LMFDB - Embedded newform 5082.2.a.cb.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5082,2,Mod(1,5082)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5082, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5082.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5082 = 2 \cdot 3 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5082.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:	$40.5799743072$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.69264.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 13x^{2} - 6x + 22 $ x^4 - 13x^2 - 6x + 22
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.13846$ of defining polynomial
Character	$\chi$	$=$	5082.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.87051 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.87051 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.87051 q^{10} +1.00000 q^{12} -6.11032 q^{13} +1.00000 q^{14} +3.87051 q^{15} +1.00000 q^{16} +1.87051 q^{17} -1.00000 q^{18} +5.11032 q^{19} +3.87051 q^{20} -1.00000 q^{21} -3.60256 q^{23} -1.00000 q^{24} +9.98083 q^{25} +6.11032 q^{26} +1.00000 q^{27} -1.00000 q^{28} +3.83341 q^{29} -3.87051 q^{30} -2.97187 q^{31} -1.00000 q^{32} -1.87051 q^{34} -3.87051 q^{35} +1.00000 q^{36} +7.33461 q^{37} -5.11032 q^{38} -6.11032 q^{39} -3.87051 q^{40} -9.67578 q^{41} +1.00000 q^{42} +10.9898 q^{43} +3.87051 q^{45} +3.60256 q^{46} +0.889678 q^{47} +1.00000 q^{48} +1.00000 q^{49} -9.98083 q^{50} +1.87051 q^{51} -6.11032 q^{52} +6.42700 q^{53} -1.00000 q^{54} +1.00000 q^{56} +5.11032 q^{57} -3.83341 q^{58} -3.35014 q^{59} +3.87051 q^{60} +5.50120 q^{61} +2.97187 q^{62} -1.00000 q^{63} +1.00000 q^{64} -23.6500 q^{65} -7.70392 q^{67} +1.87051 q^{68} -3.60256 q^{69} +3.87051 q^{70} +12.4360 q^{71} -1.00000 q^{72} -1.46410 q^{73} -7.33461 q^{74} +9.98083 q^{75} +5.11032 q^{76} +6.11032 q^{78} +13.5026 q^{79} +3.87051 q^{80} +1.00000 q^{81} +9.67578 q^{82} +7.32564 q^{83} -1.00000 q^{84} +7.23981 q^{85} -10.9898 q^{86} +3.83341 q^{87} -11.3923 q^{89} -3.87051 q^{90} +6.11032 q^{91} -3.60256 q^{92} -2.97187 q^{93} -0.889678 q^{94} +19.7795 q^{95} -1.00000 q^{96} -1.05770 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 + 4 * q^5 - 4 * q^6 - 4 * q^7 - 4 * q^8 + 4 * q^9	$ 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} + 4 q^{9} - 4 q^{10} + 4 q^{12} - 6 q^{13} + 4 q^{14} + 4 q^{15} + 4 q^{16} - 4 q^{17} - 4 q^{18} + 2 q^{19} + 4 q^{20} - 4 q^{21} + 4 q^{23} - 4 q^{24} + 10 q^{25} + 6 q^{26} + 4 q^{27} - 4 q^{28} + 6 q^{29} - 4 q^{30} + 2 q^{31} - 4 q^{32} + 4 q^{34} - 4 q^{35} + 4 q^{36} + 4 q^{37} - 2 q^{38} - 6 q^{39} - 4 q^{40} - 4 q^{41} + 4 q^{42} - 2 q^{43} + 4 q^{45} - 4 q^{46} + 22 q^{47} + 4 q^{48} + 4 q^{49} - 10 q^{50} - 4 q^{51} - 6 q^{52} + 14 q^{53} - 4 q^{54} + 4 q^{56} + 2 q^{57} - 6 q^{58} + 12 q^{59} + 4 q^{60} + 6 q^{61} - 2 q^{62} - 4 q^{63} + 4 q^{64} - 14 q^{65} - 10 q^{67} - 4 q^{68} + 4 q^{69} + 4 q^{70} + 22 q^{71} - 4 q^{72} + 8 q^{73} - 4 q^{74} + 10 q^{75} + 2 q^{76} + 6 q^{78} - 6 q^{79} + 4 q^{80} + 4 q^{81} + 4 q^{82} + 20 q^{83} - 4 q^{84} + 22 q^{85} + 2 q^{86} + 6 q^{87} - 4 q^{89} - 4 q^{90} + 6 q^{91} + 4 q^{92} + 2 q^{93} - 22 q^{94} + 10 q^{95} - 4 q^{96} + 12 q^{97} - 4 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 + 4 * q^5 - 4 * q^6 - 4 * q^7 - 4 * q^8 + 4 * q^9 - 4 * q^10 + 4 * q^12 - 6 * q^13 + 4 * q^14 + 4 * q^15 + 4 * q^16 - 4 * q^17 - 4 * q^18 + 2 * q^19 + 4 * q^20 - 4 * q^21 + 4 * q^23 - 4 * q^24 + 10 * q^25 + 6 * q^26 + 4 * q^27 - 4 * q^28 + 6 * q^29 - 4 * q^30 + 2 * q^31 - 4 * q^32 + 4 * q^34 - 4 * q^35 + 4 * q^36 + 4 * q^37 - 2 * q^38 - 6 * q^39 - 4 * q^40 - 4 * q^41 + 4 * q^42 - 2 * q^43 + 4 * q^45 - 4 * q^46 + 22 * q^47 + 4 * q^48 + 4 * q^49 - 10 * q^50 - 4 * q^51 - 6 * q^52 + 14 * q^53 - 4 * q^54 + 4 * q^56 + 2 * q^57 - 6 * q^58 + 12 * q^59 + 4 * q^60 + 6 * q^61 - 2 * q^62 - 4 * q^63 + 4 * q^64 - 14 * q^65 - 10 * q^67 - 4 * q^68 + 4 * q^69 + 4 * q^70 + 22 * q^71 - 4 * q^72 + 8 * q^73 - 4 * q^74 + 10 * q^75 + 2 * q^76 + 6 * q^78 - 6 * q^79 + 4 * q^80 + 4 * q^81 + 4 * q^82 + 20 * q^83 - 4 * q^84 + 22 * q^85 + 2 * q^86 + 6 * q^87 - 4 * q^89 - 4 * q^90 + 6 * q^91 + 4 * q^92 + 2 * q^93 - 22 * q^94 + 10 * q^95 - 4 * q^96 + 12 * q^97 - 4 * 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
$5$	3.87051	1.73094	0.865472	−	0.500958i	$-0.167019\pi$
$5$	3.87051	1.73094	0.865472	+	0.500958i	$0.167019\pi$
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−3.87051	−1.22396
$11$	0	0
$11$	0	0
$12$	1.00000	0.288675
$13$	−6.11032	−1.69470	−0.847349	−	0.531036i	$-0.821803\pi$
$13$	−6.11032	−1.69470	−0.847349	+	0.531036i	$0.821803\pi$
$14$	1.00000	0.267261
$15$	3.87051	0.999361
$16$	1.00000	0.250000
$17$	1.87051	0.453665	0.226832	−	0.973934i	$-0.427163\pi$
$17$	1.87051	0.453665	0.226832	+	0.973934i	$0.427163\pi$
$18$	−1.00000	−0.235702
$19$	5.11032	1.17239	0.586194	−	0.810171i	$-0.300626\pi$
$19$	5.11032	1.17239	0.586194	+	0.810171i	$0.300626\pi$
$20$	3.87051	0.865472
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−3.60256	−0.751185	−0.375593	−	0.926785i	$-0.622561\pi$
$23$	−3.60256	−0.751185	−0.375593	+	0.926785i	$0.622561\pi$
$24$	−1.00000	−0.204124
$25$	9.98083	1.99617
$26$	6.11032	1.19833
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	3.83341	0.711846	0.355923	−	0.934515i	$-0.384167\pi$
$29$	3.83341	0.711846	0.355923	+	0.934515i	$0.384167\pi$
$30$	−3.87051	−0.706655
$31$	−2.97187	−0.533763	−0.266881	−	0.963729i	$-0.585993\pi$
$31$	−2.97187	−0.533763	−0.266881	+	0.963729i	$0.585993\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.87051	−0.320789
$35$	−3.87051	−0.654235
$36$	1.00000	0.166667
$37$	7.33461	1.20580	0.602901	−	0.797816i	$-0.294011\pi$
$37$	7.33461	1.20580	0.602901	+	0.797816i	$0.294011\pi$
$38$	−5.11032	−0.829004
$39$	−6.11032	−0.978435
$40$	−3.87051	−0.611981
$41$	−9.67578	−1.51110	−0.755552	−	0.655089i	$-0.772631\pi$
