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Properties

Label 525.2.bv.a
Level $525$
Weight $2$
Character orbit 525.bv
Analytic conductor $4.192$
Analytic rank $0$
Dimension $640$
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] = [525,2,Mod(52,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([0, 3, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.52");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.bv (of order \(60\), degree \(16\), 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: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(640\)
Relative dimension: \(40\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 640 q + 12 q^{5} - 8 q^{7} + 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 640 q + 12 q^{5} - 8 q^{7} + 24 q^{8} + 12 q^{10} + 8 q^{15} - 80 q^{16} - 72 q^{22} + 48 q^{23} - 12 q^{25} - 36 q^{28} - 80 q^{29} + 32 q^{30} - 24 q^{32} + 36 q^{33} - 64 q^{35} + 160 q^{36} - 4 q^{37} - 192 q^{38} - 12 q^{40} - 16 q^{42} + 120 q^{43} + 60 q^{47} + 288 q^{50} - 432 q^{52} - 136 q^{53} - 16 q^{57} - 4 q^{58} - 240 q^{59} - 20 q^{60} - 4 q^{63} + 120 q^{64} + 4 q^{65} - 8 q^{67} - 132 q^{68} + 76 q^{70} - 12 q^{72} - 36 q^{73} - 48 q^{75} - 60 q^{77} - 80 q^{78} + 12 q^{80} - 80 q^{81} - 252 q^{82} - 160 q^{84} + 72 q^{85} + 24 q^{87} + 152 q^{88} + 56 q^{92} - 96 q^{93} + 172 q^{95} - 488 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
52.1 −1.00045 − 2.60626i 0.0523360 − 0.998630i −4.30539 + 3.87659i 1.46116 + 1.69263i −2.65504 + 0.862676i 1.56547 − 2.13291i 9.43590 + 4.80783i −0.994522 − 0.104528i 2.94961 − 5.50156i
52.2 −0.952068 − 2.48022i −0.0523360 + 0.998630i −3.75878 + 3.38442i 2.10803 − 0.745796i 2.52665 − 0.820958i −2.63239 + 0.265557i 7.23848 + 3.68819i −0.994522 − 0.104528i −3.85673 − 4.51833i
52.3 −0.944724 − 2.46109i 0.0523360 − 0.998630i −3.67817 + 3.31184i −0.481208 − 2.18368i −2.50716 + 0.814626i 0.854940 + 2.50381i 6.92789 + 3.52994i −0.994522 − 0.104528i −4.91961 + 3.24727i
52.4 −0.872278 − 2.27236i −0.0523360 + 0.998630i −2.91647 + 2.62600i −2.11358 − 0.729921i 2.31490 − 0.752156i 0.892264 − 2.49076i 4.17372 + 2.12662i −0.994522 − 0.104528i 0.184983 + 5.43951i
52.5 −0.802228 − 2.08988i −0.0523360 + 0.998630i −2.23772 + 2.01485i 0.318271 + 2.21330i 2.12900 − 0.691753i −0.914813 + 2.48256i 2.01681 + 1.02761i −0.994522 − 0.104528i 4.37020 − 2.44072i
52.6 −0.755992 − 1.96943i 0.0523360 − 0.998630i −1.82083 + 1.63948i 0.924367 + 2.03606i −2.00629 + 0.651884i −1.10577 + 2.40360i 0.846132 + 0.431126i −0.994522 − 0.104528i 3.31106 − 3.35972i
52.7 −0.754243 − 1.96487i −0.0523360 + 0.998630i −1.80554 + 1.62572i 2.18621 + 0.469548i 2.00165 − 0.650376i 2.57259 + 0.617866i 0.805604 + 0.410476i −0.994522 − 0.104528i −0.726334 − 4.64978i
52.8 −0.753962 − 1.96414i 0.0523360 − 0.998630i −1.80309 + 1.62351i 0.485187 − 2.18279i −2.00090 + 0.650133i −0.829509 − 2.51235i 0.799114 + 0.407169i −0.994522 − 0.104528i −4.65312 + 0.692770i
