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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 41.4
Character \(\chi\) \(=\) 90.41
Dual form 90.11.h.a.11.4

$q$-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 \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\)

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 −0.956181 − 0.292777i \(-0.905421\pi\)
0.224538 − 0.974465i \(-0.427913\pi\)
\(8\) 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 −0.204676 − 0.978830i \(-0.565614\pi\)
0.745353 + 0.666670i \(0.232281\pi\)
\(12\) 16505.4 + 123316.i 0.0663315 + 0.495581i
\(13\) 248770. − 430883.i 0.670011 − 1.16049i −0.307890 − 0.951422i \(-0.599623\pi\)
0.977901 − 0.209071i \(-0.0670438\pi\)
\(14\) 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 \(-0.846297\pi\)
0.885666 − 0.464324i \(-0.153703\pi\)
\(18\) 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 0.556472 − 0.830866i \(-0.312155\pi\)
−0.441315 + 0.897352i \(0.645488\pi\)
\(24\) −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 0.231851 − 0.972751i \(-0.425522\pi\)
0.958353 + 0.285587i \(0.0921885\pi\)
\(30\) −6.08629e6 + 4.69106e6i −0.250465 + 0.193048i
\(31\) −3.92422e6 + 6.79695e6i −0.137071 + 0.237413i −0.926387 − 0.376574i \(-0.877102\pi\)
0.789316 + 0.613987i \(0.210435\pi\)
\(32\) 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 0.664290 − 0.747475i \(-0.268734\pi\)
−0.315187 + 0.949030i \(0.602067\pi\)
\(42\) −1.34031e8 + 1.79395e7i −1.02555 + 0.137266i
\(43\) −9.45742e7 − 1.63807e8i −0.643325 − 1.11427i −0.984686 − 0.174339i \(-0.944221\pi\)
0.341361 − 0.939932i \(-0.389112\pi\)
\(44\) − 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 −0.462824 − 0.886450i \(-0.653164\pi\)
0.536276 + 0.844043i \(0.319831\pi\)
\(48\) 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 \(-0.872530\pi\)
0.920882 − 0.389840i \(-0.127470\pi\)
\(54\) −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 −0.241050 − 0.970513i \(-0.577492\pi\)
−0.961014 + 0.276501i \(0.910825\pi\)
\(60\) 6.61933e7 − 1.60784e8i 0.0851251 − 0.206770i
\(61\) 2.34338e8 + 4.05885e8i 0.277456 + 0.480567i 0.970752 − 0.240085i \(-0.0771755\pi\)
−0.693296 + 0.720653i \(0.743842\pi\)
\(62\) − 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 0.100674 − 0.994920i \(-0.467900\pi\)
0.811289 − 0.584646i \(-0.198766\pi\)
\(68\) −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 \(-0.831714\pi\)
0.863471 − 0.504398i \(-0.168286\pi\)
\(72\) 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 0.948938 + 0.315463i \(0.102160\pi\)
−0.201270 + 0.979536i \(0.564507\pi\)
\(80\) 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 −0.684381 − 0.729125i \(-0.739927\pi\)
0.289250 + 0.957253i \(0.406594\pi\)
\(84\) 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 \(-0.867534\pi\)
0.914650 − 0.404247i \(-0.132466\pi\)
\(90\) 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 −0.965467 − 0.260527i \(-0.916104\pi\)
0.257111 − 0.966382i \(-0.417230\pi\)
\(98\) − 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.41.4 yes 80
3.2 odd 2 270.11.h.a.71.38 80
9.2 odd 6 inner 90.11.h.a.11.4 ✓ 80
9.7 even 3 270.11.h.a.251.38 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.4 ✓ 80 9.2 odd 6 inner
90.11.h.a.41.4 yes 80 1.1 even 1 trivial
270.11.h.a.71.38 80 3.2 odd 2
270.11.h.a.251.38 80 9.7 even 3