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Properties

Label 90.11.k.b.7.3
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 7.3
Character \(\chi\) \(=\) 90.7
Dual form 90.11.k.b.13.3

$q$-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} +(-233.816 - 66.1741i) q^{3} +(443.405 - 256.000i) q^{4} +(2804.10 - 1379.37i) q^{5} +(-5497.92 - 77.0051i) q^{6} +(4809.41 - 1288.68i) q^{7} +(8192.00 - 8192.00i) q^{8} +(50291.0 + 30945.1i) q^{9} +(53209.3 - 46570.1i) q^{10} +(148614. - 257407. i) q^{11} +(-120616. + 30515.0i) q^{12} +(-425840. - 114103. i) q^{13} +(97569.5 - 56331.8i) q^{14} +(-746922. + 136962. i) q^{15} +(131072. - 227023. i) q^{16} +(1.92539e6 + 1.92539e6i) q^{17} +(1.28041e6 + 381825. i) q^{18} +1.54585e6i q^{19} +(890230. - 1.32947e6i) q^{20} +(-1.20980e6 - 16944.6i) q^{21} +(1.74069e6 - 6.49634e6i) q^{22} +(6.21259e6 + 1.66466e6i) q^{23} +(-2.45752e6 + 1.37332e6i) q^{24} +(5.96028e6 - 7.73579e6i) q^{25} -9.97556e6 q^{26} +(-9.71108e6 - 1.05634e7i) q^{27} +(1.80262e6 - 1.80262e6i) q^{28} +(1.60217e7 + 9.25013e6i) q^{29} +(-1.55229e7 + 7.36776e6i) q^{30} +(1.41683e7 + 2.45402e7i) q^{31} +(1.53522e6 - 5.72953e6i) q^{32} +(-5.17820e7 + 5.03515e7i) q^{33} +(5.33581e7 + 3.08063e7i) q^{34} +(1.17085e7 - 1.02476e7i) q^{35} +(3.02212e7 + 846736. i) q^{36} +(-8.48914e7 - 8.48914e7i) q^{37} +(9.05312e6 + 3.37867e7i) q^{38} +(9.20175e7 + 5.48587e7i) q^{39} +(1.16713e7 - 3.42710e7i) q^{40} +(-5.67564e7 - 9.83049e7i) q^{41} +(-2.65410e7 + 6.71471e6i) q^{42} +(-4.04039e7 - 1.50789e8i) q^{43} -1.52181e8i q^{44} +(1.83706e8 + 1.74030e7i) q^{45} +1.45534e8 q^{46} +(1.91640e7 - 5.13497e6i) q^{47} +(-4.56698e7 + 4.44082e7i) q^{48} +(-2.23161e8 + 1.28842e8i) q^{49} +(8.49663e7 - 2.03982e8i) q^{50} +(-3.22777e8 - 5.77599e8i) q^{51} +(-2.18030e8 + 5.84209e7i) q^{52} +(2.35685e8 - 2.35685e8i) q^{53} +(-2.74113e8 - 1.74007e8i) q^{54} +(6.16674e7 - 9.26789e8i) q^{55} +(2.88419e7 - 4.99556e7i) q^{56} +(1.02295e8 - 3.61445e8i) q^{57} +(4.04349e8 + 1.08345e8i) q^{58} +(6.16138e8 - 3.55728e8i) q^{59} +(-2.96127e8 + 2.51941e8i) q^{60} +(3.80145e8 - 6.58431e8i) q^{61} +(4.53385e8 + 4.53385e8i) q^{62} +(2.81748e8 + 8.40190e7i) q^{63} -1.34218e8i q^{64} +(-1.35149e9 + 2.67435e8i) q^{65} +(-8.36890e8 + 1.40376e9i) q^{66} +(5.03653e8 - 1.87966e9i) q^{67} +(1.34663e9 + 3.60828e8i) q^{68} +(-1.34245e9 - 8.00337e8i) q^{69} +(1.95892e8 - 2.92544e8i) q^{70} +6.29641e8 q^{71} +(6.65486e8 - 1.58481e8i) q^{72} +(-2.42002e9 + 2.42002e9i) q^{73} +(-2.35258e9 - 1.35826e9i) q^{74} +(-1.90552e9 + 1.41434e9i) q^{75} +(3.95737e8 + 6.85437e8i) q^{76} +(3.83031e8 - 1.42949e9i) q^{77} +(2.33245e9 + 6.60123e8i) q^{78} +(6.14062e8 + 3.54529e8i) q^{79} +(5.43883e7 - 8.17393e8i) q^{80} +(1.57158e9 + 3.11252e9i) q^{81} +(-1.81620e9 - 1.81620e9i) q^{82} +(6.37728e8 + 2.38003e9i) q^{83} +(-5.40767e8 + 3.02194e8i) q^{84} +(8.05483e9 + 2.74315e9i) q^{85} +(-1.76617e9 - 3.05909e9i) q^{86} +(-3.13401e9 - 3.22305e9i) q^{87} +(-8.91233e8 - 3.32613e9i) q^{88} -9.39654e9i q^{89} +(4.11706e9 - 6.95487e8i) q^{90} -2.19508e9 q^{91} +(3.18085e9 - 8.52305e8i) q^{92} +(-1.68885e9 - 6.67547e9i) q^{93} +(3.88783e8 - 2.24464e8i) q^{94} +(2.13230e9 + 4.33471e9i) q^{95} +(-7.38106e8 + 1.23806e9i) q^{96} +(-4.22015e9 + 1.13078e9i) q^{97} +(-4.12295e9 + 4.12295e9i) q^{98} +(1.54394e10 - 8.34638e9i) 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
\(3\) −233.816 − 66.1741i −0.962206 − 0.272321i
\(4\) 443.405 − 256.000i 0.433013 − 0.250000i
\(5\) 2804.10 − 1379.37i 0.897311 − 0.441400i
\(6\) −5497.92 − 77.0051i −0.707037 − 0.00990291i
\(7\) 4809.41 − 1288.68i 0.286155 − 0.0766751i −0.112886 − 0.993608i \(-0.536010\pi\)
0.399042 + 0.916933i \(0.369343\pi\)
\(8\) 8192.00 − 8192.00i 0.250000 − 0.250000i
\(9\) 50291.0 + 30945.1i 0.851682 + 0.524059i
\(10\) 53209.3 − 46570.1i 0.532093 − 0.465701i
\(11\) 148614. − 257407.i 0.922776 − 1.59830i 0.127678 − 0.991816i \(-0.459248\pi\)
0.795099 − 0.606480i \(-0.207419\pi\)
\(12\) −120616. + 30515.0i −0.484728 + 0.122633i
\(13\) −425840. − 114103.i −1.14691 − 0.307314i −0.365184 − 0.930935i \(-0.618994\pi\)
−0.781726 + 0.623622i \(0.785660\pi\)
\(14\) 97569.5 − 56331.8i 0.181415 − 0.104740i
\(15\) −746922. + 136962.i −0.983601 + 0.180361i
\(16\) 131072. − 227023.i 0.125000 − 0.216506i
\(17\) 1.92539e6 + 1.92539e6i 1.35605 + 1.35605i 0.878732 + 0.477316i \(0.158390\pi\)
0.477316 + 0.878732i \(0.341610\pi\)
\(18\) 1.28041e6 + 381825.i 0.677619 + 0.202070i
\(19\) 1.54585e6i 0.624308i 0.950031 + 0.312154i \(0.101051\pi\)
−0.950031 + 0.312154i \(0.898949\pi\)
\(20\) 890230. − 1.32947e6i 0.278197 − 0.415459i
\(21\) −1.20980e6 − 16944.6i −0.296221 − 0.00414893i
\(22\) 1.74069e6 − 6.49634e6i 0.337760 − 1.26054i
\(23\) 6.21259e6 + 1.66466e6i 0.965236 + 0.258634i 0.706815 − 0.707398i \(-0.250131\pi\)
0.258421 + 0.966032i \(0.416798\pi\)
\(24\) −2.45752e6 + 1.37332e6i −0.308632 + 0.172471i
\(25\) 5.96028e6 − 7.73579e6i 0.610333 − 0.792145i
\(26\) −9.97556e6 −0.839596
\(27\) −9.71108e6 − 1.05634e7i −0.676782 − 0.736184i
\(28\) 1.80262e6 − 1.80262e6i 0.104740 − 0.104740i
\(29\) 1.60217e7 + 9.25013e6i 0.781122 + 0.450981i 0.836828 − 0.547466i \(-0.184408\pi\)
