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Properties

Label 1150.2.e.c
Level $1150$
Weight $2$
Character orbit 1150.e
Analytic conductor $9.183$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1150,2,Mod(643,1150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1150, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1150.643");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.e (of order \(4\), degree \(2\), 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: \(9.18279623245\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.110166016.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 10x^{6} + 19x^{4} + 10x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{5} + 1) q^{3} + \beta_{7} q^{4} + ( - \beta_{4} + 1) q^{6} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{7} - \beta_{3} q^{8} + (2 \beta_{7} + \beta_{6} + \beta_{5} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{5} + 1) q^{3} + \beta_{7} q^{4} + ( - \beta_{4} + 1) q^{6} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{7} - \beta_{3} q^{8} + (2 \beta_{7} + \beta_{6} + \beta_{5} + \cdots + 1) q^{9}+ \cdots + ( - 5 \beta_{6} + 5 \beta_{5} + \cdots + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} + 4 q^{6} + 4 q^{12} + 4 q^{14} - 8 q^{16} + 24 q^{17} + 8 q^{18} + 12 q^{19} - 12 q^{22} - 16 q^{23} + 12 q^{26} - 8 q^{27} - 4 q^{31} + 20 q^{33} + 4 q^{34} - 4 q^{36} + 4 q^{37} + 8 q^{38} + 12 q^{41} - 8 q^{42} - 20 q^{43} - 20 q^{44} + 16 q^{47} - 4 q^{48} + 20 q^{57} - 16 q^{58} - 4 q^{62} - 4 q^{67} + 24 q^{68} - 12 q^{69} - 44 q^{71} + 8 q^{72} - 28 q^{73} - 48 q^{74} - 4 q^{77} + 4 q^{78} - 8 q^{79} - 16 q^{81} - 8 q^{82} - 28 q^{83} + 8 q^{84} + 4 q^{87} - 12 q^{88} + 40 q^{89} + 16 q^{92} + 12 q^{93} - 4 q^{96} - 8 q^{97} - 16 q^{98} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 10x^{6} + 19x^{4} + 10x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + \nu^{5} + 10\nu^{4} + 9\nu^{3} + 18\nu^{2} + 9\nu + 5 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - \nu^{5} + 10\nu^{4} - 9\nu^{3} + 18\nu^{2} - 9\nu + 5 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 9\nu^{4} + 11\nu^{2} + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + \nu^{6} + 10\nu^{5} + 9\nu^{4} + 18\nu^{3} + 9\nu^{2} + 5\nu - 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - \nu^{6} + 10\nu^{5} - 9\nu^{4} + 18\nu^{3} - 9\nu^{2} + 5\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{7} + 19\nu^{5} + 29\nu^{3} + 9\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} + \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 2\beta_{6} - 2\beta_{5} - \beta_{3} + \beta_{2} - 4\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{6} + 7\beta_{5} - 9\beta_{4} + 2\beta_{3} + 2\beta_{2} + 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{7} + 18\beta_{6} + 18\beta_{5} + 7\beta_{3} - 7\beta_{2} + 27\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 52\beta_{6} - 52\beta_{5} + 72\beta_{4} - 18\beta_{3} - 18\beta_{2} - 151 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 72\beta_{7} - 142\beta_{6} - 142\beta_{5} - 52\beta_{3} + 52\beta_{2} - 203\beta _1 - 142 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(-1\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
643.1
0.814115i
− 1.22833i
− 0.360409i
2.77462i
− 0.814115i
1.22833i
0.360409i
− 2.77462i
−0.707107 + 0.707107i −0.575666 − 0.575666i − 1.00000i 0 0.814115 −2.09689 − 2.09689i 0.707107 + 0.707107i − 2.33722i 0
