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Properties

Label 845.2.m.e
Level $845$
Weight $2$
Character orbit 845.m
Analytic conductor $6.747$
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] = [845,2,Mod(316,845)] 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("845.316"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(845, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.m (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12960000.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) 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 65)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + (\beta_{5} + 2 \beta_{4} - 1) q^{3} + ( - \beta_{5} - \beta_{4} - \beta_{2}) q^{4} + \beta_{7} q^{5} + ( - 2 \beta_{3} + \beta_1) q^{6} + ( - 3 \beta_{3} - 2 \beta_1) q^{7} + ( - \beta_{7} - 2 \beta_{6} + 2 \beta_1) q^{8}+ \cdots + ( - 2 \beta_{7} + 4 \beta_{6} - 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{4} - 8 q^{9} + 2 q^{10} + 20 q^{12} - 12 q^{14} + 6 q^{16} + 4 q^{17} - 14 q^{22} + 24 q^{23} - 8 q^{25} + 12 q^{29} - 10 q^{30} - 8 q^{35} - 4 q^{36} + 12 q^{38} + 10 q^{42} - 16 q^{43} - 30 q^{48}+ \cdots + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 5 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 13\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{6} + 8\nu^{4} - 16\nu^{2} + 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{6} + 8\nu^{4} - 24\nu^{2} + 9 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} + 8\nu^{5} - 24\nu^{3} + 9\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{7} + 8\nu^{5} - 20\nu^{3} + \nu ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 2\beta_{6} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{5} + 3\beta_{4} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{7} - 5\beta_{6} + 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{3} - 13\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(\beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
316.1
−1.40126 + 0.809017i
−0.535233 + 0.309017i
0.535233 − 0.309017i
1.40126 − 0.809017i
−1.40126 − 0.809017i
−0.535233 − 0.309017i
0.535233 + 0.309017i
1.40126 + 0.809017i
−1.40126 − 0.809017i 1.11803 − 1.93649i 0.309017 + 0.535233i 1.00000i −3.13331 + 1.80902i 0.204441 − 0.118034i 2.23607i −1.00000 − 1.73205i 0.809017 − 1.40126i
316.2 −0.535233 − 0.309017i −1.11803 + 1.93649i −0.809017 − 1.40126i − 1.00000i 1.19682 − 0.690983i 3.66854 − 2.11803i 2.23607i −1.00000 − 1.73205i −0.309017 + 0.535233i
316.3 0.535233 + 0.309017i −1.11803 + 1.93649i −0.809017 − 1.40126i 1.00000i −1.19682 + 0.690983i −3.66854 + 2.11803i − 2.23607i −1.00000 − 1.73205i −0.309017 + 0.535233i
316.4 1.40126 + 0.809017i 1.11803 − 1.93649i 0.309017 + 0.535233i − 1.00000i 3.13331 − 1.80902i −0.204441 + 0.118034i − 2.23607i −1.00000 − 1.73205i 0.809017 − 1.40126i
361.1 −1.40126 + 0.809017i 1.11803 + 1.93649i 0.309017 − 0.535233i − 1.00000i −3.13331 − 1.80902i 0.204441 + 0.118034i − 2.23607i −1.00000 + 1.73205i 0.809017 + 1.40126i
361.2 −0.535233 + 0.309017i −1.11803 − 1.93649i −0.809017 + 1.40126i 1.00000i 1.19682 + 0.690983i 3.66854 + 2.11803i − 2.23607i −1.00000 + 1.73205i −0.309017 − 0.535233i
361.3 0.535233 − 0.309017i −1.11803 − 1.93649i −0.809017 + 1.40126i − 1.00000i −1.19682 − 0.690983i −3.66854 − 2.11803i 2.23607i −1.00000 + 1.73205i −0.309017 − 0.535233i
361.4 1.40126 − 0.809017i 1.11803 + 1.93649i 0.309017 − 0.535233i 1.00000i 3.13331 + 1.80902i −0.204441 − 0.118034i 2.23607i −1.00000 + 1.73205i 0.809017 + 1.40126i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 316.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.m.e 8
13.b even 2 1 inner 845.2.m.e 8
13.c even 3 1 845.2.c.c 4
13.c even 3 1 inner 845.2.m.e 8
13.d odd 4 1 65.2.e.a ✓ 4
13.d odd 4 1 845.2.e.g 4
13.e even 6 1 845.2.c.c 4
13.e even 6 1 inner 845.2.m.e 8
13.f odd 12 1 65.2.e.a ✓ 4
13.f odd 12 1 845.2.a.b 2
13.f odd 12 1 845.2.a.e 2
13.f odd 12 1 845.2.e.g 4
39.f even 4 1 585.2.j.e 4
39.k even 12 1 585.2.j.e 4
39.k even 12 1 7605.2.a.ba 2
39.k even 12 1 7605.2.a.bf 2
52.f even 4 1 1040.2.q.n 4
52.l even 12 1 1040.2.q.n 4
65.f even 4 1 325.2.o.a 8
65.g odd 4 1 325.2.e.b 4
65.k even 4 1 325.2.o.a 8
65.o even 12 1 325.2.o.a 8
65.s odd 12 1 325.2.e.b 4
65.s odd 12 1 4225.2.a.u 2
65.s odd 12 1 4225.2.a.y 2
65.t even 12 1 325.2.o.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.e.a ✓ 4 13.d odd 4 1
65.2.e.a ✓ 4 13.f odd 12 1
325.2.e.b 4 65.g odd 4 1
325.2.e.b 4 65.s odd 12 1
325.2.o.a 8 65.f even 4 1
325.2.o.a 8 65.k even 4 1
325.2.o.a 8 65.o even 12 1
325.2.o.a 8 65.t even 12 1
585.2.j.e 4 39.f even 4 1
585.2.j.e 4 39.k even 12 1
845.2.a.b 2 13.f odd 12 1
845.2.a.e 2 13.f odd 12 1
845.2.c.c 4 13.c even 3 1
845.2.c.c 4 13.e even 6 1
845.2.e.g 4 13.d odd 4 1
845.2.e.g 4 13.f odd 12 1
845.2.m.e 8 1.a even 1 1 trivial
845.2.m.e 8 13.b even 2 1 inner
845.2.m.e 8 13.c even 3 1 inner
845.2.m.e 8 13.e even 6 1 inner
1040.2.q.n 4 52.f even 4 1
1040.2.q.n 4 52.l even 12 1
4225.2.a.u 2 65.s odd 12 1
4225.2.a.y 2 65.s odd 12 1
7605.2.a.ba 2 39.k even 12 1
7605.2.a.bf 2 39.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 3T_{2}^{6} + 8T_{2}^{4} - 3T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 3 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{4} + 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} - 18 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} - 18 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} - 2 T^{3} + \cdots + 361)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 18 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( (T^{4} - 12 T^{3} + \cdots + 961)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 6 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} - 9 T^{2} + 81)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 178 T^{6} + \cdots + 25411681 \) Copy content Toggle raw display
$43$ \( (T^{4} + 8 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 192 T^{2} + 4096)^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{8} \) Copy content Toggle raw display
$59$ \( T^{8} - 162 T^{6} + \cdots + 6561 \) Copy content Toggle raw display
$61$ \( (T^{4} + 2 T^{3} + \cdots + 32041)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 122 T^{6} + \cdots + 707281 \) Copy content Toggle raw display
$71$ \( T^{8} - 42 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$73$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{2} + 80)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 81 T^{2} + 6561)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 42 T^{6} + \cdots + 130321 \) Copy content Toggle raw display
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