LMFDB - Embedded newform 5070.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, 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("5070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.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:	$40.4841538248$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -5.12311 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -5.12311 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.12311 q^{11} +1.00000 q^{12} -5.12311 q^{14} -1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +6.00000 q^{19} -1.00000 q^{20} -5.12311 q^{21} -3.12311 q^{22} -3.12311 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -5.12311 q^{28} +2.00000 q^{29} -1.00000 q^{30} +5.12311 q^{31} +1.00000 q^{32} -3.12311 q^{33} -2.00000 q^{34} +5.12311 q^{35} +1.00000 q^{36} +3.12311 q^{37} +6.00000 q^{38} -1.00000 q^{40} -9.12311 q^{41} -5.12311 q^{42} +10.2462 q^{43} -3.12311 q^{44} -1.00000 q^{45} -3.12311 q^{46} +10.2462 q^{47} +1.00000 q^{48} +19.2462 q^{49} +1.00000 q^{50} -2.00000 q^{51} +11.3693 q^{53} +1.00000 q^{54} +3.12311 q^{55} -5.12311 q^{56} +6.00000 q^{57} +2.00000 q^{58} +7.12311 q^{59} -1.00000 q^{60} +10.0000 q^{61} +5.12311 q^{62} -5.12311 q^{63} +1.00000 q^{64} -3.12311 q^{66} -13.1231 q^{67} -2.00000 q^{68} -3.12311 q^{69} +5.12311 q^{70} +6.24621 q^{71} +1.00000 q^{72} -4.87689 q^{73} +3.12311 q^{74} +1.00000 q^{75} +6.00000 q^{76} +16.0000 q^{77} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -9.12311 q^{82} -10.2462 q^{83} -5.12311 q^{84} +2.00000 q^{85} +10.2462 q^{86} +2.00000 q^{87} -3.12311 q^{88} -5.12311 q^{89} -1.00000 q^{90} -3.12311 q^{92} +5.12311 q^{93} +10.2462 q^{94} -6.00000 q^{95} +1.00000 q^{96} +4.87689 q^{97} +19.2462 q^{98} -3.12311 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} + 2 q^{12} - 2 q^{14} - 2 q^{15} + 2 q^{16} - 4 q^{17} + 2 q^{18} + 12 q^{19} - 2 q^{20} - 2 q^{21} + 2 q^{22} + 2 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{27} - 2 q^{28} + 4 q^{29} - 2 q^{30} + 2 q^{31} + 2 q^{32} + 2 q^{33} - 4 q^{34} + 2 q^{35} + 2 q^{36} - 2 q^{37} + 12 q^{38} - 2 q^{40} - 10 q^{41} - 2 q^{42} + 4 q^{43} + 2 q^{44} - 2 q^{45} + 2 q^{46} + 4 q^{47} + 2 q^{48} + 22 q^{49} + 2 q^{50} - 4 q^{51} - 2 q^{53} + 2 q^{54} - 2 q^{55} - 2 q^{56} + 12 q^{57} + 4 q^{58} + 6 q^{59} - 2 q^{60} + 20 q^{61} + 2 q^{62} - 2 q^{63} + 2 q^{64} + 2 q^{66} - 18 q^{67} - 4 q^{68} + 2 q^{69} + 2 q^{70} - 4 q^{71} + 2 q^{72} - 18 q^{73} - 2 q^{74} + 2 q^{75} + 12 q^{76} + 32 q^{77} + 16 q^{79} - 2 q^{80} + 2 q^{81} - 10 q^{82} - 4 q^{83} - 2 q^{84} + 4 q^{85} + 4 q^{86} + 4 q^{87} + 2 q^{88} - 2 q^{89} - 2 q^{90} + 2 q^{92} + 2 q^{93} + 4 q^{94} - 12 q^{95} + 2 q^{96} + 18 q^{97} + 22 q^{98} + 2 q^{99}+O(q^{100}) \) 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^14 - 2 * q^15 + 2 * q^16 - 4 * q^17 + 2 * q^18 + 12 * q^19 - 2 * q^20 - 2 * q^21 + 2 * q^22 + 2 * q^23 + 2 * q^24 + 2 * q^25 + 2 * q^27 - 2 * q^28 + 4 * q^29 - 2 * q^30 + 2 * q^31 + 2 * q^32 + 2 * q^33 - 4 * q^34 + 2 * q^35 + 2 * q^36 - 2 * q^37 + 12 * q^38 - 2 * q^40 - 10 * q^41 - 2 * q^42 + 4 * q^43 + 2 * q^44 - 2 * q^45 + 2 * q^46 + 4 * q^47 + 2 * q^48 + 22 * q^49 + 2 * q^50 - 4 * q^51 - 2 * q^53 + 2 * q^54 - 2 * q^55 - 2 * q^56 + 12 * q^57 + 4 * q^58 + 6 * q^59 - 2 * q^60 + 20 * q^61 + 2 * q^62 - 2 * q^63 + 2 * q^64 + 2 * q^66 - 18 * q^67 - 4 * q^68 + 2 * q^69 + 2 * q^70 - 4 * q^71 + 2 * q^72 - 18 * q^73 - 2 * q^74 + 2 * q^75 + 12 * q^76 + 32 * q^77 + 16 * q^79 - 2 * q^80 + 2 * q^81 - 10 * q^82 - 4 * q^83 - 2 * q^84 + 4 * q^85 + 4 * q^86 + 4 * q^87 + 2 * q^88 - 2 * q^89 - 2 * q^90 + 2 * q^92 + 2 * q^93 + 4 * q^94 - 12 * q^95 + 2 * q^96 + 18 * q^97 + 22 * 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
$7$	−5.12311	−1.93635	−0.968176	−	0.250270i	$-0.919480\pi$
$7$	−5.12311	−1.93635	−0.968176	+	0.250270i	$0.919480\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−3.12311	−0.941652	−0.470826	−	0.882226i	$-0.656044\pi$
$11$	−3.12311	−0.941652	−0.470826	+	0.882226i	$0.656044\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−5.12311	−1.36921
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	0.235702
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	−1.00000	−0.223607
$21$	−5.12311	−1.11795
$22$	−3.12311	−0.665848
$23$	−3.12311	−0.651213	−0.325606	−	0.945505i	$-0.605568\pi$
$23$	−3.12311	−0.651213	−0.325606	+	0.945505i	$0.605568\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	−5.12311	−0.968176
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	−1.00000	−0.182574
$31$	5.12311	0.920137	0.460068	−	0.887883i	$-0.347825\pi$
$31$	5.12311	0.920137	0.460068	+	0.887883i	$0.347825\pi$
$32$	1.00000	0.176777
$33$	−3.12311	−0.543663
$34$	−2.00000	−0.342997
$35$	5.12311	0.865963
$36$	1.00000	0.166667
$37$	3.12311	0.513435	0.256718	−	0.966486i	$-0.417359\pi$
$37$	3.12311	0.513435	0.256718	+	0.966486i	$0.417359\pi$
$38$	6.00000	0.973329
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−9.12311	−1.42479	−0.712395	−	0.701779i	$-0.752389\pi$
$41$	−9.12311	−1.42479	−0.712395	+	0.701779i	$0.752389\pi$
$42$	−5.12311	−0.790512
$43$	10.2462	1.56253	0.781266	−	0.624198i	$-0.214574\pi$
$43$	10.2462	1.56253	0.781266	+	0.624198i	$0.214574\pi$
$44$	−3.12311	−0.470826
$45$	−1.00000	−0.149071
$46$	−3.12311	−0.460477
$47$	10.2462	1.49456	0.747282	−	0.664507i	$-0.231359\pi$
$47$	10.2462	1.49456	0.747282	+	0.664507i	$0.231359\pi$
$48$	1.00000	0.144338
$49$	19.2462	2.74946
$50$	1.00000	0.141421
$51$	−2.00000	−0.280056
$52$	0	0
$53$	11.3693	1.56170	0.780848	−	0.624721i	$-0.214787\pi$
$53$	11.3693	1.56170	0.780848	+	0.624721i	$0.214787\pi$
$54$	1.00000	0.136083
$55$	3.12311	0.421119
$56$	−5.12311	−0.684604
$57$	6.00000	0.794719
