↑

Properties

Label 1575.2.g.c
Level $1575$
Weight $2$
Character orbit 1575.g
Analytic conductor $12.576$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1575,2,Mod(1574,1575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1575, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1575.1574");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.g (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: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 315)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + \beta_{3} q^{4} + ( - \beta_{6} + \beta_{5} + \beta_1) q^{7} - \beta_{5} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + \beta_{3} q^{4} + ( - \beta_{6} + \beta_{5} + \beta_1) q^{7} - \beta_{5} q^{8} + (3 \beta_{7} + \beta_{4}) q^{11} + ( - 4 \beta_{6} - 2 \beta_{5}) q^{13} + ( - \beta_{7} + \beta_{3} + 3) q^{14} + ( - 2 \beta_{3} - 1) q^{16} - 2 \beta_{2} q^{17} + ( - 4 \beta_{7} - 2 \beta_{4}) q^{19} + ( - 3 \beta_{2} - 5 \beta_1) q^{22} + ( - \beta_{6} - \beta_{5}) q^{23} + (4 \beta_{3} + 6) q^{26} + ( - 3 \beta_{6} - 3 \beta_{5} + \beta_{2}) q^{28} + (\beta_{7} + \beta_{4}) q^{29} + (2 \beta_{7} + 4 \beta_{4}) q^{31} + (5 \beta_{6} + 4 \beta_{5}) q^{32} + (4 \beta_{7} + 2 \beta_{4}) q^{34} + (2 \beta_{2} + 2 \beta_1) q^{37} + (4 \beta_{2} + 6 \beta_1) q^{38} + 6 \beta_{3} q^{41} + ( - 4 \beta_{2} - 2 \beta_1) q^{43} + (5 \beta_{7} + \beta_{4}) q^{44} + (\beta_{3} + 1) q^{46} + ( - 2 \beta_{2} + 6 \beta_1) q^{47} + ( - 2 \beta_{7} + 2 \beta_{4} + 5) q^{49} - 6 \beta_{6} q^{52} + ( - \beta_{6} + 5 \beta_{5}) q^{53} + ( - \beta_{4} + \beta_{3} - 3) q^{56} + ( - \beta_{2} - \beta_1) q^{58} + (2 \beta_{3} - 6) q^{59} + (4 \beta_{7} + 8 \beta_{4}) q^{61} - 2 \beta_{2} q^{62} + ( - \beta_{3} - 4) q^{64} + (4 \beta_{2} - 4 \beta_1) q^{67} - 6 \beta_1 q^{68} + ( - 3 \beta_{7} + 7 \beta_{4}) q^{71} + (4 \beta_{6} + 2 \beta_{5}) q^{73} + ( - 6 \beta_{7} - 2 \beta_{4}) q^{74} - 6 \beta_{7} q^{76} + ( - 3 \beta_{6} - \beta_{5} + \cdots - 6 \beta_1) q^{77}+ \cdots + ( - 5 \beta_{6} + 2 \beta_{2} + 6 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{14} - 8 q^{16} + 48 q^{26} + 8 q^{46} + 40 q^{49} - 24 q^{56} - 48 q^{59} - 32 q^{64} + 32 q^{79} - 48 q^{89} + 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{3} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{24}^{7} - \zeta_{24}^{5} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24}^{5} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{6} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + \beta_{5} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1574.1
0.965926 − 0.258819i
0.965926 + 0.258819i
0.258819 − 0.965926i
0.258819 + 0.965926i
−0.258819 + 0.965926i
−0.258819 − 0.965926i
−0.965926 + 0.258819i
−0.965926 − 0.258819i
−1.93185 0 1.73205 0 0 −2.44949 − 1.00000i 0.517638 0 0
1574.2 −1.93185 0 1.73205 0 0 −2.44949 + 1.00000i 0.517638 0 0
1574.3 −0.517638 0 −1.73205 0 0 −2.44949 − 1.00000i 1.93185 0 0
1574.4 −0.517638 0 −1.73205 0 0 −2.44949 + 1.00000i 1.93185 0 0
1574.5 0.517638 0 −1.73205 0 0 2.44949 − 1.00000i −1.93185 0 0
1574.6 0.517638 0 −1.73205 0 0 2.44949 + 1.00000i −1.93185 0 0
1574.7 1.93185 0 1.73205 0 0 2.44949 − 1.00000i −0.517638 0 0
1574.8 1.93185 0 1.73205 0 0 2.44949 + 1.00000i −0.517638 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1574.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
21.c even 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.2.g.c 8
3.b odd 2 1 1575.2.g.a 8
5.b even 2 1 inner 1575.2.g.c 8
5.c odd 4 1 315.2.b.a ✓ 4
5.c odd 4 1 1575.2.b.b 4
7.b odd 2 1 1575.2.g.a 8
15.d odd 2 1 1575.2.g.a 8
15.e even 4 1 315.2.b.b yes 4
15.e even 4 1 1575.2.b.c 4
20.e even 4 1 5040.2.f.a 4
21.c even 2 1 inner 1575.2.g.c 8
35.c odd 2 1 1575.2.g.a 8
35.f even 4 1 315.2.b.b yes 4
35.f even 4 1 1575.2.b.c 4
60.l odd 4 1 5040.2.f.c 4
105.g even 2 1 inner 1575.2.g.c 8
105.k odd 4 1 315.2.b.a ✓ 4
105.k odd 4 1 1575.2.b.b 4
140.j odd 4 1 5040.2.f.c 4
420.w even 4 1 5040.2.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.b.a ✓ 4 5.c odd 4 1
315.2.b.a ✓ 4 105.k odd 4 1
315.2.b.b yes 4 15.e even 4 1
315.2.b.b yes 4 35.f even 4 1
1575.2.b.b 4 5.c odd 4 1
1575.2.b.b 4 105.k odd 4 1
1575.2.b.c 4 15.e even 4 1
1575.2.b.c 4 35.f even 4 1
1575.2.g.a 8 3.b odd 2 1
1575.2.g.a 8 7.b odd 2 1
1575.2.g.a 8 15.d odd 2 1
1575.2.g.a 8 35.c odd 2 1
1575.2.g.c 8 1.a even 1 1 trivial
1575.2.g.c 8 5.b even 2 1 inner
1575.2.g.c 8 21.c even 2 1 inner
1575.2.g.c 8 105.g even 2 1 inner
5040.2.f.a 4 20.e even 4 1
5040.2.f.a 4 420.w even 4 1
5040.2.f.c 4 60.l odd 4 1
5040.2.f.c 4 140.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1575, [\chi])\):

\( T_{2}^{4} - 4T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{59}^{2} + 12T_{59} + 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 10 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 28 T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 48 T^{2} + 144)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 48 T^{2} + 144)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 48 T^{2} + 144)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 32 T^{2} + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 104 T^{2} + 1936)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 96 T^{2} + 576)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 124 T^{2} + 2116)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 12 T + 24)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 192 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 128 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 316 T^{2} + 20164)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 48 T^{2} + 144)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 8 T - 32)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 96 T^{2} + 576)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 12 T - 12)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 336 T^{2} + 24336)^{2} \) Copy content Toggle raw display
show more
show less