LMFDB - Embedded newform 8034.2.a.w.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8034,2,Mod(1,8034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8034, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8034 = 2 \cdot 3 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8034.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:	$64.1518129839$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} - 8 x^{10} + 94 x^{9} - 7 x^{8} - 580 x^{7} + 180 x^{6} + 1787 x^{5} - 308 x^{4} + \cdots + 768 $ x^12 - 6x^11 - 8x^10 + 94x^9 - 7x^8 - 580x^7 + 180x^6 + 1787x^5 - 308x^4 - 2790x^3 - 352x^2 + 1768*x + 768
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.02386$ of defining polynomial
Character	$\chi$	$=$	8034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.812175 q^{5} -1.00000 q^{6} -2.80411 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.812175 q^{5} -1.00000 q^{6} -2.80411 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.812175 q^{10} +4.04933 q^{11} +1.00000 q^{12} +1.00000 q^{13} +2.80411 q^{14} -0.812175 q^{15} +1.00000 q^{16} +2.29591 q^{17} -1.00000 q^{18} -1.01847 q^{19} -0.812175 q^{20} -2.80411 q^{21} -4.04933 q^{22} -8.19340 q^{23} -1.00000 q^{24} -4.34037 q^{25} -1.00000 q^{26} +1.00000 q^{27} -2.80411 q^{28} +6.94281 q^{29} +0.812175 q^{30} +1.13390 q^{31} -1.00000 q^{32} +4.04933 q^{33} -2.29591 q^{34} +2.27743 q^{35} +1.00000 q^{36} +5.96324 q^{37} +1.01847 q^{38} +1.00000 q^{39} +0.812175 q^{40} -10.6535 q^{41} +2.80411 q^{42} -8.47462 q^{43} +4.04933 q^{44} -0.812175 q^{45} +8.19340 q^{46} -4.10776 q^{47} +1.00000 q^{48} +0.863023 q^{49} +4.34037 q^{50} +2.29591 q^{51} +1.00000 q^{52} +12.4604 q^{53} -1.00000 q^{54} -3.28876 q^{55} +2.80411 q^{56} -1.01847 q^{57} -6.94281 q^{58} +0.884938 q^{59} -0.812175 q^{60} -4.79433 q^{61} -1.13390 q^{62} -2.80411 q^{63} +1.00000 q^{64} -0.812175 q^{65} -4.04933 q^{66} +0.580237 q^{67} +2.29591 q^{68} -8.19340 q^{69} -2.27743 q^{70} -8.92029 q^{71} -1.00000 q^{72} +15.9992 q^{73} -5.96324 q^{74} -4.34037 q^{75} -1.01847 q^{76} -11.3547 q^{77} -1.00000 q^{78} -9.00058 q^{79} -0.812175 q^{80} +1.00000 q^{81} +10.6535 q^{82} -4.73672 q^{83} -2.80411 q^{84} -1.86468 q^{85} +8.47462 q^{86} +6.94281 q^{87} -4.04933 q^{88} +10.1598 q^{89} +0.812175 q^{90} -2.80411 q^{91} -8.19340 q^{92} +1.13390 q^{93} +4.10776 q^{94} +0.827178 q^{95} -1.00000 q^{96} +4.77835 q^{97} -0.863023 q^{98} +4.04933 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 4 q^{5} - 12 q^{6} - 12 q^{8} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^2 + 12 * q^3 + 12 * q^4 - 4 * q^5 - 12 * q^6 - 12 * q^8 + 12 * q^9	\( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 4 q^{5} - 12 q^{6} - 12 q^{8} + 12 q^{9} + 4 q^{10} - 9 q^{11} + 12 q^{12} + 12 q^{13} - 4 q^{15} + 12 q^{16} - 20 q^{17} - 12 q^{18} + 4 q^{19} - 4 q^{20} + 9 q^{22} - 30 q^{23} - 12 q^{24} + 14 q^{25} - 12 q^{26} + 12 q^{27} - 29 q^{29} + 4 q^{30} + 6 q^{31} - 12 q^{32} - 9 q^{33} + 20 q^{34} - 22 q^{35} + 12 q^{36} + 7 q^{37} - 4 q^{38} + 12 q^{39} + 4 q^{40} - 8 q^{41} - 8 q^{43} - 9 q^{44} - 4 q^{45} + 30 q^{46} - 16 q^{47} + 12 q^{48} + 10 q^{49} - 14 q^{50} - 20 q^{51} + 12 q^{52} - 9 q^{53} - 12 q^{54} - 20 q^{55} + 4 q^{57} + 29 q^{58} - 29 q^{59} - 4 q^{60} - 26 q^{61} - 6 q^{62} + 12 q^{64} - 4 q^{65} + 9 q^{66} + 12 q^{67} - 20 q^{68} - 30 q^{69} + 22 q^{70} - 35 q^{71} - 12 q^{72} + 18 q^{73} - 7 q^{74} + 14 q^{75} + 4 q^{76} - 25 q^{77} - 12 q^{78} - 37 q^{79} - 4 q^{80} + 12 q^{81} + 8 q^{82} - 24 q^{83} - 17 q^{85} + 8 q^{86} - 29 q^{87} + 9 q^{88} + 15 q^{89} + 4 q^{90} - 30 q^{92} + 6 q^{93} + 16 q^{94} - 54 q^{95} - 12 q^{96} - 11 q^{97} - 10 q^{98} - 9 q^{99}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^3 + 12 * q^4 - 4 * q^5 - 12 * q^6 - 12 * q^8 + 12 * q^9 + 4 * q^10 - 9 * q^11 + 12 * q^12 + 12 * q^13 - 4 * q^15 + 12 * q^16 - 20 * q^17 - 12 * q^18 + 4 * q^19 - 4 * q^20 + 9 * q^22 - 30 * q^23 - 12 * q^24 + 14 * q^25 - 12 * q^26 + 12 * q^27 - 29 * q^29 + 4 * q^30 + 6 * q^31 - 12 * q^32 - 9 * q^33 + 20 * q^34 - 22 * q^35 + 12 * q^36 + 7 * q^37 - 4 * q^38 + 12 * q^39 + 4 * q^40 - 8 * q^41 - 8 * q^43 - 9 * q^44 - 4 * q^45 + 30 * q^46 - 16 * q^47 + 12 * q^48 + 10 * q^49 - 14 * q^50 - 20 * q^51 + 12 * q^52 - 9 * q^53 - 12 * q^54 - 20 * q^55 + 4 * q^57 + 29 * q^58 - 29 * q^59 - 4 * q^60 - 26 * q^61 - 6 * q^62 + 12 * q^64 - 4 * q^65 + 9 * q^66 + 12 * q^67 - 20 * q^68 - 30 * q^69 + 22 * q^70 - 35 * q^71 - 12 * q^72 + 18 * q^73 - 7 * q^74 + 14 * q^75 + 4 * q^76 - 25 * q^77 - 12 * q^78 - 37 * q^79 - 4 * q^80 + 12 * q^81 + 8 * q^82 - 24 * q^83 - 17 * q^85 + 8 * q^86 - 29 * q^87 + 9 * q^88 + 15 * q^89 + 4 * q^90 - 30 * q^92 + 6 * q^93 + 16 * q^94 - 54 * q^95 - 12 * q^96 - 11 * q^97 - 10 * q^98 - 9 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−0.812175	−0.363216	−0.181608	−	0.983371i	$-0.558130\pi$
$5$	−0.812175	−0.363216	−0.181608	+	0.983371i	$0.558130\pi$
$6$	−1.00000	−0.408248
$7$	−2.80411	−1.05985	−0.529927	−	0.848043i	$-0.677781\pi$
$7$	−2.80411	−1.05985	−0.529927	+	0.848043i	$0.677781\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0.812175	0.256832
$11$	4.04933	1.22092	0.610459	−	0.792048i	$-0.290985\pi$
$11$	4.04933	1.22092	0.610459	+	0.792048i	$0.290985\pi$
$12$	1.00000	0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	2.80411	0.749429
$15$	−0.812175	−0.209703
$16$	1.00000	0.250000
$17$	2.29591	0.556841	0.278420	−	0.960459i	$-0.410189\pi$
$17$	2.29591	0.556841	0.278420	+	0.960459i	$0.410189\pi$
$18$	−1.00000	−0.235702
$19$	−1.01847	−0.233654	−0.116827	−	0.993152i	$-0.537272\pi$
$19$	−1.01847	−0.233654	−0.116827	+	0.993152i	$0.537272\pi$
$20$	−0.812175	−0.181608
$21$	−2.80411	−0.611907
$22$	−4.04933	−0.863319
$23$	−8.19340	−1.70844	−0.854221	−	0.519910i	$-0.825965\pi$
$23$	−8.19340	−1.70844	−0.854221	+	0.519910i	$0.825965\pi$
$24$	−1.00000	−0.204124
$25$	−4.34037	−0.868074
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	−2.80411	−0.529927
$29$	6.94281	1.28925	0.644624	−	0.764500i	$-0.277014\pi$
$29$	6.94281	1.28925	0.644624	+	0.764500i	$0.277014\pi$
$30$	0.812175	0.148282
$31$	1.13390	0.203655	0.101827	−	0.994802i	$-0.467531\pi$
