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Properties

Label 90.11.h.a.11.8
Level $90$
Weight $11$
Character 90.11
Analytic conductor $57.182$
Analytic rank $0$
Dimension $80$
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(11,90)] 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("90.11"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 90.h (of order \(6\), 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: \(57.1821527406\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 11.8
Character \(\chi\) \(=\) 90.11
Dual form 90.11.h.a.41.8

$q$-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+(-19.5959 - 11.3137i) q^{2} +(-113.066 + 215.093i) q^{3} +(256.000 + 443.405i) q^{4} +(1210.31 - 698.771i) q^{5} +(4649.14 - 2935.74i) q^{6} +(12618.1 - 21855.1i) q^{7} -11585.2i q^{8} +(-33481.0 - 48639.6i) q^{9} -31622.8 q^{10} +(-3739.35 - 2158.91i) q^{11} +(-124318. + 4929.62i) q^{12} +(-313498. - 542994. i) q^{13} +(-494525. + 285514. i) q^{14} +(13455.8 + 339336. i) q^{15} +(-131072. + 227023. i) q^{16} -923930. i q^{17} +(105797. + 1.33193e6i) q^{18} +823484. q^{19} +(619677. + 357771. i) q^{20} +(3.27420e6 + 5.18513e6i) q^{21} +(48850.6 + 84611.8i) q^{22} +(-5.04952e6 + 2.91534e6i) q^{23} +(2.49190e6 + 1.30990e6i) q^{24} +(976562. - 1.69146e6i) q^{25} +1.41873e7i q^{26} +(1.42476e7 - 1.70203e6i) q^{27} +1.29209e7 q^{28} +(1.45949e7 + 8.42639e6i) q^{29} +(3.57547e6 - 6.80184e6i) q^{30} +(-5.23296e6 - 9.06375e6i) q^{31} +(5.13695e6 - 2.96582e6i) q^{32} +(887162. - 560207. i) q^{33} +(-1.04531e7 + 1.81053e7i) q^{34} -3.52685e7i q^{35} +(1.29959e7 - 2.72974e7i) q^{36} -5.71361e7 q^{37} +(-1.61369e7 - 9.31666e6i) q^{38} +(1.52240e8 - 6.03682e6i) q^{39} +(-8.09543e6 - 1.40217e7i) q^{40} +(-4.92682e7 + 2.84450e7i) q^{41} +(-5.49795e6 - 1.38651e8i) q^{42} +(-8.61401e7 + 1.49199e8i) q^{43} -2.21073e6i q^{44} +(-7.45102e7 - 3.54733e7i) q^{45} +1.31933e8 q^{46} +(1.69160e8 + 9.76647e7i) q^{47} +(-3.40113e7 - 5.38614e7i) q^{48} +(-1.77193e8 - 3.06907e8i) q^{49} +(-3.82733e7 + 2.20971e7i) q^{50} +(1.98731e8 + 1.04465e8i) q^{51} +(1.60511e8 - 2.78013e8i) q^{52} -3.12545e8i q^{53} +(-2.98451e8 - 1.27840e8i) q^{54} -6.03435e6 q^{55} +(-2.53197e8 - 1.46183e8i) q^{56} +(-9.31084e7 + 1.77126e8i) q^{57} +(-1.90667e8 - 3.30246e8i) q^{58} +(-2.36288e8 + 1.36421e8i) q^{59} +(-1.47019e8 + 9.28364e7i) q^{60} +(-5.46494e8 + 9.46555e8i) q^{61} +2.36817e8i q^{62} +(-1.48549e9 + 1.17994e8i) q^{63} -1.34218e8 q^{64} +(-7.58858e8 - 4.38127e8i) q^{65} +(-2.37228e7 + 940683. i) q^{66} +(9.65279e8 + 1.67191e9i) q^{67} +(4.09675e8 - 2.36526e8i) q^{68} +(-5.61387e7 - 