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Properties

Label 4600.2.e.l.4049.1
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $4$
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 = [4,0,0,0,0,0,0,0,-12,0,4,0,0,0,0,0,0,0,-4] 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: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.1
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.l.4049.4

$q$-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)
 
\(f(q)\) \(=\) \(q-3.23607i q^{3} -4.23607i q^{7} -7.47214 q^{9} +1.00000 q^{11} +2.23607i q^{13} +6.47214i q^{17} -1.00000 q^{19} -13.7082 q^{21} +1.00000i q^{23} +14.4721i q^{27} +1.76393 q^{29} +0.472136 q^{31} -3.23607i q^{33} +11.2361i q^{37} +7.23607 q^{39} -5.94427 q^{41} -2.52786i q^{43} +11.7082i q^{47} -10.9443 q^{49} +20.9443 q^{51} -1.23607i q^{53} +3.23607i q^{57} -3.23607 q^{59} -7.23607 q^{61} +31.6525i q^{63} +12.9443i q^{67} +3.23607 q^{69} +10.0000 q^{71} +9.47214i q^{73} -4.23607i q^{77} -15.1803 q^{79} +24.4164 q^{81} -9.00000i q^{83} -5.70820i q^{87} -2.00000 q^{89} +9.47214 q^{91} -1.52786i q^{93} +6.18034i q^{97} -7.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} + 4 q^{11} - 4 q^{19} - 28 q^{21} + 16 q^{29} - 16 q^{31} + 20 q^{39} + 12 q^{41} - 8 q^{49} + 48 q^{51} - 4 q^{59} - 20 q^{61} + 4 q^{69} + 40 q^{71} - 16 q^{79} + 44 q^{81} - 8 q^{89} + 20 q^{91}+ \cdots - 12 q^{99}+O(q^{100}) \) 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
\(3\) − 3.23607i − 1.86834i −0.356822 − 0.934172i \(-0.616140\pi\)
0.356822 − 0.934172i \(-0.383860\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 4.23607i − 1.60108i −0.599277 − 0.800542i \(-0.704545\pi\)
0.599277 − 0.800542i \(-0.295455\pi\)
\(8\) 0 0
\(9\) −7.47214 −2.49071
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 − 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) 2.23607i 0.620174i 0.950708 + 0.310087i \(0.100358\pi\)
−0.950708 + 0.310087i \(0.899642\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.47214i 1.56972i 0.619671 + 0.784862i \(0.287266\pi\)
−0.619671 + 0.784862i \(0.712734\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) −13.7082 −2.99138
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 14.4721i 2.78516i
\(28\) 0 0
\(29\) 1.76393 0.327554 0.163777 − 0.986497i \(-0.447632\pi\)
0.163777 + 0.986497i \(0.447632\pi\)
\(30\) 0 0
\(31\) 0.472136 0.0847981 0.0423991 − 0.999101i \(-0.486500\pi\)
0.0423991 + 0.999101i \(0.486500\pi\)
\(32\) 0 0
\(33\) − 3.23607i − 0.563327i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 11.2361i 1.84720i 0.383360 + 0.923599i \(0.374767\pi\)
−0.383360 + 0.923599i \(0.625233\pi\)
\(38\) 0 0
\(39\) 7.23607 1.15870
\(40\) 0 0
\(41\) −5.94427 −0.928339 −0.464170 − 0.885746i \(-0.653647\pi\)
−0.464170 + 0.885746i \(0.653647\pi\)
\(42\) 0 0
\(43\) − 2.52786i − 0.385496i −0.981248 − 0.192748i \(-0.938260\pi\)
0.981248 − 0.192748i \(-0.0617399\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.7082i 1.70782i 0.520423 + 0.853909i \(0.325774\pi\)
−0.520423 + 0.853909i \(0.674226\pi\)
\(48\) 0 0
\(49\) −10.9443 −1.56347
\(50\) 0 0
\(51\) 20.9443 2.93278
\(52\) 0 0
\(53\) − 1.23607i − 0.169787i −0.996390 − 0.0848935i \(-0.972945\pi\)
0.996390 − 0.0848935i \(-0.0270550\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.23607i 0.428628i
\(58\) 0 0
\(59\) −3.23607 −0.421300 −0.210650 − 0.977562i \(-0.567558\pi\)
−0.210650 + 0.977562i \(0.567558\pi\)
\(60\) 0 0
\(61\) −7.23607 −0.926484 −0.463242 − 0.886232i \(-0.653314\pi\)
−0.463242 + 0.886232i \(0.653314\pi\)
\(62\) 0 0
\(63\) 31.6525i 3.98784i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.9443i 1.58139i 0.612207 + 0.790697i \(0.290282\pi\)
−0.612207 + 0.790697i \(0.709718\pi\)
\(68\) 0 0
\(69\) 3.23607 0.389577
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 − 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 0 0
\(73\) 9.47214i 1.10863i 0.832307 + 0.554315i \(0.187020\pi\)
−0.832307 + 0.554315i \(0.812980\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 4.23607i − 0.482745i
\(78\) 0 0
\(79\) −15.1803 −1.70792 −0.853961 − 0.520337i \(-0.825806\pi\)
−0.853961 + 0.520337i \(0.825806\pi\)
\(80\) 0 0
\(81\) 24.4164 2.71293
\(82\) 0 0
\(83\) − 9.00000i − 0.987878i −0.869496 − 0.493939i \(-0.835557\pi\)
0.869496 − 0.493939i \(-0.164443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 5.70820i − 0.611984i
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 9.47214 0.992950
\(92\) 0 0
\(93\) − 1.52786i − 0.158432i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.18034i 0.627518i 0.949503 + 0.313759i \(0.101588\pi\)
−0.949503 + 0.313759i \(0.898412\pi\)
\(98\) 0 0
\(99\) −7.47214 −0.750978
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.l.4049.1 4
5.2 odd 4 4600.2.a.q.1.1 ✓ 2
5.3 odd 4 4600.2.a.u.1.2 yes 2
5.4 even 2 inner 4600.2.e.l.4049.4 4
20.3 even 4 9200.2.a.bn.1.1 2
20.7 even 4 9200.2.a.bz.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.q.1.1 ✓ 2 5.2 odd 4
4600.2.a.u.1.2 yes 2 5.3 odd 4
4600.2.e.l.4049.1 4 1.1 even 1 trivial
4600.2.e.l.4049.4 4 5.4 even 2 inner
9200.2.a.bn.1.1 2 20.3 even 4
9200.2.a.bz.1.2 2 20.7 even 4