LMFDB - Embedded newform 6422.2.a.bd.1.3

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:	$0$
Dimension:	$6$
Coefficient field:	6.6.316645497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 12x^{4} + 24x^{3} + 13x^{2} - 24x + 5 $ x^6 - 2x^5 - 12x^4 + 24x^3 + 13x^2 - 24*x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.260281$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +0.260281 q^{3} +1.00000 q^{4} -4.36375 q^{5} +0.260281 q^{6} +0.775135 q^{7} +1.00000 q^{8} -2.93225 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.260281 q^{3} +1.00000 q^{4} -4.36375 q^{5} +0.260281 q^{6} +0.775135 q^{7} +1.00000 q^{8} -2.93225 q^{9} -4.36375 q^{10} -4.36375 q^{11} +0.260281 q^{12} +0.775135 q^{14} -1.13580 q^{15} +1.00000 q^{16} -6.93885 q^{17} -2.93225 q^{18} -1.00000 q^{19} -4.36375 q^{20} +0.201753 q^{21} -4.36375 q^{22} -4.08036 q^{23} +0.260281 q^{24} +14.0423 q^{25} -1.54405 q^{27} +0.775135 q^{28} +2.54367 q^{29} -1.13580 q^{30} +6.41220 q^{31} +1.00000 q^{32} -1.13580 q^{33} -6.93885 q^{34} -3.38249 q^{35} -2.93225 q^{36} -6.64765 q^{37} -1.00000 q^{38} -4.36375 q^{40} +1.20663 q^{41} +0.201753 q^{42} +8.33493 q^{43} -4.36375 q^{44} +12.7956 q^{45} -4.08036 q^{46} -1.36375 q^{47} +0.260281 q^{48} -6.39917 q^{49} +14.0423 q^{50} -1.80605 q^{51} +3.52493 q^{53} -1.54405 q^{54} +19.0423 q^{55} +0.775135 q^{56} -0.260281 q^{57} +2.54367 q^{58} -12.4214 q^{59} -1.13580 q^{60} +2.03105 q^{61} +6.41220 q^{62} -2.27289 q^{63} +1.00000 q^{64} -1.13580 q^{66} -11.5249 q^{67} -6.93885 q^{68} -1.06204 q^{69} -3.38249 q^{70} +12.1096 q^{71} -2.93225 q^{72} -2.30351 q^{73} -6.64765 q^{74} +3.65495 q^{75} -1.00000 q^{76} -3.38249 q^{77} +8.16942 q^{79} -4.36375 q^{80} +8.39487 q^{81} +1.20663 q^{82} +2.75361 q^{83} +0.201753 q^{84} +30.2794 q^{85} +8.33493 q^{86} +0.662069 q^{87} -4.36375 q^{88} -3.34501 q^{89} +12.7956 q^{90} -4.08036 q^{92} +1.66897 q^{93} -1.36375 q^{94} +4.36375 q^{95} +0.260281 q^{96} -2.12923 q^{97} -6.39917 q^{98} +12.7956 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 6 q^{8} + 10 q^{9}+O(q^{10}) $ 6 * q + 6 * q^2 + 2 * q^3 + 6 * q^4 + 2 * q^5 + 2 * q^6 + q^7 + 6 * q^8 + 10 * q^9	$ 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 6 q^{8} + 10 q^{9} + 2 q^{10} + 2 q^{11} + 2 q^{12} + q^{14} - 9 q^{15} + 6 q^{16} - 2 q^{17} + 10 q^{18} - 6 q^{19} + 2 q^{20} + q^{21} + 2 q^{22} + 8 q^{23} + 2 q^{24} + 16 q^{25} - 4 q^{27} + q^{28} + 20 q^{29} - 9 q^{30} + 3 q^{31} + 6 q^{32} - 9 q^{33} - 2 q^{34} + 7 q^{35} + 10 q^{36} - 9 q^{37} - 6 q^{38} + 2 q^{40} + 3 q^{41} + q^{42} + 13 q^{43} + 2 q^{44} + 38 q^{45} + 8 q^{46} + 20 q^{47} + 2 q^{48} - 7 q^{49} + 16 q^{50} + 2 q^{51} + 25 q^{53} - 4 q^{54} + 46 q^{55} + q^{56} - 2 q^{57} + 20 q^{58} - 9 q^{60} + 6 q^{61} + 3 q^{62} + 46 q^{63} + 6 q^{64} - 9 q^{66} - 32 q^{67} - 2 q^{68} - 29 q^{69} + 7 q^{70} + 39 q^{71} + 10 q^{72} + 7 q^{73} - 9 q^{74} - 15 q^{75} - 6 q^{76} + 7 q^{77} + 18 q^{79} + 2 q^{80} + 54 q^{81} + 3 q^{82} - 7 q^{83} + q^{84} - 2 q^{85} + 13 q^{86} + 12 q^{87} + 2 q^{88} + 9 q^{89} + 38 q^{90} + 8 q^{92} - 47 q^{93} + 20 q^{94} - 2 q^{95} + 2 q^{96} - 4 q^{97} - 7 q^{98} + 38 q^{99}+O(q^{100}) $ 6 * q + 6 * q^2 + 2 * q^3 + 6 * q^4 + 2 * q^5 + 2 * q^6 + q^7 + 6 * q^8 + 10 * q^9 + 2 * q^10 + 2 * q^11 + 2 * q^12 + q^14 - 9 * q^15 + 6 * q^16 - 2 * q^17 + 10 * q^18 - 6 * q^19 + 2 * q^20 + q^21 + 2 * q^22 + 8 * q^23 + 2 * q^24 + 16 * q^25 - 4 * q^27 + q^28 + 20 * q^29 - 9 * q^30 + 3 * q^31 + 6 * q^32 - 9 * q^33 - 2 * q^34 + 7 * q^35 + 10 * q^36 - 9 * q^37 - 6 * q^38 + 2 * q^40 + 3 * q^41 + q^42 + 13 * q^43 + 2 * q^44 + 38 * q^45 + 8 * q^46 + 20 * q^47 + 2 * q^48 - 7 * q^49 + 16 * q^50 + 2 * q^51 + 25 * q^53 - 4 * q^54 + 46 * q^55 + q^56 - 2 * q^57 + 20 * q^58 - 9 * q^60 + 6 * q^61 + 3 * q^62 + 46 * q^63 + 6 * q^64 - 9 * q^66 - 32 * q^67 - 2 * q^68 - 29 * q^69 + 7 * q^70 + 39 * q^71 + 10 * q^72 + 7 * q^73 - 9 * q^74 - 15 * q^75 - 6 * q^76 + 7 * q^77 + 18 * q^79 + 2 * q^80 + 54 * q^81 + 3 * q^82 - 7 * q^83 + q^84 - 2 * q^85 + 13 * q^86 + 12 * q^87 + 2 * q^88 + 9 * q^89 + 38 * q^90 + 8 * q^92 - 47 * q^93 + 20 * q^94 - 2 * q^95 + 2 * q^96 - 4 * q^97 - 7 * q^98 + 38 * 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
$3$	0.260281	0.150273	0.0751366	−	0.997173i	$-0.476061\pi$
$3$	0.260281	0.150273	0.0751366	+	0.997173i	$0.476061\pi$
$4$	1.00000	0.500000
$5$	−4.36375	−1.95153	−0.975764	−	0.218824i	$-0.929778\pi$
$5$	−4.36375	−1.95153	−0.975764	+	0.218824i	$0.929778\pi$
$6$	0.260281	0.106259
$7$	0.775135	0.292973	0.146487	−	0.989213i	$-0.453203\pi$
$7$	0.775135	0.292973	0.146487	+	0.989213i	$0.453203\pi$
$8$	1.00000	0.353553
$9$	−2.93225	−0.977418
$10$	−4.36375	−1.37994
$11$	−4.36375	−1.31572	−0.657860	−	0.753140i	$-0.728538\pi$
$11$	−4.36375	−1.31572	−0.657860	+	0.753140i	$0.728538\pi$
$12$	0.260281	0.0751366
$13$	0	0
$13$	0	0
$14$	0.775135	0.207163
$15$	−1.13580	−0.293262
$16$	1.00000	0.250000
$17$	−6.93885	−1.68292	−0.841459	−	0.540321i	$-0.818303\pi$
$17$	−6.93885	−1.68292	−0.841459	+	0.540321i	$0.818303\pi$
$18$	−2.93225	−0.691139
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−4.36375	−0.975764
$21$	0.201753	0.0440260
$22$	−4.36375	−0.930355
$23$	−4.08036	−0.850813	−0.425407	−	0.905002i	$-0.639869\pi$
$23$	−4.08036	−0.850813	−0.425407	+	0.905002i	$0.639869\pi$
$24$	0.260281	0.0531296
$25$	14.0423	2.80846
$26$	0	0
$27$	−1.54405	−0.297153
$28$	0.775135	0.146487
$29$	2.54367	0.472348	0.236174	−	0.971711i	$-0.424106\pi$
$29$	2.54367	0.472348	0.236174	+	0.971711i	$0.424106\pi$
$30$	−1.13580	−0.207368
$31$	6.41220	1.15167	0.575833	−	0.817567i	$-0.304678\pi$
$31$	6.41220	1.15167	0.575833	+	0.817567i	$0.304678\pi$
$32$	1.00000	0.176777
$33$	−1.13580	−0.197717
$34$	−6.93885	−1.19000
$35$	−3.38249	−0.571746
$36$	−2.93225	−0.488709
$37$	−6.64765	−1.09287	−0.546434	−	0.837502i	$-0.684015\pi$
$37$	−6.64765	−1.09287	−0.546434	+	0.837502i	$0.684015\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−4.36375	−0.689970
