LMFDB - Embedded newform 4114.2.a.bg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4114,2,Mod(1,4114)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4114, 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("4114.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4114 = 2 \cdot 11^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4114.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.8504553916$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 12x^{6} + 28x^{5} + 51x^{4} - 80x^{3} - 92x^{2} + 67x + 59 $ x^8 - 3x^7 - 12x^6 + 28x^5 + 51x^4 - 80x^3 - 92x^2 + 67*x + 59
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 374)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.17063$ of defining polynomial
Character	$\chi$	$=$	4114.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.17063 q^{3} +1.00000 q^{4} -2.18713 q^{5} +3.17063 q^{6} +3.13019 q^{7} -1.00000 q^{8} +7.05290 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.17063 q^{3} +1.00000 q^{4} -2.18713 q^{5} +3.17063 q^{6} +3.13019 q^{7} -1.00000 q^{8} +7.05290 q^{9} +2.18713 q^{10} -3.17063 q^{12} +6.73925 q^{13} -3.13019 q^{14} +6.93458 q^{15} +1.00000 q^{16} +1.00000 q^{17} -7.05290 q^{18} -2.67597 q^{19} -2.18713 q^{20} -9.92468 q^{21} -9.30814 q^{23} +3.17063 q^{24} -0.216473 q^{25} -6.73925 q^{26} -12.8503 q^{27} +3.13019 q^{28} +0.265413 q^{29} -6.93458 q^{30} -4.70773 q^{31} -1.00000 q^{32} -1.00000 q^{34} -6.84612 q^{35} +7.05290 q^{36} -2.39407 q^{37} +2.67597 q^{38} -21.3677 q^{39} +2.18713 q^{40} +4.58120 q^{41} +9.92468 q^{42} +5.82575 q^{43} -15.4256 q^{45} +9.30814 q^{46} -4.44029 q^{47} -3.17063 q^{48} +2.79809 q^{49} +0.216473 q^{50} -3.17063 q^{51} +6.73925 q^{52} +2.88030 q^{53} +12.8503 q^{54} -3.13019 q^{56} +8.48452 q^{57} -0.265413 q^{58} -3.36981 q^{59} +6.93458 q^{60} +11.3670 q^{61} +4.70773 q^{62} +22.0769 q^{63} +1.00000 q^{64} -14.7396 q^{65} +10.7683 q^{67} +1.00000 q^{68} +29.5127 q^{69} +6.84612 q^{70} -7.10839 q^{71} -7.05290 q^{72} -12.0894 q^{73} +2.39407 q^{74} +0.686355 q^{75} -2.67597 q^{76} +21.3677 q^{78} +4.81158 q^{79} -2.18713 q^{80} +19.5847 q^{81} -4.58120 q^{82} +8.36392 q^{83} -9.92468 q^{84} -2.18713 q^{85} -5.82575 q^{86} -0.841526 q^{87} +1.26739 q^{89} +15.4256 q^{90} +21.0951 q^{91} -9.30814 q^{92} +14.9265 q^{93} +4.44029 q^{94} +5.85269 q^{95} +3.17063 q^{96} -3.38343 q^{97} -2.79809 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - 5 q^{3} + 8 q^{4} + 2 q^{5} + 5 q^{6} - 11 q^{7} - 8 q^{8} + 11 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 - 5 * q^3 + 8 * q^4 + 2 * q^5 + 5 * q^6 - 11 * q^7 - 8 * q^8 + 11 * q^9	$ 8 q - 8 q^{2} - 5 q^{3} + 8 q^{4} + 2 q^{5} + 5 q^{6} - 11 q^{7} - 8 q^{8} + 11 q^{9} - 2 q^{10} - 5 q^{12} + 5 q^{13} + 11 q^{14} + 2 q^{15} + 8 q^{16} + 8 q^{17} - 11 q^{18} - 15 q^{19} + 2 q^{20} - 12 q^{23} + 5 q^{24} + 24 q^{25} - 5 q^{26} - 17 q^{27} - 11 q^{28} - 22 q^{29} - 2 q^{30} - 7 q^{31} - 8 q^{32} - 8 q^{34} - 10 q^{35} + 11 q^{36} + 15 q^{37} + 15 q^{38} - 35 q^{39} - 2 q^{40} - 17 q^{41} - 8 q^{43} + 3 q^{45} + 12 q^{46} + 18 q^{47} - 5 q^{48} + q^{49} - 24 q^{50} - 5 q^{51} + 5 q^{52} + 20 q^{53} + 17 q^{54} + 11 q^{56} - 9 q^{57} + 22 q^{58} + 6 q^{59} + 2 q^{60} + 5 q^{61} + 7 q^{62} + 14 q^{63} + 8 q^{64} - 3 q^{65} + 13 q^{67} + 8 q^{68} + 46 q^{69} + 10 q^{70} - 18 q^{71} - 11 q^{72} + 4 q^{73} - 15 q^{74} - 49 q^{75} - 15 q^{76} + 35 q^{78} - 21 q^{79} + 2 q^{80} + 8 q^{81} + 17 q^{82} - 38 q^{83} + 2 q^{85} + 8 q^{86} + 46 q^{87} + 12 q^{89} - 3 q^{90} - 5 q^{91} - 12 q^{92} + 3 q^{93} - 18 q^{94} - 63 q^{95} + 5 q^{96} - 66 q^{97} - q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 - 5 * q^3 + 8 * q^4 + 2 * q^5 + 5 * q^6 - 11 * q^7 - 8 * q^8 + 11 * q^9 - 2 * q^10 - 5 * q^12 + 5 * q^13 + 11 * q^14 + 2 * q^15 + 8 * q^16 + 8 * q^17 - 11 * q^18 - 15 * q^19 + 2 * q^20 - 12 * q^23 + 5 * q^24 + 24 * q^25 - 5 * q^26 - 17 * q^27 - 11 * q^28 - 22 * q^29 - 2 * q^30 - 7 * q^31 - 8 * q^32 - 8 * q^34 - 10 * q^35 + 11 * q^36 + 15 * q^37 + 15 * q^38 - 35 * q^39 - 2 * q^40 - 17 * q^41 - 8 * q^43 + 3 * q^45 + 12 * q^46 + 18 * q^47 - 5 * q^48 + q^49 - 24 * q^50 - 5 * q^51 + 5 * q^52 + 20 * q^53 + 17 * q^54 + 11 * q^56 - 9 * q^57 + 22 * q^58 + 6 * q^59 + 2 * q^60 + 5 * q^61 + 7 * q^62 + 14 * q^63 + 8 * q^64 - 3 * q^65 + 13 * q^67 + 8 * q^68 + 46 * q^69 + 10 * q^70 - 18 * q^71 - 11 * q^72 + 4 * q^73 - 15 * q^74 - 49 * q^75 - 15 * q^76 + 35 * q^78 - 21 * q^79 + 2 * q^80 + 8 * q^81 + 17 * q^82 - 38 * q^83 + 2 * q^85 + 8 * q^86 + 46 * q^87 + 12 * q^89 - 3 * q^90 - 5 * q^91 - 12 * q^92 + 3 * q^93 - 18 * q^94 - 63 * q^95 + 5 * q^96 - 66 * q^97 - 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
$3$	−3.17063	−1.83056	−0.915282	−	0.402813i	$-0.868033\pi$
$3$	−3.17063	−1.83056	−0.915282	+	0.402813i	$0.868033\pi$
$4$	1.00000	0.500000
$5$	−2.18713	−0.978113	−0.489057	−	0.872252i	$-0.662659\pi$
$5$	−2.18713	−0.978113	−0.489057	+	0.872252i	$0.662659\pi$
$6$	3.17063	1.29440
$7$	3.13019	1.18310	0.591550	−	0.806268i	$-0.298516\pi$
$7$	3.13019	1.18310	0.591550	+	0.806268i	$0.298516\pi$
$8$	−1.00000	−0.353553
$9$	7.05290	2.35097
$10$	2.18713	0.691630
$11$	0	0
$11$	0	0
$12$	−3.17063	−0.915282
$13$	6.73925	1.86913	0.934565	−	0.355792i	$-0.115789\pi$
$13$	6.73925	1.86913	0.934565	+	0.355792i	$0.115789\pi$
$14$	−3.13019	−0.836578
$15$	6.93458	1.79050
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−7.05290	−1.66239
$19$	−2.67597	−0.613910	−0.306955	−	0.951724i	$-0.599310\pi$
$19$	−2.67597	−0.613910	−0.306955	+	0.951724i	$0.599310\pi$
$20$	−2.18713	−0.489057
$21$	−9.92468	−2.16574
$22$	0	0
$23$	−9.30814	−1.94088	−0.970441	−	0.241338i	$-0.922414\pi$
$23$	−9.30814	−1.94088	−0.970441	+	0.241338i	$0.922414\pi$
$24$	3.17063	0.647202
$25$	−0.216473	−0.0432945
$26$	−6.73925	−1.32167
$27$	−12.8503	−2.47303
$28$	3.13019	0.591550
$29$	0.265413	0.0492859	0.0246430	−	0.999696i	$-0.492155\pi$
$29$	0.265413	0.0492859	0.0246430	+	0.999696i	$0.492155\pi$
$30$	−6.93458	−1.26607
