# Properties

 Label 3024.2.k.l 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: $$x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} - 45 x^{12} + 306 x^{11} - 378 x^{10} + 1704 x^{9} + 917 x^{8} - 12522 x^{7} + 27090 x^{6} - 35292 x^{5} + 30948 x^{4} - 18000 x^{3} + 7200 x^{2} - 1920 x + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{14}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} + \beta_{12} q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} + \beta_{12} q^{7} -\beta_{10} q^{11} + ( \beta_{4} + \beta_{15} ) q^{13} -\beta_{5} q^{17} -\beta_{4} q^{19} -\beta_{14} q^{23} + ( 2 + \beta_{3} - \beta_{7} ) q^{25} + ( \beta_{8} + \beta_{9} - \beta_{10} + \beta_{13} - \beta_{14} ) q^{29} + ( -\beta_{4} + \beta_{6} + \beta_{11} + \beta_{12} ) q^{31} + ( -\beta_{2} + \beta_{5} + \beta_{8} - \beta_{10} + \beta_{14} ) q^{35} + ( -1 - \beta_{3} - \beta_{7} + \beta_{11} - \beta_{12} ) q^{37} + ( -\beta_{1} + 2 \beta_{5} + \beta_{8} - \beta_{9} ) q^{41} + ( -1 + \beta_{7} - \beta_{11} + \beta_{12} ) q^{43} + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{47} + ( -\beta_{3} + \beta_{4} + \beta_{11} - \beta_{15} ) q^{49} + ( -\beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{13} + \beta_{14} ) q^{53} + ( -\beta_{4} + \beta_{6} + \beta_{11} + \beta_{12} + \beta_{15} ) q^{55} + ( -\beta_{2} + \beta_{8} - \beta_{9} ) q^{59} + ( \beta_{4} - 2 \beta_{11} - 2 \beta_{12} ) q^{61} + ( \beta_{8} + \beta_{9} - \beta_{10} - \beta_{13} + 2 \beta_{14} ) q^{65} + ( -2 + \beta_{3} + \beta_{7} ) q^{67} + ( 2 \beta_{8} + 2 \beta_{9} + \beta_{13} + \beta_{14} ) q^{71} + ( 2 \beta_{4} - \beta_{6} - \beta_{11} - \beta_{12} + 3 \beta_{15} ) q^{73} + ( -\beta_{1} - 2 \beta_{2} + \beta_{8} - \beta_{10} - \beta_{14} ) q^{77} + ( -3 - \beta_{3} - \beta_{11} + \beta_{12} ) q^{79} + ( -\beta_{1} + 3 \beta_{2} - \beta_{5} + \beta_{8} - \beta_{9} ) q^{83} + ( 1 + \beta_{3} - \beta_{7} + \beta_{11} - \beta_{12} ) q^{85} + ( -3 \beta_{1} + 2 \beta_{2} - \beta_{5} + \beta_{8} - \beta_{9} ) q^{89} + ( -2 - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{15} ) q^{91} + ( -\beta_{8} - \beta_{9} - \beta_{10} - \beta_{13} ) q^{95} + ( \beta_{4} - \beta_{6} - \beta_{11} - \beta_{12} - \beta_{15} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 2q^{7} + O(q^{10})$$ $$16q + 2q^{7} + 36q^{25} - 8q^{37} - 20q^{43} + 2q^{49} - 44q^{67} - 40q^{79} + 16q^{85} - 36q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} - 45 x^{12} + 306 x^{11} - 378 x^{10} + 1704 x^{9} + 917 x^{8} - 12522 x^{7} + 27090 x^{6} - 35292 x^{5} + 30948 x^{4} - 18000 x^{3} + 7200 x^{2} - 1920 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$9625967735304871 \nu^{15} - 135922610320481566 \nu^{14} + 586463746974274502 \nu^{13} - 1443574468406641356 \nu^{12} + 1516003722014862581 \nu^{11} + 8077218483857604690 \nu^{10} - 24300598042312983470 \nu^{9} + 30275770334978045064 \nu^{8} - 110340558648795826381 \nu^{7} - 280563829185536375770 \nu^{6} + 1148647844033093893974 \nu^{5} - 1796800164972733552596 \nu^{4} + 1839630656591715546764 \nu^{3} - 1185368258007122055024 \nu^{2} + 409906873135003778880 \nu - 90521992849087319232$$$$)/ 7684522302142264288$$ $$\beta_{2}$$ $$=$$ $$($$$$63679977 \nu^{15} - 438014494 \nu^{14} + 1443496954 \nu^{13} - 3073658820 \nu^{12} - 1502246501 \nu^{11} + 23237736362 \nu^{10} - 39143035490 \nu^{9} + 117868348872 \nu^{8} - 24801774291 \nu^{7} - 909061030402 \nu^{6} + 2381970760186 \nu^{5} - 3267629790492 \nu^{4} + 2995055019604 \nu^{3} - 1800396653968 \nu^{2} + 608118237376 \nu - 128641878336$$$$)/ 24698853056$$ $$\beta_{3}$$ $$=$$ $$($$$$12896919611658579 \nu^{15} - 63670839763550169 \nu^{14} + 164374780549634918 \nu^{13} - 292350862854969940 \nu^{12} - 878506007414906579 \nu^{11} + 2985818238940592013 \nu^{10} - 1657319496483625186 \nu^{9} + 20479970216212369304 \nu^{8} + 32986629793581824055 \nu^{7} - 126785162227774315461 \nu^{6} + 209888009648871349274 \nu^{5} - 242879832966233604796 \nu^{4} + 168246773779286763616 \nu^{3} - 76254427680148237152 \nu^{2} + 23390361071012248832 \nu + 8748872216183229304$$$$)/ 1921130575535566072$$ $$\beta_{4}$$ $$=$$ $$($$$$19830533427325021 \nu^{15} - 109347424813408518 \nu^{14} + 303747509202031774 \nu^{13} - 565356612235773536 \nu^{12} - 1170314294538647425 \nu^{11} + 5505988466081180250 \nu^{10} - 4823158399067954190 \nu^{9} + 31380199749231410196 \nu^{8} + 33529795552648846041 \nu^{7} - 232097741244840672690 \nu^{6} + 425247294085105447534 \nu^{5} - 490058282289866822840 \nu^{4} + 371056209408619765668 \nu^{3} - 170642159006615384472 \nu^{2} + 53009605304413211488 \nu - 9139436721701545248$$$$)/ 1921130575535566072$$ $$\beta_{5}$$ $$=$$ $$($$$$166495347726429999 \nu^{15} - 718993239773640506 \nu^{14} + 1522520126318531510 \nu^{13} - 2051377156717453308 \nu^{12} - 14580993899421032339 \nu^{11} + 32832904336136207694 \nu^{10} + 10087746234878269970 \nu^{9} + 232014507612254250840 \nu^{8} + 586657112311307922859 \nu^{7} - 1501322878517262329270 \nu^{6} + 1406323934504147906966 \nu^{5} - 577936068553120419876 \nu^{4} - 648330092363001340596 \nu^{3} + 875864394993863957520 \nu^{2} - 354073431298430664384 \nu + 98191886152776007488$$$$)/ 15369044604284528576$$ $$\beta_{6}$$ $$=$$ $$($$$$103445914180111363 \nu^{15} - 562758338401057656 \nu^{14} + 1549632586804565914 \nu^{13} - 2866347090965366816 \nu^{12} - 6236140769180355679 \nu^{11} + 28122382173670077108 \nu^{10} - 23575723806692084026 \nu^{9} + 163419010013194163900 \nu^{8} + 186231486934515420959 \nu^{7} - 1186408888646239693836 \nu^{6} + 