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Properties

Label 9.10.c.a.4.2
Level $9$
Weight $10$
Character 9.4
Analytic conductor $4.635$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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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] = [9,10,Mod(4,9)] 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("9.4"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 9.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.63532252547\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 1984 x^{14} - 13748 x^{13} + 1552498 x^{12} - 9136628 x^{11} + 609566956 x^{10} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{28}\cdot 17^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 4.2
Root \(0.500000 + 15.6774i\) of defining polynomial
Character \(\chi\) \(=\) 9.4
Dual form 9.10.c.a.7.2

$q$-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+(-12.8270 - 22.2170i) q^{2} +(-90.7998 - 106.950i) q^{3} +(-73.0637 + 126.550i) q^{4} +(-353.226 + 611.805i) q^{5} +(-1211.43 + 3389.15i) q^{6} +(2284.21 + 3956.36i) q^{7} -9386.09 q^{8} +(-3193.79 + 19422.2i) q^{9} +18123.3 q^{10} +(-35716.7 - 61863.1i) q^{11} +(20168.8 - 3676.53i) q^{12} +(13201.0 - 22864.9i) q^{13} +(58599.1 - 101497. i) q^{14} +(97505.6 - 17774.1i) q^{15} +(157804. + 273325. i) q^{16} -314759. q^{17} +(472469. - 178171. i) q^{18} -904019. q^{19} +(-51616.0 - 89401.5i) q^{20} +(215729. - 603534. i) q^{21} +(-916276. + 1.58704e6i) q^{22} +(55921.2 - 96858.3i) q^{23} +(852255. + 1.00385e6i) q^{24} +(727026. + 1.25925e6i) q^{25} -677318. q^{26} +(2.36720e6 - 1.42195e6i) q^{27} -667571. q^{28} +(-2.19402e6 - 3.80015e6i) q^{29} +(-1.64559e6 - 1.93829e6i) q^{30} +(4.94392e6 - 8.56313e6i) q^{31} +(1.64546e6 - 2.85003e6i) q^{32} +(-3.37322e6 + 9.43708e6i) q^{33} +(4.03742e6 + 6.99301e6i) q^{34} -3.22736e6 q^{35} +(-2.22452e6 - 1.82323e6i) q^{36} -6.32122e6 q^{37} +(1.15959e7 + 2.00846e7i) q^{38} +(-3.64406e6 + 664268. i) q^{39} +(3.31541e6 - 5.74246e6i) q^{40} +(-2.68019e6 + 4.64223e6i) q^{41} +(-1.61759e7 + 2.94867e6i) q^{42} +(-1.12275e7 - 1.94466e7i) q^{43} +1.04384e7 q^{44} +(-1.07544e7 - 8.81438e6i) q^{45} -2.86920e6 q^{46} +(2.23644e7 + 3.87363e7i) q^{47} +(1.49036e7 - 4.16950e7i) q^{48} +(9.74159e6 - 1.68729e7i) q^{49} +(1.86511e7 - 3.23047e7i) q^{50} +(2.85801e7 + 3.36636e7i) q^{51} +(1.92903e6 + 3.34118e6i) q^{52} -4.00972e7 q^{53} +(-6.19556e7 - 3.43528e7i) q^{54} +5.04642e7 q^{55} +(-2.14398e7 - 3.71348e7i) q^{56} +(8.20848e7 + 9.66852e7i) q^{57} +(-5.62853e7 + 9.74890e7i) q^{58} +(4.79118e7 - 8.29857e7i) q^{59} +(-4.87481e6 + 1.36380e7i) q^{60} +(-1.01113e7 - 1.75132e7i) q^{61} -2.53663e8 q^{62} +(-8.41364e7 + 3.17284e7i) q^{63} +7.71659e7 