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Properties

Label 975.2.bb.i
Level $975$
Weight $2$
Character orbit 975.bb
Analytic conductor $7.785$
Analytic rank $0$
Dimension $8$
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(724,975)]
 
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, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.724");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bb (of order \(6\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.1731891456.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) 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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{5} - \beta_{4} + 3 \beta_{2} + 2) q^{4} + ( - \beta_{5} + \beta_{4} - \beta_{2}) q^{6} + (\beta_{7} - \beta_{6} + \beta_{3} + \beta_1) q^{7} + ( - 4 \beta_{7} + \beta_{6}) q^{8}+ \cdots + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 10 q^{4} - 2 q^{6} + 4 q^{9} + 8 q^{11} + 40 q^{14} - 6 q^{16} + 12 q^{19} - 12 q^{21} + 18 q^{24} + 50 q^{26} + 2 q^{29} + 4 q^{31} - 36 q^{34} - 10 q^{36} + 2 q^{39} - 2 q^{41} + 40 q^{44} - 4 q^{46}+ \cdots + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9\nu^{6} - 65\nu^{4} + 585\nu^{2} - 1296 ) / 1040 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 181\nu ) / 260 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 116 ) / 65 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -29\nu^{6} + 325\nu^{4} - 1885\nu^{2} + 4176 ) / 1040 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{7} - 65\nu^{5} + 585\nu^{3} - 256\nu ) / 1040 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{7} - 65\nu^{5} + 377\nu^{3} - 256\nu ) / 832 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + 5\beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{7} + 5\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{5} + 29\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -36\beta_{7} + 29\beta_{6} - 36\beta_{3} - 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 65\beta_{4} - 116 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -260\beta_{3} - 181\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
724.1
−2.21837 − 1.28078i
−1.35234 − 0.780776i
1.35234 + 0.780776i
2.21837 + 1.28078i
−2.21837 + 1.28078i
−1.35234 + 0.780776i
1.35234 − 0.780776i
2.21837 − 1.28078i
−2.21837 − 1.28078i 0.866025 + 0.500000i 2.28078 + 3.95042i 0 −1.28078 − 2.21837i −3.08440 + 1.78078i − 6.56155i 0.500000 + 0.866025i 0
724.2 −1.35234 − 0.780776i −0.866025 − 0.500000i 0.219224 + 0.379706i 0 0.780776 + 1.35234i −0.486319 + 0.280776i 2.43845i 0.500000 + 0.866025i 0
724.3 1.35234 + 0.780776i 0.866025 + 0.500000i 0.219224 + 0.379706i 0 0.780776 + 1.35234i 0.486319 − 0.280776i − 2.43845i 0.500000 + 0.866025i 0
724.4 2.21837 + 1.28078i −0.866025 − 0.500000i 2.28078 + 3.95042i 0 −1.28078 − 2.21837i 3.08440 − 1.78078i 6.56155i 0.500000 + 0.866025i 0
874.1 −2.21837 + 1.28078i 0.866025 − 0.500000i 2.28078 − 3.95042i 0 −1.28078 + 2.21837i −3.08440 − 1.78078i 6.56155i 0.500000 − 0.866025i 0
874.2 −1.35234 + 0.780776i −0.866025 + 0.500000i 0.219224 − 0.379706i 0 0.780776 − 1.35234i −0.486319 − 0.280776i − 2.43845i 0.500000 − 0.866025i 0
874.3 1.35234 − 0.780776i 0.866025 − 0.500000i 0.219224 − 0.379706i 0 0.780776 − 1.35234i 0.486319 + 0.280776i 2.43845i 0.500000 − 0.866025i 0
874.4 2.21837 − 1.28078i −0.866025 + 0.500000i 2.28078 − 3.95042i 0 −1.28078 + 2.21837i 3.08440 + 1.78078i − 6.56155i 0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 724.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n 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.bb.i 8
