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Properties

Label 90.11.k.b.7.17
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 7.17
Character \(\chi\) \(=\) 90.7
Dual form 90.11.k.b.13.17

$q$-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} +(51.0847 + 237.570i) q^{3} +(443.405 - 256.000i) q^{4} +(2874.90 + 1224.99i) q^{5} +(2507.83 + 4893.25i) q^{6} +(2311.84 - 619.455i) q^{7} +(8192.00 - 8192.00i) q^{8} +(-53829.7 + 24272.3i) q^{9} +(70008.9 + 9937.28i) q^{10} +(-23303.1 + 40362.2i) q^{11} +(83469.0 + 92261.9i) q^{12} +(126849. + 33989.1i) q^{13} +(46900.7 - 27078.1i) q^{14} +(-144157. + 745566. i) q^{15} +(131072. - 227023. i) q^{16} +(1.09845e6 + 1.09845e6i) q^{17} +(-1.03438e6 + 845755. i) q^{18} +2.23790e6i q^{19} +(1.58834e6 - 192808. i) q^{20} +(265263. + 517578. i) q^{21} +(-272945. + 1.01864e6i) q^{22} +(-1.12938e6 - 302616. i) q^{23} +(2.36466e6 + 1.52769e6i) q^{24} +(6.76443e6 + 7.04343e6i) q^{25} +2.97152e6 q^{26} +(-8.51625e6 - 1.15484e7i) q^{27} +(866500. - 866500. i) q^{28} +(388401. + 224243. i) q^{29} +(1.21559e6 + 1.71396e7i) q^{30} +(-5.98502e6 - 1.03664e7i) q^{31} +(1.53522e6 - 5.72953e6i) q^{32} +(-1.07793e7 - 3.47422e6i) q^{33} +(3.04411e7 + 1.75752e7i) q^{34} +(7.40512e6 + 1.05110e6i) q^{35} +(-1.76546e7 + 2.45429e7i) q^{36} +(-8.02234e6 - 8.02234e6i) q^{37} +(1.31061e7 + 4.89125e7i) q^{38} +(-1.59474e6 + 3.18718e7i) q^{39} +(3.35863e7 - 1.35160e7i) q^{40} +(7.85470e7 + 1.36047e8i) q^{41} +(8.82885e6 + 9.75890e6i) q^{42} +(-1.22806e7 - 4.58320e7i) q^{43} +2.38624e7i q^{44} +(-1.84488e8 + 3.83973e6i) q^{45} -2.64564e7 q^{46} +(2.33686e7 - 6.26160e6i) q^{47} +(6.06296e7 + 1.95413e7i) q^{48} +(-2.39670e8 + 1.38373e8i) q^{49} +(1.89095e8 + 1.14329e8i) q^{50} +(-2.04844e8 + 3.17072e8i) q^{51} +(6.49467e7 - 1.74024e7i) q^{52} +(-8.65044e7 + 8.65044e7i) q^{53} +(-2.53766e8 - 2.02531e8i) q^{54} +(-1.16437e8 + 8.74911e7i) q^{55} +(1.38640e7 - 2.40132e7i) q^{56} +(-5.31658e8 + 1.14323e8i) q^{57} +(9.80231e6 + 2.62652e6i) q^{58} +(-5.34647e8 + 3.08679e8i) q^{59} +(1.26945e8 + 3.67492e8i) q^{60} +(-2.29630e8 + 3.97730e8i) q^{61} +(-1.91521e8 - 1.91521e8i) q^{62} +(-1.09410e8 + 8.94588e7i) q^{63} -1.34218e8i q^{64} +(3.23042e8 + 2.53104e8i) q^{65} +(-2.55942e8 - 1.28063e7i) q^{66} +(-1.11785e8 + 4.17186e8i) q^{67} +(7.68260e8 + 2.05855e8i) q^{68} +(1.41985e7 - 2.83765e8i) q^{69} +(1.68005e8 - 2.03940e7i) q^{70} +2.02305e9 q^{71} +(-2.42134e8 + 6.39812e8i) q^{72} +(1.14924e9 - 1.14924e9i) q^{73} +(-2.22322e8 - 1.28357e8i) q^{74} +(-1.32775e9 + 1.96684e9i) q^{75} +(5.72903e8 + 