# Properties

 Label 315.2.r.b Level 315 Weight 2 Character orbit 315.r Analytic conductor 2.515 Analytic rank 0 Dimension 84 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.r (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$84$$ Relative dimension: $$42$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$84q - 88q^{4} + 3q^{5} + 6q^{6} - 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$84q - 88q^{4} + 3q^{5} + 6q^{6} - 2q^{9} + 6q^{10} + 12q^{11} + 8q^{14} + 4q^{15} + 72q^{16} + 8q^{19} - 10q^{20} - 26q^{21} + 18q^{24} - 5q^{25} - 40q^{26} - 10q^{29} + 5q^{30} + 12q^{31} - 12q^{34} + 4q^{35} - 6q^{36} - 56q^{39} + 4q^{40} - 30q^{41} - 4q^{44} + 33q^{45} + 4q^{46} + 8q^{49} + 42q^{50} - 52q^{51} + 24q^{54} - 54q^{55} - 18q^{56} - 84q^{59} - 21q^{60} - 44q^{61} - 28q^{64} - 16q^{65} + 4q^{66} - 32q^{69} + q^{70} - 4q^{71} + 54q^{74} + 66q^{75} + 24q^{76} - 48q^{79} - 9q^{80} + 38q^{81} + 56q^{84} + q^{85} + 46q^{86} + 46q^{89} + 17q^{90} - 44q^{91} + 16q^{94} + 50q^{95} + 78q^{96} - 102q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
184.1 2.71825i 1.11099 + 1.32879i −5.38890 −2.07547 + 0.832131i 3.61200 3.01996i −2.57523 + 0.606776i 9.21190i −0.531386 + 2.95256i 2.26194 + 5.64164i
184.2 2.61859i −0.455156 1.67118i −4.85701 −1.02882 1.98533i −4.37613 + 1.19187i 2.61471 + 0.404081i 7.48135i −2.58567 + 1.52129i −5.19877 + 2.69405i
184.3 2.61386i 1.67535 0.439552i −4.83226 2.17617 + 0.514103i −1.14893 4.37913i 1.37276 2.26175i 7.40313i 2.61359 1.47280i 1.34379 5.68819i
184.4 2.42945i −1.70733 0.291573i −3.90223 1.12953 + 1.92981i −0.708362 + 4.14788i 0.958372 + 2.46607i 4.62136i 2.82997 + 0.995624i 4.68837 2.74414i
184.5 2.37032i 0.187779 + 1.72184i −3.61843 2.03667 0.923024i 4.08132 0.445097i 0.987053 + 2.45474i 3.83621i −2.92948 + 0.646652i −2.18786 4.82757i
184.6 2.19537i −1.11942 + 1.32170i −2.81965 −0.0446370 + 2.23562i 2.90163 + 2.45754i 0.555178 2.58685i 1.79943i −0.493802 2.95908i 4.90802 + 0.0979947i
184.7 2.12646i −1.54581 0.781334i −2.52185 −2.23351 + 0.106887i −1.66148 + 3.28710i −2.47990 0.921996i 1.10970i 1.77903 + 2.41558i 0.227292 + 4.74948i
184.8 2.02504i −0.248494 1.71413i −2.10077 2.21568 0.301270i −3.47118 + 0.503210i −2.64524 + 0.0518041i 0.204070i −2.87650 + 0.851904i −0.610083 4.48683i
184.9 1.80957i 0.982481 1.42644i −1.27455 −1.92051 + 1.14527i −2.58125 1.77787i 0.174487 2.63999i 1.31275i −1.06946 2.80290i 2.07246 + 3.47530i
184.10 1.80560i 1.73197 0.0166637i −1.26019 −1.93209 1.12562i −0.0300880 3.12725i 2.01307 + 1.71685i 1.33580i 2.99944 0.0577222i −2.03243 + 3.48858i
184.11 1.64358i −1.45666 + 0.937087i −0.701356 −0.289890 2.21720i 1.54018 + 2.39414i −1.64618 + 2.07125i 2.13443i 1.24373 2.73004i −3.64414 + 0.476457i
