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Properties

Label 90.11.k.a.13.20
Level $90$
Weight $11$
Character 90.13
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 13.20
Character \(\chi\) \(=\) 90.13
Dual form 90.11.k.a.7.20

$q$-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+(-21.8564 - 5.85641i) q^{2} +(99.4483 - 221.718i) q^{3} +(443.405 + 256.000i) q^{4} +(2332.72 - 2079.43i) q^{5} +(-3472.06 + 4263.56i) q^{6} +(-14654.4 - 3926.63i) q^{7} +(-8192.00 - 8192.00i) q^{8} +(-39269.1 - 44099.0i) q^{9} +(-63162.9 + 31787.5i) q^{10} +(-46415.7 - 80394.3i) q^{11} +(100856. - 72852.3i) q^{12} +(145153. - 38893.5i) q^{13} +(297296. + 171644. i) q^{14} +(-229062. - 724003. i) q^{15} +(131072. + 227023. i) q^{16} +(-1.77443e6 + 1.77443e6i) q^{17} +(600019. + 1.19382e6i) q^{18} +2.76506e6i q^{19} +(1.56668e6 - 324853. i) q^{20} +(-2.32796e6 + 2.85865e6i) q^{21} +(543658. + 2.02896e6i) q^{22} +(3.05717e6 - 819166. i) q^{23} +(-2.63100e6 + 1.00164e6i) q^{24} +(1.11757e6 - 9.70147e6i) q^{25} -3.40029e6 q^{26} +(-1.36828e7 + 4.32110e6i) q^{27} +(-5.49261e6 - 5.49261e6i) q^{28} +(-1.91807e7 + 1.10740e7i) q^{29} +(766425. + 1.71656e7i) q^{30} +(8.59491e6 - 1.48868e7i) q^{31} +(-1.53522e6 - 5.72953e6i) q^{32} +(-2.24409e7 + 2.29613e6i) q^{33} +(4.91745e7 - 2.83909e7i) q^{34} +(-4.23498e7 + 2.13130e7i) q^{35} +(-6.12275e6 - 2.96066e7i) q^{36} +(-9.36260e6 + 9.36260e6i) q^{37} +(1.61933e7 - 6.04342e7i) q^{38} +(5.81177e6 - 3.60509e7i) q^{39} +(-3.61444e7 - 2.07498e6i) q^{40} +(-1.57869e7 + 2.73438e7i) q^{41} +(6.76222e7 - 4.88463e7i) q^{42} +(1.14579e7 - 4.27613e7i) q^{43} -4.75297e7i q^{44} +(-1.83305e8 - 2.12135e7i) q^{45} -7.16161e7 q^{46} +(1.26971e8 + 3.40217e7i) q^{47} +(6.33701e7 - 6.48398e6i) q^{48} +(-4.52983e7 - 2.61530e7i) q^{49} +(-8.12418e7 + 2.05494e8i) q^{50} +(2.16960e8 + 5.69888e8i) q^{51} +(7.43181e7 + 1.99135e7i) q^{52} +(-2.72683e8 - 2.72683e8i) q^{53} +(3.24363e8 - 1.43117e7i) q^{54} +(-2.75449e8 - 9.10195e7i) q^{55} +(8.78817e7 + 1.52216e8i) q^{56} +(6.13064e8 + 2.74980e8i) q^{57} +(4.84074e8 - 1.29707e8i) q^{58} +(-6.99718e8 - 4.03982e8i) q^{59} +(8.37774e7 - 3.79667e8i) q^{60} +(-1.04310e7 - 1.80670e7i) q^{61} +(-2.75037e8 + 2.75037e8i) q^{62} +(4.02303e8 + 8.00439e8i) q^{63} +1.34218e8i q^{64} +(2.57724e8 - 3.92563e8i) q^{65} +(5.03924e8 + 8.12376e7i) q^{66} +(3.57696e8 + 1.33494e9i) q^{67} +(-1.24105e9 + 3.32537e8i) q^{68} +(1.22406e8 - 7.59295e8i) q^{69} +(1.05043e9 - 2.17809e8i) q^{70} +1.86850e9 q^{71} +(-3.95670e7 + 6.82951e8i) q^{72} +(2.74111e9 + 2.74111e9i) q^{73} +(2.59464e8 - 1.49802e8i) q^{74} +(-2.03985e9 - 1.21258e9i) q^{75} +(-7.07855e8 + 