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Properties

Label 135.4.q.a.2.16
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 2.16
Character \(\chi\) \(=\) 135.2
Dual form 135.4.q.a.68.16

$q$-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+(-1.61530 - 2.30689i) q^{2} +(4.58656 - 2.44202i) q^{3} +(0.0236138 - 0.0648784i) q^{4} +(9.09040 - 6.50881i) q^{5} +(-13.0422 - 6.63610i) q^{6} +(-10.8081 - 5.03990i) q^{7} +(-21.9497 + 5.88140i) q^{8} +(15.0731 - 22.4009i) q^{9} +(-29.6989 - 10.4569i) q^{10} +(13.0034 - 15.4969i) q^{11} +(-0.0501282 - 0.355234i) q^{12} +(24.4400 + 17.1131i) q^{13} +(5.83185 + 33.0741i) q^{14} +(25.7991 - 52.0520i) q^{15} +(48.6001 + 40.7803i) q^{16} +(44.8735 + 12.0238i) q^{17} +(-76.0242 + 1.41234i) q^{18} +(-112.010 - 64.6691i) q^{19} +(-0.207622 - 0.743469i) q^{20} +(-61.8795 + 3.27778i) q^{21} +(-56.7541 - 4.96534i) q^{22} +(41.1788 + 88.3083i) q^{23} +(-86.3111 + 80.5770i) q^{24} +(40.2709 - 118.335i) q^{25} -84.0234i q^{26} +(14.4301 - 139.552i) q^{27} +(-0.582201 + 0.582201i) q^{28} +(-39.4184 + 223.553i) q^{29} +(-161.752 + 24.5641i) q^{30} +(-205.932 - 74.9531i) q^{31} +(-0.272310 + 3.11251i) q^{32} +(21.7973 - 102.832i) q^{33} +(-44.7467 - 122.940i) q^{34} +(-131.054 + 24.5331i) q^{35} +(-1.09740 - 1.50689i) q^{36} +(8.56223 - 31.9547i) q^{37} +(31.7457 + 362.856i) q^{38} +(153.886 + 18.8072i) q^{39} +(-161.251 + 196.331i) q^{40} +(-2.55185 + 0.449961i) q^{41} +(107.516 + 137.455i) q^{42} +(332.741 - 29.1110i) q^{43} +(-0.698353 - 1.20958i) q^{44} +(-8.78300 - 301.741i) q^{45} +(137.201 - 237.640i) q^{46} +(88.6147 - 190.035i) q^{47} +(322.494 + 68.3590i) q^{48} +(-129.062 - 153.810i) q^{49} +(-338.037 + 98.2469i) q^{50} +(235.177 - 54.4340i) q^{51} +(1.68739 - 1.18153i) q^{52} +(526.460 + 526.460i) q^{53} +(-345.241 + 192.130i) q^{54} +(17.3402 - 225.510i) q^{55} +(266.876 + 47.0575i) q^{56} +(-671.665 - 23.0778i) q^{57} +(579.385 - 270.172i) q^{58} +(475.835 - 399.273i) q^{59} +(-2.76784 - 2.90295i) q^{60} +(405.503 - 147.591i) q^{61} +(159.734 + 596.135i) q^{62} +(-275.810 + 166.145i) q^{63} +(447.165 - 258.171i) q^{64} +(333.556 - 3.51051i) q^{65} +(-272.432 + 115.821i) q^{66} +(-387.073 + 552.798i) q^{67} +(1.83972 - 2.62739i) q^{68} +(404.520 + 304.472i) q^{69} +(268.287 + 262.698i) q^{70} +(203.197 - 117.316i) q^{71} +(-199.101 + 580.345i) q^{72} +(157.792 + 588.886i) q^{73} +(-87.5466 + 31.8644i) q^{74} +(-104.272 - 641.095i) q^{75} +(-6.84061 + 5.73996i) q^{76} +(-218.645 + 101.956i) q^{77} +(-205.187 - 385.378i) q^{78} +(480.075 + 84.6502i) q^{79} +(707.225 + 54.3809i) q^{80} +(-274.604 - 675.302i) q^{81} +(5.16003 + 5.16003i) q^{82} +(422.183 - 295.615i) q^{83} +(-1.24855 + 4.09205i) q^{84} +(486.179 - 182.772i) q^{85} +(-604.633 - 720.573i) q^{86} +(365.126 + 1121.60i) q^{87} +(-194.278 + 416.630i) q^{88} +(-23.7070 + 41.0616i) q^{89} +(-681.898 + 507.665i) q^{90} +(-177.902 - 308.135i) q^{91} +(6.70169 - 0.586322i) q^{92} +(-1127.56 + 159.113i) q^{93} +(-581.530 + 102.539i) q^{94} +(-1439.14 + 141.184i) q^{95} +(6.35186 + 14.9407i) q^{96} +(-143.177 - 1636.52i) q^{97} +(-146.349 + 546.181i) q^{98} +(-151.143 - 524.875i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.61530 − 2.30689i −0.571096 − 0.815610i 0.424878 − 0.905251i \(-0.360317\pi\)
−0.995974 + 0.0896409i \(0.971428\pi\)
\(3\) 4.58656 − 2.44202i 0.882684 − 0.469967i
\(4\) 0.0236138 − 0.0648784i 0.00295173 − 0.00810980i
\(5\) 9.09040 − 6.50881i 0.813070 − 0.582165i
\(6\) −13.0422 − 6.63610i −0.887407 − 0.451529i
\(7\) −10.8081 − 5.03990i −0.583582 − 0.272129i 0.108325 − 0.994116i \(-0.465451\pi\)
−0.691907 + 0.721987i \(0.743229\pi\)
\(8\) −21.9497 + 5.88140i −0.970049 + 0.259924i
\(9\) 15.0731 − 22.4009i 0.558262 − 0.829665i
\(10\) −29.6989 − 10.4569i −0.939161 − 0.330676i
\(11\) 13.0034 − 15.4969i 0.356426 − 0.424772i −0.557801 − 0.829975i \(-0.688355\pi\)
0.914227 + 0.405203i \(0.132799\pi\)
\(12\) −0.0501282 − 0.355234i −0.00120590 − 0.00854561i
\(13\) 24.4400 + 17.1131i 0.521419 + 0.365102i 0.804461 − 0.594006i \(-0.202455\pi\)
−0.283041 + 0.959108i \(0.591343\pi\)
\(14\) 5.83185 + 33.0741i 0.111331 + 0.631387i
\(15\) 25.7991 − 52.0520i 0.444086 − 0.895984i
\(16\) 48.6001 + 40.7803i 0.759376 + 0.637192i
\(17\) 44.8735 + 12.0238i 0.640201 + 0.171541i 0.564295 − 0.825573i \(-0.309148\pi\)
0.0759068 + 0.997115i \(0.475815\pi\)
\(18\) −76.0242 + 1.41234i −0.995504 + 0.0184940i
\(19\) −112.010 − 64.6691i −1.35247 − 0.780848i −0.363873 − 0.931448i \(-0.618546\pi\)
−0.988595 + 0.150601i \(0.951879\pi\)
\(20\) −0.207622 − 0.743469i −0.00232128 − 0.00831223i
\(21\) −61.8795 + 3.27778i −0.643011 + 0.0340605i
\(22\) −56.7541 − 4.96534i −0.550001 − 0.0481189i
\(23\) 41.1788 + 88.3083i 0.373321 + 0.800589i 0.999777 + 0.0211324i \(0.00672715\pi\)
−0.626456 + 0.779457i \(0.715495\pi\)
\(24\) −86.3111 + 80.5770i −0.734091 + 0.685321i
\(25\) 40.2709 − 118.335i 0.322167 − 0.946683i
\(26\) − 84.0234i − 0.633782i
\(27\) 14.4301 − 139.552i 0.102854 − 0.994696i
\(28\) −0.582201 + 0.582201i −0.00392949 + 0.00392949i
