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Properties

Label 432.8.i.c.289.6
Level $432$
Weight $8$
Character 432.289
Analytic conductor $134.950$
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] = [432,8,Mod(145,432)] 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("432.145"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 432.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(134.950331009\)
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: \( 2^{16}\cdot 3^{21} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 289.6
Root \(0.500000 + 6.17443i\) of defining polynomial
Character \(\chi\) \(=\) 432.289
Dual form 432.8.i.c.145.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(246.026 + 426.130i) q^{5} +(382.311 - 662.182i) q^{7} +(36.3512 - 62.9621i) q^{11} +(-3010.77 - 5214.80i) q^{13} +5989.93 q^{17} -18676.2 q^{19} +(-12139.5 - 21026.3i) q^{23} +(-81995.2 + 142020. i) q^{25} +(43378.1 - 75133.0i) q^{29} +(105890. + 183406. i) q^{31} +376234. q^{35} -327978. q^{37} +(196036. + 339545. i) q^{41} +(-343611. + 595152. i) q^{43} +(-320755. + 555563. i) q^{47} +(119448. + 206890. i) q^{49} +814485. q^{53} +35773.3 q^{55} +(1.25863e6 + 2.18002e6i) q^{59} +(221621. - 383858. i) q^{61} +(1.48145e6 - 2.56595e6i) q^{65} +(296048. + 512770. i) q^{67} +1.48821e6 q^{71} -5.41341e6 q^{73} +(-27794.9 - 48142.2i) q^{77} +(-444736. + 770305. i) q^{79} +(1.69323e6 - 2.93276e6i) q^{83} +(1.47368e6 + 2.55249e6i) q^{85} -1.17388e6 q^{89} -4.60420e6 q^{91} +(-4.59483e6 - 7.95847e6i) q^{95} +(4.30014e6 - 7.44806e6i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 180 q^{5} + 84 q^{7} - 8460 q^{11} - 1848 q^{13} - 30564 q^{17} - 24432 q^{19} - 51588 q^{23} + 4746 q^{25} + 414648 q^{29} - 8196 q^{31} + 2210616 q^{35} + 139344 q^{37} + 1731582 q^{41} - 408372 q^{43}+ \cdots + 9977226 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 246.026 + 426.130i 0.880210 + 1.52457i 0.851107 + 0.524992i \(0.175932\pi\)
0.0291025 + 0.999576i \(0.490735\pi\)
\(6\) 0 0
\(7\) 382.311 − 662.182i 0.421283 − 0.729683i −0.574783 − 0.818306i \(-0.694913\pi\)
0.996065 + 0.0886232i \(0.0282467\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 36.3512 − 62.9621i 0.00823463 − 0.0142628i −0.861879 − 0.507114i \(-0.830712\pi\)
0.870113 + 0.492852i \(0.164045\pi\)
\(12\) 0 0
\(13\) −3010.77 − 5214.80i −0.380080 − 0.658318i 0.610993 − 0.791636i \(-0.290770\pi\)
−0.991073 + 0.133318i \(0.957437\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 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 \(-0.601111\pi\)
−0.312335 + 0.949972i \(0.601111\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −12139.5 − 21026.3i −0.208043 − 0.360342i 0.743055 − 0.669231i \(-0.233376\pi\)
−0.951098 + 0.308889i \(0.900043\pi\)
\(24\) 0 0
\(25\) −81995.2 + 142020.i −1.04954 + 1.81785i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 43378.1 − 75133.0i 0.330276 − 0.572055i −0.652290 − 0.757970i \(-0.726191\pi\)
0.982566 + 0.185914i \(0.0595247\pi\)
\(30\) 0 0
\(31\) 105890. + 183406.i 0.638392 + 1.10573i 0.985786 + 0.168008i \(0.0537334\pi\)
