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Properties

Label 90.11.k.b.7.1
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 7.1
Character \(\chi\) \(=\) 90.7
Dual form 90.11.k.b.13.1

$q$-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} +(-242.911 + 6.56726i) q^{3} +(443.405 - 256.000i) q^{4} +(-1346.00 + 2820.27i) q^{5} +(-5270.71 + 1566.12i) q^{6} +(-3123.77 + 837.012i) q^{7} +(8192.00 - 8192.00i) q^{8} +(58962.7 - 3190.52i) q^{9} +(-12902.0 + 69523.7i) q^{10} +(-19551.5 + 33864.2i) q^{11} +(-106027. + 65097.2i) q^{12} +(262508. + 70338.8i) q^{13} +(-63372.5 + 36588.1i) q^{14} +(308436. - 693915. i) q^{15} +(131072. - 227023. i) q^{16} +(518218. + 518218. i) q^{17} +(1.27003e6 - 415043. i) q^{18} +1.02739e6i q^{19} +(125167. + 1.59510e6i) q^{20} +(753302. - 223834. i) q^{21} +(-229003. + 854652. i) q^{22} +(5.31774e6 + 1.42488e6i) q^{23} +(-1.93613e6 + 2.04373e6i) q^{24} +(-6.14221e6 - 7.59215e6i) q^{25} +6.14942e6 q^{26} +(-1.43018e7 + 1.16224e6i) q^{27} +(-1.17082e6 + 1.17082e6i) q^{28} +(-2.79676e7 - 1.61471e7i) q^{29} +(2.67746e6 - 1.69728e7i) q^{30} +(5.33240e6 + 9.23599e6i) q^{31} +(1.53522e6 - 5.72953e6i) q^{32} +(4.52689e6 - 8.35441e6i) q^{33} +(1.43613e7 + 8.29149e6i) q^{34} +(1.84399e6 - 9.93649e6i) q^{35} +(2.53276e7 - 1.65092e7i) q^{36} +(1.27072e7 + 1.27072e7i) q^{37} +(6.01680e6 + 2.24550e7i) q^{38} +(-6.42281e7 - 1.53621e7i) q^{39} +(1.20772e7 + 3.41301e7i) q^{40} +(4.62860e7 + 8.01698e7i) q^{41} +(1.51536e7 - 9.30385e6i) q^{42} +(-4.56424e7 - 1.70340e8i) q^{43} +2.00208e7i q^{44} +(-7.03655e7 + 1.70585e8i) q^{45} +1.24571e8 q^{46} +(-3.21917e8 + 8.62573e7i) q^{47} +(-3.03479e7 + 5.60073e7i) q^{48} +(-2.35573e8 + 1.36008e8i) q^{49} +(-1.78709e8 - 1.29966e8i) q^{50} +(-1.29284e8 - 1.22478e8i) q^{51} +(1.34404e8 - 3.60135e7i) q^{52} +(-5.47941e8 + 5.47941e8i) q^{53} +(-3.05779e8 + 1.09159e8i) q^{54} +(-6.91900e7 - 1.00722e8i) q^{55} +(-1.87331e7 + 3.24467e7i) q^{56} +(-6.74712e6 - 2.49564e8i) q^{57} +(-7.05835e8 - 1.89128e8i) q^{58} +(-2.50481e8 + 1.44615e8i) q^{59} +(-4.08799e7 - 3.86645e8i) q^{60} +(9.60234e7 - 1.66317e8i) q^{61} +(1.70637e8 + 1.70637e8i) q^{62} +(-1.81516e8 + 5.93190e7i) q^{63} -1.34218e8i q^{64} +(-5.51710e8 + 6.45668e8i) q^{65} +(5.00148e7 - 2.09109e8i) q^{66} +(1.13336e7 - 4.22976e7i) q^{67} +(3.62445e8 + 9.71167e7i) q^{68} +(-1.30110e9 - 3.11197e8i) q^{69} +(-1.78892e7 - 2.27975e8i) q^{70} -7.40304e8 q^{71} +(4.56886e8 - 5.09160e8i) q^{72} +(-9.14452e8 + 9.14452e8i) q^{73} +(3.52153e8 + 2.03316e8i) q^{74} +(1.54187e9 + 1.80388e9i) q^{75} +(2.63011e8 + 