↑

Properties

Label 520.2.cz.c
Level $520$
Weight $2$
Character orbit 520.cz
Analytic conductor $4.152$
Analytic rank $0$
Dimension $304$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [520,2,Mod(19,520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(520, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 6, 6, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("520.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 520.cz (of order \(12\), 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: \(4.15222090511\)
Analytic rank: \(0\)
Dimension: \(304\)
Relative dimension: \(76\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 304 q - 12 q^{4} - 16 q^{6} + 160 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 304 q - 12 q^{4} - 16 q^{6} + 160 q^{9} - 6 q^{10} - 24 q^{11} - 32 q^{14} - 20 q^{16} + 32 q^{19} - 6 q^{20} - 52 q^{24} + 8 q^{26} - 6 q^{30} + 16 q^{34} - 44 q^{35} - 48 q^{36} - 32 q^{40} - 24 q^{41} - 16 q^{44} - 144 q^{46} - 24 q^{49} + 8 q^{50} - 48 q^{54} - 24 q^{56} + 72 q^{59} + 8 q^{60} - 56 q^{65} - 72 q^{66} - 88 q^{70} + 60 q^{74} - 48 q^{75} - 116 q^{76} + 44 q^{80} - 8 q^{81} + 164 q^{84} + 36 q^{86} + 136 q^{89} + 48 q^{91} + 60 q^{94} - 148 q^{96} - 120 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.41413 + 0.0153763i 2.78902 + 1.61024i 1.99953 − 0.0434881i 1.56358 + 1.59850i −3.96879 − 2.23420i 0.955178 − 3.56477i −2.82692 + 0.0922432i 3.68574 + 6.38389i −2.23569 − 2.23645i
19.2 −1.40483 + 0.162638i −0.243322 − 0.140482i 1.94710 − 0.456959i −0.479781 − 2.18399i 0.364675 + 0.157780i −1.17116 + 4.37082i −2.66102 + 0.958622i −1.46053 − 2.52971i 1.02921 + 2.99010i
19.3 −1.38645 + 0.278825i −2.07279 − 1.19672i 1.84451 − 0.773155i −2.17849 − 0.504167i 3.20750 + 1.08126i 0.180261 − 0.672745i −2.34176 + 1.58624i 1.36429 + 2.36303i 3.16095 + 0.0915882i
19.4 −1.38608 − 0.280684i 1.09988 + 0.635018i 1.84243 + 0.778100i −1.22771 + 1.86889i −1.34629 − 1.18890i −0.483677 + 1.80511i −2.33536 − 1.59565i −0.693505 − 1.20119i 2.22627 − 2.24583i
19.5 −1.38458 + 0.287979i 1.30304 + 0.752308i 1.83414 − 0.797461i −2.15715 + 0.588831i −2.02081 − 0.666385i 0.262070 − 0.978057i −2.30986 + 1.63234i −0.368066 − 0.637508i 2.81718 − 1.43650i
19.6 −1.37478 − 0.331632i 2.18653 + 1.26239i 1.78004 + 0.911841i −0.190942 − 2.22790i −2.58734 − 2.46063i 0.700421 − 2.61401i −2.14477 − 1.84390i 1.68727 + 2.92243i −0.476339 + 3.12620i
19.7 −1.36677 − 0.363235i −0.793743 − 0.458267i 1.73612 + 0.992917i 1.96532 − 1.06655i 0.918405 + 0.914661i 0.00465034 − 0.0173553i −2.01222 − 1.98771i −1.07998 − 1.87058i −3.07355 + 0.743854i
19.8 −1.36073 − 0.385251i −0.636025 − 0.367209i 1.70316 + 1.04844i 1.08643 + 1.95440i 0.723990 + 0.744702i −0.165423 + 0.617369i −1.91363 − 2.08279i −1.23031 − 2.13097i −0.725399 − 3.07795i
19.9 −1.34307 + 0.442894i 1.30304 + 0.752308i 1.60769 − 1.18968i 2.15715 − 0.588831i −2.08326 − 0.433298i −0.262070 + 0.978057i −1.63234 + 2.30986i −0.368066 − 0.637508i −2.63641 + 1.74623i
19.10 −1.34012 + 0.451758i −2.07279 − 1.19672i 1.59183 − 1.21082i 2.17849 + 0.504167i 3.31841 + 0.667352i −0.180261 + 0.672745i −1.58624 + 2.34176i 1.36429 + 2.36303i −3.14719 + 0.308508i
