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Properties

Label 9.9.d.a.2.5
Level $9$
Weight $9$
Character 9.2
Analytic conductor $3.666$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9,9,Mod(2,9)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9.2"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 9.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.66640749055\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} + 427 x^{12} - 1362 x^{11} + 135762 x^{10} - 371244 x^{9} + 18261508 x^{8} + \cdots + 872385888256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{21} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 2.5
Root \(-2.00397 - 3.47098i\) of defining polynomial
Character \(\chi\) \(=\) 9.2
Dual form 9.9.d.a.5.5

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(6.01192 + 3.47098i) q^{2} +(-29.7082 - 75.3553i) q^{3} +(-103.905 - 179.968i) q^{4} +(331.396 - 191.331i) q^{5} +(82.9534 - 556.147i) q^{6} +(467.516 - 809.762i) q^{7} -3219.75i q^{8} +(-4795.84 + 4477.35i) q^{9} +2656.43 q^{10} +(4890.10 + 2823.30i) q^{11} +(-10474.7 + 13176.3i) q^{12} +(17663.1 + 30593.3i) q^{13} +(5621.34 - 3245.48i) q^{14} +(-24263.0 - 19288.3i) q^{15} +(-15423.9 + 26714.9i) q^{16} -152914. i q^{17} +(-44373.0 + 10271.2i) q^{18} +191248. q^{19} +(-68867.1 - 39760.4i) q^{20} +(-74908.9 - 11173.2i) q^{21} +(19599.3 + 33947.0i) q^{22} +(132313. - 76390.7i) q^{23} +(-242625. + 95653.1i) q^{24} +(-122097. + 211478. i) q^{25} +245233. i q^{26} +(479868. + 228378. i) q^{27} -194308. q^{28} +(401612. + 231871. i) q^{29} +(-78918.0 - 200176. i) q^{30} +(-393956. - 682351. i) q^{31} +(-899280. + 519200. i) q^{32} +(67474.4 - 452371. i) q^{33} +(530763. - 919308. i) q^{34} -357802. i q^{35} +(1.30409e6 + 397881. i) q^{36} -1.10561e6 q^{37} +(1.14977e6 + 663818. i) q^{38} +(1.78063e6 - 2.23988e6i) q^{39} +(-616039. - 1.06701e6i) q^{40} +(-3.14915e6 + 1.81816e6i) q^{41} +(-411564. - 327180. i) q^{42} +(-1.51318e6 + 2.62091e6i) q^{43} -1.17342e6i q^{44} +(-732664. + 2.40137e6i) q^{45} +1.06060e6 q^{46} +(4.75308e6 + 2.74419e6i) q^{47} +(2.47133e6 + 368617. i) q^{48} +(2.44526e6 + 4.23531e6i) q^{49} +(-1.46808e6 + 847594. i) q^{50} +(-1.15229e7 + 4.54281e6i) q^{51} +(3.67054e6 - 6.35757e6i) q^{52} -1.41381e7i q^{53} +(2.09223e6 + 3.03860e6i) q^{54} +2.16075e6 q^{55} +(-2.60723e6 - 1.50528e6i) q^{56} +(-5.68163e6 - 1.44115e7i) q^{57} +(1.60964e6 + 2.78797e6i) q^{58} +(7.58082e6 - 4.37679e6i) q^{59} +(-950238. + 6.37071e6i) q^{60} +(-3.47102e6 + 6.01198e6i) q^{61} -5.46965e6i q^{62} +(1.38345e6 + 5.97672e6i) q^{63} +688484. q^{64} +(1.17069e7 + 6.75900e6i) q^{65} +(1.97582e6 - 2.48541e6i) q^{66} +(7.08899e6 + 1.22785e7i) q^{67} +(-2.75197e7 + 1.58885e7i) q^{68} +(-9.68722e6 - 