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Properties

Label 900.2.s.d.49.6
Level $900$
Weight $2$
Character 900.49
Analytic conductor $7.187$
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] = [900,2,Mod(49,900)] 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("900.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.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,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.18653618192\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 49.6
Root \(-0.263711 - 1.38941i\) of defining polynomial
Character \(\chi\) \(=\) 900.49
Dual form 900.2.s.d.349.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.07141 + 1.36091i) q^{3} +(0.0748933 - 0.0432397i) q^{7} +(-0.704170 + 2.91619i) q^{9} +(0.456760 + 0.791132i) q^{11} +(2.27331 + 1.31249i) q^{13} +2.08648i q^{17} -4.93847 q^{19} +(0.139087 + 0.0555960i) q^{21} +(7.34128 + 4.23849i) q^{23} +(-4.72313 + 2.16611i) q^{27} +(1.19899 + 2.07671i) q^{29} +(-1.81249 + 3.13933i) q^{31} +(-0.587286 + 1.46924i) q^{33} -5.85199i q^{37} +(0.649447 + 4.49999i) q^{39} +(-3.32497 + 5.75902i) q^{41} +(7.14469 - 4.12499i) q^{43} +(2.32831 - 1.34425i) q^{47} +(-3.49626 + 6.05570i) q^{49} +(-2.83952 + 2.23547i) q^{51} +5.73642i q^{53} +(-5.29112 - 6.72083i) q^{57} +(6.16922 - 10.6854i) q^{59} +(3.16823 + 5.48753i) q^{61} +(0.0733573 + 0.248851i) q^{63} +(-5.33946 - 3.08274i) q^{67} +(2.09729 + 14.5320i) q^{69} -12.3905 q^{71} -5.31349i q^{73} +(0.0684166 + 0.0395003i) q^{77} +(-6.72394 - 11.6462i) q^{79} +(-8.00829 - 4.10698i) q^{81} +(5.26457 - 3.03950i) q^{83} +(-1.54162 + 3.85673i) q^{87} +8.13440 q^{89} +0.227007 q^{91} +(-6.21428 + 0.896857i) q^{93} +(9.61459 - 5.55098i) q^{97} +(-2.62873 + 0.774907i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 10 q^{9} + 6 q^{11} + 16 q^{19} + 26 q^{21} + 18 q^{29} - 4 q^{31} + 34 q^{39} + 18 q^{41} + 18 q^{49} + 6 q^{51} + 30 q^{59} + 2 q^{61} - 18 q^{69} - 48 q^{71} - 14 q^{79} - 62 q^{81} - 12 q^{89}+ \cdots - 66 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(-1\)

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
\(3\) 1.07141 + 1.36091i 0.618578 + 0.785724i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.0748933 − 0.0432397i 0.0283070 − 0.0163431i −0.485780 − 0.874081i \(-0.661464\pi\)
0.514087 + 0.857738i \(0.328131\pi\)
\(8\) 0 0
\(9\) −0.704170 + 2.91619i −0.234723 + 0.972062i
\(10\) 0 0
\(11\) 0.456760 + 0.791132i 0.137718 + 0.238535i 0.926633 − 0.375968i \(-0.122690\pi\)
−0.788914 + 0.614503i \(0.789356\pi\)
\(12\) 0 0
\(13\) 2.27331 + 1.31249i 0.630501 + 0.364020i 0.780946 − 0.624598i \(-0.214737\pi\)
−0.150445 + 0.988618i \(0.548071\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.08648i 0.506046i 0.967460 + 0.253023i \(0.0814248\pi\)
−0.967460 + 0.253023i \(0.918575\pi\)
\(18\) 0 0
\(19\) −4.93847 −1.13296 −0.566482 − 0.824074i \(-0.691696\pi\)
−0.566482 + 0.824074i \(0.691696\pi\)
\(20\) 0 0
\(21\) 0.139087 + 0.0555960i 0.0303512 + 0.0121320i
\(22\) 0 0
\(23\) 7.34128 + 4.23849i 1.53076 + 0.883786i 0.999327 + 0.0366878i \(0.0116807\pi\)
0.531436 + 0.847098i \(0.321653\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −4.72313 + 2.16611i −0.908967 + 0.416868i
\(28\) 0 0
\(29\) 1.19899 + 2.07671i 0.222647 + 0.385636i 0.955611 − 0.294632i \(-0.0951970\pi\)
−0.732964 + 0.680267i \(0.761864\pi\)
\(30\) 0 0
\(31\) −1.81249 + 3.13933i −0.325533 + 0.563840i −0.981620 − 0.190845i \(-0.938877\pi\)
0.656087 + 0.754685i \(0.272211\pi\)
\(32\) 0 0
\(33\) −0.587286 + 1.46924i −0.102233 + 0.255761i
\(34\) 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
\(39\) 0.649447 + 4.49999i 0.103995 + 0.720575i
\(40\) 0 0
\(41\) −3.32497 + 5.75902i −0.519273 + 0.899407i 0.480476 + 0.877008i \(0.340464\pi\)
−0.999749 + 0.0223994i \(0.992869\pi\)
