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Properties

Label 4900.2.e.t.2549.5
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 2549.5
Root \(1.91223i\) of defining polynomial
Character \(\chi\) \(=\) 4900.2549
Dual form 4900.2.e.t.2549.2

$q$-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)
 
\(f(q)\) \(=\) \(q+2.56885i q^{3} -3.59899 q^{9} -1.56885 q^{11} -5.56885i q^{13} +7.16784i q^{17} -3.16784 q^{19} -5.73669i q^{23} -1.53871i q^{27} -1.96986 q^{29} +0.969861 q^{31} -4.03014i q^{33} -6.70655i q^{37} +14.3055 q^{39} -8.87439 q^{41} +4.59899i q^{43} +0.401012i q^{47} -18.4131 q^{51} -9.53871i q^{53} -8.13770i q^{57} +4.56885 q^{59} +15.3055 q^{61} +0.862301i q^{67} +14.7367 q^{69} -12.1377 q^{71} -4.00000i q^{73} +12.8744 q^{79} -6.84425 q^{81} -17.1076i q^{83} -5.06028i q^{87} +5.59899 q^{89} +2.49143i q^{93} +0.233174i q^{97} +5.64627 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 16 q^{9} + 8 q^{11} + 4 q^{19} - 6 q^{31} + 28 q^{39} + 22 q^{41} - 6 q^{51} + 10 q^{59} + 34 q^{61} + 48 q^{69} - 38 q^{71} + 2 q^{79} + 46 q^{81} + 28 q^{89} - 42 q^{99}+O(q^{100}) \) 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
\(3\) 2.56885i 1.48313i 0.670883 + 0.741563i \(0.265915\pi\)
−0.670883 + 0.741563i \(0.734085\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −3.59899 −1.19966
\(10\) 0 0
\(11\) −1.56885 −0.473026 −0.236513 − 0.971628i \(-0.576005\pi\)
−0.236513 + 0.971628i \(0.576005\pi\)
\(12\) 0 0
\(13\) − 5.56885i − 1.54452i −0.635306 − 0.772260i \(-0.719126\pi\)
0.635306 − 0.772260i \(-0.280874\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.16784i 1.73846i 0.494411 + 0.869228i \(0.335384\pi\)
−0.494411 + 0.869228i \(0.664616\pi\)
\(18\) 0 0
\(19\) −3.16784 −0.726752 −0.363376 − 0.931643i \(-0.618376\pi\)
−0.363376 + 0.931643i \(0.618376\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 5.73669i − 1.19618i −0.801428 − 0.598091i \(-0.795926\pi\)
0.801428 − 0.598091i \(-0.204074\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.53871i − 0.296125i
\(28\) 0 0
\(29\) −1.96986 −0.365794 −0.182897 − 0.983132i \(-0.558547\pi\)
−0.182897 + 0.983132i \(0.558547\pi\)
\(30\) 0 0
\(31\) 0.969861 0.174192 0.0870961 − 0.996200i \(-0.472241\pi\)
0.0870961 + 0.996200i \(0.472241\pi\)
\(32\) 0 0
\(33\) − 4.03014i − 0.701557i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 6.70655i − 1.10255i −0.834324 − 0.551275i \(-0.814142\pi\)
0.834324 − 0.551275i \(-0.185858\pi\)
\(38\) 0 0
\(39\) 14.3055 2.29072
\(40\) 0 0
\(41\) −8.87439 −1.38595 −0.692973 − 0.720963i \(-0.743700\pi\)
−0.692973 + 0.720963i \(0.743700\pi\)
\(42\) 0 0
\(43\) 4.59899i 0.701339i 0.936499 + 0.350670i \(0.114046\pi\)
−0.936499 + 0.350670i \(0.885954\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.401012i 0.0584936i 0.999572 + 0.0292468i \(0.00931087\pi\)
