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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 11.4
Character \(\chi\) \(=\) 90.11
Dual form 90.11.h.a.41.4

$q$-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} +(-192.466 - 148.344i) q^{3} +(256.000 + 443.405i) q^{4} +(-1210.31 + 698.771i) q^{5} +(2093.22 + 5084.44i) q^{6} +(-12296.7 + 21298.5i) q^{7} -11585.2i q^{8} +(15037.0 + 57102.3i) q^{9} +31622.8 q^{10} +(87076.5 + 50273.7i) q^{11} +(16505.4 - 123316. i) q^{12} +(248770. + 430883. i) q^{13} +(481931. - 278243. i) q^{14} +(336601. + 45052.7i) q^{15} +(-131072. + 227023. i) q^{16} +1.31855e6i q^{17} +(351376. - 1.28910e6i) q^{18} -3.30546e6 q^{19} +(-619677. - 357771. i) q^{20} +(5.52621e6 - 2.27509e6i) q^{21} +(-1.13756e6 - 1.97032e6i) q^{22} +(741187. - 427924. i) q^{23} +(-1.71860e6 + 2.22976e6i) q^{24} +(976562. - 1.69146e6i) q^{25} -1.12581e7i q^{26} +(5.57670e6 - 1.32209e7i) q^{27} -1.25918e7 q^{28} +(2.44124e7 + 1.40945e7i) q^{29} +(-6.08629e6 - 4.69106e6i) q^{30} +(-3.92422e6 - 6.79695e6i) q^{31} +(5.13695e6 - 2.96582e6i) q^{32} +(-9.30142e6 - 2.25932e7i) q^{33} +(1.49177e7 - 2.58381e7i) q^{34} -3.43704e7i q^{35} +(-2.14700e7 + 2.12857e7i) q^{36} +8.06236e7 q^{37} +(6.47735e7 + 3.73970e7i) q^{38} +(1.60393e7 - 1.19834e8i) q^{39} +(8.09543e6 + 1.40217e7i) q^{40} +(4.04458e7 - 2.33514e7i) q^{41} +(-1.34031e8 - 1.79395e7i) q^{42} +(-9.45742e7 + 1.63807e8i) q^{43} +5.14802e7i q^{44} +(-5.81008e7 - 5.86039e7i) q^{45} -1.93656e7 q^{46} +(1.68457e7 + 9.72589e6i) q^{47} +(5.89045e7 - 2.42504e7i) q^{48} +(-1.61181e8 - 2.79173e8i) q^{49} +(-3.82733e7 + 2.20971e7i) q^{50} +(1.95599e8 - 2.53775e8i) q^{51} +(-1.27370e8 + 2.20612e8i) q^{52} +3.26059e8i q^{53} +(-2.58858e8 + 1.95982e8i) q^{54} -1.40519e8 q^{55} +(2.46749e8 + 1.42460e8i) q^{56} +(6.36187e8 + 4.90346e8i) q^{57} +(-3.18923e8 - 5.52391e8i) q^{58} +(-8.59385e8 + 4.96166e8i) q^{59} +(6.61933e7 + 1.60784e8i) q^{60} +(2.34338e8 - 4.05885e8i) q^{61} +1.77590e8i q^{62} +(-1.40110e9 - 3.81906e8i) q^{63} -1.34218e8 q^{64} +(-6.02177e8 - 3.47667e8i) q^{65} +(-7.33435e7 + 5.47969e8i) q^{66} +(1.23126e9 + 2.13261e9i) q^{67} +(-5.84650e8 + 3.37548e8i) q^{68} +(-2.06133e8 - 2.75901e7i) q^{69} +(-3.88856e8 + 6.73519e8i) q^{70} +1.82010e9i q^{71} +(6.61544e8 - 1.74207e8i) q^{72} -1.52969e9 q^{73} +(-1.57989e9 - 9.12152e8i) q^{74} +(-4.38872e8 + 1.80680e8i) q^{75} +(-8.46197e8 - 1.46566e9i) q^{76} +(-2.14151e9 + 1.23640e9i) q^{77} +(-1.67007e9 + 2.16679e9i) q^{78} +(2.30062e9 - 3.98479e9i) q^{79} -3.66357e8i q^{80} +(-3.03456e9 + 1.71729e9i) q^{81} -1.05676e9 q^{82} +(-1.55643e9 - 8.98608e8i) q^{83} +(2.42349e9 + 1.86793e9i) q^{84} +(-9.21363e8 - 1.59585e9i) q^{85} +(3.70654e9 - 2.13997e9i) q^{86} +(-2.60771e9 - 6.33416e9i) q^{87} +(5.82432e8 - 1.00880e9i) q^{88} +4.51468e9i q^{89} +(4.75511e8 + 1.80573e9i) q^{90} -1.22362e10 q^{91} +(3.79488e8 + 2.19097e8i) q^{92} +(-2.53011e8 + 1.89031e9i) q^{93} +(-2.20072e8 - 3.81175e8i) q^{94} +(4.00062e9 - 2.30976e9i) q^{95} +(-1.42865e9 - 1.91219e8i) q^{96} +(-6.08289e9 + 1.05359e10i) q^{97} +7.29421e9i q^{98} +(-1.56137e9 + 5.72823e9i) 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
\(3\) −192.466 − 148.344i −0.792039 − 0.610470i
\(4\) 256.000 + 443.405i 0.250000 + 0.433013i
\(5\) −1210.31 + 698.771i −0.387298 + 0.223607i
\(6\) 2093.22 + 5084.44i 0.269189 + 0.653863i
\(7\) −12296.7 + 21298.5i −0.731643 + 1.26724i 0.224538 + 0.974465i \(0.427913\pi\)
−0.956181 + 0.292777i \(0.905421\pi\)
\(8\) − 11585.2i − 0.353553i
\(9\) 15037.0 + 57102.3i 0.254652 + 0.967033i
\(10\) 31622.8 0.316228
\(11\) 87076.5 + 50273.7i 0.540677 + 0.312160i 0.745353 − 0.666670i \(-0.232281\pi\)
−0.204676 + 0.978830i \(0.565614\pi\)
\(12\) 16505.4 − 123316.i 0.0663315 − 0.495581i
\(13\) 248770. + 430883.i 0.670011 + 1.16049i 0.977901 + 0.209071i \(0.0670438\pi\)
−0.307890 + 0.951422i \(0.599623\pi\)
\(14\) 481931. − 278243.i 0.896075 − 0.517349i
\(15\) 336601. + 45052.7i 0.443261 + 0.0593287i
\(16\) −131072. + 227023.i −0.125000 + 0.216506i
\(17\) 1.31855e6i 0.928648i 0.885666 + 0.464324i \(0.153703\pi\)
−0.885666 + 0.464324i \(0.846297\pi\)
\(18\) 351376. − 1.28910e6i 0.185956 − 0.682217i
\(19\) −3.30546e6 −1.33495 −0.667473 − 0.744634i \(-0.732624\pi\)
−0.667473 + 0.744634i \(0.732624\pi\)
\(20\) −619677. − 357771.i −0.193649 − 0.111803i
\(21\) 5.52621e6 − 2.27509e6i 1.35310 − 0.557060i
\(22\) −1.13756e6 − 1.97032e6i −0.220730 − 0.382316i
\(23\) 741187. − 427924.i 0.115156 − 0.0664856i −0.441315 − 0.897352i \(-0.645488\pi\)
0.556472 + 0.830866i \(0.312155\pi\)
\(24\) −1.71860e6 + 2.22976e6i −0.215834 + 0.280028i
\(25\) 976562. − 1.69146e6i 0.100000 − 0.173205i
\(26\) − 1.12581e7i − 0.947538i
\(27\) 5.57670e6 − 1.32209e7i 0.388650 − 0.921385i
\(28\) −1.25918e7 −0.731643
\(29\) 2.44124e7 + 1.40945e7i 1.19020 + 0.687165i 0.958353 − 0.285587i \(-0.0921885\pi\)
