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Properties

Label 522.2.k.h.451.1
Level $522$
Weight $2$
Character 522.451
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 451.1
Root \(-1.02179 - 1.28129i\) of defining polynomial
Character \(\chi\) \(=\) 522.451
Dual form 522.2.k.h.397.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.900969 + 0.433884i) q^{2} +(0.623490 + 0.781831i) q^{4} +(-2.60002 - 1.25211i) q^{5} +(-1.89765 + 2.37957i) q^{7} +(0.222521 + 0.974928i) q^{8} +(-1.79927 - 2.25622i) q^{10} +(-1.22038 + 5.34685i) q^{11} +(0.0239308 - 0.104847i) q^{13} +(-2.74218 + 1.32056i) q^{14} +(-0.222521 + 0.974928i) q^{16} -0.816005 q^{17} +(1.27358 + 1.59701i) q^{19} +(-0.642153 - 2.81346i) q^{20} +(-3.41944 + 4.28784i) q^{22} +(-8.25746 + 3.97659i) q^{23} +(2.07491 + 2.60185i) q^{25} +(0.0670525 - 0.0840812i) q^{26} -3.04359 q^{28} +(5.37657 - 0.304047i) q^{29} +(3.10086 + 1.49330i) q^{31} +(-0.623490 + 0.781831i) q^{32} +(-0.735195 - 0.354051i) q^{34} +(7.91340 - 3.81089i) q^{35} +(-1.31456 - 5.75946i) q^{37} +(0.454534 + 1.99144i) q^{38} +(0.642153 - 2.81346i) q^{40} +2.43376 q^{41} +(-4.94123 + 2.37957i) q^{43} +(-4.94123 + 2.37957i) q^{44} -9.16509 q^{46} +(1.31510 - 5.76182i) q^{47} +(-0.503658 - 2.20667i) q^{49} +(0.740526 + 3.24446i) q^{50} +(0.0968937 - 0.0466615i) q^{52} +(3.55945 + 1.71414i) q^{53} +(9.86785 - 12.3739i) q^{55} +(-2.74218 - 1.32056i) q^{56} +(4.97605 + 2.05887i) q^{58} -1.13359 q^{59} +(5.09132 - 6.38431i) q^{61} +(2.14586 + 2.69083i) q^{62} +(-0.900969 + 0.433884i) q^{64} +(-0.193501 + 0.242642i) q^{65} +(-0.212822 - 0.932434i) q^{67} +(-0.508771 - 0.637978i) q^{68} +8.78321 q^{70} +(0.531778 - 2.32987i) q^{71} +(-8.01331 + 3.85900i) q^{73} +(1.31456 - 5.75946i) q^{74} +(-0.454534 + 1.99144i) q^{76} +(-10.4074 - 13.0504i) q^{77} +(0.934726 + 4.09530i) q^{79} +(1.79927 - 2.25622i) q^{80} +(2.19274 + 1.05597i) q^{82} +(9.64738 + 12.0974i) q^{83} +(2.12163 + 1.02172i) q^{85} -5.48435 q^{86} -5.48435 q^{88} +(4.99275 + 2.40438i) q^{89} +(0.204080 + 0.255908i) q^{91} +(-8.25746 - 3.97659i) q^{92} +(3.68482 - 4.62062i) q^{94} +(-1.31170 - 5.74692i) q^{95} +(7.43305 + 9.32075i) q^{97} +(0.503658 - 2.20667i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} - 2 q^{4} + q^{7} + 2 q^{8} - 7 q^{10} + 2 q^{11} + q^{13} - q^{14} - 2 q^{16} + 12 q^{17} - 6 q^{19} - 7 q^{20} - 2 q^{22} - 35 q^{23} - 6 q^{25} - 8 q^{26} - 6 q^{28} + 14 q^{29} - 8 q^{31}+ \cdots - 37 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) −2.60002 − 1.25211i −1.16277 − 0.559959i −0.249922 − 0.968266i \(-0.580405\pi\)
−0.912844 + 0.408307i \(0.866119\pi\)
\(6\) 0 0
\(7\) −1.89765 + 2.37957i −0.717242 + 0.899394i −0.998178 − 0.0603330i \(-0.980784\pi\)
