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Properties

Label 9.8.c.a.7.5
Level $9$
Weight $8$
Character 9.7
Analytic conductor $2.811$
Analytic rank $0$
Dimension $12$
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] = [9,8,Mod(4,9)] 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("9.4"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 9.c (of order \(3\), 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: \(2.81146522936\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 375 x^{10} - 1820 x^{9} + 50808 x^{8} - 192378 x^{7} + 3002887 x^{6} + \cdots + 754412211 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{15} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 7.5
Root \(0.500000 - 9.08282i\) of defining polynomial
Character \(\chi\) \(=\) 9.7
Dual form 9.8.c.a.4.5

$q$-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+(7.11595 - 12.3252i) q^{2} +(45.7172 + 9.84574i) q^{3} +(-37.2735 - 64.5595i) q^{4} +(-145.304 - 251.673i) q^{5} +(446.672 - 493.411i) q^{6} +(-555.940 + 962.916i) q^{7} +760.739 q^{8} +(1993.12 + 900.239i) q^{9} -4135.89 q^{10} +(-2245.36 + 3889.07i) q^{11} +(-1068.40 - 3318.46i) q^{12} +(-1218.29 - 2110.14i) q^{13} +(7912.08 + 13704.1i) q^{14} +(-4164.96 - 12936.4i) q^{15} +(10184.4 - 17639.9i) q^{16} +15905.4 q^{17} +(25278.6 - 18159.5i) q^{18} -49949.6 q^{19} +(-10831.9 + 18761.5i) q^{20} +(-34896.6 + 38548.2i) q^{21} +(31955.7 + 55348.9i) q^{22} +(-34692.5 - 60089.2i) q^{23} +(34778.8 + 7490.04i) q^{24} +(-3163.73 + 5479.74i) q^{25} -34677.2 q^{26} +(82256.4 + 60780.2i) q^{27} +82887.2 q^{28} +(47035.8 - 81468.4i) q^{29} +(-189081. - 40720.9i) q^{30} +(9963.58 + 17257.4i) q^{31} +(-96255.8 - 166720. i) q^{32} +(-140942. + 155690. i) q^{33} +(113182. - 196037. i) q^{34} +323120. q^{35} +(-16171.5 - 162230. i) q^{36} +331750. q^{37} +(-355439. + 615638. i) q^{38} +(-34920.9 - 108465. i) q^{39} +(-110538. - 191457. i) q^{40} +(-121133. - 209809. i) q^{41} +(226791. + 704414. i) q^{42} +(-415713. + 720036. i) q^{43} +334769. q^{44} +(-63041.7 - 632423. i) q^{45} -987481. q^{46} +(80005.3 - 138573. i) q^{47} +(639279. - 706172. i) q^{48} +(-206366. - 357437. i) q^{49} +(45025.8 + 77987.1i) q^{50} +(727150. + 156600. i) q^{51} +(-90819.7 + 157304. i) q^{52} +311589. q^{53} +(1.33446e6 - 581317. i) q^{54} +1.30503e6 q^{55} +(-422925. + 732527. i) q^{56} +(-2.28356e6 - 491791. i) q^{57} +(-669408. - 1.15945e6i) q^{58} +(156177. + 270506. i) q^{59} +(-679926. + 751072. i) q^{60} +(28723.9 - 49751.3i) q^{61} +283601. q^{62} +(-1.97491e6 + 1.41873e6i) q^{63} -132603. q^{64} +(-354044. + 613221. i) q^{65} +(915973. + 2.84502e6i) q^{66} +(2.05100e6 + 3.55243e6i) q^{67} +(-592849. - 