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Properties

Label 312.2.q.b
Level $312$
Weight $2$
Character orbit 312.q
Analytic conductor $2.491$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(79\) \(145\) \(157\) \(209\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\) \(1\)

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
0.500000 + 0.866025i
0.500000 − 0.866025i
0 0.500000 − 0.866025i 0 −2.00000 0 −0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0
289.1 0 0.500000 + 0.866025i 0 −2.00000 0 −0.500000 + 0.866025i 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.b ✓ 2
3.b odd 2 1 936.2.t.d 2
4.b odd 2 1 624.2.q.a 2
12.b even 2 1 1872.2.t.l 2
13.c even 3 1 inner 312.2.q.b ✓ 2
13.c even 3 1 4056.2.a.c 1
13.e even 6 1 4056.2.a.h 1
13.f odd 12 2 4056.2.c.a 2
39.i odd 6 1 936.2.t.d 2
52.i odd 6 1 8112.2.a.bf 1
52.j odd 6 1 624.2.q.a 2
52.j odd 6 1 8112.2.a.t 1
156.p even 6 1 1872.2.t.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.q.b ✓ 2 1.a even 1 1 trivial
312.2.q.b ✓ 2 13.c even 3 1 inner
624.2.q.a 2 4.b odd 2 1
624.2.q.a 2 52.j odd 6 1
936.2.t.d 2 3.b odd 2 1
936.2.t.d 2 39.i odd 6 1
1872.2.t.l 2 12.b even 2 1
1872.2.t.l 2 156.p even 6 1
4056.2.a.c 1 13.c even 3 1
4056.2.a.h 1 13.e even 6 1
4056.2.c.a 2 13.f odd 12 2
8112.2.a.t 1 52.j odd 6 1
8112.2.a.bf 1 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 \(S_{2}^{\mathrm{new}}(312, [\chi])\):

\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( (T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$13$ \( T^{2} - 5T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$29$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$31$ \( (T - 7)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$43$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$47$ \( (T - 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$67$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$73$ \( (T - 1)^{2} \) Copy content Toggle raw display
$79$ \( (T - 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$97$ \( T^{2} - 19T + 361 \) Copy content Toggle raw display
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