Properties

Label 43.4.a.a
Level 43
Weight 4
Character orbit 43.a
Self dual yes
Analytic conductor 2.537
Analytic rank 1
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 43 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.53708213025\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.45868.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{3} ) q^{2} + ( -3 + \beta_{2} - \beta_{3} ) q^{3} + ( 1 + \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{4} + ( -6 - 3 \beta_{1} - \beta_{3} ) q^{5} + ( -7 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{6} + ( -6 + 8 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{7} + ( -19 - \beta_{1} + 11 \beta_{2} + 8 \beta_{3} ) q^{8} + ( -1 + 5 \beta_{1} - 10 \beta_{2} + 7 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{3} ) q^{2} + ( -3 + \beta_{2} - \beta_{3} ) q^{3} + ( 1 + \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{4} + ( -6 - 3 \beta_{1} - \beta_{3} ) q^{5} + ( -7 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{6} + ( -6 + 8 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{7} + ( -19 - \beta_{1} + 11 \beta_{2} + 8 \beta_{3} ) q^{8} + ( -1 + 5 \beta_{1} - 10 \beta_{2} + 7 \beta_{3} ) q^{9} + ( -2 - \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{10} + ( -12 - 19 \beta_{1} + 5 \beta_{2} + 6 \beta_{3} ) q^{11} + ( 17 - 4 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{12} + ( -2 + 7 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{13} + ( 30 + 6 \beta_{1} - 14 \beta_{2} - 8 \beta_{3} ) q^{14} + ( 25 + 11 \beta_{1} - 19 \beta_{2} + 8 \beta_{3} ) q^{15} + ( 53 - 11 \beta_{1} + \beta_{2} - 22 \beta_{3} ) q^{16} + ( -65 + 15 \beta_{1} + 38 \beta_{2} + 5 \beta_{3} ) q^{17} + ( 77 + 17 \beta_{1} - 26 \beta_{2} - 12 \beta_{3} ) q^{18} + ( 30 - 21 \beta_{1} - 17 \beta_{3} ) q^{19} + ( 14 + 15 \beta_{1} + 10 \beta_{2} + 9 \beta_{3} ) q^{20} + ( -22 - 40 \beta_{1} + 38 \beta_{2} - 6 \beta_{3} ) q^{21} + ( 50 + \beta_{1} + \beta_{2} - 35 \beta_{3} ) q^{22} + ( -20 - 12 \beta_{1} - 11 \beta_{2} + 29 \beta_{3} ) q^{23} + ( 53 + 20 \beta_{1} - 23 \beta_{2} + 25 \beta_{3} ) q^{24} + ( -36 + 34 \beta_{1} + 9 \beta_{2} + 19 \beta_{3} ) q^{25} + ( -12 - \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{26} + ( -51 - 59 \beta_{1} + 45 \beta_{2} - 6 \beta_{3} ) q^{27} + ( -18 - 58 \beta_{1} + 50 \beta_{2} + 52 \beta_{3} ) q^{28} + ( -9 + 42 \beta_{1} - 105 \beta_{2} - 57 \beta_{3} ) q^{29} + ( 77 + 27 \beta_{1} - 35 \beta_{2} + 20 \beta_{3} ) q^{30} + ( 15 + 79 \beta_{1} - 8 \beta_{2} - \beta_{3} ) q^{31} + ( -79 - 15 \beta_{1} - 11 \beta_{2} + 54 \beta_{3} ) q^{32} + ( -8 + 60 \beta_{1} - 60 \beta_{2} + 10 \beta_{3} ) q^{33} + ( 29 - 33 \beta_{1} - 30 \beta_{2} - 