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Properties

Label 45.5.k.a
Level $45$
Weight $5$
Character orbit 45.k
Analytic conductor $4.652$
Analytic rank $0$
Dimension $88$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,5,Mod(7,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([8, 3]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.7");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 45.k (of order \(12\), degree \(4\), 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: \(4.65164833877\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(22\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 2 q^{2} + 6 q^{3} - 2 q^{5} + 120 q^{6} - 2 q^{7} - 72 q^{8} - 8 q^{10} - 160 q^{11} + 186 q^{12} - 2 q^{13} + 270 q^{15} + 2044 q^{16} - 908 q^{17} - 1656 q^{18} - 834 q^{20} - 1428 q^{21} + 62 q^{22}+ \cdots + 193724 q^{98}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −7.37360 + 1.97575i −7.93026 + 4.25569i 36.6100 − 21.1368i −3.41993 − 24.7650i 50.0664 − 47.0480i 1.13081 − 0.303000i −141.821 + 141.821i 44.7781 − 67.4975i 74.1466 + 175.850i
7.2 −7.18652 + 1.92562i 5.92105 − 6.77799i 34.0817 − 19.6771i −17.5700 + 17.7847i −29.4999 + 60.1119i 7.39767 − 1.98220i −122.864 + 122.864i −10.8823 − 80.2657i 92.0204 − 161.643i
7.3 −5.96640 + 1.59869i 4.37368 + 7.86581i 19.1857 − 11.0769i 17.4821 + 17.8711i −38.6701 − 39.9384i 57.8267 − 15.4946i −26.8779 + 26.8779i −42.7419 + 68.8050i −132.876 − 78.6774i
7.4 −5.12943 + 1.37443i −2.02987 − 8.76810i 10.5656 − 6.10008i 22.8406 − 10.1640i 22.4632 + 42.1855i −44.8951 + 12.0296i 14.2684 − 14.2684i −72.7592 + 35.5962i −103.190 + 83.5282i
7.5 −5.09412 + 1.36497i 8.84729 + 1.65091i 10.2306 − 5.90662i 5.54978 − 24.3762i −47.3226 + 3.66629i −32.3351 + 8.66418i 15.6131 − 15.6131i 75.5490 + 29.2122i 5.00143 + 131.751i
7.6 −4.24313 + 1.13694i −7.57398 + 4.86156i 2.85511 − 1.64840i 3.64389 + 24.7330i 26.6101 − 29.2394i −60.5059 + 16.2125i 39.4585 − 39.4585i 33.7305 − 73.6428i −43.5815 − 100.803i
7.7 −4.24152 + 1.13651i −6.48766 − 6.23781i 2.84239 − 1.64106i −24.8558 + 2.68162i 34.6068 + 19.0845i 33.8543 − 9.07123i 39.4891 − 39.4891i 3.17942 + 80.9376i 102.378 − 39.6230i
7.8 −2.72485 + 0.730121i −0.405345 + 8.99087i −6.96468 + 4.02106i −20.8337 − 13.8187i −5.45992 − 24.7947i 37.5459 − 10.0604i 47.9575 − 47.9575i −80.6714 − 7.28880i 66.8580 + 22.4427i
7.9 −1.67997 + 0.450147i 8.99424 + 0.321935i −11.2367 + 6.48753i −21.1623 + 13.3100i −15.2550 + 3.50789i −38.8993 + 10.4230i 35.6343 − 35.6343i 80.7927 + 5.79111i 29.5606 − 31.8866i
7.10 −1.49904 + 0.401665i 6.08169 − 6.63423i −11.7706 + 6.79578i 19.7780 + 15.2915i −6.45194 + 12.3878i 55.2269 − 14.7980i 32.4729 − 32.4729i −7.02599 − 80.6947i −35.7900 − 14.9784i
