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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 815.7
Character \(\chi\) \(=\) 864.815
Dual form 864.2.bh.b.335.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.24100 - 1.20827i) q^{3} +(-3.07462 + 1.11907i) q^{5} +(-1.33825 + 0.235969i) q^{7} +(0.0801499 + 2.99893i) q^{9} +(0.982057 - 2.69818i) q^{11} +(-2.36880 + 2.82303i) q^{13} +(5.16774 + 2.32622i) q^{15} +(1.94027 + 1.12022i) q^{17} +(1.22444 + 2.12079i) q^{19} +(1.94588 + 1.32413i) q^{21} +(1.08869 - 6.17426i) q^{23} +(4.37075 - 3.66750i) q^{25} +(3.52406 - 3.81851i) q^{27} +(5.00298 - 4.19800i) q^{29} +(-4.65080 - 0.820062i) q^{31} +(-4.47887 + 2.16184i) q^{33} +(3.85054 - 2.22311i) q^{35} +(3.70698 + 2.14023i) q^{37} +(6.35067 - 0.641210i) q^{39} +(7.28211 - 8.67848i) q^{41} +(3.24005 + 1.17928i) q^{43} +(-3.60244 - 9.13088i) q^{45} +(2.21737 + 12.5753i) q^{47} +(-4.84262 + 1.76257i) q^{49} +(-1.05434 - 3.73456i) q^{51} +4.14210 q^{53} +9.39487i q^{55} +(1.04297 - 4.11136i) q^{57} +(-3.92440 - 10.7822i) q^{59} +(9.59806 - 1.69240i) q^{61} +(-0.814916 - 3.99440i) q^{63} +(4.12400 - 11.3306i) q^{65} +(6.64683 + 5.57735i) q^{67} +(-8.81126 + 6.34681i) q^{69} +(3.19451 - 5.53306i) q^{71} +(-2.87137 - 4.97336i) q^{73} +(-9.85543 - 0.729710i) q^{75} +(-0.677549 + 3.84257i) q^{77} +(-5.31972 - 6.33979i) q^{79} +(-8.98715 + 0.480728i) q^{81} +(1.56022 + 1.85940i) q^{83} +(-7.21919 - 1.27294i) q^{85} +(-11.2810 - 0.835263i) q^{87} +(1.06132 - 0.612754i) q^{89} +(2.50390 - 4.33688i) q^{91} +(4.78077 + 6.63713i) q^{93} +(-6.13801 - 5.15040i) q^{95} +(16.9633 + 6.17414i) q^{97} +(8.17036 + 2.72886i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 192 q + 12 q^{3} - 12 q^{9} + 30 q^{11} - 18 q^{17} + 6 q^{19} - 12 q^{25} - 18 q^{27} - 30 q^{33} + 18 q^{35} + 18 q^{41} + 42 q^{43} - 12 q^{49} + 18 q^{51} - 36 q^{57} + 84 q^{59} - 12 q^{65} - 30 q^{67}+ \cdots + 12 q^{99}+O(q^{100}) \) 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
\(3\) −1.24100 − 1.20827i −0.716490 − 0.697597i
\(4\) 0 0
\(5\) −3.07462 + 1.11907i −1.37501 + 0.500464i −0.920662 − 0.390360i \(-0.872351\pi\)
−0.454350 + 0.890823i \(0.650128\pi\)
\(6\) 0 0
\(7\) −1.33825 + 0.235969i −0.505810 + 0.0891880i −0.420730 − 0.907186i \(-0.638226\pi\)
−0.0850807 + 0.996374i \(0.527115\pi\)
\(8\) 0 0
\(9\) 0.0801499 + 2.99893i 0.0267166 + 0.999643i
\(10\) 0 0
\(11\) 0.982057 − 2.69818i 0.296101 − 0.813531i −0.699041 − 0.715082i \(-0.746389\pi\)
0.995142 − 0.0984497i \(-0.0313883\pi\)
