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Properties

Label 4640.2.a.o.1.3
Level $4640$
Weight $2$
Character 4640.1
Self dual yes
Analytic conductor $37.051$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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] = [4640,2,Mod(1,4640)] 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("4640.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4640, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4640 = 2^{5} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4640.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-4,0,0,0,12,0,0,0,16,0,0,0,4,0,0,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(37.0505865379\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{6}, \sqrt{26})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 16x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3.77425\) of defining polynomial
Character \(\chi\) \(=\) 4640.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.44949 q^{3} -1.00000 q^{5} +2.44949 q^{7} +3.00000 q^{9} -5.09902 q^{11} +4.00000 q^{13} -2.44949 q^{15} +7.24500 q^{17} +2.44949 q^{19} +6.00000 q^{21} -5.09902 q^{23} +1.00000 q^{25} +1.00000 q^{29} +7.34847 q^{31} -12.4900 q^{33} -2.44949 q^{35} +3.24500 q^{37} +9.79796 q^{39} -8.00000 q^{41} +12.6475 q^{43} -3.00000 q^{45} -7.74855 q^{47} -1.00000 q^{49} +17.7465 q^{51} +8.00000 q^{53} +5.09902 q^{55} +6.00000 q^{57} -12.4475 q^{59} +10.4900 q^{61} +7.34847 q^{63} -4.00000 q^{65} +9.99800 q^{67} -12.4900 q^{69} +7.54851 q^{71} -13.2450 q^{73} +2.44949 q^{75} -12.4900 q^{77} +4.69894 q^{79} -9.00000 q^{81} -2.84957 q^{83} -7.24500 q^{85} +2.44949 q^{87} -6.00000 q^{89} +9.79796 q^{91} +18.0000 q^{93} -2.44949 q^{95} -1.24500 q^{97} -15.2971 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} + 12 q^{9} + 16 q^{13} + 4 q^{17} + 24 q^{21} + 4 q^{25} + 4 q^{29} - 12 q^{37} - 32 q^{41} - 12 q^{45} - 4 q^{49} + 32 q^{53} + 24 q^{57} - 8 q^{61} - 16 q^{65} - 28 q^{73} - 36 q^{81} - 4 q^{85}+ \cdots + 20 q^{97}+O(q^{100}) \) 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
\(3\) 2.44949 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.44949 0.925820 0.462910 − 0.886405i \(-0.346805\pi\)
0.462910 + 0.886405i \(0.346805\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) −5.09902 −1.53741 −0.768706 − 0.639602i \(-0.779099\pi\)
−0.768706 + 0.639602i \(0.779099\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) −2.44949 −0.632456
\(16\) 0 0
\(17\) 7.24500 1.75717 0.878585 − 0.477586i \(-0.158488\pi\)
0.878585 + 0.477586i \(0.158488\pi\)
\(18\) 0 0
\(19\) 2.44949 0.561951 0.280976 − 0.959715i \(-0.409342\pi\)
0.280976 + 0.959715i \(0.409342\pi\)
\(20\) 0 0
\(21\) 6.00000 1.30931
\(22\) 0 0
\(23\) −5.09902 −1.06322 −0.531610 − 0.846990i \(-0.678413\pi\)
−0.531610 + 0.846990i \(0.678413\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 7.34847 1.31982 0.659912 − 0.751343i \(-0.270594\pi\)
0.659912 + 0.751343i \(0.270594\pi\)
\(32\) 0 0
\(33\) −12.4900 −2.17423
\(34\) 0 0
\(35\) −2.44949 −0.414039
\(36\) 0 0
\(37\) 3.24500 0.533474 0.266737 − 0.963769i \(-0.414054\pi\)
0.266737 + 0.963769i \(0.414054\pi\)
\(38\) 0 0
\(39\) 9.79796 1.56893
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 − 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 12.6475 1.92873 0.964365 − 0.264575i \(-0.0852318\pi\)
0.964365 + 0.264575i \(0.0852318\pi\)
\(44\) 0 0
\(45\) −3.00000 −0.447214
\(46\) 0 0
\(47\) −7.74855 −1.13024 −0.565121 − 0.825008i \(-0.691171\pi\)
−0.565121 + 0.825008i \(0.691171\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 17.7465 2.48501
\(52\) 0 0
\(53\) 8.00000 1.09888 0.549442 − 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) 0 0
\(55\) 5.09902 0.687552
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) 0 0
\(59\) −12.4475 −1.62053 −0.810263 − 0.586067i \(-0.800675\pi\)
−0.810263 + 0.586067i \(0.800675\pi\)
\(60\) 0 0
\(61\) 10.4900 1.34311 0.671553 − 0.740956i \(-0.265627\pi\)
0.671553 + 0.740956i \(0.265627\pi\)
\(62\) 0 0
\(63\) 7.34847 0.925820
\(64\) 0 0
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 9.99800 1.22145 0.610725 − 0.791843i \(-0.290878\pi\)
0.610725 + 0.791843i \(0.290878\pi\)
\(68\) 0 0
\(69\) −12.4900 −1.50362
\(70\) 0 0
\(71\) 7.54851 0.895843 0.447922 − 0.894073i \(-0.352164\pi\)
0.447922 + 0.894073i \(0.352164\pi\)
\(72\) 0 0
\(73\) −13.2450 −1.55021 −0.775105 − 0.631833i \(-0.782303\pi\)
−0.775105 + 0.631833i \(0.782303\pi\)
\(74\) 0 0
\(75\) 2.44949 0.282843
\(76\) 0 0
\(77\) −12.4900 −1.42337
\(78\) 0 0
\(79\) 4.69894 0.528672 0.264336 − 0.964431i \(-0.414847\pi\)
0.264336 + 0.964431i \(0.414847\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) −2.84957 −0.312781 −0.156390 − 0.987695i \(-0.549986\pi\)
−0.156390 + 0.987695i \(0.549986\pi\)
\(84\) 0 0
\(85\) −7.24500 −0.785830
\(86\) 0 0
\(87\) 2.44949 0.262613
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 9.79796 1.02711
\(92\) 0 0
\(93\) 18.0000 1.86651
\(94\) 0 0
\(95\) −2.44949 −0.251312
\(96\) 0 0
\(97\) −1.24500 −0.126410 −0.0632052 − 0.998001i \(-0.520132\pi\)
−0.0632052 + 0.998001i \(0.520132\pi\)
\(98\) 0 0
\(99\) −15.2971 −1.53741
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 4640.2.a.o.1.3 yes 4
4.3 odd 2 inner 4640.2.a.o.1.2 ✓ 4
8.3 odd 2 9280.2.a.cd.1.3 4
8.5 even 2 9280.2.a.cd.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4640.2.a.o.1.2 ✓ 4 4.3 odd 2 inner
4640.2.a.o.1.3 yes 4 1.1 even 1 trivial
9280.2.a.cd.1.2 4 8.5 even 2
9280.2.a.cd.1.3 4 8.3 odd 2