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Properties

Label 560.2.q.l.401.1
Level $560$
Weight $2$
Character 560.401
Analytic conductor $4.472$
Analytic rank $0$
Dimension $6$
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 = [6,0,0,0,-3,0,-6] 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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.11337408.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 18x^{4} + 81x^{2} + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(-2.78499i\) of defining polynomial
Character \(\chi\) \(=\) 560.401
Dual form 560.2.q.l.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)
 
\(f(q)\) \(=\) \(q+(-1.64497 - 2.84918i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(-2.64497 - 0.0641892i) q^{7} +(-3.91187 + 6.77556i) q^{9} +(2.91187 + 5.04351i) q^{11} +2.75615 q^{13} +3.28995 q^{15} +(-1.00000 - 1.73205i) q^{17} +(-0.378076 + 0.654846i) q^{19} +(4.16802 + 7.64158i) q^{21} +(-0.266897 + 0.462279i) q^{23} +(-0.500000 - 0.866025i) q^{25} +15.8698 q^{27} -0.823739 q^{29} +(-1.28995 - 2.23425i) q^{31} +(9.57989 - 16.5929i) q^{33} +(1.37808 - 2.25852i) q^{35} +(-2.37808 + 4.11895i) q^{37} +(-4.53379 - 7.85276i) q^{39} +6.06759 q^{41} -0.710055 q^{43} +(-3.91187 - 6.77556i) q^{45} +(-6.44566 + 11.1642i) q^{47} +(6.99176 + 0.339557i) q^{49} +(-3.28995 + 5.69835i) q^{51} +(4.20181 + 7.27776i) q^{53} -5.82374 q^{55} +2.48770 q^{57} +(4.00000 + 6.92820i) q^{59} +(-4.70181 + 8.14378i) q^{61} +(10.7817 - 17.6701i) q^{63} +(-1.37808 + 2.38690i) q^{65} +(5.93492 + 10.2796i) q^{67} +1.75615 q^{69} +(-1.75615 - 3.04174i) q^{73} +(-1.64497 + 2.84918i) q^{75} +(-7.37808 - 13.5268i) q^{77} +(-4.75615 + 8.23790i) q^{79} +(-14.3698 - 24.8893i) q^{81} +6.71005 q^{83} +2.00000 q^{85} +(1.35503 + 2.34698i) q^{87} +(-0.878076 + 1.52087i) q^{89} +(-7.28995 - 0.176915i) q^{91} +(-4.24385 + 7.35056i) q^{93} +(-0.378076 - 0.654846i) q^{95} -2.00000 q^{97} -45.5634 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{5} - 6 q^{7} - 9 q^{9} + 3 q^{11} + 6 q^{13} - 6 q^{17} + 3 q^{19} + 3 q^{23} - 3 q^{25} + 36 q^{27} + 24 q^{29} + 12 q^{31} + 18 q^{33} + 3 q^{35} - 9 q^{37} - 18 q^{39} + 18 q^{41} - 24 q^{43}+ \cdots - 126 q^{99}+O(q^{100}) \) 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
\(3\) −1.64497 − 2.84918i −0.949725 − 1.64497i −0.746002 − 0.665944i \(-0.768029\pi\)
−0.203724 − 0.979028i \(-0.565304\pi\)
\(4\) 0 0
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i
\(6\) 0 0
\(7\) −2.64497 − 0.0641892i −0.999706 − 0.0242612i
\(8\) 0 0
\(9\) −3.91187 + 6.77556i −1.30396 + 2.25852i
