# Properties

 Label 1350.2.q.f Level 1350 Weight 2 Character orbit 1350.q 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.q (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.7798042729$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 450) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{5} q^{2} + ( -\zeta_{24}^{2} + \zeta_{24}^{6} ) q^{4} + ( 4 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{7} + \zeta_{24}^{3} q^{8} +O(q^{10})$$ $$q -\zeta_{24}^{5} q^{2} + ( -\zeta_{24}^{2} + \zeta_{24}^{6} ) q^{4} + ( 4 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{7} + \zeta_{24}^{3} q^{8} + ( 1 + \zeta_{24}^{4} ) q^{11} + ( 4 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{13} + ( -2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{14} + ( 1 - \zeta_{24}^{4} ) q^{16} + ( -3 \zeta_{24} + 3 \zeta_{24}^{5} ) q^{17} -7 \zeta_{24}^{6} q^{19} + ( \zeta_{24} - 2 \zeta_{24}^{5} ) q^{22} + 6 \zeta_{24}^{7} q^{23} + ( 2 - 4 \zeta_{24}^{4} ) q^{26} + ( -2 \zeta_{24}^{3} + 4 \zeta_{24}^{7} ) q^{28} + ( -2 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{29} + 8 \zeta_{24}^{4} q^{31} -\zeta_{24} q^{32} + 3 \zeta_{24}^{2} q^{34} + ( 4 \zeta_{24} + 4 \zeta_{24}^{5} ) q^{37} + ( -7 \zeta_{24}^{3} + 7 \zeta_{24}^{7} ) q^{38} + ( 14 - 7 \zeta_{24}^{4} ) q^{41} + ( -3 \zeta_{24}^{3} - 3 \zeta_{24}^{7} ) q^{43} + ( -2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{44} + 6 q^{46} + 6 \zeta_{24}^{5} q^{47} + ( 5 \zeta_{24}^{2} - 5 \zeta_{24}^{6} ) q^{49} + ( -4 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{52} + ( 2 + 2 \zeta_{24}^{4} ) q^{56} + ( 4 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{58} + ( -7 \zeta_{24}^{2} - 7 \zeta_{24}^{6} ) q^{59} + ( 4 - 4 \zeta_{24}^{4} ) q^{61} + ( 8 \zeta_{24} - 8 \zeta_{24}^{5} ) q^{62} + \zeta_{24}^{6} q^{64} + ( \zeta_{24} - 2 \zeta_{24}^{5} ) q^{67} -3 \zeta_{24}^{7} q^{68} + ( 8 - 16 \zeta_{24}^{4} ) q^{71} + ( -5 \zeta_{24}^{3} + 10 \zeta_{24}^{7} ) q^{73} + ( 4 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{74} + 7 \zeta_{24}^{4} q^{76} + 6 \zeta_{24} q^{77} -4 \zeta_{24}^{2} q^{79} + ( -7 \zeta_{24} - 7 \zeta_{24}^{5} ) q^{82} + ( 12 \zeta_{24}^{3} - 12 \zeta_{24}^{7} ) q^{83} + ( -6 + 3 \zeta_{24}^{4} ) q^{86} + ( \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{88} + ( 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{89} + 12 q^{91} -6 \zeta_{24}^{5} q^{92} + ( 6 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{94} + ( -10 \zeta_{24} + 5 \zeta_{24}^{5} ) q^{97} -5 \zeta_{24}^{3} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 12q^{11} + 4q^{16} + 32q^{31} + 84q^{41} + 48q^{46} + 24q^{56} + 16q^{61} + 28q^{76} - 36q^{86} + 96q^{91} + O(q^{100})$$

