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Properties

Label 135.4.q.a.113.7
Level $135$
Weight $4$
Character 135.113
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 113.7
Character \(\chi\) \(=\) 135.113
Dual form 135.4.q.a.92.7

$q$-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+(-4.61482 - 0.403745i) q^{2} +(-4.94146 - 1.60687i) q^{3} +(13.2551 + 2.33724i) q^{4} +(6.75211 + 8.91117i) q^{5} +(22.1552 + 9.41050i) q^{6} +(-24.2564 - 16.9845i) q^{7} +(-24.4296 - 6.54590i) q^{8} +(21.8360 + 15.8805i) q^{9} +(-27.5620 - 43.8496i) q^{10} +(17.9023 - 49.1861i) q^{11} +(-61.7440 - 32.8486i) q^{12} +(-2.35980 - 26.9726i) q^{13} +(105.082 + 88.1740i) q^{14} +(-19.0462 - 54.8839i) q^{15} +(8.91241 + 3.24385i) q^{16} +(-5.36714 + 1.43812i) q^{17} +(-94.3574 - 82.1020i) q^{18} +(-51.6306 + 29.8089i) q^{19} +(68.6727 + 133.900i) q^{20} +(92.5701 + 122.905i) q^{21} +(-102.474 + 219.757i) q^{22} +(106.393 + 151.946i) q^{23} +(110.200 + 71.6015i) q^{24} +(-33.8179 + 120.338i) q^{25} +125.427i q^{26} +(-82.3835 - 113.560i) q^{27} +(-281.825 - 281.825i) q^{28} +(-108.280 + 90.8580i) q^{29} +(65.7358 + 260.969i) q^{30} +(3.14361 - 17.8283i) q^{31} +(143.555 + 66.9408i) q^{32} +(-167.499 + 214.284i) q^{33} +(25.3491 - 4.46972i) q^{34} +(-12.4301 - 330.834i) q^{35} +(252.322 + 261.534i) q^{36} +(37.0904 + 138.423i) q^{37} +(250.301 - 116.717i) q^{38} +(-31.6806 + 137.076i) q^{39} +(-106.620 - 261.895i) q^{40} +(-11.4918 + 13.6954i) q^{41} +(-377.572 - 604.560i) q^{42} +(-134.800 - 289.079i) q^{43} +(352.257 - 610.126i) q^{44} +(5.92488 + 301.811i) q^{45} +(-429.640 - 744.158i) q^{46} +(-265.014 + 378.480i) q^{47} +(-38.8278 - 30.3504i) q^{48} +(182.586 + 501.652i) q^{49} +(204.650 - 541.687i) q^{50} +(28.8324 + 1.51787i) q^{51} +(31.7620 - 363.041i) q^{52} +(-226.069 + 226.069i) q^{53} +(334.336 + 557.323i) q^{54} +(559.184 - 172.580i) q^{55} +(481.396 + 573.706i) q^{56} +(303.029 - 64.3360i) q^{57} +(536.379 - 375.576i) q^{58} +(-619.701 + 225.553i) q^{59} +(-124.183 - 772.009i) q^{60} +(133.431 + 756.726i) q^{61} +(-21.7053 + 81.0053i) q^{62} +(-259.939 - 756.078i) q^{63} +(-701.164 - 404.817i) q^{64} +(224.424 - 203.151i) q^{65} +(859.493 - 921.257i) q^{66} +(173.481 - 15.1776i) q^{67} +(-74.5035 + 6.51821i) q^{68} +(-281.582 - 921.792i) q^{69} +(-76.2099 + 1531.76i) q^{70} +(188.189 + 108.651i) q^{71} +(-429.492 - 530.892i) q^{72} +(38.4360 - 143.445i) q^{73} +(-115.278 - 653.774i) q^{74} +(360.477 - 540.306i) q^{75} +(-754.041 + 274.449i) q^{76} +(-1269.65 + 889.016i) q^{77} +(201.544 - 619.791i) q^{78} +(-306.064 - 364.753i) q^{79} +(31.2711 + 101.323i) q^{80} +(224.618 + 693.533i) q^{81} +(58.5621 - 58.5621i) q^{82} +(-18.4531 + 210.920i) q^{83} +(939.771 + 1845.48i) q^{84} +(-49.0549 - 38.1172i) q^{85} +(505.364 + 1388.48i) q^{86} +(681.059 - 274.979i) q^{87} +(-759.313 + 1084.41i) q^{88} +(-253.493 - 439.064i) q^{89} +(94.5124 - 1395.20i) q^{90} +(-400.877 + 694.339i) q^{91} +(1055.13 + 2262.73i) q^{92} +(-44.1817 + 83.0464i) q^{93} +(1375.80 - 1639.62i) q^{94} +(-614.248 - 258.816i) q^{95} +(-601.806 - 561.459i) q^{96} +(-1624.50 + 757.516i) q^{97} +(-640.065 - 2388.76i) q^{98} +(1172.01 - 789.728i) 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 \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(82\)
\(\chi(n)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.61482 − 0.403745i −1.63159 − 0.142745i −0.765773 − 0.643111i \(-0.777643\pi\)
−0.865814 + 0.500366i \(0.833199\pi\)
\(3\) −4.94146 − 1.60687i −0.950983 − 0.309242i
\(4\) 13.2551 + 2.33724i 1.65689 + 0.292155i
\(5\) 6.75211 + 8.91117i 0.603928 + 0.797039i
\(6\) 22.1552 + 9.41050i 1.50747 + 0.640303i
\(7\) −24.2564 − 16.9845i −1.30972 − 0.917078i −0.310286 − 0.950643i \(-0.600425\pi\)
−0.999437 + 0.0335653i \(0.989314\pi\)
\(8\) −24.4296 − 6.54590i −1.07965 − 0.289291i
\(9\) 21.8360 + 15.8805i 0.808739 + 0.588167i
\(10\) −27.5620 − 43.8496i −0.871587 − 1.38665i
\(11\) 17.9023 − 49.1861i 0.490703 − 1.34820i −0.409334 − 0.912384i \(-0.634239\pi\)
0.900038 − 0.435812i \(-0.143539\pi\)
\(12\) −61.7440 − 32.8486i −1.48533 − 0.790214i
\(13\) −2.35980 − 26.9726i −0.0503454 − 0.575451i −0.978912 − 0.204284i \(-0.934513\pi\)
0.928566 − 0.371167i \(-0.121042\pi\)
\(14\) 105.082 + 88.1740i 2.00602 + 1.68325i
\(15\) −19.0462 − 54.8839i −0.327847 − 0.944731i
\(16\) 8.91241 + 3.24385i 0.139256 + 0.0506852i
\(17\) −5.36714 + 1.43812i −0.0765720 + 0.0205174i −0.296902 − 0.954908i \(-0.595953\pi\)
0.220330 + 0.975425i \(0.429287\pi\)
\(18\) −94.3574 − 82.1020i −1.23557 − 1.07509i
\(19\) −51.6306 + 29.8089i −0.623414 + 0.359928i −0.778197 − 0.628020i \(-0.783866\pi\)
0.154783 + 0.987949i \(0.450532\pi\)
\(20\) 68.6727 + 133.900i 0.767784 + 1.49705i
\(21\) 92.5701 + 122.905i 0.961926 + 1.27715i
\(22\) −102.474 + 219.757i −0.993074 + 2.12965i
\(23\) 106.393 + 151.946i 0.964546 + 1.37751i 0.925696 + 0.378269i \(0.123481\pi\)
0.0388503 + 0.999245i \(0.487630\pi\)
\(24\) 110.200 + 71.6015i 0.937267 + 0.608983i
\(25\) −33.8179 + 120.338i −0.270543 + 0.962708i
\(26\) 125.427i 0.946085i
\(27\) −82.3835 − 113.560i −0.587212 − 0.809433i