\(10\) 0 0
\(11\) 2.91187 + 5.04351i 0.877962 + 1.52067i 0.853574 + 0.520972i \(0.174430\pi\)
0.0243876 + 0.999703i \(0.492236\pi\)
\(12\) 0 0
\(13\) 2.75615 0.764419 0.382209 − 0.924076i \(-0.375163\pi\)
0.382209 + 0.924076i \(0.375163\pi\)
\(14\) 0 0
\(15\) 3.28995 0.849460
\(16\) 0 0
\(17\) −1.00000 − 1.73205i −0.242536 − 0.420084i 0.718900 − 0.695113i \(-0.244646\pi\)
−0.961436 + 0.275029i \(0.911312\pi\)
\(18\) 0 0
\(19\) −0.378076 + 0.654846i −0.0867365 + 0.150232i −0.906130 − 0.423000i \(-0.860977\pi\)
0.819393 + 0.573232i \(0.194310\pi\)
\(20\) 0 0
\(21\) 4.16802 + 7.64158i 0.909537 + 1.66753i
\(22\) 0 0
\(23\) −0.266897 + 0.462279i −0.0556518 + 0.0963918i −0.892509 − 0.451029i \(-0.851057\pi\)
0.836857 + 0.547421i \(0.184390\pi\)
\(24\) 0 0
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) 0 0
\(27\) 15.8698 3.05415
\(28\) 0 0
\(29\) −0.823739 −0.152964 −0.0764822 − 0.997071i \(-0.524369\pi\)
−0.0764822 + 0.997071i \(0.524369\pi\)
\(30\) 0 0
\(31\) −1.28995 − 2.23425i −0.231681 − 0.401283i 0.726622 − 0.687038i \(-0.241089\pi\)
−0.958303 + 0.285754i \(0.907756\pi\)
\(32\) 0 0
\(33\) 9.57989 − 16.5929i 1.66764 − 2.88845i
\(34\) 0 0
\(35\) 1.37808 − 2.25852i 0.232937 − 0.381759i
\(36\) 0 0
\(37\) −2.37808 + 4.11895i −0.390953 + 0.677151i −0.992576 − 0.121630i \(-0.961188\pi\)
0.601622 + 0.798781i \(0.294521\pi\)
\(38\) 0 0
\(39\) −4.53379 − 7.85276i −0.725988 − 1.25745i
\(40\) 0 0
\(41\) 6.06759 0.947598 0.473799 − 0.880633i \(-0.342882\pi\)
0.473799 + 0.880633i \(0.342882\pi\)
\(42\) 0 0
\(43\) −0.710055 −0.108282 −0.0541412 − 0.998533i \(-0.517242\pi\)
−0.0541412 + 0.998533i \(0.517242\pi\)
\(44\) 0 0
\(45\) −3.91187 − 6.77556i −0.583147 − 1.01004i
\(46\) 0 0
\(47\) −6.44566 + 11.1642i −0.940197 + 1.62847i −0.175102 + 0.984550i \(0.556026\pi\)
−0.765095 + 0.643918i \(0.777308\pi\)
\(48\) 0 0
\(49\) 6.99176 + 0.339557i 0.998823 + 0.0485082i
\(50\) 0 0
\(51\) −3.28995 + 5.69835i −0.460684 + 0.797929i
\(52\) 0 0
\(53\) 4.20181 + 7.27776i 0.577164 + 0.999677i 0.995803 + 0.0915241i \(0.0291738\pi\)
−0.418639 + 0.908153i \(0.637493\pi\)
\(54\) 0 0
\(55\) −5.82374 −0.785273
\(56\) 0 0
\(57\) 2.48770 0.329504
\(58\) 0 0
\(59\) 4.00000 + 6.92820i 0.520756 + 0.901975i 0.999709 + 0.0241347i \(0.00768307\pi\)
−0.478953 + 0.877841i \(0.658984\pi\)
\(60\) 0 0
\(61\) −4.70181 + 8.14378i −0.602006 + 1.04270i 0.390511 + 0.920598i \(0.372298\pi\)
−0.992517 + 0.122106i \(0.961035\pi\)
\(62\) 0 0
\(63\) 10.7817 − 17.6701i 1.35837 − 2.22622i
\(64\) 0 0
\(65\) −1.37808 + 2.38690i −0.170929 + 0.296058i
\(66\) 0 0