118 \beta_{3} ) q^{34} + ( -96 - 14 \beta_{1} + 24 \beta_{2} - 30 \beta_{3} ) q^{35} + ( -113 - 26 \beta_{1} + 99 \beta_{2} + 83 \beta_{3} ) q^{36} + ( -115 - 22 \beta_{1} + 33 \beta_{2} + 7 \beta_{3} ) q^{37} + ( -166 - 17 \beta_{1} + 72 \beta_{2} + 81 \beta_{3} ) q^{38} + ( 26 - 20 \beta_{1} + 14 \beta_{2} + 4 \beta_{3} ) q^{39} + ( 54 + 7 \beta_{1} - 90 \beta_{2} + \beta_{3} ) q^{40} + ( -104 - 66 \beta_{1} - 9 \beta_{2} - 95 \beta_{3} ) q^{41} + ( -102 - 44 \beta_{1} + 58 \beta_{2} - 42 \beta_{3} ) q^{42} + 43 q^{43} + ( -236 + 116 \beta_{1} + 64 \beta_{2} + 106 \beta_{3} ) q^{44} + ( -115 - 25 \beta_{1} + 147 \beta_{2} - 52 \beta_{3} ) q^{45} + ( 274 + 40 \beta_{1} - 75 \beta_{2} - 96 \beta_{3} ) q^{46} + ( 17 + 98 \beta_{1} - 147 \beta_{2} - 113 \beta_{3} ) q^{47} + ( 57 + 80 \beta_{1} - 71 \beta_{2} - 7 \beta_{3} ) q^{48} + ( 93 - 140 \beta_{1} + 4 \beta_{2} + 64 \beta_{3} ) q^{49} + ( 170 + 10 \beta_{1} - 91 \beta_{2} - 102 \beta_{3} ) q^{50} + ( 426 + 59 \beta_{1} - 114 \beta_{2} + 131 \beta_{3} ) q^{51} + ( 70 - 50 \beta_{1} - 6 \beta_{2} - 10 \beta_{3} ) q^{52} + ( -306 - 99 \beta_{1} + 103 \beta_{2} + 52 \beta_{3} ) q^{53} + ( -87 - 51 \beta_{1} + 77 \beta_{2} - 78 \beta_{3} ) q^{54} + ( 304 + 78 \beta_{1} + 66 \beta_{2} + 50 \beta_{3} ) q^{55} + ( 94 - 46 \beta_{1} + 14 \beta_{2} - 160 \beta_{3} ) q^{56} + ( 59 + 97 \beta_{1} - 101 \beta_{2} + 4 \beta_{3} ) q^{57} + ( -237 + 48 \beta_{1} + 129 \beta_{2} + 267 \beta_{3} ) q^{58} + ( 222 - 94 \beta_{1} + 22 \beta_{2} + 12 \beta_{3} ) q^{59} + ( -47 - 33 \beta_{1} + 65 \beta_{2} - 12 \beta_{3} ) q^{60} + ( 98 - 50 \beta_{1} + 30 \beta_{2} + 86 \beta_{3} ) q^{61} + ( -7 + 7 \beta_{1} - 76 \beta_{2} + 26 \beta_{3} ) q^{62} + ( 514 + 30 \beta_{1} - 172 \beta_{2} + 56 \beta_{3} ) q^{63} + ( 109 + 153 \beta_{1} - 155 \beta_{2} - 54 \beta_{3} ) q^{64} + ( -76 - 12 \beta_{1} - 34 \beta_{2} - 12 \beta_{3} ) q^{65} + ( 208 + 70 \beta_{1} - 90 \beta_{2} + 22 \beta_{3} ) q^{66} + ( -26 + 89 \beta_{1} + 125 \beta_{2} - 14 \beta_{3} ) q^{67} + ( -393 - 208 \beta_{1} + 83 \beta_{2} + 373 \beta_{3} ) q^{68} + ( -319 - 55 \beta_{1} + 93 \beta_{2} - 60 \beta_{3} ) q^{69} + ( -192 - 54 \beta_{1} + 104 \beta_{2} - 30 \beta_{3} ) q^{70} + ( 50 + 92 \beta_{1} + 176 \beta_{2} + 116 \beta_{3} ) q^{71} + ( -37 - 152 \beta_{1} - 15 \beta_{2} - 365 \beta_{3} ) q^{72} + ( 122 - 94 \beta_{1} + 234 \beta_{2} + 130 \beta_{3} ) q^{73} + ( 105 - 26 \beta_{1} + \beta_{2} - 169 \beta_{3} ) q^{74} + ( 15 - 113 \beta_{1} + 115 \beta_{2} + 16 \beta_{3} ) q^{75} + ( 430 + 177 \beta_{1} - 226 \beta_{2} - 345 \beta_{3} ) q^{76} + ( -548 + 282 \beta_{1} - 134 \beta_{2} - 242 \beta_{3} ) q^{77} + ( -22 - 10 \beta_{1} + 8 \beta_{2} ) q^{78} + ( 