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Properties

Label 90.11.h.a.11.5
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 11.5
Character \(\chi\) \(=\) 90.11
Dual form 90.11.h.a.41.5

$q$-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} +(-164.534 + 178.823i) q^{3} +(256.000 + 443.405i) q^{4} +(1210.31 - 698.771i) q^{5} +(5247.34 - 1642.72i) q^{6} +(-3815.60 + 6608.81i) q^{7} -11585.2i q^{8} +(-4906.41 - 58844.8i) q^{9} -31622.8 q^{10} +(-110548. - 63824.8i) q^{11} +(-121412. - 27176.3i) q^{12} +(203701. + 352820. i) q^{13} +(149540. - 86337.1i) q^{14} +(-74179.7 + 331402. i) q^{15} +(-131072. + 227023. i) q^{16} +913327. i q^{17} +(-569607. + 1.20863e6i) q^{18} -402654. q^{19} +(619677. + 357771. i) q^{20} +(-554014. - 1.76969e6i) q^{21} +(1.44419e6 + 2.50141e6i) q^{22} +(9.28358e6 - 5.35988e6i) q^{23} +(2.07171e6 + 1.90616e6i) q^{24} +(976562. - 1.69146e6i) q^{25} -9.21843e6i q^{26} +(1.13301e7 + 8.80457e6i) q^{27} -3.90717e6 q^{28} +(-1.10257e7 - 6.36570e6i) q^{29} +(5.20301e6 - 5.65488e6i) q^{30} +(1.49529e7 + 2.58991e7i) q^{31} +(5.13695e6 - 2.96582e6i) q^{32} +(2.96022e7 - 9.26718e6i) q^{33} +(1.03331e7 - 1.78975e7i) q^{34} +1.06649e7i q^{35} +(2.48360e7 - 1.72398e7i) q^{36} +5.56536e7 q^{37} +(7.89037e6 + 4.55551e6i) q^{38} +(-9.66079e7 - 2.16243e7i) q^{39} +(-8.09543e6 - 1.40217e7i) q^{40} +(3.60900e7 - 2.08366e7i) q^{41} +(-9.16533e6 + 4.09466e7i) q^{42} +(4.82911e7 - 8.36426e7i) q^{43} -6.53566e7i q^{44} +(-4.70573e7 - 6.77918e7i) q^{45} -2.42560e8 q^{46} +(-2.50589e8 - 1.44678e8i) q^{47} +(-1.90313e7 - 6.07917e7i) q^{48} +(1.12120e8 + 1.94198e8i) q^{49} +(-3.82733e7 + 2.20971e7i) q^{50} +(-1.63324e8 - 1.50273e8i) q^{51} +(-1.04295e8 + 1.80644e8i) q^{52} +2.20997e8i q^{53} +(-1.22411e8 - 3.00719e8i) q^{54} -1.78396e8 q^{55} +(7.65646e7 + 4.42046e7i) q^{56} +(6.62501e7 - 7.20038e7i) q^{57} +(1.44039e8 + 2.49484e8i) q^{58} +(6.64653e7 - 3.83738e7i) q^{59} +(-1.65935e8 + 5.19473e7i) q^{60} +(-5.20326e8 + 9.01231e8i) q^{61} -6.76690e8i q^{62} +(4.07615e8 + 1.92103e8i) q^{63} -1.34218e8 q^{64} +(4.93080e8 + 2.84680e8i) q^{65} +(-6.84928e8 - 1.53312e8i) q^{66} +(-1.98217e8 - 3.43322e8i) q^{67} +(-4.04974e8 + 2.33812e8i) q^{68} +(-5.68990e8 + 2.54200e9i) q^{69} +(1.20660e8 - 2.08989e8i) q^{70} +2.66602e9i q^{71} +(-6.81731e8 + 5.68419e7i) q^{72} -2.46835e9 q^{73} +(-1.09058e9 - 6.29649e8i) q^{74} +(1.41794e8 + 4.52933e8i) q^{75} +(-1.03079e8 - 