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Properties

Label 2376.2.b.b
Level $2376$
Weight $2$
Character orbit 2376.b
Analytic conductor $18.972$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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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] = [2376,2,Mod(593,2376)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2376.593"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2376, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2376 = 2^{3} \cdot 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2376.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.9724555203\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 32x^{10} + 345x^{8} + 1476x^{6} + 2804x^{4} + 2344x^{2} + 676 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + \beta_{5} q^{7} - \beta_{8} q^{11} + \beta_{9} q^{13} + ( - \beta_{4} - 1) q^{17} + (\beta_{10} + \beta_{9}) q^{19} + (\beta_{11} + \beta_{5} - \beta_1) q^{23} + ( - \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{25}+ \cdots + ( - \beta_{3} + 2 \beta_{2} - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{11} - 8 q^{17} - 4 q^{25} + 16 q^{29} - 20 q^{35} + 4 q^{41} - 4 q^{49} - 16 q^{55} - 4 q^{65} - 20 q^{67} - 4 q^{77} + 4 q^{83} + 12 q^{91} + 8 q^{95} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 32x^{10} + 345x^{8} + 1476x^{6} + 2804x^{4} + 2344x^{2} + 676 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -9\nu^{10} - 275\nu^{8} - 2701\nu^{6} - 9369\nu^{4} - 12740\nu^{2} - 5310 ) / 244 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9\nu^{10} + 275\nu^{8} + 2701\nu^{6} + 9247\nu^{4} + 11154\nu^{2} + 3724 ) / 122 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -27\nu^{10} - 825\nu^{8} - 8103\nu^{6} - 27863\nu^{4} - 35292\nu^{2} - 14222 ) / 244 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 223\nu^{11} + 6902\nu^{9} + 69785\nu^{7} + 258922\nu^{5} + 381698\nu^{3} + 191472\nu ) / 6344 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 42\nu^{10} + 1263\nu^{8} + 12076\nu^{6} + 39025\nu^{4} + 42902\nu^{2} + 12702 ) / 244 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 324 \nu^{11} + 637 \nu^{10} - 9900 \nu^{9} + 19552 \nu^{8} - 97480 \nu^{7} + 194519 \nu^{6} + \cdots + 348868 ) / 6344 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 324 \nu^{11} + 637 \nu^{10} + 9900 \nu^{9} + 19552 \nu^{8} + 97480 \nu^{7} + 194519 \nu^{6} + \cdots + 348868 ) / 6344 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -605\nu^{11} - 18398\nu^{9} - 179683\nu^{7} - 611322\nu^{5} - 742558\nu^{3} - 250200\nu ) / 6344 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 340\nu^{11} + 10477\nu^{9} + 104898\nu^{7} + 377547\nu^{5} + 502910\nu^{3} + 190718\nu ) / 3172 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1073\nu^{11} + 32698\nu^{9} + 320135\nu^{7} + 1092166\nu^{5} + 1322566\nu^{3} + 443848\nu ) / 6344 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} - \beta_{3} + \beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + 2\beta_{8} - 2\beta_{7} - \beta_{5} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 13\beta_{4} + 12\beta_{3} - 15\beta_{2} + 65 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 16\beta_{11} - 2\beta_{10} + \beta_{9} - 30\beta_{8} + 30\beta_{7} + 19\beta_{5} + 104\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{8} + 3\beta_{7} + \beta_{6} - 161\beta_{4} - 154\beta_{3} + 196\beta_{2} - 800 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -231\beta_{11} + 40\beta_{10} - 38\beta_{9} + 399\beta_{8} - 399\beta_{7} - 273\beta_{5} - 1277\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -78\beta_{8} - 78\beta_{7} - 38\beta_{6} + 1997\beta_{4} + 2046\beta_{3} - 2501\beta_{2} + 10075 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3248\beta_{11} - 622\beta_{10} + 861\beta_{9} - 5196\beta_{8} + 5196\beta_{7} + 3703\beta_{5} + 15910\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1483\beta_{8} + 1483\beta_{7} + 861\beta_{6} - 24819\beta_{4} - 27376\beta_{3} + 31770\beta_{2} - 127520 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 45105 \beta_{11} + 9056 \beta_{10} - 15918 \beta_{9} + 67367 \beta_{8} - 67367 \beta_{7} + \cdots - 199011 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2376\mathbb{Z}\right)^\times\).