2151136076115084440370 \nu^{5} - 2454209646766613232680 \nu^{4} + 1862112167002892038148 \nu^{3} - 855237181615557275160 \nu^{2} + 265381678914536967264 \nu - 45979845120272208160$$$$)/ 7684522302142264288$$ $$\beta_{7}$$ $$=$$ $$($$$$-14107663960345207 \nu^{15} + 69657074983267533 \nu^{14} - 179866152511286382 \nu^{13} + 319984785621266012 \nu^{12} + 960645259076306143 \nu^{11} - 3266450045541282705 \nu^{10} + 1814833707151853938 \nu^{9} - 22399813683111549176 \nu^{8} - 36089568444875126467 \nu^{7} + 138694161338870264505 \nu^{6} - 229594268069864059746 \nu^{5} + 265675330140966521196 \nu^{4} - 184034753095094222816 \nu^{3} + 83409227248484142816 \nu^{2} - 25584805500528726272 \nu - 860136715502795952$$$$)/ 960565287767783036$$ $$\beta_{8}$$ $$=$$ $$($$$$285056008857833765 \nu^{15} - 1663183207330582570 \nu^{14} + 4839202878549981066 \nu^{13} - 9392053185843428652 \nu^{12} - 14537311885819777697 \nu^{11} + 85078276272927929526 \nu^{10} - 92360559260180901442 \nu^{9} + 467665489437841643328 \nu^{8} + 338398304276928343289 \nu^{7} - 3540699862484537086286 \nu^{6} + 7075088896396222306842 \nu^{5} - 8768944295897186435364 \nu^{4} + 7215792024074813908276 \nu^{3} - 3724989584265231860832 \nu^{2} + 1309729105070966195648 \nu - 260758962053897874240$$$$)/ 15369044604284528576$$ $$\beta_{9}$$ $$=$$ $$($$$$425315347188680289 \nu^{15} - 2264341257780881178 \nu^{14} + 6097823575132527650 \nu^{13} - 11056175268959929980 \nu^{12} - 26936081144636182573 \nu^{11} + 112446815018011162998 \nu^{10} - 82678831114224238618 \nu^{9} + 662301871168381505280 \nu^{8} + 839647715163391821573 \nu^{7} - 4796235394155472638766 \nu^{6} + 8208443510027441366546 \nu^{5} - 9170802974279233461588 \nu^{4} + 6579060527466609194276 \nu^{3} - 2927487349506788131872 \nu^{2} + 990675147416249758400 \nu - 173395593155735587392$$$$)/ 15369044604284528576$$ $$\beta_{10}$$ $$=$$ $$($$$$-247812487370796629 \nu^{15} + 1371551676913492126 \nu^{14} - 3815159516715513114 \nu^{13} + 7124583038670140172 \nu^{12} + 14487934719168682481 \nu^{11} - 69074359242044407882 \nu^{10} + 60745062947036955490 \nu^{9} - 393392250764883891936 \nu^{8} - 407759785279442730137 \nu^{7} + 2923298596889134896290 \nu^{6} - 5313429374717505153882 \nu^{5} + 6239123689728757613220 \nu^{4} - 4799054844668716030132 \nu^{3} + 2314691850042689622464 \nu^{2} - 800570606298078334400 \nu + 151122658785155736384$$$$)/ 7684522302142264288$$ $$\beta_{11}$$ $$=$$ $$($$$$-291135571090866401 \nu^{15} + 1644440103187941264 \nu^{14} - 4636454425014531870 \nu^{13} + 8715983809605597256 \nu^{12} + 16523970977232350709 \nu^{11} - 83914631234874493860 \nu^{10} + 78843857616753617150 \nu^{9} - 461702437330773630492 \nu^{8} - 433930421707137283701 \nu^{7} + 3532577508249729418284 \nu^{6} - 6588772098844568088486 \nu^{5} + 