## Character values

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

 $$n$$ $$1001$$ $$1027$$ $$\chi(n)$$ $$\zeta_{24}^{4}$$ $$\zeta_{24}^{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1
 0.258819 − 0.965926i −0.258819 + 0.965926i 0.258819 + 0.965926i −0.258819 − 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i 0.965926 − 0.258819i −0.965926 + 0.258819i
−0.965926 + 0.258819i 0 0.866025 0.500000i 0 0 −0.896575 3.34607i −0.707107 + 0.707107i 0 0
143.2 0.965926 0.258819i 0 0.866025 0.500000i 0 0 0.896575 + 3.34607i 0.707107 0.707107i 0 0
557.1 −0.965926 0.258819i 0 0.866025 + 0.500000i 0 0 −0.896575 + 3.34607i −0.707107 0.707107i 0 0
557.2 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0 0 0.896575 3.34607i 0.707107 + 0.707107i 0 0
1007.1 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0 0 3.34607 0.896575i 0.707107 + 0.707107i 0 0
1007.2 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0 0 −3.34607 + 0.896575i −0.707107 0.707107i 0 0
1043.1 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0 0 3.34607 + 0.896575i 0.707107 0.707107i 0 0
1043.2 0.258819 0.965926i 0 −0.866025 0.500000i 0 0 −3.34607 0.896575i −0.707107 + 0.707107i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1043.2 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
5.c odd 4 2 inner
9.d odd 6 1 inner
45.h odd 6 1 inner
45.l even 12 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.2.q.f 8
3.b odd 2 1 450.2.p.f 8
5.b even 2 1 inner 1350.2.q.f 8
5.c odd 4 2 inner 1350.2.q.f 8
9.c even 3 1 450.2.p.f 8
9.d odd 6 1 inner 1350.2.q.f 8
15.d odd 2 1 450.2.p.f 8
15.e even 4 2 450.2.p.f 8
45.h odd 6 1 inner 1350.2.q.f 8
45.j even 6 1 450.2.p.f 8
45.k odd 12 2 450.2.p.f 8
45.l even 12 2 inner 1350.2.q.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.p.f 8 3.b odd 2 1
450.2.p.f 8 9.c even 3 1
450.2.p.f 8 15.d odd 2 1
450.2.p.f 8 15.e even 4 2
450.2.p.f 8 45.j even 6 1
450.2.p.f 8 45.k odd 12 2
1350.2.q.f 8 1.a even 1 1 trivial
1350.2.q.f 8 5.b even 2 1 inner
1350.2.q.f 8 5.c odd 4 2 inner
1350.2.q.f 8 9.d odd 6 1 inner
1350.2.q.f 8 45.h odd 6 1 inner
1350.2.q.f 8 45.l even 12 2 inner

## Hecke kernels

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

 $$T_{7}^{8} - 144 T_{7}^{4} + 20736$$ $$T_{11}^{2} - 3 T_{11} + 3$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{4} + T^{8}$$
$3$ 
$5$ 
$7$ $$( 1 + 23 T^{4} + 2401 T^{8} )( 1 + 71 T^{4} + 2401 T^{8} )$$
$11$ $$( 1 - 3 T + 14 T^{2} - 33 T^{3} + 121 T^{4} )^{4}$$
$13$ $$1 + 142 T^{4} - 8397 T^{8} + 4055662 T^{12} + 815730721 T^{16}$$
$17$ $$( 1 + 47 T^{4} + 83521 T^{8} )^{2}$$
$19$ $$( 1 + 11 T^{2} + 361 T^{4} )^{4}$$
$23$ $$1 + 958 T^{4} + 637923 T^{8} + 268087678 T^{12} + 78310985281 T^{16}$$
$29$ $$( 1 - 46 T^{2} + 1275 T^{4} - 38686 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4} )^{4}$$
$37$ $$( 1 - 2062 T^{4} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 - 21 T + 188 T^{2} - 861 T^{3} + 1681 T^{4} )^{4}$$
$43$ $$1 + 217 T^{4} - 3371712 T^{8} + 741879817 T^{12} + 11688200277601 T^{16}$$
$47$ $$1 + 1054 T^{4} - 3768765 T^{8} + 5143183774 T^{12} + 23811286661761 T^{16}$$
$53$ $$( 1 + 2809 T^{4} )^{4}$$
$59$ $$( 1 + 29 T^{2} - 2640 T^{4} + 100949 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} )^{4}$$
$67$ $$1 - 8183 T^{4} + 46810368 T^{8} - 164896623143 T^{12} + 406067677556641 T^{16}$$
$71$ $$( 1 + 50 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$( 1 - 5617 T^{4} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 + 11 T^{2} + 6241 T^{4} )^{2}( 1 + 131 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$1 + 13294 T^{4} + 129272115 T^{8} + 630910919374 T^{12} + 2252292232139041 T^{16}$$
$89$ $$( 1 + 130 T^{2} + 7921 T^{4} )^{4}$$
$97$ $$1 + 4657 T^{4} - 66841632 T^{8} + 412280861617 T^{12} + 7837433594376961 T^{16}$$