$31$	−4.70773	−0.845534	−0.422767	−	0.906238i	$-0.638941\pi$
$31$	−4.70773	−0.845534	−0.422767	+	0.906238i	$0.638941\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.00000	−0.171499
$35$	−6.84612	−1.15721
$36$	7.05290	1.17548
$37$	−2.39407	−0.393583	−0.196792	−	0.980445i	$-0.563052\pi$
$37$	−2.39407	−0.393583	−0.196792	+	0.980445i	$0.563052\pi$
$38$	2.67597	0.434100
$39$	−21.3677	−3.42156
$40$	2.18713	0.345815
$41$	4.58120	0.715464	0.357732	−	0.933824i	$-0.383550\pi$
$41$	4.58120	0.715464	0.357732	+	0.933824i	$0.383550\pi$
$42$	9.92468	1.53141
$43$	5.82575	0.888418	0.444209	−	0.895923i	$-0.353485\pi$
$43$	5.82575	0.888418	0.444209	+	0.895923i	$0.353485\pi$
$44$	0	0
$45$	−15.4256	−2.29951
$46$	9.30814	1.37241
$47$	−4.44029	−0.647683	−0.323842	−	0.946111i	$-0.604974\pi$
$47$	−4.44029	−0.647683	−0.323842	+	0.946111i	$0.604974\pi$
$48$	−3.17063	−0.457641
$49$	2.79809	0.399726
$50$	0.216473	0.0306138
$51$	−3.17063	−0.443977
$52$	6.73925	0.934565
$53$	2.88030	0.395640	0.197820	−	0.980238i	$-0.436614\pi$
$53$	2.88030	0.395640	0.197820	+	0.980238i	$0.436614\pi$
$54$	12.8503	1.74870
$55$	0	0
$56$	−3.13019	−0.418289
$57$	8.48452	1.12380
$58$	−0.265413	−0.0348504
$59$	−3.36981	−0.438712	−0.219356	−	0.975645i	$-0.570396\pi$
$59$	−3.36981	−0.438712	−0.219356	+	0.975645i	$0.570396\pi$
$60$	6.93458	0.895250
$61$	11.3670	1.45539	0.727697	−	0.685898i	$-0.240591\pi$
$61$	11.3670	1.45539	0.727697	+	0.685898i	$0.240591\pi$
$62$	4.70773	0.597883
$63$	22.0769	2.78143
$64$	1.00000	0.125000
$65$	−14.7396	−1.82822
$66$	0	0
$67$	10.7683	1.31556	0.657780	−	0.753210i	$-0.271495\pi$
$67$	10.7683	1.31556	0.657780	+	0.753210i	$0.271495\pi$
$68$	1.00000	0.121268
$69$	29.5127	3.55291
$70$	6.84612	0.818268
$71$	−7.10839	−0.843611	−0.421805	−	0.906686i	$-0.638603\pi$
$71$	−7.10839	−0.843611	−0.421805	+	0.906686i	$0.638603\pi$
$72$	−7.05290	−0.831193
$73$	−12.0894	−1.41496	−0.707479	−	0.706734i	$-0.750168\pi$
$73$	−12.0894	−1.41496	−0.707479	+	0.706734i	$0.750168\pi$
$74$	2.39407	0.278306
$75$	0.686355	0.0792534
$76$	−2.67597	−0.306955
$77$	0	0
$78$	21.3677	2.41941
$79$	4.81158	0.541345	0.270673	−	0.962671i	$-0.412754\pi$
$79$	4.81158	0.541345	0.270673	+	0.962671i	$0.412754\pi$
$80$	−2.18713	−0.244528
$81$	19.5847	2.17608
$82$	−4.58120	−0.505909
$83$	8.36392	0.918059	0.459030	−	0.888421i	$-0.348197\pi$
$83$	8.36392	0.918059	0.459030	+	0.888421i	$0.348197\pi$
$84$	−9.92468	−1.08287
$85$	−2.18713	−0.237227
$86$	−5.82575	−0.628206
$87$	−0.841526	−0.0902211
$88$	0	0
$89$	1.26739	0.134343	0.0671717	−	0.997741i	$-0.478602\pi$
$89$	1.26739	0.134343	0.0671717	+	0.997741i	$0.478602\pi$
$90$	15.4256	1.62600
$91$	21.0951	2.21137
$92$	−9.30814	−0.970441
$93$	14.9265	1.54780
$94$	4.44029	0.457981
$95$	5.85269	0.600473
$96$	3.17063	0.323601
$97$	−3.38343	−0.343535	−0.171767	−	0.985138i	$-0.554948\pi$
$97$	−3.38343	−0.343535	−0.171767	+	0.985138i	$0.554948\pi$
$98$	−2.79809	−0.282649
$99$	0	0
$100$	−0.216473	−0.0216473
$101$	−0.299809	−0.0298321	−0.0149161	−	0.999889i	$-0.504748\pi$
$101$	−0.299809	−0.0298321	−0.0149161	+	0.999889i	$0.504748\pi$
$102$	3.17063	0.313939
$103$	−17.8095	−1.75483	−0.877413	−	0.479736i	$-0.840732\pi$
$103$	−17.8095	−1.75483	−0.877413	+	0.479736i	$0.840732\pi$
$104$	−6.73925	−0.660837
$105$	21.7065	2.11834
$106$	−2.88030	−0.279760
$107$	−6.34289	−0.613191	−0.306595	−	0.951840i	$-0.599190\pi$
$107$	−6.34289	−0.613191	−0.306595	+	0.951840i	$0.599190\pi$
$108$	−12.8503	−1.23652
$109$	−13.2725	−1.27128	−0.635639	−	0.771987i	$-0.719263\pi$
$109$	−13.2725	−1.27128	−0.635639	+	0.771987i	$0.719263\pi$
$110$	0	0
$111$	7.59073	0.720480
$112$	3.13019	0.295775
$113$	4.07891	0.383711	0.191856	−	0.981423i	$-0.438549\pi$
$113$	4.07891	0.383711	0.191856	+	0.981423i	$0.438549\pi$
$114$	−8.48452	−0.794648
$115$	20.3581	1.89840
$116$	0.265413	0.0246430
$117$	47.5312	4.39427
$118$	3.36981	0.310216
$119$	3.13019	0.286944
$120$	−6.93458	−0.633037
$121$	0	0
$122$	−11.3670	−1.02912
$123$	−14.5253	−1.30970
$124$	−4.70773	−0.422767
$125$	11.4091	1.02046
$126$	−22.0769	−1.96677
$127$	5.36270	0.475863	0.237931	−	0.971282i	$-0.423531\pi$
$127$	5.36270	0.475863	0.237931	+	0.971282i	$0.423531\pi$
$128$	−1.00000	−0.0883883
$129$	−18.4713	−1.62631
$130$	14.7396	1.29275
$131$	−14.0676	−1.22909	−0.614547	−	0.788880i	$-0.710661\pi$
$131$	−14.0676	−1.22909	−0.614547	+	0.788880i	$0.710661\pi$
$132$	0	0
$133$	−8.37630	−0.726317
$134$	−10.7683	−0.930242
$135$	28.1052	2.41891
$136$	−1.00000	−0.0857493
$137$	−6.36682	−0.543954	−0.271977	−	0.962304i	$-0.587678\pi$
$137$	−6.36682	−0.543954	−0.271977	+	0.962304i	$0.587678\pi$
$138$	−29.5127	−2.51229
$139$	−2.59451	−0.220064	−0.110032	−	0.993928i	$-0.535095\pi$
$139$	−2.59451	−0.220064	−0.110032	+	0.993928i	$0.535095\pi$
$140$	−6.84612	−0.578603
$141$	14.0785	1.18563
$142$	7.10839	0.596523
$143$	0	0
$144$	7.05290	0.587742
$145$	−0.580492	−0.0482072
$146$	12.0894	1.00053
$147$	−8.87170	−0.731725
$148$	−2.39407	−0.196792
$149$	−7.10073	−0.581714	−0.290857	−	0.956767i	$-0.593940\pi$
$149$	−7.10073	−0.581714	−0.290857	+	0.956767i	$0.593940\pi$
$150$	−0.686355	−0.0560406
$151$	−11.9065	−0.968938	−0.484469	−	0.874808i	$-0.660987\pi$
$151$	−11.9065	−0.968938	−0.484469	+	0.874808i	$0.660987\pi$
$152$	2.67597	0.217050
$153$	7.05290	0.570193
$154$	0	0
$155$	10.2964	0.827028
$156$	−21.3677	−1.71078
$157$	15.3398	1.22425	0.612125	−	0.790761i	$-0.290315\pi$
$157$	15.3398	1.22425	0.612125	+	0.790761i	$0.290315\pi$
$158$	−4.81158	−0.382789
$159$	−9.13238	−0.724244
$160$	2.18713	0.172908
$161$	−29.1363	−2.29626
$162$	−19.5847	−1.53872
$163$	12.0373	0.942834	0.471417	−	0.881911i	$-0.343743\pi$
$163$	12.0373	0.942834	0.471417	+	0.881911i	$0.343743\pi$
$164$	4.58120	0.357732
$165$	0	0
$166$	−8.36392	−0.649166
$167$	10.0831	0.780253	0.390127	−	0.920761i	$-0.372431\pi$
$167$	10.0831	0.780253	0.390127	+	0.920761i	$0.372431\pi$
$168$	9.92468	0.765705
$169$	32.4174	2.49365
$170$	2.18713	0.167745
$171$	−18.8734	−1.44328
$172$	5.82575	0.444209