9.92297e8i) q^{76} +(-2.88705e7 + 1.07746e8i) q^{77} +(1.51799e8 + 7.05943e8i) q^{78} +(-5.99873e8 - 3.46337e8i) q^{79} +(6.54919e8 - 4.92107e8i) q^{80} +(2.30849e9 - 2.61315e9i) q^{81} +(2.51350e9 + 2.51350e9i) q^{82} +(1.00853e9 + 3.76387e9i) q^{83} +(2.50119e8 + 1.61589e8i) q^{84} +(1.81234e9 + 4.50351e9i) q^{85} +(-5.36822e8 - 9.29802e8i) q^{86} +(-3.34321e7 + 1.03728e8i) q^{87} +(1.39748e8 + 5.21546e8i) q^{88} +1.69928e9i q^{89} +(-4.00976e9 + 1.16436e9i) q^{90} +3.14309e8 q^{91} +(-5.78242e8 + 1.54940e8i) q^{92} +(2.15699e9 - 1.95142e9i) q^{93} +(4.74083e8 - 2.73712e8i) q^{94} +(-2.74140e9 + 6.43374e9i) q^{95} +(1.43959e9 + 7.20311e7i) q^{96} +(-1.07792e10 + 2.88827e9i) q^{97} +(-4.42795e9 + 4.42795e9i) q^{98} +(2.74715e8 - 2.73831e9i) 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
\(3\) 51.0847 + 237.570i 0.210225 + 0.977653i
\(4\) 443.405 − 256.000i 0.433013 − 0.250000i
\(5\) 2874.90 + 1224.99i 0.919967 + 0.391996i
\(6\) 2507.83 + 4893.25i 0.322509 + 0.629276i
\(7\) 2311.84 − 619.455i 0.137552 − 0.0368570i −0.189386 − 0.981903i \(-0.560650\pi\)
0.326938 + 0.945046i \(0.393983\pi\)
\(8\) 8192.00 − 8192.00i 0.250000 − 0.250000i
\(9\) −53829.7 + 24272.3i −0.911611 + 0.411054i
\(10\) 70008.9 + 9937.28i 0.700089 + 0.0993728i
\(11\) −23303.1 + 40362.2i −0.144694 + 0.250617i −0.929259 − 0.369430i \(-0.879553\pi\)
0.784565 + 0.620047i \(0.212886\pi\)
\(12\) 83469.0 + 92261.9i 0.335443 + 0.370780i
\(13\) 126849. + 33989.1i 0.341641 + 0.0915425i 0.425560 − 0.904930i \(-0.360077\pi\)
−0.0839190 + 0.996473i \(0.526744\pi\)
\(14\) 46900.7 − 27078.1i 0.0872045 − 0.0503476i
\(15\) −144157. + 745566.i −0.189836 + 0.981816i
\(16\) 131072. − 227023.i 0.125000 − 0.216506i
\(17\) 1.09845e6 + 1.09845e6i 0.773633 + 0.773633i 0.978740 − 0.205106i \(-0.0657541\pi\)
−0.205106 + 0.978740i \(0.565754\pi\)
\(18\) −1.03438e6 + 845755.i −0.547414 + 0.447592i
\(19\) 2.23790e6i 0.903802i 0.892068 + 0.451901i \(0.149254\pi\)
−0.892068 + 0.451901i \(0.850746\pi\)
\(20\) 1.58834e6 − 192808.i 0.496356 − 0.0602524i
\(21\) 265263. + 517578.i 0.0649502 + 0.126730i
\(22\) −272945. + 1.01864e6i −0.0529617 + 0.197656i
\(23\) −1.12938e6 − 302616.i −0.175469 − 0.0470168i 0.170015 − 0.985442i \(-0.445618\pi\)
−0.345484 + 0.938425i \(0.612285\pi\)
\(24\) 2.36466e6 + 1.52769e6i 0.296970 + 0.191857i
\(25\) 6.76443e6 + 7.04343e6i 0.692678 + 0.721247i
\(26\) 2.97152e6 0.250099
\(27\) −8.51625e6 − 1.15484e7i −0.593512 − 0.804825i
\(28\) 866500. − 866500.i 0.0503476 − 0.0503476i
\(29\) 388401. + 224243.i 0.0189361 + 0.0109328i 0.509438 − 0.860507i \(-0.329853\pi\)