$31$	1.13390	0.203655	0.101827	+	0.994802i	$0.467531\pi$
$32$	−1.00000	−0.176777
$33$	4.04933	0.704897
$34$	−2.29591	−0.393746
$35$	2.27743	0.384955
$36$	1.00000	0.166667
$37$	5.96324	0.980351	0.490175	−	0.871624i	$-0.336933\pi$
$37$	5.96324	0.980351	0.490175	+	0.871624i	$0.336933\pi$
$38$	1.01847	0.165218
$39$	1.00000	0.160128
$40$	0.812175	0.128416
$41$	−10.6535	−1.66379	−0.831896	−	0.554932i	$-0.812744\pi$
$41$	−10.6535	−1.66379	−0.831896	+	0.554932i	$0.812744\pi$
$42$	2.80411	0.432683
$43$	−8.47462	−1.29237	−0.646184	−	0.763182i	$-0.723636\pi$
$43$	−8.47462	−1.29237	−0.646184	+	0.763182i	$0.723636\pi$
$44$	4.04933	0.610459
$45$	−0.812175	−0.121072
$46$	8.19340	1.20805
$47$	−4.10776	−0.599179	−0.299589	−	0.954068i	$-0.596850\pi$
$47$	−4.10776	−0.599179	−0.299589	+	0.954068i	$0.596850\pi$
$48$	1.00000	0.144338
$49$	0.863023	0.123289
$50$	4.34037	0.613821
$51$	2.29591	0.321492
$52$	1.00000	0.138675
$53$	12.4604	1.71157	0.855784	−	0.517334i	$-0.173075\pi$
$53$	12.4604	1.71157	0.855784	+	0.517334i	$0.173075\pi$
$54$	−1.00000	−0.136083
$55$	−3.28876	−0.443456
$56$	2.80411	0.374715
$57$	−1.01847	−0.134900
$58$	−6.94281	−0.911636
$59$	0.884938	0.115209	0.0576045	−	0.998339i	$-0.481654\pi$
$59$	0.884938	0.115209	0.0576045	+	0.998339i	$0.481654\pi$
$60$	−0.812175	−0.104851
$61$	−4.79433	−0.613851	−0.306926	−	0.951733i	$-0.599300\pi$
$61$	−4.79433	−0.613851	−0.306926	+	0.951733i	$0.599300\pi$
$62$	−1.13390	−0.144006
$63$	−2.80411	−0.353284
$64$	1.00000	0.125000
$65$	−0.812175	−0.100738
$66$	−4.04933	−0.498438
$67$	0.580237	0.0708873	0.0354436	−	0.999372i	$-0.488716\pi$
$67$	0.580237	0.0708873	0.0354436	+	0.999372i	$0.488716\pi$
$68$	2.29591	0.278420
$69$	−8.19340	−0.986369
$70$	−2.27743	−0.272204
$71$	−8.92029	−1.05864	−0.529322	−	0.848421i	$-0.677554\pi$
$71$	−8.92029	−1.05864	−0.529322	+	0.848421i	$0.677554\pi$
$72$	−1.00000	−0.117851
$73$	15.9992	1.87256	0.936282	−	0.351249i	$-0.114243\pi$
$73$	15.9992	1.87256	0.936282	+	0.351249i	$0.114243\pi$
$74$	−5.96324	−0.693213
$75$	−4.34037	−0.501183
$76$	−1.01847	−0.116827
$77$	−11.3547	−1.29399
$78$	−1.00000	−0.113228
$79$	−9.00058	−1.01264	−0.506322	−	0.862344i	$-0.668995\pi$
$79$	−9.00058	−1.01264	−0.506322	+	0.862344i	$0.668995\pi$
$80$	−0.812175	−0.0908039
$81$	1.00000	0.111111
$82$	10.6535	1.17648
$83$	−4.73672	−0.519923	−0.259962	−	0.965619i	$-0.583710\pi$
$83$	−4.73672	−0.519923	−0.259962	+	0.965619i	$0.583710\pi$
$84$	−2.80411	−0.305953
$85$	−1.86468	−0.202253
$86$	8.47462	0.913842
$87$	6.94281	0.744348
$88$	−4.04933	−0.431660
$89$	10.1598	1.07694	0.538469	−	0.842645i	$-0.319003\pi$
$89$	10.1598	1.07694	0.538469	+	0.842645i	$0.319003\pi$
$90$	0.812175	0.0856107
$91$	−2.80411	−0.293950
$92$	−8.19340	−0.854221
$93$	1.13390	0.117580
$94$	4.10776	0.423683
$95$	0.827178	0.0848667
$96$	−1.00000	−0.102062
$97$	4.77835	0.485168	0.242584	−	0.970130i	$-0.422005\pi$
$97$	4.77835	0.485168	0.242584	+	0.970130i	$0.422005\pi$
$98$	−0.863023	−0.0871785
$99$	4.04933	0.406973
$100$	−4.34037	−0.434037
$101$	−15.0451	−1.49704	−0.748522	−	0.663110i	$-0.769236\pi$
$101$	−15.0451	−1.49704	−0.748522	+	0.663110i	$0.769236\pi$
$102$	−2.29591	−0.227329
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	−1.00000	−0.0980581
$105$	2.27743	0.222254
$106$	−12.4604	−1.21026
$107$	0.0930370	0.00899423	0.00449712	−	0.999990i	$-0.498569\pi$
$107$	0.0930370	0.00899423	0.00449712	+	0.999990i	$0.498569\pi$
$108$	1.00000	0.0962250
$109$	−7.16487	−0.686270	−0.343135	−	0.939286i	$-0.611489\pi$
$109$	−7.16487	−0.686270	−0.343135	+	0.939286i	$0.611489\pi$
$110$	3.28876	0.313571
$111$	5.96324	0.566006
$112$	−2.80411	−0.264963
$113$	−5.38039	−0.506145	−0.253072	−	0.967447i	$-0.581441\pi$
$113$	−5.38039	−0.506145	−0.253072	+	0.967447i	$0.581441\pi$
$114$	1.01847	0.0953888
$115$	6.65447	0.620533
$116$	6.94281	0.644624
$117$	1.00000	0.0924500
$118$	−0.884938	−0.0814651
$119$	−6.43799	−0.590170
$120$	0.812175	0.0741411
$121$	5.39704	0.490640
$122$	4.79433	0.434058
$123$	−10.6535	−0.960591
$124$	1.13390	0.101827
$125$	7.58601	0.678514
$126$	2.80411	0.249810
$127$	−10.4196	−0.924593	−0.462296	−	0.886725i	$-0.652974\pi$
$127$	−10.4196	−0.924593	−0.462296	+	0.886725i	$0.652974\pi$
$128$	−1.00000	−0.0883883
$129$	−8.47462	−0.746148
$130$	0.812175	0.0712324
$131$	10.0961	0.882104	0.441052	−	0.897482i	$-0.354605\pi$
$131$	10.0961	0.882104	0.441052	+	0.897482i	$0.354605\pi$
$132$	4.04933	0.352449
$133$	2.85591	0.247639
$134$	−0.580237	−0.0501249
$135$	−0.812175	−0.0699009
$136$	−2.29591	−0.196873
$137$	6.33088	0.540883	0.270442	−	0.962736i	$-0.412830\pi$
$137$	6.33088	0.540883	0.270442	+	0.962736i	$0.412830\pi$
$138$	8.19340	0.697468
$139$	−7.51896	−0.637750	−0.318875	−	0.947797i	$-0.603305\pi$
$139$	−7.51896	−0.637750	−0.318875	+	0.947797i	$0.603305\pi$
$140$	2.27743	0.192478
$141$	−4.10776	−0.345936
$142$	8.92029	0.748574
$143$	4.04933	0.338622
$144$	1.00000	0.0833333
$145$	−5.63878	−0.468275
$146$	−15.9992	−1.32410
$147$	0.863023	0.0711810
$148$	5.96324	0.490175
$149$	5.49662	0.450300	0.225150	−	0.974324i	$-0.427713\pi$
$149$	5.49662	0.450300	0.225150	+	0.974324i	$0.427713\pi$
$150$	4.34037	0.354390
$151$	6.71314	0.546308	0.273154	−	0.961970i	$-0.411933\pi$
$151$	6.71314	0.546308	0.273154	+	0.961970i	$0.411933\pi$
$152$	1.01847	0.0826091
$153$	2.29591	0.185614
$154$	11.3547	0.914992
$155$	−0.920926	−0.0739706
$156$	1.00000	0.0800641
$157$	−8.77574	−0.700381	−0.350190	−	0.936679i	$-0.613883\pi$
$157$	−8.77574	−0.700381	−0.350190	+	0.936679i	$0.613883\pi$
$158$	9.00058	0.716048
$159$	12.4604	0.988174
$160$	0.812175	0.0642080
$161$	22.9752	1.81070
$162$	−1.00000	−0.0785674
$163$	19.3491	1.51554	0.757769	−	0.652523i	$-0.226289\pi$
$163$	19.3491	1.51554	0.757769	+	0.652523i	$0.226289\pi$
$164$	−10.6535	−0.831896
$165$	−3.28876	−0.256030
$166$	4.73672	0.367641
$167$	−10.1438	−0.784952	−0.392476	−	0.919762i	$-0.628381\pi$
$167$	−10.1438	−0.784952	−0.392476	+	0.919762i	$0.628381\pi$
$168$	2.80411	0.216342
$169$	1.00000	0.0769231
$170$	1.86468	0.143015
$171$	−1.01847	−0.0778846
$172$	−8.47462	−0.646184
$173$	−15.1679	−1.15319	−0.576596	−	0.817029i	$-0.695619\pi$
$173$	−15.1679	−1.15319	−0.576596	+	0.817029i	$0.695619\pi$
