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Properties

Label 90.11.k.b.7.2
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 7.2
Character \(\chi\) \(=\) 90.7
Dual form 90.11.k.b.13.2

$q$-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} +(-242.823 + 9.27376i) q^{3} +(443.405 - 256.000i) q^{4} +(-2921.40 - 1109.53i) q^{5} +(-5252.93 + 1624.76i) q^{6} +(71.8804 - 19.2603i) q^{7} +(8192.00 - 8192.00i) q^{8} +(58877.0 - 4503.76i) q^{9} +(-70349.1 - 7141.51i) q^{10} +(26325.6 - 45597.4i) q^{11} +(-105295. + 66274.7i) q^{12} +(-402841. - 107941. i) q^{13} +(1458.25 - 841.922i) q^{14} +(719672. + 242328. i) q^{15} +(131072. - 227023. i) q^{16} +(-1.67124e6 - 1.67124e6i) q^{17} +(1.26046e6 - 443244. i) q^{18} -2.63310e6i q^{19} +(-1.57940e6 + 255905. i) q^{20} +(-17275.6 + 5343.44i) q^{21} +(308347. - 1.15077e6i) q^{22} +(-2.65101e6 - 710337. i) q^{23} +(-1.91324e6 + 2.06518e6i) q^{24} +(7.30350e6 + 6.48277e6i) q^{25} -9.43680e6 q^{26} +(-1.42549e7 + 1.63963e6i) q^{27} +(26941.5 - 26941.5i) q^{28} +(1.97851e7 + 1.14230e7i) q^{29} +(1.71486e7 + 1.08172e6i) q^{30} +(1.80964e7 + 3.13438e7i) q^{31} +(1.53522e6 - 5.72953e6i) q^{32} +(-5.96961e6 + 1.13162e7i) q^{33} +(-4.63147e7 - 2.67398e7i) q^{34} +(-231361. - 23486.7i) q^{35} +(2.49534e7 - 1.70695e7i) q^{36} +(-5.48806e7 - 5.48806e7i) q^{37} +(-1.54205e7 - 5.75501e7i) q^{38} +(9.88200e7 + 2.24747e7i) q^{39} +(-3.30214e7 + 1.48428e7i) q^{40} +(4.60256e7 + 7.97187e7i) q^{41} +(-346289. + 217961. i) q^{42} +(5.88913e7 + 2.19785e8i) q^{43} -2.69575e7i q^{44} +(-1.77000e8 - 5.21687e7i) q^{45} -6.21017e7 q^{46} +(2.20396e8 - 5.90548e7i) q^{47} +(-2.97219e7 + 5.63420e7i) q^{48} +(-2.44626e8 + 1.41235e8i) q^{49} +(1.97594e8 + 9.89178e7i) q^{50} +(4.21313e8 + 3.90316e8i) q^{51} +(-2.06254e8 + 5.52657e7i) q^{52} +(-2.56159e8 + 2.56159e8i) q^{53} +(-3.01959e8 + 1.19319e8i) q^{54} +(-1.27499e8 + 1.03999e8i) q^{55} +(431064. - 746625. i) q^{56} +(2.44187e7 + 6.39377e8i) q^{57} +(4.99330e8 + 1.33795e8i) q^{58} +(-5.12459e8 + 2.95868e8i) q^{59} +(3.81142e8 - 7.67867e7i) q^{60} +(-1.00330e8 + 1.73776e8i) q^{61} +(5.79083e8 + 5.79083e8i) q^{62} +(4.14536e6 - 1.45772e6i) q^{63} -1.34218e8i q^{64} +(1.05709e9 + 7.62303e8i) q^{65} +(-6.42019e7 + 2.82292e8i) q^{66} +(-3.59433e8 + 1.34142e9i) q^{67} +(-1.16887e9 - 3.13198e8i) q^{68} +(6.50314e8 + 1.47901e8i) q^{69} +(-5.19427e6 + 841610. i) q^{70} +1.72421e9 q^{71} +(4.45425e8 - 5.19215e8i) q^{72} +(3.36383e8 - 3.36383e8i) q^{73} +(-1.52090e9 - 8.78089e8i) q^{74} +(-1.83358e9 - 1.50643e9i) q^{75} +(-6.74073e8 - 1.16753e9i) q^{76} +(1.01408e6 - 3.78460e6i) q^{77} +(2.29147e9 - 8.75146e7i) q^{78} +(-1.82534e9 - 1.05386e9i) q^{79} +(-6.34803e8 + 5.17797e8i) q^{80} +(3.44622e9 - 5.30336e8i) q^{81} +(1.47282e9 + 1.47282e9i) q^{82} +(-2.79071e8 - 1.04151e9i) q^{83} +(-6.29217e6 + 6.79186e6i) q^{84} +(3.02805e9 + 6.73664e9i) q^{85} +(2.57431e9 + 4.45883e9i) q^{86} +(-4.91022e9 - 2.59027e9i) q^{87} +(-1.57874e8 - 5.89193e8i) q^{88} -3.54555e9i q^{89} +(-4.17411e9 - 1.03635e8i) q^{90} -3.10353e7 q^{91} +(-1.35732e9 + 3.63693e8i) q^{92} +(-4.68489e9 - 7.44318e9i) q^{93} +(4.47120e9 - 2.58145e9i) q^{94} +(-2.92151e9 + 7.69233e9i) q^{95} +(-3.19653e8 + 1.40550e9i) q^{96} +(7.10376e9 - 1.90345e9i) q^{97} +(-4.51952e9 + 4.51952e9i) q^{98} +(1.34462e9 - 2.80320e9i) 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
\(3\) −242.823 + 9.27376i −0.999272 + 0.0381636i
\(4\) 443.405 − 256.000i 0.433013 − 0.250000i
\(5\) −2921.40 − 1109.53i −0.934847 − 0.355050i
\(6\) −5252.93 + 1624.76i −0.675531 + 0.208946i
\(7\) 71.8804 − 19.2603i 0.00427681 − 0.00114597i −0.256680 − 0.966496i \(-0.582629\pi\)
0.260957 + 0.965350i \(0.415962\pi\)
\(8\) 8192.00 − 8192.00i 0.250000 − 0.250000i
\(9\) 58877.0 − 4503.76i 0.997087 − 0.0762717i
\(10\) −70349.1 − 7141.51i −0.703491 − 0.0714151i
\(11\) 26325.6 − 45597.4i 0.163462 − 0.283124i −0.772646 − 0.634837i \(-0.781067\pi\)
0.936108 + 0.351713i \(0.114401\pi\)
\(12\) −105295. + 66274.7i −0.423156 + 0.266343i
\(13\) −402841. − 107941.i −1.08497 − 0.290716i −0.328338 − 0.944560i \(-0.606489\pi\)
−0.756629 + 0.653844i \(0.773155\pi\)
\(14\) 1458.25 − 841.922i 0.00271139 − 0.00156542i
\(15\) 719672. + 242328.i 0.947716 + 0.319115i
\(16\) 131072. − 227023.i 0.125000 − 0.216506i
\(17\) −1.67124e6 − 1.67124e6i −1.17705 − 1.17705i −0.980493 − 0.196552i \(-0.937025\pi\)
−0.196552 − 0.980493i \(-0.562975\pi\)
\(18\) 1.26046e6 − 443244.i 0.667064 − 0.234574i
\(19\) − 2.63310e6i − 1.06341i −0.846931 − 0.531703i \(-0.821552\pi\)
0.846931 − 0.531703i \(-0.178448\pi\)
\(20\) −1.57940e6 + 255905.i −0.493563 + 0.0799704i
\(21\) −17275.6 + 5343.44i −0.00422996 + 0.00130835i
\(22\) 308347. − 1.15077e6i 0.0598311 − 0.223293i
\(23\) −2.65101e6 − 710337.i −0.411882 − 0.110363i 0.0469271 − 0.998898i \(-0.485057\pi\)
−0.458809 + 0.888535i \(0.651724\pi\)
\(24\) −1.91324e6 + 2.06518e6i −0.240277 + 0.259359i
\(25\) 7.30350e6 + 6.48277e6i 0.747878 + 0.663836i
\(26\) −9.43680e6 −0.794251
\(27\) −1.42549e7 + 1.63963e6i −0.993450 + 0.114269i
\(28\) 26941.5 − 26941.5i 0.00156542 − 0.00156542i
\(29\) 1.97851e7 + 1.14230e7i 0.964605 + 0.556915i 0.897587 − 0.440837i \(-0.145318\pi\)
