# Properties

 Label 3024.2.q.g Level $3024$ Weight $2$ Character orbit 3024.q Analytic conductor $24.147$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3024 = 2^{4} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3024.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.1467615712$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.309123.1 Defining polynomial: $$x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 126) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} + ( -\beta_{3} - \beta_{4} + \beta_{5} ) q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{5} + ( -\beta_{3} - \beta_{4} + \beta_{5} ) q^{7} + ( \beta_{2} + \beta_{3} ) q^{11} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{13} + ( 2 - 2 \beta_{4} + 2 \beta_{5} ) q^{17} + ( 3 \beta_{1} + 2 \beta_{4} - 3 \beta_{5} ) q^{19} + ( -2 + 2 \beta_{4} + \beta_{5} ) q^{23} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{25} + ( 2 \beta_{2} - 3 \beta_{5} ) q^{29} + ( 6 - \beta_{1} - \beta_{3} ) q^{31} + ( -4 - \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{35} + \beta_{4} q^{37} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{41} + ( 2 - 3 \beta_{2} - 2 \beta_{4} - 3 \beta_{5} ) q^{43} + ( 3 - 3 \beta_{1} + 3 \beta_{3} ) q^{47} + ( 1 + \beta_{2} - \beta_{3} - 3 \beta_{5} ) q^{49} + ( -5 + \beta_{2} + 5 \beta_{4} + \beta_{5} ) q^{53} + ( 4 + \beta_{1} - 2 \beta_{3} ) q^{55} + ( 5 - \beta_{1} + 2 \beta_{3} ) q^{59} + ( -3 + \beta_{1} - 2 \beta_{3} ) q^{61} + ( -2 + \beta_{1} + 5 \beta_{3} ) q^{65} + ( 2 + \beta_{1} + 4 \beta_{3} ) q^{67} + ( 4 + 7 \beta_{1} - 2 \beta_{3} ) q^{71} + ( 8 - 4 \beta_{2} - 8 \beta_{4} + \beta_{5} ) q^{73} + ( 1 + \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + \beta_{5} ) q^{77} + ( -4 \beta_{1} - \beta_{3} ) q^{79} + ( 1 - 3 \beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{83} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{85} + ( -4 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{89} + ( 8 - \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{91} + ( 3 + 3 \beta_{1} + 2 \beta_{3} ) q^{95} + ( 8 + 2 \beta_{2} - 8 \beta_{4} - 2 \beta_{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - q^{5} - 2q^{7} + O(q^{10})$$ $$6q - q^{5} - 2q^{7} - q^{11} + 8q^{13} + 4q^{17} + 3q^{19} - 7q^{23} + 2q^{25} + 5q^{29} + 40q^{31} - 13q^{35} + 3q^{37} + 6q^{43} + 18q^{47} + 12q^{49} - 15q^{53} + 26q^{55} + 28q^{59} - 16q^{61} - 24q^{65} + 2q^{67} + 14q^{71} + 19q^{73} - 10q^{77} + 10q^{79} + 2q^{83} - 2q^{85} + 9q^{89} + 46q^{91} + 8q^{95} + 28q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 8 \nu^{3} + 5 \nu^{2} - 18 \nu + 6$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} + 6 \nu^{2} - 5 \nu + 3$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 9$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 22 \nu^{2} + 30 \nu - 9$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_{1} + 2$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 4 \beta_{1} - 4$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$7 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + \beta_{1} - 10$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$16 \beta_{5} + 11 \beta_{4} + 8 \beta_{3} + 10 \beta_{2} - 17 \beta_{1} + 5$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-14 \beta_{5} - 16 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} - 23 \beta_{1} + 47$$$$)/3$$