1.78539e8i) q^{76} +(8.43612e8 - 4.87060e8i) q^{77} +(1.64847e9 + 1.51674e9i) q^{78} +(-5.47612e8 + 9.48492e8i) q^{79} +3.66357e8i q^{80} +(-3.43864e9 + 5.77433e8i) q^{81} -9.42957e8 q^{82} +(-2.26886e9 - 1.30993e9i) q^{83} +(6.42861e8 - 6.98693e8i) q^{84} +(6.38206e8 + 1.10541e9i) q^{85} +(-1.89262e9 + 1.09270e9i) q^{86} +(2.95244e9 - 9.24282e8i) q^{87} +(-7.39425e8 + 1.28072e9i) q^{88} +1.74674e9i q^{89} +(1.55154e8 + 1.86084e9i) q^{90} -3.10896e9 q^{91} +(4.75319e9 + 2.74426e9i) q^{92} +(-7.09162e9 - 1.58736e9i) q^{93} +(3.27368e9 + 5.67018e9i) q^{94} +(-4.87335e8 + 2.81363e8i) q^{95} +(-3.14844e8 + 1.40658e9i) q^{96} +(-7.50989e9 + 1.30075e10i) q^{97} -5.07397e9i q^{98} +(-3.21337e9 + 6.81831e9i) 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
\(3\) −164.534 + 178.823i −0.677093 + 0.735898i
\(4\) 256.000 + 443.405i 0.250000 + 0.433013i
\(5\) 1210.31 − 698.771i 0.387298 − 0.223607i
\(6\) 5247.34 − 1642.72i 0.674812 − 0.211255i
\(7\) −3815.60 + 6608.81i −0.227024 + 0.393218i −0.956925 − 0.290336i \(-0.906233\pi\)
0.729901 + 0.683553i \(0.239566\pi\)
\(8\) − 11585.2i − 0.353553i
\(9\) −4906.41 − 58844.8i −0.0830904 − 0.996542i
\(10\) −31622.8 −0.316228
\(11\) −110548. − 63824.8i −0.686415 − 0.396302i 0.115853 − 0.993266i \(-0.463040\pi\)
−0.802268 + 0.596965i \(0.796373\pi\)
\(12\) −121412. − 27176.3i −0.487926 − 0.109215i
\(13\) 203701. + 352820.i 0.548625 + 0.950246i 0.998369 + 0.0570886i \(0.0181817\pi\)
−0.449744 + 0.893157i \(0.648485\pi\)
\(14\) 149540. − 86337.1i 0.278047 − 0.160530i
\(15\) −74179.7 + 331402.i −0.0976853 + 0.436414i
\(16\) −131072. + 227023.i −0.125000 + 0.216506i
\(17\) 913327.i 0.643252i 0.946867 + 0.321626i \(0.104229\pi\)
−0.946867 + 0.321626i \(0.895771\pi\)
\(18\) −569607. + 1.20863e6i −0.301449 + 0.639632i
\(19\) −402654. −0.162616 −0.0813081 − 0.996689i \(-0.525910\pi\)
−0.0813081 + 0.996689i \(0.525910\pi\)
\(20\) 619677. + 357771.i 0.193649 + 0.111803i
\(21\) −554014. − 1.76969e6i −0.135651 − 0.433312i
\(22\) 1.44419e6 + 2.50141e6i 0.280228 + 0.485369i
\(23\) 9.28358e6 − 5.35988e6i 1.44237 − 0.832752i 0.444361 − 0.895848i \(-0.353431\pi\)
0.998007 + 0.0630961i \(0.0200975\pi\)
\(24\) 2.07171e6 + 1.90616e6i 0.260179 + 0.239388i
\(25\) 976562. − 1.69146e6i 0.100000 − 0.173205i
\(26\) − 9.21843e6i − 0.775873i
\(27\) 1.13301e7 + 8.80457e6i 0.789613 + 0.613605i
\(28\) −3.90717e6 −0.227024
\(29\) −1.10257e7 − 6.36570e6i −0.537548 − 0.310353i 0.206537 − 0.978439i \(-0.433781\pi\)
