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Properties

Label 90.11.h.a.11.20
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 11.20
Character \(\chi\) \(=\) 90.11
Dual form 90.11.h.a.41.20

$q$-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} +(239.500 + 41.0931i) q^{3} +(256.000 + 443.405i) q^{4} +(1210.31 - 698.771i) q^{5} +(-4228.31 - 3514.89i) q^{6} +(-5111.91 + 8854.09i) q^{7} -11585.2i q^{8} +(55671.7 + 19683.6i) q^{9} -31622.8 q^{10} +(-83878.2 - 48427.1i) q^{11} +(43091.2 + 116715. i) q^{12} +(-57466.9 - 99535.5i) q^{13} +(200345. - 115669. i) q^{14} +(318584. - 117621. i) q^{15} +(-131072. + 227023. i) q^{16} +1.26845e6i q^{17} +(-868243. - 1.01557e6i) q^{18} +1.73688e6 q^{19} +(619677. + 357771. i) q^{20} +(-1.58815e6 + 1.91049e6i) q^{21} +(1.09578e6 + 1.89795e6i) q^{22} +(6.63805e6 - 3.83248e6i) q^{23} +(476074. - 2.77467e6i) q^{24} +(976562. - 1.69146e6i) q^{25} +2.60065e6i q^{26} +(1.25245e7 + 7.00196e6i) q^{27} -5.23460e6 q^{28} +(2.45691e7 + 1.41850e7i) q^{29} +(-7.57366e6 - 1.29948e6i) q^{30} +(1.05043e6 + 1.81940e6i) q^{31} +(5.13695e6 - 2.96582e6i) q^{32} +(-1.80988e7 - 1.50451e7i) q^{33} +(1.43508e7 - 2.48564e7i) q^{34} +1.42882e7i q^{35} +(5.52414e6 + 2.97241e7i) q^{36} -1.00798e8 q^{37} +(-3.40357e7 - 1.96505e7i) q^{38} +(-9.67310e6 - 2.62003e7i) q^{39} +(-8.09543e6 - 1.40217e7i) q^{40} +(-1.09096e8 + 6.29867e7i) q^{41} +(5.27360e7 - 1.94700e7i) q^{42} +(-4.70164e7 + 8.14347e7i) q^{43} -4.95894e7i q^{44} +(8.11342e7 - 1.50786e7i) q^{45} -1.73438e8 q^{46} +(1.27799e8 + 7.37849e7i) q^{47} +(-4.07209e7 + 4.89860e7i) q^{48} +(8.89743e7 + 1.54108e8i) q^{49} +(-3.82733e7 + 2.20971e7i) q^{50} +(-5.21244e7 + 3.03793e8i) q^{51} +(2.94230e7 - 5.09622e7i) q^{52} -5.19853e6i q^{53} +(-1.66211e8 - 2.78909e8i) q^{54} -1.35358e8 q^{55} +(1.02577e8 + 5.92227e7i) q^{56} +(4.15982e8 + 7.13737e7i) q^{57} +(-3.20969e8 - 5.55935e8i) q^{58} +(3.10574e8 - 1.79310e8i) q^{59} +(1.33711e8 + 1.11151e8i) q^{60} +(3.95553e8 - 6.85118e8i) q^{61} -4.75372e7i q^{62} +(-4.58870e8 + 3.92302e8i) q^{63} -1.34218e8 q^{64} +(-1.39105e8 - 8.03124e7i) q^{65} +(1.84447e8 + 4.99588e8i) q^{66} +(5.84667e8 + 1.01267e9i) q^{67} +(-5.62435e8 + 3.24722e8i) q^{68} +(1.74730e9 - 6.45102e8i) q^{69} +(1.61653e8 - 2.79991e8i) q^{70} +2.00363e9i q^{71} +(2.28039e8 - 6.44970e8i) q^{72} +3.30750e9 q^{73} +(1.97523e9 + 1.14040e9i) q^{74} +(3.03394e8 - 3.64974e8i) q^{75} +(4.44640e8 + 