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Properties

Label 585.4.c.e
Level $585$
Weight $4$
Character orbit 585.c
Analytic conductor $34.516$
Analytic rank $0$
Dimension $22$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [585,4,Mod(469,585)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(585, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("585.469"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 585.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22,0,0,-110,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(34.5161173534\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: no (minimal twist has level 195)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 110 q^{4} - 8 q^{5} - 58 q^{10} - 200 q^{11} + 300 q^{14} + 1022 q^{16} - 88 q^{19} + 296 q^{20} - 346 q^{25} - 78 q^{26} + 560 q^{29} + 512 q^{31} - 156 q^{34} - 36 q^{35} + 10 q^{40} - 1400 q^{41}+ \cdots - 2376 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
469.1 − 5.54491i 0 −22.7460 −4.72366 − 10.1335i 0 28.6020i 81.7651i 0 −56.1891 + 26.1922i
469.2 − 5.38219i 0 −20.9680 −0.249293 + 11.1776i 0 12.4545i 69.7962i 0 60.1598 + 1.34174i
469.3 − 5.17836i 0 −18.8154 −8.43850 + 7.33429i 0 − 32.8406i 56.0060i 0 37.9796 + 43.6976i
469.4 − 4.60776i 0 −13.2315 3.07988 − 10.7478i 0 15.5130i 24.1055i 0 −49.5231 − 14.1914i
469.5 − 3.33847i 0 −3.14536 3.44675 − 10.6358i 0 − 22.0097i − 16.2070i 0 −35.5072 − 11.5068i
469.6 − 3.05864i 0 −1.35528 8.72741 − 6.98802i 0 23.3910i − 20.3238i 0 −21.3738 − 26.6940i
469.7 − 2.70833i 0 0.664940 10.2491 + 4.46731i 0 5.41772i − 23.4675i 0 12.0990 − 27.7578i
469.8 − 1.71052i 0 5.07413 −11.1257 − 1.10435i 0 − 20.6809i − 22.3635i 0 −1.88901 + 19.0306i
469.9 − 1.66800i 0 5.21778 −5.41931 + 9.77912i 0 1.83536i − 22.0473i 0 16.3116 + 9.03941i
469.10 − 1.29959i 0 6.31107 −8.47931 + 7.28706i 0 34.4577i − 18.5985i 0 9.47018 + 11.0196i
469.11 − 0.0799779i 0 7.99360 8.93264 − 6.72368i 0 − 28.1840i − 1.27914i 0 −0.537746 − 0.714414i
469.12 0.0799779i 0 7.99360 8.93264 + 6.72368i 0 28.1840i 1.27914i 0 −0.537746 + 0.714414i
469.13 1.29959i 0 6.31107 −8.47931 − 7.28706i 0 − 34.4577i 18.5985i 0 9.47018 − 11.0196i
469.14 1.66800i 0 5.21778 −5.41931 − 9.77912i 0 − 1.83536i 22.0473i 0 16.3116 − 9.03941i
469.15 1.71052i 0 5.07413 −11.1257 + 1.10435i 0 20.6809i 22.3635i 0 −1.88901 − 19.0306i
469.16 2.70833i 0 0.664940 10.2491 − 4.46731i 0 − 5.41772i 23.4675i 0 12.0990 + 27.7578i
469.17 3.05864i 0 −1.35528 8.72741 + 6.98802i 0 − 23.3910i 20.3238i 0 −21.3738 + 26.6940i
469.18 3.33847i 0 −3.14536 3.44675 + 10.6358i 0 22.0097i 16.2070i 0 −35.5072 + 11.5068i
469.19 4.60776i 0 −13.2315 3.07988 + 10.7478i 0 − 15.5130i − 24.1055i 0 −49.5231 + 14.1914i
469.20 5.17836i 0 −18.8154 −8.43850 − 7.33429i 0 32.8406i − 56.0060i 0 37.9796 − 43.6976i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 469.22
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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.4.c.e 22
3.b odd 2 1 195.4.c.c ✓ 22
5.b even 2 1 inner 585.4.c.e 22
15.d odd 2 1 195.4.c.c ✓ 22
15.e even 4 1 975.4.a.bb 11
15.e even 4 1 975.4.a.bc 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.4.c.c ✓ 22 3.b odd 2 1
195.4.c.c ✓ 22 15.d odd 2 1
585.4.c.e 22 1.a even 1 1 trivial
585.4.c.e 22 5.b even 2 1 inner
975.4.a.bb 11 15.e even 4 1
975.4.a.bc 11 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(585, [\chi])\):

\( T_{2}^{22} + 143 T_{2}^{20} + 8605 T_{2}^{18} + 284071 T_{2}^{16} + 5635879 T_{2}^{14} + 69455253 T_{2}^{12} + \cdots + 34105600 \) Copy content Toggle raw display
\( T_{7}^{22} + 5766 T_{7}^{20} + 14286905 T_{7}^{18} + 19928171956 T_{7}^{16} + 17215001304064 T_{7}^{14} + \cdots + 34\!\cdots\!84 \) Copy content Toggle raw display