LMFDB - Embedded newform 6042.2.a.bb.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6042,2,Mod(1,6042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6042, 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("6042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6042 = 2 \cdot 3 \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6042.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:	$48.2456129013$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 16x^{7} + 76x^{6} + 30x^{5} - 366x^{4} + 300x^{3} + 101x^{2} - 106x + 16 $ x^9 - 4x^8 - 16x^7 + 76x^6 + 30x^5 - 366x^4 + 300x^3 + 101x^2 - 106x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$3.30066$ of defining polynomial
Character	$\chi$	$=$	6042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.85114 q^{5} -1.00000 q^{6} +2.30066 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.85114 q^{5} -1.00000 q^{6} +2.30066 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.85114 q^{10} -3.88254 q^{11} -1.00000 q^{12} -5.31030 q^{13} +2.30066 q^{14} -2.85114 q^{15} +1.00000 q^{16} -4.68330 q^{17} +1.00000 q^{18} -1.00000 q^{19} +2.85114 q^{20} -2.30066 q^{21} -3.88254 q^{22} -2.42384 q^{23} -1.00000 q^{24} +3.12899 q^{25} -5.31030 q^{26} -1.00000 q^{27} +2.30066 q^{28} -0.677181 q^{29} -2.85114 q^{30} +1.10086 q^{31} +1.00000 q^{32} +3.88254 q^{33} -4.68330 q^{34} +6.55950 q^{35} +1.00000 q^{36} -0.702636 q^{37} -1.00000 q^{38} +5.31030 q^{39} +2.85114 q^{40} -8.92591 q^{41} -2.30066 q^{42} -4.76516 q^{43} -3.88254 q^{44} +2.85114 q^{45} -2.42384 q^{46} -7.56937 q^{47} -1.00000 q^{48} -1.70697 q^{49} +3.12899 q^{50} +4.68330 q^{51} -5.31030 q^{52} +1.00000 q^{53} -1.00000 q^{54} -11.0697 q^{55} +2.30066 q^{56} +1.00000 q^{57} -0.677181 q^{58} -9.48144 q^{59} -2.85114 q^{60} -12.2484 q^{61} +1.10086 q^{62} +2.30066 q^{63} +1.00000 q^{64} -15.1404 q^{65} +3.88254 q^{66} +7.32764 q^{67} -4.68330 q^{68} +2.42384 q^{69} +6.55950 q^{70} +5.61549 q^{71} +1.00000 q^{72} +9.90668 q^{73} -0.702636 q^{74} -3.12899 q^{75} -1.00000 q^{76} -8.93241 q^{77} +5.31030 q^{78} -12.7705 q^{79} +2.85114 q^{80} +1.00000 q^{81} -8.92591 q^{82} +0.151868 q^{83} -2.30066 q^{84} -13.3527 q^{85} -4.76516 q^{86} +0.677181 q^{87} -3.88254 q^{88} -7.05177 q^{89} +2.85114 q^{90} -12.2172 q^{91} -2.42384 q^{92} -1.10086 q^{93} -7.56937 q^{94} -2.85114 q^{95} -1.00000 q^{96} +10.6297 q^{97} -1.70697 q^{98} -3.88254 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 5 q^{5} - 9 q^{6} - 5 q^{7} + 9 q^{8} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 - 5 * q^5 - 9 * q^6 - 5 * q^7 + 9 * q^8 + 9 * q^9	\( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 5 q^{5} - 9 q^{6} - 5 q^{7} + 9 q^{8} + 9 q^{9} - 5 q^{10} + 3 q^{11} - 9 q^{12} - 4 q^{13} - 5 q^{14} + 5 q^{15} + 9 q^{16} - 16 q^{17} + 9 q^{18} - 9 q^{19} - 5 q^{20} + 5 q^{21} + 3 q^{22} - 9 q^{24} + 8 q^{25} - 4 q^{26} - 9 q^{27} - 5 q^{28} - 2 q^{29} + 5 q^{30} - 10 q^{31} + 9 q^{32} - 3 q^{33} - 16 q^{34} - q^{35} + 9 q^{36} - 14 q^{37} - 9 q^{38} + 4 q^{39} - 5 q^{40} - 21 q^{41} + 5 q^{42} - 4 q^{43} + 3 q^{44} - 5 q^{45} - 8 q^{47} - 9 q^{48} - 14 q^{49} + 8 q^{50} + 16 q^{51} - 4 q^{52} + 9 q^{53} - 9 q^{54} - 14 q^{55} - 5 q^{56} + 9 q^{57} - 2 q^{58} - 12 q^{59} + 5 q^{60} - 13 q^{61} - 10 q^{62} - 5 q^{63} + 9 q^{64} - 13 q^{65} - 3 q^{66} + 22 q^{67} - 16 q^{68} - q^{70} - 13 q^{71} + 9 q^{72} - 17 q^{73} - 14 q^{74} - 8 q^{75} - 9 q^{76} - 25 q^{77} + 4 q^{78} - 5 q^{80} + 9 q^{81} - 21 q^{82} - 24 q^{83} + 5 q^{84} - 16 q^{85} - 4 q^{86} + 2 q^{87} + 3 q^{88} - 23 q^{89} - 5 q^{90} - 11 q^{91} + 10 q^{93} - 8 q^{94} + 5 q^{95} - 9 q^{96} - 29 q^{97} - 14 q^{98} + 3 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 - 5 * q^5 - 9 * q^6 - 5 * q^7 + 9 * q^8 + 9 * q^9 - 5 * q^10 + 3 * q^11 - 9 * q^12 - 4 * q^13 - 5 * q^14 + 5 * q^15 + 9 * q^16 - 16 * q^17 + 9 * q^18 - 9 * q^19 - 5 * q^20 + 5 * q^21 + 3 * q^22 - 9 * q^24 + 8 * q^25 - 4 * q^26 - 9 * q^27 - 5 * q^28 - 2 * q^29 + 5 * q^30 - 10 * q^31 + 9 * q^32 - 3 * q^33 - 16 * q^34 - q^35 + 9 * q^36 - 14 * q^37 - 9 * q^38 + 4 * q^39 - 5 * q^40 - 21 * q^41 + 5 * q^42 - 4 * q^43 + 3 * q^44 - 5 * q^45 - 8 * q^47 - 9 * q^48 - 14 * q^49 + 8 * q^50 + 16 * q^51 - 4 * q^52 + 9 * q^53 - 9 * q^54 - 14 * q^55 - 5 * q^56 + 9 * q^57 - 2 * q^58 - 12 * q^59 + 5 * q^60 - 13 * q^61 - 10 * q^62 - 5 * q^63 + 9 * q^64 - 13 * q^65 - 3 * q^66 + 22 * q^67 - 16 * q^68 - q^70 - 13 * q^71 + 9 * q^72 - 17 * q^73 - 14 * q^74 - 8 * q^75 - 9 * q^76 - 25 * q^77 + 4 * q^78 - 5 * q^80 + 9 * q^81 - 21 * q^82 - 24 * q^83 + 5 * q^84 - 16 * q^85 - 4 * q^86 + 2 * q^87 + 3 * q^88 - 23 * q^89 - 5 * q^90 - 11 * q^91 + 10 * q^93 - 8 * q^94 + 5 * q^95 - 9 * q^96 - 29 * q^97 - 14 * q^98 + 3 * 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
$5$	2.85114	1.27507	0.637534	−	0.770422i	$-0.279955\pi$
$5$	2.85114	1.27507	0.637534	+	0.770422i	$0.279955\pi$
$6$	−1.00000	−0.408248
$7$	2.30066	0.869567	0.434784	−	0.900535i	$-0.356825\pi$
$7$	2.30066	0.869567	0.434784	+	0.900535i	$0.356825\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	2.85114	0.901609
$11$	−3.88254	−1.17063	−0.585315	−	0.810806i	$-0.699029\pi$
$11$	−3.88254	−1.17063	−0.585315	+	0.810806i	$0.699029\pi$
$12$	−1.00000	−0.288675
$13$	−5.31030	−1.47281	−0.736406	−	0.676540i	$-0.763479\pi$
$13$	−5.31030	−1.47281	−0.736406	+	0.676540i	$0.763479\pi$
$14$	2.30066	0.614877
$15$	−2.85114	−0.736161
$16$	1.00000	0.250000
$17$	−4.68330	−1.13587	−0.567934	−	0.823074i	$-0.692257\pi$
$17$	−4.68330	−1.13587	−0.567934	+	0.823074i	$0.692257\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	2.85114	0.637534
$21$	−2.30066	−0.502045
$22$	−3.88254	−0.827761
$23$	−2.42384	−0.505405	−0.252702	−	0.967544i	$-0.581319\pi$
$23$	−2.42384	−0.505405	−0.252702	+	0.967544i	$0.581319\pi$
$24$	−1.00000	−0.204124
$25$	3.12899	0.625798
$26$	−5.31030	−1.04144
$27$	−1.00000	−0.192450
$28$	2.30066	0.434784
$29$	−0.677181	−0.125749	−0.0628747	−	0.998021i	$-0.520027\pi$
$29$	−0.677181	−0.125749	−0.0628747	+	0.998021i	$0.520027\pi$
$30$	−2.85114	−0.520544
$31$	1.10086	0.197721	0.0988604	−	0.995101i	$-0.468480\pi$
