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Properties

Label 825.2.cm.a
Level $825$
Weight $2$
Character orbit 825.cm
Analytic conductor $6.588$
Analytic rank $0$
Dimension $480$
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] = [825,2,Mod(13,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 19, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.cm (of order \(20\), degree \(8\), 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: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(480\)
Relative dimension: \(60\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 480 q + 12 q^{5} + 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 480 q + 12 q^{5} + 20 q^{7} + 40 q^{10} - 16 q^{12} - 12 q^{15} + 120 q^{16} - 20 q^{17} + 40 q^{20} - 32 q^{22} - 48 q^{23} + 28 q^{25} - 60 q^{28} + 40 q^{30} - 16 q^{33} - 60 q^{35} - 480 q^{36} - 12 q^{37} - 84 q^{38} - 8 q^{42} + 100 q^{43} + 140 q^{44} + 48 q^{48} + 60 q^{49} - 24 q^{53} - 40 q^{55} + 144 q^{58} - 32 q^{60} - 80 q^{64} + 40 q^{65} + 32 q^{67} + 20 q^{69} + 8 q^{70} - 220 q^{74} - 40 q^{75} - 24 q^{77} + 24 q^{78} + 40 q^{79} + 120 q^{80} + 120 q^{81} - 48 q^{82} - 200 q^{85} - 20 q^{87} - 104 q^{88} + 120 q^{92} - 72 q^{93} - 80 q^{95} - 88 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −2.76595 − 0.438083i −0.156434 − 0.987688i 5.55643 + 1.80539i 2.23511 − 0.0655755i 2.80042i 0.783247 + 4.94523i −9.58749 − 4.88507i −0.951057 + 0.309017i −6.21091 − 0.797783i
13.2 −2.61350 − 0.413938i 0.156434 + 0.987688i 4.75692 + 1.54562i −2.10292 − 0.760098i − 2.64608i 0.186506 + 1.17755i −7.07708 − 3.60595i −0.951057 + 0.309017i 5.18133 + 2.85699i
13.3 −2.59688 − 0.411305i −0.156434 − 0.987688i 4.67250 + 1.51819i −2.10372 − 0.757866i 2.62925i −0.556198 − 3.51170i −6.82412 − 3.47706i −0.951057 + 0.309017i 5.15139 + 2.83336i
13.4 −2.57057 − 0.407138i 0.156434 + 0.987688i 4.53993 + 1.47511i 2.19709 + 0.415685i − 2.60261i −0.485751 − 3.06691i −6.43174 − 3.27713i −0.951057 + 0.309017i −5.47852 − 1.96306i
13.5 −2.49806 − 0.395654i 0.156434 + 0.987688i 4.18166 + 1.35870i −0.212967 + 2.22590i − 2.52920i 0.485286 + 3.06398i −5.40140 − 2.75215i −0.951057 + 0.309017i 1.41269 − 5.47618i
13.6 −2.44768 − 0.387674i −0.156434 − 0.987688i 3.93871 + 1.27977i −1.77907 + 1.35458i 2.47819i 0.324376 + 2.04803i −4.72840 − 2.40924i −0.951057 + 0.309017i 4.87973 − 2.62588i
13.7 −2.29832 − 0.364018i −0.156434 − 0.987688i 3.24765 + 1.05522i 1.35916 − 1.77558i 2.32697i −0.766473 − 4.83932i −2.93332 − 1.49460i −0.951057 + 0.309017i −3.77012 + 3.58610i
13.8 −2.24198 − 0.355095i −0.156434 − 0.987688i 2.99826 + 0.974195i −0.306891 − 2.21491i 2.26993i 0.273450 + 1.72649i −2.33108 − 1.18774i −0.951057 + 0.309017i −0.0984576 + 5.07475i
