
$416$	4.43662	0.217523
$417$	4.62553	0.226513
$418$	0	0
$419$	−8.25971	−0.403513	−0.201757	−	0.979436i	$-0.564665\pi$
$419$	−8.25971	−0.403513	−0.201757	+	0.979436i	$0.564665\pi$
$420$	−6.16612	−0.300876
$421$	−22.1872	−1.08134	−0.540668	−	0.841236i	$-0.681829\pi$
$421$	−22.1872	−1.08134	−0.540668	+	0.841236i	$0.681829\pi$
$422$	−14.3068	−0.696442
$423$	4.47289	0.217479
$424$	−0.184825	−0.00897589
$425$	−9.90468	−0.480448
$426$	−11.1426	−0.539861
$427$	18.8987	0.914573
$428$	−4.07253	−0.196853
$429$	0	0
$430$	−13.4494	−0.648586
$431$	−31.0552	−1.49588	−0.747938	−	0.663769i	$-0.768956\pi$
$431$	−31.0552	−1.49588	−0.747938	+	0.663769i	$0.768956\pi$
$432$	−1.00000	−0.0481125
$433$	34.6759	1.66642	0.833209	−	0.552959i	$-0.186501\pi$
$433$	34.6759	1.66642	0.833209	+	0.552959i	$0.186501\pi$
$434$	−5.22035	−0.250585
$435$	18.3226	0.878501
$436$	−1.18409	−0.0567074
$437$	−2.86705	−0.137150
$438$	10.6294	0.507891
$439$	25.4154	1.21301	0.606507	−	0.795078i	$-0.292570\pi$
$439$	25.4154	1.21301	0.606507	+	0.795078i	$0.292570\pi$
$440$	0	0
$441$	3.77965	0.179983
$442$	29.8348	1.41910
$443$	−25.4034	−1.20695	−0.603477	−	0.797380i	$-0.706219\pi$
$443$	−25.4034	−1.20695	−0.603477	+	0.797380i	$0.706219\pi$
$444$	−4.53985	−0.215452
$445$	−15.3365	−0.727019
$446$	3.28324	0.155466
$447$	−5.87806	−0.278023
$448$	−3.28324	−0.155118
$449$	−12.1820	−0.574902	−0.287451	−	0.957795i	$-0.592808\pi$
$449$	−12.1820	−0.574902	−0.287451	+	0.957795i	$0.592808\pi$
$450$	−1.47289	−0.0694325
$451$	0	0
$452$	−19.9302	−0.937436
$453$	−7.66080	−0.359936
$454$	−16.3963	−0.769515
$455$	27.3567	1.28250
$456$	−1.69324	−0.0792931
$457$	−11.5833	−0.541844	−0.270922	−	0.962601i	$-0.587329\pi$
$457$	−11.5833	−0.541844	−0.270922	+	0.962601i	$0.587329\pi$
$458$	−6.50359	−0.303893
$459$	−6.72468	−0.313881
$460$	3.18000	0.148268
$461$	8.69225	0.404838	0.202419	−	0.979299i	$-0.435120\pi$
$461$	8.69225	0.404838	0.202419	+	0.979299i	$0.435120\pi$
$462$	0	0
$463$	−10.0559	−0.467339	−0.233669	−	0.972316i	$-0.575073\pi$
$463$	−10.0559	−0.467339	−0.233669	+	0.972316i	$0.575073\pi$
$464$	9.75612	0.452917
$465$	2.98612	0.138478
$466$	17.1119	0.792693
$467$	11.3058	0.523169	0.261584	−	0.965181i	$-0.415755\pi$
$467$	11.3058	0.523169	0.261584	+	0.965181i	$0.415755\pi$
$468$	4.43662	0.205083
$469$	27.7571	1.28170
$470$	−8.40035	−0.387479
$471$	−1.38647	−0.0638853
$472$	−13.4760	−0.620282
$473$	0	0
$474$	10.7876	0.495489
$475$	−2.49394	−0.114430
$476$	−22.0787	−1.01198
$477$	−0.184825	−0.00846255
$478$	−4.87324	−0.222897
$479$	−20.6908	−0.945385	−0.472693	−	0.881227i	$-0.656718\pi$
$479$	−20.6908	−0.945385	−0.472693	+	0.881227i	$0.656718\pi$
$480$	1.87806	0.0857214
$481$	20.1416	0.918378
$482$	3.18965	0.145284
$483$	−5.55930	−0.252957
$484$	−11.0000	−0.500000
$485$	−19.2107	−0.872313
$486$	−1.00000	−0.0453609
$487$	−3.77891	−0.171239	−0.0856194	−	0.996328i	$-0.527287\pi$
$487$	−3.77891	−0.171239	−0.0856194	+	0.996328i	$0.527287\pi$
$488$	−5.75612	−0.260567
$489$	9.20165	0.416113
$490$	−7.09841	−0.320674
$491$	−31.7638	−1.43348	−0.716740	−	0.697341i	$-0.754367\pi$
$491$	−31.7638	−1.43348	−0.716740	+	0.697341i	$0.754367\pi$
$492$	−3.33821	−0.150498
$493$	65.6068	2.95478
$494$	7.51225	0.337992
$495$	0	0
$496$	1.59000	0.0713932
$497$	−36.5838	−1.64101
