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Properties

Label 640.2.o.a.447.1
Level $640$
Weight $2$
Character 640.447
Analytic conductor $5.110$
Analytic rank $1$
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] = [640,2,Mod(63,640)] 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("640.63"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(640, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 640.o (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-4,0,-4,0,-4,0,0,0,-8,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.11042572936\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 447.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 640.447
Dual form 640.2.o.a.63.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.00000 + 2.00000i) q^{3} +(-2.00000 + 1.00000i) q^{5} +(-2.00000 + 2.00000i) q^{7} -5.00000i q^{9} -4.00000 q^{11} +(3.00000 + 3.00000i) q^{13} +(2.00000 - 6.00000i) q^{15} +(-3.00000 - 3.00000i) q^{17} -8.00000i q^{21} +(6.00000 + 6.00000i) q^{23} +(3.00000 - 4.00000i) q^{25} +(4.00000 + 4.00000i) q^{27} -2.00000 q^{29} -4.00000i q^{31} +(8.00000 - 8.00000i) q^{33} +(2.00000 - 6.00000i) q^{35} +(3.00000 - 3.00000i) q^{37} -12.0000 q^{39} +(6.00000 - 6.00000i) q^{43} +(5.00000 + 10.0000i) q^{45} +(-6.00000 + 6.00000i) q^{47} -1.00000i q^{49} +12.0000 q^{51} +(-3.00000 - 3.00000i) q^{53} +(8.00000 - 4.00000i) q^{55} -8.00000i q^{59} +6.00000i q^{61} +(10.0000 + 10.0000i) q^{63} +(-9.00000 - 3.00000i) q^{65} +(-6.00000 - 6.00000i) q^{67} -24.0000 q^{69} -12.0000i q^{71} +(-5.00000 + 5.00000i) q^{73} +(2.00000 + 14.0000i) q^{75} +(8.00000 - 8.00000i) q^{77} -8.00000 q^{79} -1.00000 q^{81} +(-6.00000 + 6.00000i) q^{83} +(9.00000 + 3.00000i) q^{85} +(4.00000 - 4.00000i) q^{87} -12.0000 q^{91} +(8.00000 + 8.00000i) q^{93} +(11.0000 + 11.0000i) q^{97} +20.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} - 4 q^{5} - 4 q^{7} - 8 q^{11} + 6 q^{13} + 4 q^{15} - 6 q^{17} + 12 q^{23} + 6 q^{25} + 8 q^{27} - 4 q^{29} + 16 q^{33} + 4 q^{35} + 6 q^{37} - 24 q^{39} + 12 q^{43} + 10 q^{45} - 12 q^{47}+ \cdots + 22 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(-1\)

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
\(3\) −2.00000 + 2.00000i −1.15470 + 1.15470i −0.169102 + 0.985599i \(0.554087\pi\)
−0.985599 + 0.169102i \(0.945913\pi\)
\(4\) 0 0
\(5\) −2.00000 + 1.00000i −0.894427 + 0.447214i
\(6\) 0 0
\(7\) −2.00000 + 2.00000i −0.755929 + 0.755929i −0.975579 − 0.219650i \(-0.929509\pi\)
0.219650 + 0.975579i \(0.429509\pi\)
\(8\) 0 0
\(9\) − 5.00000i − 1.66667i
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 3.00000 + 3.00000i 0.832050 + 0.832050i 0.987797 − 0.155747i \(-0.0497784\pi\)
−0.155747 + 0.987797i \(0.549778\pi\)
\(14\) 0 0
\(15\) 2.00000 − 6.00000i 0.516398 − 1.54919i
\(16\) 0 0
\(17\) −3.00000 − 3.00000i −0.727607 − 0.727607i 0.242536 − 0.970143i \(-0.422021\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) − 8.00000i − 1.74574i
\(22\) 0 0
\(23\) 6.00000 + 6.00000i 1.25109 + 1.25109i 0.955233 + 0.295853i \(0.0956039\pi\)
0.295853 + 0.955233i \(0.404396\pi\)
\(24\) 0 0
\(25\) 3.00000 − 4.00000i 0.600000 − 0.800000i
\(26\) 0 0
\(27\) 4.00000 + 4.00000i 0.769800 + 0.769800i
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) − 4.00000i − 0.718421i −0.933257 − 0.359211i \(-0.883046\pi\)
0.933257 − 0.359211i \(-0.116954\pi\)
\(32\) 0 0
\(33\) 8.00000 − 8.00000i 1.39262 − 1.39262i
\(34\) 0 0
\(35\) 2.00000 − 6.00000i 0.338062 − 1.01419i
