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Properties

Label 4600.2.e.l.4049.3
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 4049.3
Root \(0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.l.4049.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.23607i q^{3} +0.236068i q^{7} +1.47214 q^{9} +1.00000 q^{11} -2.23607i q^{13} -2.47214i q^{17} -1.00000 q^{19} -0.291796 q^{21} +1.00000i q^{23} +5.52786i q^{27} +6.23607 q^{29} -8.47214 q^{31} +1.23607i q^{33} +6.76393i q^{37} +2.76393 q^{39} +11.9443 q^{41} -11.4721i q^{43} -1.70820i q^{47} +6.94427 q^{49} +3.05573 q^{51} +3.23607i q^{53} -1.23607i q^{57} +1.23607 q^{59} -2.76393 q^{61} +0.347524i q^{63} -4.94427i q^{67} -1.23607 q^{69} +10.0000 q^{71} +0.527864i q^{73} +0.236068i q^{77} +7.18034 q^{79} -2.41641 q^{81} -9.00000i q^{83} +7.70820i q^{87} -2.00000 q^{89} +0.527864 q^{91} -10.4721i q^{93} -16.1803i q^{97} +1.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} + 4 q^{11} - 4 q^{19} - 28 q^{21} + 16 q^{29} - 16 q^{31} + 20 q^{39} + 12 q^{41} - 8 q^{49} + 48 q^{51} - 4 q^{59} - 20 q^{61} + 4 q^{69} + 40 q^{71} - 16 q^{79} + 44 q^{81} - 8 q^{89} + 20 q^{91}+ \cdots - 12 q^{99}+O(q^{100}) \) 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
\(3\) 1.23607i 0.713644i 0.934172 + 0.356822i \(0.116140\pi\)
−0.934172 + 0.356822i \(0.883860\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.236068i 0.0892253i 0.999004 + 0.0446127i \(0.0142054\pi\)
−0.999004 + 0.0446127i \(0.985795\pi\)
\(8\) 0 0
\(9\) 1.47214 0.490712
\(10\) 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 \(-0.899642\pi\)
0.950708 − 0.310087i \(-0.100358\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2.47214i − 0.599581i −0.954005 − 0.299791i \(-0.903083\pi\)
0.954005 − 0.299791i \(-0.0969168\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) −0.291796 −0.0636751
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.52786i 1.06384i
\(28\) 0 0
\(29\) 6.23607 1.15801 0.579004 − 0.815324i \(-0.303441\pi\)
0.579004 + 0.815324i \(0.303441\pi\)
\(30\) 0 0
\(31\) −8.47214 −1.52164 −0.760820 − 0.648963i \(-0.775203\pi\)
−0.760820 + 0.648963i \(0.775203\pi\)
\(32\) 0 0
\(33\) 1.23607i 0.215172i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.76393i 1.11198i 0.831188 + 0.555992i \(0.187661\pi\)
−0.831188 + 0.555992i \(0.812339\pi\)
\(38\) 0 0
\(39\) 2.76393 0.442583
\(40\) 0 0
\(41\) 11.9443 1.86538 0.932691 − 0.360677i \(-0.117454\pi\)
0.932691 + 0.360677i \(0.117454\pi\)
\(42\) 0 0
\(43\) − 11.4721i − 1.74948i −0.484589 − 0.874742i \(-0.661031\pi\)
0.484589 − 0.874742i \(-0.338969\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 1.70820i − 0.249167i −0.992209 − 0.124584i \(-0.960241\pi\)
0.992209 − 0.124584i \(-0.0397595\pi\)
\(48\) 0 0
\(49\) 6.94427 0.992039
\(50\) 0 0
\(51\) 3.05573 0.427888
\(52\) 0 0
\(53\) 3.23607i 0.444508i 0.974989 + 0.222254i \(0.0713414\pi\)
−0.974989 + 0.222254i \(0.928659\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.23607i − 0.163721i
\(58\) 0 0
\(59\) 1.23607 0.160922 0.0804612 − 0.996758i \(-0.474361\pi\)
0.0804612 + 0.996758i \(0.474361\pi\)
\(60\) 0 0
\(61\) −2.76393 −0.353885 −0.176943 − 0.984221i \(-0.556621\pi\)
−0.176943 + 0.984221i \(0.556621\pi\)
\(62\) 0 0
\(63\) 0.347524i 0.0437839i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.94427i − 0.604039i −0.953302 − 0.302019i \(-0.902339\pi\)
0.953302 − 0.302019i \(-0.0976608\pi\)
\(68\) 0 0
\(69\) −1.23607 −0.148805
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 − 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 0 0
\(73\) 0.527864i 0.0617818i 0.999523 + 0.0308909i \(0.00983445\pi\)
−0.999523 + 0.0308909i \(0.990166\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.236068i 0.0269024i
\(78\) 0 0
\(79\) 7.18034 0.807851 0.403926 − 0.914792i \(-0.367645\pi\)
0.403926 + 0.914792i \(0.367645\pi\)
\(80\) 0 0
\(81\) −2.41641 −0.268490
\(82\) 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
\(87\) 7.70820i 0.826406i
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 0.527864 0.0553352
\(92\) 0 0
\(93\) − 10.4721i − 1.08591i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 16.1803i − 1.64286i −0.570306 − 0.821432i \(-0.693175\pi\)
0.570306 − 0.821432i \(-0.306825\pi\)
\(98\) 0 0
\(99\) 1.47214 0.147955
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.4049.3 4
5.2 odd 4 4600.2.a.q.1.2 ✓ 2
5.3 odd 4 4600.2.a.u.1.1 yes 2
5.4 even 2 inner 4600.2.e.l.4049.2 4
20.3 even 4 9200.2.a.bn.1.2 2
20.7 even 4 9200.2.a.bz.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.q.1.2 ✓ 2 5.2 odd 4
4600.2.a.u.1.1 yes 2 5.3 odd 4
4600.2.e.l.4049.2 4 5.4 even 2 inner
4600.2.e.l.4049.3 4 1.1 even 1 trivial
9200.2.a.bn.1.2 2 20.3 even 4
9200.2.a.bz.1.1 2 20.7 even 4