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Properties

Label 550.3.c.b.549.10
Level $550$
Weight $3$
Character 550.549
Analytic conductor $14.986$
Analytic rank $0$
Dimension $16$
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] = [550,3,Mod(549,550)] 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("550.549"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(550, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 550.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,32,0,0,0,0,-80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.9864145398\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.393016351129600000000.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 42 x^{14} - 148 x^{13} + 402 x^{12} - 928 x^{11} + 1834 x^{10} - 2940 x^{9} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{26} \)
Twist minimal: no (minimal twist has level 110)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 549.10
Root \(0.827153 + 0.342618i\) of defining polynomial
Character \(\chi\) \(=\) 550.549
Dual form 550.3.c.b.549.15

$q$-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)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} -4.76766i q^{3} +2.00000 q^{4} -6.74249i q^{6} +9.73055 q^{7} +2.82843 q^{8} -13.7306 q^{9} +(10.0795 - 4.40491i) q^{11} -9.53532i q^{12} +16.6335 q^{13} +13.7611 q^{14} +4.00000 q^{16} -12.8351 q^{17} -19.4180 q^{18} -4.69221i q^{19} -46.3920i q^{21} +(14.2546 - 6.22949i) q^{22} +32.4557i q^{23} -13.4850i q^{24} +23.5233 q^{26} +22.5538i q^{27} +19.4611 q^{28} -29.4791i q^{29} -23.4043 q^{31} +5.65685 q^{32} +(-21.0011 - 48.0557i) q^{33} -18.1516 q^{34} -27.4611 q^{36} -11.1821i q^{37} -6.63579i q^{38} -79.3026i q^{39} +69.0178i q^{41} -65.6081i q^{42} -65.2097 q^{43} +(20.1590 - 8.80982i) q^{44} +45.8992i q^{46} +45.7097i q^{47} -19.0706i q^{48} +45.6836 q^{49} +61.1934i q^{51} +33.2669 q^{52} +1.00392i q^{53} +31.8958i q^{54} +27.5222 q^{56} -22.3709 q^{57} -41.6898i q^{58} -94.0820 q^{59} -113.061i q^{61} -33.0987 q^{62} -133.606 q^{63} +8.00000 q^{64} +(-29.7001 - 67.9611i) q^{66} -13.0292i q^{67} -25.6702 q^{68} +154.738 q^{69} +36.0057 q^{71} -38.8359 q^{72} -83.2194 q^{73} -15.8139i q^{74} -9.38442i q^{76} +(98.0793 - 42.8622i) q^{77} -112.151i q^{78} +0.104581i q^{79} -16.0465 q^{81} +97.6059i q^{82} +41.9202 q^{83} -92.7839i q^{84} -92.2204 q^{86} -140.546 q^{87} +(28.5092 - 12.4590i) q^{88} +145.516 q^{89} +161.853 q^{91} +64.9113i q^{92} +111.584i q^{93} +64.6433i q^{94} -26.9700i q^{96} +89.1978i q^{97} +64.6064 q^{98} +(-138.398 + 60.4820i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4} - 80 q^{9} + 64 q^{14} + 64 q^{16} + 160 q^{26} - 128 q^{31} - 224 q^{34} - 160 q^{36} + 464 q^{49} + 128 q^{56} - 160 q^{59} + 128 q^{64} - 352 q^{66} - 224 q^{69} - 512 q^{71} + 176 q^{81}+ \cdots - 1408 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/550\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 0.707107
\(3\) − 4.76766i − 1.58922i −0.607120 − 0.794610i \(-0.707675\pi\)
0.607120 − 0.794610i \(-0.292325\pi\)
\(4\) 2.00000 0.500000
\(5\) 0 0
\(6\) − 6.74249i − 1.12375i
\(7\) 9.73055 1.39008 0.695039 − 0.718972i \(-0.255387\pi\)
0.695039 + 0.718972i \(0.255387\pi\)
\(8\) 2.82843 0.353553
\(9\) −13.7306 −1.52562
\(10\) 0 0
\(11\) 10.0795 − 4.40491i 0.916320 − 0.400447i
\(12\) − 9.53532i − 0.794610i
\(13\) 16.6335 1.27950 0.639748 − 0.768585i \(-0.279039\pi\)
0.639748 + 0.768585i \(0.279039\pi\)
\(14\) 13.7611 0.982934
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) −12.8351 −0.755007 −0.377503 − 0.926008i \(-0.623217\pi\)
−0.377503 + 0.926008i \(0.623217\pi\)
\(18\) −19.4180 −1.07878
\(19\) − 4.69221i − 0.246958i −0.992347 − 0.123479i \(-0.960595\pi\)
0.992347 − 0.123479i \(-0.0394052\pi\)
\(20\) 0 0
\(21\) − 46.3920i − 2.20914i