$41$	−9.67578	−1.51110	−0.755552	+	0.655089i	$0.772631\pi$
$42$	1.00000	0.154303
$43$	10.9898	1.67593	0.837964	−	0.545726i	$-0.183746\pi$
$43$	10.9898	1.67593	0.837964	+	0.545726i	$0.183746\pi$
$44$	0	0
$45$	3.87051	0.576981
$46$	3.60256	0.531168
$47$	0.889678	0.129773	0.0648864	−	0.997893i	$-0.479331\pi$
$47$	0.889678	0.129773	0.0648864	+	0.997893i	$0.479331\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	−9.98083	−1.41150
$51$	1.87051	0.261923
$52$	−6.11032	−0.847349
$53$	6.42700	0.882817	0.441408	−	0.897306i	$-0.354479\pi$
$53$	6.42700	0.882817	0.441408	+	0.897306i	$0.354479\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	1.00000	0.133631
$57$	5.11032	0.676879
$58$	−3.83341	−0.503351
$59$	−3.35014	−0.436151	−0.218075	−	0.975932i	$-0.569978\pi$
$59$	−3.35014	−0.436151	−0.218075	+	0.975932i	$0.569978\pi$
$60$	3.87051	0.499680
$61$	5.50120	0.704357	0.352178	−	0.935933i	$-0.385441\pi$
$61$	5.50120	0.704357	0.352178	+	0.935933i	$0.385441\pi$
$62$	2.97187	0.377427
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−23.6500	−2.93343
$66$	0	0
$67$	−7.70392	−0.941183	−0.470592	−	0.882351i	$-0.655959\pi$
$67$	−7.70392	−0.941183	−0.470592	+	0.882351i	$0.655959\pi$
$68$	1.87051	0.226832
$69$	−3.60256	−0.433697
$70$	3.87051	0.462614
$71$	12.4360	1.47588	0.737939	−	0.674868i	$-0.235799\pi$
$71$	12.4360	1.47588	0.737939	+	0.674868i	$0.235799\pi$
$72$	−1.00000	−0.117851
$73$	−1.46410	−0.171360	−0.0856801	−	0.996323i	$-0.527306\pi$
$73$	−1.46410	−0.171360	−0.0856801	+	0.996323i	$0.527306\pi$
$74$	−7.33461	−0.852631
$75$	9.98083	1.15249
$76$	5.11032	0.586194
$77$	0	0
$78$	6.11032	0.691858
$79$	13.5026	1.51916	0.759582	−	0.650411i	$-0.225403\pi$
$79$	13.5026	1.51916	0.759582	+	0.650411i	$0.225403\pi$
$80$	3.87051	0.432736
$81$	1.00000	0.111111
$82$	9.67578	1.06851
$83$	7.32564	0.804094	0.402047	−	0.915619i	$-0.368299\pi$
$83$	7.32564	0.804094	0.402047	+	0.915619i	$0.368299\pi$
$84$	−1.00000	−0.109109
$85$	7.23981	0.785268
$86$	−10.9898	−1.18506
$87$	3.83341	0.410985
$88$	0	0
$89$	−11.3923	−1.20758	−0.603791	−	0.797143i	$-0.706344\pi$
$89$	−11.3923	−1.20758	−0.603791	+	0.797143i	$0.706344\pi$
$90$	−3.87051	−0.407987
$91$	6.11032	0.640536
$92$	−3.60256	−0.375593
$93$	−2.97187	−0.308168
$94$	−0.889678	−0.0917633
$95$	19.7795	2.02934
$96$	−1.00000	−0.102062
$97$	−1.05770	−0.107393	−0.0536964	−	0.998557i	$-0.517100\pi$
$97$	−1.05770	−0.107393	−0.0536964	+	0.998557i	$0.517100\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	9.98083	0.998083
$101$	1.46410	0.145684	0.0728418	−	0.997344i	$-0.476793\pi$
$101$	1.46410	0.145684	0.0728418	+	0.997344i	$0.476793\pi$
$102$	−1.87051	−0.185208
$103$	12.7897	1.26021	0.630106	−	0.776509i	$-0.283012\pi$
$103$	12.7897	1.26021	0.630106	+	0.776509i	$0.283012\pi$
$104$	6.11032	0.599166
$105$	−3.87051	−0.377723
$106$	−6.42700	−0.624246
$107$	11.6462	1.12588	0.562941	−	0.826497i	$-0.309670\pi$
$107$	11.6462	1.12588	0.562941	+	0.826497i	$0.309670\pi$
$108$	1.00000	0.0962250
$109$	−5.96530	−0.571372	−0.285686	−	0.958323i	$-0.592221\pi$
$109$	−5.96530	−0.571372	−0.285686	+	0.958323i	$0.592221\pi$
$110$	0	0
$111$	7.33461	0.696170
$112$	−1.00000	−0.0944911
$113$	16.9040	1.59019	0.795096	−	0.606484i	$-0.207421\pi$
$113$	16.9040	1.59019	0.795096	+	0.606484i	$0.207421\pi$
$114$	−5.11032	−0.478626
$115$	−13.9437	−1.30026
$116$	3.83341	0.355923
$117$	−6.11032	−0.564899
$118$	3.35014	0.308405
$119$	−1.87051	−0.171469
$120$	−3.87051	−0.353327
$121$	0	0
$122$	−5.50120	−0.498055
$123$	−9.67578	−0.872436
$124$	−2.97187	−0.266881
$125$	19.2783	1.72431
$126$	1.00000	0.0890871
$127$	15.2051	1.34924	0.674618	−	0.738167i	$-0.264308\pi$
$127$	15.2051	1.34924	0.674618	+	0.738167i	$0.264308\pi$
$128$	−1.00000	−0.0883883
$129$	10.9898	0.967597
$130$	23.6500	2.07425
$131$	12.4333	1.08630	0.543151	−	0.839635i	$-0.317231\pi$
$131$	12.4333	1.08630	0.543151	+	0.839635i	$0.317231\pi$
$132$	0	0
$133$	−5.11032	−0.443121
$134$	7.70392	0.665517
$135$	3.87051	0.333120
$136$	−1.87051	−0.160395
$137$	−5.20512	−0.444703	−0.222352	−	0.974967i	$-0.571373\pi$
$137$	−5.20512	−0.444703	−0.222352	+	0.974967i	$0.571373\pi$
$138$	3.60256	0.306670
$139$	−12.6692	−1.07459	−0.537295	−	0.843395i	$-0.680554\pi$
$139$	−12.6692	−1.07459	−0.537295	+	0.843395i	$0.680554\pi$
$140$	−3.87051	−0.327118
$141$	0.889678	0.0749244
$142$	−12.4360	−1.04360
$143$	0	0
$144$	1.00000	0.0833333
$145$	14.8372	1.23217
$146$	1.46410	0.121170
$147$	1.00000	0.0824786
$148$	7.33461	0.602901
$149$	20.5026	1.67964	0.839820	−	0.542864i	$-0.182660\pi$
$149$	20.5026	1.67964	0.839820	+	0.542864i	$0.182660\pi$
$150$	−9.98083	−0.814931
$151$	−13.9823	−1.13786	−0.568930	−	0.822386i	$-0.692642\pi$
$151$	−13.9823	−1.13786	−0.568930	+	0.822386i	$0.692642\pi$
$152$	−5.11032	−0.414502
$153$	1.87051	0.151222
$154$	0	0
$155$	−11.5026	−0.923913
$156$	−6.11032	−0.489217
$157$	−24.6309	−1.96576	−0.982879	−	0.184252i	$-0.941014\pi$
$157$	−24.6309	−1.96576	−0.982879	+	0.184252i	$0.941014\pi$
$158$	−13.5026	−1.07421
$159$	6.42700	0.509694
$160$	−3.87051	−0.305990
$161$	3.60256	0.283921
$162$	−1.00000	−0.0785674
$163$	19.3310	1.51412	0.757059	−	0.653346i	$-0.226635\pi$
$163$	19.3310	1.51412	0.757059	+	0.653346i	$0.226635\pi$
$164$	−9.67578	−0.755552
$165$	0	0
$166$	−7.32564	−0.568580
$167$	12.6692	0.980374	0.490187	−	0.871617i	$-0.336929\pi$
$167$	12.6692	0.980374	0.490187	+	0.871617i	$0.336929\pi$
$168$	1.00000	0.0771517
$169$	24.3360	1.87200
$170$	−7.23981	−0.555268
$171$	5.11032	0.390796
$172$	10.9898	0.837964
$173$	10.8128	0.822083	0.411041	−	0.911617i	$-0.365165\pi$
$173$	10.8128	0.822083	0.411041	+	0.911617i	$0.365165\pi$
$174$	−3.83341	−0.290610
$175$	−9.98083	−0.754480
$176$	0	0
$177$	−3.35014	−0.251812
$178$	11.3923	0.853889
$179$	−3.66415	−0.273871	−0.136936	−	0.990580i	$-0.543725\pi$
$179$	−3.66415	−0.273871	−0.136936	+	0.990580i	$0.543725\pi$