7.11 −0.777665 + 0.208375i −8.98887 + 0.447392i −13.2951 + 7.67591i 20.3088 − 14.5793i 6.89710 − 2.22097i 66.5008 − 17.8188i 17.8483 − 17.8483i 80.5997 − 8.04310i −12.7555 + 15.5696i
7.12 0.540591 − 0.144851i 0.247757 + 8.99659i −13.5851 + 7.84339i 23.0808 − 9.60597i 1.43710 + 4.82759i −71.0919 + 19.0490i −12.5397 + 12.5397i −80.8772 + 4.45794i 11.0859 − 8.53618i
7.13 0.873962 − 0.234177i 2.00485 − 8.77386i −13.1474 + 7.59068i −13.2061 − 21.2273i −0.302473 − 8.13751i −26.6745 + 7.14742i −19.9493 + 19.9493i −72.9611 − 35.1806i −16.5126 − 15.4593i
7.14 1.99166 − 0.533664i −6.71025 − 5.99772i −10.1745 + 5.87424i 3.24189 + 24.7889i −16.5653 − 8.36441i −22.3091 + 5.97771i −40.4572 + 40.4572i 9.05478 + 80.4923i 19.6857 + 47.6411i
7.15 2.86666 − 0.768118i 4.92601 + 7.53222i −6.22870 + 3.59614i −6.53092 + 24.1319i 19.9068 + 17.8085i 27.3211 − 7.32067i −48.6699 + 48.6699i −32.4688 + 74.2077i −0.185764 + 74.1943i
7.16 3.53742 − 0.947849i −7.88777 + 4.33395i −2.24148 + 1.29412i −24.4029 − 5.43125i −23.7944 + 22.8074i −20.1905 + 5.41004i −48.1356 + 48.1356i 43.4337 − 68.3704i −91.4713 + 3.91765i
7.17 3.53838 − 0.948107i 8.84647 + 1.65531i −2.23516 + 1.29047i 7.26089 − 23.9224i 32.8716 − 2.53029i 59.6290 − 15.9776i −48.1298 + 48.1298i 75.5199 + 29.2872i 3.01086 − 91.5306i
7.18 5.46311 − 1.46384i 7.72308 − 4.62104i 13.8464 − 7.99420i 22.0807 + 11.7237i 35.4276 − 36.5506i −85.6591 + 22.9523i −0.0464312 + 0.0464312i 38.2920 − 71.3773i 137.791 + 31.7253i
7.19 6.10168 − 1.63494i −6.85270 − 5.83443i 20.7010 − 11.9517i 10.5860 − 22.6481i −51.3519 − 24.3961i −11.1590 + 2.99004i 35.3028 − 35.3028i 12.9189 + 79.9631i 27.5642 − 155.499i
7.20 6.12130 − 1.64020i 2.33917 − 8.69070i 20.9237 − 12.0803i −20.7596 + 13.9297i 0.0642868 − 57.0351i 76.3600 − 20.4606i 36.5683 − 36.5683i −70.0566 − 40.6580i −104.228 + 119.318i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.22
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
9.c even 3 1 inner
45.k odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.5.k.a ✓ 88
3.b odd 2 1 135.5.l.a 88
5.c odd 4 1 inner 45.5.k.a ✓ 88
9.c even 3 1 inner 45.5.k.a ✓ 88
9.d odd 6 1 135.5.l.a 88
15.e even 4 1 135.5.l.a 88
45.k odd 12 1 inner 45.5.k.a ✓ 88
45.l even 12 1 135.5.l.a 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.5.k.a ✓ 88 1.a even 1 1 trivial
45.5.k.a ✓ 88 5.c odd 4 1 inner
45.5.k.a ✓ 88 9.c even 3 1 inner
45.5.k.a ✓ 88 45.k odd 12 1 inner
135.5.l.a 88 3.b odd 2 1
135.5.l.a 88 9.d odd 6 1
135.5.l.a 88 15.e even 4 1
135.5.l.a 88 45.l even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(45, [\chi])\).