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Properties

Label 135.4.q.a.2.1
Level $135$
Weight $4$
Character 135.2
Analytic conductor $7.965$
Analytic rank $0$
Dimension $624$
Inner twists $4$

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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] = [135,4,Mod(2,135)] 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("135.2"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(135, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([2, 9])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 135.q (of order \(36\), degree \(12\), 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: \(7.96525785077\)
Analytic rank: \(0\)
Dimension: \(624\)
Relative dimension: \(52\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 2.1
Character \(\chi\) \(=\) 135.2
Dual form 135.4.q.a.68.1

$q$-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+(-3.16779 - 4.52407i) q^{2} +(-1.14475 + 5.06848i) q^{3} +(-7.69615 + 21.1450i) q^{4} +(10.9280 + 2.36206i) q^{5} +(26.5565 - 10.8769i) q^{6} +(-9.94919 - 4.63938i) q^{7} +(77.3637 - 20.7295i) q^{8} +(-24.3791 - 11.6043i) q^{9} +(-23.9314 - 56.9214i) q^{10} +(-18.4174 + 21.9490i) q^{11} +(-98.3629 - 63.2136i) q^{12} +(21.2545 + 14.8825i) q^{13} +(10.5280 + 59.7074i) q^{14} +(-24.4819 + 52.6843i) q^{15} +(-200.953 - 168.619i) q^{16} +(-34.5394 - 9.25481i) q^{17} +(24.7289 + 147.053i) q^{18} +(-101.099 - 58.3695i) q^{19} +(-134.049 + 212.893i) q^{20} +(34.9040 - 45.1164i) q^{21} +(157.641 + 13.7918i) q^{22} +(-18.3388 - 39.3277i) q^{23} +(16.5050 + 415.847i) q^{24} +(113.841 + 51.6251i) q^{25} -143.301i q^{26} +(86.7245 - 110.281i) q^{27} +(174.670 - 174.670i) q^{28} +(-21.1660 + 120.039i) q^{29} +(315.901 - 56.1347i) q^{30} +(-187.174 - 68.1257i) q^{31} +(-70.4257 + 804.970i) q^{32} +(-90.1647 - 118.474i) q^{33} +(67.5441 + 185.576i) q^{34} +(-97.7660 - 74.1997i) q^{35} +(432.999 - 426.187i) q^{36} +(-28.5650 + 106.606i) q^{37} +(56.1923 + 642.281i) q^{38} +(-99.7631 + 90.6912i) q^{39} +(894.393 - 43.7941i) q^{40} +(-360.934 + 63.6425i) q^{41} +(-314.678 - 14.9891i) q^{42} +(-539.012 + 47.1574i) q^{43} +(-322.368 - 558.358i) q^{44} +(-239.004 - 184.397i) q^{45} +(-119.828 + 207.548i) q^{46} +(159.262 - 341.538i) q^{47} +(1084.69 - 825.499i) q^{48} +(-143.014 - 170.437i) q^{49} +(-127.069 - 678.563i) q^{50} +(86.4470 - 164.468i) q^{51} +(-478.269 + 334.888i) q^{52} +(429.053 + 429.053i) q^{53} +(-773.642 - 43.0010i) q^{54} +(-253.110 + 196.355i) q^{55} +(-865.878 - 152.678i) q^{56} +(411.579 - 445.600i) q^{57} +(610.112 - 284.500i) q^{58} +(-190.352 + 159.724i) q^{59} +(-925.593 - 923.136i) q^{60} +(-334.316 + 121.681i) q^{61} +(284.721 + 1062.59i) q^{62} +(188.715 + 228.558i) q^{63} +(2047.39 - 