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Properties

Label 152.4.c.a
Level $152$
Weight $4$
Character orbit 152.c
Analytic conductor $8.968$
Analytic rank $0$
Dimension $54$
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] = [152,4,Mod(77,152)]
 
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([0, 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("152.77");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 152.c (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: \(8.96829032087\)
Analytic rank: \(0\)
Dimension: \(54\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 54 q + 2 q^{2} + 10 q^{4} - 30 q^{6} + 28 q^{7} - 40 q^{8} - 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 54 q + 2 q^{2} + 10 q^{4} - 30 q^{6} + 28 q^{7} - 40 q^{8} - 486 q^{9} + 72 q^{10} + 164 q^{12} + 16 q^{14} - 166 q^{16} + 52 q^{17} - 206 q^{18} + 40 q^{20} + 52 q^{22} + 604 q^{23} + 246 q^{24} - 1306 q^{25} - 650 q^{26} + 186 q^{28} + 204 q^{30} + 772 q^{32} - 700 q^{34} - 264 q^{36} + 1224 q^{39} - 1284 q^{40} + 236 q^{41} - 478 q^{42} + 1172 q^{44} + 188 q^{46} - 940 q^{47} - 1420 q^{48} + 2646 q^{49} - 1154 q^{50} + 1968 q^{52} - 1606 q^{54} + 1168 q^{55} + 872 q^{56} + 1190 q^{58} + 1876 q^{60} + 1968 q^{62} - 1300 q^{63} - 962 q^{64} - 1744 q^{65} + 488 q^{66} - 2226 q^{68} + 176 q^{70} - 1704 q^{71} + 284 q^{72} + 740 q^{73} - 896 q^{74} - 1768 q^{78} + 600 q^{79} + 2468 q^{80} + 5606 q^{81} + 2736 q^{82} - 7312 q^{84} + 1752 q^{86} - 1968 q^{87} + 1848 q^{88} - 1068 q^{89} + 5332 q^{90} - 490 q^{92} + 3648 q^{94} - 1520 q^{95} + 2538 q^{96} - 2396 q^{97} + 3338 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1 −2.82663 − 0.100834i − 2.68502i 7.97966 + 0.570042i − 12.4827i −0.270743 + 7.58957i 20.3829 −22.4981 − 2.41592i 19.7906 −1.25868 + 35.2839i
77.2 −2.82663 + 0.100834i 2.68502i 7.97966 − 0.570042i 12.4827i −0.270743 − 7.58957i 20.3829 −22.4981 + 2.41592i 19.7906 −1.25868 − 35.2839i
77.3 −2.80727 − 0.345292i 7.95893i 7.76155 + 1.93866i 3.66598i 2.74816 − 22.3429i −26.4722 −21.1194 − 8.12234i −36.3446 1.26583 − 10.2914i
77.4 −2.80727 + 0.345292i − 7.95893i 7.76155 − 1.93866i − 3.66598i 2.74816 + 22.3429i −26.4722 −21.1194 + 8.12234i −36.3446 1.26583 + 10.2914i
77.5 −2.59795 − 1.11832i − 6.93161i 5.49871 + 5.81069i 14.8732i −7.75177 + 18.0080i 4.64366 −7.78717 − 21.2452i −21.0472 16.6331 − 38.6400i
77.6 −2.59795 + 1.11832i 6.93161i 5.49871 − 5.81069i − 14.8732i −7.75177 − 18.0080i 4.64366 −7.78717 + 21.2452i −21.0472 16.6331 + 38.6400i
77.7 −2.56639 − 1.18897i 1.07093i 5.17270 + 6.10272i − 1.57093i 1.27330 − 2.74842i −11.0483 −6.01922 − 21.8121i 25.8531 −1.86779 + 4.03162i
77.8 −2.56639 + 1.18897i − 1.07093i 5.17270 − 6.10272i 1.57093i 1.27330 + 2.74842i −11.0483 −6.01922 + 21.8121i 25.8531 −1.86779 − 4.03162i
77.9 −2.40624 − 1.48661i 8.81566i 3.57997 + 7.15428i − 13.4319i 13.1055 − 21.2126i 29.7009 2.02136 − 22.5369i −50.7158 −19.9680 + 32.3204i
77.10 −2.40624 + 1.48661i − 8.81566i 3.57997 − 7.15428i 13.4319i 13.1055 + 21.2126i 29.7009 2.02136 + 22.5369i −50.7158 −19.9680 − 32.3204i
77.11 −2.30161 − 1.64396i 0.0723755i 2.59482 + 7.56749i − 17.7875i 0.118982 − 0.166580i −16.6344 6.46834 − 21.6832i 26.9948 −29.2419 + 40.9400i
77.12 −2.30161 + 1.64396i − 0.0723755i 2.59482 − 7.56749i 17.7875i 0.118982 + 0.166580i −16.6344 6.46834 + 21.6832i 26.9948 −29.2419 − 40.9400i
77.13 −2.23369 − 1.73512i − 5.63582i 1.97875 + 7.75142i 5.02492i −9.77879 + 12.5887i 4.57520 9.02969 − 20.7476i −4.76245 8.71882 − 11.2241i
77.14 −2.23369 + 1.73512i 5.63582i 1.97875 − 7.75142i − 5.02492i −9.77879 − 12.5887i 4.57520 9.02969 + 20.7476i −4.76245 8.71882 + 11.2241i
77.15 −1.91742 − 2.07930i 5.84368i −0.646977 + 7.97380i 15.7567i 12.1508 − 11.2048i 10.6999 17.8204 − 13.9439i −7.14861 32.7630 − 30.2123i
77.16 −1.91742 + 2.07930i − 5.84368i −0.646977 − 7.97380i − 15.7567i 12.1508 + 11.2048i 10.6999 17.8204 + 13.9439i −7.14861 32.7630 + 30.2123i
77.17 −1.68684 − 2.27037i − 9.27176i −2.30915 + 7.65949i − 17.0402i −21.0503 + 15.6400i −2.27120 21.2850 − 7.67770i −58.9655 −38.6876 + 28.7441i
77.18 −1.68684 + 2.27037i 9.27176i −2.30915 − 7.65949i 17.0402i −21.0503 − 15.6400i −2.27120 21.2850 + 7.67770i −58.9655 −38.6876 − 28.7441i
77.19 −1.27962 − 2.52241i − 0.0163393i −4.72515 + 6.45546i 11.6827i −0.0412144 + 0.0209080i −31.5071 22.3297 + 3.65828i 26.9997 29.4686 − 14.9494i
77.20 −1.27962 + 2.52241i 0.0163393i −4.72515 − 6.45546i − 11.6827i −0.0412144 − 0.0209080i −31.5071 22.3297 − 3.65828i 26.9997 29.4686 + 14.9494i
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 77.54
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 152.4.c.a ✓ 54
4.b odd 2 1 608.4.c.a 54
8.b even 2 1 inner 152.4.c.a ✓ 54
8.d odd 2 1 608.4.c.a 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.4.c.a ✓ 54 1.a even 1 1 trivial
152.4.c.a ✓ 54 8.b even 2 1 inner
608.4.c.a 54 4.b odd 2 1
608.4.c.a 54 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(152, [\chi])\).