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Properties

Label 825.2.cl.a
Level $825$
Weight $2$
Character orbit 825.cl
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(28,825)]
 
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, 7, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.28");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.cl (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.

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 480 q - 8 q^{5} - 20 q^{10} + 24 q^{12} + 20 q^{13} + 8 q^{15} - 480 q^{16} - 20 q^{17} + 40 q^{20} - 32 q^{22} + 32 q^{23} + 8 q^{25} + 20 q^{28} - 100 q^{32} - 16 q^{33} + 60 q^{35} + 120 q^{36} + 48 q^{37} + 56 q^{38} - 8 q^{42} + 80 q^{47} - 32 q^{48} - 80 q^{49} - 80 q^{52} - 24 q^{53} - 40 q^{55} + 40 q^{57} + 24 q^{58} - 32 q^{60} - 120 q^{62} + 20 q^{63} + 32 q^{67} + 68 q^{70} - 220 q^{74} + 40 q^{75} - 24 q^{77} + 24 q^{78} - 20 q^{79} + 120 q^{80} + 120 q^{81} + 32 q^{82} + 200 q^{83} + 280 q^{85} - 164 q^{88} - 80 q^{92} - 72 q^{93} + 40 q^{94} + 32 q^{97} + 160 q^{98} + 20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1 −1.92758 − 1.92758i 0.453990 − 0.891007i 5.43111i −1.86705 + 1.23049i −2.59259 + 0.842382i 3.09439 + 0.490103i 6.61372 − 6.61372i −0.587785 − 0.809017i 5.97076 + 1.22702i
28.2 −1.92339 − 1.92339i −0.453990 + 0.891007i 5.39887i −1.75654 − 1.38368i 2.58696 − 0.840553i 1.53931 + 0.243803i 6.53736 − 6.53736i −0.587785 − 0.809017i 0.717145 + 6.03987i
28.3 −1.90055 − 1.90055i 0.453990 − 0.891007i 5.22418i −0.249646 − 2.22209i −2.55623 + 0.830571i −1.60026 − 0.253457i 6.12771 − 6.12771i −0.587785 − 0.809017i −3.74873 + 4.69765i
28.4 −1.77041 − 1.77041i 0.453990 − 0.891007i 4.26869i −0.901948 + 2.04609i −2.38119 + 0.773697i −3.95500 − 0.626411i 4.01650 − 4.01650i −0.587785 − 0.809017i 5.21923 − 2.02560i
28.5 −1.72522 − 1.72522i −0.453990 + 0.891007i 3.95275i 1.90742 − 1.16694i 2.32041 − 0.753947i 2.50876 + 0.397349i 3.36891 − 3.36891i −0.587785 − 0.809017i −5.30394 − 1.27750i
28.6 −1.68690 − 1.68690i 0.453990 − 0.891007i 3.69123i 2.14590 − 0.628590i −2.26887 + 0.737200i −1.12748 − 0.178576i 2.85293 − 2.85293i −0.587785 − 0.809017i −4.68027 − 2.55954i
28.7 −1.67559 − 1.67559i −0.453990 + 0.891007i 3.61522i −0.853900 + 2.06660i 2.25367 − 0.732261i −0.693833 − 0.109892i 2.70645 − 2.70645i −0.587785 − 0.809017i 4.89358 − 2.03200i
28.8 −1.54240 − 1.54240i 0.453990 − 0.891007i 2.75797i −0.695861 − 2.12504i −2.07452 + 0.674052i 0.252647 + 0.0400153i 1.16909 − 1.16909i −0.587785 − 0.809017i −2.20435 + 4.35094i
28.9 −1.54151 − 1.54151i −0.453990 + 0.891007i 2.75251i −0.454183 − 2.18946i 2.07333 − 0.673665i −4.05878 − 0.642847i 1.15999 − 1.15999i −0.587785 − 0.809017i −2.67494 + 4.07520i
28.10 −1.50249 − 1.50249i 0.453990 − 0.891007i 2.51493i 2.23134 + 0.145271i −2.02084 + 0.656611i 2.10714 + 0.333738i 0.773675 − 0.773675i −0.587785 − 0.809017i −3.13430 − 3.57083i
28.11 −1.49329 − 1.49329i −0.453990 + 0.891007i 2.45982i −2.23487 + 0.0733316i 2.00847 − 0.652591i 0.514635 + 0.0815102i 0.686644 − 0.686644i −0.587785 − 0.809017i 3.44680 + 3.22779i
28.12 −1.45039 − 1.45039i −0.453990 + 0.891007i 2.20725i 1.69983 + 1.45278i 1.95077 − 0.633843i −2.92402 − 0.463120i 0.300592 − 0.300592i −0.587785 − 0.809017i −0.358322 − 4.57251i
28.13 −1.25857 − 1.25857i 0.453990 − 0.891007i 1.16800i −2.07857 + 0.824340i −1.69277 + 0.550016i 2.84202 + 0.450132i −1.04713 + 1.04713i −0.587785 − 0.809017i 3.65352 + 1.57854i
28.14 −1.20642 − 1.20642i −0.453990 + 0.891007i 0.910880i 1.90918 + 1.16406i 1.62263 − 0.527223i −1.29416 − 0.204975i −1.31393 + 1.31393i −0.587785 − 0.809017i −0.898932 − 3.70760i
28.15 −1.19868 − 1.19868i 0.453990 − 0.891007i 0.873661i −2.15483 − 0.597256i −1.61222 + 0.523842i −3.64676 − 0.577591i −1.35012 + 1.35012i −0.587785 − 0.809017i 1.86703 + 3.29887i
28.16 −1.13338 − 1.13338i 0.453990 − 0.891007i 0.569113i −0.241143 + 2.22303i −1.52440 + 0.495306i −1.51873 − 0.240543i −1.62174 + 1.62174i −0.587785 − 0.809017i 2.79285 − 2.24623i
28.17 −0.973360 − 0.973360i −0.453990 + 0.891007i − 0.105142i −1.10774 − 1.94240i 1.30917 − 0.425374i 2.23481 + 0.353959i −2.04906 + 2.04906i −0.587785 − 0.809017i −0.812430 + 2.96888i
28.18 −0.877508 − 0.877508i 0.453990 − 0.891007i − 0.459960i 1.09966 + 1.94699i −1.18025 + 0.383485i 3.64692 + 0.577616i −2.15863 + 2.15863i −0.587785 − 0.809017i 0.743536 − 2.67345i
28.19 −0.860319 − 0.860319i −0.453990 + 0.891007i − 0.519702i 1.94212 + 1.10823i 1.15713 − 0.375973i 4.41254 + 0.698877i −2.16775 + 2.16775i −0.587785 − 0.809017i −0.717417 − 2.62427i
28.20 −0.812094 − 0.812094i −0.453990 + 0.891007i − 0.681006i −2.21700 + 0.291388i 1.09226 − 0.354898i −4.80921 − 0.761704i −2.17723 + 2.17723i −0.587785 − 0.809017i 2.03705 + 1.56378i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
275.bf 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.cl.a ✓ 480
11.d odd 10 1 825.2.cx.a yes 480
25.f odd 20 1 825.2.cx.a yes 480
275.bf even 20 1 inner 825.2.cl.a ✓ 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.cl.a ✓ 480 1.a even 1 1 trivial
825.2.cl.a ✓ 480 275.bf even 20 1 inner
825.2.cx.a yes 480 11.d odd 10 1
825.2.cx.a yes 480 25.f odd 20 1

Hecke kernels

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