LMFDB - Embedded newform 6270.2.a.bv.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6270,2,Mod(1,6270)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6270, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6270.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6270 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6270.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:	$50.0662020673$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 23x^{4} - 21x^{3} + 38x^{2} + 14x - 12 $ x^6 - x^5 - 23x^4 - 21x^3 + 38x^2 + 14x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.15029$ of defining polynomial
Character	$\chi$	$=$	6270.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -4.58292 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -4.58292 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -6.26262 q^{13} +4.58292 q^{14} +1.00000 q^{15} +1.00000 q^{16} +5.89254 q^{17} -1.00000 q^{18} +1.00000 q^{19} -1.00000 q^{20} +4.58292 q^{21} +1.00000 q^{22} -4.86963 q^{23} +1.00000 q^{24} +1.00000 q^{25} +6.26262 q^{26} -1.00000 q^{27} -4.58292 q^{28} -4.17489 q^{29} -1.00000 q^{30} +0.286713 q^{31} -1.00000 q^{32} +1.00000 q^{33} -5.89254 q^{34} +4.58292 q^{35} +1.00000 q^{36} -11.6600 q^{37} -1.00000 q^{38} +6.26262 q^{39} +1.00000 q^{40} +5.89254 q^{41} -4.58292 q^{42} -9.16584 q^{43} -1.00000 q^{44} -1.00000 q^{45} +4.86963 q^{46} -6.99095 q^{47} -1.00000 q^{48} +14.0032 q^{49} -1.00000 q^{50} -5.89254 q^{51} -6.26262 q^{52} +0.0379527 q^{53} +1.00000 q^{54} +1.00000 q^{55} +4.58292 q^{56} -1.00000 q^{57} +4.17489 q^{58} -7.89254 q^{59} +1.00000 q^{60} -2.09209 q^{61} -0.286713 q^{62} -4.58292 q^{63} +1.00000 q^{64} +6.26262 q^{65} -1.00000 q^{66} -10.4711 q^{67} +5.89254 q^{68} +4.86963 q^{69} -4.58292 q^{70} -14.5968 q^{71} -1.00000 q^{72} -0.815502 q^{73} +11.6600 q^{74} -1.00000 q^{75} +1.00000 q^{76} +4.58292 q^{77} -6.26262 q^{78} -14.2429 q^{79} -1.00000 q^{80} +1.00000 q^{81} -5.89254 q^{82} -7.02291 q^{83} +4.58292 q^{84} -5.89254 q^{85} +9.16584 q^{86} +4.17489 q^{87} +1.00000 q^{88} -16.2429 q^{89} +1.00000 q^{90} +28.7011 q^{91} -4.86963 q^{92} -0.286713 q^{93} +6.99095 q^{94} -1.00000 q^{95} +1.00000 q^{96} -4.20849 q^{97} -14.0032 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} - 6 q^{3} + 6 q^{4} - 6 q^{5} + 6 q^{6} - 6 q^{8} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^2 - 6 * q^3 + 6 * q^4 - 6 * q^5 + 6 * q^6 - 6 * q^8 + 6 * q^9	$ 6 q - 6 q^{2} - 6 q^{3} + 6 q^{4} - 6 q^{5} + 6 q^{6} - 6 q^{8} + 6 q^{9} + 6 q^{10} - 6 q^{11} - 6 q^{12} + 2 q^{13} + 6 q^{15} + 6 q^{16} - 4 q^{17} - 6 q^{18} + 6 q^{19} - 6 q^{20} + 6 q^{22} - 4 q^{23} + 6 q^{24} + 6 q^{25} - 2 q^{26} - 6 q^{27} + 2 q^{29} - 6 q^{30} + 4 q^{31} - 6 q^{32} + 6 q^{33} + 4 q^{34} + 6 q^{36} - 6 q^{37} - 6 q^{38} - 2 q^{39} + 6 q^{40} - 4 q^{41} - 6 q^{44} - 6 q^{45} + 4 q^{46} - 14 q^{47} - 6 q^{48} + 18 q^{49} - 6 q^{50} + 4 q^{51} + 2 q^{52} + 6 q^{54} + 6 q^{55} - 6 q^{57} - 2 q^{58} - 8 q^{59} + 6 q^{60} + 10 q^{61} - 4 q^{62} + 6 q^{64} - 2 q^{65} - 6 q^{66} - 6 q^{67} - 4 q^{68} + 4 q^{69} - 18 q^{71} - 6 q^{72} - 2 q^{73} + 6 q^{74} - 6 q^{75} + 6 q^{76} + 2 q^{78} + 6 q^{79} - 6 q^{80} + 6 q^{81} + 4 q^{82} - 28 q^{83} + 4 q^{85} - 2 q^{87} + 6 q^{88} - 6 q^{89} + 6 q^{90} + 42 q^{91} - 4 q^{92} - 4 q^{93} + 14 q^{94} - 6 q^{95} + 6 q^{96} - 8 q^{97} - 18 q^{98} - 6 q^{99}+O(q^{100}) $ 6 * q - 6 * q^2 - 6 * q^3 + 6 * q^4 - 6 * q^5 + 6 * q^6 - 6 * q^8 + 6 * q^9 + 6 * q^10 - 6 * q^11 - 6 * q^12 + 2 * q^13 + 6 * q^15 + 6 * q^16 - 4 * q^17 - 6 * q^18 + 6 * q^19 - 6 * q^20 + 6 * q^22 - 4 * q^23 + 6 * q^24 + 6 * q^25 - 2 * q^26 - 6 * q^27 + 2 * q^29 - 6 * q^30 + 4 * q^31 - 6 * q^32 + 6 * q^33 + 4 * q^34 + 6 * q^36 - 6 * q^37 - 6 * q^38 - 2 * q^39 + 6 * q^40 - 4 * q^41 - 6 * q^44 - 6 * q^45 + 4 * q^46 - 14 * q^47 - 6 * q^48 + 18 * q^49 - 6 * q^50 + 4 * q^51 + 2 * q^52 + 6 * q^54 + 6 * q^55 - 6 * q^57 - 2 * q^58 - 8 * q^59 + 6 * q^60 + 10 * q^61 - 4 * q^62 + 6 * q^64 - 2 * q^65 - 6 * q^66 - 6 * q^67 - 4 * q^68 + 4 * q^69 - 18 * q^71 - 6 * q^72 - 2 * q^73 + 6 * q^74 - 6 * q^75 + 6 * q^76 + 2 * q^78 + 6 * q^79 - 6 * q^80 + 6 * q^81 + 4 * q^82 - 28 * q^83 + 4 * q^85 - 2 * q^87 + 6 * q^88 - 6 * q^89 + 6 * q^90 + 42 * q^91 - 4 * q^92 - 4 * q^93 + 14 * q^94 - 6 * q^95 + 6 * q^96 - 8 * q^97 - 18 * q^98 - 6 * 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.00000	0.408248
$7$	−4.58292	−1.73218	−0.866091	−	0.499887i	$-0.833375\pi$
$7$	−4.58292	−1.73218	−0.866091	+	0.499887i	$0.833375\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.00000	−0.288675
$13$	−6.26262	−1.73694	−0.868469	−	0.495744i	$-0.834896\pi$
$13$	−6.26262	−1.73694	−0.868469	+	0.495744i	$0.834896\pi$
$14$	4.58292	1.22484
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	5.89254	1.42915	0.714576	−	0.699558i	$-0.246620\pi$
$17$	5.89254	1.42915	0.714576	+	0.699558i	$0.246620\pi$
$18$	−1.00000	−0.235702
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−1.00000	−0.223607
$21$	4.58292	1.00008
$22$	1.00000	0.213201
$23$	−4.86963	−1.01539	−0.507694	−	0.861537i	$-0.669502\pi$
$23$	−4.86963	−1.01539	−0.507694	+	0.861537i	$0.669502\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	6.26262	1.22820
$27$	−1.00000	−0.192450
$28$	−4.58292	−0.866091
$29$	−4.17489	−0.775258	−0.387629	−	0.921815i	$-0.626706\pi$
$29$	−4.17489	−0.775258	−0.387629	+	0.921815i	$0.626706\pi$
$30$	−1.00000	−0.182574
$31$	0.286713	0.0514951	0.0257475	−	0.999668i	$-0.491803\pi$
$31$	0.286713	0.0514951	0.0257475	+	0.999668i	$0.491803\pi$
$32$	−1.00000	−0.176777
$33$	1.00000	0.174078
$34$	−5.89254	−1.01056
$35$	4.58292	0.774655
$36$	1.00000	0.166667
$37$	−11.6600	−1.91689	−0.958443	−	0.285284i	$-0.907912\pi$
$37$	−11.6600	−1.91689	−0.958443	+	0.285284i	$0.907912\pi$
$38$	−1.00000	−0.162221
$39$	6.26262	1.00282
$40$	1.00000	0.158114
$41$	5.89254	0.920260	0.460130	−	0.887851i	$-0.347803\pi$
$41$	5.89254	0.920260	0.460130	+	0.887851i	$0.347803\pi$
$42$	−4.58292	−0.707160
$43$	−9.16584	−1.39778	−0.698889	−	0.715230i	$-0.746322\pi$
