LMFDB - Embedded newform 8034.2.a.z.1.13

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:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 36 x^{12} + 108 x^{11} + 434 x^{10} - 1239 x^{9} - 2404 x^{8} + 6204 x^{7} + \cdots + 1552 $ x^14 - 3x^13 - 36x^12 + 108x^11 + 434x^10 - 1239x^9 - 2404x^8 + 6204x^7 + 6361x^6 - 14618x^5 - 7015x^4 + 15763x^3 + 1118x^2 - 6316*x + 1552
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-2.41893$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.41893 q^{5} +1.00000 q^{6} -0.257176 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.41893 q^{5} +1.00000 q^{6} -0.257176 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.41893 q^{10} +1.56925 q^{11} -1.00000 q^{12} +1.00000 q^{13} +0.257176 q^{14} -2.41893 q^{15} +1.00000 q^{16} +4.26992 q^{17} -1.00000 q^{18} +1.38482 q^{19} +2.41893 q^{20} +0.257176 q^{21} -1.56925 q^{22} +8.56848 q^{23} +1.00000 q^{24} +0.851240 q^{25} -1.00000 q^{26} -1.00000 q^{27} -0.257176 q^{28} +10.3349 q^{29} +2.41893 q^{30} +7.55065 q^{31} -1.00000 q^{32} -1.56925 q^{33} -4.26992 q^{34} -0.622092 q^{35} +1.00000 q^{36} -2.82902 q^{37} -1.38482 q^{38} -1.00000 q^{39} -2.41893 q^{40} +10.3277 q^{41} -0.257176 q^{42} -6.51464 q^{43} +1.56925 q^{44} +2.41893 q^{45} -8.56848 q^{46} +9.53966 q^{47} -1.00000 q^{48} -6.93386 q^{49} -0.851240 q^{50} -4.26992 q^{51} +1.00000 q^{52} -12.5020 q^{53} +1.00000 q^{54} +3.79591 q^{55} +0.257176 q^{56} -1.38482 q^{57} -10.3349 q^{58} -4.00889 q^{59} -2.41893 q^{60} +12.4048 q^{61} -7.55065 q^{62} -0.257176 q^{63} +1.00000 q^{64} +2.41893 q^{65} +1.56925 q^{66} +9.28134 q^{67} +4.26992 q^{68} -8.56848 q^{69} +0.622092 q^{70} -0.286350 q^{71} -1.00000 q^{72} -15.7947 q^{73} +2.82902 q^{74} -0.851240 q^{75} +1.38482 q^{76} -0.403573 q^{77} +1.00000 q^{78} +9.06214 q^{79} +2.41893 q^{80} +1.00000 q^{81} -10.3277 q^{82} -6.84929 q^{83} +0.257176 q^{84} +10.3286 q^{85} +6.51464 q^{86} -10.3349 q^{87} -1.56925 q^{88} -14.0981 q^{89} -2.41893 q^{90} -0.257176 q^{91} +8.56848 q^{92} -7.55065 q^{93} -9.53966 q^{94} +3.34978 q^{95} +1.00000 q^{96} +13.3988 q^{97} +6.93386 q^{98} +1.56925 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{2} - 14 q^{3} + 14 q^{4} - 3 q^{5} + 14 q^{6} + 4 q^{7} - 14 q^{8} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^2 - 14 * q^3 + 14 * q^4 - 3 * q^5 + 14 * q^6 + 4 * q^7 - 14 * q^8 + 14 * q^9	\( 14 q - 14 q^{2} - 14 q^{3} + 14 q^{4} - 3 q^{5} + 14 q^{6} + 4 q^{7} - 14 q^{8} + 14 q^{9} + 3 q^{10} + 5 q^{11} - 14 q^{12} + 14 q^{13} - 4 q^{14} + 3 q^{15} + 14 q^{16} - 7 q^{17} - 14 q^{18} + 31 q^{19} - 3 q^{20} - 4 q^{21} - 5 q^{22} + 14 q^{23} + 14 q^{24} + 11 q^{25} - 14 q^{26} - 14 q^{27} + 4 q^{28} + 9 q^{29} - 3 q^{30} + 23 q^{31} - 14 q^{32} - 5 q^{33} + 7 q^{34} - 2 q^{35} + 14 q^{36} + 12 q^{37} - 31 q^{38} - 14 q^{39} + 3 q^{40} - 5 q^{41} + 4 q^{42} - 2 q^{43} + 5 q^{44} - 3 q^{45} - 14 q^{46} - 11 q^{47} - 14 q^{48} + 52 q^{49} - 11 q^{50} + 7 q^{51} + 14 q^{52} + 6 q^{53} + 14 q^{54} + 30 q^{55} - 4 q^{56} - 31 q^{57} - 9 q^{58} - 4 q^{59} + 3 q^{60} + 12 q^{61} - 23 q^{62} + 4 q^{63} + 14 q^{64} - 3 q^{65} + 5 q^{66} + 24 q^{67} - 7 q^{68} - 14 q^{69} + 2 q^{70} + 20 q^{71} - 14 q^{72} + 2 q^{73} - 12 q^{74} - 11 q^{75} + 31 q^{76} - 28 q^{77} + 14 q^{78} + 59 q^{79} - 3 q^{80} + 14 q^{81} + 5 q^{82} + q^{83} - 4 q^{84} - 29 q^{85} + 2 q^{86} - 9 q^{87} - 5 q^{88} + 6 q^{89} + 3 q^{90} + 4 q^{91} + 14 q^{92} - 23 q^{93} + 11 q^{94} - 58 q^{95} + 14 q^{96} - 6 q^{97} - 52 q^{98} + 5 q^{99}+O(q^{100}) \) 14 * q - 14 * q^2 - 14 * q^3 + 14 * q^4 - 3 * q^5 + 14 * q^6 + 4 * q^7 - 14 * q^8 + 14 * q^9 + 3 * q^10 + 5 * q^11 - 14 * q^12 + 14 * q^13 - 4 * q^14 + 3 * q^15 + 14 * q^16 - 7 * q^17 - 14 * q^18 + 31 * q^19 - 3 * q^20 - 4 * q^21 - 5 * q^22 + 14 * q^23 + 14 * q^24 + 11 * q^25 - 14 * q^26 - 14 * q^27 + 4 * q^28 + 9 * q^29 - 3 * q^30 + 23 * q^31 - 14 * q^32 - 5 * q^33 + 7 * q^34 - 2 * q^35 + 14 * q^36 + 12 * q^37 - 31 * q^38 - 14 * q^39 + 3 * q^40 - 5 * q^41 + 4 * q^42 - 2 * q^43 + 5 * q^44 - 3 * q^45 - 14 * q^46 - 11 * q^47 - 14 * q^48 + 52 * q^49 - 11 * q^50 + 7 * q^51 + 14 * q^52 + 6 * q^53 + 14 * q^54 + 30 * q^55 - 4 * q^56 - 31 * q^57 - 9 * q^58 - 4 * q^59 + 3 * q^60 + 12 * q^61 - 23 * q^62 + 4 * q^63 + 14 * q^64 - 3 * q^65 + 5 * q^66 + 24 * q^67 - 7 * q^68 - 14 * q^69 + 2 * q^70 + 20 * q^71 - 14 * q^72 + 2 * q^73 - 12 * q^74 - 11 * q^75 + 31 * q^76 - 28 * q^77 + 14 * q^78 + 59 * q^79 - 3 * q^80 + 14 * q^81 + 5 * q^82 + q^83 - 4 * q^84 - 29 * q^85 + 2 * q^86 - 9 * q^87 - 5 * q^88 + 6 * q^89 + 3 * q^90 + 4 * q^91 + 14 * q^92 - 23 * q^93 + 11 * q^94 - 58 * q^95 + 14 * q^96 - 6 * q^97 - 52 * q^98 + 5 * 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
$5$	2.41893	1.08178	0.540890	−	0.841093i	$-0.318088\pi$
$5$	2.41893	1.08178	0.540890	+	0.841093i	$0.318088\pi$
$6$	1.00000	0.408248
$7$	−0.257176	−0.0972034	−0.0486017	−	0.998818i	$-0.515476\pi$
$7$	−0.257176	−0.0972034	−0.0486017	+	0.998818i	$0.515476\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−2.41893	−0.764934
$11$	1.56925	0.473146	0.236573	−	0.971614i	$-0.423976\pi$
$11$	1.56925	0.473146	0.236573	+	0.971614i	$0.423976\pi$
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0.257176	0.0687332
$15$	−2.41893	−0.624566
$16$	1.00000	0.250000
$17$	4.26992	1.03561	0.517804	−	0.855500i	$-0.326750\pi$
$17$	4.26992	1.03561	0.517804	+	0.855500i	$0.326750\pi$
$18$	−1.00000	−0.235702
$19$	1.38482	0.317699	0.158849	−	0.987303i	$-0.449222\pi$
$19$	1.38482	0.317699	0.158849	+	0.987303i	$0.449222\pi$
$20$	2.41893	0.540890
$21$	0.257176	0.0561204
$22$	−1.56925	−0.334565
$23$	8.56848	1.78665	0.893326	−	0.449409i	$-0.148365\pi$
$23$	8.56848	1.78665	0.893326	+	0.449409i	$0.148365\pi$
$24$	1.00000	0.204124
$25$	0.851240	0.170248
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	−0.257176	−0.0486017
$29$	10.3349	1.91915	0.959573	−	0.281459i	$-0.0908182\pi$
$29$	10.3349	1.91915	0.959573	+	0.281459i	$0.0908182\pi$
$30$	2.41893	0.441635
$31$	7.55065	1.35614	0.678069	−	0.734998i	$-0.262817\pi$
$31$	7.55065	1.35614	0.678069	+	0.734998i	$0.262817\pi$
$32$	−1.00000	−0.176777
