Properties

Label 1575.2.bc.a
Level $1575$
Weight $2$
Character orbit 1575.bc
Analytic conductor $12.576$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1575,2,Mod(899,1575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1575, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1575.899");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.bc (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: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 63)
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.

\(f(q)\) \(=\) \( q + (\beta_{7} - \beta_{4}) q^{2} + ( - 2 \beta_{3} + 3 \beta_1) q^{7} + 2 \beta_{7} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} - \beta_{4}) q^{2} + ( - 2 \beta_{3} + 3 \beta_1) q^{7} + 2 \beta_{7} q^{8} + \beta_{5} q^{11} + ( - 3 \beta_{3} + 6 \beta_1) q^{13} + ( - 2 \beta_{6} + 3 \beta_{5}) q^{14} + ( - 4 \beta_{2} + 4) q^{16} + ( - 4 \beta_{7} + 2 \beta_{4}) q^{17} + ( - \beta_{2} - 1) q^{19} - 2 \beta_{3} q^{22} + ( - 4 \beta_{7} + 4 \beta_{4}) q^{23} + ( - 3 \beta_{6} + 6 \beta_{5}) q^{26} + (2 \beta_{6} - 2 \beta_{5}) q^{29} + ( - \beta_{2} + 2) q^{31} + (8 \beta_{2} - 4) q^{34} + \beta_1 q^{37} + ( - 2 \beta_{7} + \beta_{4}) q^{38} + (3 \beta_{6} + 3 \beta_{5}) q^{41} + \beta_{3} q^{43} + 8 \beta_{2} q^{46} + ( - 5 \beta_{7} - 5 \beta_{4}) q^{47} + ( - 3 \beta_{2} + 8) q^{49} + 2 \beta_{4} q^{53} + (2 \beta_{6} + 4 \beta_{5}) q^{56} + 4 \beta_1 q^{58} + ( - 4 \beta_{6} + 2 \beta_{5}) q^{59} + ( - 2 \beta_{2} - 2) q^{61} + (\beta_{7} - 2 \beta_{4}) q^{62} + 8 q^{64} + ( - 11 \beta_{3} + 11 \beta_1) q^{67} + ( - 5 \beta_{6} + 5 \beta_{5}) q^{71} + ( - \beta_{3} - \beta_1) q^{73} + \beta_{5} q^{74} + (3 \beta_{7} - 2 \beta_{4}) q^{77} + ( - 5 \beta_{2} + 5) q^{79} + ( - 12 \beta_{3} + 6 \beta_1) q^{82} + ( - 3 \beta_{7} + 6 \beta_{4}) q^{83} + \beta_{6} q^{86} + ( - 4 \beta_{3} + 4 \beta_1) q^{88} + ( - 2 \beta_{6} + 4 \beta_{5}) q^{89} + ( - 3 \beta_{2} + 15) q^{91} + (10 \beta_{2} - 20) q^{94} + (6 \beta_{3} - 12 \beta_1) q^{97} + (5 \beta_{7} - 8 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{16} - 12 q^{19} + 12 q^{31} + 32 q^{46} + 52 q^{49} - 24 q^{61} + 64 q^{64} + 20 q^{79} + 108 q^{91} - 120 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

