LMFDB - Embedded newform 8034.2.a.bd.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:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 5 x^{15} - 36 x^{14} + 196 x^{13} + 498 x^{12} - 3101 x^{11} - 3150 x^{10} + 25368 x^{9} + \cdots - 66432 $ x^16 - 5x^15 - 36x^14 + 196x^13 + 498x^12 - 3101x^11 - 3150x^10 + 25368x^9 + 6763x^8 - 113788x^7 + 19731x^6 + 270913x^5 - 122680x^4 - 296326x^3 + 185524x^2 + 94528*x - 66432
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.719032$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.719032 q^{5} +1.00000 q^{6} -5.05986 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.719032 q^{5} +1.00000 q^{6} -5.05986 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.719032 q^{10} -4.16214 q^{11} +1.00000 q^{12} -1.00000 q^{13} -5.05986 q^{14} -0.719032 q^{15} +1.00000 q^{16} -6.58304 q^{17} +1.00000 q^{18} +4.38354 q^{19} -0.719032 q^{20} -5.05986 q^{21} -4.16214 q^{22} +4.96384 q^{23} +1.00000 q^{24} -4.48299 q^{25} -1.00000 q^{26} +1.00000 q^{27} -5.05986 q^{28} -4.34377 q^{29} -0.719032 q^{30} +1.33812 q^{31} +1.00000 q^{32} -4.16214 q^{33} -6.58304 q^{34} +3.63820 q^{35} +1.00000 q^{36} -0.572753 q^{37} +4.38354 q^{38} -1.00000 q^{39} -0.719032 q^{40} +11.8845 q^{41} -5.05986 q^{42} +0.420896 q^{43} -4.16214 q^{44} -0.719032 q^{45} +4.96384 q^{46} +4.13504 q^{47} +1.00000 q^{48} +18.6022 q^{49} -4.48299 q^{50} -6.58304 q^{51} -1.00000 q^{52} +6.15067 q^{53} +1.00000 q^{54} +2.99271 q^{55} -5.05986 q^{56} +4.38354 q^{57} -4.34377 q^{58} +8.11233 q^{59} -0.719032 q^{60} +6.14150 q^{61} +1.33812 q^{62} -5.05986 q^{63} +1.00000 q^{64} +0.719032 q^{65} -4.16214 q^{66} +2.81198 q^{67} -6.58304 q^{68} +4.96384 q^{69} +3.63820 q^{70} -1.58126 q^{71} +1.00000 q^{72} -1.70341 q^{73} -0.572753 q^{74} -4.48299 q^{75} +4.38354 q^{76} +21.0599 q^{77} -1.00000 q^{78} +5.14872 q^{79} -0.719032 q^{80} +1.00000 q^{81} +11.8845 q^{82} +9.81682 q^{83} -5.05986 q^{84} +4.73342 q^{85} +0.420896 q^{86} -4.34377 q^{87} -4.16214 q^{88} +2.78100 q^{89} -0.719032 q^{90} +5.05986 q^{91} +4.96384 q^{92} +1.33812 q^{93} +4.13504 q^{94} -3.15191 q^{95} +1.00000 q^{96} +4.58746 q^{97} +18.6022 q^{98} -4.16214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{2} + 16 q^{3} + 16 q^{4} + 5 q^{5} + 16 q^{6} + 4 q^{7} + 16 q^{8} + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^2 + 16 * q^3 + 16 * q^4 + 5 * q^5 + 16 * q^6 + 4 * q^7 + 16 * q^8 + 16 * q^9	\( 16 q + 16 q^{2} + 16 q^{3} + 16 q^{4} + 5 q^{5} + 16 q^{6} + 4 q^{7} + 16 q^{8} + 16 q^{9} + 5 q^{10} + 18 q^{11} + 16 q^{12} - 16 q^{13} + 4 q^{14} + 5 q^{15} + 16 q^{16} + 17 q^{17} + 16 q^{18} + 8 q^{19} + 5 q^{20} + 4 q^{21} + 18 q^{22} + 9 q^{23} + 16 q^{24} + 17 q^{25} - 16 q^{26} + 16 q^{27} + 4 q^{28} + 14 q^{29} + 5 q^{30} + 12 q^{31} + 16 q^{32} + 18 q^{33} + 17 q^{34} + 16 q^{35} + 16 q^{36} + 31 q^{37} + 8 q^{38} - 16 q^{39} + 5 q^{40} + 29 q^{41} + 4 q^{42} + 30 q^{43} + 18 q^{44} + 5 q^{45} + 9 q^{46} - q^{47} + 16 q^{48} + 36 q^{49} + 17 q^{50} + 17 q^{51} - 16 q^{52} + 12 q^{53} + 16 q^{54} + 30 q^{55} + 4 q^{56} + 8 q^{57} + 14 q^{58} + 38 q^{59} + 5 q^{60} + 12 q^{62} + 4 q^{63} + 16 q^{64} - 5 q^{65} + 18 q^{66} + 28 q^{67} + 17 q^{68} + 9 q^{69} + 16 q^{70} + 32 q^{71} + 16 q^{72} + 20 q^{73} + 31 q^{74} + 17 q^{75} + 8 q^{76} + 26 q^{77} - 16 q^{78} + 13 q^{79} + 5 q^{80} + 16 q^{81} + 29 q^{82} + 39 q^{83} + 4 q^{84} + 31 q^{85} + 30 q^{86} + 14 q^{87} + 18 q^{88} + 9 q^{89} + 5 q^{90} - 4 q^{91} + 9 q^{92} + 12 q^{93} - q^{94} - 20 q^{95} + 16 q^{96} + 35 q^{97} + 36 q^{98} + 18 q^{99}+O(q^{100}) \) 16 * q + 16 * q^2 + 16 * q^3 + 16 * q^4 + 5 * q^5 + 16 * q^6 + 4 * q^7 + 16 * q^8 + 16 * q^9 + 5 * q^10 + 18 * q^11 + 16 * q^12 - 16 * q^13 + 4 * q^14 + 5 * q^15 + 16 * q^16 + 17 * q^17 + 16 * q^18 + 8 * q^19 + 5 * q^20 + 4 * q^21 + 18 * q^22 + 9 * q^23 + 16 * q^24 + 17 * q^25 - 16 * q^26 + 16 * q^27 + 4 * q^28 + 14 * q^29 + 5 * q^30 + 12 * q^31 + 16 * q^32 + 18 * q^33 + 17 * q^34 + 16 * q^35 + 16 * q^36 + 31 * q^37 + 8 * q^38 - 16 * q^39 + 5 * q^40 + 29 * q^41 + 4 * q^42 + 30 * q^43 + 18 * q^44 + 5 * q^45 + 9 * q^46 - q^47 + 16 * q^48 + 36 * q^49 + 17 * q^50 + 17 * q^51 - 16 * q^52 + 12 * q^53 + 16 * q^54 + 30 * q^55 + 4 * q^56 + 8 * q^57 + 14 * q^58 + 38 * q^59 + 5 * q^60 + 12 * q^62 + 4 * q^63 + 16 * q^64 - 5 * q^65 + 18 * q^66 + 28 * q^67 + 17 * q^68 + 9 * q^69 + 16 * q^70 + 32 * q^71 + 16 * q^72 + 20 * q^73 + 31 * q^74 + 17 * q^75 + 8 * q^76 + 26 * q^77 - 16 * q^78 + 13 * q^79 + 5 * q^80 + 16 * q^81 + 29 * q^82 + 39 * q^83 + 4 * q^84 + 31 * q^85 + 30 * q^86 + 14 * q^87 + 18 * q^88 + 9 * q^89 + 5 * q^90 - 4 * q^91 + 9 * q^92 + 12 * q^93 - q^94 - 20 * q^95 + 16 * q^96 + 35 * q^97 + 36 * q^98 + 18 * 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
$5$	−0.719032	−0.321561	−0.160780	−	0.986990i	$-0.551401\pi$
$5$	−0.719032	−0.321561	−0.160780	+	0.986990i	$0.551401\pi$
$6$	1.00000	0.408248
$7$	−5.05986	−1.91245	−0.956224	−	0.292637i	$-0.905467\pi$
$7$	−5.05986	−1.91245	−0.956224	+	0.292637i	$0.905467\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−0.719032	−0.227378
$11$	−4.16214	−1.25493	−0.627467	−	0.778643i	$-0.715908\pi$
$11$	−4.16214	−1.25493	−0.627467	+	0.778643i	$0.715908\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−5.05986	−1.35230
$15$	−0.719032	−0.185653
$16$	1.00000	0.250000
$17$	−6.58304	−1.59662	−0.798311	−	0.602245i	$-0.794273\pi$
$17$	−6.58304	−1.59662	−0.798311	+	0.602245i	$0.794273\pi$
$18$	1.00000	0.235702
$19$	4.38354	1.00565	0.502827	−	0.864387i	$-0.332293\pi$
$19$	4.38354	1.00565	0.502827	+	0.864387i	$0.332293\pi$
$20$	−0.719032	−0.160780
$21$	−5.05986	−1.10415
$22$	−4.16214	−0.887372
$23$	4.96384	1.03503	0.517517	−	0.855673i	$-0.326857\pi$
$23$	4.96384	1.03503	0.517517	+	0.855673i	$0.326857\pi$
$24$	1.00000	0.204124
$25$	−4.48299	−0.896599
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	−5.05986	−0.956224
$29$	−4.34377	−0.806617	−0.403308	−	0.915064i	$-0.632140\pi$
$29$	−4.34377	−0.806617	−0.403308	+	0.915064i	$0.632140\pi$
$30$	−0.719032	−0.131277
$31$	1.33812	0.240333	0.120166	−	0.992754i	$-0.461657\pi$
