# Properties

 Label 675.3.i.a Level $675$ Weight $3$ Character orbit 675.i Analytic conductor $18.392$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$18.3924178443$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{2} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{7} + ( -10 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{8} +O(q^{10})$$ $$q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{2} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{7} + ( -10 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{8} + ( 2 - \zeta_{12}^{2} ) q^{11} + 4 \zeta_{12} q^{13} + ( 2 + 2 \zeta_{12}^{2} ) q^{14} + 11 \zeta_{12}^{2} q^{16} + ( -18 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{17} -11 q^{19} -3 \zeta_{12} q^{22} + ( -16 \zeta_{12} + 32 \zeta_{12}^{3} ) q^{23} + ( 4 - 8 \zeta_{12}^{2} ) q^{26} + 2 \zeta_{12}^{3} q^{28} + ( 52 - 26 \zeta_{12}^{2} ) q^{29} + ( -32 + 32 \zeta_{12}^{2} ) q^{31} + ( -9 \zeta_{12} + 18 \zeta_{12}^{3} ) q^{32} + 27 \zeta_{12}^{2} q^{34} + 34 \zeta_{12}^{3} q^{37} + ( 11 \zeta_{12} + 11 \zeta_{12}^{3} ) q^{38} + ( 7 + 7 \zeta_{12}^{2} ) q^{41} + ( -61 \zeta_{12} + 61 \zeta_{12}^{3} ) q^{43} + ( 1 - 2 \zeta_{12}^{2} ) q^{44} + 48 q^{46} + ( -28 \zeta_{12} - 28 \zeta_{12}^{3} ) q^{47} + ( -45 + 45 \zeta_{12}^{2} ) q^{49} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{52} + ( 20 - 10 \zeta_{12}^{2} ) q^{56} -78 \zeta_{12} q^{58} + ( 29 + 29 \zeta_{12}^{2} ) q^{59} -56 \zeta_{12}^{2} q^{61} + ( 64 \zeta_{12} - 32 \zeta_{12}^{3} ) q^{62} + 71 q^{64} -31 \zeta_{12} q^{67} + ( -9 \zeta_{12} + 18 \zeta_{12}^{3} ) q^{68} + ( -18 + 36 \zeta_{12}^{2} ) q^{71} + 65 \zeta_{12}^{3} q^{73} + ( 68 - 34 \zeta_{12}^{2} ) q^{74} + ( -11 + 11 \zeta_{12}^{2} ) q^{76} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{77} + 38 \zeta_{12}^{2} q^{79} -21 \zeta_{12}^{3} q^{82} + ( 28 \zeta_{12} + 28 \zeta_{12}^{3} ) q^{83} + ( 61 + 61 \zeta_{12}^{2} ) q^{86} + ( -15 \zeta_{12} + 15 \zeta_{12}^{3} ) q^{88} + ( -72 + 144 \zeta_{12}^{2} ) q^{89} -8 q^{91} + ( 16 \zeta_{12} + 16 \zeta_{12}^{3} ) q^{92} + ( -84 + 84 \zeta_{12}^{2} ) q^{94} + ( 115 \zeta_{12} - 115 \zeta_{12}^{3} ) q^{97} + ( 90 \zeta_{12} - 45 \zeta_{12}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} + O(q^{10})$$ $$4 q + 2 q^{4} + 6 q^{11} + 12 q^{14} + 22 q^{16} - 44 q^{19} + 156 q^{29} - 64 q^{31} + 54 q^{34} + 42 q^{41} + 192 q^{46} - 90 q^{49} + 60 q^{56} + 174 q^{59} - 112 q^{61} + 284 q^{64} + 204 q^{74} - 22 q^{76} + 76 q^{79} + 366 q^{86} - 32 q^{91} - 168 q^{94} + O(q^{100})$$

