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Properties

Label 522.2.k.h.181.1
Level $522$
Weight $2$
Character 522.181
Analytic conductor $4.168$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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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 181.1
Root \(2.06920 - 0.996473i\) of defining polynomial
Character \(\chi\) \(=\) 522.181
Dual form 522.2.k.h.199.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.222521 + 0.974928i) q^{2} +(-0.900969 + 0.433884i) q^{4} +(-0.110081 - 0.482295i) q^{5} +(-2.82760 - 1.36170i) q^{7} +(-0.623490 - 0.781831i) q^{8} +(0.445708 - 0.214642i) q^{10} +(-0.870839 + 1.09200i) q^{11} +(-3.56353 + 4.46852i) q^{13} +(0.698359 - 3.05971i) q^{14} +(0.623490 - 0.781831i) q^{16} -5.31800 q^{17} +(-4.47471 + 2.15491i) q^{19} +(0.308439 + 0.386771i) q^{20} +(-1.25840 - 0.606013i) q^{22} +(-0.181045 + 0.793210i) q^{23} +(4.28435 - 2.06324i) q^{25} +(-5.14944 - 2.47984i) q^{26} +3.13840 q^{28} +(-5.38501 - 0.0414712i) q^{29} +(-1.41596 - 6.20373i) q^{31} +(0.900969 + 0.433884i) q^{32} +(-1.18337 - 5.18466i) q^{34} +(-0.345477 + 1.51363i) q^{35} +(5.56630 + 6.97992i) q^{37} +(-3.09660 - 3.88301i) q^{38} +(-0.308439 + 0.386771i) q^{40} +4.01226 q^{41} +(0.310799 - 1.36170i) q^{43} +(0.310799 - 1.36170i) q^{44} -0.813609 q^{46} +(6.42079 - 8.05142i) q^{47} +(1.77665 + 2.22785i) q^{49} +(2.96486 + 3.71782i) q^{50} +(1.27181 - 5.57215i) q^{52} +(0.944288 + 4.13720i) q^{53} +(0.622528 + 0.299794i) q^{55} +(0.698359 + 3.05971i) q^{56} +(-1.15784 - 5.25922i) q^{58} -11.1598 q^{59} +(-4.38454 - 2.11149i) q^{61} +(5.73311 - 2.76092i) q^{62} +(-0.222521 + 0.974928i) q^{64} +(2.54742 + 1.22677i) q^{65} +(4.45072 + 5.58102i) q^{67} +(4.79135 - 2.30739i) q^{68} -1.55256 q^{70} +(-3.76423 + 4.72019i) q^{71} +(-2.73238 + 11.9713i) q^{73} +(-5.56630 + 6.97992i) q^{74} +(3.09660 - 3.88301i) q^{76} +(3.94935 - 1.90191i) q^{77} +(-5.86220 - 7.35096i) q^{79} +(-0.445708 - 0.214642i) q^{80} +(0.892813 + 3.91167i) q^{82} +(11.4508 - 5.51441i) q^{83} +(0.585409 + 2.56484i) q^{85} +1.39672 q^{86} +1.39672 q^{88} +(0.398796 + 1.74724i) q^{89} +(16.1610 - 7.78272i) q^{91} +(-0.181045 - 0.793210i) q^{92} +(9.27831 + 4.46820i) q^{94} +(1.53188 + 1.92092i) q^{95} +(-3.42035 + 1.64715i) q^{97} +(-1.77665 + 2.22785i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} - 2 q^{4} + q^{7} + 2 q^{8} - 7 q^{10} + 2 q^{11} + q^{13} - q^{14} - 2 q^{16} + 12 q^{17} - 6 q^{19} - 7 q^{20} - 2 q^{22} - 35 q^{23} - 6 q^{25} - 8 q^{26} - 6 q^{28} + 14 q^{29} - 8 q^{31}+ \cdots - 37 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(379\) \(407\)
\(\chi(n)\) \(e\left(\frac{3}{7}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.222521 + 0.974928i 0.157346 + 0.689378i
\(3\) 0 0
\(4\) −0.900969 + 0.433884i −0.450484 + 0.216942i
