LMFDB - Embedded newform 4026.2.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4026,2,Mod(1,4026)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4026, 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("4026.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4026 = 2 \cdot 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4026.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:	$32.1477718538$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 22x^{6} + 42x^{5} + 182x^{4} - 111x^{3} - 538x^{2} - 256x - 32 $ x^8 - 3x^7 - 22x^6 + 42x^5 + 182x^4 - 111x^3 - 538x^2 - 256*x - 32
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			1.2
Root			$-0.326029$ of defining polynomial
Character	$\chi$	$=$	4026.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.31671 q^{5} +1.00000 q^{6} +2.32603 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.31671 q^{5} +1.00000 q^{6} +2.32603 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.31671 q^{10} -1.00000 q^{11} +1.00000 q^{12} +6.73691 q^{13} +2.32603 q^{14} -2.31671 q^{15} +1.00000 q^{16} -3.34873 q^{17} +1.00000 q^{18} +2.81157 q^{19} -2.31671 q^{20} +2.32603 q^{21} -1.00000 q^{22} -1.47833 q^{23} +1.00000 q^{24} +0.367132 q^{25} +6.73691 q^{26} +1.00000 q^{27} +2.32603 q^{28} -2.52579 q^{29} -2.31671 q^{30} +5.79504 q^{31} +1.00000 q^{32} -1.00000 q^{33} -3.34873 q^{34} -5.38873 q^{35} +1.00000 q^{36} -1.41994 q^{37} +2.81157 q^{38} +6.73691 q^{39} -2.31671 q^{40} +1.17910 q^{41} +2.32603 q^{42} +9.72732 q^{43} -1.00000 q^{44} -2.31671 q^{45} -1.47833 q^{46} +1.19121 q^{47} +1.00000 q^{48} -1.58959 q^{49} +0.367132 q^{50} -3.34873 q^{51} +6.73691 q^{52} +7.59985 q^{53} +1.00000 q^{54} +2.31671 q^{55} +2.32603 q^{56} +2.81157 q^{57} -2.52579 q^{58} +1.78517 q^{59} -2.31671 q^{60} +1.00000 q^{61} +5.79504 q^{62} +2.32603 q^{63} +1.00000 q^{64} -15.6074 q^{65} -1.00000 q^{66} +7.31260 q^{67} -3.34873 q^{68} -1.47833 q^{69} -5.38873 q^{70} -2.54207 q^{71} +1.00000 q^{72} +9.03818 q^{73} -1.41994 q^{74} +0.367132 q^{75} +2.81157 q^{76} -2.32603 q^{77} +6.73691 q^{78} -2.64676 q^{79} -2.31671 q^{80} +1.00000 q^{81} +1.17910 q^{82} -15.4467 q^{83} +2.32603 q^{84} +7.75802 q^{85} +9.72732 q^{86} -2.52579 q^{87} -1.00000 q^{88} +4.75943 q^{89} -2.31671 q^{90} +15.6702 q^{91} -1.47833 q^{92} +5.79504 q^{93} +1.19121 q^{94} -6.51360 q^{95} +1.00000 q^{96} -17.4671 q^{97} -1.58959 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 5 q^{5} + 8 q^{6} + 13 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 + 5 * q^5 + 8 * q^6 + 13 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 5 q^{5} + 8 q^{6} + 13 q^{7} + 8 q^{8} + 8 q^{9} + 5 q^{10} - 8 q^{11} + 8 q^{12} + 10 q^{13} + 13 q^{14} + 5 q^{15} + 8 q^{16} + 4 q^{17} + 8 q^{18} + 11 q^{19} + 5 q^{20} + 13 q^{21} - 8 q^{22} + 2 q^{23} + 8 q^{24} + 23 q^{25} + 10 q^{26} + 8 q^{27} + 13 q^{28} + 10 q^{29} + 5 q^{30} + 9 q^{31} + 8 q^{32} - 8 q^{33} + 4 q^{34} - 3 q^{35} + 8 q^{36} + 9 q^{37} + 11 q^{38} + 10 q^{39} + 5 q^{40} + 3 q^{41} + 13 q^{42} + 16 q^{43} - 8 q^{44} + 5 q^{45} + 2 q^{46} - 16 q^{47} + 8 q^{48} + 17 q^{49} + 23 q^{50} + 4 q^{51} + 10 q^{52} + 7 q^{53} + 8 q^{54} - 5 q^{55} + 13 q^{56} + 11 q^{57} + 10 q^{58} - 14 q^{59} + 5 q^{60} + 8 q^{61} + 9 q^{62} + 13 q^{63} + 8 q^{64} + 22 q^{65} - 8 q^{66} + 8 q^{67} + 4 q^{68} + 2 q^{69} - 3 q^{70} + 11 q^{71} + 8 q^{72} + 14 q^{73} + 9 q^{74} + 23 q^{75} + 11 q^{76} - 13 q^{77} + 10 q^{78} + 22 q^{79} + 5 q^{80} + 8 q^{81} + 3 q^{82} - 16 q^{83} + 13 q^{84} + 3 q^{85} + 16 q^{86} + 10 q^{87} - 8 q^{88} + q^{89} + 5 q^{90} + 15 q^{91} + 2 q^{92} + 9 q^{93} - 16 q^{94} - 9 q^{95} + 8 q^{96} + 24 q^{97} + 17 q^{98} - 8 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 + 5 * q^5 + 8 * q^6 + 13 * q^7 + 8 * q^8 + 8 * q^9 + 5 * q^10 - 8 * q^11 + 8 * q^12 + 10 * q^13 + 13 * q^14 + 5 * q^15 + 8 * q^16 + 4 * q^17 + 8 * q^18 + 11 * q^19 + 5 * q^20 + 13 * q^21 - 8 * q^22 + 2 * q^23 + 8 * q^24 + 23 * q^25 + 10 * q^26 + 8 * q^27 + 13 * q^28 + 10 * q^29 + 5 * q^30 + 9 * q^31 + 8 * q^32 - 8 * q^33 + 4 * q^34 - 3 * q^35 + 8 * q^36 + 9 * q^37 + 11 * q^38 + 10 * q^39 + 5 * q^40 + 3 * q^41 + 13 * q^42 + 16 * q^43 - 8 * q^44 + 5 * q^45 + 2 * q^46 - 16 * q^47 + 8 * q^48 + 17 * q^49 + 23 * q^50 + 4 * q^51 + 10 * q^52 + 7 * q^53 + 8 * q^54 - 5 * q^55 + 13 * q^56 + 11 * q^57 + 10 * q^58 - 14 * q^59 + 5 * q^60 + 8 * q^61 + 9 * q^62 + 13 * q^63 + 8 * q^64 + 22 * q^65 - 8 * q^66 + 8 * q^67 + 4 * q^68 + 2 * q^69 - 3 * q^70 + 11 * q^71 + 8 * q^72 + 14 * q^73 + 9 * q^74 + 23 * q^75 + 11 * q^76 - 13 * q^77 + 10 * q^78 + 22 * q^79 + 5 * q^80 + 8 * q^81 + 3 * q^82 - 16 * q^83 + 13 * q^84 + 3 * q^85 + 16 * q^86 + 10 * q^87 - 8 * q^88 + q^89 + 5 * q^90 + 15 * q^91 + 2 * q^92 + 9 * q^93 - 16 * q^94 - 9 * q^95 + 8 * q^96 + 24 * q^97 + 17 * q^98 - 8 * 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
$5$	−2.31671	−1.03606	−0.518031	−	0.855362i	$-0.673335\pi$
$5$	−2.31671	−1.03606	−0.518031	+	0.855362i	$0.673335\pi$
$6$	1.00000	0.408248
$7$	2.32603	0.879156	0.439578	−	0.898204i	$-0.355128\pi$
$7$	2.32603	0.879156	0.439578	+	0.898204i	$0.355128\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−2.31671	−0.732607
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	6.73691	1.86848	0.934241	−	0.356641i	$-0.116078\pi$
$13$	6.73691	1.86848	0.934241	+	0.356641i	$0.116078\pi$
$14$	2.32603	0.621657
$15$	−2.31671	−0.598171
$16$	1.00000	0.250000
$17$	−3.34873	−0.812186	−0.406093	−	0.913832i	$-0.633109\pi$
$17$	−3.34873	−0.812186	−0.406093	+	0.913832i	$0.633109\pi$
$18$	1.00000	0.235702
$19$	2.81157	0.645020	0.322510	−	0.946566i	$-0.395473\pi$
$19$	2.81157	0.645020	0.322510	+	0.946566i	$0.395473\pi$
$20$	−2.31671	−0.518031
$21$	2.32603	0.507581
$22$	−1.00000	−0.213201
$23$	−1.47833	−0.308253	−0.154126	−	0.988051i	$-0.549256\pi$
$23$	−1.47833	−0.308253	−0.154126	+	0.988051i	$0.549256\pi$
$24$	1.00000	0.204124
$25$	0.367132	0.0734265
$26$	6.73691	1.32122
$27$	1.00000	0.192450
$28$	2.32603	0.439578
$29$	−2.52579	−0.469027	−0.234513	−	0.972113i	$-0.575350\pi$
$29$	−2.52579	−0.469027	−0.234513	+	0.972113i	$0.575350\pi$
$30$	−2.31671	−0.422971
