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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [567,2,Mod(80,567)] 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("567.80"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(567, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,16,0,0,-8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 80.3
Character \(\chi\) \(=\) 567.80
Dual form 567.2.p.e.404.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.88023 + 1.08555i) q^{2} +(1.35685 - 2.35014i) q^{4} +(-0.618749 - 1.07170i) q^{5} +(1.89307 + 1.84832i) q^{7} +1.54953i q^{8} +(2.32678 + 1.34337i) q^{10} +(-0.968115 - 0.558941i) q^{11} -5.39915i q^{13} +(-5.56586 - 1.42025i) q^{14} +(1.03161 + 1.78680i) q^{16} +(-0.913955 + 1.58302i) q^{17} +(-3.40985 + 1.96868i) q^{19} -3.35820 q^{20} +2.42704 q^{22} +(-0.167225 + 0.0965473i) q^{23} +(1.73430 - 3.00390i) q^{25} +(5.86107 + 10.1517i) q^{26} +(6.91242 - 1.94106i) q^{28} -8.48635i q^{29} +(-4.27901 - 2.47049i) q^{31} +(-6.56319 - 3.78926i) q^{32} -3.96859i q^{34} +(0.809523 - 3.17245i) q^{35} +(-5.81453 - 10.0711i) q^{37} +(4.27420 - 7.40314i) q^{38} +(1.66063 - 0.958767i) q^{40} +11.3840 q^{41} +3.42704 q^{43} +(-2.62718 + 1.51680i) q^{44} +(0.209614 - 0.363063i) q^{46} +(4.03316 + 6.98563i) q^{47} +(0.167404 + 6.99800i) q^{49} +7.53070i q^{50} +(-12.6887 - 7.32585i) q^{52} +(2.54245 + 1.46788i) q^{53} +1.38338i q^{55} +(-2.86403 + 2.93336i) q^{56} +(9.21238 + 15.9563i) q^{58} +(6.10123 - 10.5676i) q^{59} +(6.61605 - 3.81978i) q^{61} +10.7274 q^{62} +12.3273 q^{64} +(-5.78629 + 3.34072i) q^{65} +(0.587063 - 1.01682i) q^{67} +(2.48020 + 4.29584i) q^{68} +(1.92178 + 6.84373i) q^{70} -11.7500i q^{71} +(0.662676 + 0.382596i) q^{73} +(21.8653 + 12.6240i) q^{74} +10.6848i q^{76} +(-0.799602 - 2.84750i) q^{77} +(0.436035 + 0.755235i) q^{79} +(1.27661 - 2.21116i) q^{80} +(-21.4046 + 12.3580i) q^{82} -10.6014 q^{83} +2.26203 q^{85} +(-6.44364 + 3.72024i) q^{86} +(0.866094 - 1.50012i) q^{88} +(0.705529 + 1.22201i) q^{89} +(9.97938 - 10.2210i) q^{91} +0.524001i q^{92} +(-15.1666 - 8.75641i) q^{94} +(4.21967 + 2.43623i) q^{95} +4.92098i q^{97} +(-7.91146 - 12.9761i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 16 q^{4} - 8 q^{7} - 28 q^{16} + 24 q^{22} - 16 q^{25} - 16 q^{28} - 48 q^{31} - 4 q^{37} + 56 q^{43} + 12 q^{46} - 4 q^{49} + 48 q^{52} + 36 q^{58} + 12 q^{61} - 80 q^{64} - 20 q^{67} + 120 q^{70}+ \cdots + 72 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.88023 + 1.08555i −1.32953 + 0.767602i −0.985226 − 0.171258i \(-0.945217\pi\)
−0.344300 + 0.938860i \(0.611884\pi\)
\(3\) 0 0
\(4\) 1.35685 − 2.35014i 0.678426 − 1.17507i
\(5\) −0.618749 − 1.07170i −0.276713 − 0.479281i 0.693853 − 0.720117i \(-0.255912\pi\)