$173$	−2.16081	−0.164283	−0.0821416	−	0.996621i	$-0.526176\pi$
$173$	−2.16081	−0.164283	−0.0821416	+	0.996621i	$0.526176\pi$
$174$	0.841526	0.0637960
$175$	−0.677600	−0.0512218
$176$	0	0
$177$	10.6844	0.803091
$178$	−1.26739	−0.0949952
$179$	−15.9803	−1.19442	−0.597210	−	0.802085i	$-0.703724\pi$
$179$	−15.9803	−1.19442	−0.597210	+	0.802085i	$0.703724\pi$
$180$	−15.4256	−1.14976
$181$	−18.9553	−1.40894	−0.704470	−	0.709734i	$-0.748815\pi$
$181$	−18.9553	−1.40894	−0.704470	+	0.709734i	$0.748815\pi$
$182$	−21.0951	−1.56367
$183$	−36.0406	−2.66419
$184$	9.30814	0.686205
$185$	5.23615	0.384969
$186$	−14.9265	−1.09446
$187$	0	0
$188$	−4.44029	−0.323842
$189$	−40.2238	−2.92585
$190$	−5.85269	−0.424599
$191$	1.84977	0.133845	0.0669224	−	0.997758i	$-0.478682\pi$
$191$	1.84977	0.133845	0.0669224	+	0.997758i	$0.478682\pi$
$192$	−3.17063	−0.228821
$193$	10.2797	0.739951	0.369976	−	0.929041i	$-0.379366\pi$
$193$	10.2797	0.739951	0.369976	+	0.929041i	$0.379366\pi$
$194$	3.38343	0.242916
$195$	46.7338	3.34668
$196$	2.79809	0.199863
$197$	−22.1718	−1.57967	−0.789836	−	0.613318i	$-0.789834\pi$
$197$	−22.1718	−1.57967	−0.789836	+	0.613318i	$0.789834\pi$
$198$	0	0
$199$	−4.70436	−0.333483	−0.166742	−	0.986001i	$-0.553325\pi$
$199$	−4.70436	−0.333483	−0.166742	+	0.986001i	$0.553325\pi$
$200$	0.216473	0.0153069
$201$	−34.1424	−2.40822
$202$	0.299809	0.0210945
$203$	0.830793	0.0583102
$204$	−3.17063	−0.221989
$205$	−10.0197	−0.699804
$206$	17.8095	1.24085
$207$	−65.6494	−4.56295
$208$	6.73925	0.467283
$209$	0	0
$210$	−21.7065	−1.49789
$211$	8.21067	0.565246	0.282623	−	0.959231i	$-0.408796\pi$
$211$	8.21067	0.565246	0.282623	+	0.959231i	$0.408796\pi$
$212$	2.88030	0.197820
$213$	22.5381	1.54428
$214$	6.34289	0.433591
$215$	−12.7417	−0.868973
$216$	12.8503	0.874350
$217$	−14.7361	−1.00035
$218$	13.2725	0.898929
$219$	38.3311	2.59017
$220$	0	0
$221$	6.73925	0.453331
$222$	−7.59073	−0.509456
$223$	9.46231	0.633643	0.316821	−	0.948485i	$-0.397384\pi$
$223$	9.46231	0.633643	0.316821	+	0.948485i	$0.397384\pi$
$224$	−3.13019	−0.209145
$225$	−1.52676	−0.101784
$226$	−4.07891	−0.271325
$227$	20.7526	1.37740	0.688701	−	0.725046i	$-0.258181\pi$
$227$	20.7526	1.37740	0.688701	+	0.725046i	$0.258181\pi$
$228$	8.48452	0.561901
$229$	−10.0404	−0.663485	−0.331743	−	0.943370i	$-0.607637\pi$
$229$	−10.0404	−0.663485	−0.331743	+	0.943370i	$0.607637\pi$
$230$	−20.3581	−1.34237
$231$	0	0
$232$	−0.265413	−0.0174252
$233$	3.01860	0.197755	0.0988777	−	0.995100i	$-0.468475\pi$
$233$	3.01860	0.197755	0.0988777	+	0.995100i	$0.468475\pi$
$234$	−47.5312	−3.10721
$235$	9.71148	0.633507
$236$	−3.36981	−0.219356
$237$	−15.2558	−0.990968
$238$	−3.13019	−0.202900
$239$	−5.77527	−0.373571	−0.186786	−	0.982401i	$-0.559807\pi$
$239$	−5.77527	−0.373571	−0.186786	+	0.982401i	$0.559807\pi$
$240$	6.93458	0.447625
$241$	−6.44556	−0.415195	−0.207597	−	0.978214i	$-0.566564\pi$
$241$	−6.44556	−0.415195	−0.207597	+	0.978214i	$0.566564\pi$
$242$	0	0
$243$	−23.5452	−1.51042
$244$	11.3670	0.727697
$245$	−6.11977	−0.390978
$246$	14.5253	0.926100
$247$	−18.0340	−1.14748
$248$	4.70773	0.298941
$249$	−26.5189	−1.68057
$250$	−11.4091	−0.721574
$251$	5.18558	0.327311	0.163656	−	0.986518i	$-0.447671\pi$
$251$	5.18558	0.327311	0.163656	+	0.986518i	$0.447671\pi$
$252$	22.0769	1.39072
$253$	0	0
$254$	−5.36270	−0.336486
$255$	6.93458	0.434260
$256$	1.00000	0.0625000
$257$	18.3066	1.14193	0.570967	−	0.820973i	$-0.306568\pi$
$257$	18.3066	1.14193	0.570967	+	0.820973i	$0.306568\pi$
$258$	18.4713	1.14997
$259$	−7.49391	−0.465649
$260$	−14.7396	−0.914111
$261$	1.87193	0.115870
$262$	14.0676	0.869100
$263$	3.40382	0.209889	0.104944	−	0.994478i	$-0.466534\pi$
$263$	3.40382	0.209889	0.104944	+	0.994478i	$0.466534\pi$
$264$	0	0
$265$	−6.29959	−0.386981
$266$	8.37630	0.513584
$267$	−4.01844	−0.245924
$268$	10.7683	0.657780
$269$	16.4085	1.00044	0.500222	−	0.865897i	$-0.333252\pi$
$269$	16.4085	1.00044	0.500222	+	0.865897i	$0.333252\pi$
$270$	−28.1052	−1.71043
$271$	16.6704	1.01266	0.506328	−	0.862341i	$-0.331003\pi$
$271$	16.6704	1.01266	0.506328	+	0.862341i	$0.331003\pi$
$272$	1.00000	0.0606339
$273$	−66.8848	−4.04805
$274$	6.36682	0.384634
$275$	0	0
$276$	29.5127	1.77646
$277$	5.50874	0.330988	0.165494	−	0.986211i	$-0.447078\pi$
$277$	5.50874	0.330988	0.165494	+	0.986211i	$0.447078\pi$
$278$	2.59451	0.155608
$279$	−33.2032	−1.98782
$280$	6.84612	0.409134
$281$	−1.88017	−0.112162	−0.0560808	−	0.998426i	$-0.517860\pi$
$281$	−1.88017	−0.112162	−0.0560808	+	0.998426i	$0.517860\pi$
$282$	−14.0785	−0.838364
$283$	−1.59878	−0.0950375	−0.0475188	−	0.998870i	$-0.515131\pi$
$283$	−1.59878	−0.0950375	−0.0475188	+	0.998870i	$0.515131\pi$
$284$	−7.10839	−0.421805
$285$	−18.5567	−1.09921
$286$	0	0
$287$	14.3400	0.846465
$288$	−7.05290	−0.415596
$289$	1.00000	0.0588235
$290$	0.580492	0.0340877
$291$	10.7276	0.628863
$292$	−12.0894	−0.707479
$293$	10.5854	0.618406	0.309203	−	0.950996i	$-0.399938\pi$
$293$	10.5854	0.618406	0.309203	+	0.950996i	$0.399938\pi$
$294$	8.87170	0.517408
$295$	7.37021	0.429110
$296$	2.39407	0.139153
$297$	0	0
$298$	7.10073	0.411334
$299$	−62.7299	−3.62776
$300$	0.686355	0.0396267
$301$	18.2357	1.05109
$302$	11.9065	0.685143
$303$	0.950585	0.0546097
$304$	−2.67597	−0.153477
$305$	−24.8611	−1.42354
$306$	−7.05290	−0.403188
$307$	7.76348	0.443085	0.221543	−	0.975151i	$-0.428891\pi$
$307$	7.76348	0.443085	0.221543	+	0.975151i	$0.428891\pi$
$308$	0	0
$309$	56.4675	3.21232
$310$	−10.2964	−0.584797
$311$	−25.0298	−1.41931	−0.709655	−	0.704549i	$-0.751149\pi$
$311$	−25.0298	−1.41931	−0.709655	+	0.704549i	$0.751149\pi$
$312$	21.3677	1.20971
$313$	28.2272	1.59549	0.797747	−	0.602992i	$-0.206025\pi$
$313$	28.2272	1.59549	0.797747	+	0.602992i	$0.206025\pi$
$314$	−15.3398	−0.865675
$315$	−48.2850	−2.72055
$316$	4.81158	0.270673
$317$	−7.03316	−0.395021	−0.197511	−	0.980301i	$-0.563286\pi$
$317$	−7.03316	−0.395021	−0.197511	+	0.980301i	$0.563286\pi$
$318$	9.13238	0.512118
$319$	0	0
$320$	−2.18713	−0.122264