$31$	1.10086	0.197721	0.0988604	+	0.995101i	$0.468480\pi$
$32$	1.00000	0.176777
$33$	3.88254	0.675864
$34$	−4.68330	−0.803179
$35$	6.55950	1.10876
$36$	1.00000	0.166667
$37$	−0.702636	−0.115513	−0.0577563	−	0.998331i	$-0.518395\pi$
$37$	−0.702636	−0.115513	−0.0577563	+	0.998331i	$0.518395\pi$
$38$	−1.00000	−0.162221
$39$	5.31030	0.850329
$40$	2.85114	0.450805
$41$	−8.92591	−1.39399	−0.696996	−	0.717075i	$-0.745481\pi$
$41$	−8.92591	−1.39399	−0.696996	+	0.717075i	$0.745481\pi$
$42$	−2.30066	−0.354999
$43$	−4.76516	−0.726680	−0.363340	−	0.931657i	$-0.618364\pi$
$43$	−4.76516	−0.726680	−0.363340	+	0.931657i	$0.618364\pi$
$44$	−3.88254	−0.585315
$45$	2.85114	0.425023
$46$	−2.42384	−0.357375
$47$	−7.56937	−1.10411	−0.552053	−	0.833809i	$-0.686155\pi$
$47$	−7.56937	−1.10411	−0.552053	+	0.833809i	$0.686155\pi$
$48$	−1.00000	−0.144338
$49$	−1.70697	−0.243853
$50$	3.12899	0.442506
$51$	4.68330	0.655793
$52$	−5.31030	−0.736406
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	−1.00000	−0.136083
$55$	−11.0697	−1.49263
$56$	2.30066	0.307438
$57$	1.00000	0.132453
$58$	−0.677181	−0.0889183
$59$	−9.48144	−1.23438	−0.617189	−	0.786815i	$-0.711729\pi$
$59$	−9.48144	−1.23438	−0.617189	+	0.786815i	$0.711729\pi$
$60$	−2.85114	−0.368080
$61$	−12.2484	−1.56825	−0.784126	−	0.620602i	$-0.786888\pi$
$61$	−12.2484	−1.56825	−0.784126	+	0.620602i	$0.786888\pi$
$62$	1.10086	0.139810
$63$	2.30066	0.289856
$64$	1.00000	0.125000
$65$	−15.1404	−1.87794
$66$	3.88254	0.477908
$67$	7.32764	0.895214	0.447607	−	0.894230i	$-0.352276\pi$
$67$	7.32764	0.895214	0.447607	+	0.894230i	$0.352276\pi$
$68$	−4.68330	−0.567934
$69$	2.42384	0.291796
$70$	6.55950	0.784010
$71$	5.61549	0.666436	0.333218	−	0.942850i	$-0.391865\pi$
$71$	5.61549	0.666436	0.333218	+	0.942850i	$0.391865\pi$
$72$	1.00000	0.117851
$73$	9.90668	1.15949	0.579744	−	0.814798i	$-0.303152\pi$
$73$	9.90668	1.15949	0.579744	+	0.814798i	$0.303152\pi$
$74$	−0.702636	−0.0816798
$75$	−3.12899	−0.361305
$76$	−1.00000	−0.114708
$77$	−8.93241	−1.01794
$78$	5.31030	0.601273
$79$	−12.7705	−1.43679	−0.718395	−	0.695635i	$-0.755123\pi$
$79$	−12.7705	−1.43679	−0.718395	+	0.695635i	$0.755123\pi$
$80$	2.85114	0.318767
$81$	1.00000	0.111111
$82$	−8.92591	−0.985702
$83$	0.151868	0.0166697	0.00833485	−	0.999965i	$-0.497347\pi$
$83$	0.151868	0.0166697	0.00833485	+	0.999965i	$0.497347\pi$
$84$	−2.30066	−0.251022
$85$	−13.3527	−1.44831
$86$	−4.76516	−0.513841
$87$	0.677181	0.0726014
$88$	−3.88254	−0.413880
$89$	−7.05177	−0.747486	−0.373743	−	0.927532i	$-0.621926\pi$
$89$	−7.05177	−0.747486	−0.373743	+	0.927532i	$0.621926\pi$
$90$	2.85114	0.300536
$91$	−12.2172	−1.28071
$92$	−2.42384	−0.252702
$93$	−1.10086	−0.114154
$94$	−7.56937	−0.780720
$95$	−2.85114	−0.292521
$96$	−1.00000	−0.102062
$97$	10.6297	1.07928	0.539639	−	0.841897i	$-0.318561\pi$
$97$	10.6297	1.07928	0.539639	+	0.841897i	$0.318561\pi$
$98$	−1.70697	−0.172430
$99$	−3.88254	−0.390210
$100$	3.12899	0.312899
$101$	6.73722	0.670378	0.335189	−	0.942151i	$-0.391200\pi$
$101$	6.73722	0.670378	0.335189	+	0.942151i	$0.391200\pi$
$102$	4.68330	0.463716
$103$	0.873544	0.0860729	0.0430364	−	0.999074i	$-0.486297\pi$
$103$	0.873544	0.0860729	0.0430364	+	0.999074i	$0.486297\pi$
$104$	−5.31030	−0.520718
$105$	−6.55950	−0.640141
$106$	1.00000	0.0971286
$107$	11.3312	1.09543	0.547714	−	0.836665i	$-0.315498\pi$
$107$	11.3312	1.09543	0.547714	+	0.836665i	$0.315498\pi$
$108$	−1.00000	−0.0962250
$109$	18.8656	1.80699	0.903497	−	0.428595i	$-0.140991\pi$
$109$	18.8656	1.80699	0.903497	+	0.428595i	$0.140991\pi$
$110$	−11.0697	−1.05545
$111$	0.702636	0.0666913
$112$	2.30066	0.217392
$113$	−8.22979	−0.774194	−0.387097	−	0.922039i	$-0.626522\pi$
$113$	−8.22979	−0.774194	−0.387097	+	0.922039i	$0.626522\pi$
$114$	1.00000	0.0936586
$115$	−6.91069	−0.644425
$116$	−0.677181	−0.0628747
$117$	−5.31030	−0.490938
$118$	−9.48144	−0.872837
$119$	−10.7747	−0.987713
$120$	−2.85114	−0.260272
$121$	4.07413	0.370376
$122$	−12.2484	−1.10892
$123$	8.92591	0.804822
$124$	1.10086	0.0988604
$125$	−5.33450	−0.477133
$126$	2.30066	0.204959
$127$	5.82928	0.517265	0.258633	−	0.965976i	$-0.416728\pi$
$127$	5.82928	0.517265	0.258633	+	0.965976i	$0.416728\pi$
$128$	1.00000	0.0883883
$129$	4.76516	0.419549
$130$	−15.1404	−1.32790
$131$	−1.24132	−0.108455	−0.0542274	−	0.998529i	$-0.517270\pi$
$131$	−1.24132	−0.108455	−0.0542274	+	0.998529i	$0.517270\pi$
$132$	3.88254	0.337932
$133$	−2.30066	−0.199492
$134$	7.32764	0.633012
$135$	−2.85114	−0.245387
$136$	−4.68330	−0.401590
$137$	−1.78682	−0.152658	−0.0763291	−	0.997083i	$-0.524320\pi$
$137$	−1.78682	−0.152658	−0.0763291	+	0.997083i	$0.524320\pi$
$138$	2.42384	0.206331
$139$	0.785094	0.0665908	0.0332954	−	0.999446i	$-0.489400\pi$
$139$	0.785094	0.0665908	0.0332954	+	0.999446i	$0.489400\pi$
$140$	6.55950	0.554379
$141$	7.56937	0.637456
$142$	5.61549	0.471242
$143$	20.6175	1.72412
$144$	1.00000	0.0833333
$145$	−1.93074	−0.160339
$146$	9.90668	0.819882
$147$	1.70697	0.140788
$148$	−0.702636	−0.0577563
$149$	10.1111	0.828333	0.414166	−	0.910201i	$-0.364073\pi$
$149$	10.1111	0.828333	0.414166	+	0.910201i	$0.364073\pi$
$150$	−3.12899	−0.255481
$151$	−10.6022	−0.862797	−0.431398	−	0.902162i	$-0.641980\pi$
$151$	−10.6022	−0.862797	−0.431398	+	0.902162i	$0.641980\pi$
$152$	−1.00000	−0.0811107
$153$	−4.68330	−0.378622
$154$	−8.93241	−0.719794
$155$	3.13871	0.252108
$156$	5.31030	0.425164
$157$	−23.1208	−1.84524	−0.922620	−	0.385710i	$-0.873957\pi$
$157$	−23.1208	−1.84524	−0.922620	+	0.385710i	$0.873957\pi$
$158$	−12.7705	−1.01596
$159$	−1.00000	−0.0793052
$160$	2.85114	0.225402
$161$	−5.57642	−0.439483
$162$	1.00000	0.0785674
$163$	14.9841	1.17364	0.586821	−	0.809717i	$-0.300379\pi$
$163$	14.9841	1.17364	0.586821	+	0.809717i	$0.300379\pi$
$164$	−8.92591	−0.696996
$165$	11.0697	0.861772
$166$	0.151868	0.0117873
$167$	4.68395	0.362455	0.181227	−	0.983441i	$-0.441993\pi$
$167$	4.68395	0.362455	0.181227	+	0.983441i	$0.441993\pi$
$168$	−2.30066	−0.177500
$169$	15.1993	1.16918
$170$	−13.3527	−1.02411
$171$	−1.00000	−0.0764719
$172$	−4.76516	−0.363340
$173$	18.5127	1.40750	0.703748	−	0.710450i	$-0.251508\pi$
