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Properties

Label 2352.2.h.n
Level $2352$
Weight $2$
Character orbit 2352.h
Analytic conductor $18.781$
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] = [2352,2,Mod(2255,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, 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("2352.2255");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.h (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: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.8275904784.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 4x^{4} - 18x^{3} + 45x^{2} - 81x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 336)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - \beta_{7} q^{5} + (\beta_{7} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} - \beta_{7} q^{5} + (\beta_{7} - \beta_{2}) q^{9} + (\beta_{5} + \beta_{3} + \beta_1) q^{11} + 2 q^{13} + ( - \beta_{6} + \beta_{5} - \beta_1) q^{15} + ( - 2 \beta_{7} - \beta_{4} + \beta_{2}) q^{17} + ( - \beta_{5} + \beta_{3}) q^{19} + ( - \beta_{5} - \beta_{3}) q^{23} + ( - 2 \beta_{4} - 2 \beta_{2} - 4) q^{25} + ( - 2 \beta_{6} - \beta_{5} + \cdots + \beta_1) q^{27}+ \cdots + ( - 6 \beta_{6} + 3 \beta_{5} + 4 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{9} + 16 q^{13} - 24 q^{25} + 22 q^{33} - 20 q^{37} + 34 q^{45} - 22 q^{57} - 60 q^{61} - 26 q^{69} - 28 q^{73} - 14 q^{81} - 68 q^{85} - 6 q^{93} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 4x^{4} - 18x^{3} + 45x^{2} - 81x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4\nu^{7} - 3\nu^{6} + 2\nu^{5} - 6\nu^{4} - 20\nu^{3} - 90\nu^{2} + 54\nu - 81 ) / 54 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8\nu^{7} - 3\nu^{6} + 13\nu^{5} + 3\nu^{4} + 5\nu^{3} - 87\nu^{2} + 45\nu - 270 ) / 54 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{7} - 6\nu^{6} + 11\nu^{5} - 3\nu^{4} + 7\nu^{3} - 57\nu^{2} + 99\nu - 189 ) / 18 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{7} + 6\nu^{6} - 11\nu^{5} + 3\nu^{4} - 7\nu^{3} + 57\nu^{2} - 63\nu + 171 ) / 18 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{7} + 5\nu^{6} - 11\nu^{5} + 7\nu^{4} - 7\nu^{3} + 53\nu^{2} - 75\nu + 180 ) / 18 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{7} - 9\nu^{6} + 17\nu^{5} - 9\nu^{4} + 19\nu^{3} - 105\nu^{2} + 144\nu - 297 ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\nu^{7} - 15\nu^{6} + 38\nu^{5} - 12\nu^{4} + 25\nu^{3} - 192\nu^{2} + 225\nu - 621 ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - \beta_{6} - 2\beta_{5} + \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{6} + \beta_{5} - \beta_{3} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} - \beta_{6} + 7\beta_{5} - \beta_{4} + 4\beta_{3} + 2\beta_{2} - \beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6\beta_{7} - 8\beta_{6} - \beta_{5} + 2\beta_{4} - 3\beta_{2} - 2\beta _1 + 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{7} + 10\beta_{4} + 2\beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -18\beta_{7} - 24\beta_{5} - \beta_{4} + 5\beta_{3} + 30\beta_{2} - 6\beta _1 + 29 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2255.1
−1.37009 + 1.05965i
−1.37009 − 1.05965i
0.906034 − 1.47618i
0.906034 + 1.47618i
1.73142 − 0.0465589i
1.73142 + 0.0465589i
0.232633 − 1.71636i
0.232633 + 1.71636i
0 −1.60273 − 0.656712i 0 0.670944i 0 0 0 2.13746 + 2.10506i 0
2255.2 0 −1.60273 + 0.656712i 0 − 0.670944i 0 0 0 2.13746 − 2.10506i 0
2255.3 0 −0.825391 − 1.52274i 0 − 3.94333i 0 0 0 −1.63746 + 2.51371i 0
2255.4 0 −0.825391 + 1.52274i 0 3.94333i 0 0 0 −1.63746 − 2.51371i 0
2255.5 0 0.825391 − 1.52274i 0 3.94333i 0 0 0 −1.63746 − 2.51371i 0
2255.6 0 0.825391 + 1.52274i 0 − 3.94333i 0 0 0 −1.63746 + 2.51371i 0
2255.7 0 1.60273 − 0.656712i 0 − 0.670944i 0 0 0 2.13746 − 2.10506i 0
2255.8 0 1.60273 + 0.656712i 0 0.670944i 0 0 0 2.13746 + 2.10506i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2255.8
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 2352.2.h.n 8
3.b odd 2 1 inner 2352.2.h.n 8
4.b odd 2 1 inner 2352.2.h.n 8
7.b odd 2 1 2352.2.h.m 8
7.c even 3 1 336.2.bj.e ✓ 8
7.c even 3 1 336.2.bj.g yes 8
12.b even 2 1 inner 2352.2.h.n 8
21.c even 2 1 2352.2.h.m 8
21.h odd 6 1 336.2.bj.e ✓ 8
21.h odd 6 1 336.2.bj.g yes 8
28.d even 2 1 2352.2.h.m 8
28.g odd 6 1 336.2.bj.e ✓ 8
28.g odd 6 1 336.2.bj.g yes 8
84.h odd 2 1 2352.2.h.m 8
84.n even 6 1 336.2.bj.e ✓ 8
84.n even 6 1 336.2.bj.g yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bj.e ✓ 8 7.c even 3 1
336.2.bj.e ✓ 8 21.h odd 6 1
336.2.bj.e ✓ 8 28.g odd 6 1
336.2.bj.e ✓ 8 84.n even 6 1
336.2.bj.g yes 8 7.c even 3 1
336.2.bj.g yes 8 21.h odd 6 1
336.2.bj.g yes 8 28.g odd 6 1
336.2.bj.g yes 8 84.n even 6 1
2352.2.h.m 8 7.b odd 2 1
2352.2.h.m 8 21.c even 2 1
2352.2.h.m 8 28.d even 2 1
2352.2.h.m 8 84.h odd 2 1
2352.2.h.n 8 1.a even 1 1 trivial
2352.2.h.n 8 3.b odd 2 1 inner
2352.2.h.n 8 4.b odd 2 1 inner
2352.2.h.n 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 \(S_{2}^{\mathrm{new}}(2352, [\chi])\):

\( T_{5}^{4} + 16T_{5}^{2} + 7 \) Copy content Toggle raw display
\( T_{11}^{4} - 40T_{11}^{2} + 343 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display
\( T_{47}^{4} - 117T_{47}^{2} + 2268 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{6} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} + 16 T^{2} + 7)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 40 T^{2} + 343)^{2} \) Copy content Toggle raw display
$13$ \( (T - 2)^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 43 T^{2} + 448)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 11 T^{2} + 16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 13 T^{2} + 28)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 85 T^{2} + 1792)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 5 T - 8)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 100 T^{2} + 448)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 92 T^{2} + 64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 117 T^{2} + 2268)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 232 T^{2} + 12943)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 160 T^{2} + 5887)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 15 T + 42)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 267 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 124 T^{2} + 1792)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 7 T - 2)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 27)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 31 T^{2} + 112)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 247 T^{2} + 10108)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - T - 14)^{4} \) Copy content Toggle raw display
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