7.70102e6i) q^{69} +(1.24193e6 - 2.15108e6i) q^{70} +7.97888e6i q^{71} +(1.44159e7 + 1.54414e7i) q^{72} -4.61414e6 q^{73} +(-6.64682e6 - 3.83754e6i) q^{74} +(1.95633e7 + 2.91801e6i) q^{75} +(-1.98715e7 - 3.44184e7i) q^{76} +(4.57241e6 - 2.63988e6i) q^{77} +(1.84796e7 - 7.28544e6i) q^{78} +(-1.37621e6 + 2.38366e6i) q^{79} +1.18043e7i q^{80} +(2.95344e6 - 4.29453e7i) q^{81} -2.52433e7 q^{82} +(3.52291e7 + 2.03395e7i) q^{83} +(5.77256e6 + 1.46422e7i) q^{84} +(-2.92573e7 - 5.06751e7i) q^{85} +(-1.81943e7 + 1.05045e7i) q^{86} +(5.54150e6 - 3.71520e7i) q^{87} +(9.09033e6 - 1.57449e7i) q^{88} +2.90621e7i q^{89} +(-1.27398e7 + 1.18938e7i) q^{90} +3.30311e7 q^{91} +(-2.74958e7 - 1.58747e7i) q^{92} +(-3.97150e7 + 4.99581e7i) q^{93} +(1.90501e7 + 3.29957e7i) q^{94} +(6.33787e7 - 3.65917e7i) q^{95} +(6.58405e7 + 5.23410e7i) q^{96} +(-4.58061e7 + 7.93386e7i) q^{97} +3.39498e7i q^{98} +(-3.60931e7 + 8.35459e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 3 q^{2} - 93 q^{3} + 767 q^{4} + 438 q^{5} - 2259 q^{6} + 922 q^{7} + 17973 q^{9} - 516 q^{10} - 28677 q^{11} - 55110 q^{12} + 1684 q^{13} + 120966 q^{14} - 75276 q^{15} - 65281 q^{16} + 243324 q^{18}+ \cdots - 511060752 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.01192 + 3.47098i 0.375745 + 0.216937i 0.675965 − 0.736933i \(-0.263727\pi\)
−0.300220 + 0.953870i \(0.597060\pi\)
\(3\) −29.7082 − 75.3553i −0.366768 − 0.930312i
\(4\) −103.905 − 179.968i −0.405877 − 0.703000i
\(5\) 331.396 − 191.331i 0.530233 − 0.306130i −0.210878 − 0.977512i \(-0.567632\pi\)
0.741111 + 0.671382i \(0.234299\pi\)
\(6\) 82.9534 − 556.147i 0.0640072 − 0.429126i
\(7\) 467.516 − 809.762i 0.194717 − 0.337260i −0.752091 − 0.659060i \(-0.770954\pi\)
0.946808 + 0.321800i \(0.104288\pi\)
\(8\) − 3219.75i − 0.786071i
\(9\) −4795.84 + 4477.35i −0.730962 + 0.682418i
\(10\) 2656.43 0.265643
\(11\) 4890.10 + 2823.30i 0.334001 + 0.192835i 0.657616 − 0.753353i \(-0.271565\pi\)
−0.323615 + 0.946189i \(0.604898\pi\)
\(12\) −10474.7 + 13176.3i −0.505146 + 0.635431i
\(13\) 17663.1 + 30593.3i 0.618433 + 1.07116i 0.989772 + 0.142660i \(0.0455654\pi\)
−0.371339 + 0.928497i \(0.621101\pi\)
\(14\) 5621.34 − 3245.48i 0.146328 − 0.0844826i
\(15\) −24263.0 − 19288.3i −0.479270 − 0.381004i
\(16\) −15423.9 + 26714.9i −0.235350 + 0.407637i
\(17\) − 152914.i − 1.83085i −0.402491 − 0.915424i \(-0.631856\pi\)
0.402491 − 0.915424i \(-0.368144\pi\)
\(18\) −44373.0 + 10271.2i −0.422697 + 0.0978431i
\(19\) 191248. 1.46751 0.733756 − 0.679413i \(-0.237766\pi\)
0.733756 + 0.679413i \(0.237766\pi\)
\(20\) −68867.1 − 39760.4i −0.430419 − 0.248503i
\(21\) −74908.9 − 11173.2i −0.385173 − 0.0574515i
\(22\) 19599.3 + 33947.0i 0.0836661 + 0.144914i
\(23\) 132313. − 76390.7i 0.472814 − 0.272979i −0.244603 − 0.969623i \(-0.578658\pi\)
