LMFDB - Embedded newform 4830.2.a.bq.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4830,2,Mod(1,4830)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4830, 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("4830.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4830.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:	$38.5677441763$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	4830.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.23607 q^{11} +1.00000 q^{12} -1.23607 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +6.47214 q^{17} -1.00000 q^{18} +4.47214 q^{19} +1.00000 q^{20} -1.00000 q^{21} -1.23607 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.23607 q^{26} +1.00000 q^{27} -1.00000 q^{28} -5.70820 q^{29} -1.00000 q^{30} +4.47214 q^{31} -1.00000 q^{32} +1.23607 q^{33} -6.47214 q^{34} -1.00000 q^{35} +1.00000 q^{36} +8.47214 q^{37} -4.47214 q^{38} -1.23607 q^{39} -1.00000 q^{40} -0.472136 q^{41} +1.00000 q^{42} -5.23607 q^{43} +1.23607 q^{44} +1.00000 q^{45} -1.00000 q^{46} -2.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +6.47214 q^{51} -1.23607 q^{52} +10.0000 q^{53} -1.00000 q^{54} +1.23607 q^{55} +1.00000 q^{56} +4.47214 q^{57} +5.70820 q^{58} -8.94427 q^{59} +1.00000 q^{60} -2.00000 q^{61} -4.47214 q^{62} -1.00000 q^{63} +1.00000 q^{64} -1.23607 q^{65} -1.23607 q^{66} -13.2361 q^{67} +6.47214 q^{68} +1.00000 q^{69} +1.00000 q^{70} -3.70820 q^{71} -1.00000 q^{72} -2.94427 q^{73} -8.47214 q^{74} +1.00000 q^{75} +4.47214 q^{76} -1.23607 q^{77} +1.23607 q^{78} -4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +0.472136 q^{82} +8.47214 q^{83} -1.00000 q^{84} +6.47214 q^{85} +5.23607 q^{86} -5.70820 q^{87} -1.23607 q^{88} +2.76393 q^{89} -1.00000 q^{90} +1.23607 q^{91} +1.00000 q^{92} +4.47214 q^{93} +2.00000 q^{94} +4.47214 q^{95} -1.00000 q^{96} -7.70820 q^{97} -1.00000 q^{98} +1.23607 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 2 q^{11} + 2 q^{12} + 2 q^{13} + 2 q^{14} + 2 q^{15} + 2 q^{16} + 4 q^{17} - 2 q^{18} + 2 q^{20} - 2 q^{21} + 2 q^{22} + 2 q^{23} - 2 q^{24} + 2 q^{25} - 2 q^{26} + 2 q^{27} - 2 q^{28} + 2 q^{29} - 2 q^{30} - 2 q^{32} - 2 q^{33} - 4 q^{34} - 2 q^{35} + 2 q^{36} + 8 q^{37} + 2 q^{39} - 2 q^{40} + 8 q^{41} + 2 q^{42} - 6 q^{43} - 2 q^{44} + 2 q^{45} - 2 q^{46} - 4 q^{47} + 2 q^{48} + 2 q^{49} - 2 q^{50} + 4 q^{51} + 2 q^{52} + 20 q^{53} - 2 q^{54} - 2 q^{55} + 2 q^{56} - 2 q^{58} + 2 q^{60} - 4 q^{61} - 2 q^{63} + 2 q^{64} + 2 q^{65} + 2 q^{66} - 22 q^{67} + 4 q^{68} + 2 q^{69} + 2 q^{70} + 6 q^{71} - 2 q^{72} + 12 q^{73} - 8 q^{74} + 2 q^{75} + 2 q^{77} - 2 q^{78} - 8 q^{79} + 2 q^{80} + 2 q^{81} - 8 q^{82} + 8 q^{83} - 2 q^{84} + 4 q^{85} + 6 q^{86} + 2 q^{87} + 2 q^{88} + 10 q^{89} - 2 q^{90} - 2 q^{91} + 2 q^{92} + 4 q^{94} - 2 q^{96} - 2 q^{97} - 2 q^{98} - 2 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 - 2 * q^11 + 2 * q^12 + 2 * q^13 + 2 * q^14 + 2 * q^15 + 2 * q^16 + 4 * q^17 - 2 * q^18 + 2 * q^20 - 2 * q^21 + 2 * q^22 + 2 * q^23 - 2 * q^24 + 2 * q^25 - 2 * q^26 + 2 * q^27 - 2 * q^28 + 2 * q^29 - 2 * q^30 - 2 * q^32 - 2 * q^33 - 4 * q^34 - 2 * q^35 + 2 * q^36 + 8 * q^37 + 2 * q^39 - 2 * q^40 + 8 * q^41 + 2 * q^42 - 6 * q^43 - 2 * q^44 + 2 * q^45 - 2 * q^46 - 4 * q^47 + 2 * q^48 + 2 * q^49 - 2 * q^50 + 4 * q^51 + 2 * q^52 + 20 * q^53 - 2 * q^54 - 2 * q^55 + 2 * q^56 - 2 * q^58 + 2 * q^60 - 4 * q^61 - 2 * q^63 + 2 * q^64 + 2 * q^65 + 2 * q^66 - 22 * q^67 + 4 * q^68 + 2 * q^69 + 2 * q^70 + 6 * q^71 - 2 * q^72 + 12 * q^73 - 8 * q^74 + 2 * q^75 + 2 * q^77 - 2 * q^78 - 8 * q^79 + 2 * q^80 + 2 * q^81 - 8 * q^82 + 8 * q^83 - 2 * q^84 + 4 * q^85 + 6 * q^86 + 2 * q^87 + 2 * q^88 + 10 * q^89 - 2 * q^90 - 2 * q^91 + 2 * q^92 + 4 * q^94 - 2 * q^96 - 2 * q^97 - 2 * q^98 - 2 * 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
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	1.23607	0.372689	0.186344	−	0.982485i	$-0.440336\pi$
$11$	1.23607	0.372689	0.186344	+	0.982485i	$0.440336\pi$
$12$	1.00000	0.288675
$13$	−1.23607	−0.342824	−0.171412	−	0.985199i	$-0.554833\pi$
$13$	−1.23607	−0.342824	−0.171412	+	0.985199i	$0.554833\pi$
$14$	1.00000	0.267261
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	6.47214	1.56972	0.784862	−	0.619671i	$-0.212734\pi$
$17$	6.47214	1.56972	0.784862	+	0.619671i	$0.212734\pi$
$18$	−1.00000	−0.235702
$19$	4.47214	1.02598	0.512989	−	0.858395i	$-0.328538\pi$
$19$	4.47214	1.02598	0.512989	+	0.858395i	$0.328538\pi$
$20$	1.00000	0.223607
$21$	−1.00000	−0.218218
$22$	−1.23607	−0.263531
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	1.23607	0.242413
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	−5.70820	−1.05999	−0.529993	−	0.848002i	$-0.677806\pi$
$29$	−5.70820	−1.05999	−0.529993	+	0.848002i	$0.677806\pi$
$30$	−1.00000	−0.182574
$31$	4.47214	0.803219	0.401610	−	0.915811i	$-0.368451\pi$
$31$	4.47214	0.803219	0.401610	+	0.915811i	$0.368451\pi$
$32$	−1.00000	−0.176777
$33$	1.23607	0.215172
$34$	−6.47214	−1.10996
$35$	−1.00000	−0.169031
$36$	1.00000	0.166667
$37$	8.47214	1.39281	0.696405	−	0.717649i	$-0.254782\pi$
$37$	8.47214	1.39281	0.696405	+	0.717649i	$0.254782\pi$
$38$	−4.47214	−0.725476
$39$	−1.23607	−0.197929
$40$	−1.00000	−0.158114
$41$	−0.472136	−0.0737352	−0.0368676	−	0.999320i	$-0.511738\pi$
$41$	−0.472136	−0.0737352	−0.0368676	+	0.999320i	$0.511738\pi$
$42$	1.00000	0.154303
$43$	−5.23607	−0.798493	−0.399246	−	0.916844i	$-0.630728\pi$
$43$	−5.23607	−0.798493	−0.399246	+	0.916844i	$0.630728\pi$
$44$	1.23607	0.186344
$45$	1.00000	0.149071
$46$	−1.00000	−0.147442
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	6.47214	0.906280
$52$	−1.23607	−0.171412
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	−1.00000	−0.136083
$55$	1.23607	0.166671
$56$	1.00000	0.133631
$57$	4.47214	0.592349
$58$	5.70820	0.749524
