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 415.1
Root \(-0.974928 + 0.222521i\) of defining polynomial
Character \(\chi\) \(=\) 522.415
Dual form 522.2.n.a.361.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.433884 - 0.900969i) q^{2} +(-0.623490 + 0.781831i) q^{4} +(-1.63762 + 0.788637i) q^{5} +(0.882702 + 1.10687i) q^{7} +(0.974928 + 0.222521i) q^{8} +(1.42108 + 1.13327i) q^{10} +(-3.44878 + 0.787162i) q^{11} +(-1.23911 - 5.42888i) q^{13} +(0.614269 - 1.27554i) q^{14} +(-0.222521 - 0.974928i) q^{16} -7.46337i q^{17} +(-3.93791 - 3.14038i) q^{19} +(0.404459 - 1.77205i) q^{20} +(2.20558 + 2.76571i) q^{22} +(1.61216 + 0.776374i) q^{23} +(-1.05759 + 1.32618i) q^{25} +(-4.35362 + 3.47190i) q^{26} -1.41574 q^{28} +(-4.21464 + 3.35213i) q^{29} +(-2.07731 - 4.31359i) q^{31} +(-0.781831 + 0.623490i) q^{32} +(-6.72427 + 3.23824i) q^{34} +(-2.31845 - 1.11651i) q^{35} +(1.04717 + 0.239009i) q^{37} +(-1.12079 + 4.91049i) q^{38} +(-1.77205 + 0.404459i) q^{40} -1.71164i q^{41} +(-0.881405 + 1.83026i) q^{43} +(1.53485 - 3.18715i) q^{44} -1.78936i q^{46} +(0.377517 - 0.0861658i) q^{47} +(1.11164 - 4.87041i) q^{49} +(1.65372 + 0.377450i) q^{50} +(5.01704 + 2.41608i) q^{52} +(-2.42678 + 1.16867i) q^{53} +(5.02701 - 4.00891i) q^{55} +(0.614269 + 1.27554i) q^{56} +(4.84883 + 2.34282i) q^{58} +3.81302 q^{59} +(-10.5585 + 8.42008i) q^{61} +(-2.98510 + 3.74319i) q^{62} +(0.900969 + 0.433884i) q^{64} +(6.31060 + 7.91325i) q^{65} +(-0.659012 + 2.88732i) q^{67} +(5.83510 + 4.65334i) q^{68} +2.57329i q^{70} +(-1.39711 - 6.12116i) q^{71} +(-0.416685 + 0.865255i) q^{73} +(-0.239009 - 1.04717i) q^{74} +(4.91049 - 1.12079i) q^{76} +(-3.91554 - 3.12254i) q^{77} +(2.71230 + 0.619064i) q^{79} +(1.13327 + 1.42108i) q^{80} +(-1.54214 + 0.742654i) q^{82} +(5.65640 - 7.09290i) q^{83} +(5.88589 + 12.2222i) q^{85} +2.03143 q^{86} -3.53747 q^{88} +(-2.30413 - 4.78459i) q^{89} +(4.91532 - 6.16362i) q^{91} +(-1.61216 + 0.776374i) q^{92} +(-0.241431 - 0.302745i) q^{94} +(8.92543 + 2.03717i) q^{95} +(-8.59586 - 6.85497i) q^{97} +(-4.87041 + 1.11164i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{4} + 2 q^{5} + 4 q^{7} - 26 q^{13} - 2 q^{16} - 2 q^{20} + 4 q^{22} + 16 q^{23} + 22 q^{25} - 14 q^{26} - 4 q^{28} - 18 q^{29} + 28 q^{31} - 4 q^{34} - 4 q^{35} - 28 q^{37} - 22 q^{38} - 14 q^{40}+ \cdots - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(379\) \(407\)
\(\chi(n)\) \(e\left(\frac{5}{14}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.433884 − 0.900969i −0.306802 − 0.637081i
\(3\) 0 0
\(4\) −0.623490 + 0.781831i −0.311745 + 0.390916i
\(5\) −1.63762 + 0.788637i −0.732367 + 0.352689i −0.762615 − 0.646853i \(-0.776085\pi\)
0.0302478 + 0.999542i \(0.490370\pi\)
\(6\) 0 0
