LMFDB - Embedded newform 5070.2.a.bq.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:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.24698$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +0.307979 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +0.307979 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +0.335126 q^{11} +1.00000 q^{12} -0.307979 q^{14} +1.00000 q^{15} +1.00000 q^{16} -6.85086 q^{17} -1.00000 q^{18} +5.80194 q^{19} +1.00000 q^{20} +0.307979 q^{21} -0.335126 q^{22} +2.35690 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +0.307979 q^{28} -5.91185 q^{29} -1.00000 q^{30} +0.0609989 q^{31} -1.00000 q^{32} +0.335126 q^{33} +6.85086 q^{34} +0.307979 q^{35} +1.00000 q^{36} +7.07606 q^{37} -5.80194 q^{38} -1.00000 q^{40} +3.40581 q^{41} -0.307979 q^{42} -0.0489173 q^{43} +0.335126 q^{44} +1.00000 q^{45} -2.35690 q^{46} +7.00000 q^{47} +1.00000 q^{48} -6.90515 q^{49} -1.00000 q^{50} -6.85086 q^{51} +12.1739 q^{53} -1.00000 q^{54} +0.335126 q^{55} -0.307979 q^{56} +5.80194 q^{57} +5.91185 q^{58} -13.1347 q^{59} +1.00000 q^{60} +2.14675 q^{61} -0.0609989 q^{62} +0.307979 q^{63} +1.00000 q^{64} -0.335126 q^{66} +11.9608 q^{67} -6.85086 q^{68} +2.35690 q^{69} -0.307979 q^{70} +9.56465 q^{71} -1.00000 q^{72} +2.31336 q^{73} -7.07606 q^{74} +1.00000 q^{75} +5.80194 q^{76} +0.103211 q^{77} -0.0760644 q^{79} +1.00000 q^{80} +1.00000 q^{81} -3.40581 q^{82} +3.84117 q^{83} +0.307979 q^{84} -6.85086 q^{85} +0.0489173 q^{86} -5.91185 q^{87} -0.335126 q^{88} -4.41119 q^{89} -1.00000 q^{90} +2.35690 q^{92} +0.0609989 q^{93} -7.00000 q^{94} +5.80194 q^{95} -1.00000 q^{96} +6.13169 q^{97} +6.90515 q^{98} +0.335126 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 6 * q^7 - 3 * q^8 + 3 * q^9	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{10} + 3 q^{12} - 6 q^{14} + 3 q^{15} + 3 q^{16} - 7 q^{17} - 3 q^{18} + 13 q^{19} + 3 q^{20} + 6 q^{21} + 3 q^{23} - 3 q^{24} + 3 q^{25} + 3 q^{27} + 6 q^{28} - 14 q^{29} - 3 q^{30} + 10 q^{31} - 3 q^{32} + 7 q^{34} + 6 q^{35} + 3 q^{36} + 6 q^{37} - 13 q^{38} - 3 q^{40} - 3 q^{41} - 6 q^{42} + 9 q^{43} + 3 q^{45} - 3 q^{46} + 21 q^{47} + 3 q^{48} + 5 q^{49} - 3 q^{50} - 7 q^{51} + 3 q^{53} - 3 q^{54} - 6 q^{56} + 13 q^{57} + 14 q^{58} + 7 q^{59} + 3 q^{60} - 21 q^{61} - 10 q^{62} + 6 q^{63} + 3 q^{64} + 23 q^{67} - 7 q^{68} + 3 q^{69} - 6 q^{70} + 7 q^{71} - 3 q^{72} + 9 q^{73} - 6 q^{74} + 3 q^{75} + 13 q^{76} - 21 q^{77} + 15 q^{79} + 3 q^{80} + 3 q^{81} + 3 q^{82} + 20 q^{83} + 6 q^{84} - 7 q^{85} - 9 q^{86} - 14 q^{87} + 3 q^{89} - 3 q^{90} + 3 q^{92} + 10 q^{93} - 21 q^{94} + 13 q^{95} - 3 q^{96} + 16 q^{97} - 5 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 6 * q^7 - 3 * q^8 + 3 * q^9 - 3 * q^10 + 3 * q^12 - 6 * q^14 + 3 * q^15 + 3 * q^16 - 7 * q^17 - 3 * q^18 + 13 * q^19 + 3 * q^20 + 6 * q^21 + 3 * q^23 - 3 * q^24 + 3 * q^25 + 3 * q^27 + 6 * q^28 - 14 * q^29 - 3 * q^30 + 10 * q^31 - 3 * q^32 + 7 * q^34 + 6 * q^35 + 3 * q^36 + 6 * q^37 - 13 * q^38 - 3 * q^40 - 3 * q^41 - 6 * q^42 + 9 * q^43 + 3 * q^45 - 3 * q^46 + 21 * q^47 + 3 * q^48 + 5 * q^49 - 3 * q^50 - 7 * q^51 + 3 * q^53 - 3 * q^54 - 6 * q^56 + 13 * q^57 + 14 * q^58 + 7 * q^59 + 3 * q^60 - 21 * q^61 - 10 * q^62 + 6 * q^63 + 3 * q^64 + 23 * q^67 - 7 * q^68 + 3 * q^69 - 6 * q^70 + 7 * q^71 - 3 * q^72 + 9 * q^73 - 6 * q^74 + 3 * q^75 + 13 * q^76 - 21 * q^77 + 15 * q^79 + 3 * q^80 + 3 * q^81 + 3 * q^82 + 20 * q^83 + 6 * q^84 - 7 * q^85 - 9 * q^86 - 14 * q^87 + 3 * q^89 - 3 * q^90 + 3 * q^92 + 10 * q^93 - 21 * q^94 + 13 * q^95 - 3 * q^96 + 16 * q^97 - 5 * q^98

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
$7$	0.307979	0.116405	0.0582025	−	0.998305i	$-0.481463\pi$
$7$	0.307979	0.116405	0.0582025	+	0.998305i	$0.481463\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	0.335126	0.101044	0.0505221	−	0.998723i	$-0.483911\pi$
$11$	0.335126	0.101044	0.0505221	+	0.998723i	$0.483911\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−0.307979	−0.0823107
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−6.85086	−1.66158	−0.830788	−	0.556589i	$-0.812110\pi$
$17$	−6.85086	−1.66158	−0.830788	+	0.556589i	$0.812110\pi$
$18$	−1.00000	−0.235702
$19$	5.80194	1.33106	0.665528	−	0.746373i	$-0.268206\pi$
$19$	5.80194	1.33106	0.665528	+	0.746373i	$0.268206\pi$
$20$	1.00000	0.223607
$21$	0.307979	0.0672064
$22$	−0.335126	−0.0714490
$23$	2.35690	0.491447	0.245723	−	0.969340i	$-0.420975\pi$
$23$	2.35690	0.491447	0.245723	+	0.969340i	$0.420975\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0.307979	0.0582025
$29$	−5.91185	−1.09780	−0.548902	−	0.835887i	$-0.684954\pi$
$29$	−5.91185	−1.09780	−0.548902	+	0.835887i	$0.684954\pi$
$30$	−1.00000	−0.182574
$31$	0.0609989	0.0109557	0.00547787	−	0.999985i	$-0.498256\pi$
$31$	0.0609989	0.0109557	0.00547787	+	0.999985i	$0.498256\pi$
$32$	−1.00000	−0.176777
$33$	0.335126	0.0583379
$34$	6.85086	1.17491
$35$	0.307979	0.0520579
$36$	1.00000	0.166667
$37$	7.07606	1.16330	0.581649	−	0.813440i	$-0.302408\pi$
$37$	7.07606	1.16330	0.581649	+	0.813440i	$0.302408\pi$
$38$	−5.80194	−0.941199
$39$	0	0
$40$	−1.00000	−0.158114
$41$	3.40581	0.531899	0.265949	−	0.963987i	$-0.414315\pi$
$41$	3.40581	0.531899	0.265949	+	0.963987i	$0.414315\pi$
$42$	−0.307979	−0.0475221
$43$	−0.0489173	−0.00745982	−0.00372991	−	0.999993i	$-0.501187\pi$
$43$	−0.0489173	−0.00745982	−0.00372991	+	0.999993i	$0.501187\pi$
$44$	0.335126	0.0505221
$45$	1.00000	0.149071
$46$	−2.35690	−0.347505
$47$	7.00000	1.02105	0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000	1.02105	0.510527	+	0.859861i	$0.329450\pi$
$48$	1.00000	0.144338
$49$	−6.90515	−0.986450
$50$	−1.00000	−0.141421
$51$	−6.85086	−0.959312
$52$	0	0
$53$	12.1739	1.67221	0.836107	−	0.548567i	$-0.184826\pi$
$53$	12.1739	1.67221	0.836107	+	0.548567i	$0.184826\pi$
$54$	−1.00000	−0.136083
$55$	0.335126	0.0451883