$43$	−9.16584	−1.39778	−0.698889	+	0.715230i	$0.746322\pi$
$44$	−1.00000	−0.150756
$45$	−1.00000	−0.149071
$46$	4.86963	0.717988
$47$	−6.99095	−1.01973	−0.509867	−	0.860253i	$-0.670306\pi$
$47$	−6.99095	−1.01973	−0.509867	+	0.860253i	$0.670306\pi$
$48$	−1.00000	−0.144338
$49$	14.0032	2.00045
$50$	−1.00000	−0.141421
$51$	−5.89254	−0.825121
$52$	−6.26262	−0.868469
$53$	0.0379527	0.00521320	0.00260660	−	0.999997i	$-0.499170\pi$
$53$	0.0379527	0.00521320	0.00260660	+	0.999997i	$0.499170\pi$
$54$	1.00000	0.136083
$55$	1.00000	0.134840
$56$	4.58292	0.612419
$57$	−1.00000	−0.132453
$58$	4.17489	0.548190
$59$	−7.89254	−1.02752	−0.513761	−	0.857934i	$-0.671748\pi$
$59$	−7.89254	−1.02752	−0.513761	+	0.857934i	$0.671748\pi$
$60$	1.00000	0.129099
$61$	−2.09209	−0.267864	−0.133932	−	0.990991i	$-0.542760\pi$
$61$	−2.09209	−0.267864	−0.133932	+	0.990991i	$0.542760\pi$
$62$	−0.286713	−0.0364125
$63$	−4.58292	−0.577394
$64$	1.00000	0.125000
$65$	6.26262	0.776782
$66$	−1.00000	−0.123091
$67$	−10.4711	−1.27925	−0.639624	−	0.768688i	$-0.720910\pi$
$67$	−10.4711	−1.27925	−0.639624	+	0.768688i	$0.720910\pi$
$68$	5.89254	0.714576
$69$	4.86963	0.586235
$70$	−4.58292	−0.547764
$71$	−14.5968	−1.73232	−0.866160	−	0.499768i	$-0.833419\pi$
$71$	−14.5968	−1.73232	−0.866160	+	0.499768i	$0.833419\pi$
$72$	−1.00000	−0.117851
$73$	−0.815502	−0.0954473	−0.0477236	−	0.998861i	$-0.515197\pi$
$73$	−0.815502	−0.0954473	−0.0477236	+	0.998861i	$0.515197\pi$
$74$	11.6600	1.35544
$75$	−1.00000	−0.115470
$76$	1.00000	0.114708
$77$	4.58292	0.522272
$78$	−6.26262	−0.709102
$79$	−14.2429	−1.60245	−0.801225	−	0.598363i	$-0.795818\pi$
$79$	−14.2429	−1.60245	−0.801225	+	0.598363i	$0.795818\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−5.89254	−0.650722
$83$	−7.02291	−0.770864	−0.385432	−	0.922736i	$-0.625948\pi$
$83$	−7.02291	−0.770864	−0.385432	+	0.922736i	$0.625948\pi$
$84$	4.58292	0.500038
$85$	−5.89254	−0.639136
$86$	9.16584	0.988378
$87$	4.17489	0.447596
$88$	1.00000	0.106600
$89$	−16.2429	−1.72174	−0.860871	−	0.508823i	$-0.830081\pi$
$89$	−16.2429	−1.72174	−0.860871	+	0.508823i	$0.830081\pi$
$90$	1.00000	0.105409
$91$	28.7011	3.00869
$92$	−4.86963	−0.507694
$93$	−0.286713	−0.0297307
$94$	6.99095	0.721061
$95$	−1.00000	−0.102598
$96$	1.00000	0.102062
$97$	−4.20849	−0.427307	−0.213653	−	0.976910i	$-0.568536\pi$
$97$	−4.20849	−0.427307	−0.213653	+	0.976910i	$0.568536\pi$
$98$	−14.0032	−1.41453
$99$	−1.00000	−0.100504
$100$	1.00000	0.100000
$101$	−5.60583	−0.557801	−0.278900	−	0.960320i	$-0.589970\pi$
$101$	−5.60583	−0.557801	−0.278900	+	0.960320i	$0.589970\pi$
$102$	5.89254	0.583449
$103$	15.8684	1.56356	0.781782	−	0.623551i	$-0.214311\pi$
$103$	15.8684	1.56356	0.781782	+	0.623551i	$0.214311\pi$
$104$	6.26262	0.614100
$105$	−4.58292	−0.447247
$106$	−0.0379527	−0.00368629
$107$	−6.69474	−0.647205	−0.323603	−	0.946193i	$-0.604894\pi$
$107$	−6.69474	−0.647205	−0.323603	+	0.946193i	$0.604894\pi$
$108$	−1.00000	−0.0962250
$109$	−1.89254	−0.181273	−0.0906363	−	0.995884i	$-0.528890\pi$
$109$	−1.89254	−0.181273	−0.0906363	+	0.995884i	$0.528890\pi$
$110$	−1.00000	−0.0953463
$111$	11.6600	1.10671
$112$	−4.58292	−0.433045
$113$	5.04453	0.474549	0.237275	−	0.971443i	$-0.423746\pi$
$113$	5.04453	0.474549	0.237275	+	0.971443i	$0.423746\pi$
$114$	1.00000	0.0936586
$115$	4.86963	0.454096
$116$	−4.17489	−0.387629
$117$	−6.26262	−0.578979
$118$	7.89254	0.726567
$119$	−27.0051	−2.47555
$120$	−1.00000	−0.0912871
$121$	1.00000	0.0909091
$122$	2.09209	0.189408
$123$	−5.89254	−0.531313
$124$	0.286713	0.0257475
$125$	−1.00000	−0.0894427
$126$	4.58292	0.408279
$127$	14.6466	1.29967	0.649836	−	0.760075i	$-0.274838\pi$
$127$	14.6466	1.29967	0.649836	+	0.760075i	$0.274838\pi$
$128$	−1.00000	−0.0883883
$129$	9.16584	0.807008
$130$	−6.26262	−0.549268
$131$	6.05413	0.528952	0.264476	−	0.964392i	$-0.414801\pi$
$131$	6.05413	0.528952	0.264476	+	0.964392i	$0.414801\pi$
$132$	1.00000	0.0870388
$133$	−4.58292	−0.397390
$134$	10.4711	0.904566
$135$	1.00000	0.0860663
$136$	−5.89254	−0.505281
$137$	3.07704	0.262889	0.131445	−	0.991324i	$-0.458038\pi$
$137$	3.07704	0.262889	0.131445	+	0.991324i	$0.458038\pi$
$138$	−4.86963	−0.414531
$139$	2.77319	0.235219	0.117609	−	0.993060i	$-0.462477\pi$
$139$	2.77319	0.235219	0.117609	+	0.993060i	$0.462477\pi$
$140$	4.58292	0.387328
$141$	6.99095	0.588744
$142$	14.5968	1.22493
$143$	6.26262	0.523706
$144$	1.00000	0.0833333
$145$	4.17489	0.346706
$146$	0.815502	0.0674914
$147$	−14.0032	−1.15496
$148$	−11.6600	−0.958443
$149$	−3.79938	−0.311257	−0.155629	−	0.987816i	$-0.549740\pi$
$149$	−3.79938	−0.311257	−0.155629	+	0.987816i	$0.549740\pi$
$150$	1.00000	0.0816497
$151$	−1.96749	−0.160112	−0.0800560	−	0.996790i	$-0.525510\pi$
$151$	−1.96749	−0.160112	−0.0800560	+	0.996790i	$0.525510\pi$
$152$	−1.00000	−0.0811107
$153$	5.89254	0.476384
$154$	−4.58292	−0.369302
$155$	−0.286713	−0.0230293
$156$	6.26262	0.501411
$157$	5.02291	0.400872	0.200436	−	0.979707i	$-0.435764\pi$
$157$	5.02291	0.400872	0.200436	+	0.979707i	$0.435764\pi$
$158$	14.2429	1.13310
$159$	−0.0379527	−0.00300984
$160$	1.00000	0.0790569
$161$	22.3172	1.75884
$162$	−1.00000	−0.0785674
$163$	−12.2962	−0.963113	−0.481557	−	0.876415i	$-0.659928\pi$
$163$	−12.2962	−0.963113	−0.481557	+	0.876415i	$0.659928\pi$
$164$	5.89254	0.460130
$165$	−1.00000	−0.0778499
$166$	7.02291	0.545083
$167$	18.3317	1.41855	0.709274	−	0.704933i	$-0.249023\pi$
$167$	18.3317	1.41855	0.709274	+	0.704933i	$0.249023\pi$
$168$	−4.58292	−0.353580
$169$	26.2204	2.01695
$170$	5.89254	0.451937
$171$	1.00000	0.0764719
$172$	−9.16584	−0.698889
$173$	−11.4894	−0.873525	−0.436763	−	0.899577i	$-0.643875\pi$
$173$	−11.4894	−0.873525	−0.436763	+	0.899577i	$0.643875\pi$
$174$	−4.17489	−0.316498
$175$	−4.58292	−0.346436
$176$	−1.00000	−0.0753778
$177$	7.89254	0.593240
$178$	16.2429	1.21746
$179$	1.27330	0.0951711	0.0475855	−	0.998867i	$-0.484847\pi$
$179$	1.27330	0.0951711	0.0475855	+	0.998867i	$0.484847\pi$
$180$	−1.00000	−0.0745356
$181$	−12.0724	−0.897331	−0.448665	−	0.893700i	$-0.648100\pi$
