LMFDB - Embedded newform 6422.2.a.bk.1.9

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			1.9
Root			$-0.288819$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +2.71446 q^{3} +1.00000 q^{4} -0.231615 q^{5} +2.71446 q^{6} -3.53580 q^{7} +1.00000 q^{8} +4.36830 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.71446 q^{3} +1.00000 q^{4} -0.231615 q^{5} +2.71446 q^{6} -3.53580 q^{7} +1.00000 q^{8} +4.36830 q^{9} -0.231615 q^{10} -4.96470 q^{11} +2.71446 q^{12} -3.53580 q^{14} -0.628709 q^{15} +1.00000 q^{16} +0.380770 q^{17} +4.36830 q^{18} +1.00000 q^{19} -0.231615 q^{20} -9.59779 q^{21} -4.96470 q^{22} -8.03063 q^{23} +2.71446 q^{24} -4.94635 q^{25} +3.71421 q^{27} -3.53580 q^{28} +3.83940 q^{29} -0.628709 q^{30} -9.50697 q^{31} +1.00000 q^{32} -13.4765 q^{33} +0.380770 q^{34} +0.818943 q^{35} +4.36830 q^{36} +0.944243 q^{37} +1.00000 q^{38} -0.231615 q^{40} +8.53011 q^{41} -9.59779 q^{42} +3.48940 q^{43} -4.96470 q^{44} -1.01176 q^{45} -8.03063 q^{46} -0.0285542 q^{47} +2.71446 q^{48} +5.50187 q^{49} -4.94635 q^{50} +1.03359 q^{51} -3.99333 q^{53} +3.71421 q^{54} +1.14990 q^{55} -3.53580 q^{56} +2.71446 q^{57} +3.83940 q^{58} -10.4363 q^{59} -0.628709 q^{60} +6.70446 q^{61} -9.50697 q^{62} -15.4454 q^{63} +1.00000 q^{64} -13.4765 q^{66} -14.5167 q^{67} +0.380770 q^{68} -21.7988 q^{69} +0.818943 q^{70} -1.08474 q^{71} +4.36830 q^{72} -5.82413 q^{73} +0.944243 q^{74} -13.4267 q^{75} +1.00000 q^{76} +17.5542 q^{77} -0.0332485 q^{79} -0.231615 q^{80} -3.02283 q^{81} +8.53011 q^{82} -0.0511721 q^{83} -9.59779 q^{84} -0.0881919 q^{85} +3.48940 q^{86} +10.4219 q^{87} -4.96470 q^{88} -16.0134 q^{89} -1.01176 q^{90} -8.03063 q^{92} -25.8063 q^{93} -0.0285542 q^{94} -0.231615 q^{95} +2.71446 q^{96} -2.09080 q^{97} +5.50187 q^{98} -21.6873 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.71446	1.56720	0.783598	−	0.621269i	$-0.213382\pi$
$3$	2.71446	1.56720	0.783598	+	0.621269i	$0.213382\pi$
$4$	1.00000	0.500000
$5$	−0.231615	−0.103581	−0.0517906	−	0.998658i	$-0.516493\pi$
$5$	−0.231615	−0.103581	−0.0517906	+	0.998658i	$0.516493\pi$
$6$	2.71446	1.10817
$7$	−3.53580	−1.33641	−0.668203	−	0.743979i	$-0.732936\pi$
$7$	−3.53580	−1.33641	−0.668203	+	0.743979i	$0.732936\pi$
$8$	1.00000	0.353553
$9$	4.36830	1.45610
$10$	−0.231615	−0.0732430
$11$	−4.96470	−1.49691	−0.748456	−	0.663184i	$-0.769205\pi$
$11$	−4.96470	−1.49691	−0.748456	+	0.663184i	$0.769205\pi$
$12$	2.71446	0.783598
$13$	0	0
$13$	0	0
$14$	−3.53580	−0.944982
$15$	−0.628709	−0.162332
$16$	1.00000	0.250000
$17$	0.380770	0.0923503	0.0461751	−	0.998933i	$-0.485297\pi$
$17$	0.380770	0.0923503	0.0461751	+	0.998933i	$0.485297\pi$
$18$	4.36830	1.02962
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.231615	−0.0517906
$21$	−9.59779	−2.09441
$22$	−4.96470	−1.05848
$23$	−8.03063	−1.67450	−0.837251	−	0.546819i	$-0.815839\pi$
$23$	−8.03063	−1.67450	−0.837251	+	0.546819i	$0.815839\pi$
$24$	2.71446	0.554087
$25$	−4.94635	−0.989271
$26$	0	0
$27$	3.71421	0.714800
$28$	−3.53580	−0.668203
$29$	3.83940	0.712959	0.356480	−	0.934303i	$-0.383977\pi$
$29$	3.83940	0.712959	0.356480	+	0.934303i	$0.383977\pi$
$30$	−0.628709	−0.114786
$31$	−9.50697	−1.70750	−0.853751	−	0.520681i	$-0.825678\pi$
$31$	−9.50697	−1.70750	−0.853751	+	0.520681i	$0.825678\pi$
$32$	1.00000	0.176777
$33$	−13.4765	−2.34595
$34$	0.380770	0.0653015
$35$	0.818943	0.138427
$36$	4.36830	0.728051
$37$	0.944243	0.155233	0.0776163	−	0.996983i	$-0.475269\pi$
$37$	0.944243	0.155233	0.0776163	+	0.996983i	$0.475269\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	−0.231615	−0.0366215
$41$	8.53011	1.33218	0.666089	−	0.745872i	$-0.267967\pi$
$41$	8.53011	1.33218	0.666089	+	0.745872i	$0.267967\pi$
$42$	−9.59779	−1.48097
$43$	3.48940	0.532128	0.266064	−	0.963955i	$-0.414277\pi$
$43$	3.48940	0.532128	0.266064	+	0.963955i	$0.414277\pi$
$44$	−4.96470	−0.748456
$45$	−1.01176	−0.150825
$46$	−8.03063	−1.18405
$47$	−0.0285542	−0.00416506	−0.00208253	−	0.999998i	$-0.500663\pi$
$47$	−0.0285542	−0.00416506	−0.00208253	+	0.999998i	$0.500663\pi$
$48$	2.71446	0.391799
$49$	5.50187	0.785982
$50$	−4.94635	−0.699520
$51$	1.03359	0.144731
$52$	0	0
$53$	−3.99333	−0.548527	−0.274263	−	0.961655i	$-0.588434\pi$
$53$	−3.99333	−0.548527	−0.274263	+	0.961655i	$0.588434\pi$
$54$	3.71421	0.505440
$55$	1.14990	0.155052
$56$	−3.53580	−0.472491
$57$	2.71446	0.359539
$58$	3.83940	0.504138
$59$	−10.4363	−1.35869	−0.679343	−	0.733821i	$-0.737735\pi$
$59$	−10.4363	−1.35869	−0.679343	+	0.733821i	$0.737735\pi$
$60$	−0.628709	−0.0811660
$61$	6.70446	0.858418	0.429209	−	0.903205i	$-0.358792\pi$
$61$	6.70446	0.858418	0.429209	+	0.903205i	$0.358792\pi$
$62$	−9.50697	−1.20739
$63$	−15.4454	−1.94594
$64$	1.00000	0.125000
$65$	0	0
$66$	−13.4765	−1.65884
$67$	−14.5167	−1.77350	−0.886748	−	0.462252i	$-0.847041\pi$
$67$	−14.5167	−1.77350	−0.886748	+	0.462252i	$0.847041\pi$
$68$	0.380770	0.0461751
$69$	−21.7988	−2.62427
$70$	0.818943	0.0978824
$71$	−1.08474	−0.128735	−0.0643676	−	0.997926i	$-0.520503\pi$
$71$	−1.08474	−0.128735	−0.0643676	+	0.997926i	$0.520503\pi$
$72$	4.36830	0.514810
$73$	−5.82413	−0.681663	−0.340831	−	0.940124i	$-0.610709\pi$
$73$	−5.82413	−0.681663	−0.340831	+	0.940124i	$0.610709\pi$
$74$	0.944243	0.109766
$75$	−13.4267	−1.55038
$76$	1.00000	0.114708
$77$	17.5542	2.00048
$78$	0	0
$79$	−0.0332485	−0.00374075	−0.00187038	−	0.999998i	$-0.500595\pi$