$56$	−0.307979	−0.0411554
$57$	5.80194	0.768485
$58$	5.91185	0.776264
$59$	−13.1347	−1.70999	−0.854994	−	0.518638i	$-0.826439\pi$
$59$	−13.1347	−1.70999	−0.854994	+	0.518638i	$0.826439\pi$
$60$	1.00000	0.129099
$61$	2.14675	0.274863	0.137432	−	0.990511i	$-0.456115\pi$
$61$	2.14675	0.274863	0.137432	+	0.990511i	$0.456115\pi$
$62$	−0.0609989	−0.00774687
$63$	0.307979	0.0388016
$64$	1.00000	0.125000
$65$	0	0
$66$	−0.335126	−0.0412511
$67$	11.9608	1.46124	0.730620	−	0.682784i	$-0.239231\pi$
$67$	11.9608	1.46124	0.730620	+	0.682784i	$0.239231\pi$
$68$	−6.85086	−0.830788
$69$	2.35690	0.283737
$70$	−0.307979	−0.0368105
$71$	9.56465	1.13511	0.567557	−	0.823334i	$-0.307889\pi$
$71$	9.56465	1.13511	0.567557	+	0.823334i	$0.307889\pi$
$72$	−1.00000	−0.117851
$73$	2.31336	0.270758	0.135379	−	0.990794i	$-0.456775\pi$
$73$	2.31336	0.270758	0.135379	+	0.990794i	$0.456775\pi$
$74$	−7.07606	−0.822576
$75$	1.00000	0.115470
$76$	5.80194	0.665528
$77$	0.103211	0.0117620
$78$	0	0
$79$	−0.0760644	−0.00855792	−0.00427896	−	0.999991i	$-0.501362\pi$
$79$	−0.0760644	−0.00855792	−0.00427896	+	0.999991i	$0.501362\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−3.40581	−0.376109
$83$	3.84117	0.421623	0.210811	−	0.977527i	$-0.432389\pi$
$83$	3.84117	0.421623	0.210811	+	0.977527i	$0.432389\pi$
$84$	0.307979	0.0336032
$85$	−6.85086	−0.743080
$86$	0.0489173	0.00527489
$87$	−5.91185	−0.633817
$88$	−0.335126	−0.0357245
$89$	−4.41119	−0.467585	−0.233793	−	0.972286i	$-0.575114\pi$
$89$	−4.41119	−0.467585	−0.233793	+	0.972286i	$0.575114\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	2.35690	0.245723
$93$	0.0609989	0.00632529
$94$	−7.00000	−0.721995
$95$	5.80194	0.595266
$96$	−1.00000	−0.102062
$97$	6.13169	0.622578	0.311289	−	0.950315i	$-0.399239\pi$
$97$	6.13169	0.622578	0.311289	+	0.950315i	$0.399239\pi$
$98$	6.90515	0.697525
$99$	0.335126	0.0336814
$100$	1.00000	0.100000
$101$	1.32975	0.132315	0.0661575	−	0.997809i	$-0.478926\pi$
$101$	1.32975	0.132315	0.0661575	+	0.997809i	$0.478926\pi$
$102$	6.85086	0.678336
$103$	8.92154	0.879066	0.439533	−	0.898227i	$-0.355144\pi$
$103$	8.92154	0.879066	0.439533	+	0.898227i	$0.355144\pi$
$104$	0	0
$105$	0.307979	0.0300556
$106$	−12.1739	−1.18243
$107$	14.4644	1.39833	0.699164	−	0.714961i	$-0.253556\pi$
$107$	14.4644	1.39833	0.699164	+	0.714961i	$0.253556\pi$
$108$	1.00000	0.0962250
$109$	7.13706	0.683607	0.341803	−	0.939772i	$-0.388962\pi$
$109$	7.13706	0.683607	0.341803	+	0.939772i	$0.388962\pi$
$110$	−0.335126	−0.0319530
$111$	7.07606	0.671630
$112$	0.307979	0.0291012
$113$	−16.3424	−1.53737	−0.768683	−	0.639630i	$-0.779088\pi$
$113$	−16.3424	−1.53737	−0.768683	+	0.639630i	$0.779088\pi$
$114$	−5.80194	−0.543401
$115$	2.35690	0.219782
$116$	−5.91185	−0.548902
$117$	0	0
$118$	13.1347	1.20914
$119$	−2.10992	−0.193416
$120$	−1.00000	−0.0912871
$121$	−10.8877	−0.989790
$122$	−2.14675	−0.194358
$123$	3.40581	0.307092
$124$	0.0609989	0.00547787
$125$	1.00000	0.0894427
$126$	−0.307979	−0.0274369
$127$	−0.911854	−0.0809140	−0.0404570	−	0.999181i	$-0.512881\pi$
$127$	−0.911854	−0.0809140	−0.0404570	+	0.999181i	$0.512881\pi$
$128$	−1.00000	−0.0883883
$129$	−0.0489173	−0.00430693
$130$	0	0
$131$	13.0368	1.13903	0.569517	−	0.821980i	$-0.307130\pi$
$131$	13.0368	1.13903	0.569517	+	0.821980i	$0.307130\pi$
$132$	0.335126	0.0291689
$133$	1.78687	0.154941
$134$	−11.9608	−1.03325
$135$	1.00000	0.0860663
$136$	6.85086	0.587456
$137$	12.8877	1.10107	0.550535	−	0.834812i	$-0.314424\pi$
$137$	12.8877	1.10107	0.550535	+	0.834812i	$0.314424\pi$
$138$	−2.35690	−0.200632
$139$	−2.01208	−0.170663	−0.0853313	−	0.996353i	$-0.527195\pi$
$139$	−2.01208	−0.170663	−0.0853313	+	0.996353i	$0.527195\pi$
$140$	0.307979	0.0260289
$141$	7.00000	0.589506
$142$	−9.56465	−0.802647
$143$	0	0
$144$	1.00000	0.0833333
$145$	−5.91185	−0.490953
$146$	−2.31336	−0.191455
$147$	−6.90515	−0.569527
$148$	7.07606	0.581649
$149$	3.44935	0.282582	0.141291	−	0.989968i	$-0.454875\pi$
$149$	3.44935	0.282582	0.141291	+	0.989968i	$0.454875\pi$
$150$	−1.00000	−0.0816497
$151$	2.84117	0.231211	0.115605	−	0.993295i	$-0.463119\pi$
$151$	2.84117	0.231211	0.115605	+	0.993295i	$0.463119\pi$
$152$	−5.80194	−0.470599
$153$	−6.85086	−0.553859
$154$	−0.103211	−0.00831702
$155$	0.0609989	0.00489955
$156$	0	0
$157$	16.2228	1.29472	0.647361	−	0.762184i	$-0.275873\pi$
$157$	16.2228	1.29472	0.647361	+	0.762184i	$0.275873\pi$
$158$	0.0760644	0.00605136
$159$	12.1739	0.965453
$160$	−1.00000	−0.0790569
$161$	0.725873	0.0572068
$162$	−1.00000	−0.0785674
$163$	−0.0924579	−0.00724186	−0.00362093	−	0.999993i	$-0.501153\pi$
$163$	−0.0924579	−0.00724186	−0.00362093	+	0.999993i	$0.501153\pi$
$164$	3.40581	0.265949
$165$	0.335126	0.0260895
$166$	−3.84117	−0.298132
$167$	−9.64071	−0.746021	−0.373010	−	0.927827i	$-0.621674\pi$
$167$	−9.64071	−0.746021	−0.373010	+	0.927827i	$0.621674\pi$
$168$	−0.307979	−0.0237611
$169$	0	0
$170$	6.85086	0.525437
$171$	5.80194	0.443685
$172$	−0.0489173	−0.00372991
$173$	0.719169	0.0546774	0.0273387	−	0.999626i	$-0.491297\pi$
$173$	0.719169	0.0546774	0.0273387	+	0.999626i	$0.491297\pi$
$174$	5.91185	0.448176
$175$	0.307979	0.0232810
$176$	0.335126	0.0252610
$177$	−13.1347	−0.987262
$178$	4.41119	0.330633
$179$	−2.01208	−0.150390	−0.0751950	−	0.997169i	$-0.523958\pi$
$179$	−2.01208	−0.150390	−0.0751950	+	0.997169i	$0.523958\pi$
$180$	1.00000	0.0745356
$181$	−2.57135	−0.191127	−0.0955635	−	0.995423i	$-0.530465\pi$
$181$	−2.57135	−0.191127	−0.0955635	+	0.995423i	$0.530465\pi$
$182$	0	0
$183$	2.14675	0.158692
$184$	−2.35690	−0.173753
$185$	7.07606	0.520243
$186$	−0.0609989	−0.00447266
$187$	−2.29590	−0.167893
$188$	7.00000	0.510527
$189$	0.307979	0.0224021
$190$	−5.80194	−0.420917
$191$	3.80194	0.275099	0.137549	−	0.990495i	$-0.456077\pi$
$191$	3.80194	0.275099	0.137549	+	0.990495i	$0.456077\pi$
$192$	1.00000	0.0721688
