Properties

Label 1078.2.c.b
Level $1078$
Weight $2$
Character orbit 1078.c
Analytic conductor $8.608$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1078.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.60787333789\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} - 102 x^{7} + 144 x^{6} - 432 x^{5} + 502 x^{4} + 288 x^{3} + 72 x^{2} + 12 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 154)
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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} + \beta_{14} q^{3} - q^{4} + ( -\beta_{8} + \beta_{11} ) q^{5} -\beta_{13} q^{6} -\beta_{9} q^{8} + ( -2 - \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{9} q^{2} + \beta_{14} q^{3} - q^{4} + ( -\beta_{8} + \beta_{11} ) q^{5} -\beta_{13} q^{6} -\beta_{9} q^{8} + ( -2 - \beta_{2} ) q^{9} + ( \beta_{6} + \beta_{10} ) q^{10} + ( -1 - \beta_{3} + \beta_{9} - \beta_{12} ) q^{11} -\beta_{14} q^{12} + ( -\beta_{4} - \beta_{10} ) q^{13} + ( -2 - 3 \beta_{1} ) q^{15} + q^{16} + ( 2 \beta_{4} - \beta_{6} - 2 \beta_{10} - \beta_{13} ) q^{17} + ( -2 \beta_{9} - \beta_{15} ) q^{18} + ( -\beta_{4} + 2 \beta_{6} + \beta_{10} - \beta_{13} ) q^{19} + ( \beta_{8} - \beta_{11} ) q^{20} + ( \beta_{1} + \beta_{7} - \beta_{9} ) q^{22} + ( -2 + \beta_{2} - \beta_{3} ) q^{23} + \beta_{13} q^{24} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{25} + ( \beta_{5} + \beta_{11} ) q^{26} + ( -3 \beta_{5} - 2 \beta_{8} + 4 \beta_{11} - \beta_{14} ) q^{27} + ( \beta_{7} - 2 \beta_{9} + \beta_{12} ) q^{29} + ( \beta_{9} - 3 \beta_{12} ) q^{30} + ( -2 \beta_{5} - 3 \beta_{8} + 2 \beta_{11} - \beta_{14} ) q^{31} + \beta_{9} q^{32} + ( -\beta_{5} - 2 \beta_{6} + \beta_{8} - 3 \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} ) q^{33} + ( -2 \beta_{5} - \beta_{8} + 2 \beta_{11} - \beta_{14} ) q^{34} + ( 2 + \beta_{2} ) q^{36} + ( 2 - \beta_{3} ) q^{37} + ( \beta_{5} + 2 \beta_{8} - \beta_{11} - \beta_{14} ) q^{38} + ( \beta_{7} - \beta_{9} + 4 \beta_{12} + \beta_{15} ) q^{39} + ( -\beta_{6} - \beta_{10} ) q^{40} + ( -2 \beta_{4} + \beta_{10} - 3 \beta_{13} ) q^{41} + ( 3 \beta_{7} + \beta_{9} + 3 \beta_{12} + \beta_{15} ) q^{43} + ( 1 + \beta_{3} - \beta_{9} + \beta_{12} ) q^{44} + ( 3 \beta_{8} - 6 \beta_{11} + \beta_{14} ) q^{45} + ( \beta_{7} - 2 \beta_{9} + \beta_{15} ) q^{46} -2 \beta_{5} q^{47} + \beta_{14} q^{48} + ( \beta_{9} - 2 \beta_{12} - \beta_{15} ) q^{50} + ( \beta_{7} - 7 \beta_{9} + \beta_{12} - 3 \beta_{15} ) q^{51} + ( \beta_{4} + \beta_{10} ) q^{52} + ( 4 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{53} + ( -3 \beta_{4} + 2 \beta_{6} + 4 \beta_{10} + \beta_{13} ) q^{54} + ( -\beta_{4} + \beta_{5} + 2 \beta_{8} - 3 \beta_{11} + 2 \beta_{13} - \beta_{14} ) q^{55} + ( \beta_{7} - 4 \beta_{9} - 