# Properties

 Label 3648.2.k.g Level $3648$ Weight $2$ Character orbit 3648.k Analytic conductor $29.129$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3648 = 2^{6} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3648.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$29.1294266574$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 912) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + \beta_{3} q^{5} + \beta_{1} q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + \beta_{3} q^{5} + \beta_{1} q^{7} + q^{9} -\beta_{1} q^{11} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{13} -\beta_{3} q^{15} + ( 2 + \beta_{3} ) q^{17} + ( -4 - \beta_{2} ) q^{19} -\beta_{1} q^{21} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{23} + ( 3 + \beta_{3} ) q^{25} - q^{27} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{29} + ( 2 + 2 \beta_{3} ) q^{31} + \beta_{1} q^{33} + ( -\beta_{1} - 2 \beta_{2} ) q^{35} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{37} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{39} -4 \beta_{2} q^{41} + ( 3 \beta_{1} - 4 \beta_{2} ) q^{43} + \beta_{3} q^{45} + ( -\beta_{1} + 2 \beta_{2} ) q^{47} + ( 3 + \beta_{3} ) q^{49} + ( -2 - \beta_{3} ) q^{51} + 4 \beta_{1} q^{53} + ( \beta_{1} + 2 \beta_{2} ) q^{55} + ( 4 + \beta_{2} ) q^{57} + 4 q^{59} + ( -2 - \beta_{3} ) q^{61} + \beta_{1} q^{63} + ( 4 \beta_{1} - 8 \beta_{2} ) q^{65} + ( -4 + 4 \beta_{3} ) q^{67} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{69} -4 q^{71} + ( 6 - 3 \beta_{3} ) q^{73} + ( -3 - \beta_{3} ) q^{75} + ( 4 - \beta_{3} ) q^{77} + ( -2 + 2 \beta_{3} ) q^{79} + q^{81} -2 \beta_{2} q^{83} + ( 8 + 3 \beta_{3} ) q^{85} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{87} + ( -8 \beta_{1} + 4 \beta_{2} ) q^{89} -4 q^{91} + ( -2 - 2 \beta_{3} ) q^{93} + ( 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{95} + ( 8 \beta_{1} - 4 \beta_{2} ) q^{97} -\beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} + 2q^{5} + 4q^{9} + O(q^{10})$$ $$4q - 4q^{3} + 2q^{5} + 4q^{9} - 2q^{15} + 10q^{17} - 16q^{19} + 14q^{25} - 4q^{27} + 12q^{31} + 2q^{45} + 14q^{49} - 10q^{51} + 16q^{57} + 16q^{59} - 10q^{61} - 8q^{67} - 16q^{71} + 18q^{73} - 14q^{75} + 14q^{77} - 4q^{79} + 4q^{81} + 38q^{85} - 16q^{91} - 12q^{93} - 8q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} - \nu + 3$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 6$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 5 \nu + 3$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{2} + \beta_{1} + 2$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{2} - 2 \beta_{1} + 4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3648\mathbb{Z}\right)^\times$$.

 $$n$$ $$1217$$ $$1921$$ $$2053$$ $$2623$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2431.1
 −1.18614 + 1.26217i −1.18614 − 1.26217i 1.68614 + 0.396143i 1.68614 − 0.396143i
0 −1.00000 0 −2.37228 0 2.52434i 0 1.00000 0
2431.2 0 −1.00000 0 −2.37228 0 2.52434i 0 1.00000 0
2431.3 0 −1.00000 0 3.37228 0 0.792287i 0 1.00000 0
2431.4 0 −1.00000 0 3.37228 0 0.792287i 0 1.00000 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
76.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3648.2.k.g 4
4.b odd 2 1 3648.2.k.h 4
8.b even 2 1 912.2.k.h yes 4
8.d odd 2 1 912.2.k.g 4
19.b odd 2 1 3648.2.k.h 4
24.f even 2 1 2736.2.k.n 4
24.h odd 2 1 2736.2.k.o 4
76.d even 2 1 inner 3648.2.k.g 4
152.b even 2 1 912.2.k.h yes 4
152.g odd 2 1 912.2.k.g 4
456.l odd 2 1 2736.2.k.o 4
456.p even 2 1 2736.2.k.n 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.k.g 4 8.d odd 2 1
912.2.k.g 4 152.g odd 2 1
912.2.k.h yes 4 8.b even 2 1
912.2.k.h yes 4 152.b even 2 1
2736.2.k.n 4 24.f even 2 1
2736.2.k.n 4 456.p even 2 1
2736.2.k.o 4 24.h odd 2 1
2736.2.k.o 4 456.l odd 2 1
3648.2.k.g 4 1.a even 1 1 trivial
3648.2.k.g 4 76.d even 2 1 inner
3648.2.k.h 4 4.b odd 2 1
3648.2.k.h 4 19.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3648, [\chi])$$:

 $$T_{5}^{2} - T_{5} - 8$$ $$T_{31}^{2} - 6 T_{31} - 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$( -8 - T + T^{2} )^{2}$$
$7$ $$4 + 7 T^{2} + T^{4}$$
$11$ $$4 + 7 T^{2} + T^{4}$$
$13$ $$64 + 28 T^{2} + T^{4}$$
$17$ $$( -2 - 5 T + T^{2} )^{2}$$
$19$ $$( 19 + 8 T + T^{2} )^{2}$$
$23$ $$256 + 76 T^{2} + T^{4}$$
$29$ $$1024 + 112 T^{2} + T^{4}$$
$31$ $$( -24 - 6 T + T^{2} )^{2}$$
$37$ $$64 + 28 T^{2} + T^{4}$$
$41$ $$( 48 + T^{2} )^{2}$$
$43$ $$36 + 87 T^{2} + T^{4}$$
$47$ $$16 + 19 T^{2} + T^{4}$$
$53$ $$1024 + 112 T^{2} + T^{4}$$
$59$ $$( -4 + T )^{4}$$
$61$ $$( -2 + 5 T + T^{2} )^{2}$$
$67$ $$( -128 + 4 T + T^{2} )^{2}$$
$71$ $$( 4 + T )^{4}$$
$73$ $$( -54 - 9 T + T^{2} )^{2}$$
$79$ $$( -32 + 2 T + T^{2} )^{2}$$
$83$ $$( 12 + T^{2} )^{2}$$
$89$ $$( 176 + T^{2} )^{2}$$
$97$ $$( 176 + T^{2} )^{2}$$