Properties

Label 9.4.c.a
Level 9
Weight 4
Character orbit 9.c
Analytic conductor 0.531
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 9.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.531017190052\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} - \beta_{3} ) q^{2} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{3} + ( -4 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{4} + ( -7 + 8 \beta_{1} + \beta_{2} - \beta_{3} ) q^{5} + ( 12 - 15 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{6} + ( -2 \beta_{1} - 3 \beta_{3} ) q^{7} + ( 16 - \beta_{2} ) q^{8} + ( -3 + 21 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} - \beta_{3} ) q^{2} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{3} + ( -4 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{4} + ( -7 + 8 \beta_{1} + \beta_{2} - \beta_{3} ) q^{5} + ( 12 - 15 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{6} + ( -2 \beta_{1} - 3 \beta_{3} ) q^{7} + ( 16 - \beta_{2} ) q^{8} + ( -3 + 21 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{9} + ( 6 + 6 \beta_{2} ) q^{10} + ( -37 \beta_{1} + 8 \beta_{3} ) q^{11} + ( -52 + 22 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{12} + ( 13 + 2 \beta_{1} + 15 \beta_{2} - 15 \beta_{3} ) q^{13} + ( -34 + 26 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{14} + ( 9 - 24 \beta_{1} - 15 \beta_{2} + 9 \beta_{3} ) q^{15} + ( -\beta_{1} + 9 \beta_{3} ) q^{16} + ( 54 + 9 \beta_{2} ) q^{17} + ( 72 + 27 \beta_{2} - 18 \beta_{3} ) q^{18} + ( -52 - 27 \beta_{2} ) q^{19} + ( 16 \beta_{1} - 20 \beta_{3} ) q^{20} + ( 34 - 46 \beta_{1} + 8 \beta_{2} - 7 \beta_{3} ) q^{21} + ( 6 - 27 \beta_{1} - 21 \beta_{2} + 21 \beta_{3} ) q^{22} + ( -7 + 26 \beta_{1} + 19 \beta_{2} - 19 \beta_{3} ) q^{23} + ( -24 + 9 \beta_{1} + 18 \beta_{2} + 15 \beta_{3} ) q^{24} + ( 53 \beta_{1} + 15 \beta_{3} ) q^{25} + ( -146 - 28 \beta_{2} ) q^{26} + ( -117 - 18 \beta_{1} - 18 \beta_{2} + 36 \beta_{3} ) q^{27} + ( 92 + 18 \beta_{2} ) q^{28} + ( 26 \beta_{1} - \beta_{3} ) q^{29} + ( 42 + 48 \beta_{1} - 6 \beta_{2} + 12 \beta_{3} ) q^{30} + ( -23 + 20 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{31} + ( 216 - 207 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} ) q^{32} + ( -6 + 165 \beta_{1} + 21 \beta_{2} - 66 \beta_{3} ) q^{33} + ( -117 \beta_{1} - 63 \beta_{3} ) q^{34} + ( 11 + 19 \beta_{2} ) q^{35} + ( -12 - 141 \beta_{1} - 60 \beta_{2} - 15 \beta_{3} ) q^{36} + ( 2 + 54 \beta_{2} ) q^{37} + ( 241 \beta_{1} + 79 \beta_{3} ) q^{38} + ( 253 - 124 \beta_{1} + 11 \beta_{2} + 