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Properties

Label 90.11.h.a.11.13
Level $90$
Weight $11$
Character 90.11
Analytic conductor $57.182$
Analytic rank $0$
Dimension $80$
Inner twists $2$

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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] = [90,11,Mod(11,90)] 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("90.11"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 90.h (of order \(6\), degree \(2\), 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: \(57.1821527406\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 11.13
Character \(\chi\) \(=\) 90.11
Dual form 90.11.h.a.41.13

$q$-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+(-19.5959 - 11.3137i) q^{2} +(105.689 + 218.812i) q^{3} +(256.000 + 443.405i) q^{4} +(1210.31 - 698.771i) q^{5} +(404.512 - 5483.56i) q^{6} +(-8869.13 + 15361.8i) q^{7} -11585.2i q^{8} +(-36708.8 + 46252.0i) q^{9} -31622.8 q^{10} +(-15696.6 - 9062.45i) q^{11} +(-69966.2 + 102879. i) q^{12} +(-40262.4 - 69736.6i) q^{13} +(347598. - 200686. i) q^{14} +(280816. + 190978. i) q^{15} +(-131072. + 227023. i) q^{16} -2.53262e6i q^{17} +(1.24262e6 - 491039. i) q^{18} -2.06614e6 q^{19} +(619677. + 357771. i) q^{20} +(-4.29872e6 - 317108. i) q^{21} +(205060. + 355174. i) q^{22} +(62543.2 - 36109.3i) q^{23} +(2.53499e6 - 1.22443e6i) q^{24} +(976562. - 1.69146e6i) q^{25} +1.82207e6i q^{26} +(-1.40002e7 - 3.14402e6i) q^{27} -9.08199e6 q^{28} +(-1.16643e7 - 6.73441e6i) q^{29} +(-3.34217e6 - 6.91946e6i) q^{30} +(1.82535e6 + 3.16160e6i) q^{31} +(5.13695e6 - 2.96582e6i) q^{32} +(324020. - 4.39242e6i) q^{33} +(-2.86533e7 + 4.96290e7i) q^{34} +2.47900e7i q^{35} +(-2.99058e7 - 4.43633e6i) q^{36} +2.49721e7 q^{37} +(4.04879e7 + 2.33757e7i) q^{38} +(1.10039e7 - 1.61803e7i) q^{39} +(-8.09543e6 - 1.40217e7i) q^{40} +(1.76629e8 - 1.01977e8i) q^{41} +(8.06496e7 + 5.48485e7i) q^{42} +(-8.73241e7 + 1.51250e8i) q^{43} -9.27995e6i q^{44} +(-1.21093e7 + 8.16302e7i) q^{45} -1.63412e6 q^{46} +(1.02658e8 + 5.92693e7i) q^{47} +(-6.35284e7 - 4.68637e6i) q^{48} +(-1.60853e7 - 2.78606e7i) q^{49} +(-3.82733e7 + 2.20971e7i) q^{50} +(5.54169e8 - 2.67670e8i) q^{51} +(2.06144e7 - 3.57051e7i) q^{52} -1.02801e8i q^{53} +(2.38777e8 + 2.20004e8i) q^{54} -2.53303e7 q^{55} +(1.77970e8 + 1.02751e8i) q^{56} +(-2.18368e8 - 4.52097e8i) q^{57} +(1.52382e8 + 2.63934e8i) q^{58} +(-8.09584e8 + 4.67413e8i) q^{59} +(-1.27918e7 + 1.73405e8i) q^{60} +(2.40616e8 - 4.16759e8i) q^{61} -8.26058e7i q^{62} +(-3.84939e8 - 9.74128e8i) q^{63} -1.34218e8 q^{64} +(-9.74598e7 - 5.62685e7i) q^{65} +(-5.60440e7 + 8.24076e7i) q^{66} +(-8.24447e8 - 1.42798e9i) q^{67} +(1.12298e9 - 6.48351e8i) q^{68} +(1.45113e7 + 9.86887e6i) q^{69} +(2.80467e8 - 4.85782e8i) q^{70} -1.63717e9i q^{71} +(5.35841e8 + 4.25280e8i) q^{72} -6.40707e7 q^{73} +(-4.89352e8 - 2.82527e8i) q^{74} +(4.73323e8 + 3.49162e7i) q^{75} +(-5.28931e8 - 9.16136e8i) q^{76} +(2.78431e8 - 1.60752e8i) q^{77} +(-3.98692e8 + 1.92572e8i) q^{78} +(2.96682e9 - 5.13868e9i) q^{79} +3.66357e8i q^{80} +(-7.91716e8 - 3.39571e9i) q^{81} -4.61495e9 q^{82} +(1.30648e9 + 7.54297e8i) q^{83} +(-9.59864e8 - 1.98725e9i) q^{84} +(-1.76972e9 - 3.06525e9i) q^{85} +(3.42239e9 - 1.97592e9i) q^{86} +(2.40783e8 - 3.26406e9i) q^{87} +(-1.04991e8 + 1.81849e8i) q^{88} -6.86120e9i q^{89} +(1.16083e9 - 1.46262e9i) q^{90} +1.42837e9 q^{91} +(3.20221e7 + 1.84880e7i) q^{92} +(-4.98878e8 + 7.33554e8i) q^{93} +(-1.34111e9 - 2.32287e9i) q^{94} +(-2.50066e9 + 1.44376e9i) q^{95} +(1.19188e9 + 8.10575e8i) q^{96} +(3.09809e9 - 5.36605e9i) q^{97} +7.27939e8i q^{98} +(9.95361e8 - 3.93329e8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 44 q^{3} + 20480 q^{4} - 13568 q^{6} - 24476 q^{7} - 103396 q^{9} + 1944 q^{11} + 45056 q^{12} + 561100 q^{13} - 175680 q^{14} - 687500 q^{15} - 10485760 q^{16} - 3888896 q^{18} + 5932480 q^{19}+ \cdots - 78067269744 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −19.5959 − 11.3137i −0.612372 − 0.353553i
\(3\) 105.689 + 218.812i 0.434933 + 0.900463i
\(4\) 256.000 + 443.405i 0.250000 + 0.433013i
\(5\) 1210.31 − 698.771i 0.387298 − 0.223607i
\(6\) 404.512 − 5483.56i 0.0520206 − 0.705191i
\(7\) −8869.13 + 15361.8i −0.527705 + 0.914011i 0.471774 + 0.881720i \(0.343614\pi\)
−0.999478 + 0.0322916i \(0.989719\pi\)
\(8\) − 11585.2i − 0.353553i
\(9\) −36708.8 + 46252.0i −0.621666 + 0.783282i
\(10\) −31622.8 −0.316228
\(11\) −15696.6 − 9062.45i −0.0974637 − 0.0562707i 0.450476 − 0.892789i \(-0.351254\pi\)
−0.547940 + 0.836518i \(0.684588\pi\)
\(12\) −69966.2 + 102879.i −0.281179 + 0.413447i
\(13\) −40262.4 − 69736.6i −0.108438 − 0.187821i 0.806699 − 0.590962i \(-0.201252\pi\)
−0.915138 + 0.403141i \(0.867918\pi\)
\(14\) 347598. − 200686.i 0.646303 − 0.373143i
\(15\) 280816. + 190978.i 0.369798 + 0.251494i
\(16\) −131072. + 227023.i −0.125000 + 0.216506i
\(17\) − 2.53262e6i − 1.78372i −0.452316 − 0.891858i \(-0.649402\pi\)
0.452316 − 0.891858i \(-0.350598\pi\)
\(18\) 1.24262e6 − 491039.i 0.657623 − 0.259868i
\(19\) −2.06614e6 −0.834433 −0.417216 − 0.908807i \(-0.636994\pi\)
−0.417216 + 0.908807i \(0.636994\pi\)
\(20\) 619677. + 357771.i 0.193649 + 0.111803i
\(21\) −4.29872e6 − 317108.i −1.05255 − 0.0776446i
\(22\) 205060. + 355174.i 0.0397894 + 0.0689172i