$79$	−0.0332485	−0.00374075	−0.00187038	+	0.999998i	$0.500595\pi$
$80$	−0.231615	−0.0258953
$81$	−3.02283	−0.335870
$82$	8.53011	0.941993
$83$	−0.0511721	−0.00561687	−0.00280844	−	0.999996i	$-0.500894\pi$
$83$	−0.0511721	−0.00561687	−0.00280844	+	0.999996i	$0.500894\pi$
$84$	−9.59779	−1.04720
$85$	−0.0881919	−0.00956576
$86$	3.48940	0.376272
$87$	10.4219	1.11735
$88$	−4.96470	−0.529238
$89$	−16.0134	−1.69742	−0.848709	−	0.528860i	$-0.822620\pi$
$89$	−16.0134	−1.69742	−0.848709	+	0.528860i	$0.822620\pi$
$90$	−1.01176	−0.106649
$91$	0	0
$92$	−8.03063	−0.837251
$93$	−25.8063	−2.67599
$94$	−0.0285542	−0.00294514
$95$	−0.231615	−0.0237632
$96$	2.71446	0.277044
$97$	−2.09080	−0.212289	−0.106144	−	0.994351i	$-0.533851\pi$
$97$	−2.09080	−0.212289	−0.106144	+	0.994351i	$0.533851\pi$
$98$	5.50187	0.555773
$99$	−21.6873	−2.17966
$100$	−4.94635	−0.494635
$101$	11.1746	1.11192	0.555959	−	0.831210i	$-0.312351\pi$
$101$	11.1746	1.11192	0.555959	+	0.831210i	$0.312351\pi$
$102$	1.03359	0.102340
$103$	13.2833	1.30884	0.654419	−	0.756132i	$-0.272913\pi$
$103$	13.2833	1.30884	0.654419	+	0.756132i	$0.272913\pi$
$104$	0	0
$105$	2.22299	0.216942
$106$	−3.99333	−0.387867
$107$	−1.16202	−0.112337	−0.0561685	−	0.998421i	$-0.517888\pi$
$107$	−1.16202	−0.112337	−0.0561685	+	0.998421i	$0.517888\pi$
$108$	3.71421	0.357400
$109$	17.8789	1.71248	0.856242	−	0.516575i	$-0.172793\pi$
$109$	17.8789	1.71248	0.856242	+	0.516575i	$0.172793\pi$
$110$	1.14990	0.109638
$111$	2.56311	0.243280
$112$	−3.53580	−0.334102
$113$	−16.7156	−1.57247	−0.786237	−	0.617926i	$-0.787973\pi$
$113$	−16.7156	−1.57247	−0.786237	+	0.617926i	$0.787973\pi$
$114$	2.71446	0.254233
$115$	1.86001	0.173447
$116$	3.83940	0.356480
$117$	0	0
$118$	−10.4363	−0.960736
$119$	−1.34633	−0.123417
$120$	−0.628709	−0.0573931
$121$	13.6482	1.24075
$122$	6.70446	0.606993
$123$	23.1547	2.08778
$124$	−9.50697	−0.853751
$125$	2.30372	0.206051
$126$	−15.4454	−1.37599
$127$	−6.94033	−0.615855	−0.307927	−	0.951410i	$-0.599635\pi$
$127$	−6.94033	−0.615855	−0.307927	+	0.951410i	$0.599635\pi$
$128$	1.00000	0.0883883
$129$	9.47184	0.833949
$130$	0	0
$131$	6.94142	0.606474	0.303237	−	0.952915i	$-0.401933\pi$
$131$	6.94142	0.606474	0.303237	+	0.952915i	$0.401933\pi$
$132$	−13.4765	−1.17298
$133$	−3.53580	−0.306593
$134$	−14.5167	−1.25405
$135$	−0.860266	−0.0740399
$136$	0.380770	0.0326508
$137$	16.4969	1.40942	0.704711	−	0.709495i	$-0.251077\pi$
$137$	16.4969	1.40942	0.704711	+	0.709495i	$0.251077\pi$
$138$	−21.7988	−1.85564
$139$	−21.7713	−1.84662	−0.923308	−	0.384061i	$-0.874525\pi$
$139$	−21.7713	−1.84662	−0.923308	+	0.384061i	$0.874525\pi$
$140$	0.818943	0.0692133
$141$	−0.0775093	−0.00652746
$142$	−1.08474	−0.0910295
$143$	0	0
$144$	4.36830	0.364025
$145$	−0.889262	−0.0738492
$146$	−5.82413	−0.482008
$147$	14.9346	1.23179
$148$	0.944243	0.0776163
$149$	12.3771	1.01397	0.506985	−	0.861955i	$-0.330760\pi$
$149$	12.3771	1.01397	0.506985	+	0.861955i	$0.330760\pi$
$150$	−13.4267	−1.09628
$151$	−5.47967	−0.445930	−0.222965	−	0.974826i	$-0.571573\pi$
$151$	−5.47967	−0.445930	−0.222965	+	0.974826i	$0.571573\pi$
$152$	1.00000	0.0811107
$153$	1.66332	0.134471
$154$	17.5542	1.41456
$155$	2.20195	0.176865
$156$	0	0
$157$	14.2183	1.13474	0.567371	−	0.823462i	$-0.307961\pi$
$157$	14.2183	1.13474	0.567371	+	0.823462i	$0.307961\pi$
$158$	−0.0332485	−0.00264511
$159$	−10.8398	−0.859649
$160$	−0.231615	−0.0183108
$161$	28.3947	2.23781
$162$	−3.02283	−0.237496
$163$	19.2893	1.51085	0.755426	−	0.655234i	$-0.227430\pi$
$163$	19.2893	1.51085	0.755426	+	0.655234i	$0.227430\pi$
$164$	8.53011	0.666089
$165$	3.12135	0.242997
$166$	−0.0511721	−0.00397173
$167$	3.99223	0.308928	0.154464	−	0.987998i	$-0.450635\pi$
$167$	3.99223	0.308928	0.154464	+	0.987998i	$0.450635\pi$
$168$	−9.59779	−0.740486
$169$	0	0
$170$	−0.0881919	−0.00676401
$171$	4.36830	0.334053
$172$	3.48940	0.266064
$173$	0.804379	0.0611558	0.0305779	−	0.999532i	$-0.490265\pi$
$173$	0.804379	0.0611558	0.0305779	+	0.999532i	$0.490265\pi$
$174$	10.4219	0.790083
$175$	17.4893	1.32207
$176$	−4.96470	−0.374228
$177$	−28.3288	−2.12933
$178$	−16.0134	−1.20026
$179$	−11.1147	−0.830750	−0.415375	−	0.909650i	$-0.636350\pi$
$179$	−11.1147	−0.830750	−0.415375	+	0.909650i	$0.636350\pi$
$180$	−1.01176	−0.0754124
$181$	−7.85674	−0.583987	−0.291993	−	0.956420i	$-0.594319\pi$
$181$	−7.85674	−0.583987	−0.291993	+	0.956420i	$0.594319\pi$
$182$	0	0
$183$	18.1990	1.34531
$184$	−8.03063	−0.592026
$185$	−0.218701	−0.0160792
$186$	−25.8063	−1.89221
$187$	−1.89041	−0.138240
$188$	−0.0285542	−0.00208253
$189$	−13.1327	−0.955263
$190$	−0.231615	−0.0168031
$191$	−4.63516	−0.335389	−0.167694	−	0.985839i	$-0.553632\pi$
$191$	−4.63516	−0.335389	−0.167694	+	0.985839i	$0.553632\pi$
$192$	2.71446	0.195899
$193$	−10.8030	−0.777618	−0.388809	−	0.921318i	$-0.627113\pi$
$193$	−10.8030	−0.777618	−0.388809	+	0.921318i	$0.627113\pi$
$194$	−2.09080	−0.150111
$195$	0	0
$196$	5.50187	0.392991
$197$	−23.3190	−1.66141	−0.830706	−	0.556711i	$-0.812063\pi$
$197$	−23.3190	−1.66141	−0.830706	+	0.556711i	$0.812063\pi$
$198$	−21.6873	−1.54125
$199$	4.59229	0.325539	0.162769	−	0.986664i	$-0.447957\pi$
$199$	4.59229	0.325539	0.162769	+	0.986664i	$0.447957\pi$
$200$	−4.94635	−0.349760
$201$	−39.4050	−2.77942
$202$	11.1746	0.786245
$203$	−13.5754	−0.952803
$204$	1.03359	0.0723655
$205$	−1.97570	−0.137989
$206$	13.2833	0.925488
$207$	−35.0802	−2.43824
$208$	0	0
$209$	−4.96470	−0.343415
