LMFDB - Embedded newform 6422.2.a.bo.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, 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("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.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:	$51.2799281781$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 31 x^{13} - 4 x^{12} + 373 x^{11} + 85 x^{10} - 2208 x^{9} - 636 x^{8} + 6791 x^{7} + \cdots + 392 $ x^15 - 31x^13 - 4x^12 + 373x^11 + 85x^10 - 2208x^9 - 636x^8 + 6791x^7 + 2005x^6 - 10779x^5 - 2515x^4 + 8205x^3 + 686x^2 - 2352*x + 392
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$3.26068$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +3.26068 q^{3} +1.00000 q^{4} +1.62641 q^{5} -3.26068 q^{6} -2.16959 q^{7} -1.00000 q^{8} +7.63201 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +3.26068 q^{3} +1.00000 q^{4} +1.62641 q^{5} -3.26068 q^{6} -2.16959 q^{7} -1.00000 q^{8} +7.63201 q^{9} -1.62641 q^{10} -5.74676 q^{11} +3.26068 q^{12} +2.16959 q^{14} +5.30320 q^{15} +1.00000 q^{16} -2.37226 q^{17} -7.63201 q^{18} +1.00000 q^{19} +1.62641 q^{20} -7.07434 q^{21} +5.74676 q^{22} -0.857434 q^{23} -3.26068 q^{24} -2.35478 q^{25} +15.1035 q^{27} -2.16959 q^{28} -6.11235 q^{29} -5.30320 q^{30} -6.96656 q^{31} -1.00000 q^{32} -18.7383 q^{33} +2.37226 q^{34} -3.52865 q^{35} +7.63201 q^{36} -2.58017 q^{37} -1.00000 q^{38} -1.62641 q^{40} -5.83807 q^{41} +7.07434 q^{42} -2.20473 q^{43} -5.74676 q^{44} +12.4128 q^{45} +0.857434 q^{46} -9.14883 q^{47} +3.26068 q^{48} -2.29287 q^{49} +2.35478 q^{50} -7.73517 q^{51} -9.71683 q^{53} -15.1035 q^{54} -9.34661 q^{55} +2.16959 q^{56} +3.26068 q^{57} +6.11235 q^{58} +11.3000 q^{59} +5.30320 q^{60} -13.0261 q^{61} +6.96656 q^{62} -16.5583 q^{63} +1.00000 q^{64} +18.7383 q^{66} -1.87585 q^{67} -2.37226 q^{68} -2.79581 q^{69} +3.52865 q^{70} +14.6714 q^{71} -7.63201 q^{72} -9.17690 q^{73} +2.58017 q^{74} -7.67818 q^{75} +1.00000 q^{76} +12.4681 q^{77} +0.501689 q^{79} +1.62641 q^{80} +26.3515 q^{81} +5.83807 q^{82} -10.2135 q^{83} -7.07434 q^{84} -3.85827 q^{85} +2.20473 q^{86} -19.9304 q^{87} +5.74676 q^{88} +15.3020 q^{89} -12.4128 q^{90} -0.857434 q^{92} -22.7157 q^{93} +9.14883 q^{94} +1.62641 q^{95} -3.26068 q^{96} +10.6832 q^{97} +2.29287 q^{98} -43.8593 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 15 q^{2} + 15 q^{4} - q^{5} - 18 q^{7} - 15 q^{8} + 17 q^{9}+O(q^{10}) $ 15 * q - 15 * q^2 + 15 * q^4 - q^5 - 18 * q^7 - 15 * q^8 + 17 * q^9	$ 15 q - 15 q^{2} + 15 q^{4} - q^{5} - 18 q^{7} - 15 q^{8} + 17 q^{9} + q^{10} - 4 q^{11} + 18 q^{14} - 23 q^{15} + 15 q^{16} + 2 q^{17} - 17 q^{18} + 15 q^{19} - q^{20} - 2 q^{21} + 4 q^{22} + 17 q^{23} + 8 q^{25} + 12 q^{27} - 18 q^{28} - 20 q^{29} + 23 q^{30} - 30 q^{31} - 15 q^{32} - 36 q^{33} - 2 q^{34} + 32 q^{35} + 17 q^{36} - 35 q^{37} - 15 q^{38} + q^{40} - 15 q^{41} + 2 q^{42} + q^{43} - 4 q^{44} + 11 q^{45} - 17 q^{46} + 29 q^{49} - 8 q^{50} - q^{51} - q^{53} - 12 q^{54} - 6 q^{55} + 18 q^{56} + 20 q^{58} + 7 q^{59} - 23 q^{60} - 2 q^{61} + 30 q^{62} - 42 q^{63} + 15 q^{64} + 36 q^{66} - 34 q^{67} + 2 q^{68} - 12 q^{69} - 32 q^{70} - 4 q^{71} - 17 q^{72} - 12 q^{73} + 35 q^{74} + 31 q^{75} + 15 q^{76} - 20 q^{77} + 23 q^{79} - q^{80} + 7 q^{81} + 15 q^{82} + 3 q^{83} - 2 q^{84} - 46 q^{85} - q^{86} + 22 q^{87} + 4 q^{88} - 17 q^{89} - 11 q^{90} + 17 q^{92} - 60 q^{93} - q^{95} - 18 q^{97} - 29 q^{98} + 6 q^{99}+O(q^{100}) $ 15 * q - 15 * q^2 + 15 * q^4 - q^5 - 18 * q^7 - 15 * q^8 + 17 * q^9 + q^10 - 4 * q^11 + 18 * q^14 - 23 * q^15 + 15 * q^16 + 2 * q^17 - 17 * q^18 + 15 * q^19 - q^20 - 2 * q^21 + 4 * q^22 + 17 * q^23 + 8 * q^25 + 12 * q^27 - 18 * q^28 - 20 * q^29 + 23 * q^30 - 30 * q^31 - 15 * q^32 - 36 * q^33 - 2 * q^34 + 32 * q^35 + 17 * q^36 - 35 * q^37 - 15 * q^38 + q^40 - 15 * q^41 + 2 * q^42 + q^43 - 4 * q^44 + 11 * q^45 - 17 * q^46 + 29 * q^49 - 8 * q^50 - q^51 - q^53 - 12 * q^54 - 6 * q^55 + 18 * q^56 + 20 * q^58 + 7 * q^59 - 23 * q^60 - 2 * q^61 + 30 * q^62 - 42 * q^63 + 15 * q^64 + 36 * q^66 - 34 * q^67 + 2 * q^68 - 12 * q^69 - 32 * q^70 - 4 * q^71 - 17 * q^72 - 12 * q^73 + 35 * q^74 + 31 * q^75 + 15 * q^76 - 20 * q^77 + 23 * q^79 - q^80 + 7 * q^81 + 15 * q^82 + 3 * q^83 - 2 * q^84 - 46 * q^85 - q^86 + 22 * q^87 + 4 * q^88 - 17 * q^89 - 11 * q^90 + 17 * q^92 - 60 * q^93 - q^95 - 18 * q^97 - 29 * q^98 + 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	3.26068	1.88255	0.941276	−	0.337638i	$-0.109628\pi$
$3$	3.26068	1.88255	0.941276	+	0.337638i	$0.109628\pi$
$4$	1.00000	0.500000
$5$	1.62641	0.727354	0.363677	−	0.931525i	$-0.381521\pi$
$5$	1.62641	0.727354	0.363677	+	0.931525i	$0.381521\pi$
$6$	−3.26068	−1.33117
$7$	−2.16959	−0.820029	−0.410014	−	0.912079i	$-0.634476\pi$
$7$	−2.16959	−0.820029	−0.410014	+	0.912079i	$0.634476\pi$
$8$	−1.00000	−0.353553
$9$	7.63201	2.54400
$10$	−1.62641	−0.514317
$11$	−5.74676	−1.73271	−0.866357	−	0.499425i	$-0.833545\pi$
$11$	−5.74676	−1.73271	−0.866357	+	0.499425i	$0.833545\pi$
$12$	3.26068	0.941276
$13$	0	0
$13$	0	0
$14$	2.16959	0.579848
$15$	5.30320	1.36928
$16$	1.00000	0.250000
$17$	−2.37226	−0.575358	−0.287679	−	0.957727i	$-0.592884\pi$
$17$	−2.37226	−0.575358	−0.287679	+	0.957727i	$0.592884\pi$
$18$	−7.63201	−1.79888
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	1.62641	0.363677
$21$	−7.07434	−1.54375
$22$	5.74676	1.22521
$23$	−0.857434	−0.178787	−0.0893937	−	0.995996i	$-0.528493\pi$
$23$	−0.857434	−0.178787	−0.0893937	+	0.995996i	$0.528493\pi$
$24$	−3.26068	−0.665583
$25$	−2.35478	−0.470956
$26$	0	0
$27$	15.1035	2.90666
$28$	−2.16959	−0.410014
$29$	−6.11235	−1.13504	−0.567518	−	0.823361i	$-0.692096\pi$
$29$	−6.11235	−1.13504	−0.567518	+	0.823361i	$0.692096\pi$
$30$	−5.30320	−0.968228
$31$	−6.96656	−1.25123	−0.625615	−	0.780132i	$-0.715152\pi$
$31$	−6.96656	−1.25123	−0.625615	+	0.780132i	$0.715152\pi$
$32$	−1.00000	−0.176777
$33$	−18.7383	−3.26192
$34$	2.37226	0.406839
