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Properties

Label 2960.2.a.w.1.5
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 1.5
Root \(3.29298\) 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)
 
\(f(q)\) \(=\) \(q+2.29298 q^{3} -1.00000 q^{5} -3.82710 q^{7} +2.25774 q^{9} +4.41809 q^{11} -3.67583 q^{13} -2.29298 q^{15} -2.28688 q^{17} +2.39037 q^{19} -8.77545 q^{21} +0.265251 q^{23} +1.00000 q^{25} -1.70198 q^{27} -6.58595 q^{29} -2.34076 q^{31} +10.1306 q^{33} +3.82710 q^{35} +1.00000 q^{37} -8.42859 q^{39} -4.41809 q^{41} -7.71249 q^{43} -2.25774 q^{45} -10.9285 q^{47} +7.64669 q^{49} -5.24377 q^{51} -0.109574 q^{53} -4.41809 q^{55} +5.48105 q^{57} +2.00504 q^{59} +3.96271 q^{61} -8.64059 q^{63} +3.67583 q^{65} -6.80664 q^{67} +0.608215 q^{69} +5.79485 q^{71} -0.140654 q^{73} +2.29298 q^{75} -16.9085 q^{77} +6.62418 q^{79} -10.6758 q^{81} -13.9904 q^{83} +2.28688 q^{85} -15.1014 q^{87} +14.8139 q^{89} +14.0678 q^{91} -5.36731 q^{93} -2.39037 q^{95} -8.94394 q^{97} +9.97490 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{3} - 5 q^{5} - 11 q^{7} + 6 q^{9} + 5 q^{11} + 4 q^{13} + 3 q^{15} + 4 q^{19} + 3 q^{21} - 4 q^{23} + 5 q^{25} - 3 q^{27} - 4 q^{29} - 8 q^{31} + 5 q^{33} + 11 q^{35} + 5 q^{37} - 2 q^{39} - 5 q^{41}+ \cdots + 10 q^{99}+O(q^{100}) \) 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
\(3\) 2.29298 1.32385 0.661925 − 0.749570i \(-0.269740\pi\)
0.661925 + 0.749570i \(0.269740\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.82710 −1.44651 −0.723254 − 0.690582i \(-0.757354\pi\)
−0.723254 + 0.690582i \(0.757354\pi\)
\(8\) 0 0
\(9\) 2.25774 0.752580
\(10\) 0 0
\(11\) 4.41809 1.33210 0.666052 − 0.745905i \(-0.267983\pi\)
0.666052 + 0.745905i \(0.267983\pi\)
\(12\) 0 0
\(13\) −3.67583 −1.01949 −0.509746 − 0.860325i \(-0.670261\pi\)
−0.509746 + 0.860325i \(0.670261\pi\)
\(14\) 0 0
\(15\) −2.29298 −0.592044
\(16\) 0 0
\(17\) −2.28688 −0.554651 −0.277325 − 0.960776i \(-0.589448\pi\)
−0.277325 + 0.960776i \(0.589448\pi\)
\(18\) 0 0
\(19\) 2.39037 0.548387 0.274194 − 0.961674i \(-0.411589\pi\)
0.274194 + 0.961674i \(0.411589\pi\)
\(20\) 0 0
\(21\) −8.77545 −1.91496
\(22\) 0 0
\(23\) 0.265251 0.0553087 0.0276544 − 0.999618i \(-0.491196\pi\)
0.0276544 + 0.999618i \(0.491196\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.70198 −0.327547
\(28\) 0 0
\(29\) −6.58595 −1.22298 −0.611490 − 0.791252i \(-0.709430\pi\)
−0.611490 + 0.791252i \(0.709430\pi\)
\(30\) 0 0
\(31\) −2.34076 −0.420413 −0.210207 − 0.977657i \(-0.567414\pi\)
−0.210207 + 0.977657i \(0.567414\pi\)
\(32\) 0 0
\(33\) 10.1306 1.76351
\(34\) 0 0
\(35\) 3.82710 0.646898
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) −8.42859 −1.34965