\(22\) 14.2546 − 6.22949i 0.647936 − 0.283158i
\(23\) 32.4557i 1.41112i 0.708652 + 0.705558i \(0.249304\pi\)
−0.708652 + 0.705558i \(0.750696\pi\)
\(24\) − 13.4850i − 0.561874i
\(25\) 0 0
\(26\) 23.5233 0.904741
\(27\) 22.5538i 0.835325i
\(28\) 19.4611 0.695039
\(29\) − 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 \(0.623213\pi\)
\(32\) 5.65685 0.176777
\(33\) −21.0011 − 48.0557i −0.636398 − 1.45623i
\(34\) −18.1516 −0.533870
\(35\) 0 0
\(36\) −27.4611 −0.762810
\(37\) − 11.1821i − 0.302219i −0.988517 − 0.151109i \(-0.951715\pi\)
0.988517 − 0.151109i \(-0.0482846\pi\)
\(38\) − 6.63579i − 0.174626i
\(39\) − 79.3026i − 2.03340i
\(40\) 0 0
\(41\) 69.0178i 1.68336i 0.539976 + 0.841681i \(0.318433\pi\)
−0.539976 + 0.841681i \(0.681567\pi\)
\(42\) − 65.6081i − 1.56210i
\(43\) −65.2097 −1.51650 −0.758252 − 0.651961i \(-0.773946\pi\)
−0.758252 + 0.651961i \(0.773946\pi\)
\(44\) 20.1590 − 8.80982i 0.458160 − 0.200223i
\(45\) 0 0
\(46\) 45.8992i 0.997810i
\(47\) 45.7097i 0.972547i 0.873807 + 0.486274i \(0.161644\pi\)
−0.873807 + 0.486274i \(0.838356\pi\)
\(48\) − 19.0706i − 0.397305i
\(49\) 45.6836 0.932319
\(50\) 0 0
\(51\) 61.1934i 1.19987i
\(52\) 33.2669 0.639748
\(53\) 1.00392i 0.0189419i 0.999955 + 0.00947094i \(0.00301474\pi\)
−0.999955 + 0.00947094i \(0.996985\pi\)
\(54\) 31.8958i 0.590664i
\(55\) 0 0
\(56\) 27.5222 0.491467
\(57\) −22.3709 −0.392471
\(58\) − 41.6898i − 0.718789i
\(59\) −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 \(-0.377394\pi\)
\(62\) −33.0987 −0.533851
\(63\) −133.606 −2.12073
\(64\) 8.00000 0.125000
\(65\) 0 0
\(66\) −29.7001 − 67.9611i −0.450001 − 1.02971i
\(67\) − 13.0292i − 0.194466i −0.995262 − 0.0972329i \(-0.969001\pi\)
0.995262 − 0.0972329i \(-0.0309992\pi\)
\(68\) −25.6702 −0.377503
\(69\) 154.738 2.24257
\(70\) 0 0
\(71\) 36.0057 0.507122 0.253561 − 0.967319i \(-0.418398\pi\)
0.253561 + 0.967319i \(0.418398\pi\)
\(72\) −38.8359 −0.539388
\(73\) −83.2194 −1.13999 −0.569996 − 0.821648i \(-0.693055\pi\)
−0.569996 + 0.821648i \(0.693055\pi\)
\(74\) − 15.8139i − 0.213701i
\(75\) 0 0
\(76\) − 9.38442i − 0.123479i
\(77\) 98.0793 − 42.8622i 1.27376 − 0.556652i
\(78\) − 112.151i − 1.43783i
\(79\) 0.104581i 0.00132381i 1.00000 0.000661904i \(0.000210690\pi\)
−1.00000 0.000661904i \(0.999789\pi\)
\(80\) 0 0
\(81\) −16.0465 −0.198105
\(82\) 97.6059i 1.19032i
\(83\) 41.9202 0.505062 0.252531 − 0.967589i \(-0.418737\pi\)
0.252531 + 0.967589i \(0.418737\pi\)
\(84\) − 92.7839i − 1.10457i
\(85\) 0 0
\(86\) −92.2204 −1.07233
\(87\) −140.546 −1.61548
\(88\) 28.5092 − 12.4590i 0.323968 − 0.141579i
\(89\) 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
\(92\) 64.9113i 0.705558i
\(93\) 111.584i 1.19983i
\(94\) 64.6433i 0.687695i
\(95\) 0 0
\(96\) − 26.9700i − 0.280937i
\(97\) 89.1978i 0.919565i 0.888032 + 0.459782i \(0.152073\pi\)
−0.888032 + 0.459782i \(0.847927\pi\)
\(98\) 64.6064 0.659249
\(99\) −138.398 + 60.4820i −1.39796 + 0.610929i
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 550.3.c.b.549.10 16
5.2 odd 4 550.3.d.f.351.5 8
5.3 odd 4 110.3.d.a.21.4 ✓ 8
5.4 even 2 inner 550.3.c.b.549.7 16
11.10 odd 2 inner 550.3.c.b.549.2 16
15.8 even 4 990.3.b.b.901.7 8
20.3 even 4 880.3.j.c.241.2 8
55.32 even 4 550.3.d.f.351.1 8
55.43 even 4 110.3.d.a.21.8 yes 8
55.54 odd 2 inner 550.3.c.b.549.15 16
165.98 odd 4 990.3.b.b.901.4 8
220.43 odd 4 880.3.j.c.241.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.3.d.a.21.4 ✓ 8 5.3 odd 4
110.3.d.a.21.8 yes 8 55.43 even 4
550.3.c.b.549.2 16 11.10 odd 2 inner
550.3.c.b.549.7 16 5.4 even 2 inner
550.3.c.b.549.10 16 1.1 even 1 trivial
550.3.c.b.549.15 16 55.54 odd 2 inner
550.3.d.f.351.1 8 55.32 even 4
550.3.d.f.351.5 8 5.2 odd 4
880.3.j.c.241.1 8 220.43 odd 4
880.3.j.c.241.2 8 20.3 even 4
990.3.b.b.901.4 8 165.98 odd 4
990.3.b.b.901.7 8 15.8 even 4