$181$	−12.0724	−0.897331	−0.448665	+	0.893700i	$0.648100\pi$
$182$	−28.7011	−2.12747
$183$	2.09209	0.154651
$184$	4.86963	0.358994
$185$	11.6600	0.857257
$186$	0.286713	0.0210228
$187$	−5.89254	−0.430905
$188$	−6.99095	−0.509867
$189$	4.58292	0.333359
$190$	1.00000	0.0725476
$191$	−0.748873	−0.0541866	−0.0270933	−	0.999633i	$-0.508625\pi$
$191$	−0.748873	−0.0541866	−0.0270933	+	0.999633i	$0.508625\pi$
$192$	−1.00000	−0.0721688
$193$	12.1562	0.875025	0.437513	−	0.899212i	$-0.355860\pi$
$193$	12.1562	0.875025	0.437513	+	0.899212i	$0.355860\pi$
$194$	4.20849	0.302152
$195$	−6.26262	−0.448475
$196$	14.0032	1.00023
$197$	19.9649	1.42244	0.711220	−	0.702970i	$-0.248143\pi$
$197$	19.9649	1.42244	0.711220	+	0.702970i	$0.248143\pi$
$198$	1.00000	0.0710669
$199$	20.6883	1.46656	0.733278	−	0.679929i	$-0.237990\pi$
$199$	20.6883	1.46656	0.733278	+	0.679929i	$0.237990\pi$
$200$	−1.00000	−0.0707107
$201$	10.4711	0.738575
$202$	5.60583	0.394425
$203$	19.1332	1.34289
$204$	−5.89254	−0.412560
$205$	−5.89254	−0.411553
$206$	−15.8684	−1.10561
$207$	−4.86963	−0.338463
$208$	−6.26262	−0.434234
$209$	−1.00000	−0.0691714
$210$	4.58292	0.316252
$211$	15.7764	1.08609	0.543045	−	0.839704i	$-0.317271\pi$
$211$	15.7764	1.08609	0.543045	+	0.839704i	$0.317271\pi$
$212$	0.0379527	0.00260660
$213$	14.5968	1.00015
$214$	6.69474	0.457643
$215$	9.16584	0.625105
$216$	1.00000	0.0680414
$217$	−1.31398	−0.0891988
$218$	1.89254	0.128179
$219$	0.815502	0.0551065
$220$	1.00000	0.0674200
$221$	−36.9027	−2.48235
$222$	−11.6600	−0.782565
$223$	−1.86656	−0.124994	−0.0624971	−	0.998045i	$-0.519906\pi$
$223$	−1.86656	−0.124994	−0.0624971	+	0.998045i	$0.519906\pi$
$224$	4.58292	0.306209
$225$	1.00000	0.0666667
$226$	−5.04453	−0.335557
$227$	7.86058	0.521725	0.260863	−	0.965376i	$-0.415993\pi$
$227$	7.86058	0.521725	0.260863	+	0.965376i	$0.415993\pi$
$228$	−1.00000	−0.0662266
$229$	−9.52652	−0.629530	−0.314765	−	0.949170i	$-0.601926\pi$
$229$	−9.52652	−0.629530	−0.314765	+	0.949170i	$0.601926\pi$
$230$	−4.86963	−0.321094
$231$	−4.58292	−0.301534
$232$	4.17489	0.274095
$233$	−6.88294	−0.450916	−0.225458	−	0.974253i	$-0.572388\pi$
$233$	−6.88294	−0.450916	−0.225458	+	0.974253i	$0.572388\pi$
$234$	6.26262	0.409400
$235$	6.99095	0.456039
$236$	−7.89254	−0.513761
$237$	14.2429	0.925175
$238$	27.0051	1.75048
$239$	10.2251	0.661405	0.330702	−	0.943735i	$-0.392714\pi$
$239$	10.2251	0.661405	0.330702	+	0.943735i	$0.392714\pi$
$240$	1.00000	0.0645497
$241$	−3.03677	−0.195615	−0.0978076	−	0.995205i	$-0.531183\pi$
$241$	−3.03677	−0.195615	−0.0978076	+	0.995205i	$0.531183\pi$
$242$	−1.00000	−0.0642824
$243$	−1.00000	−0.0641500
$244$	−2.09209	−0.133932
$245$	−14.0032	−0.894630
$246$	5.89254	0.375695
$247$	−6.26262	−0.398481
$248$	−0.286713	−0.0182063
$249$	7.02291	0.445059
$250$	1.00000	0.0632456
$251$	−28.7966	−1.81762	−0.908812	−	0.417206i	$-0.863009\pi$
$251$	−28.7966	−1.81762	−0.908812	+	0.417206i	$0.863009\pi$
$252$	−4.58292	−0.288697
$253$	4.86963	0.306151
$254$	−14.6466	−0.919006
$255$	5.89254	0.369005
$256$	1.00000	0.0625000
$257$	−27.8582	−1.73775	−0.868874	−	0.495034i	$-0.835156\pi$
$257$	−27.8582	−1.73775	−0.868874	+	0.495034i	$0.835156\pi$
$258$	−9.16584	−0.570641
$259$	53.4367	3.32039
$260$	6.26262	0.388391
$261$	−4.17489	−0.258419
$262$	−6.05413	−0.374026
$263$	−20.2070	−1.24602	−0.623008	−	0.782216i	$-0.714089\pi$
$263$	−20.2070	−1.24602	−0.623008	+	0.782216i	$0.714089\pi$
$264$	−1.00000	−0.0615457
$265$	−0.0379527	−0.00233142
$266$	4.58292	0.280997
$267$	16.2429	0.994048
$268$	−10.4711	−0.639624
$269$	6.17925	0.376756	0.188378	−	0.982097i	$-0.439677\pi$
$269$	6.17925	0.376756	0.188378	+	0.982097i	$0.439677\pi$
$270$	−1.00000	−0.0608581
$271$	−1.41228	−0.0857900	−0.0428950	−	0.999080i	$-0.513658\pi$
$271$	−1.41228	−0.0857900	−0.0428950	+	0.999080i	$0.513658\pi$
$272$	5.89254	0.357288
$273$	−28.7011	−1.73707
$274$	−3.07704	−0.185891
$275$	−1.00000	−0.0603023
$276$	4.86963	0.293118
$277$	−2.06307	−0.123958	−0.0619791	−	0.998077i	$-0.519741\pi$
$277$	−2.06307	−0.123958	−0.0619791	+	0.998077i	$0.519741\pi$
$278$	−2.77319	−0.166325
$279$	0.286713	0.0171650
$280$	−4.58292	−0.273882
$281$	−3.52139	−0.210068	−0.105034	−	0.994469i	$-0.533495\pi$
$281$	−3.52139	−0.210068	−0.105034	+	0.994469i	$0.533495\pi$
$282$	−6.99095	−0.416305
$283$	−22.5711	−1.34171	−0.670855	−	0.741589i	$-0.734073\pi$
$283$	−22.5711	−1.34171	−0.670855	+	0.741589i	$0.734073\pi$
$284$	−14.5968	−0.866160
$285$	1.00000	0.0592349
$286$	−6.26262	−0.370316
$287$	−27.0051	−1.59406
$288$	−1.00000	−0.0589256
$289$	17.7221	1.04247
$290$	−4.17489	−0.245158
$291$	4.20849	0.246706
$292$	−0.815502	−0.0477236
$293$	6.51651	0.380699	0.190349	−	0.981716i	$-0.439038\pi$
$293$	6.51651	0.380699	0.190349	+	0.981716i	$0.439038\pi$
$294$	14.0032	0.816682
$295$	7.89254	0.459522
$296$	11.6600	0.677722
$297$	1.00000	0.0580259
$298$	3.79938	0.220092
$299$	30.4967	1.76367
$300$	−1.00000	−0.0577350
$301$	42.0063	2.42121
$302$	1.96749	0.113216
$303$	5.60583	0.322046
$304$	1.00000	0.0573539
$305$	2.09209	0.119792
$306$	−5.89254	−0.336854
$307$	12.4257	0.709172	0.354586	−	0.935023i	$-0.384622\pi$
$307$	12.4257	0.709172	0.354586	+	0.935023i	$0.384622\pi$
$308$	4.58292	0.261136
$309$	−15.8684	−0.902724
$310$	0.286713	0.0162842
$311$	−11.8846	−0.673915	−0.336958	−	0.941520i	$-0.609398\pi$
$311$	−11.8846	−0.673915	−0.336958	+	0.941520i	$0.609398\pi$
$312$	−6.26262	−0.354551
$313$	30.5530	1.72695	0.863477	−	0.504387i	$-0.168282\pi$
$313$	30.5530	1.72695	0.863477	+	0.504387i	$0.168282\pi$
$314$	−5.02291	−0.283459
$315$	4.58292	0.258218
$316$	−14.2429	−0.801225
$317$	−22.7045	−1.27521	−0.637606	−	0.770363i	$-0.720075\pi$
$317$	−22.7045	−1.27521	−0.637606	+	0.770363i	$0.720075\pi$
$318$	0.0379527	0.00212828
$319$	4.17489	0.233749
$320$	−1.00000	−0.0559017
$321$	6.69474	0.373664
$322$	−22.3172	−1.24369
$323$	5.89254	0.327870
$324$	1.00000	0.0555556
$325$	−6.26262	−0.347388
$326$	12.2962	0.681024
$327$	1.89254	0.104658
$328$	−5.89254	−0.325361
$329$	32.0390	1.76637
$330$	1.00000	0.0550482
