Properties

Label 242.4.c.r
Level $242$
Weight $4$
Character orbit 242.c
Analytic conductor $14.278$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 242 = 2 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 242.c (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.2784622214\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - x^{7} + 71 x^{6} - 141 x^{5} + 2911 x^{4} + 2710 x^{3} + 75340 x^{2} + 169400 x + 5856400\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 22)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{4} q^{2} + ( -1 + \beta_{4} + \beta_{7} ) q^{3} + 4 \beta_{3} q^{4} + ( \beta_{1} + 8 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} + \beta_{5} ) q^{5} + ( 2 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{6} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{5} + 3 \beta_{7} ) q^{7} + ( 8 + 8 \beta_{2} + 8 \beta_{3} - 8 \beta_{4} ) q^{8} + ( -5 - 2 \beta_{1} - 7 \beta_{3} - 21 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{9} +O(q^{10})\) \( q + 2 \beta_{4} q^{2} + ( -1 + \beta_{4} + \beta_{7} ) q^{3} + 4 \beta_{3} q^{4} + ( \beta_{1} + 8 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} + \beta_{5} ) q^{5} + ( 2 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{6} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{5} + 3 \beta_{7} ) q^{7} + ( 8 + 8 \beta_{2} + 8 \beta_{3} - 8 \beta_{4} ) q^{8} + ( -5 - 2 \beta_{1} - 7 \beta_{3} - 21 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{9} + ( -8 - 2 \beta_{1} + 8 \beta_{2} - 8 \beta_{4} + 2 \beta_{7} ) q^{10} + ( 8 + 4 \beta_{2} - 4 \beta_{4} + 4 \beta_{6} ) q^{12} + ( -7 + 3 \beta_{1} - 4 \beta_{3} - 6 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{13} + ( 2 + 4 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} + 2 \beta_{7} ) q^{14} + ( -50 - 7 \beta_{1} - 50 \beta_{2} - 12 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} ) q^{15} + 16 \beta_{2} q^{16} + ( -3 - \beta_{1} - \beta_{2} + 25 \beta_{3} - 25 \beta_{4} - \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{17} + ( -14 + 6 \beta_{1} - 14 \beta_{2} - 56 \beta_{3} + 4 \beta_{5} - 4 \beta_{7} ) q^{18} + ( 34 - 21 \beta_{2} - 21 \beta_{3} - 36 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 6 \beta_{7} ) q^{19} + ( -12 - 4 \beta_{1} - 16 \beta_{3} - 16 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} ) q^{20} + ( 66 + 3 \beta_{1} + 144 \beta_{2} - 144 \beta_{4} + 2 \beta_{6} - 3 \beta_{7} ) q^{21} + ( 42 - 6 \beta_{1} - 70 \beta_{2} + 70 \beta_{4} - 6 \beta_{6} + 6 \beta_{7} ) q^{23} + ( -8 - 8 \beta_{3} + 8 \beta_{4} + 8 \beta_{5} ) q^{24} + ( -\beta_{2} - \beta_{3} - 9 \beta_{4} + 9 \beta_{5} - 9 \beta_{6} - 7 \beta_{7} ) q^{25} + ( -8 + 6 \beta_{1} - 8 \beta_{2} - 20 \beta_{3} - 6 \beta_{5} + 6 \beta_{7} ) q^{26} + ( -4 + 2 \beta_{1} + 31 \beta_{2} - 78 \beta_{3} + 78 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{27} + ( 4 + 8 \beta_{1} + 8 \beta_{3} - 8 \beta_{4} + 8 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{28} + ( 50 - 10 \beta_{1} + 50 \beta_{2} - 120 \beta_{3} - \beta_{5} + \beta_{7} ) q^{29} + ( 72 - 24 \beta_{2} - 24 \beta_{3} - 76 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 18 \beta_{7} ) q^{30} + ( -95 + 21 \beta_{1} - 74 \beta_{3} - 20 \beta_{4} + 25 \beta_{5} - 21 \beta_{6} ) q^{31} -32 q^{32} + ( 46 + 8 \beta_{1} + 50 \beta_{2} - 50 \beta_{4} - 6 \beta_{6} - 8 \beta_{7} ) q^{34} + ( 76 - 16 \beta_{1} + 60 \beta_{3} - 214 \beta_{4} + 3 \beta_{5} + 16 \beta_{6} ) q^{35} + ( -92 - 112 \beta_{2} - 112 \beta_{3} + 84 \beta_{4} + 8 \beta_{5} - 8 \beta_{6} + 4 \beta_{7} ) q^{36} + ( -48 + 32 \beta_{1} - 48 \beta_{2} + 80 \beta_{3} + 5 \beta_{5} - 5 \beta_{7} ) q^{37} + ( 12 - 16 \beta_{1} - 42 \beta_{2} - 114 \beta_{3} + 114 \beta_{4} - 16 \beta_{5} + 12 \beta_{6} + 12 \beta_{7} ) q^{38} + ( -13 + 9 \beta_{1} + 344 \beta_{2} + 180 \beta_{3} - 180 \beta_{4} + 9 \beta_{5} - 13 \beta_{6} - 13 \beta_{7} ) q^{39} + ( -32 + 8 \beta_{1} - 32 \beta_{2} - 64 \beta_{3} + 8 \beta_{5} - 8 \beta_{7} ) q^{40} + ( -11 + 46 \beta_{2} + 46 \beta_{3} + 35 \beta_{4} - 24 \beta_{5} + 24 \beta_{6} + 7 \beta_{7} ) q^{41} + ( -294 + 6 \beta_{1} - 288 \beta_{3} + 128 \beta_{4} + 10 \beta_{5} - 6 \beta_{6} ) q^{42} + ( 255 + 7 \beta_{1} + 167 \beta_{2} - 167 \beta_{4} - 14 \beta_{6} - 7 \beta_{7} ) q^{43} + ( 313 + 28 \beta_{1} - 18 \beta_{2} + 18 \beta_{4} - 5 \beta_{6} - 28 \beta_{7} ) q^{45} + ( 152 - 12 \beta_{1} + 140 \beta_{3} + 96 \beta_{4} - 24 \beta_{5} + 12 \beta_{6} ) q^{46} + ( -29 - 198 \beta_{2} - 198 \beta_{3} + 20 \beta_{4} + 9 \beta_{5} - 9 \beta_{6} - 21 \beta_{7} ) q^{47} + ( -16 - 16 \beta_{1} - 16 \beta_{2} ) q^{48} + ( 2 - 7 \beta_{1} + 177 \beta_{2} - 62 \beta_{3} + 62 \beta_{4} - 7 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{49} + ( -14 - 4 \beta_{1} - 2 \beta_{2} - 20 \beta_{3} + 20 \beta_{4} - 4 \beta_{5} - 14 \beta_{6} - 14 \beta_{7} ) q^{50} + ( 234 + 32 \beta_{1} + 234 \beta_{2} + 169 \beta_{3} + 30 \beta_{5} - 30 \beta_{7} ) q^{51} + ( -12 - 40 \beta_{2} - 40 \beta_{3} + 24 \beta_{4} - 12 \beta_{5} + 12 \beta_{6} + 24 \beta_{7} ) q^{52} + ( 89 - 39 \beta_{1} + 50 \beta_{3} + 22 \beta_{4} - 47 \beta_{5} + 39 \beta_{6} ) q^{53} + ( -226 + 4 \beta_{1} - 156 \beta_{2} + 156 \beta_{4} - 8 \beta_{6} - 4 \beta_{7} ) q^{54} + ( 24 - 24 \beta_{1} + 16 \beta_{2} - 16 \beta_{4} + 8 \beta_{6} + 24 \beta_{7} ) q^{56} + ( -39 + 47 \beta_{1} + 8 \beta_{3} - 259 \beta_{4} + 18 \beta_{5} - 47 \beta_{6} ) q^{57} + ( -338 - 240 \beta_{2} - 240 \beta_{3} + 340 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 18 \beta_{7} ) q^{58} + ( -69 - 40 \beta_{1} - 69 \beta_{2} + 203 \beta_{3} - 14 \beta_{5} + 14 \beta_{7} ) q^{59} + ( -36 + 28 \beta_{1} - 48 \beta_{2} - 200 \beta_{3} + 200 \beta_{4} + 28 \beta_{5} - 36 \beta_{6} - 36 \beta_{7} ) q^{60} + ( 28 - \beta_{1} + 228 \beta_{2} - 18 \beta_{3} + 18 \beta_{4} - \beta_{5} + 28 \beta_{6} + 28 \beta_{7} ) q^{61} + ( -148 - 50 \beta_{1} - 148 \beta_{2} - 188 \beta_{3} - 42 \beta_{5} + 42 \beta_{7} ) q^{62} + ( -539 - 228 \beta_{2} - 228 \beta_{3} + 468 \beta_{4} + 71 \beta_{5} - 71 \beta_{6} - 48 \beta_{7} ) q^{63} -64 \beta_{4} q^{64} + ( 69 - 27 \beta_{1} + 106 \beta_{2} - 106 \beta_{4} + 53 \beta_{6} + 27 \beta_{7} ) q^{65} + ( 91 - 59 \beta_{1} - 43 \beta_{2} + 43 \beta_{4} + 51 \beta_{6} + 59 \beta_{7} ) q^{67} + ( -116 + 16 \beta_{1} - 100 \beta_{3} + 104 \beta_{4} + 4 \beta_{5} - 16 \beta_{6} ) q^{68} + ( 