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Properties

Label 90.11.k.b.7.12
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 7.12
Character \(\chi\) \(=\) 90.7
Dual form 90.11.k.b.13.12

$q$-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} +(-101.430 - 220.819i) q^{3} +(443.405 - 256.000i) q^{4} +(-3032.36 + 755.269i) q^{5} +(-3510.10 - 4232.29i) q^{6} +(-7026.82 + 1882.83i) q^{7} +(8192.00 - 8192.00i) q^{8} +(-38473.0 + 44795.3i) q^{9} +(-61853.3 + 34266.2i) q^{10} +(-14557.8 + 25214.8i) q^{11} +(-101504. - 71946.2i) q^{12} +(-91628.2 - 24551.7i) q^{13} +(-142554. + 82303.8i) q^{14} +(474349. + 592995. i) q^{15} +(131072. - 227023. i) q^{16} +(323896. + 323896. i) q^{17} +(-578542. + 1.20438e6i) q^{18} +754003. i q^{19} +(-1.15121e6 + 1.11117e6i) q^{20} +(1.12849e6 + 1.36068e6i) q^{21} +(-170513. + 636362. i) q^{22} +(-1.52055e6 - 407431. i) q^{23} +(-2.63986e6 - 978035. i) q^{24} +(8.62476e6 - 4.58049e6i) q^{25} -2.14645e6 q^{26} +(1.37939e7 + 3.95198e6i) q^{27} +(-2.63372e6 + 2.63372e6i) q^{28} +(8.82897e6 + 5.09741e6i) q^{29} +(1.38404e7 + 1.01828e7i) q^{30} +(677625. + 1.17368e6i) q^{31} +(1.53522e6 - 5.72953e6i) q^{32} +(7.04451e6 + 657098. i) q^{33} +(8.97607e6 + 5.18233e6i) q^{34} +(1.98858e7 - 1.10166e7i) q^{35} +(-5.59152e6 + 2.97115e7i) q^{36} +(4.84815e7 + 4.84815e7i) q^{37} +(4.41575e6 + 1.64798e7i) q^{38} +(3.87235e6 + 2.27235e7i) q^{39} +(-1.86539e7 + 3.10282e7i) q^{40} +(-1.48198e7 - 2.56686e7i) q^{41} +(3.26335e7 + 2.31306e7i) q^{42} +(-4.87871e7 - 1.82076e8i) q^{43} +1.49072e7i q^{44} +(8.28313e7 - 1.64893e8i) q^{45} -3.56199e7 q^{46} +(1.66596e8 - 4.46392e7i) q^{47} +(-6.34257e7 - 5.91623e6i) q^{48} +(-1.98800e8 + 1.14777e8i) q^{49} +(1.61681e8 - 1.50623e8i) q^{50} +(3.86696e7 - 1.04375e8i) q^{51} +(-4.69136e7 + 1.25705e7i) q^{52} +(2.49088e8 - 2.49088e8i) q^{53} +(3.24631e8 + 5.59316e6i) q^{54} +(2.51004e7 - 8.74555e7i) q^{55} +(-4.21396e7 + 7.29878e7i) q^{56} +(1.66498e8 - 7.64785e7i) q^{57} +(2.22822e8 + 5.97050e7i) q^{58} +(1.03683e9 - 5.98613e8i) q^{59} +(3.62136e8 + 1.41504e8i) q^{60} +(7.44219e7 - 1.28902e8i) q^{61} +(2.16840e7 + 2.16840e7i) q^{62} +(1.86001e8 - 3.87206e8i) q^{63} -1.34218e8i q^{64} +(2.96393e8 + 5.24558e6i) q^{65} +(1.57816e8 - 2.68937e7i) q^{66} +(9.08225e7 - 3.38954e8i) q^{67} +(2.26534e8 + 6.06997e7i) q^{68} +(6.42611e7 + 3.77093e8i) q^{69} +(3.70114e8 - 3.57242e8i) q^{70} -6.71664e8 q^{71} +(5.17922e7 + 6.82133e8i) q^{72} +(-1.00757e9 + 1.00757e9i) q^{73} +(1.34356e9 + 7.75703e8i) q^{74} +(-1.88627e9 - 1.43991e9i) q^{75} +(1.93025e8 + 