1.02684e6i) q^{68} +(-994422. - 3.08869e6i) q^{69} +(2.29930e6 - 3.98251e6i) q^{70} -403110. q^{71} +(1.51625e6 + 684847. i) q^{72} -823496. q^{73} +(2.36072e6 - 4.08889e6i) q^{74} +(-198589. + 219369. i) q^{75} +(1.86180e6 + 3.22472e6i) q^{76} +(-2.49656e6 - 4.32418e6i) q^{77} +(-1.58534e6 - 341422. i) q^{78} +(-489414. + 847689. i) q^{79} -5.91931e6 q^{80} +(3.16211e6 + 3.58858e6i) q^{81} -3.44791e6 q^{82} +(1.85204e6 - 3.20782e6i) q^{83} +(3.78937e6 + 816086. i) q^{84} +(-2.31111e6 - 4.00296e6i) q^{85} +(5.91638e6 + 1.02475e7i) q^{86} +(2.95246e6 - 3.26140e6i) q^{87} +(-1.70813e6 + 2.95857e6i) q^{88} +2.09023e6 q^{89} +(-8.24334e6 - 3.72329e6i) q^{90} +2.70918e6 q^{91} +(-2.58622e6 + 4.47947e6i) q^{92} +(285595. + 887060. i) q^{93} +(-1.13863e6 - 1.97216e6i) q^{94} +(7.25786e6 + 1.25710e7i) q^{95} +(-2.75906e6 - 8.56968e6i) q^{96} +(-1.75125e6 + 3.03325e6i) q^{97} -5.87396e6 q^{98} +(-7.97637e6 + 5.73004e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 9 q^{2} + 24 q^{3} - 321 q^{4} - 180 q^{5} - 1233 q^{6} - 84 q^{7} + 5922 q^{8} + 990 q^{9} + 252 q^{10} - 8460 q^{11} + 8052 q^{12} - 1848 q^{13} - 16272 q^{14} - 1188 q^{15} - 12417 q^{16} + 30564 q^{17}+ \cdots - 49382676 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.11595 − 12.3252i 0.628967 − 1.08940i −0.358792 − 0.933417i \(-0.616811\pi\)
0.987759 − 0.155985i \(-0.0498553\pi\)
\(3\) 45.7172 + 9.84574i 0.977586 + 0.210535i
\(4\) −37.2735 − 64.5595i −0.291199 − 0.504371i
\(5\) −145.304 − 251.673i −0.519854 − 0.900413i −0.999734 − 0.0230788i \(-0.992653\pi\)
0.479880 − 0.877334i \(-0.340680\pi\)
\(6\) 446.672 − 493.411i 0.844227 − 0.932566i
\(7\) −555.940 + 962.916i −0.612611 + 1.06107i 0.378188 + 0.925729i \(0.376547\pi\)
−0.990799 + 0.135344i \(0.956786\pi\)
\(8\) 760.739 0.525316
\(9\) 1993.12 + 900.239i 0.911350 + 0.411632i
\(10\) −4135.89 −1.30788
\(11\) −2245.36 + 3889.07i −0.508640 + 0.880991i 0.491310 + 0.870985i \(0.336518\pi\)
−0.999950 + 0.0100060i \(0.996815\pi\)
\(12\) −1068.40 − 3318.46i −0.178484 − 0.554374i
\(13\) −1218.29 − 2110.14i −0.153797 − 0.266385i 0.778823 − 0.627244i \(-0.215817\pi\)
−0.932620 + 0.360859i \(0.882484\pi\)
\(14\) 7912.08 + 13704.1i 0.770624 + 1.33476i
\(15\) −4164.96 − 12936.4i −0.318633 − 0.989679i
\(16\) 10184.4 − 17639.9i 0.621605 − 1.07665i
\(17\) 15905.4 0.785187 0.392593 − 0.919712i \(-0.371578\pi\)
0.392593 + 0.919712i \(0.371578\pi\)
\(18\) 25278.6 − 18159.5i 1.02164 − 0.733924i
\(19\) −49949.6 −1.67069 −0.835343 − 0.549730i \(-0.814731\pi\)
−0.835343 + 0.549730i \(0.814731\pi\)
\(20\) −10831.9 + 18761.5i −0.302762 + 0.524399i
\(21\) −34896.6 + 38548.2i −0.822273 + 0.908314i
\(22\) 31955.7 + 55348.9i 0.639836 + 1.10823i
