# Properties

 Label 1350.2.j.g Level 1350 Weight 2 Character orbit 1350.j 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.j (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.7798042729$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 450) 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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{24}^{2} + \zeta_{24}^{6} ) q^{2} + ( 1 - \zeta_{24}^{4} ) q^{4} + ( -\zeta_{24} - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{7} + \zeta_{24}^{6} q^{8} +O(q^{10})$$ $$q + ( -\zeta_{24}^{2} + \zeta_{24}^{6} ) q^{2} + ( 1 - \zeta_{24}^{4} ) q^{4} + ( -\zeta_{24} - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{7} + \zeta_{24}^{6} q^{8} + ( -2 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{11} + ( -2 \zeta_{24} - 2 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{13} + ( 2 + 2 \zeta_{24} - \zeta_{24}^{3} - 2 \zeta_{24}^{4} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{14} -\zeta_{24}^{4} q^{16} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{17} + ( -5 + \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{19} + ( -4 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{22} + ( 2 \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{23} + ( 2 + \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{26} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{28} + ( \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{29} + ( -2 + 2 \zeta_{24} - \zeta_{24}^{3} + 2 \zeta_{24}^{4} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{31} + \zeta_{24}^{2} q^{32} + ( -2 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{34} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 4 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{37} + ( -\zeta_{24} + 5 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 5 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{38} + ( 9 - 9 \zeta_{24}^{4} ) q^{41} + ( \zeta_{24} + 5 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 5 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{43} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{44} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{46} + ( 2 \zeta_{24} - 6 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 6 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{47} + ( 3 + 8 \zeta_{24} - 4 \zeta_{24}^{3} - 3 \zeta_{24}^{4} - 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{49} + ( -\zeta_{24} - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{52} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 6 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{53} + ( \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{56} + ( 2 \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{58} + ( 3 - 2 \zeta_{24} + \zeta_{24}^{3} - 3 \zeta_{24}^{4} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{59} -8 \zeta_{24}^{4} q^{61} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{62} - q^{64} + ( -6 \zeta_{24} - 7 \zeta_{24}^{2} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{67} + ( -4 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{68} + ( -6 + 3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{71} -\zeta_{24}^{6} q^{73} + ( -3 \zeta_{24} + 6 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 6 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{74} + ( -5 + 2 \zeta_{24} - \zeta_{24}^{3} + 5 \zeta_{24}^{4} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{76} + ( -8 \zeta_{24} - 12 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{77} + ( 6 \zeta_{24} - 12 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 12 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{79} + 9 \zeta_{24}^{6} q^{82} + ( -\zeta_{24} + 3 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 3 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{83} + ( -5 - 2 \zeta_{24} + \zeta_{24}^{3} + 5 \zeta_{24}^{4} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{86} + ( -2 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{88} -9 q^{89} + ( 10 + 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{91} + ( \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{92} + ( 6 - 4 \zeta_{24} + 2 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{94} + ( 4 \zeta_{24} + \zeta_{24}^{2} + 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{97} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 3 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{4} + O(q^{10})$$ $$8q + 4q^{4} + 8q^{14} - 4q^{16} - 40q^{19} + 16q^{26} - 8q^{31} + 36q^{41} + 12q^{49} - 8q^{56} + 12q^{59} - 32q^{61} - 8q^{64} - 48q^{71} - 16q^{74} - 20q^{76} + 8q^{79} - 20q^{86} - 72q^{89} + 80q^{91} + 24q^{94} + O(q^{100})$$

## Character values

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

 $$n$$ $$1001$$ $$1027$$ $$\chi(n)$$ $$-1 + \zeta_{24}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 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 0.258819 + 0.965926i
−0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 −3.85337 + 2.22474i 1.00000i 0 0
199.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 0.389270 0.224745i 1.00000i 0 0
199.3 0.866025 0.500000i 0 0.500000 0.866025i 0 0 −0.389270 + 0.224745i 1.00000i 0 0
199.4 0.866025 0.500000i 0 0.500000 0.866025i 0 0 3.85337 2.22474i 1.00000i 0 0
1099.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 −3.85337 2.22474i 1.00000i 0 0
1099.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 0.389270 + 0.224745i 1.00000i 0 0
1099.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 −0.389270 0.224745i 1.00000i 0 0
1099.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 3.85337 + 2.22474i 1.00000i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1099.4 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
9.c even 3 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.2.j.g 8
3.b odd 2 1 450.2.j.f 8
5.b even 2 1 inner 1350.2.j.g 8
5.c odd 4 1 1350.2.e.k 4
5.c odd 4 1 1350.2.e.n 4
9.c even 3 1 inner 1350.2.j.g 8
9.c even 3 1 4050.2.c.w 4
9.d odd 6 1 450.2.j.f 8
9.d odd 6 1 4050.2.c.y 4
15.d odd 2 1 450.2.j.f 8
15.e even 4 1 450.2.e.l 4
15.e even 4 1 450.2.e.m yes 4
45.h odd 6 1 450.2.j.f 8
45.h odd 6 1 4050.2.c.y 4
45.j even 6 1 inner 1350.2.j.g 8
45.j even 6 1 4050.2.c.w 4
45.k odd 12 1 1350.2.e.k 4
45.k odd 12 1 1350.2.e.n 4
45.k odd 12 1 4050.2.a.bl 2
45.k odd 12 1 4050.2.a.by 2
45.l even 12 1 450.2.e.l 4
45.l even 12 1 450.2.e.m yes 4
45.l even 12 1 4050.2.a.br 2
45.l even 12 1 4050.2.a.bu 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.e.l 4 15.e even 4 1
450.2.e.l 4 45.l even 12 1
450.2.e.m yes 4 15.e even 4 1
450.2.e.m yes 4 45.l even 12 1
450.2.j.f 8 3.b odd 2 1
450.2.j.f 8 9.d odd 6 1
450.2.j.f 8 15.d odd 2 1
450.2.j.f 8 45.h odd 6 1
1350.2.e.k 4 5.c odd 4 1
1350.2.e.k 4 45.k odd 12 1
1350.2.e.n 4 5.c odd 4 1
1350.2.e.n 4 45.k odd 12 1
1350.2.j.g 8 1.a even 1 1 trivial
1350.2.j.g 8 5.b even 2 1 inner
1350.2.j.g 8 9.c even 3 1 inner
1350.2.j.g 8 45.j even 6 1 inner
4050.2.a.bl 2 45.k odd 12 1
4050.2.a.br 2 45.l even 12 1
4050.2.a.bu 2 45.l even 12 1
4050.2.a.by 2 45.k odd 12 1
4050.2.c.w 4 9.c even 3 1
4050.2.c.w 4 45.j even 6 1
4050.2.c.y 4 9.d odd 6 1
4050.2.c.y 4 45.h odd 6 1

