# Properties

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

 $$\operatorname{Tr}(f)(q) =$$ $$32q - q^{3} - 32q^{4} - 16q^{5} - 2q^{6} + q^{7} + q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - q^{3} - 32q^{4} - 16q^{5} - 2q^{6} + q^{7} + q^{9} + 3q^{11} + 12q^{12} + 6q^{13} - 15q^{14} - q^{15} + 32q^{16} - 3q^{17} - 13q^{18} + 16q^{20} - q^{21} - 21q^{22} - 9q^{23} - 4q^{24} - 16q^{25} + 12q^{26} + 23q^{27} - 31q^{28} + 18q^{29} - 2q^{30} + 19q^{33} - 30q^{34} + q^{35} + 18q^{36} - q^{37} - 30q^{38} + 21q^{39} + 6q^{41} + 19q^{42} - 19q^{43} + 21q^{44} - 8q^{45} + 6q^{46} - 30q^{47} - 35q^{48} + 5q^{49} + 36q^{51} + 21q^{52} - 24q^{53} - 59q^{54} + 30q^{56} + 27q^{57} + 30q^{59} + 3q^{60} - 32q^{63} + 76q^{64} + 26q^{66} - 50q^{67} - 3q^{68} - 50q^{69} + 9q^{70} - 14q^{72} + 12q^{73} + 60q^{74} + 2q^{75} + 54q^{76} - 27q^{77} - 42q^{78} + 4q^{79} - 16q^{80} - 23q^{81} - 24q^{82} - 42q^{83} - 72q^{84} - 3q^{85} + 51q^{86} + 34q^{87} + 42q^{88} + 30q^{89} + 41q^{90} - 57q^{91} + 6q^{92} - 33q^{93} + 15q^{96} - 42q^{97} + 6q^{98} - q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1 2.62557i −1.73024 0.0792089i −4.89362 −0.500000 + 0.866025i −0.207969 + 4.54286i 1.43974 + 2.21972i 7.59741i 2.98745 + 0.274101i 2.27381 + 1.31279i
101.2 2.34793i −0.0139204 + 1.73199i −3.51278 −0.500000 + 0.866025i 4.06660 + 0.0326841i 2.22052 1.43851i 3.55190i −2.99961 0.0482201i 2.03337 + 1.17397i
101.3 2.21457i 1.62808 0.591077i −2.90433 −0.500000 + 0.866025i −1.30898 3.60549i −0.612213 2.57395i 2.00271i 2.30126 1.92463i 1.91788 + 1.10729i
101.4 1.93560i −1.06817 1.36345i −1.74654 −0.500000 + 0.866025i −2.63910 + 2.06755i −2.29351 1.31902i 0.490599i −0.718011 + 2.91281i 1.67628 + 0.967799i
101.5 1.23655i 1.51263 + 0.843766i 0.470942 −0.500000 + 0.866025i 1.04336 1.87045i 2.44060 + 1.02150i 3.05545i 1.57612 + 2.55262i 1.07088 + 0.618275i
101.6 1.09551i −1.60475 + 0.651761i 0.799850 −0.500000 + 0.866025i 0.714013 + 1.75802i −2.64482 0.0701258i 3.06727i 2.15042 2.09182i 0.948743 + 0.547757i
101.7 0.524306i 1.63490 0.571919i 1.72510 −0.500000 + 0.866025i −0.299860 0.857190i −1.53835 + 2.15255i 1.95309i 2.34582 1.87006i 0.454062 + 0.262153i
101.8 0.103991i 0.570662 1.63534i 1.98919 −0.500000 + 0.866025i −0.170061 0.0593436i −0.107447 2.64357i 0.414839i −2.34869 1.86646i 0.0900587 + 0.0519954i
101.9 0.396951i −1.27680 1.17037i 1.84243 −0.500000 + 0.866025i 0.464581 0.506829i 2.43622 1.03190i 1.52526i 0.260457 + 2.98867i −0.343770 0.198476i
101.10 0.465802i −1.08628 + 1.34907i 1.78303 −0.500000 + 0.866025i −0.628402 0.505990i 1.73784 + 1.99497i 1.76214i −0.640003 2.93094i −0.403396 0.232901i
101.11 0.645959i 0.803605 + 1.53435i 1.58274 −0.500000 + 0.866025i −0.991125 + 0.519096i −2.64433 + 0.0866018i 2.31430i −1.70844 + 2.46602i −0.559417 0.322980i
101.12 1.57681i −1.21339 1.23600i −0.486323 −0.500000 + 0.866025i 1.94893 1.91328i −2.52230 + 0.798755i 2.38678i −0.0553747 + 2.99949i −1.36556 0.788404i
101.13 1.71080i 1.72161 + 0.189893i −0.926832 −0.500000 + 0.866025i −0.324870 + 2.94533i 1.44091 2.21896i 1.83597i 2.92788 + 0.653845i −1.48159 0.855399i
101.14 2.32447i −1.01455 + 1.40381i −3.40314 −0.500000 + 0.866025i −3.26312 2.35828i −1.17052 2.37274i 3.26155i −0.941384 2.84847i −2.01305 1.16223i
101.15 2.44435i 0.807396 + 1.53235i −3.97484 −0.500000 + 0.866025i −3.74561 + 1.97356i 0.510801 + 2.59597i 4.82720i −1.69622 + 2.47444i −2.11687 1.22217i
101.16 2.51890i −0.170788 1.72361i −4.34486 −0.500000 + 0.866025i 4.34160 0.430199i 1.80687 + 1.93267i 5.90648i −2.94166 + 0.588745i −2.18143 1.25945i
131.1 2.51890i −0.170788 + 1.72361i −4.34486 −0.500000 0.866025i 4.34160 + 0.430199i 1.80687 1.93267i 5.90648i −2.94166 0.588745i −2.18143 + 1.25945i
131.2 2.44435i 0.807396 1.53235i −3.97484 −0.500000 0.866025i −3.74561 1.97356i 0.510801 2.59597i 4.82720i −1.69622 2.47444i −2.11687 + 1.22217i
131.3 2.32447i −1.01455 1.40381i −3.40314 −0.500000 0.866025i −3.26312 + 2.35828i −1.17052 + 2.37274i 3.26155i −0.941384 + 2.84847i −2.01305 + 1.16223i
131.4 1.71080i 1.72161 0.189893i −0.926832 −0.500000 0.866025i −0.324870 2.94533i 1.44091 + 2.21896i 1.83597i 2.92788 0.653845i −1.48159 + 0.855399i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 131.16 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.i 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.t.c 32
3.b odd 2 1 945.2.t.c 32
7.d odd 6 1 315.2.be.c yes 32
9.c even 3 1 945.2.be.c 32
9.d odd 6 1 315.2.be.c yes 32
21.g even 6 1 945.2.be.c 32
63.i even 6 1 inner 315.2.t.c 32
63.t odd 6 1 945.2.t.c 32

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