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Properties

Label 560.2.q.e.401.1
Level $560$
Weight $2$
Character 560.401
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,-1] 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 401.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 560.401
Dual form 560.2.q.e.81.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-0.500000 + 2.59808i) q^{7} +(1.00000 - 1.73205i) q^{9} +(-1.00000 - 1.73205i) q^{11} +4.00000 q^{13} -1.00000 q^{15} +(3.00000 - 5.19615i) q^{19} +(2.50000 - 0.866025i) q^{21} +(1.50000 - 2.59808i) q^{23} +(-0.500000 - 0.866025i) q^{25} -5.00000 q^{27} -3.00000 q^{29} +(-1.00000 + 1.73205i) q^{33} +(2.00000 + 1.73205i) q^{35} +(6.00000 - 10.3923i) q^{37} +(-2.00000 - 3.46410i) q^{39} -7.00000 q^{41} +9.00000 q^{43} +(-1.00000 - 1.73205i) q^{45} +(-6.50000 - 2.59808i) q^{49} +(3.00000 + 5.19615i) q^{53} -2.00000 q^{55} -6.00000 q^{57} +(-5.00000 - 8.66025i) q^{59} +(-2.50000 + 4.33013i) q^{61} +(4.00000 + 3.46410i) q^{63} +(2.00000 - 3.46410i) q^{65} +(5.50000 + 9.52628i) q^{67} -3.00000 q^{69} +10.0000 q^{71} +(4.00000 + 6.92820i) q^{73} +(-0.500000 + 0.866025i) q^{75} +(5.00000 - 1.73205i) q^{77} +(3.00000 - 5.19615i) q^{79} +(-0.500000 - 0.866025i) q^{81} +3.00000 q^{83} +(1.50000 + 2.59808i) q^{87} +(-8.50000 + 14.7224i) q^{89} +(-2.00000 + 10.3923i) q^{91} +(-3.00000 - 5.19615i) q^{95} -2.00000 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + q^{5} - q^{7} + 2 q^{9} - 2 q^{11} + 8 q^{13} - 2 q^{15} + 6 q^{19} + 5 q^{21} + 3 q^{23} - q^{25} - 10 q^{27} - 6 q^{29} - 2 q^{33} + 4 q^{35} + 12 q^{37} - 4 q^{39} - 14 q^{41} + 18 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(-0.259881\pi\)
−0.973494 + 0.228714i \(0.926548\pi\)
\(4\) 0 0
\(5\) 0.500000 − 0.866025i 0.223607 − 0.387298i
\(6\) 0 0
\(7\) −0.500000 + 2.59808i −0.188982 + 0.981981i
\(8\) 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 \(-0.264158\pi\)
−0.976478 + 0.215615i \(0.930824\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) 3.00000 − 5.19615i 0.688247 − 1.19208i −0.284157 − 0.958778i \(-0.591714\pi\)
0.972404 − 0.233301i \(-0.0749529\pi\)
\(20\) 0 0
\(21\) 2.50000 − 0.866025i 0.545545 − 0.188982i
\(22\) 0 0
\(23\) 1.50000 − 2.59808i 0.312772 − 0.541736i −0.666190 − 0.745782i \(-0.732076\pi\)
0.978961 + 0.204046i \(0.0654092\pi\)
\(24\) 0 0
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 − 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(32\) 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
\(37\) 6.00000 − 10.3923i 0.986394 − 1.70848i 0.350823 − 0.936442i \(-0.385902\pi\)
0.635571 − 0.772043i \(-0.280765\pi\)
\(38\) 0 0
\(39\) −2.00000 − 3.46410i −0.320256 − 0.554700i
\(40\) 0 0
\(41\) −7.00000 −1.09322 −0.546608 − 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) 0 0
\(43\) 9.00000 1.37249 0.686244 − 0.727372i \(-0.259258\pi\)
0.686244 + 0.727372i \(0.259258\pi\)
\(44\) 0 0
\(45\) −1.00000 − 1.73205i −0.149071 − 0.258199i
\(46\) 0 0
