LMFDB - Embedded newform 1110.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1110,2,Mod(1,1110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1110, 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("1110.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1110 = 2 \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1110.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:	$8.86339462436$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	1110.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -6.12311 q^{11} -1.00000 q^{12} +5.68466 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -5.00000 q^{17} +1.00000 q^{18} +0.561553 q^{19} -1.00000 q^{20} +1.00000 q^{21} -6.12311 q^{22} -6.56155 q^{23} -1.00000 q^{24} +1.00000 q^{25} +5.68466 q^{26} -1.00000 q^{27} -1.00000 q^{28} -6.68466 q^{29} +1.00000 q^{30} -3.56155 q^{31} +1.00000 q^{32} +6.12311 q^{33} -5.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} +0.561553 q^{38} -5.68466 q^{39} -1.00000 q^{40} +8.68466 q^{41} +1.00000 q^{42} -12.6847 q^{43} -6.12311 q^{44} -1.00000 q^{45} -6.56155 q^{46} -8.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} +1.00000 q^{50} +5.00000 q^{51} +5.68466 q^{52} +5.24621 q^{53} -1.00000 q^{54} +6.12311 q^{55} -1.00000 q^{56} -0.561553 q^{57} -6.68466 q^{58} +4.24621 q^{59} +1.00000 q^{60} -13.8078 q^{61} -3.56155 q^{62} -1.00000 q^{63} +1.00000 q^{64} -5.68466 q^{65} +6.12311 q^{66} +13.1231 q^{67} -5.00000 q^{68} +6.56155 q^{69} +1.00000 q^{70} +1.12311 q^{71} +1.00000 q^{72} -6.56155 q^{73} +1.00000 q^{74} -1.00000 q^{75} +0.561553 q^{76} +6.12311 q^{77} -5.68466 q^{78} +14.2462 q^{79} -1.00000 q^{80} +1.00000 q^{81} +8.68466 q^{82} -7.43845 q^{83} +1.00000 q^{84} +5.00000 q^{85} -12.6847 q^{86} +6.68466 q^{87} -6.12311 q^{88} +5.68466 q^{89} -1.00000 q^{90} -5.68466 q^{91} -6.56155 q^{92} +3.56155 q^{93} -8.00000 q^{94} -0.561553 q^{95} -1.00000 q^{96} +3.31534 q^{97} -6.00000 q^{98} -6.12311 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} - 2 q^{12} - q^{13} - 2 q^{14} + 2 q^{15} + 2 q^{16} - 10 q^{17} + 2 q^{18} - 3 q^{19} - 2 q^{20} + 2 q^{21} - 4 q^{22} - 9 q^{23} - 2 q^{24} + 2 q^{25} - q^{26} - 2 q^{27} - 2 q^{28} - q^{29} + 2 q^{30} - 3 q^{31} + 2 q^{32} + 4 q^{33} - 10 q^{34} + 2 q^{35} + 2 q^{36} + 2 q^{37} - 3 q^{38} + q^{39} - 2 q^{40} + 5 q^{41} + 2 q^{42} - 13 q^{43} - 4 q^{44} - 2 q^{45} - 9 q^{46} - 16 q^{47} - 2 q^{48} - 12 q^{49} + 2 q^{50} + 10 q^{51} - q^{52} - 6 q^{53} - 2 q^{54} + 4 q^{55} - 2 q^{56} + 3 q^{57} - q^{58} - 8 q^{59} + 2 q^{60} - 7 q^{61} - 3 q^{62} - 2 q^{63} + 2 q^{64} + q^{65} + 4 q^{66} + 18 q^{67} - 10 q^{68} + 9 q^{69} + 2 q^{70} - 6 q^{71} + 2 q^{72} - 9 q^{73} + 2 q^{74} - 2 q^{75} - 3 q^{76} + 4 q^{77} + q^{78} + 12 q^{79} - 2 q^{80} + 2 q^{81} + 5 q^{82} - 19 q^{83} + 2 q^{84} + 10 q^{85} - 13 q^{86} + q^{87} - 4 q^{88} - q^{89} - 2 q^{90} + q^{91} - 9 q^{92} + 3 q^{93} - 16 q^{94} + 3 q^{95} - 2 q^{96} + 19 q^{97} - 12 q^{98} - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 - 2 * q^10 - 4 * q^11 - 2 * q^12 - q^13 - 2 * q^14 + 2 * q^15 + 2 * q^16 - 10 * q^17 + 2 * q^18 - 3 * q^19 - 2 * q^20 + 2 * q^21 - 4 * q^22 - 9 * q^23 - 2 * q^24 + 2 * q^25 - q^26 - 2 * q^27 - 2 * q^28 - q^29 + 2 * q^30 - 3 * q^31 + 2 * q^32 + 4 * q^33 - 10 * q^34 + 2 * q^35 + 2 * q^36 + 2 * q^37 - 3 * q^38 + q^39 - 2 * q^40 + 5 * q^41 + 2 * q^42 - 13 * q^43 - 4 * q^44 - 2 * q^45 - 9 * q^46 - 16 * q^47 - 2 * q^48 - 12 * q^49 + 2 * q^50 + 10 * q^51 - q^52 - 6 * q^53 - 2 * q^54 + 4 * q^55 - 2 * q^56 + 3 * q^57 - q^58 - 8 * q^59 + 2 * q^60 - 7 * q^61 - 3 * q^62 - 2 * q^63 + 2 * q^64 + q^65 + 4 * q^66 + 18 * q^67 - 10 * q^68 + 9 * q^69 + 2 * q^70 - 6 * q^71 + 2 * q^72 - 9 * q^73 + 2 * q^74 - 2 * q^75 - 3 * q^76 + 4 * q^77 + q^78 + 12 * q^79 - 2 * q^80 + 2 * q^81 + 5 * q^82 - 19 * q^83 + 2 * q^84 + 10 * q^85 - 13 * q^86 + q^87 - 4 * q^88 - q^89 - 2 * q^90 + q^91 - 9 * q^92 + 3 * q^93 - 16 * q^94 + 3 * q^95 - 2 * q^96 + 19 * q^97 - 12 * q^98 - 4 * 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
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−6.12311	−1.84619	−0.923093	−	0.384577i	$-0.874347\pi$
$11$	−6.12311	−1.84619	−0.923093	+	0.384577i	$0.874347\pi$
$12$	−1.00000	−0.288675
$13$	5.68466	1.57664	0.788320	−	0.615265i	$-0.210951\pi$
$13$	5.68466	1.57664	0.788320	+	0.615265i	$0.210951\pi$
$14$	−1.00000	−0.267261
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	1.00000	0.235702
$19$	0.561553	0.128829	0.0644145	−	0.997923i	$-0.479482\pi$
$19$	0.561553	0.128829	0.0644145	+	0.997923i	$0.479482\pi$
$20$	−1.00000	−0.223607
$21$	1.00000	0.218218
$22$	−6.12311	−1.30545
$23$	−6.56155	−1.36818	−0.684089	−	0.729398i	$-0.739800\pi$
$23$	−6.56155	−1.36818	−0.684089	+	0.729398i	$0.739800\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	5.68466	1.11485
$27$	−1.00000	−0.192450
$28$	−1.00000	−0.188982
$29$	−6.68466	−1.24131	−0.620655	−	0.784084i	$-0.713133\pi$
$29$	−6.68466	−1.24131	−0.620655	+	0.784084i	$0.713133\pi$
$30$	1.00000	0.182574
$31$	−3.56155	−0.639674	−0.319837	−	0.947473i	$-0.603628\pi$
$31$	−3.56155	−0.639674	−0.319837	+	0.947473i	$0.603628\pi$
$32$	1.00000	0.176777
$33$	6.12311	1.06590
$34$	−5.00000	−0.857493
$35$	1.00000	0.169031
$36$	1.00000	0.166667
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0.561553	0.0910959
$39$	−5.68466	−0.910274
$40$	−1.00000	−0.158114
$41$	8.68466	1.35632	0.678158	−	0.734916i	$-0.262779\pi$
$41$	8.68466	1.35632	0.678158	+	0.734916i	$0.262779\pi$
$42$	1.00000	0.154303
$43$	−12.6847	−1.93439	−0.967196	−	0.254031i	$-0.918244\pi$
$43$	−12.6847	−1.93439	−0.967196	+	0.254031i	$0.918244\pi$
$44$	−6.12311	−0.923093
$45$	−1.00000	−0.149071
$46$	−6.56155	−0.967448
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	−1.00000	−0.144338
$49$	−6.00000	−0.857143
$50$	1.00000	0.141421
$51$	5.00000	0.700140
$52$	5.68466	0.788320
$53$	5.24621	0.720623	0.360311	−	0.932832i	$-0.382670\pi$
