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Properties

Label 640.2.l.a.481.4
Level $640$
Weight $2$
Character 640.481
Analytic conductor $5.110$
Analytic rank $0$
Dimension $16$
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(161,640)] 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("640.161"); 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([0, 3, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 640.l (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.11042572936\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 481.4
Root \(-0.296075 - 1.38287i\) of defining polynomial
Character \(\chi\) \(=\) 640.481
Dual form 640.2.l.a.161.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.120009 + 0.120009i) q^{3} +(0.707107 + 0.707107i) q^{5} +2.66881i q^{7} +2.97120i q^{9} +(-3.49714 - 3.49714i) q^{11} +(-2.94072 + 2.94072i) q^{13} -0.169718 q^{15} +1.85116 q^{17} +(-3.44856 + 3.44856i) q^{19} +(-0.320281 - 0.320281i) q^{21} -0.707288i q^{23} +1.00000i q^{25} +(-0.716597 - 0.716597i) q^{27} +(3.49909 - 3.49909i) q^{29} -6.84272 q^{31} +0.839377 q^{33} +(-1.88714 + 1.88714i) q^{35} +(0.0975060 + 0.0975060i) q^{37} -0.705826i q^{39} +10.2052i q^{41} +(4.43844 + 4.43844i) q^{43} +(-2.10095 + 2.10095i) q^{45} +1.89428 q^{47} -0.122561 q^{49} +(-0.222155 + 0.222155i) q^{51} +(7.43897 + 7.43897i) q^{53} -4.94571i q^{55} -0.827717i q^{57} +(0.959574 + 0.959574i) q^{59} +(-6.49825 + 6.49825i) q^{61} -7.92956 q^{63} -4.15881 q^{65} +(3.49691 - 3.49691i) q^{67} +(0.0848809 + 0.0848809i) q^{69} +7.86777i q^{71} -15.6564i q^{73} +(-0.120009 - 0.120009i) q^{75} +(9.33322 - 9.33322i) q^{77} +6.70212 q^{79} -8.74159 q^{81} +(-3.87327 + 3.87327i) q^{83} +(1.30896 + 1.30896i) q^{85} +0.839845i q^{87} -10.5055i q^{89} +(-7.84824 - 7.84824i) q^{91} +(0.821187 - 0.821187i) q^{93} -4.87701 q^{95} +4.79937 q^{97} +(10.3907 - 10.3907i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{11} + 8 q^{15} - 8 q^{19} + 24 q^{27} + 16 q^{29} + 16 q^{37} + 8 q^{43} + 40 q^{47} - 16 q^{49} - 32 q^{51} - 16 q^{53} - 8 q^{59} - 16 q^{61} - 40 q^{63} + 40 q^{67} - 16 q^{69} - 16 q^{77}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) −0.120009 + 0.120009i −0.0692872 + 0.0692872i −0.740901 − 0.671614i \(-0.765601\pi\)
0.671614 + 0.740901i \(0.265601\pi\)
\(4\) 0 0
\(5\) 0.707107 + 0.707107i 0.316228 + 0.316228i
\(6\) 0 0
\(7\) 2.66881i 1.00872i 0.863495 + 0.504358i \(0.168271\pi\)
−0.863495 + 0.504358i \(0.831729\pi\)
\(8\) 0 0
\(9\) 2.97120i 0.990399i
\(10\) 0 0
\(11\) −3.49714 − 3.49714i −1.05443 − 1.05443i −0.998431 − 0.0559977i \(-0.982166\pi\)
−0.0559977 − 0.998431i \(-0.517834\pi\)
\(12\) 0 0
\(13\) −2.94072 + 2.94072i −0.815610 + 0.815610i −0.985468 − 0.169858i \(-0.945669\pi\)