\(47\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) −6.50000 − 2.59808i −0.928571 − 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.00000 + 5.19615i 0.412082 + 0.713746i 0.995117 − 0.0987002i \(-0.0314685\pi\)
−0.583036 + 0.812447i \(0.698135\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) −5.00000 − 8.66025i −0.650945 − 1.12747i −0.982894 − 0.184172i \(-0.941040\pi\)
0.331949 − 0.943297i \(-0.392294\pi\)
\(60\) 0 0
\(61\) −2.50000 + 4.33013i −0.320092 + 0.554416i −0.980507 − 0.196485i \(-0.937047\pi\)
0.660415 + 0.750901i \(0.270381\pi\)
\(62\) 0 0
\(63\) 4.00000 + 3.46410i 0.503953 + 0.436436i
\(64\) 0 0
\(65\) 2.00000 − 3.46410i 0.248069 − 0.429669i
\(66\) 0 0
\(67\) 5.50000 + 9.52628i 0.671932 + 1.16382i 0.977356 + 0.211604i \(0.0678686\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) 0 0
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 − 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 0 0
\(73\) 4.00000 + 6.92820i 0.468165 + 0.810885i 0.999338 − 0.0363782i \(-0.0115821\pi\)
−0.531174 + 0.847263i \(0.678249\pi\)
\(74\) 0 0
\(75\) −0.500000 + 0.866025i −0.0577350 + 0.100000i
\(76\) 0 0
\(77\) 5.00000 − 1.73205i 0.569803 − 0.197386i
\(78\) 0 0
\(79\) 3.00000 − 5.19615i 0.337526 − 0.584613i −0.646440 − 0.762964i \(-0.723743\pi\)
0.983967 + 0.178352i \(0.0570765\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) 3.00000 0.329293 0.164646 − 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.50000 + 2.59808i 0.160817 + 0.278543i
\(88\) 0 0
\(89\) −8.50000 + 14.7224i −0.900998 + 1.56057i −0.0747975 + 0.997199i \(0.523831\pi\)
−0.826201 + 0.563376i \(0.809502\pi\)
\(90\) 0 0
\(91\) −2.00000 + 10.3923i −0.209657 + 1.08941i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.00000 − 5.19615i −0.307794 − 0.533114i
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 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.e.401.1 2
4.3 odd 2 280.2.q.b.121.1 yes 2
7.2 even 3 3920.2.a.v.1.1 1
7.4 even 3 inner 560.2.q.e.81.1 2
7.5 odd 6 3920.2.a.q.1.1 1
12.11 even 2 2520.2.bi.d.1801.1 2
20.3 even 4 1400.2.bh.c.849.1 4
20.7 even 4 1400.2.bh.c.849.2 4
20.19 odd 2 1400.2.q.c.401.1 2
28.3 even 6 1960.2.q.d.361.1 2
28.11 odd 6 280.2.q.b.81.1 ✓ 2
28.19 even 6 1960.2.a.l.1.1 1
28.23 odd 6 1960.2.a.c.1.1 1
28.27 even 2 1960.2.q.d.961.1 2
84.11 even 6 2520.2.bi.d.361.1 2
140.19 even 6 9800.2.a.o.1.1 1
140.39 odd 6 1400.2.q.c.1201.1 2
140.67 even 12 1400.2.bh.c.249.1 4
140.79 odd 6 9800.2.a.z.1.1 1
140.123 even 12 1400.2.bh.c.249.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.q.b.81.1 ✓ 2 28.11 odd 6
280.2.q.b.121.1 yes 2 4.3 odd 2
560.2.q.e.81.1 2 7.4 even 3 inner
560.2.q.e.401.1 2 1.1 even 1 trivial
1400.2.q.c.401.1 2 20.19 odd 2
1400.2.q.c.1201.1 2 140.39 odd 6
1400.2.bh.c.249.1 4 140.67 even 12
1400.2.bh.c.249.2 4 140.123 even 12
1400.2.bh.c.849.1 4 20.3 even 4
1400.2.bh.c.849.2 4 20.7 even 4
1960.2.a.c.1.1 1 28.23 odd 6
1960.2.a.l.1.1 1 28.19 even 6
1960.2.q.d.361.1 2 28.3 even 6
1960.2.q.d.961.1 2 28.27 even 2
2520.2.bi.d.361.1 2 84.11 even 6
2520.2.bi.d.1801.1 2 12.11 even 2
3920.2.a.q.1.1 1 7.5 odd 6
3920.2.a.v.1.1 1 7.2 even 3
9800.2.a.o.1.1 1 140.19 even 6
9800.2.a.z.1.1 1 140.79 odd 6