−0.999572 + 0.0292468i \(0.990689\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −18.4131 −2.57835
\(52\) 0 0
\(53\) − 9.53871i − 1.31024i −0.755524 − 0.655121i \(-0.772617\pi\)
0.755524 − 0.655121i \(-0.227383\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 8.13770i − 1.07786i
\(58\) 0 0
\(59\) 4.56885 0.594814 0.297407 − 0.954751i \(-0.403878\pi\)
0.297407 + 0.954751i \(0.403878\pi\)
\(60\) 0 0
\(61\) 15.3055 1.95967 0.979837 − 0.199800i \(-0.0640294\pi\)
0.979837 + 0.199800i \(0.0640294\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0.862301i 0.105347i 0.998612 + 0.0526734i \(0.0167742\pi\)
−0.998612 + 0.0526734i \(0.983226\pi\)
\(68\) 0 0
\(69\) 14.7367 1.77409
\(70\) 0 0
\(71\) −12.1377 −1.44048 −0.720240 − 0.693725i \(-0.755968\pi\)
−0.720240 + 0.693725i \(0.755968\pi\)
\(72\) 0 0
\(73\) − 4.00000i − 0.468165i −0.972217 − 0.234082i \(-0.924791\pi\)
0.972217 − 0.234082i \(-0.0752085\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 12.8744 1.44848 0.724241 − 0.689547i \(-0.242190\pi\)
0.724241 + 0.689547i \(0.242190\pi\)
\(80\) 0 0
\(81\) −6.84425 −0.760472
\(82\) 0 0
\(83\) − 17.1076i − 1.87780i −0.344192 − 0.938899i \(-0.611847\pi\)
0.344192 − 0.938899i \(-0.388153\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 5.06028i − 0.542519i
\(88\) 0 0
\(89\) 5.59899 0.593492 0.296746 − 0.954957i \(-0.404099\pi\)
0.296746 + 0.954957i \(0.404099\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 2.49143i 0.258349i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.233174i 0.0236752i 0.999930 + 0.0118376i \(0.00376811\pi\)
−0.999930 + 0.0118376i \(0.996232\pi\)
\(98\) 0 0
\(99\) 5.64627 0.567472
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.2549.5 6
5.2 odd 4 4900.2.a.ba.1.3 3
5.3 odd 4 4900.2.a.bc.1.1 3
5.4 even 2 inner 4900.2.e.t.2549.2 6
7.2 even 3 700.2.r.d.249.2 12
7.4 even 3 700.2.r.d.149.5 12
7.6 odd 2 4900.2.e.s.2549.2 6
35.2 odd 12 700.2.i.e.501.1 yes 6
35.4 even 6 700.2.r.d.149.2 12
35.9 even 6 700.2.r.d.249.5 12
35.13 even 4 4900.2.a.bb.1.3 3
35.18 odd 12 700.2.i.d.401.3 ✓ 6
35.23 odd 12 700.2.i.d.501.3 yes 6
35.27 even 4 4900.2.a.bd.1.1 3
35.32 odd 12 700.2.i.e.401.1 yes 6
35.34 odd 2 4900.2.e.s.2549.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
700.2.i.d.401.3 ✓ 6 35.18 odd 12
700.2.i.d.501.3 yes 6 35.23 odd 12
700.2.i.e.401.1 yes 6 35.32 odd 12
700.2.i.e.501.1 yes 6 35.2 odd 12
700.2.r.d.149.2 12 35.4 even 6
700.2.r.d.149.5 12 7.4 even 3
700.2.r.d.249.2 12 7.2 even 3
700.2.r.d.249.5 12 35.9 even 6
4900.2.a.ba.1.3 3 5.2 odd 4
4900.2.a.bb.1.3 3 35.13 even 4
4900.2.a.bc.1.1 3 5.3 odd 4
4900.2.a.bd.1.1 3 35.27 even 4
4900.2.e.s.2549.2 6 7.6 odd 2
4900.2.e.s.2549.5 6 35.34 odd 2
4900.2.e.t.2549.2 6 5.4 even 2 inner
4900.2.e.t.2549.5 6 1.1 even 1 trivial