$331$	−31.3772	−1.72465	−0.862323	−	0.506359i	$-0.830991\pi$
$331$	−31.3772	−1.72465	−0.862323	+	0.506359i	$0.830991\pi$
$332$	−7.02291	−0.385432
$333$	−11.6600	−0.638962
$334$	−18.3317	−1.00307
$335$	10.4711	0.572097
$336$	4.58292	0.250019
$337$	11.3049	0.615815	0.307907	−	0.951416i	$-0.400371\pi$
$337$	11.3049	0.615815	0.307907	+	0.951416i	$0.400371\pi$
$338$	−26.2204	−1.42620
$339$	−5.04453	−0.273981
$340$	−5.89254	−0.319568
$341$	−0.286713	−0.0155264
$342$	−1.00000	−0.0540738
$343$	−32.0950	−1.73297
$344$	9.16584	0.494189
$345$	−4.86963	−0.262172
$346$	11.4894	0.617676
$347$	−6.38230	−0.342620	−0.171310	−	0.985217i	$-0.554800\pi$
$347$	−6.38230	−0.342620	−0.171310	+	0.985217i	$0.554800\pi$
$348$	4.17489	0.223798
$349$	24.5109	1.31204	0.656020	−	0.754743i	$-0.272239\pi$
$349$	24.5109	1.31204	0.656020	+	0.754743i	$0.272239\pi$
$350$	4.58292	0.244967
$351$	6.26262	0.334274
$352$	1.00000	0.0533002
$353$	−10.5900	−0.563649	−0.281824	−	0.959466i	$-0.590940\pi$
$353$	−10.5900	−0.563649	−0.281824	+	0.959466i	$0.590940\pi$
$354$	−7.89254	−0.419484
$355$	14.5968	0.774717
$356$	−16.2429	−0.860871
$357$	27.0051	1.42926
$358$	−1.27330	−0.0672961
$359$	−6.40052	−0.337806	−0.168903	−	0.985633i	$-0.554023\pi$
$359$	−6.40052	−0.337806	−0.168903	+	0.985633i	$0.554023\pi$
$360$	1.00000	0.0527046
$361$	1.00000	0.0526316
$362$	12.0724	0.634509
$363$	−1.00000	−0.0524864
$364$	28.7011	1.50435
$365$	0.815502	0.0426853
$366$	−2.09209	−0.109355
$367$	30.4479	1.58937	0.794683	−	0.607025i	$-0.207637\pi$
$367$	30.4479	1.58937	0.794683	+	0.607025i	$0.207637\pi$
$368$	−4.86963	−0.253847
$369$	5.89254	0.306753
$370$	−11.6600	−0.606173
$371$	−0.173934	−0.00903021
$372$	−0.286713	−0.0148654
$373$	15.6478	0.810214	0.405107	−	0.914269i	$-0.367234\pi$
$373$	15.6478	0.810214	0.405107	+	0.914269i	$0.367234\pi$
$374$	5.89254	0.304696
$375$	1.00000	0.0516398
$376$	6.99095	0.360531
$377$	26.1458	1.34658
$378$	−4.58292	−0.235720
$379$	−7.55606	−0.388129	−0.194064	−	0.980989i	$-0.562167\pi$
$379$	−7.55606	−0.388129	−0.194064	+	0.980989i	$0.562167\pi$
$380$	−1.00000	−0.0512989
$381$	−14.6466	−0.750366
$382$	0.748873	0.0383157
$383$	−18.1524	−0.927546	−0.463773	−	0.885954i	$-0.653505\pi$
$383$	−18.1524	−0.927546	−0.463773	+	0.885954i	$0.653505\pi$
$384$	1.00000	0.0510310
$385$	−4.58292	−0.233567
$386$	−12.1562	−0.618736
$387$	−9.16584	−0.465926
$388$	−4.20849	−0.213653
$389$	−6.02366	−0.305412	−0.152706	−	0.988272i	$-0.548799\pi$
$389$	−6.02366	−0.305412	−0.152706	+	0.988272i	$0.548799\pi$
$390$	6.26262	0.317120
$391$	−28.6945	−1.45114
$392$	−14.0032	−0.707267
$393$	−6.05413	−0.305391
$394$	−19.9649	−1.00582
$395$	14.2429	0.716637
$396$	−1.00000	−0.0502519
$397$	−25.2693	−1.26823	−0.634115	−	0.773239i	$-0.718636\pi$
$397$	−25.2693	−1.26823	−0.634115	+	0.773239i	$0.718636\pi$
$398$	−20.6883	−1.03701
$399$	4.58292	0.229433
$400$	1.00000	0.0500000
$401$	23.2525	1.16117	0.580587	−	0.814198i	$-0.302823\pi$
$401$	23.2525	1.16117	0.580587	+	0.814198i	$0.302823\pi$
$402$	−10.4711	−0.522251
$403$	−1.79557	−0.0894438
$404$	−5.60583	−0.278900
$405$	−1.00000	−0.0496904
$406$	−19.1332	−0.949565
$407$	11.6600	0.577963
$408$	5.89254	0.291724
$409$	35.0717	1.73418	0.867092	−	0.498149i	$-0.165987\pi$
$409$	35.0717	1.73418	0.867092	+	0.498149i	$0.165987\pi$
$410$	5.89254	0.291012
$411$	−3.07704	−0.151779
$412$	15.8684	0.781782
$413$	36.1709	1.77985
$414$	4.86963	0.239329
$415$	7.02291	0.344741
$416$	6.26262	0.307050
$417$	−2.77319	−0.135804
$418$	1.00000	0.0489116
$419$	18.9578	0.926147	0.463074	−	0.886320i	$-0.346747\pi$
$419$	18.9578	0.926147	0.463074	+	0.886320i	$0.346747\pi$
$420$	−4.58292	−0.223624
$421$	16.3135	0.795070	0.397535	−	0.917587i	$-0.369866\pi$
$421$	16.3135	0.795070	0.397535	+	0.917587i	$0.369866\pi$
$422$	−15.7764	−0.767981
$423$	−6.99095	−0.339912
$424$	−0.0379527	−0.00184315
$425$	5.89254	0.285830
$426$	−14.5968	−0.707216
$427$	9.58786	0.463989
$428$	−6.69474	−0.323603
$429$	−6.26262	−0.302362
$430$	−9.16584	−0.442016
$431$	−6.08144	−0.292933	−0.146466	−	0.989216i	$-0.546790\pi$
$431$	−6.08144	−0.292933	−0.146466	+	0.989216i	$0.546790\pi$
$432$	−1.00000	−0.0481125
$433$	−8.97014	−0.431077	−0.215539	−	0.976495i	$-0.569151\pi$
$433$	−8.97014	−0.431077	−0.215539	+	0.976495i	$0.569151\pi$
$434$	1.31398	0.0630731
$435$	−4.17489	−0.200171
$436$	−1.89254	−0.0906363
$437$	−4.86963	−0.232946
$438$	−0.815502	−0.0389662
$439$	−3.20052	−0.152752	−0.0763761	−	0.997079i	$-0.524335\pi$
$439$	−3.20052	−0.152752	−0.0763761	+	0.997079i	$0.524335\pi$
$440$	−1.00000	−0.0476731
$441$	14.0032	0.666818
$442$	36.9027	1.75528
$443$	−22.0575	−1.04798	−0.523992	−	0.851723i	$-0.675558\pi$
$443$	−22.0575	−1.04798	−0.523992	+	0.851723i	$0.675558\pi$
$444$	11.6600	0.553357
$445$	16.2429	0.769987
$446$	1.86656	0.0883842
$447$	3.79938	0.179704
$448$	−4.58292	−0.216523
$449$	−24.5686	−1.15947	−0.579733	−	0.814807i	$-0.696843\pi$
$449$	−24.5686	−1.15947	−0.579733	+	0.814807i	$0.696843\pi$
$450$	−1.00000	−0.0471405
$451$	−5.89254	−0.277469
$452$	5.04453	0.237275
$453$	1.96749	0.0924407
$454$	−7.86058	−0.368916
$455$	−28.7011	−1.34553
$456$	1.00000	0.0468293
$457$	24.5247	1.14722	0.573608	−	0.819130i	$-0.305543\pi$
$457$	24.5247	1.14722	0.573608	+	0.819130i	$0.305543\pi$
$458$	9.52652	0.445145
$459$	−5.89254	−0.275040
$460$	4.86963	0.227048
$461$	38.2397	1.78100	0.890501	−	0.454981i	$-0.150354\pi$
$461$	38.2397	1.78100	0.890501	+	0.454981i	$0.150354\pi$
$462$	4.58292	0.213217
$463$	1.44003	0.0669238	0.0334619	−	0.999440i	$-0.489347\pi$
$463$	1.44003	0.0669238	0.0334619	+	0.999440i	$0.489347\pi$
$464$	−4.17489	−0.193815
$465$	0.286713	0.0132960
$466$	6.88294	0.318846
$467$	5.44764	0.252087	0.126043	−	0.992025i	$-0.459772\pi$
$467$	5.44764	0.252087	0.126043	+	0.992025i	$0.459772\pi$
$468$	−6.26262	−0.289490
$469$	47.9882	2.21589
$470$	−6.99095	−0.322468
$471$	−5.02291	−0.231443
$472$	7.89254	0.363284
$473$	9.16584	0.421446
$474$	−14.2429	−0.654197
$475$	1.00000	0.0458831
$476$	−27.0051	−1.23777
$477$	0.0379527	0.00173773
$478$	−10.2251	−0.467684
