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Properties

Label 616.2.a.h.1.3
Level $616$
Weight $2$
Character 616.1
Self dual yes
Analytic conductor $4.919$
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] = [616,2,Mod(1,616)] 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("616.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(616, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 616 = 2^{3} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 616.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,1,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.91878476451\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.11348.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.77571\) of defining polynomial
Character \(\chi\) \(=\) 616.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.12631 q^{3} +4.42512 q^{5} -1.00000 q^{7} -1.73143 q^{9} +1.00000 q^{11} +5.85774 q^{13} +4.98405 q^{15} -1.55143 q^{17} -7.40916 q^{19} -1.12631 q^{21} +4.98405 q^{23} +14.5817 q^{25} -5.32905 q^{27} -9.10285 q^{29} -1.43262 q^{31} +1.12631 q^{33} -4.42512 q^{35} -0.731430 q^{37} +6.59762 q^{39} -1.55143 q^{41} +2.59762 q^{43} -7.66178 q^{45} -0.953807 q^{47} +1.00000 q^{49} -1.74738 q^{51} +2.00000 q^{53} +4.42512 q^{55} -8.34500 q^{57} -8.22916 q^{59} +3.60512 q^{61} +1.73143 q^{63} +25.9212 q^{65} +9.32905 q^{67} +5.61357 q^{69} -0.478813 q^{71} +6.16405 q^{73} +16.4234 q^{75} -1.00000 q^{77} -9.74738 q^{79} -0.807862 q^{81} +10.5589 q^{83} -6.86524 q^{85} -10.2526 q^{87} +4.11881 q^{89} -5.85774 q^{91} -1.61357 q^{93} -32.7864 q^{95} -11.3290 q^{97} -1.73143 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} + 5 q^{5} - 4 q^{7} + 9 q^{9} + 4 q^{11} + 4 q^{13} - 3 q^{15} + 10 q^{17} + 6 q^{19} - q^{21} - 3 q^{23} + 17 q^{25} + 13 q^{27} - 4 q^{29} + q^{31} + q^{33} - 5 q^{35} + 13 q^{37} + 8 q^{39}+ \cdots + 9 q^{99}+O(q^{100}) \) 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
\(3\) 1.12631 0.650274 0.325137 − 0.945667i \(-0.394590\pi\)
0.325137 + 0.945667i \(0.394590\pi\)
\(4\) 0 0
\(5\) 4.42512 1.97897 0.989486 − 0.144626i \(-0.0461980\pi\)
0.989486 + 0.144626i \(0.0461980\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −1.73143 −0.577143
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 5.85774 1.62464 0.812322 − 0.583209i \(-0.198203\pi\)
0.812322 + 0.583209i \(0.198203\pi\)
\(14\) 0 0
\(15\) 4.98405 1.28688
\(16\) 0 0
\(17\) −1.55143 −0.376276 −0.188138 − 0.982143i \(-0.560245\pi\)
−0.188138 + 0.982143i \(0.560245\pi\)
\(18\) 0 0
\(19\) −7.40916 −1.69978 −0.849889 − 0.526961i \(-0.823331\pi\)
−0.849889 + 0.526961i \(0.823331\pi\)
\(20\) 0 0
\(21\) −1.12631 −0.245781
\(22\) 0 0
\(23\) 4.98405 1.03925 0.519623 − 0.854396i \(-0.326073\pi\)
0.519623 + 0.854396i \(0.326073\pi\)
\(24\) 0 0
\(25\) 14.5817 2.91633
\(26\) 0 0
\(27\) −5.32905 −1.02558
\(28\) 0 0
\(29\) −9.10285 −1.69036 −0.845179 − 0.534484i \(-0.820506\pi\)
