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Properties

Label 4368.2.h.q
Level $4368$
Weight $2$
Character orbit 4368.h
Analytic conductor $34.879$
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] = [4368,2,Mod(337,4368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4368.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4368.h (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: \(34.8786556029\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 15x^{6} + 67x^{4} + 77x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 273)
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.

\(f(q)\) \(=\) \( q + q^{3} + ( - \beta_{7} + \beta_{2}) q^{5} - \beta_{2} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + ( - \beta_{7} + \beta_{2}) q^{5} - \beta_{2} q^{7} + q^{9} + (\beta_{3} + \beta_{2}) q^{11} + ( - \beta_{6} + \beta_{5} + \beta_{2} - 1) q^{13} + ( - \beta_{7} + \beta_{2}) q^{15} + ( - \beta_{5} + \beta_{4} + 2) q^{17} + (\beta_{7} - \beta_{5} + \cdots - 2 \beta_{2}) q^{19}+ \cdots + (\beta_{3} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} + 8 q^{9} - 6 q^{13} + 20 q^{17} + 6 q^{23} - 34 q^{25} + 8 q^{27} - 18 q^{29} + 6 q^{35} - 6 q^{39} - 34 q^{43} - 8 q^{49} + 20 q^{51} - 10 q^{53} - 16 q^{55} - 20 q^{61} - 10 q^{65} + 6 q^{69} - 34 q^{75} + 4 q^{77} + 2 q^{79} + 8 q^{81} - 18 q^{87} + 6 q^{91} + 78 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 15x^{6} + 67x^{4} + 77x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 8\nu^{3} + 9\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 10\nu^{3} + 23\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 8\nu^{4} + 9\nu^{2} + 2\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} - 8\nu^{4} - 9\nu^{2} + 2\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{4} + 7\nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 12\nu^{5} + 41\nu^{3} + 36\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{5} - 7\beta_{4} + 2\beta_{3} - 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - 7\beta _1 + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 47\beta_{5} + 47\beta_{4} - 16\beta_{3} + 20\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{6} - \beta_{5} + \beta_{4} + 47\beta _1 - 164 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4\beta_{7} - 313\beta_{5} - 313\beta_{4} + 110\beta_{3} - 158\beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times\).

\(n\) \(1093\) \(1249\) \(1457\) \(2017\) \(3823\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
− 2.60520i
2.54814i
0.233455i
− 1.29051i
1.29051i
− 0.233455i
− 2.54814i
2.60520i
0 1.00000 0 − 3.78706i 0 1.00000i 0 1.00000 0
337.2 0 1.00000 0 − 3.49301i 0 1.00000i 0 1.00000 0
337.3 0 1.00000 0 − 2.94550i 0 − 1.00000i 0 1.00000 0
337.4 0 1.00000 0 − 1.33457i 0 − 1.00000i 0 1.00000 0
337.5 0 1.00000 0 1.33457i 0 1.00000i 0 1.00000 0
337.6 0 1.00000 0 2.94550i 0 1.00000i 0 1.00000 0
337.7 0 1.00000 0 3.49301i 0 − 1.00000i 0 1.00000 0
337.8 0 1.00000 0 3.78706i 0 − 1.00000i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4368.2.h.q 8
4.b odd 2 1 273.2.c.c ✓ 8
12.b even 2 1 819.2.c.d 8
13.b even 2 1 inner 4368.2.h.q 8
28.d even 2 1 1911.2.c.l 8
52.b odd 2 1 273.2.c.c ✓ 8
52.f even 4 1 3549.2.a.v 4
52.f even 4 1 3549.2.a.x 4
156.h even 2 1 819.2.c.d 8
364.h even 2 1 1911.2.c.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.c.c ✓ 8 4.b odd 2 1
273.2.c.c ✓ 8 52.b odd 2 1
819.2.c.d 8 12.b even 2 1
819.2.c.d 8 156.h even 2 1
1911.2.c.l 8 28.d even 2 1
1911.2.c.l 8 364.h even 2 1
3549.2.a.v 4 52.f even 4 1
3549.2.a.x 4 52.f even 4 1
4368.2.h.q 8 1.a even 1 1 trivial
4368.2.h.q 8 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4368, [\chi])\):

\( T_{5}^{8} + 37T_{5}^{6} + 468T_{5}^{4} + 2240T_{5}^{2} + 2704 \) Copy content Toggle raw display
\( T_{11}^{8} + 44T_{11}^{6} + 576T_{11}^{4} + 2320T_{11}^{2} + 1024 \) Copy content Toggle raw display
\( T_{17}^{4} - 10T_{17}^{3} + 8T_{17}^{2} + 112T_{17} - 160 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T - 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 37 T^{6} + \cdots + 2704 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + 44 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( T^{8} + 6 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( (T^{4} - 10 T^{3} + \cdots - 160)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 97 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( (T^{4} - 3 T^{3} + \cdots + 1352)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 9 T^{3} + \cdots - 440)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 89 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$37$ \( T^{8} + 176 T^{6} + \cdots + 262144 \) Copy content Toggle raw display
$41$ \( T^{8} + 248 T^{6} + \cdots + 1784896 \) Copy content Toggle raw display
$43$ \( (T^{4} + 17 T^{3} + \cdots - 896)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 241 T^{6} + \cdots + 6739216 \) Copy content Toggle raw display
$53$ \( (T^{4} + 5 T^{3} + \cdots + 712)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 248 T^{6} + \cdots + 1364224 \) Copy content Toggle raw display
$61$ \( (T^{4} + 10 T^{3} + \cdots + 320)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 452 T^{6} + \cdots + 66064384 \) Copy content Toggle raw display
$71$ \( T^{8} + 44 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$73$ \( T^{8} + 305 T^{6} + \cdots + 5234944 \) Copy content Toggle raw display
$79$ \( (T^{4} - T^{3} - 32 T^{2} + \cdots - 80)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 241 T^{6} + \cdots + 6739216 \) Copy content Toggle raw display
$89$ \( T^{8} + 253 T^{6} + \cdots + 150544 \) Copy content Toggle raw display
$97$ \( T^{8} + 553 T^{6} + \cdots + 82882816 \) Copy content Toggle raw display
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