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Properties

Label 322.2.e.b
Level $322$
Weight $2$
Character orbit 322.e
Analytic conductor $2.571$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [322,2,Mod(93,322)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(322, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("322.93");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.e (of order \(3\), degree \(2\), 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: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.1767277521.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + x^{6} - 10x^{5} + 38x^{4} - 40x^{3} + 64x^{2} - 38x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + (\beta_{6} - \beta_{5} + \beta_{2} + 1) q^{3} + (\beta_{5} - 1) q^{4} + ( - \beta_{7} + \beta_{5} + \cdots + \beta_{3}) q^{5} + ( - \beta_{2} - 1) q^{6} + ( - \beta_{6} + \beta_{3}) q^{7}+ \cdots + ( - 4 \beta_{3} - 6 \beta_{2} + \cdots - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 5 q^{3} - 4 q^{4} + 5 q^{5} - 10 q^{6} + q^{7} + 8 q^{8} - 7 q^{9} + 5 q^{10} + 2 q^{11} + 5 q^{12} - 14 q^{13} + q^{14} + 2 q^{15} - 4 q^{16} - 3 q^{17} - 7 q^{18} + 9 q^{19} - 10 q^{20}+ \cdots - 82 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + x^{6} - 10x^{5} + 38x^{4} - 40x^{3} + 64x^{2} - 38x + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -14\nu^{7} + 688\nu^{6} - 619\nu^{5} - 193\nu^{4} - 6480\nu^{3} + 17846\nu^{2} - 10595\nu + 23150 ) / 8102 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 74\nu^{7} - 743\nu^{6} + 957\nu^{5} - 716\nu^{4} + 8788\nu^{3} - 23147\nu^{2} + 13756\nu - 21668 ) / 8102 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -139\nu^{7} + 465\nu^{6} - 648\nu^{5} + 688\nu^{4} - 5887\nu^{3} + 15724\nu^{2} - 9416\nu - 2218 ) / 8102 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -1269\nu^{7} + 2176\nu^{6} - 262\nu^{5} + 11731\nu^{4} - 43953\nu^{3} + 36565\nu^{2} - 52069\nu + 14432 ) / 8102 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -962\nu^{7} + 1557\nu^{6} - 288\nu^{5} + 9308\nu^{4} - 33224\nu^{3} + 25443\nu^{2} - 49196\nu + 18369 ) / 4051 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4270 \nu^{7} + 7290 \nu^{6} - 2449 \nu^{5} + 42410 \nu^{4} - 149399 \nu^{3} + 128118 \nu^{2} + \cdots + 88979 ) / 8102 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4805\nu^{7} - 8118\nu^{6} + 2087\nu^{5} - 47477\nu^{4} + 167278\nu^{3} - 139939\nu^{2} + 265634\nu - 98045 ) / 8102 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{3} - \beta_{2} - \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -5\beta_{7} - 5\beta_{6} - \beta_{5} - \beta_{4} + 2\beta_{3} - 4\beta_{2} - 4\beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} - 6\beta_{5} + 2\beta_{4} + \beta_{3} + 3\beta_{2} + \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{7} + 5\beta_{6} - 38\beta_{5} + 16\beta_{4} - 23\beta_{3} - 17\beta_{2} - 17\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -70\beta_{7} - 70\beta_{6} - 11\beta_{5} - 14\beta_{4} + 4\beta_{3} - 14\beta_{2} - 8\beta _1 - 17 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 17\beta_{7} + 37\beta_{6} - 60\beta_{5} + 35\beta_{4} - 16\beta_{3} + 50\beta_{2} + 46\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -44\beta_{7} - 38\beta_{6} - 22\beta_{5} - 7\beta_{4} - 301\beta_{3} - 283\beta_{2} - 97\beta _1 - 565 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\)
\(\chi(n)\) \(-\beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1
0.373419 + 0.0835272i
2.11692 + 0.978886i
0.0512865 − 1.21608i
−1.54162 + 1.88572i
0.373419 − 0.0835272i
2.11692 − 0.978886i
0.0512865 + 1.21608i
−1.54162 − 1.88572i
