# Properties

 Label 350.2.m.b Level 350 Weight 2 Character orbit 350.m Analytic conductor 2.795 Analytic rank 0 Dimension 40 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 350.m (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.79476407074$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$10$$ over $$\Q(\zeta_{10})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$40q + 10q^{4} + 6q^{5} - 2q^{6} + 20q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q + 10q^{4} + 6q^{5} - 2q^{6} + 20q^{9} - 4q^{10} - 6q^{11} + 10q^{12} + 10q^{14} - 12q^{15} - 10q^{16} - 2q^{19} + 4q^{20} - 2q^{21} - 10q^{22} - 10q^{23} - 8q^{24} - 10q^{25} + 12q^{26} - 30q^{27} + 4q^{29} - 22q^{30} - 24q^{31} - 60q^{33} - 4q^{35} - 20q^{36} + 10q^{37} + 10q^{38} + 36q^{39} - 6q^{40} - 34q^{41} + 6q^{44} + 112q^{45} - 6q^{46} + 30q^{47} + 10q^{48} - 40q^{49} - 16q^{50} + 44q^{51} + 10q^{53} + 20q^{54} + 34q^{55} - 10q^{56} + 20q^{58} + 12q^{59} + 2q^{60} + 2q^{61} + 10q^{64} - 106q^{65} + 10q^{66} - 30q^{67} + 84q^{69} + 4q^{70} + 16q^{71} - 110q^{73} - 60q^{74} + 10q^{75} + 32q^{76} + 20q^{77} - 20q^{78} + 4q^{79} - 4q^{80} - 20q^{81} + 10q^{83} + 2q^{84} - 42q^{85} - 14q^{86} - 20q^{87} + 20q^{88} - 38q^{90} + 2q^{91} - 30q^{92} + 6q^{94} + 64q^{95} - 2q^{96} + 30q^{97} - 72q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1 −0.951057 0.309017i −1.02238 + 1.40719i 0.809017 + 0.587785i −0.814517 2.08244i 1.40719 1.02238i 1.00000i −0.587785 0.809017i −0.00786102 0.0241937i 0.131142 + 2.23222i
29.2 −0.951057 0.309017i −0.909805 + 1.25224i 0.809017 + 0.587785i 2.02853 0.940783i 1.25224 0.909805i 1.00000i −0.587785 0.809017i 0.186694 + 0.574585i −2.21996 + 0.267888i
29.3 −0.951057 0.309017i 0.474883 0.653621i 0.809017 + 0.587785i −1.51703 + 1.64275i −0.653621 + 0.474883i 1.00000i −0.587785 0.809017i 0.725345 + 2.23238i 1.95042 1.09356i
29.4 −0.951057 0.309017i 1.20334 1.65626i 0.809017 + 0.587785i −1.62675 1.53418i −1.65626 + 1.20334i 1.00000i −0.587785 0.809017i −0.368112 1.13293i 1.07304 + 1.96178i
29.5 −0.951057 0.309017i 1.95978 2.69740i 0.809017 + 0.587785i 2.12074 + 0.708831i −2.69740 + 1.95978i 1.00000i −0.587785 0.809017i −2.50820 7.71945i −1.79791 1.32948i
29.6 0.951057 + 0.309017i −1.58093 + 2.17596i 0.809017 + 0.587785i 0.479579 + 2.18403i −2.17596 + 1.58093i 1.00000i 0.587785 + 0.809017i −1.30843 4.02693i −0.218797 + 2.22534i
29.7 0.951057 + 0.309017i −0.832973 + 1.14649i 0.809017 + 0.587785i 1.21900 1.87457i −1.14649 + 0.832973i 1.00000i 0.587785 + 0.809017i 0.306458 + 0.943181i 1.73862 1.40613i
29.8 0.951057 + 0.309017i 0.454807 0.625988i 0.809017 + 0.587785i −1.27366 + 1.83787i 0.625988 0.454807i 1.00000i 0.587785 + 0.809017i 0.742040 + 2.28376i −1.77926 + 1.35434i
29.9 0.951057 + 0.309017i 0.578028 0.795587i 0.809017 + 0.587785i 1.99915 1.00169i 0.795587 0.578028i 1.00000i 0.587785 + 0.809017i 0.628209 + 1.93343i 2.21085 0.334894i
29.10 0.951057 + 0.309017i 1.91132 2.63070i 0.809017 + 0.587785i −2.23309 0.115392i 2.63070 1.91132i 1.00000i 0.587785 + 0.809017i −2.34041 7.20305i −2.08814 0.799807i
169.1 −0.951057 + 0.309017i −1.02238 1.40719i 0.809017 0.587785i −0.814517 + 2.08244i 1.40719 + 1.02238i 1.00000i −0.587785 + 0.809017i −0.00786102 + 0.0241937i 0.131142 2.23222i
169.2 −0.951057 + 0.309017i −0.909805 1.25224i 0.809017 0.587785i 2.02853 + 0.940783i 1.25224 + 0.909805i 1.00000i −0.587785 + 0.809017i 0.186694 0.574585i −2.21996 0.267888i
169.3 −0.951057 + 0.309017i 0.474883 + 0.653621i 0.809017 0.587785i −1.51703 1.64275i −0.653621 0.474883i 1.00000i −0.587785 + 0.809017i 0.725345 2.23238i 1.95042 + 1.09356i
169.4 −0.951057 + 0.309017i 1.20334 + 1.65626i 0.809017 0.587785i −1.62675 + 1.53418i −1.65626 1.20334i 1.00000i −0.587785 + 0.809017i −0.368112 + 1.13293i 1.07304 1.96178i
169.5 −0.951057 + 0.309017i 1.95978 + 2.69740i 0.809017 0.587785i 2.12074 0.708831i −2.69740 1.95978i 1.00000i −0.587785 + 0.809017i −2.50820 + 7.71945i −1.79791 + 1.32948i
169.6 0.951057 0.309017i −1.58093 2.17596i 0.809017 0.587785i 0.479579 2.18403i −2.17596 1.58093i 1.00000i 0.587785 0.809017i −1.30843 + 4.02693i −0.218797 2.22534i
169.7 0.951057 0.309017i −0.832973 1.14649i 0.809017 0.587785i 1.21900 + 1.87457i −1.14649 0.832973i 1.00000i 0.587785 0.809017i 0.306458 0.943181i 1.73862 + 1.40613i
169.8 0.951057 0.309017i 0.454807 + 0.625988i 0.809017 0.587785i −1.27366 1.83787i 0.625988 + 0.454807i 1.00000i 0.587785 0.809017i 0.742040 2.28376i −1.77926 1.35434i
169.9 0.951057 0.309017i 0.578028 + 0.795587i 0.809017 0.587785i 1.99915 + 1.00169i 0.795587 + 0.578028i 1.00000i 0.587785 0.809017i 0.628209 1.93343i 2.21085 + 0.334894i
169.10 0.951057 0.309017i 1.91132 + 2.63070i 0.809017 0.587785i −2.23309 + 0.115392i 2.63070 + 1.91132i 1.00000i 0.587785 0.809017i −2.34041 + 7.20305i −2.08814 + 0.799807i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 309.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.m.b 40
25.e even 10 1 inner 350.2.m.b 40
25.f odd 20 1 8750.2.a.be 20
25.f odd 20 1 8750.2.a.bf 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.m.b 40 1.a even 1 1 trivial
350.2.m.b 40 25.e even 10 1 inner
8750.2.a.be 20 25.f odd 20 1
8750.2.a.bf 20 25.f odd 20 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{40} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(350, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database