# Properties

 Label 4600.2.e.p Level $4600$ Weight $2$ Character orbit 4600.e Analytic conductor $36.731$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.431642176.2 Defining polynomial: $$x^{6} + 19x^{4} + 97x^{2} + 64$$ x^6 + 19*x^4 + 97*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + ( - \beta_{2} + \beta_1) q^{7} + (\beta_{3} - 3) q^{9}+O(q^{10})$$ q + b1 * q^3 + (-b2 + b1) * q^7 + (b3 - 3) * q^9 $$q + \beta_1 q^{3} + ( - \beta_{2} + \beta_1) q^{7} + (\beta_{3} - 3) q^{9} + (\beta_{5} + 2) q^{11} - \beta_{4} q^{13} + ( - \beta_{4} - 3 \beta_{2}) q^{17} + ( - \beta_{3} + 4) q^{19} + (\beta_{5} + \beta_{3} - 6) q^{21} + \beta_{2} q^{23} + ( - \beta_{4} + 2 \beta_{2} - 3 \beta_1) q^{27} + (\beta_{5} - \beta_{3} - 5) q^{29} + (\beta_{5} - 3) q^{31} + (\beta_{4} + 6 \beta_{2} + 2 \beta_1) q^{33} + (\beta_{4} - 3 \beta_{2} + 3 \beta_1) q^{37} + (3 \beta_{5} - \beta_{3} - 2) q^{39} + (\beta_{3} + 3) q^{41} + 8 \beta_{2} q^{43} - 2 \beta_{4} q^{47} + (2 \beta_{5} + \beta_{3}) q^{49} + (6 \beta_{5} - \beta_{3} - 2) q^{51} + (\beta_{4} - \beta_{2} + 3 \beta_1) q^{53} + (\beta_{4} - 2 \beta_{2} + 7 \beta_1) q^{57} + (\beta_{5} - \beta_{3} + 5) q^{59} + ( - \beta_{5} - 2 \beta_{3} + 4) q^{61} + (5 \beta_{2} - 6 \beta_1) q^{63} + (\beta_{4} - 3 \beta_{2} + 3 \beta_1) q^{67} - \beta_{5} q^{69} + ( - 2 \beta_{5} - \beta_{3} - 7) q^{71} + ( - 2 \beta_{4} + 6 \beta_{2} - 4 \beta_1) q^{73} + (\beta_{4} + 4 \beta_{2} + \beta_1) q^{77} + 4 \beta_{5} q^{79} + (\beta_{5} - \beta_{3} + 7) q^{81} + (\beta_{4} - 5 \beta_{2} - \beta_1) q^{83} + (2 \beta_{4} + 4 \beta_{2} - 2 \beta_1) q^{87} + ( - 4 \beta_{5} + 4) q^{89} + (3 \beta_{5} - 2) q^{91} + (\beta_{4} + 6 \beta_{2} - 3 \beta_1) q^{93} + (2 \beta_{2} - 3 \beta_1) q^{97} + ( - 6 \beta_{5} + 3 \beta_{3} - 4) q^{99}+O(q^{100})$$ q + b1 * q^3 + (-b2 + b1) * q^7 + (b3 - 3) * q^9 + (b5 + 2) * q^11 - b4 * q^13 + (-b4 - 3*b2) * q^17 + (-b3 + 4) * q^19 + (b5 + b3 - 6) * q^21 + b2 * q^23 + (-b4 + 2*b2 - 3*b1) * q^27 + (b5 - b3 - 5) * q^29 + (b5 - 3) * q^31 + (b4 + 6*b2 + 2*b1) * q^33 + (b4 - 3*b2 + 3*b1) * q^37 + (3*b5 - b3 - 2) * q^39 + (b3 + 3) * q^41 + 8*b2 * q^43 - 2*b4 * q^47 + (2*b5 + b3) * q^49 + (6*b5 - b3 - 2) * q^51 + (b4 - b2 + 3*b1) * q^53 + (b4 - 2*b2 + 7*b1) * q^57 + (b5 - b3 + 5) * q^59 + (-b5 - 2*b3 + 4) * q^61 + (5*b2 - 6*b1) * q^63 + (b4 - 3*b2 + 3*b1) * q^67 - b5 * q^69 + (-2*b5 - b3 - 7) * q^71 + (-2*b4 + 6*b2 - 4*b1) * q^73 + (b4 + 4*b2 + b1) * q^77 + 4*b5 * q^79 + (b5 - b3 + 7) * q^81 + (b4 - 5*b2 - b1) * q^83 + (2*b4 + 4*b2 - 2*b1) * q^87 + (-4*b5 + 4) * q^89 + (3*b5 - 2) * q^91 + (b4 + 6*b2 - 3*b1) * q^93 + (2*b2 - 3*b1) * q^97 + (-6*b5 + 3*b3 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 20 q^{9}+O(q^{10})$$ 6 * q - 20 * q^9 $$6 q - 20 q^{9} + 14 q^{11} + 26 q^{19} - 36 q^{21} - 26 q^{29} - 16 q^{31} - 4 q^{39} + 16 q^{41} + 2 q^{49} + 2 q^{51} + 34 q^{59} + 26 q^{61} - 2 q^{69} - 44 q^{71} + 8 q^{79} + 46 q^{81} + 16 q^{89} - 6 q^{91} - 42 q^{99}+O(q^{100})$$ 6 * q - 20 * q^9 + 14 * q^11 + 26 * q^19 - 36 * q^21 - 26 * q^29 - 16 * q^31 - 4 * q^39 + 16 * q^41 + 2 * q^49 + 2 * q^51 + 34 * q^59 + 26 * q^61 - 2 * q^69 - 44 * q^71 + 8 * q^79 + 46 * q^81 + 16 * q^89 - 6 * q^91 - 42 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 19x^{4} + 97x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} + 11\nu^{3} + 17\nu ) / 8$$ (v^5 + 11*v^3 + 17*v) / 8 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 6$$ v^2 + 6 $$\beta_{4}$$ $$=$$ $$( \nu^{5} + 7\nu^{3} - 19\nu ) / 4$$ (v^5 + 7*v^3 - 19*v) / 4 $$\beta_{5}$$ $$=$$ $$\nu^{4} + 10\nu^{2} + 8$$ v^4 + 10*v^2 + 8
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 6$$ b3 - 6 $$\nu^{3}$$ $$=$$ $$-\beta_{4} + 2\beta_{2} - 9\beta_1$$ -b4 + 2*b2 - 9*b1 $$\nu^{4}$$ $$=$$ $$\beta_{5} - 10\beta_{3} + 52$$ b5 - 10*b3 + 52 $$\nu^{5}$$ $$=$$ $$11\beta_{4} - 14\beta_{2} + 82\beta_1$$ 11*b4 - 14*b2 + 82*b1