## Character values

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

 $$n$$ $$326$$ $$352$$ $$\chi(n)$$ $$1 - \zeta_{12}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
224.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−0.866025 + 1.50000i 0 0.500000 + 0.866025i 0 0 −1.73205 1.00000i −8.66025 0 0
224.2 0.866025 1.50000i 0 0.500000 + 0.866025i 0 0 1.73205 + 1.00000i 8.66025 0 0
449.1 −0.866025 1.50000i 0 0.500000 0.866025i 0 0 −1.73205 + 1.00000i −8.66025 0 0
449.2 0.866025 + 1.50000i 0 0.500000 0.866025i 0 0 1.73205 1.00000i 8.66025 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
5.b even 2 1 inner
9.d odd 6 1 inner
45.h 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.i.a 4
3.b odd 2 1 225.3.i.a 4
5.b even 2 1 inner 675.3.i.a 4
5.c odd 4 1 27.3.d.a 2
5.c odd 4 1 675.3.j.a 2
9.c even 3 1 225.3.i.a 4
9.d odd 6 1 inner 675.3.i.a 4
15.d odd 2 1 225.3.i.a 4
15.e even 4 1 9.3.d.a 2
15.e even 4 1 225.3.j.a 2
20.e even 4 1 432.3.q.a 2
40.i odd 4 1 1728.3.q.a 2
40.k even 4 1 1728.3.q.b 2
45.h odd 6 1 inner 675.3.i.a 4
45.j even 6 1 225.3.i.a 4
45.k odd 12 1 9.3.d.a 2
45.k odd 12 1 81.3.b.a 2
45.k odd 12 1 225.3.j.a 2
45.l even 12 1 27.3.d.a 2
45.l even 12 1 81.3.b.a 2
45.l even 12 1 675.3.j.a 2
60.l odd 4 1 144.3.q.a 2
105.k odd 4 1 441.3.r.a 2
105.w odd 12 1 441.3.j.b 2
105.w odd 12 1 441.3.n.a 2
105.x even 12 1 441.3.j.a 2
105.x even 12 1 441.3.n.b 2
120.q odd 4 1 576.3.q.a 2
120.w even 4 1 576.3.q.b 2
180.v odd 12 1 432.3.q.a 2
180.v odd 12 1 1296.3.e.a 2
180.x even 12 1 144.3.q.a 2
180.x even 12 1 1296.3.e.a 2
315.bs even 12 1 441.3.n.a 2
315.bt odd 12 1 441.3.n.b 2
315.cb even 12 1 441.3.r.a 2
315.cg even 12 1 441.3.j.b 2
315.ch odd 12 1 441.3.j.a 2
360.bo even 12 1 576.3.q.a 2
360.br even 12 1 1728.3.q.a 2
360.bt odd 12 1 1728.3.q.b 2
360.bu odd 12 1 576.3.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.3.d.a 2 15.e even 4 1
9.3.d.a 2 45.k odd 12 1
27.3.d.a 2 5.c odd 4 1
27.3.d.a 2 45.l even 12 1
81.3.b.a 2 45.k odd 12 1
81.3.b.a 2 45.l even 12 1
144.3.q.a 2 60.l odd 4 1
144.3.q.a 2 180.x even 12 1
225.3.i.a 4 3.b odd 2 1
225.3.i.a 4 9.c even 3 1
225.3.i.a 4 15.d odd 2 1
225.3.i.a 4 45.j even 6 1
225.3.j.a 2 15.e even 4 1
225.3.j.a 2 45.k odd 12 1
432.3.q.a 2 20.e even 4 1
432.3.q.a 2 180.v odd 12 1
441.3.j.a 2 105.x even 12 1
441.3.j.a 2 315.ch odd 12 1
441.3.j.b 2 105.w odd 12 1
441.3.j.b 2 315.cg even 12 1
441.3.n.a 2 105.w odd 12 1
441.3.n.a 2 315.bs even 12 1
441.3.n.b 2 105.x even 12 1
441.3.n.b 2 315.bt odd 12 1
441.3.r.a 2 105.k odd 4 1
441.3.r.a 2 315.cb even 12 1
576.3.q.a 2 120.q odd 4 1
576.3.q.a 2 360.bo even 12 1
576.3.q.b 2 120.w even 4 1
576.3.q.b 2 360.bu odd 12 1
675.3.i.a 4 1.a even 1 1 trivial
675.3.i.a 4 5.b even 2 1 inner
675.3.i.a 4 9.d odd 6 1 inner
675.3.i.a 4 45.h odd 6 1 inner
675.3.j.a 2 5.c odd 4 1
675.3.j.a 2 45.l even 12 1
1296.3.e.a 2 180.v odd 12 1
1296.3.e.a 2 180.x even 12 1
1728.3.q.a 2 40.i odd 4 1
1728.3.q.a 2 360.br even 12 1
1728.3.q.b 2 40.k even 4 1
1728.3.q.b 2 360.bt odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 3 T_{2}^{2} + 9$$ acting on $$S_{3}^{\mathrm{new}}(675, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 + 3 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$16 - 4 T^{2} + T^{4}$$
$11$ $$( 3 - 3 T + T^{2} )^{2}$$
$13$ $$256 - 16 T^{2} + T^{4}$$
$17$ $$( -243 + T^{2} )^{2}$$
$19$ $$( 11 + T )^{4}$$
$23$ $$589824 + 768 T^{2} + T^{4}$$
$29$ $$( 2028 - 78 T + T^{2} )^{2}$$
$31$ $$( 1024 + 32 T + T^{2} )^{2}$$
$37$ $$( 1156 + T^{2} )^{2}$$
$41$ $$( 147 - 21 T + T^{2} )^{2}$$
$43$ $$13845841 - 3721 T^{2} + T^{4}$$
$47$ $$5531904 + 2352 T^{2} + T^{4}$$
$53$ $$T^{4}$$
$59$ $$( 2523 - 87 T + T^{2} )^{2}$$
$61$ $$( 3136 + 56 T + T^{2} )^{2}$$
$67$ $$923521 - 961 T^{2} + T^{4}$$
$71$ $$( 972 + T^{2} )^{2}$$
$73$ $$( 4225 + T^{2} )^{2}$$
$79$ $$( 1444 - 38 T + T^{2} )^{2}$$
$83$ $$5531904 + 2352 T^{2} + T^{4}$$
$89$ $$( 15552 + T^{2} )^{2}$$
$97$ $$174900625 - 13225 T^{2} + T^{4}$$