$174$	−6.94281	−0.526333
$175$	12.1709	0.920032
$176$	4.04933	0.305229
$177$	0.884938	0.0665160
$178$	−10.1598	−0.761510
$179$	−20.7858	−1.55360	−0.776802	−	0.629745i	$-0.783159\pi$
$179$	−20.7858	−1.55360	−0.776802	+	0.629745i	$0.783159\pi$
$180$	−0.812175	−0.0605359
$181$	8.49683	0.631565	0.315782	−	0.948832i	$-0.397733\pi$
$181$	8.49683	0.631565	0.315782	+	0.948832i	$0.397733\pi$
$182$	2.80411	0.207854
$183$	−4.79433	−0.354407
$184$	8.19340	0.604025
$185$	−4.84319	−0.356079
$186$	−1.13390	−0.0831417
$187$	9.29690	0.679857
$188$	−4.10776	−0.299589
$189$	−2.80411	−0.203969
$190$	−0.827178	−0.0600098
$191$	−24.5573	−1.77690	−0.888452	−	0.458970i	$-0.848218\pi$
$191$	−24.5573	−1.77690	−0.888452	+	0.458970i	$0.848218\pi$
$192$	1.00000	0.0721688
$193$	20.6351	1.48535	0.742675	−	0.669652i	$-0.233557\pi$
$193$	20.6351	1.48535	0.742675	+	0.669652i	$0.233557\pi$
$194$	−4.77835	−0.343066
$195$	−0.812175	−0.0581610
$196$	0.863023	0.0616445
$197$	12.8639	0.916515	0.458258	−	0.888819i	$-0.348474\pi$
$197$	12.8639	0.916515	0.458258	+	0.888819i	$0.348474\pi$
$198$	−4.04933	−0.287773
$199$	23.5266	1.66776	0.833879	−	0.551948i	$-0.186115\pi$
$199$	23.5266	1.66776	0.833879	+	0.551948i	$0.186115\pi$
$200$	4.34037	0.306911
$201$	0.580237	0.0409268
$202$	15.0451	1.05857
$203$	−19.4684	−1.36641
$204$	2.29591	0.160746
$205$	8.65247	0.604315
$206$	1.00000	0.0696733
$207$	−8.19340	−0.569481
$208$	1.00000	0.0693375
$209$	−4.12413	−0.285272
$210$	−2.27743	−0.157157
$211$	5.66546	0.390027	0.195013	−	0.980801i	$-0.437525\pi$
$211$	5.66546	0.390027	0.195013	+	0.980801i	$0.437525\pi$
$212$	12.4604	0.855784
$213$	−8.92029	−0.611208
$214$	−0.0930370	−0.00635988
$215$	6.88287	0.469408
$216$	−1.00000	−0.0680414
$217$	−3.17958	−0.215844
$218$	7.16487	0.485266
$219$	15.9992	1.08113
$220$	−3.28876	−0.221728
$221$	2.29591	0.154440
$222$	−5.96324	−0.400227
$223$	2.11119	0.141376	0.0706878	−	0.997498i	$-0.477481\pi$
$223$	2.11119	0.141376	0.0706878	+	0.997498i	$0.477481\pi$
$224$	2.80411	0.187357
$225$	−4.34037	−0.289358
$226$	5.38039	0.357898
$227$	16.6256	1.10348	0.551739	−	0.834017i	$-0.313964\pi$
$227$	16.6256	1.10348	0.551739	+	0.834017i	$0.313964\pi$
$228$	−1.01847	−0.0674500
$229$	−22.6417	−1.49620	−0.748101	−	0.663585i	$-0.769034\pi$
$229$	−22.6417	−1.49620	−0.748101	+	0.663585i	$0.769034\pi$
$230$	−6.65447	−0.438783
$231$	−11.3547	−0.747088
$232$	−6.94281	−0.455818
$233$	−0.805363	−0.0527611	−0.0263806	−	0.999652i	$-0.508398\pi$
$233$	−0.805363	−0.0527611	−0.0263806	+	0.999652i	$0.508398\pi$
$234$	−1.00000	−0.0653720
$235$	3.33622	0.217631
$236$	0.884938	0.0576045
$237$	−9.00058	−0.584651
$238$	6.43799	0.417313
$239$	−27.6450	−1.78821	−0.894103	−	0.447861i	$-0.852186\pi$
$239$	−27.6450	−1.78821	−0.894103	+	0.447861i	$0.852186\pi$
$240$	−0.812175	−0.0524257
$241$	5.25526	0.338521	0.169260	−	0.985571i	$-0.445862\pi$
$241$	5.25526	0.338521	0.169260	+	0.985571i	$0.445862\pi$
$242$	−5.39704	−0.346935
$243$	1.00000	0.0641500
$244$	−4.79433	−0.306926
$245$	−0.700926	−0.0447805
$246$	10.6535	0.679240
$247$	−1.01847	−0.0648039
$248$	−1.13390	−0.0720028
$249$	−4.73672	−0.300178
$250$	−7.58601	−0.479782
$251$	−11.7511	−0.741721	−0.370860	−	0.928689i	$-0.620937\pi$
$251$	−11.7511	−0.741721	−0.370860	+	0.928689i	$0.620937\pi$
$252$	−2.80411	−0.176642
$253$	−33.1777	−2.08587
$254$	10.4196	0.653786
$255$	−1.86468	−0.116771
$256$	1.00000	0.0625000
$257$	4.55065	0.283862	0.141931	−	0.989877i	$-0.454669\pi$
$257$	4.55065	0.283862	0.141931	+	0.989877i	$0.454669\pi$
$258$	8.47462	0.527607
$259$	−16.7216	−1.03903
$260$	−0.812175	−0.0503689
$261$	6.94281	0.429749
$262$	−10.0961	−0.623742
$263$	2.07134	0.127724	0.0638620	−	0.997959i	$-0.479658\pi$
$263$	2.07134	0.127724	0.0638620	+	0.997959i	$0.479658\pi$
$264$	−4.04933	−0.249219
$265$	−10.1200	−0.621668
$266$	−2.85591	−0.175107
$267$	10.1598	0.621771
$268$	0.580237	0.0354436
$269$	−16.0087	−0.976067	−0.488034	−	0.872825i	$-0.662286\pi$
$269$	−16.0087	−0.976067	−0.488034	+	0.872825i	$0.662286\pi$
$270$	0.812175	0.0494274
$271$	−22.3206	−1.35588	−0.677940	−	0.735118i	$-0.737127\pi$
$271$	−22.3206	−1.35588	−0.677940	+	0.735118i	$0.737127\pi$
$272$	2.29591	0.139210
$273$	−2.80411	−0.169712
$274$	−6.33088	−0.382462
$275$	−17.5756	−1.05985
$276$	−8.19340	−0.493185
$277$	1.43459	0.0861962	0.0430981	−	0.999071i	$-0.486277\pi$
$277$	1.43459	0.0861962	0.0430981	+	0.999071i	$0.486277\pi$
$278$	7.51896	0.450958
$279$	1.13390	0.0678849
$280$	−2.27743	−0.136102
$281$	−27.6330	−1.64845	−0.824223	−	0.566266i	$-0.808388\pi$
$281$	−27.6330	−1.64845	−0.824223	+	0.566266i	$0.808388\pi$
$282$	4.10776	0.244614
$283$	−29.2077	−1.73621	−0.868107	−	0.496376i	$-0.834664\pi$
$283$	−29.2077	−1.73621	−0.868107	+	0.496376i	$0.834664\pi$
$284$	−8.92029	−0.529322
$285$	0.827178	0.0489978
$286$	−4.04933	−0.239442
$287$	29.8735	1.76338
$288$	−1.00000	−0.0589256
$289$	−11.7288	−0.689928
$290$	5.63878	0.331120
$291$	4.77835	0.280112
$292$	15.9992	0.936282
$293$	−25.8130	−1.50801	−0.754005	−	0.656868i	$-0.771881\pi$
$293$	−25.8130	−1.50801	−0.754005	+	0.656868i	$0.771881\pi$
$294$	−0.863023	−0.0503325
$295$	−0.718724	−0.0418457
$296$	−5.96324	−0.346606
$297$	4.04933	0.234966
$298$	−5.49662	−0.318410
$299$	−8.19340	−0.473837
$300$	−4.34037	−0.250592
$301$	23.7637	1.36972
$302$	−6.71314	−0.386298
$303$	−15.0451	−0.864319
$304$	−1.01847	−0.0584134
$305$	3.89384	0.222960
$306$	−2.29591	−0.131249
$307$	−4.20527	−0.240008	−0.120004	−	0.992773i	$-0.538291\pi$
$307$	−4.20527	−0.240008	−0.120004	+	0.992773i	$0.538291\pi$
$308$	−11.3547	−0.646997
$309$	−1.00000	−0.0568880
$310$	0.920926	0.0523051
$311$	6.72177	0.381156	0.190578	−	0.981672i	$-0.438964\pi$
$311$	6.72177	0.381156	0.190578	+	0.981672i	$0.438964\pi$
$312$	−1.00000	−0.0566139
$313$	−20.8705	−1.17967	−0.589835	−	0.807524i	$-0.700807\pi$
$313$	−20.8705	−1.17967	−0.589835	+	0.807524i	$0.700807\pi$
$314$	8.77574	0.495244
$315$	2.27743	0.128318
$316$	−9.00058	−0.506322
$317$	−11.3847	−0.639431	−0.319715	−	0.947514i	$-0.603587\pi$
$317$	−11.3847	−0.639431	−0.319715	+	0.947514i	$0.603587\pi$
$318$	−12.4604	−0.698745
$319$	28.1137	1.57407
$320$	−0.812175	−0.0454019
