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Properties

Label 550.3.d.f.351.1
Level $550$
Weight $3$
Character 550.351
Analytic conductor $14.986$
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] = [550,3,Mod(351,550)] 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("550.351"); 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([0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 550.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.9864145398\)
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 351.1
Root \(-1.09132 - 0.437016i\) of defining polynomial
Character \(\chi\) \(=\) 550.351
Dual form 550.3.d.f.351.5

$q$-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)
 
\(f(q)\) \(=\) \(q-1.41421i q^{2} -4.76766 q^{3} -2.00000 q^{4} +6.74249i q^{6} -9.73055i q^{7} +2.82843i q^{8} +13.7306 q^{9} +(10.0795 + 4.40491i) q^{11} +9.53532 q^{12} +16.6335i q^{13} -13.7611 q^{14} +4.00000 q^{16} +12.8351i q^{17} -19.4180i q^{18} -4.69221i q^{19} +46.3920i q^{21} +(6.22949 - 14.2546i) q^{22} +32.4557 q^{23} -13.4850i q^{24} +23.5233 q^{26} -22.5538 q^{27} +19.4611i q^{28} -29.4791i q^{29} -23.4043 q^{31} -5.65685i q^{32} +(-48.0557 - 21.0011i) q^{33} +18.1516 q^{34} -27.4611 q^{36} +11.1821 q^{37} -6.63579 q^{38} -79.3026i q^{39} -69.0178i q^{41} +65.6081 q^{42} -65.2097i q^{43} +(-20.1590 - 8.80982i) q^{44} -45.8992i q^{46} -45.7097 q^{47} -19.0706 q^{48} -45.6836 q^{49} -61.1934i q^{51} -33.2669i q^{52} +1.00392 q^{53} +31.8958i q^{54} +27.5222 q^{56} +22.3709i q^{57} -41.6898 q^{58} +94.0820 q^{59} +113.061i q^{61} +33.0987i q^{62} -133.606i q^{63} -8.00000 q^{64} +(-29.7001 + 67.9611i) q^{66} +13.0292 q^{67} -25.6702i q^{68} -154.738 q^{69} +36.0057 q^{71} +38.8359i q^{72} -83.2194i q^{73} -15.8139i q^{74} +9.38442i q^{76} +(42.8622 - 98.0793i) q^{77} -112.151 q^{78} +0.104581i q^{79} -16.0465 q^{81} -97.6059 q^{82} +41.9202i q^{83} -92.7839i q^{84} -92.2204 q^{86} +140.546i q^{87} +(-12.4590 + 28.5092i) q^{88} -145.516 q^{89} +161.853 q^{91} -64.9113 q^{92} +111.584 q^{93} +64.6433i q^{94} +26.9700i q^{96} -89.1978 q^{97} +64.6064i q^{98} +(138.398 + 60.4820i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} - 16 q^{4} + 40 q^{9} - 16 q^{12} - 32 q^{14} + 32 q^{16} + 136 q^{23} + 80 q^{26} - 64 q^{27} - 64 q^{31} - 88 q^{33} + 112 q^{34} - 80 q^{36} + 48 q^{37} + 208 q^{42} - 152 q^{47} + 32 q^{48}+ \cdots + 704 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.41421i − 0.707107i
\(3\) −4.76766 −1.58922 −0.794610 − 0.607120i \(-0.792325\pi\)
−0.794610 + 0.607120i \(0.792325\pi\)
\(4\) −2.00000 −0.500000
\(5\) 0 0
\(6\) 6.74249i 1.12375i
\(7\) − 9.73055i − 1.39008i −0.718972 − 0.695039i \(-0.755387\pi\)
0.718972 − 0.695039i \(-0.244613\pi\)
\(8\) 2.82843i 0.353553i
\(9\) 13.7306 1.52562
\(10\) 0 0
\(11\) 10.0795 + 4.40491i 0.916320 + 0.400447i
\(12\) 9.53532 0.794610
\(13\) 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 \(0.123217\pi\)
−0.926008 + 0.377503i \(0.876783\pi\)
\(18\) − 19.4180i − 1.07878i
\(19\) − 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
\(22\) 6.22949 − 14.2546i 0.283158 − 0.647936i
\(23\) 32.4557 1.41112 0.705558 − 0.708652i \(-0.250696\pi\)
0.705558 + 0.708652i \(0.250696\pi\)
