# Properties

 Label 1350.2.f.a Level 1350 Weight 2 Character orbit 1350.f Analytic conductor 10.780 Analytic rank 0 Dimension 8 CM no Inner twists 4

# 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: $$2^{2}$$ Twist minimal: no (minimal twist has level 270) 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{3} q^{2} + \zeta_{24}^{6} q^{4} + ( -1 - \zeta_{24}^{2} + \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{7} + ( \zeta_{24} - \zeta_{24}^{5} ) q^{8} +O(q^{10})$$ $$q -\zeta_{24}^{3} q^{2} + \zeta_{24}^{6} q^{4} + ( -1 - \zeta_{24}^{2} + \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{7} + ( \zeta_{24} - \zeta_{24}^{5} ) q^{8} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{11} + ( -2 + 4 \zeta_{24}^{2} + 4 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{13} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{14} - q^{16} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{17} + ( -2 + 4 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{19} + ( 2 - \zeta_{24}^{2} + \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{22} + ( 2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{23} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{26} + ( -\zeta_{24}^{2} - \zeta_{24}^{4} ) q^{28} + ( -\zeta_{24} - \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{29} + ( -2 - 10 \zeta_{24}^{2} + 5 \zeta_{24}^{6} ) q^{31} + \zeta_{24}^{3} q^{32} + ( -2 + 4 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{34} + ( -6 - 6 \zeta_{24}^{2} + 6 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{37} + ( -4 \zeta_{24} + 2 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{38} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{41} + ( 2 + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{43} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{44} -2 q^{46} + 6 \zeta_{24}^{3} q^{47} + ( 1 - 2 \zeta_{24}^{4} + 5 \zeta_{24}^{6} ) q^{49} + ( -2 - 4 \zeta_{24}^{2} + 4 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{52} + ( -2 \zeta_{24} + 3 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{53} + ( \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{56} + ( -3 + 4 \zeta_{24}^{2} + 4 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{58} + ( -5 \zeta_{24} - 5 \zeta_{24}^{3} - \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{59} + ( 6 - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{61} + ( 5 \zeta_{24} + 2 \zeta_{24}^{3} + 5 \zeta_{24}^{5} ) q^{62} -\zeta_{24}^{6} q^{64} + ( -2 - 6 \zeta_{24}^{2} + 6 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{67} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{68} + ( 8 \zeta_{24} - 8 \zeta_{24}^{3} + 8 \zeta_{24}^{7} ) q^{71} + ( -9 + \zeta_{24}^{2} + \zeta_{24}^{4} - 9 \zeta_{24}^{6} ) q^{73} + ( 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{7} ) q^{74} + ( 4 - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{76} + ( -2 \zeta_{24} - \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{77} + ( -2 + 4 \zeta_{24}^{4} ) q^{79} + ( -6 - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{82} + ( -9 \zeta_{24} - 4 \zeta_{24}^{3} + 9 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{83} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{86} + ( 3 - \zeta_{24}^{2} - \zeta_{24}^{4} + 3 \zeta_{24}^{6} ) q^{88} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{89} + ( -6 - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{91} + 2 \zeta_{24}^{3} q^{92} -6 \zeta_{24}^{6} q^{94} + ( 4 - 7 \zeta_{24}^{2} + 7 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{97} + ( 5 \zeta_{24} - \zeta_{24}^{3} - 5 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{7} + O(q^{10})$$ $$8q - 4q^{7} - 8q^{16} + 20q^{22} - 4q^{28} - 16q^{31} - 24q^{37} + 24q^{43} - 16q^{46} - 8q^{58} + 48q^{61} + 8q^{67} - 68q^{73} + 32q^{76} - 40q^{82} + 20q^{88} - 48q^{91} + 60q^{97} + O(q^{100})$$