1.22604e9i) q^{76} +(3.64514e8 + 1.36039e9i) q^{77} +(-3.38153e8 + 7.53907e8i) q^{78} +(-4.86726e9 + 2.81011e9i) q^{79} +(7.77834e8 + 2.57028e8i) q^{80} +(-4.02664e8 + 3.46346e9i) q^{81} +(5.05182e8 - 5.05182e8i) q^{82} +(-7.82332e8 + 2.91970e9i) q^{83} +(-1.76404e9 + 6.71582e8i) q^{84} +(-4.49451e8 + 7.82906e9i) q^{85} +(-5.00855e8 + 8.67507e8i) q^{86} +(5.47817e8 + 5.35400e9i) q^{87} +(-2.78353e8 + 1.03883e9i) q^{88} -2.36379e9i q^{89} +(3.88215e9 + 1.53716e9i) q^{90} -2.27984e9 q^{91} +(1.56527e9 + 4.19413e8i) q^{92} +(-2.44593e9 - 3.38612e9i) q^{93} +(-2.57588e9 - 1.48718e9i) q^{94} +(5.74974e9 + 6.45011e9i) q^{95} +(-1.42302e9 - 2.29405e8i) q^{96} +(1.36821e10 + 3.66612e9i) q^{97} +(8.36895e8 + 8.36895e8i) q^{98} +(-1.72261e9 + 5.20390e9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q - 960 q^{2} + 252 q^{3} - 16128 q^{6} - 1830 q^{7} - 983040 q^{8} - 105792 q^{10} + 217740 q^{11} + 129024 q^{12} + 410742 q^{15} + 15728640 q^{16} + 2438244 q^{17} + 1427904 q^{18} + 1692672 q^{20}+ \cdots - 74083744896 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 99.4483 − 221.718i 0.409252 − 0.912421i
\(4\) 443.405 + 256.000i 0.433013 + 0.250000i
\(5\) 2332.72 − 2079.43i 0.746471 − 0.665418i
\(6\) −3472.06 + 4263.56i −0.446509 + 0.548297i
\(7\) −14654.4 − 3926.63i −0.871921 − 0.233631i −0.205003 − 0.978761i \(-0.565720\pi\)
−0.666918 + 0.745131i \(0.732387\pi\)
\(8\) −8192.00 − 8192.00i −0.250000 − 0.250000i
\(9\) −39269.1 − 44099.0i −0.665025 − 0.746821i
\(10\) −63162.9 + 31787.5i −0.631629 + 0.317875i
\(11\) −46415.7 − 80394.3i −0.288205 − 0.499185i 0.685176 − 0.728377i \(-0.259725\pi\)
−0.973381 + 0.229192i \(0.926392\pi\)
\(12\) 100856. − 72852.3i 0.405317 − 0.292777i
\(13\) 145153. − 38893.5i 0.390938 − 0.104752i −0.0579950 − 0.998317i \(-0.518471\pi\)
0.448933 + 0.893565i \(0.351804\pi\)
\(14\) 297296. + 171644.i 0.552776 + 0.319145i
\(15\) −229062. − 724003.i −0.301646 − 0.953420i
\(16\) 131072. + 227023.i 0.125000 + 0.216506i
\(17\) −1.77443e6 + 1.77443e6i −1.24973 + 1.24973i −0.293884 + 0.955841i \(0.594948\pi\)
−0.955841 + 0.293884i \(0.905052\pi\)
\(18\) 600019. + 1.19382e6i 0.317543 + 0.631796i
\(19\) 2.76506e6i 1.11670i 0.829606 + 0.558350i \(0.188565\pi\)
−0.829606 + 0.558350i \(0.811435\pi\)
\(20\) 1.56668e6 − 324853.i 0.489586 − 0.101516i
\(21\) −2.32796e6 + 2.85865e6i −0.570005 + 0.699946i
\(22\) 543658. + 2.02896e6i 0.105490 + 0.393695i
\(23\) 3.05717e6 − 819166.i 0.474985 − 0.127272i −0.0133814 − 0.999910i \(-0.504260\pi\)
0.488367 + 0.872639i \(0.337593\pi\)
\(24\) −2.63100e6 + 1.00164e6i −0.330418 + 0.125792i
\(25\) 1.11757e6 − 9.70147e6i 0.114439 − 0.993430i
\(26\) −3.40029e6 −0.286187
