# Properties

 Label 675.3.j.a Level $675$ Weight $3$ Character orbit 675.j Analytic conductor $18.392$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$675 = 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 675.j (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.3924178443$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 9) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( 2 - 2 \zeta_{6} ) q^{7} + ( -5 + 10 \zeta_{6} ) q^{8} +O(q^{10})$$ $$q + ( -1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( 2 - 2 \zeta_{6} ) q^{7} + ( -5 + 10 \zeta_{6} ) q^{8} + ( 1 + \zeta_{6} ) q^{11} -4 \zeta_{6} q^{13} + ( -4 + 2 \zeta_{6} ) q^{14} + ( 11 - 11 \zeta_{6} ) q^{16} + ( 9 - 18 \zeta_{6} ) q^{17} + 11 q^{19} -3 \zeta_{6} q^{22} + ( -32 + 16 \zeta_{6} ) q^{23} + ( -4 + 8 \zeta_{6} ) q^{26} -2 q^{28} + ( -26 - 26 \zeta_{6} ) q^{29} -32 \zeta_{6} q^{31} + ( 18 - 9 \zeta_{6} ) q^{32} + ( -27 + 27 \zeta_{6} ) q^{34} + 34 q^{37} + ( -11 - 11 \zeta_{6} ) q^{38} + ( 14 - 7 \zeta_{6} ) q^{41} + ( -61 + 61 \zeta_{6} ) q^{43} + ( 1 - 2 \zeta_{6} ) q^{44} + 48 q^{46} + ( -28 - 28 \zeta_{6} ) q^{47} + 45 \zeta_{6} q^{49} + ( -4 + 4 \zeta_{6} ) q^{52} + ( 10 + 10 \zeta_{6} ) q^{56} + 78 \zeta_{6} q^{58} + ( -58 + 29 \zeta_{6} ) q^{59} + ( -56 + 56 \zeta_{6} ) q^{61} + ( -32 + 64 \zeta_{6} ) q^{62} -71 q^{64} -31 \zeta_{6} q^{67} + ( -18 + 9 \zeta_{6} ) q^{68} + ( 18 - 36 \zeta_{6} ) q^{71} -65 q^{73} + ( -34 - 34 \zeta_{6} ) q^{74} -11 \zeta_{6} q^{76} + ( 4 - 2 \zeta_{6} ) q^{77} + ( -38 + 38 \zeta_{6} ) q^{79} -21 q^{82} + ( -28 - 28 \zeta_{6} ) q^{83} + ( 122 - 61 \zeta_{6} ) q^{86} + ( -15 + 15 \zeta_{6} ) q^{88} + ( -72 + 144 \zeta_{6} ) q^{89} -8 q^{91} + ( 16 + 16 \zeta_{6} ) q^{92} + 84 \zeta_{6} q^{94} + ( -115 + 115 \zeta_{6} ) q^{97} + ( 45 - 90 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} - q^{4} + 2 q^{7} + O(q^{10})$$ $$2 q - 3 q^{2} - q^{4} + 2 q^{7} + 3 q^{11} - 4 q^{13} - 6 q^{14} + 11 q^{16} + 22 q^{19} - 3 q^{22} - 48 q^{23} - 4 q^{28} - 78 q^{29} - 32 q^{31} + 27 q^{32} - 27 q^{34} + 68 q^{37} - 33 q^{38} + 21 q^{41} - 61 q^{43} + 96 q^{46} - 84 q^{47} + 45 q^{49} - 4 q^{52} + 30 q^{56} + 78 q^{58} - 87 q^{59} - 56 q^{61} - 142 q^{64} - 31 q^{67} - 27 q^{68} - 130 q^{73} - 102 q^{74} - 11 q^{76} + 6 q^{77} - 38 q^{79} - 42 q^{82} - 84 q^{83} + 183 q^{86} - 15 q^{88} - 16 q^{91} + 48 q^{92} + 84 q^{94} - 115 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$\chi(n)$$ $$\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.50000 0.866025i 0 −0.500000 0.866025i 0 0 1.00000 1.73205i 8.66025i 0 0
476.1 −1.50000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 1.00000 + 1.73205i 8.66025i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.3.j.a 2
3.b odd 2 1 225.3.j.a 2
5.b even 2 1 27.3.d.a 2