\(42\) 0 0
\(43\) 7.14469 − 4.12499i 1.08955 − 0.629055i 0.156097 − 0.987742i \(-0.450109\pi\)
0.933458 + 0.358687i \(0.116776\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.32831 − 1.34425i 0.339619 − 0.196079i −0.320485 − 0.947254i \(-0.603846\pi\)
0.660103 + 0.751175i \(0.270512\pi\)
\(48\) 0 0
\(49\) −3.49626 + 6.05570i −0.499466 + 0.865100i
\(50\) 0 0
\(51\) −2.83952 + 2.23547i −0.397612 + 0.313028i
\(52\) 0 0
\(53\) 5.73642i 0.787958i 0.919120 + 0.393979i \(0.128902\pi\)
−0.919120 + 0.393979i \(0.871098\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.29112 − 6.72083i −0.700826 − 0.890196i
\(58\) 0 0
\(59\) 6.16922 − 10.6854i 0.803164 − 1.39112i −0.114360 − 0.993439i \(-0.536482\pi\)
0.917524 − 0.397681i \(-0.130185\pi\)
\(60\) 0 0
\(61\) 3.16823 + 5.48753i 0.405650 + 0.702606i 0.994397 − 0.105711i \(-0.0337119\pi\)
−0.588747 + 0.808317i \(0.700379\pi\)
\(62\) 0 0
\(63\) 0.0733573 + 0.248851i 0.00924215 + 0.0313523i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −5.33946 − 3.08274i −0.652319 − 0.376617i 0.137025 − 0.990568i \(-0.456246\pi\)
−0.789344 + 0.613951i \(0.789579\pi\)
\(68\) 0 0
\(69\) 2.09729 + 14.5320i 0.252484 + 1.74945i
\(70\) 0 0
\(71\) −12.3905 −1.47048 −0.735241 − 0.677806i \(-0.762931\pi\)
−0.735241 + 0.677806i \(0.762931\pi\)
\(72\) 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 −0.944624 − 0.328155i \(-0.893573\pi\)
0.188121 − 0.982146i \(-0.439760\pi\)
\(80\) 0 0
\(81\) −8.00829 − 4.10698i −0.889810 − 0.456331i
\(82\) 0 0
\(83\) 5.26457 − 3.03950i 0.577862 − 0.333629i −0.182422 − 0.983220i \(-0.558394\pi\)
0.760283 + 0.649592i \(0.225060\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.54162 + 3.85673i −0.165279 + 0.413484i
\(88\) 0 0
\(89\) 8.13440 0.862244 0.431122 − 0.902294i \(-0.358118\pi\)
0.431122 + 0.902294i \(0.358118\pi\)
\(90\) 0 0
\(91\) 0.227007 0.0237968
\(92\) 0 0
\(93\) −6.21428 + 0.896857i −0.644390 + 0.0929998i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.61459 − 5.55098i 0.976213 − 0.563617i 0.0750885 − 0.997177i \(-0.476076\pi\)
0.901125 + 0.433560i \(0.142743\pi\)
\(98\) 0 0
\(99\) −2.62873 + 0.774907i −0.264197 + 0.0778811i
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 900.2.s.d.49.6 16
3.2 odd 2 2700.2.s.d.1549.5 16
5.2 odd 4 900.2.i.d.301.4 ✓ 8
5.3 odd 4 900.2.i.e.301.1 yes 8
5.4 even 2 inner 900.2.s.d.49.3 16
9.2 odd 6 2700.2.s.d.2449.4 16
9.4 even 3 8100.2.d.q.649.5 8
9.5 odd 6 8100.2.d.s.649.5 8
9.7 even 3 inner 900.2.s.d.349.3 16
15.2 even 4 2700.2.i.e.901.3 8
15.8 even 4 2700.2.i.d.901.2 8
15.14 odd 2 2700.2.s.d.1549.4 16
45.2 even 12 2700.2.i.e.1801.3 8
45.4 even 6 8100.2.d.q.649.4 8
45.7 odd 12 900.2.i.d.601.4 yes 8
45.13 odd 12 8100.2.a.z.1.3 4
45.14 odd 6 8100.2.d.s.649.4 8
45.22 odd 12 8100.2.a.x.1.2 4
45.23 even 12 8100.2.a.ba.1.3 4
45.29 odd 6 2700.2.s.d.2449.5 16
45.32 even 12 8100.2.a.y.1.2 4
45.34 even 6 inner 900.2.s.d.349.6 16
45.38 even 12 2700.2.i.d.1801.2 8
45.43 odd 12 900.2.i.e.601.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
900.2.i.d.301.4 ✓ 8 5.2 odd 4
900.2.i.d.601.4 yes 8 45.7 odd 12
900.2.i.e.301.1 yes 8 5.3 odd 4
900.2.i.e.601.1 yes 8 45.43 odd 12
900.2.s.d.49.3 16 5.4 even 2 inner
900.2.s.d.49.6 16 1.1 even 1 trivial
900.2.s.d.349.3 16 9.7 even 3 inner
900.2.s.d.349.6 16 45.34 even 6 inner
2700.2.i.d.901.2 8 15.8 even 4
2700.2.i.d.1801.2 8 45.38 even 12
2700.2.i.e.901.3 8 15.2 even 4
2700.2.i.e.1801.3 8 45.2 even 12
2700.2.s.d.1549.4 16 15.14 odd 2
2700.2.s.d.1549.5 16 3.2 odd 2
2700.2.s.d.2449.4 16 9.2 odd 6
2700.2.s.d.2449.5 16 45.29 odd 6
8100.2.a.x.1.2 4 45.22 odd 12
8100.2.a.y.1.2 4 45.32 even 12
8100.2.a.z.1.3 4 45.13 odd 12
8100.2.a.ba.1.3 4 45.23 even 12
8100.2.d.q.649.4 8 45.4 even 6
8100.2.d.q.649.5 8 9.4 even 3
8100.2.d.s.649.4 8 45.14 odd 6
8100.2.d.s.649.5 8 9.5 odd 6