$31$	1.33812	0.240333	0.120166	+	0.992754i	$0.461657\pi$
$32$	1.00000	0.176777
$33$	−4.16214	−0.724536
$34$	−6.58304	−1.12898
$35$	3.63820	0.614968
$36$	1.00000	0.166667
$37$	−0.572753	−0.0941600	−0.0470800	−	0.998891i	$-0.514992\pi$
$37$	−0.572753	−0.0941600	−0.0470800	+	0.998891i	$0.514992\pi$
$38$	4.38354	0.711105
$39$	−1.00000	−0.160128
$40$	−0.719032	−0.113689
$41$	11.8845	1.85605	0.928023	−	0.372524i	$-0.121508\pi$
$41$	11.8845	1.85605	0.928023	+	0.372524i	$0.121508\pi$
$42$	−5.05986	−0.780753
$43$	0.420896	0.0641860	0.0320930	−	0.999485i	$-0.489783\pi$
$43$	0.420896	0.0641860	0.0320930	+	0.999485i	$0.489783\pi$
$44$	−4.16214	−0.627467
$45$	−0.719032	−0.107187
$46$	4.96384	0.731879
$47$	4.13504	0.603157	0.301579	−	0.953441i	$-0.402486\pi$
$47$	4.13504	0.603157	0.301579	+	0.953441i	$0.402486\pi$
$48$	1.00000	0.144338
$49$	18.6022	2.65746
$50$	−4.48299	−0.633991
$51$	−6.58304	−0.921810
$52$	−1.00000	−0.138675
$53$	6.15067	0.844859	0.422429	−	0.906396i	$-0.361177\pi$
$53$	6.15067	0.844859	0.422429	+	0.906396i	$0.361177\pi$
$54$	1.00000	0.136083
$55$	2.99271	0.403537
$56$	−5.05986	−0.676152
$57$	4.38354	0.580615
$58$	−4.34377	−0.570364
$59$	8.11233	1.05614	0.528068	−	0.849202i	$-0.322917\pi$
$59$	8.11233	1.05614	0.528068	+	0.849202i	$0.322917\pi$
$60$	−0.719032	−0.0928266
$61$	6.14150	0.786338	0.393169	−	0.919466i	$-0.371379\pi$
$61$	6.14150	0.786338	0.393169	+	0.919466i	$0.371379\pi$
$62$	1.33812	0.169941
$63$	−5.05986	−0.637482
$64$	1.00000	0.125000
$65$	0.719032	0.0891849
$66$	−4.16214	−0.512324
$67$	2.81198	0.343538	0.171769	−	0.985137i	$-0.445052\pi$
$67$	2.81198	0.343538	0.171769	+	0.985137i	$0.445052\pi$
$68$	−6.58304	−0.798311
$69$	4.96384	0.597577
$70$	3.63820	0.434848
$71$	−1.58126	−0.187661	−0.0938306	−	0.995588i	$-0.529911\pi$
$71$	−1.58126	−0.187661	−0.0938306	+	0.995588i	$0.529911\pi$
$72$	1.00000	0.117851
$73$	−1.70341	−0.199369	−0.0996843	−	0.995019i	$-0.531783\pi$
$73$	−1.70341	−0.199369	−0.0996843	+	0.995019i	$0.531783\pi$
$74$	−0.572753	−0.0665812
$75$	−4.48299	−0.517651
$76$	4.38354	0.502827
$77$	21.0599	2.39999
$78$	−1.00000	−0.113228
$79$	5.14872	0.579276	0.289638	−	0.957136i	$-0.406465\pi$
$79$	5.14872	0.579276	0.289638	+	0.957136i	$0.406465\pi$
$80$	−0.719032	−0.0803902
$81$	1.00000	0.111111
$82$	11.8845	1.31242
$83$	9.81682	1.07754	0.538768	−	0.842454i	$-0.318890\pi$
$83$	9.81682	1.07754	0.538768	+	0.842454i	$0.318890\pi$
$84$	−5.05986	−0.552076
$85$	4.73342	0.513411
$86$	0.420896	0.0453864
$87$	−4.34377	−0.465701
$88$	−4.16214	−0.443686
$89$	2.78100	0.294786	0.147393	−	0.989078i	$-0.452912\pi$
$89$	2.78100	0.294786	0.147393	+	0.989078i	$0.452912\pi$
$90$	−0.719032	−0.0757926
$91$	5.05986	0.530418
$92$	4.96384	0.517517
$93$	1.33812	0.138756
$94$	4.13504	0.426497
$95$	−3.15191	−0.323379
$96$	1.00000	0.102062
$97$	4.58746	0.465786	0.232893	−	0.972502i	$-0.425181\pi$
$97$	4.58746	0.465786	0.232893	+	0.972502i	$0.425181\pi$
$98$	18.6022	1.87910
$99$	−4.16214	−0.418311
$100$	−4.48299	−0.448299
$101$	−3.90926	−0.388986	−0.194493	−	0.980904i	$-0.562306\pi$
$101$	−3.90926	−0.388986	−0.194493	+	0.980904i	$0.562306\pi$
$102$	−6.58304	−0.651818
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	−1.00000	−0.0980581
$105$	3.63820	0.355052
$106$	6.15067	0.597405
$107$	−15.3484	−1.48378	−0.741891	−	0.670520i	$-0.766071\pi$
$107$	−15.3484	−1.48378	−0.741891	+	0.670520i	$0.766071\pi$
$108$	1.00000	0.0962250
$109$	4.97155	0.476189	0.238094	−	0.971242i	$-0.423477\pi$
$109$	4.97155	0.476189	0.238094	+	0.971242i	$0.423477\pi$
$110$	2.99271	0.285344
$111$	−0.572753	−0.0543633
$112$	−5.05986	−0.478112
$113$	10.2644	0.965593	0.482797	−	0.875733i	$-0.339621\pi$
$113$	10.2644	0.965593	0.482797	+	0.875733i	$0.339621\pi$
$114$	4.38354	0.410557
$115$	−3.56916	−0.332826
$116$	−4.34377	−0.403308
$117$	−1.00000	−0.0924500
$118$	8.11233	0.746800
$119$	33.3093	3.05346
$120$	−0.719032	−0.0656383
$121$	6.32343	0.574857
$122$	6.14150	0.556025
$123$	11.8845	1.07159
$124$	1.33812	0.120166
$125$	6.81857	0.609872
$126$	−5.05986	−0.450768
$127$	−1.24622	−0.110584	−0.0552922	−	0.998470i	$-0.517609\pi$
$127$	−1.24622	−0.110584	−0.0552922	+	0.998470i	$0.517609\pi$
$128$	1.00000	0.0883883
$129$	0.420896	0.0370578
$130$	0.719032	0.0630632
$131$	−1.71906	−0.150195	−0.0750976	−	0.997176i	$-0.523927\pi$
$131$	−1.71906	−0.150195	−0.0750976	+	0.997176i	$0.523927\pi$
$132$	−4.16214	−0.362268
$133$	−22.1801	−1.92326
$134$	2.81198	0.242918
$135$	−0.719032	−0.0618844
$136$	−6.58304	−0.564491
$137$	−18.4342	−1.57494	−0.787469	−	0.616354i	$-0.788609\pi$
$137$	−18.4342	−1.57494	−0.787469	+	0.616354i	$0.788609\pi$
$138$	4.96384	0.422551
$139$	10.2656	0.870716	0.435358	−	0.900257i	$-0.356622\pi$
$139$	10.2656	0.870716	0.435358	+	0.900257i	$0.356622\pi$
$140$	3.63820	0.307484
$141$	4.13504	0.348233
$142$	−1.58126	−0.132696
$143$	4.16214	0.348056
$144$	1.00000	0.0833333
$145$	3.12330	0.259376
$146$	−1.70341	−0.140975
$147$	18.6022	1.53428
$148$	−0.572753	−0.0470800
$149$	2.46130	0.201637	0.100819	−	0.994905i	$-0.467854\pi$
$149$	2.46130	0.201637	0.100819	+	0.994905i	$0.467854\pi$
$150$	−4.48299	−0.366035
$151$	0.872672	0.0710170	0.0355085	−	0.999369i	$-0.488695\pi$
$151$	0.872672	0.0710170	0.0355085	+	0.999369i	$0.488695\pi$
$152$	4.38354	0.355552
$153$	−6.58304	−0.532207
$154$	21.0599	1.69705
$155$	−0.962147	−0.0772815
$156$	−1.00000	−0.0800641
$157$	−24.0354	−1.91824	−0.959118	−	0.283005i	$-0.908669\pi$
$157$	−24.0354	−1.91824	−0.959118	+	0.283005i	$0.908669\pi$
$158$	5.14872	0.409610
$159$	6.15067	0.487780
$160$	−0.719032	−0.0568444
$161$	−25.1164	−1.97945
$162$	1.00000	0.0785674
$163$	−2.99717	−0.234757	−0.117378	−	0.993087i	$-0.537449\pi$
$163$	−2.99717	−0.234757	−0.117378	+	0.993087i	$0.537449\pi$
$164$	11.8845	0.928023
$165$	2.99271	0.232982
$166$	9.81682	0.761933
$167$	1.93462	0.149705	0.0748526	−	0.997195i	$-0.476151\pi$
$167$	1.93462	0.149705	0.0748526	+	0.997195i	$0.476151\pi$
$168$	−5.05986	−0.390377
$169$	1.00000	0.0769231
$170$	4.73342	0.363036
$171$	4.38354	0.335218
$172$	0.420896	0.0320930
$173$	16.3654	1.24424	0.622120	−	0.782922i	$-0.286272\pi$
$173$	16.3654	1.24424	0.622120	+	0.782922i	$0.286272\pi$
