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Properties

Label 4600.2.e.q.4049.3
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 4049.3
Root \(-1.86081i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.q.4049.4

$q$-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)
 
\(f(q)\) \(=\) \(q-1.46260i q^{3} +1.53740i q^{7} +0.860806 q^{9} -0.860806 q^{11} -0.139194i q^{13} +5.50761i q^{17} -5.25901 q^{19} +2.24860 q^{21} +1.00000i q^{23} -5.64681i q^{27} -9.76663 q^{29} +6.78600 q^{31} +1.25901i q^{33} -12.0900i q^{37} -0.203585 q^{39} -9.98062 q^{41} -11.4432i q^{43} -2.32340i q^{47} +4.63640 q^{49} +8.05543 q^{51} -0.149606i q^{53} +7.69182i q^{57} +11.0152 q^{59} +4.43281 q^{61} +1.32340i q^{63} -10.4972i q^{67} +1.46260 q^{69} +7.31299 q^{71} -7.11982i q^{73} -1.32340i q^{77} -6.79641 q^{79} -5.67660 q^{81} -10.6468i q^{83} +14.2847i q^{87} +17.2936 q^{89} +0.213997 q^{91} -9.92520i q^{93} +1.93561i q^{97} -0.740987 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9} + 6 q^{11} - 14 q^{19} - 12 q^{21} + 2 q^{29} + 20 q^{31} - 14 q^{39} - 20 q^{41} - 12 q^{49} + 18 q^{51} - 20 q^{59} - 26 q^{61} + 4 q^{69} + 20 q^{71} - 28 q^{79} - 50 q^{81} + 40 q^{89}+ \cdots - 22 q^{99}+O(q^{100}) \) 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
\(3\) − 1.46260i − 0.844432i −0.906495 − 0.422216i \(-0.861252\pi\)
0.906495 − 0.422216i \(-0.138748\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.53740i 0.581083i 0.956862 + 0.290542i \(0.0938355\pi\)
−0.956862 + 0.290542i \(0.906165\pi\)
\(8\) 0 0
\(9\) 0.860806 0.286935
\(10\) 0 0
\(11\) −0.860806 −0.259543 −0.129771 − 0.991544i \(-0.541424\pi\)
−0.129771 + 0.991544i \(0.541424\pi\)
\(12\) 0 0
\(13\) − 0.139194i − 0.0386055i −0.999814 − 0.0193028i \(-0.993855\pi\)
0.999814 − 0.0193028i \(-0.00614464\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.50761i 1.33579i 0.744254 + 0.667896i \(0.232805\pi\)
−0.744254 + 0.667896i \(0.767195\pi\)
\(18\) 0 0
\(19\) −5.25901 −1.20650 −0.603250 − 0.797552i \(-0.706128\pi\)
−0.603250 + 0.797552i \(0.706128\pi\)
\(20\) 0 0
\(21\) 2.24860 0.490685
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 5.64681i − 1.08673i
\(28\) 0 0
\(29\) −9.76663 −1.81362 −0.906809 − 0.421543i \(-0.861489\pi\)
−0.906809 + 0.421543i \(0.861489\pi\)
\(30\) 0 0
\(31\) 6.78600 1.21880 0.609401 − 0.792862i \(-0.291410\pi\)
0.609401 + 0.792862i \(0.291410\pi\)
\(32\) 0 0
\(33\) 1.25901i 0.219166i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 12.0900i − 1.98759i −0.111232 − 0.993795i \(-0.535480\pi\)
0.111232 − 0.993795i \(-0.464520\pi\)
\(38\) 0 0
\(39\) −0.203585 −0.0325997
\(40\) 0 0
\(41\) −9.98062 −1.55871 −0.779356 − 0.626582i \(-0.784454\pi\)