$33$	−1.56925	−0.273171
$34$	−4.26992	−0.732285
$35$	−0.622092	−0.105153
$36$	1.00000	0.166667
$37$	−2.82902	−0.465088	−0.232544	−	0.972586i	$-0.574705\pi$
$37$	−2.82902	−0.465088	−0.232544	+	0.972586i	$0.574705\pi$
$38$	−1.38482	−0.224647
$39$	−1.00000	−0.160128
$40$	−2.41893	−0.382467
$41$	10.3277	1.61291	0.806455	−	0.591296i	$-0.201384\pi$
$41$	10.3277	1.61291	0.806455	+	0.591296i	$0.201384\pi$
$42$	−0.257176	−0.0396831
$43$	−6.51464	−0.993474	−0.496737	−	0.867901i	$-0.665469\pi$
$43$	−6.51464	−0.993474	−0.496737	+	0.867901i	$0.665469\pi$
$44$	1.56925	0.236573
$45$	2.41893	0.360593
$46$	−8.56848	−1.26335
$47$	9.53966	1.39150	0.695751	−	0.718283i	$-0.255072\pi$
$47$	9.53966	1.39150	0.695751	+	0.718283i	$0.255072\pi$
$48$	−1.00000	−0.144338
$49$	−6.93386	−0.990552
$50$	−0.851240	−0.120384
$51$	−4.26992	−0.597908
$52$	1.00000	0.138675
$53$	−12.5020	−1.71728	−0.858640	−	0.512580i	$-0.828690\pi$
$53$	−12.5020	−1.71728	−0.858640	+	0.512580i	$0.828690\pi$
$54$	1.00000	0.136083
$55$	3.79591	0.511840
$56$	0.257176	0.0343666
$57$	−1.38482	−0.183423
$58$	−10.3349	−1.35704
$59$	−4.00889	−0.521913	−0.260957	−	0.965351i	$-0.584038\pi$
$59$	−4.00889	−0.521913	−0.260957	+	0.965351i	$0.584038\pi$
$60$	−2.41893	−0.312283
$61$	12.4048	1.58827	0.794134	−	0.607743i	$-0.207925\pi$
$61$	12.4048	1.58827	0.794134	+	0.607743i	$0.207925\pi$
$62$	−7.55065	−0.958934
$63$	−0.257176	−0.0324011
$64$	1.00000	0.125000
$65$	2.41893	0.300032
$66$	1.56925	0.193161
$67$	9.28134	1.13390	0.566948	−	0.823753i	$-0.308124\pi$
$67$	9.28134	1.13390	0.566948	+	0.823753i	$0.308124\pi$
$68$	4.26992	0.517804
$69$	−8.56848	−1.03152
$70$	0.622092	0.0743542
$71$	−0.286350	−0.0339834	−0.0169917	−	0.999856i	$-0.505409\pi$
$71$	−0.286350	−0.0339834	−0.0169917	+	0.999856i	$0.505409\pi$
$72$	−1.00000	−0.117851
$73$	−15.7947	−1.84863	−0.924313	−	0.381635i	$-0.875361\pi$
$73$	−15.7947	−1.84863	−0.924313	+	0.381635i	$0.875361\pi$
$74$	2.82902	0.328867
$75$	−0.851240	−0.0982928
$76$	1.38482	0.158849
$77$	−0.403573	−0.0459914
$78$	1.00000	0.113228
$79$	9.06214	1.01957	0.509785	−	0.860302i	$-0.329725\pi$
$79$	9.06214	1.01957	0.509785	+	0.860302i	$0.329725\pi$
$80$	2.41893	0.270445
$81$	1.00000	0.111111
$82$	−10.3277	−1.14050
$83$	−6.84929	−0.751808	−0.375904	−	0.926659i	$-0.622668\pi$
$83$	−6.84929	−0.751808	−0.375904	+	0.926659i	$0.622668\pi$
$84$	0.257176	0.0280602
$85$	10.3286	1.12030
$86$	6.51464	0.702492
$87$	−10.3349	−1.10802
$88$	−1.56925	−0.167282
$89$	−14.0981	−1.49440	−0.747199	−	0.664600i	$-0.768602\pi$
$89$	−14.0981	−1.49440	−0.747199	+	0.664600i	$0.768602\pi$
$90$	−2.41893	−0.254978
$91$	−0.257176	−0.0269594
$92$	8.56848	0.893326
$93$	−7.55065	−0.782966
$94$	−9.53966	−0.983940
$95$	3.34978	0.343680
$96$	1.00000	0.102062
$97$	13.3988	1.36044	0.680220	−	0.733008i	$-0.261884\pi$
$97$	13.3988	1.36044	0.680220	+	0.733008i	$0.261884\pi$
$98$	6.93386	0.700426
$99$	1.56925	0.157715
$100$	0.851240	0.0851240
$101$	−3.46177	−0.344459	−0.172229	−	0.985057i	$-0.555097\pi$
$101$	−3.46177	−0.344459	−0.172229	+	0.985057i	$0.555097\pi$
$102$	4.26992	0.422785
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	−1.00000	−0.0980581
$105$	0.622092	0.0607099
$106$	12.5020	1.21430
$107$	−14.7757	−1.42842	−0.714209	−	0.699932i	$-0.753213\pi$
$107$	−14.7757	−1.42842	−0.714209	+	0.699932i	$0.753213\pi$
$108$	−1.00000	−0.0962250
$109$	−7.20934	−0.690530	−0.345265	−	0.938505i	$-0.612211\pi$
$109$	−7.20934	−0.690530	−0.345265	+	0.938505i	$0.612211\pi$
$110$	−3.79591	−0.361926
$111$	2.82902	0.268519
$112$	−0.257176	−0.0243008
$113$	9.35198	0.879760	0.439880	−	0.898057i	$-0.355021\pi$
$113$	9.35198	0.879760	0.439880	+	0.898057i	$0.355021\pi$
$114$	1.38482	0.129700
$115$	20.7266	1.93276
$116$	10.3349	0.959573
$117$	1.00000	0.0924500
$118$	4.00889	0.369048
$119$	−1.09812	−0.100664
$120$	2.41893	0.220817
$121$	−8.53746	−0.776133
$122$	−12.4048	−1.12307
$123$	−10.3277	−0.931214
$124$	7.55065	0.678069
$125$	−10.0356	−0.897609
$126$	0.257176	0.0229111
$127$	−10.9282	−0.969723	−0.484862	−	0.874591i	$-0.661130\pi$
$127$	−10.9282	−0.969723	−0.484862	+	0.874591i	$0.661130\pi$
$128$	−1.00000	−0.0883883
$129$	6.51464	0.573582
$130$	−2.41893	−0.212155
$131$	19.8225	1.73190	0.865949	−	0.500132i	$-0.166715\pi$
$131$	19.8225	1.73190	0.865949	+	0.500132i	$0.166715\pi$
$132$	−1.56925	−0.136586
$133$	−0.356141	−0.0308814
$134$	−9.28134	−0.801786
$135$	−2.41893	−0.208189
$136$	−4.26992	−0.366142
$137$	−7.15001	−0.610867	−0.305433	−	0.952213i	$-0.598801\pi$
$137$	−7.15001	−0.610867	−0.305433	+	0.952213i	$0.598801\pi$
$138$	8.56848	0.729398
$139$	3.85007	0.326559	0.163279	−	0.986580i	$-0.447793\pi$
$139$	3.85007	0.326559	0.163279	+	0.986580i	$0.447793\pi$
$140$	−0.622092	−0.0525763
$141$	−9.53966	−0.803384
$142$	0.286350	0.0240299
$143$	1.56925	0.131227
$144$	1.00000	0.0833333
$145$	24.9995	2.07609
$146$	15.7947	1.30718
$147$	6.93386	0.571895
$148$	−2.82902	−0.232544
$149$	−10.1252	−0.829490	−0.414745	−	0.909938i	$-0.636129\pi$
$149$	−10.1252	−0.829490	−0.414745	+	0.909938i	$0.636129\pi$
$150$	0.851240	0.0695035
$151$	15.0201	1.22232	0.611159	−	0.791508i	$-0.290704\pi$
$151$	15.0201	1.22232	0.611159	+	0.791508i	$0.290704\pi$
$152$	−1.38482	−0.112323
$153$	4.26992	0.345202
$154$	0.403573	0.0325208
$155$	18.2645	1.46704
$156$	−1.00000	−0.0800641
$157$	22.4063	1.78822	0.894109	−	0.447849i	$-0.147810\pi$
$157$	22.4063	1.78822	0.894109	+	0.447849i	$0.147810\pi$
$158$	−9.06214	−0.720946
$159$	12.5020	0.991472
$160$	−2.41893	−0.191234
$161$	−2.20361	−0.173669
$162$	−1.00000	−0.0785674
$163$	3.77020	0.295305	0.147652	−	0.989039i	$-0.452828\pi$
$163$	3.77020	0.295305	0.147652	+	0.989039i	$0.452828\pi$
$164$	10.3277	0.806455
$165$	−3.79591	−0.295511
$166$	6.84929	0.531608
$167$	11.1628	0.863801	0.431900	−	0.901921i	$-0.357843\pi$
$167$	11.1628	0.863801	0.431900	+	0.901921i	$0.357843\pi$
$168$	−0.257176	−0.0198416
$169$	1.00000	0.0769231
$170$	−10.3286	−0.792171
$171$	1.38482	0.105900
$172$	−6.51464	−0.496737
$173$	−13.9420	−1.05999	−0.529994	−	0.848001i	$-0.677806\pi$
$173$	−13.9420	−1.05999	−0.529994	+	0.848001i	$0.677806\pi$
$174$	10.3349	0.783488
$175$	−0.218919	−0.0165487
$176$	1.56925	0.118287
