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Properties

Label 4600.2.e.q.4049.1
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $6$
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] = [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 4049.1
Root \(2.11491i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.q.4049.6

$q$-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 \(-0.746952\pi\)
0.700303 − 0.713846i \(-0.253048\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.527166i 0.199250i 0.995025 + 0.0996250i \(0.0317643\pi\)
−0.995025 + 0.0996250i \(0.968236\pi\)
\(8\) 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 \(-0.806697\pi\)
0.821204 − 0.570635i \(-0.193303\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.39905i − 1.06693i −0.845823 − 0.533464i \(-0.820890\pi\)
0.845823 − 0.533464i \(-0.179110\pi\)
\(18\) 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 \(0.295663\pi\)
−0.598753 + 0.800934i \(0.704337\pi\)
\(38\) 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 \(0.110442\pi\)
−0.940410 + 0.340044i \(0.889558\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.642074i 0.0936561i 0.998903 + 0.0468280i \(0.0149113\pi\)
−0.998903 + 0.0468280i \(0.985089\pi\)
\(48\) 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 \(0.0861176\pi\)
−0.963625 + 0.267258i \(0.913882\pi\)
\(54\) 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 \(-0.823778\pi\)
0.850628 − 0.525768i \(-0.176222\pi\)
\(68\) 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 \(0.110578\pi\)
−0.940264 + 0.340447i \(0.889422\pi\)
\(74\) 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 \(-0.916668\pi\)
0.965927 − 0.258816i \(-0.0833323\pi\)
\(84\) 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 \(-0.933908\pi\)
0.978522 − 0.206144i \(-0.0660917\pi\)
\(98\) 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.4049.1 6
5.2 odd 4 4600.2.a.v.1.1 3
5.3 odd 4 920.2.a.i.1.3 ✓ 3
5.4 even 2 inner 4600.2.e.q.4049.6 6
15.8 even 4 8280.2.a.bl.1.1 3
20.3 even 4 1840.2.a.q.1.1 3
20.7 even 4 9200.2.a.ci.1.3 3
40.3 even 4 7360.2.a.cf.1.3 3
40.13 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 5.3 odd 4
1840.2.a.q.1.1 3 20.3 even 4
4600.2.a.v.1.1 3 5.2 odd 4
4600.2.e.q.4049.1 6 1.1 even 1 trivial
4600.2.e.q.4049.6 6 5.4 even 2 inner
7360.2.a.bw.1.1 3 40.13 odd 4
7360.2.a.cf.1.3 3 40.3 even 4
8280.2.a.bl.1.1 3 15.8 even 4
9200.2.a.ci.1.3 3 20.7 even 4