0.717417 + 0.696644i \(0.245324\pi\)
\(24\) −242625. + 95653.1i −0.731292 + 0.288306i
\(25\) −122097. + 211478.i −0.312568 + 0.541384i
\(26\) 245233.i 0.536643i
\(27\) 479868. + 228378.i 0.902956 + 0.429733i
\(28\) −194308. −0.316125
\(29\) 401612. + 231871.i 0.567825 + 0.327834i 0.756280 − 0.654248i \(-0.227015\pi\)
−0.188455 + 0.982082i \(0.560348\pi\)
\(30\) −78918.0 − 200176.i −0.0974296 − 0.247131i
\(31\) −393956. − 682351.i −0.426580 − 0.738858i 0.569987 − 0.821654i \(-0.306948\pi\)
−0.996567 + 0.0827958i \(0.973615\pi\)
\(32\) −899280. + 519200.i −0.857621 + 0.495147i
\(33\) 67474.4 − 452371.i 0.0568962 − 0.381451i
\(34\) 530763. − 919308.i 0.397178 − 0.687932i
\(35\) − 357802.i − 0.238435i
\(36\) 1.30409e6 + 397881.i 0.776421 + 0.236888i
\(37\) −1.10561e6 −0.589921 −0.294960 − 0.955509i \(-0.595306\pi\)
−0.294960 + 0.955509i \(0.595306\pi\)
\(38\) 1.14977e6 + 663818.i 0.551410 + 0.318357i
\(39\) 1.78063e6 − 2.23988e6i 0.769689 − 0.968202i
\(40\) −616039. − 1.06701e6i −0.240640 − 0.416801i
\(41\) −3.14915e6 + 1.81816e6i −1.11444 + 0.643425i −0.939977 − 0.341238i \(-0.889154\pi\)
−0.174468 + 0.984663i \(0.555820\pi\)
\(42\) −411564. − 327180.i −0.132264 − 0.105145i
\(43\) −1.51318e6 + 2.62091e6i −0.442607 + 0.766617i −0.997882 − 0.0650494i \(-0.979280\pi\)
0.555275 + 0.831667i \(0.312613\pi\)
\(44\) − 1.17342e6i − 0.313070i
\(45\) −732664. + 2.40137e6i −0.178671 + 0.585611i
\(46\) 1.06060e6 0.236876
\(47\) 4.75308e6 + 2.74419e6i 0.974055 + 0.562371i 0.900470 − 0.434918i \(-0.143223\pi\)
0.0735846 + 0.997289i \(0.476556\pi\)
\(48\) 2.47133e6 + 368617.i 0.465549 + 0.0694400i
\(49\) 2.44526e6 + 4.23531e6i 0.424170 + 0.734685i
\(50\) −1.46808e6 + 847594.i −0.234892 + 0.135615i
\(51\) −1.15229e7 + 4.54281e6i −1.70326 + 0.671497i
\(52\) 3.67054e6 − 6.35757e6i 0.502015 − 0.869516i
\(53\) − 1.41381e7i − 1.79179i −0.444269 − 0.895894i \(-0.646536\pi\)
0.444269 − 0.895894i \(-0.353464\pi\)
\(54\) 2.09223e6 + 3.03860e6i 0.246056 + 0.357354i
\(55\) 2.16075e6 0.236131
\(56\) −2.60723e6 − 1.50528e6i −0.265111 − 0.153062i
\(57\) −5.68163e6 − 1.44115e7i −0.538237 − 1.36524i
\(58\) 1.60964e6 + 2.78797e6i 0.142238 + 0.246364i
\(59\) 7.58082e6 − 4.37679e6i 0.625617 − 0.361200i −0.153436 − 0.988159i \(-0.549034\pi\)
0.779052 + 0.626959i \(0.215701\pi\)
\(60\) −950238. + 6.37071e6i −0.0733208 + 0.491567i
\(61\) −3.47102e6 + 6.01198e6i −0.250690 + 0.434209i −0.963716 − 0.266929i \(-0.913991\pi\)
0.713026 + 0.701138i \(0.247324\pi\)
\(62\) − 5.46965e6i − 0.370163i
\(63\) 1.38345e6 + 5.97672e6i 0.0878217 + 0.379403i
\(64\) 688484. 0.0410368
\(65\) 1.17069e7 + 6.75900e6i 0.655827 + 0.378642i
\(66\) 1.97582e6 − 2.48541e6i 0.104129 − 0.130985i