0.0670174 + 0.997752i \(0.478652\pi\)
\(30\) 1.71486e7 + 1.08172e6i 0.705704 + 0.0445153i
\(31\) 1.80964e7 + 3.13438e7i 0.632095 + 1.09482i 0.987123 + 0.159965i \(0.0511383\pi\)
−0.355027 + 0.934856i \(0.615528\pi\)
\(32\) 1.53522e6 − 5.72953e6i 0.0457532 − 0.170753i
\(33\) −5.96961e6 + 1.13162e7i −0.152537 + 0.289156i
\(34\) −4.63147e7 − 2.67398e7i −1.01935 − 0.588523i
\(35\) −231361. − 23486.7i −0.00440504 − 0.000447179i
\(36\) 2.49534e7 − 1.70695e7i 0.412683 − 0.282298i
\(37\) −5.48806e7 − 5.48806e7i −0.791425 − 0.791425i 0.190300 − 0.981726i \(-0.439054\pi\)
−0.981726 + 0.190300i \(0.939054\pi\)
\(38\) −1.54205e7 − 5.75501e7i −0.194617 − 0.726320i
\(39\) 9.88200e7 + 2.24747e7i 1.09527 + 0.249098i
\(40\) −3.30214e7 + 1.48428e7i −0.322474 + 0.144949i
\(41\) 4.60256e7 + 7.97187e7i 0.397265 + 0.688083i 0.993387 − 0.114811i \(-0.0366261\pi\)
−0.596123 + 0.802893i \(0.703293\pi\)
\(42\) −346289. + 217961.i −0.00264967 + 0.00166776i
\(43\) 5.88913e7 + 2.19785e8i 0.400598 + 1.49505i 0.812032 + 0.583613i \(0.198362\pi\)
−0.411433 + 0.911440i \(0.634972\pi\)
\(44\) − 2.69575e7i − 0.163462i
\(45\) −1.77000e8 − 5.21687e7i −0.959204 − 0.282714i
\(46\) −6.21017e7 −0.301519
\(47\) 2.20396e8 − 5.90548e7i 0.960978 − 0.257493i 0.255964 − 0.966686i \(-0.417607\pi\)
0.705014 + 0.709193i \(0.250941\pi\)
\(48\) −2.97219e7 + 5.63420e7i −0.116646 + 0.221119i
\(49\) −2.44626e8 + 1.41235e8i −0.866008 + 0.499990i
\(50\) 1.97594e8 + 9.89178e7i 0.632301 + 0.316537i
\(51\) 4.21313e8 + 3.90316e8i 1.22111 + 1.13127i
\(52\) −2.06254e8 + 5.52657e7i −0.542484 + 0.145358i
\(53\) −2.56159e8 + 2.56159e8i −0.612535 + 0.612535i −0.943606 − 0.331071i \(-0.892590\pi\)
0.331071 + 0.943606i \(0.392590\pi\)
\(54\) −3.01959e8 + 1.19319e8i −0.657626 + 0.259861i
\(55\) −1.27499e8 + 1.03999e8i −0.253335 + 0.206640i
\(56\) 431064. − 746625.i 0.000782711 − 0.00135570i
\(57\) 2.44187e7 + 6.39377e8i 0.0405834 + 1.06263i
\(58\) 4.99330e8 + 1.33795e8i 0.760760 + 0.203845i
\(59\) −5.12459e8 + 2.95868e8i −0.716802 + 0.413846i −0.813574 − 0.581461i \(-0.802481\pi\)
0.0967727 + 0.995307i \(0.469148\pi\)
\(60\) 3.81142e8 − 7.67867e7i 0.490152 − 0.0987483i
\(61\) −1.00330e8 + 1.73776e8i −0.118790 + 0.205751i −0.919288 − 0.393584i \(-0.871235\pi\)
0.800498 + 0.599335i \(0.204568\pi\)
\(62\) 5.79083e8 + 5.79083e8i 0.632095 + 0.632095i
\(63\) 4.14536e6 − 1.45772e6i 0.00417695 − 0.00146883i
\(64\) − 1.34218e8i − 0.125000i
\(65\) 1.05709e9 + 7.62303e8i 0.911060 + 0.656993i
\(66\) −6.42019e7 + 2.82292e8i −0.0512658 + 0.225413i
