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Properties

Label 348.5.d.a
Level $348$
Weight $5$
Character orbit 348.d
Analytic conductor $35.973$
Analytic rank $0$
Dimension $38$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [348,5,Mod(233,348)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(348, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("348.233");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 348 = 2^{2} \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 348.d (of order \(2\), degree \(1\), 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: \(35.9727471532\)
Analytic rank: \(0\)
Dimension: \(38\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 38 q - 18 q^{3} + 108 q^{7} - 126 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q - 18 q^{3} + 108 q^{7} - 126 q^{9} - 72 q^{13} + 74 q^{15} + 352 q^{19} + 832 q^{21} - 5750 q^{25} - 1800 q^{27} - 3080 q^{31} - 1716 q^{33} + 5840 q^{37} + 274 q^{39} + 2540 q^{43} + 4584 q^{45} + 6466 q^{49} - 3244 q^{51} + 4884 q^{55} - 3804 q^{57} + 1872 q^{61} + 4484 q^{63} - 18864 q^{67} + 7412 q^{69} + 3012 q^{73} + 2040 q^{75} + 1168 q^{79} - 2154 q^{81} + 27440 q^{85} + 16092 q^{91} + 17200 q^{93} - 10980 q^{97} - 2058 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
233.1 0 −8.99479 − 0.306065i 0 − 13.6332i 0 37.9849 0 80.8126 + 5.50598i 0
233.2 0 −8.99479 + 0.306065i 0 13.6332i 0 37.9849 0 80.8126 − 5.50598i 0
233.3 0 −8.99361 − 0.339207i 0 − 33.2276i 0 −62.7775 0 80.7699 + 6.10138i 0
233.4 0 −8.99361 + 0.339207i 0 33.2276i 0 −62.7775 0 80.7699 − 6.10138i 0
233.5 0 −8.20482 − 3.69877i 0 − 26.8042i 0 −63.0211 0 53.6381 + 60.6956i 0
233.6 0 −8.20482 + 3.69877i 0 26.8042i 0 −63.0211 0 53.6381 − 60.6956i 0
233.7 0 −7.56021 − 4.88295i 0 23.3912i 0 69.3007 0 33.3135 + 73.8323i 0
233.8 0 −7.56021 + 4.88295i 0 − 23.3912i 0 69.3007 0 33.3135 − 73.8323i 0
233.9 0 −6.30154 − 6.42578i 0 − 45.9287i 0 61.6810 0 −1.58126 + 80.9846i 0
233.10 0 −6.30154 + 6.42578i 0 45.9287i 0 61.6810 0 −1.58126 − 80.9846i 0
233.11 0 −6.25641 − 6.46972i 0 32.7047i 0 −15.1314 0 −2.71467 + 80.9545i 0
233.12 0 −6.25641 + 6.46972i 0 − 32.7047i 0 −15.1314 0 −2.71467 − 80.9545i 0
233.13 0 −6.11052 − 6.60768i 0 0.759659i 0 −1.24824 0 −6.32298 + 80.7528i 0
233.14 0 −6.11052 + 6.60768i 0 − 0.759659i 0 −1.24824 0 −6.32298 − 80.7528i 0
233.15 0 −2.08921 − 8.75415i 0 − 30.6843i 0 37.9166 0 −72.2704 + 36.5785i 0
233.16 0 −2.08921 + 8.75415i 0 30.6843i 0 37.9166 0 −72.2704 − 36.5785i 0
233.17 0 −1.94159 − 8.78807i 0 − 14.9471i 0 −35.0181 0 −73.4605 + 34.1257i 0
233.18 0 −1.94159 + 8.78807i 0 14.9471i 0 −35.0181 0 −73.4605 − 34.1257i 0
233.19 0 −1.00804 − 8.94337i 0 17.6112i 0 −87.3105 0 −78.9677 + 18.0306i 0
233.20 0 −1.00804 + 8.94337i 0 − 17.6112i 0 −87.3105 0 −78.9677 − 18.0306i 0
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 233.38
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 348.5.d.a ✓ 38
3.b odd 2 1 inner 348.5.d.a ✓ 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
348.5.d.a ✓ 38 1.a even 1 1 trivial
348.5.d.a ✓ 38 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(348, [\chi])\).