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Properties

Label 409.4.a.b
Level $409$
Weight $4$
Character orbit 409.a
Self dual yes
Analytic conductor $24.132$
Analytic rank $0$
Dimension $55$
CM no
Inner twists $1$

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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] = [409,4,Mod(1,409)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(409, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("409.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 409 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 409.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [55] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(24.1317811923\)
Analytic rank: \(0\)
Dimension: \(55\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 55 q + 13 q^{2} + 10 q^{3} + 257 q^{4} + 60 q^{5} + 48 q^{6} + 72 q^{7} + 156 q^{8} + 639 q^{9} + 21 q^{10} + 362 q^{11} + 21 q^{12} + 48 q^{13} + 279 q^{14} + 512 q^{15} + 1237 q^{16} + 270 q^{17} + 360 q^{18}+ \cdots + 9542 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.56180 −9.91254 22.9336 −7.80335 55.1315 −8.66658 −83.0575 71.2584 43.4007
1.2 −5.48444 −0.630452 22.0791 7.79317 3.45768 −22.6135 −77.2161 −26.6025 −42.7412
1.3 −5.12250 1.94641 18.2400 18.8054 −9.97048 21.6967 −52.4542 −23.2115 −96.3307
1.4 −4.97665 9.59756 16.7671 19.5487 −47.7637 2.16181 −43.6306 65.1131 −97.2869
1.5 −4.78278 3.07984 14.8750 5.71204 −14.7302 3.01507 −32.8817 −17.5146 −27.3194
1.6 −4.64033 −5.52229 13.5327 0.0788332 25.6253 −15.7391 −25.6736 3.49569 −0.365813
1.7 −4.57064 1.95051 12.8907 −16.1275 −8.91507 −15.3517 −22.3537 −23.1955 73.7130
1.8 −4.47187 1.31591 11.9976 −9.17027 −5.88457 18.8363 −17.8770 −25.2684 41.0083
1.9 −4.25237 9.13461 10.0827 −9.27771 −38.8437 11.1015 −8.85626 56.4411 39.4523
1.10 −3.55576 −8.04228 4.64342 −14.6669 28.5964 13.9637 11.9352 37.6783 52.1520
1.11 −3.54858 −7.52569 4.59245 −0.783443 26.7056 32.5189 12.0920 29.6361 2.78011
1.12 −3.45765 0.157438 3.95532 20.7369 −0.544365 −33.3825 13.9851 −26.9752 −71.7008
1.13 −3.34123 −9.52658 3.16380 6.80130 31.8305 −4.95006 16.1588 63.7558 −22.7247
1.14 −3.02509 −2.40064 1.15120 −0.974294 7.26217 14.7311 20.7183 −21.2369 2.94733
1.15 −2.86560 8.66194 0.211673 9.44070 −24.8217 25.9777 22.3182 48.0292 −27.0533
1.16 −2.66240 7.89812 −0.911626 −19.4001 −21.0279 −15.1748 23.7263 35.3803 51.6508
1.17 −2.60837 −1.54968 −1.19641 −6.19392 4.04214 −7.41637 23.9876 −24.5985 16.1560
1.18 −2.59940 3.24735 −1.24311 −9.11442 −8.44117 −26.8531 24.0266 −16.4547 23.6920
1.19 −2.13164 5.85292 −3.45611 14.3057 −12.4763 0.342042 24.4203 7.25671 −30.4947
1.20 −1.71596 −10.0008 −5.05546 6.71523 17.1609 19.0277 22.4027 73.0150 −11.5231
See all 55 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.55
Significant digits:
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Atkin-Lehner signs

\( p \) Sign
\(409\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 409.4.a.b ✓ 55
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
409.4.a.b ✓ 55 1.a even 1 1 trivial