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Properties

Label 350.2.j.f
Level $350$
Weight $2$
Character orbit 350.j
Analytic conductor $2.795$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [350,2,Mod(149,350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 2])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("350.149"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 350.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,2,0,12,0,0,12,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} + 3 q^{6} + (\zeta_{12}^{3} - 3 \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + ( - 6 \zeta_{12}^{2} + 6) q^{9} + 2 \zeta_{12}^{2} q^{11}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 12 q^{6} + 12 q^{9} + 4 q^{11} - 8 q^{14} - 2 q^{16} - 12 q^{19} - 30 q^{21} + 6 q^{24} - 36 q^{29} + 8 q^{31} + 16 q^{34} + 24 q^{36} - 28 q^{41} - 4 q^{44} - 6 q^{46} + 26 q^{49} + 24 q^{51}+ \cdots + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i −2.59808 + 1.50000i 0.500000 + 0.866025i 0 3.00000 2.59808 + 0.500000i − 1.00000i 3.00000 − 5.19615i 0
149.2 0.866025 + 0.500000i 2.59808 − 1.50000i 0.500000 + 0.866025i 0 3.00000 −2.59808 − 0.500000i 1.00000i 3.00000 − 5.19615i 0
249.1 −0.866025 + 0.500000i −2.59808 − 1.50000i 0.500000 − 0.866025i 0 3.00000 2.59808 − 0.500000i 1.00000i 3.00000 + 5.19615i 0
249.2 0.866025 − 0.500000i 2.59808 + 1.50000i 0.500000 − 0.866025i 0 3.00000 −2.59808 + 0.500000i − 1.00000i 3.00000 + 5.19615i 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.j.f 4
5.b even 2 1 inner 350.2.j.f 4
5.c odd 4 1 70.2.e.a ✓ 2
5.c odd 4 1 350.2.e.l 2
7.c even 3 1 inner 350.2.j.f 4
7.c even 3 1 2450.2.c.s 2
7.d odd 6 1 2450.2.c.a 2
15.e even 4 1 630.2.k.f 2
20.e even 4 1 560.2.q.i 2
35.f even 4 1 490.2.e.f 2
35.i odd 6 1 2450.2.c.a 2
35.j even 6 1 inner 350.2.j.f 4
35.j even 6 1 2450.2.c.s 2
35.k even 12 1 490.2.a.e 1
35.k even 12 1 490.2.e.f 2
35.k even 12 1 2450.2.a.q 1
35.l odd 12 1 70.2.e.a ✓ 2
35.l odd 12 1 350.2.e.l 2
35.l odd 12 1 490.2.a.k 1
35.l odd 12 1 2450.2.a.b 1
105.w odd 12 1 4410.2.a.h 1
105.x even 12 1 630.2.k.f 2
105.x even 12 1 4410.2.a.r 1
140.w even 12 1 560.2.q.i 2
140.w even 12 1 3920.2.a.b 1
140.x odd 12 1 3920.2.a.bk 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.a ✓ 2 5.c odd 4 1
70.2.e.a ✓ 2 35.l odd 12 1
350.2.e.l 2 5.c odd 4 1
350.2.e.l 2 35.l odd 12 1
350.2.j.f 4 1.a even 1 1 trivial
350.2.j.f 4 5.b even 2 1 inner
350.2.j.f 4 7.c even 3 1 inner
350.2.j.f 4 35.j even 6 1 inner
490.2.a.e 1 35.k even 12 1
490.2.a.k 1 35.l odd 12 1
490.2.e.f 2 35.f even 4 1
490.2.e.f 2 35.k even 12 1
560.2.q.i 2 20.e even 4 1
560.2.q.i 2 140.w even 12 1
630.2.k.f 2 15.e even 4 1
630.2.k.f 2 105.x even 12 1
2450.2.a.b 1 35.l odd 12 1
2450.2.a.q 1 35.k even 12 1
2450.2.c.a 2 7.d odd 6 1
2450.2.c.a 2 35.i odd 6 1
2450.2.c.s 2 7.c even 3 1
2450.2.c.s 2 35.j even 6 1
3920.2.a.b 1 140.w even 12 1
3920.2.a.bk 1 140.x odd 12 1
4410.2.a.h 1 105.w odd 12 1
4410.2.a.r 1 105.x even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{4} - 9T_{3}^{2} + 81 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} + 4 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 13T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$29$ \( (T + 9)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$41$ \( (T + 7)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$53$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$71$ \( (T - 2)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$79$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
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