# Properties

 Label 315.2.bh.c Level 315 Weight 2 Character orbit 315.bh Analytic conductor 2.515 Analytic rank 0 Dimension 64 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.bh (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$32$$ over $$\Q(\zeta_{6})$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$64q + 34q^{4} - 10q^{5} - 18q^{6} - 14q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q + 34q^{4} - 10q^{5} - 18q^{6} - 14q^{9} - 4q^{10} + 18q^{11} - 6q^{14} - 14q^{15} - 46q^{16} + 48q^{19} - 2q^{20} - 2q^{21} - 12q^{24} + 18q^{25} - 12q^{26} - 30q^{29} - 4q^{30} - 4q^{31} + 34q^{34} + 8q^{35} - 42q^{36} - 8q^{39} - 6q^{40} + 28q^{41} + 68q^{44} - 6q^{45} - 24q^{46} + 32q^{49} - 58q^{50} + 62q^{51} + 54q^{54} - 12q^{55} + 18q^{56} + 16q^{59} - 66q^{60} + 40q^{61} - 100q^{64} - 18q^{65} - 146q^{66} - 20q^{69} - 4q^{70} - 176q^{71} - 20q^{74} + 60q^{75} - 22q^{79} + 64q^{80} - 58q^{81} - 4q^{84} - 14q^{85} + 60q^{86} - 200q^{89} + 8q^{90} - 16q^{91} - 42q^{94} + 68q^{95} + 210q^{96} + 90q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
169.1 −2.39459 1.38252i 1.22002 + 1.22945i 2.82270 + 4.88906i 2.16705 + 0.551264i −1.22171 4.63073i 0.866025 + 0.500000i 10.0797i −0.0231036 + 2.99991i −4.42706 4.31603i
169.2 −2.27796 1.31518i 0.801507 1.53544i 2.45940 + 4.25981i −2.22338 + 0.237830i −3.84519 + 2.44355i 0.866025 + 0.500000i 7.67750i −1.71517 2.46134i 5.37757 + 2.38239i
169.3 −2.11868 1.22322i 0.00483785 + 1.73204i 1.99254 + 3.45119i −2.11025 0.739493i 2.10842 3.67557i −0.866025 0.500000i 4.85641i −2.99995 + 0.0167587i 3.56638 + 4.14805i
169.4 −2.08930 1.20626i −1.72484 0.157901i 1.91013 + 3.30844i −0.387093 2.20231i 3.41324 + 2.41051i 0.866025 + 0.500000i 4.39141i 2.95013 + 0.544707i −1.84780 + 5.06823i
169.5 −2.01113 1.16112i −0.0358898 1.73168i 1.69642 + 2.93828i 1.53191 1.62888i −1.93852 + 3.52430i −0.866025 0.500000i 3.23452i −2.99742 + 0.124299i −4.97219 + 1.49716i
169.6 −1.85508 1.07103i 1.44646 0.952757i 1.29422 + 2.24165i 1.12216 + 1.93410i −3.70374 + 0.218235i −0.866025 0.500000i 1.26045i 1.18451 2.75626i −0.0102188 4.78979i
169.7 −1.46667 0.846784i −1.23864 1.21069i 0.434088 + 0.751862i 2.19600 + 0.421388i 0.791480 + 2.82455i 0.866025 + 0.500000i 1.91682i 0.0684429 + 2.99922i −2.86400 2.47758i
169.8 −1.39315 0.804334i 1.72917 + 0.0997919i 0.293908 + 0.509063i −2.18712 + 0.465298i −2.32873 1.52986i 0.866025 + 0.500000i 2.27174i 2.98008 + 0.345115i 3.42124 + 1.11095i
169.9 −1.36108 0.785823i −1.05946 1.37023i 0.235034 + 0.407091i −2.16120 + 0.573754i 0.365260 + 2.69755i −0.866025 0.500000i 2.40451i −0.755074 + 2.90342i 3.39245 + 0.917396i
