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Properties

Label 430.3.s.b
Level $430$
Weight $3$
Character orbit 430.s
Analytic conductor $11.717$
Analytic rank $0$
Dimension $264$
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] = [430,3,Mod(47,430)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(430, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([7, 8]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("430.47");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 430.s (of order \(28\), degree \(12\), 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: \(11.7166513675\)
Analytic rank: \(0\)
Dimension: \(264\)
Relative dimension: \(22\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 264 q + 44 q^{2} - 10 q^{3} - 4 q^{5} - 8 q^{6} - 24 q^{7} - 88 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 264 q + 44 q^{2} - 10 q^{3} - 4 q^{5} - 8 q^{6} - 24 q^{7} - 88 q^{8} + 4 q^{10} - 20 q^{12} - 20 q^{13} + 72 q^{15} + 176 q^{16} + 44 q^{17} + 116 q^{18} + 168 q^{21} + 42 q^{23} - 12 q^{25} + 40 q^{26} - 34 q^{27} + 48 q^{28} + 72 q^{30} - 40 q^{31} - 176 q^{32} + 288 q^{33} - 16 q^{35} + 1616 q^{36} - 32 q^{37} - 84 q^{38} + 20 q^{40} - 152 q^{41} + 56 q^{42} - 154 q^{43} - 256 q^{45} + 112 q^{46} - 100 q^{47} - 16 q^{48} - 36 q^{50} + 480 q^{51} + 16 q^{52} - 492 q^{53} - 148 q^{55} - 96 q^{56} + 180 q^{57} + 32 q^{58} - 120 q^{60} + 304 q^{61} + 124 q^{62} + 20 q^{63} - 8 q^{65} + 180 q^{66} + 20 q^{67} + 108 q^{68} + 260 q^{70} + 184 q^{71} + 232 q^{72} - 402 q^{73} + 482 q^{75} + 168 q^{76} + 190 q^{77} - 152 q^{78} + 16 q^{80} + 544 q^{81} - 212 q^{82} - 278 q^{83} - 736 q^{85} - 168 q^{86} + 68 q^{87} - 116 q^{90} + 676 q^{91} - 56 q^{92} + 16 q^{93} - 80 q^{95} + 32 q^{96} - 56 q^{97} - 660 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −1.40532 + 0.158342i −5.52534 − 0.622556i 1.94986 − 0.445042i 4.99695 − 0.174506i 7.86345 5.22839 + 5.22839i −2.66971 + 0.934170i 21.3674 + 4.87698i −6.99469 + 1.03646i
47.2 −1.40532 + 0.158342i −5.02414 − 0.566085i 1.94986 − 0.445042i −4.05569 − 2.92427i 7.15017 0.0512395 + 0.0512395i −2.66971 + 0.934170i 16.1472 + 3.68549i 6.16258 + 3.46736i
47.3 −1.40532 + 0.158342i −4.87221 − 0.548966i 1.94986 − 0.445042i −4.47380 + 2.23273i 6.93394 −4.79819 − 4.79819i −2.66971 + 0.934170i 14.6627 + 3.34666i 5.93359 − 3.84610i
47.4 −1.40532 + 0.158342i −3.96797 − 0.447082i 1.94986 − 0.445042i 3.24073 + 3.80758i 5.64706 −6.18188 − 6.18188i −2.66971 + 0.934170i 6.77052 + 1.54533i −5.15716 − 4.83773i
47.5 −1.40532 + 0.158342i −3.64246 − 0.410406i 1.94986 − 0.445042i −2.80041 + 4.14218i 5.18381 8.95118 + 8.95118i −2.66971 + 0.934170i 4.32472 + 0.987089i 3.27960 − 6.26452i
47.6 −1.40532 + 0.158342i −3.43175 − 0.386665i 1.94986 − 0.445042i −0.721455 − 4.94768i 4.88393 2.60452 + 2.60452i −2.66971 + 0.934170i 2.85303 + 0.651185i 1.79730 + 6.83884i
47.7 −1.40532 + 0.158342i −2.92390 − 0.329444i 1.94986 − 0.445042i 3.00206 − 3.99845i 4.16118 −8.63495 − 8.63495i −2.66971 + 0.934170i −0.333695 − 0.0761637i −3.58574 + 6.09446i