0.169858 + 0.985468i \(0.445669\pi\)
\(14\) 0 0
\(15\) −0.169718 −0.0438211
\(16\) 0 0
\(17\) 1.85116 0.448971 0.224486 − 0.974477i \(-0.427930\pi\)
0.224486 + 0.974477i \(0.427930\pi\)
\(18\) 0 0
\(19\) −3.44856 + 3.44856i −0.791155 + 0.791155i −0.981682 − 0.190527i \(-0.938980\pi\)
0.190527 + 0.981682i \(0.438980\pi\)
\(20\) 0 0
\(21\) −0.320281 − 0.320281i −0.0698911 − 0.0698911i
\(22\) 0 0
\(23\) − 0.707288i − 0.147480i −0.997278 − 0.0737399i \(-0.976507\pi\)
0.997278 − 0.0737399i \(-0.0234935\pi\)
\(24\) 0 0
\(25\) 1.00000i 0.200000i
\(26\) 0 0
\(27\) −0.716597 − 0.716597i −0.137909 − 0.137909i
\(28\) 0 0
\(29\) 3.49909 − 3.49909i 0.649766 − 0.649766i −0.303171 − 0.952936i \(-0.598045\pi\)
0.952936 + 0.303171i \(0.0980452\pi\)
\(30\) 0 0
\(31\) −6.84272 −1.22899 −0.614494 − 0.788921i \(-0.710640\pi\)
−0.614494 + 0.788921i \(0.710640\pi\)
\(32\) 0 0
\(33\) 0.839377 0.146117
\(34\) 0 0
\(35\) −1.88714 + 1.88714i −0.318984 + 0.318984i
\(36\) 0 0
\(37\) 0.0975060 + 0.0975060i 0.0160299 + 0.0160299i 0.715076 − 0.699046i \(-0.246392\pi\)
−0.699046 + 0.715076i \(0.746392\pi\)
\(38\) 0 0
\(39\) − 0.705826i − 0.113023i
\(40\) 0 0
\(41\) 10.2052i 1.59379i 0.604117 + 0.796896i \(0.293526\pi\)
−0.604117 + 0.796896i \(0.706474\pi\)
\(42\) 0 0
\(43\) 4.43844 + 4.43844i 0.676855 + 0.676855i 0.959287 − 0.282432i \(-0.0911412\pi\)
−0.282432 + 0.959287i \(0.591141\pi\)
\(44\) 0 0
\(45\) −2.10095 + 2.10095i −0.313192 + 0.313192i
\(46\) 0 0
\(47\) 1.89428 0.276310 0.138155 − 0.990411i \(-0.455883\pi\)
0.138155 + 0.990411i \(0.455883\pi\)
\(48\) 0 0
\(49\) −0.122561 −0.0175087
\(50\) 0 0
\(51\) −0.222155 + 0.222155i −0.0311079 + 0.0311079i
\(52\) 0 0
\(53\) 7.43897 + 7.43897i 1.02182 + 1.02182i 0.999757 + 0.0220650i \(0.00702407\pi\)
0.0220650 + 0.999757i \(0.492976\pi\)
\(54\) 0 0
\(55\) − 4.94571i − 0.666879i
\(56\) 0 0
\(57\) − 0.827717i − 0.109634i
\(58\) 0 0
\(59\) 0.959574 + 0.959574i 0.124926 + 0.124926i 0.766805 − 0.641880i \(-0.221845\pi\)
−0.641880 + 0.766805i \(0.721845\pi\)
\(60\) 0 0
\(61\) −6.49825 + 6.49825i −0.832015 + 0.832015i −0.987792 − 0.155777i \(-0.950212\pi\)
0.155777 + 0.987792i \(0.450212\pi\)
\(62\) 0 0
\(63\) −7.92956 −0.999031
\(64\) 0 0
\(65\) −4.15881 −0.515837
\(66\) 0 0
\(67\) 3.49691 − 3.49691i 0.427216 − 0.427216i −0.460463 − 0.887679i \(-0.652317\pi\)
0.887679 + 0.460463i \(0.152317\pi\)
\(68\) 0 0
\(69\) 0.0848809 + 0.0848809i 0.0102185 + 0.0102185i
\(70\) 0 0
\(71\) 7.86777i 0.933733i 0.884328 + 0.466866i \(0.154617\pi\)
−0.884328 + 0.466866i \(0.845383\pi\)
\(72\) 0 0
\(73\) − 15.6564i − 1.83244i −0.400675 − 0.916220i \(-0.631224\pi\)
0.400675 − 0.916220i \(-0.368776\pi\)
\(74\) 0 0
\(75\) −0.120009 − 0.120009i −0.0138574 − 0.0138574i
