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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [108,3,Mod(5,108)] 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("108.5"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(108, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 5])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 108.k (of order \(18\), degree \(6\), 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: \(2.94278685509\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 29.2
Character \(\chi\) \(=\) 108.29
Dual form 108.3.k.a.41.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.59609 - 1.50344i) q^{3} +(-0.298552 - 0.355800i) q^{5} +(-10.1488 + 3.69384i) q^{7} +(4.47934 + 7.80612i) q^{9} +(-10.2314 + 12.1933i) q^{11} +(-3.11819 - 17.6841i) q^{13} +(0.240142 + 1.37254i) q^{15} +(-22.6442 + 13.0736i) q^{17} +(1.77864 - 3.08069i) q^{19} +(31.9005 + 5.66850i) q^{21} +(1.41115 - 3.87710i) q^{23} +(4.30374 - 24.4077i) q^{25} +(0.107276 - 26.9998i) q^{27} +(41.0456 + 7.23745i) q^{29} +(-6.62361 - 2.41080i) q^{31} +(44.8934 - 16.2726i) q^{33} +(4.34420 + 2.50812i) q^{35} +(4.92909 + 8.53743i) q^{37} +(-18.4919 + 50.5976i) q^{39} +(-42.8820 + 7.56125i) q^{41} +(-27.2523 - 22.8674i) q^{43} +(1.44010 - 3.92428i) q^{45} +(-5.51742 - 15.1590i) q^{47} +(51.8166 - 43.4793i) q^{49} +(78.4418 + 0.103888i) q^{51} +75.6950i q^{53} +7.39296 q^{55} +(-9.24914 + 5.32367i) q^{57} +(-18.4421 - 21.9784i) q^{59} +(-55.9422 + 20.3613i) q^{61} +(-74.2943 - 62.6764i) q^{63} +(-5.36107 + 6.38908i) q^{65} +(-4.03483 - 22.8827i) q^{67} +(-9.49246 + 7.94372i) q^{69} +(-32.2368 + 18.6119i) q^{71} +(-26.0280 + 45.0817i) q^{73} +(-47.8685 + 56.8942i) q^{75} +(58.7957 - 161.540i) q^{77} +(-20.9619 + 118.881i) q^{79} +(-40.8710 + 69.9325i) q^{81} +(115.091 + 20.2937i) q^{83} +(11.4121 + 4.15365i) q^{85} +(-95.6769 - 80.4987i) q^{87} +(-117.767 - 67.9930i) q^{89} +(96.9682 + 167.954i) q^{91} +(13.5710 + 16.2168i) q^{93} +(-1.62713 + 0.286906i) q^{95} +(-72.3238 - 60.6869i) q^{97} +(-141.012 - 25.2495i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q - 9 q^{5} + 6 q^{9} + 36 q^{11} + 45 q^{15} + 42 q^{21} - 18 q^{23} - 9 q^{25} - 18 q^{29} + 45 q^{31} - 153 q^{33} - 243 q^{35} - 123 q^{39} - 198 q^{41} + 90 q^{43} - 333 q^{45} - 243 q^{47} + 72 q^{49}+ \cdots - 945 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(55\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\) \(1\)

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
\(3\) −2.59609 − 1.50344i −0.865362 − 0.501147i
\(4\) 0 0
