Properties

Label 2496.2.d.m
Level $2496$
Weight $2$
Character orbit 2496.d
Analytic conductor $19.931$
Analytic rank $0$
Dimension $8$
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] = [2496,2,Mod(1535,2496)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2496, 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("2496.1535");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2496.d (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: \(19.9306603445\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.121550625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 156)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + (\beta_{6} + \beta_{3}) q^{5} + ( - \beta_{5} + \beta_{4}) q^{7} - \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + (\beta_{6} + \beta_{3}) q^{5} + ( - \beta_{5} + \beta_{4}) q^{7} - \beta_{6} q^{9} - q^{13} + (2 \beta_{5} - \beta_{2}) q^{15} + (\beta_{6} + \beta_{3}) q^{17} + ( - \beta_{7} + \beta_{5} - \beta_{4}) q^{19} + ( - \beta_{6} - 3) q^{21} + (2 \beta_{5} + 2 \beta_{4}) q^{23} + ( - \beta_{6} + \beta_{3} - 5) q^{25} + (\beta_{5} + \beta_{2}) q^{27} + (\beta_{6} + \beta_{3} - \beta_1) q^{29} + ( - \beta_{7} + \beta_{5} - \beta_{4}) q^{31} + ( - \beta_{7} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{2}) q^{35} + ( - \beta_{6} + \beta_{3}) q^{37} - \beta_{4} q^{39} + (\beta_{6} + \beta_{3} + \beta_1) q^{41} + (2 \beta_{7} - \beta_{5} + \beta_{4}) q^{43} + ( - \beta_{6} - 2 \beta_{3} + \beta_1 + 5) q^{45} + (\beta_{5} + \beta_{4}) q^{47} + ( - \beta_{6} + \beta_{3} + 1) q^{49} + (2 \beta_{5} - \beta_{2}) q^{51} + (\beta_{6} + \beta_{3} + \beta_1) q^{53} + (\beta_{6} - \beta_{3} - \beta_1 + 4) q^{57} + 2 q^{61} + (\beta_{5} - 3 \beta_{4} + \beta_{2}) q^{63} + ( - \beta_{6} - \beta_{3}) q^{65} + ( - \beta_{7} + \beta_{5} - \beta_{4}) q^{67} + ( - 2 \beta_{6} + 6) q^{69} + (\beta_{5} + \beta_{4}) q^{71} + (2 \beta_{6} - 2 \beta_{3} - 2) q^{73} + (4 \beta_{5} - 5 \beta_{4} + \beta_{2}) q^{75} + (\beta_{7} + \beta_{5} - \beta_{4}) q^{79} + (\beta_{6} + 2 \beta_{3} - \beta_1 + 4) q^{81} + (4 \beta_{5} + 4 \beta_{4}) q^{83} + ( - \beta_{6} + \beta_{3} - 10) q^{85} + ( - 3 \beta_{7} + \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{87} + (3 \beta_{6} + 3 \beta_{3} - \beta_1) q^{89} + (\beta_{5} - \beta_{4}) q^{91} + (\beta_{6} - \beta_{3} - \beta_1 + 4) q^{93} + (2 \beta_{7} + 4 \beta_{2}) q^{95} + (2 \beta_{6} - 2 \beta_{3} + 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{9} - 8 q^{13} - 22 q^{21} - 36 q^{25} + 4 q^{37} + 38 q^{45} + 12 q^{49} + 28 q^{57} + 16 q^{61} + 52 q^{69} - 24 q^{73} + 34 q^{81} - 76 q^{85} + 28 q^{93} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -8\nu^{7} + 105\nu^{6} - 49\nu^{5} - 220\nu^{4} - 1061\nu^{3} + 1915\nu^{2} + 1038\nu - 1060 ) / 204 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -163\nu^{7} + 297\nu^{6} + 538\nu^{5} + 2071\nu^{4} - 5827\nu^{3} - 2464\nu^{2} - 4032\nu + 6376 ) / 2448 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 83\nu^{7} - 165\nu^{6} - 110\nu^{5} - 599\nu^{4} + 2255\nu^{3} - 1540\nu^{2} + 1560\nu + 2200 ) / 816 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 83\nu^{7} - 165\nu^{6} - 110\nu^{5} - 599\nu^{4} + 2255\nu^{3} - 1540\nu^{2} - 72\nu + 2200 ) / 816 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -377\nu^{7} + 747\nu^{6} + 974\nu^{5} + 2357\nu^{4} - 11705\nu^{3} + 1600\nu^{2} + 6624\nu - 4792 ) / 2448 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 71\nu^{7} - 101\nu^{6} - 226\nu^{5} - 555\nu^{4} + 1879\nu^{3} + 304\nu^{2} - 1184\nu + 712 ) / 272 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 109\nu^{7} - 145\nu^{6} - 380\nu^{5} - 1057\nu^{4} + 2639\nu^{3} + 1378\nu^{2} + 116\nu + 1072 ) / 408 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{6} - 6\beta_{5} - \beta_{3} + \beta _1 + 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{7} + 