184.12 1.62167i 1.32290 + 1.11801i −0.629802 0.321745 2.21280i 1.81304 2.14529i −1.53778 2.15296i 2.22200i 0.500103 + 2.95802i −3.58842 0.521762i
184.13 1.36154i 1.72744 0.126307i 0.146205 1.24734 + 1.85584i −0.171973 2.35198i −1.93746 + 1.80173i 2.92215i 2.96809 0.436376i 2.52681 1.69830i
184.14 1.08755i −0.137009 + 1.72662i 0.817229 −0.940709 + 2.02856i 1.87779 + 0.149005i −1.34619 + 2.27767i 3.06389i −2.96246 0.473127i 2.20617 + 1.02307i
184.15 1.07565i −0.564624 1.63744i 0.842975 0.440420 + 2.19227i −1.76131 + 0.607338i 2.63194 + 0.269937i 3.05805i −2.36240 + 1.84907i 2.35811 0.473738i
184.16 0.872874i 0.554444 + 1.64091i 1.23809 2.16932 + 0.542260i 1.43231 0.483960i 2.41635 1.07761i 2.82644i −2.38518 + 1.81959i 0.473324 1.89354i
184.17 0.724192i −1.70912 + 0.280903i 1.47555 −2.21636 + 0.296230i 0.203428 + 1.23773i 2.56533 + 0.647375i 2.51696i 2.84219 0.960196i 0.214528 + 1.60507i
184.18 0.672284i −1.42402 + 0.985984i 1.54803 2.23061 + 0.156193i 0.662861 + 0.957346i −1.87376 1.86790i 2.38529i 1.05567 2.80812i 0.105006 1.49960i
184.19 0.286902i −1.02114 1.39903i 1.91769 −0.815278 2.08214i −0.401384 + 0.292966i −0.0701001 2.64482i 1.12399i −0.914560 + 2.85720i −0.597370 + 0.233905i
184.20 0.257756i 0.745098 1.56359i 1.93356 −1.95987 1.07653i −0.403026 0.192053i −2.46150 + 0.970066i 1.01390i −1.88966 2.33006i −0.277481 + 0.505168i
See all 84 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 214.42 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
63.h even 3 1 inner
315.r 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.r.b 84
3.b odd 2 1 945.2.r.b 84
5.b even 2 1 inner 315.2.r.b 84
7.c even 3 1 315.2.bo.b yes 84
9.c even 3 1 315.2.bo.b yes 84
9.d odd 6 1 945.2.bo.b 84
15.d odd 2 1 945.2.r.b 84
21.h odd 6 1 945.2.bo.b 84
35.j even 6 1 315.2.bo.b yes 84
45.h odd 6 1 945.2.bo.b 84
45.j even 6 1 315.2.bo.b yes 84
63.h even 3 1 inner 315.2.r.b 84
63.j odd 6 1 945.2.r.b 84
105.o odd 6 1 945.2.bo.b 84
315.r even 6 1 inner 315.2.r.b 84
315.br odd 6 1 945.2.r.b 84

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.r.b 84 1.a even 1 1 trivial
315.2.r.b 84 5.b even 2 1 inner
315.2.r.b 84 63.h even 3 1 inner
315.2.r.b 84 315.r even 6 1 inner
315.2.bo.b yes 84 7.c even 3 1
315.2.bo.b yes 84 9.c even 3 1
315.2.bo.b yes 84 35.j even 6 1
315.2.bo.b yes 84 45.j even 6 1
945.2.r.b 84 3.b odd 2 1
945.2.r.b 84 15.d odd 2 1
945.2.r.b 84 63.j odd 6 1
945.2.r.b 84 315.br odd 6 1
945.2.bo.b 84 9.d odd 6 1
945.2.bo.b 84 21.h odd 6 1
945.2.bo.b 84 45.h odd 6 1
945.2.bo.b 84 105.o odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{42} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database