Properties

 Label 3024.2.q.h 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: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 \beta_{4} + \beta_{5} ) q^{5} + ( -1 - \beta_{1} + \beta_{2} + \beta_{5} ) q^{7} +O(q^{10})$$ $$q + ( 2 - 2 \beta_{4} + \beta_{5} ) q^{5} + ( -1 - \beta_{1} + \beta_{2} + \beta_{5} ) q^{7} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{11} + ( -\beta_{1} - \beta_{4} + \beta_{5} ) q^{13} + ( 2 \beta_{2} - 2 \beta_{5} ) q^{17} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{19} + ( -3 + \beta_{2} + 3 \beta_{4} - \beta_{5} ) q^{23} + ( -3 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 3 \beta_{5} ) q^{25} + ( 1 - \beta_{2} - \beta_{4} - 3 \beta_{5} ) q^{29} + ( -5 - 2 \beta_{1} + \beta_{3} ) q^{31} + ( -5 + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{35} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{37} + ( \beta_{1} - \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{41} + ( -5 - \beta_{2} + 5 \beta_{4} + 2 \beta_{5} ) q^{43} + ( -2 - 2 \beta_{1} - \beta_{3} ) q^{47} + ( 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{49} + ( -2 + \beta_{2} + 2 \beta_{4} + 4 \beta_{5} ) q^{53} + ( -1 + \beta_{1} + 3 \beta_{3} ) q^{55} + ( 5 \beta_{1} - \beta_{3} ) q^{59} + ( -2 - 5 \beta_{1} + 3 \beta_{3} ) q^{61} + ( -5 - 2 \beta_{1} - \beta_{3} ) q^{65} + ( 3 + \beta_{1} + 3 \beta_{3} ) q^{67} + ( 1 - \beta_{1} - 3 \beta_{3} ) q^{71} + ( -9 + 3 \beta_{2} + 9 \beta_{4} + \beta_{5} ) q^{73} + ( -8 - \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{77} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{79} + ( 3 - 2 \beta_{2} - 3 \beta_{4} - \beta_{5} ) q^{83} + ( 4 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} + 8 \beta_{4} - 4 \beta_{5} ) q^{85} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{89} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{91} + ( \beta_{1} + \beta_{3} ) q^{95} + ( -10 + 2 \beta_{2} + 10 \beta_{4} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 5q^{5} - 4q^{7} + O(q^{10})$$ $$6q + 5q^{5} - 4q^{7} - q^{11} - 2q^{13} + 4q^{17} + 3q^{19} - 7q^{23} - 2q^{25} + 5q^{29} - 28q^{31} - 19q^{35} - 9q^{37} + 12q^{41} - 18q^{43} - 6q^{47} - 12q^{49} - 9q^{53} - 14q^{55} - 8q^{59} - 8q^{61} - 24q^{65} + 10q^{67} + 14q^{71} - 25q^{73} - 52q^{77} + 14q^{79} + 8q^{83} + 14q^{85} + 9q^{89} - 4q^{91} - 4q^{95} - 28q^{97} + O(q^{100})$$