$173$	18.5127	1.40750	0.703748	+	0.710450i	$0.251508\pi$
$174$	0.677181	0.0513370
$175$	7.19874	0.544174
$176$	−3.88254	−0.292658
$177$	9.48144	0.712668
$178$	−7.05177	−0.528552
$179$	4.84724	0.362300	0.181150	−	0.983455i	$-0.442018\pi$
$179$	4.84724	0.362300	0.181150	+	0.983455i	$0.442018\pi$
$180$	2.85114	0.212511
$181$	7.29939	0.542559	0.271280	−	0.962501i	$-0.412553\pi$
$181$	7.29939	0.542559	0.271280	+	0.962501i	$0.412553\pi$
$182$	−12.2172	−0.905599
$183$	12.2484	0.905431
$184$	−2.42384	−0.178688
$185$	−2.00331	−0.147286
$186$	−1.10086	−0.0807192
$187$	18.1831	1.32968
$188$	−7.56937	−0.552053
$189$	−2.30066	−0.167348
$190$	−2.85114	−0.206843
$191$	8.59434	0.621865	0.310932	−	0.950432i	$-0.399359\pi$
$191$	8.59434	0.621865	0.310932	+	0.950432i	$0.399359\pi$
$192$	−1.00000	−0.0721688
$193$	13.8304	0.995531	0.497765	−	0.867312i	$-0.334154\pi$
$193$	13.8304	0.995531	0.497765	+	0.867312i	$0.334154\pi$
$194$	10.6297	0.763165
$195$	15.1404	1.08423
$196$	−1.70697	−0.121926
$197$	7.33878	0.522866	0.261433	−	0.965222i	$-0.415805\pi$
$197$	7.33878	0.522866	0.261433	+	0.965222i	$0.415805\pi$
$198$	−3.88254	−0.275920
$199$	14.0955	0.999202	0.499601	−	0.866256i	$-0.333480\pi$
$199$	14.0955	0.999202	0.499601	+	0.866256i	$0.333480\pi$
$200$	3.12899	0.221253
$201$	−7.32764	−0.516852
$202$	6.73722	0.474029
$203$	−1.55796	−0.109348
$204$	4.68330	0.327897
$205$	−25.4490	−1.77744
$206$	0.873544	0.0608627
$207$	−2.42384	−0.168468
$208$	−5.31030	−0.368203
$209$	3.88254	0.268561
$210$	−6.55950	−0.452648
$211$	18.3072	1.26032	0.630159	−	0.776466i	$-0.282990\pi$
$211$	18.3072	1.26032	0.630159	+	0.776466i	$0.282990\pi$
$212$	1.00000	0.0686803
$213$	−5.61549	−0.384767
$214$	11.3312	0.774585
$215$	−13.5861	−0.926567
$216$	−1.00000	−0.0680414
$217$	2.53271	0.171932
$218$	18.8656	1.27774
$219$	−9.90668	−0.669431
$220$	−11.0697	−0.746317
$221$	24.8697	1.67292
$222$	0.702636	0.0471578
$223$	9.37695	0.627927	0.313964	−	0.949435i	$-0.398343\pi$
$223$	9.37695	0.627927	0.313964	+	0.949435i	$0.398343\pi$
$224$	2.30066	0.153719
$225$	3.12899	0.208599
$226$	−8.22979	−0.547438
$227$	−17.5345	−1.16380	−0.581902	−	0.813259i	$-0.697691\pi$
$227$	−17.5345	−1.16380	−0.581902	+	0.813259i	$0.697691\pi$
$228$	1.00000	0.0662266
$229$	−29.1866	−1.92871	−0.964354	−	0.264617i	$-0.914754\pi$
$229$	−29.1866	−1.92871	−0.964354	+	0.264617i	$0.914754\pi$
$230$	−6.91069	−0.455678
$231$	8.93241	0.587709
$232$	−0.677181	−0.0444591
$233$	−16.8939	−1.10675	−0.553377	−	0.832931i	$-0.686661\pi$
$233$	−16.8939	−1.10675	−0.553377	+	0.832931i	$0.686661\pi$
$234$	−5.31030	−0.347145
$235$	−21.5813	−1.40781
$236$	−9.48144	−0.617189
$237$	12.7705	0.829532
$238$	−10.7747	−0.698419
$239$	10.0635	0.650956	0.325478	−	0.945550i	$-0.394475\pi$
$239$	10.0635	0.650956	0.325478	+	0.945550i	$0.394475\pi$
$240$	−2.85114	−0.184040
$241$	−9.54393	−0.614779	−0.307389	−	0.951584i	$-0.599455\pi$
$241$	−9.54393	−0.614779	−0.307389	+	0.951584i	$0.599455\pi$
$242$	4.07413	0.261895
$243$	−1.00000	−0.0641500
$244$	−12.2484	−0.784126
$245$	−4.86680	−0.310929
$246$	8.92591	0.569095
$247$	5.31030	0.337886
$248$	1.10086	0.0699049
$249$	−0.151868	−0.00962425
$250$	−5.33450	−0.337384
$251$	7.56184	0.477299	0.238650	−	0.971106i	$-0.423295\pi$
$251$	7.56184	0.477299	0.238650	+	0.971106i	$0.423295\pi$
$252$	2.30066	0.144928
$253$	9.41065	0.591642
$254$	5.82928	0.365762
$255$	13.3527	0.836181
$256$	1.00000	0.0625000
$257$	−10.8235	−0.675150	−0.337575	−	0.941299i	$-0.609607\pi$
$257$	−10.8235	−0.675150	−0.337575	+	0.941299i	$0.609607\pi$
$258$	4.76516	0.296666
$259$	−1.61653	−0.100446
$260$	−15.1404	−0.938968
$261$	−0.677181	−0.0419165
$262$	−1.24132	−0.0766891
$263$	−13.5275	−0.834144	−0.417072	−	0.908873i	$-0.636944\pi$
$263$	−13.5275	−0.834144	−0.417072	+	0.908873i	$0.636944\pi$
$264$	3.88254	0.238954
$265$	2.85114	0.175144
$266$	−2.30066	−0.141062
$267$	7.05177	0.431561
$268$	7.32764	0.447607
$269$	−13.5528	−0.826332	−0.413166	−	0.910656i	$-0.635577\pi$
$269$	−13.5528	−0.826332	−0.413166	+	0.910656i	$0.635577\pi$
$270$	−2.85114	−0.173515
$271$	14.7819	0.897935	0.448968	−	0.893548i	$-0.351792\pi$
$271$	14.7819	0.897935	0.448968	+	0.893548i	$0.351792\pi$
$272$	−4.68330	−0.283967
$273$	12.2172	0.739418
$274$	−1.78682	−0.107946
$275$	−12.1484	−0.732579
$276$	2.42384	0.145898
$277$	−31.9582	−1.92018	−0.960091	−	0.279688i	$-0.909769\pi$
$277$	−31.9582	−1.92018	−0.960091	+	0.279688i	$0.909769\pi$
$278$	0.785094	0.0470868
$279$	1.10086	0.0659070
$280$	6.55950	0.392005
$281$	0.0806046	0.00480847	0.00240423	−	0.999997i	$-0.499235\pi$
$281$	0.0806046	0.00480847	0.00240423	+	0.999997i	$0.499235\pi$
$282$	7.56937	0.450749
$283$	−2.92216	−0.173705	−0.0868523	−	0.996221i	$-0.527681\pi$
$283$	−2.92216	−0.173705	−0.0868523	+	0.996221i	$0.527681\pi$
$284$	5.61549	0.333218
$285$	2.85114	0.168887
$286$	20.6175	1.21914
$287$	−20.5355	−1.21217
$288$	1.00000	0.0589256
$289$	4.93331	0.290195
$290$	−1.93074	−0.113377
$291$	−10.6297	−0.623121
$292$	9.90668	0.579744
$293$	3.81546	0.222902	0.111451	−	0.993770i	$-0.464450\pi$
$293$	3.81546	0.222902	0.111451	+	0.993770i	$0.464450\pi$
$294$	1.70697	0.0995524
$295$	−27.0329	−1.57392
$296$	−0.702636	−0.0408399
$297$	3.88254	0.225288
$298$	10.1111	0.585720
$299$	12.8713	0.744367
$300$	−3.12899	−0.180652
$301$	−10.9630	−0.631897
$302$	−10.6022	−0.610089
$303$	−6.73722	−0.387043
$304$	−1.00000	−0.0573539
$305$	−34.9220	−1.99963
$306$	−4.68330	−0.267726
$307$	18.6224	1.06284	0.531419	−	0.847109i	$-0.321659\pi$
$307$	18.6224	1.06284	0.531419	+	0.847109i	$0.321659\pi$
$308$	−8.93241	−0.508971
$309$	−0.873544	−0.0496942
$310$	3.13871	0.178267
$311$	11.3928	0.646024	0.323012	−	0.946395i	$-0.395305\pi$
$311$	11.3928	0.646024	0.323012	+	0.946395i	$0.395305\pi$
$312$	5.31030	0.300637
$313$	−5.80697	−0.328229	−0.164115	−	0.986441i	$-0.552477\pi$
$313$	−5.80697	−0.328229	−0.164115	+	0.986441i	$0.552477\pi$
$314$	−23.1208	−1.30478
$315$	6.55950	0.369586
$316$	−12.7705	−0.718395
$317$	12.4078	0.696892	0.348446	−	0.937329i	$-0.386710\pi$
$317$	12.4078	0.696892	0.348446	+	0.937329i	$0.386710\pi$
$318$	−1.00000	−0.0560772
$319$	2.62918	0.147206