$177$	4.00889	0.301327
$178$	14.0981	1.05670
$179$	−1.46855	−0.109765	−0.0548824	−	0.998493i	$-0.517478\pi$
$179$	−1.46855	−0.109765	−0.0548824	+	0.998493i	$0.517478\pi$
$180$	2.41893	0.180297
$181$	−16.8574	−1.25300	−0.626499	−	0.779422i	$-0.715513\pi$
$181$	−16.8574	−1.25300	−0.626499	+	0.779422i	$0.715513\pi$
$182$	0.257176	0.0190631
$183$	−12.4048	−0.916987
$184$	−8.56848	−0.631677
$185$	−6.84321	−0.503123
$186$	7.55065	0.553641
$187$	6.70056	0.489993
$188$	9.53966	0.695751
$189$	0.257176	0.0187068
$190$	−3.34978	−0.243018
$191$	10.5816	0.765660	0.382830	−	0.923819i	$-0.374949\pi$
$191$	10.5816	0.765660	0.382830	+	0.923819i	$0.374949\pi$
$192$	−1.00000	−0.0721688
$193$	6.61573	0.476210	0.238105	−	0.971239i	$-0.423474\pi$
$193$	6.61573	0.476210	0.238105	+	0.971239i	$0.423474\pi$
$194$	−13.3988	−0.961976
$195$	−2.41893	−0.173223
$196$	−6.93386	−0.495276
$197$	−18.2594	−1.30093	−0.650463	−	0.759538i	$-0.725425\pi$
$197$	−18.2594	−1.30093	−0.650463	+	0.759538i	$0.725425\pi$
$198$	−1.56925	−0.111522
$199$	−5.58471	−0.395890	−0.197945	−	0.980213i	$-0.563427\pi$
$199$	−5.58471	−0.395890	−0.197945	+	0.980213i	$0.563427\pi$
$200$	−0.851240	−0.0601918
$201$	−9.28134	−0.654655
$202$	3.46177	0.243569
$203$	−2.65789	−0.186548
$204$	−4.26992	−0.298954
$205$	24.9819	1.74481
$206$	1.00000	0.0696733
$207$	8.56848	0.595551
$208$	1.00000	0.0693375
$209$	2.17312	0.150318
$210$	−0.622092	−0.0429284
$211$	−15.0750	−1.03780	−0.518902	−	0.854834i	$-0.673659\pi$
$211$	−15.0750	−1.03780	−0.518902	+	0.854834i	$0.673659\pi$
$212$	−12.5020	−0.858640
$213$	0.286350	0.0196204
$214$	14.7757	1.01004
$215$	−15.7585	−1.07472
$216$	1.00000	0.0680414
$217$	−1.94185	−0.131821
$218$	7.20934	0.488278
$219$	15.7947	1.06730
$220$	3.79591	0.255920
$221$	4.26992	0.287226
$222$	−2.82902	−0.189871
$223$	−22.1745	−1.48492	−0.742458	−	0.669892i	$-0.766340\pi$
$223$	−22.1745	−1.48492	−0.742458	+	0.669892i	$0.766340\pi$
$224$	0.257176	0.0171833
$225$	0.851240	0.0567493
$226$	−9.35198	−0.622084
$227$	15.2505	1.01221	0.506107	−	0.862471i	$-0.331084\pi$
$227$	15.2505	1.01221	0.506107	+	0.862471i	$0.331084\pi$
$228$	−1.38482	−0.0917117
$229$	−2.61103	−0.172542	−0.0862708	−	0.996272i	$-0.527495\pi$
$229$	−2.61103	−0.172542	−0.0862708	+	0.996272i	$0.527495\pi$
$230$	−20.7266	−1.36667
$231$	0.403573	0.0265531
$232$	−10.3349	−0.678521
$233$	12.0453	0.789112	0.394556	−	0.918872i	$-0.370898\pi$
$233$	12.0453	0.789112	0.394556	+	0.918872i	$0.370898\pi$
$234$	−1.00000	−0.0653720
$235$	23.0758	1.50530
$236$	−4.00889	−0.260957
$237$	−9.06214	−0.588650
$238$	1.09812	0.0711805
$239$	−16.0683	−1.03937	−0.519685	−	0.854358i	$-0.673951\pi$
$239$	−16.0683	−1.03937	−0.519685	+	0.854358i	$0.673951\pi$
$240$	−2.41893	−0.156141
$241$	3.41064	0.219699	0.109849	−	0.993948i	$-0.464963\pi$
$241$	3.41064	0.219699	0.109849	+	0.993948i	$0.464963\pi$
$242$	8.53746	0.548809
$243$	−1.00000	−0.0641500
$244$	12.4048	0.794134
$245$	−16.7725	−1.07156
$246$	10.3277	0.658467
$247$	1.38482	0.0881137
$248$	−7.55065	−0.479467
$249$	6.84929	0.434056
$250$	10.0356	0.634705
$251$	18.3812	1.16021	0.580106	−	0.814541i	$-0.303011\pi$
$251$	18.3812	1.16021	0.580106	+	0.814541i	$0.303011\pi$
$252$	−0.257176	−0.0162006
$253$	13.4461	0.845348
$254$	10.9282	0.685698
$255$	−10.3286	−0.646805
$256$	1.00000	0.0625000
$257$	27.4438	1.71190	0.855950	−	0.517059i	$-0.172973\pi$
$257$	27.4438	1.71190	0.855950	+	0.517059i	$0.172973\pi$
$258$	−6.51464	−0.405584
$259$	0.727555	0.0452081
$260$	2.41893	0.150016
$261$	10.3349	0.639716
$262$	−19.8225	−1.22464
$263$	−24.1083	−1.48658	−0.743290	−	0.668969i	$-0.766736\pi$
$263$	−24.1083	−1.48658	−0.743290	+	0.668969i	$0.766736\pi$
$264$	1.56925	0.0965805
$265$	−30.2415	−1.85772
$266$	0.356141	0.0218364
$267$	14.0981	0.862791
$268$	9.28134	0.566948
$269$	−11.7471	−0.716233	−0.358117	−	0.933677i	$-0.616581\pi$
$269$	−11.7471	−0.716233	−0.358117	+	0.933677i	$0.616581\pi$
$270$	2.41893	0.147212
$271$	20.2824	1.23207	0.616035	−	0.787719i	$-0.288738\pi$
$271$	20.2824	1.23207	0.616035	+	0.787719i	$0.288738\pi$
$272$	4.26992	0.258902
$273$	0.257176	0.0155650
$274$	7.15001	0.431948
$275$	1.33581	0.0805522
$276$	−8.56848	−0.515762
$277$	−26.9165	−1.61725	−0.808627	−	0.588322i	$-0.799789\pi$
$277$	−26.9165	−1.61725	−0.808627	+	0.588322i	$0.799789\pi$
$278$	−3.85007	−0.230912
$279$	7.55065	0.452046
$280$	0.622092	0.0371771
$281$	−18.6951	−1.11526	−0.557628	−	0.830091i	$-0.688289\pi$
$281$	−18.6951	−1.11526	−0.557628	+	0.830091i	$0.688289\pi$
$282$	9.53966	0.568078
$283$	−23.9981	−1.42654	−0.713270	−	0.700889i	$-0.752787\pi$
$283$	−23.9981	−1.42654	−0.713270	+	0.700889i	$0.752787\pi$
$284$	−0.286350	−0.0169917
$285$	−3.34978	−0.198424
$286$	−1.56925	−0.0927916
$287$	−2.65603	−0.156780
$288$	−1.00000	−0.0589256
$289$	1.23219	0.0724819
$290$	−24.9995	−1.46802
$291$	−13.3988	−0.785450
$292$	−15.7947	−0.924313
$293$	−16.2925	−0.951815	−0.475908	−	0.879495i	$-0.657880\pi$
$293$	−16.2925	−0.951815	−0.475908	+	0.879495i	$0.657880\pi$
$294$	−6.93386	−0.404391
$295$	−9.69724	−0.564595
$296$	2.82902	0.164433
$297$	−1.56925	−0.0910570
$298$	10.1252	0.586538
$299$	8.56848	0.495528
$300$	−0.851240	−0.0491464
$301$	1.67541	0.0965690
$302$	−15.0201	−0.864309
$303$	3.46177	0.198873
$304$	1.38482	0.0794247
$305$	30.0063	1.71816
$306$	−4.26992	−0.244095
$307$	−9.48598	−0.541393	−0.270697	−	0.962665i	$-0.587254\pi$
$307$	−9.48598	−0.541393	−0.270697	+	0.962665i	$0.587254\pi$
$308$	−0.403573	−0.0229957
$309$	1.00000	0.0568880
$310$	−18.2645	−1.03736
$311$	5.00459	0.283784	0.141892	−	0.989882i	$-0.454681\pi$
$311$	5.00459	0.283784	0.141892	+	0.989882i	$0.454681\pi$
$312$	1.00000	0.0566139
$313$	17.0166	0.961835	0.480917	−	0.876766i	$-0.340304\pi$
$313$	17.0166	0.961835	0.480917	+	0.876766i	$0.340304\pi$
$314$	−22.4063	−1.26446
$315$	−0.622092	−0.0350509
$316$	9.06214	0.509785
$317$	−23.2420	−1.30540	−0.652701	−	0.757615i	$-0.726364\pi$
$317$	−23.2420	−1.30540	−0.652701	+	0.757615i	$0.726364\pi$
$318$	−12.5020	−0.701076
$319$	16.2181	0.908037
$320$	2.41893	0.135223
$321$	14.7757	0.824698
$322$	2.20361	0.122802
$323$	5.91305	0.329011
$324$	1.00000	0.0555556
$325$	0.851240	0.0472183