\(27\) −1.36828e7 + 4.32110e6i −0.953578 + 0.301145i
\(28\) −5.49261e6 − 5.49261e6i −0.319145 − 0.319145i
\(29\) −1.91807e7 + 1.10740e7i −0.935134 + 0.539900i −0.888432 − 0.459009i \(-0.848205\pi\)
−0.0467026 + 0.998909i \(0.514871\pi\)
\(30\) 766425. + 1.71656e7i 0.0315401 + 0.706403i
\(31\) 8.59491e6 − 1.48868e7i 0.300215 − 0.519988i −0.675969 − 0.736930i \(-0.736275\pi\)
0.976185 + 0.216942i \(0.0696082\pi\)
\(32\) −1.53522e6 − 5.72953e6i −0.0457532 − 0.170753i
\(33\) −2.24409e7 + 2.29613e6i −0.573416 + 0.0586715i
\(34\) 4.91745e7 − 2.83909e7i 1.08229 − 0.624863i
\(35\) −4.23498e7 + 2.13130e7i −0.806326 + 0.405793i
\(36\) −6.12275e6 − 2.96066e7i −0.101259 − 0.489639i
\(37\) −9.36260e6 + 9.36260e6i −0.135017 + 0.135017i −0.771385 − 0.636368i \(-0.780436\pi\)
0.636368 + 0.771385i \(0.280436\pi\)
\(38\) 1.61933e7 − 6.04342e7i 0.204370 − 0.762720i
\(39\) 5.81177e6 − 3.60509e7i 0.0644148 − 0.399570i
\(40\) −3.61444e7 − 2.07498e6i −0.352972 − 0.0202634i
\(41\) −1.57869e7 + 2.73438e7i −0.136263 + 0.236015i −0.926079 − 0.377329i \(-0.876843\pi\)
0.789816 + 0.613344i \(0.210176\pi\)
\(42\) 6.76222e7 − 4.88463e7i 0.517420 − 0.373754i
\(43\) 1.14579e7 − 4.27613e7i 0.0779401 − 0.290877i −0.915944 − 0.401307i \(-0.868556\pi\)
0.993884 + 0.110430i \(0.0352228\pi\)
\(44\) − 4.75297e7i − 0.288205i
\(45\) −1.83305e8 − 2.12135e7i −0.993370 − 0.114961i
\(46\) −7.16161e7 −0.347713
\(47\) 1.26971e8 + 3.40217e7i 0.553623 + 0.148343i 0.524775 − 0.851241i \(-0.324149\pi\)
0.0288481 + 0.999584i \(0.490816\pi\)
\(48\) 6.33701e7 − 6.48398e6i 0.248702 − 0.0254470i
\(49\) −4.52983e7 − 2.61530e7i −0.160362 − 0.0925850i
\(50\) −8.12418e7 + 2.05494e8i −0.259974 + 0.657582i
\(51\) 2.16960e8 + 5.69888e8i 0.628823 + 1.65173i
\(52\) 7.43181e7 + 1.99135e7i 0.195469 + 0.0523758i
\(53\) −2.72683e8 − 2.72683e8i −0.652047 − 0.652047i 0.301439 − 0.953486i \(-0.402533\pi\)
−0.953486 + 0.301439i \(0.902533\pi\)
\(54\) 3.24363e8 − 1.43117e7i 0.706419 − 0.0311689i
\(55\) −2.75449e8 − 9.10195e7i −0.547303 − 0.180851i
\(56\) 8.78817e7 + 1.52216e8i 0.159573 + 0.276388i
\(57\) 6.13064e8 + 2.74980e8i 1.01890 + 0.457012i
\(58\) 4.84074e8 − 1.29707e8i 0.737517 − 0.197617i
\(59\) −6.99718e8 − 4.03982e8i −0.978730 − 0.565070i −0.0768438 − 0.997043i \(-0.524484\pi\)
−0.901887 + 0.431973i \(0.857818\pi\)
\(60\) 8.37774e7 − 3.79667e8i 0.107738 − 0.488254i
\(61\) −1.04310e7 − 1.80670e7i −0.0123503 − 0.0213913i 0.859784 − 0.510658i \(-0.170598\pi\)
−0.872134 + 0.489266i \(0.837265\pi\)
\(62\) −2.75037e8 + 2.75037e8i −0.300215 + 0.300215i
\(63\) 4.02303e8 + 8.00439e8i 0.405369 + 0.806539i
\(64\) 1.34218e8i 0.125000i
\(65\) 2.57724e8 − 3.92563e8i 0.222121 − 0.338331i
