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Properties

Label 825.2.bi.e
Level $825$
Weight $2$
Character orbit 825.bi
Analytic conductor $6.588$
Analytic rank $0$
Dimension $48$
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] = [825,2,Mod(101,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.bi (of order \(10\), degree \(4\), 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: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{3} - 4 q^{4} - 20 q^{6} - 10 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{3} - 4 q^{4} - 20 q^{6} - 10 q^{7} + 2 q^{9} - 8 q^{12} + 32 q^{16} + 30 q^{18} - 100 q^{19} + 82 q^{22} + 100 q^{24} - 14 q^{27} - 30 q^{28} + 10 q^{31} + 46 q^{33} - 28 q^{34} + 14 q^{36} - 6 q^{37} + 52 q^{42} + 20 q^{46} + 80 q^{48} - 26 q^{49} - 30 q^{51} - 40 q^{52} + 70 q^{57} - 92 q^{58} + 70 q^{61} + 20 q^{63} + 18 q^{64} + 76 q^{66} - 20 q^{67} + 42 q^{69} + 80 q^{72} - 90 q^{73} + 108 q^{78} - 100 q^{79} + 38 q^{81} + 34 q^{82} + 70 q^{84} - 74 q^{88} - 86 q^{91} - 76 q^{93} + 10 q^{94} - 30 q^{96} - 6 q^{97} - 28 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 −0.848632 + 2.61182i 1.68513 + 0.400435i −4.48340 − 3.25738i 0 −2.47592 + 4.06143i 0.0372155 − 0.0512227i 7.86897 − 5.71714i 2.67930 + 1.34957i 0
101.2 −0.664086 + 2.04385i −0.340970 + 1.69816i −2.11826 − 1.53901i 0 −3.24434 − 1.82461i 1.16392 − 1.60200i 1.07501 − 0.781038i −2.76748 − 1.15804i 0
101.3 −0.491121 + 1.51151i −0.957081 − 1.44360i −0.425442 − 0.309102i 0 2.65207 − 0.737658i 1.78891 − 2.46222i −1.89539 + 1.37708i −1.16799 + 2.76329i 0
101.4 −0.212587 + 0.654275i −1.65165 + 0.521602i 1.23515 + 0.897390i 0 0.00984690 − 1.19152i 0.891460 − 1.22699i −1.96284 + 1.42608i 2.45586 − 1.72300i 0
101.5 −0.140144 + 0.431318i 1.58256 + 0.703911i 1.45164 + 1.05468i 0 −0.525396 + 0.583940i −2.72178 + 3.74621i −1.39214 + 1.01145i 2.00902 + 2.22797i 0
101.6 −0.0403345 + 0.124137i 1.72819 − 0.115636i 1.60425 + 1.16556i 0 −0.0553509 + 0.219196i 1.50339 − 2.06924i −0.420589 + 0.305576i 2.97326 − 0.399681i 0
101.7 0.0403345 − 0.124137i 0.644015 − 1.60787i 1.60425 + 1.16556i 0 −0.173620 − 0.144799i 1.50339 − 2.06924i 0.420589 − 0.305576i −2.17049 − 2.07099i 0
101.8 0.140144 − 0.431318i −0.180419 − 1.72263i 1.45164 + 1.05468i 0 −0.768285 − 0.163597i −2.72178 + 3.74621i 1.39214 − 1.01145i −2.93490 + 0.621591i 0
101.9 0.212587 − 0.654275i −1.00646 + 1.40962i 1.23515 + 0.897390i 0 0.708322 + 0.958169i 0.891460 − 1.22699i 1.96284 − 1.42608i −0.974079 − 2.83746i 0
101.10 0.491121 − 1.51151i 1.07720 + 1.35634i −0.425442 − 0.309102i 0 2.57916 − 0.962071i 1.78891 − 2.46222i 1.89539 − 1.37708i −0.679300 + 2.92208i 0
101.11 0.664086 − 2.04385i −1.72041 − 0.200478i −2.11826 − 1.53901i 0 −1.55225 + 3.38312i 1.16392 − 1.60200i −1.07501 + 0.781038i 2.91962 + 0.689809i 0