$180$	3.87051	0.288491
$181$	−16.3627	−1.21623	−0.608117	−	0.793848i	$-0.708075\pi$
$181$	−16.3627	−1.21623	−0.608117	+	0.793848i	$0.708075\pi$
$182$	−6.11032	−0.452927
$183$	5.50120	0.406660
$184$	3.60256	0.265584
$185$	28.3887	2.08718
$186$	2.97187	0.217908
$187$	0	0
$188$	0.889678	0.0648864
$189$	−1.00000	−0.0727393
$190$	−19.7795	−1.43496
$191$	−2.45903	−0.177929	−0.0889647	−	0.996035i	$-0.528356\pi$
$191$	−2.45903	−0.177929	−0.0889647	+	0.996035i	$0.528356\pi$
$192$	1.00000	0.0721688
$193$	7.79631	0.561191	0.280595	−	0.959826i	$-0.409468\pi$
$193$	7.79631	0.561191	0.280595	+	0.959826i	$0.409468\pi$
$194$	1.05770	0.0759381
$195$	−23.6500	−1.69362
$196$	1.00000	0.0714286
$197$	−26.9823	−1.92241	−0.961203	−	0.275842i	$-0.911043\pi$
$197$	−26.9823	−1.92241	−0.961203	+	0.275842i	$0.911043\pi$
$198$	0	0
$199$	13.1052	0.929002	0.464501	−	0.885573i	$-0.346234\pi$
$199$	13.1052	0.929002	0.464501	+	0.885573i	$0.346234\pi$
$200$	−9.98083	−0.705751
$201$	−7.70392	−0.543392
$202$	−1.46410	−0.103014
$203$	−3.83341	−0.269053
$204$	1.87051	0.130962
$205$	−37.4502	−2.61563
$206$	−12.7897	−0.891104
$207$	−3.60256	−0.250395
$208$	−6.11032	−0.423675
$209$	0	0
$210$	3.87051	0.267090
$211$	−1.43090	−0.0985071	−0.0492535	−	0.998786i	$-0.515684\pi$
$211$	−1.43090	−0.0985071	−0.0492535	+	0.998786i	$0.515684\pi$
$212$	6.42700	0.441408
$213$	12.4360	0.852098
$214$	−11.6462	−0.796119
$215$	42.5361	2.90094
$216$	−1.00000	−0.0680414
$217$	2.97187	0.201743
$218$	5.96530	0.404021
$219$	−1.46410	−0.0989348
$220$	0	0
$221$	−11.4294	−0.768825
$222$	−7.33461	−0.492267
$223$	−5.53830	−0.370872	−0.185436	−	0.982656i	$-0.559370\pi$
$223$	−5.53830	−0.370872	−0.185436	+	0.982656i	$0.559370\pi$
$224$	1.00000	0.0668153
$225$	9.98083	0.665389
$226$	−16.9040	−1.12443
$227$	−12.4539	−0.826594	−0.413297	−	0.910596i	$-0.635623\pi$
$227$	−12.4539	−0.826594	−0.413297	+	0.910596i	$0.635623\pi$
$228$	5.11032	0.338439
$229$	−29.7965	−1.96901	−0.984504	−	0.175363i	$-0.943890\pi$
$229$	−29.7965	−1.96901	−0.984504	+	0.175363i	$0.943890\pi$
$230$	13.9437	0.919422
$231$	0	0
$232$	−3.83341	−0.251676
$233$	−13.3140	−0.872230	−0.436115	−	0.899891i	$-0.643646\pi$
$233$	−13.3140	−0.872230	−0.436115	+	0.899891i	$0.643646\pi$
$234$	6.11032	0.399444
$235$	3.44351	0.224630
$236$	−3.35014	−0.218075
$237$	13.5026	0.877090
$238$	1.87051	0.121247
$239$	−23.5783	−1.52515	−0.762575	−	0.646900i	$-0.776065\pi$
$239$	−23.5783	−1.52515	−0.762575	+	0.646900i	$0.776065\pi$
$240$	3.87051	0.249840
$241$	−26.2386	−1.69018	−0.845088	−	0.534628i	$-0.820452\pi$
$241$	−26.2386	−1.69018	−0.845088	+	0.534628i	$0.820452\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	5.50120	0.352178
$245$	3.87051	0.247278
$246$	9.67578	0.616905
$247$	−31.2257	−1.98684
$248$	2.97187	0.188714
$249$	7.32564	0.464244
$250$	−19.2783	−1.21927
$251$	−21.7989	−1.37593	−0.687967	−	0.725742i	$-0.741497\pi$
$251$	−21.7989	−1.37593	−0.687967	+	0.725742i	$0.741497\pi$
$252$	−1.00000	−0.0629941
$253$	0	0
$254$	−15.2051	−0.954054
$255$	7.23981	0.453375
$256$	1.00000	0.0625000
$257$	22.1309	1.38049	0.690244	−	0.723576i	$-0.257503\pi$
$257$	22.1309	1.38049	0.690244	+	0.723576i	$0.257503\pi$
$258$	−10.9898	−0.684195
$259$	−7.33461	−0.455750
$260$	−23.6500	−1.46671
$261$	3.83341	0.237282
$262$	−12.4333	−0.768132
$263$	−6.22064	−0.383581	−0.191791	−	0.981436i	$-0.561429\pi$
$263$	−6.22064	−0.383581	−0.191791	+	0.981436i	$0.561429\pi$
$264$	0	0
$265$	24.8758	1.52811
$266$	5.11032	0.313334
$267$	−11.3923	−0.697198
$268$	−7.70392	−0.470592
$269$	7.81424	0.476442	0.238221	−	0.971211i	$-0.423436\pi$
$269$	7.81424	0.476442	0.238221	+	0.971211i	$0.423436\pi$
$270$	−3.87051	−0.235552
$271$	−26.6321	−1.61779	−0.808893	−	0.587956i	$-0.799933\pi$
$271$	−26.6321	−1.61779	−0.808893	+	0.587956i	$0.799933\pi$
$272$	1.87051	0.113416
$273$	6.11032	0.369814
$274$	5.20512	0.314453
$275$	0	0
$276$	−3.60256	−0.216849
$277$	19.2810	1.15848	0.579242	−	0.815156i	$-0.303349\pi$
$277$	19.2810	1.15848	0.579242	+	0.815156i	$0.303349\pi$
$278$	12.6692	0.759849
$279$	−2.97187	−0.177921
$280$	3.87051	0.231307
$281$	15.3205	0.913945	0.456972	−	0.889481i	$-0.348934\pi$
$281$	15.3205	0.913945	0.456972	+	0.889481i	$0.348934\pi$
$282$	−0.889678	−0.0529796
$283$	−10.6486	−0.632995	−0.316497	−	0.948593i	$-0.602507\pi$
$283$	−10.6486	−0.632995	−0.316497	+	0.948593i	$0.602507\pi$
$284$	12.4360	0.737939
$285$	19.7795	1.17164
$286$	0	0
$287$	9.67578	0.571143
$288$	−1.00000	−0.0589256
$289$	−13.5012	−0.794188
$290$	−14.8372	−0.871273
$291$	−1.05770	−0.0620032
$292$	−1.46410	−0.0856801
$293$	17.2577	1.00821	0.504104	−	0.863643i	$-0.331823\pi$
$293$	17.2577	1.00821	0.504104	+	0.863643i	$0.331823\pi$
$294$	−1.00000	−0.0583212
$295$	−12.9667	−0.754952
$296$	−7.33461	−0.426316
$297$	0	0
$298$	−20.5026	−1.18769
$299$	22.0128	1.27303
$300$	9.98083	0.576243
$301$	−10.9898	−0.633441
$302$	13.9823	0.804588
$303$	1.46410	0.0841104
$304$	5.11032	0.293097
$305$	21.2924	1.21920
$306$	−1.87051	−0.106930
$307$	17.7410	1.01253	0.506267	−	0.862377i	$-0.331025\pi$
$307$	17.7410	1.01253	0.506267	+	0.862377i	$0.331025\pi$
$308$	0	0
$309$	12.7897	0.727583
$310$	11.5026	0.653305
$311$	7.11032	0.403189	0.201595	−	0.979469i	$-0.435388\pi$
$311$	7.11032	0.403189	0.201595	+	0.979469i	$0.435388\pi$
$312$	6.11032	0.345929
$313$	2.82977	0.159948	0.0799739	−	0.996797i	$-0.474516\pi$
$313$	2.82977	0.159948	0.0799739	+	0.996797i	$0.474516\pi$
$314$	24.6309	1.39000
$315$	−3.87051	−0.218078
$316$	13.5026	0.759582
$317$	26.4924	1.48796	0.743981	−	0.668200i	$-0.232935\pi$
$317$	26.4924	1.48796	0.743981	+	0.668200i	$0.232935\pi$
$318$	−6.42700	−0.360408
$319$	0	0
$320$	3.87051	0.216368
$321$	11.6462	0.650029
$322$	−3.60256	−0.200763
$323$	9.55890	0.531871
$324$	1.00000	0.0555556
$325$	−60.9861	−3.38290
$326$	−19.3310	−1.07064
$327$	−5.96530	−0.329882
$328$	9.67578	0.534256
$329$	−0.889678	−0.0490495
