# Properties

 Label 1050.3.f.d Level $1050$ Weight $3$ Character orbit 1050.f Analytic conductor $28.610$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1050.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.6104277578$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 2 x^{11} + 103 x^{10} + 14 x^{9} + 7600 x^{8} + 1474 x^{7} + 229961 x^{6} + 442438 x^{5} + 4765612 x^{4} + 4021754 x^{3} + 21002899 x^{2} - 8701742 x + 69739201$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{2} + \beta_{2} q^{3} + 2 q^{4} -\beta_{4} q^{6} + ( 1 - \beta_{3} ) q^{7} -2 \beta_{5} q^{8} -3 q^{9} +O(q^{10})$$ $$q -\beta_{5} q^{2} + \beta_{2} q^{3} + 2 q^{4} -\beta_{4} q^{6} + ( 1 - \beta_{3} ) q^{7} -2 \beta_{5} q^{8} -3 q^{9} + ( -1 - 2 \beta_{5} + \beta_{6} + \beta_{8} ) q^{11} + 2 \beta_{2} q^{12} + ( -\beta_{4} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{13} + ( 1 + \beta_{2} - \beta_{5} + \beta_{11} ) q^{14} + 4 q^{16} + ( -2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{17} + 3 \beta_{5} q^{18} + ( -4 \beta_{2} + \beta_{4} - \beta_{7} - \beta_{9} ) q^{19} + ( -1 + \beta_{1} + \beta_{2} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{21} + ( 5 - \beta_{1} + \beta_{3} + \beta_{6} - \beta_{10} + \beta_{11} ) q^{22} + ( 1 - 2 \beta_{1} - 4 \beta_{5} + \beta_{6} + \beta_{8} ) q^{23} -2 \beta_{4} q^{24} + ( 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{8} - \beta_{9} ) q^{26} -3 \beta_{2} q^{27} + ( 2 - 2 \beta_{3} ) q^{28} + ( 3 + 4 \beta_{1} + 2 \beta_{3} + 5 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{29} + ( 8 \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{31} -4 \beta_{5} q^{32} + ( -\beta_{2} - 2 \beta_{3} + \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{33} + ( 2 \beta_{3} + 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{34} -6 q^{36} + ( -5 + 2 \beta_{1} + \beta_{3} + 2 \beta_{5} - \beta_{8} - 2 \beta_{10} + 2 \beta_{11} ) q^{37} + ( -2 \beta_{2} + 4 \beta_{4} + 2 \beta_{7} + \beta_{9} ) q^{38} + ( -1 + \beta_{1} + 3 \beta_{3} + 2 \beta_{6} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{39} + ( -2 \beta_{3} - 2 \beta_{4} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{41} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} + \beta_{10} ) q^{42} + ( 10 - 3 \beta_{1} - \beta_{3} - 6 \beta_{5} - 3 \beta_{6} - 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{43} + ( -2 - 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} ) q^{44} + ( 7 + \beta_{1} + 3 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} - \beta_{10} + \beta_{11} ) q^{46} + ( 6 \beta_{2} - \beta_{3} + 5 \beta_{4} - 3 \beta_{7} - \beta_{8} - 3 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{47} + 4 \beta_{2} q^{48} + ( 8 + 3 \beta_{1} + 6 \beta_{2} - \beta_{4} - 10 \beta_{5} - \beta_{6} - 3 \beta_{7} - 4 