169.10 −1.09189 0.630403i 1.01772 + 1.40151i −0.205184 0.355388i 0.0910240 + 2.23421i −0.227721 2.17188i −0.866025 0.500000i 3.03901i −0.928486 + 2.85270i 1.30907 2.49690i
169.11 −1.03279 0.596282i −0.463032 + 1.66901i −0.288895 0.500381i 1.52125 1.63884i 1.47342 1.44764i 0.866025 + 0.500000i 3.07418i −2.57120 1.54561i −2.54835 + 0.785481i
169.12 −0.814131 0.470039i −1.50236 + 0.861916i −0.558127 0.966705i −0.721394 2.11650i 1.62826 + 0.00445709i −0.866025 0.500000i 2.92952i 1.51420 2.58983i −0.407530 + 2.06219i
169.13 −0.719518 0.415414i 1.08800 1.34768i −0.654862 1.13425i 1.82483 1.29228i −1.34268 + 0.517712i 0.866025 + 0.500000i 2.74981i −0.632505 2.93256i −1.84983 + 0.171758i
169.14 −0.293339 0.169359i 0.530399 + 1.64884i −0.942635 1.63269i −2.22372 0.234648i 0.123660 0.573497i 0.866025 + 0.500000i 1.31601i −2.43735 + 1.74909i 0.612564 + 0.445440i
169.15 −0.101400 0.0585435i −1.22623 1.22326i −0.993145 1.72018i −1.51285 1.64660i 0.0527266 + 0.195826i 0.866025 + 0.500000i 0.466742i 0.00728996 + 2.99999i 0.0570063 + 0.255533i
169.16 −0.0813489 0.0469668i −1.71108 + 0.268687i −0.995588 1.72441i −1.28952 + 1.82678i 0.151814 + 0.0585067i 0.866025 + 0.500000i 0.374905i 2.85561 0.919491i 0.190699 0.0880417i
169.17 0.0813489 + 0.0469668i 1.71108 0.268687i −0.995588 1.72441i 2.22680 0.203371i 0.151814 + 0.0585067i −0.866025 0.500000i 0.374905i 2.85561 0.919491i 0.190699 + 0.0880417i
169.18 0.101400 + 0.0585435i 1.22623 + 1.22326i −0.993145 1.72018i −0.669567 2.13347i 0.0527266 + 0.195826i −0.866025 0.500000i 0.466742i 0.00728996 + 2.99999i 0.0570063 0.255533i
169.19 0.293339 + 0.169359i −0.530399 1.64884i −0.942635 1.63269i 0.908650 2.04312i 0.123660 0.573497i −0.866025 0.500000i 1.31601i −2.43735 + 1.74909i 0.612564 0.445440i
169.20 0.719518 + 0.415414i −1.08800 + 1.34768i −0.654862 1.13425i −2.03156 + 0.934211i −1.34268 + 0.517712i −0.866025 0.500000i 2.74981i −0.632505 2.93256i −1.84983 0.171758i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 274.32 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
9.c even 3 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.bh.c 64
3.b odd 2 1 945.2.bh.c 64
5.b even 2 1 inner 315.2.bh.c 64
9.c even 3 1 inner 315.2.bh.c 64
9.d odd 6 1 945.2.bh.c 64
15.d odd 2 1 945.2.bh.c 64
45.h odd 6 1 945.2.bh.c 64
45.j even 6 1 inner 315.2.bh.c 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bh.c 64 1.a even 1 1 trivial
315.2.bh.c 64 5.b even 2 1 inner
315.2.bh.c 64 9.c even 3 1 inner
315.2.bh.c 64 45.j even 6 1 inner
945.2.bh.c 64 3.b odd 2 1
945.2.bh.c 64 9.d odd 6 1
945.2.bh.c 64 15.d odd 2 1
945.2.bh.c 64 45.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{64} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database