$53$	5.24621	0.720623	0.360311	+	0.932832i	$0.382670\pi$
$54$	−1.00000	−0.136083
$55$	6.12311	0.825639
$56$	−1.00000	−0.133631
$57$	−0.561553	−0.0743795
$58$	−6.68466	−0.877739
$59$	4.24621	0.552810	0.276405	−	0.961041i	$-0.410857\pi$
$59$	4.24621	0.552810	0.276405	+	0.961041i	$0.410857\pi$
$60$	1.00000	0.129099
$61$	−13.8078	−1.76790	−0.883952	−	0.467579i	$-0.845126\pi$
$61$	−13.8078	−1.76790	−0.883952	+	0.467579i	$0.845126\pi$
$62$	−3.56155	−0.452318
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−5.68466	−0.705095
$66$	6.12311	0.753702
$67$	13.1231	1.60324	0.801621	−	0.597832i	$-0.203971\pi$
$67$	13.1231	1.60324	0.801621	+	0.597832i	$0.203971\pi$
$68$	−5.00000	−0.606339
$69$	6.56155	0.789918
$70$	1.00000	0.119523
$71$	1.12311	0.133288	0.0666441	−	0.997777i	$-0.478771\pi$
$71$	1.12311	0.133288	0.0666441	+	0.997777i	$0.478771\pi$
$72$	1.00000	0.117851
$73$	−6.56155	−0.767972	−0.383986	−	0.923339i	$-0.625449\pi$
$73$	−6.56155	−0.767972	−0.383986	+	0.923339i	$0.625449\pi$
$74$	1.00000	0.116248
$75$	−1.00000	−0.115470
$76$	0.561553	0.0644145
$77$	6.12311	0.697793
$78$	−5.68466	−0.643661
$79$	14.2462	1.60282	0.801412	−	0.598113i	$-0.204082\pi$
$79$	14.2462	1.60282	0.801412	+	0.598113i	$0.204082\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	8.68466	0.959060
$83$	−7.43845	−0.816476	−0.408238	−	0.912876i	$-0.633857\pi$
$83$	−7.43845	−0.816476	−0.408238	+	0.912876i	$0.633857\pi$
$84$	1.00000	0.109109
$85$	5.00000	0.542326
$86$	−12.6847	−1.36782
$87$	6.68466	0.716671
$88$	−6.12311	−0.652725
$89$	5.68466	0.602573	0.301286	−	0.953534i	$-0.402584\pi$
$89$	5.68466	0.602573	0.301286	+	0.953534i	$0.402584\pi$
$90$	−1.00000	−0.105409
$91$	−5.68466	−0.595914
$92$	−6.56155	−0.684089
$93$	3.56155	0.369316
$94$	−8.00000	−0.825137
$95$	−0.561553	−0.0576141
$96$	−1.00000	−0.102062
$97$	3.31534	0.336622	0.168311	−	0.985734i	$-0.446169\pi$
$97$	3.31534	0.336622	0.168311	+	0.985734i	$0.446169\pi$
$98$	−6.00000	−0.606092
$99$	−6.12311	−0.615395
$100$	1.00000	0.100000
$101$	10.8769	1.08229	0.541146	−	0.840929i	$-0.317991\pi$
$101$	10.8769	1.08229	0.541146	+	0.840929i	$0.317991\pi$
$102$	5.00000	0.495074
$103$	−16.4924	−1.62505	−0.812523	−	0.582929i	$-0.801907\pi$
$103$	−16.4924	−1.62505	−0.812523	+	0.582929i	$0.801907\pi$
$104$	5.68466	0.557427
$105$	−1.00000	−0.0975900
$106$	5.24621	0.509557
$107$	−10.8078	−1.04483	−0.522413	−	0.852693i	$-0.674968\pi$
$107$	−10.8078	−1.04483	−0.522413	+	0.852693i	$0.674968\pi$
$108$	−1.00000	−0.0962250
$109$	−2.12311	−0.203357	−0.101678	−	0.994817i	$-0.532421\pi$
$109$	−2.12311	−0.203357	−0.101678	+	0.994817i	$0.532421\pi$
$110$	6.12311	0.583815
$111$	−1.00000	−0.0949158
$112$	−1.00000	−0.0944911
$113$	−10.6847	−1.00513	−0.502564	−	0.864540i	$-0.667610\pi$
$113$	−10.6847	−1.00513	−0.502564	+	0.864540i	$0.667610\pi$
$114$	−0.561553	−0.0525942
$115$	6.56155	0.611868
$116$	−6.68466	−0.620655
$117$	5.68466	0.525547
$118$	4.24621	0.390895
$119$	5.00000	0.458349
$120$	1.00000	0.0912871
$121$	26.4924	2.40840
$122$	−13.8078	−1.25010
$123$	−8.68466	−0.783069
$124$	−3.56155	−0.319837
$125$	−1.00000	−0.0894427
$126$	−1.00000	−0.0890871
$127$	14.5616	1.29213	0.646064	−	0.763283i	$-0.276414\pi$
$127$	14.5616	1.29213	0.646064	+	0.763283i	$0.276414\pi$
$128$	1.00000	0.0883883
$129$	12.6847	1.11682
$130$	−5.68466	−0.498578
$131$	7.12311	0.622349	0.311174	−	0.950353i	$-0.399278\pi$
$131$	7.12311	0.622349	0.311174	+	0.950353i	$0.399278\pi$
$132$	6.12311	0.532948
$133$	−0.561553	−0.0486928
$134$	13.1231	1.13366
$135$	1.00000	0.0860663
$136$	−5.00000	−0.428746
$137$	−4.24621	−0.362778	−0.181389	−	0.983411i	$-0.558059\pi$
$137$	−4.24621	−0.362778	−0.181389	+	0.983411i	$0.558059\pi$
$138$	6.56155	0.558556
$139$	13.8078	1.17116	0.585580	−	0.810615i	$-0.300867\pi$
$139$	13.8078	1.17116	0.585580	+	0.810615i	$0.300867\pi$
$140$	1.00000	0.0845154
$141$	8.00000	0.673722
$142$	1.12311	0.0942489
$143$	−34.8078	−2.91077
$144$	1.00000	0.0833333
$145$	6.68466	0.555131
$146$	−6.56155	−0.543038
$147$	6.00000	0.494872
$148$	1.00000	0.0821995
$149$	7.75379	0.635215	0.317608	−	0.948222i	$-0.397121\pi$
$149$	7.75379	0.635215	0.317608	+	0.948222i	$0.397121\pi$
$150$	−1.00000	−0.0816497
$151$	−15.4384	−1.25636	−0.628182	−	0.778067i	$-0.716200\pi$
$151$	−15.4384	−1.25636	−0.628182	+	0.778067i	$0.716200\pi$
$152$	0.561553	0.0455479
$153$	−5.00000	−0.404226
$154$	6.12311	0.493414
$155$	3.56155	0.286071
$156$	−5.68466	−0.455137
$157$	−18.0540	−1.44086	−0.720432	−	0.693526i	$-0.756056\pi$
$157$	−18.0540	−1.44086	−0.720432	+	0.693526i	$0.756056\pi$
$158$	14.2462	1.13337
$159$	−5.24621	−0.416052
$160$	−1.00000	−0.0790569
$161$	6.56155	0.517123
$162$	1.00000	0.0785674
$163$	19.4924	1.52676	0.763382	−	0.645947i	$-0.223537\pi$
$163$	19.4924	1.52676	0.763382	+	0.645947i	$0.223537\pi$
$164$	8.68466	0.678158
$165$	−6.12311	−0.476683
$166$	−7.43845	−0.577335
$167$	19.0540	1.47444	0.737220	−	0.675652i	$-0.236138\pi$
$167$	19.0540	1.47444	0.737220	+	0.675652i	$0.236138\pi$
$168$	1.00000	0.0771517
$169$	19.3153	1.48580
$170$	5.00000	0.383482
$171$	0.561553	0.0429430
$172$	−12.6847	−0.967196
$173$	−2.12311	−0.161417	−0.0807084	−	0.996738i	$-0.525718\pi$
$173$	−2.12311	−0.161417	−0.0807084	+	0.996738i	$0.525718\pi$
$174$	6.68466	0.506763
$175$	−1.00000	−0.0755929
$176$	−6.12311	−0.461546
$177$	−4.24621	−0.319165
$178$	5.68466	0.426083
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	−1.00000	−0.0745356
$181$	20.2462	1.50489	0.752445	−	0.658656i	$-0.228875\pi$
$181$	20.2462	1.50489	0.752445	+	0.658656i	$0.228875\pi$
$182$	−5.68466	−0.421375
$183$	13.8078	1.02070
$184$	−6.56155	−0.483724
$185$	−1.00000	−0.0735215
$186$	3.56155	0.261146
$187$	30.6155	2.23883
$188$	−8.00000	−0.583460
$189$	1.00000	0.0727393
$190$	−0.561553	−0.0407393
$191$	14.3693	1.03973	0.519864	−	0.854249i	$-0.325983\pi$
$191$	14.3693	1.03973	0.519864	+	0.854249i	$0.325983\pi$