$35$	−3.52865	−0.596451
$36$	7.63201	1.27200
$37$	−2.58017	−0.424177	−0.212088	−	0.977250i	$-0.568027\pi$
$37$	−2.58017	−0.424177	−0.212088	+	0.977250i	$0.568027\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−1.62641	−0.257158
$41$	−5.83807	−0.911753	−0.455877	−	0.890043i	$-0.650674\pi$
$41$	−5.83807	−0.911753	−0.455877	+	0.890043i	$0.650674\pi$
$42$	7.07434	1.09159
$43$	−2.20473	−0.336218	−0.168109	−	0.985768i	$-0.553766\pi$
$43$	−2.20473	−0.336218	−0.168109	+	0.985768i	$0.553766\pi$
$44$	−5.74676	−0.866357
$45$	12.4128	1.85039
$46$	0.857434	0.126422
$47$	−9.14883	−1.33449	−0.667247	−	0.744837i	$-0.732527\pi$
$47$	−9.14883	−1.33449	−0.667247	+	0.744837i	$0.732527\pi$
$48$	3.26068	0.470638
$49$	−2.29287	−0.327553
$50$	2.35478	0.333016
$51$	−7.73517	−1.08314
$52$	0	0
$53$	−9.71683	−1.33471	−0.667355	−	0.744740i	$-0.732573\pi$
$53$	−9.71683	−1.33471	−0.667355	+	0.744740i	$0.732573\pi$
$54$	−15.1035	−2.05532
$55$	−9.34661	−1.26030
$56$	2.16959	0.289924
$57$	3.26068	0.431887
$58$	6.11235	0.802591
$59$	11.3000	1.47114	0.735568	−	0.677451i	$-0.236915\pi$
$59$	11.3000	1.47114	0.735568	+	0.677451i	$0.236915\pi$
$60$	5.30320	0.684641
$61$	−13.0261	−1.66782	−0.833909	−	0.551902i	$-0.813902\pi$
$61$	−13.0261	−1.66782	−0.833909	+	0.551902i	$0.813902\pi$
$62$	6.96656	0.884753
$63$	−16.5583	−2.08615
$64$	1.00000	0.125000
$65$	0	0
$66$	18.7383	2.30653
$67$	−1.87585	−0.229171	−0.114586	−	0.993413i	$-0.536554\pi$
$67$	−1.87585	−0.229171	−0.114586	+	0.993413i	$0.536554\pi$
$68$	−2.37226	−0.287679
$69$	−2.79581	−0.336576
$70$	3.52865	0.421755
$71$	14.6714	1.74118	0.870590	−	0.492010i	$-0.163738\pi$
$71$	14.6714	1.74118	0.870590	+	0.492010i	$0.163738\pi$
$72$	−7.63201	−0.899440
$73$	−9.17690	−1.07408	−0.537038	−	0.843558i	$-0.680457\pi$
$73$	−9.17690	−1.07408	−0.537038	+	0.843558i	$0.680457\pi$
$74$	2.58017	0.299938
$75$	−7.67818	−0.886600
$76$	1.00000	0.114708
$77$	12.4681	1.42088
$78$	0	0
$79$	0.501689	0.0564445	0.0282222	−	0.999602i	$-0.491015\pi$
$79$	0.501689	0.0564445	0.0282222	+	0.999602i	$0.491015\pi$
$80$	1.62641	0.181838
$81$	26.3515	2.92794
$82$	5.83807	0.644707
$83$	−10.2135	−1.12108	−0.560539	−	0.828128i	$-0.689406\pi$
$83$	−10.2135	−1.12108	−0.560539	+	0.828128i	$0.689406\pi$
$84$	−7.07434	−0.771873
$85$	−3.85827	−0.418489
$86$	2.20473	0.237742
$87$	−19.9304	−2.13676
$88$	5.74676	0.612607
$89$	15.3020	1.62201	0.811005	−	0.585040i	$-0.198921\pi$
$89$	15.3020	1.62201	0.811005	+	0.585040i	$0.198921\pi$
$90$	−12.4128	−1.30842
$91$	0	0
$92$	−0.857434	−0.0893937
$93$	−22.7157	−2.35551
$94$	9.14883	0.943629
$95$	1.62641	0.166866
$96$	−3.26068	−0.332791
$97$	10.6832	1.08471	0.542355	−	0.840149i	$-0.317533\pi$
$97$	10.6832	1.08471	0.542355	+	0.840149i	$0.317533\pi$
$98$	2.29287	0.231615
$99$	−43.8593	−4.40803
$100$	−2.35478	−0.235478
$101$	11.2649	1.12090	0.560451	−	0.828187i	$-0.310628\pi$
$101$	11.2649	1.12090	0.560451	+	0.828187i	$0.310628\pi$
$102$	7.73517	0.765896
$103$	−10.1622	−1.00131	−0.500653	−	0.865648i	$-0.666907\pi$
$103$	−10.1622	−1.00131	−0.500653	+	0.865648i	$0.666907\pi$
$104$	0	0
$105$	−11.5058	−1.12285
$106$	9.71683	0.943782
$107$	7.29201	0.704946	0.352473	−	0.935822i	$-0.385341\pi$
$107$	7.29201	0.704946	0.352473	+	0.935822i	$0.385341\pi$
$108$	15.1035	1.45333
$109$	10.8152	1.03591	0.517956	−	0.855407i	$-0.326693\pi$
$109$	10.8152	1.03591	0.517956	+	0.855407i	$0.326693\pi$
$110$	9.34661	0.891164
$111$	−8.41309	−0.798535
$112$	−2.16959	−0.205007
$113$	11.7689	1.10712	0.553562	−	0.832808i	$-0.313268\pi$
$113$	11.7689	1.10712	0.553562	+	0.832808i	$0.313268\pi$
$114$	−3.26068	−0.305390
$115$	−1.39454	−0.130042
$116$	−6.11235	−0.567518
$117$	0	0
$118$	−11.3000	−1.04025
$119$	5.14684	0.471810
$120$	−5.30320	−0.484114
$121$	22.0253	2.00230
$122$	13.0261	1.17933
$123$	−19.0361	−1.71642
$124$	−6.96656	−0.625615
$125$	−11.9619	−1.06991
$126$	16.5583	1.47513
$127$	1.62191	0.143921	0.0719607	−	0.997407i	$-0.477074\pi$
$127$	1.62191	0.143921	0.0719607	+	0.997407i	$0.477074\pi$
$128$	−1.00000	−0.0883883
$129$	−7.18890	−0.632947
$130$	0	0
$131$	−2.66662	−0.232984	−0.116492	−	0.993192i	$-0.537165\pi$
$131$	−2.66662	−0.232984	−0.116492	+	0.993192i	$0.537165\pi$
$132$	−18.7383	−1.63096
$133$	−2.16959	−0.188127
$134$	1.87585	0.162049
$135$	24.5645	2.11417
$136$	2.37226	0.203420
$137$	1.71171	0.146241	0.0731205	−	0.997323i	$-0.476704\pi$
$137$	1.71171	0.146241	0.0731205	+	0.997323i	$0.476704\pi$
$138$	2.79581	0.237995
$139$	13.1761	1.11759	0.558793	−	0.829307i	$-0.311265\pi$
$139$	13.1761	1.11759	0.558793	+	0.829307i	$0.311265\pi$
$140$	−3.52865	−0.298225
$141$	−29.8314	−2.51225
$142$	−14.6714	−1.23120
$143$	0	0
$144$	7.63201	0.636000
$145$	−9.94121	−0.825572
$146$	9.17690	0.759486
$147$	−7.47631	−0.616635
$148$	−2.58017	−0.212088
$149$	8.83515	0.723803	0.361902	−	0.932216i	$-0.382128\pi$
$149$	8.83515	0.723803	0.361902	+	0.932216i	$0.382128\pi$
$150$	7.67818	0.626921
$151$	−4.14897	−0.337638	−0.168819	−	0.985647i	$-0.553995\pi$
$151$	−4.14897	−0.337638	−0.168819	+	0.985647i	$0.553995\pi$
$152$	−1.00000	−0.0811107
$153$	−18.1051	−1.46371
$154$	−12.4681	−1.00471
$155$	−11.3305	−0.910087
$156$	0	0
$157$	16.5956	1.32447	0.662237	−	0.749295i	$-0.269607\pi$
$157$	16.5956	1.32447	0.662237	+	0.749295i	$0.269607\pi$
$158$	−0.501689	−0.0399123
$159$	−31.6834	−2.51266
$160$	−1.62641	−0.128579
$161$	1.86028	0.146611
$162$	−26.3515	−2.07037
$163$	−5.66163	−0.443453	−0.221727	−	0.975109i	$-0.571169\pi$
$163$	−5.66163	−0.443453	−0.221727	+	0.975109i	$0.571169\pi$
$164$	−5.83807	−0.455877
$165$	−30.4763	−2.37257
$166$	10.2135	0.792722
$167$	24.0152	1.85835	0.929176	−	0.369636i	$-0.120518\pi$
$167$	24.0152	1.85835	0.929176	+	0.369636i	$0.120518\pi$
$168$	7.07434	0.545797
$169$	0	0
$170$	3.85827	0.295916
$171$	7.63201	0.583634
$172$	−2.20473	−0.168109
$173$	−18.4544	−1.40306	−0.701531	−	0.712639i	$-0.747500\pi$
$173$	−18.4544	−1.40306	−0.701531	+	0.712639i	$0.747500\pi$
