# Properties

 Label 315.2.be.c Level 315 Weight 2 Character orbit 315.be Analytic conductor 2.515 Analytic rank 0 Dimension 32 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.be (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + q^{3} + 16q^{4} - 32q^{5} + 2q^{6} + q^{7} + 7q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + q^{3} + 16q^{4} - 32q^{5} + 2q^{6} + q^{7} + 7q^{9} + 15q^{12} - 6q^{13} - 6q^{14} - q^{15} - 16q^{16} + 3q^{17} + 41q^{18} - 16q^{20} - 17q^{21} - 21q^{22} - 26q^{24} + 32q^{25} - 12q^{26} - 23q^{27} - 31q^{28} + 18q^{29} - 2q^{30} + 24q^{31} - 19q^{33} + 30q^{34} - q^{35} + 18q^{36} - q^{37} - 60q^{38} - 36q^{39} - 6q^{41} + 44q^{42} - 19q^{43} - 21q^{44} - 7q^{45} + 6q^{46} - 15q^{47} + 35q^{48} + 23q^{49} - 9q^{51} + 24q^{53} - 58q^{54} + 33q^{56} + 27q^{57} + 15q^{59} - 15q^{60} - 9q^{61} - 11q^{63} + 76q^{64} + 6q^{65} + 22q^{66} + 25q^{67} - 6q^{68} + 50q^{69} + 6q^{70} + 61q^{72} + 12q^{73} + q^{75} - 54q^{76} - 27q^{77} - 42q^{78} - 2q^{79} + 16q^{80} + 43q^{81} - 24q^{82} + 42q^{83} - 36q^{84} - 3q^{85} - 55q^{87} - 84q^{88} - 30q^{89} - 41q^{90} - 57q^{91} + 6q^{92} - 48q^{93} + 24q^{94} - 9q^{96} + 42q^{97} - 6q^{98} - q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
236.1 −2.27381 + 1.31279i −0.933716 + 1.45883i 2.44681 4.23800i −1.00000 0.207969 4.54286i −2.64220 0.136993i 7.59741i −1.25635 2.72426i 2.27381 1.31279i
236.2 −2.03337 + 1.17397i 1.49299 + 0.878053i 1.75639 3.04216i −1.00000 −4.06660 0.0326841i 0.135526 2.64228i 3.55190i 1.45805 + 2.62185i 2.03337 1.17397i
236.3 −1.91788 + 1.10729i 0.302150 1.70549i 1.45217 2.51523i −1.00000 1.30898 + 3.60549i 2.53521 0.756781i 2.00271i −2.81741 1.03063i 1.91788 1.10729i
236.4 −1.67628 + 0.967799i −1.71487 + 0.243339i 0.873269 1.51255i −1.00000 2.63910 2.06755i 2.28906 + 1.32673i 0.490599i 2.88157 0.834589i 1.67628 0.967799i
236.5 −1.07088 + 0.618275i 1.48704 0.888095i −0.235471 + 0.407847i −1.00000 −1.04336 + 1.87045i −2.10494 1.60288i 3.05545i 1.42257 2.64127i 1.07088 0.618275i
236.6 −0.948743 + 0.547757i −0.237931 + 1.71563i −0.399925 + 0.692690i −1.00000 −0.714013 1.75802i 1.38314 + 2.25542i 3.06727i −2.88678 0.816404i 0.948743 0.547757i
236.7 −0.454062 + 0.262153i 0.322156 1.70183i −0.862552 + 1.49398i −1.00000 0.299860 + 0.857190i −1.09499 + 2.40853i 1.95309i −2.79243 1.09651i 0.454062 0.262153i
236.8 −0.0900587 + 0.0519954i −1.13092 1.31188i −0.994593 + 1.72269i −1.00000 0.170061 + 0.0593436i 2.34312 1.22873i 0.414839i −0.442054 + 2.96725i 0.0900587 0.0519954i
236.9 0.343770 0.198476i −1.65197 + 0.520559i −0.921215 + 1.59559i −1.00000 −0.464581 + 0.506829i −0.324456 2.62578i 1.52526i 2.45804 1.71990i −0.343770 + 0.198476i
