Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [522,2,Mod(91,522)] 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("522.91"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.n (of order \(14\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,2,2,0,4] 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: \(4.16819098551\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{14})\)
Coefficient field: \(\Q(\zeta_{28})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label 91.1
Root \(0.781831 - 0.623490i\) of defining polynomial
Character \(\chi\) \(=\) 522.91
Dual form 522.2.n.a.109.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.974928 - 0.222521i) q^{2} +(0.900969 + 0.433884i) q^{4} +(0.136585 - 0.598418i) q^{5} +(-2.59161 + 1.24805i) q^{7} +(-0.781831 - 0.623490i) q^{8} +(-0.266321 + 0.553021i) q^{10} +(3.39687 - 2.70891i) q^{11} +(-0.298199 - 0.373929i) q^{13} +(2.80435 - 0.640075i) q^{14} +(0.623490 + 0.781831i) q^{16} +0.259558i q^{17} +(3.65470 - 7.58906i) q^{19} +(0.382702 - 0.479894i) q^{20} +(-3.91449 + 1.88512i) q^{22} +(-0.0317259 - 0.139000i) q^{23} +(4.16540 + 2.00595i) q^{25} +(0.207515 + 0.430910i) q^{26} -2.87647 q^{28} +(-1.07561 - 5.27665i) q^{29} +(6.46089 + 1.47465i) q^{31} +(-0.433884 - 0.900969i) q^{32} +(0.0577572 - 0.253051i) q^{34} +(0.392883 + 1.72133i) q^{35} +(-7.50895 - 5.98819i) q^{37} +(-5.25179 + 6.58554i) q^{38} +(-0.479894 + 0.382702i) q^{40} +4.28236i q^{41} +(3.17741 - 0.725223i) q^{43} +(4.23582 - 0.966799i) q^{44} +0.142575i q^{46} +(3.97456 - 3.16960i) q^{47} +(0.794384 - 0.996126i) q^{49} +(-3.61460 - 2.88254i) q^{50} +(-0.106426 - 0.466282i) q^{52} +(2.06111 - 9.03032i) q^{53} +(-1.15710 - 2.40274i) q^{55} +(2.80435 + 0.640075i) q^{56} +(-0.125524 + 5.38370i) q^{58} +10.2463 q^{59} +(-4.31279 - 8.95559i) q^{61} +(-5.97076 - 2.87536i) q^{62} +(0.222521 + 0.974928i) q^{64} +(-0.264495 + 0.127374i) q^{65} +(-1.16176 + 1.45680i) q^{67} +(-0.112618 + 0.233854i) q^{68} -1.76560i q^{70} +(-5.97581 - 7.49342i) q^{71} +(2.90704 - 0.663513i) q^{73} +(5.98819 + 7.50895i) q^{74} +(6.58554 - 5.25179i) q^{76} +(-5.42249 + 11.2599i) q^{77} +(10.5977 + 8.45137i) q^{79} +(0.553021 - 0.266321i) q^{80} +(0.952915 - 4.17499i) q^{82} +(0.950401 + 0.457689i) q^{83} +(0.155324 + 0.0354518i) q^{85} -3.25912 q^{86} -4.34475 q^{88} +(-2.51138 - 0.573205i) q^{89} +(1.23950 + 0.596912i) q^{91} +(0.0317259 - 0.139000i) q^{92} +(-4.58021 + 2.20571i) q^{94} +(-4.04225 - 3.22359i) q^{95} +(-7.22194 + 14.9965i) q^{97} +(-0.996126 + 0.794384i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{4} + 2 q^{5} + 4 q^{7} - 26 q^{13} - 2 q^{16} - 2 q^{20} + 4 q^{22} + 16 q^{23} + 22 q^{25} - 14 q^{26} - 4 q^{28} - 18 q^{29} + 28 q^{31} - 4 q^{34} - 4 q^{35} - 28 q^{37} - 22 q^{38} - 14 q^{40}+ \cdots - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(379\) \(407\)
\(\chi(n)\) \(e\left(\frac{1}{14}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.974928 − 0.222521i −0.689378 − 0.157346i
