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Properties

Label 550.3.d.f.351.2
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 351.2
Root \(1.51954 + 1.14412i\) of defining polynomial
Character \(\chi\) \(=\) 550.351
Dual form 550.3.d.f.351.6

$q$-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)
 
\(f(q)\) \(=\) \(q-1.41421i q^{2} +1.35781 q^{3} -2.00000 q^{4} -1.92023i q^{6} -12.4389i q^{7} +2.82843i q^{8} -7.15636 q^{9} +(-7.23901 + 8.28232i) q^{11} -2.71561 q^{12} -1.28012i q^{13} -17.5912 q^{14} +4.00000 q^{16} +3.33028i q^{17} +10.1206i q^{18} +1.88632i q^{19} -16.8896i q^{21} +(11.7130 + 10.2375i) q^{22} -32.3565 q^{23} +3.84046i q^{24} -1.81036 q^{26} -21.9372 q^{27} +24.8777i q^{28} +27.9138i q^{29} +16.5112 q^{31} -5.65685i q^{32} +(-9.82918 + 11.2458i) q^{33} +4.70973 q^{34} +14.3127 q^{36} +22.4573 q^{37} +2.66765 q^{38} -1.73816i q^{39} -52.3967i q^{41} -23.8855 q^{42} +15.7171i q^{43} +(14.4780 - 16.5646i) q^{44} +45.7590i q^{46} -87.6835 q^{47} +5.43123 q^{48} -105.726 q^{49} +4.52188i q^{51} +2.56024i q^{52} -74.2161 q^{53} +31.0239i q^{54} +35.1824 q^{56} +2.56125i q^{57} +39.4761 q^{58} -26.8351 q^{59} -47.4483i q^{61} -23.3503i q^{62} +89.0171i q^{63} -8.00000 q^{64} +(15.9039 + 13.9006i) q^{66} +79.2475 q^{67} -6.66057i q^{68} -43.9339 q^{69} -74.9405 q^{71} -20.2412i q^{72} -64.0267i q^{73} -31.7594i q^{74} -3.77263i q^{76} +(103.023 + 90.0451i) q^{77} -2.45812 q^{78} +151.980i q^{79} +34.6207 q^{81} -74.1001 q^{82} -151.469i q^{83} +33.7792i q^{84} +22.2273 q^{86} +37.9016i q^{87} +(-23.4259 - 20.4750i) q^{88} +127.627 q^{89} -15.9233 q^{91} +64.7130 q^{92} +22.4190 q^{93} +124.003i q^{94} -7.68092i q^{96} +88.1768 q^{97} +149.519i q^{98} +(51.8050 - 59.2712i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} - 16 q^{4} + 40 q^{9} - 16 q^{12} - 32 q^{14} + 32 q^{16} + 136 q^{23} + 80 q^{26} - 64 q^{27} - 64 q^{31} - 88 q^{33} + 112 q^{34} - 80 q^{36} + 48 q^{37} + 208 q^{42} - 152 q^{47} + 32 q^{48}+ \cdots + 704 q^{99}+O(q^{100}) \) 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
\(3\) 1.35781 0.452602 0.226301 − 0.974057i \(-0.427337\pi\)
0.226301 + 0.974057i \(0.427337\pi\)
\(4\) −2.00000 −0.500000
\(5\) 0 0
\(6\) − 1.92023i − 0.320038i
\(7\) − 12.4389i − 1.77698i −0.458894 − 0.888491i \(-0.651754\pi\)
0.458894 − 0.888491i \(-0.348246\pi\)
\(8\) 2.82843i 0.353553i
\(9\) −7.15636 −0.795151
\(10\) 0 0
\(11\) −7.23901 + 8.28232i −0.658092 + 0.752938i
\(12\) −2.71561 −0.226301
\(13\) − 1.28012i − 0.0984708i −0.998787 − 0.0492354i \(-0.984322\pi\)
0.998787 − 0.0492354i \(-0.0156785\pi\)
\(14\) −17.5912 −1.25652
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 3.33028i 0.195899i 0.995191 + 0.0979495i \(0.0312284\pi\)
−0.995191 + 0.0979495i \(0.968772\pi\)
\(18\) 10.1206i 0.562257i
\(19\) 1.88632i 0.0992798i 0.998767 + 0.0496399i \(0.0158074\pi\)
−0.998767 + 0.0496399i \(0.984193\pi\)
\(20\) 0 0
\(21\) − 16.8896i − 0.804266i
\(22\) 11.7130 + 10.2375i 0.532407 + 0.465341i
\(23\) −32.3565 −1.40680 −0.703402 − 0.710792i \(-0.748337\pi\)
−0.703402 + 0.710792i \(0.748337\pi\)
\(24\) 3.84046i 0.160019i
\(25\) 0 0
\(26\) −1.81036 −0.0696294
\(27\) −21.9372 −0.812490
\(28\) 24.8777i 0.888491i
\(29\) 27.9138i 0.962546i 0.876571 + 0.481273i \(0.159825\pi\)
−0.876571 + 0.481273i \(0.840175\pi\)
\(30\) 0 0
\(31\) 16.5112 0.532618 0.266309 − 0.963888i \(-0.414196\pi\)
0.266309 + 0.963888i \(0.414196\pi\)
\(32\) − 5.65685i − 0.176777i
\(33\) −9.82918 + 11.2458i −0.297854 + 0.340781i
\(34\) 4.70973 0.138522
