LMFDB - Embedded newform 6422.2.a.bk.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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 12x^{7} + 14x^{6} + 54x^{5} - 11x^{4} - 84x^{3} - 48x^{2} - 4x + 1 $ x^9 - 2x^8 - 12x^7 + 14x^6 + 54x^5 - 11x^4 - 84x^3 - 48x^2 - 4x + 1
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			1.3
Root			$-1.89727$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -2.42044 q^{3} +1.00000 q^{4} +4.26312 q^{5} -2.42044 q^{6} -3.45222 q^{7} +1.00000 q^{8} +2.85851 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.42044 q^{3} +1.00000 q^{4} +4.26312 q^{5} -2.42044 q^{6} -3.45222 q^{7} +1.00000 q^{8} +2.85851 q^{9} +4.26312 q^{10} -1.33908 q^{11} -2.42044 q^{12} -3.45222 q^{14} -10.3186 q^{15} +1.00000 q^{16} -7.65615 q^{17} +2.85851 q^{18} +1.00000 q^{19} +4.26312 q^{20} +8.35589 q^{21} -1.33908 q^{22} -5.48074 q^{23} -2.42044 q^{24} +13.1742 q^{25} +0.342457 q^{27} -3.45222 q^{28} +10.4707 q^{29} -10.3186 q^{30} +3.81990 q^{31} +1.00000 q^{32} +3.24115 q^{33} -7.65615 q^{34} -14.7172 q^{35} +2.85851 q^{36} -0.372222 q^{37} +1.00000 q^{38} +4.26312 q^{40} +1.97487 q^{41} +8.35589 q^{42} +7.12837 q^{43} -1.33908 q^{44} +12.1862 q^{45} -5.48074 q^{46} -9.88338 q^{47} -2.42044 q^{48} +4.91785 q^{49} +13.1742 q^{50} +18.5312 q^{51} -6.96187 q^{53} +0.342457 q^{54} -5.70864 q^{55} -3.45222 q^{56} -2.42044 q^{57} +10.4707 q^{58} +4.79021 q^{59} -10.3186 q^{60} -12.8877 q^{61} +3.81990 q^{62} -9.86823 q^{63} +1.00000 q^{64} +3.24115 q^{66} -11.9728 q^{67} -7.65615 q^{68} +13.2658 q^{69} -14.7172 q^{70} -3.75267 q^{71} +2.85851 q^{72} -0.285907 q^{73} -0.372222 q^{74} -31.8873 q^{75} +1.00000 q^{76} +4.62279 q^{77} -16.7293 q^{79} +4.26312 q^{80} -9.40444 q^{81} +1.97487 q^{82} -5.62619 q^{83} +8.35589 q^{84} -32.6391 q^{85} +7.12837 q^{86} -25.3436 q^{87} -1.33908 q^{88} +3.60658 q^{89} +12.1862 q^{90} -5.48074 q^{92} -9.24582 q^{93} -9.88338 q^{94} +4.26312 q^{95} -2.42044 q^{96} +1.75855 q^{97} +4.91785 q^{98} -3.82777 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} - 5 q^{3} + 9 q^{4} + q^{5} - 5 q^{6} - 13 q^{7} + 9 q^{8} + 10 q^{9}+O(q^{10}) $ 9 * q + 9 * q^2 - 5 * q^3 + 9 * q^4 + q^5 - 5 * q^6 - 13 * q^7 + 9 * q^8 + 10 * q^9	\( 9 q + 9 q^{2} - 5 q^{3} + 9 q^{4} + q^{5} - 5 q^{6} - 13 q^{7} + 9 q^{8} + 10 q^{9} + q^{10} - 13 q^{11} - 5 q^{12} - 13 q^{14} - q^{15} + 9 q^{16} - 12 q^{17} + 10 q^{18} + 9 q^{19} + q^{20} + 18 q^{21} - 13 q^{22} - 22 q^{23} - 5 q^{24} + 4 q^{25} - 26 q^{27} - 13 q^{28} + 12 q^{29} - q^{30} + q^{31} + 9 q^{32} - 28 q^{33} - 12 q^{34} - 18 q^{35} + 10 q^{36} - 25 q^{37} + 9 q^{38} + q^{40} + 11 q^{41} + 18 q^{42} - 10 q^{43} - 13 q^{44} - q^{45} - 22 q^{46} - 12 q^{47} - 5 q^{48} + 2 q^{49} + 4 q^{50} + 35 q^{51} + 9 q^{53} - 26 q^{54} + 18 q^{55} - 13 q^{56} - 5 q^{57} + 12 q^{58} + 10 q^{59} - q^{60} + 32 q^{61} + q^{62} - 63 q^{63} + 9 q^{64} - 28 q^{66} - 73 q^{67} - 12 q^{68} + 2 q^{69} - 18 q^{70} - 51 q^{71} + 10 q^{72} - 14 q^{73} - 25 q^{74} - 49 q^{75} + 9 q^{76} + 18 q^{77} - 28 q^{79} + q^{80} + 29 q^{81} + 11 q^{82} - 22 q^{83} + 18 q^{84} - 51 q^{85} - 10 q^{86} - 20 q^{87} - 13 q^{88} + 3 q^{89} - q^{90} - 22 q^{92} - 59 q^{93} - 12 q^{94} + q^{95} - 5 q^{96} + 2 q^{98} + 25 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 - 5 * q^3 + 9 * q^4 + q^5 - 5 * q^6 - 13 * q^7 + 9 * q^8 + 10 * q^9 + q^10 - 13 * q^11 - 5 * q^12 - 13 * q^14 - q^15 + 9 * q^16 - 12 * q^17 + 10 * q^18 + 9 * q^19 + q^20 + 18 * q^21 - 13 * q^22 - 22 * q^23 - 5 * q^24 + 4 * q^25 - 26 * q^27 - 13 * q^28 + 12 * q^29 - q^30 + q^31 + 9 * q^32 - 28 * q^33 - 12 * q^34 - 18 * q^35 + 10 * q^36 - 25 * q^37 + 9 * q^38 + q^40 + 11 * q^41 + 18 * q^42 - 10 * q^43 - 13 * q^44 - q^45 - 22 * q^46 - 12 * q^47 - 5 * q^48 + 2 * q^49 + 4 * q^50 + 35 * q^51 + 9 * q^53 - 26 * q^54 + 18 * q^55 - 13 * q^56 - 5 * q^57 + 12 * q^58 + 10 * q^59 - q^60 + 32 * q^61 + q^62 - 63 * q^63 + 9 * q^64 - 28 * q^66 - 73 * q^67 - 12 * q^68 + 2 * q^69 - 18 * q^70 - 51 * q^71 + 10 * q^72 - 14 * q^73 - 25 * q^74 - 49 * q^75 + 9 * q^76 + 18 * q^77 - 28 * q^79 + q^80 + 29 * q^81 + 11 * q^82 - 22 * q^83 + 18 * q^84 - 51 * q^85 - 10 * q^86 - 20 * q^87 - 13 * q^88 + 3 * q^89 - q^90 - 22 * q^92 - 59 * q^93 - 12 * q^94 + q^95 - 5 * q^96 + 2 * q^98 + 25 * 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
$3$	−2.42044	−1.39744	−0.698720	−	0.715395i	$-0.746247\pi$
$3$	−2.42044	−1.39744	−0.698720	+	0.715395i	$0.746247\pi$
$4$	1.00000	0.500000
$5$	4.26312	1.90652	0.953262	−	0.302144i	$-0.0977025\pi$
$5$	4.26312	1.90652	0.953262	+	0.302144i	$0.0977025\pi$
$6$	−2.42044	−0.988139
$7$	−3.45222	−1.30482	−0.652409	−	0.757867i	$-0.726242\pi$
$7$	−3.45222	−1.30482	−0.652409	+	0.757867i	$0.726242\pi$
$8$	1.00000	0.353553
$9$	2.85851	0.952838
$10$	4.26312	1.34812
$11$	−1.33908	−0.403747	−0.201873	−	0.979412i	$-0.564703\pi$
$11$	−1.33908	−0.403747	−0.201873	+	0.979412i	$0.564703\pi$
$12$	−2.42044	−0.698720
$13$	0	0
$13$	0	0
$14$	−3.45222	−0.922646
$15$	−10.3186	−2.66425
$16$	1.00000	0.250000
$17$	−7.65615	−1.85689	−0.928445	−	0.371471i	$-0.878854\pi$
$17$	−7.65615	−1.85689	−0.928445	+	0.371471i	$0.878854\pi$
$18$	2.85851	0.673758
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	4.26312	0.953262
$21$	8.35589	1.82340
$22$	−1.33908	−0.285492
$23$	−5.48074	−1.14281	−0.571407	−	0.820667i	$-0.693602\pi$
$23$	−5.48074	−1.14281	−0.571407	+	0.820667i	$0.693602\pi$
$24$	−2.42044	−0.494070
$25$	13.1742	2.63483
$26$	0	0
$27$	0.342457	0.0659060
$28$	−3.45222	−0.652409
$29$	10.4707	1.94435	0.972177	−	0.234248i	$-0.0752629\pi$
$29$	10.4707	1.94435	0.972177	+	0.234248i	$0.0752629\pi$
$30$	−10.3186	−1.88391
$31$	3.81990	0.686074	0.343037	−	0.939322i	$-0.388544\pi$
$31$	3.81990	0.686074	0.343037	+	0.939322i	$0.388544\pi$
$32$	1.00000	0.176777
$33$	3.24115	0.564212
$34$	−7.65615	−1.31302
$35$	−14.7172	−2.48767
$36$	2.85851	0.476419
$37$	−0.372222	−0.0611929	−0.0305964	−	0.999532i	$-0.509741\pi$
