# Properties

 Label 315.2.k.c Level 315 Weight 2 Character orbit 315.k 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.k (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.

 $$\operatorname{Tr}(f)(q) =$$ $$36q - q^{3} - 22q^{4} + 36q^{5} - 4q^{6} - q^{7} + 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q - q^{3} - 22q^{4} + 36q^{5} - 4q^{6} - q^{7} + 3q^{9} - 2q^{11} + 5q^{12} + 2q^{13} - 6q^{14} - q^{15} - 30q^{16} - 5q^{17} + 3q^{18} - 2q^{19} - 22q^{20} - 11q^{21} - 19q^{22} + 6q^{23} + 16q^{24} + 36q^{25} - 4q^{26} + 17q^{27} + 5q^{28} - 8q^{29} - 4q^{30} + 10q^{32} - 5q^{33} + 10q^{34} - q^{35} - 44q^{36} - 15q^{37} + 44q^{38} - 8q^{39} - 4q^{41} - 30q^{42} - 29q^{43} - 7q^{44} + 3q^{45} - 24q^{46} - 23q^{47} - 19q^{48} - 7q^{49} - 21q^{51} + 14q^{52} - 2q^{55} + 33q^{56} + 21q^{57} + 40q^{58} - 5q^{59} + 5q^{60} - 3q^{61} - 12q^{62} + 11q^{63} + 128q^{64} + 2q^{65} - 30q^{66} - 35q^{67} + 34q^{68} - 50q^{69} - 6q^{70} + 24q^{71} + 5q^{72} - 10q^{73} - 44q^{74} - q^{75} + 10q^{76} + 5q^{77} + 66q^{78} - 28q^{79} - 30q^{80} + 47q^{81} - 8q^{82} - 22q^{83} - 2q^{84} - 5q^{85} - 38q^{86} + 45q^{87} + 100q^{88} - 4q^{89} + 3q^{90} + 7q^{91} - 50q^{92} - 28q^{93} - 2q^{94} - 2q^{95} + 79q^{96} + 16q^{97} + 16q^{98} - 89q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1 −1.37850 2.38764i −1.12222 1.31932i −2.80055 + 4.85069i 1.00000 −1.60307 + 4.49816i 2.15266 + 1.53820i 9.92824 −0.481224 + 2.96115i −1.37850 2.38764i
16.2 −1.29282 2.23923i 1.72292 0.177629i −2.34278 + 4.05782i 1.00000 −2.62518 3.62838i −2.13970 1.55618i 6.94392 2.93690 0.612079i −1.29282 2.23923i
16.3 −1.04330 1.80705i 0.853418 + 1.50721i −1.17696 + 2.03855i 1.00000 1.83323 3.11464i 2.35024 1.21507i 0.738471 −1.54336 + 2.57256i −1.04330 1.80705i
16.4 −0.949927 1.64532i −1.72850 + 0.110854i −0.804724 + 1.39382i 1.00000 1.82434 + 2.73864i 0.880383 2.49498i −0.741992 2.97542 0.383223i −0.949927 1.64532i
16.5 −0.926279 1.60436i −0.957532 + 1.44331i −0.715985 + 1.24012i 1.00000 3.20253 + 0.199324i −2.45383 + 0.989310i −1.05231 −1.16627 2.76402i −0.926279 1.60436i
16.6 −0.750300 1.29956i −0.590164 1.62841i −0.125900 + 0.218065i 1.00000 −1.67341 + 1.98874i −2.32993 + 1.25356i −2.62335 −2.30341 + 1.92205i −0.750300 1.29956i
16.7 −0.415616 0.719867i 1.70907 + 0.281183i 0.654527 1.13367i 1.00000 −0.507903 1.34717i 0.762251 + 2.53357i −2.75059 2.84187 + 0.961126i −0.415616 0.719867i
16.8 −0.349525 0.605394i 1.42094 0.990413i 0.755665 1.30885i 1.00000 −1.09625 0.514057i −1.62523 2.08773i −2.45459 1.03816 2.81464i −0.349525 0.605394i
16.9 −0.0127903 0.0221535i −1.71905 0.211800i 0.999673 1.73148i 1.00000 0.0172951 + 0.0407920i 0.170066 + 2.64028i −0.102306 2.91028 + 0.728190i −0.0127903 0.0221535i
16.10 0.129832 + 0.224875i −0.458568 1.67024i 0.966288 1.67366i 1.00000 0.316059 0.319971i 2.52961 0.775284i 1.02114 −2.57943 + 1.53184i 0.129832 + 0.224875i
16.11 0.195497 + 0.338610i 0.811590 + 1.53014i 0.923562 1.59966i 1.00000 −0.359457 + 0.573950i 0.0590728 2.64509i 1.50420 −1.68264 + 2.48369i 0.195497 + 0.338610i
16.12 0.588830 + 1.01988i −0.114571 + 1.72826i 0.306558 0.530973i 1.00000 −1.83009 + 0.900802i −1.15510 + 2.38028i 3.07736 −2.97375 0.396015i 0.588830 + 1.01988i
16.13 0.712894 + 1.23477i 1.38205 1.04400i −0.0164351 + 0.0284664i 1.00000 2.27435 + 0.962251i −0.526228 + 2.59289i 2.80471 0.820129 2.88572i 0.712894 + 1.23477i
16.14 0.792206 + 1.37214i −1.71008 + 0.274985i −0.255182 + 0.441987i 1.00000 −1.73206 2.12863i 2.64012 0.172564i 2.36020 2.84877 0.940495i 0.792206 + 1.37214i
16.15 0.845194 + 1.46392i 0.216636 1.71845i −0.428706 + 0.742540i 1.00000 2.69877 1.13529i −2.15586 1.53371i 1.93142 −2.90614 0.744556i 0.845194 + 1.46392i
16.16 1.17969 + 2.04328i 1.66846 + 0.465033i −1.78334 + 3.08884i 1.00000 1.01807 + 3.95772i −0.0799028 2.64454i −3.69640 2.56749 + 1.55177i 1.17969 + 2.04328i
16.17 1.32610 + 2.29687i −0.540247 + 1.64564i −2.51707 + 4.35969i 1.00000 −4.49624 + 0.941405i 2.58884 + 0.545800i −8.04712 −2.41627 1.77810i 1.32610 + 2.29687i
16.18 1.34882 + 2.33623i −1.34415 1.09237i −2.63865 + 4.57028i 1.00000 0.739006 4.61365i −2.16746 + 1.51728i −8.84100 0.613465 + 2.93661i 1.34882 + 2.33623i
256.1 −1.37850 + 2.38764i −1.12222 + 1.31932i −2.80055 4.85069i 1.00000 −1.60307 4.49816i 2.15266 1.53820i 9.92824 −0.481224 2.96115i −1.37850 + 2.38764i
256.2 −1.29282 + 2.23923i 1.72292 + 0.177629i −2.34278 4.05782i 1.00000 −2.62518 + 3.62838i −2.13970 + 1.55618i 6.94392 2.93690 + 0.612079i −1.29282 + 2.23923i
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 256.18 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.g 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.k.c 36
3.b odd 2 1 945.2.k.c 36
7.c even 3 1 315.2.l.c yes 36
9.c even 3 1 315.2.l.c yes 36
9.d odd 6 1 945.2.l.c 36
21.h odd 6 1 945.2.l.c 36
63.g even 3 1 inner 315.2.k.c 36
63.n odd 6 1 945.2.k.c 36

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{36} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database