7675972396318969309952 \nu^{4} - 5826047463340864722956 \nu^{3} + 2685058836161561509704 \nu^{2} - 835637634442003547424 \nu + 150396920982869709856$$$$)/ 7684522302142264288$$ $$\beta_{12}$$ $$=$$ $$($$$$-335980506902322613 \nu^{15} + 1865845786412506188 \nu^{14} - 5208086892130408390 \nu^{13} + 9732783733294230104 \nu^{12} + 19578299564138657673 \nu^{11} - 94297253750326461456 \nu^{10} + 84608950357766551158 \nu^{9} - 532911706689001127612 \nu^{8} - 548638306586091491705 \nu^{7} + 3973440416942672673336 \nu^{6} - 7318593594721039106686 \nu^{5} + 8520502969779199259536 \nu^{4} - 6411064036377998369420 \nu^{3} + 2950204768948573461960 \nu^{2} - 916968500027046048032 \nu + 150233508340123593888$$$$)/ 7684522302142264288$$ $$\beta_{13}$$ $$=$$ $$($$$$-21499912175367 \nu^{15} + 118714615744902 \nu^{14} - 331040469619086 \nu^{13} + 619813991867076 \nu^{12} + 1253112709022811 \nu^{11} - 5960180317929130 \nu^{10} + 5331726217263510 \nu^{9} - 34284127789537632 \nu^{8} - 35996761740085251 \nu^{7} + 250693834742223922 \nu^{6} - 464511509659530078 \nu^{5} + 545001311541427884 \nu^{4} - 418925624772450876 \nu^{3} + 201983454004697152 \nu^{2} - 69828260949961536 \nu + 13175314775391168$$$$)/ 434694100132496$$ $$\beta_{14}$$ $$=$$ $$($$$$1240747642399240641 \nu^{15} - 6856827026809858242 \nu^{14} + 19103269614095631970 \nu^{13} - 35733762908496343068 \nu^{12} - 72396225461829865421 \nu^{11} + 344643919038718439934 \nu^{10} - 306400368700789644986 \nu^{9} + 1975303852222086794016 \nu^{8} + 2064376738135099866693 \nu^{7} - 14529482345426408592374 \nu^{6} + 26733031893110706133426 \nu^{5} - 31374365399252977991988 \nu^{4} + 24122384513063254660324 \nu^{3} - 11632056514056489451456 \nu^{2} + 4021991012936718836416 \nu - 759002784629311903296$$$$)/ 15369044604284528576$$ $$\beta_{15}$$ $$=$$ $$($$$$-738158905111162601 \nu^{15} + 4103720694856443000 \nu^{14} - 11459302404703427278 \nu^{13} + 21409340134734606944 \nu^{12} + 42981643978667404477 \nu^{11} - 207566655142818235692 \nu^{10} + 186530862938561573742 \nu^{9} - 1169554041448852139572 \nu^{8} - 1198325909011618735165 \nu^{7} + 8745715316358408665652 \nu^{6} - 16122917299412198591318 \nu^{5} + 18688582627295661656696 \nu^{4} - 14133556987400189799500 \nu^{3} + 6504653598592364971464 \nu^{2} - 2021950457437052554016 \nu + 347623021430852021600$$$$)/ 7684522302142264288$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{12} + 2 \beta_{11} + 2 \beta_{10} + 3 \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - 2 \beta_{4} - \beta_{2} + 4$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$-4 \beta_{14} - 3 \beta_{13} + 8 \beta_{9} - 24 \beta_{2} + 12 \beta_{1}$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$6 \beta_{15} - 4 \beta_{12} - 4 \beta_{11} + 11 \beta_{9} - 11 \beta_{8} + 