# Properties

 Label 1350.2.q.g Level 1350 Weight 2 Character orbit 1350.q 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.q (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.7798042729$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 90) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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.

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

## Character values

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

 $$n$$ $$1001$$ $$1027$$ $$\chi(n)$$ $$1 - \zeta_{24}^{4}$$ $$-\zeta_{24}^{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1
 0.258819 + 0.965926i −0.258819 − 0.965926i 0.258819 − 0.965926i −0.258819 + 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i 0.965926 + 0.258819i −0.965926 − 0.258819i
−0.965926 + 0.258819i 0 0.866025 0.500000i 0 0 −1.18034 4.40508i −0.707107 + 0.707107i 0 0
143.2 0.965926 0.258819i 0 0.866025 0.500000i 0 0 −0.283763 1.05902i 0.707107 0.707107i 0 0
557.1 −0.965926 0.258819i 0 0.866025 + 0.500000i 0 0 −1.18034 + 4.40508i −0.707107 0.707107i 0 0
557.2 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0 0 −0.283763 + 1.05902i 0.707107 + 0.707107i 0 0
1007.1 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0 0 4.40508 1.18034i 0.707107 + 0.707107i 0 0
1007.2 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0 0 1.05902 0.283763i −0.707107 0.707107i 0 0
1043.1 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0 0 4.40508 + 1.18034i 0.707107 0.707107i 0 0
1043.2 0.258819 0.965926i 0 −0.866025 0.500000i 0 0 1.05902 + 0.283763i −0.707107 + 0.707107i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1043.2 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.c odd 4 1 inner
9.d odd 6 1 inner
45.l even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.2.q.g 8
3.b odd 2 1 450.2.p.a 8
5.b even 2 1 270.2.m.a 8
5.c odd 4 1 270.2.m.a 8
5.c odd 4 1 inner 1350.2.q.g 8
9.c even 3 1 450.2.p.a 8
9.d odd 6 1 inner 1350.2.q.g 8
15.d odd 2 1 90.2.l.a 8
15.e even 4 1 90.2.l.a 8
15.e even 4 1 450.2.p.a 8
45.h odd 6 1 270.2.m.a 8
45.h odd 6 1 810.2.f.b 8
45.j even 6 1 90.2.l.a 8
45.j even 6 1 810.2.f.b 8
45.k odd 12 1 90.2.l.a 8
45.k odd 12 1 450.2.p.a 8
45.k odd 12 1 810.2.f.b 8
45.l even 12 1 270.2.m.a 8
45.l even 12 1 810.2.f.b 8
45.l even 12 1 inner 1350.2.q.g 8
60.h even 2 1 720.2.cu.a 8
60.l odd 4 1 720.2.cu.a 8
180.p odd 6 1 720.2.cu.a 8
180.x even 12 1 720.2.cu.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.l.a 8 15.d odd 2 1
90.2.l.a 8 15.e even 4 1
90.2.l.a 8 45.j even 6 1
90.2.l.a 8 45.k odd 12 1
270.2.m.a 8 5.b even 2 1
270.2.m.a 8 5.c odd 4 1
270.2.m.a 8 45.h odd 6 1
270.2.m.a 8 45.l even 12 1
450.2.p.a 8 3.b odd 2 1
450.2.p.a 8 9.c even 3 1
450.2.p.a 8 15.e even 4 1
450.2.p.a 8 45.k odd 12 1
720.2.cu.a 8 60.h even 2 1
720.2.cu.a 8 60.l odd 4 1
720.2.cu.a 8 180.p odd 6 1
720.2.cu.a 8 180.x even 12 1
810.2.f.b 8 45.h odd 6 1
810.2.f.b 8 45.j even 6 1
810.2.f.b 8 45.k odd 12 1
810.2.f.b 8 45.l even 12 1
1350.2.q.g 8 1.a even 1 1 trivial
1350.2.q.g 8 5.c odd 4 1 inner
1350.2.q.g 8 9.d odd 6 1 inner
1350.2.q.g 8 45.l even 12 1 inner