\(3\) 0 0
\(4\) 0.900969 + 0.433884i 0.450484 + 0.216942i
\(5\) 0.136585 − 0.598418i 0.0610826 − 0.267621i −0.935160 − 0.354225i \(-0.884745\pi\)
0.996243 + 0.0866048i \(0.0276017\pi\)
\(6\) 0 0
\(7\) −2.59161 + 1.24805i −0.979537 + 0.471720i −0.853946 − 0.520362i \(-0.825797\pi\)
−0.125591 + 0.992082i \(0.540083\pi\)
\(8\) −0.781831 − 0.623490i −0.276419 − 0.220437i
\(9\) 0 0
\(10\) −0.266321 + 0.553021i −0.0842181 + 0.174881i
\(11\) 3.39687 − 2.70891i 1.02419 − 0.816767i 0.0409677 − 0.999160i \(-0.486956\pi\)
0.983226 + 0.182394i \(0.0583845\pi\)
\(12\) 0 0
\(13\) −0.298199 − 0.373929i −0.0827054 − 0.103709i 0.738757 − 0.673972i \(-0.235413\pi\)
−0.821463 + 0.570262i \(0.806842\pi\)
\(14\) 2.80435 − 0.640075i 0.749495 − 0.171067i
\(15\) 0 0
\(16\) 0.623490 + 0.781831i 0.155872 + 0.195458i
\(17\) 0.259558i 0.0629522i 0.999505 + 0.0314761i \(0.0100208\pi\)
−0.999505 + 0.0314761i \(0.989979\pi\)
\(18\) 0 0
\(19\) 3.65470 − 7.58906i 0.838446 − 1.74105i 0.186992 − 0.982361i \(-0.440126\pi\)
0.651453 − 0.758689i \(-0.274160\pi\)
\(20\) 0.382702 − 0.479894i 0.0855749 − 0.107308i
\(21\) 0 0
\(22\) −3.91449 + 1.88512i −0.834572 + 0.401908i
\(23\) −0.0317259 − 0.139000i −0.00661531 − 0.0289836i 0.971513 − 0.236988i \(-0.0761603\pi\)
−0.978128 + 0.208005i \(0.933303\pi\)
\(24\) 0 0
\(25\) 4.16540 + 2.00595i 0.833079 + 0.401190i
\(26\) 0.207515 + 0.430910i 0.0406971 + 0.0845083i
\(27\) 0 0
\(28\) −2.87647 −0.543602
\(29\) −1.07561 − 5.27665i −0.199736 − 0.979850i
\(30\) 0 0
\(31\) 6.46089 + 1.47465i 1.16041 + 0.264856i 0.759030 − 0.651056i \(-0.225674\pi\)
0.401380 + 0.915912i \(0.368531\pi\)
\(32\) −0.433884 − 0.900969i −0.0767005 − 0.159270i
\(33\) 0 0
\(34\) 0.0577572 − 0.253051i 0.00990528 − 0.0433979i
\(35\) 0.392883 + 1.72133i 0.0664093 + 0.290958i
\(36\) 0 0
\(37\) −7.50895 − 5.98819i −1.23446 − 0.984452i −0.999924 − 0.0123461i \(-0.996070\pi\)
−0.234541 − 0.972106i \(-0.575359\pi\)
\(38\) −5.25179 + 6.58554i −0.851953 + 1.06832i
\(39\) 0 0
\(40\) −0.479894 + 0.382702i −0.0758779 + 0.0605106i
\(41\) 4.28236i 0.668792i 0.942433 + 0.334396i \(0.108532\pi\)
−0.942433 + 0.334396i \(0.891468\pi\)
\(42\) 0 0
\(43\) 3.17741 − 0.725223i 0.484550 − 0.110595i 0.0267346 − 0.999643i \(-0.491489\pi\)
0.457816 + 0.889047i \(0.348632\pi\)
\(44\) 4.23582 − 0.966799i 0.638574 − 0.145750i
\(45\) 0 0
\(46\) 0.142575i 0.0210215i
\(47\) 3.97456 − 3.16960i 0.579749 − 0.462334i −0.289178 − 0.957275i \(-0.593382\pi\)
0.868926 + 0.494941i \(0.164810\pi\)
\(48\) 0 0
\(49\) 0.794384 − 0.996126i 0.113483 − 0.142304i
\(50\) −3.61460 − 2.88254i −0.511181 − 0.407653i
\(51\) 0 0
\(52\) −0.106426 − 0.466282i −0.0147586 − 0.0646617i
\(53\) 2.06111 − 9.03032i 0.283116 − 1.24041i −0.610659 − 0.791894i \(-0.709095\pi\)
0.893774 − 0.448517i \(-0.148048\pi\)
\(54\) 0 0
\(55\) −1.15710 − 2.40274i −0.156023 − 0.323985i