$59$	−8.94427	−1.16445	−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427	−1.16445	−0.582223	+	0.813029i	$0.697817\pi$
$60$	1.00000	0.129099
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−4.47214	−0.567962
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−1.23607	−0.153315
$66$	−1.23607	−0.152149
$67$	−13.2361	−1.61704	−0.808522	−	0.588467i	$-0.799732\pi$
$67$	−13.2361	−1.61704	−0.808522	+	0.588467i	$0.799732\pi$
$68$	6.47214	0.784862
$69$	1.00000	0.120386
$70$	1.00000	0.119523
$71$	−3.70820	−0.440083	−0.220041	−	0.975491i	$-0.570619\pi$
$71$	−3.70820	−0.440083	−0.220041	+	0.975491i	$0.570619\pi$
$72$	−1.00000	−0.117851
$73$	−2.94427	−0.344601	−0.172300	−	0.985044i	$-0.555120\pi$
$73$	−2.94427	−0.344601	−0.172300	+	0.985044i	$0.555120\pi$
$74$	−8.47214	−0.984866
$75$	1.00000	0.115470
$76$	4.47214	0.512989
$77$	−1.23607	−0.140863
$78$	1.23607	0.139957
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	0.472136	0.0521387
$83$	8.47214	0.929938	0.464969	−	0.885327i	$-0.346066\pi$
$83$	8.47214	0.929938	0.464969	+	0.885327i	$0.346066\pi$
$84$	−1.00000	−0.109109
$85$	6.47214	0.702002
$86$	5.23607	0.564620
$87$	−5.70820	−0.611984
$88$	−1.23607	−0.131765
$89$	2.76393	0.292976	0.146488	−	0.989212i	$-0.453203\pi$
$89$	2.76393	0.292976	0.146488	+	0.989212i	$0.453203\pi$
$90$	−1.00000	−0.105409
$91$	1.23607	0.129575
$92$	1.00000	0.104257
$93$	4.47214	0.463739
$94$	2.00000	0.206284
$95$	4.47214	0.458831
$96$	−1.00000	−0.102062
$97$	−7.70820	−0.782650	−0.391325	−	0.920253i	$-0.627983\pi$
$97$	−7.70820	−0.782650	−0.391325	+	0.920253i	$0.627983\pi$
$98$	−1.00000	−0.101015
$99$	1.23607	0.124230
$100$	1.00000	0.100000
$101$	−18.1803	−1.80901	−0.904506	−	0.426461i	$-0.859760\pi$
$101$	−18.1803	−1.80901	−0.904506	+	0.426461i	$0.859760\pi$
$102$	−6.47214	−0.640837
$103$	9.52786	0.938808	0.469404	−	0.882983i	$-0.344469\pi$
$103$	9.52786	0.938808	0.469404	+	0.882983i	$0.344469\pi$
$104$	1.23607	0.121206
$105$	−1.00000	−0.0975900
$106$	−10.0000	−0.971286
$107$	19.4164	1.87705	0.938527	−	0.345204i	$-0.112190\pi$
$107$	19.4164	1.87705	0.938527	+	0.345204i	$0.112190\pi$
$108$	1.00000	0.0962250
$109$	16.4721	1.57774	0.788872	−	0.614557i	$-0.210665\pi$
$109$	16.4721	1.57774	0.788872	+	0.614557i	$0.210665\pi$
$110$	−1.23607	−0.117854
$111$	8.47214	0.804140
$112$	−1.00000	−0.0944911
$113$	6.94427	0.653262	0.326631	−	0.945152i	$-0.394087\pi$
$113$	6.94427	0.653262	0.326631	+	0.945152i	$0.394087\pi$
$114$	−4.47214	−0.418854
$115$	1.00000	0.0932505
$116$	−5.70820	−0.529993
$117$	−1.23607	−0.114275
$118$	8.94427	0.823387
$119$	−6.47214	−0.593300
$120$	−1.00000	−0.0912871
$121$	−9.47214	−0.861103
$122$	2.00000	0.181071
$123$	−0.472136	−0.0425711
$124$	4.47214	0.401610
$125$	1.00000	0.0894427
$126$	1.00000	0.0890871
$127$	9.23607	0.819569	0.409784	−	0.912182i	$-0.365604\pi$
$127$	9.23607	0.819569	0.409784	+	0.912182i	$0.365604\pi$
$128$	−1.00000	−0.0883883
$129$	−5.23607	−0.461010
$130$	1.23607	0.108410
$131$	8.94427	0.781465	0.390732	−	0.920504i	$-0.372222\pi$
$131$	8.94427	0.781465	0.390732	+	0.920504i	$0.372222\pi$
$132$	1.23607	0.107586
$133$	−4.47214	−0.387783
$134$	13.2361	1.14342
$135$	1.00000	0.0860663
$136$	−6.47214	−0.554981
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	−1.00000	−0.0851257
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	−1.00000	−0.0845154
$141$	−2.00000	−0.168430
$142$	3.70820	0.311186
$143$	−1.52786	−0.127766
$144$	1.00000	0.0833333
$145$	−5.70820	−0.474041
$146$	2.94427	0.243670
$147$	1.00000	0.0824786
$148$	8.47214	0.696405
$149$	16.4721	1.34945	0.674725	−	0.738069i	$-0.264262\pi$
$149$	16.4721	1.34945	0.674725	+	0.738069i	$0.264262\pi$
$150$	−1.00000	−0.0816497
$151$	14.4721	1.17773	0.588863	−	0.808233i	$-0.299576\pi$
$151$	14.4721	1.17773	0.588863	+	0.808233i	$0.299576\pi$
$152$	−4.47214	−0.362738
$153$	6.47214	0.523241
$154$	1.23607	0.0996052
$155$	4.47214	0.359211
$156$	−1.23607	−0.0989646
$157$	13.4164	1.07075	0.535373	−	0.844616i	$-0.320171\pi$
$157$	13.4164	1.07075	0.535373	+	0.844616i	$0.320171\pi$
$158$	4.00000	0.318223
$159$	10.0000	0.793052
$160$	−1.00000	−0.0790569
$161$	−1.00000	−0.0788110
$162$	−1.00000	−0.0785674
$163$	−19.4164	−1.52081	−0.760405	−	0.649449i	$-0.775000\pi$
$163$	−19.4164	−1.52081	−0.760405	+	0.649449i	$0.775000\pi$
$164$	−0.472136	−0.0368676
$165$	1.23607	0.0962278
$166$	−8.47214	−0.657565
$167$	7.52786	0.582524	0.291262	−	0.956643i	$-0.405925\pi$
$167$	7.52786	0.582524	0.291262	+	0.956643i	$0.405925\pi$
$168$	1.00000	0.0771517
$169$	−11.4721	−0.882472
$170$	−6.47214	−0.496390
$171$	4.47214	0.341993
$172$	−5.23607	−0.399246
$173$	8.00000	0.608229	0.304114	−	0.952636i	$-0.401639\pi$
$173$	8.00000	0.608229	0.304114	+	0.952636i	$0.401639\pi$
$174$	5.70820	0.432738
$175$	−1.00000	−0.0755929
$176$	1.23607	0.0931721
$177$	−8.94427	−0.672293
$178$	−2.76393	−0.207165
$179$	19.4164	1.45125	0.725625	−	0.688090i	$-0.241551\pi$
$179$	19.4164	1.45125	0.725625	+	0.688090i	$0.241551\pi$
$180$	1.00000	0.0745356
$181$	−19.8885	−1.47830	−0.739152	−	0.673539i	$-0.764773\pi$
$181$	−19.8885	−1.47830	−0.739152	+	0.673539i	$0.764773\pi$
$182$	−1.23607	−0.0916235
$183$	−2.00000	−0.147844
$184$	−1.00000	−0.0737210
$185$	8.47214	0.622884
$186$	−4.47214	−0.327913
$187$	8.00000	0.585018
$188$	−2.00000	−0.145865
$189$	−1.00000	−0.0727393
$190$	−4.47214	−0.324443
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000	0.0721688
$193$	13.4164	0.965734	0.482867	−	0.875694i	$-0.339595\pi$
$193$	13.4164	0.965734	0.482867	+	0.875694i	$0.339595\pi$
$194$	7.70820	0.553417
$195$	−1.23607	−0.0885167
$196$	1.00000	0.0714286
$197$	1.05573	0.0752175	0.0376088	−	0.999293i	$-0.488026\pi$
$197$	1.05573	0.0752175	0.0376088	+	0.999293i	$0.488026\pi$
$198$	−1.23607	−0.0878435
$199$	−5.52786	−0.391860	−0.195930	−	0.980618i	$-0.562773\pi$