\(5\) −0.298552 − 0.355800i −0.0597103 − 0.0711600i 0.735363 − 0.677674i \(-0.237012\pi\)
−0.795073 + 0.606514i \(0.792567\pi\)
\(6\) 0 0
\(7\) −10.1488 + 3.69384i −1.44982 + 0.527692i −0.942542 − 0.334089i \(-0.891571\pi\)
−0.507281 + 0.861781i \(0.669349\pi\)
\(8\) 0 0
\(9\) 4.47934 + 7.80612i 0.497704 + 0.867347i
\(10\) 0 0
\(11\) −10.2314 + 12.1933i −0.930125 + 1.10848i 0.0637495 + 0.997966i \(0.479694\pi\)
−0.993875 + 0.110514i \(0.964750\pi\)
\(12\) 0 0
\(13\) −3.11819 − 17.6841i −0.239861 − 1.36032i −0.832131 − 0.554579i \(-0.812879\pi\)
0.592271 − 0.805739i \(-0.298232\pi\)
\(14\) 0 0
\(15\) 0.240142 + 1.37254i 0.0160095 + 0.0915028i
\(16\) 0 0
\(17\) −22.6442 + 13.0736i −1.33201 + 0.769038i −0.985608 − 0.169047i \(-0.945931\pi\)
−0.346405 + 0.938085i \(0.612598\pi\)
\(18\) 0 0
\(19\) 1.77864 − 3.08069i 0.0936126 − 0.162142i −0.815416 − 0.578875i \(-0.803492\pi\)
0.909029 + 0.416733i \(0.136825\pi\)
\(20\) 0 0
\(21\) 31.9005 + 5.66850i 1.51907 + 0.269928i
\(22\) 0 0
\(23\) 1.41115 − 3.87710i 0.0613543 − 0.168570i −0.905228 − 0.424926i \(-0.860300\pi\)
0.966582 + 0.256357i \(0.0825222\pi\)
\(24\) 0 0
\(25\) 4.30374 − 24.4077i 0.172150 − 0.976310i
\(26\) 0 0
\(27\) 0.107276 − 26.9998i 0.00397319 − 0.999992i
\(28\) 0 0
\(29\) 41.0456 + 7.23745i 1.41537 + 0.249567i 0.828442 − 0.560075i \(-0.189228\pi\)
0.586924 + 0.809642i \(0.300339\pi\)
\(30\) 0 0
\(31\) −6.62361 − 2.41080i −0.213665 − 0.0777677i 0.232970 − 0.972484i \(-0.425156\pi\)
−0.446635 + 0.894716i \(0.647378\pi\)
\(32\) 0 0
\(33\) 44.8934 − 16.2726i 1.36041 − 0.493108i
\(34\) 0 0
\(35\) 4.34420 + 2.50812i 0.124120 + 0.0716606i
\(36\) 0 0
\(37\) 4.92909 + 8.53743i 0.133219 + 0.230741i 0.924916 − 0.380173i \(-0.124135\pi\)
−0.791697 + 0.610914i \(0.790802\pi\)
\(38\) 0 0
\(39\) −18.4919 + 50.5976i −0.474152 + 1.29737i
\(40\) 0 0
\(41\) −42.8820 + 7.56125i −1.04590 + 0.184421i −0.670094 − 0.742277i \(-0.733746\pi\)
−0.375808 + 0.926697i \(0.622635\pi\)
\(42\) 0 0
\(43\) −27.2523 − 22.8674i −0.633773 − 0.531799i 0.268326 − 0.963328i \(-0.413530\pi\)
−0.902099 + 0.431529i \(0.857974\pi\)
\(44\) 0 0
\(45\) 1.44010 − 3.92428i 0.0320023 − 0.0872062i
\(46\) 0 0
\(47\) −5.51742 − 15.1590i −0.117392 − 0.322531i 0.867055 − 0.498212i \(-0.166010\pi\)
−0.984447 + 0.175680i \(0.943788\pi\)
\(48\) 0 0
\(49\) 51.8166 − 43.4793i 1.05748 − 0.887332i
\(50\) 0 0
\(51\) 78.4418 + 0.103888i 1.53807 + 0.00203703i
\(52\) 0 0
\(53\) 75.6950i 1.42821i 0.700040 + 0.714104i \(0.253166\pi\)
−0.700040 + 0.714104i \(0.746834\pi\)
\(54\) 0 0
\(55\) 7.39296 0.134417
\(56\) 0 0
\(57\) −9.24914 + 5.32367i −0.162266 + 0.0933977i