−0.779356 + 0.626582i \(0.784454\pi\)
\(42\) 0 0
\(43\) − 11.4432i − 1.74508i −0.488547 − 0.872538i \(-0.662473\pi\)
0.488547 − 0.872538i \(-0.337527\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 2.32340i − 0.338903i −0.985538 − 0.169452i \(-0.945800\pi\)
0.985538 − 0.169452i \(-0.0541997\pi\)
\(48\) 0 0
\(49\) 4.63640 0.662342
\(50\) 0 0
\(51\) 8.05543 1.12799
\(52\) 0 0
\(53\) − 0.149606i − 0.0205500i −0.999947 − 0.0102750i \(-0.996729\pi\)
0.999947 − 0.0102750i \(-0.00327069\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 7.69182i 1.01881i
\(58\) 0 0
\(59\) 11.0152 1.43406 0.717030 − 0.697042i \(-0.245501\pi\)
0.717030 + 0.697042i \(0.245501\pi\)
\(60\) 0 0
\(61\) 4.43281 0.567563 0.283782 − 0.958889i \(-0.408411\pi\)
0.283782 + 0.958889i \(0.408411\pi\)
\(62\) 0 0
\(63\) 1.32340i 0.166733i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 10.4972i − 1.28244i −0.767358 − 0.641219i \(-0.778429\pi\)
0.767358 − 0.641219i \(-0.221571\pi\)
\(68\) 0 0
\(69\) 1.46260 0.176076
\(70\) 0 0
\(71\) 7.31299 0.867892 0.433946 − 0.900939i \(-0.357121\pi\)
0.433946 + 0.900939i \(0.357121\pi\)
\(72\) 0 0
\(73\) − 7.11982i − 0.833312i −0.909064 − 0.416656i \(-0.863202\pi\)
0.909064 − 0.416656i \(-0.136798\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.32340i − 0.150816i
\(78\) 0 0
\(79\) −6.79641 −0.764656 −0.382328 − 0.924027i \(-0.624878\pi\)
−0.382328 + 0.924027i \(0.624878\pi\)
\(80\) 0 0
\(81\) −5.67660 −0.630733
\(82\) 0 0
\(83\) − 10.6468i − 1.16864i −0.811524 − 0.584320i \(-0.801361\pi\)
0.811524 − 0.584320i \(-0.198639\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 14.2847i 1.53148i
\(88\) 0 0
\(89\) 17.2936 1.83312 0.916560 − 0.399897i \(-0.130954\pi\)
0.916560 + 0.399897i \(0.130954\pi\)
\(90\) 0 0
\(91\) 0.213997 0.0224330
\(92\) 0 0
\(93\) − 9.92520i − 1.02919i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.93561i 0.196531i 0.995160 + 0.0982657i \(0.0313295\pi\)
−0.995160 + 0.0982657i \(0.968671\pi\)
\(98\) 0 0
\(99\) −0.740987 −0.0744720
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.4049.3 6
5.2 odd 4 4600.2.a.v.1.2 3
5.3 odd 4 920.2.a.i.1.2 ✓ 3
5.4 even 2 inner 4600.2.e.q.4049.4 6
15.8 even 4 8280.2.a.bl.1.2 3
20.3 even 4 1840.2.a.q.1.2 3
20.7 even 4 9200.2.a.ci.1.2 3
40.3 even 4 7360.2.a.cf.1.2 3
40.13 odd 4 7360.2.a.bw.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.i.1.2 ✓ 3 5.3 odd 4
1840.2.a.q.1.2 3 20.3 even 4
4600.2.a.v.1.2 3 5.2 odd 4
4600.2.e.q.4049.3 6 1.1 even 1 trivial
4600.2.e.q.4049.4 6 5.4 even 2 inner
7360.2.a.bw.1.2 3 40.13 odd 4
7360.2.a.cf.1.2 3 40.3 even 4
8280.2.a.bl.1.2 3 15.8 even 4
9200.2.a.ci.1.2 3 20.7 even 4