Properties

Label 1216.2.a.t
Level $1216$
Weight $2$
Character orbit 1216.a
Self dual yes
Analytic conductor $9.710$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,2,Mod(1,1216)]
 
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, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.a (trivial)

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: yes
Analytic conductor: \(9.70980888579\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 608)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + ( - \beta + 1) q^{5} - 3 q^{7} + (\beta + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + ( - \beta + 1) q^{5} - 3 q^{7} + (\beta + 1) q^{9} + ( - \beta - 1) q^{11} - \beta q^{13} - 4 q^{15} + ( - 2 \beta - 3) q^{17} + q^{19} - 3 \beta q^{21} + (\beta - 4) q^{23} - \beta q^{25} + ( - \beta + 4) q^{27} + 3 \beta q^{29} + (2 \beta - 6) q^{31} + ( - 2 \beta - 4) q^{33} + (3 \beta - 3) q^{35} + (2 \beta - 4) q^{37} + ( - \beta - 4) q^{39} + 4 q^{41} + (\beta + 7) q^{43} + ( - \beta - 3) q^{45} + ( - 3 \beta - 1) q^{47} + 2 q^{49} + ( - 5 \beta - 8) q^{51} + (\beta + 6) q^{53} + (\beta + 3) q^{55} + \beta q^{57} + ( - \beta - 6) q^{59} + (5 \beta - 7) q^{61} + ( - 3 \beta - 3) q^{63} + 4 q^{65} + ( - \beta - 2) q^{67} + ( - 3 \beta + 4) q^{69} + ( - 4 \beta - 2) q^{71} + (4 \beta - 3) q^{73} + ( - \beta - 4) q^{75} + (3 \beta + 3) q^{77} - 10 q^{79} - 7 q^{81} + (6 \beta - 8) q^{83} + (3 \beta + 5) q^{85} + (3 \beta + 12) q^{87} + (6 \beta - 6) q^{89} + 3 \beta q^{91} + ( - 4 \beta + 8) q^{93} + ( - \beta + 1) q^{95} + ( - 2 \beta + 4) q^{97} + ( - 3 \beta - 5) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + q^{5} - 6 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + q^{5} - 6 q^{7} + 3 q^{9} - 3 q^{11} - q^{13} - 8 q^{15} - 8 q^{17} + 2 q^{19} - 3 q^{21} - 7 q^{23} - q^{25} + 7 q^{27} + 3 q^{29} - 10 q^{31} - 10 q^{33} - 3 q^{35} - 6 q^{37} - 9 q^{39} + 8 q^{41} + 15 q^{43} - 7 q^{45} - 5 q^{47} + 4 q^{49} - 21 q^{51} + 13 q^{53} + 7 q^{55} + q^{57} - 13 q^{59} - 9 q^{61} - 9 q^{63} + 8 q^{65} - 5 q^{67} + 5 q^{69} - 8 q^{71} - 2 q^{73} - 9 q^{75} + 9 q^{77} - 20 q^{79} - 14 q^{81} - 10 q^{83} + 13 q^{85} + 27 q^{87} - 6 q^{89} + 3 q^{91} + 12 q^{93} + q^{95} + 6 q^{97} - 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.56155
2.56155
0 −1.56155 0 2.56155 0 −3.00000 0 −0.561553 0
1.2 0 2.56155 0 −1.56155 0 −3.00000 0 3.56155 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.a.t 2
4.b odd 2 1 1216.2.a.s 2
8.b even 2 1 608.2.a.g 2
8.d odd 2 1 608.2.a.h yes 2
24.f even 2 1 5472.2.a.bf 2
24.h odd 2 1 5472.2.a.bc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.a.g 2 8.b even 2 1
608.2.a.h yes 2 8.d odd 2 1
1216.2.a.s 2 4.b odd 2 1
1216.2.a.t 2 1.a even 1 1 trivial
5472.2.a.bc 2 24.h odd 2 1
5472.2.a.bf 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1216))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$5$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$7$ \( (T + 3)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$13$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 8T - 1 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 7T + 8 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T - 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 10T + 8 \) Copy content Toggle raw display
$37$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$41$ \( (T - 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 15T + 52 \) Copy content Toggle raw display
$47$ \( T^{2} + 5T - 32 \) Copy content Toggle raw display
$53$ \( T^{2} - 13T + 38 \) Copy content Toggle raw display
$59$ \( T^{2} + 13T + 38 \) Copy content Toggle raw display
$61$ \( T^{2} + 9T - 86 \) Copy content Toggle raw display
$67$ \( T^{2} + 5T + 2 \) Copy content Toggle raw display
$71$ \( T^{2} + 8T - 52 \) Copy content Toggle raw display
$73$ \( T^{2} + 2T - 67 \) Copy content Toggle raw display
$79$ \( (T + 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 10T - 128 \) Copy content Toggle raw display
$89$ \( T^{2} + 6T - 144 \) Copy content Toggle raw display
$97$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
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