# Properties

 Label 3330.2.h.m Level $3330$ Weight $2$ Character orbit 3330.h Analytic conductor $26.590$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3330.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$26.5901838731$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.279290944.1 Defining polynomial: $$x^{6} + 26 x^{4} + 169 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{37}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1110) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} - q^{4} + \beta_{2} q^{5} + ( -1 + \beta_{3} ) q^{7} -\beta_{2} q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} - q^{4} + \beta_{2} q^{5} + ( -1 + \beta_{3} ) q^{7} -\beta_{2} q^{8} - q^{10} + ( -1 - \beta_{3} ) q^{11} + ( -\beta_{1} + 3 \beta_{2} ) q^{13} + ( -\beta_{1} - \beta_{2} ) q^{14} + q^{16} + ( \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} ) q^{17} + ( \beta_{1} - 3 \beta_{2} ) q^{19} -\beta_{2} q^{20} + ( \beta_{1} - \beta_{2} ) q^{22} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{23} - q^{25} + ( -3 - \beta_{3} ) q^{26} + ( 1 - \beta_{3} ) q^{28} -2 \beta_{2} q^{29} + ( -2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{31} + \beta_{2} q^{32} + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} ) q^{34} + ( -\beta_{1} - \beta_{2} ) q^{35} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{37} + ( 3 + \beta_{3} ) q^{38} + q^{40} + 2 q^{41} + ( 2 \beta_{1} + \beta_{4} + \beta_{5} ) q^{43} + ( 1 + \beta_{3} ) q^{44} + ( -1 + \beta_{3} - \beta_{4} + \beta_{5} ) q^{46} + ( 4 - 2 \beta_{3} ) q^{47} + ( 2 - \beta_{3} - \beta_{4} + \beta_{5} ) q^{49} -\beta_{2} q^{50} + ( \beta_{1} - 3 \beta_{2} ) q^{52} + ( -1 - \beta_{3} ) q^{53} + ( \beta_{1} - \beta_{2} ) q^{55} + ( \beta_{1} + \beta_{2} ) q^{56} + 2 q^{58} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{59} + 2 \beta_{2} q^{61} + ( 2 + \beta_{4} - \beta_{5} ) q^{62} - q^{64} + ( -3 - \beta_{3} ) q^{65} + 4 \beta_{3} q^{67} + ( -\beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} ) q^{68} + ( 1 - \beta_{3} ) q^{70} + ( 2 + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{71} + ( -5 - \beta_{3} - \beta_{4} + \beta_{5} ) q^{73} + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{74} + ( -\beta_{1} + 3 \beta_{2} ) q^{76} + ( -7 - \beta_{3} + \beta_{4} - \beta_{5} ) q^{77} + ( -6 \beta_{2} + \beta_{4} + \beta_{5} ) q^{79} + \beta_{2} q^{80} + 2 \beta_{2} q^{82} + ( -3 - \beta_{3} - \beta_{4} + \beta_{5} ) q^{83} + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} ) q^{85} + ( 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{86} + ( -\beta_{1} + \beta_{2} ) q^{88} + ( 3 \beta_{1} + 5 \beta_{2} ) q^{89} + ( -3 \beta_{1} + 5 \beta_{2} - \beta_{4} - \beta_{5} ) q^{91} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{92} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{94} + ( 3 + \beta_{3} ) q^{95} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{97} + ( \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{4} - 6q^{7} + O(q^{10})$$ $$6q - 6q^{4} - 6q^{7} - 6q^{10} - 6q^{11} + 6q^{16} - 6q^{25} - 18q^{26} + 6q^{28} + 10q^{34} - 2q^{37} + 18q^{38} + 6q^{40} + 12q^{41} + 6q^{44} - 2q^{46} + 24q^{47} + 16q^{49} - 6q^{53} + 12q^{58} + 8q^{62} - 6q^{64} - 18q^{65} + 6q^{70} + 20q^{71} - 26q^{73} + 4q^{74} - 46q^{77} - 14q^{83} + 10q^{85} + 4q^{86} + 18q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 26 x^{4} + 169 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 13 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{4} + 13 \nu^{2}$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} + \nu^{4} + 21 \nu^{3} + 17 \nu^{2} + 100 \nu + 32$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{5} - \nu^{4} + 21 \nu^{3} - 17 \nu^{2} + 100 \nu - 32$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} - \beta_{3} - 8$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{2} - 13 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$13 \beta_{5} - 13 \beta_{4} + 17 \beta_{3} + 104$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{5} + 4 \beta_{4} - 84 \beta_{2} + 173 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times$$.