4.55549e8i) q^{76} +(3.27297e7 - 1.22149e8i) q^{77} +(-1.49376e9 + 4.03848e7i) q^{78} +(-3.43216e8 - 1.98156e8i) q^{79} +(4.63844e8 + 6.75231e8i) q^{80} +(3.46643e9 - 3.76244e8i) q^{81} +(1.48115e9 + 1.48115e9i) q^{82} +(-1.49036e9 - 5.56209e9i) q^{83} +(2.76716e8 - 2.92094e8i) q^{84} +(-2.15904e9 + 7.63995e8i) q^{85} +(-1.99516e9 - 3.45571e9i) q^{86} +(6.89968e9 + 3.73864e9i) q^{87} +(1.17250e8 + 4.37582e8i) q^{88} +5.67071e9i q^{89} +(-5.38921e8 + 4.14047e9i) q^{90} -8.78889e8 q^{91} +(2.72268e9 - 7.29540e8i) q^{92} +(-1.35596e9 - 2.20851e9i) q^{93} +(-6.53078e9 + 3.77055e9i) q^{94} +(-2.89751e9 - 1.38286e9i) q^{95} +(-3.35295e8 + 1.40185e9i) q^{96} +(-4.47584e9 + 1.19930e9i) q^{97} +(-4.35227e9 + 4.35227e9i) q^{98} +(-1.04477e9 + 2.05911e9i) 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
\(3\) −242.911 + 6.56726i −0.999635 + 0.0270258i
\(4\) 443.405 − 256.000i 0.433013 − 0.250000i
\(5\) −1346.00 + 2820.27i −0.430719 + 0.902486i
\(6\) −5270.71 + 1566.12i −0.677817 + 0.201405i
\(7\) −3123.77 + 837.012i −0.185861 + 0.0498014i −0.350549 − 0.936544i \(-0.614005\pi\)
0.164688 + 0.986346i \(0.447338\pi\)
\(8\) 8192.00 − 8192.00i 0.250000 − 0.250000i
\(9\) 58962.7 − 3190.52i 0.998539 − 0.0540318i
\(10\) −12902.0 + 69523.7i −0.129020 + 0.695237i
\(11\) −19551.5 + 33864.2i −0.121400 + 0.210270i −0.920320 − 0.391167i \(-0.872072\pi\)
0.798920 + 0.601437i \(0.205405\pi\)
\(12\) −106027. + 65097.2i −0.426098 + 0.261611i
\(13\) 262508. + 70338.8i 0.707011 + 0.189443i 0.594369 − 0.804193i \(-0.297402\pi\)
0.112642 + 0.993636i \(0.464069\pi\)
\(14\) −63372.5 + 36588.1i −0.117831 + 0.0680299i
\(15\) 308436. − 693915.i 0.406171 − 0.913797i
\(16\) 131072. − 227023.i 0.125000 − 0.216506i
\(17\) 518218. + 518218.i 0.364979 + 0.364979i 0.865642 − 0.500663i \(-0.166910\pi\)
−0.500663 + 0.865642i \(0.666910\pi\)
\(18\) 1.27003e6 − 415043.i 0.672126 − 0.219650i
\(19\) 1.02739e6i 0.414922i 0.978243 + 0.207461i \(0.0665200\pi\)
−0.978243 + 0.207461i \(0.933480\pi\)
\(20\) 125167. + 1.59510e6i 0.0391147 + 0.498468i
\(21\) 753302. − 223834.i 0.184447 − 0.0548062i
\(22\) −229003. + 854652.i −0.0444353 + 0.165835i
\(23\) 5.31774e6 + 1.42488e6i 0.826205 + 0.221381i 0.647057 − 0.762441i \(-0.275999\pi\)
0.179148 + 0.983822i \(0.442666\pi\)
\(24\) −1.93613e6 + 2.04373e6i −0.243152 + 0.256665i
\(25\) −6.14221e6 − 7.59215e6i −0.628962 − 0.777436i
\(26\) 6.14942e6 0.517568
\(27\) −1.43018e7 + 1.16224e6i −0.996714 + 0.0809984i
\(28\) −1.17082e6 + 1.17082e6i −0.0680299 + 0.0680299i
\(29\) −2.79676e7 − 1.61471e7i −1.36353 − 0.787235i −0.373438 − 0.927655i \(-0.621821\pi\)
