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Properties

Label 348.4.m.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{3} + 41 q^{5} - 6 q^{7} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{3} + 41 q^{5} - 6 q^{7} - 72 q^{9} + 99 q^{11} - 70 q^{13} - 123 q^{15} - 102 q^{17} - 18 q^{19} + 18 q^{21} + 126 q^{23} - 411 q^{25} + 216 q^{27} - 952 q^{29} - 132 q^{31} - 108 q^{33} + 832 q^{35} - 386 q^{37} + 189 q^{39} - 326 q^{41} + 542 q^{43} - 387 q^{45} - 1218 q^{47} - 296 q^{49} - 30 q^{51} + 665 q^{53} + 1291 q^{55} + 1272 q^{57} - 850 q^{59} + 690 q^{61} + 135 q^{63} + 1231 q^{65} - 68 q^{67} - 840 q^{69} - 4054 q^{71} + 2433 q^{73} - 4332 q^{75} + 31 q^{77} - 1658 q^{79} - 648 q^{81} + 3168 q^{83} + 966 q^{85} - 630 q^{87} - 1994 q^{89} - 4046 q^{91} + 396 q^{93} + 1240 q^{95} - 335 q^{97} - 1062 q^{99}+O(q^{100}) \) 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 −3.93523 + 17.2414i 0 −11.3305 + 5.45648i 0 5.61141 − 7.03648i 0
25.2 0 2.70291 − 1.30165i 0 −3.54783 + 15.5441i 0 24.8367 − 11.9607i 0 5.61141 − 7.03648i 0
25.3 0 2.70291 − 1.30165i 0 −0.674364 + 2.95458i 0 −5.05881 + 2.43619i 0 5.61141 − 7.03648i 0
25.4 0 2.70291 − 1.30165i 0 0.218076 − 0.955454i 0 −22.5648 + 10.8666i 0 5.61141 − 7.03648i 0
25.5 0 2.70291 − 1.30165i 0 0.364673 − 1.59774i 0 −0.0422075 + 0.0203261i 0 5.61141 − 7.03648i 0
25.6 0 2.70291 − 1.30165i 0 2.27128 − 9.95111i 0 25.3452 − 12.2056i 0 5.61141 − 7.03648i 0
25.7 0 2.70291 − 1.30165i 0 3.54057 − 15.5122i 0 18.3464 − 8.83516i 0 5.61141 − 7.03648i 0
25.8 0 2.70291 − 1.30165i 0 4.64539 − 20.3528i 0 −28.3291 + 13.6426i 0 5.61141 − 7.03648i 0
49.1 0 −1.87047 − 2.34549i 0 −18.0188 + 8.67742i 0 6.40272 + 8.02876i 0 −2.00269 + 8.77435i 0
49.2 0 −1.87047 − 2.34549i 0 −8.69346 + 4.18655i 0 −12.0179 − 15.0699i 0 −2.00269 + 8.77435i 0
49.3 0 −1.87047 − 2.34549i 0 −4.19035 + 2.01797i 0 −22.4215 − 28.1157i 0 −2.00269 + 8.77435i 0
49.4 0 −1.87047 − 2.34549i 0 −0.635767 + 0.306169i 0 17.8052 + 22.3270i 0 −2.00269 + 8.77435i 0
49.5 0 −1.87047 − 2.34549i 0 0.410340 − 0.197609i 0 1.84816 + 2.31751i 0 −2.00269 + 8.77435i 0
49.6 0 −1.87047 − 2.34549i 0 4.46281 − 2.14917i 0 −0.902240 − 1.13137i 0 −2.00269 + 8.77435i 0
49.7 0 −1.87047 − 2.34549i 0 15.7220 − 7.57130i 0 9.94677 + 12.4729i 0 −2.00269 + 8.77435i 0
49.8 0 −1.87047 − 2.34549i 0 18.0559 − 8.69527i 0 −4.03169 − 5.05558i 0 −2.00269 + 8.77435i 0
169.1 0 0.667563 + 2.92478i 0 −9.22300 − 11.5653i 0 −3.91473 − 17.1516i 0 −8.10872 + 3.90495i 0
169.2 0 0.667563 + 2.92478i 0 −6.05182 − 7.58874i 0 1.43500 + 6.28716i 0 −8.10872 + 3.90495i 0
169.3 0 0.667563 + 2.92478i 0 −3.98898 − 5.00202i 0 7.38534 + 32.3573i 0 −8.10872 + 3.90495i 0
169.4 0 0.667563 + 2.92478i 0 −0.802340 − 1.00610i 0 −3.53949 − 15.5075i 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.b ✓ 48
29.d even 7 1 inner 348.4.m.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
348.4.m.b ✓ 48 1.a even 1 1 trivial
348.4.m.b ✓ 48 29.d even 7 1 inner