# Properties

 Label 4600.2.e.a Level $4600$ Weight $2$ Character orbit 4600.e Analytic conductor $36.731$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} + 2 i q^{7} -6 q^{9} +O(q^{10})$$ $$q + 3 i q^{3} + 2 i q^{7} -6 q^{9} -5 i q^{13} + 6 i q^{17} -6 q^{19} -6 q^{21} + i q^{23} -9 i q^{27} -9 q^{29} + 3 q^{31} + 8 i q^{37} + 15 q^{39} + 3 q^{41} -8 i q^{43} -7 i q^{47} + 3 q^{49} -18 q^{51} -2 i q^{53} -18 i q^{57} -4 q^{59} -10 q^{61} -12 i q^{63} -8 i q^{67} -3 q^{69} + 7 q^{71} + 9 i q^{73} + 6 q^{79} + 9 q^{81} -14 i q^{83} -27 i q^{87} -16 q^{89} + 10 q^{91} + 9 i q^{93} -6 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 12 q^{9} + O(q^{10})$$ $$2 q - 12 q^{9} - 12 q^{19} - 12 q^{21} - 18 q^{29} + 6 q^{31} + 30 q^{39} + 6 q^{41} + 6 q^{49} - 36 q^{51} - 8 q^{59} - 20 q^{61} - 6 q^{69} + 14 q^{71} + 12 q^{79} + 18 q^{81} - 32 q^{89} + 20 q^{91} + O(q^{100})$$

## 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
 − 1.00000i 1.00000i
0 3.00000i 0 0 0 2.00000i 0 −6.00000 0
4049.2 0 3.00000i 0 0 0 2.00000i 0 −6.00000 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
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.a 2
5.b even 2 1 inner 4600.2.e.a 2
5.c odd 4 1 184.2.a.d 1
5.c odd 4 1 4600.2.a.a 1
15.e even 4 1 1656.2.a.c 1
20.e even 4 1 368.2.a.a 1
20.e even 4 1 9200.2.a.bj 1
35.f even 4 1 9016.2.a.b 1
40.i odd 4 1 1472.2.a.a 1
40.k even 4 1 1472.2.a.m 1
60.l odd 4 1 3312.2.a.i 1
115.e even 4 1 4232.2.a.j 1
460.k odd 4 1 8464.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.2.a.d 1 5.c odd 4 1
368.2.a.a 1 20.e even 4 1
1472.2.a.a 1 40.i odd 4 1
1472.2.a.m 1 40.k even 4 1
1656.2.a.c 1 15.e even 4 1
3312.2.a.i 1 60.l odd 4 1
4232.2.a.j 1 115.e even 4 1
4600.2.a.a 1 5.c odd 4 1
4600.2.e.a 2 1.a even 1 1 trivial
4600.2.e.a 2 5.b even 2 1 inner
8464.2.a.b 1 460.k odd 4 1
9016.2.a.b 1 35.f even 4 1
9200.2.a.bj 1 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 $$S_{2}^{\mathrm{new}}(4600, [\chi])$$:

 $$T_{3}^{2} + 9$$ $$T_{7}^{2} + 4$$ $$T_{11}$$ $$T_{13}^{2} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$25 + T^{2}$$
$17$ $$36 + T^{2}$$
$19$ $$( 6 + T )^{2}$$
$23$ $$1 + T^{2}$$
$29$ $$( 9 + T )^{2}$$
$31$ $$( -3 + T )^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$( -3 + T )^{2}$$
$43$ $$64 + T^{2}$$
$47$ $$49 + T^{2}$$
$53$ $$4 + T^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$( 10 + T )^{2}$$
$67$ $$64 + T^{2}$$
$71$ $$( -7 + T )^{2}$$
$73$ $$81 + T^{2}$$
$79$ $$( -6 + T )^{2}$$
$83$ $$196 + T^{2}$$
$89$ $$( 16 + T )^{2}$$
$97$ $$36 + T^{2}$$