# Properties

 Label 4600.2.e.m 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 920) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} -2 \beta_{1} q^{7} + ( -2 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} -2 \beta_{1} q^{7} + ( -2 + \beta_{3} ) q^{9} -4 q^{11} + ( -\beta_{1} - 2 \beta_{2} ) q^{13} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{17} -4 q^{19} + ( 10 - 2 \beta_{3} ) q^{21} -\beta_{2} q^{23} + ( \beta_{1} + 4 \beta_{2} ) q^{27} + ( 7 - \beta_{3} ) q^{29} + ( 3 + \beta_{3} ) q^{31} -4 \beta_{1} q^{33} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{37} + ( 3 + \beta_{3} ) q^{39} + ( -1 - \beta_{3} ) q^{41} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{43} + ( 3 \beta_{1} + 4 \beta_{2} ) q^{47} + ( -13 + 4 \beta_{3} ) q^{49} -8 q^{51} + ( -4 \beta_{1} - 6 \beta_{2} ) q^{53} -4 \beta_{1} q^{57} -4 \beta_{3} q^{59} + ( 8 - 2 \beta_{3} ) q^{61} + ( 4 \beta_{1} - 8 \beta_{2} ) q^{63} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{67} + ( -1 + \beta_{3} ) q^{69} + ( -1 - 3 \beta_{3} ) q^{71} + ( \beta_{1} - 14 \beta_{2} ) q^{73} + 8 \beta_{1} q^{77} + ( 4 - 4 \beta_{3} ) q^{79} -7 q^{81} -12 \beta_{2} q^{83} + ( 7 \beta_{1} - 4 \beta_{2} ) q^{87} -10 q^{89} + ( -6 - 2 \beta_{3} ) q^{91} + ( 3 \beta_{1} + 4 \beta_{2} ) q^{93} + ( 4 \beta_{1} - 6 \beta_{2} ) q^{97} + ( 8 - 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{9} + O(q^{10})$$ $$4 q - 6 q^{9} - 16 q^{11} - 16 q^{19} + 36 q^{21} + 26 q^{29} + 14 q^{31} + 14 q^{39} - 6 q^{41} - 44 q^{49} - 32 q^{51} - 8 q^{59} + 28 q^{61} - 2 q^{69} - 10 q^{71} + 8 q^{79} - 28 q^{81} - 40 q^{89} - 28 q^{91} + 24 q^{99} + O(q^{100})$$

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 5.12311i 0 −3.56155 0
4049.2 0 1.56155i 0 0 0 3.12311i 0 0.561553 0
4049.3 0 1.56155i 0 0 0 3.12311i 0 0.561553 0
4049.4 0 2.56155i 0 0 0 5.12311i 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.m 4
5.b even 2 1 inner 4600.2.e.m 4
5.c odd 4 1 920.2.a.f 2
5.c odd 4 1 4600.2.a.r 2
15.e even 4 1 8280.2.a.bb 2
20.e even 4 1 1840.2.a.k 2
20.e even 4 1 9200.2.a.bx 2
40.i odd 4 1 7360.2.a.bj 2
40.k even 4 1 7360.2.a.bm 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.f 2 5.c odd 4 1
1840.2.a.k 2 20.e even 4 1
4600.2.a.r 2 5.c odd 4 1
4600.2.e.m 4 1.a even 1 1 trivial
4600.2.e.m 4 5.b even 2 1 inner
7360.2.a.bj 2 40.i odd 4 1
7360.2.a.bm 2 40.k even 4 1
8280.2.a.bb 2 15.e even 4 1
9200.2.a.bx 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 $$S_{2}^{\mathrm{new}}(4600, [\chi])$$:

 $$T_{3}^{4} + 9 T_{3}^{2} + 16$$ $$T_{7}^{4} + 36 T_{7}^{2} + 256$$ $$T_{11} + 4$$ $$T_{13}^{4} + 13 T_{13}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$16 + 9 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$256 + 36 T^{2} + T^{4}$$
$11$ $$( 4 + T )^{4}$$
$13$ $$4 + 13 T^{2} + T^{4}$$
$17$ $$256 + 36 T^{2} + T^{4}$$
$19$ $$( 4 + T )^{4}$$
$23$ $$( 1 + T^{2} )^{2}$$
$29$ $$( 38 - 13 T + T^{2} )^{2}$$
$31$ $$( 8 - 7 T + T^{2} )^{2}$$
$37$ $$64 + 52 T^{2} + T^{4}$$
$41$ $$( -2 + 3 T + T^{2} )^{2}$$
$43$ $$64 + 84 T^{2} + T^{4}$$
$47$ $$1024 + 89 T^{2} + T^{4}$$
$53$ $$2704 + 168 T^{2} + T^{4}$$
$59$ $$( -64 + 4 T + T^{2} )^{2}$$
$61$ $$( 32 - 14 T + T^{2} )^{2}$$
$67$ $$4096 + 144 T^{2} + T^{4}$$
$71$ $$( -32 + 5 T + T^{2} )^{2}$$
$73$ $$42436 + 429 T^{2} + T^{4}$$
$79$ $$( -64 - 4 T + T^{2} )^{2}$$
$83$ $$( 144 + T^{2} )^{2}$$
$89$ $$( 10 + T )^{4}$$
$97$ $$16 + 264 T^{2} + T^{4}$$