$498$	−14.6656	−0.657182
$499$	1.35812	0.0607980	0.0303990	−	0.999538i	$-0.490322\pi$
$499$	1.35812	0.0607980	0.0303990	+	0.999538i	$0.490322\pi$
$500$	12.1565	0.543654
$501$	1.00000	0.0446767
$502$	3.31394	0.147908
$503$	11.5578	0.515338	0.257669	−	0.966233i	$-0.417046\pi$
$503$	11.5578	0.515338	0.257669	+	0.966233i	$0.417046\pi$
$504$	−3.28324	−0.146247
$505$	−17.9574	−0.799092
$506$	0	0
$507$	−6.68359	−0.296829
$508$	6.42583	0.285100
$509$	−32.0787	−1.42186	−0.710932	−	0.703261i	$-0.751727\pi$
$509$	−32.0787	−1.42186	−0.710932	+	0.703261i	$0.751727\pi$
$510$	12.6294	0.559238
$511$	34.8987	1.54383
$512$	1.00000	0.0441942
$513$	−1.69324	−0.0747582
$514$	0.905670	0.0399474
$515$	−35.3139	−1.55612
$516$	−7.16130	−0.315259
$517$	0	0
$518$	−14.9054	−0.654907
$519$	−5.75612	−0.252666
$520$	−8.33224	−0.365393
$521$	4.32955	0.189681	0.0948405	−	0.995492i	$-0.469766\pi$
$521$	4.32955	0.189681	0.0948405	+	0.995492i	$0.469766\pi$
$522$	9.75612	0.427014
$523$	21.6125	0.945050	0.472525	−	0.881317i	$-0.343343\pi$
$523$	21.6125	0.945050	0.472525	+	0.881317i	$0.343343\pi$
$524$	1.24697	0.0544742
$525$	−4.83583	−0.211053
$526$	27.0926	1.18129
$527$	10.6922	0.465762
$528$	0	0
$529$	−20.1330	−0.875346
$530$	0.347113	0.0150776
$531$	−13.4760	−0.584808
$532$	−5.55930	−0.241026
$533$	14.8104	0.641508
$534$	−8.16612	−0.353383
$535$	7.64847	0.330672
$536$	−8.45418	−0.365165
$537$	7.38647	0.318750
$538$	25.4628	1.09778
$539$	0	0
$540$	1.87806	0.0808189
$541$	−15.1970	−0.653369	−0.326685	−	0.945133i	$-0.605931\pi$
$541$	−15.1970	−0.653369	−0.326685	+	0.945133i	$0.605931\pi$
$542$	−20.2743	−0.870857
$543$	16.1968	0.695072
$544$	6.72468	0.288318
$545$	2.22378	0.0952565
$546$	14.5665	0.623388
$547$	−25.0268	−1.07007	−0.535035	−	0.844830i	$-0.679702\pi$
$547$	−25.0268	−1.07007	−0.535035	+	0.844830i	$0.679702\pi$
$548$	21.0020	0.897159
$549$	−5.75612	−0.245665
$550$	0	0
$551$	16.5194	0.703751
$552$	1.69324	0.0720689
$553$	35.4181	1.50613
$554$	−31.3205	−1.33068
$555$	8.52613	0.361914
$556$	−4.62553	−0.196166
$557$	−31.4652	−1.33322	−0.666612	−	0.745405i	$-0.732256\pi$
$557$	−31.4652	−1.33322	−0.666612	+	0.745405i	$0.732256\pi$
$558$	1.59000	0.0673101
$559$	31.7720	1.34381
$560$	6.16612	0.260566
$561$	0	0
$562$	−11.6784	−0.492622
$563$	30.9520	1.30447	0.652235	−	0.758017i	$-0.273832\pi$
$563$	30.9520	1.30447	0.652235	+	0.758017i	$0.273832\pi$
$564$	−4.47289	−0.188342
$565$	37.4301	1.57469
$566$	−23.2842	−0.978708
$567$	−3.28324	−0.137883
$568$	11.1426	0.467533
$569$	−36.8385	−1.54435	−0.772176	−	0.635409i	$-0.780831\pi$
$569$	−36.8385	−1.54435	−0.772176	+	0.635409i	$0.780831\pi$
$570$	3.18000	0.133196
$571$	−23.5617	−0.986024	−0.493012	−	0.870022i	$-0.664104\pi$
$571$	−23.5617	−0.986024	−0.493012	+	0.870022i	$0.664104\pi$
$572$	0	0
$573$	7.07677	0.295636
$574$	−10.9601	−0.457467
$575$	2.49394	0.104005
$576$	1.00000	0.0416667
$577$	−7.21219	−0.300247	−0.150124	−	0.988667i	$-0.547967\pi$
$577$	−7.21219	−0.300247	−0.150124	+	0.988667i	$0.547967\pi$
$578$	28.2213	1.17385
$579$	10.8732	0.451876
$580$	−18.3226	−0.760804
$581$	−48.1507	−1.99763
$582$	−10.2290	−0.424006
$583$	0	0
$584$	−10.6294	−0.439846
$585$	−8.33224	−0.344496
$586$	10.2439	0.423171
$587$	1.28880	0.0531945	0.0265972	−	0.999646i	$-0.491533\pi$
