↑

Properties

Label 550.2.f.b.307.1
Level $550$
Weight $2$
Character 550.307
Analytic conductor $4.392$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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] = [550,2,Mod(43,550)] 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("550.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(550, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-4,0,0,0,0,0,0,0,0,4,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.39177211117\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 110)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 307.1
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 550.307
Dual form 550.2.f.b.43.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.707107 + 0.707107i) q^{2} +(-1.70711 + 1.70711i) q^{3} -1.00000i q^{4} -2.41421i q^{6} +(2.12132 - 2.12132i) q^{7} +(0.707107 + 0.707107i) q^{8} -2.82843i q^{9} +(-1.41421 + 3.00000i) q^{11} +(1.70711 + 1.70711i) q^{12} +(3.00000 + 3.00000i) q^{13} +3.00000i q^{14} -1.00000 q^{16} +(-5.12132 + 5.12132i) q^{17} +(2.00000 + 2.00000i) q^{18} +3.00000 q^{19} +7.24264i q^{21} +(-1.12132 - 3.12132i) q^{22} +(0.171573 - 0.171573i) q^{23} -2.41421 q^{24} -4.24264 q^{26} +(-0.292893 - 0.292893i) q^{27} +(-2.12132 - 2.12132i) q^{28} +1.24264 q^{29} -7.24264 q^{31} +(0.707107 - 0.707107i) q^{32} +(-2.70711 - 7.53553i) q^{33} -7.24264i q^{34} -2.82843 q^{36} +(-0.121320 - 0.121320i) q^{37} +(-2.12132 + 2.12132i) q^{38} -10.2426 q^{39} +1.75736i q^{41} +(-5.12132 - 5.12132i) q^{42} +(1.24264 + 1.24264i) q^{43} +(3.00000 + 1.41421i) q^{44} +0.242641i q^{46} +(-4.41421 - 4.41421i) q^{47} +(1.70711 - 1.70711i) q^{48} -2.00000i q^{49} -17.4853i q^{51} +(3.00000 - 3.00000i) q^{52} +(-9.53553 + 9.53553i) q^{53} +0.414214 q^{54} +3.00000 q^{56} +(-5.12132 + 5.12132i) q^{57} +(-0.878680 + 0.878680i) q^{58} +1.41421i q^{59} +7.24264i q^{61} +(5.12132 - 5.12132i) q^{62} +(-6.00000 - 6.00000i) q^{63} +1.00000i q^{64} +(7.24264 + 3.41421i) q^{66} +(4.00000 + 4.00000i) q^{67} +(5.12132 + 5.12132i) q^{68} +0.585786i q^{69} -1.24264 q^{71} +(2.00000 - 2.00000i) q^{72} +(6.00000 + 6.00000i) q^{73} +0.171573 q^{74} -3.00000i q^{76} +(3.36396 + 9.36396i) q^{77} +(7.24264 - 7.24264i) q^{78} -10.2426 q^{79} +9.48528 q^{81} +(-1.24264 - 1.24264i) q^{82} +(-7.24264 - 7.24264i) q^{83} +7.24264 q^{84} -1.75736 q^{86} +(-2.12132 + 2.12132i) q^{87} +(-3.12132 + 1.12132i) q^{88} +5.48528i q^{89} +12.7279 q^{91} +(-0.171573 - 0.171573i) q^{92} +(12.3640 - 12.3640i) q^{93} +6.24264 q^{94} +2.41421i q^{96} +(2.24264 + 2.24264i) q^{97} +(1.41421 + 1.41421i) q^{98} +(8.48528 + 4.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 4 q^{12} + 12 q^{13} - 4 q^{16} - 12 q^{17} + 8 q^{18} + 12 q^{19} + 4 q^{22} + 12 q^{23} - 4 q^{24} - 4 q^{27} - 12 q^{29} - 12 q^{31} - 8 q^{33} + 8 q^{37} - 24 q^{39} - 12 q^{42} - 12 q^{43}+ \cdots - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(177\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.707107 + 0.707107i −0.500000 + 0.500000i