$210$	2.22299	0.153401
$211$	−7.85209	−0.540560	−0.270280	−	0.962782i	$-0.587116\pi$
$211$	−7.85209	−0.540560	−0.270280	+	0.962782i	$0.587116\pi$
$212$	−3.99333	−0.274263
$213$	−2.94449	−0.201753
$214$	−1.16202	−0.0794343
$215$	−0.808196	−0.0551185
$216$	3.71421	0.252720
$217$	33.6147	2.28192
$218$	17.8789	1.21091
$219$	−15.8094	−1.06830
$220$	1.14990	0.0775260
$221$	0	0
$222$	2.56311	0.172025
$223$	−5.06468	−0.339156	−0.169578	−	0.985517i	$-0.554240\pi$
$223$	−5.06468	−0.339156	−0.169578	+	0.985517i	$0.554240\pi$
$224$	−3.53580	−0.236245
$225$	−21.6072	−1.44048
$226$	−16.7156	−1.11191
$227$	−21.8852	−1.45257	−0.726286	−	0.687393i	$-0.758755\pi$
$227$	−21.8852	−1.45257	−0.726286	+	0.687393i	$0.758755\pi$
$228$	2.71446	0.179770
$229$	17.9767	1.18794	0.593968	−	0.804488i	$-0.297560\pi$
$229$	17.9767	1.18794	0.593968	+	0.804488i	$0.297560\pi$
$230$	1.86001	0.122646
$231$	47.6501	3.13515
$232$	3.83940	0.252069
$233$	10.5596	0.691780	0.345890	−	0.938275i	$-0.387577\pi$
$233$	10.5596	0.691780	0.345890	+	0.938275i	$0.387577\pi$
$234$	0	0
$235$	0.00661358	0.000431422 0
$236$	−10.4363	−0.679343
$237$	−0.0902519	−0.00586249
$238$	−1.34633	−0.0872693
$239$	25.1092	1.62418	0.812090	−	0.583532i	$-0.198330\pi$
$239$	25.1092	1.62418	0.812090	+	0.583532i	$0.198330\pi$
$240$	−0.628709	−0.0405830
$241$	14.3045	0.921431	0.460715	−	0.887548i	$-0.347593\pi$
$241$	14.3045	0.921431	0.460715	+	0.887548i	$0.347593\pi$
$242$	13.6482	0.877341
$243$	−19.3480	−1.24117
$244$	6.70446	0.429209
$245$	−1.27431	−0.0814130
$246$	23.1547	1.47629
$247$	0	0
$248$	−9.50697	−0.603693
$249$	−0.138905	−0.00880274
$250$	2.30372	0.145700
$251$	−5.70881	−0.360337	−0.180168	−	0.983636i	$-0.557664\pi$
$251$	−5.70881	−0.360337	−0.180168	+	0.983636i	$0.557664\pi$
$252$	−15.4454	−0.972971
$253$	39.8696	2.50658
$254$	−6.94033	−0.435475
$255$	−0.239394	−0.0149914
$256$	1.00000	0.0625000
$257$	−19.8557	−1.23856	−0.619281	−	0.785170i	$-0.712576\pi$
$257$	−19.8557	−1.23856	−0.619281	+	0.785170i	$0.712576\pi$
$258$	9.47184	0.589691
$259$	−3.33865	−0.207454
$260$	0	0
$261$	16.7717	1.03814
$262$	6.94142	0.428842
$263$	25.0831	1.54669	0.773346	−	0.633984i	$-0.218582\pi$
$263$	25.0831	1.54669	0.773346	+	0.633984i	$0.218582\pi$
$264$	−13.4765	−0.829420
$265$	0.924915	0.0568171
$266$	−3.53580	−0.216794
$267$	−43.4678	−2.66019
$268$	−14.5167	−0.886748
$269$	−0.796459	−0.0485610	−0.0242805	−	0.999705i	$-0.507729\pi$
$269$	−0.796459	−0.0485610	−0.0242805	+	0.999705i	$0.507729\pi$
$270$	−0.860266	−0.0523541
$271$	−20.1732	−1.22543	−0.612717	−	0.790302i	$-0.709924\pi$
$271$	−20.1732	−1.22543	−0.612717	+	0.790302i	$0.709924\pi$
$272$	0.380770	0.0230876
$273$	0	0
$274$	16.4969	0.996611
$275$	24.5572	1.48085
$276$	−21.7988	−1.31214
$277$	−30.0692	−1.80669	−0.903343	−	0.428919i	$-0.858895\pi$
$277$	−30.0692	−1.80669	−0.903343	+	0.428919i	$0.858895\pi$
$278$	−21.7713	−1.30575
$279$	−41.5293	−2.48630
$280$	0.818943	0.0489412
$281$	25.5552	1.52449	0.762247	−	0.647286i	$-0.224096\pi$
$281$	25.5552	1.52449	0.762247	+	0.647286i	$0.224096\pi$
$282$	−0.0775093	−0.00461561
$283$	0.0956736	0.00568721	0.00284360	−	0.999996i	$-0.499095\pi$
$283$	0.0956736	0.00568721	0.00284360	+	0.999996i	$0.499095\pi$
$284$	−1.08474	−0.0643676
$285$	−0.628709	−0.0372415
$286$	0	0
$287$	−30.1607	−1.78033
$288$	4.36830	0.257405
$289$	−16.8550	−0.991471
$290$	−0.889262	−0.0522193
$291$	−5.67540	−0.332698
$292$	−5.82413	−0.340831
$293$	17.0515	0.996159	0.498079	−	0.867131i	$-0.334039\pi$
$293$	17.0515	0.996159	0.498079	+	0.867131i	$0.334039\pi$
$294$	14.9346	0.871005
$295$	2.41719	0.140734
$296$	0.944243	0.0548830
$297$	−18.4399	−1.06999
$298$	12.3771	0.716985
$299$	0	0
$300$	−13.4267	−0.775190
$301$	−12.3378	−0.711140
$302$	−5.47967	−0.315320
$303$	30.3331	1.74259
$304$	1.00000	0.0573539
$305$	−1.55285	−0.0889161
$306$	1.66332	0.0950856
$307$	−28.2284	−1.61108	−0.805540	−	0.592541i	$-0.798125\pi$
$307$	−28.2284	−1.61108	−0.805540	+	0.592541i	$0.798125\pi$
$308$	17.5542	1.00024
$309$	36.0569	2.05121
$310$	2.20195	0.125063
$311$	20.9050	1.18542	0.592708	−	0.805418i	$-0.298059\pi$
$311$	20.9050	1.18542	0.592708	+	0.805418i	$0.298059\pi$
$312$	0	0
$313$	24.2057	1.36819	0.684094	−	0.729394i	$-0.260198\pi$
$313$	24.2057	1.36819	0.684094	+	0.729394i	$0.260198\pi$
$314$	14.2183	0.802384
$315$	3.57739	0.201563
$316$	−0.0332485	−0.00187038
$317$	5.12258	0.287713	0.143856	−	0.989599i	$-0.454050\pi$
$317$	5.12258	0.287713	0.143856	+	0.989599i	$0.454050\pi$
$318$	−10.8398	−0.607863
$319$	−19.0615	−1.06724
$320$	−0.231615	−0.0129477
$321$	−3.15427	−0.176054
$322$	28.3947	1.58237
$323$	0.380770	0.0211866
$324$	−3.02283	−0.167935
$325$	0	0
$326$	19.2893	1.06833
$327$	48.5315	2.68380
$328$	8.53011	0.470996
$329$	0.100962	0.00556621
$330$	3.12135	0.171825
$331$	1.81606	0.0998196	0.0499098	−	0.998754i	$-0.484107\pi$
$331$	1.81606	0.0998196	0.0499098	+	0.998754i	$0.484107\pi$
$332$	−0.0511721	−0.00280844
$333$	4.12474	0.226034
$334$	3.99223	0.218445
$335$	3.36228	0.183701
$336$	−9.59779	−0.523602
$337$	15.6528	0.852662	0.426331	−	0.904567i	$-0.359806\pi$
$337$	15.6528	0.852662	0.426331	+	0.904567i	$0.359806\pi$
$338$	0	0
$339$	−45.3739	−2.46437
$340$	−0.0881919	−0.00478288
$341$	47.1992	2.55598
$342$	4.36830	0.236211
$343$	5.29708	0.286016
$344$	3.48940	0.188136
$345$	5.04893	0.271825
$346$	0.804379	0.0432437
$347$	13.6159	0.730940	0.365470	−	0.930823i	$-0.380908\pi$