$321$	20.1110	1.12249
$322$	29.1363	1.62370
$323$	−2.67597	−0.148895
$324$	19.5847	1.08804
$325$	−1.45886	−0.0809231
$326$	−12.0373	−0.666684
$327$	42.0823	2.32716
$328$	−4.58120	−0.252955
$329$	−13.8990	−0.766274
$330$	0	0
$331$	−6.28983	−0.345720	−0.172860	−	0.984946i	$-0.555301\pi$
$331$	−6.28983	−0.345720	−0.172860	+	0.984946i	$0.555301\pi$
$332$	8.36392	0.459030
$333$	−16.8852	−0.925302
$334$	−10.0831	−0.551722
$335$	−23.5517	−1.28677
$336$	−9.92468	−0.541435
$337$	−3.14508	−0.171323	−0.0856617	−	0.996324i	$-0.527300\pi$
$337$	−3.14508	−0.171323	−0.0856617	+	0.996324i	$0.527300\pi$
$338$	−32.4174	−1.76328
$339$	−12.9327	−0.702409
$340$	−2.18713	−0.118614
$341$	0	0
$342$	18.8734	1.02055
$343$	−13.1528	−0.710184
$344$	−5.82575	−0.314103
$345$	−64.5480	−3.47515
$346$	2.16081	0.116166
$347$	1.63807	0.0879363	0.0439681	−	0.999033i	$-0.486000\pi$
$347$	1.63807	0.0879363	0.0439681	+	0.999033i	$0.486000\pi$
$348$	−0.841526	−0.0451106
$349$	−16.5935	−0.888227	−0.444114	−	0.895970i	$-0.646481\pi$
$349$	−16.5935	−0.888227	−0.444114	+	0.895970i	$0.646481\pi$
$350$	0.677600	0.0362193
$351$	−86.6011	−4.62242
$352$	0	0
$353$	−32.4264	−1.72589	−0.862943	−	0.505302i	$-0.831381\pi$
$353$	−32.4264	−1.72589	−0.862943	+	0.505302i	$0.831381\pi$
$354$	−10.6844	−0.567871
$355$	15.5470	0.825147
$356$	1.26739	0.0671717
$357$	−9.92468	−0.525270
$358$	15.9803	0.844583
$359$	−21.2509	−1.12158	−0.560790	−	0.827958i	$-0.689502\pi$
$359$	−21.2509	−1.12158	−0.560790	+	0.827958i	$0.689502\pi$
$360$	15.4256	0.813001
$361$	−11.8392	−0.623115
$362$	18.9553	0.996271
$363$	0	0
$364$	21.0951	1.10568
$365$	26.4411	1.38399
$366$	36.0406	1.88387
$367$	−12.5342	−0.654280	−0.327140	−	0.944976i	$-0.606085\pi$
$367$	−12.5342	−0.654280	−0.327140	+	0.944976i	$0.606085\pi$
$368$	−9.30814	−0.485221
$369$	32.3108	1.68203
$370$	−5.23615	−0.272214
$371$	9.01589	0.468082
$372$	14.9265	0.773902
$373$	−29.7257	−1.53914	−0.769570	−	0.638562i	$-0.779529\pi$
$373$	−29.7257	−1.53914	−0.769570	+	0.638562i	$0.779529\pi$
$374$	0	0
$375$	−36.1740	−1.86802
$376$	4.44029	0.228991
$377$	1.78868	0.0921218
$378$	40.2238	2.06889
$379$	6.45498	0.331570	0.165785	−	0.986162i	$-0.446984\pi$
$379$	6.45498	0.331570	0.165785	+	0.986162i	$0.446984\pi$
$380$	5.85269	0.300237
$381$	−17.0032	−0.871098
$382$	−1.84977	−0.0946425
$383$	−25.2092	−1.28813	−0.644064	−	0.764972i	$-0.722753\pi$
$383$	−25.2092	−1.28813	−0.644064	+	0.764972i	$0.722753\pi$
$384$	3.17063	0.161801
$385$	0	0
$386$	−10.2797	−0.523225
$387$	41.0884	2.08864
$388$	−3.38343	−0.171767
$389$	2.23486	0.113312	0.0566560	−	0.998394i	$-0.481956\pi$
$389$	2.23486	0.113312	0.0566560	+	0.998394i	$0.481956\pi$
$390$	−46.7338	−2.36646
$391$	−9.30814	−0.470733
$392$	−2.79809	−0.141325
$393$	44.6032	2.24994
$394$	22.1718	1.11700
$395$	−10.5235	−0.529497
$396$	0	0
$397$	12.7471	0.639758	0.319879	−	0.947458i	$-0.396358\pi$
$397$	12.7471	0.639758	0.319879	+	0.947458i	$0.396358\pi$
$398$	4.70436	0.235808
$399$	26.5582	1.32957
$400$	−0.216473	−0.0108236
$401$	−25.5730	−1.27705	−0.638526	−	0.769600i	$-0.720456\pi$
$401$	−25.5730	−1.27705	−0.638526	+	0.769600i	$0.720456\pi$
$402$	34.1424	1.70287
$403$	−31.7266	−1.58041
$404$	−0.299809	−0.0149161
$405$	−42.8343	−2.12845
$406$	−0.830793	−0.0412315
$407$	0	0
$408$	3.17063	0.156970
$409$	5.59118	0.276466	0.138233	−	0.990400i	$-0.455858\pi$
$409$	5.59118	0.276466	0.138233	+	0.990400i	$0.455858\pi$
$410$	10.0197	0.494836
$411$	20.1868	0.995744
$412$	−17.8095	−0.877413
$413$	−10.5482	−0.519041
$414$	65.6494	3.22649
$415$	−18.2930	−0.897966
$416$	−6.73925	−0.330419
$417$	8.22624	0.402841
$418$	0	0
$419$	23.4783	1.14699	0.573495	−	0.819209i	$-0.305587\pi$
$419$	23.4783	1.14699	0.573495	+	0.819209i	$0.305587\pi$
$420$	21.7065	1.05917
$421$	−18.5295	−0.903070	−0.451535	−	0.892253i	$-0.649123\pi$
$421$	−18.5295	−0.903070	−0.451535	+	0.892253i	$0.649123\pi$
$422$	−8.21067	−0.399689
$423$	−31.3169	−1.52268
$424$	−2.88030	−0.139880
$425$	−0.216473	−0.0105005
$426$	−22.5381	−1.09197
$427$	35.5808	1.72188
$428$	−6.34289	−0.306595
$429$	0	0
$430$	12.7417	0.614457
$431$	2.02369	0.0974779	0.0487389	−	0.998812i	$-0.484480\pi$
$431$	2.02369	0.0974779	0.0487389	+	0.998812i	$0.484480\pi$
$432$	−12.8503	−0.618259
$433$	−26.5586	−1.27633	−0.638163	−	0.769901i	$-0.720305\pi$
$433$	−26.5586	−1.27633	−0.638163	+	0.769901i	$0.720305\pi$
$434$	14.7361	0.707355
$435$	1.84053	0.0882465
$436$	−13.2725	−0.635639
$437$	24.9083	1.19153
$438$	−38.3311	−1.83153
$439$	−7.57359	−0.361468	−0.180734	−	0.983532i	$-0.557847\pi$
$439$	−7.57359	−0.361468	−0.180734	+	0.983532i	$0.557847\pi$
$440$	0	0
$441$	19.7346	0.939744
$442$	−6.73925	−0.320553
$443$	−35.8411	−1.70286	−0.851431	−	0.524467i	$-0.824265\pi$
$443$	−35.8411	−1.70286	−0.851431	+	0.524467i	$0.824265\pi$
$444$	7.59073	0.360240
$445$	−2.77195	−0.131403
$446$	−9.46231	−0.448053
$447$	22.5138	1.06487
$448$	3.13019	0.147888
$449$	1.77258	0.0836534	0.0418267	−	0.999125i	$-0.486682\pi$
$449$	1.77258	0.0836534	0.0418267	+	0.999125i	$0.486682\pi$
$450$	1.52676	0.0719722
$451$	0	0
$452$	4.07891	0.191856
$453$	37.7512	1.77370
$454$	−20.7526	−0.973970
$455$	−46.1377	−2.16297
$456$	−8.48452	−0.397324
$457$	19.5049	0.912400	0.456200	−	0.889877i	$-0.349210\pi$
$457$	19.5049	0.912400	0.456200	+	0.889877i	$0.349210\pi$
$458$	10.0404	0.469155
$459$	−12.8503	−0.599799
$460$	20.3581	0.949201
$461$	10.4672	0.487505	0.243753	−	0.969837i	$-0.421621\pi$
$461$	10.4672	0.487505	0.243753	+	0.969837i	$0.421621\pi$
$462$	0	0
$463$	−16.7184	−0.776969	−0.388485	−	0.921455i	$-0.627001\pi$
$463$	−16.7184	−0.776969	−0.388485	+	0.921455i	$0.627001\pi$
$464$	0.265413	0.0123215
$465$	−32.6461	−1.51393
$466$	−3.01860	−0.139834
$467$	14.3914	0.665953	0.332976	−	0.942935i	$-0.391947\pi$
$467$	14.3914	0.665953	0.332976	+	0.942935i	$0.391947\pi$
$468$	47.5312	2.19713
$469$	33.7069	1.55644
$470$	−9.71148	−0.447957
$471$	−48.6369	−2.24107
$472$	3.36981	0.155108
$473$	0	0
$474$	15.2558	0.700720
$475$	0.579274	0.0265789