$192$	−1.00000	−0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	3.31534	0.238028
$195$	5.68466	0.407087
$196$	−6.00000	−0.428571
$197$	−2.31534	−0.164961	−0.0824806	−	0.996593i	$-0.526284\pi$
$197$	−2.31534	−0.164961	−0.0824806	+	0.996593i	$0.526284\pi$
$198$	−6.12311	−0.435150
$199$	−16.4924	−1.16912	−0.584558	−	0.811352i	$-0.698732\pi$
$199$	−16.4924	−1.16912	−0.584558	+	0.811352i	$0.698732\pi$
$200$	1.00000	0.0707107
$201$	−13.1231	−0.925633
$202$	10.8769	0.765296
$203$	6.68466	0.469171
$204$	5.00000	0.350070
$205$	−8.68466	−0.606563
$206$	−16.4924	−1.14908
$207$	−6.56155	−0.456059
$208$	5.68466	0.394160
$209$	−3.43845	−0.237842
$210$	−1.00000	−0.0690066
$211$	4.43845	0.305555	0.152778	−	0.988261i	$-0.451178\pi$
$211$	4.43845	0.305555	0.152778	+	0.988261i	$0.451178\pi$
$212$	5.24621	0.360311
$213$	−1.12311	−0.0769539
$214$	−10.8078	−0.738804
$215$	12.6847	0.865087
$216$	−1.00000	−0.0680414
$217$	3.56155	0.241774
$218$	−2.12311	−0.143795
$219$	6.56155	0.443389
$220$	6.12311	0.412820
$221$	−28.4233	−1.91196
$222$	−1.00000	−0.0671156
$223$	−19.8078	−1.32643	−0.663213	−	0.748431i	$-0.730808\pi$
$223$	−19.8078	−1.32643	−0.663213	+	0.748431i	$0.730808\pi$
$224$	−1.00000	−0.0668153
$225$	1.00000	0.0666667
$226$	−10.6847	−0.710733
$227$	−10.9309	−0.725507	−0.362754	−	0.931885i	$-0.618163\pi$
$227$	−10.9309	−0.725507	−0.362754	+	0.931885i	$0.618163\pi$
$228$	−0.561553	−0.0371897
$229$	−5.12311	−0.338544	−0.169272	−	0.985569i	$-0.554142\pi$
$229$	−5.12311	−0.338544	−0.169272	+	0.985569i	$0.554142\pi$
$230$	6.56155	0.432656
$231$	−6.12311	−0.402871
$232$	−6.68466	−0.438869
$233$	11.3693	0.744829	0.372414	−	0.928067i	$-0.378530\pi$
$233$	11.3693	0.744829	0.372414	+	0.928067i	$0.378530\pi$
$234$	5.68466	0.371618
$235$	8.00000	0.521862
$236$	4.24621	0.276405
$237$	−14.2462	−0.925391
$238$	5.00000	0.324102
$239$	7.80776	0.505042	0.252521	−	0.967591i	$-0.418740\pi$
$239$	7.80776	0.505042	0.252521	+	0.967591i	$0.418740\pi$
$240$	1.00000	0.0645497
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	26.4924	1.70300
$243$	−1.00000	−0.0641500
$244$	−13.8078	−0.883952
$245$	6.00000	0.383326
$246$	−8.68466	−0.553714
$247$	3.19224	0.203117
$248$	−3.56155	−0.226159
$249$	7.43845	0.471392
$250$	−1.00000	−0.0632456
$251$	25.6155	1.61684	0.808419	−	0.588608i	$-0.200324\pi$
$251$	25.6155	1.61684	0.808419	+	0.588608i	$0.200324\pi$
$252$	−1.00000	−0.0629941
$253$	40.1771	2.52591
$254$	14.5616	0.913673
$255$	−5.00000	−0.313112
$256$	1.00000	0.0625000
$257$	10.8078	0.674170	0.337085	−	0.941474i	$-0.390559\pi$
$257$	10.8078	0.674170	0.337085	+	0.941474i	$0.390559\pi$
$258$	12.6847	0.789712
$259$	−1.00000	−0.0621370
$260$	−5.68466	−0.352548
$261$	−6.68466	−0.413770
$262$	7.12311	0.440067
$263$	−6.43845	−0.397012	−0.198506	−	0.980100i	$-0.563609\pi$
$263$	−6.43845	−0.397012	−0.198506	+	0.980100i	$0.563609\pi$
$264$	6.12311	0.376851
$265$	−5.24621	−0.322272
$266$	−0.561553	−0.0344310
$267$	−5.68466	−0.347895
$268$	13.1231	0.801621
$269$	−16.8078	−1.02479	−0.512394	−	0.858751i	$-0.671241\pi$
$269$	−16.8078	−1.02479	−0.512394	+	0.858751i	$0.671241\pi$
$270$	1.00000	0.0608581
$271$	−16.4924	−1.00184	−0.500922	−	0.865493i	$-0.667006\pi$
$271$	−16.4924	−1.00184	−0.500922	+	0.865493i	$0.667006\pi$
$272$	−5.00000	−0.303170
$273$	5.68466	0.344051
$274$	−4.24621	−0.256523
$275$	−6.12311	−0.369237
$276$	6.56155	0.394959
$277$	27.6847	1.66341	0.831705	−	0.555218i	$-0.187365\pi$
$277$	27.6847	1.66341	0.831705	+	0.555218i	$0.187365\pi$
$278$	13.8078	0.828135
$279$	−3.56155	−0.213225
$280$	1.00000	0.0597614
$281$	22.1771	1.32297	0.661487	−	0.749957i	$-0.269926\pi$
$281$	22.1771	1.32297	0.661487	+	0.749957i	$0.269926\pi$
$282$	8.00000	0.476393
$283$	9.93087	0.590329	0.295164	−	0.955446i	$-0.404626\pi$
$283$	9.93087	0.590329	0.295164	+	0.955446i	$0.404626\pi$
$284$	1.12311	0.0666441
$285$	0.561553	0.0332635
$286$	−34.8078	−2.05823
$287$	−8.68466	−0.512639
$288$	1.00000	0.0589256
$289$	8.00000	0.470588
$290$	6.68466	0.392537
$291$	−3.31534	−0.194349
$292$	−6.56155	−0.383986
$293$	−16.6155	−0.970690	−0.485345	−	0.874323i	$-0.661306\pi$
$293$	−16.6155	−0.970690	−0.485345	+	0.874323i	$0.661306\pi$
$294$	6.00000	0.349927
$295$	−4.24621	−0.247224
$296$	1.00000	0.0581238
$297$	6.12311	0.355299
$298$	7.75379	0.449165
$299$	−37.3002	−2.15713
$300$	−1.00000	−0.0577350
$301$	12.6847	0.731132
$302$	−15.4384	−0.888383
$303$	−10.8769	−0.624861
$304$	0.561553	0.0322073
$305$	13.8078	0.790630
$306$	−5.00000	−0.285831
$307$	−0.246211	−0.0140520	−0.00702601	−	0.999975i	$-0.502236\pi$
$307$	−0.246211	−0.0140520	−0.00702601	+	0.999975i	$0.502236\pi$
$308$	6.12311	0.348896
$309$	16.4924	0.938221
$310$	3.56155	0.202283
$311$	−21.1771	−1.20084	−0.600421	−	0.799684i	$-0.705000\pi$
$311$	−21.1771	−1.20084	−0.600421	+	0.799684i	$0.705000\pi$
$312$	−5.68466	−0.321830
$313$	−24.8769	−1.40613	−0.703063	−	0.711128i	$-0.748185\pi$
$313$	−24.8769	−1.40613	−0.703063	+	0.711128i	$0.748185\pi$
$314$	−18.0540	−1.01884
$315$	1.00000	0.0563436
$316$	14.2462	0.801412
$317$	16.9309	0.950932	0.475466	−	0.879734i	$-0.342279\pi$
$317$	16.9309	0.950932	0.475466	+	0.879734i	$0.342279\pi$
$318$	−5.24621	−0.294193
$319$	40.9309	2.29169
$320$	−1.00000	−0.0559017
$321$	10.8078	0.603231
$322$	6.56155	0.365661
$323$	−2.80776	−0.156228
$324$	1.00000	0.0555556
$325$	5.68466	0.315328
$326$	19.4924	1.07959
$327$	2.12311	0.117408
$328$	8.68466	0.479530
$329$	8.00000	0.441054
$330$	−6.12311	−0.337066
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	−7.43845	−0.408238
$333$	1.00000	0.0547997
$334$	19.0540	1.04259
$335$	−13.1231	−0.716992
$336$	1.00000	0.0545545
$337$	4.56155	0.248484	0.124242	−	0.992252i	$-0.460350\pi$
$337$	4.56155	0.248484	0.124242	+	0.992252i	$0.460350\pi$
$338$	19.3153	1.05062
$339$	10.6847	0.580311
$340$	5.00000	0.271163
$341$	21.8078	1.18096
$342$	0.561553	0.0303653
$343$	13.0000	0.701934
$344$	−12.6847	−0.683911
$345$	−6.56155	−0.353262
