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Properties

Label 4600.2.e.n.4049.4
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,-6,0,-8,0,0,0,0,0,0,0,8] 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: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.4
Root \(2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.n.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)
 
\(f(q)\) \(=\) \(q+2.56155i q^{3} +1.56155i q^{7} -3.56155 q^{9} -2.00000 q^{11} -0.561553i q^{13} +5.56155i q^{17} +2.00000 q^{19} -4.00000 q^{21} +1.00000i q^{23} -1.43845i q^{27} -0.123106 q^{29} -8.12311 q^{31} -5.12311i q^{33} -3.56155i q^{37} +1.43845 q^{39} -4.12311 q^{41} +10.2462i q^{43} +3.68466i q^{47} +4.56155 q^{49} -14.2462 q^{51} -4.43845i q^{53} +5.12311i q^{57} +5.56155 q^{59} -9.12311 q^{61} -5.56155i q^{63} -11.5616i q^{67} -2.56155 q^{69} -5.00000 q^{71} -3.43845i q^{73} -3.12311i q^{77} +9.12311 q^{79} -7.00000 q^{81} +4.68466i q^{83} -0.315342i q^{87} -8.00000 q^{89} +0.876894 q^{91} -20.8078i q^{93} -3.12311i q^{97} +7.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{9} - 8 q^{11} + 8 q^{19} - 16 q^{21} + 16 q^{29} - 16 q^{31} + 14 q^{39} + 10 q^{49} - 24 q^{51} + 14 q^{59} - 20 q^{61} - 2 q^{69} - 20 q^{71} + 20 q^{79} - 28 q^{81} - 32 q^{89} + 20 q^{91}+ \cdots + 12 q^{99}+O(q^{100}) \) 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
\(3\) 2.56155i 1.47891i 0.673204 + 0.739457i \(0.264917\pi\)
−0.673204 + 0.739457i \(0.735083\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.56155i 0.590211i 0.955465 + 0.295106i \(0.0953549\pi\)
−0.955465 + 0.295106i \(0.904645\pi\)
\(8\) 0 0
\(9\) −3.56155 −1.18718
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) − 0.561553i − 0.155747i −0.996963 − 0.0778734i \(-0.975187\pi\)
0.996963 − 0.0778734i \(-0.0248130\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.56155i 1.34887i 0.738332 + 0.674437i \(0.235614\pi\)
−0.738332 + 0.674437i \(0.764386\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.43845i − 0.276829i
\(28\) 0 0
\(29\) −0.123106 −0.0228601 −0.0114301 − 0.999935i \(-0.503638\pi\)
−0.0114301 + 0.999935i \(0.503638\pi\)
\(30\) 0 0
\(31\) −8.12311 −1.45895 −0.729476 − 0.684006i \(-0.760236\pi\)
−0.729476 + 0.684006i \(0.760236\pi\)
\(32\) 0 0
\(33\) − 5.12311i − 0.891818i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 3.56155i − 0.585516i −0.956187 − 0.292758i \(-0.905427\pi\)
0.956187 − 0.292758i \(-0.0945730\pi\)
\(38\) 0 0
\(39\) 1.43845 0.230336
\(40\) 0 0
\(41\) −4.12311 −0.643921 −0.321960 − 0.946753i \(-0.604342\pi\)
−0.321960 + 0.946753i \(0.604342\pi\)
\(42\) 0 0
\(43\) 10.2462i 1.56253i 0.624198 + 0.781266i \(0.285426\pi\)
−0.624198 + 0.781266i \(0.714574\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.68466i 0.537463i 0.963215 + 0.268731i \(0.0866044\pi\)
−0.963215 + 0.268731i \(0.913396\pi\)
\(48\) 0 0
\(49\) 4.56155 0.651650
\(50\) 0 0
\(51\) −14.2462 −1.99487
\(52\) 0 0
\(53\) − 4.43845i − 0.609668i −0.952406 − 0.304834i \(-0.901399\pi\)
0.952406 − 0.304834i \(-0.0986009\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.12311i 0.678572i
\(58\) 0 0
\(59\) 5.56155 0.724053 0.362026 − 0.932168i \(-0.382085\pi\)
0.362026 + 0.932168i \(0.382085\pi\)
\(60\) 0 0
\(61\) −9.12311 −1.16809 −0.584047 − 0.811720i \(-0.698532\pi\)
−0.584047 + 0.811720i \(0.698532\pi\)
\(62\) 0 0
\(63\) − 5.56155i − 0.700690i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 11.5616i − 1.41247i −0.707978 − 0.706234i \(-0.750393\pi\)
0.707978 − 0.706234i \(-0.249607\pi\)
\(68\) 0 0
\(69\) −2.56155 −0.308375
\(70\) 0 0
\(71\) −5.00000 −0.593391 −0.296695 − 0.954972i \(-0.595885\pi\)
−0.296695 + 0.954972i \(0.595885\pi\)
\(72\) 0 0
\(73\) − 3.43845i − 0.402440i −0.979546 − 0.201220i \(-0.935509\pi\)
0.979546 − 0.201220i \(-0.0644906\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 3.12311i − 0.355911i
\(78\) 0 0
\(79\) 9.12311 1.02643 0.513215 − 0.858260i \(-0.328454\pi\)
0.513215 + 0.858260i \(0.328454\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 4.68466i 0.514208i 0.966384 + 0.257104i \(0.0827683\pi\)
−0.966384 + 0.257104i \(0.917232\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 0.315342i − 0.0338082i
\(88\) 0 0
\(89\) −8.00000 −0.847998 −0.423999 − 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) 0.876894 0.0919235
\(92\) 0 0
\(93\) − 20.8078i − 2.15766i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 3.12311i − 0.317103i −0.987351 − 0.158552i \(-0.949318\pi\)
0.987351 − 0.158552i \(-0.0506824\pi\)
\(98\) 0 0
\(99\) 7.12311 0.715899
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.n.4049.4 4
5.2 odd 4 4600.2.a.t.1.2 2
5.3 odd 4 920.2.a.e.1.1 ✓ 2
5.4 even 2 inner 4600.2.e.n.4049.1 4
15.8 even 4 8280.2.a.bf.1.2 2
20.3 even 4 1840.2.a.o.1.2 2
20.7 even 4 9200.2.a.bq.1.1 2
40.3 even 4 7360.2.a.bl.1.1 2
40.13 odd 4 7360.2.a.bp.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.e.1.1 ✓ 2 5.3 odd 4
1840.2.a.o.1.2 2 20.3 even 4
4600.2.a.t.1.2 2 5.2 odd 4
4600.2.e.n.4049.1 4 5.4 even 2 inner
4600.2.e.n.4049.4 4 1.1 even 1 trivial
7360.2.a.bl.1.1 2 40.3 even 4
7360.2.a.bp.1.2 2 40.13 odd 4
8280.2.a.bf.1.2 2 15.8 even 4
9200.2.a.bq.1.1 2 20.7 even 4