2 \beta_{12} ) q^{57} + ( 1 - \beta_{1} + \beta_{3} ) q^{58} + ( \beta_{5} + 2 \beta_{11} + 3 \beta_{14} ) q^{59} + ( 2 + 3 \beta_{1} ) q^{60} + ( -2 \beta_{4} - 3 \beta_{6} - \beta_{10} - \beta_{13} ) q^{61} + ( -2 \beta_{4} + 3 \beta_{6} + 2 \beta_{10} + \beta_{13} ) q^{62} - q^{64} + ( \beta_{7} + 2 \beta_{9} + 4 \beta_{12} + 2 \beta_{15} ) q^{65} + ( -\beta_{4} - \beta_{6} - 2 \beta_{8} + \beta_{10} + 3 \beta_{11} + \beta_{13} - \beta_{14} ) q^{66} + ( -1 + \beta_{1} + 2 \beta_{3} ) q^{67} + ( -2 \beta_{4} + \beta_{6} + 2 \beta_{10} + \beta_{13} ) q^{68} + ( 2 \beta_{5} + 3 \beta_{8} - 3 \beta_{11} ) q^{69} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{71} + ( 2 \beta_{9} + \beta_{15} ) q^{72} + ( 2 \beta_{4} - \beta_{6} - 5 \beta_{10} ) q^{73} + ( \beta_{7} + 2 \beta_{9} ) q^{74} + ( -3 \beta_{5} + 2 \beta_{8} - 2 \beta_{11} - \beta_{14} ) q^{75} + ( \beta_{4} - 2 \beta_{6} - \beta_{10} + \beta_{13} ) q^{76} + ( -3 - 4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{78} + ( \beta_{7} - 3 \beta_{9} + 2 \beta_{12} + 3 \beta_{15} ) q^{79} + ( -\beta_{8} + \beta_{11} ) q^{80} + ( -1 - 4 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{81} + ( 2 \beta_{5} - \beta_{11} - 3 \beta_{14} ) q^{82} + ( -2 \beta_{4} - \beta_{6} + 2 \beta_{10} - 3 \beta_{13} ) q^{83} + ( -2 \beta_{7} + 4 \beta_{9} - 6 \beta_{12} ) q^{85} + ( -4 - 3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{86} + ( -\beta_{4} + \beta_{6} + 4 \beta_{10} + 2 \beta_{13} ) q^{87} + ( -\beta_{1} - \beta_{7} + \beta_{9} ) q^{88} + ( \beta_{5} - \beta_{8} - 4 \beta_{11} + 3 \beta_{14} ) q^{89} + ( -3 \beta_{6} - 6 \beta_{10} - \beta_{13} ) q^{90} + ( 2 - \beta_{2} + \beta_{3} ) q^{92} + ( 4 - 3 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{93} -2 \beta_{4} q^{94} + ( \beta_{7} - 5 \beta_{9} - 3 \beta_{12} ) q^{95} -\beta_{13} q^{96} + ( 4 \beta_{5} - 2 \beta_{8} + 3 \beta_{11} - 2 \beta_{14} ) q^{97} + ( 4 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{7} - 5 \beta_{9} + 5 \beta_{12} - \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} - 32 q^{9} + O(q^{10}) \) \( 16 q - 16 q^{4} - 32 q^{9} - 16 q^{11} - 8 q^{15} + 16 q^{16} - 8 q^{22} - 32 q^{23} + 32 q^{36} + 32 q^{37} + 16 q^{44} + 56 q^{53} + 24 q^{58} + 8 q^{60} - 16 q^{64} - 24 q^{67} + 8 q^{71} - 16 q^{78} + 16 q^{81} - 40 q^{86} + 8 q^{88} + 32 q^{92} + 88 q^{93} + 56 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} - 102 x^{7} + 144 x^{6} - 432 x^{5} + 502 x^{4} + 288 x^{3} + 72 x^{2} + 12 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-73568941 \nu^{15} - 2828429898 \nu^{14} + 10706428236 \nu^{13} - 20127265496 \nu^{12} + 35741021050 \nu^{11} - 17927851065 \nu^{10} + 58772639836 \nu^{9} - 308269732922 \nu^{8} + 50382616322 \nu^{7} - 35812385940 \nu^{6} - 245517720058 \nu^{5} + 1525793905272 \nu^{4} + 747026251899 \nu^{3} + 178512937629 \nu^{2} + 28424647318 \nu - 10982429182739\)\()/ 3707507912227 \)
