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Properties

Label 990.3.b.b.901.4
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 901.4
Root \(1.09132 - 0.437016i\) of defining polynomial
Character \(\chi\) \(=\) 990.901
Dual form 990.3.b.b.901.7

$q$-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)
 
\(f(q)\) \(=\) \(q-1.41421i q^{2} -2.00000 q^{4} +2.23607 q^{5} +9.73055i q^{7} +2.82843i q^{8} -3.16228i q^{10} +(-10.0795 - 4.40491i) q^{11} -16.6335i q^{13} +13.7611 q^{14} +4.00000 q^{16} +12.8351i q^{17} -4.69221i q^{19} -4.47214 q^{20} +(-6.22949 + 14.2546i) q^{22} +32.4557 q^{23} +5.00000 q^{25} -23.5233 q^{26} -19.4611i q^{28} +29.4791i q^{29} -23.4043 q^{31} -5.65685i q^{32} +18.1516 q^{34} +21.7582i q^{35} -11.1821 q^{37} -6.63579 q^{38} +6.32456i q^{40} +69.0178i q^{41} +65.2097i q^{43} +(20.1590 + 8.80982i) q^{44} -45.8992i q^{46} -45.7097 q^{47} -45.6836 q^{49} -7.07107i q^{50} +33.2669i q^{52} +1.00392 q^{53} +(-22.5385 - 9.84968i) q^{55} -27.5222 q^{56} +41.6898 q^{58} -94.0820 q^{59} +113.061i q^{61} +33.0987i q^{62} -8.00000 q^{64} -37.1935i q^{65} -13.0292 q^{67} -25.6702i q^{68} +30.7707 q^{70} -36.0057 q^{71} +83.2194i q^{73} +15.8139i q^{74} +9.38442i q^{76} +(42.8622 - 98.0793i) q^{77} +0.104581i q^{79} +8.94427 q^{80} +97.6059 q^{82} +41.9202i q^{83} +28.7002i q^{85} +92.2204 q^{86} +(12.4590 - 28.5092i) q^{88} +145.516 q^{89} +161.853 q^{91} -64.9113 q^{92} +64.6433i q^{94} -10.4921i q^{95} +89.1978 q^{97} +64.6064i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} + 32 q^{14} + 32 q^{16} + 136 q^{23} + 40 q^{25} - 80 q^{26} - 64 q^{31} + 112 q^{34} - 48 q^{37} - 152 q^{47} - 232 q^{49} - 352 q^{53} - 64 q^{56} - 64 q^{58} - 80 q^{59} - 64 q^{64} + 24 q^{67}+ \cdots - 32 q^{97}+O(q^{100}) \) 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
\(7\) 9.73055i 1.39008i 0.718972 + 0.695039i \(0.244613\pi\)
−0.718972 + 0.695039i \(0.755387\pi\)
\(8\) 2.82843i 0.353553i
\(9\) 0 0
\(10\) − 3.16228i − 0.316228i
\(11\) −10.0795 − 4.40491i −0.916320 − 0.400447i
\(12\) 0 0
\(13\) − 16.6335i − 1.27950i −0.768585 − 0.639748i \(-0.779039\pi\)
0.768585 − 0.639748i \(-0.220961\pi\)
\(14\) 13.7611 0.982934
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 12.8351i 0.755007i 0.926008 + 0.377503i \(0.123217\pi\)
−0.926008 + 0.377503i \(0.876783\pi\)
\(18\) 0 0
\(19\) − 4.69221i − 0.246958i −0.992347 − 0.123479i \(-0.960595\pi\)
0.992347 − 0.123479i \(-0.0394052\pi\)
\(20\) −4.47214 −0.223607
\(21\) 0 0
\(22\) −6.22949 + 14.2546i −0.283158 + 0.647936i
\(23\) 32.4557 1.41112 0.705558 − 0.708652i \(-0.250696\pi\)
0.705558 + 0.708652i \(0.250696\pi\)
\(24\) 0 0
\(25\) 5.00000 0.200000
\(26\) −23.5233 −0.904741
\(27\) 0 0
\(28\) − 19.4611i − 0.695039i
\(29\) 29.4791i 1.01652i 0.861203 + 0.508260i \(0.169711\pi\)
−0.861203 + 0.508260i \(0.830289\pi\)
\(30\) 0 0
\(31\) −23.4043 −0.754979 −0.377489 − 0.926014i \(-0.623213\pi\)
−0.377489 + 0.926014i \(0.623213\pi\)
\(32\) − 5.65685i − 0.176777i
\(33\) 0 0
\(34\) 18.1516 0.533870
\(35\) 21.7582i 0.621662i
\(36\) 0 0
\(37\) −11.1821 −0.302219 −0.151109 − 0.988517i \(-0.548285\pi\)
−0.151109 + 0.988517i \(0.548285\pi\)
\(38\) −6.63579 −0.174626
\(39\) 0 0
