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 1.2
Root \(-1.38679\) 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)
 
\(f(q)\) \(=\) \(q-2.38679 q^{3} -1.00000 q^{5} -4.78404 q^{7} +2.69675 q^{9} +3.12105 q^{11} -2.81780 q^{13} +2.38679 q^{15} +6.37246 q^{17} -0.114347 q^{19} +11.4185 q^{21} -5.62219 q^{23} +1.00000 q^{25} +0.723803 q^{27} +2.77357 q^{29} +6.67866 q^{31} -7.44929 q^{33} +4.78404 q^{35} +1.00000 q^{37} +6.72548 q^{39} -3.12105 q^{41} +8.57034 q^{43} -2.69675 q^{45} -3.40396 q^{47} +15.8870 q^{49} -15.2097 q^{51} -10.2438 q^{53} -3.12105 q^{55} +0.272922 q^{57} +9.11059 q^{59} -5.55466 q^{61} -12.9013 q^{63} +2.81780 q^{65} +7.84948 q^{67} +13.4190 q^{69} +4.33996 q^{71} +3.22811 q^{73} -2.38679 q^{75} -14.9312 q^{77} -15.3847 q^{79} -9.81780 q^{81} -5.68074 q^{83} -6.37246 q^{85} -6.61992 q^{87} -9.95042 q^{89} +13.4805 q^{91} -15.9405 q^{93} +0.114347 q^{95} -5.62970 q^{97} +8.41669 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{3} - 5 q^{5} - 11 q^{7} + 6 q^{9} + 5 q^{11} + 4 q^{13} + 3 q^{15} + 4 q^{19} + 3 q^{21} - 4 q^{23} + 5 q^{25} - 3 q^{27} - 4 q^{29} - 8 q^{31} + 5 q^{33} + 11 q^{35} + 5 q^{37} - 2 q^{39} - 5 q^{41}+ \cdots + 10 q^{99}+O(q^{100}) \) 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
\(3\) −2.38679 −1.37801 −0.689006 − 0.724756i \(-0.741953\pi\)
−0.689006 + 0.724756i \(0.741953\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.78404 −1.80820 −0.904098 − 0.427325i \(-0.859456\pi\)
−0.904098 + 0.427325i \(0.859456\pi\)
\(8\) 0 0
\(9\) 2.69675 0.898915
\(10\) 0 0
\(11\) 3.12105 0.941033 0.470516 − 0.882391i \(-0.344068\pi\)
0.470516 + 0.882391i \(0.344068\pi\)
\(12\) 0 0
\(13\) −2.81780 −0.781517 −0.390758 − 0.920493i \(-0.627787\pi\)
−0.390758 + 0.920493i \(0.627787\pi\)
\(14\) 0 0
\(15\) 2.38679 0.616265
\(16\) 0 0
\(17\) 6.37246 1.54555 0.772774 − 0.634681i \(-0.218868\pi\)
0.772774 + 0.634681i \(0.218868\pi\)
\(18\) 0 0
\(19\) −0.114347 −0.0262330 −0.0131165 − 0.999914i \(-0.504175\pi\)
−0.0131165 + 0.999914i \(0.504175\pi\)
\(20\) 0 0
\(21\) 11.4185 2.49171
\(22\) 0 0
\(23\) −5.62219 −1.17231 −0.586153 − 0.810200i \(-0.699358\pi\)
−0.586153 + 0.810200i \(0.699358\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0.723803 0.139296
\(28\) 0 0
\(29\) 2.77357 0.515039 0.257520 − 0.966273i \(-0.417095\pi\)
0.257520 + 0.966273i \(0.417095\pi\)
\(30\) 0 0
\(31\) 6.67866 1.19952 0.599761 − 0.800179i \(-0.295262\pi\)
0.599761 + 0.800179i \(0.295262\pi\)
\(32\) 0 0
\(33\) −7.44929 −1.29675
\(34\) 0 0
\(35\) 4.78404 0.808650
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) 6.72548 1.07694
\(40\) 0 0