$346$	−2.12311	−0.114139
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	6.68466	0.358335
$349$	14.2462	0.762582	0.381291	−	0.924455i	$-0.375480\pi$
$349$	14.2462	0.762582	0.381291	+	0.924455i	$0.375480\pi$
$350$	−1.00000	−0.0534522
$351$	−5.68466	−0.303425
$352$	−6.12311	−0.326363
$353$	−29.8078	−1.58651	−0.793254	−	0.608891i	$-0.791615\pi$
$353$	−29.8078	−1.58651	−0.793254	+	0.608891i	$0.791615\pi$
$354$	−4.24621	−0.225684
$355$	−1.12311	−0.0596083
$356$	5.68466	0.301286
$357$	−5.00000	−0.264628
$358$	−4.00000	−0.211407
$359$	−20.8769	−1.10184	−0.550920	−	0.834558i	$-0.685723\pi$
$359$	−20.8769	−1.10184	−0.550920	+	0.834558i	$0.685723\pi$
$360$	−1.00000	−0.0527046
$361$	−18.6847	−0.983403
$362$	20.2462	1.06412
$363$	−26.4924	−1.39049
$364$	−5.68466	−0.297957
$365$	6.56155	0.343447
$366$	13.8078	0.721743
$367$	−17.0000	−0.887393	−0.443696	−	0.896177i	$-0.646333\pi$
$367$	−17.0000	−0.887393	−0.443696	+	0.896177i	$0.646333\pi$
$368$	−6.56155	−0.342045
$369$	8.68466	0.452105
$370$	−1.00000	−0.0519875
$371$	−5.24621	−0.272370
$372$	3.56155	0.184658
$373$	−30.4924	−1.57884	−0.789419	−	0.613855i	$-0.789618\pi$
$373$	−30.4924	−1.57884	−0.789419	+	0.613855i	$0.789618\pi$
$374$	30.6155	1.58309
$375$	1.00000	0.0516398
$376$	−8.00000	−0.412568
$377$	−38.0000	−1.95710
$378$	1.00000	0.0514344
$379$	−2.87689	−0.147776	−0.0738881	−	0.997267i	$-0.523541\pi$
$379$	−2.87689	−0.147776	−0.0738881	+	0.997267i	$0.523541\pi$
$380$	−0.561553	−0.0288071
$381$	−14.5616	−0.746011
$382$	14.3693	0.735198
$383$	−27.6847	−1.41462	−0.707310	−	0.706904i	$-0.750091\pi$
$383$	−27.6847	−1.41462	−0.707310	+	0.706904i	$0.750091\pi$
$384$	−1.00000	−0.0510310
$385$	−6.12311	−0.312062
$386$	−14.0000	−0.712581
$387$	−12.6847	−0.644797
$388$	3.31534	0.168311
$389$	20.9309	1.06124	0.530619	−	0.847611i	$-0.321960\pi$
$389$	20.9309	1.06124	0.530619	+	0.847611i	$0.321960\pi$
$390$	5.68466	0.287854
$391$	32.8078	1.65916
$392$	−6.00000	−0.303046
$393$	−7.12311	−0.359313
$394$	−2.31534	−0.116645
$395$	−14.2462	−0.716805
$396$	−6.12311	−0.307698
$397$	27.1231	1.36127	0.680635	−	0.732623i	$-0.261704\pi$
$397$	27.1231	1.36127	0.680635	+	0.732623i	$0.261704\pi$
$398$	−16.4924	−0.826690
$399$	0.561553	0.0281128
$400$	1.00000	0.0500000
$401$	−14.8078	−0.739464	−0.369732	−	0.929138i	$-0.620551\pi$
$401$	−14.8078	−0.739464	−0.369732	+	0.929138i	$0.620551\pi$
$402$	−13.1231	−0.654521
$403$	−20.2462	−1.00854
$404$	10.8769	0.541146
$405$	−1.00000	−0.0496904
$406$	6.68466	0.331754
$407$	−6.12311	−0.303511
$408$	5.00000	0.247537
$409$	−0.876894	−0.0433596	−0.0216798	−	0.999765i	$-0.506901\pi$
$409$	−0.876894	−0.0433596	−0.0216798	+	0.999765i	$0.506901\pi$
$410$	−8.68466	−0.428905
$411$	4.24621	0.209450
$412$	−16.4924	−0.812523
$413$	−4.24621	−0.208942
$414$	−6.56155	−0.322483
$415$	7.43845	0.365139
$416$	5.68466	0.278713
$417$	−13.8078	−0.676169
$418$	−3.43845	−0.168180
$419$	−10.5616	−0.515966	−0.257983	−	0.966150i	$-0.583058\pi$
$419$	−10.5616	−0.515966	−0.257983	+	0.966150i	$0.583058\pi$
$420$	−1.00000	−0.0487950
$421$	30.9848	1.51011	0.755054	−	0.655662i	$-0.227610\pi$
$421$	30.9848	1.51011	0.755054	+	0.655662i	$0.227610\pi$
$422$	4.43845	0.216060
$423$	−8.00000	−0.388973
$424$	5.24621	0.254779
$425$	−5.00000	−0.242536
$426$	−1.12311	−0.0544146
$427$	13.8078	0.668205
$428$	−10.8078	−0.522413
$429$	34.8078	1.68053
$430$	12.6847	0.611709
$431$	−29.0000	−1.39688	−0.698440	−	0.715668i	$-0.746122\pi$
$431$	−29.0000	−1.39688	−0.698440	+	0.715668i	$0.746122\pi$
$432$	−1.00000	−0.0481125
$433$	−30.4233	−1.46205	−0.731025	−	0.682351i	$-0.760958\pi$
$433$	−30.4233	−1.46205	−0.731025	+	0.682351i	$0.760958\pi$
$434$	3.56155	0.170960
$435$	−6.68466	−0.320505
$436$	−2.12311	−0.101678
$437$	−3.68466	−0.176261
$438$	6.56155	0.313523
$439$	−14.0540	−0.670760	−0.335380	−	0.942083i	$-0.608865\pi$
$439$	−14.0540	−0.670760	−0.335380	+	0.942083i	$0.608865\pi$
$440$	6.12311	0.291908
$441$	−6.00000	−0.285714
$442$	−28.4233	−1.35196
$443$	33.8617	1.60882	0.804410	−	0.594075i	$-0.202482\pi$
$443$	33.8617	1.60882	0.804410	+	0.594075i	$0.202482\pi$
$444$	−1.00000	−0.0474579
$445$	−5.68466	−0.269479
$446$	−19.8078	−0.937925
$447$	−7.75379	−0.366742
$448$	−1.00000	−0.0472456
$449$	−1.61553	−0.0762415	−0.0381207	−	0.999273i	$-0.512137\pi$
$449$	−1.61553	−0.0762415	−0.0381207	+	0.999273i	$0.512137\pi$
$450$	1.00000	0.0471405
$451$	−53.1771	−2.50401
$452$	−10.6847	−0.502564
$453$	15.4384	0.725362
$454$	−10.9309	−0.513011
$455$	5.68466	0.266501
$456$	−0.561553	−0.0262971
$457$	−30.9309	−1.44689	−0.723443	−	0.690385i	$-0.757441\pi$
$457$	−30.9309	−1.44689	−0.723443	+	0.690385i	$0.757441\pi$
$458$	−5.12311	−0.239387
$459$	5.00000	0.233380
$460$	6.56155	0.305934
$461$	0.438447	0.0204205	0.0102103	−	0.999948i	$-0.496750\pi$
$461$	0.438447	0.0204205	0.0102103	+	0.999948i	$0.496750\pi$
$462$	−6.12311	−0.284873
$463$	4.87689	0.226649	0.113324	−	0.993558i	$-0.463850\pi$
$463$	4.87689	0.226649	0.113324	+	0.993558i	$0.463850\pi$
$464$	−6.68466	−0.310327
$465$	−3.56155	−0.165163
$466$	11.3693	0.526673
$467$	2.43845	0.112838	0.0564189	−	0.998407i	$-0.482032\pi$
$467$	2.43845	0.112838	0.0564189	+	0.998407i	$0.482032\pi$
$468$	5.68466	0.262773
$469$	−13.1231	−0.605969
$470$	8.00000	0.369012
$471$	18.0540	0.831883
$472$	4.24621	0.195448
$473$	77.6695	3.57125
$474$	−14.2462	−0.654350
$475$	0.561553	0.0257658
$476$	5.00000	0.229175
$477$	5.24621	0.240208
$478$	7.80776	0.357119
$479$	−6.56155	−0.299805	−0.149903	−	0.988701i	$-0.547896\pi$
$479$	−6.56155	−0.299805	−0.149903	+	0.988701i	$0.547896\pi$
$480$	1.00000	0.0456435
$481$	5.68466	0.259198
$482$	−2.00000	−0.0910975
$483$	−6.56155	−0.298561
$484$	26.4924	1.20420
$485$	−3.31534	−0.150542
$486$	−1.00000	−0.0453609
$487$	16.7386	0.758500	0.379250	−	0.925294i	$-0.376182\pi$
$487$	16.7386	0.758500	0.379250	+	0.925294i	$0.376182\pi$
$488$	−13.8078	−0.625048
$489$	−19.4924	−0.881478
$490$	6.00000	0.271052
