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Properties

Label 90.11.h.a.11.3
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 11.3
Character \(\chi\) \(=\) 90.11
Dual form 90.11.h.a.41.3

$q$-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+(-19.5959 - 11.3137i) q^{2} +(-208.723 + 124.434i) q^{3} +(256.000 + 443.405i) q^{4} +(-1210.31 + 698.771i) q^{5} +(5497.92 - 76.9600i) q^{6} +(4668.95 - 8086.86i) q^{7} -11585.2i q^{8} +(28081.6 - 51944.3i) q^{9} +31622.8 q^{10} +(122023. + 70450.3i) q^{11} +(-108608. - 60693.8i) q^{12} +(64447.3 + 111626. i) q^{13} +(-182985. + 105646. i) q^{14} +(165668. - 296453. i) q^{15} +(-131072. + 227023. i) q^{16} -649462. i q^{17} +(-1.13797e6 + 700190. i) q^{18} -2.91300e6 q^{19} +(-619677. - 357771. i) q^{20} +(31759.9 + 2.26889e6i) q^{21} +(-1.59411e6 - 2.76108e6i) q^{22} +(-6.52780e6 + 3.76883e6i) q^{23} +(1.44159e6 + 2.41811e6i) q^{24} +(976562. - 1.69146e6i) q^{25} -2.91655e6i q^{26} +(602352. + 1.43363e7i) q^{27} +4.78100e6 q^{28} +(-8.14303e6 - 4.70138e6i) q^{29} +(-6.60040e6 + 3.93494e6i) q^{30} +(2.14084e7 + 3.70805e7i) q^{31} +(5.13695e6 - 2.96582e6i) q^{32} +(-3.42355e7 + 479229. i) q^{33} +(-7.34782e6 + 1.27268e7i) q^{34} +1.30501e7i q^{35} +(3.02212e7 - 846240. i) q^{36} -3.24211e7 q^{37} +(5.70829e7 + 3.29568e7i) q^{38} +(-2.73417e7 - 1.52795e7i) q^{39} +(8.09543e6 + 1.40217e7i) q^{40} +(-5.27755e7 + 3.04699e7i) q^{41} +(2.50472e7 - 4.48203e7i) q^{42} +(9.09869e7 - 1.57594e8i) q^{43} +7.21411e7i q^{44} +(2.30988e6 + 8.24912e7i) q^{45} +1.70558e8 q^{46} +(-6.78491e7 - 3.91727e7i) q^{47} +(-891600. - 6.36948e7i) q^{48} +(9.76394e7 + 1.69116e8i) q^{49} +(-3.82733e7 + 2.20971e7i) q^{50} +(8.08149e7 + 1.35558e8i) q^{51} +(-3.29970e7 + 5.71526e7i) q^{52} +2.63419e7i q^{53} +(1.50393e8 - 2.87747e8i) q^{54} -1.96915e8 q^{55} +(-9.36882e7 - 5.40909e7i) q^{56} +(6.08010e8 - 3.62475e8i) q^{57} +(1.06380e8 + 1.84256e8i) q^{58} +(-2.65312e8 + 1.53178e8i) q^{59} +(1.73860e8 - 2.43369e6i) q^{60} +(5.79677e8 - 1.00403e9i) q^{61} -9.68834e8i q^{62} +(-2.88955e8 - 4.69617e8i) q^{63} -1.34218e8 q^{64} +(-1.56002e8 - 9.00679e7i) q^{65} +(6.76298e8 + 3.77939e8i) q^{66} +(-6.79816e8 - 1.17748e9i) q^{67} +(2.87975e8 - 1.66262e8i) q^{68} +(8.93533e8 - 1.59892e9i) q^{69} +(1.47645e8 - 2.55729e8i) q^{70} +1.84456e8i q^{71} +(-6.01787e8 - 3.25331e8i) q^{72} -6.46986e8 q^{73} +(6.35320e8 + 3.66802e8i) q^{74} +(6.64294e6 + 4.74563e8i) q^{75} +(-7.45728e8 - 1.29164e9i) q^{76} +(1.13944e9 - 6.57858e8i) q^{77} +(3.62917e8 + 6.08752e8i) q^{78} +(1.76767e9 - 3.06170e9i) q^{79} -3.66357e8i q^{80} +(-1.90964e9 - 2.91735e9i) q^{81} +1.37891e9 q^{82} +(-1.89918e9 - 1.09649e9i) q^{83} +(-9.97905e8 + 5.94918e8i) q^{84} +(4.53825e8 + 7.86048e8i) q^{85} +(-3.56594e9 + 2.05880e9i) q^{86} +(2.28465e9 - 3.19805e7i) q^{87} +(8.16183e8 - 1.41367e9i) q^{88} +2.30978e9i q^{89} +(8.88017e8 - 1.64262e9i) q^{90} +1.20361e9 q^{91} +(-3.34223e9 - 1.92964e9i) q^{92} +(-9.08248e9 - 5.07562e9i) q^{93} +(8.86377e8 + 1.53525e9i) q^{94} +(3.52563e9 - 2.03552e9i) q^{95} +(-7.03152e8 + 1.25824e9i) q^{96} +(7.32287e9 - 1.26836e10i) q^{97} -4.41866e9i q^{98} +(7.08610e9 - 4.36007e9i) 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
\(3\) −208.723 + 124.434i −0.858942 + 0.512072i
\(4\) 256.000 + 443.405i 0.250000 + 0.433013i
\(5\) −1210.31 + 698.771i −0.387298 + 0.223607i
\(6\) 5497.92 − 76.9600i 0.707038 − 0.00989712i
\(7\) 4668.95 − 8086.86i 0.277798 − 0.481160i −0.693039 − 0.720900i \(-0.743729\pi\)
0.970837 + 0.239740i \(0.0770621\pi\)
\(8\) − 11585.2i − 0.353553i
\(9\) 28081.6 − 51944.3i 0.475564 − 0.879681i
\(10\) 31622.8 0.316228
\(11\) 122023. + 70450.3i 0.757670 + 0.437441i 0.828458 − 0.560050i \(-0.189218\pi\)
−0.0707887 + 0.997491i \(0.522552\pi\)
\(12\) −108608. − 60693.8i −0.436469 − 0.243915i
\(13\) 64447.3 + 111626.i 0.173575 + 0.300641i 0.939667 − 0.342090i \(-0.111135\pi\)
−0.766092 + 0.642731i \(0.777801\pi\)
\(14\) −182985. + 105646.i −0.340232 + 0.196433i
\(15\) 165668. − 296453.i 0.218164 − 0.390390i
\(16\) −131072. + 227023.i −0.125000 + 0.216506i
\(17\) − 649462.i − 0.457414i −0.973495 − 0.228707i \(-0.926550\pi\)
0.973495 − 0.228707i \(-0.0734497\pi\)
\(18\) −1.13797e6 + 700190.i −0.602236 + 0.370555i
\(19\) −2.91300e6 −1.17645 −0.588224 − 0.808698i \(-0.700173\pi\)
−0.588224 + 0.808698i \(0.700173\pi\)
\(20\) −619677. − 357771.i −0.193649 − 0.111803i
\(21\) 31759.9 + 2.26889e6i 0.00777647 + 0.555541i
\(22\) −1.59411e6 − 2.76108e6i −0.309317 − 0.535753i
\(23\) −6.52780e6 + 3.76883e6i −1.01421 + 0.585554i −0.912421 − 0.409252i \(-0.865790\pi\)
−0.101788 + 0.994806i \(0.532456\pi\)
\(24\) 1.44159e6 + 2.41811e6i 0.181045 + 0.303682i
\(25\) 976562. − 1.69146e6i 0.100000 − 0.173205i
\(26\) − 2.91655e6i − 0.245473i
\(27\) 602352. + 1.43363e7i 0.0419789 + 0.999118i
\(28\) 4.78100e6 0.277798
\(29\) −8.14303e6 − 4.70138e6i −0.397005 − 0.229211i 0.288186 − 0.957574i \(-0.406948\pi\)
