# 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 $4$

# Related objects

## 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.

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{6}^{\mathrm{new}}(69, [\chi])$$.