$320$	2.85114	0.159383
$321$	−11.3312	−0.632446
$322$	−5.57642	−0.310762
$323$	4.68330	0.260586
$324$	1.00000	0.0555556
$325$	−16.6159	−0.921684
$326$	14.9841	0.829890
$327$	−18.8656	−1.04327
$328$	−8.92591	−0.492851
$329$	−17.4145	−0.960094
$330$	11.0697	0.609365
$331$	22.7711	1.25161	0.625805	−	0.779979i	$-0.284770\pi$
$331$	22.7711	1.25161	0.625805	+	0.779979i	$0.284770\pi$
$332$	0.151868	0.00833485
$333$	−0.702636	−0.0385042
$334$	4.68395	0.256294
$335$	20.8921	1.14146
$336$	−2.30066	−0.125511
$337$	0.465355	0.0253495	0.0126747	−	0.999920i	$-0.495965\pi$
$337$	0.465355	0.0253495	0.0126747	+	0.999920i	$0.495965\pi$
$338$	15.1993	0.826733
$339$	8.22979	0.446981
$340$	−13.3527	−0.724154
$341$	−4.27415	−0.231458
$342$	−1.00000	−0.0540738
$343$	−20.0318	−1.08161
$344$	−4.76516	−0.256920
$345$	6.91069	0.372059
$346$	18.5127	0.995250
$347$	−5.31871	−0.285523	−0.142762	−	0.989757i	$-0.545598\pi$
$347$	−5.31871	−0.285523	−0.142762	+	0.989757i	$0.545598\pi$
$348$	0.677181	0.0363007
$349$	10.9375	0.585469	0.292734	−	0.956194i	$-0.405435\pi$
$349$	10.9375	0.585469	0.292734	+	0.956194i	$0.405435\pi$
$350$	7.19874	0.384789
$351$	5.31030	0.283443
$352$	−3.88254	−0.206940
$353$	21.0775	1.12184	0.560921	−	0.827869i	$-0.310447\pi$
$353$	21.0775	1.12184	0.560921	+	0.827869i	$0.310447\pi$
$354$	9.48144	0.503933
$355$	16.0105	0.849751
$356$	−7.05177	−0.373743
$357$	10.7747	0.570256
$358$	4.84724	0.256185
$359$	−22.1833	−1.17079	−0.585396	−	0.810748i	$-0.699061\pi$
$359$	−22.1833	−1.17079	−0.585396	+	0.810748i	$0.699061\pi$
$360$	2.85114	0.150268
$361$	1.00000	0.0526316
$362$	7.29939	0.383647
$363$	−4.07413	−0.213837
$364$	−12.2172	−0.640355
$365$	28.2453	1.47843
$366$	12.2484	0.640236
$367$	−12.5683	−0.656062	−0.328031	−	0.944667i	$-0.606385\pi$
$367$	−12.5683	−0.656062	−0.328031	+	0.944667i	$0.606385\pi$
$368$	−2.42384	−0.126351
$369$	−8.92591	−0.464664
$370$	−2.00331	−0.104147
$371$	2.30066	0.119444
$372$	−1.10086	−0.0570771
$373$	−4.87119	−0.252221	−0.126110	−	0.992016i	$-0.540249\pi$
$373$	−4.87119	−0.252221	−0.126110	+	0.992016i	$0.540249\pi$
$374$	18.1831	0.940226
$375$	5.33450	0.275473
$376$	−7.56937	−0.390360
$377$	3.59604	0.185205
$378$	−2.30066	−0.118333
$379$	−6.71567	−0.344961	−0.172480	−	0.985013i	$-0.555178\pi$
$379$	−6.71567	−0.344961	−0.172480	+	0.985013i	$0.555178\pi$
$380$	−2.85114	−0.146260
$381$	−5.82928	−0.298643
$382$	8.59434	0.439725
$383$	−13.6868	−0.699364	−0.349682	−	0.936869i	$-0.613710\pi$
$383$	−13.6868	−0.699364	−0.349682	+	0.936869i	$0.613710\pi$
$384$	−1.00000	−0.0510310
$385$	−25.4675	−1.29795
$386$	13.8304	0.703946
$387$	−4.76516	−0.242227
$388$	10.6297	0.539639
$389$	4.10300	0.208031	0.104015	−	0.994576i	$-0.466831\pi$
$389$	4.10300	0.208031	0.104015	+	0.994576i	$0.466831\pi$
$390$	15.1404	0.766664
$391$	11.3516	0.574073
$392$	−1.70697	−0.0862149
$393$	1.24132	0.0626164
$394$	7.33878	0.369722
$395$	−36.4104	−1.83201
$396$	−3.88254	−0.195105
$397$	−32.6562	−1.63897	−0.819484	−	0.573102i	$-0.805740\pi$
$397$	−32.6562	−1.63897	−0.819484	+	0.573102i	$0.805740\pi$
$398$	14.0955	0.706543
$399$	2.30066	0.115177
$400$	3.12899	0.156450
$401$	−10.3448	−0.516595	−0.258297	−	0.966065i	$-0.583161\pi$
$401$	−10.3448	−0.516595	−0.258297	+	0.966065i	$0.583161\pi$
$402$	−7.32764	−0.365469
$403$	−5.84592	−0.291206
$404$	6.73722	0.335189
$405$	2.85114	0.141674
$406$	−1.55796	−0.0773204
$407$	2.72801	0.135223
$408$	4.68330	0.231858
$409$	−13.0082	−0.643214	−0.321607	−	0.946873i	$-0.604223\pi$
$409$	−13.0082	−0.643214	−0.321607	+	0.946873i	$0.604223\pi$
$410$	−25.4490	−1.25684
$411$	1.78682	0.0881372
$412$	0.873544	0.0430364
$413$	−21.8135	−1.07337
$414$	−2.42384	−0.119125
$415$	0.432997	0.0212550
$416$	−5.31030	−0.260359
$417$	−0.785094	−0.0384462
$418$	3.88254	0.189901
$419$	2.23225	0.109053	0.0545263	−	0.998512i	$-0.482635\pi$
$419$	2.23225	0.109053	0.0545263	+	0.998512i	$0.482635\pi$
$420$	−6.55950	−0.320071
$421$	−14.9636	−0.729280	−0.364640	−	0.931149i	$-0.618808\pi$
$421$	−14.9636	−0.729280	−0.364640	+	0.931149i	$0.618808\pi$
$422$	18.3072	0.891179
$423$	−7.56937	−0.368035
$424$	1.00000	0.0485643
$425$	−14.6540	−0.710824
$426$	−5.61549	−0.272071
$427$	−28.1795	−1.36370
$428$	11.3312	0.547714
$429$	−20.6175	−0.995421
$430$	−13.5861	−0.655182
$431$	−4.69061	−0.225939	−0.112969	−	0.993598i	$-0.536036\pi$
$431$	−4.69061	−0.225939	−0.112969	+	0.993598i	$0.536036\pi$
$432$	−1.00000	−0.0481125
$433$	−6.79861	−0.326720	−0.163360	−	0.986567i	$-0.552233\pi$
$433$	−6.79861	−0.326720	−0.163360	+	0.986567i	$0.552233\pi$
$434$	2.53271	0.121574
$435$	1.93074	0.0925718
$436$	18.8656	0.903497
$437$	2.42384	0.115948
$438$	−9.90668	−0.473359
$439$	24.6535	1.17665	0.588324	−	0.808625i	$-0.299788\pi$
$439$	24.6535	1.17665	0.588324	+	0.808625i	$0.299788\pi$
$440$	−11.0697	−0.527726
$441$	−1.70697	−0.0812842
$442$	24.8697	1.18293
$443$	−32.8075	−1.55873	−0.779366	−	0.626568i	$-0.784459\pi$
$443$	−32.8075	−1.55873	−0.779366	+	0.626568i	$0.784459\pi$
$444$	0.702636	0.0333456
$445$	−20.1056	−0.953096
$446$	9.37695	0.444012
$447$	−10.1111	−0.478238
$448$	2.30066	0.108696
$449$	−27.1478	−1.28118	−0.640592	−	0.767881i	$-0.721311\pi$
$449$	−27.1478	−1.28118	−0.640592	+	0.767881i	$0.721311\pi$
$450$	3.12899	0.147502
$451$	34.6552	1.63185
$452$	−8.22979	−0.387097
$453$	10.6022	0.498136
$454$	−17.5345	−0.822934
$455$	−34.8329	−1.63299
$456$	1.00000	0.0468293
$457$	−31.5466	−1.47569	−0.737845	−	0.674970i	$-0.764156\pi$
$457$	−31.5466	−1.47569	−0.737845	+	0.674970i	$0.764156\pi$
$458$	−29.1866	−1.36380
$459$	4.68330	0.218598
$460$	−6.91069	−0.322213
$461$	5.80607	0.270416	0.135208	−	0.990817i	$-0.456830\pi$
$461$	5.80607	0.270416	0.135208	+	0.990817i	$0.456830\pi$
$462$	8.93241	0.415573
$463$	−11.5999	−0.539095	−0.269547	−	0.962987i	$-0.586874\pi$
$463$	−11.5999	−0.539095	−0.269547	+	0.962987i	$0.586874\pi$
$464$	−0.677181	−0.0314373
$465$	−3.13871	−0.145554
$466$	−16.8939	−0.782594
$467$	−0.942222	−0.0436008	−0.0218004	−	0.999762i	$-0.506940\pi$
$467$	−0.942222	−0.0436008	−0.0218004	+	0.999762i	$0.506940\pi$
$468$	−5.31030	−0.245469
$469$	16.8584	0.778449
$470$	−21.5813	−0.995472