\(5\) −0.110081 − 0.482295i −0.0492296 − 0.215689i 0.944330 − 0.329000i \(-0.106712\pi\)
−0.993560 + 0.113311i \(0.963854\pi\)
\(6\) 0 0
\(7\) −2.82760 − 1.36170i −1.06873 − 0.514674i −0.185033 − 0.982732i \(-0.559239\pi\)
−0.883697 + 0.468059i \(0.844954\pi\)
\(8\) −0.623490 − 0.781831i −0.220437 − 0.276419i
\(9\) 0 0
\(10\) 0.445708 − 0.214642i 0.140945 − 0.0678756i
\(11\) −0.870839 + 1.09200i −0.262568 + 0.329250i −0.895587 − 0.444887i \(-0.853244\pi\)
0.633019 + 0.774136i \(0.281815\pi\)
\(12\) 0 0
\(13\) −3.56353 + 4.46852i −0.988344 + 1.23934i −0.0174472 + 0.999848i \(0.505554\pi\)
−0.970897 + 0.239497i \(0.923018\pi\)
\(14\) 0.698359 − 3.05971i 0.186644 − 0.817742i
\(15\) 0 0
\(16\) 0.623490 − 0.781831i 0.155872 − 0.195458i
\(17\) −5.31800 −1.28980 −0.644902 − 0.764265i \(-0.723102\pi\)
−0.644902 + 0.764265i \(0.723102\pi\)
\(18\) 0 0
\(19\) −4.47471 + 2.15491i −1.02657 + 0.494370i −0.869872 − 0.493277i \(-0.835799\pi\)
−0.156697 + 0.987647i \(0.550085\pi\)
\(20\) 0.308439 + 0.386771i 0.0689691 + 0.0864845i
\(21\) 0 0
\(22\) −1.25840 − 0.606013i −0.268292 − 0.129202i
\(23\) −0.181045 + 0.793210i −0.0377505 + 0.165396i −0.990289 − 0.139021i \(-0.955604\pi\)
0.952539 + 0.304417i \(0.0984616\pi\)
\(24\) 0 0
\(25\) 4.28435 − 2.06324i 0.856871 − 0.412647i
\(26\) −5.14944 − 2.47984i −1.00989 − 0.486337i
\(27\) 0 0
\(28\) 3.13840 0.593101
\(29\) −5.38501 − 0.0414712i −0.999970 − 0.00770101i
\(30\) 0 0
\(31\) −1.41596 − 6.20373i −0.254314 − 1.11422i −0.927226 − 0.374501i \(-0.877814\pi\)
0.672912 − 0.739722i \(-0.265043\pi\)
\(32\) 0.900969 + 0.433884i 0.159270 + 0.0767005i
\(33\) 0 0
\(34\) −1.18337 − 5.18466i −0.202946 − 0.889162i
\(35\) −0.345477 + 1.51363i −0.0583962 + 0.255851i
\(36\) 0 0
\(37\) 5.56630 + 6.97992i 0.915095 + 1.14749i 0.988655 + 0.150202i \(0.0479923\pi\)
−0.0735606 + 0.997291i \(0.523436\pi\)
\(38\) −3.09660 − 3.88301i −0.502335 − 0.629908i
\(39\) 0 0
\(40\) −0.308439 + 0.386771i −0.0487685 + 0.0611538i
\(41\) 4.01226 0.626610 0.313305 − 0.949652i \(-0.398564\pi\)
0.313305 + 0.949652i \(0.398564\pi\)
\(42\) 0 0
\(43\) 0.310799 − 1.36170i 0.0473964 − 0.207657i −0.945685 − 0.325084i \(-0.894607\pi\)
0.993082 + 0.117427i \(0.0374646\pi\)
\(44\) 0.310799 − 1.36170i 0.0468547 − 0.205284i
\(45\) 0 0
\(46\) −0.813609 −0.119960
\(47\) 6.42079 − 8.05142i 0.936569 − 1.17442i −0.0478986 − 0.998852i \(-0.515252\pi\)
0.984467 − 0.175568i \(-0.0561761\pi\)
\(48\) 0 0
\(49\) 1.77665 + 2.22785i 0.253807 + 0.318264i
\(50\) 2.96486 + 3.71782i 0.419295 + 0.525780i
\(51\) 0 0
\(52\) 1.27181 − 5.57215i 0.176368 − 0.772719i
\(53\) 0.944288 + 4.13720i 0.129708 + 0.568288i 0.997456 + 0.0712835i \(0.0227095\pi\)
−0.867748 + 0.497004i \(0.834433\pi\)
\(54\) 0 0
\(55\) 0.622528 + 0.299794i 0.0839416 + 0.0404241i
\(56\) 0.698359 + 3.05971i 0.0933221 + 0.408871i
\(57\) 0 0