$31$	5.79504	1.04082	0.520410	−	0.853917i	$-0.325779\pi$
$31$	5.79504	1.04082	0.520410	+	0.853917i	$0.325779\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	−3.34873	−0.574302
$35$	−5.38873	−0.910861
$36$	1.00000	0.166667
$37$	−1.41994	−0.233436	−0.116718	−	0.993165i	$-0.537237\pi$
$37$	−1.41994	−0.233436	−0.116718	+	0.993165i	$0.537237\pi$
$38$	2.81157	0.456098
$39$	6.73691	1.07877
$40$	−2.31671	−0.366304
$41$	1.17910	0.184145	0.0920725	−	0.995752i	$-0.470651\pi$
$41$	1.17910	0.184145	0.0920725	+	0.995752i	$0.470651\pi$
$42$	2.32603	0.358914
$43$	9.72732	1.48340	0.741701	−	0.670730i	$-0.234019\pi$
$43$	9.72732	1.48340	0.741701	+	0.670730i	$0.234019\pi$
$44$	−1.00000	−0.150756
$45$	−2.31671	−0.345354
$46$	−1.47833	−0.217968
$47$	1.19121	0.173755	0.0868777	−	0.996219i	$-0.472311\pi$
$47$	1.19121	0.173755	0.0868777	+	0.996219i	$0.472311\pi$
$48$	1.00000	0.144338
$49$	−1.58959	−0.227084
$50$	0.367132	0.0519204
$51$	−3.34873	−0.468916
$52$	6.73691	0.934241
$53$	7.59985	1.04392	0.521960	−	0.852970i	$-0.325201\pi$
$53$	7.59985	1.04392	0.521960	+	0.852970i	$0.325201\pi$
$54$	1.00000	0.136083
$55$	2.31671	0.312385
$56$	2.32603	0.310829
$57$	2.81157	0.372402
$58$	−2.52579	−0.331652
$59$	1.78517	0.232409	0.116204	−	0.993225i	$-0.462927\pi$
$59$	1.78517	0.232409	0.116204	+	0.993225i	$0.462927\pi$
$60$	−2.31671	−0.299086
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	5.79504	0.735970
$63$	2.32603	0.293052
$64$	1.00000	0.125000
$65$	−15.6074	−1.93587
$66$	−1.00000	−0.123091
$67$	7.31260	0.893377	0.446688	−	0.894690i	$-0.352603\pi$
$67$	7.31260	0.893377	0.446688	+	0.894690i	$0.352603\pi$
$68$	−3.34873	−0.406093
$69$	−1.47833	−0.177970
$70$	−5.38873	−0.644076
$71$	−2.54207	−0.301688	−0.150844	−	0.988558i	$-0.548199\pi$
$71$	−2.54207	−0.301688	−0.150844	+	0.988558i	$0.548199\pi$
$72$	1.00000	0.117851
$73$	9.03818	1.05784	0.528920	−	0.848672i	$-0.322597\pi$
$73$	9.03818	1.05784	0.528920	+	0.848672i	$0.322597\pi$
$74$	−1.41994	−0.165064
$75$	0.367132	0.0423928
$76$	2.81157	0.322510
$77$	−2.32603	−0.265076
$78$	6.73691	0.762805
$79$	−2.64676	−0.297784	−0.148892	−	0.988853i	$-0.547571\pi$
$79$	−2.64676	−0.297784	−0.148892	+	0.988853i	$0.547571\pi$
$80$	−2.31671	−0.259016
$81$	1.00000	0.111111
$82$	1.17910	0.130210
$83$	−15.4467	−1.69549	−0.847747	−	0.530401i	$-0.822041\pi$
$83$	−15.4467	−1.69549	−0.847747	+	0.530401i	$0.822041\pi$
$84$	2.32603	0.253791
$85$	7.75802	0.841476
$86$	9.72732	1.04892
$87$	−2.52579	−0.270793
$88$	−1.00000	−0.106600
$89$	4.75943	0.504498	0.252249	−	0.967662i	$-0.418830\pi$
$89$	4.75943	0.504498	0.252249	+	0.967662i	$0.418830\pi$
$90$	−2.31671	−0.244202
$91$	15.6702	1.64269
$92$	−1.47833	−0.154126
$93$	5.79504	0.600917
$94$	1.19121	0.122864
$95$	−6.51360	−0.668281
$96$	1.00000	0.102062
$97$	−17.4671	−1.77351	−0.886756	−	0.462237i	$-0.847047\pi$
$97$	−17.4671	−1.77351	−0.886756	+	0.462237i	$0.847047\pi$
$98$	−1.58959	−0.160573
$99$	−1.00000	−0.100504
$100$	0.367132	0.0367132
$101$	−8.14281	−0.810240	−0.405120	−	0.914263i	$-0.632770\pi$
$101$	−8.14281	−0.810240	−0.405120	+	0.914263i	$0.632770\pi$
$102$	−3.34873	−0.331574
$103$	3.27977	0.323166	0.161583	−	0.986859i	$-0.448340\pi$
$103$	3.27977	0.323166	0.161583	+	0.986859i	$0.448340\pi$
$104$	6.73691	0.660608
$105$	−5.38873	−0.525886
$106$	7.59985	0.738163
$107$	10.9839	1.06185	0.530925	−	0.847419i	$-0.321844\pi$
$107$	10.9839	1.06185	0.530925	+	0.847419i	$0.321844\pi$
$108$	1.00000	0.0962250
$109$	7.69913	0.737443	0.368722	−	0.929540i	$-0.379796\pi$
$109$	7.69913	0.737443	0.368722	+	0.929540i	$0.379796\pi$
$110$	2.31671	0.220889
$111$	−1.41994	−0.134774
$112$	2.32603	0.219789
$113$	16.3922	1.54205	0.771025	−	0.636804i	$-0.219744\pi$
$113$	16.3922	1.54205	0.771025	+	0.636804i	$0.219744\pi$
$114$	2.81157	0.263328
$115$	3.42485	0.319369
$116$	−2.52579	−0.234513
$117$	6.73691	0.622828
$118$	1.78517	0.164338
$119$	−7.78924	−0.714038
$120$	−2.31671	−0.211485
$121$	1.00000	0.0909091
$122$	1.00000	0.0905357
$123$	1.17910	0.106316
$124$	5.79504	0.520410
$125$	10.7330	0.959989
$126$	2.32603	0.207219
$127$	−11.4528	−1.01627	−0.508135	−	0.861278i	$-0.669665\pi$
$127$	−11.4528	−1.01627	−0.508135	+	0.861278i	$0.669665\pi$
$128$	1.00000	0.0883883
$129$	9.72732	0.856443
$130$	−15.6074	−1.36886
$131$	14.4872	1.26575	0.632876	−	0.774253i	$-0.281874\pi$
$131$	14.4872	1.26575	0.632876	+	0.774253i	$0.281874\pi$
$132$	−1.00000	−0.0870388
$133$	6.53981	0.567073
$134$	7.31260	0.631713
$135$	−2.31671	−0.199390
$136$	−3.34873	−0.287151
$137$	5.68823	0.485978	0.242989	−	0.970029i	$-0.421872\pi$
$137$	5.68823	0.485978	0.242989	+	0.970029i	$0.421872\pi$
$138$	−1.47833	−0.125844
$139$	−7.63922	−0.647950	−0.323975	−	0.946066i	$-0.605019\pi$
$139$	−7.63922	−0.647950	−0.323975	+	0.946066i	$0.605019\pi$
$140$	−5.38873	−0.455431
$141$	1.19121	0.100318
$142$	−2.54207	−0.213326
$143$	−6.73691	−0.563369
$144$	1.00000	0.0833333
$145$	5.85151	0.485941
$146$	9.03818	0.748005
$147$	−1.58959	−0.131107
$148$	−1.41994	−0.116718
$149$	−15.7816	−1.29288	−0.646439	−	0.762966i	$-0.723743\pi$
$149$	−15.7816	−1.29288	−0.646439	+	0.762966i	$0.723743\pi$
$150$	0.367132	0.0299762
$151$	−11.5779	−0.942193	−0.471097	−	0.882082i	$-0.656142\pi$
$151$	−11.5779	−0.942193	−0.471097	+	0.882082i	$0.656142\pi$
$152$	2.81157	0.228049
$153$	−3.34873	−0.270729
$154$	−2.32603	−0.187437
$155$	−13.4254	−1.07835
$156$	6.73691	0.539385
$157$	−7.89429	−0.630033	−0.315016	−	0.949086i	$-0.602010\pi$
$157$	−7.89429	−0.630033	−0.315016	+	0.949086i	$0.602010\pi$
$158$	−2.64676	−0.210565
$159$	7.59985	0.602708
$160$	−2.31671	−0.183152
$161$	−3.43864	−0.271002
$162$	1.00000	0.0785674
$163$	23.2664	1.82237	0.911184	−	0.412000i	$-0.135170\pi$
$163$	23.2664	1.82237	0.911184	+	0.412000i	$0.135170\pi$
$164$	1.17910	0.0920725
$165$	2.31671	0.180355
$166$	−15.4467	−1.19890
$167$	9.43400	0.730025	0.365013	−	0.931003i	$-0.381065\pi$
$167$	9.43400	0.730025	0.365013	+	0.931003i	$0.381065\pi$
$168$	2.32603	0.179457
$169$	32.3860	2.49123
$170$	7.75802	0.595013
$171$	2.81157	0.215007
$172$	9.72732	0.741701
$173$	11.2868	0.858120	0.429060	−	0.903276i	$-0.358845\pi$
$173$	11.2868	0.858120	0.429060	+	0.903276i	$0.358845\pi$