0.280936 + 0.959727i \(0.409355\pi\)
\(8\) 0.222521 + 0.974928i 0.0786730 + 0.344689i
\(9\) 0 0
\(10\) −1.79927 − 2.25622i −0.568980 − 0.713478i
\(11\) −1.22038 + 5.34685i −0.367959 + 1.61214i 0.364417 + 0.931236i \(0.381268\pi\)
−0.732377 + 0.680900i \(0.761589\pi\)
\(12\) 0 0
\(13\) 0.0239308 − 0.104847i 0.00663720 − 0.0290795i −0.971501 − 0.237036i \(-0.923824\pi\)
0.978138 + 0.207956i \(0.0666812\pi\)
\(14\) −2.74218 + 1.32056i −0.732878 + 0.352935i
\(15\) 0 0
\(16\) −0.222521 + 0.974928i −0.0556302 + 0.243732i
\(17\) −0.816005 −0.197910 −0.0989551 − 0.995092i \(-0.531550\pi\)
−0.0989551 + 0.995092i \(0.531550\pi\)
\(18\) 0 0
\(19\) 1.27358 + 1.59701i 0.292178 + 0.366380i 0.906156 − 0.422943i \(-0.139003\pi\)
−0.613978 + 0.789323i \(0.710432\pi\)
\(20\) −0.642153 − 2.81346i −0.143590 − 0.629108i
\(21\) 0 0
\(22\) −3.41944 + 4.28784i −0.729027 + 0.914171i
\(23\) −8.25746 + 3.97659i −1.72180 + 0.829175i −0.732946 + 0.680286i \(0.761855\pi\)
−0.988854 + 0.148889i \(0.952430\pi\)
\(24\) 0 0
\(25\) 2.07491 + 2.60185i 0.414981 + 0.520370i
\(26\) 0.0670525 − 0.0840812i 0.0131501 − 0.0164897i
\(27\) 0 0
\(28\) −3.04359 −0.575184
\(29\) 5.37657 − 0.304047i 0.998405 − 0.0564602i
\(30\) 0 0
\(31\) 3.10086 + 1.49330i 0.556931 + 0.268204i 0.691106 − 0.722754i \(-0.257124\pi\)
−0.134174 + 0.990958i \(0.542838\pi\)
\(32\) −0.623490 + 0.781831i −0.110218 + 0.138210i
\(33\) 0 0
\(34\) −0.735195 − 0.354051i −0.126085 − 0.0607193i
\(35\) 7.91340 − 3.81089i 1.33761 − 0.644158i
\(36\) 0 0
\(37\) −1.31456 − 5.75946i −0.216112 − 0.946850i −0.960320 − 0.278901i \(-0.910030\pi\)
0.744207 − 0.667949i \(-0.232827\pi\)
\(38\) 0.454534 + 1.99144i 0.0737351 + 0.323055i
\(39\) 0 0
\(40\) 0.642153 − 2.81346i 0.101533 − 0.444846i
\(41\) 2.43376 0.380090 0.190045 − 0.981775i \(-0.439137\pi\)
0.190045 + 0.981775i \(0.439137\pi\)
\(42\) 0 0
\(43\) −4.94123 + 2.37957i −0.753531 + 0.362881i −0.770890 − 0.636968i \(-0.780188\pi\)
0.0173595 + 0.999849i \(0.494474\pi\)
\(44\) −4.94123 + 2.37957i −0.744919 + 0.358734i
\(45\) 0 0
\(46\) −9.16509 −1.35132
\(47\) 1.31510 − 5.76182i 0.191827 − 0.840448i −0.783800 − 0.621013i \(-0.786721\pi\)
0.975627 − 0.219435i \(-0.0704214\pi\)
\(48\) 0 0
\(49\) −0.503658 − 2.20667i −0.0719512 − 0.315239i
\(50\) 0.740526 + 3.24446i 0.104726 + 0.458835i
\(51\) 0 0
\(52\) 0.0968937 − 0.0466615i 0.0134367 − 0.00647079i
\(53\) 3.55945 + 1.71414i 0.488929 + 0.235456i 0.662071 − 0.749441i \(-0.269678\pi\)
−0.173142 + 0.984897i \(0.555392\pi\)
\(54\) 0 0
\(55\) 9.86785 − 12.3739i 1.33058 − 1.66849i
\(56\) −2.74218 − 1.32056i −0.366439 − 0.176468i
\(57\) 0 0
\(58\) 4.97605 + 2.05887i 0.653387 + 0.270343i
\(59\) −1.13359 −0.147581 −0.0737904 − 0.997274i \(-0.523510\pi\)
