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: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_1) q^{2} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{3} + (3 \beta_{3} - 3 \beta_{2} + \beta_1 - 4) q^{4} + ( - \beta_{3} + \beta_{2} + 8 \beta_1 - 7) q^{5} + ( - 3 \beta_{3} + 3 \beta_{2} - 15 \beta_1 + 12) q^{6} + ( - 3 \beta_{3} - 2 \beta_1) q^{7} + ( - \beta_{2} + 16) q^{8} + (3 \beta_{3} - 6 \beta_{2} + 21 \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_1) q^{2} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{3} + (3 \beta_{3} - 3 \beta_{2} + \beta_1 - 4) q^{4} + ( - \beta_{3} + \beta_{2} + 8 \beta_1 - 7) q^{5} + ( - 3 \beta_{3} + 3 \beta_{2} - 15 \beta_1 + 12) q^{6} + ( - 3 \beta_{3} - 2 \beta_1) q^{7} + ( - \beta_{2} + 16) q^{8} + (3 \beta_{3} - 6 \beta_{2} + 21 \beta_1 - 3) q^{9} + (6 \beta_{2} + 6) q^{10} + (8 \beta_{3} - 37 \beta_1) q^{11} + ( - 2 \beta_{3} - 5 \beta_{2} + 22 \beta_1 - 52) q^{12} + ( - 15 \beta_{3} + 15 \beta_{2} + 2 \beta_1 + 13) q^{13} + (8 \beta_{3} - 8 \beta_{2} + 26 \beta_1 - 34) q^{14} + (9 \beta_{3} - 15 \beta_{2} - 24 \beta_1 + 9) q^{15} + (9 \beta_{3} - \beta_1) q^{16} + (9 \beta_{2} + 54) q^{17} + ( - 18 \beta_{3} + 27 \beta_{2} + 72) q^{18} + ( - 27 \beta_{2} - 52) q^{19} + ( - 20 \beta_{3} + 16 \beta_1) q^{20} + ( - 7 \beta_{3} + 8 \beta_{2} - 46 \beta_1 + 34) q^{21} + (21 \beta_{3} - 21 \beta_{2} - 27 \beta_1 + 6) q^{22} + ( - 19 \beta_{3} + 19 \beta_{2} + 26 \beta_1 - 7) q^{23} + (15 \beta_{3} + 18 \beta_{2} + 9 \beta_1 - 24) q^{24} + (15 \beta_{3} + 53 \beta_1) q^{25} + ( - 28 \beta_{2} - 146) q^{26} + (36 \beta_{3} - 18 \beta_{2} - 18 \beta_1 - 117) q^{27} + (18 \beta_{2} + 92) q^{28} + ( - \beta_{3} + 26 \beta_1) q^{29} + (12 \beta_{3} - 6 \beta_{2} + 48 \beta_1 + 42) q^{30} + (3 \beta_{3} - 3 \beta_{2} + 20 \beta_1 - 23) q^{31} + ( - 9 \beta_{3} + 9 \beta_{2} - 207 \beta_1 + 216) q^{32} + ( - 66 \beta_{3} + 21 \beta_{2} + 165 \beta_1 - 6) q^{33} + ( - 63 \beta_{3} - 117 \beta_1) q^{34} + (19 \beta_{2} + 11) q^{35} + ( - 15 \beta_{3} - 60 \beta_{2} - 141 \beta_1 - 12) q^{36} + (54 \beta_{2} + 2) q^{37} + (79 \beta_{3} + 241 \beta_1) q^{38} + (17 \beta_{3} + 11 \beta_{2} - 124 \beta_1 + 253) q^{39} + ( - 24 \beta_{3} + 24 \beta_{2} + 144 \beta_1 - 120) q^{40} + (98 \beta_{3} - 98 \beta_{2} + 17 \beta_1 - 115) q^{41} + (18 \beta_{3} - 60 \beta_{2} + 12 \beta_1 - 162) q^{42} + (6 \beta_{3} - 47 \beta_1) q^{43} + (79 \beta_{2} - 76) q^{44} + ( - 45 \beta_{3} + 45 \beta_{2} + 72 \beta_1 - 171) q^{45} + ( - 12 \beta_{2} - 138) q^{46} + ( - 91 \beta_{3} - 154 \beta_1) q^{47} + (7 \beta_{3} - 17 \beta_{2} + 145 \beta_1 - 88) q^{48} + (21 \beta_{3} - 21 \beta_{2} - 267 \beta_1 + 246) q^{49} + ( - 83 \beta_{3} + 83 \beta_{2} - 173 \beta_1 + 256) q^{50} + (63 \beta_{3} + 36 \beta_{2} + 117 \beta_1 + 18) q^{51} + (54 \beta_{3} + 358 \beta_1) q^{52} + ( - 162 \beta_{2} - 54) q^{53} + (81 \beta_{3} + 54 \beta_{2} - 27 \beta_1 + 324) q^{54} + ( - 93 \beta_{2} + 267) q^{55} + ( - 