−0.970566 + 0.240836i \(0.922578\pi\)
\(6\) 0 0
\(7\) 1.89307 + 1.84832i 0.715512 + 0.698600i
\(8\) 1.54953i 0.547840i
\(9\) 0 0
\(10\) 2.32678 + 1.34337i 0.735793 + 0.424811i
\(11\) −0.968115 − 0.558941i −0.291898 − 0.168527i 0.346900 − 0.937902i \(-0.387234\pi\)
−0.638797 + 0.769375i \(0.720568\pi\)
\(12\) 0 0
\(13\) − 5.39915i − 1.49746i −0.662878 − 0.748728i \(-0.730665\pi\)
0.662878 − 0.748728i \(-0.269335\pi\)
\(14\) −5.56586 − 1.42025i −1.48754 − 0.379579i
\(15\) 0 0
\(16\) 1.03161 + 1.78680i 0.257902 + 0.446700i
\(17\) −0.913955 + 1.58302i −0.221667 + 0.383938i −0.955314 − 0.295592i \(-0.904483\pi\)
0.733648 + 0.679530i \(0.237816\pi\)
\(18\) 0 0
\(19\) −3.40985 + 1.96868i −0.782272 + 0.451645i −0.837235 − 0.546843i \(-0.815829\pi\)
0.0549627 + 0.998488i \(0.482496\pi\)
\(20\) −3.35820 −0.750916
\(21\) 0 0
\(22\) 2.42704 0.517447
\(23\) −0.167225 + 0.0965473i −0.0348688 + 0.0201315i −0.517333 − 0.855784i \(-0.673075\pi\)
0.482464 + 0.875916i \(0.339742\pi\)
\(24\) 0 0
\(25\) 1.73430 − 3.00390i 0.346860 − 0.600779i
\(26\) 5.86107 + 10.1517i 1.14945 + 1.99091i
\(27\) 0 0
\(28\) 6.91242 − 1.94106i 1.30632 − 0.366827i
\(29\) − 8.48635i − 1.57588i −0.615755 − 0.787938i \(-0.711149\pi\)
0.615755 − 0.787938i \(-0.288851\pi\)
\(30\) 0 0
\(31\) −4.27901 − 2.47049i −0.768533 − 0.443713i 0.0638179 − 0.997962i \(-0.479672\pi\)
−0.832351 + 0.554249i \(0.813006\pi\)
\(32\) −6.56319 − 3.78926i −1.16022 − 0.669853i
\(33\) 0 0
\(34\) − 3.96859i − 0.680607i
\(35\) 0.809523 − 3.17245i 0.136834 − 0.536243i
\(36\) 0 0
\(37\) −5.81453 − 10.0711i −0.955903 − 1.65567i −0.732289 − 0.680994i \(-0.761548\pi\)
−0.223614 − 0.974678i \(-0.571785\pi\)
\(38\) 4.27420 − 7.40314i 0.693367 − 1.20095i
\(39\) 0 0
\(40\) 1.66063 − 0.958767i 0.262569 − 0.151594i
\(41\) 11.3840 1.77788 0.888942 − 0.458020i \(-0.151441\pi\)
0.888942 + 0.458020i \(0.151441\pi\)
\(42\) 0 0
\(43\) 3.42704 0.522619 0.261310 − 0.965255i \(-0.415846\pi\)
0.261310 + 0.965255i \(0.415846\pi\)
\(44\) −2.62718 + 1.51680i −0.396062 + 0.228666i
\(45\) 0 0
\(46\) 0.209614 − 0.363063i 0.0309060 − 0.0535307i
\(47\) 4.03316 + 6.98563i 0.588296 + 1.01896i 0.994456 + 0.105157i \(0.0335345\pi\)
−0.406159 + 0.913802i \(0.633132\pi\)
\(48\) 0 0
\(49\) 0.167404 + 6.99800i 0.0239149 + 0.999714i
\(50\) 7.53070i 1.06500i
\(51\) 0 0
\(52\) −12.6887 − 7.32585i −1.75961 − 1.01591i
\(53\) 2.54245 + 1.46788i 0.349232 + 0.201629i 0.664347 − 0.747424i \(-0.268710\pi\)
−0.315115 + 0.949053i \(0.602043\pi\)
\(54\) 0 0
\(55\) 1.38338i 0.186534i
\(56\) −2.86403 + 2.93336i −0.382722 + 0.391986i
\(57\) 0 0
\(58\) 9.21238 + 15.9563i 1.20965 + 2.09517i
\(59\) 6.10123 − 10.5676i 0.794313 − 1.37579i −0.128962 − 0.991650i \(-0.541165\pi\)
0.923275 − 0.384140i \(-0.125502\pi\)
