Properties

Label 1575.2.bk.d
Level $1575$
Weight $2$
Character orbit 1575.bk
Analytic conductor $12.576$
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] = [1575,2,Mod(26,1575)]
 
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, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1575.26");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.bk (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: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.796594176.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 5x^{6} - 2x^{5} + 63x^{4} - 64x^{3} + 46x^{2} - 16x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 315)
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.

\(f(q)\) \(=\) \( q - 2 \beta_{2} q^{4} + (\beta_{7} + \beta_{6}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{2} q^{4} + (\beta_{7} + \beta_{6}) q^{7} + 2 \beta_{3} q^{11} - \beta_{7} q^{13} + (4 \beta_{2} - 4) q^{16} + \beta_{4} q^{17} + ( - \beta_{2} - 1) q^{19} + (\beta_{4} - \beta_1) q^{23} - 2 \beta_{7} q^{28} + \beta_{5} q^{29} + ( - 3 \beta_{2} + 6) q^{31} + ( - \beta_{7} + \beta_{6}) q^{37} + ( - 2 \beta_{5} - 4 \beta_{3}) q^{41} + ( - \beta_{7} - 2 \beta_{6}) q^{43} + ( - 4 \beta_{5} - 4 \beta_{3}) q^{44} + ( - \beta_{4} - \beta_1) q^{47} + 7 \beta_{2} q^{49} - 2 \beta_{6} q^{52} + ( - 2 \beta_{4} - 4 \beta_1) q^{53} + (6 \beta_{5} + 3 \beta_{3}) q^{59} + ( - 6 \beta_{2} - 6) q^{61} + 8 q^{64} + ( - 6 \beta_{7} - 3 \beta_{6}) q^{67} + ( - 2 \beta_{4} - 2 \beta_1) q^{68} + 8 \beta_{5} q^{71} + 5 \beta_{6} q^{73} + (4 \beta_{2} - 2) q^{76} - 2 \beta_1 q^{77} + (7 \beta_{2} - 7) q^{79} - 4 \beta_1 q^{83} + ( - 7 \beta_{5} + 7 \beta_{3}) q^{89} + ( - 7 \beta_{2} + 7) q^{91} + ( - 4 \beta_{4} - 2 \beta_1) q^{92} - 2 \beta_{7} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 16 q^{16} - 12 q^{19} + 36 q^{31} + 28 q^{49} - 72 q^{61} + 64 q^{64} - 28 q^{79} + 28 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 5x^{6} - 2x^{5} + 63x^{4} - 64x^{3} + 46x^{2} - 16x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 27\nu^{7} + 32\nu^{6} - 177\nu^{5} - 186\nu^{4} + 245\nu^{3} - 228\nu^{2} + 84\nu + 14516 ) / 3886 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -1816\nu^{7} + 3101\nu^{6} + 9746\nu^{5} + 7113\nu^{4} - 110750\nu^{3} + 85499\nu^{2} - 79052\nu + 27404 ) / 19430 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5433 \nu^{7} + 14718 \nu^{6} + 22663 \nu^{5} - 14386 \nu^{4} - 368815 \nu^{3} + 583442 \nu^{2} - 282446 \nu + 98912 ) / 19430 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6733 \nu^{7} - 11738 \nu^{6} - 35503 \nu^{5} - 24794 \nu^{4} + 412275 \nu^{3} - 318082 \nu^{2} + 295126 \nu - 102352 ) / 19430 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -117\nu^{7} + 132\nu^{6} + 767\nu^{5} + 806\nu^{4} - 7055\nu^{3} + 988\nu^{2} - 364\nu - 572 ) / 290 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10112 \nu^{7} - 27667 \nu^{6} - 42542 \nu^{5} + 28569 \nu^{4} + 688330 \nu^{3} - 1090353 \nu^{2} + 528004 \nu - 184948 ) / 19430 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 