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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 = [2,0,0,0,-2,0,0,0,-6] 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: \(1\)
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: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 449.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 640.449
Dual form 640.2.f.a.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)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 2.00000i) q^{5} -3.00000 q^{9} -6.00000 q^{13} +8.00000i q^{17} +(-3.00000 + 4.00000i) q^{25} -4.00000i q^{29} -2.00000 q^{37} -10.0000 q^{41} +(3.00000 + 6.00000i) q^{45} +7.00000 q^{49} -14.0000 q^{53} -12.0000i q^{61} +(6.00000 + 12.0000i) q^{65} -16.0000i q^{73} +9.00000 q^{81} +(16.0000 - 8.00000i) q^{85} -10.0000 q^{89} -8.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 6 q^{9} - 12 q^{13} - 6 q^{25} - 4 q^{37} - 20 q^{41} + 6 q^{45} + 14 q^{49} - 28 q^{53} + 12 q^{65} + 18 q^{81} + 32 q^{85} - 20 q^{89}+O(q^{100}) \) 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
\(5\) −1.00000 − 2.00000i −0.447214 − 0.894427i
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 8.00000i 1.94029i 0.242536 + 0.970143i \(0.422021\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −3.00000 + 4.00000i −0.600000 + 0.800000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 4.00000i − 0.742781i −0.928477 − 0.371391i \(-0.878881\pi\)
0.928477 − 0.371391i \(-0.121119\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 3.00000 + 6.00000i 0.447214 + 0.894427i
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −14.0000 −1.92305 −0.961524 − 0.274721i \(-0.911414\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) − 12.0000i − 1.53644i −0.640184 − 0.768221i \(-0.721142\pi\)
0.640184 − 0.768221i \(-0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.00000 + 12.0000i 0.744208 + 1.48842i
\(66\) 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
\(73\) − 16.0000i − 1.87266i −0.351123 − 0.936329i \(-0.614200\pi\)
0.351123 − 0.936329i \(-0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 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
\(85\) 16.0000 − 8.00000i 1.73544 − 0.867722i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 8.00000i − 0.812277i −0.913812 − 0.406138i \(-0.866875\pi\)
0.913812 − 0.406138i \(-0.133125\pi\)
\(98\) 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.f.a.449.1 ✓ 2
4.3 odd 2 CM 640.2.f.a.449.1 ✓ 2
5.2 odd 4 3200.2.d.d.1601.2 2
5.3 odd 4 3200.2.d.f.1601.1 2
5.4 even 2 640.2.f.b.449.1 yes 2
8.3 odd 2 640.2.f.b.449.2 yes 2
8.5 even 2 640.2.f.b.449.2 yes 2
16.3 odd 4 1280.2.c.a.769.2 2
16.5 even 4 1280.2.c.d.769.1 2
16.11 odd 4 1280.2.c.d.769.1 2
16.13 even 4 1280.2.c.a.769.2 2
20.3 even 4 3200.2.d.f.1601.1 2
20.7 even 4 3200.2.d.d.1601.2 2
20.19 odd 2 640.2.f.b.449.1 yes 2
40.3 even 4 3200.2.d.f.1601.2 2
40.13 odd 4 3200.2.d.f.1601.2 2
40.19 odd 2 inner 640.2.f.a.449.2 yes 2
40.27 even 4 3200.2.d.d.1601.1 2
40.29 even 2 inner 640.2.f.a.449.2 yes 2
40.37 odd 4 3200.2.d.d.1601.1 2
80.3 even 4 6400.2.a.o.1.1 1
80.13 odd 4 6400.2.a.o.1.1 1
80.19 odd 4 1280.2.c.a.769.1 2
80.27 even 4 6400.2.a.n.1.1 1
80.29 even 4 1280.2.c.a.769.1 2
80.37 odd 4 6400.2.a.n.1.1 1
80.43 even 4 6400.2.a.k.1.1 1
80.53 odd 4 6400.2.a.k.1.1 1
80.59 odd 4 1280.2.c.d.769.2 2
80.67 even 4 6400.2.a.j.1.1 1
80.69 even 4 1280.2.c.d.769.2 2
80.77 odd 4 6400.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.f.a.449.1 ✓ 2 1.1 even 1 trivial
640.2.f.a.449.1 ✓ 2 4.3 odd 2 CM
640.2.f.a.449.2 yes 2 40.19 odd 2 inner
640.2.f.a.449.2 yes 2 40.29 even 2 inner
640.2.f.b.449.1 yes 2 5.4 even 2
640.2.f.b.449.1 yes 2 20.19 odd 2
640.2.f.b.449.2 yes 2 8.3 odd 2
640.2.f.b.449.2 yes 2 8.5 even 2
1280.2.c.a.769.1 2 80.19 odd 4
1280.2.c.a.769.1 2 80.29 even 4
1280.2.c.a.769.2 2 16.3 odd 4
1280.2.c.a.769.2 2 16.13 even 4
1280.2.c.d.769.1 2 16.5 even 4
1280.2.c.d.769.1 2 16.11 odd 4
1280.2.c.d.769.2 2 80.59 odd 4
1280.2.c.d.769.2 2 80.69 even 4
3200.2.d.d.1601.1 2 40.27 even 4
3200.2.d.d.1601.1 2 40.37 odd 4
3200.2.d.d.1601.2 2 5.2 odd 4
3200.2.d.d.1601.2 2 20.7 even 4
3200.2.d.f.1601.1 2 5.3 odd 4
3200.2.d.f.1601.1 2 20.3 even 4
3200.2.d.f.1601.2 2 40.3 even 4
3200.2.d.f.1601.2 2 40.13 odd 4
6400.2.a.j.1.1 1 80.67 even 4
6400.2.a.j.1.1 1 80.77 odd 4
6400.2.a.k.1.1 1 80.43 even 4
6400.2.a.k.1.1 1 80.53 odd 4
6400.2.a.n.1.1 1 80.27 even 4
6400.2.a.n.1.1 1 80.37 odd 4
6400.2.a.o.1.1 1 80.3 even 4
6400.2.a.o.1.1 1 80.13 odd 4