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Properties

Label 129.2.d.a
Level $129$
Weight $2$
Character orbit 129.d
Analytic conductor $1.030$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [129,2,Mod(128,129)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(129, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("129.128");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 129 = 3 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 129.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.03007018607\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 6x^{10} + 29x^{8} + 88x^{6} + 261x^{4} + 486x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_{6} q^{3} + \beta_{9} q^{4} + \beta_{11} q^{5} - \beta_{4} q^{6} + \beta_{8} q^{7} + ( - \beta_{11} - \beta_{6} + \beta_1) q^{8} + ( - \beta_{9} - \beta_{7} - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + \beta_{6} q^{3} + \beta_{9} q^{4} + \beta_{11} q^{5} - \beta_{4} q^{6} + \beta_{8} q^{7} + ( - \beta_{11} - \beta_{6} + \beta_1) q^{8} + ( - \beta_{9} - \beta_{7} - \beta_{2} - 1) q^{9} - \beta_{9} q^{10} + (\beta_{5} + \beta_{4}) q^{11} + ( - \beta_{10} - \beta_{8} + \cdots + \beta_1) q^{12}+ \cdots + ( - \beta_{9} + 3 \beta_{7} - \beta_{5} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{4} - 12 q^{9} - 4 q^{10} + 12 q^{13} - 8 q^{15} - 4 q^{16} - 4 q^{21} - 16 q^{24} + 16 q^{31} - 20 q^{36} - 44 q^{40} + 32 q^{43} - 24 q^{49} + 36 q^{52} + 40 q^{54} + 12 q^{57} - 28 q^{58} + 32 q^{60} - 28 q^{64} + 36 q^{66} - 8 q^{67} - 40 q^{78} + 56 q^{79} - 44 q^{81} + 52 q^{84} + 48 q^{87} + 20 q^{90} + 8 q^{96} - 4 q^{97} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 6x^{10} + 29x^{8} + 88x^{6} + 261x^{4} + 486x^{2} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{11} + 21\nu^{9} + 52\nu^{7} + 452\nu^{5} + 495\nu^{3} + 2349\nu ) / 1944 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{10} + 21\nu^{8} + 52\nu^{6} + 452\nu^{4} + 495\nu^{2} + 2349 ) / 648 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{10} + 9\nu^{8} + 20\nu^{6} + 40\nu^{4} + 93\nu^{2} + 189 ) / 108 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} + 6\nu^{9} + 29\nu^{7} + 88\nu^{5} + 261\nu^{3} + 486\nu ) / 243 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{10} - 3\nu^{8} - 4\nu^{6} + 4\nu^{4} - 54\nu^{2} + 81 ) / 162 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -5\nu^{11} + 15\nu^{9} + 44\nu^{7} + 136\nu^{5} + 63\nu^{3} + 1215\nu ) / 972 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4\nu^{10} + 15\nu^{8} + 62\nu^{6} + 172\nu^{4} + 414\nu^{2} + 567 ) / 162 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -17\nu^{11} - 75\nu^{9} - 412\nu^{7} - 956\nu^{5} - 1737\nu^{3} - 1539\nu ) / 1944 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -7\nu^{11} - 45\nu^{9} - 140\nu^{7} - 460\nu^{5} - 1119\nu^{3} - 1917\nu ) / 648 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + 2\beta_{6} - \beta_{3} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} - \beta_{5} + 2\beta_{4} - \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} - \beta_{10} - 2\beta_{8} - \beta_{6} + 8\beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 5\beta_{9} + 11\beta_{7} - 8\beta_{4} - 3\beta_{2} + 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 6\beta_{11} - 8\beta_{10} + 6\beta_{8} + 5\beta_{6} - 10\beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -12\beta_{9} - 10\beta_{7} + 20\beta_{5} + 8\beta_{4} + 4\beta_{2} - 21 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -38\beta_{11} + 16\beta_{10} + 22\beta_{8} - 42\beta_{6} - 30\beta_{3} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 8\beta_{9} - 90\beta_{7} - 32\beta_{5} + 8\beta_{4} - 29\beta_{2} + 73 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -34\beta_{11} - 37\beta_{10} - 130\beta_{8} - 84\beta_{6} + 27\beta_{3} + 168\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/129\mathbb{Z}\right)^\times\).

