Properties

Label 224.3.v.b
Level 224
Weight 3
Character orbit 224.v
Analytic conductor 6.104
Analytic rank 0
Dimension 240
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 224.v (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(60\) over \(\Q(\zeta_{8})\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240q - 8q^{2} - 8q^{4} - 4q^{7} - 8q^{8} - 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 240q - 8q^{2} - 8q^{4} - 4q^{7} - 8q^{8} - 8q^{9} - 8q^{11} + 12q^{14} - 112q^{16} - 176q^{18} - 4q^{21} - 192q^{22} + 128q^{23} - 8q^{25} + 56q^{28} - 8q^{29} - 16q^{30} - 8q^{32} + 92q^{35} + 192q^{36} - 8q^{37} - 8q^{39} - 424q^{42} + 128q^{43} - 16q^{44} - 8q^{46} - 320q^{50} - 80q^{51} - 192q^{53} + 608q^{56} - 8q^{57} - 712q^{58} + 264q^{60} + 496q^{63} - 272q^{64} - 16q^{65} + 304q^{67} + 320q^{70} + 504q^{71} - 8q^{72} + 232q^{74} + 164q^{77} + 560q^{78} - 1000q^{84} - 208q^{85} - 8q^{86} - 800q^{88} + 188q^{91} + 1560q^{92} + 64q^{93} - 16q^{95} - 376q^{98} + 64q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.96919 + 0.349710i −0.313666 0.129925i 3.75541 1.37729i −3.56967 8.61795i 0.663104 + 0.146154i −6.10386 + 3.42679i −6.91345 + 4.02544i −6.28246 6.28246i 10.0431 + 15.7220i
13.2 −1.96919 + 0.349710i 0.313666 + 0.129925i 3.75541 1.37729i 3.56967 + 8.61795i −0.663104 0.146154i −3.42679 + 6.10386i −6.91345 + 4.02544i −6.28246 6.28246i −10.0431 15.7220i
13.3 −1.91889 0.563790i −4.49480 1.86181i 3.36428 + 2.16370i 0.432309 + 1.04369i 7.57535 + 6.10672i −6.93508 0.951145i −5.23581 6.04866i 10.3729 + 10.3729i −0.241133 2.24645i
13.4 −1.91889 0.563790i 4.49480 + 1.86181i 3.36428 + 2.16370i −0.432309 1.04369i −7.57535 6.10672i 0.951145 + 6.93508i −5.23581 6.04866i 10.3729 + 10.3729i 0.241133 + 2.24645i
13.5 −1.91469 0.577879i −2.09916 0.869500i 3.33211 + 2.21292i −0.289108 0.697967i 3.51678 + 2.87788i 4.51230 + 5.35156i −5.10118 6.16263i −2.71353 2.71353i 0.150212 + 1.50346i
13.6 −1.91469 0.577879i 2.09916 + 0.869500i 3.33211 + 2.21292i 0.289108 + 0.697967i −3.51678 2.87788i −5.35156 4.51230i −5.10118 6.16263i −2.71353 2.71353i −0.150212 1.50346i
13.7 −1.89176 + 0.649047i −5.08830 2.10764i 3.15748 2.45567i −0.976625 2.35778i 10.9938 + 0.684602i 4.79964 + 5.09544i −4.37933 + 6.69489i 15.0847 + 15.0847i 3.37785 + 3.82647i
13.8 −1.89176 + 0.649047i 5.08830 + 2.10764i 3.15748 2.45567i 0.976625 + 2.35778i −10.9938 0.684602i −5.09544 4.79964i −4.37933 + 6.69489i 15.0847 + 15.0847i −3.37785 3.82647i
13.9 −1.84088 + 0.781766i −3.58432 1.48467i 2.77768 2.87828i 3.04365 + 7.34802i 7.75897 0.0689922i −1.24702 6.88803i −2.86325 + 7.47006i 4.27913 + 4.27913i −11.3474 11.1474i
13.10 −1.84088 + 0.781766i 3.58432 + 1.48467i 2.77768 2.87828i −3.04365 7.34802i −7.75897 + 0.0689922i 6.88803 + 1.24702i −2.86325 + 7.47006i 4.27913 + 4.27913i 11.3474 + 11.1474i
13.11 −1.62574 1.16489i −3.33013 1.37939i 1.28608 + 3.78761i −3.52027 8.49868i 3.80711 + 6.12175i 1.37310 6.86401i 2.32130 7.65582i 2.82310 + 2.82310i −4.17695 + 17.9174i
13.12 −1.62574 1.16489i 3.33013 + 1.37939i 1.28608 + 3.78761i 3.52027 + 8.49868i −3.80711 6.12175i 6.86401 1.37310i 2.32130 7.65582i 2.82310 + 2.82310i 4.17695 17.9174i
13.13 −1.46638 + 1.36005i −1.65755 0.686579i 0.300514 3.98870i 0.205220 + 0.495445i 3.36437 1.24757i −0.193440 + 6.99733i 4.98417 + 6.25764i −4.08789 4.08789i −0.974760 0.447398i
13.14 −1.46638 + 1.36005i 1.65755 + 0.686579i 0.300514 3.98870i −0.205220 0.495445i −3.36437 + 1.24757i −6.99733 + 0.193440i 4.98417 + 6.25764i −4.08789 4.08789i 0.974760 + 0.447398i
13.15 −1.41192 + 1.41651i −2.79421 1.15740i −0.0129803 3.99998i −1.86072 4.49217i 5.58465 2.32386i 0.944293 6.93602i 5.68432 + 5.62925i 0.104065 + 0.104065i 8.99036 + 3.70685i
13.16 −1.41192 + 1.41651i 2.79421 + 1.15740i −0.0129803 3.99998i 1.86072 + 4.49217i −5.58465 + 2.32386i 6.93602 0.944293i 5.68432 + 5.62925i 0.104065 + 0.104065i −8.99036 3.70685i
13.17 −1.33217 1.49175i −2.62896 1.08895i −0.450649 + 3.97453i 2.38974 + 5.76934i 1.87778 + 5.37243i −3.62994 + 5.98528i 6.52936 4.62250i −0.638332 0.638332i 5.42289 11.2506i
13.18 −1.33217 1.49175i 2.62896 + 1.08895i −0.450649 + 3.97453i −2.38974 5.76934i −1.87778 5.37243i −5.98528 + 3.62994i 6.52936 4.62250i −0.638332 0.638332i −5.42289 + 11.2506i
13.19 −1.28270 1.53450i −0.416824 0.172654i −0.709381 + 3.93659i 1.21861 + 2.94198i 0.269721 + 0.861080i −3.48320 6.07185i 6.95063 3.96091i −6.22003 6.22003i 2.95137 5.64363i
13.20 −1.28270 1.53450i 0.416824 + 0.172654i −0.709381 + 3.93659i −1.21861 2.94198i −0.269721 0.861080i 6.07185 + 3.48320i 6.95063 3.96091i −6.22003 6.22003i −2.95137 + 5.64363i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
32.g even 8 1 inner
224.v odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.v.b 240
7.b odd 2 1 inner 224.3.v.b 240
32.g even 8 1 inner 224.3.v.b 240
224.v odd 8 1 inner 224.3.v.b 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.v.b 240 1.a even 1 1 trivial
224.3.v.b 240 7.b odd 2 1 inner
224.3.v.b 240 32.g even 8 1 inner
224.3.v.b 240 224.v odd 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{240} + \cdots\) acting on \(S_{3}^{\mathrm{new}}(224, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database