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Properties

Label 136.4.h.a
Level $136$
Weight $4$
Character orbit 136.h
Analytic conductor $8.024$
Analytic rank $0$
Dimension $52$
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] = [136,4,Mod(101,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.101");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 136.h (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: \(8.02425976078\)
Analytic rank: \(0\)
Dimension: \(52\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 2 q^{2} + 10 q^{4} - 2 q^{8} + 428 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q - 2 q^{2} + 10 q^{4} - 2 q^{8} + 428 q^{9} - 232 q^{15} - 78 q^{16} - 28 q^{17} - 2 q^{18} + 1052 q^{25} + 448 q^{26} - 368 q^{30} + 958 q^{32} - 344 q^{33} - 198 q^{34} + 138 q^{36} - 524 q^{38} + 936 q^{47} - 1964 q^{49} - 1038 q^{50} - 1424 q^{52} - 1384 q^{55} + 2320 q^{60} - 2078 q^{64} - 1888 q^{66} - 874 q^{68} + 2472 q^{70} - 4010 q^{72} + 436 q^{76} + 1884 q^{81} - 2264 q^{84} - 1420 q^{86} + 1976 q^{87} - 224 q^{89} + 80 q^{94} + 5746 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 −2.76967 − 0.573530i −3.72680 7.34213 + 3.17697i 7.30821 10.3220 + 2.13743i 5.81849i −18.5132 − 13.0101i −13.1110 −20.2413 − 4.19147i
101.2 −2.76967 − 0.573530i 3.72680 7.34213 + 3.17697i −7.30821 −10.3220 − 2.13743i − 5.81849i −18.5132 − 13.0101i −13.1110 20.2413 + 4.19147i
101.3 −2.76967 + 0.573530i −3.72680 7.34213 − 3.17697i 7.30821 10.3220 − 2.13743i − 5.81849i −18.5132 + 13.0101i −13.1110 −20.2413 + 4.19147i
101.4 −2.76967 + 0.573530i 3.72680 7.34213 − 3.17697i −7.30821 −10.3220 + 2.13743i 5.81849i −18.5132 + 13.0101i −13.1110 20.2413 − 4.19147i
101.5 −2.64160 − 1.01092i −6.38011 5.95609 + 5.34088i −16.9716 16.8537 + 6.44977i 31.4494i −10.3344 − 20.1296i 13.7058 44.8322 + 17.1569i
101.6 −2.64160 − 1.01092i 6.38011 5.95609 + 5.34088i 16.9716 −16.8537 − 6.44977i − 31.4494i −10.3344 − 20.1296i 13.7058 −44.8322 − 17.1569i
101.7 −2.64160 + 1.01092i −6.38011 5.95609 − 5.34088i −16.9716 16.8537 − 6.44977i − 31.4494i −10.3344 + 20.1296i 13.7058 44.8322 − 17.1569i
101.8 −2.64160 + 1.01092i 6.38011 5.95609 − 5.34088i 16.9716 −16.8537 + 6.44977i 31.4494i −10.3344 + 20.1296i 13.7058 −44.8322 + 17.1569i
101.9 −2.37821 − 1.53106i −9.79402 3.31172 + 7.28234i 7.09681 23.2922 + 14.9952i − 25.9378i 3.27375 − 22.3893i 68.9229 −16.8777 − 10.8656i
101.10 −2.37821 − 1.53106i 9.79402 3.31172 + 7.28234i −7.09681 −23.2922 − 14.9952i 25.9378i 3.27375 − 22.3893i 68.9229 16.8777 + 10.8656i
101.11 −2.37821 + 1.53106i −9.79402 3.31172 − 7.28234i 7.09681 23.2922 − 14.9952i 25.9378i 3.27375 + 22.3893i 68.9229 −16.8777 + 10.8656i
101.12 −2.37821 + 1.53106i 9.79402 3.31172 − 7.28234i −7.09681 −23.2922 + 14.9952i − 25.9378i 3.27375 + 22.3893i 68.9229 16.8777 − 10.8656i
101.13 −1.98225 − 2.01760i −0.0741700 −0.141403 + 7.99875i 17.3656 0.147023 + 0.149645i 22.3324i 16.4186 − 15.5702i −26.9945 −34.4229 − 35.0368i
101.14 −1.98225 − 2.01760i 0.0741700 −0.141403 + 7.99875i −17.3656 −0.147023 − 0.149645i − 22.3324i 16.4186 − 15.5702i −26.9945 34.4229 + 35.0368i
101.15 −1.98225 + 2.01760i −0.0741700 −0.141403 − 7.99875i 17.3656 0.147023 − 0.149645i − 22.3324i 16.4186 + 15.5702i −26.9945 −34.4229 + 35.0368i
101.16 −1.98225 + 2.01760i 0.0741700 −0.141403 − 7.99875i −17.3656 −0.147023 + 0.149645i 22.3324i 16.4186 + 15.5702i −26.9945 34.4229 − 35.0368i
101.17 −1.64670 − 2.29965i −3.96792 −2.57674 + 7.57367i 5.58507 6.53398 + 9.12481i 3.60918i 21.6599 − 6.54599i −11.2556 −9.19694 − 12.8437i
101.18 −1.64670 − 2.29965i 3.96792 −2.57674 + 7.57367i −5.58507 −6.53398 − 9.12481i − 3.60918i 21.6599 − 6.54599i −11.2556 9.19694 + 12.8437i
101.19 −1.64670 + 2.29965i −3.96792 −2.57674 − 7.57367i 5.58507 6.53398 − 9.12481i − 3.60918i 21.6599 + 6.54599i −11.2556 −9.19694 + 12.8437i
101.20 −1.64670 + 2.29965i 3.96792 −2.57674 − 7.57367i −5.58507 −6.53398 + 9.12481i 3.60918i 21.6599 + 6.54599i −11.2556 9.19694 − 12.8437i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.52
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.4.h.a ✓ 52
4.b odd 2 1 544.4.h.a 52
8.b even 2 1 inner 136.4.h.a ✓ 52
8.d odd 2 1 544.4.h.a 52
17.b even 2 1 inner 136.4.h.a ✓ 52
68.d odd 2 1 544.4.h.a 52
136.e odd 2 1 544.4.h.a 52
136.h even 2 1 inner 136.4.h.a ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.h.a ✓ 52 1.a even 1 1 trivial
136.4.h.a ✓ 52 8.b even 2 1 inner
136.4.h.a ✓ 52 17.b even 2 1 inner
136.4.h.a ✓ 52 136.h even 2 1 inner
544.4.h.a 52 4.b odd 2 1
544.4.h.a 52 8.d odd 2 1
544.4.h.a 52 68.d odd 2 1
544.4.h.a 52 136.e odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(136, [\chi])\).