# Properties

 Label 315.2.bo.b Level 315 Weight 2 Character orbit 315.bo 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.bo (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.

 $$\operatorname{Tr}(f)(q) =$$ $$84q + 44q^{4} - 6q^{5} + 6q^{6} - 14q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$84q + 44q^{4} - 6q^{5} + 6q^{6} - 14q^{9} + 6q^{10} - 24q^{11} - 10q^{14} + 4q^{15} - 36q^{16} + 8q^{19} - 10q^{20} - 14q^{21} + 18q^{24} + 10q^{25} - 40q^{26} - 10q^{29} - 28q^{30} - 6q^{31} - 12q^{34} + 4q^{35} - 6q^{36} + 4q^{39} - 8q^{40} - 30q^{41} - 4q^{44} - 30q^{45} + 4q^{46} + 8q^{49} + 42q^{50} + 14q^{51} + 18q^{54} - 54q^{55} + 48q^{56} + 42q^{59} + 66q^{60} + 22q^{61} - 28q^{64} + 8q^{65} - 38q^{66} - 32q^{69} - 26q^{70} - 4q^{71} - 108q^{74} + 6q^{75} + 24q^{76} + 24q^{79} - 9q^{80} - 106q^{81} - 64q^{84} + q^{85} - 92q^{86} + 46q^{89} + 17q^{90} - 44q^{91} - 8q^{94} - 25q^{95} + 54q^{96} - 102q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −2.35408 1.35913i −0.595272 + 1.62655i 2.69445 + 4.66693i 0.317086 + 2.21347i 3.61200 3.01996i 1.81310 1.92683i 9.21190i −2.29130 1.93648i 2.26194 5.64164i
4.2 −2.26777 1.30930i 1.21970 1.22977i 2.42851 + 4.20630i 2.23375 0.101685i −4.37613 + 1.19187i −0.957411 + 2.46645i 7.48135i −0.0246457 2.99990i −5.19877 2.69405i
4.3 −2.26367 1.30693i 1.21834 + 1.23112i 2.41613 + 4.18486i −1.53331 1.62756i −1.14893 4.37913i −2.64512 + 0.0579738i 7.40313i −0.0313079 + 2.99984i 1.34379 + 5.68819i
4.4 −2.10397 1.21472i −0.601157 1.62438i 1.95111 + 3.37943i −2.23603 0.0133001i −0.708362 + 4.14788i 1.65650 + 2.06301i 4.62136i −2.27722 + 1.95301i 4.68837 + 2.74414i
4.5 −2.05276 1.18516i −1.39727 + 1.02354i 1.80922 + 3.13366i −0.218973 2.22532i 4.08132 0.445097i 1.63234 + 2.08218i 3.83621i 0.904722 2.86033i −2.18786 + 4.82757i
4.6 −1.90125 1.09768i −1.70434 0.308593i 1.40982 + 2.44189i −1.91379 + 1.15647i 2.90163 + 2.45754i −2.51786 0.812625i 1.79943i 2.80954 + 1.05190i 4.90802 0.0979947i
4.7 −1.84157 1.06323i −0.0962482 1.72937i 1.26093 + 2.18399i 1.02419 + 1.98772i −1.66148 + 3.28710i 0.441480 2.60866i 1.10970i −2.98147 + 0.332898i 0.227292 4.74948i
4.8 −1.75373 1.01252i 1.36024 1.07227i 1.05039 + 1.81932i −0.846932 2.06947i −3.47118 + 0.503210i 1.36749 2.26495i 0.204070i 0.700480 2.91708i −0.610083 + 4.48683i
4.9 −1.56714 0.904786i 1.72657 + 0.137634i 0.637277 + 1.10380i −0.0315838 + 2.23584i −2.58125 1.77787i −2.37354 1.16889i 1.31275i 2.96211 + 0.475270i 2.07246 3.47530i
4.10 −1.56370 0.902800i 0.880417 + 1.49160i 0.630096 + 1.09136i 1.94086 + 1.11043i −0.0300880 3.12725i 0.480299 + 2.60179i 1.33580i −1.44973 + 2.62646i −2.03243 3.48858i
4.11 −1.42338 0.821790i −1.53987 0.792964i 0.350678 + 0.607392i 2.06509 0.857547i 1.54018 + 2.39414i 2.61685 0.390007i 2.13443i 1.74242 + 2.44213i −3.64414 0.476457i
4.12 −1.40440 0.810833i −0.306778 + 1.70467i 0.314901 + 0.545425i 1.75547 1.38504i 1.81304 2.14529i −1.09563 2.40824i 2.22200i −2.81177 1.04591i −3.58842 + 0.521762i
4.13 −1.17913 0.680771i 0.973105 + 1.43285i −0.0731024 0.126617i −2.23088 0.152304i −0.171973 2.35198i 2.52908 0.777026i 2.92215i −1.10613 + 2.78863i 2.52681 + 1.69830i
4.14 −0.941848 0.543776i −1.56380 + 0.744658i −0.408615 0.707741i −1.28643 + 1.82896i 1.87779 + 0.149005i 2.64561 0.0270009i 3.06389i 1.89097 2.32900i 2.20617 1.02307i
4.15 −0.931541 0.537825i 1.13575 1.30770i −0.421488 0.730038i −2.11877 + 0.714718i −1.76131 + 0.607338i −1.08220 + 2.41430i 3.05805i −0.420144 2.97043i 2.35811 + 0.473738i
4.16 −0.755931 0.436437i −1.14385 + 1.30062i −0.619046 1.07222i −1.55427 1.60756i 1.43231 0.483960i −2.14141 + 1.55382i 2.82644i −0.383218 2.97542i 0.473324 + 1.89354i
4.17 −0.627169 0.362096i −1.09783 1.33969i −0.737773 1.27786i 0.851637 + 2.06754i 0.203428 + 1.23773i −0.722021 + 2.54533i 2.51696i −0.589540 + 2.94150i 0.214528 1.60507i
4.18 −0.582215 0.336142i −1.56590 0.740246i −0.774017 1.34064i −1.25057 1.85366i 0.662861 + 0.957346i −0.680768 2.55667i 2.38529i 1.90407 + 2.31830i 0.105006 + 1.49960i
4.19 −0.248464 0.143451i 0.701026 1.58384i −0.958844 1.66077i 2.21083 0.335020i −0.401384 + 0.292966i −2.25543 1.38312i 1.12399i −2.01713 2.22063i −0.597370 0.233905i
4.20 −0.223223 0.128878i 1.72666 0.136523i −0.966781 1.67451i 1.91223 + 1.15903i −0.403026 0.192053i 2.07085 1.64669i 1.01390i 2.96272 0.471459i −0.277481 0.505168i
See all 84 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 79.42 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.g even 3 1 inner
315.bo 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.bo.b yes 84
3.b odd 2 1 945.2.bo.b 84
5.b even 2 1 inner 315.2.bo.b yes 84
7.c even 3 1 315.2.r.b 84
9.c even 3 1 315.2.r.b 84
9.d odd 6 1 945.2.r.b 84
15.d odd 2 1 945.2.bo.b 84
21.h odd 6 1 945.2.r.b 84
35.j even 6 1 315.2.r.b 84
45.h odd 6 1 945.2.r.b 84
45.j even 6 1 315.2.r.b 84
63.g even 3 1 inner 315.2.bo.b yes 84
63.n odd 6 1 945.2.bo.b 84
105.o odd 6 1 945.2.r.b 84
315.v odd 6 1 945.2.bo.b 84
315.bo even 6 1 inner 315.2.bo.b yes 84

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{84} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database