$193$	−8.86294	−0.637968	−0.318984	−	0.947760i	$-0.603342\pi$
$193$	−8.86294	−0.637968	−0.318984	+	0.947760i	$0.603342\pi$
$194$	−6.13169	−0.440229
$195$	0	0
$196$	−6.90515	−0.493225
$197$	−22.6504	−1.61377	−0.806887	−	0.590706i	$-0.798849\pi$
$197$	−22.6504	−1.61377	−0.806887	+	0.590706i	$0.798849\pi$
$198$	−0.335126	−0.0238163
$199$	−5.46681	−0.387532	−0.193766	−	0.981048i	$-0.562070\pi$
$199$	−5.46681	−0.387532	−0.193766	+	0.981048i	$0.562070\pi$
$200$	−1.00000	−0.0707107
$201$	11.9608	0.843648
$202$	−1.32975	−0.0935608
$203$	−1.82072	−0.127790
$204$	−6.85086	−0.479656
$205$	3.40581	0.237872
$206$	−8.92154	−0.621593
$207$	2.35690	0.163816
$208$	0	0
$209$	1.94438	0.134495
$210$	−0.307979	−0.0212525
$211$	11.0422	0.760177	0.380089	−	0.924950i	$-0.375893\pi$
$211$	11.0422	0.760177	0.380089	+	0.924950i	$0.375893\pi$
$212$	12.1739	0.836107
$213$	9.56465	0.655359
$214$	−14.4644	−0.988767
$215$	−0.0489173	−0.00333613
$216$	−1.00000	−0.0680414
$217$	0.0187864	0.00127530
$218$	−7.13706	−0.483383
$219$	2.31336	0.156322
$220$	0.335126	0.0225942
$221$	0	0
$222$	−7.07606	−0.474914
$223$	18.3884	1.23138	0.615688	−	0.787990i	$-0.288878\pi$
$223$	18.3884	1.23138	0.615688	+	0.787990i	$0.288878\pi$
$224$	−0.307979	−0.0205777
$225$	1.00000	0.0666667
$226$	16.3424	1.08708
$227$	12.0073	0.796952	0.398476	−	0.917179i	$-0.369539\pi$
$227$	12.0073	0.796952	0.398476	+	0.917179i	$0.369539\pi$
$228$	5.80194	0.384243
$229$	25.4601	1.68245	0.841226	−	0.540684i	$-0.181835\pi$
$229$	25.4601	1.68245	0.841226	+	0.540684i	$0.181835\pi$
$230$	−2.35690	−0.155409
$231$	0.103211	0.00679082
$232$	5.91185	0.388132
$233$	−2.10992	−0.138225	−0.0691126	−	0.997609i	$-0.522017\pi$
$233$	−2.10992	−0.138225	−0.0691126	+	0.997609i	$0.522017\pi$
$234$	0	0
$235$	7.00000	0.456630
$236$	−13.1347	−0.854994
$237$	−0.0760644	−0.00494091
$238$	2.10992	0.136766
$239$	3.95838	0.256046	0.128023	−	0.991771i	$-0.459137\pi$
$239$	3.95838	0.256046	0.128023	+	0.991771i	$0.459137\pi$
$240$	1.00000	0.0645497
$241$	−23.1564	−1.49164	−0.745819	−	0.666149i	$-0.767942\pi$
$241$	−23.1564	−1.49164	−0.745819	+	0.666149i	$0.767942\pi$
$242$	10.8877	0.699887
$243$	1.00000	0.0641500
$244$	2.14675	0.137432
$245$	−6.90515	−0.441154
$246$	−3.40581	−0.217147
$247$	0	0
$248$	−0.0609989	−0.00387344
$249$	3.84117	0.243424
$250$	−1.00000	−0.0632456
$251$	−5.54048	−0.349712	−0.174856	−	0.984594i	$-0.555946\pi$
$251$	−5.54048	−0.349712	−0.174856	+	0.984594i	$0.555946\pi$
$252$	0.307979	0.0194008
$253$	0.789856	0.0496578
$254$	0.911854	0.0572148
$255$	−6.85086	−0.429017
$256$	1.00000	0.0625000
$257$	6.60819	0.412207	0.206104	−	0.978530i	$-0.433922\pi$
$257$	6.60819	0.412207	0.206104	+	0.978530i	$0.433922\pi$
$258$	0.0489173	0.00304546
$259$	2.17928	0.135414
$260$	0	0
$261$	−5.91185	−0.365935
$262$	−13.0368	−0.805418
$263$	−0.660563	−0.0407320	−0.0203660	−	0.999793i	$-0.506483\pi$
$263$	−0.660563	−0.0407320	−0.0203660	+	0.999793i	$0.506483\pi$
$264$	−0.335126	−0.0206256
$265$	12.1739	0.747837
$266$	−1.78687	−0.109560
$267$	−4.41119	−0.269960
$268$	11.9608	0.730620
$269$	−26.7928	−1.63359	−0.816794	−	0.576929i	$-0.804251\pi$
$269$	−26.7928	−1.63359	−0.816794	+	0.576929i	$0.804251\pi$
$270$	−1.00000	−0.0608581
$271$	14.0151	0.851355	0.425677	−	0.904875i	$-0.360036\pi$
$271$	14.0151	0.851355	0.425677	+	0.904875i	$0.360036\pi$
$272$	−6.85086	−0.415394
$273$	0	0
$274$	−12.8877	−0.778574
$275$	0.335126	0.0202088
$276$	2.35690	0.141868
$277$	−23.4252	−1.40748	−0.703742	−	0.710456i	$-0.748489\pi$
$277$	−23.4252	−1.40748	−0.703742	+	0.710456i	$0.748489\pi$
$278$	2.01208	0.120677
$279$	0.0609989	0.00365191
$280$	−0.307979	−0.0184052
$281$	8.40044	0.501128	0.250564	−	0.968100i	$-0.419384\pi$
$281$	8.40044	0.501128	0.250564	+	0.968100i	$0.419384\pi$
$282$	−7.00000	−0.416844
$283$	−31.6142	−1.87927	−0.939633	−	0.342183i	$-0.888834\pi$
$283$	−31.6142	−1.87927	−0.939633	+	0.342183i	$0.888834\pi$
$284$	9.56465	0.567557
$285$	5.80194	0.343677
$286$	0	0
$287$	1.04892	0.0619156
$288$	−1.00000	−0.0589256
$289$	29.9342	1.76084
$290$	5.91185	0.347156
$291$	6.13169	0.359446
$292$	2.31336	0.135379
$293$	−17.5254	−1.02385	−0.511923	−	0.859031i	$-0.671067\pi$
$293$	−17.5254	−1.02385	−0.511923	+	0.859031i	$0.671067\pi$
$294$	6.90515	0.402716
$295$	−13.1347	−0.764730
$296$	−7.07606	−0.411288
$297$	0.335126	0.0194460
$298$	−3.44935	−0.199816
$299$	0	0
$300$	1.00000	0.0577350
$301$	−0.0150655	−0.000868360 0
$302$	−2.84117	−0.163491
$303$	1.32975	0.0763921
$304$	5.80194	0.332764
$305$	2.14675	0.122923
$306$	6.85086	0.391637
$307$	20.6732	1.17988	0.589942	−	0.807446i	$-0.299151\pi$
$307$	20.6732	1.17988	0.589942	+	0.807446i	$0.299151\pi$
$308$	0.103211	0.00588102
$309$	8.92154	0.507529
$310$	−0.0609989	−0.00346451
$311$	−22.0248	−1.24891	−0.624455	−	0.781061i	$-0.714679\pi$
$311$	−22.0248	−1.24891	−0.624455	+	0.781061i	$0.714679\pi$
$312$	0	0
$313$	21.7192	1.22764	0.613820	−	0.789446i	$-0.289632\pi$
$313$	21.7192	1.22764	0.613820	+	0.789446i	$0.289632\pi$
$314$	−16.2228	−0.915506
$315$	0.307979	0.0173526
$316$	−0.0760644	−0.00427896
$317$	−16.4644	−0.924734	−0.462367	−	0.886689i	$-0.653000\pi$
$317$	−16.4644	−0.924734	−0.462367	+	0.886689i	$0.653000\pi$
$318$	−12.1739	−0.682678
$319$	−1.98121	−0.110927
$320$	1.00000	0.0559017
$321$	14.4644	0.807325
$322$	−0.725873	−0.0404513
$323$	−39.7482	−2.21165
$324$	1.00000	0.0555556
$325$	0	0
$326$	0.0924579	0.00512077
$327$	7.13706	0.394681
$328$	−3.40581	−0.188055
$329$	2.15585	0.118856
$330$	−0.335126	−0.0184481
$331$	26.4373	1.45312	0.726562	−	0.687101i	$-0.241117\pi$
$331$	26.4373	1.45312	0.726562	+	0.687101i	$0.241117\pi$
$332$	3.84117	0.210811
$333$	7.07606	0.387766
$334$	9.64071	0.527516
$335$	11.9608	0.653487
$336$	0.307979	0.0168016
$337$	27.4698	1.49638	0.748188	−	0.663487i	$-0.230924\pi$
$337$	27.4698	1.49638	0.748188	+	0.663487i	$0.230924\pi$
$338$	0	0
