# Properties

 Label 3024.2.k.k Level 3024 Weight 2 Character orbit 3024.k Analytic conductor 24.147 Analytic rank 0 Dimension 16 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.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.1467615712$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 24 x^{14} + 230 x^{12} - 1052 x^{10} + 2139 x^{8} - 1244 x^{6} + 1134 x^{4} - 104 x^{2} + 169$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{20}$$ Twist minimal: no (minimal twist has level 1512) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{9} q^{5} + \beta_{1} q^{7} +O(q^{10})$$ $$q -\beta_{9} q^{5} + \beta_{1} q^{7} + \beta_{10} q^{11} -\beta_{5} q^{13} + \beta_{11} q^{17} + \beta_{3} q^{19} + ( \beta_{13} + \beta_{15} ) q^{23} + ( -1 + \beta_{8} ) q^{25} + ( -\beta_{10} - \beta_{15} ) q^{29} + ( \beta_{4} - \beta_{5} ) q^{31} + ( \beta_{7} - \beta_{11} + \beta_{15} ) q^{35} + ( -1 + \beta_{1} - \beta_{2} - \beta_{6} + \beta_{8} ) q^{37} + ( -\beta_{9} + \beta_{11} + \beta_{14} ) q^{41} + ( -1 - \beta_{2} + 2 \beta_{8} ) q^{43} + ( \beta_{7} - 2 \beta_{9} - 2 \beta_{11} + \beta_{12} - \beta_{14} ) q^{47} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} ) q^{49} + ( -2 \beta_{10} - 2 \beta_{15} ) q^{53} + ( \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{55} + ( -\beta_{7} + 2 \beta_{9} - \beta_{12} - \beta_{14} ) q^{59} + ( \beta_{1} + 2 \beta_{3} + \beta_{6} ) q^{61} + ( -\beta_{7} - \beta_{10} + \beta_{12} - \beta_{15} ) q^{65} + ( 1 + \beta_{1} + \beta_{2} - \beta_{6} + 2 \beta_{8} ) q^{67} + ( \beta_{7} + 2 \beta_{10} - \beta_{12} + \beta_{15} ) q^{71} + ( 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{73} + ( -\beta_{9} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{77} + ( 3 - \beta_{1} + \beta_{2} + \beta_{6} ) q^{79} + ( 2 \beta_{9} - 2 \beta_{11} + \beta_{14} ) q^{83} + ( 1 - \beta_{1} + \beta_{6} + \beta_{8} ) q^{85} + ( -\beta_{7} - \beta_{9} + \beta_{11} - \beta_{12} - \beta_{14} ) q^{89} + ( -2 + \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{8} ) q^{91} + ( \beta_{13} - \beta_{15} ) q^{95} + ( 3 \beta_{1} + 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 2q^{7} + O(q^{10})$$ $$16q + 2q^{7} - 12q^{25} - 8q^{37} - 8q^{43} + 2q^{49} + 28q^{67} + 44q^{79} + 16q^{85} - 18q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 24 x^{14} + 230 x^{12} - 1052 x^{10} + 2139 x^{8} - 1244 x^{6} + 1134 x^{4} - 104 x^{2} + 169$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-305389 \nu^{14} + 7316240 \nu^{12} - 69689225 \nu^{10} + 319235369 \nu^{8} - 695756145 \nu^{6} + 747498217 \nu^{4} - 1154216584 \nu^{2} + 237296969$$$$)/ 192947092$$ $$\beta_{2}$$ $$=$$ $$($$$$-2694 \nu^{14} + 61469 \nu^{12} - 552649 \nu^{10} + 2285409 \nu^{8} - 3789335 \nu^{6} + 779859 \nu^{4} - 572101 \nu^{2} - 10061233$$$$)/1663337$$ $$\beta_{3}$$ $$=$$ $$($$$$2694 \nu^{14} - 61469 \nu^{12} + 552649 \nu^{10} - 2285409 \nu^{8} + 3789335 \nu^{6} - 779859 \nu^{4} + 3898775 \nu^{2} + 81211$$$$)/1663337$$ $$\beta_{4}$$ $$=$$ $$($$$$-11520 \nu^{14} + 293628 \nu^{12} - 3040296 \nu^{10} + 15624254 \nu^{8} - 39217560 \nu^{6} + 38849348 \nu^{4} - 16270592 \nu^{2} + 6370193$$$$)/3710521$$ $$\beta_{5}$$ $$=$$ $$($$$$-1139534 \nu^{14} + 28712659 \nu^{12} - 295900503 \nu^{10} + 1529685268 \nu^{8} - 3939691239 \nu^{6} + 4125515812 \nu^{4} - 1455012369 \nu^{2} + 691999581$$$$)/ 192947092$$ $$\beta_{6}$$ $$=$$ $$($$$$-1429545 \nu^{14} + 33944011 \nu^{12} - 320576374 \nu^{10} + 1427099575 \nu^{8} - 2709051014 \nu^{6} + 1088455103 \nu^{4} - 1464953121 \nu^{2} - 44292248$$$$)/ 192947092$$ $$\beta_{7}$$ $$=$$ $$($$$$-458582 \nu^{15} + 16643678 \nu^{13} - 243643736 \nu^{11} + 1832626360 \nu^{9} - 7303260208 \nu^{7} + 13830825760 \nu^{5} - 8231666126 \nu^{3} + 2564873142 \nu$$$$)/ 627078049$$ $$\beta_{8}$$ $$=$$ $$($$$$88700 \nu^{14} - 2103563 \nu^{12} + 19736021 \nu^{10} - 86408618 \nu^{8} + 154953525 \nu^{6} - 27036722 \nu^{4} + 23842785 \nu^{2} + 15626169$$$$)/6653348$$ $$\beta_{9}$$ $$=$$ $$($$$$4302859 \nu^{15} - 113264303 \nu^{13} + 1228806350 \nu^{11} - 6828867199 \nu^{9} + 19868045926 \nu^{7} - 27619863991 \nu^{5} + 18464795563 \nu^{3} - 10467201680 \nu$$$$)/ 2508312196$$ $$\beta_{10}$$ $$=$$ $$($$$$-403473 \nu^{15} + 10071623 \nu^{13} - 101747496 \nu^{11} + 504132793 \nu^{9} - 1172875960 \nu^{7} + 854627793 \nu^{5} + 21762817 \nu^{3} + 225114682 \nu$$$$)/86493524$$ $$\beta_{11}$$ $$=$$ $$($$$$3300067 \nu^{15} - 75367115 \nu^{13} + 666756659 \nu^{11} - 2588222763 \nu^{9} + 3036067465 \nu^{7} + 3953470823 \nu^{5} - 407547934 \nu^{3} + 1131065416 \nu$$$$)/ 627078049$$ $$\beta_{12}$$ $$=$$ $$($$$$6600134 \nu^{15} - 150734230 \nu^{13} + 1333513318 \nu^{11} - 5176445526 \nu^{9} + 6072134930 \nu^{7} + 7906941646 \nu^{5} - 815095868 \nu^{3} + 4770443028 \nu$$$$)/ 627078049$$ $$\beta_{13}$$ $$=$$ $$($$$$-248063 \nu^{15} + 6021151 \nu^{13} - 58426783 \nu^{11} + 271352435 \nu^{9} - 561779405 \nu^{7} + 316469269 \nu^{5} - 175583178 \nu^{3} - 8003320 \nu$$$$)/21623381$$ $$\beta_{14}$$ $$=$$ $$($$$$-10755599 \nu^{15} + 252302706 \nu^{13} - 2332684002 \nu^{11} + 9941641297 \nu^{9} - 16587964242 \nu^{7} + 138651525 \nu^{5} - 6407093879 \nu^{3} + 1098391515 \nu$$$$)/ 627078049$$ $$\beta_{15}$$ $$=$$ $$($$$$-1723809 \nu^{15} + 41505147 \nu^{13} - 399318104 \nu^{11} + 1835444329 \nu^{9} - 3743153672 \nu^{7} + 2039706321 \nu^{5} - 1309963695 \nu^{3} - 140493626 \nu$$$$)/86493524$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{12} - 2 \beta_{11}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 6$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{15} - \beta_{14} + 7 \beta_{12} - 10 \beta_{11} - 3 \beta_{10} + 4 \beta_{9} - \beta_{7}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{8} + 4 \beta_{6} + \beta_{5} - 4 \beta_{4} + 13 \beta_{3} + 5 \beta_{2} + 2 \beta_{1} + 29$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$30 \beta_{15} - 2 \beta_{14} - 5 \beta_{13} + 51 \beta_{12} - 43 \beta_{11} - 30 \beta_{10} + 24 \beta_{9} - 20 \beta_{7}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$13 \beta_{8} + 78 \beta_{6} + 35 \beta_{5} - 113 \beta_{4} + 231 \beta_{3} + 35 \beta_{2} + 36 \beta_{1} + 212$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$245 \beta_{15} + 7 \beta_{14} - 77 \beta_{13} + 314 \beta_{12} - 99 \beta_{11} - 217 \beta_{10} + 96 \beta_{9} - 217 \beta_{7}$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$20 \beta_{8} + 266 \beta_{6} + 194 \beta_{5} - 559 \beta_{4} + 864 \beta_{3} + 2 \beta_{2} + 178 \beta_{1} + 29$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$1728 \beta_{15} + 88 \beta_{14} - 771 \beta_{13} + 1556 \beta_{12} + 629 \beta_{11} - 1344 \beta_{10} - 128 \beta_{9} - 1868 \beta_{7}$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$-319 \beta_{8} + 2600 \beta_{6} + 3285 \beta_{5} - 8921 \beta_{4} + 11245 \beta_{3} - 1473 \beta_{2} + 3398 \beta_{1} - 8758$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$10560 \beta_{15} + 286 \beta_{14} - 5995 \beta_{13} + 4951 \beta_{12} + 12093 \beta_{11} - 7172 \beta_{10} - 7564 \beta_{9} - 14166 \beta_{7}$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$-3605 \beta_{8} + 2573 \beta_{6} + 11182 \beta_{5} - 29241 \beta_{4} + 31239 \beta_{3} - 9280 \beta_{2} + 14687 \beta_{1} - 56567$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$53248 \beta_{15} - 2884 \beta_{14} - 36868 \beta_{13} - 10003 \beta_{12} + 123074 \beta_{11} - 30784 \beta_{10} - 98032 \beta_{9} - 97052 \beta_{7}$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$-41422 \beta_{8} - 36017 \beta_{6} + 58447 \beta_{5} - 148066 \beta_{4} + 133427 \beta_{3} - 83574 \beta_{2} + 112843 \beta_{1} - 517223$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$176793 \beta_{15} - 58799 \beta_{14} - 158604 \beta_{13} - 357861 \beta_{12} + 1003586 \beta_{11} - 72873 \beta_{10} - 927572 \beta_{9} - 595917 \beta_{7}$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1889.1
 2.62616 + 0.500000i 2.62616 − 0.500000i −0.415570 − 0.500000i −0.415570 + 0.500000i −2.32849 − 0.500000i −2.32849 + 0.500000i −0.713245 − 0.500000i −0.713245 + 0.500000i 0.713245 + 0.500000i 0.713245 − 0.500000i 2.32849 + 0.500000i 2.32849 − 0.500000i 0.415570 + 0.500000i 0.415570 − 0.500000i −2.62616 − 0.500000i −2.62616 + 0.500000i