$347$	13.6159	0.730940	0.365470	+	0.930823i	$0.380908\pi$
$348$	10.4219	0.558673
$349$	−7.46105	−0.399381	−0.199690	−	0.979859i	$-0.563994\pi$
$349$	−7.46105	−0.399381	−0.199690	+	0.979859i	$0.563994\pi$
$350$	17.4893	0.934843
$351$	0	0
$352$	−4.96470	−0.264619
$353$	−26.9124	−1.43240	−0.716201	−	0.697894i	$-0.754121\pi$
$353$	−26.9124	−1.43240	−0.716201	+	0.697894i	$0.754121\pi$
$354$	−28.3288	−1.50566
$355$	0.251242	0.0133346
$356$	−16.0134	−0.848709
$357$	−3.65455	−0.193419
$358$	−11.1147	−0.587429
$359$	−16.2131	−0.855692	−0.427846	−	0.903852i	$-0.640728\pi$
$359$	−16.2131	−0.855692	−0.427846	+	0.903852i	$0.640728\pi$
$360$	−1.01176	−0.0533246
$361$	1.00000	0.0526316
$362$	−7.85674	−0.412941
$363$	37.0476	1.94449
$364$	0	0
$365$	1.34895	0.0706075
$366$	18.1990	0.951277
$367$	22.0589	1.15147	0.575733	−	0.817638i	$-0.304717\pi$
$367$	22.0589	1.15147	0.575733	+	0.817638i	$0.304717\pi$
$368$	−8.03063	−0.418625
$369$	37.2621	1.93979
$370$	−0.218701	−0.0113697
$371$	14.1196	0.733054
$372$	−25.8063	−1.33800
$373$	−23.4846	−1.21599	−0.607993	−	0.793942i	$-0.708025\pi$
$373$	−23.4846	−1.21599	−0.607993	+	0.793942i	$0.708025\pi$
$374$	−1.89041	−0.0977506
$375$	6.25337	0.322922
$376$	−0.0285542	−0.00147257
$377$	0	0
$378$	−13.1327	−0.675473
$379$	1.94171	0.0997390	0.0498695	−	0.998756i	$-0.484119\pi$
$379$	1.94171	0.0997390	0.0498695	+	0.998756i	$0.484119\pi$
$380$	−0.231615	−0.0118816
$381$	−18.8393	−0.965164
$382$	−4.63516	−0.237156
$383$	−4.16353	−0.212746	−0.106373	−	0.994326i	$-0.533924\pi$
$383$	−4.16353	−0.212746	−0.106373	+	0.994326i	$0.533924\pi$
$384$	2.71446	0.138522
$385$	−4.06580	−0.207213
$386$	−10.8030	−0.549859
$387$	15.2428	0.774833
$388$	−2.09080	−0.106144
$389$	−35.2307	−1.78627	−0.893134	−	0.449791i	$-0.851498\pi$
$389$	−35.2307	−1.78627	−0.893134	+	0.449791i	$0.851498\pi$
$390$	0	0
$391$	−3.05782	−0.154641
$392$	5.50187	0.277886
$393$	18.8422	0.950464
$394$	−23.3190	−1.17480
$395$	0.00770085	0.000387472 0
$396$	−21.6873	−1.08983
$397$	10.5783	0.530908	0.265454	−	0.964123i	$-0.414478\pi$
$397$	10.5783	0.530908	0.265454	+	0.964123i	$0.414478\pi$
$398$	4.59229	0.230191
$399$	−9.59779	−0.480491
$400$	−4.94635	−0.247318
$401$	24.1362	1.20530	0.602652	−	0.798004i	$-0.294111\pi$
$401$	24.1362	1.20530	0.602652	+	0.798004i	$0.294111\pi$
$402$	−39.4050	−1.96534
$403$	0	0
$404$	11.1746	0.555959
$405$	0.700132	0.0347899
$406$	−13.5754	−0.673734
$407$	−4.68788	−0.232370
$408$	1.03359	0.0511701
$409$	5.67719	0.280719	0.140359	−	0.990101i	$-0.455174\pi$
$409$	5.67719	0.280719	0.140359	+	0.990101i	$0.455174\pi$
$410$	−1.97570	−0.0975728
$411$	44.7801	2.20884
$412$	13.2833	0.654419
$413$	36.9005	1.81576
$414$	−35.0802	−1.72410
$415$	0.0118522	0.000581803 0
$416$	0	0
$417$	−59.0973	−2.89401
$418$	−4.96470	−0.242831
$419$	16.2509	0.793908	0.396954	−	0.917839i	$-0.370067\pi$
$419$	16.2509	0.793908	0.396954	+	0.917839i	$0.370067\pi$
$420$	2.22299	0.108471
$421$	22.4923	1.09621	0.548105	−	0.836410i	$-0.315349\pi$
$421$	22.4923	1.09621	0.548105	+	0.836410i	$0.315349\pi$
$422$	−7.85209	−0.382234
$423$	−0.124734	−0.00606475
$424$	−3.99333	−0.193933
$425$	−1.88342	−0.0913595
$426$	−2.94449	−0.142661
$427$	−23.7056	−1.14720
$428$	−1.16202	−0.0561685
$429$	0	0
$430$	−0.808196	−0.0389747
$431$	2.04216	0.0983675	0.0491837	−	0.998790i	$-0.484338\pi$
$431$	2.04216	0.0983675	0.0491837	+	0.998790i	$0.484338\pi$
$432$	3.71421	0.178700
$433$	−7.02291	−0.337500	−0.168750	−	0.985659i	$-0.553973\pi$
$433$	−7.02291	−0.337500	−0.168750	+	0.985659i	$0.553973\pi$
$434$	33.6147	1.61356
$435$	−2.41387	−0.115736
$436$	17.8789	0.856242
$437$	−8.03063	−0.384157
$438$	−15.8094	−0.755401
$439$	−7.92343	−0.378165	−0.189082	−	0.981961i	$-0.560551\pi$
$439$	−7.92343	−0.378165	−0.189082	+	0.981961i	$0.560551\pi$
$440$	1.14990	0.0548192
$441$	24.0338	1.14447
$442$	0	0
$443$	15.0553	0.715299	0.357650	−	0.933856i	$-0.383578\pi$
$443$	15.0553	0.715299	0.357650	+	0.933856i	$0.383578\pi$
$444$	2.56311	0.121640
$445$	3.70894	0.175821
$446$	−5.06468	−0.239820
$447$	33.5971	1.58909
$448$	−3.53580	−0.167051
$449$	38.6989	1.82631	0.913156	−	0.407611i	$-0.133638\pi$
$449$	38.6989	1.82631	0.913156	+	0.407611i	$0.133638\pi$
$450$	−21.6072	−1.01857
$451$	−42.3494	−1.99416
$452$	−16.7156	−0.786237
$453$	−14.8744	−0.698859
$454$	−21.8852	−1.02712
$455$	0	0
$456$	2.71446	0.127116
$457$	−4.55096	−0.212885	−0.106442	−	0.994319i	$-0.533946\pi$
$457$	−4.55096	−0.212885	−0.106442	+	0.994319i	$0.533946\pi$
$458$	17.9767	0.839998
$459$	1.41426	0.0660120
$460$	1.86001	0.0867235
$461$	22.8868	1.06595	0.532973	−	0.846132i	$-0.321075\pi$
$461$	22.8868	1.06595	0.532973	+	0.846132i	$0.321075\pi$
$462$	47.6501	2.21688
$463$	−27.8763	−1.29552	−0.647762	−	0.761843i	$-0.724295\pi$
$463$	−27.8763	−1.29552	−0.647762	+	0.761843i	$0.724295\pi$
$464$	3.83940	0.178240
$465$	5.97712	0.277182
$466$	10.5596	0.489162
$467$	−26.4598	−1.22441	−0.612207	−	0.790697i	$-0.709718\pi$
$467$	−26.4598	−1.22441	−0.612207	+	0.790697i	$0.709718\pi$
$468$	0	0
$469$	51.3281	2.37011
$470$	0.00661358	0.000305062 0
$471$	38.5950	1.77836
$472$	−10.4363	−0.480368
$473$	−17.3238	−0.796550
$474$	−0.0902519	−0.00414541
$475$	−4.94635	−0.226954
$476$	−1.34633	−0.0617087
$477$	−17.4441	−0.798710
$478$	25.1092	1.14847
$479$	−3.04282	−0.139030	−0.0695151	−	0.997581i	$-0.522145\pi$
$479$	−3.04282	−0.139030	−0.0695151	+	0.997581i	$0.522145\pi$
$480$	−0.628709	−0.0286965