$37$	−0.372222	−0.0611929	−0.0305964	+	0.999532i	$0.509741\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	4.26312	0.674058
$41$	1.97487	0.308422	0.154211	−	0.988038i	$-0.450716\pi$
$41$	1.97487	0.308422	0.154211	+	0.988038i	$0.450716\pi$
$42$	8.35589	1.28934
$43$	7.12837	1.08707	0.543533	−	0.839388i	$-0.317086\pi$
$43$	7.12837	1.08707	0.543533	+	0.839388i	$0.317086\pi$
$44$	−1.33908	−0.201873
$45$	12.1862	1.81661
$46$	−5.48074	−0.808091
$47$	−9.88338	−1.44164	−0.720820	−	0.693123i	$-0.756234\pi$
$47$	−9.88338	−1.44164	−0.720820	+	0.693123i	$0.756234\pi$
$48$	−2.42044	−0.349360
$49$	4.91785	0.702550
$50$	13.1742	1.86311
$51$	18.5312	2.59489
$52$	0	0
$53$	−6.96187	−0.956286	−0.478143	−	0.878282i	$-0.658690\pi$
$53$	−6.96187	−0.956286	−0.478143	+	0.878282i	$0.658690\pi$
$54$	0.342457	0.0466026
$55$	−5.70864	−0.769753
$56$	−3.45222	−0.461323
$57$	−2.42044	−0.320595
$58$	10.4707	1.37487
$59$	4.79021	0.623632	0.311816	−	0.950143i	$-0.399063\pi$
$59$	4.79021	0.623632	0.311816	+	0.950143i	$0.399063\pi$
$60$	−10.3186	−1.33213
$61$	−12.8877	−1.65010	−0.825050	−	0.565060i	$-0.808853\pi$
$61$	−12.8877	−1.65010	−0.825050	+	0.565060i	$0.808853\pi$
$62$	3.81990	0.485127
$63$	−9.86823	−1.24328
$64$	1.00000	0.125000
$65$	0	0
$66$	3.24115	0.398958
$67$	−11.9728	−1.46271	−0.731354	−	0.681998i	$-0.761111\pi$
$67$	−11.9728	−1.46271	−0.731354	+	0.681998i	$0.761111\pi$
$68$	−7.65615	−0.928445
$69$	13.2658	1.59701
$70$	−14.7172	−1.75905
$71$	−3.75267	−0.445360	−0.222680	−	0.974892i	$-0.571481\pi$
$71$	−3.75267	−0.445360	−0.222680	+	0.974892i	$0.571481\pi$
$72$	2.85851	0.336879
$73$	−0.285907	−0.0334629	−0.0167314	−	0.999860i	$-0.505326\pi$
$73$	−0.285907	−0.0334629	−0.0167314	+	0.999860i	$0.505326\pi$
$74$	−0.372222	−0.0432699
$75$	−31.8873	−3.68202
$76$	1.00000	0.114708
$77$	4.62279	0.526816
$78$	0	0
$79$	−16.7293	−1.88219	−0.941095	−	0.338142i	$-0.890201\pi$
$79$	−16.7293	−1.88219	−0.941095	+	0.338142i	$0.890201\pi$
$80$	4.26312	0.476631
$81$	−9.40444	−1.04494
$82$	1.97487	0.218088
$83$	−5.62619	−0.617555	−0.308777	−	0.951134i	$-0.599920\pi$
$83$	−5.62619	−0.617555	−0.308777	+	0.951134i	$0.599920\pi$
$84$	8.35589	0.911702
$85$	−32.6391	−3.54020
$86$	7.12837	0.768672
$87$	−25.3436	−2.71712
$88$	−1.33908	−0.142746
$89$	3.60658	0.382297	0.191148	−	0.981561i	$-0.438779\pi$
$89$	3.60658	0.382297	0.191148	+	0.981561i	$0.438779\pi$
$90$	12.1862	1.28454
$91$	0	0
$92$	−5.48074	−0.571407
$93$	−9.24582	−0.958746
$94$	−9.88338	−1.01939
$95$	4.26312	0.437387
$96$	−2.42044	−0.247035
$97$	1.75855	0.178554	0.0892770	−	0.996007i	$-0.471544\pi$
$97$	1.75855	0.178554	0.0892770	+	0.996007i	$0.471544\pi$
$98$	4.91785	0.496778
$99$	−3.82777	−0.384705
$100$	13.1742	1.31742
$101$	−0.369761	−0.0367926	−0.0183963	−	0.999831i	$-0.505856\pi$
$101$	−0.369761	−0.0367926	−0.0183963	+	0.999831i	$0.505856\pi$
$102$	18.5312	1.83487
$103$	5.90899	0.582230	0.291115	−	0.956688i	$-0.405974\pi$
$103$	5.90899	0.582230	0.291115	+	0.956688i	$0.405974\pi$
$104$	0	0
$105$	35.6221	3.47637
$106$	−6.96187	−0.676196
$107$	−4.96404	−0.479892	−0.239946	−	0.970786i	$-0.577130\pi$
$107$	−4.96404	−0.479892	−0.239946	+	0.970786i	$0.577130\pi$
$108$	0.342457	0.0329530
$109$	2.84555	0.272555	0.136277	−	0.990671i	$-0.456486\pi$
$109$	2.84555	0.272555	0.136277	+	0.990671i	$0.456486\pi$
$110$	−5.70864	−0.544297
$111$	0.900939	0.0855134
$112$	−3.45222	−0.326205
$113$	−9.64635	−0.907452	−0.453726	−	0.891141i	$-0.649905\pi$
$113$	−9.64635	−0.907452	−0.453726	+	0.891141i	$0.649905\pi$
$114$	−2.42044	−0.226695
$115$	−23.3651	−2.17880
$116$	10.4707	0.972177
$117$	0	0
$118$	4.79021	0.440974
$119$	26.4308	2.42290
$120$	−10.3186	−0.941956
$121$	−9.20688	−0.836989
$122$	−12.8877	−1.16680
$123$	−4.78004	−0.431002
$124$	3.81990	0.343037
$125$	34.8475	3.11685
$126$	−9.86823	−0.879132
$127$	2.62078	0.232557	0.116278	−	0.993217i	$-0.462904\pi$
$127$	2.62078	0.232557	0.116278	+	0.993217i	$0.462904\pi$
$128$	1.00000	0.0883883
$129$	−17.2538	−1.51911
$130$	0	0
$131$	−2.48227	−0.216877	−0.108439	−	0.994103i	$-0.534585\pi$
$131$	−2.48227	−0.216877	−0.108439	+	0.994103i	$0.534585\pi$
$132$	3.24115	0.282106
$133$	−3.45222	−0.299346
$134$	−11.9728	−1.03429
$135$	1.45994	0.125651
$136$	−7.65615	−0.656510
$137$	11.5126	0.983588	0.491794	−	0.870712i	$-0.336341\pi$
$137$	11.5126	0.983588	0.491794	+	0.870712i	$0.336341\pi$
$138$	13.2658	1.12926
$139$	−3.58058	−0.303701	−0.151850	−	0.988404i	$-0.548523\pi$
$139$	−3.58058	−0.303701	−0.151850	+	0.988404i	$0.548523\pi$
$140$	−14.7172	−1.24383
$141$	23.9221	2.01460
$142$	−3.75267	−0.314917
$143$	0	0
$144$	2.85851	0.238210
$145$	44.6377	3.70696
$146$	−0.285907	−0.0236618
$147$	−11.9033	−0.981772
$148$	−0.372222	−0.0305964
$149$	12.1274	0.993516	0.496758	−	0.867889i	$-0.334524\pi$
$149$	12.1274	0.993516	0.496758	+	0.867889i	$0.334524\pi$
$150$	−31.8873	−2.60358
$151$	−4.64190	−0.377753	−0.188876	−	0.982001i	$-0.560485\pi$
$151$	−4.64190	−0.377753	−0.188876	+	0.982001i	$0.560485\pi$
$152$	1.00000	0.0811107
$153$	−21.8852	−1.76931
$154$	4.62279	0.372515
$155$	16.2847	1.30802
$156$	0	0
$157$	−5.82934	−0.465232	−0.232616	−	0.972569i	$-0.574729\pi$
$157$	−5.82934	−0.465232	−0.232616	+	0.972569i	$0.574729\pi$
$158$	−16.7293	−1.33091
$159$	16.8508	1.33635
$160$	4.26312	0.337029
$161$	18.9208	1.49116
$162$	−9.40444	−0.738883
$163$	−16.6082	−1.30086	−0.650428	−	0.759568i	$-0.725410\pi$
$163$	−16.6082	−1.30086	−0.650428	+	0.759568i	$0.725410\pi$
$164$	1.97487	0.154211
$165$	13.8174	1.07568
$166$	−5.62619	−0.436677
$167$	0.642355	0.0497070	0.0248535	−	0.999691i	$-0.492088\pi$
$167$	0.642355	0.0497070	0.0248535	+	0.999691i	$0.492088\pi$
$168$	8.35589	0.644671
$169$	0	0
$170$	−32.6391	−2.50330
$171$	2.85851	0.218596
$172$	7.12837	0.543533
$173$	−2.90373	−0.220766	−0.110383	−	0.993889i	$-0.535208\pi$
$173$	−2.90373	−0.220766	−0.110383	+	0.993889i	$0.535208\pi$
$174$	−25.3436	−1.92129
$175$	−45.4802	−3.43798
$176$	−1.33908	−0.100937
