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 4.2
Root \(0.500000 - 6.17443i\) of defining polynomial
Character \(\chi\) \(=\) 9.4
Dual form 9.8.c.a.7.2

$q$-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+(-6.09721 - 10.5607i) q^{2} +(-33.3118 + 32.8226i) q^{3} +(-10.3519 + 17.9301i) q^{4} +(-246.026 + 426.130i) q^{5} +(549.738 + 151.669i) q^{6} +(-382.311 - 662.182i) q^{7} -1308.41 q^{8} +(32.3523 - 2186.76i) q^{9} +6000.29 q^{10} +(36.3512 + 62.9621i) q^{11} +(-243.670 - 937.060i) q^{12} +(-3010.77 + 5214.80i) q^{13} +(-4662.06 + 8074.93i) q^{14} +(-5791.12 - 22270.4i) q^{15} +(9302.72 + 16112.8i) q^{16} -5989.93 q^{17} +(-23290.9 + 12991.5i) q^{18} +18676.2 q^{19} +(-5093.69 - 8822.53i) q^{20} +(34470.0 + 9510.03i) q^{21} +(443.281 - 767.786i) q^{22} +(-12139.5 + 21026.3i) q^{23} +(43585.6 - 42945.6i) q^{24} +(-81995.2 - 142020. i) q^{25} +73429.1 q^{26} +(70697.5 + 73906.8i) q^{27} +15830.6 q^{28} +(-43378.1 - 75133.0i) q^{29} +(-199881. + 196945. i) q^{30} +(-105890. + 183406. i) q^{31} +(29702.8 - 51446.7i) q^{32} +(-3277.50 - 904.240i) q^{33} +(36521.9 + 63257.8i) q^{34} +376234. q^{35} +(38873.9 + 23217.3i) q^{36} -327978. q^{37} +(-113873. - 197233. i) q^{38} +(-70869.3 - 272536. i) q^{39} +(321904. - 557554. i) q^{40} +(-196036. + 339545. i) q^{41} +(-109739. - 422011. i) q^{42} +(343611. + 595152. i) q^{43} -1505.22 q^{44} +(923884. + 551787. i) q^{45} +296069. q^{46} +(-320755. - 555563. i) q^{47} +(-838754. - 231406. i) q^{48} +(119448. - 206890. i) q^{49} +(-999884. + 1.73185e6i) q^{50} +(199535. - 196605. i) q^{51} +(-62334.5 - 107966. i) q^{52} -814485. q^{53} +(349449. - 1.19724e6i) q^{54} -35773.3 q^{55} +(500221. + 866408. i) q^{56} +(-622137. + 613001. i) q^{57} +(-528971. + 916204. i) q^{58} +(1.25863e6 - 2.18002e6i) q^{59} +(459259. + 126706. i) q^{60} +(221621. + 383858. i) q^{61} +2.58252e6 q^{62} +(-1.46040e6 + 814600. i) q^{63} +1.65708e6 q^{64} +(-1.48145e6 - 2.56595e6i) q^{65} +(10434.2 + 40126.0i) q^{66} +(-296048. + 512770. i) q^{67} +(62007.4 - 107400. i) q^{68} +(-285748. - 1.09887e6i) q^{69} +(-2.29398e6 - 3.97329e6i) q^{70} +1.48821e6 q^{71} +(-42330.2 + 2.86119e6i) q^{72} -5.41341e6 q^{73} +(1.99975e6 + 3.46367e6i) q^{74} +(7.39287e6 + 2.03964e6i) q^{75} +(-193334. + 334865. i) q^{76} +(27794.9 - 48142.2i) q^{77} +(-2.44606e6 + 2.41013e6i) q^{78} +(444736. + 770305. i) q^{79} -9.15485e6 q^{80} +(-4.78088e6 - 141493. i) q^{81} +4.78109e6 q^{82} +(1.69323e6 + 