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Properties

Label 360.6.k.d
Level $360$
Weight $6$
Character orbit 360.k
Analytic conductor $57.738$
Analytic rank $0$
Dimension $40$
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] = [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: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 34 q^{4} - 392 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 34 q^{4} - 392 q^{7} + 50 q^{10} - 686 q^{16} - 396 q^{22} - 25000 q^{25} + 6748 q^{28} - 7160 q^{31} - 11352 q^{34} - 1550 q^{40} - 66068 q^{46} + 125304 q^{49} + 94928 q^{52} - 48400 q^{55} - 60996 q^{58} - 131698 q^{64} + 69500 q^{70} + 105136 q^{73} + 477524 q^{76} - 166136 q^{79} + 250920 q^{82} - 289564 q^{88} - 487268 q^{94} - 147376 q^{97}+O(q^{100}) \) 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 −5.56095 − 1.03724i 0 29.8483 + 11.5361i 25.0000i 0 135.790 −154.019 − 95.1115i 0 25.9311 − 139.024i
181.2 −5.56095 + 1.03724i 0 29.8483 − 11.5361i − 25.0000i 0 135.790 −154.019 + 95.1115i 0 25.9311 + 139.024i
181.3 −5.52229 − 1.22652i 0 28.9913 + 13.5464i − 25.0000i 0 −179.406 −143.483 − 110.366i 0 −30.6631 + 138.057i
181.4 −5.52229 + 1.22652i 0 28.9913 − 13.5464i 25.0000i 0 −179.406 −143.483 + 110.366i 0 −30.6631 − 138.057i
181.5 −5.03482 − 2.57887i 0 18.6989 + 25.9683i − 25.0000i 0 −9.61505 −27.1766 − 178.968i 0 −64.4717 + 125.871i
181.6 −5.03482 + 2.57887i 0 18.6989 − 25.9683i 25.0000i 0 −9.61505 −27.1766 + 178.968i 0 −64.4717 − 125.871i
181.7 −4.80099 − 2.99174i 0 14.0990 + 28.7266i 25.0000i 0 −172.452 18.2532 − 180.097i 0 74.7934 − 120.025i
181.8 −4.80099 + 2.99174i 0 14.0990 − 28.7266i − 25.0000i 0 −172.452 18.2532 + 180.097i 0 74.7934 + 120.025i
181.9 −4.30369 − 3.67127i 0 5.04355 + 31.6000i − 25.0000i 0 121.553 94.3064 − 154.513i 0 −91.7818 + 107.592i
181.10 −4.30369 + 3.67127i 0 5.04355 − 31.6000i 25.0000i 0 121.553 94.3064 + 154.513i 0 −91.7818 − 107.592i
181.11 −3.78360 − 4.20528i 0 −3.36881 + 31.8222i 25.0000i 0 222.362 146.568 − 106.235i 0 105.132 − 94.5899i
181.12 −3.78360 + 4.20528i 0 −3.36881 − 31.8222i − 25.0000i 0 222.362 146.568 + 106.235i 0 105.132 + 94.5899i
181.13 −2.78662 − 4.92288i 0 −16.4695 + 27.4364i 25.0000i 0 −32.0839 180.960 + 4.62293i 0 123.072 − 69.6654i
181.14 −2.78662 + 4.92288i 0 −16.4695 − 27.4364i − 25.0000i 0 −32.0839 180.960 − 4.62293i 0 123.072 + 69.6654i
181.15 −1.67970 − 5.40172i 0 −26.3572 + 18.1466i − 25.0000i 0 −214.441 142.295 + 111.894i 0 −135.043 + 41.9925i
181.16 −1.67970 + 5.40172i 0 −26.3572 − 18.1466i 25.0000i 0 −214.441 142.295 − 111.894i 0 −135.043 − 41.9925i
181.17 −1.57486 − 5.43321i 0 −27.0396 + 17.1131i − 25.0000i 0 75.3357 135.563 + 119.962i 0 −135.830 + 39.3715i
181.18 −1.57486 + 5.43321i 0 −27.0396 − 17.1131i 25.0000i 0 75.3357 135.563 − 119.962i 0 −135.830 − 39.3715i
181.19 −0.164688 − 5.65446i 0 −31.9458 + 1.86245i 25.0000i 0 −45.0423 15.7922 + 180.329i 0 141.361 − 4.11721i
181.20 −0.164688 + 5.65446i 0 −31.9458 − 1.86245i − 25.0000i 0 −45.0423 15.7922 − 180.329i 0 141.361 + 4.11721i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 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.d ✓ 40
3.b odd 2 1 inner 360.6.k.d ✓ 40
8.b even 2 1 inner 360.6.k.d ✓ 40
24.h odd 2 1 inner 360.6.k.d ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.6.k.d ✓ 40 1.a even 1 1 trivial
360.6.k.d ✓ 40 3.b odd 2 1 inner
360.6.k.d ✓ 40 8.b even 2 1 inner
360.6.k.d ✓ 40 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} + 98 T_{7}^{9} - 94896 T_{7}^{8} - 7891024 T_{7}^{7} + 2894868560 T_{7}^{6} + \cdots + 25\!\cdots\!00 \) acting on \(S_{6}^{\mathrm{new}}(360, [\chi])\). Copy content Toggle raw display