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{49} + ( 7 - \beta_{1} + 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - \beta_{10} + \beta_{11} ) q^{51} + ( -2 \beta_{4} + 2 \beta_{7} - 2 \beta_{10} - 2 \beta_{11} ) q^{52} + ( -14 - 4 \beta_{1} - 6 \beta_{3} + 16 \beta_{5} - 4 \beta_{6} + 2 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} ) q^{53} + 3 \beta_{4} q^{54} + ( 2 + 2 \beta_{2} - 2 \beta_{5} + 2 \beta_{11} ) q^{56} + ( 10 + 2 \beta_{1} - 3 \beta_{5} + \beta_{6} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{57} + ( -9 - 5 \beta_{1} - 5 \beta_{3} - 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{8} + \beta_{10} - \beta_{11} ) q^{58} + ( 2 \beta_{2} + 5 \beta_{3} - 13 \beta_{4} - \beta_{7} + 5 \beta_{8} - \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{59} + ( -12 \beta_{2} - 3 \beta_{3} - 16 \beta_{4} + 4 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} ) q^{61} + ( -6 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{7} - 4 \beta_{8} + \beta_{10} + \beta_{11} ) q^{62} + ( -3 + 3 \beta_{3} ) q^{63} + 8 q^{64} + ( 4 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{66} + ( 2 + 3 \beta_{1} + 3 \beta_{3} - 22 \beta_{5} + 5 \beta_{6} + 2 \beta_{8} - 3 \beta_{10} + 3 \beta_{11} ) q^{67} + ( -4 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{10} - 2 \beta_{11} ) q^{68} + ( -\beta_{2} - 4 \beta_{3} - \beta_{7} - 4 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{69} + ( 23 + 2 \beta_{1} + 6 \beta_{3} + 7 \beta_{6} + \beta_{8} - 4 \beta_{10} + 4 \beta_{11} ) q^{71} + 6 \beta_{5} q^{72} + ( -16 \beta_{2} - \beta_{3} + 21 \beta_{4} + 3 \beta_{7} - \beta_{8} + 4 \beta_{9} - 5 \beta_{10} - 5 \beta_{11} ) q^{73} + ( -4 - 2 \beta_{1} - 6 \beta_{3} + 9 \beta_{5} - 2 \beta_{6} + 4 \beta_{8} + \beta_{10} - \beta_{11} ) q^{74} + ( -8 \beta_{2} + 2 \beta_{4} - 2 \beta_{7} - 2 \beta_{9} ) q^{76} + ( 33 + 2 \beta_{1} + 12 \beta_{2} + 5 \beta_{3} - 9 \beta_{4} + 7 \beta_{5} + 5 \beta_{6} - 3 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 6 \beta_{11} ) q^{77} + ( -3 - 3 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{8} + \beta_{10} - \beta_{11} ) q^{78} + ( -18 - \beta_{1} + 2 \beta_{3} - 6 \beta_{5} + \beta_{6} - \beta_{8} + \beta_{10} - \beta_{11} ) q^{79} + 9 q^{81} + ( 8 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{82} + ( 20 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{8} + 6 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{83} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{7} - 2 \beta_{8} ) q^{84} + ( 8 + 6 \beta_{1} - 2 \beta_{3} - 5 \beta_{5} + 2 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} ) q^{86} + ( 7 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} + \beta_{7} + 4 \beta_{8} - \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{87} + ( 10 - 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{6} - 