−0.845179 + 0.534484i \(0.820506\pi\)
\(30\) 0 0
\(31\) −1.43262 −0.257306 −0.128653 − 0.991690i \(-0.541065\pi\)
−0.128653 + 0.991690i \(0.541065\pi\)
\(32\) 0 0
\(33\) 1.12631 0.196065
\(34\) 0 0
\(35\) −4.42512 −0.747981
\(36\) 0 0
\(37\) −0.731430 −0.120246 −0.0601232 − 0.998191i \(-0.519149\pi\)
−0.0601232 + 0.998191i \(0.519149\pi\)
\(38\) 0 0
\(39\) 6.59762 1.05646
\(40\) 0 0
\(41\) −1.55143 −0.242292 −0.121146 − 0.992635i \(-0.538657\pi\)
−0.121146 + 0.992635i \(0.538657\pi\)
\(42\) 0 0
\(43\) 2.59762 0.396133 0.198067 − 0.980189i \(-0.436534\pi\)
0.198067 + 0.980189i \(0.436534\pi\)
\(44\) 0 0
\(45\) −7.66178 −1.14215
\(46\) 0 0
\(47\) −0.953807 −0.139127 −0.0695635 − 0.997578i \(-0.522161\pi\)
−0.0695635 + 0.997578i \(0.522161\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.74738 −0.244683
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 4.42512 0.596683
\(56\) 0 0
\(57\) −8.34500 −1.10532
\(58\) 0 0
\(59\) −8.22916 −1.07135 −0.535673 − 0.844426i \(-0.679942\pi\)
−0.535673 + 0.844426i \(0.679942\pi\)
\(60\) 0 0
\(61\) 3.60512 0.461589 0.230794 − 0.973003i \(-0.425868\pi\)
0.230794 + 0.973003i \(0.425868\pi\)
\(62\) 0 0
\(63\) 1.73143 0.218140
\(64\) 0 0
\(65\) 25.9212 3.21513
\(66\) 0 0
\(67\) 9.32905 1.13972 0.569862 − 0.821740i \(-0.306997\pi\)
0.569862 + 0.821740i \(0.306997\pi\)
\(68\) 0 0
\(69\) 5.61357 0.675795
\(70\) 0 0
\(71\) −0.478813 −0.0568247 −0.0284123 − 0.999596i \(-0.509045\pi\)
−0.0284123 + 0.999596i \(0.509045\pi\)
\(72\) 0 0
\(73\) 6.16405 0.721448 0.360724 − 0.932673i \(-0.382530\pi\)
0.360724 + 0.932673i \(0.382530\pi\)
\(74\) 0 0
\(75\) 16.4234 1.89642
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −9.74738 −1.09667 −0.548333 − 0.836260i \(-0.684737\pi\)
−0.548333 + 0.836260i \(0.684737\pi\)
\(80\) 0 0
\(81\) −0.807862 −0.0897624
\(82\) 0 0
\(83\) 10.5589 1.15899 0.579497 − 0.814975i \(-0.303249\pi\)
0.579497 + 0.814975i \(0.303249\pi\)
\(84\) 0 0
\(85\) −6.86524 −0.744640
\(86\) 0 0
\(87\) −10.2526 −1.09920
\(88\) 0 0
\(89\) 4.11881 0.436592 0.218296 − 0.975883i \(-0.429950\pi\)
0.218296 + 0.975883i \(0.429950\pi\)
\(90\) 0 0
\(91\) −5.85774 −0.614058
\(92\) 0 0
\(93\) −1.61357 −0.167320
\(94\) 0 0
\(95\) −32.7864 −3.36382
\(96\) 0 0
\(97\) −11.3290 −1.15029 −0.575145 − 0.818051i \(-0.695054\pi\)
−0.575145 + 0.818051i \(0.695054\pi\)
\(98\) 0 0
\(99\) −1.73143 −0.174015
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 616.2.a.h.1.3 ✓ 4
3.2 odd 2 5544.2.a.bm.1.1 4
4.3 odd 2 1232.2.a.s.1.2 4
7.6 odd 2 4312.2.a.z.1.2 4
8.3 odd 2 4928.2.a.ch.1.3 4
8.5 even 2 4928.2.a.cc.1.2 4
11.10 odd 2 6776.2.a.bb.1.3 4
28.27 even 2 8624.2.a.cy.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
616.2.a.h.1.3 ✓ 4 1.1 even 1 trivial
1232.2.a.s.1.2 4 4.3 odd 2
4312.2.a.z.1.2 4 7.6 odd 2
4928.2.a.cc.1.2 4 8.5 even 2
4928.2.a.ch.1.3 4 8.3 odd 2
5544.2.a.bm.1.1 4 3.2 odd 2
6776.2.a.bb.1.3 4 11.10 odd 2
8624.2.a.cy.1.3 4 28.27 even 2