$177$	−11.5944	−0.871488
$178$	3.60658	0.270325
$179$	−9.75377	−0.729031	−0.364516	−	0.931197i	$-0.618765\pi$
$179$	−9.75377	−0.729031	−0.364516	+	0.931197i	$0.618765\pi$
$180$	12.1862	0.908304
$181$	−19.6557	−1.46100	−0.730500	−	0.682913i	$-0.760713\pi$
$181$	−19.6557	−1.46100	−0.730500	+	0.682913i	$0.760713\pi$
$182$	0	0
$183$	31.1938	2.30591
$184$	−5.48074	−0.404046
$185$	−1.58683	−0.116666
$186$	−9.24582	−0.677936
$187$	10.2522	0.749713
$188$	−9.88338	−0.720820
$189$	−1.18224	−0.0859953
$190$	4.26312	0.309279
$191$	−6.78083	−0.490644	−0.245322	−	0.969442i	$-0.578894\pi$
$191$	−6.78083	−0.490644	−0.245322	+	0.969442i	$0.578894\pi$
$192$	−2.42044	−0.174680
$193$	1.61415	0.116189	0.0580946	−	0.998311i	$-0.481497\pi$
$193$	1.61415	0.116189	0.0580946	+	0.998311i	$0.481497\pi$
$194$	1.75855	0.126257
$195$	0	0
$196$	4.91785	0.351275
$197$	−13.6604	−0.973264	−0.486632	−	0.873607i	$-0.661775\pi$
$197$	−13.6604	−0.973264	−0.486632	+	0.873607i	$0.661775\pi$
$198$	−3.82777	−0.272028
$199$	10.2721	0.728173	0.364087	−	0.931365i	$-0.381381\pi$
$199$	10.2721	0.728173	0.364087	+	0.931365i	$0.381381\pi$
$200$	13.1742	0.931555
$201$	28.9794	2.04405
$202$	−0.369761	−0.0260163
$203$	−36.1471	−2.53703
$204$	18.5312	1.29745
$205$	8.41909	0.588015
$206$	5.90899	0.411699
$207$	−15.6668	−1.08892
$208$	0	0
$209$	−1.33908	−0.0926258
$210$	35.6221	2.45816
$211$	−23.1474	−1.59353	−0.796767	−	0.604287i	$-0.793458\pi$
$211$	−23.1474	−1.59353	−0.796767	+	0.604287i	$0.793458\pi$
$212$	−6.96187	−0.478143
$213$	9.08311	0.622364
$214$	−4.96404	−0.339335
$215$	30.3891	2.07252
$216$	0.342457	0.0233013
$217$	−13.1871	−0.895201
$218$	2.84555	0.192725
$219$	0.692020	0.0467624
$220$	−5.70864	−0.384876
$221$	0	0
$222$	0.900939	0.0604671
$223$	−25.0578	−1.67799	−0.838996	−	0.544138i	$-0.816857\pi$
$223$	−25.0578	−1.67799	−0.838996	+	0.544138i	$0.816857\pi$
$224$	−3.45222	−0.230661
$225$	37.6586	2.51057
$226$	−9.64635	−0.641665
$227$	−4.19334	−0.278322	−0.139161	−	0.990270i	$-0.544440\pi$
$227$	−4.19334	−0.278322	−0.139161	+	0.990270i	$0.544440\pi$
$228$	−2.42044	−0.160297
$229$	−12.8108	−0.846560	−0.423280	−	0.905999i	$-0.639121\pi$
$229$	−12.8108	−0.846560	−0.423280	+	0.905999i	$0.639121\pi$
$230$	−23.3651	−1.54065
$231$	−11.1892	−0.736193
$232$	10.4707	0.687433
$233$	−7.83899	−0.513550	−0.256775	−	0.966471i	$-0.582660\pi$
$233$	−7.83899	−0.513550	−0.256775	+	0.966471i	$0.582660\pi$
$234$	0	0
$235$	−42.1340	−2.74852
$236$	4.79021	0.311816
$237$	40.4921	2.63025
$238$	26.4308	1.71325
$239$	9.37460	0.606392	0.303196	−	0.952928i	$-0.401946\pi$
$239$	9.37460	0.606392	0.303196	+	0.952928i	$0.401946\pi$
$240$	−10.3186	−0.666063
$241$	13.7745	0.887293	0.443647	−	0.896202i	$-0.353685\pi$
$241$	13.7745	0.887293	0.443647	+	0.896202i	$0.353685\pi$
$242$	−9.20688	−0.591840
$243$	21.7355	1.39433
$244$	−12.8877	−0.825050
$245$	20.9654	1.33943
$246$	−4.78004	−0.304764
$247$	0	0
$248$	3.81990	0.242564
$249$	13.6178	0.862995
$250$	34.8475	2.20395
$251$	−3.87734	−0.244736	−0.122368	−	0.992485i	$-0.539049\pi$
$251$	−3.87734	−0.244736	−0.122368	+	0.992485i	$0.539049\pi$
$252$	−9.86823	−0.621640
$253$	7.33913	0.461407
$254$	2.62078	0.164442
$255$	79.0008	4.94722
$256$	1.00000	0.0625000
$257$	−29.2867	−1.82686	−0.913428	−	0.407000i	$-0.866575\pi$
$257$	−29.2867	−1.82686	−0.913428	+	0.407000i	$0.866575\pi$
$258$	−17.2538	−1.07417
$259$	1.28499	0.0798456
$260$	0	0
$261$	29.9305	1.85265
$262$	−2.48227	−0.153355
$263$	−1.25741	−0.0775353	−0.0387676	−	0.999248i	$-0.512343\pi$
$263$	−1.25741	−0.0775353	−0.0387676	+	0.999248i	$0.512343\pi$
$264$	3.24115	0.199479
$265$	−29.6793	−1.82318
$266$	−3.45222	−0.211669
$267$	−8.72950	−0.534237
$268$	−11.9728	−0.731354
$269$	14.9946	0.914236	0.457118	−	0.889406i	$-0.348882\pi$
$269$	14.9946	0.914236	0.457118	+	0.889406i	$0.348882\pi$
$270$	1.45994	0.0888489
$271$	11.4029	0.692677	0.346339	−	0.938110i	$-0.387425\pi$
$271$	11.4029	0.692677	0.346339	+	0.938110i	$0.387425\pi$
$272$	−7.65615	−0.464222
$273$	0	0
$274$	11.5126	0.695502
$275$	−17.6412	−1.06381
$276$	13.2658	0.798507
$277$	14.0472	0.844013	0.422007	−	0.906593i	$-0.361326\pi$
$277$	14.0472	0.844013	0.422007	+	0.906593i	$0.361326\pi$
$278$	−3.58058	−0.214749
$279$	10.9192	0.653717
$280$	−14.7172	−0.879523
$281$	0.227399	0.0135655	0.00678274	−	0.999977i	$-0.497841\pi$
$281$	0.227399	0.0135655	0.00678274	+	0.999977i	$0.497841\pi$
$282$	23.9221	1.42454
$283$	32.7518	1.94689	0.973446	−	0.228916i	$-0.0735181\pi$
$283$	32.7518	1.94689	0.973446	+	0.228916i	$0.0735181\pi$
$284$	−3.75267	−0.222680
$285$	−10.3186	−0.611222
$286$	0	0
$287$	−6.81768	−0.402435
$288$	2.85851	0.168440
$289$	41.6167	2.44804
$290$	44.6377	2.62121
$291$	−4.25647	−0.249519
$292$	−0.285907	−0.0167314
$293$	−5.32509	−0.311095	−0.155547	−	0.987828i	$-0.549714\pi$
$293$	−5.32509	−0.311095	−0.155547	+	0.987828i	$0.549714\pi$
$294$	−11.9033	−0.694217
$295$	20.4212	1.18897
$296$	−0.372222	−0.0216350
$297$	−0.458577	−0.0266093
$298$	12.1274	0.702522
$299$	0	0
$300$	−31.8873	−1.84101
$301$	−24.6087	−1.41842
$302$	−4.64190	−0.267111
$303$	0.894983	0.0514155
$304$	1.00000	0.0573539
$305$	−54.9417	−3.14595
$306$	−21.8852	−1.25109
$307$	−21.0743	−1.20277	−0.601386	−	0.798958i	$-0.705385\pi$
$307$	−21.0743	−1.20277	−0.601386	+	0.798958i	$0.705385\pi$
$308$	4.62279	0.263408
$309$	−14.3023	−0.813632
$310$	16.2847	0.924907
$311$	18.1377	1.02849	0.514247	−	0.857642i	$-0.328072\pi$
$311$	18.1377	1.02849	0.514247	+	0.857642i	$0.328072\pi$
$312$	0	0
$313$	−2.89367	−0.163560	−0.0817801	−	0.996650i	$-0.526061\pi$
$313$	−2.89367	−0.163560	−0.0817801	+	0.996650i	$0.526061\pi$
$314$	−5.82934	−0.328969
$315$	−42.0694	−2.37034
$316$	−16.7293	−0.941095
$317$	−29.2571	−1.64324	−0.821620	−	0.570035i	$-0.806930\pi$
$317$	−29.2571	−1.64324	−0.821620	+	0.570035i	$0.806930\pi$
$318$	16.8508	0.944944
$319$	−14.0210	−0.785026
$320$	4.26312	0.238316
$321$	12.0151	0.670620
$322$	18.9208	1.05441
$323$	−7.65615	−0.426000