$58$	2.00000	0.262613
$59$	7.12311	0.927349	0.463675	−	0.886006i	$-0.346531\pi$
$59$	7.12311	0.927349	0.463675	+	0.886006i	$0.346531\pi$
$60$	−1.00000	−0.129099
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	5.12311	0.650635
$63$	−5.12311	−0.645451
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.12311	−0.384428
$67$	−13.1231	−1.60324	−0.801621	−	0.597832i	$-0.796029\pi$
$67$	−13.1231	−1.60324	−0.801621	+	0.597832i	$0.796029\pi$
$68$	−2.00000	−0.242536
$69$	−3.12311	−0.375978
$70$	5.12311	0.612328
$71$	6.24621	0.741289	0.370644	−	0.928775i	$-0.379137\pi$
$71$	6.24621	0.741289	0.370644	+	0.928775i	$0.379137\pi$
$72$	1.00000	0.117851
$73$	−4.87689	−0.570797	−0.285399	−	0.958409i	$-0.592126\pi$
$73$	−4.87689	−0.570797	−0.285399	+	0.958409i	$0.592126\pi$
$74$	3.12311	0.363054
$75$	1.00000	0.115470
$76$	6.00000	0.688247
$77$	16.0000	1.82337
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−9.12311	−1.00748
$83$	−10.2462	−1.12467	−0.562334	−	0.826910i	$-0.690096\pi$
$83$	−10.2462	−1.12467	−0.562334	+	0.826910i	$0.690096\pi$
$84$	−5.12311	−0.558977
$85$	2.00000	0.216930
$86$	10.2462	1.10488
$87$	2.00000	0.214423
$88$	−3.12311	−0.332924
$89$	−5.12311	−0.543048	−0.271524	−	0.962432i	$-0.587528\pi$
$89$	−5.12311	−0.543048	−0.271524	+	0.962432i	$0.587528\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−3.12311	−0.325606
$93$	5.12311	0.531241
$94$	10.2462	1.05682
$95$	−6.00000	−0.615587
$96$	1.00000	0.102062
$97$	4.87689	0.495174	0.247587	−	0.968866i	$-0.420362\pi$
$97$	4.87689	0.495174	0.247587	+	0.968866i	$0.420362\pi$
$98$	19.2462	1.94416
$99$	−3.12311	−0.313884
$100$	1.00000	0.100000
$101$	−4.24621	−0.422514	−0.211257	−	0.977431i	$-0.567756\pi$
$101$	−4.24621	−0.422514	−0.211257	+	0.977431i	$0.567756\pi$
$102$	−2.00000	−0.198030
$103$	4.87689	0.480535	0.240267	−	0.970707i	$-0.422765\pi$
$103$	4.87689	0.480535	0.240267	+	0.970707i	$0.422765\pi$
$104$	0	0
$105$	5.12311	0.499964
$106$	11.3693	1.10429
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	1.00000	0.0962250
$109$	−11.1231	−1.06540	−0.532700	−	0.846304i	$-0.678823\pi$
$109$	−11.1231	−1.06540	−0.532700	+	0.846304i	$0.678823\pi$
$110$	3.12311	0.297776
$111$	3.12311	0.296432
$112$	−5.12311	−0.484088
$113$	−4.24621	−0.399450	−0.199725	−	0.979852i	$-0.564005\pi$
$113$	−4.24621	−0.399450	−0.199725	+	0.979852i	$0.564005\pi$
$114$	6.00000	0.561951
$115$	3.12311	0.291231
$116$	2.00000	0.185695
$117$	0	0
$118$	7.12311	0.655735
$119$	10.2462	0.939269
$120$	−1.00000	−0.0912871
$121$	−1.24621	−0.113292
$122$	10.0000	0.905357
$123$	−9.12311	−0.822603
$124$	5.12311	0.460068
$125$	−1.00000	−0.0894427
$126$	−5.12311	−0.456403
$127$	4.87689	0.432754	0.216377	−	0.976310i	$-0.430576\pi$
$127$	4.87689	0.432754	0.216377	+	0.976310i	$0.430576\pi$
$128$	1.00000	0.0883883
$129$	10.2462	0.902129
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	−3.12311	−0.271831
$133$	−30.7386	−2.66538
$134$	−13.1231	−1.13366
$135$	−1.00000	−0.0860663
$136$	−2.00000	−0.171499
$137$	22.4924	1.92166	0.960829	−	0.277143i	$-0.0893876\pi$
$137$	22.4924	1.92166	0.960829	+	0.277143i	$0.0893876\pi$
$138$	−3.12311	−0.265856
$139$	16.4924	1.39887	0.699435	−	0.714697i	$-0.253435\pi$
$139$	16.4924	1.39887	0.699435	+	0.714697i	$0.253435\pi$
$140$	5.12311	0.432981
$141$	10.2462	0.862887
$142$	6.24621	0.524170
$143$	0	0
$144$	1.00000	0.0833333
$145$	−2.00000	−0.166091
$146$	−4.87689	−0.403615
$147$	19.2462	1.58740
$148$	3.12311	0.256718
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	1.00000	0.0816497
$151$	−11.3693	−0.925222	−0.462611	−	0.886561i	$-0.653087\pi$
$151$	−11.3693	−0.925222	−0.462611	+	0.886561i	$0.653087\pi$
$152$	6.00000	0.486664
$153$	−2.00000	−0.161690
$154$	16.0000	1.28932
$155$	−5.12311	−0.411498
$156$	0	0
$157$	−3.36932	−0.268901	−0.134450	−	0.990920i	$-0.542927\pi$
$157$	−3.36932	−0.268901	−0.134450	+	0.990920i	$0.542927\pi$
$158$	8.00000	0.636446
$159$	11.3693	0.901645
$160$	−1.00000	−0.0790569
$161$	16.0000	1.26098
$162$	1.00000	0.0785674
$163$	1.12311	0.0879684	0.0439842	−	0.999032i	$-0.485995\pi$
$163$	1.12311	0.0879684	0.0439842	+	0.999032i	$0.485995\pi$
$164$	−9.12311	−0.712395
$165$	3.12311	0.243133
$166$	−10.2462	−0.795260
$167$	−5.75379	−0.445242	−0.222621	−	0.974905i	$-0.571461\pi$
$167$	−5.75379	−0.445242	−0.222621	+	0.974905i	$0.571461\pi$
$168$	−5.12311	−0.395256
$169$	0	0
$170$	2.00000	0.153393
$171$	6.00000	0.458831
$172$	10.2462	0.781266
$173$	−14.8769	−1.13107	−0.565535	−	0.824725i	$-0.691330\pi$
$173$	−14.8769	−1.13107	−0.565535	+	0.824725i	$0.691330\pi$
$174$	2.00000	0.151620
$175$	−5.12311	−0.387270
$176$	−3.12311	−0.235413
$177$	7.12311	0.535405
$178$	−5.12311	−0.383993
$179$	16.4924	1.23270	0.616351	−	0.787472i	$-0.288610\pi$
$179$	16.4924	1.23270	0.616351	+	0.787472i	$0.288610\pi$
$180$	−1.00000	−0.0745356
$181$	3.75379	0.279017	0.139508	−	0.990221i	$-0.455448\pi$
$181$	3.75379	0.279017	0.139508	+	0.990221i	$0.455448\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	−3.12311	−0.230238
$185$	−3.12311	−0.229615
$186$	5.12311	0.375644
$187$	6.24621	0.456768
$188$	10.2462	0.747282
$189$	−5.12311	−0.372651
$190$	−6.00000	−0.435286
$191$	−16.4924	−1.19335	−0.596675	−	0.802483i	$-0.703512\pi$
$191$	−16.4924	−1.19335	−0.596675	+	0.802483i	$0.703512\pi$
$192$	1.00000	0.0721688
$193$	−16.8769	−1.21483	−0.607413	−	0.794386i	$-0.707793\pi$
$193$	−16.8769	−1.21483	−0.607413	+	0.794386i	$0.707793\pi$
$194$	4.87689	0.350141
$195$	0	0
$196$	19.2462	1.37473
$197$	0.246211	0.0175418	0.00877091	−	0.999962i	$-0.497208\pi$
$197$	0.246211	0.0175418	0.00877091	+	0.999962i	$0.497208\pi$
$198$	−3.12311	−0.221949
$199$	−1.75379	−0.124323	−0.0621614	−	0.998066i	$-0.519799\pi$