$587$	1.28880	0.0531945	0.0265972	+	0.999646i	$0.491533\pi$
$588$	−3.77965	−0.155870
$589$	2.69225	0.110932
$590$	25.3087	1.04194
$591$	13.1613	0.541383
$592$	4.53985	0.186587
$593$	−7.80439	−0.320488	−0.160244	−	0.987077i	$-0.551228\pi$
$593$	−7.80439	−0.320488	−0.160244	+	0.987077i	$0.551228\pi$
$594$	0	0
$595$	41.4652	1.69991
$596$	5.87806	0.240775
$597$	−4.87324	−0.199448
$598$	−7.51225	−0.307199
$599$	−13.6784	−0.558883	−0.279441	−	0.960163i	$-0.590149\pi$
$599$	−13.6784	−0.558883	−0.279441	+	0.960163i	$0.590149\pi$
$600$	1.47289	0.0601303
$601$	43.7556	1.78483	0.892414	−	0.451217i	$-0.149010\pi$
$601$	43.7556	1.78483	0.892414	+	0.451217i	$0.149010\pi$
$602$	−23.5122	−0.958288
$603$	−8.45418	−0.344281
$604$	7.66080	0.311714
$605$	20.6587	0.839895
$606$	−9.56165	−0.388416
$607$	−4.33574	−0.175982	−0.0879911	−	0.996121i	$-0.528045\pi$
$607$	−4.33574	−0.175982	−0.0879911	+	0.996121i	$0.528045\pi$
$608$	1.69324	0.0686698
$609$	32.0317	1.29799
$610$	10.8104	0.437698
$611$	19.8445	0.802822
$612$	6.72468	0.271829
$613$	38.2030	1.54301	0.771503	−	0.636226i	$-0.219505\pi$
$613$	38.2030	1.54301	0.771503	+	0.636226i	$0.219505\pi$
$614$	12.6414	0.510164
$615$	6.26936	0.252805
$616$	0	0
$617$	−23.4810	−0.945311	−0.472655	−	0.881247i	$-0.656704\pi$
$617$	−23.4810	−0.945311	−0.472655	+	0.881247i	$0.656704\pi$
$618$	−18.8034	−0.756384
$619$	−8.46559	−0.340261	−0.170130	−	0.985422i	$-0.554419\pi$
$619$	−8.46559	−0.340261	−0.170130	+	0.985422i	$0.554419\pi$
$620$	−2.98612	−0.119925
$621$	1.69324	0.0679472
$622$	34.2352	1.37271
$623$	−26.8113	−1.07417
$624$	−4.43662	−0.177607
$625$	−15.4662	−0.618647
$626$	32.9713	1.31780
$627$	0	0
$628$	1.38647	0.0553263
$629$	30.5291	1.21727
$630$	6.16612	0.245664
$631$	12.4729	0.496538	0.248269	−	0.968691i	$-0.420138\pi$
$631$	12.4729	0.496538	0.248269	+	0.968691i	$0.420138\pi$
$632$	−10.7876	−0.429106
$633$	14.3068	0.568643
$634$	7.50260	0.297966
$635$	−12.0681	−0.478908
$636$	0.184825	0.00732879
$637$	16.7689	0.664407
$638$	0	0
$639$	11.1426	0.440794
$640$	−1.87806	−0.0742369
$641$	35.1792	1.38950	0.694748	−	0.719253i	$-0.255516\pi$
$641$	35.1792	1.38950	0.694748	+	0.719253i	$0.255516\pi$
$642$	4.07253	0.160730
$643$	−18.3571	−0.723935	−0.361967	−	0.932191i	$-0.617895\pi$
$643$	−18.3571	−0.723935	−0.361967	+	0.932191i	$0.617895\pi$
$644$	5.55930	0.219067
$645$	13.4494	0.529568
$646$	11.3865	0.447995
$647$	−20.6826	−0.813117	−0.406558	−	0.913625i	$-0.633271\pi$
$647$	−20.6826	−0.813117	−0.406558	+	0.913625i	$0.633271\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	−6.53463	−0.256309
$651$	5.22035	0.204602
$652$	−9.20165	−0.360364
$653$	1.88190	0.0736443	0.0368221	−	0.999322i	$-0.488277\pi$
$653$	1.88190	0.0736443	0.0368221	+	0.999322i	$0.488277\pi$
$654$	1.18409	0.0463014
$655$	−2.34189	−0.0915052
$656$	3.33821	0.130335
$657$	−10.6294	−0.414691
$658$	−14.6855	−0.572502
$659$	29.2479	1.13934	0.569669	−	0.821874i	$-0.307071\pi$
$659$	29.2479	1.13934	0.569669	+	0.821874i	$0.307071\pi$
$660$	0	0
$661$	−46.5230	−1.80954	−0.904768	−	0.425906i	$-0.859956\pi$
$661$	−46.5230	−1.80954	−0.904768	+	0.425906i	$0.859956\pi$
$662$	−8.35095	−0.324569
$663$	−29.8348	−1.15869
$664$	14.6656	0.569137
$665$	10.4407	0.404873
$666$	4.53985	0.175916
$667$	−16.5194	−0.639635
