Properties

Label 640.2.f.d.449.1
Level $640$
Weight $2$
Character 640.449
Analytic conductor $5.110$
Analytic rank $0$
Dimension $4$
CM discriminant -40
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [640,2,Mod(449,640)] 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("640.449"); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 640.f (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,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == 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: \(\Q(\sqrt{-2}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 449.1
Root \(-2.28825i\) of defining polynomial
Character \(\chi\) \(=\) 640.449
Dual form 640.2.f.d.449.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.23607 q^{5} -2.82843i q^{7} -3.00000 q^{9} +6.32456i q^{11} +4.47214 q^{13} +6.32456i q^{19} +8.48528i q^{23} +5.00000 q^{25} +6.32456i q^{35} -4.47214 q^{37} -2.00000 q^{41} +6.70820 q^{45} -2.82843i q^{47} -1.00000 q^{49} -13.4164 q^{53} -14.1421i q^{55} +6.32456i q^{59} +8.48528i q^{63} -10.0000 q^{65} +17.8885 q^{77} +9.00000 q^{81} +14.0000 q^{89} -12.6491i q^{91} -14.1421i q^{95} -18.9737i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} + 20 q^{25} - 8 q^{41} - 4 q^{49} - 40 q^{65} + 36 q^{81} + 56 q^{89}+O(q^{100}) \) 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
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) −2.23607 −1.00000
\(6\) 0 0
\(7\) − 2.82843i − 1.06904i −0.845154 − 0.534522i \(-0.820491\pi\)
0.845154 − 0.534522i \(-0.179509\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 6.32456i 1.90693i 0.301511 + 0.953463i \(0.402509\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 4.47214 1.24035 0.620174 − 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 6.32456i 1.45095i 0.688247 + 0.725476i \(0.258380\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.48528i 1.76930i 0.466252 + 0.884652i \(0.345604\pi\)
−0.466252 + 0.884652i \(0.654396\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.32456i 1.06904i
\(36\) 0 0
\(37\) −4.47214 −0.735215 −0.367607 − 0.929981i \(-0.619823\pi\)
−0.367607 + 0.929981i \(0.619823\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 6.70820 1.00000
\(46\) 0 0
\(47\) − 2.82843i − 0.412568i −0.978492 − 0.206284i \(-0.933863\pi\)
0.978492 − 0.206284i \(-0.0661372\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −13.4164 −1.84289 −0.921443 − 0.388514i \(-0.872988\pi\)
−0.921443 + 0.388514i \(0.872988\pi\)
\(54\) 0 0
\(55\) − 14.1421i − 1.90693i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.32456i 0.823387i 0.911322 + 0.411693i \(0.135063\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 8.48528i 1.06904i
\(64\) 0 0
\(65\) −10.0000 −1.24035
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 17.8885 2.03859
\(78\) 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
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.0000 1.48400 0.741999 − 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) − 12.6491i − 1.32599i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 14.1421i − 1.45095i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) − 18.9737i − 1.90693i
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.f.d.449.1 ✓ 4
4.3 odd 2 inner 640.2.f.d.449.2 yes 4
5.2 odd 4 3200.2.d.w.1601.4 4
5.3 odd 4 3200.2.d.w.1601.2 4
5.4 even 2 inner 640.2.f.d.449.4 yes 4
8.3 odd 2 inner 640.2.f.d.449.4 yes 4
8.5 even 2 inner 640.2.f.d.449.3 yes 4
16.3 odd 4 1280.2.c.l.769.3 4
16.5 even 4 1280.2.c.l.769.2 4
16.11 odd 4 1280.2.c.l.769.1 4
16.13 even 4 1280.2.c.l.769.4 4
20.3 even 4 3200.2.d.w.1601.3 4
20.7 even 4 3200.2.d.w.1601.1 4
20.19 odd 2 inner 640.2.f.d.449.3 yes 4
40.3 even 4 3200.2.d.w.1601.4 4
40.13 odd 4 3200.2.d.w.1601.1 4
40.19 odd 2 CM 640.2.f.d.449.1 ✓ 4
40.27 even 4 3200.2.d.w.1601.2 4
40.29 even 2 inner 640.2.f.d.449.2 yes 4
40.37 odd 4 3200.2.d.w.1601.3 4
80.3 even 4 6400.2.a.cl.1.1 4
80.13 odd 4 6400.2.a.cl.1.4 4
80.19 odd 4 1280.2.c.l.769.2 4
80.27 even 4 6400.2.a.cl.1.4 4
80.29 even 4 1280.2.c.l.769.1 4
80.37 odd 4 6400.2.a.cl.1.1 4
80.43 even 4 6400.2.a.cl.1.2 4
80.53 odd 4 6400.2.a.cl.1.3 4
80.59 odd 4 1280.2.c.l.769.4 4
80.67 even 4 6400.2.a.cl.1.3 4
80.69 even 4 1280.2.c.l.769.3 4
80.77 odd 4 6400.2.a.cl.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.f.d.449.1 ✓ 4 1.1 even 1 trivial
640.2.f.d.449.1 ✓ 4 40.19 odd 2 CM
640.2.f.d.449.2 yes 4 4.3 odd 2 inner
640.2.f.d.449.2 yes 4 40.29 even 2 inner
640.2.f.d.449.3 yes 4 8.5 even 2 inner
640.2.f.d.449.3 yes 4 20.19 odd 2 inner
640.2.f.d.449.4 yes 4 5.4 even 2 inner
640.2.f.d.449.4 yes 4 8.3 odd 2 inner
1280.2.c.l.769.1 4 16.11 odd 4
1280.2.c.l.769.1 4 80.29 even 4
1280.2.c.l.769.2 4 16.5 even 4
1280.2.c.l.769.2 4 80.19 odd 4
1280.2.c.l.769.3 4 16.3 odd 4
1280.2.c.l.769.3 4 80.69 even 4
1280.2.c.l.769.4 4 16.13 even 4
1280.2.c.l.769.4 4 80.59 odd 4
3200.2.d.w.1601.1 4 20.7 even 4
3200.2.d.w.1601.1 4 40.13 odd 4
3200.2.d.w.1601.2 4 5.3 odd 4
3200.2.d.w.1601.2 4 40.27 even 4
3200.2.d.w.1601.3 4 20.3 even 4
3200.2.d.w.1601.3 4 40.37 odd 4
3200.2.d.w.1601.4 4 5.2 odd 4
3200.2.d.w.1601.4 4 40.3 even 4
6400.2.a.cl.1.1 4 80.3 even 4
6400.2.a.cl.1.1 4 80.37 odd 4
6400.2.a.cl.1.2 4 80.43 even 4
6400.2.a.cl.1.2 4 80.77 odd 4
6400.2.a.cl.1.3 4 80.53 odd 4
6400.2.a.cl.1.3 4 80.67 even 4
6400.2.a.cl.1.4 4 80.13 odd 4
6400.2.a.cl.1.4 4 80.27 even 4