Properties

Label 54.7.f.a
Level $54$
Weight $7$
Character orbit 54.f
Analytic conductor $12.423$
Analytic rank $0$
Dimension $108$
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] = [54,7,Mod(5,54)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(54, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("54.5");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 54.f (of order \(18\), degree \(6\), 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: \(12.4229205155\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(18\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 108 q - 432 q^{5} + 336 q^{6} - 2496 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 108 q - 432 q^{5} + 336 q^{6} - 2496 q^{9} - 378 q^{11} + 960 q^{12} + 4752 q^{14} + 5112 q^{15} - 22080 q^{18} + 27648 q^{20} + 16080 q^{21} + 10800 q^{22} - 59832 q^{23} + 31968 q^{25} + 3078 q^{27} + 155628 q^{29} + 149184 q^{30} - 36720 q^{31} - 57294 q^{33} + 82080 q^{34} - 536544 q^{35} - 120576 q^{36} + 245808 q^{38} + 81696 q^{39} - 56646 q^{41} + 222528 q^{42} - 148770 q^{43} + 327132 q^{45} - 721872 q^{47} - 98304 q^{48} + 311256 q^{49} - 311040 q^{50} - 1066716 q^{51} + 372240 q^{54} + 152064 q^{56} + 1674126 q^{57} + 856116 q^{59} + 354816 q^{60} + 51408 q^{61} - 1675032 q^{63} + 1769472 q^{64} - 4448952 q^{65} - 786240 q^{66} + 1608822 q^{67} - 252288 q^{68} + 3773592 q^{69} + 1188000 q^{70} + 855360 q^{71} - 233472 q^{72} + 514080 q^{73} - 856224 q^{74} - 1831308 q^{75} - 915840 q^{76} - 6362280 q^{77} - 2385792 q^{78} - 1325376 q^{79} + 3791448 q^{81} + 3439260 q^{83} - 283392 q^{84} + 4968000 q^{85} + 1530576 q^{86} + 5122692 q^{87} + 691200 q^{88} - 964710 q^{89} + 469440 q^{90} + 902880 q^{91} - 525312 q^{92} - 4105752 q^{93} - 4421088 q^{94} - 1857708 q^{95} - 491520 q^{96} - 286578 q^{97} + 4393440 q^{98} + 3499164 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −3.63616 4.33340i −26.9889 0.772588i −5.55674 + 31.5138i 23.6620 + 65.0107i 94.7881 + 119.763i 15.7287 + 89.2017i 156.767 90.5097i 727.806 + 41.7027i 195.679 338.926i
5.2 −3.63616 4.33340i −18.7528 19.4250i −5.55674 + 31.5138i −70.4704 193.616i −15.9881 + 151.896i −93.1758 528.426i 156.767 90.5097i −25.6615 + 728.548i −582.774 + 1009.39i
5.3 −3.63616 4.33340i −15.9087 + 21.8154i −5.55674 + 31.5138i −37.7226 103.642i 152.382 10.3855i −61.0019 345.959i 156.767 90.5097i −222.826 694.111i −311.957 + 540.326i
5.4 −3.63616 4.33340i −5.10601 26.5128i −5.55674 + 31.5138i −10.7717 29.5950i −96.3244 + 118.531i 95.4997 + 541.606i 156.767 90.5097i −676.857 + 270.749i −89.0795 + 154.290i
5.5 −3.63616 4.33340i −0.750158 + 26.9896i −5.55674 + 31.5138i 81.1361 + 222.920i 119.684 94.8876i 8.32537 + 47.2155i 156.767 90.5097i −727.875 40.4929i 670.977 1162.17i