$668$	−1.00000	−0.0386912
$669$	−3.28324	−0.126937
$670$	15.8775	0.613400
$671$	0	0
$672$	3.28324	0.126654
$673$	34.4048	1.32621	0.663103	−	0.748528i	$-0.269239\pi$
$673$	34.4048	1.32621	0.663103	+	0.748528i	$0.269239\pi$
$674$	−3.42682	−0.131996
$675$	1.47289	0.0566914
$676$	6.68359	0.257061
$677$	−27.3768	−1.05218	−0.526088	−	0.850430i	$-0.676342\pi$
$677$	−27.3768	−1.05218	−0.526088	+	0.850430i	$0.676342\pi$
$678$	19.9302	0.765413
$679$	−33.5843	−1.28885
$680$	−12.6294	−0.484314
$681$	16.3963	0.628306
$682$	0	0
$683$	19.4064	0.742565	0.371282	−	0.928520i	$-0.378918\pi$
$683$	19.4064	0.742565	0.371282	+	0.928520i	$0.378918\pi$
$684$	1.69324	0.0647425
$685$	−39.4430	−1.50704
$686$	10.5732	0.403686
$687$	6.50359	0.248127
$688$	7.16130	0.273022
$689$	−0.819998	−0.0312394
$690$	−3.18000	−0.121061
$691$	−18.8075	−0.715470	−0.357735	−	0.933823i	$-0.616451\pi$
$691$	−18.8075	−0.715470	−0.357735	+	0.933823i	$0.616451\pi$
$692$	5.75612	0.218815
$693$	0	0
$694$	−24.3747	−0.925251
$695$	8.68702	0.329518
$696$	−9.75612	−0.369805
$697$	22.4484	0.850293
$698$	−16.0521	−0.607581
$699$	−17.1119	−0.647231
$700$	4.83583	0.182777
$701$	18.4071	0.695225	0.347612	−	0.937638i	$-0.386992\pi$
$701$	18.4071	0.695225	0.347612	+	0.937638i	$0.386992\pi$
$702$	−4.43662	−0.167449
$703$	7.68705	0.289922
$704$	0	0
$705$	8.40035	0.316376
$706$	−3.67837	−0.138437
$707$	−31.3932	−1.18066
$708$	13.4760	0.506458
$709$	−30.0232	−1.12754	−0.563772	−	0.825931i	$-0.690650\pi$
$709$	−30.0232	−1.12754	−0.563772	+	0.825931i	$0.690650\pi$
$710$	−20.9265	−0.785357
$711$	−10.7876	−0.404565
$712$	8.16612	0.306038
$713$	−2.69225	−0.100825
$714$	22.0787	0.826275
$715$	0	0
$716$	−7.38647	−0.276045
$717$	4.87324	0.181994
$718$	−14.9361	−0.557411
$719$	−12.0884	−0.450820	−0.225410	−	0.974264i	$-0.572372\pi$
$719$	−12.0884	−0.450820	−0.225410	+	0.974264i	$0.572372\pi$
$720$	−1.87806	−0.0699912
$721$	−61.7360	−2.29917
$722$	−16.1330	−0.600406
$723$	−3.18965	−0.118624
$724$	−16.1968	−0.601950
$725$	−14.3696	−0.533675
$726$	11.0000	0.408248
$727$	5.41890	0.200976	0.100488	−	0.994938i	$-0.467960\pi$
$727$	5.41890	0.200976	0.100488	+	0.994938i	$0.467960\pi$
$728$	−14.5665	−0.539869
$729$	1.00000	0.0370370
$730$	19.9626	0.738849
$731$	48.1574	1.78117
$732$	5.75612	0.212752
$733$	12.1436	0.448534	0.224267	−	0.974528i	$-0.428001\pi$
$733$	12.1436	0.448534	0.224267	+	0.974528i	$0.428001\pi$
$734$	−6.32260	−0.233371
$735$	7.09841	0.261829
$736$	−1.69324	−0.0624135
$737$	0	0
$738$	3.33821	0.122881
$739$	27.9845	1.02943	0.514714	−	0.857362i	$-0.327898\pi$
$739$	27.9845	1.02943	0.514714	+	0.857362i	$0.327898\pi$
$740$	−8.52613	−0.313427
$741$	−7.51225	−0.275969
$742$	0.606824	0.0222772
$743$	−8.71331	−0.319660	−0.159830	−	0.987145i	$-0.551095\pi$
$743$	−8.71331	−0.319660	−0.159830	+	0.987145i	$0.551095\pi$
$744$	−1.59000	−0.0582923
$745$	−11.0394	−0.404451
$746$	33.2383	1.21694
$747$	14.6656	0.536587
$748$	0	0
$749$	13.3711	0.488569
$750$	−12.1565	−0.443892
$751$	44.5125	1.62428	0.812142	−	0.583460i	$-0.198302\pi$
$751$	44.5125	1.62428	0.812142	+	0.583460i	$0.198302\pi$
$752$	4.47289	0.163109
$753$	−3.31394	−0.120767
$754$	43.2842	1.57632
$755$	−14.3875	−0.523613
$756$	3.28324	0.119410
$757$	12.1535	0.441728	0.220864	−	0.975305i	$-0.429112\pi$
