# Properties

 Label 4600.2.e.o Level $4600$ Weight $2$ Character orbit 4600.e Analytic conductor $36.731$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4600 = 2^{3} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4600.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.7311849298$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 184) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -2 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -2 + \beta_{3} ) q^{9} + ( 2 - 2 \beta_{3} ) q^{11} + ( -\beta_{1} + 2 \beta_{2} ) q^{13} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{17} + ( -2 + 2 \beta_{3} ) q^{19} -\beta_{2} q^{23} + ( \beta_{1} + 4 \beta_{2} ) q^{27} + ( -1 - \beta_{3} ) q^{29} + ( -5 + \beta_{3} ) q^{31} + ( 2 \beta_{1} - 8 \beta_{2} ) q^{33} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{37} + ( 7 - 3 \beta_{3} ) q^{39} + ( 3 - 5 \beta_{3} ) q^{41} -8 \beta_{2} q^{43} + ( 3 \beta_{1} - 4 \beta_{2} ) q^{47} + 7 q^{49} + 8 q^{51} + 2 \beta_{2} q^{53} + ( -2 \beta_{1} + 8 \beta_{2} ) q^{57} -4 \beta_{3} q^{59} + ( 6 - 4 \beta_{3} ) q^{61} -2 \beta_{1} q^{67} + ( -1 + \beta_{3} ) q^{69} + ( 11 + \beta_{3} ) q^{71} + ( -3 \beta_{1} - 10 \beta_{2} ) q^{73} + ( 2 - 2 \beta_{3} ) q^{79} -7 q^{81} + ( 4 \beta_{1} + 8 \beta_{2} ) q^{83} + ( -\beta_{1} - 4 \beta_{2} ) q^{87} + ( 4 - 6 \beta_{3} ) q^{89} + ( -5 \beta_{1} + 4 \beta_{2} ) q^{93} + ( -6 \beta_{1} - 2 \beta_{2} ) q^{97} + ( -12 + 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 6q^{9} + O(q^{10})$$ $$4q - 6q^{9} + 4q^{11} - 4q^{19} - 6q^{29} - 18q^{31} + 22q^{39} + 2q^{41} + 28q^{49} + 32q^{51} - 8q^{59} + 16q^{61} - 2q^{69} + 46q^{71} + 4q^{79} - 28q^{81} + 4q^{89} - 40q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9 x^{2} + 16$$:

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$$.

 $$n$$ $$1151$$ $$1201$$ $$2301$$ $$2577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4049.1
 − 2.56155i − 1.56155i 1.56155i 2.56155i
0 2.56155i 0 0 0 0 0 −3.56155 0
4049.2 0 1.56155i 0 0 0 0 0 0.561553 0
4049.3 0 1.56155i 0 0 0 0 0 0.561553 0
4049.4 0 2.56155i 0 0 0 0 0 −3.56155 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4600.2.e.o 4
5.b even 2 1 inner 4600.2.e.o 4
5.c odd 4 1 184.2.a.e 2
5.c odd 4 1 4600.2.a.s 2
15.e even 4 1 1656.2.a.j 2
20.e even 4 1 368.2.a.i 2
20.e even 4 1 9200.2.a.br 2
35.f even 4 1 9016.2.a.w 2
40.i odd 4 1 1472.2.a.u 2
40.k even 4 1 1472.2.a.p 2
60.l odd 4 1 3312.2.a.t 2
115.e even 4 1 4232.2.a.o 2
460.k odd 4 1 8464.2.a.bd 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.2.a.e 2 5.c odd 4 1
368.2.a.i 2 20.e even 4 1
1472.2.a.p 2 40.k even 4 1
1472.2.a.u 2 40.i odd 4 1
1656.2.a.j 2 15.e even 4 1
3312.2.a.t 2 60.l odd 4 1
4232.2.a.o 2 115.e even 4 1
4600.2.a.s 2 5.c odd 4 1
4600.2.e.o 4 1.a even 1 1 trivial
4600.2.e.o 4 5.b even 2 1 inner
8464.2.a.bd 2 460.k odd 4 1
9016.2.a.w 2 35.f even 4 1
9200.2.a.br 2 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4600, [\chi])$$:

 $$T_{3}^{4} + 9 T_{3}^{2} + 16$$ $$T_{7}$$ $$T_{11}^{2} - 2 T_{11} - 16$$ $$T_{13}^{4} + 21 T_{13}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$16 + 9 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -16 - 2 T + T^{2} )^{2}$$
$13$ $$4 + 21 T^{2} + T^{4}$$
$17$ $$256 + 36 T^{2} + T^{4}$$
$19$ $$( -16 + 2 T + T^{2} )^{2}$$
$23$ $$( 1 + T^{2} )^{2}$$
$29$ $$( -2 + 3 T + T^{2} )^{2}$$
$31$ $$( 16 + 9 T + T^{2} )^{2}$$
$37$ $$( 68 + T^{2} )^{2}$$
$41$ $$( -106 - T + T^{2} )^{2}$$
$43$ $$( 64 + T^{2} )^{2}$$
$47$ $$64 + 137 T^{2} + T^{4}$$
$53$ $$( 4 + T^{2} )^{2}$$
$59$ $$( -64 + 4 T + T^{2} )^{2}$$
$61$ $$( -52 - 8 T + T^{2} )^{2}$$
$67$ $$256 + 36 T^{2} + T^{4}$$
$71$ $$( 128 - 23 T + T^{2} )^{2}$$
$73$ $$1156 + 221 T^{2} + T^{4}$$
$79$ $$( -16 - 2 T + T^{2} )^{2}$$
$83$ $$1024 + 208 T^{2} + T^{4}$$
$89$ $$( -152 - 2 T + T^{2} )^{2}$$
$97$ $$23104 + 308 T^{2} + T^{4}$$