# Properties

 Label 315.2.l.b Level 315 Weight 2 Character orbit 315.l 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.l (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(\zeta_{3})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 2q^{2} - 5q^{3} + 14q^{4} + 12q^{5} + 3q^{6} - 11q^{7} + 12q^{8} - 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 2q^{2} - 5q^{3} + 14q^{4} + 12q^{5} + 3q^{6} - 11q^{7} + 12q^{8} - 3q^{9} + q^{10} + q^{11} - 7q^{12} - 4q^{13} + 8q^{14} - q^{15} + 10q^{16} - 7q^{17} + 18q^{18} - 2q^{19} + 7q^{20} - 17q^{21} + 19q^{22} + q^{23} + 18q^{24} - 12q^{25} + 11q^{26} - 11q^{27} - 28q^{28} - 16q^{31} - 34q^{32} + 7q^{33} + q^{34} - 7q^{35} - 25q^{36} + 17q^{37} - 35q^{38} - 17q^{39} + 6q^{40} + 20q^{41} - 36q^{42} + 31q^{43} - 7q^{44} + 6q^{45} - 10q^{46} + 62q^{47} - 36q^{48} - 11q^{49} - q^{50} + 14q^{51} - 4q^{52} + 8q^{53} - 51q^{54} + 2q^{55} + 5q^{57} + 45q^{58} + 42q^{59} - 23q^{60} - 10q^{61} + 14q^{62} + 18q^{63} - 56q^{64} - 8q^{65} + 4q^{66} - 86q^{67} - 48q^{68} + 26q^{69} - 5q^{70} + 24q^{71} - 6q^{72} - 18q^{73} + 9q^{74} + 4q^{75} - 13q^{76} + 35q^{77} + 19q^{78} - 80q^{79} + 5q^{80} + 21q^{81} + 5q^{82} - 60q^{83} + 35q^{84} + 7q^{85} + 12q^{86} + 68q^{87} + 50q^{88} - 4q^{89} + 12q^{90} - 33q^{91} - 18q^{92} + 7q^{93} + 22q^{94} - 4q^{95} + 6q^{97} - 15q^{98} - q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1 −2.56008 −1.18219 + 1.26587i 4.55403 0.500000 + 0.866025i 3.02651 3.24073i −0.992363 2.45259i −6.53853 −0.204852 2.99300i −1.28004 2.21710i
121.2 −1.71927 −0.324473 1.70139i 0.955889 0.500000 + 0.866025i 0.557856 + 2.92514i −2.52983 0.774581i 1.79511 −2.78943 + 1.10411i −0.859635 1.48893i
121.3 −1.61038 0.678306 + 1.59371i 0.593327 0.500000 + 0.866025i −1.09233 2.56648i −2.03107 + 1.69552i 2.26528 −2.07980 + 2.16204i −0.805191 1.39463i
121.4 −1.03554 −0.388944 1.68782i −0.927661 0.500000 + 0.866025i 0.402766 + 1.74780i 2.63139 + 0.275284i 3.03170 −2.69744 + 1.31293i −0.517769 0.896802i
121.5 −0.609814 1.67160 0.453588i −1.62813 0.500000 + 0.866025i −1.01937 + 0.276604i 0.731085 + 2.54274i 2.21248 2.58852 1.51644i −0.304907 0.528114i
121.6 −0.308078 −1.71752 + 0.223878i −1.90509 0.500000 + 0.866025i 0.529131 0.0689719i −2.36933 + 1.17741i 1.20307 2.89976 0.769029i −0.154039 0.266804i
121.7 0.297462 −1.63105 0.582822i −1.91152 0.500000 + 0.866025i −0.485175 0.173367i 1.24794 2.33295i −1.16353 2.32064 + 1.90122i 0.148731 + 0.257610i
121.8 0.518491 1.09609 1.34112i −1.73117 0.500000 + 0.866025i 0.568312 0.695356i −0.619045 2.57231i −1.93458 −0.597182 2.93996i 0.259245 + 0.449026i
121.9 1.19807 −0.773328 + 1.54983i −0.564635 0.500000 + 0.866025i −0.926499 + 1.85680i 0.433740 + 2.60996i −3.07261 −1.80393 2.39705i 0.599034 + 1.03756i
121.10 2.16352 1.43178 0.974681i 2.68083 0.500000 + 0.866025i 3.09769 2.10874i −2.41085 + 1.08987i 1.47299 1.09999 2.79106i 1.08176 + 1.87367i
121.11 2.32555 −1.66925 0.462165i 3.40817 0.500000 + 0.866025i −3.88192 1.07479i 2.02670 + 1.70073i 3.27475 2.57281 + 1.54294i 1.16277 + 2.01398i
121.12 2.34008 0.308978 + 1.70427i 3.47596 0.500000 + 0.866025i 0.723033 + 3.98812i −1.61838 2.09305i 3.45385 −2.80906 + 1.05316i 1.17004 + 2.02657i
151.1 −2.56008 −1.18219 1.26587i 4.55403 0.500000 0.866025i 3.02651 + 3.24073i −0.992363 + 2.45259i −6.53853 −0.204852 + 2.99300i −1.28004 + 2.21710i
151.2 −1.71927 −0.324473 + 1.70139i 0.955889 0.500000 0.866025i 0.557856 2.92514i −2.52983 + 0.774581i 1.79511 −2.78943 1.10411i −0.859635 + 1.48893i
151.3 −1.61038 0.678306 1.59371i 0.593327 0.500000 0.866025i −1.09233 + 2.56648i −2.03107 1.69552i 2.26528 −2.07980 2.16204i −0.805191 + 1.39463i
151.4 −1.03554 −0.388944 + 1.68782i −0.927661 0.500000 0.866025i 0.402766 1.74780i 2.63139 0.275284i 3.03170 −2.69744 1.31293i −0.517769 + 0.896802i
151.5 −0.609814 1.67160 + 0.453588i −1.62813 0.500000 0.866025i −1.01937 0.276604i 0.731085 2.54274i 2.21248 2.58852 + 1.51644i −0.304907 + 0.528114i
151.6 −0.308078 −1.71752 0.223878i −1.90509 0.500000 0.866025i 0.529131 + 0.0689719i −2.36933 1.17741i 1.20307 2.89976 + 0.769029i −0.154039 + 0.266804i
151.7 0.297462 −1.63105 + 0.582822i −1.91152 0.500000 0.866025i −0.485175 + 0.173367i 1.24794 + 2.33295i −1.16353 2.32064 1.90122i 0.148731 0.257610i
151.8 0.518491 1.09609 + 1.34112i −1.73117 0.500000 0.866025i 0.568312 + 0.695356i −0.619045 + 2.57231i −1.93458 −0.597182 + 2.93996i 0.259245 0.449026i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 151.12 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.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.l.b yes 24
3.b odd 2 1 945.2.l.b 24
7.c even 3 1 315.2.k.b 24
9.c even 3 1 315.2.k.b 24
9.d odd 6 1 945.2.k.b 24
21.h odd 6 1 945.2.k.b 24
63.h even 3 1 inner 315.2.l.b yes 24
63.j odd 6 1 945.2.l.b 24

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database