236.10 0.403396 0.232901i 0.625194 + 1.61528i −0.891514 + 1.54415i −1.00000 0.628402 + 0.505990i −2.59662 0.507526i 1.76214i −2.21827 + 2.01973i −0.403396 + 0.232901i
236.11 0.559417 0.322980i 1.73059 + 0.0712304i −0.791368 + 1.37069i −1.00000 0.991125 0.519096i 1.24717 + 2.33336i 2.31430i 2.98985 + 0.246540i −0.559417 + 0.322980i
236.12 1.36556 0.788404i −1.67710 + 0.432827i 0.243162 0.421168i −1.00000 −1.94893 + 1.91328i 0.569408 + 2.58375i 2.38678i 2.62532 1.45179i −1.36556 + 0.788404i
236.13 1.48159 0.855399i 1.02526 1.39601i 0.463416 0.802660i −1.00000 0.324870 2.94533i 1.20122 2.35734i 1.83597i −0.897694 2.86254i −1.48159 + 0.855399i
236.14 2.01305 1.16223i 0.708464 + 1.58053i 1.70157 2.94720i −1.00000 3.26312 + 2.35828i 2.64011 0.172663i 3.26155i −1.99616 + 2.23950i −2.01305 + 1.16223i
236.15 2.11687 1.22217i 1.73076 + 0.0669518i 1.98742 3.44231i −1.00000 3.74561 1.97356i −2.50358 + 0.855621i 4.82720i 2.99103 + 0.231754i −2.11687 + 1.22217i
236.16 2.18143 1.25945i −1.57808 0.713898i 2.17243 3.76276i −1.00000 −4.34160 + 0.430199i −2.57718 0.598455i 5.90648i 1.98070 + 2.25318i −2.18143 + 1.25945i
311.1 −2.27381 1.31279i −0.933716 1.45883i 2.44681 + 4.23800i −1.00000 0.207969 + 4.54286i −2.64220 + 0.136993i 7.59741i −1.25635 + 2.72426i 2.27381 + 1.31279i
311.2 −2.03337 1.17397i 1.49299 0.878053i 1.75639 + 3.04216i −1.00000 −4.06660 + 0.0326841i 0.135526 + 2.64228i 3.55190i 1.45805 2.62185i 2.03337 + 1.17397i
311.3 −1.91788 1.10729i 0.302150 + 1.70549i 1.45217 + 2.51523i −1.00000 1.30898 3.60549i 2.53521 + 0.756781i 2.00271i −2.81741 + 1.03063i 1.91788 + 1.10729i
311.4 −1.67628 0.967799i −1.71487 0.243339i 0.873269 + 1.51255i −1.00000 2.63910 + 2.06755i 2.28906 1.32673i 0.490599i 2.88157 + 0.834589i 1.67628 + 0.967799i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 311.16 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.s 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.be.c yes 32
3.b odd 2 1 945.2.be.c 32
7.d odd 6 1 315.2.t.c 32
9.c even 3 1 945.2.t.c 32
9.d odd 6 1 315.2.t.c 32
21.g even 6 1 945.2.t.c 32
63.k odd 6 1 945.2.be.c 32
63.s even 6 1 inner 315.2.be.c yes 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.t.c 32 7.d odd 6 1
315.2.t.c 32 9.d odd 6 1
315.2.be.c yes 32 1.a even 1 1 trivial
315.2.be.c yes 32 63.s even 6 1 inner
945.2.t.c 32 9.c even 3 1
945.2.t.c 32 21.g even 6 1
945.2.be.c 32 3.b odd 2 1
945.2.be.c 32 63.k odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{32} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database