\(n\) \(353\) \(1189\) \(1729\) \(1783\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
593.1
− 3.60213i
− 3.50476i
− 1.77423i
− 1.26840i
− 1.17302i
− 0.780160i
0.780160i
1.17302i
1.26840i
1.77423i
3.50476i
3.60213i
0 0 0 − 3.60213i 0 − 2.55568i 0 0 0
593.2 0 0 0 − 3.50476i 0 − 0.922271i 0 0 0
593.3 0 0 0 − 1.77423i 0 3.46105i 0 0 0
593.4 0 0 0 − 1.26840i 0 − 0.159883i 0 0 0
593.5 0 0 0 − 1.17302i 0 0.309683i 0 0 0
593.6 0 0 0 − 0.780160i 0 − 4.95153i 0 0 0
593.7 0 0 0 0.780160i 0 4.95153i 0 0 0
593.8 0 0 0 1.17302i 0 − 0.309683i 0 0 0
593.9 0 0 0 1.26840i 0 0.159883i 0 0 0
593.10 0 0 0 1.77423i 0 − 3.46105i 0 0 0
593.11 0 0 0 3.50476i 0 0.922271i 0 0 0
593.12 0 0 0 3.60213i 0 2.55568i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 593.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2376.2.b.b ✓ 12
3.b odd 2 1 2376.2.b.c yes 12
4.b odd 2 1 4752.2.b.k 12
11.b odd 2 1 2376.2.b.c yes 12
12.b even 2 1 4752.2.b.j 12
33.d even 2 1 inner 2376.2.b.b ✓ 12
44.c even 2 1 4752.2.b.j 12
132.d odd 2 1 4752.2.b.k 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2376.2.b.b ✓ 12 1.a even 1 1 trivial
2376.2.b.b ✓ 12 33.d even 2 1 inner
2376.2.b.c yes 12 3.b odd 2 1
2376.2.b.c yes 12 11.b odd 2 1
4752.2.b.j 12 12.b even 2 1
4752.2.b.j 12 44.c even 2 1
4752.2.b.k 12 4.b odd 2 1
4752.2.b.k 12 132.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2376, [\chi])\):

\( T_{5}^{12} + 32T_{5}^{10} + 345T_{5}^{8} + 1476T_{5}^{6} + 2804T_{5}^{4} + 2344T_{5}^{2} + 676 \) Copy content Toggle raw display
\( T_{17}^{6} + 4T_{17}^{5} - 49T_{17}^{4} - 72T_{17}^{3} + 600T_{17}^{2} - 220T_{17} - 242 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 32 T^{10} + \cdots + 676 \) Copy content Toggle raw display
$7$ \( T^{12} + 44 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{12} + 4 T^{11} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( T^{12} + 68 T^{10} + \cdots + 165649 \) Copy content Toggle raw display
$17$ \( (T^{6} + 4 T^{5} + \cdots - 242)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 96 T^{10} + \cdots + 2226064 \) Copy content Toggle raw display
$23$ \( T^{12} + 116 T^{10} + \cdots + 1308736 \) Copy content Toggle raw display
$29$ \( (T^{6} - 8 T^{5} + \cdots - 4616)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 152 T^{4} + \cdots - 50336)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 158 T^{4} + \cdots - 20396)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 2 T^{5} + \cdots + 352)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 232 T^{10} + \cdots + 350464 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 1920893584 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 43608298276 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 3584656384 \) Copy content Toggle raw display
$61$ \( T^{12} + 468 T^{10} + \cdots + 10975969 \) Copy content Toggle raw display
$67$ \( (T^{6} + 10 T^{5} + \cdots + 25681)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 1982742784 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 3262237456 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 32977470409 \) Copy content Toggle raw display
$83$ \( (T^{6} - 2 T^{5} - 179 T^{4} + \cdots + 88)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 423551052864 \) Copy content Toggle raw display
$97$ \( (T^{6} + 8 T^{5} + \cdots + 1525)^{2} \) Copy content Toggle raw display
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