$324$	−9.40444	−0.522469
$325$	0	0
$326$	−16.6082	−0.919843
$327$	−6.88748	−0.380879
$328$	1.97487	0.109044
$329$	34.1196	1.88108
$330$	13.8174	0.760623
$331$	−15.3836	−0.845561	−0.422780	−	0.906232i	$-0.638946\pi$
$331$	−15.3836	−0.845561	−0.422780	+	0.906232i	$0.638946\pi$
$332$	−5.62619	−0.308777
$333$	−1.06400	−0.0583069
$334$	0.642355	0.0351481
$335$	−51.0414	−2.78869
$336$	8.35589	0.455851
$337$	29.0192	1.58077	0.790387	−	0.612608i	$-0.209880\pi$
$337$	29.0192	1.58077	0.790387	+	0.612608i	$0.209880\pi$
$338$	0	0
$339$	23.3484	1.26811
$340$	−32.6391	−1.77010
$341$	−5.11513	−0.277000
$342$	2.85851	0.154571
$343$	7.18804	0.388118
$344$	7.12837	0.384336
$345$	56.5536	3.04475
$346$	−2.90373	−0.156105
$347$	−17.5755	−0.943502	−0.471751	−	0.881732i	$-0.656378\pi$
$347$	−17.5755	−0.943502	−0.471751	+	0.881732i	$0.656378\pi$
$348$	−25.3436	−1.35856
$349$	−9.23659	−0.494423	−0.247212	−	0.968962i	$-0.579514\pi$
$349$	−9.23659	−0.494423	−0.247212	+	0.968962i	$0.579514\pi$
$350$	−45.4802	−2.43102
$351$	0	0
$352$	−1.33908	−0.0713730
$353$	3.64224	0.193857	0.0969286	−	0.995291i	$-0.469098\pi$
$353$	3.64224	0.193857	0.0969286	+	0.995291i	$0.469098\pi$
$354$	−11.5944	−0.616235
$355$	−15.9981	−0.849090
$356$	3.60658	0.191148
$357$	−63.9740	−3.38586
$358$	−9.75377	−0.515503
$359$	−23.7376	−1.25282	−0.626411	−	0.779493i	$-0.715477\pi$
$359$	−23.7376	−1.25282	−0.626411	+	0.779493i	$0.715477\pi$
$360$	12.1862	0.642268
$361$	1.00000	0.0526316
$362$	−19.6557	−1.03308
$363$	22.2847	1.16964
$364$	0	0
$365$	−1.21886	−0.0637978
$366$	31.1938	1.63053
$367$	−17.6979	−0.923822	−0.461911	−	0.886926i	$-0.652836\pi$
$367$	−17.6979	−0.923822	−0.461911	+	0.886926i	$0.652836\pi$
$368$	−5.48074	−0.285703
$369$	5.64519	0.293877
$370$	−1.58683	−0.0824951
$371$	24.0339	1.24778
$372$	−9.24582	−0.479373
$373$	6.36063	0.329341	0.164670	−	0.986349i	$-0.447344\pi$
$373$	6.36063	0.329341	0.164670	+	0.986349i	$0.447344\pi$
$374$	10.2522	0.530127
$375$	−84.3461	−4.35561
$376$	−9.88338	−0.509696
$377$	0	0
$378$	−1.18224	−0.0608079
$379$	18.4046	0.945380	0.472690	−	0.881229i	$-0.343283\pi$
$379$	18.4046	0.945380	0.472690	+	0.881229i	$0.343283\pi$
$380$	4.26312	0.218693
$381$	−6.34344	−0.324984
$382$	−6.78083	−0.346938
$383$	27.9086	1.42606	0.713032	−	0.701131i	$-0.247321\pi$
$383$	27.9086	1.42606	0.713032	+	0.701131i	$0.247321\pi$
$384$	−2.42044	−0.123517
$385$	19.7075	1.00439
$386$	1.61415	0.0821582
$387$	20.3766	1.03580
$388$	1.75855	0.0892770
$389$	28.7297	1.45665	0.728326	−	0.685231i	$-0.240299\pi$
$389$	28.7297	1.45665	0.728326	+	0.685231i	$0.240299\pi$
$390$	0	0
$391$	41.9614	2.12208
$392$	4.91785	0.248389
$393$	6.00819	0.303073
$394$	−13.6604	−0.688202
$395$	−71.3189	−3.58844
$396$	−3.82777	−0.192353
$397$	23.9557	1.20230	0.601152	−	0.799135i	$-0.294709\pi$
$397$	23.9557	1.20230	0.601152	+	0.799135i	$0.294709\pi$
$398$	10.2721	0.514896
$399$	8.35589	0.418318
$400$	13.1742	0.658709
$401$	−32.4110	−1.61853	−0.809263	−	0.587446i	$-0.800134\pi$
$401$	−32.4110	−1.61853	−0.809263	+	0.587446i	$0.800134\pi$
$402$	28.9794	1.44536
$403$	0	0
$404$	−0.369761	−0.0183963
$405$	−40.0922	−1.99220
$406$	−36.1471	−1.79395
$407$	0.498433	0.0247064
$408$	18.5312	0.917433
$409$	18.1815	0.899017	0.449508	−	0.893276i	$-0.351599\pi$
$409$	18.1815	0.899017	0.449508	+	0.893276i	$0.351599\pi$
$410$	8.41909	0.415789
$411$	−27.8655	−1.37451
$412$	5.90899	0.291115
$413$	−16.5369	−0.813726
$414$	−15.6668	−0.769980
$415$	−23.9851	−1.17738
$416$	0	0
$417$	8.66656	0.424403
$418$	−1.33908	−0.0654963
$419$	8.30127	0.405544	0.202772	−	0.979226i	$-0.435005\pi$
$419$	8.30127	0.405544	0.202772	+	0.979226i	$0.435005\pi$
$420$	35.6221	1.73818
$421$	−24.7031	−1.20396	−0.601978	−	0.798512i	$-0.705621\pi$
$421$	−24.7031	−1.20396	−0.601978	+	0.798512i	$0.705621\pi$
$422$	−23.1474	−1.12680
$423$	−28.2518	−1.37365
$424$	−6.96187	−0.338098
$425$	−100.863	−4.89260
$426$	9.08311	0.440078
$427$	44.4912	2.15308
$428$	−4.96404	−0.239946
$429$	0	0
$430$	30.3891	1.46549
$431$	22.8634	1.10129	0.550645	−	0.834740i	$-0.314382\pi$
$431$	22.8634	1.10129	0.550645	+	0.834740i	$0.314382\pi$
$432$	0.342457	0.0164765
$433$	−33.6946	−1.61926	−0.809629	−	0.586942i	$-0.800332\pi$
$433$	−33.6946	−1.61926	−0.809629	+	0.586942i	$0.800332\pi$
$434$	−13.1871	−0.633003
$435$	−108.043	−5.18025
$436$	2.84555	0.136277
$437$	−5.48074	−0.262180
$438$	0.692020	0.0330660
$439$	−11.1403	−0.531696	−0.265848	−	0.964015i	$-0.585652\pi$
$439$	−11.1403	−0.531696	−0.265848	+	0.964015i	$0.585652\pi$
$440$	−5.70864	−0.272149
$441$	14.0577	0.669417
$442$	0	0
$443$	−9.66888	−0.459383	−0.229691	−	0.973264i	$-0.573772\pi$
$443$	−9.66888	−0.459383	−0.229691	+	0.973264i	$0.573772\pi$
$444$	0.900939	0.0427567
$445$	15.3753	0.728858
$446$	−25.0578	−1.18652
$447$	−29.3536	−1.38838
$448$	−3.45222	−0.163102
$449$	−8.98693	−0.424120	−0.212060	−	0.977257i	$-0.568017\pi$
$449$	−8.98693	−0.424120	−0.212060	+	0.977257i	$0.568017\pi$
$450$	37.6586	1.77524
$451$	−2.64450	−0.124525
$452$	−9.64635	−0.453726
$453$	11.2354	0.527886
$454$	−4.19334	−0.196803
$455$	0	0
$456$	−2.42044	−0.113347
$457$	−19.9978	−0.935456	−0.467728	−	0.883873i	$-0.654927\pi$
$457$	−19.9978	−0.935456	−0.467728	+	0.883873i	$0.654927\pi$
$458$	−12.8108	−0.598608
$459$	−2.62191	−0.122380
$460$	−23.3651	−1.08940
$461$	−7.21475	−0.336024	−0.168012	−	0.985785i	$-0.553735\pi$
$461$	−7.21475	−0.336024	−0.168012	+	0.985785i	$0.553735\pi$
$462$	−11.1892	−0.520567
$463$	−33.4131	−1.55284	−0.776419	−	0.630217i	$-0.782966\pi$
$463$	−33.4131	−1.55284	−0.776419	+	0.630217i	$0.782966\pi$
$464$	10.4707	0.486088
$465$	−39.4160	−1.82787
$466$	−7.83899	−0.363134
$467$	28.8447	1.33477	0.667387	−	0.744711i	$-0.267413\pi$
$467$	28.8447	1.33477	0.667387	+	0.744711i	$0.267413\pi$
$468$	0	0
$469$	41.3327	1.90857
$470$	−42.1340	−1.94350
$471$	14.1095	0.650134
$472$	4.79021	0.220487
$473$	−9.54543	−0.438900
$474$	40.4921	1.85987
$475$	13.1742	0.604473
$476$	26.4308	1.21145