$321$	0.0930370	0.00519282
$322$	−22.9752	−1.28036
$323$	−2.33833	−0.130108
$324$	1.00000	0.0555556
$325$	−4.34037	−0.240761
$326$	−19.3491	−1.07165
$327$	−7.16487	−0.396218
$328$	10.6535	0.588239
$329$	11.5186	0.635042
$330$	3.28876	0.181040
$331$	−1.90596	−0.104761	−0.0523805	−	0.998627i	$-0.516681\pi$
$331$	−1.90596	−0.104761	−0.0523805	+	0.998627i	$0.516681\pi$
$332$	−4.73672	−0.259962
$333$	5.96324	0.326784
$334$	10.1438	0.555045
$335$	−0.471254	−0.0257474
$336$	−2.80411	−0.152977
$337$	22.3056	1.21506	0.607532	−	0.794295i	$-0.292159\pi$
$337$	22.3056	1.21506	0.607532	+	0.794295i	$0.292159\pi$
$338$	−1.00000	−0.0543928
$339$	−5.38039	−0.292223
$340$	−1.86468	−0.101127
$341$	4.59154	0.248646
$342$	1.01847	0.0550727
$343$	17.2087	0.929185
$344$	8.47462	0.456921
$345$	6.65447	0.358265
$346$	15.1679	0.815430
$347$	−34.3314	−1.84301	−0.921503	−	0.388371i	$-0.873038\pi$
$347$	−34.3314	−1.84301	−0.921503	+	0.388371i	$0.873038\pi$
$348$	6.94281	0.372174
$349$	−25.3383	−1.35633	−0.678165	−	0.734910i	$-0.737224\pi$
$349$	−25.3383	−1.35633	−0.678165	+	0.734910i	$0.737224\pi$
$350$	−12.1709	−0.650561
$351$	1.00000	0.0533761
$352$	−4.04933	−0.215830
$353$	−9.54966	−0.508277	−0.254139	−	0.967168i	$-0.581792\pi$
$353$	−9.54966	−0.508277	−0.254139	+	0.967168i	$0.581792\pi$
$354$	−0.884938	−0.0470339
$355$	7.24483	0.384516
$356$	10.1598	0.538469
$357$	−6.43799	−0.340735
$358$	20.7858	1.09856
$359$	−3.57808	−0.188844	−0.0944218	−	0.995532i	$-0.530100\pi$
$359$	−3.57808	−0.188844	−0.0944218	+	0.995532i	$0.530100\pi$
$360$	0.812175	0.0428054
$361$	−17.9627	−0.945406
$362$	−8.49683	−0.446584
$363$	5.39704	0.283271
$364$	−2.80411	−0.146975
$365$	−12.9941	−0.680145
$366$	4.79433	0.250604
$367$	−3.17092	−0.165521	−0.0827604	−	0.996569i	$-0.526374\pi$
$367$	−3.17092	−0.165521	−0.0827604	+	0.996569i	$0.526374\pi$
$368$	−8.19340	−0.427110
$369$	−10.6535	−0.554597
$370$	4.84319	0.251786
$371$	−34.9403	−1.81401
$372$	1.13390	0.0587901
$373$	26.5049	1.37237	0.686185	−	0.727427i	$-0.259284\pi$
$373$	26.5049	1.37237	0.686185	+	0.727427i	$0.259284\pi$
$374$	−9.29690	−0.480731
$375$	7.58601	0.391740
$376$	4.10776	0.211842
$377$	6.94281	0.357573
$378$	2.80411	0.144228
$379$	−26.3016	−1.35102	−0.675512	−	0.737349i	$-0.736078\pi$
$379$	−26.3016	−1.35102	−0.675512	+	0.737349i	$0.736078\pi$
$380$	0.827178	0.0424333
$381$	−10.4196	−0.533814
$382$	24.5573	1.25646
$383$	−10.3778	−0.530279	−0.265140	−	0.964210i	$-0.585418\pi$
$383$	−10.3778	−0.530279	−0.265140	+	0.964210i	$0.585418\pi$
$384$	−1.00000	−0.0510310
$385$	9.22204	0.469999
$386$	−20.6351	−1.05030
$387$	−8.47462	−0.430789
$388$	4.77835	0.242584
$389$	−11.6907	−0.592743	−0.296372	−	0.955073i	$-0.595777\pi$
$389$	−11.6907	−0.592743	−0.296372	+	0.955073i	$0.595777\pi$
$390$	0.812175	0.0411261
$391$	−18.8113	−0.951330
$392$	−0.863023	−0.0435893
$393$	10.0961	0.509283
$394$	−12.8639	−0.648074
$395$	7.31005	0.367808
$396$	4.04933	0.203486
$397$	−10.2444	−0.514151	−0.257075	−	0.966391i	$-0.582759\pi$
$397$	−10.2444	−0.514151	−0.257075	+	0.966391i	$0.582759\pi$
$398$	−23.5266	−1.17928
$399$	2.85591	0.142974
$400$	−4.34037	−0.217019
$401$	−29.0137	−1.44887	−0.724437	−	0.689341i	$-0.757900\pi$
$401$	−29.0137	−1.44887	−0.724437	+	0.689341i	$0.757900\pi$
$402$	−0.580237	−0.0289396
$403$	1.13390	0.0564837
$404$	−15.0451	−0.748522
$405$	−0.812175	−0.0403573
$406$	19.4684	0.966200
$407$	24.1471	1.19693
$408$	−2.29591	−0.113665
$409$	29.3234	1.44995	0.724973	−	0.688777i	$-0.241852\pi$
$409$	29.3234	1.44995	0.724973	+	0.688777i	$0.241852\pi$
$410$	−8.65247	−0.427315
$411$	6.33088	0.312279
$412$	−1.00000	−0.0492665
$413$	−2.48146	−0.122105
$414$	8.19340	0.402684
$415$	3.84705	0.188844
$416$	−1.00000	−0.0490290
$417$	−7.51896	−0.368205
$418$	4.12413	0.201718
$419$	−10.6366	−0.519632	−0.259816	−	0.965658i	$-0.583662\pi$
$419$	−10.6366	−0.519632	−0.259816	+	0.965658i	$0.583662\pi$
$420$	2.27743	0.111127
$421$	−6.31510	−0.307779	−0.153889	−	0.988088i	$-0.549180\pi$
$421$	−6.31510	−0.307779	−0.153889	+	0.988088i	$0.549180\pi$
$422$	−5.66546	−0.275790
$423$	−4.10776	−0.199726
$424$	−12.4604	−0.605131
$425$	−9.96512	−0.483379
$426$	8.92029	0.432190
$427$	13.4438	0.650592
$428$	0.0930370	0.00449712
$429$	4.04933	0.195503
$430$	−6.88287	−0.331921
$431$	−32.7901	−1.57944	−0.789721	−	0.613466i	$-0.789775\pi$
$431$	−32.7901	−1.57944	−0.789721	+	0.613466i	$0.789775\pi$
$432$	1.00000	0.0481125
$433$	−23.0398	−1.10722	−0.553611	−	0.832776i	$-0.686750\pi$
$433$	−23.0398	−1.10722	−0.553611	+	0.832776i	$0.686750\pi$
$434$	3.17958	0.152625
$435$	−5.63878	−0.270359
$436$	−7.16487	−0.343135
$437$	8.34476	0.399184
$438$	−15.9992	−0.764471
$439$	37.5925	1.79419	0.897096	−	0.441837i	$-0.145673\pi$
$439$	37.5925	1.79419	0.897096	+	0.441837i	$0.145673\pi$
$440$	3.28876	0.156785
$441$	0.863023	0.0410963
$442$	−2.29591	−0.109205
$443$	34.6459	1.64607	0.823037	−	0.567988i	$-0.192278\pi$
$443$	34.6459	1.64607	0.823037	+	0.567988i	$0.192278\pi$
$444$	5.96324	0.283003
$445$	−8.25154	−0.391161
$446$	−2.11119	−0.0999676
$447$	5.49662	0.259981
$448$	−2.80411	−0.132482
$449$	−11.6519	−0.549888	−0.274944	−	0.961460i	$-0.588659\pi$
$449$	−11.6519	−0.549888	−0.274944	+	0.961460i	$0.588659\pi$
$450$	4.34037	0.204607
$451$	−43.1394	−2.03135
$452$	−5.38039	−0.253072
$453$	6.71314	0.315411
$454$	−16.6256	−0.780277
$455$	2.27743	0.106767
$456$	1.01847	0.0476944
$457$	26.5669	1.24275	0.621374	−	0.783514i	$-0.286575\pi$
$457$	26.5669	1.24275	0.621374	+	0.783514i	$0.286575\pi$
$458$	22.6417	1.05797
$459$	2.29591	0.107164
$460$	6.65447	0.310266
$461$	−10.3671	−0.482844	−0.241422	−	0.970420i	$-0.577614\pi$
$461$	−10.3671	−0.482844	−0.241422	+	0.970420i	$0.577614\pi$
$462$	11.3547	0.528271
$463$	29.0583	1.35046	0.675228	−	0.737609i	$-0.264045\pi$
$463$	29.0583	1.35046	0.675228	+	0.737609i	$0.264045\pi$
$464$	6.94281	0.322312
$465$	−0.920926	−0.0427069
$466$	0.805363	0.0373077
$467$	5.95392	0.275514	0.137757	−	0.990466i	$-0.456011\pi$
$467$	5.95392	0.275514	0.137757	+	0.990466i	$0.456011\pi$
$468$	1.00000	0.0462250
$469$	−1.62705	−0.0751301
$470$	−3.33622	−0.153888
$471$	−8.77574	−0.404365
$472$	−0.884938	−0.0407326
$473$	−34.3165	−1.57787
