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Properties

Label 7200.2.k.l.3601.1
Level $7200$
Weight $2$
Character 7200.3601
Analytic conductor $57.492$
Analytic rank $0$
Dimension $4$
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] = [7200,2,Mod(3601,7200)] 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("7200.3601"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7200.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, -16] 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: \(57.4922894553\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{53}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3601.1
Root \(1.22474 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 7200.3601
Dual form 7200.2.k.l.3601.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.44949 q^{7} -3.46410i q^{11} +4.89898 q^{17} +3.46410i q^{19} +2.44949 q^{23} -4.00000 q^{31} +8.48528i q^{37} -4.24264i q^{43} +7.34847 q^{47} -1.00000 q^{49} -5.65685i q^{53} +10.3923i q^{59} +3.46410i q^{61} -4.24264i q^{67} -12.0000 q^{71} +4.89898 q^{73} +8.48528i q^{77} -4.00000 q^{79} -9.89949i q^{83} +6.00000 q^{89} +4.89898 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{31} - 4 q^{49} - 48 q^{71} - 16 q^{79} + 24 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(6401\) \(6751\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.44949 −0.925820 −0.462910 − 0.886405i \(-0.653195\pi\)
−0.462910 + 0.886405i \(0.653195\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 3.46410i − 1.04447i −0.852803 − 0.522233i \(-0.825099\pi\)
0.852803 − 0.522233i \(-0.174901\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.89898 1.18818 0.594089 − 0.804400i \(-0.297513\pi\)
0.594089 + 0.804400i \(0.297513\pi\)
\(18\) 0 0
\(19\) 3.46410i 0.794719i 0.917663 + 0.397360i \(0.130073\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.44949 0.510754 0.255377 − 0.966842i \(-0.417800\pi\)
0.255377 + 0.966842i \(0.417800\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.48528i 1.39497i 0.716599 + 0.697486i \(0.245698\pi\)
−0.716599 + 0.697486i \(0.754302\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) − 4.24264i − 0.646997i −0.946229 − 0.323498i \(-0.895141\pi\)
0.946229 − 0.323498i \(-0.104859\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.34847 1.07188 0.535942 − 0.844255i \(-0.319956\pi\)
0.535942 + 0.844255i \(0.319956\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 5.65685i − 0.777029i −0.921443 − 0.388514i \(-0.872988\pi\)
0.921443 − 0.388514i \(-0.127012\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.3923i 1.35296i 0.736460 + 0.676481i \(0.236496\pi\)
−0.736460 + 0.676481i \(0.763504\pi\)
\(60\) 0 0
\(61\) 3.46410i 0.443533i 0.975100 + 0.221766i \(0.0711822\pi\)
−0.975100 + 0.221766i \(0.928818\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.24264i − 0.518321i −0.965834 − 0.259161i \(-0.916554\pi\)
0.965834 − 0.259161i \(-0.0834459\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 4.89898 0.573382 0.286691 − 0.958023i \(-0.407445\pi\)
0.286691 + 0.958023i \(0.407445\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.48528i 0.966988i
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 9.89949i − 1.08661i −0.839535 − 0.543305i \(-0.817173\pi\)