$41$	1.20663	0.188444	0.0942221	−	0.995551i	$-0.469964\pi$
$41$	1.20663	0.188444	0.0942221	+	0.995551i	$0.469964\pi$
$42$	0.201753	0.0311311
$43$	8.33493	1.27106	0.635532	−	0.772074i	$-0.280781\pi$
$43$	8.33493	1.27106	0.635532	+	0.772074i	$0.280781\pi$
$44$	−4.36375	−0.657860
$45$	12.7956	1.90746
$46$	−4.08036	−0.601616
$47$	−1.36375	−0.198923	−0.0994617	−	0.995041i	$-0.531712\pi$
$47$	−1.36375	−0.198923	−0.0994617	+	0.995041i	$0.531712\pi$
$48$	0.260281	0.0375683
$49$	−6.39917	−0.914167
$50$	14.0423	1.98588
$51$	−1.80605	−0.252897
$52$	0	0
$53$	3.52493	0.484186	0.242093	−	0.970253i	$-0.422166\pi$
$53$	3.52493	0.484186	0.242093	+	0.970253i	$0.422166\pi$
$54$	−1.54405	−0.210119
$55$	19.0423	2.56767
$56$	0.775135	0.103582
$57$	−0.260281	−0.0344750
$58$	2.54367	0.334001
$59$	−12.4214	−1.61712	−0.808562	−	0.588410i	$-0.799754\pi$
$59$	−12.4214	−1.61712	−0.808562	+	0.588410i	$0.799754\pi$
$60$	−1.13580	−0.146631
$61$	2.03105	0.260049	0.130024	−	0.991511i	$-0.458494\pi$
$61$	2.03105	0.260049	0.130024	+	0.991511i	$0.458494\pi$
$62$	6.41220	0.814351
$63$	−2.27289	−0.286357
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.13580	−0.139807
$67$	−11.5249	−1.40799	−0.703995	−	0.710205i	$-0.748602\pi$
$67$	−11.5249	−1.40799	−0.703995	+	0.710205i	$0.748602\pi$
$68$	−6.93885	−0.841459
$69$	−1.06204	−0.127854
$70$	−3.38249	−0.404285
$71$	12.1096	1.43714	0.718570	−	0.695455i	$-0.244797\pi$
$71$	12.1096	1.43714	0.718570	+	0.695455i	$0.244797\pi$
$72$	−2.93225	−0.345569
$73$	−2.30351	−0.269605	−0.134802	−	0.990872i	$-0.543040\pi$
$73$	−2.30351	−0.269605	−0.134802	+	0.990872i	$0.543040\pi$
$74$	−6.64765	−0.772774
$75$	3.65495	0.422037
$76$	−1.00000	−0.114708
$77$	−3.38249	−0.385471
$78$	0	0
$79$	8.16942	0.919132	0.459566	−	0.888144i	$-0.348005\pi$
$79$	8.16942	0.919132	0.459566	+	0.888144i	$0.348005\pi$
$80$	−4.36375	−0.487882
$81$	8.39487	0.932764
$82$	1.20663	0.133250
$83$	2.75361	0.302248	0.151124	−	0.988515i	$-0.451711\pi$
$83$	2.75361	0.302248	0.151124	+	0.988515i	$0.451711\pi$
$84$	0.201753	0.0220130
$85$	30.2794	3.28426
$86$	8.33493	0.898779
$87$	0.662069	0.0709813
$88$	−4.36375	−0.465177
$89$	−3.34501	−0.354570	−0.177285	−	0.984160i	$-0.556731\pi$
$89$	−3.34501	−0.354570	−0.177285	+	0.984160i	$0.556731\pi$
$90$	12.7956	1.34878
$91$	0	0
$92$	−4.08036	−0.425407
$93$	1.66897	0.173064
$94$	−1.36375	−0.140660
$95$	4.36375	0.447711
$96$	0.260281	0.0265648
$97$	−2.12923	−0.216190	−0.108095	−	0.994141i	$-0.534475\pi$
$97$	−2.12923	−0.216190	−0.108095	+	0.994141i	$0.534475\pi$
$98$	−6.39917	−0.646413
$99$	12.7956	1.28601
$100$	14.0423	1.40423
$101$	11.5122	1.14550	0.572751	−	0.819729i	$-0.305876\pi$
$101$	11.5122	1.14550	0.572751	+	0.819729i	$0.305876\pi$
$102$	−1.80605	−0.178826
$103$	8.59033	0.846431	0.423215	−	0.906029i	$-0.360901\pi$
$103$	8.59033	0.846431	0.423215	+	0.906029i	$0.360901\pi$
$104$	0	0
$105$	−0.880398	−0.0859181
$106$	3.52493	0.342371
$107$	12.2990	1.18899	0.594495	−	0.804099i	$-0.297352\pi$
$107$	12.2990	1.18899	0.594495	+	0.804099i	$0.297352\pi$
$108$	−1.54405	−0.148576
$109$	−8.17344	−0.782874	−0.391437	−	0.920205i	$-0.628022\pi$
$109$	−8.17344	−0.782874	−0.391437	+	0.920205i	$0.628022\pi$
$110$	19.0423	1.81561
$111$	−1.73026	−0.164229
$112$	0.775135	0.0732433
$113$	4.05116	0.381101	0.190551	−	0.981677i	$-0.438973\pi$
$113$	4.05116	0.381101	0.190551	+	0.981677i	$0.438973\pi$
$114$	−0.260281	−0.0243775
$115$	17.8057	1.66039
$116$	2.54367	0.236174
$117$	0	0
$118$	−12.4214	−1.14348
$119$	−5.37854	−0.493050
$120$	−1.13580	−0.103684
$121$	8.04232	0.731120
$122$	2.03105	0.183882
$123$	0.314063	0.0283181
$124$	6.41220	0.575833
$125$	−39.4584	−3.52927
$126$	−2.27289	−0.202485
$127$	6.03228	0.535279	0.267639	−	0.963519i	$-0.413756\pi$
$127$	6.03228	0.535279	0.267639	+	0.963519i	$0.413756\pi$
$128$	1.00000	0.0883883
$129$	2.16942	0.191007
$130$	0	0
$131$	12.8272	1.12071	0.560357	−	0.828251i	$-0.310664\pi$
$131$	12.8272	1.12071	0.560357	+	0.828251i	$0.310664\pi$
$132$	−1.13580	−0.0988587
$133$	−0.775135	−0.0672127
$134$	−11.5249	−0.995599
$135$	6.73786	0.579902
$136$	−6.93885	−0.595001
$137$	13.5152	1.15468	0.577339	−	0.816504i	$-0.304091\pi$
$137$	13.5152	1.15468	0.577339	+	0.816504i	$0.304091\pi$
$138$	−1.06204	−0.0904067
$139$	2.17364	0.184365	0.0921827	−	0.995742i	$-0.470616\pi$
$139$	2.17364	0.184365	0.0921827	+	0.995742i	$0.470616\pi$
$140$	−3.38249	−0.285873
$141$	−0.354958	−0.0298929
$142$	12.1096	1.01621
$143$	0	0
$144$	−2.93225	−0.244354
$145$	−11.1000	−0.921801
$146$	−2.30351	−0.190639
$147$	−1.66558	−0.137375
$148$	−6.64765	−0.546434
$149$	−10.4236	−0.853935	−0.426968	−	0.904267i	$-0.640418\pi$
$149$	−10.4236	−0.853935	−0.426968	+	0.904267i	$0.640418\pi$
$150$	3.65495	0.298425
$151$	7.19913	0.585857	0.292928	−	0.956134i	$-0.405370\pi$
$151$	7.19913	0.585857	0.292928	+	0.956134i	$0.405370\pi$
$152$	−1.00000	−0.0811107
$153$	20.3465	1.64491
$154$	−3.38249	−0.272569
$155$	−27.9813	−2.24751
$156$	0	0
$157$	20.5120	1.63704	0.818519	−	0.574479i	$-0.194795\pi$
$157$	20.5120	1.63704	0.818519	+	0.574479i	$0.194795\pi$
$158$	8.16942	0.649924
$159$	0.917471	0.0727602
$160$	−4.36375	−0.344985
$161$	−3.16283	−0.249266
$162$	8.39487	0.659564
$163$	−2.11182	−0.165411	−0.0827053	−	0.996574i	$-0.526356\pi$
$163$	−2.11182	−0.165411	−0.0827053	+	0.996574i	$0.526356\pi$
$164$	1.20663	0.0942221
$165$	4.95635	0.385851
$166$	2.75361	0.213722
$167$	−20.3048	−1.57123	−0.785615	−	0.618715i	$-0.787654\pi$
$167$	−20.3048	−1.57123	−0.785615	+	0.618715i	$0.787654\pi$
$168$	0.201753	0.0155656
$169$	0	0
$170$	30.2794	2.32232
$171$	2.93225	0.224235
$172$	8.33493	0.635532
$173$	23.1299	1.75854	0.879268	−	0.476327i	$-0.158032\pi$
$173$	23.1299	1.75854	0.879268	+	0.476327i	$0.158032\pi$
$174$	0.662069	0.0501914
$175$	10.8847	0.822805
$176$	−4.36375	−0.328930
$177$	−3.23304	−0.243011
$178$	−3.34501	−0.250719
$179$	−13.1550	−0.983253	−0.491626	−	0.870806i	$-0.663597\pi$
