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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 = [22] 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: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 145.7
Character \(\chi\) \(=\) 432.145
Dual form 432.8.i.f.289.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(53.7582 - 93.1119i) q^{5} +(-322.045 - 557.798i) q^{7} +(-1999.99 - 3464.08i) q^{11} +(5581.68 - 9667.76i) q^{13} +18899.1 q^{17} +50434.9 q^{19} +(-10797.9 + 18702.5i) q^{23} +(33282.6 + 57647.2i) q^{25} +(54370.0 + 94171.6i) q^{29} +(28762.7 - 49818.5i) q^{31} -69250.2 q^{35} -265161. q^{37} +(203350. - 352213. i) q^{41} +(226994. + 393166. i) q^{43} +(-450753. - 780726. i) q^{47} +(204346. - 353937. i) q^{49} +474776. q^{53} -430062. q^{55} +(388284. - 672527. i) q^{59} +(844796. + 1.46323e6i) q^{61} +(-600122. - 1.03944e6i) q^{65} +(-763249. + 1.32199e6i) q^{67} -1.36632e6 q^{71} +5.13496e6 q^{73} +(-1.28817e6 + 2.23118e6i) q^{77} +(-852263. - 1.47616e6i) q^{79} +(-3.95066e6 - 6.84274e6i) q^{83} +(1.01598e6 - 1.75973e6i) q^{85} -6.21721e6 q^{89} -7.19021e6 q^{91} +(2.71129e6 - 4.69609e6i) q^{95} +(2.00221e6 + 3.46794e6i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q + 125 q^{5} + 1245 q^{7} + 1699 q^{11} - 4937 q^{13} - 26540 q^{17} - 28976 q^{19} - 18239 q^{23} - 109168 q^{25} + 3525 q^{29} - 23753 q^{31} - 613122 q^{35} - 108420 q^{37} - 75063 q^{41} + 604385 q^{43}+ \cdots + 1704887 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 53.7582 − 93.1119i 0.192331 − 0.333127i −0.753691 − 0.657229i \(-0.771729\pi\)
0.946022 + 0.324101i \(0.105062\pi\)
\(6\) 0 0
\(7\) −322.045 − 557.798i −0.354873 − 0.614658i 0.632223 − 0.774786i \(-0.282143\pi\)
−0.987096 + 0.160128i \(0.948809\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1999.99 − 3464.08i −0.453057 − 0.784717i 0.545517 − 0.838099i \(-0.316333\pi\)
−0.998574 + 0.0533823i \(0.983000\pi\)
\(12\) 0 0
\(13\) 5581.68 − 9667.76i 0.704634 − 1.22046i −0.262190 − 0.965016i \(-0.584445\pi\)
0.966824 − 0.255445i \(-0.0822221\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 18899.1 0.932977 0.466488 − 0.884527i \(-0.345519\pi\)
0.466488 + 0.884527i \(0.345519\pi\)
\(18\) 0 0
\(19\) 50434.9 1.68692 0.843458 − 0.537196i \(-0.180516\pi\)
0.843458 + 0.537196i \(0.180516\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −10797.9 + 18702.5i −0.185051 + 0.320518i −0.943594 − 0.331106i \(-0.892578\pi\)
0.758543 + 0.651623i \(0.225912\pi\)
\(24\) 0 0
\(25\) 33282.6 + 57647.2i 0.426018 + 0.737884i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 54370.0 + 94171.6i 0.413968 + 0.717013i 0.995319 − 0.0966392i \(-0.0308093\pi\)
−0.581352 + 0.813652i \(0.697476\pi\)
\(30\) 0 0
\(31\) 28762.7 − 49818.5i 0.173406 − 0.300348i −0.766202 − 0.642599i \(-0.777856\pi\)
0.939609 + 0.342251i \(0.111189\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −69250.2 −0.273013
\(36\) 0 0
\(37\) −265161. −0.860603 −0.430302 − 0.902685i \(-0.641593\pi\)
