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Properties

Label 2700.2.s.d.2449.4
Level $2700$
Weight $2$
Character 2700.2449
Analytic conductor $21.560$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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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] = [2700,2,Mod(1549,2700)] 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("2700.1549"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2700, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2700.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.5596085457\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.1333317747165888577536.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 3x^{14} + 5x^{12} + 15x^{10} + 45x^{8} + 60x^{6} + 80x^{4} + 192x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{10} \)
Twist minimal: no (minimal twist has level 900)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 2449.4
Root \(0.263711 - 1.38941i\) of defining polynomial
Character \(\chi\) \(=\) 2700.2449
Dual form 2700.2.s.d.1549.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.0748933 - 0.0432397i) q^{7} +(-0.456760 + 0.791132i) q^{11} +(-2.27331 + 1.31249i) q^{13} -2.08648i q^{17} -4.93847 q^{19} +(7.34128 - 4.23849i) q^{23} +(-1.19899 + 2.07671i) q^{29} +(-1.81249 - 3.13933i) q^{31} -5.85199i q^{37} +(3.32497 + 5.75902i) q^{41} +(-7.14469 - 4.12499i) q^{43} +(2.32831 + 1.34425i) q^{47} +(-3.49626 - 6.05570i) q^{49} -5.73642i q^{53} +(-6.16922 - 10.6854i) q^{59} +(3.16823 - 5.48753i) q^{61} +(5.33946 - 3.08274i) q^{67} +12.3905 q^{71} -5.31349i q^{73} +(0.0684166 - 0.0395003i) q^{77} +(-6.72394 + 11.6462i) q^{79} +(5.26457 + 3.03950i) q^{83} -8.13440 q^{89} +0.227007 q^{91} +(-9.61459 - 5.55098i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{11} + 16 q^{19} - 18 q^{29} - 4 q^{31} - 18 q^{41} + 18 q^{49} - 30 q^{59} + 2 q^{61} + 48 q^{71} - 14 q^{79} + 12 q^{89} - 44 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1351\) \(2377\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(-1\)

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 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −0.0748933 − 0.0432397i −0.0283070 − 0.0163431i 0.485780 − 0.874081i \(-0.338536\pi\)
−0.514087 + 0.857738i \(0.671869\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.456760 + 0.791132i −0.137718 + 0.238535i −0.926633 − 0.375968i \(-0.877310\pi\)
0.788914 + 0.614503i \(0.210644\pi\)
\(12\) 0 0
\(13\) −2.27331 + 1.31249i −0.630501 + 0.364020i −0.780946 − 0.624598i \(-0.785263\pi\)
0.150445 + 0.988618i \(0.451929\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2.08648i − 0.506046i −0.967460 − 0.253023i \(-0.918575\pi\)
0.967460 − 0.253023i \(-0.0814248\pi\)
\(18\) 0 0
\(19\) −4.93847 −1.13296 −0.566482 − 0.824074i \(-0.691696\pi\)
−0.566482 + 0.824074i \(0.691696\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.34128 − 4.23849i 1.53076 − 0.883786i 0.531436 − 0.847098i \(-0.321653\pi\)
0.999327 − 0.0366878i \(-0.0116807\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.19899 + 2.07671i −0.222647 + 0.385636i −0.955611 − 0.294632i \(-0.904803\pi\)
0.732964 + 0.680267i \(0.238136\pi\)
\(30\) 0 0
\(31\) −1.81249 − 3.13933i −0.325533 − 0.563840i 0.656087 − 0.754685i \(-0.272211\pi\)
−0.981620 + 0.190845i \(0.938877\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 5.85199i − 0.962062i −0.876704 − 0.481031i \(-0.840262\pi\)
0.876704 − 0.481031i \(-0.159738\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.32497 + 5.75902i 0.519273 + 0.899407i 0.999749 + 0.0223994i \(0.00713054\pi\)
−0.480476 + 0.877008i \(0.659536\pi\)