\(3\) −1.70711 + 1.70711i −0.985599 + 0.985599i −0.999898 − 0.0142992i \(-0.995448\pi\)
0.0142992 + 0.999898i \(0.495448\pi\)
\(4\) − 1.00000i − 0.500000i
\(5\) 0 0
\(6\) − 2.41421i − 0.985599i
\(7\) 2.12132 − 2.12132i 0.801784 − 0.801784i −0.181591 − 0.983374i \(-0.558125\pi\)
0.983374 + 0.181591i \(0.0581245\pi\)
\(8\) 0.707107 + 0.707107i 0.250000 + 0.250000i
\(9\) − 2.82843i − 0.942809i
\(10\) 0 0
\(11\) −1.41421 + 3.00000i −0.426401 + 0.904534i
\(12\) 1.70711 + 1.70711i 0.492799 + 0.492799i
\(13\) 3.00000 + 3.00000i 0.832050 + 0.832050i 0.987797 − 0.155747i \(-0.0497784\pi\)
−0.155747 + 0.987797i \(0.549778\pi\)
\(14\) 3.00000i 0.801784i
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −5.12132 + 5.12132i −1.24210 + 1.24210i −0.282975 + 0.959127i \(0.591322\pi\)
−0.959127 + 0.282975i \(0.908678\pi\)
\(18\) 2.00000 + 2.00000i 0.471405 + 0.471405i
\(19\) 3.00000 0.688247 0.344124 − 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) 7.24264i 1.58047i
\(22\) −1.12132 − 3.12132i −0.239066 − 0.665468i
\(23\) 0.171573 − 0.171573i 0.0357754 − 0.0357754i −0.688993 − 0.724768i \(-0.741947\pi\)
0.724768 + 0.688993i \(0.241947\pi\)
\(24\) −2.41421 −0.492799
\(25\) 0 0
\(26\) −4.24264 −0.832050
\(27\) −0.292893 − 0.292893i −0.0563673 − 0.0563673i
\(28\) −2.12132 − 2.12132i −0.400892 − 0.400892i
\(29\) 1.24264 0.230753 0.115376 − 0.993322i \(-0.463193\pi\)
0.115376 + 0.993322i \(0.463193\pi\)
\(30\) 0 0
\(31\) −7.24264 −1.30082 −0.650408 − 0.759585i \(-0.725402\pi\)
−0.650408 + 0.759585i \(0.725402\pi\)
\(32\) 0.707107 − 0.707107i 0.125000 − 0.125000i
\(33\) −2.70711 − 7.53553i −0.471247 − 1.31177i
\(34\) − 7.24264i − 1.24210i
\(35\) 0 0
\(36\) −2.82843 −0.471405
\(37\) −0.121320 − 0.121320i −0.0199449 − 0.0199449i 0.697064 − 0.717009i \(-0.254489\pi\)
−0.717009 + 0.697064i \(0.754489\pi\)
\(38\) −2.12132 + 2.12132i −0.344124 + 0.344124i
\(39\) −10.2426 −1.64014
\(40\) 0 0
\(41\) 1.75736i 0.274453i 0.990540 + 0.137227i \(0.0438189\pi\)
−0.990540 + 0.137227i \(0.956181\pi\)
\(42\) −5.12132 − 5.12132i −0.790237 − 0.790237i
\(43\) 1.24264 + 1.24264i 0.189501 + 0.189501i 0.795480 − 0.605979i \(-0.207219\pi\)
−0.605979 + 0.795480i \(0.707219\pi\)
\(44\) 3.00000 + 1.41421i 0.452267 + 0.213201i
\(45\) 0 0
\(46\) 0.242641i 0.0357754i
\(47\) −4.41421 − 4.41421i −0.643879 − 0.643879i 0.307628 − 0.951507i \(-0.400465\pi\)
−0.951507 + 0.307628i \(0.900465\pi\)
\(48\) 1.70711 − 1.70711i 0.246400 − 0.246400i
\(49\) − 2.00000i − 0.285714i
\(50\) 0 0
\(51\) − 17.4853i − 2.44843i
\(52\) 3.00000 − 3.00000i 0.416025 − 0.416025i
\(53\) −9.53553 + 9.53553i −1.30981 + 1.30981i −0.388254 + 0.921552i \(0.626922\pi\)
−0.921552 + 0.388254i \(0.873078\pi\)
\(54\) 0.414214 0.0563673
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) −5.12132 + 5.12132i −0.678335 + 0.678335i
\(58\) −0.878680 + 0.878680i −0.115376 + 0.115376i
\(59\) 1.41421i 0.184115i 0.995754 + 0.0920575i \(0.0293443\pi\)
−0.995754 + 0.0920575i \(0.970656\pi\)
\(60\) 0 0