1.41574e9i) q^{69} +(-3.99018e8 + 6.91119e8i) q^{70} -1.43431e9i q^{71} +(-5.63501e8 + 3.87885e8i) q^{72} -2.06481e9 q^{73} +(1.11963e9 + 6.46421e8i) q^{74} +(2.53404e8 + 4.01299e8i) q^{75} +(2.10812e8 + 3.65137e8i) q^{76} +(-9.43666e7 + 5.44826e7i) q^{77} +(-3.05159e9 - 1.60411e9i) q^{78} +(-2.26318e9 + 3.91995e9i) q^{79} +3.66357e8i q^{80} +(-1.24483e9 + 3.25700e9i) q^{81} +1.28727e9 q^{82} +(-1.80642e9 - 1.04294e9i) q^{83} +(-1.46092e9 + 2.77919e9i) q^{84} +(-6.45616e8 - 1.11824e9i) q^{85} +(3.37599e9 - 1.94913e9i) q^{86} +(-3.46265e9 + 2.18653e9i) q^{87} +(-2.50115e7 + 4.33212e7i) q^{88} +6.10466e9i q^{89} +(1.05876e9 + 1.53812e9i) q^{90} -1.58229e10 q^{91} +(-2.58535e9 - 1.49265e9i) q^{92} +(2.54122e9 - 1.00767e8i) q^{93} +(-2.20990e9 - 3.82766e9i) q^{94} +(9.96669e8 - 5.75427e8i) q^{95} +(5.71108e7 + 1.44026e9i) q^{96} +(7.57854e9 - 1.31264e10i) q^{97} +8.01884e9i q^{98} +(2.01885e7 + 2.54163e8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 44 q^{3} + 20480 q^{4} - 13568 q^{6} - 24476 q^{7} - 103396 q^{9} + 1944 q^{11} + 45056 q^{12} + 561100 q^{13} - 175680 q^{14} - 687500 q^{15} - 10485760 q^{16} - 3888896 q^{18} + 5932480 q^{19}+ \cdots - 78067269744 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(11\) \(37\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −19.5959 − 11.3137i −0.612372 − 0.353553i
\(3\) −113.066 + 215.093i −0.465294 + 0.885156i
\(4\) 256.000 + 443.405i 0.250000 + 0.433013i
\(5\) 1210.31 − 698.771i 0.387298 − 0.223607i
\(6\) 4649.14 − 2935.74i 0.597883 − 0.377539i
\(7\) 12618.1 − 21855.1i 0.750762 − 1.30036i −0.196692 − 0.980465i \(-0.563020\pi\)
0.947454 − 0.319893i \(-0.103647\pi\)
\(8\) − 11585.2i − 0.353553i
\(9\) −33481.0 − 48639.6i −0.567004 − 0.823715i
\(10\) −31622.8 −0.316228
\(11\) −3739.35 − 2158.91i −0.0232184 − 0.0134052i 0.488346 − 0.872650i \(-0.337600\pi\)
−0.511564 + 0.859245i \(0.670934\pi\)
\(12\) −124318. + 4929.62i −0.499607 + 0.0198110i
\(13\) −313498. − 542994.i −0.844341 − 1.46244i −0.886192 − 0.463318i \(-0.846659\pi\)
0.0418507 − 0.999124i \(-0.486675\pi\)
\(14\) −494525. + 285514.i −0.919492 + 0.530869i
\(15\) 13455.8 + 339336.i 0.0177195 + 0.446862i
\(16\) −131072. + 227023.i −0.125000 + 0.216506i
\(17\) − 923930.i − 0.650721i −0.945590 − 0.325360i \(-0.894514\pi\)
0.945590 − 0.325360i \(-0.105486\pi\)
\(18\) 105797. + 1.33193e6i 0.0559900 + 0.704887i
\(19\) 823484. 0.332573 0.166287 − 0.986077i \(-0.446822\pi\)
0.166287 + 0.986077i \(0.446822\pi\)
\(20\) 619677. + 357771.i 0.193649 + 0.111803i
\(21\) 3.27420e6 + 5.18513e6i 0.801695 + 1.26959i
\(22\) 48850.6 + 84611.8i 0.00947888 + 0.0164179i