614 + 346 \beta_{2} + 346 \beta_{3} - 526 \beta_{4} - 88 \beta_{5} + 88 \beta_{6} + 106 \beta_{7} ) q^{69} + ( 120 - 6 \beta_{1} + 120 \beta_{2} - 308 \beta_{3} + 32 \beta_{5} - 32 \beta_{7} ) q^{70} + ( 32 + 19 \beta_{1} - 334 \beta_{2} - 72 \beta_{3} + 72 \beta_{4} + 19 \beta_{5} + 32 \beta_{6} + 32 \beta_{7} ) q^{71} + ( 8 - 24 \beta_{1} - 224 \beta_{2} - 56 \beta_{3} + 56 \beta_{4} - 24 \beta_{5} + 8 \beta_{6} + 8 \beta_{7} ) q^{72} + ( 205 - 17 \beta_{1} + 205 \beta_{2} + 418 \beta_{3} - 62 \beta_{5} + 62 \beta_{7} ) q^{73} + ( 246 + 160 \beta_{2} + 160 \beta_{3} - 256 \beta_{4} + 10 \beta_{5} - 10 \beta_{6} + 54 \beta_{7} ) q^{74} + ( 462 - \beta_{1} + 461 \beta_{3} - \beta_{4} - 4 \beta_{5} + \beta_{6} ) q^{75} + ( -120 + 8 \beta_{1} - 228 \beta_{2} + 228 \beta_{4} + 24 \beta_{6} - 8 \beta_{7} ) q^{76} + ( -354 + 8 \beta_{1} + 360 \beta_{2} - 360 \beta_{4} - 26 \beta_{6} - 8 \beta_{7} ) q^{78} + ( -47 - 55 \beta_{1} - 102 \beta_{3} - 104 \beta_{4} - 19 \beta_{5} + 55 \beta_{6} ) q^{79} + ( -80 - 128 \beta_{2} - 128 \beta_{3} + 64 \beta_{4} + 16 \beta_{5} - 16 \beta_{6} ) q^{80} + ( -174 - 30 \beta_{1} - 174 \beta_{2} - 446 \beta_{3} - 24 \beta_{5} + 24 \beta_{7} ) q^{81} + ( 14 + 34 \beta_{1} + 92 \beta_{2} + 162 \beta_{3} - 162 \beta_{4} + 34 \beta_{5} + 14 \beta_{6} + 14 \beta_{7} ) q^{82} + ( 92 - 104 \beta_{1} - 214 \beta_{2} + 231 \beta_{3} - 231 \beta_{4} - 104 \beta_{5} + 92 \beta_{6} + 92 \beta_{7} ) q^{83} + ( -576 - 20 \beta_{1} - 576 \beta_{2} - 320 \beta_{3} - 12 \beta_{5} + 12 \beta_{7} ) q^{84} + ( 281 + 116 \beta_{2} + 116 \beta_{3} - 226 \beta_{4} - 55 \beta_{5} + 55 \beta_{6} + 11 \beta_{7} ) q^{85} + ( -348 + 14 \beta_{1} - 334 \beta_{3} + 538 \beta_{4} - 14 \beta_{5} - 14 \beta_{6} ) q^{86} + ( 122 - 39 \beta_{1} - 196 \beta_{2} + 196 \beta_{4} - 110 \beta_{6} + 39 \beta_{7} ) q^{87} + ( -521 + 61 \beta_{1} - 435 \beta_{2} + 435 \beta_{4} - 26 \beta_{6} - 61 \beta_{7} ) q^{89} + ( -20 + 56 \beta_{1} + 36 \beta_{3} + 636 \beta_{4} + 46 \beta_{5} - 56 \beta_{6} ) q^{90} + ( 210 + 898 \beta_{2} + 898 \beta_{3} - 238 \beta_{4} + 28 \beta_{5} - 28 \beta_{6} - 31 \beta_{7} ) q^{91} + ( 280 + 48 \beta_{1} + 280 \beta_{2} + 472 \beta_{3} + 24 \beta_{5} - 24 \beta_{7} ) q^{92} + ( -69 - 51 \beta_{1} - 122 \beta_{2} + 914 \beta_{3} - 914 \beta_{4} - 51 \beta_{5} - 69 \beta_{6} - 69 \beta_{7} ) q^{93} + ( -42 + 24 \beta_{1} - 396 \beta_{2} - 356 \beta_{3} + 356 \beta_{4} + 24 \beta_{5} - 42 \beta_{6} - 42 \beta_{7} ) q^{94} + ( -40 + 59 \beta_{1} - 40 \beta_{2} + 440 \beta_{3} + 44 \beta_{5} - 44 \beta_{7} ) q^{95} + ( 32 - 32 \beta_{4} - 32 \beta_{7} ) q^{96} + ( 819 - 10 \beta_{1} + 809 \beta_{3} - 941 \beta_{4} + 10 \beta_{5} + 10 \beta_{6} ) q^{97} + ( -474 + 10 \beta_{1} - 124 \beta_{2} + 124 \beta_{4} + 4 \beta_{6} - 10 \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{2} - 7q^{3} - 8q^{4} - 30q^{5} - 6q^{6} + 4q^{7} + 16q^{8} - 81q^{9} + O(q^{10}) \) \( 8q + 4q^{2} - 7q^{3} - 8q^{4} - 30q^{5} - 6q^{6} + 4q^{7} + 16q^{8} - 81q^{9} - 100q^{10} + 32q^{12} - 48q^{13} - 8q^{14} - 279q^{15} - 32q^{16} - 109q^{17} + 42q^{18} + 288q^{19} - 120q^{20} - 50q^{21} + 628q^{23} - 24q^{24} + 38q^{25} - 14q^{26} + 242q^{27} - 4q^{28} + 