47.8 −1.40532 + 0.158342i −1.93476 − 0.217995i 1.94986 − 0.445042i 0.891978 + 4.91979i 2.75348 −0.550747 − 0.550747i −2.66971 + 0.934170i −5.07858 − 1.15915i −2.03252 − 6.77265i
47.9 −1.40532 + 0.158342i −1.15455 − 0.130087i 1.94986 − 0.445042i −4.56815 − 2.03273i 1.64311 −5.50596 − 5.50596i −2.66971 + 0.934170i −7.45828 − 1.70230i 6.74158 + 2.13332i
47.10 −1.40532 + 0.158342i −0.838180 − 0.0944403i 1.94986 − 0.445042i 4.73130 − 1.61703i 1.19287 0.286793 + 0.286793i −2.66971 + 0.934170i −8.08072 − 1.84437i −6.39296 + 3.02160i
47.11 −1.40532 + 0.158342i −0.457874 − 0.0515900i 1.94986 − 0.445042i 0.629508 − 4.96021i 0.651629 2.31903 + 2.31903i −2.66971 + 0.934170i −8.56736 − 1.95544i −0.0992526 + 7.07037i
47.12 −1.40532 + 0.158342i −0.240076 − 0.0270500i 1.94986 − 0.445042i 4.75984 + 1.53100i 0.341667 6.49278 + 6.49278i −2.66971 + 0.934170i −8.71745 − 1.98970i −6.93152 − 1.39786i
47.13 −1.40532 + 0.158342i 0.530430 + 0.0597651i 1.94986 − 0.445042i −3.95215 + 3.06276i −0.754888 −6.32448 − 6.32448i −2.66971 + 0.934170i −8.49657 − 1.93929i 5.06908 − 4.92995i
47.14 −1.40532 + 0.158342i 1.52609 + 0.171949i 1.94986 − 0.445042i −4.87159 − 1.12586i −2.17188 4.44341 + 4.44341i −2.66971 + 0.934170i −6.47496 − 1.47787i 7.02443 + 0.810825i
47.15 −1.40532 + 0.158342i 2.13765 + 0.240856i 1.94986 − 0.445042i −0.322640 − 4.98958i −3.04223 7.84568 + 7.84568i −2.66971 + 0.934170i −4.26280 − 0.972956i 1.24347 + 6.96087i
47.16 −1.40532 + 0.158342i 2.87792 + 0.324263i 1.94986 − 0.445042i −3.54933 + 3.52168i −4.09574 −0.616714 − 0.616714i −2.66971 + 0.934170i −0.597086 − 0.136281i 4.43033 − 5.51110i
47.17 −1.40532 + 0.158342i 2.97227 + 0.334894i 1.94986 − 0.445042i 4.82371 − 1.31599i −4.23002 −2.83364 − 2.83364i −2.66971 + 0.934170i −0.0521151 − 0.0118949i −6.57049 + 2.61318i
47.18 −1.40532 + 0.158342i 3.27052 + 0.368500i 1.94986 − 0.445042i 1.66130 + 4.71594i −4.65449 5.14766 + 5.14766i −2.66971 + 0.934170i 1.78618 + 0.407685i −3.08139 − 6.36436i
47.19 −1.40532 + 0.158342i 3.88582 + 0.437827i 1.94986 − 0.445042i −0.746864 − 4.94390i −5.53015 −5.99336 − 5.99336i −2.66971 + 0.934170i 6.13355 + 1.39994i 1.83241 + 6.82951i
47.20 −1.40532 + 0.158342i 3.91413 + 0.441017i 1.94986 − 0.445042i 0.104644 + 4.99890i −5.57045 4.41667 + 4.41667i −2.66971 + 0.934170i 6.35160 + 1.44971i −0.938594 − 7.00850i
See next 80 embeddings (of 264 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
43.e even 7 1 inner
215.s odd 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.3.s.b ✓ 264
5.c odd 4 1 inner 430.3.s.b ✓ 264
43.e even 7 1 inner 430.3.s.b ✓ 264
215.s odd 28 1 inner 430.3.s.b ✓ 264
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.3.s.b ✓ 264 1.a even 1 1 trivial
430.3.s.b ✓ 264 5.c odd 4 1 inner
430.3.s.b ✓ 264 43.e even 7 1 inner
430.3.s.b ✓ 264 215.s odd 28 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{264} + 10 T_{3}^{263} + 50 T_{3}^{262} + 202 T_{3}^{261} - 1931 T_{3}^{260} + \cdots + 11\!\cdots\!41 \) acting on \(S_{3}^{\mathrm{new}}(430, [\chi])\). Copy content Toggle raw display