5.c odd 4 2 675.3.i.a 4
9.c even 3 1 225.3.j.a 2
9.d odd 6 1 inner 675.3.j.a 2
15.d odd 2 1 9.3.d.a 2
15.e even 4 2 225.3.i.a 4
20.d odd 2 1 432.3.q.a 2
40.e odd 2 1 1728.3.q.b 2
40.f even 2 1 1728.3.q.a 2
45.h odd 6 1 27.3.d.a 2
45.h odd 6 1 81.3.b.a 2
45.j even 6 1 9.3.d.a 2
45.j even 6 1 81.3.b.a 2
45.k odd 12 2 225.3.i.a 4
45.l even 12 2 675.3.i.a 4
60.h even 2 1 144.3.q.a 2
105.g even 2 1 441.3.r.a 2
105.o odd 6 1 441.3.j.a 2
105.o odd 6 1 441.3.n.b 2
105.p even 6 1 441.3.j.b 2
105.p even 6 1 441.3.n.a 2
120.i odd 2 1 576.3.q.b 2
120.m even 2 1 576.3.q.a 2
180.n even 6 1 432.3.q.a 2
180.n even 6 1 1296.3.e.a 2
180.p odd 6 1 144.3.q.a 2
180.p odd 6 1 1296.3.e.a 2
315.q odd 6 1 441.3.n.a 2
315.r even 6 1 441.3.n.b 2
315.bg odd 6 1 441.3.r.a 2
315.bn odd 6 1 441.3.j.b 2
315.bo even 6 1 441.3.j.a 2
360.z odd 6 1 576.3.q.a 2
360.bd even 6 1 1728.3.q.b 2
360.bh odd 6 1 1728.3.q.a 2
360.bk even 6 1 576.3.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.3.d.a 2 15.d odd 2 1
9.3.d.a 2 45.j even 6 1
27.3.d.a 2 5.b even 2 1
27.3.d.a 2 45.h odd 6 1
81.3.b.a 2 45.h odd 6 1
81.3.b.a 2 45.j even 6 1
144.3.q.a 2 60.h even 2 1
144.3.q.a 2 180.p odd 6 1
225.3.i.a 4 15.e even 4 2
225.3.i.a 4 45.k odd 12 2
225.3.j.a 2 3.b odd 2 1
225.3.j.a 2 9.c even 3 1
432.3.q.a 2 20.d odd 2 1
432.3.q.a 2 180.n even 6 1
441.3.j.a 2 105.o odd 6 1
441.3.j.a 2 315.bo even 6 1
441.3.j.b 2 105.p even 6 1
441.3.j.b 2 315.bn odd 6 1
441.3.n.a 2 105.p even 6 1
441.3.n.a 2 315.q odd 6 1
441.3.n.b 2 105.o odd 6 1
441.3.n.b 2 315.r even 6 1
441.3.r.a 2 105.g even 2 1
441.3.r.a 2 315.bg odd 6 1
576.3.q.a 2 120.m even 2 1
576.3.q.a 2 360.z odd 6 1
576.3.q.b 2 120.i odd 2 1
576.3.q.b 2 360.bk even 6 1
675.3.i.a 4 5.c odd 4 2
675.3.i.a 4 45.l even 12 2
675.3.j.a 2 1.a even 1 1 trivial
675.3.j.a 2 9.d odd 6 1 inner
1296.3.e.a 2 180.n even 6 1
1296.3.e.a 2 180.p odd 6 1
1728.3.q.a 2 40.f even 2 1
1728.3.q.a 2 360.bh odd 6 1
1728.3.q.b 2 40.e odd 2 1
1728.3.q.b 2 360.bd even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3 T_{2} + 3$$ acting on $$S_{3}^{\mathrm{new}}(675, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 + 3 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 - 2 T + T^{2}$$
$11$ $$3 - 3 T + T^{2}$$
$13$ $$16 + 4 T + T^{2}$$
$17$ $$243 + T^{2}$$
$19$ $$( -11 + T )^{2}$$
$23$ $$768 + 48 T + T^{2}$$
$29$ $$2028 + 78 T + T^{2}$$
$31$ $$1024 + 32 T + T^{2}$$
$37$ $$( -34 + T )^{2}$$
$41$ $$147 - 21 T + T^{2}$$
$43$ $$3721 + 61 T + T^{2}$$
$47$ $$2352 + 84 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$2523 + 87 T + T^{2}$$
$61$ $$3136 + 56 T + T^{2}$$
$67$ $$961 + 31 T + T^{2}$$
$71$ $$972 + T^{2}$$
$73$ $$( 65 + T )^{2}$$
$79$ $$1444 + 38 T + T^{2}$$
$83$ $$2352 + 84 T + T^{2}$$
$89$ $$15552 + T^{2}$$
$97$ $$13225 + 115 T + T^{2}$$