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Properties

Label 2960.2.a.w.1.4
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 1.4
Root \(2.10563\) 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)
 
\(f(q)\) \(=\) \(q+1.10563 q^{3} -1.00000 q^{5} -2.46164 q^{7} -1.77758 q^{9} -1.71497 q^{11} +6.49255 q^{13} -1.10563 q^{15} +3.32980 q^{17} -0.734568 q^{19} -2.72166 q^{21} +2.08603 q^{23} +1.00000 q^{25} -5.28224 q^{27} -4.21126 q^{29} -7.46459 q^{31} -1.89612 q^{33} +2.46164 q^{35} +1.00000 q^{37} +7.17836 q^{39} +1.71497 q^{41} -1.81885 q^{43} +1.77758 q^{45} +0.882270 q^{47} -0.940340 q^{49} +3.68152 q^{51} -7.03066 q^{53} +1.71497 q^{55} -0.812159 q^{57} -0.387867 q^{59} -11.8224 q^{61} +4.37577 q^{63} -6.49255 q^{65} -12.1086 q^{67} +2.30638 q^{69} -13.7486 q^{71} +16.6719 q^{73} +1.10563 q^{75} +4.22163 q^{77} -8.23253 q^{79} -0.507447 q^{81} +4.80275 q^{83} -3.32980 q^{85} -4.65609 q^{87} -1.52506 q^{89} -15.9823 q^{91} -8.25307 q^{93} +0.734568 q^{95} -18.1588 q^{97} +3.04850 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{3} - 5 q^{5} - 11 q^{7} + 6 q^{9} + 5 q^{11} + 4 q^{13} + 3 q^{15} + 4 q^{19} + 3 q^{21} - 4 q^{23} + 5 q^{25} - 3 q^{27} - 4 q^{29} - 8 q^{31} + 5 q^{33} + 11 q^{35} + 5 q^{37} - 2 q^{39} - 5 q^{41}+ \cdots + 10 q^{99}+O(q^{100}) \) 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
\(3\) 1.10563 0.638335 0.319168 − 0.947698i \(-0.396597\pi\)
0.319168 + 0.947698i \(0.396597\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.46164 −0.930412 −0.465206 − 0.885203i \(-0.654020\pi\)
−0.465206 + 0.885203i \(0.654020\pi\)
\(8\) 0 0
\(9\) −1.77758 −0.592528
\(10\) 0 0
\(11\) −1.71497 −0.517083 −0.258541 − 0.966000i \(-0.583242\pi\)
−0.258541 + 0.966000i \(0.583242\pi\)
\(12\) 0 0
\(13\) 6.49255 1.80071 0.900355 − 0.435156i \(-0.143307\pi\)
0.900355 + 0.435156i \(0.143307\pi\)
\(14\) 0 0
\(15\) −1.10563 −0.285472
\(16\) 0 0
\(17\) 3.32980 0.807594 0.403797 − 0.914849i \(-0.367690\pi\)
0.403797 + 0.914849i \(0.367690\pi\)
\(18\) 0 0
\(19\) −0.734568 −0.168521 −0.0842607 − 0.996444i \(-0.526853\pi\)
−0.0842607 + 0.996444i \(0.526853\pi\)
\(20\) 0 0
\(21\) −2.72166 −0.593915
\(22\) 0 0
\(23\) 2.08603 0.434968 0.217484 − 0.976064i \(-0.430215\pi\)
0.217484 + 0.976064i \(0.430215\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.28224 −1.01657
\(28\) 0 0
\(29\) −4.21126 −0.782011 −0.391006 − 0.920388i \(-0.627873\pi\)
−0.391006 + 0.920388i \(0.627873\pi\)
\(30\) 0 0
\(31\) −7.46459 −1.34068 −0.670340 − 0.742054i \(-0.733852\pi\)
−0.670340 + 0.742054i \(0.733852\pi\)
\(32\) 0 0
\(33\) −1.89612 −0.330072
\(34\) 0 0
\(35\) 2.46164 0.416093
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) 7.17836 1.14946