$339$	−16.3424	−0.887598
$340$	−6.85086	−0.371540
$341$	0.0204423	0.00110701
$342$	−5.80194	−0.313733
$343$	−4.28249	−0.231233
$344$	0.0489173	0.00263745
$345$	2.35690	0.126891
$346$	−0.719169	−0.0386627
$347$	−33.1323	−1.77863	−0.889317	−	0.457291i	$-0.848820\pi$
$347$	−33.1323	−1.77863	−0.889317	+	0.457291i	$0.848820\pi$
$348$	−5.91185	−0.316909
$349$	26.3739	1.41176	0.705881	−	0.708331i	$-0.250551\pi$
$349$	26.3739	1.41176	0.705881	+	0.708331i	$0.250551\pi$
$350$	−0.307979	−0.0164621
$351$	0	0
$352$	−0.335126	−0.0178623
$353$	21.0291	1.11926	0.559632	−	0.828741i	$-0.310942\pi$
$353$	21.0291	1.11926	0.559632	+	0.828741i	$0.310942\pi$
$354$	13.1347	0.698100
$355$	9.56465	0.507639
$356$	−4.41119	−0.233793
$357$	−2.10992	−0.111669
$358$	2.01208	0.106342
$359$	−30.8877	−1.63019	−0.815095	−	0.579327i	$-0.803315\pi$
$359$	−30.8877	−1.63019	−0.815095	+	0.579327i	$0.803315\pi$
$360$	−1.00000	−0.0527046
$361$	14.6625	0.771710
$362$	2.57135	0.135147
$363$	−10.8877	−0.571456
$364$	0	0
$365$	2.31336	0.121087
$366$	−2.14675	−0.112213
$367$	11.9584	0.624222	0.312111	−	0.950046i	$-0.398964\pi$
$367$	11.9584	0.624222	0.312111	+	0.950046i	$0.398964\pi$
$368$	2.35690	0.122862
$369$	3.40581	0.177300
$370$	−7.07606	−0.367867
$371$	3.74930	0.194654
$372$	0.0609989	0.00316265
$373$	25.0315	1.29608	0.648040	−	0.761606i	$-0.275589\pi$
$373$	25.0315	1.29608	0.648040	+	0.761606i	$0.275589\pi$
$374$	2.29590	0.118718
$375$	1.00000	0.0516398
$376$	−7.00000	−0.360997
$377$	0	0
$378$	−0.307979	−0.0158407
$379$	−34.4989	−1.77209	−0.886044	−	0.463602i	$-0.846557\pi$
$379$	−34.4989	−1.77209	−0.886044	+	0.463602i	$0.846557\pi$
$380$	5.80194	0.297633
$381$	−0.911854	−0.0467157
$382$	−3.80194	−0.194524
$383$	10.4644	0.534707	0.267353	−	0.963599i	$-0.413851\pi$
$383$	10.4644	0.534707	0.267353	+	0.963599i	$0.413851\pi$
$384$	−1.00000	−0.0510310
$385$	0.103211	0.00526014
$386$	8.86294	0.451112
$387$	−0.0489173	−0.00248661
$388$	6.13169	0.311289
$389$	25.2218	1.27879	0.639397	−	0.768877i	$-0.279184\pi$
$389$	25.2218	1.27879	0.639397	+	0.768877i	$0.279184\pi$
$390$	0	0
$391$	−16.1468	−0.816576
$392$	6.90515	0.348763
$393$	13.0368	0.657621
$394$	22.6504	1.14111
$395$	−0.0760644	−0.00382722
$396$	0.335126	0.0168407
$397$	21.3860	1.07333	0.536665	−	0.843795i	$-0.319684\pi$
$397$	21.3860	1.07333	0.536665	+	0.843795i	$0.319684\pi$
$398$	5.46681	0.274027
$399$	1.78687	0.0894555
$400$	1.00000	0.0500000
$401$	−11.8062	−0.589576	−0.294788	−	0.955563i	$-0.595249\pi$
$401$	−11.8062	−0.589576	−0.294788	+	0.955563i	$0.595249\pi$
$402$	−11.9608	−0.596549
$403$	0	0
$404$	1.32975	0.0661575
$405$	1.00000	0.0496904
$406$	1.82072	0.0903610
$407$	2.37137	0.117544
$408$	6.85086	0.339168
$409$	−5.65950	−0.279844	−0.139922	−	0.990163i	$-0.544685\pi$
$409$	−5.65950	−0.279844	−0.139922	+	0.990163i	$0.544685\pi$
$410$	−3.40581	−0.168201
$411$	12.8877	0.635703
$412$	8.92154	0.439533
$413$	−4.04520	−0.199051
$414$	−2.35690	−0.115835
$415$	3.84117	0.188555
$416$	0	0
$417$	−2.01208	−0.0985321
$418$	−1.94438	−0.0951026
$419$	17.0084	0.830913	0.415456	−	0.909613i	$-0.363622\pi$
$419$	17.0084	0.830913	0.415456	+	0.909613i	$0.363622\pi$
$420$	0.307979	0.0150278
$421$	−10.2851	−0.501267	−0.250634	−	0.968082i	$-0.580639\pi$
$421$	−10.2851	−0.501267	−0.250634	+	0.968082i	$0.580639\pi$
$422$	−11.0422	−0.537526
$423$	7.00000	0.340352
$424$	−12.1739	−0.591217
$425$	−6.85086	−0.332315
$426$	−9.56465	−0.463409
$427$	0.661154	0.0319955
$428$	14.4644	0.699164
$429$	0	0
$430$	0.0489173	0.00235900
$431$	−35.9657	−1.73241	−0.866203	−	0.499693i	$-0.833446\pi$
$431$	−35.9657	−1.73241	−0.866203	+	0.499693i	$0.833446\pi$
$432$	1.00000	0.0481125
$433$	7.63342	0.366839	0.183419	−	0.983035i	$-0.441283\pi$
$433$	7.63342	0.366839	0.183419	+	0.983035i	$0.441283\pi$
$434$	−0.0187864	−0.000901774 0
$435$	−5.91185	−0.283452
$436$	7.13706	0.341803
$437$	13.6746	0.654143
$438$	−2.31336	−0.110536
$439$	−34.4373	−1.64360	−0.821801	−	0.569775i	$-0.807030\pi$
$439$	−34.4373	−1.64360	−0.821801	+	0.569775i	$0.807030\pi$
$440$	−0.335126	−0.0159765
$441$	−6.90515	−0.328817
$442$	0	0
$443$	33.6431	1.59843	0.799216	−	0.601044i	$-0.205248\pi$
$443$	33.6431	1.59843	0.799216	+	0.601044i	$0.205248\pi$
$444$	7.07606	0.335815
$445$	−4.41119	−0.209110
$446$	−18.3884	−0.870714
$447$	3.44935	0.163149
$448$	0.307979	0.0145506
$449$	−27.9729	−1.32012	−0.660060	−	0.751213i	$-0.729469\pi$
$449$	−27.9729	−1.32012	−0.660060	+	0.751213i	$0.729469\pi$
$450$	−1.00000	−0.0471405
$451$	1.14138	0.0537453
$452$	−16.3424	−0.768683
$453$	2.84117	0.133490
$454$	−12.0073	−0.563530
$455$	0	0
$456$	−5.80194	−0.271701
$457$	−6.18060	−0.289116	−0.144558	−	0.989496i	$-0.546176\pi$
$457$	−6.18060	−0.289116	−0.144558	+	0.989496i	$0.546176\pi$
$458$	−25.4601	−1.18967
$459$	−6.85086	−0.319771
$460$	2.35690	0.109891
$461$	38.1062	1.77478	0.887391	−	0.461017i	$-0.152515\pi$
$461$	38.1062	1.77478	0.887391	+	0.461017i	$0.152515\pi$
$462$	−0.103211	−0.00480183
$463$	−2.06829	−0.0961218	−0.0480609	−	0.998844i	$-0.515304\pi$
$463$	−2.06829	−0.0961218	−0.0480609	+	0.998844i	$0.515304\pi$
$464$	−5.91185	−0.274451
$465$	0.0609989	0.00282876
$466$	2.10992	0.0977400
$467$	−4.27114	−0.197645	−0.0988225	−	0.995105i	$-0.531508\pi$
$467$	−4.27114	−0.197645	−0.0988225	+	0.995105i	$0.531508\pi$
$468$	0	0
$469$	3.68366	0.170096
$470$	−7.00000	−0.322886
$471$	16.2228	0.747508
$472$	13.1347	0.604572
$473$	−0.0163935	−0.000753772 0
$474$	0.0760644	0.00349375
$475$	5.80194	0.266211
$476$	−2.10992	−0.0967079
$477$	12.1739	0.557405
$478$	−3.95838	−0.181052
$479$	−38.7560	−1.77081	−0.885404	−	0.464823i	$-0.846118\pi$
$479$	−38.7560	−1.77081	−0.885404	+	0.464823i	$0.846118\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	23.1564	1.05475
$483$	0.725873	0.0330284
$484$	−10.8877	−0.494895
$485$	6.13169	0.278426
$486$	−1.00000	−0.0453609
$487$	−33.4620	−1.51631	−0.758155	−	0.652075i	$-0.773899\pi$
