# Properties

 Label 1350.2.j.f 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: 8.0.303595776.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 90) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{3} q^{4} -\beta_{7} q^{7} + ( \beta_{1} + \beta_{5} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{3} q^{4} -\beta_{7} q^{7} + ( \beta_{1} + \beta_{5} ) q^{8} + ( -2 + \beta_{2} - 2 \beta_{3} ) q^{11} + ( 2 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{13} + ( -1 + \beta_{2} - \beta_{3} - \beta_{6} ) q^{14} + ( -1 - \beta_{3} ) q^{16} + ( -4 \beta_{1} + \beta_{4} - 4 \beta_{5} ) q^{17} + ( -1 - \beta_{6} ) q^{19} + ( \beta_{1} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{22} + ( \beta_{1} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{23} + 2 \beta_{6} q^{26} + ( \beta_{1} + \beta_{4} + \beta_{5} ) q^{28} + ( 1 + \beta_{2} + \beta_{3} ) q^{29} + ( 2 - 2 \beta_{2} + 2 \beta_{6} ) q^{31} + \beta_{5} q^{32} + ( 4 + \beta_{2} + 4 \beta_{3} ) q^{34} + ( 4 \beta_{1} + 4 \beta_{5} ) q^{37} + ( -\beta_{1} - \beta_{7} ) q^{38} + 3 \beta_{3} q^{41} + ( 9 \beta_{1} + \beta_{7} ) q^{43} + ( -1 + \beta_{6} ) q^{44} + ( 2 + \beta_{6} ) q^{46} + ( 4 \beta_{1} - \beta_{7} ) q^{47} + ( -1 + \beta_{2} - 2 \beta_{3} - \beta_{6} ) q^{49} + 2 \beta_{7} q^{52} + ( 2 \beta_{1} + 4 \beta_{4} + 2 \beta_{5} ) q^{53} + ( -1 + \beta_{2} - \beta_{3} ) q^{56} + ( \beta_{1} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{58} + ( 1 - \beta_{2} + 2 \beta_{3} + \beta_{6} ) q^{59} + ( 1 - 3 \beta_{2} + \beta_{3} ) q^{61} -2 \beta_{4} q^{62} - q^{64} -7 \beta_{5} q^{67} + ( \beta_{1} + \beta_{4} - 4 \beta_{5} + \beta_{7} ) q^{68} + 6 q^{71} + ( -4 \beta_{1} + 3 \beta_{4} - 4 \beta_{5} ) q^{73} + ( -4 - 4 \beta_{3} ) q^{74} + ( -1 + \beta_{2} - \beta_{6} ) q^{76} + ( 2 \beta_{1} + 2 \beta_{4} + 10 \beta_{5} + 2 \beta_{7} ) q^{77} + ( 2 + 2 \beta_{3} ) q^{79} + ( -3 \beta_{1} - 3 \beta_{5} ) q^{82} + ( 4 \beta_{1} - \beta_{7} ) q^{83} + ( 1 - \beta_{2} - 8 \beta_{3} + \beta_{6} ) q^{86} + ( -\beta_{1} + \beta_{7} ) q^{88} + ( 6 - 3 \beta_{6} ) q^{89} + ( -16 + 2 \beta_{6} ) q^{91} + ( 2 \beta_{1} + \beta_{7} ) q^{92} + ( -1 + \beta_{2} - 5 \beta_{3} - \beta_{6} ) q^{94} + ( -5 \beta_{1} + \beta_{7} ) q^{97} + ( 2 \beta_{1} + \beta_{4} + 2 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{4} + O(q^{10})$$ $$8q + 4q^{4} - 6q^{11} + 2q^{14} - 4q^{16} - 4q^{19} - 8q^{26} + 6q^{29} + 4q^{31} + 18q^{34} - 12q^{41} - 12q^{44} + 12q^{46} + 6q^{49} - 2q^{56} - 6q^{59} - 2q^{61} - 8q^{64} + 48q^{71} - 16q^{74} - 2q^{76} + 8q^{79} + 34q^{86} + 60q^{89} - 136q^{91} + 18q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 32 \nu^{5} + 16 \nu^{3} + 45 \nu$$$$)/432$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} - 16 \nu^{4} - 32 \nu^{2} - 51$$$$)/48$$ $$\beta_{3}$$ $$=$$ $$($$$$-5 \nu^{6} - 16 \nu^{4} - 80 \nu^{2} - 225$$$$)/144$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + 8 \nu^{5} + 40 \nu^{3} + 165 \nu$$$$)/72$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 13 \nu$$$$)/48$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} - 5 \nu^{4} - 7 \nu^{2} - 27$$$$)/9$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{5} + 10 \nu^{3} - 3 \nu$$$$)/18$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} + 2 \beta_{5} + \beta_{4}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{6} - 7 \beta_{3} - \beta_{2} - 6$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{7} - 11 \beta_{5} + 2 \beta_{4} - 9 \beta_{1}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$-5 \beta_{6} + 13 \beta_{3} - 5 \beta_{2}$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{7} - \beta_{5} - 2 \beta_{4} + 45 \beta_{1}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$-16 \beta_{6} - 16 \beta_{3} + 32 \beta_{2} - 39$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$13 \beta_{7} + 118 \beta_{5} - 13 \beta_{4}$$$$)/3$$

