# Properties

 Label 1350.2.f.d Level 1350 Weight 2 Character orbit 1350.f Analytic conductor 10.780 Analytic rank 0 Dimension 8 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$1350 = 2 \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1350.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.7798042729$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{3} q^{2} + \zeta_{24}^{6} q^{4} + ( -\zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{7} + ( \zeta_{24} - \zeta_{24}^{5} ) q^{8} +O(q^{10})$$ $$q -\zeta_{24}^{3} q^{2} + \zeta_{24}^{6} q^{4} + ( -\zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{7} + ( \zeta_{24} - \zeta_{24}^{5} ) q^{8} + ( 2 \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{14} - q^{16} -6 \zeta_{24}^{3} q^{17} + \zeta_{24}^{6} q^{19} + ( 6 \zeta_{24} - 6 \zeta_{24}^{5} ) q^{23} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{28} + ( 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{29} + 7 q^{31} + \zeta_{24}^{3} q^{32} + 6 \zeta_{24}^{6} q^{34} + ( -5 \zeta_{24}^{3} + 10 \zeta_{24}^{7} ) q^{37} + ( \zeta_{24} - \zeta_{24}^{5} ) q^{38} + ( \zeta_{24} + \zeta_{24}^{5} ) q^{43} -6 q^{46} + 4 \zeta_{24}^{6} q^{49} + ( 12 \zeta_{24} - 12 \zeta_{24}^{5} ) q^{53} + ( -1 + 2 \zeta_{24}^{4} ) q^{56} + ( -6 \zeta_{24} - 6 \zeta_{24}^{5} ) q^{58} + ( 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{59} + 5 q^{61} -7 \zeta_{24}^{3} q^{62} -\zeta_{24}^{6} q^{64} + ( 6 \zeta_{24}^{3} - 12 \zeta_{24}^{7} ) q^{67} + ( 6 \zeta_{24} - 6 \zeta_{24}^{5} ) q^{68} + ( -6 + 12 \zeta_{24}^{4} ) q^{71} + ( 7 \zeta_{24} + 7 \zeta_{24}^{5} ) q^{73} + ( 10 \zeta_{24}^{2} - 5 \zeta_{24}^{6} ) q^{74} - q^{76} -13 \zeta_{24}^{6} q^{79} + ( 6 \zeta_{24} - 6 \zeta_{24}^{5} ) q^{83} + ( 1 - 2 \zeta_{24}^{4} ) q^{86} + ( -12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{89} + 6 \zeta_{24}^{3} q^{92} + ( -5 \zeta_{24}^{3} + 10 \zeta_{24}^{7} ) q^{97} + ( 4 \zeta_{24} - 4 \zeta_{24}^{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 8q^{16} + 56q^{31} - 48q^{46} + 40q^{61} - 8q^{76} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1027$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{24}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 0.965926 − 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i −0.965926 + 0.258819i 0.965926 + 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i −0.965926 − 0.258819i
−0.707107 + 0.707107i 0 1.00000i 0 0 −1.22474 1.22474i 0.707107 + 0.707107i 0 0
107.2 −0.707107 + 0.707107i 0 1.00000i 0 0 1.22474 + 1.22474i 0.707107 + 0.707107i 0 0
107.3 0.707107 0.707107i 0 1.00000i 0 0 −1.22474 1.22474i −0.707107 0.707107i 0 0
107.4 0.707107 0.707107i 0 1.00000i 0 0 1.22474 + 1.22474i −0.707107 0.707107i 0 0
593.1 −0.707107 0.707107i 0 1.00000i 0 0 −1.22474 + 1.22474i 0.707107 0.707107i 0 0
593.2 −0.707107 0.707107i 0 1.00000i 0 0 1.22474 1.22474i 0.707107 0.707107i 0 0
593.3 0.707107 + 0.707107i 0 1.00000i 0 0 −1.22474 + 1.22474i −0.707107 + 0.707107i 0 0
593.4 0.707107 + 0.707107i 0 1.00000i 0 0 1.22474 1.22474i −0.707107 + 0.707107i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 593.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.2.f.d 8
3.b odd 2 1 inner 1350.2.f.d 8
5.b even 2 1 inner 1350.2.f.d 8
5.c odd 4 2 inner 1350.2.f.d 8
15.d odd 2 1 inner 1350.2.f.d 8
15.e even 4 2 inner 1350.2.f.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1350.2.f.d 8 1.a even 1 1 trivial
1350.2.f.d 8 3.b odd 2 1 inner
1350.2.f.d 8 5.b even 2 1 inner
1350.2.f.d 8 5.c odd 4 2 inner
1350.2.f.d 8 15.d odd 2 1 inner
1350.2.f.d 8 15.e even 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1350, [\chi])$$:

 $$T_{7}^{4} + 9$$ $$T_{29}^{2} - 108$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ 
$5$ 
$7$ $$( 1 + 23 T^{4} + 2401 T^{8} )^{2}$$
$11$ $$( 1 - 11 T^{2} )^{8}$$
$13$ $$( 1 + 169 T^{4} )^{4}$$
$17$ $$( 1 - 8 T + 32 T^{2} - 136 T^{3} + 289 T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 136 T^{3} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 37 T^{2} + 361 T^{4} )^{4}$$
$23$ $$( 1 - 958 T^{4} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 50 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 - 7 T + 31 T^{2} )^{8}$$
$37$ $$( 1 - 2737 T^{4} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 - 41 T^{2} )^{8}$$
$43$ $$( 1 + 3191 T^{4} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 + 2209 T^{4} )^{4}$$
$53$ $$( 1 - 4174 T^{4} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 + 10 T^{2} + 3481 T^{4} )^{4}$$
$61$ $$( 1 - 5 T + 61 T^{2} )^{8}$$
$67$ $$( 1 - 8302 T^{4} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 - 34 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$( 1 - 10657 T^{4} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 + 11 T^{2} + 6241 T^{4} )^{4}$$
$83$ $$( 1 + 3122 T^{4} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 70 T^{2} + 7921 T^{4} )^{4}$$
$97$ $$( 1 - 4657 T^{4} + 88529281 T^{8} )^{2}$$