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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 89.5
Character \(\chi\) \(=\) 522.89
Dual form 522.2.r.b.305.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.111964 - 0.993712i) q^{2} +(-0.974928 - 0.222521i) q^{4} +(0.139983 + 0.175534i) q^{5} +(-0.379369 - 1.66212i) q^{7} +(-0.330279 + 0.943883i) q^{8} +(0.190103 - 0.119450i) q^{10} +(0.0727648 + 0.207950i) q^{11} +(2.64550 - 5.49343i) q^{13} +(-1.69415 + 0.190885i) q^{14} +(0.900969 + 0.433884i) q^{16} +(-0.952520 - 0.952520i) q^{17} +(-1.56527 - 2.49112i) q^{19} +(-0.0974138 - 0.202282i) q^{20} +(0.214789 - 0.0490243i) q^{22} +(-4.95521 - 3.95165i) q^{23} +(1.10139 - 4.82550i) q^{25} +(-5.16269 - 3.24393i) q^{26} +1.70487i q^{28} +(-0.642473 + 5.34670i) q^{29} +(4.79649 + 0.540435i) q^{31} +(0.532032 - 0.846724i) q^{32} +(-1.05318 + 0.839882i) q^{34} +(0.238653 - 0.299262i) q^{35} +(-3.05702 - 1.06970i) q^{37} +(-2.65071 + 1.27651i) q^{38} +(-0.211917 + 0.0741529i) q^{40} +(4.15163 - 4.15163i) q^{41} +(0.0820565 + 0.728272i) q^{43} +(-0.0246672 - 0.218928i) q^{44} +(-4.48161 + 4.48161i) q^{46} +(-5.32959 + 1.86490i) q^{47} +(3.68805 - 1.77607i) q^{49} +(-4.67184 - 1.63475i) q^{50} +(-3.80157 + 4.76702i) q^{52} +(0.534396 - 0.426167i) q^{53} +(-0.0263163 + 0.0418822i) q^{55} +(1.69415 + 0.190885i) q^{56} +(5.24115 + 1.23707i) q^{58} +1.13448i q^{59} +(3.30933 + 2.07939i) q^{61} +(1.07407 - 4.70582i) q^{62} +(-0.781831 - 0.623490i) q^{64} +(1.33461 - 0.304615i) q^{65} +(0.282777 + 0.587192i) q^{67} +(0.716683 + 1.14059i) q^{68} +(-0.270659 - 0.270659i) q^{70} +(9.79839 + 4.71866i) q^{71} +(-7.38764 + 0.832387i) q^{73} +(-1.40525 + 2.91803i) q^{74} +(0.971703 + 2.77697i) q^{76} +(0.318033 - 0.199834i) q^{77} +(-4.94162 + 14.1223i) q^{79} +(0.0499595 + 0.218887i) q^{80} +(-3.66069 - 4.59036i) q^{82} +(3.09277 + 0.705905i) q^{83} +(0.0338623 - 0.300536i) q^{85} +0.732880 q^{86} -0.220313 q^{88} +(-0.794897 + 7.05491i) q^{89} +(-10.1344 - 2.31310i) q^{91} +(3.95165 + 4.95521i) q^{92} +(1.25645 + 5.50488i) q^{94} +(0.218163 - 0.623473i) q^{95} +(5.55617 - 3.49117i) q^{97} +(-1.35197 - 3.86372i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 8 q^{7} + 4 q^{10} + 12 q^{16} - 40 q^{19} - 20 q^{25} - 28 q^{31} - 12 q^{37} - 4 q^{40} - 72 q^{43} + 16 q^{46} + 52 q^{49} + 4 q^{52} + 140 q^{55} + 44 q^{58} + 24 q^{61} + 140 q^{67} - 20 q^{70}+ \cdots + 28 q^{97}+O(q^{100}) \) 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
\(5\) 0.139983 + 0.175534i 0.0626025 + 0.0785010i 0.812147 − 0.583453i \(-0.198299\pi\)
−0.749544 + 0.661954i \(0.769727\pi\)
\(6\) 0 0