 $$n$$ $$371$$ $$631$$ $$667$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2071.1
 − 3.44055i − 0.309984i 3.75054i 3.44055i 0.309984i − 3.75054i
1.00000i 0 −1.00000 1.00000i 0 −4.44055 1.00000i 0 −1.00000
2071.2 1.00000i 0 −1.00000 1.00000i 0 −1.30998 1.00000i 0 −1.00000
2071.3 1.00000i 0 −1.00000 1.00000i 0 2.75054 1.00000i 0 −1.00000
2071.4 1.00000i 0 −1.00000 1.00000i 0 −4.44055 1.00000i 0 −1.00000
2071.5 1.00000i 0 −1.00000 1.00000i 0 −1.30998 1.00000i 0 −1.00000
2071.6 1.00000i 0 −1.00000 1.00000i 0 2.75054 1.00000i 0 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2071.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3330.2.h.m 6
3.b odd 2 1 1110.2.h.f 6
37.b even 2 1 inner 3330.2.h.m 6
111.d odd 2 1 1110.2.h.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.h.f 6 3.b odd 2 1
1110.2.h.f 6 111.d odd 2 1
3330.2.h.m 6 1.a even 1 1 trivial
3330.2.h.m 6 37.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3330, [\chi])$$:

 $$T_{7}^{3} + 3 T_{7}^{2} - 10 T_{7} - 16$$ $$T_{11}^{3} + 3 T_{11}^{2} - 10 T_{11} - 8$$ $$T_{13}^{6} + 53 T_{13}^{4} + 340 T_{13}^{2} + 64$$ $$T_{41} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{3}$$
$3$ $$T^{6}$$
$5$ $$( 1 + T^{2} )^{3}$$
$7$ $$( -16 - 10 T + 3 T^{2} + T^{3} )^{2}$$
$11$ $$( -8 - 10 T + 3 T^{2} + T^{3} )^{2}$$
$13$ $$64 + 340 T^{2} + 53 T^{4} + T^{6}$$
$17$ $$55696 + 4664 T^{2} + 121 T^{4} + T^{6}$$
$19$ $$64 + 340 T^{2} + 53 T^{4} + T^{6}$$
$23$ $$16384 + 2880 T^{2} + 113 T^{4} + T^{6}$$
$29$ $$( 4 + T^{2} )^{3}$$
$31$ $$256 + 2576 T^{2} + 120 T^{4} + T^{6}$$
$37$ $$50653 + 2738 T + 1147 T^{2} + 116 T^{3} + 31 T^{4} + 2 T^{5} + T^{6}$$
$41$ $$( -2 + T )^{6}$$
$43$ $$1024 + 6528 T^{2} + 164 T^{4} + T^{6}$$
$47$ $$( 176 - 4 T - 12 T^{2} + T^{3} )^{2}$$
$53$ $$( -8 - 10 T + 3 T^{2} + T^{3} )^{2}$$
$59$ $$30976 + 6416 T^{2} + 184 T^{4} + T^{6}$$
$61$ $$( 4 + T^{2} )^{3}$$
$67$ $$( -256 - 208 T + T^{3} )^{2}$$
$71$ $$( 1888 - 192 T - 10 T^{2} + T^{3} )^{2}$$
$73$ $$( -452 - 28 T + 13 T^{2} + T^{3} )^{2}$$
$79$ $$30976 + 12816 T^{2} + 248 T^{4} + T^{6}$$
$83$ $$( -352 - 68 T + 7 T^{2} + T^{3} )^{2}$$
$89$ $$322624 + 18804 T^{2} + 309 T^{4} + T^{6}$$
$97$ $$12544 + 2704 T^{2} + 152 T^{4} + T^{6}$$