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Properties

Label 415.4.b.a
Level $415$
Weight $4$
Character orbit 415.b
Analytic conductor $24.486$
Analytic rank $0$
Dimension $124$
Inner twists $2$

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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] = [415,4,Mod(84,415)] 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("415.84"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(415, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 415 = 5 \cdot 83 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 415.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.4857926524\)
Analytic rank: \(0\)
Dimension: \(124\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 124 q - 504 q^{4} + 6 q^{5} - 1088 q^{9} + 62 q^{10} + 60 q^{11} + 124 q^{14} - 120 q^{15} + 2080 q^{16} + 124 q^{19} - 206 q^{20} + 24 q^{21} + 368 q^{24} - 430 q^{25} - 176 q^{26} + 208 q^{29} - 52 q^{30}+ \cdots + 2668 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
84.1 − 5.59070i 9.32458i −23.2559 −3.95795 + 10.4563i 52.1309 14.9769i 85.2909i −59.9478 58.4581 + 22.1277i
84.2 − 5.58218i − 4.95572i −23.1608 0.142604 − 11.1794i −27.6637 − 27.5915i 84.6302i 2.44088 −62.4056 − 0.796043i
84.3 − 5.43165i − 5.71216i −21.5028 10.3260 + 4.28658i −31.0264 16.9337i 73.3426i −5.62872 23.2832 − 56.0869i
84.4 − 5.36371i 4.49952i −20.7693 5.77941 − 9.57070i 24.1341 33.2479i 68.4909i 6.75431 −51.3344 − 30.9991i
84.5 − 5.32811i 1.15648i −20.3887 −4.45888 + 10.2527i 6.16183 − 32.9123i 66.0084i 25.6626 54.6276 + 23.7574i
84.6 − 5.22457i − 9.01343i −19.2961 −9.65647 + 5.63494i −47.0912 8.67367i 59.0173i −54.2418 29.4401 + 50.4509i
84.7 − 5.17649i − 1.39860i −18.7961 −11.1737 − 0.386298i −7.23982 10.8286i 55.8857i 25.0439 −1.99967 + 57.8404i
84.8 − 5.09381i 2.66448i −17.9469 6.10790 + 9.36448i 13.5724 − 10.1095i 50.6679i 19.9006 47.7009 − 31.1125i
84.9 − 5.06299i 4.85010i −17.6339 10.1684 − 4.64797i 24.5560 − 6.05924i 48.7763i 3.47650 −23.5327 − 51.4825i
84.10 − 5.04149i 0.170131i −17.4166 −9.56621 − 5.78685i 0.857713 10.4782i 47.4735i 26.9711 −29.1743 + 48.2279i
84.11 − 4.98044i 7.53464i −16.8048 −6.99634 − 8.72073i 37.5259 − 14.6496i 43.8520i −29.7709 −43.4331 + 34.8449i
84.12 − 4.82727i − 2.71547i −15.3025 3.18919 + 10.7158i −13.1083 13.1614i 35.2511i 19.6262 51.7282 − 15.3951i
84.13 − 4.78707i − 8.94432i −14.9160 2.73811 − 10.8399i −42.8171 26.2645i 33.1076i −53.0009 −51.8912 − 13.1075i
84.14 − 4.54206i − 3.36464i −12.6303 10.9753 − 2.13132i −15.2824 − 11.6054i 21.0312i 15.6792 −9.68060 − 49.8505i
84.15 − 4.53810i 9.82601i −12.5943 10.9214 + 2.39235i 44.5914 − 18.0658i 20.8496i −69.5504 10.8567 − 49.5623i
84.16 − 4.48211i − 9.62254i −12.0894 9.86295 + 5.26518i −43.1293 − 31.5948i 18.3290i −65.5932 23.5992 − 44.2069i
84.17 − 4.41429i 4.21802i −11.4859 −8.85731 + 6.82261i 18.6196 21.5411i 15.3879i 9.20830 30.1170 + 39.0987i
84.18 − 4.35680i − 6.61019i −10.9817 −4.93226 + 10.0336i −28.7993 − 14.5063i 12.9909i −16.6946 43.7144 + 21.4889i
84.19 − 4.19599i − 7.96065i −9.60634 −6.13690 − 9.34550i −33.4028 − 20.2067i 6.74017i −36.3720 −39.2136 + 25.7504i
84.20 − 4.12476i − 3.06656i −9.01365 5.77616 − 9.57267i −12.6488 7.17192i 4.18107i 17.5962 −39.4850 − 23.8253i
See next 80 embeddings (of 124 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 84.124
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 415.4.b.a ✓ 124
5.b even 2 1 inner 415.4.b.a ✓ 124
5.c odd 4 1 2075.4.a.m 62
5.c odd 4 1 2075.4.a.n 62
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
415.4.b.a ✓ 124 1.a even 1 1 trivial
415.4.b.a ✓ 124 5.b even 2 1 inner
2075.4.a.m 62 5.c odd 4 1
2075.4.a.n 62 5.c odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(415, [\chi])\).