# Properties

 Label 950.2.f.d Level $950$ Weight $2$ Character orbit 950.f Analytic conductor $7.586$ Analytic rank $0$ Dimension $32$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32q - 8q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 8q^{6} - 24q^{11} - 32q^{16} + 32q^{26} + 56q^{36} - 64q^{61} + 72q^{66} + 4q^{76} - 32q^{81} + 8q^{96} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
493.1 −0.707107 0.707107i −2.10096 + 2.10096i 1.00000i 0 2.97120 −0.978377 + 0.978377i 0.707107 0.707107i 5.82806i 0
493.2 −0.707107 0.707107i −2.10096 + 2.10096i 1.00000i 0 2.97120 0.978377 0.978377i 0.707107 0.707107i 5.82806i 0
493.3 −0.707107 0.707107i −0.267657 + 0.267657i 1.00000i 0 0.378524 −0.709827 + 0.709827i 0.707107 0.707107i 2.85672i 0
493.4 −0.707107 0.707107i −0.267657 + 0.267657i 1.00000i 0 0.378524 0.709827 0.709827i 0.707107 0.707107i 2.85672i 0
493.5 −0.707107 0.707107i 1.16084 1.16084i 1.00000i 0 −1.64167 −2.19318 + 2.19318i 0.707107 0.707107i 0.304919i 0
493.6 −0.707107 0.707107i 1.16084 1.16084i 1.00000i 0 −1.64167 2.19318 2.19318i 0.707107 0.707107i 0.304919i 0
493.7 −0.707107 0.707107i 1.91489 1.91489i 1.00000i 0 −2.70806 −2.95447 + 2.95447i 0.707107 0.707107i 4.33358i 0
493.8 −0.707107 0.707107i 1.91489 1.91489i 1.00000i 0 −2.70806 2.95447 2.95447i 0.707107 0.707107i 4.33358i 0
493.9 0.707107 + 0.707107i −1.91489 + 1.91489i 1.00000i 0 −2.70806 −2.95447 + 2.95447i −0.707107 + 0.707107i 4.33358i 0
493.10 0.707107 + 0.707107i −1.91489 + 1.91489i 1.00000i 0 −2.70806 2.95447 2.95447i −0.707107 + 0.707107i 4.33358i 0
493.11 0.707107 + 0.707107i −1.16084 + 1.16084i 1.00000i 0 −1.64167 −2.19318 + 2.19318i −0.707107 + 0.707107i 0.304919i 0
493.12 0.707107 + 0.707107i −1.16084 + 1.16084i 1.00000i 0 −1.64167 2.19318 2.19318i −0.707107 + 0.707107i 0.304919i 0
493.13 0.707107 + 0.707107i 0.267657 0.267657i 1.00000i 0 0.378524 −0.709827 + 0.709827i −0.707107 + 0.707107i 2.85672i 0
493.14 0.707107 + 0.707107i 0.267657 0.267657i 1.00000i 0 0.378524 0.709827 0.709827i −0.707107 + 0.707107i 2.85672i 0
493.15 0.707107 + 0.707107i 2.10096 2.10096i 1.00000i 0 2.97120 −0.978377 + 0.978377i −0.707107 + 0.707107i 5.82806i 0
493.16 0.707107 + 0.707107i 2.10096 2.10096i 1.00000i 0 2.97120 0.978377 0.978377i −0.707107 + 0.707107i 5.82806i 0
607.1 −0.707107 + 0.707107i −2.10096 2.10096i 1.00000i 0 2.97120 −0.978377 0.978377i 0.707107 + 0.707107i 5.82806i 0
607.2 −0.707107 + 0.707107i −2.10096 2.10096i 1.00000i 0 2.97120 0.978377 + 0.978377i 0.707107 + 0.707107i 5.82806i 0
607.3 −0.707107 + 0.707107i −0.267657 0.267657i 1.00000i 0 0.378524 −0.709827 0.709827i 0.707107 + 0.707107i 2.85672i 0
607.4 −0.707107 + 0.707107i −0.267657 0.267657i 1.00000i 0 0.378524 0.709827 + 0.709827i 0.707107 + 0.707107i 2.85672i 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 607.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
19.b odd 2 1 inner
95.d odd 2 1 inner
95.g even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.f.d 32
5.b even 2 1 inner 950.2.f.d 32
5.c odd 4 2 inner 950.2.f.d 32
19.b odd 2 1 inner 950.2.f.d 32
95.d odd 2 1 inner 950.2.f.d 32
95.g even 4 2 inner 950.2.f.d 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.f.d 32 1.a even 1 1 trivial
950.2.f.d 32 5.b even 2 1 inner
950.2.f.d 32 5.c odd 4 2 inner
950.2.f.d 32 19.b odd 2 1 inner
950.2.f.d 32 95.d odd 2 1 inner
950.2.f.d 32 95.g even 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} + 139 T_{3}^{12} + 5151 T_{3}^{8} + 30550 T_{3}^{4} + 625$$ acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$.