## Character values

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

 $$n$$ $$757$$ $$785$$ $$1135$$ $$2593$$ $$\chi(n)$$ $$1$$ $$-\beta_{4}$$ $$1$$ $$-\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2305.1
 0.5 − 1.41036i 0.5 + 2.05195i 0.5 + 0.224437i 0.5 + 1.41036i 0.5 − 2.05195i 0.5 − 0.224437i
0 0 0 −1.59097 2.75564i 0 2.56238 + 0.658939i 0 0 0
2305.2 0 0 0 0.296790 + 0.514055i 0 −2.32383 1.26483i 0 0 0
2305.3 0 0 0 0.794182 + 1.37556i 0 −1.23855 + 2.33795i 0 0 0
2881.1 0 0 0 −1.59097 + 2.75564i 0 2.56238 0.658939i 0 0 0
2881.2 0 0 0 0.296790 0.514055i 0 −2.32383 + 1.26483i 0 0 0
2881.3 0 0 0 0.794182 1.37556i 0 −1.23855 2.33795i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2881.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.q.g 6
3.b odd 2 1 1008.2.q.g 6
4.b odd 2 1 378.2.e.d 6
7.c even 3 1 3024.2.t.h 6
9.c even 3 1 3024.2.t.h 6
9.d odd 6 1 1008.2.t.h 6
12.b even 2 1 126.2.e.c 6
21.h odd 6 1 1008.2.t.h 6
28.d even 2 1 2646.2.e.p 6
28.f even 6 1 2646.2.f.m 6
28.f even 6 1 2646.2.h.o 6
28.g odd 6 1 378.2.h.c 6
28.g odd 6 1 2646.2.f.l 6
36.f odd 6 1 378.2.h.c 6
36.f odd 6 1 1134.2.g.l 6
36.h even 6 1 126.2.h.d yes 6
36.h even 6 1 1134.2.g.m 6
63.h even 3 1 inner 3024.2.q.g 6
63.j odd 6 1 1008.2.q.g 6
84.h odd 2 1 882.2.e.o 6
84.j odd 6 1 882.2.f.o 6
84.j odd 6 1 882.2.h.p 6
84.n even 6 1 126.2.h.d yes 6
84.n even 6 1 882.2.f.n 6
252.n even 6 1 2646.2.f.m 6
252.o even 6 1 882.2.f.n 6
252.o even 6 1 1134.2.g.m 6
252.r odd 6 1 882.2.e.o 6
252.r odd 6 1 7938.2.a.bw 3
252.s odd 6 1 882.2.h.p 6
252.u odd 6 1 378.2.e.d 6
252.u odd 6 1 7938.2.a.ca 3
252.bb even 6 1 126.2.e.c 6
252.bb even 6 1 7938.2.a.bv 3
252.bi even 6 1 2646.2.h.o 6
252.bj even 6 1 2646.2.e.p 6
252.bj even 6 1 7938.2.a.bz 3
252.bl odd 6 1 1134.2.g.l 6
252.bl odd 6 1 2646.2.f.l 6
252.bn odd 6 1 882.2.f.o 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.c 6 12.b even 2 1
126.2.e.c 6 252.bb even 6 1
126.2.h.d yes 6 36.h even 6 1
126.2.h.d yes 6 84.n even 6 1
378.2.e.d 6 4.b odd 2 1
378.2.e.d 6 252.u odd 6 1
378.2.h.c 6 28.g odd 6 1
378.2.h.c 6 36.f odd 6 1
882.2.e.o 6 84.h odd 2 1
882.2.e.o 6 252.r odd 6 1
882.2.f.n 6 84.n even 6 1
882.2.f.n 6 252.o even 6 1
882.2.f.o 6 84.j odd 6 1
882.2.f.o 6 252.bn odd 6 1
882.2.h.p 6 84.j odd 6 1
882.2.h.p 6 252.s odd 6 1
1008.2.q.g 6 3.b odd 2 1
1008.2.q.g 6 63.j odd 6 1
1008.2.t.h 6 9.d odd 6 1
1008.2.t.h 6 21.h odd 6 1
1134.2.g.l 6 36.f odd 6 1
1134.2.g.l 6 252.bl odd 6 1
1134.2.g.m 6 36.h even 6 1
1134.2.g.m 6 252.o even 6 1
2646.2.e.p 6 28.d even 2 1
2646.2.e.p 6 252.bj even 6 1
2646.2.f.l 6 28.g odd 6 1
2646.2.f.l 6 252.bl odd 6 1
2646.2.f.m 6 28.f even 6 1
2646.2.f.m 6 252.n even 6 1
2646.2.h.o 6 28.f even 6 1
2646.2.h.o 6 252.bi even 6 1
3024.2.q.g 6 1.a even 1 1 trivial
3024.2.q.g 6 63.h even 3 1 inner
3024.2.t.h 6 7.c even 3 1
3024.2.t.h 6 9.c even 3 1
7938.2.a.bv 3 252.bb even 6 1
7938.2.a.bw 3 252.r odd 6 1
7938.2.a.bz 3 252.bj even 6 1
7938.2.a.ca 3 252.u odd 6 1