$179$	−13.1550	−0.983253	−0.491626	+	0.870806i	$0.663597\pi$
$180$	12.7956	0.953730
$181$	−0.551157	−0.0409672	−0.0204836	−	0.999790i	$-0.506521\pi$
$181$	−0.551157	−0.0409672	−0.0204836	+	0.999790i	$0.506521\pi$
$182$	0	0
$183$	0.528643	0.0390784
$184$	−4.08036	−0.300808
$185$	29.0087	2.13276
$186$	1.66897	0.122375
$187$	30.2794	2.21425
$188$	−1.36375	−0.0994617
$189$	−1.19685	−0.0870579
$190$	4.36375	0.316580
$191$	−5.61275	−0.406124	−0.203062	−	0.979166i	$-0.565089\pi$
$191$	−5.61275	−0.406124	−0.203062	+	0.979166i	$0.565089\pi$
$192$	0.260281	0.0187841
$193$	7.88071	0.567266	0.283633	−	0.958933i	$-0.408460\pi$
$193$	7.88071	0.567266	0.283633	+	0.958933i	$0.408460\pi$
$194$	−2.12923	−0.152870
$195$	0	0
$196$	−6.39917	−0.457083
$197$	−6.81454	−0.485516	−0.242758	−	0.970087i	$-0.578052\pi$
$197$	−6.81454	−0.485516	−0.242758	+	0.970087i	$0.578052\pi$
$198$	12.7956	0.909345
$199$	−24.2470	−1.71882	−0.859411	−	0.511286i	$-0.829169\pi$
$199$	−24.2470	−1.71882	−0.859411	+	0.511286i	$0.829169\pi$
$200$	14.0423	0.992942
$201$	−2.99971	−0.211583
$202$	11.5122	0.809993
$203$	1.97169	0.138386
$204$	−1.80605	−0.126449
$205$	−5.26544	−0.367754
$206$	8.59033	0.598517
$207$	11.9646	0.831600
$208$	0	0
$209$	4.36375	0.301847
$210$	−0.880398	−0.0607533
$211$	−5.05776	−0.348190	−0.174095	−	0.984729i	$-0.555700\pi$
$211$	−5.05776	−0.348190	−0.174095	+	0.984729i	$0.555700\pi$
$212$	3.52493	0.242093
$213$	3.15188	0.215964
$214$	12.2990	0.840743
$215$	−36.3716	−2.48052
$216$	−1.54405	−0.105059
$217$	4.97032	0.337407
$218$	−8.17344	−0.553575
$219$	−0.599558	−0.0405144
$220$	19.0423	1.28383
$221$	0	0
$222$	−1.73026	−0.116127
$223$	14.0800	0.942864	0.471432	−	0.881902i	$-0.343737\pi$
$223$	14.0800	0.942864	0.471432	+	0.881902i	$0.343737\pi$
$224$	0.775135	0.0517909
$225$	−41.1756	−2.74504
$226$	4.05116	0.269479
$227$	−18.6540	−1.23811	−0.619056	−	0.785347i	$-0.712485\pi$
$227$	−18.6540	−1.23811	−0.619056	+	0.785347i	$0.712485\pi$
$228$	−0.260281	−0.0172375
$229$	−5.65636	−0.373783	−0.186891	−	0.982381i	$-0.559841\pi$
$229$	−5.65636	−0.373783	−0.186891	+	0.982381i	$0.559841\pi$
$230$	17.8057	1.17407
$231$	−0.880398	−0.0579260
$232$	2.54367	0.167000
$233$	−19.9409	−1.30637	−0.653185	−	0.757198i	$-0.726568\pi$
$233$	−19.9409	−1.30637	−0.653185	+	0.757198i	$0.726568\pi$
$234$	0	0
$235$	5.95107	0.388205
$236$	−12.4214	−0.808562
$237$	2.12634	0.138121
$238$	−5.37854	−0.348639
$239$	3.43781	0.222373	0.111187	−	0.993800i	$-0.464535\pi$
$239$	3.43781	0.222373	0.111187	+	0.993800i	$0.464535\pi$
$240$	−1.13580	−0.0733156
$241$	−5.43845	−0.350321	−0.175161	−	0.984540i	$-0.556044\pi$
$241$	−5.43845	−0.350321	−0.175161	+	0.984540i	$0.556044\pi$
$242$	8.04232	0.516980
$243$	6.81718	0.437322
$244$	2.03105	0.130024
$245$	27.9244	1.78402
$246$	0.314063	0.0200239
$247$	0	0
$248$	6.41220	0.407175
$249$	0.716713	0.0454198
$250$	−39.4584	−2.49557
$251$	−25.9262	−1.63644	−0.818222	−	0.574902i	$-0.805040\pi$
$251$	−25.9262	−1.63644	−0.818222	+	0.574902i	$0.805040\pi$
$252$	−2.27289	−0.143179
$253$	17.8057	1.11943
$254$	6.03228	0.378499
$255$	7.88115	0.493537
$256$	1.00000	0.0625000
$257$	−1.88939	−0.117857	−0.0589283	−	0.998262i	$-0.518768\pi$
$257$	−1.88939	−0.117857	−0.0589283	+	0.998262i	$0.518768\pi$
$258$	2.16942	0.135062
$259$	−5.15283	−0.320181
$260$	0	0
$261$	−7.45870	−0.461682
$262$	12.8272	0.792464
$263$	−4.19046	−0.258395	−0.129197	−	0.991619i	$-0.541240\pi$
$263$	−4.19046	−0.258395	−0.129197	+	0.991619i	$0.541240\pi$
$264$	−1.13580	−0.0699037
$265$	−15.3819	−0.944904
$266$	−0.775135	−0.0475266
$267$	−0.870641	−0.0532824
$268$	−11.5249	−0.703995
$269$	5.06288	0.308689	0.154345	−	0.988017i	$-0.450673\pi$
$269$	5.06288	0.308689	0.154345	+	0.988017i	$0.450673\pi$
$270$	6.73786	0.410053
$271$	23.4633	1.42529	0.712647	−	0.701522i	$-0.247496\pi$
$271$	23.4633	1.42529	0.712647	+	0.701522i	$0.247496\pi$
$272$	−6.93885	−0.420730
$273$	0	0
$274$	13.5152	0.816481
$275$	−61.2772	−3.69515
$276$	−1.06204	−0.0639272
$277$	7.03193	0.422508	0.211254	−	0.977431i	$-0.432245\pi$
$277$	7.03193	0.422508	0.211254	+	0.977431i	$0.432245\pi$
$278$	2.17364	0.130366
$279$	−18.8022	−1.12566
$280$	−3.38249	−0.202143
$281$	20.4814	1.22182	0.610910	−	0.791700i	$-0.290804\pi$
$281$	20.4814	1.22182	0.610910	+	0.791700i	$0.290804\pi$
$282$	−0.354958	−0.0211374
$283$	−26.9345	−1.60109	−0.800544	−	0.599274i	$-0.795456\pi$
$283$	−26.9345	−1.60109	−0.800544	+	0.599274i	$0.795456\pi$
$284$	12.1096	0.718570
$285$	1.13580	0.0672790
$286$	0	0
$287$	0.935302	0.0552091
$288$	−2.93225	−0.172785
$289$	31.1476	1.83221
$290$	−11.1000	−0.651812
$291$	−0.554197	−0.0324876
$292$	−2.30351	−0.134802
$293$	−9.68946	−0.566064	−0.283032	−	0.959110i	$-0.591340\pi$
$293$	−9.68946	−0.566064	−0.283032	+	0.959110i	$0.591340\pi$
$294$	−1.66558	−0.0971386
$295$	54.2038	3.15587
$296$	−6.64765	−0.386387
$297$	6.73786	0.390970
$298$	−10.4236	−0.603823
$299$	0	0
$300$	3.65495	0.211018
$301$	6.46069	0.372388
$302$	7.19913	0.414263
$303$	2.99639	0.172138
$304$	−1.00000	−0.0573539
$305$	−8.86298	−0.507493
$306$	20.3465	1.16313
$307$	−12.5815	−0.718064	−0.359032	−	0.933325i	$-0.616893\pi$
$307$	−12.5815	−0.718064	−0.359032	+	0.933325i	$0.616893\pi$
$308$	−3.38249	−0.192736
$309$	2.23590	0.127196
$310$	−27.9813	−1.58923
$311$	−16.7407	−0.949280	−0.474640	−	0.880180i	$-0.657422\pi$
$311$	−16.7407	−0.949280	−0.474640	+	0.880180i	$0.657422\pi$
$312$	0	0
$313$	−11.5101	−0.650590	−0.325295	−	0.945613i	$-0.605464\pi$
$313$	−11.5101	−0.650590	−0.325295	+	0.945613i	$0.605464\pi$
$314$	20.5120	1.15756
$315$	9.91833	0.558835
$316$	8.16942	0.459566
$317$	4.14425	0.232764	0.116382	−	0.993205i	$-0.462870\pi$
$317$	4.14425	0.232764	0.116382	+	0.993205i	$0.462870\pi$
$318$	0.917471	0.0514492
$319$	−11.1000	−0.621478
$320$	−4.36375	−0.243941
$321$	3.20119	0.178673
$322$	−3.16283	−0.176257
$323$	6.93885	0.386088
$324$	8.39487	0.466382
$325$	0	0
$326$	−2.11182	−0.116963
$327$	−2.12739	−0.117645
$328$	1.20663	0.0666251
$329$	−1.05709	−0.0582793