$330$	0	0
$331$	16.7759	0.922087	0.461043	−	0.887378i	$-0.347475\pi$
$331$	16.7759	0.922087	0.461043	+	0.887378i	$0.347475\pi$
$332$	7.32564	0.402047
$333$	7.33461	0.401934
$334$	−12.6692	−0.693229
$335$	−29.8181	−1.62913
$336$	−1.00000	−0.0545545
$337$	16.7476	0.912299	0.456149	−	0.889903i	$-0.349228\pi$
$337$	16.7476	0.912299	0.456149	+	0.889903i	$0.349228\pi$
$338$	−24.3360	−1.32371
$339$	16.9040	0.918097
$340$	7.23981	0.392634
$341$	0	0
$342$	−5.11032	−0.276335
$343$	−1.00000	−0.0539949
$344$	−10.9898	−0.592530
$345$	−13.9437	−0.750705
$346$	−10.8128	−0.581300
$347$	−12.2771	−0.659069	−0.329535	−	0.944144i	$-0.606892\pi$
$347$	−12.2771	−0.659069	−0.329535	+	0.944144i	$0.606892\pi$
$348$	3.83341	0.205492
$349$	−25.2975	−1.35414	−0.677072	−	0.735916i	$-0.736752\pi$
$349$	−25.2975	−1.35414	−0.677072	+	0.735916i	$0.736752\pi$
$350$	9.98083	0.533498
$351$	−6.11032	−0.326145
$352$	0	0
$353$	4.79729	0.255334	0.127667	−	0.991817i	$-0.459251\pi$
$353$	4.79729	0.255334	0.127667	+	0.991817i	$0.459251\pi$
$354$	3.35014	0.178058
$355$	48.1335	2.55466
$356$	−11.3923	−0.603791
$357$	−1.87051	−0.0989978
$358$	3.66415	0.193656
$359$	8.53466	0.450442	0.225221	−	0.974308i	$-0.427690\pi$
$359$	8.53466	0.450442	0.225221	+	0.974308i	$0.427690\pi$
$360$	−3.87051	−0.203994
$361$	7.11539	0.374494
$362$	16.3627	0.860007
$363$	0	0
$364$	6.11032	0.320268
$365$	−5.66682	−0.296615
$366$	−5.50120	−0.287552
$367$	15.1666	0.791690	0.395845	−	0.918317i	$-0.370452\pi$
$367$	15.1666	0.791690	0.395845	+	0.918317i	$0.370452\pi$
$368$	−3.60256	−0.187796
$369$	−9.67578	−0.503701
$370$	−28.3887	−1.47586
$371$	−6.42700	−0.333673
$372$	−2.97187	−0.154084
$373$	5.90539	0.305769	0.152885	−	0.988244i	$-0.451144\pi$
$373$	5.90539	0.305769	0.152885	+	0.988244i	$0.451144\pi$
$374$	0	0
$375$	19.2783	0.995529
$376$	−0.889678	−0.0458816
$377$	−23.4234	−1.20636
$378$	1.00000	0.0514344
$379$	−22.9964	−1.18124	−0.590622	−	0.806949i	$-0.701117\pi$
$379$	−22.9964	−1.18124	−0.590622	+	0.806949i	$0.701117\pi$
$380$	19.7795	1.01467
$381$	15.2051	0.778982
$382$	2.45903	0.125815
$383$	−32.7594	−1.67393	−0.836963	−	0.547259i	$-0.815671\pi$
$383$	−32.7594	−1.67393	−0.836963	+	0.547259i	$0.815671\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−7.79631	−0.396822
$387$	10.9898	0.558643
$388$	−1.05770	−0.0536964
$389$	−20.5783	−1.04336	−0.521679	−	0.853142i	$-0.674694\pi$
$389$	−20.5783	−1.04336	−0.521679	+	0.853142i	$0.674694\pi$
$390$	23.6500	1.19757
$391$	−6.73861	−0.340786
$392$	−1.00000	−0.0505076
$393$	12.4333	0.627177
$394$	26.9823	1.35935
$395$	52.2620	2.62959
$396$	0	0
$397$	−25.8579	−1.29777	−0.648885	−	0.760886i	$-0.724765\pi$
$397$	−25.8579	−1.29777	−0.648885	+	0.760886i	$0.724765\pi$
$398$	−13.1052	−0.656904
$399$	−5.11032	−0.255836
$400$	9.98083	0.499041
$401$	−24.4655	−1.22175	−0.610875	−	0.791727i	$-0.709182\pi$
$401$	−24.4655	−1.22175	−0.610875	+	0.791727i	$0.709182\pi$
$402$	7.70392	0.384236
$403$	18.1591	0.904567
$404$	1.46410	0.0728418
$405$	3.87051	0.192327
$406$	3.83341	0.190249
$407$	0	0
$408$	−1.87051	−0.0926039
$409$	−3.46046	−0.171109	−0.0855543	−	0.996334i	$-0.527266\pi$
$409$	−3.46046	−0.171109	−0.0855543	+	0.996334i	$0.527266\pi$
$410$	37.4502	1.84953
$411$	−5.20512	−0.256749
$412$	12.7897	0.630106
$413$	3.35014	0.164849
$414$	3.60256	0.177056
$415$	28.3540	1.39184
$416$	6.11032	0.299583
$417$	−12.6692	−0.620414
$418$	0	0
$419$	24.2308	1.18375	0.591876	−	0.806029i	$-0.298388\pi$
$419$	24.2308	1.18375	0.591876	+	0.806029i	$0.298388\pi$
$420$	−3.87051	−0.188861
$421$	−28.1323	−1.37109	−0.685543	−	0.728032i	$-0.740435\pi$
$421$	−28.1323	−1.37109	−0.685543	+	0.728032i	$0.740435\pi$
$422$	1.43090	0.0696550
$423$	0.889678	0.0432576
$424$	−6.42700	−0.312123
$425$	18.6692	0.905590
$426$	−12.4360	−0.602525
$427$	−5.50120	−0.266222
$428$	11.6462	0.562941
$429$	0	0
$430$	−42.5361	−2.05127
$431$	−19.9333	−0.960152	−0.480076	−	0.877227i	$-0.659391\pi$
$431$	−19.9333	−0.960152	−0.480076	+	0.877227i	$0.659391\pi$
$432$	1.00000	0.0481125
$433$	−26.2807	−1.26297	−0.631486	−	0.775387i	$-0.717555\pi$
$433$	−26.2807	−1.26297	−0.631486	+	0.775387i	$0.717555\pi$
$434$	−2.97187	−0.142654
$435$	14.8372	0.711391
$436$	−5.96530	−0.285686
$437$	−18.4102	−0.880681
$438$	1.46410	0.0699575
$439$	10.7447	0.512815	0.256407	−	0.966569i	$-0.417461\pi$
$439$	10.7447	0.512815	0.256407	+	0.966569i	$0.417461\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	11.4294	0.543641
$443$	6.68715	0.317716	0.158858	−	0.987301i	$-0.449219\pi$
$443$	6.68715	0.317716	0.158858	+	0.987301i	$0.449219\pi$
$444$	7.33461	0.348085
$445$	−44.0940	−2.09026
$446$	5.53830	0.262246
$447$	20.5026	0.969741
$448$	−1.00000	−0.0472456
$449$	21.1731	0.999220	0.499610	−	0.866251i	$-0.333477\pi$
$449$	21.1731	0.999220	0.499610	+	0.866251i	$0.333477\pi$
$450$	−9.98083	−0.470501
$451$	0	0
$452$	16.9040	0.795096
$453$	−13.9823	−0.656944
$454$	12.4539	0.584490
$455$	23.6500	1.10873
$456$	−5.11032	−0.239313
$457$	−14.6139	−0.683611	−0.341805	−	0.939771i	$-0.611038\pi$
$457$	−14.6139	−0.683611	−0.341805	+	0.939771i	$0.611038\pi$
$458$	29.7965	1.39230
$459$	1.87051	0.0873078
$460$	−13.9437	−0.650130
$461$	−5.23981	−0.244042	−0.122021	−	0.992527i	$-0.538938\pi$
$461$	−5.23981	−0.244042	−0.122021	+	0.992527i	$0.538938\pi$
$462$	0	0
$463$	−6.27425	−0.291589	−0.145794	−	0.989315i	$-0.546574\pi$
$463$	−6.27425	−0.291589	−0.145794	+	0.989315i	$0.546574\pi$
$464$	3.83341	0.177962
$465$	−11.5026	−0.533422
$466$	13.3140	0.616760
$467$	10.7194	0.496037	0.248018	−	0.968755i	$-0.420221\pi$
$467$	10.7194	0.496037	0.248018	+	0.968755i	$0.420221\pi$
$468$	−6.11032	−0.282450
$469$	7.70392	0.355734
$470$	−3.44351	−0.158837
$471$	−24.6309	−1.13493
$472$	3.35014	0.154203
$473$	0	0
$474$	−13.5026	−0.620196
$475$	51.0053	2.34028
$476$	−1.87051	−0.0857346
$477$	6.42700	0.294272
$478$	23.5783	1.07844
