Properties

Label 441.2.u.c
Level $441$
Weight $2$
Character orbit 441.u
Analytic conductor $3.521$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.u (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{7})\)
Twist minimal: no (minimal twist has level 147)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24q - q^{2} - 3q^{4} - 3q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q - q^{2} - 3q^{4} - 3q^{8} - 30q^{10} - 9q^{11} + 21q^{14} - 29q^{16} - 5q^{17} + 26q^{19} + 13q^{20} + 11q^{22} - 4q^{23} - 28q^{25} + 22q^{26} - 7q^{28} - 6q^{29} + 36q^{31} - 14q^{32} + 46q^{34} + 7q^{35} - 22q^{37} + 45q^{38} + 35q^{40} + 11q^{41} + 6q^{43} - 82q^{44} - 16q^{46} - 29q^{47} - 42q^{49} + 48q^{50} - 50q^{52} - 28q^{53} + 23q^{55} - 21q^{56} + 39q^{58} + 15q^{59} - 32q^{61} + 8q^{62} + 29q^{64} + 21q^{65} - 34q^{67} + 22q^{68} - 24q^{71} - 15q^{73} - 6q^{74} + 7q^{76} + 21q^{77} - 34q^{79} - 8q^{80} + 14q^{82} - 14q^{83} + 20q^{85} + 100q^{86} - 108q^{88} - 10q^{89} + 84q^{91} + 21q^{92} + 99q^{94} - 18q^{95} - 64q^{97} - 91q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 −0.540253 + 2.36700i 0 −3.50889 1.68979i −1.42001 + 1.78063i 0 −1.01369 2.44386i 2.86791 3.59625i 0 −3.44760 4.32315i
64.2 −0.271791 + 1.19079i 0 0.457820 + 0.220474i 0.164209 0.205912i 0 2.60345 + 0.471214i −1.91005 + 2.39513i 0 0.200568 + 0.251504i
64.3 0.243969 1.06890i 0 0.718913 + 0.346210i 2.19062 2.74696i 0 0.626501 2.57050i 1.91263 2.39836i 0 −2.40177 3.01173i
64.4 0.345553 1.51397i 0 −0.370753 0.178545i −0.0888147 + 0.111370i 0 −0.524245 + 2.59329i 1.53801 1.92860i 0 0.137921 + 0.172947i
127.1 −1.72952 0.832892i 0 1.05054 + 1.31734i 0.379489 + 1.66265i 0 0.548424 + 2.58829i 0.134579 + 0.589630i 0 0.728476 3.19166i
127.2 −1.28533 0.618981i 0 0.0219496 + 0.0275239i −0.813628 3.56474i 0 −0.766737 2.53222i 0.623724 + 2.73271i 0 −1.16073 + 5.08547i
127.3 0.0441061 + 0.0212404i 0 −1.24549 1.56179i 0.763028 + 3.34304i 0 1.92953 1.81022i −0.0435471 0.190792i 0 −0.0373533 + 0.163656i
127.4 2.06977 + 0.996749i 0 2.04346 + 2.56242i 0.349558 + 1.53152i 0 −0.354322 + 2.62192i 0.653025 + 2.86109i 0 −0.803031 + 3.51831i
190.1 −1.18466 + 1.48552i 0 −0.358304 1.56983i 0.295313 + 0.142215i 0 −2.01260 1.71740i −0.667291 0.321350i 0 −0.561110 + 0.270216i
190.2 −0.159899 + 0.200507i 0 0.430406 + 1.88573i 1.50958 + 0.726973i 0 2.63685 0.216906i −0.909048 0.437774i 0 −0.387144 + 0.186439i
190.3 0.309296 0.387845i 0 0.390282 + 1.70994i −0.734303 0.353621i 0 −2.35634 + 1.20320i 1.67779 + 0.807983i 0 −0.364267 + 0.175422i
190.4 1.65876 2.08002i 0 −1.12995 4.95062i −2.59504 1.24971i 0 −1.31683 + 2.29477i −7.37774 3.55293i 0 −6.90396 + 3.32477i
253.1 −1.18466 1.48552i 0 −0.358304 + 1.56983i 0.295313 0.142215i 0 −2.01260 + 1.71740i −0.667291 + 0.321350i 0 −0.561110 0.270216i
253.2 −0.159899 0.200507i 0 0.430406 1.88573i 1.50958 0.726973i 0 2.63685 + 0.216906i −0.909048 + 0.437774i 0 −0.387144 0.186439i
253.3 0.309296 + 0.387845i 0 0.390282 1.70994i −0.734303 + 0.353621i 0 −2.35634 1.20320i 1.67779 0.807983i 0 −0.364267 0.175422i
253.4 1.65876 + 2.08002i 0 −1.12995 + 4.95062i −2.59504 + 1.24971i 0 −1.31683 2.29477i −7.37774 + 3.55293i 0 −6.90396 3.32477i
316.1 −1.72952 + 0.832892i 0 1.05054 1.31734i 0.379489 1.66265i 0 0.548424 2.58829i 0.134579 0.589630i 0 0.728476 + 3.19166i
316.2 −1.28533 + 0.618981i 0 0.0219496 0.0275239i −0.813628 + 3.56474i 0 −0.766737 + 2.53222i 0.623724 2.73271i 0 −1.16073 5.08547i
316.3 0.0441061 0.0212404i 0 −1.24549 + 1.56179i 0.763028 3.34304i 0 1.92953 + 1.81022i −0.0435471 + 0.190792i 0 −0.0373533 0.163656i
316.4 2.06977 0.996749i 0 2.04346 2.56242i 0.349558 1.53152i 0 −0.354322 2.62192i 0.653025 2.86109i 0 −0.803031 3.51831i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 379.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.u.c 24
3.b odd 2 1 147.2.i.a 24
49.e even 7 1 inner 441.2.u.c 24
147.k even 14 1 7203.2.a.b 12
147.l odd 14 1 147.2.i.a 24
147.l odd 14 1 7203.2.a.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.i.a 24 3.b odd 2 1
147.2.i.a 24 147.l odd 14 1
441.2.u.c 24 1.a even 1 1 trivial
441.2.u.c 24 49.e even 7 1 inner
7203.2.a.a 12 147.l odd 14 1
7203.2.a.b 12 147.k even 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{24} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\).