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Properties

Label 6080.2.a.ch.1.3
Level $6080$
Weight $2$
Character 6080.1
Self dual yes
Analytic conductor $48.549$
Analytic rank $0$
Dimension $4$
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] = [6080,2,Mod(1,6080)] 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("6080.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6080, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6080.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,2,0,4,0,-4,0,8,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(48.5490444289\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.11344.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.51658\) of defining polynomial
Character \(\chi\) \(=\) 6080.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.53844 q^{3} +1.00000 q^{5} -5.03316 q^{7} -0.633188 q^{9} -3.03316 q^{11} +4.57160 q^{13} +1.53844 q^{15} -1.07689 q^{17} +1.00000 q^{19} -7.74324 q^{21} -4.11005 q^{23} +1.00000 q^{25} -5.58946 q^{27} +1.07689 q^{29} -5.58946 q^{31} -4.66635 q^{33} -5.03316 q^{35} -0.0947438 q^{37} +7.03316 q^{39} +10.6663 q^{41} +5.03316 q^{43} -0.633188 q^{45} +12.2995 q^{47} +18.3327 q^{49} -1.65673 q^{51} +4.09474 q^{53} -3.03316 q^{55} +1.53844 q^{57} -1.39997 q^{59} +5.69951 q^{61} +3.18694 q^{63} +4.57160 q^{65} +5.28168 q^{67} -6.32308 q^{69} +5.67692 q^{71} +9.07689 q^{73} +1.53844 q^{75} +15.2664 q^{77} +5.39997 q^{79} -6.69951 q^{81} +1.95627 q^{83} -1.07689 q^{85} +1.65673 q^{87} -2.18949 q^{89} -23.0096 q^{91} -8.59907 q^{93} +1.00000 q^{95} -2.16106 q^{97} +1.92056 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 4 q^{5} - 4 q^{7} + 8 q^{9} + 4 q^{11} - 2 q^{13} + 2 q^{15} + 4 q^{17} + 4 q^{19} + 4 q^{21} + 8 q^{23} + 4 q^{25} - 4 q^{27} - 4 q^{29} - 4 q^{31} + 8 q^{33} - 4 q^{35} + 6 q^{37} + 12 q^{39}+ \cdots - 4 q^{99}+O(q^{100}) \) 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
\(3\) 1.53844 0.888221 0.444111 − 0.895972i \(-0.353520\pi\)
0.444111 + 0.895972i \(0.353520\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −5.03316 −1.90236 −0.951178 − 0.308644i \(-0.900125\pi\)
−0.951178 + 0.308644i \(0.900125\pi\)
\(8\) 0 0
\(9\) −0.633188 −0.211063
\(10\) 0 0
\(11\) −3.03316 −0.914532 −0.457266 − 0.889330i \(-0.651171\pi\)
−0.457266 + 0.889330i \(0.651171\pi\)
\(12\) 0 0
\(13\) 4.57160 1.26793 0.633967 − 0.773360i \(-0.281425\pi\)
0.633967 + 0.773360i \(0.281425\pi\)
\(14\) 0 0
\(15\) 1.53844 0.397225
\(16\) 0 0
\(17\) −1.07689 −0.261184 −0.130592 − 0.991436i \(-0.541688\pi\)
−0.130592 + 0.991436i \(0.541688\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −7.74324 −1.68971
\(22\) 0 0
\(23\) −4.11005 −0.857004 −0.428502 − 0.903541i \(-0.640959\pi\)
−0.428502 + 0.903541i \(0.640959\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.58946 −1.07569
\(28\) 0 0
\(29\) 1.07689 0.199973 0.0999866 − 0.994989i \(-0.468120\pi\)
0.0999866 + 0.994989i \(0.468120\pi\)
\(30\) 0 0
\(31\) −5.58946 −1.00390 −0.501948 − 0.864898i \(-0.667383\pi\)
−0.501948 + 0.864898i \(0.667383\pi\)
\(32\) 0 0
\(33\) −4.66635 −0.812307
\(34\) 0 0