$474$	9.00058	0.413411
$475$	4.42055	0.202829
$476$	−6.43799	−0.295085
$477$	12.4604	0.570523
$478$	27.6450	1.26445
$479$	−2.33530	−0.106703	−0.0533513	−	0.998576i	$-0.516990\pi$
$479$	−2.33530	−0.106703	−0.0533513	+	0.998576i	$0.516990\pi$
$480$	0.812175	0.0370705
$481$	5.96324	0.271900
$482$	−5.25526	−0.239370
$483$	22.9752	1.04541
$484$	5.39704	0.245320
$485$	−3.88086	−0.176221
$486$	−1.00000	−0.0453609
$487$	17.3845	0.787769	0.393884	−	0.919160i	$-0.371131\pi$
$487$	17.3845	0.787769	0.393884	+	0.919160i	$0.371131\pi$
$488$	4.79433	0.217029
$489$	19.3491	0.874996
$490$	0.700926	0.0316646
$491$	−19.7595	−0.891733	−0.445867	−	0.895099i	$-0.647104\pi$
$491$	−19.7595	−0.891733	−0.445867	+	0.895099i	$0.647104\pi$
$492$	−10.6535	−0.480295
$493$	15.9401	0.717906
$494$	1.01847	0.0458233
$495$	−3.28876	−0.147819
$496$	1.13390	0.0509137
$497$	25.0135	1.12201
$498$	4.73672	0.212258
$499$	−0.921867	−0.0412684	−0.0206342	−	0.999787i	$-0.506569\pi$
$499$	−0.921867	−0.0412684	−0.0206342	+	0.999787i	$0.506569\pi$
$500$	7.58601	0.339257
$501$	−10.1438	−0.453192
$502$	11.7511	0.524476
$503$	−12.8080	−0.571078	−0.285539	−	0.958367i	$-0.592173\pi$
$503$	−12.8080	−0.571078	−0.285539	+	0.958367i	$0.592173\pi$
$504$	2.80411	0.124905
$505$	12.2193	0.543750
$506$	33.1777	1.47493
$507$	1.00000	0.0444116
$508$	−10.4196	−0.462296
$509$	5.13942	0.227801	0.113900	−	0.993492i	$-0.463666\pi$
$509$	5.13942	0.227801	0.113900	+	0.993492i	$0.463666\pi$
$510$	1.86468	0.0825696
$511$	−44.8635	−1.98464
$512$	−1.00000	−0.0441942
$513$	−1.01847	−0.0449667
$514$	−4.55065	−0.200721
$515$	0.812175	0.0357887
$516$	−8.47462	−0.373074
$517$	−16.6337	−0.731548
$518$	16.7216	0.734704
$519$	−15.1679	−0.665796
$520$	0.812175	0.0356162
$521$	6.97500	0.305580	0.152790	−	0.988259i	$-0.451174\pi$
$521$	6.97500	0.305580	0.152790	+	0.988259i	$0.451174\pi$
$522$	−6.94281	−0.303879
$523$	−0.406703	−0.0177839	−0.00889194	−	0.999960i	$-0.502830\pi$
$523$	−0.406703	−0.0177839	−0.00889194	+	0.999960i	$0.502830\pi$
$524$	10.0961	0.441052
$525$	12.1709	0.531180
$526$	−2.07134	−0.0903145
$527$	2.60334	0.113403
$528$	4.04933	0.176224
$529$	44.1318	1.91877
$530$	10.1200	0.439586
$531$	0.884938	0.0384030
$532$	2.85591	0.123819
$533$	−10.6535	−0.461453
$534$	−10.1598	−0.439658
$535$	−0.0755623	−0.00326684
$536$	−0.580237	−0.0250624
$537$	−20.7858	−0.896973
$538$	16.0087	0.690184
$539$	3.49466	0.150526
$540$	−0.812175	−0.0349504
$541$	43.1680	1.85594	0.927969	−	0.372657i	$-0.121553\pi$
$541$	43.1680	1.85594	0.927969	+	0.372657i	$0.121553\pi$
$542$	22.3206	0.958751
$543$	8.49683	0.364634
$544$	−2.29591	−0.0984365
$545$	5.81913	0.249264
$546$	2.80411	0.120005
$547$	−31.5402	−1.34856	−0.674280	−	0.738476i	$-0.735546\pi$
$547$	−31.5402	−1.34856	−0.674280	+	0.738476i	$0.735546\pi$
$548$	6.33088	0.270442
$549$	−4.79433	−0.204617
$550$	17.5756	0.749425
$551$	−7.07107	−0.301238
$552$	8.19340	0.348734
$553$	25.2386	1.07325
$554$	−1.43459	−0.0609499
$555$	−4.84319	−0.205582
$556$	−7.51896	−0.318875
$557$	−1.08304	−0.0458899	−0.0229449	−	0.999737i	$-0.507304\pi$
$557$	−1.08304	−0.0458899	−0.0229449	+	0.999737i	$0.507304\pi$
$558$	−1.13390	−0.0480019
$559$	−8.47462	−0.358438
$560$	2.27743	0.0962388
$561$	9.29690	0.392516
$562$	27.6330	1.16563
$563$	19.7982	0.834393	0.417196	−	0.908816i	$-0.363013\pi$
$563$	19.7982	0.834393	0.417196	+	0.908816i	$0.363013\pi$
$564$	−4.10776	−0.172968
$565$	4.36982	0.183840
$566$	29.2077	1.22769
$567$	−2.80411	−0.117761
$568$	8.92029	0.374287
$569$	−27.6820	−1.16049	−0.580246	−	0.814442i	$-0.697043\pi$
$569$	−27.6820	−1.16049	−0.580246	+	0.814442i	$0.697043\pi$
$570$	−0.827178	−0.0346467
$571$	20.4317	0.855042	0.427521	−	0.904005i	$-0.359387\pi$
$571$	20.4317	0.855042	0.427521	+	0.904005i	$0.359387\pi$
$572$	4.04933	0.169311
$573$	−24.5573	−1.02590
$574$	−29.8735	−1.24689
$575$	35.5624	1.48305
$576$	1.00000	0.0416667
$577$	−6.45607	−0.268770	−0.134385	−	0.990929i	$-0.542906\pi$
$577$	−6.45607	−0.268770	−0.134385	+	0.990929i	$0.542906\pi$
$578$	11.7288	0.487853
$579$	20.6351	0.857567
$580$	−5.63878	−0.234137
$581$	13.2823	0.551042
$582$	−4.77835	−0.198069
$583$	50.4562	2.08968
$584$	−15.9992	−0.662052
$585$	−0.812175	−0.0335793
$586$	25.8130	1.06632
$587$	−7.18362	−0.296500	−0.148250	−	0.988950i	$-0.547364\pi$
$587$	−7.18362	−0.296500	−0.148250	+	0.988950i	$0.547364\pi$
$588$	0.863023	0.0355905
$589$	−1.15485	−0.0475847
$590$	0.718724	0.0295894
$591$	12.8639	0.529150
$592$	5.96324	0.245088
$593$	43.5311	1.78761	0.893804	−	0.448458i	$-0.148027\pi$
$593$	43.5311	1.78761	0.893804	+	0.448458i	$0.148027\pi$
$594$	−4.04933	−0.166146
$595$	5.22877	0.214359
$596$	5.49662	0.225150
$597$	23.5266	0.962880
$598$	8.19340	0.335053
$599$	−11.6674	−0.476716	−0.238358	−	0.971177i	$-0.576609\pi$
$599$	−11.6674	−0.476716	−0.238358	+	0.971177i	$0.576609\pi$
$600$	4.34037	0.177195
$601$	43.9848	1.79418	0.897090	−	0.441849i	$-0.145677\pi$
$601$	43.9848	1.79418	0.897090	+	0.441849i	$0.145677\pi$
$602$	−23.7637	−0.968538
$603$	0.580237	0.0236291
$604$	6.71314	0.273154
$605$	−4.38334	−0.178208
$606$	15.0451	0.611166
$607$	6.80863	0.276354	0.138177	−	0.990408i	$-0.455876\pi$
$607$	6.80863	0.276354	0.138177	+	0.990408i	$0.455876\pi$
$608$	1.01847	0.0413045
$609$	−19.4684	−0.788899
$610$	−3.89384	−0.157657
$611$	−4.10776	−0.166182
$612$	2.29591	0.0928068
$613$	−34.8280	−1.40669	−0.703344	−	0.710850i	$-0.748311\pi$
$613$	−34.8280	−1.40669	−0.703344	+	0.710850i	$0.748311\pi$
$614$	4.20527	0.169711
$615$	8.65247	0.348901
$616$	11.3547	0.457496
$617$	36.1839	1.45671	0.728354	−	0.685201i	$-0.240286\pi$
$617$	36.1839	1.45671	0.728354	+	0.685201i	$0.240286\pi$
$618$	1.00000	0.0402259
$619$	−13.4757	−0.541632	−0.270816	−	0.962631i	$-0.587294\pi$
$619$	−13.4757	−0.541632	−0.270816	+	0.962631i	$0.587294\pi$
$620$	−0.920926	−0.0369853
$621$	−8.19340	−0.328790
$622$	−6.72177	−0.269518
$623$	−28.4892	−1.14140
$624$	1.00000	0.0400320
$625$	15.5407	0.621628
$626$	20.8705	0.834153
$627$	−4.12413	−0.164702
$628$	−8.77574	−0.350190
$629$	13.6911	0.545899
$630$	−2.27743	−0.0907348
$631$	15.6495	0.622998	0.311499	−	0.950246i	$-0.399169\pi$
$631$	15.6495	0.622998	0.311499	+	0.950246i	$0.399169\pi$
