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Properties

Label 4600.2.e.o.4049.4
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,-6,0,4,0,0,0,0,0,0,0,-4] 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 184)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.4
Root \(2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.o.4049.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.56155i q^{3} -3.56155 q^{9} +5.12311 q^{11} -4.56155i q^{13} -3.12311i q^{17} -5.12311 q^{19} +1.00000i q^{23} -1.43845i q^{27} +0.561553 q^{29} -6.56155 q^{31} +13.1231i q^{33} -8.24621i q^{37} +11.6847 q^{39} +10.8078 q^{41} +8.00000i q^{43} +11.6847i q^{47} +7.00000 q^{49} +8.00000 q^{51} -2.00000i q^{53} -13.1231i q^{57} +6.24621 q^{59} +12.2462 q^{61} -5.12311i q^{67} -2.56155 q^{69} +9.43845 q^{71} +2.31534i q^{73} +5.12311 q^{79} -7.00000 q^{81} +2.24621i q^{83} +1.43845i q^{87} +13.3693 q^{89} -16.8078i q^{93} -13.3693i q^{97} -18.2462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{9} + 4 q^{11} - 4 q^{19} - 6 q^{29} - 18 q^{31} + 22 q^{39} + 2 q^{41} + 28 q^{49} + 32 q^{51} - 8 q^{59} + 16 q^{61} - 2 q^{69} + 46 q^{71} + 4 q^{79} - 28 q^{81} + 4 q^{89} - 40 q^{99}+O(q^{100}) \) 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
\(3\) 2.56155i 1.47891i 0.673204 + 0.739457i \(0.264917\pi\)
−0.673204 + 0.739457i \(0.735083\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −3.56155 −1.18718
\(10\) 0 0
\(11\) 5.12311 1.54467 0.772337 − 0.635213i \(-0.219088\pi\)
0.772337 + 0.635213i \(0.219088\pi\)
\(12\) 0 0
\(13\) − 4.56155i − 1.26515i −0.774500 − 0.632574i \(-0.781999\pi\)
0.774500 − 0.632574i \(-0.218001\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.12311i − 0.757464i −0.925506 − 0.378732i \(-0.876360\pi\)
0.925506 − 0.378732i \(-0.123640\pi\)
\(18\) 0 0
\(19\) −5.12311 −1.17532 −0.587661 − 0.809108i \(-0.699951\pi\)
−0.587661 + 0.809108i \(0.699951\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.43845i − 0.276829i
\(28\) 0 0
\(29\) 0.561553 0.104278 0.0521389 − 0.998640i \(-0.483396\pi\)
0.0521389 + 0.998640i \(0.483396\pi\)
\(30\) 0 0
\(31\) −6.56155 −1.17849 −0.589245 − 0.807955i \(-0.700575\pi\)
−0.589245 + 0.807955i \(0.700575\pi\)
\(32\) 0 0
\(33\) 13.1231i 2.28444i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 8.24621i − 1.35567i −0.735215 − 0.677834i \(-0.762919\pi\)
0.735215 − 0.677834i \(-0.237081\pi\)
\(38\) 0 0
\(39\) 11.6847 1.87104
\(40\) 0 0
\(41\) 10.8078 1.68789 0.843945 − 0.536430i \(-0.180228\pi\)
0.843945 + 0.536430i \(0.180228\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.6847i 1.70438i 0.523230 + 0.852191i \(0.324727\pi\)
−0.523230 + 0.852191i \(0.675273\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 8.00000 1.12022
\(52\) 0 0
\(53\) − 2.00000i − 0.274721i −0.990521 − 0.137361i \(-0.956138\pi\)
0.990521 − 0.137361i \(-0.0438619\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 13.1231i − 1.73820i
\(58\) 0 0
\(59\) 6.24621 0.813187 0.406594 − 0.913609i \(-0.366716\pi\)
0.406594 + 0.913609i \(0.366716\pi\)
\(60\) 0 0
\(61\) 12.2462 1.56797 0.783983 − 0.620782i \(-0.213185\pi\)
0.783983 + 0.620782i \(0.213185\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 5.12311i − 0.625887i −0.949772 − 0.312943i \(-0.898685\pi\)
0.949772 − 0.312943i \(-0.101315\pi\)
\(68\) 0 0
\(69\) −2.56155 −0.308375
\(70\) 0 0
\(71\) 9.43845 1.12014 0.560069 − 0.828446i \(-0.310775\pi\)
0.560069 + 0.828446i \(0.310775\pi\)
\(72\) 0 0
\(73\) 2.31534i 0.270990i 0.990778 + 0.135495i \(0.0432625\pi\)
−0.990778 + 0.135495i \(0.956737\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.12311 0.576394 0.288197 − 0.957571i \(-0.406944\pi\)
0.288197 + 0.957571i \(0.406944\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 2.24621i 0.246554i 0.992372 + 0.123277i \(0.0393403\pi\)
−0.992372 + 0.123277i \(0.960660\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.43845i 0.154218i
\(88\) 0 0
\(89\) 13.3693 1.41714 0.708572 − 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 16.8078i − 1.74288i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 13.3693i − 1.35745i −0.734393 − 0.678724i \(-0.762533\pi\)
0.734393 − 0.678724i \(-0.237467\pi\)
\(98\) 0 0
\(99\) −18.2462 −1.83381
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.o.4049.4 4
5.2 odd 4 4600.2.a.s.1.2 2
5.3 odd 4 184.2.a.e.1.1 ✓ 2
5.4 even 2 inner 4600.2.e.o.4049.1 4
15.8 even 4 1656.2.a.j.1.1 2
20.3 even 4 368.2.a.i.1.2 2
20.7 even 4 9200.2.a.br.1.1 2
35.13 even 4 9016.2.a.w.1.2 2
40.3 even 4 1472.2.a.p.1.1 2
40.13 odd 4 1472.2.a.u.1.2 2
60.23 odd 4 3312.2.a.t.1.2 2
115.68 even 4 4232.2.a.o.1.1 2
460.183 odd 4 8464.2.a.bd.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.a.e.1.1 ✓ 2 5.3 odd 4
368.2.a.i.1.2 2 20.3 even 4
1472.2.a.p.1.1 2 40.3 even 4
1472.2.a.u.1.2 2 40.13 odd 4
1656.2.a.j.1.1 2 15.8 even 4
3312.2.a.t.1.2 2 60.23 odd 4
4232.2.a.o.1.1 2 115.68 even 4
4600.2.a.s.1.2 2 5.2 odd 4
4600.2.e.o.4049.1 4 5.4 even 2 inner
4600.2.e.o.4049.4 4 1.1 even 1 trivial
8464.2.a.bd.1.2 2 460.183 odd 4
9016.2.a.w.1.2 2 35.13 even 4
9200.2.a.br.1.1 2 20.7 even 4