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Properties

Label 4600.2.e.t.4049.5
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $8$
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 = [8,0,0,0,0,0,0,0,-12,0,2,0,0,0,0,0,0,0,30] 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: \(8\)
Coefficient field: 8.0.61734359296.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 13x^{6} + 38x^{4} + 25x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.5
Root \(0.491317i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.t.4049.4

$q$-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)
 
\(f(q)\) \(=\) \(q+0.329452i q^{3} -4.07069i q^{7} +2.89146 q^{9} -3.08806 q^{11} +6.32062i q^{13} +0.982634i q^{17} +7.08806 q^{19} +1.34110 q^{21} -1.00000i q^{23} +1.94095i q^{27} +7.63270 q^{29} -5.92047 q^{31} -1.01737i q^{33} +11.4825i q^{37} -2.08234 q^{39} -5.74124 q^{41} -1.74696i q^{43} +2.24992i q^{47} -9.57054 q^{49} -0.323730 q^{51} -1.31781i q^{53} +2.33517i q^{57} -7.92901 q^{59} +9.51722 q^{61} -11.7703i q^{63} -3.34110i q^{67} +0.329452 q^{69} -3.92619 q^{71} +1.66744i q^{73} +12.5705i q^{77} +15.9231 q^{79} +8.03493 q^{81} +11.9116i q^{83} +2.51461i q^{87} +10.4998 q^{89} +25.7293 q^{91} -1.95051i q^{93} +2.84125i q^{97} -8.92901 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{9} + 2 q^{11} + 30 q^{19} + 16 q^{21} - 2 q^{29} - 24 q^{31} + 6 q^{39} - 18 q^{41} + 6 q^{49} - 4 q^{51} + 10 q^{59} + 28 q^{61} - 4 q^{71} + 42 q^{79} + 88 q^{81} + 32 q^{89} + 66 q^{91} + 2 q^{99}+O(q^{100}) \) 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
\(3\) 0.329452i 0.190209i 0.995467 + 0.0951045i \(0.0303185\pi\)
−0.995467 + 0.0951045i \(0.969681\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 4.07069i − 1.53858i −0.638901 − 0.769289i \(-0.720611\pi\)
0.638901 − 0.769289i \(-0.279389\pi\)
\(8\) 0 0
\(9\) 2.89146 0.963821
\(10\) 0 0
\(11\) −3.08806 −0.931085 −0.465542 − 0.885026i \(-0.654141\pi\)
−0.465542 + 0.885026i \(0.654141\pi\)
\(12\) 0 0
\(13\) 6.32062i 1.75302i 0.481380 + 0.876512i \(0.340136\pi\)
−0.481380 + 0.876512i \(0.659864\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.982634i 0.238324i 0.992875 + 0.119162i \(0.0380207\pi\)
−0.992875 + 0.119162i \(0.961979\pi\)
\(18\) 0 0
\(19\) 7.08806 1.62611 0.813056 − 0.582185i \(-0.197802\pi\)
0.813056 + 0.582185i \(0.197802\pi\)
\(20\) 0 0
\(21\) 1.34110 0.292651
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.94095i 0.373536i
\(28\) 0 0
\(29\) 7.63270 1.41736 0.708679 − 0.705531i \(-0.249292\pi\)
0.708679 + 0.705531i \(0.249292\pi\)
\(30\) 0 0
\(31\) −5.92047 −1.06335 −0.531674 − 0.846949i \(-0.678437\pi\)
−0.531674 + 0.846949i \(0.678437\pi\)
\(32\) 0 0
\(33\) − 1.01737i − 0.177101i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 11.4825i 1.88771i 0.330362 + 0.943854i \(0.392829\pi\)
−0.330362 + 0.943854i \(0.607171\pi\)
\(38\) 0 0
\(39\) −2.08234 −0.333441
\(40\) 0 0
\(41\) −5.74124 −0.896631 −0.448316 − 0.893875i \(-0.647976\pi\)
−0.448316 + 0.893875i \(0.647976\pi\)
\(42\) 0 0
\(43\) − 1.74696i − 0.266409i −0.991089 − 0.133205i \(-0.957473\pi\)
0.991089 − 0.133205i \(-0.0425268\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.24992i 0.328185i 0.986445 + 0.164093i \(0.0524696\pi\)
−0.986445 + 0.164093i \(0.947530\pi\)
\(48\) 0 0
\(49\) −9.57054 −1.36722
\(50\) 0 0
\(51\) −0.323730 −0.0453313
\(52\) 0 0
\(53\) − 1.31781i − 0.181015i −0.995896 − 0.0905073i \(-0.971151\pi\)
0.995896 − 0.0905073i \(-0.0288489\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.33517i 0.309301i
\(58\) 0 0
\(59\) −7.92901 −1.03227 −0.516134 − 0.856508i \(-0.672629\pi\)
−0.516134 + 0.856508i \(0.672629\pi\)
\(60\) 0 0
\(61\) 9.51722 1.21855 0.609277 − 0.792957i \(-0.291460\pi\)
0.609277 + 0.792957i \(0.291460\pi\)
\(62\) 0 0
\(63\) − 11.7703i − 1.48291i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 3.34110i − 0.408180i −0.978952 − 0.204090i \(-0.934576\pi\)
0.978952 − 0.204090i \(-0.0654235\pi\)
\(68\) 0 0
\(69\) 0.329452 0.0396613
\(70\) 0 0
\(71\) −3.92619 −0.465954 −0.232977 − 0.972482i \(-0.574847\pi\)
−0.232977 + 0.972482i \(0.574847\pi\)
\(72\) 0 0
\(73\) 1.66744i 0.195159i 0.995228 + 0.0975793i \(0.0311100\pi\)
−0.995228 + 0.0975793i \(0.968890\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.5705i 1.43255i
\(78\) 0 0
\(79\) 15.9231 1.79149 0.895743 − 0.444572i \(-0.146644\pi\)
0.895743 + 0.444572i \(0.146644\pi\)
\(80\) 0 0
\(81\) 8.03493 0.892771
\(82\) 0 0
\(83\) 11.9116i 1.30747i 0.756723 + 0.653736i \(0.226799\pi\)
−0.756723 + 0.653736i \(0.773201\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.51461i 0.269594i
\(88\) 0 0
\(89\) 10.4998 1.11298 0.556491 − 0.830854i \(-0.312147\pi\)
0.556491 + 0.830854i \(0.312147\pi\)
\(90\) 0 0
\(91\) 25.7293 2.69716
\(92\) 0 0
\(93\) − 1.95051i − 0.202258i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.84125i 0.288485i 0.989542 + 0.144242i \(0.0460745\pi\)
−0.989542 + 0.144242i \(0.953925\pi\)
\(98\) 0 0
\(99\) −8.92901 −0.897399
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.t.4049.5 8
5.2 odd 4 4600.2.a.bb.1.2 yes 4
5.3 odd 4 4600.2.a.ba.1.3 ✓ 4
5.4 even 2 inner 4600.2.e.t.4049.4 8
20.3 even 4 9200.2.a.cp.1.2 4
20.7 even 4 9200.2.a.cn.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.ba.1.3 ✓ 4 5.3 odd 4
4600.2.a.bb.1.2 yes 4 5.2 odd 4
4600.2.e.t.4049.4 8 5.4 even 2 inner
4600.2.e.t.4049.5 8 1.1 even 1 trivial
9200.2.a.cn.1.3 4 20.7 even 4
9200.2.a.cp.1.2 4 20.3 even 4