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Properties

Label 560.2.bj.d.433.8
Level $560$
Weight $2$
Character 560.433
Analytic conductor $4.472$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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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(97,560)] 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("560.97"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(560, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.bj (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,0,0,-4] 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: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 433.8
Character \(\chi\) \(=\) 560.433
Dual form 560.2.bj.d.97.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.730185 - 0.730185i) q^{3} +(-1.51201 - 1.64737i) q^{5} +(-2.41329 - 1.08445i) q^{7} +1.93366i q^{9} -5.95977 q^{11} +(-0.921623 + 0.921623i) q^{13} +(-2.30693 - 0.0988432i) q^{15} +(2.02728 + 2.02728i) q^{17} -3.29475 q^{19} +(-2.55400 + 0.970294i) q^{21} +(-0.0544054 - 0.0544054i) q^{23} +(-0.427677 + 4.98168i) q^{25} +(3.60248 + 3.60248i) q^{27} -3.78901i q^{29} -4.88150i q^{31} +(-4.35174 + 4.35174i) q^{33} +(1.86240 + 5.61529i) q^{35} +(-3.20809 + 3.20809i) q^{37} +1.34591i q^{39} -10.6672i q^{41} +(1.60483 + 1.60483i) q^{43} +(3.18546 - 2.92370i) q^{45} +(-2.64854 - 2.64854i) q^{47} +(4.64792 + 5.23420i) q^{49} +2.96059 q^{51} +(-9.16786 - 9.16786i) q^{53} +(9.01121 + 9.81797i) q^{55} +(-2.40577 + 2.40577i) q^{57} -13.3231 q^{59} -11.2091i q^{61} +(2.09697 - 4.66648i) q^{63} +(2.91176 + 0.124758i) q^{65} +(6.43325 - 6.43325i) q^{67} -0.0794520 q^{69} +8.51221 q^{71} +(-8.66633 + 8.66633i) q^{73} +(3.32526 + 3.94983i) q^{75} +(14.3826 + 6.46310i) q^{77} +1.76209i q^{79} -0.540019 q^{81} +(2.36897 - 2.36897i) q^{83} +(0.274428 - 6.40496i) q^{85} +(-2.76668 - 2.76668i) q^{87} -9.73989 q^{89} +(3.22360 - 1.22468i) q^{91} +(-3.56440 - 3.56440i) q^{93} +(4.98168 + 5.42768i) q^{95} +(-2.05040 - 2.05040i) q^{97} -11.5242i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 4 q^{7} - 8 q^{11} + 8 q^{15} + 16 q^{21} + 32 q^{23} + 8 q^{25} - 12 q^{35} - 8 q^{37} - 16 q^{43} + 24 q^{51} - 16 q^{53} - 20 q^{63} - 48 q^{65} + 32 q^{67} + 32 q^{71} - 40 q^{77} - 72 q^{81}+ \cdots + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(1\) \(1\)

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
\(3\) 0.730185 − 0.730185i 0.421573 − 0.421573i −0.464172 − 0.885745i \(-0.653648\pi\)
0.885745 + 0.464172i \(0.153648\pi\)
