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Properties

Label 90.11.k.b.7.9
Level $90$
Weight $11$
Character 90.7
Analytic conductor $57.182$
Analytic rank $0$
Dimension $120$
Inner twists $4$

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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(7,90)] 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("90.7"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([8, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 90.k (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [120,960] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(57.1821527406\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(30\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 7.9
Character \(\chi\) \(=\) 90.7
Dual form 90.11.k.b.13.9

$q$-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+(21.8564 - 5.85641i) q^{2} +(-156.929 + 185.532i) q^{3} +(443.405 - 256.000i) q^{4} +(-1734.18 - 2599.66i) q^{5} +(-2343.36 + 4974.11i) q^{6} +(-12056.6 + 3230.55i) q^{7} +(8192.00 - 8192.00i) q^{8} +(-9795.42 - 58230.9i) q^{9} +(-53127.6 - 46663.2i) q^{10} +(141276. - 244697. i) q^{11} +(-22087.0 + 122440. i) q^{12} +(222194. + 59536.6i) q^{13} +(-244594. + 141216. i) q^{14} +(754465. + 86217.0i) q^{15} +(131072. - 227023. i) q^{16} +(763689. + 763689. i) q^{17} +(-555116. - 1.21535e6i) q^{18} +3.76206e6i q^{19} +(-1.43446e6 - 708754. i) q^{20} +(1.29266e6 - 2.74385e6i) q^{21} +(1.65474e6 - 6.17557e6i) q^{22} +(-9.22770e6 - 2.47255e6i) q^{23} +(234316. + 2.80544e6i) q^{24} +(-3.75087e6 + 9.01656e6i) q^{25} +5.20502e6 q^{26} +(1.23409e7 + 7.32076e6i) q^{27} +(-4.51892e6 + 4.51892e6i) q^{28} +(-2.89894e7 - 1.67371e7i) q^{29} +(1.69948e7 - 2.53406e6i) q^{30} +(938620. + 1.62574e6i) q^{31} +(1.53522e6 - 5.72953e6i) q^{32} +(2.32289e7 + 6.46114e7i) q^{33} +(2.11640e7 + 1.22190e7i) q^{34} +(2.93066e7 + 2.57407e7i) q^{35} +(-1.92504e7 - 2.33122e7i) q^{36} +(3.68961e7 + 3.68961e7i) q^{37} +(2.20322e7 + 8.22252e7i) q^{38} +(-4.59146e7 + 3.18810e7i) q^{39} +(-3.55028e7 - 7.09004e6i) q^{40} +(-6.17418e7 - 1.06940e8i) q^{41} +(1.21837e7 - 6.75410e7i) q^{42} +(-1.67609e7 - 6.25524e7i) q^{43} -1.44667e8i q^{44} +(-1.34394e8 + 1.26448e8i) q^{45} -2.16165e8 q^{46} +(-3.65498e7 + 9.79349e6i) q^{47} +(2.15511e7 + 5.99447e7i) q^{48} +(-1.09706e8 + 6.33390e7i) q^{49} +(-2.91758e7 + 2.19036e8i) q^{50} +(-2.61534e8 + 2.18438e7i) q^{51} +(1.13763e8 - 3.04827e7i) q^{52} +(1.19005e8 - 1.19005e8i) q^{53} +(3.12601e8 + 8.77322e7i) q^{54} +(-8.81129e8 + 5.70789e7i) q^{55} +(-7.23027e7 + 1.25232e8i) q^{56} +(-6.97984e8 - 5.90378e8i) q^{57} +(-7.31624e8 - 1.96038e8i) q^{58} +(-5.81549e8 + 3.35757e8i) q^{59} +(3.56605e8 - 1.54914e8i) q^{60} +(-7.09977e8 + 1.22972e9i) q^{61} +(3.00358e7 + 3.00358e7i) q^{62} +(3.06217e8 + 