Properties

Label 63.4.o.a
Level $63$
Weight $4$
Character orbit 63.o
Analytic conductor $3.717$
Analytic rank $0$
Dimension $44$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 63.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.71712033036\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44q - 6q^{2} + 78q^{4} + 5q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 44q - 6q^{2} + 78q^{4} + 5q^{7} - 18q^{11} + 204q^{14} + 66q^{15} - 242q^{16} - 210q^{18} - 57q^{21} - 34q^{22} - 102q^{23} - 352q^{25} + 300q^{28} + 246q^{29} - 144q^{30} + 1068q^{32} - 1818q^{36} + 328q^{37} - 798q^{39} - 1158q^{42} - 170q^{43} + 968q^{46} - 79q^{49} + 288q^{50} - 204q^{51} + 1212q^{56} + 324q^{57} - 538q^{58} + 5316q^{60} + 783q^{63} - 808q^{64} + 4350q^{65} - 590q^{67} + 384q^{70} + 4284q^{72} - 5304q^{74} - 2787q^{77} + 612q^{78} - 302q^{79} - 3720q^{81} + 7050q^{84} - 612q^{85} - 13692q^{86} + 1294q^{88} + 210q^{91} - 10194q^{92} - 4266q^{93} + 6336q^{95} + 1014q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1 −4.67796 2.70082i −2.75680 4.40455i 10.5889 + 18.3405i 3.43953 + 5.95744i 1.00030 + 28.0499i 14.2961 + 11.7738i 71.1815i −11.8001 + 24.2849i 37.1582i
20.2 −4.67796 2.70082i 2.75680 + 4.40455i 10.5889 + 18.3405i −3.43953 5.95744i −1.00030 28.0499i −17.3445 6.49383i 71.1815i −11.8001 + 24.2849i 37.1582i
20.3 −3.38393 1.95371i −4.35656 + 2.83202i 3.63397 + 6.29422i 5.82670 + 10.0921i 20.2752 1.07189i −2.49865 18.3509i 2.86048i 10.9593 24.6758i 45.5347i
20.4 −3.38393 1.95371i 4.35656 2.83202i 3.63397 + 6.29422i −5.82670 10.0921i −20.2752 + 1.07189i 17.1417 7.01157i 2.86048i 10.9593 24.6758i 45.5347i
20.5 −3.28475 1.89645i −4.97559 + 1.49782i 3.19307 + 5.53055i −9.97590 17.2788i 19.1841 + 4.51602i −4.13417 + 18.0529i 6.12125i 22.5131 14.9051i 75.6753i
20.6 −3.28475 1.89645i 4.97559 1.49782i 3.19307 + 5.53055i 9.97590 + 17.2788i −19.1841 4.51602i −13.5672 + 12.6068i 6.12125i 22.5131 14.9051i 75.6753i
20.7 −2.31807 1.33834i −3.10623 4.16549i −0.417710 0.723496i −0.223284 0.386739i 1.62562 + 13.8131i −7.50759 16.9303i 23.6495i −7.70264 + 25.8780i 1.19532i
20.8 −2.31807 1.33834i 3.10623 + 4.16549i −0.417710 0.723496i 0.223284 + 0.386739i −1.62562 13.8131i 18.4159 1.96340i 23.6495i −7.70264 + 25.8780i 1.19532i
20.9 −1.10556 0.638294i −0.213646 + 5.19176i −3.18516 5.51686i 1.59510 + 2.76280i 3.55007 5.60342i −13.1775 + 13.0136i 18.3450i −26.9087 2.21839i 4.07258i
20.10 −1.10556 0.638294i 0.213646 5.19176i −3.18516 5.51686i −1.59510 2.76280i −3.55007 + 5.60342i −4.68134 + 17.9188i 18.3450i −26.9087 2.21839i 4.07258i
20.11 −0.0847887 0.0489528i −5.17293 0.490712i −3.99521 6.91990i 9.06347 + 15.6984i 0.414584 + 0.294836i 12.7516 + 13.4312i 1.56555i 26.5184 + 5.07683i 1.77473i
20.12 −0.0847887 0.0489528i 5.17293 + 0.490712i −3.99521 6.91990i −9.06347 15.6984i −0.414584 0.294836i −18.0075 4.32762i 1.56555i 26.5184 + 5.07683i 1.77473i
20.13 0.628557 + 0.362898i −2.99138 + 4.24872i −3.73661 6.47200i −5.53318 9.58374i −3.42210 + 1.58500i 13.3114 12.8765i 11.2304i −9.10332 25.4191i 8.03191i
20.14 0.628557 + 0.362898i 2.99138 4.24872i −3.73661 6.47200i 5.53318 + 9.58374i 3.42210 1.58500i 4.49570 17.9663i 11.2304i −9.10332 25.4191i 8.03191i
20.15 1.93743 + 1.11857i −5.00204 1.40697i −1.49759 2.59390i −2.75758 4.77627i −8.11730 8.32104i −17.8859 4.80585i 24.5978i 23.0409 + 14.0754i 12.3382i
20.16 1.93743 + 1.11857i 5.00204 + 1.40697i −1.49759 2.59390i 2.75758 + 4.77627i 8.11730 + 8.32104i 13.1049 + 13.0867i 24.5978i 23.0409 + 14.0754i 12.3382i
20.17 2.59186 + 1.49641i −1.03772 5.09148i 0.478502 + 0.828790i −7.80147 13.5125i 4.92933 14.7493i 17.7116 + 5.41283i 21.0785i −24.8463 + 10.5670i 46.6969i
20.18 2.59186 + 1.49641i 1.03772 + 5.09148i 0.478502 + 0.828790i 7.80147 + 13.5125i −4.92933 + 14.7493i −13.5435 12.6323i 21.0785i −24.8463 + 10.5670i 46.6969i
20.19 3.92583 + 2.26658i −3.93727 + 3.39086i 6.27474 + 10.8682i 0.0687529 + 0.119084i −23.1427 + 4.38780i 2.53789 + 18.3455i 20.6235i 4.00416 26.7014i 0.623335i
20.20 3.92583 + 2.26658i 3.93727 3.39086i 6.27474 + 10.8682i −0.0687529 0.119084i 23.1427 4.38780i −17.1567 + 6.97489i 20.6235i 4.00416 26.7014i 0.623335i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.d odd 6 1 inner
63.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.4.o.a 44
3.b odd 2 1 189.4.o.a 44
7.b odd 2 1 inner 63.4.o.a 44
9.c even 3 1 189.4.o.a 44
9.c even 3 1 567.4.c.c 44
9.d odd 6 1 inner 63.4.o.a 44
9.d odd 6 1 567.4.c.c 44
21.c even 2 1 189.4.o.a 44
63.l odd 6 1 189.4.o.a 44
63.l odd 6 1 567.4.c.c 44
63.o even 6 1 inner 63.4.o.a 44
63.o even 6 1 567.4.c.c 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.o.a 44 1.a even 1 1 trivial
63.4.o.a 44 7.b odd 2 1 inner
63.4.o.a 44 9.d odd 6 1 inner
63.4.o.a 44 63.o even 6 1 inner
189.4.o.a 44 3.b odd 2 1
189.4.o.a 44 9.c even 3 1
189.4.o.a 44 21.c even 2 1
189.4.o.a 44 63.l odd 6 1
567.4.c.c 44 9.c even 3 1
567.4.c.c 44 9.d odd 6 1
567.4.c.c 44 63.l odd 6 1
567.4.c.c 44 63.o even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(63, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database