\(67\) 5.93492 + 10.2796i 0.725066 + 1.25585i 0.958947 + 0.283585i \(0.0915239\pi\)
−0.233881 + 0.972265i \(0.575143\pi\)
\(68\) 0 0
\(69\) 1.75615 0.211416
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −1.75615 − 3.04174i −0.205542 − 0.356009i 0.744763 − 0.667329i \(-0.232562\pi\)
−0.950305 + 0.311320i \(0.899229\pi\)
\(74\) 0 0
\(75\) −1.64497 + 2.84918i −0.189945 + 0.328995i
\(76\) 0 0
\(77\) −7.37808 − 13.5268i −0.840810 − 1.54153i
\(78\) 0 0
\(79\) −4.75615 + 8.23790i −0.535109 + 0.926836i 0.464049 + 0.885809i \(0.346396\pi\)
−0.999158 + 0.0410263i \(0.986937\pi\)
\(80\) 0 0
\(81\) −14.3698 − 24.8893i −1.59665 − 2.76548i
\(82\) 0 0
\(83\) 6.71005 0.736524 0.368262 − 0.929722i \(-0.379953\pi\)
0.368262 + 0.929722i \(0.379953\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) 1.35503 + 2.34698i 0.145274 + 0.251622i
\(88\) 0 0
\(89\) −0.878076 + 1.52087i −0.0930758 + 0.161212i −0.908804 − 0.417223i \(-0.863003\pi\)
0.815728 + 0.578436i \(0.196337\pi\)
\(90\) 0 0
\(91\) −7.28995 − 0.176915i −0.764194 − 0.0185458i
\(92\) 0 0
\(93\) −4.24385 + 7.35056i −0.440067 + 0.762218i
\(94\) 0 0
\(95\) −0.378076 − 0.654846i −0.0387898 − 0.0671858i
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) −45.5634 −4.57929
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.l.401.1 6
4.3 odd 2 280.2.q.e.121.3 yes 6
7.2 even 3 3920.2.a.cc.1.3 3
7.4 even 3 inner 560.2.q.l.81.1 6
7.5 odd 6 3920.2.a.cb.1.1 3
12.11 even 2 2520.2.bi.q.1801.3 6
20.3 even 4 1400.2.bh.i.849.1 12
20.7 even 4 1400.2.bh.i.849.6 12
20.19 odd 2 1400.2.q.j.401.1 6
28.3 even 6 1960.2.q.w.361.1 6
28.11 odd 6 280.2.q.e.81.3 ✓ 6
28.19 even 6 1960.2.a.v.1.3 3
28.23 odd 6 1960.2.a.w.1.1 3
28.27 even 2 1960.2.q.w.961.1 6
84.11 even 6 2520.2.bi.q.361.3 6
140.19 even 6 9800.2.a.cf.1.1 3
140.39 odd 6 1400.2.q.j.1201.1 6
140.67 even 12 1400.2.bh.i.249.1 12
140.79 odd 6 9800.2.a.ce.1.3 3
140.123 even 12 1400.2.bh.i.249.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.q.e.81.3 ✓ 6 28.11 odd 6
280.2.q.e.121.3 yes 6 4.3 odd 2
560.2.q.l.81.1 6 7.4 even 3 inner
560.2.q.l.401.1 6 1.1 even 1 trivial
1400.2.q.j.401.1 6 20.19 odd 2
1400.2.q.j.1201.1 6 140.39 odd 6
1400.2.bh.i.249.1 12 140.67 even 12
1400.2.bh.i.249.6 12 140.123 even 12
1400.2.bh.i.849.1 12 20.3 even 4
1400.2.bh.i.849.6 12 20.7 even 4
1960.2.a.v.1.3 3 28.19 even 6
1960.2.a.w.1.1 3 28.23 odd 6
1960.2.q.w.361.1 6 28.3 even 6
1960.2.q.w.961.1 6 28.27 even 2
2520.2.bi.q.361.3 6 84.11 even 6
2520.2.bi.q.1801.3 6 12.11 even 2
3920.2.a.cb.1.1 3 7.5 odd 6
3920.2.a.cc.1.3 3 7.2 even 3
9800.2.a.ce.1.3 3 140.79 odd 6
9800.2.a.cf.1.1 3 140.19 even 6