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Properties

Label 168.4.j.a
Level $168$
Weight $4$
Character orbit 168.j
Analytic conductor $9.912$
Analytic rank $0$
Dimension $72$
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] = [168,4,Mod(155,168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(168, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.155");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 168.j (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: \(9.91232088096\)
Analytic rank: \(0\)
Dimension: \(72\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 12 q^{4} + 30 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 12 q^{4} + 30 q^{6} + 12 q^{10} - 78 q^{12} + 180 q^{16} - 424 q^{18} + 48 q^{19} - 192 q^{22} - 354 q^{24} + 1800 q^{25} + 264 q^{27} - 168 q^{28} + 636 q^{30} + 232 q^{33} + 1272 q^{34} + 828 q^{36} + 1068 q^{40} - 70 q^{42} - 864 q^{43} - 1896 q^{46} - 2970 q^{48} - 3528 q^{49} - 3204 q^{52} + 1826 q^{54} - 344 q^{57} + 2616 q^{58} + 3620 q^{60} + 1524 q^{64} - 4568 q^{66} + 1632 q^{67} + 252 q^{70} - 2496 q^{72} + 3304 q^{75} - 5676 q^{76} + 2372 q^{78} + 920 q^{81} + 4200 q^{82} - 490 q^{84} + 7392 q^{88} - 9152 q^{90} - 7728 q^{94} - 4846 q^{96} - 5312 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1 −2.79941 − 0.404075i 4.65040 − 2.31814i 7.67345 + 2.26235i 0.943399 −13.9551 + 4.61033i − 7.00000i −20.5670 − 9.43390i 16.2524 − 21.5606i −2.64097 − 0.381204i
155.2 −2.79941 + 0.404075i 4.65040 + 2.31814i 7.67345 − 2.26235i 0.943399 −13.9551 − 4.61033i 7.00000i −20.5670 + 9.43390i 16.2524 + 21.5606i −2.64097 + 0.381204i
155.3 −2.78806 − 0.476159i −0.879587 + 5.12116i 7.54655 + 2.65512i −19.9344 4.89083 − 13.8593i 7.00000i −19.7760 − 10.9960i −25.4527 − 9.00902i 55.5783 + 9.49195i
155.4 −2.78806 + 0.476159i −0.879587 − 5.12116i 7.54655 − 2.65512i −19.9344 4.89083 + 13.8593i − 7.00000i −19.7760 + 10.9960i −25.4527 + 9.00902i 55.5783 − 9.49195i
155.5 −2.76688 − 0.586823i −5.19530 + 0.0938990i 7.31128 + 3.24734i −5.69382 14.4299 + 2.78891i 7.00000i −18.3238 − 13.2754i 26.9824 − 0.975668i 15.7541 + 3.34126i
155.6 −2.76688 + 0.586823i −5.19530 − 0.0938990i 7.31128 − 3.24734i −5.69382 14.4299 − 2.78891i − 7.00000i −18.3238 + 13.2754i 26.9824 + 0.975668i 15.7541 − 3.34126i
155.7 −2.75552 − 0.638067i −1.27909 − 5.03626i 7.18574 + 3.51641i 2.88024 0.311090 + 14.6936i 7.00000i −17.5567 − 14.2745i −23.7278 + 12.8837i −7.93656 − 1.83779i
155.8 −2.75552 + 0.638067i −1.27909 + 5.03626i 7.18574 − 3.51641i 2.88024 0.311090 − 14.6936i − 7.00000i −17.5567 + 14.2745i −23.7278 − 12.8837i −7.93656 + 1.83779i
155.9 −2.70573 − 0.824030i −4.29750 − 2.92087i 6.64195 + 4.45920i 19.2105 9.22100 + 11.4444i − 7.00000i −14.2968 − 17.5386i 9.93706 + 25.1049i −51.9785 − 15.8301i
155.10 −2.70573 + 0.824030i −4.29750 + 2.92087i 6.64195 − 4.45920i 19.2105 9.22100 − 11.4444i 7.00000i −14.2968 + 17.5386i 9.93706 − 25.1049i −51.9785 + 15.8301i
155.11 −2.35023 − 1.57366i 4.29823 + 2.91979i 3.04717 + 7.39694i −11.5502 −5.50707 − 13.6262i − 7.00000i 4.47874 − 22.1797i 9.94960 + 25.0999i 27.1457 + 18.1762i
155.12 −2.35023 + 1.57366i 4.29823 − 2.91979i 3.04717 − 7.39694i −11.5502 −5.50707 + 13.6262i 7.00000i 4.47874 + 22.1797i 9.94960 − 25.0999i 27.1457 − 18.1762i
155.13 −2.31370 − 1.62690i −3.47955 + 3.85911i 2.70640 + 7.52831i 3.79912 14.3290 − 3.26794i − 7.00000i 5.98600 − 21.8213i −2.78544 − 26.8559i −8.79002 − 6.18079i
155.14 −2.31370 + 1.62690i −3.47955 − 3.85911i 2.70640 − 7.52831i 3.79912 14.3290 + 3.26794i 7.00000i 5.98600 + 21.8213i −2.78544 + 26.8559i −8.79002 + 6.18079i
155.15 −2.29928 − 1.64722i 2.84075 − 4.35088i 2.57336 + 7.57481i −12.1896 −13.6985 + 5.32455i 7.00000i 6.56048 − 21.6555i −10.8603 − 24.7195i 28.0272 + 20.0788i
155.16 −2.29928 + 1.64722i 2.84075 + 4.35088i 2.57336 − 7.57481i −12.1896 −13.6985 − 5.32455i − 7.00000i 6.56048 + 21.6555i −10.8603 + 24.7195i 28.0272 − 20.0788i
155.17 −2.17240 − 1.81126i 5.13222 − 0.812590i 1.43867 + 7.86958i 20.6866 −12.6211 − 7.53052i 7.00000i 11.1285 − 19.7017i 25.6794 − 8.34079i −44.9396 − 37.4688i
155.18 −2.17240 + 1.81126i 5.13222 + 0.812590i 1.43867 − 7.86958i 20.6866 −12.6211 + 7.53052i − 7.00000i 11.1285 + 19.7017i 25.6794 + 8.34079i −44.9396 + 37.4688i
155.19 −1.77952 − 2.19848i −4.56313 + 2.48552i −1.66661 + 7.82448i 4.54875 13.5846 + 5.60890i 7.00000i 20.1677 − 10.2598i 14.6443 − 22.6836i −8.09460 − 10.0003i
155.20 −1.77952 + 2.19848i −4.56313 − 2.48552i −1.66661 − 7.82448i 4.54875 13.5846 − 5.60890i − 7.00000i 20.1677 + 10.2598i 14.6443 + 22.6836i −8.09460 + 10.0003i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 155.72
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.4.j.a ✓ 72
3.b odd 2 1 inner 168.4.j.a ✓ 72
4.b odd 2 1 672.4.j.a 72
8.b even 2 1 672.4.j.a 72
8.d odd 2 1 inner 168.4.j.a ✓ 72
12.b even 2 1 672.4.j.a 72
24.f even 2 1 inner 168.4.j.a ✓ 72
24.h odd 2 1 672.4.j.a 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.j.a ✓ 72 1.a even 1 1 trivial
168.4.j.a ✓ 72 3.b odd 2 1 inner
168.4.j.a ✓ 72 8.d odd 2 1 inner
168.4.j.a ✓ 72 24.f even 2 1 inner
672.4.j.a 72 4.b odd 2 1
672.4.j.a 72 8.b even 2 1
672.4.j.a 72 12.b even 2 1
672.4.j.a 72 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(168, [\chi])\).