$174$	−2.52579	−0.191479
$175$	0.853960	0.0645533
$176$	−1.00000	−0.0753778
$177$	1.78517	0.134181
$178$	4.75943	0.356734
$179$	−1.24086	−0.0927462	−0.0463731	−	0.998924i	$-0.514766\pi$
$179$	−1.24086	−0.0927462	−0.0463731	+	0.998924i	$0.514766\pi$
$180$	−2.31671	−0.172677
$181$	−3.88160	−0.288517	−0.144258	−	0.989540i	$-0.546080\pi$
$181$	−3.88160	−0.288517	−0.144258	+	0.989540i	$0.546080\pi$
$182$	15.6702	1.16156
$183$	1.00000	0.0739221
$184$	−1.47833	−0.108984
$185$	3.28958	0.241854
$186$	5.79504	0.424913
$187$	3.34873	0.244883
$188$	1.19121	0.0868777
$189$	2.32603	0.169194
$190$	−6.51360	−0.472546
$191$	11.5303	0.834306	0.417153	−	0.908836i	$-0.363028\pi$
$191$	11.5303	0.834306	0.417153	+	0.908836i	$0.363028\pi$
$192$	1.00000	0.0721688
$193$	−8.73792	−0.628969	−0.314484	−	0.949263i	$-0.601832\pi$
$193$	−8.73792	−0.628969	−0.314484	+	0.949263i	$0.601832\pi$
$194$	−17.4671	−1.25406
$195$	−15.6074	−1.11767
$196$	−1.58959	−0.113542
$197$	−23.2550	−1.65685	−0.828426	−	0.560099i	$-0.810763\pi$
$197$	−23.2550	−1.65685	−0.828426	+	0.560099i	$0.810763\pi$
$198$	−1.00000	−0.0710669
$199$	−21.0823	−1.49448	−0.747241	−	0.664553i	$-0.768622\pi$
$199$	−21.0823	−1.49448	−0.747241	+	0.664553i	$0.768622\pi$
$200$	0.367132	0.0259602
$201$	7.31260	0.515791
$202$	−8.14281	−0.572926
$203$	−5.87505	−0.412348
$204$	−3.34873	−0.234458
$205$	−2.73164	−0.190786
$206$	3.27977	0.228513
$207$	−1.47833	−0.102751
$208$	6.73691	0.467121
$209$	−2.81157	−0.194481
$210$	−5.38873	−0.371858
$211$	14.7561	1.01585	0.507927	−	0.861400i	$-0.330412\pi$
$211$	14.7561	1.01585	0.507927	+	0.861400i	$0.330412\pi$
$212$	7.59985	0.521960
$213$	−2.54207	−0.174180
$214$	10.9839	0.750842
$215$	−22.5354	−1.53690
$216$	1.00000	0.0680414
$217$	13.4794	0.915043
$218$	7.69913	0.521451
$219$	9.03818	0.610744
$220$	2.31671	0.156192
$221$	−22.5601	−1.51756
$222$	−1.41994	−0.0952998
$223$	6.63858	0.444552	0.222276	−	0.974984i	$-0.428651\pi$
$223$	6.63858	0.444552	0.222276	+	0.974984i	$0.428651\pi$
$224$	2.32603	0.155414
$225$	0.367132	0.0244755
$226$	16.3922	1.09039
$227$	24.0039	1.59320	0.796598	−	0.604509i	$-0.206631\pi$
$227$	24.0039	1.59320	0.796598	+	0.604509i	$0.206631\pi$
$228$	2.81157	0.186201
$229$	2.55902	0.169105	0.0845523	−	0.996419i	$-0.473054\pi$
$229$	2.55902	0.169105	0.0845523	+	0.996419i	$0.473054\pi$
$230$	3.42485	0.225828
$231$	−2.32603	−0.153041
$232$	−2.52579	−0.165826
$233$	6.77289	0.443707	0.221854	−	0.975080i	$-0.428789\pi$
$233$	6.77289	0.443707	0.221854	+	0.975080i	$0.428789\pi$
$234$	6.73691	0.440406
$235$	−2.75968	−0.180021
$236$	1.78517	0.116204
$237$	−2.64676	−0.171926
$238$	−7.78924	−0.504901
$239$	13.3814	0.865573	0.432786	−	0.901497i	$-0.357530\pi$
$239$	13.3814	0.865573	0.432786	+	0.901497i	$0.357530\pi$
$240$	−2.31671	−0.149543
$241$	12.4268	0.800480	0.400240	−	0.916410i	$-0.368927\pi$
$241$	12.4268	0.800480	0.400240	+	0.916410i	$0.368927\pi$
$242$	1.00000	0.0642824
$243$	1.00000	0.0641500
$244$	1.00000	0.0640184
$245$	3.68261	0.235273
$246$	1.17910	0.0751769
$247$	18.9413	1.20521
$248$	5.79504	0.367985
$249$	−15.4467	−0.978894
$250$	10.7330	0.678814
$251$	−0.933334	−0.0589115	−0.0294557	−	0.999566i	$-0.509377\pi$
$251$	−0.933334	−0.0589115	−0.0294557	+	0.999566i	$0.509377\pi$
$252$	2.32603	0.146526
$253$	1.47833	0.0929417
$254$	−11.4528	−0.718611
$255$	7.75802	0.485826
$256$	1.00000	0.0625000
$257$	−9.97589	−0.622279	−0.311139	−	0.950364i	$-0.600711\pi$
$257$	−9.97589	−0.622279	−0.311139	+	0.950364i	$0.600711\pi$
$258$	9.72732	0.605597
$259$	−3.30281	−0.205227
$260$	−15.6074	−0.967933
$261$	−2.52579	−0.156342
$262$	14.4872	0.895022
$263$	18.3659	1.13249	0.566244	−	0.824237i	$-0.308396\pi$
$263$	18.3659	1.13249	0.566244	+	0.824237i	$0.308396\pi$
$264$	−1.00000	−0.0615457
$265$	−17.6066	−1.08157
$266$	6.53981	0.400981
$267$	4.75943	0.291272
$268$	7.31260	0.446688
$269$	−13.2661	−0.808846	−0.404423	−	0.914572i	$-0.632528\pi$
$269$	−13.2661	−0.808846	−0.404423	+	0.914572i	$0.632528\pi$
$270$	−2.31671	−0.140990
$271$	−9.93145	−0.603293	−0.301646	−	0.953420i	$-0.597536\pi$
$271$	−9.93145	−0.603293	−0.301646	+	0.953420i	$0.597536\pi$
$272$	−3.34873	−0.203046
$273$	15.6702	0.948407
$274$	5.68823	0.343639
$275$	−0.367132	−0.0221389
$276$	−1.47833	−0.0889849
$277$	−25.7449	−1.54686	−0.773431	−	0.633881i	$-0.781461\pi$
$277$	−25.7449	−1.54686	−0.773431	+	0.633881i	$0.781461\pi$
$278$	−7.63922	−0.458170
$279$	5.79504	0.346940
$280$	−5.38873	−0.322038
$281$	−8.14683	−0.485999	−0.243000	−	0.970026i	$-0.578131\pi$
$281$	−8.14683	−0.485999	−0.243000	+	0.970026i	$0.578131\pi$
$282$	1.19121	0.0709353
$283$	−30.3814	−1.80599	−0.902993	−	0.429656i	$-0.858635\pi$
$283$	−30.3814	−1.80599	−0.902993	+	0.429656i	$0.858635\pi$
$284$	−2.54207	−0.150844
$285$	−6.51360	−0.385832
$286$	−6.73691	−0.398362
$287$	2.74263	0.161892
$288$	1.00000	0.0589256
$289$	−5.78602	−0.340354
$290$	5.85151	0.343612
$291$	−17.4671	−1.02394
$292$	9.03818	0.528920
$293$	−3.76071	−0.219703	−0.109852	−	0.993948i	$-0.535038\pi$
$293$	−3.76071	−0.219703	−0.109852	+	0.993948i	$0.535038\pi$
$294$	−1.58959	−0.0927067
$295$	−4.13571	−0.240790
$296$	−1.41994	−0.0825321
$297$	−1.00000	−0.0580259
$298$	−15.7816	−0.914203
$299$	−9.95937	−0.575965
$300$	0.367132	0.0211964
$301$	22.6260	1.30414
$302$	−11.5779	−0.666231
$303$	−8.14281	−0.467792
$304$	2.81157	0.161255
$305$	−2.31671	−0.132654
$306$	−3.34873	−0.191434
$307$	−11.0669	−0.631619	−0.315810	−	0.948823i	$-0.602276\pi$
$307$	−11.0669	−0.631619	−0.315810	+	0.948823i	$0.602276\pi$
$308$	−2.32603	−0.132538
$309$	3.27977	0.186580
$310$	−13.4254	−0.762512
$311$	7.26313	0.411854	0.205927	−	0.978567i	$-0.433979\pi$
$311$	7.26313	0.411854	0.205927	+	0.978567i	$0.433979\pi$
$312$	6.73691	0.381402
$313$	18.9628	1.07184	0.535922	−	0.844268i	$-0.319964\pi$
$313$	18.9628	1.07184	0.535922	+	0.844268i	$0.319964\pi$
$314$	−7.89429	−0.445500
$315$	−5.38873	−0.303620
$316$	−2.64676	−0.148892
$317$	24.6712	1.38567	0.692836	−	0.721095i	$-0.256361\pi$
$317$	24.6712	1.38567	0.692836	+	0.721095i	$0.256361\pi$
$318$	7.59985	0.426179
$319$	2.52579	0.141417
$320$	−2.31671	−0.129508
$321$	10.9839	0.613060
$322$	−3.43864	−0.191628