$632$	9.00058	0.358024
$633$	5.66546	0.225182
$634$	11.3847	0.452146
$635$	8.46256	0.335827
$636$	12.4604	0.494087
$637$	0.863023	0.0341942
$638$	−28.1137	−1.11303
$639$	−8.92029	−0.352881
$640$	0.812175	0.0321040
$641$	−17.3158	−0.683934	−0.341967	−	0.939712i	$-0.611093\pi$
$641$	−17.3158	−0.683934	−0.341967	+	0.939712i	$0.611093\pi$
$642$	−0.0930370	−0.00367188
$643$	44.8856	1.77011	0.885057	−	0.465483i	$-0.154119\pi$
$643$	44.8856	1.77011	0.885057	+	0.465483i	$0.154119\pi$
$644$	22.9752	0.905349
$645$	6.88287	0.271013
$646$	2.33833	0.0920002
$647$	34.9463	1.37388	0.686940	−	0.726715i	$-0.258954\pi$
$647$	34.9463	1.37388	0.686940	+	0.726715i	$0.258954\pi$
$648$	−1.00000	−0.0392837
$649$	3.58340	0.140661
$650$	4.34037	0.170243
$651$	−3.17958	−0.124618
$652$	19.3491	0.757769
$653$	−36.4184	−1.42516	−0.712581	−	0.701590i	$-0.752474\pi$
$653$	−36.4184	−1.42516	−0.712581	+	0.701590i	$0.752474\pi$
$654$	7.16487	0.280169
$655$	−8.19983	−0.320394
$656$	−10.6535	−0.415948
$657$	15.9992	0.624188
$658$	−11.5186	−0.449042
$659$	−27.5734	−1.07411	−0.537053	−	0.843549i	$-0.680462\pi$
$659$	−27.5734	−1.07411	−0.537053	+	0.843549i	$0.680462\pi$
$660$	−3.28876	−0.128015
$661$	−32.6336	−1.26930	−0.634651	−	0.772799i	$-0.718856\pi$
$661$	−32.6336	−1.26930	−0.634651	+	0.772799i	$0.718856\pi$
$662$	1.90596	0.0740773
$663$	2.29591	0.0891659
$664$	4.73672	0.183821
$665$	−2.31950	−0.0899462
$666$	−5.96324	−0.231071
$667$	−56.8852	−2.20261
$668$	−10.1438	−0.392476
$669$	2.11119	0.0816232
$670$	0.471254	0.0182061
$671$	−19.4138	−0.749462
$672$	2.80411	0.108171
$673$	−41.2898	−1.59161	−0.795803	−	0.605556i	$-0.792951\pi$
$673$	−41.2898	−1.59161	−0.795803	+	0.605556i	$0.792951\pi$
$674$	−22.3056	−0.859181
$675$	−4.34037	−0.167061
$676$	1.00000	0.0384615
$677$	−8.12620	−0.312315	−0.156158	−	0.987732i	$-0.549911\pi$
$677$	−8.12620	−0.312315	−0.156158	+	0.987732i	$0.549911\pi$
$678$	5.38039	0.206633
$679$	−13.3990	−0.514207
$680$	1.86468	0.0715073
$681$	16.6256	0.637094
$682$	−4.59154	−0.175819
$683$	−2.65217	−0.101482	−0.0507412	−	0.998712i	$-0.516158\pi$
$683$	−2.65217	−0.101482	−0.0507412	+	0.998712i	$0.516158\pi$
$684$	−1.01847	−0.0389423
$685$	−5.14178	−0.196457
$686$	−17.2087	−0.657033
$687$	−22.6417	−0.863833
$688$	−8.47462	−0.323092
$689$	12.4604	0.474703
$690$	−6.65447	−0.253331
$691$	−5.60099	−0.213072	−0.106536	−	0.994309i	$-0.533976\pi$
$691$	−5.60099	−0.213072	−0.106536	+	0.994309i	$0.533976\pi$
$692$	−15.1679	−0.576596
$693$	−11.3547	−0.431331
$694$	34.3314	1.30320
$695$	6.10671	0.231641
$696$	−6.94281	−0.263167
$697$	−24.4594	−0.926467
$698$	25.3383	0.959070
$699$	−0.805363	−0.0304616
$700$	12.1709	0.460016
$701$	−3.74962	−0.141621	−0.0708106	−	0.997490i	$-0.522559\pi$
$701$	−3.74962	−0.141621	−0.0708106	+	0.997490i	$0.522559\pi$
$702$	−1.00000	−0.0377426
$703$	−6.07340	−0.229063
$704$	4.04933	0.152615
$705$	3.33622	0.125649
$706$	9.54966	0.359406
$707$	42.1881	1.58665
$708$	0.884938	0.0332580
$709$	18.7137	0.702809	0.351405	−	0.936224i	$-0.385704\pi$
$709$	18.7137	0.702809	0.351405	+	0.936224i	$0.385704\pi$
$710$	−7.24483	−0.271894
$711$	−9.00058	−0.337548
$712$	−10.1598	−0.380755
$713$	−9.29051	−0.347932
$714$	6.43799	0.240936
$715$	−3.28876	−0.122993
$716$	−20.7858	−0.776802
$717$	−27.6450	−1.03242
$718$	3.57808	0.133533
$719$	−49.5279	−1.84708	−0.923540	−	0.383501i	$-0.874718\pi$
$719$	−49.5279	−1.84708	−0.923540	+	0.383501i	$0.874718\pi$
$720$	−0.812175	−0.0302680
$721$	2.80411	0.104430
$722$	17.9627	0.668503
$723$	5.25526	0.195445
$724$	8.49683	0.315782
$725$	−30.1344	−1.11916
$726$	−5.39704	−0.200303
$727$	25.3400	0.939807	0.469904	−	0.882718i	$-0.344289\pi$
$727$	25.3400	0.939807	0.469904	+	0.882718i	$0.344289\pi$
$728$	2.80411	0.103927
$729$	1.00000	0.0370370
$730$	12.9941	0.480935
$731$	−19.4570	−0.719643
$732$	−4.79433	−0.177204
$733$	−47.0633	−1.73832	−0.869162	−	0.494528i	$-0.835341\pi$
$733$	−47.0633	−1.73832	−0.869162	+	0.494528i	$0.835341\pi$
$734$	3.17092	0.117041
$735$	−0.700926	−0.0258540
$736$	8.19340	0.302013
$737$	2.34957	0.0865475
$738$	10.6535	0.392159
$739$	−48.7133	−1.79195	−0.895973	−	0.444108i	$-0.853521\pi$
$739$	−48.7133	−1.79195	−0.895973	+	0.444108i	$0.853521\pi$
$740$	−4.84319	−0.178039
$741$	−1.01847	−0.0374145
$742$	34.9403	1.28270
$743$	3.11576	0.114306	0.0571530	−	0.998365i	$-0.481798\pi$
$743$	3.11576	0.114306	0.0571530	+	0.998365i	$0.481798\pi$
$744$	−1.13390	−0.0415709
$745$	−4.46421	−0.163556
$746$	−26.5049	−0.970412
$747$	−4.73672	−0.173308
$748$	9.29690	0.339928
$749$	−0.260886	−0.00953257
$750$	−7.58601	−0.277002
$751$	−10.5836	−0.386200	−0.193100	−	0.981179i	$-0.561854\pi$
$751$	−10.5836	−0.386200	−0.193100	+	0.981179i	$0.561854\pi$
$752$	−4.10776	−0.149795
$753$	−11.7511	−0.428233
$754$	−6.94281	−0.252842
$755$	−5.45224	−0.198428
$756$	−2.80411	−0.101984
$757$	−44.7518	−1.62653	−0.813266	−	0.581893i	$-0.802312\pi$
$757$	−44.7518	−1.62653	−0.813266	+	0.581893i	$0.802312\pi$
$758$	26.3016	0.955319
$759$	−33.1777	−1.20428
$760$	−0.827178	−0.0300049
$761$	−32.1122	−1.16407	−0.582034	−	0.813164i	$-0.697743\pi$
$761$	−32.1122	−1.16407	−0.582034	+	0.813164i	$0.697743\pi$
$762$	10.4196	0.377464
$763$	20.0911	0.727346
$764$	−24.5573	−0.888452
$765$	−1.86468	−0.0674178
$766$	10.3778	0.374964
$767$	0.884938	0.0319532
$768$	1.00000	0.0360844
$769$	−10.9588	−0.395185	−0.197593	−	0.980284i	$-0.563312\pi$
$769$	−10.9588	−0.395185	−0.197593	+	0.980284i	$0.563312\pi$
$770$	−9.22204	−0.332339
$771$	4.55065	0.163888
$772$	20.6351	0.742675
$773$	2.54269	0.0914540	0.0457270	−	0.998954i	$-0.485440\pi$
$773$	2.54269	0.0914540	0.0457270	+	0.998954i	$0.485440\pi$
$774$	8.47462	0.304614
$775$	−4.92156	−0.176788
$776$	−4.77835	−0.171533
$777$	−16.7216	−0.599883
$778$	11.6907	0.419133
$779$	10.8503	0.388751
$780$	−0.812175	−0.0290805
$781$	−36.1212	−1.29252
$782$	18.8113	0.672692
$783$	6.94281	0.248116
$784$	0.863023	0.0308223
$785$	7.12744	0.254389
$786$	−10.0961	−0.360117
$787$	−29.6963	−1.05856	−0.529279	−	0.848448i	$-0.677538\pi$
$787$	−29.6963	−1.05856	−0.529279	+	0.848448i	$0.677538\pi$
$788$	12.8639	0.458258
$789$	2.07134	0.0737415
$790$	−7.31005	−0.260080
