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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [864,2,Mod(47,864)] 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("864.47"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(864, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 9, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.bh (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [192] 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: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(32\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 815.6
Character \(\chi\) \(=\) 864.815
Dual form 864.2.bh.b.335.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.40807 - 1.00863i) q^{3} +(2.25586 - 0.821067i) q^{5} +(-1.95474 + 0.344674i) q^{7} +(0.965315 + 2.84045i) q^{9} +(-0.553700 + 1.52128i) q^{11} +(-3.57715 + 4.26309i) q^{13} +(-4.00457 - 1.11922i) q^{15} +(-6.40600 - 3.69851i) q^{17} +(1.39557 + 2.41719i) q^{19} +(3.10006 + 1.48630i) q^{21} +(-0.465506 + 2.64002i) q^{23} +(0.584547 - 0.490493i) q^{25} +(1.50575 - 4.97320i) q^{27} +(0.138481 - 0.116199i) q^{29} +(1.76926 + 0.311969i) q^{31} +(2.31406 - 1.58358i) q^{33} +(-4.12663 + 2.38251i) q^{35} +(3.55469 + 2.05230i) q^{37} +(9.33677 - 2.39468i) q^{39} +(-2.85334 + 3.40048i) q^{41} +(-11.1139 - 4.04511i) q^{43} +(4.50982 + 5.61508i) q^{45} +(2.08588 + 11.8296i) q^{47} +(-2.87563 + 1.04664i) q^{49} +(5.28965 + 11.6691i) q^{51} +10.1087 q^{53} +3.88642i q^{55} +(0.473009 - 4.81119i) q^{57} +(1.96718 + 5.40478i) q^{59} +(-7.60780 + 1.34146i) q^{61} +(-2.86597 - 5.21963i) q^{63} +(-4.56929 + 12.5540i) q^{65} +(-1.38817 - 1.16481i) q^{67} +(3.31828 - 3.24780i) q^{69} +(-1.25119 + 2.16713i) q^{71} +(-3.79038 - 6.56514i) q^{73} +(-1.31781 + 0.101054i) q^{75} +(0.557996 - 3.16455i) q^{77} +(6.77140 + 8.06984i) q^{79} +(-7.13633 + 5.48386i) q^{81} +(-0.718131 - 0.855836i) q^{83} +(-17.4878 - 3.08357i) q^{85} +(-0.312194 + 0.0239400i) q^{87} +(9.87804 - 5.70309i) q^{89} +(5.52304 - 9.56619i) q^{91} +(-2.17658 - 2.22381i) q^{93} +(5.13289 + 4.30701i) q^{95} +(-5.25587 - 1.91298i) q^{97} +(-4.85561 - 0.104246i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 192 q + 12 q^{3} - 12 q^{9} + 30 q^{11} - 18 q^{17} + 6 q^{19} - 12 q^{25} - 18 q^{27} - 30 q^{33} + 18 q^{35} + 18 q^{41} + 42 q^{43} - 12 q^{49} + 18 q^{51} - 36 q^{57} + 84 q^{59} - 12 q^{65} - 30 q^{67}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{18}\right)\) \(-1\)

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
\(3\) −1.40807 − 1.00863i −0.812949 − 0.582335i
\(4\) 0 0
\(5\) 2.25586 − 0.821067i 1.00885 − 0.367192i 0.215861 − 0.976424i \(-0.430744\pi\)
0.792992 + 0.609232i \(0.208522\pi\)
