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Properties

Label 72.7.j.a
Level $72$
Weight $7$
Character orbit 72.j
Analytic conductor $16.564$
Analytic rank $0$
Dimension $140$
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] = [72,7,Mod(5,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 5]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.5");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 72.j (of order \(6\), 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: \(16.5638940206\)
Analytic rank: \(0\)
Dimension: \(140\)
Relative dimension: \(70\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 140 q - 3 q^{2} - q^{4} + 317 q^{6} - 2 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 140 q - 3 q^{2} - q^{4} + 317 q^{6} - 2 q^{7} - 4 q^{9} + 124 q^{10} - 6946 q^{12} - 7428 q^{14} + 1454 q^{15} - q^{16} + 20878 q^{18} - 35346 q^{20} + 127 q^{22} - 6 q^{23} + 50209 q^{24} - 193752 q^{25} + 8188 q^{28} - 21344 q^{30} - 2 q^{31} - 180093 q^{32} + 45534 q^{33} + 20061 q^{34} + 39831 q^{36} + 274239 q^{38} + 125738 q^{39} - 58376 q^{40} + 31314 q^{41} - 213688 q^{42} - 22560 q^{46} - 6 q^{47} - 114411 q^{48} - 974808 q^{49} + 723249 q^{50} - 79338 q^{52} - 231173 q^{54} + 62492 q^{55} + 1294542 q^{56} + 125032 q^{57} - 221138 q^{58} - 1156344 q^{60} + 275294 q^{63} - 804478 q^{64} - 6 q^{65} + 1782258 q^{66} + 2100195 q^{68} - 86464 q^{70} + 1113091 q^{72} - 8 q^{73} - 1886976 q^{74} + 432081 q^{76} + 1658428 q^{78} - 2 q^{79} + 597124 q^{81} - 965382 q^{82} + 4997288 q^{84} - 1041879 q^{86} - 1337970 q^{87} - 536717 q^{88} - 4261994 q^{90} - 737184 q^{92} - 72534 q^{94} + 93744 q^{95} + 1053622 q^{96} - 2 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −7.98824 − 0.433656i 19.9076 − 18.2398i 63.6239 + 6.92830i 54.0624 − 93.6388i −166.936 + 137.070i −182.538 − 316.165i −505.238 − 82.9358i 63.6224 − 726.218i −472.470 + 724.565i
5.2 −7.97469 − 0.635864i −7.07752 + 26.0559i 63.1914 + 10.1416i 48.9289 − 84.7473i 73.0091 − 203.287i 6.45730 + 11.1844i −497.483 − 121.058i −628.817 − 368.822i −444.080 + 644.721i
5.3 −7.95659 + 0.832232i 24.0024 + 12.3647i 62.6148 − 13.2435i 0.173829 − 0.301081i −201.268 − 78.4053i 167.050 + 289.340i −487.179 + 157.483i 423.229 + 593.564i −1.13252 + 2.54024i
5.4 −7.93602 − 1.00978i −16.2129 − 21.5903i 61.9607 + 16.0272i 92.8213 − 160.771i 106.864 + 187.713i 288.471 + 499.646i −475.537 − 189.758i −203.285 + 700.083i −898.975 + 1182.15i
5.5 −7.83990 + 1.59247i −26.3328 + 5.96513i 58.9281 − 24.9696i −104.839 + 181.586i 196.947 − 88.7002i 128.782 + 223.056i −422.227 + 289.600i 657.834 − 314.158i 532.756 − 1590.57i
5.6 −7.83675 − 1.60790i 6.22898 + 26.2717i 58.8293 + 25.2014i −96.7705 + 167.611i −6.57280 − 215.900i −278.012 − 481.531i −420.509 − 292.089i −651.400 + 327.291i 1027.87 − 1157.93i
5.7 −7.59729 + 2.50624i −12.5996 − 23.8799i 51.4376 − 38.0812i −40.6546 + 70.4159i 155.572 + 149.845i −158.685 − 274.850i −295.345 + 418.229i −411.498 + 601.756i 132.386 − 636.860i
