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Properties

Label 90.11.k.b.7.13
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 7.13
Character \(\chi\) \(=\) 90.7
Dual form 90.11.k.b.13.13

$q$-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} +(-60.1960 - 235.426i) q^{3} +(443.405 - 256.000i) q^{4} +(2718.24 - 1541.69i) q^{5} +(-2694.42 - 4793.04i) q^{6} +(11257.1 - 3016.33i) q^{7} +(8192.00 - 8192.00i) q^{8} +(-51801.9 + 28343.4i) q^{9} +(50382.2 - 49614.9i) q^{10} +(-102571. + 177659. i) q^{11} +(-86960.3 - 88978.9i) q^{12} +(-494704. - 132555. i) q^{13} +(228375. - 131852. i) q^{14} +(-526581. - 547141. i) q^{15} +(131072. - 227023. i) q^{16} +(-1.49151e6 - 1.49151e6i) q^{17} +(-966212. + 922858. i) q^{18} +554669. i q^{19} +(810608. - 1.37946e6i) q^{20} +(-1.38776e6 - 2.46865e6i) q^{21} +(-1.20140e6 + 4.48369e6i) q^{22} +(2.28380e6 + 611943. i) q^{23} +(-2.42174e6 - 1.43548e6i) q^{24} +(5.01202e6 - 8.38136e6i) q^{25} -1.15887e7 q^{26} +(9.79105e6 + 1.04894e7i) q^{27} +(4.21928e6 - 4.21928e6i) q^{28} +(-3.03160e7 - 1.75030e7i) q^{29} +(-1.47134e7 - 8.87466e6i) q^{30} +(6.88201e6 + 1.19200e7i) q^{31} +(1.53522e6 - 5.72953e6i) q^{32} +(4.79999e7 + 1.34536e7i) q^{33} +(-4.13339e7 - 2.38641e7i) q^{34} +(2.59493e7 - 2.55541e7i) q^{35} +(-1.57133e7 + 2.58289e7i) q^{36} +(-5.95665e7 - 5.95665e7i) q^{37} +(3.24837e6 + 1.21231e7i) q^{38} +(-1.42782e6 + 1.24445e8i) q^{39} +(9.63830e6 - 3.48973e7i) q^{40} +(6.98036e6 + 1.20903e7i) q^{41} +(-4.47888e7 - 4.58285e7i) q^{42} +(-4.08123e7 - 1.52314e8i) q^{43} +1.05033e8i q^{44} +(-9.71131e7 + 1.56907e8i) q^{45} +5.34995e7 q^{46} +(-1.70628e8 + 4.57197e7i) q^{47} +(-6.13372e7 - 1.71919e7i) q^{48} +(-1.27006e8 + 7.33272e7i) q^{49} +(6.04600e7 - 2.12539e8i) q^{50} +(-2.61357e8 + 4.40923e8i) q^{51} +(-2.53288e8 + 6.78684e7i) q^{52} +(1.53388e8 - 1.53388e8i) q^{53} +(2.75427e8 + 1.71919e8i) q^{54} +(-4.91884e6 + 6.41052e8i) q^{55} +(6.75084e7 - 1.16928e8i) q^{56} +(1.30584e8 - 3.33889e7i) q^{57} +(-7.65104e8 - 2.05009e8i) q^{58} +(-7.44145e7 + 4.29632e7i) q^{59} +(-3.73557e8 - 1.07800e8i) q^{60} +(-6.03944e8 + 1.04606e9i) q^{61} +(2.20224e8 + 2.20224e8i) q^{62} +(-4.97646e8 + 4.75317e8i) q^{63} -1.34218e8i q^{64} +(-1.54908e9 + 4.02362e8i) q^{65} +(1.12790e9 + 1.29408e7i) q^{66} +(-5.11779e7 + 1.90999e8i) q^{67} +(-1.04317e9 - 2.79516e8i) q^{68} +(6.59152e6 - 5.74503e8i) q^{69} +(4.17503e8 - 7.10490e8i) q^{70} +2.29741e9 q^{71} +(-1.92172e8 + 6.56550e8i) q^{72} +(2.89872e9 - 2.89872e9i) q^{73} +(-1.65075e9 - 9.53063e8i) q^{74} +(-2.27489e9 - 6.75435e8i) q^{75} +(1.41995e8 + 2.45943e8i) q^{76} +(-6.18779e8 + 2.30932e9i) q^{77} +(6.97596e8 + 2.72829e9i) q^{78} +(6.81886e8 + 3.93687e8i) q^{79} +(6.28559e6 - 8.19176e8i) q^{80} +(1.88008e9 - 2.93649e9i) q^{81} +(2.23372e8 + 2.23372e8i) q^{82} +(-1.20543e9 - 4.49873e9i) q^{83} +(-1.24731e9 - 7.39344e8i) q^{84} +(-6.35372e9 - 1.75483e9i) q^{85} +(-1.78402e9 - 3.09001e9i) q^{86} +(-2.29575e9 + 8.19079e9i) q^{87} +(6.15117e8 + 2.29565e9i) q^{88} +8.02416e9i q^{89} +(-1.20364e9 + 3.99815e9i) q^{90} -5.96876e9 q^{91} +(1.16931e9 - 3.13315e8i) q^{92} +(2.39201e9 - 2.33774e9i) q^{93} +(-3.46157e9 + 1.99854e9i) q^{94} +(8.55128e8 + 1.50772e9i) q^{95} +(-1.44129e9 - 1.65366e7i) q^{96} +(-9.21088e9 + 2.46805e9i) q^{97} +(-2.34647e9 + 2.34647e9i) q^{98} +(2.77930e8 - 1.21103e10i) 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
\(3\) −60.1960 − 235.426i −0.247720 − 0.968832i
\(4\) 443.405 − 256.000i 0.433013 − 0.250000i
\(5\) 2718.24 − 1541.69i 0.869836 − 0.493340i
\(6\) −2694.42 − 4793.04i −0.346505 − 0.616388i
\(7\) 11257.1 − 3016.33i 0.669787 − 0.179469i 0.0921281 − 0.995747i \(-0.470633\pi\)
0.577659 + 0.816278i \(0.303966\pi\)
\(8\) 8192.00 − 8192.00i 0.250000 − 0.250000i
\(9\) −51801.9 + 28343.4i −0.877269 + 0.479998i
\(10\) 50382.2 − 49614.9i 0.503822 − 0.496149i
\(11\) −102571. + 177659.i −0.636888 + 1.10312i 0.349224 + 0.937039i \(0.386445\pi\)
−0.986112 + 0.166083i \(0.946888\pi\)
\(12\) −86960.3 − 88978.9i −0.349474 − 0.357586i
\(13\) −494704. − 132555.i −1.33238 − 0.357010i −0.478778 − 0.877936i \(-0.658920\pi\)
−0.853602 + 0.520925i \(0.825587\pi\)
\(14\) 228375. − 131852.i 0.424628 − 0.245159i
\(15\) −526581. − 547141.i −0.693440 − 0.720515i
\(16\) 131072. − 227023.i 0.125000 − 0.216506i
\(17\) −1.49151e6 − 1.49151e6i −1.05046 − 1.05046i −0.998657 − 0.0518074i \(-0.983502\pi\)
−0.0518074 − 0.998657i \(-0.516498\pi\)
\(18\) −966212. + 922858.i −0.511340 + 0.488396i
\(19\) 554669.i 0.224009i 0.993708 + 0.112005i \(0.0357272\pi\)
−0.993708 + 0.112005i \(0.964273\pi\)
\(20\) 810608. − 1.37946e6i 0.253315 − 0.431082i
\(21\) −1.38776e6 − 2.46865e6i −0.339795 − 0.604453i
\(22\) −1.20140e6 + 4.48369e6i −0.233117 + 0.870005i
\(23\) 2.28380e6 + 611943.i 0.354829 + 0.0950762i 0.431830 − 0.901955i \(-0.357868\pi\)
−0.0770012 + 0.997031i \(0.524535\pi\)
\(24\) −2.42174e6 − 1.43548e6i −0.304138 − 0.180278i
\(25\) 5.01202e6 − 8.38136e6i 0.513230 − 0.858251i
\(26\) −1.15887e7 −0.975370
\(27\) 9.79105e6 + 1.04894e7i 0.682355 + 0.731021i
\(28\) 4.21928e6 − 4.21928e6i 0.245159 − 0.245159i
\(29\) −3.03160e7 − 1.75030e7i −1.47803 − 0.853339i −0.478335 − 0.878177i \(-0.658760\pi\)
