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Properties

Label 90.11.h.a.11.16
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 11.16
Character \(\chi\) \(=\) 90.11
Dual form 90.11.h.a.41.16

$q$-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+(-19.5959 - 11.3137i) q^{2} +(213.450 + 116.138i) q^{3} +(256.000 + 443.405i) q^{4} +(-1210.31 + 698.771i) q^{5} +(-2868.79 - 4690.75i) q^{6} +(-12940.6 + 22413.8i) q^{7} -11585.2i q^{8} +(32072.8 + 49579.4i) q^{9} +31622.8 q^{10} +(53638.1 + 30967.9i) q^{11} +(3146.87 + 124376. i) q^{12} +(37959.2 + 65747.3i) q^{13} +(507166. - 292813. i) q^{14} +(-339494. + 8589.62i) q^{15} +(-131072. + 227023. i) q^{16} +866449. i q^{17} +(-67568.0 - 1.33442e6i) q^{18} +3.81151e6 q^{19} +(-619677. - 357771. i) q^{20} +(-5.36528e6 + 3.28132e6i) q^{21} +(-700725. - 1.21369e6i) q^{22} +(-1.05354e7 + 6.08261e6i) q^{23} +(1.34549e6 - 2.47287e6i) q^{24} +(976562. - 1.69146e6i) q^{25} -1.71784e6i q^{26} +(1.08786e6 + 1.43076e7i) q^{27} -1.32512e7 q^{28} +(-1.23594e7 - 7.13569e6i) q^{29} +(6.74988e6 + 3.67262e6i) q^{30} +(8.75637e6 + 1.51665e7i) q^{31} +(5.13695e6 - 2.96582e6i) q^{32} +(7.85247e6 + 1.28395e7i) q^{33} +(9.80275e6 - 1.69789e7i) q^{34} -3.61701e7i q^{35} +(-1.37731e7 + 2.69136e7i) q^{36} +1.16632e7 q^{37} +(-7.46900e7 - 4.31223e7i) q^{38} +(466612. + 1.84423e7i) q^{39} +(8.09543e6 + 1.40217e7i) q^{40} +(1.28946e8 - 7.44471e7i) q^{41} +(1.42261e8 - 3.59939e6i) q^{42} +(4.99447e7 - 8.65067e7i) q^{43} +3.17112e7i q^{44} +(-7.34626e7 - 3.75948e7i) q^{45} +2.75268e8 q^{46} +(-3.16872e8 - 1.82946e8i) q^{47} +(-5.43434e7 + 3.32356e7i) q^{48} +(-1.93681e8 - 3.35466e8i) q^{49} +(-3.82733e7 + 2.20971e7i) q^{50} +(-1.00628e8 + 1.84943e8i) q^{51} +(-1.94351e7 + 3.36626e7i) q^{52} +9.47585e6i q^{53} +(1.40555e8 - 2.92678e8i) q^{54} -8.65580e7 q^{55} +(2.59669e8 + 1.49920e8i) q^{56} +(8.13566e8 + 4.42662e8i) q^{57} +(1.61462e8 + 2.79661e8i) q^{58} +(5.16273e8 - 2.98070e8i) q^{59} +(-9.07192e7 - 1.48334e8i) q^{60} +(-6.45295e8 + 1.11768e9i) q^{61} -3.96268e8i q^{62} +(-1.52631e9 + 7.72843e7i) q^{63} -1.34218e8 q^{64} +(-9.18846e7 - 5.30496e7i) q^{65} +(-8.61363e6 - 3.40443e8i) q^{66} +(-1.95210e8 - 3.38114e8i) q^{67} +(-3.84188e8 + 2.21811e8i) q^{68} +(-2.95520e9 + 7.47703e7i) q^{69} +(-4.09218e8 + 7.08787e8i) q^{70} -1.89256e8i q^{71} +(5.74390e8 - 3.71571e8i) q^{72} -3.42543e9 q^{73} +(-2.28552e8 - 1.31954e8i) q^{74} +(4.04890e8 - 2.47625e8i) q^{75} +(9.75746e8 + 1.69004e9i) q^{76} +(-1.38822e9 + 8.01488e8i) q^{77} +(1.99507e8 - 3.66672e8i) q^{78} +(-1.01333e9 + 1.75514e9i) q^{79} -3.66357e8i q^{80} +(-1.42946e9 + 3.18030e9i) q^{81} -3.36909e9 q^{82} +(-3.87487e9 - 2.23716e9i) q^{83} +(-2.82847e9 - 1.53897e9i) q^{84} +(-6.05450e8 - 1.04867e9i) q^{85} +(-1.95742e9 + 1.13012e9i) q^{86} +(-1.80938e9 - 2.95851e9i) q^{87} +(3.58771e8 - 6.21410e8i) q^{88} +3.09752e9i q^{89} +(1.01423e9 + 1.56784e9i) q^{90} -1.96486e9 q^{91} +(-5.39412e9 - 3.11430e9i) q^{92} +(1.07637e8 + 4.25424e9i) q^{93} +(4.13959e9 + 7.16998e9i) q^{94} +(-4.61310e9 + 2.66337e9i) q^{95} +(1.44093e9 - 3.64572e7i) q^{96} +(1.68792e9 - 2.92356e9i) q^{97} +8.76502e9i q^{98} +(1.84947e8 + 3.65257e9i) 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
\(3\) 213.450 + 116.138i 0.878395 + 0.477936i
\(4\) 256.000 + 443.405i 0.250000 + 0.433013i
\(5\) −1210.31 + 698.771i −0.387298 + 0.223607i
\(6\) −2868.79 − 4690.75i −0.368929 − 0.603234i
\(7\) −12940.6 + 22413.8i −0.769954 + 1.33360i 0.167633 + 0.985849i \(0.446387\pi\)
−0.937587 + 0.347750i \(0.886946\pi\)
\(8\) − 11585.2i − 0.353553i
\(9\) 32072.8 + 49579.4i 0.543155 + 0.839632i
\(10\) 31622.8 0.316228
\(11\) 53638.1 + 30967.9i 0.333050 + 0.192287i 0.657194 − 0.753721i \(-0.271743\pi\)
−0.324144 + 0.946008i \(0.605076\pi\)
\(12\) 3146.87 + 124376.i 0.0126466 + 0.499840i
\(13\) 37959.2 + 65747.3i 0.102235 + 0.177077i 0.912605 − 0.408842i \(-0.134067\pi\)
−0.810370 + 0.585918i \(0.800734\pi\)
\(14\) 507166. − 292813.i 0.942997 − 0.544440i
\(15\) −339494. + 8589.62i −0.447071 + 0.0113114i
\(16\) −131072. + 227023.i −0.125000 + 0.216506i
\(17\) 866449.i 0.610237i 0.952314 + 0.305118i \(0.0986960\pi\)
−0.952314 + 0.305118i \(0.901304\pi\)
\(18\) −67568.0 − 1.33442e6i −0.0357584 − 0.706202i
\(19\) 3.81151e6 1.53932 0.769660 − 0.638454i \(-0.220426\pi\)
0.769660 + 0.638454i \(0.220426\pi\)
\(20\) −619677. − 357771.i −0.193649 − 0.111803i
\(21\) −5.36528e6 + 3.28132e6i −1.31370 + 0.803438i
\(22\) −700725. − 1.21369e6i −0.135967 − 0.235502i
\(23\) −1.05354e7 + 6.08261e6i −1.63686 + 0.945042i −0.654956 + 0.755667i \(0.727313\pi\)
−0.981905 + 0.189375i \(0.939354\pi\)
\(24\) 1.34549e6 − 2.47287e6i 0.168976 − 0.310559i
\(25\) 976562. − 1.69146e6i 0.100000 − 0.173205i
\(26\) − 1.71784e6i − 0.144582i
\(27\) 1.08786e6 + 1.43076e7i 0.0758147 + 0.997122i
\(28\) −1.32512e7 −0.769954
\(29\) −1.23594e7 − 7.13569e6i −0.602569 − 0.347893i 0.167483 − 0.985875i \(-0.446436\pi\)
