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Properties

Label 315.2.bb.a
Level $315$
Weight $2$
Character orbit 315.bb
Analytic conductor $2.515$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,2,Mod(89,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.89");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.bb (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: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12745506816.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + 2 \beta_{4} q^{4} + (\beta_{5} - \beta_{2} + \beta_1) q^{5} + (\beta_{6} + \beta_{3}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{4} q^{4} + (\beta_{5} - \beta_{2} + \beta_1) q^{5} + (\beta_{6} + \beta_{3}) q^{7} - 2 \beta_{2} q^{11} + \beta_{3} q^{13} + (4 \beta_{4} - 4) q^{16} + (\beta_{2} - 2 \beta_1) q^{17} + (\beta_{4} + 1) q^{19} + ( - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{2}) q^{20} + (\beta_{7} - 2 \beta_{5} + \cdots + 2 \beta_1) q^{23}+ \cdots - 2 \beta_{3} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} - 16 q^{16} + 12 q^{19} + 8 q^{25} + 36 q^{31} - 28 q^{49} - 72 q^{61} - 64 q^{64} + 28 q^{79} - 56 q^{85} + 28 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 203\nu ) / 165 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 148 ) / 55 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -8\nu^{6} + 55\nu^{4} - 440\nu^{2} + 576 ) / 495 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -8\nu^{7} + 55\nu^{5} - 440\nu^{3} + 81\nu ) / 495 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -23\nu^{6} + 220\nu^{4} - 1265\nu^{2} + 1656 ) / 495 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{7} - 55\nu^{5} + 341\nu^{3} - 81\nu ) / 297 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 4\beta_{4} + \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{7} - 5\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{6} - 23\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -24\beta_{7} - 31\beta_{5} - 24\beta_{2} - 31\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -55\beta_{3} - 148 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -165\beta_{2} - 203\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(-1\) \(1 - \beta_{4}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1
−1.00781 + 0.581861i
−2.23256 + 1.28897i
2.23256 − 1.28897i
1.00781 − 0.581861i
−1.00781 − 0.581861i
−2.23256 − 1.28897i
2.23256 + 1.28897i
1.00781 + 0.581861i
0 0 1.00000 + 1.73205i −2.23256 + 0.125246i 0 −1.32288 + 2.29129i 0 0 0
89.2 0 0 1.00000 + 1.73205i −1.00781 − 1.99607i 0 1.32288 − 2.29129i 0 0 0
89.3 0 0 1.00000 + 1.73205i 1.00781 + 1.99607i 0 1.32288 − 2.29129i 0 0 0
89.4 0 0 1.00000 + 1.73205i 2.23256 − 0.125246i 0 −1.32288 + 2.29129i 0 0 0
269.1 0 0 1.00000 − 1.73205i −2.23256 − 0.125246i 0 −1.32288 − 2.29129i 0 0 0
269.2 0 0 1.00000 − 1.73205i −1.00781 + 1.99607i 0 1.32288 + 2.29129i 0 0 0
269.3 0 0 1.00000 − 1.73205i 1.00781 − 1.99607i 0 1.32288 + 2.29129i 0 0 0
269.4 0 0 1.00000 − 1.73205i 2.23256 + 0.125246i 0 −1.32288 − 2.29129i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.4
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
7.d odd 6 1 inner
15.d odd 2 1 inner
21.g even 6 1 inner
35.i odd 6 1 inner
105.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.bb.a ✓ 8
3.b odd 2 1 inner 315.2.bb.a ✓ 8
5.b even 2 1 inner 315.2.bb.a ✓ 8
5.c odd 4 2 1575.2.bk.d 8
7.c even 3 1 2205.2.g.a 8
7.d odd 6 1 inner 315.2.bb.a ✓ 8
7.d odd 6 1 2205.2.g.a 8
15.d odd 2 1 inner 315.2.bb.a ✓ 8
15.e even 4 2 1575.2.bk.d 8
21.g even 6 1 inner 315.2.bb.a ✓ 8
21.g even 6 1 2205.2.g.a 8
21.h odd 6 1 2205.2.g.a 8
35.i odd 6 1 inner 315.2.bb.a ✓ 8
35.i odd 6 1 2205.2.g.a 8
35.j even 6 1 2205.2.g.a 8
35.k even 12 2 1575.2.bk.d 8
105.o odd 6 1 2205.2.g.a 8
105.p even 6 1 inner 315.2.bb.a ✓ 8
105.p even 6 1 2205.2.g.a 8
105.w odd 12 2 1575.2.bk.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bb.a ✓ 8 1.a even 1 1 trivial
315.2.bb.a ✓ 8 3.b odd 2 1 inner
315.2.bb.a ✓ 8 5.b even 2 1 inner
315.2.bb.a ✓ 8 7.d odd 6 1 inner
315.2.bb.a ✓ 8 15.d odd 2 1 inner
315.2.bb.a ✓ 8 21.g even 6 1 inner
315.2.bb.a ✓ 8 35.i odd 6 1 inner
315.2.bb.a ✓ 8 105.p even 6 1 inner
1575.2.bk.d 8 5.c odd 4 2
1575.2.bk.d 8 15.e even 4 2
1575.2.bk.d 8 35.k even 12 2
1575.2.bk.d 8 105.w odd 12 2
2205.2.g.a 8 7.c even 3 1
2205.2.g.a 8 7.d odd 6 1
2205.2.g.a 8 21.g even 6 1
2205.2.g.a 8 21.h odd 6 1
2205.2.g.a 8 35.i odd 6 1
2205.2.g.a 8 35.j even 6 1
2205.2.g.a 8 105.o odd 6 1
2205.2.g.a 8 105.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 7)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 14 T^{2} + 196)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 3 T + 3)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 42 T^{2} + 1764)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 9 T + 27)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 21 T^{2} + 441)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 24)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 21)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 14 T^{2} + 196)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 168 T^{2} + 28224)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 54 T^{2} + 2916)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 18 T + 108)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 189 T^{2} + 35721)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 128)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 175 T^{2} + 30625)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 7 T + 49)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 224)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 294 T^{2} + 86436)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 28)^{4} \) Copy content Toggle raw display
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