$481$	0	0
$482$	14.3045	0.651550
$483$	77.0763	3.50709
$484$	13.6482	0.620373
$485$	0.484261	0.0219891
$486$	−19.3480	−0.877643
$487$	−34.9069	−1.58178	−0.790891	−	0.611957i	$-0.790382\pi$
$487$	−34.9069	−1.58178	−0.790891	+	0.611957i	$0.790382\pi$
$488$	6.70446	0.303497
$489$	52.3600	2.36780
$490$	−1.27431	−0.0575677
$491$	−21.8529	−0.986207	−0.493104	−	0.869971i	$-0.664138\pi$
$491$	−21.8529	−0.986207	−0.493104	+	0.869971i	$0.664138\pi$
$492$	23.1547	1.04389
$493$	1.46193	0.0658420
$494$	0	0
$495$	5.02310	0.225772
$496$	−9.50697	−0.426876
$497$	3.83543	0.172043
$498$	−0.138905	−0.00622447
$499$	−12.9680	−0.580529	−0.290265	−	0.956946i	$-0.593743\pi$
$499$	−12.9680	−0.580529	−0.290265	+	0.956946i	$0.593743\pi$
$500$	2.30372	0.103026
$501$	10.8367	0.484150
$502$	−5.70881	−0.254797
$503$	14.2382	0.634850	0.317425	−	0.948283i	$-0.397182\pi$
$503$	14.2382	0.634850	0.317425	+	0.948283i	$0.397182\pi$
$504$	−15.4454	−0.687995
$505$	−2.58821	−0.115174
$506$	39.8696	1.77242
$507$	0	0
$508$	−6.94033	−0.307927
$509$	18.1749	0.805588	0.402794	−	0.915291i	$-0.368039\pi$
$509$	18.1749	0.805588	0.402794	+	0.915291i	$0.368039\pi$
$510$	−0.239394	−0.0106005
$511$	20.5930	0.910979
$512$	1.00000	0.0441942
$513$	3.71421	0.163986
$514$	−19.8557	−0.875795
$515$	−3.07660	−0.135571
$516$	9.47184	0.416975
$517$	0.141763	0.00623473
$518$	−3.33865	−0.146692
$519$	2.18346	0.0958431
$520$	0	0
$521$	−16.3670	−0.717051	−0.358525	−	0.933520i	$-0.616720\pi$
$521$	−16.3670	−0.717051	−0.358525	+	0.933520i	$0.616720\pi$
$522$	16.7717	0.734077
$523$	−6.31638	−0.276196	−0.138098	−	0.990419i	$-0.544099\pi$
$523$	−6.31638	−0.276196	−0.138098	+	0.990419i	$0.544099\pi$
$524$	6.94142	0.303237
$525$	47.4741	2.07194
$526$	25.0831	1.09368
$527$	−3.61997	−0.157688
$528$	−13.4765	−0.586489
$529$	41.4910	1.80396
$530$	0.924915	0.0401757
$531$	−45.5888	−1.97838
$532$	−3.53580	−0.153296
$533$	0	0
$534$	−43.4678	−1.88104
$535$	0.269142	0.0116360
$536$	−14.5167	−0.627026
$537$	−30.1704	−1.30195
$538$	−0.796459	−0.0343378
$539$	−27.3151	−1.17655
$540$	−0.860266	−0.0370199
$541$	24.0687	1.03480	0.517398	−	0.855745i	$-0.326901\pi$
$541$	24.0687	1.03480	0.517398	+	0.855745i	$0.326901\pi$
$542$	−20.1732	−0.866513
$543$	−21.3268	−0.915221
$544$	0.380770	0.0163254
$545$	−4.14101	−0.177381
$546$	0	0
$547$	−11.8701	−0.507530	−0.253765	−	0.967266i	$-0.581669\pi$
$547$	−11.8701	−0.507530	−0.253765	+	0.967266i	$0.581669\pi$
$548$	16.4969	0.704711
$549$	29.2871	1.24994
$550$	24.5572	1.04712
$551$	3.83940	0.163564
$552$	−21.7988	−0.927820
$553$	0.117560	0.00499916
$554$	−30.0692	−1.27752
$555$	−0.593654	−0.0251992
$556$	−21.7713	−0.923308
$557$	4.70856	0.199508	0.0997541	−	0.995012i	$-0.468194\pi$
$557$	4.70856	0.199508	0.0997541	+	0.995012i	$0.468194\pi$
$558$	−41.5293	−1.75808
$559$	0	0
$560$	0.818943	0.0346067
$561$	−5.13144	−0.216650
$562$	25.5552	1.07798
$563$	−12.4418	−0.524361	−0.262180	−	0.965019i	$-0.584442\pi$
$563$	−12.4418	−0.524361	−0.262180	+	0.965019i	$0.584442\pi$
$564$	−0.0775093	−0.00326373
$565$	3.87158	0.162879
$566$	0.0956736	0.00402146
$567$	10.6881	0.448859
$568$	−1.08474	−0.0455148
$569$	−43.5622	−1.82622	−0.913112	−	0.407709i	$-0.866328\pi$
$569$	−43.5622	−1.82622	−0.913112	+	0.407709i	$0.866328\pi$
$570$	−0.628709	−0.0263337
$571$	23.7652	0.994543	0.497272	−	0.867595i	$-0.334335\pi$
$571$	23.7652	0.994543	0.497272	+	0.867595i	$0.334335\pi$
$572$	0	0
$573$	−12.5820	−0.525620
$574$	−30.1607	−1.25888
$575$	39.7223	1.65654
$576$	4.36830	0.182013
$577$	−18.3437	−0.763657	−0.381828	−	0.924233i	$-0.624705\pi$
$577$	−18.3437	−0.763657	−0.381828	+	0.924233i	$0.624705\pi$
$578$	−16.8550	−0.701076
$579$	−29.3244	−1.21868
$580$	−0.889262	−0.0369246
$581$	0.180934	0.00750642
$582$	−5.67540	−0.235253
$583$	19.8257	0.821096
$584$	−5.82413	−0.241004
$585$	0	0
$586$	17.0515	0.704391
$587$	−7.03387	−0.290319	−0.145159	−	0.989408i	$-0.546370\pi$
$587$	−7.03387	−0.290319	−0.145159	+	0.989408i	$0.546370\pi$
$588$	14.9346	0.615893
$589$	−9.50697	−0.391728
$590$	2.41719	0.0995142
$591$	−63.2986	−2.60376
$592$	0.944243	0.0388081
$593$	43.0804	1.76910	0.884551	−	0.466444i	$-0.154465\pi$
$593$	43.0804	1.76910	0.884551	+	0.466444i	$0.154465\pi$
$594$	−18.4399	−0.756599
$595$	0.311829	0.0127837
$596$	12.3771	0.506985
$597$	12.4656	0.510183
$598$	0	0
$599$	−25.8861	−1.05768	−0.528839	−	0.848722i	$-0.677372\pi$
$599$	−25.8861	−1.05768	−0.528839	+	0.848722i	$0.677372\pi$
$600$	−13.4267	−0.548142
$601$	−17.6477	−0.719863	−0.359931	−	0.932979i	$-0.617200\pi$
$601$	−17.6477	−0.719863	−0.359931	+	0.932979i	$0.617200\pi$
$602$	−12.3378	−0.502852
$603$	−63.4133	−2.58239
$604$	−5.47967	−0.222965
$605$	−3.16113	−0.128518
$606$	30.3331	1.23220
$607$	30.2432	1.22753	0.613767	−	0.789487i	$-0.289653\pi$
$607$	30.2432	1.22753	0.613767	+	0.789487i	$0.289653\pi$
$608$	1.00000	0.0405554
$609$	−36.8498	−1.49323
$610$	−1.55285	−0.0628731
$611$	0	0
$612$	1.66332	0.0672357
$613$	−33.1735	−1.33987	−0.669933	−	0.742422i	$-0.733677\pi$
$613$	−33.1735	−1.33987	−0.669933	+	0.742422i	$0.733677\pi$
$614$	−28.2284	−1.13921
$615$	−5.36296	−0.216255
$616$	17.5542	0.707278
$617$	29.6645	1.19425	0.597124	−	0.802149i	$-0.296310\pi$
$617$	29.6645	1.19425	0.597124	+	0.802149i	$0.296310\pi$
$618$	36.0569	1.45042
$619$	−34.3447	−1.38043	−0.690215	−	0.723604i	$-0.742484\pi$
$619$	−34.3447	−1.38043	−0.690215	+	0.723604i	$0.742484\pi$
$620$	2.20195	0.0884326