$477$	−19.9006	−0.911186
$478$	9.37460	0.428784
$479$	14.3762	0.656866	0.328433	−	0.944527i	$-0.393480\pi$
$479$	14.3762	0.656866	0.328433	+	0.944527i	$0.393480\pi$
$480$	−10.3186	−0.470978
$481$	0	0
$482$	13.7745	0.627411
$483$	−45.7965	−2.08381
$484$	−9.20688	−0.418494
$485$	7.49692	0.340418
$486$	21.7355	0.985941
$487$	33.7402	1.52892	0.764458	−	0.644674i	$-0.223007\pi$
$487$	33.7402	1.52892	0.764458	+	0.644674i	$0.223007\pi$
$488$	−12.8877	−0.583398
$489$	40.1991	1.81787
$490$	20.9654	0.947119
$491$	−1.60000	−0.0722069	−0.0361034	−	0.999348i	$-0.511495\pi$
$491$	−1.60000	−0.0722069	−0.0361034	+	0.999348i	$0.511495\pi$
$492$	−4.78004	−0.215501
$493$	−80.1650	−3.61045
$494$	0	0
$495$	−16.3182	−0.733450
$496$	3.81990	0.171518
$497$	12.9551	0.581114
$498$	13.6178	0.610230
$499$	−16.6948	−0.747362	−0.373681	−	0.927557i	$-0.621904\pi$
$499$	−16.6948	−0.747362	−0.373681	+	0.927557i	$0.621904\pi$
$500$	34.8475	1.55843
$501$	−1.55478	−0.0694625
$502$	−3.87734	−0.173054
$503$	11.1251	0.496046	0.248023	−	0.968754i	$-0.420219\pi$
$503$	11.1251	0.496046	0.248023	+	0.968754i	$0.420219\pi$
$504$	−9.86823	−0.439566
$505$	−1.57634	−0.0701460
$506$	7.33913	0.326264
$507$	0	0
$508$	2.62078	0.116278
$509$	−23.6814	−1.04966	−0.524830	−	0.851207i	$-0.675871\pi$
$509$	−23.6814	−1.04966	−0.524830	+	0.851207i	$0.675871\pi$
$510$	79.0008	3.49822
$511$	0.987015	0.0436630
$512$	1.00000	0.0441942
$513$	0.342457	0.0151199
$514$	−29.2867	−1.29178
$515$	25.1907	1.11004
$516$	−17.2538	−0.759555
$517$	13.2346	0.582057
$518$	1.28499	0.0564593
$519$	7.02829	0.308507
$520$	0	0
$521$	2.99516	0.131220	0.0656101	−	0.997845i	$-0.479101\pi$
$521$	2.99516	0.131220	0.0656101	+	0.997845i	$0.479101\pi$
$522$	29.9305	1.31002
$523$	−9.03248	−0.394963	−0.197481	−	0.980307i	$-0.563276\pi$
$523$	−9.03248	−0.394963	−0.197481	+	0.980307i	$0.563276\pi$
$524$	−2.48227	−0.108439
$525$	110.082	4.80437
$526$	−1.25741	−0.0548257
$527$	−29.2457	−1.27396
$528$	3.24115	0.141053
$529$	7.03855	0.306024
$530$	−29.6793	−1.28919
$531$	13.6929	0.594220
$532$	−3.45222	−0.149673
$533$	0	0
$534$	−8.72950	−0.377763
$535$	−21.1623	−0.914926
$536$	−11.9728	−0.517145
$537$	23.6084	1.01878
$538$	14.9946	0.646462
$539$	−6.58538	−0.283652
$540$	1.45994	0.0628257
$541$	26.5602	1.14191	0.570955	−	0.820981i	$-0.306573\pi$
$541$	26.5602	1.14191	0.570955	+	0.820981i	$0.306573\pi$
$542$	11.4029	0.489797
$543$	47.5755	2.04166
$544$	−7.65615	−0.328255
$545$	12.1309	0.519632
$546$	0	0
$547$	−17.3207	−0.740581	−0.370290	−	0.928916i	$-0.620742\pi$
$547$	−17.3207	−0.740581	−0.370290	+	0.928916i	$0.620742\pi$
$548$	11.5126	0.491794
$549$	−36.8396	−1.57228
$550$	−17.6412	−0.752224
$551$	10.4707	0.446065
$552$	13.2658	0.564630
$553$	57.7532	2.45592
$554$	14.0472	0.596808
$555$	3.84081	0.163033
$556$	−3.58058	−0.151850
$557$	−22.3034	−0.945023	−0.472512	−	0.881324i	$-0.656653\pi$
$557$	−22.3034	−0.945023	−0.472512	+	0.881324i	$0.656653\pi$
$558$	10.9192	0.462248
$559$	0	0
$560$	−14.7172	−0.621917
$561$	−24.8147	−1.04768
$562$	0.227399	0.00959225
$563$	21.0624	0.887672	0.443836	−	0.896108i	$-0.353617\pi$
$563$	21.0624	0.887672	0.443836	+	0.896108i	$0.353617\pi$
$564$	23.9221	1.00730
$565$	−41.1235	−1.73008
$566$	32.7518	1.37666
$567$	32.4662	1.36345
$568$	−3.75267	−0.157459
$569$	−5.09369	−0.213538	−0.106769	−	0.994284i	$-0.534051\pi$
$569$	−5.09369	−0.213538	−0.106769	+	0.994284i	$0.534051\pi$
$570$	−10.3186	−0.432199
$571$	45.0658	1.88595	0.942973	−	0.332870i	$-0.108017\pi$
$571$	45.0658	1.88595	0.942973	+	0.332870i	$0.108017\pi$
$572$	0	0
$573$	16.4126	0.685645
$574$	−6.81768	−0.284565
$575$	−72.2043	−3.01113
$576$	2.85851	0.119105
$577$	35.9953	1.49851	0.749253	−	0.662284i	$-0.230413\pi$
$577$	35.9953	1.49851	0.749253	+	0.662284i	$0.230413\pi$
$578$	41.6167	1.73102
$579$	−3.90695	−0.162367
$580$	44.6377	1.85348
$581$	19.4229	0.805796
$582$	−4.25647	−0.176436
$583$	9.32247	0.386097
$584$	−0.285907	−0.0118309
$585$	0	0
$586$	−5.32509	−0.219977
$587$	−14.8259	−0.611930	−0.305965	−	0.952043i	$-0.598979\pi$
$587$	−14.8259	−0.611930	−0.305965	+	0.952043i	$0.598979\pi$
$588$	−11.9033	−0.490886
$589$	3.81990	0.157396
$590$	20.4212	0.840728
$591$	33.0642	1.36008
$592$	−0.372222	−0.0152982
$593$	−27.2166	−1.11765	−0.558825	−	0.829285i	$-0.688748\pi$
$593$	−27.2166	−1.11765	−0.558825	+	0.829285i	$0.688748\pi$
$594$	−0.458577	−0.0188156
$595$	112.677	4.61932
$596$	12.1274	0.496758
$597$	−24.8631	−1.01758
$598$	0	0
$599$	39.8287	1.62736	0.813678	−	0.581316i	$-0.197462\pi$
$599$	39.8287	1.62736	0.813678	+	0.581316i	$0.197462\pi$
$600$	−31.8873	−1.30179
$601$	−27.2007	−1.10954	−0.554770	−	0.832004i	$-0.687194\pi$
$601$	−27.2007	−1.10954	−0.554770	+	0.832004i	$0.687194\pi$
$602$	−24.6087	−1.00298
$603$	−34.2244	−1.39372
$604$	−4.64190	−0.188876
$605$	−39.2500	−1.59574
$606$	0.894983	0.0363562
$607$	1.47868	0.0600180	0.0300090	−	0.999550i	$-0.490446\pi$
$607$	1.47868	0.0600180	0.0300090	+	0.999550i	$0.490446\pi$
$608$	1.00000	0.0405554
$609$	87.4917	3.54534
$610$	−54.9417	−2.22453
$611$	0	0
$612$	−21.8852	−0.884657
$613$	−38.8821	−1.57043	−0.785217	−	0.619221i	$-0.787449\pi$
$613$	−38.8821	−1.57043	−0.785217	+	0.619221i	$0.787449\pi$
$614$	−21.0743	−0.850489
$615$	−20.3779	−0.821715
$616$	4.62279	0.186258
$617$	20.0518	0.807257	0.403628	−	0.914923i	$-0.367749\pi$
$617$	20.0518	0.807257	0.403628	+	0.914923i	$0.367749\pi$
$618$	−14.3023	−0.575325
$619$	14.2107	0.571177	0.285588	−	0.958352i	$-0.407811\pi$
$619$	14.2107	0.571177	0.285588	+	0.958352i	$0.407811\pi$
$620$	16.2847	0.654008
$621$	−1.87692	−0.0753183
$622$	18.1377	0.727255
$623$	−12.4507	−0.498828
$624$	0	0
$625$	82.6880	3.30752
$626$	−2.89367	−0.115654
$627$	3.24115	0.129439
$628$	−5.82934	−0.232616
$629$	2.84979	0.113628
$630$	−42.0694	−1.67609
$631$	21.6239	0.860833	0.430417	−	0.902630i	$-0.358367\pi$
$631$	21.6239	0.860833	0.430417	+	0.902630i	$0.358367\pi$
$632$	−16.7293	−0.665455
$633$	56.0269	2.22687