$479$	−41.2232	−1.88354	−0.941769	−	0.336261i	$-0.890837\pi$
$479$	−41.2232	−1.88354	−0.941769	+	0.336261i	$0.890837\pi$
$480$	−3.87051	−0.176664
$481$	−44.8168	−2.04347
$482$	26.2386	1.19513
$483$	3.60256	0.163922
$484$	0	0
$485$	−4.09382	−0.185891
$486$	−1.00000	−0.0453609
$487$	41.1563	1.86497	0.932485	−	0.361208i	$-0.117635\pi$
$487$	41.1563	1.86497	0.932485	+	0.361208i	$0.117635\pi$
$488$	−5.50120	−0.249028
$489$	19.3310	0.874177
$490$	−3.87051	−0.174852
$491$	0.633360	0.0285832	0.0142916	−	0.999898i	$-0.495451\pi$
$491$	0.633360	0.0285832	0.0142916	+	0.999898i	$0.495451\pi$
$492$	−9.67578	−0.436218
$493$	7.17042	0.322939
$494$	31.2257	1.40491
$495$	0	0
$496$	−2.97187	−0.133441
$497$	−12.4360	−0.557829
$498$	−7.32564	−0.328270
$499$	−10.3744	−0.464421	−0.232210	−	0.972666i	$-0.574596\pi$
$499$	−10.3744	−0.464421	−0.232210	+	0.972666i	$0.574596\pi$
$500$	19.2783	0.862154
$501$	12.6692	0.566019
$502$	21.7989	0.972932
$503$	28.3540	1.26424	0.632120	−	0.774870i	$-0.282185\pi$
$503$	28.3540	1.26424	0.632120	+	0.774870i	$0.282185\pi$
$504$	1.00000	0.0445435
$505$	5.66682	0.252170
$506$	0	0
$507$	24.3360	1.08080
$508$	15.2051	0.674618
$509$	−9.16678	−0.406310	−0.203155	−	0.979147i	$-0.565120\pi$
$509$	−9.16678	−0.406310	−0.203155	+	0.979147i	$0.565120\pi$
$510$	−7.23981	−0.320584
$511$	1.46410	0.0647680
$512$	−1.00000	−0.0441942
$513$	5.11032	0.225626
$514$	−22.1309	−0.976153
$515$	49.5028	2.18135
$516$	10.9898	0.483799
$517$	0	0
$518$	7.33461	0.322264
$519$	10.8128	0.474630
$520$	23.6500	1.03712
$521$	41.4424	1.81563	0.907813	−	0.419374i	$-0.137751\pi$
$521$	41.4424	1.81563	0.907813	+	0.419374i	$0.137751\pi$
$522$	−3.83341	−0.167784
$523$	22.8618	0.999677	0.499838	−	0.866119i	$-0.333393\pi$
$523$	22.8618	0.999677	0.499838	+	0.866119i	$0.333393\pi$
$524$	12.4333	0.543151
$525$	−9.98083	−0.435599
$526$	6.22064	0.271233
$527$	−5.55890	−0.242149
$528$	0	0
$529$	−10.0216	−0.435721
$530$	−24.8758	−1.08053
$531$	−3.35014	−0.145384
$532$	−5.11032	−0.221561
$533$	59.1221	2.56086
$534$	11.3923	0.492993
$535$	45.0768	1.94884
$536$	7.70392	0.332758
$537$	−3.66415	−0.158120
$538$	−7.81424	−0.336896
$539$	0	0
$540$	3.87051	0.166560
$541$	−23.7337	−1.02039	−0.510196	−	0.860058i	$-0.670427\pi$
$541$	−23.7337	−1.02039	−0.510196	+	0.860058i	$0.670427\pi$
$542$	26.6321	1.14395
$543$	−16.3627	−0.702193
$544$	−1.87051	−0.0801974
$545$	−23.0887	−0.989013
$546$	−6.11032	−0.261498
$547$	−13.6874	−0.585232	−0.292616	−	0.956230i	$-0.594526\pi$
$547$	−13.6874	−0.585232	−0.292616	+	0.956230i	$0.594526\pi$
$548$	−5.20512	−0.222352
$549$	5.50120	0.234786
$550$	0	0
$551$	19.5900	0.834560
$552$	3.60256	0.153335
$553$	−13.5026	−0.574190
$554$	−19.2810	−0.819171
$555$	28.3887	1.20503
$556$	−12.6692	−0.537295
$557$	−13.7398	−0.582173	−0.291086	−	0.956697i	$-0.594017\pi$
$557$	−13.7398	−0.582173	−0.291086	+	0.956697i	$0.594017\pi$
$558$	2.97187	0.125809
$559$	−67.1512	−2.84019
$560$	−3.87051	−0.163559
$561$	0	0
$562$	−15.3205	−0.646257
$563$	−0.210186	−0.00885829	−0.00442914	−	0.999990i	$-0.501410\pi$
$563$	−0.210186	−0.00885829	−0.00442914	+	0.999990i	$0.501410\pi$
$564$	0.889678	0.0374622
$565$	65.4269	2.75253
$566$	10.6486	0.447595
$567$	−1.00000	−0.0419961
$568$	−12.4360	−0.521802
$569$	12.9103	0.541227	0.270613	−	0.962688i	$-0.412773\pi$
$569$	12.9103	0.541227	0.270613	+	0.962688i	$0.412773\pi$
$570$	−19.7795	−0.828474
$571$	−15.4517	−0.646633	−0.323316	−	0.946291i	$-0.604798\pi$
$571$	−15.4517	−0.646633	−0.323316	+	0.946291i	$0.604798\pi$
$572$	0	0
$573$	−2.45903	−0.102728
$574$	−9.67578	−0.403859
$575$	−35.9565	−1.49949
$576$	1.00000	0.0416667
$577$	6.03203	0.251117	0.125558	−	0.992086i	$-0.459928\pi$
$577$	6.03203	0.251117	0.125558	+	0.992086i	$0.459928\pi$
$578$	13.5012	0.561576
$579$	7.79631	0.324004
$580$	14.8372	0.616083
$581$	−7.32564	−0.303919
$582$	1.05770	0.0438429
$583$	0	0
$584$	1.46410	0.0605850
$585$	−23.6500	−0.977809
$586$	−17.2577	−0.712911
$587$	13.2999	0.548946	0.274473	−	0.961595i	$-0.411497\pi$
$587$	13.2999	0.548946	0.274473	+	0.961595i	$0.411497\pi$
$588$	1.00000	0.0412393
$589$	−15.1872	−0.625777
$590$	12.9667	0.533832
$591$	−26.9823	−1.10990
$592$	7.33461	0.301451
$593$	−0.139433	−0.00572583	−0.00286292	−	0.999996i	$-0.500911\pi$
$593$	−0.139433	−0.00572583	−0.00286292	+	0.999996i	$0.500911\pi$
$594$	0	0
$595$	−7.23981	−0.296803
$596$	20.5026	0.839820
$597$	13.1052	0.536360
$598$	−22.0128	−0.900170
$599$	8.67169	0.354316	0.177158	−	0.984182i	$-0.443310\pi$
$599$	8.67169	0.354316	0.177158	+	0.984182i	$0.443310\pi$
$600$	−9.98083	−0.407466
$601$	7.42460	0.302856	0.151428	−	0.988468i	$-0.451613\pi$
$601$	7.42460	0.302856	0.151428	+	0.988468i	$0.451613\pi$
$602$	10.9898	0.447911
$603$	−7.70392	−0.313728
$604$	−13.9823	−0.568930
$605$	0	0
$606$	−1.46410	−0.0594751
$607$	0.373135	0.0151451	0.00757253	−	0.999971i	$-0.497590\pi$
$607$	0.373135	0.0151451	0.00757253	+	0.999971i	$0.497590\pi$
$608$	−5.11032	−0.207251
$609$	−3.83341	−0.155338
$610$	−21.2924	−0.862106
$611$	−5.43622	−0.219926
$612$	1.87051	0.0756108
$613$	14.2064	0.573789	0.286895	−	0.957962i	$-0.407377\pi$
$613$	14.2064	0.573789	0.286895	+	0.957962i	$0.407377\pi$
$614$	−17.7410	−0.715969
$615$	−37.4502	−1.51014
$616$	0	0
$617$	−10.7245	−0.431753	−0.215876	−	0.976421i	$-0.569261\pi$
$617$	−10.7245	−0.431753	−0.215876	+	0.976421i	$0.569261\pi$
$618$	−12.7897	−0.514479
$619$	48.3824	1.94465	0.972325	−	0.233631i	$-0.0750606\pi$
$619$	48.3824	1.94465	0.972325	+	0.233631i	$0.0750606\pi$
$620$	−11.5026	−0.461957
$621$	−3.60256	−0.144566
$622$	−7.11032	−0.285098
$623$	11.3923	0.456423
$624$	−6.11032	−0.244609
$625$	24.7128	0.988513
$626$	−2.82977	−0.113100
$627$	0	0
$628$	−24.6309	−0.982879
$629$	13.7194	0.547030
$630$	3.87051	0.154205
$631$	−14.2742	−0.568249	−0.284124	−	0.958787i	$-0.591703\pi$
$631$	−14.2742	−0.568249	−0.284124	+	0.958787i	$0.591703\pi$
$632$	−13.5026	−0.537106
