↑

Properties

Label 522.2.k.h.397.2
Level $522$
Weight $2$
Character 522.397
Analytic conductor $4.168$
Analytic rank $0$
Dimension $12$
Inner twists $2$

Related objects

Downloads

Learn more

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(181,522)] 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("522.181"); 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, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.k (of order \(7\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,2,0,-2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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 \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 13 x^{10} - 9 x^{9} - 5 x^{8} + 35 x^{7} + 197 x^{6} - 140 x^{5} - 80 x^{4} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label 397.2
Root \(1.52179 - 1.90827i\) of defining polynomial
Character \(\chi\) \(=\) 522.397
Dual form 522.2.k.h.451.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.900969 - 0.433884i) q^{2} +(0.623490 - 0.781831i) q^{4} +(1.07557 - 0.517965i) q^{5} +(1.27416 + 1.59774i) q^{7} +(0.222521 - 0.974928i) q^{8} +(0.744314 - 0.933340i) q^{10} +(0.819415 + 3.59009i) q^{11} +(-0.479858 - 2.10239i) q^{13} +(1.84121 + 0.886679i) q^{14} +(-0.222521 - 0.974928i) q^{16} +6.53517 q^{17} +(3.31337 - 4.15484i) q^{19} +(0.265643 - 1.16386i) q^{20} +(2.29595 + 2.87903i) q^{22} +(-5.30987 - 2.55710i) q^{23} +(-2.22890 + 2.79495i) q^{25} +(-1.34453 - 1.68599i) q^{26} +2.04359 q^{28} +(-1.75040 - 5.09275i) q^{29} +(-8.10571 + 3.90350i) q^{31} +(-0.623490 - 0.781831i) q^{32} +(5.88799 - 2.83551i) q^{34} +(2.19801 + 1.05851i) q^{35} +(-0.406764 + 1.78215i) q^{37} +(1.18253 - 5.18100i) q^{38} +(-0.265643 - 1.16386i) q^{40} +8.32895 q^{41} +(3.31774 + 1.59774i) q^{43} +(3.31774 + 1.59774i) q^{44} -5.89351 q^{46} +(-0.220911 - 0.967876i) q^{47} +(0.628344 - 2.75296i) q^{49} +(-0.795484 + 3.48524i) q^{50} +(-1.94290 - 0.935653i) q^{52} +(-5.10353 + 2.45773i) q^{53} +(2.74087 + 3.43695i) q^{55} +(1.84121 - 0.886679i) q^{56} +(-3.78671 - 3.82894i) q^{58} -2.94918 q^{59} +(-1.12786 - 1.41429i) q^{61} +(-5.60932 + 7.03387i) q^{62} +(-0.900969 - 0.433884i) q^{64} +(-1.60508 - 2.01271i) q^{65} +(-1.34482 + 5.89206i) q^{67} +(4.07461 - 5.10940i) q^{68} +2.43961 q^{70} +(0.836003 + 3.66277i) q^{71} +(-11.4647 - 5.52110i) q^{73} +(0.406764 + 1.78215i) q^{74} +(-1.18253 - 5.18100i) q^{76} +(-4.69197 + 5.88354i) q^{77} +(-3.14487 + 13.7786i) q^{79} +(-0.744314 - 0.933340i) q^{80} +(7.50412 - 3.61380i) q^{82} +(-1.27960 + 1.60457i) q^{83} +(7.02900 - 3.38499i) q^{85} +3.68242 q^{86} +3.68242 q^{88} +(-14.8768 + 7.16429i) q^{89} +(2.74767 - 3.44546i) q^{91} +(-5.30987 + 2.55710i) q^{92} +(-0.618980 - 0.776176i) q^{94} +(1.41169 - 6.18501i) q^{95} +(-2.86572 + 3.59350i) q^{97} +(-0.628344 - 2.75296i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} - 2 q^{4} + q^{7} + 2 q^{8} - 7 q^{10} + 2 q^{11} + q^{13} - q^{14} - 2 q^{16} + 12 q^{17} - 6 q^{19} - 7 q^{20} - 2 q^{22} - 35 q^{23} - 6 q^{25} - 8 q^{26} - 6 q^{28} + 14 q^{29} - 8 q^{31}+ \cdots - 37 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(379\) \(407\)