528q^{29} + 558q^{30} - 522q^{31} - 256q^{32} + 208q^{34} - 17q^{35} - 84q^{36} - 406q^{37} + 544q^{38} - 1429q^{39} - 40q^{40} - 329q^{41} - 1480q^{42} + 1442q^{43} + 2652q^{45} + 1044q^{46} + 666q^{47} - 112q^{48} - 114q^{49} + 34q^{50} + 1158q^{51} + 28q^{52} + 414q^{53} - 1144q^{54} + 48q^{56} - 593q^{57} - 1056q^{58} - 888q^{59} + 844q^{60} - 302q^{61} - 646q^{62} - 2061q^{63} - 128q^{64} - 138q^{65} + 578q^{67} - 436q^{68} + 1930q^{69} + 1394q^{70} + 1090q^{71} + 648q^{72} + 253q^{73} + 812q^{74} + 2763q^{75} - 128q^{76} - 4152q^{78} - 674q^{79} + 80q^{80} - 230q^{81} - 722q^{82} - 428q^{83} - 2860q^{84} + 1046q^{85} - 984q^{86} + 2122q^{87} - 2202q^{89} + 1366q^{90} - 2217q^{91} + 832q^{92} - 3721q^{93} + 2138q^{94} - 973q^{95} + 224q^{96} + 3012q^{97} - 3292q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} + 71 x^{6} - 141 x^{5} + 2911 x^{4} + 2710 x^{3} + 75340 x^{2} + 169400 x + 5856400\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(22554025143 \nu^{7} + 151013926876 \nu^{6} - 1356924294556 \nu^{5} + 28230146638036 \nu^{4} - 111071208987556 \nu^{3} + 8011043329394989 \nu^{2} - 4059658693406480 \nu + 48229141584676800\)\()/ 342693986325717620 \)
\(\beta_{3}\)\(=\)\((\)\(14112299722 \nu^{7} + 955320989289 \nu^{6} + 1256420838616 \nu^{5} - 5250112809736 \nu^{4} + 12754848436776 \nu^{3} - 1313471371492594 \nu^{2} + 8498101996878245 \nu - 66484077419717320\)\()/ 171346993162858810 \)
\(\beta_{4}\)\(=\)\((\)\(-123788106187 \nu^{7} + 5732748190805 \nu^{6} - 11662655347575 \nu^{5} + 374269556401525 \nu^{4} - 772584075715295 \nu^{3} + 13504262211521188 \nu^{2} + 13624366122837560 \nu + 426480737760227920\)\()/ 685387972651435240 \)
\(\beta_{5}\)\(=\)\((\)\(15778904729 \nu^{7} - 268932734519 \nu^{6} + 2855478562109 \nu^{5} - 16065997834439 \nu^{4} + 722720174659769 \nu^{3} - 523534449789100 \nu^{2} + 4037135429586600 \nu - 12007762986133200\)\()/ 31153998756883420 \)
\(\beta_{6}\)\(=\)\((\)\(-15728307097 \nu^{7} - 23011705883 \nu^{6} - 3489045673392 \nu^{5} - 619537249643 \nu^{4} - 38031215360567 \nu^{3} - 53736302216310 \nu^{2} + 582451195267480 \nu - 22785233088495405\)\()/ 7788499689220855 \)
\(\beta_{7}\)\(=\)\((\)\(254952731119 \nu^{7} - 130622718559 \nu^{6} + 16218883337689 \nu^{5} - 18738040845679 \nu^{4} + 629078544513089 \nu^{3} + 1043207365589370 \nu^{2} + 20338656497650260 \nu + 32952393866979400\)\()/ 31153998756883420 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{7} - \beta_{6} + \beta_{5} - 8 \beta_{4} + 8 \beta_{3} + 54 \beta_{2} + \beta_{1} - 1\)
\(\nu^{3}\)\(=\)\(8 \beta_{6} + 47 \beta_{5} + 46 \beta_{4} + 8 \beta_{3} - 8 \beta_{1} + 16\)
\(\nu^{4}\)\(=\)\(109 \beta_{7} + 16 \beta_{6} - 16 \beta_{5} + 2226 \beta_{4} - 3034 \beta_{3} - 3034 \beta_{2} - 2210\)
\(\nu^{5}\)\(=\)\(-824 \beta_{7} - 3143 \beta_{6} - 1608 \beta_{4} + 1608 \beta_{2} + 824 \beta_{1} - 7549\)
\(\nu^{6}\)\(=\)\(-2432 \beta_{7} + 2432 \beta_{5} + 176314 \beta_{3} + 63048 \beta_{2} - 6725 \beta_{1} + 63048\)
\(\nu^{7}\)\(=\)\(185471 \beta_{7} + 185471 \beta_{6} - 119991 \beta_{5} + 185128 \beta_{4} - 185128 \beta_{3} - 513934 \beta_{2} - 119991 \beta_{1} + 185471\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/242\mathbb{Z}\right)^\times\).