$326$	−3.77020	−0.208812
$327$	7.20934	0.398678
$328$	−10.3277	−0.570250
$329$	−2.45337	−0.135259
$330$	3.79591	0.208958
$331$	−0.342089	−0.0188029	−0.00940146	−	0.999956i	$-0.502993\pi$
$331$	−0.342089	−0.0188029	−0.00940146	+	0.999956i	$0.502993\pi$
$332$	−6.84929	−0.375904
$333$	−2.82902	−0.155029
$334$	−11.1628	−0.610799
$335$	22.4509	1.22663
$336$	0.257176	0.0140301
$337$	25.7894	1.40484	0.702418	−	0.711764i	$-0.252104\pi$
$337$	25.7894	1.40484	0.702418	+	0.711764i	$0.252104\pi$
$338$	−1.00000	−0.0543928
$339$	−9.35198	−0.507930
$340$	10.3286	0.560150
$341$	11.8488	0.641651
$342$	−1.38482	−0.0748823
$343$	3.58345	0.193488
$344$	6.51464	0.351246
$345$	−20.7266	−1.11588
$346$	13.9420	0.749525
$347$	18.4618	0.991079	0.495540	−	0.868585i	$-0.334970\pi$
$347$	18.4618	0.991079	0.495540	+	0.868585i	$0.334970\pi$
$348$	−10.3349	−0.554010
$349$	−20.4645	−1.09544	−0.547719	−	0.836662i	$-0.684504\pi$
$349$	−20.4645	−1.09544	−0.547719	+	0.836662i	$0.684504\pi$
$350$	0.218919	0.0117017
$351$	−1.00000	−0.0533761
$352$	−1.56925	−0.0836412
$353$	10.0583	0.535351	0.267676	−	0.963509i	$-0.413744\pi$
$353$	10.0583	0.535351	0.267676	+	0.963509i	$0.413744\pi$
$354$	−4.00889	−0.213070
$355$	−0.692661	−0.0367626
$356$	−14.0981	−0.747199
$357$	1.09812	0.0581187
$358$	1.46855	0.0776155
$359$	14.8765	0.785149	0.392575	−	0.919720i	$-0.371584\pi$
$359$	14.8765	0.785149	0.392575	+	0.919720i	$0.371584\pi$
$360$	−2.41893	−0.127489
$361$	−17.0823	−0.899068
$362$	16.8574	0.886004
$363$	8.53746	0.448100
$364$	−0.257176	−0.0134797
$365$	−38.2063	−1.99981
$366$	12.4048	0.648407
$367$	−10.7987	−0.563688	−0.281844	−	0.959460i	$-0.590946\pi$
$367$	−10.7987	−0.563688	−0.281844	+	0.959460i	$0.590946\pi$
$368$	8.56848	0.446663
$369$	10.3277	0.537636
$370$	6.84321	0.355761
$371$	3.21521	0.166925
$372$	−7.55065	−0.391483
$373$	27.3303	1.41511	0.707554	−	0.706659i	$-0.249799\pi$
$373$	27.3303	1.41511	0.707554	+	0.706659i	$0.249799\pi$
$374$	−6.70056	−0.346478
$375$	10.0356	0.518235
$376$	−9.53966	−0.491970
$377$	10.3349	0.532276
$378$	−0.257176	−0.0132277
$379$	8.62300	0.442934	0.221467	−	0.975168i	$-0.428916\pi$
$379$	8.62300	0.442934	0.221467	+	0.975168i	$0.428916\pi$
$380$	3.34978	0.171840
$381$	10.9282	0.559870
$382$	−10.5816	−0.541403
$383$	5.34302	0.273015	0.136508	−	0.990639i	$-0.456412\pi$
$383$	5.34302	0.273015	0.136508	+	0.990639i	$0.456412\pi$
$384$	1.00000	0.0510310
$385$	−0.976216	−0.0497526
$386$	−6.61573	−0.336732
$387$	−6.51464	−0.331158
$388$	13.3988	0.680220
$389$	−30.3993	−1.54131	−0.770654	−	0.637254i	$-0.780070\pi$
$389$	−30.3993	−1.54131	−0.770654	+	0.637254i	$0.780070\pi$
$390$	2.41893	0.122487
$391$	36.5867	1.85027
$392$	6.93386	0.350213
$393$	−19.8225	−0.999912
$394$	18.2594	0.919893
$395$	21.9207	1.10295
$396$	1.56925	0.0788577
$397$	−35.9897	−1.80627	−0.903136	−	0.429354i	$-0.858741\pi$
$397$	−35.9897	−1.80627	−0.903136	+	0.429354i	$0.858741\pi$
$398$	5.58471	0.279936
$399$	0.356141	0.0178294
$400$	0.851240	0.0425620
$401$	−15.2475	−0.761423	−0.380712	−	0.924694i	$-0.624321\pi$
$401$	−15.2475	−0.761423	−0.380712	+	0.924694i	$0.624321\pi$
$402$	9.28134	0.462911
$403$	7.55065	0.376125
$404$	−3.46177	−0.172229
$405$	2.41893	0.120198
$406$	2.65789	0.131909
$407$	−4.43943	−0.220054
$408$	4.26992	0.211392
$409$	18.7585	0.927550	0.463775	−	0.885953i	$-0.346495\pi$
$409$	18.7585	0.927550	0.463775	+	0.885953i	$0.346495\pi$
$410$	−24.9819	−1.23377
$411$	7.15001	0.352684
$412$	−1.00000	−0.0492665
$413$	1.03099	0.0507317
$414$	−8.56848	−0.421118
$415$	−16.5680	−0.813291
$416$	−1.00000	−0.0490290
$417$	−3.85007	−0.188539
$418$	−2.17312	−0.106291
$419$	12.8880	0.629621	0.314811	−	0.949154i	$-0.398059\pi$
$419$	12.8880	0.629621	0.314811	+	0.949154i	$0.398059\pi$
$420$	0.622092	0.0303550
$421$	−13.2438	−0.645462	−0.322731	−	0.946491i	$-0.604601\pi$
$421$	−13.2438	−0.645462	−0.322731	+	0.946491i	$0.604601\pi$
$422$	15.0750	0.733839
$423$	9.53966	0.463834
$424$	12.5020	0.607150
$425$	3.63473	0.176310
$426$	−0.286350	−0.0138737
$427$	−3.19021	−0.154385
$428$	−14.7757	−0.714209
$429$	−1.56925	−0.0757640
$430$	15.7585	0.759942
$431$	−13.7422	−0.661937	−0.330969	−	0.943642i	$-0.607375\pi$
$431$	−13.7422	−0.661937	−0.330969	+	0.943642i	$0.607375\pi$
$432$	−1.00000	−0.0481125
$433$	21.5048	1.03346	0.516729	−	0.856149i	$-0.327150\pi$
$433$	21.5048	1.03346	0.516729	+	0.856149i	$0.327150\pi$
$434$	1.94185	0.0932116
$435$	−24.9995	−1.19863
$436$	−7.20934	−0.345265
$437$	11.8658	0.567617
$438$	−15.7947	−0.754699
$439$	2.91356	0.139057	0.0695283	−	0.997580i	$-0.477851\pi$
$439$	2.91356	0.139057	0.0695283	+	0.997580i	$0.477851\pi$
$440$	−3.79591	−0.180963
$441$	−6.93386	−0.330184
$442$	−4.26992	−0.203099
$443$	−29.2420	−1.38933	−0.694664	−	0.719335i	$-0.744447\pi$
$443$	−29.2420	−1.38933	−0.694664	+	0.719335i	$0.744447\pi$
$444$	2.82902	0.134259
$445$	−34.1024	−1.61661
$446$	22.1745	1.04999
$447$	10.1252	0.478906
$448$	−0.257176	−0.0121504
$449$	34.2300	1.61541	0.807706	−	0.589585i	$-0.200709\pi$
$449$	34.2300	1.61541	0.807706	+	0.589585i	$0.200709\pi$
$450$	−0.851240	−0.0401279
$451$	16.2067	0.763142
$452$	9.35198	0.439880
$453$	−15.0201	−0.705706
$454$	−15.2505	−0.715743
$455$	−0.622092	−0.0291641
$456$	1.38482	0.0648500
$457$	−4.90047	−0.229234	−0.114617	−	0.993410i	$-0.536564\pi$
$457$	−4.90047	−0.229234	−0.114617	+	0.993410i	$0.536564\pi$
$458$	2.61103	0.122005
$459$	−4.26992	−0.199303
$460$	20.7266	0.966382
$461$	12.2479	0.570443	0.285221	−	0.958462i	$-0.407933\pi$
$461$	12.2479	0.570443	0.285221	+	0.958462i	$0.407933\pi$
$462$	−0.403573	−0.0187759
$463$	33.1408	1.54018	0.770091	−	0.637934i	$-0.220211\pi$
$463$	33.1408	1.54018	0.770091	+	0.637934i	$0.220211\pi$
$464$	10.3349	0.479787
$465$	−18.2645	−0.846997
$466$	−12.0453	−0.557987
$467$	−23.1797	−1.07263	−0.536314	−	0.844019i	$-0.680184\pi$
$467$	−23.1797	−1.07263	−0.536314	+	0.844019i	$0.680184\pi$
$468$	1.00000	0.0462250
$469$	−2.38694	−0.110219
$470$	−23.0758	−1.06441
$471$	−22.4063	−1.03243
$472$	4.00889	0.184524
$473$	−10.2231	−0.470058
$474$	9.06214	0.416238
$475$	1.17881	0.0540876
$476$	−1.09812	−0.0503322
$477$	−12.5020	−0.572427
$478$	16.0683	0.734946
