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Properties

Label 90.11.g.a.73.1
Level $90$
Weight $11$
Character 90.73
Analytic conductor $57.182$
Analytic rank $0$
Dimension $2$
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] = [90,11,Mod(37,90)] 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("90.37"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 90.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-32,0,0,-5850] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(57.1821527406\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 73.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 90.73
Dual form 90.11.g.a.37.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-16.0000 + 16.0000i) q^{2} -512.000i q^{4} +(-2925.00 + 1100.00i) q^{5} +(6953.00 - 6953.00i) q^{7} +(8192.00 + 8192.00i) q^{8} +(29200.0 - 64400.0i) q^{10} -75242.0 q^{11} +(109857. + 109857. i) q^{13} +222496. i q^{14} -262144. q^{16} +(1.52893e6 - 1.52893e6i) q^{17} +4.03868e6i q^{19} +(563200. + 1.49760e6i) q^{20} +(1.20387e6 - 1.20387e6i) q^{22} +(712423. + 712423. i) q^{23} +(7.34562e6 - 6.43500e6i) q^{25} -3.51542e6 q^{26} +(-3.55994e6 - 3.55994e6i) q^{28} +446120. i q^{29} -2.90807e7 q^{31} +(4.19430e6 - 4.19430e6i) q^{32} +4.89257e7i q^{34} +(-1.26892e7 + 2.79858e7i) q^{35} +(-911847. + 911847. i) q^{37} +(-6.46189e7 - 6.46189e7i) q^{38} +(-3.29728e7 - 1.49504e7i) q^{40} +1.63946e8 q^{41} +(1.18423e8 + 1.18423e8i) q^{43} +3.85239e7i q^{44} -2.27975e7 q^{46} +(-2.76320e8 + 2.76320e8i) q^{47} +1.85787e8i q^{49} +(-1.45700e7 + 2.20490e8i) q^{50} +(5.62468e7 - 5.62468e7i) q^{52} +(-3.08460e8 - 3.08460e8i) q^{53} +(2.20083e8 - 8.27662e7i) q^{55} +1.13918e8 q^{56} +(-7.13792e6 - 7.13792e6i) q^{58} -9.40888e8i q^{59} -1.35361e9 q^{61} +(4.65291e8 - 4.65291e8i) q^{62} +1.34218e8i q^{64} +(-4.42174e8 - 2.00489e8i) q^{65} +(8.53571e8 - 8.53571e8i) q^{67} +(-7.82811e8 - 7.82811e8i) q^{68} +(-2.44746e8 - 6.50801e8i) q^{70} -2.82701e9 q^{71} +(-2.75330e9 - 2.75330e9i) q^{73} -2.91791e7i q^{74} +2.06780e9 q^{76} +(-5.23158e8 + 5.23158e8i) q^{77} -3.32450e9i q^{79} +(7.66771e8 - 2.88358e8i) q^{80} +(-2.62313e9 + 2.62313e9i) q^{82} +(-1.34634e9 - 1.34634e9i) q^{83} +(-2.79029e9 + 6.15393e9i) q^{85} -3.78953e9 q^{86} +(-6.16382e8 - 6.16382e8i) q^{88} -2.66745e9i q^{89} +1.52767e9 q^{91} +(3.64761e8 - 3.64761e8i) q^{92} -8.84225e9i q^{94} +(-4.44255e9 - 1.18131e10i) q^{95} +(-5.26563e8 + 5.26563e8i) q^{97} +(-2.97259e9 - 2.97259e9i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{2} - 5850 q^{5} + 13906 q^{7} + 16384 q^{8} + 58400 q^{10} - 150484 q^{11} + 219714 q^{13} - 524288 q^{16} + 3057854 q^{17} + 1126400 q^{20} + 2407744 q^{22} + 1424846 q^{23} + 14691250 q^{25}+ \cdots - 5945178592 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.0000 + 16.0000i −0.500000 + 0.500000i
