Properties

Label 828.2.u.c
Level $828$
Weight $2$
Character orbit 828.u
Analytic conductor $6.612$
Analytic rank $0$
Dimension $240$
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] = [828,2,Mod(19,828)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(828, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 0, 15]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("828.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 828 = 2^{2} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 828.u (of order \(22\), degree \(10\), 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: \(6.61161328736\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(24\) over \(\Q(\zeta_{22})\)
Twist minimal: no (minimal twist has level 276)
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 4 q^{2} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 4 q^{2} + 4 q^{8} - 8 q^{16} + 24 q^{25} + 40 q^{26} - 32 q^{29} - 36 q^{32} - 22 q^{34} + 110 q^{38} - 22 q^{40} + 16 q^{41} + 154 q^{44} - 88 q^{46} - 40 q^{49} + 142 q^{50} - 70 q^{52} + 110 q^{56} - 46 q^{58} - 40 q^{62} - 48 q^{64} - 72 q^{70} - 22 q^{74} + 110 q^{76} + 192 q^{77} - 198 q^{80} + 172 q^{82} - 200 q^{85} - 220 q^{86} + 176 q^{88} + 88 q^{89} - 154 q^{92} + 126 q^{94} - 88 q^{97} - 228 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.41418 + 0.0101703i 0 1.99979 0.0287651i 1.32696 2.06479i 0 0.0202392 + 0.140767i −2.82777 + 0.0610173i 0 −1.85556 + 2.93347i
19.2 −1.37328 + 0.337778i 0 1.77181 0.927729i −0.802068 + 1.24804i 0 0.333353 + 2.31852i −2.11983 + 1.87251i 0 0.679905 1.98484i
19.3 −1.36812 0.358110i 0 1.74351 + 0.979876i −0.606007 + 0.942966i 0 −0.442225 3.07574i −2.03444 1.96496i 0 1.16678 1.07308i
19.4 −1.31756 + 0.513838i 0 1.47194 1.35403i −0.897443 + 1.39645i 0 −0.549207 3.81982i −1.24362 + 2.54035i 0 0.464889 2.30105i
19.5 −1.12363 0.858753i 0 0.525087 + 1.92984i −0.628133 + 0.977395i 0 0.140631 + 0.978108i 1.06725 2.61935i 0 1.54513 0.558818i
19.6 −1.04496 + 0.952920i 0 0.183889 1.99153i −1.44457 + 2.24780i 0 0.629892 + 4.38099i 1.70561 + 2.25630i 0 −0.632448 3.72542i
19.7 −0.881142 + 1.10616i 0 −0.447177 1.94937i 0.505840 0.787103i 0 −0.334710 2.32796i 2.55034 + 1.22302i 0 0.424944 + 1.25309i
19.8 −0.631897 1.26519i 0 −1.20141 + 1.59894i 1.27480 1.98363i 0 0.0657938 + 0.457606i 2.78213 + 0.509647i 0 −3.31522 0.359416i
19.9 −0.569834 + 1.29433i 0 −1.35058 1.47511i 2.32731 3.62137i 0 0.331623 + 2.30649i 2.67888 0.907528i 0 3.36106 + 5.07589i
19.10 −0.542362 1.30608i 0 −1.41169 + 1.41674i 1.27480 1.98363i 0 −0.0657938 0.457606i 2.61601 + 1.07539i 0 −3.28219 0.589147i
19.11 −0.298132 + 1.38243i 0 −1.82223 0.824294i −0.197236 + 0.306906i 0 −0.608715 4.23371i 1.68280 2.27337i 0 −0.365474 0.364164i
19.12 0.000698111 1.41421i 0 −2.00000 + 0.00197456i −2.33778 + 3.63766i 0 0.00906423 + 0.0630431i −0.00418866 2.82842i 0 −5.14605 3.30358i
19.13 0.0868188 1.41155i 0 −1.98492 0.245097i −0.628133 + 0.977395i 0 −0.140631 0.978108i −0.518295 + 2.78053i 0 1.32510 + 0.971495i
19.14 0.186283 + 1.40189i 0 −1.93060 + 0.522296i 1.47832 2.30032i 0 0.464774 + 3.23258i −1.09184 2.60919i 0 3.50018 + 1.64394i
19.15 0.625288 1.26847i 0 −1.21803 1.58632i −0.606007 + 0.942966i 0 0.442225 + 3.07574i −2.77381 + 0.553129i 0 0.817195 + 1.35833i
19.16 0.933775 1.06210i 0 −0.256128 1.98353i 1.32696 2.06479i 0 −0.0202392 0.140767i −2.34588 1.58014i 0 −0.953939 3.33742i
19.17 0.937490 + 1.05883i 0 −0.242227 + 1.98528i 1.47832 2.30032i 0 −0.464774 3.23258i −2.32915 + 1.60470i 0 3.82155 0.591235i
19.18 1.06833 + 0.926640i 0 0.282675 + 1.97992i −2.33778 + 3.63766i 0 −0.00906423 0.0630431i −1.53269 + 2.37716i 0 −5.86833 + 1.71995i
19.19 1.15458 0.816661i 0 0.666131 1.88581i −0.802068 + 1.24804i 0 −0.333353 2.31852i −0.770961 2.72133i 0 0.0931716 + 2.09599i
19.20 1.24001 + 0.679987i 0 1.07524 + 1.68638i −0.197236 + 0.306906i 0 0.608715 + 4.23371i 0.186585 + 2.82227i 0 −0.453266 + 0.246447i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
23.d odd 22 1 inner
92.h even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 828.2.u.c 240
3.b odd 2 1 276.2.m.a 240
4.b odd 2 1 inner 828.2.u.c 240
12.b even 2 1 276.2.m.a 240
23.d odd 22 1 inner 828.2.u.c 240
69.g even 22 1 276.2.m.a 240
92.h even 22 1 inner 828.2.u.c 240
276.j odd 22 1 276.2.m.a 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.2.m.a 240 3.b odd 2 1
276.2.m.a 240 12.b even 2 1
276.2.m.a 240 69.g even 22 1
276.2.m.a 240 276.j odd 22 1
828.2.u.c 240 1.a even 1 1 trivial
828.2.u.c 240 4.b odd 2 1 inner
828.2.u.c 240 23.d odd 22 1 inner
828.2.u.c 240 92.h even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{120} - 36 T_{5}^{118} + 924 T_{5}^{116} + 44 T_{5}^{115} - 20280 T_{5}^{114} - 5368 T_{5}^{113} + 405027 T_{5}^{112} + 111276 T_{5}^{111} - 7122858 T_{5}^{110} - 2026992 T_{5}^{109} + 111162417 T_{5}^{108} + \cdots + 37\!\cdots\!29 \) acting on \(S_{2}^{\mathrm{new}}(828, [\chi])\). Copy content Toggle raw display