7.70140e8i) q^{76} +(8.57557e8 - 4.95111e8i) q^{77} +(-1.06869e8 + 6.22857e8i) q^{78} +(-1.40038e8 + 2.42553e8i) q^{79} +3.66357e8i q^{80} +(2.71189e9 + 2.19164e9i) q^{81} +2.85045e9 q^{82} +(3.27351e9 + 1.88996e9i) q^{83} +(-1.25369e9 - 2.15106e8i) q^{84} +(8.86353e8 + 1.53521e9i) q^{85} +(1.84266e9 - 1.06386e9i) q^{86} +(5.30140e9 + 4.40693e9i) q^{87} +(-5.61040e8 + 9.71749e8i) q^{88} +6.97588e9i q^{89} +(-1.76049e9 - 6.22451e8i) q^{90} +1.17506e9 q^{91} +(3.39868e9 + 1.96223e9i) q^{92} +(1.76814e8 + 4.78913e8i) q^{93} +(-1.66956e9 - 2.89177e9i) q^{94} +(2.10215e9 - 1.21368e9i) q^{95} +(1.35218e9 - 4.99221e8i) q^{96} +(5.07737e9 - 8.79427e9i) q^{97} -4.02652e9i q^{98} +(-3.71642e9 - 4.34705e9i) 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
\(3\) 239.500 + 41.0931i 0.985598 + 0.169108i
\(4\) 256.000 + 443.405i 0.250000 + 0.433013i
\(5\) 1210.31 − 698.771i 0.387298 − 0.223607i
\(6\) −4228.31 − 3514.89i −0.543764 − 0.452018i
\(7\) −5111.91 + 8854.09i −0.304154 + 0.526810i −0.977073 − 0.212906i \(-0.931707\pi\)
0.672919 + 0.739716i \(0.265040\pi\)
\(8\) − 11585.2i − 0.353553i
\(9\) 55671.7 + 19683.6i 0.942805 + 0.333344i
\(10\) −31622.8 −0.316228
\(11\) −83878.2 − 48427.1i −0.520818 − 0.300694i 0.216451 − 0.976293i \(-0.430552\pi\)
−0.737269 + 0.675599i \(0.763885\pi\)
\(12\) 43091.2 + 116715.i 0.173174 + 0.469053i
\(13\) −57466.9 − 99535.5i −0.154775 − 0.268078i 0.778202 − 0.628014i \(-0.216132\pi\)
−0.932977 + 0.359936i \(0.882799\pi\)
\(14\) 200345. − 115669.i 0.372511 − 0.215069i
\(15\) 318584. − 117621.i 0.419534 − 0.154891i
\(16\) −131072. + 227023.i −0.125000 + 0.216506i
\(17\) 1.26845e6i 0.893362i 0.894693 + 0.446681i \(0.147394\pi\)
−0.894693 + 0.446681i \(0.852606\pi\)
\(18\) −868243. − 1.01557e6i −0.459493 − 0.537463i
\(19\) 1.73688e6 0.701457 0.350728 − 0.936477i \(-0.385934\pi\)
0.350728 + 0.936477i \(0.385934\pi\)
\(20\) 619677. + 357771.i 0.193649 + 0.111803i
\(21\) −1.58815e6 + 1.91049e6i −0.388861 + 0.467788i
\(22\) 1.09578e6 + 1.89795e6i 0.212623 + 0.368274i
\(23\) 6.63805e6 − 3.83248e6i 1.03134 − 0.595444i 0.113971 − 0.993484i \(-0.463643\pi\)
0.917368 + 0.398040i \(0.130310\pi\)
\(24\) 476074. − 2.77467e6i 0.0597885 − 0.348461i
\(25\) 976562. − 1.69146e6i 0.100000 − 0.173205i
\(26\) 2.60065e6i 0.218885i
\(27\) 1.25245e7 + 7.00196e6i 0.872856 + 0.487978i
\(28\) −5.23460e6 −0.304154
\(29\) 2.45691e7 + 1.41850e7i 1.19784 + 0.691574i 0.960074 − 0.279748i \(-0.0902508\pi\)
