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Properties

Label 312.2.q.d
Level $312$
Weight $2$
Character orbit 312.q
Analytic conductor $2.491$
Analytic rank $0$
Dimension $4$
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] = [312,2,Mod(217,312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(312, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("312.217");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 312.q (of order \(3\), 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 \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} - q^{5} + ( - \beta_{2} - \beta_1) q^{7} - \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} - q^{5} + ( - \beta_{2} - \beta_1) q^{7} - \beta_1 q^{9} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{11} + ( - \beta_{3} - \beta_{2}) q^{13} + ( - \beta_1 + 1) q^{15} + (\beta_{2} + 2 \beta_1) q^{17} + ( - \beta_{2} + 3 \beta_1) q^{19} + ( - \beta_{3} + 1) q^{21} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{23} - 4 q^{25} + q^{27} + (3 \beta_1 - 3) q^{29} + (2 \beta_{3} - 2) q^{31} + (\beta_{2} + \beta_1) q^{33} + (\beta_{2} + \beta_1) q^{35} + ( - \beta_{3} - \beta_{2} - 6 \beta_1 + 6) q^{37} + \beta_{2} q^{39} + (\beta_{3} + \beta_{2} - 8 \beta_1 + 8) q^{41} + ( - \beta_{2} - \beta_1) q^{43} + \beta_1 q^{45} + (\beta_{3} - 5) q^{47} + (2 \beta_{3} + 2 \beta_{2} + 7 \beta_1 - 7) q^{49} + (\beta_{3} - 2) q^{51} + 3 q^{53} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{55} + ( - \beta_{3} - 3) q^{57} + (2 \beta_{2} - 2 \beta_1) q^{59} + ( - \beta_{2} + 4 \beta_1) q^{61} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{63} + (\beta_{3} + \beta_{2}) q^{65} + (\beta_{3} + \beta_{2} - 7 \beta_1 + 7) q^{67} + (\beta_{2} + \beta_1) q^{69} + (\beta_{2} + 13 \beta_1) q^{71} + 7 q^{73} + ( - 4 \beta_1 + 4) q^{75} + (2 \beta_{3} - 14) q^{77} + 12 q^{79} + (\beta_1 - 1) q^{81} + ( - 3 \beta_{3} - 1) q^{83} + ( - \beta_{2} - 2 \beta_1) q^{85} - 3 \beta_1 q^{87} + (4 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 2) q^{89} + (\beta_{3} - 13) q^{91} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots + 2) q^{93}+ \cdots + (\beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 4 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 4 q^{5} - 2 q^{7} - 2 q^{9} + 2 q^{11} + 2 q^{15} + 4 q^{17} + 6 q^{19} + 4 q^{21} + 2 q^{23} - 16 q^{25} + 4 q^{27} - 6 q^{29} - 8 q^{31} + 2 q^{33} + 2 q^{35} + 12 q^{37} + 16 q^{41} - 2 q^{43} + 2 q^{45} - 20 q^{47} - 14 q^{49} - 8 q^{51} + 12 q^{53} - 2 q^{55} - 12 q^{57} - 4 q^{59} + 8 q^{61} - 2 q^{63} + 14 q^{67} + 2 q^{69} + 26 q^{71} + 28 q^{73} + 8 q^{75} - 56 q^{77} + 48 q^{79} - 2 q^{81} - 4 q^{83} - 4 q^{85} - 6 q^{87} + 4 q^{89} - 52 q^{91} + 4 q^{93} - 6 q^{95} - 8 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 4\nu^{2} - 4\nu + 9 ) / 12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 4\nu^{2} + 28\nu - 9 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 5 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 7\beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} - 5 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
217.1
1.15139 + 1.99426i
−0.651388 − 1.12824i
1.15139 − 1.99426i
−0.651388 + 1.12824i
0 −0.500000 + 0.866025i 0 −1.00000 0 −2.30278 − 3.98852i 0 −0.500000 − 0.866025i 0
217.2 0 −0.500000 + 0.866025i 0 −1.00000 0 1.30278 + 2.25647i 0 −0.500000 − 0.866025i 0
289.1 0 −0.500000 − 0.866025i 0 −1.00000 0 −2.30278 + 3.98852i 0 −0.500000 + 0.866025i 0
289.2 0 −0.500000 − 0.866025i 0 −1.00000 0 1.30278 − 2.25647i 0 −0.500000 + 0.866025i 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
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 312.2.q.d ✓ 4
3.b odd 2 1 936.2.t.e 4
4.b odd 2 1 624.2.q.i 4
12.b even 2 1 1872.2.t.q 4
13.c even 3 1 inner 312.2.q.d ✓ 4
13.c even 3 1 4056.2.a.v 2
13.e even 6 1 4056.2.a.w 2
13.f odd 12 2 4056.2.c.l 4
39.i odd 6 1 936.2.t.e 4
52.i odd 6 1 8112.2.a.bn 2
52.j odd 6 1 624.2.q.i 4
52.j odd 6 1 8112.2.a.bl 2
156.p even 6 1 1872.2.t.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.q.d ✓ 4 1.a even 1 1 trivial
312.2.q.d ✓ 4 13.c even 3 1 inner
624.2.q.i 4 4.b odd 2 1
624.2.q.i 4 52.j odd 6 1
936.2.t.e 4 3.b odd 2 1
936.2.t.e 4 39.i odd 6 1
1872.2.t.q 4 12.b even 2 1
1872.2.t.q 4 156.p even 6 1
4056.2.a.v 2 13.c even 3 1
4056.2.a.w 2 13.e even 6 1
4056.2.c.l 4 13.f odd 12 2
8112.2.a.bl 2 52.j odd 6 1
8112.2.a.bn 2 52.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(312, [\chi])\):

\( T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 2T_{7}^{3} + 16T_{7}^{2} - 24T_{7} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$13$ \( T^{4} + 13T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{4} - 6 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$29$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T - 48)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 12 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$41$ \( T^{4} - 16 T^{3} + \cdots + 2601 \) Copy content Toggle raw display
$43$ \( T^{4} + 2 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$47$ \( (T^{2} + 10 T + 12)^{2} \) Copy content Toggle raw display
$53$ \( (T - 3)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 4 T^{3} + \cdots + 2304 \) Copy content Toggle raw display
$61$ \( T^{4} - 8 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$67$ \( T^{4} - 14 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$71$ \( T^{4} - 26 T^{3} + \cdots + 24336 \) Copy content Toggle raw display
$73$ \( (T - 7)^{4} \) Copy content Toggle raw display
$79$ \( (T - 12)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 2 T - 116)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 4 T^{3} + \cdots + 41616 \) Copy content Toggle raw display
$97$ \( T^{4} + 8 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
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