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 1.1
Root \(2.11491\) 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)
 
\(f(q)\) \(=\) \(q-1.11491 q^{3} +1.47283 q^{5} -1.00000 q^{7} -1.75698 q^{9} +2.00000 q^{11} -0.715853 q^{13} -1.64207 q^{15} -1.00000 q^{17} +1.28415 q^{19} +1.11491 q^{21} +2.22982 q^{23} -2.83076 q^{25} +5.30359 q^{27} +9.17548 q^{29} -6.75698 q^{31} -2.22982 q^{33} -1.47283 q^{35} -8.45963 q^{37} +0.798110 q^{39} -6.77643 q^{41} +4.39905 q^{43} -2.58774 q^{45} +2.22982 q^{47} +1.00000 q^{49} +1.11491 q^{51} -6.64832 q^{53} +2.94567 q^{55} -1.43171 q^{57} +5.28415 q^{59} -0.885092 q^{61} +1.75698 q^{63} -1.05433 q^{65} +4.90454 q^{67} -2.48604 q^{69} -16.4596 q^{71} +11.2361 q^{73} +3.15604 q^{75} -2.00000 q^{77} +10.6894 q^{79} -0.642074 q^{81} +2.37737 q^{83} -1.47283 q^{85} -10.2298 q^{87} +7.51396 q^{89} +0.715853 q^{91} +7.53341 q^{93} +1.89134 q^{95} +5.85021 q^{97} -3.51396 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - q^{5} - 3 q^{7} + 2 q^{9} + 6 q^{11} - 4 q^{13} - 4 q^{15} - 3 q^{17} + 2 q^{19} - 3 q^{21} - 6 q^{23} - 4 q^{25} + 6 q^{27} + 4 q^{29} - 13 q^{31} + 6 q^{33} + q^{35} - 14 q^{39} - 5 q^{41}+ \cdots + 4 q^{99}+O(q^{100}) \) 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
\(3\) −1.11491 −0.643692 −0.321846 − 0.946792i \(-0.604303\pi\)
−0.321846 + 0.946792i \(0.604303\pi\)
\(4\) 0 0
\(5\) 1.47283 0.658671 0.329336 − 0.944213i \(-0.393175\pi\)
0.329336 + 0.944213i \(0.393175\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −1.75698 −0.585660
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) −0.715853 −0.198542 −0.0992709 − 0.995060i \(-0.531651\pi\)
−0.0992709 + 0.995060i \(0.531651\pi\)
\(14\) 0 0
\(15\) −1.64207 −0.423982
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) 1.28415 0.294604 0.147302 − 0.989092i \(-0.452941\pi\)
0.147302 + 0.989092i \(0.452941\pi\)
\(20\) 0 0
\(21\) 1.11491 0.243293
\(22\) 0 0
\(23\) 2.22982 0.464949 0.232474 − 0.972603i \(-0.425318\pi\)
0.232474 + 0.972603i \(0.425318\pi\)
\(24\) 0 0
\(25\) −2.83076 −0.566152
\(26\) 0 0
\(27\) 5.30359 1.02068
\(28\) 0 0
\(29\) 9.17548 1.70384 0.851922 − 0.523668i \(-0.175437\pi\)
0.851922 + 0.523668i \(0.175437\pi\)
\(30\) 0 0
\(31\) −6.75698 −1.21359 −0.606795 − 0.794859i \(-0.707545\pi\)
−0.606795 + 0.794859i \(0.707545\pi\)
\(32\) 0 0
\(33\) −2.22982 −0.388161
\(34\) 0 0
\(35\) −1.47283 −0.248954
\(36\) 0 0
\(37\) −8.45963 −1.39075 −0.695377 − 0.718645i \(-0.744763\pi\)
−0.695377 + 0.718645i \(0.744763\pi\)
\(38\) 0 0