\(24\) − 13.4850i − 0.561874i
\(25\) 0 0
\(26\) 23.5233 0.904741
\(27\) −22.5538 −0.835325
\(28\) 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
\(33\) −48.0557 − 21.0011i −1.45623 − 0.636398i
\(34\) 18.1516 0.533870
\(35\) 0 0
\(36\) −27.4611 −0.762810
\(37\) 11.1821 0.302219 0.151109 − 0.988517i \(-0.451715\pi\)
0.151109 + 0.988517i \(0.451715\pi\)
\(38\) −6.63579 −0.174626
\(39\) − 79.3026i − 2.03340i
\(40\) 0 0
\(41\) − 69.0178i − 1.68336i −0.539976 − 0.841681i \(-0.681567\pi\)
0.539976 − 0.841681i \(-0.318433\pi\)
\(42\) 65.6081 1.56210
\(43\) − 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 \(0.661644\pi\)
\(48\) −19.0706 −0.397305
\(49\) −45.6836 −0.932319
\(50\) 0 0
\(51\) − 61.1934i − 1.19987i
\(52\) − 33.2669i − 0.639748i
\(53\) 1.00392 0.0189419 0.00947094 − 0.999955i \(-0.496985\pi\)
0.00947094 + 0.999955i \(0.496985\pi\)
\(54\) 31.8958i 0.590664i
\(55\) 0 0
\(56\) 27.5222 0.491467
\(57\) 22.3709i 0.392471i
\(58\) −41.6898 −0.718789
\(59\) 94.0820 1.59461 0.797305 − 0.603576i \(-0.206258\pi\)
0.797305 + 0.603576i \(0.206258\pi\)
\(60\) 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
\(63\) − 133.606i − 2.12073i
\(64\) −8.00000 −0.125000
\(65\) 0 0
\(66\) −29.7001 + 67.9611i −0.450001 + 1.02971i
\(67\) 13.0292 0.194466 0.0972329 − 0.995262i \(-0.469001\pi\)
0.0972329 + 0.995262i \(0.469001\pi\)
\(68\) − 25.6702i − 0.377503i
\(69\) −154.738 −2.24257
\(70\) 0 0
\(71\) 36.0057 0.507122 0.253561 − 0.967319i \(-0.418398\pi\)
0.253561 + 0.967319i \(0.418398\pi\)
\(72\) 38.8359i 0.539388i
\(73\) − 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
\(78\) −112.151 −1.43783
\(79\) 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
\(82\) −97.6059 −1.19032
\(83\) 41.9202i 0.505062i 0.967589 + 0.252531i \(0.0812630\pi\)
−0.967589 + 0.252531i \(0.918737\pi\)
\(84\) − 92.7839i − 1.10457i
\(85\) 0 0
\(86\) −92.2204 −1.07233
\(87\) 140.546i 1.61548i
\(88\) −12.4590 + 28.5092i −0.141579 + 0.323968i
\(89\) −145.516 −1.63501 −0.817505 − 0.575921i \(-0.804643\pi\)
−0.817505 + 0.575921i \(0.804643\pi\)
\(90\) 0 0
\(91\) 161.853 1.77860
\(92\) −64.9113 −0.705558
\(93\) 111.584 1.19983
\(94\) 64.6433i 0.687695i
\(95\) 0 0
\(96\) 26.9700i 0.280937i
\(97\) −89.1978 −0.919565 −0.459782 − 0.888032i \(-0.652073\pi\)
−0.459782 + 0.888032i \(0.652073\pi\)
\(98\) 64.6064i 0.659249i
\(99\) 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.d.f.351.1 8
5.2 odd 4 550.3.c.b.549.15 16
5.3 odd 4 550.3.c.b.549.2 16
5.4 even 2 110.3.d.a.21.8 yes 8
11.10 odd 2 inner 550.3.d.f.351.5 8
15.14 odd 2 990.3.b.b.901.4 8
20.19 odd 2 880.3.j.c.241.1 8
55.32 even 4 550.3.c.b.549.7 16
55.43 even 4 550.3.c.b.549.10 16
55.54 odd 2 110.3.d.a.21.4 ✓ 8
165.164 even 2 990.3.b.b.901.7 8
220.219 even 2 880.3.j.c.241.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.3.d.a.21.4 ✓ 8 55.54 odd 2
110.3.d.a.21.8 yes 8 5.4 even 2
550.3.c.b.549.2 16 5.3 odd 4
550.3.c.b.549.7 16 55.32 even 4
550.3.c.b.549.10 16 55.43 even 4
550.3.c.b.549.15 16 5.2 odd 4
550.3.d.f.351.1 8 1.1 even 1 trivial
550.3.d.f.351.5 8 11.10 odd 2 inner
880.3.j.c.241.1 8 20.19 odd 2
880.3.j.c.241.2 8 220.219 even 2
990.3.b.b.901.4 8 15.14 odd 2
990.3.b.b.901.7 8 165.164 even 2