$479$	−11.7802	−0.538250	−0.269125	−	0.963105i	$-0.586734\pi$
$479$	−11.7802	−0.538250	−0.269125	+	0.963105i	$0.586734\pi$
$480$	−1.00000	−0.0456435
$481$	73.0219	3.32951
$482$	3.03677	0.138321
$483$	−22.3172	−1.01547
$484$	1.00000	0.0454545
$485$	4.20849	0.191097
$486$	1.00000	0.0453609
$487$	−7.04113	−0.319064	−0.159532	−	0.987193i	$-0.550998\pi$
$487$	−7.04113	−0.319064	−0.159532	+	0.987193i	$0.550998\pi$
$488$	2.09209	0.0947042
$489$	12.2962	0.556054
$490$	14.0032	0.632599
$491$	0.545709	0.0246275	0.0123138	−	0.999924i	$-0.496080\pi$
$491$	0.545709	0.0246275	0.0123138	+	0.999924i	$0.496080\pi$
$492$	−5.89254	−0.265656
$493$	−24.6007	−1.10796
$494$	6.26262	0.281768
$495$	1.00000	0.0449467
$496$	0.286713	0.0128738
$497$	66.8959	3.00069
$498$	−7.02291	−0.314704
$499$	13.7032	0.613441	0.306721	−	0.951800i	$-0.400768\pi$
$499$	13.7032	0.613441	0.306721	+	0.951800i	$0.400768\pi$
$500$	−1.00000	−0.0447214
$501$	−18.3317	−0.818999
$502$	28.7966	1.28525
$503$	−34.3523	−1.53169	−0.765847	−	0.643022i	$-0.777680\pi$
$503$	−34.3523	−1.53169	−0.765847	+	0.643022i	$0.777680\pi$
$504$	4.58292	0.204140
$505$	5.60583	0.249456
$506$	−4.86963	−0.216482
$507$	−26.2204	−1.16449
$508$	14.6466	0.649836
$509$	−8.86568	−0.392964	−0.196482	−	0.980507i	$-0.562952\pi$
$509$	−8.86568	−0.392964	−0.196482	+	0.980507i	$0.562952\pi$
$510$	−5.89254	−0.260926
$511$	3.73738	0.165332
$512$	−1.00000	−0.0441942
$513$	−1.00000	−0.0441511
$514$	27.8582	1.22877
$515$	−15.8684	−0.699247
$516$	9.16584	0.403504
$517$	6.99095	0.307462
$518$	−53.4367	−2.34787
$519$	11.4894	0.504330
$520$	−6.26262	−0.274634
$521$	24.0678	1.05443	0.527216	−	0.849732i	$-0.323236\pi$
$521$	24.0678	1.05443	0.527216	+	0.849732i	$0.323236\pi$
$522$	4.17489	0.182730
$523$	−40.7211	−1.78061	−0.890304	−	0.455366i	$-0.849508\pi$
$523$	−40.7211	−1.78061	−0.890304	+	0.455366i	$0.849508\pi$
$524$	6.05413	0.264476
$525$	4.58292	0.200015
$526$	20.2070	0.881066
$527$	1.68947	0.0735943
$528$	1.00000	0.0435194
$529$	0.713340	0.0310148
$530$	0.0379527	0.00164856
$531$	−7.89254	−0.342507
$532$	−4.58292	−0.198695
$533$	−36.9027	−1.59844
$534$	−16.2429	−0.702898
$535$	6.69474	0.289439
$536$	10.4711	0.452283
$537$	−1.27330	−0.0549470
$538$	−6.17925	−0.266407
$539$	−14.0032	−0.603159
$540$	1.00000	0.0430331
$541$	−40.1487	−1.72613	−0.863063	−	0.505096i	$-0.831457\pi$
$541$	−40.1487	−1.72613	−0.863063	+	0.505096i	$0.831457\pi$
$542$	1.41228	0.0606627
$543$	12.0724	0.518074
$544$	−5.89254	−0.252641
$545$	1.89254	0.0810676
$546$	28.7011	1.22829
$547$	30.4516	1.30202	0.651009	−	0.759070i	$-0.274346\pi$
$547$	30.4516	1.30202	0.651009	+	0.759070i	$0.274346\pi$
$548$	3.07704	0.131445
$549$	−2.09209	−0.0892880
$550$	1.00000	0.0426401
$551$	−4.17489	−0.177856
$552$	−4.86963	−0.207265
$553$	65.2740	2.77573
$554$	2.06307	0.0876516
$555$	−11.6600	−0.494938
$556$	2.77319	0.117609
$557$	−5.05483	−0.214180	−0.107090	−	0.994249i	$-0.534153\pi$
$557$	−5.05483	−0.214180	−0.107090	+	0.994249i	$0.534153\pi$
$558$	−0.286713	−0.0121375
$559$	57.4022	2.42785
$560$	4.58292	0.193664
$561$	5.89254	0.248783
$562$	3.52139	0.148541
$563$	15.1466	0.638355	0.319177	−	0.947695i	$-0.396593\pi$
$563$	15.1466	0.638355	0.319177	+	0.947695i	$0.396593\pi$
$564$	6.99095	0.294372
$565$	−5.04453	−0.212225
$566$	22.5711	0.948732
$567$	−4.58292	−0.192465
$568$	14.5968	0.612467
$569$	−39.9258	−1.67378	−0.836888	−	0.547374i	$-0.815628\pi$
$569$	−39.9258	−1.67378	−0.836888	+	0.547374i	$0.815628\pi$
$570$	−1.00000	−0.0418854
$571$	40.7883	1.70694	0.853468	−	0.521146i	$-0.174495\pi$
$571$	40.7883	1.70694	0.853468	+	0.521146i	$0.174495\pi$
$572$	6.26262	0.261853
$573$	0.748873	0.0312846
$574$	27.0051	1.12717
$575$	−4.86963	−0.203078
$576$	1.00000	0.0416667
$577$	−2.02445	−0.0842788	−0.0421394	−	0.999112i	$-0.513417\pi$
$577$	−2.02445	−0.0842788	−0.0421394	+	0.999112i	$0.513417\pi$
$578$	−17.7221	−0.737140
$579$	−12.1562	−0.505196
$580$	4.17489	0.173353
$581$	32.1854	1.33528
$582$	−4.20849	−0.174447
$583$	−0.0379527	−0.00157184
$584$	0.815502	0.0337457
$585$	6.26262	0.258927
$586$	−6.51651	−0.269195
$587$	33.1667	1.36893	0.684467	−	0.729044i	$-0.260035\pi$
$587$	33.1667	1.36893	0.684467	+	0.729044i	$0.260035\pi$
$588$	−14.0032	−0.577481
$589$	0.286713	0.0118138
$590$	−7.89254	−0.324931
$591$	−19.9649	−0.821246
$592$	−11.6600	−0.479221
$593$	−15.2306	−0.625444	−0.312722	−	0.949845i	$-0.601241\pi$
$593$	−15.2306	−0.625444	−0.312722	+	0.949845i	$0.601241\pi$
$594$	−1.00000	−0.0410305
$595$	27.0051	1.10710
$596$	−3.79938	−0.155629
$597$	−20.6883	−0.846716
$598$	−30.4967	−1.24710
$599$	−44.1484	−1.80386	−0.901928	−	0.431886i	$-0.857848\pi$
$599$	−44.1484	−1.80386	−0.901928	+	0.431886i	$0.857848\pi$
$600$	1.00000	0.0408248
$601$	−15.8446	−0.646315	−0.323157	−	0.946345i	$-0.604744\pi$
$601$	−15.8446	−0.646315	−0.323157	+	0.946345i	$0.604744\pi$
$602$	−42.0063	−1.71205
$603$	−10.4711	−0.426416
$604$	−1.96749	−0.0800560
$605$	−1.00000	−0.0406558
$606$	−5.60583	−0.227721
$607$	30.1059	1.22196	0.610980	−	0.791646i	$-0.290775\pi$
$607$	30.1059	1.22196	0.610980	+	0.791646i	$0.290775\pi$
$608$	−1.00000	−0.0405554
$609$	−19.1332	−0.775317
$610$	−2.09209	−0.0847061
$611$	43.7816	1.77122
$612$	5.89254	0.238192
$613$	4.40762	0.178022	0.0890111	−	0.996031i	$-0.471629\pi$
$613$	4.40762	0.178022	0.0890111	+	0.996031i	$0.471629\pi$
$614$	−12.4257	−0.501460
$615$	5.89254	0.237610
$616$	−4.58292	−0.184651
$617$	3.71724	0.149651	0.0748253	−	0.997197i	$-0.476160\pi$
$617$	3.71724	0.149651	0.0748253	+	0.997197i	$0.476160\pi$
$618$	15.8684	0.638323
$619$	−46.1233	−1.85385	−0.926927	−	0.375243i	$-0.877559\pi$
$619$	−46.1233	−1.85385	−0.926927	+	0.375243i	$0.877559\pi$
$620$	−0.286713	−0.0115147
$621$	4.86963	0.195412
$622$	11.8846	0.476530
$623$	74.4399	2.98237
$624$	6.26262	0.250705
$625$	1.00000	0.0400000
$626$	−30.5530	−1.22114
$627$	1.00000	0.0399362
$628$	5.02291	0.200436
$629$	−68.7068	−2.73952
$630$	−4.58292	−0.182588
$631$	−12.7522	−0.507658	−0.253829	−	0.967249i	$-0.581690\pi$
$631$	−12.7522	−0.507658	−0.253829	+	0.967249i	$0.581690\pi$