0.231851 + 0.972751i \(0.425522\pi\)
\(30\) −6.08629e6 − 4.69106e6i −0.250465 − 0.193048i
\(31\) −3.92422e6 − 6.79695e6i −0.137071 − 0.237413i 0.789316 − 0.613987i \(-0.210435\pi\)
−0.926387 + 0.376574i \(0.877102\pi\)
\(32\) 5.13695e6 − 2.96582e6i 0.153093 − 0.0883883i
\(33\) −9.30142e6 − 2.25932e7i −0.237673 − 0.577310i
\(34\) 1.49177e7 − 2.58381e7i 0.328326 − 0.568678i
\(35\) − 3.43704e7i − 0.654401i
\(36\) −2.14700e7 + 2.12857e7i −0.355074 + 0.352026i
\(37\) 8.06236e7 1.16266 0.581331 − 0.813667i \(-0.302532\pi\)
0.581331 + 0.813667i \(0.302532\pi\)
\(38\) 6.47735e7 + 3.73970e7i 0.817484 + 0.471975i
\(39\) 1.60393e7 − 1.19834e8i 0.177771 − 1.32818i
\(40\) 8.09543e6 + 1.40217e7i 0.0790569 + 0.136931i
\(41\) 4.04458e7 − 2.33514e7i 0.349104 − 0.201555i −0.315187 − 0.949030i \(-0.602067\pi\)
0.664290 + 0.747475i \(0.268734\pi\)
\(42\) −1.34031e8 − 1.79395e7i −1.02555 − 0.137266i
\(43\) −9.45742e7 + 1.63807e8i −0.643325 + 1.11427i 0.341361 + 0.939932i \(0.389112\pi\)
−0.984686 + 0.174339i \(0.944221\pi\)
\(44\) 5.14802e7i 0.312160i
\(45\) −5.81008e7 − 5.86039e7i −0.314862 − 0.317588i
\(46\) −1.93656e7 −0.0940249
\(47\) 1.68457e7 + 9.72589e6i 0.0734515 + 0.0424072i 0.536276 − 0.844043i \(-0.319831\pi\)
−0.462824 + 0.886450i \(0.653164\pi\)
\(48\) 5.89045e7 − 2.42504e7i 0.231176 − 0.0951728i
\(49\) −1.61181e8 − 2.79173e8i −0.570602 − 0.988311i
\(50\) −3.82733e7 + 2.20971e7i −0.122474 + 0.0707107i
\(51\) 1.95599e8 − 2.53775e8i 0.566912 − 0.735525i
\(52\) −1.27370e8 + 2.20612e8i −0.335005 + 0.580246i
\(53\) 3.26059e8i 0.779681i 0.920882 + 0.389840i \(0.127470\pi\)
−0.920882 + 0.389840i \(0.872530\pi\)
\(54\) −2.58858e8 + 1.95982e8i −0.563757 + 0.426823i
\(55\) −1.40519e8 −0.279204
\(56\) 2.46749e8 + 1.42460e8i 0.448038 + 0.258675i
\(57\) 6.36187e8 + 4.90346e8i 1.05733 + 0.814945i
\(58\) −3.18923e8 − 5.52391e8i −0.485899 − 0.841601i
\(59\) −8.59385e8 + 4.96166e8i −1.20206 + 0.694012i −0.961014 − 0.276501i \(-0.910825\pi\)
−0.241050 + 0.970513i \(0.577492\pi\)
\(60\) 6.61933e7 + 1.60784e8i 0.0851251 + 0.206770i
\(61\) 2.34338e8 − 4.05885e8i 0.277456 − 0.480567i −0.693296 − 0.720653i \(-0.743842\pi\)
0.970752 + 0.240085i \(0.0771755\pi\)
\(62\) 1.77590e8i 0.193847i
\(63\) −1.40110e9 − 3.81906e8i −1.41178 − 0.384816i
\(64\) −1.34218e8 −0.125000
\(65\) −6.02177e8 − 3.47667e8i −0.518988 − 0.299638i
\(66\) −7.33435e7 + 5.47969e8i −0.0585655 + 0.437559i
