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Properties

Label 4600.2.e.m.4049.2
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $4$
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 = [4,0,0,0,0,0,0,0,-6,0,-16,0,0,0,0,0,0,0,-16] 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: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.2
Root \(-1.56155i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.m.4049.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.56155i q^{3} +3.12311i q^{7} +0.561553 q^{9} -4.00000 q^{11} +3.56155i q^{13} -5.12311i q^{17} -4.00000 q^{19} +4.87689 q^{21} +1.00000i q^{23} -5.56155i q^{27} +4.43845 q^{29} +5.56155 q^{31} +6.24621i q^{33} -1.12311i q^{37} +5.56155 q^{39} -3.56155 q^{41} -0.876894i q^{43} -8.68466i q^{47} -2.75379 q^{49} -8.00000 q^{51} +12.2462i q^{53} +6.24621i q^{57} -10.2462 q^{59} +2.87689 q^{61} +1.75379i q^{63} +10.2462i q^{67} +1.56155 q^{69} -8.68466 q^{71} +12.4384i q^{73} -12.4924i q^{77} -6.24621 q^{79} -7.00000 q^{81} +12.0000i q^{83} -6.93087i q^{87} -10.0000 q^{89} -11.1231 q^{91} -8.68466i q^{93} -0.246211i q^{97} -2.24621 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{9} - 16 q^{11} - 16 q^{19} + 36 q^{21} + 26 q^{29} + 14 q^{31} + 14 q^{39} - 6 q^{41} - 44 q^{49} - 32 q^{51} - 8 q^{59} + 28 q^{61} - 2 q^{69} - 10 q^{71} + 8 q^{79} - 28 q^{81} - 40 q^{89}+ \cdots + 24 q^{99}+O(q^{100}) \) 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
\(3\) − 1.56155i − 0.901563i −0.892634 − 0.450781i \(-0.851145\pi\)
0.892634 − 0.450781i \(-0.148855\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.12311i 1.18042i 0.807249 + 0.590211i \(0.200956\pi\)
−0.807249 + 0.590211i \(0.799044\pi\)
\(8\) 0 0
\(9\) 0.561553 0.187184
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 3.56155i 0.987797i 0.869520 + 0.493899i \(0.164429\pi\)
−0.869520 + 0.493899i \(0.835571\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 5.12311i − 1.24254i −0.783598 − 0.621268i \(-0.786618\pi\)
0.783598 − 0.621268i \(-0.213382\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 4.87689 1.06423
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 5.56155i − 1.07032i
\(28\) 0 0
\(29\) 4.43845 0.824199 0.412099 − 0.911139i \(-0.364796\pi\)
0.412099 + 0.911139i \(0.364796\pi\)
\(30\) 0 0
\(31\) 5.56155 0.998884 0.499442 − 0.866347i \(-0.333538\pi\)
0.499442 + 0.866347i \(0.333538\pi\)
\(32\) 0 0
\(33\) 6.24621i 1.08733i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 1.12311i − 0.184637i −0.995730 − 0.0923187i \(-0.970572\pi\)
0.995730 − 0.0923187i \(-0.0294279\pi\)
\(38\) 0 0
\(39\) 5.56155 0.890561
\(40\) 0 0
\(41\) −3.56155 −0.556221 −0.278111 − 0.960549i \(-0.589708\pi\)
−0.278111 + 0.960549i \(0.589708\pi\)
\(42\) 0 0
\(43\) − 0.876894i − 0.133725i −0.997762 − 0.0668626i \(-0.978701\pi\)
0.997762 − 0.0668626i \(-0.0212989\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 8.68466i − 1.26679i −0.773830 − 0.633394i \(-0.781661\pi\)
0.773830 − 0.633394i \(-0.218339\pi\)
\(48\) 0 0
\(49\) −2.75379 −0.393398
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) 0 0
\(53\) 12.2462i 1.68215i 0.540921 + 0.841073i \(0.318076\pi\)
−0.540921 + 0.841073i \(0.681924\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.24621i 0.827331i
\(58\) 0 0
\(59\) −10.2462 −1.33394 −0.666972 − 0.745083i \(-0.732410\pi\)
−0.666972 + 0.745083i \(0.732410\pi\)
\(60\) 0 0
\(61\) 2.87689 0.368349 0.184174 − 0.982894i \(-0.441039\pi\)
0.184174 + 0.982894i \(0.441039\pi\)
\(62\) 0 0
\(63\) 1.75379i 0.220957i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 10.2462i 1.25177i 0.779914 + 0.625887i \(0.215263\pi\)
−0.779914 + 0.625887i \(0.784737\pi\)
\(68\) 0 0
\(69\) 1.56155 0.187989
\(70\) 0 0
\(71\) −8.68466 −1.03068 −0.515340 − 0.856986i \(-0.672334\pi\)
−0.515340 + 0.856986i \(0.672334\pi\)
\(72\) 0 0
\(73\) 12.4384i 1.45581i 0.685678 + 0.727905i \(0.259506\pi\)
−0.685678 + 0.727905i \(0.740494\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 12.4924i − 1.42364i
\(78\) 0 0
\(79\) −6.24621 −0.702754 −0.351377 − 0.936234i \(-0.614286\pi\)
−0.351377 + 0.936234i \(0.614286\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 6.93087i − 0.743067i
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −11.1231 −1.16602
\(92\) 0 0
\(93\) − 8.68466i − 0.900557i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 0.246211i − 0.0249990i −0.999922 − 0.0124995i \(-0.996021\pi\)
0.999922 − 0.0124995i \(-0.00397881\pi\)
\(98\) 0 0
\(99\) −2.24621 −0.225753
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.m.4049.2 4
5.2 odd 4 920.2.a.f.1.1 ✓ 2
5.3 odd 4 4600.2.a.r.1.2 2
5.4 even 2 inner 4600.2.e.m.4049.3 4
15.2 even 4 8280.2.a.bb.1.1 2
20.3 even 4 9200.2.a.bx.1.1 2
20.7 even 4 1840.2.a.k.1.2 2
40.27 even 4 7360.2.a.bm.1.1 2
40.37 odd 4 7360.2.a.bj.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.f.1.1 ✓ 2 5.2 odd 4
1840.2.a.k.1.2 2 20.7 even 4
4600.2.a.r.1.2 2 5.3 odd 4
4600.2.e.m.4049.2 4 1.1 even 1 trivial
4600.2.e.m.4049.3 4 5.4 even 2 inner
7360.2.a.bj.1.2 2 40.37 odd 4
7360.2.a.bm.1.1 2 40.27 even 4
8280.2.a.bb.1.1 2 15.2 even 4
9200.2.a.bx.1.1 2 20.3 even 4