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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [522,2,Mod(89,522)] 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("522.89"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(28)) chi = DirichletCharacter(H, H._module([14, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.r (of order \(28\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.16819098551\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(6\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label 89.3
Character \(\chi\) \(=\) 522.89
Dual form 522.2.r.b.305.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.111964 + 0.993712i) q^{2} +(-0.974928 - 0.222521i) q^{4} +(1.39309 + 1.74688i) q^{5} +(1.11420 + 4.88164i) q^{7} +(0.330279 - 0.943883i) q^{8} +(-1.89187 + 1.18874i) q^{10} +(-1.59799 - 4.56680i) q^{11} +(-1.01899 + 2.11596i) q^{13} +(-4.97570 + 0.560627i) q^{14} +(0.900969 + 0.433884i) q^{16} +(2.31969 + 2.31969i) q^{17} +(1.01864 + 1.62115i) q^{19} +(-0.969443 - 2.01307i) q^{20} +(4.71701 - 1.07663i) q^{22} +(-3.88923 - 3.10156i) q^{23} +(0.00172093 - 0.00753988i) q^{25} +(-1.98857 - 1.24950i) q^{26} -5.00718i q^{28} +(-3.22824 + 4.31027i) q^{29} +(-1.26777 - 0.142843i) q^{31} +(-0.532032 + 0.846724i) q^{32} +(-2.56482 + 2.04538i) q^{34} +(-6.97544 + 8.74693i) q^{35} +(3.42473 + 1.19836i) q^{37} +(-1.72501 + 0.830721i) q^{38} +(2.10895 - 0.737955i) q^{40} +(-1.09544 + 1.09544i) q^{41} +(1.34290 + 11.9185i) q^{43} +(0.541719 + 4.80789i) q^{44} +(3.51751 - 3.51751i) q^{46} +(7.09754 - 2.48354i) q^{47} +(-16.2822 + 7.84110i) q^{49} +(0.00729979 + 0.00255431i) q^{50} +(1.46429 - 1.83616i) q^{52} +(1.86280 - 1.48553i) q^{53} +(5.75149 - 9.15345i) q^{55} +(4.97570 + 0.560627i) q^{56} +(-3.92172 - 3.69054i) q^{58} -9.07862i q^{59} +(6.43952 + 4.04622i) q^{61} +(0.283890 - 1.24380i) q^{62} +(-0.781831 - 0.623490i) q^{64} +(-5.11587 + 1.16766i) q^{65} +(-4.09981 - 8.51334i) q^{67} +(-1.74535 - 2.77771i) q^{68} +(-7.91093 - 7.91093i) q^{70} +(-7.27265 - 3.50232i) q^{71} +(-2.57864 + 0.290543i) q^{73} +(-1.57428 + 3.26902i) q^{74} +(-0.632358 - 1.80717i) q^{76} +(20.5130 - 12.8892i) q^{77} +(5.08472 - 14.5313i) q^{79} +(0.497187 + 2.17832i) q^{80} +(-0.965899 - 1.21120i) q^{82} +(15.9175 + 3.63306i) q^{83} +(-0.820679 + 7.28373i) q^{85} -11.9939 q^{86} -4.83831 q^{88} +(-0.976249 + 8.66445i) q^{89} +(-11.4647 - 2.61675i) q^{91} +(3.10156 + 3.88923i) q^{92} +(1.67325 + 7.33098i) q^{94} +(-1.41290 + 4.03784i) q^{95} +(11.4495 - 7.19420i) q^{97} +(-5.96876 - 17.0577i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 8 q^{7} + 4 q^{10} + 12 q^{16} - 40 q^{19} - 20 q^{25} - 28 q^{31} - 12 q^{37} - 4 q^{40} - 72 q^{43} + 16 q^{46} + 52 q^{49} + 4 q^{52} + 140 q^{55} + 44 q^{58} + 24 q^{61} + 140 q^{67} - 20 q^{70}+ \cdots + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/522\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(407\)
