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Properties

Label 560.2.q.h.81.1
Level $560$
Weight $2$
Character 560.81
Analytic conductor $4.472$
Analytic rank $0$
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] = [560,2,Mod(81,560)] 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("560.81"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(560, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1,0,1,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 81.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 560.81
Dual form 560.2.q.h.401.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{5} +(2.50000 - 0.866025i) q^{7} +(1.00000 + 1.73205i) q^{9} +(1.00000 - 1.73205i) q^{11} +1.00000 q^{15} +(-2.00000 + 3.46410i) q^{17} +(-1.00000 - 1.73205i) q^{19} +(0.500000 - 2.59808i) q^{21} +(0.500000 + 0.866025i) q^{23} +(-0.500000 + 0.866025i) q^{25} +5.00000 q^{27} +9.00000 q^{29} +(2.00000 - 3.46410i) q^{31} +(-1.00000 - 1.73205i) q^{33} +(2.00000 + 1.73205i) q^{35} +(-2.00000 - 3.46410i) q^{37} +1.00000 q^{41} -9.00000 q^{43} +(-1.00000 + 1.73205i) q^{45} +(5.50000 - 4.33013i) q^{49} +(2.00000 + 3.46410i) q^{51} +(5.00000 - 8.66025i) q^{53} +2.00000 q^{55} -2.00000 q^{57} +(-5.00000 + 8.66025i) q^{59} +(-4.50000 - 7.79423i) q^{61} +(4.00000 + 3.46410i) q^{63} +(2.50000 - 4.33013i) q^{67} +1.00000 q^{69} -14.0000 q^{71} +(-6.00000 + 10.3923i) q^{73} +(0.500000 + 0.866025i) q^{75} +(1.00000 - 5.19615i) q^{77} +(7.00000 + 12.1244i) q^{79} +(-0.500000 + 0.866025i) q^{81} -11.0000 q^{83} -4.00000 q^{85} +(4.50000 - 7.79423i) q^{87} +(7.50000 + 12.9904i) q^{89} +(-2.00000 - 3.46410i) q^{93} +(1.00000 - 1.73205i) q^{95} -18.0000 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + q^{5} + 5 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{15} - 4 q^{17} - 2 q^{19} + q^{21} + q^{23} - q^{25} + 10 q^{27} + 18 q^{29} + 4 q^{31} - 2 q^{33} + 4 q^{35} - 4 q^{37} + 2 q^{41} - 18 q^{43}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(1\)

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
\(3\) 0.500000 − 0.866025i 0.288675 − 0.500000i −0.684819 − 0.728714i \(-0.740119\pi\)
0.973494 + 0.228714i \(0.0734519\pi\)
\(4\) 0 0
\(5\) 0.500000 + 0.866025i 0.223607 + 0.387298i
\(6\) 0 0
\(7\) 2.50000 − 0.866025i 0.944911 − 0.327327i
\(8\) 0 0
\(9\) 1.00000 + 1.73205i 0.333333 + 0.577350i
\(10\) 0 0
\(11\) 1.00000 − 1.73205i 0.301511 − 0.522233i −0.674967 − 0.737848i \(-0.735842\pi\)
0.976478 + 0.215615i \(0.0691756\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −2.00000 + 3.46410i −0.485071 + 0.840168i −0.999853 − 0.0171533i \(-0.994540\pi\)
0.514782 + 0.857321i \(0.327873\pi\)
\(18\) 0 0
\(19\) −1.00000 − 1.73205i −0.229416 − 0.397360i 0.728219 − 0.685344i \(-0.240348\pi\)
−0.957635 + 0.287984i \(0.907015\pi\)
\(20\) 0 0
\(21\) 0.500000 − 2.59808i 0.109109 − 0.566947i
\(22\) 0 0
\(23\) 0.500000 + 0.866025i 0.104257 + 0.180579i 0.913434 − 0.406986i \(-0.133420\pi\)
−0.809177 + 0.587565i \(0.800087\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 9.00000 1.67126 0.835629 − 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) 2.00000 − 3.46410i 0.359211 − 0.622171i −0.628619 − 0.777714i \(-0.716379\pi\)
0.987829 + 0.155543i \(0.0497126\pi\)
\(32\) 0 0
\(33\) −1.00000 − 1.73205i −0.174078 − 0.301511i
\(34\) 0 0
\(35\) 2.00000 + 1.73205i 0.338062 + 0.292770i
\(36\) 0 0
\(37\) −2.00000 − 3.46410i −0.328798 − 0.569495i 0.653476 − 0.756948i \(-0.273310\pi\)
−0.982274 + 0.187453i \(0.939977\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.00000 0.156174 0.0780869 − 0.996947i \(-0.475119\pi\)