$491$	−12.1771	−0.549544	−0.274772	−	0.961509i	$-0.588602\pi$
$491$	−12.1771	−0.549544	−0.274772	+	0.961509i	$0.588602\pi$
$492$	−8.68466	−0.391535
$493$	33.4233	1.50531
$494$	3.19224	0.143625
$495$	6.12311	0.275213
$496$	−3.56155	−0.159918
$497$	−1.12311	−0.0503782
$498$	7.43845	0.333325
$499$	32.8078	1.46868	0.734339	−	0.678783i	$-0.237492\pi$
$499$	32.8078	1.46868	0.734339	+	0.678783i	$0.237492\pi$
$500$	−1.00000	−0.0447214
$501$	−19.0540	−0.851269
$502$	25.6155	1.14328
$503$	−18.2462	−0.813558	−0.406779	−	0.913527i	$-0.633348\pi$
$503$	−18.2462	−0.813558	−0.406779	+	0.913527i	$0.633348\pi$
$504$	−1.00000	−0.0445435
$505$	−10.8769	−0.484015
$506$	40.1771	1.78609
$507$	−19.3153	−0.857824
$508$	14.5616	0.646064
$509$	23.0540	1.02185	0.510925	−	0.859625i	$-0.329303\pi$
$509$	23.0540	1.02185	0.510925	+	0.859625i	$0.329303\pi$
$510$	−5.00000	−0.221404
$511$	6.56155	0.290266
$512$	1.00000	0.0441942
$513$	−0.561553	−0.0247932
$514$	10.8078	0.476710
$515$	16.4924	0.726743
$516$	12.6847	0.558411
$517$	48.9848	2.15435
$518$	−1.00000	−0.0439375
$519$	2.12311	0.0931940
$520$	−5.68466	−0.249289
$521$	−18.4384	−0.807803	−0.403902	−	0.914802i	$-0.632346\pi$
$521$	−18.4384	−0.807803	−0.403902	+	0.914802i	$0.632346\pi$
$522$	−6.68466	−0.292580
$523$	30.7386	1.34411	0.672053	−	0.740503i	$-0.265413\pi$
$523$	30.7386	1.34411	0.672053	+	0.740503i	$0.265413\pi$
$524$	7.12311	0.311174
$525$	1.00000	0.0436436
$526$	−6.43845	−0.280730
$527$	17.8078	0.775718
$528$	6.12311	0.266474
$529$	20.0540	0.871912
$530$	−5.24621	−0.227881
$531$	4.24621	0.184270
$532$	−0.561553	−0.0243464
$533$	49.3693	2.13842
$534$	−5.68466	−0.245999
$535$	10.8078	0.467260
$536$	13.1231	0.566832
$537$	4.00000	0.172613
$538$	−16.8078	−0.724634
$539$	36.7386	1.58244
$540$	1.00000	0.0430331
$541$	−20.4233	−0.878066	−0.439033	−	0.898471i	$-0.644679\pi$
$541$	−20.4233	−0.878066	−0.439033	+	0.898471i	$0.644679\pi$
$542$	−16.4924	−0.708410
$543$	−20.2462	−0.868848
$544$	−5.00000	−0.214373
$545$	2.12311	0.0909439
$546$	5.68466	0.243281
$547$	−39.7386	−1.69910	−0.849551	−	0.527507i	$-0.823127\pi$
$547$	−39.7386	−1.69910	−0.849551	+	0.527507i	$0.823127\pi$
$548$	−4.24621	−0.181389
$549$	−13.8078	−0.589301
$550$	−6.12311	−0.261090
$551$	−3.75379	−0.159917
$552$	6.56155	0.279278
$553$	−14.2462	−0.605811
$554$	27.6847	1.17621
$555$	1.00000	0.0424476
$556$	13.8078	0.585580
$557$	7.61553	0.322680	0.161340	−	0.986899i	$-0.448418\pi$
$557$	7.61553	0.322680	0.161340	+	0.986899i	$0.448418\pi$
$558$	−3.56155	−0.150773
$559$	−72.1080	−3.04984
$560$	1.00000	0.0422577
$561$	−30.6155	−1.29259
$562$	22.1771	0.935484
$563$	4.93087	0.207811	0.103906	−	0.994587i	$-0.466866\pi$
$563$	4.93087	0.207811	0.103906	+	0.994587i	$0.466866\pi$
$564$	8.00000	0.336861
$565$	10.6847	0.449507
$566$	9.93087	0.417426
$567$	−1.00000	−0.0419961
$568$	1.12311	0.0471245
$569$	2.31534	0.0970642	0.0485321	−	0.998822i	$-0.484546\pi$
$569$	2.31534	0.0970642	0.0485321	+	0.998822i	$0.484546\pi$
$570$	0.561553	0.0235209
$571$	−5.80776	−0.243047	−0.121524	−	0.992589i	$-0.538778\pi$
$571$	−5.80776	−0.243047	−0.121524	+	0.992589i	$0.538778\pi$
$572$	−34.8078	−1.45539
$573$	−14.3693	−0.600287
$574$	−8.68466	−0.362491
$575$	−6.56155	−0.273636
$576$	1.00000	0.0416667
$577$	−14.4924	−0.603327	−0.301664	−	0.953414i	$-0.597542\pi$
$577$	−14.4924	−0.603327	−0.301664	+	0.953414i	$0.597542\pi$
$578$	8.00000	0.332756
$579$	14.0000	0.581820
$580$	6.68466	0.277565
$581$	7.43845	0.308599
$582$	−3.31534	−0.137425
$583$	−32.1231	−1.33040
$584$	−6.56155	−0.271519
$585$	−5.68466	−0.235032
$586$	−16.6155	−0.686381
$587$	−12.3002	−0.507683	−0.253842	−	0.967246i	$-0.581694\pi$
$587$	−12.3002	−0.507683	−0.253842	+	0.967246i	$0.581694\pi$
$588$	6.00000	0.247436
$589$	−2.00000	−0.0824086
$590$	−4.24621	−0.174814
$591$	2.31534	0.0952404
$592$	1.00000	0.0410997
$593$	4.87689	0.200270	0.100135	−	0.994974i	$-0.468073\pi$
$593$	4.87689	0.200270	0.100135	+	0.994974i	$0.468073\pi$
$594$	6.12311	0.251234
$595$	−5.00000	−0.204980
$596$	7.75379	0.317608
$597$	16.4924	0.674990
$598$	−37.3002	−1.52532
$599$	26.7386	1.09251	0.546255	−	0.837619i	$-0.316053\pi$
$599$	26.7386	1.09251	0.546255	+	0.837619i	$0.316053\pi$
$600$	−1.00000	−0.0408248
$601$	−42.8617	−1.74837	−0.874183	−	0.485596i	$-0.838603\pi$
$601$	−42.8617	−1.74837	−0.874183	+	0.485596i	$0.838603\pi$
$602$	12.6847	0.516988
$603$	13.1231	0.534414
$604$	−15.4384	−0.628182
$605$	−26.4924	−1.07707
$606$	−10.8769	−0.441844
$607$	−26.4924	−1.07529	−0.537647	−	0.843170i	$-0.680687\pi$
$607$	−26.4924	−1.07529	−0.537647	+	0.843170i	$0.680687\pi$
$608$	0.561553	0.0227740
$609$	−6.68466	−0.270876
$610$	13.8078	0.559060
$611$	−45.4773	−1.83981
$612$	−5.00000	−0.202113
$613$	8.43845	0.340826	0.170413	−	0.985373i	$-0.445490\pi$
$613$	8.43845	0.340826	0.170413	+	0.985373i	$0.445490\pi$
$614$	−0.246211	−0.00993628
$615$	8.68466	0.350199
$616$	6.12311	0.246707
$617$	14.4924	0.583443	0.291721	−	0.956503i	$-0.405772\pi$
$617$	14.4924	0.583443	0.291721	+	0.956503i	$0.405772\pi$
$618$	16.4924	0.663423
$619$	−35.8078	−1.43924	−0.719618	−	0.694370i	$-0.755683\pi$
$619$	−35.8078	−1.43924	−0.719618	+	0.694370i	$0.755683\pi$
$620$	3.56155	0.143035
$621$	6.56155	0.263306
$622$	−21.1771	−0.849124
$623$	−5.68466	−0.227751
$624$	−5.68466	−0.227568
$625$	1.00000	0.0400000
$626$	−24.8769	−0.994281
$627$	3.43845	0.137318
$628$	−18.0540	−0.720432
$629$	−5.00000	−0.199363
$630$	1.00000	0.0398410
$631$	15.3153	0.609694	0.304847	−	0.952401i	$-0.401395\pi$
$631$	15.3153	0.609694	0.304847	+	0.952401i	$0.401395\pi$
$632$	14.2462	0.566684
$633$	−4.43845	−0.176412
$634$	16.9309	0.672411
$635$	−14.5616	−0.577858
$636$	−5.24621	−0.208026
$637$	−34.1080	−1.35141
$638$	40.9309	1.62047
$639$	1.12311	0.0444294
$640$	−1.00000	−0.0395285
$641$	1.94602	0.0768634	0.0384317	−	0.999261i	$-0.487764\pi$
$641$	1.94602	0.0768634	0.0384317	+	0.999261i	$0.487764\pi$
$642$	10.8078	0.426548
