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Properties

Label 169.14.a.c
Level $169$
Weight $14$
Character orbit 169.a
Self dual yes
Analytic conductor $181.220$
Analytic rank $1$
Dimension $14$
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] = [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 = [14,-65] 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: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 5 x^{13} - 83806 x^{12} + 371578 x^{11} + 2652253571 x^{10} - 14037350343 x^{9} + \cdots - 28\!\cdots\!92 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{24}\cdot 3^{3}\cdot 5\cdot 13^{6} \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 5) q^{2} + (\beta_{3} + \beta_1 - 52) q^{3} + ( - \beta_{3} + \beta_{2} + \cdots + 3807) q^{4} + (\beta_{5} - 3 \beta_{3} - 18 \beta_1 - 1490) q^{5} + (\beta_{4} + 4 \beta_{3} + \cdots + 10340) q^{6}+ \cdots + (22371 \beta_{13} + \cdots - 1188826971250) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 65 q^{2} - 728 q^{3} + 53249 q^{4} - 20930 q^{5} + 143910 q^{6} - 173992 q^{7} - 1308489 q^{8} + 5231838 q^{9} - 2769243 q^{10} + 10986144 q^{11} - 22602732 q^{12} - 11865878 q^{14} - 75354448 q^{15}+ \cdots - 16619904586272 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 5 x^{13} - 83806 x^{12} + 371578 x^{11} + 2652253571 x^{10} - 14037350343 x^{9} + \cdots - 28\!\cdots\!92 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 59\!\cdots\!53 \nu^{13} + \cdots - 18\!\cdots\!68 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 59\!\cdots\!53 \nu^{13} + \cdots + 97\!\cdots\!32 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 20\!\cdots\!47 \nu^{13} + \cdots - 16\!\cdots\!32 ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\!\cdots\!21 \nu^{13} + \cdots - 32\!\cdots\!24 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 35\!\cdots\!27 \nu^{13} + \cdots + 84\!\cdots\!12 ) / 77\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 73\!\cdots\!29 \nu^{13} + \cdots - 24\!\cdots\!24 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 53\!\cdots\!39 \nu^{13} + \cdots - 27\!\cdots\!84 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 77\!\cdots\!73 \nu^{13} + \cdots + 32\!\cdots\!88 ) / 77\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12\!\cdots\!49 \nu^{13} + \cdots + 11\!\cdots\!44 ) / 77\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 96\!\cdots\!12 \nu^{13} + \cdots - 52\!\cdots\!72 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 64\!\cdots\!29 \nu^{13} + \cdots - 23\!\cdots\!24 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 42\!\cdots\!59 \nu^{13} + \cdots - 99\!\cdots\!96 ) / 38\!\cdots\!00 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + 11974 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - 3\beta_{5} - \beta_{4} - 63\beta_{3} + 13\beta_{2} + 20500\beta _1 + 2838 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{13} - 3 \beta_{12} + 4 \beta_{11} + 4 \beta_{10} + 6 \beta_{9} + \beta_{8} + \cdots + 245584891 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 52 \beta_{13} + 82 \beta_{12} - 48 \beta_{11} + 404 \beta_{10} - 1620 \beta_{9} + 30 \beta_{8} + \cdots + 1523338250 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 71856 \beta_{13} - 93332 \beta_{12} + 175400 \beta_{11} + 181132 \beta_{10} + 184024 \beta_{9} + \cdots + 5824832503642 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1388548 \beta_{13} + 3641266 \beta_{12} - 804736 \beta_{11} + 20501964 \beta_{10} + \cdots + 80589575157324 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2144167418 \beta_{13} - 2193480523 \beta_{12} + 5672437068 \beta_{11} + 6451954216 \beta_{10} + \cdots + 14\!\cdots\!03 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 17548633728 \beta_{13} + 146863071112 \beta_{12} - 13527744848 \beta_{11} + 792856589240 \beta_{10} + \cdots + 33\!\cdots\!08 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 61612837037552 \beta_{13} - 44455069295728 \beta_{12} + 163869472444688 \beta_{11} + \cdots + 39\!\cdots\!10 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 344051200820744 \beta_{13} + \cdots + 12\!\cdots\!78 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 17\!\cdots\!98 \beta_{13} + \cdots + 10\!\cdots\!23 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 36\!\cdots\!32 \beta_{13} + \cdots + 45\!\cdots\!10 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−160.740
−159.709
−113.537
−108.421
−85.9030
−28.9774
−2.62814
22.6132
27.4573
79.0934
86.5912
126.590
148.045
174.526
−165.740 −902.967 19277.9 67212.0 149658. −275625. −1.83738e6 −778974. −1.11397e7
1.2 −164.709 511.369 18937.1 −48546.3 −84227.1 432193. −1.76982e6 −1.33283e6 7.99602e6
1.3 −118.537 −2145.81 5859.08 −45885.5 254358. −355362. 276538. 3.01017e6 5.43914e6
1.4 −113.421 1813.01 4672.28 10940.5 −205633. −179544. 399209. 1.69269e6 −1.24088e6
1.5 −90.9030 −217.527 71.3521 2568.18 19773.9 154716. 738191. −1.54700e6 −233455.
1.6 −33.9774 −1595.69 −7037.54 28205.3 54217.3 354344. 517460. 951897. −958341.
1.7 −7.62814 374.529 −8133.81 −42407.7 −2856.96 −357265. 124536. −1.45405e6 323492.
1.8 17.6132 899.397 −7881.78 51640.9 15841.2 −151038. −283110. −785408. 909560.
1.9 22.4573 2284.08 −7687.67 −17999.0 51294.3 563042. −356614. 3.62272e6 −404209.
1.10 74.0934 −1917.97 −2702.17 22075.6 −142109. −501994. −807186. 2.08427e6 1.63565e6
1.11 81.5912 −964.287 −1534.87 −41512.4 −78677.3 316788. −793627. −664474. −3.38705e6
1.12 121.590 711.452 6592.10 32981.4 86505.3 94225.6 −194532. −1.08816e6 4.01021e6
1.13 143.045 1731.14 12269.8 −41380.4 247630. −375678. 583312. 1.40251e6 −5.91926e6
1.14 169.526 −1308.74 20547.2 1177.50 −221866. 107205. 2.09453e6 118470. 199618.
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \( +1 \)

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 169.14.a.c 14
13.b even 2 1 169.14.a.e 14
13.e even 6 2 13.14.c.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.14.c.a ✓ 28 13.e even 6 2
169.14.a.c 14 1.a even 1 1 trivial
169.14.a.e 14 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 65 T_{2}^{13} - 81856 T_{2}^{12} - 4621032 T_{2}^{11} + 2534857336 T_{2}^{10} + \cdots - 60\!\cdots\!52 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(169))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots - 60\!\cdots\!52 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots - 76\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{14} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots - 59\!\cdots\!63 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 20\!\cdots\!61 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 12\!\cdots\!97 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 84\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 94\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 46\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 14\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 47\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 42\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 32\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 81\!\cdots\!16 \) Copy content Toggle raw display
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