\(60\) 0 0
\(61\) 6.61605 − 3.81978i 0.847098 − 0.489072i −0.0125724 − 0.999921i \(-0.504002\pi\)
0.859671 + 0.510848i \(0.170669\pi\)
\(62\) 10.7274 1.36238
\(63\) 0 0
\(64\) 12.3273 1.54092
\(65\) −5.78629 + 3.34072i −0.717701 + 0.414365i
\(66\) 0 0
\(67\) 0.587063 − 1.01682i 0.0717212 − 0.124225i −0.827935 − 0.560825i \(-0.810484\pi\)
0.899656 + 0.436600i \(0.143817\pi\)
\(68\) 2.48020 + 4.29584i 0.300769 + 0.520947i
\(69\) 0 0
\(70\) 1.92178 + 6.84373i 0.229696 + 0.817983i
\(71\) − 11.7500i − 1.39447i −0.716843 − 0.697234i \(-0.754414\pi\)
0.716843 − 0.697234i \(-0.245586\pi\)
\(72\) 0 0
\(73\) 0.662676 + 0.382596i 0.0775603 + 0.0447795i 0.538279 − 0.842767i \(-0.319075\pi\)
−0.460718 + 0.887546i \(0.652408\pi\)
\(74\) 21.8653 + 12.6240i 2.54179 + 1.46751i
\(75\) 0 0
\(76\) 10.6848i 1.22563i
\(77\) −0.799602 − 2.84750i −0.0911231 − 0.324503i
\(78\) 0 0
\(79\) 0.436035 + 0.755235i 0.0490578 + 0.0849706i 0.889512 − 0.456913i \(-0.151045\pi\)
−0.840454 + 0.541883i \(0.817711\pi\)
\(80\) 1.27661 − 2.21116i 0.142730 − 0.247215i
\(81\) 0 0
\(82\) −21.4046 + 12.3580i −2.36374 + 1.36471i
\(83\) −10.6014 −1.16365 −0.581825 − 0.813314i \(-0.697661\pi\)
−0.581825 + 0.813314i \(0.697661\pi\)
\(84\) 0 0
\(85\) 2.26203 0.245352
\(86\) −6.44364 + 3.72024i −0.694835 + 0.401163i
\(87\) 0 0
\(88\) 0.866094 − 1.50012i 0.0923260 − 0.159913i
\(89\) 0.705529 + 1.22201i 0.0747859 + 0.129533i 0.900993 − 0.433833i \(-0.142839\pi\)
−0.826207 + 0.563366i \(0.809506\pi\)
\(90\) 0 0
\(91\) 9.97938 − 10.2210i 1.04612 − 1.07145i
\(92\) 0.524001i 0.0546309i
\(93\) 0 0
\(94\) −15.1666 − 8.75641i −1.56431 − 0.903155i
\(95\) 4.21967 + 2.43623i 0.432929 + 0.249952i
\(96\) 0 0
\(97\) 4.92098i 0.499649i 0.968291 + 0.249825i \(0.0803730\pi\)
−0.968291 + 0.249825i \(0.919627\pi\)
\(98\) −7.91146 − 12.9761i −0.799178 − 1.31079i
\(99\) 0 0
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 567.2.p.e.80.3 ✓ 32
3.2 odd 2 inner 567.2.p.e.80.14 yes 32
7.5 odd 6 inner 567.2.p.e.404.14 yes 32
9.2 odd 6 567.2.s.g.458.3 32
9.4 even 3 567.2.i.g.269.3 32
9.5 odd 6 567.2.i.g.269.14 32
9.7 even 3 567.2.s.g.458.14 32
21.5 even 6 inner 567.2.p.e.404.3 yes 32
63.5 even 6 567.2.s.g.26.14 32
63.40 odd 6 567.2.s.g.26.3 32
63.47 even 6 567.2.i.g.215.14 32
63.61 odd 6 567.2.i.g.215.3 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.i.g.215.3 32 63.61 odd 6
567.2.i.g.215.14 32 63.47 even 6
567.2.i.g.269.3 32 9.4 even 3
567.2.i.g.269.14 32 9.5 odd 6
567.2.p.e.80.3 ✓ 32 1.1 even 1 trivial
567.2.p.e.80.14 yes 32 3.2 odd 2 inner
567.2.p.e.404.3 yes 32 21.5 even 6 inner
567.2.p.e.404.14 yes 32 7.5 odd 6 inner
567.2.s.g.26.3 32 63.40 odd 6
567.2.s.g.26.14 32 63.5 even 6
567.2.s.g.458.3 32 9.2 odd 6
567.2.s.g.458.14 32 9.7 even 3