−0.0557061 + 0.998447i \(0.517741\pi\)
\(30\) −1.55229e7 + 7.36776e6i −0.638803 + 0.303200i
\(31\) 1.41683e7 + 2.45402e7i 0.494890 + 0.857175i 0.999983 − 0.00588999i \(-0.00187485\pi\)
−0.505092 + 0.863065i \(0.668542\pi\)
\(32\) 1.53522e6 − 5.72953e6i 0.0457532 − 0.170753i
\(33\) −5.17820e7 + 5.03515e7i −1.32315 + 1.28660i
\(34\) 5.33581e7 + 3.08063e7i 1.17437 + 0.678024i
\(35\) 1.17085e7 − 1.02476e7i 0.222926 − 0.195110i
\(36\) 3.02212e7 + 846736.i 0.499804 + 0.0140035i
\(37\) −8.48914e7 − 8.48914e7i −1.22421 − 1.22421i −0.966122 − 0.258085i \(-0.916909\pi\)
−0.258085 − 0.966122i \(-0.583091\pi\)
\(38\) 9.05312e6 + 3.37867e7i 0.114256 + 0.426411i
\(39\) 9.20175e7 + 5.48587e7i 1.01988 + 0.608027i
\(40\) 1.16713e7 − 3.42710e7i 0.113978 − 0.334678i
\(41\) −5.67564e7 − 9.83049e7i −0.489886 − 0.848508i 0.510046 − 0.860147i \(-0.329628\pi\)
−0.999932 + 0.0116391i \(0.996295\pi\)
\(42\) −2.65410e7 + 6.71471e6i −0.203082 + 0.0513784i
\(43\) −4.04039e7 − 1.50789e8i −0.274840 − 1.02572i −0.955948 − 0.293535i \(-0.905168\pi\)
0.681108 − 0.732183i \(-0.261498\pi\)
\(44\) − 1.52181e8i − 0.922776i
\(45\) 1.83706e8 + 1.74030e7i 0.995543 + 0.0943110i
\(46\) 1.45534e8 0.706602
\(47\) 1.91640e7 − 5.13497e6i 0.0835596 − 0.0223897i −0.216797 − 0.976217i \(-0.569561\pi\)
0.300357 + 0.953827i \(0.402894\pi\)
\(48\) −4.56698e7 + 4.44082e7i −0.179235 + 0.174284i
\(49\) −2.23161e8 + 1.28842e8i −0.790020 + 0.456118i
\(50\) 8.49663e7 − 2.03982e8i 0.271892 − 0.652744i
\(51\) −3.22777e8 − 5.77599e8i −0.935517 − 1.67408i
\(52\) −2.18030e8 + 5.84209e7i −0.573455 + 0.153657i
\(53\) 2.35685e8 − 2.35685e8i 0.563576 − 0.563576i −0.366745 − 0.930321i \(-0.619528\pi\)
0.930321 + 0.366745i \(0.119528\pi\)
\(54\) −2.74113e8 − 1.74007e8i −0.596982 − 0.378963i
\(55\) 6.16674e7 − 9.26789e8i 0.122530 − 1.84148i
\(56\) 2.88419e7 − 4.99556e7i 0.0523701 − 0.0907076i
\(57\) 1.02295e8 − 3.61445e8i 0.170012 − 0.600713i
\(58\) 4.04349e8 + 1.08345e8i 0.616051 + 0.165070i
\(59\) 6.16138e8 − 3.55728e8i 0.861823 − 0.497574i −0.00279931 − 0.999996i \(-0.500891\pi\)
0.864622 + 0.502422i \(0.167558\pi\)
\(60\) −2.96127e8 + 2.51941e8i −0.380821 + 0.323999i
\(61\) 3.80145e8 − 6.58431e8i 0.450091 − 0.779581i −0.548300 − 0.836282i \(-0.684725\pi\)
0.998391 + 0.0567008i \(0.0180581\pi\)
\(62\) 4.53385e8 + 4.53385e8i 0.494890 + 0.494890i
\(63\) 2.81748e8 + 8.40190e7i 0.283896 + 0.0846593i
\(64\) − 1.34218e8i − 0.125000i
\(65\) −1.35149e9 + 2.67435e8i −1.16478 + 0.230490i
\(66\) −8.36890e8 + 1.40376e9i −0.668265 + 1.12092i
