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Properties

Label 90.11.k.b.7.20
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 7.20
Character \(\chi\) \(=\) 90.7
Dual form 90.11.k.b.13.20

$q$-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} +(92.2595 - 224.805i) q^{3} +(443.405 - 256.000i) q^{4} +(-2292.39 - 2123.81i) q^{5} +(699.913 - 5453.73i) q^{6} +(25517.9 - 6837.50i) q^{7} +(8192.00 - 8192.00i) q^{8} +(-42025.4 - 41480.8i) q^{9} +(-62541.3 - 32993.8i) q^{10} +(84880.3 - 147017. i) q^{11} +(-16641.7 - 123298. i) q^{12} +(-542922. - 145476. i) q^{13} +(517686. - 298886. i) q^{14} +(-688938. + 319398. i) q^{15} +(131072. - 227023. i) q^{16} +(1.09286e6 + 1.09286e6i) q^{17} +(-1.16145e6 - 660503. i) q^{18} -2.35972e6i q^{19} +(-1.56015e6 - 354858. i) q^{20} +(817166. - 6.36737e6i) q^{21} +(994187. - 3.71036e6i) q^{22} +(1.12261e7 + 3.00802e6i) q^{23} +(-1.08581e6 - 2.59739e6i) q^{24} +(744459. + 9.73721e6i) q^{25} -1.27183e7 q^{26} +(-1.32023e7 + 5.62051e6i) q^{27} +(9.56436e6 - 9.56436e6i) q^{28} +(-1.23521e7 - 7.13146e6i) q^{29} +(-1.31872e7 + 1.10156e7i) q^{30} +(-1.56289e7 - 2.70700e7i) q^{31} +(1.53522e6 - 5.72953e6i) q^{32} +(-2.52191e7 - 3.26452e7i) q^{33} +(3.02862e7 + 1.74858e7i) q^{34} +(-7.30185e7 - 3.85210e7i) q^{35} +(-2.92533e7 - 7.63428e6i) q^{36} +(4.30969e7 + 4.30969e7i) q^{37} +(-1.38195e7 - 5.15750e7i) q^{38} +(-8.27933e7 + 1.08630e8i) q^{39} +(-3.61775e7 + 1.38096e6i) q^{40} +(-6.68501e7 - 1.15788e8i) q^{41} +(-1.94296e7 - 1.43953e8i) q^{42} +(3.41621e7 + 1.27495e8i) q^{43} -8.69174e7i q^{44} +(8.24107e6 + 1.84344e8i) q^{45} +2.62978e8 q^{46} +(6.30596e6 - 1.68968e6i) q^{47} +(-3.89433e7 - 5.04107e7i) q^{48} +(3.59781e8 - 2.07720e8i) q^{49} +(7.32962e7 + 2.08461e8i) q^{50} +(3.46507e8 - 1.44853e8i) q^{51} +(-2.77976e8 + 7.44835e7i) q^{52} +(-4.67760e8 + 4.67760e8i) q^{53} +(-2.55639e8 + 2.00162e8i) q^{54} +(-5.06815e8 + 1.56750e8i) q^{55} +(1.53030e8 - 2.65055e8i) q^{56} +(-5.30476e8 - 2.17707e8i) q^{57} +(-3.11736e8 - 8.35295e7i) q^{58} +(-5.25035e8 + 3.03129e8i) q^{59} +(-2.23713e8 + 3.17991e8i) q^{60} +(2.01568e8 - 3.49126e8i) q^{61} +(-5.00124e8 - 5.00124e8i) q^{62} +(-1.35602e9 - 7.71153e8i) q^{63} -1.34218e8i q^{64} +(9.35625e8 + 1.48655e9i) q^{65} +(-7.42383e8 - 5.65814e8i) q^{66} +(-5.84742e8 + 2.18229e9i) q^{67} +(7.64351e8 + 2.04807e8i) q^{68} +(1.71193e9 - 2.24616e9i) q^{69} +(-1.82152e9 - 4.14306e8i) q^{70} -3.16551e9 q^{71} +(-6.84082e8 + 4.46151e6i) q^{72} +(1.41464e9 - 1.41464e9i) q^{73} +(1.19434e9 + 6.89550e8i) q^{74} +(2.25765e9 + 7.30992e8i) q^{75} +(-6.04088e8 - 1.04631e9i) q^{76} +(1.16074e9 - 4.33193e9i) q^{77} +(-1.17338e9 + 2.85913e9i) q^{78} +(3.03726e9 + 1.75356e9i) q^{79} +(-7.82623e8 + 2.42053e8i) q^{80} +(4.54789e7 + 3.48649e9i) q^{81} +(-2.13920e9 - 2.13920e9i) q^{82} +(-3.40982e8 - 1.27256e9i) q^{83} +(-1.26771e9 - 3.03252e9i) q^{84} +(-1.84228e8 - 4.82629e9i) q^{85} +(1.49332e9 + 2.58651e9i) q^{86} +(-2.74278e9 + 2.11886e9i) q^{87} +(-5.09024e8 - 1.89970e9i) q^{88} -3.73345e9i q^{89} +(1.25971e9 + 3.98083e9i) q^{90} -1.48489e10 q^{91} +(5.74776e9 - 1.54011e9i) q^{92} +(-7.52738e9 + 1.01598e9i) q^{93} +(1.27930e8 - 7.38605e7i) q^{94} +(-5.01161e9 + 5.40939e9i) q^{95} +(-1.14639e9 - 8.73728e8i) q^{96} +(4.58868e9 - 1.22953e9i) q^{97} +(6.64703e9 - 6.64703e9i) q^{98} +(-9.66550e9 + 2.65754e9i) 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
\(3\) 92.2595 − 224.805i 0.379669 − 0.925123i
\(4\) 443.405 − 256.000i 0.433013 − 0.250000i
\(5\) −2292.39 − 2123.81i −0.733564 − 0.679620i
\(6\) 699.913 − 5453.73i 0.0900094 − 0.701355i
\(7\) 25517.9 − 6837.50i 1.51829 − 0.406824i 0.599111 − 0.800666i \(-0.295521\pi\)
0.919178 + 0.393841i \(0.128854\pi\)
\(8\) 8192.00 − 8192.00i 0.250000 − 0.250000i
\(9\) −42025.4 − 41480.8i −0.711703 − 0.702480i
\(10\) −62541.3 − 32993.8i −0.625413 − 0.329938i
\(11\) 84880.3 − 147017.i 0.527040 − 0.912860i −0.472464 − 0.881350i \(-0.656635\pi\)
0.999503 − 0.0315096i \(-0.0100315\pi\)
\(12\) −16641.7 − 123298.i −0.0668792 − 0.495507i
\(13\) −542922. − 145476.i −1.46225 − 0.391808i −0.561982 − 0.827150i \(-0.689961\pi\)
−0.900265 + 0.435342i \(0.856628\pi\)
\(14\) 517686. − 298886.i 0.962557 − 0.555733i
\(15\) −688938. + 319398.i −0.907243 + 0.420606i
\(16\) 131072. − 227023.i 0.125000 − 0.216506i
\(17\) 1.09286e6 + 1.09286e6i 0.769697 + 0.769697i 0.978053 − 0.208356i \(-0.0668113\pi\)
−0.208356 + 0.978053i \(0.566811\pi\)
\(18\) −1.16145e6 − 660503.i −0.614665 − 0.349552i
\(19\) − 2.35972e6i − 0.952999i −0.879175 − 0.476500i \(-0.841905\pi\)
0.879175 − 0.476500i \(-0.158095\pi\)
\(20\) −1.56015e6 − 354858.i −0.487548 − 0.110893i
\(21\) 817166. − 6.36737e6i 0.200085 − 1.55906i
\(22\) 994187. − 3.71036e6i 0.192910 − 0.719950i
\(23\) 1.12261e7 + 3.00802e6i 1.74417 + 0.467350i 0.983367 − 0.181630i \(-0.0581372\pi\)
0.760806 + 0.648979i \(0.224804\pi\)
\(24\) −1.08581e6 − 2.59739e6i −0.136363 − 0.326198i
\(25\) 744459. + 9.73721e6i 0.0762326 + 0.997090i
\(26\) −1.27183e7 −1.07044
\(27\) −1.32023e7 + 5.62051e6i −0.920092 + 0.391703i
\(28\) 9.56436e6 − 9.56436e6i 0.555733 − 0.555733i
\(29\) −1.23521e7 − 7.13146e6i −0.602212 − 0.347687i 0.167699 − 0.985838i \(-0.446366\pi\)