## 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
 − 3.07912i − 2.95759i − 0.878468i 0.878468i 2.95759i 3.07912i
0 3.07912i 0 0 0 2.07912i 0 −6.48097 0
4049.2 0 2.95759i 0 0 0 3.95759i 0 −5.74732 0
4049.3 0 0.878468i 0 0 0 0.121532i 0 2.22829 0
4049.4 0 0.878468i 0 0 0 0.121532i 0 2.22829 0
4049.5 0 2.95759i 0 0 0 3.95759i 0 −5.74732 0
4049.6 0 3.07912i 0 0 0 2.07912i 0 −6.48097 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4049.6 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.p 6
5.b even 2 1 inner 4600.2.e.p 6
5.c odd 4 1 920.2.a.h 3
5.c odd 4 1 4600.2.a.x 3
15.e even 4 1 8280.2.a.bj 3
20.e even 4 1 1840.2.a.s 3
20.e even 4 1 9200.2.a.ce 3
40.i odd 4 1 7360.2.a.by 3
40.k even 4 1 7360.2.a.cc 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.h 3 5.c odd 4 1
1840.2.a.s 3 20.e even 4 1
4600.2.a.x 3 5.c odd 4 1
4600.2.e.p 6 1.a even 1 1 trivial
4600.2.e.p 6 5.b even 2 1 inner
7360.2.a.by 3 40.i odd 4 1
7360.2.a.cc 3 40.k even 4 1
8280.2.a.bj 3 15.e even 4 1
9200.2.a.ce 3 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 $$S_{2}^{\mathrm{new}}(4600, [\chi])$$:

 $$T_{3}^{6} + 19T_{3}^{4} + 97T_{3}^{2} + 64$$ T3^6 + 19*T3^4 + 97*T3^2 + 64 $$T_{7}^{6} + 20T_{7}^{4} + 68T_{7}^{2} + 1$$ T7^6 + 20*T7^4 + 68*T7^2 + 1 $$T_{11}^{3} - 7T_{11}^{2} + 7T_{11} + 14$$ T11^3 - 7*T11^2 + 7*T11 + 14 $$T_{13}^{6} + 47T_{13}^{4} + 629T_{13}^{2} + 2500$$ T13^6 + 47*T13^4 + 629*T13^2 + 2500

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 19 T^{4} + 97 T^{2} + 64$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 20 T^{4} + 68 T^{2} + 1$$
$11$ $$(T^{3} - 7 T^{2} + 7 T + 14)^{2}$$
$13$ $$T^{6} + 47 T^{4} + 629 T^{2} + \cdots + 2500$$
$17$ $$T^{6} + 80 T^{4} + 1760 T^{2} + \cdots + 6889$$
$19$ $$(T^{3} - 13 T^{2} + 33 T + 62)^{2}$$
$23$ $$(T^{2} + 1)^{3}$$
$29$ $$(T^{3} + 13 T^{2} + 26 T - 76)^{2}$$
$31$ $$(T^{3} + 8 T^{2} + 12 T - 1)^{2}$$
$37$ $$T^{6} + 209 T^{4} + 13424 T^{2} + \cdots + 246016$$
$41$ $$(T^{3} - 8 T^{2} - 2 T + 1)^{2}$$
$43$ $$(T^{2} + 64)^{3}$$
$47$ $$T^{6} + 188 T^{4} + 10064 T^{2} + \cdots + 160000$$
$53$ $$T^{6} + 201 T^{4} + 9400 T^{2} + \cdots + 90000$$
$59$ $$(T^{3} - 17 T^{2} + 66 T - 36)^{2}$$
$61$ $$(T^{3} - 13 T^{2} - 51 T + 720)^{2}$$
$67$ $$T^{6} + 209 T^{4} + 13424 T^{2} + \cdots + 246016$$
$71$ $$(T^{3} + 22 T^{2} + 96 T - 225)^{2}$$
$73$ $$T^{6} + 496 T^{4} + 81856 T^{2} + \cdots + 4494400$$
$79$ $$(T^{3} - 4 T^{2} - 144 T + 512)^{2}$$
$83$ $$T^{6} + 145 T^{4} + 2680 T^{2} + \cdots + 1296$$
$89$ $$(T^{3} - 8 T^{2} - 128 T + 64)^{2}$$
$97$ $$T^{6} + 171 T^{4} + 6261 T^{2} + \cdots + 2500$$