\(23\) −34692.5 − 60089.2i −0.594550 − 1.02979i −0.993610 − 0.112867i \(-0.963997\pi\)
0.399060 − 0.916925i \(-0.369337\pi\)
\(24\) 34778.8 + 7490.04i 0.513542 + 0.110597i
\(25\) −3163.73 + 5479.74i −0.0404957 + 0.0701406i
\(26\) −34677.2 −0.386934
\(27\) 82256.4 + 60780.2i 0.804260 + 0.594277i
\(28\) 82887.2 0.713566
\(29\) 47035.8 − 81468.4i 0.358126 − 0.620292i −0.629522 − 0.776983i \(-0.716749\pi\)
0.987648 + 0.156691i \(0.0500826\pi\)
\(30\) −189081. − 40720.9i −1.27857 − 0.275355i
\(31\) 9963.58 + 17257.4i 0.0600689 + 0.104042i 0.894496 − 0.447076i \(-0.147535\pi\)
−0.834427 + 0.551118i \(0.814201\pi\)
\(32\) −96255.8 − 166720.i −0.519280 − 0.899420i
\(33\) −140942. + 155690.i −0.682719 + 0.754158i
\(34\) 113182. − 196037.i 0.493857 − 0.855385i
\(35\) 323120. 1.27387
\(36\) −16171.5 − 162230.i −0.0577687 − 0.579526i
\(37\) 331750. 1.07673 0.538363 − 0.842713i \(-0.319043\pi\)
0.538363 + 0.842713i \(0.319043\pi\)
\(38\) −355439. + 615638.i −1.05081 + 1.82005i
\(39\) −34920.9 − 108465.i −0.0942669 − 0.292794i
\(40\) −110538. − 191457.i −0.273087 − 0.473001i
\(41\) −121133. − 209809.i −0.274486 − 0.475423i 0.695520 − 0.718507i \(-0.255174\pi\)
−0.970005 + 0.243084i \(0.921841\pi\)
\(42\) 226791. + 704414.i 0.472338 + 1.46709i
\(43\) −415713. + 720036.i −0.797359 + 1.38107i 0.123971 + 0.992286i \(0.460437\pi\)
−0.921330 + 0.388781i \(0.872896\pi\)
\(44\) 334769. 0.592462
\(45\) −63041.7 − 632423.i −0.103130 − 1.03458i
\(46\) −987481. −1.49581
\(47\) 80005.3 − 138573.i 0.112403 − 0.194687i −0.804336 − 0.594175i \(-0.797479\pi\)
0.916738 + 0.399488i \(0.130812\pi\)
\(48\) 639279. − 706172.i 0.834346 − 0.921651i
\(49\) −206366. − 357437.i −0.250583 − 0.434023i
\(50\) 45025.8 + 77987.1i 0.0509409 + 0.0882323i
\(51\) 727150. + 156600.i 0.767588 + 0.165309i
\(52\) −90819.7 + 157304.i −0.0895712 + 0.155142i
\(53\) 311589. 0.287486 0.143743 − 0.989615i \(-0.454086\pi\)
0.143743 + 0.989615i \(0.454086\pi\)
\(54\) 1.33446e6 − 581317.i 1.15326 − 0.502383i
\(55\) 1.30503e6 1.05767
\(56\) −422925. + 732527.i −0.321814 + 0.557398i
\(57\) −2.28356e6 − 491791.i −1.63324 − 0.351738i
\(58\) −669408. − 1.15945e6i −0.450498 − 0.780286i
\(59\) 156177. + 270506.i 0.0989997 + 0.171473i 0.911271 − 0.411807i \(-0.135102\pi\)
−0.812271 + 0.583280i \(0.801769\pi\)
\(60\) −679926. + 751072.i −0.406380 + 0.448903i
\(61\) 28723.9 − 49751.3i 0.0162028 − 0.0280640i −0.857810 − 0.513966i \(-0.828176\pi\)
0.874013 + 0.485902i \(0.161509\pi\)
\(62\) 283601. 0.151125
\(63\) −1.97491e6 + 1.41873e6i −0.995074 + 0.714838i
\(64\) −132603. −0.0632302
\(65\) −354044. + 613221.i −0.159904 + 0.276962i
\(66\) 915973. + 2.84502e6i 0.392174 + 1.21810i