$323$	−9.41520	−0.523876
$324$	1.00000	0.0555556
$325$	2.47334	0.137196
$326$	23.2664	1.28861
$327$	7.69913	0.425763
$328$	1.17910	0.0651051
$329$	2.77078	0.152758
$330$	2.31671	0.127531
$331$	−27.1064	−1.48990	−0.744950	−	0.667120i	$-0.767527\pi$
$331$	−27.1064	−1.48990	−0.744950	+	0.667120i	$0.767527\pi$
$332$	−15.4467	−0.847747
$333$	−1.41994	−0.0778120
$334$	9.43400	0.516206
$335$	−16.9412	−0.925594
$336$	2.32603	0.126895
$337$	−28.5619	−1.55587	−0.777933	−	0.628347i	$-0.783732\pi$
$337$	−28.5619	−1.55587	−0.777933	+	0.628347i	$0.783732\pi$
$338$	32.3860	1.76156
$339$	16.3922	0.890303
$340$	7.75802	0.420738
$341$	−5.79504	−0.313819
$342$	2.81157	0.152033
$343$	−19.9796	−1.07880
$344$	9.72732	0.524462
$345$	3.42485	0.184388
$346$	11.2868	0.606782
$347$	−30.8843	−1.65796	−0.828978	−	0.559281i	$-0.811077\pi$
$347$	−30.8843	−1.65796	−0.828978	+	0.559281i	$0.811077\pi$
$348$	−2.52579	−0.135396
$349$	5.53946	0.296521	0.148260	−	0.988948i	$-0.452633\pi$
$349$	5.53946	0.296521	0.148260	+	0.988948i	$0.452633\pi$
$350$	0.853960	0.0456461
$351$	6.73691	0.359590
$352$	−1.00000	−0.0533002
$353$	−33.7973	−1.79885	−0.899425	−	0.437074i	$-0.856015\pi$
$353$	−33.7973	−1.79885	−0.899425	+	0.437074i	$0.856015\pi$
$354$	1.78517	0.0948806
$355$	5.88923	0.312568
$356$	4.75943	0.252249
$357$	−7.78924	−0.412250
$358$	−1.24086	−0.0655815
$359$	−1.24214	−0.0655579	−0.0327789	−	0.999463i	$-0.510436\pi$
$359$	−1.24214	−0.0655579	−0.0327789	+	0.999463i	$0.510436\pi$
$360$	−2.31671	−0.122101
$361$	−11.0950	−0.583950
$362$	−3.88160	−0.204012
$363$	1.00000	0.0524864
$364$	15.6702	0.821344
$365$	−20.9388	−1.09599
$366$	1.00000	0.0522708
$367$	3.96096	0.206761	0.103380	−	0.994642i	$-0.467034\pi$
$367$	3.96096	0.206761	0.103380	+	0.994642i	$0.467034\pi$
$368$	−1.47833	−0.0770632
$369$	1.17910	0.0613817
$370$	3.28958	0.171017
$371$	17.6775	0.917769
$372$	5.79504	0.300459
$373$	−11.2901	−0.584581	−0.292290	−	0.956330i	$-0.594417\pi$
$373$	−11.2901	−0.584581	−0.292290	+	0.956330i	$0.594417\pi$
$374$	3.34873	0.173159
$375$	10.7330	0.554250
$376$	1.19121	0.0614318
$377$	−17.0160	−0.876368
$378$	2.32603	0.119638
$379$	−1.17428	−0.0603185	−0.0301593	−	0.999545i	$-0.509601\pi$
$379$	−1.17428	−0.0603185	−0.0301593	+	0.999545i	$0.509601\pi$
$380$	−6.51360	−0.334140
$381$	−11.4528	−0.586743
$382$	11.5303	0.589944
$383$	13.7988	0.705086	0.352543	−	0.935796i	$-0.385317\pi$
$383$	13.7988	0.705086	0.352543	+	0.935796i	$0.385317\pi$
$384$	1.00000	0.0510310
$385$	5.38873	0.274635
$386$	−8.73792	−0.444748
$387$	9.72732	0.494468
$388$	−17.4671	−0.886756
$389$	−19.1636	−0.971632	−0.485816	−	0.874061i	$-0.661477\pi$
$389$	−19.1636	−0.971632	−0.485816	+	0.874061i	$0.661477\pi$
$390$	−15.6074	−0.790314
$391$	4.95052	0.250359
$392$	−1.58959	−0.0802863
$393$	14.4872	0.730782
$394$	−23.2550	−1.17157
$395$	6.13178	0.308523
$396$	−1.00000	−0.0502519
$397$	−8.75564	−0.439433	−0.219716	−	0.975564i	$-0.570513\pi$
$397$	−8.75564	−0.439433	−0.219716	+	0.975564i	$0.570513\pi$
$398$	−21.0823	−1.05676
$399$	6.53981	0.327400
$400$	0.367132	0.0183566
$401$	6.31267	0.315240	0.157620	−	0.987500i	$-0.449618\pi$
$401$	6.31267	0.315240	0.157620	+	0.987500i	$0.449618\pi$
$402$	7.31260	0.364719
$403$	39.0406	1.94475
$404$	−8.14281	−0.405120
$405$	−2.31671	−0.115118
$406$	−5.87505	−0.291574
$407$	1.41994	0.0703836
$408$	−3.34873	−0.165787
$409$	−35.0417	−1.73270	−0.866351	−	0.499436i	$-0.833541\pi$
$409$	−35.0417	−1.73270	−0.866351	+	0.499436i	$0.833541\pi$
$410$	−2.73164	−0.134906
$411$	5.68823	0.280580
$412$	3.27977	0.161583
$413$	4.15235	0.204324
$414$	−1.47833	−0.0726559
$415$	35.7854	1.75664
$416$	6.73691	0.330304
$417$	−7.63922	−0.374094
$418$	−2.81157	−0.137519
$419$	13.0223	0.636182	0.318091	−	0.948060i	$-0.396958\pi$
$419$	13.0223	0.636182	0.318091	+	0.948060i	$0.396958\pi$
$420$	−5.38873	−0.262943
$421$	28.1361	1.37127	0.685634	−	0.727947i	$-0.259525\pi$
$421$	28.1361	1.37127	0.685634	+	0.727947i	$0.259525\pi$
$422$	14.7561	0.718317
$423$	1.19121	0.0579185
$424$	7.59985	0.369081
$425$	−1.22943	−0.0596359
$426$	−2.54207	−0.123164
$427$	2.32603	0.112564
$428$	10.9839	0.530925
$429$	−6.73691	−0.325261
$430$	−22.5354	−1.08675
$431$	25.2728	1.21735	0.608673	−	0.793421i	$-0.291702\pi$
$431$	25.2728	1.21735	0.608673	+	0.793421i	$0.291702\pi$
$432$	1.00000	0.0481125
$433$	0.646235	0.0310561	0.0155280	−	0.999879i	$-0.495057\pi$
$433$	0.646235	0.0310561	0.0155280	+	0.999879i	$0.495057\pi$
$434$	13.4794	0.647033
$435$	5.85151	0.280558
$436$	7.69913	0.368722
$437$	−4.15643	−0.198829
$438$	9.03818	0.431861
$439$	−25.1067	−1.19828	−0.599138	−	0.800646i	$-0.704490\pi$
$439$	−25.1067	−1.19828	−0.599138	+	0.800646i	$0.704490\pi$
$440$	2.31671	0.110445
$441$	−1.58959	−0.0756947
$442$	−22.5601	−1.07307
$443$	−34.4261	−1.63563	−0.817817	−	0.575479i	$-0.804816\pi$
$443$	−34.4261	−1.63563	−0.817817	+	0.575479i	$0.804816\pi$
$444$	−1.41994	−0.0673872
$445$	−11.0262	−0.522692
$446$	6.63858	0.314346
$447$	−15.7816	−0.746443
$448$	2.32603	0.109895
$449$	−32.4631	−1.53203	−0.766013	−	0.642825i	$-0.777762\pi$
$449$	−32.4631	−1.53203	−0.766013	+	0.642825i	$0.777762\pi$
$450$	0.367132	0.0173068
$451$	−1.17910	−0.0555218
$452$	16.3922	0.771025
$453$	−11.5779	−0.543975
$454$	24.0039	1.12656
$455$	−36.3034	−1.70193
$456$	2.81157	0.131664
$457$	6.03825	0.282457	0.141229	−	0.989977i	$-0.454895\pi$
$457$	6.03825	0.282457	0.141229	+	0.989977i	$0.454895\pi$
$458$	2.55902	0.119575
$459$	−3.34873	−0.156305
$460$	3.42485	0.159685
$461$	−16.1571	−0.752509	−0.376255	−	0.926516i	$-0.622788\pi$
$461$	−16.1571	−0.752509	−0.376255	+	0.926516i	$0.622788\pi$
$462$	−2.32603	−0.108217
$463$	−30.7875	−1.43082	−0.715408	−	0.698707i	$-0.753759\pi$
$463$	−30.7875	−1.43082	−0.715408	+	0.698707i	$0.753759\pi$
$464$	−2.52579	−0.117257
$465$	−13.4254	−0.622588
$466$	6.77289	0.313748
$467$	8.26907	0.382647	0.191323	−	0.981527i	$-0.438722\pi$
$467$	8.26907	0.382647	0.191323	+	0.981527i	$0.438722\pi$
$468$	6.73691	0.311414
$469$	17.0093	0.785418
$470$	−2.75968	−0.127294
$471$	−7.89429	−0.363750
$472$	1.78517	0.0821690
$473$	−9.72732	−0.447263
$474$	−2.64676	−0.121570
$475$	1.03222	0.0473615
$476$	−7.78924	−0.357019
