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Properties

Label 990.3.b.b.901.2
Level $990$
Weight $3$
Character 990.901
Analytic conductor $26.976$
Analytic rank $0$
Dimension $8$
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] = [990,3,Mod(901,990)] 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("990.901"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(990, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 990.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-16,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(26.9755461717\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.4956160000.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 19x^{4} - 30x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 110)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 901.2
Root \(-1.51954 + 1.14412i\) of defining polynomial
Character \(\chi\) \(=\) 990.901
Dual form 990.3.b.b.901.5

$q$-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)
 
\(f(q)\) \(=\) \(q-1.41421i q^{2} -2.00000 q^{4} -2.23607 q^{5} +12.4389i q^{7} +2.82843i q^{8} +3.16228i q^{10} +(7.23901 - 8.28232i) q^{11} +1.28012i q^{13} +17.5912 q^{14} +4.00000 q^{16} +3.33028i q^{17} +1.88632i q^{19} +4.47214 q^{20} +(-11.7130 - 10.2375i) q^{22} -32.3565 q^{23} +5.00000 q^{25} +1.81036 q^{26} -24.8777i q^{28} -27.9138i q^{29} +16.5112 q^{31} -5.65685i q^{32} +4.70973 q^{34} -27.8142i q^{35} -22.4573 q^{37} +2.66765 q^{38} -6.32456i q^{40} +52.3967i q^{41} -15.7171i q^{43} +(-14.4780 + 16.5646i) q^{44} +45.7590i q^{46} -87.6835 q^{47} -105.726 q^{49} -7.07107i q^{50} -2.56024i q^{52} -74.2161 q^{53} +(-16.1869 + 18.5198i) q^{55} -35.1824 q^{56} -39.4761 q^{58} +26.8351 q^{59} -47.4483i q^{61} -23.3503i q^{62} -8.00000 q^{64} -2.86244i q^{65} -79.2475 q^{67} -6.66057i q^{68} -39.3352 q^{70} +74.9405 q^{71} +64.0267i q^{73} +31.7594i q^{74} -3.77263i q^{76} +(103.023 + 90.0451i) q^{77} +151.980i q^{79} -8.94427 q^{80} +74.1001 q^{82} -151.469i q^{83} -7.44674i q^{85} -22.2273 q^{86} +(23.4259 + 20.4750i) q^{88} -127.627 q^{89} -15.9233 q^{91} +64.7130 q^{92} +124.003i q^{94} -4.21793i q^{95} -88.1768 q^{97} +149.519i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} + 32 q^{14} + 32 q^{16} + 136 q^{23} + 40 q^{25} - 80 q^{26} - 64 q^{31} + 112 q^{34} - 48 q^{37} - 152 q^{47} - 232 q^{49} - 352 q^{53} - 64 q^{56} - 64 q^{58} - 80 q^{59} - 64 q^{64} + 24 q^{67}+ \cdots - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/990\mathbb{Z}\right)^\times\).