$199$	−5.52786	−0.391860	−0.195930	+	0.980618i	$0.562773\pi$
$200$	−1.00000	−0.0707107
$201$	−13.2361	−0.933600
$202$	18.1803	1.27916
$203$	5.70820	0.400637
$204$	6.47214	0.453140
$205$	−0.472136	−0.0329754
$206$	−9.52786	−0.663838
$207$	1.00000	0.0695048
$208$	−1.23607	−0.0857059
$209$	5.52786	0.382370
$210$	1.00000	0.0690066
$211$	−20.9443	−1.44186	−0.720932	−	0.693006i	$-0.756286\pi$
$211$	−20.9443	−1.44186	−0.720932	+	0.693006i	$0.756286\pi$
$212$	10.0000	0.686803
$213$	−3.70820	−0.254082
$214$	−19.4164	−1.32728
$215$	−5.23607	−0.357097
$216$	−1.00000	−0.0680414
$217$	−4.47214	−0.303588
$218$	−16.4721	−1.11563
$219$	−2.94427	−0.198955
$220$	1.23607	0.0833357
$221$	−8.00000	−0.538138
$222$	−8.47214	−0.568613
$223$	−20.1803	−1.35138	−0.675688	−	0.737188i	$-0.736153\pi$
$223$	−20.1803	−1.35138	−0.675688	+	0.737188i	$0.736153\pi$
$224$	1.00000	0.0668153
$225$	1.00000	0.0666667
$226$	−6.94427	−0.461926
$227$	−2.94427	−0.195418	−0.0977091	−	0.995215i	$-0.531151\pi$
$227$	−2.94427	−0.195418	−0.0977091	+	0.995215i	$0.531151\pi$
$228$	4.47214	0.296174
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	−1.00000	−0.0659380
$231$	−1.23607	−0.0813273
$232$	5.70820	0.374762
$233$	15.5279	1.01726	0.508632	−	0.860984i	$-0.330151\pi$
$233$	15.5279	1.01726	0.508632	+	0.860984i	$0.330151\pi$
$234$	1.23607	0.0808043
$235$	−2.00000	−0.130466
$236$	−8.94427	−0.582223
$237$	−4.00000	−0.259828
$238$	6.47214	0.419526
$239$	9.23607	0.597432	0.298716	−	0.954342i	$-0.403442\pi$
$239$	9.23607	0.597432	0.298716	+	0.954342i	$0.403442\pi$
$240$	1.00000	0.0645497
$241$	13.8885	0.894640	0.447320	−	0.894374i	$-0.352379\pi$
$241$	13.8885	0.894640	0.447320	+	0.894374i	$0.352379\pi$
$242$	9.47214	0.608892
$243$	1.00000	0.0641500
$244$	−2.00000	−0.128037
$245$	1.00000	0.0638877
$246$	0.472136	0.0301023
$247$	−5.52786	−0.351730
$248$	−4.47214	−0.283981
$249$	8.47214	0.536900
$250$	−1.00000	−0.0632456
$251$	10.6525	0.672378	0.336189	−	0.941794i	$-0.390862\pi$
$251$	10.6525	0.672378	0.336189	+	0.941794i	$0.390862\pi$
$252$	−1.00000	−0.0629941
$253$	1.23607	0.0777109
$254$	−9.23607	−0.579522
$255$	6.47214	0.405301
$256$	1.00000	0.0625000
$257$	28.8328	1.79854	0.899271	−	0.437392i	$-0.144098\pi$
$257$	28.8328	1.79854	0.899271	+	0.437392i	$0.144098\pi$
$258$	5.23607	0.325983
$259$	−8.47214	−0.526433
$260$	−1.23607	−0.0766577
$261$	−5.70820	−0.353329
$262$	−8.94427	−0.552579
$263$	7.05573	0.435075	0.217537	−	0.976052i	$-0.430198\pi$
$263$	7.05573	0.435075	0.217537	+	0.976052i	$0.430198\pi$
$264$	−1.23607	−0.0760747
$265$	10.0000	0.614295
$266$	4.47214	0.274204
$267$	2.76393	0.169150
$268$	−13.2361	−0.808522
$269$	−15.7082	−0.957746	−0.478873	−	0.877884i	$-0.658955\pi$
$269$	−15.7082	−0.957746	−0.478873	+	0.877884i	$0.658955\pi$
$270$	−1.00000	−0.0608581
$271$	1.41641	0.0860407	0.0430203	−	0.999074i	$-0.486302\pi$
$271$	1.41641	0.0860407	0.0430203	+	0.999074i	$0.486302\pi$
$272$	6.47214	0.392431
$273$	1.23607	0.0748102
$274$	−14.0000	−0.845771
$275$	1.23607	0.0745377
$276$	1.00000	0.0601929
$277$	−6.65248	−0.399709	−0.199854	−	0.979826i	$-0.564047\pi$
$277$	−6.65248	−0.399709	−0.199854	+	0.979826i	$0.564047\pi$
$278$	−8.00000	−0.479808
$279$	4.47214	0.267740
$280$	1.00000	0.0597614
$281$	24.1803	1.44248	0.721239	−	0.692686i	$-0.243573\pi$
$281$	24.1803	1.44248	0.721239	+	0.692686i	$0.243573\pi$
$282$	2.00000	0.119098
$283$	−13.7082	−0.814868	−0.407434	−	0.913235i	$-0.633576\pi$
$283$	−13.7082	−0.814868	−0.407434	+	0.913235i	$0.633576\pi$
$284$	−3.70820	−0.220041
$285$	4.47214	0.264906
$286$	1.52786	0.0903445
$287$	0.472136	0.0278693
$288$	−1.00000	−0.0589256
$289$	24.8885	1.46403
$290$	5.70820	0.335197
$291$	−7.70820	−0.451863
$292$	−2.94427	−0.172300
$293$	−1.05573	−0.0616763	−0.0308381	−	0.999524i	$-0.509818\pi$
$293$	−1.05573	−0.0616763	−0.0308381	+	0.999524i	$0.509818\pi$
$294$	−1.00000	−0.0583212
$295$	−8.94427	−0.520756
$296$	−8.47214	−0.492433
$297$	1.23607	0.0717239
$298$	−16.4721	−0.954205
$299$	−1.23607	−0.0714837
$300$	1.00000	0.0577350
$301$	5.23607	0.301802
$302$	−14.4721	−0.832778
$303$	−18.1803	−1.04443
$304$	4.47214	0.256495
$305$	−2.00000	−0.114520
$306$	−6.47214	−0.369987
$307$	26.4721	1.51084	0.755422	−	0.655238i	$-0.227432\pi$
$307$	26.4721	1.51084	0.755422	+	0.655238i	$0.227432\pi$
$308$	−1.23607	−0.0704315
$309$	9.52786	0.542021
$310$	−4.47214	−0.254000
$311$	−20.7639	−1.17741	−0.588707	−	0.808346i	$-0.700363\pi$
$311$	−20.7639	−1.17741	−0.588707	+	0.808346i	$0.700363\pi$
$312$	1.23607	0.0699786
$313$	1.81966	0.102853	0.0514266	−	0.998677i	$-0.483623\pi$
$313$	1.81966	0.102853	0.0514266	+	0.998677i	$0.483623\pi$
$314$	−13.4164	−0.757132
$315$	−1.00000	−0.0563436
$316$	−4.00000	−0.225018
$317$	−17.4164	−0.978203	−0.489101	−	0.872227i	$-0.662675\pi$
$317$	−17.4164	−0.978203	−0.489101	+	0.872227i	$0.662675\pi$
$318$	−10.0000	−0.560772
$319$	−7.05573	−0.395045
$320$	1.00000	0.0559017
$321$	19.4164	1.08372
$322$	1.00000	0.0557278
$323$	28.9443	1.61050
$324$	1.00000	0.0555556
$325$	−1.23607	−0.0685647
$326$	19.4164	1.07538
$327$	16.4721	0.910911
$328$	0.472136	0.0260693
$329$	2.00000	0.110264
$330$	−1.23607	−0.0680433
$331$	−28.9443	−1.59092	−0.795461	−	0.606005i	$-0.792771\pi$
$331$	−28.9443	−1.59092	−0.795461	+	0.606005i	$0.792771\pi$
$332$	8.47214	0.464969
$333$	8.47214	0.464270
$334$	−7.52786	−0.411906
$335$	−13.2361	−0.723164
$336$	−1.00000	−0.0545545
$337$	−20.7639	−1.13108	−0.565542	−	0.824720i	$-0.691333\pi$
$337$	−20.7639	−1.13108	−0.565542	+	0.824720i	$0.691333\pi$
$338$	11.4721	0.624002
$339$	6.94427	0.377161
$340$	6.47214	0.351001
$341$	5.52786	0.299351
$342$	−4.47214	−0.241825
$343$	−1.00000	−0.0539949
$344$	5.23607	0.282310
$345$	1.00000	0.0538382
$346$	−8.00000	−0.430083
$347$	−21.8885	−1.17504	−0.587519	−	0.809210i	$-0.699895\pi$
$347$	−21.8885	−1.17504	−0.587519	+	0.809210i	$0.699895\pi$
