# Properties

 Label 1350.2.f.f 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ 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.

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

## 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 0.258819 + 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 −1.44949 1.44949i 0.707107 + 0.707107i 0 0
107.2 −0.707107 + 0.707107i 0 1.00000i 0 0 3.44949 + 3.44949i 0.707107 + 0.707107i 0 0
107.3 0.707107 0.707107i 0 1.00000i 0 0 −1.44949 1.44949i −0.707107 0.707107i 0 0
107.4 0.707107 0.707107i 0 1.00000i 0 0 3.44949 + 3.44949i −0.707107 0.707107i 0 0
593.1 −0.707107 0.707107i 0 1.00000i 0 0 −1.44949 + 1.44949i 0.707107 0.707107i 0 0
593.2 −0.707107 0.707107i 0 1.00000i 0 0 3.44949 3.44949i 0.707107 0.707107i 0 0
593.3 0.707107 + 0.707107i 0 1.00000i 0 0 −1.44949 + 1.44949i −0.707107 + 0.707107i 0 0
593.4 0.707107 + 0.707107i 0 1.00000i 0 0 3.44949 3.44949i −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.f 8
3.b odd 2 1 inner 1350.2.f.f 8
5.b even 2 1 270.2.f.a 8
5.c odd 4 1 270.2.f.a 8
5.c odd 4 1 inner 1350.2.f.f 8
15.d odd 2 1 270.2.f.a 8
15.e even 4 1 270.2.f.a 8
15.e even 4 1 inner 1350.2.f.f 8
20.d odd 2 1 2160.2.w.f 8
20.e even 4 1 2160.2.w.f 8
45.h odd 6 1 810.2.m.a 8
45.h odd 6 1 810.2.m.h 8
45.j even 6 1 810.2.m.a 8
45.j even 6 1 810.2.m.h 8
45.k odd 12 1 810.2.m.a 8
45.k odd 12 1 810.2.m.h 8
45.l even 12 1 810.2.m.a 8
45.l even 12 1 810.2.m.h 8
60.h even 2 1 2160.2.w.f 8
60.l odd 4 1 2160.2.w.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.f.a 8 5.b even 2 1
270.2.f.a 8 5.c odd 4 1
270.2.f.a 8 15.d odd 2 1
270.2.f.a 8 15.e even 4 1
810.2.m.a 8 45.h odd 6 1
810.2.m.a 8 45.j even 6 1
810.2.m.a 8 45.k odd 12 1
810.2.m.a 8 45.l even 12 1
810.2.m.h 8 45.h odd 6 1
810.2.m.h 8 45.j even 6 1
810.2.m.h 8 45.k odd 12 1
810.2.m.h 8 45.l even 12 1
1350.2.f.f 8 1.a even 1 1 trivial
1350.2.f.f 8 3.b odd 2 1 inner
1350.2.f.f 8 5.c odd 4 1 inner
1350.2.f.f 8 15.e even 4 1 inner
2160.2.w.f 8 20.d odd 2 1
2160.2.w.f 8 20.e even 4 1
2160.2.w.f 8 60.h even 2 1
2160.2.w.f 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 $$S_{2}^{\mathrm{new}}(1350, [\chi])$$:

 $$T_{7}^{4} - 4 T_{7}^{3} + 8 T_{7}^{2} + 40 T_{7} + 100$$ $$T_{29}^{4} - 10 T_{29}^{2} + 1$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ 
$5$ 
$7$ $$( 1 - 2 T + 7 T^{2} )^{4}( 1 - 10 T^{2} + 49 T^{4} )^{2}$$
$11$ $$( 1 - 22 T^{2} + 267 T^{4} - 2662 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 + 12 T + 72 T^{2} + 336 T^{3} + 1343 T^{4} + 4368 T^{5} + 12168 T^{6} + 26364 T^{7} + 28561 T^{8} )^{2}$$
$17$ $$1 - 802 T^{4} + 296739 T^{8} - 66983842 T^{12} + 6975757441 T^{16}$$
$19$ $$( 1 - 56 T^{2} + 1410 T^{4} - 20216 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$1 - 1522 T^{4} + 1068819 T^{8} - 425918002 T^{12} + 78310985281 T^{16}$$
$29$ $$( 1 + 106 T^{2} + 4467 T^{4} + 89146 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - 2 T + 57 T^{2} - 62 T^{3} + 961 T^{4} )^{4}$$
$37$ $$( 1 + 1369 T^{4} )^{4}$$
$41$ $$( 1 - 136 T^{2} + 7890 T^{4} - 228616 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 24 T + 288 T^{2} - 2688 T^{3} + 20327 T^{4} - 115584 T^{5} + 532512 T^{6} - 1908168 T^{7} + 3418801 T^{8} )^{2}$$
$47$ $$1 - 3026 T^{4} + 4575795 T^{8} - 14765914706 T^{12} + 23811286661761 T^{16}$$
$53$ $$( 1 - 3854 T^{4} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 + 110 T^{2} + 3481 T^{4} )^{4}$$
$61$ $$( 1 + 12 T + 152 T^{2} + 732 T^{3} + 3721 T^{4} )^{4}$$
$67$ $$( 1 + 4 T + 8 T^{2} + 268 T^{3} + 4489 T^{4} )^{4}$$
$71$ $$( 1 - 88 T^{2} + 2418 T^{4} - 443608 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 - 8 T + 32 T^{2} - 552 T^{3} + 9506 T^{4} - 40296 T^{5} + 170528 T^{6} - 3112136 T^{7} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 - 286 T^{2} + 32715 T^{4} - 1784926 T^{6} + 38950081 T^{8} )^{2}$$
$83$ $$1 - 20132 T^{4} + 192702054 T^{8} - 955430918372 T^{12} + 2252292232139041 T^{16}$$
$89$ $$( 1 + 40 T^{2} - 5358 T^{4} + 316840 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 + 24 T + 288 T^{2} + 3768 T^{3} + 45698 T^{4} + 365496 T^{5} + 2709792 T^{6} + 21904152 T^{7} + 88529281 T^{8} )^{2}$$