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Properties

Label 539.2.p.a
Level $539$
Weight $2$
Character orbit 539.p
Analytic conductor $4.304$
Analytic rank $0$
Dimension $324$
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] = [539,2,Mod(76,539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(539, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([1, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("539.76");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.p (of order \(14\), degree \(6\), 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: \(4.30393666895\)
Analytic rank: \(0\)
Dimension: \(324\)
Relative dimension: \(54\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 324 q - 14 q^{3} + 42 q^{4} - 14 q^{5} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 324 q - 14 q^{3} + 42 q^{4} - 14 q^{5} + 40 q^{9} - 6 q^{11} - 14 q^{12} - 12 q^{14} - 2 q^{15} - 74 q^{16} - 70 q^{20} - 35 q^{22} - 24 q^{23} + 16 q^{25} - 42 q^{26} - 56 q^{27} - 7 q^{33} + 14 q^{34} - 10 q^{36} - 14 q^{37} - 70 q^{38} + 106 q^{42} + 43 q^{44} - 126 q^{45} + 14 q^{47} - 48 q^{49} + 58 q^{53} + 105 q^{55} + 74 q^{56} + 42 q^{58} + 14 q^{59} + 98 q^{60} + 22 q^{64} - 105 q^{66} - 36 q^{67} + 154 q^{69} - 22 q^{70} + 50 q^{71} - 14 q^{75} + 12 q^{77} + 12 q^{81} - 14 q^{82} - 84 q^{86} + 37 q^{88} + 42 q^{89} - 282 q^{91} - 154 q^{92} - 32 q^{93} + 80 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1 −1.20486 − 2.50192i 0.0714649 + 0.0163114i −3.56092 + 4.46525i 2.80665 + 0.640599i −0.0452954 − 0.198452i −2.58012 + 0.585648i 10.0475 + 2.29328i −2.69807 − 1.29932i −1.77889 − 7.79383i
76.2 −1.13634 − 2.35963i −0.493691 − 0.112682i −3.02961 + 3.79901i −0.771563 − 0.176104i 0.295113 + 1.29297i 2.35955 + 1.19689i 7.30026 + 1.66624i −2.47187 − 1.19039i 0.461216 + 2.02072i
76.3 −1.08952 − 2.26242i −2.76552 − 0.631211i −2.68449 + 3.36624i −2.05211 − 0.468380i 1.58503 + 6.94447i 0.329681 + 2.62513i 5.64439 + 1.28830i 4.54675 + 2.18960i 1.17615 + 5.15303i
76.4 −1.07750 − 2.23745i 2.24592 + 0.512616i −2.59821 + 3.25806i 1.95946 + 0.447235i −1.27303 − 5.57749i 2.09098 + 1.62105i 5.24709 + 1.19761i 2.07847 + 1.00094i −1.11066 − 4.86611i
76.5 −1.06439 − 2.21023i −1.25487 − 0.286416i −2.50521 + 3.14143i −1.23425 − 0.281709i 0.702626 + 3.07841i −0.216465 − 2.63688i 4.82647 + 1.10161i −1.21024 − 0.582823i 0.691081 + 3.02782i
76.6 −1.04985 − 2.18004i 2.33507 + 0.532964i −2.40341 + 3.01379i −4.16803 − 0.951326i −1.28959 − 5.65008i 2.40252 − 1.10811i 4.37542 + 0.998660i 2.46559 + 1.18737i 2.30189 + 10.0852i
76.7 −1.01224 − 2.10194i −3.15380 − 0.719834i −2.14656 + 2.69170i 2.19250 + 0.500424i 1.67936 + 7.35776i −1.38056 − 2.25700i 3.28165 + 0.749015i 6.72539 + 3.23878i −1.16748 − 5.11506i
76.8 −1.00364 − 2.08409i 2.98209 + 0.680642i −2.08914 + 2.61970i −0.377041 − 0.0860571i −1.57444 − 6.89805i −1.63031 + 2.08377i 3.04609 + 0.695251i 5.72667 + 2.75782i 0.199064 + 0.872156i