$174$	19.9304	1.51092
$175$	5.10892	0.386198
$176$	−5.74676	−0.433178
$177$	36.8457	2.76949
$178$	−15.3020	−1.14693
$179$	23.5583	1.76083	0.880414	−	0.474206i	$-0.157265\pi$
$179$	23.5583	1.76083	0.880414	+	0.474206i	$0.157265\pi$
$180$	12.4128	0.925195
$181$	−5.48960	−0.408039	−0.204019	−	0.978967i	$-0.565401\pi$
$181$	−5.48960	−0.408039	−0.204019	+	0.978967i	$0.565401\pi$
$182$	0	0
$183$	−42.4738	−3.13975
$184$	0.857434	0.0632109
$185$	−4.19642	−0.308527
$186$	22.7157	1.66559
$187$	13.6328	0.996930
$188$	−9.14883	−0.667247
$189$	−32.7684	−2.38355
$190$	−1.62641	−0.117992
$191$	8.80339	0.636991	0.318495	−	0.947924i	$-0.396822\pi$
$191$	8.80339	0.636991	0.318495	+	0.947924i	$0.396822\pi$
$192$	3.26068	0.235319
$193$	−2.00765	−0.144514	−0.0722569	−	0.997386i	$-0.523020\pi$
$193$	−2.00765	−0.144514	−0.0722569	+	0.997386i	$0.523020\pi$
$194$	−10.6832	−0.767006
$195$	0	0
$196$	−2.29287	−0.163776
$197$	−5.37364	−0.382856	−0.191428	−	0.981507i	$-0.561312\pi$
$197$	−5.37364	−0.382856	−0.191428	+	0.981507i	$0.561312\pi$
$198$	43.8593	3.11695
$199$	7.46561	0.529223	0.264611	−	0.964355i	$-0.414756\pi$
$199$	7.46561	0.529223	0.264611	+	0.964355i	$0.414756\pi$
$200$	2.35478	0.166508
$201$	−6.11654	−0.431427
$202$	−11.2649	−0.792598
$203$	13.2613	0.930762
$204$	−7.73517	−0.541570
$205$	−9.49511	−0.663167
$206$	10.1622	0.708031
$207$	−6.54394	−0.454835
$208$	0	0
$209$	−5.74676	−0.397512
$210$	11.5058	0.793975
$211$	−17.7848	−1.22436	−0.612179	−	0.790719i	$-0.709707\pi$
$211$	−17.7848	−1.22436	−0.612179	+	0.790719i	$0.709707\pi$
$212$	−9.71683	−0.667355
$213$	47.8388	3.27786
$214$	−7.29201	−0.498472
$215$	−3.58579	−0.244549
$216$	−15.1035	−1.02766
$217$	15.1146	1.02604
$218$	−10.8152	−0.732501
$219$	−29.9229	−2.02200
$220$	−9.34661	−0.630148
$221$	0	0
$222$	8.41309	0.564650
$223$	−20.9787	−1.40484	−0.702419	−	0.711764i	$-0.747897\pi$
$223$	−20.9787	−1.40484	−0.702419	+	0.711764i	$0.747897\pi$
$224$	2.16959	0.144962
$225$	−17.9717	−1.19811
$226$	−11.7689	−0.782855
$227$	11.3430	0.752865	0.376432	−	0.926444i	$-0.377151\pi$
$227$	11.3430	0.752865	0.376432	+	0.926444i	$0.377151\pi$
$228$	3.26068	0.215944
$229$	16.2798	1.07580	0.537901	−	0.843008i	$-0.319218\pi$
$229$	16.2798	1.07580	0.537901	+	0.843008i	$0.319218\pi$
$230$	1.39454	0.0919533
$231$	40.6545	2.67487
$232$	6.11235	0.401296
$233$	8.21854	0.538414	0.269207	−	0.963082i	$-0.413238\pi$
$233$	8.21854	0.538414	0.269207	+	0.963082i	$0.413238\pi$
$234$	0	0
$235$	−14.8798	−0.970649
$236$	11.3000	0.735568
$237$	1.63585	0.106260
$238$	−5.14684	−0.333620
$239$	−21.7546	−1.40719	−0.703594	−	0.710603i	$-0.748422\pi$
$239$	−21.7546	−1.40719	−0.703594	+	0.710603i	$0.748422\pi$
$240$	5.30320	0.342320
$241$	−25.1966	−1.62306	−0.811528	−	0.584313i	$-0.801364\pi$
$241$	−25.1966	−1.62306	−0.811528	+	0.584313i	$0.801364\pi$
$242$	−22.0253	−1.41584
$243$	40.6133	2.60534
$244$	−13.0261	−0.833909
$245$	−3.72915	−0.238247
$246$	19.0361	1.21369
$247$	0	0
$248$	6.96656	0.442377
$249$	−33.3029	−2.11049
$250$	11.9619	0.756538
$251$	−9.04188	−0.570719	−0.285359	−	0.958421i	$-0.592113\pi$
$251$	−9.04188	−0.570719	−0.285359	+	0.958421i	$0.592113\pi$
$252$	−16.5583	−1.04308
$253$	4.92747	0.309787
$254$	−1.62191	−0.101768
$255$	−12.5806	−0.787827
$256$	1.00000	0.0625000
$257$	−29.7766	−1.85741	−0.928707	−	0.370814i	$-0.879079\pi$
$257$	−29.7766	−1.85741	−0.928707	+	0.370814i	$0.879079\pi$
$258$	7.18890	0.447561
$259$	5.59791	0.347837
$260$	0	0
$261$	−46.6495	−2.88753
$262$	2.66662	0.164745
$263$	−11.0125	−0.679063	−0.339531	−	0.940595i	$-0.610268\pi$
$263$	−11.0125	−0.679063	−0.339531	+	0.940595i	$0.610268\pi$
$264$	18.7383	1.15326
$265$	−15.8036	−0.970806
$266$	2.16959	0.133026
$267$	49.8949	3.05352
$268$	−1.87585	−0.114586
$269$	1.59662	0.0973474	0.0486737	−	0.998815i	$-0.484501\pi$
$269$	1.59662	0.0973474	0.0486737	+	0.998815i	$0.484501\pi$
$270$	−24.5645	−1.49495
$271$	3.46533	0.210504	0.105252	−	0.994446i	$-0.466435\pi$
$271$	3.46533	0.210504	0.105252	+	0.994446i	$0.466435\pi$
$272$	−2.37226	−0.143839
$273$	0	0
$274$	−1.71171	−0.103408
$275$	13.5324	0.816033
$276$	−2.79581	−0.168288
$277$	−8.47516	−0.509223	−0.254612	−	0.967043i	$-0.581948\pi$
$277$	−8.47516	−0.509223	−0.254612	+	0.967043i	$0.581948\pi$
$278$	−13.1761	−0.790252
$279$	−53.1688	−3.18313
$280$	3.52865	0.210877
$281$	−12.2406	−0.730215	−0.365107	−	0.930965i	$-0.618968\pi$
$281$	−12.2406	−0.730215	−0.365107	+	0.930965i	$0.618968\pi$
$282$	29.8314	1.77643
$283$	−6.99508	−0.415814	−0.207907	−	0.978149i	$-0.566665\pi$
$283$	−6.99508	−0.415814	−0.207907	+	0.978149i	$0.566665\pi$
$284$	14.6714	0.870590
$285$	5.30320	0.314135
$286$	0	0
$287$	12.6662	0.747664
$288$	−7.63201	−0.449720
$289$	−11.3724	−0.668964
$290$	9.94121	0.583768
$291$	34.8343	2.04202
$292$	−9.17690	−0.537038
$293$	33.2345	1.94158	0.970792	−	0.239925i	$-0.0771227\pi$
$293$	33.2345	1.94158	0.970792	+	0.239925i	$0.0771227\pi$
$294$	7.47631	0.436027
$295$	18.3785	1.07004
$296$	2.58017	0.149969
$297$	−86.7960	−5.03642
$298$	−8.83515	−0.511806
$299$	0	0
$300$	−7.67818	−0.443300
$301$	4.78336	0.275708
$302$	4.14897	0.238746
$303$	36.7313	2.11016
$304$	1.00000	0.0573539
$305$	−21.1858	−1.21309
$306$	18.1051	1.03500
$307$	14.6529	0.836286	0.418143	−	0.908381i	$-0.362681\pi$
$307$	14.6529	0.836286	0.418143	+	0.908381i	$0.362681\pi$
$308$	12.4681	0.710438
$309$	−33.1355	−1.88501
$310$	11.3305	0.643529
$311$	−12.5519	−0.711751	−0.355876	−	0.934533i	$-0.615817\pi$
$311$	−12.5519	−0.711751	−0.355876	+	0.934533i	$0.615817\pi$
$312$	0	0
$313$	31.5843	1.78525	0.892626	−	0.450798i	$-0.148861\pi$
$313$	31.5843	1.78525	0.892626	+	0.450798i	$0.148861\pi$
$314$	−16.5956	−0.936544
$315$	−26.9307	−1.51737
$316$	0.501689	0.0282222
$317$	1.68495	0.0946363	0.0473182	−	0.998880i	$-0.484933\pi$
$317$	1.68495	0.0946363	0.0473182	+	0.998880i	$0.484933\pi$
$318$	31.6834	1.77672
$319$	35.1262	1.96669
$320$	1.62641	0.0909192
$321$	23.7769	1.32710
