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Properties

Label 6400.2.a.cm.1.1
Level $6400$
Weight $2$
Character 6400.1
Self dual yes
Analytic conductor $51.104$
Analytic rank $1$
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] = [6400,2,Mod(1,6400)] 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("6400.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6400 = 2^{8} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6400.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, -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: yes
Analytic conductor: \(51.1042572936\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.93185\) of defining polynomial
Character \(\chi\) \(=\) 6400.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.41421 q^{3} -2.44949 q^{7} -1.00000 q^{9} -3.46410 q^{11} +4.89898 q^{17} +3.46410 q^{19} +3.46410 q^{21} -2.44949 q^{23} +5.65685 q^{27} -4.00000 q^{31} +4.89898 q^{33} +8.48528 q^{37} -4.24264 q^{43} +7.34847 q^{47} -1.00000 q^{49} -6.92820 q^{51} +5.65685 q^{53} -4.89898 q^{57} +10.3923 q^{59} +3.46410 q^{61} +2.44949 q^{63} +4.24264 q^{67} +3.46410 q^{69} -12.0000 q^{71} +4.89898 q^{73} +8.48528 q^{77} -4.00000 q^{79} -5.00000 q^{81} -9.89949 q^{83} +6.00000 q^{89} +5.65685 q^{93} -4.89898 q^{97} +3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} - 16 q^{31} - 4 q^{49} - 48 q^{71} - 16 q^{79} - 20 q^{81} + 24 q^{89}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.41421 −0.816497 −0.408248 − 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) 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
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −3.46410 −1.04447 −0.522233 − 0.852803i \(-0.674901\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 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
\(19\) 3.46410 0.794719 0.397360 − 0.917663i \(-0.369927\pi\)
0.397360 + 0.917663i \(0.369927\pi\)
\(20\) 0 0
\(21\) 3.46410 0.755929
\(22\) 0 0
\(23\) −2.44949 −0.510754 −0.255377 − 0.966842i \(-0.582200\pi\)
−0.255377 + 0.966842i \(0.582200\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 4.89898 0.852803
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.48528 1.39497 0.697486 − 0.716599i \(-0.254302\pi\)
0.697486 + 0.716599i \(0.254302\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −4.24264 −0.646997 −0.323498 − 0.946229i \(-0.604859\pi\)
−0.323498 + 0.946229i \(0.604859\pi\)
\(44\) 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
\(51\) −6.92820 −0.970143
\(52\) 0 0
\(53\) 5.65685 0.777029 0.388514 − 0.921443i \(-0.372988\pi\)
0.388514 + 0.921443i \(0.372988\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.89898 −0.648886
\(58\) 0 0
\(59\) 10.3923 1.35296 0.676481 − 0.736460i \(-0.263504\pi\)
0.676481 + 0.736460i \(0.263504\pi\)
\(60\) 0 0
\(61\) 3.46410 0.443533 0.221766 − 0.975100i \(-0.428818\pi\)
0.221766 + 0.975100i \(0.428818\pi\)
\(62\) 0 0
\(63\) 2.44949 0.308607
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.24264 0.518321 0.259161 − 0.965834i \(-0.416554\pi\)
0.259161 + 0.965834i \(0.416554\pi\)
\(68\) 0 0
\(69\) 3.46410 0.417029
\(70\) 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
\(77\) 8.48528 0.966988
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) −9.89949 −1.08661 −0.543305 − 0.839535i \(-0.682827\pi\)
−0.543305 + 0.839535i \(0.682827\pi\)