$199$	−1.75379	−0.124323	−0.0621614	+	0.998066i	$0.519799\pi$
$200$	1.00000	0.0707107
$201$	−13.1231	−0.925633
$202$	−4.24621	−0.298762
$203$	−10.2462	−0.719143
$204$	−2.00000	−0.140028
$205$	9.12311	0.637185
$206$	4.87689	0.339789
$207$	−3.12311	−0.217071
$208$	0	0
$209$	−18.7386	−1.29618
$210$	5.12311	0.353528
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	11.3693	0.780848
$213$	6.24621	0.427983
$214$	−8.00000	−0.546869
$215$	−10.2462	−0.698786
$216$	1.00000	0.0680414
$217$	−26.2462	−1.78171
$218$	−11.1231	−0.753352
$219$	−4.87689	−0.329550
$220$	3.12311	0.210560
$221$	0	0
$222$	3.12311	0.209609
$223$	15.3693	1.02921	0.514603	−	0.857429i	$-0.327939\pi$
$223$	15.3693	1.02921	0.514603	+	0.857429i	$0.327939\pi$
$224$	−5.12311	−0.342302
$225$	1.00000	0.0666667
$226$	−4.24621	−0.282454
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	6.00000	0.397360
$229$	3.12311	0.206381	0.103190	−	0.994662i	$-0.467095\pi$
$229$	3.12311	0.206381	0.103190	+	0.994662i	$0.467095\pi$
$230$	3.12311	0.205931
$231$	16.0000	1.05272
$232$	2.00000	0.131306
$233$	24.2462	1.58842	0.794211	−	0.607642i	$-0.207884\pi$
$233$	24.2462	1.58842	0.794211	+	0.607642i	$0.207884\pi$
$234$	0	0
$235$	−10.2462	−0.668389
$236$	7.12311	0.463675
$237$	8.00000	0.519656
$238$	10.2462	0.664163
$239$	28.4924	1.84302	0.921511	−	0.388353i	$-0.126956\pi$
$239$	28.4924	1.84302	0.921511	+	0.388353i	$0.126956\pi$
$240$	−1.00000	−0.0645497
$241$	2.24621	0.144691	0.0723456	−	0.997380i	$-0.476952\pi$
$241$	2.24621	0.144691	0.0723456	+	0.997380i	$0.476952\pi$
$242$	−1.24621	−0.0801095
$243$	1.00000	0.0641500
$244$	10.0000	0.640184
$245$	−19.2462	−1.22960
$246$	−9.12311	−0.581668
$247$	0	0
$248$	5.12311	0.325318
$249$	−10.2462	−0.649327
$250$	−1.00000	−0.0632456
$251$	−9.75379	−0.615654	−0.307827	−	0.951442i	$-0.599602\pi$
$251$	−9.75379	−0.615654	−0.307827	+	0.951442i	$0.599602\pi$
$252$	−5.12311	−0.322725
$253$	9.75379	0.613215
$254$	4.87689	0.306004
$255$	2.00000	0.125245
$256$	1.00000	0.0625000
$257$	−4.24621	−0.264871	−0.132436	−	0.991192i	$-0.542280\pi$
$257$	−4.24621	−0.264871	−0.132436	+	0.991192i	$0.542280\pi$
$258$	10.2462	0.637901
$259$	−16.0000	−0.994192
$260$	0	0
$261$	2.00000	0.123797
$262$	4.00000	0.247121
$263$	−2.63068	−0.162215	−0.0811074	−	0.996705i	$-0.525846\pi$
$263$	−2.63068	−0.162215	−0.0811074	+	0.996705i	$0.525846\pi$
$264$	−3.12311	−0.192214
$265$	−11.3693	−0.698412
$266$	−30.7386	−1.88471
$267$	−5.12311	−0.313529
$268$	−13.1231	−0.801621
$269$	0.246211	0.0150118	0.00750588	−	0.999972i	$-0.497611\pi$
$269$	0.246211	0.0150118	0.00750588	+	0.999972i	$0.497611\pi$
$270$	−1.00000	−0.0608581
$271$	14.8769	0.903707	0.451853	−	0.892092i	$-0.350763\pi$
$271$	14.8769	0.903707	0.451853	+	0.892092i	$0.350763\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	22.4924	1.35882
$275$	−3.12311	−0.188330
$276$	−3.12311	−0.187989
$277$	−27.8617	−1.67405	−0.837025	−	0.547165i	$-0.815707\pi$
$277$	−27.8617	−1.67405	−0.837025	+	0.547165i	$0.815707\pi$
$278$	16.4924	0.989150
$279$	5.12311	0.306712
$280$	5.12311	0.306164
$281$	5.12311	0.305619	0.152809	−	0.988256i	$-0.451168\pi$
$281$	5.12311	0.305619	0.152809	+	0.988256i	$0.451168\pi$
$282$	10.2462	0.610153
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	6.24621	0.370644
$285$	−6.00000	−0.355409
$286$	0	0
$287$	46.7386	2.75889
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	−2.00000	−0.117444
$291$	4.87689	0.285889
$292$	−4.87689	−0.285399
$293$	−20.7386	−1.21156	−0.605782	−	0.795631i	$-0.707140\pi$
$293$	−20.7386	−1.21156	−0.605782	+	0.795631i	$0.707140\pi$
$294$	19.2462	1.12246
$295$	−7.12311	−0.414723
$296$	3.12311	0.181527
$297$	−3.12311	−0.181221
$298$	14.0000	0.810998
$299$	0	0
$300$	1.00000	0.0577350
$301$	−52.4924	−3.02561
$302$	−11.3693	−0.654231
$303$	−4.24621	−0.243938
$304$	6.00000	0.344124
$305$	−10.0000	−0.572598
$306$	−2.00000	−0.114332
$307$	22.8769	1.30565	0.652827	−	0.757507i	$-0.273583\pi$
$307$	22.8769	1.30565	0.652827	+	0.757507i	$0.273583\pi$
$308$	16.0000	0.911685
$309$	4.87689	0.277437
$310$	−5.12311	−0.290973
$311$	24.4924	1.38884	0.694419	−	0.719571i	$-0.255661\pi$
$311$	24.4924	1.38884	0.694419	+	0.719571i	$0.255661\pi$
$312$	0	0
$313$	−0.246211	−0.0139167	−0.00695834	−	0.999976i	$-0.502215\pi$
$313$	−0.246211	−0.0139167	−0.00695834	+	0.999976i	$0.502215\pi$
$314$	−3.36932	−0.190142
$315$	5.12311	0.288654
$316$	8.00000	0.450035
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	11.3693	0.637560
$319$	−6.24621	−0.349721
$320$	−1.00000	−0.0559017
$321$	−8.00000	−0.446516
$322$	16.0000	0.891645
$323$	−12.0000	−0.667698
$324$	1.00000	0.0555556
$325$	0	0
$326$	1.12311	0.0622031
$327$	−11.1231	−0.615109
$328$	−9.12311	−0.503739
$329$	−52.4924	−2.89400
$330$	3.12311	0.171921
$331$	24.2462	1.33269	0.666346	−	0.745643i	$-0.267857\pi$
$331$	24.2462	1.33269	0.666346	+	0.745643i	$0.267857\pi$
$332$	−10.2462	−0.562334
$333$	3.12311	0.171145
$334$	−5.75379	−0.314833
$335$	13.1231	0.716992
$336$	−5.12311	−0.279488
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	0	0
$339$	−4.24621	−0.230623
$340$	2.00000	0.108465
$341$	−16.0000	−0.866449
$342$	6.00000	0.324443
$343$	−62.7386	−3.38757
$344$	10.2462	0.552439
$345$	3.12311	0.168142
$346$	−14.8769	−0.799787
$347$	2.24621	0.120583	0.0602915	−	0.998181i	$-0.480797\pi$
$347$	2.24621	0.120583	0.0602915	+	0.998181i	$0.480797\pi$
$348$	2.00000	0.107211
$349$	−5.36932	−0.287413	−0.143706	−	0.989620i	$-0.545902\pi$
$349$	−5.36932	−0.287413	−0.143706	+	0.989620i	$0.545902\pi$
$350$	−5.12311	−0.273842
$351$	0	0
$352$	−3.12311	−0.166462
$353$	−4.24621	−0.226003	−0.113002	−	0.993595i	$-0.536046\pi$
$353$	−4.24621	−0.226003	−0.113002	+	0.993595i	$0.536046\pi$
$354$	7.12311	0.378589