320 - 29 \beta_{1} - 156 \beta_{2} + 135 \beta_{3} ) q^{79} + ( 22 - 29 \beta_{1} - 90 \beta_{2} + 69 \beta_{3} ) q^{80} + ( 496 + 189 \beta_{1} - 117 \beta_{2} - 36 \beta_{3} ) q^{81} + ( -638 - 86 \beta_{1} + 351 \beta_{2} + 190 \beta_{3} ) q^{82} + ( -212 - 41 \beta_{1} - 91 \beta_{2} - 276 \beta_{3} ) q^{83} + ( -174 + 220 \beta_{1} - 134 \beta_{2} + 14 \beta_{3} ) q^{84} + ( 87 + 39 \beta_{1} - 539 \beta_{2} + 38 \beta_{3} ) q^{85} + ( -43 + 43 \beta_{3} ) q^{86} + ( -96 - 327 \beta_{1} + 204 \beta_{2} - 87 \beta_{3} ) q^{87} + ( 556 + 34 \beta_{1} - 442 \beta_{2} - 338 \beta_{3} ) q^{88} + ( -222 + 198 \beta_{1} - 48 \beta_{2} + 228 \beta_{3} ) q^{89} + ( -595 - 199 \beta_{1} + 181 \beta_{2} - 106 \beta_{3} ) q^{90} + ( 232 - 154 \beta_{1} + 78 \beta_{2} + 70 \beta_{3} ) q^{91} + ( -732 + 75 \beta_{1} + 336 \beta_{2} + 405 \beta_{3} ) q^{92} + ( -12 - 259 \beta_{1} + 272 \beta_{2} - 29 \beta_{3} ) q^{93} + ( -627 + 34 \beta_{1} + 241 \beta_{2} + 503 \beta_{3} ) q^{94} + ( 271 + 62 \beta_{1} + 3 \beta_{2} + 101 \beta_{3} ) q^{95} + ( -395 - 96 \beta_{1} + 125 \beta_{2} - 51 \beta_{3} ) q^{96} + ( -468 + 170 \beta_{1} - 63 \beta_{2} - 241 \beta_{3} ) q^{97} + ( 411 + 60 \beta_{1} - 52 \beta_{2} - 103 \beta_{3} ) q^{98} + ( -112 + 133 \beta_{1} + 257 \beta_{2} - 294 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 11q^{3} + 2q^{4} - 27q^{5} - 27q^{6} - 20q^{7} - 66q^{8} - 9q^{9} + O(q^{10}) \) \( 4q - 4q^{2} - 11q^{3} + 2q^{4} - 27q^{5} - 27q^{6} - 20q^{7} - 66q^{8} - 9q^{9} - 3q^{10} - 62q^{11} + 61q^{12} - 2q^{13} + 112q^{14} + 92q^{15} + 202q^{16} - 207q^{17} + 299q^{18} + 99q^{19} + 81q^{20} - 90q^{21} + 202q^{22} - 103q^{23} + 209q^{24} - 101q^{25} - 50q^{26} - 218q^{27} - 80q^{28} - 99q^{29} + 300q^{30} + 131q^{31} - 342q^{32} - 32q^{33} + 53q^{34} - 374q^{35} - 379q^{36} - 449q^{37} - 609q^{38} + 98q^{39} + 133q^{40} - 491q^{41} - 394q^{42} + 172q^{43} - 764q^{44} - 338q^{45} + 1061q^{46} + 19q^{47} + 237q^{48} + 236q^{49} + 599q^{50} + 1649q^{51} + 224q^{52} - 1220q^{53} - 322q^{54} + 1360q^{55} + 344q^{56} + 232q^{57} - 771q^{58} + 816q^{59} - 156q^{60} + 372q^{61} - 97q^{62} + 1914q^{63} + 434q^{64} - 350q^{65} + 812q^{66} + 110q^{67} - 1697q^{68} - 1238q^{69} - 718q^{70} + 468q^{71} - 315q^{72} + 628q^{73} + 395q^{74} + 62q^{75} + 1671q^{76} - 2044q^{77} - 90q^{78} + 1095q^{79} - 31q^{80} + 2056q^{81} - 2287q^{82} - 980q^{83} - 610q^{84} - 152q^{85} - 172q^{86} - 507q^{87} + 1816q^{88} - 738q^{89} - 2398q^{90} + 852q^{91} - 2517q^{92} - 35q^{93} - 2233q^{94} + 1149q^{95} - 1551q^{96} - 1765q^{97} + 1652q^{98} - 58q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 10 x^{2} + 11 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + \nu^{2} - 8 \nu - 2 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{3} + \nu^{2} + 19 \nu - 11 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 2 \beta_{2} - \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(-\beta_{3} + \beta_{2} + 9 \beta_{1} - 3\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.0844804