$330$	4.95635	0.272838
$331$	−0.316380	−0.0173898	−0.00869492	−	0.999962i	$-0.502768\pi$
$331$	−0.316380	−0.0173898	−0.00869492	+	0.999962i	$0.502768\pi$
$332$	2.75361	0.151124
$333$	19.4926	1.06819
$334$	−20.3048	−1.11103
$335$	50.2917	2.74773
$336$	0.201753	0.0110065
$337$	17.7103	0.964739	0.482370	−	0.875968i	$-0.339776\pi$
$337$	17.7103	0.964739	0.482370	+	0.875968i	$0.339776\pi$
$338$	0	0
$339$	1.05444	0.0572693
$340$	30.2794	1.64213
$341$	−27.9813	−1.51527
$342$	2.93225	0.158558
$343$	−10.3862	−0.560800
$344$	8.33493	0.449389
$345$	4.63447	0.249512
$346$	23.1299	1.24347
$347$	22.4138	1.20324	0.601619	−	0.798783i	$-0.294523\pi$
$347$	22.4138	1.20324	0.601619	+	0.798783i	$0.294523\pi$
$348$	0.662069	0.0354906
$349$	20.0965	1.07574	0.537871	−	0.843027i	$-0.319229\pi$
$349$	20.0965	1.07574	0.537871	+	0.843027i	$0.319229\pi$
$350$	10.8847	0.581811
$351$	0	0
$352$	−4.36375	−0.232589
$353$	11.7682	0.626360	0.313180	−	0.949694i	$-0.398606\pi$
$353$	11.7682	0.626360	0.313180	+	0.949694i	$0.398606\pi$
$354$	−3.23304	−0.171834
$355$	−52.8431	−2.80462
$356$	−3.34501	−0.177285
$357$	−1.39993	−0.0740922
$358$	−13.1550	−0.695265
$359$	7.47763	0.394654	0.197327	−	0.980338i	$-0.436774\pi$
$359$	7.47763	0.394654	0.197327	+	0.980338i	$0.436774\pi$
$360$	12.7956	0.674389
$361$	1.00000	0.0526316
$362$	−0.551157	−0.0289682
$363$	2.09326	0.109868
$364$	0	0
$365$	10.0519	0.526142
$366$	0.528643	0.0276326
$367$	16.4233	0.857289	0.428645	−	0.903473i	$-0.358991\pi$
$367$	16.4233	0.857289	0.428645	+	0.903473i	$0.358991\pi$
$368$	−4.08036	−0.212703
$369$	−3.53815	−0.184189
$370$	29.0087	1.50809
$371$	2.73230	0.141854
$372$	1.66897	0.0865322
$373$	−12.8349	−0.664567	−0.332283	−	0.943180i	$-0.607819\pi$
$373$	−12.8349	−0.664567	−0.332283	+	0.943180i	$0.607819\pi$
$374$	30.2794	1.56571
$375$	−10.2703	−0.530354
$376$	−1.36375	−0.0703301
$377$	0	0
$378$	−1.19685	−0.0615592
$379$	36.0892	1.85378	0.926889	−	0.375334i	$-0.122472\pi$
$379$	36.0892	1.85378	0.926889	+	0.375334i	$0.122472\pi$
$380$	4.36375	0.223856
$381$	1.57009	0.0804380
$382$	−5.61275	−0.287173
$383$	32.8134	1.67669	0.838344	−	0.545142i	$-0.183524\pi$
$383$	32.8134	1.67669	0.838344	+	0.545142i	$0.183524\pi$
$384$	0.260281	0.0132824
$385$	14.7604	0.752258
$386$	7.88071	0.401117
$387$	−24.4401	−1.24236
$388$	−2.12923	−0.108095
$389$	−2.97899	−0.151041	−0.0755204	−	0.997144i	$-0.524062\pi$
$389$	−2.97899	−0.151041	−0.0755204	+	0.997144i	$0.524062\pi$
$390$	0	0
$391$	28.3130	1.43185
$392$	−6.39917	−0.323207
$393$	3.33866	0.168413
$394$	−6.81454	−0.343312
$395$	−35.6493	−1.79371
$396$	12.7956	0.643004
$397$	20.3722	1.02245	0.511225	−	0.859447i	$-0.329192\pi$
$397$	20.3722	1.02245	0.511225	+	0.859447i	$0.329192\pi$
$398$	−24.2470	−1.21539
$399$	−0.201753	−0.0101003
$400$	14.0423	0.702116
$401$	−13.8861	−0.693438	−0.346719	−	0.937969i	$-0.612704\pi$
$401$	−13.8861	−0.693438	−0.346719	+	0.937969i	$0.612704\pi$
$402$	−2.99971	−0.149612
$403$	0	0
$404$	11.5122	0.572751
$405$	−36.6331	−1.82032
$406$	1.97169	0.0978533
$407$	29.0087	1.43791
$408$	−1.80605	−0.0894128
$409$	31.1280	1.53918	0.769590	−	0.638538i	$-0.220461\pi$
$409$	31.1280	1.53918	0.769590	+	0.638538i	$0.220461\pi$
$410$	−5.26544	−0.260041
$411$	3.51774	0.173517
$412$	8.59033	0.423215
$413$	−9.62824	−0.473775
$414$	11.9646	0.588030
$415$	−12.0161	−0.589846
$416$	0	0
$417$	0.565756	0.0277052
$418$	4.36375	0.213438
$419$	−28.0907	−1.37232	−0.686160	−	0.727451i	$-0.740705\pi$
$419$	−28.0907	−1.37232	−0.686160	+	0.727451i	$0.740705\pi$
$420$	−0.880398	−0.0429590
$421$	10.8442	0.528512	0.264256	−	0.964453i	$-0.414874\pi$
$421$	10.8442	0.528512	0.264256	+	0.964453i	$0.414874\pi$
$422$	−5.05776	−0.246208
$423$	3.99886	0.194431
$424$	3.52493	0.171186
$425$	−97.4375	−4.72641
$426$	3.15188	0.152709
$427$	1.57434	0.0761874
$428$	12.2990	0.594495
$429$	0	0
$430$	−36.3716	−1.75399
$431$	−6.90864	−0.332777	−0.166389	−	0.986060i	$-0.553211\pi$
$431$	−6.90864	−0.332777	−0.166389	+	0.986060i	$0.553211\pi$
$432$	−1.54405	−0.0742882
$433$	39.3767	1.89232	0.946161	−	0.323697i	$-0.104926\pi$
$433$	39.3767	1.89232	0.946161	+	0.323697i	$0.104926\pi$
$434$	4.97032	0.238583
$435$	−2.88911	−0.138522
$436$	−8.17344	−0.391437
$437$	4.08036	0.195190
$438$	−0.599558	−0.0286480
$439$	−36.9430	−1.76319	−0.881597	−	0.472003i	$-0.843531\pi$
$439$	−36.9430	−1.76319	−0.881597	+	0.472003i	$0.843531\pi$
$440$	19.0423	0.907807
$441$	18.7640	0.893523
$442$	0	0
$443$	10.0111	0.475642	0.237821	−	0.971309i	$-0.423567\pi$
$443$	10.0111	0.475642	0.237821	+	0.971309i	$0.423567\pi$
$444$	−1.73026	−0.0821143
$445$	14.5968	0.691953
$446$	14.0800	0.666706
$447$	−2.71306	−0.128324
$448$	0.775135	0.0366217
$449$	19.3184	0.911694	0.455847	−	0.890058i	$-0.349336\pi$
$449$	19.3184	0.911694	0.455847	+	0.890058i	$0.349336\pi$
$450$	−41.1756	−1.94104
$451$	−5.26544	−0.247940
$452$	4.05116	0.190551
$453$	1.87380	0.0880386
$454$	−18.6540	−0.875478
$455$	0	0
$456$	−0.260281	−0.0121888
$457$	−8.71798	−0.407810	−0.203905	−	0.978991i	$-0.565363\pi$
$457$	−8.71798	−0.407810	−0.203905	+	0.978991i	$0.565363\pi$
$458$	−5.65636	−0.264304
$459$	10.7139	0.500084
$460$	17.8057	0.830193
$461$	−4.77897	−0.222579	−0.111290	−	0.993788i	$-0.535498\pi$
$461$	−4.77897	−0.222579	−0.111290	+	0.993788i	$0.535498\pi$
$462$	−0.880398	−0.0409598
$463$	−24.7123	−1.14848	−0.574239	−	0.818688i	$-0.694702\pi$
$463$	−24.7123	−1.14848	−0.574239	+	0.818688i	$0.694702\pi$
$464$	2.54367	0.118087
$465$	−7.28298	−0.337740
$466$	−19.9409	−0.923744
$467$	−28.5590	−1.32155	−0.660776	−	0.750583i	$-0.729773\pi$
$467$	−28.5590	−1.32155	−0.660776	+	0.750583i	$0.729773\pi$
$468$	0	0
$469$	−8.93334	−0.412503
$470$	5.95107	0.274502
$471$	5.33889	0.246003
$472$	−12.4214	−0.571740
$473$	−36.3716	−1.67237
$474$	2.12634	0.0976662
$475$	−14.0423	−0.644306
$476$	−5.37854	−0.246525
$477$	−10.3360	−0.473252
$478$	3.43781	0.157242
$479$	33.7512	1.54213	0.771066	−	0.636755i	$-0.219724\pi$