$348$	−5.70820	−0.305992
$349$	13.5279	0.724130	0.362065	−	0.932153i	$-0.382072\pi$
$349$	13.5279	0.724130	0.362065	+	0.932153i	$0.382072\pi$
$350$	1.00000	0.0534522
$351$	−1.23607	−0.0659764
$352$	−1.23607	−0.0658826
$353$	31.8885	1.69726	0.848628	−	0.528990i	$-0.177429\pi$
$353$	31.8885	1.69726	0.848628	+	0.528990i	$0.177429\pi$
$354$	8.94427	0.475383
$355$	−3.70820	−0.196811
$356$	2.76393	0.146488
$357$	−6.47214	−0.342542
$358$	−19.4164	−1.02619
$359$	20.9443	1.10540	0.552698	−	0.833381i	$-0.313598\pi$
$359$	20.9443	1.10540	0.552698	+	0.833381i	$0.313598\pi$
$360$	−1.00000	−0.0527046
$361$	1.00000	0.0526316
$362$	19.8885	1.04532
$363$	−9.47214	−0.497158
$364$	1.23607	0.0647876
$365$	−2.94427	−0.154110
$366$	2.00000	0.104542
$367$	−4.58359	−0.239262	−0.119631	−	0.992818i	$-0.538171\pi$
$367$	−4.58359	−0.239262	−0.119631	+	0.992818i	$0.538171\pi$
$368$	1.00000	0.0521286
$369$	−0.472136	−0.0245784
$370$	−8.47214	−0.440445
$371$	−10.0000	−0.519174
$372$	4.47214	0.231869
$373$	−28.4721	−1.47423	−0.737116	−	0.675767i	$-0.763813\pi$
$373$	−28.4721	−1.47423	−0.737116	+	0.675767i	$0.763813\pi$
$374$	−8.00000	−0.413670
$375$	1.00000	0.0516398
$376$	2.00000	0.103142
$377$	7.05573	0.363388
$378$	1.00000	0.0514344
$379$	9.52786	0.489414	0.244707	−	0.969597i	$-0.421308\pi$
$379$	9.52786	0.489414	0.244707	+	0.969597i	$0.421308\pi$
$380$	4.47214	0.229416
$381$	9.23607	0.473178
$382$	0	0
$383$	−17.8885	−0.914062	−0.457031	−	0.889451i	$-0.651087\pi$
$383$	−17.8885	−0.914062	−0.457031	+	0.889451i	$0.651087\pi$
$384$	−1.00000	−0.0510310
$385$	−1.23607	−0.0629959
$386$	−13.4164	−0.682877
$387$	−5.23607	−0.266164
$388$	−7.70820	−0.391325
$389$	−8.47214	−0.429554	−0.214777	−	0.976663i	$-0.568903\pi$
$389$	−8.47214	−0.429554	−0.214777	+	0.976663i	$0.568903\pi$
$390$	1.23607	0.0625907
$391$	6.47214	0.327310
$392$	−1.00000	−0.0505076
$393$	8.94427	0.451179
$394$	−1.05573	−0.0531868
$395$	−4.00000	−0.201262
$396$	1.23607	0.0621148
$397$	19.7082	0.989126	0.494563	−	0.869142i	$-0.335328\pi$
$397$	19.7082	0.989126	0.494563	+	0.869142i	$0.335328\pi$
$398$	5.52786	0.277087
$399$	−4.47214	−0.223887
$400$	1.00000	0.0500000
$401$	−33.1246	−1.65416	−0.827082	−	0.562081i	$-0.810001\pi$
$401$	−33.1246	−1.65416	−0.827082	+	0.562081i	$0.810001\pi$
$402$	13.2361	0.660155
$403$	−5.52786	−0.275363
$404$	−18.1803	−0.904506
$405$	1.00000	0.0496904
$406$	−5.70820	−0.283293
$407$	10.4721	0.519085
$408$	−6.47214	−0.320418
$409$	7.88854	0.390063	0.195032	−	0.980797i	$-0.437519\pi$
$409$	7.88854	0.390063	0.195032	+	0.980797i	$0.437519\pi$
$410$	0.472136	0.0233171
$411$	14.0000	0.690569
$412$	9.52786	0.469404
$413$	8.94427	0.440119
$414$	−1.00000	−0.0491473
$415$	8.47214	0.415881
$416$	1.23607	0.0606032
$417$	8.00000	0.391762
$418$	−5.52786	−0.270377
$419$	16.1803	0.790461	0.395231	−	0.918582i	$-0.370665\pi$
$419$	16.1803	0.790461	0.395231	+	0.918582i	$0.370665\pi$
$420$	−1.00000	−0.0487950
$421$	19.8885	0.969308	0.484654	−	0.874706i	$-0.338945\pi$
$421$	19.8885	0.969308	0.484654	+	0.874706i	$0.338945\pi$
$422$	20.9443	1.01955
$423$	−2.00000	−0.0972433
$424$	−10.0000	−0.485643
$425$	6.47214	0.313945
$426$	3.70820	0.179663
$427$	2.00000	0.0967868
$428$	19.4164	0.938527
$429$	−1.52786	−0.0737660
$430$	5.23607	0.252506
$431$	−12.3607	−0.595393	−0.297696	−	0.954661i	$-0.596218\pi$
$431$	−12.3607	−0.595393	−0.297696	+	0.954661i	$0.596218\pi$
$432$	1.00000	0.0481125
$433$	39.1246	1.88021	0.940104	−	0.340887i	$-0.110727\pi$
$433$	39.1246	1.88021	0.940104	+	0.340887i	$0.110727\pi$
$434$	4.47214	0.214669
$435$	−5.70820	−0.273687
$436$	16.4721	0.788872
$437$	4.47214	0.213931
$438$	2.94427	0.140683
$439$	−14.9443	−0.713251	−0.356626	−	0.934247i	$-0.616073\pi$
$439$	−14.9443	−0.713251	−0.356626	+	0.934247i	$0.616073\pi$
$440$	−1.23607	−0.0589272
$441$	1.00000	0.0476190
$442$	8.00000	0.380521
$443$	−24.9443	−1.18514	−0.592569	−	0.805520i	$-0.701886\pi$
$443$	−24.9443	−1.18514	−0.592569	+	0.805520i	$0.701886\pi$
$444$	8.47214	0.402070
$445$	2.76393	0.131023
$446$	20.1803	0.955567
$447$	16.4721	0.779105
$448$	−1.00000	−0.0472456
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	−1.00000	−0.0471405
$451$	−0.583592	−0.0274803
$452$	6.94427	0.326631
$453$	14.4721	0.679960
$454$	2.94427	0.138182
$455$	1.23607	0.0579478
$456$	−4.47214	−0.209427
$457$	38.6525	1.80809	0.904043	−	0.427441i	$-0.140585\pi$
$457$	38.6525	1.80809	0.904043	+	0.427441i	$0.140585\pi$
$458$	14.0000	0.654177
$459$	6.47214	0.302093
$460$	1.00000	0.0466252
$461$	−0.291796	−0.0135903	−0.00679515	−	0.999977i	$-0.502163\pi$
$461$	−0.291796	−0.0135903	−0.00679515	+	0.999977i	$0.502163\pi$
$462$	1.23607	0.0575071
$463$	−28.0689	−1.30447	−0.652236	−	0.758016i	$-0.726169\pi$
$463$	−28.0689	−1.30447	−0.652236	+	0.758016i	$0.726169\pi$
$464$	−5.70820	−0.264997
$465$	4.47214	0.207390
$466$	−15.5279	−0.719314
$467$	33.4164	1.54633	0.773163	−	0.634207i	$-0.218673\pi$
$467$	33.4164	1.54633	0.773163	+	0.634207i	$0.218673\pi$
$468$	−1.23607	−0.0571373
$469$	13.2361	0.611185
$470$	2.00000	0.0922531
$471$	13.4164	0.618195
$472$	8.94427	0.411693
$473$	−6.47214	−0.297589
$474$	4.00000	0.183726
$475$	4.47214	0.205196
$476$	−6.47214	−0.296650
$477$	10.0000	0.457869
$478$	−9.23607	−0.422448
$479$	12.5836	0.574959	0.287480	−	0.957787i	$-0.407183\pi$
$479$	12.5836	0.574959	0.287480	+	0.957787i	$0.407183\pi$
$480$	−1.00000	−0.0456435
$481$	−10.4721	−0.477488
$482$	−13.8885	−0.632606
$483$	−1.00000	−0.0455016
$484$	−9.47214	−0.430552
$485$	−7.70820	−0.350012
$486$	−1.00000	−0.0453609
$487$	−15.1246	−0.685362	−0.342681	−	0.939452i	$-0.611335\pi$
$487$	−15.1246	−0.685362	−0.342681	+	0.939452i	$0.611335\pi$
$488$	2.00000	0.0905357
$489$	−19.4164	−0.878040
$490$	−1.00000	−0.0451754
$491$	−1.52786	−0.0689515	−0.0344758	−	0.999406i	$-0.510976\pi$
$491$	−1.52786	−0.0689515	−0.0344758	+	0.999406i	$0.510976\pi$