\(76\) 0 0
\(77\) 9.33322 − 9.33322i 1.06362 − 1.06362i
\(78\) 0 0
\(79\) 6.70212 0.754047 0.377024 − 0.926204i \(-0.376948\pi\)
0.377024 + 0.926204i \(0.376948\pi\)
\(80\) 0 0
\(81\) −8.74159 −0.971288
\(82\) 0 0
\(83\) −3.87327 + 3.87327i −0.425147 + 0.425147i −0.886971 − 0.461825i \(-0.847195\pi\)
0.461825 + 0.886971i \(0.347195\pi\)
\(84\) 0 0
\(85\) 1.30896 + 1.30896i 0.141977 + 0.141977i
\(86\) 0 0
\(87\) 0.839845i 0.0900408i
\(88\) 0 0
\(89\) − 10.5055i − 1.11358i −0.830653 − 0.556790i \(-0.812033\pi\)
0.830653 − 0.556790i \(-0.187967\pi\)
\(90\) 0 0
\(91\) −7.84824 − 7.84824i −0.822719 − 0.822719i
\(92\) 0 0
\(93\) 0.821187 − 0.821187i 0.0851531 − 0.0851531i
\(94\) 0 0
\(95\) −4.87701 −0.500370
\(96\) 0 0
\(97\) 4.79937 0.487303 0.243651 − 0.969863i \(-0.421655\pi\)
0.243651 + 0.969863i \(0.421655\pi\)
\(98\) 0 0
\(99\) 10.3907 − 10.3907i 1.04430 − 1.04430i
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.l.a.481.4 16
4.3 odd 2 640.2.l.b.481.5 16
8.3 odd 2 80.2.l.a.21.7 ✓ 16
8.5 even 2 320.2.l.a.241.5 16
16.3 odd 4 640.2.l.b.161.5 16
16.5 even 4 320.2.l.a.81.5 16
16.11 odd 4 80.2.l.a.61.7 yes 16
16.13 even 4 inner 640.2.l.a.161.4 16
24.5 odd 2 2880.2.t.c.2161.7 16
24.11 even 2 720.2.t.c.181.2 16
32.3 odd 8 5120.2.a.s.1.5 8
32.13 even 8 5120.2.a.t.1.5 8
32.19 odd 8 5120.2.a.v.1.4 8
32.29 even 8 5120.2.a.u.1.4 8
40.3 even 4 400.2.q.h.149.6 16
40.13 odd 4 1600.2.q.g.49.5 16
40.19 odd 2 400.2.l.h.101.2 16
40.27 even 4 400.2.q.g.149.3 16
40.29 even 2 1600.2.l.i.1201.4 16
40.37 odd 4 1600.2.q.h.49.4 16
48.5 odd 4 2880.2.t.c.721.6 16
48.11 even 4 720.2.t.c.541.2 16
80.27 even 4 400.2.q.h.349.6 16
80.37 odd 4 1600.2.q.g.849.5 16
80.43 even 4 400.2.q.g.349.3 16
80.53 odd 4 1600.2.q.h.849.4 16
80.59 odd 4 400.2.l.h.301.2 16
80.69 even 4 1600.2.l.i.401.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.l.a.21.7 ✓ 16 8.3 odd 2
80.2.l.a.61.7 yes 16 16.11 odd 4
320.2.l.a.81.5 16 16.5 even 4
320.2.l.a.241.5 16 8.5 even 2
400.2.l.h.101.2 16 40.19 odd 2
400.2.l.h.301.2 16 80.59 odd 4
400.2.q.g.149.3 16 40.27 even 4
400.2.q.g.349.3 16 80.43 even 4
400.2.q.h.149.6 16 40.3 even 4
400.2.q.h.349.6 16 80.27 even 4
640.2.l.a.161.4 16 16.13 even 4 inner
640.2.l.a.481.4 16 1.1 even 1 trivial
640.2.l.b.161.5 16 16.3 odd 4
640.2.l.b.481.5 16 4.3 odd 2
720.2.t.c.181.2 16 24.11 even 2
720.2.t.c.541.2 16 48.11 even 4
1600.2.l.i.401.4 16 80.69 even 4
1600.2.l.i.1201.4 16 40.29 even 2
1600.2.q.g.49.5 16 40.13 odd 4
1600.2.q.g.849.5 16 80.37 odd 4
1600.2.q.h.49.4 16 40.37 odd 4
1600.2.q.h.849.4 16 80.53 odd 4
2880.2.t.c.721.6 16 48.5 odd 4
2880.2.t.c.2161.7 16 24.5 odd 2
5120.2.a.s.1.5 8 32.3 odd 8
5120.2.a.t.1.5 8 32.13 even 8
5120.2.a.u.1.4 8 32.29 even 8
5120.2.a.v.1.4 8 32.19 odd 8