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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [702,2,Mod(181,702)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("702.181"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(702, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 702 = 2 \cdot 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 702.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.60549822189\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 234)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 415.12
Character \(\chi\) \(=\) 702.415
Dual form 702.2.t.a.181.12

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(0.515076 + 0.297379i) q^{5} +(1.45217 - 0.838409i) q^{7} -1.00000i q^{8} +0.594758 q^{10} +(-0.416337 + 0.240372i) q^{11} +(3.56065 + 0.567277i) q^{13} +(0.838409 - 1.45217i) q^{14} +(-0.500000 - 0.866025i) q^{16} +2.09349 q^{17} +0.480744i q^{19} +(0.515076 - 0.297379i) q^{20} +(-0.240372 + 0.416337i) q^{22} +(1.83339 - 3.17553i) q^{23} +(-2.32313 - 4.02378i) q^{25} +(3.36725 - 1.28905i) q^{26} -1.67682i q^{28} +(-1.23339 - 2.13629i) q^{29} +(0.993791 + 0.573765i) q^{31} +(-0.866025 - 0.500000i) q^{32} +(1.81302 - 1.04675i) q^{34} +0.997301 q^{35} +3.65012i q^{37} +(0.240372 + 0.416337i) q^{38} +(0.297379 - 0.515076i) q^{40} +(8.58235 + 4.95502i) q^{41} +(-3.45822 - 5.98981i) q^{43} +0.480744i q^{44} -3.66679i q^{46} +(-5.40488 + 3.12051i) q^{47} +(-2.09414 + 3.62716i) q^{49} +(-4.02378 - 2.32313i) q^{50} +(2.27160 - 2.79997i) q^{52} +5.08592 q^{53} -0.285927 q^{55} +(-0.838409 - 1.45217i) q^{56} +(-2.13629 - 1.23339i) q^{58} +(8.13185 + 4.69493i) q^{59} +(-3.90635 - 6.76599i) q^{61} +1.14753 q^{62} -1.00000 q^{64} +(1.66531 + 1.35105i) q^{65} +(-12.4551 - 7.19096i) q^{67} +(1.04675 - 1.81302i) q^{68} +(0.863688 - 0.498651i) q^{70} +6.51028i q^{71} +5.91514i q^{73} +(1.82506 + 3.16110i) q^{74} +(0.416337 + 0.240372i) q^{76} +(-0.403061 + 0.698121i) q^{77} +(1.02895 + 1.78219i) q^{79} -0.594758i q^{80} +9.91005 q^{82} +(-9.57834 + 5.53006i) q^{83} +(1.07831 + 0.622560i) q^{85} +(-5.98981 - 3.45822i) q^{86} +(0.240372 + 0.416337i) q^{88} -9.48720i q^{89} +(5.64626 - 2.16150i) q^{91} +(-1.83339 - 3.17553i) q^{92} +(-3.12051 + 5.40488i) q^{94} +(-0.142963 + 0.247620i) q^{95} +(-8.41374 + 4.85767i) q^{97} +4.18828i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 14 q^{4} + 2 q^{13} - 8 q^{14} - 14 q^{16} - 16 q^{17} + 8 q^{23} + 14 q^{25} - 8 q^{26} + 16 q^{29} + 68 q^{35} - 4 q^{43} + 10 q^{49} - 2 q^{52} + 120 q^{53} + 8 q^{56} + 28 q^{61} - 68 q^{62}+ \cdots - 8 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/702\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(677\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 − 0.500000i 0.612372 − 0.353553i
\(3\) 0 0