$643$	−34.1231	−1.34568	−0.672842	−	0.739786i	$-0.734927\pi$
$643$	−34.1231	−1.34568	−0.672842	+	0.739786i	$0.734927\pi$
$644$	6.56155	0.258561
$645$	−12.6847	−0.499458
$646$	−2.80776	−0.110470
$647$	−36.8078	−1.44706	−0.723531	−	0.690292i	$-0.757482\pi$
$647$	−36.8078	−1.44706	−0.723531	+	0.690292i	$0.757482\pi$
$648$	1.00000	0.0392837
$649$	−26.0000	−1.02059
$650$	5.68466	0.222971
$651$	−3.56155	−0.139588
$652$	19.4924	0.763382
$653$	−28.4924	−1.11499	−0.557497	−	0.830179i	$-0.688238\pi$
$653$	−28.4924	−1.11499	−0.557497	+	0.830179i	$0.688238\pi$
$654$	2.12311	0.0830200
$655$	−7.12311	−0.278323
$656$	8.68466	0.339079
$657$	−6.56155	−0.255991
$658$	8.00000	0.311872
$659$	−18.2462	−0.710771	−0.355386	−	0.934720i	$-0.615650\pi$
$659$	−18.2462	−0.710771	−0.355386	+	0.934720i	$0.615650\pi$
$660$	−6.12311	−0.238342
$661$	32.8617	1.27817	0.639087	−	0.769135i	$-0.279312\pi$
$661$	32.8617	1.27817	0.639087	+	0.769135i	$0.279312\pi$
$662$	−12.0000	−0.466393
$663$	28.4233	1.10387
$664$	−7.43845	−0.288668
$665$	0.561553	0.0217761
$666$	1.00000	0.0387492
$667$	43.8617	1.69833
$668$	19.0540	0.737220
$669$	19.8078	0.765812
$670$	−13.1231	−0.506990
$671$	84.5464	3.26388
$672$	1.00000	0.0385758
$673$	−18.5616	−0.715495	−0.357748	−	0.933818i	$-0.616455\pi$
$673$	−18.5616	−0.715495	−0.357748	+	0.933818i	$0.616455\pi$
$674$	4.56155	0.175704
$675$	−1.00000	−0.0384900
$676$	19.3153	0.742898
$677$	35.9309	1.38094	0.690468	−	0.723363i	$-0.257405\pi$
$677$	35.9309	1.38094	0.690468	+	0.723363i	$0.257405\pi$
$678$	10.6847	0.410342
$679$	−3.31534	−0.127231
$680$	5.00000	0.191741
$681$	10.9309	0.418872
$682$	21.8078	0.835062
$683$	−30.4384	−1.16469	−0.582347	−	0.812940i	$-0.697866\pi$
$683$	−30.4384	−1.16469	−0.582347	+	0.812940i	$0.697866\pi$
$684$	0.561553	0.0214715
$685$	4.24621	0.162239
$686$	13.0000	0.496342
$687$	5.12311	0.195459
$688$	−12.6847	−0.483598
$689$	29.8229	1.13616
$690$	−6.56155	−0.249794
$691$	−5.31534	−0.202205	−0.101103	−	0.994876i	$-0.532237\pi$
$691$	−5.31534	−0.202205	−0.101103	+	0.994876i	$0.532237\pi$
$692$	−2.12311	−0.0807084
$693$	6.12311	0.232598
$694$	−28.0000	−1.06287
$695$	−13.8078	−0.523758
$696$	6.68466	0.253381
$697$	−43.4233	−1.64477
$698$	14.2462	0.539227
$699$	−11.3693	−0.430027
$700$	−1.00000	−0.0377964
$701$	9.36932	0.353874	0.176937	−	0.984222i	$-0.443381\pi$
$701$	9.36932	0.353874	0.176937	+	0.984222i	$0.443381\pi$
$702$	−5.68466	−0.214554
$703$	0.561553	0.0211794
$704$	−6.12311	−0.230773
$705$	−8.00000	−0.301297
$706$	−29.8078	−1.12183
$707$	−10.8769	−0.409068
$708$	−4.24621	−0.159582
$709$	−4.50758	−0.169286	−0.0846428	−	0.996411i	$-0.526975\pi$
$709$	−4.50758	−0.169286	−0.0846428	+	0.996411i	$0.526975\pi$
$710$	−1.12311	−0.0421494
$711$	14.2462	0.534275
$712$	5.68466	0.213042
$713$	23.3693	0.875188
$714$	−5.00000	−0.187120
$715$	34.8078	1.30174
$716$	−4.00000	−0.149487
$717$	−7.80776	−0.291586
$718$	−20.8769	−0.779119
$719$	−2.00000	−0.0745874	−0.0372937	−	0.999304i	$-0.511874\pi$
$719$	−2.00000	−0.0745874	−0.0372937	+	0.999304i	$0.511874\pi$
$720$	−1.00000	−0.0372678
$721$	16.4924	0.614210
$722$	−18.6847	−0.695371
$723$	2.00000	0.0743808
$724$	20.2462	0.752445
$725$	−6.68466	−0.248262
$726$	−26.4924	−0.983226
$727$	50.9848	1.89092	0.945462	−	0.325734i	$-0.105611\pi$
$727$	50.9848	1.89092	0.945462	+	0.325734i	$0.105611\pi$
$728$	−5.68466	−0.210687
$729$	1.00000	0.0370370
$730$	6.56155	0.242854
$731$	63.4233	2.34580
$732$	13.8078	0.510350
$733$	−22.3002	−0.823676	−0.411838	−	0.911257i	$-0.635113\pi$
$733$	−22.3002	−0.823676	−0.411838	+	0.911257i	$0.635113\pi$
$734$	−17.0000	−0.627481
$735$	−6.00000	−0.221313
$736$	−6.56155	−0.241862
$737$	−80.3542	−2.95988
$738$	8.68466	0.319687
$739$	−39.1771	−1.44115	−0.720576	−	0.693376i	$-0.756123\pi$
$739$	−39.1771	−1.44115	−0.720576	+	0.693376i	$0.756123\pi$
$740$	−1.00000	−0.0367607
$741$	−3.19224	−0.117270
$742$	−5.24621	−0.192594
$743$	21.4233	0.785944	0.392972	−	0.919550i	$-0.371447\pi$
$743$	21.4233	0.785944	0.392972	+	0.919550i	$0.371447\pi$
$744$	3.56155	0.130573
$745$	−7.75379	−0.284077
$746$	−30.4924	−1.11641
$747$	−7.43845	−0.272159
$748$	30.6155	1.11941
$749$	10.8078	0.394907
$750$	1.00000	0.0365148
$751$	22.7386	0.829745	0.414872	−	0.909880i	$-0.363826\pi$
$751$	22.7386	0.829745	0.414872	+	0.909880i	$0.363826\pi$
$752$	−8.00000	−0.291730
$753$	−25.6155	−0.933482
$754$	−38.0000	−1.38388
$755$	15.4384	0.561863
$756$	1.00000	0.0363696
$757$	−14.9460	−0.543223	−0.271611	−	0.962407i	$-0.587556\pi$
$757$	−14.9460	−0.543223	−0.271611	+	0.962407i	$0.587556\pi$
$758$	−2.87689	−0.104494
$759$	−40.1771	−1.45834
$760$	−0.561553	−0.0203697
$761$	17.3153	0.627681	0.313840	−	0.949476i	$-0.398384\pi$
$761$	17.3153	0.627681	0.313840	+	0.949476i	$0.398384\pi$
$762$	−14.5616	−0.527509
$763$	2.12311	0.0768616
$764$	14.3693	0.519864
$765$	5.00000	0.180775
$766$	−27.6847	−1.00029
$767$	24.1383	0.871582
$768$	−1.00000	−0.0360844
$769$	−9.50758	−0.342852	−0.171426	−	0.985197i	$-0.554837\pi$
$769$	−9.50758	−0.342852	−0.171426	+	0.985197i	$0.554837\pi$
$770$	−6.12311	−0.220661
$771$	−10.8078	−0.389232
$772$	−14.0000	−0.503871
$773$	−3.49242	−0.125614	−0.0628069	−	0.998026i	$-0.520005\pi$
$773$	−3.49242	−0.125614	−0.0628069	+	0.998026i	$0.520005\pi$
$774$	−12.6847	−0.455941
$775$	−3.56155	−0.127935
$776$	3.31534	0.119014
$777$	1.00000	0.0358748
$778$	20.9309	0.750408
$779$	4.87689	0.174733
$780$	5.68466	0.203543
$781$	−6.87689	−0.246075
$782$	32.8078	1.17320
$783$	6.68466	0.238890
$784$	−6.00000	−0.214286
$785$	18.0540	0.644374
$786$	−7.12311	−0.254073
$787$	26.8769	0.958058	0.479029	−	0.877799i	$-0.340989\pi$
$787$	26.8769	0.958058	0.479029	+	0.877799i	$0.340989\pi$
$788$	−2.31534	−0.0824806
$789$	6.43845	0.229215
$790$	−14.2462	−0.506857
$791$	10.6847	0.379903
$792$	−6.12311	−0.217575
$793$	−78.4924	−2.78735
$794$	27.1231	0.962563
$795$	5.24621	0.186064
$796$	−16.4924	−0.584558
$797$	−25.6155	−0.907349	−0.453674	−	0.891168i	$-0.649887\pi$
