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 \(3\) and degree \(2\))

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: \( 3 \)
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.

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

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

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
9.c Even 1 yes

Hecke kernels

There are no other newforms in \(S_{4}^{\mathrm{new}}(9, [\chi])\).