Properties

Label 230.5.f.b
Level $230$
Weight $5$
Character orbit 230.f
Analytic conductor $23.775$
Analytic rank $0$
Dimension $44$
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] = [230,5,Mod(47,230)] 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("230.47"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 230.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [44,88] 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: \(23.7750915093\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 88 q^{2} + 24 q^{5} - 80 q^{7} - 704 q^{8} - 136 q^{10} - 632 q^{11} + 500 q^{13} + 12 q^{15} - 2816 q^{16} + 120 q^{17} + 2648 q^{18} - 736 q^{20} - 856 q^{21} - 1264 q^{22} + 2052 q^{25} + 2000 q^{26}+ \cdots - 27984 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 2.00000 + 2.00000i −11.2624 + 11.2624i 8.00000i −23.2954 − 9.07330i −45.0495 2.86636 + 2.86636i −16.0000 + 16.0000i − 172.682i −28.4442 − 64.7374i
47.2 2.00000 + 2.00000i −11.0455 + 11.0455i 8.00000i 3.15467 + 24.8002i −44.1818 −2.46420 − 2.46420i −16.0000 + 16.0000i − 163.004i −43.2910 + 55.9097i
47.3 2.00000 + 2.00000i −9.95475 + 9.95475i 8.00000i 24.5881 + 4.51957i −39.8190 −57.5399 − 57.5399i −16.0000 + 16.0000i − 117.194i 40.1370 + 58.2153i
47.4 2.00000 + 2.00000i −9.27707 + 9.27707i 8.00000i 23.6357 − 8.14565i −37.1083 15.7230 + 15.7230i −16.0000 + 16.0000i − 91.1281i 63.5628 + 30.9802i
47.5 2.00000 + 2.00000i −8.96878 + 8.96878i 8.00000i 1.96751 − 24.9225i −35.8751 53.1255 + 53.1255i −16.0000 + 16.0000i − 79.8782i 53.7799 − 45.9099i
47.6 2.00000 + 2.00000i −5.57799 + 5.57799i 8.00000i −24.4867 − 5.04022i −22.3120 −57.8690 − 57.8690i −16.0000 + 16.0000i 18.7720i −38.8929 − 59.0537i
47.7 2.00000 + 2.00000i −4.66373 + 4.66373i 8.00000i −12.4209 + 21.6961i −18.6549 −24.9944 − 24.9944i −16.0000 + 16.0000i 37.4993i −68.2341 + 18.5503i
47.8 2.00000 + 2.00000i −4.44879 + 4.44879i 8.00000i 18.7807 + 16.5011i −17.7952 10.7782 + 10.7782i −16.0000 + 16.0000i 41.4165i 4.55931 + 70.5635i
47.9 2.00000 + 2.00000i −4.05030 + 4.05030i 8.00000i −24.3623 + 5.61047i −16.2012 52.2645 + 52.2645i −16.0000 + 16.0000i 48.1901i −59.9456 − 37.5037i
47.10 2.00000 + 2.00000i −3.68907 + 3.68907i 8.00000i 6.02861 − 24.2622i −14.7563 −4.81567 − 4.81567i −16.0000 + 16.0000i 53.7815i 60.5817 − 36.4673i
47.11 2.00000 + 2.00000i −0.310392 + 0.310392i 8.00000i 6.21241 + 24.2158i −1.24157 40.7961 + 40.7961i −16.0000 + 16.0000i 80.8073i −36.0068 + 60.8565i
47.12 2.00000 + 2.00000i 1.43653 − 1.43653i 8.00000i 23.6990 − 7.95979i 5.74612 −17.1725 − 17.1725i −16.0000 + 16.0000i 76.8728i 63.3175 + 31.4784i
47.13 2.00000 + 2.00000i 2.64936 − 2.64936i 8.00000i 22.4224 + 11.0560i 10.5974 −62.5314 − 62.5314i −16.0000 + 16.0000i 66.9618i 22.7329 + 66.9568i
47.14 2.00000 + 2.00000i 3.59966 − 3.59966i 8.00000i −10.4188 − 22.7255i 14.3986 7.45573 + 7.45573i −16.0000 + 16.0000i 55.0849i 24.6134 − 66.2886i
47.15 2.00000 + 2.00000i 4.87410 − 4.87410i 8.00000i 23.2684 − 9.14227i 19.4964 49.4466 + 49.4466i −16.0000 + 16.0000i 33.4862i 64.8214 + 28.2523i
47.16 2.00000 + 2.00000i 5.56222 − 5.56222i 8.00000i −17.4021 + 17.9490i 22.2489 −33.3744 − 33.3744i −16.0000 + 16.0000i 19.1234i −70.7022 + 1.09393i
47.17 2.00000 + 2.00000i 7.12415 − 7.12415i 8.00000i −12.4603 + 21.6735i 28.4966 −34.8776 − 34.8776i −16.0000 + 16.0000i − 20.5070i −68.2676 + 18.4264i
47.18 2.00000 + 2.00000i 7.50341 − 7.50341i 8.00000i −23.5955 − 8.26153i 30.0136 38.1189 + 38.1189i −16.0000 + 16.0000i − 31.6022i −30.6679 − 63.7140i
47.19 2.00000 + 2.00000i 7.61108 − 7.61108i 8.00000i −1.36901 + 24.9625i 30.4443 45.0835 + 45.0835i −16.0000 + 16.0000i − 34.8570i −52.6630 + 47.1870i
47.20 2.00000 + 2.00000i 9.67771 − 9.67771i 8.00000i 14.1269 − 20.6259i 38.7108 −43.7928 − 43.7928i −16.0000 + 16.0000i − 106.316i 69.5058 − 12.9980i
See all 44 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 47.22
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.5.f.b ✓ 44
5.c odd 4 1 inner 230.5.f.b ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.5.f.b ✓ 44 1.a even 1 1 trivial
230.5.f.b ✓ 44 5.c odd 4 1 inner