$621$	−29.8274	−1.19693
$622$	20.9050	0.838215
$623$	56.6202	2.26844
$624$	0	0
$625$	24.1982	0.967928
$626$	24.2057	0.967455
$627$	−13.4765	−0.538199
$628$	14.2183	0.567371
$629$	0.359539	0.0143358
$630$	3.57739	0.142527
$631$	21.2514	0.846006	0.423003	−	0.906128i	$-0.360976\pi$
$631$	21.2514	0.846006	0.423003	+	0.906128i	$0.360976\pi$
$632$	−0.0332485	−0.00132256
$633$	−21.3142	−0.847163
$634$	5.12258	0.203443
$635$	1.60748	0.0637910
$636$	−10.8398	−0.429824
$637$	0	0
$638$	−19.0615	−0.754651
$639$	−4.73848	−0.187451
$640$	−0.231615	−0.00915538
$641$	−20.7030	−0.817717	−0.408859	−	0.912598i	$-0.634073\pi$
$641$	−20.7030	−0.817717	−0.408859	+	0.912598i	$0.634073\pi$
$642$	−3.15427	−0.124489
$643$	1.31799	0.0519765	0.0259882	−	0.999662i	$-0.491727\pi$
$643$	1.31799	0.0519765	0.0259882	+	0.999662i	$0.491727\pi$
$644$	28.3947	1.11891
$645$	−2.19382	−0.0863815
$646$	0.380770	0.0149812
$647$	0.953841	0.0374993	0.0187497	−	0.999824i	$-0.494031\pi$
$647$	0.953841	0.0374993	0.0187497	+	0.999824i	$0.494031\pi$
$648$	−3.02283	−0.118748
$649$	51.8129	2.03383
$650$	0	0
$651$	91.2459	3.57621
$652$	19.2893	0.755426
$653$	2.36191	0.0924286	0.0462143	−	0.998932i	$-0.485284\pi$
$653$	2.36191	0.0924286	0.0462143	+	0.998932i	$0.485284\pi$
$654$	48.5315	1.89773
$655$	−1.60773	−0.0628194
$656$	8.53011	0.333045
$657$	−25.4416	−0.992570
$658$	0.100962	0.00393591
$659$	−23.2883	−0.907182	−0.453591	−	0.891210i	$-0.649857\pi$
$659$	−23.2883	−0.907182	−0.453591	+	0.891210i	$0.649857\pi$
$660$	3.12135	0.121498
$661$	−40.0869	−1.55920	−0.779599	−	0.626279i	$-0.784577\pi$
$661$	−40.0869	−1.55920	−0.779599	+	0.626279i	$0.784577\pi$
$662$	1.81606	0.0705831
$663$	0	0
$664$	−0.0511721	−0.00198586
$665$	0.818943	0.0317572
$666$	4.12474	0.159830
$667$	−30.8328	−1.19385
$668$	3.99223	0.154464
$669$	−13.7479	−0.531524
$670$	3.36228	0.129896
$671$	−33.2856	−1.28498
$672$	−9.59779	−0.370243
$673$	0.810634	0.0312476	0.0156238	−	0.999878i	$-0.495027\pi$
$673$	0.810634	0.0312476	0.0156238	+	0.999878i	$0.495027\pi$
$674$	15.6528	0.602923
$675$	−18.3718	−0.707131
$676$	0	0
$677$	−48.7701	−1.87439	−0.937194	−	0.348808i	$-0.886587\pi$
$677$	−48.7701	−1.87439	−0.937194	+	0.348808i	$0.886587\pi$
$678$	−45.3739	−1.74257
$679$	7.39266	0.283704
$680$	−0.0881919	−0.00338201
$681$	−59.4065	−2.27646
$682$	47.1992	1.80735
$683$	18.7243	0.716465	0.358232	−	0.933632i	$-0.383380\pi$
$683$	18.7243	0.716465	0.358232	+	0.933632i	$0.383380\pi$
$684$	4.36830	0.167026
$685$	−3.82091	−0.145990
$686$	5.29708	0.202244
$687$	48.7972	1.86173
$688$	3.48940	0.133032
$689$	0	0
$690$	5.04893	0.192210
$691$	38.6969	1.47210	0.736050	−	0.676927i	$-0.236689\pi$
$691$	38.6969	1.47210	0.736050	+	0.676927i	$0.236689\pi$
$692$	0.804379	0.0305779
$693$	76.6819	2.91291
$694$	13.6159	0.516853
$695$	5.04255	0.191275
$696$	10.4219	0.395042
$697$	3.24801	0.123027
$698$	−7.46105	−0.282405
$699$	28.6635	1.08415
$700$	17.4893	0.661034
$701$	−19.7902	−0.747464	−0.373732	−	0.927537i	$-0.621922\pi$
$701$	−19.7902	−0.747464	−0.373732	+	0.927537i	$0.621922\pi$
$702$	0	0
$703$	0.944243	0.0356128
$704$	−4.96470	−0.187114
$705$	0.0179523	0.000676123 0
$706$	−26.9124	−1.01286
$707$	−39.5113	−1.48597
$708$	−28.3288	−1.06466
$709$	20.5586	0.772094	0.386047	−	0.922479i	$-0.373840\pi$
$709$	20.5586	0.772094	0.386047	+	0.922479i	$0.373840\pi$
$710$	0.251242	0.00942895
$711$	−0.145240	−0.00544691
$712$	−16.0134	−0.600128
$713$	76.3470	2.85922
$714$	−3.65455	−0.136768
$715$	0	0
$716$	−11.1147	−0.415375
$717$	68.1580	2.54541
$718$	−16.2131	−0.605066
$719$	19.2535	0.718035	0.359018	−	0.933331i	$-0.383112\pi$
$719$	19.2535	0.718035	0.359018	+	0.933331i	$0.383112\pi$
$720$	−1.01176	−0.0377062
$721$	−46.9669	−1.74914
$722$	1.00000	0.0372161
$723$	38.8289	1.44406
$724$	−7.85674	−0.291993
$725$	−18.9911	−0.705310
$726$	37.0476	1.37496
$727$	26.6100	0.986910	0.493455	−	0.869771i	$-0.335734\pi$
$727$	26.6100	0.986910	0.493455	+	0.869771i	$0.335734\pi$
$728$	0	0
$729$	−43.4509	−1.60929
$730$	1.34895	0.0499270
$731$	1.32866	0.0491422
$732$	18.1990	0.672655
$733$	−30.5736	−1.12926	−0.564631	−	0.825343i	$-0.690982\pi$
$733$	−30.5736	−1.12926	−0.564631	+	0.825343i	$0.690982\pi$
$734$	22.0589	0.814210
$735$	−3.45908	−0.127590
$736$	−8.03063	−0.296013
$737$	72.0710	2.65477
$738$	37.2621	1.37164
$739$	33.1721	1.22026	0.610128	−	0.792303i	$-0.291118\pi$
$739$	33.1721	1.22026	0.610128	+	0.792303i	$0.291118\pi$
$740$	−0.218701	−0.00803959
$741$	0	0
$742$	14.1196	0.518348
$743$	−6.08989	−0.223416	−0.111708	−	0.993741i	$-0.535632\pi$
$743$	−6.08989	−0.223416	−0.111708	+	0.993741i	$0.535632\pi$
$744$	−25.8063	−0.946105
$745$	−2.86671	−0.105028
$746$	−23.4846	−0.859832
$747$	−0.223535	−0.00817873
$748$	−1.89041	−0.0691201
$749$	4.10868	0.150128
$750$	6.25337	0.228341
$751$	27.0177	0.985891	0.492946	−	0.870060i	$-0.335920\pi$
$751$	27.0177	0.985891	0.492946	+	0.870060i	$0.335920\pi$
$752$	−0.0285542	−0.00104127
$753$	−15.4963	−0.564718
$754$	0	0
$755$	1.26917	0.0461899
$756$	−13.1327	−0.477632
$757$	23.2816	0.846185	0.423092	−	0.906087i	$-0.360945\pi$
$757$	23.2816	0.846185	0.423092	+	0.906087i	$0.360945\pi$
$758$	1.94171	0.0705261
$759$	108.225	3.92830
$760$	−0.231615	−0.00840155
$761$	−49.5768	−1.79716	−0.898579	−	0.438811i	$-0.855400\pi$
$761$	−49.5768	−1.79716	−0.898579	+	0.438811i	$0.855400\pi$
$762$	−18.8393	−0.682474
$763$	−63.2160	−2.28857
$764$	−4.63516	−0.167694