$479$	33.7512	1.54213	0.771066	+	0.636755i	$0.219724\pi$
$480$	−1.13580	−0.0518420
$481$	0	0
$482$	−5.43845	−0.247715
$483$	−0.823223	−0.0374579
$484$	8.04232	0.365560
$485$	9.29141	0.421901
$486$	6.81718	0.309234
$487$	−5.50881	−0.249628	−0.124814	−	0.992180i	$-0.539833\pi$
$487$	−5.50881	−0.249628	−0.124814	+	0.992180i	$0.539833\pi$
$488$	2.03105	0.0919412
$489$	−0.549666	−0.0248568
$490$	27.9244	1.26149
$491$	22.4320	1.01234	0.506171	−	0.862433i	$-0.331060\pi$
$491$	22.4320	1.01234	0.506171	+	0.862433i	$0.331060\pi$
$492$	0.314063	0.0141591
$493$	−17.6502	−0.794924
$494$	0	0
$495$	−55.8369	−2.50968
$496$	6.41220	0.287916
$497$	9.38654	0.421044
$498$	0.716713	0.0321167
$499$	−2.39076	−0.107025	−0.0535125	−	0.998567i	$-0.517042\pi$
$499$	−2.39076	−0.107025	−0.0535125	+	0.998567i	$0.517042\pi$
$500$	−39.4584	−1.76463
$501$	−5.28494	−0.236114
$502$	−25.9262	−1.15714
$503$	−22.0472	−0.983035	−0.491517	−	0.870868i	$-0.663558\pi$
$503$	−22.0472	−0.983035	−0.491517	+	0.870868i	$0.663558\pi$
$504$	−2.27289	−0.101243
$505$	−50.2362	−2.23548
$506$	17.8057	0.791558
$507$	0	0
$508$	6.03228	0.267639
$509$	−3.86985	−0.171528	−0.0857641	−	0.996315i	$-0.527333\pi$
$509$	−3.86985	−0.171528	−0.0857641	+	0.996315i	$0.527333\pi$
$510$	7.88115	0.348983
$511$	−1.78553	−0.0789871
$512$	1.00000	0.0441942
$513$	1.54405	0.0681715
$514$	−1.88939	−0.0833372
$515$	−37.4861	−1.65183
$516$	2.16942	0.0955035
$517$	5.95107	0.261728
$518$	−5.15283	−0.226402
$519$	6.02028	0.264261
$520$	0	0
$521$	41.7278	1.82813	0.914064	−	0.405570i	$-0.132927\pi$
$521$	41.7278	1.82813	0.914064	+	0.405570i	$0.132927\pi$
$522$	−7.45870	−0.326458
$523$	−8.08529	−0.353545	−0.176773	−	0.984252i	$-0.556566\pi$
$523$	−8.08529	−0.353545	−0.176773	+	0.984252i	$0.556566\pi$
$524$	12.8272	0.560357
$525$	2.83308	0.123646
$526$	−4.19046	−0.182713
$527$	−44.4933	−1.93816
$528$	−1.13580	−0.0494294
$529$	−6.35068	−0.276117
$530$	−15.3819	−0.668148
$531$	36.4226	1.58061
$532$	−0.775135	−0.0336064
$533$	0	0
$534$	−0.870641	−0.0376763
$535$	−53.6698	−2.32035
$536$	−11.5249	−0.497799
$537$	−3.42400	−0.147757
$538$	5.06288	0.218276
$539$	27.9244	1.20279
$540$	6.73786	0.289951
$541$	−33.6272	−1.44574	−0.722872	−	0.690982i	$-0.757179\pi$
$541$	−33.6272	−1.44574	−0.722872	+	0.690982i	$0.757179\pi$
$542$	23.4633	1.00784
$543$	−0.143456	−0.00615627
$544$	−6.93885	−0.297501
$545$	35.6669	1.52780
$546$	0	0
$547$	24.6548	1.05417	0.527083	−	0.849814i	$-0.323286\pi$
$547$	24.6548	1.05417	0.527083	+	0.849814i	$0.323286\pi$
$548$	13.5152	0.577339
$549$	−5.95555	−0.254177
$550$	−61.2772	−2.61287
$551$	−2.54367	−0.108364
$552$	−1.06204	−0.0452034
$553$	6.33240	0.269281
$554$	7.03193	0.298758
$555$	7.55041	0.320497
$556$	2.17364	0.0921827
$557$	8.69700	0.368504	0.184252	−	0.982879i	$-0.441014\pi$
$557$	8.69700	0.368504	0.184252	+	0.982879i	$0.441014\pi$
$558$	−18.8022	−0.795961
$559$	0	0
$560$	−3.38249	−0.142936
$561$	7.88115	0.332742
$562$	20.4814	0.863957
$563$	44.6483	1.88170	0.940851	−	0.338821i	$-0.110028\pi$
$563$	44.6483	1.88170	0.940851	+	0.338821i	$0.110028\pi$
$564$	−0.354958	−0.0149464
$565$	−17.6783	−0.743730
$566$	−26.9345	−1.13214
$567$	6.50716	0.273275
$568$	12.1096	0.508106
$569$	−41.8543	−1.75462	−0.877312	−	0.479920i	$-0.840666\pi$
$569$	−41.8543	−1.75462	−0.877312	+	0.479920i	$0.840666\pi$
$570$	1.13580	0.0475734
$571$	1.94613	0.0814429	0.0407214	−	0.999171i	$-0.487034\pi$
$571$	1.94613	0.0814429	0.0407214	+	0.999171i	$0.487034\pi$
$572$	0	0
$573$	−1.46089	−0.0610296
$574$	0.935302	0.0390387
$575$	−57.2977	−2.38948
$576$	−2.93225	−0.122177
$577$	17.3159	0.720870	0.360435	−	0.932784i	$-0.382628\pi$
$577$	17.3159	0.720870	0.360435	+	0.932784i	$0.382628\pi$
$578$	31.1476	1.29557
$579$	2.05120	0.0852448
$580$	−11.1000	−0.460901
$581$	2.13442	0.0885507
$582$	−0.554197	−0.0229722
$583$	−15.3819	−0.637054
$584$	−2.30351	−0.0953197
$585$	0	0
$586$	−9.68946	−0.400268
$587$	45.0778	1.86056	0.930279	−	0.366852i	$-0.119564\pi$
$587$	45.0778	1.86056	0.930279	+	0.366852i	$0.119564\pi$
$588$	−1.66558	−0.0686874
$589$	−6.41220	−0.264210
$590$	54.2038	2.23153
$591$	−1.77369	−0.0729600
$592$	−6.64765	−0.273217
$593$	−39.5863	−1.62561	−0.812807	−	0.582533i	$-0.802062\pi$
$593$	−39.5863	−1.62561	−0.812807	+	0.582533i	$0.802062\pi$
$594$	6.73786	0.276458
$595$	23.4706	0.962202
$596$	−10.4236	−0.426968
$597$	−6.31102	−0.258293
$598$	0	0
$599$	15.9168	0.650341	0.325171	−	0.945655i	$-0.394578\pi$
$599$	15.9168	0.650341	0.325171	+	0.945655i	$0.394578\pi$
$600$	3.65495	0.149213
$601$	21.8645	0.891870	0.445935	−	0.895065i	$-0.352871\pi$
$601$	21.8645	0.891870	0.445935	+	0.895065i	$0.352871\pi$
$602$	6.46069	0.263318
$603$	33.7939	1.37619
$604$	7.19913	0.292928
$605$	−35.0947	−1.42680
$606$	2.99639	0.121720
$607$	−5.59196	−0.226971	−0.113485	−	0.993540i	$-0.536201\pi$
$607$	−5.59196	−0.226971	−0.113485	+	0.993540i	$0.536201\pi$
$608$	−1.00000	−0.0405554
$609$	0.513193	0.0207956
$610$	−8.86298	−0.358852
$611$	0	0
$612$	20.3465	0.822457
$613$	45.9423	1.85559	0.927796	−	0.373088i	$-0.121701\pi$
$613$	45.9423	1.85559	0.927796	+	0.373088i	$0.121701\pi$
$614$	−12.5815	−0.507748
$615$	−1.37049	−0.0552636
$616$	−3.38249	−0.136285
$617$	−10.6161	−0.427389	−0.213695	−	0.976900i	$-0.568550\pi$
$617$	−10.6161	−0.427389	−0.213695	+	0.976900i	$0.568550\pi$
$618$	2.23590	0.0899410
$619$	−2.47379	−0.0994299	−0.0497149	−	0.998763i	$-0.515831\pi$
$619$	−2.47379	−0.0994299	−0.0497149	+	0.998763i	$0.515831\pi$
$620$	−27.9813	−1.12375
$621$	6.30028	0.252822
$622$	−16.7407	−0.671242
$623$	−2.59283	−0.103880
$624$	0	0
$625$	101.975	4.07901
$626$	−11.5101	−0.460037
$627$	1.13580	0.0453595
$628$	20.5120	0.818519
$629$	46.1271	1.83921
$630$	9.91833	0.395156
$631$	−28.5339	−1.13592	−0.567958	−	0.823058i	$-0.692266\pi$
$631$	−28.5339	−1.13592	−0.567958	+	0.823058i	$0.692266\pi$
$632$	8.16942	0.324962
$633$	−1.31644	−0.0523236
$634$	4.14425	0.164589
$635$	−26.3234	−1.04461
$636$	0.917471	0.0363801
$637$	0	0
$638$	−11.1000	−0.439452
