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Properties

Label 3200.2.f.q.449.4
Level $3200$
Weight $2$
Character 3200.449
Analytic conductor $25.552$
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] = [3200,2,Mod(449,3200)] 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("3200.449"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,28,0,0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(49)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.5521286468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 640)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.4
Root \(1.58114 + 1.58114i\) of defining polynomial
Character \(\chi\) \(=\) 3200.449
Dual form 3200.2.f.q.449.3

$q$-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)
 
\(f(q)\) \(=\) \(q+3.16228 q^{3} +3.16228i q^{7} +7.00000 q^{9} +6.00000 q^{13} -2.00000i q^{17} -6.32456i q^{19} +10.0000i q^{21} -3.16228i q^{23} +12.6491 q^{27} +4.00000i q^{29} -6.32456 q^{31} +2.00000 q^{37} +18.9737 q^{39} +3.16228 q^{43} +9.48683i q^{47} -3.00000 q^{49} -6.32456i q^{51} -6.00000 q^{53} -20.0000i q^{57} -6.32456i q^{59} +2.00000i q^{61} +22.1359i q^{63} +9.48683 q^{67} -10.0000i q^{69} +6.32456 q^{71} +14.0000i q^{73} -12.6491 q^{79} +19.0000 q^{81} -3.16228 q^{83} +12.6491i q^{87} -10.0000 q^{89} +18.9737i q^{91} -20.0000 q^{93} +2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 28 q^{9} + 24 q^{13} + 8 q^{37} - 12 q^{49} - 24 q^{53} + 76 q^{81} - 40 q^{89} - 80 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1151\) \(2177\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) 3.16228 1.82574 0.912871 − 0.408248i \(-0.133860\pi\)
0.912871 + 0.408248i \(0.133860\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.16228i 1.19523i 0.801784 + 0.597614i \(0.203885\pi\)
−0.801784 + 0.597614i \(0.796115\pi\)
\(8\) 0 0
\(9\) 7.00000 2.33333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 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
\(19\) − 6.32456i − 1.45095i −0.688247 − 0.725476i \(-0.741620\pi\)
0.688247 − 0.725476i \(-0.258380\pi\)
\(20\) 0 0
\(21\) 10.0000i 2.18218i
\(22\) 0 0
\(23\) − 3.16228i − 0.659380i −0.944089 − 0.329690i \(-0.893056\pi\)
0.944089 − 0.329690i \(-0.106944\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 12.6491 2.43432
\(28\) 0 0
\(29\) 4.00000i 0.742781i 0.928477 + 0.371391i \(0.121119\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) −6.32456 −1.13592 −0.567962 − 0.823055i \(-0.692268\pi\)
−0.567962 + 0.823055i \(0.692268\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 18.9737 3.03822
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 3.16228 0.482243 0.241121 − 0.970495i \(-0.422485\pi\)
0.241121 + 0.970495i \(0.422485\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.48683i 1.38380i 0.721995 + 0.691898i \(0.243225\pi\)
−0.721995 + 0.691898i \(0.756775\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) − 6.32456i − 0.885615i
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 20.0000i − 2.64906i
\(58\) 0 0
