# Properties

 Label 1350.3.k.a Level $1350$ Weight $3$ Character orbit 1350.k Analytic conductor $36.785$ 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: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1350.k (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.7848356886$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 18) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{7} - \beta_{4}) q^{2} - 2 \beta_{2} q^{4} + (3 \beta_{7} + 3 \beta_{4} + \beta_1) q^{7} - 2 \beta_{7} q^{8}+O(q^{10})$$ q + (b7 - b4) * q^2 - 2*b2 * q^4 + (3*b7 + 3*b4 + b1) * q^7 - 2*b7 * q^8 $$q + (\beta_{7} - \beta_{4}) q^{2} - 2 \beta_{2} q^{4} + (3 \beta_{7} + 3 \beta_{4} + \beta_1) q^{7} - 2 \beta_{7} q^{8} + (3 \beta_{6} - 3 \beta_{2} - 3) q^{11} + (12 \beta_{7} - 6 \beta_{4} + 5 \beta_{3} - 5 \beta_1) q^{13} + (\beta_{5} - 6 \beta_{2} + 12) q^{14} + (4 \beta_{2} - 4) q^{16} + ( - 6 \beta_{7} + 6 \beta_{3} - 12 \beta_1) q^{17} + (6 \beta_{6} + 6 \beta_{5} + 10) q^{19} + ( - 6 \beta_{7} + 3 \beta_{4} - 6 \beta_{3} + 6 \beta_1) q^{22} + ( - 3 \beta_{4} + 3 \beta_{3} + 3 \beta_1) q^{23} + (5 \beta_{6} - 5 \beta_{5} - 24 \beta_{2} + 12) q^{26} + (6 \beta_{7} - 12 \beta_{4} - 2 \beta_{3}) q^{28} + (6 \beta_{6} + 3 \beta_{2} + 3) q^{29} + ( - 18 \beta_{6} + 9 \beta_{5} + 19 \beta_{2}) q^{31} + 4 \beta_{4} q^{32} + (6 \beta_{6} - 12 \beta_{5} + 12 \beta_{2} - 12) q^{34} + ( - 6 \beta_{7} + 12 \beta_{4} + 32 \beta_{3}) q^{37} + (10 \beta_{7} - 10 \beta_{4} - 24 \beta_{3} + 12 \beta_1) q^{38} + ( - 18 \beta_{5} - 21 \beta_{2} + 42) q^{41} + (9 \beta_{7} + 9 \beta_{4} + 23 \beta_1) q^{43} + ( - 6 \beta_{6} + 6 \beta_{5} + 12 \beta_{2} - 6) q^{44} + (3 \beta_{6} + 3 \beta_{5} - 6) q^{46} + (21 \beta_{7} - 21 \beta_{4} - 18 \beta_{3} + 9 \beta_1) q^{47} + (12 \beta_{6} - 6 \beta_{5} + 6 \beta_{2}) q^{49} + ( - 12 \beta_{7} - 12 \beta_{4} + 10 \beta_1) q^{52} + (30 \beta_{7} - 30 \beta_{3} + 60 \beta_1) q^{53} + ( - 2 \beta_{6} - 12 \beta_{2} - 12) q^{56} + (6 \beta_{7} - 3 \beta_{4} - 12 \beta_{3} + 12 \beta_1) q^{58} + ( - 39 \beta_{5} - 21 \beta_{2} + 42) q^{59} + (18 \beta_{6} - 36 \beta_{5} - 31 \beta_{2} + 31) q^{61} + (19 \beta_{7} + 18 \beta_{3} - 36 \beta_1) q^{62} + 8 q^{64} + (18 \beta_{7} - 9 \beta_{4} - 53 \beta_{3} + 53 \beta_1) q^{67} + (12 \beta_{4} + 12 \beta_{3} + 12 \beta_1) q^{68} + (24 \beta_{6} - 24 \beta_{5} - 60 \beta_{2} + 30) q^{71} + (18 \beta_{7} - 36 \beta_{4} + 52 \beta_{3}) q^{73} + (32 \beta_{6} + 12 \beta_{2} + 12) q^{74} + ( - 24 \beta_{6} + 12 \beta_{5} - 20 \beta_{2}) q^{76} + ( - 24 \beta_{4} + 15 \beta_{3} + 15 \beta_1) q^{77} + (15 \beta_{6} - 30 \beta_{5} + 7 \beta_{2} - 7) q^{79} + (21 \beta_{7} - 42 \beta_{4} + 36 \beta_{3}) q^{82} + (15 \beta_{7} - 15 \beta_{4} - 126 \beta_{3} + 63 \beta_1) q^{83} + (23 \beta_{5} - 18 \beta_{2} + 36) q^{86} + (6 \beta_{7} + 6 \beta_{4} - 12 \beta_1) q^{88} + (66 \beta_{6} - 66 \beta_{5} + 60 \beta_{2} - 30) q^{89} + ( - 9 \beta_{6} - 9 \beta_{5} + 103) q^{91} + ( - 6 \beta_{7} + 6 \beta_{4} - 12 \beta_{3} + 6 \beta_1) q^{92} + ( - 18 \beta_{6} + 9 \beta_{5} - 42 \beta_{2}) q^{94} + (42 \beta_{7} + 42 \beta_{4} + 7 \beta_1) q^{97} + (6 \beta_{7} - 12 \beta_{3} + 24 \beta_1) q^{98}+O(q^{100})$$ q + (b7 - b4) * q^2 - 2*b2 * q^4 + (3*b7 + 3*b4 + b1) * q^7 - 2*b7 * q^8 + (3*b6 - 3*b2 - 3) * q^11 + (12*b7 - 6*b4 + 5*b3 - 5*b1) * q^13 + (b5 - 6*b2 + 12) * q^14 + (4*b2 - 4) * q^16 + (-6*b7 + 6*b3 - 12*b1) * q^17 + (6*b6 + 6*b5 + 10) * q^19 + (-6*b7 + 3*b4 - 6*b3 + 6*b1) * q^22 + (-3*b4 + 3*b3 + 3*b1) * q^23 + (5*b6 - 5*b5 - 24*b2 + 12) * q^26 + (6*b7 - 12*b4 - 2*b3) * q^28 + (6*b6 + 3*b2 + 3) * q^29 + (-18*b6 + 9*b5 + 19*b2) * q^31 + 4*b4 * q^32 + (6*b6 - 12*b5 + 12*b2 - 12) * q^34 + (-6*b7 + 12*b4 + 32*b3) * q^37 + (10*b7 - 10*b4 - 24*b3 + 12*b1) * q^38 + (-18*b5 - 21*b2 + 42) * q^41 + (9*b7 + 9*b4 + 23*b1) * q^43 + (-6*b6 + 6*b5 + 12*b2 - 6) * q^44 + (3*b6 + 3*b5 - 6) * q^46 + (21*b7 - 21*b4 - 18*b3 + 9*b1) * q^47 + (12*b6 - 6*b5 + 6*b2) * q^49 + (-12*b7 - 12*b4 + 10*b1) * q^52 + (30*b7 - 30*b3 + 60*b1) * q^53 + (-2*b6 - 12*b2 - 12) * q^56 + (6*b7 - 3*b4 - 12*b3 + 12*b1) * q^58 + (-39*b5 - 21*b2 + 42) * q^59 + (18*b6 - 36*b5 - 31*b2 + 31) * q^61 + (19*b7 + 18*b3 - 36*b1) * q^62 + 8 * q^64 + (18*b7 - 9*b4 - 53*b3 + 53*b1) * q^67 + (12*b4 + 12*b3 + 12*b1) * q^68 + (24*b6 - 24*b5 - 60*b2 + 30) * q^71 + (18*b7 - 36*b4 + 52*b3) * q^73 + (32*b6 + 12*b2 + 12) * q^74 + (-24*b6 + 12*b5 - 20*b2) * q^76 + (-24*b4 + 15*b3 + 15*b1) * q^77 + (15*b6 - 30*b5 + 7*b2 - 7) * q^79 + (21*b7 - 42*b4 + 36*b3) * q^82 + (15*b7 - 15*b4 - 126*b3 + 63*b1) * q^83 + (23*b5 - 18*b2 + 36) * q^86 + (6*b7 + 6*b4 - 12*b1) * q^88 + (66*b6 - 66*b5 + 60*b2 - 30) * q^89 + (-9*b6 - 9*b5 + 103) * q^91 + (-6*b7 + 6*b4 - 12*b3 + 6*b1) * q^92 + (-18*b6 + 9*b5 - 42*b2) * q^94 + (42*b7 + 42*b4 + 7*b1) * q^97 + (6*b7 - 12*b3 + 24*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{4}+O(q^{10})$$ 8 * q - 8 * q^4 $$8 q - 8 q^{4} - 36 q^{11} + 72 q^{14} - 16 q^{16} + 80 q^{19} + 36 q^{29} + 76 q^{31} - 48 q^{34} + 252 q^{41} - 48 q^{46} + 24 q^{49} - 144 q^{56} + 252 q^{59} + 124 q^{61} + 64 q^{64} + 144 q^{74} - 80 q^{76} - 28 q^{79} + 216 q^{86} + 824 q^{91} - 168 q^{94}+O(q^{100})$$ 8 * q - 8 * q^4 - 36 * q^11 + 72 * q^14 - 16 * q^16 + 80 * q^19 + 36 * q^29 + 76 * q^31 - 48 * q^34 + 252 * q^41 - 48 * q^46 + 24 * q^49 - 144 * q^56 + 252 * q^59 + 124 * q^61 + 64 * q^64 + 144 * q^74 - 80 * q^76 - 28 * q^79 + 216 * q^86 + 824 * q^91 - 168 * q^94