−0.999691 + 0.0248381i \(0.992093\pi\)
\(30\) −1.47134e7 − 8.87466e6i −0.605492 − 0.365212i
\(31\) 6.88201e6 + 1.19200e7i 0.240385 + 0.416358i 0.960824 − 0.277160i \(-0.0893931\pi\)
−0.720439 + 0.693518i \(0.756060\pi\)
\(32\) 1.53522e6 − 5.72953e6i 0.0457532 − 0.170753i
\(33\) 4.79999e7 + 1.34536e7i 1.22651 + 0.343771i
\(34\) −4.13339e7 − 2.38641e7i −0.909729 − 0.525232i
\(35\) 2.59493e7 − 2.55541e7i 0.494066 − 0.486542i
\(36\) −1.57133e7 + 2.58289e7i −0.259869 + 0.427163i
\(37\) −5.95665e7 − 5.95665e7i −0.859000 − 0.859000i 0.132220 − 0.991220i \(-0.457789\pi\)
−0.991220 + 0.132220i \(0.957789\pi\)
\(38\) 3.24837e6 + 1.21231e7i 0.0409966 + 0.153001i
\(39\) −1.42782e6 + 1.24445e8i −0.0158252 + 1.37929i
\(40\) 9.63830e6 − 3.48973e7i 0.0941240 − 0.340794i
\(41\) 6.98036e6 + 1.20903e7i 0.0602502 + 0.104356i 0.894577 − 0.446913i \(-0.147477\pi\)
−0.834327 + 0.551270i \(0.814143\pi\)
\(42\) −4.47888e7 − 4.58285e7i −0.342707 − 0.350662i
\(43\) −4.08123e7 − 1.52314e8i −0.277619 − 1.03609i −0.954066 − 0.299596i \(-0.903148\pi\)
0.676447 − 0.736491i \(-0.263519\pi\)
\(44\) 1.05033e8i 0.636888i
\(45\) −9.71131e7 + 1.56907e8i −0.526278 + 0.850312i
\(46\) 5.34995e7 0.259753
\(47\) −1.70628e8 + 4.57197e7i −0.743980 + 0.199349i −0.610846 − 0.791749i \(-0.709171\pi\)
−0.133134 + 0.991098i \(0.542504\pi\)
\(48\) −6.13372e7 − 1.71919e7i −0.240723 − 0.0674709i
\(49\) −1.27006e8 + 7.33272e7i −0.449620 + 0.259588i
\(50\) 6.04600e7 − 2.12539e8i 0.193472 − 0.680124i
\(51\) −2.61357e8 + 4.40923e8i −0.757502 + 1.27794i
\(52\) −2.53288e8 + 6.78684e7i −0.666190 + 0.178505i
\(53\) 1.53388e8 − 1.53388e8i 0.366785 − 0.366785i −0.499518 − 0.866303i \(-0.666490\pi\)
0.866303 + 0.499518i \(0.166490\pi\)
\(54\) 2.75427e8 + 1.71919e8i 0.599843 + 0.374417i
\(55\) −4.91884e6 + 6.41052e8i −0.00977348 + 1.27374i
\(56\) 6.75084e7 − 1.16928e8i 0.122580 − 0.212314i
\(57\) 1.30584e8 − 3.33889e7i 0.217027 − 0.0554917i
\(58\) −7.65104e8 − 2.05009e8i −1.16568 − 0.312344i
\(59\) −7.44145e7 + 4.29632e7i −0.104087 + 0.0600948i −0.551140 − 0.834413i \(-0.685807\pi\)
0.447053 + 0.894508i \(0.352474\pi\)
\(60\) −3.73557e8 − 1.07800e8i −0.480397 − 0.138632i
\(61\) −6.03944e8 + 1.04606e9i −0.715068 + 1.23853i 0.247865 + 0.968795i \(0.420271\pi\)
−0.962933 + 0.269740i \(0.913062\pi\)
\(62\) 2.20224e8 + 2.20224e8i 0.240385 + 0.240385i
\(63\) −4.97646e8 + 4.75317e8i −0.501439 + 0.478939i
\(64\) − 1.34218e8i − 0.125000i
\(65\) −1.54908e9 + 4.02362e8i −1.33508 + 0.346777i
\(66\) 1.12790e9 + 1.29408e7i 0.900636 + 0.0103334i