$471$	23.1208	1.06535
$472$	−9.48144	−0.436418
$473$	18.5009	0.850674
$474$	12.7705	0.586567
$475$	−3.12899	−0.143568
$476$	−10.7747	−0.493857
$477$	1.00000	0.0457869
$478$	10.0635	0.460296
$479$	−18.3353	−0.837761	−0.418880	−	0.908041i	$-0.637577\pi$
$479$	−18.3353	−0.837761	−0.418880	+	0.908041i	$0.637577\pi$
$480$	−2.85114	−0.130136
$481$	3.73121	0.170128
$482$	−9.54393	−0.434714
$483$	5.57642	0.253736
$484$	4.07413	0.185188
$485$	30.3066	1.37615
$486$	−1.00000	−0.0453609
$487$	19.6082	0.888535	0.444267	−	0.895894i	$-0.353464\pi$
$487$	19.6082	0.888535	0.444267	+	0.895894i	$0.353464\pi$
$488$	−12.2484	−0.554461
$489$	−14.9841	−0.677602
$490$	−4.86680	−0.219860
$491$	9.64175	0.435126	0.217563	−	0.976046i	$-0.430189\pi$
$491$	9.64175	0.435126	0.217563	+	0.976046i	$0.430189\pi$
$492$	8.92591	0.402411
$493$	3.17144	0.142835
$494$	5.31030	0.238922
$495$	−11.0697	−0.497545
$496$	1.10086	0.0494302
$497$	12.9193	0.579511
$498$	−0.151868	−0.00680538
$499$	30.4665	1.36387	0.681934	−	0.731414i	$-0.261139\pi$
$499$	30.4665	1.36387	0.681934	+	0.731414i	$0.261139\pi$
$500$	−5.33450	−0.238566
$501$	−4.68395	−0.209263
$502$	7.56184	0.337502
$503$	2.09002	0.0931892	0.0465946	−	0.998914i	$-0.485163\pi$
$503$	2.09002	0.0931892	0.0465946	+	0.998914i	$0.485163\pi$
$504$	2.30066	0.102479
$505$	19.2087	0.854778
$506$	9.41065	0.418354
$507$	−15.1993	−0.675025
$508$	5.82928	0.258633
$509$	−38.1574	−1.69130	−0.845649	−	0.533740i	$-0.820786\pi$
$509$	−38.1574	−1.69130	−0.845649	+	0.533740i	$0.820786\pi$
$510$	13.3527	0.591269
$511$	22.7919	1.00825
$512$	1.00000	0.0441942
$513$	1.00000	0.0441511
$514$	−10.8235	−0.477403
$515$	2.49060	0.109749
$516$	4.76516	0.209775
$517$	29.3884	1.29250
$518$	−1.61653	−0.0710261
$519$	−18.5127	−0.812618
$520$	−15.1404	−0.663951
$521$	−4.91969	−0.215536	−0.107768	−	0.994176i	$-0.534370\pi$
$521$	−4.91969	−0.215536	−0.107768	+	0.994176i	$0.534370\pi$
$522$	−0.677181	−0.0296394
$523$	7.51569	0.328638	0.164319	−	0.986407i	$-0.447457\pi$
$523$	7.51569	0.328638	0.164319	+	0.986407i	$0.447457\pi$
$524$	−1.24132	−0.0542274
$525$	−7.19874	−0.314179
$526$	−13.5275	−0.589829
$527$	−5.15567	−0.224585
$528$	3.88254	0.168966
$529$	−17.1250	−0.744566
$530$	2.85114	0.123846
$531$	−9.48144	−0.411459
$532$	−2.30066	−0.0997462
$533$	47.3993	2.05309
$534$	7.05177	0.305160
$535$	32.3068	1.39675
$536$	7.32764	0.316506
$537$	−4.84724	−0.209174
$538$	−13.5528	−0.584305
$539$	6.62738	0.285461
$540$	−2.85114	−0.122693
$541$	41.4252	1.78101	0.890505	−	0.454974i	$-0.150352\pi$
$541$	41.4252	1.78101	0.890505	+	0.454974i	$0.150352\pi$
$542$	14.7819	0.634936
$543$	−7.29939	−0.313247
$544$	−4.68330	−0.200795
$545$	53.7883	2.30404
$546$	12.2172	0.522848
$547$	−35.7937	−1.53043	−0.765214	−	0.643776i	$-0.777367\pi$
$547$	−35.7937	−1.53043	−0.765214	+	0.643776i	$0.777367\pi$
$548$	−1.78682	−0.0763291
$549$	−12.2484	−0.522751
$550$	−12.1484	−0.518011
$551$	0.677181	0.0288489
$552$	2.42384	0.103165
$553$	−29.3805	−1.24939
$554$	−31.9582	−1.35777
$555$	2.00331	0.0850359
$556$	0.785094	0.0332954
$557$	−34.9067	−1.47905	−0.739523	−	0.673131i	$-0.764949\pi$
$557$	−34.9067	−1.47905	−0.739523	+	0.673131i	$0.764949\pi$
$558$	1.10086	0.0466033
$559$	25.3044	1.07026
$560$	6.55950	0.277189
$561$	−18.1831	−0.767692
$562$	0.0806046	0.00340010
$563$	37.9790	1.60062	0.800312	−	0.599583i	$-0.204667\pi$
$563$	37.9790	1.60062	0.800312	+	0.599583i	$0.204667\pi$
$564$	7.56937	0.318728
$565$	−23.4643	−0.987150
$566$	−2.92216	−0.122828
$567$	2.30066	0.0966186
$568$	5.61549	0.235621
$569$	−1.27055	−0.0532644	−0.0266322	−	0.999645i	$-0.508478\pi$
$569$	−1.27055	−0.0532644	−0.0266322	+	0.999645i	$0.508478\pi$
$570$	2.85114	0.119421
$571$	8.82843	0.369458	0.184729	−	0.982789i	$-0.440859\pi$
$571$	8.82843	0.369458	0.184729	+	0.982789i	$0.440859\pi$
$572$	20.6175	0.862060
$573$	−8.59434	−0.359034
$574$	−20.5355	−0.857134
$575$	−7.58416	−0.316281
$576$	1.00000	0.0416667
$577$	−45.8078	−1.90700	−0.953501	−	0.301390i	$-0.902550\pi$
$577$	−45.8078	−1.90700	−0.953501	+	0.301390i	$0.902550\pi$
$578$	4.93331	0.205199
$579$	−13.8304	−0.574770
$580$	−1.93074	−0.0801695
$581$	0.349397	0.0144954
$582$	−10.6297	−0.440613
$583$	−3.88254	−0.160798
$584$	9.90668	0.409941
$585$	−15.1404	−0.625979
$586$	3.81546	0.157615
$587$	31.6332	1.30564	0.652821	−	0.757512i	$-0.273585\pi$
$587$	31.6332	1.30564	0.652821	+	0.757512i	$0.273585\pi$
$588$	1.70697	0.0703942
$589$	−1.10086	−0.0453603
$590$	−27.0329	−1.11293
$591$	−7.33878	−0.301877
$592$	−0.702636	−0.0288782
$593$	−42.0943	−1.72860	−0.864302	−	0.502973i	$-0.832240\pi$
$593$	−42.0943	−1.72860	−0.864302	+	0.502973i	$0.832240\pi$
$594$	3.88254	0.159303
$595$	−30.7201	−1.25940
$596$	10.1111	0.414166
$597$	−14.0955	−0.576890
$598$	12.8713	0.526347
$599$	−1.31918	−0.0539002	−0.0269501	−	0.999637i	$-0.508580\pi$
$599$	−1.31918	−0.0539002	−0.0269501	+	0.999637i	$0.508580\pi$
$600$	−3.12899	−0.127741
$601$	−27.1549	−1.10767	−0.553836	−	0.832626i	$-0.686837\pi$
$601$	−27.1549	−1.10767	−0.553836	+	0.832626i	$0.686837\pi$
$602$	−10.9630	−0.446819
$603$	7.32764	0.298405
$604$	−10.6022	−0.431398
$605$	11.6159	0.472254
$606$	−6.73722	−0.273681
$607$	14.1378	0.573835	0.286917	−	0.957955i	$-0.407369\pi$
$607$	14.1378	0.573835	0.286917	+	0.957955i	$0.407369\pi$
$608$	−1.00000	−0.0405554
$609$	1.55796	0.0631318
$610$	−34.9220	−1.41395
$611$	40.1956	1.62614
$612$	−4.68330	−0.189311
$613$	35.7472	1.44382	0.721908	−	0.691989i	$-0.243265\pi$
$613$	35.7472	1.44382	0.721908	+	0.691989i	$0.243265\pi$
$614$	18.6224	0.751540
$615$	25.4490	1.02620
$616$	−8.93241	−0.359897
$617$	−5.73794	−0.231001	−0.115500	−	0.993307i	$-0.536847\pi$
$617$	−5.73794	−0.231001	−0.115500	+	0.993307i	$0.536847\pi$
$618$	−0.873544	−0.0351391
$619$	−13.3656	−0.537210	−0.268605	−	0.963250i	$-0.586563\pi$
$619$	−13.3656	−0.537210	−0.268605	+	0.963250i	$0.586563\pi$
$620$	3.13871	0.126054
$621$	2.42384	0.0972652
$622$	11.3928	0.456808
$623$	−16.2237	−0.649989
$624$	5.31030	0.212582
$625$	−30.8544	−1.23417
$626$	−5.80697	−0.232093
$627$	−3.88254	−0.155054
$628$	−23.1208	−0.922620
$629$	3.29066	0.131207
$630$	6.55950	0.261337