$322$	−1.86028	−0.103669
$323$	−2.37226	−0.131996
$324$	26.3515	1.46397
$325$	0	0
$326$	5.66163	0.313569
$327$	35.2650	1.95016
$328$	5.83807	0.322353
$329$	19.8492	1.09432
$330$	30.4763	1.67766
$331$	−17.9971	−0.989211	−0.494605	−	0.869118i	$-0.664687\pi$
$331$	−17.9971	−0.989211	−0.494605	+	0.869118i	$0.664687\pi$
$332$	−10.2135	−0.560539
$333$	−19.6919	−1.07911
$334$	−24.0152	−1.31405
$335$	−3.05091	−0.166689
$336$	−7.07434	−0.385937
$337$	−20.1053	−1.09521	−0.547603	−	0.836738i	$-0.684460\pi$
$337$	−20.1053	−1.09521	−0.547603	+	0.836738i	$0.684460\pi$
$338$	0	0
$339$	38.3745	2.08422
$340$	−3.85827	−0.209244
$341$	40.0351	2.16802
$342$	−7.63201	−0.412692
$343$	20.1617	1.08863
$344$	2.20473	0.118871
$345$	−4.54715	−0.244810
$346$	18.4544	0.992115
$347$	−23.8002	−1.27766	−0.638830	−	0.769348i	$-0.720581\pi$
$347$	−23.8002	−1.27766	−0.638830	+	0.769348i	$0.720581\pi$
$348$	−19.9304	−1.06838
$349$	−23.4250	−1.25391	−0.626955	−	0.779056i	$-0.715699\pi$
$349$	−23.4250	−1.25391	−0.626955	+	0.779056i	$0.715699\pi$
$350$	−5.10892	−0.273083
$351$	0	0
$352$	5.74676	0.306303
$353$	−28.9495	−1.54083	−0.770413	−	0.637545i	$-0.779950\pi$
$353$	−28.9495	−1.54083	−0.770413	+	0.637545i	$0.779950\pi$
$354$	−36.8457	−1.95833
$355$	23.8618	1.26645
$356$	15.3020	0.811005
$357$	16.7822	0.888206
$358$	−23.5583	−1.24509
$359$	−3.55616	−0.187687	−0.0938435	−	0.995587i	$-0.529915\pi$
$359$	−3.55616	−0.187687	−0.0938435	+	0.995587i	$0.529915\pi$
$360$	−12.4128	−0.654211
$361$	1.00000	0.0526316
$362$	5.48960	0.288527
$363$	71.8173	3.76943
$364$	0	0
$365$	−14.9254	−0.781233
$366$	42.4738	2.22014
$367$	3.81299	0.199037	0.0995184	−	0.995036i	$-0.468270\pi$
$367$	3.81299	0.199037	0.0995184	+	0.995036i	$0.468270\pi$
$368$	−0.857434	−0.0446968
$369$	−44.5562	−2.31950
$370$	4.19642	0.218161
$371$	21.0816	1.09450
$372$	−22.7157	−1.17775
$373$	11.2239	0.581152	0.290576	−	0.956852i	$-0.406153\pi$
$373$	11.2239	0.581152	0.290576	+	0.956852i	$0.406153\pi$
$374$	−13.6328	−0.704936
$375$	−39.0039	−2.01415
$376$	9.14883	0.471815
$377$	0	0
$378$	32.7684	1.68542
$379$	6.19845	0.318393	0.159197	−	0.987247i	$-0.449110\pi$
$379$	6.19845	0.318393	0.159197	+	0.987247i	$0.449110\pi$
$380$	1.62641	0.0834332
$381$	5.28853	0.270939
$382$	−8.80339	−0.450420
$383$	3.03258	0.154957	0.0774787	−	0.996994i	$-0.475313\pi$
$383$	3.03258	0.154957	0.0774787	+	0.996994i	$0.475313\pi$
$384$	−3.26068	−0.166396
$385$	20.2783	1.03348
$386$	2.00765	0.102187
$387$	−16.8265	−0.855338
$388$	10.6832	0.542355
$389$	−21.5300	−1.09161	−0.545807	−	0.837911i	$-0.683777\pi$
$389$	−21.5300	−1.09161	−0.545807	+	0.837911i	$0.683777\pi$
$390$	0	0
$391$	2.03406	0.102867
$392$	2.29287	0.115807
$393$	−8.69500	−0.438605
$394$	5.37364	0.270720
$395$	0.815954	0.0410551
$396$	−43.8593	−2.20401
$397$	−4.67613	−0.234688	−0.117344	−	0.993091i	$-0.537438\pi$
$397$	−4.67613	−0.234688	−0.117344	+	0.993091i	$0.537438\pi$
$398$	−7.46561	−0.374217
$399$	−7.07434	−0.354160
$400$	−2.35478	−0.117739
$401$	−21.2131	−1.05933	−0.529665	−	0.848207i	$-0.677682\pi$
$401$	−21.2131	−1.05933	−0.529665	+	0.848207i	$0.677682\pi$
$402$	6.11654	0.305065
$403$	0	0
$404$	11.2649	0.560451
$405$	42.8584	2.12965
$406$	−13.2613	−0.658148
$407$	14.8276	0.734977
$408$	7.73517	0.382948
$409$	11.9831	0.592528	0.296264	−	0.955106i	$-0.404259\pi$
$409$	11.9831	0.592528	0.296264	+	0.955106i	$0.404259\pi$
$410$	9.49511	0.468930
$411$	5.58132	0.275306
$412$	−10.1622	−0.500653
$413$	−24.5164	−1.20637
$414$	6.54394	0.321617
$415$	−16.6114	−0.815420
$416$	0	0
$417$	42.9631	2.10391
$418$	5.74676	0.281083
$419$	−38.6604	−1.88869	−0.944343	−	0.328962i	$-0.893301\pi$
$419$	−38.6604	−1.88869	−0.944343	+	0.328962i	$0.893301\pi$
$420$	−11.5058	−0.561425
$421$	−9.90414	−0.482698	−0.241349	−	0.970438i	$-0.577590\pi$
$421$	−9.90414	−0.482698	−0.241349	+	0.970438i	$0.577590\pi$
$422$	17.7848	0.865752
$423$	−69.8239	−3.39495
$424$	9.71683	0.471891
$425$	5.58616	0.270968
$426$	−47.8388	−2.31780
$427$	28.2613	1.36766
$428$	7.29201	0.352473
$429$	0	0
$430$	3.58579	0.172922
$431$	23.0774	1.11160	0.555799	−	0.831317i	$-0.312412\pi$
$431$	23.0774	1.11160	0.555799	+	0.831317i	$0.312412\pi$
$432$	15.1035	0.726666
$433$	−23.7544	−1.14156	−0.570781	−	0.821103i	$-0.693359\pi$
$433$	−23.7544	−1.14156	−0.570781	+	0.821103i	$0.693359\pi$
$434$	−15.1146	−0.725523
$435$	−32.4151	−1.55418
$436$	10.8152	0.517956
$437$	−0.857434	−0.0410166
$438$	29.9229	1.42977
$439$	18.2272	0.869935	0.434968	−	0.900446i	$-0.356760\pi$
$439$	18.2272	0.869935	0.434968	+	0.900446i	$0.356760\pi$
$440$	9.34661	0.445582
$441$	−17.4992	−0.833295
$442$	0	0
$443$	35.0384	1.66472	0.832362	−	0.554232i	$-0.186988\pi$
$443$	35.0384	1.66472	0.832362	+	0.554232i	$0.186988\pi$
$444$	−8.41309	−0.399268
$445$	24.8874	1.17977
$446$	20.9787	0.993371
$447$	28.8085	1.36260
$448$	−2.16959	−0.102504
$449$	11.4714	0.541367	0.270683	−	0.962668i	$-0.412750\pi$
$449$	11.4714	0.541367	0.270683	+	0.962668i	$0.412750\pi$
$450$	17.9717	0.847195
$451$	33.5500	1.57981
$452$	11.7689	0.553562
$453$	−13.5284	−0.635622
$454$	−11.3430	−0.532356
$455$	0	0
$456$	−3.26068	−0.152695
$457$	−19.3028	−0.902949	−0.451475	−	0.892284i	$-0.649102\pi$
$457$	−19.3028	−0.902949	−0.451475	+	0.892284i	$0.649102\pi$
$458$	−16.2798	−0.760707
$459$	−35.8294	−1.67237
$460$	−1.39454	−0.0650208
$461$	−38.7275	−1.80372	−0.901859	−	0.432031i	$-0.857797\pi$
$461$	−38.7275	−1.80372	−0.901859	+	0.432031i	$0.857797\pi$
$462$	−40.6545	−1.89142
$463$	20.1438	0.936162	0.468081	−	0.883686i	$-0.344946\pi$
$463$	20.1438	0.936162	0.468081	+	0.883686i	$0.344946\pi$
$464$	−6.11235	−0.283759
$465$	−36.9451	−1.71329
$466$	−8.21854	−0.380716
$467$	36.1954	1.67492	0.837462	−	0.546496i	$-0.184039\pi$
$467$	36.1954	1.67492	0.837462	+	0.546496i	$0.184039\pi$
$468$	0	0
$469$	4.06983	0.187927
$470$	14.8798	0.686352
$471$	54.1129	2.49339
$472$	−11.3000	−0.520125
$473$	12.6700	0.582569
$474$	−1.63585	−0.0751369