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 + 2.05195i 0.5 − 1.41036i 0.5 + 0.224437i 0.5 − 2.05195i 0.5 + 1.41036i 0.5 − 0.224437i
0 0 0 −0.230252 0.398809i 0 −0.0665372 2.64491i 0 0 0
2305.2 0 0 0 0.880438 + 1.52496i 0 0.710533 + 2.54856i 0 0 0
2305.3 0 0 0 1.84981 + 3.20397i 0 −2.64400 + 0.0963576i 0 0 0
2881.1 0 0 0 −0.230252 + 0.398809i 0 −0.0665372 + 2.64491i 0 0 0
2881.2 0 0 0 0.880438 1.52496i 0 0.710533 2.54856i 0 0 0
2881.3 0 0 0 1.84981 3.20397i 0 −2.64400 0.0963576i 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.h 6
3.b odd 2 1 1008.2.q.h 6
4.b odd 2 1 378.2.e.c 6
7.c even 3 1 3024.2.t.g 6
9.c even 3 1 3024.2.t.g 6
9.d odd 6 1 1008.2.t.g 6
12.b even 2 1 126.2.e.d 6
21.h odd 6 1 1008.2.t.g 6
28.d even 2 1 2646.2.e.o 6
28.f even 6 1 2646.2.f.n 6
28.f even 6 1 2646.2.h.p 6
28.g odd 6 1 378.2.h.d 6
28.g odd 6 1 2646.2.f.o 6
36.f odd 6 1 378.2.h.d 6
36.f odd 6 1 1134.2.g.n 6
36.h even 6 1 126.2.h.c yes 6
36.h even 6 1 1134.2.g.k 6
63.h even 3 1 inner 3024.2.q.h 6
63.j odd 6 1 1008.2.q.h 6
84.h odd 2 1 882.2.e.p 6
84.j odd 6 1 882.2.f.m 6
84.j odd 6 1 882.2.h.o 6
84.n even 6 1 126.2.h.c yes 6
84.n even 6 1 882.2.f.l 6
252.n even 6 1 2646.2.f.n 6
252.o even 6 1 882.2.f.l 6
252.o even 6 1 1134.2.g.k 6
252.r odd 6 1 882.2.e.p 6
252.r odd 6 1 7938.2.a.by 3
252.s odd 6 1 882.2.h.o 6
252.u odd 6 1 378.2.e.c 6
252.u odd 6 1 7938.2.a.bu 3
252.bb even 6 1 126.2.e.d 6
252.bb even 6 1 7938.2.a.cb 3
252.bi even 6 1 2646.2.h.p 6
252.bj even 6 1 2646.2.e.o 6
252.bj even 6 1 7938.2.a.bx 3
252.bl odd 6 1 1134.2.g.n 6
252.bl odd 6 1 2646.2.f.o 6
252.bn odd 6 1 882.2.f.m 6

