↑

Properties

Label 4600.2.a.s.1.2
Level $4600$
Weight $2$
Character 4600.1
Self dual yes
Analytic conductor $36.731$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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] = [4600,2,Mod(1,4600)] 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("4600.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1,0,0,0,0,0,3,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 184)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 4600.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.56155 q^{3} +3.56155 q^{9} +5.12311 q^{11} -4.56155 q^{13} +3.12311 q^{17} +5.12311 q^{19} +1.00000 q^{23} +1.43845 q^{27} -0.561553 q^{29} -6.56155 q^{31} +13.1231 q^{33} +8.24621 q^{37} -11.6847 q^{39} +10.8078 q^{41} +8.00000 q^{43} -11.6847 q^{47} -7.00000 q^{49} +8.00000 q^{51} -2.00000 q^{53} +13.1231 q^{57} -6.24621 q^{59} +12.2462 q^{61} +5.12311 q^{67} +2.56155 q^{69} +9.43845 q^{71} +2.31534 q^{73} -5.12311 q^{79} -7.00000 q^{81} +2.24621 q^{83} -1.43845 q^{87} -13.3693 q^{89} -16.8078 q^{93} +13.3693 q^{97} +18.2462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 3 q^{9} + 2 q^{11} - 5 q^{13} - 2 q^{17} + 2 q^{19} + 2 q^{23} + 7 q^{27} + 3 q^{29} - 9 q^{31} + 18 q^{33} - 11 q^{39} + q^{41} + 16 q^{43} - 11 q^{47} - 14 q^{49} + 16 q^{51} - 4 q^{53}+ \cdots + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 0
\(3\) 2.56155 1.47891 0.739457 − 0.673204i \(-0.235083\pi\)
0.739457 + 0.673204i \(0.235083\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 3.56155 1.18718
\(10\) 0 0
\(11\) 5.12311 1.54467 0.772337 − 0.635213i \(-0.219088\pi\)
0.772337 + 0.635213i \(0.219088\pi\)
\(12\) 0 0
\(13\) −4.56155 −1.26515 −0.632574 − 0.774500i \(-0.718001\pi\)
−0.632574 + 0.774500i \(0.718001\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.12311 0.757464 0.378732 − 0.925506i \(-0.376360\pi\)
0.378732 + 0.925506i \(0.376360\pi\)
\(18\) 0 0
\(19\) 5.12311 1.17532 0.587661 − 0.809108i \(-0.300049\pi\)
0.587661 + 0.809108i \(0.300049\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.43845 0.276829
\(28\) 0 0
\(29\) −0.561553 −0.104278 −0.0521389 − 0.998640i \(-0.516604\pi\)
−0.0521389 + 0.998640i \(0.516604\pi\)
\(30\) 0 0
\(31\) −6.56155 −1.17849 −0.589245 − 0.807955i \(-0.700575\pi\)
−0.589245 + 0.807955i \(0.700575\pi\)
\(32\) 0 0
\(33\) 13.1231 2.28444
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.24621 1.35567 0.677834 − 0.735215i \(-0.262919\pi\)
0.677834 + 0.735215i \(0.262919\pi\)
\(38\) 0 0
\(39\) −11.6847 −1.87104
\(40\) 0 0
\(41\) 10.8078 1.68789 0.843945 − 0.536430i \(-0.180228\pi\)
0.843945 + 0.536430i \(0.180228\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.6847 −1.70438 −0.852191 − 0.523230i \(-0.824727\pi\)
−0.852191 + 0.523230i \(0.824727\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 8.00000 1.12022
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 13.1231 1.73820
\(58\) 0 0
\(59\) −6.24621 −0.813187 −0.406594 − 0.913609i \(-0.633284\pi\)
−0.406594 + 0.913609i \(0.633284\pi\)
\(60\) 0 0
\(61\) 12.2462 1.56797 0.783983 − 0.620782i \(-0.213185\pi\)
0.783983 + 0.620782i \(0.213185\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.12311 0.625887 0.312943 − 0.949772i \(-0.398685\pi\)
0.312943 + 0.949772i \(0.398685\pi\)
\(68\) 0 0
\(69\) 2.56155 0.308375
\(70\) 0 0
\(71\) 9.43845 1.12014 0.560069 − 0.828446i \(-0.310775\pi\)
0.560069 + 0.828446i \(0.310775\pi\)
\(72\) 0 0
\(73\) 2.31534 0.270990 0.135495 − 0.990778i \(-0.456737\pi\)
0.135495 + 0.990778i \(0.456737\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −5.12311 −0.576394 −0.288197 − 0.957571i \(-0.593056\pi\)
−0.288197 + 0.957571i \(0.593056\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 2.24621 0.246554 0.123277 − 0.992372i \(-0.460660\pi\)
0.123277 + 0.992372i \(0.460660\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.43845 −0.154218
\(88\) 0 0
\(89\) −13.3693 −1.41714 −0.708572 − 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −16.8078 −1.74288
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.3693 1.35745 0.678724 − 0.734393i \(-0.262533\pi\)
0.678724 + 0.734393i \(0.262533\pi\)
\(98\) 0 0
\(99\) 18.2462 1.83381
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 4600.2.a.s.1.2 2
4.3 odd 2 9200.2.a.br.1.1 2
5.2 odd 4 4600.2.e.o.4049.1 4
5.3 odd 4 4600.2.e.o.4049.4 4
5.4 even 2 184.2.a.e.1.1 ✓ 2
15.14 odd 2 1656.2.a.j.1.1 2
20.19 odd 2 368.2.a.i.1.2 2
35.34 odd 2 9016.2.a.w.1.2 2
40.19 odd 2 1472.2.a.p.1.1 2
40.29 even 2 1472.2.a.u.1.2 2
60.59 even 2 3312.2.a.t.1.2 2
115.114 odd 2 4232.2.a.o.1.1 2
460.459 even 2 8464.2.a.bd.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.a.e.1.1 ✓ 2 5.4 even 2
368.2.a.i.1.2 2 20.19 odd 2
1472.2.a.p.1.1 2 40.19 odd 2
1472.2.a.u.1.2 2 40.29 even 2
1656.2.a.j.1.1 2 15.14 odd 2
3312.2.a.t.1.2 2 60.59 even 2
4232.2.a.o.1.1 2 115.114 odd 2
4600.2.a.s.1.2 2 1.1 even 1 trivial
4600.2.e.o.4049.1 4 5.2 odd 4
4600.2.e.o.4049.4 4 5.3 odd 4
8464.2.a.bd.1.2 2 460.459 even 2
9016.2.a.w.1.2 2 35.34 odd 2
9200.2.a.br.1.1 2 4.3 odd 2