$797$	−25.6155	−0.907349	−0.453674	+	0.891168i	$0.649887\pi$
$798$	0.561553	0.0198788
$799$	40.0000	1.41510
$800$	1.00000	0.0353553
$801$	5.68466	0.200858
$802$	−14.8078	−0.522880
$803$	40.1771	1.41782
$804$	−13.1231	−0.462816
$805$	−6.56155	−0.231264
$806$	−20.2462	−0.713142
$807$	16.8078	0.591661
$808$	10.8769	0.382648
$809$	−25.4384	−0.894368	−0.447184	−	0.894442i	$-0.647573\pi$
$809$	−25.4384	−0.894368	−0.447184	+	0.894442i	$0.647573\pi$
$810$	−1.00000	−0.0351364
$811$	2.73863	0.0961664	0.0480832	−	0.998843i	$-0.484689\pi$
$811$	2.73863	0.0961664	0.0480832	+	0.998843i	$0.484689\pi$
$812$	6.68466	0.234586
$813$	16.4924	0.578415
$814$	−6.12311	−0.214615
$815$	−19.4924	−0.682790
$816$	5.00000	0.175035
$817$	−7.12311	−0.249206
$818$	−0.876894	−0.0306599
$819$	−5.68466	−0.198638
$820$	−8.68466	−0.303281
$821$	23.4384	0.818007	0.409004	−	0.912533i	$-0.365876\pi$
$821$	23.4384	0.818007	0.409004	+	0.912533i	$0.365876\pi$
$822$	4.24621	0.148104
$823$	31.5464	1.09964	0.549819	−	0.835284i	$-0.314697\pi$
$823$	31.5464	1.09964	0.549819	+	0.835284i	$0.314697\pi$
$824$	−16.4924	−0.574541
$825$	6.12311	0.213179
$826$	−4.24621	−0.147745
$827$	−10.4384	−0.362980	−0.181490	−	0.983393i	$-0.558092\pi$
$827$	−10.4384	−0.362980	−0.181490	+	0.983393i	$0.558092\pi$
$828$	−6.56155	−0.228030
$829$	−22.1231	−0.768367	−0.384184	−	0.923257i	$-0.625517\pi$
$829$	−22.1231	−0.768367	−0.384184	+	0.923257i	$0.625517\pi$
$830$	7.43845	0.258192
$831$	−27.6847	−0.960370
$832$	5.68466	0.197080
$833$	30.0000	1.03944
$834$	−13.8078	−0.478124
$835$	−19.0540	−0.659390
$836$	−3.43845	−0.118921
$837$	3.56155	0.123105
$838$	−10.5616	−0.364843
$839$	28.2462	0.975168	0.487584	−	0.873076i	$-0.337878\pi$
$839$	28.2462	0.975168	0.487584	+	0.873076i	$0.337878\pi$
$840$	−1.00000	−0.0345033
$841$	15.6847	0.540850
$842$	30.9848	1.06781
$843$	−22.1771	−0.763819
$844$	4.43845	0.152778
$845$	−19.3153	−0.664468
$846$	−8.00000	−0.275046
$847$	−26.4924	−0.910290
$848$	5.24621	0.180156
$849$	−9.93087	−0.340827
$850$	−5.00000	−0.171499
$851$	−6.56155	−0.224927
$852$	−1.12311	−0.0384770
$853$	7.93087	0.271548	0.135774	−	0.990740i	$-0.456648\pi$
$853$	7.93087	0.271548	0.135774	+	0.990740i	$0.456648\pi$
$854$	13.8078	0.472492
$855$	−0.561553	−0.0192047
$856$	−10.8078	−0.369402
$857$	−8.75379	−0.299024	−0.149512	−	0.988760i	$-0.547770\pi$
$857$	−8.75379	−0.299024	−0.149512	+	0.988760i	$0.547770\pi$
$858$	34.8078	1.18832
$859$	−2.80776	−0.0957997	−0.0478998	−	0.998852i	$-0.515253\pi$
$859$	−2.80776	−0.0957997	−0.0478998	+	0.998852i	$0.515253\pi$
$860$	12.6847	0.432543
$861$	8.68466	0.295972
$862$	−29.0000	−0.987744
$863$	20.1922	0.687352	0.343676	−	0.939088i	$-0.388328\pi$
$863$	20.1922	0.687352	0.343676	+	0.939088i	$0.388328\pi$
$864$	−1.00000	−0.0340207
$865$	2.12311	0.0721878
$866$	−30.4233	−1.03383
$867$	−8.00000	−0.271694
$868$	3.56155	0.120887
$869$	−87.2311	−2.95911
$870$	−6.68466	−0.226631
$871$	74.6004	2.52774
$872$	−2.12311	−0.0718974
$873$	3.31534	0.112207
$874$	−3.68466	−0.124635
$875$	1.00000	0.0338062
$876$	6.56155	0.221694
$877$	46.9309	1.58474	0.792371	−	0.610039i	$-0.208846\pi$
$877$	46.9309	1.58474	0.792371	+	0.610039i	$0.208846\pi$
$878$	−14.0540	−0.474299
$879$	16.6155	0.560428
$880$	6.12311	0.206410
$881$	34.9309	1.17685	0.588425	−	0.808551i	$-0.299748\pi$
$881$	34.9309	1.17685	0.588425	+	0.808551i	$0.299748\pi$
$882$	−6.00000	−0.202031
$883$	−14.3693	−0.483566	−0.241783	−	0.970330i	$-0.577732\pi$
$883$	−14.3693	−0.483566	−0.241783	+	0.970330i	$0.577732\pi$
$884$	−28.4233	−0.955979
$885$	4.24621	0.142735
$886$	33.8617	1.13761
$887$	−17.8078	−0.597926	−0.298963	−	0.954265i	$-0.596641\pi$
$887$	−17.8078	−0.597926	−0.298963	+	0.954265i	$0.596641\pi$
$888$	−1.00000	−0.0335578
$889$	−14.5616	−0.488379
$890$	−5.68466	−0.190550
$891$	−6.12311	−0.205132
$892$	−19.8078	−0.663213
$893$	−4.49242	−0.150333
$894$	−7.75379	−0.259325
$895$	4.00000	0.133705
$896$	−1.00000	−0.0334077
$897$	37.3002	1.24542
$898$	−1.61553	−0.0539109
$899$	23.8078	0.794033
$900$	1.00000	0.0333333
$901$	−26.2311	−0.873883
$902$	−53.1771	−1.77060
$903$	−12.6847	−0.422119
$904$	−10.6847	−0.355366
$905$	−20.2462	−0.673007
$906$	15.4384	0.512908
$907$	45.4384	1.50876	0.754379	−	0.656439i	$-0.227938\pi$
$907$	45.4384	1.50876	0.754379	+	0.656439i	$0.227938\pi$
$908$	−10.9309	−0.362754
$909$	10.8769	0.360764
$910$	5.68466	0.188445
$911$	20.4924	0.678944	0.339472	−	0.940616i	$-0.389752\pi$
$911$	20.4924	0.678944	0.339472	+	0.940616i	$0.389752\pi$
$912$	−0.561553	−0.0185949
$913$	45.5464	1.50737
$914$	−30.9309	−1.02310
$915$	−13.8078	−0.456471
$916$	−5.12311	−0.169272
$917$	−7.12311	−0.235226
$918$	5.00000	0.165025
$919$	6.73863	0.222287	0.111144	−	0.993804i	$-0.464549\pi$
$919$	6.73863	0.222287	0.111144	+	0.993804i	$0.464549\pi$
$920$	6.56155	0.216328
$921$	0.246211	0.00811294
$922$	0.438447	0.0144395
$923$	6.38447	0.210147
$924$	−6.12311	−0.201435
$925$	1.00000	0.0328798
$926$	4.87689	0.160265
$927$	−16.4924	−0.541682
$928$	−6.68466	−0.219435
$929$	−5.80776	−0.190547	−0.0952733	−	0.995451i	$-0.530372\pi$
$929$	−5.80776	−0.190547	−0.0952733	+	0.995451i	$0.530372\pi$
$930$	−3.56155	−0.116788
$931$	−3.36932	−0.110425
$932$	11.3693	0.372414
$933$	21.1771	0.693307
$934$	2.43845	0.0797884
$935$	−30.6155	−1.00123
$936$	5.68466	0.185809
$937$	−0.630683	−0.0206035	−0.0103018	−	0.999947i	$-0.503279\pi$
$937$	−0.630683	−0.0206035	−0.0103018	+	0.999947i	$0.503279\pi$
$938$	−13.1231	−0.428485
$939$	24.8769	0.811827
$940$	8.00000	0.260931
$941$	−8.24621	−0.268819	−0.134409	−	0.990926i	$-0.542914\pi$
$941$	−8.24621	−0.268819	−0.134409	+	0.990926i	$0.542914\pi$
$942$	18.0540	0.588230
$943$	−56.9848	−1.85568
$944$	4.24621	0.138202
$945$	−1.00000	−0.0325300
$946$	77.6695	2.52525
$947$	16.3002	0.529685	0.264842	−	0.964292i	$-0.414680\pi$
$947$	16.3002	0.529685	0.264842	+	0.964292i	$0.414680\pi$
$948$	−14.2462	−0.462695