$492$	−0.472136	−0.0212855
$493$	−36.9443	−1.66389
$494$	5.52786	0.248710
$495$	1.23607	0.0555571
$496$	4.47214	0.200805
$497$	3.70820	0.166336
$498$	−8.47214	−0.379645
$499$	17.8885	0.800801	0.400401	−	0.916340i	$-0.368871\pi$
$499$	17.8885	0.800801	0.400401	+	0.916340i	$0.368871\pi$
$500$	1.00000	0.0447214
$501$	7.52786	0.336320
$502$	−10.6525	−0.475443
$503$	−37.3050	−1.66335	−0.831673	−	0.555266i	$-0.812616\pi$
$503$	−37.3050	−1.66335	−0.831673	+	0.555266i	$0.812616\pi$
$504$	1.00000	0.0445435
$505$	−18.1803	−0.809015
$506$	−1.23607	−0.0549499
$507$	−11.4721	−0.509495
$508$	9.23607	0.409784
$509$	−2.18034	−0.0966419	−0.0483209	−	0.998832i	$-0.515387\pi$
$509$	−2.18034	−0.0966419	−0.0483209	+	0.998832i	$0.515387\pi$
$510$	−6.47214	−0.286591
$511$	2.94427	0.130247
$512$	−1.00000	−0.0441942
$513$	4.47214	0.197450
$514$	−28.8328	−1.27176
$515$	9.52786	0.419848
$516$	−5.23607	−0.230505
$517$	−2.47214	−0.108724
$518$	8.47214	0.372244
$519$	8.00000	0.351161
$520$	1.23607	0.0542052
$521$	−31.1246	−1.36359	−0.681797	−	0.731541i	$-0.738801\pi$
$521$	−31.1246	−1.36359	−0.681797	+	0.731541i	$0.738801\pi$
$522$	5.70820	0.249841
$523$	12.1803	0.532609	0.266305	−	0.963889i	$-0.414197\pi$
$523$	12.1803	0.532609	0.266305	+	0.963889i	$0.414197\pi$
$524$	8.94427	0.390732
$525$	−1.00000	−0.0436436
$526$	−7.05573	−0.307644
$527$	28.9443	1.26083
$528$	1.23607	0.0537930
$529$	1.00000	0.0434783
$530$	−10.0000	−0.434372
$531$	−8.94427	−0.388148
$532$	−4.47214	−0.193892
$533$	0.583592	0.0252782
$534$	−2.76393	−0.119607
$535$	19.4164	0.839445
$536$	13.2361	0.571711
$537$	19.4164	0.837880
$538$	15.7082	0.677229
$539$	1.23607	0.0532412
$540$	1.00000	0.0430331
$541$	−35.3050	−1.51788	−0.758939	−	0.651161i	$-0.774282\pi$
$541$	−35.3050	−1.51788	−0.758939	+	0.651161i	$0.774282\pi$
$542$	−1.41641	−0.0608399
$543$	−19.8885	−0.853499
$544$	−6.47214	−0.277491
$545$	16.4721	0.705589
$546$	−1.23607	−0.0528988
$547$	16.3607	0.699532	0.349766	−	0.936837i	$-0.386261\pi$
$547$	16.3607	0.699532	0.349766	+	0.936837i	$0.386261\pi$
$548$	14.0000	0.598050
$549$	−2.00000	−0.0853579
$550$	−1.23607	−0.0527061
$551$	−25.5279	−1.08752
$552$	−1.00000	−0.0425628
$553$	4.00000	0.170097
$554$	6.65248	0.282637
$555$	8.47214	0.359622
$556$	8.00000	0.339276
$557$	−2.36068	−0.100025	−0.0500126	−	0.998749i	$-0.515926\pi$
$557$	−2.36068	−0.100025	−0.0500126	+	0.998749i	$0.515926\pi$
$558$	−4.47214	−0.189321
$559$	6.47214	0.273742
$560$	−1.00000	−0.0422577
$561$	8.00000	0.337760
$562$	−24.1803	−1.01999
$563$	0.472136	0.0198982	0.00994908	−	0.999951i	$-0.496833\pi$
$563$	0.472136	0.0198982	0.00994908	+	0.999951i	$0.496833\pi$
$564$	−2.00000	−0.0842152
$565$	6.94427	0.292148
$566$	13.7082	0.576199
$567$	−1.00000	−0.0419961
$568$	3.70820	0.155593
$569$	−20.1803	−0.846004	−0.423002	−	0.906129i	$-0.639024\pi$
$569$	−20.1803	−0.846004	−0.423002	+	0.906129i	$0.639024\pi$
$570$	−4.47214	−0.187317
$571$	21.5279	0.900913	0.450457	−	0.892798i	$-0.351261\pi$
$571$	21.5279	0.900913	0.450457	+	0.892798i	$0.351261\pi$
$572$	−1.52786	−0.0638832
$573$	0	0
$574$	−0.472136	−0.0197066
$575$	1.00000	0.0417029
$576$	1.00000	0.0416667
$577$	1.41641	0.0589658	0.0294829	−	0.999565i	$-0.490614\pi$
$577$	1.41641	0.0589658	0.0294829	+	0.999565i	$0.490614\pi$
$578$	−24.8885	−1.03523
$579$	13.4164	0.557567
$580$	−5.70820	−0.237020
$581$	−8.47214	−0.351483
$582$	7.70820	0.319515
$583$	12.3607	0.511927
$584$	2.94427	0.121835
$585$	−1.23607	−0.0511051
$586$	1.05573	0.0436117
$587$	−0.360680	−0.0148868	−0.00744342	−	0.999972i	$-0.502369\pi$
$587$	−0.360680	−0.0148868	−0.00744342	+	0.999972i	$0.502369\pi$
$588$	1.00000	0.0412393
$589$	20.0000	0.824086
$590$	8.94427	0.368230
$591$	1.05573	0.0434269
$592$	8.47214	0.348203
$593$	−16.8328	−0.691241	−0.345620	−	0.938374i	$-0.612332\pi$
$593$	−16.8328	−0.691241	−0.345620	+	0.938374i	$0.612332\pi$
$594$	−1.23607	−0.0507165
$595$	−6.47214	−0.265332
$596$	16.4721	0.674725
$597$	−5.52786	−0.226240
$598$	1.23607	0.0505466
$599$	41.5967	1.69960	0.849799	−	0.527108i	$-0.176724\pi$
$599$	41.5967	1.69960	0.849799	+	0.527108i	$0.176724\pi$
$600$	−1.00000	−0.0408248
$601$	−20.8328	−0.849788	−0.424894	−	0.905243i	$-0.639689\pi$
$601$	−20.8328	−0.849788	−0.424894	+	0.905243i	$0.639689\pi$
$602$	−5.23607	−0.213406
$603$	−13.2361	−0.539014
$604$	14.4721	0.588863
$605$	−9.47214	−0.385097
$606$	18.1803	0.738526
$607$	−47.0132	−1.90821	−0.954103	−	0.299480i	$-0.903187\pi$
$607$	−47.0132	−1.90821	−0.954103	+	0.299480i	$0.903187\pi$
$608$	−4.47214	−0.181369
$609$	5.70820	0.231308
$610$	2.00000	0.0809776
$611$	2.47214	0.100012
$612$	6.47214	0.261621
$613$	−1.63932	−0.0662115	−0.0331058	−	0.999452i	$-0.510540\pi$
$613$	−1.63932	−0.0662115	−0.0331058	+	0.999452i	$0.510540\pi$
$614$	−26.4721	−1.06833
$615$	−0.472136	−0.0190384
$616$	1.23607	0.0498026
$617$	−1.41641	−0.0570224	−0.0285112	−	0.999593i	$-0.509077\pi$
$617$	−1.41641	−0.0570224	−0.0285112	+	0.999593i	$0.509077\pi$
$618$	−9.52786	−0.383267
$619$	−22.3607	−0.898752	−0.449376	−	0.893343i	$-0.648354\pi$
$619$	−22.3607	−0.898752	−0.449376	+	0.893343i	$0.648354\pi$
$620$	4.47214	0.179605
$621$	1.00000	0.0401286
$622$	20.7639	0.832558
$623$	−2.76393	−0.110735
$624$	−1.23607	−0.0494823
$625$	1.00000	0.0400000
$626$	−1.81966	−0.0727282
$627$	5.52786	0.220762
$628$	13.4164	0.535373
$629$	54.8328	2.18633
$630$	1.00000	0.0398410
$631$	7.05573	0.280884	0.140442	−	0.990089i	$-0.455148\pi$
$631$	7.05573	0.280884	0.140442	+	0.990089i	$0.455148\pi$
$632$	4.00000	0.159111
$633$	−20.9443	−0.832460
$634$	17.4164	0.691694
$635$	9.23607	0.366522
$636$	10.0000	0.396526
$637$	−1.23607	−0.0489748
$638$	7.05573	0.279339
$639$	−3.70820	−0.146694
$640$	−1.00000	−0.0395285
$641$	42.0689	1.66162	0.830811	−	0.556555i	$-0.187877\pi$
$641$	42.0689	1.66162	0.830811	+	0.556555i	$0.187877\pi$