\(41\) −3.12105 −0.487427 −0.243713 − 0.969847i \(-0.578366\pi\)
−0.243713 + 0.969847i \(0.578366\pi\)
\(42\) 0 0
\(43\) 8.57034 1.30696 0.653482 − 0.756942i \(-0.273307\pi\)
0.653482 + 0.756942i \(0.273307\pi\)
\(44\) 0 0
\(45\) −2.69675 −0.402007
\(46\) 0 0
\(47\) −3.40396 −0.496518 −0.248259 − 0.968694i \(-0.579858\pi\)
−0.248259 + 0.968694i \(0.579858\pi\)
\(48\) 0 0
\(49\) 15.8870 2.26957
\(50\) 0 0
\(51\) −15.2097 −2.12978
\(52\) 0 0
\(53\) −10.2438 −1.40709 −0.703546 − 0.710650i \(-0.748401\pi\)
−0.703546 + 0.710650i \(0.748401\pi\)
\(54\) 0 0
\(55\) −3.12105 −0.420843
\(56\) 0 0
\(57\) 0.272922 0.0361494
\(58\) 0 0
\(59\) 9.11059 1.18610 0.593049 − 0.805167i \(-0.297924\pi\)
0.593049 + 0.805167i \(0.297924\pi\)
\(60\) 0 0
\(61\) −5.55466 −0.711201 −0.355601 − 0.934638i \(-0.615724\pi\)
−0.355601 + 0.934638i \(0.615724\pi\)
\(62\) 0 0
\(63\) −12.9013 −1.62541
\(64\) 0 0
\(65\) 2.81780 0.349505
\(66\) 0 0
\(67\) 7.84948 0.958967 0.479484 − 0.877551i \(-0.340824\pi\)
0.479484 + 0.877551i \(0.340824\pi\)
\(68\) 0 0
\(69\) 13.4190 1.61545
\(70\) 0 0
\(71\) 4.33996 0.515059 0.257529 − 0.966270i \(-0.417092\pi\)
0.257529 + 0.966270i \(0.417092\pi\)
\(72\) 0 0
\(73\) 3.22811 0.377822 0.188911 − 0.981994i \(-0.439504\pi\)
0.188911 + 0.981994i \(0.439504\pi\)
\(74\) 0 0
\(75\) −2.38679 −0.275602
\(76\) 0 0
\(77\) −14.9312 −1.70157
\(78\) 0 0
\(79\) −15.3847 −1.73091 −0.865457 − 0.500983i \(-0.832972\pi\)
−0.865457 + 0.500983i \(0.832972\pi\)
\(80\) 0 0
\(81\) −9.81780 −1.09087
\(82\) 0 0
\(83\) −5.68074 −0.623542 −0.311771 − 0.950157i \(-0.600922\pi\)
−0.311771 + 0.950157i \(0.600922\pi\)
\(84\) 0 0
\(85\) −6.37246 −0.691190
\(86\) 0 0
\(87\) −6.61992 −0.709730
\(88\) 0 0
\(89\) −9.95042 −1.05474 −0.527371 − 0.849635i \(-0.676822\pi\)
−0.527371 + 0.849635i \(0.676822\pi\)
\(90\) 0 0
\(91\) 13.4805 1.41314
\(92\) 0 0
\(93\) −15.9405 −1.65296
\(94\) 0 0
\(95\) 0.114347 0.0117318
\(96\) 0 0
\(97\) −5.62970 −0.571610 −0.285805 − 0.958288i \(-0.592261\pi\)
−0.285805 + 0.958288i \(0.592261\pi\)
\(98\) 0 0
\(99\) 8.41669 0.845909
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.1.2 5
4.3 odd 2 185.2.a.e.1.1 ✓ 5
12.11 even 2 1665.2.a.p.1.5 5
20.3 even 4 925.2.b.f.149.9 10
20.7 even 4 925.2.b.f.149.2 10
20.19 odd 2 925.2.a.f.1.5 5
28.27 even 2 9065.2.a.k.1.1 5
60.59 even 2 8325.2.a.ch.1.1 5
148.147 odd 2 6845.2.a.f.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.e.1.1 ✓ 5 4.3 odd 2
925.2.a.f.1.5 5 20.19 odd 2
925.2.b.f.149.2 10 20.7 even 4
925.2.b.f.149.9 10 20.3 even 4
1665.2.a.p.1.5 5 12.11 even 2
2960.2.a.w.1.2 5 1.1 even 1 trivial
6845.2.a.f.1.5 5 148.147 odd 2
8325.2.a.ch.1.1 5 60.59 even 2
9065.2.a.k.1.1 5 28.27 even 2