\(29\) −39.4184 + 223.553i −0.252407 + 1.43147i 0.550233 + 0.835011i \(0.314539\pi\)
−0.802641 + 0.596463i \(0.796572\pi\)
\(30\) −161.752 + 24.5641i −0.984389 + 0.149492i
\(31\) −205.932 − 74.9531i −1.19311 − 0.434257i −0.332297 − 0.943175i \(-0.607824\pi\)
−0.860815 + 0.508918i \(0.830046\pi\)
\(32\) −0.272310 + 3.11251i −0.00150431 + 0.0171944i
\(33\) 21.7973 − 102.832i 0.114983 − 0.542447i
\(34\) −44.7467 − 122.940i −0.225706 − 0.620121i
\(35\) −131.054 + 24.5331i −0.632918 + 0.118481i
\(36\) −1.09740 − 1.50689i −0.00508058 − 0.00697634i
\(37\) 8.56223 − 31.9547i 0.0380438 − 0.141982i −0.944292 − 0.329110i \(-0.893251\pi\)
0.982335 + 0.187128i \(0.0599180\pi\)
\(38\) 31.7457 + 362.856i 0.135522 + 1.54902i
\(39\) 153.886 + 18.8072i 0.631834 + 0.0772196i
\(40\) −161.251 + 196.331i −0.637399 + 0.776065i
\(41\) −2.55185 + 0.449961i −0.00972031 + 0.00171395i −0.178506 − 0.983939i \(-0.557126\pi\)
0.168786 + 0.985653i \(0.446015\pi\)
\(42\) 107.516 + 137.455i 0.395001 + 0.504994i
\(43\) 332.741 − 29.1110i 1.18006 − 0.103242i 0.519805 − 0.854285i \(-0.326005\pi\)
0.660253 + 0.751044i \(0.270449\pi\)
\(44\) −0.698353 − 1.20958i −0.00239274 − 0.00414435i
\(45\) −8.78300 − 301.741i −0.0290954 − 0.999577i
\(46\) 137.201 − 237.640i 0.439766 − 0.761698i
\(47\) 88.6147 − 190.035i 0.275017 − 0.589775i −0.719484 − 0.694509i \(-0.755621\pi\)
0.994500 + 0.104734i \(0.0333992\pi\)
\(48\) 322.494 + 68.3590i 0.969749 + 0.205558i
\(49\) −129.062 − 153.810i −0.376273 − 0.448425i
\(50\) −338.037 + 98.2469i −0.956112 + 0.277884i
\(51\) 235.177 − 54.4340i 0.645714 − 0.149457i
\(52\) 1.68739 − 1.18153i 0.00449999 − 0.00315092i
\(53\) 526.460 + 526.460i 1.36443 + 1.36443i 0.868178 + 0.496253i \(0.165291\pi\)
0.496253 + 0.868178i \(0.334709\pi\)
\(54\) −345.241 + 192.130i −0.870024 + 0.484178i
\(55\) 17.3402 − 225.510i 0.0425119 − 0.552868i
\(56\) 266.876 + 47.0575i 0.636836 + 0.112291i
\(57\) −671.665 − 23.0778i −1.56077 − 0.0536267i
\(58\) 579.385 − 270.172i 1.31167 − 0.611643i
\(59\) 475.835 − 399.273i 1.04997 − 0.881032i 0.0568821 − 0.998381i \(-0.481884\pi\)
0.993091 + 0.117349i \(0.0374397\pi\)
\(60\) −2.76784 − 2.90295i −0.00595543 − 0.00624615i
\(61\) 405.503 − 147.591i 0.851137 − 0.309788i 0.120633 − 0.992697i \(-0.461508\pi\)
0.730504 + 0.682909i \(0.239285\pi\)
\(62\) 159.734 + 596.135i 0.327197 + 1.22112i
\(63\) −275.810 + 166.145i −0.551568 + 0.332258i
\(64\) 447.165 − 258.171i 0.873370 − 0.504240i
\(65\) 333.556 − 3.51051i 0.636500 − 0.00669885i
\(66\) −272.432 + 115.821i −0.508091 + 0.216009i