$633$	−1.43090	−0.0568731
$634$	−26.4924	−1.05215
$635$	58.8515	2.33545
$636$	6.42700	0.254847
$637$	−6.11032	−0.242100
$638$	0	0
$639$	12.4360	0.491959
$640$	−3.87051	−0.152995
$641$	36.9732	1.46035	0.730177	−	0.683258i	$-0.239437\pi$
$641$	36.9732	1.46035	0.730177	+	0.683258i	$0.239437\pi$
$642$	−11.6462	−0.459640
$643$	33.5901	1.32467	0.662333	−	0.749210i	$-0.269567\pi$
$643$	33.5901	1.32467	0.662333	+	0.749210i	$0.269567\pi$
$644$	3.60256	0.141961
$645$	42.5361	1.67486
$646$	−9.55890	−0.376090
$647$	−5.26424	−0.206959	−0.103479	−	0.994632i	$-0.532998\pi$
$647$	−5.26424	−0.206959	−0.103479	+	0.994632i	$0.532998\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	60.9861	2.39207
$651$	2.97187	0.116477
$652$	19.3310	0.757059
$653$	−9.52551	−0.372762	−0.186381	−	0.982478i	$-0.559676\pi$
$653$	−9.52551	−0.372762	−0.186381	+	0.982478i	$0.559676\pi$
$654$	5.96530	0.233262
$655$	48.1232	1.88033
$656$	−9.67578	−0.377776
$657$	−1.46410	−0.0571200
$658$	0.889678	0.0346833
$659$	−1.26405	−0.0492405	−0.0246203	−	0.999697i	$-0.507838\pi$
$659$	−1.26405	−0.0492405	−0.0246203	+	0.999697i	$0.507838\pi$
$660$	0	0
$661$	20.3345	0.790922	0.395461	−	0.918483i	$-0.370585\pi$
$661$	20.3345	0.790922	0.395461	+	0.918483i	$0.370585\pi$
$662$	−16.7759	−0.652014
$663$	−11.4294	−0.443881
$664$	−7.32564	−0.284290
$665$	−19.7795	−0.767018
$666$	−7.33461	−0.284210
$667$	−13.8101	−0.534728
$668$	12.6692	0.490187
$669$	−5.53830	−0.214123
$670$	29.8181	1.15197
$671$	0	0
$672$	1.00000	0.0385758
$673$	35.4747	1.36745	0.683725	−	0.729739i	$-0.260359\pi$
$673$	35.4747	1.36745	0.683725	+	0.729739i	$0.260359\pi$
$674$	−16.7476	−0.645093
$675$	9.98083	0.384162
$676$	24.3360	0.936001
$677$	−6.85036	−0.263281	−0.131640	−	0.991298i	$-0.542024\pi$
$677$	−6.85036	−0.263281	−0.131640	+	0.991298i	$0.542024\pi$
$678$	−16.9040	−0.649193
$679$	1.05770	0.0405906
$680$	−7.23981	−0.277634
$681$	−12.4539	−0.477234
$682$	0	0
$683$	−6.72834	−0.257453	−0.128726	−	0.991680i	$-0.541089\pi$
$683$	−6.72834	−0.257453	−0.128726	+	0.991680i	$0.541089\pi$
$684$	5.11032	0.195398
$685$	−20.1464	−0.769756
$686$	1.00000	0.0381802
$687$	−29.7965	−1.13681
$688$	10.9898	0.418982
$689$	−39.2711	−1.49611
$690$	13.9437	0.530829
$691$	32.3824	1.23188	0.615941	−	0.787792i	$-0.288776\pi$
$691$	32.3824	1.23188	0.615941	+	0.787792i	$0.288776\pi$
$692$	10.8128	0.411041
$693$	0	0
$694$	12.2771	0.466032
$695$	−49.0363	−1.86005
$696$	−3.83341	−0.145305
$697$	−18.0986	−0.685534
$698$	25.2975	0.957525
$699$	−13.3140	−0.503582
$700$	−9.98083	−0.377240
$701$	−36.7975	−1.38982	−0.694911	−	0.719096i	$-0.744556\pi$
$701$	−36.7975	−1.38982	−0.694911	+	0.719096i	$0.744556\pi$
$702$	6.11032	0.230619
$703$	37.4822	1.41367
$704$	0	0
$705$	3.44351	0.129690
$706$	−4.79729	−0.180548
$707$	−1.46410	−0.0550632
$708$	−3.35014	−0.125906
$709$	−34.7613	−1.30549	−0.652745	−	0.757578i	$-0.726383\pi$
$709$	−34.7613	−1.30549	−0.652745	+	0.757578i	$0.726383\pi$
$710$	−48.1335	−1.80642
$711$	13.5026	0.506388
$712$	11.3923	0.426945
$713$	10.7063	0.400955
$714$	1.87051	0.0700020
$715$	0	0
$716$	−3.66415	−0.136936
$717$	−23.5783	−0.880546
$718$	−8.53466	−0.318511
$719$	40.3004	1.50295	0.751475	−	0.659762i	$-0.229343\pi$
$719$	40.3004	1.50295	0.751475	+	0.659762i	$0.229343\pi$
$720$	3.87051	0.144245
$721$	−12.7897	−0.476315
$722$	−7.11539	−0.264807
$723$	−26.2386	−0.975823
$724$	−16.3627	−0.608117
$725$	38.2606	1.42096
$726$	0	0
$727$	−23.6130	−0.875758	−0.437879	−	0.899034i	$-0.644270\pi$
$727$	−23.6130	−0.875758	−0.437879	+	0.899034i	$0.644270\pi$
$728$	−6.11032	−0.226464
$729$	1.00000	0.0370370
$730$	5.66682	0.209738
$731$	20.5565	0.760309
$732$	5.50120	0.203330
$733$	−7.20609	−0.266163	−0.133082	−	0.991105i	$-0.542487\pi$
$733$	−7.20609	−0.266163	−0.133082	+	0.991105i	$0.542487\pi$
$734$	−15.1666	−0.559809
$735$	3.87051	0.142766
$736$	3.60256	0.132792
$737$	0	0
$738$	9.67578	0.356170
$739$	17.7536	0.653078	0.326539	−	0.945184i	$-0.394118\pi$
$739$	17.7536	0.653078	0.326539	+	0.945184i	$0.394118\pi$
$740$	28.3887	1.04359
$741$	−31.2257	−1.14711
$742$	6.42700	0.235943
$743$	30.0964	1.10413	0.552065	−	0.833801i	$-0.313840\pi$
$743$	30.0964	1.10413	0.552065	+	0.833801i	$0.313840\pi$
$744$	2.97187	0.108954
$745$	79.3556	2.90736
$746$	−5.90539	−0.216212
$747$	7.32564	0.268031
$748$	0	0
$749$	−11.6462	−0.425544
$750$	−19.2783	−0.703945
$751$	−24.7228	−0.902149	−0.451074	−	0.892486i	$-0.648959\pi$
$751$	−24.7228	−0.902149	−0.451074	+	0.892486i	$0.648959\pi$
$752$	0.889678	0.0324432
$753$	−21.7989	−0.794396
$754$	23.4234	0.853028
$755$	−54.1184	−1.96957
$756$	−1.00000	−0.0363696
$757$	−51.0785	−1.85648	−0.928239	−	0.371983i	$-0.878678\pi$
$757$	−51.0785	−1.85648	−0.928239	+	0.371983i	$0.878678\pi$
$758$	22.9964	0.835265
$759$	0	0
$760$	−19.7795	−0.717479
$761$	−30.9551	−1.12212	−0.561061	−	0.827775i	$-0.689607\pi$
$761$	−30.9551	−1.12212	−0.561061	+	0.827775i	$0.689607\pi$
$762$	−15.2051	−0.550823
$763$	5.96530	0.215958
$764$	−2.45903	−0.0889647
$765$	7.23981	0.261756
$766$	32.7594	1.18365
$767$	20.4704	0.739144
$768$	1.00000	0.0360844
$769$	−7.94737	−0.286590	−0.143295	−	0.989680i	$-0.545770\pi$
$769$	−7.94737	−0.286590	−0.143295	+	0.989680i	$0.545770\pi$
$770$	0	0
$771$	22.1309	0.797026
$772$	7.79631	0.280595
$773$	0.861795	0.0309966	0.0154983	−	0.999880i	$-0.495067\pi$
$773$	0.861795	0.0309966	0.0154983	+	0.999880i	$0.495067\pi$
$774$	−10.9898	−0.395020
$775$	−29.6617	−1.06548
$776$	1.05770	0.0379691
$777$	−7.33461	−0.263128
$778$	20.5783	0.737766
$779$	−49.4464	−1.77160
$780$	−23.6500	−0.846808
$781$	0	0
$782$	6.73861	0.240972
$783$	3.83341	0.136995
$784$	1.00000	0.0357143
$785$	−95.3340	−3.40262
$786$	−12.4333	−0.443481
$787$	−11.2846	−0.402254	−0.201127	−	0.979565i	$-0.564460\pi$
$787$	−11.2846	−0.402254	−0.201127	+	0.979565i	$0.564460\pi$
$788$	−26.9823	−0.961203
$789$	−6.22064	−0.221461
$790$	−52.2620	−1.85940