101.12 0.848632 − 2.61182i 0.139896 − 1.72639i −4.48340 − 3.25738i 0 −4.39031 − 1.83046i 0.0372155 − 0.0512227i −7.86897 + 5.71714i −2.96086 − 0.483031i 0
326.1 −2.09233 − 1.52016i 1.46636 + 0.921840i 1.44890 + 4.45925i 0 −1.66676 − 4.15790i −0.326781 + 0.106178i 2.14883 − 6.61341i 1.30042 + 2.70350i 0
326.2 −1.54632 − 1.12347i 0.613479 − 1.61977i 0.510897 + 1.57238i 0 −2.76839 + 1.81546i 0.0795268 − 0.0258398i −0.204777 + 0.630238i −2.24729 − 1.98739i 0
326.3 −1.45632 − 1.05808i −0.960940 − 1.44104i 0.383300 + 1.17968i 0 −0.125297 + 3.11536i 0.145486 − 0.0472713i −0.422546 + 1.30046i −1.15319 + 2.76950i 0
326.4 −1.37228 − 0.997022i 0.579977 + 1.63206i 0.271073 + 0.834277i 0 0.831309 − 2.81790i −3.82974 + 1.24436i −0.588527 + 1.81130i −2.32725 + 1.89312i 0
326.5 −0.860692 − 0.625329i −1.72070 + 0.197973i −0.268280 − 0.825682i 0 1.60479 + 0.905610i 2.18340 − 0.709430i −0.942926 + 2.90203i 2.92161 − 0.681304i 0
326.6 −0.131900 − 0.0958309i 1.46543 − 0.923314i −0.609820 − 1.87683i 0 −0.281772 − 0.0186487i −3.41501 + 1.10960i −0.200186 + 0.616109i 1.29498 − 2.70611i 0
326.7 0.131900 + 0.0958309i −0.642850 − 1.60834i −0.609820 − 1.87683i 0 0.0693364 − 0.273744i −3.41501 + 1.10960i 0.200186 − 0.616109i −2.17349 + 2.06784i 0
326.8 0.860692 + 0.625329i 1.27571 + 1.17157i −0.268280 − 0.825682i 0 0.365379 + 1.80609i 2.18340 − 0.709430i 0.942926 − 2.90203i 0.254870 + 2.98915i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.bi.e 48
3.b odd 2 1 inner 825.2.bi.e 48
5.b even 2 1 165.2.p.b ✓ 48
5.c odd 4 1 825.2.bs.g 48
5.c odd 4 1 825.2.bs.h 48
11.d odd 10 1 inner 825.2.bi.e 48
15.d odd 2 1 165.2.p.b ✓ 48
15.e even 4 1 825.2.bs.g 48
15.e even 4 1 825.2.bs.h 48
33.f even 10 1 inner 825.2.bi.e 48
55.h odd 10 1 165.2.p.b ✓ 48
55.l even 20 1 825.2.bs.g 48
55.l even 20 1 825.2.bs.h 48
165.r even 10 1 165.2.p.b ✓ 48
165.u odd 20 1 825.2.bs.g 48
165.u odd 20 1 825.2.bs.h 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.p.b ✓ 48 5.b even 2 1
165.2.p.b ✓ 48 15.d odd 2 1
165.2.p.b ✓ 48 55.h odd 10 1
165.2.p.b ✓ 48 165.r even 10 1
825.2.bi.e 48 1.a even 1 1 trivial
825.2.bi.e 48 3.b odd 2 1 inner
825.2.bi.e 48 11.d odd 10 1 inner
825.2.bi.e 48 33.f even 10 1 inner
825.2.bs.g 48 5.c odd 4 1
825.2.bs.g 48 15.e even 4 1
825.2.bs.g 48 55.l even 20 1
825.2.bs.g 48 165.u odd 20 1
825.2.bs.h 48 5.c odd 4 1
825.2.bs.h 48 15.e even 4 1
825.2.bs.h 48 55.l even 20 1
825.2.bs.h 48 165.u odd 20 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{48} + 14 T_{2}^{46} + 135 T_{2}^{44} + 1083 T_{2}^{42} + 8563 T_{2}^{40} + 45065 T_{2}^{38} + \cdots + 1 \) Copy content Toggle raw display
\( T_{7}^{24} + 5 T_{7}^{23} - 2 T_{7}^{22} - 40 T_{7}^{21} + 184 T_{7}^{20} + 1105 T_{7}^{19} - 2729 T_{7}^{18} + \cdots + 1 \) Copy content Toggle raw display