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Properties

Label 4900.2.e.s.2549.4
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 2549.4
Root \(2.19869i\) of defining polynomial
Character \(\chi\) \(=\) 4900.2549
Dual form 4900.2.e.s.2549.3

$q$-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)
 
\(f(q)\) \(=\) \(q+0.364448i q^{3} +2.86718 q^{9} +1.36445 q^{11} +2.63555i q^{13} +2.23163i q^{17} -6.23163 q^{19} +6.59607i q^{23} +2.13828i q^{27} -5.50273 q^{29} -4.50273 q^{31} +0.497270i q^{33} +2.09334i q^{37} -0.960522 q^{39} -9.32497 q^{41} -1.86718i q^{43} -6.86718i q^{47} -0.813312 q^{51} -10.1383i q^{53} -2.27110i q^{57} -1.63555 q^{59} -0.0394782 q^{61} +6.72890i q^{67} -2.40393 q^{69} -6.27110 q^{71} +4.00000i q^{73} -5.32497 q^{79} +7.82224 q^{81} +14.7738i q^{83} -2.00546i q^{87} +0.867178 q^{89} -1.64101i q^{93} -16.0988i q^{97} +3.91211 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 16 q^{9} + 8 q^{11} - 4 q^{19} + 6 q^{31} + 28 q^{39} - 22 q^{41} - 6 q^{51} - 10 q^{59} - 34 q^{61} - 48 q^{69} - 38 q^{71} + 2 q^{79} + 46 q^{81} - 28 q^{89} - 42 q^{99}+O(q^{100}) \) 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
\(3\) 0.364448i 0.210414i 0.994450 + 0.105207i \(0.0335505\pi\)
−0.994450 + 0.105207i \(0.966449\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 2.86718 0.955726
\(10\) 0 0
\(11\) 1.36445 0.411397 0.205698 − 0.978615i \(-0.434053\pi\)
0.205698 + 0.978615i \(0.434053\pi\)
\(12\) 0 0
\(13\) 2.63555i 0.730971i 0.930817 + 0.365485i \(0.119097\pi\)
−0.930817 + 0.365485i \(0.880903\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.23163i 0.541249i 0.962685 + 0.270624i \(0.0872301\pi\)
−0.962685 + 0.270624i \(0.912770\pi\)
\(18\) 0 0
\(19\) −6.23163 −1.42963 −0.714816 − 0.699312i \(-0.753490\pi\)
−0.714816 + 0.699312i \(0.753490\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.59607i 1.37538i 0.726006 + 0.687688i \(0.241374\pi\)
−0.726006 + 0.687688i \(0.758626\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.13828i 0.411512i
\(28\) 0 0
\(29\) −5.50273 −1.02183 −0.510916 − 0.859631i \(-0.670694\pi\)
−0.510916 + 0.859631i \(0.670694\pi\)
\(30\) 0 0
\(31\) −4.50273 −0.808714 −0.404357 − 0.914601i \(-0.632505\pi\)
−0.404357 + 0.914601i \(0.632505\pi\)
\(32\) 0 0
\(33\) 0.497270i 0.0865637i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.09334i 0.344144i 0.985084 + 0.172072i \(0.0550461\pi\)
−0.985084 + 0.172072i \(0.944954\pi\)
\(38\) 0 0
\(39\) −0.960522 −0.153807
\(40\) 0 0
\(41\) −9.32497 −1.45632 −0.728158 − 0.685410i \(-0.759623\pi\)
−0.728158 + 0.685410i \(0.759623\pi\)
\(42\) 0 0
\(43\) − 1.86718i − 0.284742i −0.989813 − 0.142371i \(-0.954527\pi\)
0.989813 − 0.142371i \(-0.0454726\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 6.86718i − 1.00168i −0.865540 − 0.500840i \(-0.833024\pi\)
