# Properties

 Label 1350.3.i.b Level $1350$ Weight $3$ Character orbit 1350.i Analytic conductor $36.785$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1350 = 2 \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1350.i (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.7848356886$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - \beta_1) q^{2} + ( - 2 \beta_{2} + 2) q^{4} + (3 \beta_{3} - \beta_{2} + 3 \beta_1) q^{7} + 2 \beta_{3} q^{8}+O(q^{10})$$ q + (b3 - b1) * q^2 + (-2*b2 + 2) * q^4 + (3*b3 - b2 + 3*b1) * q^7 + 2*b3 * q^8 $$q + (\beta_{3} - \beta_1) q^{2} + ( - 2 \beta_{2} + 2) q^{4} + (3 \beta_{3} - \beta_{2} + 3 \beta_1) q^{7} + 2 \beta_{3} q^{8} + (3 \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 6) q^{11} + ( - 12 \beta_{3} - 5 \beta_{2} + 6 \beta_1 + 5) q^{13} + ( - 6 \beta_{2} + \beta_1 - 6) q^{14} - 4 \beta_{2} q^{16} + ( - 6 \beta_{3} + 12 \beta_{2} - 6) q^{17} + ( - 6 \beta_{3} + 12 \beta_1 - 10) q^{19} + ( - 6 \beta_{3} - 6 \beta_{2} + 3 \beta_1 + 6) q^{22} + (3 \beta_{2} + 3 \beta_1 + 3) q^{23} + (5 \beta_{3} + 24 \beta_{2} - 12) q^{26} + ( - 6 \beta_{3} + 12 \beta_1 - 2) q^{28} + ( - 6 \beta_{3} + 3 \beta_{2} + 6 \beta_1 - 6) q^{29} + ( - 18 \beta_{3} - 19 \beta_{2} + 9 \beta_1 + 19) q^{31} + 4 \beta_1 q^{32} + ( - 6 \beta_{3} + 12 \beta_{2} - 6 \beta_1) q^{34} + ( - 6 \beta_{3} + 12 \beta_1 - 32) q^{37} + ( - 10 \beta_{3} + 12 \beta_{2} + 10 \beta_1 - 24) q^{38} + (21 \beta_{2} + 18 \beta_1 + 21) q^{41} + ( - 9 \beta_{3} + 23 \beta_{2} - 9 \beta_1) q^{43} + (6 \beta_{3} + 12 \beta_{2} - 6) q^{44} + (3 \beta_{3} - 6 \beta_1 - 6) q^{46} + (21 \beta_{3} - 9 \beta_{2} - 21 \beta_1 + 18) q^{47} + ( - 12 \beta_{3} + 6 \beta_{2} + 6 \beta_1 - 6) q^{49} + ( - 12 \beta_{3} - 10 \beta_{2} - 12 \beta_1) q^{52} + ( - 30 \beta_{3} + 60 \beta_{2} - 30) q^{53} + ( - 2 \beta_{3} + 12 \beta_{2} + 2 \beta_1 - 24) q^{56} + ( - 6 \beta_{3} + 12 \beta_{2} + 3 \beta_1 - 12) q^{58} + ( - 21 \beta_{2} - 39 \beta_1 - 21) q^{59} + (18 \beta_{3} + 31 \beta_{2} + 18 \beta_1) q^{61} + (19 \beta_{3} + 36 \beta_{2} - 18) q^{62} - 8 q^{64} + (18 \beta_{3} - 53 \beta_{2} - 9 \beta_1 + 53) q^{67} + (12 \beta_{2} - 12 \beta_1 + 12) q^{68} + (24 \beta_{3} + 60 \beta_{2} - 30) q^{71} + ( - 18 \beta_{3} + 36 \beta_1 + 52) q^{73} + ( - 32 \beta_{3} + 12 \beta_{2} + 32 \beta_1 - 24) q^{74} + ( - 24 \beta_{3} + 20 \beta_{2} + 12 \beta_1 - 20) q^{76} + ( - 15 \beta_{2} - 24 \beta_1 - 15) q^{77} + ( - 15 \beta_{3} + 7 \beta_{2} - 15 \beta_1) q^{79} + (21 \beta_{3} - 42 \beta_1 - 36) q^{82} + ( - 15 \beta_{3} + 63 \beta_{2} + 15 \beta_1 - 126) q^{83} + (18 \beta_{2} - 23 \beta_1 + 18) q^{86} + ( - 6 \beta_{3} - 12 \beta_{2} - 6 \beta_1) q^{88} + ( - 66 \beta_{3} + 60 \beta_{2} - 30) q^{89} + ( - 9 \beta_{3} + 18 \beta_1 + 103) q^{91} + ( - 6 \beta_{3} - 6 \beta_{2} + 6 \beta_1 + 12) q^{92} + (18 \beta_{3} - 42 \beta_{2} - 9 \beta_1 + 42) q^{94} + (42 \beta_{3} - 7 \beta_{2} + 42 \beta_1) q^{97} + ( - 6 \beta_{3} + 24 \beta_{2} - 12) q^{98}+O(q^{100})$$ q + (b3 - b1) * q^2 + (-2*b2 + 2) * q^4 + (3*b3 - b2 + 3*b1) * q^7 + 2*b3 * q^8 + (3*b3 + 3*b2 - 3*b1 - 6) * q^11 + (-12*b3 - 5*b2 + 6*b1 + 5) * q^13 + (-6*b2 + b1 - 6) * q^14 - 4*b2 * q^16 + (-6*b3 + 12*b2 - 6) * q^17 + (-6*b3 + 12*b1 - 10) * q^19 + (-6*b3 - 6*b2 + 3*b1 + 6) * q^22 + (3*b2 + 3*b1 + 3) * q^23 + (5*b3 + 24*b2 - 12) * q^26 + (-6*b3 + 12*b1 - 2) * q^28 + (-6*b3 + 3*b2 + 6*b1 - 6) * q^29 + (-18*b3 - 19*b2 + 9*b1 + 19) * q^31 + 4*b1 * q^32 + (-6*b3 + 12*b2 - 6*b1) * q^34 + (-6*b3 + 12*b1 - 32) * q^37 + (-10*b3 + 12*b2 + 10*b1 - 24) * q^38 + (21*b2 + 18*b1 + 21) * q^41 + (-9*b3 + 23*b2 - 9*b1) * q^43 + (6*b3 + 12*b2 - 6) * q^44 + (3*b3 - 6*b1 - 6) * q^46 + (21*b3 - 9*b2 - 21*b1 + 18) * q^47 + (-12*b3 + 6*b2 + 6*b1 - 6) * q^49 + (-12*b3 - 10*b2 - 12*b1) * q^52 + (-30*b3 + 60*b2 - 30) * q^53 + (-2*b3 + 12*b2 + 2*b1 - 24) * q^56 + (-6*b3 + 12*b2 + 3*b1 - 12) * q^58 + (-21*b2 - 39*b1 - 21) * q^59 + (18*b3 + 31*b2 + 18*b1) * q^61 + (19*b3 + 36*b2 - 18) * q^62 - 8 * q^64 + (18*b3 - 53*b2 - 9*b1 + 53) * q^67 + (12*b2 - 12*b1 + 12) * q^68 + (24*b3 + 60*b2 - 30) * q^71 + (-18*b3 + 36*b1 + 52) * q^73 + (-32*b3 + 12*b2 + 32*b1 - 24) * q^74 + (-24*b3 + 20*b2 + 12*b1 - 20) * q^76 + (-15*b2 - 24*b1 - 15) * q^77 + (-15*b3 + 7*b2 - 15*b1) * q^79 + (21*b3 - 42*b1 - 36) * q^82 + (-15*b3 + 63*b2 + 15*b1 - 126) * q^83 + (18*b2 - 23*b1 + 18) * q^86 + (-6*b3 - 12*b2 - 6*b1) * q^88 + (-66*b3 + 60*b2 - 30) * q^89 + (-9*b3 + 18*b1 + 103) * q^91 + (-6*b3 - 6*b2 + 6*b1 + 12) * q^92 + (18*b3 - 42*b2 - 9*b1 + 42) * q^94 + (42*b3 - 7*b2 + 42*b1) * q^97 + (-6*b3 + 24*b2 - 12) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} - 2 q^{7}+O(q^{10})$$ 4 * q + 4 * q^4 - 2 * q^7 $$4 q + 4 q^{4} - 2 q^{7} - 18 q^{11} + 10 q^{13} - 36 q^{14} - 8 q^{16} - 40 q^{19} + 12 q^{22} + 18 q^{23} - 8 q^{28} - 18 q^{29} + 38 q^{31} + 24 q^{34} - 128 q^{37} - 72 q^{38} + 126 q^{41} + 46 q^{43} - 24 q^{46} + 54 q^{47} - 12 q^{49} - 20 q^{52} - 72 q^{56} - 24 q^{58} - 126 q^{59} + 62 q^{61} - 32 q^{64} + 106 q^{67} + 72 q^{68} + 208 q^{73} - 72 q^{74} - 40 q^{76} - 90 q^{77} + 14 q^{79} - 144 q^{82} - 378 q^{83} + 108 q^{86} - 24 q^{88} + 412 q^{91} + 36 q^{92} + 84 q^{94} - 14 q^{97}+O(q^{100})$$ 4 * q + 4 * q^4 - 2 * q^7 - 18 * q^11 + 10 * q^13 - 36 * q^14 - 8 * q^16 - 40 * q^19 + 12 * q^22 + 18 * q^23 - 8 * q^28 - 18 * q^29 + 38 * q^31 + 24 * q^34 - 128 * q^37 - 72 * q^38 + 126 * q^41 + 46 * q^43 - 24 * q^46 + 54 * q^47 - 12 * q^49 - 20 * q^52 - 72 * q^56 - 24 * q^58 - 126 * q^59 + 62 * q^61 - 32 * q^64 + 106 * q^67 + 72 * q^68 + 208 * q^73 - 72 * q^74 - 40 * q^76 - 90 * q^77 + 14 * q^79 - 144 * q^82 - 378 * q^83 + 108 * q^86 - 24 * q^88 + 412 * q^91 + 36 * q^92 + 84 * q^94 - 14 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 1.22474 − 0.707107i −1.22474 + 0.707107i 1.22474 + 0.707107i −1.22474 − 0.707107i