5.8 −7.59716 − 2.50663i 12.6861 − 23.8341i 51.4336 + 38.0866i −106.929 + 185.206i −156.122 + 149.272i 267.810 + 463.860i −295.280 − 418.275i −407.126 − 604.723i 1276.60 − 1139.01i
5.9 −7.46395 − 2.87915i −26.9713 + 1.24421i 47.4210 + 42.9797i −7.54460 + 13.0676i 204.895 + 68.3678i −156.472 − 271.018i −230.202 − 457.330i 725.904 − 67.1160i 93.9362 − 75.8140i
5.10 −7.32138 + 3.22450i −26.7644 + 3.55934i 43.2052 − 47.2156i 100.826 − 174.637i 184.475 − 112.361i −187.285 − 324.386i −164.075 + 484.998i 703.662 − 190.527i −175.073 + 1603.70i
5.11 −7.03719 + 3.80499i −10.4599 + 24.8916i 35.0441 − 53.5529i −8.24990 + 14.2892i −21.1036 − 214.967i 142.080 + 246.089i −42.8438 + 510.204i −510.180 − 520.728i 3.68567 − 131.947i
5.12 −6.81381 + 4.19189i 10.4599 − 24.8916i 28.8561 − 57.1255i 8.24990 − 14.2892i 33.0707 + 213.453i 142.080 + 246.089i 42.8438 + 510.204i −510.180 − 520.728i 3.68567 + 131.947i
5.13 −6.72146 − 4.33843i −13.8891 − 23.1537i 26.3561 + 58.3211i −24.0551 + 41.6647i −7.09581 + 215.883i −101.031 − 174.991i 75.8702 − 506.347i −343.188 + 643.166i 342.445 − 175.687i
5.14 −6.53010 − 4.62145i −21.8009 + 15.9287i 21.2844 + 60.3571i 17.6699 − 30.6052i 215.975 − 3.26399i 194.799 + 337.401i 139.948 − 492.502i 221.556 − 694.517i −256.827 + 118.194i
5.15 −6.45319 + 4.72825i 26.7644 − 3.55934i 19.2873 − 61.0246i −100.826 + 174.637i −155.886 + 149.518i −187.285 − 324.386i 164.075 + 484.998i 703.662 − 190.527i −175.073 − 1603.70i
5.16 −6.38046 − 4.82594i 27.0000 − 0.0492591i 17.4206 + 61.5835i −22.8039 + 39.4975i −172.510 − 129.986i −41.8302 − 72.4520i 186.046 − 477.002i 728.995 − 2.65999i 336.112 − 141.962i
5.17 −6.34800 − 4.86856i 15.8845 + 21.8331i 16.5943 + 61.8112i 122.026 − 211.355i 5.46045 − 215.931i −28.5227 − 49.4028i 195.591 − 473.168i −224.364 + 693.615i −1803.62 + 747.594i
5.18 −5.96911 + 5.32633i 12.5996 + 23.8799i 7.26052 − 63.5868i 40.6546 − 70.4159i −202.401 − 75.4318i −158.685 − 274.850i 295.345 + 418.229i −411.498 + 601.756i 132.386 + 636.860i
5.19 −5.29907 + 5.99332i 26.3328 − 5.96513i −7.83975 − 63.5180i 104.839 − 181.586i −103.788 + 189.431i 128.782 + 223.056i 422.227 + 289.600i 657.834 − 314.158i 532.756 + 1590.57i
5.20 −5.06626 − 6.19136i 1.22852 + 26.9720i −12.6659 + 62.7342i −68.8118 + 119.186i 160.770 − 144.254i 221.876 + 384.301i 452.579 − 239.408i −725.981 + 66.2712i 1086.54 − 177.787i
See next 80 embeddings (of 140 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.70
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
9.d odd 6 1 inner
72.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.7.j.a ✓ 140
8.b even 2 1 inner 72.7.j.a ✓ 140
9.d odd 6 1 inner 72.7.j.a ✓ 140
72.j odd 6 1 inner 72.7.j.a ✓ 140
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.7.j.a ✓ 140 1.a even 1 1 trivial
72.7.j.a ✓ 140 8.b even 2 1 inner
72.7.j.a ✓ 140 9.d odd 6 1 inner
72.7.j.a ✓ 140 72.j odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(72, [\chi])\).