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Properties

Label 5328.2.e.e
Level $5328$
Weight $2$
Character orbit 5328.e
Analytic conductor $42.544$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5328,2,Mod(2591,5328)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5328, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5328.2591");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5328 = 2^{4} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5328.e (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: \(42.5442941969\)
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: \( x^{12} + 4x^{10} + 2x^{8} - 70x^{6} + 37x^{4} + 116x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{5} + \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{5} + \beta_{2} q^{7} + (\beta_1 - 1) q^{13} + ( - \beta_{9} - \beta_{6}) q^{17} + \beta_{5} q^{19} + \beta_{11} q^{23} + (\beta_{3} - 1) q^{25} + (\beta_{9} - \beta_{6}) q^{29} + \beta_{4} q^{31} + \beta_{10} q^{35} + q^{37} - \beta_{7} q^{41} + ( - 2 \beta_{5} - \beta_{4}) q^{43} + (\beta_{10} + \beta_{8}) q^{47} + (2 \beta_1 - 3) q^{49} + ( - \beta_{7} - 2 \beta_{6}) q^{53} + (\beta_{11} - \beta_{10}) q^{59} + ( - 2 \beta_{3} + 2) q^{61} + (\beta_{9} - 2 \beta_{7} + 2 \beta_{6}) q^{65} + ( - 2 \beta_{4} + \beta_{2}) q^{67} + ( - \beta_{10} + \beta_{8}) q^{71} + (\beta_{3} - 8) q^{73} + ( - \beta_{5} + 2 \beta_{2}) q^{79} + ( - \beta_{10} - \beta_{8}) q^{83} + ( - \beta_{3} + 2 \beta_1 - 4) q^{85} + ( - 2 \beta_{7} + \beta_{6}) q^{89} + (2 \beta_{4} - 4 \beta_{2}) q^{91} + (\beta_{10} + \beta_{8}) q^{95} + ( - 2 \beta_{3} + \beta_1 - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 16 q^{13} - 12 q^{25} + 12 q^{37} - 44 q^{49} + 24 q^{61} - 96 q^{73} - 56 q^{85} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 4x^{10} + 2x^{8} - 70x^{6} + 37x^{4} + 116x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 268\nu^{10} + 2038\nu^{8} + 1592\nu^{6} - 22806\nu^{4} - 29656\nu^{2} + 144341 ) / 46825 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -408\nu^{10} + 252\nu^{8} + 9038\nu^{6} + 55686\nu^{4} - 109724\nu^{2} - 80526 ) / 46825 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -908\nu^{10} - 4948\nu^{8} - 6512\nu^{6} + 66086\nu^{4} + 88176\nu^{2} - 126876 ) / 46825 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2318\nu^{10} - 4628\nu^{8} + 18938\nu^{6} + 196556\nu^{4} - 320214\nu^{2} - 240726 ) / 46825 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3014\nu^{10} - 14114\nu^{8} - 22936\nu^{6} + 159338\nu^{4} - 96432\nu^{2} - 87228 ) / 46825 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2521\nu^{11} + 9736\nu^{9} + 299\nu^{7} - 197407\nu^{5} + 74668\nu^{3} + 544802\nu ) / 140475 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 103\nu^{11} + 322\nu^{9} - 730\nu^{7} - 10009\nu^{5} + 5683\nu^{3} + 30713\nu ) / 5619 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1642\nu^{11} + 2842\nu^{9} - 11492\nu^{7} - 118064\nu^{5} + 333796\nu^{3} + 156334\nu ) / 46825 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -6832\nu^{11} - 34342\nu^{9} - 50648\nu^{7} + 387094\nu^{5} - 10246\nu^{3} - 841604\nu ) / 140475 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 638\nu^{11} + 2140\nu^{9} - 12\nu^{7} - 45486\nu^{5} + 56150\nu^{3} + 25054\nu ) / 9365 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -5082\nu^{11} - 16142\nu^{9} + 3777\nu^{7} + 350694\nu^{5} - 468771\nu^{3} - 211129\nu ) / 46825 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{10} + 2\beta_{8} - 3\beta_{7} + 9\beta_{6} ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + \beta_{4} + 3\beta_{3} - 3\beta_{2} + 3\beta _1 - 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 20\beta_{11} + 25\beta_{10} - 9\beta_{9} + 13\beta_{8} + 12\beta_{7} - 36\beta_{6} ) / 36 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{5} - 8\beta_{4} + 3\beta_{3} + 24\beta_{2} + 12 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -70\beta_{11} - 119\beta_{10} - 36\beta_{9} + 4\beta_{8} + 48\beta_{7} - 126\beta_{6} ) / 36 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -16\beta_{5} + 