\(23\) 62543.2 − 36109.3i 0.00971719 − 0.00561022i −0.495134 − 0.868817i \(-0.664881\pi\)
0.504851 + 0.863207i \(0.331548\pi\)
\(24\) 2.53499e6 − 1.22443e6i 0.318362 − 0.153772i
\(25\) 976562. − 1.69146e6i 0.100000 − 0.173205i
\(26\) 1.82207e6i 0.153355i
\(27\) −1.40002e7 − 3.14402e6i −0.975700 − 0.219112i
\(28\) −9.08199e6 −0.527705
\(29\) −1.16643e7 − 6.73441e6i −0.568683 − 0.328329i 0.187940 − 0.982180i \(-0.439819\pi\)
−0.756623 + 0.653851i \(0.773152\pi\)
\(30\) −3.34217e6 − 6.91946e6i −0.137538 − 0.284751i
\(31\) 1.82535e6 + 3.16160e6i 0.0637584 + 0.110433i 0.896143 − 0.443766i \(-0.146358\pi\)
−0.832384 + 0.554199i \(0.813025\pi\)
\(32\) 5.13695e6 − 2.96582e6i 0.153093 − 0.0883883i
\(33\) 324020. − 4.39242e6i 0.00827947 − 0.112236i
\(34\) −2.86533e7 + 4.96290e7i −0.630639 + 1.09230i
\(35\) 2.47900e7i 0.471993i
\(36\) −2.99058e7 − 4.43633e6i −0.494588 − 0.0733688i
\(37\) 2.49721e7 0.360120 0.180060 − 0.983656i \(-0.442371\pi\)
0.180060 + 0.983656i \(0.442371\pi\)
\(38\) 4.04879e7 + 2.33757e7i 0.510984 + 0.295017i
\(39\) 1.10039e7 − 1.61803e7i 0.121962 − 0.179334i
\(40\) −8.09543e6 − 1.40217e7i −0.0790569 − 0.136931i
\(41\) 1.76629e8 − 1.01977e8i 1.52455 − 0.880202i 0.524977 − 0.851116i \(-0.324074\pi\)
0.999577 − 0.0290858i \(-0.00925960\pi\)
\(42\) 8.06496e7 + 5.48485e7i 0.617101 + 0.419680i
\(43\) −8.73241e7 + 1.51250e8i −0.594007 + 1.02885i 0.399679 + 0.916655i \(0.369122\pi\)
−0.993686 + 0.112195i \(0.964212\pi\)
\(44\) − 9.27995e6i − 0.0562707i
\(45\) −1.21093e7 + 8.16302e7i −0.0656231 + 0.442373i
\(46\) −1.63412e6 −0.00793405
\(47\) 1.02658e8 + 5.92693e7i 0.447612 + 0.258429i 0.706821 − 0.707392i \(-0.250129\pi\)
−0.259209 + 0.965821i \(0.583462\pi\)
\(48\) −6.35284e7 − 4.68637e6i −0.249323 − 0.0183921i
\(49\) −1.60853e7 − 2.78606e7i −0.0569442 − 0.0986303i
\(50\) −3.82733e7 + 2.20971e7i −0.122474 + 0.0707107i
\(51\) 5.54169e8 − 2.67670e8i 1.60617 − 0.775797i
\(52\) 2.06144e7 − 3.57051e7i 0.0542192 − 0.0939104i
\(53\) − 1.02801e8i − 0.245820i −0.992418 − 0.122910i \(-0.960777\pi\)
0.992418 − 0.122910i \(-0.0392227\pi\)
\(54\) 2.38777e8 + 2.20004e8i 0.520024 + 0.479140i
\(55\) −2.53303e7 −0.0503300
\(56\) 1.77970e8 + 1.02751e8i 0.323152 + 0.186572i
\(57\) −2.18368e8 − 4.52097e8i −0.362923 − 0.751376i
\(58\) 1.52382e8 + 2.63934e8i 0.232164 + 0.402120i
\(59\) −8.09584e8 + 4.67413e8i −1.13240 + 0.653794i −0.944539 − 0.328400i \(-0.893490\pi\)
−0.187866 + 0.982195i \(0.560157\pi\)
\(60\) −1.27918e7 + 1.73405e8i −0.0164503 + 0.223001i
\(61\) 2.40616e8 − 4.16759e8i 0.284889 − 0.493442i −0.687693 − 0.726001i \(-0.741377\pi\)
0.972582 + 0.232559i \(0.0747100\pi\)
\(62\) − 8.26058e7i − 0.0901680i
