# Properties

 Label 1350.2.q.d 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 450) 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{24}^{7} q^{2} -\zeta_{24}^{2} q^{4} + ( -3 \zeta_{24} + 2 \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{7} + ( \zeta_{24} - \zeta_{24}^{5} ) q^{8} +O(q^{10})$$ $$q + \zeta_{24}^{7} q^{2} -\zeta_{24}^{2} q^{4} + ( -3 \zeta_{24} + 2 \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{7} + ( \zeta_{24} - \zeta_{24}^{5} ) q^{8} + ( -4 + 2 \zeta_{24}^{4} ) q^{11} + ( -2 \zeta_{24} + 3 \zeta_{24}^{3} + \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{13} + ( 3 - \zeta_{24}^{2} - 3 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{14} + \zeta_{24}^{4} q^{16} + ( 3 - 6 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{19} + ( -2 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{22} + ( 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{23} + ( 1 - 2 \zeta_{24}^{4} + 3 \zeta_{24}^{6} ) q^{26} + ( -\zeta_{24} + 3 \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{28} + ( \zeta_{24}^{2} - 9 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{29} + ( 5 + 3 \zeta_{24}^{2} - 5 \zeta_{24}^{4} - 6 \zeta_{24}^{6} ) q^{31} + ( -\zeta_{24}^{3} + \zeta_{24}^{7} ) q^{32} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{37} + ( 2 \zeta_{24} + 6 \zeta_{24}^{3} - 3 \zeta_{24}^{7} ) q^{38} + ( 1 + \zeta_{24}^{4} ) q^{41} + ( 2 \zeta_{24} - 4 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{43} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{44} + ( -3 - 6 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{46} -6 \zeta_{24}^{7} q^{47} + ( -6 + 5 \zeta_{24}^{2} - 6 \zeta_{24}^{4} ) q^{49} + ( -3 \zeta_{24} + 2 \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{52} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{53} + ( 2 - 3 \zeta_{24}^{2} - \zeta_{24}^{4} + 3 \zeta_{24}^{6} ) q^{56} + ( -2 \zeta_{24} + 9 \zeta_{24}^{3} + \zeta_{24}^{5} - 9 \zeta_{24}^{7} ) q^{58} + ( 9 + 2 \zeta_{24}^{2} - 9 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{59} + 4 \zeta_{24}^{4} q^{61} + ( 3 \zeta_{24} + 5 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{62} -\zeta_{24}^{6} q^{64} + ( -2 \zeta_{24}^{3} - 9 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{67} + ( 1 - 2 \zeta_{24}^{4} - 9 \zeta_{24}^{6} ) q^{71} + ( 3 \zeta_{24} + 3 \zeta_{24}^{5} ) q^{73} + ( -3 \zeta_{24}^{2} + 3 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{74} + ( -2 - 3 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{76} + ( 12 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{77} + ( -10 \zeta_{24}^{2} + 10 \zeta_{24}^{6} ) q^{79} + ( -\zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{82} + ( -12 \zeta_{24} + 6 \zeta_{24}^{3} - 3 \zeta_{24}^{7} ) q^{83} + ( 2 + 3 \zeta_{24}^{2} + 2 \zeta_{24}^{4} ) q^{86} + ( -2 \zeta_{24} + 4 \zeta_{24}^{5} ) q^{88} + ( -10 \zeta_{24}^{2} + 5 \zeta_{24}^{6} ) q^{89} + ( -12 + 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{91} + ( 3 \zeta_{24} - 6 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{92} + 6 \zeta_{24}^{2} q^{94} + ( 6 \zeta_{24} + 10 \zeta_{24}^{3} - 5 \zeta_{24}^{7} ) q^{97} + ( -5 \zeta_{24} + 6 \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 12 \zeta_{24}^{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 24q^{11} + 12q^{14} + 4q^{16} - 36q^{29} + 20q^{31} + 12q^{41} - 24q^{46} - 72q^{49} + 12q^{56} + 36q^{59} + 16q^{61} + 12q^{74} - 8q^{76} + 24q^{86} - 96q^{91} + O(q^{100})$$

