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Properties

Label 152.6.b.b
Level $152$
Weight $6$
Character orbit 152.b
Analytic conductor $24.378$
Analytic rank $0$
Dimension $96$
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] = [152,6,Mod(75,152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(152, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("152.75");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 152.b (of order \(2\), degree \(1\), 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: \(24.3783406116\)
Analytic rank: \(0\)
Dimension: \(96\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 82 q^{4} - 110 q^{6} - 6168 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 82 q^{4} - 110 q^{6} - 6168 q^{9} + 944 q^{11} - 3270 q^{16} - 3832 q^{17} + 5240 q^{19} - 1356 q^{20} + 12622 q^{24} - 62504 q^{25} - 6 q^{26} - 110 q^{28} - 14656 q^{30} + 7720 q^{35} - 90984 q^{36} + 22390 q^{38} - 35306 q^{42} + 45096 q^{43} - 22100 q^{44} - 210840 q^{49} + 55038 q^{54} - 36336 q^{57} + 64402 q^{58} + 132976 q^{62} - 182030 q^{64} + 101048 q^{66} - 9030 q^{68} - 4336 q^{73} + 80748 q^{74} - 104304 q^{76} - 31392 q^{80} - 20624 q^{81} + 145904 q^{82} + 52152 q^{83} - 12226 q^{92} - 80058 q^{96} - 752768 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
75.1 −5.63325 − 0.516227i − 19.3062i 31.4670 + 5.81607i − 63.3569i −9.96640 + 108.757i − 60.7576i −174.259 − 49.0075i −129.731 −32.7066 + 356.906i
75.2 −5.63325 + 0.516227i 19.3062i 31.4670 − 5.81607i 63.3569i −9.96640 − 108.757i 60.7576i −174.259 + 49.0075i −129.731 −32.7066 − 356.906i
75.3 −5.61333 − 0.700358i − 4.49700i 31.0190 + 7.86269i 93.3850i −3.14951 + 25.2431i − 95.0367i −168.613 − 65.8603i 222.777 65.4030 − 524.201i
75.4 −5.61333 + 0.700358i 4.49700i 31.0190 − 7.86269i − 93.3850i −3.14951 − 25.2431i 95.0367i −168.613 + 65.8603i 222.777 65.4030 + 524.201i
75.5 −5.51636 − 1.25289i 8.00180i 28.8605 + 13.8228i − 16.6238i 10.0254 − 44.1408i − 226.892i −141.887 − 112.411i 178.971 −20.8278 + 91.7031i
75.6 −5.51636 + 1.25289i − 8.00180i 28.8605 − 13.8228i 16.6238i 10.0254 + 44.1408i 226.892i −141.887 + 112.411i 178.971 −20.8278 − 91.7031i
75.7 −5.47893 − 1.40761i − 26.7990i 28.0372 + 15.4244i − 21.0026i −37.7227 + 146.830i 145.610i −131.902 − 123.975i −475.186 −29.5636 + 115.072i
75.8 −5.47893 + 1.40761i 26.7990i 28.0372 − 15.4244i 21.0026i −37.7227 − 146.830i − 145.610i −131.902 + 123.975i −475.186 −29.5636 − 115.072i
75.9 −5.44850 − 1.52113i − 19.2136i 27.3724 + 16.5757i 61.7747i −29.2264 + 104.686i 34.9784i −123.925 − 131.950i −126.164 93.9670 − 336.579i
75.10 −5.44850 + 1.52113i 19.2136i 27.3724 − 16.5757i − 61.7747i −29.2264 − 104.686i − 34.9784i −123.925 + 131.950i −126.164 93.9670 + 336.579i
75.11 −5.42318 − 1.60907i 14.2121i 26.8218 + 17.4525i − 9.24558i 22.8682 − 77.0746i 124.323i −117.377 − 137.806i 41.0170 −14.8768 + 50.1404i
75.12 −5.42318 + 1.60907i − 14.2121i 26.8218 − 17.4525i 9.24558i 22.8682 + 77.0746i − 124.323i −117.377 + 137.806i 41.0170 −14.8768 − 50.1404i
75.13 −5.05036 − 2.54831i 27.1983i 19.0122 + 25.7398i 36.9812i 69.3098 − 137.361i 150.607i −30.4253 − 178.444i −496.748 94.2398 − 186.768i
75.14 −5.05036 + 2.54831i − 27.1983i 19.0122 − 25.7398i − 36.9812i 69.3098 + 137.361i − 150.607i −30.4253 + 178.444i −496.748 94.2398 + 186.768i
75.15 −4.93864 − 2.75858i 1.79162i 16.7804 + 27.2473i − 26.9679i 4.94234 − 8.84819i 21.5599i −7.70850 − 180.855i 239.790 −74.3932 + 133.185i
75.16 −4.93864 + 2.75858i − 1.79162i 16.7804 − 27.2473i 26.9679i 4.94234 + 8.84819i − 21.5599i −7.70850 + 180.855i 239.790 −74.3932 − 133.185i
75.17 −4.93566 − 2.76392i 28.5036i 16.7215 + 27.2836i − 107.944i 78.7817 − 140.684i − 87.6251i −7.12184 − 180.879i −569.454 −298.348 + 532.773i
75.18 −4.93566 + 2.76392i − 28.5036i 16.7215 − 27.2836i 107.944i 78.7817 + 140.684i 87.6251i −7.12184 + 180.879i −569.454 −298.348 − 532.773i
75.19 −4.33851 − 3.63006i − 26.7852i 5.64529 + 31.4981i 46.5237i −97.2319 + 116.208i − 208.957i 89.8480 − 157.148i −474.446 168.884 − 201.844i
75.20 −4.33851 + 3.63006i 26.7852i 5.64529 − 31.4981i − 46.5237i −97.2319 − 116.208i 208.957i 89.8480 + 157.148i −474.446 168.884 + 201.844i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 75.96
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
19.b odd 2 1 inner
152.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 152.6.b.b ✓ 96
4.b odd 2 1 608.6.b.b 96
8.b even 2 1 608.6.b.b 96
8.d odd 2 1 inner 152.6.b.b ✓ 96
19.b odd 2 1 inner 152.6.b.b ✓ 96
76.d even 2 1 608.6.b.b 96
152.b even 2 1 inner 152.6.b.b ✓ 96
152.g odd 2 1 608.6.b.b 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.6.b.b ✓ 96 1.a even 1 1 trivial
152.6.b.b ✓ 96 8.d odd 2 1 inner
152.6.b.b ✓ 96 19.b odd 2 1 inner
152.6.b.b ✓ 96 152.b even 2 1 inner
608.6.b.b 96 4.b odd 2 1
608.6.b.b 96 8.b even 2 1
608.6.b.b 96 76.d even 2 1
608.6.b.b 96 152.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} + 7374 T_{3}^{46} + 25211281 T_{3}^{44} + 53082818500 T_{3}^{42} + 77113215958634 T_{3}^{40} + \cdots + 11\!\cdots\!84 \) acting on \(S_{6}^{\mathrm{new}}(152, [\chi])\). Copy content Toggle raw display