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Properties

Label 348.4.m.a
Level $348$
Weight $4$
Character orbit 348.m
Analytic conductor $20.533$
Analytic rank $0$
Dimension $48$
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] = [348,4,Mod(25,348)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(348, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 0, 8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("348.25");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 348 = 2^{2} \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 348.m (of order \(7\), degree \(6\), 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: \(20.5326646820\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 24 q^{3} - 21 q^{5} - 6 q^{7} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 24 q^{3} - 21 q^{5} - 6 q^{7} - 72 q^{9} - 99 q^{11} - 50 q^{13} - 63 q^{15} + 226 q^{17} + 34 q^{19} - 18 q^{21} - 82 q^{23} - 195 q^{25} - 216 q^{27} + 402 q^{29} - 88 q^{31} - 108 q^{33} + 1176 q^{35} + 14 q^{37} - 213 q^{39} + 802 q^{41} - 514 q^{43} - 63 q^{45} - 1082 q^{47} - 152 q^{49} + 426 q^{51} - 427 q^{53} - 1765 q^{55} + 480 q^{57} + 1970 q^{59} - 42 q^{61} + 135 q^{63} - 3085 q^{65} - 320 q^{67} - 1044 q^{69} - 1770 q^{71} - 855 q^{73} + 4812 q^{75} - 395 q^{77} + 1042 q^{79} - 648 q^{81} + 2124 q^{83} - 4806 q^{85} - 264 q^{87} + 5774 q^{89} + 1122 q^{91} - 264 q^{93} - 5848 q^{95} - 631 q^{97} + 1062 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 0 −2.70291 + 1.30165i 0 −4.78489 + 20.9640i 0 −7.98961 + 3.84759i 0 5.61141 − 7.03648i 0
25.2 0 −2.70291 + 1.30165i 0 −2.24804 + 9.84932i 0 −22.2842 + 10.7315i 0 5.61141 − 7.03648i 0
25.3 0 −2.70291 + 1.30165i 0 −1.69360 + 7.42016i 0 11.9551 − 5.75729i 0 5.61141 − 7.03648i 0
25.4 0 −2.70291 + 1.30165i 0 −0.678227 + 2.97151i 0 27.3843 − 13.1876i 0 5.61141 − 7.03648i 0
25.5 0 −2.70291 + 1.30165i 0 0.248698 − 1.08962i 0 −1.68005 + 0.809067i 0 5.61141 − 7.03648i 0
25.6 0 −2.70291 + 1.30165i 0 1.67034 − 7.31824i 0 −23.3314 + 11.2358i 0 5.61141 − 7.03648i 0
25.7 0 −2.70291 + 1.30165i 0 3.19457 − 13.9963i 0 −8.94742 + 4.30885i 0 5.61141 − 7.03648i 0
25.8 0 −2.70291 + 1.30165i 0 4.18339 − 18.3286i 0 26.0962 − 12.5673i 0 5.61141 − 7.03648i 0
49.1 0 1.87047 + 2.34549i 0 −18.5989 + 8.95675i 0 −22.5249 − 28.2454i 0 −2.00269 + 8.77435i 0
49.2 0 1.87047 + 2.34549i 0 −11.4946 + 5.53553i 0 18.7669 + 23.5329i 0 −2.00269 + 8.77435i 0
49.3 0 1.87047 + 2.34549i 0 −10.9781 + 5.28677i 0 2.95054 + 3.69986i 0 −2.00269 + 8.77435i 0
49.4 0 1.87047 + 2.34549i 0 −5.71274 + 2.75111i 0 3.04397 + 3.81702i 0 −2.00269 + 8.77435i 0
49.5 0 1.87047 + 2.34549i 0 1.02354 − 0.492913i 0 −12.8463 − 16.1087i 0 −2.00269 + 8.77435i 0
49.6 0 1.87047 + 2.34549i 0 8.27101 − 3.98311i 0 −3.44589 − 4.32101i 0 −2.00269 + 8.77435i 0
49.7 0 1.87047 + 2.34549i 0 11.9368 − 5.74847i 0 −9.05067 − 11.3492i 0 −2.00269 + 8.77435i 0
49.8 0 1.87047 + 2.34549i 0 14.4307 − 6.94946i 0 19.7359 + 24.7481i 0 −2.00269 + 8.77435i 0
169.1 0 −0.667563 − 2.92478i 0 −13.5361 − 16.9737i 0 0.471340 + 2.06507i 0 −8.10872 + 3.90495i 0
169.2 0 −0.667563 − 2.92478i 0 −5.74107 − 7.19907i 0 4.41308 + 19.3350i 0 −8.10872 + 3.90495i 0
169.3 0 −0.667563 − 2.92478i 0 −4.59117 − 5.75715i 0 −7.23694 − 31.7071i 0 −8.10872 + 3.90495i 0
169.4 0 −0.667563 − 2.92478i 0 −4.24829 − 5.32719i 0 1.89895 + 8.31984i 0 −8.10872 + 3.90495i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 348.4.m.a ✓ 48
29.d even 7 1 inner 348.4.m.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
348.4.m.a ✓ 48 1.a even 1 1 trivial
348.4.m.a ✓ 48 29.d even 7 1 inner