\(42\) 0 0
\(43\) −7.14469 − 4.12499i −1.08955 − 0.629055i −0.156097 − 0.987742i \(-0.549891\pi\)
−0.933458 + 0.358687i \(0.883224\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.32831 + 1.34425i 0.339619 + 0.196079i 0.660103 − 0.751175i \(-0.270512\pi\)
−0.320485 + 0.947254i \(0.603846\pi\)
\(48\) 0 0
\(49\) −3.49626 − 6.05570i −0.499466 − 0.865100i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 5.73642i − 0.787958i −0.919120 − 0.393979i \(-0.871098\pi\)
0.919120 − 0.393979i \(-0.128902\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.16922 − 10.6854i −0.803164 − 1.39112i −0.917524 − 0.397681i \(-0.869815\pi\)
0.114360 − 0.993439i \(-0.463518\pi\)
\(60\) 0 0
\(61\) 3.16823 − 5.48753i 0.405650 − 0.702606i −0.588747 − 0.808317i \(-0.700379\pi\)
0.994397 + 0.105711i \(0.0337119\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.33946 − 3.08274i 0.652319 − 0.376617i −0.137025 − 0.990568i \(-0.543754\pi\)
0.789344 + 0.613951i \(0.210421\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.3905 1.47048 0.735241 − 0.677806i \(-0.237069\pi\)
0.735241 + 0.677806i \(0.237069\pi\)
\(72\) 0 0
\(73\) − 5.31349i − 0.621897i −0.950427 − 0.310948i \(-0.899353\pi\)
0.950427 − 0.310948i \(-0.100647\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.0684166 − 0.0395003i 0.00779679 − 0.00450148i
\(78\) 0 0
\(79\) −6.72394 + 11.6462i −0.756503 + 1.31030i 0.188121 + 0.982146i \(0.439760\pi\)
−0.944624 + 0.328155i \(0.893573\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.26457 + 3.03950i 0.577862 + 0.333629i 0.760283 − 0.649592i \(-0.225060\pi\)
−0.182422 + 0.983220i \(0.558394\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.13440 −0.862244 −0.431122 − 0.902294i \(-0.641882\pi\)
−0.431122 + 0.902294i \(0.641882\pi\)
\(90\) 0 0
\(91\) 0.227007 0.0237968
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.61459 − 5.55098i −0.976213 − 0.563617i −0.0750885 − 0.997177i \(-0.523924\pi\)
−0.901125 + 0.433560i \(0.857257\pi\)
\(98\) 0 0
\(99\) 0 0
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 2700.2.s.d.2449.4 16
3.2 odd 2 900.2.s.d.349.3 16
5.2 odd 4 2700.2.i.e.1801.3 8
5.3 odd 4 2700.2.i.d.1801.2 8
5.4 even 2 inner 2700.2.s.d.2449.5 16
9.2 odd 6 8100.2.d.q.649.5 8
9.4 even 3 inner 2700.2.s.d.1549.5 16
9.5 odd 6 900.2.s.d.49.6 16
9.7 even 3 8100.2.d.s.649.5 8
15.2 even 4 900.2.i.d.601.4 yes 8
15.8 even 4 900.2.i.e.601.1 yes 8
15.14 odd 2 900.2.s.d.349.6 16
45.2 even 12 8100.2.a.x.1.2 4
45.4 even 6 inner 2700.2.s.d.1549.4 16
45.7 odd 12 8100.2.a.y.1.2 4
45.13 odd 12 2700.2.i.d.901.2 8
45.14 odd 6 900.2.s.d.49.3 16
45.22 odd 12 2700.2.i.e.901.3 8
45.23 even 12 900.2.i.e.301.1 yes 8
45.29 odd 6 8100.2.d.q.649.4 8
45.32 even 12 900.2.i.d.301.4 ✓ 8
45.34 even 6 8100.2.d.s.649.4 8
45.38 even 12 8100.2.a.z.1.3 4
45.43 odd 12 8100.2.a.ba.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
900.2.i.d.301.4 ✓ 8 45.32 even 12
900.2.i.d.601.4 yes 8 15.2 even 4
900.2.i.e.301.1 yes 8 45.23 even 12
900.2.i.e.601.1 yes 8 15.8 even 4
900.2.s.d.49.3 16 45.14 odd 6
900.2.s.d.49.6 16 9.5 odd 6
900.2.s.d.349.3 16 3.2 odd 2
900.2.s.d.349.6 16 15.14 odd 2
2700.2.i.d.901.2 8 45.13 odd 12
2700.2.i.d.1801.2 8 5.3 odd 4
2700.2.i.e.901.3 8 45.22 odd 12
2700.2.i.e.1801.3 8 5.2 odd 4
2700.2.s.d.1549.4 16 45.4 even 6 inner
2700.2.s.d.1549.5 16 9.4 even 3 inner
2700.2.s.d.2449.4 16 1.1 even 1 trivial
2700.2.s.d.2449.5 16 5.4 even 2 inner
8100.2.a.x.1.2 4 45.2 even 12
8100.2.a.y.1.2 4 45.7 odd 12
8100.2.a.z.1.3 4 45.38 even 12
8100.2.a.ba.1.3 4 45.43 odd 12
8100.2.d.q.649.4 8 45.29 odd 6
8100.2.d.q.649.5 8 9.2 odd 6
8100.2.d.s.649.4 8 45.34 even 6
8100.2.d.s.649.5 8 9.7 even 3