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Properties

Label 1216.3.e.c
Level $1216$
Weight $3$
Character orbit 1216.e
Analytic conductor $33.134$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1216,3,Mod(1025,1216)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1216, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1216.1025"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1216.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-14,0,-22,0,-46,0,6,0,0,0,0,0,-34] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(33.1336001462\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - 7 q^{5} - 11 q^{7} - 23 q^{9} + 3 q^{11} + 2 \beta q^{13} - 7 \beta q^{15} - 17 q^{17} - 19 q^{19} - 11 \beta q^{21} - 2 q^{23} + 24 q^{25} - 14 \beta q^{27} + 7 \beta q^{29} - \beta q^{31} + \cdots - 69 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{5} - 22 q^{7} - 46 q^{9} + 6 q^{11} - 34 q^{17} - 38 q^{19} - 4 q^{23} + 48 q^{25} + 154 q^{35} - 128 q^{39} - 42 q^{43} + 322 q^{45} + 10 q^{47} + 144 q^{49} - 42 q^{55} - 46 q^{61} + 506 q^{63}+ \cdots - 138 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1025.1
− 1.41421i
1.41421i
0 − 5.65685i 0 −7.00000 0 −11.0000 0 −23.0000 0
1025.2 0 5.65685i 0 −7.00000 0 −11.0000 0 −23.0000 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
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.3.e.c 2
4.b odd 2 1 1216.3.e.d 2
8.b even 2 1 304.3.e.f 2
8.d odd 2 1 152.3.e.a ✓ 2
19.b odd 2 1 inner 1216.3.e.c 2
24.f even 2 1 1368.3.o.a 2
24.h odd 2 1 2736.3.o.b 2
76.d even 2 1 1216.3.e.d 2
152.b even 2 1 152.3.e.a ✓ 2
152.g odd 2 1 304.3.e.f 2
456.l odd 2 1 1368.3.o.a 2
456.p even 2 1 2736.3.o.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.3.e.a ✓ 2 8.d odd 2 1
152.3.e.a ✓ 2 152.b even 2 1
304.3.e.f 2 8.b even 2 1
304.3.e.f 2 152.g odd 2 1
1216.3.e.c 2 1.a even 1 1 trivial
1216.3.e.c 2 19.b odd 2 1 inner
1216.3.e.d 2 4.b odd 2 1
1216.3.e.d 2 76.d even 2 1
1368.3.o.a 2 24.f even 2 1
1368.3.o.a 2 456.l odd 2 1
2736.3.o.b 2 24.h odd 2 1
2736.3.o.b 2 456.p even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1216, [\chi])\):

\( T_{3}^{2} + 32 \) Copy content Toggle raw display
\( T_{5} + 7 \) Copy content Toggle raw display
\( T_{7} + 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 32 \) Copy content Toggle raw display
$5$ \( (T + 7)^{2} \) Copy content Toggle raw display
$7$ \( (T + 11)^{2} \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 128 \) Copy content Toggle raw display
$17$ \( (T + 17)^{2} \) Copy content Toggle raw display
$19$ \( (T + 19)^{2} \) Copy content Toggle raw display
$23$ \( (T + 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 1568 \) Copy content Toggle raw display
$31$ \( T^{2} + 32 \) Copy content Toggle raw display
$37$ \( T^{2} + 1568 \) Copy content Toggle raw display
$41$ \( T^{2} + 1568 \) Copy content Toggle raw display
$43$ \( (T + 21)^{2} \) Copy content Toggle raw display
$47$ \( (T - 5)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 32 \) Copy content Toggle raw display
$59$ \( T^{2} + 1152 \) Copy content Toggle raw display
$61$ \( (T + 23)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1568 \) Copy content Toggle raw display
$71$ \( T^{2} + 8192 \) Copy content Toggle raw display
$73$ \( (T - 39)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 9248 \) Copy content Toggle raw display
$83$ \( (T + 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 14112 \) Copy content Toggle raw display
$97$ \( T^{2} + 28800 \) Copy content Toggle raw display
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