## Hecke kernels

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

 $$T_{5}^{6} + T_{5}^{5} + 7 T_{5}^{4} - 12 T_{5}^{3} + 33 T_{5}^{2} - 18 T_{5} + 9$$ $$T_{11}^{6} + T_{11}^{5} + 7 T_{11}^{4} - 12 T_{11}^{3} + 33 T_{11}^{2} - 18 T_{11} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + T - 8 T^{2} - 17 T^{3} + 23 T^{4} + 52 T^{5} - 11 T^{6} + 260 T^{7} + 575 T^{8} - 2125 T^{9} - 5000 T^{10} + 3125 T^{11} + 15625 T^{12}$$
$7$ $$1 + 2 T - 4 T^{2} - 31 T^{3} - 28 T^{4} + 98 T^{5} + 343 T^{6}$$
$11$ $$1 + T - 26 T^{2} - 23 T^{3} + 407 T^{4} + 202 T^{5} - 4853 T^{6} + 2222 T^{7} + 49247 T^{8} - 30613 T^{9} - 380666 T^{10} + 161051 T^{11} + 1771561 T^{12}$$
$13$ $$1 - 8 T + 24 T^{2} - 42 T^{3} - 32 T^{4} + 1408 T^{5} - 7901 T^{6} + 18304 T^{7} - 5408 T^{8} - 92274 T^{9} + 685464 T^{10} - 2970344 T^{11} + 4826809 T^{12}$$
$17$ $$1 - 4 T - 23 T^{2} + 68 T^{3} + 410 T^{4} - 220 T^{5} - 8111 T^{6} - 3740 T^{7} + 118490 T^{8} + 334084 T^{9} - 1920983 T^{10} - 5679428 T^{11} + 24137569 T^{12}$$
$19$ $$1 - 3 T - 12 T^{2} + 67 T^{3} - 153 T^{4} - 54 T^{5} + 6315 T^{6} - 1026 T^{7} - 55233 T^{8} + 459553 T^{9} - 1563852 T^{10} - 7428297 T^{11} + 47045881 T^{12}$$
$23$ $$1 + 7 T - 32 T^{2} - 83 T^{3} + 2423 T^{4} + 3946 T^{5} - 46865 T^{6} + 90758 T^{7} + 1281767 T^{8} - 1009861 T^{9} - 8954912 T^{10} + 45054401 T^{11} + 148035889 T^{12}$$
$29$ $$1 - 5 T + 4 T^{2} - 251 T^{3} + 197 T^{4} + 3418 T^{5} + 20293 T^{6} + 99122 T^{7} + 165677 T^{8} - 6121639 T^{9} + 2829124 T^{10} - 102555745 T^{11} + 594823321 T^{12}$$
$31$ $$( 1 - 20 T + 214 T^{2} - 1441 T^{3} + 6634 T^{4} - 19220 T^{5} + 29791 T^{6} )^{2}$$
$37$ $$( 1 - 11 T + 37 T^{2} )^{3}( 1 + 10 T + 37 T^{2} )^{3}$$
$41$ $$1 - 90 T^{2} - 18 T^{3} + 4410 T^{4} + 810 T^{5} - 194177 T^{6} + 33210 T^{7} + 7413210 T^{8} - 1240578 T^{9} - 254318490 T^{10} + 4750104241 T^{12}$$
$43$ $$( 1 - 18 T + 198 T^{2} - 1519 T^{3} + 8514 T^{4} - 33282 T^{5} + 79507 T^{6} )( 1 + 12 T - 6 T^{2} - 547 T^{3} - 258 T^{4} + 22188 T^{5} + 79507 T^{6} )$$
$47$ $$( 1 - 9 T + 87 T^{2} - 657 T^{3} + 4089 T^{4} - 19881 T^{5} + 103823 T^{6} )^{2}$$
$53$ $$1 + 15 T + 33 T^{3} + 13635 T^{4} + 60360 T^{5} - 225155 T^{6} + 3199080 T^{7} + 38300715 T^{8} + 4912941 T^{9} + 6272932395 T^{11} + 22164361129 T^{12}$$
$59$ $$( 1 - 14 T + 216 T^{2} - 1589 T^{3} + 12744 T^{4} - 48734 T^{5} + 205379 T^{6} )^{2}$$
$61$ $$( 1 + 8 T + 178 T^{2} + 883 T^{3} + 10858 T^{4} + 29768 T^{5} + 226981 T^{6} )^{2}$$
$67$ $$( 1 - T + 89 T^{2} + 77 T^{3} + 5963 T^{4} - 4489 T^{5} + 300763 T^{6} )^{2}$$
$71$ $$( 1 - 7 T + 15 T^{2} + 599 T^{3} + 1065 T^{4} - 35287 T^{5} + 357911 T^{6} )^{2}$$
$73$ $$1 - 19 T + 134 T^{2} - 27 T^{3} - 5759 T^{4} + 41986 T^{5} - 314903 T^{6} + 3064978 T^{7} - 30689711 T^{8} - 10503459 T^{9} + 3805364294 T^{10} - 39388360267 T^{11} + 151334226289 T^{12}$$
$79$ $$( 1 - 5 T + 163 T^{2} - 469 T^{3} + 12877 T^{4} - 31205 T^{5} + 493039 T^{6} )^{2}$$
$83$ $$1 - 2 T - 182 T^{2} - 2 T^{3} + 18788 T^{4} + 13564 T^{5} - 1721225 T^{6} + 1125812 T^{7} + 129430532 T^{8} - 1143574 T^{9} - 8637414422 T^{10} - 7878081286 T^{11} + 326940373369 T^{12}$$
$89$ $$1 - 9 T - 144 T^{2} + 1197 T^{3} + 16101 T^{4} - 73314 T^{5} - 1141967 T^{6} - 6524946 T^{7} + 127536021 T^{8} + 843847893 T^{9} - 9034882704 T^{10} - 50256535041 T^{11} + 496981290961 T^{12}$$
$97$ $$1 - 28 T + 281 T^{2} - 2724 T^{3} + 45178 T^{4} - 388196 T^{5} + 2169217 T^{6} - 37655012 T^{7} + 425079802 T^{8} - 2486121252 T^{9} + 24876727961 T^{10} - 240445527196 T^{11} + 832972004929 T^{12}$$