2 \beta_{10} + 2 \beta_{11} ) q^{88} + ( -18 \beta_{2} - \beta_{3} + 23 \beta_{4} - 3 \beta_{7} - \beta_{8} + 5 \beta_{10} + 5 \beta_{11} ) q^{89} + ( -13 + 3 \beta_{1} + 24 \beta_{2} + 7 \beta_{3} - 4 \beta_{4} - 30 \beta_{5} + 3 \beta_{6} + 7 \beta_{8} - 4 \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{91} + ( 2 - 4 \beta_{1} - 8 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} ) q^{92} + ( -24 - \beta_{3} - 8 \beta_{5} - 2 \beta_{6} - \beta_{8} + 4 \beta_{10} - 4 \beta_{11} ) q^{93} + ( -8 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + 6 \beta_{7} - 4 \beta_{8} + 3 \beta_{9} + \beta_{10} + \beta_{11} ) q^{94} -4 \beta_{4} q^{96} + ( -16 \beta_{2} + 13 \beta_{3} - 24 \beta_{4} - 4 \beta_{7} + 13 \beta_{8} - 8 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{97} + ( 19 - 2 \beta_{1} + 5 \beta_{2} - 6 \beta_{3} - 4 \beta_{4} - 7 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + 4 \beta_{10} - \beta_{11} ) q^{98} + ( 3 + 6 \beta_{5} - 3 \beta_{6} - 3 \beta_{8} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 24 q^{4} + 10 q^{7} - 36 q^{9} + O(q^{10})$$ $$12 q + 24 q^{4} + 10 q^{7} - 36 q^{9} - 16 q^{11} + 8 q^{14} + 48 q^{16} - 6 q^{21} + 48 q^{22} + 20 q^{28} + 48 q^{29} - 72 q^{36} - 64 q^{37} - 12 q^{39} - 24 q^{42} + 108 q^{43} - 32 q^{44} + 80 q^{46} + 118 q^{49} + 72 q^{51} - 176 q^{53} + 16 q^{56} + 132 q^{57} - 128 q^{58} - 30 q^{63} + 96 q^{64} + 4 q^{67} + 248 q^{71} - 64 q^{74} + 396 q^{77} - 48 q^{78} - 208 q^{79} + 108 q^{81} - 12 q^{84} + 128 q^{86} + 96 q^{88} - 158 q^{91} - 252 q^{93} + 240 q^{98} + 48 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} + 103 x^{10} + 14 x^{9} + 7600 x^{8} + 1474 x^{7} + 229961 x^{6} + 442438 x^{5} + 4765612 x^{4} + 4021754 x^{3} + 21002899 x^{2} - 8701742 x + 69739201$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-123263485324786117 \nu^{11} + 392128266733045979 \nu^{10} - 6083651291835521143 \nu^{9} + 10019123877835491302 \nu^{8} - 876689117601446842783 \nu^{7} + 1923228388418524380623 \nu^{6} - 24193492971107383597753 \nu^{5} - 36430399774596190914310 \nu^{4} - 1635592998788363776126285 \nu^{3} - 27765130373466532370125 \nu^{2} - 318802697202801394837807 \nu - 43640716670021836400283934$$$$)/$$$$25\!\cdots\!10$$ $$\beta_{2}$$ $$=$$ $$($$$$-24280460053617088 \nu^{11} - 134114608273836392 \nu^{10} - 2083185333447793936 \nu^{9} - 18454165661498903902 \nu^{8} - 182261081586826460528 \nu^{7} - 1361584676095656827128 \nu^{6} - 5339499743496833485408 \nu^{5} - 48009258941143298532488 \nu^{4} - 178759842954027859325680 \nu^{3} - 904631032972096968422224 \nu^{2} - 587388760026044469925040 \nu - 1838092903954916334438895$$$$)/$$$$15\!