Properties

Label 430.2.r.a
Level 430
Weight 2
Character orbit 430.r
Analytic conductor 3.434
Analytic rank 0
Dimension 264
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 430.r (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.43356728692\)
Analytic rank: \(0\)
Dimension: \(264\)
Relative dimension: \(22\) over \(\Q(\zeta_{28})\)
Coefficient ring index: multiple of None
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.

\(\operatorname{Tr}(f)(q) = \) \( 264q - 8q^{6} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 264q - 8q^{6} - 40q^{13} + 44q^{16} - 8q^{17} - 16q^{21} - 24q^{23} - 8q^{25} + 48q^{31} - 56q^{33} + 32q^{35} + 256q^{36} - 100q^{38} + 40q^{41} - 168q^{43} + 48q^{47} - 280q^{51} - 16q^{52} + 24q^{53} - 8q^{56} - 40q^{57} - 16q^{60} - 84q^{62} + 84q^{65} + 144q^{66} + 40q^{67} + 20q^{68} - 252q^{73} - 56q^{77} - 56q^{78} + 48q^{81} - 56q^{82} - 44q^{83} + 20q^{86} - 144q^{87} + 4q^{90} + 24q^{92} + 28q^{95} + 8q^{96} - 60q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1 −0.846724 0.532032i −2.43675 + 1.53111i 0.433884 + 0.900969i 0.235258 2.22366i 2.87785 −1.03661 + 1.03661i 0.111964 0.993712i 2.29179 4.75895i −1.38226 + 1.75766i
27.2 −0.846724 0.532032i −2.36912 + 1.48862i 0.433884 + 0.900969i 1.52080 + 1.63926i 2.79798 2.44635 2.44635i 0.111964 0.993712i 2.09509 4.35051i −0.415555 2.19711i
27.3 −0.846724 0.532032i −1.58096 + 0.993385i 0.433884 + 0.900969i −2.11822 + 0.716352i 1.86715 −0.600259 + 0.600259i 0.111964 0.993712i 0.210980 0.438105i 2.17467 + 0.520406i
27.4 −0.846724 0.532032i −1.25844 + 0.790731i 0.433884 + 0.900969i 1.73201 1.41427i 1.48625 1.74142 1.74142i 0.111964 0.993712i −0.343232 + 0.712728i −2.21897 + 0.276012i
27.5 −0.846724 0.532032i −0.0459382 + 0.0288649i 0.433884 + 0.900969i −1.90179 1.17609i 0.0542540 −0.499604 + 0.499604i 0.111964 0.993712i −1.30037 + 2.70025i 0.984578 + 2.00764i
27.6 −0.846724 0.532032i 0.296314 0.186186i 0.433884 + 0.900969i −0.486637 + 2.18247i −0.349954 0.0102848 0.0102848i 0.111964 0.993712i −1.24851 + 2.59257i 1.57319 1.58905i
27.7 −0.846724 0.532032i 0.306612 0.192657i 0.433884 + 0.900969i 1.17132 1.90473i −0.362116 −3.59081 + 3.59081i 0.111964 0.993712i −1.24476 + 2.58476i −2.00516 + 0.989606i
27.8 −0.846724 0.532032i 1.28668 0.808477i 0.433884 + 0.900969i 2.21966 + 0.270414i −1.51960 2.16897 2.16897i 0.111964 0.993712i −0.299731 + 0.622397i −1.73557 1.40989i
27.9 −0.846724 0.532032i 1.93498 1.21583i 0.433884 + 0.900969i 0.121846 2.23275i −2.28525 1.40881 1.40881i 0.111964 0.993712i 0.964257 2.00230i −1.29106 + 1.82569i
27.10 −0.846724 0.532032i 1.99342 1.25255i 0.433884 + 0.900969i 1.05240 + 1.97293i −2.35427 −2.68327 + 2.68327i 0.111964 0.993712i 1.10318 2.29079i 0.158565 2.23044i
27.11 −0.846724 0.532032i 2.25003 1.41379i 0.433884 + 0.900969i −2.02218 + 0.954350i −2.65734 2.89927 2.89927i 0.111964 0.993712i 1.76219 3.65923i 2.21997 + 0.267794i
27.12 0.846724 + 0.532032i −2.83739 + 1.78285i 0.433884 + 0.900969i −0.608113 2.15179i −3.35102 1.61190 1.61190i −0.111964 + 0.993712i 3.57059 7.41440i 0.629917 2.14551i
27.13 0.846724 + 0.532032i −1.90831 + 1.19907i 0.433884 + 0.900969i 2.05629 + 0.878446i −2.25376 2.27342 2.27342i −0.111964 + 0.993712i 0.902233 1.87351i 1.27375 + 1.83781i
27.14 0.846724 + 0.532032i −1.65275 + 1.03849i 0.433884 + 0.900969i −1.53740 + 1.62370i −1.95193 0.349245 0.349245i −0.111964 + 0.993712i 0.351458 0.729810i −2.16561 + 0.556884i
27.15 0.846724 + 0.532032i −1.05896 + 0.665389i 0.433884 + 0.900969i −1.40404 1.74031i −1.25066 −1.58379 + 1.58379i −0.111964 + 0.993712i −0.622995 + 1.29366i −0.262931 2.22056i
27.16 0.846724 + 0.532032i −0.527328 + 0.331342i 0.433884 + 0.900969i 1.70699 + 1.44436i −0.622786 −1.95690 + 1.95690i −0.111964 + 0.993712i −1.13336 + 2.35345i 0.676901 + 2.13115i
27.17 0.846724 + 0.532032i −0.126989 + 0.0797926i 0.433884 + 0.900969i 1.11185 1.94005i −0.149977 −0.830650 + 0.830650i −0.111964 + 0.993712i −1.29189 + 2.68264i 1.97360 1.05114i
27.18 0.846724 + 0.532032i 0.919405 0.577701i 0.433884 + 0.900969i −1.69116 + 1.46286i 1.08584 −2.01843 + 2.01843i −0.111964 + 0.993712i −0.790083 + 1.64062i −2.21024 + 0.338890i
27.19 0.846724 + 0.532032i 0.969909 0.609434i 0.433884 + 0.900969i 1.59901 1.56306i 1.14548 2.66216 2.66216i −0.111964 + 0.993712i −0.732338 + 1.52072i 2.18552 0.472759i
27.20 0.846724 + 0.532032i 1.04232 0.654933i 0.433884 + 0.900969i −0.0867089 + 2.23439i 1.23100 2.75433 2.75433i −0.111964 + 0.993712i −0.644158 + 1.33761i −1.26218 + 1.84578i
See next 80 embeddings (of 264 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 383.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
43.f odd 14 1 inner
215.r even 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.r.a 264
5.c odd 4 1 inner 430.2.r.a 264
43.f odd 14 1 inner 430.2.r.a 264
215.r even 28 1 inner 430.2.r.a 264
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.r.a 264 1.a even 1 1 trivial
430.2.r.a 264 5.c odd 4 1 inner
430.2.r.a 264 43.f odd 14 1 inner
430.2.r.a 264 215.r even 28 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(430, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database