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Properties

Label 2960.2.a.w.1.3
Level $2960$
Weight $2$
Character 2960.1
Self dual yes
Analytic conductor $23.636$
Analytic rank $1$
Dimension $5$
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] = [2960,2,Mod(1,2960)] 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("2960.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2960, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2960.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,-3,0,-5,0,-11,0,6,0,5,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(23.6357189983\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.973904.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 19x + 6 \) 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 185)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.383115\) of defining polynomial
Character \(\chi\) \(=\) 2960.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.38311 q^{3} -1.00000 q^{5} +2.62521 q^{7} -1.08699 q^{9} +1.64446 q^{11} +2.44254 q^{13} +1.38311 q^{15} -0.578749 q^{17} -5.20156 q^{19} -3.63096 q^{21} -8.22913 q^{23} +1.00000 q^{25} +5.65278 q^{27} +0.766229 q^{29} -4.21452 q^{31} -2.27447 q^{33} -2.62521 q^{35} +1.00000 q^{37} -3.37831 q^{39} -1.64446 q^{41} +1.91893 q^{43} +1.08699 q^{45} +9.56543 q^{47} -0.108279 q^{49} +0.800477 q^{51} +7.74217 q^{53} -1.64446 q^{55} +7.19435 q^{57} +13.0359 q^{59} -3.86379 q^{61} -2.85359 q^{63} -2.44254 q^{65} -11.4566 q^{67} +11.3818 q^{69} +2.54690 q^{71} -9.79732 q^{73} -1.38311 q^{75} +4.31704 q^{77} +1.81364 q^{79} -4.55746 q^{81} -10.9822 q^{83} +0.578749 q^{85} -1.05978 q^{87} -8.85915 q^{89} +6.41217 q^{91} +5.82917 q^{93} +5.20156 q^{95} -10.5605 q^{97} -1.78751 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{3} - 5 q^{5} - 11 q^{7} + 6 q^{9} + 5 q^{11} + 4 q^{13} + 3 q^{15} + 4 q^{19} + 3 q^{21} - 4 q^{23} + 5 q^{25} - 3 q^{27} - 4 q^{29} - 8 q^{31} + 5 q^{33} + 11 q^{35} + 5 q^{37} - 2 q^{39} - 5 q^{41}+ \cdots + 10 q^{99}+O(q^{100}) \) 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
\(3\) −1.38311 −0.798542 −0.399271 − 0.916833i \(-0.630737\pi\)
−0.399271 + 0.916833i \(0.630737\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.62521 0.992236 0.496118 − 0.868255i \(-0.334758\pi\)
0.496118 + 0.868255i \(0.334758\pi\)
\(8\) 0 0
\(9\) −1.08699 −0.362331
\(10\) 0 0
\(11\) 1.64446 0.495823 0.247911 − 0.968783i \(-0.420256\pi\)
0.247911 + 0.968783i \(0.420256\pi\)
\(12\) 0 0
\(13\) 2.44254 0.677438 0.338719 − 0.940888i \(-0.390006\pi\)
0.338719 + 0.940888i \(0.390006\pi\)
\(14\) 0 0
\(15\) 1.38311 0.357119
\(16\) 0 0
\(17\) −0.578749 −0.140367 −0.0701837 − 0.997534i \(-0.522359\pi\)
−0.0701837 + 0.997534i \(0.522359\pi\)
\(18\) 0 0
\(19\) −5.20156 −1.19332 −0.596660 − 0.802494i \(-0.703506\pi\)
−0.596660 + 0.802494i \(0.703506\pi\)
\(20\) 0 0
\(21\) −3.63096 −0.792341
\(22\) 0 0
\(23\) −8.22913 −1.71589 −0.857946 − 0.513739i \(-0.828260\pi\)
−0.857946 + 0.513739i \(0.828260\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.65278 1.08788
\(28\) 0 0