$479$	−7.47120	−0.341368	−0.170684	−	0.985326i	$-0.554598\pi$
$479$	−7.47120	−0.341368	−0.170684	+	0.985326i	$0.554598\pi$
$480$	2.41893	0.110409
$481$	−2.82902	−0.128992
$482$	−3.41064	−0.155351
$483$	2.20361	0.100268
$484$	−8.53746	−0.388066
$485$	32.4107	1.47170
$486$	1.00000	0.0453609
$487$	−13.0822	−0.592813	−0.296406	−	0.955062i	$-0.595788\pi$
$487$	−13.0822	−0.592813	−0.296406	+	0.955062i	$0.595788\pi$
$488$	−12.4048	−0.561537
$489$	−3.77020	−0.170494
$490$	16.7725	0.757707
$491$	−29.2439	−1.31976	−0.659879	−	0.751372i	$-0.729392\pi$
$491$	−29.2439	−1.31976	−0.659879	+	0.751372i	$0.729392\pi$
$492$	−10.3277	−0.465607
$493$	44.1293	1.98748
$494$	−1.38482	−0.0623058
$495$	3.79591	0.170613
$496$	7.55065	0.339034
$497$	0.0736422	0.00330331
$498$	−6.84929	−0.306924
$499$	20.8225	0.932142	0.466071	−	0.884747i	$-0.345669\pi$
$499$	20.8225	0.932142	0.466071	+	0.884747i	$0.345669\pi$
$500$	−10.0356	−0.448805
$501$	−11.1628	−0.498715
$502$	−18.3812	−0.820393
$503$	12.1548	0.541957	0.270978	−	0.962585i	$-0.412653\pi$
$503$	12.1548	0.541957	0.270978	+	0.962585i	$0.412653\pi$
$504$	0.257176	0.0114555
$505$	−8.37379	−0.372629
$506$	−13.4461	−0.597751
$507$	−1.00000	−0.0444116
$508$	−10.9282	−0.484862
$509$	5.21062	0.230957	0.115478	−	0.993310i	$-0.463160\pi$
$509$	5.21062	0.230957	0.115478	+	0.993310i	$0.463160\pi$
$510$	10.3286	0.457360
$511$	4.06201	0.179693
$512$	−1.00000	−0.0441942
$513$	−1.38482	−0.0611411
$514$	−27.4438	−1.21050
$515$	−2.41893	−0.106591
$516$	6.51464	0.286791
$517$	14.9701	0.658384
$518$	−0.727555	−0.0319670
$519$	13.9420	0.611984
$520$	−2.41893	−0.106077
$521$	−33.9850	−1.48891	−0.744456	−	0.667672i	$-0.767291\pi$
$521$	−33.9850	−1.48891	−0.744456	+	0.667672i	$0.767291\pi$
$522$	−10.3349	−0.452347
$523$	37.6173	1.64489	0.822444	−	0.568846i	$-0.192610\pi$
$523$	37.6173	1.64489	0.822444	+	0.568846i	$0.192610\pi$
$524$	19.8225	0.865949
$525$	0.218919	0.00955439
$526$	24.1083	1.05117
$527$	32.2407	1.40443
$528$	−1.56925	−0.0682928
$529$	50.4189	2.19213
$530$	30.2415	1.31361
$531$	−4.00889	−0.173971
$532$	−0.356141	−0.0154407
$533$	10.3277	0.447341
$534$	−14.0981	−0.610085
$535$	−35.7414	−1.54523
$536$	−9.28134	−0.400893
$537$	1.46855	0.0633728
$538$	11.7471	0.506453
$539$	−10.8809	−0.468676
$540$	−2.41893	−0.104094
$541$	29.9167	1.28622	0.643110	−	0.765774i	$-0.277644\pi$
$541$	29.9167	1.28622	0.643110	+	0.765774i	$0.277644\pi$
$542$	−20.2824	−0.871205
$543$	16.8574	0.723419
$544$	−4.26992	−0.183071
$545$	−17.4389	−0.747001
$546$	−0.257176	−0.0110061
$547$	31.6336	1.35255	0.676276	−	0.736648i	$-0.263592\pi$
$547$	31.6336	1.35255	0.676276	+	0.736648i	$0.263592\pi$
$548$	−7.15001	−0.305433
$549$	12.4048	0.529422
$550$	−1.33581	−0.0569590
$551$	14.3120	0.609710
$552$	8.56848	0.364699
$553$	−2.33057	−0.0991057
$554$	26.9165	1.14357
$555$	6.84321	0.290478
$556$	3.85007	0.163279
$557$	13.8220	0.585659	0.292829	−	0.956165i	$-0.405403\pi$
$557$	13.8220	0.585659	0.292829	+	0.956165i	$0.405403\pi$
$558$	−7.55065	−0.319645
$559$	−6.51464	−0.275540
$560$	−0.622092	−0.0262882
$561$	−6.70056	−0.282898
$562$	18.6951	0.788606
$563$	0.435298	0.0183456	0.00917281	−	0.999958i	$-0.497080\pi$
$563$	0.435298	0.0183456	0.00917281	+	0.999958i	$0.497080\pi$
$564$	−9.53966	−0.401692
$565$	22.6218	0.951707
$566$	23.9981	1.00872
$567$	−0.257176	−0.0108004
$568$	0.286350	0.0120150
$569$	10.2054	0.427831	0.213916	−	0.976852i	$-0.431378\pi$
$569$	10.2054	0.427831	0.213916	+	0.976852i	$0.431378\pi$
$570$	3.34978	0.140307
$571$	22.7820	0.953396	0.476698	−	0.879067i	$-0.341834\pi$
$571$	22.7820	0.953396	0.476698	+	0.879067i	$0.341834\pi$
$572$	1.56925	0.0656136
$573$	−10.5816	−0.442054
$574$	2.65603	0.110860
$575$	7.29384	0.304174
$576$	1.00000	0.0416667
$577$	16.3459	0.680490	0.340245	−	0.940337i	$-0.389490\pi$
$577$	16.3459	0.680490	0.340245	+	0.940337i	$0.389490\pi$
$578$	−1.23219	−0.0512525
$579$	−6.61573	−0.274940
$580$	24.9995	1.03805
$581$	1.76147	0.0730782
$582$	13.3988	0.555397
$583$	−19.6187	−0.812524
$584$	15.7947	0.653588
$585$	2.41893	0.100011
$586$	16.2925	0.673035
$587$	−45.6887	−1.88577	−0.942887	−	0.333112i	$-0.891901\pi$
$587$	−45.6887	−1.88577	−0.942887	+	0.333112i	$0.891901\pi$
$588$	6.93386	0.285948
$589$	10.4563	0.430843
$590$	9.69724	0.399229
$591$	18.2594	0.751089
$592$	−2.82902	−0.116272
$593$	−5.09918	−0.209398	−0.104699	−	0.994504i	$-0.533388\pi$
$593$	−5.09918	−0.209398	−0.104699	+	0.994504i	$0.533388\pi$
$594$	1.56925	0.0643870
$595$	−2.65628	−0.108897
$596$	−10.1252	−0.414745
$597$	5.58471	0.228567
$598$	−8.56848	−0.350391
$599$	−2.52897	−0.103331	−0.0516654	−	0.998664i	$-0.516453\pi$
$599$	−2.52897	−0.103331	−0.0516654	+	0.998664i	$0.516453\pi$
$600$	0.851240	0.0347517
$601$	−13.9478	−0.568943	−0.284471	−	0.958685i	$-0.591818\pi$
$601$	−13.9478	−0.568943	−0.284471	+	0.958685i	$0.591818\pi$
$602$	−1.67541	−0.0682846
$603$	9.28134	0.377965
$604$	15.0201	0.611159
$605$	−20.6516	−0.839605
$606$	−3.46177	−0.140625
$607$	−26.2324	−1.06474	−0.532371	−	0.846511i	$-0.678699\pi$
$607$	−26.2324	−1.06474	−0.532371	+	0.846511i	$0.678699\pi$
$608$	−1.38482	−0.0561617
$609$	2.65789	0.107703
$610$	−30.0063	−1.21492
$611$	9.53966	0.385933
$612$	4.26992	0.172601
$613$	−18.5862	−0.750688	−0.375344	−	0.926886i	$-0.622475\pi$
$613$	−18.5862	−0.750688	−0.375344	+	0.926886i	$0.622475\pi$
$614$	9.48598	0.382823
$615$	−24.9819	−1.00737
$616$	0.403573	0.0162604
$617$	8.44285	0.339896	0.169948	−	0.985453i	$-0.445640\pi$
$617$	8.44285	0.339896	0.169948	+	0.985453i	$0.445640\pi$
$618$	−1.00000	−0.0402259
$619$	33.6205	1.35132	0.675662	−	0.737212i	$-0.263858\pi$
$619$	33.6205	1.35132	0.675662	+	0.737212i	$0.263858\pi$
$620$	18.2645	0.733521
$621$	−8.56848	−0.343841
$622$	−5.00459	−0.200666
$623$	3.62570	0.145261
$624$	−1.00000	−0.0400320
$625$	−28.5316	−1.14126
$626$	−17.0166	−0.680120
$627$	−2.17312	−0.0867861
$628$	22.4063	0.894109
$629$	−12.0797	−0.481648
$630$	0.622092	0.0247847
$631$	−27.1072	−1.07912	−0.539560	−	0.841947i	$-0.681409\pi$
$631$	−27.1072	−1.07912	−0.539560	+	0.841947i	$0.681409\pi$
$632$	−9.06214	−0.360473
$633$	15.0750	0.599177
$634$	23.2420	0.923059
$635$	−26.4347	−1.04903