46 \beta_{3} - 10 \beta_1) q^{56} + ( - 79 \beta_{3} + 2 \beta_{2} - 241 \beta_1 - 164) q^{57} + ( - 24 \beta_{3} + 24 \beta_{2} - 18 \beta_1 + 42) q^{58} + ( - 136 \beta_{3} + 136 \beta_{2} + 467 \beta_1 - 331) q^{59} + (12 \beta_{3} + 24 \beta_{2} - 336 \beta_1 + 168) q^{60} + (105 \beta_{3} - 272 \beta_1) q^{61} + (26 \beta_{2} + 70) q^{62} + ( - 57 \beta_{3} + 78 \beta_{2} + 24 \beta_1 + 192) q^{63} + ( - 153 \beta_{2} - 440) q^{64} + (107 \beta_{3} - 136 \beta_1) q^{65} + ( - 48 \beta_{3} + 33 \beta_{2} + 222 \beta_1 - 330) q^{66} + (66 \beta_{3} - 66 \beta_{2} + 461 \beta_1 - 527) q^{67} + (171 \beta_{3} - 171 \beta_{2} + 261 \beta_1 - 432) q^{68} + (45 \beta_{3} - 33 \beta_{2} - 204 \beta_1 + 297) q^{69} + ( - 30 \beta_{3} - 144 \beta_1) q^{70} + (144 \beta_{2} + 756) q^{71} + (27 \beta_{3} - 99 \beta_{2} + 333 \beta_1) q^{72} + (243 \beta_{2} - 106) q^{73} + ( - 56 \beta_{3} - 380 \beta_1) q^{74} + (121 \beta_{3} - 83 \beta_{2} + 187 \beta_1 - 256) q^{75} + ( - 183 \beta_{3} + 183 \beta_{2} - 673 \beta_1 + 856) q^{76} + (71 \beta_{3} - 71 \beta_{2} - 118 \beta_1 + 47) q^{77} + ( - 174 \beta_{3} - 90 \beta_{2} - 342 \beta_1 - 78) q^{78} + ( - 309 \beta_{3} + 556 \beta_1) q^{79} + ( - 64 \beta_{2} + 16) q^{80} + ( - 135 \beta_{3} - 135 \beta_{2} + 351 \beta_1 - 351) q^{81} + (213 \beta_{2} + 1014) q^{82} + (107 \beta_{3} - 460 \beta_1) q^{83} + (110 \beta_{3} + 56 \beta_{2} + 218 \beta_1 + 52) q^{84} + (18 \beta_{3} - 18 \beta_{2} + 288 \beta_1 - 306) q^{85} + (35 \beta_{3} - 35 \beta_{2} - \beta_1 - 34) q^{86} + (51 \beta_{3} - 24 \beta_{2} - 42 \beta_1 - 42) q^{87} + (165 \beta_{3} - 693 \beta_1) q^{88} + (72 \beta_{2} - 162) q^{89} + (144 \beta_{3} - 18 \beta_{2} + 144 \beta_1 - 306) q^{90} + ( - 69 \beta_{2} - 425) q^{91} + ( - 2 \beta_{3} + 430 \beta_1) q^{92} + (17 \beta_{3} - 43 \beta_{2} - 16 \beta_1 - 71) q^{93} + (336 \beta_{3} - 336 \beta_{2} + 882 \beta_1 - 1218) q^{94} + ( - 164 \beta_{3} + 164 \beta_{2} + 16 \beta_1 + 148) q^{95} + ( - 198 \beta_{3} + 423 \beta_{2} + 342 \beta_1 + 360) q^{96} + ( - 102 \beta_{3} - 317 \beta_1) q^{97} + ( - 225 \beta_{2} - 324) q^{98} + (279 \beta_{3} - 81 \beta_{2} - 1080 \beta_1 + 504) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} - 3 q^{3} - 5 q^{4} - 15 q^{5} + 9 q^{6} - 7 q^{7} + 66 q^{8} + 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{2} - 3 q^{3} - 5 q^{4} - 15 q^{5} + 9 q^{6} - 7 q^{7} + 66 q^{8} + 45 q^{9} + 12 q^{10} - 66 q^{11} - 156 q^{12} + 11 q^{13} - 60 q^{14} + 27 q^{15} + 7 q^{16} + 198 q^{17} + 216 q^{18} - 154 q^{19} + 12 q^{20} + 21 q^{21} + 33 q^{22} - 33 q^{23} - 99 q^{24} + 121 q^{25} - 528 q^{26} - 432 q^{27} + 332 q^{28} + 51 q^{29} + 288 q^{30} - 43 q^{31} + 423 q^{32} + 198 q^{33} - 297 q^{34} + 6 q^{35} - 225 q^{36} - 100 q^{37} + 561 q^{38} + 759 q^{39} - 264 q^{40} - 132 q^{41} - 486 q^{42} - 88 q^{43} - 462 q^{44} - 675 q^{45} - 528 q^{46} - 399 q^{47} - 21 q^{48} + 513 q^{49} + 429 q^{50} + 297 q^{51} + 770 q^{52} + 