76.9 −0.883051 − 1.83368i 0.861327 + 0.196592i −1.33561 + 1.67480i 4.08698 + 0.932827i −0.400110 − 1.75299i 1.98134 − 1.75337i 0.282043 + 0.0643745i −1.99967 − 0.962991i −1.89851 − 8.31793i
76.10 −0.856141 − 1.77779i −1.16665 − 0.266280i −1.18060 + 1.48042i 2.81823 + 0.643242i 0.525424 + 2.30203i −1.63441 + 2.08055i −0.204822 − 0.0467492i −1.41274 − 0.680342i −1.26925 − 5.56094i
76.11 −0.842046 − 1.74853i 1.63999 + 0.374318i −1.10132 + 1.38102i −0.472311 − 0.107802i −0.726445 − 3.18276i −2.64560 − 0.0279441i −0.442010 − 0.100886i −0.153443 − 0.0738941i 0.209213 + 0.916622i
76.12 −0.830698 − 1.72496i 1.19405 + 0.272533i −1.03846 + 1.30218i −0.203997 − 0.0465610i −0.521782 − 2.28608i −0.808464 − 2.51920i −0.624259 − 0.142483i −1.35144 − 0.650817i 0.0891439 + 0.390565i
76.13 −0.743826 − 1.54457i −2.50374 − 0.571463i −0.585441 + 0.734119i 2.07578 + 0.473784i 0.979684 + 4.29228i 2.63798 + 0.202641i −1.77336 − 0.404757i 3.23926 + 1.55994i −0.812228 − 3.55861i
76.14 −0.660894 − 1.37236i 0.228024 + 0.0520451i −0.199611 + 0.250304i −2.93571 − 0.670057i −0.0792753 − 0.347328i 1.21395 + 2.35081i −2.49460 − 0.569376i −2.65362 − 1.27792i 1.02063 + 4.47169i
76.15 −0.636546 − 1.32180i 0.314620 + 0.0718100i −0.0949871 + 0.119110i 0.166617 + 0.0380292i −0.105352 − 0.461576i −0.0729806 + 2.64474i −2.64271 − 0.603181i −2.60908 − 1.25647i −0.0557922 − 0.244441i
76.16 −0.627090 − 1.30217i −1.71706 − 0.391908i −0.0554150 + 0.0694883i −2.96698 − 0.677193i 0.566423 + 2.48166i −2.62859 − 0.300822i −2.69288 − 0.614633i 0.0918039 + 0.0442104i 0.978744 + 4.28816i
76.17 −0.577230 − 1.19863i −1.76362 − 0.402535i 0.143459 − 0.179891i −3.39480 − 0.774842i 0.535524 + 2.34629i 1.34701 − 2.27718i −2.89248 − 0.660190i 0.245419 + 0.118187i 1.03083 + 4.51638i
76.18 −0.486488 − 1.01020i 2.59725 + 0.592805i 0.463141 − 0.580760i −0.0107786 − 0.00246015i −0.664677 − 2.91214i 2.23578 − 1.41466i −2.99825 − 0.684332i 3.69138 + 1.77767i 0.00275842 + 0.0120854i
76.19 −0.401187 − 0.833074i −2.14020 − 0.488488i 0.713919 − 0.895226i 0.858535 + 0.195955i 0.451676 + 1.97892i 2.58458 + 0.565653i −2.83512 − 0.647098i 1.63895 + 0.789276i −0.181188 − 0.793837i
76.20 −0.374937 − 0.778564i −2.02457 − 0.462094i 0.781395 − 0.979838i 1.26225 + 0.288101i 0.399315 + 1.74951i −2.33380 − 1.24634i −2.74079 − 0.625568i 1.18244 + 0.569431i −0.248960 − 1.09076i
See next 80 embeddings (of 324 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.54
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
49.f odd 14 1 inner
539.p even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.p.a ✓ 324
11.b odd 2 1 inner 539.2.p.a ✓ 324
49.f odd 14 1 inner 539.2.p.a ✓ 324
539.p even 14 1 inner 539.2.p.a ✓ 324
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
539.2.p.a ✓ 324 1.a even 1 1 trivial
539.2.p.a ✓ 324 11.b odd 2 1 inner
539.2.p.a ✓ 324 49.f odd 14 1 inner
539.2.p.a ✓ 324 539.p even 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(539, [\chi])\).