$636$	12.5020	0.495736
$637$	−6.93386	−0.274730
$638$	−16.2181	−0.642079
$639$	−0.286350	−0.0113278
$640$	−2.41893	−0.0956168
$641$	−7.16304	−0.282923	−0.141462	−	0.989944i	$-0.545180\pi$
$641$	−7.16304	−0.282923	−0.141462	+	0.989944i	$0.545180\pi$
$642$	−14.7757	−0.583149
$643$	43.7819	1.72659	0.863294	−	0.504701i	$-0.168397\pi$
$643$	43.7819	1.72659	0.863294	+	0.504701i	$0.168397\pi$
$644$	−2.20361	−0.0868343
$645$	15.7585	0.620490
$646$	−5.91305	−0.232646
$647$	−40.6556	−1.59834	−0.799169	−	0.601107i	$-0.794727\pi$
$647$	−40.6556	−1.59834	−0.799169	+	0.601107i	$0.794727\pi$
$648$	−1.00000	−0.0392837
$649$	−6.29095	−0.246941
$650$	−0.851240	−0.0333884
$651$	1.94185	0.0761070
$652$	3.77020	0.147652
$653$	−42.1850	−1.65083	−0.825413	−	0.564530i	$-0.809058\pi$
$653$	−42.1850	−1.65083	−0.825413	+	0.564530i	$0.809058\pi$
$654$	−7.20934	−0.281908
$655$	47.9493	1.87353
$656$	10.3277	0.403227
$657$	−15.7947	−0.616209
$658$	2.45337	0.0956423
$659$	32.0508	1.24852	0.624261	−	0.781216i	$-0.285400\pi$
$659$	32.0508	1.24852	0.624261	+	0.781216i	$0.285400\pi$
$660$	−3.79591	−0.147755
$661$	26.4456	1.02861	0.514307	−	0.857606i	$-0.328049\pi$
$661$	26.4456	1.02861	0.514307	+	0.857606i	$0.328049\pi$
$662$	0.342089	0.0132957
$663$	−4.26992	−0.165830
$664$	6.84929	0.265804
$665$	−0.861482	−0.0334069
$666$	2.82902	0.109622
$667$	88.5546	3.42885
$668$	11.1628	0.431900
$669$	22.1745	0.857317
$670$	−22.4509	−0.867356
$671$	19.4662	0.751483
$672$	−0.257176	−0.00992078
$673$	−29.5184	−1.13785	−0.568925	−	0.822390i	$-0.692640\pi$
$673$	−29.5184	−1.13785	−0.568925	+	0.822390i	$0.692640\pi$
$674$	−25.7894	−0.993370
$675$	−0.851240	−0.0327643
$676$	1.00000	0.0384615
$677$	22.0651	0.848030	0.424015	−	0.905655i	$-0.360620\pi$
$677$	22.0651	0.848030	0.424015	+	0.905655i	$0.360620\pi$
$678$	9.35198	0.359161
$679$	−3.44584	−0.132239
$680$	−10.3286	−0.396086
$681$	−15.2505	−0.584402
$682$	−11.8488	−0.453716
$683$	−1.67975	−0.0642737	−0.0321369	−	0.999483i	$-0.510231\pi$
$683$	−1.67975	−0.0642737	−0.0321369	+	0.999483i	$0.510231\pi$
$684$	1.38482	0.0529498
$685$	−17.2954	−0.660823
$686$	−3.58345	−0.136817
$687$	2.61103	0.0996169
$688$	−6.51464	−0.248368
$689$	−12.5020	−0.476288
$690$	20.7266	0.789048
$691$	−34.2647	−1.30349	−0.651745	−	0.758438i	$-0.725963\pi$
$691$	−34.2647	−1.30349	−0.651745	+	0.758438i	$0.725963\pi$
$692$	−13.9420	−0.529994
$693$	−0.403573	−0.0153305
$694$	−18.4618	−0.700799
$695$	9.31306	0.353265
$696$	10.3349	0.391744
$697$	44.0982	1.67034
$698$	20.4645	0.774592
$699$	−12.0453	−0.455594
$700$	−0.218919	−0.00827434
$701$	−9.73279	−0.367602	−0.183801	−	0.982963i	$-0.558840\pi$
$701$	−9.73279	−0.367602	−0.183801	+	0.982963i	$0.558840\pi$
$702$	1.00000	0.0377426
$703$	−3.91767	−0.147758
$704$	1.56925	0.0591433
$705$	−23.0758	−0.869085
$706$	−10.0583	−0.378550
$707$	0.890284	0.0334826
$708$	4.00889	0.150663
$709$	43.2474	1.62419	0.812096	−	0.583524i	$-0.198327\pi$
$709$	43.2474	1.62419	0.812096	+	0.583524i	$0.198327\pi$
$710$	0.692661	0.0259951
$711$	9.06214	0.339857
$712$	14.0981	0.528350
$713$	64.6977	2.42295
$714$	−1.09812	−0.0410961
$715$	3.79591	0.141959
$716$	−1.46855	−0.0548824
$717$	16.0683	0.600081
$718$	−14.8765	−0.555184
$719$	36.1888	1.34961	0.674807	−	0.737995i	$-0.264227\pi$
$719$	36.1888	1.34961	0.674807	+	0.737995i	$0.264227\pi$
$720$	2.41893	0.0901483
$721$	0.257176	0.00957773
$722$	17.0823	0.635737
$723$	−3.41064	−0.126843
$724$	−16.8574	−0.626499
$725$	8.79750	0.326731
$726$	−8.53746	−0.316855
$727$	39.2177	1.45450	0.727252	−	0.686370i	$-0.240797\pi$
$727$	39.2177	1.45450	0.727252	+	0.686370i	$0.240797\pi$
$728$	0.257176	0.00953157
$729$	1.00000	0.0370370
$730$	38.2063	1.41408
$731$	−27.8170	−1.02885
$732$	−12.4048	−0.458493
$733$	−14.5729	−0.538261	−0.269131	−	0.963104i	$-0.586736\pi$
$733$	−14.5729	−0.538261	−0.269131	+	0.963104i	$0.586736\pi$
$734$	10.7987	0.398588
$735$	16.7725	0.618665
$736$	−8.56848	−0.315839
$737$	14.5647	0.536499
$738$	−10.3277	−0.380166
$739$	−13.0285	−0.479261	−0.239631	−	0.970864i	$-0.577026\pi$
$739$	−13.0285	−0.479261	−0.239631	+	0.970864i	$0.577026\pi$
$740$	−6.84321	−0.251561
$741$	−1.38482	−0.0508725
$742$	−3.21521	−0.118034
$743$	13.9766	0.512751	0.256376	−	0.966577i	$-0.417472\pi$
$743$	13.9766	0.512751	0.256376	+	0.966577i	$0.417472\pi$
$744$	7.55065	0.276820
$745$	−24.4922	−0.897326
$746$	−27.3303	−1.00063
$747$	−6.84929	−0.250603
$748$	6.70056	0.244997
$749$	3.79995	0.138847
$750$	−10.0356	−0.366447
$751$	−6.09636	−0.222459	−0.111230	−	0.993795i	$-0.535479\pi$
$751$	−6.09636	−0.222459	−0.111230	+	0.993795i	$0.535479\pi$
$752$	9.53966	0.347875
$753$	−18.3812	−0.669848
$754$	−10.3349	−0.376376
$755$	36.3326	1.32228
$756$	0.257176	0.00935340
$757$	35.0595	1.27426	0.637130	−	0.770756i	$-0.280121\pi$
$757$	35.0595	1.27426	0.637130	+	0.770756i	$0.280121\pi$
$758$	−8.62300	−0.313202
$759$	−13.4461	−0.488062
$760$	−3.34978	−0.121509
$761$	−18.3039	−0.663515	−0.331758	−	0.943365i	$-0.607642\pi$
$761$	−18.3039	−0.663515	−0.331758	+	0.943365i	$0.607642\pi$
$762$	−10.9282	−0.395888
$763$	1.85407	0.0671218
$764$	10.5816	0.382830
$765$	10.3286	0.373433
$766$	−5.34302	−0.193051
$767$	−4.00889	−0.144753
$768$	−1.00000	−0.0360844
$769$	−28.0075	−1.00998	−0.504989	−	0.863126i	$-0.668503\pi$
$769$	−28.0075	−1.00998	−0.504989	+	0.863126i	$0.668503\pi$
$770$	0.976216	0.0351804
$771$	−27.4438	−0.988365
$772$	6.61573	0.238105
$773$	16.9295	0.608910	0.304455	−	0.952527i	$-0.401526\pi$
$773$	16.9295	0.608910	0.304455	+	0.952527i	$0.401526\pi$
$774$	6.51464	0.234164
$775$	6.42742	0.230880
$776$	−13.3988	−0.480988
$777$	−0.727555	−0.0261009
$778$	30.3993	1.08987
$779$	14.3019	0.512419
$780$	−2.41893	−0.0866117
$781$	−0.449354	−0.0160791
$782$	−36.5867	−1.30834
$783$	−10.3349	−0.369340
$784$	−6.93386	−0.247638
$785$	54.1994	1.93446
$786$	19.8225	0.707044
$787$	4.30348	0.153402	0.0767012	−	0.997054i	$-0.475561\pi$
$787$	4.30348	0.153402	0.0767012	+	0.997054i	$0.475561\pi$
$788$	−18.2594	−0.650463
$789$	24.1083	0.858278
$790$	−21.9207	−0.779904
$791$	−2.40510	−0.0855156
$792$	−1.56925	−0.0557608
$793$	12.4048	0.440506
$794$	35.9897	1.27723
$795$	30.2415	1.07255