$487$	−33.4620	−1.51631	−0.758155	+	0.652075i	$0.773899\pi$
$488$	−2.14675	−0.0971789
$489$	−0.0924579	−0.00418109
$490$	6.90515	0.311943
$491$	35.1487	1.58624	0.793119	−	0.609067i	$-0.208456\pi$
$491$	35.1487	1.58624	0.793119	+	0.609067i	$0.208456\pi$
$492$	3.40581	0.153546
$493$	40.5013	1.82408
$494$	0	0
$495$	0.335126	0.0150628
$496$	0.0609989	0.00273893
$497$	2.94571	0.132133
$498$	−3.84117	−0.172127
$499$	−9.61463	−0.430410	−0.215205	−	0.976569i	$-0.569042\pi$
$499$	−9.61463	−0.430410	−0.215205	+	0.976569i	$0.569042\pi$
$500$	1.00000	0.0447214
$501$	−9.64071	−0.430715
$502$	5.54048	0.247284
$503$	18.5265	0.826055	0.413028	−	0.910719i	$-0.364471\pi$
$503$	18.5265	0.826055	0.413028	+	0.910719i	$0.364471\pi$
$504$	−0.307979	−0.0137185
$505$	1.32975	0.0591730
$506$	−0.789856	−0.0351134
$507$	0	0
$508$	−0.911854	−0.0404570
$509$	5.43296	0.240812	0.120406	−	0.992725i	$-0.461580\pi$
$509$	5.43296	0.240812	0.120406	+	0.992725i	$0.461580\pi$
$510$	6.85086	0.303361
$511$	0.712464	0.0315175
$512$	−1.00000	−0.0441942
$513$	5.80194	0.256162
$514$	−6.60819	−0.291475
$515$	8.92154	0.393130
$516$	−0.0489173	−0.00215347
$517$	2.34588	0.103172
$518$	−2.17928	−0.0957519
$519$	0.719169	0.0315680
$520$	0	0
$521$	−38.8001	−1.69986	−0.849932	−	0.526892i	$-0.823357\pi$
$521$	−38.8001	−1.69986	−0.849932	+	0.526892i	$0.823357\pi$
$522$	5.91185	0.258755
$523$	32.0629	1.40201	0.701007	−	0.713155i	$-0.252734\pi$
$523$	32.0629	1.40201	0.701007	+	0.713155i	$0.252734\pi$
$524$	13.0368	0.569517
$525$	0.307979	0.0134413
$526$	0.660563	0.0288019
$527$	−0.417895	−0.0182038
$528$	0.335126	0.0145845
$529$	−17.4450	−0.758480
$530$	−12.1739	−0.528800
$531$	−13.1347	−0.569996
$532$	1.78687	0.0774707
$533$	0	0
$534$	4.41119	0.190891
$535$	14.4644	0.625351
$536$	−11.9608	−0.516627
$537$	−2.01208	−0.0868277
$538$	26.7928	1.15512
$539$	−2.31409	−0.0996750
$540$	1.00000	0.0430331
$541$	31.8334	1.36862	0.684312	−	0.729189i	$-0.260103\pi$
$541$	31.8334	1.36862	0.684312	+	0.729189i	$0.260103\pi$
$542$	−14.0151	−0.601999
$543$	−2.57135	−0.110347
$544$	6.85086	0.293728
$545$	7.13706	0.305718
$546$	0	0
$547$	−44.4010	−1.89845	−0.949225	−	0.314597i	$-0.898131\pi$
$547$	−44.4010	−1.89845	−0.949225	+	0.314597i	$0.898131\pi$
$548$	12.8877	0.550535
$549$	2.14675	0.0916211
$550$	−0.335126	−0.0142898
$551$	−34.3002	−1.46124
$552$	−2.35690	−0.100316
$553$	−0.0234262	−0.000996184 0
$554$	23.4252	0.995241
$555$	7.07606	0.300362
$556$	−2.01208	−0.0853313
$557$	−7.73423	−0.327710	−0.163855	−	0.986484i	$-0.552393\pi$
$557$	−7.73423	−0.327710	−0.163855	+	0.986484i	$0.552393\pi$
$558$	−0.0609989	−0.00258229
$559$	0	0
$560$	0.307979	0.0130145
$561$	−2.29590	−0.0969328
$562$	−8.40044	−0.354351
$563$	44.5803	1.87884	0.939418	−	0.342774i	$-0.111367\pi$
$563$	44.5803	1.87884	0.939418	+	0.342774i	$0.111367\pi$
$564$	7.00000	0.294753
$565$	−16.3424	−0.687531
$566$	31.6142	1.32884
$567$	0.307979	0.0129339
$568$	−9.56465	−0.401324
$569$	24.9065	1.04413	0.522067	−	0.852905i	$-0.325161\pi$
$569$	24.9065	1.04413	0.522067	+	0.852905i	$0.325161\pi$
$570$	−5.80194	−0.243016
$571$	−0.873690	−0.0365628	−0.0182814	−	0.999833i	$-0.505819\pi$
$571$	−0.873690	−0.0365628	−0.0182814	+	0.999833i	$0.505819\pi$
$572$	0	0
$573$	3.80194	0.158828
$574$	−1.04892	−0.0437810
$575$	2.35690	0.0982894
$576$	1.00000	0.0416667
$577$	−20.0871	−0.836236	−0.418118	−	0.908393i	$-0.637310\pi$
$577$	−20.0871	−0.836236	−0.418118	+	0.908393i	$0.637310\pi$
$578$	−29.9342	−1.24510
$579$	−8.86294	−0.368331
$580$	−5.91185	−0.245476
$581$	1.18300	0.0490790
$582$	−6.13169	−0.254167
$583$	4.07979	0.168967
$584$	−2.31336	−0.0957273
$585$	0	0
$586$	17.5254	0.723968
$587$	−16.0901	−0.664108	−0.332054	−	0.943260i	$-0.607742\pi$
$587$	−16.0901	−0.664108	−0.332054	+	0.943260i	$0.607742\pi$
$588$	−6.90515	−0.284764
$589$	0.353912	0.0145827
$590$	13.1347	0.540746
$591$	−22.6504	−0.931713
$592$	7.07606	0.290824
$593$	4.85517	0.199378	0.0996889	−	0.995019i	$-0.468215\pi$
$593$	4.85517	0.199378	0.0996889	+	0.995019i	$0.468215\pi$
$594$	−0.335126	−0.0137504
$595$	−2.10992	−0.0864981
$596$	3.44935	0.141291
$597$	−5.46681	−0.223742
$598$	0	0
$599$	−34.4319	−1.40685	−0.703425	−	0.710770i	$-0.748347\pi$
$599$	−34.4319	−1.40685	−0.703425	+	0.710770i	$0.748347\pi$
$600$	−1.00000	−0.0408248
$601$	−24.5244	−1.00037	−0.500185	−	0.865919i	$-0.666735\pi$
$601$	−24.5244	−1.00037	−0.500185	+	0.865919i	$0.666735\pi$
$602$	0.0150655	0.000614024 0
$603$	11.9608	0.487080
$604$	2.84117	0.115605
$605$	−10.8877	−0.442648
$606$	−1.32975	−0.0540174
$607$	17.6420	0.716068	0.358034	−	0.933708i	$-0.383447\pi$
$607$	17.6420	0.716068	0.358034	+	0.933708i	$0.383447\pi$
$608$	−5.80194	−0.235300
$609$	−1.82072	−0.0737795
$610$	−2.14675	−0.0869194
$611$	0	0
$612$	−6.85086	−0.276929
$613$	44.5018	1.79741	0.898706	−	0.438551i	$-0.144508\pi$
$613$	44.5018	1.79741	0.898706	+	0.438551i	$0.144508\pi$
$614$	−20.6732	−0.834304
$615$	3.40581	0.137336
$616$	−0.103211	−0.00415851
$617$	32.1618	1.29479	0.647393	−	0.762156i	$-0.275859\pi$
$617$	32.1618	1.29479	0.647393	+	0.762156i	$0.275859\pi$
$618$	−8.92154	−0.358877
$619$	24.8340	0.998162	0.499081	−	0.866555i	$-0.333671\pi$
$619$	24.8340	0.998162	0.499081	+	0.866555i	$0.333671\pi$
$620$	0.0609989	0.00244978
$621$	2.35690	0.0945790
$622$	22.0248	0.883112
$623$	−1.35855	−0.0544292
$624$	0	0
$625$	1.00000	0.0400000
$626$	−21.7192	−0.868073
$627$	1.94438	0.0776510
$628$	16.2228	0.647361
$629$	−48.4771	−1.93291
$630$	−0.307979	−0.0122702
$631$	10.5418	0.419663	0.209831	−	0.977738i	$-0.432708\pi$
$631$	10.5418	0.419663	0.209831	+	0.977738i	$0.432708\pi$
$632$	0.0760644	0.00302568
$633$	11.0422	0.438889
$634$	16.4644	0.653886
$635$	−0.911854	−0.0361858
$636$	12.1739	0.482726
$637$	0	0
$638$	1.98121	0.0784370
$639$	9.56465	0.378372
$640$	−1.00000	−0.0395285