−0.770052 + 0.637982i \(0.779770\pi\)
\(30\) 6.74988e6 + 3.67262e6i 0.277773 + 0.151136i
\(31\) 8.75637e6 + 1.51665e7i 0.305855 + 0.529757i 0.977451 − 0.211161i \(-0.0677243\pi\)
−0.671596 + 0.740917i \(0.734391\pi\)
\(32\) 5.13695e6 − 2.96582e6i 0.153093 − 0.0883883i
\(33\) 7.85247e6 + 1.28395e7i 0.200649 + 0.328080i
\(34\) 9.80275e6 − 1.69789e7i 0.215751 − 0.373692i
\(35\) − 3.61701e7i − 0.688668i
\(36\) −1.37731e7 + 2.69136e7i −0.227783 + 0.445101i
\(37\) 1.16632e7 0.168194 0.0840969 − 0.996458i \(-0.473199\pi\)
0.0840969 + 0.996458i \(0.473199\pi\)
\(38\) −7.46900e7 − 4.31223e7i −0.942637 − 0.544232i
\(39\) 466612. + 1.84423e7i 0.00517170 + 0.204405i
\(40\) 8.09543e6 + 1.40217e7i 0.0790569 + 0.136931i
\(41\) 1.28946e8 − 7.44471e7i 1.11299 − 0.642582i 0.173384 − 0.984854i \(-0.444530\pi\)
0.939601 + 0.342272i \(0.111196\pi\)
\(42\) 1.42261e8 − 3.59939e6i 1.08853 − 0.0275412i
\(43\) 4.99447e7 − 8.65067e7i 0.339740 − 0.588447i −0.644644 − 0.764483i \(-0.722994\pi\)
0.984384 + 0.176036i \(0.0563275\pi\)
\(44\) 3.17112e7i 0.192287i
\(45\) −7.34626e7 − 3.75948e7i −0.398111 − 0.203735i
\(46\) 2.75268e8 1.33649
\(47\) −3.16872e8 − 1.82946e8i −1.38164 − 0.797689i −0.389284 − 0.921118i \(-0.627277\pi\)
−0.992353 + 0.123429i \(0.960611\pi\)
\(48\) −5.43434e7 + 3.32356e7i −0.213275 + 0.130436i
\(49\) −1.93681e8 − 3.35466e8i −0.685658 − 1.18759i
\(50\) −3.82733e7 + 2.20971e7i −0.122474 + 0.0707107i
\(51\) −1.00628e8 + 1.84943e8i −0.291654 + 0.536029i
\(52\) −1.94351e7 + 3.36626e7i −0.0511176 + 0.0885383i
\(53\) 9.47585e6i 0.0226589i 0.999936 + 0.0113295i \(0.00360636\pi\)
−0.999936 + 0.0113295i \(0.996394\pi\)
\(54\) 1.40555e8 − 2.92678e8i 0.306109 − 0.637415i
\(55\) −8.65580e7 −0.171986
\(56\) 2.59669e8 + 1.49920e8i 0.471499 + 0.272220i
\(57\) 8.13566e8 + 4.42662e8i 1.35213 + 0.735696i
\(58\) 1.61462e8 + 2.79661e8i 0.245998 + 0.426081i
\(59\) 5.16273e8 − 2.98070e8i 0.722136 − 0.416926i −0.0934022 − 0.995628i \(-0.529774\pi\)
0.815538 + 0.578703i \(0.196441\pi\)
\(60\) −9.07192e7 − 1.48334e8i −0.116666 − 0.190759i
\(61\) −6.45295e8 + 1.11768e9i −0.764028 + 1.32334i 0.176730 + 0.984259i \(0.443448\pi\)
−0.940759 + 0.339077i \(0.889885\pi\)
\(62\) − 3.96268e8i − 0.432544i
\(63\) −1.52631e9 + 7.72843e7i −1.53794 + 0.0778732i
\(64\) −1.34218e8 −0.125000
\(65\) −9.18846e7 − 5.30496e7i −0.0791910 − 0.0457210i
\(66\) −8.61363e6 − 3.40443e8i −0.00687807 − 0.271847i