$796$	−5.58471	−0.197945
$797$	9.26665	0.328242	0.164121	−	0.986440i	$-0.447521\pi$
$797$	9.26665	0.328242	0.164121	+	0.986440i	$0.447521\pi$
$798$	−0.356141	−0.0126073
$799$	40.7335	1.44105
$800$	−0.851240	−0.0300959
$801$	−14.0981	−0.498133
$802$	15.2475	0.538408
$803$	−24.7858	−0.874670
$804$	−9.28134	−0.327328
$805$	−5.33038	−0.187871
$806$	−7.55065	−0.265960
$807$	11.7471	0.413517
$808$	3.46177	0.121785
$809$	−25.8177	−0.907701	−0.453850	−	0.891078i	$-0.649950\pi$
$809$	−25.8177	−0.907701	−0.453850	+	0.891078i	$0.649950\pi$
$810$	−2.41893	−0.0849927
$811$	27.7188	0.973339	0.486669	−	0.873586i	$-0.338212\pi$
$811$	27.7188	0.973339	0.486669	+	0.873586i	$0.338212\pi$
$812$	−2.65789	−0.0932738
$813$	−20.2824	−0.711336
$814$	4.43943	0.155602
$815$	9.11986	0.319455
$816$	−4.26992	−0.149477
$817$	−9.02158	−0.315625
$818$	−18.7585	−0.655877
$819$	−0.257176	−0.00898645
$820$	24.9819	0.872406
$821$	8.58538	0.299632	0.149816	−	0.988714i	$-0.452132\pi$
$821$	8.58538	0.299632	0.149816	+	0.988714i	$0.452132\pi$
$822$	−7.15001	−0.249385
$823$	−25.0310	−0.872527	−0.436263	−	0.899819i	$-0.643698\pi$
$823$	−25.0310	−0.872527	−0.436263	+	0.899819i	$0.643698\pi$
$824$	1.00000	0.0348367
$825$	−1.33581	−0.0465068
$826$	−1.03099	−0.0358727
$827$	3.21644	0.111847	0.0559233	−	0.998435i	$-0.482190\pi$
$827$	3.21644	0.111847	0.0559233	+	0.998435i	$0.482190\pi$
$828$	8.56848	0.297775
$829$	−27.9871	−0.972034	−0.486017	−	0.873949i	$-0.661551\pi$
$829$	−27.9871	−0.972034	−0.486017	+	0.873949i	$0.661551\pi$
$830$	16.5680	0.575083
$831$	26.9165	0.933721
$832$	1.00000	0.0346688
$833$	−29.6070	−1.02582
$834$	3.85007	0.133317
$835$	27.0020	0.934442
$836$	2.17312	0.0751589
$837$	−7.55065	−0.260989
$838$	−12.8880	−0.445209
$839$	28.7827	0.993688	0.496844	−	0.867840i	$-0.334492\pi$
$839$	28.7827	0.993688	0.496844	+	0.867840i	$0.334492\pi$
$840$	−0.622092	−0.0214642
$841$	77.8106	2.68312
$842$	13.2438	0.456411
$843$	18.6951	0.643894
$844$	−15.0750	−0.518902
$845$	2.41893	0.0832138
$846$	−9.53966	−0.327980
$847$	2.19563	0.0754427
$848$	−12.5020	−0.429320
$849$	23.9981	0.823614
$850$	−3.63473	−0.124670
$851$	−24.2404	−0.830950
$852$	0.286350	0.00981018
$853$	6.26327	0.214450	0.107225	−	0.994235i	$-0.465803\pi$
$853$	6.26327	0.214450	0.107225	+	0.994235i	$0.465803\pi$
$854$	3.19021	0.109167
$855$	3.34978	0.114560
$856$	14.7757	0.505022
$857$	−38.6866	−1.32151	−0.660755	−	0.750601i	$-0.729764\pi$
$857$	−38.6866	−1.32151	−0.660755	+	0.750601i	$0.729764\pi$
$858$	1.56925	0.0535732
$859$	3.86346	0.131820	0.0659098	−	0.997826i	$-0.479005\pi$
$859$	3.86346	0.131820	0.0659098	+	0.997826i	$0.479005\pi$
$860$	−15.7585	−0.537360
$861$	2.65603	0.0905171
$862$	13.7422	0.468060
$863$	1.04093	0.0354336	0.0177168	−	0.999843i	$-0.494360\pi$
$863$	1.04093	0.0354336	0.0177168	+	0.999843i	$0.494360\pi$
$864$	1.00000	0.0340207
$865$	−33.7247	−1.14667
$866$	−21.5048	−0.730764
$867$	−1.23219	−0.0418475
$868$	−1.94185	−0.0659106
$869$	14.2208	0.482406
$870$	24.9995	0.847562
$871$	9.28134	0.314486
$872$	7.20934	0.244139
$873$	13.3988	0.453480
$874$	−11.8658	−0.401366
$875$	2.58091	0.0872506
$876$	15.7947	0.533652
$877$	−22.7979	−0.769831	−0.384915	−	0.922952i	$-0.625769\pi$
$877$	−22.7979	−0.769831	−0.384915	+	0.922952i	$0.625769\pi$
$878$	−2.91356	−0.0983279
$879$	16.2925	0.549531
$880$	3.79591	0.127960
$881$	26.3482	0.887694	0.443847	−	0.896103i	$-0.353613\pi$
$881$	26.3482	0.887694	0.443847	+	0.896103i	$0.353613\pi$
$882$	6.93386	0.233475
$883$	22.9812	0.773379	0.386690	−	0.922210i	$-0.373618\pi$
$883$	22.9812	0.773379	0.386690	+	0.922210i	$0.373618\pi$
$884$	4.26992	0.143613
$885$	9.69724	0.325969
$886$	29.2420	0.982403
$887$	27.5466	0.924924	0.462462	−	0.886639i	$-0.346966\pi$
$887$	27.5466	0.924924	0.462462	+	0.886639i	$0.346966\pi$
$888$	−2.82902	−0.0949356
$889$	2.81048	0.0942604
$890$	34.1024	1.14312
$891$	1.56925	0.0525718
$892$	−22.1745	−0.742458
$893$	13.2107	0.442078
$894$	−10.1252	−0.338638
$895$	−3.55233	−0.118741
$896$	0.257176	0.00859165
$897$	−8.56848	−0.286093
$898$	−34.2300	−1.14227
$899$	78.0354	2.60263
$900$	0.851240	0.0283747
$901$	−53.3824	−1.77843
$902$	−16.2067	−0.539623
$903$	−1.67541	−0.0557541
$904$	−9.35198	−0.311042
$905$	−40.7769	−1.35547
$906$	15.0201	0.499009
$907$	−26.5420	−0.881312	−0.440656	−	0.897676i	$-0.645254\pi$
$907$	−26.5420	−0.881312	−0.440656	+	0.897676i	$0.645254\pi$
$908$	15.2505	0.506107
$909$	−3.46177	−0.114820
$910$	0.622092	0.0206221
$911$	32.1818	1.06623	0.533115	−	0.846043i	$-0.321021\pi$
$911$	32.1818	1.06623	0.533115	+	0.846043i	$0.321021\pi$
$912$	−1.38482	−0.0458558
$913$	−10.7482	−0.355715
$914$	4.90047	0.162093
$915$	−30.0063	−0.991978
$916$	−2.61103	−0.0862708
$917$	−5.09787	−0.168346
$918$	4.26992	0.140928
$919$	−37.8115	−1.24729	−0.623643	−	0.781709i	$-0.714348\pi$
$919$	−37.8115	−1.24729	−0.623643	+	0.781709i	$0.714348\pi$
$920$	−20.7266	−0.683336
$921$	9.48598	0.312574
$922$	−12.2479	−0.403364
$923$	−0.286350	−0.00942531
$924$	0.403573	0.0132766
$925$	−2.40817	−0.0791803
$926$	−33.1408	−1.08907
$927$	−1.00000	−0.0328443
$928$	−10.3349	−0.339260
$929$	−3.13069	−0.102715	−0.0513573	−	0.998680i	$-0.516355\pi$
$929$	−3.13069	−0.102715	−0.0513573	+	0.998680i	$0.516355\pi$
$930$	18.2645	0.598918
$931$	−9.60212	−0.314697
$932$	12.0453	0.394556
$933$	−5.00459	−0.163843
$934$	23.1797	0.758462
$935$	16.2082	0.530065
$936$	−1.00000	−0.0326860
$937$	32.1463	1.05017	0.525087	−	0.851049i	$-0.324033\pi$
$937$	32.1463	1.05017	0.525087	+	0.851049i	$0.324033\pi$
$938$	2.38694	0.0779363
$939$	−17.0166	−0.555316
$940$	23.0758	0.752649
$941$	46.2552	1.50788	0.753938	−	0.656946i	$-0.228152\pi$
$941$	46.2552	1.50788	0.753938	+	0.656946i	$0.228152\pi$
$942$	22.4063	0.730037
$943$	88.4924	2.88171
$944$	−4.00889	−0.130478
$945$	0.622092	0.0202366
$946$	10.2231	0.332381
$947$	15.2368	0.495129	0.247564	−	0.968871i	$-0.420370\pi$
$947$	15.2368	0.495129	0.247564	+	0.968871i	$0.420370\pi$
$948$	−9.06214	−0.294325
$949$	−15.7947	−0.512717
$950$	−1.17881	−0.0382457
$951$	23.2420	0.753674
$952$	1.09812	0.0355903
$953$	−7.86823	−0.254877	−0.127438	−	0.991846i	$-0.540676\pi$
$953$	−7.86823	−0.254877	−0.127438	+	0.991846i	$0.540676\pi$