0.237768 + 0.971322i \(0.423584\pi\)
\(30\) −7.57366e6 − 1.29948e6i −0.311673 − 0.0534765i
\(31\) 1.05043e6 + 1.81940e6i 0.0366910 + 0.0635507i 0.883788 − 0.467888i \(-0.154985\pi\)
−0.847097 + 0.531439i \(0.821652\pi\)
\(32\) 5.13695e6 − 2.96582e6i 0.153093 − 0.0883883i
\(33\) −1.80988e7 − 1.50451e7i −0.462467 − 0.384438i
\(34\) 1.43508e7 − 2.48564e7i 0.315851 − 0.547070i
\(35\) 1.42882e7i 0.272043i
\(36\) 5.52414e6 + 2.97241e7i 0.0913592 + 0.491583i
\(37\) −1.00798e8 −1.45359 −0.726796 − 0.686853i \(-0.758991\pi\)
−0.726796 + 0.686853i \(0.758991\pi\)
\(38\) −3.40357e7 − 1.96505e7i −0.429553 − 0.248002i
\(39\) −9.67310e6 − 2.62003e7i −0.107212 − 0.290391i
\(40\) −8.09543e6 − 1.40217e7i −0.0790569 − 0.136931i
\(41\) −1.09096e8 + 6.29867e7i −0.941651 + 0.543662i −0.890477 − 0.455027i \(-0.849629\pi\)
−0.0511734 + 0.998690i \(0.516296\pi\)
\(42\) 5.27360e7 − 1.94700e7i 0.403516 − 0.148977i
\(43\) −4.70164e7 + 8.14347e7i −0.319821 + 0.553946i −0.980450 − 0.196766i \(-0.936956\pi\)
0.660630 + 0.750712i \(0.270289\pi\)
\(44\) − 4.95894e7i − 0.300694i
\(45\) 8.11342e7 − 1.50786e7i 0.439685 − 0.0817141i
\(46\) −1.73438e8 −0.842085
\(47\) 1.27799e8 + 7.37849e7i 0.557236 + 0.321720i 0.752035 − 0.659123i \(-0.229072\pi\)
−0.194800 + 0.980843i \(0.562406\pi\)
\(48\) −4.07209e7 + 4.89860e7i −0.159813 + 0.192250i
\(49\) 8.89743e7 + 1.54108e8i 0.314981 + 0.545563i
\(50\) −3.82733e7 + 2.20971e7i −0.122474 + 0.0707107i
\(51\) −5.21244e7 + 3.03793e8i −0.151074 + 0.880495i
\(52\) 2.94230e7 − 5.09622e7i 0.0773875 − 0.134039i
\(53\) − 5.19853e6i − 0.0124309i −0.999981 − 0.00621544i \(-0.998022\pi\)
0.999981 − 0.00621544i \(-0.00197845\pi\)
\(54\) −1.66211e8 − 2.78909e8i −0.361986 − 0.607426i
\(55\) −1.35358e8 −0.268949
\(56\) 1.02577e8 + 5.92227e7i 0.186255 + 0.107535i
\(57\) 4.15982e8 + 7.13737e7i 0.691354 + 0.118622i
\(58\) −3.20969e8 − 5.55935e8i −0.489017 − 0.847002i
\(59\) 3.10574e8 − 1.79310e8i 0.434415 − 0.250810i −0.266811 − 0.963749i \(-0.585970\pi\)
0.701226 + 0.712939i \(0.252636\pi\)
\(60\) 1.33711e8 + 1.11151e8i 0.171953 + 0.142941i
\(61\) 3.95553e8 − 6.85118e8i 0.468334 − 0.811178i −0.531011 − 0.847365i \(-0.678188\pi\)
0.999345 + 0.0361868i \(0.0115211\pi\)
\(62\) − 4.75372e7i − 0.0518889i
\(63\) −4.58870e8 + 3.92302e8i −0.462367 + 0.395291i
\(64\) −1.34218e8 −0.125000
\(65\) −1.39105e8 − 8.03124e7i −0.119888 − 0.0692175i
\(66\) 1.84447e8 + 4.99588e8i 0.147283 + 0.398926i