2 \beta_{6} - 7 \beta_{5} + 22 \beta_{4} - 15 \beta_{2} + 8 \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$67 \beta_{15} - 112 \beta_{12} + 8 \beta_{11} + 4 \beta_{7} + 65 \beta_{6} + 128 \beta_{4} - 44 \beta_{3} + 204$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$84 \beta_{15} - 128 \beta_{14} - 4 \beta_{13} - 370 \beta_{12} + 250 \beta_{11} + 130 \beta_{10} + 185 \beta_{9} + 435 \beta_{8} + 65 \beta_{7} + 44 \beta_{6} + 65 \beta_{5} + 250 \beta_{4} - 128 \beta_{3} + 233 \beta_{2} - 128 \beta_{1} + 636$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-500 \beta_{14} - 187 \beta_{13} + 176 \beta_{10} + 796 \beta_{9} + 796 \beta_{8}$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$1090 \beta_{15} - 1768 \beta_{14} - 266 \beta_{13} + 2966 \beta_{12} - 4558 \beta_{11} + 1374 \beta_{10} + 5245 \beta_{9} + 2279 \beta_{8} - 687 \beta_{7} + 678 \beta_{6} - 687 \beta_{5} + 2966 \beta_{4} + 1768 \beta_{3} - 3223 \beta_{2} + 1768 \beta_{1} - 8588$$$$)/8$$ $$\nu^{8}$$ $$=$$ $$($$$$9979 \beta_{15} + 2712 \beta_{12} - 17760 \beta_{11} - 1356 \beta_{7} + 7817 \beta_{6} + 23184 \beta_{4} + 5932 \beta_{3} - 28012$$$$)/8$$ $$\nu^{9}$$ $$=$$ $$($$$$13880 \beta_{15} - 10236 \beta_{12} - 10236 \beta_{11} - 18053 \beta_{9} + 18053 \beta_{8} + 9304 \beta_{6} + 7817 \beta_{5} + 36106 \beta_{4} + 42321 \beta_{2} - 23184 \beta_{1}$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$-72212 \beta_{14} - 20739 \beta_{13} + 37216 \beta_{10} - 18608 \beta_{9} + 278408 \beta_{8} + 55824 \beta_{5} + 396360 \beta_{2} - 216636 \beta_{1}$$$$)/8$$ $$\nu^{11}$$ $$=$$ $$($$$$-175310 \beta_{15} - 297016 \beta_{14} - 68102 \beta_{13} + 705502 \beta_{12} - 445702 \beta_{11} + 185902 \beta_{10} + 352751 \beta_{9} + 798453 \beta_{8} - 92951 \beta_{7} - 121706 \beta_{6} + 92951 \beta_{5} - 445702 \beta_{4} + 297016 \beta_{3} + 542559 \beta_{2} - 297016 \beta_{1} - 1423612$$$$)/8$$ $$\nu^{12}$$ $$=$$ $$409507 \beta_{12} - 409507 \beta_{11} - 60853 \beta_{7} + 222851 \beta_{3} - 1060651$$ $$\nu^{13}$$ $$=$$ $$($$$$2205436 \beta_{15} + 3762880 \beta_{14} + 909452 \beta_{13} + 5542138 \beta_{12} - 8818194 \beta_{11} - 2266082 \beta_{10} - 9951235 \beta_{9} - 4409097 \beta_{8} - 1133041 \beta_{7} + 1557444 \beta_{6} + 1133041 \beta_{5} + 5542138 \beta_{4} + 3762880 \beta_{3} + 6875993 \beta_{2} - 3762880 \beta_{1} - 17998140$$$$)/8$$ $$\nu^{14}$$ $$=$$ $$($$$$11084276 \beta_{14} + 2912299 \beta_{13} - 6229776 \beta_{10} - 44295104 \beta_{9} + 3114888 \beta_{8} + 9344664 \beta_{5} + 60798800 \beta_{2} - 33252828 \beta_{1}$$$$)/8$$ $$\nu^{15}$$ $$=$$ $$($$$$-27689690 \beta_{15} + 20590108 \beta_{12} + 20590108 \beta_{11} - 34586683 \beta_{9} + 34586683 \beta_{8} - 19720302 \beta_{6} + 13996575 \beta_{5} - 69173366 \beta_{4} + 86648007 \beta_{2} - 47409992 \beta_{1}$$$$)/4$$

