# Properties

 Label 99.2.p.a Level 99 Weight 2 Character orbit 99.p Analytic conductor 0.791 Analytic rank 0 Dimension 80 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 99.p (of order $$30$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.790518980011$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$10$$ over $$\Q(\zeta_{30})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$80q - 15q^{2} - 3q^{3} + 5q^{4} - 6q^{5} - 15q^{6} - 5q^{7} - q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q - 15q^{2} - 3q^{3} + 5q^{4} - 6q^{5} - 15q^{6} - 5q^{7} - q^{9} - 3q^{11} - 54q^{12} - 5q^{13} - 9q^{14} + 5q^{16} - 50q^{19} - 3q^{20} - 11q^{22} - 42q^{23} - 5q^{24} - 2q^{25} + 3q^{27} - 20q^{28} + 30q^{29} + 50q^{30} - 6q^{31} + 4q^{33} - 10q^{34} - 17q^{36} - 6q^{37} + 9q^{38} + 85q^{39} + 15q^{40} - 15q^{41} + 19q^{42} - 12q^{45} - 40q^{46} - 21q^{47} + 70q^{48} - q^{49} + 60q^{50} - 45q^{51} - 5q^{52} - 18q^{55} + 90q^{56} + 60q^{57} - 29q^{58} + 81q^{59} + 43q^{60} - 5q^{61} + 15q^{63} - 8q^{64} - 39q^{66} + 10q^{67} + 180q^{68} - 20q^{69} + 30q^{70} + 5q^{72} - 20q^{73} - 15q^{74} - 30q^{75} + 33q^{77} + 152q^{78} - 5q^{79} - 73q^{81} - 2q^{82} - 60q^{83} - 135q^{84} - 5q^{85} - 48q^{86} - 59q^{88} - 70q^{90} + 52q^{91} - 213q^{92} - 34q^{93} - 5q^{94} - 135q^{95} - 145q^{96} + 27q^{97} + 35q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −1.66735 1.85178i 1.13023 + 1.31247i −0.439972 + 4.18606i 1.05074 + 0.946093i 0.545914 4.28128i 1.27299 + 2.85917i 4.45339 3.23558i −0.445156 + 2.96679i 3.52320i
2.2 −1.65440 1.83740i −0.642641 1.60842i −0.429934 + 4.09055i −1.51955 1.36821i −1.89212 + 3.84176i −0.441031 0.990572i 4.22672 3.07089i −2.17403 + 2.06727i 5.05558i
2.3 −1.02662 1.14018i −0.926210 + 1.46360i −0.0369992 + 0.352024i −2.07594 1.86918i 2.61964 0.446523i −1.07503 2.41456i −2.04313 + 1.48442i −1.28427 2.71121i 4.28588i
2.4 −0.906431 1.00669i 1.04844 1.37868i 0.0172421 0.164047i 2.90247 + 2.61340i −2.33825 + 0.194222i −0.686294 1.54144i −2.37263 + 1.72381i −0.801540 2.89094i 5.29077i
2.5 −0.536887 0.596273i 1.73203 + 0.00818115i 0.141763 1.34878i −2.14661 1.93281i −0.925027 1.03716i 0.542079 + 1.21753i −2.17861 + 1.58285i 2.99987 + 0.0283400i 2.31767i
2.6 0.0350855 + 0.0389664i −0.778272 + 1.54735i 0.208770 1.98631i 1.98818 + 1.79017i −0.0876008 + 0.0239631i 1.00434 + 2.25577i 0.169565 0.123196i −1.78858 2.40852i 0.140281i
2.7 0.148911 + 0.165382i −1.37407 1.05449i 0.203880 1.93979i 0.397470 + 0.357884i −0.0302197 0.384271i −1.30165 2.92356i 0.711251 0.516754i 0.776111 + 2.89787i 0.119027i
2.8 0.780898 + 0.867275i 0.189274 1.72168i 0.0666926 0.634537i −1.15400 1.03907i 1.64097 1.18030i 1.76474 + 3.96367i 2.49070 1.80960i −2.92835 0.651738i 1.81224i
2.9 1.13824 + 1.26415i 0.775984 + 1.54850i −0.0934131 + 0.888766i −1.30752 1.17729i −1.07427 + 2.74352i −1.14161 2.56410i 1.52254 1.10619i −1.79570 + 2.40322i 2.99294i