5.b even 2 1 inner 975.2.bb.i 8
5.c odd 4 1 39.2.e.b ✓ 4
5.c odd 4 1 975.2.i.k 4
13.c even 3 1 inner 975.2.bb.i 8
15.e even 4 1 117.2.g.c 4
20.e even 4 1 624.2.q.h 4
60.l odd 4 1 1872.2.t.r 4
65.f even 4 1 507.2.j.g 8
65.h odd 4 1 507.2.e.g 4
65.k even 4 1 507.2.j.g 8
65.n even 6 1 inner 975.2.bb.i 8
65.o even 12 1 507.2.b.d 4
65.o even 12 1 507.2.j.g 8
65.q odd 12 1 39.2.e.b ✓ 4
65.q odd 12 1 507.2.a.g 2
65.q odd 12 1 975.2.i.k 4
65.r odd 12 1 507.2.a.d 2
65.r odd 12 1 507.2.e.g 4
65.t even 12 1 507.2.b.d 4
65.t even 12 1 507.2.j.g 8
195.bc odd 12 1 1521.2.b.h 4
195.bf even 12 1 1521.2.a.m 2
195.bl even 12 1 117.2.g.c 4
195.bl even 12 1 1521.2.a.g 2
195.bn odd 12 1 1521.2.b.h 4
260.bg even 12 1 8112.2.a.bo 2
260.bj even 12 1 624.2.q.h 4
260.bj even 12 1 8112.2.a.bk 2
780.cj odd 12 1 1872.2.t.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b ✓ 4 5.c odd 4 1
39.2.e.b ✓ 4 65.q odd 12 1
117.2.g.c 4 15.e even 4 1
117.2.g.c 4 195.bl even 12 1
507.2.a.d 2 65.r odd 12 1
507.2.a.g 2 65.q odd 12 1
507.2.b.d 4 65.o even 12 1
507.2.b.d 4 65.t even 12 1
507.2.e.g 4 65.h odd 4 1
507.2.e.g 4 65.r odd 12 1
507.2.j.g 8 65.f even 4 1
507.2.j.g 8 65.k even 4 1
507.2.j.g 8 65.o even 12 1
507.2.j.g 8 65.t even 12 1
624.2.q.h 4 20.e even 4 1
624.2.q.h 4 260.bj even 12 1
975.2.i.k 4 5.c odd 4 1
975.2.i.k 4 65.q odd 12 1
975.2.bb.i 8 1.a even 1 1 trivial
975.2.bb.i 8 5.b even 2 1 inner
975.2.bb.i 8 13.c even 3 1 inner
975.2.bb.i 8 65.n even 6 1 inner
1521.2.a.g 2 195.bl even 12 1
1521.2.a.m 2 195.bf even 12 1
1521.2.b.h 4 195.bc odd 12 1
1521.2.b.h 4 195.bn odd 12 1
1872.2.t.r 4 60.l odd 4 1
1872.2.t.r 4 780.cj odd 12 1
8112.2.a.bk 2 260.bj even 12 1
8112.2.a.bo 2 260.bg even 12 1

Hecke kernels

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

\( T_{2}^{8} - 9T_{2}^{6} + 65T_{2}^{4} - 144T_{2}^{2} + 256 \) Copy content Toggle raw display
\( T_{7}^{8} - 13T_{7}^{6} + 165T_{7}^{4} - 52T_{7}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 9 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 13 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T + 4)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 25 T^{2} + 169)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 9 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( (T^{4} - 6 T^{3} + 44 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - T^{3} + 39 T^{2} + \cdots + 1444)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - T - 4)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} - 69 T^{6} + \cdots + 456976 \) Copy content Toggle raw display
$41$ \( (T^{4} + T^{3} + 5 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 21 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$47$ \( (T^{2} + 68)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 137 T^{2} + 64)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 14 T^{3} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 16 T^{3} + \cdots + 2209)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 21 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$71$ \( (T^{2} + 14 T + 196)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 106 T^{2} + 361)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 15 T + 52)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 84 T^{2} + 64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 18 T^{3} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 93 T^{6} + \cdots + 2085136 \) Copy content Toggle raw display
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