\(3\) 0 0
\(4\) − 512.000i − 0.500000i
\(5\) −2925.00 + 1100.00i −0.936000 + 0.352000i
\(6\) 0 0
\(7\) 6953.00 − 6953.00i 0.413697 − 0.413697i −0.469328 − 0.883024i \(-0.655504\pi\)
0.883024 + 0.469328i \(0.155504\pi\)
\(8\) 8192.00 + 8192.00i 0.250000 + 0.250000i
\(9\) 0 0
\(10\) 29200.0 − 64400.0i 0.292000 − 0.644000i
\(11\) −75242.0 −0.467194 −0.233597 − 0.972334i \(-0.575050\pi\)
−0.233597 + 0.972334i \(0.575050\pi\)
\(12\) 0 0
\(13\) 109857. + 109857.i 0.295877 + 0.295877i 0.839396 − 0.543520i \(-0.182909\pi\)
−0.543520 + 0.839396i \(0.682909\pi\)
\(14\) 222496.i 0.413697i
\(15\) 0 0
\(16\) −262144. −0.250000
\(17\) 1.52893e6 − 1.52893e6i 1.07682 − 1.07682i 0.0800247 − 0.996793i \(-0.474500\pi\)
0.996793 − 0.0800247i \(-0.0254999\pi\)
\(18\) 0 0
\(19\) 4.03868e6i 1.63107i 0.578711 + 0.815533i \(0.303556\pi\)
−0.578711 + 0.815533i \(0.696444\pi\)
\(20\) 563200. + 1.49760e6i 0.176000 + 0.468000i
\(21\) 0 0
\(22\) 1.20387e6 − 1.20387e6i 0.233597 − 0.233597i
\(23\) 712423. + 712423.i 0.110688 + 0.110688i 0.760281 − 0.649594i \(-0.225061\pi\)
−0.649594 + 0.760281i \(0.725061\pi\)
\(24\) 0 0
\(25\) 7.34562e6 − 6.43500e6i 0.752192 − 0.658944i
\(26\) −3.51542e6 −0.295877
\(27\) 0 0
\(28\) −3.55994e6 − 3.55994e6i −0.206848 − 0.206848i
\(29\) 446120.i 0.0217501i 0.999941 + 0.0108751i \(0.00346171\pi\)
−0.999941 + 0.0108751i \(0.996538\pi\)
\(30\) 0 0
\(31\) −2.90807e7 −1.01577 −0.507886 − 0.861424i \(-0.669573\pi\)
−0.507886 + 0.861424i \(0.669573\pi\)
\(32\) 4.19430e6 − 4.19430e6i 0.125000 − 0.125000i
\(33\) 0 0
\(34\) 4.89257e7i 1.07682i
\(35\) −1.26892e7 + 2.79858e7i −0.241599 + 0.532841i
\(36\) 0 0
\(37\) −911847. + 911847.i −0.0131496 + 0.0131496i −0.713651 − 0.700501i \(-0.752960\pi\)
0.700501 + 0.713651i \(0.252960\pi\)
\(38\) −6.46189e7 − 6.46189e7i −0.815533 − 0.815533i
\(39\) 0 0
\(40\) −3.29728e7 − 1.49504e7i −0.322000 − 0.146000i
\(41\) 1.63946e8 1.41508 0.707540 − 0.706674i \(-0.249805\pi\)
0.707540 + 0.706674i \(0.249805\pi\)
\(42\) 0 0
\(43\) 1.18423e8 + 1.18423e8i 0.805551 + 0.805551i 0.983957 − 0.178406i \(-0.0570941\pi\)
−0.178406 + 0.983957i \(0.557094\pi\)
\(44\) 3.85239e7i 0.233597i
\(45\) 0 0
\(46\) −2.27975e7 −0.110688
\(47\) −2.76320e8 + 2.76320e8i −1.20482 + 1.20482i −0.232142 + 0.972682i \(0.574573\pi\)
−0.972682 + 0.232142i \(0.925427\pi\)
\(48\) 0 0
\(49\) 1.85787e8i 0.657710i
\(50\) −1.45700e7 + 2.20490e8i −0.0466240 + 0.705568i
\(51\) 0 0
\(52\) 5.62468e7 − 5.62468e7i 0.147938 − 0.147938i
\(53\) −3.08460e8 − 3.08460e8i −0.737598 − 0.737598i 0.234515 − 0.972113i \(-0.424650\pi\)
−0.972113 + 0.234515i \(0.924650\pi\)
\(54\) 0 0
