# 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: $$x^{2} + 1$$ 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 $$q + 3 i q^{3} + 2 i q^{7} - 6 q^{9} - 5 i q^{13} + 6 i q^{17} - 6 q^{19} - 6 q^{21} + i q^{23} - 9 i q^{27} - 9 q^{29} + 3 q^{31} + 8 i q^{37} + 15 q^{39} + 3 q^{41} - 8 i q^{43} - 7 i q^{47} + 3 q^{49} - 18 q^{51} - 2 i q^{53} - 18 i q^{57} - 4 q^{59} - 10 q^{61} - 12 i q^{63} - 8 i q^{67} - 3 q^{69} + 7 q^{71} + 9 i q^{73} + 6 q^{79} + 9 q^{81} - 14 i q^{83} - 27 i q^{87} - 16 q^{89} + 10 q^{91} + 9 i q^{93} - 6 i q^{97} +O(q^{100})$$ 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 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 12 q^{9}+O(q^{10})$$ 2 * q - 12 * q^9 $$2 q - 12 q^{9} - 12 q^{19} - 12 q^{21} - 18 q^{29} + 6 q^{31} + 30 q^{39} + 6 q^{41} + 6 q^{49} - 36 q^{51} - 8 q^{59} - 20 q^{61} - 6 q^{69} + 14 q^{71} + 12 q^{79} + 18 q^{81} - 32 q^{89} + 20 q^{91}+O(q^{100})$$ 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

## 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 $$S_{2}^{\mathrm{new}}(4600, [\chi])$$:

 $$T_{3}^{2} + 9$$ T3^2 + 9 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11}$$ T11 $$T_{13}^{2} + 25$$ T13^2 + 25

## Hecke characteristic polynomials

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