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Properties

Label 1600.2.c.k.449.2
Level $1600$
Weight $2$
Character 1600.449
Analytic conductor $12.776$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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(449,1600)] 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("1600.449"); 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, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,6,0,-8,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1600.449
Dual form 1600.2.c.k.449.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.00000i q^{7} +3.00000 q^{9} -4.00000 q^{11} +2.00000i q^{13} -2.00000i q^{17} +4.00000 q^{19} +4.00000i q^{23} -2.00000 q^{29} -8.00000 q^{31} +6.00000i q^{37} -6.00000 q^{41} +8.00000i q^{43} -4.00000i q^{47} -9.00000 q^{49} -6.00000i q^{53} -4.00000 q^{59} +2.00000 q^{61} +12.0000i q^{63} +8.00000i q^{67} -6.00000i q^{73} -16.0000i q^{77} +9.00000 q^{81} +16.0000i q^{83} +6.00000 q^{89} -8.00000 q^{91} +14.0000i q^{97} -12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{9} - 8 q^{11} + 8 q^{19} - 4 q^{29} - 16 q^{31} - 12 q^{41} - 18 q^{49} - 8 q^{59} + 4 q^{61} + 18 q^{81} + 12 q^{89} - 16 q^{91} - 24 q^{99}+O(q^{100}) \) 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
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2.00000i − 0.485071i −0.970143 − 0.242536i \(-0.922021\pi\)
0.970143 − 0.242536i \(-0.0779791\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 4.00000i − 0.583460i −0.956501 − 0.291730i \(-0.905769\pi\)
0.956501 − 0.291730i \(-0.0942309\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 6.00000i − 0.824163i −0.911147 − 0.412082i \(-0.864802\pi\)
0.911147 − 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 12.0000i 1.51186i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) − 6.00000i − 0.702247i −0.936329 − 0.351123i \(-0.885800\pi\)
0.936329 − 0.351123i \(-0.114200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 16.0000i − 1.82337i
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 16.0000i 1.75623i 0.478451 + 0.878114i \(0.341198\pi\)
−0.478451 + 0.878114i \(0.658802\pi\)
\(84\) 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
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) 0 0
\(99\) −12.0000 −1.20605
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.c.k.449.2 2
4.3 odd 2 1600.2.c.m.449.1 2
5.2 odd 4 320.2.a.c.1.1 1
5.3 odd 4 1600.2.a.o.1.1 1
5.4 even 2 inner 1600.2.c.k.449.1 2
8.3 odd 2 400.2.c.d.49.1 2
8.5 even 2 200.2.c.b.49.2 2
15.2 even 4 2880.2.a.t.1.1 1
20.3 even 4 1600.2.a.k.1.1 1
20.7 even 4 320.2.a.d.1.1 1
20.19 odd 2 1600.2.c.m.449.2 2
24.5 odd 2 1800.2.f.a.649.2 2
24.11 even 2 3600.2.f.t.2449.1 2
40.3 even 4 400.2.a.e.1.1 1
40.13 odd 4 200.2.a.c.1.1 1
40.19 odd 2 400.2.c.d.49.2 2
40.27 even 4 80.2.a.a.1.1 1
40.29 even 2 200.2.c.b.49.1 2
40.37 odd 4 40.2.a.a.1.1 ✓ 1
60.47 odd 4 2880.2.a.bg.1.1 1
80.27 even 4 1280.2.d.a.641.1 2
80.37 odd 4 1280.2.d.j.641.1 2
80.67 even 4 1280.2.d.a.641.2 2
80.77 odd 4 1280.2.d.j.641.2 2
120.29 odd 2 1800.2.f.a.649.1 2
120.53 even 4 1800.2.a.v.1.1 1
120.59 even 2 3600.2.f.t.2449.2 2
120.77 even 4 360.2.a.a.1.1 1
120.83 odd 4 3600.2.a.h.1.1 1
120.107 odd 4 720.2.a.e.1.1 1
280.13 even 4 9800.2.a.x.1.1 1
280.27 odd 4 3920.2.a.s.1.1 1
280.37 odd 12 1960.2.q.h.361.1 2
280.117 even 12 1960.2.q.i.361.1 2
280.157 even 12 1960.2.q.i.961.1 2
280.237 even 4 1960.2.a.g.1.1 1
280.277 odd 12 1960.2.q.h.961.1 2
360.77 even 12 3240.2.q.x.2161.1 2
360.157 odd 12 3240.2.q.k.2161.1 2
360.277 odd 12 3240.2.q.k.1081.1 2
360.317 even 12 3240.2.q.x.1081.1 2
440.197 even 4 4840.2.a.f.1.1 1
440.307 odd 4 9680.2.a.q.1.1 1
520.77 odd 4 6760.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.a.a.1.1 ✓ 1 40.37 odd 4
80.2.a.a.1.1 1 40.27 even 4
200.2.a.c.1.1 1 40.13 odd 4
200.2.c.b.49.1 2 40.29 even 2
200.2.c.b.49.2 2 8.5 even 2
320.2.a.c.1.1 1 5.2 odd 4
320.2.a.d.1.1 1 20.7 even 4
360.2.a.a.1.1 1 120.77 even 4
400.2.a.e.1.1 1 40.3 even 4
400.2.c.d.49.1 2 8.3 odd 2
400.2.c.d.49.2 2 40.19 odd 2
720.2.a.e.1.1 1 120.107 odd 4
1280.2.d.a.641.1 2 80.27 even 4
1280.2.d.a.641.2 2 80.67 even 4
1280.2.d.j.641.1 2 80.37 odd 4
1280.2.d.j.641.2 2 80.77 odd 4
1600.2.a.k.1.1 1 20.3 even 4
1600.2.a.o.1.1 1 5.3 odd 4
1600.2.c.k.449.1 2 5.4 even 2 inner
1600.2.c.k.449.2 2 1.1 even 1 trivial
1600.2.c.m.449.1 2 4.3 odd 2
1600.2.c.m.449.2 2 20.19 odd 2
1800.2.a.v.1.1 1 120.53 even 4
1800.2.f.a.649.1 2 120.29 odd 2
1800.2.f.a.649.2 2 24.5 odd 2
1960.2.a.g.1.1 1 280.237 even 4
1960.2.q.h.361.1 2 280.37 odd 12
1960.2.q.h.961.1 2 280.277 odd 12
1960.2.q.i.361.1 2 280.117 even 12
1960.2.q.i.961.1 2 280.157 even 12
2880.2.a.t.1.1 1 15.2 even 4
2880.2.a.bg.1.1 1 60.47 odd 4
3240.2.q.k.1081.1 2 360.277 odd 12
3240.2.q.k.2161.1 2 360.157 odd 12
3240.2.q.x.1081.1 2 360.317 even 12
3240.2.q.x.2161.1 2 360.77 even 12
3600.2.a.h.1.1 1 120.83 odd 4
3600.2.f.t.2449.1 2 24.11 even 2
3600.2.f.t.2449.2 2 120.59 even 2
3920.2.a.s.1.1 1 280.27 odd 4
4840.2.a.f.1.1 1 440.197 even 4
6760.2.a.i.1.1 1 520.77 odd 4
9680.2.a.q.1.1 1 440.307 odd 4
9800.2.a.x.1.1 1 280.13 even 4