\(\chi(n)\) \(e\left(\frac{6}{7}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.900969 − 0.433884i 0.637081 − 0.306802i
\(3\) 0 0
\(4\) 0.623490 − 0.781831i 0.311745 − 0.390916i
\(5\) 1.07557 − 0.517965i 0.481007 − 0.231641i −0.177636 − 0.984096i \(-0.556845\pi\)
0.658643 + 0.752455i \(0.271131\pi\)
\(6\) 0 0
\(7\) 1.27416 + 1.59774i 0.481585 + 0.603889i 0.961965 − 0.273171i \(-0.0880726\pi\)
−0.480380 + 0.877061i \(0.659501\pi\)
\(8\) 0.222521 − 0.974928i 0.0786730 − 0.344689i
\(9\) 0 0
\(10\) 0.744314 − 0.933340i 0.235373 − 0.295148i
\(11\) 0.819415 + 3.59009i 0.247063 + 1.08245i 0.934431 + 0.356144i \(0.115909\pi\)
−0.687368 + 0.726309i \(0.741234\pi\)
\(12\) 0 0
\(13\) −0.479858 − 2.10239i −0.133089 − 0.583099i −0.996858 − 0.0792116i \(-0.974760\pi\)
0.863769 − 0.503888i \(-0.168097\pi\)
\(14\) 1.84121 + 0.886679i 0.492083 + 0.236975i
\(15\) 0 0
\(16\) −0.222521 − 0.974928i −0.0556302 − 0.243732i
\(17\) 6.53517 1.58501 0.792506 − 0.609864i \(-0.208776\pi\)
0.792506 + 0.609864i \(0.208776\pi\)
\(18\) 0 0
\(19\) 3.31337 − 4.15484i 0.760140 − 0.953186i −0.239704 − 0.970846i \(-0.577050\pi\)
0.999844 + 0.0176605i \(0.00562181\pi\)
\(20\) 0.265643 − 1.16386i 0.0593995 − 0.260246i
\(21\) 0 0
\(22\) 2.29595 + 2.87903i 0.489498 + 0.613811i
\(23\) −5.30987 − 2.55710i −1.10718 − 0.533192i −0.211274 − 0.977427i \(-0.567761\pi\)
−0.895910 + 0.444235i \(0.853476\pi\)
\(24\) 0 0
\(25\) −2.22890 + 2.79495i −0.445779 + 0.558989i
\(26\) −1.34453 − 1.68599i −0.263684 − 0.330650i
\(27\) 0 0
\(28\) 2.04359 0.386202
\(29\) −1.75040 − 5.09275i −0.325040 − 0.945700i
\(30\) 0 0
\(31\) −8.10571 + 3.90350i −1.45583 + 0.701090i −0.983596 − 0.180385i \(-0.942266\pi\)
−0.472232 + 0.881475i \(0.656551\pi\)
\(32\) −0.623490 − 0.781831i −0.110218 − 0.138210i
\(33\) 0 0
\(34\) 5.88799 − 2.83551i 1.00978 − 0.486285i
\(35\) 2.19801 + 1.05851i 0.371532 + 0.178920i
\(36\) 0 0
\(37\) −0.406764 + 1.78215i −0.0668717 + 0.292984i −0.997295 − 0.0735026i \(-0.976582\pi\)
0.930423 + 0.366487i \(0.119439\pi\)
\(38\) 1.18253 − 5.18100i 0.191832 − 0.840469i
\(39\) 0 0
\(40\) −0.265643 − 1.16386i −0.0420018 − 0.184022i
\(41\) 8.32895 1.30076 0.650382 − 0.759608i \(-0.274609\pi\)
0.650382 + 0.759608i \(0.274609\pi\)
\(42\) 0 0
\(43\) 3.31774 + 1.59774i 0.505951 + 0.243653i 0.669405 − 0.742898i \(-0.266549\pi\)
−0.163454 + 0.986551i \(0.552263\pi\)
\(44\) 3.31774 + 1.59774i 0.500168 + 0.240868i
\(45\) 0 0
\(46\) −5.89351 −0.868951
\(47\) −0.220911 − 0.967876i −0.0322232 − 0.141179i 0.956257 − 0.292526i \(-0.0944960\pi\)
−0.988481 + 0.151347i \(0.951639\pi\)
\(48\) 0 0
\(49\) 0.628344 − 2.75296i 0.0897635 − 0.393280i
\(50\) −0.795484 + 3.48524i −0.112498 + 0.492888i
\(51\) 0 0
\(52\) −1.94290 − 0.935653i −0.269432 − 0.129752i
\(53\) −5.10353 + 2.45773i −0.701023 + 0.337595i −0.750215 − 0.661194i \(-0.770050\pi\)
0.0491913 + 0.998789i \(0.484336\pi\)
