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Properties

Label 27.8.c.a.10.5
Level $27$
Weight $8$
Character 27.10
Analytic conductor $8.434$
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] = [27,8,Mod(10,27)] 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("27.10"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 27.c (of order \(3\), degree \(2\), not 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: \(8.43439568807\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{21} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 10.5
Root \(0.500000 - 6.17443i\) of defining polynomial
Character \(\chi\) \(=\) 27.10
Dual form 27.8.c.a.19.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(6.09721 + 10.5607i) q^{2} +(-10.3519 + 17.9301i) q^{4} +(246.026 - 426.130i) q^{5} +(-382.311 - 662.182i) q^{7} +1308.41 q^{8} +6000.29 q^{10} +(-36.3512 - 62.9621i) q^{11} +(-3010.77 + 5214.80i) q^{13} +(4662.06 - 8074.93i) q^{14} +(9302.72 + 16112.8i) q^{16} +5989.93 q^{17} +18676.2 q^{19} +(5093.69 + 8822.53i) q^{20} +(443.281 - 767.786i) q^{22} +(12139.5 - 21026.3i) q^{23} +(-81995.2 - 142020. i) q^{25} -73429.1 q^{26} +15830.6 q^{28} +(43378.1 + 75133.0i) q^{29} +(-105890. + 183406. i) q^{31} +(-29702.8 + 51446.7i) q^{32} +(36521.9 + 63257.8i) q^{34} -376234. q^{35} -327978. q^{37} +(113873. + 197233. i) q^{38} +(321904. - 557554. i) q^{40} +(196036. - 339545. i) q^{41} +(343611. + 595152. i) q^{43} +1505.22 q^{44} +296069. q^{46} +(320755. + 555563. i) q^{47} +(119448. - 206890. i) q^{49} +(999884. - 1.73185e6i) q^{50} +(-62334.5 - 107966. i) q^{52} +814485. q^{53} -35773.3 q^{55} +(-500221. - 866408. i) q^{56} +(-528971. + 916204. i) q^{58} +(-1.25863e6 + 2.18002e6i) q^{59} +(221621. + 383858. i) q^{61} -2.58252e6 q^{62} +1.65708e6 q^{64} +(1.48145e6 + 2.56595e6i) q^{65} +(-296048. + 512770. i) q^{67} +(-62007.4 + 107400. i) q^{68} +(-2.29398e6 - 3.97329e6i) q^{70} -1.48821e6 q^{71} -5.41341e6 q^{73} +(-1.99975e6 - 3.46367e6i) q^{74} +(-193334. + 334865. i) q^{76} +(-27794.9 + 48142.2i) q^{77} +(444736. + 770305. i) q^{79} +9.15485e6 q^{80} +4.78109e6 q^{82} +(-1.69323e6 - 2.93276e6i) q^{83} +(1.47368e6 - 2.55249e6i) q^{85} +(-4.19014e6 + 7.25754e6i) q^{86} +(-47562.4 - 82380.4i) q^{88} -1.17388e6 q^{89} +4.60420e6 q^{91} +(251335. + 435325. i) q^{92} +(-3.91142e6 + 6.77477e6i) q^{94} +(4.59483e6 - 7.95847e6i) q^{95} +(4.30014e6 + 7.44806e6i) q^{97} +2.91320e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 9 q^{2} - 321 q^{4} + 180 q^{5} - 84 q^{7} - 5922 q^{8} + 252 q^{10} + 8460 q^{11} - 1848 q^{13} + 16272 q^{14} - 12417 q^{16} - 30564 q^{17} + 24432 q^{19} + 40788 q^{20} - 35001 q^{22} + 51588 q^{23}+ \cdots + 95833314 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