2.93276e6i) q^{83} +(-527347. + 519602. i) q^{84} +(1.47368e6 - 2.55249e6i) q^{85} +(4.19014e6 - 7.25754e6i) q^{86} +(3.91107e6 + 1.07904e6i) q^{87} +(-47562.4 - 82380.4i) q^{88} +1.17388e6 q^{89} +(194123. - 1.31212e7i) q^{90} +4.60420e6 q^{91} +(-251335. - 435325. i) q^{92} +(-2.49250e6 - 9.58516e6i) q^{93} +(-3.91142e6 + 6.77477e6i) q^{94} +(-4.59483e6 + 7.95847e6i) q^{95} +(699163. + 2.68871e6i) q^{96} +(4.30014e6 + 7.44806e6i) q^{97} -2.91320e6 q^{98} +(138859. - 77454.3i) 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 \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.09721 − 10.5607i −0.538922 − 0.933441i −0.998962 − 0.0455426i \(-0.985498\pi\)
0.460040 − 0.887898i \(-0.347835\pi\)
\(3\) −33.3118 + 32.8226i −0.712318 + 0.701857i
\(4\) −10.3519 + 17.9301i −0.0808745 + 0.140079i
\(5\) −246.026 + 426.130i −0.880210 + 1.52457i −0.0291025 + 0.999576i \(0.509265\pi\)
−0.851107 + 0.524992i \(0.824068\pi\)
\(6\) 549.738 + 151.669i 1.03903 + 0.286660i
\(7\) −382.311 − 662.182i −0.421283 − 0.729683i 0.574783 − 0.818306i \(-0.305087\pi\)
−0.996065 + 0.0886232i \(0.971753\pi\)
\(8\) −1308.41 −0.903504
\(9\) 32.3523 − 2186.76i 0.0147930 − 0.999891i
\(10\) 6000.29 1.89746
\(11\) 36.3512 + 62.9621i 0.00823463 + 0.0142628i 0.870113 − 0.492852i \(-0.164045\pi\)
−0.861879 + 0.507114i \(0.830712\pi\)
\(12\) −243.670 − 937.060i −0.0407069 − 0.156543i
\(13\) −3010.77 + 5214.80i −0.380080 + 0.658318i −0.991073 − 0.133318i \(-0.957437\pi\)
0.610993 + 0.791636i \(0.290770\pi\)
\(14\) −4662.06 + 8074.93i −0.454077 + 0.786485i
\(15\) −5791.12 − 22270.4i −0.443040 − 1.70376i
\(16\) 9302.72 + 16112.8i 0.567793 + 0.983446i
\(17\) −5989.93 −0.295700 −0.147850 − 0.989010i \(-0.547235\pi\)
−0.147850 + 0.989010i \(0.547235\pi\)
\(18\) −23290.9 + 12991.5i −0.941311 + 0.525055i
\(19\) 18676.2 0.624670 0.312335 − 0.949972i \(-0.398889\pi\)
0.312335 + 0.949972i \(0.398889\pi\)
\(20\) −5093.69 − 8822.53i −0.142373 − 0.246597i
\(21\) 34470.0 + 9510.03i 0.812220 + 0.224086i
\(22\) 443.281 − 767.786i 0.00887565 − 0.0153731i
\(23\) −12139.5 + 21026.3i −0.208043 + 0.360342i −0.951098 − 0.308889i \(-0.900043\pi\)
0.743055 + 0.669231i \(0.233376\pi\)
\(24\) 43585.6 − 42945.6i 0.643582 − 0.634131i
\(25\) −81995.2 − 142020.i −1.04954 − 1.81785i
\(26\) 73429.1 0.819335
\(27\) 70697.5 + 73906.8i 0.691243 + 0.722622i
\(28\) 15830.6 0.136284
\(29\) −43378.1 − 75133.0i −0.330276 − 0.572055i 0.652290 − 0.757970i \(-0.273809\pi\)
−0.982566 + 0.185914i \(0.940475\pi\)