108 q^{53} + 1215 q^{54} + 1254 q^{55} - 66 q^{56} - 1221 q^{57} + 60 q^{58} - 798 q^{59} - 36 q^{60} - 439 q^{61} + 228 q^{62} + 603 q^{63} - 1454 q^{64} - 165 q^{65} - 990 q^{66} - 988 q^{67} - 693 q^{68} + 891 q^{69} - 318 q^{70} + 2736 q^{71} + 891 q^{72} - 910 q^{73} - 816 q^{74} - 363 q^{75} + 1529 q^{76} + 165 q^{77} - 990 q^{78} + 803 q^{79} + 192 q^{80} - 567 q^{81} + 3630 q^{82} - 813 q^{83} + 642 q^{84} - 594 q^{85} - 33 q^{86} - 153 q^{87} - 1221 q^{88} - 792 q^{89} - 756 q^{90} - 1562 q^{91} + 858 q^{92} - 213 q^{93} - 2100 q^{94} + 132 q^{95} + 1080 q^{96} - 736 q^{97} - 846 q^{98} + 297 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} + \nu^{2} + 2\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 8\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 11 ) / 3 \) Copy content Toggle raw display

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

$p$ $F_p(T)$
$2$ \( T^{4} + 3 T^{3} + 15 T^{2} - 18 T + 36 \) Copy content Toggle raw display
$3$ \( T^{4} + 3 T^{3} - 18 T^{2} + 81 T + 729 \) Copy content Toggle raw display
$5$ \( T^{4} + 15 T^{3} + 177 T^{2} + \cdots + 2304 \) Copy content Toggle raw display
$7$ \( T^{4} + 7 T^{3} + 111 T^{2} + \cdots + 3844 \) Copy content Toggle raw display
$11$ \( T^{4} + 66 T^{3} + 3795 T^{2} + \cdots + 314721 \) Copy content Toggle raw display
$13$ \( T^{4} - 11 T^{3} + 1947 T^{2} + \cdots + 3334276 \) Copy content Toggle raw display
$17$ \( (T^{2} - 99 T + 1782)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 77 T - 4532)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 33 T^{3} + 3795 T^{2} + \cdots + 7322436 \) Copy content Toggle raw display
$29$ \( T^{4} - 51 T^{3} + 1959 T^{2} + \cdots + 412164 \) Copy content Toggle raw display
$31$ \( T^{4} + 43 T^{3} + 1461 T^{2} + \cdots + 150544 \) Copy content Toggle raw display
$37$ \( (T^{2} + 50 T - 23432)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 132 T^{3} + \cdots + 5606565129 \) Copy content Toggle raw display
$43$ \( T^{4} + 88 T^{3} + 6105 T^{2} + \cdots + 2686321 \) Copy content Toggle raw display
$47$ \( T^{4} + 399 T^{3} + \cdots + 813276324 \) Copy content Toggle raw display
$53$ \( (T^{2} - 54 T - 215784)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 798 T^{3} + \cdots + 43678881 \) Copy content Toggle raw display
$61$ \( T^{4} + 439 T^{3} + \cdots + 1829786176 \) Copy content Toggle raw display
$67$ \( T^{4} + 988 T^{3} + \cdots + 43305193801 \) Copy content Toggle raw display
$71$ \( (T^{2} - 1368 T + 296784)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 455 T - 435398)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 803 T^{3} + \cdots + 392522298256 \) Copy content Toggle raw display
$83$ \( T^{4} + 813 T^{3} + \cdots + 5010940944 \) Copy content Toggle raw display
$89$ \( (T^{2} + 396 T - 3564)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 736 T^{3} + \cdots + 2459267281 \) Copy content Toggle raw display
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