\(67\) 5.84667e8 + 1.01267e9i 0.433046 + 0.750058i 0.997134 − 0.0756568i \(-0.0241053\pi\)
−0.564088 + 0.825715i \(0.690772\pi\)
\(68\) −5.62435e8 + 3.24722e8i −0.386837 + 0.223340i
\(69\) 1.74730e9 − 6.45102e8i 1.11718 − 0.412461i
\(70\) 1.61653e8 − 2.79991e8i 0.0961819 − 0.166592i
\(71\) 2.00363e9i 1.11052i 0.831677 + 0.555260i \(0.187381\pi\)
−0.831677 + 0.555260i \(0.812619\pi\)
\(72\) 2.28039e8 − 6.44970e8i 0.117855 − 0.333332i
\(73\) 3.30750e9 1.59546 0.797728 − 0.603017i \(-0.206035\pi\)
0.797728 + 0.603017i \(0.206035\pi\)
\(74\) 1.97523e9 + 1.14040e9i 0.890140 + 0.513922i
\(75\) 3.03394e8 − 3.64974e8i 0.127850 − 0.153800i
\(76\) 4.44640e8 + 7.70140e8i 0.175364 + 0.303740i
\(77\) 8.57557e8 − 4.95111e8i 0.316818 − 0.182915i
\(78\) −1.06869e8 + 6.22857e8i −0.0370151 + 0.215732i
\(79\) −1.40038e8 + 2.42553e8i −0.0455103 + 0.0788262i −0.887883 − 0.460069i \(-0.847825\pi\)
0.842373 + 0.538895i \(0.181158\pi\)
\(80\) 3.66357e8i 0.111803i
\(81\) 2.71189e9 + 2.19164e9i 0.777764 + 0.628557i
\(82\) 2.85045e9 0.768855
\(83\) 3.27351e9 + 1.88996e9i 0.831042 + 0.479802i 0.854209 − 0.519929i \(-0.174042\pi\)
−0.0231674 + 0.999732i \(0.507375\pi\)
\(84\) −1.25369e9 − 2.15106e8i −0.299773 − 0.0514347i
\(85\) 8.86353e8 + 1.53521e9i 0.199762 + 0.345997i
\(86\) 1.84266e9 − 1.06386e9i 0.391699 − 0.226148i
\(87\) 5.30140e9 + 4.40693e9i 1.06364 + 0.884178i
\(88\) −5.61040e8 + 9.71749e8i −0.106312 + 0.184137i
\(89\) 6.97588e9i 1.24925i 0.780925 + 0.624625i \(0.214748\pi\)
−0.780925 + 0.624625i \(0.785252\pi\)
\(90\) −1.76049e9 − 6.22451e8i −0.298141 − 0.105413i
\(91\) 1.17506e9 0.188302
\(92\) 3.39868e9 + 1.96223e9i 0.515670 + 0.297722i
\(93\) 1.76814e8 + 4.78913e8i 0.0254157 + 0.0688401i
\(94\) −1.66956e9 − 2.89177e9i −0.227491 − 0.394025i
\(95\) 2.10215e9 − 1.21368e9i 0.271673 − 0.156851i
\(96\) 1.35218e9 − 4.99221e8i 0.165835 − 0.0612261i
\(97\) 5.07737e9 − 8.79427e9i 0.591263 − 1.02410i −0.402800 − 0.915288i \(-0.631963\pi\)
0.994063 − 0.108809i \(-0.0347037\pi\)
\(98\) − 4.02652e9i − 0.445450i
\(99\) −3.71642e9 − 4.34705e9i −0.390795 − 0.457108i
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.11.20 ✓ 80
3.2 odd 2 270.11.h.a.251.30 80
9.4 even 3 270.11.h.a.71.30 80
9.5 odd 6 inner 90.11.h.a.41.20 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.20 ✓ 80 1.1 even 1 trivial
90.11.h.a.41.20 yes 80 9.5 odd 6 inner
270.11.h.a.71.30 80 9.4 even 3
270.11.h.a.251.30 80 3.2 odd 2