$475$	−2.35478	−0.108045
$476$	5.14684	0.235905
$477$	−74.1589	−3.39550
$478$	21.7546	0.995032
$479$	−11.3987	−0.520821	−0.260410	−	0.965498i	$-0.583858\pi$
$479$	−11.3987	−0.520821	−0.260410	+	0.965498i	$0.583858\pi$
$480$	−5.30320	−0.242057
$481$	0	0
$482$	25.1966	1.14767
$483$	6.06577	0.276002
$484$	22.0253	1.00115
$485$	17.3752	0.788968
$486$	−40.6133	−1.84225
$487$	−10.4535	−0.473694	−0.236847	−	0.971547i	$-0.576114\pi$
$487$	−10.4535	−0.473694	−0.236847	+	0.971547i	$0.576114\pi$
$488$	13.0261	0.589663
$489$	−18.4608	−0.834824
$490$	3.72915	0.168466
$491$	4.13572	0.186642	0.0933212	−	0.995636i	$-0.470252\pi$
$491$	4.13572	0.186642	0.0933212	+	0.995636i	$0.470252\pi$
$492$	−19.0361	−0.858212
$493$	14.5001	0.653051
$494$	0	0
$495$	−71.3334	−3.20620
$496$	−6.96656	−0.312808
$497$	−31.8310	−1.42782
$498$	33.3029	1.49234
$499$	−7.26427	−0.325193	−0.162597	−	0.986693i	$-0.551987\pi$
$499$	−7.26427	−0.325193	−0.162597	+	0.986693i	$0.551987\pi$
$500$	−11.9619	−0.534953
$501$	78.3058	3.49845
$502$	9.04188	0.403559
$503$	10.9089	0.486404	0.243202	−	0.969976i	$-0.421802\pi$
$503$	10.9089	0.486404	0.243202	+	0.969976i	$0.421802\pi$
$504$	16.5583	0.737567
$505$	18.3214	0.815293
$506$	−4.92747	−0.219053
$507$	0	0
$508$	1.62191	0.0719607
$509$	7.82669	0.346912	0.173456	−	0.984842i	$-0.444507\pi$
$509$	7.82669	0.346912	0.173456	+	0.984842i	$0.444507\pi$
$510$	12.5806	0.557078
$511$	19.9101	0.880773
$512$	−1.00000	−0.0441942
$513$	15.1035	0.666834
$514$	29.7766	1.31339
$515$	−16.5279	−0.728304
$516$	−7.18890	−0.316474
$517$	52.5761	2.31230
$518$	−5.59791	−0.245958
$519$	−60.1738	−2.64134
$520$	0	0
$521$	20.3312	0.890728	0.445364	−	0.895350i	$-0.353074\pi$
$521$	20.3312	0.890728	0.445364	+	0.895350i	$0.353074\pi$
$522$	46.6495	2.04179
$523$	−14.7478	−0.644876	−0.322438	−	0.946591i	$-0.604502\pi$
$523$	−14.7478	−0.644876	−0.322438	+	0.946591i	$0.604502\pi$
$524$	−2.66662	−0.116492
$525$	16.6585	0.727037
$526$	11.0125	0.480170
$527$	16.5265	0.719905
$528$	−18.7383	−0.815481
$529$	−22.2648	−0.968035
$530$	15.8036	0.686463
$531$	86.2418	3.74257
$532$	−2.16959	−0.0940637
$533$	0	0
$534$	−49.8949	−2.15916
$535$	11.8598	0.512745
$536$	1.87585	0.0810244
$537$	76.8159	3.31485
$538$	−1.59662	−0.0688350
$539$	13.1766	0.567556
$540$	24.5645	1.05709
$541$	−15.8933	−0.683305	−0.341652	−	0.939826i	$-0.610987\pi$
$541$	−15.8933	−0.683305	−0.341652	+	0.939826i	$0.610987\pi$
$542$	−3.46533	−0.148849
$543$	−17.8998	−0.768154
$544$	2.37226	0.101710
$545$	17.5900	0.753475
$546$	0	0
$547$	−28.3684	−1.21295	−0.606473	−	0.795104i	$-0.707416\pi$
$547$	−28.3684	−1.21295	−0.606473	+	0.795104i	$0.707416\pi$
$548$	1.71171	0.0731205
$549$	−99.4151	−4.24293
$550$	−13.5324	−0.577022
$551$	−6.11235	−0.260395
$552$	2.79581	0.118998
$553$	−1.08846	−0.0462861
$554$	8.47516	0.360075
$555$	−13.6832	−0.580818
$556$	13.1761	0.558793
$557$	14.5484	0.616434	0.308217	−	0.951316i	$-0.400268\pi$
$557$	14.5484	0.616434	0.308217	+	0.951316i	$0.400268\pi$
$558$	53.1688	2.25081
$559$	0	0
$560$	−3.52865	−0.149113
$561$	44.4522	1.87677
$562$	12.2406	0.516340
$563$	−14.0073	−0.590338	−0.295169	−	0.955445i	$-0.595376\pi$
$563$	−14.0073	−0.590338	−0.295169	+	0.955445i	$0.595376\pi$
$564$	−29.8314	−1.25613
$565$	19.1411	0.805271
$566$	6.99508	0.294025
$567$	−57.1720	−2.40100
$568$	−14.6714	−0.615600
$569$	42.9027	1.79857	0.899286	−	0.437360i	$-0.144087\pi$
$569$	42.9027	1.79857	0.899286	+	0.437360i	$0.144087\pi$
$570$	−5.30320	−0.222127
$571$	−6.44158	−0.269572	−0.134786	−	0.990875i	$-0.543035\pi$
$571$	−6.44158	−0.269572	−0.134786	+	0.990875i	$0.543035\pi$
$572$	0	0
$573$	28.7050	1.19917
$574$	−12.6662	−0.528678
$575$	2.01907	0.0842010
$576$	7.63201	0.318000
$577$	21.8688	0.910409	0.455205	−	0.890387i	$-0.349566\pi$
$577$	21.8688	0.910409	0.455205	+	0.890387i	$0.349566\pi$
$578$	11.3724	0.473029
$579$	−6.54630	−0.272055
$580$	−9.94121	−0.412786
$581$	22.1591	0.919316
$582$	−34.8343	−1.44393
$583$	55.8403	2.31267
$584$	9.17690	0.379743
$585$	0	0
$586$	−33.2345	−1.37291
$587$	14.3289	0.591416	0.295708	−	0.955278i	$-0.404444\pi$
$587$	14.3289	0.591416	0.295708	+	0.955278i	$0.404444\pi$
$588$	−7.47631	−0.308318
$589$	−6.96656	−0.287052
$590$	−18.3785	−0.756630
$591$	−17.5217	−0.720746
$592$	−2.58017	−0.106044
$593$	12.7015	0.521588	0.260794	−	0.965394i	$-0.416016\pi$
$593$	12.7015	0.521588	0.260794	+	0.965394i	$0.416016\pi$
$594$	86.7960	3.56128
$595$	8.37088	0.343173
$596$	8.83515	0.361902
$597$	24.3429	0.996289
$598$	0	0
$599$	−10.6707	−0.435994	−0.217997	−	0.975949i	$-0.569952\pi$
$599$	−10.6707	−0.435994	−0.217997	+	0.975949i	$0.569952\pi$
$600$	7.67818	0.313460
$601$	40.8176	1.66499	0.832493	−	0.554035i	$-0.186913\pi$
$601$	40.8176	1.66499	0.832493	+	0.554035i	$0.186913\pi$
$602$	−4.78336	−0.194955
$603$	−14.3165	−0.583013
$604$	−4.14897	−0.168819
$605$	35.8222	1.45638
$606$	−36.7313	−1.49211
$607$	−13.2038	−0.535924	−0.267962	−	0.963430i	$-0.586350\pi$
$607$	−13.2038	−0.535924	−0.267962	+	0.963430i	$0.586350\pi$
$608$	−1.00000	−0.0405554
$609$	43.2408	1.75221
$610$	21.1858	0.857787
$611$	0	0
$612$	−18.1051	−0.731856
$613$	−37.7335	−1.52404	−0.762020	−	0.647553i	$-0.775792\pi$
$613$	−37.7335	−1.52404	−0.762020	+	0.647553i	$0.775792\pi$
$614$	−14.6529	−0.591343
$615$	−30.9605	−1.24845
$616$	−12.4681	−0.502355
$617$	−24.0955	−0.970050	−0.485025	−	0.874500i	$-0.661189\pi$
$617$	−24.0955	−0.970050	−0.485025	+	0.874500i	$0.661189\pi$
$618$	33.1355	1.33290
$619$	19.8318	0.797109	0.398555	−	0.917145i	$-0.369512\pi$
$619$	19.8318	0.797109	0.398555	+	0.917145i	$0.369512\pi$
$620$	−11.3305	−0.455044
$621$	−12.9502	−0.519675
$622$	12.5519	0.503284
$623$	−33.1991	−1.33009
$624$	0	0
$625$	−7.68109	−0.307244
$626$	−31.5843	−1.26236
$627$	−18.7383	−0.748337
$628$	16.5956	0.662237
$629$	6.12083	0.244053
$630$	26.9307	1.07294
$631$	16.2529	0.647016	0.323508	−	0.946225i	$-0.395138\pi$
$631$	16.2529	0.647016	0.323508	+	0.946225i	$0.395138\pi$
