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Properties

Label 360.6.k.c
Level $360$
Weight $6$
Character orbit 360.k
Analytic conductor $57.738$
Analytic rank $0$
Dimension $22$
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] = [360,6,Mod(181,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.181");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 360.k (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: \(57.7381751327\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: no (minimal twist has level 120)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 2 q^{2} - 34 q^{4} + 196 q^{7} + 248 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 2 q^{2} - 34 q^{4} + 196 q^{7} + 248 q^{8} - 100 q^{10} - 392 q^{14} - 3254 q^{16} - 3468 q^{17} - 1900 q^{20} - 3060 q^{22} + 1672 q^{23} - 13750 q^{25} + 756 q^{26} + 4708 q^{28} + 12724 q^{31} + 10612 q^{32} - 23964 q^{34} - 35848 q^{38} - 4250 q^{40} + 33620 q^{41} + 16708 q^{44} + 32140 q^{46} + 21208 q^{47} + 105870 q^{49} - 1250 q^{50} - 32392 q^{52} + 24200 q^{55} + 13180 q^{56} + 22440 q^{58} + 140084 q^{62} + 93878 q^{64} + 135504 q^{68} - 62800 q^{70} - 200312 q^{71} - 96644 q^{73} - 281548 q^{74} - 139348 q^{76} - 223196 q^{79} - 32400 q^{80} + 226452 q^{82} + 574912 q^{86} + 315788 q^{88} + 221788 q^{89} + 169176 q^{92} - 207164 q^{94} - 35800 q^{95} + 189764 q^{97} - 325446 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1 −5.55206 − 1.08382i 0 29.6507 + 12.0349i 25.0000i 0 142.727 −151.578 − 98.9544i 0 27.0955 − 138.801i
181.2 −5.55206 + 1.08382i 0 29.6507 − 12.0349i − 25.0000i 0 142.727 −151.578 + 98.9544i 0 27.0955 + 138.801i
181.3 −4.37610 − 3.58465i 0 6.30053 + 31.3736i 25.0000i 0 −142.542 84.8917 − 159.879i 0 89.6163 − 109.403i
181.4 −4.37610 + 3.58465i 0 6.30053 − 31.3736i − 25.0000i 0 −142.542 84.8917 + 159.879i 0 89.6163 + 109.403i
181.5 −4.06587 − 3.93302i 0 1.06266 + 31.9824i − 25.0000i 0 −136.900 121.467 − 134.216i 0 −98.3256 + 101.647i
181.6 −4.06587 + 3.93302i 0 1.06266 − 31.9824i 25.0000i 0 −136.900 121.467 + 134.216i 0 −98.3256 − 101.647i
181.7 −2.96961 − 4.81471i 0 −14.3628 + 28.5956i − 25.0000i 0 210.071 180.332 − 15.7648i 0 −120.368 + 74.2402i
181.8 −2.96961 + 4.81471i 0 −14.3628 − 28.5956i 25.0000i 0 210.071 180.332 + 15.7648i 0 −120.368 − 74.2402i
181.9 −2.22082 − 5.20269i 0 −22.1359 + 23.1085i 25.0000i 0 66.4714 169.386 + 63.8460i 0 130.067 − 55.5206i
181.10 −2.22082 + 5.20269i 0 −22.1359 − 23.1085i − 25.0000i 0 66.4714 169.386 − 63.8460i 0 130.067 + 55.5206i
181.11 0.692125 − 5.61435i 0 −31.0419 − 7.77167i 25.0000i 0 −200.556 −65.1178 + 168.901i 0 140.359 + 17.3031i
181.12 0.692125 + 5.61435i 0 −31.0419 + 7.77167i − 25.0000i 0 −200.556 −65.1178 − 168.901i 0 140.359 − 17.3031i
181.13 1.43965 − 5.47059i 0 −27.8548 − 15.7515i − 25.0000i 0 188.016 −126.271 + 129.706i 0 −136.765 − 35.9913i
181.14 1.43965 + 5.47059i 0 −27.8548 + 15.7515i 25.0000i 0 188.016 −126.271 − 129.706i 0 −136.765 + 35.9913i
181.15 3.29517 − 4.59803i 0 −10.2837 − 30.3026i − 25.0000i 0 −187.086 −173.219 − 52.5671i 0 −114.951 − 82.3792i
181.16 3.29517 + 4.59803i 0 −10.2837 + 30.3026i 25.0000i 0 −187.086 −173.219 + 52.5671i 0 −114.951 + 82.3792i
181.17 4.09104 − 3.90683i 0 1.47329 − 31.9661i 25.0000i 0 138.004 −118.859 − 136.531i 0 97.6709 + 102.276i
181.18 4.09104 + 3.90683i 0 1.47329 + 31.9661i − 25.0000i 0 138.004 −118.859 + 136.531i 0 97.6709 − 102.276i
181.19 5.00985 − 2.62705i 0 18.1973 − 26.3222i − 25.0000i 0 6.87318 22.0159 − 179.676i 0 −65.6761 − 125.246i
181.20 5.00985 + 2.62705i 0 18.1973 + 26.3222i 25.0000i 0 6.87318 22.0159 + 179.676i 0 −65.6761 + 125.246i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.22
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 360.6.k.c 22
3.b odd 2 1 120.6.k.b ✓ 22
8.b even 2 1 inner 360.6.k.c 22
12.b even 2 1 480.6.k.b 22
24.f even 2 1 480.6.k.b 22
24.h odd 2 1 120.6.k.b ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.6.k.b ✓ 22 3.b odd 2 1
120.6.k.b ✓ 22 24.h odd 2 1
360.6.k.c 22 1.a even 1 1 trivial
360.6.k.c 22 8.b even 2 1 inner
480.6.k.b 22 12.b even 2 1
480.6.k.b 22 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{11} - 98 T_{7}^{10} - 114104 T_{7}^{9} + 10938688 T_{7}^{8} + 4656454032 T_{7}^{7} + \cdots - 33\!\cdots\!76 \) acting on \(S_{6}^{\mathrm{new}}(360, [\chi])\). Copy content Toggle raw display