\(63\) −3.84939e8 − 9.74128e8i −0.387873 − 0.981552i
\(64\) −1.34218e8 −0.125000
\(65\) −9.74598e7 − 5.62685e7i −0.0839961 − 0.0484951i
\(66\) −5.60440e7 + 8.24076e7i −0.0447517 + 0.0658033i
\(67\) −8.24447e8 − 1.42798e9i −0.610645 − 1.05767i −0.991132 − 0.132882i \(-0.957577\pi\)
0.380487 − 0.924786i \(-0.375756\pi\)
\(68\) 1.12298e9 − 6.48351e8i 0.772371 − 0.445929i
\(69\) 1.45113e7 + 9.86887e6i 0.00927812 + 0.00630990i
\(70\) 2.80467e8 − 4.85782e8i 0.166875 − 0.289036i
\(71\) − 1.63717e9i − 0.907405i −0.891153 − 0.453703i \(-0.850103\pi\)
0.891153 − 0.453703i \(-0.149897\pi\)
\(72\) 5.35841e8 + 4.25280e8i 0.276932 + 0.219792i
\(73\) −6.40707e7 −0.0309062 −0.0154531 − 0.999881i \(-0.504919\pi\)
−0.0154531 + 0.999881i \(0.504919\pi\)
\(74\) −4.89352e8 − 2.82527e8i −0.220527 − 0.127321i
\(75\) 4.73323e8 + 3.49162e7i 0.199458 + 0.0147136i
\(76\) −5.28931e8 − 9.16136e8i −0.208608 − 0.361320i
\(77\) 2.78431e8 − 1.60752e8i 0.102864 − 0.0593886i
\(78\) −3.98692e8 + 1.92572e8i −0.138091 + 0.0666992i
\(79\) 2.96682e9 − 5.13868e9i 0.964175 − 1.67000i 0.252359 − 0.967634i \(-0.418794\pi\)
0.711816 − 0.702366i \(-0.247873\pi\)
\(80\) 3.66357e8i 0.111803i
\(81\) −7.91716e8 − 3.39571e9i −0.227062 − 0.973880i
\(82\) −4.61495e9 −1.24479
\(83\) 1.30648e9 + 7.54297e8i 0.331675 + 0.191493i 0.656584 − 0.754253i \(-0.272001\pi\)
−0.324910 + 0.945745i \(0.605334\pi\)
\(84\) −9.59864e8 − 1.98725e9i −0.229516 − 0.475178i
\(85\) −1.76972e9 − 3.06525e9i −0.398851 − 0.690830i
\(86\) 3.42239e9 − 1.97592e9i 0.727507 − 0.420026i
\(87\) 2.40783e8 − 3.26406e9i 0.0483092 − 0.654879i
\(88\) −1.04991e8 + 1.81849e8i −0.0198947 + 0.0344586i
\(89\) − 6.86120e9i − 1.22871i −0.789029 − 0.614356i \(-0.789416\pi\)
0.789029 − 0.614356i \(-0.210584\pi\)
\(90\) 1.16083e9 − 1.46262e9i 0.196588 − 0.247696i
\(91\) 1.42837e9 0.228894
\(92\) 3.20221e7 + 1.84880e7i 0.00485860 + 0.00280511i
\(93\) −4.98878e8 + 7.33554e8i −0.0717099 + 0.105443i
\(94\) −1.34111e9 − 2.32287e9i −0.182737 − 0.316509i
\(95\) −2.50066e9 + 1.44376e9i −0.323174 + 0.186585i
\(96\) 1.19188e9 + 8.10575e8i 0.146176 + 0.0994116i
\(97\) 3.09809e9 − 5.36605e9i 0.360774 − 0.624879i −0.627314 − 0.778766i \(-0.715846\pi\)
0.988088 + 0.153887i \(0.0491792\pi\)
\(98\) 7.27939e8i 0.0805313i
\(99\) 9.95361e8 − 3.93329e8i 0.104666 − 0.0413600i
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 90.11.h.a.11.13 ✓ 80
3.2 odd 2 270.11.h.a.251.35 80
9.4 even 3 270.11.h.a.71.35 80
9.5 odd 6 inner 90.11.h.a.41.13 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.13 ✓ 80 1.1 even 1 trivial
90.11.h.a.41.13 yes 80 9.5 odd 6 inner
270.11.h.a.71.35 80 9.4 even 3
270.11.h.a.251.35 80 3.2 odd 2