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Properties

Label 90.11.h.a.11.14
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 11.14
Character \(\chi\) \(=\) 90.11
Dual form 90.11.h.a.41.14

$q$-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} +(157.398 - 185.135i) q^{3} +(256.000 + 443.405i) q^{4} +(1210.31 - 698.771i) q^{5} +(-5178.91 + 1847.14i) q^{6} +(9074.69 - 15717.8i) q^{7} -11585.2i q^{8} +(-9501.01 - 58279.6i) q^{9} -31622.8 q^{10} +(-172310. - 99483.2i) q^{11} +(122384. + 22396.3i) q^{12} +(21651.3 + 37501.2i) q^{13} +(-355654. + 205337. i) q^{14} +(61132.3 - 334055. i) q^{15} +(-131072. + 227023. i) q^{16} +534451. i q^{17} +(-473178. + 1.24953e6i) q^{18} +230992. q^{19} +(619677. + 357771. i) q^{20} +(-1.48159e6 - 4.15399e6i) q^{21} +(2.25105e6 + 3.89893e6i) q^{22} +(-3.36337e6 + 1.94184e6i) q^{23} +(-2.14483e6 - 1.82349e6i) q^{24} +(976562. - 1.69146e6i) q^{25} -979827. i q^{26} +(-1.22850e7 - 7.41410e6i) q^{27} +9.29248e6 q^{28} +(-9.28138e6 - 5.35861e6i) q^{29} +(-4.97735e6 + 5.85449e6i) q^{30} +(-1.27400e7 - 2.20663e7i) q^{31} +(5.13695e6 - 2.96582e6i) q^{32} +(-4.55390e7 + 1.62422e7i) q^{33} +(6.04662e6 - 1.04731e7i) q^{34} -2.53645e7i q^{35} +(2.34092e7 - 1.91324e7i) q^{36} -994838. q^{37} +(-4.52651e6 - 2.61338e6i) q^{38} +(1.03506e7 + 1.89418e6i) q^{39} +(-8.09543e6 - 1.40217e7i) q^{40} +(1.50855e8 - 8.70964e7i) q^{41} +(-1.79640e7 + 9.81635e7i) q^{42} +(3.71908e7 - 6.44164e7i) q^{43} -1.01871e8i q^{44} +(-5.22233e7 - 6.38972e7i) q^{45} +8.78776e7 q^{46} +(-3.86328e7 - 2.23046e7i) q^{47} +(2.13996e7 + 5.99990e7i) q^{48} +(-2.34624e7 - 4.06381e7i) q^{49} +(-3.82733e7 + 2.20971e7i) q^{50} +(9.89456e7 + 8.41212e7i) q^{51} +(-1.10855e7 + 1.92006e7i) q^{52} +2.68397e8i q^{53} +(1.56856e8 + 2.84275e8i) q^{54} -2.78064e8 q^{55} +(-1.82095e8 - 1.05132e8i) q^{56} +(3.63576e7 - 4.27648e7i) q^{57} +(1.21251e8 + 2.10014e8i) q^{58} +(-7.04916e8 + 4.06983e8i) q^{59} +(1.63772e8 - 5.84118e7i) q^{60} +(2.76428e8 - 4.78787e8i) q^{61} +5.76546e8i q^{62} +(-1.00225e9 - 3.79534e8i) q^{63} -1.34218e8 q^{64} +(5.24095e7 + 3.02586e7i) q^{65} +(1.07614e9 + 1.96934e8i) q^{66} +(-2.41671e8 - 4.18587e8i) q^{67} +(-2.36978e8 + 1.36819e8i) q^{68} +(-1.69883e8 + 9.28318e8i) q^{69} +(-2.86967e8 + 4.97041e8i) q^{70} +2.30753e9i q^{71} +(-6.75183e8 + 1.10072e8i) q^{72} -1.39626e9 q^{73} +(1.94948e7 + 1.12553e7i) q^{74} +(-1.59439e8 - 4.47027e8i) q^{75} +(5.91341e7 + 