\(30\) −199881. + 196945.i −1.35159 + 1.33175i
\(31\) −105890. + 183406.i −0.638392 + 1.10573i 0.347394 + 0.937719i \(0.387067\pi\)
−0.985786 + 0.168008i \(0.946267\pi\)
\(32\) 29702.8 − 51446.7i 0.160241 − 0.277545i
\(33\) −3277.50 − 904.240i −0.0158761 − 0.00438011i
\(34\) 36521.9 + 63257.8i 0.159359 + 0.276018i
\(35\) 376234. 1.48327
\(36\) 38873.9 + 23217.3i 0.138867 + 0.0829378i
\(37\) −327978. −1.06448 −0.532242 − 0.846592i \(-0.678650\pi\)
−0.532242 + 0.846592i \(0.678650\pi\)
\(38\) −113873. − 197233.i −0.336648 − 0.583092i
\(39\) −70869.3 − 272536.i −0.191308 − 0.735694i
\(40\) 321904. − 557554.i 0.795273 − 1.37745i
\(41\) −196036. + 339545.i −0.444214 + 0.769402i −0.997997 − 0.0632592i \(-0.979851\pi\)
0.553783 + 0.832661i \(0.313184\pi\)
\(42\) −109739. − 422011.i −0.228553 − 0.878925i
\(43\) 343611. + 595152.i 0.659064 + 1.14153i 0.980858 + 0.194723i \(0.0623809\pi\)
−0.321794 + 0.946810i \(0.604286\pi\)
\(44\) −1505.22 −0.00266388
\(45\) 923884. + 551787.i 1.51138 + 0.902666i
\(46\) 296069. 0.448477
\(47\) −320755. − 555563.i −0.450641 − 0.780533i 0.547785 − 0.836619i \(-0.315471\pi\)
−0.998426 + 0.0560862i \(0.982138\pi\)
\(48\) −838754. − 231406.i −1.09469 − 0.302017i
\(49\) 119448. − 206890.i 0.145042 − 0.251220i
\(50\) −999884. + 1.73185e6i −1.13124 + 1.95936i
\(51\) 199535. − 196605.i 0.210632 − 0.207539i
\(52\) −62334.5 − 107966.i −0.0614776 − 0.106482i
\(53\) −814485. −0.751480 −0.375740 − 0.926725i \(-0.622611\pi\)
−0.375740 + 0.926725i \(0.622611\pi\)
\(54\) 349449. − 1.19724e6i 0.301999 − 1.03467i
\(55\) −35773.3 −0.0289928
\(56\) 500221. + 866408.i 0.380631 + 0.659272i
\(57\) −622137. + 613001.i −0.444963 + 0.438429i
\(58\) −528971. + 916204.i −0.355986 + 0.616587i
\(59\) 1.25863e6 − 2.18002e6i 0.797843 − 1.38190i −0.123176 − 0.992385i \(-0.539308\pi\)
0.921018 − 0.389519i \(-0.127359\pi\)
\(60\) 459259. + 126706.i 0.274491 + 0.0757301i
\(61\) 221621. + 383858.i 0.125013 + 0.216529i 0.921738 − 0.387813i \(-0.126769\pi\)
−0.796725 + 0.604342i \(0.793436\pi\)
\(62\) 2.58252e6 1.37617
\(63\) −1.46040e6 + 814600.i −0.735835 + 0.410442i
\(64\) 1.65708e6 0.790157
\(65\) −1.48145e6 − 2.56595e6i −0.669101 − 1.15892i
\(66\) 10434.2 + 40126.0i 0.00446742 + 0.0171799i
\(67\) −296048. + 512770.i −0.120254 + 0.208286i −0.919868 − 0.392228i \(-0.871704\pi\)
0.799614 + 0.600515i \(0.205038\pi\)
\(68\) 62007.4 − 107400.i 0.0239145 − 0.0414212i
\(69\) −285748. − 1.09887e6i −0.104715 − 0.402695i