−0.430302 + 0.902685i \(0.641593\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 203350. − 352213.i 0.460788 − 0.798108i −0.538213 − 0.842809i \(-0.680900\pi\)
0.999000 + 0.0447012i \(0.0142336\pi\)
\(42\) 0 0
\(43\) 226994. + 393166.i 0.435387 + 0.754113i 0.997327 − 0.0730652i \(-0.0232781\pi\)
−0.561940 + 0.827178i \(0.689945\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −450753. − 780726.i −0.633280 − 1.09687i −0.986877 − 0.161475i \(-0.948375\pi\)
0.353597 − 0.935398i \(-0.384958\pi\)
\(48\) 0 0
\(49\) 204346. − 353937.i 0.248130 − 0.429774i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 474776. 0.438050 0.219025 − 0.975719i \(-0.429712\pi\)
0.219025 + 0.975719i \(0.429712\pi\)
\(54\) 0 0
\(55\) −430062. −0.348548
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 388284. − 672527.i 0.246132 − 0.426312i −0.716318 − 0.697774i \(-0.754174\pi\)
0.962449 + 0.271462i \(0.0875071\pi\)
\(60\) 0 0
\(61\) 844796. + 1.46323e6i 0.476538 + 0.825388i 0.999639 − 0.0268828i \(-0.00855809\pi\)
−0.523100 + 0.852271i \(0.675225\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −600122. − 1.03944e6i −0.271046 − 0.469465i
\(66\) 0 0
\(67\) −763249. + 1.32199e6i −0.310031 + 0.536989i −0.978369 − 0.206869i \(-0.933673\pi\)
0.668338 + 0.743858i \(0.267006\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.36632e6 −0.453052 −0.226526 − 0.974005i \(-0.572737\pi\)
−0.226526 + 0.974005i \(0.572737\pi\)
\(72\) 0 0
\(73\) 5.13496e6 1.54493 0.772463 − 0.635060i \(-0.219025\pi\)
0.772463 + 0.635060i \(0.219025\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.28817e6 + 2.23118e6i −0.321555 + 0.556950i
\(78\) 0 0
\(79\) −852263. − 1.47616e6i −0.194482 − 0.336852i 0.752249 − 0.658879i \(-0.228969\pi\)
−0.946731 + 0.322027i \(0.895636\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.95066e6 − 6.84274e6i −0.758396 − 1.31358i −0.943668 − 0.330894i \(-0.892650\pi\)
0.185272 − 0.982687i \(-0.440684\pi\)
\(84\) 0 0
\(85\) 1.01598e6 − 1.75973e6i 0.179440 − 0.310800i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.21721e6 −0.934825 −0.467412 − 0.884039i \(-0.654814\pi\)
−0.467412 + 0.884039i \(0.654814\pi\)
\(90\) 0 0
\(91\) −7.19021e6 −1.00022
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.71129e6 − 4.69609e6i 0.324446 − 0.561957i
\(96\) 0 0
\(97\) 2.00221e6 + 3.46794e6i 0.222746 + 0.385807i 0.955641 − 0.294535i \(-0.0951647\pi\)
−0.732895 + 0.680342i \(0.761831\pi\)
\(98\) 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.f.145.7 22
3.2 odd 2 144.8.i.f.49.4 22
4.3 odd 2 216.8.i.b.145.7 22
9.2 odd 6 144.8.i.f.97.4 22
9.7 even 3 inner 432.8.i.f.289.7 22
12.11 even 2 72.8.i.b.49.8 yes 22
36.7 odd 6 216.8.i.b.73.7 22
36.11 even 6 72.8.i.b.25.8 ✓ 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.8.i.b.25.8 ✓ 22 36.11 even 6
72.8.i.b.49.8 yes 22 12.11 even 2
144.8.i.f.49.4 22 3.2 odd 2
144.8.i.f.97.4 22 9.2 odd 6
216.8.i.b.73.7 22 36.7 odd 6
216.8.i.b.145.7 22 4.3 odd 2
432.8.i.f.145.7 22 1.1 even 1 trivial
432.8.i.f.289.7 22 9.7 even 3 inner