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) 0.515076 + 0.297379i 0.230349 + 0.132992i 0.610733 − 0.791837i \(-0.290875\pi\)
−0.380384 + 0.924829i \(0.624208\pi\)
\(6\) 0 0
\(7\) 1.45217 − 0.838409i 0.548868 − 0.316889i −0.199798 − 0.979837i \(-0.564028\pi\)
0.748665 + 0.662948i \(0.230695\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) 0.594758 0.188079
\(11\) −0.416337 + 0.240372i −0.125530 + 0.0724750i −0.561450 − 0.827510i \(-0.689756\pi\)
0.435920 + 0.899985i \(0.356423\pi\)
\(12\) 0 0
\(13\) 3.56065 + 0.567277i 0.987545 + 0.157334i
\(14\) 0.838409 − 1.45217i 0.224074 − 0.388108i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 2.09349 0.507746 0.253873 − 0.967238i \(-0.418295\pi\)
0.253873 + 0.967238i \(0.418295\pi\)
\(18\) 0 0
\(19\) 0.480744i 0.110290i 0.998478 + 0.0551452i \(0.0175622\pi\)
−0.998478 + 0.0551452i \(0.982438\pi\)
\(20\) 0.515076 − 0.297379i 0.115174 − 0.0664960i
\(21\) 0 0
\(22\) −0.240372 + 0.416337i −0.0512475 + 0.0887633i
\(23\) 1.83339 − 3.17553i 0.382289 − 0.662144i −0.609100 − 0.793093i \(-0.708469\pi\)
0.991389 + 0.130950i \(0.0418026\pi\)
\(24\) 0 0
\(25\) −2.32313 − 4.02378i −0.464626 − 0.804756i
\(26\) 3.36725 − 1.28905i 0.660372 − 0.252803i
\(27\) 0 0
\(28\) − 1.67682i − 0.316889i
\(29\) −1.23339 − 2.13629i −0.229035 − 0.396700i 0.728488 − 0.685059i \(-0.240224\pi\)
−0.957522 + 0.288359i \(0.906890\pi\)
\(30\) 0 0
\(31\) 0.993791 + 0.573765i 0.178490 + 0.103051i 0.586583 − 0.809889i \(-0.300473\pi\)
−0.408093 + 0.912940i \(0.633806\pi\)
\(32\) −0.866025 − 0.500000i −0.153093 − 0.0883883i
\(33\) 0 0
\(34\) 1.81302 − 1.04675i 0.310930 − 0.179515i
\(35\) 0.997301 0.168575
\(36\) 0 0
\(37\) 3.65012i 0.600076i 0.953927 + 0.300038i \(0.0969994\pi\)
−0.953927 + 0.300038i \(0.903001\pi\)
\(38\) 0.240372 + 0.416337i 0.0389935 + 0.0675388i
\(39\) 0 0
\(40\) 0.297379 − 0.515076i 0.0470198 − 0.0814406i
\(41\) 8.58235 + 4.95502i 1.34034 + 0.773845i 0.986857 − 0.161597i \(-0.0516645\pi\)
0.353481 + 0.935442i \(0.384998\pi\)
\(42\) 0 0
\(43\) −3.45822 − 5.98981i −0.527374 − 0.913438i −0.999491 − 0.0319023i \(-0.989843\pi\)
0.472117 − 0.881536i \(-0.343490\pi\)
\(44\) 0.480744i 0.0724750i
\(45\) 0 0
\(46\) − 3.66679i − 0.540638i
\(47\) −5.40488 + 3.12051i −0.788383 + 0.455173i −0.839393 − 0.543525i \(-0.817089\pi\)
0.0510099 + 0.998698i \(0.483756\pi\)
\(48\) 0 0
\(49\) −2.09414 + 3.62716i −0.299163 + 0.518165i
\(50\) −4.02378 − 2.32313i −0.569049 − 0.328540i
\(51\) 0 0
\(52\) 2.27160 − 2.79997i 0.315014 − 0.388286i
\(53\) 5.08592 0.698605 0.349302 − 0.937010i \(-0.386419\pi\)
0.349302 + 0.937010i \(0.386419\pi\)
\(54\) 0 0
\(55\) −0.285927 −0.0385544
\(56\) −0.838409 − 1.45217i −0.112037 − 0.194054i
\(57\) 0 0
\(58\) −2.13629 − 1.23339i −0.280509 − 0.161952i
\(59\) 8.13185 + 4.69493i 1.05868 + 0.611227i 0.925066 − 0.379807i \(-0.124009\pi\)
