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Properties

Label 4600.2.a.be.1.4
Level $4600$
Weight $2$
Character 4600.1
Self dual yes
Analytic conductor $36.731$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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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(1,4600)] 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("4600.1"); 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, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,0,0,0,0,2,0,13,0,-1] 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: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.13955077.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 14x^{3} - x^{2} + 32x + 16 \) 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)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.31091\) of defining polynomial
Character \(\chi\) \(=\) 4600.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.31091 q^{3} +4.66212 q^{7} -1.28151 q^{9} +2.23020 q^{11} +2.80072 q^{13} -7.63271 q^{17} -1.36222 q^{19} +6.11163 q^{21} +1.00000 q^{23} -5.61268 q^{27} +8.94362 q^{29} -1.58140 q^{31} +2.92360 q^{33} +1.40251 q^{37} +3.67149 q^{39} +10.7134 q^{41} +7.26391 q^{47} +14.7353 q^{49} -10.0058 q^{51} +8.38212 q^{53} -1.78575 q^{57} -4.88331 q^{59} +4.33282 q^{61} -5.97454 q^{63} -8.54355 q^{67} +1.31091 q^{69} +8.81604 q^{71} +5.26391 q^{73} +10.3974 q^{77} +7.08222 q^{79} -3.51322 q^{81} +4.59749 q^{83} +11.7243 q^{87} -4.70241 q^{89} +13.0573 q^{91} -2.07308 q^{93} -16.5160 q^{97} -2.85802 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{7} + 13 q^{9} - q^{11} - 4 q^{13} - 4 q^{17} + 7 q^{19} + 6 q^{21} + 5 q^{23} - 3 q^{27} + 4 q^{29} + 19 q^{31} - 17 q^{33} - 15 q^{37} + 19 q^{39} + 25 q^{41} + 11 q^{47} + 25 q^{49} + 19 q^{51}+ \cdots - 65 q^{99}+O(q^{100}) \) 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
\(3\) 1.31091 0.756856 0.378428 − 0.925631i \(-0.376465\pi\)
0.378428 + 0.925631i \(0.376465\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.66212 1.76211 0.881057 − 0.473010i \(-0.156832\pi\)
0.881057 + 0.473010i \(0.156832\pi\)
\(8\) 0 0
\(9\) −1.28151 −0.427169
\(10\) 0 0
\(11\) 2.23020 0.672430 0.336215 − 0.941785i \(-0.390853\pi\)
0.336215 + 0.941785i \(0.390853\pi\)
\(12\) 0 0
\(13\) 2.80072 0.776779 0.388389 − 0.921495i \(-0.373032\pi\)
0.388389 + 0.921495i \(0.373032\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.63271 −1.85120 −0.925602 − 0.378498i \(-0.876441\pi\)
−0.925602 + 0.378498i \(0.876441\pi\)
\(18\) 0 0
\(19\) −1.36222 −0.312515 −0.156258 − 0.987716i \(-0.549943\pi\)
−0.156258 + 0.987716i \(0.549943\pi\)
\(20\) 0 0
\(21\) 6.11163 1.33367
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.61268 −1.08016
\(28\) 0 0
\(29\) 8.94362 1.66079 0.830395 − 0.557176i \(-0.188115\pi\)
0.830395 + 0.557176i \(0.188115\pi\)
\(30\) 0 0
\(31\) −1.58140 −0.284028 −0.142014 − 0.989865i \(-0.545358\pi\)
−0.142014 + 0.989865i \(0.545358\pi\)
\(32\) 0 0
\(33\) 2.92360 0.508933
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.40251 0.230572 0.115286 − 0.993332i \(-0.463222\pi\)
0.115286 + 0.993332i \(0.463222\pi\)
\(38\) 0 0
\(39\) 3.67149 0.587909
\(40\) 0 0
\(41\) 10.7134 1.67316 0.836578 − 0.547848i \(-0.184553\pi\)
0.836578 + 0.547848i \(0.184553\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.26391 1.05955 0.529775 − 0.848138i \(-0.322276\pi\)
0.529775 + 0.848138i \(0.322276\pi\)
\(48\) 0 0
\(49\) 14.7353 2.10505
\(50\) 0 0
\(51\) −10.0058 −1.40109
\(52\) 0 0
\(53\) 8.38212 1.15137 0.575686 − 0.817671i \(-0.304735\pi\)
0.575686 + 0.817671i \(0.304735\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.78575 −0.236529
\(58\) 0 0
\(59\) −4.88331 −0.635752 −0.317876 − 0.948132i \(-0.602970\pi\)
−0.317876 + 0.948132i \(0.602970\pi\)
\(60\) 0 0
\(61\) 4.33282 0.554760 0.277380 − 0.960760i \(-0.410534\pi\)
0.277380 + 0.960760i \(0.410534\pi\)
\(62\) 0 0
\(63\) −5.97454 −0.752721
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −8.54355 −1.04376 −0.521880 − 0.853019i \(-0.674769\pi\)
−0.521880 + 0.853019i \(0.674769\pi\)
\(68\) 0 0
\(69\) 1.31091 0.157815
\(70\) 0 0
\(71\) 8.81604 1.04627 0.523136 − 0.852249i \(-0.324762\pi\)
0.523136 + 0.852249i \(0.324762\pi\)
\(72\) 0 0
\(73\) 5.26391 0.616095 0.308047 − 0.951371i \(-0.400325\pi\)
0.308047 + 0.951371i \(0.400325\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.3974 1.18490
\(78\) 0 0
\(79\) 7.08222 0.796812 0.398406 − 0.917209i \(-0.369563\pi\)
0.398406 + 0.917209i \(0.369563\pi\)
\(80\) 0 0
\(81\) −3.51322 −0.390357
\(82\) 0 0
\(83\) 4.59749 0.504640 0.252320 − 0.967644i \(-0.418806\pi\)
0.252320 + 0.967644i \(0.418806\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 11.7243 1.25698
\(88\) 0 0
\(89\) −4.70241 −0.498454 −0.249227 − 0.968445i \(-0.580177\pi\)
−0.249227 + 0.968445i \(0.580177\pi\)
\(90\) 0 0
\(91\) 13.0573 1.36877
\(92\) 0 0
\(93\) −2.07308 −0.214968
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −16.5160 −1.67695 −0.838474 − 0.544942i \(-0.816552\pi\)
−0.838474 + 0.544942i \(0.816552\pi\)
\(98\) 0 0
\(99\) −2.85802 −0.287241
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.a.be.1.4 5
4.3 odd 2 9200.2.a.cu.1.2 5
5.2 odd 4 4600.2.e.u.4049.4 10
5.3 odd 4 4600.2.e.u.4049.7 10
5.4 even 2 920.2.a.j.1.2 ✓ 5
15.14 odd 2 8280.2.a.bs.1.1 5
20.19 odd 2 1840.2.a.v.1.4 5
40.19 odd 2 7360.2.a.cp.1.2 5
40.29 even 2 7360.2.a.co.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.j.1.2 ✓ 5 5.4 even 2
1840.2.a.v.1.4 5 20.19 odd 2
4600.2.a.be.1.4 5 1.1 even 1 trivial
4600.2.e.u.4049.4 10 5.2 odd 4
4600.2.e.u.4049.7 10 5.3 odd 4
7360.2.a.co.1.4 5 40.29 even 2
7360.2.a.cp.1.2 5 40.19 odd 2
8280.2.a.bs.1.1 5 15.14 odd 2
9200.2.a.cu.1.2 5 4.3 odd 2