$174$	−4.34377	−0.329300
$175$	22.6833	1.71470
$176$	−4.16214	−0.313733
$177$	8.11233	0.609760
$178$	2.78100	0.208445
$179$	−20.3709	−1.52259	−0.761296	−	0.648405i	$-0.775436\pi$
$179$	−20.3709	−1.52259	−0.761296	+	0.648405i	$0.775436\pi$
$180$	−0.719032	−0.0535935
$181$	−17.8561	−1.32723	−0.663615	−	0.748074i	$-0.730979\pi$
$181$	−17.8561	−1.32723	−0.663615	+	0.748074i	$0.730979\pi$
$182$	5.05986	0.375062
$183$	6.14150	0.453992
$184$	4.96384	0.365940
$185$	0.411828	0.0302782
$186$	1.33812	0.0981154
$187$	27.3996	2.00365
$188$	4.13504	0.301579
$189$	−5.05986	−0.368051
$190$	−3.15191	−0.228663
$191$	7.99297	0.578351	0.289176	−	0.957276i	$-0.406619\pi$
$191$	7.99297	0.578351	0.289176	+	0.957276i	$0.406619\pi$
$192$	1.00000	0.0721688
$193$	−25.4753	−1.83375	−0.916877	−	0.399169i	$-0.869299\pi$
$193$	−25.4753	−1.83375	−0.916877	+	0.399169i	$0.869299\pi$
$194$	4.58746	0.329360
$195$	0.719032	0.0514909
$196$	18.6022	1.32873
$197$	−0.124351	−0.00885964	−0.00442982	−	0.999990i	$-0.501410\pi$
$197$	−0.124351	−0.00885964	−0.00442982	+	0.999990i	$0.501410\pi$
$198$	−4.16214	−0.295791
$199$	−16.3494	−1.15898	−0.579491	−	0.814979i	$-0.696748\pi$
$199$	−16.3494	−1.15898	−0.579491	+	0.814979i	$0.696748\pi$
$200$	−4.48299	−0.316996
$201$	2.81198	0.198342
$202$	−3.90926	−0.275055
$203$	21.9788	1.54261
$204$	−6.58304	−0.460905
$205$	−8.54532	−0.596831
$206$	1.00000	0.0696733
$207$	4.96384	0.345011
$208$	−1.00000	−0.0693375
$209$	−18.2449	−1.26203
$210$	3.63820	0.251060
$211$	8.05692	0.554661	0.277331	−	0.960775i	$-0.410550\pi$
$211$	8.05692	0.554661	0.277331	+	0.960775i	$0.410550\pi$
$212$	6.15067	0.422429
$213$	−1.58126	−0.108346
$214$	−15.3484	−1.04919
$215$	−0.302638	−0.0206397
$216$	1.00000	0.0680414
$217$	−6.77068	−0.459624
$218$	4.97155	0.336716
$219$	−1.70341	−0.115105
$220$	2.99271	0.201769
$221$	6.58304	0.442823
$222$	−0.572753	−0.0384407
$223$	−16.1985	−1.08473	−0.542364	−	0.840143i	$-0.682471\pi$
$223$	−16.1985	−1.08473	−0.542364	+	0.840143i	$0.682471\pi$
$224$	−5.05986	−0.338076
$225$	−4.48299	−0.298866
$226$	10.2644	0.682777
$227$	27.3918	1.81806	0.909028	−	0.416734i	$-0.136825\pi$
$227$	27.3918	1.81806	0.909028	+	0.416734i	$0.136825\pi$
$228$	4.38354	0.290307
$229$	10.8229	0.715199	0.357600	−	0.933875i	$-0.383595\pi$
$229$	10.8229	0.715199	0.357600	+	0.933875i	$0.383595\pi$
$230$	−3.56916	−0.235344
$231$	21.0599	1.38564
$232$	−4.34377	−0.285182
$233$	23.7920	1.55866	0.779331	−	0.626612i	$-0.215559\pi$
$233$	23.7920	1.55866	0.779331	+	0.626612i	$0.215559\pi$
$234$	−1.00000	−0.0653720
$235$	−2.97322	−0.193952
$236$	8.11233	0.528068
$237$	5.14872	0.334445
$238$	33.3093	2.15912
$239$	3.41325	0.220785	0.110392	−	0.993888i	$-0.464789\pi$
$239$	3.41325	0.220785	0.110392	+	0.993888i	$0.464789\pi$
$240$	−0.719032	−0.0464133
$241$	8.08232	0.520628	0.260314	−	0.965524i	$-0.416174\pi$
$241$	8.08232	0.520628	0.260314	+	0.965524i	$0.416174\pi$
$242$	6.32343	0.406486
$243$	1.00000	0.0641500
$244$	6.14150	0.393169
$245$	−13.3756	−0.854533
$246$	11.8845	0.757727
$247$	−4.38354	−0.278918
$248$	1.33812	0.0849704
$249$	9.81682	0.622115
$250$	6.81857	0.431244
$251$	−11.4284	−0.721357	−0.360679	−	0.932690i	$-0.617455\pi$
$251$	−11.4284	−0.721357	−0.360679	+	0.932690i	$0.617455\pi$
$252$	−5.05986	−0.318741
$253$	−20.6602	−1.29890
$254$	−1.24622	−0.0781950
$255$	4.73342	0.296418
$256$	1.00000	0.0625000
$257$	−11.8568	−0.739606	−0.369803	−	0.929110i	$-0.620575\pi$
$257$	−11.8568	−0.739606	−0.369803	+	0.929110i	$0.620575\pi$
$258$	0.420896	0.0262038
$259$	2.89805	0.180076
$260$	0.719032	0.0445925
$261$	−4.34377	−0.268872
$262$	−1.71906	−0.106204
$263$	−25.2244	−1.55541	−0.777703	−	0.628632i	$-0.783615\pi$
$263$	−25.2244	−1.55541	−0.777703	+	0.628632i	$0.783615\pi$
$264$	−4.16214	−0.256162
$265$	−4.42252	−0.271673
$266$	−22.1801	−1.35995
$267$	2.78100	0.170195
$268$	2.81198	0.171769
$269$	28.9069	1.76248	0.881241	−	0.472667i	$-0.156709\pi$
$269$	28.9069	1.76248	0.881241	+	0.472667i	$0.156709\pi$
$270$	−0.719032	−0.0437589
$271$	17.1387	1.04110	0.520551	−	0.853831i	$-0.325727\pi$
$271$	17.1387	1.04110	0.520551	+	0.853831i	$0.325727\pi$
$272$	−6.58304	−0.399156
$273$	5.05986	0.306237
$274$	−18.4342	−1.11365
$275$	18.6589	1.12517
$276$	4.96384	0.298788
$277$	26.9333	1.61826	0.809132	−	0.587627i	$-0.199938\pi$
$277$	26.9333	1.61826	0.809132	+	0.587627i	$0.199938\pi$
$278$	10.2656	0.615689
$279$	1.33812	0.0801109
$280$	3.63820	0.217424
$281$	12.8692	0.767712	0.383856	−	0.923393i	$-0.374596\pi$
$281$	12.8692	0.767712	0.383856	+	0.923393i	$0.374596\pi$
$282$	4.13504	0.246238
$283$	0.684428	0.0406850	0.0203425	−	0.999793i	$-0.493524\pi$
$283$	0.684428	0.0406850	0.0203425	+	0.999793i	$0.493524\pi$
$284$	−1.58126	−0.0938306
$285$	−3.15191	−0.186703
$286$	4.16214	0.246113
$287$	−60.1338	−3.54959
$288$	1.00000	0.0589256
$289$	26.3364	1.54920
$290$	3.12330	0.183407
$291$	4.58746	0.268922
$292$	−1.70341	−0.0996843
$293$	10.1832	0.594907	0.297453	−	0.954736i	$-0.403863\pi$
$293$	10.1832	0.594907	0.297453	+	0.954736i	$0.403863\pi$
$294$	18.6022	1.08490
$295$	−5.83302	−0.339612
$296$	−0.572753	−0.0332906
$297$	−4.16214	−0.241512
$298$	2.46130	0.142579
$299$	−4.96384	−0.287067
$300$	−4.48299	−0.258826
$301$	−2.12967	−0.122752
$302$	0.872672	0.0502166
$303$	−3.90926	−0.224581
$304$	4.38354	0.251414
$305$	−4.41593	−0.252855
$306$	−6.58304	−0.376327
$307$	2.73429	0.156054	0.0780270	−	0.996951i	$-0.475138\pi$
$307$	2.73429	0.156054	0.0780270	+	0.996951i	$0.475138\pi$
$308$	21.0599	1.20000
$309$	1.00000	0.0568880
$310$	−0.962147	−0.0546463
$311$	14.9858	0.849768	0.424884	−	0.905248i	$-0.360315\pi$
$311$	14.9858	0.849768	0.424884	+	0.905248i	$0.360315\pi$
$312$	−1.00000	−0.0566139
$313$	1.55027	0.0876265	0.0438133	−	0.999040i	$-0.486049\pi$
$313$	1.55027	0.0876265	0.0438133	+	0.999040i	$0.486049\pi$
$314$	−24.0354	−1.35640
$315$	3.63820	0.204989
$316$	5.14872	0.289638
$317$	7.27441	0.408571	0.204286	−	0.978911i	$-0.434513\pi$
$317$	7.27441	0.408571	0.204286	+	0.978911i	$0.434513\pi$
$318$	6.15067	0.344912
$319$	18.0794	1.01225
$320$	−0.719032	−0.0401951
$321$	−15.3484	−0.856662
$322$	−25.1164	−1.39968
$323$	−28.8571	−1.60565
