Properties

Label 975.2.bt.d
Level $975$
Weight $2$
Character orbit 975.bt
Analytic conductor $7.785$
Analytic rank $0$
Dimension $8$
CM discriminant -3
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,2,Mod(68,975)]
 
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, 9, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.68");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bt (of order \(12\), degree \(4\), 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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{24}^{7} - 2 \zeta_{24}^{3}) q^{3} - 2 \zeta_{24}^{2} q^{4} + ( - 2 \zeta_{24}^{5} + 4 \zeta_{24}) q^{7} + 3 \zeta_{24}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{24}^{7} - 2 \zeta_{24}^{3}) q^{3} - 2 \zeta_{24}^{2} q^{4} + ( - 2 \zeta_{24}^{5} + 4 \zeta_{24}) q^{7} + 3 \zeta_{24}^{2} q^{9} + (2 \zeta_{24}^{5} + 2 \zeta_{24}) q^{12} + (3 \zeta_{24}^{7} - 4 \zeta_{24}^{3}) q^{13} + 4 \zeta_{24}^{4} q^{16} + 7 \zeta_{24}^{2} q^{19} - 6 q^{21} + ( - 3 \zeta_{24}^{5} - 3 \zeta_{24}) q^{27} + (4 \zeta_{24}^{7} - 8 \zeta_{24}^{3}) q^{28} + 7 q^{31} - 6 \zeta_{24}^{4} q^{36} + ( - 8 \zeta_{24}^{5} + 4 \zeta_{24}) q^{37} + ( - 2 \zeta_{24}^{6} + 7 \zeta_{24}^{2}) q^{39} + ( - 6 \zeta_{24}^{7} - 6 \zeta_{24}^{3}) q^{43} + ( - 4 \zeta_{24}^{7} - 4 \zeta_{24}^{3}) q^{48} + ( - 5 \zeta_{24}^{6} + 5 \zeta_{24}^{2}) q^{49} + (2 \zeta_{24}^{5} + 6 \zeta_{24}) q^{52} + ( - 7 \zeta_{24}^{5} - 7 \zeta_{24}) q^{57} + ( - 13 \zeta_{24}^{4} + 13) q^{61} + ( - 6 \zeta_{24}^{7} + 12 \zeta_{24}^{3}) q^{63} - 8 \zeta_{24}^{6} q^{64} + (4 \zeta_{24}^{5} - 2 \zeta_{24}) q^{67} + ( - 18 \zeta_{24}^{7} + 9 \zeta_{24}^{3}) q^{73} - 14 \zeta_{24}^{4} q^{76} - 17 \zeta_{24}^{6} q^{79} + 9 \zeta_{24}^{4} q^{81} + 12 \zeta_{24}^{2} q^{84} + (4 \zeta_{24}^{4} - 14) q^{91} + (7 \zeta_{24}^{7} - 14 \zeta_{24}^{3}) q^{93} + ( - 11 \zeta_{24}^{5} + 22 \zeta_{24}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{16} - 48 q^{21} + 56 q^{31} - 24 q^{36} + 52 q^{61} - 56 q^{76} + 36 q^{81} - 96 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(-1 + \zeta_{24}^{4}\) \(-1\) \(\zeta_{24}^{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
−0.258819 + 0.965926i
0.258819 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
0 −0.448288 + 1.67303i 1.73205 + 1.00000i 0 0 0.896575 + 3.34607i 0 −2.59808 1.50000i 0
68.2 0 0.448288 1.67303i 1.73205 + 1.00000i 0 0 −0.896575 3.34607i 0 −2.59808 1.50000i 0
107.1 0 −1.67303 0.448288i −1.73205 1.00000i 0 0 3.34607 0.896575i 0 2.59808 + 1.50000i 0
107.2 0 1.67303 + 0.448288i −1.73205 1.00000i 0 0 −3.34607 + 0.896575i 0 2.59808 + 1.50000i 0
893.1 0 −1.67303 + 0.448288i −1.73205 + 1.00000i 0 0 3.34607 + 0.896575i 0 2.59808 1.50000i 0
893.2 0 1.67303 0.448288i −1.73205 + 1.00000i 0 0 −3.34607 0.896575i 0 2.59808 1.50000i 0
932.1 0 −0.448288 1.67303i 1.73205 1.00000i 0 0 0.896575 3.34607i 0 −2.59808 + 1.50000i 0
932.2 0 0.448288 + 1.67303i 1.73205 1.00000i 0 0 −0.896575 + 3.34607i 0 −2.59808 + 1.50000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
13.c even 3 1 inner
15.d odd 2 1 inner
15.e even 4 2 inner
39.i odd 6 1 inner
65.n even 6 1 inner
65.q odd 12 2 inner
195.x odd 6 1 inner
195.bl even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bt.d 8
3.b odd 2 1 CM 975.2.bt.d 8
5.b even 2 1 inner 975.2.bt.d 8
5.c odd 4 2 inner 975.2.bt.d 8
13.c even 3 1 inner 975.2.bt.d 8
15.d odd 2 1 inner 975.2.bt.d 8
15.e even 4 2 inner 975.2.bt.d 8
39.i odd 6 1 inner 975.2.bt.d 8
65.n even 6 1 inner 975.2.bt.d 8
65.q odd 12 2 inner 975.2.bt.d 8
195.x odd 6 1 inner 975.2.bt.d 8
195.bl even 12 2 inner 975.2.bt.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.bt.d 8 1.a even 1 1 trivial
975.2.bt.d 8 3.b odd 2 1 CM
975.2.bt.d 8 5.b even 2 1 inner
975.2.bt.d 8 5.c odd 4 2 inner
975.2.bt.d 8 13.c even 3 1 inner
975.2.bt.d 8 15.d odd 2 1 inner
975.2.bt.d 8 15.e even 4 2 inner
975.2.bt.d 8 39.i odd 6 1 inner
975.2.bt.d 8 65.n even 6 1 inner
975.2.bt.d 8 65.q odd 12 2 inner
975.2.bt.d 8 195.x odd 6 1 inner
975.2.bt.d 8 195.bl even 12 2 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
\( T_{7}^{8} - 144T_{7}^{4} + 20736 \) Copy content Toggle raw display
\( T_{59} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 9T^{4} + 81 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 144 T^{4} + 20736 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 337 T^{4} + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} - 49 T^{2} + 2401)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T - 7)^{8} \) Copy content Toggle raw display
$37$ \( T^{8} - 2304 T^{4} + \cdots + 5308416 \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} - 11664 T^{4} + \cdots + 136048896 \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{2} - 13 T + 169)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} - 144 T^{4} + 20736 \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} + 59049)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 289)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} - 131769 T^{4} + \cdots + 17363069361 \) Copy content Toggle raw display
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