$757$	12.1535	0.441728	0.220864	+	0.975305i	$0.429112\pi$
$758$	8.24771	0.299570
$759$	0	0
$760$	−3.18000	−0.115351
$761$	−6.20972	−0.225102	−0.112551	−	0.993646i	$-0.535902\pi$
$761$	−6.20972	−0.225102	−0.112551	+	0.993646i	$0.535902\pi$
$762$	−6.42583	−0.232783
$763$	3.88763	0.140742
$764$	−7.07677	−0.256028
$765$	−12.6294	−0.456616
$766$	19.0701	0.689029
$767$	−59.7878	−2.15881
$768$	−1.00000	−0.0360844
$769$	42.2300	1.52285	0.761426	−	0.648252i	$-0.224500\pi$
$769$	42.2300	1.52285	0.761426	+	0.648252i	$0.224500\pi$
$770$	0	0
$771$	−0.905670	−0.0326169
$772$	−10.8732	−0.391336
$773$	−21.4868	−0.772828	−0.386414	−	0.922325i	$-0.626286\pi$
$773$	−21.4868	−0.772828	−0.386414	+	0.922325i	$0.626286\pi$
$774$	7.16130	0.257408
$775$	−2.34189	−0.0841231
$776$	10.2290	0.367200
$777$	14.9054	0.534729
$778$	11.0610	0.396556
$779$	5.65237	0.202517
$780$	8.33224	0.298342
$781$	0	0
$782$	−11.3865	−0.407179
$783$	−9.75612	−0.348655
$784$	3.77965	0.134987
$785$	−2.60388	−0.0929365
$786$	−1.24697	−0.0444780
$787$	−34.1291	−1.21657	−0.608285	−	0.793718i	$-0.708142\pi$
$787$	−34.1291	−1.21657	−0.608285	+	0.793718i	$0.708142\pi$
$788$	−13.1613	−0.468852
$789$	−27.0926	−0.964522
$790$	20.2597	0.720808
$791$	65.4355	2.32662
$792$	0	0
$793$	−25.5377	−0.906871
$794$	26.0158	0.923267
$795$	−0.347113	−0.0123108
$796$	4.87324	0.172727
$797$	2.57899	0.0913525	0.0456762	−	0.998956i	$-0.485456\pi$
$797$	2.57899	0.0913525	0.0456762	+	0.998956i	$0.485456\pi$
$798$	5.55930	0.196797
$799$	30.0787	1.06411
$800$	−1.47289	−0.0520744
$801$	8.16612	0.288536
$802$	22.7405	0.802995
$803$	0	0
$804$	8.45418	0.298156
$805$	−10.4407	−0.367986
$806$	7.05423	0.248475
$807$	−25.4628	−0.896334
$808$	9.56165	0.336378
$809$	−12.6697	−0.445443	−0.222722	−	0.974882i	$-0.571494\pi$
$809$	−12.6697	−0.445443	−0.222722	+	0.974882i	$0.571494\pi$
$810$	−1.87806	−0.0659884
$811$	14.0293	0.492636	0.246318	−	0.969189i	$-0.420779\pi$
$811$	14.0293	0.492636	0.246318	+	0.969189i	$0.420779\pi$
$812$	−32.0317	−1.12409
$813$	20.2743	0.711052
$814$	0	0
$815$	17.2813	0.605336
$816$	−6.72468	−0.235411
$817$	12.1258	0.424227
$818$	−28.1513	−0.984285
$819$	−14.5665	−0.508994
$820$	−6.26936	−0.218935
$821$	−49.8065	−1.73826	−0.869129	−	0.494585i	$-0.835320\pi$
$821$	−49.8065	−1.73826	−0.869129	+	0.494585i	$0.835320\pi$
$822$	−21.0020	−0.732527
$823$	37.9460	1.32271	0.661357	−	0.750071i	$-0.269981\pi$
$823$	37.9460	1.32271	0.661357	+	0.750071i	$0.269981\pi$
$824$	18.8034	0.655047
$825$	0	0
$826$	44.2448	1.53947
$827$	34.4683	1.19858	0.599290	−	0.800532i	$-0.295450\pi$
$827$	34.4683	1.19858	0.599290	+	0.800532i	$0.295450\pi$
$828$	−1.69324	−0.0588440
$829$	−12.6201	−0.438313	−0.219156	−	0.975690i	$-0.570330\pi$
$829$	−12.6201	−0.438313	−0.219156	+	0.975690i	$0.570330\pi$
$830$	−27.5429	−0.956030
$831$	31.3205	1.08650
$832$	4.43662	0.153812
$833$	25.4169	0.880644
$834$	4.62553	0.160169
$835$	1.87806	0.0649930
$836$	0	0
$837$	−1.59000	−0.0549585
$838$	−8.25971	−0.285327
$839$	−0.303821	−0.0104891	−0.00524453	−	0.999986i	$-0.501669\pi$
$839$	−0.303821	−0.0104891	−0.00524453	+	0.999986i	$0.501669\pi$
$840$	−6.16612	−0.212751
$841$	66.1819	2.28214
$842$	−22.1872	−0.764621
$843$	11.6784	0.402224
$844$	−14.3068	−0.492459
$845$	−12.5522	−0.431809
$846$	4.47289	0.153781
