Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4600,2,Mod(4049,4600)] 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("4600.4049"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,-6,0,6,0,0,0,0,0,0,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.3356224.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 8x^{4} + 16x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.6
Root \(-2.11491i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.q.4049.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.47283i q^{3} -0.527166i q^{7} -3.11491 q^{9} +3.11491 q^{11} +4.11491i q^{13} +4.39905i q^{17} +3.70265 q^{19} +1.30359 q^{21} -1.00000i q^{23} -0.284147i q^{27} +9.10170 q^{29} +4.83076 q^{31} +7.70265i q^{33} -9.74378i q^{37} -10.1755 q^{39} +6.93246 q^{41} -4.45963i q^{43} -0.642074i q^{47} +6.72210 q^{49} -10.8781 q^{51} -3.89134i q^{53} +9.15604i q^{57} -8.79811 q^{59} -3.45339 q^{61} +1.64207i q^{63} +8.60719i q^{67} +2.47283 q^{69} +12.3642 q^{71} -5.81756i q^{73} -1.64207i q^{77} +3.17548 q^{79} -8.64207 q^{81} +4.71585i q^{83} +22.5070i q^{87} +5.43171 q^{89} +2.16924 q^{91} +11.9457i q^{93} +4.06058i q^{97} -9.70265 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9} + 6 q^{11} - 14 q^{19} - 12 q^{21} + 2 q^{29} + 20 q^{31} - 14 q^{39} - 20 q^{41} - 12 q^{49} + 18 q^{51} - 20 q^{59} - 26 q^{61} + 4 q^{69} + 20 q^{71} - 28 q^{79} - 50 q^{81} + 40 q^{89}+ \cdots - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 2.47283i 1.42769i 0.700303 + 0.713846i \(0.253048\pi\)
−0.700303 + 0.713846i \(0.746952\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 0.527166i − 0.199250i −0.995025 − 0.0996250i \(-0.968236\pi\)
0.995025 − 0.0996250i \(-0.0317643\pi\)
\(8\) 0 0
\(9\) −3.11491 −1.03830
\(10\) 0 0
\(11\) 3.11491 0.939180 0.469590 − 0.882885i \(-0.344402\pi\)
0.469590 + 0.882885i \(0.344402\pi\)
\(12\) 0 0
\(13\) 4.11491i 1.14127i 0.821204 + 0.570635i \(0.193303\pi\)
−0.821204 + 0.570635i \(0.806697\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.39905i 1.06693i 0.845823 + 0.533464i \(0.179110\pi\)
−0.845823 + 0.533464i \(0.820890\pi\)
\(18\) 0 0
\(19\) 3.70265 0.849446 0.424723 − 0.905323i \(-0.360372\pi\)
0.424723 + 0.905323i \(0.360372\pi\)
\(20\) 0 0
\(21\) 1.30359 0.284468
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 0.284147i − 0.0546842i
\(28\) 0 0
\(29\) 9.10170 1.69014 0.845072 − 0.534653i \(-0.179558\pi\)
0.845072 + 0.534653i \(0.179558\pi\)
\(30\) 0 0
\(31\) 4.83076 0.867630 0.433815 − 0.901002i \(-0.357167\pi\)
0.433815 + 0.901002i \(0.357167\pi\)
\(32\) 0 0
\(33\) 7.70265i 1.34086i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 9.74378i − 1.60187i −0.598753 − 0.800934i \(-0.704337\pi\)
0.598753 − 0.800934i \(-0.295663\pi\)
\(38\) 0 0
\(39\) −10.1755 −1.62938
\(40\) 0 0
\(41\) 6.93246 1.08267 0.541334 − 0.840807i \(-0.317919\pi\)
0.541334 + 0.840807i \(0.317919\pi\)
\(42\) 0 0
\(43\) − 4.45963i − 0.680087i −0.940410 − 0.340044i \(-0.889558\pi\)
0.940410 − 0.340044i \(-0.110442\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 0.642074i − 0.0936561i −0.998903 − 0.0468280i \(-0.985089\pi\)
0.998903 − 0.0468280i \(-0.0149113\pi\)
\(48\) 0 0
\(49\) 6.72210 0.960299
\(50\) 0 0
\(51\) −10.8781 −1.52324
\(52\) 0 0
\(53\) − 3.89134i − 0.534516i −0.963625 − 0.267258i \(-0.913882\pi\)
0.963625 − 0.267258i \(-0.0861176\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 9.15604i 1.21275i
\(58\) 0 0
\(59\) −8.79811 −1.14542 −0.572708 − 0.819759i \(-0.694107\pi\)
−0.572708 + 0.819759i \(0.694107\pi\)
\(60\) 0 0
\(61\) −3.45339 −0.442161 −0.221080 − 0.975256i \(-0.570958\pi\)
−0.221080 + 0.975256i \(0.570958\pi\)
\(62\) 0 0
\(63\) 1.64207i 0.206882i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.60719i 1.05154i 0.850628 + 0.525768i \(0.176222\pi\)
−0.850628 + 0.525768i \(0.823778\pi\)
\(68\) 0 0
\(69\) 2.47283 0.297694
\(70\) 0 0
\(71\) 12.3642 1.46736 0.733678 − 0.679497i \(-0.237802\pi\)
0.733678 + 0.679497i \(0.237802\pi\)
\(72\) 0 0
\(73\) − 5.81756i − 0.680893i −0.940264 − 0.340447i \(-0.889422\pi\)
0.940264 − 0.340447i \(-0.110578\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.64207i − 0.187132i
\(78\) 0 0
\(79\) 3.17548 0.357270 0.178635 − 0.983915i \(-0.442832\pi\)
0.178635 + 0.983915i \(0.442832\pi\)
\(80\) 0 0
\(81\) −8.64207 −0.960230
\(82\) 0 0
\(83\) 4.71585i 0.517632i 0.965927 + 0.258816i \(0.0833323\pi\)
−0.965927 + 0.258816i \(0.916668\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 22.5070i 2.41300i
\(88\) 0 0
\(89\) 5.43171 0.575760 0.287880 − 0.957667i \(-0.407050\pi\)
0.287880 + 0.957667i \(0.407050\pi\)
\(90\) 0 0
\(91\) 2.16924 0.227398
\(92\) 0 0
\(93\) 11.9457i 1.23871i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.06058i 0.412289i 0.978522 + 0.206144i \(0.0660917\pi\)
−0.978522 + 0.206144i \(0.933908\pi\)
\(98\) 0 0
\(99\) −9.70265 −0.975153
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 4600.2.e.q.4049.6 6
5.2 odd 4 920.2.a.i.1.3 ✓ 3
5.3 odd 4 4600.2.a.v.1.1 3
5.4 even 2 inner 4600.2.e.q.4049.1 6
15.2 even 4 8280.2.a.bl.1.1 3
20.3 even 4 9200.2.a.ci.1.3 3
20.7 even 4 1840.2.a.q.1.1 3
40.27 even 4 7360.2.a.cf.1.3 3
40.37 odd 4 7360.2.a.bw.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.i.1.3 ✓ 3 5.2 odd 4
1840.2.a.q.1.1 3 20.7 even 4
4600.2.a.v.1.1 3 5.3 odd 4
4600.2.e.q.4049.1 6 5.4 even 2 inner
4600.2.e.q.4049.6 6 1.1 even 1 trivial
7360.2.a.bw.1.1 3 40.37 odd 4
7360.2.a.cf.1.3 3 40.27 even 4
8280.2.a.bl.1.1 3 15.2 even 4
9200.2.a.ci.1.3 3 20.3 even 4