17 \beta_{3} ) q^{39} + ( -120 + 144 \beta_{1} + 24 \beta_{2} - 24 \beta_{3} ) q^{40} + ( -115 + 17 \beta_{1} - 98 \beta_{2} + 98 \beta_{3} ) q^{41} + ( -162 + 12 \beta_{1} - 60 \beta_{2} + 18 \beta_{3} ) q^{42} + ( -47 \beta_{1} + 6 \beta_{3} ) q^{43} + ( -76 + 79 \beta_{2} ) q^{44} + ( -171 + 72 \beta_{1} + 45 \beta_{2} - 45 \beta_{3} ) q^{45} + ( -138 - 12 \beta_{2} ) q^{46} + ( -154 \beta_{1} - 91 \beta_{3} ) q^{47} + ( -88 + 145 \beta_{1} - 17 \beta_{2} + 7 \beta_{3} ) q^{48} + ( 246 - 267 \beta_{1} - 21 \beta_{2} + 21 \beta_{3} ) q^{49} + ( 256 - 173 \beta_{1} + 83 \beta_{2} - 83 \beta_{3} ) q^{50} + ( 18 + 117 \beta_{1} + 36 \beta_{2} + 63 \beta_{3} ) q^{51} + ( 358 \beta_{1} + 54 \beta_{3} ) q^{52} + ( -54 - 162 \beta_{2} ) q^{53} + ( 324 - 27 \beta_{1} + 54 \beta_{2} + 81 \beta_{3} ) q^{54} + ( 267 - 93 \beta_{2} ) q^{55} + ( -10 \beta_{1} - 46 \beta_{3} ) q^{56} + ( -164 - 241 \beta_{1} + 2 \beta_{2} - 79 \beta_{3} ) q^{57} + ( 42 - 18 \beta_{1} + 24 \beta_{2} - 24 \beta_{3} ) q^{58} + ( -331 + 467 \beta_{1} + 136 \beta_{2} - 136 \beta_{3} ) q^{59} + ( 168 - 336 \beta_{1} + 24 \beta_{2} + 12 \beta_{3} ) q^{60} + ( -272 \beta_{1} + 105 \beta_{3} ) q^{61} + ( 70 + 26 \beta_{2} ) q^{62} + ( 192 + 24 \beta_{1} + 78 \beta_{2} - 57 \beta_{3} ) q^{63} + ( -440 - 153 \beta_{2} ) q^{64} + ( -136 \beta_{1} + 107 \beta_{3} ) q^{65} + ( -330 + 222 \beta_{1} + 33 \beta_{2} - 48 \beta_{3} ) q^{66} + ( -527 + 461 \beta_{1} - 66 \beta_{2} + 66 \beta_{3} ) q^{67} + ( -432 + 261 \beta_{1} - 171 \beta_{2} + 171 \beta_{3} ) q^{68} + ( 297 - 204 \beta_{1} - 33 \beta_{2} + 45 \beta_{3} ) q^{69} + ( -144 \beta_{1} - 30 \beta_{3} ) q^{70} + ( 756 + 144 \beta_{2} ) q^{71} + ( 333 \beta_{1} - 99 \beta_{2} + 27 \beta_{3} ) q^{72} + ( -106 + 243 \beta_{2} ) q^{73} + ( -380 \beta_{1} - 56 \beta_{3} ) q^{74} + ( -256 + 187 \beta_{1} - 83 \beta_{2} + 121 \beta_{3} ) q^{75} + ( 856 - 673 \beta_{1} + 183 \beta_{2} - 183 \beta_{3} ) q^{76} + ( 47 - 118 \beta_{1} - 71 \beta_{2} + 71 \beta_{3} ) q^{77} + ( -78 - 342 \beta_{1} - 90 \beta_{2} - 174 \beta_{3} ) q^{78} + ( 556 \beta_{1} - 309 \beta_{3} ) q^{79} + ( 16 - 64 \beta_{2} ) q^{80} + ( -351 + 351 \beta_{1} - 135 \beta_{2} - 135 \beta_{3} ) q^{81} + ( 1014 + 213 \beta_{2} ) q^{82} + ( -460 \beta_{1} + 107 \beta_{3} ) q^{83} + ( 52 + 218 \beta_{1} + 56 \beta_{2} + 110 \beta_{3} ) q^{84} + ( -306 + 288 \beta_{1} - 18 \beta_{2} + 18 \beta_{3} ) q^{85} + ( -34 - \beta_{1} - 35 \beta_{2} + 35 \beta_{3} ) q^{86} + ( -42 - 42 \beta_{1} - 24 \beta_{2} + 51 \beta_{3} ) q^{87} + ( -693 \beta_{1} + 165 \beta_{3} ) q^{88} + ( -162 + 72 \beta_{2} ) q^{89} + ( -306 + 144 \beta_{1} - 18 \beta_{2} + 144 \beta_{3} ) q^{90} + ( -425 - 69 \beta_{2} ) q^{91} + ( 430 \beta_{1} - 2 \beta_{3} ) q^{92} + ( -71 - 16 \beta_{1} - 43 \beta_{2} + 17 \beta_{3} ) q^{93} + ( -1218 + 882 \beta_{1} - 336 \beta_{2} + 336 \beta_{3} ) q^{94} + ( 148 + 16 \beta_{1} + 164 \beta_{2} - 164 \beta_{3} ) q^{95} + ( 360 + 342 \beta_{1} + 423 \beta_{2} - 198 \beta_{3} ) q^{96} + ( -317 \beta_{1} - 102 \beta_{3} ) q^{97} + ( -324 - 225 \beta_{2} ) q^{98} + ( 504 - 1080 \beta_{1} - 81 \beta_{2} + 279 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 3q^{2} - 3q^{3} - 5q^{4} - 15q^{5} + 9q^{6} - 7q^{7} + 66q^{8} + 45q^{9} + O(q^{10}) \) \( 4q - 3q^{2} - 3q^{3} - 5q^{4} - 15q^{5} + 9q^{6} - 7q^{7} + 66q^{8} + 45q^{9} + 12q^{10} - 66q^{11} - 156q^{12} + 11q^{13} - 60q^{14} + 27q^{15} + 7q^{16} + 198q^{17} + 216q^{18} - 154q^{19} + 12q^{20} + 21q^{21} + 33q^{22} - 33q^{23} - 99q^{24} + 121q^{25} - 528q^{26} - 432q^{27} + 332q^{28} + 51q^{29} + 288q^{30} - 43q^{31} + 423q^{32} + 198q^{33} - 297q^{34} + 6q^{35} - 225q^{36} - 100q^{37} + 561q^{38} + 759q^{39} - 264q^{40} - 132q^{41} - 486q^{42} - 88q^{43} - 462q^{44} - 675q^{45} - 528q^{46} - 399q^{47} - 21q^{48} + 513q^{49} + 429q^{50} + 297q^{51} + 770q^{52} + 108q^{53} + 1215q^{54} + 1254q^{55} - 66q^{56} - 1221q^{57} + 60q^{58} - 798q^{59} - 36q^{60} - 439q^{61} + 228q^{62} + 603q^{63} - 1454q^{64} - 165q^{65} - 990q^{66} - 988q^{67} - 693q^{68} + 891q^{69} - 318q^{70} + 2736q^{71} + 891q^{72} - 910q^{73} - 816q^{74} - 363q^{75} + 1529q^{76} + 165q^{77} - 990q^{78} + 803q^{79} + 192q^{80} - 567q^{81} + 3630q^{82} - 813q^{83} + 642q^{84} - 594q^{85} - 33q^{86} - 153q^{87} - 1221q^{88} - 792q^{89} - 756q^{90} - 1562q^{91} + 858q^{92} - 213q^{93} - 2100q^{94} + 132q^{95} + 1080q^{96} - 736q^{97} - 846q^{98} + 297q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 3 \)\()/6\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 5 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{3} + \nu^{2} + 2 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 2 \beta_{1} + 2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 1\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(4 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 11\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1 + \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
1.68614 + 0.396143i
−1.18614 1.26217i
1.68614 0.396143i
−1.18614 + 1.26217i
