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Properties

Label 3.29.b.a
Level $3$
Weight $29$
Character orbit 3.b
Analytic conductor $14.901$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3,29,Mod(2,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 29, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.2");
 
S:= CuspForms(chi, 29);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 29 \)
Character orbit: \([\chi]\) \(=\) 3.b (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: \(14.9005422744\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 43751258x^{6} + 536382928179456x^{4} + 1860360285955844341760x^{2} + 187193718217769921334476800 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{31}\cdot 3^{40}\cdot 5^{2}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} - 18 \beta_1 + 761535) q^{3} + (\beta_{3} + 15 \beta_{2} + \cdots - 125325866) q^{4}+ \cdots + ( - 81 \beta_{7} + 45 \beta_{6} + \cdots - 200070911655) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} - 18 \beta_1 + 761535) q^{3} + (\beta_{3} + 15 \beta_{2} + \cdots - 125325866) q^{4}+ \cdots + (14\!\cdots\!06 \beta_{7} + \cdots + 25\!\cdots\!20) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6092280 q^{3} - 1002606928 q^{4} + 56151925488 q^{6} + 401156726320 q^{7} - 1600567293240 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6092280 q^{3} - 1002606928 q^{4} + 56151925488 q^{6} + 401156726320 q^{7} - 1600567293240 q^{9} + 191378354280480 q^{10} - 34\!\cdots\!20 q^{12}+ \cdots + 20\!\cdots\!60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 43751258x^{6} + 536382928179456x^{4} + 1860360285955844341760x^{2} + 187193718217769921334476800 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 6\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3642205 \nu^{7} - 9562383776 \nu^{6} + 198701833202610 \nu^{5} + \cdots - 48\!\cdots\!80 ) / 15\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3642205 \nu^{7} + 9562383776 \nu^{6} - 198701833202610 \nu^{5} + \cdots + 44\!\cdots\!68 ) / 10\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7346327485 \nu^{7} + 22538538560032 \nu^{6} + \cdots + 15\!\cdots\!00 ) / 15\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4951772825 \nu^{7} + 248621978176 \nu^{6} + \cdots + 12\!\cdots\!80 ) / 25\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 43091647217 \nu^{7} + 92997328862560 \nu^{6} + \cdots - 41\!\cdots\!80 ) / 50\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 90494641877 \nu^{7} + 204670779899888 \nu^{6} + \cdots - 43\!\cdots\!00 ) / 37\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 15\beta_{2} + 3\beta _1 - 393761322 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -15\beta_{7} + 45\beta_{6} - 431\beta_{5} - 89\beta_{4} + 518\beta_{3} - 240544\beta_{2} - 692379628\beta_1 ) / 216 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 17413 \beta_{7} - 52239 \beta_{6} + 17413 \beta_{5} + 692889 \beta_{4} - 116840713 \beta_{3} + \cdots + 34\!\cdots\!06 ) / 162 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2184764475 \beta_{7} - 6554293425 \beta_{6} + 66878145627 \beta_{5} + 20467750477 \beta_{4} + \cdots + 70\!\cdots\!68 \beta_1 ) / 972 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 122928563659 \beta_{7} + 368785690977 \beta_{6} - 122928563659 \beta_{5} - 3916429117047 \beta_{4} + \cdots - 12\!\cdots\!74 ) / 27 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 98\!\cdots\!25 \beta_{7} + \cdots - 28\!\cdots\!88 \beta_1 ) / 162 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
− 5066.09i
− 3497.44i
− 2397.95i
− 322.019i
322.019i
2397.95i
3497.44i
5066.09i
− 30396.6i 3.93308e6 + 2.72170e6i −6.55515e8 5.86690e9i 8.27304e10 − 1.19552e11i 5.38370e11 1.17659e13i 8.06144e12 + 2.14094e13i 1.78334e14
2.2 − 20984.7i −668986. − 4.73595e6i −1.71921e8 − 4.58251e9i −9.93824e10 + 1.40385e10i −5.85864e11 − 2.02533e12i −2.19817e13 + 6.33657e12i −9.61624e13
2.3 − 14387.7i −3.94388e6 + 2.70602e6i 6.14299e7 2.19810e9i 3.89334e10 + 5.67434e10i 2.28443e11 − 4.74600e12i 8.23166e12 − 2.13445e13i 3.16256e13
2.4 − 1932.12i 3.72593e6 + 2.99904e6i 2.64702e8 − 9.37189e9i 5.79449e9 − 7.19893e9i 1.96297e10 − 1.03008e12i 4.88833e12 + 2.23484e13i −1.81076e13
2.5 1932.12i 3.72593e6 − 2.99904e6i 2.64702e8 9.37189e9i 5.79449e9 + 7.19893e9i 1.96297e10 1.03008e12i 4.88833e12 − 2.23484e13i −1.81076e13
2.6 14387.7i −3.94388e6 − 2.70602e6i 6.14299e7 − 2.19810e9i 3.89334e10 − 5.67434e10i 2.28443e11 4.74600e12i 8.23166e12 + 2.13445e13i 3.16256e13
2.7 20984.7i −668986. + 4.73595e6i −1.71921e8 4.58251e9i −9.93824e10 − 1.40385e10i −5.85864e11 2.02533e12i −2.19817e13 − 6.33657e12i −9.61624e13
2.8 30396.6i 3.93308e6 − 2.72170e6i −6.55515e8 − 5.86690e9i 8.27304e10 + 1.19552e11i 5.38370e11 − 1.17659e13i 8.06144e12 − 2.14094e13i 1.78334e14
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.29.b.a ✓ 8
3.b odd 2 1 inner 3.29.b.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.29.b.a ✓ 8 1.a even 1 1 trivial
3.29.b.a ✓ 8 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{29}^{\mathrm{new}}(3, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 27\!\cdots\!41 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 41\!\cdots\!16)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 17\!\cdots\!16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 56\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 24\!\cdots\!24)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 81\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 92\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 54\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 41\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
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