\(40\) 0 0
\(41\) −4.41809 −0.689990 −0.344995 − 0.938605i \(-0.612119\pi\)
−0.344995 + 0.938605i \(0.612119\pi\)
\(42\) 0 0
\(43\) −7.71249 −1.17614 −0.588072 − 0.808809i \(-0.700113\pi\)
−0.588072 + 0.808809i \(0.700113\pi\)
\(44\) 0 0
\(45\) −2.25774 −0.336564
\(46\) 0 0
\(47\) −10.9285 −1.59409 −0.797045 − 0.603920i \(-0.793605\pi\)
−0.797045 + 0.603920i \(0.793605\pi\)
\(48\) 0 0
\(49\) 7.64669 1.09238
\(50\) 0 0
\(51\) −5.24377 −0.734275
\(52\) 0 0
\(53\) −0.109574 −0.0150512 −0.00752559 − 0.999972i \(-0.502395\pi\)
−0.00752559 + 0.999972i \(0.502395\pi\)
\(54\) 0 0
\(55\) −4.41809 −0.595735
\(56\) 0 0
\(57\) 5.48105 0.725983
\(58\) 0 0
\(59\) 2.00504 0.261034 0.130517 − 0.991446i \(-0.458336\pi\)
0.130517 + 0.991446i \(0.458336\pi\)
\(60\) 0 0
\(61\) 3.96271 0.507374 0.253687 − 0.967286i \(-0.418357\pi\)
0.253687 + 0.967286i \(0.418357\pi\)
\(62\) 0 0
\(63\) −8.64059 −1.08861
\(64\) 0 0
\(65\) 3.67583 0.455931
\(66\) 0 0
\(67\) −6.80664 −0.831563 −0.415782 − 0.909464i \(-0.636492\pi\)
−0.415782 + 0.909464i \(0.636492\pi\)
\(68\) 0 0
\(69\) 0.608215 0.0732205
\(70\) 0 0
\(71\) 5.79485 0.687722 0.343861 − 0.939020i \(-0.388265\pi\)
0.343861 + 0.939020i \(0.388265\pi\)
\(72\) 0 0
\(73\) −0.140654 −0.0164623 −0.00823116 − 0.999966i \(-0.502620\pi\)
−0.00823116 + 0.999966i \(0.502620\pi\)
\(74\) 0 0
\(75\) 2.29298 0.264770
\(76\) 0 0
\(77\) −16.9085 −1.92690
\(78\) 0 0
\(79\) 6.62418 0.745278 0.372639 − 0.927976i \(-0.378453\pi\)
0.372639 + 0.927976i \(0.378453\pi\)
\(80\) 0 0
\(81\) −10.6758 −1.18620
\(82\) 0 0
\(83\) −13.9904 −1.53565 −0.767825 − 0.640660i \(-0.778661\pi\)
−0.767825 + 0.640660i \(0.778661\pi\)
\(84\) 0 0
\(85\) 2.28688 0.248047
\(86\) 0 0
\(87\) −15.1014 −1.61904
\(88\) 0 0
\(89\) 14.8139 1.57027 0.785136 − 0.619323i \(-0.212593\pi\)
0.785136 + 0.619323i \(0.212593\pi\)
\(90\) 0 0
\(91\) 14.0678 1.47470
\(92\) 0 0
\(93\) −5.36731 −0.556565
\(94\) 0 0
\(95\) −2.39037 −0.245246
\(96\) 0 0
\(97\) −8.94394 −0.908119 −0.454060 − 0.890971i \(-0.650025\pi\)
−0.454060 + 0.890971i \(0.650025\pi\)
\(98\) 0 0
\(99\) 9.97490 1.00252
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.1.5 5
4.3 odd 2 185.2.a.e.1.5 ✓ 5
12.11 even 2 1665.2.a.p.1.1 5
20.3 even 4 925.2.b.f.149.1 10
20.7 even 4 925.2.b.f.149.10 10
20.19 odd 2 925.2.a.f.1.1 5
28.27 even 2 9065.2.a.k.1.5 5
60.59 even 2 8325.2.a.ch.1.5 5
148.147 odd 2 6845.2.a.f.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.e.1.5 ✓ 5 4.3 odd 2
925.2.a.f.1.1 5 20.19 odd 2
925.2.b.f.149.1 10 20.3 even 4
925.2.b.f.149.10 10 20.7 even 4
1665.2.a.p.1.1 5 12.11 even 2
2960.2.a.w.1.5 5 1.1 even 1 trivial
6845.2.a.f.1.1 5 148.147 odd 2
8325.2.a.ch.1.5 5 60.59 even 2
9065.2.a.k.1.5 5 28.27 even 2