## Character values

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

 $$n$$ $$1001$$ $$1027$$ $$\chi(n)$$ $$\beta_{3}$$ $$-1$$

## 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
 −0.396143 + 1.68614i 1.26217 − 1.18614i −1.26217 + 1.18614i 0.396143 − 1.68614i −0.396143 − 1.68614i 1.26217 + 1.18614i −1.26217 − 1.18614i 0.396143 + 1.68614i
−0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 −2.92048 + 1.68614i 1.00000i 0 0
199.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 2.05446 1.18614i 1.00000i 0 0
199.3 0.866025 0.500000i 0 0.500000 0.866025i 0 0 −2.05446 + 1.18614i 1.00000i 0 0
199.4 0.866025 0.500000i 0 0.500000 0.866025i 0 0 2.92048 1.68614i 1.00000i 0 0
1099.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 −2.92048 1.68614i 1.00000i 0 0
1099.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 2.05446 + 1.18614i 1.00000i 0 0
1099.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 −2.05446 1.18614i 1.00000i 0 0
1099.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 2.92048 + 1.68614i 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.f 8
3.b odd 2 1 450.2.j.g 8
5.b even 2 1 inner 1350.2.j.f 8
5.c odd 4 1 270.2.e.c 4
5.c odd 4 1 1350.2.e.l 4
9.c even 3 1 inner 1350.2.j.f 8
9.c even 3 1 4050.2.c.ba 4
9.d odd 6 1 450.2.j.g 8
9.d odd 6 1 4050.2.c.v 4
15.d odd 2 1 450.2.j.g 8
15.e even 4 1 90.2.e.c 4
15.e even 4 1 450.2.e.j 4
20.e even 4 1 2160.2.q.f 4
45.h odd 6 1 450.2.j.g 8
45.h odd 6 1 4050.2.c.v 4
45.j even 6 1 inner 1350.2.j.f 8
45.j even 6 1 4050.2.c.ba 4
45.k odd 12 1 270.2.e.c 4
45.k odd 12 1 810.2.a.k 2
45.k odd 12 1 1350.2.e.l 4
45.k odd 12 1 4050.2.a.bo 2
45.l even 12 1 90.2.e.c 4
45.l even 12 1 450.2.e.j 4
45.l even 12 1 810.2.a.i 2
45.l even 12 1 4050.2.a.bw 2
60.l odd 4 1 720.2.q.f 4
180.v odd 12 1 720.2.q.f 4
180.v odd 12 1 6480.2.a.be 2
180.x even 12 1 2160.2.q.f 4
180.x even 12 1 6480.2.a.bn 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.c 4 15.e even 4 1
90.2.e.c 4 45.l even 12 1
270.2.e.c 4 5.c odd 4 1
270.2.e.c 4 45.k odd 12 1
450.2.e.j 4 15.e even 4 1
450.2.e.j 4 45.l even 12 1
450.2.j.g 8 3.b odd 2 1
450.2.j.g 8 9.d odd 6 1
450.2.j.g 8 15.d odd 2 1
450.2.j.g 8 45.h odd 6 1
720.2.q.f 4 60.l odd 4 1
720.2.q.f 4 180.v odd 12 1
810.2.a.i 2 45.l even 12 1
810.2.a.k 2 45.k odd 12 1
1350.2.e.l 4 5.c odd 4 1
1350.2.e.l 4 45.k odd 12 1
1350.2.j.f 8 1.a even 1 1 trivial
1350.2.j.f 8 5.b even 2 1 inner
1350.2.j.f 8 9.c even 3 1 inner
1350.2.j.f 8 45.j even 6 1 inner
2160.2.q.f 4 20.e even 4 1
2160.2.q.f 4 180.x even 12 1
4050.2.a.bo 2 45.k odd 12 1
4050.2.a.bw 2 45.l even 12 1
4050.2.c.v 4 9.d odd 6 1
4050.2.c.v 4 45.h odd 6 1
4050.2.c.ba 4 9.c even 3 1
4050.2.c.ba 4 45.j even 6 1
6480.2.a.be 2 180.v odd 12 1
6480.2.a.bn 2 180.x even 12 1

