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Properties

Label 4600.2.e.s.4049.6
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $6$
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] = [4600,2,Mod(4049,4600)] 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("4600.4049"); 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, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.6
Root \(0.445042i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.s.4049.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.24698i q^{3} -4.49396i q^{7} -2.04892 q^{9} +3.38404 q^{11} +3.04892i q^{13} -4.49396i q^{17} -7.20775 q^{19} +10.0978 q^{21} -1.00000i q^{23} +2.13706i q^{27} +5.51573 q^{29} -1.29590 q^{31} +7.60388i q^{33} +5.82371i q^{37} -6.85086 q^{39} +3.63102 q^{41} -10.7138i q^{43} -2.06100i q^{47} -13.1957 q^{49} +10.0978 q^{51} -2.98792i q^{53} -16.1957i q^{57} +9.31767 q^{59} -13.3056 q^{61} +9.20775i q^{63} +8.19567i q^{67} +2.24698 q^{69} +3.48858 q^{71} -8.72348i q^{73} -15.2078i q^{77} +9.92154 q^{79} -10.9487 q^{81} -15.7560i q^{83} +12.3937i q^{87} +0.121998 q^{89} +13.7017 q^{91} -2.91185i q^{93} -15.6039i q^{97} -6.93362 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{9} - 8 q^{19} + 24 q^{21} + 8 q^{29} + 20 q^{31} - 14 q^{39} - 8 q^{41} - 6 q^{49} + 24 q^{51} + 22 q^{59} - 8 q^{61} + 4 q^{69} + 8 q^{71} + 8 q^{79} - 2 q^{81} + 40 q^{89} + 28 q^{91} - 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.24698i 1.29729i 0.761089 + 0.648647i \(0.224665\pi\)
−0.761089 + 0.648647i \(0.775335\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 4.49396i − 1.69856i −0.527945 − 0.849278i \(-0.677037\pi\)
0.527945 − 0.849278i \(-0.322963\pi\)
\(8\) 0 0
\(9\) −2.04892 −0.682972
\(10\) 0 0
\(11\) 3.38404 1.02033 0.510164 − 0.860077i \(-0.329585\pi\)
0.510164 + 0.860077i \(0.329585\pi\)
\(12\) 0 0
\(13\) 3.04892i 0.845618i 0.906219 + 0.422809i \(0.138956\pi\)
−0.906219 + 0.422809i \(0.861044\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.49396i − 1.08995i −0.838454 − 0.544973i \(-0.816540\pi\)
0.838454 − 0.544973i \(-0.183460\pi\)
\(18\) 0 0
\(19\) −7.20775 −1.65357 −0.826786 − 0.562517i \(-0.809833\pi\)
−0.826786 + 0.562517i \(0.809833\pi\)
\(20\) 0 0
\(21\) 10.0978 2.20353
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.13706i 0.411278i
\(28\) 0 0
\(29\) 5.51573 1.02425 0.512123 − 0.858912i \(-0.328859\pi\)
0.512123 + 0.858912i \(0.328859\pi\)
\(30\) 0 0
\(31\) −1.29590 −0.232750 −0.116375 − 0.993205i \(-0.537127\pi\)
−0.116375 + 0.993205i \(0.537127\pi\)
\(32\) 0 0
\(33\) 7.60388i 1.32366i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.82371i 0.957412i 0.877975 + 0.478706i \(0.158894\pi\)
−0.877975 + 0.478706i \(0.841106\pi\)
\(38\) 0 0
\(39\) −6.85086 −1.09701
\(40\) 0 0
\(41\) 3.63102 0.567070 0.283535 − 0.958962i \(-0.408493\pi\)
0.283535 + 0.958962i \(0.408493\pi\)
\(42\) 0 0
\(43\) − 10.7138i − 1.63384i −0.576752 − 0.816919i \(-0.695680\pi\)
0.576752 − 0.816919i \(-0.304320\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 2.06100i − 0.300628i −0.988638 − 0.150314i \(-0.951972\pi\)
0.988638 − 0.150314i \(-0.0480284\pi\)
\(48\) 0 0
\(49\) −13.1957 −1.88510
\(50\) 0 0
\(51\) 10.0978 1.41398
\(52\) 0 0
\(53\) − 2.98792i − 0.410422i −0.978718 − 0.205211i \(-0.934212\pi\)
0.978718 − 0.205211i \(-0.0657881\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 16.1957i − 2.14517i
\(58\) 0 0
\(59\) 9.31767 1.21306 0.606528 − 0.795062i \(-0.292562\pi\)
0.606528 + 0.795062i \(0.292562\pi\)
\(60\) 0 0
\(61\) −13.3056 −1.70361 −0.851803 − 0.523863i \(-0.824491\pi\)
−0.851803 + 0.523863i \(0.824491\pi\)
\(62\) 0 0
\(63\) 9.20775i 1.16007i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.19567i 1.00126i 0.865661 + 0.500630i \(0.166898\pi\)
−0.865661 + 0.500630i \(0.833102\pi\)
\(68\) 0 0
\(69\) 2.24698 0.270505
\(70\) 0 0
\(71\) 3.48858 0.414019 0.207009 − 0.978339i \(-0.433627\pi\)
0.207009 + 0.978339i \(0.433627\pi\)
\(72\) 0 0
\(73\) − 8.72348i − 1.02101i −0.859876 − 0.510503i \(-0.829459\pi\)
0.859876 − 0.510503i \(-0.170541\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 15.2078i − 1.73308i
\(78\) 0 0
\(79\) 9.92154 1.11626 0.558130 − 0.829753i \(-0.311519\pi\)
0.558130 + 0.829753i \(0.311519\pi\)
\(80\) 0 0
\(81\) −10.9487 −1.21652
\(82\) 0 0
\(83\) − 15.7560i − 1.72945i −0.502249 − 0.864723i \(-0.667494\pi\)
0.502249 − 0.864723i \(-0.332506\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 12.3937i 1.32875i
\(88\) 0 0
\(89\) 0.121998 0.0129317 0.00646587 − 0.999979i \(-0.497942\pi\)
0.00646587 + 0.999979i \(0.497942\pi\)
\(90\) 0 0
\(91\) 13.7017 1.43633
\(92\) 0 0
\(93\) − 2.91185i − 0.301945i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 15.6039i − 1.58433i −0.610305 − 0.792167i \(-0.708953\pi\)
0.610305 − 0.792167i \(-0.291047\pi\)
\(98\) 0 0
\(99\) −6.93362 −0.696855
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.e.s.4049.6 6
5.2 odd 4 4600.2.a.z.1.3 yes 3
5.3 odd 4 4600.2.a.w.1.1 ✓ 3
5.4 even 2 inner 4600.2.e.s.4049.1 6
20.3 even 4 9200.2.a.ch.1.3 3
20.7 even 4 9200.2.a.cb.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.w.1.1 ✓ 3 5.3 odd 4
4600.2.a.z.1.3 yes 3 5.2 odd 4
4600.2.e.s.4049.1 6 5.4 even 2 inner
4600.2.e.s.4049.6 6 1.1 even 1 trivial
9200.2.a.cb.1.1 3 20.7 even 4
9200.2.a.ch.1.3 3 20.3 even 4