$476$	3.13019	0.143472
$477$	20.3145	0.930137
$478$	5.77527	0.264155
$479$	8.02891	0.366850	0.183425	−	0.983034i	$-0.441282\pi$
$479$	8.02891	0.366850	0.183425	+	0.983034i	$0.441282\pi$
$480$	−6.93458	−0.316519
$481$	−16.1343	−0.735659
$482$	6.44556	0.293587
$483$	92.3803	4.20345
$484$	0	0
$485$	7.39998	0.336016
$486$	23.5452	1.06803
$487$	−39.4450	−1.78743	−0.893713	−	0.448640i	$-0.851909\pi$
$487$	−39.4450	−1.78743	−0.893713	+	0.448640i	$0.851909\pi$
$488$	−11.3670	−0.514560
$489$	−38.1658	−1.72592
$490$	6.11977	0.276463
$491$	19.2025	0.866598	0.433299	−	0.901250i	$-0.357349\pi$
$491$	19.2025	0.866598	0.433299	+	0.901250i	$0.357349\pi$
$492$	−14.5253	−0.654851
$493$	0.265413	0.0119536
$494$	18.0340	0.811389
$495$	0	0
$496$	−4.70773	−0.211383
$497$	−22.2506	−0.998076
$498$	26.5189	1.18834
$499$	−17.4617	−0.781693	−0.390847	−	0.920456i	$-0.627818\pi$
$499$	−17.4617	−0.781693	−0.390847	+	0.920456i	$0.627818\pi$
$500$	11.4091	0.510230
$501$	−31.9698	−1.42830
$502$	−5.18558	−0.231444
$503$	−32.8207	−1.46340	−0.731701	−	0.681626i	$-0.761273\pi$
$503$	−32.8207	−1.46340	−0.731701	+	0.681626i	$0.761273\pi$
$504$	−22.0769	−0.983384
$505$	0.655721	0.0291792
$506$	0	0
$507$	−102.784	−4.56479
$508$	5.36270	0.237931
$509$	27.0717	1.19993	0.599965	−	0.800026i	$-0.295181\pi$
$509$	27.0717	1.19993	0.599965	+	0.800026i	$0.295181\pi$
$510$	−6.93458	−0.307068
$511$	−37.8421	−1.67404
$512$	−1.00000	−0.0441942
$513$	34.3869	1.51822
$514$	−18.3066	−0.807470
$515$	38.9517	1.71642
$516$	−18.4713	−0.813153
$517$	0	0
$518$	7.49391	0.329263
$519$	6.85113	0.300731
$520$	14.7396	0.646374
$521$	−16.7321	−0.733048	−0.366524	−	0.930409i	$-0.619452\pi$
$521$	−16.7321	−0.733048	−0.366524	+	0.930409i	$0.619452\pi$
$522$	−1.87193	−0.0819322
$523$	−16.7704	−0.733319	−0.366660	−	0.930355i	$-0.619499\pi$
$523$	−16.7704	−0.733319	−0.366660	+	0.930355i	$0.619499\pi$
$524$	−14.0676	−0.614547
$525$	2.14842	0.0937647
$526$	−3.40382	−0.148414
$527$	−4.70773	−0.205072
$528$	0	0
$529$	63.6415	2.76702
$530$	6.29959	0.273637
$531$	−23.7670	−1.03140
$532$	−8.37630	−0.363159
$533$	30.8738	1.33729
$534$	4.01844	0.173895
$535$	13.8727	0.599770
$536$	−10.7683	−0.465121
$537$	50.6675	2.18646
$538$	−16.4085	−0.707421
$539$	0	0
$540$	28.1052	1.20945
$541$	−7.28013	−0.312997	−0.156499	−	0.987678i	$-0.550021\pi$
$541$	−7.28013	−0.312997	−0.156499	+	0.987678i	$0.550021\pi$
$542$	−16.6704	−0.716056
$543$	60.1004	2.57915
$544$	−1.00000	−0.0428746
$545$	29.0287	1.24345
$546$	66.8848	2.86241
$547$	16.9936	0.726594	0.363297	−	0.931673i	$-0.381651\pi$
$547$	16.9936	0.726594	0.363297	+	0.931673i	$0.381651\pi$
$548$	−6.36682	−0.271977
$549$	80.1703	3.42159
$550$	0	0
$551$	−0.710237	−0.0302571
$552$	−29.5127	−1.25614
$553$	15.0612	0.640466
$554$	−5.50874	−0.234044
$555$	−16.6019	−0.704711
$556$	−2.59451	−0.110032
$557$	11.1310	0.471635	0.235817	−	0.971797i	$-0.424223\pi$
$557$	11.1310	0.471635	0.235817	+	0.971797i	$0.424223\pi$
$558$	33.2032	1.40560
$559$	39.2611	1.66057
$560$	−6.84612	−0.289302
$561$	0	0
$562$	1.88017	0.0793103
$563$	14.6407	0.617030	0.308515	−	0.951219i	$-0.400168\pi$
$563$	14.6407	0.617030	0.308515	+	0.951219i	$0.400168\pi$
$564$	14.0785	0.592813
$565$	−8.92109	−0.375313
$566$	1.59878	0.0672017
$567$	61.3039	2.57452
$568$	7.10839	0.298261
$569$	−39.8108	−1.66896	−0.834478	−	0.551041i	$-0.814231\pi$
$569$	−39.8108	−1.66896	−0.834478	+	0.551041i	$0.814231\pi$
$570$	18.5567	0.777256
$571$	−36.2347	−1.51637	−0.758187	−	0.652037i	$-0.773915\pi$
$571$	−36.2347	−1.51637	−0.758187	+	0.652037i	$0.773915\pi$
$572$	0	0
$573$	−5.86494	−0.245012
$574$	−14.3400	−0.598541
$575$	2.01496	0.0840296
$576$	7.05290	0.293871
$577$	7.06413	0.294084	0.147042	−	0.989130i	$-0.453025\pi$
$577$	7.06413	0.294084	0.147042	+	0.989130i	$0.453025\pi$
$578$	−1.00000	−0.0415945
$579$	−32.5932	−1.35453
$580$	−0.580492	−0.0241036
$581$	26.1807	1.08616
$582$	−10.7276	−0.444673
$583$	0	0
$584$	12.0894	0.500263
$585$	−103.957	−4.29809
$586$	−10.5854	−0.437279
$587$	15.0239	0.620104	0.310052	−	0.950720i	$-0.399654\pi$
$587$	15.0239	0.620104	0.310052	+	0.950720i	$0.399654\pi$
$588$	−8.87170	−0.365863
$589$	12.5978	0.519082
$590$	−7.37021	−0.303427
$591$	70.2985	2.89169
$592$	−2.39407	−0.0983959
$593$	30.4162	1.24904	0.624522	−	0.781008i	$-0.285294\pi$
$593$	30.4162	1.24904	0.624522	+	0.781008i	$0.285294\pi$
$594$	0	0
$595$	−6.84612	−0.280664
$596$	−7.10073	−0.290857
$597$	14.9158	0.610463
$598$	62.7299	2.56522
$599$	−32.0577	−1.30984	−0.654922	−	0.755696i	$-0.727299\pi$
$599$	−32.0577	−1.30984	−0.654922	+	0.755696i	$0.727299\pi$
$600$	−0.686355	−0.0280203
$601$	−33.4493	−1.36443	−0.682213	−	0.731154i	$-0.738982\pi$
$601$	−33.4493	−1.36443	−0.682213	+	0.731154i	$0.738982\pi$
$602$	−18.2357	−0.743231
$603$	75.9480	3.09284
$604$	−11.9065	−0.484469
$605$	0	0
$606$	−0.950585	−0.0386149
$607$	38.5586	1.56505	0.782523	−	0.622621i	$-0.213932\pi$
$607$	38.5586	1.56505	0.782523	+	0.622621i	$0.213932\pi$
$608$	2.67597	0.108525
$609$	−2.63414	−0.106741
$610$	24.8611	1.00660
$611$	−29.9242	−1.21060
$612$	7.05290	0.285097
$613$	23.2238	0.938001	0.469001	−	0.883198i	$-0.344614\pi$
$613$	23.2238	0.938001	0.469001	+	0.883198i	$0.344614\pi$
$614$	−7.76348	−0.313309
$615$	31.7687	1.28104
$616$	0	0
$617$	−11.3379	−0.456446	−0.228223	−	0.973609i	$-0.573292\pi$
$617$	−11.3379	−0.456446	−0.228223	+	0.973609i	$0.573292\pi$
$618$	−56.4675	−2.27145
$619$	−42.8349	−1.72168	−0.860840	−	0.508875i	$-0.830061\pi$
$619$	−42.8349	−1.72168	−0.860840	+	0.508875i	$0.830061\pi$
$620$	10.2964	0.413514
$621$	119.612	4.79987
$622$	25.0298	1.00360
$623$	3.96718	0.158942
$624$	−21.3677	−0.855391
$625$	−23.8708	−0.954831
$626$	−28.2272	−1.12818
$627$	0	0
$628$	15.3398	0.612125
$629$	−2.39407	−0.0954580
$630$	48.2850	1.92372
$631$	−37.2319	−1.48218	−0.741089	−	0.671407i	$-0.765690\pi$
$631$	−37.2319	−1.48218	−0.741089	+	0.671407i	$0.765690\pi$
$632$	−4.81158	−0.191395
$633$	−26.0330	−1.03472
$634$	7.03316	0.279322
$635$	−11.7289	−0.465448
