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Properties

Label 1600.2.f.c.1249.3
Level $1600$
Weight $2$
Character 1600.1249
Analytic conductor $12.776$
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] = [1600,2,Mod(1249,1600)] 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("1600.1249"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-4,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,0,0, 0,0] 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: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.3
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1600.1249
Dual form 1600.2.f.c.1249.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.46410i q^{7} -2.00000 q^{9} -3.00000i q^{11} -3.46410 q^{13} -3.00000i q^{17} +1.00000i q^{19} -3.46410i q^{21} +5.00000 q^{27} -10.3923i q^{29} +6.92820 q^{31} +3.00000i q^{33} +10.3923 q^{37} +3.46410 q^{39} -9.00000 q^{41} +4.00000 q^{43} +10.3923i q^{47} -5.00000 q^{49} +3.00000i q^{51} -1.00000i q^{57} -12.0000i q^{59} -3.46410i q^{61} -6.92820i q^{63} +11.0000 q^{67} +10.3923 q^{71} +7.00000i q^{73} +10.3923 q^{77} +10.3923 q^{79} +1.00000 q^{81} -15.0000 q^{83} +10.3923i q^{87} +3.00000 q^{89} -12.0000i q^{91} -6.92820 q^{93} -14.0000i q^{97} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 8 q^{9} + 20 q^{27} - 36 q^{41} + 16 q^{43} - 20 q^{49} + 44 q^{67} + 4 q^{81} - 60 q^{83} + 12 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) −1.00000 −0.577350 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.46410i 1.30931i 0.755929 + 0.654654i \(0.227186\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) − 3.00000i − 0.904534i −0.891883 − 0.452267i \(-0.850615\pi\)
0.891883 − 0.452267i \(-0.149385\pi\)
\(12\) 0 0
\(13\) −3.46410 −0.960769 −0.480384 − 0.877058i \(-0.659503\pi\)
−0.480384 + 0.877058i \(0.659503\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.00000i − 0.727607i −0.931476 − 0.363803i \(-0.881478\pi\)
0.931476 − 0.363803i \(-0.118522\pi\)
\(18\) 0 0
\(19\) 1.00000i 0.229416i 0.993399 + 0.114708i \(0.0365932\pi\)
−0.993399 + 0.114708i \(0.963407\pi\)
\(20\) 0 0
\(21\) − 3.46410i − 0.755929i
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) − 10.3923i − 1.92980i −0.262613 − 0.964901i \(-0.584584\pi\)
0.262613 − 0.964901i \(-0.415416\pi\)
\(30\) 0 0
\(31\) 6.92820 1.24434 0.622171 − 0.782881i \(-0.286251\pi\)
0.622171 + 0.782881i \(0.286251\pi\)
\(32\) 0 0
\(33\) 3.00000i 0.522233i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.3923 1.70848 0.854242 − 0.519875i \(-0.174022\pi\)
0.854242 + 0.519875i \(0.174022\pi\)
\(38\) 0 0
\(39\) 3.46410 0.554700
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 − 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.3923i 1.51587i 0.652328 + 0.757937i \(0.273792\pi\)
−0.652328 + 0.757937i \(0.726208\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 3.00000i 0.420084i
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.00000i − 0.132453i
\(58\) 0 0
\(59\) − 12.0000i − 1.56227i −0.624364 − 0.781133i \(-0.714642\pi\)