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

Character values

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

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(-1\) \(1 - \beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
899.1
0.258819 + 0.965926i
−0.965926 + 0.258819i
−0.258819 0.965926i
0.965926 0.258819i
0.258819 0.965926i
−0.965926 0.258819i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.707107 1.22474i 0 0 0 0 −2.59808 0.500000i −2.82843 0 0
899.2 −0.707107 1.22474i 0 0 0 0 2.59808 + 0.500000i −2.82843 0 0
899.3 0.707107 + 1.22474i 0 0 0 0 −2.59808 0.500000i 2.82843 0 0
899.4 0.707107 + 1.22474i 0 0 0 0 2.59808 + 0.500000i 2.82843 0 0
1349.1 −0.707107 + 1.22474i 0 0 0 0 −2.59808 + 0.500000i −2.82843 0 0
1349.2 −0.707107 + 1.22474i 0 0 0 0 2.59808 0.500000i −2.82843 0 0
1349.3 0.707107 1.22474i 0 0 0 0 −2.59808 + 0.500000i 2.82843 0 0
1349.4 0.707107 1.22474i 0 0 0 0 2.59808 0.500000i 2.82843 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 899.4
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 1575.2.bc.a 8
3.b odd 2 1 inner 1575.2.bc.a 8
5.b even 2 1 inner 1575.2.bc.a 8
5.c odd 4 1 63.2.p.a 4
5.c odd 4 1 1575.2.bk.c 4
7.d odd 6 1 inner 1575.2.bc.a 8
15.d odd 2 1 inner 1575.2.bc.a 8
15.e even 4 1 63.2.p.a 4
15.e even 4 1 1575.2.bk.c 4
20.e even 4 1 1008.2.bt.b 4
21.g even 6 1 inner 1575.2.bc.a 8
35.f even 4 1 441.2.p.a 4
35.i odd 6 1 inner 1575.2.bc.a 8
35.k even 12 1 63.2.p.a 4
35.k even 12 1 441.2.c.a 4
35.k even 12 1 1575.2.bk.c 4
35.l odd 12 1 441.2.c.a 4
35.l odd 12 1 441.2.p.a 4
45.k odd 12 1 567.2.i.d 4
45.k odd 12 1 567.2.s.d 4
45.l even 12 1 567.2.i.d 4
45.l even 12 1 567.2.s.d 4
60.l odd 4 1 1008.2.bt.b 4
105.k odd 4 1 441.2.p.a 4
105.p even 6 1 inner 1575.2.bc.a 8
105.w odd 12 1 63.2.p.a 4
105.w odd 12 1 441.2.c.a 4
105.w odd 12 1 1575.2.bk.c 4
105.x even 12 1 441.2.c.a 4
105.x even 12 1 441.2.p.a 4
140.w even 12 1 7056.2.k.b 4
140.x odd 12 1 1008.2.bt.b 4
140.x odd 12 1 7056.2.k.b 4
315.bs even 12 1 567.2.s.d 4
315.bu odd 12 1 567.2.s.d 4
315.bw odd 12 1 567.2.i.d 4
315.cg even 12 1 567.2.i.d 4
420.bp odd 12 1 7056.2.k.b 4
420.br even 12 1 1008.2.bt.b 4
420.br even 12 1 7056.2.k.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.p.a 4 5.c odd 4 1
63.2.p.a 4 15.e even 4 1
63.2.p.a 4 35.k even 12 1
63.2.p.a 4 105.w odd 12 1
441.2.c.a 4 35.k even 12 1
441.2.c.a 4 35.l odd 12 1
441.2.c.a 4 105.w odd 12 1
441.2.c.a 4 105.x even 12 1
441.2.p.a 4 35.f even 4 1
441.2.p.a 4 35.l odd 12 1
441.2.p.a 4 105.k odd 4 1
441.2.p.a 4 105.x even 12 1
567.2.i.d 4 45.k odd 12 1
567.2.i.d 4 45.l even 12 1
567.2.i.d 4 315.bw odd 12 1
567.2.i.d 4 315.cg even 12 1
567.2.s.d 4 45.k odd 12 1
567.2.s.d 4 45.l even 12 1
567.2.s.d 4 315.bs even 12 1
567.2.s.d 4 315.bu odd 12 1
1008.2.bt.b 4 20.e even 4 1
1008.2.bt.b 4 60.l odd 4 1
1008.2.bt.b 4 140.x odd 12 1
1008.2.bt.b 4 420.br even 12 1
1575.2.bc.a 8 1.a even 1 1 trivial
1575.2.bc.a 8 3.b odd 2 1 inner
1575.2.bc.a 8 5.b even 2 1 inner
1575.2.bc.a 8 7.d odd 6 1 inner
1575.2.bc.a 8 15.d odd 2 1 inner
1575.2.bc.a 8 21.g even 6 1 inner
1575.2.bc.a 8 35.i odd 6 1 inner
1575.2.bc.a 8 105.p even 6 1 inner
1575.2.bk.c 4 5.c odd 4 1
1575.2.bk.c 4 15.e even 4 1
1575.2.bk.c 4 35.k even 12 1
1575.2.bk.c 4 105.w odd 12 1
7056.2.k.b 4 140.w even 12 1
7056.2.k.b 4 140.x odd 12 1
7056.2.k.b 4 420.bp odd 12 1
7056.2.k.b 4 420.br even 12 1

Hecke kernels

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

\( T_{2}^{4} + 2T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} - 2T_{11}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 13 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 27)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 24 T^{2} + 576)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 3 T + 3)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 3 T + 3)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 54)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 150 T^{2} + 22500)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 24 T^{2} + 576)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 6 T + 12)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 121 T^{2} + 14641)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 50)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 5 T + 25)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 54)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 24 T^{2} + 576)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
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