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Properties

Label 975.2.b.l
Level $975$
Weight $2$
Character orbit 975.b
Analytic conductor $7.785$
Analytic rank $0$
Dimension $6$
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] = [975,2,Mod(376,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.376");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.b (of order \(2\), degree \(1\), 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: \(6\)
Coefficient field: 6.0.50922496.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 8x^{4} + 13x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + q^{3} + ( - \beta_{5} - 1) q^{4} + \beta_{2} q^{6} + ( - \beta_{3} - \beta_1) q^{7} + (\beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + q^{3} + ( - \beta_{5} - 1) q^{4} + \beta_{2} q^{6} + ( - \beta_{3} - \beta_1) q^{7} + (\beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1) q^{8} + q^{9} + ( - \beta_{4} + \beta_{2}) q^{11} + ( - \beta_{5} - 1) q^{12} + ( - \beta_{5} + \beta_{2} + \beta_1 - 1) q^{13} + ( - \beta_{3} + \beta_1 - 2) q^{14} + ( - \beta_{3} + \beta_1 + 1) q^{16} + (\beta_{3} - \beta_1 - 2) q^{17} + \beta_{2} q^{18} + ( - 2 \beta_{4} + \beta_{3} + \cdots + \beta_1) q^{19}+ \cdots + ( - \beta_{4} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9} - 6 q^{12} - 4 q^{13} - 8 q^{14} + 10 q^{16} - 16 q^{17} - 12 q^{22} - 24 q^{23} - 14 q^{26} + 6 q^{27} + 12 q^{29} - 6 q^{36} - 16 q^{38} - 4 q^{39} - 8 q^{42} + 8 q^{43} + 10 q^{48} - 22 q^{49} - 16 q^{51} + 32 q^{52} + 8 q^{53} - 32 q^{56} + 4 q^{61} + 32 q^{62} + 10 q^{64} - 12 q^{66} + 40 q^{68} - 24 q^{69} - 52 q^{74} - 24 q^{77} - 14 q^{78} + 32 q^{79} + 6 q^{81} - 20 q^{82} + 12 q^{87} + 28 q^{88} + 24 q^{91} + 24 q^{92} + 12 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 8x^{4} + 13x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 8\nu^{3} + 11\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{2} + \nu - 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - 6\nu^{3} - \nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{4} + 7\nu^{2} + 6 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{4} - 5\beta_{3} + 2\beta_{2} - 5\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{5} + 7\beta_{3} - 7\beta _1 + 30 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -16\beta_{4} + 29\beta_{3} - 12\beta_{2} + 29\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
376.1
− 0.634243i
2.43255i
− 1.29632i
1.29632i
− 2.43255i
0.634243i
− 2.51912i 1.00000 −4.34596 0 − 2.51912i 1.26849i 5.90976i 1.00000 0
376.2 − 1.61036i 1.00000 −0.593272 0 − 1.61036i − 4.86509i − 2.26534i 1.00000 0
376.3 − 0.246506i 1.00000 1.93923 0 − 0.246506i 2.59264i − 0.971044i 1.00000 0
376.4 0.246506i 1.00000 1.93923 0 0.246506i − 2.59264i 0.971044i 1.00000 0
376.5 1.61036i 1.00000 −0.593272 0 1.61036i 4.86509i 2.26534i 1.00000 0
376.6 2.51912i 1.00000 −4.34596 0 2.51912i − 1.26849i − 5.90976i 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 376.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.b.l 6
5.b even 2 1 975.2.b.j 6
5.c odd 4 2 195.2.h.c ✓ 12
13.b even 2 1 inner 975.2.b.l 6
15.e even 4 2 585.2.h.g 12
20.e even 4 2 3120.2.r.n 12
65.d even 2 1 975.2.b.j 6
65.h odd 4 2 195.2.h.c ✓ 12
195.s even 4 2 585.2.h.g 12
260.p even 4 2 3120.2.r.n 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.h.c ✓ 12 5.c odd 4 2
195.2.h.c ✓ 12 65.h odd 4 2
585.2.h.g 12 15.e even 4 2
585.2.h.g 12 195.s even 4 2
975.2.b.j 6 5.b even 2 1
975.2.b.j 6 65.d even 2 1
975.2.b.l 6 1.a even 1 1 trivial
975.2.b.l 6 13.b even 2 1 inner
3120.2.r.n 12 20.e even 4 2
3120.2.r.n 12 260.p even 4 2

Hecke kernels

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

\( T_{2}^{6} + 9T_{2}^{4} + 17T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{17}^{3} + 8T_{17}^{2} - 12T_{17} - 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 9 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 32 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{6} + 20 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{6} + 4 T^{5} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( (T^{3} + 8 T^{2} + \cdots - 128)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 64 T^{4} + \cdots + 6400 \) Copy content Toggle raw display
$23$ \( (T + 4)^{6} \) Copy content Toggle raw display
$29$ \( (T^{3} - 6 T^{2} + \cdots + 104)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 160 T^{4} + \cdots + 43264 \) Copy content Toggle raw display
$37$ \( T^{6} + 132 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( T^{6} + 76 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$43$ \( (T^{3} - 4 T^{2} - 28 T - 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 20 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$53$ \( (T^{3} - 4 T^{2} + \cdots + 160)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 164 T^{4} + \cdots + 141376 \) Copy content Toggle raw display
$61$ \( (T^{3} - 2 T^{2} + \cdots - 256)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 240 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$71$ \( T^{6} + 468 T^{4} + \cdots + 3356224 \) Copy content Toggle raw display
$73$ \( T^{6} + 52 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$79$ \( (T^{3} - 16 T^{2} + \cdots - 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 228 T^{4} + \cdots + 12544 \) Copy content Toggle raw display
$89$ \( T^{6} + 140 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$97$ \( T^{6} + 340 T^{4} + \cdots + 640000 \) Copy content Toggle raw display
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