\(58\) −1.15784 − 5.25922i −0.152032 − 0.690569i
\(59\) −11.1598 −1.45288 −0.726438 − 0.687232i \(-0.758826\pi\)
−0.726438 + 0.687232i \(0.758826\pi\)
\(60\) 0 0
\(61\) −4.38454 − 2.11149i −0.561383 − 0.270348i 0.131598 − 0.991303i \(-0.457989\pi\)
−0.692982 + 0.720955i \(0.743703\pi\)
\(62\) 5.73311 − 2.76092i 0.728106 − 0.350637i
\(63\) 0 0
\(64\) −0.222521 + 0.974928i −0.0278151 + 0.121866i
\(65\) 2.54742 + 1.22677i 0.315969 + 0.152163i
\(66\) 0 0
\(67\) 4.45072 + 5.58102i 0.543742 + 0.681830i 0.975460 − 0.220178i \(-0.0706639\pi\)
−0.431718 + 0.902009i \(0.642092\pi\)
\(68\) 4.79135 − 2.30739i 0.581036 − 0.279812i
\(69\) 0 0
\(70\) −1.55256 −0.185566
\(71\) −3.76423 + 4.72019i −0.446732 + 0.560184i −0.953304 − 0.302014i \(-0.902341\pi\)
0.506572 + 0.862198i \(0.330913\pi\)
\(72\) 0 0
\(73\) −2.73238 + 11.9713i −0.319801 + 1.40114i 0.518103 + 0.855319i \(0.326639\pi\)
−0.837903 + 0.545819i \(0.816219\pi\)
\(74\) −5.56630 + 6.97992i −0.647070 + 0.811400i
\(75\) 0 0
\(76\) 3.09660 − 3.88301i 0.355204 − 0.445412i
\(77\) 3.94935 − 1.90191i 0.450070 − 0.216743i
\(78\) 0 0
\(79\) −5.86220 − 7.35096i −0.659549 − 0.827048i 0.333745 − 0.942663i \(-0.391687\pi\)
−0.993294 + 0.115615i \(0.963116\pi\)
\(80\) −0.445708 − 0.214642i −0.0498316 − 0.0239977i
\(81\) 0 0
\(82\) 0.892813 + 3.91167i 0.0985947 + 0.431971i
\(83\) 11.4508 − 5.51441i 1.25689 − 0.605285i 0.317538 − 0.948246i \(-0.397144\pi\)
0.939350 + 0.342960i \(0.111430\pi\)
\(84\) 0 0
\(85\) 0.585409 + 2.56484i 0.0634965 + 0.278196i
\(86\) 1.39672 0.150612
\(87\) 0 0
\(88\) 1.39672 0.148891
\(89\) 0.398796 + 1.74724i 0.0422722 + 0.185207i 0.991656 − 0.128911i \(-0.0411483\pi\)
−0.949384 + 0.314118i \(0.898291\pi\)
\(90\) 0 0
\(91\) 16.1610 − 7.78272i 1.69413 − 0.815851i
\(92\) −0.181045 − 0.793210i −0.0188753 − 0.0826979i
\(93\) 0 0
\(94\) 9.27831 + 4.46820i 0.956985 + 0.460860i
\(95\) 1.53188 + 1.92092i 0.157168 + 0.197082i
\(96\) 0 0
\(97\) −3.42035 + 1.64715i −0.347283 + 0.167243i −0.599391 − 0.800456i \(-0.704591\pi\)
0.252108 + 0.967699i \(0.418876\pi\)
\(98\) −1.77665 + 2.22785i −0.179469 + 0.225047i
\(99\) 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.181.1 12
3.2 odd 2 58.2.d.b.7.2 ✓ 12
12.11 even 2 464.2.u.h.65.1 12
29.25 even 7 inner 522.2.k.h.199.1 12
87.2 even 28 1682.2.b.i.1681.5 12
87.5 odd 14 1682.2.a.q.1.5 6
87.53 odd 14 1682.2.a.t.1.2 6
87.56 even 28 1682.2.b.i.1681.8 12
87.83 odd 14 58.2.d.b.25.2 yes 12
348.83 even 14 464.2.u.h.257.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.7.2 ✓ 12 3.2 odd 2
58.2.d.b.25.2 yes 12 87.83 odd 14
464.2.u.h.65.1 12 12.11 even 2
464.2.u.h.257.1 12 348.83 even 14
522.2.k.h.181.1 12 1.1 even 1 trivial
522.2.k.h.199.1 12 29.25 even 7 inner
1682.2.a.q.1.5 6 87.5 odd 14
1682.2.a.t.1.2 6 87.53 odd 14
1682.2.b.i.1681.5 12 87.2 even 28
1682.2.b.i.1681.8 12 87.56 even 28