\(58\) 0 0
\(59\) −18.4421 − 21.9784i −0.312578 − 0.372516i 0.586767 − 0.809756i \(-0.300400\pi\)
−0.899345 + 0.437240i \(0.855956\pi\)
\(60\) 0 0
\(61\) −55.9422 + 20.3613i −0.917085 + 0.333792i −0.757078 − 0.653324i \(-0.773374\pi\)
−0.160007 + 0.987116i \(0.551152\pi\)
\(62\) 0 0
\(63\) −74.2943 − 62.6764i −1.17927 − 0.994864i
\(64\) 0 0
\(65\) −5.36107 + 6.38908i −0.0824780 + 0.0982935i
\(66\) 0 0
\(67\) −4.03483 − 22.8827i −0.0602214 − 0.341532i 0.939779 − 0.341784i \(-0.111031\pi\)
−1.00000 0.000251397i \(0.999920\pi\)
\(68\) 0 0
\(69\) −9.49246 + 7.94372i −0.137572 + 0.115126i
\(70\) 0 0
\(71\) −32.2368 + 18.6119i −0.454039 + 0.262139i −0.709534 − 0.704671i \(-0.751095\pi\)
0.255496 + 0.966810i \(0.417761\pi\)
\(72\) 0 0
\(73\) −26.0280 + 45.0817i −0.356547 + 0.617558i −0.987382 − 0.158360i \(-0.949380\pi\)
0.630834 + 0.775918i \(0.282713\pi\)
\(74\) 0 0
\(75\) −47.8685 + 56.8942i −0.638246 + 0.758590i
\(76\) 0 0
\(77\) 58.7957 − 161.540i 0.763580 − 2.09792i
\(78\) 0 0
\(79\) −20.9619 + 118.881i −0.265341 + 1.50482i 0.502722 + 0.864448i \(0.332332\pi\)
−0.768063 + 0.640374i \(0.778779\pi\)
\(80\) 0 0
\(81\) −40.8710 + 69.9325i −0.504581 + 0.863364i
\(82\) 0 0
\(83\) 115.091 + 20.2937i 1.38664 + 0.244502i 0.816642 − 0.577145i \(-0.195833\pi\)
0.569997 + 0.821647i \(0.306944\pi\)
\(84\) 0 0
\(85\) 11.4121 + 4.15365i 0.134260 + 0.0488665i
\(86\) 0 0
\(87\) −95.6769 − 80.4987i −1.09974 − 0.925272i
\(88\) 0 0
\(89\) −117.767 − 67.9930i −1.32323 − 0.763966i −0.338985 − 0.940792i \(-0.610084\pi\)
−0.984242 + 0.176826i \(0.943417\pi\)
\(90\) 0 0
\(91\) 96.9682 + 167.954i 1.06558 + 1.84565i
\(92\) 0 0
\(93\) 13.5710 + 16.2168i 0.145925 + 0.174375i
\(94\) 0 0
\(95\) −1.62713 + 0.286906i −0.0171276 + 0.00302006i
\(96\) 0 0
\(97\) −72.3238 − 60.6869i −0.745606 − 0.625638i 0.188731 − 0.982029i \(-0.439563\pi\)
−0.934337 + 0.356391i \(0.884007\pi\)
\(98\) 0 0
\(99\) −141.012 − 25.2495i −1.42436 − 0.255046i
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 108.3.k.a.29.2 ✓ 36
3.2 odd 2 324.3.k.a.89.4 36
4.3 odd 2 432.3.bc.b.353.5 36
27.11 odd 18 2916.3.c.b.1457.20 36
27.13 even 9 324.3.k.a.233.4 36
27.14 odd 18 inner 108.3.k.a.41.2 yes 36
27.16 even 9 2916.3.c.b.1457.17 36
108.95 even 18 432.3.bc.b.257.5 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.3.k.a.29.2 ✓ 36 1.1 even 1 trivial
108.3.k.a.41.2 yes 36 27.14 odd 18 inner
324.3.k.a.89.4 36 3.2 odd 2
324.3.k.a.233.4 36 27.13 even 9
432.3.bc.b.257.5 36 108.95 even 18
432.3.bc.b.353.5 36 4.3 odd 2
2916.3.c.b.1457.17 36 27.16 even 9
2916.3.c.b.1457.20 36 27.11 odd 18