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

Hecke kernels

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

 $$T_{5}^{6} - 5 T_{5}^{5} + 21 T_{5}^{4} - 26 T_{5}^{3} + 31 T_{5}^{2} + 12 T_{5} + 9$$ $$T_{11}^{6} + T_{11}^{5} + 27 T_{11}^{4} - 92 T_{11}^{3} + 643 T_{11}^{2} - 858 T_{11} + 1089$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 5 T + 6 T^{2} - T^{3} + 31 T^{4} - 68 T^{5} + 29 T^{6} - 340 T^{7} + 775 T^{8} - 125 T^{9} + 3750 T^{10} - 15625 T^{11} + 15625 T^{12}$$
$7$ $$1 + 4 T + 14 T^{2} + 55 T^{3} + 98 T^{4} + 196 T^{5} + 343 T^{6}$$
$11$ $$1 + T - 6 T^{2} - 103 T^{3} - 83 T^{4} + 352 T^{5} + 5027 T^{6} + 3872 T^{7} - 10043 T^{8} - 137093 T^{9} - 87846 T^{10} + 161051 T^{11} + 1771561 T^{12}$$
$13$ $$1 + 2 T - 32 T^{2} - 26 T^{3} + 730 T^{4} + 230 T^{5} - 10729 T^{6} + 2990 T^{7} + 123370 T^{8} - 57122 T^{9} - 913952 T^{10} + 742586 T^{11} + 4826809 T^{12}$$
$17$ $$1 - 4 T + 9 T^{2} - 92 T^{3} + 58 T^{4} + 20 T^{5} + 5393 T^{6} + 340 T^{7} + 16762 T^{8} - 451996 T^{9} + 751689 T^{10} - 5679428 T^{11} + 24137569 T^{12}$$
$19$ $$1 - 3 T - 42 T^{2} + 61 T^{3} + 1311 T^{4} - 726 T^{5} - 27501 T^{6} - 13794 T^{7} + 473271 T^{8} + 418399 T^{9} - 5473482 T^{10} - 7428297 T^{11} + 47045881 T^{12}$$
$23$ $$1 + 7 T - 24 T^{2} - 127 T^{3} + 1417 T^{4} + 3484 T^{5} - 22393 T^{6} + 80132 T^{7} + 749593 T^{8} - 1545209 T^{9} - 6716184 T^{10} + 45054401 T^{11} + 148035889 T^{12}$$
$29$ $$1 - 5 T - 30 T^{2} + 371 T^{3} - 185 T^{4} - 6020 T^{5} + 44357 T^{6} - 174580 T^{7} - 155585 T^{8} + 9048319 T^{9} - 21218430 T^{10} - 102555745 T^{11} + 594823321 T^{12}$$
$31$ $$( 1 + 14 T + 138 T^{2} + 841 T^{3} + 4278 T^{4} + 13454 T^{5} + 29791 T^{6} )^{2}$$
$37$ $$1 + 9 T - 21 T^{2} - 268 T^{3} + 1293 T^{4} + 4875 T^{5} - 42882 T^{6} + 180375 T^{7} + 1770117 T^{8} - 13575004 T^{9} - 39357381 T^{10} + 624095613 T^{11} + 2565726409 T^{12}$$
$41$ $$1 - 12 T - 18 T^{2} + 78 T^{3} + 7470 T^{4} - 24546 T^{5} - 158105 T^{6} - 1006386 T^{7} + 12557070 T^{8} + 5375838 T^{9} - 50863698 T^{10} - 1390274412 T^{11} + 4750104241 T^{12}$$
$43$ $$1 + 18 T + 114 T^{2} + 682 T^{3} + 7188 T^{4} + 33492 T^{5} + 63039 T^{6} + 1440156 T^{7} + 13290612 T^{8} + 54223774 T^{9} + 389743314 T^{10} + 2646151974 T^{11} + 6321363049 T^{12}$$
$47$ $$( 1 + 3 T + 117 T^{2} + 309 T^{3} + 5499 T^{4} + 6627 T^{5} + 103823 T^{6} )^{2}$$
$53$ $$1 + 9 T - 36 T^{2} - 873 T^{3} - 1179 T^{4} + 26334 T^{5} + 272077 T^{6} + 1395702 T^{7} - 3311811 T^{8} - 129969621 T^{9} - 284057316 T^{10} + 3763759437 T^{11} + 22164361129 T^{12}$$
$59$ $$( 1 + 4 T + 76 T^{2} + 649 T^{3} + 4484 T^{4} + 13924 T^{5} + 205379 T^{6} )^{2}$$
$61$ $$( 1 + 4 T + 48 T^{2} - 229 T^{3} + 2928 T^{4} + 14884 T^{5} + 226981 T^{6} )^{2}$$
$67$ $$( 1 - 5 T + 143 T^{2} - 521 T^{3} + 9581 T^{4} - 22445 T^{5} + 300763 T^{6} )^{2}$$
$71$ $$( 1 - 7 T + 163 T^{2} - 895 T^{3} + 11573 T^{4} - 35287 T^{5} + 357911 T^{6} )^{2}$$
$73$ $$1 + 25 T + 254 T^{2} + 2073 T^{3} + 20533 T^{4} + 115046 T^{5} + 366817 T^{6} + 8398358 T^{7} + 109420357 T^{8} + 806432241 T^{9} + 7213153214 T^{10} + 51826789825 T^{11} + 151334226289 T^{12}$$
$79$ $$( 1 - 7 T + 93 T^{2} - 335 T^{3} + 7347 T^{4} - 43687 T^{5} + 493039 T^{6} )^{2}$$
$83$ $$1 - 8 T - 180 T^{2} + 518 T^{3} + 29404 T^{4} - 32420 T^{5} - 2713585 T^{6} - 2690860 T^{7} + 202564156 T^{8} + 296185666 T^{9} - 8542497780 T^{10} - 31512325144 T^{11} + 326940373369 T^{12}$$
$89$ $$1 - 9 T - 180 T^{2} + 729 T^{3} + 31041 T^{4} - 54846 T^{5} - 2925911 T^{6} - 4881294 T^{7} + 245875761 T^{8} + 513922401 T^{9} - 11293603380 T^{10} - 50256535041 T^{11} + 496981290961 T^{12}$$
$97$ $$1 + 28 T + 257 T^{2} + 2820 T^{3} + 59506 T^{4} + 545924 T^{5} + 3126001 T^{6} + 52954628 T^{7} + 559891954 T^{8} + 2573737860 T^{9} + 22752025217 T^{10} + 240445527196 T^{11} + 832972004929 T^{12}$$