\(84\) 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
\(93\) 5.65685 0.586588
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.89898 −0.497416 −0.248708 − 0.968579i \(-0.580006\pi\)
−0.248708 + 0.968579i \(0.580006\pi\)
\(98\) 0 0
\(99\) 3.46410 0.348155
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 6400.2.a.cm.1.1 4
4.3 odd 2 6400.2.a.co.1.4 4
5.2 odd 4 1280.2.c.k.769.3 4
5.3 odd 4 1280.2.c.k.769.1 4
5.4 even 2 inner 6400.2.a.cm.1.4 4
8.3 odd 2 6400.2.a.co.1.2 4
8.5 even 2 inner 6400.2.a.cm.1.3 4
16.3 odd 4 200.2.d.e.101.4 4
16.5 even 4 800.2.d.f.401.4 4
16.11 odd 4 200.2.d.e.101.3 4
16.13 even 4 800.2.d.f.401.2 4
20.3 even 4 1280.2.c.i.769.3 4
20.7 even 4 1280.2.c.i.769.1 4
20.19 odd 2 6400.2.a.co.1.1 4
40.3 even 4 1280.2.c.i.769.2 4
40.13 odd 4 1280.2.c.k.769.4 4
40.19 odd 2 6400.2.a.co.1.3 4
40.27 even 4 1280.2.c.i.769.4 4
40.29 even 2 inner 6400.2.a.cm.1.2 4
40.37 odd 4 1280.2.c.k.769.2 4
48.5 odd 4 7200.2.k.l.3601.4 4
48.11 even 4 1800.2.k.m.901.2 4
48.29 odd 4 7200.2.k.l.3601.3 4
48.35 even 4 1800.2.k.m.901.1 4
80.3 even 4 40.2.f.a.29.3 yes 4
80.13 odd 4 160.2.f.a.49.4 4
80.19 odd 4 200.2.d.e.101.1 4
80.27 even 4 40.2.f.a.29.4 yes 4
80.29 even 4 800.2.d.f.401.3 4
80.37 odd 4 160.2.f.a.49.3 4
80.43 even 4 40.2.f.a.29.1 ✓ 4
80.53 odd 4 160.2.f.a.49.1 4
80.59 odd 4 200.2.d.e.101.2 4
80.67 even 4 40.2.f.a.29.2 yes 4
80.69 even 4 800.2.d.f.401.1 4
80.77 odd 4 160.2.f.a.49.2 4
240.29 odd 4 7200.2.k.l.3601.1 4
240.53 even 4 1440.2.d.c.1009.4 4
240.59 even 4 1800.2.k.m.901.3 4
240.77 even 4 1440.2.d.c.1009.3 4
240.83 odd 4 360.2.d.b.109.2 4
240.107 odd 4 360.2.d.b.109.1 4
240.149 odd 4 7200.2.k.l.3601.2 4
240.173 even 4 1440.2.d.c.1009.1 4
240.179 even 4 1800.2.k.m.901.4 4
240.197 even 4 1440.2.d.c.1009.2 4
240.203 odd 4 360.2.d.b.109.4 4
240.227 odd 4 360.2.d.b.109.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.f.a.29.1 ✓ 4 80.43 even 4
40.2.f.a.29.2 yes 4 80.67 even 4
40.2.f.a.29.3 yes 4 80.3 even 4
40.2.f.a.29.4 yes 4 80.27 even 4
160.2.f.a.49.1 4 80.53 odd 4
160.2.f.a.49.2 4 80.77 odd 4
160.2.f.a.49.3 4 80.37 odd 4
160.2.f.a.49.4 4 80.13 odd 4
200.2.d.e.101.1 4 80.19 odd 4
200.2.d.e.101.2 4 80.59 odd 4
200.2.d.e.101.3 4 16.11 odd 4
200.2.d.e.101.4 4 16.3 odd 4
360.2.d.b.109.1 4 240.107 odd 4
360.2.d.b.109.2 4 240.83 odd 4
360.2.d.b.109.3 4 240.227 odd 4
360.2.d.b.109.4 4 240.203 odd 4
800.2.d.f.401.1 4 80.69 even 4
800.2.d.f.401.2 4 16.13 even 4
800.2.d.f.401.3 4 80.29 even 4
800.2.d.f.401.4 4 16.5 even 4
1280.2.c.i.769.1 4 20.7 even 4
1280.2.c.i.769.2 4 40.3 even 4
1280.2.c.i.769.3 4 20.3 even 4
1280.2.c.i.769.4 4 40.27 even 4
1280.2.c.k.769.1 4 5.3 odd 4
1280.2.c.k.769.2 4 40.37 odd 4
1280.2.c.k.769.3 4 5.2 odd 4
1280.2.c.k.769.4 4 40.13 odd 4
1440.2.d.c.1009.1 4 240.173 even 4
1440.2.d.c.1009.2 4 240.197 even 4
1440.2.d.c.1009.3 4 240.77 even 4
1440.2.d.c.1009.4 4 240.53 even 4
1800.2.k.m.901.1 4 48.35 even 4
1800.2.k.m.901.2 4 48.11 even 4
1800.2.k.m.901.3 4 240.59 even 4
1800.2.k.m.901.4 4 240.179 even 4
6400.2.a.cm.1.1 4 1.1 even 1 trivial
6400.2.a.cm.1.2 4 40.29 even 2 inner
6400.2.a.cm.1.3 4 8.5 even 2 inner
6400.2.a.cm.1.4 4 5.4 even 2 inner
6400.2.a.co.1.1 4 20.19 odd 2
6400.2.a.co.1.2 4 8.3 odd 2
6400.2.a.co.1.3 4 40.19 odd 2
6400.2.a.co.1.4 4 4.3 odd 2
7200.2.k.l.3601.1 4 240.29 odd 4
7200.2.k.l.3601.2 4 240.149 odd 4
7200.2.k.l.3601.3 4 48.29 odd 4
7200.2.k.l.3601.4 4 48.5 odd 4