$847$	36.1156	1.24095
$848$	−0.184825	−0.00634691
$849$	23.2842	0.799112
$850$	−9.90468	−0.339728
$851$	−7.68705	−0.263509
$852$	−11.1426	−0.381739
$853$	−18.1416	−0.621157	−0.310578	−	0.950548i	$-0.600523\pi$
$853$	−18.1416	−0.621157	−0.310578	+	0.950548i	$0.600523\pi$
$854$	18.8987	0.646701
$855$	−3.18000	−0.108754
$856$	−4.07253	−0.139196
$857$	−5.83536	−0.199332	−0.0996660	−	0.995021i	$-0.531777\pi$
$857$	−5.83536	−0.199332	−0.0996660	+	0.995021i	$0.531777\pi$
$858$	0	0
$859$	−31.3216	−1.06868	−0.534340	−	0.845270i	$-0.679440\pi$
$859$	−31.3216	−1.06868	−0.534340	+	0.845270i	$0.679440\pi$
$860$	−13.4494	−0.458619
$861$	10.9601	0.373520
$862$	−31.0552	−1.05774
$863$	−52.3206	−1.78101	−0.890507	−	0.454969i	$-0.849650\pi$
$863$	−52.3206	−1.78101	−0.890507	+	0.454969i	$0.849650\pi$
$864$	−1.00000	−0.0340207
$865$	−10.8104	−0.367563
$866$	34.6759	1.17834
$867$	−28.2213	−0.958446
$868$	−5.22035	−0.177190
$869$	0	0
$870$	18.3226	0.621194
$871$	−37.5080	−1.27091
$872$	−1.18409	−0.0400982
$873$	10.2290	0.346199
$874$	−2.86705	−0.0969794
$875$	−39.9126	−1.34929
$876$	10.6294	0.359133
$877$	−10.6861	−0.360843	−0.180421	−	0.983589i	$-0.557746\pi$
$877$	−10.6861	−0.360843	−0.180421	+	0.983589i	$0.557746\pi$
$878$	25.4154	0.857730
$879$	−10.2439	−0.345517
$880$	0	0
$881$	−12.6695	−0.426845	−0.213423	−	0.976960i	$-0.568461\pi$
$881$	−12.6695	−0.426845	−0.213423	+	0.976960i	$0.568461\pi$
$882$	3.77965	0.127267
$883$	18.6707	0.628318	0.314159	−	0.949370i	$-0.398277\pi$
$883$	18.6707	0.628318	0.314159	+	0.949370i	$0.398277\pi$
$884$	29.8348	1.00345
$885$	−25.3087	−0.850743
$886$	−25.4034	−0.853446
$887$	−11.5157	−0.386659	−0.193330	−	0.981134i	$-0.561929\pi$
$887$	−11.5157	−0.386659	−0.193330	+	0.981134i	$0.561929\pi$
$888$	−4.53985	−0.152348
$889$	−21.0975	−0.707589
$890$	−15.3365	−0.514080
$891$	0	0
$892$	3.28324	0.109931
$893$	7.57365	0.253443
$894$	−5.87806	−0.196592
$895$	13.8722	0.463698
$896$	−3.28324	−0.109685
$897$	7.51225	0.250827
$898$	−12.1820	−0.406517
$899$	15.5122	0.517362
$900$	−1.47289	−0.0490962
$901$	−1.24289	−0.0414066
$902$	0	0
$903$	23.5122	0.782439
$904$	−19.9302	−0.662867
$905$	30.4186	1.01115
$906$	−7.66080	−0.254513
$907$	1.93711	0.0643208	0.0321604	−	0.999483i	$-0.489761\pi$
$907$	1.93711	0.0643208	0.0321604	+	0.999483i	$0.489761\pi$
$908$	−16.3963	−0.544129
$909$	9.56165	0.317140
$910$	27.3567	0.906867
$911$	−10.6481	−0.352789	−0.176394	−	0.984320i	$-0.556443\pi$
$911$	−10.6481	−0.352789	−0.176394	+	0.984320i	$0.556443\pi$
$912$	−1.69324	−0.0560687
$913$	0	0
$914$	−11.5833	−0.383141
$915$	−10.8104	−0.357379
$916$	−6.50359	−0.214885
$917$	−4.09410	−0.135199
$918$	−6.72468	−0.221948
$919$	32.0735	1.05801	0.529004	−	0.848620i	$-0.322566\pi$
$919$	32.0735	1.05801	0.529004	+	0.848620i	$0.322566\pi$
$920$	3.18000	0.104842
$921$	−12.6414	−0.416547
$922$	8.69225	0.286264
$923$	49.4355	1.62719
$924$	0	0
$925$	−6.68669	−0.219857
$926$	−10.0559	−0.330459
$927$	18.8034	0.617585
$928$	9.75612	0.320260
$929$	4.27606	0.140293	0.0701465	−	0.997537i	$-0.477653\pi$
$929$	4.27606	0.140293	0.0701465	+	0.997537i	$0.477653\pi$
$930$	2.98612	0.0979187
$931$	6.39984	0.209746
$932$	17.1119	0.560519
$933$	−34.2352	−1.12081
$934$	11.3058	0.369936
$935$	0	0
$936$	4.43662	0.145015
