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Properties

Label 169.10.a.d
Level $169$
Weight $10$
Character orbit 169.a
Self dual yes
Analytic conductor $87.041$
Analytic rank $1$
Dimension $9$
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] = [169,10,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, 10, 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, 10); N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 169.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,15] 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: \(87.0410563117\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3 x^{8} - 3184 x^{7} + 4328 x^{6} + 3323368 x^{5} - 2832720 x^{4} - 1268725952 x^{3} + \cdots + 390142272000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 13^{2} \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{2} + (\beta_{3} - 18) q^{3} + (\beta_{2} - 2 \beta_1 + 200) q^{4} + (\beta_{4} - \beta_{3} + 7 \beta_1 + 124) q^{5} + (\beta_{6} + 2 \beta_{3} - \beta_{2} + \cdots + 230) q^{6}+ \cdots + ( - 22876 \beta_{8} - 19573 \beta_{7} + \cdots - 125070211) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 15 q^{2} - 161 q^{3} + 1793 q^{4} + 1140 q^{5} + 2118 q^{6} - 1939 q^{7} + 7239 q^{8} + 33654 q^{9} - 46923 q^{10} - 5433 q^{11} + 4356 q^{12} - 4950 q^{14} - 347428 q^{15} - 400127 q^{16} - 248589 q^{17}+ \cdots - 1131074634 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3 x^{8} - 3184 x^{7} + 4328 x^{6} + 3323368 x^{5} - 2832720 x^{4} - 1268725952 x^{3} + \cdots + 390142272000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 708 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 60199 \nu^{8} - 23080813 \nu^{7} - 543953414 \nu^{6} + 83217824628 \nu^{5} + \cdots + 10\!\cdots\!20 ) / 417575199150080 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 31848991 \nu^{8} + 959555337 \nu^{7} - 116006648294 \nu^{6} - 2042879267332 \nu^{5} + \cdots + 21\!\cdots\!00 ) / 730756598512640 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 136245217 \nu^{8} + 7231100859 \nu^{7} - 384065441318 \nu^{6} - 20404537289644 \nu^{5} + \cdots + 80\!\cdots\!00 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2326141 \nu^{8} + 73562703 \nu^{7} - 8347836590 \nu^{6} - 162190499836 \nu^{5} + \cdots - 43\!\cdots\!92 ) / 41757519915008 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 295739873 \nu^{8} + 3080314837 \nu^{7} + 1144549491446 \nu^{6} - 9922631049620 \nu^{5} + \cdots - 41\!\cdots\!40 ) / 584605278810112 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 862743299 \nu^{8} - 13119464927 \nu^{7} - 2428839028066 \nu^{6} + 28600740670332 \nu^{5} + \cdots + 45\!\cdots\!00 ) / 14\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 708 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{4} + 21\beta_{3} + 9\beta_{2} + 1111\beta _1 + 1373 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -44\beta_{6} + 19\beta_{5} + 37\beta_{4} + 327\beta_{3} + 1507\beta_{2} + 7017\beta _1 + 780567 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 48 \beta_{8} + 112 \beta_{7} - 500 \beta_{6} + 1929 \beta_{5} - 577 \beta_{4} + 59573 \beta_{3} + \cdots + 4825349 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2096 \beta_{8} + 3280 \beta_{7} - 103676 \beta_{6} + 47251 \beta_{5} + 95109 \beta_{4} + \cdots + 986992551 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 103888 \beta_{8} + 300208 \beta_{7} - 1582868 \beta_{6} + 3287801 \beta_{5} + 958607 \beta_{4} + \cdots + 12016457973 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 7583344 \beta_{8} + 10086288 \beta_{7} - 186733116 \beta_{6} + 94197483 \beta_{5} + 182981037 \beta_{4} + \cdots + 1364356391391 \) 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
41.1215
29.1723
24.2574
14.0491
−4.38428
−7.18744
−26.9303
−32.5507
−34.5476
−39.1215 72.1497 1018.49 1007.31 −2822.60 380.109 −19814.6 −14477.4 −39407.4
1.2 −27.1723 −80.4157 226.333 −1952.16 2185.08 1080.19 7762.23 −13216.3 53044.7
1.3 −22.2574 −256.527 −16.6063 2488.59 5709.64 −167.896 11765.4 46123.2 −55389.7
1.4 −12.0491 193.813 −366.820 26.3134 −2335.27 −2696.19 10589.0 17880.6 −317.053
1.5 6.38428 −5.82434 −471.241 1063.81 −37.1842 6475.20 −6277.28 −19649.1 6791.63
1.6 9.18744 −148.535 −427.591 −1229.19 −1364.65 −8333.07 −8632.44 2379.53 −11293.1
1.7 28.9303 190.754 324.962 −1752.49 5518.56 4404.48 −5411.06 16704.0 −50700.2
1.8 34.5507 57.7993 681.751 2017.22 1997.01 −11757.7 5865.01 −16342.2 69696.2
1.9 36.5476 −184.214 823.724 −529.393 −6732.57 8675.83 11392.7 14251.8 −19348.0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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 169.10.a.d 9
13.b even 2 1 169.10.a.c 9
13.e even 6 2 13.10.c.a ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.10.c.a ✓ 18 13.e even 6 2
169.10.a.c 9 13.b even 2 1
169.10.a.d 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} - 15 T_{2}^{8} - 3088 T_{2}^{7} + 39912 T_{2}^{6} + 3108520 T_{2}^{5} - 29769792 T_{2}^{4} + \cdots - 610864349184 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(169))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + \cdots - 610864349184 \) Copy content Toggle raw display
$3$ \( T^{9} + \cdots + 50\!\cdots\!40 \) Copy content Toggle raw display
$5$ \( T^{9} + \cdots - 31\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{9} + \cdots - 45\!\cdots\!20 \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots - 53\!\cdots\!48 \) Copy content Toggle raw display
$13$ \( T^{9} \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots - 44\!\cdots\!43 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots - 24\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( T^{9} + \cdots - 32\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots - 43\!\cdots\!61 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots - 96\!\cdots\!43 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots + 20\!\cdots\!75 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots - 34\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots - 79\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots + 65\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots - 47\!\cdots\!85 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots + 24\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots + 48\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots - 35\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots - 73\!\cdots\!92 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots - 16\!\cdots\!76 \) Copy content Toggle raw display
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