\(28\) −281.825 − 281.825i −1.90214 − 1.90214i
\(29\) −108.280 + 90.8580i −0.693350 + 0.581790i −0.919873 − 0.392216i \(-0.871709\pi\)
0.226523 + 0.974006i \(0.427264\pi\)
\(30\) 65.7358 + 260.969i 0.400056 + 1.58821i
\(31\) 3.14361 − 17.8283i 0.0182132 − 0.103292i −0.974346 − 0.225055i \(-0.927744\pi\)
0.992559 + 0.121763i \(0.0388548\pi\)
\(32\) 143.555 + 66.9408i 0.793037 + 0.369799i
\(33\) −167.499 + 214.284i −0.883569 + 1.13037i
\(34\) 25.3491 − 4.46972i 0.127863 − 0.0225456i
\(35\) −12.4301 − 330.834i −0.0600307 − 1.59775i
\(36\) 252.322 + 261.534i 1.16816 + 1.21081i
\(37\) 37.0904 + 138.423i 0.164801 + 0.615044i 0.998066 + 0.0621704i \(0.0198022\pi\)
−0.833265 + 0.552874i \(0.813531\pi\)
\(38\) 250.301 − 116.717i 1.06853 − 0.498265i
\(39\) −31.6806 + 137.076i −0.130076 + 0.562813i
\(40\) −106.620 − 261.895i −0.421453 − 1.03523i
\(41\) −11.4918 + 13.6954i −0.0437736 + 0.0521674i −0.787487 − 0.616331i \(-0.788618\pi\)
0.743713 + 0.668499i \(0.233063\pi\)
\(42\) −377.572 − 604.560i −1.38716 − 2.22109i
\(43\) −134.800 − 289.079i −0.478065 − 1.02521i −0.986447 − 0.164082i \(-0.947534\pi\)
0.508382 − 0.861132i \(-0.330244\pi\)
\(44\) 352.257 − 610.126i 1.20692 − 2.09045i
\(45\) 5.92488 + 301.811i 0.0196273 + 0.999807i
\(46\) −429.640 − 744.158i −1.37711 − 2.38522i
\(47\) −265.014 + 378.480i −0.822475 + 1.17462i 0.159796 + 0.987150i \(0.448916\pi\)
−0.982271 + 0.187466i \(0.939972\pi\)
\(48\) −38.8278 − 30.3504i −0.116757 − 0.0912647i
\(49\) 182.586 + 501.652i 0.532322 + 1.46254i
\(50\) 204.650 − 541.687i 0.578837 − 1.53212i
\(51\) 28.8324 + 1.51787i 0.0791635 + 0.00416754i
\(52\) 31.7620 − 363.041i 0.0847038 − 0.968169i
\(53\) −226.069 + 226.069i −0.585905 + 0.585905i −0.936520 − 0.350615i \(-0.885973\pi\)
0.350615 + 0.936520i \(0.385973\pi\)
\(54\) 334.336 + 557.323i 0.842544 + 1.40448i
\(55\) 559.184 − 172.580i 1.37091 − 0.423103i
\(56\) 481.396 + 573.706i 1.14874 + 1.36901i
\(57\) 303.029 − 64.3360i 0.704161 − 0.149500i
\(58\) 536.379 − 375.576i 1.21431 − 0.850269i
\(59\) −619.701 + 225.553i −1.36743 + 0.497703i −0.918343 − 0.395786i \(-0.870472\pi\)
−0.449085 + 0.893489i \(0.648250\pi\)
\(60\) −124.183 − 772.009i −0.267200 − 1.66110i
\(61\) 133.431 + 756.726i 0.280068 + 1.58834i 0.722390 + 0.691486i \(0.243044\pi\)
−0.442322 + 0.896856i \(0.645845\pi\)
\(62\) −21.7053 + 81.0053i −0.0444609 + 0.165930i
\(63\) −259.939 − 756.078i −0.519829 − 1.51201i
\(64\) −701.164 − 404.817i −1.36946 − 0.790659i
\(65\) 224.424 − 203.151i 0.428252 − 0.387658i
\(66\) 859.493 − 921.257i 1.60297 − 1.71817i