\(7\) −0.379369 − 1.66212i −0.143388 − 0.628223i −0.994634 − 0.103457i \(-0.967010\pi\)
0.851246 − 0.524767i \(-0.175847\pi\)
\(8\) −0.330279 + 0.943883i −0.116771 + 0.333713i
\(9\) 0 0
\(10\) 0.190103 − 0.119450i 0.0601159 − 0.0377733i
\(11\) 0.0727648 + 0.207950i 0.0219394 + 0.0626992i 0.954332 − 0.298747i \(-0.0965687\pi\)
−0.932393 + 0.361446i \(0.882283\pi\)
\(12\) 0 0
\(13\) 2.64550 − 5.49343i 0.733729 − 1.52360i −0.114179 − 0.993460i \(-0.536424\pi\)
0.847908 − 0.530143i \(-0.177862\pi\)
\(14\) −1.69415 + 0.190885i −0.452780 + 0.0510161i
\(15\) 0 0
\(16\) 0.900969 + 0.433884i 0.225242 + 0.108471i
\(17\) −0.952520 − 0.952520i −0.231020 − 0.231020i 0.582098 − 0.813118i \(-0.302232\pi\)
−0.813118 + 0.582098i \(0.802232\pi\)
\(18\) 0 0
\(19\) −1.56527 − 2.49112i −0.359098 − 0.571502i 0.617501 − 0.786570i \(-0.288145\pi\)
−0.976599 + 0.215069i \(0.931002\pi\)
\(20\) −0.0974138 − 0.202282i −0.0217824 − 0.0452316i
\(21\) 0 0
\(22\) 0.214789 − 0.0490243i 0.0457932 − 0.0104520i
\(23\) −4.95521 − 3.95165i −1.03323 − 0.823976i −0.0486362 − 0.998817i \(-0.515488\pi\)
−0.984597 + 0.174841i \(0.944059\pi\)
\(24\) 0 0
\(25\) 1.10139 − 4.82550i 0.220278 − 0.965099i
\(26\) −5.16269 − 3.24393i −1.01249 − 0.636187i
\(27\) 0 0
\(28\) 1.70487i 0.322190i
\(29\) −0.642473 + 5.34670i −0.119304 + 0.992858i
\(30\) 0 0
\(31\) 4.79649 + 0.540435i 0.861475 + 0.0970650i 0.531637 − 0.846972i \(-0.321577\pi\)
0.329838 + 0.944037i \(0.393006\pi\)
\(32\) 0.532032 − 0.846724i 0.0940509 − 0.149681i
\(33\) 0 0
\(34\) −1.05318 + 0.839882i −0.180619 + 0.144039i
\(35\) 0.238653 − 0.299262i 0.0403397 − 0.0505844i
\(36\) 0 0
\(37\) −3.05702 − 1.06970i −0.502572 − 0.175857i 0.0670737 − 0.997748i \(-0.478634\pi\)
−0.569645 + 0.821891i \(0.692919\pi\)
\(38\) −2.65071 + 1.27651i −0.430002 + 0.207078i
\(39\) 0 0
\(40\) −0.211917 + 0.0741529i −0.0335070 + 0.0117246i
\(41\) 4.15163 − 4.15163i 0.648376 − 0.648376i −0.304225 − 0.952600i \(-0.598397\pi\)
0.952600 + 0.304225i \(0.0983973\pi\)
\(42\) 0 0
\(43\) 0.0820565 + 0.728272i 0.0125135 + 0.111060i 0.998416 − 0.0562593i \(-0.0179174\pi\)
−0.985903 + 0.167320i \(0.946489\pi\)
\(44\) −0.0246672 − 0.218928i −0.00371873 − 0.0330046i
\(45\) 0 0
\(46\) −4.48161 + 4.48161i −0.660777 + 0.660777i
\(47\) −5.32959 + 1.86490i −0.777401 + 0.272024i −0.689674 − 0.724120i \(-0.742246\pi\)
−0.0877271 + 0.996145i \(0.527960\pi\)
\(48\) 0 0
\(49\) 3.68805 − 1.77607i 0.526864 − 0.253724i
\(50\) −4.67184 − 1.63475i −0.660698 − 0.231188i
\(51\) 0 0
\(52\) −3.80157 + 4.76702i −0.527183 + 0.661067i
\(53\) 0.534396 − 0.426167i 0.0734050 − 0.0585385i −0.586100 − 0.810239i \(-0.699337\pi\)
0.659505 + 0.751700i \(0.270766\pi\)
\(54\) 0 0