## 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
 0.453016 + 0.121385i 0.453016 − 0.121385i −0.651359 + 2.43091i −0.651359 − 2.43091i 0.924776 − 0.247793i 0.924776 + 0.247793i 0.916156 − 3.41914i 0.916156 + 3.41914i 3.41914 + 0.916156i 3.41914 − 0.916156i 0.247793 + 0.924776i 0.247793 − 0.924776i −2.43091 − 0.651359i −2.43091 + 0.651359i 0.121385 − 0.453016i 0.121385 + 0.453016i
0 0 0 −4.29876 0 −0.832811 2.51126i 0 0 0
1889.2 0 0 0 −4.29876 0 −0.832811 + 2.51126i 0 0 0
1889.3 0 0 0 −2.85593 0 2.30652 1.29613i 0 0 0
1889.4 0 0 0 −2.85593 0 2.30652 + 1.29613i 0 0 0
1889.5 0 0 0 −1.22223 0 1.48321 2.19091i 0 0 0
1889.6 0 0 0 −1.22223 0 1.48321 + 2.19091i 0 0 0
1889.7 0 0 0 −0.933004 0 −2.45692 0.981597i 0 0 0
1889.8 0 0 0 −0.933004 0 −2.45692 + 0.981597i 0 0 0
1889.9 0 0 0 0.933004 0 −2.45692 0.981597i 0 0 0
1889.10 0 0 0 0.933004 0 −2.45692 + 0.981597i 0 0 0
1889.11 0 0 0 1.22223 0 1.48321 2.19091i 0 0 0
1889.12 0 0 0 1.22223 0 1.48321 + 2.19091i 0 0 0
1889.13 0 0 0 2.85593 0 2.30652 1.29613i 0 0 0
1889.14 0 0 0 2.85593 0 2.30652 + 1.29613i 0 0 0
1889.15 0 0 0 4.29876 0 −0.832811 2.51126i 0 0 0
1889.16 0 0 0 4.29876 0 −0.832811 + 2.51126i 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.l 16
3.b odd 2 1 inner 3024.2.k.l 16
4.b odd 2 1 1512.2.k.b 16
7.b odd 2 1 inner 3024.2.k.l 16
12.b even 2 1 1512.2.k.b 16
21.c even 2 1 inner 3024.2.k.l 16
28.d even 2 1 1512.2.k.b 16
84.h odd 2 1 1512.2.k.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.k.b 16 4.b odd 2 1
1512.2.k.b 16 12.b even 2 1
1512.2.k.b 16 28.d even 2 1
1512.2.k.b 16 84.h odd 2 1
3024.2.k.l 16 1.a even 1 1 trivial
3024.2.k.l 16 3.b odd 2 1 inner
3024.2.k.l 16 7.b odd 2 1 inner
3024.2.k.l 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 $$S_{2}^{\mathrm{new}}(3024, [\chi])$$:

 $$T_{5}^{8} - 29 T_{5}^{6} + 215 T_{5}^{4} - 391 T_{5}^{2} + 196$$ $$T_{11}^{8} + 33 T_{11}^{6} + 380 T_{11}^{4} + 1764 T_{11}^{2} + 2704$$ $$T_{13}^{8} + 55 T_{13}^{6} + 920 T_{13}^{4} + 5036 T_{13}^{2} + 7744$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 11 T^{2} + 45 T^{4} + 34 T^{6} - 214 T^{8} + 850 T^{10} + 28125 T^{12} + 171875 T^{14} + 390625 T^{16} )^{2}$$
$7$ $$( 1 - T + 7 T^{3} + 14 T^{4} + 49 T^{5} - 343 T^{7} + 2401 T^{8} )^{2}$$
$11$ $$( 1 - 55 T^{2} + 1590 T^{4} - 29597 T^{6} + 386186 T^{8} - 3581237 T^{10} + 23279190 T^{12} - 97435855 T^{14} + 214358881 T^{16} )^{2}$$
$13$ $$( 1 - 49 T^{2} + 1362 T^{4} - 26411 T^{6} + 392258 T^{8} - 4463459 T^{10} + 38900082 T^{12} - 236513641 T^{14} + 815730721 T^{16} )^{2}$$
$17$ $$( 1 + 87 T^{2} + 3537 T^{4} + 92514 T^{6} + 1788926 T^{8} + 26736546 T^{10} + 295413777 T^{12} + 2099968503 T^{14} + 6975757441 T^{16} )^{2}$$