\(n\) \(44\) \(46\)
\(\chi(n)\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
128.1
−0.583090 − 1.63095i
−0.583090 + 1.63095i
1.36973 − 1.06011i
1.36973 + 1.06011i
−0.885347 − 1.48868i
−0.885347 + 1.48868i
0.885347 − 1.48868i
0.885347 + 1.48868i
−1.36973 − 1.06011i
−1.36973 + 1.06011i
0.583090 − 1.63095i
0.583090 + 1.63095i
−2.26382 0.583090 − 1.63095i 3.12489 1.38036 −1.32001 + 3.69218i 0.790787i −2.54654 −2.32001 − 1.90198i −3.12489
128.2 −2.26382 0.583090 + 1.63095i 3.12489 1.38036 −1.32001 − 3.69218i − 0.790787i −2.54654 −2.32001 + 1.90198i −3.12489
128.3 −1.27932 −1.36973 − 1.06011i −0.363328 −0.284000 1.75233 + 1.35623i 3.33016i 3.02346 0.752332 + 2.90413i 0.363328
128.4 −1.27932 −1.36973 + 1.06011i −0.363328 −0.284000 1.75233 − 1.35623i − 3.33016i 3.02346 0.752332 − 2.90413i 0.363328
128.5 −0.488306 0.885347 − 1.48868i −1.76156 −3.60749 −0.432320 + 0.726930i − 3.90956i 1.83679 −1.43232 − 2.63599i 1.76156
128.6 −0.488306 0.885347 + 1.48868i −1.76156 −3.60749 −0.432320 − 0.726930i 3.90956i 1.83679 −1.43232 + 2.63599i 1.76156
128.7 0.488306 −0.885347 − 1.48868i −1.76156 3.60749 −0.432320 − 0.726930i − 3.90956i −1.83679 −1.43232 + 2.63599i 1.76156
128.8 0.488306 −0.885347 + 1.48868i −1.76156 3.60749 −0.432320 + 0.726930i 3.90956i −1.83679 −1.43232 − 2.63599i 1.76156
128.9 1.27932 1.36973 − 1.06011i −0.363328 0.284000 1.75233 − 1.35623i 3.33016i −3.02346 0.752332 − 2.90413i 0.363328
128.10 1.27932 1.36973 + 1.06011i −0.363328 0.284000 1.75233 + 1.35623i − 3.33016i −3.02346 0.752332 + 2.90413i 0.363328
128.11 2.26382 −0.583090 − 1.63095i 3.12489 −1.38036 −1.32001 − 3.69218i 0.790787i 2.54654 −2.32001 + 1.90198i −3.12489
128.12 2.26382 −0.583090 + 1.63095i 3.12489 −1.38036 −1.32001 + 3.69218i − 0.790787i 2.54654 −2.32001 − 1.90198i −3.12489
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 128.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
43.b odd 2 1 inner
129.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 129.2.d.a ✓ 12
3.b odd 2 1 inner 129.2.d.a ✓ 12
4.b odd 2 1 2064.2.l.h 12
12.b even 2 1 2064.2.l.h 12
43.b odd 2 1 inner 129.2.d.a ✓ 12
129.d even 2 1 inner 129.2.d.a ✓ 12
172.d even 2 1 2064.2.l.h 12
516.h odd 2 1 2064.2.l.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
129.2.d.a ✓ 12 1.a even 1 1 trivial
129.2.d.a ✓ 12 3.b odd 2 1 inner
129.2.d.a ✓ 12 43.b odd 2 1 inner
129.2.d.a ✓ 12 129.d even 2 1 inner
2064.2.l.h 12 4.b odd 2 1
2064.2.l.h 12 12.b even 2 1
2064.2.l.h 12 172.d even 2 1
2064.2.l.h 12 516.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(129, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 7 T^{4} + 10 T^{2} - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + 6 T^{10} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( (T^{6} - 15 T^{4} + 26 T^{2} - 2)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 27 T^{4} + \cdots + 106)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 35 T^{4} + \cdots + 1325)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} - 3 T^{2} - 7 T + 17)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + 42 T^{4} + \cdots + 212)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 95 T^{4} + \cdots + 27136)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 58 T^{4} + \cdots + 848)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 55 T^{4} + \cdots - 6050)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 4 T^{2} - 25 T - 22)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 132 T^{4} + \cdots + 1696)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 154 T^{4} + \cdots + 84800)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 16 T^{5} + \cdots + 79507)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 225 T^{4} + \cdots + 154548)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 134 T^{4} + \cdots + 13568)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 76 T^{4} + \cdots + 13568)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 124 T^{4} + \cdots + 42400)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 2 T^{2} - 7 T - 4)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} - 340 T^{4} + \cdots - 516128)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 200 T^{4} + \cdots + 122536)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 14 T^{2} + \cdots + 20)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 151 T^{4} + \cdots + 33125)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 332 T^{4} + \cdots - 800)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + T^{2} - 101 T + 227)^{4} \) Copy content Toggle raw display
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