\(40\) 6.32456i 0.158114i
\(41\) 69.0178i 1.68336i 0.539976 + 0.841681i \(0.318433\pi\)
−0.539976 + 0.841681i \(0.681567\pi\)
\(42\) 0 0
\(43\) 65.2097i 1.51650i 0.651961 + 0.758252i \(0.273946\pi\)
−0.651961 + 0.758252i \(0.726054\pi\)
\(44\) 20.1590 + 8.80982i 0.458160 + 0.200223i
\(45\) 0 0
\(46\) − 45.8992i − 0.997810i
\(47\) −45.7097 −0.972547 −0.486274 − 0.873807i \(-0.661644\pi\)
−0.486274 + 0.873807i \(0.661644\pi\)
\(48\) 0 0
\(49\) −45.6836 −0.932319
\(50\) − 7.07107i − 0.141421i
\(51\) 0 0
\(52\) 33.2669i 0.639748i
\(53\) 1.00392 0.0189419 0.00947094 − 0.999955i \(-0.496985\pi\)
0.00947094 + 0.999955i \(0.496985\pi\)
\(54\) 0 0
\(55\) −22.5385 − 9.84968i −0.409791 − 0.179085i
\(56\) −27.5222 −0.491467
\(57\) 0 0
\(58\) 41.6898 0.718789
\(59\) −94.0820 −1.59461 −0.797305 − 0.603576i \(-0.793742\pi\)
−0.797305 + 0.603576i \(0.793742\pi\)
\(60\) 0 0
\(61\) 113.061i 1.85346i 0.375723 + 0.926732i \(0.377394\pi\)
−0.375723 + 0.926732i \(0.622606\pi\)
\(62\) 33.0987i 0.533851i
\(63\) 0 0
\(64\) −8.00000 −0.125000
\(65\) − 37.1935i − 0.572208i
\(66\) 0 0
\(67\) −13.0292 −0.194466 −0.0972329 − 0.995262i \(-0.530999\pi\)
−0.0972329 + 0.995262i \(0.530999\pi\)
\(68\) − 25.6702i − 0.377503i
\(69\) 0 0
\(70\) 30.7707 0.439582
\(71\) −36.0057 −0.507122 −0.253561 − 0.967319i \(-0.581602\pi\)
−0.253561 + 0.967319i \(0.581602\pi\)
\(72\) 0 0
\(73\) 83.2194i 1.13999i 0.821648 + 0.569996i \(0.193055\pi\)
−0.821648 + 0.569996i \(0.806945\pi\)
\(74\) 15.8139i 0.213701i
\(75\) 0 0
\(76\) 9.38442i 0.123479i
\(77\) 42.8622 − 98.0793i 0.556652 − 1.27376i
\(78\) 0 0
\(79\) 0.104581i 0.00132381i 1.00000 0.000661904i \(0.000210690\pi\)
−1.00000 0.000661904i \(0.999789\pi\)
\(80\) 8.94427 0.111803
\(81\) 0 0
\(82\) 97.6059 1.19032
\(83\) 41.9202i 0.505062i 0.967589 + 0.252531i \(0.0812630\pi\)
−0.967589 + 0.252531i \(0.918737\pi\)
\(84\) 0 0
\(85\) 28.7002i 0.337649i
\(86\) 92.2204 1.07233
\(87\) 0 0
\(88\) 12.4590 − 28.5092i 0.141579 − 0.323968i
\(89\) 145.516 1.63501 0.817505 − 0.575921i \(-0.195357\pi\)
0.817505 + 0.575921i \(0.195357\pi\)
\(90\) 0 0
\(91\) 161.853 1.77860
\(92\) −64.9113 −0.705558
\(93\) 0 0
\(94\) 64.6433i 0.687695i
\(95\) − 10.4921i − 0.110443i
\(96\) 0 0
\(97\) 89.1978 0.919565 0.459782 − 0.888032i \(-0.347927\pi\)
0.459782 + 0.888032i \(0.347927\pi\)
\(98\) 64.6064i 0.659249i
\(99\) 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.901.4 8
3.2 odd 2 110.3.d.a.21.8 yes 8
11.10 odd 2 inner 990.3.b.b.901.7 8
12.11 even 2 880.3.j.c.241.1 8
15.2 even 4 550.3.c.b.549.2 16
15.8 even 4 550.3.c.b.549.15 16
15.14 odd 2 550.3.d.f.351.1 8
33.32 even 2 110.3.d.a.21.4 ✓ 8
132.131 odd 2 880.3.j.c.241.2 8
165.32 odd 4 550.3.c.b.549.10 16
165.98 odd 4 550.3.c.b.549.7 16
165.164 even 2 550.3.d.f.351.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.3.d.a.21.4 ✓ 8 33.32 even 2
110.3.d.a.21.8 yes 8 3.2 odd 2
550.3.c.b.549.2 16 15.2 even 4
550.3.c.b.549.7 16 165.98 odd 4
550.3.c.b.549.10 16 165.32 odd 4
550.3.c.b.549.15 16 15.8 even 4
550.3.d.f.351.1 8 15.14 odd 2
550.3.d.f.351.5 8 165.164 even 2
880.3.j.c.241.1 8 12.11 even 2
880.3.j.c.241.2 8 132.131 odd 2
990.3.b.b.901.4 8 1.1 even 1 trivial
990.3.b.b.901.7 8 11.10 odd 2 inner