\(35\) −5.03316 −0.850759
\(36\) 0 0
\(37\) −0.0947438 −0.0155758 −0.00778789 − 0.999970i \(-0.502479\pi\)
−0.00778789 + 0.999970i \(0.502479\pi\)
\(38\) 0 0
\(39\) 7.03316 1.12621
\(40\) 0 0
\(41\) 10.6663 1.66580 0.832902 − 0.553421i \(-0.186678\pi\)
0.832902 + 0.553421i \(0.186678\pi\)
\(42\) 0 0
\(43\) 5.03316 0.767550 0.383775 − 0.923427i \(-0.374624\pi\)
0.383775 + 0.923427i \(0.374624\pi\)
\(44\) 0 0
\(45\) −0.633188 −0.0943901
\(46\) 0 0
\(47\) 12.2995 1.79407 0.897036 − 0.441958i \(-0.145716\pi\)
0.897036 + 0.441958i \(0.145716\pi\)
\(48\) 0 0
\(49\) 18.3327 2.61896
\(50\) 0 0
\(51\) −1.65673 −0.231989
\(52\) 0 0
\(53\) 4.09474 0.562456 0.281228 − 0.959641i \(-0.409258\pi\)
0.281228 + 0.959641i \(0.409258\pi\)
\(54\) 0 0
\(55\) −3.03316 −0.408991
\(56\) 0 0
\(57\) 1.53844 0.203772
\(58\) 0 0
\(59\) −1.39997 −0.182261 −0.0911304 − 0.995839i \(-0.529048\pi\)
−0.0911304 + 0.995839i \(0.529048\pi\)
\(60\) 0 0
\(61\) 5.69951 0.729747 0.364874 − 0.931057i \(-0.381112\pi\)
0.364874 + 0.931057i \(0.381112\pi\)
\(62\) 0 0
\(63\) 3.18694 0.401516
\(64\) 0 0
\(65\) 4.57160 0.567038
\(66\) 0 0
\(67\) 5.28168 0.645260 0.322630 − 0.946525i \(-0.395433\pi\)
0.322630 + 0.946525i \(0.395433\pi\)
\(68\) 0 0
\(69\) −6.32308 −0.761210
\(70\) 0 0
\(71\) 5.67692 0.673726 0.336863 − 0.941554i \(-0.390634\pi\)
0.336863 + 0.941554i \(0.390634\pi\)
\(72\) 0 0
\(73\) 9.07689 1.06237 0.531185 − 0.847256i \(-0.321747\pi\)
0.531185 + 0.847256i \(0.321747\pi\)
\(74\) 0 0
\(75\) 1.53844 0.177644
\(76\) 0 0
\(77\) 15.2664 1.73977
\(78\) 0 0
\(79\) 5.39997 0.607544 0.303772 − 0.952745i \(-0.401754\pi\)
0.303772 + 0.952745i \(0.401754\pi\)
\(80\) 0 0
\(81\) −6.69951 −0.744390
\(82\) 0 0
\(83\) 1.95627 0.214729 0.107364 − 0.994220i \(-0.465759\pi\)
0.107364 + 0.994220i \(0.465759\pi\)
\(84\) 0 0
\(85\) −1.07689 −0.116805
\(86\) 0 0
\(87\) 1.65673 0.177621
\(88\) 0 0
\(89\) −2.18949 −0.232085 −0.116043 − 0.993244i \(-0.537021\pi\)
−0.116043 + 0.993244i \(0.537021\pi\)
\(90\) 0 0
\(91\) −23.0096 −2.41206
\(92\) 0 0
\(93\) −8.59907 −0.891682
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −2.16106 −0.219423 −0.109711 − 0.993963i \(-0.534993\pi\)
−0.109711 + 0.993963i \(0.534993\pi\)
\(98\) 0 0
\(99\) 1.92056 0.193024
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 6080.2.a.ch.1.3 4
4.3 odd 2 6080.2.a.cc.1.2 4
8.3 odd 2 95.2.a.b.1.3 ✓ 4
8.5 even 2 1520.2.a.t.1.2 4
24.11 even 2 855.2.a.m.1.2 4
40.3 even 4 475.2.b.e.324.4 8
40.19 odd 2 475.2.a.i.1.2 4
40.27 even 4 475.2.b.e.324.5 8
40.29 even 2 7600.2.a.cf.1.3 4
56.27 even 2 4655.2.a.y.1.3 4
120.59 even 2 4275.2.a.bo.1.3 4
152.75 even 2 1805.2.a.p.1.2 4
760.379 even 2 9025.2.a.bf.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.a.b.1.3 ✓ 4 8.3 odd 2
475.2.a.i.1.2 4 40.19 odd 2
475.2.b.e.324.4 8 40.3 even 4
475.2.b.e.324.5 8 40.27 even 4
855.2.a.m.1.2 4 24.11 even 2
1520.2.a.t.1.2 4 8.5 even 2
1805.2.a.p.1.2 4 152.75 even 2
4275.2.a.bo.1.3 4 120.59 even 2
4655.2.a.y.1.3 4 56.27 even 2
6080.2.a.cc.1.2 4 4.3 odd 2
6080.2.a.ch.1.3 4 1.1 even 1 trivial
7600.2.a.cf.1.3 4 40.29 even 2
9025.2.a.bf.1.3 4 760.379 even 2