# Properties

 Label 891.2.u.c Level $891$ Weight $2$ Character orbit 891.u Analytic conductor $7.115$ Analytic rank $0$ Dimension $32$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$891 = 3^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 891.u (of order $$30$$, degree $$8$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.11467082010$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$4$$ over $$\Q(\zeta_{30})$$ Twist minimal: no (minimal twist has level 99) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 4 q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32 q + 4 q^{4} + 20 q^{16} + 48 q^{22} + 32 q^{25} + 80 q^{28} - 16 q^{31} - 40 q^{34} - 24 q^{37} - 60 q^{40} - 80 q^{46} + 24 q^{49} + 40 q^{52} + 32 q^{55} - 12 q^{58} + 72 q^{64} - 96 q^{67} - 76 q^{70} - 40 q^{73} - 24 q^{82} + 100 q^{85} + 12 q^{88} - 144 q^{91} + 80 q^{94} - 60 q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1 −0.254665 2.42297i 0 −3.84965 + 0.818267i −3.77497 0.396765i 0 0.273322 0.246100i 1.45728 + 4.48505i 0 9.24768i
107.2 −0.0265163 0.252286i 0 1.89335 0.402444i 2.90572 + 0.305404i 0 −2.02056 + 1.81932i −0.308515 0.949513i 0 0.741170i
107.3 0.0265163 + 0.252286i 0 1.89335 0.402444i −2.90572 0.305404i 0 −2.02056 + 1.81932i 0.308515 + 0.949513i 0 0.741170i
107.4 0.254665 + 2.42297i 0 −3.84965 + 0.818267i 3.77497 + 0.396765i 0 0.273322 0.246100i −1.45728 4.48505i 0 9.24768i
134.1 −2.29943 + 0.488759i 0 3.22139 1.43426i −0.467414 + 2.19901i 0 4.02888 + 0.423453i −2.90269 + 2.10893i 0 5.28492i
134.2 −0.673250 + 0.143104i 0 −1.39430 + 0.620784i 0.00833908 0.0392323i 0 −0.245496 0.0258027i 1.96356 1.42661i 0 0.0276065i
134.3 0.673250 0.143104i 0 −1.39430 + 0.620784i −0.00833908 + 0.0392323i 0 −0.245496 0.0258027i −1.96356 + 1.42661i 0 0.0276065i
134.4 2.29943 0.488759i 0 3.22139 1.43426i 0.467414 2.19901i 0 4.02888 + 0.423453i 2.90269 2.10893i 0 5.28492i
215.1 −1.57299 + 1.74698i 0 −0.368594 3.50694i 1.67069 1.50430i 0 −1.64772 + 3.70084i 2.90269 + 2.10893i 0 5.28492i
215.2 −0.460557 + 0.511500i 0 0.159537 + 1.51789i −0.0298066 + 0.0268380i 0 0.100402 0.225507i −1.96356 1.42661i 0 0.0276065i
215.3 0.460557 0.511500i 0 0.159537 + 1.51789i 0.0298066 0.0268380i 0 0.100402 0.225507i 1.96356 + 1.42661i 0 0.0276065i
215.4 1.57299 1.74698i 0 −0.368594 3.50694i −1.67069 + 1.50430i 0 −1.64772 + 3.70084i −2.90269 2.10893i 0 5.28492i
431.1 −1.57299 1.74698i 0 −0.368594 + 3.50694i 1.67069 + 1.50430i 0 −1.64772 3.70084i 2.90269 2.10893i 0 5.28492i
431.2 −0.460557 0.511500i 0 0.159537 1.51789i −0.0298066 0.0268380i 0 0.100402 + 0.225507i −1.96356 + 1.42661i 0 0.0276065i
431.3 0.460557 + 0.511500i 0 0.159537 1.51789i 0.0298066 + 0.0268380i 0 0.100402 + 0.225507i 1.96356 1.42661i 0 0.0276065i
431.4 1.57299 + 1.74698i 0 −0.368594 + 3.50694i −1.67069 1.50430i 0 −1.64772 3.70084i −2.90269 + 2.10893i 0 5.28492i
458.1 −0.254665 + 2.42297i 0 −3.84965 0.818267i −3.77497 + 0.396765i 0 0.273322 + 0.246100i 1.45728 4.48505i 0 9.24768i
458.2 −0.0265163 + 0.252286i 0 1.89335 + 0.402444i 2.90572 0.305404i 0 −2.02056 1.81932i −0.308515 + 0.949513i 0 0.741170i
458.3 0.0265163 0.252286i 0 1.89335 + 0.402444i −2.90572 + 0.305404i 0 −2.02056 1.81932i 0.308515 0.949513i 0 0.741170i
458.4 0.254665 2.42297i 0 −3.84965 0.818267i 3.77497 0.396765i 0 0.273322 + 0.246100i −1.45728 + 4.48505i 0 9.24768i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 755.4 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
9.c even 3 1 inner
9.d odd 6 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner
99.o odd 30 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 891.2.u.c 32
3.b odd 2 1 inner 891.2.u.c 32
9.c even 3 1 99.2.j.a 16
9.c even 3 1 inner 891.2.u.c 32
9.d odd 6 1 99.2.j.a 16
9.d odd 6 1 inner 891.2.u.c 32
11.d odd 10 1 inner 891.2.u.c 32
33.f even 10 1 inner 891.2.u.c 32
36.f odd 6 1 1584.2.cd.c 16
36.h even 6 1 1584.2.cd.c 16
99.m even 15 1 1089.2.d.g 16
99.n odd 30 1 1089.2.d.g 16
99.o odd 30 1 99.2.j.a 16
99.o odd 30 1 inner 891.2.u.c 32
99.o odd 30 1 1089.2.d.g 16
99.p even 30 1 99.2.j.a 16
99.p even 30 1 inner 891.2.u.c 32
99.p even 30 1 1089.2.d.g 16
396.bb odd 30 1 1584.2.cd.c 16
396.bf even 30 1 1584.2.cd.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.j.a 16 9.c even 3 1
99.2.j.a 16 9.d odd 6 1
99.2.j.a 16 99.o odd 30 1
99.2.j.a 16 99.p even 30 1
891.2.u.c 32 1.a even 1 1 trivial
891.2.u.c 32 3.b odd 2 1 inner
891.2.u.c 32 9.c even 3 1 inner
891.2.u.c 32 9.d odd 6 1 inner
891.2.u.c 32 11.d odd 10 1 inner
891.2.u.c 32 33.f even 10 1 inner
891.2.u.c 32 99.o odd 30 1 inner
891.2.u.c 32 99.p even 30 1 inner
1089.2.d.g 16 99.m even 15 1
1089.2.d.g 16 99.n odd 30 1
1089.2.d.g 16 99.o odd 30 1
1089.2.d.g 16 99.p even 30 1
1584.2.cd.c 16 36.f odd 6 1
1584.2.cd.c 16 36.h even 6 1
1584.2.cd.c 16 396.bb odd 30 1
1584.2.cd.c 16 396.bf even 30 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{32} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(891, [\chi])$$.