Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [320,6,Mod(129,320)] 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("320.129"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,90,0,0,0,-306,0,504] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.1
Root \(0.500000 + 1.65831i\) of defining polynomial
Character \(\chi\) \(=\) 320.129
Dual form 320.6.c.g.129.2

$q$-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)
 
\(f(q)\) \(=\) \(q-19.8997i q^{3} +(45.0000 - 33.1662i) q^{5} +59.6992i q^{7} -153.000 q^{9} +252.000 q^{11} +119.398i q^{13} +(-660.000 - 895.489i) q^{15} -689.858i q^{17} -220.000 q^{19} +1188.00 q^{21} -2434.40i q^{23} +(925.000 - 2984.96i) q^{25} -1790.98i q^{27} +6930.00 q^{29} -6752.00 q^{31} -5014.74i q^{33} +(1980.00 + 2686.47i) q^{35} -13969.6i q^{37} +2376.00 q^{39} -198.000 q^{41} -417.895i q^{43} +(-6885.00 + 5074.44i) q^{45} +10540.2i q^{47} +13243.0 q^{49} -13728.0 q^{51} +5823.99i q^{53} +(11340.0 - 8357.89i) q^{55} +4377.94i q^{57} -24660.0 q^{59} +5698.00 q^{61} -9133.98i q^{63} +(3960.00 + 5372.93i) q^{65} -43640.1i q^{67} -48444.0 q^{69} -53352.0 q^{71} +70922.7i q^{73} +(-59400.0 - 18407.3i) q^{75} +15044.2i q^{77} -51920.0 q^{79} -72819.0 q^{81} -61841.8i q^{83} +(-22880.0 - 31043.6i) q^{85} -137905. i q^{87} -9990.00 q^{89} -7128.00 q^{91} +134363. i q^{93} +(-9900.00 + 7296.57i) q^{95} -101250. i q^{97} -38556.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 90 q^{5} - 306 q^{9} + 504 q^{11} - 1320 q^{15} - 440 q^{19} + 2376 q^{21} + 1850 q^{25} + 13860 q^{29} - 13504 q^{31} + 3960 q^{35} + 4752 q^{39} - 396 q^{41} - 13770 q^{45} + 26486 q^{49} - 27456 q^{51}+ \cdots - 77112 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 0
\(3\) − 19.8997i − 1.27657i −0.769800 − 0.638285i \(-0.779644\pi\)
0.769800 − 0.638285i \(-0.220356\pi\)
\(4\) 0 0
\(5\) 45.0000 − 33.1662i 0.804984 − 0.593296i
\(6\) 0 0
\(7\) 59.6992i 0.460494i 0.973132 + 0.230247i \(0.0739534\pi\)
−0.973132 + 0.230247i \(0.926047\pi\)
\(8\) 0 0
\(9\) −153.000 −0.629630
\(10\) 0 0
\(11\) 252.000 0.627941 0.313970 − 0.949433i \(-0.398341\pi\)
0.313970 + 0.949433i \(0.398341\pi\)
\(12\) 0 0
\(13\) 119.398i 0.195948i 0.995189 + 0.0979739i \(0.0312362\pi\)
−0.995189 + 0.0979739i \(0.968764\pi\)
\(14\) 0 0
\(15\) −660.000 − 895.489i −0.757383 − 1.02762i
\(16\) 0 0
\(17\) − 689.858i − 0.578945i −0.957186 − 0.289473i \(-0.906520\pi\)
0.957186 − 0.289473i \(-0.0934799\pi\)
\(18\) 0 0
\(19\) −220.000 −0.139810 −0.0699051 − 0.997554i \(-0.522270\pi\)
−0.0699051 + 0.997554i \(0.522270\pi\)
\(20\) 0 0
\(21\) 1188.00 0.587852
\(22\) 0 0
\(23\) − 2434.40i − 0.959561i −0.877388 − 0.479781i \(-0.840716\pi\)
0.877388 − 0.479781i \(-0.159284\pi\)
\(24\) 0 0
\(25\) 925.000 − 2984.96i 0.296000 − 0.955188i
\(26\) 0 0
\(27\) − 1790.98i − 0.472804i
\(28\) 0 0
\(29\) 6930.00 1.53016 0.765082 − 0.643932i \(-0.222698\pi\)
0.765082 + 0.643932i \(0.222698\pi\)
\(30\) 0 0
\(31\) −6752.00 −1.26191 −0.630955 − 0.775820i \(-0.717337\pi\)
−0.630955 + 0.775820i \(0.717337\pi\)
\(32\) 0 0
\(33\) − 5014.74i − 0.801610i
\(34\) 0 0
\(35\) 1980.00 + 2686.47i 0.273209 + 0.370690i
\(36\) 0 0
\(37\) − 13969.6i − 1.67757i −0.544464 − 0.838785i \(-0.683267\pi\)
0.544464 − 0.838785i \(-0.316733\pi\)
\(38\) 0 0
\(39\) 2376.00 0.250141
\(40\) 0 0
\(41\) −198.000 −0.0183952 −0.00919762 − 0.999958i \(-0.502928\pi\)
−0.00919762 + 0.999958i \(0.502928\pi\)
\(42\) 0 0
