# Properties

 Label 1050.3.e.e Level $1050$ Weight $3$ Character orbit 1050.e Analytic conductor $28.610$ Analytic rank $0$ Dimension $24$ 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.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.6104277578$$ Analytic rank: $$0$$ Dimension: $$24$$ Twist minimal: no (minimal twist has level 210) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 48 q^{4} - 44 q^{9}+O(q^{10})$$ 24 * q - 48 * q^4 - 44 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 48 q^{4} - 44 q^{9} + 96 q^{16} + 80 q^{19} - 28 q^{21} + 224 q^{31} - 128 q^{34} + 88 q^{36} + 92 q^{39} - 144 q^{46} + 168 q^{49} - 284 q^{51} + 144 q^{54} - 192 q^{64} + 224 q^{66} + 152 q^{69} - 160 q^{76} + 72 q^{79} - 212 q^{81} + 56 q^{84} + 168 q^{91} + 128 q^{94} + 876 q^{99}+O(q^{100})$$ 24 * q - 48 * q^4 - 44 * q^9 + 96 * q^16 + 80 * q^19 - 28 * q^21 + 224 * q^31 - 128 * q^34 + 88 * q^36 + 92 * q^39 - 144 * q^46 + 168 * q^49 - 284 * q^51 + 144 * q^54 - 192 * q^64 + 224 * q^66 + 152 * q^69 - 160 * q^76 + 72 * q^79 - 212 * q^81 + 56 * q^84 + 168 * q^91 + 128 * q^94 + 876 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
701.1 1.41421i −1.28756 2.70965i −2.00000 0 −3.83202 + 1.82088i −2.64575 2.82843i −5.68438 + 6.97766i 0
701.2 1.41421i 1.28756 2.70965i −2.00000 0 −3.83202 1.82088i 2.64575 2.82843i −5.68438 6.97766i 0
701.3 1.41421i −1.28756 + 2.70965i −2.00000 0 −3.83202 1.82088i −2.64575 2.82843i −5.68438 6.97766i 0
701.4 1.41421i 1.28756 + 2.70965i −2.00000 0 −3.83202 + 1.82088i 2.64575 2.82843i −5.68438 + 6.97766i 0
701.5 1.41421i −2.22285 + 2.01468i −2.00000 0 2.84919 + 3.14358i 2.64575 2.82843i 0.882103 8.95667i 0
701.6 1.41421i 2.22285 + 2.01468i −2.00000 0 2.84919 3.14358i −2.64575 2.82843i 0.882103 + 8.95667i 0
701.7 1.41421i −2.22285 2.01468i −2.00000 0 2.84919 3.14358i 2.64575 2.82843i 0.882103 + 8.95667i 0
701.8 1.41421i 2.22285 2.01468i −2.00000 0 2.84919 + 3.14358i −2.64575 2.82843i 0.882103 8.95667i 0
701.9 1.41421i −2.46556 + 1.70910i −2.00000 0 2.41703 + 3.48683i −2.64575 2.82843i 3.15796 8.42777i 0
701.10 1.41421i 2.46556 + 1.70910i −2.00000 0 2.41703 3.48683i 2.64575 2.82843i 3.15796 + 8.42777i 0
701.11 1.41421i −2.46556 1.70910i −2.00000 0 2.41703 3.48683i −2.64575 2.82843i 3.15796 + 8.42777i 0
701.12 1.41421i 2.46556 1.70910i −2.00000 0 2.41703 + 3.48683i 2.64575 2.82843i 3.15796 8.42777i 0
701.13 1.41421i −0.554262 + 2.94835i −2.00000 0 4.16960 + 0.783844i 2.64575 2.82843i −8.38559 3.26832i 0
701.14 1.41421i 0.554262 + 2.94835i −2.00000 0 4.16960 0.783844i −2.64575 2.82843i −8.38559 + 3.26832i 0
701.15 1.41421i −0.554262 2.94835i −2.00000 0 4.16960 0.783844i 2.64575 2.82843i −8.38559 + 3.26832i 0
701.16 1.41421i 0.554262 2.94835i −2.00000 0 4.16960 + 0.783844i −2.64575 2.82843i −8.38559 3.26832i 0
701.17 1.41421i −2.79991 1.07726i −2.00000 0 −1.52347 + 3.95968i 2.64575 2.82843i 6.67904 + 6.03245i 0
701.18 1.41421i 2.79991 1.07726i −2.00000 0 −1.52347 3.95968i −2.64575 2.82843i 6.67904 6.03245i 0
701.19 1.41421i −2.79991 + 1.07726i −2.00000 0 −1.52347 3.95968i 2.64575 2.82843i 6.67904 6.03245i 0
701.20 1.41421i 2.79991 + 1.07726i −2.00000 0 −1.52347 + 3.95968i −2.64575 2.82843i 6.67904 + 6.03245i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 701.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.e.e 24
3.b odd 2 1 inner 1050.3.e.e 24
5.b even 2 1 inner 1050.3.e.e 24
5.c odd 4 2 210.3.c.a 24
15.d odd 2 1 inner 1050.3.e.e 24
15.e even 4 2 210.3.c.a 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.c.a 24 5.c odd 4 2
210.3.c.a 24 15.e even 4 2
1050.3.e.e 24 1.a even 1 1 trivial
1050.3.e.e 24 3.b odd 2 1 inner
1050.3.e.e 24 5.b even 2 1 inner
1050.3.e.e 24 15.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1050, [\chi])$$:

 $$T_{11}^{12} + 526T_{11}^{10} + 76657T_{11}^{8} + 4209048T_{11}^{6} + 95989320T_{11}^{4} + 817879680T_{11}^{2} + 2289048336$$ T11^12 + 526*T11^10 + 76657*T11^8 + 4209048*T11^6 + 95989320*T11^4 + 817879680*T11^2 + 2289048336 $$T_{13}^{12} - 1082 T_{13}^{10} + 379213 T_{13}^{8} - 48657988 T_{13}^{6} + 2156635604 T_{13}^{4} - 30447344608 T_{13}^{2} + 989983296$$ T13^12 - 1082*T13^10 + 379213*T13^8 - 48657988*T13^6 + 2156635604*T13^4 - 30447344608*T13^2 + 989983296