$632$	14.2429	0.566552
$633$	−15.7764	−0.627054
$634$	22.7045	0.901710
$635$	−14.6466	−0.581231
$636$	−0.0379527	−0.00150492
$637$	−87.6965	−3.47466
$638$	−4.17489	−0.165286
$639$	−14.5968	−0.577440
$640$	1.00000	0.0395285
$641$	30.3718	1.19961	0.599807	−	0.800145i	$-0.295244\pi$
$641$	30.3718	1.19961	0.599807	+	0.800145i	$0.295244\pi$
$642$	−6.69474	−0.264220
$643$	−43.5767	−1.71850	−0.859248	−	0.511559i	$-0.829068\pi$
$643$	−43.5767	−1.71850	−0.859248	+	0.511559i	$0.829068\pi$
$644$	22.3172	0.879419
$645$	−9.16584	−0.360905
$646$	−5.89254	−0.231839
$647$	44.3654	1.74418	0.872092	−	0.489342i	$-0.162763\pi$
$647$	44.3654	1.74418	0.872092	+	0.489342i	$0.162763\pi$
$648$	−1.00000	−0.0392837
$649$	7.89254	0.309809
$650$	6.26262	0.245640
$651$	1.31398	0.0514990
$652$	−12.2962	−0.481557
$653$	44.5316	1.74266	0.871328	−	0.490701i	$-0.163259\pi$
$653$	44.5316	1.74266	0.871328	+	0.490701i	$0.163259\pi$
$654$	−1.89254	−0.0740043
$655$	−6.05413	−0.236555
$656$	5.89254	0.230065
$657$	−0.815502	−0.0318158
$658$	−32.0390	−1.24901
$659$	−1.07199	−0.0417589	−0.0208794	−	0.999782i	$-0.506647\pi$
$659$	−1.07199	−0.0417589	−0.0208794	+	0.999782i	$0.506647\pi$
$660$	−1.00000	−0.0389249
$661$	8.46161	0.329118	0.164559	−	0.986367i	$-0.447380\pi$
$661$	8.46161	0.329118	0.164559	+	0.986367i	$0.447380\pi$
$662$	31.3772	1.21951
$663$	36.9027	1.43318
$664$	7.02291	0.272542
$665$	4.58292	0.177718
$666$	11.6600	0.451814
$667$	20.3302	0.787189
$668$	18.3317	0.709274
$669$	1.86656	0.0721654
$670$	−10.4711	−0.404534
$671$	2.09209	0.0807641
$672$	−4.58292	−0.176790
$673$	43.4777	1.67594	0.837971	−	0.545715i	$-0.183742\pi$
$673$	43.4777	1.67594	0.837971	+	0.545715i	$0.183742\pi$
$674$	−11.3049	−0.435447
$675$	−1.00000	−0.0384900
$676$	26.2204	1.00848
$677$	−7.38076	−0.283666	−0.141833	−	0.989891i	$-0.545300\pi$
$677$	−7.38076	−0.283666	−0.141833	+	0.989891i	$0.545300\pi$
$678$	5.04453	0.193734
$679$	19.2872	0.740173
$680$	5.89254	0.225969
$681$	−7.86058	−0.301218
$682$	0.286713	0.0109788
$683$	51.6190	1.97515	0.987573	−	0.157163i	$-0.0502348\pi$
$683$	51.6190	1.97515	0.987573	+	0.157163i	$0.0502348\pi$
$684$	1.00000	0.0382360
$685$	−3.07704	−0.117568
$686$	32.0950	1.22539
$687$	9.52652	0.363460
$688$	−9.16584	−0.349445
$689$	−0.237683	−0.00905501
$690$	4.86963	0.185384
$691$	28.2621	1.07514	0.537571	−	0.843218i	$-0.319342\pi$
$691$	28.2621	1.07514	0.537571	+	0.843218i	$0.319342\pi$
$692$	−11.4894	−0.436763
$693$	4.58292	0.174091
$694$	6.38230	0.242269
$695$	−2.77319	−0.105193
$696$	−4.17489	−0.158249
$697$	34.7221	1.31519
$698$	−24.5109	−0.927753
$699$	6.88294	0.260337
$700$	−4.58292	−0.173218
$701$	5.59670	0.211385	0.105692	−	0.994399i	$-0.466294\pi$
$701$	5.59670	0.211385	0.105692	+	0.994399i	$0.466294\pi$
$702$	−6.26262	−0.236367
$703$	−11.6600	−0.439764
$704$	−1.00000	−0.0376889
$705$	−6.99095	−0.263294
$706$	10.5900	0.398560
$707$	25.6911	0.966212
$708$	7.89254	0.296620
$709$	−22.9859	−0.863252	−0.431626	−	0.902053i	$-0.642060\pi$
$709$	−22.9859	−0.863252	−0.431626	+	0.902053i	$0.642060\pi$
$710$	−14.5968	−0.547807
$711$	−14.2429	−0.534150
$712$	16.2429	0.608728
$713$	−1.39619	−0.0522875
$714$	−27.0051	−1.01064
$715$	−6.26262	−0.234209
$716$	1.27330	0.0475855
$717$	−10.2251	−0.381862
$718$	6.40052	0.238865
$719$	−7.46150	−0.278267	−0.139133	−	0.990274i	$-0.544432\pi$
$719$	−7.46150	−0.278267	−0.139133	+	0.990274i	$0.544432\pi$
$720$	−1.00000	−0.0372678
$721$	−72.7239	−2.70838
$722$	−1.00000	−0.0372161
$723$	3.03677	0.112939
$724$	−12.0724	−0.448665
$725$	−4.17489	−0.155052
$726$	1.00000	0.0371135
$727$	29.5600	1.09632	0.548160	−	0.836374i	$-0.315329\pi$
$727$	29.5600	1.09632	0.548160	+	0.836374i	$0.315329\pi$
$728$	−28.7011	−1.06373
$729$	1.00000	0.0370370
$730$	−0.815502	−0.0301831
$731$	−54.0101	−1.99764
$732$	2.09209	0.0773257
$733$	−12.9060	−0.476692	−0.238346	−	0.971180i	$-0.576605\pi$
$733$	−12.9060	−0.476692	−0.238346	+	0.971180i	$0.576605\pi$
$734$	−30.4479	−1.12385
$735$	14.0032	0.516515
$736$	4.86963	0.179497
$737$	10.4711	0.385708
$738$	−5.89254	−0.216907
$739$	−6.80747	−0.250417	−0.125208	−	0.992130i	$-0.539960\pi$
$739$	−6.80747	−0.250417	−0.125208	+	0.992130i	$0.539960\pi$
$740$	11.6600	0.428629
$741$	6.26262	0.230063
$742$	0.173934	0.00638533
$743$	−2.19760	−0.0806223	−0.0403111	−	0.999187i	$-0.512835\pi$
$743$	−2.19760	−0.0806223	−0.0403111	+	0.999187i	$0.512835\pi$
$744$	0.286713	0.0105114
$745$	3.79938	0.139198
$746$	−15.6478	−0.572908
$747$	−7.02291	−0.256955
$748$	−5.89254	−0.215453
$749$	30.6815	1.12108
$750$	−1.00000	−0.0365148
$751$	−51.1031	−1.86478	−0.932389	−	0.361458i	$-0.882279\pi$
$751$	−51.1031	−1.86478	−0.932389	+	0.361458i	$0.882279\pi$
$752$	−6.99095	−0.254934
$753$	28.7966	1.04941
$754$	−26.1458	−0.952173
$755$	1.96749	0.0716042
$756$	4.58292	0.166679
$757$	−17.4174	−0.633048	−0.316524	−	0.948585i	$-0.602516\pi$
$757$	−17.4174	−0.633048	−0.316524	+	0.948585i	$0.602516\pi$
$758$	7.55606	0.274448
$759$	−4.86963	−0.176757
$760$	1.00000	0.0362738
$761$	−30.4339	−1.10323	−0.551614	−	0.834099i	$-0.685988\pi$
$761$	−30.4339	−1.10323	−0.551614	+	0.834099i	$0.685988\pi$
$762$	14.6466	0.530589
$763$	8.67337	0.313997
$764$	−0.748873	−0.0270933
$765$	−5.89254	−0.213045
$766$	18.1524	0.655874
$767$	49.4280	1.78474
$768$	−1.00000	−0.0360844
$769$	−45.0328	−1.62392	−0.811962	−	0.583711i	$-0.801600\pi$
$769$	−45.0328	−1.62392	−0.811962	+	0.583711i	$0.801600\pi$
$770$	4.58292	0.165157
$771$	27.8582	1.00329
$772$	12.1562	0.437513
$773$	−27.4534	−0.987429	−0.493714	−	0.869624i	$-0.664361\pi$
$773$	−27.4534	−0.987429	−0.493714	+	0.869624i	$0.664361\pi$
$774$	9.16584	0.329459
$775$	0.286713	0.0102990
$776$	4.20849	0.151076
$777$	−53.4367	−1.91703
$778$	6.02366	0.215959
$779$	5.89254	0.211122
$780$	−6.26262	−0.224238
$781$	14.5968	0.522314
$782$	28.6945	1.02611
$783$	4.17489	0.149199
$784$	14.0032	0.500113
$785$	−5.02291	−0.179275
$786$	6.05413	0.215944
$787$	−28.7766	−1.02577	−0.512887	−	0.858456i	$-0.671424\pi$
$787$	−28.7766	−1.02577	−0.512887	+	0.858456i	$0.671424\pi$
