Properties

Label 69.6.c.b
Level $69$
Weight $6$
Character orbit 69.c
Analytic conductor $11.066$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 69.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.0664835671\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32q - 408q^{4} - 528q^{6} - 444q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q - 408q^{4} - 528q^{6} - 444q^{9} - 2484q^{12} + 520q^{13} + 4936q^{16} + 7188q^{18} + 18660q^{24} + 36032q^{25} - 22032q^{27} + 6544q^{31} - 33912q^{36} - 63912q^{39} + 54328q^{46} + 88284q^{48} - 207664q^{49} + 46296q^{52} - 38628q^{54} - 139296q^{55} - 184144q^{58} + 486584q^{64} - 113580q^{69} + 37176q^{70} - 15504q^{72} - 93896q^{73} + 249840q^{75} + 368028q^{78} - 339372q^{81} - 23512q^{82} + 259584q^{85} + 509928q^{87} + 82740q^{93} - 562000q^{94} + 1404q^{96} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1 9.90483i 13.3502 8.04819i −66.1056 −91.0951 −79.7159 132.231i 106.321i 337.810i 113.453 214.889i 902.281i
68.2 9.90483i 13.3502 8.04819i −66.1056 91.0951 −79.7159 132.231i 106.321i 337.810i 113.453 214.889i 902.281i
68.3 9.59549i −8.65480 12.9651i −60.0734 −42.3345 −124.407 + 83.0470i 238.780i 269.378i −93.1890 + 224.421i 406.221i
68.4 9.59549i −8.65480 12.9651i −60.0734 42.3345 −124.407 + 83.0470i 238.780i 269.378i −93.1890 + 224.421i 406.221i
68.5 8.75944i 10.6432 + 11.3896i −44.7279 −43.1688 99.7663 93.2285i 151.444i 111.489i −16.4449 + 242.443i 378.135i
68.6 8.75944i 10.6432 + 11.3896i −44.7279 43.1688 99.7663 93.2285i 151.444i 111.489i −16.4449 + 242.443i 378.135i
68.7 7.06404i −15.4896 + 1.75248i −17.9007 −82.3579 12.3796 + 109.419i 96.3127i 99.5980i 236.858 54.2905i 581.779i
68.8 7.06404i −15.4896 + 1.75248i −17.9007 82.3579 12.3796 + 109.419i 96.3127i 99.5980i 236.858 54.2905i 581.779i
68.9 5.41836i 3.03549 15.2901i 2.64141 −41.1824 −82.8470 16.4474i 67.8449i 187.700i −224.572 92.8256i 223.141i
68.10 5.41836i 3.03549 15.2901i 2.64141 41.1824 −82.8470 16.4474i 67.8449i 187.700i −224.572 92.8256i 223.141i
68.11 2.78145i −1.49341 + 15.5168i 24.2635 −76.0392 43.1591 + 4.15386i 165.966i 156.494i −238.539 46.3458i 211.500i
68.12 2.78145i −1.49341 + 15.5168i 24.2635 76.0392 43.1591 + 4.15386i 165.966i 156.494i −238.539 46.3458i 211.500i
68.13 1.57119i −12.8969 8.75610i 29.5314 −37.1969 −13.7575 + 20.2635i 122.242i 96.6776i 89.6613 + 225.854i 58.4435i
68.14 1.57119i −12.8969 8.75610i 29.5314 37.1969 −13.7575 + 20.2635i 122.242i 96.6776i 89.6613 + 225.854i 58.4435i
68.15 1.27619i 11.5059 + 10.5173i 30.3713 −80.0597 13.4221 14.6838i 196.848i 79.5978i 21.7727 + 242.023i 102.171i
68.16 1.27619i 11.5059 + 10.5173i 30.3713 80.0597 13.4221 14.6838i 196.848i 79.5978i 21.7727 + 242.023i 102.171i
68.17 1.27619i 11.5059 10.5173i 30.3713 −80.0597 13.4221 + 14.6838i 196.848i 79.5978i 21.7727 242.023i 102.171i
68.18 1.27619i 11.5059 10.5173i 30.3713 80.0597 13.4221 + 14.6838i 196.848i 79.5978i 21.7727 242.023i 102.171i
68.19 1.57119i −12.8969 + 8.75610i 29.5314 −37.1969 −13.7575 20.2635i 122.242i 96.6776i 89.6613 225.854i 58.4435i
68.20 1.57119i −12.8969 + 8.75610i 29.5314 37.1969 −13.7575 20.2635i 122.242i 96.6776i 89.6613 225.854i 58.4435i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.b odd 2 1 inner
69.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.6.c.b 32
3.b odd 2 1 inner 69.6.c.b 32
23.b odd 2 1 inner 69.6.c.b 32
69.c even 2 1 inner 69.6.c.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.6.c.b 32 1.a even 1 1 trivial
69.6.c.b 32 3.b odd 2 1 inner
69.6.c.b 32 23.b odd 2 1 inner
69.6.c.b 32 69.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{16} + \cdots\) acting on \(S_{6}^{\mathrm{new}}(69, [\chi])\).