$791$	−16.9040	−0.601036
$792$	0	0
$793$	−33.6141	−1.19367
$794$	25.8579	0.917663
$795$	24.8758	0.882252
$796$	13.1052	0.464501
$797$	−42.9800	−1.52243	−0.761216	−	0.648499i	$-0.775397\pi$
$797$	−42.9800	−1.52243	−0.761216	+	0.648499i	$0.775397\pi$
$798$	5.11032	0.180903
$799$	1.66415	0.0588734
$800$	−9.98083	−0.352876
$801$	−11.3923	−0.402527
$802$	24.4655	0.863908
$803$	0	0
$804$	−7.70392	−0.271696
$805$	13.9437	0.491452
$806$	−18.1591	−0.639625
$807$	7.81424	0.275074
$808$	−1.46410	−0.0515069
$809$	35.7337	1.25633	0.628166	−	0.778080i	$-0.283806\pi$
$809$	35.7337	1.25633	0.628166	+	0.778080i	$0.283806\pi$
$810$	−3.87051	−0.135996
$811$	1.90806	0.0670009	0.0335005	−	0.999439i	$-0.489334\pi$
$811$	1.90806	0.0670009	0.0335005	+	0.999439i	$0.489334\pi$
$812$	−3.83341	−0.134526
$813$	−26.6321	−0.934029
$814$	0	0
$815$	74.8207	2.62085
$816$	1.87051	0.0654809
$817$	56.1614	1.96484
$818$	3.46046	0.120992
$819$	6.11032	0.213512
$820$	−37.4502	−1.30782
$821$	−8.59626	−0.300012	−0.150006	−	0.988685i	$-0.547929\pi$
$821$	−8.59626	−0.300012	−0.150006	+	0.988685i	$0.547929\pi$
$822$	5.20512	0.181549
$823$	−37.5131	−1.30762	−0.653812	−	0.756657i	$-0.726831\pi$
$823$	−37.5131	−1.30762	−0.653812	+	0.756657i	$0.726831\pi$
$824$	−12.7897	−0.445552
$825$	0	0
$826$	−3.35014	−0.116566
$827$	−14.0075	−0.487087	−0.243544	−	0.969890i	$-0.578310\pi$
$827$	−14.0075	−0.487087	−0.243544	+	0.969890i	$0.578310\pi$
$828$	−3.60256	−0.125198
$829$	−17.2446	−0.598930	−0.299465	−	0.954107i	$-0.596808\pi$
$829$	−17.2446	−0.598930	−0.299465	+	0.954107i	$0.596808\pi$
$830$	−28.3540	−0.984180
$831$	19.2810	0.668851
$832$	−6.11032	−0.211837
$833$	1.87051	0.0648092
$834$	12.6692	0.438699
$835$	49.0363	1.69697
$836$	0	0
$837$	−2.97187	−0.102723
$838$	−24.2308	−0.837038
$839$	35.1952	1.21507	0.607536	−	0.794292i	$-0.292158\pi$
$839$	35.1952	1.21507	0.607536	+	0.794292i	$0.292158\pi$
$840$	3.87051	0.133545
$841$	−14.3050	−0.493275
$842$	28.1323	0.969505
$843$	15.3205	0.527666
$844$	−1.43090	−0.0492535
$845$	94.1928	3.24033
$846$	−0.889678	−0.0305878
$847$	0	0
$848$	6.42700	0.220704
$849$	−10.6486	−0.365460
$850$	−18.6692	−0.640349
$851$	−26.4234	−0.905781
$852$	12.4360	0.426049
$853$	−21.5805	−0.738901	−0.369451	−	0.929250i	$-0.620454\pi$
$853$	−21.5805	−0.738901	−0.369451	+	0.929250i	$0.620454\pi$
$854$	5.50120	0.188247
$855$	19.7795	0.676446
$856$	−11.6462	−0.398060
$857$	−15.7873	−0.539285	−0.269643	−	0.962960i	$-0.586906\pi$
$857$	−15.7873	−0.539285	−0.269643	+	0.962960i	$0.586906\pi$
$858$	0	0
$859$	−29.8695	−1.01914	−0.509568	−	0.860431i	$-0.670195\pi$
$859$	−29.8695	−1.01914	−0.509568	+	0.860431i	$0.670195\pi$
$860$	42.5361	1.45047
$861$	9.67578	0.329750
$862$	19.9333	0.678930
$863$	−30.9415	−1.05326	−0.526631	−	0.850094i	$-0.676545\pi$
$863$	−30.9415	−1.05326	−0.526631	+	0.850094i	$0.676545\pi$
$864$	−1.00000	−0.0340207
$865$	41.8511	1.42298
$866$	26.2807	0.893056
$867$	−13.5012	−0.458525
$868$	2.97187	0.100872
$869$	0	0
$870$	−14.8372	−0.503029
$871$	47.0734	1.59502
$872$	5.96530	0.202011
$873$	−1.05770	−0.0357976
$874$	18.4102	0.622735
$875$	−19.2783	−0.651727
$876$	−1.46410	−0.0494674
$877$	17.9342	0.605597	0.302798	−	0.953055i	$-0.402079\pi$
$877$	17.9342	0.605597	0.302798	+	0.953055i	$0.402079\pi$
$878$	−10.7447	−0.362615
$879$	17.2577	0.582089
$880$	0	0
$881$	57.0950	1.92358	0.961789	−	0.273790i	$-0.0882775\pi$
$881$	57.0950	1.92358	0.961789	+	0.273790i	$0.0882775\pi$
$882$	−1.00000	−0.0336718
$883$	−56.5217	−1.90211	−0.951054	−	0.309026i	$-0.899997\pi$
$883$	−56.5217	−1.90211	−0.951054	+	0.309026i	$0.899997\pi$
$884$	−11.4294	−0.384412
$885$	−12.9667	−0.435872
$886$	−6.68715	−0.224659
$887$	41.6895	1.39980	0.699899	−	0.714242i	$-0.253228\pi$
$887$	41.6895	1.39980	0.699899	+	0.714242i	$0.253228\pi$
$888$	−7.33461	−0.246133
$889$	−15.2051	−0.509963
$890$	44.0940	1.47803
$891$	0	0
$892$	−5.53830	−0.185436
$893$	4.54654	0.152144
$894$	−20.5026	−0.685710
$895$	−14.1821	−0.474056
$896$	1.00000	0.0334077
$897$	22.0128	0.734986
$898$	−21.1731	−0.706555
$899$	−11.3924	−0.379957
$900$	9.98083	0.332694
$901$	12.0218	0.400503
$902$	0	0
$903$	−10.9898	−0.365717
$904$	−16.9040	−0.562217
$905$	−63.3321	−2.10523
$906$	13.9823	0.464529
$907$	48.8171	1.62094	0.810472	−	0.585777i	$-0.199210\pi$
$907$	48.8171	1.62094	0.810472	+	0.585777i	$0.199210\pi$
$908$	−12.4539	−0.413297
$909$	1.46410	0.0485612
$910$	−23.6500	−0.783991
$911$	11.8543	0.392749	0.196375	−	0.980529i	$-0.437083\pi$
$911$	11.8543	0.392749	0.196375	+	0.980529i	$0.437083\pi$
$912$	5.11032	0.169220
$913$	0	0
$914$	14.6139	0.483386
$915$	21.2924	0.703906
$916$	−29.7965	−0.984504
$917$	−12.4333	−0.410584
$918$	−1.87051	−0.0617360
$919$	−5.15151	−0.169933	−0.0849664	−	0.996384i	$-0.527078\pi$
$919$	−5.15151	−0.169933	−0.0849664	+	0.996384i	$0.527078\pi$
$920$	13.9437	0.459711
$921$	17.7410	0.584586
$922$	5.23981	0.172564
$923$	−75.9878	−2.50117
$924$	0	0
$925$	73.2055	2.40698
$926$	6.27425	0.206185
$927$	12.7897	0.420070
$928$	−3.83341	−0.125838
$929$	−16.8312	−0.552213	−0.276107	−	0.961127i	$-0.589044\pi$
$929$	−16.8312	−0.552213	−0.276107	+	0.961127i	$0.589044\pi$
$930$	11.5026	0.377186
$931$	5.11032	0.167484
$932$	−13.3140	−0.436115
$933$	7.11032	0.232781
$934$	−10.7194	−0.350751
$935$	0	0
$936$	6.11032	0.199722
$937$	−11.6632	−0.381019	−0.190510	−	0.981685i	$-0.561014\pi$
$937$	−11.6632	−0.381019	−0.190510	+	0.981685i	$0.561014\pi$
$938$	−7.70392	−0.251542
$939$	2.82977	0.0923459
$940$	3.44351	0.112315
$941$	52.3890	1.70783	0.853917	−	0.520410i	$-0.174221\pi$
$941$	52.3890	1.70783	0.853917	+	0.520410i	$0.174221\pi$
$942$	24.6309	0.802517
$943$	34.8576	1.13512
$944$	−3.35014	−0.109038
$945$	−3.87051	−0.125908
$946$	0	0
$947$	21.3799	0.694755	0.347377	−	0.937725i	$-0.387072\pi$
$947$	21.3799	0.694755	0.347377	+	0.937725i	$0.387072\pi$
$948$	13.5026	0.438545
$949$	8.94613	0.290404
$950$	−51.0053	−1.65483
