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Properties

Label 1520.2.a.q.1.3
Level $1520$
Weight $2$
Character 1520.1
Self dual yes
Analytic conductor $12.137$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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] = [1520,2,Mod(1,1520)] 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("1520.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1520, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,1,0,-3,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(12.1372611072\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) 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 760)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.34292\) of defining polynomial
Character \(\chi\) \(=\) 1520.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.34292 q^{3} -1.00000 q^{5} -1.19656 q^{7} +2.48929 q^{9} -4.97858 q^{11} -6.63565 q^{13} -2.34292 q^{15} +1.48929 q^{17} -1.00000 q^{19} -2.80344 q^{21} +0.510711 q^{23} +1.00000 q^{25} -1.19656 q^{27} -7.88240 q^{29} +2.97858 q^{31} -11.6644 q^{33} +1.19656 q^{35} -7.14637 q^{37} -15.5468 q^{39} +1.66442 q^{41} +6.39312 q^{43} -2.48929 q^{45} +9.95715 q^{47} -5.56825 q^{49} +3.48929 q^{51} -11.4219 q^{53} +4.97858 q^{55} -2.34292 q^{57} +11.8396 q^{59} +3.66442 q^{61} -2.97858 q^{63} +6.63565 q^{65} -7.61423 q^{67} +1.19656 q^{69} +13.8396 q^{73} +2.34292 q^{75} +5.95715 q^{77} -12.6858 q^{79} -10.2713 q^{81} +8.68585 q^{83} -1.48929 q^{85} -18.4679 q^{87} -4.87819 q^{89} +7.93994 q^{91} +6.97858 q^{93} +1.00000 q^{95} -6.81079 q^{97} -12.3931 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} - 3 q^{5} + q^{7} - 11 q^{13} - q^{15} - 3 q^{17} - 3 q^{19} - 13 q^{21} + 9 q^{23} + 3 q^{25} + q^{27} - 7 q^{29} - 6 q^{31} - 8 q^{33} - q^{35} - 20 q^{37} - 3 q^{39} - 22 q^{41} + 10 q^{43}+ \cdots - 28 q^{99}+O(q^{100}) \) 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
\(3\) 2.34292 1.35269 0.676344 − 0.736586i \(-0.263563\pi\)
0.676344 + 0.736586i \(0.263563\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.19656 −0.452256 −0.226128 − 0.974098i \(-0.572607\pi\)
−0.226128 + 0.974098i \(0.572607\pi\)
\(8\) 0 0
\(9\) 2.48929 0.829763
\(10\) 0 0
\(11\) −4.97858 −1.50110 −0.750549 − 0.660815i \(-0.770211\pi\)
−0.750549 + 0.660815i \(0.770211\pi\)
\(12\) 0 0
\(13\) −6.63565 −1.84040 −0.920200 − 0.391449i \(-0.871974\pi\)
−0.920200 + 0.391449i \(0.871974\pi\)
\(14\) 0 0
\(15\) −2.34292 −0.604940
\(16\) 0 0
\(17\) 1.48929 0.361206 0.180603 − 0.983556i \(-0.442195\pi\)
0.180603 + 0.983556i \(0.442195\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −2.80344 −0.611761
\(22\) 0 0
\(23\) 0.510711 0.106491 0.0532453 − 0.998581i \(-0.483043\pi\)
0.0532453 + 0.998581i \(0.483043\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.19656 −0.230278
\(28\) 0 0
\(29\) −7.88240 −1.46373 −0.731863 − 0.681452i \(-0.761349\pi\)
−0.731863 + 0.681452i \(0.761349\pi\)