\(\chi(n)\) \(e\left(\frac{1}{28}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.111964 + 0.993712i −0.0791708 + 0.702661i
\(3\) 0 0
\(4\) −0.974928 − 0.222521i −0.487464 − 0.111260i
\(5\) 1.39309 + 1.74688i 0.623007 + 0.781227i 0.988764 − 0.149484i \(-0.0477612\pi\)
−0.365757 + 0.930710i \(0.619190\pi\)
\(6\) 0 0
\(7\) 1.11420 + 4.88164i 0.421129 + 1.84509i 0.525838 + 0.850585i \(0.323752\pi\)
−0.104709 + 0.994503i \(0.533391\pi\)
\(8\) 0.330279 − 0.943883i 0.116771 − 0.333713i
\(9\) 0 0
\(10\) −1.89187 + 1.18874i −0.598261 + 0.375912i
\(11\) −1.59799 − 4.56680i −0.481813 − 1.37694i −0.886602 − 0.462533i \(-0.846941\pi\)
0.404789 − 0.914410i \(-0.367345\pi\)
\(12\) 0 0
\(13\) −1.01899 + 2.11596i −0.282618 + 0.586862i −0.993156 − 0.116797i \(-0.962737\pi\)
0.710538 + 0.703659i \(0.248452\pi\)
\(14\) −4.97570 + 0.560627i −1.32981 + 0.149834i
\(15\) 0 0
\(16\) 0.900969 + 0.433884i 0.225242 + 0.108471i
\(17\) 2.31969 + 2.31969i 0.562607 + 0.562607i 0.930047 − 0.367441i \(-0.119766\pi\)
−0.367441 + 0.930047i \(0.619766\pi\)
\(18\) 0 0
\(19\) 1.01864 + 1.62115i 0.233691 + 0.371918i 0.942897 − 0.333084i \(-0.108089\pi\)
−0.709206 + 0.705001i \(0.750946\pi\)
\(20\) −0.969443 − 2.01307i −0.216774 − 0.450136i
\(21\) 0 0
\(22\) 4.71701 − 1.07663i 1.00567 − 0.229537i
\(23\) −3.88923 − 3.10156i −0.810960 − 0.646719i 0.127603 − 0.991825i \(-0.459272\pi\)
−0.938564 + 0.345106i \(0.887843\pi\)
\(24\) 0 0
\(25\) 0.00172093 − 0.00753988i 0.000344186 − 0.00150798i
\(26\) −1.98857 − 1.24950i −0.389990 − 0.245047i
\(27\) 0 0
\(28\) − 5.00718i − 0.946269i
\(29\) −3.22824 + 4.31027i −0.599469 + 0.800398i
\(30\) 0 0
\(31\) −1.26777 − 0.142843i −0.227698 − 0.0256554i −0.00262174 − 0.999997i \(-0.500835\pi\)
−0.225076 + 0.974341i \(0.572263\pi\)
\(32\) −0.532032 + 0.846724i −0.0940509 + 0.149681i
\(33\) 0 0
\(34\) −2.56482 + 2.04538i −0.439863 + 0.350779i
\(35\) −6.97544 + 8.74693i −1.17906 + 1.47850i
\(36\) 0 0
\(37\) 3.42473 + 1.19836i 0.563022 + 0.197010i 0.596759 − 0.802421i \(-0.296455\pi\)
−0.0337365 + 0.999431i \(0.510741\pi\)
\(38\) −1.72501 + 0.830721i −0.279834 + 0.134761i
\(39\) 0 0
\(40\) 2.10895 − 0.737955i 0.333455 − 0.116681i
\(41\) −1.09544 + 1.09544i −0.171078 + 0.171078i −0.787453 − 0.616375i \(-0.788601\pi\)
0.616375 + 0.787453i \(0.288601\pi\)
\(42\) 0 0
\(43\) 1.34290 + 11.9185i 0.204790 + 1.81756i 0.504369 + 0.863488i \(0.331725\pi\)
−0.299579 + 0.954071i \(0.596846\pi\)
\(44\) 0.541719 + 4.80789i 0.0816672 + 0.724817i
\(45\) 0 0
\(46\) 3.51751 − 3.51751i 0.518629 − 0.518629i
\(47\) 7.09754 − 2.48354i 1.03528 − 0.362261i 0.241460 − 0.970411i \(-0.422374\pi\)
0.793822 + 0.608150i \(0.208088\pi\)
\(48\) 0 0
\(49\) −16.2822 + 7.84110i −2.32603 + 1.12016i
\(50\) 0.00729979 + 0.00255431i 0.00103235 + 0.000361234i
