# Properties

 Label 315.2.bl.i Level 315 Weight 2 Character orbit 315.bl Analytic conductor 2.515 Analytic rank 0 Dimension 24 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 6q^{2} - 5q^{3} + 18q^{4} + 12q^{5} - q^{6} + q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 6q^{2} - 5q^{3} + 18q^{4} + 12q^{5} - q^{6} + q^{9} + 9q^{11} + 18q^{12} - 3q^{13} + 9q^{14} + 2q^{15} - 18q^{16} - 18q^{17} + 2q^{18} - 18q^{20} + 4q^{21} - 9q^{22} + 9q^{23} + 7q^{24} - 12q^{25} + 18q^{26} + 4q^{27} - 9q^{28} + 9q^{29} - 5q^{30} + 42q^{31} + 18q^{32} - 13q^{33} + 39q^{34} + 9q^{35} - 21q^{36} + 12q^{38} - 21q^{39} + 6q^{40} + 33q^{41} + 26q^{42} + 18q^{43} - q^{45} - 30q^{46} + 17q^{48} - 6q^{50} - 12q^{51} - 129q^{52} - 52q^{54} + 6q^{56} + 6q^{57} - 15q^{58} - 12q^{59} + 15q^{60} + 15q^{61} - 12q^{62} - 83q^{63} - 60q^{64} - 3q^{65} - 29q^{66} - 15q^{67} - 9q^{68} - 61q^{69} + 18q^{70} + 61q^{72} - 18q^{74} + 7q^{75} - 54q^{76} - 57q^{77} - 66q^{78} + 21q^{79} - 36q^{80} + q^{81} + 30q^{83} - 42q^{84} - 9q^{85} - 102q^{86} - 10q^{87} - 9q^{88} - 102q^{89} + 37q^{90} + 42q^{91} - 3q^{92} - 6q^{93} + 156q^{94} - 18q^{95} + 42q^{96} + 45q^{97} - 3q^{98} + 23q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1 −2.29184 + 1.32320i 1.70895 0.281919i 2.50170 4.33307i 0.500000 0.866025i −3.54362 + 2.90740i −1.30676 2.30052i 7.94816i 2.84104 0.963574i 2.64639i
41.2 −2.21303 + 1.27769i −0.920618 + 1.46713i 2.26501 3.92311i 0.500000 0.866025i 0.162815 4.42307i 1.57885 + 2.12303i 6.46517i −1.30493 2.70133i 2.55539i
41.3 −1.41561 + 0.817305i −1.73161 + 0.0392468i 0.335974 0.581925i 0.500000 0.866025i 2.41921 1.47081i −2.43457 + 1.03581i 2.17085i 2.99692 0.135920i 1.63461i
41.4 −1.13192 + 0.653515i −0.957531 + 1.44331i −0.145837 + 0.252598i 0.500000 0.866025i 0.140627 2.25947i 0.709992 2.54871i 2.99529i −1.16627 2.76402i 1.30703i
41.5 −0.334847 + 0.193324i −0.454500 1.67136i −0.925251 + 1.60258i 0.500000 0.866025i 0.475302 + 0.471783i 1.56056 + 2.13651i 1.48879i −2.58686 + 1.51926i 0.386648i
41.6 0.552767 0.319140i 0.662870 + 1.60019i −0.796299 + 1.37923i 0.500000 0.866025i 0.877097 + 0.672983i 2.58723 0.553395i 2.29308i −2.12121 + 2.12143i 0.638280i
41.7 0.963349 0.556190i −1.73166 0.0368078i −0.381305 + 0.660440i 0.500000 0.866025i −1.68867 + 0.927673i −1.36360 + 2.26729i 3.07307i 2.99729 + 0.127477i 1.11238i
41.8 1.02342 0.590871i 1.63294 + 0.577510i −0.301744 + 0.522636i 0.500000 0.866025i 2.01241 0.373820i −2.07834 + 1.63723i 3.07665i 2.33296 + 1.88607i 1.18174i
41.9 1.58089 0.912729i 0.791176 1.54079i 0.666147 1.15380i 0.500000 0.866025i −0.155560 3.15796i 2.11583 + 1.58847i 1.21887i −1.74808 2.43808i 1.82546i
