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Properties

Label 17.12.c.a
Level $17$
Weight $12$
Character orbit 17.c
Analytic conductor $13.062$
Analytic rank $0$
Dimension $32$
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] = [17,12,Mod(4,17)] 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("17.4"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(17, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 17.c (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: \(13.0618340695\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 528 q^{3} - 32772 q^{4} - 15496 q^{5} + 39058 q^{6} - 38368 q^{7} + 178138 q^{10} - 24816 q^{11} + 1322202 q^{12} - 1533752 q^{13} - 757692 q^{14} + 22082308 q^{16} + 17346416 q^{17} + 16775420 q^{18}+ \cdots + 229465269792 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 − 85.4879i 62.9008 − 62.9008i −5260.19 −835.746 + 835.746i −5377.26 − 5377.26i −4912.00 − 4912.00i 274603.i 169234.i 71446.2 + 71446.2i
4.2 − 66.0051i −572.909 + 572.909i −2308.67 −7460.40 + 7460.40i 37814.9 + 37814.9i −24652.1 − 24652.1i 17205.8i − 479304.i 492425. + 492425.i
4.3 − 58.9332i 500.660 − 500.660i −1425.12 2487.06 − 2487.06i −29505.5 − 29505.5i −731.266 − 731.266i − 36708.3i − 324174.i −146570. − 146570.i
4.4 − 58.2592i −297.283 + 297.283i −1346.14 8591.36 − 8591.36i 17319.5 + 17319.5i −6474.40 − 6474.40i − 40889.9i 393.067i −500526. − 500526.i
4.5 − 48.9428i −20.4617 + 20.4617i −347.397 −2628.43 + 2628.43i 1001.45 + 1001.45i 46739.6 + 46739.6i − 83232.3i 176310.i 128643. + 128643.i
4.6 − 32.2416i 149.760 − 149.760i 1008.48 −6170.50 + 6170.50i −4828.50 − 4828.50i −42929.1 − 42929.1i − 98545.8i 132291.i 198947. + 198947.i
4.7 − 5.06273i −258.651 + 258.651i 2022.37 2155.39 − 2155.39i 1309.48 + 1309.48i −31309.1 − 31309.1i − 20607.2i 43346.3i −10912.2 − 10912.2i
4.8 3.27628i 288.856 − 288.856i 2037.27 5645.17 − 5645.17i 946.373 + 946.373i −182.666 − 182.666i 13384.5i 10271.8i 18495.2 + 18495.2i
4.9 4.94429i −400.826 + 400.826i 2023.55 −1207.06 + 1207.06i −1981.80 − 1981.80i 50168.6 + 50168.6i 20130.9i − 144177.i −5968.07 − 5968.07i
4.10 21.1462i 485.546 − 485.546i 1600.84 −8779.09 + 8779.09i 10267.5 + 10267.5i 43052.7 + 43052.7i 77159.1i − 294363.i −185644. − 185644.i
4.11 44.2190i −276.864 + 276.864i 92.6798 −5753.13 + 5753.13i −12242.6 − 12242.6i −23295.1 − 23295.1i 94658.7i 23839.8i −254398. − 254398.i
4.12 49.5175i 136.167 − 136.167i −403.984 −1140.26 + 1140.26i 6742.65 + 6742.65i −14036.7 − 14036.7i 81407.6i 140064.i −56462.7 − 56462.7i
4.13 63.2626i 72.6870 − 72.6870i −1954.16 5432.90 − 5432.90i 4598.37 + 4598.37i 37054.9 + 37054.9i 5936.57i 166580.i 343700. + 343700.i
4.14 66.7457i −522.898 + 522.898i −2406.98 7524.98 − 7524.98i −34901.2 − 34901.2i −23035.1 − 23035.1i − 23960.6i − 369698.i 502260. + 502260.i
4.15 77.1558i 534.614 − 534.614i −3905.01 775.823 − 775.823i 41248.6 + 41248.6i −52299.8 − 52299.8i − 143279.i − 394478.i 59859.2 + 59859.2i
4.16 88.6653i −145.298 + 145.298i −5813.53 −6386.08 + 6386.08i −12882.9 − 12882.9i 27657.5 + 27657.5i − 333872.i 134924.i −566224. − 566224.i
13.1 − 88.6653i −145.298 − 145.298i −5813.53 −6386.08 − 6386.08i −12882.9 + 12882.9i 27657.5 − 27657.5i 333872.i − 134924.i −566224. + 566224.i
13.2 − 77.1558i 534.614 + 534.614i −3905.01 775.823 + 775.823i 41248.6 − 41248.6i −52299.8 + 52299.8i 143279.i 394478.i 59859.2 − 59859.2i
13.3 − 66.7457i −522.898 − 522.898i −2406.98 7524.98 + 7524.98i −34901.2 + 34901.2i −23035.1 + 23035.1i 23960.6i 369698.i 502260. − 502260.i
13.4 − 63.2626i 72.6870 + 72.6870i −1954.16 5432.90 + 5432.90i 4598.37 − 4598.37i 37054.9 − 37054.9i − 5936.57i − 166580.i 343700. − 343700.i
See all 32 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 4.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.12.c.a ✓ 32
17.c even 4 1 inner 17.12.c.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.12.c.a ✓ 32 1.a even 1 1 trivial
17.12.c.a ✓ 32 17.c even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(17, [\chi])\).