\(51\) 0 0
\(52\) 1.46429 − 1.83616i 0.203061 − 0.254630i
\(53\) 1.86280 − 1.48553i 0.255875 − 0.204054i −0.487148 − 0.873320i \(-0.661963\pi\)
0.743023 + 0.669266i \(0.233391\pi\)
\(54\) 0 0
\(55\) 5.75149 − 9.15345i 0.775531 − 1.23425i
\(56\) 4.97570 + 0.560627i 0.664906 + 0.0749169i
\(57\) 0 0
\(58\) −3.92172 − 3.69054i −0.514947 − 0.484592i
\(59\) − 9.07862i − 1.18194i −0.806695 − 0.590968i \(-0.798746\pi\)
0.806695 − 0.590968i \(-0.201254\pi\)
\(60\) 0 0
\(61\) 6.43952 + 4.04622i 0.824496 + 0.518065i 0.876963 − 0.480557i \(-0.159566\pi\)
−0.0524670 + 0.998623i \(0.516708\pi\)
\(62\) 0.283890 − 1.24380i 0.0360541 − 0.157963i
\(63\) 0 0
\(64\) −0.781831 − 0.623490i −0.0977289 − 0.0779362i
\(65\) −5.11587 + 1.16766i −0.634546 + 0.144831i
\(66\) 0 0
\(67\) −4.09981 − 8.51334i −0.500871 − 1.04007i −0.986172 − 0.165723i \(-0.947004\pi\)
0.485301 − 0.874347i \(-0.338710\pi\)
\(68\) −1.74535 − 2.77771i −0.211655 − 0.336846i
\(69\) 0 0
\(70\) −7.91093 − 7.91093i −0.945537 − 0.945537i
\(71\) −7.27265 − 3.50232i −0.863104 − 0.415649i −0.0506795 − 0.998715i \(-0.516139\pi\)
−0.812425 + 0.583066i \(0.801853\pi\)
\(72\) 0 0
\(73\) −2.57864 + 0.290543i −0.301807 + 0.0340054i −0.261569 − 0.965185i \(-0.584240\pi\)
−0.0402373 + 0.999190i \(0.512811\pi\)
\(74\) −1.57428 + 3.26902i −0.183006 + 0.380016i
\(75\) 0 0
\(76\) −0.632358 − 1.80717i −0.0725364 − 0.207297i
\(77\) 20.5130 − 12.8892i 2.33767 − 1.46886i
\(78\) 0 0
\(79\) 5.08472 − 14.5313i 0.572076 − 1.63490i −0.185865 − 0.982575i \(-0.559509\pi\)
0.757941 − 0.652323i \(-0.226205\pi\)
\(80\) 0.497187 + 2.17832i 0.0555872 + 0.243543i
\(81\) 0 0
\(82\) −0.965899 − 1.21120i −0.106666 − 0.133755i
\(83\) 15.9175 + 3.63306i 1.74717 + 0.398780i 0.972373 − 0.233432i \(-0.0749958\pi\)
0.774795 + 0.632212i \(0.217853\pi\)
\(84\) 0 0
\(85\) −0.820679 + 7.28373i −0.0890151 + 0.790031i
\(86\) −11.9939 −1.29334
\(87\) 0 0
\(88\) −4.83831 −0.515766
\(89\) −0.976249 + 8.66445i −0.103482 + 0.918429i 0.829500 + 0.558507i \(0.188626\pi\)
−0.932982 + 0.359923i \(0.882803\pi\)
\(90\) 0 0
\(91\) −11.4647 − 2.61675i −1.20183 − 0.274310i
\(92\) 3.10156 + 3.88923i 0.323360 + 0.405480i
\(93\) 0 0
\(94\) 1.67325 + 7.33098i 0.172582 + 0.756133i
\(95\) −1.41290 + 4.03784i −0.144961 + 0.414274i
\(96\) 0 0
\(97\) 11.4495 − 7.19420i 1.16252 − 0.730460i 0.194394 − 0.980923i \(-0.437726\pi\)
0.968126 + 0.250463i \(0.0805830\pi\)
\(98\) −5.96876 − 17.0577i −0.602936 − 1.72309i
\(99\) 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 522.2.r.b.89.3 ✓ 72
3.2 odd 2 inner 522.2.r.b.89.4 yes 72
29.15 odd 28 inner 522.2.r.b.305.4 yes 72
87.44 even 28 inner 522.2.r.b.305.3 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
522.2.r.b.89.3 ✓ 72 1.1 even 1 trivial
522.2.r.b.89.4 yes 72 3.2 odd 2 inner
522.2.r.b.305.3 yes 72 87.44 even 28 inner
522.2.r.b.305.4 yes 72 29.15 odd 28 inner