Properties

Label 975.2.n.e
Level $975$
Weight $2$
Character orbit 975.n
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(749,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.749");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.n (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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

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

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{12}^{2} + 1) q^{3} - 2 \zeta_{12}^{3} q^{4} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} - 2) q^{7} - 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{12}^{2} + 1) q^{3} - 2 \zeta_{12}^{3} q^{4} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} - 2) q^{7} - 3 q^{9} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{12} + ( - 3 \zeta_{12}^{2} + 4) q^{13} - 4 q^{16} + ( - 3 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12} - 3) q^{19} + (4 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 5 \zeta_{12} - 4) q^{21} + (6 \zeta_{12}^{2} - 3) q^{27} + (6 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 6) q^{28} + ( - 6 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12} - 6) q^{31} + 6 \zeta_{12}^{3} q^{36} + ( - 3 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12} - 3) q^{37} + ( - 5 \zeta_{12}^{2} - 2) q^{39} + ( - 7 \zeta_{12}^{3} + 14 \zeta_{12}) q^{43} + (8 \zeta_{12}^{2} - 4) q^{48} + (7 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 5) q^{49} + ( - 2 \zeta_{12}^{3} - 6 \zeta_{12}) q^{52} + ( - 7 \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} + 7) q^{57} + ( - 5 \zeta_{12}^{3} + 10 \zeta_{12}) q^{61} + (6 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 3 \zeta_{12} + 6) q^{63} + 8 \zeta_{12}^{3} q^{64} + (2 \zeta_{12}^{3} + 9 \zeta_{12}^{2} - 9 \zeta_{12} - 2) q^{67} + ( - \zeta_{12}^{3} - 8 \zeta_{12}^{2} - 8 \zeta_{12} - 1) q^{73} + ( - 4 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 10 \zeta_{12} + 4) q^{76} + (10 \zeta_{12}^{3} - 20 \zeta_{12}) q^{79} + 9 q^{81} + ( - 2 \zeta_{12}^{3} + 10 \zeta_{12}^{2} + 10 \zeta_{12} - 2) q^{84} + (\zeta_{12}^{3} + 5 \zeta_{12}^{2} - 10 \zeta_{12} - 11) q^{91} + ( - 4 \zeta_{12}^{3} + 7 \zeta_{12}^{2} - 7 \zeta_{12} + 4) q^{93} + ( - 3 \zeta_{12}^{3} - 11 \zeta_{12}^{2} + 11 \zeta_{12} + 3) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{7} - 12 q^{9} + 10 q^{13} - 16 q^{16} - 2 q^{19} - 6 q^{21} - 20 q^{28} - 14 q^{31} - 20 q^{37} - 18 q^{39} + 30 q^{57} + 30 q^{63} + 10 q^{67} - 20 q^{73} - 4 q^{76} + 36 q^{81} + 12 q^{84} - 34 q^{91} + 30 q^{93} - 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
749.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
0 1.73205i 2.00000i 0 0 −3.36603 3.36603i 0 −3.00000 0
749.2 0 1.73205i 2.00000i 0 0 −1.63397 1.63397i 0 −3.00000 0
824.1 0 1.73205i 2.00000i 0 0 −1.63397 + 1.63397i 0 −3.00000 0
824.2 0 1.73205i 2.00000i 0 0 −3.36603 + 3.36603i 0 −3.00000 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.g odd 4 1 inner
195.n even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.n.e 4
3.b odd 2 1 CM 975.2.n.e 4
5.b even 2 1 975.2.n.j 4
5.c odd 4 1 975.2.o.e 4
5.c odd 4 1 975.2.o.h yes 4
13.d odd 4 1 975.2.n.j 4
15.d odd 2 1 975.2.n.j 4
15.e even 4 1 975.2.o.e 4
15.e even 4 1 975.2.o.h yes 4
39.f even 4 1 975.2.n.j 4
65.f even 4 1 975.2.o.e 4
65.g odd 4 1 inner 975.2.n.e 4
65.k even 4 1 975.2.o.h yes 4
195.j odd 4 1 975.2.o.h yes 4
195.n even 4 1 inner 975.2.n.e 4
195.u odd 4 1 975.2.o.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.n.e 4 1.a even 1 1 trivial
975.2.n.e 4 3.b odd 2 1 CM
975.2.n.e 4 65.g odd 4 1 inner
975.2.n.e 4 195.n even 4 1 inner
975.2.n.j 4 5.b even 2 1
975.2.n.j 4 13.d odd 4 1
975.2.n.j 4 15.d odd 2 1
975.2.n.j 4 39.f even 4 1
975.2.o.e 4 5.c odd 4 1
975.2.o.e 4 15.e even 4 1
975.2.o.e 4 65.f even 4 1
975.2.o.e 4 195.u odd 4 1
975.2.o.h yes 4 5.c odd 4 1
975.2.o.h yes 4 15.e even 4 1
975.2.o.h yes 4 65.k even 4 1
975.2.o.h yes 4 195.j odd 4 1

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
\( T_{7}^{4} + 10T_{7}^{3} + 50T_{7}^{2} + 110T_{7} + 121 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{37}^{4} + 20T_{37}^{3} + 200T_{37}^{2} + 520T_{37} + 676 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 10 T^{3} + 50 T^{2} + \cdots + 121 \) 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
$19$ \( T^{4} + 2 T^{3} + 2 T^{2} - 74 T + 1369 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 14 T^{3} + 98 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$37$ \( T^{4} + 20 T^{3} + 200 T^{2} + \cdots + 676 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 147)^{2} \) 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
$61$ \( (T^{2} - 75)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 11881 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 20 T^{3} + 200 T^{2} + \cdots + 2116 \) Copy content Toggle raw display
$79$ \( (T^{2} - 300)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 10 T^{3} + 50 T^{2} + \cdots + 28561 \) Copy content Toggle raw display
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