$765$	−0.385249	−0.0139287
$766$	−4.16353	−0.150434
$767$	0	0
$768$	2.71446	0.0979497
$769$	−49.1905	−1.77385	−0.886927	−	0.461909i	$-0.847165\pi$
$769$	−49.1905	−1.77385	−0.886927	+	0.461909i	$0.847165\pi$
$770$	−4.06580	−0.146521
$771$	−53.8974	−1.94107
$772$	−10.8030	−0.388809
$773$	21.7610	0.782687	0.391344	−	0.920245i	$-0.372010\pi$
$773$	21.7610	0.782687	0.391344	+	0.920245i	$0.372010\pi$
$774$	15.2428	0.547890
$775$	47.0249	1.68918
$776$	−2.09080	−0.0750554
$777$	−9.06264	−0.325121
$778$	−35.2307	−1.26308
$779$	8.53011	0.305623
$780$	0	0
$781$	5.38542	0.192705
$782$	−3.05782	−0.109347
$783$	14.2604	0.509623
$784$	5.50187	0.196495
$785$	−3.29316	−0.117538
$786$	18.8422	0.672080
$787$	4.25032	0.151507	0.0757537	−	0.997127i	$-0.475864\pi$
$787$	4.25032	0.151507	0.0757537	+	0.997127i	$0.475864\pi$
$788$	−23.3190	−0.830706
$789$	68.0872	2.42397
$790$	0.00770085	0.000273984 0
$791$	59.1031	2.10146
$792$	−21.6873	−0.770625
$793$	0	0
$794$	10.5783	0.375409
$795$	2.51065	0.0890435
$796$	4.59229	0.162769
$797$	−3.28860	−0.116488	−0.0582441	−	0.998302i	$-0.518550\pi$
$797$	−3.28860	−0.116488	−0.0582441	+	0.998302i	$0.518550\pi$
$798$	−9.59779	−0.339758
$799$	−0.0108726	−0.000384645 0
$800$	−4.94635	−0.174880
$801$	−69.9514	−2.47161
$802$	24.1362	0.852279
$803$	28.9150	1.02039
$804$	−39.4050	−1.38971
$805$	−6.57663	−0.231796
$806$	0	0
$807$	−2.16196	−0.0761045
$808$	11.1746	0.393122
$809$	−20.8979	−0.734730	−0.367365	−	0.930077i	$-0.619740\pi$
$809$	−20.8979	−0.734730	−0.367365	+	0.930077i	$0.619740\pi$
$810$	0.700132	0.0246001
$811$	5.70238	0.200238	0.100119	−	0.994975i	$-0.468078\pi$
$811$	5.70238	0.200238	0.100119	+	0.994975i	$0.468078\pi$
$812$	−13.5754	−0.476402
$813$	−54.7594	−1.92050
$814$	−4.68788	−0.164310
$815$	−4.46768	−0.156496
$816$	1.03359	0.0361827
$817$	3.48940	0.122079
$818$	5.67719	0.198498
$819$	0	0
$820$	−1.97570	−0.0689944
$821$	8.29970	0.289662	0.144831	−	0.989456i	$-0.453736\pi$
$821$	8.29970	0.289662	0.144831	+	0.989456i	$0.453736\pi$
$822$	44.7801	1.56188
$823$	24.6141	0.857994	0.428997	−	0.903306i	$-0.358867\pi$
$823$	24.6141	0.857994	0.428997	+	0.903306i	$0.358867\pi$
$824$	13.2833	0.462744
$825$	66.6595	2.32078
$826$	36.9005	1.28393
$827$	−4.08885	−0.142183	−0.0710917	−	0.997470i	$-0.522648\pi$
$827$	−4.08885	−0.142183	−0.0710917	+	0.997470i	$0.522648\pi$
$828$	−35.0802	−1.21912
$829$	−24.6629	−0.856578	−0.428289	−	0.903642i	$-0.640883\pi$
$829$	−24.6629	−0.856578	−0.428289	+	0.903642i	$0.640883\pi$
$830$	0.0118522	0.000411397 0
$831$	−81.6218	−2.83143
$832$	0	0
$833$	2.09495	0.0725856
$834$	−59.0973	−2.04637
$835$	−0.924658	−0.0319991
$836$	−4.96470	−0.171708
$837$	−35.3109	−1.22052
$838$	16.2509	0.561378
$839$	23.6508	0.816517	0.408258	−	0.912866i	$-0.366136\pi$
$839$	23.6508	0.816517	0.408258	+	0.912866i	$0.366136\pi$
$840$	2.22299	0.0767004
$841$	−14.2590	−0.491689
$842$	22.4923	0.775137
$843$	69.3686	2.38918
$844$	−7.85209	−0.270280
$845$	0	0
$846$	−0.124734	−0.00428843
$847$	−48.2573	−1.65814
$848$	−3.99333	−0.137132
$849$	0.259702	0.00891296
$850$	−1.88342	−0.0646009
$851$	−7.58286	−0.259937
$852$	−2.94449	−0.100877
$853$	−22.8299	−0.781682	−0.390841	−	0.920458i	$-0.627816\pi$
$853$	−22.8299	−0.781682	−0.390841	+	0.920458i	$0.627816\pi$
$854$	−23.7056	−0.811190
$855$	−1.01176	−0.0346016
$856$	−1.16202	−0.0397172
$857$	−33.2805	−1.13684	−0.568420	−	0.822738i	$-0.692445\pi$
$857$	−33.2805	−1.13684	−0.568420	+	0.822738i	$0.692445\pi$
$858$	0	0
$859$	41.7476	1.42441	0.712205	−	0.701972i	$-0.247697\pi$
$859$	41.7476	1.42441	0.712205	+	0.701972i	$0.247697\pi$
$860$	−0.808196	−0.0275593
$861$	−81.8702	−2.79013
$862$	2.04216	0.0695563
$863$	32.8718	1.11897	0.559485	−	0.828841i	$-0.310999\pi$
$863$	32.8718	1.11897	0.559485	+	0.828841i	$0.310999\pi$
$864$	3.71421	0.126360
$865$	−0.186306	−0.00633460
$866$	−7.02291	−0.238648
$867$	−45.7523	−1.55383
$868$	33.6147	1.14096
$869$	0.165069	0.00559958
$870$	−2.41387	−0.0818378
$871$	0	0
$872$	17.8789	0.605454
$873$	−9.13326	−0.309114
$874$	−8.03063	−0.271640
$875$	−8.14550	−0.275368
$876$	−15.8094	−0.534149
$877$	10.2238	0.345232	0.172616	−	0.984989i	$-0.444778\pi$
$877$	10.2238	0.345232	0.172616	+	0.984989i	$0.444778\pi$
$878$	−7.92343	−0.267403
$879$	46.2856	1.56118
$880$	1.14990	0.0387630
$881$	−30.5640	−1.02973	−0.514863	−	0.857272i	$-0.672157\pi$
$881$	−30.5640	−1.02973	−0.514863	+	0.857272i	$0.672157\pi$
$882$	24.0338	0.809262
$883$	−10.4048	−0.350151	−0.175075	−	0.984555i	$-0.556017\pi$
$883$	−10.4048	−0.350151	−0.175075	+	0.984555i	$0.556017\pi$
$884$	0	0
$885$	6.56138	0.220558
$886$	15.0553	0.505793
$887$	20.2600	0.680265	0.340132	−	0.940378i	$-0.389528\pi$
$887$	20.2600	0.680265	0.340132	+	0.940378i	$0.389528\pi$
$888$	2.56311	0.0860124
$889$	24.5396	0.823032
$890$	3.70894	0.124324
$891$	15.0074	0.502768
$892$	−5.06468	−0.169578
$893$	−0.0285542	−0.000955530 0
$894$	33.5971	1.12366
$895$	2.57432	0.0860502
$896$	−3.53580	−0.118123
$897$	0	0
$898$	38.6989	1.29140
$899$	−36.5011	−1.21738
$900$	−21.6072	−0.720239
$901$	−1.52054	−0.0506566
$902$	−42.3494	−1.41008
$903$	−33.4905	−1.11449
$904$	−16.7156	−0.555953
$905$	1.81974	0.0604901
$906$	−14.8744	−0.494168
$907$	−48.8981	−1.62363	−0.811817	−	0.583913i	$-0.801521\pi$
$907$	−48.8981	−1.62363	−0.811817	+	0.583913i	$0.801521\pi$
$908$	−21.8852	−0.726286
$909$	48.8142	1.61907
$910$	0	0
$911$	39.1289	1.29640	0.648200	−	0.761470i	$-0.275522\pi$