\(7\) 0.882702 + 1.10687i 0.333630 + 0.418359i 0.920144 − 0.391580i \(-0.128071\pi\)
−0.586514 + 0.809939i \(0.699500\pi\)
\(8\) 0.974928 + 0.222521i 0.344689 + 0.0786730i
\(9\) 0 0
\(10\) 1.42108 + 1.13327i 0.449383 + 0.358371i
\(11\) −3.44878 + 0.787162i −1.03985 + 0.237338i −0.708160 − 0.706052i \(-0.750475\pi\)
−0.331686 + 0.943390i \(0.607618\pi\)
\(12\) 0 0
\(13\) −1.23911 − 5.42888i −0.343666 − 1.50570i −0.791269 − 0.611468i \(-0.790579\pi\)
0.447603 − 0.894232i \(-0.352278\pi\)
\(14\) 0.614269 − 1.27554i 0.164170 − 0.340903i
\(15\) 0 0
\(16\) −0.222521 − 0.974928i −0.0556302 − 0.243732i
\(17\) − 7.46337i − 1.81013i −0.425269 − 0.905067i \(-0.639820\pi\)
0.425269 − 0.905067i \(-0.360180\pi\)
\(18\) 0 0
\(19\) −3.93791 − 3.14038i −0.903418 − 0.720452i 0.0571884 − 0.998363i \(-0.481786\pi\)
−0.960607 + 0.277911i \(0.910358\pi\)
\(20\) 0.404459 − 1.77205i 0.0904398 − 0.396243i
\(21\) 0 0
\(22\) 2.20558 + 2.76571i 0.470231 + 0.589651i
\(23\) 1.61216 + 0.776374i 0.336158 + 0.161885i 0.594346 − 0.804210i \(-0.297411\pi\)
−0.258188 + 0.966095i \(0.583125\pi\)
\(24\) 0 0
\(25\) −1.05759 + 1.32618i −0.211518 + 0.265236i
\(26\) −4.35362 + 3.47190i −0.853816 + 0.680895i
\(27\) 0 0
\(28\) −1.41574 −0.267551
\(29\) −4.21464 + 3.35213i −0.782639 + 0.622476i
\(30\) 0 0
\(31\) −2.07731 − 4.31359i −0.373097 − 0.774743i 0.626894 − 0.779105i \(-0.284326\pi\)
−0.999990 + 0.00436147i \(0.998612\pi\)
\(32\) −0.781831 + 0.623490i −0.138210 + 0.110218i
\(33\) 0 0
\(34\) −6.72427 + 3.23824i −1.15320 + 0.555353i
\(35\) −2.31845 − 1.11651i −0.391890 − 0.188724i
\(36\) 0 0
\(37\) 1.04717 + 0.239009i 0.172153 + 0.0392929i 0.307729 − 0.951474i \(-0.400431\pi\)
−0.135575 + 0.990767i \(0.543288\pi\)
\(38\) −1.12079 + 4.91049i −0.181816 + 0.796587i
\(39\) 0 0
\(40\) −1.77205 + 0.404459i −0.280186 + 0.0639506i
\(41\) − 1.71164i − 0.267314i −0.991028 − 0.133657i \(-0.957328\pi\)
0.991028 − 0.133657i \(-0.0426720\pi\)
\(42\) 0 0
\(43\) −0.881405 + 1.83026i −0.134413 + 0.279112i −0.957302 − 0.289091i \(-0.906647\pi\)
0.822888 + 0.568203i \(0.192361\pi\)
\(44\) 1.53485 − 3.18715i 0.231388 − 0.480481i
\(45\) 0 0
\(46\) − 1.78936i − 0.263827i
\(47\) 0.377517 − 0.0861658i 0.0550665 − 0.0125686i −0.194899 − 0.980823i \(-0.562438\pi\)
0.249965 + 0.968255i \(0.419581\pi\)
\(48\) 0 0
\(49\) 1.11164 − 4.87041i 0.158806 − 0.695774i
\(50\) 1.65372 + 0.377450i 0.233871 + 0.0533795i
\(51\) 0 0
\(52\) 5.01704 + 2.41608i 0.695738 + 0.335050i
\(53\) −2.42678 + 1.16867i −0.333343 + 0.160530i −0.593067 − 0.805153i \(-0.702083\pi\)
0.259723 + 0.965683i \(0.416369\pi\)
\(54\) 0 0
\(55\) 5.02701 − 4.00891i 0.677842 − 0.540561i
\(56\) 0.614269 + 1.27554i 0.0820851 + 0.170451i
\(57\) 0 0
\(58\) 4.84883 + 2.34282i 0.636683 + 0.307628i
\(59\) 3.81302 0.496413 0.248206 − 0.968707i \(-0.420159\pi\)