$642$	−19.4164	−0.766304
$643$	32.1803	1.26907	0.634534	−	0.772895i	$-0.281192\pi$
$643$	32.1803	1.26907	0.634534	+	0.772895i	$0.281192\pi$
$644$	−1.00000	−0.0394055
$645$	−5.23607	−0.206170
$646$	−28.9443	−1.13880
$647$	2.00000	0.0786281	0.0393141	−	0.999227i	$-0.487483\pi$
$647$	2.00000	0.0786281	0.0393141	+	0.999227i	$0.487483\pi$
$648$	−1.00000	−0.0392837
$649$	−11.0557	−0.433975
$650$	1.23607	0.0484826
$651$	−4.47214	−0.175277
$652$	−19.4164	−0.760405
$653$	27.3050	1.06853	0.534263	−	0.845319i	$-0.320589\pi$
$653$	27.3050	1.06853	0.534263	+	0.845319i	$0.320589\pi$
$654$	−16.4721	−0.644111
$655$	8.94427	0.349482
$656$	−0.472136	−0.0184338
$657$	−2.94427	−0.114867
$658$	−2.00000	−0.0779681
$659$	−0.875388	−0.0341003	−0.0170501	−	0.999855i	$-0.505427\pi$
$659$	−0.875388	−0.0341003	−0.0170501	+	0.999855i	$0.505427\pi$
$660$	1.23607	0.0481139
$661$	−26.9443	−1.04801	−0.524005	−	0.851715i	$-0.675563\pi$
$661$	−26.9443	−1.04801	−0.524005	+	0.851715i	$0.675563\pi$
$662$	28.9443	1.12495
$663$	−8.00000	−0.310694
$664$	−8.47214	−0.328783
$665$	−4.47214	−0.173422
$666$	−8.47214	−0.328289
$667$	−5.70820	−0.221023
$668$	7.52786	0.291262
$669$	−20.1803	−0.780217
$670$	13.2361	0.511354
$671$	−2.47214	−0.0954358
$672$	1.00000	0.0385758
$673$	41.7771	1.61039	0.805194	−	0.593011i	$-0.202061\pi$
$673$	41.7771	1.61039	0.805194	+	0.593011i	$0.202061\pi$
$674$	20.7639	0.799797
$675$	1.00000	0.0384900
$676$	−11.4721	−0.441236
$677$	39.8885	1.53304	0.766521	−	0.642220i	$-0.221986\pi$
$677$	39.8885	1.53304	0.766521	+	0.642220i	$0.221986\pi$
$678$	−6.94427	−0.266693
$679$	7.70820	0.295814
$680$	−6.47214	−0.248195
$681$	−2.94427	−0.112825
$682$	−5.52786	−0.211673
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	4.47214	0.170996
$685$	14.0000	0.534913
$686$	1.00000	0.0381802
$687$	−14.0000	−0.534133
$688$	−5.23607	−0.199623
$689$	−12.3607	−0.470904
$690$	−1.00000	−0.0380693
$691$	−19.4164	−0.738635	−0.369317	−	0.929303i	$-0.620409\pi$
$691$	−19.4164	−0.738635	−0.369317	+	0.929303i	$0.620409\pi$
$692$	8.00000	0.304114
$693$	−1.23607	−0.0469543
$694$	21.8885	0.830878
$695$	8.00000	0.303457
$696$	5.70820	0.216369
$697$	−3.05573	−0.115744
$698$	−13.5279	−0.512037
$699$	15.5279	0.587318
$700$	−1.00000	−0.0377964
$701$	−6.94427	−0.262282	−0.131141	−	0.991364i	$-0.541864\pi$
$701$	−6.94427	−0.262282	−0.131141	+	0.991364i	$0.541864\pi$
$702$	1.23607	0.0466524
$703$	37.8885	1.42899
$704$	1.23607	0.0465861
$705$	−2.00000	−0.0753244
$706$	−31.8885	−1.20014
$707$	18.1803	0.683742
$708$	−8.94427	−0.336146
$709$	−29.4164	−1.10476	−0.552378	−	0.833594i	$-0.686280\pi$
$709$	−29.4164	−1.10476	−0.552378	+	0.833594i	$0.686280\pi$
$710$	3.70820	0.139166
$711$	−4.00000	−0.150012
$712$	−2.76393	−0.103583
$713$	4.47214	0.167483
$714$	6.47214	0.242214
$715$	−1.52786	−0.0571389
$716$	19.4164	0.725625
$717$	9.23607	0.344927
$718$	−20.9443	−0.781633
$719$	−40.1803	−1.49847	−0.749237	−	0.662302i	$-0.769580\pi$
$719$	−40.1803	−1.49847	−0.749237	+	0.662302i	$0.769580\pi$
$720$	1.00000	0.0372678
$721$	−9.52786	−0.354836
$722$	−1.00000	−0.0372161
$723$	13.8885	0.516521
$724$	−19.8885	−0.739152
$725$	−5.70820	−0.211997
$726$	9.47214	0.351544
$727$	17.5279	0.650072	0.325036	−	0.945702i	$-0.394623\pi$
$727$	17.5279	0.650072	0.325036	+	0.945702i	$0.394623\pi$
$728$	−1.23607	−0.0458117
$729$	1.00000	0.0370370
$730$	2.94427	0.108972
$731$	−33.8885	−1.25341
$732$	−2.00000	−0.0739221
$733$	−22.9443	−0.847466	−0.423733	−	0.905787i	$-0.639281\pi$
$733$	−22.9443	−0.847466	−0.423733	+	0.905787i	$0.639281\pi$
$734$	4.58359	0.169183
$735$	1.00000	0.0368856
$736$	−1.00000	−0.0368605
$737$	−16.3607	−0.602653
$738$	0.472136	0.0173796
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	8.47214	0.311442
$741$	−5.52786	−0.203071
$742$	10.0000	0.367112
$743$	2.83282	0.103926	0.0519630	−	0.998649i	$-0.483452\pi$
$743$	2.83282	0.103926	0.0519630	+	0.998649i	$0.483452\pi$
$744$	−4.47214	−0.163956
$745$	16.4721	0.603492
$746$	28.4721	1.04244
$747$	8.47214	0.309979
$748$	8.00000	0.292509
$749$	−19.4164	−0.709460
$750$	−1.00000	−0.0365148
$751$	−14.1115	−0.514934	−0.257467	−	0.966287i	$-0.582888\pi$
$751$	−14.1115	−0.514934	−0.257467	+	0.966287i	$0.582888\pi$
$752$	−2.00000	−0.0729325
$753$	10.6525	0.388198
$754$	−7.05573	−0.256954
$755$	14.4721	0.526695
$756$	−1.00000	−0.0363696
$757$	−19.5279	−0.709752	−0.354876	−	0.934913i	$-0.615477\pi$
$757$	−19.5279	−0.709752	−0.354876	+	0.934913i	$0.615477\pi$
$758$	−9.52786	−0.346068
$759$	1.23607	0.0448664
$760$	−4.47214	−0.162221
$761$	−15.5279	−0.562885	−0.281442	−	0.959578i	$-0.590813\pi$
$761$	−15.5279	−0.562885	−0.281442	+	0.959578i	$0.590813\pi$
$762$	−9.23607	−0.334587
$763$	−16.4721	−0.596331
$764$	0	0
$765$	6.47214	0.234001
$766$	17.8885	0.646339
$767$	11.0557	0.399199
$768$	1.00000	0.0360844
$769$	35.4164	1.27715	0.638574	−	0.769560i	$-0.279525\pi$
$769$	35.4164	1.27715	0.638574	+	0.769560i	$0.279525\pi$
$770$	1.23607	0.0445448
$771$	28.8328	1.03839
$772$	13.4164	0.482867
$773$	−0.111456	−0.00400880	−0.00200440	−	0.999998i	$-0.500638\pi$
$773$	−0.111456	−0.00400880	−0.00200440	+	0.999998i	$0.500638\pi$
$774$	5.23607	0.188207
$775$	4.47214	0.160644
$776$	7.70820	0.276708
$777$	−8.47214	−0.303936
$778$	8.47214	0.303741
$779$	−2.11146	−0.0756508
$780$	−1.23607	−0.0442583
$781$	−4.58359	−0.164014
$782$	−6.47214	−0.231443
$783$	−5.70820	−0.203995
$784$	1.00000	0.0357143
$785$	13.4164	0.478852
$786$	−8.94427	−0.319032
$787$	1.34752	0.0480340	0.0240170	−	0.999712i	$-0.492354\pi$
$787$	1.34752	0.0480340	0.0240170	+	0.999712i	$0.492354\pi$
$788$	1.05573	0.0376088
$789$	7.05573	0.251191
$790$	4.00000	0.142314
$791$	−6.94427	−0.246910
$792$	−1.23607	−0.0439218
$793$	2.47214	0.0877881
$794$	−19.7082	−0.699418
$795$	10.0000	0.354663
$796$	−5.52786	−0.195930
$797$	10.0000	0.354218	0.177109	−	0.984191i	$-0.443325\pi$
