Properties

Label 975.2.bc.c
Level $975$
Weight $2$
Character orbit 975.bc
Analytic conductor $7.785$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,2,Mod(751,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.751");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bc (of order \(6\), 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: \(7.78541419707\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) 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 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{3} - 2 \zeta_{6} q^{4} + ( - \zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{3} - 2 \zeta_{6} q^{4} + ( - \zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} + ( - 2 \zeta_{6} - 2) q^{11} + 2 q^{12} + (\zeta_{6} - 4) q^{13} + (4 \zeta_{6} - 4) q^{16} + ( - 2 \zeta_{6} + 4) q^{19} + (2 \zeta_{6} - 1) q^{21} + (6 \zeta_{6} - 6) q^{23} + q^{27} + ( - 2 \zeta_{6} - 2) q^{28} + (6 \zeta_{6} - 6) q^{29} + (2 \zeta_{6} - 1) q^{31} + ( - 2 \zeta_{6} + 4) q^{33} + (2 \zeta_{6} - 2) q^{36} + ( - 4 \zeta_{6} + 3) q^{39} + ( - 4 \zeta_{6} - 4) q^{41} - \zeta_{6} q^{43} + (8 \zeta_{6} - 4) q^{44} + ( - 4 \zeta_{6} + 2) q^{47} - 4 \zeta_{6} q^{48} + (4 \zeta_{6} - 4) q^{49} + (6 \zeta_{6} + 2) q^{52} - 12 q^{53} + (4 \zeta_{6} - 2) q^{57} + (2 \zeta_{6} - 4) q^{59} - \zeta_{6} q^{61} + ( - \zeta_{6} - 1) q^{63} + 8 q^{64} + ( - 5 \zeta_{6} - 5) q^{67} - 6 \zeta_{6} q^{69} + (6 \zeta_{6} - 12) q^{71} + ( - 2 \zeta_{6} + 1) q^{73} + ( - 4 \zeta_{6} - 4) q^{76} - 6 q^{77} - 11 q^{79} + (\zeta_{6} - 1) q^{81} + (16 \zeta_{6} - 8) q^{83} + ( - 2 \zeta_{6} + 4) q^{84} - 6 \zeta_{6} q^{87} + (4 \zeta_{6} + 4) q^{89} + (5 \zeta_{6} - 7) q^{91} + 12 q^{92} + ( - \zeta_{6} - 1) q^{93} + ( - 3 \zeta_{6} + 6) q^{97} + (4 \zeta_{6} - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 2 q^{4} + 3 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - 2 q^{4} + 3 q^{7} - q^{9} - 6 q^{11} + 4 q^{12} - 7 q^{13} - 4 q^{16} + 6 q^{19} - 6 q^{23} + 2 q^{27} - 6 q^{28} - 6 q^{29} + 6 q^{33} - 2 q^{36} + 2 q^{39} - 12 q^{41} - q^{43} - 4 q^{48} - 4 q^{49} + 10 q^{52} - 24 q^{53} - 6 q^{59} - q^{61} - 3 q^{63} + 16 q^{64} - 15 q^{67} - 6 q^{69} - 18 q^{71} - 12 q^{76} - 12 q^{77} - 22 q^{79} - q^{81} + 6 q^{84} - 6 q^{87} + 12 q^{89} - 9 q^{91} + 24 q^{92} - 3 q^{93} + 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(\zeta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
751.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −0.500000 0.866025i −1.00000 + 1.73205i 0 0 1.50000 + 0.866025i 0 −0.500000 + 0.866025i 0
901.1 0 −0.500000 + 0.866025i −1.00000 1.73205i 0 0 1.50000 0.866025i 0 −0.500000 0.866025i 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
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bc.c 2
5.b even 2 1 39.2.j.a 2
5.c odd 4 2 975.2.w.d 4
13.e even 6 1 inner 975.2.bc.c 2
15.d odd 2 1 117.2.q.a 2
20.d odd 2 1 624.2.bv.b 2
60.h even 2 1 1872.2.by.f 2
65.d even 2 1 507.2.j.b 2
65.g odd 4 2 507.2.e.f 4
65.l even 6 1 39.2.j.a 2
65.l even 6 1 507.2.b.c 2
65.n even 6 1 507.2.b.c 2
65.n even 6 1 507.2.j.b 2
65.r odd 12 2 975.2.w.d 4
65.s odd 12 2 507.2.a.e 2
65.s odd 12 2 507.2.e.f 4
195.x odd 6 1 1521.2.b.f 2
195.y odd 6 1 117.2.q.a 2
195.y odd 6 1 1521.2.b.f 2
195.bh even 12 2 1521.2.a.h 2
260.w odd 6 1 624.2.bv.b 2
260.bc even 12 2 8112.2.a.bu 2
780.cb even 6 1 1872.2.by.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.j.a 2 5.b even 2 1
39.2.j.a 2 65.l even 6 1
117.2.q.a 2 15.d odd 2 1
117.2.q.a 2 195.y odd 6 1
507.2.a.e 2 65.s odd 12 2
507.2.b.c 2 65.l even 6 1
507.2.b.c 2 65.n even 6 1
507.2.e.f 4 65.g odd 4 2
507.2.e.f 4 65.s odd 12 2
507.2.j.b 2 65.d even 2 1
507.2.j.b 2 65.n even 6 1
624.2.bv.b 2 20.d odd 2 1
624.2.bv.b 2 260.w odd 6 1
975.2.w.d 4 5.c odd 4 2
975.2.w.d 4 65.r odd 12 2
975.2.bc.c 2 1.a even 1 1 trivial
975.2.bc.c 2 13.e even 6 1 inner
1521.2.a.h 2 195.bh even 12 2
1521.2.b.f 2 195.x odd 6 1
1521.2.b.f 2 195.y odd 6 1
1872.2.by.f 2 60.h even 2 1
1872.2.by.f 2 780.cb even 6 1
8112.2.a.bu 2 260.bc even 12 2

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$13$ \( T^{2} + 7T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 3 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$43$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 12 \) Copy content Toggle raw display
$53$ \( (T + 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$61$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} + 15T + 75 \) Copy content Toggle raw display
$71$ \( T^{2} + 18T + 108 \) Copy content Toggle raw display
$73$ \( T^{2} + 3 \) Copy content Toggle raw display
$79$ \( (T + 11)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 192 \) Copy content Toggle raw display
$89$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
$97$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
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