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Properties

Label 1216.2.i.n
Level $1216$
Weight $2$
Character orbit 1216.i
Analytic conductor $9.710$
Analytic rank $0$
Dimension $6$
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] = [1216,2,Mod(577,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.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: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.2696112.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 5x^{4} + 18x^{2} - 8x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_{3} - \beta_{2} - \beta_1) q^{3} + ( - \beta_{2} - \beta_1) q^{5} + (\beta_{3} - \beta_{2} - 1) q^{7} + (2 \beta_{5} + 2 \beta_{4} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - \beta_{3} - \beta_{2} - \beta_1) q^{3} + ( - \beta_{2} - \beta_1) q^{5} + (\beta_{3} - \beta_{2} - 1) q^{7} + (2 \beta_{5} + 2 \beta_{4} - 2) q^{9} + (\beta_{2} - 1) q^{11} + (\beta_{5} + \beta_{4} + 2 \beta_1 - 1) q^{13} + (2 \beta_{4} + \beta_1 - 2) q^{15} + ( - \beta_{5} - 3 \beta_{4} + \cdots + 2 \beta_1) q^{17}+ \cdots - 4 \beta_{5} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{3} + q^{5} - 4 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{3} + q^{5} - 4 q^{7} - 6 q^{9} - 8 q^{11} - q^{13} - 5 q^{15} - 11 q^{17} - 6 q^{21} - q^{23} + 6 q^{25} - 38 q^{27} - 3 q^{29} - 12 q^{31} - 6 q^{33} + 12 q^{35} + 24 q^{37} - 2 q^{39} + 19 q^{41} + 5 q^{43} + 12 q^{45} - 17 q^{47} + 22 q^{49} + 23 q^{51} - 5 q^{53} - 10 q^{55} - 29 q^{57} + 13 q^{59} - 3 q^{61} + 40 q^{63} + 42 q^{65} - 9 q^{67} - 10 q^{69} + 3 q^{71} + 11 q^{73} + 24 q^{75} - 20 q^{77} + 19 q^{79} - 23 q^{81} - 24 q^{83} + 17 q^{85} + 66 q^{87} - 3 q^{89} + 44 q^{91} + 42 q^{93} + 13 q^{95} - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 5x^{4} + 18x^{2} - 8x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 5\nu^{4} + 25\nu^{3} - 18\nu^{2} + 8\nu - 40 ) / 82 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{5} - 10\nu^{4} + 9\nu^{3} - 36\nu^{2} + 16\nu - 121 ) / 41 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -10\nu^{5} + 9\nu^{4} - 45\nu^{3} - 25\nu^{2} - 162\nu + 72 ) / 82 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -26\nu^{5} + 7\nu^{4} - 117\nu^{3} - 65\nu^{2} - 454\nu - 26 ) / 82 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 3\beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + 4\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{5} + 13\beta_{4} - 2\beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{5} + 11\beta_{4} + 7\beta_{3} - 18\beta_{2} - 18\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(-1 + \beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
−0.906803 + 1.57063i
1.17146 − 2.02903i
0.235342 − 0.407624i
−0.906803 − 1.57063i
1.17146 + 2.02903i
0.235342 + 0.407624i
0 −1.05139 − 1.82106i 0 −0.906803 − 1.57063i 0 −2.52444 0 −0.710831 + 1.23120i 0
577.2 0 −0.0731827 − 0.126756i 0 1.17146 + 2.02903i 0 3.83221 0 1.48929 − 2.57952i 0
577.3 0 1.62457 + 2.81384i 0 0.235342 + 0.407624i 0 −3.30777 0 −3.77846 + 6.54448i 0
961.1 0 −1.05139 + 1.82106i 0 −0.906803 + 1.57063i 0 −2.52444 0 −0.710831 − 1.23120i 0
961.2 0 −0.0731827 + 0.126756i 0 1.17146 − 2.02903i 0 3.83221 0 1.48929 + 2.57952i 0