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{24}^{4}$$ v^4 $$\beta_{3}$$ $$=$$ $$\zeta_{24}^{6}$$ v^6 $$\beta_{4}$$ $$=$$ $$\zeta_{24}^{7} + \zeta_{24}$$ v^7 + v $$\beta_{5}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}$$ -v^7 + v $$\beta_{6}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3}$$ -v^7 + v^5 + v^3 $$\beta_{7}$$ $$=$$ $$-\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}$$ -v^5 + v^3 + v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{5} + \beta_{4} ) / 2$$ (b5 + b4) / 2 $$\zeta_{24}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{7} + \beta_{6} - \beta_{5} ) / 2$$ (b7 + b6 - b5) / 2 $$\zeta_{24}^{4}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{24}^{5}$$ $$=$$ $$( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2$$ (-b7 + b6 + b4) / 2 $$\zeta_{24}^{6}$$ $$=$$ $$\beta_{3}$$ b3 $$\zeta_{24}^{7}$$ $$=$$ $$( -\beta_{5} + \beta_{4} ) / 2$$ (-b5 + b4) / 2

## Character values

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

 $$n$$ $$1001$$ $$1027$$ $$\chi(n)$$ $$\beta_{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 0.258819 + 0.965926i −0.965926 + 0.258819i −0.258819 − 0.965926i 0.965926 − 0.258819i 0.258819 − 0.965926i −0.965926 − 0.258819i −0.258819 + 0.965926i 0.965926 + 0.258819i
−0.707107 1.22474i 0 −1.00000 + 1.73205i 0 0 −7.22999 + 4.17423i 2.82843 0 0
449.2 −0.707107 1.22474i 0 −1.00000 + 1.73205i 0 0 −5.49794 + 3.17423i 2.82843 0 0
449.3 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 0 0 5.49794 3.17423i −2.82843 0 0
449.4 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 0 0 7.22999 4.17423i −2.82843 0 0
899.1 −0.707107 + 1.22474i 0 −1.00000 1.73205i 0 0 −7.22999 4.17423i 2.82843 0 0
899.2 −0.707107 + 1.22474i 0 −1.00000 1.73205i 0 0 −5.49794 3.17423i 2.82843 0 0
899.3 0.707107 1.22474i 0 −1.00000 1.73205i 0 0 5.49794 + 3.17423i −2.82843 0 0
899.4 0.707107 1.22474i 0 −1.00000 1.73205i 0 0 7.22999 + 4.17423i −2.82843 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 899.4 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.d odd 6 1 inner
45.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.3.k.a 8
3.b odd 2 1 450.3.k.a 8
5.b even 2 1 inner 1350.3.k.a 8
5.c odd 4 1 54.3.d.a 4
5.c odd 4 1 1350.3.i.b 4
9.c even 3 1 450.3.k.a 8
9.d odd 6 1 inner 1350.3.k.a 8
15.d odd 2 1 450.3.k.a 8
15.e even 4 1 18.3.d.a 4
15.e even 4 1 450.3.i.b 4
20.e even 4 1 432.3.q.d 4
40.i odd 4 1 1728.3.q.d 4
40.k even 4 1 1728.3.q.c 4
45.h odd 6 1 inner 1350.3.k.a 8
45.j even 6 1 450.3.k.a 8
45.k odd 12 1 18.3.d.a 4
45.k odd 12 1 162.3.b.a 4
45.k odd 12 1 450.3.i.b 4
45.l even 12 1 54.3.d.a 4