\(\beta_{2}\)\(=\)\((\)\(-973484628 \nu^{15} + 16798064640 \nu^{14} - 57412057192 \nu^{13} + 106920971791 \nu^{12} - 159815035576 \nu^{11} + 58809142080 \nu^{10} - 208743822856 \nu^{9} + 1110018585936 \nu^{8} - 324206168264 \nu^{7} + 209105071620 \nu^{6} + 1201975196104 \nu^{5} - 5438378020432 \nu^{4} - 2697240781348 \nu^{3} - 647205045504 \nu^{2} - 103443810800 \nu + 17707406591301\)\()/ 3707507912227 \)
\(\beta_{3}\)\(=\)\((\)\(-4604477275 \nu^{15} + 26065356324 \nu^{14} - 75543186530 \nu^{13} + 146897647868 \nu^{12} - 132924225826 \nu^{11} - 13305240150 \nu^{10} - 16121748394 \nu^{9} + 141233089912 \nu^{8} - 557871539990 \nu^{7} + 316601789265 \nu^{6} + 1303176243718 \nu^{5} - 490404558308 \nu^{4} - 369763065215 \nu^{3} - 98321373186 \nu^{2} - 17113540344 \nu + 1817177370827\)\()/ 3707507912227 \)
\(\beta_{4}\)\(=\)\((\)\(-275511061668 \nu^{15} + 1657750582996 \nu^{14} - 4984224043414 \nu^{13} + 9981450357924 \nu^{12} - 9469500994272 \nu^{11} - 4940688808597 \nu^{10} + 20095109174076 \nu^{9} - 36700380318006 \nu^{8} + 25948536491036 \nu^{7} + 28399264833487 \nu^{6} - 41476514830836 \nu^{5} + 119515269818808 \nu^{4} - 137980607533908 \nu^{3} - 79280602131237 \nu^{2} - 12412885174910 \nu - 1555184012514\)\()/ 3707507912227 \)
\(\beta_{5}\)\(=\)\((\)\(-464283294072 \nu^{15} + 2897231896860 \nu^{14} - 9050241140038 \nu^{13} + 18865423115472 \nu^{12} - 20243277302808 \nu^{11} - 3659327351235 \nu^{10} + 34524757354698 \nu^{9} - 69586426065626 \nu^{8} + 59597927208868 \nu^{7} + 33784753952673 \nu^{6} - 76446349441200 \nu^{5} + 218895832838984 \nu^{4} - 286618252654908 \nu^{3} - 66966547472787 \nu^{2} - 10425321369434 \nu - 1395125725490\)\()/ 3707507912227 \)
\(\beta_{6}\)\(=\)\((\)\(476129408728 \nu^{15} - 2861932186210 \nu^{14} + 8584036352598 \nu^{13} - 17121557060007 \nu^{12} + 16021649705480 \nu^{11} + 9119986400225 \nu^{10} - 35176977398796 \nu^{9} + 63105295993728 \nu^{8} - 43746959330988 \nu^{7} - 50114785825646 \nu^{6} + 71179229667320 \nu^{5} - 205114133957706 \nu^{4} + 238865093218054 \nu^{3} + 137040061864945 \nu^{2} + 21456435065178 \nu + 2687120456379\)\()/ 3707507912227 \)
\(\beta_{7}\)\(=\)\((\)\(770100281810 \nu^{15} - 4733263012640 \nu^{14} + 14562049553051 \nu^{13} - 29920520825502 \nu^{12} + 30799143245438 \nu^{11} + 8829439453946 \nu^{10} - 56098808754161 \nu^{9} + 110050939842954 \nu^{8} - 89459573480504 \nu^{7} - 62707181505525 \nu^{6} + 117223638772823 \nu^{5} - 351034936647114 \nu^{4} + 441822991616242 \nu^{3} + 152138648268578 \nu^{2} + 39169305360240 \nu + 4887214330899\)\()/ 3707507912227 \)
