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Properties

Label 975.2.w.d
Level $975$
Weight $2$
Character orbit 975.w
Analytic conductor $7.785$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [975,2,Mod(49,975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("975.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(975, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.w (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,4,0,0,0,0,2,0,-12,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.78541419707\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
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.

\(f(q)\) \(=\) \( q - \zeta_{12} q^{3} + 2 \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{7} + \zeta_{12}^{2} q^{9} + ( - 2 \zeta_{12}^{2} - 2) q^{11} - 2 \zeta_{12}^{3} q^{12} + ( - 3 \zeta_{12}^{3} - \zeta_{12}) q^{13} + \cdots + ( - 4 \zeta_{12}^{2} + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 2 q^{9} - 12 q^{11} - 8 q^{16} - 12 q^{19} + 12 q^{29} - 4 q^{36} - 4 q^{39} - 24 q^{41} + 8 q^{49} + 12 q^{59} - 2 q^{61} - 32 q^{64} + 12 q^{69} - 36 q^{71} - 24 q^{76} + 44 q^{79} - 2 q^{81}+ \cdots - 18 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(\zeta_{12}^{2}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 − 0.500000i
0 −0.866025 + 0.500000i 1.00000 − 1.73205i 0 0 −0.866025 + 1.50000i 0 0.500000 − 0.866025i 0
49.2 0 0.866025 − 0.500000i 1.00000 − 1.73205i 0 0 0.866025 − 1.50000i 0 0.500000 − 0.866025i 0
199.1 0 −0.866025 − 0.500000i 1.00000 + 1.73205i 0 0 −0.866025 − 1.50000i 0 0.500000 + 0.866025i 0
199.2 0 0.866025 + 0.500000i 1.00000 + 1.73205i 0 0 0.866025 + 1.50000i 0 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.e even 6 1 inner
65.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.w.d 4
5.b even 2 1 inner 975.2.w.d 4
5.c odd 4 1 39.2.j.a ✓ 2
5.c odd 4 1 975.2.bc.c 2
13.e even 6 1 inner 975.2.w.d 4
15.e even 4 1 117.2.q.a 2
20.e even 4 1 624.2.bv.b 2
60.l odd 4 1 1872.2.by.f 2
65.f even 4 1 507.2.e.f 4
65.h odd 4 1 507.2.j.b 2
65.k even 4 1 507.2.e.f 4
65.l even 6 1 inner 975.2.w.d 4
65.o even 12 1 507.2.a.e 2
65.o even 12 1 507.2.e.f 4
65.q odd 12 1 507.2.b.c 2
65.q odd 12 1 507.2.j.b 2
65.r odd 12 1 39.2.j.a ✓ 2
65.r odd 12 1 507.2.b.c 2
65.r odd 12 1 975.2.bc.c 2
65.t even 12 1 507.2.a.e 2
65.t even 12 1 507.2.e.f 4
195.bc odd 12 1 1521.2.a.h 2
195.bf even 12 1 117.2.q.a 2
195.bf even 12 1 1521.2.b.f 2
195.bl even 12 1 1521.2.b.f 2
195.bn odd 12 1 1521.2.a.h 2
260.be odd 12 1 8112.2.a.bu 2
260.bg even 12 1 624.2.bv.b 2
260.bl odd 12 1 8112.2.a.bu 2
780.cw odd 12 1 1872.2.by.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.j.a ✓ 2 5.c odd 4 1
39.2.j.a ✓ 2 65.r odd 12 1
117.2.q.a 2 15.e even 4 1
117.2.q.a 2 195.bf even 12 1
507.2.a.e 2 65.o even 12 1
507.2.a.e 2 65.t even 12 1
507.2.b.c 2 65.q odd 12 1
507.2.b.c 2 65.r odd 12 1
507.2.e.f 4 65.f even 4 1
507.2.e.f 4 65.k even 4 1
507.2.e.f 4 65.o even 12 1
507.2.e.f 4 65.t even 12 1
507.2.j.b 2 65.h odd 4 1
507.2.j.b 2 65.q odd 12 1
624.2.bv.b 2 20.e even 4 1
624.2.bv.b 2 260.bg even 12 1
975.2.w.d 4 1.a even 1 1 trivial
975.2.w.d 4 5.b even 2 1 inner
975.2.w.d 4 13.e even 6 1 inner
975.2.w.d 4 65.l even 6 1 inner
975.2.bc.c 2 5.c odd 4 1
975.2.bc.c 2 65.r odd 12 1
1521.2.a.h 2 195.bc odd 12 1
1521.2.a.h 2 195.bn odd 12 1
1521.2.b.f 2 195.bf even 12 1
1521.2.b.f 2 195.bl even 12 1
1872.2.by.f 2 60.l odd 4 1
1872.2.by.f 2 780.cw odd 12 1
8112.2.a.bu 2 260.be odd 12 1
8112.2.a.bu 2 260.bl odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{7}^{4} + 3T_{7}^{2} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$11$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 23T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$29$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 12 T + 48)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$47$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 6 T + 12)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$71$ \( (T^{2} + 18 T + 108)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$79$ \( (T - 11)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 12 T + 48)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
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