7\beta_{6} - \beta_{5} - 9\beta_{4} + 7\beta_{3} - 10\beta_{2} + \beta _1 + 40 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{7} - 2\beta_{6} - 15\beta_{5} - 21\beta_{4} + 16\beta_{3} + 6\beta_{2} + 2\beta _1 - 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -33\beta_{7} - 49\beta_{6} - 121\beta_{5} + 55\beta_{4} - 29\beta_{3} - 22\beta_{2} + 29\beta _1 + 116 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -63\beta_{7} + 70\beta_{6} + 27\beta_{5} - 63\beta_{4} + 70\beta_{3} - 36\beta_{2} + 10\beta _1 + 162 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -93\beta_{7} - 117\beta_{6} - 465\beta_{5} - 161\beta_{4} + 247\beta_{3} + 186\beta_{2} + 117\beta _1 - 468 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1535.1
2.25820 0.369600i
2.25820 + 0.369600i
−0.862555 0.141174i
−0.862555 + 0.141174i
−1.44918 + 1.77086i
−1.44918 1.77086i
0.553538 + 0.676408i
0.553538 0.676408i
0 −1.70466 0.306808i 0 2.09201i 0 0.613616i 0 2.81174 + 1.04601i 0
1535.2 0 −1.70466 + 0.306808i 0 2.09201i 0 0.613616i 0 2.81174 1.04601i 0
1535.3 0 −0.586627 1.62968i 0 3.82407i 0 3.25937i 0 −2.31174 + 1.91203i 0
1535.4 0 −0.586627 + 1.62968i 0 3.82407i 0 3.25937i 0 −2.31174 1.91203i 0
1535.5 0 0.586627 1.62968i 0 3.82407i 0 3.25937i 0 −2.31174 1.91203i 0
1535.6 0 0.586627 + 1.62968i 0 3.82407i 0 3.25937i 0 −2.31174 + 1.91203i 0
1535.7 0 1.70466 0.306808i 0 2.09201i 0 0.613616i 0 2.81174 1.04601i 0
1535.8 0 1.70466 + 0.306808i 0 2.09201i 0 0.613616i 0 2.81174 + 1.04601i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1535.8
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 2496.2.d.m 8
3.b odd 2 1 inner 2496.2.d.m 8
4.b odd 2 1 inner 2496.2.d.m 8
8.b even 2 1 156.2.c.c 8
8.d odd 2 1 156.2.c.c 8
12.b even 2 1 inner 2496.2.d.m 8
24.f even 2 1 156.2.c.c 8
24.h odd 2 1 156.2.c.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.c.c 8 8.b even 2 1
156.2.c.c 8 8.d odd 2 1
156.2.c.c 8 24.f even 2 1
156.2.c.c 8 24.h odd 2 1
2496.2.d.m 8 1.a even 1 1 trivial
2496.2.d.m 8 3.b odd 2 1 inner
2496.2.d.m 8 4.b odd 2 1 inner
2496.2.d.m 8 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 \(S_{2}^{\mathrm{new}}(2496, [\chi])\):

\( T_{5}^{4} + 19T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{7}^{4} + 11T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{23}^{4} - 52T_{23}^{2} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{6} - 8 T^{4} - 9 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} + 19 T^{2} + 64)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 11 T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T + 1)^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 19 T^{2} + 64)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 28)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 52 T^{2} + 256)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 48)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 28)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - T - 26)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 124 T^{2} + 64)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 179 T^{2} + 6724)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 13 T^{2} + 16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 124 T^{2} + 64)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T - 2)^{8} \) Copy content Toggle raw display
$67$ \( (T^{2} + 28)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 13 T^{2} + 16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 6 T - 96)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 44 T^{2} + 64)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 208 T^{2} + 4096)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 220 T^{2} + 1600)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 10 T - 80)^{4} \) Copy content Toggle raw display
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