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Properties

Label 550.3.c.b.549.12
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 549.12
Root \(0.312289 - 0.129354i\) of defining polynomial
Character \(\chi\) \(=\) 550.549
Dual form 550.3.c.b.549.13

$q$-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)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} -1.35781i q^{3} +2.00000 q^{4} -1.92023i q^{6} +12.4389 q^{7} +2.82843 q^{8} +7.15636 q^{9} +(-7.23901 + 8.28232i) q^{11} -2.71561i q^{12} -1.28012 q^{13} +17.5912 q^{14} +4.00000 q^{16} -3.33028 q^{17} +10.1206 q^{18} -1.88632i q^{19} -16.8896i q^{21} +(-10.2375 + 11.7130i) q^{22} +32.3565i q^{23} -3.84046i q^{24} -1.81036 q^{26} -21.9372i q^{27} +24.8777 q^{28} -27.9138i q^{29} +16.5112 q^{31} +5.65685 q^{32} +(11.2458 + 9.82918i) q^{33} -4.70973 q^{34} +14.3127 q^{36} +22.4573i q^{37} -2.66765i q^{38} +1.73816i q^{39} -52.3967i q^{41} -23.8855i q^{42} +15.7171 q^{43} +(-14.4780 + 16.5646i) q^{44} +45.7590i q^{46} -87.6835i q^{47} -5.43123i q^{48} +105.726 q^{49} +4.52188i q^{51} -2.56024 q^{52} +74.2161i q^{53} -31.0239i q^{54} +35.1824 q^{56} -2.56125 q^{57} -39.4761i q^{58} +26.8351 q^{59} -47.4483i q^{61} +23.3503 q^{62} +89.0171 q^{63} +8.00000 q^{64} +(15.9039 + 13.9006i) q^{66} +79.2475i q^{67} -6.66057 q^{68} +43.9339 q^{69} -74.9405 q^{71} +20.2412 q^{72} -64.0267 q^{73} +31.7594i q^{74} -3.77263i q^{76} +(-90.0451 + 103.023i) q^{77} +2.45812i q^{78} -151.980i q^{79} +34.6207 q^{81} -74.1001i q^{82} -151.469 q^{83} -33.7792i q^{84} +22.2273 q^{86} -37.9016 q^{87} +(-20.4750 + 23.4259i) q^{88} -127.627 q^{89} -15.9233 q^{91} +64.7130i q^{92} -22.4190i q^{93} -124.003i q^{94} -7.68092i q^{96} +88.1768i q^{97} +149.519 q^{98} +(-51.8050 + 59.2712i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4} - 80 q^{9} + 64 q^{14} + 64 q^{16} + 160 q^{26} - 128 q^{31} - 224 q^{34} - 160 q^{36} + 464 q^{49} + 128 q^{56} - 160 q^{59} + 128 q^{64} - 352 q^{66} - 224 q^{69} - 512 q^{71} + 176 q^{81}+ \cdots - 1408 q^{99}+O(q^{100}) \) 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
\(3\) − 1.35781i − 0.452602i −0.974057 − 0.226301i \(-0.927337\pi\)
0.974057 − 0.226301i \(-0.0726634\pi\)
\(4\) 2.00000 0.500000
\(5\) 0 0
\(6\) − 1.92023i − 0.320038i
\(7\) 12.4389 1.77698 0.888491 − 0.458894i \(-0.151754\pi\)
0.888491 + 0.458894i \(0.151754\pi\)
\(8\) 2.82843 0.353553
\(9\) 7.15636 0.795151
\(10\) 0 0
\(11\) −7.23901 + 8.28232i −0.658092 + 0.752938i
\(12\) − 2.71561i − 0.226301i
\(13\) −1.28012 −0.0984708 −0.0492354 − 0.998787i \(-0.515678\pi\)
−0.0492354 + 0.998787i \(0.515678\pi\)
\(14\) 17.5912 1.25652
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) −3.33028 −0.195899 −0.0979495 − 0.995191i \(-0.531228\pi\)
−0.0979495 + 0.995191i \(0.531228\pi\)
\(18\) 10.1206 0.562257
\(19\) − 1.88632i − 0.0992798i −0.998767 − 0.0496399i \(-0.984193\pi\)
0.998767 − 0.0496399i \(-0.0158074\pi\)
\(20\) 0 0
\(21\) − 16.8896i − 0.804266i
\(22\) −10.2375 + 11.7130i −0.465341 + 0.532407i
\(23\) 32.3565i 1.40680i 0.710792 + 0.703402i \(0.248337\pi\)
−0.710792 + 0.703402i \(0.751663\pi\)
\(24\) − 3.84046i − 0.160019i
\(25\) 0 0
\(26\) −1.81036 −0.0696294
\(27\) − 21.9372i − 0.812490i
\(28\) 24.8777 0.888491
\(29\) − 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 \(0.414196\pi\)
\(32\) 5.65685 0.176777
\(33\) 11.2458 + 9.82918i 0.340781 + 0.297854i