6.70420e8i) q^{63} -1.34218e8i q^{64} +(-2.30548e8 - 6.80876e8i) q^{65} +(8.86091e8 + 1.27614e9i) q^{66} +(-3.08636e8 + 1.15185e9i) q^{67} +(5.34128e8 + 1.43119e8i) q^{68} +(1.90683e9 - 1.32402e9i) q^{69} +(7.91284e8 + 3.90967e8i) q^{70} -1.38118e9 q^{71} +(-5.57271e8 - 3.96783e8i) q^{72} +(3.58478e8 - 3.58478e8i) q^{73} +(1.02249e9 + 5.90338e8i) q^{74} +(-1.08424e9 - 2.11087e9i) q^{75} +(9.63088e8 + 1.66812e9i) q^{76} +(-9.12798e8 + 3.40661e9i) q^{77} +(-8.16821e8 + 9.65700e8i) q^{78} +(3.15863e9 + 1.82364e9i) q^{79} +(-8.17487e8 + 5.29562e7i) q^{80} +(-3.29488e9 + 1.14079e9i) q^{81} +(-1.97574e9 - 1.97574e9i) q^{82} +(8.09728e8 + 3.02194e9i) q^{83} +(-1.29255e8 - 1.54756e9i) q^{84} +(6.60960e8 - 3.30971e9i) q^{85} +(-7.32664e8 - 1.26901e9i) q^{86} +(7.65456e9 - 2.75194e9i) q^{87} +(-8.47227e8 - 3.16189e9i) q^{88} +4.41863e9i q^{89} +(-2.19683e9 + 3.55075e9i) q^{90} -2.87123e9 q^{91} +(-4.72458e9 + 1.26595e9i) q^{92} +(-4.48924e8 - 8.09815e7i) q^{93} +(-7.41493e8 + 4.28101e8i) q^{94} +(9.78010e9 - 6.52409e9i) q^{95} +(8.22091e8 + 1.18396e9i) q^{96} +(-6.73424e9 + 1.80443e9i) q^{97} +(-2.02685e9 + 2.02685e9i) q^{98} +(-1.56328e10 - 5.82971e9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 960 q^{2} - 128 q^{3} - 8192 q^{6} + 1830 q^{7} + 983040 q^{8} + 105792 q^{10} + 217740 q^{11} - 65536 q^{12} - 3385658 q^{15} + 15728640 q^{16} + 2438244 q^{17} + 2534464 q^{18} + 1692672 q^{20}+ \cdots + 76966514304 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 21.8564 − 5.85641i 0.683013 − 0.183013i
\(3\) −156.929 + 185.532i −0.645799 + 0.763507i
\(4\) 443.405 − 256.000i 0.433013 − 0.250000i
\(5\) −1734.18 − 2599.66i −0.554937 − 0.831892i
\(6\) −2343.36 + 4974.11i −0.301358 + 0.639675i
\(7\) −12056.6 + 3230.55i −0.717354 + 0.192214i −0.598991 − 0.800756i \(-0.704431\pi\)
−0.118363 + 0.992970i \(0.537765\pi\)
\(8\) 8192.00 − 8192.00i 0.250000 − 0.250000i
\(9\) −9795.42 − 58230.9i −0.165886 − 0.986145i
\(10\) −53127.6 − 46663.2i −0.531276 − 0.466632i
\(11\) 141276. − 244697.i 0.877213 − 1.51938i 0.0228268 − 0.999739i \(-0.492733\pi\)
0.854386 − 0.519638i \(-0.173933\pi\)
\(12\) −22087.0 + 122440.i −0.0887626 + 0.492058i
\(13\) 222194. + 59536.6i 0.598432 + 0.160349i 0.545304 − 0.838238i \(-0.316414\pi\)
0.0531279 + 0.998588i \(0.483081\pi\)
\(14\) −244594. + 141216.i −0.454784 + 0.262570i
\(15\) 754465. + 86217.0i 0.993534 + 0.113537i
\(16\) 131072. − 227023.i 0.125000 − 0.216506i
\(17\) 763689. + 763689.i 0.537863 + 0.537863i 0.922901 − 0.385038i \(-0.125812\pi\)
−0.385038 + 0.922901i \(0.625812\pi\)
\(18\) −555116. − 1.21535e6i −0.293779 − 0.643190i
\(19\) 3.76206e6i 1.51935i 0.650302 + 0.759676i \(0.274642\pi\)
−0.650302 + 0.759676i \(0.725358\pi\)
\(20\) −1.43446e6 − 708754.i −0.448268 − 0.221485i
\(21\) 1.29266e6 − 2.74385e6i 0.316510 − 0.671837i
