Properties

Label 825.2.bs.h
Level $825$
Weight $2$
Character orbit 825.bs
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(74,825)]
 
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, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.74");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.bs (of order \(10\), degree \(4\), not 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{3} + 4 q^{4} - 20 q^{6} + 50 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{3} + 4 q^{4} - 20 q^{6} + 50 q^{8} - 2 q^{9} + 36 q^{12} + 32 q^{16} + 60 q^{17} + 30 q^{18} + 100 q^{19} - 28 q^{23} - 100 q^{24} + 12 q^{27} + 10 q^{31} + 28 q^{33} + 28 q^{34} + 14 q^{36} - 60 q^{38} - 20 q^{42} + 20 q^{46} + 2 q^{47} - 116 q^{48} + 26 q^{49} - 30 q^{51} - 10 q^{53} + 70 q^{57} + 70 q^{61} + 110 q^{62} + 70 q^{63} - 18 q^{64} + 76 q^{66} + 90 q^{68} - 42 q^{69} - 50 q^{72} - 14 q^{77} - 148 q^{78} + 100 q^{79} + 38 q^{81} - 30 q^{83} - 70 q^{84} - 86 q^{91} - 192 q^{92} - 10 q^{94} - 30 q^{96} + 28 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1 −1.52016 2.09233i −0.921840 1.46636i −1.44890 + 4.45925i 0 −1.66676 + 4.15790i 0.106178 0.326781i 6.61341 2.14883i −1.30042 + 2.70350i 0
74.2 −1.52016 2.09233i −0.116121 1.72815i −1.44890 + 4.45925i 0 −3.43934 + 2.87004i −0.106178 + 0.326781i 6.61341 2.14883i −2.97303 + 0.401348i 0
74.3 −1.12347 1.54632i −1.67101 + 0.455760i −0.510897 + 1.57238i 0 2.58208 + 2.07189i 0.0258398 0.0795268i −0.630238 + 0.204777i 2.58457 1.52316i 0
74.4 −1.12347 1.54632i 1.61977 0.613479i −0.510897 + 1.57238i 0 −2.76839 1.81546i −0.0258398 + 0.0795268i −0.630238 + 0.204777i 2.24729 1.98739i 0
74.5 −0.625329 0.860692i −0.197973 + 1.72070i 0.268280 0.825682i 0 1.60479 0.905610i −0.709430 + 2.18340i −2.90203 + 0.942926i −2.92161 0.681304i 0
74.6 −0.625329 0.860692i 1.17157 + 1.27571i 0.268280 0.825682i 0 0.365379 1.80609i 0.709430 2.18340i −2.90203 + 0.942926i −0.254870 + 2.98915i 0
74.7 0.0958309 + 0.131900i −0.923314 + 1.46543i 0.609820 1.87683i 0 −0.281772 + 0.0186487i −1.10960 + 3.41501i 0.616109 0.200186i −1.29498 2.70611i 0
74.8 0.0958309 + 0.131900i 1.60834 + 0.642850i 0.609820 1.87683i 0 0.0693364 + 0.273744i 1.10960 3.41501i 0.616109 0.200186i 2.17349 + 2.06784i 0
74.9 0.997022 + 1.37228i −0.979464 + 1.42851i −0.271073 + 0.834277i 0 −2.93687 + 0.0801565i 1.24436 3.82974i 1.81130 0.588527i −1.08130 2.79836i 0
74.10 0.997022 + 1.37228i 1.63206 + 0.579977i −0.271073 + 0.834277i 0 0.831309 + 2.81790i −1.24436 + 3.82974i 1.81130 0.588527i 2.32725 + 1.89312i 0
74.11 1.05808 + 1.45632i −1.44104 0.960940i −0.383300 + 1.17968i 0 −0.125297 3.11536i 0.0472713 0.145486i 1.30046 0.422546i 1.15319 + 2.76950i 0
74.12 1.05808 + 1.45632i 0.600999 1.62444i −0.383300 + 1.17968i 0 3.00160 0.843534i −0.0472713 + 0.145486i 1.30046 0.422546i −2.27760 1.95257i 0
149.1 −2.04385 + 0.664086i 0.200478 + 1.72041i 2.11826 1.53901i 0 −1.55225 3.38312i −1.60200 + 1.16392i −0.781038 + 1.07501i −2.91962 + 0.689809i 0
149.2 −2.04385 + 0.664086i 1.69816 0.340970i 2.11826 1.53901i 0 −3.24434 + 1.82461i 1.60200 1.16392i −0.781038 + 1.07501i 2.76748 1.15804i 0
149.3 −0.654275 + 0.212587i −1.40962 + 1.00646i −1.23515 + 0.897390i 0 0.708322 0.958169i −1.22699 + 0.891460i 1.42608 1.96284i 0.974079 2.83746i 0
149.4 −0.654275 + 0.212587i 0.521602 1.65165i −1.23515 + 0.897390i 0 0.00984690 + 1.19152i 1.22699 0.891460i 1.42608 1.96284i −2.45586 1.72300i 0
149.5 −0.431318 + 0.140144i 0.703911 + 1.58256i −1.45164 + 1.05468i 0 −0.525396 0.583940i −3.74621 + 2.72178i 1.01145 1.39214i −2.00902 + 2.22797i 0
149.6 −0.431318 + 0.140144i 1.72263 + 0.180419i −1.45164 + 1.05468i 0 −0.768285 + 0.163597i 3.74621 2.72178i 1.01145 1.39214i 2.93490 + 0.621591i 0
149.7 0.124137 0.0403345i −1.60787 + 0.644015i −1.60425 + 1.16556i 0 −0.173620 + 0.144799i 2.06924 1.50339i −0.305576 + 0.420589i 2.17049 2.07099i 0
149.8 0.124137 0.0403345i 0.115636 1.72819i −1.60425 + 1.16556i 0 −0.0553509 0.219196i −2.06924 + 1.50339i −0.305576 + 0.420589i −2.97326 0.399681i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 74.12
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 7 T_{2}^{22} - 15 T_{2}^{21} + 43 T_{2}^{20} + 105 T_{2}^{19} - 128 T_{2}^{18} - 270 T_{2}^{17} + 886 T_{2}^{16} + 1425 T_{2}^{15} - 1264 T_{2}^{14} - 4970 T_{2}^{13} + 6374 T_{2}^{12} + 6395 T_{2}^{11} - 8398 T_{2}^{10} - 12920 T_{2}^{9} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\). Copy content Toggle raw display