0.0780869 + 0.996947i \(0.475119\pi\)
\(42\) 0 0
\(43\) −9.00000 −1.37249 −0.686244 − 0.727372i \(-0.740742\pi\)
−0.686244 + 0.727372i \(0.740742\pi\)
\(44\) 0 0
\(45\) −1.00000 + 1.73205i −0.149071 + 0.258199i
\(46\) 0 0
\(47\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 5.50000 − 4.33013i 0.785714 − 0.618590i
\(50\) 0 0
\(51\) 2.00000 + 3.46410i 0.280056 + 0.485071i
\(52\) 0 0
\(53\) 5.00000 − 8.66025i 0.686803 − 1.18958i −0.286064 − 0.958211i \(-0.592347\pi\)
0.972867 − 0.231367i \(-0.0743197\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) −5.00000 + 8.66025i −0.650945 + 1.12747i 0.331949 + 0.943297i \(0.392294\pi\)
−0.982894 + 0.184172i \(0.941040\pi\)
\(60\) 0 0
\(61\) −4.50000 − 7.79423i −0.576166 − 0.997949i −0.995914 − 0.0903080i \(-0.971215\pi\)
0.419748 − 0.907641i \(-0.362118\pi\)
\(62\) 0 0
\(63\) 4.00000 + 3.46410i 0.503953 + 0.436436i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.50000 − 4.33013i 0.305424 − 0.529009i −0.671932 − 0.740613i \(-0.734535\pi\)
0.977356 + 0.211604i \(0.0678686\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −14.0000 −1.66149 −0.830747 − 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) 0 0
\(73\) −6.00000 + 10.3923i −0.702247 + 1.21633i 0.265429 + 0.964130i \(0.414486\pi\)
−0.967676 + 0.252197i \(0.918847\pi\)
\(74\) 0 0
\(75\) 0.500000 + 0.866025i 0.0577350 + 0.100000i
\(76\) 0 0
\(77\) 1.00000 − 5.19615i 0.113961 − 0.592157i
\(78\) 0 0
\(79\) 7.00000 + 12.1244i 0.787562 + 1.36410i 0.927457 + 0.373930i \(0.121990\pi\)
−0.139895 + 0.990166i \(0.544677\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) −11.0000 −1.20741 −0.603703 − 0.797209i \(-0.706309\pi\)
−0.603703 + 0.797209i \(0.706309\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 0 0
\(87\) 4.50000 − 7.79423i 0.482451 − 0.835629i
\(88\) 0 0
\(89\) 7.50000 + 12.9904i 0.794998 + 1.37698i 0.922840 + 0.385183i \(0.125862\pi\)
−0.127842 + 0.991795i \(0.540805\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −2.00000 − 3.46410i −0.207390 − 0.359211i
\(94\) 0 0
\(95\) 1.00000 − 1.73205i 0.102598 − 0.177705i
\(96\) 0 0
\(97\) −18.0000 −1.82762 −0.913812 − 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
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 560.2.q.h.81.1 2
4.3 odd 2 280.2.q.a.81.1 ✓ 2
7.2 even 3 inner 560.2.q.h.401.1 2
7.3 odd 6 3920.2.a.y.1.1 1
7.4 even 3 3920.2.a.m.1.1 1
12.11 even 2 2520.2.bi.a.361.1 2
20.3 even 4 1400.2.bh.b.249.1 4
20.7 even 4 1400.2.bh.b.249.2 4
20.19 odd 2 1400.2.q.e.1201.1 2
28.3 even 6 1960.2.a.e.1.1 1
28.11 odd 6 1960.2.a.i.1.1 1
28.19 even 6 1960.2.q.k.961.1 2
28.23 odd 6 280.2.q.a.121.1 yes 2
28.27 even 2 1960.2.q.k.361.1 2
84.23 even 6 2520.2.bi.a.1801.1 2
140.23 even 12 1400.2.bh.b.849.2 4
140.39 odd 6 9800.2.a.r.1.1 1
140.59 even 6 9800.2.a.bc.1.1 1
140.79 odd 6 1400.2.q.e.401.1 2
140.107 even 12 1400.2.bh.b.849.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.q.a.81.1 ✓ 2 4.3 odd 2
280.2.q.a.121.1 yes 2 28.23 odd 6
560.2.q.h.81.1 2 1.1 even 1 trivial
560.2.q.h.401.1 2 7.2 even 3 inner
1400.2.q.e.401.1 2 140.79 odd 6
1400.2.q.e.1201.1 2 20.19 odd 2
1400.2.bh.b.249.1 4 20.3 even 4
1400.2.bh.b.249.2 4 20.7 even 4
1400.2.bh.b.849.1 4 140.107 even 12
1400.2.bh.b.849.2 4 140.23 even 12
1960.2.a.e.1.1 1 28.3 even 6
1960.2.a.i.1.1 1 28.11 odd 6
1960.2.q.k.361.1 2 28.27 even 2
1960.2.q.k.961.1 2 28.19 even 6
2520.2.bi.a.361.1 2 12.11 even 2
2520.2.bi.a.1801.1 2 84.23 even 6
3920.2.a.m.1.1 1 7.4 even 3
3920.2.a.y.1.1 1 7.3 odd 6
9800.2.a.r.1.1 1 140.39 odd 6
9800.2.a.bc.1.1 1 140.59 even 6