$324$	1.00000	0.0555556
$325$	4.48299	0.248672
$326$	−2.99717	−0.165998
$327$	4.97155	0.274928
$328$	11.8845	0.656211
$329$	−20.9227	−1.15351
$330$	2.99271	0.164743
$331$	23.1124	1.27037	0.635187	−	0.772358i	$-0.280923\pi$
$331$	23.1124	1.27037	0.635187	+	0.772358i	$0.280923\pi$
$332$	9.81682	0.538768
$333$	−0.572753	−0.0313867
$334$	1.93462	0.105858
$335$	−2.02190	−0.110468
$336$	−5.05986	−0.276038
$337$	34.5121	1.88000	0.939998	−	0.341181i	$-0.110827\pi$
$337$	34.5121	1.88000	0.939998	+	0.341181i	$0.110827\pi$
$338$	1.00000	0.0543928
$339$	10.2644	0.557485
$340$	4.73342	0.256706
$341$	−5.56943	−0.301601
$342$	4.38354	0.237035
$343$	−58.7055	−3.16980
$344$	0.420896	0.0226932
$345$	−3.56916	−0.192157
$346$	16.3654	0.879810
$347$	−23.1693	−1.24379	−0.621897	−	0.783099i	$-0.713638\pi$
$347$	−23.1693	−1.24379	−0.621897	+	0.783099i	$0.713638\pi$
$348$	−4.34377	−0.232850
$349$	−0.474357	−0.0253917	−0.0126959	−	0.999919i	$-0.504041\pi$
$349$	−0.474357	−0.0253917	−0.0126959	+	0.999919i	$0.504041\pi$
$350$	22.6833	1.21247
$351$	−1.00000	−0.0533761
$352$	−4.16214	−0.221843
$353$	−8.50488	−0.452669	−0.226335	−	0.974050i	$-0.572674\pi$
$353$	−8.50488	−0.452669	−0.226335	+	0.974050i	$0.572674\pi$
$354$	8.11233	0.431165
$355$	1.13698	0.0603445
$356$	2.78100	0.147393
$357$	33.3093	1.76291
$358$	−20.3709	−1.07663
$359$	−27.4529	−1.44891	−0.724454	−	0.689323i	$-0.757908\pi$
$359$	−27.4529	−1.44891	−0.724454	+	0.689323i	$0.757908\pi$
$360$	−0.719032	−0.0378963
$361$	0.215465	0.0113402
$362$	−17.8561	−0.938494
$363$	6.32343	0.331894
$364$	5.05986	0.265209
$365$	1.22480	0.0641091
$366$	6.14150	0.321021
$367$	−0.266030	−0.0138867	−0.00694333	−	0.999976i	$-0.502210\pi$
$367$	−0.266030	−0.0138867	−0.00694333	+	0.999976i	$0.502210\pi$
$368$	4.96384	0.258758
$369$	11.8845	0.618682
$370$	0.411828	0.0214099
$371$	−31.1215	−1.61575
$372$	1.33812	0.0693781
$373$	5.87066	0.303971	0.151986	−	0.988383i	$-0.451433\pi$
$373$	5.87066	0.303971	0.151986	+	0.988383i	$0.451433\pi$
$374$	27.3996	1.41680
$375$	6.81857	0.352110
$376$	4.13504	0.213248
$377$	4.34377	0.223715
$378$	−5.05986	−0.260251
$379$	12.8691	0.661039	0.330519	−	0.943799i	$-0.392776\pi$
$379$	12.8691	0.661039	0.330519	+	0.943799i	$0.392776\pi$
$380$	−3.15191	−0.161689
$381$	−1.24622	−0.0638460
$382$	7.99297	0.408956
$383$	−4.82830	−0.246715	−0.123357	−	0.992362i	$-0.539366\pi$
$383$	−4.82830	−0.246715	−0.123357	+	0.992362i	$0.539366\pi$
$384$	1.00000	0.0510310
$385$	−15.1427	−0.771744
$386$	−25.4753	−1.29666
$387$	0.420896	0.0213953
$388$	4.58746	0.232893
$389$	−6.13474	−0.311044	−0.155522	−	0.987832i	$-0.549706\pi$
$389$	−6.13474	−0.311044	−0.155522	+	0.987832i	$0.549706\pi$
$390$	0.719032	0.0364096
$391$	−32.6772	−1.65256
$392$	18.6022	0.939552
$393$	−1.71906	−0.0867152
$394$	−0.124351	−0.00626471
$395$	−3.70209	−0.186272
$396$	−4.16214	−0.209156
$397$	−15.3500	−0.770396	−0.385198	−	0.922834i	$-0.625867\pi$
$397$	−15.3500	−0.770396	−0.385198	+	0.922834i	$0.625867\pi$
$398$	−16.3494	−0.819524
$399$	−22.1801	−1.11040
$400$	−4.48299	−0.224150
$401$	−3.41708	−0.170641	−0.0853204	−	0.996354i	$-0.527191\pi$
$401$	−3.41708	−0.170641	−0.0853204	+	0.996354i	$0.527191\pi$
$402$	2.81198	0.140249
$403$	−1.33812	−0.0666563
$404$	−3.90926	−0.194493
$405$	−0.719032	−0.0357290
$406$	21.9788	1.09079
$407$	2.38388	0.118165
$408$	−6.58304	−0.325909
$409$	22.8253	1.12864	0.564318	−	0.825557i	$-0.309139\pi$
$409$	22.8253	1.12864	0.564318	+	0.825557i	$0.309139\pi$
$410$	−8.54532	−0.422023
$411$	−18.4342	−0.909291
$412$	1.00000	0.0492665
$413$	−41.0472	−2.01980
$414$	4.96384	0.243960
$415$	−7.05860	−0.346493
$416$	−1.00000	−0.0490290
$417$	10.2656	0.502708
$418$	−18.2449	−0.892389
$419$	−19.4497	−0.950179	−0.475090	−	0.879937i	$-0.657584\pi$
$419$	−19.4497	−0.950179	−0.475090	+	0.879937i	$0.657584\pi$
$420$	3.63820	0.177526
$421$	−37.1576	−1.81095	−0.905475	−	0.424400i	$-0.860485\pi$
$421$	−37.1576	−1.81095	−0.905475	+	0.424400i	$0.860485\pi$
$422$	8.05692	0.392205
$423$	4.13504	0.201052
$424$	6.15067	0.298703
$425$	29.5117	1.43153
$426$	−1.58126	−0.0766124
$427$	−31.0751	−1.50383
$428$	−15.3484	−0.741891
$429$	4.16214	0.200950
$430$	−0.302638	−0.0145945
$431$	−7.63825	−0.367922	−0.183961	−	0.982934i	$-0.558892\pi$
$431$	−7.63825	−0.367922	−0.183961	+	0.982934i	$0.558892\pi$
$432$	1.00000	0.0481125
$433$	27.6331	1.32796	0.663981	−	0.747749i	$-0.268865\pi$
$433$	27.6331	1.32796	0.663981	+	0.747749i	$0.268865\pi$
$434$	−6.77068	−0.325003
$435$	3.12330	0.149751
$436$	4.97155	0.238094
$437$	21.7592	1.04089
$438$	−1.70341	−0.0813919
$439$	5.34954	0.255320	0.127660	−	0.991818i	$-0.459253\pi$
$439$	5.34954	0.255320	0.127660	+	0.991818i	$0.459253\pi$
$440$	2.99271	0.142672
$441$	18.6022	0.885818
$442$	6.58304	0.313123
$443$	−34.6311	−1.64537	−0.822687	−	0.568495i	$-0.807526\pi$
$443$	−34.6311	−1.64537	−0.822687	+	0.568495i	$0.807526\pi$
$444$	−0.572753	−0.0271817
$445$	−1.99963	−0.0947915
$446$	−16.1985	−0.767019
$447$	2.46130	0.116415
$448$	−5.05986	−0.239056
$449$	25.1255	1.18575	0.592874	−	0.805296i	$-0.297993\pi$
$449$	25.1255	1.18575	0.592874	+	0.805296i	$0.297993\pi$
$450$	−4.48299	−0.211330
$451$	−49.4649	−2.32921
$452$	10.2644	0.482797
$453$	0.872672	0.0410017
$454$	27.3918	1.28556
$455$	−3.63820	−0.170561
$456$	4.38354	0.205278
$457$	21.6769	1.01400	0.507001	−	0.861946i	$-0.330754\pi$
$457$	21.6769	1.01400	0.507001	+	0.861946i	$0.330754\pi$
$458$	10.8229	0.505722
$459$	−6.58304	−0.307270
$460$	−3.56916	−0.166413
$461$	12.4659	0.580597	0.290298	−	0.956936i	$-0.406245\pi$
$461$	12.4659	0.580597	0.290298	+	0.956936i	$0.406245\pi$
$462$	21.0599	0.979793
$463$	−24.4769	−1.13754	−0.568769	−	0.822497i	$-0.692580\pi$
$463$	−24.4769	−1.13754	−0.568769	+	0.822497i	$0.692580\pi$
$464$	−4.34377	−0.201654
$465$	−0.962147	−0.0446185
$466$	23.7920	1.10214
$467$	29.0244	1.34309	0.671544	−	0.740965i	$-0.265631\pi$
$467$	29.0244	1.34309	0.671544	+	0.740965i	$0.265631\pi$
$468$	−1.00000	−0.0462250
$469$	−14.2282	−0.656998
$470$	−2.97322	−0.137145
$471$	−24.0354	−1.10749
$472$	8.11233	0.373400
$473$	−1.75183	−0.0805492
$474$	5.14872	0.236488
$475$	−19.6514	−0.901668
$476$	33.3093	1.52673