−2.18614 3.78651i 3.55842 + 3.78651i −5.55842 + 9.62747i −2.31386 + 4.00772i 6.55842 21.7518i −6.05842 10.4935i 13.6277 −1.67527 + 26.9480i 20.2337
4.2 0.686141 + 1.18843i −5.05842 1.18843i 3.05842 5.29734i −5.18614 + 8.98266i −2.05842 6.82701i 2.55842 + 4.43132i 19.3723 24.1753 + 12.0232i −14.2337
7.1 −2.18614 + 3.78651i 3.55842 3.78651i −5.55842 9.62747i −2.31386 4.00772i 6.55842 + 21.7518i −6.05842 + 10.4935i 13.6277 −1.67527 26.9480i 20.2337
7.2 0.686141 1.18843i −5.05842 + 1.18843i 3.05842 + 5.29734i −5.18614 8.98266i −2.05842 + 6.82701i 2.55842 4.43132i 19.3723 24.1753 12.0232i −14.2337
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.4.c.a 4
3.b odd 2 1 27.4.c.a 4
4.b odd 2 1 144.4.i.c 4
5.b even 2 1 225.4.e.b 4
5.c odd 4 2 225.4.k.b 8
9.c even 3 1 inner 9.4.c.a 4
9.c even 3 1 81.4.a.d 2
9.d odd 6 1 27.4.c.a 4
9.d odd 6 1 81.4.a.a 2
12.b even 2 1 432.4.i.c 4
36.f odd 6 1 144.4.i.c 4
36.f odd 6 1 1296.4.a.u 2
36.h even 6 1 432.4.i.c 4
36.h even 6 1 1296.4.a.i 2
45.h odd 6 1 2025.4.a.n 2
45.j even 6 1 225.4.e.b 4
45.j even 6 1 2025.4.a.g 2
45.k odd 12 2 225.4.k.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.c.a 4 1.a even 1 1 trivial
9.4.c.a 4 9.c even 3 1 inner
27.4.c.a 4 3.b odd 2 1
27.4.c.a 4 9.d odd 6 1
81.4.a.a 2 9.d odd 6 1
81.4.a.d 2 9.c even 3 1
144.4.i.c 4 4.b odd 2 1
144.4.i.c 4 36.f odd 6 1
225.4.e.b 4 5.b even 2 1
225.4.e.b 4 45.j even 6 1
225.4.k.b 8 5.c odd 4 2
225.4.k.b 8 45.k odd 12 2
432.4.i.c 4 12.b even 2 1
432.4.i.c 4 36.h even 6 1
1296.4.a.i 2 36.h even 6 1
1296.4.a.u 2 36.f odd 6 1
2025.4.a.g 2 45.j even 6 1
2025.4.a.n 2 45.h odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(9, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T - T^{2} - 18 T^{3} - 36 T^{4} - 144 T^{5} - 64 T^{6} + 1536 T^{7} + 4096 T^{8} \)
$3$ \( 1 + 3 T - 18 T^{2} + 81 T^{3} + 729 T^{4} \)
$5$ \( 1 + 15 T - 73 T^{2} + 720 T^{3} + 45054 T^{4} + 90000 T^{5} - 1140625 T^{6} + 29296875 T^{7} + 244140625 T^{8} \)
$7$ \( 1 + 7 T - 575 T^{2} - 434 T^{3} + 254920 T^{4} - 148862 T^{5} - 67648175 T^{6} + 282475249 T^{7} + 13841287201 T^{8} \)
$11$ \( 1 + 66 T + 1133 T^{2} + 37026 T^{3} + 2818332 T^{4} + 49281606 T^{5} + 2007178613 T^{6} + 155624547606 T^{7} + 3138428376721 T^{8} \)
$13$ \( 1 - 11 T - 2447 T^{2} + 20086 T^{3} + 1501978 T^{4} + 44128942 T^{5} - 11811201623 T^{6} - 116649493103 T^{7} + 23298085122481 T^{8} \)
$17$ \( ( 1 - 99 T + 11608 T^{2} - 486387 T^{3} + 24137569 T^{4} )^{2} \)
$19$ \( ( 1 + 77 T + 9186 T^{2} + 528143 T^{3} + 47045881 T^{4} )^{2} \)