\(36\) 0 0
\(37\) 3.00000 − 3.00000i 0.493197 − 0.493197i −0.416115 − 0.909312i \(-0.636609\pi\)
0.909312 + 0.416115i \(0.136609\pi\)
\(38\) 0 0
\(39\) −12.0000 −1.92154
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 6.00000 − 6.00000i 0.914991 − 0.914991i −0.0816682 − 0.996660i \(-0.526025\pi\)
0.996660 + 0.0816682i \(0.0260248\pi\)
\(44\) 0 0
\(45\) 5.00000 + 10.0000i 0.745356 + 1.49071i
\(46\) 0 0
\(47\) −6.00000 + 6.00000i −0.875190 + 0.875190i −0.993032 − 0.117842i \(-0.962402\pi\)
0.117842 + 0.993032i \(0.462402\pi\)
\(48\) 0 0
\(49\) − 1.00000i − 0.142857i
\(50\) 0 0
\(51\) 12.0000 1.68034
\(52\) 0 0
\(53\) −3.00000 − 3.00000i −0.412082 − 0.412082i 0.470381 − 0.882463i \(-0.344116\pi\)
−0.882463 + 0.470381i \(0.844116\pi\)
\(54\) 0 0
\(55\) 8.00000 − 4.00000i 1.07872 − 0.539360i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 8.00000i − 1.04151i −0.853706 − 0.520756i \(-0.825650\pi\)
0.853706 − 0.520756i \(-0.174350\pi\)
\(60\) 0 0
\(61\) 6.00000i 0.768221i 0.923287 + 0.384111i \(0.125492\pi\)
−0.923287 + 0.384111i \(0.874508\pi\)
\(62\) 0 0
\(63\) 10.0000 + 10.0000i 1.25988 + 1.25988i
\(64\) 0 0
\(65\) −9.00000 − 3.00000i −1.11631 − 0.372104i
\(66\) 0 0
\(67\) −6.00000 − 6.00000i −0.733017 − 0.733017i 0.238200 − 0.971216i \(-0.423443\pi\)
−0.971216 + 0.238200i \(0.923443\pi\)
\(68\) 0 0
\(69\) −24.0000 −2.88926
\(70\) 0 0
\(71\) − 12.0000i − 1.42414i −0.702109 − 0.712069i \(-0.747758\pi\)
0.702109 − 0.712069i \(-0.252242\pi\)
\(72\) 0 0
\(73\) −5.00000 + 5.00000i −0.585206 + 0.585206i −0.936329 − 0.351123i \(-0.885800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 2.00000 + 14.0000i 0.230940 + 1.61658i
\(76\) 0 0
\(77\) 8.00000 − 8.00000i 0.911685 − 0.911685i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) −6.00000 + 6.00000i −0.658586 + 0.658586i −0.955045 − 0.296460i \(-0.904194\pi\)
0.296460 + 0.955045i \(0.404194\pi\)
\(84\) 0 0
\(85\) 9.00000 + 3.00000i 0.976187 + 0.325396i
\(86\) 0 0
\(87\) 4.00000 − 4.00000i 0.428845 − 0.428845i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −12.0000 −1.25794
\(92\) 0 0
\(93\) 8.00000 + 8.00000i 0.829561 + 0.829561i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 11.0000 + 11.0000i 1.11688 + 1.11688i 0.992196 + 0.124684i \(0.0397918\pi\)
0.124684 + 0.992196i \(0.460208\pi\)
\(98\) 0 0
\(99\) 20.0000i 2.01008i
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.o.a.447.1 yes 2
4.3 odd 2 640.2.o.g.447.1 yes 2
5.3 odd 4 640.2.o.b.63.1 yes 2
8.3 odd 2 640.2.o.b.447.1 yes 2
8.5 even 2 640.2.o.h.447.1 yes 2
16.3 odd 4 1280.2.n.l.767.1 2
16.5 even 4 1280.2.n.k.767.1 2
16.11 odd 4 1280.2.n.a.767.1 2
16.13 even 4 1280.2.n.b.767.1 2
20.3 even 4 640.2.o.h.63.1 yes 2
40.3 even 4 inner 640.2.o.a.63.1 ✓ 2
40.13 odd 4 640.2.o.g.63.1 yes 2
80.3 even 4 1280.2.n.b.1023.1 2
80.13 odd 4 1280.2.n.l.1023.1 2
80.43 even 4 1280.2.n.k.1023.1 2
80.53 odd 4 1280.2.n.a.1023.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.o.a.63.1 ✓ 2 40.3 even 4 inner
640.2.o.a.447.1 yes 2 1.1 even 1 trivial
640.2.o.b.63.1 yes 2 5.3 odd 4
640.2.o.b.447.1 yes 2 8.3 odd 2
640.2.o.g.63.1 yes 2 40.13 odd 4
640.2.o.g.447.1 yes 2 4.3 odd 2
640.2.o.h.63.1 yes 2 20.3 even 4
640.2.o.h.447.1 yes 2 8.5 even 2
1280.2.n.a.767.1 2 16.11 odd 4
1280.2.n.a.1023.1 2 80.53 odd 4
1280.2.n.b.767.1 2 16.13 even 4
1280.2.n.b.1023.1 2 80.3 even 4
1280.2.n.k.767.1 2 16.5 even 4
1280.2.n.k.1023.1 2 80.43 even 4
1280.2.n.l.767.1 2 16.3 odd 4
1280.2.n.l.1023.1 2 80.13 odd 4