0.133611 + 0.991034i \(0.457343\pi\)
\(60\) 0 0
\(61\) −3.90635 − 6.76599i −0.500157 − 0.866297i −1.00000 0.000180927i \(-0.999942\pi\)
0.499843 − 0.866116i \(-0.333391\pi\)
\(62\) 1.14753 0.145737
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 1.66531 + 1.35105i 0.206556 + 0.167577i
\(66\) 0 0
\(67\) −12.4551 − 7.19096i −1.52163 − 0.878516i −0.999674 − 0.0255454i \(-0.991868\pi\)
−0.521960 − 0.852970i \(-0.674799\pi\)
\(68\) 1.04675 − 1.81302i 0.126936 − 0.219860i
\(69\) 0 0
\(70\) 0.863688 − 0.498651i 0.103231 − 0.0596002i
\(71\) 6.51028i 0.772628i 0.922367 + 0.386314i \(0.126252\pi\)
−0.922367 + 0.386314i \(0.873748\pi\)
\(72\) 0 0
\(73\) 5.91514i 0.692315i 0.938176 + 0.346157i \(0.112514\pi\)
−0.938176 + 0.346157i \(0.887486\pi\)
\(74\) 1.82506 + 3.16110i 0.212159 + 0.367470i
\(75\) 0 0
\(76\) 0.416337 + 0.240372i 0.0477571 + 0.0275726i
\(77\) −0.403061 + 0.698121i −0.0459330 + 0.0795583i
\(78\) 0 0
\(79\) 1.02895 + 1.78219i 0.115766 + 0.200512i 0.918086 − 0.396382i \(-0.129734\pi\)
−0.802320 + 0.596894i \(0.796401\pi\)
\(80\) − 0.594758i − 0.0664960i
\(81\) 0 0
\(82\) 9.91005 1.09438
\(83\) −9.57834 + 5.53006i −1.05136 + 0.607003i −0.923030 − 0.384729i \(-0.874295\pi\)
−0.128330 + 0.991732i \(0.540962\pi\)
\(84\) 0 0
\(85\) 1.07831 + 0.622560i 0.116959 + 0.0675261i
\(86\) −5.98981 − 3.45822i −0.645898 − 0.372909i
\(87\) 0 0
\(88\) 0.240372 + 0.416337i 0.0256238 + 0.0443817i
\(89\) − 9.48720i − 1.00564i −0.864391 − 0.502821i \(-0.832296\pi\)
0.864391 − 0.502821i \(-0.167704\pi\)
\(90\) 0 0
\(91\) 5.64626 − 2.16150i 0.591889 − 0.226586i
\(92\) −1.83339 − 3.17553i −0.191144 − 0.331072i
\(93\) 0 0
\(94\) −3.12051 + 5.40488i −0.321856 + 0.557471i
\(95\) −0.142963 + 0.247620i −0.0146677 + 0.0254053i
\(96\) 0 0
\(97\) −8.41374 + 4.85767i −0.854286 + 0.493222i −0.862095 − 0.506747i \(-0.830848\pi\)
0.00780887 + 0.999970i \(0.497514\pi\)
\(98\) 4.18828i 0.423080i
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 702.2.t.a.415.12 28
3.2 odd 2 234.2.t.a.103.7 yes 28
9.2 odd 6 234.2.t.a.25.14 yes 28
9.4 even 3 2106.2.b.d.649.10 14
9.5 odd 6 2106.2.b.c.649.5 14
9.7 even 3 inner 702.2.t.a.181.3 28
13.12 even 2 inner 702.2.t.a.415.3 28
39.38 odd 2 234.2.t.a.103.14 yes 28
117.25 even 6 inner 702.2.t.a.181.12 28
117.38 odd 6 234.2.t.a.25.7 ✓ 28
117.77 odd 6 2106.2.b.c.649.10 14
117.103 even 6 2106.2.b.d.649.5 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
234.2.t.a.25.7 ✓ 28 117.38 odd 6
234.2.t.a.25.14 yes 28 9.2 odd 6
234.2.t.a.103.7 yes 28 3.2 odd 2
234.2.t.a.103.14 yes 28 39.38 odd 2
702.2.t.a.181.3 28 9.7 even 3 inner
702.2.t.a.181.12 28 117.25 even 6 inner
702.2.t.a.415.3 28 13.12 even 2 inner
702.2.t.a.415.12 28 1.1 even 1 trivial
2106.2.b.c.649.5 14 9.5 odd 6
2106.2.b.c.649.10 14 117.77 odd 6
2106.2.b.d.649.5 14 117.103 even 6
2106.2.b.d.649.10 14 9.4 even 3