Properties

Label 2178.4.a.by
Level $2178$
Weight $4$
Character orbit 2178.a
Self dual yes
Analytic conductor $128.506$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2178.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(128.506159993\)
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: \(\Z[a_1, \ldots, a_{17}]\)
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.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + ( -6 + \beta_{2} ) q^{5} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{7} + 8 q^{8} +O(q^{10})\) \( q + 2 q^{2} + 4 q^{4} + ( -6 + \beta_{2} ) q^{5} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{7} + 8 q^{8} + ( -12 + 2 \beta_{2} ) q^{10} + ( 10 + 6 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{13} + ( -2 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{14} + 16 q^{16} + ( 14 - 3 \beta_{1} - 4 \beta_{3} ) q^{17} + ( -7 + 6 \beta_{1} - 7 \beta_{2} + 5 \beta_{3} ) q^{19} + ( -24 + 4 \beta_{2} ) q^{20} + ( -76 + 6 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} ) q^{23} + ( -6 - 7 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{25} + ( 20 + 12 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{26} + ( -4 - 4 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{28} + ( 143 - 9 \beta_{1} + 6 \beta_{2} - 5 \beta_{3} ) q^{29} + ( 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} + ( 93 - 27 \beta_{1} + 10 \beta_{2} - 5 \beta_{3} ) q^{37} + ( -14 + 12 \beta_{1} - 14 \beta_{2} + 10 \beta_{3} ) q^{38} + ( -48 + 8 \beta_{2} ) q^{40} + ( 3 - 7 \beta_{1} - 22 \beta_{2} - 2 \beta_{3} ) q^{41} + ( 188 - 14 \beta_{1} + 12 \beta_{2} - 19 \beta_{3} ) q^{43} + ( -152 + 12 \beta_{1} + 4 \beta_{2} - 16 \beta_{3} ) q^{46} + ( -93 + 21 \beta_{1} + 20 \beta_{2} - 11 \beta_{3} ) q^{47} + ( 209 - 2 \beta_{1} + 9 \beta_{2} - 4 \beta_{3} ) q^{49} + ( -12 - 14 \beta_{1} - 12 \beta_{2} - 6 \beta_{3} ) q^{50} + ( 40 + 24 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} ) q^{52} + ( 34 - 8 \beta_{1} + 21 \beta_{2} + 18 \beta_{3} ) q^{53} + ( -8 - 8 \beta_{1} - 16 \beta_{2} - 8 \beta_{3} ) q^{56} + ( 286 - 18 \beta_{1} + 12 \beta_{2} - 10 \beta_{3} ) q^{58} + ( -255 - 26 \beta_{1} + 5 \beta_{2} + 9 \beta_{3} ) q^{59} + ( 230 - 28 \beta_{1} - 13 \beta_{2} - 14 \beta_{3} ) q^{61} + ( 96 - 8 \beta_{1} + 14 \beta_{2} + 28 \beta_{3} ) q^{62} + 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} + ( -362 + 38 \beta_{1} + 6 \beta_{2} + 26 \beta_{3} ) q^{70} + ( 300 + 32 \beta_{1} + 23 \beta_{2} + 28 \beta_{3} ) q^{71} + ( 297 - 45 \beta_{1} - 54 \beta_{2} - 8 \beta_{3} ) q^{73} + ( 186 - 54 \beta_{1} + 20 \beta_{2} - 10 \beta_{3} ) q^{74} + ( -28 + 24 \beta_{1} - 28 \beta_{2} + 20 \beta_{3} ) q^{76} + ( 144 - 36 \beta_{1} - 41 \beta_{2} - 14 \beta_{3} ) q^{79} + ( -96 + 16 \beta_{2} ) q^{80} + ( 6 - 14 \beta_{1} - 44 \beta_{2} - 4 \beta_{3} ) q^{82} + ( 384 + 92 \beta_{1} + 5 \beta_{2} - 17 \beta_{3} ) q^{83} + ( -181 + 11 \beta_{1} + 24 \beta_{2} + 31 \beta_{3} ) q^{85} + ( 376 - 28 \beta_{1} + 24 \beta_{2} - 38 \beta_{3} ) q^{86} + ( 298 + 26 \beta_{1} + 76 \beta_{2} - 15 \beta_{3} ) q^{89} + ( -547 - 31 \beta_{1} + 45 \beta_{2} - 73 \beta_{3} ) q^{91} + ( -304 + 24 \beta_{1} + 8 \beta_{2} - 32 \beta_{3} ) q^{92} + ( -186 + 42 \beta_{1} + 40 \beta_{2} - 22 \beta_{3} ) q^{94} + ( -447 + 15 \beta_{1} - 33 \beta_{2} - 11 \beta_{3} ) q^{95} + ( 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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{2} + 16q^{4} - 25q^{5} - 3q^{7} + 32q^{8} + O(q^{10}) \) \( 4q + 8q^{2} + 16q^{4} - 25q^{5} - 3q^{7} + 32q^{8} - 50q^{10} + 41q^{13} - 6q^{14} + 64q^{16} + 52q^{17} - 16q^{19} - 100q^{20} - 314q^{23} - 21q^{25} + 82q^{26} - 12q^{28} + 561q^{29} + 199q^{31} + 128q^{32} + 104q^{34} - 714q^{35} + 357q^{37} - 32q^{38} - 200q^{40} + 32q^{41} + 721q^{43} - 628q^{46} - 403q^{47} + 823q^{49} - 42q^{50} + 164q^{52} + 133q^{53} - 24q^{56} + 1122q^{58} - 1016q^{59} + 919q^{61} + 398q^{62} + 256q^{64} + 69q^{65} + 289q^{67} + 208q^{68} - 1428q^{70} + 1205q^{71} + 1234q^{73} + 714q^{74} - 64q^{76} + 603q^{79} - 400q^{80} + 64q^{82} + 1514q^{83} - 717q^{85} + 1442q^{86} + 1101q^{89} - 2306q^{91} - 1256q^{92} - 806q^{94} - 1766q^{95} + 2116q^{97} + 1646q^{98} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{2} + 9 \nu + 46 \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 5 \nu^{2} - 50 \nu + 200 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 6 \nu^{2} - 61 \nu - 346 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{2} + 11 \beta_{1} + 5\)\()/11\)
\(\nu^{2}\)\(=\)\((\)\(9 \beta_{3} - 9 \beta_{2} + 11 \beta_{1} + 551\)\()/11\)
\(\nu^{3}\)\(=\)\((\)\(95 \beta_{3} - 7 \beta_{2} + 605 \beta_{1} + 805\)\()/11\)