$951$	26.4924	0.859076
$952$	1.87051	0.0606235
$953$	−17.9731	−0.582206	−0.291103	−	0.956692i	$-0.594022\pi$
$953$	−17.9731	−0.582206	−0.291103	+	0.956692i	$0.594022\pi$
$954$	−6.42700	−0.208082
$955$	−9.51770	−0.307986
$956$	−23.5783	−0.762575
$957$	0	0
$958$	41.2232	1.33186
$959$	5.20512	0.168082
$960$	3.87051	0.124920
$961$	−22.1680	−0.715097
$962$	44.8168	1.44495
$963$	11.6462	0.375294
$964$	−26.2386	−0.845088
$965$	30.1757	0.971389
$966$	−3.60256	−0.115910
$967$	−48.4415	−1.55777	−0.778886	−	0.627165i	$-0.784215\pi$
$967$	−48.4415	−1.55777	−0.778886	+	0.627165i	$0.784215\pi$
$968$	0	0
$969$	9.55890	0.307076
$970$	4.09382	0.131445
$971$	−43.5783	−1.39849	−0.699246	−	0.714881i	$-0.746481\pi$
$971$	−43.5783	−1.39849	−0.699246	+	0.714881i	$0.746481\pi$
$972$	1.00000	0.0320750
$973$	12.6692	0.406157
$974$	−41.1563	−1.31873
$975$	−60.9861	−1.95312
$976$	5.50120	0.176089
$977$	−10.8759	−0.347951	−0.173975	−	0.984750i	$-0.555661\pi$
$977$	−10.8759	−0.347951	−0.173975	+	0.984750i	$0.555661\pi$
$978$	−19.3310	−0.618136
$979$	0	0
$980$	3.87051	0.123639
$981$	−5.96530	−0.190457
$982$	−0.633360	−0.0202113
$983$	5.03314	0.160532	0.0802661	−	0.996773i	$-0.474423\pi$
$983$	5.03314	0.160532	0.0802661	+	0.996773i	$0.474423\pi$
$984$	9.67578	0.308453
$985$	−104.435	−3.32758
$986$	−7.17042	−0.228353
$987$	−0.889678	−0.0283188
$988$	−31.2257	−0.993422
$989$	−39.5914	−1.25893
$990$	0	0
$991$	−47.7334	−1.51630	−0.758151	−	0.652079i	$-0.773897\pi$
$991$	−47.7334	−1.51630	−0.758151	+	0.652079i	$0.773897\pi$
$992$	2.97187	0.0943568
$993$	16.7759	0.532367
$994$	12.4360	0.394445
$995$	50.7237	1.60805
$996$	7.32564	0.232122
$997$	17.2856	0.547441	0.273721	−	0.961809i	$-0.411746\pi$
$997$	17.2856	0.547441	0.273721	+	0.961809i	$0.411746\pi$
$998$	10.3744	0.328395
$999$	7.33461	0.232057

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5082.2.a.cb.1.4	✓	4
11.10	odd	2		5082.2.a.cg.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5082.2.a.cb.1.4	✓	4	1.1	even	1	trivial
5082.2.a.cg.1.4	yes	4	11.10	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:	\( N \)	\(=\)	\( 5082 = 2 \cdot 3 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5082.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.87051 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.87051 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.87051 q^{10} +1.00000 q^{12} -6.11032 q^{13} +1.00000 q^{14} +3.87051 q^{15} +1.00000 q^{16} +1.87051 q^{17} -1.00000 q^{18} +5.11032 q^{19} +3.87051 q^{20} -1.00000 q^{21} -3.60256 q^{23} -1.00000 q^{24} +9.98083 q^{25} +6.11032 q^{26} +1.00000 q^{27} -1.00000 q^{28} +3.83341 q^{29} -3.87051 q^{30} -2.97187 q^{31} -1.00000 q^{32} -1.87051 q^{34} -3.87051 q^{35} +1.00000 q^{36} +7.33461 q^{37} -5.11032 q^{38} -6.11032 q^{39} -3.87051 q^{40} -9.67578 q^{41} +1.00000 q^{42} +10.9898 q^{43} +3.87051 q^{45} +3.60256 q^{46} +0.889678 q^{47} +1.00000 q^{48} +1.00000 q^{49} -9.98083 q^{50} +1.87051 q^{51} -6.11032 q^{52} +6.42700 q^{53} -1.00000 q^{54} +1.00000 q^{56} +5.11032 q^{57} -3.83341 q^{58} -3.35014 q^{59} +3.87051 q^{60} +5.50120 q^{61} +2.97187 q^{62} -1.00000 q^{63} +1.00000 q^{64} -23.6500 q^{65} -7.70392 q^{67} +1.87051 q^{68} -3.60256 q^{69} +3.87051 q^{70} +12.4360 q^{71} -1.00000 q^{72} -1.46410 q^{73} -7.33461 q^{74} +9.98083 q^{75} +5.11032 q^{76} +6.11032 q^{78} +13.5026 q^{79} +3.87051 q^{80} +1.00000 q^{81} +9.67578 q^{82} +7.32564 q^{83} -1.00000 q^{84} +7.23981 q^{85} -10.9898 q^{86} +3.83341 q^{87} -11.3923 q^{89} -3.87051 q^{90} +6.11032 q^{91} -3.60256 q^{92} -2.97187 q^{93} -0.889678 q^{94} +19.7795 q^{95} -1.00000 q^{96} -1.05770 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 + 4 * q^5 - 4 * q^6 - 4 * q^7 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} + 4 q^{9} - 4 q^{10} + 4 q^{12} - 6 q^{13} + 4 q^{14} + 4 q^{15} + 4 q^{16} - 4 q^{17} - 4 q^{18} + 2 q^{19} + 4 q^{20} - 4 q^{21} + 4 q^{23} - 4 q^{24} + 10 q^{25} + 6 q^{26} + 4 q^{27} - 4 q^{28} + 6 q^{29} - 4 q^{30} + 2 q^{31} - 4 q^{32} + 4 q^{34} - 4 q^{35} + 4 q^{36} + 4 q^{37} - 2 q^{38} - 6 q^{39} - 4 q^{40} - 4 q^{41} + 4 q^{42} - 2 q^{43} + 4 q^{45} - 4 q^{46} + 22 q^{47} + 4 q^{48} + 4 q^{49} - 10 q^{50} - 4 q^{51} - 6 q^{52} + 14 q^{53} - 4 q^{54} + 4 q^{56} + 2 q^{57} - 6 q^{58} + 12 q^{59} + 4 q^{60} + 6 q^{61} - 2 q^{62} - 4 q^{63} + 4 q^{64} - 14 q^{65} - 10 q^{67} - 4 q^{68} + 4 q^{69} + 4 q^{70} + 22 q^{71} - 4 q^{72} + 8 q^{73} - 4 q^{74} + 10 q^{75} + 2 q^{76} + 6 q^{78} - 6 q^{79} + 4 q^{80} + 4 q^{81} + 4 q^{82} + 20 q^{83} - 4 q^{84} + 22 q^{85} + 2 q^{86} + 6 q^{87} - 4 q^{89} - 4 q^{90} + 6 q^{91} + 4 q^{92} + 2 q^{93} - 22 q^{94} + 10 q^{95} - 4 q^{96} + 12 q^{97} - 4 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 + 4 * q^5 - 4 * q^6 - 4 * q^7 - 4 * q^8 + 4 * q^9 - 4 * q^10 + 4 * q^12 - 6 * q^13 + 4 * q^14 + 4 * q^15 + 4 * q^16 - 4 * q^17 - 4 * q^18 + 2 * q^19 + 4 * q^20 - 4 * q^21 + 4 * q^23 - 4 * q^24 + 10 * q^25 + 6 * q^26 + 4 * q^27 - 4 * q^28 + 6 * q^29 - 4 * q^30 + 2 * q^31 - 4 * q^32 + 4 * q^34 - 4 * q^35 + 4 * q^36 + 4 * q^37 - 2 * q^38 - 6 * q^39 - 4 * q^40 - 4 * q^41 + 4 * q^42 - 2 * q^43 + 4 * q^45 - 4 * q^46 + 22 * q^47 + 4 * q^48 + 4 * q^49 - 10 * q^50 - 4 * q^51 - 6 * q^52 + 14 * q^53 - 4 * q^54 + 4 * q^56 + 2 * q^57 - 6 * q^58 + 12 * q^59 + 4 * q^60 + 6 * q^61 - 2 * q^62 - 4 * q^63 + 4 * q^64 - 14 * q^65 - 10 * q^67 - 4 * q^68 + 4 * q^69 + 4 * q^70 + 22 * q^71 - 4 * q^72 + 8 * q^73 - 4 * q^74 + 10 * q^75 + 2 * q^76 + 6 * q^78 - 6 * q^79 + 4 * q^80 + 4 * q^81 + 4 * q^82 + 20 * q^83 - 4 * q^84 + 22 * q^85 + 2 * q^86 + 6 * q^87 - 4 * q^89 - 4 * q^90 + 6 * q^91 + 4 * q^92 + 2 * q^93 - 22 * q^94 + 10 * q^95 - 4 * q^96 + 12 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

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Database

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