## Character values

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

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

## 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 −1.22474 4.57081i −0.707107 + 0.707107i 0 0
143.2 0.965926 0.258819i 0 0.866025 0.500000i 0 0 1.22474 + 4.57081i 0.707107 0.707107i 0 0
557.1 −0.965926 0.258819i 0 0.866025 + 0.500000i 0 0 −1.22474 + 4.57081i −0.707107 0.707107i 0 0
557.2 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0 0 1.22474 4.57081i 0.707107 + 0.707107i 0 0
1007.1 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0 0 −1.22474 + 0.328169i 0.707107 + 0.707107i 0 0
1007.2 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0 0 1.22474 0.328169i −0.707107 0.707107i 0 0
1043.1 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0 0 −1.22474 0.328169i 0.707107 0.707107i 0 0
1043.2 0.258819 0.965926i 0 −0.866025 0.500000i 0 0 1.22474 + 0.328169i −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.b even 2 1 inner
45.l even 12 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.2.q.d 8
3.b odd 2 1 450.2.p.c 8
5.b even 2 1 inner 1350.2.q.d 8
5.c odd 4 2 1350.2.q.a 8
9.c even 3 1 450.2.p.e yes 8
9.d odd 6 1 1350.2.q.a 8
15.d odd 2 1 450.2.p.c 8
15.e even 4 2 450.2.p.e yes 8
45.h odd 6 1 1350.2.q.a 8
45.j even 6 1 450.2.p.e yes 8
45.k odd 12 2 450.2.p.c 8
45.l even 12 2 inner 1350.2.q.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.p.c 8 3.b odd 2 1
450.2.p.c 8 15.d odd 2 1
450.2.p.c 8 45.k odd 12 2
450.2.p.e yes 8 9.c even 3 1
450.2.p.e yes 8 15.e even 4 2
450.2.p.e yes 8 45.j even 6 1
1350.2.q.a 8 5.c odd 4 2
1350.2.q.a 8 9.d odd 6 1
1350.2.q.a 8 45.h odd 6 1
1350.2.q.d 8 1.a even 1 1 trivial
1350.2.q.d 8 5.b even 2 1 inner
1350.2.q.d 8 45.l even 12 2 inner

## Hecke kernels

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

 $$T_{7}^{8} + 36 T_{7}^{6} + 396 T_{7}^{4} - 1296 T_{7}^{2} + 1296$$ $$T_{11}^{2} + 6 T_{11} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{4} + T^{8}$$
$3$ 1
$5$ 1
$7$ $$1 + 36 T^{2} + 634 T^{4} + 7272 T^{6} + 59571 T^{8} + 356328 T^{10} + 1522234 T^{12} + 4235364 T^{14} + 5764801 T^{16}$$
$11$ $$( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} )^{4}$$
$13$ $$1 + 36 T^{2} + 682 T^{4} + 9000 T^{6} + 106947 T^{8} + 1521000 T^{10} + 19478602 T^{12} + 173765124 T^{14} + 815730721 T^{16}$$
$17$ $$( 1 + 289 T^{4} )^{4}$$
$19$ $$( 1 - 14 T^{2} + 339 T^{4} - 5054 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$1 + 108 T^{2} + 5818 T^{4} + 208440 T^{6} + 5501811 T^{8} + 110264760 T^{10} + 1628114938 T^{12} + 15987876012 T^{14} + 78310985281 T^{16}$$
$29$ $$( 1 + 18 T + 188 T^{2} + 1404 T^{3} + 8259 T^{4} + 40716 T^{5} + 158108 T^{6} + 439002 T^{7} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - 10 T + 40 T^{2} + 20 T^{3} - 461 T^{4} + 620 T^{5} + 38440 T^{6} - 297910 T^{7} + 923521 T^{8} )^{2}$$
$37$ $$1 - 644 T^{4} - 1762266 T^{8} - 1206959684 T^{12} + 3512479453921 T^{16}$$
$41$ $$( 1 - 3 T + 44 T^{2} - 123 T^{3} + 1681 T^{4} )^{4}$$
$43$ $$1 + 72 T^{2} + 1633 T^{4} - 6840 T^{6} - 214704 T^{8} - 12647160 T^{10} + 5582902033 T^{12} + 455138139528 T^{14} + 11688200277601 T^{16}$$
$47$ $$1 + 1054 T^{4} - 3768765 T^{8} + 5143183774 T^{12} + 23811286661761 T^{16}$$
$53$ $$1 + 508 T^{4} - 3205722 T^{8} + 4008364348 T^{12} + 62259690411361 T^{16}$$
$59$ $$( 1 - 18 T + 137 T^{2} - 1242 T^{3} + 12372 T^{4} - 73278 T^{5} + 476897 T^{6} - 3696822 T^{7} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} )^{4}$$
$67$ $$1 + 216 T^{2} + 26737 T^{4} + 2415960 T^{6} + 174766032 T^{8} + 10845244440 T^{10} + 538780522177 T^{12} + 19539010548504 T^{14} + 406067677556641 T^{16}$$
$71$ $$( 1 - 116 T^{2} + 12474 T^{4} - 584756 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 + 3503 T^{4} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 + 58 T^{2} - 2877 T^{4} + 361978 T^{6} + 38950081 T^{8} )^{2}$$
$83$ $$1 + 432 T^{2} + 91513 T^{4} + 12659760 T^{6} + 1239875616 T^{8} + 87213086640 T^{10} + 4343053329673 T^{12} + 141238241295408 T^{14} + 2252292232139041 T^{16}$$
$89$ $$( 1 + 103 T^{2} + 7921 T^{4} )^{4}$$
$97$ $$1 - 360 T^{2} + 65929 T^{4} - 8182440 T^{6} + 834546960 T^{8} - 76988577960 T^{10} + 5836646967049 T^{12} - 299869921774440 T^{14} + 7837433594376961 T^{16}$$