−0.769911 + 0.638151i \(0.779700\pi\)
\(30\) −1.31872e7 + 1.10156e7i −0.542682 + 0.453316i
\(31\) −1.56289e7 − 2.70700e7i −0.545908 − 0.945541i −0.998549 − 0.0538479i \(-0.982851\pi\)
0.452641 − 0.891693i \(-0.350482\pi\)
\(32\) 1.53522e6 − 5.72953e6i 0.0457532 − 0.170753i
\(33\) −2.52191e7 − 3.26452e7i −0.644407 − 0.834161i
\(34\) 3.02862e7 + 1.74858e7i 0.666577 + 0.384848i
\(35\) −7.30185e7 − 3.85210e7i −1.39025 − 0.733429i
\(36\) −2.92533e7 − 7.63428e6i −0.483797 − 0.126257i
\(37\) 4.30969e7 + 4.30969e7i 0.621495 + 0.621495i 0.945914 − 0.324419i \(-0.105169\pi\)
−0.324419 + 0.945914i \(0.605169\pi\)
\(38\) −1.38195e7 − 5.15750e7i −0.174411 − 0.650911i
\(39\) −8.27933e7 + 1.08630e8i −0.917640 + 1.20400i
\(40\) −3.61775e7 + 1.38096e6i −0.353296 + 0.0134860i
\(41\) −6.68501e7 − 1.15788e8i −0.577010 − 0.999410i −0.995820 − 0.0913363i \(-0.970886\pi\)
0.418811 − 0.908074i \(-0.362447\pi\)
\(42\) −1.94296e7 − 1.43953e8i −0.148668 − 1.10148i
\(43\) 3.41621e7 + 1.27495e8i 0.232382 + 0.867260i 0.979312 + 0.202357i \(0.0648602\pi\)
−0.746930 + 0.664902i \(0.768473\pi\)
\(44\) − 8.69174e7i − 0.527040i
\(45\) 8.24107e6 + 1.84344e8i 0.0446603 + 0.999002i
\(46\) 2.62978e8 1.27682
\(47\) 6.30596e6 − 1.68968e6i 0.0274955 − 0.00736740i −0.245045 − 0.969512i \(-0.578803\pi\)
0.272541 + 0.962144i \(0.412136\pi\)
\(48\) −3.89433e7 − 5.04107e7i −0.152836 − 0.197841i
\(49\) 3.59781e8 − 2.07720e8i 1.27367 − 0.735355i
\(50\) 7.32962e7 + 2.08461e8i 0.234548 + 0.667074i
\(51\) 3.46507e8 − 1.44853e8i 1.00429 − 0.419834i
\(52\) −2.77976e8 + 7.44835e7i −0.731124 + 0.195904i
\(53\) −4.67760e8 + 4.67760e8i −1.11852 + 1.11852i −0.126562 + 0.991959i \(0.540394\pi\)
−0.991959 + 0.126562i \(0.959606\pi\)
\(54\) −2.55639e8 + 2.00162e8i −0.556748 + 0.435927i
\(55\) −5.06815e8 + 1.56750e8i −1.00702 + 0.311454i
\(56\) 1.53030e8 − 2.65055e8i 0.277866 − 0.481279i
\(57\) −5.30476e8 − 2.17707e8i −0.881641 − 0.361824i
\(58\) −3.11736e8 − 8.35295e7i −0.474949 − 0.127262i
\(59\) −5.25035e8 + 3.03129e8i −0.734392 + 0.424001i −0.820027 − 0.572325i \(-0.806042\pi\)
0.0856347 + 0.996327i \(0.472708\pi\)
\(60\) −2.23713e8 + 3.17991e8i −0.287696 + 0.408939i
\(61\) 2.01568e8 − 3.49126e8i 0.238656 − 0.413364i −0.721673 − 0.692234i \(-0.756627\pi\)
0.960329 + 0.278870i \(0.0899599\pi\)
\(62\) −5.00124e8 − 5.00124e8i −0.545908 − 0.545908i
\(63\) −1.35602e9 − 7.71153e8i −1.36636 − 0.777030i
\(64\) − 1.34218e8i − 0.125000i
\(65\) 9.35625e8 + 1.48655e9i 0.806371 + 1.28119i
\(66\) −7.42383e8 − 5.65814e8i −0.592800 − 0.451808i
