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Properties

Label 80.12.j.a
Level $80$
Weight $12$
Character orbit 80.j
Analytic conductor $61.467$
Analytic rank $0$
Dimension $260$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [80,12,Mod(43,80)] 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("80.43"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 3])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 80.j (of order \(4\), degree \(2\), 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: \(61.4674544448\)
Analytic rank: \(0\)
Dimension: \(260\)
Relative dimension: \(130\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 260 q - 2 q^{2} - 3956 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} - 74840 q^{8} - 14880348 q^{9} - 4098 q^{10} - 4 q^{11} - 114212 q^{12} - 4 q^{13} + 708588 q^{15} + 5740152 q^{16} - 4 q^{17} - 6791722 q^{18}+ \cdots - 189350900660 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −45.2481 + 0.780472i 201.994i 2046.78 − 70.6297i −705.591 + 6952.00i −157.651 − 9139.85i 18488.6 − 18488.6i −92557.9 + 4793.32i 136345. 26500.8 − 315115.i
43.2 −45.2246 − 1.65339i − 660.049i 2042.53 + 149.548i −6629.07 + 2209.87i −1091.32 + 29850.5i 54915.2 − 54915.2i −92125.5 − 10140.3i −258517. 303451. − 88980.2i
43.3 −45.2180 − 1.82444i 429.304i 2041.34 + 164.995i 5272.30 − 4585.95i 783.240 − 19412.3i 1887.77 − 1887.77i −92004.5 − 11185.1i −7154.98 −246770. + 197749.i
43.4 −45.2124 + 1.95953i − 400.326i 2040.32 − 177.190i −2945.53 − 6336.56i 784.450 + 18099.7i −19284.5 + 19284.5i −91900.6 + 12009.2i 16885.9 145591. + 280719.i
43.5 −45.2056 − 2.11039i − 177.435i 2039.09 + 190.803i 6763.40 − 1756.30i −374.457 + 8021.06i −41152.5 + 41152.5i −91775.7 − 12928.6i 145664. −309450. + 65121.3i
43.6 −45.0581 + 4.21476i 166.345i 2012.47 − 379.818i −1133.92 − 6895.10i −701.103 − 7495.19i 59238.9 − 59238.9i −89077.4 + 25596.0i 149476. 80153.5 + 305901.i
43.7 −44.9593 − 5.16327i − 139.132i 1994.68 + 464.274i −5972.08 + 3628.00i −718.373 + 6255.26i −56253.4 + 56253.4i −87282.4 − 31172.5i 157789. 287233. − 132277.i
43.8 −44.3359 − 9.07354i − 728.725i 1883.34 + 804.567i 2131.92 − 6654.55i −6612.12 + 32308.7i −2579.41 + 2579.41i −76199.3 − 52759.8i −353893. −154901. + 275691.i
43.9 −43.6623 + 11.8996i 613.176i 1764.80 − 1039.13i 3522.49 + 6034.91i −7296.54 − 26772.7i −52076.2 + 52076.2i −64690.1 + 66371.1i −198838. −225613. − 221582.i
43.10 −43.6367 + 11.9933i − 219.501i 1760.32 − 1046.69i 5720.85 + 4012.48i 2632.53 + 9578.30i 37491.7 − 37491.7i −64261.4 + 66786.2i 128966. −297762. − 106480.i
43.11 −43.5097 + 12.4460i − 736.922i 1738.20 − 1083.04i 5257.45 + 4602.98i 9171.70 + 32063.3i −18835.9 + 18835.9i −62149.0 + 68756.4i −365907. −286039. − 134840.i
43.12 −43.4378 − 12.6949i 810.756i 1725.68 + 1102.88i −2601.69 − 6485.32i 10292.5 − 35217.4i −28299.4 + 28299.4i −60958.7 − 69813.9i −480179. 30681.0 + 314736.i
43.13 −43.2490 + 13.3237i 538.459i 1692.96 − 1152.48i −6958.96 − 633.204i −7174.27 − 23287.8i 578.185 − 578.185i −57863.5 + 72399.9i −112791. 309405. − 65333.8i
43.14 −43.1911 − 13.5105i 326.557i 1682.93 + 1167.06i −6985.70 + 167.856i 4411.95 − 14104.4i −12381.5 + 12381.5i −56920.1 − 73144.0i 70507.3 303987. + 87130.3i
43.15 −42.3976 − 15.8253i − 209.575i 1547.12 + 1341.91i 4210.05 + 5577.06i −3316.58 + 8885.47i 15334.6 − 15334.6i −44358.2 − 81377.4i 133225. −90237.6 − 303079.i
43.16 −41.9762 + 16.9115i 792.470i 1476.00 − 1419.76i 6967.42 + 532.099i −13401.9 − 33264.9i 40707.4 − 40707.4i −37946.5 + 84557.6i −450861. −301465. + 95494.2i
43.17 −41.7502 − 17.4621i 750.121i 1438.15 + 1458.09i −2602.99 + 6484.79i 13098.7 − 31317.7i 58461.1 − 58461.1i −34581.8 − 85988.6i −385535. 221913. − 225288.i
43.18 −41.4489 + 18.1657i − 263.479i 1388.02 − 1505.89i −4359.86 − 5460.75i 4786.27 + 10920.9i −12369.2 + 12369.2i −30176.3 + 87631.8i 107726. 279909. + 147142.i
43.19 −40.9882 + 19.1824i − 409.401i 1312.07 − 1572.50i −5145.70 + 4727.56i 7853.29 + 16780.7i −1805.56 + 1805.56i −23615.2 + 89622.9i 9537.42 120228. − 292481.i
43.20 −40.8323 − 19.5121i 493.884i 1286.55 + 1593.45i 5328.96 + 4519.99i 9636.73 − 20166.4i −20138.0 + 20138.0i −21441.3 − 90167.7i −66774.3 −129399. − 288541.i
See next 80 embeddings (of 260 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.130
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.12.j.a ✓ 260
5.c odd 4 1 80.12.s.a yes 260
16.f odd 4 1 80.12.s.a yes 260
80.j even 4 1 inner 80.12.j.a ✓ 260
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.12.j.a ✓ 260 1.a even 1 1 trivial
80.12.j.a ✓ 260 80.j even 4 1 inner
80.12.s.a yes 260 5.c odd 4 1
80.12.s.a yes 260 16.f odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(80, [\chi])\).