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Properties

Label 325.2.o.b
Level $325$
Weight $2$
Character orbit 325.o
Analytic conductor $2.595$
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] = [325,2,Mod(74,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("325.74");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.o (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: \(2.59513806569\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.592240896.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 7x^{6} + 40x^{4} - 63x^{2} + 81 \) 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 65)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{6} + \beta_{4} - 2 \beta_{3} + 1) q^{4} + (\beta_{6} + \beta_{4} - \beta_{3}) q^{6} + (\beta_{7} - \beta_{2}) q^{7} + 3 \beta_{7} q^{8} + (2 \beta_{3} - 2) q^{9}+ \cdots + (4 \beta_{4} - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{4} + 2 q^{6} - 8 q^{9} + 8 q^{11} - 4 q^{14} + 6 q^{16} - 8 q^{19} - 8 q^{21} - 12 q^{24} - 26 q^{26} + 4 q^{29} - 32 q^{31} + 68 q^{34} + 12 q^{36} - 12 q^{41} - 28 q^{44} - 6 q^{46} - 24 q^{49}+ \cdots - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 7x^{6} + 40x^{4} - 63x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 97\nu ) / 120 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{6} + 40\nu^{4} - 280\nu^{2} + 441 ) / 360 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 57 ) / 40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -7\nu^{7} + 40\nu^{5} - 280\nu^{3} + 81\nu ) / 360 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -19\nu^{6} + 160\nu^{4} - 760\nu^{2} + 1197 ) / 360 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{7} + 40\nu^{5} - 190\nu^{3} + 81\nu ) / 270 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} - 4\beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{7} - 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{6} - 19\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 21\beta_{7} - 19\beta_{5} - 21\beta_{2} - 19\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -40\beta_{4} - 57 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -120\beta_{2} - 97\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(-1 + \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1
−1.99426 − 1.15139i
−1.12824 − 0.651388i
1.12824 + 0.651388i
1.99426 + 1.15139i
−1.99426 + 1.15139i
−1.12824 + 0.651388i
1.12824 − 0.651388i
1.99426 − 1.15139i
−1.99426 − 1.15139i −0.866025 − 0.500000i 1.65139 + 2.86029i 0 1.15139 + 1.99426i 0.866025 − 0.500000i − 3.00000i −1.00000 − 1.73205i 0
74.2 −1.12824 − 0.651388i 0.866025 + 0.500000i −0.151388 − 0.262211i 0 −0.651388 − 1.12824i −0.866025 + 0.500000i 3.00000i −1.00000 − 1.73205i 0
74.3 1.12824 + 0.651388i −0.866025 − 0.500000i −0.151388 − 0.262211i 0 −0.651388 − 1.12824i 0.866025 − 0.500000i − 3.00000i −1.00000 − 1.73205i 0
74.4 1.99426 + 1.15139i 0.866025 + 0.500000i 1.65139 + 2.86029i 0 1.15139 + 1.99426i −0.866025 + 0.500000i 3.00000i −1.00000 − 1.73205i 0
224.1 −1.99426 + 1.15139i −0.866025 + 0.500000i 1.65139 − 2.86029i 0 1.15139 − 1.99426i 0.866025 + 0.500000i 3.00000i −1.00000 + 1.73205i 0
224.2 −1.12824 + 0.651388i 0.866025 − 0.500000i −0.151388 + 0.262211i 0 −0.651388 + 1.12824i −0.866025 − 0.500000i − 3.00000i −1.00000 + 1.73205i 0
224.3 1.12824 − 0.651388i −0.866025 + 0.500000i −0.151388 + 0.262211i 0 −0.651388 + 1.12824i 0.866025 + 0.500000i 3.00000i −1.00000 + 1.73205i 0
224.4 1.99426 − 1.15139i 0.866025 − 0.500000i 1.65139 − 2.86029i 0 1.15139 − 1.99426i −0.866025 − 0.500000i − 3.00000i −1.00000 + 1.73205i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 74.4
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 325.2.o.b 8
5.b even 2 1 inner 325.2.o.b 8
5.c odd 4 1 65.2.e.b ✓ 4
5.c odd 4 1 325.2.e.a 4
13.c even 3 1 inner 325.2.o.b 8
15.e even 4 1 585.2.j.d 4
20.e even 4 1 1040.2.q.o 4
65.f even 4 1 845.2.m.d 8
65.h odd 4 1 845.2.e.d 4
65.k even 4 1 845.2.m.d 8
65.n even 6 1 inner 325.2.o.b 8
65.o even 12 1 845.2.c.d 4
65.o even 12 1 845.2.m.d 8
65.q odd 12 1 65.2.e.b ✓ 4
65.q odd 12 1 325.2.e.a 4
65.q odd 12 1 845.2.a.c 2
65.q odd 12 1 4225.2.a.x 2
65.r odd 12 1 845.2.a.f 2
65.r odd 12 1 845.2.e.d 4
65.r odd 12 1 4225.2.a.t 2
65.t even 12 1 845.2.c.d 4
65.t even 12 1 845.2.m.d 8
195.bf even 12 1 7605.2.a.bb 2
195.bl even 12 1 585.2.j.d 4
195.bl even 12 1 7605.2.a.bg 2
260.bj even 12 1 1040.2.q.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.e.b ✓ 4 5.c odd 4 1
65.2.e.b ✓ 4 65.q odd 12 1
325.2.e.a 4 5.c odd 4 1
325.2.e.a 4 65.q odd 12 1
325.2.o.b 8 1.a even 1 1 trivial
325.2.o.b 8 5.b even 2 1 inner
325.2.o.b 8 13.c even 3 1 inner
325.2.o.b 8 65.n even 6 1 inner
585.2.j.d 4 15.e even 4 1
585.2.j.d 4 195.bl even 12 1
845.2.a.c 2 65.q odd 12 1
845.2.a.f 2 65.r odd 12 1
845.2.c.d 4 65.o even 12 1
845.2.c.d 4 65.t even 12 1
845.2.e.d 4 65.h odd 4 1
845.2.e.d 4 65.r odd 12 1
845.2.m.d 8 65.f even 4 1
845.2.m.d 8 65.k even 4 1
845.2.m.d 8 65.o even 12 1
845.2.m.d 8 65.t even 12 1
1040.2.q.o 4 20.e even 4 1
1040.2.q.o 4 260.bj even 12 1
4225.2.a.t 2 65.r odd 12 1
4225.2.a.x 2 65.q odd 12 1
7605.2.a.bb 2 195.bf even 12 1
7605.2.a.bg 2 195.bl even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 7T_{2}^{6} + 40T_{2}^{4} - 63T_{2}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 7 T^{6} + \cdots + 81 \) 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
$7$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{3} + 25 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 13)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} - 58 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( (T^{4} + 4 T^{3} + 25 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 9 T^{2} + 81)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 2 T^{3} + \cdots + 2601)^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} - 13 T^{2} + 169)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 3 T + 9)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} - 122 T^{6} + \cdots + 3418801 \) Copy content Toggle raw display
$47$ \( (T^{4} + 112 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 136 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 117 T^{2} + 13689)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 49 T^{2} + 2401)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 12 T^{3} + \cdots + 6561)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 232 T^{2} + 144)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 48)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 112 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 2 T^{3} + \cdots + 2601)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 314 T^{6} + \cdots + 294499921 \) Copy content Toggle raw display
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