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Properties

Label 230.5.f.a
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
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 88 q^{2} + 24 q^{5} - 80 q^{7} + 704 q^{8} - 184 q^{10} + 8 q^{11} + 20 q^{13} + 396 q^{15} - 2816 q^{16} + 1080 q^{17} - 2648 q^{18} + 544 q^{20} - 3096 q^{21} - 16 q^{22} - 1884 q^{25} - 80 q^{26} - 3828 q^{27} + 640 q^{28} - 2520 q^{30} - 1580 q^{31} + 5632 q^{32} + 3644 q^{33} + 8208 q^{35} + 10592 q^{36} + 3104 q^{37} - 4064 q^{38} - 704 q^{40} + 4124 q^{41} + 6192 q^{42} - 960 q^{43} - 11316 q^{45} + 2424 q^{47} + 7832 q^{50} + 14840 q^{51} + 160 q^{52} - 3116 q^{53} - 2572 q^{55} - 2560 q^{56} - 9408 q^{57} - 3928 q^{58} + 6912 q^{60} + 19136 q^{61} + 3160 q^{62} + 4564 q^{63} - 9220 q^{65} - 14576 q^{66} - 5152 q^{67} - 8640 q^{68} - 23672 q^{70} + 7900 q^{71} - 21184 q^{72} + 16424 q^{73} + 24156 q^{75} + 16256 q^{76} - 27012 q^{77} - 1808 q^{78} - 1536 q^{80} - 116684 q^{81} - 8248 q^{82} + 11184 q^{83} - 14620 q^{85} + 3840 q^{86} + 8312 q^{87} + 128 q^{88} + 14544 q^{90} + 10296 q^{91} - 7488 q^{93} + 19536 q^{95} + 41292 q^{97} + 51024 q^{98}+O(q^{100}) \) 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 −11.8046 + 11.8046i 8.00000i −2.32340 − 24.8918i 47.2184 62.0285 + 62.0285i 16.0000 − 16.0000i − 197.698i −45.1368 + 54.4304i
47.2 −2.00000 − 2.00000i −10.5012 + 10.5012i 8.00000i 10.9451 + 22.4768i 42.0050 −20.3725 − 20.3725i 16.0000 − 16.0000i − 139.552i 23.0634 − 66.8437i
47.3 −2.00000 − 2.00000i −10.3237 + 10.3237i 8.00000i −22.1733 + 11.5475i 41.2946 6.42895 + 6.42895i 16.0000 − 16.0000i − 132.156i 67.4416 + 21.2517i
47.4 −2.00000 − 2.00000i −9.43718 + 9.43718i 8.00000i −10.0800 − 22.8778i 37.7487 −36.9388 − 36.9388i 16.0000 − 16.0000i − 97.1208i −25.5957 + 65.9156i
47.5 −2.00000 − 2.00000i −6.81573 + 6.81573i 8.00000i 24.3590 − 5.62491i 27.2629 −23.8398 − 23.8398i 16.0000 − 16.0000i − 11.9084i −59.9678 − 37.4682i
47.6 −2.00000 − 2.00000i −6.79892 + 6.79892i 8.00000i 24.9298 − 1.87172i 27.1957 46.9898 + 46.9898i 16.0000 − 16.0000i − 11.4506i −53.6031 − 46.1162i
47.7 −2.00000 − 2.00000i −5.15211 + 5.15211i 8.00000i −17.4200 + 17.9317i 20.6085 −7.30291 − 7.30291i 16.0000 − 16.0000i 27.9115i 70.7033 − 1.02335i
47.8 −2.00000 − 2.00000i −2.34537 + 2.34537i 8.00000i −23.4912 − 8.55343i 9.38149 −59.8209 − 59.8209i 16.0000 − 16.0000i 69.9985i 29.8756 + 64.0894i
47.9 −2.00000 − 2.00000i −2.20663 + 2.20663i 8.00000i −2.16754 − 24.9059i 8.82650 −2.30518 − 2.30518i 16.0000 − 16.0000i 71.2616i −45.4766 + 54.1468i
47.10 −2.00000 − 2.00000i −1.91399 + 1.91399i 8.00000i −2.47793 + 24.8769i 7.65597 57.2690 + 57.2690i 16.0000 − 16.0000i 73.6733i 54.7096 − 44.7979i
47.11 −2.00000 − 2.00000i −1.72340 + 1.72340i 8.00000i 4.34818 + 24.6190i 6.89361 −48.9296 − 48.9296i 16.0000 − 16.0000i 75.0598i 40.5416 − 57.9343i
47.12 −2.00000 − 2.00000i −0.665037 + 0.665037i 8.00000i 13.6244 − 20.9613i 2.66015 18.0918 + 18.0918i 16.0000 − 16.0000i 80.1155i −69.1714 + 14.6738i
47.13 −2.00000 − 2.00000i −0.545965 + 0.545965i 8.00000i −21.0759 − 13.4464i 2.18386 44.0481 + 44.0481i 16.0000 − 16.0000i 80.4038i 15.2589 + 69.0447i
47.14 −2.00000 − 2.00000i 3.09361 − 3.09361i 8.00000i 15.7881 + 19.3839i −12.3744 −30.9151 − 30.9151i 16.0000 − 16.0000i 61.8591i 7.19166 − 70.3440i
47.15 −2.00000 − 2.00000i 4.47696 − 4.47696i 8.00000i −20.9409 + 13.6558i −17.9078 −15.4160 − 15.4160i 16.0000 − 16.0000i 40.9136i 69.1933 + 14.5701i
47.16 −2.00000 − 2.00000i 6.39678 − 6.39678i 8.00000i 22.1785 + 11.5375i −25.5871 18.6881 + 18.6881i 16.0000 − 16.0000i − 0.837704i −21.2821 − 67.4320i
47.17 −2.00000 − 2.00000i 6.64524 − 6.64524i 8.00000i −21.5847 − 12.6135i −26.5810 27.2438 + 27.2438i 16.0000 − 16.0000i − 7.31852i 17.9425 + 68.3964i
47.18 −2.00000 − 2.00000i 6.67173 − 6.67173i 8.00000i 5.03883 − 24.4869i −26.6869 −43.7544 − 43.7544i 16.0000 − 16.0000i − 8.02391i −59.0515 + 38.8962i
47.19 −2.00000 − 2.00000i 7.00362 − 7.00362i 8.00000i 23.3343 − 8.97266i −28.0145 −20.2451 − 20.2451i 16.0000 − 16.0000i − 17.1014i −64.6140 − 28.7233i
47.20 −2.00000 − 2.00000i 11.2589 − 11.2589i 8.00000i 19.3240 − 15.8613i −45.0357 67.6527 + 67.6527i 16.0000 − 16.0000i − 172.526i −70.3707 − 6.92536i
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.a ✓ 44
5.c odd 4 1 inner 230.5.f.a ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.5.f.a ✓ 44 1.a even 1 1 trivial
230.5.f.a ✓ 44 5.c odd 4 1 inner