$788$	19.9649	0.711220
$789$	20.2070	0.719387
$790$	−14.2429	−0.506739
$791$	−23.1187	−0.822006
$792$	1.00000	0.0355335
$793$	13.1019	0.465263
$794$	25.2693	0.896774
$795$	0.0379527	0.00134604
$796$	20.6883	0.733278
$797$	39.9525	1.41519	0.707594	−	0.706619i	$-0.249780\pi$
$797$	39.9525	1.41519	0.707594	+	0.706619i	$0.249780\pi$
$798$	−4.58292	−0.162234
$799$	−41.1945	−1.45736
$800$	−1.00000	−0.0353553
$801$	−16.2429	−0.573914
$802$	−23.2525	−0.821074
$803$	0.815502	0.0287784
$804$	10.4711	0.369287
$805$	−22.3172	−0.786576
$806$	1.79557	0.0632463
$807$	−6.17925	−0.217520
$808$	5.60583	0.197212
$809$	56.3312	1.98050	0.990250	−	0.139303i	$-0.0444863\pi$
$809$	56.3312	1.98050	0.990250	+	0.139303i	$0.0444863\pi$
$810$	1.00000	0.0351364
$811$	30.2911	1.06366	0.531832	−	0.846850i	$-0.321504\pi$
$811$	30.2911	1.06366	0.531832	+	0.846850i	$0.321504\pi$
$812$	19.1332	0.671444
$813$	1.41228	0.0495309
$814$	−11.6600	−0.408681
$815$	12.2962	0.430717
$816$	−5.89254	−0.206280
$817$	−9.16584	−0.320672
$818$	−35.0717	−1.22625
$819$	28.7011	1.00290
$820$	−5.89254	−0.205776
$821$	24.5504	0.856815	0.428408	−	0.903586i	$-0.359075\pi$
$821$	24.5504	0.856815	0.428408	+	0.903586i	$0.359075\pi$
$822$	3.07704	0.107324
$823$	−28.0041	−0.976161	−0.488080	−	0.872799i	$-0.662303\pi$
$823$	−28.0041	−0.976161	−0.488080	+	0.872799i	$0.662303\pi$
$824$	−15.8684	−0.552804
$825$	1.00000	0.0348155
$826$	−36.1709	−1.25855
$827$	−42.5075	−1.47813	−0.739066	−	0.673633i	$-0.764733\pi$
$827$	−42.5075	−1.47813	−0.739066	+	0.673633i	$0.764733\pi$
$828$	−4.86963	−0.169231
$829$	13.8336	0.480462	0.240231	−	0.970716i	$-0.422777\pi$
$829$	13.8336	0.480462	0.240231	+	0.970716i	$0.422777\pi$
$830$	−7.02291	−0.243769
$831$	2.06307	0.0715673
$832$	−6.26262	−0.217117
$833$	82.5143	2.85895
$834$	2.77319	0.0960277
$835$	−18.3317	−0.634394
$836$	−1.00000	−0.0345857
$837$	−0.286713	−0.00991023
$838$	−18.9578	−0.654885
$839$	−5.68950	−0.196423	−0.0982116	−	0.995166i	$-0.531312\pi$
$839$	−5.68950	−0.196423	−0.0982116	+	0.995166i	$0.531312\pi$
$840$	4.58292	0.158126
$841$	−11.5703	−0.398975
$842$	−16.3135	−0.562199
$843$	3.52139	0.121283
$844$	15.7764	0.543045
$845$	−26.2204	−0.902009
$846$	6.99095	0.240354
$847$	−4.58292	−0.157471
$848$	0.0379527	0.00130330
$849$	22.5711	0.774636
$850$	−5.89254	−0.202113
$851$	56.7798	1.94638
$852$	14.5968	0.500077
$853$	−6.03863	−0.206759	−0.103379	−	0.994642i	$-0.532966\pi$
$853$	−6.03863	−0.206759	−0.103379	+	0.994642i	$0.532966\pi$
$854$	−9.58786	−0.328090
$855$	−1.00000	−0.0341993
$856$	6.69474	0.228822
$857$	−47.0833	−1.60833	−0.804167	−	0.594404i	$-0.797388\pi$
$857$	−47.0833	−1.60833	−0.804167	+	0.594404i	$0.797388\pi$
$858$	6.26262	0.213802
$859$	32.5962	1.11217	0.556084	−	0.831126i	$-0.312303\pi$
$859$	32.5962	1.11217	0.556084	+	0.831126i	$0.312303\pi$
$860$	9.16584	0.312553
$861$	27.0051	0.920330
$862$	6.08144	0.207135
$863$	−47.3784	−1.61278	−0.806389	−	0.591386i	$-0.798581\pi$
$863$	−47.3784	−1.61278	−0.806389	+	0.591386i	$0.798581\pi$
$864$	1.00000	0.0340207
$865$	11.4894	0.390652
$866$	8.97014	0.304818
$867$	−17.7221	−0.601872
$868$	−1.31398	−0.0445994
$869$	14.2429	0.483157
$870$	4.17489	0.141542
$871$	65.5765	2.22198
$872$	1.89254	0.0640896
$873$	−4.20849	−0.142436
$874$	4.86963	0.164718
$875$	4.58292	0.154931
$876$	0.815502	0.0275533
$877$	23.3982	0.790101	0.395051	−	0.918659i	$-0.370727\pi$
$877$	23.3982	0.790101	0.395051	+	0.918659i	$0.370727\pi$
$878$	3.20052	0.108012
$879$	−6.51651	−0.219797
$880$	1.00000	0.0337100
$881$	7.73422	0.260573	0.130286	−	0.991476i	$-0.458410\pi$
$881$	7.73422	0.260573	0.130286	+	0.991476i	$0.458410\pi$
$882$	−14.0032	−0.471511
$883$	6.72784	0.226410	0.113205	−	0.993572i	$-0.463888\pi$
$883$	6.72784	0.226410	0.113205	+	0.993572i	$0.463888\pi$
$884$	−36.9027	−1.24117
$885$	−7.89254	−0.265305
$886$	22.0575	0.741036
$887$	56.1596	1.88566	0.942828	−	0.333279i	$-0.108155\pi$
$887$	56.1596	1.88566	0.942828	+	0.333279i	$0.108155\pi$
$888$	−11.6600	−0.391283
$889$	−67.1240	−2.25127
$890$	−16.2429	−0.544463
$891$	−1.00000	−0.0335013
$892$	−1.86656	−0.0624971
$893$	−6.99095	−0.233943
$894$	−3.79938	−0.127070
$895$	−1.27330	−0.0425618
$896$	4.58292	0.153105
$897$	−30.4967	−1.01825
$898$	24.5686	0.819866
$899$	−1.19699	−0.0399220
$900$	1.00000	0.0333333
$901$	0.223638	0.00745046
$902$	5.89254	0.196200
$903$	−42.0063	−1.39788
$904$	−5.04453	−0.167779
$905$	12.0724	0.401299
$906$	−1.96749	−0.0653654
$907$	−23.6610	−0.785650	−0.392825	−	0.919613i	$-0.628502\pi$
$907$	−23.6610	−0.785650	−0.392825	+	0.919613i	$0.628502\pi$
$908$	7.86058	0.260863
$909$	−5.60583	−0.185934
$910$	28.7011	0.951432
$911$	−27.8660	−0.923242	−0.461621	−	0.887077i	$-0.652732\pi$
$911$	−27.8660	−0.923242	−0.461621	+	0.887077i	$0.652732\pi$
$912$	−1.00000	−0.0331133
$913$	7.02291	0.232424
$914$	−24.5247	−0.811204
$915$	−2.09209	−0.0691622
$916$	−9.52652	−0.314765
$917$	−27.7456	−0.916241
$918$	5.89254	0.194483
$919$	18.4195	0.607604	0.303802	−	0.952735i	$-0.401744\pi$
$919$	18.4195	0.607604	0.303802	+	0.952735i	$0.401744\pi$
$920$	−4.86963	−0.160547
$921$	−12.4257	−0.409440
$922$	−38.2397	−1.25936
$923$	91.4141	3.00893
$924$	−4.58292	−0.150767
$925$	−11.6600	−0.383377
$926$	−1.44003	−0.0473223
$927$	15.8684	0.521188
$928$	4.17489	0.137048
$929$	28.9727	0.950563	0.475282	−	0.879834i	$-0.342346\pi$
$929$	28.9727	0.950563	0.475282	+	0.879834i	$0.342346\pi$
$930$	−0.286713	−0.00940167
$931$	14.0032	0.458935
$932$	−6.88294	−0.225458
$933$	11.8846	0.389085
$934$	−5.44764	−0.178252
$935$	5.89254	0.192707
$936$	6.26262	0.204700
$937$	53.9659	1.76299	0.881495	−	0.472194i	$-0.156538\pi$
$937$	53.9659	1.76299	0.881495	+	0.472194i	$0.156538\pi$
$938$	−47.9882	−1.56687
$939$	−30.5530	−0.997058
$940$	6.99095	0.228020
$941$	−0.723872	−0.0235975	−0.0117988	−	0.999930i	$-0.503756\pi$
$941$	−0.723872	−0.0235975	−0.0117988	+	0.999930i	$0.503756\pi$
$942$	5.02291	0.163655
$943$	−28.6945	−0.934422
$944$	−7.89254	−0.256880
$945$	−4.58292	−0.149082
$946$	−9.16584	−0.298007
$947$	−10.1897	−0.331121	−0.165561	−	0.986200i	$-0.552943\pi$