\(43\) − 417.895i − 0.0344664i −0.999851 − 0.0172332i \(-0.994514\pi\)
0.999851 − 0.0172332i \(-0.00548577\pi\)
\(44\) 0 0
\(45\) −6885.00 + 5074.44i −0.506842 + 0.373557i
\(46\) 0 0
\(47\) 10540.2i 0.695994i 0.937496 + 0.347997i \(0.113138\pi\)
−0.937496 + 0.347997i \(0.886862\pi\)
\(48\) 0 0
\(49\) 13243.0 0.787945
\(50\) 0 0
\(51\) −13728.0 −0.739064
\(52\) 0 0
\(53\) 5823.99i 0.284794i 0.989810 + 0.142397i \(0.0454810\pi\)
−0.989810 + 0.142397i \(0.954519\pi\)
\(54\) 0 0
\(55\) 11340.0 − 8357.89i 0.505483 − 0.372555i
\(56\) 0 0
\(57\) 4377.94i 0.178477i
\(58\) 0 0
\(59\) −24660.0 −0.922281 −0.461140 − 0.887327i \(-0.652560\pi\)
−0.461140 + 0.887327i \(0.652560\pi\)
\(60\) 0 0
\(61\) 5698.00 0.196064 0.0980320 − 0.995183i \(-0.468745\pi\)
0.0980320 + 0.995183i \(0.468745\pi\)
\(62\) 0 0
\(63\) − 9133.98i − 0.289941i
\(64\) 0 0
\(65\) 3960.00 + 5372.93i 0.116255 + 0.157735i
\(66\) 0 0
\(67\) − 43640.1i − 1.18768i −0.804583 − 0.593840i \(-0.797611\pi\)
0.804583 − 0.593840i \(-0.202389\pi\)
\(68\) 0 0
\(69\) −48444.0 −1.22495
\(70\) 0 0
\(71\) −53352.0 −1.25604 −0.628022 − 0.778196i \(-0.716135\pi\)
−0.628022 + 0.778196i \(0.716135\pi\)
\(72\) 0 0
\(73\) 70922.7i 1.55768i 0.627223 + 0.778840i \(0.284192\pi\)
−0.627223 + 0.778840i \(0.715808\pi\)
\(74\) 0 0
\(75\) −59400.0 − 18407.3i −1.21936 − 0.377865i
\(76\) 0 0
\(77\) 15044.2i 0.289163i
\(78\) 0 0
\(79\) −51920.0 −0.935981 −0.467990 − 0.883734i \(-0.655022\pi\)
−0.467990 + 0.883734i \(0.655022\pi\)
\(80\) 0 0
\(81\) −72819.0 −1.23320
\(82\) 0 0
\(83\) − 61841.8i − 0.985342i −0.870216 − 0.492671i \(-0.836021\pi\)
0.870216 − 0.492671i \(-0.163979\pi\)
\(84\) 0 0
\(85\) −22880.0 − 31043.6i −0.343486 − 0.466042i
\(86\) 0 0
\(87\) − 137905.i − 1.95336i
\(88\) 0 0
\(89\) −9990.00 −0.133687 −0.0668437 − 0.997763i \(-0.521293\pi\)
−0.0668437 + 0.997763i \(0.521293\pi\)
\(90\) 0 0
\(91\) −7128.00 −0.0902328
\(92\) 0 0
\(93\) 134363.i 1.61092i
\(94\) 0 0
\(95\) −9900.00 + 7296.57i −0.112545 + 0.0829488i
\(96\) 0 0
\(97\) − 101250.i − 1.09261i −0.837586 − 0.546305i \(-0.816034\pi\)
0.837586 − 0.546305i \(-0.183966\pi\)
\(98\) 0 0
\(99\) −38556.0 −0.395370
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 320.6.c.g.129.1 2
4.3 odd 2 320.6.c.f.129.2 2
5.4 even 2 inner 320.6.c.g.129.2 2
8.3 odd 2 5.6.b.a.4.2 yes 2
8.5 even 2 80.6.c.a.49.2 2
20.19 odd 2 320.6.c.f.129.1 2
24.5 odd 2 720.6.f.f.289.1 2
24.11 even 2 45.6.b.b.19.1 2
40.3 even 4 25.6.a.c.1.2 2
40.13 odd 4 400.6.a.t.1.1 2
40.19 odd 2 5.6.b.a.4.1 ✓ 2
40.27 even 4 25.6.a.c.1.1 2
40.29 even 2 80.6.c.a.49.1 2
40.37 odd 4 400.6.a.t.1.2 2
56.27 even 2 245.6.b.a.99.2 2
120.29 odd 2 720.6.f.f.289.2 2
120.59 even 2 45.6.b.b.19.2 2
120.83 odd 4 225.6.a.n.1.1 2
120.107 odd 4 225.6.a.n.1.2 2
280.139 even 2 245.6.b.a.99.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.6.b.a.4.1 ✓ 2 40.19 odd 2
5.6.b.a.4.2 yes 2 8.3 odd 2
25.6.a.c.1.1 2 40.27 even 4
25.6.a.c.1.2 2 40.3 even 4
45.6.b.b.19.1 2 24.11 even 2
45.6.b.b.19.2 2 120.59 even 2
80.6.c.a.49.1 2 40.29 even 2
80.6.c.a.49.2 2 8.5 even 2
225.6.a.n.1.1 2 120.83 odd 4
225.6.a.n.1.2 2 120.107 odd 4
245.6.b.a.99.1 2 280.139 even 2
245.6.b.a.99.2 2 56.27 even 2
320.6.c.f.129.1 2 20.19 odd 2
320.6.c.f.129.2 2 4.3 odd 2
320.6.c.g.129.1 2 1.1 even 1 trivial
320.6.c.g.129.2 2 5.4 even 2 inner
400.6.a.t.1.1 2 40.13 odd 4
400.6.a.t.1.2 2 40.37 odd 4
720.6.f.f.289.1 2 24.5 odd 2
720.6.f.f.289.2 2 120.29 odd 2