19.11 −1.29794 + 0.561566i −0.243322 − 0.140482i 1.36929 − 1.45776i 0.479781 + 2.18399i 0.394708 + 0.0456956i 1.17116 − 4.37082i −0.958622 + 2.66102i −1.46053 − 2.52971i −1.84918 − 2.56525i
19.12 −1.28548 − 0.589537i −1.44911 − 0.836645i 1.30489 + 1.51567i −2.22720 + 0.198930i 1.36956 + 1.92979i 1.04673 − 3.90647i −0.783862 − 2.71764i −0.100049 − 0.173290i 2.98029 + 1.05730i
19.13 −1.26999 − 0.622201i −2.57233 − 1.48514i 1.22573 + 1.58037i 0.279042 − 2.21859i 2.34278 + 3.48661i −0.308941 + 1.15298i −0.573354 − 2.76970i 2.91127 + 5.04247i −1.73479 + 2.64396i
19.14 −1.23236 + 0.693749i 2.78902 + 1.61024i 1.03743 − 1.70990i −1.56358 − 1.59850i −4.55417 − 0.0495190i −0.955178 + 3.56477i −0.0922432 + 2.82692i 3.68574 + 6.38389i 3.03586 + 0.885197i
19.15 −1.12317 − 0.859359i 1.20799 + 0.697433i 0.523005 + 1.93041i −1.58462 − 1.57765i −0.757428 − 1.82143i 0.217575 − 0.812000i 1.07149 − 2.61762i −0.527174 − 0.913092i 0.424030 + 3.13372i
19.16 −1.06004 + 0.936119i 1.09988 + 0.635018i 0.247362 − 1.98464i 1.22771 − 1.86889i −1.76037 + 0.356479i 0.483677 − 1.80511i 1.59565 + 2.33536i −0.693505 − 1.20119i 0.448083 + 3.13037i
19.17 −1.04105 − 0.957196i 2.07640 + 1.19881i 0.167552 + 1.99297i 1.66852 + 1.48864i −1.01413 − 3.23554i −0.957940 + 3.57508i 1.73323 − 2.23515i 1.37430 + 2.38036i −0.312088 − 3.14684i
19.18 −1.03705 − 0.961527i −2.16660 − 1.25089i 0.150933 + 1.99430i −1.68896 + 1.46540i 1.04410 + 3.38047i −1.22960 + 4.58893i 1.76104 − 2.21331i 1.62943 + 2.82225i 3.16056 + 0.104293i
19.19 −1.02478 + 0.974592i 2.18653 + 1.26239i 0.100343 − 1.99748i 0.190942 + 2.22790i −3.47102 + 0.837298i −0.700421 + 2.61401i 1.84390 + 2.14477i 1.68727 + 2.92243i −2.36697 − 2.09701i
19.20 −1.01559 − 0.984160i 0.637901 + 0.368292i 0.0628570 + 1.99901i 2.19106 − 0.446362i −0.285389 − 1.00183i 0.727468 − 2.71495i 1.90351 − 2.09204i −1.22872 − 2.12821i −2.66452 − 1.70304i
See next 80 embeddings (of 304 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.76
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.d odd 2 1 inner
13.f odd 12 1 inner
40.e odd 2 1 inner
65.s odd 12 1 inner
104.u even 12 1 inner
520.cz even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 520.2.cz.c ✓ 304
5.b even 2 1 inner 520.2.cz.c ✓ 304
8.d odd 2 1 inner 520.2.cz.c ✓ 304
13.f odd 12 1 inner 520.2.cz.c ✓ 304
40.e odd 2 1 inner 520.2.cz.c ✓ 304
65.s odd 12 1 inner 520.2.cz.c ✓ 304
104.u even 12 1 inner 520.2.cz.c ✓ 304
520.cz even 12 1 inner 520.2.cz.c ✓ 304
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.cz.c ✓ 304 1.a even 1 1 trivial
520.2.cz.c ✓ 304 5.b even 2 1 inner
520.2.cz.c ✓ 304 8.d odd 2 1 inner
520.2.cz.c ✓ 304 13.f odd 12 1 inner
520.2.cz.c ✓ 304 40.e odd 2 1 inner
520.2.cz.c ✓ 304 65.s odd 12 1 inner
520.2.cz.c ✓ 304 104.u even 12 1 inner
520.2.cz.c ✓ 304 520.cz even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(520, [\chi])\):

\( T_{3}^{152} - 154 T_{3}^{150} + 12381 T_{3}^{148} - 683630 T_{3}^{146} + 28926510 T_{3}^{144} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
\( T_{7}^{152} + 6 T_{7}^{150} - 1976 T_{7}^{148} - 11928 T_{7}^{146} + 2277332 T_{7}^{144} + \cdots + 96\!\cdots\!96 \) Copy content Toggle raw display