## Hecke kernels

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

 $$T_{7}^{8} - 20 T_{7}^{6} + 396 T_{7}^{4} - 80 T_{7}^{2} + 16$$ $$T_{11}^{4} + 24 T_{11}^{2} + 576$$ $$T_{19}^{2} + 10 T_{19} + 19$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ 
$5$ 
$7$ $$1 + 8 T^{2} + 46 T^{4} - 640 T^{6} - 5213 T^{8} - 31360 T^{10} + 110446 T^{12} + 941192 T^{14} + 5764801 T^{16}$$
$11$ $$( 1 + 2 T^{2} - 117 T^{4} + 242 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$1 + 32 T^{2} + 526 T^{4} + 5120 T^{6} + 46387 T^{8} + 865280 T^{10} + 15023086 T^{12} + 154457888 T^{14} + 815730721 T^{16}$$
$17$ $$( 1 - 10 T^{2} + 289 T^{4} )^{4}$$
$19$ $$( 1 + 10 T + 57 T^{2} + 190 T^{3} + 361 T^{4} )^{4}$$
$23$ $$( 1 + 40 T^{2} + 1071 T^{4} + 21160 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 52 T^{2} + 1863 T^{4} - 43732 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 + 4 T - 44 T^{2} - 8 T^{3} + 2143 T^{4} - 248 T^{5} - 42284 T^{6} + 119164 T^{7} + 923521 T^{8} )^{2}$$
$37$ $$( 1 - 8 T^{2} - 702 T^{4} - 10952 T^{6} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} )^{4}$$
$43$ $$1 + 110 T^{2} + 5977 T^{4} + 266750 T^{6} + 11699428 T^{8} + 493220750 T^{10} + 20434173577 T^{12} + 695349935390 T^{14} + 11688200277601 T^{16}$$
$47$ $$1 + 68 T^{2} + 2506 T^{4} - 156400 T^{6} - 10608173 T^{8} - 345487600 T^{10} + 12228480586 T^{12} + 732986642372 T^{14} + 23811286661761 T^{16}$$
$53$ $$( 1 - 128 T^{2} + 8850 T^{4} - 359552 T^{6} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 - 6 T - 85 T^{2} - 18 T^{3} + 9036 T^{4} - 1062 T^{5} - 295885 T^{6} - 1232274 T^{7} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{4}$$
$67$ $$( 1 + 62 T^{2} + 4489 T^{4} )^{2}( 1 - 62 T^{2} - 645 T^{4} - 278318 T^{6} + 20151121 T^{8} )$$
$71$ $$( 1 + 12 T + 124 T^{2} + 852 T^{3} + 5041 T^{4} )^{4}$$
$73$ $$( 1 - 145 T^{2} + 5329 T^{4} )^{4}$$
$79$ $$( 1 - 4 T + 70 T^{2} + 848 T^{3} - 4589 T^{4} + 66992 T^{5} + 436870 T^{6} - 1972156 T^{7} + 38950081 T^{8} )^{2}$$
$83$ $$1 + 302 T^{2} + 54841 T^{4} + 6820670 T^{6} + 646505092 T^{8} + 46987595630 T^{10} + 2602661781961 T^{12} + 98735992757438 T^{14} + 2252292232139041 T^{16}$$
$89$ $$( 1 + 9 T + 89 T^{2} )^{8}$$
$97$ $$1 + 194 T^{2} + 9793 T^{4} + 1750850 T^{6} + 332586244 T^{8} + 16473747650 T^{10} + 866967248833 T^{12} + 161596568956226 T^{14} + 7837433594376961 T^{16}$$