$477$	7.59985	0.347973
$478$	13.3814	0.612052
$479$	−12.7944	−0.584589	−0.292294	−	0.956328i	$-0.594419\pi$
$479$	−12.7944	−0.584589	−0.292294	+	0.956328i	$0.594419\pi$
$480$	−2.31671	−0.105743
$481$	−9.56598	−0.436171
$482$	12.4268	0.566025
$483$	−3.43864	−0.156463
$484$	1.00000	0.0454545
$485$	40.4661	1.83747
$486$	1.00000	0.0453609
$487$	12.4786	0.565460	0.282730	−	0.959200i	$-0.408760\pi$
$487$	12.4786	0.565460	0.282730	+	0.959200i	$0.408760\pi$
$488$	1.00000	0.0452679
$489$	23.2664	1.05214
$490$	3.68261	0.166363
$491$	−0.250763	−0.0113168	−0.00565839	−	0.999984i	$-0.501801\pi$
$491$	−0.250763	−0.0113168	−0.00565839	+	0.999984i	$0.501801\pi$
$492$	1.17910	0.0531581
$493$	8.45817	0.380937
$494$	18.9413	0.852211
$495$	2.31671	0.104128
$496$	5.79504	0.260205
$497$	−5.91292	−0.265231
$498$	−15.4467	−0.692183
$499$	−24.5144	−1.09742	−0.548708	−	0.836014i	$-0.684880\pi$
$499$	−24.5144	−1.09742	−0.548708	+	0.836014i	$0.684880\pi$
$500$	10.7330	0.479994
$501$	9.43400	0.421480
$502$	−0.933334	−0.0416567
$503$	25.6568	1.14398	0.571989	−	0.820261i	$-0.306172\pi$
$503$	25.6568	1.14398	0.571989	+	0.820261i	$0.306172\pi$
$504$	2.32603	0.103610
$505$	18.8645	0.839460
$506$	1.47833	0.0657197
$507$	32.3860	1.43831
$508$	−11.4528	−0.508135
$509$	−9.20150	−0.407849	−0.203925	−	0.978987i	$-0.565370\pi$
$509$	−9.20150	−0.407849	−0.203925	+	0.978987i	$0.565370\pi$
$510$	7.75802	0.343531
$511$	21.0231	0.930006
$512$	1.00000	0.0441942
$513$	2.81157	0.124134
$514$	−9.97589	−0.440018
$515$	−7.59827	−0.334820
$516$	9.72732	0.428221
$517$	−1.19121	−0.0523892
$518$	−3.30281	−0.145117
$519$	11.2868	0.495436
$520$	−15.6074	−0.684432
$521$	−19.0707	−0.835502	−0.417751	−	0.908562i	$-0.637182\pi$
$521$	−19.0707	−0.835502	−0.417751	+	0.908562i	$0.637182\pi$
$522$	−2.52579	−0.110551
$523$	−40.1575	−1.75596	−0.877982	−	0.478693i	$-0.841111\pi$
$523$	−40.1575	−1.75596	−0.877982	+	0.478693i	$0.841111\pi$
$524$	14.4872	0.632876
$525$	0.853960	0.0372699
$526$	18.3659	0.800790
$527$	−19.4060	−0.845339
$528$	−1.00000	−0.0435194
$529$	−20.8145	−0.904980
$530$	−17.6066	−0.764783
$531$	1.78517	0.0774696
$532$	6.53981	0.283537
$533$	7.94351	0.344072
$534$	4.75943	0.205961
$535$	−25.4464	−1.10014
$536$	7.31260	0.315856
$537$	−1.24086	−0.0535471
$538$	−13.2661	−0.571941
$539$	1.58959	0.0684684
$540$	−2.31671	−0.0996952
$541$	45.4248	1.95296	0.976482	−	0.215599i	$-0.0691704\pi$
$541$	45.4248	1.95296	0.976482	+	0.215599i	$0.0691704\pi$
$542$	−9.93145	−0.426592
$543$	−3.88160	−0.166575
$544$	−3.34873	−0.143576
$545$	−17.8366	−0.764038
$546$	15.6702	0.670625
$547$	−42.7496	−1.82784	−0.913920	−	0.405895i	$-0.866960\pi$
$547$	−42.7496	−1.82784	−0.913920	+	0.405895i	$0.866960\pi$
$548$	5.68823	0.242989
$549$	1.00000	0.0426790
$550$	−0.367132	−0.0156546
$551$	−7.10144	−0.302531
$552$	−1.47833	−0.0629218
$553$	−6.15645	−0.261799
$554$	−25.7449	−1.09380
$555$	3.28958	0.139635
$556$	−7.63922	−0.323975
$557$	−24.6334	−1.04375	−0.521876	−	0.853022i	$-0.674767\pi$
$557$	−24.6334	−1.04375	−0.521876	+	0.853022i	$0.674767\pi$
$558$	5.79504	0.245323
$559$	65.5321	2.77171
$560$	−5.38873	−0.227715
$561$	3.34873	0.141383
$562$	−8.14683	−0.343653
$563$	21.6094	0.910726	0.455363	−	0.890306i	$-0.349510\pi$
$563$	21.6094	0.910726	0.455363	+	0.890306i	$0.349510\pi$
$564$	1.19121	0.0501589
$565$	−37.9760	−1.59766
$566$	−30.3814	−1.27702
$567$	2.32603	0.0976840
$568$	−2.54207	−0.106663
$569$	−11.0008	−0.461177	−0.230588	−	0.973051i	$-0.574065\pi$
$569$	−11.0008	−0.461177	−0.230588	+	0.973051i	$0.574065\pi$
$570$	−6.51360	−0.272825
$571$	−7.51925	−0.314671	−0.157335	−	0.987545i	$-0.550290\pi$
$571$	−7.51925	−0.314671	−0.157335	+	0.987545i	$0.550290\pi$
$572$	−6.73691	−0.281684
$573$	11.5303	0.481687
$574$	2.74263	0.114475
$575$	−0.542742	−0.0226339
$576$	1.00000	0.0416667
$577$	−1.65941	−0.0690823	−0.0345411	−	0.999403i	$-0.510997\pi$
$577$	−1.65941	−0.0690823	−0.0345411	+	0.999403i	$0.510997\pi$
$578$	−5.78602	−0.240667
$579$	−8.73792	−0.363135
$580$	5.85151	0.242971
$581$	−35.9294	−1.49060
$582$	−17.4671	−0.724033
$583$	−7.59985	−0.314754
$584$	9.03818	0.374003
$585$	−15.6074	−0.645289
$586$	−3.76071	−0.155354
$587$	11.1236	0.459119	0.229559	−	0.973295i	$-0.426272\pi$
$587$	11.1236	0.459119	0.229559	+	0.973295i	$0.426272\pi$
$588$	−1.58959	−0.0655535
$589$	16.2932	0.671349
$590$	−4.13571	−0.170264
$591$	−23.2550	−0.956584
$592$	−1.41994	−0.0583590
$593$	−23.7914	−0.976995	−0.488498	−	0.872565i	$-0.662455\pi$
$593$	−23.7914	−0.976995	−0.488498	+	0.872565i	$0.662455\pi$
$594$	−1.00000	−0.0410305
$595$	18.0454	0.739789
$596$	−15.7816	−0.646439
$597$	−21.0823	−0.862840
$598$	−9.95937	−0.407269
$599$	−4.08393	−0.166865	−0.0834324	−	0.996513i	$-0.526588\pi$
$599$	−4.08393	−0.166865	−0.0834324	+	0.996513i	$0.526588\pi$
$600$	0.367132	0.0149881
$601$	−21.7429	−0.886913	−0.443456	−	0.896296i	$-0.646248\pi$
$601$	−21.7429	−0.886913	−0.443456	+	0.896296i	$0.646248\pi$
$602$	22.6260	0.922168
$603$	7.31260	0.297792
$604$	−11.5779	−0.471097
$605$	−2.31671	−0.0941875
$606$	−8.14281	−0.330779
$607$	12.3970	0.503179	0.251589	−	0.967834i	$-0.419047\pi$
$607$	12.3970	0.503179	0.251589	+	0.967834i	$0.419047\pi$
$608$	2.81157	0.114024
$609$	−5.87505	−0.238069
$610$	−2.31671	−0.0938007
$611$	8.02505	0.324659
$612$	−3.34873	−0.135364
$613$	−34.7487	−1.40349	−0.701743	−	0.712430i	$-0.747594\pi$
$613$	−34.7487	−1.40349	−0.701743	+	0.712430i	$0.747594\pi$
$614$	−11.0669	−0.446622
$615$	−2.73164	−0.110150
$616$	−2.32603	−0.0937184
$617$	29.5313	1.18889	0.594443	−	0.804138i	$-0.297373\pi$
$617$	29.5313	1.18889	0.594443	+	0.804138i	$0.297373\pi$
$618$	3.27977	0.131932
$619$	−2.82286	−0.113460	−0.0567302	−	0.998390i	$-0.518067\pi$
$619$	−2.82286	−0.113460	−0.0567302	+	0.998390i	$0.518067\pi$
$620$	−13.4254	−0.539177
$621$	−1.47833	−0.0593233
$622$	7.26313	0.291225
$623$	11.0706	0.443533
$624$	6.73691	0.269692
$625$	−26.7009	−1.06804
$626$	18.9628	0.757908
$627$	−2.81157	−0.112283
$628$	−7.89429	−0.315016
$629$	4.75498	0.189593
$630$	−5.38873	−0.214692
$631$	−31.9158	−1.27055	−0.635275	−	0.772286i	$-0.719113\pi$
$631$	−31.9158	−1.27055	−0.635275	+	0.772286i	$0.719113\pi$