\(67\) 2.05100e6 + 3.55243e6i 0.833111 + 1.44299i 0.895559 + 0.444943i \(0.146776\pi\)
−0.0624478 + 0.998048i \(0.519891\pi\)
\(68\) −592849. − 1.02684e6i −0.228646 − 0.396026i
\(69\) −994422. − 3.08869e6i −0.364417 − 1.13188i
\(70\) 2.29930e6 − 3.98251e6i 0.801223 − 1.38776i
\(71\) −403110. −0.133666 −0.0668328 − 0.997764i \(-0.521289\pi\)
−0.0668328 + 0.997764i \(0.521289\pi\)
\(72\) 1.51625e6 + 684847.i 0.478747 + 0.216237i
\(73\) −823496. −0.247760 −0.123880 − 0.992297i \(-0.539534\pi\)
−0.123880 + 0.992297i \(0.539534\pi\)
\(74\) 2.36072e6 − 4.08889e6i 0.677226 − 1.17299i
\(75\) −198589. + 219369.i −0.0543551 + 0.0600428i
\(76\) 1.86180e6 + 3.22472e6i 0.486502 + 0.842646i
\(77\) −2.49656e6 − 4.32418e6i −0.623197 − 1.07941i
\(78\) −1.58534e6 − 341422.i −0.378261 − 0.0814631i
\(79\) −489414. + 847689.i −0.111682 + 0.193438i −0.916448 − 0.400153i \(-0.868957\pi\)
0.804767 + 0.593591i \(0.202290\pi\)
\(80\) −5.91931e6 −1.29258
\(81\) 3.16211e6 + 3.58858e6i 0.661118 + 0.750282i
\(82\) −3.44791e6 −0.690569
\(83\) 1.85204e6 − 3.20782e6i 0.355530 − 0.615796i −0.631678 − 0.775231i \(-0.717634\pi\)
0.987209 + 0.159434i \(0.0509670\pi\)
\(84\) 3.78937e6 + 816086.i 0.697572 + 0.150231i
\(85\) −2.31111e6 − 4.00296e6i −0.408182 − 0.706993i
\(86\) 5.91638e6 + 1.02475e7i 1.00303 + 1.73729i
\(87\) 2.95246e6 − 3.26140e6i 0.480692 − 0.530991i
\(88\) −1.70813e6 + 2.95857e6i −0.267197 + 0.462799i
\(89\) 2.09023e6 0.314289 0.157145 − 0.987576i \(-0.449771\pi\)
0.157145 + 0.987576i \(0.449771\pi\)
\(90\) −8.24334e6 − 3.72329e6i −1.19194 − 0.538367i
\(91\) 2.70918e6 0.376872
\(92\) −2.58622e6 + 4.47947e6i −0.346265 + 0.599748i
\(93\) 285595. + 887060.i 0.0368180 + 0.114357i
\(94\) −1.13863e6 − 1.97216e6i −0.141395 − 0.244903i
\(95\) 7.25786e6 + 1.25710e7i 0.868512 + 1.50431i
\(96\) −2.75906e6 − 8.56968e6i −0.318282 − 0.988587i
\(97\) −1.75125e6 + 3.03325e6i −0.194826 + 0.337448i −0.946843 − 0.321695i \(-0.895747\pi\)
0.752018 + 0.659143i \(0.229081\pi\)
\(98\) −5.87396e6 −0.630435
\(99\) −7.97637e6 + 5.73004e6i −0.826194 + 0.593519i
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 9.8.c.a.7.5 yes 12
3.2 odd 2 27.8.c.a.19.2 12
4.3 odd 2 144.8.i.c.97.1 12
9.2 odd 6 81.8.a.c.1.5 6
9.4 even 3 inner 9.8.c.a.4.5 ✓ 12
9.5 odd 6 27.8.c.a.10.2 12
9.7 even 3 81.8.a.e.1.2 6
12.11 even 2 432.8.i.c.289.5 12
36.23 even 6 432.8.i.c.145.5 12
36.31 odd 6 144.8.i.c.49.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.5 ✓ 12 9.4 even 3 inner
9.8.c.a.7.5 yes 12 1.1 even 1 trivial
27.8.c.a.10.2 12 9.5 odd 6
27.8.c.a.19.2 12 3.2 odd 2
81.8.a.c.1.5 6 9.2 odd 6
81.8.a.e.1.2 6 9.7 even 3
144.8.i.c.49.1 12 36.31 odd 6
144.8.i.c.97.1 12 4.3 odd 2
432.8.i.c.145.5 12 36.23 even 6
432.8.i.c.289.5 12 12.11 even 2