$641$	33.5663	1.32579	0.662895	−	0.748713i	$-0.269328\pi$
$641$	33.5663	1.32579	0.662895	+	0.748713i	$0.269328\pi$
$642$	−14.4644	−0.570865
$643$	8.24160	0.325017	0.162509	−	0.986707i	$-0.448041\pi$
$643$	8.24160	0.325017	0.162509	+	0.986707i	$0.448041\pi$
$644$	0.725873	0.0286034
$645$	−0.0489173	−0.00192612
$646$	39.7482	1.56387
$647$	−3.66056	−0.143912	−0.0719558	−	0.997408i	$-0.522924\pi$
$647$	−3.66056	−0.143912	−0.0719558	+	0.997408i	$0.522924\pi$
$648$	−1.00000	−0.0392837
$649$	−4.40176	−0.172784
$650$	0	0
$651$	0.0187864	0.000736295 0
$652$	−0.0924579	−0.00362093
$653$	−44.6064	−1.74558	−0.872791	−	0.488093i	$-0.837693\pi$
$653$	−44.6064	−1.74558	−0.872791	+	0.488093i	$0.837693\pi$
$654$	−7.13706	−0.279081
$655$	13.0368	0.509391
$656$	3.40581	0.132975
$657$	2.31336	0.0902526
$658$	−2.15585	−0.0840438
$659$	−2.51035	−0.0977895	−0.0488947	−	0.998804i	$-0.515570\pi$
$659$	−2.51035	−0.0977895	−0.0488947	+	0.998804i	$0.515570\pi$
$660$	0.335126	0.0130447
$661$	40.3096	1.56786	0.783932	−	0.620847i	$-0.213211\pi$
$661$	40.3096	1.56786	0.783932	+	0.620847i	$0.213211\pi$
$662$	−26.4373	−1.02751
$663$	0	0
$664$	−3.84117	−0.149066
$665$	1.78687	0.0692919
$666$	−7.07606	−0.274192
$667$	−13.9336	−0.539512
$668$	−9.64071	−0.373010
$669$	18.3884	0.710935
$670$	−11.9608	−0.462085
$671$	0.719432	0.0277733
$672$	−0.307979	−0.0118805
$673$	42.6340	1.64342	0.821710	−	0.569906i	$-0.193020\pi$
$673$	42.6340	1.64342	0.821710	+	0.569906i	$0.193020\pi$
$674$	−27.4698	−1.05810
$675$	1.00000	0.0384900
$676$	0	0
$677$	18.5254	0.711990	0.355995	−	0.934488i	$-0.384142\pi$
$677$	18.5254	0.711990	0.355995	+	0.934488i	$0.384142\pi$
$678$	16.3424	0.627627
$679$	1.88843	0.0724712
$680$	6.85086	0.262718
$681$	12.0073	0.460121
$682$	−0.0204423	−0.000782776 0
$683$	48.1769	1.84344	0.921719	−	0.387859i	$-0.126785\pi$
$683$	48.1769	1.84344	0.921719	+	0.387859i	$0.126785\pi$
$684$	5.80194	0.221843
$685$	12.8877	0.492413
$686$	4.28249	0.163506
$687$	25.4601	0.971364
$688$	−0.0489173	−0.00186496
$689$	0	0
$690$	−2.35690	−0.0897255
$691$	37.0084	1.40786	0.703932	−	0.710267i	$-0.251426\pi$
$691$	37.0084	1.40786	0.703932	+	0.710267i	$0.251426\pi$
$692$	0.719169	0.0273387
$693$	0.103211	0.00392068
$694$	33.1323	1.25768
$695$	−2.01208	−0.0763226
$696$	5.91185	0.224088
$697$	−23.3327	−0.883790
$698$	−26.3739	−0.998266
$699$	−2.10992	−0.0798044
$700$	0.307979	0.0116405
$701$	9.25188	0.349439	0.174719	−	0.984618i	$-0.444098\pi$
$701$	9.25188	0.349439	0.174719	+	0.984618i	$0.444098\pi$
$702$	0	0
$703$	41.0549	1.54841
$704$	0.335126	0.0126305
$705$	7.00000	0.263635
$706$	−21.0291	−0.791439
$707$	0.409534	0.0154021
$708$	−13.1347	−0.493631
$709$	−15.0573	−0.565488	−0.282744	−	0.959195i	$-0.591245\pi$
$709$	−15.0573	−0.565488	−0.282744	+	0.959195i	$0.591245\pi$
$710$	−9.56465	−0.358955
$711$	−0.0760644	−0.00285264
$712$	4.41119	0.165316
$713$	0.143768	0.00538416
$714$	2.10992	0.0789616
$715$	0	0
$716$	−2.01208	−0.0751950
$717$	3.95838	0.147828
$718$	30.8877	1.15272
$719$	24.8364	0.926241	0.463120	−	0.886295i	$-0.346730\pi$
$719$	24.8364	0.926241	0.463120	+	0.886295i	$0.346730\pi$
$720$	1.00000	0.0372678
$721$	2.74764	0.102328
$722$	−14.6625	−0.545681
$723$	−23.1564	−0.861197
$724$	−2.57135	−0.0955635
$725$	−5.91185	−0.219561
$726$	10.8877	0.404080
$727$	−24.7590	−0.918260	−0.459130	−	0.888369i	$-0.651839\pi$
$727$	−24.7590	−0.918260	−0.459130	+	0.888369i	$0.651839\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−2.31336	−0.0856211
$731$	0.335126	0.0123951
$732$	2.14675	0.0793462
$733$	−15.4359	−0.570140	−0.285070	−	0.958507i	$-0.592017\pi$
$733$	−15.4359	−0.570140	−0.285070	+	0.958507i	$0.592017\pi$
$734$	−11.9584	−0.441392
$735$	−6.90515	−0.254700
$736$	−2.35690	−0.0868763
$737$	4.00836	0.147650
$738$	−3.40581	−0.125370
$739$	−20.3327	−0.747952	−0.373976	−	0.927438i	$-0.622006\pi$
$739$	−20.3327	−0.747952	−0.373976	+	0.927438i	$0.622006\pi$
$740$	7.07606	0.260121
$741$	0	0
$742$	−3.74930	−0.137641
$743$	17.8243	0.653910	0.326955	−	0.945040i	$-0.393977\pi$
$743$	17.8243	0.653910	0.326955	+	0.945040i	$0.393977\pi$
$744$	−0.0609989	−0.00223633
$745$	3.44935	0.126375
$746$	−25.0315	−0.916467
$747$	3.84117	0.140541
$748$	−2.29590	−0.0839463
$749$	4.45473	0.162772
$750$	−1.00000	−0.0365148
$751$	48.0616	1.75379	0.876896	−	0.480680i	$-0.159610\pi$
$751$	48.0616	1.75379	0.876896	+	0.480680i	$0.159610\pi$
$752$	7.00000	0.255264
$753$	−5.54048	−0.201906
$754$	0	0
$755$	2.84117	0.103401
$756$	0.307979	0.0112011
$757$	32.9748	1.19849	0.599244	−	0.800566i	$-0.295468\pi$
$757$	32.9748	1.19849	0.599244	+	0.800566i	$0.295468\pi$
$758$	34.4989	1.25306
$759$	0.789856	0.0286700
$760$	−5.80194	−0.210458
$761$	11.9946	0.434805	0.217402	−	0.976082i	$-0.430242\pi$
$761$	11.9946	0.434805	0.217402	+	0.976082i	$0.430242\pi$
$762$	0.911854	0.0330330
$763$	2.19806	0.0795752
$764$	3.80194	0.137549
$765$	−6.85086	−0.247693
$766$	−10.4644	−0.378095
$767$	0	0
$768$	1.00000	0.0360844
$769$	−18.5036	−0.667259	−0.333629	−	0.942704i	$-0.608273\pi$
$769$	−18.5036	−0.667259	−0.333629	+	0.942704i	$0.608273\pi$
$770$	−0.103211	−0.00371948
$771$	6.60819	0.237988
$772$	−8.86294	−0.318984
$773$	−23.4946	−0.845040	−0.422520	−	0.906354i	$-0.638854\pi$
$773$	−23.4946	−0.845040	−0.422520	+	0.906354i	$0.638854\pi$
$774$	0.0489173	0.00175830
$775$	0.0609989	0.00219115
$776$	−6.13169	−0.220115
$777$	2.17928	0.0781811
$778$	−25.2218	−0.904244
$779$	19.7603	0.707987
$780$	0	0
$781$	3.20536	0.114697
$782$	16.1468	0.577407
$783$	−5.91185	−0.211272
$784$	−6.90515	−0.246612
$785$	16.2228	0.579017
$786$	−13.0368	−0.465009
$787$	7.23682	0.257965	0.128982	−	0.991647i	$-0.458829\pi$
$787$	7.23682	0.257965	0.128982	+	0.991647i	$0.458829\pi$
$788$	−22.6504	−0.806887
$789$	−0.660563	−0.0235166
$790$	0.0760644	0.00270625
$791$	−5.03311	−0.178957
$792$	−0.335126	−0.0119082
$793$	0	0
$794$	−21.3860	−0.758959