\(67\) −387.073 + 552.798i −0.705798 + 1.00798i 0.292858 + 0.956156i \(0.405394\pi\)
−0.998656 + 0.0518286i \(0.983495\pi\)
\(68\) 1.83972 − 2.62739i 0.00328087 − 0.00468556i
\(69\) 404.520 + 304.472i 0.705775 + 0.531219i
\(70\) 268.287 + 262.698i 0.458091 + 0.448549i
\(71\) 203.197 − 117.316i 0.339648 − 0.196096i −0.320468 − 0.947259i \(-0.603840\pi\)
0.660117 + 0.751163i \(0.270507\pi\)
\(72\) −199.101 + 580.345i −0.325892 + 0.949921i
\(73\) 157.792 + 588.886i 0.252988 + 0.944163i 0.969199 + 0.246279i \(0.0792080\pi\)
−0.716211 + 0.697884i \(0.754125\pi\)
\(74\) −87.5466 + 31.8644i −0.137528 + 0.0500562i
\(75\) −104.272 − 641.095i −0.160538 − 0.987030i
\(76\) −6.84061 + 5.73996i −0.0103246 + 0.00866340i
\(77\) −218.645 + 101.956i −0.323596 + 0.150895i
\(78\) −205.187 − 385.378i −0.297857 − 0.559430i
\(79\) 480.075 + 84.6502i 0.683705 + 0.120556i 0.504703 − 0.863293i \(-0.331602\pi\)
0.179002 + 0.983849i \(0.442713\pi\)
\(80\) 707.225 + 54.3809i 0.988378 + 0.0759997i
\(81\) −274.604 − 675.302i −0.376686 − 0.926341i
\(82\) 5.16003 + 5.16003i 0.00694914 + 0.00694914i
\(83\) 422.183 − 295.615i 0.558320 − 0.390940i −0.260082 − 0.965587i \(-0.583749\pi\)
0.818402 + 0.574647i \(0.194861\pi\)
\(84\) −1.24855 + 4.09205i −0.00162177 + 0.00531522i
\(85\) 486.179 − 182.772i 0.620394 − 0.233228i
\(86\) −604.633 − 720.573i −0.758131 − 0.903505i
\(87\) 365.126 + 1121.60i 0.449949 + 1.38216i
\(88\) −194.278 + 416.630i −0.235342 + 0.504693i
\(89\) −23.7070 + 41.0616i −0.0282352 + 0.0489048i −0.879798 − 0.475348i \(-0.842322\pi\)
0.851563 + 0.524253i \(0.175655\pi\)
\(90\) −681.898 + 507.665i −0.798648 + 0.594585i
\(91\) −177.902 − 308.135i −0.204936 − 0.354960i
\(92\) 6.70169 − 0.586322i 0.00759456 − 0.000664438i
\(93\) −1127.56 + 159.113i −1.25723 + 0.177411i
\(94\) −581.530 + 102.539i −0.638087 + 0.112512i
\(95\) −1439.14 + 141.184i −1.55423 + 0.152476i
\(96\) 6.35186 + 14.9407i 0.00675295 + 0.0158842i
\(97\) −143.177 − 1636.52i −0.149870 − 1.71302i −0.583653 − 0.812003i \(-0.698377\pi\)
0.433783 − 0.901017i \(-0.357179\pi\)
\(98\) −146.349 + 546.181i −0.150852 + 0.562986i
\(99\) −151.143 − 524.875i −0.153439 − 0.532848i
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.2.16 ✓ 624
5.3 odd 4 inner 135.4.q.a.83.16 yes 624
27.14 odd 18 inner 135.4.q.a.122.16 yes 624
135.68 even 36 inner 135.4.q.a.68.16 yes 624
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.q.a.2.16 ✓ 624 1.1 even 1 trivial
135.4.q.a.68.16 yes 624 135.68 even 36 inner
135.4.q.a.83.16 yes 624 5.3 odd 4 inner
135.4.q.a.122.16 yes 624 27.14 odd 18 inner