$634$	−29.2571	−1.16195
$635$	11.1727	0.443375
$636$	16.8508	0.668176
$637$	0	0
$638$	−14.0210	−0.555097
$639$	−10.7271	−0.424356
$640$	4.26312	0.168515
$641$	−10.5087	−0.415068	−0.207534	−	0.978228i	$-0.566544\pi$
$641$	−10.5087	−0.415068	−0.207534	+	0.978228i	$0.566544\pi$
$642$	12.0151	0.474200
$643$	−15.4702	−0.610086	−0.305043	−	0.952339i	$-0.598671\pi$
$643$	−15.4702	−0.610086	−0.305043	+	0.952339i	$0.598671\pi$
$644$	18.9208	0.745582
$645$	−73.5549	−2.89622
$646$	−7.65615	−0.301227
$647$	11.9459	0.469642	0.234821	−	0.972039i	$-0.424550\pi$
$647$	11.9459	0.469642	0.234821	+	0.972039i	$0.424550\pi$
$648$	−9.40444	−0.369441
$649$	−6.41445	−0.251789
$650$	0	0
$651$	31.9186	1.25099
$652$	−16.6082	−0.650428
$653$	43.9326	1.71922	0.859609	−	0.510953i	$-0.170707\pi$
$653$	43.9326	1.71922	0.859609	+	0.510953i	$0.170707\pi$
$654$	−6.88748	−0.269322
$655$	−10.5822	−0.413482
$656$	1.97487	0.0771056
$657$	−0.817269	−0.0318847
$658$	34.1196	1.33012
$659$	13.9038	0.541613	0.270807	−	0.962634i	$-0.412710\pi$
$659$	13.9038	0.541613	0.270807	+	0.962634i	$0.412710\pi$
$660$	13.8174	0.537842
$661$	−11.1500	−0.433686	−0.216843	−	0.976207i	$-0.569576\pi$
$661$	−11.1500	−0.433686	−0.216843	+	0.976207i	$0.569576\pi$
$662$	−15.3836	−0.597902
$663$	0	0
$664$	−5.62619	−0.218339
$665$	−14.7172	−0.570710
$666$	−1.06400	−0.0412292
$667$	−57.3870	−2.22203
$668$	0.642355	0.0248535
$669$	60.6507	2.34489
$670$	−51.0414	−1.97190
$671$	17.2576	0.666222
$672$	8.35589	0.322335
$673$	29.7879	1.14824	0.574120	−	0.818771i	$-0.305344\pi$
$673$	29.7879	1.14824	0.574120	+	0.818771i	$0.305344\pi$
$674$	29.0192	1.11778
$675$	4.51159	0.173651
$676$	0	0
$677$	−35.9164	−1.38038	−0.690190	−	0.723628i	$-0.742473\pi$
$677$	−35.9164	−1.38038	−0.690190	+	0.723628i	$0.742473\pi$
$678$	23.3484	0.896689
$679$	−6.07092	−0.232981
$680$	−32.6391	−1.25165
$681$	10.1497	0.388938
$682$	−5.11513	−0.195868
$683$	22.2920	0.852981	0.426490	−	0.904492i	$-0.359750\pi$
$683$	22.2920	0.852981	0.426490	+	0.904492i	$0.359750\pi$
$684$	2.85851	0.109298
$685$	49.0796	1.87523
$686$	7.18804	0.274441
$687$	31.0077	1.18302
$688$	7.12837	0.271767
$689$	0	0
$690$	56.5536	2.15296
$691$	−25.7248	−0.978619	−0.489309	−	0.872110i	$-0.662751\pi$
$691$	−25.7248	−0.978619	−0.489309	+	0.872110i	$0.662751\pi$
$692$	−2.90373	−0.110383
$693$	13.2143	0.501970
$694$	−17.5755	−0.667157
$695$	−15.2644	−0.579012
$696$	−25.3436	−0.960646
$697$	−15.1199	−0.572706
$698$	−9.23659	−0.349610
$699$	18.9738	0.717655
$700$	−45.4802	−1.71899
$701$	−22.0308	−0.832090	−0.416045	−	0.909344i	$-0.636584\pi$
$701$	−22.0308	−0.832090	−0.416045	+	0.909344i	$0.636584\pi$
$702$	0	0
$703$	−0.372222	−0.0140386
$704$	−1.33908	−0.0504683
$705$	101.983	3.84089
$706$	3.64224	0.137078
$707$	1.27650	0.0480077
$708$	−11.5944	−0.435744
$709$	−7.08937	−0.266247	−0.133123	−	0.991099i	$-0.542501\pi$
$709$	−7.08937	−0.266247	−0.133123	+	0.991099i	$0.542501\pi$
$710$	−15.9981	−0.600397
$711$	−47.8209	−1.79342
$712$	3.60658	0.135162
$713$	−20.9359	−0.784054
$714$	−63.9740	−2.39417
$715$	0	0
$716$	−9.75377	−0.364516
$717$	−22.6906	−0.847397
$718$	−23.7376	−0.885879
$719$	−26.4878	−0.987829	−0.493914	−	0.869511i	$-0.664434\pi$
$719$	−26.4878	−0.987829	−0.493914	+	0.869511i	$0.664434\pi$
$720$	12.1862	0.454152
$721$	−20.3992	−0.759705
$722$	1.00000	0.0372161
$723$	−33.3403	−1.23994
$724$	−19.6557	−0.730500
$725$	137.942	5.12305
$726$	22.2847	0.827061
$727$	5.64045	0.209193	0.104596	−	0.994515i	$-0.466645\pi$
$727$	5.64045	0.209193	0.104596	+	0.994515i	$0.466645\pi$
$728$	0	0
$729$	−24.3960	−0.903557
$730$	−1.21886	−0.0451119
$731$	−54.5759	−2.01856
$732$	31.1938	1.15296
$733$	36.2681	1.33959	0.669796	−	0.742545i	$-0.266382\pi$
$733$	36.2681	1.33959	0.669796	+	0.742545i	$0.266382\pi$
$734$	−17.6979	−0.653241
$735$	−50.7454	−1.87177
$736$	−5.48074	−0.202023
$737$	16.0325	0.590563
$738$	5.64519	0.207802
$739$	−33.0586	−1.21608	−0.608040	−	0.793906i	$-0.708044\pi$
$739$	−33.0586	−1.21608	−0.608040	+	0.793906i	$0.708044\pi$
$740$	−1.58683	−0.0583329
$741$	0	0
$742$	24.0339	0.882313
$743$	21.7734	0.798787	0.399394	−	0.916780i	$-0.369221\pi$
$743$	21.7734	0.798787	0.399394	+	0.916780i	$0.369221\pi$
$744$	−9.24582	−0.338968
$745$	51.7006	1.89416
$746$	6.36063	0.232879
$747$	−16.0825	−0.588430
$748$	10.2522	0.374856
$749$	17.1370	0.626172
$750$	−84.3461	−3.07988
$751$	2.48328	0.0906162	0.0453081	−	0.998973i	$-0.485573\pi$
$751$	2.48328	0.0906162	0.0453081	+	0.998973i	$0.485573\pi$
$752$	−9.88338	−0.360410
$753$	9.38486	0.342003
$754$	0	0
$755$	−19.7890	−0.720194
$756$	−1.18224	−0.0429977
$757$	24.0547	0.874284	0.437142	−	0.899392i	$-0.355991\pi$
$757$	24.0547	0.874284	0.437142	+	0.899392i	$0.355991\pi$
$758$	18.4046	0.668485
$759$	−17.7639	−0.644789
$760$	4.26312	0.154640
$761$	−52.4081	−1.89979	−0.949897	−	0.312564i	$-0.898812\pi$
$761$	−52.4081	−1.89979	−0.949897	+	0.312564i	$0.898812\pi$
$762$	−6.34344	−0.229798
$763$	−9.82349	−0.355634
$764$	−6.78083	−0.245322
$765$	−93.2993	−3.37324
$766$	27.9086	1.00838
$767$	0	0
$768$	−2.42044	−0.0873400
$769$	−6.31234	−0.227629	−0.113814	−	0.993502i	$-0.536307\pi$
$769$	−6.31234	−0.227629	−0.113814	+	0.993502i	$0.536307\pi$
$770$	19.7075	0.710209
$771$	70.8867	2.55292
$772$	1.61415	0.0580946
$773$	23.0536	0.829182	0.414591	−	0.910008i	$-0.363925\pi$
$773$	23.0536	0.829182	0.414591	+	0.910008i	$0.363925\pi$
$774$	20.3766	0.732420
$775$	50.3240	1.80769
$776$	1.75855	0.0631284
$777$	−3.11024	−0.111579
$778$	28.7297	1.03001
$779$	1.97487	0.0707570
$780$	0	0
$781$	5.02511	0.179813
$782$	41.9614	1.50054
$783$	3.58576	0.128145
$784$	4.91785	0.175638
$785$	−24.8512	−0.886976
$786$	6.00819	0.214305
$787$	−5.09565	−0.181640	−0.0908201	−	0.995867i	$-0.528949\pi$
$787$	−5.09565	−0.181640	−0.0908201	+	0.995867i	$0.528949\pi$
$788$	−13.6604	−0.486632
$789$	3.04348	0.108351
$790$	−71.3189	−2.53741
$791$	33.3013	1.18406
$792$	−3.82777	−0.136014
$793$	0	0
$794$	23.9557	0.850157