\(55\) 2.20083e8 − 8.27662e7i 0.437293 − 0.164452i
\(56\) 1.13918e8 0.206848
\(57\) 0 0
\(58\) −7.13792e6 − 7.13792e6i −0.0108751 − 0.0108751i
\(59\) − 9.40888e8i − 1.31607i −0.752989 − 0.658034i \(-0.771388\pi\)
0.752989 − 0.658034i \(-0.228612\pi\)
\(60\) 0 0
\(61\) −1.35361e9 −1.60267 −0.801336 − 0.598215i \(-0.795877\pi\)
−0.801336 + 0.598215i \(0.795877\pi\)
\(62\) 4.65291e8 − 4.65291e8i 0.507886 − 0.507886i
\(63\) 0 0
\(64\) 1.34218e8i 0.125000i
\(65\) −4.42174e8 − 2.00489e8i −0.381089 − 0.172792i
\(66\) 0 0
\(67\) 8.53571e8 − 8.53571e8i 0.632216 − 0.632216i −0.316407 − 0.948623i \(-0.602477\pi\)
0.948623 + 0.316407i \(0.102477\pi\)
\(68\) −7.82811e8 − 7.82811e8i −0.538409 − 0.538409i
\(69\) 0 0
\(70\) −2.44746e8 − 6.50801e8i −0.145621 − 0.387220i
\(71\) −2.82701e9 −1.56688 −0.783441 − 0.621466i \(-0.786537\pi\)
−0.783441 + 0.621466i \(0.786537\pi\)
\(72\) 0 0
\(73\) −2.75330e9 − 2.75330e9i −1.32812 − 1.32812i −0.907005 − 0.421119i \(-0.861637\pi\)
−0.421119 − 0.907005i \(-0.638363\pi\)
\(74\) − 2.91791e7i − 0.0131496i
\(75\) 0 0
\(76\) 2.06780e9 0.815533
\(77\) −5.23158e8 + 5.23158e8i −0.193276 + 0.193276i
\(78\) 0 0
\(79\) − 3.32450e9i − 1.08042i −0.841532 − 0.540208i \(-0.818346\pi\)
0.841532 − 0.540208i \(-0.181654\pi\)
\(80\) 7.66771e8 − 2.88358e8i 0.234000 − 0.0880000i
\(81\) 0 0
\(82\) −2.62313e9 + 2.62313e9i −0.707540 + 0.707540i
\(83\) −1.34634e9 − 1.34634e9i −0.341794 − 0.341794i 0.515248 − 0.857041i \(-0.327700\pi\)
−0.857041 + 0.515248i \(0.827700\pi\)
\(84\) 0 0
\(85\) −2.79029e9 + 6.15393e9i −0.628861 + 1.38694i
\(86\) −3.78953e9 −0.805551
\(87\) 0 0
\(88\) −6.16382e8 − 6.16382e8i −0.116798 − 0.116798i
\(89\) − 2.66745e9i − 0.477690i −0.971058 − 0.238845i \(-0.923231\pi\)
0.971058 − 0.238845i \(-0.0767688\pi\)
\(90\) 0 0
\(91\) 1.52767e9 0.244807
\(92\) 3.64761e8 − 3.64761e8i 0.0553438 − 0.0553438i
\(93\) 0 0
\(94\) − 8.84225e9i − 1.20482i
\(95\) −4.44255e9 − 1.18131e10i −0.574135 − 1.52668i
\(96\) 0 0
\(97\) −5.26563e8 + 5.26563e8i −0.0613185 + 0.0613185i −0.737101 − 0.675783i \(-0.763806\pi\)
0.675783 + 0.737101i \(0.263806\pi\)
\(98\) −2.97259e9 − 2.97259e9i −0.328855 − 0.328855i
\(99\) 0 0
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.g.a.73.1 2
3.2 odd 2 10.11.c.b.3.1 ✓ 2
5.2 odd 4 inner 90.11.g.a.37.1 2
12.11 even 2 80.11.p.a.33.1 2
15.2 even 4 10.11.c.b.7.1 yes 2
15.8 even 4 50.11.c.b.7.1 2
15.14 odd 2 50.11.c.b.43.1 2
60.47 odd 4 80.11.p.a.17.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.11.c.b.3.1 ✓ 2 3.2 odd 2
10.11.c.b.7.1 yes 2 15.2 even 4
50.11.c.b.7.1 2 15.8 even 4
50.11.c.b.43.1 2 15.14 odd 2
80.11.p.a.17.1 2 60.47 odd 4
80.11.p.a.33.1 2 12.11 even 2
90.11.g.a.37.1 2 5.2 odd 4 inner
90.11.g.a.73.1 2 1.1 even 1 trivial