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Properties

Label 90.11.k.b.7.18
Level $90$
Weight $11$
Character 90.7
Analytic conductor $57.182$
Analytic rank $0$
Dimension $120$
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] = [90,11,Mod(7,90)] 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("90.7"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([8, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 90.k (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [120,960] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(57.1821527406\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(30\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 7.18
Character \(\chi\) \(=\) 90.7
Dual form 90.11.k.b.13.18

$q$-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+(21.8564 - 5.85641i) q^{2} +(64.1341 - 234.384i) q^{3} +(443.405 - 256.000i) q^{4} +(543.965 + 3077.29i) q^{5} +(29.0937 - 5498.39i) q^{6} +(20748.0 - 5559.42i) q^{7} +(8192.00 - 8192.00i) q^{8} +(-50822.6 - 30064.0i) q^{9} +(29911.0 + 64072.9i) q^{10} +(-112339. + 194577. i) q^{11} +(-31564.9 - 120345. i) q^{12} +(369092. + 98897.8i) q^{13} +(420919. - 243018. i) q^{14} +(756154. + 69862.7i) q^{15} +(131072. - 227023. i) q^{16} +(1.44977e6 + 1.44977e6i) q^{17} +(-1.28687e6 - 359453. i) q^{18} -1.36045e6i q^{19} +(1.02898e6 + 1.22523e6i) q^{20} +(27618.3 - 5.21956e6i) q^{21} +(-1.31580e6 + 4.91065e6i) q^{22} +(121707. + 32611.4i) q^{23} +(-1.39469e6 - 2.44546e6i) q^{24} +(-9.17383e6 + 3.34788e6i) q^{25} +8.64620e6 q^{26} +(-1.03060e7 + 9.98388e6i) q^{27} +(7.77657e6 - 7.77657e6i) q^{28} +(2.77331e7 + 1.60117e7i) q^{29} +(1.69360e7 - 2.90140e6i) q^{30} +(1.52295e7 + 2.63783e7i) q^{31} +(1.53522e6 - 5.72953e6i) q^{32} +(3.84009e7 + 3.88094e7i) q^{33} +(4.01773e7 + 2.31964e7i) q^{34} +(2.83942e7 + 6.08236e7i) q^{35} +(-3.02314e7 - 319937. i) q^{36} +(-7.19217e7 - 7.19217e7i) q^{37} +(-7.96735e6 - 2.97346e7i) q^{38} +(4.68514e7 - 8.01664e7i) q^{39} +(2.96653e7 + 2.07530e7i) q^{40} +(5.08122e7 + 8.80094e7i) q^{41} +(-2.99642e7 - 1.14242e8i) q^{42} +(-8.97840e6 - 3.35078e7i) q^{43} +1.15035e8i q^{44} +(6.48700e7 - 1.72750e8i) q^{45} +2.85107e6 q^{46} +(-8.40117e7 + 2.25109e7i) q^{47} +(-4.48044e7 - 4.52811e7i) q^{48} +(1.54943e8 - 8.94565e7i) q^{49} +(-1.80900e8 + 1.26898e8i) q^{50} +(4.32783e8 - 2.46824e8i) q^{51} +(1.88975e8 - 5.06357e7i) q^{52} +(1.18417e8 - 1.18417e8i) q^{53} +(-1.66782e8 + 2.78568e8i) q^{54} +(-6.59878e8 - 2.39857e8i) q^{55} +(1.24425e8 - 2.15511e8i) q^{56} +(-3.18868e8 - 8.72513e7i) q^{57} +(6.99916e8 + 1.87542e8i) q^{58} +(1.18548e9 - 6.84439e8i) q^{59} +(3.53168e8 - 1.62598e8i) q^{60} +(4.95824e8 - 8.58792e8i) q^{61} +(4.87345e8 + 4.87345e8i) q^{62} +(-1.22161e9 - 3.41225e8i) q^{63} -1.34218e8i q^{64} +(-1.03564e8 + 1.18960e9i) q^{65} +(1.06659e9 + 6.23343e8i) q^{66} +(-6.90438e7 + 2.57675e8i) q^{67} +(1.01398e9 + 2.71695e8i) q^{68} +(1.54492e7 - 2.64347e7i) q^{69} +(9.76803e8 + 1.16310e9i) q^{70} +1.35798e8 q^{71} +(-6.62623e8 + 1.70055e8i) q^{72} +(2.06388e9 - 2.06388e9i) q^{73} +(-1.99315e9 - 1.15075e9i) q^{74} +(1.96334e8 + 2.36491e9i) q^{75} +(-3.48275e8 - 6.03231e8i) q^{76} +(-1.24908e9 + 4.66162e9i) q^{77} +(5.54517e8 - 2.02653e9i) q^{78} +(-5.81771e8 - 3.35886e8i) q^{79} +(7.69916e8 + 2.79854e8i) q^{80} +(1.67910e9 + 3.05586e9i) q^{81} +(1.62599e9 + 1.62599e9i) q^{82} +(-1.43304e9 - 5.34819e9i) q^{83} +(-1.32396e9 - 2.32145e9i) q^{84} +(-3.67275e9 + 5.25000e9i) q^{85} +(-3.92471e8 - 6.79780e8i) q^{86} +(5.53152e9 - 5.47329e9i) q^{87} +(6.73692e8 + 2.51425e9i) q^{88} -8.97893e9i q^{89} +(4.06131e8 - 4.15560e9i) q^{90} +8.20774e9 q^{91} +(6.23141e7 - 1.66970e7i) q^{92} +(7.15938e9 - 1.87781e9i) q^{93} +(-1.70436e9 + 9.84013e8i) q^{94} +(4.18650e9 - 7.40038e8i) q^{95} +(-1.24445e9 - 7.27289e8i) q^{96} +(-6.27435e9 + 1.68121e9i) q^{97} +(2.86261e9 - 2.86261e9i) q^{98} +(1.15591e10 - 6.51154e9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 960 q^{2} - 128 q^{3} - 8192 q^{6} + 1830 q^{7} + 983040 q^{8} + 105792 q^{10} + 217740 q^{11} - 65536 q^{12} - 3385658 q^{15} + 15728640 q^{16} + 2438244 q^{17} + 2534464 q^{18} + 1692672 q^{20}+ \cdots + 76966514304 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(11\) \(37\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 21.8564 − 5.85641i 0.683013 − 0.183013i
\(3\) 64.1341 − 234.384i 0.263926 − 0.964543i
\(4\) 443.405 − 256.000i 0.433013 − 0.250000i
\(5\) 543.965 + 3077.29i 0.174069 + 0.984733i
\(6\) 29.0937 − 5498.39i 0.00374148 − 0.707097i
\(7\) 20748.0 − 5559.42i 1.23449 − 0.330780i 0.418163 − 0.908372i \(-0.362674\pi\)
0.816325 + 0.577592i \(0.196008\pi\)
\(8\) 8192.00 − 8192.00i 0.250000 − 0.250000i
\(9\) −50822.6 − 30064.0i −0.860686 − 0.509137i
\(10\) 29911.0 + 64072.9i 0.299110 + 0.640729i
\(11\) −112339. + 194577.i −0.697536 + 1.20817i 0.271782 + 0.962359i \(0.412387\pi\)
−0.969318 + 0.245809i \(0.920946\pi\)
\(12\) −31564.9 − 120345.i −0.126852 − 0.483641i
\(13\) 369092. + 98897.8i 0.994071 + 0.266361i 0.718960 − 0.695052i \(-0.244619\pi\)
0.275111 + 0.961412i \(0.411285\pi\)
\(14\) 420919. − 243018.i 0.782634 − 0.451854i
\(15\) 756154. + 69862.7i 0.995759 + 0.0920002i
\(16\) 131072. − 227023.i 0.125000 − 0.216506i
\(17\) 1.44977e6 + 1.44977e6i 1.02107 + 1.02107i 0.999773 + 0.0212964i \(0.00677936\pi\)
0.0212964 + 0.999773i \(0.493221\pi\)
\(18\) −1.28687e6 − 359453.i −0.681038 − 0.190230i
\(19\) − 1.36045e6i − 0.549433i −0.961525 − 0.274717i \(-0.911416\pi\)
0.961525 − 0.274717i \(-0.0885840\pi\)
\(20\) 1.02898e6 + 1.22523e6i 0.321557 + 0.382885i
\(21\) 27618.3 − 5.21956e6i 0.00676241 − 1.27802i
