# Properties

 Label 99.3.l.a Level $99$ Weight $3$ Character orbit 99.l Analytic conductor $2.698$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 99.l (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.69755461717$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 16 q^{4} - 16 q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32 q + 16 q^{4} - 16 q^{7} + 48 q^{10} + 8 q^{13} + 96 q^{16} - 40 q^{19} - 60 q^{22} - 188 q^{25} - 348 q^{28} - 164 q^{31} + 296 q^{34} - 36 q^{37} + 48 q^{40} + 544 q^{43} + 296 q^{46} + 196 q^{49} - 640 q^{52} - 440 q^{55} - 208 q^{58} - 432 q^{61} - 328 q^{64} + 48 q^{67} + 112 q^{70} + 712 q^{73} + 2104 q^{76} + 432 q^{79} + 676 q^{82} - 68 q^{85} - 176 q^{88} + 64 q^{91} - 1360 q^{94} + 132 q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1 −1.91284 + 2.63279i 0 −2.03659 6.26797i −0.936271 1.28867i 0 −3.36204 10.3473i 8.01778 + 2.60514i 0 5.18373
26.2 −1.75181 + 2.41117i 0 −1.50880 4.64360i 4.61811 + 6.35629i 0 0.867699 + 2.67050i 2.50164 + 0.812833i 0 −23.4161
26.3 −1.15480 + 1.58945i 0 0.0432885 + 0.133228i −5.65603 7.78486i 0 −1.61633 4.97456i −7.73579 2.51351i 0 18.9052
26.4 −0.480038 + 0.660715i 0 1.02996 + 3.16989i −0.615389 0.847011i 0 2.11067 + 6.49597i −5.69568 1.85064i 0 0.855043
26.5 0.480038 0.660715i 0 1.02996 + 3.16989i 0.615389 + 0.847011i 0 2.11067 + 6.49597i 5.69568 + 1.85064i 0 0.855043
26.6 1.15480 1.58945i 0 0.0432885 + 0.133228i 5.65603 + 7.78486i 0 −1.61633 4.97456i 7.73579 + 2.51351i 0 18.9052
26.7 1.75181 2.41117i 0 −1.50880 4.64360i −4.61811 6.35629i 0 0.867699 + 2.67050i −2.50164 0.812833i 0 −23.4161
26.8 1.91284 2.63279i 0 −2.03659 6.26797i 0.936271 + 1.28867i 0 −3.36204 10.3473i −8.01778 2.60514i 0 5.18373
53.1 −3.44320 1.11876i 0 7.36792 + 5.35311i −0.157113 + 0.0510491i 0 −8.33195 6.05352i −10.8683 14.9589i 0 0.598083
53.2 −2.84833 0.925479i 0 4.02041 + 2.92100i −1.53284 + 0.498051i 0 2.69388 + 1.95722i −1.70668 2.34904i 0 4.82698
53.3 −1.28527 0.417608i 0 −1.75856 1.27767i −4.85485 + 1.57744i 0 10.6531 + 7.73991i 4.90400 + 6.74978i 0 6.89852
53.4 −0.296118 0.0962144i 0 −3.15764 2.29416i 5.65537 1.83754i 0 −7.01499 5.09669i 1.44634 + 1.99072i 0 −1.85145
53.5 0.296118 + 0.0962144i 0 −3.15764 2.29416i −5.65537 + 1.83754i 0 −7.01499 5.09669i −1.44634 1.99072i 0 −1.85145
53.6 1.28527 + 0.417608i 0 −1.75856 1.27767i 4.85485 1.57744i 0 10.6531 + 7.73991i −4.90400 6.74978i 0 6.89852
53.7 2.84833 + 0.925479i 0 4.02041 + 2.92100i 1.53284 0.498051i 0 2.69388 + 1.95722i 1.70668 + 2.34904i 0 4.82698
53.8 3.44320 + 1.11876i 0 7.36792 + 5.35311i 0.157113 0.0510491i 0 −8.33195 6.05352i 10.8683 + 14.9589i 0 0.598083
71.1 −3.44320 + 1.11876i 0 7.36792 5.35311i −0.157113 0.0510491i 0 −8.33195 + 6.05352i −10.8683 + 14.9589i 0 0.598083
71.2 −2.84833 + 0.925479i 0 4.02041 2.92100i −1.53284 0.498051i 0 2.69388 1.95722i −1.70668 + 2.34904i 0 4.82698
71.3 −1.28527 + 0.417608i 0 −1.75856 + 1.27767i −4.85485 1.57744i 0 10.6531 7.73991i 4.90400 6.74978i 0 6.89852
71.4 −0.296118 + 0.0962144i 0 −3.15764 + 2.29416i 5.65537 + 1.83754i 0 −7.01499 + 5.09669i 1.44634 1.99072i 0 −1.85145
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 80.8 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
11.c even 5 1 inner
33.h odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.3.l.a 32
3.b odd 2 1 inner 99.3.l.a 32
11.c even 5 1 inner 99.3.l.a 32
11.c even 5 1 1089.3.b.i 16
11.d odd 10 1 1089.3.b.j 16
33.f even 10 1 1089.3.b.j 16
33.h odd 10 1 inner 99.3.l.a 32
33.h odd 10 1 1089.3.b.i 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.3.l.a 32 1.a even 1 1 trivial
99.3.l.a 32 3.b odd 2 1 inner
99.3.l.a 32 11.c even 5 1 inner
99.3.l.a 32 33.h odd 10 1 inner
1089.3.b.i 16 11.c even 5 1
1089.3.b.i 16 33.h odd 10 1
1089.3.b.j 16 11.d odd 10 1
1089.3.b.j 16 33.f even 10 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(99, [\chi])$$.