$639$	−35.5083	−1.40469
$640$	−4.36375	−0.172492
$641$	22.3729	0.883678	0.441839	−	0.897094i	$-0.354326\pi$
$641$	22.3729	0.883678	0.441839	+	0.897094i	$0.354326\pi$
$642$	3.20119	0.126341
$643$	−11.1412	−0.439365	−0.219682	−	0.975571i	$-0.570502\pi$
$643$	−11.1412	−0.439365	−0.219682	+	0.975571i	$0.570502\pi$
$644$	−3.16283	−0.124633
$645$	−9.46682	−0.372756
$646$	6.93885	0.273005
$647$	−9.46200	−0.371990	−0.185995	−	0.982551i	$-0.559551\pi$
$647$	−9.46200	−0.371990	−0.185995	+	0.982551i	$0.559551\pi$
$648$	8.39487	0.329782
$649$	54.2038	2.12768
$650$	0	0
$651$	1.29368	0.0507033
$652$	−2.11182	−0.0827053
$653$	9.39067	0.367485	0.183743	−	0.982974i	$-0.441179\pi$
$653$	9.39067	0.367485	0.183743	+	0.982974i	$0.441179\pi$
$654$	−2.12739	−0.0831875
$655$	−55.9745	−2.18710
$656$	1.20663	0.0471110
$657$	6.75446	0.263517
$658$	−1.05709	−0.0412097
$659$	−17.9190	−0.698027	−0.349013	−	0.937118i	$-0.613483\pi$
$659$	−17.9190	−0.698027	−0.349013	+	0.937118i	$0.613483\pi$
$660$	4.95635	0.192926
$661$	0.722654	0.0281080	0.0140540	−	0.999901i	$-0.495526\pi$
$661$	0.722654	0.0281080	0.0140540	+	0.999901i	$0.495526\pi$
$662$	−0.316380	−0.0122965
$663$	0	0
$664$	2.75361	0.106861
$665$	3.38249	0.131168
$666$	19.4926	0.755323
$667$	−10.3791	−0.401880
$668$	−20.3048	−0.785615
$669$	3.66475	0.141687
$670$	50.2917	1.94294
$671$	−8.86298	−0.342152
$672$	0.201753	0.00778278
$673$	40.2647	1.55209	0.776045	−	0.630677i	$-0.217223\pi$
$673$	40.2647	1.55209	0.776045	+	0.630677i	$0.217223\pi$
$674$	17.7103	0.682174
$675$	−21.6821	−0.834543
$676$	0	0
$677$	−19.1338	−0.735373	−0.367686	−	0.929950i	$-0.619850\pi$
$677$	−19.1338	−0.735373	−0.367686	+	0.929950i	$0.619850\pi$
$678$	1.05444	0.0404955
$679$	−1.65044	−0.0633380
$680$	30.2794	1.16116
$681$	−4.85529	−0.186055
$682$	−27.9813	−1.07146
$683$	34.3981	1.31621	0.658103	−	0.752928i	$-0.271359\pi$
$683$	34.3981	1.31621	0.658103	+	0.752928i	$0.271359\pi$
$684$	2.93225	0.112118
$685$	−58.9768	−2.25339
$686$	−10.3862	−0.396545
$687$	−1.47224	−0.0561695
$688$	8.33493	0.317766
$689$	0	0
$690$	4.63447	0.176431
$691$	−41.8318	−1.59136	−0.795678	−	0.605719i	$-0.792885\pi$
$691$	−41.8318	−1.59136	−0.795678	+	0.605719i	$0.792885\pi$
$692$	23.1299	0.879268
$693$	9.91833	0.376766
$694$	22.4138	0.850818
$695$	−9.48521	−0.359794
$696$	0.662069	0.0250957
$697$	−8.37263	−0.317136
$698$	20.0965	0.760665
$699$	−5.19023	−0.196312
$700$	10.8847	0.411403
$701$	9.76756	0.368916	0.184458	−	0.982840i	$-0.440947\pi$
$701$	9.76756	0.368916	0.184458	+	0.982840i	$0.440947\pi$
$702$	0	0
$703$	6.64765	0.250721
$704$	−4.36375	−0.164465
$705$	1.54895	0.0583368
$706$	11.7682	0.442903
$707$	8.92348	0.335602
$708$	−3.23304	−0.121505
$709$	−2.81576	−0.105748	−0.0528740	−	0.998601i	$-0.516838\pi$
$709$	−2.81576	−0.105748	−0.0528740	+	0.998601i	$0.516838\pi$
$710$	−52.8431	−1.98317
$711$	−23.9548	−0.898376
$712$	−3.34501	−0.125359
$713$	−26.1641	−0.979852
$714$	−1.39993	−0.0523911
$715$	0	0
$716$	−13.1550	−0.491626
$717$	0.894795	0.0334167
$718$	7.47763	0.279063
$719$	11.0765	0.413083	0.206542	−	0.978438i	$-0.433779\pi$
$719$	11.0765	0.413083	0.206542	+	0.978438i	$0.433779\pi$
$720$	12.7956	0.476865
$721$	6.65867	0.247982
$722$	1.00000	0.0372161
$723$	−1.41552	−0.0526439
$724$	−0.551157	−0.0204836
$725$	35.7191	1.32657
$726$	2.09326	0.0776882
$727$	0.407640	0.0151185	0.00755927	−	0.999971i	$-0.497594\pi$
$727$	0.407640	0.0151185	0.00755927	+	0.999971i	$0.497594\pi$
$728$	0	0
$729$	−23.4102	−0.867046
$730$	10.0519	0.372038
$731$	−57.8348	−2.13910
$732$	0.528643	0.0195392
$733$	−32.2976	−1.19294	−0.596470	−	0.802636i	$-0.703430\pi$
$733$	−32.2976	−1.19294	−0.596470	+	0.802636i	$0.703430\pi$
$734$	16.4233	0.606195
$735$	7.26818	0.268091
$736$	−4.08036	−0.150404
$737$	50.2917	1.85252
$738$	−3.53815	−0.130241
$739$	−3.62754	−0.133441	−0.0667206	−	0.997772i	$-0.521254\pi$
$739$	−3.62754	−0.133441	−0.0667206	+	0.997772i	$0.521254\pi$
$740$	29.0087	1.06638
$741$	0	0
$742$	2.73230	0.100306
$743$	−8.04553	−0.295162	−0.147581	−	0.989050i	$-0.547149\pi$
$743$	−8.04553	−0.295162	−0.147581	+	0.989050i	$0.547149\pi$
$744$	1.66897	0.0611875
$745$	45.4860	1.66648
$746$	−12.8349	−0.469920
$747$	−8.07429	−0.295423
$748$	30.2794	1.10712
$749$	9.53339	0.348342
$750$	−10.2703	−0.375017
$751$	18.8879	0.689230	0.344615	−	0.938744i	$-0.388009\pi$
$751$	18.8879	0.689230	0.344615	+	0.938744i	$0.388009\pi$
$752$	−1.36375	−0.0497309
$753$	−6.74808	−0.245914
$754$	0	0
$755$	−31.4152	−1.14332
$756$	−1.19685	−0.0435289
$757$	−21.6822	−0.788051	−0.394026	−	0.919099i	$-0.628918\pi$
$757$	−21.6822	−0.788051	−0.394026	+	0.919099i	$0.628918\pi$
$758$	36.0892	1.31082
$759$	4.63447	0.168221
$760$	4.36375	0.158290
$761$	1.28461	0.0465670	0.0232835	−	0.999729i	$-0.492588\pi$
$761$	1.28461	0.0465670	0.0232835	+	0.999729i	$0.492588\pi$
$762$	1.57009	0.0568783
$763$	−6.33552	−0.229361
$764$	−5.61275	−0.203062
$765$	−88.7869	−3.21010
$766$	32.8134	1.18560
$767$	0	0
$768$	0.260281	0.00939207
$769$	−11.4195	−0.411796	−0.205898	−	0.978573i	$-0.566012\pi$
$769$	−11.4195	−0.411796	−0.205898	+	0.978573i	$0.566012\pi$
$770$	14.7604	0.531927
$771$	−0.491771	−0.0177107
$772$	7.88071	0.283633
$773$	−35.6963	−1.28391	−0.641953	−	0.766744i	$-0.721876\pi$
$773$	−35.6963	−1.28391	−0.641953	+	0.766744i	$0.721876\pi$
$774$	−24.4401	−0.878482
$775$	90.0422	3.23441
$776$	−2.12923	−0.0764348
$777$	−1.34118	−0.0481146
$778$	−2.97899	−0.106802
$779$	−1.20663	−0.0432321
$780$	0	0
$781$	−52.8431	−1.89087
$782$	28.3130	1.01247
$783$	−3.92756	−0.140360
$784$	−6.39917	−0.228542
$785$	−89.5094	−3.19473
$786$	3.33866	0.119086
$787$	−47.2502	−1.68429	−0.842144	−	0.539253i	$-0.818707\pi$
$787$	−47.2502	−1.68429	−0.842144	+	0.539253i	$0.818707\pi$
$788$	−6.81454	−0.242758
$789$	−1.09070	−0.0388298
$790$	−35.6493	−1.26835
$791$	3.14019	0.111652
$792$	12.7956	0.454673
$793$	0	0
$794$	20.3722	0.722981
$795$	−4.00362	−0.141994
$796$	−24.2470	−0.859411
$797$	45.8247	1.62319	0.811597	−	0.584218i	$-0.198599\pi$