1.02423e8i) q^{76} +(-3.12732e9 + 1.80556e9i) q^{77} +(-1.81400e8 - 1.54222e8i) q^{78} +(-1.18146e8 + 2.04635e8i) q^{79} +3.66357e8i q^{80} +(-3.30625e9 + 1.10743e9i) q^{81} -3.94153e9 q^{82} +(-5.39921e9 - 3.11724e9i) q^{83} +(1.46261e9 - 1.72036e9i) q^{84} +(3.73459e8 + 6.46850e8i) q^{85} +(-1.45758e9 + 8.41532e8i) q^{86} +(-2.45293e9 + 8.74877e8i) q^{87} +(-1.15254e9 + 1.99625e9i) q^{88} +5.09316e9i q^{89} +(3.00448e8 + 1.84296e9i) q^{90} +7.85916e8 q^{91} +(-1.72204e9 - 9.94222e8i) q^{92} +(-6.09050e9 - 1.11456e9i) q^{93} +(5.04696e8 + 8.74160e8i) q^{94} +(2.79572e8 - 1.61411e8i) q^{95} +(2.59466e8 - 1.41784e9i) q^{96} +(-2.36897e9 + 4.10318e9i) q^{97} +1.06179e9i q^{98} +(-4.16072e9 + 1.09873e10i) 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
\(3\) 157.398 − 185.135i 0.647727 − 0.761873i
\(4\) 256.000 + 443.405i 0.250000 + 0.433013i
\(5\) 1210.31 − 698.771i 0.387298 − 0.223607i
\(6\) −5178.91 + 1847.14i −0.666013 + 0.237544i
\(7\) 9074.69 − 15717.8i 0.539935 − 0.935195i −0.458972 − 0.888451i \(-0.651782\pi\)
0.998907 − 0.0467443i \(-0.0148846\pi\)
\(8\) − 11585.2i − 0.353553i
\(9\) −9501.01 − 58279.6i −0.160901 − 0.986971i
\(10\) −31622.8 −0.316228
\(11\) −172310. − 99483.2i −1.06991 − 0.617712i −0.141752 − 0.989902i \(-0.545274\pi\)
−0.928157 + 0.372190i \(0.878607\pi\)
\(12\) 122384. + 22396.3i 0.491832 + 0.0900056i
\(13\) 21651.3 + 37501.2i 0.0583133 + 0.101002i 0.893708 − 0.448648i \(-0.148094\pi\)
−0.835395 + 0.549650i \(0.814761\pi\)
\(14\) −355654. + 205337.i −0.661283 + 0.381792i
\(15\) 61132.3 − 334055.i 0.0805035 − 0.439908i
\(16\) −131072. + 227023.i −0.125000 + 0.216506i
\(17\) 534451.i 0.376412i 0.982130 + 0.188206i \(0.0602672\pi\)
−0.982130 + 0.188206i \(0.939733\pi\)
\(18\) −473178. + 1.24953e6i −0.250416 + 0.661281i
\(19\) 230992. 0.0932888 0.0466444 − 0.998912i \(-0.485147\pi\)
0.0466444 + 0.998912i \(0.485147\pi\)
\(20\) 619677. + 357771.i 0.193649 + 0.111803i
\(21\) −1.48159e6 − 4.15399e6i −0.362769 − 1.01711i
\(22\) 2.25105e6 + 3.89893e6i 0.436788 + 0.756540i
\(23\) −3.36337e6 + 1.94184e6i −0.522558 + 0.301699i −0.737981 − 0.674822i \(-0.764221\pi\)
0.215422 + 0.976521i \(0.430887\pi\)
\(24\) −2.14483e6 − 1.82349e6i −0.269363 − 0.229006i
\(25\) 976562. − 1.69146e6i 0.100000 − 0.173205i
\(26\) − 979827.i − 0.0824674i
\(27\) −1.22850e7 − 7.41410e6i −0.856166 − 0.516701i
\(28\) 9.29248e6 0.539935
\(29\) −9.28138e6 − 5.35861e6i −0.452504 − 0.261253i 0.256383 − 0.966575i \(-0.417469\pi\)
