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Properties

Label 136.4.c.a
Level $136$
Weight $4$
Character orbit 136.c
Analytic conductor $8.024$
Analytic rank $0$
Dimension $24$
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] = [136,4,Mod(69,136)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.69");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 136.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: \(8.02425976078\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + q^{2} + 5 q^{4} - 70 q^{6} - 56 q^{7} - 95 q^{8} - 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + q^{2} + 5 q^{4} - 70 q^{6} - 56 q^{7} - 95 q^{8} - 216 q^{9} + 82 q^{10} + 72 q^{12} - 22 q^{14} + 180 q^{15} - 119 q^{16} - 408 q^{17} - 33 q^{18} - 114 q^{20} + 262 q^{22} - 20 q^{23} - 72 q^{24} - 600 q^{25} - 432 q^{26} - 396 q^{28} + 544 q^{30} + 372 q^{31} + 261 q^{32} + 280 q^{33} - 17 q^{34} - 349 q^{36} + 846 q^{38} - 12 q^{39} + 378 q^{40} - 1300 q^{42} - 1396 q^{44} + 1434 q^{46} + 548 q^{47} + 1004 q^{48} + 1384 q^{49} - 859 q^{50} - 1404 q^{52} + 1408 q^{54} - 1396 q^{55} + 3376 q^{56} - 168 q^{57} - 1250 q^{58} - 3008 q^{60} + 2038 q^{62} + 1260 q^{63} + 3257 q^{64} + 72 q^{65} - 2392 q^{66} - 85 q^{68} + 2228 q^{70} - 1024 q^{71} + 3811 q^{72} + 216 q^{73} - 3102 q^{74} - 4362 q^{76} + 2720 q^{78} + 2196 q^{79} + 6298 q^{80} + 3080 q^{81} - 2562 q^{82} - 2228 q^{84} + 3114 q^{86} - 5108 q^{87} + 2516 q^{88} - 424 q^{89} - 8070 q^{90} - 4080 q^{92} + 4012 q^{94} + 3312 q^{95} + 4716 q^{96} - 1768 q^{97} - 3447 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1 −2.82610 − 0.114660i − 1.94034i 7.97371 + 0.648080i − 19.3262i −0.222478 + 5.48359i −22.3787 −22.4602 − 2.74580i 23.2351 −2.21593 + 54.6177i
69.2 −2.82610 + 0.114660i 1.94034i 7.97371 − 0.648080i 19.3262i −0.222478 − 5.48359i −22.3787 −22.4602 + 2.74580i 23.2351 −2.21593 − 54.6177i
69.3 −2.64179 − 1.01043i − 10.1194i 5.95806 + 5.33868i − 4.18536i −10.2249 + 26.7332i −4.35411 −10.3455 − 20.1239i −75.4018 −4.22902 + 11.0568i
69.4 −2.64179 + 1.01043i 10.1194i 5.95806 − 5.33868i 4.18536i −10.2249 − 26.7332i −4.35411 −10.3455 + 20.1239i −75.4018 −4.22902 − 11.0568i
69.5 −2.45407 − 1.40626i 5.22773i 4.04488 + 6.90210i 5.37073i 7.35154 − 12.8292i 20.5579 −0.220272 − 22.6263i −0.329183 7.55263 − 13.1801i
69.6 −2.45407 + 1.40626i − 5.22773i 4.04488 − 6.90210i − 5.37073i 7.35154 + 12.8292i 20.5579 −0.220272 + 22.6263i −0.329183 7.55263 + 13.1801i
69.7 −1.80554 − 2.17716i − 1.55260i −1.48002 + 7.86190i − 2.20330i −3.38025 + 2.80328i −27.5631 19.7888 − 10.9728i 24.5894 −4.79692 + 3.97815i
69.8 −1.80554 + 2.17716i 1.55260i −1.48002 − 7.86190i 2.20330i −3.38025 − 2.80328i −27.5631 19.7888 + 10.9728i 24.5894 −4.79692 − 3.97815i
69.9 −1.04672 − 2.62762i − 7.45512i −5.80876 + 5.50075i 18.6150i −19.5892 + 7.80341i 27.6565 20.5340 + 9.50549i −28.5788 48.9132 − 19.4847i
69.10 −1.04672 + 2.62762i 7.45512i −5.80876 − 5.50075i − 18.6150i −19.5892 − 7.80341i 27.6565 20.5340 − 9.50549i −28.5788 48.9132 + 19.4847i
69.11 −0.176003 − 2.82295i 1.13231i −7.93805 + 0.993692i 2.24109i 3.19646 − 0.199290i 16.3901 4.20225 + 22.2338i 25.7179 6.32649 − 0.394438i
69.12 −0.176003 + 2.82295i − 1.13231i −7.93805 − 0.993692i − 2.24109i 3.19646 + 0.199290i 16.3901 4.20225 − 22.2338i 25.7179 6.32649 + 0.394438i
69.13 1.33006 − 2.49618i 7.46468i −4.46188 − 6.64016i − 2.41867i 18.6332 + 9.92848i −32.4110 −22.5096 + 2.30585i −28.7215 −6.03745 − 3.21698i
69.14 1.33006 + 2.49618i − 7.46468i −4.46188 + 6.64016i 2.41867i 18.6332 − 9.92848i −32.4110 −22.5096 − 2.30585i −28.7215 −6.03745 + 3.21698i
69.15 1.44576 − 2.43100i − 6.32164i −3.81955 − 7.02930i − 5.83445i −15.3679 − 9.13957i 19.4068 −22.6104 − 0.877336i −12.9631 −14.1836 − 8.43522i
69.16 1.44576 + 2.43100i 6.32164i −3.81955 + 7.02930i 5.83445i −15.3679 + 9.13957i 19.4068 −22.6104 + 0.877336i −12.9631 −14.1836 + 8.43522i
69.17 1.45603 − 2.42486i − 2.03266i −3.75992 − 7.06137i − 17.9724i −4.92893 − 2.95963i −7.12296 −22.5974 − 1.16430i 22.8683 −43.5806 − 26.1684i
69.18 1.45603 + 2.42486i 2.03266i −3.75992 + 7.06137i 17.9724i −4.92893 + 2.95963i −7.12296 −22.5974 + 1.16430i 22.8683 −43.5806 + 26.1684i
69.19 1.83440 − 2.15290i − 8.83681i −1.26995 − 7.89856i 17.9402i −19.0248 − 16.2102i −22.9766 −19.3344 − 11.7551i −51.0892 38.6235 + 32.9096i
69.20 1.83440 + 2.15290i 8.83681i −1.26995 + 7.89856i − 17.9402i −19.0248 + 16.2102i −22.9766 −19.3344 + 11.7551i −51.0892 38.6235 − 32.9096i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 69.24
Significant digits:
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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 136.4.c.a ✓ 24
4.b odd 2 1 544.4.c.b 24
8.b even 2 1 inner 136.4.c.a ✓ 24
8.d odd 2 1 544.4.c.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.c.a ✓ 24 1.a even 1 1 trivial
136.4.c.a ✓ 24 8.b even 2 1 inner
544.4.c.b 24 4.b odd 2 1
544.4.c.b 24 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 432 T_{3}^{22} + 79420 T_{3}^{20} + 8137268 T_{3}^{18} + 510601712 T_{3}^{16} + \cdots + 739617194377216 \) acting on \(S_{4}^{\mathrm{new}}(136, [\chi])\). Copy content Toggle raw display