$355$	−6.24621	−0.331514
$356$	−5.12311	−0.271524
$357$	10.2462	0.542287
$358$	16.4924	0.871652
$359$	34.2462	1.80745	0.903723	−	0.428118i	$-0.140823\pi$
$359$	34.2462	1.80745	0.903723	+	0.428118i	$0.140823\pi$
$360$	−1.00000	−0.0527046
$361$	17.0000	0.894737
$362$	3.75379	0.197295
$363$	−1.24621	−0.0654091
$364$	0	0
$365$	4.87689	0.255268
$366$	10.0000	0.522708
$367$	33.3693	1.74186	0.870932	−	0.491403i	$-0.163516\pi$
$367$	33.3693	1.74186	0.870932	+	0.491403i	$0.163516\pi$
$368$	−3.12311	−0.162803
$369$	−9.12311	−0.474930
$370$	−3.12311	−0.162363
$371$	−58.2462	−3.02399
$372$	5.12311	0.265621
$373$	−1.12311	−0.0581522	−0.0290761	−	0.999577i	$-0.509257\pi$
$373$	−1.12311	−0.0581522	−0.0290761	+	0.999577i	$0.509257\pi$
$374$	6.24621	0.322984
$375$	−1.00000	−0.0516398
$376$	10.2462	0.528408
$377$	0	0
$378$	−5.12311	−0.263504
$379$	6.00000	0.308199	0.154100	−	0.988055i	$-0.450752\pi$
$379$	6.00000	0.308199	0.154100	+	0.988055i	$0.450752\pi$
$380$	−6.00000	−0.307794
$381$	4.87689	0.249851
$382$	−16.4924	−0.843826
$383$	−18.2462	−0.932338	−0.466169	−	0.884696i	$-0.654366\pi$
$383$	−18.2462	−0.932338	−0.466169	+	0.884696i	$0.654366\pi$
$384$	1.00000	0.0510310
$385$	−16.0000	−0.815436
$386$	−16.8769	−0.859011
$387$	10.2462	0.520844
$388$	4.87689	0.247587
$389$	−16.2462	−0.823716	−0.411858	−	0.911248i	$-0.635120\pi$
$389$	−16.2462	−0.823716	−0.411858	+	0.911248i	$0.635120\pi$
$390$	0	0
$391$	6.24621	0.315884
$392$	19.2462	0.972080
$393$	4.00000	0.201773
$394$	0.246211	0.0124039
$395$	−8.00000	−0.402524
$396$	−3.12311	−0.156942
$397$	9.36932	0.470233	0.235116	−	0.971967i	$-0.424453\pi$
$397$	9.36932	0.470233	0.235116	+	0.971967i	$0.424453\pi$
$398$	−1.75379	−0.0879095
$399$	−30.7386	−1.53886
$400$	1.00000	0.0500000
$401$	23.3693	1.16701	0.583504	−	0.812110i	$-0.301681\pi$
$401$	23.3693	1.16701	0.583504	+	0.812110i	$0.301681\pi$
$402$	−13.1231	−0.654521
$403$	0	0
$404$	−4.24621	−0.211257
$405$	−1.00000	−0.0496904
$406$	−10.2462	−0.508511
$407$	−9.75379	−0.483477
$408$	−2.00000	−0.0990148
$409$	24.4924	1.21107	0.605536	−	0.795818i	$-0.292959\pi$
$409$	24.4924	1.21107	0.605536	+	0.795818i	$0.292959\pi$
$410$	9.12311	0.450558
$411$	22.4924	1.10947
$412$	4.87689	0.240267
$413$	−36.4924	−1.79567
$414$	−3.12311	−0.153492
$415$	10.2462	0.502967
$416$	0	0
$417$	16.4924	0.807637
$418$	−18.7386	−0.916537
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	5.12311	0.249982
$421$	25.3693	1.23642	0.618212	−	0.786011i	$-0.287857\pi$
$421$	25.3693	1.23642	0.618212	+	0.786011i	$0.287857\pi$
$422$	4.00000	0.194717
$423$	10.2462	0.498188
$424$	11.3693	0.552143
$425$	−2.00000	−0.0970143
$426$	6.24621	0.302630
$427$	−51.2311	−2.47924
$428$	−8.00000	−0.386695
$429$	0	0
$430$	−10.2462	−0.494116
$431$	−0.492423	−0.0237192	−0.0118596	−	0.999930i	$-0.503775\pi$
$431$	−0.492423	−0.0237192	−0.0118596	+	0.999930i	$0.503775\pi$
$432$	1.00000	0.0481125
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	−26.2462	−1.25986
$435$	−2.00000	−0.0958927
$436$	−11.1231	−0.532700
$437$	−18.7386	−0.896390
$438$	−4.87689	−0.233027
$439$	3.50758	0.167408	0.0837038	−	0.996491i	$-0.473325\pi$
$439$	3.50758	0.167408	0.0837038	+	0.996491i	$0.473325\pi$
$440$	3.12311	0.148888
$441$	19.2462	0.916486
$442$	0	0
$443$	−36.4924	−1.73381	−0.866904	−	0.498476i	$-0.833893\pi$
$443$	−36.4924	−1.73381	−0.866904	+	0.498476i	$0.833893\pi$
$444$	3.12311	0.148216
$445$	5.12311	0.242858
$446$	15.3693	0.727758
$447$	14.0000	0.662177
$448$	−5.12311	−0.242044
$449$	−37.1231	−1.75195	−0.875974	−	0.482359i	$-0.839780\pi$
$449$	−37.1231	−1.75195	−0.875974	+	0.482359i	$0.839780\pi$
$450$	1.00000	0.0471405
$451$	28.4924	1.34166
$452$	−4.24621	−0.199725
$453$	−11.3693	−0.534177
$454$	−8.00000	−0.375459
$455$	0	0
$456$	6.00000	0.280976
$457$	−6.63068	−0.310170	−0.155085	−	0.987901i	$-0.549565\pi$
$457$	−6.63068	−0.310170	−0.155085	+	0.987901i	$0.549565\pi$
$458$	3.12311	0.145933
$459$	−2.00000	−0.0933520
$460$	3.12311	0.145616
$461$	14.4924	0.674979	0.337490	−	0.941329i	$-0.390422\pi$
$461$	14.4924	0.674979	0.337490	+	0.941329i	$0.390422\pi$
$462$	16.0000	0.744387
$463$	35.8617	1.66664	0.833318	−	0.552794i	$-0.186438\pi$
$463$	35.8617	1.66664	0.833318	+	0.552794i	$0.186438\pi$
$464$	2.00000	0.0928477
$465$	−5.12311	−0.237578
$466$	24.2462	1.12318
$467$	5.75379	0.266254	0.133127	−	0.991099i	$-0.457498\pi$
$467$	5.75379	0.266254	0.133127	+	0.991099i	$0.457498\pi$
$468$	0	0
$469$	67.2311	3.10444
$470$	−10.2462	−0.472622
$471$	−3.36932	−0.155250
$472$	7.12311	0.327868
$473$	−32.0000	−1.47136
$474$	8.00000	0.367452
$475$	6.00000	0.275299
$476$	10.2462	0.469634
$477$	11.3693	0.520565
$478$	28.4924	1.30321
$479$	20.4924	0.936323	0.468161	−	0.883643i	$-0.344917\pi$
$479$	20.4924	0.936323	0.468161	+	0.883643i	$0.344917\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	2.24621	0.102312
$483$	16.0000	0.728025
$484$	−1.24621	−0.0566460
$485$	−4.87689	−0.221448
$486$	1.00000	0.0453609
$487$	7.36932	0.333936	0.166968	−	0.985962i	$-0.446602\pi$
$487$	7.36932	0.333936	0.166968	+	0.985962i	$0.446602\pi$
$488$	10.0000	0.452679
$489$	1.12311	0.0507886
$490$	−19.2462	−0.869455
$491$	10.7386	0.484628	0.242314	−	0.970198i	$-0.422094\pi$
$491$	10.7386	0.484628	0.242314	+	0.970198i	$0.422094\pi$
$492$	−9.12311	−0.411301
$493$	−4.00000	−0.180151
$494$	0	0
$495$	3.12311	0.140373
$496$	5.12311	0.230034
$497$	−32.0000	−1.43540
$498$	−10.2462	−0.459144
$499$	−1.50758	−0.0674884	−0.0337442	−	0.999431i	$-0.510743\pi$
$499$	−1.50758	−0.0674884	−0.0337442	+	0.999431i	$0.510743\pi$
$500$	−1.00000	−0.0447214
$501$	−5.75379	−0.257060
$502$	−9.75379	−0.435333
$503$	10.6307	0.473999	0.236999	−	0.971510i	$-0.423836\pi$
$503$	10.6307	0.473999	0.236999	+	0.971510i	$0.423836\pi$