\(22\) 1.65474e6 − 6.17557e6i 0.321082 − 1.19830i
\(23\) −9.22770e6 − 2.47255e6i −1.43369 − 0.384155i −0.543368 − 0.839494i \(-0.682851\pi\)
−0.890318 + 0.455339i \(0.849518\pi\)
\(24\) 234316. + 2.80544e6i 0.0294269 + 0.352327i
\(25\) −3.75087e6 + 9.01656e6i −0.384089 + 0.923296i
\(26\) 5.20502e6 0.438083
\(27\) 1.23409e7 + 7.32076e6i 0.860058 + 0.510196i
\(28\) −4.51892e6 + 4.51892e6i −0.262570 + 0.262570i
\(29\) −2.89894e7 − 1.67371e7i −1.41335 − 0.815998i −0.417648 − 0.908609i \(-0.637145\pi\)
−0.995702 + 0.0926106i \(0.970479\pi\)
\(30\) 1.69948e7 − 2.53406e6i 0.699375 − 0.104282i
\(31\) 938620. + 1.62574e6i 0.0327855 + 0.0567861i 0.881953 − 0.471338i \(-0.156229\pi\)
−0.849167 + 0.528124i \(0.822896\pi\)
\(32\) 1.53522e6 − 5.72953e6i 0.0457532 − 0.170753i
\(33\) 2.32289e7 + 6.46114e7i 0.593552 + 1.65097i
\(34\) 2.11640e7 + 1.22190e7i 0.465803 + 0.268932i
\(35\) 2.93066e7 + 2.57407e7i 0.557988 + 0.490094i
\(36\) −1.92504e7 − 2.33122e7i −0.318367 − 0.385542i
\(37\) 3.68961e7 + 3.68961e7i 0.532074 + 0.532074i 0.921189 − 0.389115i \(-0.127219\pi\)
−0.389115 + 0.921189i \(0.627219\pi\)
\(38\) 2.20322e7 + 8.22252e7i 0.278061 + 1.03774i
\(39\) −4.59146e7 + 3.18810e7i −0.508895 + 0.353354i
\(40\) −3.55028e7 − 7.09004e6i −0.346707 − 0.0692387i
\(41\) −6.17418e7 − 1.06940e8i −0.532917 − 0.923040i −0.999261 − 0.0384361i \(-0.987762\pi\)
0.466344 − 0.884603i \(-0.345571\pi\)
\(42\) 1.21837e7 − 6.75410e7i 0.0932254 − 0.516798i
\(43\) −1.67609e7 − 6.25524e7i −0.114013 − 0.425502i 0.885198 − 0.465214i \(-0.154023\pi\)
−0.999211 + 0.0397120i \(0.987356\pi\)
\(44\) − 1.44667e8i − 0.877213i
\(45\) −1.34394e8 + 1.26448e8i −0.728310 + 0.685248i
\(46\) −2.16165e8 −1.04953
\(47\) −3.65498e7 + 9.79349e6i −0.159366 + 0.0427020i −0.337620 − 0.941283i \(-0.609622\pi\)
0.178254 + 0.983985i \(0.442955\pi\)
\(48\) 2.15511e7 + 5.99447e7i 0.0845792 + 0.235258i
\(49\) −1.09706e8 + 6.33390e7i −0.388375 + 0.224229i
\(50\) −2.91758e7 + 2.19036e8i −0.0933627 + 0.700916i
\(51\) −2.61534e8 + 2.18438e7i −0.758014 + 0.0633107i
\(52\) 1.13763e8 − 3.04827e7i 0.299216 − 0.0801747i
\(53\) 1.19005e8 − 1.19005e8i 0.284568 − 0.284568i −0.550360 − 0.834928i \(-0.685509\pi\)
0.834928 + 0.550360i \(0.185509\pi\)
\(54\) 3.12601e8 + 8.77322e7i 0.680803 + 0.191069i
\(55\) −8.81129e8 + 5.70789e7i −1.75076 + 0.113413i
\(56\) −7.23027e7 + 1.25232e8i −0.131285 + 0.227392i
\(57\) −6.97984e8 − 5.90378e8i −1.16004 − 0.981196i
\(58\) −7.31624e8 − 1.96038e8i −1.11467 − 0.298676i
\(59\) −5.81549e8 + 3.35757e8i −0.813441 + 0.469640i −0.848149 − 0.529757i \(-0.822283\pi\)
0.0347085 + 0.999397i \(0.488950\pi\)
\(60\) 3.56605e8 − 1.54914e8i 0.458597 − 0.199221i
\(61\) −7.09977e8 + 1.22972e9i −0.840612 + 1.45598i 0.0487669 + 0.998810i \(0.484471\pi\)
−0.889378 + 0.457172i \(0.848862\pi\)