\(67\) 7.08899e6 + 1.22785e7i 0.351791 + 0.609321i 0.986563 − 0.163379i \(-0.0522393\pi\)
−0.634772 + 0.772700i \(0.718906\pi\)
\(68\) −2.75197e7 + 1.58885e7i −1.28709 + 0.743099i
\(69\) −9.68722e6 − 7.70102e6i −0.427369 − 0.339744i
\(70\) 1.24193e6 − 2.15108e6i 0.0517254 − 0.0895909i
\(71\) 7.97888e6i 0.313985i 0.987600 + 0.156992i \(0.0501798\pi\)
−0.987600 + 0.156992i \(0.949820\pi\)
\(72\) 1.44159e7 + 1.54414e7i 0.536430 + 0.574588i
\(73\) −4.61414e6 −0.162480 −0.0812399 − 0.996695i \(-0.525888\pi\)
−0.0812399 + 0.996695i \(0.525888\pi\)
\(74\) −6.64682e6 − 3.83754e6i −0.221660 − 0.127975i
\(75\) 1.95633e7 + 2.91801e6i 0.618297 + 0.0922235i
\(76\) −1.98715e7 − 3.44184e7i −0.595630 − 1.03166i
\(77\) 4.57241e6 − 2.63988e6i 0.130071 − 0.0750968i
\(78\) 1.84796e7 − 7.28544e6i 0.499245 − 0.196824i
\(79\) −1.37621e6 + 2.38366e6i −0.0353326 + 0.0611979i −0.883151 − 0.469089i \(-0.844582\pi\)
0.847818 + 0.530287i \(0.177916\pi\)
\(80\) 1.18043e7i 0.288191i
\(81\) 2.95344e6 − 4.29453e7i 0.0686102 − 0.997644i
\(82\) −2.52433e7 −0.558329
\(83\) 3.52291e7 + 2.03395e7i 0.742316 + 0.428577i 0.822911 − 0.568170i \(-0.192349\pi\)
−0.0805945 + 0.996747i \(0.525682\pi\)
\(84\) 5.77256e6 + 1.46422e7i 0.115945 + 0.294095i
\(85\) −2.92573e7 − 5.06751e7i −0.560478 − 0.970776i
\(86\) −1.81943e7 + 1.05045e7i −0.332614 + 0.192035i
\(87\) 5.54150e6 − 3.71520e7i 0.0967275 − 0.648493i
\(88\) 9.09033e6 − 1.57449e7i 0.151582 − 0.262548i
\(89\) 2.90621e7i 0.463199i 0.972811 + 0.231599i \(0.0743958\pi\)
−0.972811 + 0.231599i \(0.925604\pi\)
\(90\) −1.27398e7 + 1.18938e7i −0.194175 + 0.181280i
\(91\) 3.30311e7 0.481678
\(92\) −2.74958e7 − 1.58747e7i −0.383808 − 0.221592i
\(93\) −3.97150e7 + 4.99581e7i −0.530913 + 0.667843i
\(94\) 1.90501e7 + 3.29957e7i 0.243997 + 0.422616i
\(95\) 6.33787e7 − 3.65917e7i 0.778124 − 0.449250i
\(96\) 6.58405e7 + 5.23410e7i 0.775190 + 0.616250i
\(97\) −4.58061e7 + 7.93386e7i −0.517412 + 0.896185i 0.482383 + 0.875960i \(0.339771\pi\)
−0.999795 + 0.0202242i \(0.993562\pi\)
\(98\) 3.39498e7i 0.368072i
\(99\) −3.60931e7 + 8.35459e6i −0.375736 + 0.0869729i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9.9.d.a.2.5 ✓ 14
3.2 odd 2 27.9.d.a.8.3 14
4.3 odd 2 144.9.q.a.65.4 14
9.2 odd 6 81.9.b.a.80.9 14
9.4 even 3 27.9.d.a.17.3 14
9.5 odd 6 inner 9.9.d.a.5.5 yes 14
9.7 even 3 81.9.b.a.80.6 14
12.11 even 2 432.9.q.a.305.3 14
36.23 even 6 144.9.q.a.113.4 14
36.31 odd 6 432.9.q.a.17.3 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.9.d.a.2.5 ✓ 14 1.1 even 1 trivial
9.9.d.a.5.5 yes 14 9.5 odd 6 inner
27.9.d.a.8.3 14 3.2 odd 2
27.9.d.a.17.3 14 9.4 even 3
81.9.b.a.80.6 14 9.7 even 3
81.9.b.a.80.9 14 9.2 odd 6
144.9.q.a.65.4 14 4.3 odd 2
144.9.q.a.113.4 14 36.23 even 6
432.9.q.a.17.3 14 36.31 odd 6
432.9.q.a.305.3 14 12.11 even 2