↑

Properties

Label 550.3.c.b.549.2
Level $550$
Weight $3$
Character 550.549
Analytic conductor $14.986$
Analytic rank $0$
Dimension $16$
Inner twists $4$

Related objects

Downloads

Learn more

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 549.2
Root \(1.26417 - 3.05197i\) of defining polynomial
Character \(\chi\) \(=\) 550.549
Dual form 550.3.c.b.549.7

$q$-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)
 
\(f(q)\) \(=\) \(q-1.41421 q^{2} -4.76766i q^{3} +2.00000 q^{4} +6.74249i q^{6} -9.73055 q^{7} -2.82843 q^{8} -13.7306 q^{9} +(10.0795 + 4.40491i) q^{11} -9.53532i q^{12} -16.6335 q^{13} +13.7611 q^{14} +4.00000 q^{16} +12.8351 q^{17} +19.4180 q^{18} +4.69221i q^{19} +46.3920i q^{21} +(-14.2546 - 6.22949i) q^{22} +32.4557i q^{23} +13.4850i q^{24} +23.5233 q^{26} +22.5538i q^{27} -19.4611 q^{28} +29.4791i q^{29} -23.4043 q^{31} -5.65685 q^{32} +(21.0011 - 48.0557i) q^{33} -18.1516 q^{34} -27.4611 q^{36} -11.1821i q^{37} -6.63579i q^{38} +79.3026i q^{39} -69.0178i q^{41} -65.6081i q^{42} +65.2097 q^{43} +(20.1590 + 8.80982i) q^{44} -45.8992i q^{46} +45.7097i q^{47} -19.0706i q^{48} +45.6836 q^{49} -61.1934i q^{51} -33.2669 q^{52} +1.00392i q^{53} -31.8958i q^{54} +27.5222 q^{56} +22.3709 q^{57} -41.6898i q^{58} -94.0820 q^{59} +113.061i q^{61} +33.0987 q^{62} +133.606 q^{63} +8.00000 q^{64} +(-29.7001 + 67.9611i) q^{66} -13.0292i q^{67} +25.6702 q^{68} +154.738 q^{69} +36.0057 q^{71} +38.8359 q^{72} +83.2194 q^{73} +15.8139i q^{74} +9.38442i q^{76} +(-98.0793 - 42.8622i) q^{77} -112.151i q^{78} -0.104581i q^{79} -16.0465 q^{81} +97.6059i q^{82} -41.9202 q^{83} +92.7839i q^{84} -92.2204 q^{86} +140.546 q^{87} +(-28.5092 - 12.4590i) q^{88} +145.516 q^{89} +161.853 q^{91} +64.9113i q^{92} +111.584i q^{93} -64.6433i q^{94} +26.9700i q^{96} +89.1978i q^{97} -64.6064 q^{98} +(-138.398 - 60.4820i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4} - 80 q^{9} + 64 q^{14} + 64 q^{16} + 160 q^{26} - 128 q^{31} - 224 q^{34} - 160 q^{36} + 464 q^{49} + 128 q^{56} - 160 q^{59} + 128 q^{64} - 352 q^{66} - 224 q^{69} - 512 q^{71} + 176 q^{81}+ \cdots - 1408 q^{99}+O(q^{100}) \) 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 \(-0.744613\pi\)
−0.695039 + 0.718972i \(0.744613\pi\)
\(8\) −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 \(-0.720961\pi\)
−0.639748 + 0.768585i \(0.720961\pi\)
\(14\) 13.7611 0.982934
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 12.8351 0.755007 0.377503 − 0.926008i \(-0.376783\pi\)
0.377503 + 0.926008i \(0.376783\pi\)
\(18\) 19.4180 1.07878
\(19\) 4.69221i 0.246958i 0.992347 + 0.123479i \(0.0394052\pi\)
−0.992347 + 0.123479i \(0.960595\pi\)
\(20\) 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 \(0.169711\pi\)
−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 \(0.623213\pi\)
\(32\) −5.65685 −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 \(-0.681567\pi\)
0.539976 − 0.841681i \(-0.318433\pi\)
\(42\) − 65.6081i − 1.56210i
\(43\) 65.2097 1.51650 0.758252 − 0.651961i \(-0.226054\pi\)
0.758252 + 0.651961i \(0.226054\pi\)
\(44\) 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 \(0.377394\pi\)
−0.375723 + 0.926732i \(0.622606\pi\)
\(62\) 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 \(-0.306945\pi\)
0.569996 + 0.821648i \(0.306945\pi\)
\(74\) 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 \(-0.999789\pi\)
1.00000 0.000661904i \(-0.000210690\pi\)
\(80\) 0 0
\(81\) −16.0465 −0.198105
\(82\) 97.6059i 1.19032i
\(83\) −41.9202 −0.505062 −0.252531 − 0.967589i \(-0.581263\pi\)
−0.252531 + 0.967589i \(0.581263\pi\)
\(84\) 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.549.2 16
5.2 odd 4 550.3.d.f.351.1 8
5.3 odd 4 110.3.d.a.21.8 yes 8
5.4 even 2 inner 550.3.c.b.549.15 16
11.10 odd 2 inner 550.3.c.b.549.10 16
15.8 even 4 990.3.b.b.901.4 8
20.3 even 4 880.3.j.c.241.1 8
55.32 even 4 550.3.d.f.351.5 8
55.43 even 4 110.3.d.a.21.4 ✓ 8
55.54 odd 2 inner 550.3.c.b.549.7 16
165.98 odd 4 990.3.b.b.901.7 8
220.43 odd 4 880.3.j.c.241.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.3.d.a.21.4 ✓ 8 55.43 even 4
110.3.d.a.21.8 yes 8 5.3 odd 4
550.3.c.b.549.2 16 1.1 even 1 trivial
550.3.c.b.549.7 16 55.54 odd 2 inner
550.3.c.b.549.10 16 11.10 odd 2 inner
550.3.c.b.549.15 16 5.4 even 2 inner
550.3.d.f.351.1 8 5.2 odd 4
550.3.d.f.351.5 8 55.32 even 4
880.3.j.c.241.1 8 20.3 even 4
880.3.j.c.241.2 8 220.43 odd 4
990.3.b.b.901.4 8 15.8 even 4
990.3.b.b.901.7 8 165.98 odd 4