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 = [5,0,0,0,0,0,4,0,3,0,0] 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: 5.5.521397.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 9x^{3} - 3x^{2} + 18x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.61696\) 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)
 
\(f(q)\) \(=\) \(q+2.61696 q^{3} +3.83744 q^{7} +3.84849 q^{9} -0.508005 q^{11} +1.01106 q^{13} +1.44705 q^{17} -0.508005 q^{19} +10.0424 q^{21} -1.00000 q^{23} +2.22047 q^{27} +7.51040 q^{29} -0.439038 q^{31} -1.32943 q^{33} -7.02642 q^{37} +2.64590 q^{39} +5.47041 q^{41} +6.72592 q^{43} -2.64098 q^{47} +7.72592 q^{49} +3.78688 q^{51} +4.77648 q^{53} -1.32943 q^{57} +3.85345 q^{59} -9.05844 q^{61} +14.7683 q^{63} +3.45696 q^{67} -2.61696 q^{69} -2.73649 q^{71} -9.21300 q^{73} -1.94944 q^{77} +10.5504 q^{79} -5.73458 q^{81} -1.40211 q^{83} +19.6544 q^{87} +6.77086 q^{89} +3.87986 q^{91} -1.14895 q^{93} -0.313420 q^{97} -1.95506 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 4 q^{7} + 3 q^{9} + 4 q^{13} + 6 q^{17} - 5 q^{23} + 9 q^{27} + 12 q^{29} - 18 q^{31} + 6 q^{33} + 10 q^{37} + 9 q^{39} - 6 q^{41} + 10 q^{43} + 22 q^{47} + 15 q^{49} - 6 q^{51} + 10 q^{53} + 6 q^{57}+ \cdots - 6 q^{99}+O(q^{100}) \) 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
\(3\) 2.61696 1.51090 0.755452 − 0.655204i \(-0.227417\pi\)
0.755452 + 0.655204i \(0.227417\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.83744 1.45041 0.725207 − 0.688531i \(-0.241744\pi\)
0.725207 + 0.688531i \(0.241744\pi\)
\(8\) 0 0
\(9\) 3.84849 1.28283
\(10\) 0 0
\(11\) −0.508005 −0.153169 −0.0765847 − 0.997063i \(-0.524402\pi\)
−0.0765847 + 0.997063i \(0.524402\pi\)
\(12\) 0 0
\(13\) 1.01106 0.280417 0.140208 − 0.990122i \(-0.455223\pi\)
0.140208 + 0.990122i \(0.455223\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.44705 0.350961 0.175481 − 0.984483i \(-0.443852\pi\)
0.175481 + 0.984483i \(0.443852\pi\)
\(18\) 0 0
\(19\) −0.508005 −0.116544 −0.0582722 − 0.998301i \(-0.518559\pi\)
−0.0582722 + 0.998301i \(0.518559\pi\)
\(20\) 0 0
\(21\) 10.0424 2.19144
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.22047 0.427330
\(28\) 0 0
\(29\) 7.51040 1.39465 0.697323 − 0.716757i \(-0.254374\pi\)
0.697323 + 0.716757i \(0.254374\pi\)
\(30\) 0 0
\(31\) −0.439038 −0.0788536 −0.0394268 − 0.999222i \(-0.512553\pi\)
−0.0394268 + 0.999222i \(0.512553\pi\)
\(32\) 0 0
\(33\) −1.32943 −0.231424
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.02642 −1.15514 −0.577568 − 0.816343i \(-0.695998\pi\)
−0.577568 + 0.816343i \(0.695998\pi\)
\(38\) 0 0
\(39\) 2.64590 0.423683
\(40\) 0 0
\(41\) 5.47041 0.854335 0.427167 − 0.904173i \(-0.359512\pi\)
0.427167 + 0.904173i \(0.359512\pi\)
\(42\) 0 0
\(43\) 6.72592 1.02569 0.512847 − 0.858480i \(-0.328591\pi\)
0.512847 + 0.858480i \(0.328591\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.64098 −0.385227 −0.192614 − 0.981275i \(-0.561696\pi\)
−0.192614 + 0.981275i \(0.561696\pi\)
\(48\) 0 0
\(49\) 7.72592 1.10370
\(50\) 0 0
\(51\) 3.78688 0.530269
\(52\) 0 0
\(53\) 4.77648 0.656100 0.328050 − 0.944660i \(-0.393609\pi\)
0.328050 + 0.944660i \(0.393609\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.32943 −0.176087
\(58\) 0 0
\(59\) 3.85345 0.501676 0.250838 − 0.968029i \(-0.419294\pi\)
0.250838 + 0.968029i \(0.419294\pi\)
\(60\) 0 0
\(61\) −9.05844 −1.15981 −0.579907 − 0.814683i \(-0.696911\pi\)
−0.579907 + 0.814683i \(0.696911\pi\)
\(62\) 0 0
\(63\) 14.7683 1.86064
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.45696 0.422335 0.211167 − 0.977450i \(-0.432273\pi\)
0.211167 + 0.977450i \(0.432273\pi\)
\(68\) 0 0
\(69\) −2.61696 −0.315045
\(70\) 0 0
\(71\) −2.73649 −0.324762 −0.162381 − 0.986728i \(-0.551917\pi\)
−0.162381 + 0.986728i \(0.551917\pi\)
\(72\) 0 0
\(73\) −9.21300 −1.07830 −0.539150 − 0.842210i \(-0.681254\pi\)
−0.539150 + 0.842210i \(0.681254\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.94944 −0.222159
\(78\) 0 0
\(79\) 10.5504 1.18702 0.593508 − 0.804828i \(-0.297742\pi\)
0.593508 + 0.804828i \(0.297742\pi\)
\(80\) 0 0
\(81\) −5.73458 −0.637176
\(82\) 0 0
\(83\) −1.40211 −0.153901 −0.0769505 − 0.997035i \(-0.524518\pi\)
−0.0769505 + 0.997035i \(0.524518\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 19.6544 2.10718
\(88\) 0 0
\(89\) 6.77086 0.717710 0.358855 − 0.933393i \(-0.383167\pi\)
0.358855 + 0.933393i \(0.383167\pi\)
\(90\) 0 0
\(91\) 3.87986 0.406720
\(92\) 0 0
\(93\) −1.14895 −0.119140
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.313420 −0.0318230 −0.0159115 − 0.999873i \(-0.505065\pi\)
−0.0159115 + 0.999873i \(0.505065\pi\)
\(98\) 0 0
\(99\) −1.95506 −0.196490
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.bf.1.5 yes 5
4.3 odd 2 9200.2.a.ct.1.1 5
5.2 odd 4 4600.2.e.w.4049.1 10
5.3 odd 4 4600.2.e.w.4049.10 10
5.4 even 2 4600.2.a.bd.1.1 ✓ 5
20.19 odd 2 9200.2.a.cv.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bd.1.1 ✓ 5 5.4 even 2
4600.2.a.bf.1.5 yes 5 1.1 even 1 trivial
4600.2.e.w.4049.1 10 5.2 odd 4
4600.2.e.w.4049.10 10 5.3 odd 4
9200.2.a.ct.1.1 5 4.3 odd 2
9200.2.a.cv.1.5 5 20.19 odd 2