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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 = [10,0,0,0,0,0,0,0,-26,0,-2,0,0,0,0,0,0,0,-14] 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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 28x^{8} + 260x^{6} + 897x^{4} + 1056x^{2} + 256 \) 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 4049.3
Root \(-1.93283i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.u.4049.8

$q$-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)
 
\(f(q)\) \(=\) \(q-1.93283i q^{3} +2.38236i q^{7} -0.735829 q^{9} +5.33368 q^{11} +4.53752i q^{13} +1.81464i q^{17} -7.00233 q^{19} +4.60469 q^{21} +1.00000i q^{23} -4.37626i q^{27} +0.118188 q^{29} -0.884147 q^{31} -10.3091i q^{33} +7.51903i q^{37} +8.77026 q^{39} -1.45186 q^{41} +10.4389i q^{47} +1.32437 q^{49} +3.50739 q^{51} +9.42167i q^{53} +13.5343i q^{57} -7.79239 q^{59} -2.80533 q^{61} -1.75301i q^{63} -3.11134i q^{67} +1.93283 q^{69} -13.5909 q^{71} -12.4389i q^{73} +12.7067i q^{77} -6.80169 q^{79} -10.6660 q^{81} +13.5190i q^{83} -0.228437i q^{87} -2.89906 q^{89} -10.8100 q^{91} +1.70890i q^{93} -1.97774i q^{97} -3.92468 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 26 q^{9} - 2 q^{11} - 14 q^{19} + 12 q^{21} - 8 q^{29} + 38 q^{31} - 38 q^{39} + 50 q^{41} - 50 q^{49} + 38 q^{51} + 2 q^{59} - 10 q^{61} + 2 q^{71} + 4 q^{79} + 114 q^{81} - 12 q^{89} + 22 q^{91}+ \cdots + 130 q^{99}+O(q^{100}) \) 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
\(3\) − 1.93283i − 1.11592i −0.829868 − 0.557960i \(-0.811584\pi\)
0.829868 − 0.557960i \(-0.188416\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.38236i 0.900447i 0.892916 + 0.450223i \(0.148656\pi\)
−0.892916 + 0.450223i \(0.851344\pi\)
\(8\) 0 0
\(9\) −0.735829 −0.245276
\(10\) 0 0
\(11\) 5.33368 1.60816 0.804082 − 0.594519i \(-0.202657\pi\)
0.804082 + 0.594519i \(0.202657\pi\)
\(12\) 0 0
\(13\) 4.53752i 1.25848i 0.777210 + 0.629241i \(0.216634\pi\)
−0.777210 + 0.629241i \(0.783366\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.81464i 0.440115i 0.975487 + 0.220058i \(0.0706245\pi\)
−0.975487 + 0.220058i \(0.929375\pi\)
\(18\) 0 0
\(19\) −7.00233 −1.60645 −0.803223 − 0.595679i \(-0.796883\pi\)
−0.803223 + 0.595679i \(0.796883\pi\)
\(20\) 0 0
\(21\) 4.60469 1.00483
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 4.37626i − 0.842211i
\(28\) 0 0
\(29\) 0.118188 0.0219469 0.0109735 − 0.999940i \(-0.496507\pi\)
0.0109735 + 0.999940i \(0.496507\pi\)
\(30\) 0 0
\(31\) −0.884147 −0.158797 −0.0793987 − 0.996843i \(-0.525300\pi\)
−0.0793987 + 0.996843i \(0.525300\pi\)
\(32\) 0 0
\(33\) − 10.3091i − 1.79458i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.51903i 1.23612i 0.786130 + 0.618061i \(0.212081\pi\)
−0.786130 + 0.618061i \(0.787919\pi\)
\(38\) 0 0
\(39\) 8.77026 1.40436
\(40\) 0 0
\(41\) −1.45186 −0.226743 −0.113371 − 0.993553i \(-0.536165\pi\)
−0.113371 + 0.993553i \(0.536165\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.4389i 1.52267i 0.648357 + 0.761336i \(0.275456\pi\)
−0.648357 + 0.761336i \(0.724544\pi\)
\(48\) 0 0
\(49\) 1.32437 0.189195
\(50\) 0 0
\(51\) 3.50739 0.491133
\(52\) 0 0
\(53\) 9.42167i 1.29417i 0.762420 + 0.647083i \(0.224011\pi\)
−0.762420 + 0.647083i \(0.775989\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 13.5343i 1.79266i
\(58\) 0 0
\(59\) −7.79239 −1.01448 −0.507241 − 0.861804i \(-0.669335\pi\)
−0.507241 + 0.861804i \(0.669335\pi\)
\(60\) 0 0
\(61\) −2.80533 −0.359186 −0.179593 − 0.983741i \(-0.557478\pi\)
−0.179593 + 0.983741i \(0.557478\pi\)
\(62\) 0 0
\(63\) − 1.75301i − 0.220858i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 3.11134i − 0.380111i −0.981773 − 0.190055i \(-0.939133\pi\)
0.981773 − 0.190055i \(-0.0608668\pi\)
\(68\) 0 0
\(69\) 1.93283 0.232685
\(70\) 0 0
\(71\) −13.5909 −1.61294 −0.806470 − 0.591275i \(-0.798625\pi\)
−0.806470 + 0.591275i \(0.798625\pi\)
\(72\) 0 0
\(73\) − 12.4389i − 1.45586i −0.685649 − 0.727932i \(-0.740481\pi\)
0.685649 − 0.727932i \(-0.259519\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.7067i 1.44807i
\(78\) 0 0
\(79\) −6.80169 −0.765250 −0.382625 − 0.923904i \(-0.624980\pi\)
−0.382625 + 0.923904i \(0.624980\pi\)
\(80\) 0 0
\(81\) −10.6660 −1.18512
\(82\) 0 0
\(83\) 13.5190i 1.48391i 0.670451 + 0.741953i \(0.266100\pi\)
−0.670451 + 0.741953i \(0.733900\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 0.228437i − 0.0244910i
\(88\) 0 0
\(89\) −2.89906 −0.307300 −0.153650 − 0.988125i \(-0.549103\pi\)
−0.153650 + 0.988125i \(0.549103\pi\)
\(90\) 0 0
\(91\) −10.8100 −1.13320
\(92\) 0 0
\(93\) 1.70890i 0.177205i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 1.97774i − 0.200809i −0.994947 − 0.100405i \(-0.967986\pi\)
0.994947 − 0.100405i \(-0.0320138\pi\)
\(98\) 0 0
\(99\) −3.92468 −0.394445
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.u.4049.3 10
5.2 odd 4 4600.2.a.be.1.2 5
5.3 odd 4 920.2.a.j.1.4 ✓ 5
5.4 even 2 inner 4600.2.e.u.4049.8 10
15.8 even 4 8280.2.a.bs.1.4 5
20.3 even 4 1840.2.a.v.1.2 5
20.7 even 4 9200.2.a.cu.1.4 5
40.3 even 4 7360.2.a.cp.1.4 5
40.13 odd 4 7360.2.a.co.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.j.1.4 ✓ 5 5.3 odd 4
1840.2.a.v.1.2 5 20.3 even 4
4600.2.a.be.1.2 5 5.2 odd 4
4600.2.e.u.4049.3 10 1.1 even 1 trivial
4600.2.e.u.4049.8 10 5.4 even 2 inner
7360.2.a.co.1.2 5 40.13 odd 4
7360.2.a.cp.1.4 5 40.3 even 4
8280.2.a.bs.1.4 5 15.8 even 4
9200.2.a.cu.1.4 5 20.7 even 4