# Properties

 Label 3024.2.b.v Level 3024 Weight 2 Character orbit 3024.b Analytic conductor 24.147 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3024 = 2^{4} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3024.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.1467615712$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.4161798144.13 Defining polynomial: $$x^{8} + 6 x^{6} + 29 x^{4} + 42 x^{2} + 49$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}\cdot 3^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{5} + ( 1 - \beta_{5} ) q^{7} +O(q^{10})$$ $$q -\beta_{1} q^{5} + ( 1 - \beta_{5} ) q^{7} + \beta_{2} q^{11} + ( -\beta_{3} - \beta_{5} ) q^{13} + ( \beta_{1} - \beta_{2} ) q^{17} + ( -1 - \beta_{4} ) q^{19} + ( -2 \beta_{1} + \beta_{2} ) q^{23} + ( -4 - \beta_{4} ) q^{25} + \beta_{6} q^{29} + ( 1 - \beta_{4} ) q^{31} + ( -\beta_{1} - \beta_{6} ) q^{35} + q^{37} + ( -\beta_{1} + 2 \beta_{2} ) q^{41} + ( 2 \beta_{3} + \beta_{5} ) q^{43} -\beta_{7} q^{47} + ( -5 - 2 \beta_{5} ) q^{49} + ( \beta_{6} + \beta_{7} ) q^{53} + ( 3 - 2 \beta_{4} ) q^{55} + \beta_{6} q^{59} + \beta_{3} q^{61} + ( -2 \beta_{6} - \beta_{7} ) q^{65} + ( \beta_{3} + 3 \beta_{5} ) q^{67} + \beta_{1} q^{71} + ( 2 \beta_{3} - 3 \beta_{5} ) q^{73} + ( \beta_{2} - \beta_{7} ) q^{77} + ( 2 \beta_{3} - \beta_{5} ) q^{79} + \beta_{7} q^{83} + ( 6 + 3 \beta_{4} ) q^{85} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{89} + ( -6 - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{91} + ( 4 \beta_{1} - 3 \beta_{2} ) q^{95} + ( -\beta_{3} - 4 \beta_{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{7} + O(q^{10})$$ $$8q + 8q^{7} - 8q^{19} - 32q^{25} + 8q^{31} + 8q^{37} - 40q^{49} + 24q^{55} + 48q^{85} - 48q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 6 x^{6} + 29 x^{4} + 42 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-6 \nu^{7} - 29 \nu^{5} - 174 \nu^{3} - 455 \nu$$$$)/203$$ $$\beta_{2}$$ $$=$$ $$($$$$-10 \nu^{7} - 116 \nu^{5} - 493 \nu^{3} - 1232 \nu$$$$)/203$$ $$\beta_{3}$$ $$=$$ $$($$$$24 \nu^{6} + 116 \nu^{4} + 696 \nu^{2} + 602$$$$)/203$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{6} - 135$$$$)/29$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{6} - 6 \nu^{4} - 22 \nu^{2} - 21$$$$)/7$$ $$\beta_{6}$$ $$=$$ $$($$$$-54 \nu^{7} - 261 \nu^{5} - 957 \nu^{3} - 441 \nu$$$$)/203$$ $$\beta_{7}$$ $$=$$ $$($$$$90 \nu^{7} + 435 \nu^{5} + 2001 \nu^{3} + 735 \nu$$$$)/203$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} - \beta_{6} - 6 \beta_{1}$$$$)/12$$ $$\nu^{2}$$ $$=$$ $$($$$$6 \beta_{5} - 2 \beta_{4} + 9 \beta_{3} - 18$$$$)/12$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{7} + 5 \beta_{6}$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$($$$$-12 \beta_{5} - 4 \beta_{4} - 11 \beta_{3} - 22$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-11 \beta_{7} - 23 \beta_{6} - 36 \beta_{2} + 102 \beta_{1}$$$$)/12$$ $$\nu^{6}$$ $$=$$ $$($$$$29 \beta_{4} + 135$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-45 \beta_{7} - 103 \beta_{6} + 174 \beta_{2} - 444 \beta_{1}$$$$)/12$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$$.