\cdots\!63$$ $$\beta_{3}$$ $$=$$ $$($$$$870434615881970501662 \nu^{11} - 19661721884260722084800 \nu^{10} + 137965359060520615834218 \nu^{9} - 1691949821640399306610901 \nu^{8} + 6989856336202081755196606 \nu^{7} - 115764395641859048152726716 \nu^{6} + 223776682672204235096323516 \nu^{5} - 2390725121378167281451853993 \nu^{4} - 3753254696646455950781278656 \nu^{3} - 35335846019660968629987877022 \nu^{2} - 20929296682707593633327271992 \nu + 151002372998714947812646496505$$$$)/$$$$30\!\cdots\!30$$ $$\beta_{4}$$ $$=$$ $$($$$$-958718783 \nu^{11} + 2012568579 \nu^{10} - 103184081077 \nu^{9} - 19016500460 \nu^{8} - 7700992137321 \nu^{7} - 2045621186539 \nu^{6} - 255131073706781 \nu^{5} - 480510667159236 \nu^{4} - 5530500971596237 \nu^{3} - 4615659330484469 \nu^{2} - 42759904104131991 \nu + 2230068709179416$$$$)/ 27944424384218370$$ $$\beta_{5}$$ $$=$$ $$($$$$-397109320631 \nu^{11} + 851720091837 \nu^{10} - 39749913778739 \nu^{9} + 602434788466 \nu^{8} - 2891227107455349 \nu^{7} + 246516127760419 \nu^{6} - 81498625698911219 \nu^{5} - 140179083439640610 \nu^{4} - 1555747296135356705 \nu^{3} - 180817370957826635 \nu^{2} - 1012034368291331301 \nu + 5658330747746351878$$$$)/ 9027254980906654590$$ $$\beta_{6}$$ $$=$$ $$($$$$1425595836696379005232 \nu^{11} + 14092435823463248208082 \nu^{10} + 81145718752682366682358 \nu^{9} + 1638844517269036590365857 \nu^{8} + 9622073376860387200747180 \nu^{7} + 109652132417521385473933360 \nu^{6} + 241865969125713739141124170 \nu^{5} + 3165481303312438061806001293 \nu^{4} + 15859452697935593694875889196 \nu^{3} + 36238170969129129588544665232 \nu^{2} + 26804283133536654346546614466 \nu - 82177159186569453925937146247$$$$)/$$$$30\!\cdots\!30$$ $$\beta_{7}$$ $$=$$ $$($$$$479599577738276162044 \nu^{11} + 4581533870192141943359 \nu^{10} + 31703744785534734998446 \nu^{9} + 594082871930616765858982 \nu^{8} + 3106794269092375834407416 \nu^{7} + 42468146586181866049029865 \nu^{6} + 71780391763685510164526380 \nu^{5} + 1462452102116231762072350358 \nu^{4} + 3494795868521851591438683226 \nu^{3} + 22637224673600899779077464483 \nu^{2} + 4657181835408061333061384984 \nu + 51081160874206482645502640890$$$$)/$$$$60\!\cdots\!26$$ $$\beta_{8}$$ $$=$$ $$($$$$-4064494901406269284530 \nu^{11} + 7071227722139188880729 \nu^{10} - 373771122968762467485370 \nu^{9} - 441638830945824481698849 \nu^{8} - 26316498220073314131955402 \nu^{7} - 31011757701529485245679823 \nu^{6} - 649487792104519501263103812 \nu^{5} - 3237621847048438120387772353 \nu^{4} - 12185598879260007507946308796 \nu^{3} - 33997940579994499818015511497 \nu^{2} - 34095696438471300916231012624 \nu - 301535343855974745875761142929$$$$)/$$$$30\!\cdots\!30$$ $$\beta_{9}$$ $$=$$ $$($$$$4996870264807970033459 \nu^{11} - 24909673939090560254427 \nu^{10} + 614650049165745341243881 \nu^{9} - 1689480727374246126148660 \nu^{8} + 44569375117050885957261333 \nu^{7} - 107758569547000015016723303 \nu^{6} + 1594353621457658439998973233 \nu^{5} - 1312860206232351853413466992 \nu^{4} + 29466418332641001110672152321 \nu^{3} - 12786422963717956367807939263 \nu^{2} + 279054187041523986390662976003 \nu - 125559989500555004928761983118$$$$)/$$$$30\!