−0.744085 + 0.668085i \(0.767114\pi\)
\(30\) 5.20301e6 − 5.65488e6i 0.214116 − 0.232711i
\(31\) 1.49529e7 + 2.58991e7i 0.522296 + 0.904642i 0.999664 + 0.0259389i \(0.00825755\pi\)
−0.477368 + 0.878703i \(0.658409\pi\)
\(32\) 5.13695e6 − 2.96582e6i 0.153093 − 0.0883883i
\(33\) 2.96022e7 − 9.26718e6i 0.756404 − 0.236798i
\(34\) 1.03331e7 − 1.78975e7i 0.227424 − 0.393910i
\(35\) 1.06649e7i 0.203057i
\(36\) 2.48360e7 − 1.72398e7i 0.410743 − 0.285115i
\(37\) 5.56536e7 0.802574 0.401287 − 0.915952i \(-0.368563\pi\)
0.401287 + 0.915952i \(0.368563\pi\)
\(38\) 7.89037e6 + 4.55551e6i 0.0995817 + 0.0574935i
\(39\) −9.66079e7 − 2.16243e7i −1.07075 − 0.239673i
\(40\) −8.09543e6 − 1.40217e7i −0.0790569 − 0.136931i
\(41\) 3.60900e7 − 2.08366e7i 0.311507 − 0.179849i −0.336094 − 0.941829i \(-0.609106\pi\)
0.647601 + 0.761980i \(0.275772\pi\)
\(42\) −9.16533e6 + 4.09466e7i −0.0701296 + 0.313308i
\(43\) 4.82911e7 − 8.36426e7i 0.328492 − 0.568965i −0.653721 − 0.756736i \(-0.726793\pi\)
0.982213 + 0.187771i \(0.0601263\pi\)
\(44\) − 6.53566e7i − 0.396302i
\(45\) −4.70573e7 − 6.77918e7i −0.255014 − 0.367379i
\(46\) −2.42560e8 −1.17769
\(47\) −2.50589e8 − 1.44678e8i −1.09263 − 0.630829i −0.158353 − 0.987383i \(-0.550618\pi\)
−0.934275 + 0.356553i \(0.883952\pi\)
\(48\) −1.90313e7 − 6.07917e7i −0.0746899 − 0.238582i
\(49\) 1.12120e8 + 1.94198e8i 0.396920 + 0.687485i
\(50\) −3.82733e7 + 2.20971e7i −0.122474 + 0.0707107i
\(51\) −1.63324e8 − 1.50273e8i −0.473368 − 0.435542i
\(52\) −1.04295e8 + 1.80644e8i −0.274312 + 0.475123i
\(53\) 2.20997e8i 0.528455i 0.964460 + 0.264227i \(0.0851169\pi\)
−0.964460 + 0.264227i \(0.914883\pi\)
\(54\) −1.22411e8 − 3.00719e8i −0.266595 − 0.654925i
\(55\) −1.78396e8 −0.354463
\(56\) 7.65646e7 + 4.42046e7i 0.139023 + 0.0802652i
\(57\) 6.62501e7 − 7.20038e7i 0.110106 − 0.119669i
\(58\) 1.44039e8 + 2.49484e8i 0.219453 + 0.380104i
\(59\) 6.64653e7 − 3.83738e7i 0.0929683 − 0.0536753i −0.452795 − 0.891615i \(-0.649573\pi\)
0.545763 + 0.837939i \(0.316240\pi\)
\(60\) −1.65935e8 + 5.19473e7i −0.213394 + 0.0668047i
\(61\) −5.20326e8 + 9.01231e8i −0.616065 + 1.06706i 0.374132 + 0.927375i \(0.377941\pi\)
−0.990197 + 0.139680i \(0.955393\pi\)
\(62\) − 6.76690e8i − 0.738637i
\(63\) 4.07615e8 + 1.92103e8i 0.410721 + 0.193567i
\(64\) −1.34218e8 −0.125000
\(65\) 4.93080e8 + 2.84680e8i 0.424963 + 0.245352i
\(66\) −6.84928e8 − 1.53312e8i −0.546922 − 0.122421i