\(4\) 0 0
\(5\) −1.51201 − 1.64737i −0.676190 − 0.736728i
\(6\) 0 0
\(7\) −2.41329 − 1.08445i −0.912137 − 0.409885i
\(8\) 0 0
\(9\) 1.93366i 0.644553i
\(10\) 0 0
\(11\) −5.95977 −1.79694 −0.898470 − 0.439036i \(-0.855320\pi\)
−0.898470 + 0.439036i \(0.855320\pi\)
\(12\) 0 0
\(13\) −0.921623 + 0.921623i −0.255612 + 0.255612i −0.823267 − 0.567655i \(-0.807851\pi\)
0.567655 + 0.823267i \(0.307851\pi\)
\(14\) 0 0
\(15\) −2.30693 − 0.0988432i −0.595647 − 0.0255212i
\(16\) 0 0
\(17\) 2.02728 + 2.02728i 0.491689 + 0.491689i 0.908838 − 0.417149i \(-0.136971\pi\)
−0.417149 + 0.908838i \(0.636971\pi\)
\(18\) 0 0
\(19\) −3.29475 −0.755867 −0.377933 − 0.925833i \(-0.623365\pi\)
−0.377933 + 0.925833i \(0.623365\pi\)
\(20\) 0 0
\(21\) −2.55400 + 0.970294i −0.557328 + 0.211736i
\(22\) 0 0
\(23\) −0.0544054 − 0.0544054i −0.0113443 − 0.0113443i 0.701412 − 0.712756i \(-0.252553\pi\)
−0.712756 + 0.701412i \(0.752553\pi\)
\(24\) 0 0
\(25\) −0.427677 + 4.98168i −0.0855353 + 0.996335i
\(26\) 0 0
\(27\) 3.60248 + 3.60248i 0.693298 + 0.693298i
\(28\) 0 0
\(29\) − 3.78901i − 0.703602i −0.936075 − 0.351801i \(-0.885569\pi\)
0.936075 − 0.351801i \(-0.114431\pi\)
\(30\) 0 0
\(31\) − 4.88150i − 0.876744i −0.898794 − 0.438372i \(-0.855555\pi\)
0.898794 − 0.438372i \(-0.144445\pi\)
\(32\) 0 0
\(33\) −4.35174 + 4.35174i −0.757540 + 0.757540i
\(34\) 0 0
\(35\) 1.86240 + 5.61529i 0.314804 + 0.949157i
\(36\) 0 0
\(37\) −3.20809 + 3.20809i −0.527406 + 0.527406i −0.919798 − 0.392392i \(-0.871648\pi\)
0.392392 + 0.919798i \(0.371648\pi\)
\(38\) 0 0
\(39\) 1.34591i 0.215518i
\(40\) 0 0
\(41\) − 10.6672i − 1.66593i −0.553323 − 0.832967i \(-0.686641\pi\)
0.553323 − 0.832967i \(-0.313359\pi\)
\(42\) 0 0
\(43\) 1.60483 + 1.60483i 0.244734 + 0.244734i 0.818805 − 0.574071i \(-0.194637\pi\)
−0.574071 + 0.818805i \(0.694637\pi\)
\(44\) 0 0
\(45\) 3.18546 − 2.92370i 0.474860 − 0.435840i
\(46\) 0 0
\(47\) −2.64854 − 2.64854i −0.386329 − 0.386329i 0.487047 − 0.873376i \(-0.338074\pi\)
−0.873376 + 0.487047i \(0.838074\pi\)
\(48\) 0 0
\(49\) 4.64792 + 5.23420i 0.663988 + 0.747743i
\(50\) 0 0
\(51\) 2.96059 0.414565
\(52\) 0 0
\(53\) −9.16786 − 9.16786i −1.25930 − 1.25930i −0.951426 − 0.307876i \(-0.900382\pi\)
−0.307876 − 0.951426i \(-0.599618\pi\)
\(54\) 0 0
\(55\) 9.01121 + 9.81797i 1.21507 + 1.32385i
\(56\) 0 0
\(57\) −2.40577 + 2.40577i −0.318653 + 0.318653i
\(58\) 0 0
\(59\) −13.3231 −1.73452 −0.867261 − 0.497854i \(-0.834121\pi\)
−0.867261 + 0.497854i \(0.834121\pi\)
\(60\) 0 0