$504$	−5.12311	−0.228201
$505$	4.24621	0.188954
$506$	9.75379	0.433609
$507$	0	0
$508$	4.87689	0.216377
$509$	−15.7538	−0.698274	−0.349137	−	0.937072i	$-0.613525\pi$
$509$	−15.7538	−0.698274	−0.349137	+	0.937072i	$0.613525\pi$
$510$	2.00000	0.0885615
$511$	24.9848	1.10526
$512$	1.00000	0.0441942
$513$	6.00000	0.264906
$514$	−4.24621	−0.187292
$515$	−4.87689	−0.214902
$516$	10.2462	0.451064
$517$	−32.0000	−1.40736
$518$	−16.0000	−0.703000
$519$	−14.8769	−0.653023
$520$	0	0
$521$	−16.2462	−0.711759	−0.355880	−	0.934532i	$-0.615819\pi$
$521$	−16.2462	−0.711759	−0.355880	+	0.934532i	$0.615819\pi$
$522$	2.00000	0.0875376
$523$	22.7386	0.994291	0.497146	−	0.867667i	$-0.334382\pi$
$523$	22.7386	0.994291	0.497146	+	0.867667i	$0.334382\pi$
$524$	4.00000	0.174741
$525$	−5.12311	−0.223591
$526$	−2.63068	−0.114703
$527$	−10.2462	−0.446332
$528$	−3.12311	−0.135916
$529$	−13.2462	−0.575922
$530$	−11.3693	−0.493852
$531$	7.12311	0.309116
$532$	−30.7386	−1.33269
$533$	0	0
$534$	−5.12311	−0.221698
$535$	8.00000	0.345870
$536$	−13.1231	−0.566832
$537$	16.4924	0.711701
$538$	0.246211	0.0106149
$539$	−60.1080	−2.58903
$540$	−1.00000	−0.0430331
$541$	19.1231	0.822167	0.411083	−	0.911598i	$-0.365151\pi$
$541$	19.1231	0.822167	0.411083	+	0.911598i	$0.365151\pi$
$542$	14.8769	0.639017
$543$	3.75379	0.161090
$544$	−2.00000	−0.0857493
$545$	11.1231	0.476461
$546$	0	0
$547$	−44.9848	−1.92341	−0.961707	−	0.274081i	$-0.911626\pi$
$547$	−44.9848	−1.92341	−0.961707	+	0.274081i	$0.911626\pi$
$548$	22.4924	0.960829
$549$	10.0000	0.426790
$550$	−3.12311	−0.133170
$551$	12.0000	0.511217
$552$	−3.12311	−0.132928
$553$	−40.9848	−1.74285
$554$	−27.8617	−1.18373
$555$	−3.12311	−0.132568
$556$	16.4924	0.699435
$557$	28.2462	1.19683	0.598415	−	0.801186i	$-0.295797\pi$
$557$	28.2462	1.19683	0.598415	+	0.801186i	$0.295797\pi$
$558$	5.12311	0.216878
$559$	0	0
$560$	5.12311	0.216491
$561$	6.24621	0.263715
$562$	5.12311	0.216105
$563$	−32.9848	−1.39015	−0.695073	−	0.718939i	$-0.744628\pi$
$563$	−32.9848	−1.39015	−0.695073	+	0.718939i	$0.744628\pi$
$564$	10.2462	0.431443
$565$	4.24621	0.178639
$566$	4.00000	0.168133
$567$	−5.12311	−0.215150
$568$	6.24621	0.262085
$569$	−36.7386	−1.54016	−0.770082	−	0.637945i	$-0.779785\pi$
$569$	−36.7386	−1.54016	−0.770082	+	0.637945i	$0.779785\pi$
$570$	−6.00000	−0.251312
$571$	−24.4924	−1.02498	−0.512488	−	0.858694i	$-0.671276\pi$
$571$	−24.4924	−1.02498	−0.512488	+	0.858694i	$0.671276\pi$
$572$	0	0
$573$	−16.4924	−0.688981
$574$	46.7386	1.95083
$575$	−3.12311	−0.130243
$576$	1.00000	0.0416667
$577$	−2.63068	−0.109517	−0.0547584	−	0.998500i	$-0.517439\pi$
$577$	−2.63068	−0.109517	−0.0547584	+	0.998500i	$0.517439\pi$
$578$	−13.0000	−0.540729
$579$	−16.8769	−0.701380
$580$	−2.00000	−0.0830455
$581$	52.4924	2.17775
$582$	4.87689	0.202154
$583$	−35.5076	−1.47057
$584$	−4.87689	−0.201807
$585$	0	0
$586$	−20.7386	−0.856705
$587$	−16.4924	−0.680715	−0.340358	−	0.940296i	$-0.610548\pi$
$587$	−16.4924	−0.680715	−0.340358	+	0.940296i	$0.610548\pi$
$588$	19.2462	0.793700
$589$	30.7386	1.26656
$590$	−7.12311	−0.293254
$591$	0.246211	0.0101278
$592$	3.12311	0.128359
$593$	−38.4924	−1.58069	−0.790347	−	0.612659i	$-0.790100\pi$
$593$	−38.4924	−1.58069	−0.790347	+	0.612659i	$0.790100\pi$
$594$	−3.12311	−0.128143
$595$	−10.2462	−0.420054
$596$	14.0000	0.573462
$597$	−1.75379	−0.0717778
$598$	0	0
$599$	−3.50758	−0.143316	−0.0716579	−	0.997429i	$-0.522829\pi$
$599$	−3.50758	−0.143316	−0.0716579	+	0.997429i	$0.522829\pi$
$600$	1.00000	0.0408248
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	−52.4924	−2.13943
$603$	−13.1231	−0.534414
$604$	−11.3693	−0.462611
$605$	1.24621	0.0506657
$606$	−4.24621	−0.172491
$607$	−9.36932	−0.380289	−0.190144	−	0.981756i	$-0.560896\pi$
$607$	−9.36932	−0.380289	−0.190144	+	0.981756i	$0.560896\pi$
$608$	6.00000	0.243332
$609$	−10.2462	−0.415197
$610$	−10.0000	−0.404888
$611$	0	0
$612$	−2.00000	−0.0808452
$613$	−14.6307	−0.590928	−0.295464	−	0.955354i	$-0.595474\pi$
$613$	−14.6307	−0.590928	−0.295464	+	0.955354i	$0.595474\pi$
$614$	22.8769	0.923236
$615$	9.12311	0.367879
$616$	16.0000	0.644658
$617$	−8.73863	−0.351804	−0.175902	−	0.984408i	$-0.556284\pi$
$617$	−8.73863	−0.351804	−0.175902	+	0.984408i	$0.556284\pi$
$618$	4.87689	0.196177
$619$	−26.9848	−1.08461	−0.542306	−	0.840181i	$-0.682449\pi$
$619$	−26.9848	−1.08461	−0.542306	+	0.840181i	$0.682449\pi$
$620$	−5.12311	−0.205749
$621$	−3.12311	−0.125326
$622$	24.4924	0.982057
$623$	26.2462	1.05153
$624$	0	0
$625$	1.00000	0.0400000
$626$	−0.246211	−0.00984058
$627$	−18.7386	−0.748349
$628$	−3.36932	−0.134450
$629$	−6.24621	−0.249053
$630$	5.12311	0.204109
$631$	5.61553	0.223551	0.111775	−	0.993734i	$-0.464346\pi$
$631$	5.61553	0.223551	0.111775	+	0.993734i	$0.464346\pi$
$632$	8.00000	0.318223
$633$	4.00000	0.158986
$634$	−6.00000	−0.238290
$635$	−4.87689	−0.193534
$636$	11.3693	0.450823
$637$	0	0
$638$	−6.24621	−0.247290
$639$	6.24621	0.247096
$640$	−1.00000	−0.0395285
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	−8.00000	−0.315735
$643$	7.36932	0.290617	0.145309	−	0.989386i	$-0.453582\pi$
$643$	7.36932	0.290617	0.145309	+	0.989386i	$0.453582\pi$
$644$	16.0000	0.630488
$645$	−10.2462	−0.403444
$646$	−12.0000	−0.472134
$647$	11.6155	0.456654	0.228327	−	0.973585i	$-0.426675\pi$
$647$	11.6155	0.456654	0.228327	+	0.973585i	$0.426675\pi$
$648$	1.00000	0.0392837
$649$	−22.2462	−0.873240
$650$	0	0
$651$	−26.2462	−1.02867
$652$	1.12311	0.0439842
$653$	43.8617	1.71644	0.858221	−	0.513280i	$-0.171570\pi$
$653$	43.8617	1.71644	0.858221	+	0.513280i	$0.171570\pi$
$654$	−11.1231	−0.434948
$655$	−4.00000	−0.156293
$656$	−9.12311	−0.356197
$657$	−4.87689	−0.190266
$658$	−52.4924	−2.04637