\(6\) 0 0
\(7\) −1.95474 + 0.344674i −0.738823 + 0.130274i −0.530380 − 0.847760i \(-0.677951\pi\)
−0.208443 + 0.978034i \(0.566840\pi\)
\(8\) 0 0
\(9\) 0.965315 + 2.84045i 0.321772 + 0.946817i
\(10\) 0 0
\(11\) −0.553700 + 1.52128i −0.166947 + 0.458683i −0.994750 − 0.102336i \(-0.967368\pi\)
0.827803 + 0.561019i \(0.189590\pi\)
\(12\) 0 0
\(13\) −3.57715 + 4.26309i −0.992124 + 1.18237i −0.00890108 + 0.999960i \(0.502833\pi\)
−0.983223 + 0.182407i \(0.941611\pi\)
\(14\) 0 0
\(15\) −4.00457 − 1.11922i −1.03397 − 0.288982i
\(16\) 0 0
\(17\) −6.40600 − 3.69851i −1.55368 − 0.897020i −0.997837 − 0.0657345i \(-0.979061\pi\)
−0.555846 − 0.831285i \(-0.687606\pi\)
\(18\) 0 0
\(19\) 1.39557 + 2.41719i 0.320165 + 0.554543i 0.980522 − 0.196410i \(-0.0629282\pi\)
−0.660357 + 0.750952i \(0.729595\pi\)
\(20\) 0 0
\(21\) 3.10006 + 1.48630i 0.676489 + 0.324336i
\(22\) 0 0
\(23\) −0.465506 + 2.64002i −0.0970648 + 0.550482i 0.897030 + 0.441969i \(0.145720\pi\)
−0.994095 + 0.108513i \(0.965391\pi\)
\(24\) 0 0
\(25\) 0.584547 − 0.490493i 0.116909 − 0.0980986i
\(26\) 0 0
\(27\) 1.50575 − 4.97320i 0.289781 − 0.957093i
\(28\) 0 0
\(29\) 0.138481 − 0.116199i 0.0257153 − 0.0215777i −0.629839 − 0.776725i \(-0.716879\pi\)
0.655555 + 0.755148i \(0.272435\pi\)
\(30\) 0 0
\(31\) 1.76926 + 0.311969i 0.317769 + 0.0560313i 0.330258 − 0.943891i \(-0.392864\pi\)
−0.0124887 + 0.999922i \(0.503975\pi\)
\(32\) 0 0
\(33\) 2.31406 − 1.58358i 0.402826 − 0.275667i
\(34\) 0 0
\(35\) −4.12663 + 2.38251i −0.697528 + 0.402718i
\(36\) 0 0
\(37\) 3.55469 + 2.05230i 0.584387 + 0.337396i 0.762875 − 0.646546i \(-0.223787\pi\)
−0.178488 + 0.983942i \(0.557120\pi\)
\(38\) 0 0
\(39\) 9.33677 − 2.39468i 1.49508 − 0.383456i
\(40\) 0 0
\(41\) −2.85334 + 3.40048i −0.445617 + 0.531066i −0.941360 − 0.337404i \(-0.890451\pi\)
0.495743 + 0.868469i \(0.334896\pi\)
\(42\) 0 0
\(43\) −11.1139 − 4.04511i −1.69485 − 0.616874i −0.699625 − 0.714510i \(-0.746650\pi\)
−0.995222 + 0.0976364i \(0.968872\pi\)
\(44\) 0 0
\(45\) 4.50982 + 5.61508i 0.672284 + 0.837047i
\(46\) 0 0
\(47\) 2.08588 + 11.8296i 0.304257 + 1.72552i 0.626983 + 0.779033i \(0.284289\pi\)
−0.322727 + 0.946492i \(0.604599\pi\)
\(48\) 0 0
\(49\) −2.87563 + 1.04664i −0.410804 + 0.149520i
\(50\) 0 0
\(51\) 5.28965 + 11.6691i 0.740699 + 1.63400i
\(52\) 0 0
\(53\) 10.1087 1.38854 0.694271 − 0.719714i \(-0.255727\pi\)
0.694271 + 0.719714i \(0.255727\pi\)
\(54\) 0 0
\(55\) 3.88642i 0.524045i
\(56\) 0 0
\(57\) 0.473009 − 4.81119i 0.0626516 − 0.637258i
\(58\) 0 0
\(59\) 1.96718 + 5.40478i 0.256105 + 0.703642i 0.999399 + 0.0346769i \(0.0110402\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(60\) 0 0
\(61\) −7.60780 + 1.34146i −0.974079 + 0.171756i −0.637965 − 0.770065i \(-0.720224\pi\)