1182.06i) q^{64} +(197.115 + 212.841i) q^{65} +(-250.364 + 783.213i) q^{66} +(124.249 - 177.447i) q^{67} +(461.514 - 659.110i) q^{68} +(220.325 - 47.9294i) q^{69} +(-25.9825 + 677.348i) q^{70} +(518.077 - 299.112i) q^{71} +(-2126.61 - 392.387i) q^{72} +(-186.717 - 696.837i) q^{73} +(572.781 - 208.475i) q^{74} +(-391.981 + 517.905i) q^{75} +(2012.30 - 1688.52i) q^{76} +(285.068 - 132.929i) q^{77} +(726.321 + 164.045i) q^{78} +(-289.044 - 50.9662i) q^{79} +(-1797.72 - 2317.33i) q^{80} +(459.679 + 565.806i) q^{81} +(1431.28 + 1431.28i) q^{82} +(-431.527 + 302.158i) q^{83} +(685.359 + 1085.27i) q^{84} +(-355.586 - 182.721i) q^{85} +(1920.82 + 2289.14i) q^{86} +(-584.184 - 244.694i) q^{87} +(-969.844 + 2079.84i) q^{88} +(407.184 - 705.263i) q^{89} +(-77.1105 + 1665.40i) q^{90} +(-142.419 - 246.677i) q^{91} +(972.722 - 85.1021i) q^{92} +(559.562 - 870.700i) q^{93} +(-2049.65 + 361.408i) q^{94} +(-966.935 - 876.663i) q^{95} +(-3999.36 - 1278.44i) q^{96} +(22.3109 + 255.015i) q^{97} +(-318.032 + 1186.91i) q^{98} +(703.702 - 321.374i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 624 q - 12 q^{2} - 12 q^{3} - 12 q^{5} - 12 q^{7} - 18 q^{8} - 6 q^{10} - 12 q^{12} - 12 q^{13} - 12 q^{15} - 24 q^{16} - 18 q^{17} + 702 q^{18} + 756 q^{20} - 24 q^{21} - 12 q^{22} - 324 q^{23} + 420 q^{25}+ \cdots - 5832 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(82\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.16779 − 4.52407i −1.11998 − 1.59950i −0.737222 − 0.675651i \(-0.763863\pi\)
−0.382759 − 0.923848i \(-0.625026\pi\)
\(3\) −1.14475 + 5.06848i −0.220308 + 0.975430i
\(4\) −7.69615 + 21.1450i −0.962019 + 2.64312i
\(5\) 10.9280 + 2.36206i 0.977428 + 0.211269i
\(6\) 26.5565 − 10.8769i 1.80694 − 0.740081i
\(7\) −9.94919 − 4.63938i −0.537206 − 0.250503i 0.135028 − 0.990842i \(-0.456887\pi\)
−0.672234 + 0.740339i \(0.734665\pi\)
\(8\) 77.3637 − 20.7295i 3.41902 − 0.916125i
\(9\) −24.3791 − 11.6043i −0.902929 − 0.429790i
\(10\) −23.9314 − 56.9214i −0.756776 − 1.80001i
\(11\) −18.4174 + 21.9490i −0.504823 + 0.601624i −0.956923 − 0.290343i \(-0.906231\pi\)
0.452100 + 0.891967i \(0.350675\pi\)
\(12\) −98.3629 − 63.2136i −2.36624 − 1.52068i
\(13\) 21.2545 + 14.8825i 0.453456 + 0.317514i 0.777896 − 0.628393i \(-0.216287\pi\)
−0.324440 + 0.945906i \(0.605176\pi\)
\(14\) 10.5280 + 59.7074i 0.200981 + 1.13982i
\(15\) −24.4819 + 52.6843i −0.421414 + 0.906869i
\(16\) −200.953 − 168.619i −3.13989 − 2.63468i
\(17\) −34.5394 − 9.25481i −0.492767 − 0.132037i 0.00387401 − 0.999992i \(-0.498767\pi\)
−0.496641 + 0.867956i \(0.665434\pi\)
\(18\) 24.7289 + 147.053i 0.323814 + 1.92559i
\(19\) −101.099 − 58.3695i −1.22072 − 0.704784i −0.255650 − 0.966770i \(-0.582289\pi\)
−0.965072 + 0.261986i \(0.915623\pi\)
