# Properties

 Label 1350.2.f.e 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: $$2^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{24} - \zeta_{24}^{5} ) q^{2} -\zeta_{24}^{6} q^{4} + ( -\zeta_{24} + 3 \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{7} -\zeta_{24}^{3} q^{8} +O(q^{10})$$ $$q + ( \zeta_{24} - \zeta_{24}^{5} ) q^{2} -\zeta_{24}^{6} q^{4} + ( -\zeta_{24} + 3 \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{7} -\zeta_{24}^{3} q^{8} + 3 \zeta_{24}^{6} q^{11} + ( 3 \zeta_{24} - 3 \zeta_{24}^{5} ) q^{13} + ( 3 - 2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{14} - q^{16} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{17} + ( 3 - 6 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{19} + 3 \zeta_{24}^{3} q^{22} + 3 \zeta_{24}^{3} q^{23} -3 \zeta_{24}^{6} q^{26} + ( 3 \zeta_{24} - \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{28} + ( 3 + 6 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{29} -2 q^{31} + ( -\zeta_{24} + \zeta_{24}^{5} ) q^{32} + ( 3 - 6 \zeta_{24}^{4} + 3 \zeta_{24}^{6} ) q^{34} + ( 4 \zeta_{24} + 3 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{37} + ( -3 \zeta_{24} - \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{38} + 6 \zeta_{24}^{6} q^{41} + ( 9 \zeta_{24} + \zeta_{24}^{3} - 9 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{43} + 3 q^{44} + 3 q^{46} + ( 9 \zeta_{24} - 9 \zeta_{24}^{5} ) q^{47} + ( 6 - 12 \zeta_{24}^{4} + 5 \zeta_{24}^{6} ) q^{49} -3 \zeta_{24}^{3} q^{52} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{53} + ( -1 + 2 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{56} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{58} + ( -3 - 12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{59} + ( -4 - 6 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{61} + ( -2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{62} + \zeta_{24}^{6} q^{64} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{67} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{68} + ( 3 - 6 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{71} + ( 12 \zeta_{24} - 2 \zeta_{24}^{3} - 12 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{73} + ( 3 + 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{74} + ( -1 - 6 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{76} + ( -9 \zeta_{24} + 3 \zeta_{24}^{3} + 9 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{77} + ( 3 - 6 \zeta_{24}^{4} - 5 \zeta_{24}^{6} ) q^{79} + 6 \zeta_{24}^{3} q^{82} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{83} + ( 1 - 2 \zeta_{24}^{4} - 9 \zeta_{24}^{6} ) q^{86} + ( 3 \zeta_{24} - 3 \zeta_{24}^{5} ) q^{88} + ( -3 - 6 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{89} + ( 9 - 6 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{91} + ( 3 \zeta_{24} - 3 \zeta_{24}^{5} ) q^{92} -9 \zeta_{24}^{6} q^{94} + ( -5 \zeta_{24} + 6 \zeta_{24}^{3} - 5 \zeta_{24}^{5} ) q^{97} + ( -6 \zeta_{24} + 5 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 24q^{14} - 8q^{16} + 24q^{29} - 16q^{31} + 24q^{44} + 24q^{46} - 24q^{59} - 32q^{61} + 24q^{74} - 8q^{76} - 24q^{89} + 72q^{91} + O(q^{100})$$

