Properties

Label 1248.2.h.b
Level $1248$
Weight $2$
Character orbit 1248.h
Analytic conductor $9.965$
Analytic rank $0$
Dimension $12$
CM discriminant -104
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1248,2,Mod(623,1248)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1248, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1248.623");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1248.h (of order \(2\), degree \(1\), not 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: \(9.96533017226\)
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: \( x^{12} - 16x^{9} + 92x^{6} - 68x^{3} + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 312)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + \beta_{9} q^{5} + (\beta_{11} - \beta_1) q^{7} - \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} + \beta_{9} q^{5} + (\beta_{11} - \beta_1) q^{7} - \beta_{4} q^{9} - \beta_{2} q^{13} + ( - \beta_{9} - \beta_{7} - \beta_{3} - \beta_{2}) q^{15} + (\beta_{10} - 2 \beta_{5} - \beta_{4}) q^{17} + ( - \beta_{11} - \beta_{7} - 2 \beta_{3} + \beta_{2}) q^{21} + ( - \beta_{10} - 2 \beta_{5} - \beta_{4} - 5) q^{25} + ( - \beta_{8} + 1) q^{27} + 2 \beta_{2} q^{31} + (\beta_{10} + 2 \beta_{8} - \beta_{6} - \beta_{5} - 2 \beta_{4}) q^{35} + (\beta_{9} + 2 \beta_{7} - 2 \beta_1) q^{37} + ( - \beta_{11} - \beta_{9} + \beta_{7} + \beta_1) q^{39} + ( - \beta_{10} + \beta_{6} - \beta_{5} - 2 \beta_{4}) q^{43} + ( - \beta_{11} - \beta_{9} + \beta_{7} + 4 \beta_{3} + \beta_{2} + 2 \beta_1) q^{45} + (\beta_{11} + 2 \beta_{7} + 3 \beta_1) q^{47} + (\beta_{10} + 2 \beta_{6} + 4 \beta_{5} - \beta_{4} + 7) q^{49} + (\beta_{10} - \beta_{8} + \beta_{6} - 2 \beta_{4} - 5) q^{51} + (\beta_{11} + \beta_{9} - \beta_{7} + 5 \beta_{3} - \beta_{2} + \beta_1) q^{63} + ( - \beta_{10} - 2 \beta_{6} + 4 \beta_{5} - \beta_{4}) q^{65} + ( - \beta_{11} - 2 \beta_{9} - 2 \beta_{7} - 3 \beta_1) q^{71} + ( - \beta_{10} - \beta_{8} - \beta_{6} + 5 \beta_{5} - 2 \beta_{4} + 7) q^{75} + (\beta_{10} - 2 \beta_{6} - 2 \beta_{5}) q^{81} + ( - 2 \beta_{11} - \beta_{9} - 2 \beta_{7} - 2 \beta_{2} + 4 \beta_1) q^{85} + (\beta_{10} + 3 \beta_{6} + 5 \beta_{5} - 2 \beta_{4}) q^{91} + (2 \beta_{11} + 2 \beta_{9} - 2 \beta_{7} - 2 \beta_1) q^{93}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 60 q^{25} + 12 q^{27} + 84 q^{49} - 60 q^{51} + 84 q^{75}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 16x^{9} + 92x^{6} - 68x^{3} + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{10} - 49\nu^{7} + 339\nu^{4} - 737\nu ) / 210 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{9} - 21\nu^{6} + 16\nu^{3} + 372 ) / 105 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{9} + 49\nu^{6} - 269\nu^{3} + 107 ) / 70 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{11} + 67\nu^{8} - 422\nu^{5} + 521\nu^{2} ) / 90 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{10} + 31\nu^{7} - 166\nu^{4} + 53\nu ) / 30 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{11} + \nu^{10} - 31\nu^{8} - 18\nu^{7} + 166\nu^{5} + 113\nu^{4} - 23\nu^{2} - 144\nu ) / 30 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -12\nu^{11} - 17\nu^{10} + 196\nu^{8} + 266\nu^{7} - 1146\nu^{5} - 1501\nu^{4} + 1058\nu^{2} + 688\nu ) / 210 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} - 15\nu^{6} + 83\nu^{3} - 33 ) / 6 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 40\nu^{11} + 51\nu^{10} - 595\nu^{8} - 798\nu^{7} + 3050\nu^{5} + 4503\nu^{4} + 475\nu^{2} - 2064\nu ) / 630 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4\nu^{11} - \nu^{10} - 62\nu^{8} + 18\nu^{7} + 332\nu^{5} - 113\nu^{4} - 46\nu^{2} + 144\nu ) / 30 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -76\nu^{11} + 51\nu^{10} + 1183\nu^{8} - 798\nu^{7} - 6488\nu^{5} + 4503\nu^{4} + 2699\nu^{2} - 2064\nu ) / 630 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{11} + \beta_{9} - \beta_{7} + 3\beta_{5} - 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{11} + 2\beta_{10} + 3\beta_{7} + 2\beta_{6} - 3\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{8} + 3\beta_{3} - 2\beta_{2} + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{11} + \beta_{10} + 9\beta_{9} - 9\beta_{7} - 2\beta_{6} + 30\beta_{5} - 6\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 17\beta_{11} + 6\beta_{10} + 14\beta_{9} + 31\beta_{7} + 6\beta_{6} - 39\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{8} + 