2.10 1.48406 + 1.64822i −1.61485 + 0.626310i −0.305125 + 2.90307i −0.217939 0.196233i −3.42883 1.73214i 0.539627 + 1.21202i −1.64909 + 1.19814i 2.21547 2.02279i 0.650432i
29.1 −2.46099 + 1.09570i −1.72861 + 0.109172i 3.51765 3.90675i 0.588502 1.32180i 4.13447 2.16271i −0.300148 + 1.41209i −2.71136 + 8.34470i 2.97616 0.377431i 3.89775i
29.2 −1.79897 + 0.800953i 0.666085 + 1.59885i 1.25651 1.39549i −0.804185 + 1.80623i −2.47887 2.34279i 0.0659410 0.310228i 0.0743474 0.228818i −2.11266 + 2.12994i 3.89347i
29.3 −1.65764 + 0.738027i −0.00460448 1.73204i 0.864813 0.960472i −0.488756 + 1.09776i 1.28593 + 2.86770i 1.00654 4.73540i 0.396737 1.22103i −2.99996 + 0.0159503i 2.18041i
29.4 −1.31398 + 0.585020i 1.60559 0.649686i 0.0460231 0.0511138i 1.45099 3.25897i −1.72962 + 1.79297i −0.449955 + 2.11687i 0.858363 2.64177i 2.15582 2.08625i 5.13106i
29.5 −0.571041 + 0.254244i −1.42001 0.991749i −1.07681 + 1.19592i −0.518875 + 1.16541i 1.06303 + 0.205300i −1.00684 + 4.73680i 0.697171 2.14567i 1.03287 + 2.81659i 0.797419i
29.6 0.337994 0.150485i −1.17234 + 1.27500i −1.24667 + 1.38456i −0.966595 + 2.17101i −0.204377 + 0.607361i 0.327889 1.54259i −0.441671 + 1.35932i −0.251238 2.98946i 0.879247i
29.7 0.471493 0.209922i 1.16254 + 1.28394i −1.16002 + 1.28834i 0.661751 1.48632i 0.817657 + 0.361323i 0.171735 0.807949i −0.595467 + 1.83266i −0.296983 + 2.98526i 0.839703i
29.8 1.05233 0.468527i 1.16424 1.28240i −0.450383 + 0.500201i −0.381354 + 0.856535i 0.624322 1.89498i 0.180225 0.847891i −0.951517 + 2.92847i −0.289100 2.98604i 1.08003i
29.9 1.93906 0.863327i −1.27072 + 1.17698i 1.67637 1.86180i 1.21148 2.72104i −1.44788 + 3.37928i −0.664357 + 3.12555i 0.331430 1.02004i 0.229440 2.99121i 6.32217i
29.10 2.14162 0.953512i −1.25398 1.19479i 2.33911 2.59784i −1.18835 + 2.66909i −3.82481 1.36311i 0.273495 1.28669i 1.08356 3.33484i 0.144932 + 2.99650i 6.84930i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 95.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
11.d odd 10 1 inner
99.p even 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.2.p.a 80
3.b odd 2 1 297.2.t.a 80
9.c even 3 1 297.2.t.a 80
9.c even 3 1 891.2.k.a 80
9.d odd 6 1 inner 99.2.p.a 80
9.d odd 6 1 891.2.k.a 80
11.d odd 10 1 inner 99.2.p.a 80
33.f even 10 1 297.2.t.a 80
99.o odd 30 1 297.2.t.a 80
99.o odd 30 1 891.2.k.a 80
99.p even 30 1 inner 99.2.p.a 80
99.p even 30 1 891.2.k.a 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.p.a 80 1.a even 1 1 trivial
99.2.p.a 80 9.d odd 6 1 inner
99.2.p.a 80 11.d odd 10 1 inner
99.2.p.a 80 99.p even 30 1 inner
297.2.t.a 80 3.b odd 2 1
297.2.t.a 80 9.c even 3 1
297.2.t.a 80 33.f even 10 1
297.2.t.a 80 99.o odd 30 1
891.2.k.a 80 9.c even 3 1
891.2.k.a 80 9.d odd 6 1
891.2.k.a 80 99.o odd 30 1
891.2.k.a 80 99.p even 30 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(99, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database