\(n\) \(123\)
\(\chi(n)\) \(\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
2.22300 6.84169i
−2.53202 + 7.79275i
5.60402 + 4.07156i
−4.79501 3.48378i
5.60402 4.07156i
−4.79501 + 3.48378i
2.22300 + 6.84169i
−2.53202 7.79275i
1.61803 1.17557i −2.41398 7.42948i 1.23607 3.80423i −12.0690 8.76866i −12.6398 9.18334i 6.72664 20.7025i −2.47214 7.60845i −27.5264 + 19.9991i −29.8363
3.2 1.61803 1.17557i 2.34103 + 7.20496i 1.23607 3.80423i −4.37525 3.17880i 12.2578 + 8.90583i −6.84467 + 21.0657i −2.47214 7.60845i −24.5876 + 17.8639i −10.8162
9.1 −0.618034 + 1.90211i −6.91304 + 5.02262i −3.23607 2.35114i 3.93561 + 12.1126i −5.28109 16.2535i −18.9804 13.7901i 6.47214 4.70228i 14.2200 43.7646i −25.4718
9.2 −0.618034 + 1.90211i 3.48599 2.53272i −3.23607 2.35114i −2.49134 7.66756i 2.66306 + 8.19605i 21.0985 + 15.3289i 6.47214 4.70228i −2.60601 + 8.02046i 16.1243
27.1 −0.618034 1.90211i −6.91304 5.02262i −3.23607 + 2.35114i 3.93561 12.1126i −5.28109 + 16.2535i −18.9804 + 13.7901i 6.47214 + 4.70228i 14.2200 + 43.7646i −25.4718
27.2 −0.618034 1.90211i 3.48599 + 2.53272i −3.23607 + 2.35114i −2.49134 + 7.66756i 2.66306 8.19605i 21.0985 15.3289i 6.47214 + 4.70228i −2.60601 8.02046i 16.1243
81.1 1.61803 + 1.17557i −2.41398 + 7.42948i 1.23607 + 3.80423i −12.0690 + 8.76866i −12.6398 + 9.18334i 6.72664 + 20.7025i −2.47214 + 7.60845i −27.5264 19.9991i −29.8363
81.2 1.61803 + 1.17557i 2.34103 7.20496i 1.23607 + 3.80423i −4.37525 + 3.17880i 12.2578 8.90583i −6.84467 21.0657i −2.47214 + 7.60845i −24.5876 17.8639i −10.8162
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 242.4.c.r 8
11.b odd 2 1 242.4.c.n 8
11.c even 5 2 22.4.c.b 8
11.c even 5 1 242.4.a.n 4
11.c even 5 1 inner 242.4.c.r 8
11.d odd 10 1 242.4.a.o 4
11.d odd 10 1 242.4.c.n 8
11.d odd 10 2 242.4.c.q 8
33.f even 10 1 2178.4.a.bt 4
33.h odd 10 2 198.4.f.d 8
33.h odd 10 1 2178.4.a.by 4
44.g even 10 1 1936.4.a.bm 4
44.h odd 10 2 176.4.m.b 8
44.h odd 10 1 1936.4.a.bn 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.4.c.b 8 11.c even 5 2
176.4.m.b 8 44.h odd 10 2
198.4.f.d 8 33.h odd 10 2
242.4.a.n 4 11.c even 5 1
242.4.a.o 4 11.d odd 10 1
242.4.c.n 8 11.b odd 2 1
242.4.c.n 8 11.d odd 10 1
242.4.c.q 8 11.d odd 10 2
242.4.c.r 8 1.a even 1 1 trivial
242.4.c.r 8 11.c even 5 1 inner
1936.4.a.bm 4 44.g even 10 1
1936.4.a.bn 4 44.h odd 10 1
2178.4.a.bt 4 33.f even 10 1
2178.4.a.by 4 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(242, [\chi])\):

\(T_{3}^{8} + \cdots\)
\(T_{5}^{8} + \cdots\)
\(T_{7}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T + 4 T^{2} - 8 T^{3} + 16 T^{4} )^{2} \)
$3$ \( 1 + 7 T + 38 T^{2} + 98 T^{3} + 62 T^{4} + 4223 T^{5} + 33219 T^{6} + 213996 T^{7} + 1147672 T^{8} + 5777892 T^{9} + 24216651 T^{10} + 83121309 T^{11} + 32949342 T^{12} + 1406192886 T^{13} + 14721978582 T^{14} + 73222472421 T^{15} + 282429536481 T^{16} \)