\(23\) −5.04952e6 + 2.91534e6i −0.784532 + 0.452950i −0.838034 − 0.545618i \(-0.816295\pi\)
0.0535018 + 0.998568i \(0.482962\pi\)
\(24\) 2.49190e6 + 1.30990e6i 0.312950 + 0.164506i
\(25\) 976562. − 1.69146e6i 0.100000 − 0.173205i
\(26\) 1.41873e7i 1.19408i
\(27\) 1.42476e7 − 1.70203e6i 0.992940 − 0.118617i
\(28\) 1.29209e7 0.750762
\(29\) 1.45949e7 + 8.42639e6i 0.711561 + 0.410820i 0.811639 − 0.584160i \(-0.198576\pi\)
−0.100078 + 0.994980i \(0.531909\pi\)
\(30\) 3.57547e6 − 6.80184e6i 0.147139 − 0.279911i
\(31\) −5.23296e6 − 9.06375e6i −0.182784 − 0.316592i 0.760043 − 0.649872i \(-0.225178\pi\)
−0.942828 + 0.333281i \(0.891844\pi\)
\(32\) 5.13695e6 − 2.96582e6i 0.153093 − 0.0883883i
\(33\) 887162. − 560207.i 0.0226690 − 0.0143146i
\(34\) −1.04531e7 + 1.81053e7i −0.230064 + 0.398483i
\(35\) − 3.52685e7i − 0.671502i
\(36\) 1.29959e7 − 2.72974e7i 0.214928 − 0.451449i
\(37\) −5.71361e7 −0.823952 −0.411976 − 0.911195i \(-0.635161\pi\)
−0.411976 + 0.911195i \(0.635161\pi\)
\(38\) −1.61369e7 − 9.31666e6i −0.203659 − 0.117582i
\(39\) 1.52240e8 − 6.03682e6i 1.68736 − 0.0669091i
\(40\) −8.09543e6 − 1.40217e7i −0.0790569 − 0.136931i
\(41\) −4.92682e7 + 2.84450e7i −0.425253 + 0.245520i −0.697322 − 0.716758i \(-0.745625\pi\)
0.272069 + 0.962278i \(0.412292\pi\)
\(42\) −5.49795e6 − 1.38651e8i −0.0420682 − 1.06090i
\(43\) −8.61401e7 + 1.49199e8i −0.585953 + 1.01490i 0.408802 + 0.912623i \(0.365947\pi\)
−0.994756 + 0.102278i \(0.967387\pi\)
\(44\) − 2.21073e6i − 0.0134052i
\(45\) −7.45102e7 − 3.54733e7i −0.403788 − 0.192238i
\(46\) 1.31933e8 0.640568
\(47\) 1.69160e8 + 9.76647e7i 0.737579 + 0.425842i 0.821189 − 0.570657i \(-0.193311\pi\)
−0.0836092 + 0.996499i \(0.526645\pi\)
\(48\) −3.40113e7 − 5.38614e7i −0.133480 − 0.211384i
\(49\) −1.77193e8 − 3.06907e8i −0.627287 − 1.08649i
\(50\) −3.82733e7 + 2.20971e7i −0.122474 + 0.0707107i
\(51\) 1.98731e8 + 1.04465e8i 0.575989 + 0.302776i
\(52\) 1.60511e8 − 2.78013e8i 0.422171 − 0.731221i
\(53\) − 3.12545e8i − 0.747365i −0.927557 − 0.373683i \(-0.878095\pi\)
0.927557 − 0.373683i \(-0.121905\pi\)
\(54\) −2.98451e8 − 1.27840e8i −0.649987 − 0.278419i
\(55\) −6.03435e6 −0.0119899
\(56\) −2.53197e8 − 1.46183e8i −0.459746 − 0.265434i
\(57\) −9.31084e7 + 1.77126e8i −0.154744 + 0.294379i
\(58\) −1.90667e8 − 3.30246e8i −0.290494 − 0.503150i
\(59\) −2.36288e8 + 1.36421e8i −0.330508 + 0.190819i −0.656067 − 0.754703i \(-0.727781\pi\)
0.325559 + 0.945522i \(0.394448\pi\)
\(60\) −1.47019e8 + 9.28364e7i −0.189067 + 0.119388i
\(61\) −5.46494e8 + 9.46555e8i −0.647047 + 1.12072i 0.336778 + 0.941584i \(0.390663\pi\)
−0.983825 + 0.179134i \(0.942670\pi\)
\(62\) 2.36817e8i 0.258496i