\(67\) 1.23126e9 + 2.13261e9i 0.911962 + 1.57957i 0.811289 + 0.584646i \(0.198766\pi\)
0.100674 + 0.994920i \(0.467900\pi\)
\(68\) −5.84650e8 + 3.37548e8i −0.402116 + 0.232162i
\(69\) −2.06133e8 − 2.75901e7i −0.131796 − 0.0176404i
\(70\) −3.88856e8 + 6.73519e8i −0.231366 + 0.400737i
\(71\) 1.82010e9i 1.00880i 0.863471 + 0.504398i \(0.168286\pi\)
−0.863471 + 0.504398i \(0.831714\pi\)
\(72\) 6.61544e8 − 1.74207e8i 0.341898 − 0.0900332i
\(73\) −1.52969e9 −0.737884 −0.368942 − 0.929452i \(-0.620280\pi\)
−0.368942 + 0.929452i \(0.620280\pi\)
\(74\) −1.57989e9 − 9.12152e8i −0.711982 − 0.411063i
\(75\) −4.38872e8 + 1.80680e8i −0.184940 + 0.0761382i
\(76\) −8.46197e8 − 1.46566e9i −0.333736 − 0.578049i
\(77\) −2.14151e9 + 1.23640e9i −0.791164 + 0.456779i
\(78\) −1.67007e9 + 2.16679e9i −0.578444 + 0.750487i
\(79\) 2.30062e9 − 3.98479e9i 0.747668 − 1.29500i −0.201270 − 0.979536i \(-0.564507\pi\)
0.948938 − 0.315463i \(-0.102160\pi\)
\(80\) − 3.66357e8i − 0.111803i
\(81\) −3.03456e9 + 1.71729e9i −0.870304 + 0.492514i
\(82\) −1.05676e9 −0.285042
\(83\) −1.55643e9 − 8.98608e8i −0.395130 − 0.228129i 0.289250 − 0.957253i \(-0.406594\pi\)
−0.684381 + 0.729125i \(0.739927\pi\)
\(84\) 2.42349e9 + 1.86793e9i 0.579490 + 0.446646i
\(85\) −9.21363e8 − 1.59585e9i −0.207652 − 0.359664i
\(86\) 3.70654e9 − 2.13997e9i 0.787909 − 0.454899i
\(87\) −2.60771e9 − 6.33416e9i −0.523195 − 1.27085i
\(88\) 5.82432e8 − 1.00880e9i 0.110365 − 0.191158i
\(89\) 4.51468e9i 0.808495i 0.914650 + 0.404247i \(0.132466\pi\)
−0.914650 + 0.404247i \(0.867534\pi\)
\(90\) 4.75511e8 + 1.80573e9i 0.0805281 + 0.305803i
\(91\) −1.22362e10 −1.96083
\(92\) 3.79488e8 + 2.19097e8i 0.0575782 + 0.0332428i
\(93\) −2.53011e8 + 1.89031e9i −0.0363684 + 0.271718i
\(94\) −2.20072e8 − 3.81175e8i −0.0299864 − 0.0519380i
\(95\) 4.00062e9 − 2.30976e9i 0.517022 − 0.298503i
\(96\) −1.42865e9 − 1.91219e8i −0.175214 − 0.0234517i
\(97\) −6.08289e9 + 1.05359e10i −0.708356 + 1.22691i 0.257111 + 0.966382i \(0.417230\pi\)
−0.965467 + 0.260527i \(0.916104\pi\)
\(98\) 7.29421e9i 0.806953i
\(99\) −1.56137e9 + 5.72823e9i −0.164184 + 0.602344i
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.11.4 ✓ 80
3.2 odd 2 270.11.h.a.251.38 80
9.4 even 3 270.11.h.a.71.38 80
9.5 odd 6 inner 90.11.h.a.41.4 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.4 ✓ 80 1.1 even 1 trivial
90.11.h.a.41.4 yes 80 9.5 odd 6 inner
270.11.h.a.71.38 80 9.4 even 3
270.11.h.a.251.38 80 3.2 odd 2