$19$ $$( 1 - 112 T^{2} + 5952 T^{4} - 197456 T^{6} + 4482014 T^{8} - 71281616 T^{10} + 775670592 T^{12} - 5269138672 T^{14} + 16983563041 T^{16} )^{2}$$
$23$ $$( 1 - 99 T^{2} + 5182 T^{4} - 187533 T^{6} + 5005890 T^{8} - 99204957 T^{10} + 1450136062 T^{12} - 14655553011 T^{14} + 78310985281 T^{16} )^{2}$$
$29$ $$( 1 - 103 T^{2} + 5226 T^{4} - 168029 T^{6} + 4755218 T^{8} - 141312389 T^{10} + 3696250506 T^{12} - 61266802063 T^{14} + 500246412961 T^{16} )^{2}$$
$31$ $$( 1 - 94 T^{2} + 5349 T^{4} - 230570 T^{6} + 7832744 T^{8} - 221577770 T^{10} + 4939913829 T^{12} - 83425346014 T^{14} + 852891037441 T^{16} )^{2}$$
$37$ $$( 1 + 2 T + 49 T^{2} + 410 T^{3} + 1084 T^{4} + 15170 T^{5} + 67081 T^{6} + 101306 T^{7} + 1874161 T^{8} )^{4}$$
$41$ $$( 1 + 102 T^{2} + 10017 T^{4} + 548706 T^{6} + 28065644 T^{8} + 922374786 T^{10} + 28305647937 T^{12} + 484510632582 T^{14} + 7984925229121 T^{16} )^{2}$$
$43$ $$( 1 + 5 T + 109 T^{2} + 248 T^{3} + 5122 T^{4} + 10664 T^{5} + 201541 T^{6} + 397535 T^{7} + 3418801 T^{8} )^{4}$$
$47$ $$( 1 + 278 T^{2} + 35193 T^{4} + 2749546 T^{6} + 150940940 T^{8} + 6073747114 T^{10} + 171730613433 T^{12} + 2996621861462 T^{14} + 23811286661761 T^{16} )^{2}$$
$53$ $$( 1 - 142 T^{2} + 13833 T^{4} - 1028534 T^{6} + 58011620 T^{8} - 2889152006 T^{10} + 109149023673 T^{12} - 3147339280318 T^{14} + 62259690411361 T^{16} )^{2}$$
$59$ $$( 1 + 283 T^{2} + 41785 T^{4} + 4061626 T^{6} + 280646974 T^{8} + 14138520106 T^{10} + 506323929385 T^{12} + 11937091020403 T^{14} + 146830437604321 T^{16} )^{2}$$
$61$ $$( 1 - 236 T^{2} + 32440 T^{4} - 3088964 T^{6} + 216205582 T^{8} - 11494035044 T^{10} + 449159082040 T^{12} - 12158808349196 T^{14} + 191707312997281 T^{16} )^{2}$$
$67$ $$( 1 + 11 T + 198 T^{2} + 2071 T^{3} + 17498 T^{4} + 138757 T^{5} + 888822 T^{6} + 3308393 T^{7} + 20151121 T^{8} )^{4}$$
$71$ $$( 1 - 167 T^{2} + 18034 T^{4} - 1223801 T^{6} + 87222970 T^{8} - 6169180841 T^{10} + 458274255154 T^{12} - 21392747414807 T^{14} + 645753531245761 T^{16} )^{2}$$
$73$ $$( 1 - 117 T^{2} + 15114 T^{4} - 1400907 T^{6} + 111599882 T^{8} - 7465433403 T^{10} + 429211014474 T^{12} - 17706104475813 T^{14} + 806460091894081 T^{16} )^{2}$$
$79$ $$( 1 + 10 T + 261 T^{2} + 1850 T^{3} + 28244 T^{4} + 146150 T^{5} + 1628901 T^{6} + 4930390 T^{7} + 38950081 T^{8} )^{4}$$
$83$ $$( 1 + 339 T^{2} + 56745 T^{4} + 6361962 T^{6} + 567506462 T^{8} + 43827556218 T^{10} + 2693022425145 T^{12} + 110832786572091 T^{14} + 2252292232139041 T^{16} )^{2}$$
$89$ $$( 1 + 321 T^{2} + 55134 T^{4} + 6909375 T^{6} + 691071074 T^{8} + 54729159375 T^{10} + 3459230715294 T^{12} + 159530994398481 T^{14} + 3936588805702081 T^{16} )^{2}$$
$97$ $$( 1 - 537 T^{2} + 140634 T^{4} - 23455755 T^{6} + 2707648082 T^{8} - 220695198795 T^{10} + 12450226904154 T^{12} - 447305966646873 T^{14} + 7837433594376961 T^{16} )^{2}$$