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Properties

Label 2312.2.a.w
Level $2312$
Weight $2$
Character orbit 2312.a
Self dual yes
Analytic conductor $18.461$
Analytic rank $0$
Dimension $12$
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] = [2312,2,Mod(1,2312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2312, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2312.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2312 = 2^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2312.a (trivial)

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: yes
Analytic conductor: \(18.4614129473\)
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: \( x^{12} - 32x^{10} + 380x^{8} - 2000x^{6} + 4068x^{4} - 800x^{2} + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 136)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{10} q^{5} + \beta_{6} q^{7} + (\beta_{4} + \beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{10} q^{5} + \beta_{6} q^{7} + (\beta_{4} + \beta_{3} + 2) q^{9} + ( - \beta_{10} - \beta_{9} + \beta_{6}) q^{11} + (\beta_{5} + \beta_{4} + 2) q^{13} + ( - \beta_{8} - \beta_{5} + 2 \beta_{4} + 1) q^{15} + ( - \beta_{7} - 1) q^{19} + (2 \beta_{8} + \beta_{7} - 2 \beta_{4} + 1) q^{21} + ( - \beta_{11} + \beta_{9} + \cdots + \beta_1) q^{23}+ \cdots + ( - \beta_{11} - 5 \beta_{9} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 28 q^{9} + 24 q^{13} + 8 q^{15} - 8 q^{19} + 16 q^{21} + 20 q^{25} + 24 q^{33} - 32 q^{35} + 8 q^{43} + 24 q^{47} + 36 q^{49} + 8 q^{53} + 56 q^{55} - 40 q^{59} + 40 q^{67} + 56 q^{69} + 80 q^{77} + 60 q^{81} - 24 q^{83} + 24 q^{87} + 48 q^{89} + 40 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 32x^{10} + 380x^{8} - 2000x^{6} + 4068x^{4} - 800x^{2} + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} - 24\nu^{7} + 186\nu^{5} - 456\nu^{3} - 80\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} - 24\nu^{6} + 186\nu^{4} - 456\nu^{2} - 32 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{8} + 24\nu^{6} - 186\nu^{4} + 472\nu^{2} - 48 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{10} + 26\nu^{8} - 226\nu^{6} + 700\nu^{4} - 368\nu^{2} + 32 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} - 28\nu^{9} + 282\nu^{7} - 1200\nu^{5} + 1712\nu^{3} + 576\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{8} - 124\nu^{6} + 1010\nu^{4} - 2736\nu^{2} + 240 ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{10} - 29\nu^{8} + 310\nu^{6} - 1466\nu^{4} + 2656\nu^{2} - 240 ) / 32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{11} + 31\nu^{9} - 354\nu^{7} + 1774\nu^{5} - 3368\nu^{3} + 432\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3\nu^{11} - 96\nu^{9} + 1142\nu^{7} - 6024\nu^{5} + 12160\nu^{3} - 1600\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3\nu^{11} - 100\nu^{9} + 1238\nu^{7} - 6768\nu^{5} + 14048\nu^{3} - 1792\nu ) / 64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{10} + 2\beta_{2} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + 3\beta_{7} + \beta_{5} + 14\beta_{4} + 8\beta_{3} + 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 18\beta_{11} - 18\beta_{10} + 2\beta_{9} + 4\beta_{6} + 30\beta_{2} + 66\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 20\beta_{8} + 52\beta_{7} + 20\beta_{5} + 166\beta_{4} + 66\beta_{3} + 370 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 238\beta_{11} - 230\beta_{10} + 48\beta_{9} + 72\beta_{6} + 380\beta_{2} + 568\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 294\beta_{8} + 690\beta_{7} + 294\beta_{5} + 1836\beta_{4} + 568\beta_{3} + 3380 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2820\beta_{11} - 2628\beta_{10} + 780\beta_{9} + 984\beta_{6} + 4484\beta_{2} + 5084\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3824\beta_{8} + 8288\beta_{7} + 3792\beta_{5} + 19652\beta_{4} + 5084\beta_{3} + 31852 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 31732\beta_{11} - 28612\beta_{10} + 10704\beta_{9} + 12112\beta_{6} + 50968\beta_{2} + 47104\beta_1 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.24836
−2.84905
−2.67543
−2.44105
−0.399310
−0.234385
0.234385
0.399310
2.44105
2.67543
2.84905
3.24836
0 −3.24836 0 0.210351 0 −1.90844 0 7.55187 0
1.2 0 −2.84905 0 −3.59014 0 3.97054 0 5.11711 0
1.3 0 −2.67543 0 −1.02573 0 −2.33494 0 4.15793 0
1.4 0 −2.44105 0 3.53569 0 −1.74188 0 2.95871 0
1.5 0 −0.399310 0 1.18736 0 −5.11361 0 −2.84055 0
1.6 0 −0.234385 0 −3.47903 0 −2.44081 0 −2.94506 0
1.7 0 0.234385 0 3.47903 0 2.44081 0 −2.94506 0
1.8 0 0.399310 0 −1.18736 0 5.11361 0 −2.84055 0
1.9 0 2.44105 0 −3.53569 0 1.74188 0 2.95871 0
1.10 0 2.67543 0 1.02573 0 2.33494 0 4.15793 0
1.11 0 2.84905 0 3.59014 0 −3.97054 0 5.11711 0
1.12 0 3.24836 0 −0.210351 0 1.90844 0 7.55187 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(17\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2312.2.a.w 12
4.b odd 2 1 4624.2.a.bt 12
17.b even 2 1 inner 2312.2.a.w 12
17.c even 4 2 2312.2.b.n 12
17.e odd 16 2 136.2.n.c ✓ 12
51.i even 16 2 1224.2.bq.c 12
68.d odd 2 1 4624.2.a.bt 12
68.i even 16 2 272.2.v.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.n.c ✓ 12 17.e odd 16 2
272.2.v.f 12 68.i even 16 2
1224.2.bq.c 12 51.i even 16 2
2312.2.a.w 12 1.a even 1 1 trivial
2312.2.a.w 12 17.b even 2 1 inner
2312.2.b.n 12 17.c even 4 2
4624.2.a.bt 12 4.b odd 2 1
4624.2.a.bt 12 68.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2312))\):