\(61\) 7.24264i 0.927325i 0.886012 + 0.463663i \(0.153465\pi\)
−0.886012 + 0.463663i \(0.846535\pi\)
\(62\) 5.12132 − 5.12132i 0.650408 − 0.650408i
\(63\) −6.00000 − 6.00000i −0.755929 − 0.755929i
\(64\) 1.00000i 0.125000i
\(65\) 0 0
\(66\) 7.24264 + 3.41421i 0.891507 + 0.420261i
\(67\) 4.00000 + 4.00000i 0.488678 + 0.488678i 0.907889 − 0.419211i \(-0.137693\pi\)
−0.419211 + 0.907889i \(0.637693\pi\)
\(68\) 5.12132 + 5.12132i 0.621051 + 0.621051i
\(69\) 0.585786i 0.0705204i
\(70\) 0 0
\(71\) −1.24264 −0.147474 −0.0737372 − 0.997278i \(-0.523493\pi\)
−0.0737372 + 0.997278i \(0.523493\pi\)
\(72\) 2.00000 − 2.00000i 0.235702 − 0.235702i
\(73\) 6.00000 + 6.00000i 0.702247 + 0.702247i 0.964892 − 0.262646i \(-0.0845950\pi\)
−0.262646 + 0.964892i \(0.584595\pi\)
\(74\) 0.171573 0.0199449
\(75\) 0 0
\(76\) − 3.00000i − 0.344124i
\(77\) 3.36396 + 9.36396i 0.383359 + 1.06712i
\(78\) 7.24264 − 7.24264i 0.820068 − 0.820068i
\(79\) −10.2426 −1.15239 −0.576194 − 0.817313i \(-0.695463\pi\)
−0.576194 + 0.817313i \(0.695463\pi\)
\(80\) 0 0
\(81\) 9.48528 1.05392
\(82\) −1.24264 − 1.24264i −0.137227 − 0.137227i
\(83\) −7.24264 − 7.24264i −0.794983 − 0.794983i 0.187317 − 0.982300i \(-0.440021\pi\)
−0.982300 + 0.187317i \(0.940021\pi\)
\(84\) 7.24264 0.790237
\(85\) 0 0
\(86\) −1.75736 −0.189501
\(87\) −2.12132 + 2.12132i −0.227429 + 0.227429i
\(88\) −3.12132 + 1.12132i −0.332734 + 0.119533i
\(89\) 5.48528i 0.581439i 0.956808 + 0.290719i \(0.0938946\pi\)
−0.956808 + 0.290719i \(0.906105\pi\)
\(90\) 0 0
\(91\) 12.7279 1.33425
\(92\) −0.171573 − 0.171573i −0.0178877 − 0.0178877i
\(93\) 12.3640 − 12.3640i 1.28208 − 1.28208i
\(94\) 6.24264 0.643879
\(95\) 0 0
\(96\) 2.41421i 0.246400i
\(97\) 2.24264 + 2.24264i 0.227706 + 0.227706i 0.811734 − 0.584028i \(-0.198524\pi\)
−0.584028 + 0.811734i \(0.698524\pi\)
\(98\) 1.41421 + 1.41421i 0.142857 + 0.142857i
\(99\) 8.48528 + 4.00000i 0.852803 + 0.402015i
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 550.2.f.b.307.1 4
5.2 odd 4 110.2.f.b.43.1 ✓ 4
5.3 odd 4 550.2.f.a.43.2 4
5.4 even 2 110.2.f.c.87.2 yes 4
11.10 odd 2 550.2.f.a.307.2 4
15.2 even 4 990.2.m.c.703.2 4
15.14 odd 2 990.2.m.d.307.1 4
20.7 even 4 880.2.bd.c.593.1 4
20.19 odd 2 880.2.bd.b.417.1 4
55.32 even 4 110.2.f.c.43.2 yes 4
55.43 even 4 inner 550.2.f.b.43.1 4
55.54 odd 2 110.2.f.b.87.1 yes 4
165.32 odd 4 990.2.m.d.703.1 4
165.164 even 2 990.2.m.c.307.2 4
220.87 odd 4 880.2.bd.b.593.1 4
220.219 even 2 880.2.bd.c.417.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.f.b.43.1 ✓ 4 5.2 odd 4
110.2.f.b.87.1 yes 4 55.54 odd 2
110.2.f.c.43.2 yes 4 55.32 even 4
110.2.f.c.87.2 yes 4 5.4 even 2
550.2.f.a.43.2 4 5.3 odd 4
550.2.f.a.307.2 4 11.10 odd 2
550.2.f.b.43.1 4 55.43 even 4 inner
550.2.f.b.307.1 4 1.1 even 1 trivial
880.2.bd.b.417.1 4 20.19 odd 2
880.2.bd.b.593.1 4 220.87 odd 4
880.2.bd.c.417.1 4 220.219 even 2
880.2.bd.c.593.1 4 20.7 even 4
990.2.m.c.307.2 4 165.164 even 2
990.2.m.c.703.2 4 15.2 even 4
990.2.m.d.307.1 4 15.14 odd 2
990.2.m.d.703.1 4 165.32 odd 4