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Properties

Label 640.2.d.a.321.1
Level $640$
Weight $2$
Character 640.321
Analytic conductor $5.110$
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] = [640,2,Mod(321,640)] 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("640.321"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(640, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 640.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-28,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.11042572936\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 321.1
Root \(-1.58114 + 1.58114i\) of defining polynomial
Character \(\chi\) \(=\) 640.321
Dual form 640.2.d.a.321.4

$q$-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)
 
\(f(q)\) \(=\) \(q-3.16228i q^{3} -1.00000i q^{5} -3.16228 q^{7} -7.00000 q^{9} +6.00000i q^{13} -3.16228 q^{15} -2.00000 q^{17} -6.32456i q^{19} +10.0000i q^{21} -3.16228 q^{23} -1.00000 q^{25} +12.6491i q^{27} -4.00000i q^{29} +6.32456 q^{31} +3.16228i q^{35} -2.00000i q^{37} +18.9737 q^{39} -3.16228i q^{43} +7.00000i q^{45} -9.48683 q^{47} +3.00000 q^{49} +6.32456i q^{51} -6.00000i q^{53} -20.0000 q^{57} -6.32456i q^{59} +2.00000i q^{61} +22.1359 q^{63} +6.00000 q^{65} +9.48683i q^{67} +10.0000i q^{69} -6.32456 q^{71} -14.0000 q^{73} +3.16228i q^{75} -12.6491 q^{79} +19.0000 q^{81} +3.16228i q^{83} +2.00000i q^{85} -12.6491 q^{87} +10.0000 q^{89} -18.9737i q^{91} -20.0000i q^{93} -6.32456 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 28 q^{9} - 8 q^{17} - 4 q^{25} + 12 q^{49} - 80 q^{57} + 24 q^{65} - 56 q^{73} + 76 q^{81} + 40 q^{89} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/640\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 3.16228i − 1.82574i −0.408248 − 0.912871i \(-0.633860\pi\)
0.408248 − 0.912871i \(-0.366140\pi\)
\(4\) 0 0
\(5\) − 1.00000i − 0.447214i
\(6\) 0 0
\(7\) −3.16228 −1.19523 −0.597614 − 0.801784i \(-0.703885\pi\)
−0.597614 + 0.801784i \(0.703885\pi\)
\(8\) 0 0
\(9\) −7.00000 −2.33333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) −3.16228 −0.816497
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 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
\(23\) −3.16228 −0.659380 −0.329690 − 0.944089i \(-0.606944\pi\)
−0.329690 + 0.944089i \(0.606944\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 12.6491i 2.43432i
\(28\) 0 0
\(29\) − 4.00000i − 0.742781i −0.928477 − 0.371391i \(-0.878881\pi\)
0.928477 − 0.371391i \(-0.121119\pi\)
\(30\) 0 0
\(31\) 6.32456 1.13592 0.567962 − 0.823055i \(-0.307732\pi\)
0.567962 + 0.823055i \(0.307732\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.16228i 0.534522i
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 − 0.164399i \(-0.947432\pi\)
0.986394 − 0.164399i \(-0.0525685\pi\)
\(38\) 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
\(43\) − 3.16228i − 0.482243i −0.970495 − 0.241121i \(-0.922485\pi\)
0.970495 − 0.241121i \(-0.0775152\pi\)
\(44\) 0 0
\(45\) 7.00000i 1.04350i
\(46\) 0 0
