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Properties

Label 3.35.b.a
Level $3$
Weight $35$
Character orbit 3.b
Analytic conductor $21.968$
Analytic rank $0$
Dimension $10$
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] = [3,35,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, 35, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.2");
 
S:= CuspForms(chi, 35);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 35 \)
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: \(21.9676962128\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 789518143 x^{8} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{54}\cdot 3^{65}\cdot 5^{4}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} + 41 \beta_1 + 11936911) q^{3} + (\beta_{3} - 15 \beta_{2} + \cdots - 5558253328) q^{4}+ \cdots + (\beta_{9} + 9 \beta_{8} + \cdots + 478750119881211) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} + 41 \beta_1 + 11936911) q^{3} + (\beta_{3} - 15 \beta_{2} + \cdots - 5558253328) q^{4}+ \cdots + ( - 28\!\cdots\!80 \beta_{9} + \cdots + 25\!\cdots\!17) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 119369106 q^{3} - 55582533344 q^{4} - 9250224859872 q^{6} - 123569771565772 q^{7} + 47\!\cdots\!18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 119369106 q^{3} - 55582533344 q^{4} - 9250224859872 q^{6} - 123569771565772 q^{7} + 47\!\cdots\!18 q^{9}+ \cdots + 25\!\cdots\!80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 789518143 x^{8} + \cdots + 50\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 12\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 81\!\cdots\!25 \nu^{9} + \cdots - 14\!\cdots\!00 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 81\!\cdots\!25 \nu^{9} + \cdots + 25\!\cdots\!00 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 43\!\cdots\!83 \nu^{9} + \cdots + 31\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 39\!\cdots\!89 \nu^{9} + \cdots + 55\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 42\!\cdots\!65 \nu^{9} + \cdots - 11\!\cdots\!00 ) / 88\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21\!\cdots\!15 \nu^{9} + \cdots + 15\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12\!\cdots\!31 \nu^{9} + \cdots + 29\!\cdots\!00 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 29\!\cdots\!43 \nu^{9} + \cdots + 17\!\cdots\!00 ) / 66\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 15\beta_{2} + 5\beta _1 - 22738122512 ) / 144 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{8} - 14 \beta_{7} - 37 \beta_{5} - 178 \beta_{4} - 2028 \beta_{3} - 2605108 \beta_{2} + \cdots + 1041214 ) / 1728 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3054 \beta_{9} - 265564 \beta_{7} - 27352 \beta_{6} + 20268 \beta_{5} + 41646 \beta_{4} + \cdots + 51\!\cdots\!72 ) / 1296 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 119671920 \beta_{9} - 3188619079 \beta_{8} + 54693108386 \beta_{7} + 181468975747 \beta_{5} + \cdots - 51\!\cdots\!42 ) / 15552 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4613664266934 \beta_{9} + 396961680078828 \beta_{7} + 37095867257976 \beta_{6} + \cdots - 43\!\cdots\!80 ) / 3888 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 44\!\cdots\!40 \beta_{9} + \cdots + 19\!\cdots\!30 ) / 15552 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 17\!\cdots\!50 \beta_{9} + \cdots + 12\!\cdots\!84 ) / 3888 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 10\!\cdots\!00 \beta_{9} + \cdots - 66\!\cdots\!98 ) / 15552 \) 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
− 17330.4i
− 16987.2i
− 12705.0i
− 4945.08i
− 3839.21i
3839.21i
4945.08i
12705.0i
16987.2i
17330.4i
− 207965.i 7.87239e7 + 1.02371e8i −2.60695e10 − 2.95819e11i 2.12895e13 − 1.63718e13i 4.71207e13 1.84873e15i −4.28228e15 + 1.61180e16i −6.15199e16
2.2 − 203846.i −1.15096e8 − 5.85668e7i −2.43733e10 − 1.79877e11i −1.19386e13 + 2.34619e13i −2.97937e14 1.46635e15i 9.81704e15 + 1.34816e16i −3.66671e16
2.3 − 152461.i 6.59892e7 − 1.11007e8i −6.06435e9 1.02364e12i −1.69242e13 − 1.00608e13i 3.85840e14 − 1.69468e15i −7.96802e15 − 1.46506e16i 1.56064e17
2.4 − 59341.0i −8.15029e7 + 1.00172e8i 1.36585e10 7.02280e10i 5.94431e12 + 4.83646e12i 9.28487e13 − 1.82998e15i −3.39173e15 − 1.63286e16i 4.16740e15
2.5 − 46070.5i 1.11570e8 − 6.50325e7i 1.50574e10 − 1.37507e12i −2.99608e12 − 5.14011e12i −2.89657e14 − 1.48519e15i 8.21874e15 − 1.45114e16i −6.33504e16
2.6 46070.5i 1.11570e8 + 6.50325e7i 1.50574e10 1.37507e12i −2.99608e12 + 5.14011e12i −2.89657e14 1.48519e15i 8.21874e15 + 1.45114e16i −6.33504e16
2.7 59341.0i −8.15029e7 − 1.00172e8i 1.36585e10 − 7.02280e10i 5.94431e12 − 4.83646e12i 9.28487e13 1.82998e15i −3.39173e15 + 1.63286e16i 4.16740e15
2.8 152461.i 6.59892e7 + 1.11007e8i −6.06435e9 − 1.02364e12i −1.69242e13 + 1.00608e13i 3.85840e14 1.69468e15i −7.96802e15 + 1.46506e16i 1.56064e17
2.9 203846.i −1.15096e8 + 5.85668e7i −2.43733e10 1.79877e11i −1.19386e13 − 2.34619e13i −2.97937e14 − 1.46635e15i 9.81704e15 − 1.34816e16i −3.66671e16
2.10 207965.i 7.87239e7 − 1.02371e8i −2.60695e10 2.95819e11i 2.12895e13 + 1.63718e13i 4.71207e13 − 1.84873e15i −4.28228e15 − 1.61180e16i −6.15199e16
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.10
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 3.35.b.a ✓ 10
3.b odd 2 1 inner 3.35.b.a ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.35.b.a ✓ 10 1.a even 1 1 trivial
3.35.b.a ✓ 10 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{35}^{\mathrm{new}}(3, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 12\!\cdots\!49 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{5} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 57\!\cdots\!88)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots - 73\!\cdots\!12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 56\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 51\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 72\!\cdots\!92)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
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