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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8,24,Mod(5,8)] 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("8.5"); S:= CuspForms(chi, 24); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 24, names="a")
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 8.b (of order \(2\), degree \(1\), 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: \(26.8163229876\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 966 q^{2} - 2692748 q^{4} + 808474636 q^{6} - 3954653488 q^{7} - 62449515288 q^{8} - 627621192182 q^{9} + 306551586824 q^{10} + 4738963291912 q^{12} + 2554976858640 q^{14} + 34787309795152 q^{15}+ \cdots - 16\!\cdots\!62 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −2853.29 − 497.312i − 539585.i 7.89397e6 + 2.83796e6i 1.50931e8i −2.68342e8 + 1.53959e9i 7.15109e9 −2.11125e10 − 1.20233e10i −1.97009e11 7.50598e10 − 4.30650e11i
5.2 −2853.29 + 497.312i 539585.i 7.89397e6 − 2.83796e6i − 1.50931e8i −2.68342e8 − 1.53959e9i 7.15109e9 −2.11125e10 + 1.20233e10i −1.97009e11 7.50598e10 + 4.30650e11i
5.3 −2851.53 − 507.341i − 171052.i 7.87382e6 + 2.89339e6i − 1.49706e8i −8.67818e7 + 4.87760e8i −5.79730e9 −2.09845e10 − 1.22453e10i 6.48843e10 −7.59522e10 + 4.26892e11i
5.4 −2851.53 + 507.341i 171052.i 7.87382e6 − 2.89339e6i 1.49706e8i −8.67818e7 − 4.87760e8i −5.79730e9 −2.09845e10 + 1.22453e10i 6.48843e10 −7.59522e10 − 4.26892e11i
5.5 −2179.65 − 1907.29i 204533.i 1.11314e6 + 8.31443e6i 4.09062e7i 3.90104e8 − 4.45811e8i 2.70002e9 1.34317e10 − 2.02456e10i 5.23093e10 7.80199e10 − 8.91613e10i
5.6 −2179.65 + 1907.29i − 204533.i 1.11314e6 − 8.31443e6i − 4.09062e7i 3.90104e8 + 4.45811e8i 2.70002e9 1.34317e10 + 2.02456e10i 5.23093e10 7.80199e10 + 8.91613e10i
5.7 −1299.20 − 2588.57i − 383993.i −5.01275e6 + 6.72615e6i − 7.63811e7i −9.93991e8 + 4.98884e8i −2.00080e9 2.39237e10 + 4.23723e9i −5.33073e10 −1.97718e11 + 9.92345e10i
5.8 −1299.20 + 2588.57i 383993.i −5.01275e6 − 6.72615e6i 7.63811e7i −9.93991e8 − 4.98884e8i −2.00080e9 2.39237e10 − 4.23723e9i −5.33073e10 −1.97718e11 − 9.92345e10i
5.9 −410.583 − 2867.06i 545453.i −8.05145e6 + 2.35433e6i − 5.01084e7i 1.56385e9 − 2.23954e8i −8.97921e9 1.00558e10 + 2.21173e10i −2.03376e11 −1.43664e11 + 2.05737e10i
5.10 −410.583 + 2867.06i − 545453.i −8.05145e6 − 2.35433e6i 5.01084e7i 1.56385e9 + 2.23954e8i −8.97921e9 1.00558e10 − 2.21173e10i −2.03376e11 −1.43664e11 − 2.05737e10i
5.11 172.429 − 2891.17i − 182955.i −8.32914e6 − 997045.i 1.93448e8i −5.28954e8 − 3.15467e7i −3.28206e9 −4.31882e9 + 2.39091e10i 6.06707e10 5.59291e11 + 3.33561e10i
5.12 172.429 + 2891.17i 182955.i −8.32914e6 + 997045.i − 1.93448e8i −5.28954e8 + 3.15467e7i −3.28206e9 −4.31882e9 − 2.39091e10i 6.06707e10 5.59291e11 − 3.33561e10i
5.13 566.751 − 2840.32i 81213.9i −7.74619e6 − 3.21950e6i − 9.51649e7i 2.30673e8 + 4.60280e7i 8.76687e9 −1.35346e10 + 2.01770e10i 8.75475e10 −2.70298e11 − 5.39348e10i
5.14 566.751 + 2840.32i − 81213.9i −7.74619e6 + 3.21950e6i 9.51649e7i 2.30673e8 − 4.60280e7i 8.76687e9 −1.35346e10 − 2.01770e10i 8.75475e10 −2.70298e11 + 5.39348e10i
5.15 2085.75 − 2009.54i − 469027.i 312107. − 8.38280e6i − 3.92647e7i −9.42529e8 − 9.78274e8i 6.41467e8 −1.61946e10 − 1.81116e10i −1.25843e11 −7.89040e10 − 8.18964e10i
5.16 2085.75 + 2009.54i 469027.i 312107. + 8.38280e6i 3.92647e7i −9.42529e8 + 9.78274e8i 6.41467e8 −1.61946e10 + 1.81116e10i −1.25843e11 −7.89040e10 + 8.18964e10i
5.17 2118.82 − 1974.65i 107637.i 590159. − 8.36782e6i − 7.28539e7i 2.12546e8 + 2.28064e8i −7.45772e9 −1.52730e10 − 1.88952e10i 8.25574e10 −1.43861e11 − 1.54364e11i
5.18 2118.82 + 1974.65i − 107637.i 590159. + 8.36782e6i 7.28539e7i 2.12546e8 − 2.28064e8i −7.45772e9 −1.52730e10 + 1.88952e10i 8.25574e10 −1.43861e11 + 1.54364e11i
5.19 2237.26 − 1839.37i 452526.i 1.62204e6 − 8.23029e6i 1.90005e8i 8.32364e8 + 1.01242e9i 5.12753e9 −1.15096e10 − 2.13968e10i −1.10637e11 3.49490e11 + 4.25091e11i
5.20 2237.26 + 1839.37i − 452526.i 1.62204e6 + 8.23029e6i − 1.90005e8i 8.32364e8 − 1.01242e9i 5.12753e9 −1.15096e10 + 2.13968e10i −1.10637e11 3.49490e11 − 4.25091e11i
See all 22 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 5.22
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.24.b.a ✓ 22
4.b odd 2 1 32.24.b.a 22
8.b even 2 1 inner 8.24.b.a ✓ 22
8.d odd 2 1 32.24.b.a 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.24.b.a ✓ 22 1.a even 1 1 trivial
8.24.b.a ✓ 22 8.b even 2 1 inner
32.24.b.a 22 4.b odd 2 1
32.24.b.a 22 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{24}^{\mathrm{new}}(8, [\chi])\).