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

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 \(0.0145017\pi\)
−0.460040 + 0.887898i \(0.652165\pi\)
\(3\) 0 0
\(4\) −10.3519 + 17.9301i −0.0808745 + 0.140079i
\(5\) 246.026 − 426.130i 0.880210 − 1.52457i 0.0291025 − 0.999576i \(-0.490735\pi\)
0.851107 − 0.524992i \(-0.175932\pi\)
\(6\) 0 0
\(7\) −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
\(9\) 0 0
\(10\) 6000.29 1.89746
\(11\) −36.3512 − 62.9621i −0.00823463 − 0.0142628i 0.861879 − 0.507114i \(-0.169288\pi\)
−0.870113 + 0.492852i \(0.835955\pi\)
\(12\) 0 0
\(13\) −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
\(15\) 0 0
\(16\) 9302.72 + 16112.8i 0.567793 + 0.983446i
\(17\) 5989.93 0.295700 0.147850 − 0.989010i \(-0.452765\pi\)
0.147850 + 0.989010i \(0.452765\pi\)
\(18\) 0 0
\(19\) 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
\(21\) 0 0
\(22\) 443.281 − 767.786i 0.00887565 − 0.0153731i
\(23\) 12139.5 − 21026.3i 0.208043 − 0.360342i −0.743055 − 0.669231i \(-0.766624\pi\)
0.951098 + 0.308889i \(0.0999571\pi\)
\(24\) 0 0
\(25\) −81995.2 − 142020.i −1.04954 − 1.81785i
\(26\) −73429.1 −0.819335
\(27\) 0 0
\(28\) 15830.6 0.136284
\(29\) 43378.1 + 75133.0i 0.330276 + 0.572055i 0.982566 − 0.185914i \(-0.0595247\pi\)
−0.652290 + 0.757970i \(0.726191\pi\)
\(30\) 0 0
\(31\) −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
\(33\) 0 0
\(34\) 36521.9 + 63257.8i 0.159359 + 0.276018i
\(35\) −376234. −1.48327
\(36\) 0 0
\(37\) −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
\(39\) 0 0
\(40\) 321904. − 557554.i 0.795273 − 1.37745i
\(41\) 196036. − 339545.i 0.444214 − 0.769402i −0.553783 − 0.832661i \(-0.686816\pi\)
0.997997 + 0.0632592i \(0.0201495\pi\)
\(42\) 0 0
\(43\) 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
\(45\) 0 0
\(46\) 296069. 0.448477
\(47\) 320755. + 555563.i 0.450641 + 0.780533i 0.998426 − 0.0560862i \(-0.0178622\pi\)
−0.547785 + 0.836619i \(0.684529\pi\)
\(48\) 0 0
\(49\) 119448. − 206890.i 0.145042 − 0.251220i
\(50\) 999884. − 1.73185e6i 1.13124 − 1.95936i
\(51\) 0 0
\(52\) −62334.5 − 107966.i −0.0614776 − 0.106482i
\(53\) 814485. 0.751480 0.375740 − 0.926725i \(-0.377389\pi\)
0.375740 + 0.926725i \(0.377389\pi\)
\(54\) 0 0
\(55\) −35773.3 −0.0289928
\(56\) −500221. − 866408.i −0.380631 − 0.659272i
\(57\) 0 0
\(58\) −528971. + 916204.i −0.355986 + 0.616587i
\(59\) −1.25863e6 + 2.18002e6i −0.797843 + 1.38190i 0.123176 + 0.992385i \(0.460692\pi\)
−0.921018 + 0.389519i \(0.872641\pi\)
\(60\) 0 0
\(61\) 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
\(63\) 0 0
\(64\) 1.65708e6 0.790157
\(65\) 1.48145e6 + 2.56595e6i 0.669101 + 1.15892i
\(66\) 0 0
\(67\) −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
\(69\) 0 0
\(70\) −2.29398e6 − 3.97329e6i −0.799367 − 1.38454i
\(71\) −1.48821e6 −0.493469 −0.246734 − 0.969083i \(-0.579357\pi\)
−0.246734 + 0.969083i \(0.579357\pi\)
\(72\) 0 0
\(73\) −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
\(75\) 0 0
\(76\) −193334. + 334865.i −0.0505198 + 0.0875029i
\(77\) −27794.9 + 48142.2i −0.00693821 + 0.0120173i
\(78\) 0 0
\(79\) 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
\(81\) 0 0
\(82\) 4.78109e6 0.957588
\(83\) −1.69323e6 − 2.93276e6i −0.325044 − 0.562993i 0.656477 − 0.754346i \(-0.272046\pi\)
−0.981521 + 0.191353i \(0.938713\pi\)
\(84\) 0 0
\(85\) 1.47368e6 − 2.55249e6i 0.260278 − 0.450814i
\(86\) −4.19014e6 + 7.25754e6i −0.710369 + 1.23040i
\(87\) 0 0
\(88\) −47562.4 − 82380.4i −0.00744002 − 0.0128865i
\(89\) −1.17388e6 −0.176506 −0.0882531 − 0.996098i \(-0.528128\pi\)
−0.0882531 + 0.996098i \(0.528128\pi\)
\(90\) 0 0
\(91\) 4.60420e6 0.640485
\(92\) 251335. + 435325.i 0.0336508 + 0.0582849i
\(93\) 0 0
\(94\) −3.91142e6 + 6.77477e6i −0.485721 + 0.841293i
\(95\) 4.59483e6 − 7.95847e6i 0.549840 − 0.952351i
\(96\) 0 0
\(97\) 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
\(99\) 0 0
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 27.8.c.a.10.5 12
3.2 odd 2 9.8.c.a.4.2 ✓ 12
4.3 odd 2 432.8.i.c.145.6 12
9.2 odd 6 9.8.c.a.7.2 yes 12
9.4 even 3 81.8.a.c.1.2 6
9.5 odd 6 81.8.a.e.1.5 6
9.7 even 3 inner 27.8.c.a.19.5 12
12.11 even 2 144.8.i.c.49.5 12
36.7 odd 6 432.8.i.c.289.6 12
36.11 even 6 144.8.i.c.97.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.2 ✓ 12 3.2 odd 2
9.8.c.a.7.2 yes 12 9.2 odd 6
27.8.c.a.10.5 12 1.1 even 1 trivial
27.8.c.a.19.5 12 9.7 even 3 inner
81.8.a.c.1.2 6 9.4 even 3
81.8.a.e.1.5 6 9.5 odd 6
144.8.i.c.49.5 12 12.11 even 2
144.8.i.c.97.5 12 36.11 even 6
432.8.i.c.145.6 12 4.3 odd 2
432.8.i.c.289.6 12 36.7 odd 6