\(\beta_{8}\)\(=\)\((\)\(-805007695745 \nu^{15} + 5029032860754 \nu^{14} - 15724330059984 \nu^{13} + 32793740003767 \nu^{12} - 35212340782612 \nu^{11} - 6414480346779 \nu^{10} + 60443911042560 \nu^{9} - 121464878425672 \nu^{8} + 103864347546612 \nu^{7} + 58759557056628 \nu^{6} - 132943428475514 \nu^{5} + 378138116252806 \nu^{4} - 494704696217447 \nu^{3} - 115575276518547 \nu^{2} - 17991168996474 \nu - 2408034273855\)\()/ 3707507912227 \)
\(\beta_{9}\)\(=\)\((\)\(-1616321538 \nu^{15} + 9938916556 \nu^{14} - 30594542475 \nu^{13} + 62871582510 \nu^{12} - 64714538406 \nu^{11} - 18641341380 \nu^{10} + 118271636061 \nu^{9} - 231117756606 \nu^{8} + 186162436800 \nu^{7} + 134499720676 \nu^{6} - 250255719435 \nu^{5} + 736991238906 \nu^{4} - 924410484114 \nu^{3} - 318408624800 \nu^{2} - 82003950372 \nu - 10233974547\)\()/ 3810388399 \)
\(\beta_{10}\)\(=\)\((\)\(854458 \nu^{15} - 5149288 \nu^{14} + 15489103 \nu^{13} - 31002870 \nu^{12} + 29353426 \nu^{11} + 15677868 \nu^{10} - 63062803 \nu^{9} + 114195864 \nu^{8} - 80446252 \nu^{7} - 89178588 \nu^{6} + 128785525 \nu^{5} - 370469382 \nu^{4} + 434157674 \nu^{3} + 248804552 \nu^{2} + 38956914 \nu + 4877403\)\()/1828267\)
\(\beta_{11}\)\(=\)\((\)\(1570148876 \nu^{15} - 9814610316 \nu^{14} + 30686832232 \nu^{13} - 63994473524 \nu^{12} + 68737309508 \nu^{11} + 12449599248 \nu^{10} - 117627630280 \nu^{9} + 236328234138 \nu^{8} - 202589468876 \nu^{7} - 114627530376 \nu^{6} + 259195647184 \nu^{5} - 741062624316 \nu^{4} + 968316440200 \nu^{3} + 226232221584 \nu^{2} + 35218281816 \nu + 4714210963\)\()/ 1973128213 \)
\(\beta_{12}\)\(=\)\((\)\(3087253540240 \nu^{15} - 18982048227910 \nu^{14} + 58429841322972 \nu^{13} - 120068214463500 \nu^{12} + 123576350616332 \nu^{11} + 35598713867733 \nu^{10} - 225839628061730 \nu^{9} + 441382360047234 \nu^{8} - 355634599263140 \nu^{7} - 256040217912758 \nu^{6} + 477587659531454 \nu^{5} - 1407573904107408 \nu^{4} + 1765495989661296 \nu^{3} + 608117198238559 \nu^{2} + 156616076158474 \nu + 19545417290592\)\()/ 3707507912227 \)
\(\beta_{13}\)\(=\)\((\)\(3131859052620 \nu^{15} - 18867452624394 \nu^{14} + 56728752984694 \nu^{13} - 113478926578056 \nu^{12} + 107222867624596 \nu^{11} + 57967676009072 \nu^{10} - 231355635133834 \nu^{9} + 418078071142320 \nu^{8} - 293684812663328 \nu^{7} - 327591625067831 \nu^{6} + 471436953125158 \nu^{5} - 1356409325056608 \nu^{4} + 1589452776259132 \nu^{3} + 910890443850692 \nu^{2} + 142623474998120 \nu + 17856504555702\)\()/ 3707507912227 \)
\(\beta_{14}\)\(=\)\((\)\(-5329561831437 \nu^{15} + 33315128327418 \nu^{14} - 104171995568578 \nu^{13} + 217250160715424 \nu^{12} - 233356529166572 \nu^{11} - 42306147026280 \nu^{10} + 399610538807086 \nu^{9} - 802749774453640 \nu^{8} + 687914400857440 \nu^{7} + 389163059670207 \nu^{6} - 880025451774682 \nu^{5} + 2513669318791624 \nu^{4} - 3284887445773991 \nu^{3} - 767456463763692 \nu^{2} - 119471231971904 \nu - 15992012287118\)\()/ 3707507912227 \)