\(67\) −5.11779e7 + 1.90999e8i −0.0379061 + 0.141467i −0.982286 − 0.187390i \(-0.939997\pi\)
0.944379 + 0.328858i \(0.106664\pi\)
\(68\) −1.04317e9 − 2.79516e8i −0.717481 − 0.192248i
\(69\) 6.59152e6 − 5.74503e8i 0.00421444 − 0.367322i
\(70\) 4.17503e8 − 7.10490e8i 0.248410 − 0.422734i
\(71\) 2.29741e9 1.27335 0.636674 − 0.771133i \(-0.280310\pi\)
0.636674 + 0.771133i \(0.280310\pi\)
\(72\) −1.92172e8 + 6.56550e8i −0.0993177 + 0.339317i
\(73\) 2.89872e9 − 2.89872e9i 1.39827 − 1.39827i 0.593270 − 0.805004i \(-0.297837\pi\)
0.805004 − 0.593270i \(-0.202163\pi\)
\(74\) −1.65075e9 − 9.53063e8i −0.743916 − 0.429500i
\(75\) −2.27489e9 − 6.75435e8i −0.958638 − 0.284628i
\(76\) 1.41995e8 + 2.45943e8i 0.0560023 + 0.0969989i
\(77\) −6.18779e8 + 2.30932e9i −0.228603 + 0.853158i
\(78\) 6.97596e8 + 2.72829e9i 0.241619 + 0.944970i
\(79\) 6.81886e8 + 3.93687e8i 0.221603 + 0.127943i 0.606692 − 0.794937i \(-0.292496\pi\)
−0.385089 + 0.922879i \(0.625829\pi\)
\(80\) 6.28559e6 − 8.19176e8i 0.00191821 − 0.249993i
\(81\) 1.88008e9 − 2.93649e9i 0.539203 − 0.842176i
\(82\) 2.23372e8 + 2.23372e8i 0.0602502 + 0.0602502i
\(83\) −1.20543e9 − 4.49873e9i −0.306021 − 1.14209i −0.932063 − 0.362297i \(-0.881992\pi\)
0.626041 − 0.779790i \(-0.284674\pi\)
\(84\) −1.24731e9 − 7.39344e8i −0.298249 − 0.176787i
\(85\) −6.35372e9 − 1.75483e9i −1.43197 − 0.395496i
\(86\) −1.78402e9 − 3.09001e9i −0.379234 − 0.656853i
\(87\) −2.29575e9 + 8.19079e9i −0.460605 + 1.64335i
\(88\) 6.15117e8 + 2.29565e9i 0.116559 + 0.435002i
\(89\) 8.02416e9i 1.43698i 0.695539 + 0.718488i \(0.255165\pi\)
−0.695539 + 0.718488i \(0.744835\pi\)
\(90\) −1.20364e9 + 3.99815e9i −0.203837 + 0.677090i
\(91\) −5.96876e9 −0.956483
\(92\) 1.16931e9 − 3.13315e8i 0.177415 − 0.0475381i
\(93\) 2.39201e9 − 2.33774e9i 0.343833 − 0.336033i
\(94\) −3.46157e9 + 1.99854e9i −0.471665 + 0.272316i
\(95\) 8.55128e8 + 1.50772e9i 0.110513 + 0.194851i
\(96\) −1.44129e9 − 1.65366e7i −0.176765 − 0.00202810i
\(97\) −9.21088e9 + 2.46805e9i −1.07261 + 0.287405i −0.751566 − 0.659658i \(-0.770701\pi\)
−0.321046 + 0.947064i \(0.604034\pi\)
\(98\) −2.34647e9 + 2.34647e9i −0.259588 + 0.259588i
\(99\) 2.77930e8 − 1.21103e10i 0.0292253 − 1.27344i
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.7.13 ✓ 120
5.3 odd 4 inner 90.11.k.b.43.28 yes 120
9.4 even 3 inner 90.11.k.b.67.28 yes 120
45.13 odd 12 inner 90.11.k.b.13.13 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.k.b.7.13 ✓ 120 1.1 even 1 trivial
90.11.k.b.13.13 yes 120 45.13 odd 12 inner
90.11.k.b.43.28 yes 120 5.3 odd 4 inner
90.11.k.b.67.28 yes 120 9.4 even 3 inner