\(20\) −134.049 + 212.893i −1.49871 + 2.38022i
\(21\) 34.9040 − 45.1164i 0.362699 − 0.468819i
\(22\) 157.641 + 13.7918i 1.52769 + 0.133656i
\(23\) −18.3388 − 39.3277i −0.166257 − 0.356539i 0.805474 − 0.592631i \(-0.201911\pi\)
−0.971731 + 0.236093i \(0.924133\pi\)
\(24\) 16.5050 + 415.847i 0.140377 + 3.53685i
\(25\) 113.841 + 51.6251i 0.910731 + 0.413001i
\(26\) − 143.301i − 1.08091i
\(27\) 86.7245 − 110.281i 0.618153 − 0.786058i
\(28\) 174.670 − 174.670i 1.17891 − 1.17891i
\(29\) −21.1660 + 120.039i −0.135532 + 0.768642i 0.838955 + 0.544200i \(0.183167\pi\)
−0.974488 + 0.224441i \(0.927944\pi\)
\(30\) 315.901 − 56.1347i 1.92251 − 0.341625i
\(31\) −187.174 − 68.1257i −1.08443 − 0.394701i −0.262877 − 0.964829i \(-0.584671\pi\)
−0.821556 + 0.570128i \(0.806894\pi\)
\(32\) −70.4257 + 804.970i −0.389051 + 4.44687i
\(33\) −90.1647 − 118.474i −0.475626 − 0.624962i
\(34\) 67.5441 + 185.576i 0.340698 + 0.936059i
\(35\) −97.7660 − 74.1997i −0.472156 − 0.358344i
\(36\) 432.999 − 426.187i 2.00462 − 1.97309i
\(37\) −28.5650 + 106.606i −0.126921 + 0.473674i −0.999901 − 0.0140814i \(-0.995518\pi\)
0.872980 + 0.487756i \(0.162184\pi\)
\(38\) 56.1923 + 642.281i 0.239884 + 2.74189i
\(39\) −99.7631 + 90.6912i −0.409612 + 0.372364i
\(40\) 894.393 − 43.7941i 3.53540 − 0.173111i
\(41\) −360.934 + 63.6425i −1.37484 + 0.242422i −0.811765 − 0.583984i \(-0.801493\pi\)
−0.563075 + 0.826405i \(0.690382\pi\)
\(42\) −314.678 − 14.9891i −1.15609 − 0.0550684i
\(43\) −539.012 + 47.1574i −1.91159 + 0.167243i −0.980365 − 0.197191i \(-0.936818\pi\)
−0.931229 + 0.364434i \(0.881263\pi\)
\(44\) −322.368 − 558.358i −1.10452 − 1.91308i
\(45\) −239.004 − 184.397i −0.791746 − 0.610850i
\(46\) −119.828 + 207.548i −0.384079 + 0.665244i
\(47\) 159.262 − 341.538i 0.494270 − 1.05997i −0.488060 − 0.872810i \(-0.662295\pi\)
0.982330 − 0.187156i \(-0.0599271\pi\)
\(48\) 1084.69 − 825.499i 3.26169 − 2.48230i
\(49\) −143.014 − 170.437i −0.416949 − 0.496901i
\(50\) −127.069 − 678.563i −0.359407 − 1.91927i
\(51\) 86.4470 − 164.468i 0.237353 − 0.451571i
\(52\) −478.269 + 334.888i −1.27546 + 0.893088i
\(53\) 429.053 + 429.053i 1.11198 + 1.11198i 0.992883 + 0.119097i \(0.0380000\pi\)
0.119097 + 0.992883i \(0.462000\pi\)
\(54\) −773.642 − 43.0010i −1.94962 − 0.108365i
\(55\) −253.110 + 196.355i −0.620533 + 0.481391i
\(56\) −865.878 − 152.678i −2.06621 − 0.364329i
\(57\) 411.579 − 445.600i 0.956402 − 1.03546i
\(58\) 610.112 − 284.500i 1.38124 − 0.644080i
\(59\) −190.352 + 159.724i −0.420029 + 0.352446i −0.828174 − 0.560471i \(-0.810620\pi\)
0.408145 + 0.912917i \(0.366176\pi\)
\(60\) −925.593 − 923.136i −1.99156 − 1.98627i
\(61\) −334.316 + 121.681i −0.701718 + 0.255404i −0.668144 − 0.744032i \(-0.732911\pi\)
−0.0335735 + 0.999436i \(0.510689\pi\)