−0.490502 + 0.871440i \(0.663187\pi\)
\(30\) 1.21559e6 + 1.71396e7i 0.0500242 + 0.705335i
\(31\) −5.98502e6 − 1.03664e7i −0.209054 − 0.362091i 0.742363 − 0.669998i \(-0.233705\pi\)
−0.951417 + 0.307906i \(0.900372\pi\)
\(32\) 1.53522e6 − 5.72953e6i 0.0457532 − 0.170753i
\(33\) −1.07793e7 − 3.47422e6i −0.275435 − 0.0887745i
\(34\) 3.04411e7 + 1.75752e7i 0.669986 + 0.386817i
\(35\) 7.40512e6 + 1.05110e6i 0.140991 + 0.0200127i
\(36\) −1.76546e7 + 2.45429e7i −0.291976 + 0.405894i
\(37\) −8.02234e6 − 8.02234e6i −0.115689 − 0.115689i 0.646892 − 0.762581i \(-0.276068\pi\)
−0.762581 + 0.646892i \(0.776068\pi\)
\(38\) 1.31061e7 + 4.89125e7i 0.165407 + 0.617308i
\(39\) −1.59474e6 + 3.18718e7i −0.0176753 + 0.353251i
\(40\) 3.35863e7 − 1.35160e7i 0.327991 − 0.131993i
\(41\) 7.85470e7 + 1.36047e8i 0.677969 + 1.17428i 0.975591 + 0.219594i \(0.0704731\pi\)
−0.297622 + 0.954684i \(0.596194\pi\)
\(42\) 8.82885e6 + 9.75890e6i 0.0675550 + 0.0746715i
\(43\) −1.22806e7 − 4.58320e7i −0.0835370 − 0.311764i 0.911496 − 0.411309i \(-0.134928\pi\)
−0.995033 + 0.0995443i \(0.968262\pi\)
\(44\) 2.38624e7i 0.144694i
\(45\) −1.84488e8 + 3.83973e6i −0.999783 + 0.0208084i
\(46\) −2.64564e7 −0.128452
\(47\) 2.33686e7 − 6.26160e6i 0.101893 − 0.0273021i −0.207512 − 0.978232i \(-0.566537\pi\)
0.309405 + 0.950930i \(0.399870\pi\)
\(48\) 6.06296e7 + 1.95413e7i 0.237946 + 0.0766916i
\(49\) −2.39670e8 + 1.38373e8i −0.848463 + 0.489860i
\(50\) 1.89095e8 + 1.14329e8i 0.605105 + 0.365852i
\(51\) −2.04844e8 + 3.17072e8i −0.593708 + 0.918982i
\(52\) 6.49467e7 − 1.74024e7i 0.170821 − 0.0457713i
\(53\) −8.65044e7 + 8.65044e7i −0.206852 + 0.206852i −0.802928 − 0.596076i \(-0.796726\pi\)
0.596076 + 0.802928i \(0.296726\pi\)
\(54\) −2.53766e8 − 2.02531e8i −0.552669 − 0.441086i
\(55\) −1.16437e8 + 8.74911e7i −0.231355 + 0.173840i
\(56\) 1.38640e7 − 2.40132e7i 0.0251738 − 0.0436023i
\(57\) −5.31658e8 + 1.14323e8i −0.883605 + 0.190002i
\(58\) 9.80231e6 + 2.62652e6i 0.0149344 + 0.00400167i
\(59\) −5.34647e8 + 3.08679e8i −0.747837 + 0.431764i −0.824912 − 0.565261i \(-0.808775\pi\)
0.0770746 + 0.997025i \(0.475442\pi\)
\(60\) 1.26945e8 + 3.67492e8i 0.163252 + 0.472598i
\(61\) −2.29630e8 + 3.97730e8i −0.271881 + 0.470912i −0.969343 − 0.245710i \(-0.920979\pi\)
0.697463 + 0.716621i \(0.254312\pi\)
\(62\) −1.91521e8 − 1.91521e8i −0.209054 − 0.209054i
\(63\) −1.09410e8 + 8.94588e7i −0.110244 + 0.0901406i
\(64\) − 1.34218e8i − 0.125000i
\(65\) 3.23042e8 + 2.53104e8i 0.278414 + 0.218138i
\(66\) −2.55942e8 − 1.28063e7i −0.204373 − 0.0102260i