252\nu^{7} - 282\nu^{6} - 1652\nu^{5} - 1736\nu^{4} + 15240\nu^{3} - 2128\nu^{2} + 784\nu + 1237 ) / 335 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + 2\beta_{5} + 2\beta_{3} - \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 2\beta_{4} + 2\beta_{3} + 7\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{7} + 13\beta_{5} - 3\beta _1 + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 15\beta_{7} + 15\beta_{6} + 28\beta_{5} + 12\beta_{4} + 28\beta_{3} + 45\beta_{2} + 12\beta _1 - 45 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -39\beta_{6} + 35\beta_{4} - 73\beta_{3} + 131\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 155\beta_{7} + 290\beta_{5} + 58\beta _1 - 217 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -147\beta_{7} - 147\beta_{6} - 275\beta_{5} + 329\beta_{4} - 275\beta_{3} + 1231\beta_{2} + 329\beta _1 - 1231 ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
−2.12039 1.80156i
0.329101 0.387344i
0.170899 0.478682i
2.62039 + 0.935532i
−2.12039 + 1.80156i
0.329101 + 0.387344i
0.170899 + 0.478682i
2.62039 0.935532i
0 0 −1.00000 1.73205i 0 0 −2.29129 1.32288i 0 0 0
26.2 0 0 −1.00000 1.73205i 0 0 −2.29129 1.32288i 0 0 0
26.3 0 0 −1.00000 1.73205i 0 0 2.29129 + 1.32288i 0 0 0
26.4 0 0 −1.00000 1.73205i 0 0 2.29129 + 1.32288i 0 0 0
1151.1 0 0 −1.00000 + 1.73205i 0 0 −2.29129 + 1.32288i 0 0 0
1151.2 0 0 −1.00000 + 1.73205i 0 0 −2.29129 + 1.32288i 0 0 0
1151.3 0 0 −1.00000 + 1.73205i 0 0 2.29129 1.32288i 0 0 0
1151.4 0 0 −1.00000 + 1.73205i 0 0 2.29129 1.32288i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.4
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.bk.d 8
3.b odd 2 1 inner 1575.2.bk.d 8
5.b even 2 1 inner 1575.2.bk.d 8
5.c odd 4 2 315.2.bb.a 8
7.d odd 6 1 inner 1575.2.bk.d 8
15.d odd 2 1 inner 1575.2.bk.d 8
15.e even 4 2 315.2.bb.a 8
21.g even 6 1 inner 1575.2.bk.d 8
35.i odd 6 1 inner 1575.2.bk.d 8
35.k even 12 2 315.2.bb.a 8
35.k even 12 2 2205.2.g.a 8
35.l odd 12 2 2205.2.g.a 8
105.p even 6 1 inner 1575.2.bk.d 8
105.w odd 12 2 315.2.bb.a 8
105.w odd 12 2 2205.2.g.a 8
105.x even 12 2 2205.2.g.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bb.a 8 5.c odd 4 2
315.2.bb.a 8 15.e even 4 2
315.2.bb.a 8 35.k even 12 2
315.2.bb.a 8 105.w odd 12 2
1575.2.bk.d 8 1.a even 1 1 trivial
1575.2.bk.d 8 3.b odd 2 1 inner
1575.2.bk.d 8 5.b even 2 1 inner
1575.2.bk.d 8 7.d odd 6 1 inner
1575.2.bk.d 8 15.d odd 2 1 inner
1575.2.bk.d 8 21.g even 6 1 inner
1575.2.bk.d 8 35.i odd 6 1 inner
1575.2.bk.d 8 105.p even 6 1 inner
2205.2.g.a 8 35.k even 12 2
2205.2.g.a 8 35.l odd 12 2
2205.2.g.a 8 105.w odd 12 2
2205.2.g.a 8 105.x even 12 2

Hecke kernels

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

\( T_{2} \) Copy content Toggle raw display
\( T_{11}^{4} - 8T_{11}^{2} + 64 \) Copy content Toggle raw display
\( T_{37}^{4} + 21T_{37}^{2} + 441 \) 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
$5$ \( T^{8} \) 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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