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Properties

Label 4900.2.e.t.2549.6
Level $4900$
Weight $2$
Character 4900.2549
Analytic conductor $39.127$
Analytic rank $0$
Dimension $6$
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] = [4900,2,Mod(2549,4900)] 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("4900.2549"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4900.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,-16,0,8,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0, 0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(39.1266969904\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.4227136.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 9x^{4} + 22x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 700)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2549.6
Root \(0.713538i\) of defining polynomial
Character \(\chi\) \(=\) 4900.2549
Dual form 4900.2.e.t.2549.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.20440i q^{3} -7.26819 q^{9} +4.20440 q^{11} -0.204402i q^{13} -5.06379i q^{17} -1.06379 q^{19} -2.14061i q^{23} -13.6770i q^{27} +7.47259 q^{29} -8.47259 q^{31} +13.4726i q^{33} -10.6132i q^{37} +0.654985 q^{39} +10.5494 q^{41} -8.26819i q^{43} +3.26819i q^{47} +16.2264 q^{51} -5.67699i q^{53} -3.40880i q^{57} -1.20440 q^{59} +1.65498 q^{61} -12.4088i q^{67} +6.85939 q^{69} -0.591197 q^{71} +4.00000i q^{73} -6.54942 q^{79} +22.0220 q^{81} -3.88139i q^{83} +23.9452i q^{87} +9.26819 q^{89} -27.1496i q^{93} +1.33198i q^{97} -30.5584 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 16 q^{9} + 8 q^{11} + 4 q^{19} - 6 q^{31} + 28 q^{39} + 22 q^{41} - 6 q^{51} + 10 q^{59} + 34 q^{61} + 48 q^{69} - 38 q^{71} + 2 q^{79} + 46 q^{81} + 28 q^{89} - 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\) \(2451\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 3.20440i 1.85006i 0.379892 + 0.925031i \(0.375961\pi\)
−0.379892 + 0.925031i \(0.624039\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −7.26819 −2.42273
\(10\) 0 0
\(11\) 4.20440 1.26767 0.633837 − 0.773466i \(-0.281479\pi\)
0.633837 + 0.773466i \(0.281479\pi\)
\(12\) 0 0
\(13\) − 0.204402i − 0.0566908i −0.999598 − 0.0283454i \(-0.990976\pi\)
0.999598 − 0.0283454i \(-0.00902383\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 5.06379i − 1.22815i −0.789248 − 0.614074i \(-0.789529\pi\)
0.789248 − 0.614074i \(-0.210471\pi\)
\(18\) 0 0
\(19\) −1.06379 −0.244050 −0.122025 − 0.992527i \(-0.538939\pi\)
−0.122025 + 0.992527i \(0.538939\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 2.14061i − 0.446349i −0.974779 − 0.223174i \(-0.928358\pi\)
0.974779 − 0.223174i \(-0.0716419\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 13.6770i − 2.63214i
\(28\) 0 0
\(29\) 7.47259 1.38763 0.693813 − 0.720156i \(-0.255930\pi\)
0.693813 + 0.720156i \(0.255930\pi\)
\(30\) 0 0
\(31\) −8.47259 −1.52172 −0.760861 − 0.648915i \(-0.775223\pi\)
−0.760861 + 0.648915i \(0.775223\pi\)
\(32\) 0 0
\(33\) 13.4726i 2.34528i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 10.6132i − 1.74480i −0.488793 − 0.872400i \(-0.662563\pi\)