\cdots\!30$$ $$\beta_{10}$$ $$=$$ $$($$$$12114535344353879769274 \nu^{11} - 31661184705508755023301 \nu^{10} + 1167950433688588177255661 \nu^{9} - 160552294436965630401311 \nu^{8} + 82032580996509421249270200 \nu^{7} - 16676233201020924642674755 \nu^{6} + 2091465168597897250415570425 \nu^{5} + 5106708557942875782332028921 \nu^{4} + 35499390944691116110191991892 \nu^{3} + 10211156922368773869357314209 \nu^{2} - 41638296250866175510856043243 \nu - 37695005600102966351655342469$$$$)/$$$$30\!\cdots\!30$$ $$\beta_{11}$$ $$=$$ $$($$$$-13638743814637698305053 \nu^{11} + 48630342634658011184448 \nu^{10} - 1438248795267909007122842 \nu^{9} + 1806872436070400365299251 \nu^{8} - 101398587879224661405233373 \nu^{7} + 119795238669744901527476458 \nu^{6} - 2977379569085116114114088218 \nu^{5} - 2595294633816151227096427689 \nu^{4} - 49625100550412998623053677493 \nu^{3} + 4047185815529059934062126334 \nu^{2} - 108185608722019779219114798960 \nu + 131065364304835669887379586707$$$$)/$$$$30\!\cdots\!30$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{11} + \beta_{10} + \beta_{9} - \beta_{6} + \beta_{5} - \beta_{3} - \beta_{1} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{11} + \beta_{10} - \beta_{9} - 4 \beta_{7} + \beta_{6} - 10 \beta_{5} - 9 \beta_{4} + 5 \beta_{3} - 33 \beta_{2} + \beta_{1} - 34$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$6 \beta_{11} - 6 \beta_{10} - 8 \beta_{8} + 51 \beta_{6} - 7 \beta_{5} + 59 \beta_{3} + 77 \beta_{1} - 133$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-78 \beta_{11} - 234 \beta_{10} + 78 \beta_{9} - 280 \beta_{8} + 240 \beta_{7} + 78 \beta_{6} - 916 \beta_{5} + 858 \beta_{4} + 58 \beta_{3} + 1747 \beta_{2} + 98 \beta_{1} - 1845$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-3935 \beta_{11} - 2943 \beta_{10} - 1951 \beta_{9} - 2484 \beta_{8} - 972 \beta_{7} - 2943 \beta_{6} + 2055 \beta_{5} + 1596 \beta_{4} - 6939 \beta_{3} + 4824 \beta_{2} - 5427 \beta_{1} + 10251$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$-5557 \beta_{11} + 5557 \beta_{10} + 8488 \beta_{8} - 5413 \beta_{6} + 68046 \beta_{5} - 13901 \beta_{3} - 8681 \beta_{1} + 113528$$ $$\nu^{7}$$ $$=$$ $$($$$$183289 \beta_{11} + 264461 \beta_{10} + 102117 \beta_{9} + 316554 \beta_{8} + 70042 \beta_{7} - 183289 \beta_{6} + 277901 \beta_{5} - 287910 \beta_{4} + 10009 \beta_{3} - 373184 \beta_{2} - 376587 \beta_{1} + 749771$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$1153764 \beta_{11} + 379548 \beta_{10} - 394668 \beta_{9} + 341320 \beta_{8} - 797976 \beta_{7} + 379548 \beta_{6} - 4815128 \beta_{5} - 4776900 \beta_{4} + 1860164 \beta_{3} - 6610945 \beta_{2} + 720868 \beta_{1} - 