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Properties

Label 280.2.x.a.153.5
Level $280$
Weight $2$
Character 280.153
Analytic conductor $2.236$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [280,2,Mod(97,280)] 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("280.97"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(280, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 280.x (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.23581125660\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 153.5
Character \(\chi\) \(=\) 280.153
Dual form 280.2.x.a.97.5

$q$-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 \(\left(\mathbb{Z}/280\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(141\) \(241\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(1\) \(-1\)

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
\(3\) −0.730185 + 0.730185i −0.421573 + 0.421573i −0.885745 − 0.464172i \(-0.846352\pi\)
0.464172 + 0.885745i \(0.346352\pi\)
\(4\) 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 \(-0.144680\pi\)
0.898470 + 0.439036i \(0.144680\pi\)
\(12\) 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 \(-0.376635\pi\)
0.377933 + 0.925833i \(0.376635\pi\)
\(20\) 0 0
\(21\) −2.55400 + 0.970294i −0.557328 + 0.211736i
\(22\) 0 0
\(23\) 0.0544054 + 0.0544054i 0.0113443 + 0.0113443i 0.712756 − 0.701412i \(-0.247447\pi\)
−0.701412 + 0.712756i \(0.747447\pi\)
\(24\) 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 \(0.144445\pi\)
−0.898794 + 0.438372i \(0.855555\pi\)
\(32\) 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 0.574071 − 0.818805i \(-0.305363\pi\)
−0.818805 + 0.574071i \(0.805363\pi\)
\(44\) 0 0
\(45\) 3.18546 − 2.92370i 0.474860 − 0.435840i
\(46\) 0 0
\(47\) 2.64854 + 2.64854i 0.386329 + 0.386329i 0.873376 − 0.487047i \(-0.161926\pi\)
−0.487047 + 0.873376i \(0.661926\pi\)
\(48\) 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 \(-0.165879\pi\)
0.867261 + 0.497854i \(0.165879\pi\)
\(60\) 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 −0.980827 − 0.194880i \(-0.937568\pi\)
0.194880 + 0.980827i \(0.437568\pi\)
\(68\) 0 0
\(69\) −0.0794520 −0.00956489
\(70\) 0 0
\(71\) −8.51221 −1.01021 −0.505107 − 0.863057i \(-0.668547\pi\)
−0.505107 + 0.863057i \(0.668547\pi\)
\(72\) 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 \(-0.968396\pi\)
0.995075 − 0.0991251i \(-0.0316044\pi\)
\(80\) 0 0
\(81\) −0.540019 −0.0600021
\(82\) 0 0
\(83\) −2.36897 + 2.36897i −0.260028 + 0.260028i −0.825065 − 0.565037i \(-0.808862\pi\)
0.565037 + 0.825065i \(0.308862\pi\)
\(84\) 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 280.2.x.a.153.5 yes 24
4.3 odd 2 560.2.bj.d.433.8 24
5.2 odd 4 inner 280.2.x.a.97.8 yes 24
5.3 odd 4 1400.2.x.b.657.5 24
5.4 even 2 1400.2.x.b.993.8 24
7.6 odd 2 inner 280.2.x.a.153.8 yes 24
20.7 even 4 560.2.bj.d.97.5 24
28.27 even 2 560.2.bj.d.433.5 24
35.13 even 4 1400.2.x.b.657.8 24
35.27 even 4 inner 280.2.x.a.97.5 ✓ 24
35.34 odd 2 1400.2.x.b.993.5 24
140.27 odd 4 560.2.bj.d.97.8 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.x.a.97.5 ✓ 24 35.27 even 4 inner
280.2.x.a.97.8 yes 24 5.2 odd 4 inner
280.2.x.a.153.5 yes 24 1.1 even 1 trivial
280.2.x.a.153.8 yes 24 7.6 odd 2 inner
560.2.bj.d.97.5 24 20.7 even 4
560.2.bj.d.97.8 24 140.27 odd 4
560.2.bj.d.433.5 24 28.27 even 2
560.2.bj.d.433.8 24 4.3 odd 2
1400.2.x.b.657.5 24 5.3 odd 4
1400.2.x.b.657.8 24 35.13 even 4
1400.2.x.b.993.5 24 35.34 odd 2
1400.2.x.b.993.8 24 5.4 even 2