$791$	15.0872	0.536439
$792$	−4.04933	−0.143887
$793$	−4.79433	−0.170252
$794$	10.2444	0.363559
$795$	−10.1200	−0.358920
$796$	23.5266	0.833879
$797$	12.2889	0.435294	0.217647	−	0.976028i	$-0.430162\pi$
$797$	12.2889	0.435294	0.217647	+	0.976028i	$0.430162\pi$
$798$	−2.85591	−0.101098
$799$	−9.43107	−0.333647
$800$	4.34037	0.153455
$801$	10.1598	0.358979
$802$	29.0137	1.02451
$803$	64.7860	2.28625
$804$	0.580237	0.0204634
$805$	−18.6599	−0.657674
$806$	−1.13390	−0.0399400
$807$	−16.0087	−0.563533
$808$	15.0451	0.529285
$809$	38.1455	1.34112	0.670562	−	0.741854i	$-0.266053\pi$
$809$	38.1455	1.34112	0.670562	+	0.741854i	$0.266053\pi$
$810$	0.812175	0.0285369
$811$	34.7872	1.22154	0.610771	−	0.791807i	$-0.290860\pi$
$811$	34.7872	1.22154	0.610771	+	0.791807i	$0.290860\pi$
$812$	−19.4684	−0.683207
$813$	−22.3206	−0.782817
$814$	−24.1471	−0.846356
$815$	−15.7148	−0.550467
$816$	2.29591	0.0803731
$817$	8.63117	0.301966
$818$	−29.3234	−1.02527
$819$	−2.80411	−0.0979835
$820$	8.65247	0.302158
$821$	15.0268	0.524438	0.262219	−	0.965008i	$-0.415546\pi$
$821$	15.0268	0.524438	0.262219	+	0.965008i	$0.415546\pi$
$822$	−6.33088	−0.220815
$823$	38.1909	1.33125	0.665626	−	0.746286i	$-0.268165\pi$
$823$	38.1909	1.33125	0.665626	+	0.746286i	$0.268165\pi$
$824$	1.00000	0.0348367
$825$	−17.5756	−0.611903
$826$	2.48146	0.0863411
$827$	−19.1560	−0.666121	−0.333060	−	0.942905i	$-0.608081\pi$
$827$	−19.1560	−0.666121	−0.333060	+	0.942905i	$0.608081\pi$
$828$	−8.19340	−0.284740
$829$	−4.32580	−0.150241	−0.0751207	−	0.997174i	$-0.523934\pi$
$829$	−4.32580	−0.150241	−0.0751207	+	0.997174i	$0.523934\pi$
$830$	−3.84705	−0.133533
$831$	1.43459	0.0497654
$832$	1.00000	0.0346688
$833$	1.98143	0.0686524
$834$	7.51896	0.260360
$835$	8.23855	0.285107
$836$	−4.12413	−0.142636
$837$	1.13390	0.0391934
$838$	10.6366	0.367435
$839$	4.61218	0.159230	0.0796151	−	0.996826i	$-0.474631\pi$
$839$	4.61218	0.159230	0.0796151	+	0.996826i	$0.474631\pi$
$840$	−2.27743	−0.0785787
$841$	19.2026	0.662160
$842$	6.31510	0.217633
$843$	−27.6330	−0.951730
$844$	5.66546	0.195013
$845$	−0.812175	−0.0279397
$846$	4.10776	0.141228
$847$	−15.1339	−0.520007
$848$	12.4604	0.427892
$849$	−29.2077	−1.00240
$850$	9.96512	0.341801
$851$	−48.8592	−1.67487
$852$	−8.92029	−0.305604
$853$	7.79993	0.267065	0.133532	−	0.991044i	$-0.457368\pi$
$853$	7.79993	0.267065	0.133532	+	0.991044i	$0.457368\pi$
$854$	−13.4438	−0.460038
$855$	0.827178	0.0282889
$856$	−0.0930370	−0.00317994
$857$	−1.80417	−0.0616293	−0.0308146	−	0.999525i	$-0.509810\pi$
$857$	−1.80417	−0.0616293	−0.0308146	+	0.999525i	$0.509810\pi$
$858$	−4.04933	−0.138242
$859$	−35.2598	−1.20305	−0.601525	−	0.798854i	$-0.705440\pi$
$859$	−35.2598	−1.20305	−0.601525	+	0.798854i	$0.705440\pi$
$860$	6.88287	0.234704
$861$	29.8735	1.01809
$862$	32.7901	1.11683
$863$	−43.8890	−1.49400	−0.747000	−	0.664824i	$-0.768506\pi$
$863$	−43.8890	−1.49400	−0.747000	+	0.664824i	$0.768506\pi$
$864$	−1.00000	−0.0340207
$865$	12.3190	0.418857
$866$	23.0398	0.782924
$867$	−11.7288	−0.398330
$868$	−3.17958	−0.107922
$869$	−36.4463	−1.23636
$870$	5.63878	0.191172
$871$	0.580237	0.0196606
$872$	7.16487	0.242633
$873$	4.77835	0.161723
$874$	−8.34476	−0.282266
$875$	−21.2720	−0.719125
$876$	15.9992	0.540563
$877$	12.7397	0.430191	0.215095	−	0.976593i	$-0.430994\pi$
$877$	12.7397	0.430191	0.215095	+	0.976593i	$0.430994\pi$
$878$	−37.5925	−1.26868
$879$	−25.8130	−0.870650
$880$	−3.28876	−0.110864
$881$	−32.4436	−1.09305	−0.546526	−	0.837442i	$-0.684050\pi$
$881$	−32.4436	−1.09305	−0.546526	+	0.837442i	$0.684050\pi$
$882$	−0.863023	−0.0290595
$883$	1.56011	0.0525018	0.0262509	−	0.999655i	$-0.491643\pi$
$883$	1.56011	0.0525018	0.0262509	+	0.999655i	$0.491643\pi$
$884$	2.29591	0.0772199
$885$	−0.718724	−0.0241596
$886$	−34.6459	−1.16395
$887$	−42.1818	−1.41633	−0.708163	−	0.706049i	$-0.750476\pi$
$887$	−42.1818	−1.41633	−0.708163	+	0.706049i	$0.750476\pi$
$888$	−5.96324	−0.200113
$889$	29.2178	0.979933
$890$	8.25154	0.276592
$891$	4.04933	0.135658
$892$	2.11119	0.0706878
$893$	4.18365	0.140000
$894$	−5.49662	−0.183834
$895$	16.8817	0.564293
$896$	2.80411	0.0936787
$897$	−8.19340	−0.273570
$898$	11.6519	0.388829
$899$	7.87247	0.262562
$900$	−4.34037	−0.144679
$901$	28.6080	0.953071
$902$	43.1394	1.43638
$903$	23.7637	0.790808
$904$	5.38039	0.178949
$905$	−6.90091	−0.229394
$906$	−6.71314	−0.223029
$907$	6.97762	0.231688	0.115844	−	0.993267i	$-0.463043\pi$
$907$	6.97762	0.231688	0.115844	+	0.993267i	$0.463043\pi$
$908$	16.6256	0.551739
$909$	−15.0451	−0.499015
$910$	−2.27743	−0.0754959
$911$	45.6638	1.51291	0.756454	−	0.654047i	$-0.226930\pi$
$911$	45.6638	1.51291	0.756454	+	0.654047i	$0.226930\pi$
$912$	−1.01847	−0.0337250
$913$	−19.1805	−0.634783
$914$	−26.5669	−0.878755
$915$	3.89384	0.128726
$916$	−22.6417	−0.748101
$917$	−28.3107	−0.934901
$918$	−2.29591	−0.0757764
$919$	−38.2955	−1.26325	−0.631625	−	0.775274i	$-0.717612\pi$
$919$	−38.2955	−1.26325	−0.631625	+	0.775274i	$0.717612\pi$
$920$	−6.65447	−0.219391
$921$	−4.20527	−0.138569
$922$	10.3671	0.341423
$923$	−8.92029	−0.293615
$924$	−11.3547	−0.373544
$925$	−25.8827	−0.851018
$926$	−29.0583	−0.954916
$927$	−1.00000	−0.0328443
$928$	−6.94281	−0.227909
$929$	21.2377	0.696787	0.348394	−	0.937348i	$-0.386727\pi$
$929$	21.2377	0.696787	0.348394	+	0.937348i	$0.386727\pi$
$930$	0.920926	0.0301984
$931$	−0.878966	−0.0288069
$932$	−0.805363	−0.0263806
$933$	6.72177	0.220061
$934$	−5.95392	−0.194818
$935$	−7.55071	−0.246935
$936$	−1.00000	−0.0326860
$937$	15.2350	0.497706	0.248853	−	0.968541i	$-0.419946\pi$
$937$	15.2350	0.497706	0.248853	+	0.968541i	$0.419946\pi$
$938$	1.62705	0.0531250
$939$	−20.8705	−0.681083
$940$	3.33622	0.108816
$941$	10.3780	0.338315	0.169157	−	0.985589i	$-0.445895\pi$
$941$	10.3780	0.338315	0.169157	+	0.985589i	$0.445895\pi$
$942$	8.77574	0.285929
$943$	87.2881	2.84249
$944$	0.884938	0.0288023
$945$	2.27743	0.0740847
$946$	34.3165	1.11573
$947$	−14.8925	−0.483943	−0.241971	−	0.970283i	$-0.577794\pi$
$947$	−14.8925	−0.483943	−0.241971	+	0.970283i	$0.577794\pi$
$948$	−9.00058	−0.292325
$949$	15.9992	0.519356
$950$	−4.42055	−0.143422
$951$	−11.3847	−0.369176
$952$	6.43799	0.208656
