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Properties

Label 290.2.c.a.231.1
Level $290$
Weight $2$
Character 290.231
Analytic conductor $2.316$
Analytic rank $0$
Dimension $2$
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] = [290,2,Mod(231,290)] 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("290.231"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(290, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 290 = 2 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 290.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.31566165862\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 231.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 290.231
Dual form 290.2.c.a.231.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} -1.00000 q^{5} +4.00000 q^{7} +1.00000i q^{8} +3.00000 q^{9} +1.00000i q^{10} -2.00000i q^{11} -2.00000 q^{13} -4.00000i q^{14} +1.00000 q^{16} -6.00000i q^{17} -3.00000i q^{18} -2.00000i q^{19} +1.00000 q^{20} -2.00000 q^{22} +1.00000 q^{25} +2.00000i q^{26} -4.00000 q^{28} +(5.00000 - 2.00000i) q^{29} +10.0000i q^{31} -1.00000i q^{32} -6.00000 q^{34} -4.00000 q^{35} -3.00000 q^{36} +6.00000i q^{37} -2.00000 q^{38} -1.00000i q^{40} +12.0000i q^{41} -4.00000i q^{43} +2.00000i q^{44} -3.00000 q^{45} -8.00000i q^{47} +9.00000 q^{49} -1.00000i q^{50} +2.00000 q^{52} -2.00000 q^{53} +2.00000i q^{55} +4.00000i q^{56} +(-2.00000 - 5.00000i) q^{58} -12.0000 q^{59} +4.00000i q^{61} +10.0000 q^{62} +12.0000 q^{63} -1.00000 q^{64} +2.00000 q^{65} -8.00000 q^{67} +6.00000i q^{68} +4.00000i q^{70} -8.00000 q^{71} +3.00000i q^{72} +2.00000i q^{73} +6.00000 q^{74} +2.00000i q^{76} -8.00000i q^{77} +10.0000i q^{79} -1.00000 q^{80} +9.00000 q^{81} +12.0000 q^{82} -4.00000 q^{83} +6.00000i q^{85} -4.00000 q^{86} +2.00000 q^{88} +12.0000i q^{89} +3.00000i q^{90} -8.00000 q^{91} -8.00000 q^{94} +2.00000i q^{95} -2.00000i q^{97} -9.00000i q^{98} -6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{5} + 8 q^{7} + 6 q^{9} - 4 q^{13} + 2 q^{16} + 2 q^{20} - 4 q^{22} + 2 q^{25} - 8 q^{28} + 10 q^{29} - 12 q^{34} - 8 q^{35} - 6 q^{36} - 4 q^{38} - 6 q^{45} + 18 q^{49} + 4 q^{52} - 4 q^{53}+ \cdots - 16 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/290\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(117\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) −1.00000 −0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 3.00000 1.00000
\(10\) 1.00000i 0.316228i
\(11\) − 2.00000i − 0.603023i −0.953463 − 0.301511i \(-0.902509\pi\)
0.953463 − 0.301511i \(-0.0974911\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) − 4.00000i − 1.06904i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 6.00000i − 1.45521i −0.685994 − 0.727607i \(-0.740633\pi\)
0.685994 − 0.727607i \(-0.259367\pi\)
\(18\) − 3.00000i − 0.707107i
\(19\) − 2.00000i − 0.458831i −0.973329 − 0.229416i \(-0.926318\pi\)
0.973329 − 0.229416i \(-0.0736815\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.00000i 0.392232i
\(27\) 0 0
\(28\) −4.00000 −0.755929
\(29\) 5.00000 − 2.00000i 0.928477 − 0.371391i
