# Properties

 Label 1815.2.c.k Level $1815$ Weight $2$ Character orbit 1815.c Analytic conductor $14.493$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1815.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.4928479669$$ Analytic rank: $$0$$ Dimension: $$24$$ Twist minimal: no (minimal twist has level 165) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 24q^{4} - 2q^{5} + 8q^{6} - 24q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 24q^{4} - 2q^{5} + 8q^{6} - 24q^{9} + 6q^{10} - 12q^{14} + 48q^{16} - 32q^{19} - 2q^{20} + 16q^{21} - 24q^{24} + 2q^{25} + 32q^{26} + 8q^{30} - 12q^{34} - 10q^{35} + 24q^{36} - 36q^{39} - 34q^{40} + 2q^{45} + 56q^{46} - 24q^{49} - 46q^{50} + 36q^{51} - 8q^{54} + 12q^{56} - 40q^{59} - 26q^{60} + 40q^{61} + 12q^{64} - 10q^{65} - 2q^{70} + 64q^{71} + 136q^{74} + 20q^{75} + 68q^{76} - 64q^{79} + 76q^{80} + 24q^{81} - 60q^{84} - 72q^{86} + 20q^{89} - 6q^{90} + 4q^{94} - 64q^{95} + 56q^{96} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
364.1 2.72592i 1.00000i −5.43063 1.85250 1.25229i 2.72592 0.486331i 9.35163i −1.00000 −3.41365 5.04977i
364.2 2.59201i 1.00000i −4.71851 −0.0953833 + 2.23403i 2.59201 3.41835i 7.04639i −1.00000 5.79063 + 0.247234i
364.3 2.36377i 1.00000i −3.58741 0.979185 + 2.01027i −2.36377 1.71423i 3.75227i −1.00000 4.75182 2.31457i
364.4 2.21395i 1.00000i −2.90157 −2.14116 + 0.644543i −2.21395 1.59339i 1.99602i −1.00000 1.42699 + 4.74042i
364.5 2.10651i 1.00000i −2.43738 −0.823234 2.07901i 2.10651 3.31633i 0.921348i −1.00000 −4.37946 + 1.73415i
364.6 1.90743i 1.00000i −1.63829 −2.10308 + 0.759628i 1.90743 4.45734i 0.689943i −1.00000 1.44894 + 4.01149i
364.7 1.27093i 1.00000i 0.384732 1.01340 1.99324i −1.27093 0.483291i 3.03083i −1.00000 −2.53328 1.28796i
364.8 0.803366i 1.00000i 1.35460 2.13623 0.660705i 0.803366 0.508505i 2.69497i −1.00000 −0.530788 1.71617i
364.9 0.784131i 1.00000i 1.38514 −1.83964 + 1.27111i −0.784131 1.51801i 2.65439i −1.00000 0.996719 + 1.44252i
364.10 0.488299i 1.00000i 1.76156 2.10928 0.742259i 0.488299 5.10895i 1.83677i −1.00000 −0.362444 1.02996i
364.11 0.298064i 1.00000i 1.91116 −1.68093 + 1.47460i 0.298064 2.32107i 1.16578i −1.00000 0.439527 + 0.501026i
364.12 0.288813i 1.00000i 1.91659 −0.407156 2.19869i −0.288813 3.66740i 1.13116i −1.00000 −0.635009 + 0.117592i
364.13 0.288813i 1.00000i 1.91659 −0.407156 + 2.19869i −0.288813 3.66740i 1.13116i −1.00000 −0.635009 0.117592i
364.14 0.298064i 1.00000i 1.91116 −1.68093 1.47460i 0.298064 2.32107i 1.16578i −1.00000 0.439527 0.501026i
364.15 0.488299i 1.00000i 1.76156 2.10928 + 0.742259i 0.488299 5.10895i 1.83677i −1.00000 −0.362444 + 1.02996i
364.16 0.784131i 1.00000i 1.38514 −1.83964 1.27111i −0.784131 1.51801i 2.65439i −1.00000 0.996719 1.44252i
364.17 0.803366i 1.00000i 1.35460 2.13623 + 0.660705i 0.803366 0.508505i 2.69497i −1.00000 −0.530788 + 1.71617i
364.18 1.27093i 1.00000i 0.384732 1.01340 + 1.99324i −1.27093 0.483291i 3.03083i −1.00000 −2.53328 + 1.28796i
364.19 1.90743i 1.00000i −1.63829 −2.10308 0.759628i 1.90743 4.45734i 0.689943i −1.00000 1.44894 4.01149i
364.20 2.10651i 1.00000i −2.43738 −0.823234 + 2.07901i 2.10651 3.31633i 0.921348i −1.00000 −4.37946 1.73415i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 364.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1815.2.c.k 24
5.b even 2 1 inner 1815.2.c.k 24
5.c odd 4 1 9075.2.a.dx 12
5.c odd 4 1 9075.2.a.ea 12
11.b odd 2 1 1815.2.c.j 24
11.d odd 10 2 165.2.s.a 48
33.f even 10 2 495.2.ba.c 48
55.d odd 2 1 1815.2.c.j 24
55.e even 4 1 9075.2.a.dy 12
55.e even 4 1 9075.2.a.dz 12
55.h odd 10 2 165.2.s.a 48
55.l even 20 2 825.2.n.o 24
55.l even 20 2 825.2.n.p 24
165.r even 10 2 495.2.ba.c 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.s.a 48 11.d odd 10 2
165.2.s.a 48 55.h odd 10 2
495.2.ba.c 48 33.f even 10 2
495.2.ba.c 48 165.r even 10 2
825.2.n.o 24 55.l even 20 2
825.2.n.p 24 55.l even 20 2
1815.2.c.j 24 11.b odd 2 1
1815.2.c.j 24 55.d odd 2 1
1815.2.c.k 24 1.a even 1 1 trivial
1815.2.c.k 24 5.b even 2 1 inner
9075.2.a.dx 12 5.c odd 4 1
9075.2.a.dy 12 55.e even 4 1
9075.2.a.dz 12 55.e even 4 1
9075.2.a.ea 12 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1815, [\chi])$$:

 $$T_{2}^{24} + \cdots$$ $$T_{19}^{12} + \cdots$$