\(67\) 173.481 − 15.1776i 0.316329 − 0.0276752i 0.0721144 − 0.997396i \(-0.477025\pi\)
0.244215 + 0.969721i \(0.421470\pi\)
\(68\) −74.5035 + 6.51821i −0.132866 + 0.0116242i
\(69\) −281.582 − 921.792i −0.491282 − 1.60827i
\(70\) −76.2099 + 1531.76i −0.130126 + 2.61544i
\(71\) 188.189 + 108.651i 0.314563 + 0.181613i 0.648967 − 0.760817i \(-0.275202\pi\)
−0.334404 + 0.942430i \(0.608535\pi\)
\(72\) −429.492 − 530.892i −0.703002 − 0.868975i
\(73\) 38.4360 − 143.445i 0.0616246 − 0.229986i −0.928244 − 0.371971i \(-0.878682\pi\)
0.989869 + 0.141985i \(0.0453486\pi\)
\(74\) −115.278 − 653.774i −0.181092 − 1.02702i
\(75\) 360.477 − 540.306i 0.554991 − 0.831856i
\(76\) −754.041 + 274.449i −1.13808 + 0.414229i
\(77\) −1269.65 + 889.016i −1.87909 + 1.31575i
\(78\) 201.544 − 619.791i 0.292569 − 0.899711i
\(79\) −306.064 − 364.753i −0.435885 − 0.519467i 0.502726 − 0.864446i \(-0.332331\pi\)
−0.938611 + 0.344979i \(0.887886\pi\)
\(80\) 31.2711 + 101.323i 0.0437027 + 0.141603i
\(81\) 224.618 + 693.533i 0.308118 + 0.951348i
\(82\) 58.5621 − 58.5621i 0.0788671 − 0.0788671i
\(83\) −18.4531 + 210.920i −0.0244036 + 0.278934i 0.974182 + 0.225764i \(0.0724880\pi\)
−0.998585 + 0.0531696i \(0.983068\pi\)
\(84\) 939.771 + 1845.48i 1.22068 + 2.39713i
\(85\) −49.0549 − 38.1172i −0.0625971 − 0.0486398i
\(86\) 505.364 + 1388.48i 0.633660 + 1.74097i
\(87\) 681.059 − 274.979i 0.839279 − 0.338860i
\(88\) −759.313 + 1084.41i −0.919808 + 1.31362i
\(89\) −253.493 − 439.064i −0.301913 − 0.522929i 0.674656 − 0.738132i \(-0.264292\pi\)
−0.976569 + 0.215203i \(0.930959\pi\)
\(90\) 94.5124 − 1395.20i 0.110694 − 1.63407i
\(91\) −400.877 + 694.339i −0.461795 + 0.799852i
\(92\) 1055.13 + 2262.73i 1.19570 + 2.56419i
\(93\) −44.1817 + 83.0464i −0.0492627 + 0.0925969i
\(94\) 1375.80 − 1639.62i 1.50961 − 1.79908i
\(95\) −614.248 − 258.816i −0.663374 − 0.279515i
\(96\) −601.806 − 561.459i −0.639808 − 0.596913i
\(97\) −1624.50 + 757.516i −1.70044 + 0.792928i −0.703856 + 0.710342i \(0.748540\pi\)
−0.996584 + 0.0825861i \(0.973682\pi\)
\(98\) −640.065 − 2388.76i −0.659759 − 2.46225i
\(99\) 1172.01 − 789.728i 1.18982 − 0.801723i
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.113.7 yes 624
5.2 odd 4 inner 135.4.q.a.32.46 ✓ 624
27.11 odd 18 inner 135.4.q.a.38.46 yes 624
135.92 even 36 inner 135.4.q.a.92.7 yes 624
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.q.a.32.46 ✓ 624 5.2 odd 4 inner
135.4.q.a.38.46 yes 624 27.11 odd 18 inner
135.4.q.a.92.7 yes 624 135.92 even 36 inner
135.4.q.a.113.7 yes 624 1.1 even 1 trivial