3.34329e8i) q^{76} +(5.48197e7 - 2.04590e8i) q^{77} +(2.17714e8 + 4.73976e8i) q^{78} +(4.73012e9 + 2.73094e9i) q^{79} +(-2.25993e8 + 7.87411e8i) q^{80} +(-5.26446e8 - 3.44681e9i) q^{81} +(-4.74233e8 - 4.74233e8i) q^{82} +(1.22660e9 + 4.57772e9i) q^{83} +(8.48714e8 + 3.14438e8i) q^{84} +(-1.22680e9 - 7.37540e8i) q^{85} +(-2.13262e9 - 3.69381e9i) q^{86} +(2.30083e8 - 2.46663e9i) q^{87} +(8.73025e7 + 3.25817e8i) q^{88} +4.53366e9i q^{89} +(8.44716e8 - 4.08906e9i) q^{90} +6.90081e8 q^{91} +(-7.78524e8 + 2.08605e8i) q^{92} +(1.90440e8 - 2.68679e8i) q^{93} +(3.37976e9 - 1.95130e9i) q^{94} +(-5.69476e8 - 2.28641e9i) q^{95} +(-1.42090e9 + 2.42139e8i) q^{96} +(5.07990e9 - 1.36115e9i) q^{97} +(-3.67286e9 + 3.67286e9i) q^{98} +(-5.69424e8 - 1.62221e9i) 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
\(3\) −101.430 − 220.819i −0.417407 − 0.908720i
\(4\) 443.405 − 256.000i 0.433013 − 0.250000i
\(5\) −3032.36 + 755.269i −0.970354 + 0.241686i
\(6\) −3510.10 − 4232.29i −0.451401 − 0.544276i
\(7\) −7026.82 + 1882.83i −0.418089 + 0.112027i −0.461730 − 0.887021i \(-0.652771\pi\)
0.0436408 + 0.999047i \(0.486104\pi\)
\(8\) 8192.00 − 8192.00i 0.250000 − 0.250000i
\(9\) −38473.0 + 44795.3i −0.651543 + 0.758612i
\(10\) −61853.3 + 34266.2i −0.618533 + 0.342662i
\(11\) −14557.8 + 25214.8i −0.0903924 + 0.156564i −0.907676 − 0.419671i \(-0.862145\pi\)
0.817284 + 0.576235i \(0.195479\pi\)
\(12\) −101504. − 71946.2i −0.407922 − 0.289135i
\(13\) −91628.2 − 24551.7i −0.246781 − 0.0661249i 0.133308 − 0.991075i \(-0.457440\pi\)
−0.380089 + 0.924950i \(0.624107\pi\)
\(14\) −142554. + 82303.8i −0.265058 + 0.153031i
\(15\) 474349. + 592995.i 0.624658 + 0.780899i
\(16\) 131072. − 227023.i 0.125000 − 0.216506i
\(17\) 323896. + 323896.i 0.228119 + 0.228119i 0.811906 − 0.583788i \(-0.198430\pi\)
−0.583788 + 0.811906i \(0.698430\pi\)
\(18\) −578542. + 1.20438e6i −0.306177 + 0.637382i
\(19\) 754003.i 0.304513i 0.988341 + 0.152256i \(0.0486539\pi\)
−0.988341 + 0.152256i \(0.951346\pi\)
\(20\) −1.15121e6 + 1.11117e6i −0.359754 + 0.347242i
\(21\) 1.12849e6 + 1.36068e6i 0.276314 + 0.333165i
\(22\) −170513. + 636362.i −0.0330859 + 0.123478i
\(23\) −1.52055e6 − 407431.i −0.236245 − 0.0633017i 0.138754 − 0.990327i \(-0.455690\pi\)
−0.374999 + 0.927025i \(0.622357\pi\)
\(24\) −2.63986e6 − 978035.i −0.331532 − 0.122828i
\(25\) 8.62476e6 − 4.58049e6i 0.883176 − 0.469042i
\(26\) −2.14645e6 −0.180656
\(27\) 1.37939e7 + 3.95198e6i 0.961324 + 0.275420i
\(28\) −2.63372e6 + 2.63372e6i −0.153031 + 0.153031i
\(29\) 8.82897e6 + 5.09741e6i 0.430447 + 0.248519i 0.699537 − 0.714596i \(-0.253390\pi\)