7331813$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$3248552 \beta_{11} - 3248552 \beta_{10} - 4810800 \beta_{8} + 11912105 \beta_{6} - 27500585 \beta_{5} + 16722905 \beta_{3} + 26063681 \beta_{1} - 55482497$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-27264133 \beta_{11} - 80953743 \beta_{10} + 26425477 \beta_{9} - 107767100 \beta_{8} + 47508652 \beta_{7} + 27264133 \beta_{6} - 335703962 \beta_{5} + 338569053 \beta_{4} - 2865091 \beta_{3} + 428751705 \beta_{2} + 57393357 \beta_{1} - 486145062$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-1301713519 \beta_{11} - 794670131 \beta_{10} - 287626743 \beta_{9} - 1011637682 \beta_{8} - 274895074 \beta_{7} - 794670131 \beta_{6} + 2391949767 \beta_{5} + 2608917318 \beta_{4} - 2543050421 \beta_{3} + 2342795960 \beta_{2} - 1806307813 \beta_{1} + 4149103773$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
601.1
 0.854122 + 1.47938i 4.23503 + 7.33529i −3.88205 − 6.72390i 0.854122 − 1.47938i 4.23503 − 7.33529i −3.88205 + 6.72390i −1.37625 − 2.38373i −2.07881 − 3.60061i 3.24795 + 5.62562i −1.37625 + 2.38373i −2.07881 + 3.60061i 3.24795 − 5.62562i
−1.41421 1.73205i 2.00000 0 2.44949i −6.06258 + 3.49930i −2.82843 −3.00000 0
601.2 −1.41421 1.73205i 2.00000 0 2.44949i 2.03595 6.69738i −2.82843 −3.00000 0
601.3 −1.41421 1.73205i 2.00000 0 2.44949i 5.11242 + 4.78154i −2.82843 −3.00000 0
601.4 −1.41421 1.73205i 2.00000 0 2.44949i −6.06258 3.49930i −2.82843 −3.00000 0
601.5 −1.41421 1.73205i 2.00000 0 2.44949i 2.03595 + 6.69738i −2.82843 −3.00000 0
601.6 −1.41421 1.73205i 2.00000 0 2.44949i 5.11242 4.78154i −2.82843 −3.00000 0
601.7 1.41421 1.73205i 2.00000 0 2.44949i −6.84490 + 1.46536i 2.82843 −3.00000 0
601.8 1.41421 1.73205i 2.00000 0 2.44949i 3.84151 5.85173i 2.82843 −3.00000 0
601.9 1.41421 1.73205i 2.00000 0 2.44949i 6.91761 + 1.07086i 2.82843 −3.00000 0
601.10 1.41421 1.73205i 2.00000 0 2.44949i −6.84490 1.46536i 2.82843 −3.00000 0
601.11 1.41421 1.73205i 2.00000 0 2.44949i 3.84151 + 5.85173i 2.82843 −3.00000 0
601.12 1.41421 1.73205i 2.00000 0 2.44949i 6.91761 1.07086i 2.82843 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 601.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.f.d yes 12
5.b even 2 1 1050.3.f.c 12
5.c odd 4 2 1050.3.h.c 24
7.b odd 2 1 inner 1050.3.f.d yes 12
35.c odd 2 1 1050.3.f.c 12
35.f even 4 2 1050.3.h.c 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.3.f.c 12 5.b even 2 1
1050.3.f.c 12 35.c odd 2 1
1050.3.f.d yes 12 1.a even 1 1 trivial
1050.3.f.d yes 12 7.b odd 2 1 inner
1050.3.h.c 24 5.c odd 4 2
1050.3.h.c 24 35.f even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1050, [\chi])$$:

 $$T_{11}^{6} + 8 T_{11}^{5} - 462 T_{11}^{4} - 4780 T_{11}^{3} + 27217 T_{11}^{2} + 422940 T_{11} + 1141308$$ $$T_{23}^{6} - 1294 T_{23}^{4} - 7740 T_{23}^{3} + 320689 T_{23}^{2} + 1942260 T_{23} - 14928228$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T^{2} )^{6}$$