q^{64} +(9.32588e6 + 1.61529e7i) q^{65} +(2.52932e8 - 4.61065e7i) q^{66} +(-9.12489e7 + 1.58048e8i) q^{67} +(2.29975e7 - 3.98328e7i) q^{68} +(-1.54367e7 + 2.81392e6i) q^{69} +(4.13974e7 + 7.17024e7i) q^{70} +1.38543e8 q^{71} +(2.99772e7 - 1.82298e8i) q^{72} -2.17970e8 q^{73} +(8.10823e7 + 1.40439e8i) q^{74} +(6.86631e7 - 1.92095e8i) q^{75} +(6.60510e7 - 1.14404e8i) q^{76} +(1.63169e8 - 2.82617e8i) q^{77} +(6.15004e7 + 7.24395e7i) q^{78} +(-1.97294e8 - 3.41722e8i) q^{79} -2.22962e8 q^{80} +(-3.67020e8 - 1.24061e8i) q^{81} +1.37515e8 q^{82} +(2.36638e8 + 4.09870e8i) q^{83} +(6.06153e7 + 7.13970e7i) q^{84} +(1.11181e8 - 1.92571e8i) q^{85} +(-2.88030e8 + 4.98883e8i) q^{86} +(-2.07211e8 + 5.79704e8i) q^{87} +(3.35240e8 + 5.80653e8i) q^{88} +2.60507e8 q^{89} +(-5.78820e7 + 3.51994e8i) q^{90} +1.20616e8 q^{91} +(8.17161e6 + 1.41537e7i) q^{92} +(-1.36474e9 + 2.48775e8i) q^{93} +(5.73736e8 - 9.93741e8i) q^{94} +(3.19323e8 - 5.53083e8i) q^{95} +(-4.54219e8 + 8.27988e7i) q^{96} +(-2.87421e8 - 4.97828e8i) q^{97} -4.99822e8 q^{98} +(1.31559e9 - 4.96118e8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 15 q^{2} - 3 q^{3} - 1793 q^{4} + 453 q^{5} + 2439 q^{6} - 343 q^{7} - 14478 q^{8} - 15669 q^{9} + 1020 q^{10} + 99150 q^{11} - 241212 q^{12} + 32435 q^{13} + 394824 q^{14} + 723843 q^{15} - 328193 q^{16}+ \cdots + 1672014609 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −12.8270 − 22.2170i −0.566879 − 0.981862i −0.996872 − 0.0790305i \(-0.974818\pi\)
0.429994 − 0.902832i \(-0.358516\pi\)
\(3\) −90.7998 − 106.950i −0.647201 − 0.762319i
\(4\) −73.0637 + 126.550i −0.142703 + 0.247168i
\(5\) −353.226 + 611.805i −0.252748 + 0.437772i −0.964281 − 0.264880i \(-0.914668\pi\)
0.711534 + 0.702652i \(0.248001\pi\)
\(6\) −1211.43 + 3389.15i −0.381608 + 1.06760i
\(7\) 2284.21 + 3956.36i 0.359579 + 0.622809i 0.987891 − 0.155153i \(-0.0495870\pi\)
−0.628311 + 0.777962i \(0.716254\pi\)
\(8\) −9386.09 −0.810177
\(9\) −3193.79 + 19422.2i −0.162261 + 0.986748i
\(10\) 18123.3 0.573109
\(11\) −35716.7 − 61863.1i −0.735537 − 1.27399i −0.954487 − 0.298251i \(-0.903597\pi\)
0.218951 − 0.975736i \(-0.429737\pi\)
\(12\) 20168.8 − 3676.53i 0.280778 − 0.0511826i
\(13\) 13201.0 − 22864.9i 0.128193 − 0.222036i −0.794784 − 0.606893i \(-0.792416\pi\)
0.922976 + 0.384857i \(0.125749\pi\)
\(14\) 58599.1 − 101497.i 0.407675 − 0.706114i
\(15\) 97505.6 − 17774.1i 0.497301 − 0.0906520i
\(16\) 157804. + 273325.i 0.601975 + 1.04265i
\(17\) −314759. −0.914026 −0.457013 − 0.889460i \(-0.651081\pi\)
−0.457013 + 0.889460i \(0.651081\pi\)
\(18\) 472469. − 178171.i 1.06083 − 0.400048i
\(19\) −904019. −1.59143 −0.795713 − 0.605674i \(-0.792904\pi\)
−0.795713 + 0.605674i \(0.792904\pi\)
