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Properties

Label 640.2.l.a.161.4
Level $640$
Weight $2$
Character 640.161
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 161.4
Root \(-0.296075 + 1.38287i\) of defining polynomial
Character \(\chi\) \(=\) 640.161
Dual form 640.2.l.a.481.4

$q$-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 \(\left(\mathbb{Z}/640\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(1\)

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 0.671614 − 0.740901i \(-0.265601\pi\)
−0.740901 + 0.671614i \(0.765601\pi\)
\(4\) 0 0
\(5\) 0.707107 − 0.707107i 0.316228 − 0.316228i
\(6\) 0 0
\(7\) − 2.66881i − 1.00872i −0.863495 − 0.504358i \(-0.831729\pi\)
0.863495 − 0.504358i \(-0.168271\pi\)
\(8\) 0 0
\(9\) − 2.97120i − 0.990399i
\(10\) 0 0
\(11\) −3.49714 + 3.49714i −1.05443 + 1.05443i −0.0559977 + 0.998431i \(0.517834\pi\)
−0.998431 + 0.0559977i \(0.982166\pi\)
\(12\) 0 0
\(13\) −2.94072 − 2.94072i −0.815610 − 0.815610i 0.169858 − 0.985468i \(-0.445669\pi\)
−0.985468 + 0.169858i \(0.945669\pi\)
\(14\) 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 0.190527 − 0.981682i \(-0.438980\pi\)
−0.981682 + 0.190527i \(0.938980\pi\)
\(20\) 0 0
\(21\) −0.320281 + 0.320281i −0.0698911 + 0.0698911i
\(22\) 0 0
\(23\) 0.707288i 0.147480i 0.997278 + 0.0737399i \(0.0234935\pi\)
−0.997278 + 0.0737399i \(0.976507\pi\)
\(24\) 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 0.952936 − 0.303171i \(-0.0980452\pi\)
−0.303171 + 0.952936i \(0.598045\pi\)
\(30\) 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 −0.699046 − 0.715076i \(-0.746392\pi\)
0.715076 + 0.699046i \(0.246392\pi\)
\(38\) 0 0
\(39\) 0.705826i 0.113023i
\(40\) 0 0
\(41\) − 10.2052i − 1.59379i −0.604117 − 0.796896i \(-0.706474\pi\)
0.604117 − 0.796896i \(-0.293526\pi\)
\(42\) 0 0
\(43\) 4.43844 − 4.43844i 0.676855 − 0.676855i −0.282432 − 0.959287i \(-0.591141\pi\)
0.959287 + 0.282432i \(0.0911412\pi\)
\(44\) 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 0.0220650 − 0.999757i \(-0.492976\pi\)
0.999757 − 0.0220650i \(-0.00702407\pi\)
\(54\) 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 −0.641880 − 0.766805i \(-0.721845\pi\)
0.766805 + 0.641880i \(0.221845\pi\)
\(60\) 0 0
\(61\) −6.49825 − 6.49825i −0.832015 − 0.832015i 0.155777 − 0.987792i \(-0.450212\pi\)
−0.987792 + 0.155777i \(0.950212\pi\)
\(62\) 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 0.887679 − 0.460463i \(-0.152317\pi\)
−0.460463 + 0.887679i \(0.652317\pi\)
\(68\) 0 0
\(69\) 0.0848809 − 0.0848809i 0.0102185 − 0.0102185i
\(70\) 0 0
\(71\) − 7.86777i − 0.933733i −0.884328 − 0.466866i \(-0.845383\pi\)
0.884328 − 0.466866i \(-0.154617\pi\)
\(72\) 0 0
\(73\) 15.6564i 1.83244i 0.400675 + 0.916220i \(0.368776\pi\)
−0.400675 + 0.916220i \(0.631224\pi\)
\(74\) 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 0.461825 − 0.886971i \(-0.347195\pi\)
−0.886971 + 0.461825i \(0.847195\pi\)
\(84\) 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 \(0.187967\pi\)
−0.830653 + 0.556790i \(0.812033\pi\)
\(90\) 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.161.4 16
4.3 odd 2 640.2.l.b.161.5 16
8.3 odd 2 80.2.l.a.61.7 yes 16
8.5 even 2 320.2.l.a.81.5 16
16.3 odd 4 80.2.l.a.21.7 ✓ 16
16.5 even 4 inner 640.2.l.a.481.4 16
16.11 odd 4 640.2.l.b.481.5 16
16.13 even 4 320.2.l.a.241.5 16
24.5 odd 2 2880.2.t.c.721.6 16
24.11 even 2 720.2.t.c.541.2 16
32.5 even 8 5120.2.a.t.1.5 8
32.11 odd 8 5120.2.a.s.1.5 8
32.21 even 8 5120.2.a.u.1.4 8
32.27 odd 8 5120.2.a.v.1.4 8
40.3 even 4 400.2.q.g.349.3 16
40.13 odd 4 1600.2.q.h.849.4 16
40.19 odd 2 400.2.l.h.301.2 16
40.27 even 4 400.2.q.h.349.6 16
40.29 even 2 1600.2.l.i.401.4 16
40.37 odd 4 1600.2.q.g.849.5 16
48.29 odd 4 2880.2.t.c.2161.7 16
48.35 even 4 720.2.t.c.181.2 16
80.3 even 4 400.2.q.h.149.6 16
80.13 odd 4 1600.2.q.g.49.5 16
80.19 odd 4 400.2.l.h.101.2 16
80.29 even 4 1600.2.l.i.1201.4 16
80.67 even 4 400.2.q.g.149.3 16
80.77 odd 4 1600.2.q.h.49.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.l.a.21.7 ✓ 16 16.3 odd 4
80.2.l.a.61.7 yes 16 8.3 odd 2
320.2.l.a.81.5 16 8.5 even 2
320.2.l.a.241.5 16 16.13 even 4
400.2.l.h.101.2 16 80.19 odd 4
400.2.l.h.301.2 16 40.19 odd 2
400.2.q.g.149.3 16 80.67 even 4
400.2.q.g.349.3 16 40.3 even 4
400.2.q.h.149.6 16 80.3 even 4
400.2.q.h.349.6 16 40.27 even 4
640.2.l.a.161.4 16 1.1 even 1 trivial
640.2.l.a.481.4 16 16.5 even 4 inner
640.2.l.b.161.5 16 4.3 odd 2
640.2.l.b.481.5 16 16.11 odd 4
720.2.t.c.181.2 16 48.35 even 4
720.2.t.c.541.2 16 24.11 even 2
1600.2.l.i.401.4 16 40.29 even 2
1600.2.l.i.1201.4 16 80.29 even 4
1600.2.q.g.49.5 16 80.13 odd 4
1600.2.q.g.849.5 16 40.37 odd 4
1600.2.q.h.49.4 16 80.77 odd 4
1600.2.q.h.849.4 16 40.13 odd 4
2880.2.t.c.721.6 16 24.5 odd 2
2880.2.t.c.2161.7 16 48.29 odd 4
5120.2.a.s.1.5 8 32.11 odd 8
5120.2.a.t.1.5 8 32.5 even 8
5120.2.a.u.1.4 8 32.21 even 8
5120.2.a.v.1.4 8 32.27 odd 8