−0.336113 + 0.941822i \(0.609113\pi\)
\(62\) 0 0
\(63\) −2.86597 − 5.21963i −0.361079 − 0.657612i
\(64\) 0 0
\(65\) −4.56929 + 12.5540i −0.566751 + 1.55714i
\(66\) 0 0
\(67\) −1.38817 − 1.16481i −0.169592 − 0.142305i 0.554041 − 0.832489i \(-0.313085\pi\)
−0.723633 + 0.690184i \(0.757529\pi\)
\(68\) 0 0
\(69\) 3.31828 − 3.24780i 0.399474 − 0.390989i
\(70\) 0 0
\(71\) −1.25119 + 2.16713i −0.148489 + 0.257191i −0.930669 − 0.365862i \(-0.880774\pi\)
0.782180 + 0.623052i \(0.214108\pi\)
\(72\) 0 0
\(73\) −3.79038 − 6.56514i −0.443631 − 0.768391i 0.554325 − 0.832300i \(-0.312976\pi\)
−0.997956 + 0.0639093i \(0.979643\pi\)
\(74\) 0 0
\(75\) −1.31781 + 0.101054i −0.152168 + 0.0116687i
\(76\) 0 0
\(77\) 0.557996 − 3.16455i 0.0635896 − 0.360634i
\(78\) 0 0
\(79\) 6.77140 + 8.06984i 0.761842 + 0.907928i 0.997963 − 0.0637957i \(-0.0203206\pi\)
−0.236121 + 0.971724i \(0.575876\pi\)
\(80\) 0 0
\(81\) −7.13633 + 5.48386i −0.792926 + 0.609318i
\(82\) 0 0
\(83\) −0.718131 − 0.855836i −0.0788251 − 0.0939402i 0.725191 − 0.688548i \(-0.241752\pi\)
−0.804016 + 0.594608i \(0.797307\pi\)
\(84\) 0 0
\(85\) −17.4878 − 3.08357i −1.89682 − 0.334460i
\(86\) 0 0
\(87\) −0.312194 + 0.0239400i −0.0334707 + 0.00256664i
\(88\) 0 0
\(89\) 9.87804 − 5.70309i 1.04707 − 0.604526i 0.125243 − 0.992126i \(-0.460029\pi\)
0.921828 + 0.387600i \(0.126696\pi\)
\(90\) 0 0
\(91\) 5.52304 − 9.56619i 0.578972 − 1.00281i
\(92\) 0 0
\(93\) −2.17658 − 2.22381i −0.225701 − 0.230599i
\(94\) 0 0
\(95\) 5.13289 + 4.30701i 0.526623 + 0.441890i
\(96\) 0 0
\(97\) −5.25587 − 1.91298i −0.533652 − 0.194234i 0.0611161 − 0.998131i \(-0.480534\pi\)
−0.594768 + 0.803897i \(0.702756\pi\)
\(98\) 0 0
\(99\) −4.85561 − 0.104246i −0.488007 − 0.0104771i
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 864.2.bh.b.815.6 192
4.3 odd 2 216.2.v.b.59.16 yes 192
8.3 odd 2 inner 864.2.bh.b.815.5 192
8.5 even 2 216.2.v.b.59.32 yes 192
12.11 even 2 648.2.v.b.611.17 192
24.5 odd 2 648.2.v.b.611.1 192
27.11 odd 18 inner 864.2.bh.b.335.5 192
108.11 even 18 216.2.v.b.11.32 yes 192
108.43 odd 18 648.2.v.b.35.1 192
216.11 even 18 inner 864.2.bh.b.335.6 192
216.173 odd 18 216.2.v.b.11.16 ✓ 192
216.205 even 18 648.2.v.b.35.17 192
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.2.v.b.11.16 ✓ 192 216.173 odd 18
216.2.v.b.11.32 yes 192 108.11 even 18
216.2.v.b.59.16 yes 192 4.3 odd 2
216.2.v.b.59.32 yes 192 8.5 even 2
648.2.v.b.35.1 192 108.43 odd 18
648.2.v.b.35.17 192 216.205 even 18
648.2.v.b.611.1 192 24.5 odd 2
648.2.v.b.611.17 192 12.11 even 2
864.2.bh.b.335.5 192 27.11 odd 18 inner
864.2.bh.b.335.6 192 216.11 even 18 inner
864.2.bh.b.815.5 192 8.3 odd 2 inner
864.2.bh.b.815.6 192 1.1 even 1 trivial