$3$ $$( 3 + T^{2} )^{6}$$
$5$ $$T^{12}$$
$7$ $$13841287201 - 2824752490 T - 51883209 T^{2} + 111766550 T^{3} - 13649685 T^{4} - 1323980 T^{5} + 455994 T^{6} - 27020 T^{7} - 5685 T^{8} + 950 T^{9} - 9 T^{10} - 10 T^{11} + T^{12}$$
$11$ $$( 1141308 + 422940 T + 27217 T^{2} - 4780 T^{3} - 462 T^{4} + 8 T^{5} + T^{6} )^{2}$$
$13$ $$6029833104 + 17985623616 T^{2} + 4178588056 T^{4} + 69725976 T^{6} + 422609 T^{8} + 1086 T^{10} + T^{12}$$
$17$ $$33708960000 + 16778793600 T^{2} + 2196124816 T^{4} + 70118752 T^{6} + 520728 T^{8} + 1312 T^{10} + T^{12}$$
$19$ $$7157315699856 + 599825028480 T^{2} + 15981265432 T^{4} + 177793608 T^{6} + 870689 T^{8} + 1758 T^{10} + T^{12}$$
$23$ $$( -14928228 + 1942260 T + 320689 T^{2} - 7740 T^{3} - 1294 T^{4} + T^{6} )^{2}$$
$29$ $$( -186322500 - 12421500 T + 1327825 T^{2} + 54420 T^{3} - 2578 T^{4} - 24 T^{5} + T^{6} )^{2}$$
$31$ $$50797435981824 + 175331220559872 T^{2} + 5312986809472 T^{4} + 17908946208 T^{6} + 18307913 T^{8} + 7302 T^{10} + T^{12}$$
$37$ $$( 348454564 + 54469748 T + 1447065 T^{2} - 91388 T^{3} - 2838 T^{4} + 32 T^{5} + T^{6} )^{2}$$
$41$ $$2445149009608704 + 1704631335321600 T^{2} + 15577851277312 T^{4} + 41147664384 T^{6} + 33864896 T^{8} + 10416 T^{10} + T^{12}$$
$43$ $$( 822485377 - 46657694 T - 6876245 T^{2} + 420652 T^{3} - 5221 T^{4} - 54 T^{5} + T^{6} )^{2}$$
$47$ $$214949334924658944 + 26042245174774656 T^{2} + 89706769400464 T^{4} + 112819414944 T^{6} + 62528408 T^{8} + 14400 T^{10} + T^{12}$$
$53$ $$( 2495949696 + 21556992 T - 14879408 T^{2} - 589024 T^{3} - 4144 T^{4} + 88 T^{5} + T^{6} )^{2}$$
$59$ $$10372454711215720704 + 75670621026857856 T^{2} + 158049917377168 T^{4} + 142137905056 T^{6} + 61738392 T^{8} + 12736 T^{10} + T^{12}$$
$61$ $$28\!\cdots\!56$$$$+ 7458559796026152960 T^{2} + 5865892919604352 T^{4} + 2040587434272 T^{6} + 354099929 T^{8} + 30102 T^{10} + T^{12}$$
$67$ $$( -372707559 - 18596538 T + 7865787 T^{2} - 178044 T^{3} - 11309 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$71$ $$( -37762911432 - 968882016 T + 32495017 T^{2} + 762244 T^{3} - 8194 T^{4} - 124 T^{5} + T^{6} )^{2}$$
$73$ $$62\!\cdots\!44$$$$+ 48837556611592401408 T^{2} + 30156224375725072 T^{4} + 7339482832352 T^{6} + 856218888 T^{8} + 47528 T^{10} + T^{12}$$
$79$ $$( -128832956 - 51994180 T - 4985703 T^{2} - 143444 T^{3} + 1206 T^{4} + 104 T^{5} + T^{6} )^{2}$$
$83$ $$38\!\cdots\!96$$$$+ 64145943779730259968 T^{2} + 35526201938132992 T^{4} + 8565231243264 T^{6} + 970083728 T^{8} + 50904 T^{10} + T^{12}$$
$89$ $$21\!\cdots\!76$$$$+ 4187346160551077376 T^{2} + 6718843303660816 T^{4} + 3107873882784 T^{6} + 573717320 T^{8} + 43128 T^{10} + T^{12}$$
$97$ $$87\!\cdots\!56$$$$+$$$$48\!\cdots\!80$$$$T^{2} + 1127102337696340096 T^{4} + 104781616488608 T^{6} + 4827816729 T^{8} + 110294 T^{10} + T^{12}$$