$911$	39.1289	1.29640	0.648200	+	0.761470i	$0.275522\pi$
$912$	2.71446	0.0898848
$913$	0.254054	0.00840797
$914$	−4.55096	−0.150532
$915$	−4.21516	−0.139349
$916$	17.9767	0.593968
$917$	−24.5435	−0.810496
$918$	1.41426	0.0466775
$919$	55.8068	1.84090	0.920448	−	0.390866i	$-0.127824\pi$
$919$	55.8068	1.84090	0.920448	+	0.390866i	$0.127824\pi$
$920$	1.86001	0.0613228
$921$	−76.6250	−2.52488
$922$	22.8868	0.753738
$923$	0	0
$924$	47.6501	1.56757
$925$	−4.67056	−0.153567
$926$	−27.8763	−0.916073
$927$	58.0253	1.90580
$928$	3.83940	0.126035
$929$	−53.2017	−1.74549	−0.872746	−	0.488175i	$-0.837663\pi$
$929$	−53.2017	−1.74549	−0.872746	+	0.488175i	$0.837663\pi$
$930$	5.97712	0.195998
$931$	5.50187	0.180317
$932$	10.5596	0.345890
$933$	56.7459	1.85778
$934$	−26.4598	−0.865792
$935$	0.437846	0.0143191
$936$	0	0
$937$	−25.6406	−0.837642	−0.418821	−	0.908069i	$-0.637557\pi$
$937$	−25.6406	−0.837642	−0.418821	+	0.908069i	$0.637557\pi$
$938$	51.3281	1.67592
$939$	65.7055	2.14422
$940$	0.00661358	0.000215711 0
$941$	−28.4643	−0.927911	−0.463955	−	0.885859i	$-0.653570\pi$
$941$	−28.4643	−0.927911	−0.463955	+	0.885859i	$0.653570\pi$
$942$	38.5950	1.25749
$943$	−68.5021	−2.23074
$944$	−10.4363	−0.339671
$945$	3.04173	0.0989474
$946$	−17.3238	−0.563246
$947$	−6.34460	−0.206172	−0.103086	−	0.994672i	$-0.532872\pi$
$947$	−6.34460	−0.206172	−0.103086	+	0.994672i	$0.532872\pi$
$948$	−0.0902519	−0.00293124
$949$	0	0
$950$	−4.94635	−0.160481
$951$	13.9050	0.450902
$952$	−1.34633	−0.0436347
$953$	2.64922	0.0858167	0.0429084	−	0.999079i	$-0.486338\pi$
$953$	2.64922	0.0858167	0.0429084	+	0.999079i	$0.486338\pi$
$954$	−17.4441	−0.564774
$955$	1.07357	0.0347400
$956$	25.1092	0.812090
$957$	−51.7417	−1.67257
$958$	−3.04282	−0.0983092
$959$	−58.3295	−1.88356
$960$	−0.628709	−0.0202915
$961$	59.3825	1.91557
$962$	0	0
$963$	−5.07607	−0.163574
$964$	14.3045	0.460715
$965$	2.50214	0.0805466
$966$	77.0763	2.47989
$967$	−55.1123	−1.77229	−0.886146	−	0.463406i	$-0.846627\pi$
$967$	−55.1123	−1.77229	−0.886146	+	0.463406i	$0.846627\pi$
$968$	13.6482	0.438670
$969$	1.03359	0.0332036
$970$	0.484261	0.0155487
$971$	−0.794986	−0.0255123	−0.0127562	−	0.999919i	$-0.504061\pi$
$971$	−0.794986	−0.0255123	−0.0127562	+	0.999919i	$0.504061\pi$
$972$	−19.3480	−0.620587
$973$	76.9788	2.46783
$974$	−34.9069	−1.11849
$975$	0	0
$976$	6.70446	0.214605
$977$	11.8979	0.380647	0.190324	−	0.981721i	$-0.439046\pi$
$977$	11.8979	0.380647	0.190324	+	0.981721i	$0.439046\pi$
$978$	52.3600	1.67429
$979$	79.5017	2.54089
$980$	−1.27431	−0.0407065
$981$	78.1003	2.49355
$982$	−21.8529	−0.697354
$983$	−15.0146	−0.478891	−0.239445	−	0.970910i	$-0.576966\pi$
$983$	−15.0146	−0.478891	−0.239445	+	0.970910i	$0.576966\pi$
$984$	23.1547	0.738143
$985$	5.40103	0.172091
$986$	1.46193	0.0465573
$987$	0.274057	0.00872334
$988$	0	0
$989$	−28.0221	−0.891050
$990$	5.02310	0.159645
$991$	−61.5921	−1.95654	−0.978269	−	0.207338i	$-0.933520\pi$
$991$	−61.5921	−1.95654	−0.978269	+	0.207338i	$0.933520\pi$
$992$	−9.50697	−0.301847
$993$	4.92962	0.156437
$994$	3.83543	0.121652
$995$	−1.06364	−0.0337197
$996$	−0.138905	−0.00440137
$997$	25.5383	0.808807	0.404404	−	0.914581i	$-0.367479\pi$
$997$	25.5383	0.808807	0.404404	+	0.914581i	$0.367479\pi$
$998$	−12.9680	−0.410496
$999$	3.50712	0.110960

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bk.1.9	yes	9
13.12	even	2		6422.2.a.bi.1.9	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.bi.1.9	✓	9	13.12	even	2
6422.2.a.bk.1.9	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.71446 q^{3} +1.00000 q^{4} -0.231615 q^{5} +2.71446 q^{6} -3.53580 q^{7} +1.00000 q^{8} +4.36830 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.71446 q^{3} +1.00000 q^{4} -0.231615 q^{5} +2.71446 q^{6} -3.53580 q^{7} +1.00000 q^{8} +4.36830 q^{9} -0.231615 q^{10} -4.96470 q^{11} +2.71446 q^{12} -3.53580 q^{14} -0.628709 q^{15} +1.00000 q^{16} +0.380770 q^{17} +4.36830 q^{18} +1.00000 q^{19} -0.231615 q^{20} -9.59779 q^{21} -4.96470 q^{22} -8.03063 q^{23} +2.71446 q^{24} -4.94635 q^{25} +3.71421 q^{27} -3.53580 q^{28} +3.83940 q^{29} -0.628709 q^{30} -9.50697 q^{31} +1.00000 q^{32} -13.4765 q^{33} +0.380770 q^{34} +0.818943 q^{35} +4.36830 q^{36} +0.944243 q^{37} +1.00000 q^{38} -0.231615 q^{40} +8.53011 q^{41} -9.59779 q^{42} +3.48940 q^{43} -4.96470 q^{44} -1.01176 q^{45} -8.03063 q^{46} -0.0285542 q^{47} +2.71446 q^{48} +5.50187 q^{49} -4.94635 q^{50} +1.03359 q^{51} -3.99333 q^{53} +3.71421 q^{54} +1.14990 q^{55} -3.53580 q^{56} +2.71446 q^{57} +3.83940 q^{58} -10.4363 q^{59} -0.628709 q^{60} +6.70446 q^{61} -9.50697 q^{62} -15.4454 q^{63} +1.00000 q^{64} -13.4765 q^{66} -14.5167 q^{67} +0.380770 q^{68} -21.7988 q^{69} +0.818943 q^{70} -1.08474 q^{71} +4.36830 q^{72} -5.82413 q^{73} +0.944243 q^{74} -13.4267 q^{75} +1.00000 q^{76} +17.5542 q^{77} -0.0332485 q^{79} -0.231615 q^{80} -3.02283 q^{81} +8.53011 q^{82} -0.0511721 q^{83} -9.59779 q^{84} -0.0881919 q^{85} +3.48940 q^{86} +10.4219 q^{87} -4.96470 q^{88} -16.0134 q^{89} -1.01176 q^{90} -8.03063 q^{92} -25.8063 q^{93} -0.0285542 q^{94} -0.231615 q^{95} +2.71446 q^{96} -2.09080 q^{97} +5.50187 q^{98} -21.6873 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} - 5 q^{3} + 9 q^{4} + q^{5} - 5 q^{6} - 13 q^{7} + 9 q^{8} + 10 q^{9}+O(q^{10}) \) 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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