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Properties

Label 2610.2.f.c.811.2
Level $2610$
Weight $2$
Character 2610.811
Analytic conductor $20.841$
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] = [2610,2,Mod(811,2610)] 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("2610.811"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2610, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2610 = 2 \cdot 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2610.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2,2,0,8,0,0,0,0,0,-4] 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: \(20.8409549276\)
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: no (minimal twist has level 290)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 811.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2610.811
Dual form 2610.2.f.c.811.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +1.00000 q^{5} +4.00000 q^{7} -1.00000i q^{8} +1.00000i q^{10} +2.00000i q^{11} -2.00000 q^{13} +4.00000i q^{14} +1.00000 q^{16} +6.00000i q^{17} -2.00000i q^{19} -1.00000 q^{20} -2.00000 q^{22} +1.00000 q^{25} -2.00000i q^{26} -4.00000 q^{28} +(-5.00000 + 2.00000i) q^{29} +10.0000i q^{31} +1.00000i q^{32} -6.00000 q^{34} +4.00000 q^{35} +6.00000i q^{37} +2.00000 q^{38} -1.00000i q^{40} -12.0000i q^{41} -4.00000i q^{43} -2.00000i q^{44} +8.00000i q^{47} +9.00000 q^{49} +1.00000i q^{50} +2.00000 q^{52} +2.00000 q^{53} +2.00000i q^{55} -4.00000i q^{56} +(-2.00000 - 5.00000i) q^{58} +12.0000 q^{59} +4.00000i q^{61} -10.0000 q^{62} -1.00000 q^{64} -2.00000 q^{65} -8.00000 q^{67} -6.00000i q^{68} +4.00000i q^{70} +8.00000 q^{71} +2.00000i q^{73} -6.00000 q^{74} +2.00000i q^{76} +8.00000i q^{77} +10.0000i q^{79} +1.00000 q^{80} +12.0000 q^{82} +4.00000 q^{83} +6.00000i q^{85} +4.00000 q^{86} +2.00000 q^{88} -12.0000i q^{89} -8.00000 q^{91} -8.00000 q^{94} -2.00000i q^{95} -2.00000i q^{97} +9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{5} + 8 q^{7} - 4 q^{13} + 2 q^{16} - 2 q^{20} - 4 q^{22} + 2 q^{25} - 8 q^{28} - 10 q^{29} - 12 q^{34} + 8 q^{35} + 4 q^{38} + 18 q^{49} + 4 q^{52} + 4 q^{53} - 4 q^{58} + 24 q^{59}+ \cdots - 16 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2610\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1451\) \(1567\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(4\) −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
\(9\) 0 0
\(10\) 1.00000i 0.316228i
\(11\) 2.00000i 0.603023i 0.953463 + 0.301511i \(0.0974911\pi\)
−0.953463 + 0.301511i \(0.902509\pi\)
\(12\) 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 \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 0 0
\(19\) − 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
\(36\) 0 0
\(37\) 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 \(-0.613552\pi\)
0.349215 − 0.937043i \(-0.386448\pi\)
\(42\) 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
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) 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 \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 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 \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 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
\(63\) 0 0
\(64\) −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 \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 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
\(81\) 0 0
\(82\) 12.0000 1.32518
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 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 \(-0.780588\pi\)
0.771690 − 0.635999i \(-0.219412\pi\)
\(90\) 0 0
\(91\) −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
\(99\) 0 0
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 2610.2.f.c.811.2 2
3.2 odd 2 290.2.c.a.231.1 ✓ 2
12.11 even 2 2320.2.g.a.1681.2 2
15.2 even 4 1450.2.d.d.1449.2 2
15.8 even 4 1450.2.d.a.1449.1 2
15.14 odd 2 1450.2.c.b.1101.2 2
29.28 even 2 inner 2610.2.f.c.811.1 2
87.17 even 4 8410.2.a.e.1.1 1
87.41 even 4 8410.2.a.l.1.1 1
87.86 odd 2 290.2.c.a.231.2 yes 2
348.347 even 2 2320.2.g.a.1681.1 2
435.173 even 4 1450.2.d.d.1449.1 2
435.347 even 4 1450.2.d.a.1449.2 2
435.434 odd 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 3.2 odd 2
290.2.c.a.231.2 yes 2 87.86 odd 2
1450.2.c.b.1101.1 2 435.434 odd 2
1450.2.c.b.1101.2 2 15.14 odd 2
1450.2.d.a.1449.1 2 15.8 even 4
1450.2.d.a.1449.2 2 435.347 even 4
1450.2.d.d.1449.1 2 435.173 even 4
1450.2.d.d.1449.2 2 15.2 even 4
2320.2.g.a.1681.1 2 348.347 even 2
2320.2.g.a.1681.2 2 12.11 even 2
2610.2.f.c.811.1 2 29.28 even 2 inner
2610.2.f.c.811.2 2 1.1 even 1 trivial
8410.2.a.e.1.1 1 87.17 even 4
8410.2.a.l.1.1 1 87.41 even 4