$477$	6.15067	0.281620
$478$	3.41325	0.156118
$479$	18.2321	0.833044	0.416522	−	0.909126i	$-0.363249\pi$
$479$	18.2321	0.833044	0.416522	+	0.909126i	$0.363249\pi$
$480$	−0.719032	−0.0328192
$481$	0.572753	0.0261153
$482$	8.08232	0.368140
$483$	−25.1164	−1.14283
$484$	6.32343	0.287429
$485$	−3.29853	−0.149778
$486$	1.00000	0.0453609
$487$	34.2663	1.55275	0.776377	−	0.630269i	$-0.217055\pi$
$487$	34.2663	1.55275	0.776377	+	0.630269i	$0.217055\pi$
$488$	6.14150	0.278012
$489$	−2.99717	−0.135537
$490$	−13.3756	−0.604246
$491$	41.3525	1.86621	0.933106	−	0.359602i	$-0.117088\pi$
$491$	41.3525	1.86621	0.933106	+	0.359602i	$0.117088\pi$
$492$	11.8845	0.535794
$493$	28.5952	1.28786
$494$	−4.38354	−0.197225
$495$	2.99271	0.134512
$496$	1.33812	0.0600832
$497$	8.00096	0.358892
$498$	9.81682	0.439902
$499$	29.9206	1.33943	0.669716	−	0.742617i	$-0.266416\pi$
$499$	29.9206	1.33943	0.669716	+	0.742617i	$0.266416\pi$
$500$	6.81857	0.304936
$501$	1.93462	0.0864323
$502$	−11.4284	−0.510077
$503$	16.2227	0.723335	0.361667	−	0.932307i	$-0.382208\pi$
$503$	16.2227	0.723335	0.361667	+	0.932307i	$0.382208\pi$
$504$	−5.05986	−0.225384
$505$	2.81088	0.125083
$506$	−20.6602	−0.918459
$507$	1.00000	0.0444116
$508$	−1.24622	−0.0552922
$509$	−12.4638	−0.552449	−0.276225	−	0.961093i	$-0.589083\pi$
$509$	−12.4638	−0.552449	−0.276225	+	0.961093i	$0.589083\pi$
$510$	4.73342	0.209599
$511$	8.61899	0.381282
$512$	1.00000	0.0441942
$513$	4.38354	0.193538
$514$	−11.8568	−0.522981
$515$	−0.719032	−0.0316843
$516$	0.420896	0.0185289
$517$	−17.2106	−0.756922
$518$	2.89805	0.127333
$519$	16.3654	0.718362
$520$	0.719032	0.0315316
$521$	−19.6885	−0.862567	−0.431284	−	0.902216i	$-0.641939\pi$
$521$	−19.6885	−0.862567	−0.431284	+	0.902216i	$0.641939\pi$
$522$	−4.34377	−0.190121
$523$	2.92407	0.127861	0.0639303	−	0.997954i	$-0.479636\pi$
$523$	2.92407	0.127861	0.0639303	+	0.997954i	$0.479636\pi$
$524$	−1.71906	−0.0750976
$525$	22.6833	0.989981
$526$	−25.2244	−1.09984
$527$	−8.80887	−0.383720
$528$	−4.16214	−0.181134
$529$	1.63976	0.0712938
$530$	−4.42252	−0.192102
$531$	8.11233	0.352045
$532$	−22.1801	−0.961630
$533$	−11.8845	−0.514774
$534$	2.78100	0.120346
$535$	11.0360	0.477126
$536$	2.81198	0.121459
$537$	−20.3709	−0.879068
$538$	28.9069	1.24626
$539$	−77.4250	−3.33493
$540$	−0.719032	−0.0309422
$541$	24.2077	1.04077	0.520386	−	0.853931i	$-0.325788\pi$
$541$	24.2077	1.04077	0.520386	+	0.853931i	$0.325788\pi$
$542$	17.1387	0.736170
$543$	−17.8561	−0.766277
$544$	−6.58304	−0.282246
$545$	−3.57470	−0.153124
$546$	5.05986	0.216542
$547$	14.0267	0.599738	0.299869	−	0.953980i	$-0.403057\pi$
$547$	14.0267	0.599738	0.299869	+	0.953980i	$0.403057\pi$
$548$	−18.4342	−0.787469
$549$	6.14150	0.262113
$550$	18.6589	0.795616
$551$	−19.0411	−0.811178
$552$	4.96384	0.211275
$553$	−26.0518	−1.10783
$554$	26.9333	1.14429
$555$	0.411828	0.0174811
$556$	10.2656	0.435358
$557$	−2.58896	−0.109698	−0.0548488	−	0.998495i	$-0.517468\pi$
$557$	−2.58896	−0.109698	−0.0548488	+	0.998495i	$0.517468\pi$
$558$	1.33812	0.0566469
$559$	−0.420896	−0.0178020
$560$	3.63820	0.153742
$561$	27.3996	1.15681
$562$	12.8692	0.542854
$563$	−25.3440	−1.06812	−0.534060	−	0.845446i	$-0.679334\pi$
$563$	−25.3440	−1.06812	−0.534060	+	0.845446i	$0.679334\pi$
$564$	4.13504	0.174116
$565$	−7.38043	−0.310497
$566$	0.684428	0.0287686
$567$	−5.05986	−0.212494
$568$	−1.58126	−0.0663482
$569$	36.6919	1.53821	0.769103	−	0.639125i	$-0.220703\pi$
$569$	36.6919	1.53821	0.769103	+	0.639125i	$0.220703\pi$
$570$	−3.15191	−0.132019
$571$	−20.6997	−0.866255	−0.433128	−	0.901333i	$-0.642590\pi$
$571$	−20.6997	−0.866255	−0.433128	+	0.901333i	$0.642590\pi$
$572$	4.16214	0.174028
$573$	7.99297	0.333911
$574$	−60.1338	−2.50994
$575$	−22.2529	−0.928009
$576$	1.00000	0.0416667
$577$	−6.39902	−0.266395	−0.133197	−	0.991090i	$-0.542524\pi$
$577$	−6.39902	−0.266395	−0.133197	+	0.991090i	$0.542524\pi$
$578$	26.3364	1.09545
$579$	−25.4753	−1.05872
$580$	3.12330	0.129688
$581$	−49.6717	−2.06073
$582$	4.58746	0.190156
$583$	−25.5999	−1.06024
$584$	−1.70341	−0.0704874
$585$	0.719032	0.0297283
$586$	10.1832	0.420663
$587$	21.6700	0.894418	0.447209	−	0.894430i	$-0.352418\pi$
$587$	21.6700	0.894418	0.447209	+	0.894430i	$0.352418\pi$
$588$	18.6022	0.767141
$589$	5.86569	0.241692
$590$	−5.83302	−0.240142
$591$	−0.124351	−0.00511512
$592$	−0.572753	−0.0235400
$593$	43.1674	1.77267	0.886336	−	0.463043i	$-0.153242\pi$
$593$	43.1674	1.77267	0.886336	+	0.463043i	$0.153242\pi$
$594$	−4.16214	−0.170775
$595$	−23.9504	−0.981872
$596$	2.46130	0.100819
$597$	−16.3494	−0.669138
$598$	−4.96384	−0.202987
$599$	33.4536	1.36688	0.683438	−	0.730009i	$-0.260484\pi$
$599$	33.4536	1.36688	0.683438	+	0.730009i	$0.260484\pi$
$600$	−4.48299	−0.183017
$601$	4.27333	0.174313	0.0871564	−	0.996195i	$-0.472222\pi$
$601$	4.27333	0.174313	0.0871564	+	0.996195i	$0.472222\pi$
$602$	−2.12967	−0.0867991
$603$	2.81198	0.114513
$604$	0.872672	0.0355085
$605$	−4.54675	−0.184852
$606$	−3.90926	−0.158803
$607$	−26.1052	−1.05958	−0.529789	−	0.848129i	$-0.677729\pi$
$607$	−26.1052	−1.05958	−0.529789	+	0.848129i	$0.677729\pi$
$608$	4.38354	0.177776
$609$	21.9788	0.890628
$610$	−4.41593	−0.178796
$611$	−4.13504	−0.167286
$612$	−6.58304	−0.266104
$613$	−36.8444	−1.48813	−0.744065	−	0.668107i	$-0.767105\pi$
$613$	−36.8444	−1.48813	−0.744065	+	0.668107i	$0.767105\pi$
$614$	2.73429	0.110347
$615$	−8.54532	−0.344581
$616$	21.0599	0.848526
$617$	44.3694	1.78625	0.893123	−	0.449812i	$-0.148509\pi$
$617$	44.3694	1.78625	0.893123	+	0.449812i	$0.148509\pi$
$618$	1.00000	0.0402259
$619$	26.6396	1.07074	0.535368	−	0.844619i	$-0.320173\pi$
$619$	26.6396	1.07074	0.535368	+	0.844619i	$0.320173\pi$
$620$	−0.962147	−0.0386408
$621$	4.96384	0.199192
$622$	14.9858	0.600877
$623$	−14.0715	−0.563762
$624$	−1.00000	−0.0400320
$625$	17.5122	0.700488
$626$	1.55027	0.0619613
$627$	−18.2449	−0.728633
$628$	−24.0354	−0.959118
$629$	3.77046	0.150338
$630$	3.63820	0.144949
$631$	−25.4754	−1.01416	−0.507080	−	0.861899i	$-0.669275\pi$
$631$	−25.4754	−1.01416	−0.507080	+	0.861899i	$0.669275\pi$
$632$	5.14872	0.204805
$633$	8.05692	0.320234
$634$	7.27441	0.288904