$797$	45.8247	1.62319	0.811597	+	0.584218i	$0.198599\pi$
$798$	−0.201753	−0.00714197
$799$	9.46286	0.334772
$800$	14.0423	0.496471
$801$	9.80841	0.346563
$802$	−13.8861	−0.490335
$803$	10.0519	0.354725
$804$	−2.99971	−0.105792
$805$	13.8018	0.486449
$806$	0	0
$807$	1.31777	0.0463877
$808$	11.5122	0.404996
$809$	16.5686	0.582522	0.291261	−	0.956644i	$-0.405925\pi$
$809$	16.5686	0.582522	0.291261	+	0.956644i	$0.405925\pi$
$810$	−36.6331	−1.28716
$811$	15.5277	0.545252	0.272626	−	0.962120i	$-0.412108\pi$
$811$	15.5277	0.545252	0.272626	+	0.962120i	$0.412108\pi$
$812$	1.97169	0.0691928
$813$	6.10705	0.214184
$814$	29.0087	1.01675
$815$	9.21546	0.322804
$816$	−1.80605	−0.0632244
$817$	−8.33493	−0.291602
$818$	31.1280	1.08836
$819$	0	0
$820$	−5.26544	−0.183877
$821$	−2.24388	−0.0783120	−0.0391560	−	0.999233i	$-0.512467\pi$
$821$	−2.24388	−0.0783120	−0.0391560	+	0.999233i	$0.512467\pi$
$822$	3.51774	0.122695
$823$	−16.5355	−0.576391	−0.288195	−	0.957572i	$-0.593055\pi$
$823$	−16.5355	−0.576391	−0.288195	+	0.957572i	$0.593055\pi$
$824$	8.59033	0.299258
$825$	−15.9493	−0.555282
$826$	−9.62824	−0.335009
$827$	−28.8538	−1.00334	−0.501672	−	0.865058i	$-0.667282\pi$
$827$	−28.8538	−1.00334	−0.501672	+	0.865058i	$0.667282\pi$
$828$	11.9646	0.415800
$829$	−16.8384	−0.584823	−0.292411	−	0.956293i	$-0.594458\pi$
$829$	−16.8384	−0.584823	−0.292411	+	0.956293i	$0.594458\pi$
$830$	−12.0161	−0.417084
$831$	1.83028	0.0634916
$832$	0	0
$833$	44.4028	1.53847
$834$	0.565756	0.0195905
$835$	88.6050	3.06630
$836$	4.36375	0.150923
$837$	−9.90077	−0.342221
$838$	−28.0907	−0.970376
$839$	34.0552	1.17572	0.587859	−	0.808964i	$-0.299971\pi$
$839$	34.0552	1.17572	0.587859	+	0.808964i	$0.299971\pi$
$840$	−0.880398	−0.0303766
$841$	−22.5297	−0.776887
$842$	10.8442	0.373714
$843$	5.33092	0.183607
$844$	−5.05776	−0.174095
$845$	0	0
$846$	3.99886	0.137484
$847$	6.23388	0.214199
$848$	3.52493	0.121047
$849$	−7.01052	−0.240601
$850$	−97.4375	−3.34208
$851$	27.1248	0.929826
$852$	3.15188	0.107982
$853$	5.48844	0.187921	0.0939603	−	0.995576i	$-0.470047\pi$
$853$	5.48844	0.187921	0.0939603	+	0.995576i	$0.470047\pi$
$854$	1.57434	0.0538726
$855$	−12.7956	−0.437601
$856$	12.2990	0.420371
$857$	−12.5021	−0.427064	−0.213532	−	0.976936i	$-0.568497\pi$
$857$	−12.5021	−0.427064	−0.213532	+	0.976936i	$0.568497\pi$
$858$	0	0
$859$	−37.7834	−1.28915	−0.644577	−	0.764540i	$-0.722966\pi$
$859$	−37.7834	−1.28915	−0.644577	+	0.764540i	$0.722966\pi$
$860$	−36.3716	−1.24026
$861$	0.243441	0.00829645
$862$	−6.90864	−0.235309
$863$	35.2458	1.19978	0.599891	−	0.800082i	$-0.295211\pi$
$863$	35.2458	1.19978	0.599891	+	0.800082i	$0.295211\pi$
$864$	−1.54405	−0.0525297
$865$	−100.933	−3.43183
$866$	39.3767	1.33807
$867$	8.10713	0.275333
$868$	4.97032	0.168704
$869$	−35.6493	−1.20932
$870$	−2.88911	−0.0979499
$871$	0	0
$872$	−8.17344	−0.276788
$873$	6.24343	0.211308
$874$	4.08036	0.138020
$875$	−30.5856	−1.03398
$876$	−0.599558	−0.0202572
$877$	−36.9518	−1.24777	−0.623887	−	0.781515i	$-0.714447\pi$
$877$	−36.9518	−1.24777	−0.623887	+	0.781515i	$0.714447\pi$
$878$	−36.9430	−1.24677
$879$	−2.52198	−0.0850643
$880$	19.0423	0.641916
$881$	1.25780	0.0423764	0.0211882	−	0.999776i	$-0.493255\pi$
$881$	1.25780	0.0423764	0.0211882	+	0.999776i	$0.493255\pi$
$882$	18.7640	0.631816
$883$	7.95258	0.267626	0.133813	−	0.991007i	$-0.457278\pi$
$883$	7.95258	0.267626	0.133813	+	0.991007i	$0.457278\pi$
$884$	0	0
$885$	14.1082	0.474242
$886$	10.0111	0.336330
$887$	−6.38803	−0.214489	−0.107245	−	0.994233i	$-0.534203\pi$
$887$	−6.38803	−0.214489	−0.107245	+	0.994233i	$0.534203\pi$
$888$	−1.73026	−0.0580636
$889$	4.67583	0.156822
$890$	14.5968	0.489285
$891$	−36.6331	−1.22726
$892$	14.0800	0.471432
$893$	1.36375	0.0456362
$894$	−2.71306	−0.0907385
$895$	57.4053	1.91885
$896$	0.775135	0.0258954
$897$	0	0
$898$	19.3184	0.644665
$899$	16.3106	0.543987
$900$	−41.1756	−1.37252
$901$	−24.4590	−0.814846
$902$	−5.26544	−0.175320
$903$	1.68159	0.0559600
$904$	4.05116	0.134740
$905$	2.40511	0.0799487
$906$	1.87380	0.0622527
$907$	−10.0113	−0.332420	−0.166210	−	0.986090i	$-0.553153\pi$
$907$	−10.0113	−0.332420	−0.166210	+	0.986090i	$0.553153\pi$
$908$	−18.6540	−0.619056
$909$	−33.7566	−1.11964
$910$	0	0
$911$	−31.2277	−1.03462	−0.517309	−	0.855798i	$-0.673066\pi$
$911$	−31.2277	−1.03462	−0.517309	+	0.855798i	$0.673066\pi$
$912$	−0.260281	−0.00861876
$913$	−12.0161	−0.397674
$914$	−8.71798	−0.288365
$915$	−2.30686	−0.0762626
$916$	−5.65636	−0.186891
$917$	9.94277	0.328339
$918$	10.7139	0.353613
$919$	−12.4072	−0.409277	−0.204638	−	0.978838i	$-0.565602\pi$
$919$	−12.4072	−0.409277	−0.204638	+	0.978838i	$0.565602\pi$
$920$	17.8057	0.587035
$921$	−3.27472	−0.107906
$922$	−4.77897	−0.157387
$923$	0	0
$924$	−0.880398	−0.0289630
$925$	−93.3485	−3.06928
$926$	−24.7123	−0.812097
$927$	−25.1890	−0.827316
$928$	2.54367	0.0835002
$929$	−51.4106	−1.68673	−0.843363	−	0.537344i	$-0.819428\pi$
$929$	−51.4106	−1.68673	−0.843363	+	0.537344i	$0.819428\pi$
$930$	−7.28298	−0.238818
$931$	6.39917	0.209724
$932$	−19.9409	−0.653185
$933$	−4.35729	−0.142651
$934$	−28.5590	−0.934479
$935$	−132.132	−4.32117
$936$	0	0
$937$	−2.03140	−0.0663628	−0.0331814	−	0.999449i	$-0.510564\pi$
$937$	−2.03140	−0.0663628	−0.0331814	+	0.999449i	$0.510564\pi$
$938$	−8.93334	−0.291684
$939$	−2.99586	−0.0977663
$940$	5.95107	0.194102
$941$	−42.4246	−1.38300	−0.691501	−	0.722376i	$-0.743050\pi$
$941$	−42.4246	−1.38300	−0.691501	+	0.722376i	$0.743050\pi$
$942$	5.33889	0.173950
$943$	−4.92349	−0.160331
$944$	−12.4214	−0.404281
$945$	5.22275	0.169896
$946$	−36.3716	−1.18254
$947$	−7.29296	−0.236989	−0.118495	−	0.992955i	$-0.537807\pi$
$947$	−7.29296	−0.236989	−0.118495	+	0.992955i	$0.537807\pi$
$948$	2.12634	0.0690604
$949$	0	0
$950$	−14.0423	−0.455593
$951$	1.07867	0.0349782
$952$	−5.37854	−0.174320
$953$	−32.7289	−1.06019	−0.530096	−	0.847938i	$-0.677844\pi$
$953$	−32.7289	−1.06019	−0.530096	+	0.847938i	$0.677844\pi$