−0.990092 + 0.140420i \(0.955155\pi\)
\(30\) 2.67746e6 − 1.69728e7i 0.110184 − 0.698469i
\(31\) 5.33240e6 + 9.23599e6i 0.186258 + 0.322608i 0.944000 − 0.329947i \(-0.107031\pi\)
−0.757742 + 0.652554i \(0.773697\pi\)
\(32\) 1.53522e6 − 5.72953e6i 0.0457532 − 0.170753i
\(33\) 4.52689e6 − 8.35441e6i 0.115673 − 0.213474i
\(34\) 1.43613e7 + 8.29149e6i 0.316081 + 0.182490i
\(35\) 1.84399e6 − 9.93649e6i 0.0351089 − 0.189188i
\(36\) 2.53276e7 − 1.65092e7i 0.418872 − 0.273031i
\(37\) 1.27072e7 + 1.27072e7i 0.183249 + 0.183249i 0.792770 − 0.609521i \(-0.208638\pi\)
−0.609521 + 0.792770i \(0.708638\pi\)
\(38\) 6.01680e6 + 2.24550e7i 0.0759359 + 0.283397i
\(39\) −6.42281e7 − 1.53621e7i −0.711872 − 0.170266i
\(40\) 1.20772e7 + 3.41301e7i 0.117942 + 0.333301i
\(41\) 4.62860e7 + 8.01698e7i 0.399513 + 0.691977i 0.993666 − 0.112375i \(-0.0358459\pi\)
−0.594153 + 0.804352i \(0.702513\pi\)
\(42\) 1.51536e7 − 9.30385e6i 0.115950 − 0.0711896i
\(43\) −4.56424e7 − 1.70340e8i −0.310474 − 1.15871i −0.928130 − 0.372257i \(-0.878584\pi\)
0.617655 − 0.786449i \(-0.288083\pi\)
\(44\) 2.00208e7i 0.121400i
\(45\) −7.03655e7 + 1.70585e8i −0.381327 + 0.924440i
\(46\) 1.24571e8 0.604824
\(47\) −3.21917e8 + 8.62573e7i −1.40364 + 0.376103i −0.879649 − 0.475624i \(-0.842222\pi\)
−0.523986 + 0.851727i \(0.675556\pi\)
\(48\) −3.03479e7 + 5.60073e7i −0.119103 + 0.219805i
\(49\) −2.35573e8 + 1.36008e8i −0.833961 + 0.481488i
\(50\) −1.78709e8 − 1.29966e8i −0.571870 − 0.415891i
\(51\) −1.29284e8 − 1.22478e8i −0.374710 − 0.354982i
\(52\) 1.34404e8 − 3.60135e7i 0.353505 − 0.0947215i
\(53\) −5.47941e8 + 5.47941e8i −1.31025 + 1.31025i −0.389023 + 0.921228i \(0.627187\pi\)
−0.921228 + 0.389023i \(0.872813\pi\)
\(54\) −3.05779e8 + 1.09159e8i −0.665945 + 0.237734i
\(55\) −6.91900e7 − 1.00722e8i −0.137477 − 0.200129i
\(56\) −1.87331e7 + 3.24467e7i −0.0340150 + 0.0589157i
\(57\) −6.74712e6 − 2.49564e8i −0.0112136 − 0.414770i
\(58\) −7.05835e8 − 1.89128e8i −1.07538 − 0.288148i
\(59\) −2.50481e8 + 1.44615e8i −0.350360 + 0.202281i −0.664844 − 0.746982i \(-0.731502\pi\)
0.314484 + 0.949263i \(0.398169\pi\)
\(60\) −4.08799e7 − 3.86645e8i −0.0525719 − 0.497229i
\(61\) 9.60234e7 − 1.66317e8i 0.113692 − 0.196919i −0.803564 − 0.595218i \(-0.797066\pi\)
0.917256 + 0.398298i \(0.130399\pi\)
\(62\) 1.70637e8 + 1.70637e8i 0.186258 + 0.186258i
\(63\) −1.81516e8 + 5.93190e7i −0.182899 + 0.0597710i
\(64\) − 1.34218e8i − 0.125000i
\(65\) −5.51710e8 + 6.45668e8i −0.475493 + 0.556471i
\(66\) 5.00148e7 − 2.09109e8i 0.0399373 − 0.166975i