$953$	−13.2268	−0.428458	−0.214229	−	0.976783i	$-0.568724\pi$
$953$	−13.2268	−0.428458	−0.214229	+	0.976783i	$0.568724\pi$
$954$	−12.4604	−0.403420
$955$	19.9448	0.645399
$956$	−27.6450	−0.894103
$957$	28.1137	0.908787
$958$	2.33530	0.0754501
$959$	−17.7525	−0.573257
$960$	−0.812175	−0.0262128
$961$	−29.7143	−0.958525
$962$	−5.96324	−0.192263
$963$	0.0930370	0.00299808
$964$	5.25526	0.169260
$965$	−16.7593	−0.539502
$966$	−22.9752	−0.739214
$967$	−23.5141	−0.756162	−0.378081	−	0.925772i	$-0.623416\pi$
$967$	−23.5141	−0.756162	−0.378081	+	0.925772i	$0.623416\pi$
$968$	−5.39704	−0.173467
$969$	−2.33833	−0.0751179
$970$	3.88086	0.124607
$971$	13.0234	0.417941	0.208971	−	0.977922i	$-0.432989\pi$
$971$	13.0234	0.417941	0.208971	+	0.977922i	$0.432989\pi$
$972$	1.00000	0.0320750
$973$	21.0840	0.675922
$974$	−17.3845	−0.557036
$975$	−4.34037	−0.139003
$976$	−4.79433	−0.153463
$977$	38.2541	1.22386	0.611928	−	0.790913i	$-0.290394\pi$
$977$	38.2541	1.22386	0.611928	+	0.790913i	$0.290394\pi$
$978$	−19.3491	−0.618716
$979$	41.1404	1.31485
$980$	−0.700926	−0.0223902
$981$	−7.16487	−0.228757
$982$	19.7595	0.630551
$983$	14.4742	0.461656	0.230828	−	0.972995i	$-0.425857\pi$
$983$	14.4742	0.461656	0.230828	+	0.972995i	$0.425857\pi$
$984$	10.6535	0.339620
$985$	−10.4477	−0.332893
$986$	−15.9401	−0.507636
$987$	11.5186	0.366641
$988$	−1.01847	−0.0324019
$989$	69.4359	2.20793
$990$	3.28876	0.104524
$991$	55.8050	1.77270	0.886351	−	0.463013i	$-0.153232\pi$
$991$	55.8050	1.77270	0.886351	+	0.463013i	$0.153232\pi$
$992$	−1.13390	−0.0360014
$993$	−1.90596	−0.0604838
$994$	−25.0135	−0.793379
$995$	−19.1077	−0.605755
$996$	−4.73672	−0.150089
$997$	−24.8981	−0.788530	−0.394265	−	0.918997i	$-0.629001\pi$
$997$	−24.8981	−0.788530	−0.394265	+	0.918997i	$0.629001\pi$
$998$	0.921867	0.0291812
$999$	5.96324	0.188669

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.w.1.7	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.w.1.7	✓	12	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:	\( N \)	\(=\)	\( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8034.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.812175 q^{5} -1.00000 q^{6} -2.80411 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.812175 q^{5} -1.00000 q^{6} -2.80411 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.812175 q^{10} +4.04933 q^{11} +1.00000 q^{12} +1.00000 q^{13} +2.80411 q^{14} -0.812175 q^{15} +1.00000 q^{16} +2.29591 q^{17} -1.00000 q^{18} -1.01847 q^{19} -0.812175 q^{20} -2.80411 q^{21} -4.04933 q^{22} -8.19340 q^{23} -1.00000 q^{24} -4.34037 q^{25} -1.00000 q^{26} +1.00000 q^{27} -2.80411 q^{28} +6.94281 q^{29} +0.812175 q^{30} +1.13390 q^{31} -1.00000 q^{32} +4.04933 q^{33} -2.29591 q^{34} +2.27743 q^{35} +1.00000 q^{36} +5.96324 q^{37} +1.01847 q^{38} +1.00000 q^{39} +0.812175 q^{40} -10.6535 q^{41} +2.80411 q^{42} -8.47462 q^{43} +4.04933 q^{44} -0.812175 q^{45} +8.19340 q^{46} -4.10776 q^{47} +1.00000 q^{48} +0.863023 q^{49} +4.34037 q^{50} +2.29591 q^{51} +1.00000 q^{52} +12.4604 q^{53} -1.00000 q^{54} -3.28876 q^{55} +2.80411 q^{56} -1.01847 q^{57} -6.94281 q^{58} +0.884938 q^{59} -0.812175 q^{60} -4.79433 q^{61} -1.13390 q^{62} -2.80411 q^{63} +1.00000 q^{64} -0.812175 q^{65} -4.04933 q^{66} +0.580237 q^{67} +2.29591 q^{68} -8.19340 q^{69} -2.27743 q^{70} -8.92029 q^{71} -1.00000 q^{72} +15.9992 q^{73} -5.96324 q^{74} -4.34037 q^{75} -1.01847 q^{76} -11.3547 q^{77} -1.00000 q^{78} -9.00058 q^{79} -0.812175 q^{80} +1.00000 q^{81} +10.6535 q^{82} -4.73672 q^{83} -2.80411 q^{84} -1.86468 q^{85} +8.47462 q^{86} +6.94281 q^{87} -4.04933 q^{88} +10.1598 q^{89} +0.812175 q^{90} -2.80411 q^{91} -8.19340 q^{92} +1.13390 q^{93} +4.10776 q^{94} +0.827178 q^{95} -1.00000 q^{96} +4.77835 q^{97} -0.863023 q^{98} +4.04933 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 4 q^{5} - 12 q^{6} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^2 + 12 * q^3 + 12 * q^4 - 4 * q^5 - 12 * q^6 - 12 * q^8 + 12 * q^9	\( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 4 q^{5} - 12 q^{6} - 12 q^{8} + 12 q^{9} + 4 q^{10} - 9 q^{11} + 12 q^{12} + 12 q^{13} - 4 q^{15} + 12 q^{16} - 20 q^{17} - 12 q^{18} + 4 q^{19} - 4 q^{20} + 9 q^{22} - 30 q^{23} - 12 q^{24} + 14 q^{25} - 12 q^{26} + 12 q^{27} - 29 q^{29} + 4 q^{30} + 6 q^{31} - 12 q^{32} - 9 q^{33} + 20 q^{34} - 22 q^{35} + 12 q^{36} + 7 q^{37} - 4 q^{38} + 12 q^{39} + 4 q^{40} - 8 q^{41} - 8 q^{43} - 9 q^{44} - 4 q^{45} + 30 q^{46} - 16 q^{47} + 12 q^{48} + 10 q^{49} - 14 q^{50} - 20 q^{51} + 12 q^{52} - 9 q^{53} - 12 q^{54} - 20 q^{55} + 4 q^{57} + 29 q^{58} - 29 q^{59} - 4 q^{60} - 26 q^{61} - 6 q^{62} + 12 q^{64} - 4 q^{65} + 9 q^{66} + 12 q^{67} - 20 q^{68} - 30 q^{69} + 22 q^{70} - 35 q^{71} - 12 q^{72} + 18 q^{73} - 7 q^{74} + 14 q^{75} + 4 q^{76} - 25 q^{77} - 12 q^{78} - 37 q^{79} - 4 q^{80} + 12 q^{81} + 8 q^{82} - 24 q^{83} - 17 q^{85} + 8 q^{86} - 29 q^{87} + 9 q^{88} + 15 q^{89} + 4 q^{90} - 30 q^{92} + 6 q^{93} + 16 q^{94} - 54 q^{95} - 12 q^{96} - 11 q^{97} - 10 q^{98} - 9 q^{99}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^3 + 12 * q^4 - 4 * q^5 - 12 * q^6 - 12 * q^8 + 12 * q^9 + 4 * q^10 - 9 * q^11 + 12 * q^12 + 12 * q^13 - 4 * q^15 + 12 * q^16 - 20 * q^17 - 12 * q^18 + 4 * q^19 - 4 * q^20 + 9 * q^22 - 30 * q^23 - 12 * q^24 + 14 * q^25 - 12 * q^26 + 12 * q^27 - 29 * q^29 + 4 * q^30 + 6 * q^31 - 12 * q^32 - 9 * q^33 + 20 * q^34 - 22 * q^35 + 12 * q^36 + 7 * q^37 - 4 * q^38 + 12 * q^39 + 4 * q^40 - 8 * q^41 - 8 * q^43 - 9 * q^44 - 4 * q^45 + 30 * q^46 - 16 * q^47 + 12 * q^48 + 10 * q^49 - 14 * q^50 - 20 * q^51 + 12 * q^52 - 9 * q^53 - 12 * q^54 - 20 * q^55 + 4 * q^57 + 29 * q^58 - 29 * q^59 - 4 * q^60 - 26 * q^61 - 6 * q^62 + 12 * q^64 - 4 * q^65 + 9 * q^66 + 12 * q^67 - 20 * q^68 - 30 * q^69 + 22 * q^70 - 35 * q^71 - 12 * q^72 + 18 * q^73 - 7 * q^74 + 14 * q^75 + 4 * q^76 - 25 * q^77 - 12 * q^78 - 37 * q^79 - 4 * q^80 + 12 * q^81 + 8 * q^82 - 24 * q^83 - 17 * q^85 + 8 * q^86 - 29 * q^87 + 9 * q^88 + 15 * q^89 + 4 * q^90 - 30 * q^92 + 6 * q^93 + 16 * q^94 - 54 * q^95 - 12 * q^96 - 11 * q^97 - 10 * q^98 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more