961.3 0 1.62457 − 2.81384i 0 0.235342 − 0.407624i 0 −3.30777 0 −3.77846 − 6.54448i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 577.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.i.n 6
4.b odd 2 1 1216.2.i.m 6
8.b even 2 1 152.2.i.c ✓ 6
8.d odd 2 1 304.2.i.f 6
19.c even 3 1 inner 1216.2.i.n 6
24.f even 2 1 2736.2.s.y 6
24.h odd 2 1 1368.2.s.k 6
76.g odd 6 1 1216.2.i.m 6
152.k odd 6 1 304.2.i.f 6
152.k odd 6 1 5776.2.a.bk 3
152.l odd 6 1 2888.2.a.n 3
152.o even 6 1 5776.2.a.bq 3
152.p even 6 1 152.2.i.c ✓ 6
152.p even 6 1 2888.2.a.r 3
456.u even 6 1 2736.2.s.y 6
456.x odd 6 1 1368.2.s.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.i.c ✓ 6 8.b even 2 1
152.2.i.c ✓ 6 152.p even 6 1
304.2.i.f 6 8.d odd 2 1
304.2.i.f 6 152.k odd 6 1
1216.2.i.m 6 4.b odd 2 1
1216.2.i.m 6 76.g odd 6 1
1216.2.i.n 6 1.a even 1 1 trivial
1216.2.i.n 6 19.c even 3 1 inner
1368.2.s.k 6 24.h odd 2 1
1368.2.s.k 6 456.x odd 6 1
2736.2.s.y 6 24.f even 2 1
2736.2.s.y 6 456.u even 6 1
2888.2.a.n 3 152.l odd 6 1
2888.2.a.r 3 152.p even 6 1
5776.2.a.bk 3 152.k odd 6 1
5776.2.a.bq 3 152.o even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1216, [\chi])\):

\( T_{3}^{6} - T_{3}^{5} + 8T_{3}^{4} + 9T_{3}^{3} + 48T_{3}^{2} + 7T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{6} - T_{5}^{5} + 5T_{5}^{4} + 18T_{5}^{2} - 8T_{5} + 4 \) Copy content Toggle raw display
\( T_{7}^{3} + 2T_{7}^{2} - 14T_{7} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - T^{5} + 8 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} - T^{5} + 5 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( (T^{3} + 2 T^{2} - 14 T - 32)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} + 4 T^{2} + T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + T^{5} + \cdots + 5776 \) Copy content Toggle raw display
$17$ \( T^{6} + 11 T^{5} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( T^{6} + 18 T^{4} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} + T^{5} + 5 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$29$ \( T^{6} + 3 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$31$ \( (T^{3} + 6 T^{2} + \cdots - 216)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 12 T^{2} + \cdots + 292)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 19 T^{5} + \cdots + 1681 \) Copy content Toggle raw display
$43$ \( T^{6} - 5 T^{5} + \cdots + 118336 \) Copy content Toggle raw display
$47$ \( T^{6} + 17 T^{5} + \cdots + 521284 \) Copy content Toggle raw display
$53$ \( T^{6} + 5 T^{5} + \cdots + 1936 \) Copy content Toggle raw display
$59$ \( T^{6} - 13 T^{5} + \cdots + 2809 \) Copy content Toggle raw display
$61$ \( T^{6} + 3 T^{5} + \cdots + 58564 \) Copy content Toggle raw display
$67$ \( T^{6} + 9 T^{5} + \cdots + 529 \) Copy content Toggle raw display
$71$ \( T^{6} - 3 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$73$ \( T^{6} - 11 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$79$ \( T^{6} - 19 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$83$ \( (T^{3} + 12 T^{2} + \cdots - 632)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 3 T^{5} + \cdots + 295936 \) Copy content Toggle raw display
$97$ \( T^{6} + T^{5} + 18 T^{4} + \cdots + 1 \) Copy content Toggle raw display
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