\(67\) −1.95210e8 − 3.38114e8i −0.144587 − 0.250431i 0.784632 − 0.619962i \(-0.212852\pi\)
−0.929219 + 0.369530i \(0.879519\pi\)
\(68\) −3.84188e8 + 2.21811e8i −0.264240 + 0.152559i
\(69\) −2.95520e9 + 7.47703e7i −1.88948 + 0.0478061i
\(70\) −4.09218e8 + 7.08787e8i −0.243481 + 0.421721i
\(71\) − 1.89256e8i − 0.104896i −0.998624 − 0.0524479i \(-0.983298\pi\)
0.998624 − 0.0524479i \(-0.0167024\pi\)
\(72\) 5.74390e8 − 3.71571e8i 0.296855 − 0.192034i
\(73\) −3.42543e9 −1.65235 −0.826173 − 0.563416i \(-0.809487\pi\)
−0.826173 + 0.563416i \(0.809487\pi\)
\(74\) −2.28552e8 − 1.31954e8i −0.102997 − 0.0594655i
\(75\) 4.04890e8 − 2.47625e8i 0.170620 − 0.104349i
\(76\) 9.75746e8 + 1.69004e9i 0.384830 + 0.666545i
\(77\) −1.38822e9 + 8.01488e8i −0.512866 + 0.296104i
\(78\) 1.99507e8 − 3.66672e8i 0.0691011 − 0.127000i
\(79\) −1.01333e9 + 1.75514e9i −0.329318 + 0.570396i −0.982377 − 0.186911i \(-0.940152\pi\)
0.653058 + 0.757308i \(0.273486\pi\)
\(80\) − 3.66357e8i − 0.111803i
\(81\) −1.42946e9 + 3.18030e9i −0.409965 + 0.912101i
\(82\) −3.36909e9 −0.908749
\(83\) −3.87487e9 − 2.23716e9i −0.983709 − 0.567945i −0.0803210 − 0.996769i \(-0.525595\pi\)
−0.903388 + 0.428825i \(0.858928\pi\)
\(84\) −2.82847e9 − 1.53897e9i −0.676324 − 0.367988i
\(85\) −6.05450e8 − 1.04867e9i −0.136453 − 0.236344i
\(86\) −1.95742e9 + 1.13012e9i −0.416095 + 0.240233i
\(87\) −1.80938e9 − 2.95851e9i −0.363023 − 0.593577i
\(88\) 3.58771e8 − 6.21410e8i 0.0679836 − 0.117751i
\(89\) 3.09752e9i 0.554708i 0.960768 + 0.277354i \(0.0894575\pi\)
−0.960768 + 0.277354i \(0.910542\pi\)
\(90\) 1.01423e9 + 1.56784e9i 0.171761 + 0.265515i
\(91\) −1.96486e9 −0.314866
\(92\) −5.39412e9 − 3.11430e9i −0.818430 − 0.472521i
\(93\) 1.07637e8 + 4.25424e9i 0.0154721 + 0.611515i
\(94\) 4.13959e9 + 7.16998e9i 0.564051 + 0.976965i
\(95\) −4.61310e9 + 2.66337e9i −0.596176 + 0.344202i
\(96\) 1.44093e9 − 3.64572e7i 0.176720 − 0.00447124i
\(97\) 1.68792e9 − 2.92356e9i 0.196559 − 0.340451i −0.750851 − 0.660471i \(-0.770357\pi\)
0.947411 + 0.320021i \(0.103690\pi\)
\(98\) 8.76502e9i 0.969667i
\(99\) 1.84947e8 + 3.65257e9i 0.0194479 + 0.384081i
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.11.16 ✓ 80
3.2 odd 2 270.11.h.a.251.39 80
9.4 even 3 270.11.h.a.71.39 80
9.5 odd 6 inner 90.11.h.a.41.16 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.16 ✓ 80 1.1 even 1 trivial
90.11.h.a.41.16 yes 80 9.5 odd 6 inner
270.11.h.a.71.39 80 9.4 even 3
270.11.h.a.251.39 80 3.2 odd 2