\(20\) −51616.0 − 89401.5i −0.0721355 − 0.124942i
\(21\) 215729. − 603534.i 0.242059 − 0.677197i
\(22\) −916276. + 1.58704e6i −0.833920 + 1.44439i
\(23\) 55921.2 − 96858.3i 0.0416678 − 0.0721708i −0.844439 − 0.535651i \(-0.820066\pi\)
0.886107 + 0.463480i \(0.153400\pi\)
\(24\) 852255. + 1.00385e6i 0.524347 + 0.617614i
\(25\) 727026. + 1.25925e6i 0.372237 + 0.644734i
\(26\) −677318. −0.290678
\(27\) 2.36720e6 − 1.42195e6i 0.857233 − 0.514929i
\(28\) −667571. −0.205251
\(29\) −2.19402e6 − 3.80015e6i −0.576035 − 0.997722i −0.995928 − 0.0901485i \(-0.971266\pi\)
0.419893 − 0.907573i \(-0.362067\pi\)
\(30\) −1.64559e6 − 1.93829e6i −0.370917 − 0.436892i
\(31\) 4.94392e6 − 8.56313e6i 0.961488 − 1.66535i 0.242721 − 0.970096i \(-0.421960\pi\)
0.718767 − 0.695251i \(-0.244707\pi\)
\(32\) 1.64546e6 − 2.85003e6i 0.277404 − 0.480478i
\(33\) −3.37322e6 + 9.43708e6i −0.495145 + 1.38524i
\(34\) 4.03742e6 + 6.99301e6i 0.518142 + 0.897448i
\(35\) −3.22736e6 −0.363531
\(36\) −2.22452e6 − 1.82323e6i −0.220737 − 0.180917i
\(37\) −6.32122e6 −0.554489 −0.277245 − 0.960799i \(-0.589421\pi\)
−0.277245 + 0.960799i \(0.589421\pi\)
\(38\) 1.15959e7 + 2.00846e7i 0.902145 + 1.56256i
\(39\) −3.64406e6 + 664268.i −0.252229 + 0.0459783i
\(40\) 3.31541e6 − 5.74246e6i 0.204770 − 0.354673i
\(41\) −2.68019e6 + 4.64223e6i −0.148128 + 0.256566i −0.930536 − 0.366201i \(-0.880658\pi\)
0.782407 + 0.622767i \(0.213992\pi\)
\(42\) −1.61759e7 + 2.94867e6i −0.802133 + 0.146219i
\(43\) −1.12275e7 − 1.94466e7i −0.500812 − 0.867432i −1.00000 0.000938191i \(-0.999701\pi\)
0.499187 − 0.866494i \(-0.333632\pi\)
\(44\) 1.04384e7 0.419852
\(45\) −1.07544e7 − 8.81438e6i −0.390959 − 0.320432i
\(46\) −2.86920e6 −0.0944824
\(47\) 2.23644e7 + 3.87363e7i 0.668524 + 1.15792i 0.978317 + 0.207114i \(0.0664069\pi\)
−0.309793 + 0.950804i \(0.600260\pi\)
\(48\) 1.49036e7 − 4.16950e7i 0.405234 − 1.13370i
\(49\) 9.74159e6 − 1.68729e7i 0.241406 − 0.418127i
\(50\) 1.86511e7 − 3.23047e7i 0.422027 − 0.730971i
\(51\) 2.85801e7 + 3.36636e7i 0.591559 + 0.696779i
\(52\) 1.92903e6 + 3.34118e6i 0.0365868 + 0.0633702i
\(53\) −4.00972e7 −0.698027 −0.349013 − 0.937118i \(-0.613483\pi\)
−0.349013 + 0.937118i \(0.613483\pi\)
\(54\) −6.19556e7 − 3.43528e7i −0.991537 − 0.549782i
\(55\) 5.04642e7 0.743621
\(56\) −2.14398e7 − 3.71348e7i −0.291323 − 0.504586i
\(57\) 8.20848e7 + 9.66852e7i 1.02997 + 1.21317i
\(58\) −5.62853e7 + 9.74890e7i −0.653084 + 1.13117i
\(59\) 4.79118e7 − 8.29857e7i 0.514764 − 0.891598i −0.485089 − 0.874465i \(-0.661213\pi\)
0.999853 − 0.0171331i \(-0.00545391\pi\)
\(60\) −4.87481e6 + 1.36380e7i −0.0485598 + 0.135853i
\(61\) −1.01113e7 − 1.75132e7i −0.0935020 − 0.161950i 0.815480 − 0.578785i \(-0.196473\pi\)
−0.908982 + 0.416834i \(0.863139\pi\)
