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Properties

Label 4640.2.a.o.1.4
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 1.4
Root \(1.32476\) 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)
 
\(f(q)\) \(=\) \(q+2.44949 q^{3} -1.00000 q^{5} +2.44949 q^{7} +3.00000 q^{9} +5.09902 q^{11} +4.00000 q^{13} -2.44949 q^{15} -5.24500 q^{17} +2.44949 q^{19} +6.00000 q^{21} +5.09902 q^{23} +1.00000 q^{25} +1.00000 q^{29} +7.34847 q^{31} +12.4900 q^{33} -2.44949 q^{35} -9.24500 q^{37} +9.79796 q^{39} -8.00000 q^{41} -7.74855 q^{43} -3.00000 q^{45} +12.6475 q^{47} -1.00000 q^{49} -12.8476 q^{51} +8.00000 q^{53} -5.09902 q^{55} +6.00000 q^{57} -2.24945 q^{59} -14.4900 q^{61} +7.34847 q^{63} -4.00000 q^{65} -0.200040 q^{67} +12.4900 q^{69} -2.64953 q^{71} -0.755002 q^{73} +2.44949 q^{75} +12.4900 q^{77} +14.8970 q^{79} -9.00000 q^{81} +17.5465 q^{83} +5.24500 q^{85} +2.44949 q^{87} -6.00000 q^{89} +9.79796 q^{91} +18.0000 q^{93} -2.44949 q^{95} +11.2450 q^{97} +15.2971 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} + 12 q^{9} + 16 q^{13} + 4 q^{17} + 24 q^{21} + 4 q^{25} + 4 q^{29} - 12 q^{37} - 32 q^{41} - 12 q^{45} - 4 q^{49} + 32 q^{53} + 24 q^{57} - 8 q^{61} - 16 q^{65} - 28 q^{73} - 36 q^{81} - 4 q^{85}+ \cdots + 20 q^{97}+O(q^{100}) \) 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 \(-0.220901\pi\)
0.768706 + 0.639602i \(0.220901\pi\)
\(12\) 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
\(17\) −5.24500 −1.27210 −0.636049 − 0.771648i \(-0.719433\pi\)
−0.636049 + 0.771648i \(0.719433\pi\)
\(18\) 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 \(-0.321587\pi\)
0.531610 + 0.846990i \(0.321587\pi\)
\(24\) 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
\(37\) −9.24500 −1.51987 −0.759934 − 0.650000i \(-0.774769\pi\)
−0.759934 + 0.650000i \(0.774769\pi\)
\(38\) 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
\(43\) −7.74855 −1.18164 −0.590821 − 0.806802i \(-0.701196\pi\)
−0.590821 + 0.806802i \(0.701196\pi\)
\(44\) 0 0
\(45\) −3.00000 −0.447214
\(46\) 0 0
\(47\) 12.6475 1.84483 0.922416 − 0.386198i \(-0.126212\pi\)
0.922416 + 0.386198i \(0.126212\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −12.8476 −1.79902
\(52\) 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
\(59\) −2.24945 −0.292853 −0.146427 − 0.989222i \(-0.546777\pi\)
−0.146427 + 0.989222i \(0.546777\pi\)
\(60\) 0 0
\(61\) −14.4900 −1.85525 −0.927627 − 0.373508i \(-0.878155\pi\)
−0.927627 + 0.373508i \(0.878155\pi\)
\(62\) 0 0
\(63\) 7.34847 0.925820
\(64\) 0 0
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −0.200040 −0.0244388 −0.0122194 − 0.999925i \(-0.503890\pi\)
−0.0122194 + 0.999925i \(0.503890\pi\)
\(68\) 0 0
\(69\) 12.4900 1.50362
\(70\) 0 0
\(71\) −2.64953 −0.314441 −0.157221 − 0.987563i \(-0.550253\pi\)
−0.157221 + 0.987563i \(0.550253\pi\)
\(72\) 0 0
\(73\) −0.755002 −0.0883663 −0.0441832 − 0.999023i \(-0.514069\pi\)
−0.0441832 + 0.999023i \(0.514069\pi\)
\(74\) 0 0
\(75\) 2.44949 0.282843
\(76\) 0 0
\(77\) 12.4900 1.42337
\(78\) 0 0
\(79\) 14.8970 1.67604 0.838021 − 0.545639i \(-0.183713\pi\)
0.838021 + 0.545639i \(0.183713\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 17.5465 1.92598 0.962990 − 0.269538i \(-0.0868710\pi\)
0.962990 + 0.269538i \(0.0868710\pi\)
\(84\) 0 0
\(85\) 5.24500 0.568900
\(86\) 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
\(97\) 11.2450 1.14176 0.570878 − 0.821035i \(-0.306603\pi\)
0.570878 + 0.821035i \(0.306603\pi\)
\(98\) 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.1.4 yes 4
4.3 odd 2 inner 4640.2.a.o.1.1 ✓ 4
8.3 odd 2 9280.2.a.cd.1.4 4
8.5 even 2 9280.2.a.cd.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4640.2.a.o.1.1 ✓ 4 4.3 odd 2 inner
4640.2.a.o.1.4 yes 4 1.1 even 1 trivial
9280.2.a.cd.1.1 4 8.5 even 2
9280.2.a.cd.1.4 4 8.3 odd 2