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Properties

Label 575.2.w.a
Level $575$
Weight $2$
Character orbit 575.w
Analytic conductor $4.591$
Analytic rank $0$
Dimension $4640$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [575,2,Mod(17,575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(220))
 
chi = DirichletCharacter(H, H._module([143, 70]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.w (of order \(220\), degree \(80\), 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: \(4.59139811622\)
Analytic rank: \(0\)
Dimension: \(4640\)
Relative dimension: \(58\) over \(\Q(\zeta_{220})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{220}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 4640 q - 72 q^{2} - 76 q^{3} - 90 q^{4} - 88 q^{5} - 54 q^{6} - 88 q^{7} - 64 q^{8} - 90 q^{9} - 88 q^{10} - 66 q^{11} - 84 q^{12} - 64 q^{13} - 110 q^{14} - 88 q^{15} - 158 q^{16} - 88 q^{17} - 148 q^{18}+ \cdots - 220 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −1.72233 − 2.16880i −0.557434 + 1.17588i −1.28426 + 5.52274i −1.03105 − 1.98417i 3.51033 − 0.816292i −2.16856 − 2.89686i 9.18455 − 4.35400i 0.827079 + 1.01147i −2.52745 + 5.65354i
17.2 −1.69106 − 2.12942i 1.18536 − 2.50045i −1.22175 + 5.25394i 2.21064 − 0.336251i −7.32902 + 1.70429i 0.803482 + 1.07333i 8.33972 − 3.95350i −2.94813 − 3.60541i −4.45435 − 4.13877i
17.3 −1.62093 − 2.04112i −0.918010 + 1.93649i −1.08573 + 4.66900i 2.22562 + 0.215954i 5.44064 − 1.26517i 1.18098 + 1.57760i 6.57946 − 3.11904i −1.00823 − 1.23302i −3.16679 − 4.89279i
17.4 −1.49183 − 1.87855i 0.872559 − 1.84062i −0.850386 + 3.65694i −1.59538 + 1.56676i −4.75940 + 1.10675i 0.0413971 + 0.0553000i 3.80313 − 1.80290i −0.727479 − 0.889670i 5.32329 + 0.659658i
17.5 −1.46639 − 1.84651i 0.0619289 − 0.130636i −0.806304 + 3.46738i 1.66760 + 1.48966i −0.332032 + 0.0772108i −2.45024 − 3.27314i 3.32359 − 1.57557i 1.88581 + 2.30624i 0.305322 − 5.26367i
17.6 −1.46526 − 1.84508i −0.0116097 + 0.0244901i −0.804358 + 3.45901i 0.497564 − 2.18001i 0.0621973 − 0.0144634i 1.43541 + 1.91749i 3.30273 − 1.56568i 1.89857 + 2.32186i −4.75135 + 2.27622i
17.7 −1.40928 − 1.77459i −0.264530 + 0.558012i −0.710119 + 3.05375i −1.78233 + 1.35030i 1.36304 − 0.316961i −0.738552 − 0.986589i 2.32457 − 1.10198i 1.65764 + 2.02720i 4.90802 + 1.25995i
17.8 −1.29748 − 1.63382i −1.16883 + 2.46559i −0.532904 + 2.29166i −2.21265 − 0.322771i 5.54485 − 1.28940i 0.266366 + 0.355823i 0.665125 − 0.315308i −2.81393 − 3.44129i 2.34352 + 4.03385i
17.9 −1.28999 − 1.62438i 0.928097 − 1.95777i −0.521552 + 2.24285i 0.934173 − 2.03158i −4.37741 + 1.01792i −2.10297 − 2.80924i 0.567350 − 0.268956i −1.07247 − 1.31158i −4.50514 + 1.10326i