$937$	−34.8517	−1.13855	−0.569277	−	0.822146i	$-0.692777\pi$
$937$	−34.8517	−1.13855	−0.569277	+	0.822146i	$0.692777\pi$
$938$	27.7571	0.906301
$939$	−32.9713	−1.07598
$940$	−8.40035	−0.273989
$941$	−49.6544	−1.61869	−0.809344	−	0.587335i	$-0.800177\pi$
$941$	−49.6544	−1.61869	−0.809344	+	0.587335i	$0.800177\pi$
$942$	−1.38647	−0.0451737
$943$	−5.65237	−0.184067
$944$	−13.4760	−0.438606
$945$	−6.16612	−0.200584
$946$	0	0
$947$	−56.4580	−1.83464	−0.917320	−	0.398152i	$-0.869652\pi$
$947$	−56.4580	−1.83464	−0.917320	+	0.398152i	$0.869652\pi$
$948$	10.7876	0.350364
$949$	−47.1584	−1.53083
$950$	−2.49394	−0.0809142
$951$	−7.50260	−0.243288
$952$	−22.0787	−0.715575
$953$	−21.3714	−0.692286	−0.346143	−	0.938182i	$-0.612509\pi$
$953$	−21.3714	−0.692286	−0.346143	+	0.938182i	$0.612509\pi$
$954$	−0.184825	−0.00598393
$955$	13.2906	0.430074
$956$	−4.87324	−0.157612
$957$	0	0
$958$	−20.6908	−0.668488
$959$	−68.9544	−2.22665
$960$	1.87806	0.0606142
$961$	−28.4719	−0.918448
$962$	20.1416	0.649391
$963$	−4.07253	−0.131236
$964$	3.18965	0.102732
$965$	20.4206	0.657363
$966$	−5.55930	−0.178867
$967$	−58.2894	−1.87446	−0.937230	−	0.348711i	$-0.886620\pi$
$967$	−58.2894	−1.87446	−0.937230	+	0.348711i	$0.886620\pi$
$968$	−11.0000	−0.353553
$969$	−11.3865	−0.365786
$970$	−19.2107	−0.616819
$971$	−29.2403	−0.938365	−0.469182	−	0.883101i	$-0.655451\pi$
$971$	−29.2403	−0.938365	−0.469182	+	0.883101i	$0.655451\pi$
$972$	−1.00000	−0.0320750
$973$	15.1867	0.486863
$974$	−3.77891	−0.121084
$975$	6.53463	0.209276
$976$	−5.75612	−0.184249
$977$	−5.96609	−0.190872	−0.0954360	−	0.995436i	$-0.530425\pi$
$977$	−5.96609	−0.190872	−0.0954360	+	0.995436i	$0.530425\pi$
$978$	9.20165	0.294236
$979$	0	0
$980$	−7.09841	−0.226750
$981$	−1.18409	−0.0378049
$982$	−31.7638	−1.01362
$983$	−4.58429	−0.146216	−0.0731080	−	0.997324i	$-0.523292\pi$
$983$	−4.58429	−0.146216	−0.0731080	+	0.997324i	$0.523292\pi$
$984$	−3.33821	−0.106418
$985$	24.7177	0.787572
$986$	65.6068	2.08935
$987$	14.6855	0.467446
$988$	7.51225	0.238996
$989$	−12.1258	−0.385577
$990$	0	0
$991$	53.6940	1.70564	0.852822	−	0.522201i	$-0.174889\pi$
$991$	53.6940	1.70564	0.852822	+	0.522201i	$0.174889\pi$
$992$	1.59000	0.0504826
$993$	8.35095	0.265009
$994$	−36.5838	−1.16037
$995$	−9.15224	−0.290146
$996$	−14.6656	−0.464698
$997$	−21.3298	−0.675521	−0.337760	−	0.941232i	$-0.609669\pi$
$997$	−21.3298	−0.675521	−0.337760	+	0.941232i	$0.609669\pi$
$998$	1.35812	0.0429907
$999$	−4.53985	−0.143635

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1002.2.a.i.1.1	✓	4
3.2	odd	2		3006.2.a.s.1.4		4
4.3	odd	2		8016.2.a.o.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.i.1.1	✓	4	1.1	even	1	trivial
3006.2.a.s.1.4		4	3.2	odd	2
8016.2.a.o.1.1		4	4.3	odd	2

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:	\( N \)	\(=\)	\( 1002 = 2 \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.87806 q^{5} -1.00000 q^{6} -3.28324 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.87806 q^{5} -1.00000 q^{6} -3.28324 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.87806 q^{10} -1.00000 q^{12} +4.43662 q^{13} -3.28324 q^{14} +1.87806 q^{15} +1.00000 q^{16} +6.72468 q^{17} +1.00000 q^{18} +1.69324 q^{19} -1.87806 q^{20} +3.28324 q^{21} -1.69324 q^{23} -1.00000 q^{24} -1.47289 q^{25} +4.43662 q^{26} -1.00000 