−0.269090 + 0.963115i \(0.586723\pi\)
\(30\) 1.38404e7 + 1.01828e7i 0.569563 + 0.419044i
\(31\) 677625. + 1.17368e6i 0.0236691 + 0.0409960i 0.877617 − 0.479362i \(-0.159132\pi\)
−0.853948 + 0.520358i \(0.825799\pi\)
\(32\) 1.53522e6 − 5.72953e6i 0.0457532 − 0.170753i
\(33\) 7.04451e6 + 657098.i 0.180003 + 0.0167904i
\(34\) 8.97607e6 + 5.18233e6i 0.197557 + 0.114059i
\(35\) 1.98858e7 − 1.10166e7i 0.378619 − 0.209752i
\(36\) −5.59152e6 + 2.97115e7i −0.0924735 + 0.491374i
\(37\) 4.84815e7 + 4.84815e7i 0.699145 + 0.699145i 0.964226 − 0.265081i \(-0.0853988\pi\)
−0.265081 + 0.964226i \(0.585399\pi\)
\(38\) 4.41575e6 + 1.64798e7i 0.0557297 + 0.207986i
\(39\) 3.87235e6 + 2.27235e7i 0.0429193 + 0.251856i
\(40\) −1.86539e7 + 3.10282e7i −0.182167 + 0.303010i
\(41\) −1.48198e7 − 2.56686e7i −0.127915 − 0.221556i 0.794953 − 0.606670i \(-0.207495\pi\)
−0.922869 + 0.385115i \(0.874162\pi\)
\(42\) 3.26335e7 + 2.31306e7i 0.249699 + 0.176987i
\(43\) −4.87871e7 − 1.82076e8i −0.331866 − 1.23854i −0.907227 − 0.420642i \(-0.861805\pi\)
0.575360 − 0.817900i \(-0.304862\pi\)
\(44\) 1.49072e7i 0.0903924i
\(45\) 8.28313e7 − 1.64893e8i 0.448882 − 0.893591i
\(46\) −3.56199e7 −0.172943
\(47\) 1.66596e8 − 4.46392e7i 0.726398 − 0.194638i 0.123373 − 0.992360i \(-0.460629\pi\)
0.603024 + 0.797723i \(0.293962\pi\)
\(48\) −6.34257e7 − 5.91623e6i −0.248919 − 0.0232187i
\(49\) −1.98800e8 + 1.14777e8i −0.703777 + 0.406326i
\(50\) 1.61681e8 − 1.50623e8i 0.517379 − 0.481994i
\(51\) 3.86696e7 − 1.04375e8i 0.112078 − 0.302514i
\(52\) −4.69136e7 + 1.25705e7i −0.123391 + 0.0330624i
\(53\) 2.49088e8 − 2.49088e8i 0.595625 − 0.595625i −0.343520 − 0.939145i \(-0.611619\pi\)
0.939145 + 0.343520i \(0.111619\pi\)
\(54\) 3.24631e8 + 5.59316e6i 0.707002 + 0.0121812i
\(55\) 2.51004e7 − 8.74555e7i 0.0498733 − 0.173769i
\(56\) −4.21396e7 + 7.29878e7i −0.0765156 + 0.132529i
\(57\) 1.66498e8 − 7.64785e7i 0.276717 − 0.127106i
\(58\) 2.22822e8 + 5.97050e7i 0.339483 + 0.0909642i
\(59\) 1.03683e9 − 5.98613e8i 1.45026 − 0.837310i 0.451768 − 0.892136i \(-0.350794\pi\)
0.998496 + 0.0548253i \(0.0174602\pi\)
\(60\) 3.62136e8 + 1.41504e8i 0.465709 + 0.181975i
\(61\) 7.44219e7 − 1.28902e8i 0.0881153 − 0.152620i −0.818599 − 0.574365i \(-0.805249\pi\)
0.906714 + 0.421745i \(0.138582\pi\)
\(62\) 2.16840e7 + 2.16840e7i 0.0236691 + 0.0236691i
\(63\) 1.86001e8 − 3.87206e8i 0.187418 − 0.390157i
\(64\) − 1.34218e8i − 0.125000i
\(65\) 2.96393e8 + 5.24558e6i 0.255447 + 0.00452092i
\(66\) 1.57816e8 − 2.68937e7i 0.126018 − 0.0214749i
