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Properties

Label 230.5.f.b
Level $230$
Weight $5$
Character orbit 230.f
Analytic conductor $23.775$
Analytic rank $0$
Dimension $44$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [230,5,Mod(47,230)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 230.f (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 88 q^{2} + 24 q^{5} - 80 q^{7} - 704 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 88 q^{2} + 24 q^{5} - 80 q^{7} - 704 q^{8} - 136 q^{10} - 632 q^{11} + 500 q^{13} + 12 q^{15} - 2816 q^{16} + 120 q^{17} + 2648 q^{18} - 736 q^{20} - 856 q^{21} - 1264 q^{22} + 2052 q^{25} + 2000 q^{26} + 588 q^{27} + 640 q^{28} - 1704 q^{30} + 3220 q^{31} - 5632 q^{32} - 2884 q^{33} + 720 q^{35} + 10592 q^{36} + 1856 q^{37} - 1696 q^{38} - 1856 q^{40} - 5156 q^{41} - 1712 q^{42} + 960 q^{43} + 460 q^{45} - 1224 q^{47} + 4456 q^{50} + 3640 q^{51} + 4000 q^{52} + 13556 q^{53} + 12980 q^{55} + 2560 q^{56} - 3072 q^{57} - 3112 q^{58} - 6912 q^{60} - 4864 q^{61} + 6440 q^{62} - 13484 q^{63} - 4100 q^{65} - 11536 q^{66} - 1888 q^{67} - 960 q^{68} + 8824 q^{70} - 1060 q^{71} + 21184 q^{72} + 16616 q^{73} - 11044 q^{75} - 6784 q^{76} + 8892 q^{77} - 23152 q^{78} - 1536 q^{80} - 4044 q^{81} - 10312 q^{82} - 27024 q^{83} - 11068 q^{85} + 3840 q^{86} - 8392 q^{87} + 10112 q^{88} - 7184 q^{90} - 54024 q^{91} + 22528 q^{93} - 25968 q^{95} - 3252 q^{97} - 27984 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
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 990-1000
Embeddings: e.g. 1-3 or 47.22
Significant digits:
Format:

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