Properties

Label 242.4.a.n
Level $242$
Weight $4$
Character orbit 242.a
Self dual yes
Analytic conductor $14.278$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 242 = 2 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 242.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.2784622214\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.978025.2
Defining polynomial: \(x^{4} - 2 x^{3} - 99 x^{2} + 100 x + 2420\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 11 \)
Twist minimal: no (minimal twist has level 22)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 q^{2} + ( 1 - \beta_{1} ) q^{3} + 4 q^{4} + ( 6 - \beta_{2} ) q^{5} + ( -2 + 2 \beta_{1} ) q^{6} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{7} -8 q^{8} + ( 25 - \beta_{1} - 2 \beta_{2} ) q^{9} +O(q^{10})\) \( q -2 q^{2} + ( 1 - \beta_{1} ) q^{3} + 4 q^{4} + ( 6 - \beta_{2} ) q^{5} + ( -2 + 2 \beta_{1} ) q^{6} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{7} -8 q^{8} + ( 25 - \beta_{1} - 2 \beta_{2} ) q^{9} + ( -12 + 2 \beta_{2} ) q^{10} + ( 4 - 4 \beta_{1} ) q^{12} + ( 10 - 6 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{13} + ( 2 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{14} + ( 19 - 9 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} ) q^{15} + 16 q^{16} + ( -14 - 3 \beta_{1} - 4 \beta_{3} ) q^{17} + ( -50 + 2 \beta_{1} + 4 \beta_{2} ) q^{18} + ( -7 - 6 \beta_{1} - 7 \beta_{2} - 5 \beta_{3} ) q^{19} + ( 24 - 4 \beta_{2} ) q^{20} + ( -2 \beta_{1} + 11 \beta_{2} + 14 \beta_{3} ) q^{21} + ( 76 + 6 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} ) q^{23} + ( -8 + 8 \beta_{1} ) q^{24} + ( -6 + 7 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{25} + ( -20 + 12 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{26} + ( 75 - 4 \beta_{1} + 8 \beta_{2} + 6 \beta_{3} ) q^{27} + ( -4 + 4 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{28} + ( -143 - 9 \beta_{1} - 6 \beta_{2} - 5 \beta_{3} ) q^{29} + ( -38 + 18 \beta_{1} - 10 \beta_{2} - 6 \beta_{3} ) q^{30} + ( 48 + 4 \beta_{1} + 7 \beta_{2} - 14 \beta_{3} ) q^{31} -32 q^{32} + ( 28 + 6 \beta_{1} + 8 \beta_{3} ) q^{34} + ( 181 + 19 \beta_{1} - 3 \beta_{2} + 13 \beta_{3} ) q^{35} + ( 100 - 4 \beta_{1} - 8 \beta_{2} ) q^{36} + ( 93 + 27 \beta_{1} + 10 \beta_{2} + 5 \beta_{3} ) q^{37} + ( 14 + 12 \beta_{1} + 14 \beta_{2} + 10 \beta_{3} ) q^{38} + ( 251 - 13 \beta_{1} - 15 \beta_{2} - 19 \beta_{3} ) q^{39} + ( -48 + 8 \beta_{2} ) q^{40} + ( -3 - 7 \beta_{1} + 22 \beta_{2} - 2 \beta_{3} ) q^{41} + ( 4 \beta_{1} - 