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Properties

Label 675.4.b.i
Level $675$
Weight $4$
Character orbit 675.b
Analytic conductor $39.826$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [675,4,Mod(649,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.649");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.b (of order \(2\), degree \(1\), 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: \(39.8262892539\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 27)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - 10 q^{4} - 11 \beta_1 q^{7} - 2 \beta_{2} q^{8} - 4 \beta_{3} q^{11} + 29 \beta_1 q^{13} + 11 \beta_{3} q^{14} - 44 q^{16} - 12 \beta_{2} q^{17} - 29 q^{19} - 72 \beta_1 q^{22} - 20 \beta_{2} q^{23}+ \cdots + 222 \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 40 q^{4} - 176 q^{16} - 116 q^{19} - 1072 q^{31} + 864 q^{34} + 1440 q^{46} + 888 q^{49} + 3068 q^{61} + 2912 q^{64} + 1160 q^{76} + 1900 q^{79} + 1276 q^{91} + 6624 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\zeta_{8}^{3} + 3\zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -3\zeta_{8}^{3} + 3\zeta_{8} \) Copy content Toggle raw display

Display \(\zeta_{8}^j\) in terms of \(\beta_i\)

\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 6 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 6 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{8}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
−0.707107 − 0.707107i
0.707107 − 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
− 4.24264i 0 −10.0000 0 0 − 11.0000i 8.48528i 0 0
649.2 − 4.24264i 0 −10.0000 0 0 11.0000i 8.48528i 0 0
649.3 4.24264i 0 −10.0000 0 0 − 11.0000i − 8.48528i 0 0
649.4 4.24264i 0 −10.0000 0 0 11.0000i − 8.48528i 0 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 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.4.b.i 4
3.b odd 2 1 inner 675.4.b.i 4
5.b even 2 1 inner 675.4.b.i 4
5.c odd 4 1 27.4.a.c ✓ 2
5.c odd 4 1 675.4.a.n 2
15.d odd 2 1 inner 675.4.b.i 4
15.e even 4 1 27.4.a.c ✓ 2
15.e even 4 1 675.4.a.n 2
20.e even 4 1 432.4.a.q 2
35.f even 4 1 1323.4.a.t 2
40.i odd 4 1 1728.4.a.bp 2
40.k even 4 1 1728.4.a.bk 2
45.k odd 12 2 81.4.c.e 4
45.l even 12 2 81.4.c.e 4
60.l odd 4 1 432.4.a.q 2
105.k odd 4 1 1323.4.a.t 2
120.q odd 4 1 1728.4.a.bk 2
120.w even 4 1 1728.4.a.bp 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.4.a.c ✓ 2 5.c odd 4 1
27.4.a.c ✓ 2 15.e even 4 1
81.4.c.e 4 45.k odd 12 2
81.4.c.e 4 45.l even 12 2
432.4.a.q 2 20.e even 4 1
432.4.a.q 2 60.l odd 4 1
675.4.a.n 2 5.c odd 4 1
675.4.a.n 2 15.e even 4 1
675.4.b.i 4 1.a even 1 1 trivial
675.4.b.i 4 3.b odd 2 1 inner
675.4.b.i 4 5.b even 2 1 inner
675.4.b.i 4 15.d odd 2 1 inner
1323.4.a.t 2 35.f even 4 1
1323.4.a.t 2 105.k odd 4 1
1728.4.a.bk 2 40.k even 4 1
1728.4.a.bk 2 120.q odd 4 1
1728.4.a.bp 2 40.i odd 4 1
1728.4.a.bp 2 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(675, [\chi])\):

\( T_{2}^{2} + 18 \) Copy content Toggle raw display
\( T_{7}^{2} + 121 \) Copy content Toggle raw display
\( T_{11}^{2} - 288 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 121)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 288)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 841)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 2592)^{2} \) Copy content Toggle raw display
$19$ \( (T + 29)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 7200)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 73728)^{2} \) Copy content Toggle raw display
$31$ \( (T + 268)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 6889)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 73728)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 53824)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 152352)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 93312)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 83232)^{2} \) Copy content Toggle raw display
$61$ \( (T - 767)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 261121)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 508032)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 18769)^{2} \) Copy content Toggle raw display
$79$ \( (T - 475)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 332928)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 64800)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 674041)^{2} \) Copy content Toggle raw display
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