Properties

Label 441.2.u.e
Level $441$
Weight $2$
Character orbit 441.u
Analytic conductor $3.521$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $4$

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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: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 60q - 12q^{4} - 2q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 60q - 12q^{4} - 2q^{7} + 12q^{10} - 4q^{13} - 48q^{19} + 6q^{22} - 22q^{25} + 40q^{28} - 76q^{31} - 12q^{34} + 34q^{37} + 86q^{40} + 4q^{43} + 8q^{46} + 26q^{49} + 66q^{52} + 10q^{55} + 42q^{58} + 62q^{61} - 128q^{64} + 8q^{67} + 96q^{70} - 70q^{73} + 50q^{76} - 24q^{79} - 36q^{82} + 72q^{85} - 216q^{88} + 52q^{91} - 38q^{94} - 252q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 −0.605649 + 2.65352i 0 −4.87244 2.34644i 1.34310 1.68419i 0 −1.61866 + 2.09283i 5.78335 7.25209i 0 3.65560 + 4.58397i
64.2 −0.458811 + 2.01018i 0 −2.02838 0.976818i −2.04151 + 2.55997i 0 2.15910 + 1.52915i 0.323108 0.405164i 0 −4.20934 5.27834i
64.3 −0.418387 + 1.83307i 0 −1.38317 0.666101i 0.364972 0.457660i 0 −1.75476 1.98010i −0.544876 + 0.683253i 0 0.686225 + 0.860499i
64.4 −0.150352 + 0.658735i 0 1.39061 + 0.669683i 1.64892 2.06768i 0 −1.22090 + 2.34721i −1.49278 + 1.87188i 0 1.11413 + 1.39708i
64.5 −0.131442 + 0.575886i 0 1.48757 + 0.716376i 1.01472 1.27242i 0 2.21270 1.45050i −1.34467 + 1.68616i 0 0.599392 + 0.751614i
64.6 0.131442 0.575886i 0 1.48757 + 0.716376i −1.01472 + 1.27242i 0 2.21270 1.45050i 1.34467 1.68616i 0 0.599392 + 0.751614i
64.7 0.150352 0.658735i 0 1.39061 + 0.669683i −1.64892 + 2.06768i 0 −1.22090 + 2.34721i 1.49278 1.87188i 0 1.11413 + 1.39708i
64.8 0.418387 1.83307i 0 −1.38317 0.666101i −0.364972 + 0.457660i 0 −1.75476 1.98010i 0.544876 0.683253i 0 0.686225 + 0.860499i
64.9 0.458811 2.01018i 0 −2.02838 0.976818i 2.04151 2.55997i 0 2.15910 + 1.52915i −0.323108 + 0.405164i 0 −4.20934 5.27834i
64.10 0.605649 2.65352i 0 −4.87244 2.34644i −1.34310 + 1.68419i 0 −1.61866 + 2.09283i −5.78335 + 7.25209i 0 3.65560 + 4.58397i
127.1 −2.47259 1.19074i 0 3.44888 + 4.32476i 0.875748 + 3.83690i 0 2.35340 + 1.20894i −2.15666 9.44894i 0 2.40338 10.5299i
127.2 −2.10278 1.01265i 0 2.14925 + 2.69508i −0.460720 2.01854i 0 −2.61455 0.405131i −0.751559 3.29280i 0 −1.07528 + 4.71110i
127.3 −1.23234 0.593462i 0 −0.0805240 0.100974i 0.303726 + 1.33071i 0 −0.112387 2.64336i 0.648032 + 2.83922i 0 0.415434 1.82013i
127.4 −0.964230 0.464348i 0 −0.532861 0.668186i −0.00423780 0.0185670i 0 1.74036 + 1.99277i 0.679819 + 2.97848i 0 −0.00453535 + 0.0198707i
127.5 −0.0642773 0.0309543i 0 −1.24381 1.55968i −0.848708 3.71843i 0 −2.26779 + 1.36276i 0.0634200 + 0.277861i 0 −0.0605489 + 0.265282i
127.6 0.0642773 + 0.0309543i 0 −1.24381 1.55968i 0.848708 + 3.71843i 0 −2.26779 + 1.36276i −0.0634200 0.277861i 0 −0.0605489 + 0.265282i
127.7 0.964230 + 0.464348i 0 −0.532861 0.668186i 0.00423780 + 0.0185670i 0 1.74036 + 1.99277i −0.679819 2.97848i 0 −0.00453535 + 0.0198707i
127.8 1.23234 + 0.593462i 0 −0.0805240 0.100974i −0.303726 1.33071i 0 −0.112387 2.64336i −0.648032 2.83922i 0 0.415434 1.82013i
127.9 2.10278 + 1.01265i 0 2.14925 + 2.69508i 0.460720 + 2.01854i 0 −2.61455 0.405131i 0.751559 + 3.29280i 0 −1.07528 + 4.71110i
127.10 2.47259 + 1.19074i 0 3.44888 + 4.32476i −0.875748 3.83690i 0 2.35340 + 1.20894i 2.15666 + 9.44894i 0 2.40338 10.5299i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 379.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
49.e even 7 1 inner
147.l odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.u.e 60
3.b odd 2 1 inner 441.2.u.e 60
49.e even 7 1 inner 441.2.u.e 60
147.l odd 14 1 inner 441.2.u.e 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.u.e 60 1.a even 1 1 trivial
441.2.u.e 60 3.b odd 2 1 inner
441.2.u.e 60 49.e even 7 1 inner
441.2.u.e 60 147.l odd 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(15\!\cdots\!60\)\( T_{2}^{24} + \)\(14\!\cdots\!05\)\( T_{2}^{22} + \)\(20\!\cdots\!29\)\( T_{2}^{20} + \)\(18\!\cdots\!62\)\( T_{2}^{18} + \)\(21\!\cdots\!12\)\( T_{2}^{16} + \)\(21\!\cdots\!82\)\( T_{2}^{14} + \)\(12\!\cdots\!96\)\( T_{2}^{12} + 455150907666 T_{2}^{10} + 100193165226 T_{2}^{8} + 14523264252 T_{2}^{6} + 1285313022 T_{2}^{4} - 8358714 T_{2}^{2} + 35721 \)">\(T_{2}^{60} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\).