\(22\) −1.31580e6 + 4.91065e6i −0.255316 + 0.952852i
\(23\) 121707. + 32611.4i 0.0189094 + 0.00506675i 0.268261 − 0.963346i \(-0.413551\pi\)
−0.249352 + 0.968413i \(0.580218\pi\)
\(24\) −1.39469e6 − 2.44546e6i −0.175154 − 0.307117i
\(25\) −9.17383e6 + 3.34788e6i −0.939400 + 0.342823i
\(26\) 8.64620e6 0.727711
\(27\) −1.03060e7 + 9.98388e6i −0.718242 + 0.695794i
\(28\) 7.77657e6 − 7.77657e6i 0.451854 − 0.451854i
\(29\) 2.77331e7 + 1.60117e7i 1.35210 + 0.780634i 0.988543 − 0.150939i \(-0.0482297\pi\)
0.363554 + 0.931573i \(0.381563\pi\)
\(30\) 1.69360e7 − 2.90140e6i 0.696953 − 0.119399i
\(31\) 1.52295e7 + 2.63783e7i 0.531959 + 0.921380i 0.999304 + 0.0373046i \(0.0118772\pi\)
−0.467345 + 0.884075i \(0.654789\pi\)
\(32\) 1.53522e6 − 5.72953e6i 0.0457532 − 0.170753i
\(33\) 3.84009e7 + 3.88094e7i 0.981232 + 0.991671i
\(34\) 4.01773e7 + 2.31964e7i 0.884272 + 0.510535i
\(35\) 2.83942e7 + 6.08236e7i 0.540616 + 1.15806i
\(36\) −3.02314e7 − 319937.i −0.499972 − 0.00529117i
\(37\) −7.19217e7 − 7.19217e7i −1.03717 − 1.03717i −0.999282 − 0.0378910i \(-0.987936\pi\)
−0.0378910 − 0.999282i \(-0.512064\pi\)
\(38\) −7.96735e6 − 2.97346e7i −0.100553 − 0.375270i
\(39\) 4.68514e7 − 8.01664e7i 0.519278 − 0.888525i
\(40\) 2.96653e7 + 2.07530e7i 0.289701 + 0.202666i
\(41\) 5.08122e7 + 8.80094e7i 0.438580 + 0.759643i 0.997580 − 0.0695244i \(-0.0221482\pi\)
−0.559000 + 0.829168i \(0.688815\pi\)
\(42\) −2.99642e7 − 1.14242e8i −0.229275 − 0.874140i
\(43\) −8.97840e6 − 3.35078e7i −0.0610740 − 0.227931i 0.928642 − 0.370977i \(-0.120977\pi\)
−0.989716 + 0.143046i \(0.954310\pi\)
\(44\) 1.15035e8i 0.697536i
\(45\) 6.48700e7 − 1.72750e8i 0.351545 − 0.936171i
\(46\) 2.85107e6 0.0138426
\(47\) −8.40117e7 + 2.25109e7i −0.366311 + 0.0981528i −0.437279 − 0.899326i \(-0.644058\pi\)
0.0709679 + 0.997479i \(0.477391\pi\)
\(48\) −4.48044e7 − 4.52811e7i −0.175839 − 0.177710i
\(49\) 1.54943e8 − 8.94565e7i 0.548520 − 0.316688i
\(50\) −1.80900e8 + 1.26898e8i −0.578881 + 0.406075i
\(51\) 4.32783e8 − 2.46824e8i 1.25435 − 0.715378i
\(52\) 1.88975e8 − 5.06357e7i 0.497036 − 0.133180i
\(53\) 1.18417e8 − 1.18417e8i 0.283162 − 0.283162i −0.551207 − 0.834369i \(-0.685832\pi\)
0.834369 + 0.551207i \(0.185832\pi\)
\(54\) −1.66782e8 + 2.78568e8i −0.363229 + 0.606683i
\(55\) −6.59878e8 − 2.39857e8i −1.31114 − 0.476583i
\(56\) 1.24425e8 − 2.15511e8i 0.225927 − 0.391317i
\(57\) −3.18868e8 − 8.72513e7i −0.529952 − 0.145010i
\(58\) 6.99916e8 + 1.87542e8i 1.06637 + 0.285732i
\(59\) 1.18548e9 − 6.84439e8i 1.65819 − 0.957358i 0.684645 − 0.728877i \(-0.259957\pi\)
0.973548 − 0.228481i \(-0.0733760\pi\)
\(60\) 3.53168e8 − 1.62598e8i 0.454176 − 0.209102i
\(61\) 4.95824e8 − 8.58792e8i 0.587054 − 1.01681i −0.407562 − 0.913178i \(-0.633621\pi\)
0.994616 − 0.103630i \(-0.0330457\pi\)
\(62\) 4.87345e8 + 4.87345e8i 0.531959 + 0.531959i