5.6 −3.63616 4.33340i 8.16974 + 25.7343i −5.55674 + 31.5138i −59.7321 164.113i 81.8107 128.977i 83.7342 + 474.880i 156.767 90.5097i −595.511 + 420.485i −493.971 + 855.583i
5.7 −3.63616 4.33340i 17.9291 20.1878i −5.55674 + 31.5138i 20.1217 + 55.2839i −152.675 4.28805i −42.4874 240.958i 156.767 90.5097i −86.0942 723.898i 166.402 288.216i
5.8 −3.63616 4.33340i 23.0877 + 13.9985i −5.55674 + 31.5138i −1.37580 3.77997i −23.2893 150.949i −65.8860 373.658i 156.767 90.5097i 337.084 + 646.387i −11.3775 + 19.7065i
5.9 −3.63616 4.33340i 26.8457 + 2.88196i −5.55674 + 31.5138i 6.65021 + 18.2713i −85.1266 126.813i 42.6327 + 241.782i 156.767 90.5097i 712.389 + 154.737i 54.9957 95.2554i
5.10 3.63616 + 4.33340i −26.5547 4.88367i −5.55674 + 31.5138i −55.9790 153.801i −75.3939 132.830i 71.8292 + 407.363i −156.767 + 90.5097i 681.299 + 259.369i 462.933 801.824i
5.11 3.63616 + 4.33340i −18.7421 + 19.4354i −5.55674 + 31.5138i 42.0848 + 115.627i −152.370 10.5469i 49.3468 + 279.859i −156.767 + 90.5097i −26.4686 728.519i −348.032 + 602.809i
5.12 3.63616 + 4.33340i −16.4084 21.4421i −5.55674 + 31.5138i 32.0681 + 88.1063i 33.2537 149.071i −50.3203 285.381i −156.767 + 90.5097i −190.528 + 703.662i −265.196 + 459.332i
5.13 3.63616 + 4.33340i −6.50900 + 26.2037i −5.55674 + 31.5138i −50.4685 138.661i −137.219 + 67.0746i −64.1010 363.535i −156.767 + 90.5097i −644.266 341.119i 417.362 722.893i
5.14 3.63616 + 4.33340i −4.60243 26.6048i −5.55674 + 31.5138i −47.7747 131.260i 98.5543 116.684i 31.0953 + 176.350i −156.767 + 90.5097i −686.635 + 244.894i 395.086 684.309i
5.15 3.63616 + 4.33340i 12.1966 24.0882i −5.55674 + 31.5138i 54.4203 + 149.519i 148.733 34.7359i 29.4094 + 166.789i −156.767 + 90.5097i −431.487 587.589i −450.043 + 779.498i
5.16 3.63616 + 4.33340i 20.8741 + 17.1252i −5.55674 + 31.5138i −41.8999 115.119i 1.69101 + 152.726i 68.2247 + 386.921i −156.767 + 90.5097i 142.455 + 714.946i 346.503 600.160i
5.17 3.63616 + 4.33340i 21.1953 + 16.7260i −5.55674 + 31.5138i 53.1645 + 146.068i 4.58881 + 152.666i −17.0891 96.9171i −156.767 + 90.5097i 169.481 + 709.026i −439.658 + 761.510i
5.18 3.63616 + 4.33340i 25.1415 9.84405i −5.55674 + 31.5138i −34.1182 93.7391i 134.077 + 73.1537i −101.765 577.136i −156.767 + 90.5097i 535.189 494.988i 282.150 488.698i
11.1 −3.63616 + 4.33340i −26.9889 + 0.772588i −5.55674 31.5138i 23.6620 65.0107i 94.7881 119.763i 15.7287 89.2017i 156.767 + 90.5097i 727.806 41.7027i 195.679 + 338.926i
11.2 −3.63616 + 4.33340i −18.7528 + 19.4250i −5.55674 31.5138i −70.4704 + 193.616i −15.9881 151.896i −93.1758 + 528.426i 156.767 + 90.5097i −25.6615 728.548i −582.774 1009.39i
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 54.7.f.a 108
27.f odd 18 1 inner 54.7.f.a 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.7.f.a 108 1.a even 1 1 trivial
54.7.f.a 108 27.f odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(54, [\chi])\).