0.839535 − 0.543305i \(-0.182827\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.89898 0.497416 0.248708 − 0.968579i \(-0.419994\pi\)
0.248708 + 0.968579i \(0.419994\pi\)
\(98\) 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 7200.2.k.l.3601.1 4
3.2 odd 2 800.2.d.f.401.3 4
4.3 odd 2 1800.2.k.m.901.4 4
5.2 odd 4 1440.2.d.c.1009.1 4
5.3 odd 4 1440.2.d.c.1009.3 4
5.4 even 2 inner 7200.2.k.l.3601.3 4
8.3 odd 2 1800.2.k.m.901.3 4
8.5 even 2 inner 7200.2.k.l.3601.2 4
12.11 even 2 200.2.d.e.101.1 4
15.2 even 4 160.2.f.a.49.4 4
15.8 even 4 160.2.f.a.49.2 4
15.14 odd 2 800.2.d.f.401.2 4
20.3 even 4 360.2.d.b.109.3 4
20.7 even 4 360.2.d.b.109.2 4
20.19 odd 2 1800.2.k.m.901.1 4
24.5 odd 2 800.2.d.f.401.1 4
24.11 even 2 200.2.d.e.101.2 4
40.3 even 4 360.2.d.b.109.1 4
40.13 odd 4 1440.2.d.c.1009.2 4
40.19 odd 2 1800.2.k.m.901.2 4
40.27 even 4 360.2.d.b.109.4 4
40.29 even 2 inner 7200.2.k.l.3601.4 4
40.37 odd 4 1440.2.d.c.1009.4 4
48.5 odd 4 6400.2.a.cm.1.4 4
48.11 even 4 6400.2.a.co.1.1 4
48.29 odd 4 6400.2.a.cm.1.2 4
48.35 even 4 6400.2.a.co.1.3 4
60.23 odd 4 40.2.f.a.29.2 yes 4
60.47 odd 4 40.2.f.a.29.3 yes 4
60.59 even 2 200.2.d.e.101.4 4
120.29 odd 2 800.2.d.f.401.4 4
120.53 even 4 160.2.f.a.49.3 4
120.59 even 2 200.2.d.e.101.3 4
120.77 even 4 160.2.f.a.49.1 4
120.83 odd 4 40.2.f.a.29.4 yes 4
120.107 odd 4 40.2.f.a.29.1 ✓ 4
240.29 odd 4 6400.2.a.cm.1.3 4
240.53 even 4 1280.2.c.k.769.3 4
240.59 even 4 6400.2.a.co.1.4 4
240.77 even 4 1280.2.c.k.769.4 4
240.83 odd 4 1280.2.c.i.769.4 4
240.107 odd 4 1280.2.c.i.769.3 4
240.149 odd 4 6400.2.a.cm.1.1 4
240.173 even 4 1280.2.c.k.769.2 4
240.179 even 4 6400.2.a.co.1.2 4
240.197 even 4 1280.2.c.k.769.1 4
240.203 odd 4 1280.2.c.i.769.1 4
240.227 odd 4 1280.2.c.i.769.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.f.a.29.1 ✓ 4 120.107 odd 4
40.2.f.a.29.2 yes 4 60.23 odd 4
40.2.f.a.29.3 yes 4 60.47 odd 4
40.2.f.a.29.4 yes 4 120.83 odd 4
160.2.f.a.49.1 4 120.77 even 4
160.2.f.a.49.2 4 15.8 even 4
160.2.f.a.49.3 4 120.53 even 4
160.2.f.a.49.4 4 15.2 even 4
200.2.d.e.101.1 4 12.11 even 2
200.2.d.e.101.2 4 24.11 even 2
200.2.d.e.101.3 4 120.59 even 2
200.2.d.e.101.4 4 60.59 even 2
360.2.d.b.109.1 4 40.3 even 4
360.2.d.b.109.2 4 20.7 even 4
360.2.d.b.109.3 4 20.3 even 4
360.2.d.b.109.4 4 40.27 even 4
800.2.d.f.401.1 4 24.5 odd 2
800.2.d.f.401.2 4 15.14 odd 2
800.2.d.f.401.3 4 3.2 odd 2
800.2.d.f.401.4 4 120.29 odd 2
1280.2.c.i.769.1 4 240.203 odd 4
1280.2.c.i.769.2 4 240.227 odd 4
1280.2.c.i.769.3 4 240.107 odd 4
1280.2.c.i.769.4 4 240.83 odd 4
1280.2.c.k.769.1 4 240.197 even 4
1280.2.c.k.769.2 4 240.173 even 4
1280.2.c.k.769.3 4 240.53 even 4
1280.2.c.k.769.4 4 240.77 even 4
1440.2.d.c.1009.1 4 5.2 odd 4
1440.2.d.c.1009.2 4 40.13 odd 4
1440.2.d.c.1009.3 4 5.3 odd 4
1440.2.d.c.1009.4 4 40.37 odd 4
1800.2.k.m.901.1 4 20.19 odd 2
1800.2.k.m.901.2 4 40.19 odd 2
1800.2.k.m.901.3 4 8.3 odd 2
1800.2.k.m.901.4 4 4.3 odd 2
6400.2.a.cm.1.1 4 240.149 odd 4
6400.2.a.cm.1.2 4 48.29 odd 4
6400.2.a.cm.1.3 4 240.29 odd 4
6400.2.a.cm.1.4 4 48.5 odd 4
6400.2.a.co.1.1 4 48.11 even 4
6400.2.a.co.1.2 4 240.179 even 4
6400.2.a.co.1.3 4 48.35 even 4
6400.2.a.co.1.4 4 240.59 even 4
7200.2.k.l.3601.1 4 1.1 even 1 trivial
7200.2.k.l.3601.2 4 8.5 even 2 inner
7200.2.k.l.3601.3 4 5.4 even 2 inner
7200.2.k.l.3601.4 4 40.29 even 2 inner