0.248206 + 0.968707i \(0.420159\pi\)
\(60\) 0 0
\(61\) −10.5585 + 8.42008i −1.35187 + 1.07808i −0.362607 + 0.931942i \(0.618113\pi\)
−0.989264 + 0.146139i \(0.953315\pi\)
\(62\) −2.98510 + 3.74319i −0.379107 + 0.475386i
\(63\) 0 0
\(64\) 0.900969 + 0.433884i 0.112621 + 0.0542355i
\(65\) 6.31060 + 7.91325i 0.782734 + 0.981518i
\(66\) 0 0
\(67\) −0.659012 + 2.88732i −0.0805111 + 0.352742i −0.999097 − 0.0424783i \(-0.986475\pi\)
0.918586 + 0.395221i \(0.129332\pi\)
\(68\) 5.83510 + 4.65334i 0.707610 + 0.564300i
\(69\) 0 0
\(70\) 2.57329i 0.307567i
\(71\) −1.39711 − 6.12116i −0.165807 − 0.726448i −0.987643 − 0.156723i \(-0.949907\pi\)
0.821836 − 0.569725i \(-0.192950\pi\)
\(72\) 0 0
\(73\) −0.416685 + 0.865255i −0.0487693 + 0.101270i −0.923930 − 0.382563i \(-0.875042\pi\)
0.875160 + 0.483833i \(0.160756\pi\)
\(74\) −0.239009 − 1.04717i −0.0277843 − 0.121731i
\(75\) 0 0
\(76\) 4.91049 − 1.12079i 0.563272 − 0.128563i
\(77\) −3.91554 − 3.12254i −0.446217 − 0.355846i
\(78\) 0 0
\(79\) 2.71230 + 0.619064i 0.305157 + 0.0696501i 0.372357 − 0.928089i \(-0.378550\pi\)
−0.0672004 + 0.997739i \(0.521407\pi\)
\(80\) 1.13327 + 1.42108i 0.126703 + 0.158881i
\(81\) 0 0
\(82\) −1.54214 + 0.742654i −0.170300 + 0.0820124i
\(83\) 5.65640 − 7.09290i 0.620870 − 0.778547i −0.367596 − 0.929985i \(-0.619819\pi\)
0.988467 + 0.151439i \(0.0483906\pi\)
\(84\) 0 0
\(85\) 5.88589 + 12.2222i 0.638415 + 1.32568i
\(86\) 2.03143 0.219055
\(87\) 0 0
\(88\) −3.53747 −0.377096
\(89\) −2.30413 − 4.78459i −0.244238 − 0.507165i 0.742428 − 0.669926i \(-0.233674\pi\)
−0.986665 + 0.162761i \(0.947960\pi\)
\(90\) 0 0
\(91\) 4.91532 − 6.16362i 0.515266 − 0.646123i
\(92\) −1.61216 + 0.776374i −0.168079 + 0.0809426i
\(93\) 0 0
\(94\) −0.241431 − 0.302745i −0.0249017 − 0.0312258i
\(95\) 8.92543 + 2.03717i 0.915729 + 0.209009i
\(96\) 0 0
\(97\) −8.59586 − 6.85497i −0.872778 − 0.696017i 0.0809406 − 0.996719i \(-0.474208\pi\)
−0.953718 + 0.300702i \(0.902779\pi\)
\(98\) −4.87041 + 1.11164i −0.491986 + 0.112293i
\(99\) 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.415.1 12
3.2 odd 2 58.2.e.a.9.2 ✓ 12
12.11 even 2 464.2.y.c.241.2 12
29.13 even 14 inner 522.2.n.a.361.1 12
87.62 odd 14 1682.2.b.j.1681.7 12
87.68 even 28 1682.2.a.s.1.2 6
87.71 odd 14 58.2.e.a.13.2 yes 12
87.77 even 28 1682.2.a.r.1.6 6
87.83 odd 14 1682.2.b.j.1681.5 12
348.71 even 14 464.2.y.c.129.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.e.a.9.2 ✓ 12 3.2 odd 2
58.2.e.a.13.2 yes 12 87.71 odd 14
464.2.y.c.129.2 12 348.71 even 14
464.2.y.c.241.2 12 12.11 even 2
522.2.n.a.361.1 12 29.13 even 14 inner
522.2.n.a.415.1 12 1.1 even 1 trivial
1682.2.a.r.1.6 6 87.77 even 28
1682.2.a.s.1.2 6 87.68 even 28
1682.2.b.j.1681.5 12 87.83 odd 14
1682.2.b.j.1681.7 12 87.62 odd 14