\(56\) 2.80435 + 0.640075i 0.374747 + 0.0855337i
\(57\) 0 0
\(58\) −0.125524 + 5.38370i −0.0164821 + 0.706915i
\(59\) 10.2463 1.33395 0.666977 − 0.745078i \(-0.267588\pi\)
0.666977 + 0.745078i \(0.267588\pi\)
\(60\) 0 0
\(61\) −4.31279 − 8.95559i −0.552196 − 1.14665i −0.971113 − 0.238622i \(-0.923304\pi\)
0.418917 − 0.908025i \(-0.362410\pi\)
\(62\) −5.97076 − 2.87536i −0.758287 − 0.365172i
\(63\) 0 0
\(64\) 0.222521 + 0.974928i 0.0278151 + 0.121866i
\(65\) −0.264495 + 0.127374i −0.0328066 + 0.0157988i
\(66\) 0 0
\(67\) −1.16176 + 1.45680i −0.141931 + 0.177976i −0.847716 − 0.530450i \(-0.822023\pi\)
0.705785 + 0.708426i \(0.250594\pi\)
\(68\) −0.112618 + 0.233854i −0.0136570 + 0.0283590i
\(69\) 0 0
\(70\) − 1.76560i − 0.211029i
\(71\) −5.97581 − 7.49342i −0.709198 − 0.889306i 0.288475 − 0.957487i \(-0.406852\pi\)
−0.997673 + 0.0681816i \(0.978280\pi\)
\(72\) 0 0
\(73\) 2.90704 − 0.663513i 0.340243 − 0.0776583i −0.0489853 − 0.998799i \(-0.515599\pi\)
0.389229 + 0.921141i \(0.372742\pi\)
\(74\) 5.98819 + 7.50895i 0.696113 + 0.872898i
\(75\) 0 0
\(76\) 6.58554 − 5.25179i 0.755413 − 0.602422i
\(77\) −5.42249 + 11.2599i −0.617950 + 1.28319i
\(78\) 0 0
\(79\) 10.5977 + 8.45137i 1.19233 + 0.950853i 0.999538 − 0.0303860i \(-0.00967364\pi\)
0.192794 + 0.981239i \(0.438245\pi\)
\(80\) 0.553021 − 0.266321i 0.0618296 − 0.0297756i
\(81\) 0 0
\(82\) 0.952915 − 4.17499i 0.105232 − 0.461051i
\(83\) 0.950401 + 0.457689i 0.104320 + 0.0502379i 0.485316 − 0.874339i \(-0.338705\pi\)
−0.380996 + 0.924577i \(0.624419\pi\)
\(84\) 0 0
\(85\) 0.155324 + 0.0354518i 0.0168473 + 0.00384528i
\(86\) −3.25912 −0.351440
\(87\) 0 0
\(88\) −4.34475 −0.463152
\(89\) −2.51138 − 0.573205i −0.266205 − 0.0607596i 0.0873347 − 0.996179i \(-0.472165\pi\)
−0.353540 + 0.935419i \(0.615022\pi\)
\(90\) 0 0
\(91\) 1.23950 + 0.596912i 0.129935 + 0.0625733i
\(92\) 0.0317259 − 0.139000i 0.00330766 − 0.0144918i
\(93\) 0 0
\(94\) −4.58021 + 2.20571i −0.472413 + 0.227502i
\(95\) −4.04225 − 3.22359i −0.414726 − 0.330733i
\(96\) 0 0
\(97\) −7.22194 + 14.9965i −0.733277 + 1.52267i 0.115147 + 0.993348i \(0.463266\pi\)
−0.848424 + 0.529317i \(0.822448\pi\)
\(98\) −0.996126 + 0.794384i −0.100624 + 0.0802449i
\(99\) 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.n.a.91.1 12
3.2 odd 2 58.2.e.a.33.2 ✓ 12
12.11 even 2 464.2.y.c.33.2 12
29.22 even 14 inner 522.2.n.a.109.1 12
87.14 even 28 1682.2.a.r.1.4 6
87.23 odd 14 1682.2.b.j.1681.3 12
87.35 odd 14 1682.2.b.j.1681.9 12
87.44 even 28 1682.2.a.s.1.4 6
87.80 odd 14 58.2.e.a.51.2 yes 12
348.167 even 14 464.2.y.c.225.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.e.a.33.2 ✓ 12 3.2 odd 2
58.2.e.a.51.2 yes 12 87.80 odd 14
464.2.y.c.33.2 12 12.11 even 2
464.2.y.c.225.2 12 348.167 even 14
522.2.n.a.91.1 12 1.1 even 1 trivial
522.2.n.a.109.1 12 29.22 even 14 inner
1682.2.a.r.1.4 6 87.14 even 28
1682.2.a.s.1.4 6 87.44 even 28
1682.2.b.j.1681.3 12 87.23 odd 14
1682.2.b.j.1681.9 12 87.35 odd 14