$797$	10.0000	0.354218	0.177109	+	0.984191i	$0.443325\pi$
$798$	4.47214	0.158312
$799$	−12.9443	−0.457935
$800$	−1.00000	−0.0353553
$801$	2.76393	0.0976587
$802$	33.1246	1.16967
$803$	−3.63932	−0.128429
$804$	−13.2361	−0.466800
$805$	−1.00000	−0.0352454
$806$	5.52786	0.194711
$807$	−15.7082	−0.552955
$808$	18.1803	0.639582
$809$	49.1935	1.72955	0.864776	−	0.502159i	$-0.167461\pi$
$809$	49.1935	1.72955	0.864776	+	0.502159i	$0.167461\pi$
$810$	−1.00000	−0.0351364
$811$	10.4721	0.367726	0.183863	−	0.982952i	$-0.441140\pi$
$811$	10.4721	0.367726	0.183863	+	0.982952i	$0.441140\pi$
$812$	5.70820	0.200319
$813$	1.41641	0.0496756
$814$	−10.4721	−0.367048
$815$	−19.4164	−0.680127
$816$	6.47214	0.226570
$817$	−23.4164	−0.819236
$818$	−7.88854	−0.275816
$819$	1.23607	0.0431917
$820$	−0.472136	−0.0164877
$821$	13.7082	0.478420	0.239210	−	0.970968i	$-0.423112\pi$
$821$	13.7082	0.478420	0.239210	+	0.970968i	$0.423112\pi$
$822$	−14.0000	−0.488306
$823$	−27.7082	−0.965847	−0.482924	−	0.875662i	$-0.660425\pi$
$823$	−27.7082	−0.965847	−0.482924	+	0.875662i	$0.660425\pi$
$824$	−9.52786	−0.331919
$825$	1.23607	0.0430344
$826$	−8.94427	−0.311211
$827$	−11.4164	−0.396987	−0.198494	−	0.980102i	$-0.563605\pi$
$827$	−11.4164	−0.396987	−0.198494	+	0.980102i	$0.563605\pi$
$828$	1.00000	0.0347524
$829$	−45.3050	−1.57351	−0.786753	−	0.617268i	$-0.788239\pi$
$829$	−45.3050	−1.57351	−0.786753	+	0.617268i	$0.788239\pi$
$830$	−8.47214	−0.294072
$831$	−6.65248	−0.230772
$832$	−1.23607	−0.0428529
$833$	6.47214	0.224246
$834$	−8.00000	−0.277017
$835$	7.52786	0.260512
$836$	5.52786	0.191185
$837$	4.47214	0.154580
$838$	−16.1803	−0.558941
$839$	−41.8885	−1.44615	−0.723077	−	0.690768i	$-0.757273\pi$
$839$	−41.8885	−1.44615	−0.723077	+	0.690768i	$0.757273\pi$
$840$	1.00000	0.0345033
$841$	3.58359	0.123572
$842$	−19.8885	−0.685404
$843$	24.1803	0.832815
$844$	−20.9443	−0.720932
$845$	−11.4721	−0.394653
$846$	2.00000	0.0687614
$847$	9.47214	0.325466
$848$	10.0000	0.343401
$849$	−13.7082	−0.470464
$850$	−6.47214	−0.221992
$851$	8.47214	0.290421
$852$	−3.70820	−0.127041
$853$	24.6525	0.844085	0.422042	−	0.906576i	$-0.361313\pi$
$853$	24.6525	0.844085	0.422042	+	0.906576i	$0.361313\pi$
$854$	−2.00000	−0.0684386
$855$	4.47214	0.152944
$856$	−19.4164	−0.663639
$857$	33.7771	1.15380	0.576902	−	0.816814i	$-0.304262\pi$
$857$	33.7771	1.15380	0.576902	+	0.816814i	$0.304262\pi$
$858$	1.52786	0.0521604
$859$	−43.4164	−1.48135	−0.740674	−	0.671864i	$-0.765494\pi$
$859$	−43.4164	−1.48135	−0.740674	+	0.671864i	$0.765494\pi$
$860$	−5.23607	−0.178548
$861$	0.472136	0.0160904
$862$	12.3607	0.421006
$863$	0.360680	0.0122777	0.00613884	−	0.999981i	$-0.498046\pi$
$863$	0.360680	0.0122777	0.00613884	+	0.999981i	$0.498046\pi$
$864$	−1.00000	−0.0340207
$865$	8.00000	0.272008
$866$	−39.1246	−1.32951
$867$	24.8885	0.845259
$868$	−4.47214	−0.151794
$869$	−4.94427	−0.167723
$870$	5.70820	0.193526
$871$	16.3607	0.554360
$872$	−16.4721	−0.557817
$873$	−7.70820	−0.260883
$874$	−4.47214	−0.151272
$875$	−1.00000	−0.0338062
$876$	−2.94427	−0.0994777
$877$	−3.81966	−0.128981	−0.0644904	−	0.997918i	$-0.520542\pi$
$877$	−3.81966	−0.128981	−0.0644904	+	0.997918i	$0.520542\pi$
$878$	14.9443	0.504345
$879$	−1.05573	−0.0356088
$880$	1.23607	0.0416678
$881$	37.5967	1.26667	0.633333	−	0.773879i	$-0.281686\pi$
$881$	37.5967	1.26667	0.633333	+	0.773879i	$0.281686\pi$
$882$	−1.00000	−0.0336718
$883$	22.4721	0.756248	0.378124	−	0.925755i	$-0.376569\pi$
$883$	22.4721	0.756248	0.378124	+	0.925755i	$0.376569\pi$
$884$	−8.00000	−0.269069
$885$	−8.94427	−0.300658
$886$	24.9443	0.838019
$887$	8.11146	0.272356	0.136178	−	0.990684i	$-0.456518\pi$
$887$	8.11146	0.272356	0.136178	+	0.990684i	$0.456518\pi$
$888$	−8.47214	−0.284306
$889$	−9.23607	−0.309768
$890$	−2.76393	−0.0926472
$891$	1.23607	0.0414098
$892$	−20.1803	−0.675688
$893$	−8.94427	−0.299309
$894$	−16.4721	−0.550911
$895$	19.4164	0.649019
$896$	1.00000	0.0334077
$897$	−1.23607	−0.0412711
$898$	2.00000	0.0667409
$899$	−25.5279	−0.851402
$900$	1.00000	0.0333333
$901$	64.7214	2.15618
$902$	0.583592	0.0194315
$903$	5.23607	0.174245
$904$	−6.94427	−0.230963
$905$	−19.8885	−0.661118
$906$	−14.4721	−0.480805
$907$	21.5967	0.717108	0.358554	−	0.933509i	$-0.383270\pi$
$907$	21.5967	0.717108	0.358554	+	0.933509i	$0.383270\pi$
$908$	−2.94427	−0.0977091
$909$	−18.1803	−0.603004
$910$	−1.23607	−0.0409753
$911$	−52.3607	−1.73479	−0.867393	−	0.497623i	$-0.834206\pi$
$911$	−52.3607	−1.73479	−0.867393	+	0.497623i	$0.834206\pi$
$912$	4.47214	0.148087
$913$	10.4721	0.346577
$914$	−38.6525	−1.27851
$915$	−2.00000	−0.0661180
$916$	−14.0000	−0.462573
$917$	−8.94427	−0.295366
$918$	−6.47214	−0.213612
$919$	−51.7771	−1.70797	−0.853984	−	0.520299i	$-0.825821\pi$
$919$	−51.7771	−1.70797	−0.853984	+	0.520299i	$0.825821\pi$
$920$	−1.00000	−0.0329690
$921$	26.4721	0.872287
$922$	0.291796	0.00960979
$923$	4.58359	0.150871
$924$	−1.23607	−0.0406637
$925$	8.47214	0.278562
$926$	28.0689	0.922401
$927$	9.52786	0.312936
$928$	5.70820	0.187381
$929$	50.7214	1.66411	0.832057	−	0.554690i	$-0.187163\pi$
$929$	50.7214	1.66411	0.832057	+	0.554690i	$0.187163\pi$
$930$	−4.47214	−0.146647
$931$	4.47214	0.146568
$932$	15.5279	0.508632
$933$	−20.7639	−0.679781
$934$	−33.4164	−1.09342
$935$	8.00000	0.261628
$936$	1.23607	0.0404021
$937$	−10.5410	−0.344360	−0.172180	−	0.985065i	$-0.555081\pi$
$937$	−10.5410	−0.344360	−0.172180	+	0.985065i	$0.555081\pi$
$938$	−13.2361	−0.432173
$939$	1.81966	0.0593824
$940$	−2.00000	−0.0652328
$941$	−35.8885	−1.16993	−0.584967	−	0.811057i	$-0.698892\pi$
$941$	−35.8885	−1.16993	−0.584967	+	0.811057i	$0.698892\pi$
$942$	−13.4164	−0.437130
$943$	−0.472136	−0.0153749
$944$	−8.94427	−0.291111
$945$	−1.00000	−0.0325300
$946$	6.47214	0.210427
$947$	27.0557	0.879193	0.439597	−	0.898195i	$-0.355121\pi$
