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Properties

Label 1216.2.c.h
Level $1216$
Weight $2$
Character orbit 1216.c
Analytic conductor $9.710$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,2,Mod(609,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.609");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.c (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: 8.0.303595776.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_1 q^{5} + (2 \beta_{6} - \beta_{3}) q^{7} + (\beta_{7} - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_1 q^{5} + (2 \beta_{6} - \beta_{3}) q^{7} + (\beta_{7} - 5) q^{9} + ( - \beta_{4} - \beta_{2}) q^{11} + ( - \beta_{5} + 3 \beta_1) q^{13} + (2 \beta_{6} + 2 \beta_{3}) q^{15} + 5 q^{17} + \beta_{4} q^{19} + ( - 5 \beta_{5} - \beta_1) q^{21} + (\beta_{6} - \beta_{3}) q^{23} + (\beta_{7} + 2) q^{25} + (8 \beta_{4} - 3 \beta_{2}) q^{27} + ( - 3 \beta_{5} + \beta_1) q^{29} + 2 \beta_{3} q^{31} + 8 q^{33} + (5 \beta_{4} - \beta_{2}) q^{35} + 4 \beta_1 q^{37} + (3 \beta_{6} + 7 \beta_{3}) q^{39} + ( - 7 \beta_{4} - \beta_{2}) q^{43} + ( - 2 \beta_{5} - 7 \beta_1) q^{45} + ( - 3 \beta_{6} + 4 \beta_{3}) q^{47} + 4 q^{49} + 5 \beta_{2} q^{51} + (5 \beta_{5} - \beta_1) q^{53} + ( - \beta_{6} - 2 \beta_{3}) q^{55} - \beta_{7} q^{57} + ( - 2 \beta_{4} - \beta_{2}) q^{59} + ( - 4 \beta_{5} - 3 \beta_1) q^{61} - 11 \beta_{6} q^{63} + (2 \beta_{7} - 8) q^{65} + (6 \beta_{4} + \beta_{2}) q^{67} + ( - 3 \beta_{5} + \beta_1) q^{69} + ( - 4 \beta_{6} + 2 \beta_{3}) q^{71} + ( - 4 \beta_{7} - 5) q^{73} + (8 \beta_{4} + \beta_{2}) q^{75} + (6 \beta_{5} - \beta_1) q^{77} + (4 \beta_{6} - 6 \beta_{3}) q^{79} + ( - 8 \beta_{7} + 9) q^{81} - 8 \beta_{4} q^{83} + 5 \beta_1 q^{85} + ( - 7 \beta_{6} + 5 \beta_{3}) q^{87} + (4 \beta_{7} - 2) q^{89} + (16 \beta_{4} - \beta_{2}) q^{91} + (2 \beta_{5} - 6 \beta_1) q^{93} - \beta_{6} q^{95} + (2 \beta_{7} + 4) q^{97} + ( - 3 \beta_{4} + 5 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 44 q^{9} + 40 q^{17} + 12 q^{25} + 64 q^{33} + 32 q^{49} + 4 q^{57} - 72 q^{65} - 24 q^{73} + 104 q^{81} - 32 q^{89} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} + 4\nu^{4} + 20\nu^{2} + 27 ) / 36 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 8\nu^{5} + 40\nu^{3} + 165\nu ) / 72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 4\nu^{5} + 2\nu^{3} - 9\nu ) / 54 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{7} + 16\nu^{5} + 8\nu^{3} + 81\nu ) / 216 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{6} + 16\nu^{4} + 80\nu^{2} + 153 ) / 72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 5\nu^{5} + 16\nu^{3} + 18\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} + 5\nu^{4} + 7\nu^{2} + 18 ) / 9 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{4} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 2\beta_{5} + \beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{6} - 4\beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{7} - 6\beta_{5} + 5\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{6} + 15\beta_{4} + 16\beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{5} - 16\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13\beta_{6} + 35\beta_{4} - 48\beta_{3} - 13\beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
609.1
1.26217 − 1.18614i
−1.26217 − 1.18614i
−0.396143 − 1.68614i
0.396143 − 1.68614i
0.396143 + 1.68614i
−0.396143 + 1.68614i
−1.26217 + 1.18614i
1.26217 + 1.18614i
0 − 3.37228i 0 − 2.52434i 0 −3.31662 0 −8.37228 0
609.2 0 − 3.37228i 0 2.52434i 0 3.31662 0 −8.37228 0
609.3 0 − 2.37228i 0 − 0.792287i 0 3.31662 0 −2.62772 0
609.4 0 − 2.37228i 0 0.792287i 0 −3.31662 0 −2.62772 0
609.5 0 2.37228i 0 − 0.792287i 0 −3.31662 0 −2.62772 0
609.6 0 2.37228i 0 0.792287i 0 3.31662 0 −2.62772 0
609.7 0 3.37228i 0 − 2.52434i 0 3.31662 0 −8.37228 0
609.8 0 3.37228i 0 2.52434i 0 −3.31662 0 −8.37228 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 609.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.c.h ✓ 8
4.b odd 2 1 inner 1216.2.c.h ✓ 8
8.b even 2 1 inner 1216.2.c.h ✓ 8
8.d odd 2 1 inner 1216.2.c.h ✓ 8
16.e even 4 1 4864.2.a.bi 4
16.e even 4 1 4864.2.a.bl 4
16.f odd 4 1 4864.2.a.bi 4
16.f odd 4 1 4864.2.a.bl 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.c.h ✓ 8 1.a even 1 1 trivial
1216.2.c.h ✓ 8 4.b odd 2 1 inner
1216.2.c.h ✓ 8 8.b even 2 1 inner
1216.2.c.h ✓ 8 8.d odd 2 1 inner
4864.2.a.bi 4 16.e even 4 1
4864.2.a.bi 4 16.f odd 4 1
4864.2.a.bl 4 16.e even 4 1
4864.2.a.bl 4 16.f odd 4 1

Hecke kernels

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

\( T_{3}^{4} + 17T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{5}^{4} + 7T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} - 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 17 T^{2} + 64)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 7 T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 11)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 17 T^{2} + 64)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 51 T^{2} + 576)^{2} \) Copy content Toggle raw display
$17$ \( (T - 5)^{8} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 7 T^{2} + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 43 T^{2} + 256)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 112 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 101 T^{2} + 1156)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 87 T^{2} + 36)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 127 T^{2} + 3364)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 21 T^{2} + 36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 231 T^{2} + 4356)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 77 T^{2} + 484)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 44)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 6 T - 123)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 184 T^{2} + 16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 64)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 8 T - 116)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 6 T - 24)^{4} \) Copy content Toggle raw display
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