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Properties

Label 45.8.l.a
Level $45$
Weight $8$
Character orbit 45.l
Analytic conductor $14.057$
Analytic rank $0$
Dimension $160$
Inner twists $4$

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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] = [45,8,Mod(2,45)] 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("45.2"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([2, 3])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 45.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.0573261468\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(40\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 6 q^{2} - 30 q^{3} - 6 q^{5} + 1848 q^{6} - 2 q^{7} - 8 q^{10} + 17712 q^{11} + 2022 q^{12} - 2 q^{13} - 37566 q^{15} + 278524 q^{16} - 60480 q^{18} - 109062 q^{20} - 84156 q^{21} - 514 q^{22} - 6894 q^{23}+ \cdots + 17617708 q^{97}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −5.77416 − 21.5495i −15.6667 + 44.0631i −320.187 + 184.860i 206.868 − 187.964i 1040.00 + 83.1820i 909.529 − 243.708i 3813.20 + 3813.20i −1696.11 − 1380.65i −5245.02 − 3372.55i
2.2 −5.31665 − 19.8420i 14.6381 − 44.4154i −254.587 + 146.986i −268.652 + 77.1444i −959.115 − 54.3087i 249.908 − 66.9625i 2410.80 + 2410.80i −1758.45 − 1300.32i 2959.03 + 4920.44i
2.3 −5.14124 − 19.1874i 46.2628 + 6.83751i −230.872 + 133.294i 200.753 + 194.482i −106.654 − 922.815i −1312.46 + 351.673i 1946.62 + 1946.62i 2093.50 + 632.645i 2699.48 − 4851.80i
2.4 −5.03075 − 18.7750i −45.5097 + 10.7642i −216.341 + 124.905i −180.800 + 213.158i 431.045 + 800.293i −590.406 + 158.199i 1674.18 + 1674.18i 1955.26 − 979.750i 4911.60 + 2322.18i
2.5 −4.57589 − 17.0774i −33.6438 − 32.4822i −159.849 + 92.2889i 50.3558 − 274.935i −400.763 + 723.185i −970.003 + 259.912i 707.312 + 707.312i 76.8113 + 2185.65i −4925.61 + 398.123i
2.6 −4.55257 − 16.9904i 42.6271 + 19.2335i −157.097 + 90.6999i −194.608 − 200.631i 132.723 − 811.814i 178.901 − 47.9364i 664.177 + 664.177i 1447.14 + 1639.74i −2522.83 + 4219.85i
2.7 −4.34505 − 16.2160i 28.8864 − 36.7774i −133.227 + 76.9185i 234.662 − 151.851i −721.893 − 308.622i 939.647 − 251.778i 306.709 + 306.709i −518.147 − 2124.73i −3482.02 − 3145.48i
2.8 −4.25072 − 15.8639i −39.4827 − 25.0622i −122.744 + 70.8663i 174.395 + 218.429i −229.755 + 732.883i 1397.88 − 374.560i 159.479 + 159.479i 930.769 + 1979.05i 2723.84 − 3695.07i
2.9 −3.78097 − 14.1108i 8.79733 + 45.9305i −73.9665 + 42.7046i 37.4931 + 276.982i 614.851 − 297.798i 244.091 − 65.4039i −439.955 − 439.955i −2032.21 + 808.131i 3766.67 − 1576.32i
2.10 −3.59642 − 13.4220i −9.28072 + 45.8352i −56.3657 + 32.5427i −265.376 − 87.7530i 648.580 − 40.2767i −398.979 + 106.906i −618.174 − 618.174i −2014.74 − 850.768i −223.420 + 3877.48i
2.11 −2.78092 − 10.3785i −38.8782 + 25.9901i 10.8706 − 6.27615i 274.913 − 50.4783i 377.857 + 331.223i −1053.30 + 282.230i −1067.86 − 1067.86i 836.030 − 2020.90i −1288.40 − 2712.82i
2.12 −2.63469 − 9.83279i 8.06841 − 46.0641i 21.1091 − 12.1873i 156.389 + 231.662i −474.196 + 42.0296i −853.835 + 228.784i −1096.81 − 1096.81i −2056.80 − 743.328i 1865.85 − 2148.10i
2.13 −2.42149 − 9.03711i −45.7352 + 9.76191i 35.0455 − 20.2335i −125.399 − 249.800i 198.967 + 389.675i 1605.93 − 430.307i −1114.51 − 1114.51i 1996.41 − 892.925i −1953.82 + 1738.13i
2.14 −2.24463 − 8.37706i 45.7581 − 9.65388i 45.7145 − 26.3933i −156.841 + 231.357i −183.581 − 361.649i 1115.30 − 298.842i −1108.66 − 1108.66i 2000.61 − 883.486i 2290.14 + 794.556i
2.15 −1.89003 − 7.05370i 31.6411 + 34.4360i 64.6688 − 37.3366i 243.580 − 137.091i 183.098 − 288.272i 398.007 − 106.646i −1046.54 − 1046.54i −184.678 + 2179.19i −1427.37 − 1459.03i
2.16 −1.86078 − 6.94453i −21.9418 − 41.2984i 66.0873 − 38.1555i −278.887 + 18.6283i −245.969 + 229.223i −43.1320 + 11.5572i −1038.67 − 1038.67i −1224.12 + 1812.32i 648.313 + 1902.08i
2.17 −1.82710 − 6.81885i 39.2422 − 25.4372i 67.6929 − 39.0825i −64.5152 − 271.961i −245.152 − 221.110i −1055.29 + 282.763i −1029.12 − 1029.12i 892.894 − 1996.42i −1736.58 + 936.820i
2.18 −0.415134 − 1.54930i −45.0693 + 12.4802i 108.623 − 62.7137i −173.691 + 218.990i 38.0454 + 64.6450i −363.096 + 97.2914i −287.429 − 287.429i 1875.49 − 1124.95i 411.386 + 178.189i
2.19 −0.0665609 − 0.248409i 35.7462 + 30.1531i 110.794 − 63.9669i −231.026 + 157.328i 5.11099 − 10.8867i −1731.66 + 463.996i −46.5410 − 46.5410i 368.581 + 2155.72i 54.4589 + 46.9169i
2.20 0.172634 + 0.644277i −40.9813 − 22.5285i 110.466 − 63.7776i 263.251 + 93.9348i 7.43983 − 30.2925i −301.555 + 80.8015i 120.531 + 120.531i 1171.94 + 1846.49i −15.0740 + 185.823i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 2.40
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.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 45.8.l.a ✓ 160
5.c odd 4 1 inner 45.8.l.a ✓ 160
9.d odd 6 1 inner 45.8.l.a ✓ 160
45.l even 12 1 inner 45.8.l.a ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.8.l.a ✓ 160 1.a even 1 1 trivial
45.8.l.a ✓ 160 5.c odd 4 1 inner
45.8.l.a ✓ 160 9.d odd 6 1 inner
45.8.l.a ✓ 160 45.l even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(45, [\chi])\).