−0.708887 + 0.705322i \(0.750802\pi\)
\(30\) −4.97735e6 + 5.85449e6i −0.204829 + 0.240925i
\(31\) −1.27400e7 − 2.20663e7i −0.445001 − 0.770764i 0.553051 − 0.833147i \(-0.313463\pi\)
−0.998052 + 0.0623830i \(0.980130\pi\)
\(32\) 5.13695e6 − 2.96582e6i 0.153093 − 0.0883883i
\(33\) −4.55390e7 + 1.62422e7i −1.16363 + 0.415026i
\(34\) 6.04662e6 − 1.04731e7i 0.133082 − 0.230504i
\(35\) − 2.53645e7i − 0.482933i
\(36\) 2.34092e7 − 1.91324e7i 0.387146 − 0.316415i
\(37\) −994838. −0.0143464 −0.00717322 − 0.999974i \(-0.502283\pi\)
−0.00717322 + 0.999974i \(0.502283\pi\)
\(38\) −4.52651e6 − 2.61338e6i −0.0571275 − 0.0329826i
\(39\) 1.03506e7 + 1.89418e6i 0.114721 + 0.0209941i
\(40\) −8.09543e6 − 1.40217e7i −0.0790569 − 0.136931i
\(41\) 1.50855e8 − 8.70964e7i 1.30209 − 0.751763i 0.321329 − 0.946968i \(-0.395871\pi\)
0.980762 + 0.195205i \(0.0625373\pi\)
\(42\) −1.79640e7 + 9.81635e7i −0.137454 + 0.751110i
\(43\) 3.71908e7 − 6.44164e7i 0.252984 − 0.438182i −0.711362 − 0.702826i \(-0.751921\pi\)
0.964346 + 0.264644i \(0.0852546\pi\)
\(44\) − 1.01871e8i − 0.617712i
\(45\) −5.22233e7 − 6.38972e7i −0.283010 − 0.346274i
\(46\) 8.78776e7 0.426667
\(47\) −3.86328e7 − 2.23046e7i −0.168448 − 0.0972537i 0.413406 − 0.910547i \(-0.364339\pi\)
−0.581854 + 0.813293i \(0.697672\pi\)
\(48\) 2.13996e7 + 5.99990e7i 0.0839845 + 0.235471i
\(49\) −2.34624e7 − 4.06381e7i −0.0830600 − 0.143864i
\(50\) −3.82733e7 + 2.20971e7i −0.122474 + 0.0707107i
\(51\) 9.89456e7 + 8.41212e7i 0.286778 + 0.243812i
\(52\) −1.10855e7 + 1.92006e7i −0.0291566 + 0.0505008i
\(53\) 2.68397e8i 0.641798i 0.947113 + 0.320899i \(0.103985\pi\)
−0.947113 + 0.320899i \(0.896015\pi\)
\(54\) 1.56856e8 + 2.84275e8i 0.341611 + 0.619114i
\(55\) −2.78064e8 −0.552498
\(56\) −1.82095e8 − 1.05132e8i −0.330641 − 0.190896i
\(57\) 3.63576e7 − 4.27648e7i 0.0604257 − 0.0710742i
\(58\) 1.21251e8 + 2.10014e8i 0.184734 + 0.319969i
\(59\) −7.04916e8 + 4.06983e8i −0.986001 + 0.569268i −0.904076 − 0.427371i \(-0.859440\pi\)
−0.0819242 + 0.996639i \(0.526107\pi\)
\(60\) 1.63772e8 − 5.84118e7i 0.210612 − 0.0751180i
\(61\) 2.76428e8 − 4.78787e8i 0.327290 − 0.566883i −0.654683 − 0.755904i \(-0.727198\pi\)
0.981973 + 0.189020i \(0.0605312\pi\)
\(62\) 5.76546e8i 0.629326i
\(63\) −1.00225e9 − 3.79534e8i −1.00989 − 0.382427i
\(64\) −1.34218e8 −0.125000
\(65\) 5.24095e7 + 3.02586e7i 0.0451693 + 0.0260785i
\(66\) 1.07614e9 + 1.96934e8i 0.859307 + 0.157254i