$795$	71.8368	2.54779
$796$	10.2721	0.364087
$797$	10.7206	0.379742	0.189871	−	0.981809i	$-0.439193\pi$
$797$	10.7206	0.379742	0.189871	+	0.981809i	$0.439193\pi$
$798$	8.35589	0.295795
$799$	75.6687	2.67696
$800$	13.1742	0.465777
$801$	10.3095	0.364267
$802$	−32.4110	−1.14447
$803$	0.382851	0.0135105
$804$	28.9794	1.02202
$805$	80.6614	2.84294
$806$	0	0
$807$	−36.2934	−1.27759
$808$	−0.369761	−0.0130082
$809$	12.4953	0.439310	0.219655	−	0.975578i	$-0.429507\pi$
$809$	12.4953	0.439310	0.219655	+	0.975578i	$0.429507\pi$
$810$	−40.0922	−1.40870
$811$	29.7805	1.04574	0.522868	−	0.852414i	$-0.324862\pi$
$811$	29.7805	1.04574	0.522868	+	0.852414i	$0.324862\pi$
$812$	−36.1471	−1.26851
$813$	−27.6000	−0.967975
$814$	0.498433	0.0174701
$815$	−70.8027	−2.48011
$816$	18.5312	0.648723
$817$	7.12837	0.249390
$818$	18.1815	0.635701
$819$	0	0
$820$	8.41909	0.294007
$821$	3.28190	0.114539	0.0572695	−	0.998359i	$-0.481761\pi$
$821$	3.28190	0.114539	0.0572695	+	0.998359i	$0.481761\pi$
$822$	−27.8655	−0.971922
$823$	−16.2469	−0.566330	−0.283165	−	0.959071i	$-0.591384\pi$
$823$	−16.2469	−0.566330	−0.283165	+	0.959071i	$0.591384\pi$
$824$	5.90899	0.205850
$825$	42.6995	1.48660
$826$	−16.5369	−0.575391
$827$	−19.3232	−0.671935	−0.335967	−	0.941874i	$-0.609063\pi$
$827$	−19.3232	−0.671935	−0.335967	+	0.941874i	$0.609063\pi$
$828$	−15.6668	−0.544458
$829$	−29.2280	−1.01513	−0.507565	−	0.861614i	$-0.669454\pi$
$829$	−29.2280	−1.01513	−0.507565	+	0.861614i	$0.669454\pi$
$830$	−23.9851	−0.832535
$831$	−34.0003	−1.17946
$832$	0	0
$833$	−37.6518	−1.30456
$834$	8.66656	0.300098
$835$	2.73844	0.0947675
$836$	−1.33908	−0.0463129
$837$	1.30815	0.0452163
$838$	8.30127	0.286763
$839$	−0.276497	−0.00954573	−0.00477286	−	0.999989i	$-0.501519\pi$
$839$	−0.276497	−0.00954573	−0.00477286	+	0.999989i	$0.501519\pi$
$840$	35.6221	1.22908
$841$	80.6348	2.78051
$842$	−24.7031	−0.851326
$843$	−0.550405	−0.0189569
$844$	−23.1474	−0.796767
$845$	0	0
$846$	−28.2518	−0.971316
$847$	31.7842	1.09212
$848$	−6.96187	−0.239072
$849$	−79.2737	−2.72066
$850$	−100.863	−3.45959
$851$	2.04005	0.0699321
$852$	9.08311	0.311182
$853$	−15.3975	−0.527200	−0.263600	−	0.964632i	$-0.584910\pi$
$853$	−15.3975	−0.527200	−0.263600	+	0.964632i	$0.584910\pi$
$854$	44.4912	1.52246
$855$	12.1862	0.416759
$856$	−4.96404	−0.169667
$857$	−5.85624	−0.200045	−0.100023	−	0.994985i	$-0.531892\pi$
$857$	−5.85624	−0.200045	−0.100023	+	0.994985i	$0.531892\pi$
$858$	0	0
$859$	−3.12119	−0.106494	−0.0532469	−	0.998581i	$-0.516957\pi$
$859$	−3.12119	−0.106494	−0.0532469	+	0.998581i	$0.516957\pi$
$860$	30.3891	1.03626
$861$	16.5018	0.562379
$862$	22.8634	0.778729
$863$	−8.81792	−0.300165	−0.150083	−	0.988673i	$-0.547954\pi$
$863$	−8.81792	−0.300165	−0.150083	+	0.988673i	$0.547954\pi$
$864$	0.342457	0.0116506
$865$	−12.3789	−0.420896
$866$	−33.6946	−1.14499
$867$	−100.730	−3.42099
$868$	−13.1871	−0.447601
$869$	22.4018	0.759928
$870$	−108.043	−3.66299
$871$	0	0
$872$	2.84555	0.0963626
$873$	5.02685	0.170133
$874$	−5.48074	−0.185389
$875$	−120.301	−4.06693
$876$	0.692020	0.0233812
$877$	4.13056	0.139479	0.0697396	−	0.997565i	$-0.477783\pi$
$877$	4.13056	0.139479	0.0697396	+	0.997565i	$0.477783\pi$
$878$	−11.1403	−0.375966
$879$	12.8890	0.434737
$880$	−5.70864	−0.192438
$881$	46.5030	1.56672	0.783362	−	0.621565i	$-0.213503\pi$
$881$	46.5030	1.56672	0.783362	+	0.621565i	$0.213503\pi$
$882$	14.0577	0.473349
$883$	46.3797	1.56080	0.780400	−	0.625280i	$-0.215015\pi$
$883$	46.3797	1.56080	0.780400	+	0.625280i	$0.215015\pi$
$884$	0	0
$885$	−49.4282	−1.66151
$886$	−9.66888	−0.324832
$887$	6.31221	0.211943	0.105972	−	0.994369i	$-0.466205\pi$
$887$	6.31221	0.211943	0.105972	+	0.994369i	$0.466205\pi$
$888$	0.900939	0.0302335
$889$	−9.04752	−0.303444
$890$	15.3753	0.515381
$891$	12.5933	0.421890
$892$	−25.0578	−0.838996
$893$	−9.88338	−0.330735
$894$	−29.3536	−0.981732
$895$	−41.5815	−1.38992
$896$	−3.45222	−0.115331
$897$	0	0
$898$	−8.98693	−0.299898
$899$	39.9968	1.33397
$900$	37.6586	1.25529
$901$	53.3011	1.77572
$902$	−2.64450	−0.0880521
$903$	59.5639	1.98216
$904$	−9.64635	−0.320833
$905$	−83.7948	−2.78543
$906$	11.2354	0.373272
$907$	13.0280	0.432587	0.216294	−	0.976328i	$-0.430603\pi$
$907$	13.0280	0.432587	0.216294	+	0.976328i	$0.430603\pi$
$908$	−4.19334	−0.139161
$909$	−1.05697	−0.0350574
$910$	0	0
$911$	−32.2476	−1.06841	−0.534206	−	0.845354i	$-0.679389\pi$
$911$	−32.2476	−1.06841	−0.534206	+	0.845354i	$0.679389\pi$
$912$	−2.42044	−0.0801487
$913$	7.53390	0.249336
$914$	−19.9978	−0.661467
$915$	132.983	4.39628
$916$	−12.8108	−0.423280
$917$	8.56936	0.282985
$918$	−2.62191	−0.0865358
$919$	−2.47878	−0.0817673	−0.0408836	−	0.999164i	$-0.513017\pi$
$919$	−2.47878	−0.0817673	−0.0408836	+	0.999164i	$0.513017\pi$
$920$	−23.3651	−0.770323
$921$	51.0090	1.68080
$922$	−7.21475	−0.237605
$923$	0	0
$924$	−11.1892	−0.368097
$925$	−4.90371	−0.161233
$926$	−33.4131	−1.09802
$927$	16.8909	0.554771
$928$	10.4707	0.343716
$929$	13.7260	0.450335	0.225168	−	0.974320i	$-0.427707\pi$
$929$	13.7260	0.450335	0.225168	+	0.974320i	$0.427707\pi$
$930$	−39.4160	−1.29250
$931$	4.91785	0.161176
$932$	−7.83899	−0.256775
$933$	−43.9011	−1.43726
$934$	28.8447	0.943828
$935$	43.7062	1.42935
$936$	0	0
$937$	40.8248	1.33369	0.666844	−	0.745198i	$-0.267645\pi$
$937$	40.8248	1.33369	0.666844	+	0.745198i	$0.267645\pi$
$938$	41.3327	1.34956
$939$	7.00395	0.228565
$940$	−42.1340	−1.37426
$941$	−41.8623	−1.36467	−0.682336	−	0.731039i	$-0.739036\pi$
$941$	−41.8623	−1.36467	−0.682336	+	0.731039i	$0.739036\pi$
$942$	14.1095	0.459714
$943$	−10.8237	−0.352469
$944$	4.79021	0.155908
$945$	−5.04003	−0.163952
$946$	−9.54543	−0.310349
$947$	−0.193811	−0.00629802	−0.00314901	−	0.999995i	$-0.501002\pi$
$947$	−0.193811	−0.00629802	−0.00314901	+	0.999995i	$0.501002\pi$
$948$	40.4921	1.31512
$949$	0	0
$950$	13.1742	0.427427
$951$	70.8149	2.29633
$952$	26.4308	0.856626
$953$	19.7574	0.640004	0.320002	−	0.947417i	$-0.396316\pi$