## Hecke kernels

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

 $$T_{7}^{8} - 17 T_{7}^{6} + 225 T_{7}^{4} - 1088 T_{7}^{2} + 4096$$ $$T_{11}^{4} + 3 T_{11}^{3} + 15 T_{11}^{2} - 18 T_{11} + 36$$ $$T_{19}^{2} + T_{19} - 8$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ 
$5$ 
$7$ $$1 + 11 T^{2} + T^{4} + 242 T^{6} + 6070 T^{8} + 11858 T^{10} + 2401 T^{12} + 1294139 T^{14} + 5764801 T^{16}$$
$11$ $$( 1 + 3 T - 7 T^{2} - 18 T^{3} + 36 T^{4} - 198 T^{5} - 847 T^{6} + 3993 T^{7} + 14641 T^{8} )^{2}$$
$13$ $$1 - 16 T^{2} - 14 T^{4} + 1088 T^{6} + 1075 T^{8} + 183872 T^{10} - 399854 T^{12} - 77228944 T^{14} + 815730721 T^{16}$$
$17$ $$( 1 - 11 T^{2} - 60 T^{4} - 3179 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 + T + 30 T^{2} + 19 T^{3} + 361 T^{4} )^{4}$$
$23$ $$1 + 71 T^{2} + 2797 T^{4} + 84206 T^{6} + 2089006 T^{8} + 44544974 T^{10} + 782715277 T^{12} + 10510548119 T^{14} + 78310985281 T^{16}$$
$29$ $$( 1 - 3 T - 43 T^{2} + 18 T^{3} + 1602 T^{4} + 522 T^{5} - 36163 T^{6} - 73167 T^{7} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - 2 T - 26 T^{2} + 64 T^{3} - 185 T^{4} + 1984 T^{5} - 24986 T^{6} - 59582 T^{7} + 923521 T^{8} )^{2}$$
$37$ $$( 1 - 58 T^{2} + 1369 T^{4} )^{4}$$
$41$ $$( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} )^{4}$$
$43$ $$1 + 11 T^{2} - 1223 T^{4} - 25894 T^{6} - 1836194 T^{8} - 47878006 T^{10} - 4181193623 T^{12} + 69534993539 T^{14} + 11688200277601 T^{16}$$
$47$ $$1 + 131 T^{2} + 9121 T^{4} + 474482 T^{6} + 21853270 T^{8} + 1048130738 T^{10} + 44507570401 T^{12} + 1412077208099 T^{14} + 23811286661761 T^{16}$$
$53$ $$( 1 + 26 T^{2} + 2809 T^{4} )^{4}$$
$59$ $$( 1 + 3 T - 103 T^{2} - 18 T^{3} + 8532 T^{4} - 1062 T^{5} - 358543 T^{6} + 616137 T^{7} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + T - 47 T^{2} - 74 T^{3} - 1478 T^{4} - 4514 T^{5} - 174887 T^{6} + 226981 T^{7} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 + 85 T^{2} + 2736 T^{4} + 381565 T^{6} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 - 6 T + 71 T^{2} )^{8}$$
$73$ $$( 1 - 83 T^{2} + 3396 T^{4} - 442307 T^{6} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 - 2 T - 75 T^{2} - 158 T^{3} + 6241 T^{4} )^{4}$$
$83$ $$1 + 275 T^{2} + 43609 T^{4} + 5015450 T^{6} + 456585310 T^{8} + 34551435050 T^{10} + 2069609920489 T^{12} + 89908602676475 T^{14} + 2252292232139041 T^{16}$$
$89$ $$( 1 - 15 T + 160 T^{2} - 1335 T^{3} + 7921 T^{4} )^{4}$$
$97$ $$1 + 311 T^{2} + 54721 T^{4} + 7209602 T^{6} + 765422830 T^{8} + 67835145218 T^{10} + 4844410785601 T^{12} + 259054293532919 T^{14} + 7837433594376961 T^{16}$$