$632$	−2.64676	−0.105283
$633$	14.7561	0.586503
$634$	24.6712	0.979818
$635$	26.5327	1.05292
$636$	7.59985	0.301354
$637$	−10.7089	−0.424303
$638$	2.52579	0.0999968
$639$	−2.54207	−0.100563
$640$	−2.31671	−0.0915759
$641$	−1.15542	−0.0456363	−0.0228182	−	0.999740i	$-0.507264\pi$
$641$	−1.15542	−0.0456363	−0.0228182	+	0.999740i	$0.507264\pi$
$642$	10.9839	0.433499
$643$	3.88892	0.153364	0.0766820	−	0.997056i	$-0.475567\pi$
$643$	3.88892	0.153364	0.0766820	+	0.997056i	$0.475567\pi$
$644$	−3.43864	−0.135501
$645$	−22.5354	−0.887329
$646$	−9.41520	−0.370436
$647$	16.8470	0.662322	0.331161	−	0.943574i	$-0.392560\pi$
$647$	16.8470	0.662322	0.331161	+	0.943574i	$0.392560\pi$
$648$	1.00000	0.0392837
$649$	−1.78517	−0.0700739
$650$	2.47334	0.0970123
$651$	13.4794	0.528300
$652$	23.2664	0.911184
$653$	29.3701	1.14934	0.574671	−	0.818384i	$-0.305130\pi$
$653$	29.3701	1.14934	0.574671	+	0.818384i	$0.305130\pi$
$654$	7.69913	0.301060
$655$	−33.5626	−1.31140
$656$	1.17910	0.0460362
$657$	9.03818	0.352613
$658$	2.77078	0.108016
$659$	−8.76846	−0.341571	−0.170785	−	0.985308i	$-0.554630\pi$
$659$	−8.76846	−0.341571	−0.170785	+	0.985308i	$0.554630\pi$
$660$	2.31671	0.0901777
$661$	−12.1069	−0.470905	−0.235452	−	0.971886i	$-0.575657\pi$
$661$	−12.1069	−0.470905	−0.235452	+	0.971886i	$0.575657\pi$
$662$	−27.1064	−1.05352
$663$	−22.5601	−0.876161
$664$	−15.4467	−0.599448
$665$	−15.1508	−0.587523
$666$	−1.41994	−0.0550214
$667$	3.73394	0.144579
$668$	9.43400	0.365013
$669$	6.63858	0.256662
$670$	−16.9412	−0.654494
$671$	−1.00000	−0.0386046
$672$	2.32603	0.0897285
$673$	29.0617	1.12025	0.560123	−	0.828409i	$-0.310754\pi$
$673$	29.0617	1.12025	0.560123	+	0.828409i	$0.310754\pi$
$674$	−28.5619	−1.10016
$675$	0.367132	0.0141309
$676$	32.3860	1.24561
$677$	16.6272	0.639036	0.319518	−	0.947580i	$-0.396479\pi$
$677$	16.6272	0.639036	0.319518	+	0.947580i	$0.396479\pi$
$678$	16.3922	0.629540
$679$	−40.6289	−1.55919
$680$	7.75802	0.297507
$681$	24.0039	0.919833
$682$	−5.79504	−0.221903
$683$	−15.2491	−0.583492	−0.291746	−	0.956496i	$-0.594236\pi$
$683$	−15.2491	−0.583492	−0.291746	+	0.956496i	$0.594236\pi$
$684$	2.81157	0.107503
$685$	−13.1780	−0.503504
$686$	−19.9796	−0.762826
$687$	2.55902	0.0976325
$688$	9.72732	0.370851
$689$	51.1995	1.95055
$690$	3.42485	0.130382
$691$	−19.5506	−0.743739	−0.371869	−	0.928285i	$-0.621283\pi$
$691$	−19.5506	−0.743739	−0.371869	+	0.928285i	$0.621283\pi$
$692$	11.2868	0.429060
$693$	−2.32603	−0.0883585
$694$	−30.8843	−1.17235
$695$	17.6978	0.671317
$696$	−2.52579	−0.0957397
$697$	−3.94850	−0.149560
$698$	5.53946	0.209672
$699$	6.77289	0.256174
$700$	0.853960	0.0322767
$701$	−12.1800	−0.460033	−0.230016	−	0.973187i	$-0.573878\pi$
$701$	−12.1800	−0.460033	−0.230016	+	0.973187i	$0.573878\pi$
$702$	6.73691	0.254268
$703$	−3.99226	−0.150571
$704$	−1.00000	−0.0376889
$705$	−2.75968	−0.103935
$706$	−33.7973	−1.27198
$707$	−18.9404	−0.712328
$708$	1.78517	0.0670907
$709$	3.16586	0.118897	0.0594483	−	0.998231i	$-0.481066\pi$
$709$	3.16586	0.118897	0.0594483	+	0.998231i	$0.481066\pi$
$710$	5.88923	0.221019
$711$	−2.64676	−0.0992614
$712$	4.75943	0.178367
$713$	−8.56697	−0.320835
$714$	−7.78924	−0.291505
$715$	15.6074	0.583685
$716$	−1.24086	−0.0463731
$717$	13.3814	0.499739
$718$	−1.24214	−0.0463564
$719$	−10.6359	−0.396654	−0.198327	−	0.980136i	$-0.563551\pi$
$719$	−10.6359	−0.396654	−0.198327	+	0.980136i	$0.563551\pi$
$720$	−2.31671	−0.0863386
$721$	7.62885	0.284113
$722$	−11.0950	−0.412915
$723$	12.4268	0.462157
$724$	−3.88160	−0.144258
$725$	−0.927298	−0.0344390
$726$	1.00000	0.0371135
$727$	38.8432	1.44061	0.720307	−	0.693656i	$-0.244001\pi$
$727$	38.8432	1.44061	0.720307	+	0.693656i	$0.244001\pi$
$728$	15.6702	0.580778
$729$	1.00000	0.0370370
$730$	−20.9388	−0.774980
$731$	−32.5742	−1.20480
$732$	1.00000	0.0369611
$733$	46.8459	1.73029	0.865146	−	0.501520i	$-0.167225\pi$
$733$	46.8459	1.73029	0.865146	+	0.501520i	$0.167225\pi$
$734$	3.96096	0.146202
$735$	3.68261	0.135835
$736$	−1.47833	−0.0544919
$737$	−7.31260	−0.269363
$738$	1.17910	0.0434034
$739$	−12.4065	−0.456379	−0.228190	−	0.973617i	$-0.573281\pi$
$739$	−12.4065	−0.456379	−0.228190	+	0.973617i	$0.573281\pi$
$740$	3.28958	0.120927
$741$	18.9413	0.695827
$742$	17.6775	0.648961
$743$	16.9200	0.620735	0.310368	−	0.950617i	$-0.399548\pi$
$743$	16.9200	0.620735	0.310368	+	0.950617i	$0.399548\pi$
$744$	5.79504	0.212456
$745$	36.5613	1.33950
$746$	−11.2901	−0.413361
$747$	−15.4467	−0.565165
$748$	3.34873	0.122442
$749$	25.5488	0.933533
$750$	10.7330	0.391914
$751$	−49.5061	−1.80650	−0.903251	−	0.429113i	$-0.858826\pi$
$751$	−49.5061	−1.80650	−0.903251	+	0.429113i	$0.858826\pi$
$752$	1.19121	0.0434388
$753$	−0.933334	−0.0340126
$754$	−17.0160	−0.619686
$755$	26.8225	0.976171
$756$	2.32603	0.0845969
$757$	12.3869	0.450209	0.225104	−	0.974335i	$-0.427728\pi$
$757$	12.3869	0.450209	0.225104	+	0.974335i	$0.427728\pi$
$758$	−1.17428	−0.0426516
$759$	1.47833	0.0536599
$760$	−6.51360	−0.236273
$761$	−31.9129	−1.15684	−0.578420	−	0.815739i	$-0.696331\pi$
$761$	−31.9129	−1.15684	−0.578420	+	0.815739i	$0.696331\pi$
$762$	−11.4528	−0.414890
$763$	17.9084	0.648328
$764$	11.5303	0.417153
$765$	7.75802	0.280492
$766$	13.7988	0.498571
$767$	12.0265	0.434252
$768$	1.00000	0.0360844
$769$	42.3559	1.52739	0.763696	−	0.645576i	$-0.223383\pi$
$769$	42.3559	1.52739	0.763696	+	0.645576i	$0.223383\pi$
$770$	5.38873	0.194196
$771$	−9.97589	−0.359273
$772$	−8.73792	−0.314484
$773$	−17.2674	−0.621064	−0.310532	−	0.950563i	$-0.600507\pi$
$773$	−17.2674	−0.621064	−0.310532	+	0.950563i	$0.600507\pi$
$774$	9.72732	0.349641
$775$	2.12754	0.0764237
$776$	−17.4671	−0.627031
$777$	−3.30281	−0.118488
$778$	−19.1636	−0.687047
$779$	3.31514	0.118777
$780$	−15.6074	−0.558836
$781$	2.54207	0.0909623
$782$	4.95052	0.177030
$783$	−2.52579	−0.0902642
$784$	−1.58959	−0.0567710
$785$	18.2888	0.652754
$786$	14.4872	0.516741
$787$	34.5979	1.23328	0.616641	−	0.787244i	$-0.288493\pi$
$787$	34.5979	1.23328	0.616641	+	0.787244i	$0.288493\pi$
$788$	−23.2550	−0.828426
$789$	18.3659	0.653843
$790$	6.13178	0.218159
$791$	38.1288	1.35570
$792$	−1.00000	−0.0355335
$793$	6.73691	0.239235