\(59\) − 6.32456i − 0.823387i −0.911322 − 0.411693i \(-0.864937\pi\)
0.911322 − 0.411693i \(-0.135063\pi\)
\(60\) 0 0
\(61\) 2.00000i 0.256074i 0.991769 + 0.128037i \(0.0408676\pi\)
−0.991769 + 0.128037i \(0.959132\pi\)
\(62\) 0 0
\(63\) 22.1359i 2.78887i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 9.48683 1.15900 0.579501 − 0.814972i \(-0.303248\pi\)
0.579501 + 0.814972i \(0.303248\pi\)
\(68\) 0 0
\(69\) − 10.0000i − 1.20386i
\(70\) 0 0
\(71\) 6.32456 0.750587 0.375293 − 0.926906i \(-0.377542\pi\)
0.375293 + 0.926906i \(0.377542\pi\)
\(72\) 0 0
\(73\) 14.0000i 1.63858i 0.573382 + 0.819288i \(0.305631\pi\)
−0.573382 + 0.819288i \(0.694369\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −12.6491 −1.42314 −0.711568 − 0.702617i \(-0.752015\pi\)
−0.711568 + 0.702617i \(0.752015\pi\)
\(80\) 0 0
\(81\) 19.0000 2.11111
\(82\) 0 0
\(83\) −3.16228 −0.347105 −0.173553 − 0.984825i \(-0.555525\pi\)
−0.173553 + 0.984825i \(0.555525\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 12.6491i 1.35613i
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 18.9737i 1.98898i
\(92\) 0 0
\(93\) −20.0000 −2.07390
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) 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 3200.2.f.q.449.4 4
4.3 odd 2 inner 3200.2.f.q.449.1 4
5.2 odd 4 3200.2.d.j.1601.1 4
5.3 odd 4 640.2.d.a.321.3 yes 4
5.4 even 2 3200.2.f.p.449.1 4
8.3 odd 2 3200.2.f.p.449.3 4
8.5 even 2 3200.2.f.p.449.2 4
15.8 even 4 5760.2.k.t.2881.4 4
20.3 even 4 640.2.d.a.321.1 ✓ 4
20.7 even 4 3200.2.d.j.1601.4 4
20.19 odd 2 3200.2.f.p.449.4 4
40.3 even 4 640.2.d.a.321.4 yes 4
40.13 odd 4 640.2.d.a.321.2 yes 4
40.19 odd 2 inner 3200.2.f.q.449.2 4
40.27 even 4 3200.2.d.j.1601.2 4
40.29 even 2 inner 3200.2.f.q.449.3 4
40.37 odd 4 3200.2.d.j.1601.3 4
60.23 odd 4 5760.2.k.t.2881.3 4
80.3 even 4 1280.2.a.h.1.2 2
80.13 odd 4 1280.2.a.h.1.1 2
80.27 even 4 6400.2.a.bz.1.2 2
80.37 odd 4 6400.2.a.bz.1.1 2
80.43 even 4 1280.2.a.l.1.1 2
80.53 odd 4 1280.2.a.l.1.2 2
80.67 even 4 6400.2.a.ca.1.1 2
80.77 odd 4 6400.2.a.ca.1.2 2
120.53 even 4 5760.2.k.t.2881.2 4
120.83 odd 4 5760.2.k.t.2881.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.d.a.321.1 ✓ 4 20.3 even 4
640.2.d.a.321.2 yes 4 40.13 odd 4
640.2.d.a.321.3 yes 4 5.3 odd 4
640.2.d.a.321.4 yes 4 40.3 even 4
1280.2.a.h.1.1 2 80.13 odd 4
1280.2.a.h.1.2 2 80.3 even 4
1280.2.a.l.1.1 2 80.43 even 4
1280.2.a.l.1.2 2 80.53 odd 4
3200.2.d.j.1601.1 4 5.2 odd 4
3200.2.d.j.1601.2 4 40.27 even 4
3200.2.d.j.1601.3 4 40.37 odd 4
3200.2.d.j.1601.4 4 20.7 even 4
3200.2.f.p.449.1 4 5.4 even 2
3200.2.f.p.449.2 4 8.5 even 2
3200.2.f.p.449.3 4 8.3 odd 2
3200.2.f.p.449.4 4 20.19 odd 2
3200.2.f.q.449.1 4 4.3 odd 2 inner
3200.2.f.q.449.2 4 40.19 odd 2 inner
3200.2.f.q.449.3 4 40.29 even 2 inner
3200.2.f.q.449.4 4 1.1 even 1 trivial
5760.2.k.t.2881.1 4 120.83 odd 4
5760.2.k.t.2881.2 4 120.53 even 4
5760.2.k.t.2881.3 4 60.23 odd 4
5760.2.k.t.2881.4 4 15.8 even 4
6400.2.a.bz.1.1 2 80.37 odd 4
6400.2.a.bz.1.2 2 80.27 even 4
6400.2.a.ca.1.1 2 80.67 even 4
6400.2.a.ca.1.2 2 80.77 odd 4