\(67\) 5.03653e8 − 1.87966e9i 0.373042 − 1.39221i −0.483143 − 0.875542i \(-0.660505\pi\)
0.856185 − 0.516670i \(-0.172829\pi\)
\(68\) 1.34663e9 + 3.60828e8i 0.926198 + 0.248174i
\(69\) −1.34245e9 − 8.00337e8i −0.858325 − 0.511714i
\(70\) 1.95892e8 − 2.92544e8i 0.116554 − 0.174061i
\(71\) 6.29641e8 0.348981 0.174490 − 0.984659i \(-0.444172\pi\)
0.174490 + 0.984659i \(0.444172\pi\)
\(72\) 6.65486e8 − 1.58481e8i 0.343935 − 0.0819059i
\(73\) −2.42002e9 + 2.42002e9i −1.16736 + 1.16736i −0.184534 + 0.982826i \(0.559078\pi\)
−0.982826 + 0.184534i \(0.940922\pi\)
\(74\) −2.35258e9 − 1.35826e9i −1.06019 − 0.612104i
\(75\) −1.90552e9 + 1.41434e9i −0.802984 + 0.596001i
\(76\) 3.95737e8 + 6.85437e8i 0.156077 + 0.270333i
\(77\) 3.83031e8 − 1.42949e9i 0.141508 − 0.528115i
\(78\) 2.33245e9 + 6.60123e8i 0.807865 + 0.228640i
\(79\) 6.14062e8 + 3.54529e8i 0.199561 + 0.115217i 0.596451 − 0.802650i \(-0.296577\pi\)
−0.396889 + 0.917866i \(0.629910\pi\)
\(80\) 5.43883e7 − 8.17393e8i 0.0165980 − 0.249448i
\(81\) 1.57158e9 + 3.11252e9i 0.450725 + 0.892663i
\(82\) −1.81620e9 − 1.81620e9i −0.489886 − 0.489886i
\(83\) 6.37728e8 + 2.38003e9i 0.161899 + 0.604217i 0.998415 + 0.0562742i \(0.0179221\pi\)
−0.836516 + 0.547943i \(0.815411\pi\)
\(84\) −5.40767e8 + 3.02194e8i −0.129305 + 0.0722587i
\(85\) 8.05483e9 + 2.74315e9i 1.81535 + 0.618237i
\(86\) −1.76617e9 − 3.05909e9i −0.375439 − 0.650279i
\(87\) −3.13401e9 − 3.22305e9i −0.628788 − 0.646653i
\(88\) −8.91233e8 − 3.32613e9i −0.168880 − 0.630268i
\(89\) − 9.39654e9i − 1.68274i −0.540457 − 0.841372i \(-0.681749\pi\)
0.540457 − 0.841372i \(-0.318251\pi\)
\(90\) 4.11706e9 − 6.95487e8i 0.697228 − 0.117781i
\(91\) −2.19508e9 −0.351758
\(92\) 3.18085e9 − 8.52305e8i 0.482618 − 0.129317i
\(93\) −1.68885e9 − 6.67547e9i −0.242760 − 0.959549i
\(94\) 3.88783e8 − 2.24464e8i 0.0529747 − 0.0305849i
\(95\) 2.13230e9 + 4.33471e9i 0.275570 + 0.560198i
\(96\) −7.38106e8 + 1.23806e9i −0.0905237 + 0.151840i
\(97\) −4.22015e9 + 1.13078e9i −0.491438 + 0.131680i −0.496024 − 0.868309i \(-0.665207\pi\)
0.00458555 + 0.999989i \(0.498540\pi\)
\(98\) −4.12295e9 + 4.12295e9i −0.456118 + 0.456118i
\(99\) 1.54394e10 − 8.34638e9i 1.62351 − 0.877651i
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.7.3 ✓ 120
5.3 odd 4 inner 90.11.k.b.43.18 yes 120
9.4 even 3 inner 90.11.k.b.67.18 yes 120
45.13 odd 12 inner 90.11.k.b.13.3 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.k.b.7.3 ✓ 120 1.1 even 1 trivial
90.11.k.b.13.3 yes 120 45.13 odd 12 inner
90.11.k.b.43.18 yes 120 5.3 odd 4 inner
90.11.k.b.67.18 yes 120 9.4 even 3 inner