## Character values

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

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

## 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
 0.258819 + 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i −0.258819 − 0.965926i 0.258819 − 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i −0.258819 + 0.965926i
−0.707107 + 0.707107i 0 1.00000i 0 0 −3.34607 3.34607i 0.707107 + 0.707107i 0 0
107.2 −0.707107 + 0.707107i 0 1.00000i 0 0 −0.896575 0.896575i 0.707107 + 0.707107i 0 0
107.3 0.707107 0.707107i 0 1.00000i 0 0 0.896575 + 0.896575i −0.707107 0.707107i 0 0
107.4 0.707107 0.707107i 0 1.00000i 0 0 3.34607 + 3.34607i −0.707107 0.707107i 0 0
593.1 −0.707107 0.707107i 0 1.00000i 0 0 −3.34607 + 3.34607i 0.707107 0.707107i 0 0
593.2 −0.707107 0.707107i 0 1.00000i 0 0 −0.896575 + 0.896575i 0.707107 0.707107i 0 0
593.3 0.707107 + 0.707107i 0 1.00000i 0 0 0.896575 0.896575i −0.707107 + 0.707107i 0 0
593.4 0.707107 + 0.707107i 0 1.00000i 0 0 3.34607 3.34607i −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
5.b even 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.e yes 8
3.b odd 2 1 1350.2.f.b 8
5.b even 2 1 inner 1350.2.f.e yes 8
5.c odd 4 2 1350.2.f.b 8
15.d odd 2 1 1350.2.f.b 8
15.e even 4 2 inner 1350.2.f.e yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1350.2.f.b 8 3.b odd 2 1
1350.2.f.b 8 5.c odd 4 2
1350.2.f.b 8 15.d odd 2 1
1350.2.f.e yes 8 1.a even 1 1 trivial
1350.2.f.e yes 8 5.b even 2 1 inner
1350.2.f.e yes 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 $$S_{2}^{\mathrm{new}}(1350, [\chi])$$:

 $$T_{7}^{8} + 504 T_{7}^{4} + 1296$$ $$T_{29}^{2} - 6 T_{29} - 18$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ 
$5$ 
$7$ $$1 + 28 T^{4} + 3270 T^{8} + 67228 T^{12} + 5764801 T^{16}$$
$11$ $$( 1 - 13 T^{2} + 121 T^{4} )^{4}$$
$13$ $$( 1 - 49 T^{4} + 28561 T^{8} )^{2}$$
$17$ $$1 + 796 T^{4} + 309894 T^{8} + 66482716 T^{12} + 6975757441 T^{16}$$
$19$ $$( 1 - 20 T^{2} + 714 T^{4} - 7220 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$( 1 + 311 T^{4} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 6 T + 40 T^{2} - 174 T^{3} + 841 T^{4} )^{4}$$
$31$ $$( 1 + 2 T + 31 T^{2} )^{8}$$
$37$ $$1 - 1442 T^{4} + 2270595 T^{8} - 2702540162 T^{12} + 3512479453921 T^{16}$$
$41$ $$( 1 - 46 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$1 - 5444 T^{4} + 14231334 T^{8} - 18611952644 T^{12} + 11688200277601 T^{16}$$
$47$ $$( 1 - 4249 T^{4} + 4879681 T^{8} )^{2}$$
$53$ $$1 + 508 T^{4} - 3205722 T^{8} + 4008364348 T^{12} + 62259690411361 T^{16}$$
$59$ $$( 1 + 6 T + 19 T^{2} + 354 T^{3} + 3481 T^{4} )^{4}$$
$61$ $$( 1 + 8 T + 111 T^{2} + 488 T^{3} + 3721 T^{4} )^{4}$$
$67$ $$1 + 3196 T^{4} + 5515494 T^{8} + 64402982716 T^{12} + 406067677556641 T^{16}$$
$71$ $$( 1 - 158 T^{2} + 12435 T^{4} - 796478 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$1 - 7292 T^{4} + 67324998 T^{8} - 207079973372 T^{12} + 806460091894081 T^{16}$$
$79$ $$( 1 - 212 T^{2} + 21018 T^{4} - 1323092 T^{6} + 38950081 T^{8} )^{2}$$
$83$ $$1 + 4516 T^{4} + 69906534 T^{8} + 214321777636 T^{12} + 2252292232139041 T^{16}$$
$89$ $$( 1 + 6 T + 160 T^{2} + 534 T^{3} + 7921 T^{4} )^{4}$$
$97$ $$1 - 2258 T^{4} - 119271597 T^{8} - 199899116498 T^{12} + 7837433594376961 T^{16}$$