\(61\) − 11.2091i − 1.43518i −0.696465 − 0.717591i \(-0.745245\pi\)
0.696465 − 0.717591i \(-0.254755\pi\)
\(62\) 0 0
\(63\) 2.09697 − 4.66648i 0.264193 − 0.587921i
\(64\) 0 0
\(65\) 2.91176 + 0.124758i 0.361159 + 0.0154743i
\(66\) 0 0
\(67\) 6.43325 − 6.43325i 0.785947 − 0.785947i −0.194880 − 0.980827i \(-0.562432\pi\)
0.980827 + 0.194880i \(0.0624318\pi\)
\(68\) 0 0
\(69\) −0.0794520 −0.00956489
\(70\) 0 0
\(71\) 8.51221 1.01021 0.505107 − 0.863057i \(-0.331453\pi\)
0.505107 + 0.863057i \(0.331453\pi\)
\(72\) 0 0
\(73\) −8.66633 + 8.66633i −1.01432 + 1.01432i −0.0144206 + 0.999896i \(0.504590\pi\)
−0.999896 + 0.0144206i \(0.995410\pi\)
\(74\) 0 0
\(75\) 3.32526 + 3.94983i 0.383968 + 0.456087i
\(76\) 0 0
\(77\) 14.3826 + 6.46310i 1.63905 + 0.736539i
\(78\) 0 0
\(79\) 1.76209i 0.198250i 0.995075 + 0.0991251i \(0.0316044\pi\)
−0.995075 + 0.0991251i \(0.968396\pi\)
\(80\) 0 0
\(81\) −0.540019 −0.0600021
\(82\) 0 0
\(83\) 2.36897 − 2.36897i 0.260028 − 0.260028i −0.565037 − 0.825065i \(-0.691138\pi\)
0.825065 + 0.565037i \(0.191138\pi\)
\(84\) 0 0
\(85\) 0.274428 − 6.40496i 0.0297659 − 0.694716i
\(86\) 0 0
\(87\) −2.76668 − 2.76668i −0.296619 − 0.296619i
\(88\) 0 0
\(89\) −9.73989 −1.03243 −0.516213 − 0.856460i \(-0.672659\pi\)
−0.516213 + 0.856460i \(0.672659\pi\)
\(90\) 0 0
\(91\) 3.22360 − 1.22468i 0.337925 − 0.128382i
\(92\) 0 0
\(93\) −3.56440 − 3.56440i −0.369611 − 0.369611i
\(94\) 0 0
\(95\) 4.98168 + 5.42768i 0.511109 + 0.556868i
\(96\) 0 0
\(97\) −2.05040 − 2.05040i −0.208187 − 0.208187i 0.595310 − 0.803496i \(-0.297029\pi\)
−0.803496 + 0.595310i \(0.797029\pi\)
\(98\) 0 0
\(99\) − 11.5242i − 1.15822i
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.bj.d.433.8 24
4.3 odd 2 280.2.x.a.153.5 yes 24
5.2 odd 4 inner 560.2.bj.d.97.5 24
7.6 odd 2 inner 560.2.bj.d.433.5 24
20.3 even 4 1400.2.x.b.657.5 24
20.7 even 4 280.2.x.a.97.8 yes 24
20.19 odd 2 1400.2.x.b.993.8 24
28.27 even 2 280.2.x.a.153.8 yes 24
35.27 even 4 inner 560.2.bj.d.97.8 24
140.27 odd 4 280.2.x.a.97.5 ✓ 24
140.83 odd 4 1400.2.x.b.657.8 24
140.139 even 2 1400.2.x.b.993.5 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.x.a.97.5 ✓ 24 140.27 odd 4
280.2.x.a.97.8 yes 24 20.7 even 4
280.2.x.a.153.5 yes 24 4.3 odd 2
280.2.x.a.153.8 yes 24 28.27 even 2
560.2.bj.d.97.5 24 5.2 odd 4 inner
560.2.bj.d.97.8 24 35.27 even 4 inner
560.2.bj.d.433.5 24 7.6 odd 2 inner
560.2.bj.d.433.8 24 1.1 even 1 trivial
1400.2.x.b.657.5 24 20.3 even 4
1400.2.x.b.657.8 24 140.83 odd 4
1400.2.x.b.993.5 24 140.139 even 2
1400.2.x.b.993.8 24 20.19 odd 2