\(35\) 0 0
\(36\) 14.3127 0.397576
\(37\) 22.4573 0.606954 0.303477 − 0.952839i \(-0.401852\pi\)
0.303477 + 0.952839i \(0.401852\pi\)
\(38\) 2.66765 0.0702014
\(39\) − 1.73816i − 0.0445681i
\(40\) 0 0
\(41\) − 52.3967i − 1.27797i −0.769220 − 0.638984i \(-0.779355\pi\)
0.769220 − 0.638984i \(-0.220645\pi\)
\(42\) −23.8855 −0.568702
\(43\) 15.7171i 0.365513i 0.983158 + 0.182756i \(0.0585020\pi\)
−0.983158 + 0.182756i \(0.941498\pi\)
\(44\) 14.4780 − 16.5646i 0.329046 − 0.376469i
\(45\) 0 0
\(46\) 45.7590i 0.994761i
\(47\) −87.6835 −1.86561 −0.932804 − 0.360385i \(-0.882645\pi\)
−0.932804 + 0.360385i \(0.882645\pi\)
\(48\) 5.43123 0.113151
\(49\) −105.726 −2.15766
\(50\) 0 0
\(51\) 4.52188i 0.0886644i
\(52\) 2.56024i 0.0492354i
\(53\) −74.2161 −1.40030 −0.700152 − 0.713994i \(-0.746884\pi\)
−0.700152 + 0.713994i \(0.746884\pi\)
\(54\) 31.0239i 0.574517i
\(55\) 0 0
\(56\) 35.1824 0.628258
\(57\) 2.56125i 0.0449343i
\(58\) 39.4761 0.680623
\(59\) −26.8351 −0.454832 −0.227416 − 0.973798i \(-0.573028\pi\)
−0.227416 + 0.973798i \(0.573028\pi\)
\(60\) 0 0
\(61\) − 47.4483i − 0.777841i −0.921271 − 0.388921i \(-0.872848\pi\)
0.921271 − 0.388921i \(-0.127152\pi\)
\(62\) − 23.3503i − 0.376618i
\(63\) 89.0171i 1.41297i
\(64\) −8.00000 −0.125000
\(65\) 0 0
\(66\) 15.9039 + 13.9006i 0.240969 + 0.210614i
\(67\) 79.2475 1.18280 0.591399 − 0.806379i \(-0.298576\pi\)
0.591399 + 0.806379i \(0.298576\pi\)
\(68\) − 6.66057i − 0.0979495i
\(69\) −43.9339 −0.636723
\(70\) 0 0
\(71\) −74.9405 −1.05550 −0.527750 − 0.849400i \(-0.676964\pi\)
−0.527750 + 0.849400i \(0.676964\pi\)
\(72\) − 20.2412i − 0.281128i
\(73\) − 64.0267i − 0.877078i −0.898712 − 0.438539i \(-0.855496\pi\)
0.898712 − 0.438539i \(-0.144504\pi\)
\(74\) − 31.7594i − 0.429182i
\(75\) 0 0
\(76\) − 3.77263i − 0.0496399i
\(77\) 103.023 + 90.0451i 1.33796 + 1.16942i
\(78\) −2.45812 −0.0315144
\(79\) 151.980i 1.92380i 0.273400 + 0.961901i \(0.411852\pi\)
−0.273400 + 0.961901i \(0.588148\pi\)
\(80\) 0 0
\(81\) 34.6207 0.427416
\(82\) −74.1001 −0.903660
\(83\) − 151.469i − 1.82492i −0.409161 − 0.912462i \(-0.634179\pi\)
0.409161 − 0.912462i \(-0.365821\pi\)
\(84\) 33.7792i 0.402133i
\(85\) 0 0
\(86\) 22.2273 0.258457
\(87\) 37.9016i 0.435651i
\(88\) −23.4259 − 20.4750i −0.266204 − 0.232671i
\(89\) 127.627 1.43401 0.717005 − 0.697068i \(-0.245513\pi\)
0.717005 + 0.697068i \(0.245513\pi\)
\(90\) 0 0
\(91\) −15.9233 −0.174981
\(92\) 64.7130 0.703402
\(93\) 22.4190 0.241064
\(94\) 124.003i 1.31918i
\(95\) 0 0
\(96\) − 7.68092i − 0.0800095i
\(97\) 88.1768 0.909040 0.454520 − 0.890737i \(-0.349811\pi\)
0.454520 + 0.890737i \(0.349811\pi\)
\(98\) 149.519i 1.52570i
\(99\) 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.d.f.351.2 8
5.2 odd 4 550.3.c.b.549.12 16
5.3 odd 4 550.3.c.b.549.5 16
5.4 even 2 110.3.d.a.21.7 yes 8
11.10 odd 2 inner 550.3.d.f.351.6 8
15.14 odd 2 990.3.b.b.901.2 8
20.19 odd 2 880.3.j.c.241.3 8
55.32 even 4 550.3.c.b.549.4 16
55.43 even 4 550.3.c.b.549.13 16
55.54 odd 2 110.3.d.a.21.3 ✓ 8
165.164 even 2 990.3.b.b.901.5 8
220.219 even 2 880.3.j.c.241.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.3.d.a.21.3 ✓ 8 55.54 odd 2
110.3.d.a.21.7 yes 8 5.4 even 2
550.3.c.b.549.4 16 55.32 even 4
550.3.c.b.549.5 16 5.3 odd 4
550.3.c.b.549.12 16 5.2 odd 4
550.3.c.b.549.13 16 55.43 even 4
550.3.d.f.351.2 8 1.1 even 1 trivial
550.3.d.f.351.6 8 11.10 odd 2 inner
880.3.j.c.241.3 8 20.19 odd 2
880.3.j.c.241.4 8 220.219 even 2
990.3.b.b.901.2 8 15.14 odd 2
990.3.b.b.901.5 8 165.164 even 2