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Properties

Label 975.2.bp.c
Level $975$
Weight $2$
Character orbit 975.bp
Analytic conductor $7.785$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,2,Mod(149,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 6, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bp (of order \(12\), degree \(4\), 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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

$q$-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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{2} + 2) q^{3} - 2 \zeta_{12} q^{4} + ( - 2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + \cdots + 1) q^{7} + \cdots + ( - 3 \zeta_{12}^{2} + 3) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{2} + 2) q^{3} - 2 \zeta_{12} q^{4} + ( - 2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + \cdots + 1) q^{7} + \cdots + (11 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + \cdots - 8) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{3} - 2 q^{7} + 6 q^{9} - 10 q^{13} + 8 q^{16} + 16 q^{19} - 12 q^{21} - 4 q^{28} - 14 q^{31} + 2 q^{37} - 6 q^{39} + 24 q^{48} - 48 q^{49} + 18 q^{57} - 30 q^{63} - 10 q^{67} + 14 q^{73} + 32 q^{76} - 18 q^{81} + 12 q^{84} + 32 q^{91} - 6 q^{93} - 38 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(\zeta_{12}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
0 1.50000 + 0.866025i 1.73205 − 1.00000i 0 0 0.366025 + 0.0980762i 0 1.50000 + 2.59808i 0
449.1 0 1.50000 + 0.866025i −1.73205 + 1.00000i 0 0 −1.36603 + 5.09808i 0 1.50000 + 2.59808i 0
674.1 0 1.50000 − 0.866025i 1.73205 + 1.00000i 0 0 0.366025 − 0.0980762i 0 1.50000 − 2.59808i 0
899.1 0 1.50000 − 0.866025i −1.73205 − 1.00000i 0 0 −1.36603 − 5.09808i 0 1.50000 − 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
65.s odd 12 1 inner
195.bh even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bp.c 4
3.b odd 2 1 CM 975.2.bp.c 4
5.b even 2 1 975.2.bp.b 4
5.c odd 4 1 975.2.bo.a ✓ 4
5.c odd 4 1 975.2.bo.b yes 4
13.f odd 12 1 975.2.bp.b 4
15.d odd 2 1 975.2.bp.b 4
15.e even 4 1 975.2.bo.a ✓ 4
15.e even 4 1 975.2.bo.b yes 4
39.k even 12 1 975.2.bp.b 4
65.o even 12 1 975.2.bo.a ✓ 4
65.s odd 12 1 inner 975.2.bp.c 4
65.t even 12 1 975.2.bo.b yes 4
195.bc odd 12 1 975.2.bo.b yes 4
195.bh even 12 1 inner 975.2.bp.c 4
195.bn odd 12 1 975.2.bo.a ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.bo.a ✓ 4 5.c odd 4 1
975.2.bo.a ✓ 4 15.e even 4 1
975.2.bo.a ✓ 4 65.o even 12 1
975.2.bo.a ✓ 4 195.bn odd 12 1
975.2.bo.b yes 4 5.c odd 4 1
975.2.bo.b yes 4 15.e even 4 1
975.2.bo.b yes 4 65.t even 12 1
975.2.bo.b yes 4 195.bc odd 12 1
975.2.bp.b 4 5.b even 2 1
975.2.bp.b 4 13.f odd 12 1
975.2.bp.b 4 15.d odd 2 1
975.2.bp.b 4 39.k even 12 1
975.2.bp.c 4 1.a even 1 1 trivial
975.2.bp.c 4 3.b odd 2 1 CM
975.2.bp.c 4 65.s odd 12 1 inner
975.2.bp.c 4 195.bh even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{7}^{4} + 2T_{7}^{3} + 26T_{7}^{2} - 20T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 5 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 16 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 14 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$37$ \( T^{4} - 2 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 108 T^{2} + 11664 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$67$ \( T^{4} + 10 T^{3} + \cdots + 14884 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 14 T^{3} + \cdots + 9409 \) Copy content Toggle raw display
$79$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 38 T^{3} + \cdots + 28561 \) Copy content Toggle raw display
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