$632$	−0.501689	−0.0199561
$633$	−57.9906	−2.30492
$634$	−1.68495	−0.0669180
$635$	2.63790	0.104682
$636$	−31.6834	−1.25633
$637$	0	0
$638$	−35.1262	−1.39066
$639$	111.973	4.42956
$640$	−1.62641	−0.0642896
$641$	24.6980	0.975512	0.487756	−	0.872980i	$-0.337816\pi$
$641$	24.6980	0.975512	0.487756	+	0.872980i	$0.337816\pi$
$642$	−23.7769	−0.938399
$643$	21.8548	0.861869	0.430935	−	0.902383i	$-0.358184\pi$
$643$	21.8548	0.861869	0.430935	+	0.902383i	$0.358184\pi$
$644$	1.86028	0.0733054
$645$	−11.6921	−0.460377
$646$	2.37226	0.0933353
$647$	−15.3312	−0.602732	−0.301366	−	0.953509i	$-0.597443\pi$
$647$	−15.3312	−0.602732	−0.301366	+	0.953509i	$0.597443\pi$
$648$	−26.3515	−1.03518
$649$	−64.9385	−2.54906
$650$	0	0
$651$	49.2838	1.93158
$652$	−5.66163	−0.221727
$653$	−10.2053	−0.399362	−0.199681	−	0.979861i	$-0.563991\pi$
$653$	−10.2053	−0.399362	−0.199681	+	0.979861i	$0.563991\pi$
$654$	−35.2650	−1.37897
$655$	−4.33703	−0.169462
$656$	−5.83807	−0.227938
$657$	−70.0382	−2.73245
$658$	−19.8492	−0.773803
$659$	40.6231	1.58245	0.791226	−	0.611524i	$-0.209443\pi$
$659$	40.6231	1.58245	0.791226	+	0.611524i	$0.209443\pi$
$660$	−30.4763	−1.18629
$661$	−19.7559	−0.768415	−0.384207	−	0.923247i	$-0.625525\pi$
$661$	−19.7559	−0.768415	−0.384207	+	0.923247i	$0.625525\pi$
$662$	17.9971	0.699477
$663$	0	0
$664$	10.2135	0.396361
$665$	−3.52865	−0.136835
$666$	19.6919	0.763044
$667$	5.24094	0.202930
$668$	24.0152	0.929176
$669$	−68.4048	−2.64468
$670$	3.05091	0.117867
$671$	74.8578	2.88985
$672$	7.07434	0.272898
$673$	−31.8510	−1.22777	−0.613883	−	0.789397i	$-0.710393\pi$
$673$	−31.8510	−1.22777	−0.613883	+	0.789397i	$0.710393\pi$
$674$	20.1053	0.774428
$675$	−35.5654	−1.36891
$676$	0	0
$677$	−37.1853	−1.42915	−0.714573	−	0.699561i	$-0.753379\pi$
$677$	−37.1853	−1.42915	−0.714573	+	0.699561i	$0.753379\pi$
$678$	−38.3745	−1.47377
$679$	−23.1781	−0.889494
$680$	3.85827	0.147958
$681$	36.9860	1.41731
$682$	−40.0351	−1.53302
$683$	51.2806	1.96220	0.981100	−	0.193503i	$-0.0619851\pi$
$683$	51.2806	1.96220	0.981100	+	0.193503i	$0.0619851\pi$
$684$	7.63201	0.291817
$685$	2.78394	0.106369
$686$	−20.1617	−0.769779
$687$	53.0832	2.02525
$688$	−2.20473	−0.0840544
$689$	0	0
$690$	4.54715	0.173107
$691$	2.88040	0.109575	0.0547877	−	0.998498i	$-0.482552\pi$
$691$	2.88040	0.109575	0.0547877	+	0.998498i	$0.482552\pi$
$692$	−18.4544	−0.701531
$693$	95.1568	3.61471
$694$	23.8002	0.903442
$695$	21.4298	0.812880
$696$	19.9304	0.755460
$697$	13.8494	0.524584
$698$	23.4250	0.886648
$699$	26.7980	1.01359
$700$	5.10892	0.193099
$701$	−48.2704	−1.82315	−0.911575	−	0.411135i	$-0.865133\pi$
$701$	−48.2704	−1.82315	−0.911575	+	0.411135i	$0.865133\pi$
$702$	0	0
$703$	−2.58017	−0.0973129
$704$	−5.74676	−0.216589
$705$	−48.5181	−1.82730
$706$	28.9495	1.08953
$707$	−24.4403	−0.919172
$708$	36.8457	1.38475
$709$	−17.1880	−0.645508	−0.322754	−	0.946483i	$-0.604609\pi$
$709$	−17.1880	−0.645508	−0.322754	+	0.946483i	$0.604609\pi$
$710$	−23.8618	−0.895518
$711$	3.82890	0.143595
$712$	−15.3020	−0.573467
$713$	5.97336	0.223704
$714$	−16.7822	−0.628057
$715$	0	0
$716$	23.5583	0.880414
$717$	−70.9347	−2.64910
$718$	3.55616	0.132715
$719$	−24.7710	−0.923804	−0.461902	−	0.886931i	$-0.652833\pi$
$719$	−24.7710	−0.923804	−0.461902	+	0.886931i	$0.652833\pi$
$720$	12.4128	0.462597
$721$	22.0477	0.821100
$722$	−1.00000	−0.0372161
$723$	−82.1580	−3.05549
$724$	−5.48960	−0.204019
$725$	14.3933	0.534552
$726$	−71.8173	−2.66539
$727$	−24.6509	−0.914251	−0.457125	−	0.889402i	$-0.651121\pi$
$727$	−24.6509	−0.914251	−0.457125	+	0.889402i	$0.651121\pi$
$728$	0	0
$729$	53.3722	1.97675
$730$	14.9254	0.552415
$731$	5.23019	0.193445
$732$	−42.4738	−1.56988
$733$	12.0401	0.444712	0.222356	−	0.974965i	$-0.428625\pi$
$733$	12.0401	0.444712	0.222356	+	0.974965i	$0.428625\pi$
$734$	−3.81299	−0.140740
$735$	−12.1596	−0.448512
$736$	0.857434	0.0316054
$737$	10.7801	0.397089
$738$	44.5562	1.64014
$739$	−7.91189	−0.291044	−0.145522	−	0.989355i	$-0.546486\pi$
$739$	−7.91189	−0.291044	−0.145522	+	0.989355i	$0.546486\pi$
$740$	−4.19642	−0.154263
$741$	0	0
$742$	−21.0816	−0.773928
$743$	28.0158	1.02780	0.513900	−	0.857850i	$-0.328200\pi$
$743$	28.0158	1.02780	0.513900	+	0.857850i	$0.328200\pi$
$744$	22.7157	0.832797
$745$	14.3696	0.526461
$746$	−11.2239	−0.410937
$747$	−77.9495	−2.85202
$748$	13.6328	0.498465
$749$	−15.8207	−0.578076
$750$	39.0039	1.42422
$751$	23.5102	0.857899	0.428949	−	0.903329i	$-0.358884\pi$
$751$	23.5102	0.857899	0.428949	+	0.903329i	$0.358884\pi$
$752$	−9.14883	−0.333623
$753$	−29.4826	−1.07441
$754$	0	0
$755$	−6.74794	−0.245583
$756$	−32.7684	−1.19177
$757$	−27.8505	−1.01224	−0.506121	−	0.862462i	$-0.668921\pi$
$757$	−27.8505	−1.01224	−0.506121	+	0.862462i	$0.668921\pi$
$758$	−6.19845	−0.225138
$759$	16.0669	0.583191
$760$	−1.62641	−0.0589962
$761$	10.7559	0.389900	0.194950	−	0.980813i	$-0.437545\pi$
$761$	10.7559	0.389900	0.194950	+	0.980813i	$0.437545\pi$
$762$	−5.28853	−0.191583
$763$	−23.4647	−0.849478
$764$	8.80339	0.318495
$765$	−29.4464	−1.06464
$766$	−3.03258	−0.109571
$767$	0	0
$768$	3.26068	0.117659
$769$	−54.0965	−1.95077	−0.975386	−	0.220506i	$-0.929229\pi$
$769$	−54.0965	−1.95077	−0.975386	+	0.220506i	$0.929229\pi$
$770$	−20.2783	−0.730780
$771$	−97.0919	−3.49668
$772$	−2.00765	−0.0722569
$773$	25.3327	0.911153	0.455576	−	0.890197i	$-0.349433\pi$
$773$	25.3327	0.911153	0.455576	+	0.890197i	$0.349433\pi$
$774$	16.8265	0.604815
$775$	16.4047	0.589275
$776$	−10.6832	−0.383503
$777$	18.2530	0.654822
$778$	21.5300	0.771888
$779$	−5.83807	−0.209171
$780$	0	0
$781$	−84.3133	−3.01697
$782$	−2.03406	−0.0727377
$783$	−92.3177	−3.29917
$784$	−2.29287	−0.0818882
$785$	26.9913	0.963361
$786$	8.69500	0.310140
$787$	−26.9799	−0.961731	−0.480865	−	0.876794i	$-0.659677\pi$
$787$	−26.9799	−0.961731	−0.480865	+	0.876794i	$0.659677\pi$
$788$	−5.37364	−0.191428
$789$	−35.9083	−1.27837
