# Properties

 Label 845.2.l.g Level $845$ Weight $2$ Character orbit 845.l Analytic conductor $6.747$ Analytic rank $0$ Dimension $72$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$845 = 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 845.l (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.74735897080$$ Analytic rank: $$0$$ Dimension: $$72$$ Relative dimension: $$36$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$72 q - 32 q^{4} + 36 q^{9}+O(q^{10})$$ 72 * q - 32 * q^4 + 36 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$72 q - 32 q^{4} + 36 q^{9} + 26 q^{10} - 16 q^{14} + 24 q^{16} + 32 q^{25} - 40 q^{29} + 62 q^{30} - 20 q^{35} + 64 q^{36} + 12 q^{40} - 88 q^{49} + 160 q^{51} + 40 q^{55} - 40 q^{56} - 16 q^{61} - 272 q^{64} + 32 q^{66} - 60 q^{69} + 60 q^{74} + 26 q^{75} - 64 q^{79} + 60 q^{81} + 280 q^{90} - 256 q^{94} + 82 q^{95}+O(q^{100})$$ 72 * q - 32 * q^4 + 36 * q^9 + 26 * q^10 - 16 * q^14 + 24 * q^16 + 32 * q^25 - 40 * q^29 + 62 * q^30 - 20 * q^35 + 64 * q^36 + 12 * q^40 - 88 * q^49 + 160 * q^51 + 40 * q^55 - 40 * q^56 - 16 * q^61 - 272 * q^64 + 32 * q^66 - 60 * q^69 + 60 * q^74 + 26 * q^75 - 64 * q^79 + 60 * q^81 + 280 * q^90 - 256 * q^94 + 82 * q^95

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
654.1 −1.25878 + 2.18027i −2.26868 1.30982i −2.16905 3.75690i 0.117790 + 2.23296i 5.71153 3.29755i −0.615337 1.06580i 5.88628 1.93127 + 3.34505i −5.01673 2.55399i
654.2 −1.25878 + 2.18027i 2.26868 + 1.30982i −2.16905 3.75690i 0.117790 2.23296i −5.71153 + 3.29755i −0.615337 1.06580i 5.88628 1.93127 + 3.34505i 4.72019 + 3.06762i
654.3 −1.07846 + 1.86795i −1.51940 0.877227i −1.32615 2.29696i −2.22909 + 0.176470i 3.27723 1.89211i −1.93061 3.34392i 1.40696 0.0390543 + 0.0676440i 2.07435 4.35414i
654.4 −1.07846 + 1.86795i 1.51940 + 0.877227i −1.32615 2.29696i −2.22909 0.176470i −3.27723 + 1.89211i −1.93061 3.34392i 1.40696 0.0390543 + 0.0676440i 2.73362 3.97351i
654.5 −1.07395 + 1.86013i −2.05625 1.18717i −1.30673 2.26333i −1.93861 + 1.11435i 4.41661 2.54993i 1.77439 + 3.07334i 1.31766 1.31877 + 2.28417i 0.00912812 4.80284i
654.6 −1.07395 + 1.86013i 2.05625 + 1.18717i −1.30673 2.26333i −1.93861 1.11435i −4.41661 + 2.54993i 1.77439 + 3.07334i 1.31766 1.31877 + 2.28417i 4.15482 2.40932i
654.7 −0.972806 + 1.68495i −0.651549 0.376172i −0.892702 1.54621i 2.08452 + 0.809176i 1.26766 0.731885i −0.747922 1.29544i −0.417520 −1.21699 2.10789i −3.39126 + 2.72514i
654.8 −0.972806 + 1.68495i 0.651549 + 0.376172i −0.892702 1.54621i 2.08452 0.809176i −1.26766 + 0.731885i −0.747922 1.29544i −0.417520 −1.21699 2.10789i −0.664415 + 4.29949i
654.9 −0.883256 + 1.52984i −0.598760 0.345695i −0.560281 0.970435i 0.913245 + 2.04107i 1.05772 0.610673i 2.18110 + 3.77778i −1.55354 −1.26099 2.18410i −3.92915 0.405668i
654.10 −0.883256 + 1.52984i 0.598760 + 0.345695i −0.560281 0.970435i 0.913245 2.04107i −1.05772 + 0.610673i 2.18110 + 3.77778i −1.55354 −1.26099 2.18410i 2.31590 + 3.19991i
654.11 −0.788477 + 1.36568i −2.57698 1.48782i −0.243391 0.421566i −0.978285 2.01071i 4.06377 2.34622i 1.25129 + 2.16729i −2.38627 2.92720 + 5.07006i 3.51735 + 0.249373i
654.12 −0.788477 + 1.36568i 2.57698 + 1.48782i −0.243391 0.421566i −0.978285 + 2.01071i −4.06377 + 2.34622i 1.25129 + 2.16729i −2.38627 2.92720 + 5.07006i −1.97464 2.92142i
654.13 −0.443938 + 0.768923i −1.09905 0.634534i 0.605838 + 1.04934i −1.79562 1.33257i 0.975816 0.563388i −1.14064 1.97564i −2.85157 −0.694733 1.20331i 1.82179 0.789117i