\(12\) 0 0
\(13\) −2.36880 + 2.82303i −0.656987 + 0.782967i −0.986950 − 0.161027i \(-0.948519\pi\)
0.329963 + 0.943994i \(0.392964\pi\)
\(14\) 0 0
\(15\) 5.16774 + 2.32622i 1.33430 + 0.600627i
\(16\) 0 0
\(17\) 1.94027 + 1.12022i 0.470585 + 0.271692i 0.716484 − 0.697603i \(-0.245750\pi\)
−0.245900 + 0.969295i \(0.579083\pi\)
\(18\) 0 0
\(19\) 1.22444 + 2.12079i 0.280906 + 0.486544i 0.971608 − 0.236596i \(-0.0760317\pi\)
−0.690702 + 0.723139i \(0.742698\pi\)
\(20\) 0 0
\(21\) 1.94588 + 1.32413i 0.424626 + 0.288949i
\(22\) 0 0
\(23\) 1.08869 − 6.17426i 0.227007 − 1.28742i −0.631803 − 0.775129i \(-0.717685\pi\)
0.858810 − 0.512294i \(-0.171204\pi\)
\(24\) 0 0
\(25\) 4.37075 − 3.66750i 0.874151 − 0.733500i
\(26\) 0 0
\(27\) 3.52406 − 3.81851i 0.678206 − 0.734872i
\(28\) 0 0
\(29\) 5.00298 − 4.19800i 0.929030 − 0.779549i −0.0466128 − 0.998913i \(-0.514843\pi\)
0.975643 + 0.219364i \(0.0703983\pi\)
\(30\) 0 0
\(31\) −4.65080 − 0.820062i −0.835309 − 0.147287i −0.260396 − 0.965502i \(-0.583853\pi\)
−0.574913 + 0.818215i \(0.694964\pi\)
\(32\) 0 0
\(33\) −4.47887 + 2.16184i −0.779671 + 0.376328i
\(34\) 0 0
\(35\) 3.85054 − 2.22311i 0.650860 − 0.375774i
\(36\) 0 0
\(37\) 3.70698 + 2.14023i 0.609424 + 0.351851i 0.772740 − 0.634723i \(-0.218886\pi\)
−0.163316 + 0.986574i \(0.552219\pi\)
\(38\) 0 0
\(39\) 6.35067 − 0.641210i 1.01692 − 0.102676i
\(40\) 0 0
\(41\) 7.28211 − 8.67848i 1.13727 − 1.35535i 0.211454 − 0.977388i \(-0.432180\pi\)
0.925821 − 0.377963i \(-0.123375\pi\)
\(42\) 0 0
\(43\) 3.24005 + 1.17928i 0.494103 + 0.179839i 0.577040 − 0.816716i \(-0.304208\pi\)
−0.0829367 + 0.996555i \(0.526430\pi\)
\(44\) 0 0
\(45\) −3.60244 − 9.13088i −0.537021 − 1.36115i
\(46\) 0 0
\(47\) 2.21737 + 12.5753i 0.323437 + 1.83430i 0.520439 + 0.853899i \(0.325768\pi\)
−0.197002 + 0.980403i \(0.563121\pi\)
\(48\) 0 0
\(49\) −4.84262 + 1.76257i −0.691803 + 0.251796i
\(50\) 0 0
\(51\) −1.05434 − 3.73456i −0.147638 − 0.522943i
\(52\) 0 0
\(53\) 4.14210 0.568961 0.284481 − 0.958682i \(-0.408179\pi\)
0.284481 + 0.958682i \(0.408179\pi\)
\(54\) 0 0
\(55\) 9.39487i 1.26680i
\(56\) 0 0
\(57\) 1.04297 − 4.11136i 0.138145 − 0.544563i
\(58\) 0 0
\(59\) −3.92440 − 10.7822i −0.510913 − 1.40372i −0.880287 − 0.474442i \(-0.842650\pi\)
0.369373 − 0.929281i \(-0.379572\pi\)
\(60\) 0 0
\(61\) 9.59806 − 1.69240i 1.22891 − 0.216689i 0.478750 − 0.877951i \(-0.341090\pi\)
0.750155 + 0.661262i \(0.229979\pi\)
\(62\) 0 0
\(63\) −0.814916 − 3.99440i −0.102670 − 0.503247i
\(64\) 0 0
\(65\) 4.12400 − 11.3306i 0.511519 − 1.40539i
