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Properties

Label 825.2.ce.c
Level $825$
Weight $2$
Character orbit 825.ce
Analytic conductor $6.588$
Analytic rank $0$
Dimension $448$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,2,Mod(164,825)]
 
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, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.164");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.ce (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: \(448\)
Relative dimension: \(112\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 448 q - 10 q^{3} + 108 q^{4} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 448 q - 10 q^{3} + 108 q^{4} - 14 q^{9} - 40 q^{12} - 10 q^{15} - 140 q^{16} + 10 q^{22} - 36 q^{25} - 10 q^{27} - 32 q^{31} + 20 q^{33} - 28 q^{34} - 6 q^{36} - 20 q^{37} - 70 q^{42} + 82 q^{45} - 60 q^{48} + 440 q^{49} + 6 q^{55} + 40 q^{58} + 88 q^{60} + 188 q^{64} - 30 q^{66} - 140 q^{67} + 34 q^{69} - 76 q^{70} + 70 q^{75} + 60 q^{78} + 50 q^{81} - 80 q^{88} - 80 q^{91} + 120 q^{97} - 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
164.1 −1.62572 − 2.23761i −0.323152 + 1.70164i −1.74590 + 5.37332i 2.19890 + 0.406019i 4.33295 − 2.04330i −3.32952 9.60080 − 3.11949i −2.79115 − 1.09977i −2.66627 − 5.58034i
164.2 −1.62572 − 2.23761i 0.738763 − 1.56660i −1.74590 + 5.37332i −2.19890 − 0.406019i −4.70645 + 0.893784i 3.32952 9.60080 − 3.11949i −1.90846 − 2.31469i 2.66627 + 5.58034i
164.3 −1.58817 − 2.18593i −1.72150 − 0.190846i −1.63798 + 5.04118i −0.390419 − 2.20172i 2.31687 + 4.06619i 2.72685 8.48165 − 2.75585i 2.92716 + 0.657084i −4.19276 + 4.35015i
164.4 −1.58817 − 2.18593i −1.50490 − 0.857477i −1.63798 + 5.04118i 0.390419 + 2.20172i 0.515659 + 4.65144i −2.72685 8.48165 − 2.75585i 1.52947 + 2.58084i 4.19276 − 4.35015i
164.5 −1.56061 − 2.14799i 1.25445 + 1.19430i −1.56034 + 4.80223i 1.25500 − 1.85067i 0.607632 − 4.55839i 2.49583 7.70000 − 2.50188i 0.147307 + 2.99638i −5.93378 + 0.192422i
164.6 −1.56061 − 2.14799i 1.71686 − 0.228858i −1.56034 + 4.80223i −1.25500 + 1.85067i −3.17094 − 3.33065i −2.49583 7.70000 − 2.50188i 2.89525 − 0.785835i 5.93378 − 0.192422i
164.7 −1.45448 − 2.00191i 0.627496 + 1.61439i −1.27412 + 3.92135i −1.66382 + 1.49389i 2.31919 − 3.60428i 0.493454 4.99661 − 1.62350i −2.21250 + 2.02605i 5.41062 + 1.15798i
164.8 −1.45448 − 2.00191i 1.45657 − 0.937234i −1.27412 + 3.92135i 1.66382 − 1.49389i −3.99480 − 1.55274i −0.493454 4.99661 − 1.62350i 1.24318 − 2.73029i −5.41062 − 1.15798i
164.9 −1.44305 − 1.98618i −1.43787 + 0.965673i −1.24451 + 3.83020i 1.74382 + 1.39967i 3.99292 + 1.46237i 3.57771 4.73357 − 1.53803i 1.13495 − 2.77703i 0.263592 − 5.48335i
164.10 −1.44305 − 1.98618i −0.595654 − 1.62641i −1.24451 + 3.83020i −1.74382 − 1.39967i −2.37078 + 3.53006i −3.57771 4.73357 − 1.53803i −2.29039 + 1.93755i −0.263592 + 5.48335i
164.11 −1.43000 − 1.96822i −1.17081 + 1.27640i −1.21097 + 3.72700i −1.43791 + 1.71243i 4.18650 + 0.479156i 2.76385 4.43968 − 1.44254i −0.258417 − 2.98885i 5.42665 + 0.381361i
