Properties

Label 936.2.w.g
Level $936$
Weight $2$
Character orbit 936.w
Analytic conductor $7.474$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [936,2,Mod(307,936)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(936, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("936.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 936.w (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: \(7.47399762919\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{26})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 104)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{2} - 2 \beta_{2} q^{4} + \beta_1 q^{5} - \beta_{3} q^{7} + ( - 2 \beta_{2} - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{2} - 2 \beta_{2} q^{4} + \beta_1 q^{5} - \beta_{3} q^{7} + ( - 2 \beta_{2} - 2) q^{8} + ( - \beta_{3} + \beta_1) q^{10} + ( - \beta_{2} - 1) q^{11} - \beta_1 q^{13} + ( - \beta_{3} - \beta_1) q^{14} - 4 q^{16} + 3 \beta_{2} q^{17} + ( - 2 \beta_{2} + 2) q^{19} - 2 \beta_{3} q^{20} - 2 q^{22} + ( - \beta_{3} + \beta_1) q^{23} + 8 \beta_{2} q^{25} + (\beta_{3} - \beta_1) q^{26} - 2 \beta_1 q^{28} + ( - \beta_{3} - \beta_1) q^{29} + 2 \beta_1 q^{31} + (4 \beta_{2} - 4) q^{32} + (3 \beta_{2} + 3) q^{34} + 13 q^{35} - \beta_{3} q^{37} - 4 \beta_{2} q^{38} + ( - 2 \beta_{3} - 2 \beta_1) q^{40} + (6 \beta_{2} - 6) q^{41} - \beta_{2} q^{43} + (2 \beta_{2} - 2) q^{44} - 2 \beta_{3} q^{46} + \beta_{3} q^{47} - 6 \beta_{2} q^{49} + (8 \beta_{2} + 8) q^{50} + 2 \beta_{3} q^{52} + (\beta_{3} + \beta_1) q^{53} + ( - \beta_{3} - \beta_1) q^{55} + (2 \beta_{3} - 2 \beta_1) q^{56} - 2 \beta_1 q^{58} + ( - 8 \beta_{2} - 8) q^{59} + ( - 2 \beta_{3} + 2 \beta_1) q^{62} + 8 \beta_{2} q^{64} - 13 \beta_{2} q^{65} + ( - 3 \beta_{2} + 3) q^{67} + 6 q^{68} + ( - 13 \beta_{2} + 13) q^{70} + 3 \beta_1 q^{71} + ( - 6 \beta_{2} - 6) q^{73} + ( - \beta_{3} - \beta_1) q^{74} + ( - 4 \beta_{2} - 4) q^{76} + (\beta_{3} - \beta_1) q^{77} + (\beta_{3} + \beta_1) q^{79} - 4 \beta_1 q^{80} + 12 \beta_{2} q^{82} + (5 \beta_{2} - 5) q^{83} + 3 \beta_{3} q^{85} + ( - \beta_{2} - 1) q^{86} + 4 \beta_{2} q^{88} + (2 \beta_{2} + 2) q^{89} - 13 q^{91} + ( - 2 \beta_{3} - 2 \beta_1) q^{92} + (\beta_{3} + \beta_1) q^{94} + ( - 2 \beta_{3} + 2 \beta_1) q^{95} + (7 \beta_{2} - 7) q^{97} + ( - 6 \beta_{2} - 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 8 q^{8} - 4 q^{11} - 16 q^{16} + 8 q^{19} - 8 q^{22} - 16 q^{32} + 12 q^{34} + 52 q^{35} - 24 q^{41} - 8 q^{44} + 32 q^{50} - 32 q^{59} + 12 q^{67} + 24 q^{68} + 52 q^{70} - 24 q^{73} - 16 q^{76} - 20 q^{83} - 4 q^{86} + 8 q^{89} - 52 q^{91} - 28 q^{97} - 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 13\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 13\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/936\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−2.54951 + 2.54951i
2.54951 2.54951i
−2.54951 2.54951i
2.54951 + 2.54951i
1.00000 + 1.00000i 0 2.00000i −2.54951 + 2.54951i 0 −2.54951 2.54951i −2.00000 + 2.00000i 0 −5.09902
307.2 1.00000 + 1.00000i 0 2.00000i 2.54951 2.54951i 0 2.54951 + 2.54951i −2.00000 + 2.00000i 0 5.09902
811.1 1.00000 1.00000i 0 2.00000i −2.54951 2.54951i 0 −2.54951 + 2.54951i −2.00000 2.00000i 0 −5.09902
811.2 1.00000 1.00000i 0 2.00000i 2.54951 + 2.54951i 0 2.54951 2.54951i −2.00000 2.00000i 0 5.09902
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 936.2.w.g 4
3.b odd 2 1 104.2.m.a 4
8.d odd 2 1 inner 936.2.w.g 4
12.b even 2 1 416.2.u.a 4
13.d odd 4 1 inner 936.2.w.g 4
24.f even 2 1 104.2.m.a 4
24.h odd 2 1 416.2.u.a 4
39.f even 4 1 104.2.m.a 4
104.m even 4 1 inner 936.2.w.g 4
156.l odd 4 1 416.2.u.a 4
312.w odd 4 1 104.2.m.a 4
312.y even 4 1 416.2.u.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.m.a 4 3.b odd 2 1
104.2.m.a 4 24.f even 2 1
104.2.m.a 4 39.f even 4 1
104.2.m.a 4 312.w odd 4 1
416.2.u.a 4 12.b even 2 1
416.2.u.a 4 24.h odd 2 1
416.2.u.a 4 156.l odd 4 1
416.2.u.a 4 312.y even 4 1
936.2.w.g 4 1.a even 1 1 trivial
936.2.w.g 4 8.d odd 2 1 inner
936.2.w.g 4 13.d odd 4 1 inner
936.2.w.g 4 104.m even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(936, [\chi])\):

\( T_{5}^{4} + 169 \) Copy content Toggle raw display
\( T_{7}^{4} + 169 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 169 \) Copy content Toggle raw display
$7$ \( T^{4} + 169 \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 169 \) Copy content Toggle raw display
$17$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T + 8)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 26)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 26)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2704 \) Copy content Toggle raw display
$37$ \( T^{4} + 169 \) Copy content Toggle raw display
$41$ \( (T^{2} + 12 T + 72)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 169 \) Copy content Toggle raw display
$53$ \( (T^{2} + 26)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 16 T + 128)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 13689 \) Copy content Toggle raw display
$73$ \( (T^{2} + 12 T + 72)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 26)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 10 T + 50)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 4 T + 8)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 14 T + 98)^{2} \) Copy content Toggle raw display
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