−0.347394 + 0.937719i \(0.612933\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 376234. 1.48327
\(36\) 0 0
\(37\) −327978. −1.06448 −0.532242 − 0.846592i \(-0.678650\pi\)
−0.532242 + 0.846592i \(0.678650\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 196036. + 339545.i 0.444214 + 0.769402i 0.997997 − 0.0632592i \(-0.0201495\pi\)
−0.553783 + 0.832661i \(0.686816\pi\)
\(42\) 0 0
\(43\) −343611. + 595152.i −0.659064 + 1.14153i 0.321794 + 0.946810i \(0.395714\pi\)
−0.980858 + 0.194723i \(0.937619\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −320755. + 555563.i −0.450641 + 0.780533i −0.998426 − 0.0560862i \(-0.982138\pi\)
0.547785 + 0.836619i \(0.315471\pi\)
\(48\) 0 0
\(49\) 119448. + 206890.i 0.145042 + 0.251220i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 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
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.25863e6 + 2.18002e6i 0.797843 + 1.38190i 0.921018 + 0.389519i \(0.127359\pi\)
−0.123176 + 0.992385i \(0.539308\pi\)
\(60\) 0 0
\(61\) 221621. − 383858.i 0.125013 − 0.216529i −0.796725 − 0.604342i \(-0.793436\pi\)
0.921738 + 0.387813i \(0.126769\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.48145e6 − 2.56595e6i 0.669101 − 1.15892i
\(66\) 0 0
\(67\) 296048. + 512770.i 0.120254 + 0.208286i 0.919868 − 0.392228i \(-0.128296\pi\)
−0.799614 + 0.600515i \(0.794962\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.48821e6 0.493469 0.246734 − 0.969083i \(-0.420643\pi\)
0.246734 + 0.969083i \(0.420643\pi\)
\(72\) 0 0
\(73\) −5.41341e6 −1.62870 −0.814350 − 0.580374i \(-0.802906\pi\)
−0.814350 + 0.580374i \(0.802906\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −27794.9 − 48142.2i −0.00693821 − 0.0120173i
\(78\) 0 0
\(79\) −444736. + 770305.i −0.101486 + 0.175779i −0.912297 − 0.409529i \(-0.865693\pi\)
0.810811 + 0.585308i \(0.199026\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.69323e6 − 2.93276e6i 0.325044 − 0.562993i −0.656477 − 0.754346i \(-0.727954\pi\)
0.981521 + 0.191353i \(0.0612874\pi\)
\(84\) 0 0
\(85\) 1.47368e6 + 2.55249e6i 0.260278 + 0.450814i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −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
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.59483e6 − 7.95847e6i −0.549840 − 0.952351i
\(96\) 0 0
\(97\) 4.30014e6 − 7.44806e6i 0.478390 − 0.828595i −0.521303 − 0.853371i \(-0.674554\pi\)
0.999693 + 0.0247763i \(0.00788736\pi\)
\(98\) 0 0
\(99\) 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 432.8.i.c.289.6 12
3.2 odd 2 144.8.i.c.97.5 12
4.3 odd 2 27.8.c.a.19.5 12
9.4 even 3 inner 432.8.i.c.145.6 12
9.5 odd 6 144.8.i.c.49.5 12
12.11 even 2 9.8.c.a.7.2 yes 12
36.7 odd 6 81.8.a.c.1.2 6
36.11 even 6 81.8.a.e.1.5 6
36.23 even 6 9.8.c.a.4.2 ✓ 12
36.31 odd 6 27.8.c.a.10.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.2 ✓ 12 36.23 even 6
9.8.c.a.7.2 yes 12 12.11 even 2
27.8.c.a.10.5 12 36.31 odd 6
27.8.c.a.19.5 12 4.3 odd 2
81.8.a.c.1.2 6 36.7 odd 6
81.8.a.e.1.5 6 36.11 even 6
144.8.i.c.49.5 12 9.5 odd 6
144.8.i.c.97.5 12 3.2 odd 2
432.8.i.c.145.6 12 9.4 even 3 inner
432.8.i.c.289.6 12 1.1 even 1 trivial