164.12 −1.43000 − 1.96822i −0.196952 − 1.72082i −1.21097 + 3.72700i 1.43791 − 1.71243i −3.10531 + 2.84841i −2.76385 4.43968 − 1.44254i −2.92242 + 0.677836i −5.42665 − 0.381361i
164.13 −1.41506 − 1.94767i 1.52148 + 0.827705i −1.17297 + 3.61003i −1.40120 − 1.74259i −0.540897 − 4.13459i −4.95869 4.11171 − 1.33598i 1.62981 + 2.51868i −1.41121 + 5.19495i
164.14 −1.41506 − 1.94767i 1.71742 + 0.224677i −1.17297 + 3.61003i 1.40120 + 1.74259i −1.99266 − 3.66289i 4.95869 4.11171 − 1.33598i 2.89904 + 0.771727i 1.41121 − 5.19495i
164.15 −1.33952 − 1.84370i 0.0189390 + 1.73195i −0.986858 + 3.03724i −1.83047 − 1.28428i 3.16782 − 2.35490i 0.812974 2.58688 − 0.840527i −2.99928 + 0.0656026i 0.0841499 + 5.09516i
164.16 −1.33952 − 1.84370i 1.03334 − 1.39004i −0.986858 + 3.03724i 1.83047 + 1.28428i −3.94699 − 0.0431607i −0.812974 2.58688 − 0.840527i −0.864437 − 2.87276i −0.0841499 − 5.09516i
164.17 −1.31295 − 1.80712i −1.48334 + 0.894265i −0.923805 + 2.84318i 1.10782 − 1.94235i 3.56358 + 1.50644i −1.24026 2.10208 − 0.683006i 1.40058 − 2.65299i −4.96456 + 0.548243i
164.18 −1.31295 − 1.80712i −0.674410 − 1.59536i −0.923805 + 2.84318i −1.10782 + 1.94235i −1.99754 + 3.31336i 1.24026 2.10208 − 0.683006i −2.09034 + 2.15185i 4.96456 − 0.548243i
164.19 −1.16449 − 1.60278i −1.72996 + 0.0850155i −0.594834 + 1.83071i −2.23088 + 0.152161i 2.15078 + 2.67375i −3.11716 −0.141458 + 0.0459625i 2.98554 − 0.294147i 2.84171 + 3.39842i
164.20 −1.16449 − 1.60278i −1.34960 − 1.08563i −0.594834 + 1.83071i 2.23088 − 0.152161i −0.168428 + 3.42730i 3.11716 −0.141458 + 0.0459625i 0.642833 + 2.93032i −2.84171 − 3.39842i
See next 80 embeddings (of 448 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 164.112
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
25.e even 10 1 inner
33.d even 2 1 inner
75.h odd 10 1 inner
275.s odd 10 1 inner
825.ce 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.ce.c ✓ 448
3.b odd 2 1 inner 825.2.ce.c ✓ 448
11.b odd 2 1 inner 825.2.ce.c ✓ 448
25.e even 10 1 inner 825.2.ce.c ✓ 448
33.d even 2 1 inner 825.2.ce.c ✓ 448
75.h odd 10 1 inner 825.2.ce.c ✓ 448
275.s odd 10 1 inner 825.2.ce.c ✓ 448
825.ce even 10 1 inner 825.2.ce.c ✓ 448
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.ce.c ✓ 448 1.a even 1 1 trivial
825.2.ce.c ✓ 448 3.b odd 2 1 inner
825.2.ce.c ✓ 448 11.b odd 2 1 inner
825.2.ce.c ✓ 448 25.e even 10 1 inner
825.2.ce.c ✓ 448 33.d even 2 1 inner
825.2.ce.c ✓ 448 75.h odd 10 1 inner
825.2.ce.c ✓ 448 275.s odd 10 1 inner
825.2.ce.c ✓ 448 825.ce even 10 1 inner

Hecke kernels

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

\( T_{2}^{224} - 83 T_{2}^{222} + 3670 T_{2}^{220} - 114904 T_{2}^{218} + 2856773 T_{2}^{216} + \cdots + 86\!\cdots\!25 \) Copy content Toggle raw display
\( T_{23}^{224} + 604 T_{23}^{222} + 195334 T_{23}^{220} + 44881979 T_{23}^{218} + 8241208395 T_{23}^{216} + \cdots + 43\!\cdots\!36 \) Copy content Toggle raw display