$5$ \( 1 + 30 T + 306 T^{2} + 920 T^{3} - 569 T^{4} + 85310 T^{5} + 1402636 T^{6} + 18705240 T^{7} + 272699281 T^{8} + 2338155000 T^{9} + 21916187500 T^{10} + 166621093750 T^{11} - 138916015625 T^{12} + 28076171875000 T^{13} + 1167297363281250 T^{14} + 14305114746093750 T^{15} + 59604644775390625 T^{16} \)
$7$ \( 1 - 4 T - 278 T^{2} + 1486 T^{3} - 85843 T^{4} - 588616 T^{5} + 34111706 T^{6} + 31711312 T^{7} + 2545738687 T^{8} + 10876980016 T^{9} + 4013208099194 T^{10} - 23752778737912 T^{11} - 1188177617195443 T^{12} + 7054876403775298 T^{13} - 452698980219104822 T^{14} - 2234183456333136028 T^{15} + \)\(19\!\cdots\!01\)\( T^{16} \)
$11$ 1
$13$ \( 1 + 48 T - 5550 T^{2} - 362926 T^{3} + 4851591 T^{4} + 1163685872 T^{5} + 49432845496 T^{6} - 1273897050480 T^{7} - 189168222289331 T^{8} - 2798751819904560 T^{9} + 238602903535702264 T^{10} + 12340306099992958256 T^{11} + \)\(11\!\cdots\!71\)\( T^{12} - \)\(18\!\cdots\!82\)\( T^{13} - \)\(62\!\cdots\!50\)\( T^{14} + \)\(11\!\cdots\!24\)\( T^{15} + \)\(54\!\cdots\!61\)\( T^{16} \)
$17$ \( 1 + 109 T - 1340 T^{2} + 17028 T^{3} + 61854916 T^{4} + 3258771851 T^{5} + 82359392839 T^{6} + 11257525677000 T^{7} + 1063382720825544 T^{8} + 55308223651101000 T^{9} + 1987955527449468391 T^{10} + \)\(38\!\cdots\!47\)\( T^{11} + \)\(36\!\cdots\!76\)\( T^{12} + \)\(48\!\cdots\!04\)\( T^{13} - \)\(18\!\cdots\!60\)\( T^{14} + \)\(75\!\cdots\!53\)\( T^{15} + \)\(33\!\cdots\!21\)\( T^{16} \)
$19$ \( 1 - 288 T + 30566 T^{2} - 1822176 T^{3} + 176552535 T^{4} - 24586907808 T^{5} + 2222669375260 T^{6} - 119753076108480 T^{7} + 6107895510065021 T^{8} - 821386349028064320 T^{9} + \)\(10\!\cdots\!60\)\( T^{10} - \)\(79\!\cdots\!32\)\( T^{11} + \)\(39\!\cdots\!35\)\( T^{12} - \)\(27\!\cdots\!24\)\( T^{13} + \)\(31\!\cdots\!06\)\( T^{14} - \)\(20\!\cdots\!72\)\( T^{15} + \)\(48\!\cdots\!21\)\( T^{16} \)
$23$ \( ( 1 - 314 T + 61816 T^{2} - 8042722 T^{3} + 950275950 T^{4} - 97855798574 T^{5} + 9150986514424 T^{6} - 565561935699382 T^{7} + 21914624432020321 T^{8} )^{2} \)
$29$ \( 1 - 528 T + 82486 T^{2} + 8724144 T^{3} - 5230024245 T^{4} + 769204136712 T^{5} - 8600199101200 T^{6} - 17285931767133480 T^{7} + 3853264451262183881 T^{8} - \)\(42\!\cdots\!20\)\( T^{9} - \)\(51\!\cdots\!00\)\( T^{10} + \)\(11\!\cdots\!28\)\( T^{11} - \)\(18\!\cdots\!45\)\( T^{12} + \)\(75\!\cdots\!56\)\( T^{13} + \)\(17\!\cdots\!46\)\( T^{14} - \)\(27\!\cdots\!12\)\( T^{15} + \)\(12\!\cdots\!81\)\( T^{16} \)
$31$ \( 1 + 522 T + 160492 T^{2} + 35181052 T^{3} + 6294650003 T^{4} + 763981141902 T^{5} + 33910403803870 T^{6} - 9279992614495420 T^{7} - 2496218287452452949 T^{8} - \)\(27\!\cdots\!20\)\( T^{9} + \)\(30\!\cdots\!70\)\( T^{10} + \)\(20\!\cdots\!42\)\( T^{11} + \)\(49\!\cdots\!83\)\( T^{12} + \)\(82\!\cdots\!52\)\( T^{13} + \)\(11\!\cdots\!72\)\( T^{14} + \)\(10\!\cdots\!82\)\( T^{15} + \)\(62\!\cdots\!21\)\( T^{16} \)