13.9 −2.08238 − 0.329816i 0.156434 + 0.987688i 2.32540 + 0.755567i 2.03503 + 0.926646i − 2.10833i 0.329254 + 2.07883i −0.836072 − 0.426000i −0.951057 + 0.309017i −3.93206 − 2.60081i
13.10 −2.02849 − 0.321282i −0.156434 − 0.987688i 2.10945 + 0.685400i 2.22687 − 0.202586i 2.05378i −0.0578321 − 0.365137i −0.398928 − 0.203264i −0.951057 + 0.309017i −4.58228 − 0.304508i
13.11 −1.97019 − 0.312048i 0.156434 + 0.987688i 1.88217 + 0.611554i −0.708014 + 2.12102i − 1.99475i −0.601529 − 3.79791i 0.0372689 + 0.0189894i −0.951057 + 0.309017i 2.05678 − 3.95788i
13.12 −1.92034 − 0.304152i 0.156434 + 0.987688i 1.69310 + 0.550120i −0.662450 − 2.13569i − 1.94428i −0.446435 − 2.81868i 0.380730 + 0.193992i −0.951057 + 0.309017i 0.622557 + 4.30274i
13.13 −1.78948 − 0.283426i −0.156434 − 0.987688i 1.21980 + 0.396338i −1.69772 + 1.45524i 1.81179i −0.119350 − 0.753548i 1.15815 + 0.590105i −0.951057 + 0.309017i 3.45050 − 2.12296i
13.14 −1.73303 − 0.274485i 0.156434 + 0.987688i 1.02593 + 0.333346i −1.11194 − 1.94000i − 1.75463i −0.122380 − 0.772677i 1.44030 + 0.733869i −0.951057 + 0.309017i 1.39452 + 3.66728i
13.15 −1.67020 − 0.264533i −0.156434 − 0.987688i 0.817461 + 0.265609i 1.51045 + 1.64880i 1.69101i 0.0451044 + 0.284778i 1.71835 + 0.875545i −0.951057 + 0.309017i −2.08659 − 3.15338i
13.16 −1.56960 − 0.248600i −0.156434 − 0.987688i 0.499721 + 0.162369i 0.0418837 − 2.23568i 1.58916i 0.338368 + 2.13637i 2.08791 + 1.06384i −0.951057 + 0.309017i −0.621529 + 3.49870i
13.17 −1.36052 − 0.215485i 0.156434 + 0.987688i −0.0975393 − 0.0316924i 1.70379 + 1.44814i − 1.37748i 0.495636 + 3.12932i 2.58056 + 1.31486i −0.951057 + 0.309017i −2.00598 − 2.33736i
13.18 −1.20873 − 0.191444i 0.156434 + 0.987688i −0.477734 − 0.155225i 2.05294 − 0.886252i − 1.22380i 0.158274 + 0.999301i 2.72856 + 1.39027i −0.951057 + 0.309017i −2.65112 + 0.678216i
13.19 −1.15561 − 0.183031i −0.156434 − 0.987688i −0.600177 − 0.195009i −2.11231 − 0.733591i 1.17002i −0.0190767 − 0.120446i 2.74286 + 1.39756i −0.951057 + 0.309017i 2.30674 + 1.23436i
13.20 −1.12458 − 0.178116i −0.156434 − 0.987688i −0.669158 − 0.217423i 0.0189731 + 2.23599i 1.13860i −0.723246 − 4.56640i 2.74279 + 1.39752i −0.951057 + 0.309017i 0.376928 − 2.51793i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
275.bg even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.cm.a ✓ 480
11.d odd 10 1 825.2.cy.a yes 480
25.f odd 20 1 825.2.cy.a yes 480
275.bg even 20 1 inner 825.2.cm.a ✓ 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.cm.a ✓ 480 1.a even 1 1 trivial
825.2.cm.a ✓ 480 275.bg even 20 1 inner
825.2.cy.a yes 480 11.d odd 10 1
825.2.cy.a yes 480 25.f odd 20 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(825, [\chi])\).