$949$	−37.3002	−1.21082
$950$	0.561553	0.0182192
$951$	−16.9309	−0.549021
$952$	5.00000	0.162051
$953$	−1.61553	−0.0523321	−0.0261660	−	0.999658i	$-0.508330\pi$
$953$	−1.61553	−0.0523321	−0.0261660	+	0.999658i	$0.508330\pi$
$954$	5.24621	0.169852
$955$	−14.3693	−0.464980
$956$	7.80776	0.252521
$957$	−40.9309	−1.32311
$958$	−6.56155	−0.211994
$959$	4.24621	0.137117
$960$	1.00000	0.0322749
$961$	−18.3153	−0.590817
$962$	5.68466	0.183281
$963$	−10.8078	−0.348275
$964$	−2.00000	−0.0644157
$965$	14.0000	0.450676
$966$	−6.56155	−0.211115
$967$	32.6307	1.04933	0.524666	−	0.851308i	$-0.324190\pi$
$967$	32.6307	1.04933	0.524666	+	0.851308i	$0.324190\pi$
$968$	26.4924	0.851499
$969$	2.80776	0.0901984
$970$	−3.31534	−0.106449
$971$	−15.8078	−0.507295	−0.253648	−	0.967297i	$-0.581630\pi$
$971$	−15.8078	−0.507295	−0.253648	+	0.967297i	$0.581630\pi$
$972$	−1.00000	−0.0320750
$973$	−13.8078	−0.442657
$974$	16.7386	0.536340
$975$	−5.68466	−0.182055
$976$	−13.8078	−0.441976
$977$	−12.5076	−0.400153	−0.200076	−	0.979780i	$-0.564119\pi$
$977$	−12.5076	−0.400153	−0.200076	+	0.979780i	$0.564119\pi$
$978$	−19.4924	−0.623299
$979$	−34.8078	−1.11246
$980$	6.00000	0.191663
$981$	−2.12311	−0.0677855
$982$	−12.1771	−0.388586
$983$	−24.9309	−0.795171	−0.397586	−	0.917565i	$-0.630152\pi$
$983$	−24.9309	−0.795171	−0.397586	+	0.917565i	$0.630152\pi$
$984$	−8.68466	−0.276857
$985$	2.31534	0.0737729
$986$	33.4233	1.06441
$987$	−8.00000	−0.254643
$988$	3.19224	0.101559
$989$	83.2311	2.64659
$990$	6.12311	0.194605
$991$	3.42329	0.108744	0.0543722	−	0.998521i	$-0.482684\pi$
$991$	3.42329	0.108744	0.0543722	+	0.998521i	$0.482684\pi$
$992$	−3.56155	−0.113079
$993$	12.0000	0.380808
$994$	−1.12311	−0.0356227
$995$	16.4924	0.522845
$996$	7.43845	0.235696
$997$	19.3002	0.611243	0.305622	−	0.952153i	$-0.401136\pi$
$997$	19.3002	0.611243	0.305622	+	0.952153i	$0.401136\pi$
$998$	32.8078	1.03851
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1110.2.a.r.1.1	✓	2
3.2	odd	2		3330.2.a.bc.1.2		2
4.3	odd	2		8880.2.a.bp.1.2		2
5.4	even	2		5550.2.a.bw.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.r.1.1	✓	2	1.1	even	1	trivial
3330.2.a.bc.1.2		2	3.2	odd	2
5550.2.a.bw.1.1		2	5.4	even	2
8880.2.a.bp.1.2		2	4.3	odd	2

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:	\( N \)	\(=\)	\( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1110.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -6.12311 q^{11} -1.00000 q^{12} +5.68466 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -5.00000 q^{17} +1.00000 q^{18} +0.561553 q^{19} -1.00000 q^{20} +1.00000 q^{21} -6.12311 q^{22} -6.56155 q^{23} -1.00000 q^{24} +1.00000 q^{25} +5.68466 q^{26} -1.00000 q^{27} -1.00000 q^{28} -6.68466 q^{29} +1.00000 q^{30} -3.56155 q^{31} +1.00000 q^{32} +6.12311 q^{33} -5.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} +0.561553 q^{38} -5.68466 q^{39} -1.00000 q^{40} +8.68466 q^{41} +1.00000 q^{42} -12.6847 q^{43} -6.12311 q^{44} -1.00000 q^{45} -6.56155 q^{46} -8.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} +1.00000 q^{50} +5.00000 q^{51} +5.68466 q^{52} +5.24621 q^{53} -1.00000 q^{54} +6.12311 q^{55} -1.00000 q^{56} -0.561553 q^{57} -6.68466 q^{58} +4.24621 q^{59} +1.00000 q^{60} -13.8078 q^{61} -3.56155 q^{62} -1.00000 q^{63} +1.00000 q^{64} -5.68466 q^{65} +6.12311 q^{66} +13.1231 q^{67} -5.00000 q^{68} +6.56155 q^{69} +1.00000 q^{70} +1.12311 q^{71} +1.00000 q^{72} -6.56155 q^{73} +1.00000 q^{74} -1.00000 q^{75} +0.561553 q^{76} +6.12311 q^{77} -5.68466 q^{78} +14.2462 q^{79} -1.00000 q^{80} +1.00000 q^{81} +8.68466 q^{82} -7.43845 q^{83} +1.00000 q^{84} +5.00000 q^{85} -12.6847 q^{86} +6.68466 q^{87} -6.12311 q^{88} +5.68466 q^{89} -1.00000 q^{90} -5.68466 q^{91} -6.56155 q^{92} +3.56155 q^{93} -8.00000 q^{94} -0.561553 q^{95} -1.00000 q^{96} +3.31534 q^{97} -6.00000 q^{98} -6.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} - 2 q^{12} - q^{13} - 2 q^{14} + 2 q^{15} + 2 q^{16} - 10 q^{17} + 2 q^{18} - 3 q^{19} - 2 q^{20} + 2 q^{21} - 4 q^{22} - 9 q^{23} - 2 q^{24} + 2 q^{25} - q^{26} - 2 q^{27} - 2 q^{28} - q^{29} + 2 q^{30} - 3 q^{31} + 2 q^{32} + 4 q^{33} - 10 q^{34} + 2 q^{35} + 2 q^{36} + 2 q^{37} - 3 q^{38} + q^{39} - 2 q^{40} + 5 q^{41} + 2 q^{42} - 13 q^{43} - 4 q^{44} - 2 q^{45} - 9 q^{46} - 16 q^{47} - 2 q^{48} - 12 q^{49} + 2 q^{50} + 10 q^{51} - q^{52} - 6 q^{53} - 2 q^{54} + 4 q^{55} - 2 q^{56} + 3 q^{57} - q^{58} - 8 q^{59} + 2 q^{60} - 7 q^{61} - 3 q^{62} - 2 q^{63} + 2 q^{64} + q^{65} + 4 q^{66} + 18 q^{67} - 10 q^{68} + 9 q^{69} + 2 q^{70} - 6 q^{71} + 2 q^{72} - 9 q^{73} + 2 q^{74} - 2 q^{75} - 3 q^{76} + 4 q^{77} + q^{78} + 12 q^{79} - 2 q^{80} + 2 q^{81} + 5 q^{82} - 19 q^{83} + 2 q^{84} + 10 q^{85} - 13 q^{86} + q^{87} - 4 q^{88} - q^{89} - 2 q^{90} + q^{91} - 9 q^{92} + 3 q^{93} - 16 q^{94} + 3 q^{95} - 2 q^{96} + 19 q^{97} - 12 q^{98} - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 - 2 * q^10 - 4 * q^11 - 2 * q^12 - q^13 - 2 * q^14 + 2 * q^15 + 2 * q^16 - 10 * q^17 + 2 * q^18 - 3 * q^19 - 2 * q^20 + 2 * q^21 - 4 * q^22 - 9 * q^23 - 2 * q^24 + 2 * q^25 - q^26 - 2 * q^27 - 2 * q^28 - q^29 + 2 * q^30 - 3 * q^31 + 2 * q^32 + 4 * q^33 - 10 * q^34 + 2 * q^35 + 2 * q^36 + 2 * q^37 - 3 * q^38 + q^39 - 2 * q^40 + 5 * q^41 + 2 * q^42 - 13 * q^43 - 4 * q^44 - 2 * q^45 - 9 * q^46 - 16 * q^47 - 2 * q^48 - 12 * q^49 + 2 * q^50 + 10 * q^51 - q^52 - 6 * q^53 - 2 * q^54 + 4 * q^55 - 2 * q^56 + 3 * q^57 - q^58 - 8 * q^59 + 2 * q^60 - 7 * q^61 - 3 * q^62 - 2 * q^63 + 2 * q^64 + q^65 + 4 * q^66 + 18 * q^67 - 10 * q^68 + 9 * q^69 + 2 * q^70 - 6 * q^71 + 2 * q^72 - 9 * q^73 + 2 * q^74 - 2 * q^75 - 3 * q^76 + 4 * q^77 + q^78 + 12 * q^79 - 2 * q^80 + 2 * q^81 + 5 * q^82 - 19 * q^83 + 2 * q^84 + 10 * q^85 - 13 * q^86 + q^87 - 4 * q^88 - q^89 - 2 * q^90 + q^91 - 9 * q^92 + 3 * q^93 - 16 * q^94 + 3 * q^95 - 2 * q^96 + 19 * q^97 - 12 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more