$631$	−7.59717	−0.302439	−0.151219	−	0.988500i	$-0.548320\pi$
$631$	−7.59717	−0.302439	−0.151219	+	0.988500i	$0.548320\pi$
$632$	−12.7705	−0.507982
$633$	−18.3072	−0.727644
$634$	12.4078	0.492777
$635$	16.6201	0.659548
$636$	−1.00000	−0.0396526
$637$	9.06452	0.359149
$638$	2.62918	0.104090
$639$	5.61549	0.222145
$640$	2.85114	0.112701
$641$	30.0169	1.18560	0.592798	−	0.805351i	$-0.298023\pi$
$641$	30.0169	1.18560	0.592798	+	0.805351i	$0.298023\pi$
$642$	−11.3312	−0.447207
$643$	−40.1691	−1.58412	−0.792058	−	0.610446i	$-0.790990\pi$
$643$	−40.1691	−1.58412	−0.792058	+	0.610446i	$0.790990\pi$
$644$	−5.57642	−0.219742
$645$	13.5861	0.534954
$646$	4.68330	0.184262
$647$	27.4077	1.07751	0.538754	−	0.842463i	$-0.318895\pi$
$647$	27.4077	1.07751	0.538754	+	0.842463i	$0.318895\pi$
$648$	1.00000	0.0392837
$649$	36.8121	1.44500
$650$	−16.6159	−0.651729
$651$	−2.53271	−0.0992648
$652$	14.9841	0.586821
$653$	−28.0572	−1.09796	−0.548981	−	0.835835i	$-0.684984\pi$
$653$	−28.0572	−1.09796	−0.548981	+	0.835835i	$0.684984\pi$
$654$	−18.8656	−0.737702
$655$	−3.53918	−0.138287
$656$	−8.92591	−0.348498
$657$	9.90668	0.386496
$658$	−17.4145	−0.678889
$659$	−37.8346	−1.47383	−0.736913	−	0.675988i	$-0.763717\pi$
$659$	−37.8346	−1.47383	−0.736913	+	0.675988i	$0.763717\pi$
$660$	11.0697	0.430886
$661$	13.2294	0.514563	0.257282	−	0.966336i	$-0.417173\pi$
$661$	13.2294	0.514563	0.257282	+	0.966336i	$0.417173\pi$
$662$	22.7711	0.885022
$663$	−24.8697	−0.965861
$664$	0.151868	0.00589363
$665$	−6.55950	−0.254366
$666$	−0.702636	−0.0272266
$667$	1.64138	0.0635543
$668$	4.68395	0.181227
$669$	−9.37695	−0.362534
$670$	20.8921	0.807133
$671$	47.5551	1.83584
$672$	−2.30066	−0.0887498
$673$	36.7905	1.41817	0.709085	−	0.705123i	$-0.249108\pi$
$673$	36.7905	1.41817	0.709085	+	0.705123i	$0.249108\pi$
$674$	0.465355	0.0179248
$675$	−3.12899	−0.120435
$676$	15.1993	0.584588
$677$	32.5353	1.25043	0.625216	−	0.780452i	$-0.285011\pi$
$677$	32.5353	1.25043	0.625216	+	0.780452i	$0.285011\pi$
$678$	8.22979	0.316063
$679$	24.4552	0.938505
$680$	−13.3527	−0.512054
$681$	17.5345	0.671923
$682$	−4.27415	−0.163666
$683$	32.2959	1.23577	0.617884	−	0.786269i	$-0.287990\pi$
$683$	32.2959	1.23577	0.617884	+	0.786269i	$0.287990\pi$
$684$	−1.00000	−0.0382360
$685$	−5.09447	−0.194650
$686$	−20.0318	−0.764816
$687$	29.1866	1.11354
$688$	−4.76516	−0.181670
$689$	−5.31030	−0.202306
$690$	6.91069	0.263086
$691$	30.6238	1.16499	0.582493	−	0.812836i	$-0.302078\pi$
$691$	30.6238	1.16499	0.582493	+	0.812836i	$0.302078\pi$
$692$	18.5127	0.703748
$693$	−8.93241	−0.339314
$694$	−5.31871	−0.201895
$695$	2.23841	0.0849078
$696$	0.677181	0.0256685
$697$	41.8027	1.58339
$698$	10.9375	0.413989
$699$	16.8939	0.638985
$700$	7.19874	0.272087
$701$	−22.5901	−0.853217	−0.426609	−	0.904436i	$-0.640292\pi$
$701$	−22.5901	−0.853217	−0.426609	+	0.904436i	$0.640292\pi$
$702$	5.31030	0.200424
$703$	0.702636	0.0265004
$704$	−3.88254	−0.146329
$705$	21.5813	0.812799
$706$	21.0775	0.793262
$707$	15.5000	0.582939
$708$	9.48144	0.356334
$709$	18.7148	0.702849	0.351424	−	0.936216i	$-0.385697\pi$
$709$	18.7148	0.702849	0.351424	+	0.936216i	$0.385697\pi$
$710$	16.0105	0.600865
$711$	−12.7705	−0.478930
$712$	−7.05177	−0.264276
$713$	−2.66831	−0.0999291
$714$	10.7747	0.403232
$715$	58.7833	2.19837
$716$	4.84724	0.181150
$717$	−10.0635	−0.375830
$718$	−22.1833	−0.827875
$719$	26.7757	0.998564	0.499282	−	0.866439i	$-0.333597\pi$
$719$	26.7757	0.998564	0.499282	+	0.866439i	$0.333597\pi$
$720$	2.85114	0.106256
$721$	2.00973	0.0748462
$722$	1.00000	0.0372161
$723$	9.54393	0.354943
$724$	7.29939	0.271280
$725$	−2.11889	−0.0786938
$726$	−4.07413	−0.151205
$727$	−13.0044	−0.482308	−0.241154	−	0.970487i	$-0.577526\pi$
$727$	−13.0044	−0.482308	−0.241154	+	0.970487i	$0.577526\pi$
$728$	−12.2172	−0.452799
$729$	1.00000	0.0370370
$730$	28.2453	1.04541
$731$	22.3167	0.825412
$732$	12.2484	0.452715
$733$	14.1317	0.521965	0.260982	−	0.965344i	$-0.415954\pi$
$733$	14.1317	0.521965	0.260982	+	0.965344i	$0.415954\pi$
$734$	−12.5683	−0.463906
$735$	4.86680	0.179515
$736$	−2.42384	−0.0893438
$737$	−28.4499	−1.04796
$738$	−8.92591	−0.328567
$739$	7.76155	0.285513	0.142757	−	0.989758i	$-0.454403\pi$
$739$	7.76155	0.285513	0.142757	+	0.989758i	$0.454403\pi$
$740$	−2.00331	−0.0736432
$741$	−5.31030	−0.195079
$742$	2.30066	0.0844598
$743$	−40.6844	−1.49257	−0.746284	−	0.665628i	$-0.768164\pi$
$743$	−40.6844	−1.49257	−0.746284	+	0.665628i	$0.768164\pi$
$744$	−1.10086	−0.0403596
$745$	28.8281	1.05618
$746$	−4.87119	−0.178347
$747$	0.151868	0.00555657
$748$	18.1831	0.664840
$749$	26.0692	0.952549
$750$	5.33450	0.194789
$751$	35.0812	1.28013	0.640066	−	0.768320i	$-0.278907\pi$
$751$	35.0812	1.28013	0.640066	+	0.768320i	$0.278907\pi$
$752$	−7.56937	−0.276026
$753$	−7.56184	−0.275569
$754$	3.59604	0.130960
$755$	−30.2284	−1.10012
$756$	−2.30066	−0.0836742
$757$	10.2440	0.372323	0.186162	−	0.982519i	$-0.440395\pi$
$757$	10.2440	0.372323	0.186162	+	0.982519i	$0.440395\pi$
$758$	−6.71567	−0.243924
$759$	−9.41065	−0.341585
$760$	−2.85114	−0.103422
$761$	0.527779	0.0191320	0.00956599	−	0.999954i	$-0.496955\pi$
$761$	0.527779	0.0191320	0.00956599	+	0.999954i	$0.496955\pi$
$762$	−5.82928	−0.211173
$763$	43.4032	1.57130
$764$	8.59434	0.310932
$765$	−13.3527	−0.482769
$766$	−13.6868	−0.494525
$767$	50.3493	1.81801
$768$	−1.00000	−0.0360844
$769$	−40.6303	−1.46516	−0.732582	−	0.680678i	$-0.761685\pi$
$769$	−40.6303	−1.46516	−0.732582	+	0.680678i	$0.761685\pi$
$770$	−25.4675	−0.917786
$771$	10.8235	0.389798
$772$	13.8304	0.497765
$773$	−11.2000	−0.402837	−0.201419	−	0.979505i	$-0.564555\pi$
$773$	−11.2000	−0.402837	−0.201419	+	0.979505i	$0.564555\pi$
$774$	−4.76516	−0.171280
$775$	3.44459	0.123733
$776$	10.6297	0.381582
$777$	1.61653	0.0579925
$778$	4.10300	0.147100
$779$	8.92591	0.319804
$780$	15.1404	0.542113
$781$	−21.8024	−0.780151
$782$	11.3516	0.405931
$783$	0.677181	0.0242005
$784$	−1.70697	−0.0609632
$785$	−65.9206	−2.35281
$786$	1.24132	0.0442764
$787$	−9.02778	−0.321806	−0.160903	−	0.986970i	$-0.551441\pi$
$787$	−9.02778	−0.321806	−0.160903	+	0.986970i	$0.551441\pi$
$788$	7.33878	0.261433