\(66\) 5.03924e8 + 8.12376e7i 0.402388 + 0.0648690i
\(67\) 3.57696e8 + 1.33494e9i 0.264936 + 0.988753i 0.962290 + 0.272026i \(0.0876938\pi\)
−0.697354 + 0.716727i \(0.745640\pi\)
\(68\) −1.24105e9 + 3.32537e8i −0.853578 + 0.228716i
\(69\) 1.22406e8 − 7.59295e8i 0.0782632 − 0.485473i
\(70\) 1.05043e9 − 2.17809e8i 0.624996 − 0.129594i
\(71\) 1.86850e9 1.03562 0.517812 − 0.855494i \(-0.326747\pi\)
0.517812 + 0.855494i \(0.326747\pi\)
\(72\) −3.95670e7 + 6.82951e8i −0.0204489 + 0.352962i
\(73\) 2.74111e9 + 2.74111e9i 1.32225 + 1.32225i 0.911954 + 0.410292i \(0.134573\pi\)
0.410292 + 0.911954i \(0.365427\pi\)
\(74\) 2.59464e8 − 1.49802e8i 0.116928 − 0.0675084i
\(75\) −2.03985e9 − 1.21258e9i −0.859593 − 0.510980i
\(76\) −7.07855e8 + 1.22604e9i −0.279175 + 0.483545i
\(77\) 3.64514e8 + 1.36039e9i 0.134667 + 0.502584i
\(78\) −3.38153e8 + 7.53907e8i −0.117123 + 0.261123i
\(79\) −4.86726e9 + 2.81011e9i −1.58179 + 0.913247i −0.587192 + 0.809448i \(0.699766\pi\)
−0.994598 + 0.103799i \(0.966900\pi\)
\(80\) 7.77834e8 + 2.57028e8i 0.237376 + 0.0784386i
\(81\) −4.02664e8 + 3.46346e9i −0.115483 + 0.993309i
\(82\) 5.05182e8 − 5.05182e8i 0.136263 − 0.136263i
\(83\) −7.82332e8 + 2.91970e9i −0.198610 + 0.741222i 0.792693 + 0.609621i \(0.208678\pi\)
−0.991303 + 0.131601i \(0.957988\pi\)
\(84\) −1.76404e9 + 6.71582e8i −0.421806 + 0.160584i
\(85\) −4.49451e8 + 7.82906e9i −0.101295 + 1.76447i
\(86\) −5.00855e8 + 8.67507e8i −0.106468 + 0.184408i
\(87\) 5.47817e8 + 5.35400e9i 0.109910 + 1.07419i
\(88\) −2.78353e8 + 1.03883e9i −0.0527451 + 0.196848i
\(89\) − 2.36379e9i − 0.423310i −0.977344 − 0.211655i \(-0.932115\pi\)
0.977344 − 0.211655i \(-0.0678853\pi\)
\(90\) 3.88215e9 + 1.53716e9i 0.657445 + 0.260319i
\(91\) −2.27984e9 −0.365341
\(92\) 1.56527e9 + 4.19413e8i 0.237493 + 0.0636360i
\(93\) −2.44593e9 − 3.38612e9i −0.351584 − 0.486729i
\(94\) −2.57588e9 − 1.48718e9i −0.350983 − 0.202640i
\(95\) 5.74974e9 + 6.45011e9i 0.743071 + 0.833584i
\(96\) −1.42302e9 − 2.29405e8i −0.174523 − 0.0281349i
\(97\) 1.36821e10 + 3.66612e9i 1.59329 + 0.426922i 0.943008 − 0.332769i \(-0.107983\pi\)
0.650285 + 0.759691i \(0.274650\pi\)
\(98\) 8.36895e8 + 8.36895e8i 0.0925850 + 0.0925850i
\(99\) −1.72261e9 + 5.20390e9i −0.181139 + 0.547208i
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.a.13.20 yes 120
5.2 odd 4 inner 90.11.k.a.67.5 yes 120
9.7 even 3 inner 90.11.k.a.43.5 yes 120
45.7 odd 12 inner 90.11.k.a.7.20 ✓ 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.k.a.7.20 ✓ 120 45.7 odd 12 inner
90.11.k.a.13.20 yes 120 1.1 even 1 trivial
90.11.k.a.43.5 yes 120 9.7 even 3 inner
90.11.k.a.67.5 yes 120 5.2 odd 4 inner