\( T_{3}^{12} - 32T_{3}^{10} + 380T_{3}^{8} - 2000T_{3}^{6} + 4068T_{3}^{4} - 800T_{3}^{2} + 32 \) Copy content Toggle raw display
\( T_{5}^{12} - 40T_{5}^{10} + 564T_{5}^{8} - 3184T_{5}^{6} + 5636T_{5}^{4} - 3136T_{5}^{2} + 128 \) Copy content Toggle raw display
\( T_{7}^{12} - 60T_{7}^{10} + 1290T_{7}^{8} - 12816T_{7}^{6} + 64080T_{7}^{4} - 156416T_{7}^{2} + 147968 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 32 T^{10} + \cdots + 32 \) Copy content Toggle raw display
$5$ \( T^{12} - 40 T^{10} + \cdots + 128 \) Copy content Toggle raw display
$7$ \( T^{12} - 60 T^{10} + \cdots + 147968 \) Copy content Toggle raw display
$11$ \( T^{12} - 80 T^{10} + \cdots + 16928 \) Copy content Toggle raw display
$13$ \( (T^{6} - 12 T^{5} + \cdots - 512)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( (T^{6} + 4 T^{5} - 30 T^{4} + \cdots - 32)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 156 T^{10} + \cdots + 43655168 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 118210688 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 103219712 \) Copy content Toggle raw display
$37$ \( T^{12} - 88 T^{10} + \cdots + 512 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 591542408 \) Copy content Toggle raw display
$43$ \( (T^{6} - 4 T^{5} + \cdots + 12176)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 12 T^{5} + \cdots + 20992)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 4 T^{5} + \cdots - 5008)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 20 T^{5} + \cdots + 3056)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 118949888 \) Copy content Toggle raw display
$67$ \( (T^{6} - 20 T^{5} + \cdots + 28928)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} - 220 T^{10} + \cdots + 10913792 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 545424392 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 1130596352 \) Copy content Toggle raw display
$83$ \( (T^{6} + 12 T^{5} + \cdots - 16)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 24 T^{5} + \cdots + 112064)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} - 444 T^{10} + \cdots + 30451208 \) Copy content Toggle raw display
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