\(n\) \(397\) \(541\) \(551\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.41421i − 0.707107i
\(3\) 0 0
\(4\) −2.00000 −0.500000
\(5\) −2.23607 −0.447214
\(6\) 0 0
\(7\) 12.4389i 1.77698i 0.458894 + 0.888491i \(0.348246\pi\)
−0.458894 + 0.888491i \(0.651754\pi\)
\(8\) 2.82843i 0.353553i
\(9\) 0 0
\(10\) 3.16228i 0.316228i
\(11\) 7.23901 − 8.28232i 0.658092 − 0.752938i
\(12\) 0 0
\(13\) 1.28012i 0.0984708i 0.998787 + 0.0492354i \(0.0156785\pi\)
−0.998787 + 0.0492354i \(0.984322\pi\)
\(14\) 17.5912 1.25652
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 3.33028i 0.195899i 0.995191 + 0.0979495i \(0.0312284\pi\)
−0.995191 + 0.0979495i \(0.968772\pi\)
\(18\) 0 0
\(19\) 1.88632i 0.0992798i 0.998767 + 0.0496399i \(0.0158074\pi\)
−0.998767 + 0.0496399i \(0.984193\pi\)
\(20\) 4.47214 0.223607
\(21\) 0 0
\(22\) −11.7130 − 10.2375i −0.532407 − 0.465341i
\(23\) −32.3565 −1.40680 −0.703402 − 0.710792i \(-0.748337\pi\)
−0.703402 + 0.710792i \(0.748337\pi\)
\(24\) 0 0
\(25\) 5.00000 0.200000
\(26\) 1.81036 0.0696294
\(27\) 0 0
\(28\) − 24.8777i − 0.888491i
\(29\) − 27.9138i − 0.962546i −0.876571 − 0.481273i \(-0.840175\pi\)
0.876571 − 0.481273i \(-0.159825\pi\)
\(30\) 0 0
\(31\) 16.5112 0.532618 0.266309 − 0.963888i \(-0.414196\pi\)
0.266309 + 0.963888i \(0.414196\pi\)
\(32\) − 5.65685i − 0.176777i
\(33\) 0 0
\(34\) 4.70973 0.138522
\(35\) − 27.8142i − 0.794690i
\(36\) 0 0
\(37\) −22.4573 −0.606954 −0.303477 − 0.952839i \(-0.598148\pi\)
−0.303477 + 0.952839i \(0.598148\pi\)
\(38\) 2.66765 0.0702014
\(39\) 0 0
\(40\) − 6.32456i − 0.158114i
\(41\) 52.3967i 1.27797i 0.769220 + 0.638984i \(0.220645\pi\)
−0.769220 + 0.638984i \(0.779355\pi\)
\(42\) 0 0
\(43\) − 15.7171i − 0.365513i −0.983158 − 0.182756i \(-0.941498\pi\)
0.983158 − 0.182756i \(-0.0585020\pi\)
\(44\) −14.4780 + 16.5646i −0.329046 + 0.376469i
\(45\) 0 0
\(46\) 45.7590i 0.994761i
\(47\) −87.6835 −1.86561 −0.932804 − 0.360385i \(-0.882645\pi\)
−0.932804 + 0.360385i \(0.882645\pi\)
\(48\) 0 0
\(49\) −105.726 −2.15766
\(50\) − 7.07107i − 0.141421i
\(51\) 0 0
\(52\) − 2.56024i − 0.0492354i
\(53\) −74.2161 −1.40030 −0.700152 − 0.713994i \(-0.746884\pi\)
−0.700152 + 0.713994i \(0.746884\pi\)
\(54\) 0 0
\(55\) −16.1869 + 18.5198i −0.294308 + 0.336724i
\(56\) −35.1824 −0.628258
\(57\) 0 0
\(58\) −39.4761 −0.680623
\(59\) 26.8351 0.454832 0.227416 − 0.973798i \(-0.426972\pi\)
0.227416 + 0.973798i \(0.426972\pi\)
\(60\) 0 0
\(61\) − 47.4483i − 0.777841i −0.921271 − 0.388921i \(-0.872848\pi\)
0.921271 − 0.388921i \(-0.127152\pi\)
\(62\) − 23.3503i − 0.376618i
\(63\) 0 0
\(64\) −8.00000 −0.125000
\(65\) − 2.86244i − 0.0440375i
\(66\) 0 0
\(67\) −79.2475 −1.18280 −0.591399 − 0.806379i \(-0.701424\pi\)
−0.591399 + 0.806379i \(0.701424\pi\)
\(68\) − 6.66057i − 0.0979495i
\(69\) 0 0
\(70\) −39.3352 −0.561931
\(71\) 74.9405 1.05550 0.527750 − 0.849400i \(-0.323036\pi\)
0.527750 + 0.849400i \(0.323036\pi\)
\(72\) 0 0
\(73\) 64.0267i 0.877078i 0.898712 + 0.438539i \(0.144504\pi\)
−0.898712 + 0.438539i \(0.855496\pi\)
\(74\) 31.7594i 0.429182i
\(75\) 0 0
\(76\) − 3.77263i − 0.0496399i
\(77\) 103.023 + 90.0451i 1.33796 + 1.16942i
\(78\) 0 0
\(79\) 151.980i 1.92380i 0.273400 + 0.961901i \(0.411852\pi\)
−0.273400 + 0.961901i \(0.588148\pi\)
\(80\) −8.94427 −0.111803
\(81\) 0 0
\(82\) 74.1001 0.903660
\(83\) − 151.469i − 1.82492i −0.409161 − 0.912462i \(-0.634179\pi\)
0.409161 − 0.912462i \(-0.365821\pi\)
\(84\) 0 0
\(85\) − 7.44674i − 0.0876087i
\(86\) −22.2273 −0.258457
\(87\) 0 0
\(88\) 23.4259 + 20.4750i 0.266204 + 0.232671i
\(89\) −127.627 −1.43401 −0.717005 − 0.697068i \(-0.754487\pi\)
−0.717005 + 0.697068i \(0.754487\pi\)
\(90\) 0 0
\(91\) −15.9233 −0.174981
\(92\) 64.7130 0.703402
\(93\) 0 0
\(94\) 124.003i 1.31918i
\(95\) − 4.21793i − 0.0443993i
\(96\) 0 0
\(97\) −88.1768 −0.909040 −0.454520 − 0.890737i \(-0.650189\pi\)
−0.454520 + 0.890737i \(0.650189\pi\)
\(98\) 149.519i 1.52570i
\(99\) 0 0
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 990.3.b.b.901.2 8
3.2 odd 2 110.3.d.a.21.7 yes 8
11.10 odd 2 inner 990.3.b.b.901.5 8
12.11 even 2 880.3.j.c.241.3 8
15.2 even 4 550.3.c.b.549.5 16
15.8 even 4 550.3.c.b.549.12 16
15.14 odd 2 550.3.d.f.351.2 8
33.32 even 2 110.3.d.a.21.3 ✓ 8
132.131 odd 2 880.3.j.c.241.4 8
165.32 odd 4 550.3.c.b.549.13 16
165.98 odd 4 550.3.c.b.549.4 16
165.164 even 2 550.3.d.f.351.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.3.d.a.21.3 ✓ 8 33.32 even 2
110.3.d.a.21.7 yes 8 3.2 odd 2
550.3.c.b.549.4 16 165.98 odd 4
550.3.c.b.549.5 16 15.2 even 4
550.3.c.b.549.12 16 15.8 even 4
550.3.c.b.549.13 16 165.32 odd 4
550.3.d.f.351.2 8 15.14 odd 2
550.3.d.f.351.6 8 165.164 even 2
880.3.j.c.241.3 8 12.11 even 2
880.3.j.c.241.4 8 132.131 odd 2
990.3.b.b.901.2 8 1.1 even 1 trivial
990.3.b.b.901.5 8 11.10 odd 2 inner