\(67\) −5.84742e8 + 2.18229e9i −0.433102 + 1.61636i 0.312467 + 0.949929i \(0.398845\pi\)
−0.745568 + 0.666429i \(0.767822\pi\)
\(68\) 7.64351e8 + 2.04807e8i 0.525713 + 0.140864i
\(69\) 1.71193e9 − 2.24616e9i 1.09456 − 1.43614i
\(70\) −1.82152e9 − 4.14306e8i −1.08378 − 0.246508i
\(71\) −3.16551e9 −1.75449 −0.877247 − 0.480039i \(-0.840623\pi\)
−0.877247 + 0.480039i \(0.840623\pi\)
\(72\) −6.84082e8 + 4.46151e6i −0.353546 + 0.00230579i
\(73\) 1.41464e9 − 1.41464e9i 0.682387 − 0.682387i −0.278150 − 0.960538i \(-0.589721\pi\)
0.960538 + 0.278150i \(0.0897213\pi\)
\(74\) 1.19434e9 + 6.89550e8i 0.538230 + 0.310747i
\(75\) 2.25765e9 + 7.30992e8i 0.951374 + 0.308039i
\(76\) −6.04088e8 − 1.04631e9i −0.238250 − 0.412661i
\(77\) 1.16074e9 − 4.33193e9i 0.428825 − 1.60040i
\(78\) −1.17338e9 + 2.85913e9i −0.406412 + 0.990287i
\(79\) 3.03726e9 + 1.75356e9i 0.987067 + 0.569884i 0.904396 − 0.426693i \(-0.140322\pi\)
0.0826709 + 0.996577i \(0.473655\pi\)
\(80\) −7.82623e8 + 2.42053e8i −0.238838 + 0.0738688i
\(81\) 4.54789e7 + 3.48649e9i 0.0130432 + 0.999915i
\(82\) −2.13920e9 − 2.13920e9i −0.577010 − 0.577010i
\(83\) −3.40982e8 − 1.27256e9i −0.0865647 − 0.323064i 0.909041 − 0.416706i \(-0.136816\pi\)
−0.995606 + 0.0936424i \(0.970149\pi\)
\(84\) −1.26771e9 − 3.03252e9i −0.303126 − 0.725115i
\(85\) −1.84228e8 − 4.82629e9i −0.0415204 − 1.08772i
\(86\) 1.49332e9 + 2.58651e9i 0.317439 + 0.549821i
\(87\) −2.74278e9 + 2.11886e9i −0.550294 + 0.425114i
\(88\) −5.09024e8 − 1.89970e9i −0.0964550 − 0.359975i
\(89\) − 3.73345e9i − 0.668591i −0.942468 − 0.334296i \(-0.891502\pi\)
0.942468 − 0.334296i \(-0.108498\pi\)
\(90\) 1.25971e9 + 3.98083e9i 0.213334 + 0.674158i
\(91\) −1.48489e10 −2.37951
\(92\) 5.74776e9 − 1.54011e9i 0.872086 − 0.233675i
\(93\) −7.52738e9 + 1.01598e9i −1.08201 + 0.146040i
\(94\) 1.27930e8 − 7.38605e7i 0.0174315 − 0.0100641i
\(95\) −5.01161e9 + 5.40939e9i −0.647678 + 0.699086i
\(96\) −1.14639e9 − 8.73728e8i −0.140597 − 0.107157i
\(97\) 4.58868e9 − 1.22953e9i 0.534354 − 0.143180i 0.0184561 − 0.999830i \(-0.494125\pi\)
0.515898 + 0.856650i \(0.327458\pi\)
\(98\) 6.64703e9 − 6.64703e9i 0.735355 − 0.735355i
\(99\) −9.66550e9 + 2.65754e9i −1.01636 + 0.279450i
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.7.20 ✓ 120
5.3 odd 4 inner 90.11.k.b.43.26 yes 120
9.4 even 3 inner 90.11.k.b.67.26 yes 120
45.13 odd 12 inner 90.11.k.b.13.20 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.k.b.7.20 ✓ 120 1.1 even 1 trivial
90.11.k.b.13.20 yes 120 45.13 odd 12 inner
90.11.k.b.43.26 yes 120 5.3 odd 4 inner
90.11.k.b.67.26 yes 120 9.4 even 3 inner