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Properties

Label 8.30.b.a
Level $8$
Weight $30$
Character orbit 8.b
Analytic conductor $42.622$
Analytic rank $0$
Dimension $28$
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] = [8,30,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, 30); 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, 30, names="a")
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 30 \)
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: \(42.6224339064\)
Analytic rank: \(0\)
Dimension: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 8642 q^{2} - 69796972 q^{4} - 35869200308 q^{6} + 1356446145696 q^{7} - 9960937476728 q^{8} - 594796603828988 q^{9} - 62453907924808 q^{10} - 73\!\cdots\!44 q^{12} + 78\!\cdots\!40 q^{14} - 11\!\cdots\!36 q^{15}+ \cdots + 15\!\cdots\!18 q^{98}+O(q^{100}) \) 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 −23014.6 − 2683.36i 1.92933e6i 5.22470e8 + 1.23513e8i 2.40238e10i 5.17709e9 − 4.44027e10i −2.26247e12 −1.16930e13 − 4.24457e12i 6.49081e13 6.44647e13 − 5.52899e14i
5.2 −23014.6 + 2683.36i − 1.92933e6i 5.22470e8 − 1.23513e8i − 2.40238e10i 5.17709e9 + 4.44027e10i −2.26247e12 −1.16930e13 + 4.24457e12i 6.49081e13 6.44647e13 + 5.52899e14i
5.3 −21420.9 − 8832.69i − 1.37657e7i 3.80838e8 + 3.78408e8i 3.24318e9i −1.21588e11 + 2.94874e11i −1.14169e11 −4.81553e12 − 1.14697e13i −1.20865e14 2.86460e13 − 6.94719e13i
5.4 −21420.9 + 8832.69i 1.37657e7i 3.80838e8 − 3.78408e8i − 3.24318e9i −1.21588e11 − 2.94874e11i −1.14169e11 −4.81553e12 + 1.14697e13i −1.20865e14 2.86460e13 + 6.94719e13i
5.5 −21173.9 − 9409.30i 5.72019e6i 3.59801e8 + 3.98464e8i − 6.31246e9i 5.38230e10 − 1.21119e11i 2.79408e12 −3.86913e12 − 1.18225e13i 3.59098e13 −5.93958e13 + 1.33660e14i
5.6 −21173.9 + 9409.30i − 5.72019e6i 3.59801e8 − 3.98464e8i 6.31246e9i 5.38230e10 + 1.21119e11i 2.79408e12 −3.86913e12 + 1.18225e13i 3.59098e13 −5.93958e13 − 1.33660e14i
5.7 −15876.9 − 16875.8i 1.30630e7i −3.27158e7 + 5.35873e8i 6.85473e9i 2.20449e11 − 2.07401e11i −1.34577e12 9.56273e12 − 7.95592e12i −1.02013e14 1.15679e14 − 1.08832e14i
5.8 −15876.9 + 16875.8i − 1.30630e7i −3.27158e7 − 5.35873e8i − 6.85473e9i 2.20449e11 + 2.07401e11i −1.34577e12 9.56273e12 + 7.95592e12i −1.02013e14 1.15679e14 + 1.08832e14i
5.9 −13702.6 − 18684.5i − 2.53770e6i −1.61348e8 + 5.12052e8i − 1.41899e10i −4.74155e10 + 3.47731e10i −9.83944e11 1.17783e13 − 4.00175e12i 6.21905e13 −2.65131e14 + 1.94439e14i
5.10 −13702.6 + 18684.5i 2.53770e6i −1.61348e8 − 5.12052e8i 1.41899e10i −4.74155e10 − 3.47731e10i −9.83944e11 1.17783e13 + 4.00175e12i 6.21905e13 −2.65131e14 − 1.94439e14i
5.11 −8176.08 − 21680.0i − 6.50530e6i −4.03174e8 + 3.54515e8i 1.03890e10i −1.41035e11 + 5.31879e10i 3.70123e11 1.09823e13 + 5.84228e12i 2.63114e13 2.25234e14 − 8.49415e13i
5.12 −8176.08 + 21680.0i 6.50530e6i −4.03174e8 − 3.54515e8i − 1.03890e10i −1.41035e11 − 5.31879e10i 3.70123e11 1.09823e13 − 5.84228e12i 2.63114e13 2.25234e14 + 8.49415e13i
5.13 1176.52 − 23140.6i 9.62485e6i −5.34103e8 − 5.44508e7i − 1.90239e10i 2.22725e11 + 1.13238e10i −8.83863e11 −1.88841e12 + 1.22954e13i −2.40073e13 −4.40223e14 − 2.23820e13i
5.14 1176.52 + 23140.6i − 9.62485e6i −5.34103e8 + 5.44508e7i 1.90239e10i 2.22725e11 − 1.13238e10i −8.83863e11 −1.88841e12 − 1.22954e13i −2.40073e13 −4.40223e14 + 2.23820e13i
5.15 1364.59 − 23130.3i 7.47781e6i −5.33147e8 − 6.31264e7i 2.01913e10i 1.72964e11 + 1.02041e10i 1.70351e12 −2.18765e12 + 1.22457e13i 1.27128e13 4.67029e14 + 2.75527e13i
5.16 1364.59 + 23130.3i − 7.47781e6i −5.33147e8 + 6.31264e7i − 2.01913e10i 1.72964e11 − 1.02041e10i 1.70351e12 −2.18765e12 − 1.22457e13i 1.27128e13 4.67029e14 − 2.75527e13i
5.17 4985.59 − 22627.7i − 1.40185e7i −4.87159e8 − 2.25625e8i − 2.07995e10i −3.17207e11 − 6.98904e10i 2.73002e12 −7.53416e12 + 9.89843e12i −1.27888e14 −4.70646e14 − 1.03698e14i
5.18 4985.59 + 22627.7i 1.40185e7i −4.87159e8 + 2.25625e8i 2.07995e10i −3.17207e11 + 6.98904e10i 2.73002e12 −7.53416e12 − 9.89843e12i −1.27888e14 −4.70646e14 + 1.03698e14i
5.19 10885.6 − 20454.2i − 7.14793e6i −2.99879e8 − 4.45313e8i 6.52418e9i −1.46205e11 − 7.78094e10i −3.17101e12 −1.23729e13 + 1.28629e12i 1.75375e13 1.33447e14 + 7.10196e13i
5.20 10885.6 + 20454.2i 7.14793e6i −2.99879e8 + 4.45313e8i − 6.52418e9i −1.46205e11 + 7.78094e10i −3.17101e12 −1.23729e13 − 1.28629e12i 1.75375e13 1.33447e14 − 7.10196e13i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.28
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 8.30.b.a ✓ 28
4.b odd 2 1 32.30.b.a 28
8.b even 2 1 inner 8.30.b.a ✓ 28
8.d odd 2 1 32.30.b.a 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.30.b.a ✓ 28 1.a even 1 1 trivial
8.30.b.a ✓ 28 8.b even 2 1 inner
32.30.b.a 28 4.b odd 2 1
32.30.b.a 28 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{30}^{\mathrm{new}}(8, [\chi])\).