$37$ \( 1 + 406 T - 5318 T^{2} - 27663524 T^{3} - 3521516073 T^{4} + 1758227311414 T^{5} + 498571849360436 T^{6} - 45032760859024008 T^{7} - 32601870976696136623 T^{8} - \)\(22\!\cdots\!24\)\( T^{9} + \)\(12\!\cdots\!24\)\( T^{10} + \)\(22\!\cdots\!78\)\( T^{11} - \)\(23\!\cdots\!13\)\( T^{12} - \)\(92\!\cdots\!32\)\( T^{13} - \)\(89\!\cdots\!22\)\( T^{14} + \)\(34\!\cdots\!22\)\( T^{15} + \)\(43\!\cdots\!61\)\( T^{16} \)
$41$ \( 1 + 329 T - 55160 T^{2} - 17291460 T^{3} + 3867540040 T^{4} + 810769263487 T^{5} + 94427755341103 T^{6} - 42504597447646080 T^{7} - 34767930814290379080 T^{8} - \)\(29\!\cdots\!80\)\( T^{9} + \)\(44\!\cdots\!23\)\( T^{10} + \)\(26\!\cdots\!07\)\( T^{11} + \)\(87\!\cdots\!40\)\( T^{12} - \)\(26\!\cdots\!60\)\( T^{13} - \)\(59\!\cdots\!60\)\( T^{14} + \)\(24\!\cdots\!89\)\( T^{15} + \)\(50\!\cdots\!61\)\( T^{16} \)
$43$ \( ( 1 - 721 T + 420117 T^{2} - 154459221 T^{3} + 51447883420 T^{4} - 12280589284047 T^{5} + 2655712080056733 T^{6} - 362369273206463803 T^{7} + 39959630797262576401 T^{8} )^{2} \)
$47$ \( 1 - 666 T + 215830 T^{2} - 88278972 T^{3} + 49867598541 T^{4} - 15263379264354 T^{5} + 2694775262754914 T^{6} - 1173324858240942840 T^{7} + \)\(55\!\cdots\!99\)\( T^{8} - \)\(12\!\cdots\!20\)\( T^{9} + \)\(29\!\cdots\!06\)\( T^{10} - \)\(17\!\cdots\!18\)\( T^{11} + \)\(57\!\cdots\!81\)\( T^{12} - \)\(10\!\cdots\!96\)\( T^{13} + \)\(27\!\cdots\!70\)\( T^{14} - \)\(86\!\cdots\!02\)\( T^{15} + \)\(13\!\cdots\!81\)\( T^{16} \)
$53$ \( 1 - 414 T + 111938 T^{2} - 31720518 T^{3} + 42174716831 T^{4} - 4836347208574 T^{5} - 2624686559799020 T^{6} + 993826932968752496 T^{7} + \)\(38\!\cdots\!49\)\( T^{8} + \)\(14\!\cdots\!92\)\( T^{9} - \)\(58\!\cdots\!80\)\( T^{10} - \)\(15\!\cdots\!42\)\( T^{11} + \)\(20\!\cdots\!71\)\( T^{12} - \)\(23\!\cdots\!26\)\( T^{13} + \)\(12\!\cdots\!82\)\( T^{14} - \)\(67\!\cdots\!42\)\( T^{15} + \)\(24\!\cdots\!81\)\( T^{16} \)
$59$ \( 1 + 888 T - 72874 T^{2} - 366469944 T^{3} - 125594098825 T^{4} + 29314994105528 T^{5} + 26622351545153740 T^{6} + 1700698924437048800 T^{7} - \)\(25\!\cdots\!59\)\( T^{8} + \)\(34\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!40\)\( T^{10} + \)\(25\!\cdots\!92\)\( T^{11} - \)\(22\!\cdots\!25\)\( T^{12} - \)\(13\!\cdots\!56\)\( T^{13} - \)\(54\!\cdots\!54\)\( T^{14} + \)\(13\!\cdots\!92\)\( T^{15} + \)\(31\!\cdots\!61\)\( T^{16} \)
$61$ \( 1 + 302 T - 374758 T^{2} - 265079458 T^{3} + 17172817983 T^{4} + 53588613835342 T^{5} + 17468090773098500 T^{6} - 4210498173089967520 T^{7} - \)\(41\!\cdots\!19\)\( T^{8} - \)\(95\!\cdots\!20\)\( T^{9} + \)\(89\!\cdots\!00\)\( T^{10} + \)\(62\!\cdots\!22\)\( T^{11} + \)\(45\!\cdots\!43\)\( T^{12} - \)\(15\!\cdots\!58\)\( T^{13} - \)\(51\!\cdots\!98\)\( T^{14} + \)\(93\!\cdots\!22\)\( T^{15} + \)\(70\!\cdots\!41\)\( T^{16} \)