\(67\) −3.59433e8 + 1.34142e9i −0.266222 + 0.993555i 0.695276 + 0.718743i \(0.255282\pi\)
−0.961498 + 0.274812i \(0.911384\pi\)
\(68\) −1.16887e9 − 3.13198e8i −0.803937 − 0.215414i
\(69\) 6.50314e8 + 1.47901e8i 0.415794 + 0.0945641i
\(70\) −5.19427e6 + 841610.i −0.00309054 + 0.000500750i
\(71\) 1.72421e9 0.955651 0.477825 − 0.878455i \(-0.341425\pi\)
0.477825 + 0.878455i \(0.341425\pi\)
\(72\) 4.45425e8 − 5.19215e8i 0.230204 − 0.268340i
\(73\) 3.36383e8 − 3.36383e8i 0.162263 − 0.162263i −0.621305 − 0.783568i \(-0.713397\pi\)
0.783568 + 0.621305i \(0.213397\pi\)
\(74\) −1.52090e9 − 8.78089e8i −0.685395 − 0.395713i
\(75\) −1.83358e9 − 1.50643e9i −0.772668 − 0.634810i
\(76\) −6.74073e8 − 1.16753e9i −0.265851 − 0.460468i
\(77\) 1.01408e6 − 3.78460e6i 0.000374644 − 0.00139819i
\(78\) 2.29147e9 − 8.75146e7i 0.793673 − 0.0303115i
\(79\) −1.82534e9 − 1.05386e9i −0.593211 − 0.342490i 0.173155 − 0.984895i \(-0.444604\pi\)
−0.766366 + 0.642404i \(0.777937\pi\)
\(80\) −6.34803e8 + 5.17797e8i −0.193727 + 0.158019i
\(81\) 3.44622e9 − 5.30336e8i 0.988365 − 0.152099i
\(82\) 1.47282e9 + 1.47282e9i 0.397265 + 0.397265i
\(83\) −2.79071e8 − 1.04151e9i −0.0708473 − 0.264406i 0.921412 − 0.388586i \(-0.127036\pi\)
−0.992260 + 0.124181i \(0.960370\pi\)
\(84\) −6.29217e6 + 6.79186e6i −0.00150454 + 0.00162402i
\(85\) 3.02805e9 + 6.73664e9i 0.682447 + 1.51827i
\(86\) 2.57431e9 + 4.45883e9i 0.547227 + 0.947826i
\(87\) −4.91022e9 − 2.59027e9i −0.985156 − 0.519696i
\(88\) −1.57874e8 − 5.89193e8i −0.0299155 − 0.111646i
\(89\) − 3.54555e9i − 0.634942i −0.948268 − 0.317471i \(-0.897166\pi\)
0.948268 − 0.317471i \(-0.102834\pi\)
\(90\) −4.17411e9 − 1.03635e8i −0.706889 − 0.0175506i
\(91\) −3.10353e7 −0.00497335
\(92\) −1.35732e9 + 3.63693e8i −0.205941 + 0.0551817i
\(93\) −4.68489e9 − 7.44318e9i −0.673417 − 1.06990i
\(94\) 4.47120e9 − 2.58145e9i 0.609236 − 0.351742i
\(95\) −2.92151e9 + 7.69233e9i −0.377563 + 0.994122i
\(96\) −3.19653e8 + 1.40550e9i −0.0392033 + 0.172375i
\(97\) 7.10376e9 − 1.90345e9i 0.827237 − 0.221657i 0.179729 − 0.983716i \(-0.442478\pi\)
0.647508 + 0.762059i \(0.275811\pi\)
\(98\) −4.51952e9 + 4.51952e9i −0.499990 + 0.499990i
\(99\) 1.34462e9 − 2.80320e9i 0.141391 − 0.294766i
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.7.2 ✓ 120
5.3 odd 4 inner 90.11.k.b.43.15 yes 120
9.4 even 3 inner 90.11.k.b.67.15 yes 120
45.13 odd 12 inner 90.11.k.b.13.2 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.k.b.7.2 ✓ 120 1.1 even 1 trivial
90.11.k.b.13.2 yes 120 45.13 odd 12 inner
90.11.k.b.43.15 yes 120 5.3 odd 4 inner
90.11.k.b.67.15 yes 120 9.4 even 3 inner