Properties

Label 45.4.l.a
Level $45$
Weight $4$
Character orbit 45.l
Analytic conductor $2.655$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,4,Mod(2,45)]
 
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([2, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.2");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 45.l (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: \(2.65508595026\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 6 q^{2} - 6 q^{5} - 24 q^{6} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 6 q^{2} - 6 q^{5} - 24 q^{6} - 2 q^{7} - 8 q^{10} - 36 q^{11} - 138 q^{12} - 2 q^{13} - 96 q^{15} + 316 q^{16} - 480 q^{18} + 378 q^{20} + 480 q^{21} - 34 q^{22} + 306 q^{23} - 146 q^{25} + 180 q^{27} - 232 q^{28} - 1170 q^{30} - 4 q^{31} - 1770 q^{32} - 294 q^{33} - 216 q^{36} + 136 q^{37} + 114 q^{38} + 126 q^{40} + 1992 q^{41} + 1698 q^{42} - 2 q^{43} + 1134 q^{45} - 952 q^{46} + 3462 q^{47} + 4326 q^{48} + 666 q^{50} - 2496 q^{51} - 242 q^{52} + 284 q^{55} - 7128 q^{56} - 2544 q^{57} + 534 q^{58} + 1818 q^{60} + 32 q^{61} - 4038 q^{63} - 2094 q^{65} + 2892 q^{66} + 610 q^{67} - 2694 q^{68} + 498 q^{70} - 1854 q^{72} - 8 q^{73} - 6408 q^{75} + 1368 q^{76} - 6486 q^{77} + 1434 q^{78} + 3012 q^{81} - 3784 q^{82} + 2814 q^{83} - 1658 q^{85} + 12480 q^{86} + 4830 q^{87} - 1338 q^{88} + 13914 q^{90} + 992 q^{91} + 13152 q^{92} + 8310 q^{93} + 4284 q^{95} - 7932 q^{96} + 358 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −1.41347 5.27515i 4.81992 1.94123i −18.9011 + 10.9125i −3.32298 10.6751i −17.0531 22.6819i −4.22008 + 1.13077i 53.3880 + 53.3880i 19.4633 18.7131i −51.6158 + 32.6182i
2.2 −1.27001 4.73973i −2.23746 + 4.68975i −13.9239 + 8.03899i 1.75298 + 11.0421i 25.0698 + 4.64896i −21.2589 + 5.69629i 28.0284 + 28.0284i −16.9875 20.9863i 50.1101 22.3322i
2.3 −1.05047 3.92041i −3.51799 3.82410i −7.33795 + 4.23656i −9.79933 + 5.38267i −11.2965 + 17.8091i 22.8928 6.13411i 1.35786 + 1.35786i −2.24743 + 26.9063i 31.3962 + 32.7631i
2.4 −0.771317 2.87860i 0.414910 + 5.17956i −0.763180 + 0.440622i 1.89424 11.0187i 14.5898 5.18944i 31.3140 8.39055i −15.0012 15.0012i −26.6557 + 4.29810i −33.1795 + 3.04617i
2.5 −0.748271 2.79258i −0.352314 5.18419i −0.310415 + 0.179218i 11.0394 1.76941i −14.2137 + 4.86305i −18.7862 + 5.03374i −15.6218 15.6218i −26.7517 + 3.65293i −13.2017 29.5046i
2.6 −0.689083 2.57169i 5.10395 + 0.974542i 0.789441 0.455784i 6.87382 + 8.81763i −1.01082 13.7973i 1.49332 0.400135i −16.7770 16.7770i 25.1005 + 9.94802i 17.9396 23.7534i
2.7 −0.429078 1.60134i −4.91725 + 1.67949i 4.54802 2.62580i −7.92902 7.88230i 4.79931 + 7.15356i −30.4193 + 8.15082i −15.5344 15.5344i 21.3587 16.5169i −9.22008 + 16.0792i
2.8 −0.126293 0.471331i 3.65204 3.69630i 6.72200 3.88095i −11.1362 + 0.993015i −2.20341 1.25450i 5.71856 1.53228i −5.43846 5.43846i −0.325220 26.9980i 1.87445 + 5.12340i
