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Properties

Label 169.14.a.f
Level $169$
Weight $14$
Character orbit 169.a
Self dual yes
Analytic conductor $181.220$
Analytic rank $0$
Dimension $30$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [169,14,Mod(1,169)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(169, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 14, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("169.1"); S:= CuspForms(chi, 14); N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 169.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [30,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(181.220269929\)
Analytic rank: \(0\)
Dimension: \(30\)
Twist minimal: no (minimal twist has level 13)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 1460 q^{3} + 131070 q^{4} + 18234142 q^{9} + 7786950 q^{10} - 27694876 q^{12} + 61125588 q^{14} + 700365042 q^{16} - 47049798 q^{17} + 256370724 q^{22} + 834173772 q^{23} + 11926991364 q^{25} + 5605403288 q^{27}+ \cdots + 22616629029900 q^{95}+O(q^{100}) \) 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 −175.128 −1199.81 22477.8 −54580.7 210120. −282637. −2.50185e6 −154789. 9.55861e6
1.2 −170.896 −2184.38 21013.4 22940.5 373302. 158631. −2.19112e6 3.17721e6 −3.92043e6
1.3 −161.029 1238.11 17738.3 −40204.7 −199372. −5011.98 −1.53724e6 −61399.6 6.47413e6
1.4 −155.557 1920.54 16006.1 38811.6 −298754. −510847. −1.21554e6 2.09414e6 −6.03743e6
1.5 −131.328 −395.120 9055.01 34433.6 51890.3 374853. −113337. −1.43820e6 −4.52209e6
1.6 −120.050 82.5506 6219.93 28055.4 −9910.17 −450235. 236746. −1.58751e6 −3.36804e6
1.7 −113.025 1496.50 4582.61 −55447.0 −169142. 501899. 407950. 645202. 6.26688e6
1.8 −100.903 −395.015 1989.51 −4300.66 39858.3 474644. 625853. −1.43829e6 433951.
1.9 −89.1502 2459.54 −244.250 52376.5 −219269. 277531. 752093. 4.45503e6 −4.66937e6
1.10 −84.7459 −1625.76 −1010.14 11734.8 137776. −159640. 779843. 1.04876e6 −994475.
1.11 −66.1543 938.734 −3815.61 −11059.1 −62101.3 −345588. 794355. −713101. 731610.
1.12 −63.7977 −1849.62 −4121.85 −57759.0 118001. −171668. 785596. 1.82676e6 3.68489e6
1.13 −13.1427 1728.48 −8019.27 −50287.2 −22717.0 −294728. 213060. 1.39333e6 660911.
1.14 −6.53496 −1731.40 −8149.29 60806.7 11314.6 −202236. 106790. 1.40343e6 −397369.
1.15 −2.16686 246.634 −8187.30 3830.61 −534.423 44561.3 35491.7 −1.53349e6 −8300.40
1.16 2.16686 246.634 −8187.30 −3830.61 534.423 −44561.3 −35491.7 −1.53349e6 −8300.40
1.17 6.53496 −1731.40 −8149.29 −60806.7 −11314.6 202236. −106790. 1.40343e6 −397369.
1.18 13.1427 1728.48 −8019.27 50287.2 22717.0 294728. −213060. 1.39333e6 660911.
1.19 63.7977 −1849.62 −4121.85 57759.0 −118001. 171668. −785596. 1.82676e6 3.68489e6
1.20 66.1543 938.734 −3815.61 11059.1 62101.3 345588. −794355. −713101. 731610.
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.30
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Atkin-Lehner signs

\( p \) Sign
\(13\) \( -1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.14.a.f 30
13.b even 2 1 inner 169.14.a.f 30
13.f odd 12 2 13.14.e.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.14.e.a ✓ 30 13.f odd 12 2
169.14.a.f 30 1.a even 1 1 trivial
169.14.a.f 30 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{30} - 188415 T_{2}^{28} + 15763087812 T_{2}^{26} - 772961987927488 T_{2}^{24} + \cdots - 63\!\cdots\!72 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(169))\). Copy content Toggle raw display