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 1.1
Root \(-1.62871\) 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)
 
\(f(q)\) \(=\) \(q-2.62871 q^{3} -1.00000 q^{5} -2.55244 q^{7} +3.91009 q^{9} -2.46863 q^{11} +1.55854 q^{13} +2.62871 q^{15} -6.83662 q^{17} +7.66011 q^{19} +6.70960 q^{21} +7.50003 q^{23} +1.00000 q^{25} -2.39236 q^{27} +3.25741 q^{29} -0.658785 q^{31} +6.48930 q^{33} +2.55244 q^{35} +1.00000 q^{37} -4.09694 q^{39} +2.46863 q^{41} -10.9579 q^{43} -3.91009 q^{45} -3.11521 q^{47} -0.485072 q^{49} +17.9715 q^{51} +8.64184 q^{53} +2.46863 q^{55} -20.1362 q^{57} +6.23634 q^{59} +3.27808 q^{61} -9.98026 q^{63} -1.55854 q^{65} -1.47764 q^{67} -19.7154 q^{69} +8.06686 q^{71} -4.96199 q^{73} -2.62871 q^{75} +6.30102 q^{77} -12.8206 q^{79} -5.44146 q^{81} -1.14934 q^{83} +6.83662 q^{85} -8.56277 q^{87} +11.5207 q^{89} -3.97807 q^{91} +1.73175 q^{93} -7.66011 q^{95} +17.2929 q^{97} -9.65257 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{3} - 5 q^{5} - 11 q^{7} + 6 q^{9} + 5 q^{11} + 4 q^{13} + 3 q^{15} + 4 q^{19} + 3 q^{21} - 4 q^{23} + 5 q^{25} - 3 q^{27} - 4 q^{29} - 8 q^{31} + 5 q^{33} + 11 q^{35} + 5 q^{37} - 2 q^{39} - 5 q^{41}+ \cdots + 10 q^{99}+O(q^{100}) \) 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
\(3\) −2.62871 −1.51768 −0.758842 − 0.651275i \(-0.774234\pi\)
−0.758842 + 0.651275i \(0.774234\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.55244 −0.964730 −0.482365 − 0.875970i \(-0.660222\pi\)
−0.482365 + 0.875970i \(0.660222\pi\)
\(8\) 0 0
\(9\) 3.91009 1.30336
\(10\) 0 0
\(11\) −2.46863 −0.744320 −0.372160 − 0.928169i \(-0.621383\pi\)
−0.372160 + 0.928169i \(0.621383\pi\)
\(12\) 0 0
\(13\) 1.55854 0.432261 0.216131 − 0.976364i \(-0.430656\pi\)
0.216131 + 0.976364i \(0.430656\pi\)
\(14\) 0 0
\(15\) 2.62871 0.678729
\(16\) 0 0
\(17\) −6.83662 −1.65812 −0.829062 − 0.559156i \(-0.811125\pi\)
−0.829062 + 0.559156i \(0.811125\pi\)
\(18\) 0 0
\(19\) 7.66011 1.75735 0.878675 − 0.477421i \(-0.158428\pi\)
0.878675 + 0.477421i \(0.158428\pi\)
\(20\) 0 0
\(21\) 6.70960 1.46415
\(22\) 0 0
\(23\) 7.50003 1.56387 0.781933 − 0.623363i \(-0.214234\pi\)
0.781933 + 0.623363i \(0.214234\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −2.39236 −0.460410
\(28\) 0 0
\(29\) 3.25741 0.604886 0.302443 − 0.953167i \(-0.402198\pi\)
0.302443 + 0.953167i \(0.402198\pi\)
\(30\) 0 0
\(31\) −0.658785 −0.118321 −0.0591607 − 0.998248i \(-0.518842\pi\)
−0.0591607 + 0.998248i \(0.518842\pi\)
\(32\) 0 0
\(33\) 6.48930 1.12964
\(34\) 0 0
\(35\) 2.55244 0.431440
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) −4.09694 −0.656036
\(40\) 0 0