$954$	−10.3360	−0.334640
$955$	24.4926	0.792563
$956$	3.43781	0.111187
$957$	−2.88911	−0.0933915
$958$	33.7512	1.09045
$959$	10.4761	0.338290
$960$	−1.13580	−0.0366578
$961$	10.1163	0.326334
$962$	0	0
$963$	−36.0638	−1.16214
$964$	−5.43845	−0.175161
$965$	−34.3894	−1.10704
$966$	−0.823223	−0.0264868
$967$	−2.05333	−0.0660307	−0.0330154	−	0.999455i	$-0.510511\pi$
$967$	−2.05333	−0.0660307	−0.0330154	+	0.999455i	$0.510511\pi$
$968$	8.04232	0.258490
$969$	1.80605	0.0580187
$970$	9.29141	0.298329
$971$	−41.6592	−1.33691	−0.668454	−	0.743754i	$-0.733044\pi$
$971$	−41.6592	−1.33691	−0.668454	+	0.743754i	$0.733044\pi$
$972$	6.81718	0.218661
$973$	1.68486	0.0540142
$974$	−5.50881	−0.176514
$975$	0	0
$976$	2.03105	0.0650122
$977$	56.4520	1.80606	0.903030	−	0.429576i	$-0.141337\pi$
$977$	56.4520	1.80606	0.903030	+	0.429576i	$0.141337\pi$
$978$	−0.549666	−0.0175764
$979$	14.5968	0.466515
$980$	27.9244	0.892011
$981$	23.9666	0.765195
$982$	22.4320	0.715835
$983$	32.1452	1.02527	0.512636	−	0.858606i	$-0.328669\pi$
$983$	32.1452	1.02527	0.512636	+	0.858606i	$0.328669\pi$
$984$	0.314063	0.0100120
$985$	29.7370	0.947498
$986$	−17.6502	−0.562096
$987$	−0.275140	−0.00875781
$988$	0	0
$989$	−34.0095	−1.08144
$990$	−55.8369	−1.77461
$991$	35.1925	1.11793	0.558964	−	0.829192i	$-0.311199\pi$
$991$	35.1925	1.11793	0.558964	+	0.829192i	$0.311199\pi$
$992$	6.41220	0.203588
$993$	−0.0823477	−0.00261323
$994$	9.38654	0.297723
$995$	105.808	3.35433
$996$	0.716713	0.0227099
$997$	−29.2648	−0.926827	−0.463413	−	0.886142i	$-0.653375\pi$
$997$	−29.2648	−0.926827	−0.463413	+	0.886142i	$0.653375\pi$
$998$	−2.39076	−0.0756781
$999$	10.2643	0.324749

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bd.1.3		6
13.3	even	3		494.2.g.f.191.4	✓	12
13.9	even	3		494.2.g.f.419.4	yes	12
13.12	even	2		6422.2.a.bc.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.g.f.191.4	✓	12	13.3	even	3
494.2.g.f.419.4	yes	12	13.9	even	3
6422.2.a.bc.1.3		6	13.12	even	2
6422.2.a.bd.1.3		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.260281 q^{3} +1.00000 q^{4} -4.36375 q^{5} +0.260281 q^{6} +0.775135 q^{7} +1.00000 q^{8} -2.93225 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.260281 q^{3} +1.00000 q^{4} -4.36375 q^{5} +0.260281 q^{6} +0.775135 q^{7} +1.00000 q^{8} -2.93225 q^{9} -4.36375 q^{10} -4.36375 q^{11} +0.260281 q^{12} +0.775135 q^{14} -1.13580 q^{15} +1.00000 q^{16} -6.93885 q^{17} -2.93225 q^{18} -1.00000 q^{19} -4.36375 q^{20} +0.201753 q^{21} -4.36375 q^{22} -4.08036 q^{23} +0.260281 q^{24} +14.0423 q^{25} -1.54405 q^{27} +0.775135 q^{28} +2.54367 q^{29} -1.13580 q^{30} +6.41220 q^{31} +1.00000 q^{32} -1.13580 q^{33} -6.93885 q^{34} -3.38249 q^{35} -2.93225 q^{36} -6.64765 q^{37} -1.00000 q^{38} -4.36375 q^{40} +1.20663 q^{41} +0.201753 q^{42} +8.33493 q^{43} -4.36375 q^{44} +12.7956 q^{45} -4.08036 q^{46} -1.36375 q^{47} +0.260281 q^{48} -6.39917 q^{49} +14.0423 q^{50} -1.80605 q^{51} +3.52493 q^{53} -1.54405 q^{54} +19.0423 q^{55} +0.775135 q^{56} -0.260281 q^{57} +2.54367 q^{58} -12.4214 q^{59} -1.13580 q^{60} +2.03105 q^{61} +6.41220 q^{62} -2.27289 q^{63} +1.00000 q^{64} -1.13580 q^{66} -11.5249 q^{67} -6.93885 q^{68} -1.06204 q^{69} -3.38249 q^{70} +12.1096 q^{71} -2.93225 q^{72} -2.30351 q^{73} -6.64765 q^{74} +3.65495 q^{75} -1.00000 q^{76} -3.38249 q^{77} +8.16942 q^{79} -4.36375 q^{80} +8.39487 q^{81} +1.20663 q^{82} +2.75361 q^{83} +0.201753 q^{84} +30.2794 q^{85} +8.33493 q^{86} +0.662069 q^{87} -4.36375 q^{88} -3.34501 q^{89} +12.7956 q^{90} -4.08036 q^{92} +1.66897 q^{93} -1.36375 q^{94} +4.36375 q^{95} +0.260281 q^{96} -2.12923 q^{97} -6.39917 q^{98} +12.7956 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 6 q^{8} + 10 q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 + 2 * q^3 + 6 * q^4 + 2 * q^5 + 2 * q^6 + q^7 + 6 * q^8 + 10 * q^9	\( 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 6 q^{8} + 10 q^{9} + 2 q^{10} + 2 q^{11} + 2 q^{12} + q^{14} - 9 q^{15} + 6 q^{16} - 2 q^{17} + 10 q^{18} - 6 q^{19} + 2 q^{20} + q^{21} + 2 q^{22} + 8 q^{23} + 2 q^{24} + 16 q^{25} - 4 q^{27} + q^{28} + 20 q^{29} - 9 q^{30} + 3 q^{31} + 6 q^{32} - 9 q^{33} - 2 q^{34} + 7 q^{35} + 10 q^{36} - 9 q^{37} - 6 q^{38} + 2 q^{40} + 3 q^{41} + q^{42} + 13 q^{43} + 2 q^{44} + 38 q^{45} + 8 q^{46} + 20 q^{47} + 2 q^{48} - 7 q^{49} + 16 q^{50} + 2 q^{51} + 25 q^{53} - 4 q^{54} + 46 q^{55} + q^{56} - 2 q^{57} + 20 q^{58} - 9 q^{60} + 6 q^{61} + 3 q^{62} + 46 q^{63} + 6 q^{64} - 9 q^{66} - 32 q^{67} - 2 q^{68} - 29 q^{69} + 7 q^{70} + 39 q^{71} + 10 q^{72} + 7 q^{73} - 9 q^{74} - 15 q^{75} - 6 q^{76} + 7 q^{77} + 18 q^{79} + 2 q^{80} + 54 q^{81} + 3 q^{82} - 7 q^{83} + q^{84} - 2 q^{85} + 13 q^{86} + 12 q^{87} + 2 q^{88} + 9 q^{89} + 38 q^{90} + 8 q^{92} - 47 q^{93} + 20 q^{94} - 2 q^{95} + 2 q^{96} - 4 q^{97} - 7 q^{98} + 38 q^{99}+O(q^{100}) \) 6 * q + 6 * q^2 + 2 * q^3 + 6 * q^4 + 2 * q^5 + 2 * q^6 + q^7 + 6 * q^8 + 10 * q^9 + 2 * q^10 + 2 * q^11 + 2 * q^12 + q^14 - 9 * q^15 + 6 * q^16 - 2 * q^17 + 10 * q^18 - 6 * q^19 + 2 * q^20 + q^21 + 2 * q^22 + 8 * q^23 + 2 * q^24 + 16 * q^25 - 4 * q^27 + q^28 + 20 * q^29 - 9 * q^30 + 3 * q^31 + 6 * q^32 - 9 * q^33 - 2 * q^34 + 7 * q^35 + 10 * q^36 - 9 * q^37 - 6 * q^38 + 2 * q^40 + 3 * q^41 + q^42 + 13 * q^43 + 2 * q^44 + 38 * q^45 + 8 * q^46 + 20 * q^47 + 2 * q^48 - 7 * q^49 + 16 * q^50 + 2 * q^51 + 25 * q^53 - 4 * q^54 + 46 * q^55 + q^56 - 2 * q^57 + 20 * q^58 - 9 * q^60 + 6 * q^61 + 3 * q^62 + 46 * q^63 + 6 * q^64 - 9 * q^66 - 32 * q^67 - 2 * q^68 - 29 * q^69 + 7 * q^70 + 39 * q^71 + 10 * q^72 + 7 * q^73 - 9 * q^74 - 15 * q^75 - 6 * q^76 + 7 * q^77 + 18 * q^79 + 2 * q^80 + 54 * q^81 + 3 * q^82 - 7 * q^83 + q^84 - 2 * q^85 + 13 * q^86 + 12 * q^87 + 2 * q^88 + 9 * q^89 + 38 * q^90 + 8 * q^92 - 47 * q^93 + 20 * q^94 - 2 * q^95 + 2 * q^96 - 4 * q^97 - 7 * q^98 + 38 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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