0.865540 − 0.500840i \(-0.166976\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −0.813312 −0.113886
\(52\) 0 0
\(53\) − 10.1383i − 1.39260i −0.717751 − 0.696300i \(-0.754828\pi\)
0.717751 − 0.696300i \(-0.245172\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 2.27110i − 0.300815i
\(58\) 0 0
\(59\) −1.63555 −0.212931 −0.106465 − 0.994316i \(-0.533953\pi\)
−0.106465 + 0.994316i \(0.533953\pi\)
\(60\) 0 0
\(61\) −0.0394782 −0.00505467 −0.00252733 − 0.999997i \(-0.500804\pi\)
−0.00252733 + 0.999997i \(0.500804\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.72890i 0.822065i 0.911621 + 0.411033i \(0.134832\pi\)
−0.911621 + 0.411033i \(0.865168\pi\)
\(68\) 0 0
\(69\) −2.40393 −0.289399
\(70\) 0 0
\(71\) −6.27110 −0.744243 −0.372122 − 0.928184i \(-0.621370\pi\)
−0.372122 + 0.928184i \(0.621370\pi\)
\(72\) 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
\(79\) −5.32497 −0.599106 −0.299553 − 0.954080i \(-0.596838\pi\)
−0.299553 + 0.954080i \(0.596838\pi\)
\(80\) 0 0
\(81\) 7.82224 0.869138
\(82\) 0 0
\(83\) 14.7738i 1.62164i 0.585296 + 0.810819i \(0.300978\pi\)
−0.585296 + 0.810819i \(0.699022\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 2.00546i − 0.215008i
\(88\) 0 0
\(89\) 0.867178 0.0919206 0.0459603 − 0.998943i \(-0.485365\pi\)
0.0459603 + 0.998943i \(0.485365\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 1.64101i − 0.170165i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 16.0988i − 1.63459i −0.576222 − 0.817293i \(-0.695474\pi\)
0.576222 − 0.817293i \(-0.304526\pi\)
\(98\) 0 0
\(99\) 3.91211 0.393182
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.s.2549.4 6
5.2 odd 4 4900.2.a.bd.1.2 3
5.3 odd 4 4900.2.a.bb.1.2 3
5.4 even 2 inner 4900.2.e.s.2549.3 6
7.3 odd 6 700.2.r.d.149.3 12
7.5 odd 6 700.2.r.d.249.4 12
7.6 odd 2 4900.2.e.t.2549.3 6
35.3 even 12 700.2.i.d.401.2 ✓ 6
35.12 even 12 700.2.i.e.501.2 yes 6
35.13 even 4 4900.2.a.bc.1.2 3
35.17 even 12 700.2.i.e.401.2 yes 6
35.19 odd 6 700.2.r.d.249.3 12
35.24 odd 6 700.2.r.d.149.4 12
35.27 even 4 4900.2.a.ba.1.2 3
35.33 even 12 700.2.i.d.501.2 yes 6
35.34 odd 2 4900.2.e.t.2549.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
700.2.i.d.401.2 ✓ 6 35.3 even 12
700.2.i.d.501.2 yes 6 35.33 even 12
700.2.i.e.401.2 yes 6 35.17 even 12
700.2.i.e.501.2 yes 6 35.12 even 12
700.2.r.d.149.3 12 7.3 odd 6
700.2.r.d.149.4 12 35.24 odd 6
700.2.r.d.249.3 12 35.19 odd 6
700.2.r.d.249.4 12 7.5 odd 6
4900.2.a.ba.1.2 3 35.27 even 4
4900.2.a.bb.1.2 3 5.3 odd 4
4900.2.a.bc.1.2 3 35.13 even 4
4900.2.a.bd.1.2 3 5.2 odd 4
4900.2.e.s.2549.3 6 5.4 even 2 inner
4900.2.e.s.2549.4 6 1.1 even 1 trivial
4900.2.e.t.2549.3 6 7.6 odd 2
4900.2.e.t.2549.4 6 35.34 odd 2