\(39\) 0.798110 0.127800
\(40\) 0 0
\(41\) −6.77643 −1.05830 −0.529150 − 0.848528i \(-0.677489\pi\)
−0.529150 + 0.848528i \(0.677489\pi\)
\(42\) 0 0
\(43\) 4.39905 0.670850 0.335425 − 0.942067i \(-0.391120\pi\)
0.335425 + 0.942067i \(0.391120\pi\)
\(44\) 0 0
\(45\) −2.58774 −0.385758
\(46\) 0 0
\(47\) 2.22982 0.325252 0.162626 − 0.986688i \(-0.448004\pi\)
0.162626 + 0.986688i \(0.448004\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 1.11491 0.156118
\(52\) 0 0
\(53\) −6.64832 −0.913217 −0.456608 − 0.889668i \(-0.650936\pi\)
−0.456608 + 0.889668i \(0.650936\pi\)
\(54\) 0 0
\(55\) 2.94567 0.397194
\(56\) 0 0
\(57\) −1.43171 −0.189634
\(58\) 0 0
\(59\) 5.28415 0.687937 0.343969 − 0.938981i \(-0.388229\pi\)
0.343969 + 0.938981i \(0.388229\pi\)
\(60\) 0 0
\(61\) −0.885092 −0.113324 −0.0566622 − 0.998393i \(-0.518046\pi\)
−0.0566622 + 0.998393i \(0.518046\pi\)
\(62\) 0 0
\(63\) 1.75698 0.221359
\(64\) 0 0
\(65\) −1.05433 −0.130774
\(66\) 0 0
\(67\) 4.90454 0.599185 0.299592 − 0.954067i \(-0.403149\pi\)
0.299592 + 0.954067i \(0.403149\pi\)
\(68\) 0 0
\(69\) −2.48604 −0.299284
\(70\) 0 0
\(71\) −16.4596 −1.95340 −0.976699 − 0.214612i \(-0.931151\pi\)
−0.976699 + 0.214612i \(0.931151\pi\)
\(72\) 0 0
\(73\) 11.2361 1.31508 0.657541 − 0.753419i \(-0.271597\pi\)
0.657541 + 0.753419i \(0.271597\pi\)
\(74\) 0 0
\(75\) 3.15604 0.364428
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) 10.6894 1.20266 0.601328 − 0.799002i \(-0.294638\pi\)
0.601328 + 0.799002i \(0.294638\pi\)
\(80\) 0 0
\(81\) −0.642074 −0.0713415
\(82\) 0 0
\(83\) 2.37737 0.260951 0.130475 − 0.991452i \(-0.458350\pi\)
0.130475 + 0.991452i \(0.458350\pi\)
\(84\) 0 0
\(85\) −1.47283 −0.159751
\(86\) 0 0
\(87\) −10.2298 −1.09675
\(88\) 0 0
\(89\) 7.51396 0.796478 0.398239 − 0.917282i \(-0.369621\pi\)
0.398239 + 0.917282i \(0.369621\pi\)
\(90\) 0 0
\(91\) 0.715853 0.0750418
\(92\) 0 0
\(93\) 7.53341 0.781178
\(94\) 0 0
\(95\) 1.89134 0.194047
\(96\) 0 0
\(97\) 5.85021 0.593999 0.296999 − 0.954878i \(-0.404014\pi\)
0.296999 + 0.954878i \(0.404014\pi\)
\(98\) 0 0
\(99\) −3.51396 −0.353167
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.1.1 3
4.3 odd 2 7616.2.a.ba.1.3 3
8.3 odd 2 1904.2.a.p.1.1 3
8.5 even 2 952.2.a.d.1.3 ✓ 3
24.5 odd 2 8568.2.a.z.1.3 3
56.13 odd 2 6664.2.a.l.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.a.d.1.3 ✓ 3 8.5 even 2
1904.2.a.p.1.1 3 8.3 odd 2
6664.2.a.l.1.1 3 56.13 odd 2
7616.2.a.ba.1.3 3 4.3 odd 2
7616.2.a.bg.1.1 3 1.1 even 1 trivial
8568.2.a.z.1.3 3 24.5 odd 2