$659$	−38.2462	−1.48986	−0.744930	−	0.667142i	$-0.767517\pi$
$659$	−38.2462	−1.48986	−0.744930	+	0.667142i	$0.767517\pi$
$660$	3.12311	0.121567
$661$	0.876894	0.0341072	0.0170536	−	0.999855i	$-0.494571\pi$
$661$	0.876894	0.0341072	0.0170536	+	0.999855i	$0.494571\pi$
$662$	24.2462	0.942356
$663$	0	0
$664$	−10.2462	−0.397630
$665$	30.7386	1.19199
$666$	3.12311	0.121018
$667$	−6.24621	−0.241854
$668$	−5.75379	−0.222621
$669$	15.3693	0.594212
$670$	13.1231	0.506990
$671$	−31.2311	−1.20566
$672$	−5.12311	−0.197628
$673$	38.9848	1.50276	0.751378	−	0.659872i	$-0.229390\pi$
$673$	38.9848	1.50276	0.751378	+	0.659872i	$0.229390\pi$
$674$	−6.00000	−0.231111
$675$	1.00000	0.0384900
$676$	0	0
$677$	21.1231	0.811827	0.405913	−	0.913912i	$-0.366953\pi$
$677$	21.1231	0.811827	0.405913	+	0.913912i	$0.366953\pi$
$678$	−4.24621	−0.163075
$679$	−24.9848	−0.958830
$680$	2.00000	0.0766965
$681$	−8.00000	−0.306561
$682$	−16.0000	−0.612672
$683$	48.9848	1.87435	0.937177	−	0.348856i	$-0.113430\pi$
$683$	48.9848	1.87435	0.937177	+	0.348856i	$0.113430\pi$
$684$	6.00000	0.229416
$685$	−22.4924	−0.859391
$686$	−62.7386	−2.39537
$687$	3.12311	0.119154
$688$	10.2462	0.390633
$689$	0	0
$690$	3.12311	0.118895
$691$	20.7386	0.788935	0.394467	−	0.918910i	$-0.370929\pi$
$691$	20.7386	0.788935	0.394467	+	0.918910i	$0.370929\pi$
$692$	−14.8769	−0.565535
$693$	16.0000	0.607790
$694$	2.24621	0.0852650
$695$	−16.4924	−0.625593
$696$	2.00000	0.0758098
$697$	18.2462	0.691125
$698$	−5.36932	−0.203232
$699$	24.2462	0.917076
$700$	−5.12311	−0.193635
$701$	14.0000	0.528773	0.264386	−	0.964417i	$-0.414831\pi$
$701$	14.0000	0.528773	0.264386	+	0.964417i	$0.414831\pi$
$702$	0	0
$703$	18.7386	0.706741
$704$	−3.12311	−0.117706
$705$	−10.2462	−0.385895
$706$	−4.24621	−0.159808
$707$	21.7538	0.818135
$708$	7.12311	0.267703
$709$	39.6155	1.48779	0.743896	−	0.668295i	$-0.232976\pi$
$709$	39.6155	1.48779	0.743896	+	0.668295i	$0.232976\pi$
$710$	−6.24621	−0.234416
$711$	8.00000	0.300023
$712$	−5.12311	−0.191997
$713$	−16.0000	−0.599205
$714$	10.2462	0.383455
$715$	0	0
$716$	16.4924	0.616351
$717$	28.4924	1.06407
$718$	34.2462	1.27806
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	−1.00000	−0.0372678
$721$	−24.9848	−0.930484
$722$	17.0000	0.632674
$723$	2.24621	0.0835375
$724$	3.75379	0.139508
$725$	2.00000	0.0742781
$726$	−1.24621	−0.0462512
$727$	37.8617	1.40421	0.702107	−	0.712071i	$-0.252243\pi$
$727$	37.8617	1.40421	0.702107	+	0.712071i	$0.252243\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	4.87689	0.180502
$731$	−20.4924	−0.757940
$732$	10.0000	0.369611
$733$	−10.6307	−0.392653	−0.196327	−	0.980539i	$-0.562901\pi$
$733$	−10.6307	−0.392653	−0.196327	+	0.980539i	$0.562901\pi$
$734$	33.3693	1.23168
$735$	−19.2462	−0.709907
$736$	−3.12311	−0.115119
$737$	40.9848	1.50970
$738$	−9.12311	−0.335826
$739$	45.2311	1.66385	0.831926	−	0.554887i	$-0.187239\pi$
$739$	45.2311	1.66385	0.831926	+	0.554887i	$0.187239\pi$
$740$	−3.12311	−0.114808
$741$	0	0
$742$	−58.2462	−2.13829
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	5.12311	0.187822
$745$	−14.0000	−0.512920
$746$	−1.12311	−0.0411198
$747$	−10.2462	−0.374889
$748$	6.24621	0.228384
$749$	40.9848	1.49755
$750$	−1.00000	−0.0365148
$751$	9.75379	0.355921	0.177960	−	0.984038i	$-0.443050\pi$
$751$	9.75379	0.355921	0.177960	+	0.984038i	$0.443050\pi$
$752$	10.2462	0.373641
$753$	−9.75379	−0.355448
$754$	0	0
$755$	11.3693	0.413772
$756$	−5.12311	−0.186326
$757$	5.12311	0.186202	0.0931012	−	0.995657i	$-0.470322\pi$
$757$	5.12311	0.186202	0.0931012	+	0.995657i	$0.470322\pi$
$758$	6.00000	0.217930
$759$	9.75379	0.354040
$760$	−6.00000	−0.217643
$761$	5.12311	0.185712	0.0928562	−	0.995680i	$-0.470400\pi$
$761$	5.12311	0.185712	0.0928562	+	0.995680i	$0.470400\pi$
$762$	4.87689	0.176671
$763$	56.9848	2.06299
$764$	−16.4924	−0.596675
$765$	2.00000	0.0723102
$766$	−18.2462	−0.659262
$767$	0	0
$768$	1.00000	0.0360844
$769$	−32.9848	−1.18946	−0.594732	−	0.803924i	$-0.702742\pi$
$769$	−32.9848	−1.18946	−0.594732	+	0.803924i	$0.702742\pi$
$770$	−16.0000	−0.576600
$771$	−4.24621	−0.152924
$772$	−16.8769	−0.607413
$773$	−0.246211	−0.00885560	−0.00442780	−	0.999990i	$-0.501409\pi$
$773$	−0.246211	−0.00885560	−0.00442780	+	0.999990i	$0.501409\pi$
$774$	10.2462	0.368292
$775$	5.12311	0.184027
$776$	4.87689	0.175070
$777$	−16.0000	−0.573997
$778$	−16.2462	−0.582455
$779$	−54.7386	−1.96122
$780$	0	0
$781$	−19.5076	−0.698036
$782$	6.24621	0.223364
$783$	2.00000	0.0714742
$784$	19.2462	0.687365
$785$	3.36932	0.120256
$786$	4.00000	0.142675
$787$	−53.6155	−1.91119	−0.955594	−	0.294688i	$-0.904784\pi$
$787$	−53.6155	−1.91119	−0.955594	+	0.294688i	$0.904784\pi$
$788$	0.246211	0.00877091
$789$	−2.63068	−0.0936548
$790$	−8.00000	−0.284627
$791$	21.7538	0.773476
$792$	−3.12311	−0.110975
$793$	0	0
$794$	9.36932	0.332505
$795$	−11.3693	−0.403228
$796$	−1.75379	−0.0621614
$797$	31.8617	1.12860	0.564300	−	0.825570i	$-0.309146\pi$
$797$	31.8617	1.12860	0.564300	+	0.825570i	$0.309146\pi$
$798$	−30.7386	−1.08814
$799$	−20.4924	−0.724970
$800$	1.00000	0.0353553
$801$	−5.12311	−0.181016
$802$	23.3693	0.825199
$803$	15.2311	0.537492
$804$	−13.1231	−0.462816
$805$	−16.0000	−0.563926
$806$	0	0
$807$	0.246211	0.00866705
$808$	−4.24621	−0.149381
$809$	−46.4924	−1.63459	−0.817293	−	0.576222i	$-0.804526\pi$
$809$	−46.4924	−1.63459	−0.817293	+	0.576222i	$0.804526\pi$
$810$	−1.00000	−0.0351364
$811$	44.2462	1.55369	0.776847	−	0.629689i	$-0.216818\pi$
$811$	44.2462	1.55369	0.776847	+	0.629689i	$0.216818\pi$
$812$	−10.2462	−0.359572
$813$	14.8769	0.521755
$814$	−9.75379	−0.341870
$815$	−1.12311	−0.0393407
$816$	−2.00000	−0.0700140
$817$	61.4773	2.15082
$818$	24.4924	0.856357
$819$	0	0