$635$	0.896074	0.0355596
$636$	6.15067	0.243890
$637$	−18.6022	−0.737046
$638$	18.0794	0.715769
$639$	−1.58126	−0.0625537
$640$	−0.719032	−0.0284222
$641$	1.15351	0.0455609	0.0227805	−	0.999740i	$-0.492748\pi$
$641$	1.15351	0.0455609	0.0227805	+	0.999740i	$0.492748\pi$
$642$	−15.3484	−0.605751
$643$	19.6657	0.775539	0.387769	−	0.921756i	$-0.373246\pi$
$643$	19.6657	0.775539	0.387769	+	0.921756i	$0.373246\pi$
$644$	−25.1164	−0.989723
$645$	−0.302638	−0.0119163
$646$	−28.8571	−1.13537
$647$	8.90304	0.350015	0.175007	−	0.984567i	$-0.444005\pi$
$647$	8.90304	0.350015	0.175007	+	0.984567i	$0.444005\pi$
$648$	1.00000	0.0392837
$649$	−33.7647	−1.32538
$650$	4.48299	0.175837
$651$	−6.77068	−0.265364
$652$	−2.99717	−0.117378
$653$	5.70927	0.223421	0.111711	−	0.993741i	$-0.464367\pi$
$653$	5.70927	0.223421	0.111711	+	0.993741i	$0.464367\pi$
$654$	4.97155	0.194403
$655$	1.23606	0.0482969
$656$	11.8845	0.464011
$657$	−1.70341	−0.0664562
$658$	−20.9227	−0.815652
$659$	23.8704	0.929858	0.464929	−	0.885348i	$-0.346080\pi$
$659$	23.8704	0.929858	0.464929	+	0.885348i	$0.346080\pi$
$660$	2.99271	0.116491
$661$	2.49620	0.0970910	0.0485455	−	0.998821i	$-0.484541\pi$
$661$	2.49620	0.0970910	0.0485455	+	0.998821i	$0.484541\pi$
$662$	23.1124	0.898290
$663$	6.58304	0.255664
$664$	9.81682	0.380966
$665$	15.9482	0.618445
$666$	−0.572753	−0.0221937
$667$	−21.5618	−0.834875
$668$	1.93462	0.0748526
$669$	−16.1985	−0.626268
$670$	−2.02190	−0.0781128
$671$	−25.5618	−0.986802
$672$	−5.05986	−0.195188
$673$	−21.1154	−0.813940	−0.406970	−	0.913441i	$-0.633415\pi$
$673$	−21.1154	−0.813940	−0.406970	+	0.913441i	$0.633415\pi$
$674$	34.5121	1.32936
$675$	−4.48299	−0.172550
$676$	1.00000	0.0384615
$677$	1.63434	0.0628130	0.0314065	−	0.999507i	$-0.490001\pi$
$677$	1.63434	0.0628130	0.0314065	+	0.999507i	$0.490001\pi$
$678$	10.2644	0.394202
$679$	−23.2119	−0.890791
$680$	4.73342	0.181518
$681$	27.3918	1.04966
$682$	−5.56943	−0.213264
$683$	−7.51901	−0.287707	−0.143854	−	0.989599i	$-0.545949\pi$
$683$	−7.51901	−0.287707	−0.143854	+	0.989599i	$0.545949\pi$
$684$	4.38354	0.167609
$685$	13.2548	0.506438
$686$	−58.7055	−2.24138
$687$	10.8229	0.412921
$688$	0.420896	0.0160465
$689$	−6.15067	−0.234322
$690$	−3.56916	−0.135876
$691$	−29.4006	−1.11845	−0.559226	−	0.829015i	$-0.688902\pi$
$691$	−29.4006	−1.11845	−0.559226	+	0.829015i	$0.688902\pi$
$692$	16.3654	0.622120
$693$	21.0599	0.799998
$694$	−23.1693	−0.879496
$695$	−7.38129	−0.279988
$696$	−4.34377	−0.164650
$697$	−78.2361	−2.96340
$698$	−0.474357	−0.0179547
$699$	23.7920	0.899894
$700$	22.6833	0.857349
$701$	−5.34212	−0.201769	−0.100885	−	0.994898i	$-0.532167\pi$
$701$	−5.34212	−0.201769	−0.100885	+	0.994898i	$0.532167\pi$
$702$	−1.00000	−0.0377426
$703$	−2.51069	−0.0946924
$704$	−4.16214	−0.156867
$705$	−2.97322	−0.111978
$706$	−8.50488	−0.320085
$707$	19.7803	0.743915
$708$	8.11233	0.304880
$709$	15.1495	0.568953	0.284476	−	0.958683i	$-0.408180\pi$
$709$	15.1495	0.568953	0.284476	+	0.958683i	$0.408180\pi$
$710$	1.13698	0.0426700
$711$	5.14872	0.193092
$712$	2.78100	0.104222
$713$	6.64220	0.248752
$714$	33.3093	1.24657
$715$	−2.99271	−0.111921
$716$	−20.3709	−0.761296
$717$	3.41325	0.127470
$718$	−27.4529	−1.02453
$719$	−24.8160	−0.925482	−0.462741	−	0.886494i	$-0.653134\pi$
$719$	−24.8160	−0.925482	−0.462741	+	0.886494i	$0.653134\pi$
$720$	−0.719032	−0.0267967
$721$	−5.05986	−0.188439
$722$	0.215465	0.00801876
$723$	8.08232	0.300585
$724$	−17.8561	−0.663615
$725$	19.4731	0.723212
$726$	6.32343	0.234685
$727$	−19.7266	−0.731618	−0.365809	−	0.930690i	$-0.619208\pi$
$727$	−19.7266	−0.731618	−0.365809	+	0.930690i	$0.619208\pi$
$728$	5.05986	0.187531
$729$	1.00000	0.0370370
$730$	1.22480	0.0453320
$731$	−2.77078	−0.102481
$732$	6.14150	0.226996
$733$	35.8742	1.32504	0.662521	−	0.749043i	$-0.269486\pi$
$733$	35.8742	1.32504	0.662521	+	0.749043i	$0.269486\pi$
$734$	−0.266030	−0.00981935
$735$	−13.3756	−0.493365
$736$	4.96384	0.182970
$737$	−11.7039	−0.431117
$738$	11.8845	0.437474
$739$	21.9096	0.805958	0.402979	−	0.915209i	$-0.367975\pi$
$739$	21.9096	0.805958	0.402979	+	0.915209i	$0.367975\pi$
$740$	0.411828	0.0151391
$741$	−4.38354	−0.161034
$742$	−31.1215	−1.14251
$743$	37.1255	1.36200	0.681001	−	0.732282i	$-0.261545\pi$
$743$	37.1255	1.36200	0.681001	+	0.732282i	$0.261545\pi$
$744$	1.33812	0.0490577
$745$	−1.76975	−0.0648386
$746$	5.87066	0.214940
$747$	9.81682	0.359178
$748$	27.3996	1.00183
$749$	77.6606	2.83766
$750$	6.81857	0.248979
$751$	−20.5326	−0.749243	−0.374622	−	0.927178i	$-0.622227\pi$
$751$	−20.5326	−0.749243	−0.374622	+	0.927178i	$0.622227\pi$
$752$	4.13504	0.150789
$753$	−11.4284	−0.416476
$754$	4.34377	0.158191
$755$	−0.627479	−0.0228363
$756$	−5.05986	−0.184025
$757$	8.89125	0.323158	0.161579	−	0.986860i	$-0.448341\pi$
$757$	8.89125	0.323158	0.161579	+	0.986860i	$0.448341\pi$
$758$	12.8691	0.467425
$759$	−20.6602	−0.749919
$760$	−3.15191	−0.114332
$761$	22.5603	0.817809	0.408905	−	0.912577i	$-0.365911\pi$
$761$	22.5603	0.817809	0.408905	+	0.912577i	$0.365911\pi$
$762$	−1.24622	−0.0451459
$763$	−25.1554	−0.910686
$764$	7.99297	0.289176
$765$	4.73342	0.171137
$766$	−4.82830	−0.174454
$767$	−8.11233	−0.292919
$768$	1.00000	0.0360844
$769$	−18.5990	−0.670698	−0.335349	−	0.942094i	$-0.608854\pi$
$769$	−18.5990	−0.670698	−0.335349	+	0.942094i	$0.608854\pi$
$770$	−15.1427	−0.545705
$771$	−11.8568	−0.427012
$772$	−25.4753	−0.916877
$773$	−2.94340	−0.105867	−0.0529334	−	0.998598i	$-0.516857\pi$
$773$	−2.94340	−0.105867	−0.0529334	+	0.998598i	$0.516857\pi$
$774$	0.420896	0.0151288
$775$	−5.99876	−0.215482
$776$	4.58746	0.164680
$777$	2.89805	0.103967
$778$	−6.13474	−0.219941
$779$	52.0962	1.86654
$780$	0.719032	0.0257455
$781$	6.58144	0.235502
$782$	−32.6772	−1.16853
$783$	−4.34377	−0.155234
$784$	18.6022	0.664364
$785$	17.2822	0.616830
$786$	−1.71906	−0.0613169
$787$	−47.5099	−1.69355	−0.846773	−	0.531955i	$-0.821458\pi$
$787$	−47.5099	−1.69355	−0.846773	+	0.531955i	$0.821458\pi$
$788$	−0.124351	−0.00442982
$789$	−25.2244	−0.898014
$790$	−3.70209	−0.131714
$791$	−51.9364	−1.84665
$792$	−4.16214	−0.147895
$793$	−6.14150	−0.218091
$794$	−15.3500	−0.544753
$795$	−4.42252	−0.156851
