Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7616,2,Mod(1,7616)] 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("7616.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7616, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7616 = 2^{6} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7616.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,3,0,-1,0,-3,0,2,0,6] 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: \(60.8140661794\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) 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 952)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.254102\) of defining polynomial
Character \(\chi\) \(=\) 7616.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.25410 q^{3} -2.93543 q^{5} -1.00000 q^{7} -1.42723 q^{9} +2.00000 q^{11} +3.36266 q^{13} -3.68133 q^{15} -1.00000 q^{17} +5.36266 q^{19} -1.25410 q^{21} -2.50820 q^{23} +3.61676 q^{25} -5.55220 q^{27} -4.37907 q^{29} -6.42723 q^{31} +2.50820 q^{33} +2.93543 q^{35} +1.01641 q^{37} +4.21712 q^{39} +8.48763 q^{41} +6.10856 q^{43} +4.18953 q^{45} -2.50820 q^{47} +1.00000 q^{49} -1.25410 q^{51} +11.3145 q^{53} -5.87086 q^{55} +6.72532 q^{57} +9.36266 q^{59} -3.25410 q^{61} +1.42723 q^{63} -9.87086 q^{65} -7.66075 q^{67} -3.14554 q^{69} -6.98359 q^{71} -13.5040 q^{73} +4.53579 q^{75} -2.00000 q^{77} -3.52461 q^{79} -2.68133 q^{81} -14.5962 q^{83} +2.93543 q^{85} -5.49180 q^{87} +6.85446 q^{89} -3.36266 q^{91} -8.06040 q^{93} -15.7417 q^{95} -15.5316 q^{97} -2.85446 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - q^{5} - 3 q^{7} + 2 q^{9} + 6 q^{11} - 4 q^{13} - 4 q^{15} - 3 q^{17} + 2 q^{19} - 3 q^{21} - 6 q^{23} - 4 q^{25} + 6 q^{27} + 4 q^{29} - 13 q^{31} + 6 q^{33} + q^{35} - 14 q^{39} - 5 q^{41}+ \cdots + 4 q^{99}+O(q^{100}) \) 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
\(3\) 1.25410 0.724056 0.362028 − 0.932167i \(-0.382084\pi\)
0.362028 + 0.932167i \(0.382084\pi\)
\(4\) 0 0
\(5\) −2.93543 −1.31277 −0.656383 − 0.754428i \(-0.727914\pi\)
−0.656383 + 0.754428i \(0.727914\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −1.42723 −0.475743
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 3.36266 0.932634 0.466317 − 0.884618i \(-0.345581\pi\)
0.466317 + 0.884618i \(0.345581\pi\)
\(14\) 0 0
\(15\) −3.68133 −0.950515
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) 5.36266 1.23028 0.615139 − 0.788418i \(-0.289100\pi\)
0.615139 + 0.788418i \(0.289100\pi\)
\(20\) 0 0
\(21\) −1.25410 −0.273667
\(22\) 0 0
\(23\) −2.50820 −0.522997 −0.261498 − 0.965204i \(-0.584217\pi\)
−0.261498 + 0.965204i \(0.584217\pi\)
\(24\) 0 0
\(25\) 3.61676 0.723353
\(26\) 0 0
\(27\) −5.55220 −1.06852
\(28\) 0 0
\(29\) −4.37907 −0.813173 −0.406586 − 0.913612i \(-0.633281\pi\)
−0.406586 + 0.913612i \(0.633281\pi\)
\(30\) 0 0
