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Properties

Label 4600.2.e.e.4049.1
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $1$
Dimension $2$
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 = [2,0,0,0,0,0,0,0,4,0,-8,0,0,0,0,0,0,0,-12] 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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 184)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.e.4049.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} -4.00000 q^{11} -5.00000i q^{13} +2.00000i q^{17} -6.00000 q^{19} -2.00000 q^{21} +1.00000i q^{23} -5.00000i q^{27} -1.00000 q^{29} -9.00000 q^{31} +4.00000i q^{33} +4.00000i q^{37} -5.00000 q^{39} +3.00000 q^{41} +8.00000i q^{43} +5.00000i q^{47} +3.00000 q^{49} +2.00000 q^{51} +6.00000i q^{53} +6.00000i q^{57} +4.00000 q^{59} -10.0000 q^{61} -4.00000i q^{63} +4.00000i q^{67} +1.00000 q^{69} -5.00000 q^{71} -15.0000i q^{73} +8.00000i q^{77} +6.00000 q^{79} +1.00000 q^{81} +6.00000i q^{83} +1.00000i q^{87} +8.00000 q^{89} -10.0000 q^{91} +9.00000i q^{93} -10.0000i q^{97} -8.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{9} - 8 q^{11} - 12 q^{19} - 4 q^{21} - 2 q^{29} - 18 q^{31} - 10 q^{39} + 6 q^{41} + 6 q^{49} + 4 q^{51} + 8 q^{59} - 20 q^{61} + 2 q^{69} - 10 q^{71} + 12 q^{79} + 2 q^{81} + 16 q^{89} - 20 q^{91}+ \cdots - 16 q^{99}+O(q^{100}) \) 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
\(3\) − 1.00000i − 0.577350i −0.957427 − 0.288675i \(-0.906785\pi\)
0.957427 − 0.288675i \(-0.0932147\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.00000i − 0.755929i −0.925820 − 0.377964i \(-0.876624\pi\)
0.925820 − 0.377964i \(-0.123376\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) − 5.00000i − 1.38675i −0.720577 − 0.693375i \(-0.756123\pi\)
0.720577 − 0.693375i \(-0.243877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 − 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 5.00000i − 0.962250i
\(28\) 0 0
\(29\) −1.00000 −0.185695 −0.0928477 − 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) −9.00000 −1.61645 −0.808224 − 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) 0 0
\(33\) 4.00000i 0.696311i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) 0 0
\(39\) −5.00000 −0.800641
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 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
\(47\) 5.00000i 0.729325i 0.931140 + 0.364662i \(0.118816\pi\)
−0.931140 + 0.364662i \(0.881184\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.00000i 0.794719i
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) − 4.00000i − 0.503953i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −5.00000 −0.593391 −0.296695 − 0.954972i \(-0.595885\pi\)
−0.296695 + 0.954972i \(0.595885\pi\)
\(72\) 0 0
\(73\) − 15.0000i − 1.75562i −0.479012 − 0.877809i \(-0.659005\pi\)
0.479012 − 0.877809i \(-0.340995\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.00000i 0.911685i
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 − 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.00000i 0.107211i
\(88\) 0 0
\(89\) 8.00000 0.847998 0.423999 − 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) −10.0000 −1.04828
\(92\) 0 0
\(93\) 9.00000i 0.933257i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 10.0000i − 1.01535i −0.861550 − 0.507673i \(-0.830506\pi\)
0.861550 − 0.507673i \(-0.169494\pi\)
\(98\) 0 0
\(99\) −8.00000 −0.804030
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.e.4049.1 2
5.2 odd 4 184.2.a.a.1.1 ✓ 1
5.3 odd 4 4600.2.a.i.1.1 1
5.4 even 2 inner 4600.2.e.e.4049.2 2
15.2 even 4 1656.2.a.i.1.1 1
20.3 even 4 9200.2.a.o.1.1 1
20.7 even 4 368.2.a.e.1.1 1
35.27 even 4 9016.2.a.k.1.1 1
40.27 even 4 1472.2.a.e.1.1 1
40.37 odd 4 1472.2.a.l.1.1 1
60.47 odd 4 3312.2.a.r.1.1 1
115.22 even 4 4232.2.a.f.1.1 1
460.367 odd 4 8464.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.a.a.1.1 ✓ 1 5.2 odd 4
368.2.a.e.1.1 1 20.7 even 4
1472.2.a.e.1.1 1 40.27 even 4
1472.2.a.l.1.1 1 40.37 odd 4
1656.2.a.i.1.1 1 15.2 even 4
3312.2.a.r.1.1 1 60.47 odd 4
4232.2.a.f.1.1 1 115.22 even 4
4600.2.a.i.1.1 1 5.3 odd 4
4600.2.e.e.4049.1 2 1.1 even 1 trivial
4600.2.e.e.4049.2 2 5.4 even 2 inner
8464.2.a.p.1.1 1 460.367 odd 4
9016.2.a.k.1.1 1 35.27 even 4
9200.2.a.o.1.1 1 20.3 even 4