45.l even 12 1 162.3.b.a 4
45.l even 12 1 1350.3.i.b 4
60.l odd 4 1 144.3.q.c 4
120.q odd 4 1 576.3.q.e 4
120.w even 4 1 576.3.q.f 4
180.v odd 12 1 432.3.q.d 4
180.v odd 12 1 1296.3.e.g 4
180.x even 12 1 144.3.q.c 4
180.x even 12 1 1296.3.e.g 4
360.bo even 12 1 576.3.q.e 4
360.br even 12 1 1728.3.q.d 4
360.bt odd 12 1 1728.3.q.c 4
360.bu odd 12 1 576.3.q.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.d.a 4 15.e even 4 1
18.3.d.a 4 45.k odd 12 1
54.3.d.a 4 5.c odd 4 1
54.3.d.a 4 45.l even 12 1
144.3.q.c 4 60.l odd 4 1
144.3.q.c 4 180.x even 12 1
162.3.b.a 4 45.k odd 12 1
162.3.b.a 4 45.l even 12 1
432.3.q.d 4 20.e even 4 1
432.3.q.d 4 180.v odd 12 1
450.3.i.b 4 15.e even 4 1
450.3.i.b 4 45.k odd 12 1
450.3.k.a 8 3.b odd 2 1
450.3.k.a 8 9.c even 3 1
450.3.k.a 8 15.d odd 2 1
450.3.k.a 8 45.j even 6 1
576.3.q.e 4 120.q odd 4 1
576.3.q.e 4 360.bo even 12 1
576.3.q.f 4 120.w even 4 1
576.3.q.f 4 360.bu odd 12 1
1296.3.e.g 4 180.v odd 12 1
1296.3.e.g 4 180.x even 12 1
1350.3.i.b 4 5.c odd 4 1
1350.3.i.b 4 45.l even 12 1
1350.3.k.a 8 1.a even 1 1 trivial
1350.3.k.a 8 5.b even 2 1 inner
1350.3.k.a 8 9.d odd 6 1 inner
1350.3.k.a 8 45.h odd 6 1 inner
1728.3.q.c 4 40.k even 4 1
1728.3.q.c 4 360.bt odd 12 1
1728.3.q.d 4 40.i odd 4 1
1728.3.q.d 4 360.br even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} - 110T_{7}^{6} + 9291T_{7}^{4} - 308990T_{7}^{2} + 7890481$$ acting on $$S_{3}^{\mathrm{new}}(1350, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} + 2 T^{2} + 4)^{2}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8} - 110 T^{6} + 9291 T^{4} + \cdots + 7890481$$
$11$ $$(T^{4} + 18 T^{3} + 117 T^{2} + 162 T + 81)^{2}$$
$13$ $$T^{8} - 482 T^{6} + \cdots + 1330863361$$
$17$ $$(T^{4} - 360 T^{2} + 1296)^{2}$$
$19$ $$(T^{2} - 20 T - 116)^{4}$$
$23$ $$T^{8} + 90 T^{6} + 8019 T^{4} + \cdots + 6561$$
$29$ $$(T^{4} - 18 T^{3} + 63 T^{2} + 810 T + 2025)^{2}$$
$31$ $$(T^{4} - 38 T^{3} + 1569 T^{2} + \cdots + 15625)^{2}$$
$37$ $$(T^{4} + 2480 T^{2} + 652864)^{2}$$
$41$ $$(T^{4} - 126 T^{3} + 5967 T^{2} + \cdots + 455625)^{2}$$
$43$ $$T^{8} - 2030 T^{6} + \cdots + 3418801$$
$47$ $$T^{8} + 2250 T^{6} + \cdots + 166726039041$$
$53$ $$(T^{4} - 9000 T^{2} + 810000)^{2}$$
$59$ $$(T^{4} - 126 T^{3} + 3573 T^{2} + \cdots + 2954961)^{2}$$
$61$ $$(T^{4} - 62 T^{3} + 4827 T^{2} + \cdots + 966289)^{2}$$
$67$ $$T^{8} - 6590 T^{6} + \cdots + 29120366676241$$
$71$ $$(T^{4} + 7704 T^{2} + 2396304)^{2}$$
$73$ $$(T^{4} + 9296 T^{2} + 577600)^{2}$$
$79$ $$(T^{4} + 14 T^{3} + 1497 T^{2} + \cdots + 1692601)^{2}$$
$83$ $$T^{8} + 24714 T^{6} + \cdots + 17\!\cdots\!01$$
$89$ $$(T^{4} + 22824 T^{2} + 36144144)^{2}$$
$97$ $$T^{8} - 21266 T^{6} + \cdots + 12\!\cdots\!25$$