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
−7.19378
6.92695
8.19378
−5.92695
2.00000 0 4.00000 −14.9181 0 21.7679 8.00000 0 −29.8363
1.2 2.00000 0 4.00000 −12.7359 0 23.4611 8.00000 0 −25.4718
1.3 2.00000 0 4.00000 −5.40810 0 −22.1498 8.00000 0 −10.8162
1.4 2.00000 0 4.00000 8.06215 0 −26.0792 8.00000 0 16.1243
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(11\) \(1\)

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 2178.4.a.by 4
3.b odd 2 1 242.4.a.n 4
11.b odd 2 1 2178.4.a.bt 4
11.c even 5 2 198.4.f.d 8
12.b even 2 1 1936.4.a.bn 4
33.d even 2 1 242.4.a.o 4
33.f even 10 2 242.4.c.n 8
33.f even 10 2 242.4.c.q 8
33.h odd 10 2 22.4.c.b 8
33.h odd 10 2 242.4.c.r 8
132.d odd 2 1 1936.4.a.bm 4
132.o even 10 2 176.4.m.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.4.c.b 8 33.h odd 10 2
176.4.m.b 8 132.o even 10 2
198.4.f.d 8 11.c even 5 2
242.4.a.n 4 3.b odd 2 1
242.4.a.o 4 33.d even 2 1
242.4.c.n 8 33.f even 10 2
242.4.c.q 8 33.f even 10 2
242.4.c.r 8 33.h odd 10 2
1936.4.a.bm 4 132.d odd 2 1
1936.4.a.bn 4 12.b even 2 1
2178.4.a.bt 4 11.b odd 2 1
2178.4.a.by 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2178))\):

\( T_{5}^{4} + 25 T_{5}^{3} + 73 T_{5}^{2} - 1710 T_{5} - 8284 \)
\( T_{7}^{4} + 3 T_{7}^{3} - 1093 T_{7}^{2} - 1496 T_{7} + 295004 \)
\( T_{17}^{4} - 52 T_{17}^{3} - 3079 T_{17}^{2} + 174580 T_{17} - 2019455 \)

Hecke characteristic polynomials

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