3.05867
−3.18808
1.21390
−5.19893 0.759721 19.0289 −1.54763 −3.94973 −13.3169 −57.3382 −26.4228 8.04602
1.2 −1.25341 1.08716 −6.42897 −14.9226 −1.36266 2.62749 18.0854 −25.8181 18.7041
1.3 0.132290 −3.71054 −7.98250 2.43196 −0.490867 −30.9271 −2.11433 −13.2319 0.321724
1.4 2.32005 −9.13635 −2.61739 −12.9617 −21.1967 21.6165 −24.6328 56.4728 −30.0718
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 43.4.a.a 4
3.b odd 2 1 387.4.a.e 4
4.b odd 2 1 688.4.a.f 4
5.b even 2 1 1075.4.a.a 4
7.b odd 2 1 2107.4.a.b 4
43.b odd 2 1 1849.4.a.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.4.a.a 4 1.a even 1 1 trivial
387.4.a.e 4 3.b odd 2 1
688.4.a.f 4 4.b odd 2 1
1075.4.a.a 4 5.b even 2 1
1849.4.a.b 4 43.b odd 2 1
2107.4.a.b 4 7.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(43\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 4 T_{2}^{3} - 9 T_{2}^{2} - 14 T_{2} + 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(43))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T + 23 T^{2} + 82 T^{3} + 242 T^{4} + 656 T^{5} + 1472 T^{6} + 2048 T^{7} + 4096 T^{8} \)
$3$ \( 1 + 11 T + 119 T^{2} + 839 T^{3} + 4996 T^{4} + 22653 T^{5} + 86751 T^{6} + 216513 T^{7} + 531441 T^{8} \)
$5$ \( 1 + 27 T + 665 T^{2} + 9849 T^{3} + 134272 T^{4} + 1231125 T^{5} + 10390625 T^{6} + 52734375 T^{7} + 244140625 T^{8} \)
$7$ \( 1 + 20 T + 768 T^{2} + 13108 T^{3} + 314942 T^{4} + 4496044 T^{5} + 90354432 T^{6} + 807072140 T^{7} + 13841287201 T^{8} \)
$11$ \( 1 + 62 T + 3077 T^{2} + 39962 T^{3} + 1317884 T^{4} + 53189422 T^{5} + 5451093197 T^{6} + 146192756842 T^{7} + 3138428376721 T^{8} \)
$13$ \( 1 + 2 T + 8273 T^{2} + 17238 T^{3} + 26690692 T^{4} + 37871886 T^{5} + 39932190857 T^{6} + 21208998746 T^{7} + 23298085122481 T^{8} \)
$17$ \( 1 + 207 T + 14258 T^{2} - 190032 T^{3} - 71042783 T^{4} - 933627216 T^{5} + 344153458802 T^{6} + 24547690434879 T^{7} + 582622237229761 T^{8} \)
$19$ \( 1 - 99 T + 22201 T^{2} - 1662075 T^{3} + 222917608 T^{4} - 11400172425 T^{5} + 1044465604081 T^{6} - 31946082080121 T^{7} + 2213314919066161 T^{8} \)
$23$ \( 1 + 103 T + 33718 T^{2} + 2793222 T^{3} + 586593953 T^{4} + 33985132074 T^{5} + 4991474105302 T^{6} + 185518724130689 T^{7} + 21914624432020321 T^{8} \)
$29$ \( 1 + 99 T - 13567 T^{2} - 167967 T^{3} + 888213804 T^{4} - 4096547163 T^{5} - 8069967996007 T^{6} + 1436207451611031 T^{7} + 353814783205469041 T^{8} \)