\(30\) 0 0
\(31\) 10.0000i 1.79605i 0.439941 + 0.898027i \(0.354999\pi\)
−0.439941 + 0.898027i \(0.645001\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) −4.00000 −0.676123
\(36\) −3.00000 −0.500000
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) − 1.00000i − 0.158114i
\(41\) 12.0000i 1.87409i 0.349215 + 0.937043i \(0.386448\pi\)
−0.349215 + 0.937043i \(0.613552\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 − 0.304997i \(-0.901344\pi\)
0.952353 − 0.304997i \(-0.0986555\pi\)
\(44\) 2.00000i 0.301511i
\(45\) −3.00000 −0.447214
\(46\) 0 0
\(47\) − 8.00000i − 1.16692i −0.812142 − 0.583460i \(-0.801699\pi\)
0.812142 − 0.583460i \(-0.198301\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) − 1.00000i − 0.141421i
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 2.00000i 0.269680i
\(56\) 4.00000i 0.534522i
\(57\) 0 0
\(58\) −2.00000 − 5.00000i −0.262613 − 0.656532i
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 4.00000i 0.512148i 0.966657 + 0.256074i \(0.0824290\pi\)
−0.966657 + 0.256074i \(0.917571\pi\)
\(62\) 10.0000 1.27000
\(63\) 12.0000 1.51186
\(64\) −1.00000 −0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 6.00000i 0.727607i
\(69\) 0 0
\(70\) 4.00000i 0.478091i
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 3.00000i 0.353553i
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) 2.00000i 0.229416i
\(77\) − 8.00000i − 0.911685i
\(78\) 0 0
\(79\) 10.0000i 1.12509i 0.826767 + 0.562544i \(0.190177\pi\)
−0.826767 + 0.562544i \(0.809823\pi\)
\(80\) −1.00000 −0.111803
\(81\) 9.00000 1.00000
\(82\) 12.0000 1.32518
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 6.00000i 0.650791i
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) 12.0000i 1.27200i 0.771690 + 0.635999i \(0.219412\pi\)
−0.771690 + 0.635999i \(0.780588\pi\)
\(90\) 3.00000i 0.316228i
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 2.00000i 0.205196i
\(96\) 0 0
\(97\) − 2.00000i − 0.203069i −0.994832 − 0.101535i \(-0.967625\pi\)
0.994832 − 0.101535i \(-0.0323753\pi\)
\(98\) − 9.00000i − 0.909137i
\(99\) − 6.00000i − 0.603023i
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 290.2.c.a.231.1 ✓ 2
3.2 odd 2 2610.2.f.c.811.2 2
4.3 odd 2 2320.2.g.a.1681.2 2
5.2 odd 4 1450.2.d.d.1449.2 2
5.3 odd 4 1450.2.d.a.1449.1 2
5.4 even 2 1450.2.c.b.1101.2 2
29.12 odd 4 8410.2.a.l.1.1 1
29.17 odd 4 8410.2.a.e.1.1 1
29.28 even 2 inner 290.2.c.a.231.2 yes 2
87.86 odd 2 2610.2.f.c.811.1 2
116.115 odd 2 2320.2.g.a.1681.1 2
145.28 odd 4 1450.2.d.d.1449.1 2
145.57 odd 4 1450.2.d.a.1449.2 2
145.144 even 2 1450.2.c.b.1101.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
290.2.c.a.231.1 ✓ 2 1.1 even 1 trivial
290.2.c.a.231.2 yes 2 29.28 even 2 inner
1450.2.c.b.1101.1 2 145.144 even 2
1450.2.c.b.1101.2 2 5.4 even 2
1450.2.d.a.1449.1 2 5.3 odd 4
1450.2.d.a.1449.2 2 145.57 odd 4
1450.2.d.d.1449.1 2 145.28 odd 4
1450.2.d.d.1449.2 2 5.2 odd 4
2320.2.g.a.1681.1 2 116.115 odd 2
2320.2.g.a.1681.2 2 4.3 odd 2
2610.2.f.c.811.1 2 87.86 odd 2
2610.2.f.c.811.2 2 3.2 odd 2
8410.2.a.e.1.1 1 29.17 odd 4
8410.2.a.l.1.1 1 29.12 odd 4