\(67\) −1.98217e8 − 3.43322e8i −0.146814 − 0.254289i 0.783234 − 0.621727i \(-0.213569\pi\)
−0.930048 + 0.367437i \(0.880235\pi\)
\(68\) −4.04974e8 + 2.33812e8i −0.278536 + 0.160813i
\(69\) −5.68990e8 + 2.54200e9i −0.363797 + 1.62529i
\(70\) 1.20660e8 − 2.08989e8i 0.0717914 − 0.124346i
\(71\) 2.66602e9i 1.47765i 0.673898 + 0.738824i \(0.264619\pi\)
−0.673898 + 0.738824i \(0.735381\pi\)
\(72\) −6.81731e8 + 5.68419e7i −0.352331 + 0.0293769i
\(73\) −2.46835e9 −1.19067 −0.595337 − 0.803476i \(-0.702981\pi\)
−0.595337 + 0.803476i \(0.702981\pi\)
\(74\) −1.09058e9 − 6.29649e8i −0.491474 − 0.283753i
\(75\) 1.41794e8 + 4.52933e8i 0.0597519 + 0.190866i
\(76\) −1.03079e8 − 1.78539e8i −0.0406541 − 0.0704149i
\(77\) 8.43612e8 − 4.87060e8i 0.311666 − 0.179940i
\(78\) 1.64847e9 + 1.51674e9i 0.570963 + 0.525338i
\(79\) −5.47612e8 + 9.48492e8i −0.177966 + 0.308246i −0.941184 − 0.337895i \(-0.890285\pi\)
0.763218 + 0.646142i \(0.223618\pi\)
\(80\) 3.66357e8i 0.111803i
\(81\) −3.43864e9 + 5.77433e8i −0.986192 + 0.165606i
\(82\) −9.42957e8 −0.254345
\(83\) −2.26886e9 − 1.30993e9i −0.575993 − 0.332550i 0.183546 − 0.983011i \(-0.441242\pi\)
−0.759539 + 0.650461i \(0.774576\pi\)
\(84\) 6.42861e8 − 6.98693e8i 0.153717 − 0.167067i
\(85\) 6.38206e8 + 1.10541e9i 0.143836 + 0.249131i
\(86\) −1.89262e9 + 1.09270e9i −0.402319 + 0.232279i
\(87\) 2.95244e9 − 9.24282e8i 0.592358 − 0.185442i
\(88\) −7.39425e8 + 1.28072e9i −0.140114 + 0.242684i
\(89\) 1.74674e9i 0.312808i 0.987693 + 0.156404i \(0.0499902\pi\)
−0.987693 + 0.156404i \(0.950010\pi\)
\(90\) 1.55154e8 + 1.86084e9i 0.0262755 + 0.315134i
\(91\) −3.10896e9 −0.498205
\(92\) 4.75319e9 + 2.74426e9i 0.721184 + 0.416376i
\(93\) −7.09162e9 − 1.58736e9i −1.01937 − 0.228171i
\(94\) 3.27368e9 + 5.67018e9i 0.446064 + 0.772605i
\(95\) −4.87335e8 + 2.81363e8i −0.0629810 + 0.0363621i
\(96\) −3.14844e8 + 1.40658e9i −0.0386135 + 0.172508i
\(97\) −7.50989e9 + 1.30075e10i −0.874530 + 1.51473i −0.0172680 + 0.999851i \(0.505497\pi\)
−0.857262 + 0.514880i \(0.827836\pi\)
\(98\) − 5.07397e9i − 0.561330i
\(99\) −3.21337e9 + 6.81831e9i −0.337897 + 0.716970i
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.11.5 ✓ 80
3.2 odd 2 270.11.h.a.251.29 80
9.4 even 3 270.11.h.a.71.29 80
9.5 odd 6 inner 90.11.h.a.41.5 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.5 ✓ 80 1.1 even 1 trivial
90.11.h.a.41.5 yes 80 9.5 odd 6 inner
270.11.h.a.71.29 80 9.4 even 3
270.11.h.a.251.29 80 3.2 odd 2