$23$ \( 1 + 33 T - 20539 T^{2} - 89298 T^{3} + 306484632 T^{4} - 1086488766 T^{5} - 3040509124171 T^{6} + 59438037828279 T^{7} + 21914624432020321 T^{8} \)
$29$ \( 1 - 51 T - 46819 T^{2} - 32742 T^{3} + 1784077290 T^{4} - 798544638 T^{5} - 27849033065899 T^{6} - 739864444769319 T^{7} + 353814783205469041 T^{8} \)
$31$ \( 1 + 43 T - 58121 T^{2} + 16684 T^{3} + 2653813660 T^{4} + 497033044 T^{5} - 51582601443401 T^{6} + 1136903752908853 T^{7} + 787662783788549761 T^{8} \)
$37$ \( ( 1 + 50 T + 77874 T^{2} + 2532650 T^{3} + 2565726409 T^{4} )^{2} \)
$41$ \( 1 + 132 T - 45541 T^{2} - 9883764 T^{3} - 1986392520 T^{4} - 681198898644 T^{5} - 216324497239381 T^{6} + 43214415340002852 T^{7} + 22563490300366186081 T^{8} \)
$43$ \( 1 + 88 T - 152909 T^{2} + 144232 T^{3} + 18872321152 T^{4} + 11467453624 T^{5} - 966593302459541 T^{6} + 44228149850442184 T^{7} + 39959630797262576401 T^{8} \)
$47$ \( 1 + 399 T - 19927 T^{2} - 11378682 T^{3} + 4778899632 T^{4} - 1181368901286 T^{5} - 214797423860983 T^{6} + 446533058768004033 T^{7} + \)\(11\!\cdots\!41\)\( T^{8} \)
$53$ \( ( 1 - 54 T + 81970 T^{2} - 8039358 T^{3} + 22164361129 T^{4} )^{2} \)
$59$ \( 1 + 798 T + 219437 T^{2} + 5273982 T^{3} + 1228510332 T^{4} + 1083165149178 T^{5} + 9255969760580117 T^{6} + 6913070663286641322 T^{7} + \)\(17\!\cdots\!81\)\( T^{8} \)
$61$ \( 1 + 439 T - 218465 T^{2} - 18778664 T^{3} + 73809546934 T^{4} - 4262399933384 T^{5} - 11255398584775865 T^{6} + 5133730134754187899 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} \)
$67$ \( 1 + 988 T + 166519 T^{2} + 205601812 T^{3} + 271446260584 T^{4} + 61837417782556 T^{5} + 15063039340399711 T^{6} + 26880055983539407636 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} \)
$71$ \( ( 1 - 1368 T + 1012606 T^{2} - 489622248 T^{3} + 128100283921 T^{4} )^{2} \)
$73$ \( ( 1 + 455 T + 342636 T^{2} + 177002735 T^{3} + 151334226289 T^{4} )^{2} \)
$79$ \( 1 - 803 T + 285247 T^{2} + 503092348 T^{3} - 431718608228 T^{4} + 248044148165572 T^{5} + 69339967424998687 T^{6} - 96240831574042510157 T^{7} + \)\(59\!\cdots\!41\)\( T^{8} \)
$83$ \( 1 + 813 T - 553393 T^{2} + 57550644 T^{3} + 769801212072 T^{4} + 32906710080828 T^{5} - 180926514039791017 T^{6} + \)\(15\!\cdots\!39\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \)
$89$ \( ( 1 + 396 T + 1406374 T^{2} + 279167724 T^{3} + 496981290961 T^{4} )^{2} \)
$97$ \( 1 + 736 T - 1333241 T^{2} + 36498976 T^{3} + 2188025435632 T^{4} + 33311629922848 T^{5} - 1110552428823544889 T^{6} + \)\(55\!\cdots\!12\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} \)
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