−0.685191 + 0.728364i \(0.740281\pi\)
\(30\) −6.60040e6 + 3.93494e6i −0.271621 + 0.161932i
\(31\) 2.14084e7 + 3.70805e7i 0.747784 + 1.29520i 0.948883 + 0.315628i \(0.102215\pi\)
−0.201099 + 0.979571i \(0.564451\pi\)
\(32\) 5.13695e6 − 2.96582e6i 0.153093 − 0.0883883i
\(33\) −3.42355e7 + 479229.i −0.874796 + 0.0122454i
\(34\) −7.34782e6 + 1.27268e7i −0.161720 + 0.280107i
\(35\) 1.30501e7i 0.248470i
\(36\) 3.02212e7 − 846240.i 0.499804 − 0.0139953i
\(37\) −3.24211e7 −0.467540 −0.233770 − 0.972292i \(-0.575106\pi\)
−0.233770 + 0.972292i \(0.575106\pi\)
\(38\) 5.70829e7 + 3.29568e7i 0.720424 + 0.415937i
\(39\) −2.73417e7 − 1.52795e7i −0.303042 − 0.169350i
\(40\) 8.09543e6 + 1.40217e7i 0.0790569 + 0.136931i
\(41\) −5.27755e7 + 3.04699e7i −0.455526 + 0.262998i −0.710161 − 0.704039i \(-0.751378\pi\)
0.254635 + 0.967037i \(0.418045\pi\)
\(42\) 2.50472e7 − 4.48203e7i 0.191651 − 0.342948i
\(43\) 9.09869e7 − 1.57594e8i 0.618923 − 1.07201i −0.370760 − 0.928729i \(-0.620903\pi\)
0.989683 − 0.143277i \(-0.0457640\pi\)
\(44\) 7.21411e7i 0.437441i
\(45\) 2.30988e6 + 8.24912e7i 0.0125177 + 0.447038i
\(46\) 1.70558e8 0.828099
\(47\) −6.78491e7 − 3.91727e7i −0.295839 − 0.170803i 0.344733 − 0.938701i \(-0.387969\pi\)
−0.640572 + 0.767898i \(0.721303\pi\)
\(48\) −891600. − 6.36948e7i −0.00349916 − 0.249976i
\(49\) 9.76394e7 + 1.69116e8i 0.345657 + 0.598695i
\(50\) −3.82733e7 + 2.20971e7i −0.122474 + 0.0707107i
\(51\) 8.08149e7 + 1.35558e8i 0.234229 + 0.392892i
\(52\) −3.29970e7 + 5.71526e7i −0.0867877 + 0.150321i
\(53\) 2.63419e7i 0.0629895i 0.999504 + 0.0314947i \(0.0100267\pi\)
−0.999504 + 0.0314947i \(0.989973\pi\)
\(54\) 1.50393e8 − 2.87747e8i 0.327535 − 0.626674i
\(55\) −1.96915e8 −0.391259
\(56\) −9.36882e7 − 5.40909e7i −0.170116 − 0.0982164i
\(57\) 6.08010e8 − 3.62475e8i 1.01050 − 0.602427i
\(58\) 1.06380e8 + 1.84256e8i 0.162077 + 0.280725i
\(59\) −2.65312e8 + 1.53178e8i −0.371105 + 0.214257i −0.673941 − 0.738785i \(-0.735400\pi\)
0.302836 + 0.953043i \(0.402066\pi\)
\(60\) 1.73860e8 − 2.43369e6i 0.223585 − 0.00312974i
\(61\) 5.79677e8 − 1.00403e9i 0.686336 − 1.18877i −0.286679 − 0.958027i \(-0.592551\pi\)
0.973015 − 0.230742i \(-0.0741154\pi\)
\(62\) − 9.68834e8i − 1.05753i
\(63\) −2.88955e8 − 4.69617e8i −0.291157 − 0.473196i
\(64\) −1.34218e8 −0.125000
\(65\) −1.56002e8 − 9.00679e7i −0.134451 − 0.0776253i
\(66\) 6.76298e8 + 3.77939e8i 0.540030 + 0.301788i