0 0 0 −2.86833 0 −2.43500 1.03478i 0 0 0
1889.2 0 0 0 −2.86833 0 −2.43500 + 1.03478i 0 0 0
1889.3 0 0 0 −2.70790 0 0.946562 2.47063i 0 0 0
1889.4 0 0 0 −2.70790 0 0.946562 + 2.47063i 0 0 0
1889.5 0 0 0 −1.10598 0 2.64465 0.0763047i 0 0 0
1889.6 0 0 0 −1.10598 0 2.64465 + 0.0763047i 0 0 0
1889.7 0 0 0 −0.465643 0 −0.656211 2.56308i 0 0 0
1889.8 0 0 0 −0.465643 0 −0.656211 + 2.56308i 0 0 0
1889.9 0 0 0 0.465643 0 −0.656211 2.56308i 0 0 0
1889.10 0 0 0 0.465643 0 −0.656211 + 2.56308i 0 0 0
1889.11 0 0 0 1.10598 0 2.64465 0.0763047i 0 0 0
1889.12 0 0 0 1.10598 0 2.64465 + 0.0763047i 0 0 0
1889.13 0 0 0 2.70790 0 0.946562 2.47063i 0 0 0
1889.14 0 0 0 2.70790 0 0.946562 + 2.47063i 0 0 0
1889.15 0 0 0 2.86833 0 −2.43500 1.03478i 0 0 0
1889.16 0 0 0 2.86833 0 −2.43500 + 1.03478i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1889.16 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
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.k.k 16
3.b odd 2 1 inner 3024.2.k.k 16
4.b odd 2 1 1512.2.k.a 16
7.b odd 2 1 inner 3024.2.k.k 16
12.b even 2 1 1512.2.k.a 16
21.c even 2 1 inner 3024.2.k.k 16
28.d even 2 1 1512.2.k.a 16
84.h odd 2 1 1512.2.k.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.k.a 16 4.b odd 2 1
1512.2.k.a 16 12.b even 2 1
1512.2.k.a 16 28.d even 2 1
1512.2.k.a 16 84.h odd 2 1
3024.2.k.k 16 1.a even 1 1 trivial
3024.2.k.k 16 3.b odd 2 1 inner
3024.2.k.k 16 7.b odd 2 1 inner
3024.2.k.k 16 21.c even 2 1 inner

## Hecke kernels

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

 $$T_{5}^{8} - 17 T_{5}^{6} + 83 T_{5}^{4} - 91 T_{5}^{2} + 16$$ $$T_{11}^{8} + 51 T_{11}^{6} + 779 T_{11}^{4} + 3801 T_{11}^{2} + 784$$ $$T_{13}^{8} + 55 T_{13}^{6} + 956 T_{13}^{4} + 5264 T_{13}^{2} + 3136$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 23 T^{2} + 273 T^{4} + 2194 T^{6} + 12806 T^{8} + 54850 T^{10} + 170625 T^{12} + 359375 T^{14} + 390625 T^{16} )^{2}$$
$7$ $$( 1 - T - 5 T^{3} - 34 T^{4} - 35 T^{5} - 343 T^{7} + 2401 T^{8} )^{2}$$
$11$ $$( 1 - 37 T^{2} + 801 T^{4} - 12446 T^{6} + 149966 T^{8} - 1505966 T^{10} + 11727441 T^{12} - 65547757 T^{14} + 214358881 T^{16} )^{2}$$
$13$ $$( 1 - 49 T^{2} + 1398 T^{4} - 28055 T^{6} + 418226 T^{8} - 4741295 T^{10} + 39928278 T^{12} - 236513641 T^{14} + 815730721 T^{16} )^{2}$$
$17$ $$( 1 + 72 T^{2} + 2652 T^{4} + 65784 T^{6} + 1248326 T^{8} + 19011576 T^{10} + 221497692 T^{12} + 1737904968 T^{14} + 6975757441 T^{16} )^{2}$$
$19$ $$( 1 - 100 T^{2} + 4914 T^{4} - 156608 T^{6} + 3515195 T^{8} - 56535488 T^{10} + 640397394 T^{12} - 4704588100 T^{14} + 16983563041 T^{16} )^{2}$$
