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Properties

Label 29.10.a.b
Level $29$
Weight $10$
Character orbit 29.a
Self dual yes
Analytic conductor $14.936$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [29,10,Mod(1,29)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(29, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 10, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("29.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 29.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] 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: \(14.9360392488\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 4803 x^{10} + 14952 x^{9} + 8248476 x^{8} - 14809944 x^{7} - 6122244486 x^{6} + \cdots + 40\!\cdots\!38 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{2} + ( - \beta_{3} + \beta_1 + 20) q^{3} + (\beta_{2} + 3 \beta_1 + 291) q^{4} + ( - \beta_{7} - 2 \beta_{3} + \cdots + 149) q^{5} + (\beta_{5} - 10 \beta_{3} + 2 \beta_{2} + \cdots + 451) q^{6}+ \cdots + (41236 \beta_{11} + 145381 \beta_{10} + \cdots + 174542219) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 16 q^{2} + 242 q^{3} + 3498 q^{4} + 1762 q^{5} + 5446 q^{6} + 12080 q^{7} + 25350 q^{8} + 43026 q^{9} - 46678 q^{10} + 24474 q^{11} - 14210 q^{12} + 107722 q^{13} + 677768 q^{14} + 505426 q^{15}+ \cdots + 2123567852 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 4803 x^{10} + 14952 x^{9} + 8248476 x^{8} - 14809944 x^{7} - 6122244486 x^{6} + \cdots + 40\!\cdots\!38 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 802 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 40\!\cdots\!49 \nu^{11} + \cdots - 33\!\cdots\!98 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 20\!\cdots\!37 \nu^{11} + \cdots + 10\!\cdots\!86 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14\!\cdots\!55 \nu^{11} + \cdots - 99\!\cdots\!22 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\!\cdots\!41 \nu^{11} + \cdots + 11\!\cdots\!26 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 24\!\cdots\!53 \nu^{11} + \cdots - 29\!\cdots\!10 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10\!\cdots\!25 \nu^{11} + \cdots + 39\!\cdots\!78 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 23\!\cdots\!11 \nu^{11} + \cdots + 28\!\cdots\!78 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13\!\cdots\!43 \nu^{11} + \cdots - 54\!\cdots\!34 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 38\!\cdots\!67 \nu^{11} + \cdots - 22\!\cdots\!50 ) / 19\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 802 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + 3 \beta_{10} + \beta_{9} + 3 \beta_{8} + 3 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} + \cdots + 613 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 22 \beta_{11} - 47 \beta_{10} + 3 \beta_{9} - 2 \beta_{8} + 47 \beta_{7} + 68 \beta_{6} + \cdots + 1101706 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2465 \beta_{11} + 6934 \beta_{10} + 2458 \beta_{9} + 6247 \beta_{8} + 5752 \beta_{7} + 8569 \beta_{6} + \cdots + 260885 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 42992 \beta_{11} - 96284 \beta_{10} + 14428 \beta_{9} + 952 \beta_{8} + 118244 \beta_{7} + \cdots + 1729803152 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4998295 \beta_{11} + 13174201 \beta_{10} + 5060579 \beta_{9} + 11052189 \beta_{8} + 10182857 \beta_{7} + \cdots - 355340293 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 64582010 \beta_{11} - 156075269 \beta_{10} + 41345617 \beta_{9} + 11520058 \beta_{8} + \cdots + 2869577136274 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 9607745489 \beta_{11} + 23867759212 \beta_{10} + 9927622412 \beta_{9} + 18795167111 \beta_{8} + \cdots - 407253696883 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 84569969644 \beta_{11} - 233629946502 \beta_{10} + 97038898062 \beta_{9} + 38774614140 \beta_{8} + \cdots + 48\!\cdots\!16 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 18065382227179 \beta_{11} + 42678308589903 \beta_{10} + 19052616072789 \beta_{9} + 31684894303913 \beta_{8} + \cdots + 18\!\cdots\!71 \) 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
−41.5077
−41.0148
−24.9566
−21.0654
−11.2767
−7.09515
4.88142
9.20723
26.3150
29.2132
38.0522
43.2474
−40.5077 −111.357 1128.88 2140.75 4510.81 −10261.8 −24988.3 −7282.70 −86716.9
1.2 −40.0148 −3.22123 1089.19 −146.392 128.897 8248.48 −23096.1 −19672.6 5857.84
1.3 −23.9566 161.985 61.9199 2644.17 −3880.61 2059.91 10782.4 6556.14 −63345.5
1.4 −20.0654 −72.4764 −109.378 −1017.86 1454.27 −11164.8 12468.2 −14430.2 20423.8
1.5 −10.2767 52.0955 −406.390 −2693.22 −535.369 4178.53 9437.99 −16969.1 27677.4
1.6 −6.09515 250.876 −474.849 −546.084 −1529.13 6563.37 6015.00 43256.0 3328.46
1.7 5.88142 −88.8951 −477.409 959.601 −522.829 −2615.28 −5819.13 −11780.7 5643.82
1.8 10.2072 −225.321 −407.813 −1619.94 −2299.90 −263.065 −9388.73 31086.4 −16535.1
1.9 27.3150 101.195 234.111 2383.64 2764.14 3032.73 −7590.56 −9442.56 65109.1
1.10 30.2132 241.333 400.835 −389.202 7291.44 3096.10 −3358.64 38558.7 −11759.0
1.11 39.0522 −174.323 1013.08 −303.923 −6807.72 12556.4 19568.2 10705.6 −11868.9
1.12 44.2474 110.108 1445.83 350.462 4872.00 −3350.56 41319.6 −7559.17 15507.0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(29\) \( -1 \)

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 29.10.a.b ✓ 12
3.b odd 2 1 261.10.a.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.10.a.b ✓ 12 1.a even 1 1 trivial
261.10.a.e 12 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 16 T_{2}^{11} - 4693 T_{2}^{10} + 62542 T_{2}^{9} + 7898928 T_{2}^{8} + \cdots + 41\!\cdots\!60 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(29))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 41\!\cdots\!60 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 51\!\cdots\!72 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots - 14\!\cdots\!08 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots - 13\!\cdots\!72 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 31\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 78\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 47\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T - 707281)^{12} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots - 22\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots - 18\!\cdots\!12 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 35\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots - 16\!\cdots\!40 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 45\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 21\!\cdots\!08 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 25\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 22\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 96\!\cdots\!20 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 31\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 53\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 64\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 32\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 95\!\cdots\!04 \) Copy content Toggle raw display
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