\(\beta_{15}\)\(=\)\((\)\(-7516121976528 \nu^{15} + 46209561787616 \nu^{14} - 142236306189628 \nu^{13} + 292274046112479 \nu^{12} - 300793116749828 \nu^{11} - 86647872654676 \nu^{10} + 549671926014968 \nu^{9} - 1074450180412176 \nu^{8} + 866011308655208 \nu^{7} + 622075435017104 \nu^{6} - 1161717303443108 \nu^{5} + 3426608236011048 \nu^{4} - 4298074501231946 \nu^{3} - 1480448402799748 \nu^{2} - 381276303387704 \nu - 47582417802291\)\()/ 3707507912227 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{12} + \beta_{9} - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{1} + 2\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{13} + \beta_{12} + 2 \beta_{10} + 2 \beta_{9} + \beta_{4}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{14} - \beta_{13} - 2 \beta_{11} + 2 \beta_{10} + \beta_{8} + \beta_{6} - 3 \beta_{5} + 3 \beta_{4}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-5 \beta_{14} - 9 \beta_{11} + 3 \beta_{8} - 5 \beta_{5} + \beta_{2} + 6 \beta_{1} + 13\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(8 \beta_{15} - 8 \beta_{14} + 8 \beta_{13} + 35 \beta_{12} - 14 \beta_{11} - 14 \beta_{10} + 29 \beta_{9} + 8 \beta_{8} - 3 \beta_{7} - 8 \beta_{6} - 11 \beta_{5} - 11 \beta_{4} + 3 \beta_{3} + 8 \beta_{2} + 35 \beta_{1} + 64\)\()/4\)
\(\nu^{6}\)\(=\)\(8 \beta_{15} + 34 \beta_{12} + 29 \beta_{9} + \beta_{7}\)
\(\nu^{7}\)\(=\)\((\)\(51 \beta_{15} - 49 \beta_{14} - 49 \beta_{13} + 196 \beta_{12} - 82 \beta_{11} + 82 \beta_{10} + 142 \beta_{9} + 51 \beta_{8} + 2 \beta_{7} + 51 \beta_{6} - 47 \beta_{5} + 47 \beta_{4} + 2 \beta_{3} - 51 \beta_{2} - 196 \beta_{1} - 338\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(-148 \beta_{14} - 245 \beta_{11} + 150 \beta_{8} - 118 \beta_{5} + 10 \beta_{3} - 50 \beta_{2} - 188 \beta_{1} - 323\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-278 \beta_{14} + 278 \beta_{13} - 456 \beta_{11} - 456 \beta_{10} + 298 \beta_{8} - 298 \beta_{6} - 223 \beta_{5} - 223 \beta_{4}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(288 \beta_{15} + 809 \beta_{13} + 1027 \beta_{12} - 1320 \beta_{10} + 674 \beta_{9} + 70 \beta_{7} - 864 \beta_{6} - 599 \beta_{4}\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(1673 \beta_{15} + 1533 \beta_{14} + 1533 \beta_{13} + 5864 \beta_{12} + 2490 \beta_{11} - 2490 \beta_{10} + 3716 \beta_{9} - 1673 \beta_{8} + 408 \beta_{7} - 1673 \beta_{6} + 1125 \beta_{5} - 1125 \beta_{4} + 408 \beta_{3} - 1673 \beta_{2} - 5864 \beta_{1} - 9580\)\()/4\)
\(\nu^{12}\)\(=\)\(428 \beta_{3} - 1603 \beta_{2} - 5576 \beta_{1} - 9071\)