22 \beta_{2} - 28 \beta_{3} ) q^{42} + ( 188 + 14 \beta_{1} + 12 \beta_{2} + 19 \beta_{3} ) q^{43} + ( 329 + 5 \beta_{1} - 19 \beta_{2} + 9 \beta_{3} ) q^{45} + ( -152 - 12 \beta_{1} + 4 \beta_{2} + 16 \beta_{3} ) q^{46} + ( 93 + 21 \beta_{1} - 20 \beta_{2} - 11 \beta_{3} ) q^{47} + ( 16 - 16 \beta_{1} ) q^{48} + ( 209 + 2 \beta_{1} + 9 \beta_{2} + 4 \beta_{3} ) q^{49} + ( 12 - 14 \beta_{1} + 12 \beta_{2} - 6 \beta_{3} ) q^{50} + ( 35 + 2 \beta_{1} - 2 \beta_{2} - 32 \beta_{3} ) q^{51} + ( 40 - 24 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} ) q^{52} + ( -34 - 8 \beta_{1} - 21 \beta_{2} + 18 \beta_{3} ) q^{53} + ( -150 + 8 \beta_{1} - 16 \beta_{2} - 12 \beta_{3} ) q^{54} + ( 8 - 8 \beta_{1} + 16 \beta_{2} - 8 \beta_{3} ) q^{56} + ( 260 - 29 \beta_{1} + 28 \beta_{2} - 19 \beta_{3} ) q^{57} + ( 286 + 18 \beta_{1} + 12 \beta_{2} + 10 \beta_{3} ) q^{58} + ( 255 - 26 \beta_{1} - 5 \beta_{2} + 9 \beta_{3} ) q^{59} + ( 76 - 36 \beta_{1} + 20 \beta_{2} + 12 \beta_{3} ) q^{60} + ( 230 + 28 \beta_{1} - 13 \beta_{2} + 14 \beta_{3} ) q^{61} + ( -96 - 8 \beta_{1} - 14 \beta_{2} + 28 \beta_{3} ) q^{62} + ( 350 + 48 \beta_{1} - 19 \beta_{2} + 52 \beta_{3} ) q^{63} + 64 q^{64} + ( -13 - 53 \beta_{1} + 22 \beta_{2} - 5 \beta_{3} ) q^{65} + ( 72 - 51 \beta_{1} + 29 \beta_{2} - 30 \beta_{3} ) q^{67} + ( -56 - 12 \beta_{1} - 16 \beta_{3} ) q^{68} + ( -412 - 106 \beta_{1} + 30 \beta_{2} - 58 \beta_{3} ) q^{69} + ( -362 - 38 \beta_{1} + 6 \beta_{2} - 26 \beta_{3} ) q^{70} + ( -300 + 32 \beta_{1} - 23 \beta_{2} + 28 \beta_{3} ) q^{71} + ( -200 + 8 \beta_{1} + 16 \beta_{2} ) q^{72} + ( 297 + 45 \beta_{1} - 54 \beta_{2} + 8 \beta_{3} ) q^{73} + ( -186 - 54 \beta_{1} - 20 \beta_{2} - 10 \beta_{3} ) q^{74} + ( -207 - 3 \beta_{1} + 41 \beta_{2} + 42 \beta_{3} ) q^{75} + ( -28 - 24 \beta_{1} - 28 \beta_{2} - 20 \beta_{3} ) q^{76} + ( -502 + 26 \beta_{1} + 30 \beta_{2} + 38 \beta_{3} ) q^{78} + ( 144 + 36 \beta_{1} - 41 \beta_{2} + 14 \beta_{3} ) q^{79} + ( 96 - 16 \beta_{2} ) q^{80} + ( -344 - 6 \beta_{1} + 24 \beta_{3} ) q^{81} + ( 6 + 14 \beta_{1} - 44 \beta_{2} + 4 \beta_{3} ) q^{82} + ( -384 + 92 \beta_{1} - 5 \beta_{2} - 17 \beta_{3} ) q^{83} + ( -8 \beta_{1} + 44 \beta_{2} + 56 \beta_{3} ) q^{84} + ( -181 - 11 \beta_{1} + 24 \beta_{2} - 31 \beta_{3} ) q^{85} + ( -376 - 28 \beta_{1} - 24 \beta_{2} - 38 \beta_{3} ) q^{86} + ( 264 + 110 \beta_{1} + 17 \beta_{2} - 22 \beta_{3} ) q^{87} + ( -298 + 26 \beta_{1} - 76 \beta_{2} - 15 \beta_{3} ) q^{89} + ( -658 - 10 \beta_{1} + 38 \beta_{2} - 18 \beta_{3} ) q^{90} + ( -547 + 31 \beta_{1} + 45 \beta_{2} + 73 \beta_{3} ) q^{91} + ( 304 + 24 \beta_{1} - 8 \beta_{2} - 32 \beta_{3} ) q^{92} + ( -611 - 69 \beta_{1} - 13 \beta_{2} - 133 \beta_{3} ) q^{93} + ( -186 - 42 \beta_{1} + 40 \beta_{2} + 22 \beta_{3} ) q^{94} + ( 447 + 15 \beta_{1} + 33 \beta_{2} - 11 \beta_{3} ) q^{95} + ( -32 + 32 \beta_{1} ) q^{96} + ( 566 + 20 \beta_{1} + 69 \beta_{2} + 79 \beta_{3} ) q^{97} + ( -418 - 4 \beta_{1} - 18 \beta_{2} - 8 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{2} + 4q^{3} + 16q^{4} + 25q^{5} - 8q^{6} - 3q^{7} - 32q^{8} + 102q^{9} + O(q^{10}) \) \( 4q - 8q^{2} + 4q^{3} + 16q^{4} + 25q^{5} - 8q^{6} - 3q^{7} - 32q^{8} + 102q^{9} - 50q^{10} + 16q^{12} + 41q^{13} + 6q^{14} + 68q^{15} + 64q^{16} - 52q^{17} - 204q^{18} - 16q^{19} + 100q^{20} - 25q^{21} + 314q^{23} - 32q^{24} - 21q^{25} - 82q^{26} + 286q^{27} - 12q^{28} - 561q^{29} - 136q^{30} + 199q^{31} - 128q^{32} + 104q^{34} + 714q^{35} + 408q^{36} + 357q^{37} + 32q^{38} + 1038q^{39} - 200q^{40} - 32q^{41} + 50q^{42} + 721q^{43} + 1326q^{45} - 628q^{46} + 403q^{47} + 64q^{48} + 823q^{49} + 42q^{50} + 174q^{51} + 164q^{52} - 133q^{53} - 572q^{54} + 24q^{56} + 1031q^{57} + 1122q^{58} + 1016q^{59} + 272q^{60} + 919q^{61} - 398q^{62} + 1367q^{63} + 256q^{64} - 69q^{65} + 289q^{67} - 208q^{68} - 1620q^{69} - 1428q^{70} - 1205q^{71} - 816q^{72} + 1234q^{73} - 714q^{74} - 911q^{75} - 64q^{76} - 2076q^{78} + 603q^{79} + 400q^{80} - 1400q^{81} + 64q^{82} - 1514q^{83} - 100q^{84} - 717q^{85} - 1442q^{86} + 1061q^{87} - 1101q^{89} - 2652q^{90} - 2306q^{91} + 1256q^{92} - 2298q^{93} - 806q^{94} + 1766q^{95} - 128q^{96} + 2116q^{97} - 1646q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 99 x^{2} + 100 x + 2420\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{2} + 7 \nu - 54 \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} - 2 \nu^{2} + 57 \nu + 146 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - 9 \nu^{2} - 46 \nu + 400 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 11 \beta_{1} + 6\)\()/11\)
\(\nu^{2}\)\(=\)\((\)\(-7 \beta_{3} - 7 \beta_{2} + 11 \beta_{1} + 552\)\()/11\)
\(\nu^{3}\)\(=\)\((\)\(71 \beta_{3} - 17 \beta_{2} + 605 \beta_{1} + 844\)\()/11\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
8.19378
6.92695
−7.19378
−5.92695
−2.00000 −7.81182 4.00000 14.9181 15.6236 21.7679 −8.00000 34.0245 −29.8363
1.2 −2.00000 −4.30892 4.00000 −8.06215 8.61784 −26.0792 −8.00000 −8.43321 16.1243
1.3 −2.00000 7.57575 4.00000 5.40810 −15.1515 −22.1498 −8.00000 30.3919 −10.8162