$795$	12.1739	0.431764
$796$	−5.46681	−0.193766
$797$	−46.7079	−1.65448	−0.827240	−	0.561849i	$-0.810090\pi$
$797$	−46.7079	−1.65448	−0.827240	+	0.561849i	$0.810090\pi$
$798$	−1.78687	−0.0632546
$799$	−47.9560	−1.69656
$800$	−1.00000	−0.0353553
$801$	−4.41119	−0.155862
$802$	11.8062	0.416893
$803$	0.775265	0.0273585
$804$	11.9608	0.421824
$805$	0.725873	0.0255837
$806$	0	0
$807$	−26.7928	−0.943153
$808$	−1.32975	−0.0467804
$809$	−9.95838	−0.350118	−0.175059	−	0.984558i	$-0.556012\pi$
$809$	−9.95838	−0.350118	−0.175059	+	0.984558i	$0.556012\pi$
$810$	−1.00000	−0.0351364
$811$	−36.2452	−1.27274	−0.636370	−	0.771384i	$-0.719565\pi$
$811$	−36.2452	−1.27274	−0.636370	+	0.771384i	$0.719565\pi$
$812$	−1.82072	−0.0638949
$813$	14.0151	0.491530
$814$	−2.37137	−0.0831165
$815$	−0.0924579	−0.00323866
$816$	−6.85086	−0.239828
$817$	−0.283815	−0.00992944
$818$	5.65950	0.197880
$819$	0	0
$820$	3.40581	0.118936
$821$	−1.37867	−0.0481158	−0.0240579	−	0.999711i	$-0.507659\pi$
$821$	−1.37867	−0.0481158	−0.0240579	+	0.999711i	$0.507659\pi$
$822$	−12.8877	−0.449510
$823$	−23.8713	−0.832101	−0.416051	−	0.909341i	$-0.636586\pi$
$823$	−23.8713	−0.832101	−0.416051	+	0.909341i	$0.636586\pi$
$824$	−8.92154	−0.310797
$825$	0.335126	0.0116676
$826$	4.04520	0.140750
$827$	−21.9571	−0.763521	−0.381761	−	0.924261i	$-0.624682\pi$
$827$	−21.9571	−0.763521	−0.381761	+	0.924261i	$0.624682\pi$
$828$	2.35690	0.0819078
$829$	−27.4862	−0.954635	−0.477317	−	0.878731i	$-0.658391\pi$
$829$	−27.4862	−0.954635	−0.477317	+	0.878731i	$0.658391\pi$
$830$	−3.84117	−0.133329
$831$	−23.4252	−0.812611
$832$	0	0
$833$	47.3062	1.63906
$834$	2.01208	0.0696727
$835$	−9.64071	−0.333631
$836$	1.94438	0.0672477
$837$	0.0609989	0.00210843
$838$	−17.0084	−0.587544
$839$	20.5749	0.710325	0.355163	−	0.934805i	$-0.384425\pi$
$839$	20.5749	0.710325	0.355163	+	0.934805i	$0.384425\pi$
$840$	−0.307979	−0.0106263
$841$	5.95002	0.205173
$842$	10.2851	0.354449
$843$	8.40044	0.289326
$844$	11.0422	0.380089
$845$	0	0
$846$	−7.00000	−0.240665
$847$	−3.35317	−0.115216
$848$	12.1739	0.418053
$849$	−31.6142	−1.08499
$850$	6.85086	0.234982
$851$	16.6775	0.571699
$852$	9.56465	0.327679
$853$	−50.3812	−1.72502	−0.862509	−	0.506041i	$-0.831108\pi$
$853$	−50.3812	−1.72502	−0.862509	+	0.506041i	$0.831108\pi$
$854$	−0.661154	−0.0226242
$855$	5.80194	0.198422
$856$	−14.4644	−0.494384
$857$	21.6045	0.737995	0.368997	−	0.929430i	$-0.379701\pi$
$857$	21.6045	0.737995	0.368997	+	0.929430i	$0.379701\pi$
$858$	0	0
$859$	−38.2959	−1.30664	−0.653320	−	0.757082i	$-0.726624\pi$
$859$	−38.2959	−1.30664	−0.653320	+	0.757082i	$0.726624\pi$
$860$	−0.0489173	−0.00166807
$861$	1.04892	0.0357470
$862$	35.9657	1.22500
$863$	8.61894	0.293392	0.146696	−	0.989182i	$-0.453136\pi$
$863$	8.61894	0.293392	0.146696	+	0.989182i	$0.453136\pi$
$864$	−1.00000	−0.0340207
$865$	0.719169	0.0244525
$866$	−7.63342	−0.259394
$867$	29.9342	1.01662
$868$	0.0187864	0.000637651 0
$869$	−0.0254911	−0.000864727 0
$870$	5.91185	0.200431
$871$	0	0
$872$	−7.13706	−0.241691
$873$	6.13169	0.207526
$874$	−13.6746	−0.462549
$875$	0.307979	0.0104116
$876$	2.31336	0.0781610
$877$	−33.1021	−1.11778	−0.558890	−	0.829242i	$-0.688773\pi$
$877$	−33.1021	−1.11778	−0.558890	+	0.829242i	$0.688773\pi$
$878$	34.4373	1.16220
$879$	−17.5254	−0.591118
$880$	0.335126	0.0112971
$881$	20.8267	0.701669	0.350835	−	0.936437i	$-0.385898\pi$
$881$	20.8267	0.701669	0.350835	+	0.936437i	$0.385898\pi$
$882$	6.90515	0.232508
$883$	0.785807	0.0264445	0.0132223	−	0.999913i	$-0.495791\pi$
$883$	0.785807	0.0264445	0.0132223	+	0.999913i	$0.495791\pi$
$884$	0	0
$885$	−13.1347	−0.441517
$886$	−33.6431	−1.13026
$887$	−44.2392	−1.48541	−0.742704	−	0.669620i	$-0.766457\pi$
$887$	−44.2392	−1.48541	−0.742704	+	0.669620i	$0.766457\pi$
$888$	−7.07606	−0.237457
$889$	−0.280831	−0.00941878
$890$	4.41119	0.147863
$891$	0.335126	0.0112271
$892$	18.3884	0.615688
$893$	40.6136	1.35908
$894$	−3.44935	−0.115364
$895$	−2.01208	−0.0672565
$896$	−0.307979	−0.0102888
$897$	0	0
$898$	27.9729	0.933466
$899$	−0.360617	−0.0120272
$900$	1.00000	0.0333333
$901$	−83.4016	−2.77851
$902$	−1.14138	−0.0380036
$903$	−0.0150655	−0.000501348 0
$904$	16.3424	0.543541
$905$	−2.57135	−0.0854746
$906$	−2.84117	−0.0943914
$907$	8.44371	0.280369	0.140184	−	0.990125i	$-0.455230\pi$
$907$	8.44371	0.280369	0.140184	+	0.990125i	$0.455230\pi$
$908$	12.0073	0.398476
$909$	1.32975	0.0441050
$910$	0	0
$911$	−47.2801	−1.56646	−0.783230	−	0.621732i	$-0.786429\pi$
$911$	−47.2801	−1.56646	−0.783230	+	0.621732i	$0.786429\pi$
$912$	5.80194	0.192121
$913$	1.28727	0.0426025
$914$	6.18060	0.204436
$915$	2.14675	0.0709694
$916$	25.4601	0.841226
$917$	4.01507	0.132589
$918$	6.85086	0.226112
$919$	−11.1927	−0.369213	−0.184606	−	0.982813i	$-0.559101\pi$
$919$	−11.1927	−0.369213	−0.184606	+	0.982813i	$0.559101\pi$
$920$	−2.35690	−0.0777046
$921$	20.6732	0.681206
$922$	−38.1062	−1.25496
$923$	0	0
$924$	0.103211	0.00339541
$925$	7.07606	0.232660
$926$	2.06829	0.0679684
$927$	8.92154	0.293022
$928$	5.91185	0.194066
$929$	1.11828	0.0366895	0.0183447	−	0.999832i	$-0.494160\pi$
$929$	1.11828	0.0366895	0.0183447	+	0.999832i	$0.494160\pi$
$930$	−0.0609989	−0.00200023
$931$	−40.0632	−1.31302
$932$	−2.10992	−0.0691126
$933$	−22.0248	−0.721058
$934$	4.27114	0.139756
$935$	−2.29590	−0.0750839
$936$	0	0
$937$	−29.9051	−0.976959	−0.488479	−	0.872575i	$-0.662448\pi$
$937$	−29.9051	−0.976959	−0.488479	+	0.872575i	$0.662448\pi$
$938$	−3.68366	−0.120276
$939$	21.7192	0.708778
$940$	7.00000	0.228315
$941$	−32.7646	−1.06810	−0.534048	−	0.845454i	$-0.679330\pi$
$941$	−32.7646	−1.06810	−0.534048	+	0.845454i	$0.679330\pi$
$942$	−16.2228	−0.528568
$943$	8.02715	0.261400
$944$	−13.1347	−0.427497
$945$	0.307979	0.0100185
$946$	0.0163935	0.000532997 0
$947$	−6.81535	−0.221469	−0.110735	−	0.993850i	$-0.535320\pi$
$947$	−6.81535	−0.221469	−0.110735	+	0.993850i	$0.535320\pi$