$953$	19.7574	0.640004	0.320002	+	0.947417i	$0.396316\pi$
$954$	−19.9006	−0.644306
$955$	−28.9075	−0.935425
$956$	9.37460	0.303196
$957$	33.9370	1.09703
$958$	14.3762	0.464474
$959$	−39.7441	−1.28340
$960$	−10.3186	−0.333032
$961$	−16.4084	−0.529303
$962$	0	0
$963$	−14.1898	−0.457259
$964$	13.7745	0.443647
$965$	6.88132	0.221518
$966$	−45.7965	−1.47348
$967$	−44.8504	−1.44229	−0.721145	−	0.692784i	$-0.756384\pi$
$967$	−44.8504	−1.44229	−0.721145	+	0.692784i	$0.756384\pi$
$968$	−9.20688	−0.295920
$969$	18.5312	0.595309
$970$	7.49692	0.240712
$971$	36.3573	1.16676	0.583381	−	0.812199i	$-0.301730\pi$
$971$	36.3573	1.16676	0.583381	+	0.812199i	$0.301730\pi$
$972$	21.7355	0.697166
$973$	12.3610	0.396274
$974$	33.7402	1.08111
$975$	0	0
$976$	−12.8877	−0.412525
$977$	33.7310	1.07915	0.539575	−	0.841938i	$-0.318585\pi$
$977$	33.7310	1.07915	0.539575	+	0.841938i	$0.318585\pi$
$978$	40.1991	1.28543
$979$	−4.82949	−0.154351
$980$	20.9654	0.669714
$981$	8.13405	0.259700
$982$	−1.60000	−0.0510580
$983$	−17.9494	−0.572498	−0.286249	−	0.958155i	$-0.592408\pi$
$983$	−17.9494	−0.572498	−0.286249	+	0.958155i	$0.592408\pi$
$984$	−4.78004	−0.152382
$985$	−58.2359	−1.85555
$986$	−80.1650	−2.55297
$987$	−82.5844	−2.62869
$988$	0	0
$989$	−39.0688	−1.24232
$990$	−16.3182	−0.518627
$991$	−34.6496	−1.10068	−0.550340	−	0.834941i	$-0.685502\pi$
$991$	−34.6496	−1.10068	−0.550340	+	0.834941i	$0.685502\pi$
$992$	3.81990	0.121282
$993$	37.2351	1.18162
$994$	12.9551	0.410910
$995$	43.7914	1.38828
$996$	13.6178	0.431498
$997$	−0.134100	−0.00424698	−0.00212349	−	0.999998i	$-0.500676\pi$
$997$	−0.134100	−0.00424698	−0.00212349	+	0.999998i	$0.500676\pi$
$998$	−16.6948	−0.528465
$999$	−0.127470	−0.00403298

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bk.1.3	yes	9
13.12	even	2		6422.2.a.bi.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.bi.1.3	✓	9	13.12	even	2
6422.2.a.bk.1.3	yes	9	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.42044 q^{3} +1.00000 q^{4} +4.26312 q^{5} -2.42044 q^{6} -3.45222 q^{7} +1.00000 q^{8} +2.85851 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.42044 q^{3} +1.00000 q^{4} +4.26312 q^{5} -2.42044 q^{6} -3.45222 q^{7} +1.00000 q^{8} +2.85851 q^{9} +4.26312 q^{10} -1.33908 q^{11} -2.42044 q^{12} -3.45222 q^{14} -10.3186 q^{15} +1.00000 q^{16} -7.65615 q^{17} +2.85851 q^{18} +1.00000 q^{19} +4.26312 q^{20} +8.35589 q^{21} -1.33908 q^{22} -5.48074 q^{23} -2.42044 q^{24} +13.1742 q^{25} +0.342457 q^{27} -3.45222 q^{28} +10.4707 q^{29} -10.3186 q^{30} +3.81990 q^{31} +1.00000 q^{32} +3.24115 q^{33} -7.65615 q^{34} -14.7172 q^{35} +2.85851 q^{36} -0.372222 q^{37} +1.00000 q^{38} +4.26312 q^{40} +1.97487 q^{41} +8.35589 q^{42} +7.12837 q^{43} -1.33908 q^{44} +12.1862 q^{45} -5.48074 q^{46} -9.88338 q^{47} -2.42044 q^{48} +4.91785 q^{49} +13.1742 q^{50} +18.5312 q^{51} -6.96187 q^{53} +0.342457 q^{54} -5.70864 q^{55} -3.45222 q^{56} -2.42044 q^{57} +10.4707 q^{58} +4.79021 q^{59} -10.3186 q^{60} -12.8877 q^{61} +3.81990 q^{62} -9.86823 q^{63} +1.00000 q^{64} +3.24115 q^{66} -11.9728 q^{67} -7.65615 q^{68} +13.2658 q^{69} -14.7172 q^{70} -3.75267 q^{71} +2.85851 q^{72} -0.285907 q^{73} -0.372222 q^{74} -31.8873 q^{75} +1.00000 q^{76} +4.62279 q^{77} -16.7293 q^{79} +4.26312 q^{80} -9.40444 q^{81} +1.97487 q^{82} -5.62619 q^{83} +8.35589 q^{84} -32.6391 q^{85} +7.12837 q^{86} -25.3436 q^{87} -1.33908 q^{88} +3.60658 q^{89} +12.1862 q^{90} -5.48074 q^{92} -9.24582 q^{93} -9.88338 q^{94} +4.26312 q^{95} -2.42044 q^{96} +1.75855 q^{97} +4.91785 q^{98} -3.82777 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} - 5 q^{3} + 9 q^{4} + q^{5} - 5 q^{6} - 13 q^{7} + 9 q^{8} + 10 q^{9}+O(q^{10}) \) 9 * q + 9 * q^2 - 5 * q^3 + 9 * q^4 + q^5 - 5 * q^6 - 13 * q^7 + 9 * q^8 + 10 * q^9	\( 9 q + 9 q^{2} - 5 q^{3} + 9 q^{4} + q^{5} - 5 q^{6} - 13 q^{7} + 9 q^{8} + 10 q^{9} + q^{10} - 13 q^{11} - 5 q^{12} - 13 q^{14} - q^{15} + 9 q^{16} - 12 q^{17} + 10 q^{18} + 9 q^{19} + q^{20} + 18 q^{21} - 13 q^{22} - 22 q^{23} - 5 q^{24} + 4 q^{25} - 26 q^{27} - 13 q^{28} + 12 q^{29} - q^{30} + q^{31} + 9 q^{32} - 28 q^{33} - 12 q^{34} - 18 q^{35} + 10 q^{36} - 25 q^{37} + 9 q^{38} + q^{40} + 11 q^{41} + 18 q^{42} - 10 q^{43} - 13 q^{44} - q^{45} - 22 q^{46} - 12 q^{47} - 5 q^{48} + 2 q^{49} + 4 q^{50} + 35 q^{51} + 9 q^{53} - 26 q^{54} + 18 q^{55} - 13 q^{56} - 5 q^{57} + 12 q^{58} + 10 q^{59} - q^{60} + 32 q^{61} + q^{62} - 63 q^{63} + 9 q^{64} - 28 q^{66} - 73 q^{67} - 12 q^{68} + 2 q^{69} - 18 q^{70} - 51 q^{71} + 10 q^{72} - 14 q^{73} - 25 q^{74} - 49 q^{75} + 9 q^{76} + 18 q^{77} - 28 q^{79} + q^{80} + 29 q^{81} + 11 q^{82} - 22 q^{83} + 18 q^{84} - 51 q^{85} - 10 q^{86} - 20 q^{87} - 13 q^{88} + 3 q^{89} - q^{90} - 22 q^{92} - 59 q^{93} - 12 q^{94} + q^{95} - 5 q^{96} + 2 q^{98} + 25 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 - 5 * q^3 + 9 * q^4 + q^5 - 5 * q^6 - 13 * q^7 + 9 * q^8 + 10 * q^9 + q^10 - 13 * q^11 - 5 * q^12 - 13 * q^14 - q^15 + 9 * q^16 - 12 * q^17 + 10 * q^18 + 9 * q^19 + q^20 + 18 * q^21 - 13 * q^22 - 22 * q^23 - 5 * q^24 + 4 * q^25 - 26 * q^27 - 13 * q^28 + 12 * q^29 - q^30 + q^31 + 9 * q^32 - 28 * q^33 - 12 * q^34 - 18 * q^35 + 10 * q^36 - 25 * q^37 + 9 * q^38 + q^40 + 11 * q^41 + 18 * q^42 - 10 * q^43 - 13 * q^44 - q^45 - 22 * q^46 - 12 * q^47 - 5 * q^48 + 2 * q^49 + 4 * q^50 + 35 * q^51 + 9 * q^53 - 26 * q^54 + 18 * q^55 - 13 * q^56 - 5 * q^57 + 12 * q^58 + 10 * q^59 - q^60 + 32 * q^61 + q^62 - 63 * q^63 + 9 * q^64 - 28 * q^66 - 73 * q^67 - 12 * q^68 + 2 * q^69 - 18 * q^70 - 51 * q^71 + 10 * q^72 - 14 * q^73 - 25 * q^74 - 49 * q^75 + 9 * q^76 + 18 * q^77 - 28 * q^79 + q^80 + 29 * q^81 + 11 * q^82 - 22 * q^83 + 18 * q^84 - 51 * q^85 - 10 * q^86 - 20 * q^87 - 13 * q^88 + 3 * q^89 - q^90 - 22 * q^92 - 59 * q^93 - 12 * q^94 + q^95 - 5 * q^96 + 2 * q^98 + 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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