$794$	−8.75564	−0.310726
$795$	−17.6066	−0.624443
$796$	−21.0823	−0.747241
$797$	22.4877	0.796555	0.398278	−	0.917265i	$-0.369608\pi$
$797$	22.4877	0.796555	0.398278	+	0.917265i	$0.369608\pi$
$798$	6.53981	0.231507
$799$	−3.98903	−0.141122
$800$	0.367132	0.0129801
$801$	4.75943	0.168166
$802$	6.31267	0.222908
$803$	−9.03818	−0.318950
$804$	7.31260	0.257896
$805$	7.96631	0.280776
$806$	39.0406	1.37515
$807$	−13.2661	−0.466988
$808$	−8.14281	−0.286463
$809$	29.0101	1.01994	0.509970	−	0.860192i	$-0.329657\pi$
$809$	29.0101	1.01994	0.509970	+	0.860192i	$0.329657\pi$
$810$	−2.31671	−0.0814008
$811$	49.2125	1.72808	0.864042	−	0.503420i	$-0.167925\pi$
$811$	49.2125	1.72808	0.864042	+	0.503420i	$0.167925\pi$
$812$	−5.87505	−0.206174
$813$	−9.93145	−0.348311
$814$	1.41994	0.0497687
$815$	−53.9015	−1.88809
$816$	−3.34873	−0.117229
$817$	27.3491	0.956824
$818$	−35.0417	−1.22521
$819$	15.6702	0.547563
$820$	−2.73164	−0.0953929
$821$	18.3090	0.638989	0.319494	−	0.947588i	$-0.396487\pi$
$821$	18.3090	0.638989	0.319494	+	0.947588i	$0.396487\pi$
$822$	5.68823	0.198400
$823$	36.4902	1.27197	0.635984	−	0.771703i	$-0.280595\pi$
$823$	36.4902	1.27197	0.635984	+	0.771703i	$0.280595\pi$
$824$	3.27977	0.114256
$825$	−0.367132	−0.0127819
$826$	4.15235	0.144479
$827$	−6.33099	−0.220150	−0.110075	−	0.993923i	$-0.535109\pi$
$827$	−6.33099	−0.220150	−0.110075	+	0.993923i	$0.535109\pi$
$828$	−1.47833	−0.0513755
$829$	31.5361	1.09529	0.547646	−	0.836710i	$-0.315524\pi$
$829$	31.5361	1.09529	0.547646	+	0.836710i	$0.315524\pi$
$830$	35.7854	1.24213
$831$	−25.7449	−0.893081
$832$	6.73691	0.233560
$833$	5.32310	0.184434
$834$	−7.63922	−0.264524
$835$	−21.8558	−0.756352
$836$	−2.81157	−0.0972404
$837$	5.79504	0.200306
$838$	13.0223	0.449849
$839$	−12.2715	−0.423659	−0.211829	−	0.977307i	$-0.567942\pi$
$839$	−12.2715	−0.423659	−0.211829	+	0.977307i	$0.567942\pi$
$840$	−5.38873	−0.185929
$841$	−22.6204	−0.780014
$842$	28.1361	0.969633
$843$	−8.14683	−0.280592
$844$	14.7561	0.507927
$845$	−75.0288	−2.58107
$846$	1.19121	0.0409545
$847$	2.32603	0.0799233
$848$	7.59985	0.260980
$849$	−30.3814	−1.04269
$850$	−1.22943	−0.0421690
$851$	2.09913	0.0719573
$852$	−2.54207	−0.0870898
$853$	32.2858	1.10544	0.552722	−	0.833366i	$-0.313589\pi$
$853$	32.2858	1.10544	0.552722	+	0.833366i	$0.313589\pi$
$854$	2.32603	0.0795951
$855$	−6.51360	−0.222760
$856$	10.9839	0.375421
$857$	8.24691	0.281709	0.140855	−	0.990030i	$-0.455015\pi$
$857$	8.24691	0.281709	0.140855	+	0.990030i	$0.455015\pi$
$858$	−6.73691	−0.229994
$859$	−57.9424	−1.97697	−0.988485	−	0.151321i	$-0.951647\pi$
$859$	−57.9424	−1.97697	−0.988485	+	0.151321i	$0.951647\pi$
$860$	−22.5354	−0.768449
$861$	2.74263	0.0934685
$862$	25.2728	0.860793
$863$	−6.50426	−0.221408	−0.110704	−	0.993853i	$-0.535310\pi$
$863$	−6.50426	−0.221408	−0.110704	+	0.993853i	$0.535310\pi$
$864$	1.00000	0.0340207
$865$	−26.1482	−0.889066
$866$	0.646235	0.0219600
$867$	−5.78602	−0.196503
$868$	13.4794	0.457521
$869$	2.64676	0.0897853
$870$	5.85151	0.198385
$871$	49.2643	1.66926
$872$	7.69913	0.260726
$873$	−17.4671	−0.591171
$874$	−4.15643	−0.140593
$875$	24.9653	0.843980
$876$	9.03818	0.305372
$877$	13.9445	0.470871	0.235436	−	0.971890i	$-0.424348\pi$
$877$	13.9445	0.470871	0.235436	+	0.971890i	$0.424348\pi$
$878$	−25.1067	−0.847309
$879$	−3.76071	−0.126846
$880$	2.31671	0.0780962
$881$	−18.6210	−0.627357	−0.313678	−	0.949529i	$-0.601561\pi$
$881$	−18.6210	−0.627357	−0.313678	+	0.949529i	$0.601561\pi$
$882$	−1.58959	−0.0535242
$883$	13.1569	0.442764	0.221382	−	0.975187i	$-0.428943\pi$
$883$	13.1569	0.442764	0.221382	+	0.975187i	$0.428943\pi$
$884$	−22.5601	−0.758778
$885$	−4.13571	−0.139020
$886$	−34.4261	−1.15657
$887$	3.49874	0.117476	0.0587381	−	0.998273i	$-0.481292\pi$
$887$	3.49874	0.117476	0.0587381	+	0.998273i	$0.481292\pi$
$888$	−1.41994	−0.0476499
$889$	−26.6395	−0.893459
$890$	−11.0262	−0.369599
$891$	−1.00000	−0.0335013
$892$	6.63858	0.222276
$893$	3.34917	0.112076
$894$	−15.7816	−0.527815
$895$	2.87471	0.0960909
$896$	2.32603	0.0777072
$897$	−9.95937	−0.332534
$898$	−32.4631	−1.08331
$899$	−14.6370	−0.488172
$900$	0.367132	0.0122377
$901$	−25.4498	−0.847857
$902$	−1.17910	−0.0392598
$903$	22.6260	0.752947
$904$	16.3922	0.545197
$905$	8.99252	0.298922
$906$	−11.5779	−0.384649
$907$	−5.33962	−0.177299	−0.0886495	−	0.996063i	$-0.528255\pi$
$907$	−5.33962	−0.177299	−0.0886495	+	0.996063i	$0.528255\pi$
$908$	24.0039	0.796598
$909$	−8.14281	−0.270080
$910$	−36.3034	−1.20345
$911$	47.1164	1.56104	0.780518	−	0.625133i	$-0.214955\pi$
$911$	47.1164	1.56104	0.780518	+	0.625133i	$0.214955\pi$
$912$	2.81157	0.0931005
$913$	15.4467	0.511211
$914$	6.03825	0.199728
$915$	−2.31671	−0.0765880
$916$	2.55902	0.0845523
$917$	33.6976	1.11279
$918$	−3.34873	−0.110525
$919$	−17.4361	−0.575163	−0.287582	−	0.957756i	$-0.592851\pi$
$919$	−17.4361	−0.575163	−0.287582	+	0.957756i	$0.592851\pi$
$920$	3.42485	0.112914
$921$	−11.0669	−0.364665
$922$	−16.1571	−0.532105
$923$	−17.1257	−0.563699
$924$	−2.32603	−0.0765207
$925$	−0.521304	−0.0171404
$926$	−30.7875	−1.01174
$927$	3.27977	0.107722
$928$	−2.52579	−0.0829130
$929$	4.30895	0.141372	0.0706860	−	0.997499i	$-0.477481\pi$
$929$	4.30895	0.141372	0.0706860	+	0.997499i	$0.477481\pi$
$930$	−13.4254	−0.440236
$931$	−4.46925	−0.146474
$932$	6.77289	0.221854
$933$	7.26313	0.237784
$934$	8.26907	0.270572
$935$	−7.75802	−0.253714
$936$	6.73691	0.220203
$937$	40.9374	1.33737	0.668683	−	0.743548i	$-0.266858\pi$
$937$	40.9374	1.33737	0.668683	+	0.743548i	$0.266858\pi$
$938$	17.0093	0.555374
$939$	18.9628	0.618829
$940$	−2.75968	−0.0900107
$941$	−3.94068	−0.128462	−0.0642312	−	0.997935i	$-0.520460\pi$
$941$	−3.94068	−0.128462	−0.0642312	+	0.997935i	$0.520460\pi$
$942$	−7.89429	−0.257210
$943$	−1.74310	−0.0567632
$944$	1.78517	0.0581022
$945$	−5.38873	−0.175295
$946$	−9.72732	−0.316262
$947$	−41.2397	−1.34011	−0.670055	−	0.742311i	$-0.733730\pi$
$947$	−41.2397	−1.34011	−0.670055	+	0.742311i	$0.733730\pi$
$948$	−2.64676	−0.0859629
$949$	60.8894	1.97655
$950$	1.03222	0.0334896
$951$	24.6712	0.800018
$952$	−7.78924	−0.252451
$953$	−48.0777	−1.55739	−0.778695	−	0.627402i	$-0.784118\pi$