$947$	−10.1897	−0.331121	−0.165561	+	0.986200i	$0.552943\pi$
$948$	14.2429	0.462587
$949$	5.10718	0.165786
$950$	−1.00000	−0.0324443
$951$	22.7045	0.736244
$952$	27.0051	0.875239
$953$	−33.9882	−1.10099	−0.550494	−	0.834839i	$-0.685560\pi$
$953$	−33.9882	−1.10099	−0.550494	+	0.834839i	$0.685560\pi$
$954$	−0.0379527	−0.00122876
$955$	0.748873	0.0242330
$956$	10.2251	0.330702
$957$	−4.17489	−0.134955
$958$	11.7802	0.380600
$959$	−14.1018	−0.455372
$960$	1.00000	0.0322749
$961$	−30.9178	−0.997348
$962$	−73.0219	−2.35432
$963$	−6.69474	−0.215735
$964$	−3.03677	−0.0978076
$965$	−12.1562	−0.391323
$966$	22.3172	0.718043
$967$	−41.3273	−1.32899	−0.664497	−	0.747291i	$-0.731354\pi$
$967$	−41.3273	−1.32899	−0.664497	+	0.747291i	$0.731354\pi$
$968$	−1.00000	−0.0321412
$969$	−5.89254	−0.189296
$970$	−4.20849	−0.135126
$971$	−30.6303	−0.982974	−0.491487	−	0.870885i	$-0.663546\pi$
$971$	−30.6303	−0.982974	−0.491487	+	0.870885i	$0.663546\pi$
$972$	−1.00000	−0.0320750
$973$	−12.7093	−0.407442
$974$	7.04113	0.225612
$975$	6.26262	0.200564
$976$	−2.09209	−0.0669660
$977$	16.4827	0.527328	0.263664	−	0.964615i	$-0.415069\pi$
$977$	16.4827	0.527328	0.263664	+	0.964615i	$0.415069\pi$
$978$	−12.2962	−0.393189
$979$	16.2429	0.519125
$980$	−14.0032	−0.447315
$981$	−1.89254	−0.0604242
$982$	−0.545709	−0.0174143
$983$	1.06098	0.0338400	0.0169200	−	0.999857i	$-0.494614\pi$
$983$	1.06098	0.0338400	0.0169200	+	0.999857i	$0.494614\pi$
$984$	5.89254	0.187847
$985$	−19.9649	−0.636134
$986$	24.6007	0.783447
$987$	−32.0390	−1.01981
$988$	−6.26262	−0.199240
$989$	44.6343	1.41929
$990$	−1.00000	−0.0317821
$991$	49.7147	1.57924	0.789620	−	0.613597i	$-0.210278\pi$
$991$	49.7147	1.57924	0.789620	+	0.613597i	$0.210278\pi$
$992$	−0.286713	−0.00910313
$993$	31.3772	0.995725
$994$	−66.8959	−2.12181
$995$	−20.6883	−0.655864
$996$	7.02291	0.222529
$997$	47.1625	1.49365	0.746826	−	0.665019i	$-0.231577\pi$
$997$	47.1625	1.49365	0.746826	+	0.665019i	$0.231577\pi$
$998$	−13.7032	−0.433768
$999$	11.6600	0.368905

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6270.2.a.bv.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6270.2.a.bv.1.1	✓	6	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:	\( N \)	\(=\)	\( 6270 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6270.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -4.58292 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -4.58292 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -6.26262 q^{13} +4.58292 q^{14} +1.00000 q^{15} +1.00000 q^{16} +5.89254 q^{17} -1.00000 q^{18} +1.00000 q^{19} -1.00000 q^{20} +4.58292 q^{21} +1.00000 q^{22} -4.86963 q^{23} +1.00000 q^{24} +1.00000 q^{25} +6.26262 q^{26} -1.00000 q^{27} -4.58292 q^{28} -4.17489 q^{29} -1.00000 q^{30} +0.286713 q^{31} -1.00000 q^{32} +1.00000 q^{33} -5.89254 q^{34} +4.58292 q^{35} +1.00000 q^{36} -11.6600 q^{37} -1.00000 q^{38} +6.26262 q^{39} +1.00000 q^{40} +5.89254 q^{41} -4.58292 q^{42} -9.16584 q^{43} -1.00000 q^{44} -1.00000 q^{45} +4.86963 q^{46} -6.99095 q^{47} -1.00000 q^{48} +14.0032 q^{49} -1.00000 q^{50} -5.89254 q^{51} -6.26262 q^{52} +0.0379527 q^{53} +1.00000 q^{54} +1.00000 q^{55} +4.58292 q^{56} -1.00000 q^{57} +4.17489 q^{58} -7.89254 q^{59} +1.00000 q^{60} -2.09209 q^{61} -0.286713 q^{62} -4.58292 q^{63} +1.00000 q^{64} +6.26262 q^{65} -1.00000 q^{66} -10.4711 q^{67} +5.89254 q^{68} +4.86963 q^{69} -4.58292 q^{70} -14.5968 q^{71} -1.00000 q^{72} -0.815502 q^{73} +11.6600 q^{74} -1.00000 q^{75} +1.00000 q^{76} +4.58292 q^{77} -6.26262 q^{78} -14.2429 q^{79} -1.00000 q^{80} +1.00000 q^{81} -5.89254 q^{82} -7.02291 q^{83} +4.58292 q^{84} -5.89254 q^{85} +9.16584 q^{86} +4.17489 q^{87} +1.00000 q^{88} -16.2429 q^{89} +1.00000 q^{90} +28.7011 q^{91} -4.86963 q^{92} -0.286713 q^{93} +6.99095 q^{94} -1.00000 q^{95} +1.00000 q^{96} -4.20849 q^{97} -14.0032 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} - 6 q^{3} + 6 q^{4} - 6 q^{5} + 6 q^{6} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^2 - 6 * q^3 + 6 * q^4 - 6 * q^5 + 6 * q^6 - 6 * q^8 + 6 * q^9	\( 6 q - 6 q^{2} - 6 q^{3} + 6 q^{4} - 6 q^{5} + 6 q^{6} - 6 q^{8} + 6 q^{9} + 6 q^{10} - 6 q^{11} - 6 q^{12} + 2 q^{13} + 6 q^{15} + 6 q^{16} - 4 q^{17} - 6 q^{18} + 6 q^{19} - 6 q^{20} + 6 q^{22} - 4 q^{23} + 6 q^{24} + 6 q^{25} - 2 q^{26} - 6 q^{27} + 2 q^{29} - 6 q^{30} + 4 q^{31} - 6 q^{32} + 6 q^{33} + 4 q^{34} + 6 q^{36} - 6 q^{37} - 6 q^{38} - 2 q^{39} + 6 q^{40} - 4 q^{41} - 6 q^{44} - 6 q^{45} + 4 q^{46} - 14 q^{47} - 6 q^{48} + 18 q^{49} - 6 q^{50} + 4 q^{51} + 2 q^{52} + 6 q^{54} + 6 q^{55} - 6 q^{57} - 2 q^{58} - 8 q^{59} + 6 q^{60} + 10 q^{61} - 4 q^{62} + 6 q^{64} - 2 q^{65} - 6 q^{66} - 6 q^{67} - 4 q^{68} + 4 q^{69} - 18 q^{71} - 6 q^{72} - 2 q^{73} + 6 q^{74} - 6 q^{75} + 6 q^{76} + 2 q^{78} + 6 q^{79} - 6 q^{80} + 6 q^{81} + 4 q^{82} - 28 q^{83} + 4 q^{85} - 2 q^{87} + 6 q^{88} - 6 q^{89} + 6 q^{90} + 42 q^{91} - 4 q^{92} - 4 q^{93} + 14 q^{94} - 6 q^{95} + 6 q^{96} - 8 q^{97} - 18 q^{98} - 6 q^{99}+O(q^{100}) \) 6 * q - 6 * q^2 - 6 * q^3 + 6 * q^4 - 6 * q^5 + 6 * q^6 - 6 * q^8 + 6 * q^9 + 6 * q^10 - 6 * q^11 - 6 * q^12 + 2 * q^13 + 6 * q^15 + 6 * q^16 - 4 * q^17 - 6 * q^18 + 6 * q^19 - 6 * q^20 + 6 * q^22 - 4 * q^23 + 6 * q^24 + 6 * q^25 - 2 * q^26 - 6 * q^27 + 2 * q^29 - 6 * q^30 + 4 * q^31 - 6 * q^32 + 6 * q^33 + 4 * q^34 + 6 * q^36 - 6 * q^37 - 6 * q^38 - 2 * q^39 + 6 * q^40 - 4 * q^41 - 6 * q^44 - 6 * q^45 + 4 * q^46 - 14 * q^47 - 6 * q^48 + 18 * q^49 - 6 * q^50 + 4 * q^51 + 2 * q^52 + 6 * q^54 + 6 * q^55 - 6 * q^57 - 2 * q^58 - 8 * q^59 + 6 * q^60 + 10 * q^61 - 4 * q^62 + 6 * q^64 - 2 * q^65 - 6 * q^66 - 6 * q^67 - 4 * q^68 + 4 * q^69 - 18 * q^71 - 6 * q^72 - 2 * q^73 + 6 * q^74 - 6 * q^75 + 6 * q^76 + 2 * q^78 + 6 * q^79 - 6 * q^80 + 6 * q^81 + 4 * q^82 - 28 * q^83 + 4 * q^85 - 2 * q^87 + 6 * q^88 - 6 * q^89 + 6 * q^90 + 42 * q^91 - 4 * q^92 - 4 * q^93 + 14 * q^94 - 6 * q^95 + 6 * q^96 - 8 * q^97 - 18 * q^98 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more