# Properties

 Label 1050.3.q.e Level $1050$ Weight $3$ Character orbit 1050.q Analytic conductor $28.610$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1050.q (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.6104277578$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 210) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 32 q^{4} - 48 q^{9}+O(q^{10})$$ 32 * q + 32 * q^4 - 48 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 32 q^{4} - 48 q^{9} - 8 q^{11} - 16 q^{14} - 64 q^{16} + 144 q^{19} - 48 q^{21} - 144 q^{29} + 240 q^{31} - 192 q^{36} - 72 q^{39} + 16 q^{44} + 16 q^{46} + 80 q^{49} - 24 q^{51} + 32 q^{56} - 264 q^{59} + 192 q^{61} - 256 q^{64} + 144 q^{66} - 272 q^{71} + 224 q^{74} - 560 q^{79} - 144 q^{81} + 48 q^{84} - 176 q^{86} + 600 q^{89} - 544 q^{91} + 48 q^{99}+O(q^{100})$$ 32 * q + 32 * q^4 - 48 * q^9 - 8 * q^11 - 16 * q^14 - 64 * q^16 + 144 * q^19 - 48 * q^21 - 144 * q^29 + 240 * q^31 - 192 * q^36 - 72 * q^39 + 16 * q^44 + 16 * q^46 + 80 * q^49 - 24 * q^51 + 32 * q^56 - 264 * q^59 + 192 * q^61 - 256 * q^64 + 144 * q^66 - 272 * q^71 + 224 * q^74 - 560 * q^79 - 144 * q^81 + 48 * q^84 - 176 * q^86 + 600 * q^89 - 544 * q^91 + 48 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1 −1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i −2.86123 6.38854i 2.82843i −1.50000 2.59808i 0
199.2 −1.22474 + 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i 4.01037 5.73733i 2.82843i −1.50000 2.59808i 0
199.3 −1.22474 + 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −6.50174 + 2.59373i 2.82843i −1.50000 2.59808i 0
199.4 −1.22474 + 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −6.72455 1.94434i 2.82843i −1.50000 2.59808i 0
199.5 −1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 5.56601 + 4.24494i 2.82843i −1.50000 2.59808i 0
199.6 −1.22474 + 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i 6.46925 + 2.67372i 2.82843i −1.50000 2.59808i 0
199.7 −1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 4.61524 5.26304i 2.82843i −1.50000 2.59808i 0
199.8 −1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 0.325616 + 6.99242i 2.82843i −1.50000 2.59808i 0
199.9 1.22474 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −0.325616 6.99242i 2.82843i −1.50000 2.59808i 0
199.10 1.22474 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i 2.86123 + 6.38854i 2.82843i −1.50000 2.59808i 0
199.11 1.22474 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −5.56601 4.24494i 2.82843i −1.50000 2.59808i 0
199.12 1.22474 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 6.72455 + 1.94434i 2.82843i −1.50000 2.59808i 0
199.13 1.22474 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i −6.46925 2.67372i 2.82843i −1.50000 2.59808i 0
199.14 1.22474 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i 6.50174 2.59373i 2.82843i −1.50000 2.59808i 0
199.15 1.22474 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i −4.01037 + 5.73733i 2.82843i −1.50000 2.59808i 0
199.16 1.22474 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i −4.61524 + 5.26304i 2.82843i −1.50000 2.59808i 0
649.1 −1.22474 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i −2.86123 + 6.38854i 2.82843i −1.50000 + 2.59808i 0
649.2 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i 4.01037 + 5.73733i 2.82843i −1.50000 + 2.59808i 0
649.3 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i −6.50174 2.59373i 2.82843i −1.50000 + 2.59808i 0
649.4 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i −6.72455 + 1.94434i 2.82843i −1.50000 + 2.59808i 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 649.16 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
7.d odd 6 1 inner
35.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.q.e 32
5.b even 2 1 inner 1050.3.q.e 32
5.c odd 4 1 210.3.o.b 16
5.c odd 4 1 1050.3.p.i 16
7.d odd 6 1 inner 1050.3.q.e 32
15.e even 4 1 630.3.v.c 16
35.i odd 6 1 inner 1050.3.q.e 32
35.k even 12 1 210.3.o.b 16
35.k even 12 1 1050.3.p.i 16
35.k even 12 1 1470.3.f.d 16
35.l odd 12 1 1470.3.f.d 16
105.w odd 12 1 630.3.v.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.o.b 16 5.c odd 4 1
210.3.o.b 16 35.k even 12 1
630.3.v.c 16 15.e even 4 1
630.3.v.c 16 105.w odd 12 1
1050.3.p.i 16 5.c odd 4 1
1050.3.p.i 16 35.k even 12 1
1050.3.q.e 32 1.a even 1 1 trivial
1050.3.q.e 32 5.b even 2 1 inner
1050.3.q.e 32 7.d odd 6 1 inner
1050.3.q.e 32 35.i odd 6 1 inner
1470.3.f.d 16 35.k even 12 1
1470.3.f.d 16 35.l odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{16} + 4 T_{11}^{15} + 604 T_{11}^{14} + 352 T_{11}^{13} + 252064 T_{11}^{12} + 129616 T_{11}^{11} + 49322848 T_{11}^{10} - 7989152 T_{11}^{9} + 6945139504 T_{11}^{8} + 676772544 T_{11}^{7} + \cdots + 9682651996416$$ acting on $$S_{3}^{\mathrm{new}}(1050, [\chi])$$.