\(67\) −6.79816e8 − 1.17748e9i −0.503521 − 0.872123i −0.999992 − 0.00407009i \(-0.998704\pi\)
0.496471 − 0.868053i \(-0.334629\pi\)
\(68\) 2.87975e8 − 1.66262e8i 0.198066 − 0.114353i
\(69\) 8.93533e8 − 1.59892e9i 0.571301 − 1.02231i
\(70\) 1.47645e8 − 2.55729e8i 0.0878474 − 0.152156i
\(71\) 1.84456e8i 0.102235i 0.998693 + 0.0511176i \(0.0162783\pi\)
−0.998693 + 0.0511176i \(0.983722\pi\)
\(72\) −6.01787e8 − 3.25331e8i −0.311014 − 0.168137i
\(73\) −6.46986e8 −0.312091 −0.156045 − 0.987750i \(-0.549875\pi\)
−0.156045 + 0.987750i \(0.549875\pi\)
\(74\) 6.35320e8 + 3.66802e8i 0.286308 + 0.165300i
\(75\) 6.64294e6 + 4.74563e8i 0.00279933 + 0.199980i
\(76\) −7.45728e8 − 1.29164e9i −0.294112 − 0.509417i
\(77\) 1.13944e9 − 6.57858e8i 0.420958 − 0.243040i
\(78\) 3.62917e8 + 6.08752e8i 0.125700 + 0.210847i
\(79\) 1.76767e9 − 3.06170e9i 0.574468 − 0.995008i −0.421631 − 0.906768i \(-0.638542\pi\)
0.996099 − 0.0882408i \(-0.0281245\pi\)
\(80\) − 3.66357e8i − 0.111803i
\(81\) −1.90964e9 − 2.91735e9i −0.547679 − 0.836689i
\(82\) 1.37891e9 0.371935
\(83\) −1.89918e9 − 1.09649e9i −0.482144 − 0.278366i 0.239166 − 0.970979i \(-0.423126\pi\)
−0.721309 + 0.692613i \(0.756459\pi\)
\(84\) −9.97905e8 + 5.94918e8i −0.238612 + 0.142253i
\(85\) 4.53825e8 + 7.86048e8i 0.102281 + 0.177156i
\(86\) −3.56594e9 + 2.05880e9i −0.758023 + 0.437645i
\(87\) 2.28465e9 − 3.19805e7i 0.458377 − 0.00641636i
\(88\) 8.16183e8 − 1.41367e9i 0.154659 − 0.267877i
\(89\) 2.30978e9i 0.413639i 0.978379 + 0.206820i \(0.0663113\pi\)
−0.978379 + 0.206820i \(0.933689\pi\)
\(90\) 8.88017e8 − 1.64262e9i 0.150386 − 0.278180i
\(91\) 1.20361e9 0.192876
\(92\) −3.34223e9 − 1.92964e9i −0.507105 − 0.292777i
\(93\) −9.08248e9 − 5.07562e9i −1.30554 − 0.729582i
\(94\) 8.86377e8 + 1.53525e9i 0.120776 + 0.209190i
\(95\) 3.52563e9 − 2.03552e9i 0.455636 − 0.263062i
\(96\) −7.03152e8 + 1.25824e9i −0.0862369 + 0.154315i
\(97\) 7.32287e9 − 1.26836e10i 0.852752 − 1.47701i −0.0259630 − 0.999663i \(-0.508265\pi\)
0.878715 − 0.477347i \(-0.158401\pi\)
\(98\) − 4.41866e9i − 0.488832i
\(99\) 7.08610e9 − 4.36007e9i 0.745129 − 0.458477i
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.11.3 ✓ 80
3.2 odd 2 270.11.h.a.251.28 80
9.4 even 3 270.11.h.a.71.28 80
9.5 odd 6 inner 90.11.h.a.41.3 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.3 ✓ 80 1.1 even 1 trivial
90.11.h.a.41.3 yes 80 9.5 odd 6 inner
270.11.h.a.71.28 80 9.4 even 3
270.11.h.a.251.28 80 3.2 odd 2