\(34\) −4.70973 −0.138522
\(35\) 0 0
\(36\) 14.3127 0.397576
\(37\) 22.4573i 0.606954i 0.952839 + 0.303477i \(0.0981476\pi\)
−0.952839 + 0.303477i \(0.901852\pi\)
\(38\) − 2.66765i − 0.0702014i
\(39\) 1.73816i 0.0445681i
\(40\) 0 0
\(41\) − 52.3967i − 1.27797i −0.769220 − 0.638984i \(-0.779355\pi\)
0.769220 − 0.638984i \(-0.220645\pi\)
\(42\) − 23.8855i − 0.568702i
\(43\) 15.7171 0.365513 0.182756 − 0.983158i \(-0.441498\pi\)
0.182756 + 0.983158i \(0.441498\pi\)
\(44\) −14.4780 + 16.5646i −0.329046 + 0.376469i
\(45\) 0 0
\(46\) 45.7590i 0.994761i
\(47\) − 87.6835i − 1.86561i −0.360385 − 0.932804i \(-0.617355\pi\)
0.360385 − 0.932804i \(-0.382645\pi\)
\(48\) − 5.43123i − 0.113151i
\(49\) 105.726 2.15766
\(50\) 0 0
\(51\) 4.52188i 0.0886644i
\(52\) −2.56024 −0.0492354
\(53\) 74.2161i 1.40030i 0.713994 + 0.700152i \(0.246884\pi\)
−0.713994 + 0.700152i \(0.753116\pi\)
\(54\) − 31.0239i − 0.574517i
\(55\) 0 0
\(56\) 35.1824 0.628258
\(57\) −2.56125 −0.0449343
\(58\) − 39.4761i − 0.680623i
\(59\) 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 \(-0.127152\pi\)
\(62\) 23.3503 0.376618
\(63\) 89.0171 1.41297
\(64\) 8.00000 0.125000
\(65\) 0 0
\(66\) 15.9039 + 13.9006i 0.240969 + 0.210614i
\(67\) 79.2475i 1.18280i 0.806379 + 0.591399i \(0.201424\pi\)
−0.806379 + 0.591399i \(0.798576\pi\)
\(68\) −6.66057 −0.0979495
\(69\) 43.9339 0.636723
\(70\) 0 0
\(71\) −74.9405 −1.05550 −0.527750 − 0.849400i \(-0.676964\pi\)
−0.527750 + 0.849400i \(0.676964\pi\)
\(72\) 20.2412 0.281128
\(73\) −64.0267 −0.877078 −0.438539 − 0.898712i \(-0.644504\pi\)
−0.438539 + 0.898712i \(0.644504\pi\)
\(74\) 31.7594i 0.429182i
\(75\) 0 0
\(76\) − 3.77263i − 0.0496399i
\(77\) −90.0451 + 103.023i −1.16942 + 1.33796i
\(78\) 2.45812i 0.0315144i
\(79\) − 151.980i − 1.92380i −0.273400 − 0.961901i \(-0.588148\pi\)
0.273400 − 0.961901i \(-0.411852\pi\)
\(80\) 0 0
\(81\) 34.6207 0.427416
\(82\) − 74.1001i − 0.903660i
\(83\) −151.469 −1.82492 −0.912462 − 0.409161i \(-0.865821\pi\)
−0.912462 + 0.409161i \(0.865821\pi\)
\(84\) − 33.7792i − 0.402133i
\(85\) 0 0
\(86\) 22.2273 0.258457
\(87\) −37.9016 −0.435651
\(88\) −20.4750 + 23.4259i −0.232671 + 0.266204i
\(89\) −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
\(92\) 64.7130i 0.703402i
\(93\) − 22.4190i − 0.241064i
\(94\) − 124.003i − 1.31918i
\(95\) 0 0
\(96\) − 7.68092i − 0.0800095i
\(97\) 88.1768i 0.909040i 0.890737 + 0.454520i \(0.150189\pi\)
−0.890737 + 0.454520i \(0.849811\pi\)
\(98\) 149.519 1.52570
\(99\) −51.8050 + 59.2712i −0.523282 + 0.598699i
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.549.12 16
5.2 odd 4 110.3.d.a.21.7 yes 8
5.3 odd 4 550.3.d.f.351.2 8
5.4 even 2 inner 550.3.c.b.549.5 16
11.10 odd 2 inner 550.3.c.b.549.4 16
15.2 even 4 990.3.b.b.901.2 8
20.7 even 4 880.3.j.c.241.3 8
55.32 even 4 110.3.d.a.21.3 ✓ 8
55.43 even 4 550.3.d.f.351.6 8
55.54 odd 2 inner 550.3.c.b.549.13 16
165.32 odd 4 990.3.b.b.901.5 8
220.87 odd 4 880.3.j.c.241.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.3.d.a.21.3 ✓ 8 55.32 even 4
110.3.d.a.21.7 yes 8 5.2 odd 4
550.3.c.b.549.4 16 11.10 odd 2 inner
550.3.c.b.549.5 16 5.4 even 2 inner
550.3.c.b.549.12 16 1.1 even 1 trivial
550.3.c.b.549.13 16 55.54 odd 2 inner
550.3.d.f.351.2 8 5.3 odd 4
550.3.d.f.351.6 8 55.43 even 4
880.3.j.c.241.3 8 20.7 even 4
880.3.j.c.241.4 8 220.87 odd 4
990.3.b.b.901.2 8 15.2 even 4
990.3.b.b.901.5 8 165.32 odd 4