 $$n$$ $$757$$ $$785$$ $$1135$$ $$2593$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1567.1
 −1.05050 − 1.81952i 1.05050 − 1.81952i 0.629640 − 1.09057i −0.629640 − 1.09057i −0.629640 + 1.09057i 0.629640 + 1.09057i 1.05050 + 1.81952i −1.05050 + 1.81952i
0 0 0 3.63904i 0 1.00000 2.44949i 0 0 0
1567.2 0 0 0 3.63904i 0 1.00000 + 2.44949i 0 0 0
1567.3 0 0 0 2.18114i 0 1.00000 2.44949i 0 0 0
1567.4 0 0 0 2.18114i 0 1.00000 + 2.44949i 0 0 0
1567.5 0 0 0 2.18114i 0 1.00000 2.44949i 0 0 0
1567.6 0 0 0 2.18114i 0 1.00000 + 2.44949i 0 0 0
1567.7 0 0 0 3.63904i 0 1.00000 2.44949i 0 0 0
1567.8 0 0 0 3.63904i 0 1.00000 + 2.44949i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1567.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.b.v yes 8
3.b odd 2 1 inner 3024.2.b.v yes 8
4.b odd 2 1 3024.2.b.u 8
7.b odd 2 1 3024.2.b.u 8
12.b even 2 1 3024.2.b.u 8
21.c even 2 1 3024.2.b.u 8
28.d even 2 1 inner 3024.2.b.v yes 8
84.h odd 2 1 inner 3024.2.b.v yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3024.2.b.u 8 4.b odd 2 1
3024.2.b.u 8 7.b odd 2 1
3024.2.b.u 8 12.b even 2 1
3024.2.b.u 8 21.c even 2 1
3024.2.b.v yes 8 1.a even 1 1 trivial
3024.2.b.v yes 8 3.b odd 2 1 inner
3024.2.b.v yes 8 28.d even 2 1 inner
3024.2.b.v yes 8 84.h odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3024, [\chi])$$:

 $$T_{5}^{4} + 18 T_{5}^{2} + 63$$ $$T_{11}^{4} + 30 T_{11}^{2} + 63$$ $$T_{13}^{4} + 36 T_{13}^{2} + 36$$ $$T_{17}^{4} + 36 T_{17}^{2} + 252$$ $$T_{19}^{2} + 2 T_{19} - 17$$ $$T_{29}^{4} - 108 T_{29}^{2} + 2268$$ $$T_{47}^{4} - 180 T_{47}^{2} + 2268$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 2 T^{2} + 33 T^{4} - 50 T^{6} + 625 T^{8} )^{2}$$
$7$ $$( 1 - 2 T + 7 T^{2} )^{4}$$
$11$ $$( 1 - 14 T^{2} + 129 T^{4} - 1694 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - 16 T^{2} + 114 T^{4} - 2704 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$( 1 - 32 T^{2} + 762 T^{4} - 9248 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 + 2 T + 21 T^{2} + 38 T^{3} + 361 T^{4} )^{4}$$
$23$ $$( 1 - 14 T^{2} - 351 T^{4} - 7406 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 + 8 T^{2} + 1050 T^{4} + 6728 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - 2 T + 45 T^{2} - 62 T^{3} + 961 T^{4} )^{4}$$
$37$ $$( 1 - T + 37 T^{2} )^{8}$$
$41$ $$( 1 - 50 T^{2} + 3825 T^{4} - 84050 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 64 T^{2} + 3570 T^{4} - 118336 T^{6} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 + 8 T^{2} - 1398 T^{4} + 17672 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 - 4 T^{2} + 3030 T^{4} - 11236 T^{6} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 + 128 T^{2} + 10410 T^{4} + 445568 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 110 T^{2} + 3721 T^{4} )^{4}$$
$67$ $$( 1 - 136 T^{2} + 11010 T^{4} - 610504 T^{6} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 - 266 T^{2} + 27753 T^{4} - 1340906 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 - 88 T^{2} + 2226 T^{4} - 468952 T^{6} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 - 208 T^{2} + 22146 T^{4} - 1298128 T^{6} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 + 152 T^{2} + 13722 T^{4} + 1047128 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 - 146 T^{2} + 13233 T^{4} - 1156466 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 172 T^{2} + 21606 T^{4} - 1618348 T^{6} + 88529281 T^{8} )^{2}$$