25\beta_{3} - 5\beta_{2} + 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 61\beta_{11} + 36\beta_{10} + 61\beta_{9} - 61\beta_{7} - 72\beta_{6} + 183\beta_{5} + 69\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 21\beta_{11} - 58\beta_{10} + 252\beta_{9} + 273\beta_{7} - 58\beta_{6} - 309\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 139\beta_{8} + 501\beta_{3} + 16\beta_{2} - 58 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 225\beta_{11} + 475\beta_{10} + 225\beta_{9} - 225\beta_{7} - 950\beta_{6} + 336\beta_{5} + 1488\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -1051\beta_{11} - 1344\beta_{10} + 2744\beta_{9} + 1693\beta_{7} - 1344\beta_{6} - 1587\beta_{4} ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(769\) \(833\) \(1093\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
623.1
−0.598446 + 0.582311i
2.03946 + 0.378672i
2.03946 0.378672i
−0.598446 0.582311i
0.803519 0.227114i
−0.691788 + 1.95556i
−0.691788 1.95556i
0.803519 + 0.227114i
−0.205073 0.809425i
−1.34767 + 1.57689i
−1.34767 1.57689i
−0.205073 + 0.809425i
0 −1.44101 0.960984i 0 3.66851i 0 −5.27581 0 1.15302 + 2.76957i 0
623.2 0 −1.44101 0.960984i 0 3.66851i 0 5.27581 0 1.15302 + 2.76957i 0
623.3 0 −1.44101 + 0.960984i 0 3.66851i 0 5.27581 0 1.15302 2.76957i 0
623.4 0 −1.44101 + 0.960984i 0 3.66851i 0 −5.27581 0 1.15302 2.76957i 0
623.5 0 −0.111731 1.72844i 0 4.04932i 0 2.99062 0 −2.97503 + 0.386242i 0
623.6 0 −0.111731 1.72844i 0 4.04932i 0 −2.99062 0 −2.97503 + 0.386242i 0
623.7 0 −0.111731 + 1.72844i 0 4.04932i 0 −2.99062 0 −2.97503 0.386242i 0
623.8 0 −0.111731 + 1.72844i 0 4.04932i 0 2.99062 0 −2.97503 0.386242i 0
623.9 0 1.55274 0.767460i 0 0.380805i 0 2.28519 0 1.82201 2.38333i 0
623.10 0 1.55274 0.767460i 0 0.380805i 0 −2.28519 0 1.82201 2.38333i 0
623.11 0 1.55274 + 0.767460i 0 0.380805i 0 −2.28519 0 1.82201 + 2.38333i 0
623.12 0 1.55274 + 0.767460i 0 0.380805i 0 2.28519 0 1.82201 + 2.38333i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 623.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
104.h odd 2 1 CM by \(\Q(\sqrt{-26}) \)
3.b odd 2 1 inner
8.d odd 2 1 inner
13.b even 2 1 inner
24.f even 2 1 inner
39.d odd 2 1 inner
312.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1248.2.h.b 12
3.b odd 2 1 inner 1248.2.h.b 12
4.b odd 2 1 312.2.h.b 12
8.b even 2 1 312.2.h.b 12
8.d odd 2 1 inner 1248.2.h.b 12
12.b even 2 1 312.2.h.b 12
13.b even 2 1 inner 1248.2.h.b 12
24.f even 2 1 inner 1248.2.h.b 12
24.h odd 2 1 312.2.h.b 12
39.d odd 2 1 inner 1248.2.h.b 12
52.b odd 2 1 312.2.h.b 12
104.e even 2 1 312.2.h.b 12
104.h odd 2 1 CM 1248.2.h.b 12
156.h even 2 1 312.2.h.b 12
312.b odd 2 1 312.2.h.b 12
312.h even 2 1 inner 1248.2.h.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.h.b 12 4.b odd 2 1
312.2.h.b 12 8.b even 2 1
312.2.h.b 12 12.b even 2 1
312.2.h.b 12 24.h odd 2 1
312.2.h.b 12 52.b odd 2 1
312.2.h.b 12 104.e even 2 1
312.2.h.b 12 156.h even 2 1
312.2.h.b 12 312.b odd 2 1
1248.2.h.b 12 1.a even 1 1 trivial
1248.2.h.b 12 3.b odd 2 1 inner
1248.2.h.b 12 8.d odd 2 1 inner
1248.2.h.b 12 13.b even 2 1 inner
1248.2.h.b 12 24.f even 2 1 inner
1248.2.h.b 12 39.d odd 2 1 inner
1248.2.h.b 12 104.h odd 2 1 CM
1248.2.h.b 12 312.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 30T_{5}^{4} + 225T_{5}^{2} + 32 \) acting on \(S_{2}^{\mathrm{new}}(1248, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 2 T^{3} + 27)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + 30 T^{4} + 225 T^{2} + 32)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 42 T^{4} + 441 T^{2} - 1300)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( (T^{2} - 13)^{6} \) Copy content Toggle raw display
$17$ \( (T^{6} + 102 T^{4} + 2601 T^{2} + \cdots + 6656)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( (T^{2} - 52)^{6} \) Copy content Toggle raw display
$37$ \( (T^{6} - 222 T^{4} + 12321 T^{2} + \cdots - 87412)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( (T^{3} - 129 T + 218)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} + 282 T^{4} + 19881 T^{2} + \cdots + 336200)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( T^{12} \) Copy content Toggle raw display
$71$ \( (T^{6} + 426 T^{4} + 45369 T^{2} + \cdots + 397832)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} \) Copy content Toggle raw display
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