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Properties

Label 2268.3.bg.c
Level $2268$
Weight $3$
Character orbit 2268.bg
Analytic conductor $61.799$
Analytic rank $0$
Dimension $8$
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] = [2268,3,Mod(701,2268)] 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("2268.701"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2268, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2268.bg (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,8] 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: \(61.7985239569\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12745506816.5
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 252)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + (\beta_{6} + \beta_{3}) q^{7} + ( - \beta_{7} + 2 \beta_{5}) q^{11} + ( - 4 \beta_{6} + 2 \beta_{4}) q^{13} + (2 \beta_{7} - 5 \beta_{5} + \cdots - 5 \beta_1) q^{17} + ( - 6 \beta_{3} - 10) q^{19}+ \cdots + ( - 34 \beta_{6} + 24 \beta_{4} + \cdots - 24) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{13} - 80 q^{19} - 36 q^{25} - 40 q^{31} + 64 q^{37} + 16 q^{43} - 28 q^{49} - 208 q^{55} + 272 q^{61} - 168 q^{67} + 288 q^{73} - 232 q^{79} + 272 q^{85} + 224 q^{91} - 96 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 203\nu ) / 55 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 148 ) / 55 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -8\nu^{6} + 55\nu^{4} - 440\nu^{2} + 576 ) / 495 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\nu^{7} - 110\nu^{5} + 880\nu^{3} - 1152\nu ) / 495 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -23\nu^{6} + 220\nu^{4} - 1265\nu^{2} + 1656 ) / 495 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -31\nu^{7} + 275\nu^{5} - 1705\nu^{3} + 2232\nu ) / 495 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 4\beta_{4} + \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + 5\beta_{5} + 2\beta_{2} + 5\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{6} - 23\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 16\beta_{7} + 31\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -55\beta_{3} - 148 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -110\beta_{2} - 203\beta_1 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1
−2.23256 − 1.28897i
−1.00781 − 0.581861i
1.00781 + 0.581861i
2.23256 + 1.28897i
−2.23256 + 1.28897i
−1.00781 + 0.581861i
1.00781 − 0.581861i
2.23256 − 1.28897i
0 0 0 −4.46512 − 2.57794i 0 1.32288 + 2.29129i 0 0 0
701.2 0 0 0 −2.01563 − 1.16372i 0 −1.32288 − 2.29129i 0 0 0
701.3 0 0 0 2.01563 + 1.16372i 0 −1.32288 − 2.29129i 0 0 0
701.4 0 0 0 4.46512 + 2.57794i 0 1.32288 + 2.29129i 0 0 0
2213.1 0 0 0 −4.46512 + 2.57794i 0 1.32288 − 2.29129i 0 0 0
2213.2 0 0 0 −2.01563 + 1.16372i 0 −1.32288 + 2.29129i 0 0 0
2213.3 0 0 0 2.01563 − 1.16372i 0 −1.32288 + 2.29129i 0 0 0
2213.4 0 0 0 4.46512 − 2.57794i 0 1.32288 − 2.29129i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 701.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.3.bg.c 8
3.b odd 2 1 inner 2268.3.bg.c 8
9.c even 3 1 252.3.c.a ✓ 4
9.c even 3 1 inner 2268.3.bg.c 8
9.d odd 6 1 252.3.c.a ✓ 4
9.d odd 6 1 inner 2268.3.bg.c 8
36.f odd 6 1 1008.3.d.c 4
36.h even 6 1 1008.3.d.c 4
63.g even 3 1 1764.3.bk.d 8
63.h even 3 1 1764.3.bk.d 8
63.i even 6 1 1764.3.bk.e 8
63.j odd 6 1 1764.3.bk.d 8
63.k odd 6 1 1764.3.bk.e 8
63.l odd 6 1 1764.3.c.f 4
63.n odd 6 1 1764.3.bk.d 8
63.o even 6 1 1764.3.c.f 4
63.s even 6 1 1764.3.bk.e 8
63.t odd 6 1 1764.3.bk.e 8
72.j odd 6 1 4032.3.d.h 4
72.l even 6 1 4032.3.d.e 4
72.n even 6 1 4032.3.d.h 4
72.p odd 6 1 4032.3.d.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.3.c.a ✓ 4 9.c even 3 1
252.3.c.a ✓ 4 9.d odd 6 1
1008.3.d.c 4 36.f odd 6 1
1008.3.d.c 4 36.h even 6 1
1764.3.c.f 4 63.l odd 6 1
1764.3.c.f 4 63.o even 6 1
1764.3.bk.d 8 63.g even 3 1
1764.3.bk.d 8 63.h even 3 1
1764.3.bk.d 8 63.j odd 6 1
1764.3.bk.d 8 63.n odd 6 1
1764.3.bk.e 8 63.i even 6 1
1764.3.bk.e 8 63.k odd 6 1
1764.3.bk.e 8 63.s even 6 1
1764.3.bk.e 8 63.t odd 6 1
2268.3.bg.c 8 1.a even 1 1 trivial
2268.3.bg.c 8 3.b odd 2 1 inner
2268.3.bg.c 8 9.c even 3 1 inner
2268.3.bg.c 8 9.d odd 6 1 inner
4032.3.d.e 4 72.l even 6 1
4032.3.d.e 4 72.p odd 6 1
4032.3.d.h 4 72.j odd 6 1
4032.3.d.h 4 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 32T_{5}^{6} + 880T_{5}^{4} - 4608T_{5}^{2} + 20736 \) acting on \(S_{3}^{\mathrm{new}}(2268, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 32 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$7$ \( (T^{4} + 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 116 T^{6} + \cdots + 8503056 \) Copy content Toggle raw display
$13$ \( (T^{4} - 4 T^{3} + \cdots + 11664)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 704 T^{2} + 121104)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 20 T - 152)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 308 T^{6} + \cdots + 3111696 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 1272032088336 \) Copy content Toggle raw display
$31$ \( (T^{4} + 20 T^{3} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 16 T - 944)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} - 288 T^{6} + \cdots + 136048896 \) Copy content Toggle raw display
$43$ \( (T^{4} - 8 T^{3} + \cdots + 984064)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 19007367745536 \) Copy content Toggle raw display
$53$ \( (T^{2} + 2178)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 14281868906496 \) Copy content Toggle raw display
$61$ \( (T^{4} - 136 T^{3} + \cdots + 21123216)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 84 T^{3} + \cdots + 13868176)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 15668 T^{2} + 32695524)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 72 T + 1268)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 116 T^{3} + \cdots + 14470416)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 228509902503936 \) Copy content Toggle raw display
$89$ \( (T^{4} + 49536 T^{2} + 601034256)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 48 T^{3} + \cdots + 56490256)^{2} \) Copy content Toggle raw display
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