643.2 −0.707107 + 0.707107i 0.868559 + 0.868559i − 1.00000i 0 −1.22833 1.38978 + 1.38978i 0.707107 + 0.707107i − 1.49121i 0
643.3 0.707107 − 0.707107i −0.254848 − 0.254848i − 1.00000i 0 −0.360409 0.812668 + 0.812668i −0.707107 − 0.707107i − 2.87011i 0
643.4 0.707107 − 0.707107i 1.96195 + 1.96195i − 1.00000i 0 2.77462 −0.105561 − 0.105561i −0.707107 − 0.707107i 4.69853i 0
1057.1 −0.707107 − 0.707107i −0.575666 + 0.575666i 1.00000i 0 0.814115 −2.09689 + 2.09689i 0.707107 − 0.707107i 2.33722i 0
1057.2 −0.707107 − 0.707107i 0.868559 − 0.868559i 1.00000i 0 −1.22833 1.38978 − 1.38978i 0.707107 − 0.707107i 1.49121i 0
1057.3 0.707107 + 0.707107i −0.254848 + 0.254848i 1.00000i 0 −0.360409 0.812668 − 0.812668i −0.707107 + 0.707107i 2.87011i 0
1057.4 0.707107 + 0.707107i 1.96195 − 1.96195i 1.00000i 0 2.77462 −0.105561 + 0.105561i −0.707107 + 0.707107i − 4.69853i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 643.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
115.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1150.2.e.c 8
5.b even 2 1 230.2.e.a ✓ 8
5.c odd 4 1 230.2.e.b yes 8
5.c odd 4 1 1150.2.e.b 8
23.b odd 2 1 1150.2.e.b 8
115.c odd 2 1 230.2.e.b yes 8
115.e even 4 1 230.2.e.a ✓ 8
115.e even 4 1 inner 1150.2.e.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.e.a ✓ 8 5.b even 2 1
230.2.e.a ✓ 8 115.e even 4 1
230.2.e.b yes 8 5.c odd 4 1
230.2.e.b yes 8 115.c odd 2 1
1150.2.e.b 8 5.c odd 4 1
1150.2.e.b 8 23.b odd 2 1
1150.2.e.c 8 1.a even 1 1 trivial
1150.2.e.c 8 115.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1150, [\chi])\):

\( T_{3}^{8} - 4T_{3}^{7} + 8T_{3}^{6} - T_{3}^{4} + 8T_{3}^{2} + 4T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{8} - 8T_{7}^{5} + 47T_{7}^{4} - 56T_{7}^{3} + 32T_{7}^{2} + 8T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 8 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} + 50 T^{6} + \cdots + 289 \) Copy content Toggle raw display
$13$ \( T^{8} - 60 T^{5} + \cdots + 5041 \) Copy content Toggle raw display
$17$ \( T^{8} - 24 T^{7} + \cdots + 30625 \) Copy content Toggle raw display
$19$ \( (T^{4} - 6 T^{3} + 3 T^{2} + \cdots - 49)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 16 T^{7} + \cdots + 279841 \) Copy content Toggle raw display
$29$ \( T^{8} + 64 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$31$ \( (T^{4} + 2 T^{3} - 35 T^{2} + \cdots - 49)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 4 T^{7} + \cdots + 795664 \) Copy content Toggle raw display
$41$ \( (T^{4} - 6 T^{3} + \cdots - 119)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 20 T^{7} + \cdots + 3083536 \) Copy content Toggle raw display
$47$ \( T^{8} - 16 T^{7} + \cdots + 4624 \) Copy content Toggle raw display
$53$ \( T^{8} + 800 T^{5} + \cdots + 614656 \) Copy content Toggle raw display
$59$ \( T^{8} + 108 T^{6} + \cdots + 226576 \) Copy content Toggle raw display
$61$ \( T^{8} + 338 T^{6} + \cdots + 13315201 \) Copy content Toggle raw display
$67$ \( T^{8} + 4 T^{7} + \cdots + 9265936 \) Copy content Toggle raw display
$71$ \( (T^{4} + 22 T^{3} + \cdots - 1193)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 28 T^{7} + \cdots + 1882384 \) Copy content Toggle raw display
$79$ \( (T^{4} + 4 T^{3} + \cdots - 5348)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 28 T^{7} + \cdots + 10000 \) Copy content Toggle raw display
$89$ \( (T^{4} - 20 T^{3} + \cdots - 2500)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 8 T^{7} + \cdots + 246584209 \) Copy content Toggle raw display
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