## 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 −0.965926 + 0.258819i 0.258819 − 0.965926i 0.965926 + 0.258819i −0.258819 − 0.965926i −0.965926 − 0.258819i 0.258819 + 0.965926i
−0.707107 + 0.707107i 0 1.00000i 0 0 −1.36603 1.36603i 0.707107 + 0.707107i 0 0
107.2 −0.707107 + 0.707107i 0 1.00000i 0 0 0.366025 + 0.366025i 0.707107 + 0.707107i 0 0
107.3 0.707107 0.707107i 0 1.00000i 0 0 −1.36603 1.36603i −0.707107 0.707107i 0 0
107.4 0.707107 0.707107i 0 1.00000i 0 0 0.366025 + 0.366025i −0.707107 0.707107i 0 0
593.1 −0.707107 0.707107i 0 1.00000i 0 0 −1.36603 + 1.36603i 0.707107 0.707107i 0 0
593.2 −0.707107 0.707107i 0 1.00000i 0 0 0.366025 0.366025i 0.707107 0.707107i 0 0
593.3 0.707107 + 0.707107i 0 1.00000i 0 0 −1.36603 + 1.36603i −0.707107 + 0.707107i 0 0
593.4 0.707107 + 0.707107i 0 1.00000i 0 0 0.366025 0.366025i −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.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.2.f.a 8
3.b odd 2 1 inner 1350.2.f.a 8
5.b even 2 1 270.2.f.b 8
5.c odd 4 1 270.2.f.b 8
5.c odd 4 1 inner 1350.2.f.a 8
15.d odd 2 1 270.2.f.b 8
15.e even 4 1 270.2.f.b 8
15.e even 4 1 inner 1350.2.f.a 8
20.d odd 2 1 2160.2.w.b 8
20.e even 4 1 2160.2.w.b 8
45.h odd 6 1 810.2.m.d 8
45.h odd 6 1 810.2.m.e 8
45.j even 6 1 810.2.m.d 8
45.j even 6 1 810.2.m.e 8
45.k odd 12 1 810.2.m.d 8
45.k odd 12 1 810.2.m.e 8
45.l even 12 1 810.2.m.d 8
45.l even 12 1 810.2.m.e 8
60.h even 2 1 2160.2.w.b 8
60.l odd 4 1 2160.2.w.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.f.b 8 5.b even 2 1
270.2.f.b 8 5.c odd 4 1
270.2.f.b 8 15.d odd 2 1
270.2.f.b 8 15.e even 4 1
810.2.m.d 8 45.h odd 6 1
810.2.m.d 8 45.j even 6 1
810.2.m.d 8 45.k odd 12 1
810.2.m.d 8 45.l even 12 1
810.2.m.e 8 45.h odd 6 1
810.2.m.e 8 45.j even 6 1
810.2.m.e 8 45.k odd 12 1
810.2.m.e 8 45.l even 12 1
1350.2.f.a 8 1.a even 1 1 trivial
1350.2.f.a 8 3.b odd 2 1 inner
1350.2.f.a 8 5.c odd 4 1 inner
1350.2.f.a 8 15.e even 4 1 inner
2160.2.w.b 8 20.d odd 2 1
2160.2.w.b 8 20.e even 4 1
2160.2.w.b 8 60.h even 2 1
2160.2.w.b 8 60.l odd 4 1

## Hecke kernels

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

 $$T_{7}^{4} + 2 T_{7}^{3} + 2 T_{7}^{2} - 2 T_{7} + 1$$ $$T_{29}^{4} - 52 T_{29}^{2} + 484$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ 
$5$ 
$7$ $$( 1 + 2 T + 2 T^{2} + 12 T^{3} + 71 T^{4} + 84 T^{5} + 98 T^{6} + 686 T^{7} + 2401 T^{8} )^{2}$$
$11$ $$( 1 - 16 T^{2} + 231 T^{4} - 1936 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - 334 T^{4} + 28561 T^{8} )^{2}$$
$17$ $$1 - 124 T^{4} - 77946 T^{8} - 10356604 T^{12} + 6975757441 T^{16}$$
$19$ $$( 1 - 20 T^{2} + 54 T^{4} - 7220 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$( 1 + 706 T^{4} + 279841 T^{8} )^{2}$$
$29$ $$( 1 + 64 T^{2} + 2514 T^{4} + 53824 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 + 4 T - 9 T^{2} + 124 T^{3} + 961 T^{4} )^{4}$$
$37$ $$( 1 + 12 T + 72 T^{2} + 12 T^{3} - 1294 T^{4} + 444 T^{5} + 98568 T^{6} + 607836 T^{7} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 - 52 T^{2} + 2838 T^{4} - 87412 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 12 T + 72 T^{2} - 660 T^{3} + 5906 T^{4} - 28380 T^{5} + 133128 T^{6} - 954084 T^{7} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 - 1054 T^{4} + 4879681 T^{8} )^{2}$$
$53$ $$1 + 878 T^{4} + 6253683 T^{8} + 6927842318 T^{12} + 62259690411361 T^{16}$$
$59$ $$( 1 + 112 T^{2} + 8370 T^{4} + 389872 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 12 T + 146 T^{2} - 732 T^{3} + 3721 T^{4} )^{4}$$
$67$ $$( 1 - 4 T + 8 T^{2} - 60 T^{3} - 2254 T^{4} - 4020 T^{5} + 35912 T^{6} - 1203052 T^{7} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 - 28 T^{2} - 2010 T^{4} - 141148 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 + 17 T + 73 T^{2} )^{4}( 1 + 143 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 146 T^{2} + 6241 T^{4} )^{4}$$
$83$ $$1 + 6286 T^{4} + 19632339 T^{8} + 298323005806 T^{12} + 2252292232139041 T^{16}$$
$89$ $$( 1 + 244 T^{2} + 30294 T^{4} + 1932724 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 30 T + 450 T^{2} - 4080 T^{3} + 35471 T^{4} - 395760 T^{5} + 4234050 T^{6} - 27380190 T^{7} + 88529281 T^{8} )^{2}$$