\(62\) 3.00358e7 + 3.00358e7i 0.0327855 + 0.0327855i
\(63\) 3.06217e8 + 6.70420e8i 0.308550 + 0.675529i
\(64\) − 1.34218e8i − 0.125000i
\(65\) −2.30548e8 − 6.80876e8i −0.198699 − 0.586815i
\(66\) 8.86091e8 + 1.27614e9i 0.707552 + 1.01901i
\(67\) −3.08636e8 + 1.15185e9i −0.228598 + 0.853141i 0.752332 + 0.658784i \(0.228929\pi\)
−0.980931 + 0.194357i \(0.937738\pi\)
\(68\) 5.34128e8 + 1.43119e8i 0.367367 + 0.0984358i
\(69\) 1.90683e9 − 1.32402e9i 1.21918 − 0.846543i
\(70\) 7.91284e8 + 3.90967e8i 0.470806 + 0.232622i
\(71\) −1.38118e9 −0.765526 −0.382763 − 0.923847i \(-0.625027\pi\)
−0.382763 + 0.923847i \(0.625027\pi\)
\(72\) −5.57271e8 − 3.96783e8i −0.288008 − 0.205065i
\(73\) 3.58478e8 − 3.58478e8i 0.172921 − 0.172921i −0.615340 − 0.788261i \(-0.710981\pi\)
0.788261 + 0.615340i \(0.210981\pi\)
\(74\) 1.02249e9 + 5.90338e8i 0.460789 + 0.266037i
\(75\) −1.08424e9 − 2.11087e9i −0.456899 − 0.889519i
\(76\) 9.63088e8 + 1.66812e9i 0.379838 + 0.657898i
\(77\) −9.12798e8 + 3.40661e9i −0.337226 + 1.25854i
\(78\) −8.16821e8 + 9.65700e8i −0.282913 + 0.334479i
\(79\) 3.15863e9 + 1.82364e9i 1.02651 + 0.592657i 0.915983 − 0.401217i \(-0.131412\pi\)
0.110528 + 0.993873i \(0.464746\pi\)
\(80\) −8.17487e8 + 5.29562e7i −0.249477 + 0.0161610i
\(81\) −3.29488e9 + 1.14079e9i −0.944963 + 0.327176i
\(82\) −1.97574e9 − 1.97574e9i −0.532917 − 0.532917i
\(83\) 8.09728e8 + 3.02194e9i 0.205565 + 0.767178i 0.989277 + 0.146053i \(0.0466572\pi\)
−0.783712 + 0.621124i \(0.786676\pi\)
\(84\) −1.29255e8 − 1.54756e9i −0.0309065 − 0.370041i
\(85\) 6.60960e8 − 3.30971e9i 0.148964 − 0.745925i
\(86\) −7.32664e8 − 1.26901e9i −0.155745 − 0.269757i
\(87\) 7.65456e9 − 2.75194e9i 1.53576 − 0.552132i
\(88\) −8.47227e8 − 3.16189e9i −0.160541 − 0.599148i
\(89\) 4.41863e9i 0.791293i 0.918403 + 0.395647i \(0.129480\pi\)
−0.918403 + 0.395647i \(0.870520\pi\)
\(90\) −2.19683e9 + 3.55075e9i −0.372036 + 0.601323i
\(91\) −2.87123e9 −0.460109
\(92\) −4.72458e9 + 1.26595e9i −0.716843 + 0.192078i
\(93\) −4.48924e8 − 8.09815e7i −0.0645294 − 0.0116405i
\(94\) −7.41493e8 + 4.28101e8i −0.101034 + 0.0583320i
\(95\) 9.78010e9 − 6.52409e9i 1.26394 − 0.843145i
\(96\) 8.22091e8 + 1.18396e9i 0.100824 + 0.145205i
\(97\) −6.73424e9 + 1.80443e9i −0.784205 + 0.210127i −0.628638 − 0.777698i \(-0.716387\pi\)
−0.155567 + 0.987825i \(0.549721\pi\)
\(98\) −2.02685e9 + 2.02685e9i −0.224229 + 0.224229i
\(99\) −1.56328e10 − 5.82971e9i −1.64384 − 0.613015i
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.k.b.7.9 ✓ 120
5.3 odd 4 inner 90.11.k.b.43.7 yes 120
9.4 even 3 inner 90.11.k.b.67.7 yes 120
45.13 odd 12 inner 90.11.k.b.13.9 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.k.b.7.9 ✓ 120 1.1 even 1 trivial
90.11.k.b.13.9 yes 120 45.13 odd 12 inner
90.11.k.b.43.7 yes 120 5.3 odd 4 inner
90.11.k.b.67.7 yes 120 9.4 even 3 inner