Properties

Label 312.2.x.a
Level $312$
Weight $2$
Character orbit 312.x
Analytic conductor $2.491$
Analytic rank $0$
Dimension $4$
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] = [312,2,Mod(161,312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(312, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("312.161");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 312.x (of order \(4\), degree \(2\), 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: \(2.49133254306\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{3} + ( - \beta_{3} - \beta_{2}) q^{5} + (3 \beta_1 + 3) q^{7} + (2 \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{3} + ( - \beta_{3} - \beta_{2}) q^{5} + (3 \beta_1 + 3) q^{7} + (2 \beta_{2} - 1) q^{9} + (3 \beta_1 - 2) q^{13} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{15} + 4 \beta_{3} q^{17} + ( - 3 \beta_1 + 3) q^{19} + ( - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 3) q^{21} - 2 \beta_{3} q^{23} - \beta_1 q^{25} + (\beta_{2} - 5) q^{27} + 6 \beta_{2} q^{29} + ( - \beta_1 + 1) q^{31} - 6 \beta_{2} q^{35} + ( - 7 \beta_1 - 7) q^{37} + ( - 3 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 2) q^{39} + ( - \beta_{3} - \beta_{2}) q^{41} - 2 \beta_1 q^{43} + (\beta_{3} + \beta_{2} - 4 \beta_1 + 4) q^{45} + (4 \beta_{3} - 4 \beta_{2}) q^{47} + 11 \beta_1 q^{49} + (4 \beta_{3} + 8 \beta_1) q^{51} - 8 \beta_{2} q^{53} + (3 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 3) q^{57} + (4 \beta_{3} - 4 \beta_{2}) q^{59} - 8 q^{61} + ( - 6 \beta_{3} + 6 \beta_{2} - 3 \beta_1 - 3) q^{63} + (5 \beta_{3} - \beta_{2}) q^{65} + ( - 9 \beta_1 + 9) q^{67} + ( - 2 \beta_{3} - 4 \beta_1) q^{69} + (\beta_1 + 1) q^{73} + (\beta_{3} - \beta_1) q^{75} + 2 q^{79} + ( - 4 \beta_{2} - 7) q^{81} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{83} + ( - 8 \beta_1 - 8) q^{85} + (6 \beta_{2} - 12) q^{87} + (3 \beta_{3} - 3 \beta_{2}) q^{89} + (3 \beta_1 - 15) q^{91} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{93} - 6 \beta_{3} q^{95} + (9 \beta_1 - 9) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 12 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 12 q^{7} - 4 q^{9} - 8 q^{13} + 8 q^{15} + 12 q^{19} + 12 q^{21} - 20 q^{27} + 4 q^{31} - 28 q^{37} - 8 q^{39} + 16 q^{45} + 12 q^{57} - 32 q^{61} - 12 q^{63} + 36 q^{67} + 4 q^{73} + 8 q^{79} - 28 q^{81} - 32 q^{85} - 48 q^{87} - 60 q^{91} + 4 q^{93} - 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/312\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(157\) \(209\)
\(\chi(n)\) \(1\) \(-\beta_{1}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−0.707107 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
0 1.00000 1.41421i 0 1.41421 + 1.41421i 0 3.00000 + 3.00000i 0 −1.00000 2.82843i 0
161.2 0 1.00000 + 1.41421i 0 −1.41421 1.41421i 0 3.00000 + 3.00000i 0 −1.00000 + 2.82843i 0
281.1 0 1.00000 1.41421i 0 −1.41421 + 1.41421i 0 3.00000 3.00000i 0 −1.00000 2.82843i 0
281.2 0 1.00000 + 1.41421i 0 1.41421 1.41421i 0 3.00000 3.00000i 0 −1.00000 + 2.82843i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 312.2.x.a 4
3.b odd 2 1 inner 312.2.x.a 4
4.b odd 2 1 624.2.bf.a 4
12.b even 2 1 624.2.bf.a 4
13.d odd 4 1 inner 312.2.x.a 4
39.f even 4 1 inner 312.2.x.a 4
52.f even 4 1 624.2.bf.a 4
156.l odd 4 1 624.2.bf.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.x.a 4 1.a even 1 1 trivial
312.2.x.a 4 3.b odd 2 1 inner
312.2.x.a 4 13.d odd 4 1 inner
312.2.x.a 4 39.f even 4 1 inner
624.2.bf.a 4 4.b odd 2 1
624.2.bf.a 4 12.b even 2 1
624.2.bf.a 4 52.f even 4 1
624.2.bf.a 4 156.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 16 \) acting on \(S_{2}^{\mathrm{new}}(312, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 2 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 16 \) Copy content Toggle raw display
$7$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 14 T + 98)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 16 \) Copy content Toggle raw display
$43$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 4096 \) Copy content Toggle raw display
$53$ \( (T^{2} + 128)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 4096 \) Copy content Toggle raw display
$61$ \( (T + 8)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 18 T + 162)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$79$ \( (T - 2)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 256 \) Copy content Toggle raw display
$89$ \( T^{4} + 1296 \) Copy content Toggle raw display
$97$ \( (T^{2} + 18 T + 162)^{2} \) Copy content Toggle raw display
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