\(55\) −0.0263163 + 0.0418822i −0.00354849 + 0.00564739i
\(56\) 1.69415 + 0.190885i 0.226390 + 0.0255080i
\(57\) 0 0
\(58\) 5.24115 + 1.23707i 0.688197 + 0.162436i
\(59\) 1.13448i 0.147697i 0.997269 + 0.0738484i \(0.0235281\pi\)
−0.997269 + 0.0738484i \(0.976472\pi\)
\(60\) 0 0
\(61\) 3.30933 + 2.07939i 0.423716 + 0.266238i 0.726967 − 0.686672i \(-0.240929\pi\)
−0.303251 + 0.952911i \(0.598072\pi\)
\(62\) 1.07407 − 4.70582i 0.136407 − 0.597640i
\(63\) 0 0
\(64\) −0.781831 − 0.623490i −0.0977289 − 0.0779362i
\(65\) 1.33461 − 0.304615i 0.165538 − 0.0377829i
\(66\) 0 0
\(67\) 0.282777 + 0.587192i 0.0345467 + 0.0717369i 0.917528 − 0.397671i \(-0.130182\pi\)
−0.882981 + 0.469408i \(0.844467\pi\)
\(68\) 0.716683 + 1.14059i 0.0869105 + 0.138317i
\(69\) 0 0
\(70\) −0.270659 − 0.270659i −0.0323500 − 0.0323500i
\(71\) 9.79839 + 4.71866i 1.16285 + 0.560001i 0.912871 − 0.408249i \(-0.133860\pi\)
0.249984 + 0.968250i \(0.419575\pi\)
\(72\) 0 0
\(73\) −7.38764 + 0.832387i −0.864658 + 0.0974235i −0.533142 − 0.846026i \(-0.678989\pi\)
−0.331516 + 0.943450i \(0.607560\pi\)
\(74\) −1.40525 + 2.91803i −0.163357 + 0.339215i
\(75\) 0 0
\(76\) 0.971703 + 2.77697i 0.111462 + 0.318540i
\(77\) 0.318033 − 0.199834i 0.0362433 − 0.0227732i
\(78\) 0 0
\(79\) −4.94162 + 14.1223i −0.555975 + 1.58889i 0.231695 + 0.972788i \(0.425573\pi\)
−0.787670 + 0.616097i \(0.788713\pi\)
\(80\) 0.0499595 + 0.218887i 0.00558564 + 0.0244723i
\(81\) 0 0
\(82\) −3.66069 − 4.59036i −0.404256 − 0.506921i
\(83\) 3.09277 + 0.705905i 0.339476 + 0.0774832i 0.388860 − 0.921297i \(-0.372869\pi\)
−0.0493845 + 0.998780i \(0.515726\pi\)
\(84\) 0 0
\(85\) 0.0338623 − 0.300536i 0.00367288 − 0.0325977i
\(86\) 0.732880 0.0790285
\(87\) 0 0
\(88\) −0.220313 −0.0234854
\(89\) −0.794897 + 7.05491i −0.0842589 + 0.747819i 0.878553 + 0.477645i \(0.158510\pi\)
−0.962812 + 0.270173i \(0.912919\pi\)
\(90\) 0 0
\(91\) −10.1344 − 2.31310i −1.06237 − 0.242479i
\(92\) 3.95165 + 4.95521i 0.411988 + 0.516616i
\(93\) 0 0
\(94\) 1.25645 + 5.50488i 0.129593 + 0.567785i
\(95\) 0.218163 − 0.623473i 0.0223830 − 0.0639670i
\(96\) 0 0
\(97\) 5.55617 − 3.49117i 0.564144 − 0.354475i −0.219571 − 0.975597i \(-0.570466\pi\)
0.783714 + 0.621122i \(0.213323\pi\)
\(98\) −1.35197 − 3.86372i −0.136570 − 0.390294i
\(99\) 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.89.5 yes 72
3.2 odd 2 inner 522.2.r.b.89.2 ✓ 72
29.15 odd 28 inner 522.2.r.b.305.2 yes 72
87.44 even 28 inner 522.2.r.b.305.5 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
522.2.r.b.89.2 ✓ 72 3.2 odd 2 inner
522.2.r.b.89.5 yes 72 1.1 even 1 trivial
522.2.r.b.305.2 yes 72 29.15 odd 28 inner
522.2.r.b.305.5 yes 72 87.44 even 28 inner