$790$	−0.815954	−0.0290303
$791$	−25.5337	−0.907874
$792$	43.8593	1.55847
$793$	0	0
$794$	4.67613	0.165949
$795$	−51.5303	−1.82759
$796$	7.46561	0.264611
$797$	−10.1825	−0.360684	−0.180342	−	0.983604i	$-0.557720\pi$
$797$	−10.1825	−0.360684	−0.180342	+	0.983604i	$0.557720\pi$
$798$	7.07434	0.250429
$799$	21.7034	0.767811
$800$	2.35478	0.0832541
$801$	116.785	4.12639
$802$	21.2131	0.749060
$803$	52.7375	1.86107
$804$	−6.11654	−0.215714
$805$	3.02559	0.106638
$806$	0	0
$807$	5.20605	0.183261
$808$	−11.2649	−0.396299
$809$	−28.7758	−1.01170	−0.505852	−	0.862620i	$-0.668822\pi$
$809$	−28.7758	−1.01170	−0.505852	+	0.862620i	$0.668822\pi$
$810$	−42.8584	−1.50589
$811$	−24.2577	−0.851803	−0.425902	−	0.904770i	$-0.640043\pi$
$811$	−24.2577	−0.851803	−0.425902	+	0.904770i	$0.640043\pi$
$812$	13.2613	0.465381
$813$	11.2993	0.396285
$814$	−14.8276	−0.519707
$815$	−9.20815	−0.322548
$816$	−7.73517	−0.270785
$817$	−2.20473	−0.0771336
$818$	−11.9831	−0.418980
$819$	0	0
$820$	−9.49511	−0.331584
$821$	−25.8755	−0.903060	−0.451530	−	0.892256i	$-0.649122\pi$
$821$	−25.8755	−0.903060	−0.451530	+	0.892256i	$0.649122\pi$
$822$	−5.58132	−0.194671
$823$	−46.2546	−1.61233	−0.806167	−	0.591688i	$-0.798462\pi$
$823$	−46.2546	−1.61233	−0.806167	+	0.591688i	$0.798462\pi$
$824$	10.1622	0.354015
$825$	44.1247	1.53622
$826$	24.5164	0.853035
$827$	−9.67764	−0.336525	−0.168262	−	0.985742i	$-0.553816\pi$
$827$	−9.67764	−0.336525	−0.168262	+	0.985742i	$0.553816\pi$
$828$	−6.54394	−0.227418
$829$	−11.4581	−0.397955	−0.198978	−	0.980004i	$-0.563762\pi$
$829$	−11.4581	−0.397955	−0.198978	+	0.980004i	$0.563762\pi$
$830$	16.6114	0.576589
$831$	−27.6348	−0.958639
$832$	0	0
$833$	5.43929	0.188460
$834$	−42.9631	−1.48769
$835$	39.0586	1.35168
$836$	−5.74676	−0.198756
$837$	−105.219	−3.63691
$838$	38.6604	1.33550
$839$	−30.6465	−1.05803	−0.529017	−	0.848611i	$-0.677439\pi$
$839$	−30.6465	−1.05803	−0.529017	+	0.848611i	$0.677439\pi$
$840$	11.5058	0.396987
$841$	8.36087	0.288306
$842$	9.90414	0.341319
$843$	−39.9127	−1.37467
$844$	−17.7848	−0.612179
$845$	0	0
$846$	69.8239	2.40060
$847$	−47.7859	−1.64194
$848$	−9.71683	−0.333677
$849$	−22.8087	−0.782792
$850$	−5.58616	−0.191604
$851$	2.21232	0.0758375
$852$	47.8388	1.63893
$853$	−33.1040	−1.13346	−0.566729	−	0.823904i	$-0.691792\pi$
$853$	−33.1040	−1.13346	−0.566729	+	0.823904i	$0.691792\pi$
$854$	−28.2613	−0.967081
$855$	12.4128	0.424508
$856$	−7.29201	−0.249236
$857$	−2.51768	−0.0860023	−0.0430011	−	0.999075i	$-0.513692\pi$
$857$	−2.51768	−0.0860023	−0.0430011	+	0.999075i	$0.513692\pi$
$858$	0	0
$859$	10.6840	0.364534	0.182267	−	0.983249i	$-0.441656\pi$
$859$	10.6840	0.364534	0.182267	+	0.983249i	$0.441656\pi$
$860$	−3.58579	−0.122275
$861$	41.3005	1.40752
$862$	−23.0774	−0.786018
$863$	1.62815	0.0554228	0.0277114	−	0.999616i	$-0.491178\pi$
$863$	1.62815	0.0554228	0.0277114	+	0.999616i	$0.491178\pi$
$864$	−15.1035	−0.513830
$865$	−30.0145	−1.02052
$866$	23.7544	0.807206
$867$	−37.0816	−1.25936
$868$	15.1146	0.513022
$869$	−2.88309	−0.0978021
$870$	32.4151	1.09897
$871$	0	0
$872$	−10.8152	−0.366250
$873$	81.5339	2.75950
$874$	0.857434	0.0290031
$875$	25.9525	0.877353
$876$	−29.9229	−1.01100
$877$	−26.5468	−0.896420	−0.448210	−	0.893928i	$-0.647938\pi$
$877$	−26.5468	−0.896420	−0.448210	+	0.893928i	$0.647938\pi$
$878$	−18.2272	−0.615137
$879$	108.367	3.65513
$880$	−9.34661	−0.315074
$881$	45.4032	1.52967	0.764836	−	0.644226i	$-0.222820\pi$
$881$	45.4032	1.52967	0.764836	+	0.644226i	$0.222820\pi$
$882$	17.4992	0.589229
$883$	19.4299	0.653867	0.326933	−	0.945047i	$-0.393985\pi$
$883$	19.4299	0.653867	0.326933	+	0.945047i	$0.393985\pi$
$884$	0	0
$885$	59.9263	2.01440
$886$	−35.0384	−1.17714
$887$	28.6993	0.963627	0.481814	−	0.876274i	$-0.339978\pi$
$887$	28.6993	0.963627	0.481814	+	0.876274i	$0.339978\pi$
$888$	8.41309	0.282325
$889$	−3.51889	−0.118020
$890$	−24.8874	−0.834227
$891$	−151.436	−5.07329
$892$	−20.9787	−0.702419
$893$	−9.14883	−0.306154
$894$	−28.8085	−0.963502
$895$	38.3155	1.28075
$896$	2.16959	0.0724810
$897$	0	0
$898$	−11.4714	−0.382804
$899$	42.5821	1.42019
$900$	−17.9717	−0.599057
$901$	23.0509	0.767935
$902$	−33.5500	−1.11709
$903$	15.5970	0.519035
$904$	−11.7689	−0.391428
$905$	−8.92835	−0.296788
$906$	13.5284	0.449453
$907$	−20.8538	−0.692439	−0.346219	−	0.938154i	$-0.612535\pi$
$907$	−20.8538	−0.692439	−0.346219	+	0.938154i	$0.612535\pi$
$908$	11.3430	0.376432
$909$	85.9740	2.85158
$910$	0	0
$911$	−26.5845	−0.880783	−0.440392	−	0.897806i	$-0.645160\pi$
$911$	−26.5845	−0.880783	−0.440392	+	0.897806i	$0.645160\pi$
$912$	3.26068	0.107972
$913$	58.6946	1.94251
$914$	19.3028	0.638481
$915$	−69.0799	−2.28371
$916$	16.2798	0.537901
$917$	5.78549	0.191054
$918$	35.8294	1.18255
$919$	8.59486	0.283518	0.141759	−	0.989901i	$-0.454724\pi$
$919$	8.59486	0.283518	0.141759	+	0.989901i	$0.454724\pi$
$920$	1.39454	0.0459767
$921$	47.7784	1.57435
$922$	38.7275	1.27542
$923$	0	0
$924$	40.6545	1.33744
$925$	6.07573	0.199769
$926$	−20.1438	−0.661966
$927$	−77.5576	−2.54733
$928$	6.11235	0.200648
$929$	7.84269	0.257310	0.128655	−	0.991689i	$-0.458934\pi$
$929$	7.84269	0.257310	0.128655	+	0.991689i	$0.458934\pi$
$930$	36.9451	1.21148
$931$	−2.29287	−0.0751458
$932$	8.21854	0.269207
$933$	−40.9276	−1.33991
$934$	−36.1954	−1.18435
$935$	22.1726	0.725121
$936$	0	0
$937$	26.4962	0.865595	0.432797	−	0.901491i	$-0.357527\pi$
$937$	26.4962	0.865595	0.432797	+	0.901491i	$0.357527\pi$
$938$	−4.06983	−0.132885
$939$	102.986	3.36083
$940$	−14.8798	−0.485324
$941$	44.4644	1.44950	0.724750	−	0.689012i	$-0.241955\pi$
$941$	44.4644	1.44950	0.724750	+	0.689012i	$0.241955\pi$
$942$	−54.1129	−1.76309
$943$	5.00576	0.163010
$944$	11.3000	0.367784
$945$	−53.2949	−1.73368
$946$	−12.6700	−0.411938
$947$	33.0014	1.07240	0.536201	−	0.844090i	$-0.319859\pi$
$947$	33.0014	1.07240	0.536201	+	0.844090i	$0.319859\pi$
$948$	1.63585	0.0531298
$949$	0	0
$950$	2.35478	0.0763992
