# Properties

 Label 4410.2.d.d Level $4410$ Weight $2$ Character orbit 4410.d Analytic conductor $35.214$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4410.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$35.2140272914$$ Analytic rank: $$0$$ Dimension: $$24$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 24q^{2} + 24q^{4} + 24q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 24q^{2} + 24q^{4} + 24q^{8} + 24q^{16} - 32q^{23} - 16q^{25} + 24q^{32} - 32q^{46} - 16q^{50} + 32q^{53} + 24q^{64} - 32q^{85} - 32q^{92} - 32q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4409.1 1.00000 0 1.00000 −2.12190 0.705358i 0 0 1.00000 0 −2.12190 0.705358i
4409.2 1.00000 0 1.00000 −2.12190 + 0.705358i 0 0 1.00000 0 −2.12190 + 0.705358i
4409.3 1.00000 0 1.00000 −2.00118 0.997634i 0 0 1.00000 0 −2.00118 0.997634i
4409.4 1.00000 0 1.00000 −2.00118 + 0.997634i 0 0 1.00000 0 −2.00118 + 0.997634i
4409.5 1.00000 0 1.00000 −1.60367 1.55828i 0 0 1.00000 0 −1.60367 1.55828i
4409.6 1.00000 0 1.00000 −1.60367 + 1.55828i 0 0 1.00000 0 −1.60367 + 1.55828i
4409.7 1.00000 0 1.00000 −1.12606 1.93184i 0 0 1.00000 0 −1.12606 1.93184i
4409.8 1.00000 0 1.00000 −1.12606 + 1.93184i 0 0 1.00000 0 −1.12606 + 1.93184i
4409.9 1.00000 0 1.00000 −0.613159 2.15036i 0 0 1.00000 0 −0.613159 2.15036i
4409.10 1.00000 0 1.00000 −0.613159 + 2.15036i 0 0 1.00000 0 −0.613159 + 2.15036i
4409.11 1.00000 0 1.00000 −0.526370 2.17323i 0 0 1.00000 0 −0.526370 2.17323i
4409.12 1.00000 0 1.00000 −0.526370 + 2.17323i 0 0 1.00000 0 −0.526370 + 2.17323i
4409.13 1.00000 0 1.00000 0.526370 2.17323i 0 0 1.00000 0 0.526370 2.17323i
4409.14 1.00000 0 1.00000 0.526370 + 2.17323i 0 0 1.00000 0 0.526370 + 2.17323i
4409.15 1.00000 0 1.00000 0.613159 2.15036i 0 0 1.00000 0 0.613159 2.15036i
4409.16 1.00000 0 1.00000 0.613159 + 2.15036i 0 0 1.00000 0 0.613159 + 2.15036i
4409.17 1.00000 0 1.00000 1.12606 1.93184i 0 0 1.00000 0 1.12606 1.93184i
4409.18 1.00000 0 1.00000 1.12606 + 1.93184i 0 0 1.00000 0 1.12606 + 1.93184i
4409.19 1.00000 0 1.00000 1.60367 1.55828i 0 0 1.00000 0 1.60367 1.55828i
4409.20 1.00000 0 1.00000 1.60367 + 1.55828i 0 0 1.00000 0 1.60367 + 1.55828i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4409.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
15.d odd 2 1 inner
105.g even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4410.2.d.d yes 24
3.b odd 2 1 4410.2.d.c 24
5.b even 2 1 4410.2.d.c 24
7.b odd 2 1 inner 4410.2.d.d yes 24
15.d odd 2 1 inner 4410.2.d.d yes 24
21.c even 2 1 4410.2.d.c 24
35.c odd 2 1 4410.2.d.c 24
105.g even 2 1 inner 4410.2.d.d yes 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4410.2.d.c 24 3.b odd 2 1
4410.2.d.c 24 5.b even 2 1
4410.2.d.c 24 21.c even 2 1
4410.2.d.c 24 35.c odd 2 1
4410.2.d.d yes 24 1.a even 1 1 trivial
4410.2.d.d yes 24 7.b odd 2 1 inner
4410.2.d.d yes 24 15.d odd 2 1 inner
4410.2.d.d yes 24 105.g even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4410, [\chi])$$:

 $$T_{11}^{12} + 56 T_{11}^{10} + 1140 T_{11}^{8} + 10032 T_{11}^{6} + 33476 T_{11}^{4} + 13696 T_{11}^{2} + 1024$$ $$T_{23}^{6} + 8 T_{23}^{5} - 20 T_{23}^{4} - 192 T_{23}^{3} - 68 T_{23}^{2} + 832 T_{23} + 896$$