41.10 1.94805 1.12471i −0.913713 1.47144i 1.52994 2.64993i 0.500000 0.866025i −3.43490 1.83878i −2.43271 1.04016i 2.38412i −1.33026 + 2.68894i 2.24942i
41.11 1.99878 1.15400i −1.43832 + 0.965005i 1.66342 2.88113i 0.500000 0.866025i −1.76127 + 3.58865i 2.12864 1.57127i 3.06234i 1.13753 2.77597i 2.30799i
41.12 2.32000 1.33945i 0.852012 + 1.50800i 2.58825 4.48298i 0.500000 0.866025i 3.99656 + 2.35733i −1.06513 + 2.42188i 8.50953i −1.54815 + 2.56968i 2.67890i
146.1 −2.29184 1.32320i 1.70895 + 0.281919i 2.50170 + 4.33307i 0.500000 + 0.866025i −3.54362 2.90740i −1.30676 + 2.30052i 7.94816i 2.84104 + 0.963574i 2.64639i
146.2 −2.21303 1.27769i −0.920618 1.46713i 2.26501 + 3.92311i 0.500000 + 0.866025i 0.162815 + 4.42307i 1.57885 2.12303i 6.46517i −1.30493 + 2.70133i 2.55539i
146.3 −1.41561 0.817305i −1.73161 0.0392468i 0.335974 + 0.581925i 0.500000 + 0.866025i 2.41921 + 1.47081i −2.43457 1.03581i 2.17085i 2.99692 + 0.135920i 1.63461i
146.4 −1.13192 0.653515i −0.957531 1.44331i −0.145837 0.252598i 0.500000 + 0.866025i 0.140627 + 2.25947i 0.709992 + 2.54871i 2.99529i −1.16627 + 2.76402i 1.30703i
146.5 −0.334847 0.193324i −0.454500 + 1.67136i −0.925251 1.60258i 0.500000 + 0.866025i 0.475302 0.471783i 1.56056 2.13651i 1.48879i −2.58686 1.51926i 0.386648i
146.6 0.552767 + 0.319140i 0.662870 1.60019i −0.796299 1.37923i 0.500000 + 0.866025i 0.877097 0.672983i 2.58723 + 0.553395i 2.29308i −2.12121 2.12143i 0.638280i
146.7 0.963349 + 0.556190i −1.73166 + 0.0368078i −0.381305 0.660440i 0.500000 + 0.866025i −1.68867 0.927673i −1.36360 2.26729i 3.07307i 2.99729 0.127477i 1.11238i
146.8 1.02342 + 0.590871i 1.63294 0.577510i −0.301744 0.522636i 0.500000 + 0.866025i 2.01241 + 0.373820i −2.07834 1.63723i 3.07665i 2.33296 1.88607i 1.18174i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 146.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.o 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.bl.i 24
3.b odd 2 1 945.2.bl.i 24
7.b odd 2 1 315.2.bl.j yes 24
9.c even 3 1 945.2.bl.j 24
9.d odd 6 1 315.2.bl.j yes 24
21.c even 2 1 945.2.bl.j 24
63.l odd 6 1 945.2.bl.i 24
63.o even 6 1 inner 315.2.bl.i 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bl.i 24 1.a even 1 1 trivial
315.2.bl.i 24 63.o even 6 1 inner
315.2.bl.j yes 24 7.b odd 2 1
315.2.bl.j yes 24 9.d odd 6 1
945.2.bl.i 24 3.b odd 2 1
945.2.bl.i 24 63.l odd 6 1
945.2.bl.j 24 9.c even 3 1
945.2.bl.j 24 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$:

 $$T_{2}^{24} - \cdots$$ $$T_{11}^{24} - \cdots$$ $$T_{13}^{24} + \cdots$$ $$T_{17}^{12} + \cdots$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database