$796$	−16.3494	−0.579491
$797$	10.0220	0.354996	0.177498	−	0.984121i	$-0.443200\pi$
$797$	10.0220	0.354996	0.177498	+	0.984121i	$0.443200\pi$
$798$	−22.1801	−0.785168
$799$	−27.2211	−0.963014
$800$	−4.48299	−0.158498
$801$	2.78100	0.0982618
$802$	−3.41708	−0.120661
$803$	7.08982	0.250194
$804$	2.81198	0.0991708
$805$	18.0595	0.636512
$806$	−1.33812	−0.0471331
$807$	28.9069	1.01757
$808$	−3.90926	−0.137527
$809$	55.9180	1.96597	0.982987	−	0.183677i	$-0.0588000\pi$
$809$	55.9180	1.96597	0.982987	+	0.183677i	$0.0588000\pi$
$810$	−0.719032	−0.0252642
$811$	30.0469	1.05509	0.527544	−	0.849528i	$-0.323113\pi$
$811$	30.0469	1.05509	0.527544	+	0.849528i	$0.323113\pi$
$812$	21.9788	0.771306
$813$	17.1387	0.601080
$814$	2.38388	0.0835550
$815$	2.15506	0.0754885
$816$	−6.58304	−0.230453
$817$	1.84502	0.0645490
$818$	22.8253	0.798067
$819$	5.05986	0.176806
$820$	−8.54532	−0.298416
$821$	33.9687	1.18552	0.592758	−	0.805381i	$-0.298039\pi$
$821$	33.9687	1.18552	0.592758	+	0.805381i	$0.298039\pi$
$822$	−18.4342	−0.642966
$823$	−43.1058	−1.50257	−0.751287	−	0.659976i	$-0.770566\pi$
$823$	−43.1058	−1.50257	−0.751287	+	0.659976i	$0.770566\pi$
$824$	1.00000	0.0348367
$825$	18.6589	0.649618
$826$	−41.0472	−1.42822
$827$	−48.5004	−1.68653	−0.843263	−	0.537501i	$-0.819368\pi$
$827$	−48.5004	−1.68653	−0.843263	+	0.537501i	$0.819368\pi$
$828$	4.96384	0.172506
$829$	−32.4895	−1.12841	−0.564204	−	0.825636i	$-0.690817\pi$
$829$	−32.4895	−1.12841	−0.564204	+	0.825636i	$0.690817\pi$
$830$	−7.05860	−0.245008
$831$	26.9333	0.934305
$832$	−1.00000	−0.0346688
$833$	−122.459	−4.24295
$834$	10.2656	0.355468
$835$	−1.39105	−0.0481393
$836$	−18.2449	−0.631014
$837$	1.33812	0.0462520
$838$	−19.4497	−0.671878
$839$	18.9498	0.654220	0.327110	−	0.944986i	$-0.393925\pi$
$839$	18.9498	0.654220	0.327110	+	0.944986i	$0.393925\pi$
$840$	3.63820	0.125530
$841$	−10.1317	−0.349369
$842$	−37.1576	−1.28053
$843$	12.8692	0.443239
$844$	8.05692	0.277331
$845$	−0.719032	−0.0247354
$846$	4.13504	0.142166
$847$	−31.9957	−1.09938
$848$	6.15067	0.211215
$849$	0.684428	0.0234895
$850$	29.5117	1.01224
$851$	−2.84306	−0.0974588
$852$	−1.58126	−0.0541731
$853$	35.4459	1.21364	0.606822	−	0.794838i	$-0.292444\pi$
$853$	35.4459	1.21364	0.606822	+	0.794838i	$0.292444\pi$
$854$	−31.0751	−1.06337
$855$	−3.15191	−0.107793
$856$	−15.3484	−0.524596
$857$	10.4048	0.355420	0.177710	−	0.984083i	$-0.443131\pi$
$857$	10.4048	0.355420	0.177710	+	0.984083i	$0.443131\pi$
$858$	4.16214	0.142093
$859$	−30.2567	−1.03234	−0.516172	−	0.856485i	$-0.672644\pi$
$859$	−30.2567	−1.03234	−0.516172	+	0.856485i	$0.672644\pi$
$860$	−0.302638	−0.0103199
$861$	−60.1338	−2.04936
$862$	−7.63825	−0.260160
$863$	−20.4983	−0.697772	−0.348886	−	0.937165i	$-0.613440\pi$
$863$	−20.4983	−0.697772	−0.348886	+	0.937165i	$0.613440\pi$
$864$	1.00000	0.0340207
$865$	−11.7672	−0.400098
$866$	27.6331	0.939012
$867$	26.3364	0.894432
$868$	−6.77068	−0.229812
$869$	−21.4297	−0.726953
$870$	3.12330	0.105890
$871$	−2.81198	−0.0952802
$872$	4.97155	0.168358
$873$	4.58746	0.155262
$874$	21.7592	0.736017
$875$	−34.5010	−1.16635
$876$	−1.70341	−0.0575527
$877$	10.1663	0.343291	0.171646	−	0.985159i	$-0.445092\pi$
$877$	10.1663	0.343291	0.171646	+	0.985159i	$0.445092\pi$
$878$	5.34954	0.180538
$879$	10.1832	0.343470
$880$	2.99271	0.100884
$881$	−15.8827	−0.535103	−0.267552	−	0.963544i	$-0.586215\pi$
$881$	−15.8827	−0.535103	−0.267552	+	0.963544i	$0.586215\pi$
$882$	18.6022	0.626368
$883$	13.7593	0.463037	0.231518	−	0.972831i	$-0.425631\pi$
$883$	13.7593	0.463037	0.231518	+	0.972831i	$0.425631\pi$
$884$	6.58304	0.221412
$885$	−5.83302	−0.196075
$886$	−34.6311	−1.16346
$887$	21.4734	0.721007	0.360503	−	0.932758i	$-0.382605\pi$
$887$	21.4734	0.721007	0.360503	+	0.932758i	$0.382605\pi$
$888$	−0.572753	−0.0192203
$889$	6.30572	0.211487
$890$	−1.99963	−0.0670277
$891$	−4.16214	−0.139437
$892$	−16.1985	−0.542364
$893$	18.1261	0.606568
$894$	2.46130	0.0823180
$895$	14.6473	0.489606
$896$	−5.05986	−0.169038
$897$	−4.96384	−0.165738
$898$	25.1255	0.838450
$899$	−5.81246	−0.193856
$900$	−4.48299	−0.149433
$901$	−40.4901	−1.34892
$902$	−49.4649	−1.64700
$903$	−2.12967	−0.0708711
$904$	10.2644	0.341389
$905$	12.8391	0.426785
$906$	0.872672	0.0289926
$907$	−36.2857	−1.20485	−0.602423	−	0.798177i	$-0.705798\pi$
$907$	−36.2857	−1.20485	−0.602423	+	0.798177i	$0.705798\pi$
$908$	27.3918	0.909028
$909$	−3.90926	−0.129662
$910$	−3.63820	−0.120605
$911$	−27.9148	−0.924859	−0.462429	−	0.886656i	$-0.653022\pi$
$911$	−27.9148	−0.924859	−0.462429	+	0.886656i	$0.653022\pi$
$912$	4.38354	0.145154
$913$	−40.8590	−1.35224
$914$	21.6769	0.717007
$915$	−4.41593	−0.145986
$916$	10.8229	0.357600
$917$	8.69822	0.287240
$918$	−6.58304	−0.217273
$919$	8.27421	0.272941	0.136470	−	0.990644i	$-0.456424\pi$
$919$	8.27421	0.272941	0.136470	+	0.990644i	$0.456424\pi$
$920$	−3.56916	−0.117672
$921$	2.73429	0.0900978
$922$	12.4659	0.410544
$923$	1.58126	0.0520478
$924$	21.0599	0.692819
$925$	2.56765	0.0844238
$926$	−24.4769	−0.804361
$927$	1.00000	0.0328443
$928$	−4.34377	−0.142591
$929$	33.0459	1.08420	0.542100	−	0.840314i	$-0.317629\pi$
$929$	33.0459	1.08420	0.542100	+	0.840314i	$0.317629\pi$
$930$	−0.962147	−0.0315501
$931$	81.5435	2.67248
$932$	23.7920	0.779331
$933$	14.9858	0.490614
$934$	29.0244	0.949707
$935$	−19.7012	−0.644297
$936$	−1.00000	−0.0326860
$937$	−15.0522	−0.491735	−0.245867	−	0.969303i	$-0.579073\pi$
$937$	−15.0522	−0.491735	−0.245867	+	0.969303i	$0.579073\pi$
$938$	−14.2282	−0.464568
$939$	1.55027	0.0505912
$940$	−2.97322	−0.0969758
$941$	−0.257117	−0.00838176	−0.00419088	−	0.999991i	$-0.501334\pi$
$941$	−0.257117	−0.00838176	−0.00419088	+	0.999991i	$0.501334\pi$
$942$	−24.0354	−0.783117
$943$	58.9928	1.92107
$944$	8.11233	0.264034
$945$	3.63820	0.118351
$946$	−1.75183	−0.0569569
$947$	47.4815	1.54294	0.771471	−	0.636265i	$-0.219521\pi$
$947$	47.4815	1.54294	0.771471	+	0.636265i	$0.219521\pi$
$948$	5.14872	0.167223
$949$	1.70341	0.0552949
$950$	−19.6514	−0.637576
$951$	7.27441	0.235889
$952$	33.3093	1.07956
$953$	3.69940	0.119835	0.0599176	−	0.998203i	$-0.480916\pi$
$953$	3.69940	0.119835	0.0599176	+	0.998203i	$0.480916\pi$