\(\nu^{13}\)\(=\)\((\)\(-9210 \beta_{15} - 8354 \beta_{14} + 8354 \beta_{13} - 31783 \beta_{12} - 13502 \beta_{11} - 13502 \beta_{10} - 19597 \beta_{9} + 9210 \beta_{8} - 2489 \beta_{7} - 9210 \beta_{6} - 5865 \beta_{5} - 5865 \beta_{4} + 2489 \beta_{3} - 9210 \beta_{2} - 31783 \beta_{1} - 51380\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(-8782 \beta_{15} + 23857 \beta_{13} - 30180 \beta_{12} - 38504 \beta_{10} - 18481 \beta_{9} - 2459 \beta_{7} - 26346 \beta_{6} - 16480 \beta_{4}\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(45285 \beta_{14} + 45285 \beta_{13} + 73006 \beta_{11} - 73006 \beta_{10} - 50203 \beta_{8} - 50203 \beta_{6} + 31097 \beta_{5} - 31097 \beta_{4}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(981\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1077.1
−0.186243 + 0.0499037i
0.601150 2.24352i
−0.347596 + 1.29724i
1.60599 0.430324i
0.430324 1.60599i
−1.29724 + 0.347596i
2.24352 0.601150i
−0.0499037 + 0.186243i
−0.0499037 0.186243i
2.24352 + 0.601150i
−1.29724 0.347596i
0.430324 + 1.60599i
1.60599 + 0.430324i
−0.347596 1.29724i
0.601150 + 2.24352i
−0.186243 0.0499037i
1.00000i 3.12703i −1.00000 2.20288i −3.12703 0 1.00000i −6.77832 2.20288
1077.2 1.00000i 2.59518i −1.00000 3.53900i −2.59518 0 1.00000i −3.73495 −3.53900
1077.3 1.00000i 1.55924i −1.00000 1.01909i −1.55924 0 1.00000i 0.568783 1.01909
1077.4 1.00000i 1.02738i −1.00000 1.25868i −1.02738 0 1.00000i 1.94448 −1.25868
1077.5 1.00000i 1.02738i −1.00000 1.25868i 1.02738 0 1.00000i 1.94448 1.25868
1077.6 1.00000i 1.55924i −1.00000 1.01909i 1.55924 0 1.00000i 0.568783 −1.01909
1077.7 1.00000i 2.59518i −1.00000 3.53900i 2.59518 0 1.00000i −3.73495 3.53900
1077.8 1.00000i 3.12703i −1.00000 2.20288i 3.12703 0 1.00000i −6.77832 −2.20288
1077.9 1.00000i 3.12703i −1.00000 2.20288i 3.12703 0 1.00000i −6.77832 −2.20288
1077.10 1.00000i 2.59518i −1.00000 3.53900i 2.59518 0 1.00000i −3.73495 3.53900
1077.11 1.00000i 1.55924i −1.00000 1.01909i 1.55924 0 1.00000i 0.568783 −1.01909
1077.12 1.00000i 1.02738i −1.00000 1.25868i 1.02738 0 1.00000i 1.94448 1.25868
1077.13 1.00000i 1.02738i −1.00000 1.25868i −1.02738 0 1.00000i 1.94448 −1.25868
1077.14 1.00000i 1.55924i −1.00000 1.01909i −1.55924 0 1.00000i 0.568783 1.01909
1077.15 1.00000i 2.59518i −1.00000 3.53900i −2.59518 0 1.00000i −3.73495 −3.53900
1077.16 1.00000i 3.12703i −1.00000 2.20288i −3.12703 0 1.00000i −6.77832 2.20288
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1077.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.c.b 16
7.b odd 2 1 inner 1078.2.c.b 16
7.c even 3 1 154.2.i.a 16
7.c even 3 1 1078.2.i.c 16
7.d odd 6 1 154.2.i.a 16
7.d odd 6 1 1078.2.i.c 16
11.b odd 2 1 inner 1078.2.c.b 16
21.g even 6 1 1386.2.bk.c 16
21.h odd 6 1 1386.2.bk.c 16
28.f even 6 1 1232.2.bn.b 16
28.g odd 6 1 1232.2.bn.b 16
77.b even 2 1 inner 1078.2.c.b 16
