Properties

 Label 315.2.l.c Level 315 Weight 2 Character orbit 315.l Analytic conductor 2.515 Analytic rank 0 Dimension 36 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: $$36$$ Relative dimension: $$18$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36q - q^{3} + 44q^{4} - 18q^{5} - 4q^{6} - q^{7} - 9q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q - q^{3} + 44q^{4} - 18q^{5} - 4q^{6} - q^{7} - 9q^{9} + q^{11} + 8q^{12} + 2q^{13} + 9q^{14} - q^{15} + 60q^{16} - 5q^{17} - 21q^{18} - 2q^{19} - 22q^{20} - 23q^{21} - 19q^{22} - 3q^{23} - 32q^{24} - 18q^{25} - 4q^{26} + 17q^{27} + 5q^{28} - 8q^{29} + 2q^{30} - 20q^{32} - 35q^{33} + 10q^{34} - q^{35} - 44q^{36} - 15q^{37} - 22q^{38} + 7q^{39} - 4q^{41} + 57q^{42} - 29q^{43} - 7q^{44} + 6q^{45} - 24q^{46} + 46q^{47} - 19q^{48} - 7q^{49} + 42q^{51} - 7q^{52} + 21q^{54} - 2q^{55} - 12q^{56} + 21q^{57} - 20q^{58} + 10q^{59} - 13q^{60} + 6q^{61} - 12q^{62} + 2q^{63} + 128q^{64} - 4q^{65} - 12q^{66} + 70q^{67} - 17q^{68} - 50q^{69} - 3q^{70} + 24q^{71} - 10q^{72} - 10q^{73} + 22q^{74} + 2q^{75} + 10q^{76} + 35q^{77} + 66q^{78} + 56q^{79} - 30q^{80} - 49q^{81} - 8q^{82} - 22q^{83} - 86q^{84} - 5q^{85} + 19q^{86} - 42q^{87} - 50q^{88} - 4q^{89} + 3q^{90} + 7q^{91} - 50q^{92} - q^{93} + 4q^{94} + 4q^{95} - 179q^{96} + 16q^{97} + 16q^{98} - 89q^{99} + O(q^{100})$$

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 −2.69765 −0.273945 1.71025i 5.27730 −0.500000 0.866025i 0.739006 + 4.61365i −0.230272 2.63571i −8.84100 −2.84991 + 0.937027i 1.34882 + 2.33623i
121.2 −2.65219 1.69529 + 0.354953i 5.03414 −0.500000 0.866025i −4.49624 0.941405i −1.76710 + 1.96910i −8.04712 2.74802 + 1.20350i 1.32610 + 2.29687i
121.3 −2.35938 −0.431498 + 1.67744i 3.56668 −0.500000 0.866025i 1.01807 3.95772i 2.33019 + 1.25307i −3.69640 −2.62762 1.44762i 1.17969 + 2.04328i
121.4 −1.69039 −1.59654 0.671613i 0.857411 −0.500000 0.866025i 2.69877 + 1.13529i 2.40616 1.10017i 1.93142 2.09787 + 2.14451i 0.845194 + 1.46392i
121.5 −1.58441 1.09319 1.34348i 0.510363 −0.500000 0.866025i −1.73206 + 2.12863i −1.17061 + 2.37269i 2.36020 −0.609890 2.93735i 0.792206 + 1.37214i
121.6 −1.42579 −1.59516 + 0.674891i 0.0328702 −0.500000 0.866025i 2.27435 0.962251i −1.98240 1.75217i 2.80471 2.08904 2.15311i 0.712894 + 1.23477i
121.7 −1.17766 1.55400 + 0.764908i −0.613115 −0.500000 0.866025i −1.83009 0.900802i −1.48383 2.19049i 3.07736 1.82983 + 2.37733i 0.588830 + 1.01988i
121.8 −0.390993 0.919343 + 1.46793i −1.84712 −0.500000 0.866025i −0.359457 0.573950i 2.26118 + 1.37370i 1.50420 −1.30962 + 2.69906i 0.195497 + 0.338610i
121.9 −0.259663 −1.21719 1.23225i −1.93258 −0.500000 0.866025i 0.316059 + 0.319971i −0.593390 + 2.57835i 1.02114 −0.0368964 + 2.99977i 0.129832 + 0.224875i
121.10 0.0255806 0.676102 1.59464i −1.99935 −0.500000 0.866025i 0.0172951 0.0407920i −2.37158 1.17286i −0.102306 −2.08577 2.15628i −0.0127903 0.0221535i
121.11 0.699049 −1.56820 + 0.735367i −1.51133 −0.500000 0.866025i −1.09625 + 0.514057i 2.62064 0.363625i −2.45459 1.91847 2.30640i −0.349525 0.605394i
121.12 0.831231 −0.611026 + 1.62069i −1.30905 −0.500000 0.866025i −0.507903 + 1.34717i −2.57526 0.606656i −2.75059 −2.25330 1.98057i −0.415616 0.719867i
121.13 1.50060 −1.11516 1.32530i 0.251799 −0.500000 0.866025i −1.67341 1.98874i 0.0793460 2.64456i −2.62335 −0.512840 + 2.95584i −0.750300 1.29956i
121.14 1.85256 1.72871 0.107594i 1.43197 −0.500000 0.866025i 3.20253 0.199324i 0.370146 2.61973i −1.05231 2.97685 0.371996i −0.926279 1.60436i
121.15 1.89985 0.960253 1.44150i 1.60945 −0.500000 0.866025i 1.82434 2.73864i 1.72052 + 2.00992i −0.741992 −1.15583 2.76840i −0.949927 1.64532i
121.16 2.08660 0.878572 + 1.49269i 2.35391 −0.500000 0.866025i 1.83323 + 3.11464i −0.122839 + 2.64290i 0.738471 −1.45622 + 2.62286i −1.04330 1.80705i
121.17 2.58565 −1.01529 + 1.40328i 4.68556 −0.500000 0.866025i −2.62518 + 3.62838i 2.41754 1.07494i 6.94392 −0.938372 2.84947i −1.29282 2.23923i
121.18 2.75701 −0.581455 1.63154i 5.60109 −0.500000 0.866025i −1.60307 4.49816i −2.40845 + 1.09515i 9.92824 −2.32382 + 1.89733i −1.37850 2.38764i
151.1 −2.69765 −0.273945 + 1.71025i 5.27730 −0.500000 + 0.866025i 0.739006 4.61365i −0.230272 + 2.63571i −8.84100 −2.84991 0.937027i 1.34882 2.33623i
151.2 −2.65219 1.69529 0.354953i 5.03414 −0.500000 + 0.866025i −4.49624 + 0.941405i −1.76710 1.96910i −8.04712 2.74802 1.20350i 1.32610 2.29687i
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 151.18 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.c yes 36
3.b odd 2 1 945.2.l.c 36
7.c even 3 1 315.2.k.c 36
9.c even 3 1 315.2.k.c 36
9.d odd 6 1 945.2.k.c 36
21.h odd 6 1 945.2.k.c 36
63.h even 3 1 inner 315.2.l.c yes 36
63.j odd 6 1 945.2.l.c 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.k.c 36 7.c even 3 1
315.2.k.c 36 9.c even 3 1
315.2.l.c yes 36 1.a even 1 1 trivial
315.2.l.c yes 36 63.h even 3 1 inner
945.2.k.c 36 9.d odd 6 1
945.2.k.c 36 21.h odd 6 1
945.2.l.c 36 3.b odd 2 1
945.2.l.c 36 63.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{18} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database