−0.500000 + 0.866025i −0.685837 − 1.18790i −0.500000 − 0.866025i 1.57146 − 2.72186i 1.37167 −1.66774 + 2.05393i 1.00000 0.559256 − 0.968659i 1.57146 + 2.72186i
93.2 −0.500000 + 0.866025i 0.313577 + 0.543132i −0.500000 − 0.866025i 0.475705 − 0.823946i −0.627155 2.62597 + 0.322894i 1.00000 1.30334 − 2.25745i 0.475705 + 0.823946i
93.3 −0.500000 + 0.866025i 1.26826 + 2.19669i −0.500000 − 0.866025i −1.34706 + 2.33318i −2.53652 −2.28677 − 1.33067i 1.00000 −1.71698 + 2.97389i −1.34706 − 2.33318i
93.4 −0.500000 + 0.866025i 1.60400 + 2.77821i −0.500000 − 0.866025i 1.79990 − 3.11751i −3.20800 1.82854 − 1.91218i 1.00000 −3.64562 + 6.31440i 1.79990 + 3.11751i
277.1 −0.500000 − 0.866025i −0.685837 + 1.18790i −0.500000 + 0.866025i 1.57146 + 2.72186i 1.37167 −1.66774 − 2.05393i 1.00000 0.559256 + 0.968659i 1.57146 − 2.72186i
277.2 −0.500000 − 0.866025i 0.313577 − 0.543132i −0.500000 + 0.866025i 0.475705 + 0.823946i −0.627155 2.62597 − 0.322894i 1.00000 1.30334 + 2.25745i 0.475705 − 0.823946i
277.3 −0.500000 − 0.866025i 1.26826 − 2.19669i −0.500000 + 0.866025i −1.34706 − 2.33318i −2.53652 −2.28677 + 1.33067i 1.00000 −1.71698 − 2.97389i −1.34706 + 2.33318i
277.4 −0.500000 − 0.866025i 1.60400 − 2.77821i −0.500000 + 0.866025i 1.79990 + 3.11751i −3.20800 1.82854 + 1.91218i 1.00000 −3.64562 − 6.31440i 1.79990 − 3.11751i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 93.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.e.b ✓ 8
7.c even 3 1 inner 322.2.e.b ✓ 8
7.c even 3 1 2254.2.a.w 4
7.d odd 6 1 2254.2.a.ba 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.e.b ✓ 8 1.a even 1 1 trivial
322.2.e.b ✓ 8 7.c even 3 1 inner
2254.2.a.w 4 7.c even 3 1
2254.2.a.ba 4 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 5T_{3}^{7} + 22T_{3}^{6} - 37T_{3}^{5} + 71T_{3}^{4} - 37T_{3}^{3} + 142T_{3}^{2} - 77T_{3} + 49 \) acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} - 5 T^{7} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( T^{8} - 5 T^{7} + \cdots + 841 \) Copy content Toggle raw display
$7$ \( T^{8} - T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} - 2 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( (T^{4} + 7 T^{3} + 6 T^{2} + \cdots + 7)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 3 T^{7} + \cdots + 1814409 \) Copy content Toggle raw display
$19$ \( T^{8} - 9 T^{7} + \cdots + 95481 \) Copy content Toggle raw display
$23$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 2 T^{3} - 27 T^{2} + \cdots - 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 14 T^{7} + \cdots + 11449 \) Copy content Toggle raw display
$37$ \( T^{8} + 30 T^{6} + \cdots + 729 \) Copy content Toggle raw display
$41$ \( (T^{4} + 15 T^{3} + \cdots - 63)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 18 T^{3} + \cdots - 81)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 21 T^{7} + \cdots + 35721 \) Copy content Toggle raw display
$53$ \( T^{8} - 3 T^{7} + \cdots + 13689 \) Copy content Toggle raw display
$59$ \( T^{8} - 16 T^{7} + \cdots + 1320201 \) Copy content Toggle raw display
$61$ \( T^{8} - T^{7} + \cdots + 8346321 \) Copy content Toggle raw display
$67$ \( T^{8} - 17 T^{7} + \cdots + 82369 \) Copy content Toggle raw display
$71$ \( (T^{4} + T^{3} - 216 T^{2} + \cdots + 5243)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 4 T^{7} + \cdots + 19245769 \) Copy content Toggle raw display
$79$ \( T^{8} + 5 T^{7} + \cdots + 145161 \) Copy content Toggle raw display
$83$ \( (T^{4} + 4 T^{3} + \cdots + 17003)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 9 T^{7} + \cdots + 12131289 \) Copy content Toggle raw display
$97$ \( (T^{4} - 40 T^{3} + \cdots - 637)^{2} \) Copy content Toggle raw display
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