Properties

Label 450.2.w.a
Level 450
Weight 2
Character orbit 450.w
Analytic conductor 3.593
Analytic rank 0
Dimension 480
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 450.w (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.59326809096\)
Analytic rank: \(0\)
Dimension: \(480\)
Relative dimension: \(30\) over \(\Q(\zeta_{60})\)
Coefficient ring index: multiple of None
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.

\(\operatorname{Tr}(f)(q) = \) \( 480q - 4q^{3} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 480q - 4q^{3} + 4q^{12} + 8q^{15} - 60q^{16} + 8q^{18} + 12q^{20} + 24q^{23} - 48q^{25} + 8q^{27} + 24q^{30} - 16q^{33} + 24q^{37} - 36q^{38} + 40q^{39} - 44q^{42} + 12q^{45} - 48q^{47} - 8q^{48} - 48q^{50} + 24q^{55} + 28q^{57} - 12q^{58} - 60q^{59} - 24q^{60} + 20q^{63} + 24q^{65} + 12q^{67} - 144q^{68} - 140q^{69} + 16q^{72} - 168q^{75} - 432q^{77} - 76q^{78} + 40q^{81} + 48q^{82} - 60q^{83} - 60q^{84} + 24q^{85} - 44q^{87} - 52q^{90} + 24q^{92} - 72q^{93} - 60q^{95} + 36q^{97} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −0.358368 0.933580i −1.72733 0.127856i −0.743145 + 0.669131i −2.14941 + 0.616468i 0.499654 + 1.65842i −1.07243 0.287356i 0.891007 + 0.453990i 2.96731 + 0.441697i 1.34580 + 1.78573i
23.2 −0.358368 0.933580i −1.72496 0.156526i −0.743145 + 0.669131i −0.578851 2.15985i 0.472042 + 1.66649i −0.339629 0.0910032i 0.891007 + 0.453990i 2.95100 + 0.540003i −1.80895 + 1.31442i
23.3 −0.358368 0.933580i −1.48144 0.897405i −0.743145 + 0.669131i 1.69030 1.46386i −0.306899 + 1.70464i 3.92816 + 1.05255i 0.891007 + 0.453990i 1.38933 + 2.65890i −1.97238 1.05343i
23.4 −0.358368 0.933580i −1.40196 + 1.01710i −0.743145 + 0.669131i 2.16221 + 0.569949i 1.45196 + 0.944351i −4.53737 1.21579i 0.891007 + 0.453990i 0.931011 2.85188i −0.242774 2.22285i
23.5 −0.358368 0.933580i −1.02833 + 1.39375i −0.743145 + 0.669131i 0.468469 2.18644i 1.66970 + 0.460550i −0.570319 0.152817i 0.891007 + 0.453990i −0.885087 2.86646i −2.20911 + 0.346198i
23.6 −0.358368 0.933580i −0.466852 1.66795i −0.743145 + 0.669131i −0.954188 + 2.02226i −1.38986 + 1.03358i 3.63027 + 0.972729i 0.891007 + 0.453990i −2.56410 + 1.55737i 2.22989 + 0.166099i
23.7 −0.358368 0.933580i −0.0491411 + 1.73135i −0.743145 + 0.669131i 1.96089 + 1.07466i 1.63397 0.574584i 3.32195 + 0.890113i 0.891007 + 0.453990i −2.99517 0.170161i 0.300562 2.21578i
23.8 −0.358368 0.933580i 0.0371786 1.73165i −0.743145 + 0.669131i 1.91306 1.15766i −1.62996 + 0.585859i −2.33480 0.625609i 0.891007 + 0.453990i −2.99724 0.128761i −1.76635 1.37113i
23.9 −0.358368 0.933580i 0.169956 + 1.72369i −0.743145 + 0.669131i −1.60127 + 1.56075i 1.54830 0.776384i −2.59992 0.696648i 0.891007 + 0.453990i −2.94223 + 0.585904i 2.03093 + 0.935586i
23.10 −0.358368 0.933580i 0.807668 1.53221i −0.743145 + 0.669131i −1.96952 + 1.05877i −1.71989 0.204927i −3.23303 0.866288i 0.891007 + 0.453990i −1.69535 2.47504i 1.69426 + 1.45927i
23.11 −0.358368 0.933580i 0.866349 + 1.49981i −0.743145 + 0.669131i −1.65651 1.50200i 1.08972 1.34629i −1.52847 0.409552i 0.891007 + 0.453990i −1.49888 + 2.59872i −0.808595 + 2.08475i
23.12 −0.358368 0.933580i 1.34709 1.08874i −0.743145 + 0.669131i 1.80264 + 1.32305i −1.49918 0.867446i 0.216796 + 0.0580902i 0.891007 + 0.453990i 0.629292 2.93326i 0.589168 2.15705i
23.13 −0.358368 0.933580i 1.39329 1.02895i −0.743145 + 0.669131i −1.12618 1.93176i −1.45992 0.932005i 3.27132 + 0.876546i 0.891007 + 0.453990i 0.882516 2.86726i −1.39987 + 1.74366i
23.14 −0.358368 0.933580i 1.56679 + 0.738354i −0.743145 + 0.669131i 1.74716 1.39550i 0.127826 1.72733i −0.469048 0.125681i 0.891007 + 0.453990i 1.90967 + 2.31369i −1.92894 1.13101i
23.15 −0.358368 0.933580i 1.67532 + 0.439663i −0.743145 + 0.669131i −0.842800 + 2.07116i −0.189920 1.72161i 2.31653 + 0.620713i 0.891007 + 0.453990i 2.61339 + 1.47315i 2.23562 + 0.0445853i
23.16 0.358368 + 0.933580i −1.73185 0.0261705i −0.743145 + 0.669131i −1.81212 + 1.31005i −0.596208 1.62620i 4.79708 + 1.28537i −0.891007 0.453990i 2.99863 + 0.0906470i −1.87244 1.22227i
23.17 0.358368 + 0.933580i −1.69592 + 0.351926i −0.743145 + 0.669131i 2.22290 + 0.242291i −0.936315 1.45716i −0.250623 0.0671543i −0.891007 0.453990i 2.75230 1.19368i 0.570419 + 2.16209i
23.18 0.358368 + 0.933580i −1.63302 0.577277i −0.743145 + 0.669131i 0.309636 + 2.21453i −0.0462871 1.73143i −4.14621 1.11097i −0.891007 0.453990i 2.33350 + 1.88541i −1.95647 + 1.08269i
23.19 0.358368 + 0.933580i −1.58003 + 0.709574i −0.743145 + 0.669131i −0.430645 2.19421i −1.22868 1.22080i 0.475783 + 0.127486i −0.891007 0.453990i 1.99301 2.24230i 1.89414 1.18837i
23.20 0.358368 + 0.933580i −0.955668 1.44454i −0.743145 + 0.669131i 2.22864 0.182152i 1.00611 1.40987i 2.61210 + 0.699910i −0.891007 0.453990i −1.17340 + 2.76100i 0.968725 + 2.01533i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 437.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
25.f odd 20 1 inner
225.w even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.w.a 480
9.d odd 6 1 inner 450.2.w.a 480
25.f odd 20 1 inner 450.2.w.a 480
225.w even 60 1 inner 450.2.w.a 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.w.a 480 1.a even 1 1 trivial
450.2.w.a 480 9.d odd 6 1 inner
450.2.w.a 480 25.f odd 20 1 inner
450.2.w.a 480 225.w even 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(450, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database