\(67\) −2.41671e8 − 4.18587e8i −0.178999 − 0.310036i 0.762539 − 0.646942i \(-0.223953\pi\)
−0.941538 + 0.336907i \(0.890619\pi\)
\(68\) −2.36978e8 + 1.36819e8i −0.162991 + 0.0941029i
\(69\) −1.69883e8 + 9.28318e8i −0.108619 + 0.593542i
\(70\) −2.86967e8 + 4.97041e8i −0.170743 + 0.295735i
\(71\) 2.30753e9i 1.27895i 0.768810 + 0.639477i \(0.220849\pi\)
−0.768810 + 0.639477i \(0.779151\pi\)
\(72\) −6.75183e8 + 1.10072e8i −0.348947 + 0.0568869i
\(73\) −1.39626e9 −0.673524 −0.336762 − 0.941590i \(-0.609332\pi\)
−0.336762 + 0.941590i \(0.609332\pi\)
\(74\) 1.94948e7 + 1.12553e7i 0.00878536 + 0.00507223i
\(75\) −1.59439e8 − 4.47027e8i −0.0671876 − 0.188377i
\(76\) 5.91341e7 + 1.02423e8i 0.0233222 + 0.0403953i
\(77\) −3.12732e9 + 1.80556e9i −1.15536 + 0.667049i
\(78\) −1.81400e8 − 1.54222e8i −0.0628297 − 0.0534164i
\(79\) −1.18146e8 + 2.04635e8i −0.0383959 + 0.0665036i −0.884585 − 0.466379i \(-0.845558\pi\)
0.846189 + 0.532883i \(0.178891\pi\)
\(80\) 3.66357e8i 0.111803i
\(81\) −3.30625e9 + 1.10743e9i −0.948222 + 0.317608i
\(82\) −3.94153e9 −1.06315
\(83\) −5.39921e9 − 3.11724e9i −1.37069 − 0.791369i −0.379677 − 0.925119i \(-0.623965\pi\)
−0.991015 + 0.133750i \(0.957298\pi\)
\(84\) 1.46261e9 − 1.72036e9i 0.349730 − 0.411362i
\(85\) 3.73459e8 + 6.46850e8i 0.0841682 + 0.145784i
\(86\) −1.45758e9 + 8.41532e8i −0.309841 + 0.178887i
\(87\) −2.45293e9 + 8.74877e8i −0.492141 + 0.175530i
\(88\) −1.15254e9 + 1.99625e9i −0.218394 + 0.378270i
\(89\) 5.09316e9i 0.912089i 0.889957 + 0.456045i \(0.150734\pi\)
−0.889957 + 0.456045i \(0.849266\pi\)
\(90\) 3.00448e8 + 1.84296e9i 0.0508812 + 0.312108i
\(91\) 7.85916e8 0.125942
\(92\) −1.72204e9 − 9.94222e8i −0.261279 − 0.150850i
\(93\) −6.09050e9 − 1.11456e9i −0.875463 − 0.160210i
\(94\) 5.04696e8 + 8.74160e8i 0.0687687 + 0.119111i
\(95\) 2.79572e8 − 1.61411e8i 0.0361306 − 0.0208600i
\(96\) 2.59466e8 − 1.41784e9i 0.0318218 − 0.173889i
\(97\) −2.36897e9 + 4.10318e9i −0.275868 + 0.477817i −0.970354 − 0.241689i \(-0.922299\pi\)
0.694486 + 0.719506i \(0.255632\pi\)
\(98\) 1.06179e9i 0.117465i
\(99\) −4.16072e9 + 1.09873e10i −0.437515 + 1.15536i
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.11.14 ✓ 80
3.2 odd 2 270.11.h.a.251.36 80
9.4 even 3 270.11.h.a.71.36 80
9.5 odd 6 inner 90.11.h.a.41.14 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.14 ✓ 80 1.1 even 1 trivial
90.11.h.a.41.14 yes 80 9.5 odd 6 inner
270.11.h.a.71.36 80 9.4 even 3
270.11.h.a.251.36 80 3.2 odd 2