$820$	9.12311	0.318593
$821$	−27.7538	−0.968614	−0.484307	−	0.874898i	$-0.660928\pi$
$821$	−27.7538	−0.968614	−0.484307	+	0.874898i	$0.660928\pi$
$822$	22.4924	0.784513
$823$	51.1231	1.78204	0.891020	−	0.453965i	$-0.149991\pi$
$823$	51.1231	1.78204	0.891020	+	0.453965i	$0.149991\pi$
$824$	4.87689	0.169895
$825$	−3.12311	−0.108733
$826$	−36.4924	−1.26973
$827$	50.7386	1.76436	0.882178	−	0.470917i	$-0.156077\pi$
$827$	50.7386	1.76436	0.882178	+	0.470917i	$0.156077\pi$
$828$	−3.12311	−0.108535
$829$	7.75379	0.269300	0.134650	−	0.990893i	$-0.457009\pi$
$829$	7.75379	0.269300	0.134650	+	0.990893i	$0.457009\pi$
$830$	10.2462	0.355651
$831$	−27.8617	−0.966513
$832$	0	0
$833$	−38.4924	−1.33368
$834$	16.4924	0.571086
$835$	5.75379	0.199118
$836$	−18.7386	−0.648089
$837$	5.12311	0.177080
$838$	−28.0000	−0.967244
$839$	2.73863	0.0945481	0.0472741	−	0.998882i	$-0.484947\pi$
$839$	2.73863	0.0945481	0.0472741	+	0.998882i	$0.484947\pi$
$840$	5.12311	0.176764
$841$	−25.0000	−0.862069
$842$	25.3693	0.874284
$843$	5.12311	0.176449
$844$	4.00000	0.137686
$845$	0	0
$846$	10.2462	0.352272
$847$	6.38447	0.219373
$848$	11.3693	0.390424
$849$	4.00000	0.137280
$850$	−2.00000	−0.0685994
$851$	−9.75379	−0.334356
$852$	6.24621	0.213992
$853$	−21.8617	−0.748532	−0.374266	−	0.927321i	$-0.622105\pi$
$853$	−21.8617	−0.748532	−0.374266	+	0.927321i	$0.622105\pi$
$854$	−51.2311	−1.75309
$855$	−6.00000	−0.205196
$856$	−8.00000	−0.273434
$857$	−2.49242	−0.0851395	−0.0425698	−	0.999093i	$-0.513554\pi$
$857$	−2.49242	−0.0851395	−0.0425698	+	0.999093i	$0.513554\pi$
$858$	0	0
$859$	−12.0000	−0.409435	−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000	−0.409435	−0.204717	+	0.978821i	$0.565628\pi$
$860$	−10.2462	−0.349393
$861$	46.7386	1.59285
$862$	−0.492423	−0.0167720
$863$	−10.2462	−0.348785	−0.174393	−	0.984676i	$-0.555796\pi$
$863$	−10.2462	−0.348785	−0.174393	+	0.984676i	$0.555796\pi$
$864$	1.00000	0.0340207
$865$	14.8769	0.505830
$866$	18.0000	0.611665
$867$	−13.0000	−0.441503
$868$	−26.2462	−0.890854
$869$	−24.9848	−0.847553
$870$	−2.00000	−0.0678064
$871$	0	0
$872$	−11.1231	−0.376676
$873$	4.87689	0.165058
$874$	−18.7386	−0.633844
$875$	5.12311	0.173193
$876$	−4.87689	−0.164775
$877$	27.1231	0.915882	0.457941	−	0.888983i	$-0.348587\pi$
$877$	27.1231	0.915882	0.457941	+	0.888983i	$0.348587\pi$
$878$	3.50758	0.118375
$879$	−20.7386	−0.699497
$880$	3.12311	0.105280
$881$	−11.7538	−0.395995	−0.197998	−	0.980203i	$-0.563444\pi$
$881$	−11.7538	−0.395995	−0.197998	+	0.980203i	$0.563444\pi$
$882$	19.2462	0.648054
$883$	26.2462	0.883255	0.441628	−	0.897198i	$-0.354401\pi$
$883$	26.2462	0.883255	0.441628	+	0.897198i	$0.354401\pi$
$884$	0	0
$885$	−7.12311	−0.239441
$886$	−36.4924	−1.22599
$887$	−27.1231	−0.910705	−0.455352	−	0.890311i	$-0.650487\pi$
$887$	−27.1231	−0.910705	−0.455352	+	0.890311i	$0.650487\pi$
$888$	3.12311	0.104805
$889$	−24.9848	−0.837965
$890$	5.12311	0.171727
$891$	−3.12311	−0.104628
$892$	15.3693	0.514603
$893$	61.4773	2.05726
$894$	14.0000	0.468230
$895$	−16.4924	−0.551281
$896$	−5.12311	−0.171151
$897$	0	0
$898$	−37.1231	−1.23881
$899$	10.2462	0.341730
$900$	1.00000	0.0333333
$901$	−22.7386	−0.757534
$902$	28.4924	0.948694
$903$	−52.4924	−1.74684
$904$	−4.24621	−0.141227
$905$	−3.75379	−0.124780
$906$	−11.3693	−0.377720
$907$	−42.2462	−1.40276	−0.701381	−	0.712786i	$-0.747433\pi$
$907$	−42.2462	−1.40276	−0.701381	+	0.712786i	$0.747433\pi$
$908$	−8.00000	−0.265489
$909$	−4.24621	−0.140838
$910$	0	0
$911$	4.00000	0.132526	0.0662630	−	0.997802i	$-0.478892\pi$
$911$	4.00000	0.132526	0.0662630	+	0.997802i	$0.478892\pi$
$912$	6.00000	0.198680
$913$	32.0000	1.05905
$914$	−6.63068	−0.219324
$915$	−10.0000	−0.330590
$916$	3.12311	0.103190
$917$	−20.4924	−0.676719
$918$	−2.00000	−0.0660098
$919$	−38.2462	−1.26163	−0.630813	−	0.775935i	$-0.717279\pi$
$919$	−38.2462	−1.26163	−0.630813	+	0.775935i	$0.717279\pi$
$920$	3.12311	0.102966
$921$	22.8769	0.753819
$922$	14.4924	0.477283
$923$	0	0
$924$	16.0000	0.526361
$925$	3.12311	0.102687
$926$	35.8617	1.17849
$927$	4.87689	0.160178
$928$	2.00000	0.0656532
$929$	46.1080	1.51275	0.756376	−	0.654137i	$-0.226968\pi$
$929$	46.1080	1.51275	0.756376	+	0.654137i	$0.226968\pi$
$930$	−5.12311	−0.167993
$931$	115.477	3.78461
$932$	24.2462	0.794211
$933$	24.4924	0.801846
$934$	5.75379	0.188270
$935$	−6.24621	−0.204273
$936$	0	0
$937$	−3.75379	−0.122631	−0.0613155	−	0.998118i	$-0.519530\pi$
$937$	−3.75379	−0.122631	−0.0613155	+	0.998118i	$0.519530\pi$
$938$	67.2311	2.19517
$939$	−0.246211	−0.00803480
$940$	−10.2462	−0.334195
$941$	14.0000	0.456387	0.228193	−	0.973616i	$-0.426718\pi$
$941$	14.0000	0.456387	0.228193	+	0.973616i	$0.426718\pi$
$942$	−3.36932	−0.109778
$943$	28.4924	0.927841
$944$	7.12311	0.231837
$945$	5.12311	0.166655
$946$	−32.0000	−1.04041
$947$	−24.4924	−0.795897	−0.397948	−	0.917408i	$-0.630278\pi$
$947$	−24.4924	−0.795897	−0.397948	+	0.917408i	$0.630278\pi$
$948$	8.00000	0.259828
$949$	0	0
$950$	6.00000	0.194666
$951$	−6.00000	−0.194563
$952$	10.2462	0.332082
$953$	42.9848	1.39242	0.696208	−	0.717840i	$-0.254869\pi$
$953$	42.9848	1.39242	0.696208	+	0.717840i	$0.254869\pi$
$954$	11.3693	0.368095
$955$	16.4924	0.533682
$956$	28.4924	0.921511
$957$	−6.24621	−0.201911
$958$	20.4924	0.662080
$959$	−115.231	−3.72100
$960$	−1.00000	−0.0322749
$961$	−4.75379	−0.153348
$962$	0	0
$963$	−8.00000	−0.257796
$964$	2.24621	0.0723456
$965$	16.8769	0.543286
$966$	16.0000	0.514792
$967$	6.38447	0.205311	0.102655	−	0.994717i	$-0.467266\pi$
$967$	6.38447	0.205311	0.102655	+	0.994717i	$0.467266\pi$
$968$	−1.24621	−0.0400547
$969$	−12.0000	−0.385496
$970$	−4.87689	−0.156588
$971$	54.2462	1.74084	0.870422	−	0.492307i	$-0.163846\pi$