## Hecke kernels

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

 $$T_{7}^{8} - \cdots$$ $$T_{11}^{4} - 12 T_{11}^{3} + 52 T_{11}^{2} - 48 T_{11} + 16$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{4} + T^{8}$$
$3$ 
$5$ 
$7$ $$1 - 8 T + 32 T^{2} - 64 T^{3} - 7 T^{4} + 464 T^{5} - 1440 T^{6} + 2472 T^{7} - 4016 T^{8} + 17304 T^{9} - 70560 T^{10} + 159152 T^{11} - 16807 T^{12} - 1075648 T^{13} + 3764768 T^{14} - 6588344 T^{15} + 5764801 T^{16}$$
$11$ $$( 1 - 12 T + 74 T^{2} - 312 T^{3} + 1083 T^{4} - 3432 T^{5} + 8954 T^{6} - 15972 T^{7} + 14641 T^{8} )^{2}$$
$13$ $$1 + 142 T^{4} - 8397 T^{8} + 4055662 T^{12} + 815730721 T^{16}$$
$17$ $$1 + 188 T^{4} - 45306 T^{8} + 15701948 T^{12} + 6975757441 T^{16}$$
$19$ $$( 1 - 32 T^{2} + 594 T^{4} - 11552 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$1 - 967 T^{4} + 655248 T^{8} - 270606247 T^{12} + 78310985281 T^{16}$$
$29$ $$1 - 106 T^{2} + 6769 T^{4} - 295210 T^{6} + 9797332 T^{8} - 248271610 T^{10} + 4787585089 T^{12} - 63051272026 T^{14} + 500246412961 T^{16}$$
$31$ $$( 1 - 4 T - 44 T^{2} + 8 T^{3} + 2143 T^{4} + 248 T^{5} - 42284 T^{6} - 119164 T^{7} + 923521 T^{8} )^{2}$$
$37$ $$( 1 - 6 T + 18 T^{2} - 222 T^{3} + 1369 T^{4} )^{4}$$
$41$ $$( 1 + 6 T + 65 T^{2} + 318 T^{3} + 1620 T^{4} + 13038 T^{5} + 109265 T^{6} + 413526 T^{7} + 2825761 T^{8} )^{2}$$
$43$ $$1 - 1778 T^{4} - 257517 T^{8} - 6078628178 T^{12} + 11688200277601 T^{16}$$
$47$ $$1 + 4249 T^{4} + 13174320 T^{8} + 20733764569 T^{12} + 23811286661761 T^{16}$$
$53$ $$1 - 4900 T^{4} + 13820838 T^{8} - 38663356900 T^{12} + 62259690411361 T^{16}$$
$59$ $$1 - 16 T^{2} - 5906 T^{4} + 12800 T^{6} + 24961747 T^{8} + 44556800 T^{10} - 71565134066 T^{12} - 674888538256 T^{14} + 146830437604321 T^{16}$$
$61$ $$( 1 - 6 T - 89 T^{2} - 18 T^{3} + 9708 T^{4} - 1098 T^{5} - 331169 T^{6} - 1361886 T^{7} + 13845841 T^{8} )^{2}$$
$67$ $$1 + 4 T + 8 T^{2} - 304 T^{3} - 3511 T^{4} - 10624 T^{5} + 31800 T^{6} + 1654308 T^{7} - 4277600 T^{8} + 110838636 T^{9} + 142750200 T^{10} - 3195306112 T^{11} - 70750585831 T^{12} - 410438032528 T^{13} + 723667057352 T^{14} + 24242846421292 T^{15} + 406067677556641 T^{16}$$
$71$ $$( 1 - 244 T^{2} + 24582 T^{4} - 1230004 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 - 8 T + 32 T^{2} - 264 T^{3} + 578 T^{4} - 19272 T^{5} + 170528 T^{6} - 3112136 T^{7} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 + 152 T^{2} + 16863 T^{4} + 948632 T^{6} + 38950081 T^{8} )^{2}$$
$83$ $$1 - 12 T + 72 T^{2} - 288 T^{3} - 8375 T^{4} + 56640 T^{5} - 35208 T^{6} - 3433884 T^{7} + 82789344 T^{8} - 285012372 T^{9} - 242547912 T^{10} + 32386015680 T^{11} - 397463438375 T^{12} - 1134443705184 T^{13} + 23539706882568 T^{14} - 325632611875524 T^{15} + 2252292232139041 T^{16}$$
$89$ $$( 1 + 286 T^{2} + 35427 T^{4} + 2265406 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$1 + 12 T + 72 T^{2} - 744 T^{3} - 12542 T^{4} - 57996 T^{5} + 483840 T^{6} + 14345892 T^{7} + 128096019 T^{8} + 1391551524 T^{9} + 4552450560 T^{10} - 52931383308 T^{11} - 1110334242302 T^{12} - 6388981151208 T^{13} + 59973984354888 T^{14} + 969579413737356 T^{15} + 7837433594376961 T^{16}$$