\(47\) −9.48683 −1.38380 −0.691898 − 0.721995i \(-0.743225\pi\)
−0.691898 + 0.721995i \(0.743225\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 6.32456i 0.885615i
\(52\) 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
\(57\) −20.0000 −2.64906
\(58\) 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
\(63\) 22.1359 2.78887
\(64\) 0 0
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) 9.48683i 1.15900i 0.814972 + 0.579501i \(0.196752\pi\)
−0.814972 + 0.579501i \(0.803248\pi\)
\(68\) 0 0
\(69\) 10.0000i 1.20386i
\(70\) 0 0
\(71\) −6.32456 −0.750587 −0.375293 − 0.926906i \(-0.622458\pi\)
−0.375293 + 0.926906i \(0.622458\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 0 0
\(75\) 3.16228i 0.365148i
\(76\) 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
\(83\) 3.16228i 0.347105i 0.984825 + 0.173553i \(0.0555246\pi\)
−0.984825 + 0.173553i \(0.944475\pi\)
\(84\) 0 0
\(85\) 2.00000i 0.216930i
\(86\) 0 0
\(87\) −12.6491 −1.35613
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) − 18.9737i − 1.98898i
\(92\) 0 0
\(93\) − 20.0000i − 2.07390i
\(94\) 0 0
\(95\) −6.32456 −0.648886
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 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 640.2.d.a.321.1 ✓ 4
3.2 odd 2 5760.2.k.t.2881.3 4
4.3 odd 2 inner 640.2.d.a.321.3 yes 4
5.2 odd 4 3200.2.f.q.449.1 4
5.3 odd 4 3200.2.f.p.449.4 4
5.4 even 2 3200.2.d.j.1601.4 4
8.3 odd 2 inner 640.2.d.a.321.2 yes 4
8.5 even 2 inner 640.2.d.a.321.4 yes 4
12.11 even 2 5760.2.k.t.2881.4 4
16.3 odd 4 1280.2.a.h.1.1 2
16.5 even 4 1280.2.a.l.1.1 2
16.11 odd 4 1280.2.a.l.1.2 2
16.13 even 4 1280.2.a.h.1.2 2
20.3 even 4 3200.2.f.p.449.1 4
20.7 even 4 3200.2.f.q.449.4 4
20.19 odd 2 3200.2.d.j.1601.1 4
24.5 odd 2 5760.2.k.t.2881.1 4
24.11 even 2 5760.2.k.t.2881.2 4
40.3 even 4 3200.2.f.q.449.3 4
40.13 odd 4 3200.2.f.q.449.2 4
40.19 odd 2 3200.2.d.j.1601.3 4
40.27 even 4 3200.2.f.p.449.2 4
40.29 even 2 3200.2.d.j.1601.2 4
40.37 odd 4 3200.2.f.p.449.3 4
80.19 odd 4 6400.2.a.ca.1.2 2
80.29 even 4 6400.2.a.ca.1.1 2
80.59 odd 4 6400.2.a.bz.1.1 2
80.69 even 4 6400.2.a.bz.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.d.a.321.1 ✓ 4 1.1 even 1 trivial
640.2.d.a.321.2 yes 4 8.3 odd 2 inner
640.2.d.a.321.3 yes 4 4.3 odd 2 inner
640.2.d.a.321.4 yes 4 8.5 even 2 inner
1280.2.a.h.1.1 2 16.3 odd 4
1280.2.a.h.1.2 2 16.13 even 4
1280.2.a.l.1.1 2 16.5 even 4
1280.2.a.l.1.2 2 16.11 odd 4
3200.2.d.j.1601.1 4 20.19 odd 2
3200.2.d.j.1601.2 4 40.29 even 2
3200.2.d.j.1601.3 4 40.19 odd 2
3200.2.d.j.1601.4 4 5.4 even 2
3200.2.f.p.449.1 4 20.3 even 4
3200.2.f.p.449.2 4 40.27 even 4
3200.2.f.p.449.3 4 40.37 odd 4
3200.2.f.p.449.4 4 5.3 odd 4
3200.2.f.q.449.1 4 5.2 odd 4
3200.2.f.q.449.2 4 40.13 odd 4
3200.2.f.q.449.3 4 40.3 even 4
3200.2.f.q.449.4 4 20.7 even 4
5760.2.k.t.2881.1 4 24.5 odd 2
5760.2.k.t.2881.2 4 24.11 even 2
5760.2.k.t.2881.3 4 3.2 odd 2
5760.2.k.t.2881.4 4 12.11 even 2
6400.2.a.bz.1.1 2 80.59 odd 4
6400.2.a.bz.1.2 2 80.69 even 4
6400.2.a.ca.1.1 2 80.29 even 4
6400.2.a.ca.1.2 2 80.19 odd 4