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Properties

Label 845.2.d.e
Level $845$
Weight $2$
Character orbit 845.d
Analytic conductor $6.747$
Analytic rank $0$
Dimension $36$
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(844,845)] 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("845.844"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(845, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 32 q^{4} - 36 q^{9} - 26 q^{10} - 8 q^{14} - 24 q^{16} + 16 q^{25} + 40 q^{29} - 62 q^{30} + 20 q^{35} - 64 q^{36} + 6 q^{40} + 88 q^{49} + 80 q^{51} - 40 q^{55} + 40 q^{56} + 16 q^{61} - 136 q^{64}+ \cdots - 82 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
844.1 −2.51756 − 2.61964i 4.33809 −0.117790 + 2.23296i 6.59510i −1.23067 −5.88628 −3.86253 0.296543 − 5.62161i
844.2 −2.51756 2.61964i 4.33809 −0.117790 − 2.23296i − 6.59510i −1.23067 −5.88628 −3.86253 0.296543 + 5.62161i
844.3 −2.15692 − 1.75445i 2.65230 2.22909 + 0.176470i 3.78422i −3.86123 −1.40696 −0.0781086 −4.80798 − 0.380631i
844.4 −2.15692 1.75445i 2.65230 2.22909 − 0.176470i − 3.78422i −3.86123 −1.40696 −0.0781086 −4.80798 + 0.380631i
844.5 −2.14790 − 2.37435i 2.61347 1.93861 + 1.11435i 5.09986i 3.54879 −1.31766 −2.63754 −4.16394 − 2.39351i
844.6 −2.14790 2.37435i 2.61347 1.93861 − 1.11435i − 5.09986i 3.54879 −1.31766 −2.63754 −4.16394 + 2.39351i
844.7 −1.94561 − 0.752344i 1.78540 −2.08452 + 0.809176i 1.46377i −1.49584 0.417520 2.43398 4.05567 − 1.57434i
844.8 −1.94561 0.752344i 1.78540 −2.08452 − 0.809176i − 1.46377i −1.49584 0.417520 2.43398 4.05567 + 1.57434i
844.9 −1.76651 − 0.691389i 1.12056 −0.913245 + 2.04107i 1.22135i 4.36221 1.55354 2.52198 1.61326 − 3.60558i
844.10 −1.76651 0.691389i 1.12056 −0.913245 − 2.04107i − 1.22135i 4.36221 1.55354 2.52198 1.61326 + 3.60558i
844.11 −1.57695 − 2.97563i 0.486782 0.978285 − 2.01071i 4.69244i 2.50257 2.38627 −5.85440 −1.54271 + 3.17080i
844.12 −1.57695 2.97563i 0.486782 0.978285 + 2.01071i − 4.69244i 2.50257 2.38627 −5.85440 −1.54271 − 3.17080i
844.13 −0.887876 − 1.26907i −1.21168 1.79562 − 1.33257i 1.12678i −2.28128 2.85157 1.38947 −1.59429 + 1.18316i
844.14 −0.887876 1.26907i −1.21168 1.79562 + 1.33257i − 1.12678i −2.28128 2.85157 1.38947 −1.59429 − 1.18316i
844.15 −0.395078 − 2.73651i −1.84391 1.91697 + 1.15119i 1.08113i 0.629405 1.51865 −4.48846 −0.757353 − 0.454808i
844.16 −0.395078 2.73651i −1.84391 1.91697 − 1.15119i − 1.08113i 0.629405 1.51865 −4.48846 −0.757353 + 0.454808i
844.17 −0.242854 − 1.19348i −1.94102 −1.65037 + 1.50873i 0.289840i −4.78046 0.957093 1.57562 0.400799 − 0.366402i
844.18 −0.242854 1.19348i −1.94102 −1.65037 − 1.50873i − 0.289840i −4.78046 0.957093 1.57562 0.400799 + 0.366402i
844.19 0.242854 − 1.19348i −1.94102 1.65037 − 1.50873i − 0.289840i 4.78046 −0.957093 1.57562 0.400799 − 0.366402i
844.20 0.242854 1.19348i −1.94102 1.65037 + 1.50873i 0.289840i 4.78046 −0.957093 1.57562 0.400799 + 0.366402i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 844.36
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.d.e 36
5.b even 2 1 inner 845.2.d.e 36
13.b even 2 1 inner 845.2.d.e 36
13.c even 3 2 845.2.l.g 72
13.d odd 4 1 845.2.b.g ✓ 18
13.d odd 4 1 845.2.b.h yes 18
13.e even 6 2 845.2.l.g 72
13.f odd 12 2 845.2.n.h 36
13.f odd 12 2 845.2.n.i 36
65.d even 2 1 inner 845.2.d.e 36
65.f even 4 1 4225.2.a.ca 18
65.f even 4 1 4225.2.a.cb 18
65.g odd 4 1 845.2.b.g ✓ 18
65.g odd 4 1 845.2.b.h yes 18
65.k even 4 1 4225.2.a.ca 18
65.k even 4 1 4225.2.a.cb 18
65.l even 6 2 845.2.l.g 72
65.n even 6 2 845.2.l.g 72
65.s odd 12 2 845.2.n.h 36
65.s odd 12 2 845.2.n.i 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
845.2.b.g ✓ 18 13.d odd 4 1
845.2.b.g ✓ 18 65.g odd 4 1
845.2.b.h yes 18 13.d odd 4 1
845.2.b.h yes 18 65.g odd 4 1
845.2.d.e 36 1.a even 1 1 trivial
845.2.d.e 36 5.b even 2 1 inner
845.2.d.e 36 13.b even 2 1 inner
845.2.d.e 36 65.d even 2 1 inner
845.2.l.g 72 13.c even 3 2
845.2.l.g 72 13.e even 6 2
845.2.l.g 72 65.l even 6 2
845.2.l.g 72 65.n even 6 2
845.2.n.h 36 13.f odd 12 2
845.2.n.h 36 65.s odd 12 2
845.2.n.i 36 13.f odd 12 2
845.2.n.i 36 65.s odd 12 2
4225.2.a.ca 18 65.f even 4 1
4225.2.a.ca 18 65.k even 4 1
4225.2.a.cb 18 65.f even 4 1
4225.2.a.cb 18 65.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} - 26 T_{2}^{16} + 281 T_{2}^{14} - 1632 T_{2}^{12} + 5482 T_{2}^{10} - 10620 T_{2}^{8} + \cdots - 29 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\). Copy content Toggle raw display