2.9 0.0388957 + 0.145161i −5.17295 + 0.490448i 6.90864 3.98871i 9.52728 + 5.85072i −0.272400 0.731834i 17.2584 4.62436i 1.69784 + 1.69784i 26.5189 5.07413i −0.478726 + 1.61056i
2.10 0.335877 + 1.25351i 0.636882 + 5.15697i 5.46973 3.15795i −6.60187 + 9.02305i −6.25041 + 2.53045i 2.99997 0.803840i 13.1367 + 13.1367i −26.1888 + 6.56877i −13.5279 5.24487i
2.11 0.385641 + 1.43923i 4.61491 + 2.38802i 5.00553 2.88994i 4.60989 10.1857i −1.65722 + 7.56285i −24.7344 + 6.62758i 14.5184 + 14.5184i 15.5947 + 22.0410i 16.4374 + 2.70668i
2.12 0.576664 + 2.15214i −2.09209 4.75638i 2.62904 1.51788i 1.05576 11.1304i 9.02996 7.24530i 13.8105 3.70051i 17.3866 + 17.3866i −18.2463 + 19.9016i 24.5629 4.14635i
2.13 0.956489 + 3.56967i 3.31660 4.00002i −4.89944 + 2.82870i 6.37460 + 9.18501i 17.4510 + 8.01320i −4.17492 + 1.11867i 6.12165 + 6.12165i −5.00028 26.5329i −26.6902 + 31.5406i
2.14 1.11026 + 4.14355i −4.87317 1.80338i −9.00816 + 5.20086i −9.88574 + 5.22228i 2.06189 22.1945i −14.5845 + 3.90791i −7.28515 7.28515i 20.4957 + 17.5763i −32.6146 35.1640i
2.15 1.15194 + 4.29910i −2.52446 + 4.54171i −10.2271 + 5.90461i 10.7022 3.23468i −22.4333 5.62113i −7.13532 + 1.91190i −11.9883 11.9883i −14.2542 22.9307i 26.2345 + 42.2836i
2.16 1.30825 + 4.88245i 4.86054 + 1.83715i −15.1986 + 8.77490i −10.1192 4.75411i −2.61101 + 26.1348i 28.4601 7.62587i −34.1329 34.1329i 20.2497 + 17.8591i 9.97324 55.6261i
23.1 −1.41347 + 5.27515i 4.81992 + 1.94123i −18.9011 10.9125i −3.32298 + 10.6751i −17.0531 + 22.6819i −4.22008 1.13077i 53.3880 53.3880i 19.4633 + 18.7131i −51.6158 32.6182i
23.2 −1.27001 + 4.73973i −2.23746 4.68975i −13.9239 8.03899i 1.75298 11.0421i 25.0698 4.64896i −21.2589 5.69629i 28.0284 28.0284i −16.9875 + 20.9863i 50.1101 + 22.3322i
23.3 −1.05047 + 3.92041i −3.51799 + 3.82410i −7.33795 4.23656i −9.79933 5.38267i −11.2965 17.8091i 22.8928 + 6.13411i 1.35786 1.35786i −2.24743 26.9063i 31.3962 32.7631i
23.4 −0.771317 + 2.87860i 0.414910 5.17956i −0.763180 0.440622i 1.89424 + 11.0187i 14.5898 + 5.18944i 31.3140 + 8.39055i −15.0012 + 15.0012i −26.6557 4.29810i −33.1795 3.04617i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
9.d odd 6 1 inner
45.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.4.l.a 64
3.b odd 2 1 135.4.m.a 64
5.b even 2 1 225.4.p.b 64
5.c odd 4 1 inner 45.4.l.a 64
5.c odd 4 1 225.4.p.b 64
9.c even 3 1 135.4.m.a 64
9.d odd 6 1 inner 45.4.l.a 64
15.e even 4 1 135.4.m.a 64
45.h odd 6 1 225.4.p.b 64
45.k odd 12 1 135.4.m.a 64
45.l even 12 1 inner 45.4.l.a 64
45.l even 12 1 225.4.p.b 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.l.a 64 1.a even 1 1 trivial
45.4.l.a 64 5.c odd 4 1 inner
45.4.l.a 64 9.d odd 6 1 inner
45.4.l.a 64 45.l even 12 1 inner
135.4.m.a 64 3.b odd 2 1
135.4.m.a 64 9.c even 3 1
135.4.m.a 64 15.e even 4 1
135.4.m.a 64 45.k odd 12 1
225.4.p.b 64 5.b even 2 1
225.4.p.b 64 5.c odd 4 1
225.4.p.b 64 45.h odd 6 1
225.4.p.b 64 45.l even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(45, [\chi])\).