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Properties

Label 1156.2.e.e
Level $1156$
Weight $2$
Character orbit 1156.e
Analytic conductor $9.231$
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] = [1156,2,Mod(829,1156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1156, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1156.829");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1156 = 2^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1156.e (of order \(4\), degree \(2\), 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: \(9.23070647366\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.12745506816.8
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 71x^{4} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{2} - \beta_1) q^{5} + ( - 3 \beta_{7} - \beta_{6}) q^{7} + (\beta_{5} + 3 \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{2} - \beta_1) q^{5} + ( - 3 \beta_{7} - \beta_{6}) q^{7} + (\beta_{5} + 3 \beta_{3}) q^{9} + (\beta_{7} + 2 \beta_{6}) q^{11} + (\beta_{4} - 1) q^{13} + ( - 2 \beta_{5} - 7 \beta_{3}) q^{15} + (\beta_{5} + 3 \beta_{3}) q^{19} + (2 \beta_{4} + 3) q^{21} + ( - 5 \beta_{7} - \beta_{6}) q^{23} + (3 \beta_{5} + 4 \beta_{3}) q^{25} - 5 \beta_{7} q^{27} + (8 \beta_{2} + \beta_1) q^{29} + 5 \beta_{2} q^{31} + (\beta_{4} - 11) q^{33} + ( - \beta_{4} - 1) q^{35} + ( - 2 \beta_{2} - 2 \beta_1) q^{37} + (5 \beta_{2} - \beta_1) q^{39} - 6 \beta_{7} q^{41} - \beta_{3} q^{43} + (7 \beta_{7} - 4 \beta_{6}) q^{45} + ( - 2 \beta_{4} - 5) q^{47} + (5 \beta_{5} - 2 \beta_{3}) q^{49} + (\beta_{5} + 5 \beta_{3}) q^{53} + ( - 3 \beta_{4} + 12) q^{55} + ( - 5 \beta_{7} + 3 \beta_{6}) q^{57} + ( - 4 \beta_{5} - 2 \beta_{3}) q^{59} + ( - 9 \beta_{7} + 2 \beta_{6}) q^{61} + \beta_{2} q^{63} + ( - 5 \beta_{2} + 2 \beta_1) q^{65} - 2 \beta_{4} q^{67} + (4 \beta_{4} + 1) q^{69} + 6 \beta_{2} q^{71} + (\beta_{2} + 6 \beta_1) q^{73} + ( - 15 \beta_{7} + 4 \beta_{6}) q^{75} + ( - 5 \beta_{5} + 8 \beta_{3}) q^{77} - 2 \beta_{7} q^{79} + (2 \beta_{4} + 4) q^{81} + ( - \beta_{5} - 5 \beta_{3}) q^{83} + ( - 7 \beta_{5} - 2 \beta_{3}) q^{87} + ( - 4 \beta_{4} + 2) q^{89} + ( - 5 \beta_{7} - 2 \beta_{6}) q^{91} + ( - 5 \beta_{5} - 5 \beta_{3}) q^{93} + (7 \beta_{7} - 4 \beta_{6}) q^{95} + ( - 3 \beta_{2} - 5 \beta_1) q^{97} + (8 \beta_{2} - 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{13} + 32 q^{21} - 84 q^{33} - 12 q^{35} - 48 q^{47} + 84 q^{55} - 8 q^{67} + 24 q^{69} + 40 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 71x^{4} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 41\nu ) / 55 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 96\nu^{2} ) / 275 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + 41 ) / 11 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{6} - 301\nu^{2} ) / 275 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 96\nu^{3} ) / 275 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6\nu^{7} + 301\nu^{3} ) / 1375 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 6\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -5\beta_{7} + 6\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 11\beta_{4} - 41 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 55\beta_{2} - 41\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -96\beta_{5} - 301\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 480\beta_{7} - 301\beta_{6} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1156\mathbb{Z}\right)^\times\).

\(n\) \(579\) \(581\)
\(\chi(n)\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
829.1
−1.97374 − 1.97374i
−1.26663 − 1.26663i
1.26663 + 1.26663i
1.97374 + 1.97374i
−1.97374 + 1.97374i
−1.26663 + 1.26663i
1.26663 − 1.26663i
1.97374 − 1.97374i
0 −1.97374 − 1.97374i 0 2.68085 + 2.68085i 0 0.147582 − 0.147582i 0 4.79129i 0
829.2 0 −1.26663 − 1.26663i 0 0.559525 + 0.559525i 0 −3.38795 + 3.38795i 0 0.208712i 0
829.3 0 1.26663 + 1.26663i 0 −0.559525 − 0.559525i 0 3.38795 − 3.38795i 0 0.208712i 0
829.4 0 1.97374 + 1.97374i 0 −2.68085 − 2.68085i 0 −0.147582 + 0.147582i 0 4.79129i 0
905.1 0 −1.97374 + 1.97374i 0 2.68085 − 2.68085i 0 0.147582 + 0.147582i 0 − 4.79129i 0
905.2 0 −1.26663 + 1.26663i 0 0.559525 − 0.559525i 0 −3.38795 − 3.38795i 0 − 0.208712i 0
905.3 0 1.26663 − 1.26663i 0 −0.559525 + 0.559525i 0 3.38795 + 3.38795i 0 − 0.208712i 0
905.4 0 1.97374 − 1.97374i 0 −2.68085 + 2.68085i 0 −0.147582 − 0.147582i 0 − 4.79129i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 829.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner
17.c even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1156.2.e.e 8
17.b even 2 1 inner 1156.2.e.e 8
17.c even 4 2 inner 1156.2.e.e 8
17.d even 8 1 1156.2.a.b ✓ 2
17.d even 8 1 1156.2.a.d yes 2
17.d even 8 2 1156.2.b.b 4
17.e odd 16 8 1156.2.h.g 16
68.g odd 8 1 4624.2.a.k 2
68.g odd 8 1 4624.2.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1156.2.a.b ✓ 2 17.d even 8 1
1156.2.a.d yes 2 17.d even 8 1
1156.2.b.b 4 17.d even 8 2
1156.2.e.e 8 1.a even 1 1 trivial
1156.2.e.e 8 17.b even 2 1 inner
1156.2.e.e 8 17.c even 4 2 inner
1156.2.h.g 16 17.e odd 16 8
4624.2.a.k 2 68.g odd 8 1
4624.2.a.u 2 68.g odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 71T_{3}^{4} + 625 \) acting on \(S_{2}^{\mathrm{new}}(1156, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 71T^{4} + 625 \) Copy content Toggle raw display
$5$ \( T^{8} + 207T^{4} + 81 \) Copy content Toggle raw display
$7$ \( T^{8} + 527T^{4} + 1 \) Copy content Toggle raw display
$11$ \( (T^{4} + 441)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + T - 5)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 23 T^{2} + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 2151 T^{4} + 50625 \) Copy content Toggle raw display
$29$ \( T^{8} + 9927 T^{4} + 6765201 \) Copy content Toggle raw display
$31$ \( (T^{4} + 625)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 1136 T^{4} + 160000 \) Copy content Toggle raw display
$41$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 12 T + 15)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 51 T^{2} + 225)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 84)^{4} \) Copy content Toggle raw display
$61$ \( T^{8} + 46082 T^{4} + 38950081 \) Copy content Toggle raw display
$67$ \( (T^{2} + 2 T - 20)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 1171350625 \) Copy content Toggle raw display
$79$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 51 T^{2} + 225)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 84)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + 34847 T^{4} + 294499921 \) Copy content Toggle raw display
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