0.488793 − 0.872400i \(-0.337437\pi\)
\(38\) 0 0
\(39\) 0.654985 0.104881
\(40\) 0 0
\(41\) 10.5494 1.64754 0.823771 − 0.566923i \(-0.191866\pi\)
0.823771 + 0.566923i \(0.191866\pi\)
\(42\) 0 0
\(43\) − 8.26819i − 1.26089i −0.776235 − 0.630444i \(-0.782873\pi\)
0.776235 − 0.630444i \(-0.217127\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.26819i 0.476714i 0.971178 + 0.238357i \(0.0766089\pi\)
−0.971178 + 0.238357i \(0.923391\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 16.2264 2.27215
\(52\) 0 0
\(53\) − 5.67699i − 0.779795i −0.920858 − 0.389897i \(-0.872510\pi\)
0.920858 − 0.389897i \(-0.127490\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 3.40880i − 0.451507i
\(58\) 0 0
\(59\) −1.20440 −0.156800 −0.0783999 − 0.996922i \(-0.524981\pi\)
−0.0783999 + 0.996922i \(0.524981\pi\)
\(60\) 0 0
\(61\) 1.65498 0.211899 0.105950 − 0.994372i \(-0.466212\pi\)
0.105950 + 0.994372i \(0.466212\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 12.4088i − 1.51598i −0.652268 − 0.757988i \(-0.726182\pi\)
0.652268 − 0.757988i \(-0.273818\pi\)
\(68\) 0 0
\(69\) 6.85939 0.825773
\(70\) 0 0
\(71\) −0.591197 −0.0701622 −0.0350811 − 0.999384i \(-0.511169\pi\)
−0.0350811 + 0.999384i \(0.511169\pi\)
\(72\) 0 0
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −6.54942 −0.736867 −0.368433 − 0.929654i \(-0.620106\pi\)
−0.368433 + 0.929654i \(0.620106\pi\)
\(80\) 0 0
\(81\) 22.0220 2.44689
\(82\) 0 0
\(83\) − 3.88139i − 0.426038i −0.977048 − 0.213019i \(-0.931670\pi\)
0.977048 − 0.213019i \(-0.0683297\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 23.9452i 2.56719i
\(88\) 0 0
\(89\) 9.26819 0.982426 0.491213 − 0.871039i \(-0.336554\pi\)
0.491213 + 0.871039i \(0.336554\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 27.1496i − 2.81528i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.33198i 0.135242i 0.997711 + 0.0676209i \(0.0215408\pi\)
−0.997711 + 0.0676209i \(0.978459\pi\)
\(98\) 0 0
\(99\) −30.5584 −3.07123
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 4900.2.e.t.2549.6 6
5.2 odd 4 4900.2.a.bc.1.3 3
5.3 odd 4 4900.2.a.ba.1.1 3
5.4 even 2 inner 4900.2.e.t.2549.1 6
7.2 even 3 700.2.r.d.249.1 12
7.4 even 3 700.2.r.d.149.6 12
7.6 odd 2 4900.2.e.s.2549.1 6
35.2 odd 12 700.2.i.d.501.1 yes 6
35.4 even 6 700.2.r.d.149.1 12
35.9 even 6 700.2.r.d.249.6 12
35.13 even 4 4900.2.a.bd.1.3 3
35.18 odd 12 700.2.i.e.401.3 yes 6
35.23 odd 12 700.2.i.e.501.3 yes 6
35.27 even 4 4900.2.a.bb.1.1 3
35.32 odd 12 700.2.i.d.401.1 ✓ 6
35.34 odd 2 4900.2.e.s.2549.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
700.2.i.d.401.1 ✓ 6 35.32 odd 12
700.2.i.d.501.1 yes 6 35.2 odd 12
700.2.i.e.401.3 yes 6 35.18 odd 12
700.2.i.e.501.3 yes 6 35.23 odd 12
700.2.r.d.149.1 12 35.4 even 6
700.2.r.d.149.6 12 7.4 even 3
700.2.r.d.249.1 12 7.2 even 3
700.2.r.d.249.6 12 35.9 even 6
4900.2.a.ba.1.1 3 5.3 odd 4
4900.2.a.bb.1.1 3 35.27 even 4
4900.2.a.bc.1.3 3 5.2 odd 4
4900.2.a.bd.1.3 3 35.13 even 4
4900.2.e.s.2549.1 6 7.6 odd 2
4900.2.e.s.2549.6 6 35.34 odd 2
4900.2.e.t.2549.1 6 5.4 even 2 inner
4900.2.e.t.2549.6 6 1.1 even 1 trivial