52.9 −0.641755 − 1.67183i −0.0523360 + 0.998630i −0.896875 + 0.807550i −0.674040 + 2.13206i 1.70313 − 0.553379i 0.561772 − 2.58542i −1.26552 − 0.644814i −0.994522 − 0.104528i 3.99701 − 0.241379i
52.10 −0.611789 − 1.59376i −0.0523360 + 0.998630i −0.679511 + 0.611834i −1.29445 − 1.82330i 1.62360 − 0.527539i −2.56289 − 0.656955i −1.65133 − 0.841396i −0.994522 − 0.104528i −2.11398 + 3.17852i
52.11 −0.498488 − 1.29861i 0.0523360 − 0.998630i 0.0484028 − 0.0435821i −0.935202 + 2.03111i −1.32292 + 0.429841i −2.07682 − 1.63916i −2.55950 − 1.30413i −0.994522 − 0.104528i 3.10380 + 0.201975i
52.12 −0.496455 − 1.29331i 0.0523360 − 0.998630i 0.0601081 − 0.0541216i 2.05147 − 0.889639i −1.31752 + 0.428088i 2.12828 + 1.57176i −2.56850 − 1.30872i −0.994522 − 0.104528i −2.16904 − 2.21152i
52.13 −0.423361 − 1.10289i 0.0523360 − 0.998630i 0.449149 − 0.404416i −1.56576 − 1.59637i −1.12354 + 0.365060i 1.78771 − 1.95041i −2.74138 − 1.39680i −0.994522 − 0.104528i −1.09774 + 2.40271i
52.14 −0.385788 − 1.00501i 0.0523360 − 0.998630i 0.625072 − 0.562818i −2.23091 + 0.151798i −1.02383 + 0.332661i −1.22058 + 2.34738i −2.72515 − 1.38853i −0.994522 − 0.104528i 1.01322 + 2.18353i
52.15 −0.377442 − 0.983270i −0.0523360 + 0.998630i 0.661931 − 0.596006i 2.00799 − 0.983866i 1.00168 − 0.325465i −2.28125 − 1.34011i −2.71274 − 1.38221i −0.994522 − 0.104528i −1.72530 − 1.60304i
52.16 −0.221735 − 0.577641i −0.0523360 + 0.998630i 1.20179 − 1.08209i 1.83440 + 1.27866i 0.588454 − 0.191200i 2.58948 − 0.542755i −1.99414 − 1.01606i −0.994522 − 0.104528i 0.331851 − 1.34315i
52.17 −0.206383 − 0.537647i −0.0523360 + 0.998630i 1.23982 − 1.11634i −1.08016 − 1.95787i 0.547712 − 0.177962i 2.64165 − 0.147266i −1.88233 − 0.959096i −0.994522 − 0.104528i −0.829717 + 0.984816i
52.18 −0.107591 − 0.280285i 0.0523360 − 0.998630i 1.41931 − 1.27795i 1.68997 + 1.46424i −0.285532 + 0.0927749i −1.90734 − 1.83359i −1.04590 − 0.532913i −0.994522 − 0.104528i 0.228576 − 0.631214i
52.19 −0.0754782 − 0.196628i −0.0523360 + 0.998630i 1.45332 − 1.30858i −2.18626 + 0.469349i 0.200308 − 0.0650841i −2.39330 + 1.12789i −0.742319 − 0.378230i −0.994522 − 0.104528i 0.257302 + 0.394452i
52.20 −0.0670074 − 0.174560i 0.0523360 − 0.998630i 1.46031 − 1.31487i 0.297475 − 2.21619i −0.177828 + 0.0577798i −2.22491 + 1.43170i −0.660575 − 0.336580i −0.994522 − 0.104528i −0.406792 + 0.0965741i
See next 80 embeddings (of 640 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 52.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
25.f odd 20 1 inner
175.x even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.bv.a ✓ 640
7.d odd 6 1 inner 525.2.bv.a ✓ 640
25.f odd 20 1 inner 525.2.bv.a ✓ 640
175.x even 60 1 inner 525.2.bv.a ✓ 640
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.bv.a ✓ 640 1.a even 1 1 trivial
525.2.bv.a ✓ 640 7.d odd 6 1 inner
525.2.bv.a ✓ 640 25.f odd 20 1 inner
525.2.bv.a ✓ 640 175.x even 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(525, [\chi])\).