\(54\) 0 0
\(55\) 2.74087 + 3.43695i 0.369579 + 0.463438i
\(56\) 1.84121 − 0.886679i 0.246042 − 0.118487i
\(57\) 0 0
\(58\) −3.78671 − 3.82894i −0.497220 − 0.502765i
\(59\) −2.94918 −0.383951 −0.191975 − 0.981400i \(-0.561489\pi\)
−0.191975 + 0.981400i \(0.561489\pi\)
\(60\) 0 0
\(61\) −1.12786 − 1.41429i −0.144407 − 0.181081i 0.704368 − 0.709835i \(-0.251231\pi\)
−0.848775 + 0.528754i \(0.822659\pi\)
\(62\) −5.60932 + 7.03387i −0.712385 + 0.893302i
\(63\) 0 0
\(64\) −0.900969 − 0.433884i −0.112621 − 0.0542355i
\(65\) −1.60508 − 2.01271i −0.199086 − 0.249646i
\(66\) 0 0
\(67\) −1.34482 + 5.89206i −0.164296 + 0.719830i 0.823912 + 0.566717i \(0.191787\pi\)
−0.988209 + 0.153113i \(0.951070\pi\)
\(68\) 4.07461 − 5.10940i 0.494120 − 0.619606i
\(69\) 0 0
\(70\) 2.43961 0.291589
\(71\) 0.836003 + 3.66277i 0.0992153 + 0.434691i 1.00000 0.000424562i \(0.000135142\pi\)
−0.900785 + 0.434266i \(0.857008\pi\)
\(72\) 0 0
\(73\) −11.4647 − 5.52110i −1.34184 − 0.646196i −0.381329 − 0.924439i \(-0.624534\pi\)
−0.960511 + 0.278243i \(0.910248\pi\)
\(74\) 0.406764 + 1.78215i 0.0472854 + 0.207171i
\(75\) 0 0
\(76\) −1.18253 − 5.18100i −0.135645 − 0.594301i
\(77\) −4.69197 + 5.88354i −0.534700 + 0.670492i
\(78\) 0 0
\(79\) −3.14487 + 13.7786i −0.353826 + 1.55021i 0.414438 + 0.910077i \(0.363978\pi\)
−0.768264 + 0.640133i \(0.778879\pi\)
\(80\) −0.744314 − 0.933340i −0.0832169 − 0.104351i
\(81\) 0 0
\(82\) 7.50412 − 3.61380i 0.828692 − 0.399077i
\(83\) −1.27960 + 1.60457i −0.140455 + 0.176125i −0.847083 − 0.531460i \(-0.821644\pi\)
0.706629 + 0.707584i \(0.250215\pi\)
\(84\) 0 0
\(85\) 7.02900 − 3.38499i 0.762403 − 0.367154i
\(86\) 3.68242 0.397085
\(87\) 0 0
\(88\) 3.68242 0.392547
\(89\) −14.8768 + 7.16429i −1.57694 + 0.759413i −0.998416 − 0.0562548i \(-0.982084\pi\)
−0.578521 + 0.815668i \(0.696370\pi\)
\(90\) 0 0
\(91\) 2.74767 − 3.44546i 0.288034 − 0.361183i
\(92\) −5.30987 + 2.55710i −0.553592 + 0.266596i
\(93\) 0 0
\(94\) −0.618980 − 0.776176i −0.0638429 − 0.0800565i
\(95\) 1.41169 − 6.18501i 0.144836 − 0.634569i
\(96\) 0 0
\(97\) −2.86572 + 3.59350i −0.290970 + 0.364864i −0.905734 − 0.423846i \(-0.860680\pi\)
0.614764 + 0.788711i \(0.289251\pi\)
\(98\) −0.628344 − 2.75296i −0.0634724 − 0.278091i
\(99\) 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.k.h.397.2 12
3.2 odd 2 58.2.d.b.49.2 yes 12
12.11 even 2 464.2.u.h.49.1 12
29.16 even 7 inner 522.2.k.h.451.2 12
87.62 odd 14 1682.2.a.q.1.3 6
87.68 even 28 1682.2.b.i.1681.10 12
87.74 odd 14 58.2.d.b.45.2 ✓ 12
87.77 even 28 1682.2.b.i.1681.3 12
87.83 odd 14 1682.2.a.t.1.4 6
348.335 even 14 464.2.u.h.161.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.45.2 ✓ 12 87.74 odd 14
58.2.d.b.49.2 yes 12 3.2 odd 2
464.2.u.h.49.1 12 12.11 even 2
464.2.u.h.161.1 12 348.335 even 14
522.2.k.h.397.2 12 1.1 even 1 trivial
522.2.k.h.451.2 12 29.16 even 7 inner
1682.2.a.q.1.3 6 87.62 odd 14
1682.2.a.t.1.4 6 87.83 odd 14
1682.2.b.i.1681.3 12 87.77 even 28
1682.2.b.i.1681.10 12 87.68 even 28