$636$	−9.13238	−0.362122
$637$	18.8570	0.747141
$638$	0	0
$639$	−50.1348	−1.98330
$640$	2.18713	0.0864538
$641$	5.06222	0.199946	0.0999728	−	0.994990i	$-0.468124\pi$
$641$	5.06222	0.199946	0.0999728	+	0.994990i	$0.468124\pi$
$642$	−20.1110	−0.793717
$643$	−26.4199	−1.04190	−0.520951	−	0.853587i	$-0.674422\pi$
$643$	−26.4199	−1.04190	−0.520951	+	0.853587i	$0.674422\pi$
$644$	−29.1363	−1.14813
$645$	40.3991	1.59071
$646$	2.67597	0.105285
$647$	19.0552	0.749138	0.374569	−	0.927199i	$-0.377791\pi$
$647$	19.0552	0.749138	0.374569	+	0.927199i	$0.377791\pi$
$648$	−19.5847	−0.769361
$649$	0	0
$650$	1.45886	0.0572213
$651$	46.7227	1.83121
$652$	12.0373	0.471417
$653$	31.9586	1.25064	0.625318	−	0.780370i	$-0.284969\pi$
$653$	31.9586	1.25064	0.625318	+	0.780370i	$0.284969\pi$
$654$	−42.0823	−1.64555
$655$	30.7677	1.20219
$656$	4.58120	0.178866
$657$	−85.2654	−3.32652
$658$	13.8990	0.541838
$659$	−16.8240	−0.655371	−0.327685	−	0.944787i	$-0.606269\pi$
$659$	−16.8240	−0.655371	−0.327685	+	0.944787i	$0.606269\pi$
$660$	0	0
$661$	−0.894564	−0.0347945	−0.0173972	−	0.999849i	$-0.505538\pi$
$661$	−0.894564	−0.0347945	−0.0173972	+	0.999849i	$0.505538\pi$
$662$	6.28983	0.244461
$663$	−21.3677	−0.829851
$664$	−8.36392	−0.324583
$665$	18.3200	0.710420
$666$	16.8852	0.654287
$667$	−2.47050	−0.0956582
$668$	10.0831	0.390127
$669$	−30.0015	−1.15992
$670$	23.5517	0.909882
$671$	0	0
$672$	9.92468	0.382853
$673$	12.6863	0.489022	0.244511	−	0.969647i	$-0.421373\pi$
$673$	12.6863	0.489022	0.244511	+	0.969647i	$0.421373\pi$
$674$	3.14508	0.121144
$675$	2.78173	0.107069
$676$	32.4174	1.24682
$677$	46.3409	1.78103	0.890513	−	0.454957i	$-0.150345\pi$
$677$	46.3409	1.78103	0.890513	+	0.454957i	$0.150345\pi$
$678$	12.9327	0.496678
$679$	−10.5908	−0.406436
$680$	2.18713	0.0838725
$681$	−65.7990	−2.52142
$682$	0	0
$683$	−13.7926	−0.527758	−0.263879	−	0.964556i	$-0.585002\pi$
$683$	−13.7926	−0.527758	−0.263879	+	0.964556i	$0.585002\pi$
$684$	−18.8734	−0.721641
$685$	13.9251	0.532049
$686$	13.1528	0.502176
$687$	31.8343	1.21455
$688$	5.82575	0.222105
$689$	19.4111	0.739503
$690$	64.5480	2.45730
$691$	−36.5395	−1.39003	−0.695013	−	0.718997i	$-0.744601\pi$
$691$	−36.5395	−1.39003	−0.695013	+	0.718997i	$0.744601\pi$
$692$	−2.16081	−0.0821416
$693$	0	0
$694$	−1.63807	−0.0621803
$695$	5.67453	0.215247
$696$	0.841526	0.0318980
$697$	4.58120	0.173525
$698$	16.5935	0.628072
$699$	−9.57088	−0.362004
$700$	−0.677600	−0.0256109
$701$	−42.8622	−1.61888	−0.809442	−	0.587200i	$-0.800230\pi$
$701$	−42.8622	−1.61888	−0.809442	+	0.587200i	$0.800230\pi$
$702$	86.6011	3.26855
$703$	6.40648	0.241625
$704$	0	0
$705$	−30.7915	−1.15968
$706$	32.4264	1.22039
$707$	−0.938460	−0.0352944
$708$	10.6844	0.401546
$709$	−33.3423	−1.25220	−0.626099	−	0.779744i	$-0.715349\pi$
$709$	−33.3423	−1.25220	−0.626099	+	0.779744i	$0.715349\pi$
$710$	−15.5470	−0.583467
$711$	33.9356	1.27269
$712$	−1.26739	−0.0474976
$713$	43.8202	1.64108
$714$	9.92468	0.371422
$715$	0	0
$716$	−15.9803	−0.597210
$717$	18.3113	0.683846
$718$	21.2509	0.793076
$719$	21.8868	0.816240	0.408120	−	0.912928i	$-0.366184\pi$
$719$	21.8868	0.816240	0.408120	+	0.912928i	$0.366184\pi$
$720$	−15.4256	−0.574878
$721$	−55.7472	−2.07614
$722$	11.8392	0.440609
$723$	20.4365	0.760041
$724$	−18.9553	−0.704470
$725$	−0.0574546	−0.00213381
$726$	0	0
$727$	−42.7204	−1.58441	−0.792205	−	0.610255i	$-0.791067\pi$
$727$	−42.7204	−1.58441	−0.792205	+	0.610255i	$0.791067\pi$
$728$	−21.0951	−0.781837
$729$	15.8989	0.588848
$730$	−26.4411	−0.978628
$731$	5.82575	0.215473
$732$	−36.0406	−1.33210
$733$	−7.24038	−0.267429	−0.133715	−	0.991020i	$-0.542691\pi$
$733$	−7.24038	−0.267429	−0.133715	+	0.991020i	$0.542691\pi$
$734$	12.5342	0.462646
$735$	19.4035	0.715710
$736$	9.30814	0.343103
$737$	0	0
$738$	−32.3108	−1.18938
$739$	14.0488	0.516792	0.258396	−	0.966039i	$-0.416806\pi$
$739$	14.0488	0.516792	0.258396	+	0.966039i	$0.416806\pi$
$740$	5.23615	0.192485
$741$	57.1793	2.10053
$742$	−9.01589	−0.330984
$743$	1.95011	0.0715426	0.0357713	−	0.999360i	$-0.488611\pi$
$743$	1.95011	0.0715426	0.0357713	+	0.999360i	$0.488611\pi$
$744$	−14.9265	−0.547231
$745$	15.5302	0.568982
$746$	29.7257	1.08834
$747$	58.9899	2.15833
$748$	0	0
$749$	−19.8545	−0.725466
$750$	36.1740	1.32089
$751$	−31.1250	−1.13577	−0.567884	−	0.823109i	$-0.692238\pi$
$751$	−31.1250	−1.13577	−0.567884	+	0.823109i	$0.692238\pi$
$752$	−4.44029	−0.161921
$753$	−16.4416	−0.599164
$754$	−1.78868	−0.0651400
$755$	26.0411	0.947731
$756$	−40.2238	−1.46292
$757$	10.1186	0.367768	0.183884	−	0.982948i	$-0.441133\pi$
$757$	10.1186	0.367768	0.183884	+	0.982948i	$0.441133\pi$
$758$	−6.45498	−0.234456
$759$	0	0
$760$	−5.85269	−0.212299
$761$	−29.2036	−1.05863	−0.529316	−	0.848425i	$-0.677551\pi$
$761$	−29.2036	−1.05863	−0.529316	+	0.848425i	$0.677551\pi$
$762$	17.0032	0.615959
$763$	−41.5455	−1.50405
$764$	1.84977	0.0669224
$765$	−15.4256	−0.557714
$766$	25.2092	0.910844
$767$	−22.7100	−0.820010
$768$	−3.17063	−0.114410
$769$	6.58062	0.237303	0.118652	−	0.992936i	$-0.462143\pi$
$769$	6.58062	0.237303	0.118652	+	0.992936i	$0.462143\pi$
$770$	0	0
$771$	−58.0435	−2.09039
$772$	10.2797	0.369976
$773$	0.262894	0.00945563	0.00472781	−	0.999989i	$-0.498495\pi$
$773$	0.262894	0.00945563	0.00472781	+	0.999989i	$0.498495\pi$
$774$	−41.0884	−1.47689
$775$	1.01909	0.0366070
$776$	3.38343	0.121458
$777$	23.7604	0.852400
$778$	−2.23486	−0.0801237
$779$	−12.2592	−0.439230
$780$	46.7338	1.67334
$781$	0	0
$782$	9.30814	0.332859
$783$	−3.41063	−0.121886
$784$	2.79809	0.0999316
$785$	−33.5501	−1.19745
$786$	−44.6032	−1.59094
$787$	−54.8081	−1.95370	−0.976848	−	0.213934i	$-0.931372\pi$
$787$	−54.8081	−1.95370	−0.976848	+	0.213934i	$0.931372\pi$
$788$	−22.1718	−0.789836
$789$	−10.7923	−0.384215
$790$	10.5235	0.374411
$791$	12.7678	0.453969
$792$	0	0
$793$	76.6050	2.72032
$794$	−12.7471	−0.452377
$795$	19.9737	0.708393
$796$	−4.70436	−0.166742
$797$	20.5203	0.726867	0.363434	−	0.931620i	$-0.381604\pi$