\(70\) −2.29398e6 − 3.97329e6i −0.799367 − 1.38454i
\(71\) 1.48821e6 0.493469 0.246734 − 0.969083i \(-0.420643\pi\)
0.246734 + 0.969083i \(0.420643\pi\)
\(72\) −42330.2 + 2.86119e6i −0.0133655 + 0.903406i
\(73\) −5.41341e6 −1.62870 −0.814350 − 0.580374i \(-0.802906\pi\)
−0.814350 + 0.580374i \(0.802906\pi\)
\(74\) 1.99975e6 + 3.46367e6i 0.573674 + 0.993633i
\(75\) 7.39287e6 + 2.03964e6i 2.02348 + 0.558264i
\(76\) −193334. + 334865.i −0.0505198 + 0.0875029i
\(77\) 27794.9 − 48142.2i 0.00693821 − 0.0120173i
\(78\) −2.44606e6 + 2.41013e6i −0.583627 + 0.575056i
\(79\) 444736. + 770305.i 0.101486 + 0.175779i 0.912297 − 0.409529i \(-0.134307\pi\)
−0.810811 + 0.585308i \(0.800974\pi\)
\(80\) −9.15485e6 −1.99911
\(81\) −4.78088e6 − 141493.i −0.999562 − 0.0295828i
\(82\) 4.78109e6 0.957588
\(83\) 1.69323e6 + 2.93276e6i 0.325044 + 0.562993i 0.981521 − 0.191353i \(-0.0612874\pi\)
−0.656477 + 0.754346i \(0.727954\pi\)
\(84\) −527347. + 519602.i −0.0970775 + 0.0956519i
\(85\) 1.47368e6 − 2.55249e6i 0.260278 − 0.450814i
\(86\) 4.19014e6 − 7.25754e6i 0.710369 − 1.23040i
\(87\) 3.91107e6 + 1.07904e6i 0.636763 + 0.175678i
\(88\) −47562.4 − 82380.4i −0.00744002 − 0.0128865i
\(89\) 1.17388e6 0.176506 0.0882531 − 0.996098i \(-0.471872\pi\)
0.0882531 + 0.996098i \(0.471872\pi\)
\(90\) 194123. − 1.31212e7i 0.0280691 − 1.89725i
\(91\) 4.60420e6 0.640485
\(92\) −251335. − 435325.i −0.0336508 − 0.0582849i
\(93\) −2.49250e6 − 9.58516e6i −0.321325 − 1.23569i
\(94\) −3.91142e6 + 6.77477e6i −0.485721 + 0.841293i
\(95\) −4.59483e6 + 7.95847e6i −0.549840 + 0.952351i
\(96\) 699163. + 2.68871e6i 0.0806546 + 0.310166i
\(97\) 4.30014e6 + 7.44806e6i 0.478390 + 0.828595i 0.999693 − 0.0247763i \(-0.00788736\pi\)
−0.521303 + 0.853371i \(0.674554\pi\)
\(98\) −2.91320e6 −0.312665
\(99\) 138859. − 77454.3i 0.0143830 − 0.00802274i
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.4.2 ✓ 12
3.2 odd 2 27.8.c.a.10.5 12
4.3 odd 2 144.8.i.c.49.5 12
9.2 odd 6 27.8.c.a.19.5 12
9.4 even 3 81.8.a.e.1.5 6
9.5 odd 6 81.8.a.c.1.2 6
9.7 even 3 inner 9.8.c.a.7.2 yes 12
12.11 even 2 432.8.i.c.145.6 12
36.7 odd 6 144.8.i.c.97.5 12
36.11 even 6 432.8.i.c.289.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.2 ✓ 12 1.1 even 1 trivial
9.8.c.a.7.2 yes 12 9.7 even 3 inner
27.8.c.a.10.5 12 3.2 odd 2
27.8.c.a.19.5 12 9.2 odd 6
81.8.a.c.1.2 6 9.5 odd 6
81.8.a.e.1.5 6 9.4 even 3
144.8.i.c.49.5 12 4.3 odd 2
144.8.i.c.97.5 12 36.7 odd 6
432.8.i.c.145.6 12 12.11 even 2
432.8.i.c.289.6 12 36.11 even 6