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Properties

Label 167.14.a.a
Level $167$
Weight $14$
Character orbit 167.a
Self dual yes
Analytic conductor $179.076$
Analytic rank $1$
Dimension $85$
CM no
Inner twists $1$

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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] = [167,14,Mod(1,167)] 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("167.1"); S:= CuspForms(chi, 14); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(167, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 14, names="a")
 
Level: \( N \) \(=\) \( 167 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 167.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [85] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(179.075651350\)
Analytic rank: \(1\)
Dimension: \(85\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 85 q - 193 q^{2} - 2917 q^{3} + 331957 q^{4} - 109376 q^{5} - 60032 q^{6} - 1411789 q^{7} - 2881356 q^{8} + 40832500 q^{9} - 8558507 q^{10} - 11363820 q^{11} - 42931455 q^{12} - 74147467 q^{13} - 56786925 q^{14}+ \cdots + 6211433505660 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −178.438 2449.32 23647.9 −30890.9 −437051. 251132. −2.75792e6 4.40485e6 5.51210e6
1.2 −174.584 −1666.24 22287.7 33186.8 290900. −264242. −2.46090e6 1.18203e6 −5.79390e6
1.3 −173.457 −392.986 21895.4 36680.0 68166.3 409636. −2.37695e6 −1.43988e6 −6.36240e6
1.4 −163.461 −704.860 18527.5 −16742.1 115217. 286917. −1.68946e6 −1.09750e6 2.73669e6
1.5 −159.772 −136.142 17335.1 −27185.9 21751.6 −294437. −1.46080e6 −1.57579e6 4.34354e6
1.6 −158.701 1414.71 16993.9 39372.8 −224515. 293225. −1.39687e6 407078. −6.24848e6
1.7 −157.779 −1188.82 16702.1 54989.3 187570. −241207. −1.34271e6 −181031. −8.67615e6
1.8 −154.403 −2342.94 15648.2 −14336.5 361756. −515513. −1.15126e6 3.89504e6 2.21360e6
1.9 −150.381 −36.8593 14422.5 −36639.6 5542.94 253744. −936948. −1.59296e6 5.50990e6
1.10 −149.858 956.498 14265.4 −56663.3 −143339. 89224.7 −910154. −679434. 8.49145e6
1.11 −149.681 1780.17 14212.4 −51917.4 −266458. −429681. −901147. 1.57468e6 7.77105e6
1.12 −146.939 −1737.20 13399.1 −20800.5 255263. 184836. −765131. 1.42355e6 3.05641e6
1.13 −145.105 2039.85 12863.6 8583.30 −295993. 141636. −677869. 2.56666e6 −1.24548e6
1.14 −143.454 −491.527 12387.2 49284.1 70511.7 −178528. −601819. −1.35272e6 −7.07003e6
1.15 −137.541 272.835 10725.5 65608.1 −37525.9 −163860. −348458. −1.51988e6 −9.02379e6
1.16 −127.919 −758.364 8171.27 −29773.5 97009.2 −330126. 2651.95 −1.01921e6 3.80859e6
1.17 −126.366 −2268.06 7776.48 −59918.1 286607. 39726.0 52507.7 3.54978e6 7.57163e6
1.18 −119.818 2248.90 6164.28 −11784.6 −269458. −314083. 242957. 3.46324e6 1.41201e6
1.19 −111.751 1646.12 4296.39 28083.0 −183957. −352691. 435340. 1.11539e6 −3.13831e6
1.20 −106.428 1245.75 3135.01 51351.3 −132583. 342083. 538208. −42439.5 −5.46524e6
See all 85 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.85
Significant digits:
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Atkin-Lehner signs

\( p \) Sign
\(167\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 167.14.a.a ✓ 85
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
167.14.a.a ✓ 85 1.a even 1 1 trivial