−0.0737904 + 0.997274i \(0.523510\pi\)
\(60\) 0 0
\(61\) 5.09132 − 6.38431i 0.651876 − 0.817427i −0.340555 − 0.940224i \(-0.610615\pi\)
0.992432 + 0.122797i \(0.0391865\pi\)
\(62\) 2.14586 + 2.69083i 0.272525 + 0.341735i
\(63\) 0 0
\(64\) −0.900969 + 0.433884i −0.112621 + 0.0542355i
\(65\) −0.193501 + 0.242642i −0.0240008 + 0.0300961i
\(66\) 0 0
\(67\) −0.212822 − 0.932434i −0.0260003 − 0.113915i 0.960263 − 0.279098i \(-0.0900355\pi\)
−0.986263 + 0.165183i \(0.947178\pi\)
\(68\) −0.508771 − 0.637978i −0.0616975 − 0.0773662i
\(69\) 0 0
\(70\) 8.78321 1.04979
\(71\) 0.531778 − 2.32987i 0.0631104 − 0.276505i −0.933520 − 0.358525i \(-0.883280\pi\)
0.996631 + 0.0820198i \(0.0261371\pi\)
\(72\) 0 0
\(73\) −8.01331 + 3.85900i −0.937886 + 0.451662i −0.839423 − 0.543478i \(-0.817107\pi\)
−0.0984633 + 0.995141i \(0.531393\pi\)
\(74\) 1.31456 − 5.75946i 0.152814 − 0.669524i
\(75\) 0 0
\(76\) −0.454534 + 1.99144i −0.0521386 + 0.228434i
\(77\) −10.4074 − 13.0504i −1.18603 − 1.48723i
\(78\) 0 0
\(79\) 0.934726 + 4.09530i 0.105165 + 0.460758i 0.999900 + 0.0141598i \(0.00450735\pi\)
−0.894735 + 0.446598i \(0.852636\pi\)
\(80\) 1.79927 − 2.25622i 0.201165 − 0.252253i
\(81\) 0 0
\(82\) 2.19274 + 1.05597i 0.242148 + 0.116612i
\(83\) 9.64738 + 12.0974i 1.05894 + 1.32787i 0.942329 + 0.334688i \(0.108631\pi\)
0.116609 + 0.993178i \(0.462798\pi\)
\(84\) 0 0
\(85\) 2.12163 + 1.02172i 0.230123 + 0.110822i
\(86\) −5.48435 −0.591393
\(87\) 0 0
\(88\) −5.48435 −0.584634
\(89\) 4.99275 + 2.40438i 0.529231 + 0.254864i 0.679364 − 0.733802i \(-0.262256\pi\)
−0.150133 + 0.988666i \(0.547970\pi\)
\(90\) 0 0
\(91\) 0.204080 + 0.255908i 0.0213934 + 0.0268265i
\(92\) −8.25746 − 3.97659i −0.860900 − 0.414588i
\(93\) 0 0
\(94\) 3.68482 − 4.62062i 0.380060 − 0.476581i
\(95\) −1.31170 − 5.74692i −0.134577 − 0.589622i
\(96\) 0 0
\(97\) 7.43305 + 9.32075i 0.754712 + 0.946379i 0.999732 − 0.0231388i \(-0.00736597\pi\)
−0.245020 + 0.969518i \(0.578795\pi\)
\(98\) 0.503658 − 2.20667i 0.0508772 − 0.222907i
\(99\) 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.451.1 12
3.2 odd 2 58.2.d.b.45.1 ✓ 12
12.11 even 2 464.2.u.h.161.2 12
29.20 even 7 inner 522.2.k.h.397.1 12
87.20 odd 14 58.2.d.b.49.1 yes 12
87.26 even 28 1682.2.b.i.1681.7 12
87.32 even 28 1682.2.b.i.1681.6 12
87.65 odd 14 1682.2.a.t.1.1 6
87.80 odd 14 1682.2.a.q.1.6 6
348.107 even 14 464.2.u.h.49.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.45.1 ✓ 12 3.2 odd 2
58.2.d.b.49.1 yes 12 87.20 odd 14
464.2.u.h.49.2 12 348.107 even 14
464.2.u.h.161.2 12 12.11 even 2
522.2.k.h.397.1 12 29.20 even 7 inner
522.2.k.h.451.1 12 1.1 even 1 trivial
1682.2.a.q.1.6 6 87.80 odd 14
1682.2.a.t.1.1 6 87.65 odd 14
1682.2.b.i.1681.6 12 87.32 even 28
1682.2.b.i.1681.7 12 87.26 even 28