$31$ \( 1 - 131 T + 64436 T^{2} - 3696474 T^{3} + 1950279047 T^{4} - 110121656934 T^{5} + 57187187188916 T^{6} - 3463590503047901 T^{7} + 787662783788549761 T^{8} \)
$37$ \( 1 + 449 T + 266817 T^{2} + 71392239 T^{3} + 21942685768 T^{4} + 3616231082067 T^{5} + 684579423270153 T^{6} + 58352821167989573 T^{7} + 6582952005840035281 T^{8} \)
$41$ \( 1 + 491 T + 187430 T^{2} + 55413200 T^{3} + 16109516117 T^{4} + 3819133157200 T^{5} + 890312037890630 T^{6} + 160744529787434851 T^{7} + 22563490300366186081 T^{8} \)
$43$ \( ( 1 - 43 T )^{4} \)
$47$ \( 1 - 19 T + 108577 T^{2} - 15073647 T^{3} + 11126438636 T^{4} - 1564991252481 T^{5} + 1170374862776833 T^{6} - 21263478988952573 T^{7} + \)\(11\!\cdots\!41\)\( T^{8} \)
$53$ \( 1 + 1220 T + 1002413 T^{2} + 551938550 T^{3} + 246387730196 T^{4} + 82170955508350 T^{5} + 22217843732404277 T^{6} + 4025711581998602260 T^{7} + \)\(49\!\cdots\!41\)\( T^{8} \)
$59$ \( 1 - 816 T + 987944 T^{2} - 513131184 T^{3} + 321570111646 T^{4} - 105386369438736 T^{5} + 41672005127424104 T^{6} - 7069004588022430224 T^{7} + \)\(17\!\cdots\!81\)\( T^{8} \)
$61$ \( 1 - 372 T + 843468 T^{2} - 226528412 T^{3} + 278949483798 T^{4} - 51417645484172 T^{5} + 43455787121523948 T^{6} - 4350222346534300452 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} \)
$67$ \( 1 - 110 T + 855641 T^{2} - 172379298 T^{3} + 331907899304 T^{4} - 51845314804374 T^{5} + 77399900577465329 T^{6} - 2992718783592444170 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} \)
$71$ \( 1 - 468 T + 925804 T^{2} - 494793108 T^{3} + 407494264950 T^{4} - 177091896077388 T^{5} + 118595755255197484 T^{6} - 21457098336234146508 T^{7} + \)\(16\!\cdots\!41\)\( T^{8} \)
$73$ \( 1 - 628 T + 1127772 T^{2} - 544172556 T^{3} + 627880103782 T^{4} - 211692375217452 T^{5} + 170670503050398108 T^{6} - 36971356452792249364 T^{7} + \)\(22\!\cdots\!21\)\( T^{8} \)
$79$ \( 1 - 1095 T + 1663851 T^{2} - 1399726691 T^{3} + 1195813590120 T^{4} - 690119848003949 T^{5} + 404461305956071371 T^{6} - \)\(13\!\cdots\!05\)\( T^{7} + \)\(59\!\cdots\!41\)\( T^{8} \)
$83$ \( 1 + 980 T + 1568765 T^{2} + 705601262 T^{3} + 896947637720 T^{4} + 403453628795194 T^{5} + 512892614828219285 T^{6} + \)\(18\!\cdots\!40\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \)
$89$ \( 1 + 738 T + 1788008 T^{2} + 1031145174 T^{3} + 1620376904526 T^{4} + 726925382169606 T^{5} + 888606524088595688 T^{6} + \)\(25\!\cdots\!42\)\( T^{7} + \)\(24\!\cdots\!21\)\( T^{8} \)
$97$ \( 1 + 1765 T + 3796458 T^{2} + 4179032168 T^{3} + 5180997869897 T^{4} + 3814089825865064 T^{5} + 3162343231888741482 T^{6} + \)\(13\!\cdots\!05\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} \)
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