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Properties

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

$q$-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)
 
\(f(q)\) \(=\) \(q+1.41421i q^{2} -2.00000 q^{4} +2.23607 q^{5} -9.73055i q^{7} -2.82843i q^{8} +3.16228i q^{10} +(-10.0795 + 4.40491i) q^{11} +16.6335i q^{13} +13.7611 q^{14} +4.00000 q^{16} -12.8351i q^{17} +4.69221i q^{19} -4.47214 q^{20} +(-6.22949 - 14.2546i) q^{22} +32.4557 q^{23} +5.00000 q^{25} -23.5233 q^{26} +19.4611i q^{28} -29.4791i q^{29} -23.4043 q^{31} +5.65685i q^{32} +18.1516 q^{34} -21.7582i q^{35} -11.1821 q^{37} -6.63579 q^{38} -6.32456i q^{40} -69.0178i q^{41} -65.2097i q^{43} +(20.1590 - 8.80982i) q^{44} +45.8992i q^{46} -45.7097 q^{47} -45.6836 q^{49} +7.07107i q^{50} -33.2669i q^{52} +1.00392 q^{53} +(-22.5385 + 9.84968i) q^{55} -27.5222 q^{56} +41.6898 q^{58} -94.0820 q^{59} -113.061i q^{61} -33.0987i q^{62} -8.00000 q^{64} +37.1935i q^{65} -13.0292 q^{67} +25.6702i q^{68} +30.7707 q^{70} -36.0057 q^{71} -83.2194i q^{73} -15.8139i q^{74} -9.38442i q^{76} +(42.8622 + 98.0793i) q^{77} -0.104581i q^{79} +8.94427 q^{80} +97.6059 q^{82} -41.9202i q^{83} -28.7002i q^{85} +92.2204 q^{86} +(12.4590 + 28.5092i) q^{88} +145.516 q^{89} +161.853 q^{91} -64.9113 q^{92} -64.6433i q^{94} +10.4921i q^{95} +89.1978 q^{97} -64.6064i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} + 32 q^{14} + 32 q^{16} + 136 q^{23} + 40 q^{25} - 80 q^{26} - 64 q^{31} + 112 q^{34} - 48 q^{37} - 152 q^{47} - 232 q^{49} - 352 q^{53} - 64 q^{56} - 64 q^{58} - 80 q^{59} - 64 q^{64} + 24 q^{67}+ \cdots - 32 q^{97}+O(q^{100}) \) 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 \(-0.755387\pi\)
0.718972 − 0.695039i \(-0.244613\pi\)
\(8\) − 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 \(0.220961\pi\)
−0.768585 + 0.639748i \(0.779039\pi\)
\(14\) 13.7611 0.982934
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) − 12.8351i − 0.755007i −0.926008 − 0.377503i \(-0.876783\pi\)
0.926008 − 0.377503i \(-0.123217\pi\)
\(18\) 0 0
\(19\) 4.69221i 0.246958i 0.992347 + 0.123479i \(0.0394052\pi\)
−0.992347 + 0.123479i \(0.960595\pi\)
\(20\) −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 \(-0.830289\pi\)
0.861203 − 0.508260i \(-0.169711\pi\)
\(30\) 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 \(-0.681567\pi\)
0.539976 − 0.841681i \(-0.318433\pi\)
\(42\) 0 0
\(43\) − 65.2097i − 1.51650i −0.651961 − 0.758252i \(-0.726054\pi\)
0.651961 − 0.758252i \(-0.273946\pi\)
\(44\) 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 \(-0.622606\pi\)
0.375723 − 0.926732i \(-0.377394\pi\)
\(62\) − 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 \(-0.806945\pi\)
0.821648 − 0.569996i \(-0.193055\pi\)
\(74\) − 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 \(-0.999789\pi\)
1.00000 0.000661904i \(-0.000210690\pi\)
\(80\) 8.94427 0.111803
\(81\) 0 0
\(82\) 97.6059 1.19032
\(83\) − 41.9202i − 0.505062i −0.967589 − 0.252531i \(-0.918737\pi\)
0.967589 − 0.252531i \(-0.0812630\pi\)
\(84\) 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.901.7 8
3.2 odd 2 110.3.d.a.21.4 ✓ 8
11.10 odd 2 inner 990.3.b.b.901.4 8
12.11 even 2 880.3.j.c.241.2 8
15.2 even 4 550.3.c.b.549.10 16
15.8 even 4 550.3.c.b.549.7 16
15.14 odd 2 550.3.d.f.351.5 8
33.32 even 2 110.3.d.a.21.8 yes 8
132.131 odd 2 880.3.j.c.241.1 8
165.32 odd 4 550.3.c.b.549.2 16
165.98 odd 4 550.3.c.b.549.15 16
165.164 even 2 550.3.d.f.351.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.3.d.a.21.4 ✓ 8 3.2 odd 2
110.3.d.a.21.8 yes 8 33.32 even 2
550.3.c.b.549.2 16 165.32 odd 4
550.3.c.b.549.7 16 15.8 even 4
550.3.c.b.549.10 16 15.2 even 4
550.3.c.b.549.15 16 165.98 odd 4
550.3.d.f.351.1 8 165.164 even 2
550.3.d.f.351.5 8 15.14 odd 2
880.3.j.c.241.1 8 132.131 odd 2
880.3.j.c.241.2 8 12.11 even 2
990.3.b.b.901.4 8 11.10 odd 2 inner
990.3.b.b.901.7 8 1.1 even 1 trivial