\(66\) 0 0
\(67\) 6.64683 + 5.57735i 0.812039 + 0.681382i 0.951094 − 0.308903i \(-0.0999619\pi\)
−0.139054 + 0.990285i \(0.544406\pi\)
\(68\) 0 0
\(69\) −8.81126 + 6.34681i −1.06075 + 0.764066i
\(70\) 0 0
\(71\) 3.19451 − 5.53306i 0.379119 − 0.656653i −0.611816 − 0.791000i \(-0.709561\pi\)
0.990934 + 0.134348i \(0.0428939\pi\)
\(72\) 0 0
\(73\) −2.87137 − 4.97336i −0.336069 − 0.582088i 0.647621 − 0.761963i \(-0.275764\pi\)
−0.983690 + 0.179875i \(0.942431\pi\)
\(74\) 0 0
\(75\) −9.85543 − 0.729710i −1.13801 − 0.0842597i
\(76\) 0 0
\(77\) −0.677549 + 3.84257i −0.0772138 + 0.437901i
\(78\) 0 0
\(79\) −5.31972 − 6.33979i −0.598515 − 0.713283i 0.378703 − 0.925518i \(-0.376370\pi\)
−0.977219 + 0.212236i \(0.931926\pi\)
\(80\) 0 0
\(81\) −8.98715 + 0.480728i −0.998572 + 0.0534142i
\(82\) 0 0
\(83\) 1.56022 + 1.85940i 0.171257 + 0.204096i 0.844845 − 0.535011i \(-0.179692\pi\)
−0.673589 + 0.739106i \(0.735248\pi\)
\(84\) 0 0
\(85\) −7.21919 − 1.27294i −0.783032 − 0.138070i
\(86\) 0 0
\(87\) −11.2810 − 0.835263i −1.20945 − 0.0895495i
\(88\) 0 0
\(89\) 1.06132 − 0.612754i 0.112500 − 0.0649518i −0.442694 − 0.896673i \(-0.645977\pi\)
0.555194 + 0.831721i \(0.312644\pi\)
\(90\) 0 0
\(91\) 2.50390 − 4.33688i 0.262480 − 0.454628i
\(92\) 0 0
\(93\) 4.78077 + 6.63713i 0.495743 + 0.688239i
\(94\) 0 0
\(95\) −6.13801 − 5.15040i −0.629747 − 0.528420i
\(96\) 0 0
\(97\) 16.9633 + 6.17414i 1.72236 + 0.626889i 0.998041 − 0.0625705i \(-0.0199298\pi\)
0.724324 + 0.689460i \(0.242152\pi\)
\(98\) 0 0
\(99\) 8.17036 + 2.72886i 0.821152 + 0.274261i
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.815.7 192
4.3 odd 2 216.2.v.b.59.8 yes 192
8.3 odd 2 inner 864.2.bh.b.815.8 192
8.5 even 2 216.2.v.b.59.11 yes 192
12.11 even 2 648.2.v.b.611.25 192
24.5 odd 2 648.2.v.b.611.22 192
27.11 odd 18 inner 864.2.bh.b.335.8 192
108.11 even 18 216.2.v.b.11.11 yes 192
108.43 odd 18 648.2.v.b.35.22 192
216.11 even 18 inner 864.2.bh.b.335.7 192
216.173 odd 18 216.2.v.b.11.8 ✓ 192
216.205 even 18 648.2.v.b.35.25 192
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.2.v.b.11.8 ✓ 192 216.173 odd 18
216.2.v.b.11.11 yes 192 108.11 even 18
216.2.v.b.59.8 yes 192 4.3 odd 2
216.2.v.b.59.11 yes 192 8.5 even 2
648.2.v.b.35.22 192 108.43 odd 18
648.2.v.b.35.25 192 216.205 even 18
648.2.v.b.611.22 192 24.5 odd 2
648.2.v.b.611.25 192 12.11 even 2
864.2.bh.b.335.7 192 216.11 even 18 inner
864.2.bh.b.335.8 192 27.11 odd 18 inner
864.2.bh.b.815.7 192 1.1 even 1 trivial
864.2.bh.b.815.8 192 8.3 odd 2 inner