q^{27} -3.28324 q^{28} +9.75612 q^{29} +1.87806 q^{30} +1.59000 q^{31} +1.00000 q^{32} +6.72468 q^{34} +6.16612 q^{35} +1.00000 q^{36} +4.53985 q^{37} +1.69324 q^{38} -4.43662 q^{39} -1.87806 q^{40} +3.33821 q^{41} +3.28324 q^{42} +7.16130 q^{43} -1.87806 q^{45} -1.69324 q^{46} +4.47289 q^{47} -1.00000 q^{48} +3.77965 q^{49} -1.47289 q^{50} -6.72468 q^{51} +4.43662 q^{52} -0.184825 q^{53} -1.00000 q^{54} -3.28324 q^{56} -1.69324 q^{57} +9.75612 q^{58} -13.4760 q^{59} +1.87806 q^{60} -5.75612 q^{61} +1.59000 q^{62} -3.28324 q^{63} +1.00000 q^{64} -8.33224 q^{65} -8.45418 q^{67} +6.72468 q^{68} +1.69324 q^{69} +6.16612 q^{70} +11.1426 q^{71} +1.00000 q^{72} -10.6294 q^{73} +4.53985 q^{74} +1.47289 q^{75} +1.69324 q^{76} -4.43662 q^{78} -10.7876 q^{79} -1.87806 q^{80} +1.00000 q^{81} +3.33821 q^{82} +14.6656 q^{83} +3.28324 q^{84} -12.6294 q^{85} +7.16130 q^{86} -9.75612 q^{87} +8.16612 q^{89} -1.87806 q^{90} -14.5665 q^{91} -1.69324 q^{92} -1.59000 q^{93} +4.47289 q^{94} -3.18000 q^{95} -1.00000 q^{96} +10.2290 q^{97} +3.77965 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 5 q^{5} - 4 q^{6} + q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 5 * q^5 - 4 * q^6 + q^7 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 5 q^{5} - 4 q^{6} + q^{7} + 4 q^{8} + 4 q^{9} + 5 q^{10} - 4 q^{12} + 8 q^{13} + q^{14} - 5 q^{15} + 4 q^{16} + 10 q^{17} + 4 q^{18} - 2 q^{19} + 5 q^{20} - q^{21} + 2 q^{23} - 4 q^{24} + 5 q^{25} + 8 q^{26} - 4 q^{27} + q^{28} + 14 q^{29} - 5 q^{30} + q^{31} + 4 q^{32} + 10 q^{34} + 5 q^{35} + 4 q^{36} + 5 q^{37} - 2 q^{38} - 8 q^{39} + 5 q^{40} + 14 q^{41} - q^{42} + 2 q^{43} + 5 q^{45} + 2 q^{46} + 7 q^{47} - 4 q^{48} + 13 q^{49} + 5 q^{50} - 10 q^{51} + 8 q^{52} + 3 q^{53} - 4 q^{54} + q^{56} + 2 q^{57} + 14 q^{58} - 5 q^{59} - 5 q^{60} + 2 q^{61} + q^{62} + q^{63} + 4 q^{64} + 6 q^{65} - 7 q^{67} + 10 q^{68} - 2 q^{69} + 5 q^{70} + 2 q^{71} + 4 q^{72} + 2 q^{73} + 5 q^{74} - 5 q^{75} - 2 q^{76} - 8 q^{78} - 10 q^{79} + 5 q^{80} + 4 q^{81} + 14 q^{82} + 13 q^{83} - q^{84} - 6 q^{85} + 2 q^{86} - 14 q^{87} + 13 q^{89} + 5 q^{90} - 30 q^{91} + 2 q^{92} - q^{93} + 7 q^{94} - 2 q^{95} - 4 q^{96} + 5 q^{97} + 13 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 5 * q^5 - 4 * q^6 + q^7 + 4 * q^8 + 4 * q^9 + 5 * q^10 - 4 * q^12 + 8 * q^13 + q^14 - 5 * q^15 + 4 * q^16 + 10 * q^17 + 4 * q^18 - 2 * q^19 + 5 * q^20 - q^21 + 2 * q^23 - 4 * q^24 + 5 * q^25 + 8 * q^26 - 4 * q^27 + q^28 + 14 * q^29 - 5 * q^30 + q^31 + 4 * q^32 + 10 * q^34 + 5 * q^35 + 4 * q^36 + 5 * q^37 - 2 * q^38 - 8 * q^39 + 5 * q^40 + 14 * q^41 - q^42 + 2 * q^43 + 5 * q^45 + 2 * q^46 + 7 * q^47 - 4 * q^48 + 13 * q^49 + 5 * q^50 - 10 * q^51 + 8 * q^52 + 3 * q^53 - 4 * q^54 + q^56 + 2 * q^57 + 14 * q^58 - 5 * q^59 - 5 * q^60 + 2 * q^61 + q^62 + q^63 + 4 * q^64 + 6 * q^65 - 7 * q^67 + 10 * q^68 - 2 * q^69 + 5 * q^70 + 2 * q^71 + 4 * q^72 + 2 * q^73 + 5 * q^74 - 5 * q^75 - 2 * q^76 - 8 * q^78 - 10 * q^79 + 5 * q^80 + 4 * q^81 + 14 * q^82 + 13 * q^83 - q^84 - 6 * q^85 + 2 * q^86 - 14 * q^87 + 13 * q^89 + 5 * q^90 - 30 * q^91 + 2 * q^92 - q^93 + 7 * q^94 - 2 * q^95 - 4 * q^96 + 5 * q^97 + 13 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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