\(62\) 284.721 + 1062.59i 0.583220 + 2.17661i
\(63\) 188.715 + 228.558i 0.377395 + 0.457072i
\(64\) 2047.39 − 1182.06i 3.99880 − 2.30871i
\(65\) 197.115 + 212.841i 0.376140 + 0.406148i
\(66\) −250.364 + 783.213i −0.466934 + 1.46071i
\(67\) 124.249 − 177.447i 0.226560 − 0.323561i −0.689738 − 0.724059i \(-0.742274\pi\)
0.916297 + 0.400499i \(0.131163\pi\)
\(68\) 461.514 − 659.110i 0.823040 − 1.17542i
\(69\) 220.325 − 47.9294i 0.384406 − 0.0836235i
\(70\) −25.9825 + 677.348i −0.0443644 + 1.15655i
\(71\) 518.077 − 299.112i 0.865978 − 0.499973i −3.15257e−5 − 1.00000i \(-0.500010\pi\)
0.866010 + 0.500027i \(0.166677\pi\)
\(72\) −2126.61 − 392.387i −3.48088 − 0.642268i
\(73\) −186.717 − 696.837i −0.299364 − 1.11724i −0.937689 − 0.347475i \(-0.887039\pi\)
0.638326 − 0.769767i \(-0.279627\pi\)
\(74\) 572.781 − 208.475i 0.899790 − 0.327497i
\(75\) −391.981 + 517.905i −0.603495 + 0.797367i
\(76\) 2012.30 − 1688.52i 3.03719 − 2.54850i
\(77\) 285.068 − 132.929i 0.421903 − 0.196736i
\(78\) 726.321 + 164.045i 1.05435 + 0.238134i
\(79\) −289.044 − 50.9662i −0.411645 − 0.0725842i −0.0360093 − 0.999351i \(-0.511465\pi\)
−0.375636 + 0.926767i \(0.622576\pi\)
\(80\) −1797.72 − 2317.33i −2.51239 − 3.23857i
\(81\) 459.679 + 565.806i 0.630561 + 0.776140i
\(82\) 1431.28 + 1431.28i 1.92755 + 1.92755i
\(83\) −431.527 + 302.158i −0.570677 + 0.399592i −0.822992 − 0.568053i \(-0.807697\pi\)
0.252315 + 0.967645i \(0.418808\pi\)
\(84\) 685.359 + 1085.27i 0.890223 + 1.40967i
\(85\) −355.586 − 182.721i −0.453749 − 0.233163i
\(86\) 1920.82 + 2289.14i 2.40845 + 2.87028i
\(87\) −584.184 − 244.694i −0.719898 − 0.301540i
\(88\) −969.844 + 2079.84i −1.17484 + 2.51945i
\(89\) 407.184 − 705.263i 0.484959 − 0.839974i −0.514891 − 0.857255i \(-0.672168\pi\)
0.999851 + 0.0172813i \(0.00550107\pi\)
\(90\) −77.1105 + 1665.40i −0.0903129 + 1.95054i
\(91\) −142.419 − 246.677i −0.164061 − 0.284162i
\(92\) 972.722 − 85.1021i 1.10232 − 0.0964403i
\(93\) 559.562 − 870.700i 0.623913 − 0.970833i
\(94\) −2049.65 + 361.408i −2.24899 + 0.396557i
\(95\) −966.935 − 876.663i −1.04427 − 0.946776i
\(96\) −3999.36 − 1278.44i −4.25190 − 1.35917i
\(97\) 22.3109 + 255.015i 0.0233539 + 0.266936i 0.998890 + 0.0470981i \(0.0149973\pi\)
−0.975536 + 0.219838i \(0.929447\pi\)
\(98\) −318.032 + 1186.91i −0.327817 + 1.22343i
\(99\) 703.702 − 321.374i 0.714391 − 0.326256i
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 135.4.q.a.2.1 ✓ 624
5.3 odd 4 inner 135.4.q.a.83.1 yes 624
27.14 odd 18 inner 135.4.q.a.122.1 yes 624
135.68 even 36 inner 135.4.q.a.68.1 yes 624
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.q.a.2.1 ✓ 624 1.1 even 1 trivial
135.4.q.a.68.1 yes 624 135.68 even 36 inner
135.4.q.a.83.1 yes 624 5.3 odd 4 inner
135.4.q.a.122.1 yes 624 27.14 odd 18 inner