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Properties

Label 1352.2.i.l
Level $1352$
Weight $2$
Character orbit 1352.i
Analytic conductor $10.796$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1352,2,Mod(529,1352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1352, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1352.529");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1352 = 2^{3} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1352.i (of order \(3\), 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: \(10.7957743533\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.195105024.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 4x^{5} - 20x^{4} + 12x^{3} + 45x^{2} - 108x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 104)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_{2} - \beta_1 - 1) q^{3} + ( - \beta_{4} + \beta_{2} + 1) q^{5} + ( - \beta_{7} + \beta_{6} + \cdots + \beta_1) q^{7}+ \cdots + ( - 2 \beta_{7} + 2 \beta_{3} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_{2} - \beta_1 - 1) q^{3} + ( - \beta_{4} + \beta_{2} + 1) q^{5} + ( - \beta_{7} + \beta_{6} + \cdots + \beta_1) q^{7}+ \cdots + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} + 8 q^{5} - 4 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} + 8 q^{5} - 4 q^{7} - 6 q^{9} - 4 q^{11} + 12 q^{15} - 16 q^{19} + 16 q^{21} - 2 q^{23} + 20 q^{25} - 28 q^{27} - 8 q^{29} + 8 q^{31} - 8 q^{33} + 16 q^{35} - 12 q^{37} + 16 q^{41} - 6 q^{43} - 28 q^{45} + 40 q^{47} - 2 q^{49} - 68 q^{51} + 20 q^{53} + 16 q^{55} + 32 q^{57} - 4 q^{59} - 4 q^{61} - 8 q^{63} - 8 q^{67} + 18 q^{69} - 16 q^{71} + 48 q^{73} + 22 q^{75} + 60 q^{77} + 16 q^{79} - 20 q^{81} + 48 q^{83} - 20 q^{85} - 10 q^{87} - 16 q^{89} + 16 q^{93} - 16 q^{95} - 32 q^{97} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 4x^{5} - 20x^{4} + 12x^{3} + 45x^{2} - 108x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + \nu^{6} - 2\nu^{5} - 10\nu^{4} - 10\nu^{3} + 66\nu^{2} + 45\nu - 81 ) / 108 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 2\nu^{6} - 10\nu^{5} - 2\nu^{4} + 22\nu^{3} - 18\nu^{2} - 9\nu + 108 ) / 54 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{7} - 8\nu^{6} + 4\nu^{5} + 26\nu^{4} - 52\nu^{3} - 18\nu^{2} + 153\nu - 162 ) / 108 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 2\nu^{4} - 6\nu^{3} + 2\nu^{2} + 9\nu - 18 ) / 18 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{7} - 17\nu^{6} + 6\nu^{5} + 50\nu^{4} - 66\nu^{3} - 22\nu^{2} + 237\nu - 315 ) / 72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -10\nu^{7} + 25\nu^{6} - 26\nu^{5} - 52\nu^{4} + 122\nu^{3} + 36\nu^{2} - 504\nu + 621 ) / 108 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\nu^{7} - 33\nu^{6} + 10\nu^{5} + 90\nu^{4} - 142\nu^{3} - 86\nu^{2} + 519\nu - 603 ) / 72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} + 4\beta_{5} + 3\beta _1 + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{7} - 4\beta_{6} + 8\beta_{5} - \beta_{4} - 10\beta_{3} + 4\beta_{2} - 2\beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{7} + 4\beta_{6} + 14\beta_{5} - 8\beta_{4} + 4\beta_{3} - \beta_{2} - \beta _1 + 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -12\beta_{7} - 11\beta_{6} + 12\beta_{5} - 2\beta_{4} + \beta_{3} - 4\beta_{2} - 13\beta _1 + 13 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -16\beta_{7} + 8\beta_{5} - 24\beta_{4} + 72\beta_{3} + 6\beta_{2} - 27 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -44\beta_{7} - 23\beta_{6} + 28\beta_{5} + 25\beta_{4} + 85\beta_{3} + 29\beta_{2} - 25\beta _1 - 38 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1352\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1015\) \(1185\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1
0.560908 − 1.63871i
−1.58726 + 0.693255i
1.72124 − 0.193255i
1.30512 + 1.13871i
0.560908 + 1.63871i
−1.58726 − 0.693255i
1.72124 + 0.193255i
1.30512 − 1.13871i
0 −1.13871 + 1.97231i 0 1.12182 0 −1.30512 − 2.26053i 0 −1.09334 − 1.89372i 0
529.2 0 −0.193255 + 0.334727i 0 −3.17452 0 −1.72124 − 2.98127i 0 1.42531 + 2.46870i 0