$797$	20.5203	0.726867	0.363434	+	0.931620i	$0.381604\pi$
$798$	−26.5582	−0.940148
$799$	−4.44029	−0.157086
$800$	0.216473	0.00765346
$801$	8.93880	0.315837
$802$	25.5730	0.903012
$803$	0	0
$804$	−34.1424	−1.20411
$805$	63.7247	2.24600
$806$	31.7266	1.11752
$807$	−52.0253	−1.83138
$808$	0.299809	0.0105473
$809$	−35.9855	−1.26518	−0.632591	−	0.774486i	$-0.718008\pi$
$809$	−35.9855	−1.26518	−0.632591	+	0.774486i	$0.718008\pi$
$810$	42.8343	1.50504
$811$	−30.8441	−1.08308	−0.541541	−	0.840674i	$-0.682159\pi$
$811$	−30.8441	−1.08308	−0.541541	+	0.840674i	$0.682159\pi$
$812$	0.830793	0.0291551
$813$	−52.8558	−1.85373
$814$	0	0
$815$	−26.3271	−0.922198
$816$	−3.17063	−0.110994
$817$	−15.5895	−0.545409
$818$	−5.59118	−0.195491
$819$	148.782	5.19886
$820$	−10.0197	−0.349902
$821$	−32.7804	−1.14405	−0.572023	−	0.820238i	$-0.693841\pi$
$821$	−32.7804	−1.14405	−0.572023	+	0.820238i	$0.693841\pi$
$822$	−20.1868	−0.704097
$823$	17.6899	0.616632	0.308316	−	0.951284i	$-0.400235\pi$
$823$	17.6899	0.616632	0.308316	+	0.951284i	$0.400235\pi$
$824$	17.8095	0.620425
$825$	0	0
$826$	10.5482	0.367017
$827$	−54.5617	−1.89730	−0.948649	−	0.316332i	$-0.897549\pi$
$827$	−54.5617	−1.89730	−0.948649	+	0.316332i	$0.897549\pi$
$828$	−65.6494	−2.28148
$829$	−0.295501	−0.0102632	−0.00513158	−	0.999987i	$-0.501633\pi$
$829$	−0.295501	−0.0102632	−0.00513158	+	0.999987i	$0.501633\pi$
$830$	18.2930	0.634958
$831$	−17.4662	−0.605895
$832$	6.73925	0.233641
$833$	2.79809	0.0969479
$834$	−8.22624	−0.284851
$835$	−22.0530	−0.763176
$836$	0	0
$837$	60.4956	2.09103
$838$	−23.4783	−0.811044
$839$	−7.18025	−0.247890	−0.123945	−	0.992289i	$-0.539555\pi$
$839$	−7.18025	−0.247890	−0.123945	+	0.992289i	$0.539555\pi$
$840$	−21.7065	−0.748947
$841$	−28.9296	−0.997571
$842$	18.5295	0.638567
$843$	5.96133	0.205319
$844$	8.21067	0.282623
$845$	−70.9011	−2.43907
$846$	31.3169	1.07670
$847$	0	0
$848$	2.88030	0.0989100
$849$	5.06914	0.173972
$850$	0.216473	0.00742495
$851$	22.2844	0.763899
$852$	22.5381	0.772142
$853$	49.6020	1.69834	0.849170	−	0.528120i	$-0.177103\pi$
$853$	49.6020	1.69834	0.849170	+	0.528120i	$0.177103\pi$
$854$	−35.5808	−1.21755
$855$	41.2785	1.41169
$856$	6.34289	0.216796
$857$	14.8385	0.506875	0.253437	−	0.967352i	$-0.418439\pi$
$857$	14.8385	0.506875	0.253437	+	0.967352i	$0.418439\pi$
$858$	0	0
$859$	17.1317	0.584526	0.292263	−	0.956338i	$-0.405592\pi$
$859$	17.1317	0.584526	0.292263	+	0.956338i	$0.405592\pi$
$860$	−12.7417	−0.434487
$861$	−45.4670	−1.54951
$862$	−2.02369	−0.0689273
$863$	44.5067	1.51502	0.757512	−	0.652821i	$-0.226415\pi$
$863$	44.5067	1.51502	0.757512	+	0.652821i	$0.226415\pi$
$864$	12.8503	0.437175
$865$	4.72597	0.160688
$866$	26.5586	0.902499
$867$	−3.17063	−0.107680
$868$	−14.7361	−0.500176
$869$	0	0
$870$	−1.84053	−0.0623997
$871$	72.5704	2.45895
$872$	13.2725	0.449465
$873$	−23.8630	−0.807639
$874$	−24.9083	−0.842537
$875$	35.7126	1.20731
$876$	38.3311	1.29509
$877$	−14.8400	−0.501110	−0.250555	−	0.968102i	$-0.580613\pi$
$877$	−14.8400	−0.501110	−0.250555	+	0.968102i	$0.580613\pi$
$878$	7.57359	0.255596
$879$	−33.5624	−1.13203
$880$	0	0
$881$	4.10396	0.138266	0.0691330	−	0.997607i	$-0.477977\pi$
$881$	4.10396	0.138266	0.0691330	+	0.997607i	$0.477977\pi$
$882$	−19.7346	−0.664499
$883$	28.1238	0.946442	0.473221	−	0.880944i	$-0.343091\pi$
$883$	28.1238	0.946442	0.473221	+	0.880944i	$0.343091\pi$
$884$	6.73925	0.226665
$885$	−23.3682	−0.785514
$886$	35.8411	1.20411
$887$	−10.0553	−0.337623	−0.168811	−	0.985648i	$-0.553993\pi$
$887$	−10.0553	−0.337623	−0.168811	+	0.985648i	$0.553993\pi$
$888$	−7.59073	−0.254728
$889$	16.7863	0.562994
$890$	2.77195	0.0929160
$891$	0	0
$892$	9.46231	0.316821
$893$	11.8821	0.397619
$894$	−22.5138	−0.752974
$895$	34.9509	1.16828
$896$	−3.13019	−0.104572
$897$	198.893	6.64085
$898$	−1.77258	−0.0591519
$899$	−1.24949	−0.0416729
$900$	−1.52676	−0.0508920
$901$	2.88030	0.0959568
$902$	0	0
$903$	−57.8187	−1.92408
$904$	−4.07891	−0.135662
$905$	41.4577	1.37810
$906$	−37.7512	−1.25420
$907$	8.68669	0.288437	0.144218	−	0.989546i	$-0.453933\pi$
$907$	8.68669	0.288437	0.144218	+	0.989546i	$0.453933\pi$
$908$	20.7526	0.688701
$909$	−2.11453	−0.0701344
$910$	46.1377	1.52945
$911$	30.9082	1.02404	0.512018	−	0.858975i	$-0.328898\pi$
$911$	30.9082	1.02404	0.512018	+	0.858975i	$0.328898\pi$
$912$	8.48452	0.280951
$913$	0	0
$914$	−19.5049	−0.645164
$915$	78.8253	2.60588
$916$	−10.0404	−0.331743
$917$	−44.0343	−1.45414
$918$	12.8503	0.424122
$919$	50.0501	1.65100	0.825500	−	0.564403i	$-0.190894\pi$
$919$	50.0501	1.65100	0.825500	+	0.564403i	$0.190894\pi$
$920$	−20.3581	−0.671187
$921$	−24.6151	−0.811096
$922$	−10.4672	−0.344718
$923$	−47.9052	−1.57682
$924$	0	0
$925$	0.518251	0.0170400
$926$	16.7184	0.549400
$927$	−125.609	−4.12554
$928$	−0.265413	−0.00871260
$929$	14.5201	0.476389	0.238194	−	0.971217i	$-0.423444\pi$
$929$	14.5201	0.476389	0.238194	+	0.971217i	$0.423444\pi$
$930$	32.6461	1.07051
$931$	−7.48760	−0.245396
$932$	3.01860	0.0988777
$933$	79.3603	2.59814
$934$	−14.3914	−0.470900
$935$	0	0
$936$	−47.5312	−1.55361
$937$	12.2245	0.399358	0.199679	−	0.979861i	$-0.436010\pi$
$937$	12.2245	0.399358	0.199679	+	0.979861i	$0.436010\pi$
$938$	−33.7069	−1.10057
$939$	−89.4980	−2.92066
$940$	9.71148	0.316754
$941$	14.5566	0.474531	0.237266	−	0.971445i	$-0.423749\pi$
$941$	14.5566	0.474531	0.237266	+	0.971445i	$0.423749\pi$
$942$	48.6369	1.58467
$943$	−42.6425	−1.38863
$944$	−3.36981	−0.109678
$945$	87.9745	2.86181
$946$	0	0
$947$	−48.6255	−1.58012	−0.790059	−	0.613031i	$-0.789950\pi$
$947$	−48.6255	−1.58012	−0.790059	+	0.613031i	$0.789950\pi$
$948$	−15.2558	−0.495484
$949$	−81.4735	−2.64474
$950$	−0.579274	−0.0187941
$951$	22.2995	0.723112
$952$	−3.13019	−0.101450
$953$	−46.3958	−1.50291	−0.751454	−	0.659786i	$-0.770647\pi$
$953$	−46.3958	−1.50291	−0.751454	+	0.659786i	$0.770647\pi$
$954$	−20.3145	−0.657706
$955$	−4.04569	−0.130915
$956$	−5.77527	−0.186786
$957$	0	0
$958$	−8.02891	−0.259402
$959$	−19.9294	−0.643553
$960$	6.93458	0.223812