\(29\) 0.766229 0.142285 0.0711426 − 0.997466i \(-0.477335\pi\)
0.0711426 + 0.997466i \(0.477335\pi\)
\(30\) 0 0
\(31\) −4.21452 −0.756950 −0.378475 − 0.925611i \(-0.623551\pi\)
−0.378475 + 0.925611i \(0.623551\pi\)
\(32\) 0 0
\(33\) −2.27447 −0.395935
\(34\) 0 0
\(35\) −2.62521 −0.443741
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) −3.37831 −0.540962
\(40\) 0 0
\(41\) −1.64446 −0.256821 −0.128411 − 0.991721i \(-0.540988\pi\)
−0.128411 + 0.991721i \(0.540988\pi\)
\(42\) 0 0
\(43\) 1.91893 0.292634 0.146317 − 0.989238i \(-0.453258\pi\)
0.146317 + 0.989238i \(0.453258\pi\)
\(44\) 0 0
\(45\) 1.08699 0.162039
\(46\) 0 0
\(47\) 9.56543 1.39526 0.697630 − 0.716458i \(-0.254238\pi\)
0.697630 + 0.716458i \(0.254238\pi\)
\(48\) 0 0
\(49\) −0.108279 −0.0154684
\(50\) 0 0
\(51\) 0.800477 0.112089
\(52\) 0 0
\(53\) 7.74217 1.06347 0.531735 − 0.846911i \(-0.321540\pi\)
0.531735 + 0.846911i \(0.321540\pi\)
\(54\) 0 0
\(55\) −1.64446 −0.221739
\(56\) 0 0
\(57\) 7.19435 0.952915
\(58\) 0 0
\(59\) 13.0359 1.69713 0.848565 − 0.529092i \(-0.177467\pi\)
0.848565 + 0.529092i \(0.177467\pi\)
\(60\) 0 0
\(61\) −3.86379 −0.494707 −0.247354 − 0.968925i \(-0.579561\pi\)
−0.247354 + 0.968925i \(0.579561\pi\)
\(62\) 0 0
\(63\) −2.85359 −0.359518
\(64\) 0 0
\(65\) −2.44254 −0.302959
\(66\) 0 0
\(67\) −11.4566 −1.39965 −0.699824 − 0.714315i \(-0.746738\pi\)
−0.699824 + 0.714315i \(0.746738\pi\)
\(68\) 0 0
\(69\) 11.3818 1.37021
\(70\) 0 0
\(71\) 2.54690 0.302261 0.151131 − 0.988514i \(-0.451709\pi\)
0.151131 + 0.988514i \(0.451709\pi\)
\(72\) 0 0
\(73\) −9.79732 −1.14669 −0.573345 − 0.819314i \(-0.694354\pi\)
−0.573345 + 0.819314i \(0.694354\pi\)
\(74\) 0 0
\(75\) −1.38311 −0.159708
\(76\) 0 0
\(77\) 4.31704 0.491973
\(78\) 0 0
\(79\) 1.81364 0.204050 0.102025 − 0.994782i \(-0.467468\pi\)
0.102025 + 0.994782i \(0.467468\pi\)
\(80\) 0 0
\(81\) −4.55746 −0.506385
\(82\) 0 0
\(83\) −10.9822 −1.20546 −0.602728 − 0.797947i \(-0.705920\pi\)
−0.602728 + 0.797947i \(0.705920\pi\)
\(84\) 0 0
\(85\) 0.578749 0.0627742
\(86\) 0 0
\(87\) −1.05978 −0.113621
\(88\) 0 0
\(89\) −8.85915 −0.939068 −0.469534 − 0.882914i \(-0.655578\pi\)
−0.469534 + 0.882914i \(0.655578\pi\)
\(90\) 0 0
\(91\) 6.41217 0.672178
\(92\) 0 0
\(93\) 5.82917 0.604456
\(94\) 0 0
\(95\) 5.20156 0.533669
\(96\) 0 0
\(97\) −10.5605 −1.07225 −0.536126 − 0.844138i \(-0.680113\pi\)
−0.536126 + 0.844138i \(0.680113\pi\)
\(98\) 0 0
\(99\) −1.78751 −0.179652
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 2960.2.a.w.1.3 5
4.3 odd 2 185.2.a.e.1.4 ✓ 5
12.11 even 2 1665.2.a.p.1.2 5
20.3 even 4 925.2.b.f.149.3 10
20.7 even 4 925.2.b.f.149.8 10
20.19 odd 2 925.2.a.f.1.2 5
28.27 even 2 9065.2.a.k.1.4 5
60.59 even 2 8325.2.a.ch.1.4 5
148.147 odd 2 6845.2.a.f.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.e.1.4 ✓ 5 4.3 odd 2
925.2.a.f.1.2 5 20.19 odd 2
925.2.b.f.149.3 10 20.3 even 4
925.2.b.f.149.8 10 20.7 even 4
1665.2.a.p.1.2 5 12.11 even 2
2960.2.a.w.1.3 5 1.1 even 1 trivial
6845.2.a.f.1.2 5 148.147 odd 2
8325.2.a.ch.1.4 5 60.59 even 2
9065.2.a.k.1.4 5 28.27 even 2