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Properties

Label 308.4.w.a
Level $308$
Weight $4$
Character orbit 308.w
Analytic conductor $18.173$
Analytic rank $0$
Dimension $96$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [308,4,Mod(13,308)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(308, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 5, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("308.13");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 308.w (of order \(10\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.1725882818\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(24\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 10 q^{7} + 168 q^{9} + 60 q^{11} - 192 q^{15} - 112 q^{23} + 548 q^{25} - 700 q^{29} - 940 q^{35} + 288 q^{37} + 460 q^{39} + 90 q^{49} - 1420 q^{51} - 2240 q^{53} + 560 q^{57} + 320 q^{63} + 4272 q^{67}+ \cdots - 10212 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 0 −5.88297 − 8.09721i 0 18.0330 − 5.85929i 0 −18.1193 − 3.83270i 0 −22.6120 + 69.5927i 0
13.2 0 −5.22436 − 7.19072i 0 −16.6762 + 5.41844i 0 18.4891 − 1.07412i 0 −16.0690 + 49.4553i 0
13.3 0 −5.12163 − 7.04932i 0 4.34391 − 1.41142i 0 9.70945 − 15.7711i 0 −15.1184 + 46.5296i 0
13.4 0 −4.13359 − 5.68940i 0 −12.0841 + 3.92637i 0 −18.5068 + 0.704636i 0 −6.93923 + 21.3567i 0
13.5 0 −3.84999 − 5.29905i 0 4.60773 − 1.49714i 0 11.4880 + 14.5268i 0 −4.91409 + 15.1240i 0
13.6 0 −3.77642 − 5.19779i 0 2.80908 − 0.912724i 0 −2.94649 + 18.2844i 0 −4.41225 + 13.5795i 0
13.7 0 −2.54235 − 3.49924i 0 2.06273 − 0.670222i 0 −10.8266 − 15.0262i 0 2.56231 − 7.88597i 0
13.8 0 −2.29629 − 3.16057i 0 12.0476 − 3.91452i 0 16.4615 − 8.48648i 0 3.62719 − 11.1634i 0
13.9 0 −2.00983 − 2.76629i 0 −13.3670 + 4.34320i 0 −7.33070 − 17.0077i 0 4.73050 − 14.5590i 0
13.10 0 −0.802077 − 1.10396i 0 −4.83302 + 1.57034i 0 2.45890 + 18.3563i 0 7.76805 − 23.9076i 0
13.11 0 −0.468867 − 0.645340i 0 18.3612 − 5.96593i 0 −15.1893 + 10.5964i 0 8.14683 − 25.0734i 0
13.12 0 −0.129653 − 0.178452i 0 −11.1231 + 3.61412i 0 18.0478 − 4.15670i 0 8.32842 − 25.6323i 0
13.13 0 0.129653 + 0.178452i 0 11.1231 − 3.61412i 0 −1.62381 − 18.4489i 0 8.32842 − 25.6323i 0
13.14 0 0.468867 + 0.645340i 0 −18.3612 + 5.96593i 0 −5.38402 + 17.7204i 0 8.14683 − 25.0734i 0
13.15 0 0.802077 + 1.10396i 0 4.83302 − 1.57034i 0 −18.2177 + 3.33385i 0 7.76805 − 23.9076i 0
13.16 0 2.00983 + 2.76629i 0 13.3670 − 4.34320i 0 18.4406 + 1.71625i 0 4.73050 − 14.5590i 0
13.17 0 2.29629 + 3.16057i 0 −12.0476 + 3.91452i 0 2.98425 − 18.2782i 0 3.62719 − 11.1634i 0
13.18 0 2.54235 + 3.49924i 0 −2.06273 + 0.670222i 0 17.6363 + 5.65332i 0 2.56231 − 7.88597i 0
13.19 0 3.77642 + 5.19779i 0 −2.80908 + 0.912724i 0 −16.4790 + 8.45246i 0 −4.41225 + 13.5795i 0
13.20 0 3.84999 + 5.29905i 0 −4.60773 + 1.49714i 0 −17.3657 − 6.43669i 0 −4.91409 + 15.1240i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.d odd 10 1 inner
77.l even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 308.4.w.a ✓ 96
7.b odd 2 1 inner 308.4.w.a ✓ 96
11.d odd 10 1 inner 308.4.w.a ✓ 96
77.l even 10 1 inner 308.4.w.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.4.w.a ✓ 96 1.a even 1 1 trivial
308.4.w.a ✓ 96 7.b odd 2 1 inner
308.4.w.a ✓ 96 11.d odd 10 1 inner
308.4.w.a ✓ 96 77.l even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(308, [\chi])\).