34\beta_{4} - 15\beta_{3} - 81\beta_{2} - 60\beta _1 + 150 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 218\beta_{11} + 295\beta_{10} + 225\beta_{9} + 67\beta_{8} - 870\beta_{7} + 1566\beta_{6} ) / 36 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -4\beta_{5} - 20\beta_{4} + 171\beta_{3} + 72\beta_{2} + 471\beta _1 - 975 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 578\beta_{11} + 808\beta_{10} - 1593\beta_{9} + 139\beta_{8} + 4368\beta_{7} - 8622\beta_{6} ) / 36 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 185\beta_{5} - 620\beta_{4} - 624\beta_{3} + 1644\beta_{2} - 1845\beta _1 + 3981 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -8764\beta_{11} - 12782\beta_{10} + 3573\beta_{9} - 1481\beta_{8} - 12066\beta_{7} + 22428\beta_{6} ) / 36 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5328\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1333\) \(1999\) \(2369\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2591.1
0.287886 + 0.847603i
−0.287886 + 0.847603i
0.819567 + 2.05926i
−0.819567 + 2.05926i
1.49848 − 0.202559i
−1.49848 − 0.202559i
−1.49848 + 0.202559i
1.49848 + 0.202559i
−0.819567 − 2.05926i
0.819567 − 2.05926i
−0.287886 − 0.847603i
0.287886 − 0.847603i
0 0 0 − 3.10942i 0 − 1.79032i 0 0 0
2591.2 0 0 0 − 3.10942i 0 1.79032i 0 0 0
2591.3 0 0 0 − 2.70430i 0 − 4.43271i 0 0 0
2591.4 0 0 0 − 2.70430i 0 4.43271i 0 0 0
2591.5 0 0 0 − 1.00910i 0 − 3.02421i 0 0 0
2591.6 0 0 0 − 1.00910i 0 3.02421i 0 0 0
2591.7 0 0 0 1.00910i 0 − 3.02421i 0 0 0
2591.8 0 0 0 1.00910i 0 3.02421i 0 0 0
2591.9 0 0 0 2.70430i 0 − 4.43271i 0 0 0
2591.10 0 0 0 2.70430i 0 4.43271i 0 0 0
2591.11 0 0 0 3.10942i 0 − 1.79032i 0 0 0
2591.12 0 0 0 3.10942i 0 1.79032i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2591.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5328.2.e.e ✓ 12
3.b odd 2 1 inner 5328.2.e.e ✓ 12
4.b odd 2 1 inner 5328.2.e.e ✓ 12
12.b even 2 1 inner 5328.2.e.e ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5328.2.e.e ✓ 12 1.a even 1 1 trivial
5328.2.e.e ✓ 12 3.b odd 2 1 inner
5328.2.e.e ✓ 12 4.b odd 2 1 inner
5328.2.e.e ✓ 12 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5328, [\chi])\):

\( T_{5}^{6} + 18T_{5}^{4} + 88T_{5}^{2} + 72 \) Copy content Toggle raw display
\( T_{11} \) 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
$5$ \( (T^{6} + 18 T^{4} + \cdots + 72)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 32 T^{4} + \cdots + 576)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( (T^{3} + 4 T^{2} - 12 T - 8)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + 90 T^{4} + \cdots + 5832)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 76 T^{4} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 116 T^{4} + \cdots - 16200)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 106 T^{4} + \cdots + 1800)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 76 T^{4} + \cdots + 14400)^{2} \) Copy content Toggle raw display
$37$ \( (T - 1)^{12} \) Copy content Toggle raw display
$41$ \( (T^{2} + 18)^{6} \) Copy content Toggle raw display
$43$ \( (T^{6} + 252 T^{4} + \cdots + 46656)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 224 T^{4} + \cdots - 373248)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 150 T^{4} + \cdots + 5832)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 284 T^{4} + \cdots - 285768)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 6 T^{2} + \cdots + 344)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 304 T^{4} + \cdots + 788544)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 448 T^{4} + \cdots - 41472)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 24 T^{2} + \cdots + 328)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + 220 T^{4} + \cdots + 576)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 224 T^{4} + \cdots - 373248)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 210 T^{4} + \cdots + 202248)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 4 T^{2} + \cdots - 328)^{4} \) Copy content Toggle raw display
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