77.h odd 6 1 154.2.i.a 16
77.h odd 6 1 1078.2.i.c 16
77.i even 6 1 154.2.i.a 16
77.i even 6 1 1078.2.i.c 16
231.k odd 6 1 1386.2.bk.c 16
231.l even 6 1 1386.2.bk.c 16
308.m odd 6 1 1232.2.bn.b 16
308.n even 6 1 1232.2.bn.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.i.a 16 7.c even 3 1
154.2.i.a 16 7.d odd 6 1
154.2.i.a 16 77.h odd 6 1
154.2.i.a 16 77.i even 6 1
1078.2.c.b 16 1.a even 1 1 trivial
1078.2.c.b 16 7.b odd 2 1 inner
1078.2.c.b 16 11.b odd 2 1 inner
1078.2.c.b 16 77.b even 2 1 inner
1078.2.i.c 16 7.c even 3 1
1078.2.i.c 16 7.d odd 6 1
1078.2.i.c 16 77.h odd 6 1
1078.2.i.c 16 77.i even 6 1
1232.2.bn.b 16 28.f even 6 1
1232.2.bn.b 16 28.g odd 6 1
1232.2.bn.b 16 308.m odd 6 1
1232.2.bn.b 16 308.n even 6 1
1386.2.bk.c 16 21.g even 6 1
1386.2.bk.c 16 21.h odd 6 1
1386.2.bk.c 16 231.k odd 6 1
1386.2.bk.c 16 231.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 20 T_{3}^{6} + 126 T_{3}^{4} + 272 T_{3}^{2} + 169 \) acting on \(S_{2}^{\mathrm{new}}(1078, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{8} \)
$3$ \( ( 169 + 272 T^{2} + 126 T^{4} + 20 T^{6} + T^{8} )^{2} \)
$5$ \( ( 100 + 188 T^{2} + 108 T^{4} + 20 T^{6} + T^{8} )^{2} \)
$7$ \( T^{16} \)
$11$ \( ( 14641 + 10648 T + 2420 T^{2} - 440 T^{3} - 362 T^{4} - 40 T^{5} + 20 T^{6} + 8 T^{7} + T^{8} )^{2} \)
$13$ \( ( 25 - 908 T^{2} + 510 T^{4} - 44 T^{6} + T^{8} )^{2} \)
$17$ \( ( 78400 - 23168 T^{2} + 2352 T^{4} - 92 T^{6} + T^{8} )^{2} \)
$19$ \( ( 4 - 164 T^{2} + 1272 T^{4} - 80 T^{6} + T^{8} )^{2} \)
$23$ \( ( -14 - 34 T - 6 T^{2} + 8 T^{3} + T^{4} )^{4} \)
$29$ \( ( 9 + 900 T^{2} + 366 T^{4} + 36 T^{6} + T^{8} )^{2} \)
$31$ \( ( 3136 + 40640 T^{2} + 7200 T^{4} + 164 T^{6} + T^{8} )^{2} \)
$37$ \( ( -2 - 2 T + 12 T^{2} - 8 T^{3} + T^{4} )^{4} \)
$41$ \( ( 1674436 - 287612 T^{2} + 12588 T^{4} - 200 T^{6} + T^{8} )^{2} \)
$43$ \( ( 1201216 + 527488 T^{2} + 19536 T^{4} + 244 T^{6} + T^{8} )^{2} \)
$47$ \( ( 4096 + 14336 T^{2} + 1920 T^{4} + 80 T^{6} + T^{8} )^{2} \)
$53$ \( ( -1082 + 322 T + 24 T^{2} - 14 T^{3} + T^{4} )^{4} \)
$59$ \( ( 2356225 + 448508 T^{2} + 19974 T^{4} + 296 T^{6} + T^{8} )^{2} \)
$61$ \( ( 1306449 - 632556 T^{2} + 26838 T^{4} - 300 T^{6} + T^{8} )^{2} \)
$67$ \( ( 417 - 48 T - 54 T^{2} + 6 T^{3} + T^{4} )^{4} \)
$71$ \( ( 3262 - 338 T - 174 T^{2} - 2 T^{3} + T^{4} )^{4} \)
$73$ \( ( 125316 - 144972 T^{2} + 11868 T^{4} - 204 T^{6} + T^{8} )^{2} \)
$79$ \( ( 33953929 + 2113108 T^{2} + 46182 T^{4} + 400 T^{6} + T^{8} )^{2} \)
$83$ \( ( 107584 - 77504 T^{2} + 9888 T^{4} - 260 T^{6} + T^{8} )^{2} \)
$89$ \( ( 49617936 + 3072528 T^{2} + 62088 T^{4} + 456 T^{6} + T^{8} )^{2} \)
$97$ \( ( 496532089 + 14100500 T^{2} + 145902 T^{4} + 644 T^{6} + T^{8} )^{2} \)
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