\(63\) −1.48549e9 + 1.17994e8i −1.49681 + 0.118893i
\(64\) −1.34218e8 −0.125000
\(65\) −7.58858e8 − 4.38127e8i −0.654024 − 0.377601i
\(66\) −2.37228e7 + 940683.i −0.0189429 + 0.000751145i
\(67\) 9.65279e8 + 1.67191e9i 0.714955 + 1.23834i 0.962977 + 0.269584i \(0.0868862\pi\)
−0.248022 + 0.968754i \(0.579780\pi\)
\(68\) 4.09675e8 − 2.36526e8i 0.281770 − 0.162680i
\(69\) −5.61387e7 − 1.41574e9i −0.0358936 − 0.905189i
\(70\) −3.99018e8 + 6.91119e8i −0.237412 + 0.411209i
\(71\) − 1.43431e9i − 0.794970i −0.917609 − 0.397485i \(-0.869883\pi\)
0.917609 − 0.397485i \(-0.130117\pi\)
\(72\) −5.63501e8 + 3.87885e8i −0.291227 + 0.200466i
\(73\) −2.06481e9 −0.996017 −0.498008 − 0.867172i \(-0.665935\pi\)
−0.498008 + 0.867172i \(0.665935\pi\)
\(74\) 1.11963e9 + 6.46421e8i 0.504566 + 0.291311i
\(75\) 2.53404e8 + 4.01299e8i 0.106784 + 0.169107i
\(76\) 2.10812e8 + 3.65137e8i 0.0831433 + 0.144008i
\(77\) −9.43666e7 + 5.44826e7i −0.0348630 + 0.0201282i
\(78\) −3.05159e9 − 1.60411e9i −1.05695 − 0.555597i
\(79\) −2.26318e9 + 3.91995e9i −0.735503 + 1.27393i 0.219000 + 0.975725i \(0.429721\pi\)
−0.954502 + 0.298203i \(0.903613\pi\)
\(80\) 3.66357e8i 0.111803i
\(81\) −1.24483e9 + 3.25700e9i −0.357014 + 0.934099i
\(82\) 1.28727e9 0.347218
\(83\) −1.80642e9 − 1.04294e9i −0.458594 − 0.264769i 0.252859 − 0.967503i \(-0.418629\pi\)
−0.711453 + 0.702734i \(0.751963\pi\)
\(84\) −1.46092e9 + 2.77919e9i −0.349325 + 0.664542i
\(85\) −6.45616e8 − 1.11824e9i −0.145506 − 0.252023i
\(86\) 3.37599e9 − 1.94913e9i 0.717644 − 0.414332i
\(87\) −3.46265e9 + 2.18653e9i −0.694725 + 0.438691i
\(88\) −2.50115e7 + 4.33212e7i −0.00473944 + 0.00820895i
\(89\) 6.10466e9i 1.09323i 0.837384 + 0.546615i \(0.184084\pi\)
−0.837384 + 0.546615i \(0.815916\pi\)
\(90\) 1.05876e9 + 1.53812e9i 0.179302 + 0.260482i
\(91\) −1.58229e10 −2.53560
\(92\) −2.58535e9 − 1.49265e9i −0.392266 − 0.226475i
\(93\) 2.54122e9 − 1.00767e8i 0.365281 − 0.0144846i
\(94\) −2.20990e9 − 3.82766e9i −0.301116 − 0.521547i
\(95\) 9.96669e8 − 5.75427e8i 0.128805 − 0.0743656i
\(96\) 5.71108e7 + 1.44026e9i 0.00700426 + 0.176638i
\(97\) 7.57854e9 − 1.31264e10i 0.882525 − 1.52858i 0.0340000 − 0.999422i \(-0.489175\pi\)
0.848525 − 0.529156i \(-0.177491\pi\)
\(98\) 8.01884e9i 0.887118i
\(99\) 2.01885e7 + 2.54163e8i 0.00212289 + 0.0267261i
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.h.a.11.8 ✓ 80
3.2 odd 2 270.11.h.a.251.22 80
9.4 even 3 270.11.h.a.71.22 80
9.5 odd 6 inner 90.11.h.a.41.8 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.8 ✓ 80 1.1 even 1 trivial
90.11.h.a.41.8 yes 80 9.5 odd 6 inner
270.11.h.a.71.22 80 9.4 even 3
270.11.h.a.251.22 80 3.2 odd 2