\(40\) 0 0
\(41\) 1.71497 0.267833 0.133917 − 0.990993i \(-0.457245\pi\)
0.133917 + 0.990993i \(0.457245\pi\)
\(42\) 0 0
\(43\) −1.81885 −0.277372 −0.138686 − 0.990336i \(-0.544288\pi\)
−0.138686 + 0.990336i \(0.544288\pi\)
\(44\) 0 0
\(45\) 1.77758 0.264986
\(46\) 0 0
\(47\) 0.882270 0.128692 0.0643462 − 0.997928i \(-0.479504\pi\)
0.0643462 + 0.997928i \(0.479504\pi\)
\(48\) 0 0
\(49\) −0.940340 −0.134334
\(50\) 0 0
\(51\) 3.68152 0.515516
\(52\) 0 0
\(53\) −7.03066 −0.965735 −0.482867 − 0.875693i \(-0.660405\pi\)
−0.482867 + 0.875693i \(0.660405\pi\)
\(54\) 0 0
\(55\) 1.71497 0.231247
\(56\) 0 0
\(57\) −0.812159 −0.107573
\(58\) 0 0
\(59\) −0.387867 −0.0504959 −0.0252480 − 0.999681i \(-0.508038\pi\)
−0.0252480 + 0.999681i \(0.508038\pi\)
\(60\) 0 0
\(61\) −11.8224 −1.51370 −0.756848 − 0.653590i \(-0.773262\pi\)
−0.756848 + 0.653590i \(0.773262\pi\)
\(62\) 0 0
\(63\) 4.37577 0.551295
\(64\) 0 0
\(65\) −6.49255 −0.805302
\(66\) 0 0
\(67\) −12.1086 −1.47930 −0.739649 − 0.672992i \(-0.765009\pi\)
−0.739649 + 0.672992i \(0.765009\pi\)
\(68\) 0 0
\(69\) 2.30638 0.277655
\(70\) 0 0
\(71\) −13.7486 −1.63166 −0.815828 − 0.578295i \(-0.803718\pi\)
−0.815828 + 0.578295i \(0.803718\pi\)
\(72\) 0 0
\(73\) 16.6719 1.95129 0.975646 − 0.219349i \(-0.0703933\pi\)
0.975646 + 0.219349i \(0.0703933\pi\)
\(74\) 0 0
\(75\) 1.10563 0.127667
\(76\) 0 0
\(77\) 4.22163 0.481100
\(78\) 0 0
\(79\) −8.23253 −0.926232 −0.463116 − 0.886298i \(-0.653269\pi\)
−0.463116 + 0.886298i \(0.653269\pi\)
\(80\) 0 0
\(81\) −0.507447 −0.0563829
\(82\) 0 0
\(83\) 4.80275 0.527171 0.263585 − 0.964636i \(-0.415095\pi\)
0.263585 + 0.964636i \(0.415095\pi\)
\(84\) 0 0
\(85\) −3.32980 −0.361167
\(86\) 0 0
\(87\) −4.65609 −0.499185
\(88\) 0 0
\(89\) −1.52506 −0.161656 −0.0808280 − 0.996728i \(-0.525756\pi\)
−0.0808280 + 0.996728i \(0.525756\pi\)
\(90\) 0 0
\(91\) −15.9823 −1.67540
\(92\) 0 0
\(93\) −8.25307 −0.855804
\(94\) 0 0
\(95\) 0.734568 0.0753650
\(96\) 0 0
\(97\) −18.1588 −1.84375 −0.921875 − 0.387487i \(-0.873343\pi\)
−0.921875 + 0.387487i \(0.873343\pi\)
\(98\) 0 0
\(99\) 3.04850 0.306386
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.1.4 5
4.3 odd 2 185.2.a.e.1.2 ✓ 5
12.11 even 2 1665.2.a.p.1.4 5
20.3 even 4 925.2.b.f.149.7 10
20.7 even 4 925.2.b.f.149.4 10
20.19 odd 2 925.2.a.f.1.4 5
28.27 even 2 9065.2.a.k.1.2 5
60.59 even 2 8325.2.a.ch.1.2 5
148.147 odd 2 6845.2.a.f.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.e.1.2 ✓ 5 4.3 odd 2
925.2.a.f.1.4 5 20.19 odd 2
925.2.b.f.149.4 10 20.7 even 4
925.2.b.f.149.7 10 20.3 even 4
1665.2.a.p.1.4 5 12.11 even 2
2960.2.a.w.1.4 5 1.1 even 1 trivial
6845.2.a.f.1.4 5 148.147 odd 2
8325.2.a.ch.1.2 5 60.59 even 2
9065.2.a.k.1.2 5 28.27 even 2