$789$	13.5275	0.481593
$790$	−36.4104	−1.29542
$791$	−18.9340	−0.673214
$792$	−3.88254	−0.137960
$793$	65.0429	2.30974
$794$	−32.6562	−1.15893
$795$	−2.85114	−0.101119
$796$	14.0955	0.499601
$797$	16.2217	0.574603	0.287302	−	0.957840i	$-0.407242\pi$
$797$	16.2217	0.574603	0.287302	+	0.957840i	$0.407242\pi$
$798$	2.30066	0.0814424
$799$	35.4496	1.25412
$800$	3.12899	0.110627
$801$	−7.05177	−0.249162
$802$	−10.3448	−0.365288
$803$	−38.4631	−1.35733
$804$	−7.32764	−0.258426
$805$	−15.8991	−0.560371
$806$	−5.84592	−0.205914
$807$	13.5528	0.477083
$808$	6.73722	0.237015
$809$	−13.1688	−0.462991	−0.231496	−	0.972836i	$-0.574362\pi$
$809$	−13.1688	−0.462991	−0.231496	+	0.972836i	$0.574362\pi$
$810$	2.85114	0.100179
$811$	48.3600	1.69815	0.849074	−	0.528274i	$-0.177161\pi$
$811$	48.3600	1.69815	0.849074	+	0.528274i	$0.177161\pi$
$812$	−1.55796	−0.0546738
$813$	−14.7819	−0.518423
$814$	2.72801	0.0956168
$815$	42.7216	1.49647
$816$	4.68330	0.163948
$817$	4.76516	0.166712
$818$	−13.0082	−0.454821
$819$	−12.2172	−0.426903
$820$	−25.4490	−0.888718
$821$	−11.3561	−0.396330	−0.198165	−	0.980169i	$-0.563498\pi$
$821$	−11.3561	−0.396330	−0.198165	+	0.980169i	$0.563498\pi$
$822$	1.78682	0.0623224
$823$	−18.9478	−0.660477	−0.330239	−	0.943897i	$-0.607129\pi$
$823$	−18.9478	−0.660477	−0.330239	+	0.943897i	$0.607129\pi$
$824$	0.873544	0.0304314
$825$	12.1484	0.422954
$826$	−21.8135	−0.758990
$827$	−20.3954	−0.709218	−0.354609	−	0.935015i	$-0.615386\pi$
$827$	−20.3954	−0.709218	−0.354609	+	0.935015i	$0.615386\pi$
$828$	−2.42384	−0.0842341
$829$	−32.9060	−1.14287	−0.571437	−	0.820646i	$-0.693614\pi$
$829$	−32.9060	−1.14287	−0.571437	+	0.820646i	$0.693614\pi$
$830$	0.432997	0.0150296
$831$	31.9582	1.10862
$832$	−5.31030	−0.184102
$833$	7.99425	0.276984
$834$	−0.785094	−0.0271856
$835$	13.3546	0.462154
$836$	3.88254	0.134281
$837$	−1.10086	−0.0380514
$838$	2.23225	0.0771119
$839$	−57.8458	−1.99706	−0.998529	−	0.0542114i	$-0.982736\pi$
$839$	−57.8458	−1.99706	−0.998529	+	0.0542114i	$0.982736\pi$
$840$	−6.55950	−0.226324
$841$	−28.5414	−0.984187
$842$	−14.9636	−0.515679
$843$	−0.0806046	−0.00277617
$844$	18.3072	0.630159
$845$	43.3353	1.49078
$846$	−7.56937	−0.260240
$847$	9.37319	0.322067
$848$	1.00000	0.0343401
$849$	2.92216	0.100288
$850$	−14.6540	−0.502628
$851$	1.70307	0.0583806
$852$	−5.61549	−0.192384
$853$	26.1193	0.894309	0.447154	−	0.894457i	$-0.352437\pi$
$853$	26.1193	0.894309	0.447154	+	0.894457i	$0.352437\pi$
$854$	−28.1795	−0.964282
$855$	−2.85114	−0.0975069
$856$	11.3312	0.387293
$857$	9.36706	0.319973	0.159986	−	0.987119i	$-0.448855\pi$
$857$	9.36706	0.319973	0.159986	+	0.987119i	$0.448855\pi$
$858$	−20.6175	−0.703869
$859$	20.4256	0.696914	0.348457	−	0.937325i	$-0.386706\pi$
$859$	20.4256	0.696914	0.348457	+	0.937325i	$0.386706\pi$
$860$	−13.5861	−0.463283
$861$	20.5355	0.699847
$862$	−4.69061	−0.159763
$863$	−21.7622	−0.740795	−0.370397	−	0.928873i	$-0.620778\pi$
$863$	−21.7622	−0.740795	−0.370397	+	0.928873i	$0.620778\pi$
$864$	−1.00000	−0.0340207
$865$	52.7823	1.79465
$866$	−6.79861	−0.231026
$867$	−4.93331	−0.167544
$868$	2.53271	0.0859658
$869$	49.5819	1.68195
$870$	1.93074	0.0654581
$871$	−38.9120	−1.31848
$872$	18.8656	0.638869
$873$	10.6297	0.359759
$874$	2.42384	0.0819875
$875$	−12.2729	−0.414899
$876$	−9.90668	−0.334716
$877$	27.3987	0.925188	0.462594	−	0.886570i	$-0.346919\pi$
$877$	27.3987	0.925188	0.462594	+	0.886570i	$0.346919\pi$
$878$	24.6535	0.832016
$879$	−3.81546	−0.128692
$880$	−11.0697	−0.373158
$881$	3.73553	0.125853	0.0629267	−	0.998018i	$-0.479957\pi$
$881$	3.73553	0.125853	0.0629267	+	0.998018i	$0.479957\pi$
$882$	−1.70697	−0.0574766
$883$	5.34850	0.179991	0.0899957	−	0.995942i	$-0.471315\pi$
$883$	5.34850	0.179991	0.0899957	+	0.995942i	$0.471315\pi$
$884$	24.8697	0.836460
$885$	27.0329	0.908701
$886$	−32.8075	−1.10219
$887$	−54.6714	−1.83569	−0.917843	−	0.396943i	$-0.870071\pi$
$887$	−54.6714	−1.83569	−0.917843	+	0.396943i	$0.870071\pi$
$888$	0.702636	0.0235789
$889$	13.4112	0.449797
$890$	−20.1056	−0.673940
$891$	−3.88254	−0.130070
$892$	9.37695	0.313964
$893$	7.56937	0.253299
$894$	−10.1111	−0.338166
$895$	13.8202	0.461957
$896$	2.30066	0.0768596
$897$	−12.8713	−0.429760
$898$	−27.1478	−0.905934
$899$	−0.745484	−0.0248633
$900$	3.12899	0.104300
$901$	−4.68330	−0.156023
$902$	34.6552	1.15389
$903$	10.9630	0.364826
$904$	−8.22979	−0.273719
$905$	20.8116	0.691800
$906$	10.6022	0.352235
$907$	−24.6121	−0.817232	−0.408616	−	0.912706i	$-0.633989\pi$
$907$	−24.6121	−0.817232	−0.408616	+	0.912706i	$0.633989\pi$
$908$	−17.5345	−0.581902
$909$	6.73722	0.223459
$910$	−34.8329	−1.15470
$911$	1.13711	0.0376742	0.0188371	−	0.999823i	$-0.494004\pi$
$911$	1.13711	0.0376742	0.0188371	+	0.999823i	$0.494004\pi$
$912$	1.00000	0.0331133
$913$	−0.589635	−0.0195141
$914$	−31.5466	−1.04347
$915$	34.9220	1.15449
$916$	−29.1866	−0.964354
$917$	−2.85586	−0.0943087
$918$	4.68330	0.154572
$919$	33.4073	1.10200	0.551002	−	0.834504i	$-0.314246\pi$
$919$	33.4073	1.10200	0.551002	+	0.834504i	$0.314246\pi$
$920$	−6.91069	−0.227839
$921$	−18.6224	−0.613630
$922$	5.80607	0.191213
$923$	−29.8200	−0.981536
$924$	8.93241	0.293855
$925$	−2.19854	−0.0722876
$926$	−11.5999	−0.381198
$927$	0.873544	0.0286910
$928$	−0.677181	−0.0222296
$929$	−23.2940	−0.764252	−0.382126	−	0.924110i	$-0.624808\pi$
$929$	−23.2940	−0.764252	−0.382126	+	0.924110i	$0.624808\pi$
$930$	−3.13871	−0.102922
$931$	1.70697	0.0559436
$932$	−16.8939	−0.553377
$933$	−11.3928	−0.372982
$934$	−0.942222	−0.0308304
$935$	51.8426	1.69543
$936$	−5.31030	−0.173573
$937$	−0.514009	−0.0167919	−0.00839597	−	0.999965i	$-0.502673\pi$
$937$	−0.514009	−0.0167919	−0.00839597	+	0.999965i	$0.502673\pi$
$938$	16.8584	0.550446
$939$	5.80697	0.189503
$940$	−21.5813	−0.703905
$941$	−36.9378	−1.20414	−0.602069	−	0.798444i	$-0.705657\pi$
$941$	−36.9378	−1.20414	−0.602069	+	0.798444i	$0.705657\pi$
$942$	23.1208	0.753316
$943$	21.6349	0.704531
$944$	−9.48144	−0.308594
$945$	−6.55950	−0.213380
$946$	18.5009	0.601517
$947$	46.3952	1.50764	0.753820	−	0.657081i	$-0.228209\pi$
$947$	46.3952	1.50764	0.753820	+	0.657081i	$0.228209\pi$