$954$	6.15067	0.199135
$955$	−5.74720	−0.185975
$956$	3.41325	0.110392
$957$	18.0794	0.584423
$958$	18.2321	0.589051
$959$	93.2743	3.01199
$960$	−0.719032	−0.0232066
$961$	−29.2094	−0.942240
$962$	0.572753	0.0184663
$963$	−15.3484	−0.494594
$964$	8.08232	0.260314
$965$	18.3176	0.589664
$966$	−25.1164	−0.808106
$967$	−5.14972	−0.165604	−0.0828019	−	0.996566i	$-0.526387\pi$
$967$	−5.14972	−0.165604	−0.0828019	+	0.996566i	$0.526387\pi$
$968$	6.32343	0.203243
$969$	−28.8571	−0.927022
$970$	−3.29853	−0.105909
$971$	−45.4030	−1.45705	−0.728526	−	0.685018i	$-0.759794\pi$
$971$	−45.4030	−1.45705	−0.728526	+	0.685018i	$0.759794\pi$
$972$	1.00000	0.0320750
$973$	−51.9425	−1.66520
$974$	34.2663	1.09796
$975$	4.48299	0.143571
$976$	6.14150	0.196584
$977$	−4.73879	−0.151607	−0.0758036	−	0.997123i	$-0.524152\pi$
$977$	−4.73879	−0.151607	−0.0758036	+	0.997123i	$0.524152\pi$
$978$	−2.99717	−0.0958390
$979$	−11.5749	−0.369936
$980$	−13.3756	−0.427267
$981$	4.97155	0.158730
$982$	41.3525	1.31961
$983$	−34.6168	−1.10411	−0.552053	−	0.833809i	$-0.686155\pi$
$983$	−34.6168	−1.10411	−0.552053	+	0.833809i	$0.686155\pi$
$984$	11.8845	0.378864
$985$	0.0894123	0.00284891
$986$	28.5952	0.910656
$987$	−20.9227	−0.665977
$988$	−4.38354	−0.139459
$989$	2.08926	0.0664347
$990$	2.99271	0.0951146
$991$	3.29569	0.104691	0.0523455	−	0.998629i	$-0.483330\pi$
$991$	3.29569	0.104691	0.0523455	+	0.998629i	$0.483330\pi$
$992$	1.33812	0.0424852
$993$	23.1124	0.733451
$994$	8.00096	0.253775
$995$	11.7558	0.372683
$996$	9.81682	0.311058
$997$	−28.0298	−0.887712	−0.443856	−	0.896098i	$-0.646390\pi$
$997$	−28.0298	−0.887712	−0.443856	+	0.896098i	$0.646390\pi$
$998$	29.9206	0.947121
$999$	−0.572753	−0.0181211

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.bd.1.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.bd.1.7	✓	16	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.719032 q^{5} +1.00000 q^{6} -5.05986 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.719032 q^{5} +1.00000 q^{6} -5.05986 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.719032 q^{10} -4.16214 q^{11} +1.00000 q^{12} -1.00000 q^{13} -5.05986 q^{14} -0.719032 q^{15} +1.00000 q^{16} -6.58304 q^{17} +1.00000 q^{18} +4.38354 q^{19} -0.719032 q^{20} -5.05986 q^{21} -4.16214 q^{22} +4.96384 q^{23} +1.00000 q^{24} -4.48299 q^{25} -1.00000 q^{26} +1.00000 q^{27} -5.05986 q^{28} -4.34377 q^{29} -0.719032 q^{30} +1.33812 q^{31} +1.00000 q^{32} -4.16214 q^{33} -6.58304 q^{34} +3.63820 q^{35} +1.00000 q^{36} -0.572753 q^{37} +4.38354 q^{38} -1.00000 q^{39} -0.719032 q^{40} +11.8845 q^{41} -5.05986 q^{42} +0.420896 q^{43} -4.16214 q^{44} -0.719032 q^{45} +4.96384 q^{46} +4.13504 q^{47} +1.00000 q^{48} +18.6022 q^{49} -4.48299 q^{50} -6.58304 q^{51} -1.00000 q^{52} +6.15067 q^{53} +1.00000 q^{54} +2.99271 q^{55} -5.05986 q^{56} +4.38354 q^{57} -4.34377 q^{58} +8.11233 q^{59} -0.719032 q^{60} +6.14150 q^{61} +1.33812 q^{62} -5.05986 q^{63} +1.00000 q^{64} +0.719032 q^{65} -4.16214 q^{66} +2.81198 q^{67} -6.58304 q^{68} +4.96384 q^{69} +3.63820 q^{70} -1.58126 q^{71} +1.00000 q^{72} -1.70341 q^{73} -0.572753 q^{74} -4.48299 q^{75} +4.38354 q^{76} +21.0599 q^{77} -1.00000 q^{78} +5.14872 q^{79} -0.719032 q^{80} +1.00000 q^{81} +11.8845 q^{82} +9.81682 q^{83} -5.05986 q^{84} +4.73342 q^{85} +0.420896 q^{86} -4.34377 q^{87} -4.16214 q^{88} +2.78100 q^{89} -0.719032 q^{90} +5.05986 q^{91} +4.96384 q^{92} +1.33812 q^{93} +4.13504 q^{94} -3.15191 q^{95} +1.00000 q^{96} +4.58746 q^{97} +18.6022 q^{98} -4.16214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} + 16 q^{3} + 16 q^{4} + 5 q^{5} + 16 q^{6} + 4 q^{7} + 16 q^{8} + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^2 + 16 * q^3 + 16 * q^4 + 5 * q^5 + 16 * q^6 + 4 * q^7 + 16 * q^8 + 16 * q^9	\( 16 q + 16 q^{2} + 16 q^{3} + 16 q^{4} + 5 q^{5} + 16 q^{6} + 4 q^{7} + 16 q^{8} + 16 q^{9} + 5 q^{10} + 18 q^{11} + 16 q^{12} - 16 q^{13} + 4 q^{14} + 5 q^{15} + 16 q^{16} + 17 q^{17} + 16 q^{18} + 8 q^{19} + 5 q^{20} + 4 q^{21} + 18 q^{22} + 9 q^{23} + 16 q^{24} + 17 q^{25} - 16 q^{26} + 16 q^{27} + 4 q^{28} + 14 q^{29} + 5 q^{30} + 12 q^{31} + 16 q^{32} + 18 q^{33} + 17 q^{34} + 16 q^{35} + 16 q^{36} + 31 q^{37} + 8 q^{38} - 16 q^{39} + 5 q^{40} + 29 q^{41} + 4 q^{42} + 30 q^{43} + 18 q^{44} + 5 q^{45} + 9 q^{46} - q^{47} + 16 q^{48} + 36 q^{49} + 17 q^{50} + 17 q^{51} - 16 q^{52} + 12 q^{53} + 16 q^{54} + 30 q^{55} + 4 q^{56} + 8 q^{57} + 14 q^{58} + 38 q^{59} + 5 q^{60} + 12 q^{62} + 4 q^{63} + 16 q^{64} - 5 q^{65} + 18 q^{66} + 28 q^{67} + 17 q^{68} + 9 q^{69} + 16 q^{70} + 32 q^{71} + 16 q^{72} + 20 q^{73} + 31 q^{74} + 17 q^{75} + 8 q^{76} + 26 q^{77} - 16 q^{78} + 13 q^{79} + 5 q^{80} + 16 q^{81} + 29 q^{82} + 39 q^{83} + 4 q^{84} + 31 q^{85} + 30 q^{86} + 14 q^{87} + 18 q^{88} + 9 q^{89} + 5 q^{90} - 4 q^{91} + 9 q^{92} + 12 q^{93} - q^{94} - 20 q^{95} + 16 q^{96} + 35 q^{97} + 36 q^{98} + 18 q^{99}+O(q^{100}) \) 16 * q + 16 * q^2 + 16 * q^3 + 16 * q^4 + 5 * q^5 + 16 * q^6 + 4 * q^7 + 16 * q^8 + 16 * q^9 + 5 * q^10 + 18 * q^11 + 16 * q^12 - 16 * q^13 + 4 * q^14 + 5 * q^15 + 16 * q^16 + 17 * q^17 + 16 * q^18 + 8 * q^19 + 5 * q^20 + 4 * q^21 + 18 * q^22 + 9 * q^23 + 16 * q^24 + 17 * q^25 - 16 * q^26 + 16 * q^27 + 4 * q^28 + 14 * q^29 + 5 * q^30 + 12 * q^31 + 16 * q^32 + 18 * q^33 + 17 * q^34 + 16 * q^35 + 16 * q^36 + 31 * q^37 + 8 * q^38 - 16 * q^39 + 5 * q^40 + 29 * q^41 + 4 * q^42 + 30 * q^43 + 18 * q^44 + 5 * q^45 + 9 * q^46 - q^47 + 16 * q^48 + 36 * q^49 + 17 * q^50 + 17 * q^51 - 16 * q^52 + 12 * q^53 + 16 * q^54 + 30 * q^55 + 4 * q^56 + 8 * q^57 + 14 * q^58 + 38 * q^59 + 5 * q^60 + 12 * q^62 + 4 * q^63 + 16 * q^64 - 5 * q^65 + 18 * q^66 + 28 * q^67 + 17 * q^68 + 9 * q^69 + 16 * q^70 + 32 * q^71 + 16 * q^72 + 20 * q^73 + 31 * q^74 + 17 * q^75 + 8 * q^76 + 26 * q^77 - 16 * q^78 + 13 * q^79 + 5 * q^80 + 16 * q^81 + 29 * q^82 + 39 * q^83 + 4 * q^84 + 31 * q^85 + 30 * q^86 + 14 * q^87 + 18 * q^88 + 9 * q^89 + 5 * q^90 - 4 * q^91 + 9 * q^92 + 12 * q^93 - q^94 - 20 * q^95 + 16 * q^96 + 35 * q^97 + 36 * q^98 + 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more