\(31\) −6.42723 −1.15436 −0.577182 − 0.816615i \(-0.695848\pi\)
−0.577182 + 0.816615i \(0.695848\pi\)
\(32\) 0 0
\(33\) 2.50820 0.436622
\(34\) 0 0
\(35\) 2.93543 0.496179
\(36\) 0 0
\(37\) 1.01641 0.167096 0.0835481 − 0.996504i \(-0.473375\pi\)
0.0835481 + 0.996504i \(0.473375\pi\)
\(38\) 0 0
\(39\) 4.21712 0.675280
\(40\) 0 0
\(41\) 8.48763 1.32554 0.662772 − 0.748821i \(-0.269380\pi\)
0.662772 + 0.748821i \(0.269380\pi\)
\(42\) 0 0
\(43\) 6.10856 0.931547 0.465773 − 0.884904i \(-0.345776\pi\)
0.465773 + 0.884904i \(0.345776\pi\)
\(44\) 0 0
\(45\) 4.18953 0.624539
\(46\) 0 0
\(47\) −2.50820 −0.365859 −0.182930 − 0.983126i \(-0.558558\pi\)
−0.182930 + 0.983126i \(0.558558\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.25410 −0.175609
\(52\) 0 0
\(53\) 11.3145 1.55417 0.777083 − 0.629398i \(-0.216698\pi\)
0.777083 + 0.629398i \(0.216698\pi\)
\(54\) 0 0
\(55\) −5.87086 −0.791627
\(56\) 0 0
\(57\) 6.72532 0.890791
\(58\) 0 0
\(59\) 9.36266 1.21891 0.609457 − 0.792819i \(-0.291387\pi\)
0.609457 + 0.792819i \(0.291387\pi\)
\(60\) 0 0
\(61\) −3.25410 −0.416645 −0.208323 − 0.978060i \(-0.566800\pi\)
−0.208323 + 0.978060i \(0.566800\pi\)
\(62\) 0 0
\(63\) 1.42723 0.179814
\(64\) 0 0
\(65\) −9.87086 −1.22433
\(66\) 0 0
\(67\) −7.66075 −0.935910 −0.467955 − 0.883752i \(-0.655009\pi\)
−0.467955 + 0.883752i \(0.655009\pi\)
\(68\) 0 0
\(69\) −3.14554 −0.378679
\(70\) 0 0
\(71\) −6.98359 −0.828800 −0.414400 − 0.910095i \(-0.636009\pi\)
−0.414400 + 0.910095i \(0.636009\pi\)
\(72\) 0 0
\(73\) −13.5040 −1.58053 −0.790264 − 0.612767i \(-0.790057\pi\)
−0.790264 + 0.612767i \(0.790057\pi\)
\(74\) 0 0
\(75\) 4.53579 0.523748
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) −3.52461 −0.396550 −0.198275 − 0.980146i \(-0.563534\pi\)
−0.198275 + 0.980146i \(0.563534\pi\)
\(80\) 0 0
\(81\) −2.68133 −0.297926
\(82\) 0 0
\(83\) −14.5962 −1.60214 −0.801070 − 0.598571i \(-0.795735\pi\)
−0.801070 + 0.598571i \(0.795735\pi\)
\(84\) 0 0
\(85\) 2.93543 0.318392
\(86\) 0 0
\(87\) −5.49180 −0.588782
\(88\) 0 0
\(89\) 6.85446 0.726571 0.363286 − 0.931678i \(-0.381655\pi\)
0.363286 + 0.931678i \(0.381655\pi\)
\(90\) 0 0
\(91\) −3.36266 −0.352503
\(92\) 0 0
\(93\) −8.06040 −0.835824
\(94\) 0 0
\(95\) −15.7417 −1.61507
\(96\) 0 0
\(97\) −15.5316 −1.57700 −0.788499 − 0.615037i \(-0.789141\pi\)
−0.788499 + 0.615037i \(0.789141\pi\)
\(98\) 0 0
\(99\) −2.85446 −0.286884
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 7616.2.a.bg.1.2 3
4.3 odd 2 7616.2.a.ba.1.2 3
8.3 odd 2 1904.2.a.p.1.2 3
8.5 even 2 952.2.a.d.1.2 ✓ 3
24.5 odd 2 8568.2.a.z.1.1 3
56.13 odd 2 6664.2.a.l.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.a.d.1.2 ✓ 3 8.5 even 2
1904.2.a.p.1.2 3 8.3 odd 2
6664.2.a.l.1.2 3 56.13 odd 2
7616.2.a.ba.1.2 3 4.3 odd 2
7616.2.a.bg.1.2 3 1.1 even 1 trivial
8568.2.a.z.1.1 3 24.5 odd 2