\(67\) −1.11785e8 + 4.17186e8i −0.0827959 + 0.308998i −0.994888 − 0.100988i \(-0.967800\pi\)
0.912092 + 0.409986i \(0.134466\pi\)
\(68\) 7.68260e8 + 2.05855e8i 0.528401 + 0.141585i
\(69\) 1.41985e7 − 2.83765e8i 0.00907812 − 0.181432i
\(70\) 1.68005e8 − 2.03940e7i 0.0999613 − 0.0121342i
\(71\) 2.02305e9 1.12128 0.560641 − 0.828059i \(-0.310555\pi\)
0.560641 + 0.828059i \(0.310555\pi\)
\(72\) −2.42134e8 + 6.39812e8i −0.125139 + 0.330666i
\(73\) 1.14924e9 − 1.14924e9i 0.554367 − 0.554367i −0.373331 − 0.927698i \(-0.621785\pi\)
0.927698 + 0.373331i \(0.121785\pi\)
\(74\) −2.22322e8 − 1.28357e8i −0.100190 − 0.0578446i
\(75\) −1.32775e9 + 1.96684e9i −0.559511 + 0.828823i
\(76\) 5.72903e8 + 9.92297e8i 0.225950 + 0.391358i
\(77\) −2.88705e7 + 1.07746e8i −0.0106660 + 0.0398059i
\(78\) 1.51799e8 + 7.05943e8i 0.0525770 + 0.244510i
\(79\) −5.99873e8 − 3.46337e8i −0.194950 − 0.112555i 0.399348 − 0.916800i \(-0.369237\pi\)
−0.594298 + 0.804245i \(0.702570\pi\)
\(80\) 6.54919e8 − 4.92107e8i 0.199866 − 0.150179i
\(81\) 2.30849e9 − 2.61315e9i 0.662069 − 0.749443i
\(82\) 2.51350e9 + 2.51350e9i 0.677969 + 0.677969i
\(83\) 1.00853e9 + 3.76387e9i 0.256033 + 0.955530i 0.967513 + 0.252823i \(0.0813589\pi\)
−0.711479 + 0.702707i \(0.751974\pi\)
\(84\) 2.50119e8 + 1.61589e8i 0.0598068 + 0.0386381i
\(85\) 1.81234e9 + 4.50351e9i 0.408456 + 1.01498i
\(86\) −5.36822e8 − 9.29802e8i −0.114114 − 0.197651i
\(87\) −3.34321e7 + 1.03728e8i −0.00670760 + 0.0208113i
\(88\) 1.39748e8 + 5.21546e8i 0.0264808 + 0.0988279i
\(89\) 1.69928e9i 0.304310i 0.988357 + 0.152155i \(0.0486213\pi\)
−0.988357 + 0.152155i \(0.951379\pi\)
\(90\) −4.00976e9 + 1.16436e9i −0.679057 + 0.197185i
\(91\) 3.14309e8 0.0503675
\(92\) −5.78242e8 + 1.54940e8i −0.0877346 + 0.0235084i
\(93\) 2.15699e9 − 1.95142e9i 0.310051 − 0.280502i
\(94\) 4.74083e8 − 2.73712e8i 0.0645974 − 0.0372953i
\(95\) −2.74140e9 + 6.43374e9i −0.354287 + 0.831468i
\(96\) 1.43959e9 + 7.20311e7i 0.176556 + 0.00883413i
\(97\) −1.07792e10 + 2.88827e9i −1.25524 + 0.336341i −0.824359 − 0.566068i \(-0.808464\pi\)
−0.430882 + 0.902408i \(0.641797\pi\)
\(98\) −4.42795e9 + 4.42795e9i −0.489860 + 0.489860i
\(99\) 2.74715e8 − 2.73831e9i 0.0288873 − 0.287943i
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.7.17 ✓ 120
5.3 odd 4 inner 90.11.k.b.43.2 yes 120
9.4 even 3 inner 90.11.k.b.67.2 yes 120
45.13 odd 12 inner 90.11.k.b.13.17 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.k.b.7.17 ✓ 120 1.1 even 1 trivial
90.11.k.b.13.17 yes 120 45.13 odd 12 inner
90.11.k.b.43.2 yes 120 5.3 odd 4 inner
90.11.k.b.67.2 yes 120 9.4 even 3 inner