\(67\) 1.13336e7 − 4.22976e7i 0.00839449 − 0.0313287i −0.961602 − 0.274449i \(-0.911504\pi\)
0.969996 + 0.243121i \(0.0781711\pi\)
\(68\) 3.62445e8 + 9.71167e7i 0.249285 + 0.0667958i
\(69\) −1.30110e9 − 3.11197e8i −0.831886 − 0.198971i
\(70\) −1.78892e7 − 2.27975e8i −0.0106439 − 0.135643i
\(71\) −7.40304e8 −0.410316 −0.205158 − 0.978729i \(-0.565771\pi\)
−0.205158 + 0.978729i \(0.565771\pi\)
\(72\) 4.56886e8 − 5.09160e8i 0.236127 − 0.263143i
\(73\) −9.14452e8 + 9.14452e8i −0.441110 + 0.441110i −0.892385 − 0.451275i \(-0.850969\pi\)
0.451275 + 0.892385i \(0.350969\pi\)
\(74\) 3.52153e8 + 2.03316e8i 0.158698 + 0.0916246i
\(75\) 1.54187e9 + 1.80388e9i 0.649743 + 0.760154i
\(76\) 2.63011e8 + 4.55549e8i 0.103730 + 0.179666i
\(77\) 3.27297e7 − 1.22149e8i 0.0120917 − 0.0451270i
\(78\) −1.49376e9 + 4.03848e7i −0.517379 + 0.0139877i
\(79\) −3.43216e8 − 1.98156e8i −0.111540 − 0.0643978i 0.443192 − 0.896427i \(-0.353846\pi\)
−0.554732 + 0.832029i \(0.687179\pi\)
\(80\) 4.63844e8 + 6.75231e8i 0.141554 + 0.206064i
\(81\) 3.46643e9 − 3.76244e8i 0.994161 − 0.107906i
\(82\) 1.48115e9 + 1.48115e9i 0.399513 + 0.399513i
\(83\) −1.49036e9 − 5.56209e9i −0.378355 − 1.41204i −0.848380 − 0.529388i \(-0.822421\pi\)
0.470024 − 0.882654i \(-0.344245\pi\)
\(84\) 2.76716e8 − 2.92094e8i 0.0661665 − 0.0698436i
\(85\) −2.15904e9 + 7.63995e8i −0.486592 + 0.172185i
\(86\) −1.99516e9 − 3.45571e9i −0.424116 − 0.734590i
\(87\) 6.89968e9 + 3.73864e9i 1.38431 + 0.750097i
\(88\) 1.17250e8 + 4.37582e8i 0.0222177 + 0.0829175i
\(89\) 5.67071e9i 1.01552i 0.861500 + 0.507758i \(0.169526\pi\)
−0.861500 + 0.507758i \(0.830474\pi\)
\(90\) −5.38921e8 + 4.14047e9i −0.0912668 + 0.701192i
\(91\) −8.78889e8 −0.140840
\(92\) 2.72268e9 − 7.29540e8i 0.413102 − 0.110690i
\(93\) −1.35596e9 − 2.20851e9i −0.194908 − 0.317456i
\(94\) −6.53078e9 + 3.77055e9i −0.889869 + 0.513766i
\(95\) −2.89751e9 − 1.38286e9i −0.374461 − 0.178715i
\(96\) −3.35295e8 + 1.40185e9i −0.0411217 + 0.171927i
\(97\) −4.47584e9 + 1.19930e9i −0.521214 + 0.139659i −0.509828 − 0.860276i \(-0.670291\pi\)
−0.0113860 + 0.999935i \(0.503624\pi\)
\(98\) −4.35227e9 + 4.35227e9i −0.481488 + 0.481488i
\(99\) −1.04477e9 + 2.05911e9i −0.109861 + 0.216523i
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.7.1 ✓ 120
5.3 odd 4 inner 90.11.k.b.43.16 yes 120
9.4 even 3 inner 90.11.k.b.67.16 yes 120
45.13 odd 12 inner 90.11.k.b.13.1 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.k.b.7.1 ✓ 120 1.1 even 1 trivial
90.11.k.b.13.1 yes 120 45.13 odd 12 inner
90.11.k.b.43.16 yes 120 5.3 odd 4 inner
90.11.k.b.67.16 yes 120 9.4 even 3 inner