$951$	5.49408	0.178158
$952$	−5.14684	−0.166810
$953$	10.6012	0.343405	0.171703	−	0.985149i	$-0.445073\pi$
$953$	10.6012	0.343405	0.171703	+	0.985149i	$0.445073\pi$
$954$	74.1589	2.40098
$955$	14.3179	0.463318
$956$	−21.7546	−0.703594
$957$	114.535	3.70240
$958$	11.3987	0.368276
$959$	−3.71371	−0.119922
$960$	5.30320	0.171160
$961$	17.5329	0.565577
$962$	0	0
$963$	55.6527	1.79338
$964$	−25.1966	−0.811528
$965$	−3.26527	−0.105113
$966$	−6.06577	−0.195163
$967$	18.1468	0.583560	0.291780	−	0.956485i	$-0.405752\pi$
$967$	18.1468	0.583560	0.291780	+	0.956485i	$0.405752\pi$
$968$	−22.0253	−0.707919
$969$	−7.73517	−0.248490
$970$	−17.3752	−0.557885
$971$	−36.9139	−1.18462	−0.592312	−	0.805709i	$-0.701785\pi$
$971$	−36.9139	−1.18462	−0.592312	+	0.805709i	$0.701785\pi$
$972$	40.6133	1.30267
$973$	−28.5868	−0.916452
$974$	10.4535	0.334953
$975$	0	0
$976$	−13.0261	−0.416955
$977$	−37.2246	−1.19092	−0.595460	−	0.803385i	$-0.703030\pi$
$977$	−37.2246	−1.19092	−0.595460	+	0.803385i	$0.703030\pi$
$978$	18.4608	0.590310
$979$	−87.9370	−2.81048
$980$	−3.72915	−0.119123
$981$	82.5420	2.63536
$982$	−4.13572	−0.131976
$983$	−9.67066	−0.308446	−0.154223	−	0.988036i	$-0.549287\pi$
$983$	−9.67066	−0.308446	−0.154223	+	0.988036i	$0.549287\pi$
$984$	19.0361	0.606847
$985$	−8.73976	−0.278472
$986$	−14.5001	−0.461777
$987$	64.7219	2.06012
$988$	0	0
$989$	1.89041	0.0601114
$990$	71.3334	2.26712
$991$	25.5109	0.810380	0.405190	−	0.914232i	$-0.367205\pi$
$991$	25.5109	0.810380	0.405190	+	0.914232i	$0.367205\pi$
$992$	6.96656	0.221188
$993$	−58.6827	−1.86224
$994$	31.8310	1.00962
$995$	12.1422	0.384932
$996$	−33.3029	−1.05524
$997$	16.9139	0.535669	0.267834	−	0.963465i	$-0.413692\pi$
$997$	16.9139	0.535669	0.267834	+	0.963465i	$0.413692\pi$
$998$	7.26427	0.229946
$999$	−38.9695	−1.23294

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bo.1.15	✓	15
13.12	even	2		6422.2.a.bq.1.15	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.bo.1.15	✓	15	1.1	even	1	trivial
6422.2.a.bq.1.15	yes	15	13.12	even	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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +3.26068 q^{3} +1.00000 q^{4} +1.62641 q^{5} -3.26068 q^{6} -2.16959 q^{7} -1.00000 q^{8} +7.63201 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +3.26068 q^{3} +1.00000 q^{4} +1.62641 q^{5} -3.26068 q^{6} -2.16959 q^{7} -1.00000 q^{8} +7.63201 q^{9} -1.62641 q^{10} -5.74676 q^{11} +3.26068 q^{12} +2.16959 q^{14} +5.30320 q^{15} +1.00000 q^{16} -2.37226 q^{17} -7.63201 q^{18} +1.00000 q^{19} +1.62641 q^{20} -7.07434 q^{21} +5.74676 q^{22} -0.857434 q^{23} -3.26068 q^{24} -2.35478 q^{25} +15.1035 q^{27} -2.16959 q^{28} -6.11235 q^{29} -5.30320 q^{30} -6.96656 q^{31} -1.00000 q^{32} -18.7383 q^{33} +2.37226 q^{34} -3.52865 q^{35} +7.63201 q^{36} -2.58017 q^{37} -1.00000 q^{38} -1.62641 q^{40} -5.83807 q^{41} +7.07434 q^{42} -2.20473 q^{43} -5.74676 q^{44} +12.4128 q^{45} +0.857434 q^{46} -9.14883 q^{47} +3.26068 q^{48} -2.29287 q^{49} +2.35478 q^{50} -7.73517 q^{51} -9.71683 q^{53} -15.1035 q^{54} -9.34661 q^{55} +2.16959 q^{56} +3.26068 q^{57} +6.11235 q^{58} +11.3000 q^{59} +5.30320 q^{60} -13.0261 q^{61} +6.96656 q^{62} -16.5583 q^{63} +1.00000 q^{64} +18.7383 q^{66} -1.87585 q^{67} -2.37226 q^{68} -2.79581 q^{69} +3.52865 q^{70} +14.6714 q^{71} -7.63201 q^{72} -9.17690 q^{73} +2.58017 q^{74} -7.67818 q^{75} +1.00000 q^{76} +12.4681 q^{77} +0.501689 q^{79} +1.62641 q^{80} +26.3515 q^{81} +5.83807 q^{82} -10.2135 q^{83} -7.07434 q^{84} -3.85827 q^{85} +2.20473 q^{86} -19.9304 q^{87} +5.74676 q^{88} +15.3020 q^{89} -12.4128 q^{90} -0.857434 q^{92} -22.7157 q^{93} +9.14883 q^{94} +1.62641 q^{95} -3.26068 q^{96} +10.6832 q^{97} +2.29287 q^{98} -43.8593 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 15 q^{2} + 15 q^{4} - q^{5} - 18 q^{7} - 15 q^{8} + 17 q^{9}+O(q^{10}) \) 15 * q - 15 * q^2 + 15 * q^4 - q^5 - 18 * q^7 - 15 * q^8 + 17 * q^9	\( 15 q - 15 q^{2} + 15 q^{4} - q^{5} - 18 q^{7} - 15 q^{8} + 17 q^{9} + q^{10} - 4 q^{11} + 18 q^{14} - 23 q^{15} + 15 q^{16} + 2 q^{17} - 17 q^{18} + 15 q^{19} - q^{20} - 2 q^{21} + 4 q^{22} + 17 q^{23} + 8 q^{25} + 12 q^{27} - 18 q^{28} - 20 q^{29} + 23 q^{30} - 30 q^{31} - 15 q^{32} - 36 q^{33} - 2 q^{34} + 32 q^{35} + 17 q^{36} - 35 q^{37} - 15 q^{38} + q^{40} - 15 q^{41} + 2 q^{42} + q^{43} - 4 q^{44} + 11 q^{45} - 17 q^{46} + 29 q^{49} - 8 q^{50} - q^{51} - q^{53} - 12 q^{54} - 6 q^{55} + 18 q^{56} + 20 q^{58} + 7 q^{59} - 23 q^{60} - 2 q^{61} + 30 q^{62} - 42 q^{63} + 15 q^{64} + 36 q^{66} - 34 q^{67} + 2 q^{68} - 12 q^{69} - 32 q^{70} - 4 q^{71} - 17 q^{72} - 12 q^{73} + 35 q^{74} + 31 q^{75} + 15 q^{76} - 20 q^{77} + 23 q^{79} - q^{80} + 7 q^{81} + 15 q^{82} + 3 q^{83} - 2 q^{84} - 46 q^{85} - q^{86} + 22 q^{87} + 4 q^{88} - 17 q^{89} - 11 q^{90} + 17 q^{92} - 60 q^{93} - q^{95} - 18 q^{97} - 29 q^{98} + 6 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 + 15 * q^4 - q^5 - 18 * q^7 - 15 * q^8 + 17 * q^9 + q^10 - 4 * q^11 + 18 * q^14 - 23 * q^15 + 15 * q^16 + 2 * q^17 - 17 * q^18 + 15 * q^19 - q^20 - 2 * q^21 + 4 * q^22 + 17 * q^23 + 8 * q^25 + 12 * q^27 - 18 * q^28 - 20 * q^29 + 23 * q^30 - 30 * q^31 - 15 * q^32 - 36 * q^33 - 2 * q^34 + 32 * q^35 + 17 * q^36 - 35 * q^37 - 15 * q^38 + q^40 - 15 * q^41 + 2 * q^42 + q^43 - 4 * q^44 + 11 * q^45 - 17 * q^46 + 29 * q^49 - 8 * q^50 - q^51 - q^53 - 12 * q^54 - 6 * q^55 + 18 * q^56 + 20 * q^58 + 7 * q^59 - 23 * q^60 - 2 * q^61 + 30 * q^62 - 42 * q^63 + 15 * q^64 + 36 * q^66 - 34 * q^67 + 2 * q^68 - 12 * q^69 - 32 * q^70 - 4 * q^71 - 17 * q^72 - 12 * q^73 + 35 * q^74 + 31 * q^75 + 15 * q^76 - 20 * q^77 + 23 * q^79 - q^80 + 7 * q^81 + 15 * q^82 + 3 * q^83 - 2 * q^84 - 46 * q^85 - q^86 + 22 * q^87 + 4 * q^88 - 17 * q^89 - 11 * q^90 + 17 * q^92 - 60 * q^93 - q^95 - 18 * q^97 - 29 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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