1.4 −2.00000 8.54499 4.00000 12.7359 −17.0900 23.4611 −8.00000 46.0168 −25.4718
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(11\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 242.4.a.n 4
3.b odd 2 1 2178.4.a.by 4
4.b odd 2 1 1936.4.a.bn 4
11.b odd 2 1 242.4.a.o 4
11.c even 5 2 22.4.c.b 8
11.c even 5 2 242.4.c.r 8
11.d odd 10 2 242.4.c.n 8
11.d odd 10 2 242.4.c.q 8
33.d even 2 1 2178.4.a.bt 4
33.h odd 10 2 198.4.f.d 8
44.c even 2 1 1936.4.a.bm 4
44.h odd 10 2 176.4.m.b 8
    
        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 1.a even 1 1 trivial
242.4.a.o 4 11.b odd 2 1
242.4.c.n 8 11.d odd 10 2
242.4.c.q 8 11.d odd 10 2
242.4.c.r 8 11.c even 5 2
1936.4.a.bm 4 44.c even 2 1
1936.4.a.bn 4 4.b odd 2 1
2178.4.a.bt 4 33.d even 2 1
2178.4.a.by 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(242))\):

\( T_{3}^{4} - 4 T_{3}^{3} - 97 T_{3}^{2} + 242 T_{3} + 2179 \)
\( T_{5}^{4} - 25 T_{5}^{3} + 73 T_{5}^{2} + 1710 T_{5} - 8284 \)
\( T_{7}^{4} + 3 T_{7}^{3} - 1093 T_{7}^{2} - 1496 T_{7} + 295004 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T )^{4} \)
$3$ \( 1 - 4 T + 11 T^{2} - 82 T^{3} + 1315 T^{4} - 2214 T^{5} + 8019 T^{6} - 78732 T^{7} + 531441 T^{8} \)
$5$ \( 1 - 25 T + 573 T^{2} - 7665 T^{3} + 103716 T^{4} - 958125 T^{5} + 8953125 T^{6} - 48828125 T^{7} + 244140625 T^{8} \)
$7$ \( 1 + 3 T + 279 T^{2} + 1591 T^{3} + 251100 T^{4} + 545713 T^{5} + 32824071 T^{6} + 121060821 T^{7} + 13841287201 T^{8} \)
$11$ 1
$13$ \( 1 - 41 T + 5637 T^{2} - 241261 T^{3} + 15241940 T^{4} - 530050417 T^{5} + 27208722333 T^{6} - 434784474293 T^{7} + 23298085122481 T^{8} \)
$17$ \( 1 + 52 T + 16573 T^{2} + 591848 T^{3} + 112551705 T^{4} + 2907749224 T^{5} + 400031931037 T^{6} + 6166569577844 T^{7} + 582622237229761 T^{8} \)
$19$ \( 1 + 16 T + 14825 T^{2} + 500832 T^{3} + 111187933 T^{4} + 3435206688 T^{5} + 697455185825 T^{6} + 5163003164464 T^{7} + 2213314919066161 T^{8} \)
$23$ \( 1 - 314 T + 61816 T^{2} - 8042722 T^{3} + 950275950 T^{4} - 97855798574 T^{5} + 9150986514424 T^{6} - 565561935699382 T^{7} + 21914624432020321 T^{8} \)
$29$ \( 1 + 561 T + 200635 T^{2} + 47945667 T^{3} + 8699420208 T^{4} + 1169346872463 T^{5} + 119342377008835 T^{6} + 8138508892462509 T^{7} + 353814783205469041 T^{8} \)
$31$ \( 1 - 199 T + 51813 T^{2} + 256853 T^{3} + 389222024 T^{4} + 7651907723 T^{5} + 45984228223653 T^{6} - 5261484809973529 T^{7} + 787662783788549761 T^{8} \)
$37$ \( 1 - 357 T + 160529 T^{2} - 42781809 T^{3} + 12141822140 T^{4} - 2167026971277 T^{5} + 411873494710361 T^{6} - 46396341106842489 T^{7} + 6582952005840035281 T^{8} \)