$947$	27.0557	0.879193	0.439597	+	0.898195i	$0.355121\pi$
$948$	−4.00000	−0.129914
$949$	3.63932	0.118137
$950$	−4.47214	−0.145095
$951$	−17.4164	−0.564766
$952$	6.47214	0.209763
$953$	−52.8328	−1.71142	−0.855711	−	0.517453i	$-0.826880\pi$
$953$	−52.8328	−1.71142	−0.855711	+	0.517453i	$0.826880\pi$
$954$	−10.0000	−0.323762
$955$	0	0
$956$	9.23607	0.298716
$957$	−7.05573	−0.228079
$958$	−12.5836	−0.406557
$959$	−14.0000	−0.452084
$960$	1.00000	0.0322749
$961$	−11.0000	−0.354839
$962$	10.4721	0.337635
$963$	19.4164	0.625685
$964$	13.8885	0.447320
$965$	13.4164	0.431889
$966$	1.00000	0.0321745
$967$	16.8754	0.542676	0.271338	−	0.962484i	$-0.412534\pi$
$967$	16.8754	0.542676	0.271338	+	0.962484i	$0.412534\pi$
$968$	9.47214	0.304446
$969$	28.9443	0.929824
$970$	7.70820	0.247496
$971$	16.7639	0.537980	0.268990	−	0.963143i	$-0.413310\pi$
$971$	16.7639	0.537980	0.268990	+	0.963143i	$0.413310\pi$
$972$	1.00000	0.0320750
$973$	−8.00000	−0.256468
$974$	15.1246	0.484624
$975$	−1.23607	−0.0395859
$976$	−2.00000	−0.0640184
$977$	−17.4164	−0.557200	−0.278600	−	0.960407i	$-0.589870\pi$
$977$	−17.4164	−0.557200	−0.278600	+	0.960407i	$0.589870\pi$
$978$	19.4164	0.620868
$979$	3.41641	0.109189
$980$	1.00000	0.0319438
$981$	16.4721	0.525915
$982$	1.52786	0.0487561
$983$	−11.0557	−0.352623	−0.176311	−	0.984334i	$-0.556417\pi$
$983$	−11.0557	−0.352623	−0.176311	+	0.984334i	$0.556417\pi$
$984$	0.472136	0.0150511
$985$	1.05573	0.0336383
$986$	36.9443	1.17655
$987$	2.00000	0.0636607
$988$	−5.52786	−0.175865
$989$	−5.23607	−0.166497
$990$	−1.23607	−0.0392848
$991$	−58.2492	−1.85035	−0.925174	−	0.379544i	$-0.876081\pi$
$991$	−58.2492	−1.85035	−0.925174	+	0.379544i	$0.876081\pi$
$992$	−4.47214	−0.141990
$993$	−28.9443	−0.918519
$994$	−3.70820	−0.117617
$995$	−5.52786	−0.175245
$996$	8.47214	0.268450
$997$	−51.1246	−1.61913	−0.809566	−	0.587028i	$-0.800298\pi$
$997$	−51.1246	−1.61913	−0.809566	+	0.587028i	$0.800298\pi$
$998$	−17.8885	−0.566252
$999$	8.47214	0.268047

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4830.2.a.bq.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4830.2.a.bq.1.2	✓	2	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:	\( N \)	\(=\)	\( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4830.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.23607 q^{11} +1.00000 q^{12} -1.23607 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +6.47214 q^{17} -1.00000 q^{18} +4.47214 q^{19} +1.00000 q^{20} -1.00000 q^{21} -1.23607 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.23607 q^{26} +1.00000 q^{27} -1.00000 q^{28} -5.70820 q^{29} -1.00000 q^{30} +4.47214 q^{31} -1.00000 q^{32} +1.23607 q^{33} -6.47214 q^{34} -1.00000 q^{35} +1.00000 q^{36} +8.47214 q^{37} -4.47214 q^{38} -1.23607 q^{39} -1.00000 q^{40} -0.472136 q^{41} +1.00000 q^{42} -5.23607 q^{43} +1.23607 q^{44} +1.00000 q^{45} -1.00000 q^{46} -2.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +6.47214 q^{51} -1.23607 q^{52} +10.0000 q^{53} -1.00000 q^{54} +1.23607 q^{55} +1.00000 q^{56} +4.47214 q^{57} +5.70820 q^{58} -8.94427 q^{59} +1.00000 q^{60} -2.00000 q^{61} -4.47214 q^{62} -1.00000 q^{63} +1.00000 q^{64} -1.23607 q^{65} -1.23607 q^{66} -13.2361 q^{67} +6.47214 q^{68} +1.00000 q^{69} +1.00000 q^{70} -3.70820 q^{71} -1.00000 q^{72} -2.94427 q^{73} -8.47214 q^{74} +1.00000 q^{75} +4.47214 q^{76} -1.23607 q^{77} +1.23607 q^{78} -4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +0.472136 q^{82} +8.47214 q^{83} -1.00000 q^{84} +6.47214 q^{85} +5.23607 q^{86} -5.70820 q^{87} -1.23607 q^{88} +2.76393 q^{89} -1.00000 q^{90} +1.23607 q^{91} +1.00000 q^{92} +4.47214 q^{93} +2.00000 q^{94} +4.47214 q^{95} -1.00000 q^{96} -7.70820 q^{97} -1.00000 q^{98} +1.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 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	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} + 2 q^{12} + 2 q^{13} + 2 q^{14} + 2 q^{15} + 2 q^{16} + 4 q^{17} - 2 q^{18} + 2 q^{20} - 2 q^{21} + 2 q^{22} + 2 q^{23} - 2 q^{24} + 2 q^{25} - 2 q^{26} + 2 q^{27} - 2 q^{28} + 2 q^{29} - 2 q^{30} - 2 q^{32} - 2 q^{33} - 4 q^{34} - 2 q^{35} + 2 q^{36} + 8 q^{37} + 2 q^{39} - 2 q^{40} + 8 q^{41} + 2 q^{42} - 6 q^{43} - 2 q^{44} + 2 q^{45} - 2 q^{46} - 4 q^{47} + 2 q^{48} + 2 q^{49} - 2 q^{50} + 4 q^{51} + 2 q^{52} + 20 q^{53} - 2 q^{54} - 2 q^{55} + 2 q^{56} - 2 q^{58} + 2 q^{60} - 4 q^{61} - 2 q^{63} + 2 q^{64} + 2 q^{65} + 2 q^{66} - 22 q^{67} + 4 q^{68} + 2 q^{69} + 2 q^{70} + 6 q^{71} - 2 q^{72} + 12 q^{73} - 8 q^{74} + 2 q^{75} + 2 q^{77} - 2 q^{78} - 8 q^{79} + 2 q^{80} + 2 q^{81} - 8 q^{82} + 8 q^{83} - 2 q^{84} + 4 q^{85} + 6 q^{86} + 2 q^{87} + 2 q^{88} + 10 q^{89} - 2 q^{90} - 2 q^{91} + 2 q^{92} + 4 q^{94} - 2 q^{96} - 2 q^{97} - 2 q^{98} - 2 q^{99}+O(q^{100}) \) 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 - 2 * q^11 + 2 * q^12 + 2 * q^13 + 2 * q^14 + 2 * q^15 + 2 * q^16 + 4 * q^17 - 2 * q^18 + 2 * q^20 - 2 * q^21 + 2 * q^22 + 2 * q^23 - 2 * q^24 + 2 * q^25 - 2 * q^26 + 2 * q^27 - 2 * q^28 + 2 * q^29 - 2 * q^30 - 2 * q^32 - 2 * q^33 - 4 * q^34 - 2 * q^35 + 2 * q^36 + 8 * q^37 + 2 * q^39 - 2 * q^40 + 8 * q^41 + 2 * q^42 - 6 * q^43 - 2 * q^44 + 2 * q^45 - 2 * q^46 - 4 * q^47 + 2 * q^48 + 2 * q^49 - 2 * q^50 + 4 * q^51 + 2 * q^52 + 20 * q^53 - 2 * q^54 - 2 * q^55 + 2 * q^56 - 2 * q^58 + 2 * q^60 - 4 * q^61 - 2 * q^63 + 2 * q^64 + 2 * q^65 + 2 * q^66 - 22 * q^67 + 4 * q^68 + 2 * q^69 + 2 * q^70 + 6 * q^71 - 2 * q^72 + 12 * q^73 - 8 * q^74 + 2 * q^75 + 2 * q^77 - 2 * q^78 - 8 * q^79 + 2 * q^80 + 2 * q^81 - 8 * q^82 + 8 * q^83 - 2 * q^84 + 4 * q^85 + 6 * q^86 + 2 * q^87 + 2 * q^88 + 10 * q^89 - 2 * q^90 - 2 * q^91 + 2 * q^92 + 4 * q^94 - 2 * q^96 - 2 * q^97 - 2 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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