$961$	−8.83726	−0.285073
$962$	16.1343	0.520189
$963$	−44.7358	−1.44159
$964$	−6.44556	−0.207597
$965$	−22.4831	−0.723756
$966$	−92.3803	−2.97229
$967$	−28.0911	−0.903350	−0.451675	−	0.892183i	$-0.649173\pi$
$967$	−28.0911	−0.903350	−0.451675	+	0.892183i	$0.649173\pi$
$968$	0	0
$969$	8.48452	0.272562
$970$	−7.39998	−0.237599
$971$	51.6187	1.65652	0.828262	−	0.560341i	$-0.189330\pi$
$971$	51.6187	1.65652	0.828262	+	0.560341i	$0.189330\pi$
$972$	−23.5452	−0.755212
$973$	−8.12131	−0.260357
$974$	39.4450	1.26390
$975$	4.62551	0.148135
$976$	11.3670	0.363849
$977$	−15.6501	−0.500692	−0.250346	−	0.968156i	$-0.580544\pi$
$977$	−15.6501	−0.500692	−0.250346	+	0.968156i	$0.580544\pi$
$978$	38.1658	1.22041
$979$	0	0
$980$	−6.11977	−0.195489
$981$	−93.6099	−2.98873
$982$	−19.2025	−0.612777
$983$	41.4087	1.32073	0.660366	−	0.750944i	$-0.270401\pi$
$983$	41.4087	1.32073	0.660366	+	0.750944i	$0.270401\pi$
$984$	14.5253	0.463050
$985$	48.4925	1.54510
$986$	−0.265413	−0.00845247
$987$	44.0685	1.40271
$988$	−18.0340	−0.573739
$989$	−54.2269	−1.72431
$990$	0	0
$991$	49.8148	1.58242	0.791210	−	0.611545i	$-0.209451\pi$
$991$	49.8148	1.58242	0.791210	+	0.611545i	$0.209451\pi$
$992$	4.70773	0.149471
$993$	19.9427	0.632863
$994$	22.2506	0.705747
$995$	10.2890	0.326184
$996$	−26.5189	−0.840284
$997$	9.96452	0.315579	0.157790	−	0.987473i	$-0.449563\pi$
$997$	9.96452	0.315579	0.157790	+	0.987473i	$0.449563\pi$
$998$	17.4617	0.552741
$999$	30.7645	0.973345

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4114.2.a.bg.1.1		8
11.7	odd	10		374.2.g.f.137.4	✓	16
11.8	odd	10		374.2.g.f.273.4	yes	16
11.10	odd	2		4114.2.a.bi.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
374.2.g.f.137.4	✓	16	11.7	odd	10
374.2.g.f.273.4	yes	16	11.8	odd	10
4114.2.a.bg.1.1		8	1.1	even	1	trivial
4114.2.a.bi.1.1		8	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:	\( N \)	\(=\)	\( 4114 = 2 \cdot 11^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4114.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.17063 q^{3} +1.00000 q^{4} -2.18713 q^{5} +3.17063 q^{6} +3.13019 q^{7} -1.00000 q^{8} +7.05290 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.17063 q^{3} +1.00000 q^{4} -2.18713 q^{5} +3.17063 q^{6} +3.13019 q^{7} -1.00000 q^{8} +7.05290 q^{9} +2.18713 q^{10} -3.17063 q^{12} +6.73925 q^{13} -3.13019 q^{14} +6.93458 q^{15} +1.00000 q^{16} +1.00000 q^{17} -7.05290 q^{18} -2.67597 q^{19} -2.18713 q^{20} -9.92468 q^{21} -9.30814 q^{23} +3.17063 q^{24} -0.216473 q^{25} -6.73925 q^{26} -12.8503 q^{27} +3.13019 q^{28} +0.265413 q^{29} -6.93458 q^{30} -4.70773 q^{31} -1.00000 q^{32} -1.00000 q^{34} -6.84612 q^{35} +7.05290 q^{36} -2.39407 q^{37} +2.67597 q^{38} -21.3677 q^{39} +2.18713 q^{40} +4.58120 q^{41} +9.92468 q^{42} +5.82575 q^{43} -15.4256 q^{45} +9.30814 q^{46} -4.44029 q^{47} -3.17063 q^{48} +2.79809 q^{49} +0.216473 q^{50} -3.17063 q^{51} +6.73925 q^{52} +2.88030 q^{53} +12.8503 q^{54} -3.13019 q^{56} +8.48452 q^{57} -0.265413 q^{58} -3.36981 q^{59} +6.93458 q^{60} +11.3670 q^{61} +4.70773 q^{62} +22.0769 q^{63} +1.00000 q^{64} -14.7396 q^{65} +10.7683 q^{67} +1.00000 q^{68} +29.5127 q^{69} +6.84612 q^{70} -7.10839 q^{71} -7.05290 q^{72} -12.0894 q^{73} +2.39407 q^{74} +0.686355 q^{75} -2.67597 q^{76} +21.3677 q^{78} +4.81158 q^{79} -2.18713 q^{80} +19.5847 q^{81} -4.58120 q^{82} +8.36392 q^{83} -9.92468 q^{84} -2.18713 q^{85} -5.82575 q^{86} -0.841526 q^{87} +1.26739 q^{89} +15.4256 q^{90} +21.0951 q^{91} -9.30814 q^{92} +14.9265 q^{93} +4.44029 q^{94} +5.85269 q^{95} +3.17063 q^{96} -3.38343 q^{97} -2.79809 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 5 q^{3} + 8 q^{4} + 2 q^{5} + 5 q^{6} - 11 q^{7} - 8 q^{8} + 11 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 - 5 * q^3 + 8 * q^4 + 2 * q^5 + 5 * q^6 - 11 * q^7 - 8 * q^8 + 11 * q^9	\( 8 q - 8 q^{2} - 5 q^{3} + 8 q^{4} + 2 q^{5} + 5 q^{6} - 11 q^{7} - 8 q^{8} + 11 q^{9} - 2 q^{10} - 5 q^{12} + 5 q^{13} + 11 q^{14} + 2 q^{15} + 8 q^{16} + 8 q^{17} - 11 q^{18} - 15 q^{19} + 2 q^{20} - 12 q^{23} + 5 q^{24} + 24 q^{25} - 5 q^{26} - 17 q^{27} - 11 q^{28} - 22 q^{29} - 2 q^{30} - 7 q^{31} - 8 q^{32} - 8 q^{34} - 10 q^{35} + 11 q^{36} + 15 q^{37} + 15 q^{38} - 35 q^{39} - 2 q^{40} - 17 q^{41} - 8 q^{43} + 3 q^{45} + 12 q^{46} + 18 q^{47} - 5 q^{48} + q^{49} - 24 q^{50} - 5 q^{51} + 5 q^{52} + 20 q^{53} + 17 q^{54} + 11 q^{56} - 9 q^{57} + 22 q^{58} + 6 q^{59} + 2 q^{60} + 5 q^{61} + 7 q^{62} + 14 q^{63} + 8 q^{64} - 3 q^{65} + 13 q^{67} + 8 q^{68} + 46 q^{69} + 10 q^{70} - 18 q^{71} - 11 q^{72} + 4 q^{73} - 15 q^{74} - 49 q^{75} - 15 q^{76} + 35 q^{78} - 21 q^{79} + 2 q^{80} + 8 q^{81} + 17 q^{82} - 38 q^{83} + 2 q^{85} + 8 q^{86} + 46 q^{87} + 12 q^{89} - 3 q^{90} - 5 q^{91} - 12 q^{92} + 3 q^{93} - 18 q^{94} - 63 q^{95} + 5 q^{96} - 66 q^{97} - q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 - 5 * q^3 + 8 * q^4 + 2 * q^5 + 5 * q^6 - 11 * q^7 - 8 * q^8 + 11 * q^9 - 2 * q^10 - 5 * q^12 + 5 * q^13 + 11 * q^14 + 2 * q^15 + 8 * q^16 + 8 * q^17 - 11 * q^18 - 15 * q^19 + 2 * q^20 - 12 * q^23 + 5 * q^24 + 24 * q^25 - 5 * q^26 - 17 * q^27 - 11 * q^28 - 22 * q^29 - 2 * q^30 - 7 * q^31 - 8 * q^32 - 8 * q^34 - 10 * q^35 + 11 * q^36 + 15 * q^37 + 15 * q^38 - 35 * q^39 - 2 * q^40 - 17 * q^41 - 8 * q^43 + 3 * q^45 + 12 * q^46 + 18 * q^47 - 5 * q^48 + q^49 - 24 * q^50 - 5 * q^51 + 5 * q^52 + 20 * q^53 + 17 * q^54 + 11 * q^56 - 9 * q^57 + 22 * q^58 + 6 * q^59 + 2 * q^60 + 5 * q^61 + 7 * q^62 + 14 * q^63 + 8 * q^64 - 3 * q^65 + 13 * q^67 + 8 * q^68 + 46 * q^69 + 10 * q^70 - 18 * q^71 - 11 * q^72 + 4 * q^73 - 15 * q^74 - 49 * q^75 - 15 * q^76 + 35 * q^78 - 21 * q^79 + 2 * q^80 + 8 * q^81 + 17 * q^82 - 38 * q^83 + 2 * q^85 + 8 * q^86 + 46 * q^87 + 12 * q^89 - 3 * q^90 - 5 * q^91 - 12 * q^92 + 3 * q^93 - 18 * q^94 - 63 * q^95 + 5 * q^96 - 66 * q^97 - q^98

Introduction

L-functions

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Database

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