$67$ \( ( 1 - 289 T + 686735 T^{2} - 243308709 T^{3} + 267199450216 T^{4} - 73178257244967 T^{5} + 62120937078828215 T^{6} - 7862688440529239683 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} )^{2} \)
$71$ \( 1 - 1090 T - 143028 T^{2} + 705054350 T^{3} - 262816671537 T^{4} - 195394750802610 T^{5} + 164095529988407974 T^{6} + 23390610332557696900 T^{7} - \)\(62\!\cdots\!45\)\( T^{8} + \)\(83\!\cdots\!00\)\( T^{9} + \)\(21\!\cdots\!54\)\( T^{10} - \)\(89\!\cdots\!10\)\( T^{11} - \)\(43\!\cdots\!17\)\( T^{12} + \)\(41\!\cdots\!50\)\( T^{13} - \)\(30\!\cdots\!08\)\( T^{14} - \)\(82\!\cdots\!90\)\( T^{15} + \)\(26\!\cdots\!81\)\( T^{16} \)
$73$ \( 1 - 253 T + 242148 T^{2} + 153484248 T^{3} + 27413114052 T^{4} + 97831236100713 T^{5} + 36243442544494539 T^{6} + 8871101046105798936 T^{7} + \)\(23\!\cdots\!32\)\( T^{8} + \)\(34\!\cdots\!12\)\( T^{9} + \)\(54\!\cdots\!71\)\( T^{10} + \)\(57\!\cdots\!69\)\( T^{11} + \)\(62\!\cdots\!92\)\( T^{12} + \)\(13\!\cdots\!36\)\( T^{13} + \)\(83\!\cdots\!12\)\( T^{14} - \)\(34\!\cdots\!69\)\( T^{15} + \)\(52\!\cdots\!41\)\( T^{16} \)
$79$ \( 1 + 674 T - 776972 T^{2} - 654255424 T^{3} + 174326887083 T^{4} + 318069877959854 T^{5} + 120607404735943670 T^{6} - 80049450610837979100 T^{7} - \)\(13\!\cdots\!89\)\( T^{8} - \)\(39\!\cdots\!00\)\( T^{9} + \)\(29\!\cdots\!70\)\( T^{10} + \)\(38\!\cdots\!26\)\( T^{11} + \)\(10\!\cdots\!03\)\( T^{12} - \)\(19\!\cdots\!76\)\( T^{13} - \)\(11\!\cdots\!92\)\( T^{14} + \)\(47\!\cdots\!46\)\( T^{15} + \)\(34\!\cdots\!81\)\( T^{16} \)
$83$ \( 1 + 428 T + 177930 T^{2} - 134660776 T^{3} - 247636118949 T^{4} - 379957559050908 T^{5} + 90010500467785276 T^{6} + \)\(15\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!89\)\( T^{8} + \)\(86\!\cdots\!00\)\( T^{9} + \)\(29\!\cdots\!44\)\( T^{10} - \)\(71\!\cdots\!24\)\( T^{11} - \)\(26\!\cdots\!89\)\( T^{12} - \)\(82\!\cdots\!32\)\( T^{13} + \)\(62\!\cdots\!70\)\( T^{14} + \)\(85\!\cdots\!24\)\( T^{15} + \)\(11\!\cdots\!21\)\( T^{16} \)
$89$ \( ( 1 + 1101 T + 2406895 T^{2} + 1914547747 T^{3} + 2334666485008 T^{4} + 1349696810654843 T^{5} + 1196181784307576095 T^{6} + \)\(38\!\cdots\!09\)\( T^{7} + \)\(24\!\cdots\!21\)\( T^{8} )^{2} \)
$97$ \( 1 - 3012 T + 5947742 T^{2} - 10226856716 T^{3} + 16448850705271 T^{4} - 21915763249625172 T^{5} + 26024459406245646260 T^{6} - \)\(28\!\cdots\!48\)\( T^{7} + \)\(29\!\cdots\!09\)\( T^{8} - \)\(26\!\cdots\!04\)\( T^{9} + \)\(21\!\cdots\!40\)\( T^{10} - \)\(16\!\cdots\!24\)\( T^{11} + \)\(11\!\cdots\!11\)\( T^{12} - \)\(64\!\cdots\!88\)\( T^{13} + \)\(34\!\cdots\!38\)\( T^{14} - \)\(15\!\cdots\!64\)\( T^{15} + \)\(48\!\cdots\!81\)\( T^{16} \)
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