\(30\) 0 0
\(31\) 2.97858 0.534968 0.267484 − 0.963562i \(-0.413808\pi\)
0.267484 + 0.963562i \(0.413808\pi\)
\(32\) 0 0
\(33\) −11.6644 −2.03052
\(34\) 0 0
\(35\) 1.19656 0.202255
\(36\) 0 0
\(37\) −7.14637 −1.17486 −0.587428 − 0.809277i \(-0.699859\pi\)
−0.587428 + 0.809277i \(0.699859\pi\)
\(38\) 0 0
\(39\) −15.5468 −2.48948
\(40\) 0 0
\(41\) 1.66442 0.259939 0.129970 − 0.991518i \(-0.458512\pi\)
0.129970 + 0.991518i \(0.458512\pi\)
\(42\) 0 0
\(43\) 6.39312 0.974941 0.487470 − 0.873139i \(-0.337920\pi\)
0.487470 + 0.873139i \(0.337920\pi\)
\(44\) 0 0
\(45\) −2.48929 −0.371081
\(46\) 0 0
\(47\) 9.95715 1.45240 0.726200 − 0.687483i \(-0.241285\pi\)
0.726200 + 0.687483i \(0.241285\pi\)
\(48\) 0 0
\(49\) −5.56825 −0.795464
\(50\) 0 0
\(51\) 3.48929 0.488598
\(52\) 0 0
\(53\) −11.4219 −1.56892 −0.784458 − 0.620182i \(-0.787059\pi\)
−0.784458 + 0.620182i \(0.787059\pi\)
\(54\) 0 0
\(55\) 4.97858 0.671311
\(56\) 0 0
\(57\) −2.34292 −0.310328
\(58\) 0 0
\(59\) 11.8396 1.54138 0.770690 − 0.637211i \(-0.219912\pi\)
0.770690 + 0.637211i \(0.219912\pi\)
\(60\) 0 0
\(61\) 3.66442 0.469181 0.234591 − 0.972094i \(-0.424625\pi\)
0.234591 + 0.972094i \(0.424625\pi\)
\(62\) 0 0
\(63\) −2.97858 −0.375265
\(64\) 0 0
\(65\) 6.63565 0.823052
\(66\) 0 0
\(67\) −7.61423 −0.930226 −0.465113 − 0.885251i \(-0.653986\pi\)
−0.465113 + 0.885251i \(0.653986\pi\)
\(68\) 0 0
\(69\) 1.19656 0.144049
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 13.8396 1.61980 0.809899 − 0.586570i \(-0.199522\pi\)
0.809899 + 0.586570i \(0.199522\pi\)
\(74\) 0 0
\(75\) 2.34292 0.270537
\(76\) 0 0
\(77\) 5.95715 0.678881
\(78\) 0 0
\(79\) −12.6858 −1.42727 −0.713635 − 0.700518i \(-0.752952\pi\)
−0.713635 + 0.700518i \(0.752952\pi\)
\(80\) 0 0
\(81\) −10.2713 −1.14126
\(82\) 0 0
\(83\) 8.68585 0.953395 0.476698 − 0.879067i \(-0.341834\pi\)
0.476698 + 0.879067i \(0.341834\pi\)
\(84\) 0 0
\(85\) −1.48929 −0.161536
\(86\) 0 0
\(87\) −18.4679 −1.97996
\(88\) 0 0
\(89\) −4.87819 −0.517087 −0.258544 − 0.966000i \(-0.583243\pi\)
−0.258544 + 0.966000i \(0.583243\pi\)
\(90\) 0 0
\(91\) 7.93994 0.832332
\(92\) 0 0
\(93\) 6.97858 0.723645
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −6.81079 −0.691531 −0.345765 − 0.938321i \(-0.612381\pi\)
−0.345765 + 0.938321i \(0.612381\pi\)
\(98\) 0 0
\(99\) −12.3931 −1.24555
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 1520.2.a.q.1.3 3
4.3 odd 2 760.2.a.i.1.1 ✓ 3
5.4 even 2 7600.2.a.bp.1.1 3
8.3 odd 2 6080.2.a.bx.1.3 3
8.5 even 2 6080.2.a.br.1.1 3
12.11 even 2 6840.2.a.bm.1.2 3
20.3 even 4 3800.2.d.n.3649.1 6
20.7 even 4 3800.2.d.n.3649.6 6
20.19 odd 2 3800.2.a.w.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.i.1.1 ✓ 3 4.3 odd 2
1520.2.a.q.1.3 3 1.1 even 1 trivial
3800.2.a.w.1.3 3 20.19 odd 2
3800.2.d.n.3649.1 6 20.3 even 4
3800.2.d.n.3649.6 6 20.7 even 4
6080.2.a.br.1.1 3 8.5 even 2
6080.2.a.bx.1.3 3 8.3 odd 2
6840.2.a.bm.1.2 3 12.11 even 2
7600.2.a.bp.1.1 3 5.4 even 2