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Properties

Label 650.2.o.b
Level $650$
Weight $2$
Character orbit 650.o
Analytic conductor $5.190$
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] = [650,2,Mod(399,650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(650, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4])) 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("650.399"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,2,0,-4,0,0,2,0,-6] 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: \(5.19027613138\)
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 130)
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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} - 2 \zeta_{12} q^{3} + \zeta_{12}^{2} q^{4} - 2 \zeta_{12}^{2} q^{6} + (\zeta_{12}^{3} - \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} + (3 \zeta_{12}^{2} - 3) q^{11} + \cdots - 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 4 q^{6} + 2 q^{9} - 6 q^{11} - 4 q^{14} - 2 q^{16} + 10 q^{19} + 8 q^{21} + 4 q^{24} - 4 q^{26} - 16 q^{31} - 24 q^{34} - 2 q^{36} + 8 q^{39} - 12 q^{41} - 12 q^{44} - 12 q^{49} + 48 q^{51}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
399.1
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i 1.73205 + 1.00000i 0.500000 + 0.866025i 0 −1.00000 − 1.73205i 0.866025 − 0.500000i − 1.00000i 0.500000 + 0.866025i 0
399.2 0.866025 + 0.500000i −1.73205 − 1.00000i 0.500000 + 0.866025i 0 −1.00000 − 1.73205i −0.866025 + 0.500000i 1.00000i 0.500000 + 0.866025i 0
549.1 −0.866025 + 0.500000i 1.73205 − 1.00000i 0.500000 − 0.866025i 0 −1.00000 + 1.73205i 0.866025 + 0.500000i 1.00000i 0.500000 − 0.866025i 0
549.2 0.866025 − 0.500000i −1.73205 + 1.00000i 0.500000 − 0.866025i 0 −1.00000 + 1.73205i −0.866025 − 0.500000i − 1.00000i 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
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.o.b 4
5.b even 2 1 inner 650.2.o.b 4
5.c odd 4 1 130.2.e.b ✓ 2
5.c odd 4 1 650.2.e.a 2
13.c even 3 1 inner 650.2.o.b 4
15.e even 4 1 1170.2.i.f 2
20.e even 4 1 1040.2.q.c 2
65.f even 4 1 1690.2.l.i 4
65.h odd 4 1 1690.2.e.e 2
65.k even 4 1 1690.2.l.i 4
65.n even 6 1 inner 650.2.o.b 4
65.o even 12 1 1690.2.d.a 2
65.o even 12 1 1690.2.l.i 4
65.q odd 12 1 130.2.e.b ✓ 2
65.q odd 12 1 650.2.e.a 2
65.q odd 12 1 1690.2.a.a 1
65.q odd 12 1 8450.2.a.w 1
65.r odd 12 1 1690.2.a.g 1
65.r odd 12 1 1690.2.e.e 2
65.r odd 12 1 8450.2.a.k 1
65.t even 12 1 1690.2.d.a 2
65.t even 12 1 1690.2.l.i 4
195.bl even 12 1 1170.2.i.f 2
260.bj even 12 1 1040.2.q.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.e.b ✓ 2 5.c odd 4 1
130.2.e.b ✓ 2 65.q odd 12 1
650.2.e.a 2 5.c odd 4 1
650.2.e.a 2 65.q odd 12 1
650.2.o.b 4 1.a even 1 1 trivial
650.2.o.b 4 5.b even 2 1 inner
650.2.o.b 4 13.c even 3 1 inner
650.2.o.b 4 65.n even 6 1 inner
1040.2.q.c 2 20.e even 4 1
1040.2.q.c 2 260.bj even 12 1
1170.2.i.f 2 15.e even 4 1
1170.2.i.f 2 195.bl even 12 1
1690.2.a.a 1 65.q odd 12 1
1690.2.a.g 1 65.r odd 12 1
1690.2.d.a 2 65.o even 12 1
1690.2.d.a 2 65.t even 12 1
1690.2.e.e 2 65.h odd 4 1
1690.2.e.e 2 65.r odd 12 1
1690.2.l.i 4 65.f even 4 1
1690.2.l.i 4 65.k even 4 1
1690.2.l.i 4 65.o even 12 1
1690.2.l.i 4 65.t even 12 1
8450.2.a.k 1 65.r odd 12 1
8450.2.a.w 1 65.q odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$19$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T + 4)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 121 T^{2} + 14641 \) Copy content Toggle raw display
$41$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$47$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 256 T^{2} + 65536 \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$79$ \( (T - 16)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 9 T + 81)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
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