−1.22474 0.707107i 0 1.00000 + 1.73205i 0 0 3.17423 5.49794i 2.82843i 0 0
251.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i 0 0 −4.17423 + 7.22999i 2.82843i 0 0
1151.1 −1.22474 + 0.707107i 0 1.00000 1.73205i 0 0 3.17423 + 5.49794i 2.82843i 0 0
1151.2 1.22474 0.707107i 0 1.00000 1.73205i 0 0 −4.17423 7.22999i 2.82843i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d 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.i.b 4
3.b odd 2 1 450.3.i.b 4
5.b even 2 1 54.3.d.a 4
5.c odd 4 2 1350.3.k.a 8
9.c even 3 1 450.3.i.b 4
9.d odd 6 1 inner 1350.3.i.b 4
15.d odd 2 1 18.3.d.a 4
15.e even 4 2 450.3.k.a 8
20.d odd 2 1 432.3.q.d 4
40.e odd 2 1 1728.3.q.c 4
40.f even 2 1 1728.3.q.d 4
45.h odd 6 1 54.3.d.a 4
45.h odd 6 1 162.3.b.a 4
45.j even 6 1 18.3.d.a 4
45.j even 6 1 162.3.b.a 4
45.k odd 12 2 450.3.k.a 8
45.l even 12 2 1350.3.k.a 8
60.h even 2 1 144.3.q.c 4
120.i odd 2 1 576.3.q.f 4
120.m even 2 1 576.3.q.e 4
180.n even 6 1 432.3.q.d 4
180.n even 6 1 1296.3.e.g 4
180.p odd 6 1 144.3.q.c 4
180.p odd 6 1 1296.3.e.g 4
360.z odd 6 1 576.3.q.e 4
360.bd even 6 1 1728.3.q.c 4
360.bh odd 6 1 1728.3.q.d 4
360.bk even 6 1 576.3.q.f 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 2T_{7}^{3} + 57T_{7}^{2} - 106T_{7} + 2809$$ acting on $$S_{3}^{\mathrm{new}}(1350, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2T^{2} + 4$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 2 T^{3} + 57 T^{2} + \cdots + 2809$$
$11$ $$T^{4} + 18 T^{3} + 117 T^{2} + \cdots + 81$$
$13$ $$T^{4} - 10 T^{3} + 291 T^{2} + \cdots + 36481$$
$17$ $$T^{4} + 360T^{2} + 1296$$
$19$ $$(T^{2} + 20 T - 116)^{2}$$
$23$ $$T^{4} - 18 T^{3} + 117 T^{2} + \cdots + 81$$
$29$ $$T^{4} + 18 T^{3} + 63 T^{2} + \cdots + 2025$$
$31$ $$T^{4} - 38 T^{3} + 1569 T^{2} + \cdots + 15625$$
$37$ $$(T^{2} + 64 T + 808)^{2}$$
$41$ $$T^{4} - 126 T^{3} + 5967 T^{2} + \cdots + 455625$$
$43$ $$T^{4} - 46 T^{3} + 2073 T^{2} + \cdots + 1849$$
$47$ $$T^{4} - 54 T^{3} + 333 T^{2} + \cdots + 408321$$
$53$ $$T^{4} + 9000 T^{2} + 810000$$
$59$ $$T^{4} + 126 T^{3} + 3573 T^{2} + \cdots + 2954961$$
$61$ $$T^{4} - 62 T^{3} + 4827 T^{2} + \cdots + 966289$$
$67$ $$T^{4} - 106 T^{3} + 8913 T^{2} + \cdots + 5396329$$
$71$ $$T^{4} + 7704 T^{2} + \cdots + 2396304$$
$73$ $$(T^{2} - 104 T + 760)^{2}$$
$79$ $$T^{4} - 14 T^{3} + 1497 T^{2} + \cdots + 1692601$$
$83$ $$T^{4} + 378 T^{3} + \cdots + 131262849$$
$89$ $$T^{4} + 22824 T^{2} + \cdots + 36144144$$
$97$ $$T^{4} + 14 T^{3} + \cdots + 110986225$$