0.624364 − 0.781133i \(-0.285358\pi\)
\(60\) 0 0
\(61\) − 3.46410i − 0.443533i −0.975100 − 0.221766i \(-0.928818\pi\)
0.975100 − 0.221766i \(-0.0711822\pi\)
\(62\) 0 0
\(63\) − 6.92820i − 0.872872i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 11.0000 1.34386 0.671932 − 0.740613i \(-0.265465\pi\)
0.671932 + 0.740613i \(0.265465\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.3923 1.23334 0.616670 − 0.787222i \(-0.288481\pi\)
0.616670 + 0.787222i \(0.288481\pi\)
\(72\) 0 0
\(73\) 7.00000i 0.819288i 0.912245 + 0.409644i \(0.134347\pi\)
−0.912245 + 0.409644i \(0.865653\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.3923 1.18431
\(78\) 0 0
\(79\) 10.3923 1.16923 0.584613 − 0.811312i \(-0.301246\pi\)
0.584613 + 0.811312i \(0.301246\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −15.0000 −1.64646 −0.823232 − 0.567705i \(-0.807831\pi\)
−0.823232 + 0.567705i \(0.807831\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 10.3923i 1.11417i
\(88\) 0 0
\(89\) 3.00000 0.317999 0.159000 − 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) − 12.0000i − 1.25794i
\(92\) 0 0
\(93\) −6.92820 −0.718421
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 14.0000i − 1.42148i −0.703452 − 0.710742i \(-0.748359\pi\)
0.703452 − 0.710742i \(-0.251641\pi\)
\(98\) 0 0
\(99\) 6.00000i 0.603023i
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 1600.2.f.c.1249.3 4
4.3 odd 2 1600.2.f.g.1249.2 4
5.2 odd 4 1600.2.d.f.801.3 yes 4
5.3 odd 4 1600.2.d.e.801.2 yes 4
5.4 even 2 1600.2.f.g.1249.1 4
8.3 odd 2 inner 1600.2.f.c.1249.1 4
8.5 even 2 1600.2.f.g.1249.4 4
20.3 even 4 1600.2.d.e.801.3 yes 4
20.7 even 4 1600.2.d.f.801.2 yes 4
20.19 odd 2 inner 1600.2.f.c.1249.4 4
40.3 even 4 1600.2.d.e.801.1 ✓ 4
40.13 odd 4 1600.2.d.e.801.4 yes 4
40.19 odd 2 1600.2.f.g.1249.3 4
40.27 even 4 1600.2.d.f.801.4 yes 4
40.29 even 2 inner 1600.2.f.c.1249.2 4
40.37 odd 4 1600.2.d.f.801.1 yes 4
80.3 even 4 6400.2.a.bb.1.2 2
80.13 odd 4 6400.2.a.cf.1.1 2
80.27 even 4 6400.2.a.ba.1.1 2
80.37 odd 4 6400.2.a.cg.1.2 2
80.43 even 4 6400.2.a.cf.1.2 2
80.53 odd 4 6400.2.a.bb.1.1 2
80.67 even 4 6400.2.a.cg.1.1 2
80.77 odd 4 6400.2.a.ba.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1600.2.d.e.801.1 ✓ 4 40.3 even 4
1600.2.d.e.801.2 yes 4 5.3 odd 4
1600.2.d.e.801.3 yes 4 20.3 even 4
1600.2.d.e.801.4 yes 4 40.13 odd 4
1600.2.d.f.801.1 yes 4 40.37 odd 4
1600.2.d.f.801.2 yes 4 20.7 even 4
1600.2.d.f.801.3 yes 4 5.2 odd 4
1600.2.d.f.801.4 yes 4 40.27 even 4
1600.2.f.c.1249.1 4 8.3 odd 2 inner
1600.2.f.c.1249.2 4 40.29 even 2 inner
1600.2.f.c.1249.3 4 1.1 even 1 trivial
1600.2.f.c.1249.4 4 20.19 odd 2 inner
1600.2.f.g.1249.1 4 5.4 even 2
1600.2.f.g.1249.2 4 4.3 odd 2
1600.2.f.g.1249.3 4 40.19 odd 2
1600.2.f.g.1249.4 4 8.5 even 2
6400.2.a.ba.1.1 2 80.27 even 4
6400.2.a.ba.1.2 2 80.77 odd 4
6400.2.a.bb.1.1 2 80.53 odd 4
6400.2.a.bb.1.2 2 80.3 even 4
6400.2.a.cf.1.1 2 80.13 odd 4
6400.2.a.cf.1.2 2 80.43 even 4
6400.2.a.cg.1.1 2 80.67 even 4
6400.2.a.cg.1.2 2 80.37 odd 4