$948$	12.7705	0.414766
$949$	−52.6074	−1.70771
$950$	−3.12899	−0.101518
$951$	−12.4078	−0.402351
$952$	−10.7747	−0.349209
$953$	−49.8567	−1.61502	−0.807509	−	0.589856i	$-0.799185\pi$
$953$	−49.8567	−1.61502	−0.807509	+	0.589856i	$0.799185\pi$
$954$	1.00000	0.0323762
$955$	24.5037	0.792920
$956$	10.0635	0.325478
$957$	−2.62918	−0.0849895
$958$	−18.3353	−0.592386
$959$	−4.11086	−0.132747
$960$	−2.85114	−0.0920201
$961$	−29.7881	−0.960906
$962$	3.73121	0.120299
$963$	11.3312	0.365143
$964$	−9.54393	−0.307389
$965$	39.4323	1.26937
$966$	5.57642	0.179418
$967$	−0.275279	−0.00885236	−0.00442618	−	0.999990i	$-0.501409\pi$
$967$	−0.275279	−0.00885236	−0.00442618	+	0.999990i	$0.501409\pi$
$968$	4.07413	0.130948
$969$	−4.68330	−0.150449
$970$	30.3066	0.973087
$971$	−33.9332	−1.08897	−0.544484	−	0.838771i	$-0.683275\pi$
$971$	−33.9332	−1.08897	−0.544484	+	0.838771i	$0.683275\pi$
$972$	−1.00000	−0.0320750
$973$	1.80623	0.0579052
$974$	19.6082	0.628289
$975$	16.6159	0.532134
$976$	−12.2484	−0.392063
$977$	24.7550	0.791983	0.395992	−	0.918254i	$-0.370401\pi$
$977$	24.7550	0.791983	0.395992	+	0.918254i	$0.370401\pi$
$978$	−14.9841	−0.479137
$979$	27.3788	0.875030
$980$	−4.86680	−0.155464
$981$	18.8656	0.602331
$982$	9.64175	0.307681
$983$	−8.37078	−0.266986	−0.133493	−	0.991050i	$-0.542619\pi$
$983$	−8.37078	−0.266986	−0.133493	+	0.991050i	$0.542619\pi$
$984$	8.92591	0.284548
$985$	20.9239	0.666690
$986$	3.17144	0.100999
$987$	17.4145	0.554311
$988$	5.31030	0.168943
$989$	11.5500	0.367268
$990$	−11.0697	−0.351817
$991$	38.5950	1.22601	0.613005	−	0.790079i	$-0.289961\pi$
$991$	38.5950	1.22601	0.613005	+	0.790079i	$0.289961\pi$
$992$	1.10086	0.0349524
$993$	−22.7711	−0.722618
$994$	12.9193	0.409776
$995$	40.1882	1.27405
$996$	−0.151868	−0.00481213
$997$	−31.7646	−1.00600	−0.502998	−	0.864288i	$-0.667770\pi$
$997$	−31.7646	−1.00600	−0.502998	+	0.864288i	$0.667770\pi$
$998$	30.4665	0.964400
$999$	0.702636	0.0222304

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.bb.1.9	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.bb.1.9	✓	9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.85114 q^{5} -1.00000 q^{6} +2.30066 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.85114 q^{5} -1.00000 q^{6} +2.30066 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.85114 q^{10} -3.88254 q^{11} -1.00000 q^{12} -5.31030 q^{13} +2.30066 q^{14} -2.85114 q^{15} +1.00000 q^{16} -4.68330 q^{17} +1.00000 q^{18} -1.00000 q^{19} +2.85114 q^{20} -2.30066 q^{21} -3.88254 q^{22} -2.42384 q^{23} -1.00000 q^{24} +3.12899 q^{25} -5.31030 q^{26} -1.00000 q^{27} +2.30066 q^{28} -0.677181 q^{29} -2.85114 q^{30} +1.10086 q^{31} +1.00000 q^{32} +3.88254 q^{33} -4.68330 q^{34} +6.55950 q^{35} +1.00000 q^{36} -0.702636 q^{37} -1.00000 q^{38} +5.31030 q^{39} +2.85114 q^{40} -8.92591 q^{41} -2.30066 q^{42} -4.76516 q^{43} -3.88254 q^{44} +2.85114 q^{45} -2.42384 q^{46} -7.56937 q^{47} -1.00000 q^{48} -1.70697 q^{49} +3.12899 q^{50} +4.68330 q^{51} -5.31030 q^{52} +1.00000 q^{53} -1.00000 q^{54} -11.0697 q^{55} +2.30066 q^{56} +1.00000 q^{57} -0.677181 q^{58} -9.48144 q^{59} -2.85114 q^{60} -12.2484 q^{61} +1.10086 q^{62} +2.30066 q^{63} +1.00000 q^{64} -15.1404 q^{65} +3.88254 q^{66} +7.32764 q^{67} -4.68330 q^{68} +2.42384 q^{69} +6.55950 q^{70} +5.61549 q^{71} +1.00000 q^{72} +9.90668 q^{73} -0.702636 q^{74} -3.12899 q^{75} -1.00000 q^{76} -8.93241 q^{77} +5.31030 q^{78} -12.7705 q^{79} +2.85114 q^{80} +1.00000 q^{81} -8.92591 q^{82} +0.151868 q^{83} -2.30066 q^{84} -13.3527 q^{85} -4.76516 q^{86} +0.677181 q^{87} -3.88254 q^{88} -7.05177 q^{89} +2.85114 q^{90} -12.2172 q^{91} -2.42384 q^{92} -1.10086 q^{93} -7.56937 q^{94} -2.85114 q^{95} -1.00000 q^{96} +10.6297 q^{97} -1.70697 q^{98} -3.88254 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 5 q^{5} - 9 q^{6} - 5 q^{7} + 9 q^{8} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 - 5 * q^5 - 9 * q^6 - 5 * q^7 + 9 * q^8 + 9 * q^9	\( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 5 q^{5} - 9 q^{6} - 5 q^{7} + 9 q^{8} + 9 q^{9} - 5 q^{10} + 3 q^{11} - 9 q^{12} - 4 q^{13} - 5 q^{14} + 5 q^{15} + 9 q^{16} - 16 q^{17} + 9 q^{18} - 9 q^{19} - 5 q^{20} + 5 q^{21} + 3 q^{22} - 9 q^{24} + 8 q^{25} - 4 q^{26} - 9 q^{27} - 5 q^{28} - 2 q^{29} + 5 q^{30} - 10 q^{31} + 9 q^{32} - 3 q^{33} - 16 q^{34} - q^{35} + 9 q^{36} - 14 q^{37} - 9 q^{38} + 4 q^{39} - 5 q^{40} - 21 q^{41} + 5 q^{42} - 4 q^{43} + 3 q^{44} - 5 q^{45} - 8 q^{47} - 9 q^{48} - 14 q^{49} + 8 q^{50} + 16 q^{51} - 4 q^{52} + 9 q^{53} - 9 q^{54} - 14 q^{55} - 5 q^{56} + 9 q^{57} - 2 q^{58} - 12 q^{59} + 5 q^{60} - 13 q^{61} - 10 q^{62} - 5 q^{63} + 9 q^{64} - 13 q^{65} - 3 q^{66} + 22 q^{67} - 16 q^{68} - q^{70} - 13 q^{71} + 9 q^{72} - 17 q^{73} - 14 q^{74} - 8 q^{75} - 9 q^{76} - 25 q^{77} + 4 q^{78} - 5 q^{80} + 9 q^{81} - 21 q^{82} - 24 q^{83} + 5 q^{84} - 16 q^{85} - 4 q^{86} + 2 q^{87} + 3 q^{88} - 23 q^{89} - 5 q^{90} - 11 q^{91} + 10 q^{93} - 8 q^{94} + 5 q^{95} - 9 q^{96} - 29 q^{97} - 14 q^{98} + 3 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 - 5 * q^5 - 9 * q^6 - 5 * q^7 + 9 * q^8 + 9 * q^9 - 5 * q^10 + 3 * q^11 - 9 * q^12 - 4 * q^13 - 5 * q^14 + 5 * q^15 + 9 * q^16 - 16 * q^17 + 9 * q^18 - 9 * q^19 - 5 * q^20 + 5 * q^21 + 3 * q^22 - 9 * q^24 + 8 * q^25 - 4 * q^26 - 9 * q^27 - 5 * q^28 - 2 * q^29 + 5 * q^30 - 10 * q^31 + 9 * q^32 - 3 * q^33 - 16 * q^34 - q^35 + 9 * q^36 - 14 * q^37 - 9 * q^38 + 4 * q^39 - 5 * q^40 - 21 * q^41 + 5 * q^42 - 4 * q^43 + 3 * q^44 - 5 * q^45 - 8 * q^47 - 9 * q^48 - 14 * q^49 + 8 * q^50 + 16 * q^51 - 4 * q^52 + 9 * q^53 - 9 * q^54 - 14 * q^55 - 5 * q^56 + 9 * q^57 - 2 * q^58 - 12 * q^59 + 5 * q^60 - 13 * q^61 - 10 * q^62 - 5 * q^63 + 9 * q^64 - 13 * q^65 - 3 * q^66 + 22 * q^67 - 16 * q^68 - q^70 - 13 * q^71 + 9 * q^72 - 17 * q^73 - 14 * q^74 - 8 * q^75 - 9 * q^76 - 25 * q^77 + 4 * q^78 - 5 * q^80 + 9 * q^81 - 21 * q^82 - 24 * q^83 + 5 * q^84 - 16 * q^85 - 4 * q^86 + 2 * q^87 + 3 * q^88 - 23 * q^89 - 5 * q^90 - 11 * q^91 + 10 * q^93 - 8 * q^94 + 5 * q^95 - 9 * q^96 - 29 * q^97 - 14 * q^98 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more