$23$ $$( 1 - 21 T^{2} + 697 T^{4} - 11718 T^{6} + 312438 T^{8} - 6198822 T^{10} + 195049177 T^{12} - 3108753669 T^{14} + 78310985281 T^{16} )^{2}$$
$29$ $$( 1 - 172 T^{2} + 14052 T^{4} - 714740 T^{6} + 24798806 T^{8} - 601096340 T^{10} + 9938712612 T^{12} - 102309611212 T^{14} + 500246412961 T^{16} )^{2}$$
$31$ $$( 1 - 187 T^{2} + 16353 T^{4} - 883634 T^{6} + 32653478 T^{8} - 849172274 T^{10} + 15102338913 T^{12} - 165963188347 T^{14} + 852891037441 T^{16} )^{2}$$
$37$ $$( 1 + 2 T + 52 T^{2} - 100 T^{3} + 949 T^{4} - 3700 T^{5} + 71188 T^{6} + 101306 T^{7} + 1874161 T^{8} )^{4}$$
$41$ $$( 1 + 135 T^{2} + 11289 T^{4} + 646218 T^{6} + 29821310 T^{8} + 1086292458 T^{10} + 31900015929 T^{12} + 641264072535 T^{14} + 7984925229121 T^{16} )^{2}$$
$43$ $$( 1 + 2 T + 40 T^{2} - 118 T^{3} - 194 T^{4} - 5074 T^{5} + 73960 T^{6} + 159014 T^{7} + 3418801 T^{8} )^{4}$$
$47$ $$( 1 + 20 T^{2} + 4068 T^{4} + 4972 T^{6} + 7535798 T^{8} + 10983148 T^{10} + 19850542308 T^{12} + 215584306580 T^{14} + 23811286661761 T^{16} )^{2}$$
$53$ $$( 1 - 184 T^{2} + 17436 T^{4} - 1193288 T^{6} + 68587430 T^{8} - 3351945992 T^{10} + 137578426716 T^{12} - 4078242447736 T^{14} + 62259690411361 T^{16} )^{2}$$
$59$ $$( 1 + 148 T^{2} + 10276 T^{4} + 457612 T^{6} + 23078422 T^{8} + 1592947372 T^{10} + 124518001636 T^{12} + 6242718978868 T^{14} + 146830437604321 T^{16} )^{2}$$
$61$ $$( 1 - 317 T^{2} + 50098 T^{4} - 5135915 T^{6} + 370298362 T^{8} - 19110739715 T^{10} + 693648942418 T^{12} - 16331958672437 T^{14} + 191707312997281 T^{16} )^{2}$$
$67$ $$( 1 - 7 T + 114 T^{2} - 35 T^{3} + 3554 T^{4} - 2345 T^{5} + 511746 T^{6} - 2105341 T^{7} + 20151121 T^{8} )^{4}$$
$71$ $$( 1 - 341 T^{2} + 63193 T^{4} - 7537622 T^{6} + 634982182 T^{8} - 37997152502 T^{10} + 1605840357433 T^{12} - 43682196817061 T^{14} + 645753531245761 T^{16} )^{2}$$
$73$ $$( 1 - 321 T^{2} + 57726 T^{4} - 6821535 T^{6} + 583944098 T^{8} - 36351960015 T^{10} + 1639316859966 T^{12} - 48578286638769 T^{14} + 806460091894081 T^{16} )^{2}$$
$79$ $$( 1 - 11 T + 246 T^{2} - 2515 T^{3} + 26570 T^{4} - 198685 T^{5} + 1535286 T^{6} - 5423429 T^{7} + 38950081 T^{8} )^{4}$$
$83$ $$( 1 + 132 T^{2} + 12996 T^{4} + 1459932 T^{6} + 170705174 T^{8} + 10057471548 T^{10} + 616768339716 T^{12} + 43156129284708 T^{14} + 2252292232139041 T^{16} )^{2}$$
$89$ $$( 1 + 351 T^{2} + 57993 T^{4} + 6040074 T^{6} + 534549758 T^{8} + 47843426154 T^{10} + 3638610782313 T^{12} + 174440433127311 T^{14} + 3936588805702081 T^{16} )^{2}$$
$97$ $$( 1 - 285 T^{2} + 64986 T^{4} - 9080499 T^{6} + 1059084650 T^{8} - 85438415091 T^{10} + 5753163855066 T^{12} - 237397021404765 T^{14} + 7837433594376961 T^{16} )^{2}$$