$954$	12.5020	0.404767
$955$	25.5963	0.828276
$956$	−16.0683	−0.519685
$957$	−16.2181	−0.524255
$958$	7.47120	0.241384
$959$	1.83881	0.0593783
$960$	−2.41893	−0.0780707
$961$	26.0124	0.839109
$962$	2.82902	0.0912112
$963$	−14.7757	−0.476139
$964$	3.41064	0.109849
$965$	16.0030	0.515155
$966$	−2.20361	−0.0708999
$967$	−4.92886	−0.158501	−0.0792507	−	0.996855i	$-0.525253\pi$
$967$	−4.92886	−0.158501	−0.0792507	+	0.996855i	$0.525253\pi$
$968$	8.53746	0.274404
$969$	−5.91305	−0.189955
$970$	−32.4107	−1.04065
$971$	12.0048	0.385252	0.192626	−	0.981272i	$-0.438300\pi$
$971$	12.0048	0.385252	0.192626	+	0.981272i	$0.438300\pi$
$972$	−1.00000	−0.0320750
$973$	−0.990145	−0.0317426
$974$	13.0822	0.419182
$975$	−0.851240	−0.0272615
$976$	12.4048	0.397067
$977$	0.539082	0.0172468	0.00862339	−	0.999963i	$-0.497255\pi$
$977$	0.539082	0.0172468	0.00862339	+	0.999963i	$0.497255\pi$
$978$	3.77020	0.120558
$979$	−22.1235	−0.707069
$980$	−16.7725	−0.535779
$981$	−7.20934	−0.230177
$982$	29.2439	0.933210
$983$	−42.9713	−1.37057	−0.685286	−	0.728274i	$-0.740323\pi$
$983$	−42.9713	−1.37057	−0.685286	+	0.728274i	$0.740323\pi$
$984$	10.3277	0.329234
$985$	−44.1682	−1.40731
$986$	−44.1293	−1.40536
$987$	2.45337	0.0780916
$988$	1.38482	0.0440569
$989$	−55.8206	−1.77499
$990$	−3.79591	−0.120642
$991$	49.6660	1.57769	0.788846	−	0.614592i	$-0.210679\pi$
$991$	49.6660	1.57769	0.788846	+	0.614592i	$0.210679\pi$
$992$	−7.55065	−0.239734
$993$	0.342089	0.0108559
$994$	−0.0736422	−0.00233579
$995$	−13.5090	−0.428265
$996$	6.84929	0.217028
$997$	46.7267	1.47985	0.739924	−	0.672690i	$-0.234861\pi$
$997$	46.7267	1.47985	0.739924	+	0.672690i	$0.234861\pi$
$998$	−20.8225	−0.659124
$999$	2.82902	0.0895062

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.z.1.13	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.z.1.13	✓	14	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.41893 q^{5} +1.00000 q^{6} -0.257176 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.41893 q^{5} +1.00000 q^{6} -0.257176 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.41893 q^{10} +1.56925 q^{11} -1.00000 q^{12} +1.00000 q^{13} +0.257176 q^{14} -2.41893 q^{15} +1.00000 q^{16} +4.26992 q^{17} -1.00000 q^{18} +1.38482 q^{19} +2.41893 q^{20} +0.257176 q^{21} -1.56925 q^{22} +8.56848 q^{23} +1.00000 q^{24} +0.851240 q^{25} -1.00000 q^{26} -1.00000 q^{27} -0.257176 q^{28} +10.3349 q^{29} +2.41893 q^{30} +7.55065 q^{31} -1.00000 q^{32} -1.56925 q^{33} -4.26992 q^{34} -0.622092 q^{35} +1.00000 q^{36} -2.82902 q^{37} -1.38482 q^{38} -1.00000 q^{39} -2.41893 q^{40} +10.3277 q^{41} -0.257176 q^{42} -6.51464 q^{43} +1.56925 q^{44} +2.41893 q^{45} -8.56848 q^{46} +9.53966 q^{47} -1.00000 q^{48} -6.93386 q^{49} -0.851240 q^{50} -4.26992 q^{51} +1.00000 q^{52} -12.5020 q^{53} +1.00000 q^{54} +3.79591 q^{55} +0.257176 q^{56} -1.38482 q^{57} -10.3349 q^{58} -4.00889 q^{59} -2.41893 q^{60} +12.4048 q^{61} -7.55065 q^{62} -0.257176 q^{63} +1.00000 q^{64} +2.41893 q^{65} +1.56925 q^{66} +9.28134 q^{67} +4.26992 q^{68} -8.56848 q^{69} +0.622092 q^{70} -0.286350 q^{71} -1.00000 q^{72} -15.7947 q^{73} +2.82902 q^{74} -0.851240 q^{75} +1.38482 q^{76} -0.403573 q^{77} +1.00000 q^{78} +9.06214 q^{79} +2.41893 q^{80} +1.00000 q^{81} -10.3277 q^{82} -6.84929 q^{83} +0.257176 q^{84} +10.3286 q^{85} +6.51464 q^{86} -10.3349 q^{87} -1.56925 q^{88} -14.0981 q^{89} -2.41893 q^{90} -0.257176 q^{91} +8.56848 q^{92} -7.55065 q^{93} -9.53966 q^{94} +3.34978 q^{95} +1.00000 q^{96} +13.3988 q^{97} +6.93386 q^{98} +1.56925 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{2} - 14 q^{3} + 14 q^{4} - 3 q^{5} + 14 q^{6} + 4 q^{7} - 14 q^{8} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^2 - 14 * q^3 + 14 * q^4 - 3 * q^5 + 14 * q^6 + 4 * q^7 - 14 * q^8 + 14 * q^9	\( 14 q - 14 q^{2} - 14 q^{3} + 14 q^{4} - 3 q^{5} + 14 q^{6} + 4 q^{7} - 14 q^{8} + 14 q^{9} + 3 q^{10} + 5 q^{11} - 14 q^{12} + 14 q^{13} - 4 q^{14} + 3 q^{15} + 14 q^{16} - 7 q^{17} - 14 q^{18} + 31 q^{19} - 3 q^{20} - 4 q^{21} - 5 q^{22} + 14 q^{23} + 14 q^{24} + 11 q^{25} - 14 q^{26} - 14 q^{27} + 4 q^{28} + 9 q^{29} - 3 q^{30} + 23 q^{31} - 14 q^{32} - 5 q^{33} + 7 q^{34} - 2 q^{35} + 14 q^{36} + 12 q^{37} - 31 q^{38} - 14 q^{39} + 3 q^{40} - 5 q^{41} + 4 q^{42} - 2 q^{43} + 5 q^{44} - 3 q^{45} - 14 q^{46} - 11 q^{47} - 14 q^{48} + 52 q^{49} - 11 q^{50} + 7 q^{51} + 14 q^{52} + 6 q^{53} + 14 q^{54} + 30 q^{55} - 4 q^{56} - 31 q^{57} - 9 q^{58} - 4 q^{59} + 3 q^{60} + 12 q^{61} - 23 q^{62} + 4 q^{63} + 14 q^{64} - 3 q^{65} + 5 q^{66} + 24 q^{67} - 7 q^{68} - 14 q^{69} + 2 q^{70} + 20 q^{71} - 14 q^{72} + 2 q^{73} - 12 q^{74} - 11 q^{75} + 31 q^{76} - 28 q^{77} + 14 q^{78} + 59 q^{79} - 3 q^{80} + 14 q^{81} + 5 q^{82} + q^{83} - 4 q^{84} - 29 q^{85} + 2 q^{86} - 9 q^{87} - 5 q^{88} + 6 q^{89} + 3 q^{90} + 4 q^{91} + 14 q^{92} - 23 q^{93} + 11 q^{94} - 58 q^{95} + 14 q^{96} - 6 q^{97} - 52 q^{98} + 5 q^{99}+O(q^{100}) \) 14 * q - 14 * q^2 - 14 * q^3 + 14 * q^4 - 3 * q^5 + 14 * q^6 + 4 * q^7 - 14 * q^8 + 14 * q^9 + 3 * q^10 + 5 * q^11 - 14 * q^12 + 14 * q^13 - 4 * q^14 + 3 * q^15 + 14 * q^16 - 7 * q^17 - 14 * q^18 + 31 * q^19 - 3 * q^20 - 4 * q^21 - 5 * q^22 + 14 * q^23 + 14 * q^24 + 11 * q^25 - 14 * q^26 - 14 * q^27 + 4 * q^28 + 9 * q^29 - 3 * q^30 + 23 * q^31 - 14 * q^32 - 5 * q^33 + 7 * q^34 - 2 * q^35 + 14 * q^36 + 12 * q^37 - 31 * q^38 - 14 * q^39 + 3 * q^40 - 5 * q^41 + 4 * q^42 - 2 * q^43 + 5 * q^44 - 3 * q^45 - 14 * q^46 - 11 * q^47 - 14 * q^48 + 52 * q^49 - 11 * q^50 + 7 * q^51 + 14 * q^52 + 6 * q^53 + 14 * q^54 + 30 * q^55 - 4 * q^56 - 31 * q^57 - 9 * q^58 - 4 * q^59 + 3 * q^60 + 12 * q^61 - 23 * q^62 + 4 * q^63 + 14 * q^64 - 3 * q^65 + 5 * q^66 + 24 * q^67 - 7 * q^68 - 14 * q^69 + 2 * q^70 + 20 * q^71 - 14 * q^72 + 2 * q^73 - 12 * q^74 - 11 * q^75 + 31 * q^76 - 28 * q^77 + 14 * q^78 + 59 * q^79 - 3 * q^80 + 14 * q^81 + 5 * q^82 + q^83 - 4 * q^84 - 29 * q^85 + 2 * q^86 - 9 * q^87 - 5 * q^88 + 6 * q^89 + 3 * q^90 + 4 * q^91 + 14 * q^92 - 23 * q^93 + 11 * q^94 - 58 * q^95 + 14 * q^96 - 6 * q^97 - 52 * q^98 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more