\(63\) −1.22161e9 − 3.41225e8i −1.23092 − 0.343825i
\(64\) − 1.34218e8i − 0.125000i
\(65\) −1.03564e8 + 1.18960e9i −0.0892573 + 1.02526i
\(66\) 1.06659e9 + 6.23343e8i 0.851682 + 0.497746i
\(67\) −6.90438e7 + 2.57675e8i −0.0511389 + 0.190853i −0.986770 − 0.162127i \(-0.948165\pi\)
0.935631 + 0.352979i \(0.114831\pi\)
\(68\) 1.01398e9 + 2.71695e8i 0.697404 + 0.186869i
\(69\) 1.54492e7 − 2.64347e7i 0.00987778 − 0.0169017i
\(70\) 9.76803e8 + 1.16310e9i 0.581188 + 0.692032i
\(71\) 1.35798e8 0.0752663 0.0376331 − 0.999292i \(-0.488018\pi\)
0.0376331 + 0.999292i \(0.488018\pi\)
\(72\) −6.62623e8 + 1.70055e8i −0.342456 + 0.0878873i
\(73\) 2.06388e9 − 2.06388e9i 0.995565 − 0.995565i −0.00442549 − 0.999990i \(-0.501409\pi\)
0.999990 + 0.00442549i \(0.00140868\pi\)
\(74\) −1.99315e9 − 1.15075e9i −0.898218 − 0.518586i
\(75\) 1.96334e8 + 2.36491e9i 0.0827349 + 0.996572i
\(76\) −3.48275e8 − 6.03231e8i −0.137358 − 0.237912i
\(77\) −1.24908e9 + 4.66162e9i −0.461462 + 1.72220i
\(78\) 5.54517e8 − 2.02653e9i 0.192062 − 0.701908i
\(79\) −5.81771e8 − 3.35886e8i −0.189067 − 0.109158i 0.402479 − 0.915429i \(-0.368149\pi\)
−0.591546 + 0.806271i \(0.701482\pi\)
\(80\) 7.69916e8 + 2.79854e8i 0.234960 + 0.0854047i
\(81\) 1.67910e9 + 3.05586e9i 0.481560 + 0.876413i
\(82\) 1.62599e9 + 1.62599e9i 0.438580 + 0.438580i
\(83\) −1.43304e9 − 5.34819e9i −0.363805 − 1.35774i −0.869033 − 0.494753i \(-0.835258\pi\)
0.505228 − 0.862986i \(-0.331408\pi\)
\(84\) −1.32396e9 − 2.32145e9i −0.316576 − 0.555089i
\(85\) −3.67275e9 + 5.25000e9i −0.827745 + 1.18322i
\(86\) −3.92471e8 − 6.79780e8i −0.0834287 − 0.144503i
\(87\) 5.53152e9 − 5.47329e9i 1.10981 − 1.09813i
\(88\) 6.73692e8 + 2.51425e9i 0.127658 + 0.476426i
\(89\) − 8.97893e9i − 1.60796i −0.594658 − 0.803979i \(-0.702712\pi\)
0.594658 − 0.803979i \(-0.297288\pi\)
\(90\) 4.06131e8 − 4.15560e9i 0.0687787 − 0.703754i
\(91\) 8.20774e9 1.31528
\(92\) 6.23141e7 − 1.66970e7i 0.00945469 − 0.00253338i
\(93\) 7.15938e9 − 1.87781e9i 1.02911 − 0.269921i
\(94\) −1.70436e9 + 9.84013e8i −0.232232 + 0.134079i
\(95\) 4.18650e9 − 7.40038e8i 0.541045 − 0.0956392i
\(96\) −1.24445e9 − 7.27289e8i −0.152623 − 0.0891972i
\(97\) −6.27435e9 + 1.68121e9i −0.730651 + 0.195777i −0.604919 − 0.796287i \(-0.706795\pi\)
−0.125732 + 0.992064i \(0.540128\pi\)
\(98\) 2.86261e9 − 2.86261e9i 0.316688 − 0.316688i
\(99\) 1.15591e10 − 6.51154e9i 1.21548 − 0.684712i
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.k.b.7.18 ✓ 120
5.3 odd 4 inner 90.11.k.b.43.27 yes 120
9.4 even 3 inner 90.11.k.b.67.27 yes 120
45.13 odd 12 inner 90.11.k.b.13.18 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.k.b.7.18 ✓ 120 1.1 even 1 trivial
90.11.k.b.13.18 yes 120 45.13 odd 12 inner
90.11.k.b.43.27 yes 120 5.3 odd 4 inner
90.11.k.b.67.27 yes 120 9.4 even 3 inner