\(62\) −2.53663e8 −2.18019
\(63\) −8.41364e7 + 3.17284e7i −0.672901 + 0.253756i
\(64\) 7.71659e7 0.574931
\(65\) 9.32588e6 + 1.61529e7i 0.0648007 + 0.112238i
\(66\) 2.52932e8 − 4.61065e7i 1.64080 − 0.299099i
\(67\) −9.12489e7 + 1.58048e8i −0.553211 + 0.958190i 0.444829 + 0.895615i \(0.353264\pi\)
−0.998040 + 0.0625744i \(0.980069\pi\)
\(68\) 2.29975e7 − 3.98328e7i 0.130434 − 0.225918i
\(69\) −1.54367e7 + 2.81392e6i −0.0819847 + 0.0149448i
\(70\) 4.13974e7 + 7.17024e7i 0.206078 + 0.356938i
\(71\) 1.38543e8 0.647025 0.323512 − 0.946224i \(-0.395136\pi\)
0.323512 + 0.946224i \(0.395136\pi\)
\(72\) 2.99772e7 − 1.82298e8i 0.131460 − 0.799440i
\(73\) −2.17970e8 −0.898347 −0.449173 − 0.893445i \(-0.648281\pi\)
−0.449173 + 0.893445i \(0.648281\pi\)
\(74\) 8.10823e7 + 1.40439e8i 0.314328 + 0.544432i
\(75\) 6.86631e7 − 1.92095e8i 0.250581 − 0.701036i
\(76\) 6.60510e7 − 1.14404e8i 0.227101 − 0.393350i
\(77\) 1.63169e8 − 2.82617e8i 0.528967 − 0.916198i
\(78\) 6.15004e7 + 7.24395e7i 0.188127 + 0.221590i
\(79\) −1.97294e8 − 3.41722e8i −0.569890 − 0.987078i −0.996576 − 0.0826785i \(-0.973653\pi\)
0.426686 − 0.904400i \(-0.359681\pi\)
\(80\) −2.22962e8 −0.608591
\(81\) −3.67020e8 − 1.24061e8i −0.947343 − 0.320222i
\(82\) 1.37515e8 0.335883
\(83\) 2.36638e8 + 4.09870e8i 0.547310 + 0.947969i 0.998458 + 0.0555197i \(0.0176816\pi\)
−0.451147 + 0.892450i \(0.648985\pi\)
\(84\) 6.06153e7 + 7.13970e7i 0.132839 + 0.156467i
\(85\) 1.11181e8 − 1.92571e8i 0.231018 − 0.400135i
\(86\) −2.88030e8 + 4.98883e8i −0.567799 + 0.983458i
\(87\) −2.07211e8 + 5.79704e8i −0.387772 + 1.08485i
\(88\) 3.35240e8 + 5.80653e8i 0.595915 + 1.03215i
\(89\) 2.60507e8 0.440113 0.220057 − 0.975487i \(-0.429376\pi\)
0.220057 + 0.975487i \(0.429376\pi\)
\(90\) −5.78820e7 + 3.51994e8i −0.0929934 + 0.565514i
\(91\) 1.20616e8 0.184381
\(92\) 8.17161e6 + 1.41537e7i 0.0118922 + 0.0205979i
\(93\) −1.36474e9 + 2.48775e8i −1.89180 + 0.344853i
\(94\) 5.73736e8 − 9.93741e8i 0.757944 − 1.31280i
\(95\) 3.19323e8 − 5.53083e8i 0.402229 − 0.696682i
\(96\) −4.54219e8 + 8.27988e7i −0.545814 + 0.0994955i
\(97\) −2.87421e8 − 4.97828e8i −0.329644 − 0.570961i 0.652797 − 0.757533i \(-0.273595\pi\)
−0.982441 + 0.186572i \(0.940262\pi\)
\(98\) −4.99822e8 −0.547391
\(99\) 1.31559e9 − 4.96118e8i 1.37645 − 0.519070i
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 9.10.c.a.4.2 ✓ 16
3.2 odd 2 27.10.c.a.10.7 16
9.2 odd 6 27.10.c.a.19.7 16
9.4 even 3 81.10.a.c.1.7 8
9.5 odd 6 81.10.a.d.1.2 8
9.7 even 3 inner 9.10.c.a.7.2 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.10.c.a.4.2 ✓ 16 1.1 even 1 trivial
9.10.c.a.7.2 yes 16 9.7 even 3 inner
27.10.c.a.10.7 16 3.2 odd 2
27.10.c.a.19.7 16 9.2 odd 6
81.10.a.c.1.7 8 9.4 even 3
81.10.a.d.1.2 8 9.5 odd 6