\(41\) 2.46863 0.385535 0.192768 − 0.981244i \(-0.438254\pi\)
0.192768 + 0.981244i \(0.438254\pi\)
\(42\) 0 0
\(43\) −10.9579 −1.67107 −0.835535 − 0.549438i \(-0.814842\pi\)
−0.835535 + 0.549438i \(0.814842\pi\)
\(44\) 0 0
\(45\) −3.91009 −0.582882
\(46\) 0 0
\(47\) −3.11521 −0.454400 −0.227200 − 0.973848i \(-0.572957\pi\)
−0.227200 + 0.973848i \(0.572957\pi\)
\(48\) 0 0
\(49\) −0.485072 −0.0692960
\(50\) 0 0
\(51\) 17.9715 2.51651
\(52\) 0 0
\(53\) 8.64184 1.18705 0.593524 − 0.804816i \(-0.297736\pi\)
0.593524 + 0.804816i \(0.297736\pi\)
\(54\) 0 0
\(55\) 2.46863 0.332870
\(56\) 0 0
\(57\) −20.1362 −2.66710
\(58\) 0 0
\(59\) 6.23634 0.811903 0.405951 − 0.913895i \(-0.366940\pi\)
0.405951 + 0.913895i \(0.366940\pi\)
\(60\) 0 0
\(61\) 3.27808 0.419716 0.209858 − 0.977732i \(-0.432700\pi\)
0.209858 + 0.977732i \(0.432700\pi\)
\(62\) 0 0
\(63\) −9.98026 −1.25739
\(64\) 0 0
\(65\) −1.55854 −0.193313
\(66\) 0 0
\(67\) −1.47764 −0.180523 −0.0902615 − 0.995918i \(-0.528770\pi\)
−0.0902615 + 0.995918i \(0.528770\pi\)
\(68\) 0 0
\(69\) −19.7154 −2.37345
\(70\) 0 0
\(71\) 8.06686 0.957360 0.478680 − 0.877989i \(-0.341115\pi\)
0.478680 + 0.877989i \(0.341115\pi\)
\(72\) 0 0
\(73\) −4.96199 −0.580757 −0.290379 − 0.956912i \(-0.593781\pi\)
−0.290379 + 0.956912i \(0.593781\pi\)
\(74\) 0 0
\(75\) −2.62871 −0.303537
\(76\) 0 0
\(77\) 6.30102 0.718068
\(78\) 0 0
\(79\) −12.8206 −1.44243 −0.721214 − 0.692713i \(-0.756415\pi\)
−0.721214 + 0.692713i \(0.756415\pi\)
\(80\) 0 0
\(81\) −5.44146 −0.604607
\(82\) 0 0
\(83\) −1.14934 −0.126157 −0.0630784 − 0.998009i \(-0.520092\pi\)
−0.0630784 + 0.998009i \(0.520092\pi\)
\(84\) 0 0
\(85\) 6.83662 0.741536
\(86\) 0 0
\(87\) −8.56277 −0.918026
\(88\) 0 0
\(89\) 11.5207 1.22119 0.610596 − 0.791942i \(-0.290930\pi\)
0.610596 + 0.791942i \(0.290930\pi\)
\(90\) 0 0
\(91\) −3.97807 −0.417015
\(92\) 0 0
\(93\) 1.73175 0.179574
\(94\) 0 0
\(95\) −7.66011 −0.785911
\(96\) 0 0
\(97\) 17.2929 1.75583 0.877916 − 0.478815i \(-0.158933\pi\)
0.877916 + 0.478815i \(0.158933\pi\)
\(98\) 0 0
\(99\) −9.65257 −0.970120
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.1.1 5
4.3 odd 2 185.2.a.e.1.3 ✓ 5
12.11 even 2 1665.2.a.p.1.3 5
20.3 even 4 925.2.b.f.149.5 10
20.7 even 4 925.2.b.f.149.6 10
20.19 odd 2 925.2.a.f.1.3 5
28.27 even 2 9065.2.a.k.1.3 5
60.59 even 2 8325.2.a.ch.1.3 5
148.147 odd 2 6845.2.a.f.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.e.1.3 ✓ 5 4.3 odd 2
925.2.a.f.1.3 5 20.19 odd 2
925.2.b.f.149.5 10 20.3 even 4
925.2.b.f.149.6 10 20.7 even 4
1665.2.a.p.1.3 5 12.11 even 2
2960.2.a.w.1.1 5 1.1 even 1 trivial
6845.2.a.f.1.3 5 148.147 odd 2
8325.2.a.ch.1.3 5 60.59 even 2
9065.2.a.k.1.3 5 28.27 even 2