$971$	54.2462	1.74084	0.870422	+	0.492307i	$0.163846\pi$
$972$	1.00000	0.0320750
$973$	−84.4924	−2.70870
$974$	7.36932	0.236128
$975$	0	0
$976$	10.0000	0.320092
$977$	−44.7386	−1.43132	−0.715658	−	0.698451i	$-0.753873\pi$
$977$	−44.7386	−1.43132	−0.715658	+	0.698451i	$0.753873\pi$
$978$	1.12311	0.0359130
$979$	16.0000	0.511362
$980$	−19.2462	−0.614798
$981$	−11.1231	−0.355133
$982$	10.7386	0.342684
$983$	−27.5076	−0.877355	−0.438678	−	0.898644i	$-0.644553\pi$
$983$	−27.5076	−0.877355	−0.438678	+	0.898644i	$0.644553\pi$
$984$	−9.12311	−0.290834
$985$	−0.246211	−0.00784494
$986$	−4.00000	−0.127386
$987$	−52.4924	−1.67085
$988$	0	0
$989$	−32.0000	−1.01754
$990$	3.12311	0.0992588
$991$	18.7386	0.595252	0.297626	−	0.954682i	$-0.403805\pi$
$991$	18.7386	0.595252	0.297626	+	0.954682i	$0.403805\pi$
$992$	5.12311	0.162659
$993$	24.2462	0.769430
$994$	−32.0000	−1.01498
$995$	1.75379	0.0555988
$996$	−10.2462	−0.324664
$997$	−52.3542	−1.65807	−0.829036	−	0.559195i	$-0.811110\pi$
$997$	−52.3542	−1.65807	−0.829036	+	0.559195i	$0.811110\pi$
$998$	−1.50758	−0.0477215
$999$	3.12311	0.0988107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.bh.1.1		2
13.5	odd	4		390.2.b.d.181.1	✓	4
13.8	odd	4		390.2.b.d.181.4	yes	4
13.12	even	2		5070.2.a.bd.1.2		2
39.5	even	4		1170.2.b.f.181.3		4
39.8	even	4		1170.2.b.f.181.2		4
52.31	even	4		3120.2.g.o.961.4		4
52.47	even	4		3120.2.g.o.961.1		4
65.8	even	4		1950.2.f.o.649.4		4
65.18	even	4		1950.2.f.l.649.3		4
65.34	odd	4		1950.2.b.h.1351.1		4
65.44	odd	4		1950.2.b.h.1351.4		4
65.47	even	4		1950.2.f.l.649.1		4
65.57	even	4		1950.2.f.o.649.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.b.d.181.1	✓	4	13.5	odd	4
390.2.b.d.181.4	yes	4	13.8	odd	4
1170.2.b.f.181.2		4	39.8	even	4
1170.2.b.f.181.3		4	39.5	even	4
1950.2.b.h.1351.1		4	65.34	odd	4
1950.2.b.h.1351.4		4	65.44	odd	4
1950.2.f.l.649.1		4	65.47	even	4
1950.2.f.l.649.3		4	65.18	even	4
1950.2.f.o.649.2		4	65.57	even	4
1950.2.f.o.649.4		4	65.8	even	4
3120.2.g.o.961.1		4	52.47	even	4
3120.2.g.o.961.4		4	52.31	even	4
5070.2.a.bd.1.2		2	13.12	even	2
5070.2.a.bh.1.1		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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -5.12311 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -5.12311 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.12311 q^{11} +1.00000 q^{12} -5.12311 q^{14} -1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +6.00000 q^{19} -1.00000 q^{20} -5.12311 q^{21} -3.12311 q^{22} -3.12311 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -5.12311 q^{28} +2.00000 q^{29} -1.00000 q^{30} +5.12311 q^{31} +1.00000 q^{32} -3.12311 q^{33} -2.00000 q^{34} +5.12311 q^{35} +1.00000 q^{36} +3.12311 q^{37} +6.00000 q^{38} -1.00000 q^{40} -9.12311 q^{41} -5.12311 q^{42} +10.2462 q^{43} -3.12311 q^{44} -1.00000 q^{45} -3.12311 q^{46} +10.2462 q^{47} +1.00000 q^{48} +19.2462 q^{49} +1.00000 q^{50} -2.00000 q^{51} +11.3693 q^{53} +1.00000 q^{54} +3.12311 q^{55} -5.12311 q^{56} +6.00000 q^{57} +2.00000 q^{58} +7.12311 q^{59} -1.00000 q^{60} +10.0000 q^{61} +5.12311 q^{62} -5.12311 q^{63} +1.00000 q^{64} -3.12311 q^{66} -13.1231 q^{67} -2.00000 q^{68} -3.12311 q^{69} +5.12311 q^{70} +6.24621 q^{71} +1.00000 q^{72} -4.87689 q^{73} +3.12311 q^{74} +1.00000 q^{75} +6.00000 q^{76} +16.0000 q^{77} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -9.12311 q^{82} -10.2462 q^{83} -5.12311 q^{84} +2.00000 q^{85} +10.2462 q^{86} +2.00000 q^{87} -3.12311 q^{88} -5.12311 q^{89} -1.00000 q^{90} -3.12311 q^{92} +5.12311 q^{93} +10.2462 q^{94} -6.00000 q^{95} +1.00000 q^{96} +4.87689 q^{97} +19.2462 q^{98} -3.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 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	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} + 2 q^{12} - 2 q^{14} - 2 q^{15} + 2 q^{16} - 4 q^{17} + 2 q^{18} + 12 q^{19} - 2 q^{20} - 2 q^{21} + 2 q^{22} + 2 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{27} - 2 q^{28} + 4 q^{29} - 2 q^{30} + 2 q^{31} + 2 q^{32} + 2 q^{33} - 4 q^{34} + 2 q^{35} + 2 q^{36} - 2 q^{37} + 12 q^{38} - 2 q^{40} - 10 q^{41} - 2 q^{42} + 4 q^{43} + 2 q^{44} - 2 q^{45} + 2 q^{46} + 4 q^{47} + 2 q^{48} + 22 q^{49} + 2 q^{50} - 4 q^{51} - 2 q^{53} + 2 q^{54} - 2 q^{55} - 2 q^{56} + 12 q^{57} + 4 q^{58} + 6 q^{59} - 2 q^{60} + 20 q^{61} + 2 q^{62} - 2 q^{63} + 2 q^{64} + 2 q^{66} - 18 q^{67} - 4 q^{68} + 2 q^{69} + 2 q^{70} - 4 q^{71} + 2 q^{72} - 18 q^{73} - 2 q^{74} + 2 q^{75} + 12 q^{76} + 32 q^{77} + 16 q^{79} - 2 q^{80} + 2 q^{81} - 10 q^{82} - 4 q^{83} - 2 q^{84} + 4 q^{85} + 4 q^{86} + 4 q^{87} + 2 q^{88} - 2 q^{89} - 2 q^{90} + 2 q^{92} + 2 q^{93} + 4 q^{94} - 12 q^{95} + 2 q^{96} + 18 q^{97} + 22 q^{98} + 2 q^{99}+O(q^{100}) \) 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^14 - 2 * q^15 + 2 * q^16 - 4 * q^17 + 2 * q^18 + 12 * q^19 - 2 * q^20 - 2 * q^21 + 2 * q^22 + 2 * q^23 + 2 * q^24 + 2 * q^25 + 2 * q^27 - 2 * q^28 + 4 * q^29 - 2 * q^30 + 2 * q^31 + 2 * q^32 + 2 * q^33 - 4 * q^34 + 2 * q^35 + 2 * q^36 - 2 * q^37 + 12 * q^38 - 2 * q^40 - 10 * q^41 - 2 * q^42 + 4 * q^43 + 2 * q^44 - 2 * q^45 + 2 * q^46 + 4 * q^47 + 2 * q^48 + 22 * q^49 + 2 * q^50 - 4 * q^51 - 2 * q^53 + 2 * q^54 - 2 * q^55 - 2 * q^56 + 12 * q^57 + 4 * q^58 + 6 * q^59 - 2 * q^60 + 20 * q^61 + 2 * q^62 - 2 * q^63 + 2 * q^64 + 2 * q^66 - 18 * q^67 - 4 * q^68 + 2 * q^69 + 2 * q^70 - 4 * q^71 + 2 * q^72 - 18 * q^73 - 2 * q^74 + 2 * q^75 + 12 * q^76 + 32 * q^77 + 16 * q^79 - 2 * q^80 + 2 * q^81 - 10 * q^82 - 4 * q^83 - 2 * q^84 + 4 * q^85 + 4 * q^86 + 4 * q^87 + 2 * q^88 - 2 * q^89 - 2 * q^90 + 2 * q^92 + 2 * q^93 + 4 * q^94 - 12 * q^95 + 2 * q^96 + 18 * q^97 + 22 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more