\(67\) 9.08225e7 − 3.38954e8i 0.0672697 − 0.251054i −0.924100 − 0.382151i \(-0.875183\pi\)
0.991370 + 0.131097i \(0.0418500\pi\)
\(68\) 2.26534e8 + 6.06997e7i 0.155808 + 0.0417486i
\(69\) 6.42611e7 + 3.77093e8i 0.0410868 + 0.241103i
\(70\) 3.70114e8 − 3.57242e8i 0.220214 − 0.212555i
\(71\) −6.71664e8 −0.372272 −0.186136 − 0.982524i \(-0.559597\pi\)
−0.186136 + 0.982524i \(0.559597\pi\)
\(72\) 5.17922e7 + 6.82133e8i 0.0267671 + 0.352539i
\(73\) −1.00757e9 + 1.00757e9i −0.486026 + 0.486026i −0.907050 − 0.421024i \(-0.861671\pi\)
0.421024 + 0.907050i \(0.361671\pi\)
\(74\) 1.34356e9 + 7.75703e8i 0.605477 + 0.349572i
\(75\) −1.88627e9 − 1.43991e9i −0.794872 − 0.606778i
\(76\) 1.93025e8 + 3.34329e8i 0.0761282 + 0.131858i
\(77\) 5.48197e7 − 2.04590e8i 0.0202527 − 0.0755841i
\(78\) 2.17714e8 + 4.73976e8i 0.0754073 + 0.164166i
\(79\) 4.73012e9 + 2.73094e9i 1.53722 + 0.887516i 0.999000 + 0.0447132i \(0.0142374\pi\)
0.538223 + 0.842803i \(0.319096\pi\)
\(80\) −2.25993e8 + 7.87411e8i −0.0689677 + 0.240299i
\(81\) −5.26446e8 − 3.44681e9i −0.150983 − 0.988536i
\(82\) −4.74233e8 − 4.74233e8i −0.127915 − 0.127915i
\(83\) 1.22660e9 + 4.57772e9i 0.311394 + 1.16214i 0.927300 + 0.374320i \(0.122124\pi\)
−0.615905 + 0.787820i \(0.711210\pi\)
\(84\) 8.48714e8 + 3.14438e8i 0.202939 + 0.0751862i
\(85\) −1.22680e9 − 7.37540e8i −0.276489 − 0.166223i
\(86\) −2.13262e9 − 3.69381e9i −0.453338 − 0.785204i
\(87\) 2.30083e8 − 2.46663e9i 0.0461623 − 0.494889i
\(88\) 8.73025e7 + 3.25817e8i 0.0165430 + 0.0617392i
\(89\) 4.53366e9i 0.811893i 0.913897 + 0.405947i \(0.133058\pi\)
−0.913897 + 0.405947i \(0.866942\pi\)
\(90\) 8.44716e8 − 4.08906e9i 0.143053 − 0.692485i
\(91\) 6.90081e8 0.110584
\(92\) −7.78524e8 + 2.08605e8i −0.118123 + 0.0316508i
\(93\) 1.90440e8 − 2.68679e8i 0.0273743 − 0.0386206i
\(94\) 3.37976e9 − 1.95130e9i 0.460518 − 0.265880i
\(95\) −5.69476e8 − 2.28641e9i −0.0735965 − 0.295485i
\(96\) −1.42090e9 + 2.42139e8i −0.174264 + 0.0296967i
\(97\) 5.07990e9 − 1.36115e9i 0.591556 − 0.158507i 0.0493919 − 0.998779i \(-0.484272\pi\)
0.542164 + 0.840272i \(0.317605\pi\)
\(98\) −3.67286e9 + 3.67286e9i −0.406326 + 0.406326i
\(99\) −5.69424e8 − 1.62221e9i −0.0598769 − 0.170581i
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.7.12 ✓ 120
5.3 odd 4 inner 90.11.k.b.43.25 yes 120
9.4 even 3 inner 90.11.k.b.67.25 yes 120
45.13 odd 12 inner 90.11.k.b.13.12 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.k.b.7.12 ✓ 120 1.1 even 1 trivial
90.11.k.b.13.12 yes 120 45.13 odd 12 inner
90.11.k.b.43.25 yes 120 5.3 odd 4 inner
90.11.k.b.67.25 yes 120 9.4 even 3 inner