17.10 −1.28465 − 1.61765i 0.394172 − 0.831485i −0.513494 + 2.20820i −1.76135 − 1.37755i −1.85143 + 0.430531i 2.63389 + 3.51846i 0.498590 − 0.236360i 1.36304 + 1.66693i 0.0343172 + 4.61892i
17.11 −1.27308 − 1.60309i −1.31201 + 2.76761i −0.496174 + 2.13371i 0.864114 − 2.06235i 6.10703 − 1.42013i −0.123455 − 0.164916i 0.352652 − 0.167177i −4.03927 − 4.93982i −4.40623 + 1.24029i
17.12 −1.19630 − 1.50640i −0.939334 + 1.98148i −0.385130 + 1.65619i 0.808350 + 2.08484i 4.10863 − 0.955421i 1.05957 + 1.41542i −0.520807 + 0.246892i −1.14487 − 1.40012i 2.17359 − 3.71179i
17.13 −1.11932 − 1.40948i 0.881394 − 1.85926i −0.280746 + 1.20730i 1.44251 + 1.70856i −3.60715 + 0.838806i 3.08527 + 4.12144i −1.23683 + 0.586328i −0.780941 − 0.955051i 0.793546 − 3.94561i
17.14 −1.10873 − 1.39613i 1.35863 − 2.86596i −0.266918 + 1.14784i −0.900694 + 2.04664i −5.50760 + 1.28074i −0.826618 − 1.10423i −1.32348 + 0.627403i −4.46880 − 5.46511i 3.85601 − 1.01168i
17.15 −1.03807 − 1.30717i 0.199600 − 0.421046i −0.178090 + 0.765846i 2.14613 − 0.627786i −0.757576 + 0.176167i 0.568703 + 0.759699i −1.83067 + 0.867844i 1.76160 + 2.15434i −3.04847 − 2.15366i
17.16 −0.806914 − 1.01608i 0.273200 − 0.576301i 0.0716782 − 0.308240i −2.23315 − 0.114207i −0.806019 + 0.187432i −0.252593 − 0.337424i −2.71592 + 1.28750i 1.64155 + 2.00753i 1.68592 + 2.36122i
17.17 −0.759470 − 0.956341i −0.595177 + 1.25550i 0.115200 − 0.495400i −0.258974 − 2.22102i 1.65270 − 0.384319i 0.786010 + 1.04999i −2.76827 + 1.31232i 0.677004 + 0.827941i −1.92737 + 1.93447i
17.18 −0.749177 − 0.943380i 0.802958 − 1.69380i 0.124294 − 0.534506i −1.84483 − 1.26356i −2.19946 + 0.511461i −1.95000 − 2.60490i −2.77446 + 1.31525i −0.325178 − 0.397676i 0.190088 + 2.68701i
17.19 −0.696510 − 0.877060i −0.936582 + 1.97567i 0.168885 − 0.726261i 2.23429 + 0.0891024i 2.38512 − 0.554636i −2.70093 − 3.60801i −2.77865 + 1.31724i −1.12706 − 1.37833i −1.47806 − 2.02167i
17.20 −0.658728 − 0.829484i −1.09704 + 2.31415i 0.198872 − 0.855215i −2.01388 + 0.971754i 2.64220 − 0.614418i −2.82386 − 3.77224i −2.75464 + 1.30586i −2.25276 − 2.75500i 2.13265 + 1.03036i
See next 80 embeddings (of 4640 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.58
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.d odd 22 1 inner
25.f odd 20 1 inner
575.w even 220 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 575.2.w.a ✓ 4640
23.d odd 22 1 inner 575.2.w.a ✓ 4640
25.f odd 20 1 inner 575.2.w.a ✓ 4640
575.w even 220 1 inner 575.2.w.a ✓ 4640
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
575.2.w.a ✓ 4640 1.a even 1 1 trivial
575.2.w.a ✓ 4640 23.d odd 22 1 inner
575.2.w.a ✓ 4640 25.f odd 20 1 inner
575.2.w.a ✓ 4640 575.w even 220 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(575, [\chi])\).