$41$ \( 1 + 32 T + 194093 T^{2} + 3202364 T^{3} + 18031305245 T^{4} + 220710129244 T^{5} + 921961982448413 T^{6} + 10476221900606752 T^{7} + 22563490300366186081 T^{8} \)
$43$ \( 1 - 721 T + 420117 T^{2} - 154459221 T^{3} + 51447883420 T^{4} - 12280589284047 T^{5} + 2655712080056733 T^{6} - 362369273206463803 T^{7} + 39959630797262576401 T^{8} \)
$47$ \( 1 - 403 T + 357463 T^{2} - 121226987 T^{3} + 52982120160 T^{4} - 12586149471301 T^{5} + 3853170649150327 T^{6} - 451009580660415101 T^{7} + \)\(11\!\cdots\!41\)\( T^{8} \)
$53$ \( 1 + 133 T + 365887 T^{2} + 13770175 T^{3} + 65244311076 T^{4} + 2050062343475 T^{5} + 8109651600406423 T^{6} + 438868557709683689 T^{7} + \)\(49\!\cdots\!41\)\( T^{8} \)
$59$ \( 1 - 1016 T + 1078585 T^{2} - 639078152 T^{3} + 358860597453 T^{4} - 131253231779608 T^{5} + 45495290877177985 T^{6} - 8801603751753418024 T^{7} + \)\(17\!\cdots\!81\)\( T^{8} \)
$61$ \( 1 - 919 T + 1093783 T^{2} - 606559837 T^{3} + 388669851044 T^{4} - 137677558362097 T^{5} + 56352109629697663 T^{6} - 10746920259314575579 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} \)
$67$ \( 1 - 289 T + 686735 T^{2} - 243308709 T^{3} + 267199450216 T^{4} - 73178257244967 T^{5} + 62120937078828215 T^{6} - 7862688440529239683 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} \)
$71$ \( 1 + 1205 T + 1608131 T^{2} + 1210556495 T^{3} + 899763338876 T^{4} + 433271485681945 T^{5} + 206002037682161651 T^{6} + 55247443365731082355 T^{7} + \)\(16\!\cdots\!41\)\( T^{8} \)
$73$ \( 1 - 1234 T + 1514091 T^{2} - 1290056552 T^{3} + 893250631145 T^{4} - 501853929689384 T^{5} + 229133790016138299 T^{6} - 72647537998002604642 T^{7} + \)\(22\!\cdots\!21\)\( T^{8} \)
$79$ \( 1 - 603 T + 1688657 T^{2} - 808010451 T^{3} + 1205184633704 T^{4} - 398380664750589 T^{5} + 410491333377725297 T^{6} - 72270512377518846357 T^{7} + \)\(59\!\cdots\!41\)\( T^{8} \)
$83$ \( 1 + 1514 T + 2070047 T^{2} + 2046213464 T^{3} + 1753896205685 T^{4} + 1169998257940168 T^{5} + 676781939071378343 T^{6} + \)\(28\!\cdots\!42\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \)
$89$ \( 1 + 1101 T + 2406895 T^{2} + 1914547747 T^{3} + 2334666485008 T^{4} + 1349696810654843 T^{5} + 1196181784307576095 T^{6} + \)\(38\!\cdots\!09\)\( T^{7} + \)\(24\!\cdots\!21\)\( T^{8} \)
$97$ \( 1 - 2116 T + 3632073 T^{2} - 4516910940 T^{3} + 5180506457941 T^{4} - 4122462658342620 T^{5} + 3025415128858487817 T^{6} - \)\(16\!\cdots\!72\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} \)
show more
show less