$953$	−48.0777	−1.55739	−0.778695	+	0.627402i	$0.784118\pi$
$954$	7.59985	0.246054
$955$	−26.7124	−0.864394
$956$	13.3814	0.432786
$957$	2.52579	0.0816471
$958$	−12.7944	−0.413367
$959$	13.2310	0.427251
$960$	−2.31671	−0.0747714
$961$	2.58244	0.0833045
$962$	−9.56598	−0.308420
$963$	10.9839	0.353950
$964$	12.4268	0.400240
$965$	20.2432	0.651651
$966$	−3.43864	−0.110636
$967$	39.5651	1.27233	0.636164	−	0.771554i	$-0.280520\pi$
$967$	39.5651	1.27233	0.636164	+	0.771554i	$0.280520\pi$
$968$	1.00000	0.0321412
$969$	−9.41520	−0.302460
$970$	40.4661	1.29929
$971$	−21.9427	−0.704174	−0.352087	−	0.935967i	$-0.614528\pi$
$971$	−21.9427	−0.704174	−0.352087	+	0.935967i	$0.614528\pi$
$972$	1.00000	0.0320750
$973$	−17.7690	−0.569649
$974$	12.4786	0.399840
$975$	2.47334	0.0792102
$976$	1.00000	0.0320092
$977$	−8.95418	−0.286470	−0.143235	−	0.989689i	$-0.545750\pi$
$977$	−8.95418	−0.286470	−0.143235	+	0.989689i	$0.545750\pi$
$978$	23.2664	0.743979
$979$	−4.75943	−0.152112
$980$	3.68261	0.117637
$981$	7.69913	0.245814
$982$	−0.250763	−0.00800217
$983$	22.8798	0.729751	0.364875	−	0.931056i	$-0.381112\pi$
$983$	22.8798	0.729751	0.364875	+	0.931056i	$0.381112\pi$
$984$	1.17910	0.0375884
$985$	53.8751	1.71660
$986$	8.45817	0.269363
$987$	2.77078	0.0881950
$988$	18.9413	0.602604
$989$	−14.3802	−0.457263
$990$	2.31671	0.0736298
$991$	−22.9302	−0.728401	−0.364201	−	0.931321i	$-0.618658\pi$
$991$	−22.9302	−0.728401	−0.364201	+	0.931321i	$0.618658\pi$
$992$	5.79504	0.183993
$993$	−27.1064	−0.860195
$994$	−5.91292	−0.187547
$995$	48.8415	1.54838
$996$	−15.4467	−0.489447
$997$	−13.1184	−0.415463	−0.207731	−	0.978186i	$-0.566608\pi$
$997$	−13.1184	−0.415463	−0.207731	+	0.978186i	$0.566608\pi$
$998$	−24.5144	−0.775990
$999$	−1.41994	−0.0449248

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4026.2.a.bb.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4026.2.a.bb.1.2	✓	8	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:	\( N \)	\(=\)	\( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4026.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.31671 q^{5} +1.00000 q^{6} +2.32603 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.31671 q^{5} +1.00000 q^{6} +2.32603 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.31671 q^{10} -1.00000 q^{11} +1.00000 q^{12} +6.73691 q^{13} +2.32603 q^{14} -2.31671 q^{15} +1.00000 q^{16} -3.34873 q^{17} +1.00000 q^{18} +2.81157 q^{19} -2.31671 q^{20} +2.32603 q^{21} -1.00000 q^{22} -1.47833 q^{23} +1.00000 q^{24} +0.367132 q^{25} +6.73691 q^{26} +1.00000 q^{27} +2.32603 q^{28} -2.52579 q^{29} -2.31671 q^{30} +5.79504 q^{31} +1.00000 q^{32} -1.00000 q^{33} -3.34873 q^{34} -5.38873 q^{35} +1.00000 q^{36} -1.41994 q^{37} +2.81157 q^{38} +6.73691 q^{39} -2.31671 q^{40} +1.17910 q^{41} +2.32603 q^{42} +9.72732 q^{43} -1.00000 q^{44} -2.31671 q^{45} -1.47833 q^{46} +1.19121 q^{47} +1.00000 q^{48} -1.58959 q^{49} +0.367132 q^{50} -3.34873 q^{51} +6.73691 q^{52} +7.59985 q^{53} +1.00000 q^{54} +2.31671 q^{55} +2.32603 q^{56} +2.81157 q^{57} -2.52579 q^{58} +1.78517 q^{59} -2.31671 q^{60} +1.00000 q^{61} +5.79504 q^{62} +2.32603 q^{63} +1.00000 q^{64} -15.6074 q^{65} -1.00000 q^{66} +7.31260 q^{67} -3.34873 q^{68} -1.47833 q^{69} -5.38873 q^{70} -2.54207 q^{71} +1.00000 q^{72} +9.03818 q^{73} -1.41994 q^{74} +0.367132 q^{75} +2.81157 q^{76} -2.32603 q^{77} +6.73691 q^{78} -2.64676 q^{79} -2.31671 q^{80} +1.00000 q^{81} +1.17910 q^{82} -15.4467 q^{83} +2.32603 q^{84} +7.75802 q^{85} +9.72732 q^{86} -2.52579 q^{87} -1.00000 q^{88} +4.75943 q^{89} -2.31671 q^{90} +15.6702 q^{91} -1.47833 q^{92} +5.79504 q^{93} +1.19121 q^{94} -6.51360 q^{95} +1.00000 q^{96} -17.4671 q^{97} -1.58959 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 5 q^{5} + 8 q^{6} + 13 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 + 5 * q^5 + 8 * q^6 + 13 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 5 q^{5} + 8 q^{6} + 13 q^{7} + 8 q^{8} + 8 q^{9} + 5 q^{10} - 8 q^{11} + 8 q^{12} + 10 q^{13} + 13 q^{14} + 5 q^{15} + 8 q^{16} + 4 q^{17} + 8 q^{18} + 11 q^{19} + 5 q^{20} + 13 q^{21} - 8 q^{22} + 2 q^{23} + 8 q^{24} + 23 q^{25} + 10 q^{26} + 8 q^{27} + 13 q^{28} + 10 q^{29} + 5 q^{30} + 9 q^{31} + 8 q^{32} - 8 q^{33} + 4 q^{34} - 3 q^{35} + 8 q^{36} + 9 q^{37} + 11 q^{38} + 10 q^{39} + 5 q^{40} + 3 q^{41} + 13 q^{42} + 16 q^{43} - 8 q^{44} + 5 q^{45} + 2 q^{46} - 16 q^{47} + 8 q^{48} + 17 q^{49} + 23 q^{50} + 4 q^{51} + 10 q^{52} + 7 q^{53} + 8 q^{54} - 5 q^{55} + 13 q^{56} + 11 q^{57} + 10 q^{58} - 14 q^{59} + 5 q^{60} + 8 q^{61} + 9 q^{62} + 13 q^{63} + 8 q^{64} + 22 q^{65} - 8 q^{66} + 8 q^{67} + 4 q^{68} + 2 q^{69} - 3 q^{70} + 11 q^{71} + 8 q^{72} + 14 q^{73} + 9 q^{74} + 23 q^{75} + 11 q^{76} - 13 q^{77} + 10 q^{78} + 22 q^{79} + 5 q^{80} + 8 q^{81} + 3 q^{82} - 16 q^{83} + 13 q^{84} + 3 q^{85} + 16 q^{86} + 10 q^{87} - 8 q^{88} + q^{89} + 5 q^{90} + 15 q^{91} + 2 q^{92} + 9 q^{93} - 16 q^{94} - 9 q^{95} + 8 q^{96} + 24 q^{97} + 17 q^{98} - 8 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 + 5 * q^5 + 8 * q^6 + 13 * q^7 + 8 * q^8 + 8 * q^9 + 5 * q^10 - 8 * q^11 + 8 * q^12 + 10 * q^13 + 13 * q^14 + 5 * q^15 + 8 * q^16 + 4 * q^17 + 8 * q^18 + 11 * q^19 + 5 * q^20 + 13 * q^21 - 8 * q^22 + 2 * q^23 + 8 * q^24 + 23 * q^25 + 10 * q^26 + 8 * q^27 + 13 * q^28 + 10 * q^29 + 5 * q^30 + 9 * q^31 + 8 * q^32 - 8 * q^33 + 4 * q^34 - 3 * q^35 + 8 * q^36 + 9 * q^37 + 11 * q^38 + 10 * q^39 + 5 * q^40 + 3 * q^41 + 13 * q^42 + 16 * q^43 - 8 * q^44 + 5 * q^45 + 2 * q^46 - 16 * q^47 + 8 * q^48 + 17 * q^49 + 23 * q^50 + 4 * q^51 + 10 * q^52 + 7 * q^53 + 8 * q^54 - 5 * q^55 + 13 * q^56 + 11 * q^57 + 10 * q^58 - 14 * q^59 + 5 * q^60 + 8 * q^61 + 9 * q^62 + 13 * q^63 + 8 * q^64 + 22 * q^65 - 8 * q^66 + 8 * q^67 + 4 * q^68 + 2 * q^69 - 3 * q^70 + 11 * q^71 + 8 * q^72 + 14 * q^73 + 9 * q^74 + 23 * q^75 + 11 * q^76 - 13 * q^77 + 10 * q^78 + 22 * q^79 + 5 * q^80 + 8 * q^81 + 3 * q^82 - 16 * q^83 + 13 * q^84 + 3 * q^85 + 16 * q^86 + 10 * q^87 - 8 * q^88 + q^89 + 5 * q^90 + 15 * q^91 + 2 * q^92 + 9 * q^93 - 16 * q^94 - 9 * q^95 + 8 * q^96 + 24 * q^97 + 17 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more