$948$	−0.0760644	−0.00247046
$949$	0	0
$950$	−5.80194	−0.188240
$951$	−16.4644	−0.533895
$952$	2.10992	0.0683828
$953$	23.6886	0.767348	0.383674	−	0.923469i	$-0.374659\pi$
$953$	23.6886	0.767348	0.383674	+	0.923469i	$0.374659\pi$
$954$	−12.1739	−0.394145
$955$	3.80194	0.123028
$956$	3.95838	0.128023
$957$	−1.98121	−0.0640435
$958$	38.7560	1.25215
$959$	3.96913	0.128170
$960$	1.00000	0.0322749
$961$	−30.9963	−0.999880
$962$	0	0
$963$	14.4644	0.466109
$964$	−23.1564	−0.745819
$965$	−8.86294	−0.285308
$966$	−0.725873	−0.0233546
$967$	−28.9075	−0.929604	−0.464802	−	0.885415i	$-0.653874\pi$
$967$	−28.9075	−0.929604	−0.464802	+	0.885415i	$0.653874\pi$
$968$	10.8877	0.349944
$969$	−39.7482	−1.27690
$970$	−6.13169	−0.196877
$971$	−47.2717	−1.51702	−0.758511	−	0.651660i	$-0.774073\pi$
$971$	−47.2717	−1.51702	−0.758511	+	0.651660i	$0.774073\pi$
$972$	1.00000	0.0320750
$973$	−0.619678	−0.0198660
$974$	33.4620	1.07219
$975$	0	0
$976$	2.14675	0.0687159
$977$	−33.9527	−1.08624	−0.543122	−	0.839654i	$-0.682758\pi$
$977$	−33.9527	−1.08624	−0.543122	+	0.839654i	$0.682758\pi$
$978$	0.0924579	0.00295648
$979$	−1.47830	−0.0472468
$980$	−6.90515	−0.220577
$981$	7.13706	0.227869
$982$	−35.1487	−1.12164
$983$	−22.9366	−0.731564	−0.365782	−	0.930701i	$-0.619198\pi$
$983$	−22.9366	−0.731564	−0.365782	+	0.930701i	$0.619198\pi$
$984$	−3.40581	−0.108573
$985$	−22.6504	−0.721702
$986$	−40.5013	−1.28982
$987$	2.15585	0.0686215
$988$	0	0
$989$	−0.115293	−0.00366611
$990$	−0.335126	−0.0106510
$991$	−5.75781	−0.182903	−0.0914514	−	0.995810i	$-0.529151\pi$
$991$	−5.75781	−0.182903	−0.0914514	+	0.995810i	$0.529151\pi$
$992$	−0.0609989	−0.00193672
$993$	26.4373	0.838961
$994$	−2.94571	−0.0934321
$995$	−5.46681	−0.173310
$996$	3.84117	0.121712
$997$	3.86353	0.122359	0.0611796	−	0.998127i	$-0.480514\pi$
$997$	3.86353	0.122359	0.0611796	+	0.998127i	$0.480514\pi$
$998$	9.61463	0.304346
$999$	7.07606	0.223877

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.bq.1.1	✓	3
13.5	odd	4		5070.2.b.y.1351.4		6
13.8	odd	4		5070.2.b.y.1351.3		6
13.12	even	2		5070.2.a.bv.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bq.1.1	✓	3	1.1	even	1	trivial
5070.2.a.bv.1.3	yes	3	13.12	even	2
5070.2.b.y.1351.3		6	13.8	odd	4
5070.2.b.y.1351.4		6	13.5	odd	4

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +0.307979 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +0.307979 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +0.335126 q^{11} +1.00000 q^{12} -0.307979 q^{14} +1.00000 q^{15} +1.00000 q^{16} -6.85086 q^{17} -1.00000 q^{18} +5.80194 q^{19} +1.00000 q^{20} +0.307979 q^{21} -0.335126 q^{22} +2.35690 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +0.307979 q^{28} -5.91185 q^{29} -1.00000 q^{30} +0.0609989 q^{31} -1.00000 q^{32} +0.335126 q^{33} +6.85086 q^{34} +0.307979 q^{35} +1.00000 q^{36} +7.07606 q^{37} -5.80194 q^{38} -1.00000 q^{40} +3.40581 q^{41} -0.307979 q^{42} -0.0489173 q^{43} +0.335126 q^{44} +1.00000 q^{45} -2.35690 q^{46} +7.00000 q^{47} +1.00000 q^{48} -6.90515 q^{49} -1.00000 q^{50} -6.85086 q^{51} +12.1739 q^{53} -1.00000 q^{54} +0.335126 q^{55} -0.307979 q^{56} +5.80194 q^{57} +5.91185 q^{58} -13.1347 q^{59} +1.00000 q^{60} +2.14675 q^{61} -0.0609989 q^{62} +0.307979 q^{63} +1.00000 q^{64} -0.335126 q^{66} +11.9608 q^{67} -6.85086 q^{68} +2.35690 q^{69} -0.307979 q^{70} +9.56465 q^{71} -1.00000 q^{72} +2.31336 q^{73} -7.07606 q^{74} +1.00000 q^{75} +5.80194 q^{76} +0.103211 q^{77} -0.0760644 q^{79} +1.00000 q^{80} +1.00000 q^{81} -3.40581 q^{82} +3.84117 q^{83} +0.307979 q^{84} -6.85086 q^{85} +0.0489173 q^{86} -5.91185 q^{87} -0.335126 q^{88} -4.41119 q^{89} -1.00000 q^{90} +2.35690 q^{92} +0.0609989 q^{93} -7.00000 q^{94} +5.80194 q^{95} -1.00000 q^{96} +6.13169 q^{97} +6.90515 q^{98} +0.335126 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 6 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 6 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{10} + 3 q^{12} - 6 q^{14} + 3 q^{15} + 3 q^{16} - 7 q^{17} - 3 q^{18} + 13 q^{19} + 3 q^{20} + 6 q^{21} + 3 q^{23} - 3 q^{24} + 3 q^{25} + 3 q^{27} + 6 q^{28} - 14 q^{29} - 3 q^{30} + 10 q^{31} - 3 q^{32} + 7 q^{34} + 6 q^{35} + 3 q^{36} + 6 q^{37} - 13 q^{38} - 3 q^{40} - 3 q^{41} - 6 q^{42} + 9 q^{43} + 3 q^{45} - 3 q^{46} + 21 q^{47} + 3 q^{48} + 5 q^{49} - 3 q^{50} - 7 q^{51} + 3 q^{53} - 3 q^{54} - 6 q^{56} + 13 q^{57} + 14 q^{58} + 7 q^{59} + 3 q^{60} - 21 q^{61} - 10 q^{62} + 6 q^{63} + 3 q^{64} + 23 q^{67} - 7 q^{68} + 3 q^{69} - 6 q^{70} + 7 q^{71} - 3 q^{72} + 9 q^{73} - 6 q^{74} + 3 q^{75} + 13 q^{76} - 21 q^{77} + 15 q^{79} + 3 q^{80} + 3 q^{81} + 3 q^{82} + 20 q^{83} + 6 q^{84} - 7 q^{85} - 9 q^{86} - 14 q^{87} + 3 q^{89} - 3 q^{90} + 3 q^{92} + 10 q^{93} - 21 q^{94} + 13 q^{95} - 3 q^{96} + 16 q^{97} - 5 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 6 * q^7 - 3 * q^8 + 3 * q^9 - 3 * q^10 + 3 * q^12 - 6 * q^14 + 3 * q^15 + 3 * q^16 - 7 * q^17 - 3 * q^18 + 13 * q^19 + 3 * q^20 + 6 * q^21 + 3 * q^23 - 3 * q^24 + 3 * q^25 + 3 * q^27 + 6 * q^28 - 14 * q^29 - 3 * q^30 + 10 * q^31 - 3 * q^32 + 7 * q^34 + 6 * q^35 + 3 * q^36 + 6 * q^37 - 13 * q^38 - 3 * q^40 - 3 * q^41 - 6 * q^42 + 9 * q^43 + 3 * q^45 - 3 * q^46 + 21 * q^47 + 3 * q^48 + 5 * q^49 - 3 * q^50 - 7 * q^51 + 3 * q^53 - 3 * q^54 - 6 * q^56 + 13 * q^57 + 14 * q^58 + 7 * q^59 + 3 * q^60 - 21 * q^61 - 10 * q^62 + 6 * q^63 + 3 * q^64 + 23 * q^67 - 7 * q^68 + 3 * q^69 - 6 * q^70 + 7 * q^71 - 3 * q^72 + 9 * q^73 - 6 * q^74 + 3 * q^75 + 13 * q^76 - 21 * q^77 + 15 * q^79 + 3 * q^80 + 3 * q^81 + 3 * q^82 + 20 * q^83 + 6 * q^84 - 7 * q^85 - 9 * q^86 - 14 * q^87 + 3 * q^89 - 3 * q^90 + 3 * q^92 + 10 * q^93 - 21 * q^94 + 13 * q^95 - 3 * q^96 + 16 * q^97 - 5 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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