529.3 0 0.693255 − 1.20075i 0 3.44247 0 1.58726 + 2.74922i 0 0.538796 + 0.933222i 0
529.4 0 1.63871 − 2.83834i 0 2.61023 0 −0.560908 − 0.971521i 0 −3.87076 − 6.70436i 0
1329.1 0 −1.13871 − 1.97231i 0 1.12182 0 −1.30512 + 2.26053i 0 −1.09334 + 1.89372i 0
1329.2 0 −0.193255 − 0.334727i 0 −3.17452 0 −1.72124 + 2.98127i 0 1.42531 − 2.46870i 0
1329.3 0 0.693255 + 1.20075i 0 3.44247 0 1.58726 − 2.74922i 0 0.538796 − 0.933222i 0
1329.4 0 1.63871 + 2.83834i 0 2.61023 0 −0.560908 + 0.971521i 0 −3.87076 + 6.70436i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1352.2.i.l 8
13.b even 2 1 1352.2.i.k 8
13.c even 3 1 1352.2.a.l 4
13.c even 3 1 inner 1352.2.i.l 8
13.d odd 4 1 104.2.o.a ✓ 8
13.d odd 4 1 1352.2.o.f 8
13.e even 6 1 1352.2.a.k 4
13.e even 6 1 1352.2.i.k 8
13.f odd 12 1 104.2.o.a ✓ 8
13.f odd 12 2 1352.2.f.f 8
13.f odd 12 1 1352.2.o.f 8
39.f even 4 1 936.2.bi.b 8
39.k even 12 1 936.2.bi.b 8
52.f even 4 1 208.2.w.c 8
52.i odd 6 1 2704.2.a.bd 4
52.j odd 6 1 2704.2.a.be 4
52.l even 12 1 208.2.w.c 8
52.l even 12 2 2704.2.f.q 8
104.j odd 4 1 832.2.w.g 8
104.m even 4 1 832.2.w.i 8
104.u even 12 1 832.2.w.i 8
104.x odd 12 1 832.2.w.g 8
156.l odd 4 1 1872.2.by.n 8
156.v odd 12 1 1872.2.by.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.o.a ✓ 8 13.d odd 4 1
104.2.o.a ✓ 8 13.f odd 12 1
208.2.w.c 8 52.f even 4 1
208.2.w.c 8 52.l even 12 1
832.2.w.g 8 104.j odd 4 1
832.2.w.g 8 104.x odd 12 1
832.2.w.i 8 104.m even 4 1
832.2.w.i 8 104.u even 12 1
936.2.bi.b 8 39.f even 4 1
936.2.bi.b 8 39.k even 12 1
1352.2.a.k 4 13.e even 6 1
1352.2.a.l 4 13.c even 3 1
1352.2.f.f 8 13.f odd 12 2
1352.2.i.k 8 13.b even 2 1
1352.2.i.k 8 13.e even 6 1
1352.2.i.l 8 1.a even 1 1 trivial
1352.2.i.l 8 13.c even 3 1 inner
1352.2.o.f 8 13.d odd 4 1
1352.2.o.f 8 13.f odd 12 1
1872.2.by.n 8 156.l odd 4 1
1872.2.by.n 8 156.v odd 12 1
2704.2.a.bd 4 52.i odd 6 1
2704.2.a.be 4 52.j odd 6 1
2704.2.f.q 8 52.l even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1352, [\chi])\):

\( T_{3}^{8} - 2T_{3}^{7} + 11T_{3}^{6} - 2T_{3}^{5} + 61T_{3}^{4} - 40T_{3}^{3} + 92T_{3}^{2} + 32T_{3} + 16 \) Copy content Toggle raw display
\( T_{5}^{4} - 4T_{5}^{3} - 7T_{5}^{2} + 40T_{5} - 32 \) Copy content Toggle raw display
\( T_{7}^{8} + 4T_{7}^{7} + 23T_{7}^{6} + 52T_{7}^{5} + 241T_{7}^{4} + 536T_{7}^{3} + 1376T_{7}^{2} + 1280T_{7} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{4} - 4 T^{3} - 7 T^{2} + \cdots - 32)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{7} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( T^{8} + 4 T^{7} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + 34 T^{6} + \cdots + 9409 \) Copy content Toggle raw display
$19$ \( T^{8} + 16 T^{7} + \cdots + 80656 \) Copy content Toggle raw display
$23$ \( T^{8} + 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( T^{8} + 8 T^{7} + \cdots + 3721 \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{3} - 88 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 12 T^{7} + \cdots + 89401 \) Copy content Toggle raw display
$41$ \( (T^{4} - 8 T^{3} + \cdots + 169)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 6 T^{7} + \cdots + 692224 \) Copy content Toggle raw display
$47$ \( (T^{4} - 20 T^{3} + \cdots - 368)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 10 T^{3} + \cdots - 1724)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 4 T^{7} + \cdots + 43264 \) Copy content Toggle raw display
$61$ \( T^{8} + 4 T^{7} + \cdots + 896809 \) Copy content Toggle raw display
$67$ \( T^{8} + 8 T^{7} + \cdots + 55696 \) Copy content Toggle raw display
$71$ \( T^{8} + 16 T^{7} + \cdots + 1430416 \) Copy content Toggle raw display
$73$ \( (T^{4} - 24 T^{3} + \cdots + 144)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} + \cdots + 3328)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 24 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 16 T^{7} + \cdots + 186267904 \) Copy content Toggle raw display
$97$ \( T^{8} + 32 T^{7} + \cdots + 55830784 \) Copy content Toggle raw display
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