654.14 −0.443938 + 0.768923i 1.09905 + 0.634534i 0.605838 + 1.04934i −1.79562 + 1.33257i −0.975816 + 0.563388i −1.14064 1.97564i −2.85157 −0.694733 1.20331i −0.227498 1.97227i
654.15 −0.197539 + 0.342148i −2.36988 1.36825i 0.921957 + 1.59688i −1.91697 + 1.15119i 0.936289 0.540567i 0.314703 + 0.545081i −1.51865 2.24423 + 3.88712i −0.0151994 0.883291i
654.16 −0.197539 + 0.342148i 2.36988 + 1.36825i 0.921957 + 1.59688i −1.91697 1.15119i −0.936289 + 0.540567i 0.314703 + 0.545081i −1.51865 2.24423 + 3.88712i 0.772552 0.428482i
654.17 −0.121427 + 0.210318i −1.03358 0.596738i 0.970511 + 1.68097i 1.65037 + 1.50873i 0.251009 0.144920i −2.39023 4.14000i −0.957093 −0.787808 1.36452i −0.517713 + 0.163901i
654.18 −0.121427 + 0.210318i 1.03358 + 0.596738i 0.970511 + 1.68097i 1.65037 1.50873i −0.251009 + 0.144920i −2.39023 4.14000i −0.957093 −0.787808 1.36452i 0.116914 + 0.530303i
654.19 0.121427 0.210318i −1.03358 0.596738i 0.970511 + 1.68097i −1.65037 1.50873i −0.251009 + 0.144920i 2.39023 + 4.14000i 0.957093 −0.787808 1.36452i −0.517713 + 0.163901i
654.20 0.121427 0.210318i 1.03358 + 0.596738i 0.970511 + 1.68097i −1.65037 + 1.50873i 0.251009 0.144920i 2.39023 + 4.14000i 0.957093 −0.787808 1.36452i 0.116914 + 0.530303i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 699.36 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
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner
65.d even 2 1 inner
65.l even 6 1 inner
65.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.l.g 72
5.b even 2 1 inner 845.2.l.g 72
13.b even 2 1 inner 845.2.l.g 72
13.c even 3 1 845.2.d.e 36
13.c even 3 1 inner 845.2.l.g 72
13.d odd 4 1 845.2.n.h 36
13.d odd 4 1 845.2.n.i 36
13.e even 6 1 845.2.d.e 36
13.e even 6 1 inner 845.2.l.g 72
13.f odd 12 1 845.2.b.g 18
13.f odd 12 1 845.2.b.h yes 18
13.f odd 12 1 845.2.n.h 36
13.f odd 12 1 845.2.n.i 36
65.d even 2 1 inner 845.2.l.g 72
65.g odd 4 1 845.2.n.h 36
65.g odd 4 1 845.2.n.i 36
65.l even 6 1 845.2.d.e 36
65.l even 6 1 inner 845.2.l.g 72
65.n even 6 1 845.2.d.e 36
65.n even 6 1 inner 845.2.l.g 72
65.o even 12 1 4225.2.a.ca 18
65.o even 12 1 4225.2.a.cb 18
65.s odd 12 1 845.2.b.g 18
65.s odd 12 1 845.2.b.h yes 18
65.s odd 12 1 845.2.n.h 36
65.s odd 12 1 845.2.n.i 36
65.t even 12 1 4225.2.a.ca 18
65.t even 12 1 4225.2.a.cb 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
845.2.b.g 18 13.f odd 12 1
845.2.b.g 18 65.s odd 12 1
845.2.b.h yes 18 13.f odd 12 1
845.2.b.h yes 18 65.s odd 12 1
845.2.d.e 36 13.c even 3 1
845.2.d.e 36 13.e even 6 1
845.2.d.e 36 65.l even 6 1
845.2.d.e 36 65.n even 6 1
845.2.l.g 72 1.a even 1 1 trivial
845.2.l.g 72 5.b even 2 1 inner
845.2.l.g 72 13.b even 2 1 inner
845.2.l.g 72 13.c even 3 1 inner
845.2.l.g 72 13.e even 6 1 inner
845.2.l.g 72 65.d even 2 1 inner
845.2.l.g 72 65.l even 6 1 inner
845.2.l.g 72 65.n even 6 1 inner
845.2.n.h 36 13.d odd 4 1
845.2.n.h 36 13.f odd 12 1
845.2.n.h 36 65.g odd 4 1
845.2.n.h 36 65.s odd 12 1
845.2.n.i 36 13.d odd 4 1
845.2.n.i 36 13.f odd 12 1
845.2.n.i 36 65.g odd 4 1
845.2.n.i 36 65.s odd 12 1
4225.2.a.ca 18 65.o even 12 1
4225.2.a.ca 18 65.t even 12 1
4225.2.a.cb 18 65.o even 12 1
4225.2.a.cb 18 65.t even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{36} + 26 T_{2}^{34} + 395 T_{2}^{32} + 4042 T_{2}^{30} + 31047 T_{2}^{28} + 184148 T_{2}^{26} + 868966 T_{2}^{24} + 3270701 T_{2}^{22} + 9882692 T_{2}^{20} + 23616179 T_{2}^{18} + 44194422 T_{2}^{16} + \cdots + 841$$ acting on $$S_{2}^{\mathrm{new}}(845, [\chi])$$.