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Properties

Label 2352.2.h.o
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: 8.0.56070144.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{2} q^{5} + (\beta_{7} - \beta_{5} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{2} q^{5} + (\beta_{7} - \beta_{5} + 1) q^{9} + (\beta_{4} + 2 \beta_{3}) q^{11} + ( - 3 \beta_{5} + 1) q^{13} + ( - \beta_{6} + 3 \beta_{4} + \cdots - \beta_1) q^{15}+ \cdots + ( - 6 \beta_{6} - 3 \beta_{4} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} + 8 q^{13} - 24 q^{25} + 8 q^{33} + 64 q^{37} + 8 q^{45} - 16 q^{57} + 24 q^{61} - 64 q^{69} + 64 q^{73} - 8 q^{81} + 16 q^{85} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5\nu^{7} + \nu^{6} + 25\nu^{5} + 46\nu^{4} + 5\nu^{3} + 95\nu^{2} + 9\nu - 19 ) / 37 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -6\nu^{7} + 21\nu^{6} - 67\nu^{5} + 115\nu^{4} - 117\nu^{3} + 71\nu^{2} + 115\nu - 66 ) / 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{7} - 21\nu^{6} + 67\nu^{5} - 78\nu^{4} + 43\nu^{3} + 188\nu^{2} - 263\nu + 251 ) / 37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -12\nu^{7} + 42\nu^{6} - 134\nu^{5} + 230\nu^{4} - 234\nu^{3} + 142\nu^{2} + 82\nu - 58 ) / 37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} - 3\nu^{5} + 11\nu^{4} - 17\nu^{3} + 25\nu^{2} - 17\nu + 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{7} - 36\nu^{6} + 136\nu^{5} - 361\nu^{4} + 634\nu^{3} - 756\nu^{2} + 564\nu - 167 ) / 37 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -42\nu^{7} + 147\nu^{6} - 543\nu^{5} + 990\nu^{4} - 1559\nu^{3} + 1422\nu^{2} - 971\nu + 278 ) / 37 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{4} + 2\beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} + 2\beta_{5} - \beta_{4} + 2\beta_{2} - 2\beta _1 - 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} + 3\beta_{5} + 7\beta_{4} - 5\beta_{2} - 6\beta _1 - 13 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{7} - 14\beta_{6} - 8\beta_{5} + 17\beta_{4} + 4\beta_{3} - 12\beta_{2} + 2\beta _1 + 18 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 8\beta_{7} - 10\beta_{6} - 25\beta_{5} - 16\beta_{4} + 10\beta_{3} + 11\beta_{2} + 40\beta _1 + 67 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 17\beta_{7} + 37\beta_{6} + 9\beta_{5} - 54\beta_{4} - 7\beta_{3} + 32\beta_{2} + 23\beta _1 - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -8\beta_{7} + 126\beta_{6} + 154\beta_{5} - 41\beta_{4} - 84\beta_{3} + 6\beta_{2} - 154\beta _1 - 320 ) / 4 \) 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
0.500000 + 2.19293i
0.500000 − 2.19293i
0.500000 − 1.56488i
0.500000 + 1.56488i
0.500000 + 0.564882i
0.500000 − 0.564882i
0.500000 − 1.19293i
0.500000 + 1.19293i
0 −1.69293 − 0.366025i 0 3.38587i 0 0 0 2.73205 + 1.23931i 0
2255.2 0 −1.69293 + 0.366025i 0 − 3.38587i 0 0 0 2.73205 − 1.23931i 0
2255.3 0 −1.06488 − 1.36603i 0 − 2.12976i 0 0 0 −0.732051 + 2.90931i 0
2255.4 0 −1.06488 + 1.36603i 0 2.12976i 0 0 0 −0.732051 − 2.90931i 0
2255.5 0 1.06488 − 1.36603i 0 2.12976i 0 0 0 −0.732051 − 2.90931i 0
2255.6 0 1.06488 + 1.36603i 0 − 2.12976i 0 0 0 −0.732051 + 2.90931i 0
2255.7 0 1.69293 − 0.366025i 0 − 3.38587i 0 0 0 2.73205 − 1.23931i 0
2255.8 0 1.69293 + 0.366025i 0 3.38587i 0 0 0 2.73205 + 1.23931i 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.o 8
3.b odd 2 1 inner 2352.2.h.o 8
4.b odd 2 1 inner 2352.2.h.o 8
7.b odd 2 1 336.2.h.b ✓ 8
12.b even 2 1 inner 2352.2.h.o 8
21.c even 2 1 336.2.h.b ✓ 8
28.d even 2 1 336.2.h.b ✓ 8
56.e even 2 1 1344.2.h.g 8
56.h odd 2 1 1344.2.h.g 8
84.h odd 2 1 336.2.h.b ✓ 8
168.e odd 2 1 1344.2.h.g 8
168.i even 2 1 1344.2.h.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.h.b ✓ 8 7.b odd 2 1
336.2.h.b ✓ 8 21.c even 2 1
336.2.h.b ✓ 8 28.d even 2 1
336.2.h.b ✓ 8 84.h odd 2 1
1344.2.h.g 8 56.e even 2 1
1344.2.h.g 8 56.h odd 2 1
1344.2.h.g 8 168.e odd 2 1
1344.2.h.g 8 168.i even 2 1
2352.2.h.o 8 1.a even 1 1 trivial
2352.2.h.o 8 3.b odd 2 1 inner
2352.2.h.o 8 4.b odd 2 1 inner
2352.2.h.o 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 \(S_{2}^{\mathrm{new}}(2352, [\chi])\):

\( T_{5}^{4} + 16T_{5}^{2} + 52 \) Copy content Toggle raw display
\( T_{11}^{4} - 40T_{11}^{2} + 208 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 26 \) Copy content Toggle raw display
\( T_{47} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} + 16 T^{2} + 52)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 40 T^{2} + 208)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T - 26)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 40 T^{2} + 208)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 8 T^{2} + 4)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 64 T^{2} + 832)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 40 T^{2} + 208)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 96 T^{2} + 576)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 16 T + 52)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 88 T^{2} + 208)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} + 88 T^{2} + 208)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 160 T^{2} + 6292)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 6 T + 6)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 88 T^{2} + 208)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 16 T + 52)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 144)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 160 T^{2} + 6292)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 376 T^{2} + 35152)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 4 T - 44)^{4} \) Copy content Toggle raw display
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