Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [567,2,Mod(190,567)] 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("567.190"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(567, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-2,0,0,4,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(i, \sqrt{3}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 379.4
Root \(1.09445 - 0.895644i\) of defining polynomial
Character \(\chi\) \(=\) 567.379
Dual form 567.2.f.n.190.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.09445 - 1.89564i) q^{2} +(-1.39564 - 2.41733i) q^{4} +(0.456850 + 0.791288i) q^{5} +(0.500000 - 0.866025i) q^{7} -1.73205 q^{8} +2.00000 q^{10} +(1.32288 - 2.29129i) q^{11} +(-2.00000 - 3.46410i) q^{13} +(-1.09445 - 1.89564i) q^{14} +(0.895644 - 1.55130i) q^{16} -3.46410 q^{17} +5.58258 q^{19} +(1.27520 - 2.20871i) q^{20} +(-2.89564 - 5.01540i) q^{22} +(-1.73205 - 3.00000i) q^{23} +(2.08258 - 3.60713i) q^{25} -8.75560 q^{26} -2.79129 q^{28} +(-4.37780 + 7.58258i) q^{29} +(4.58258 + 7.93725i) q^{31} +(-3.69253 - 6.39564i) q^{32} +(-3.79129 + 6.56670i) q^{34} +0.913701 q^{35} +3.00000 q^{37} +(6.10985 - 10.5826i) q^{38} +(-0.791288 - 1.37055i) q^{40} +(-0.456850 - 0.791288i) q^{41} +(-0.291288 + 0.504525i) q^{43} -7.38505 q^{44} -7.58258 q^{46} +(-6.56670 + 11.3739i) q^{47} +(-0.500000 - 0.866025i) q^{49} +(-4.55855 - 7.89564i) q^{50} +(-5.58258 + 9.66930i) q^{52} +8.66025 q^{53} +2.41742 q^{55} +(-0.866025 + 1.50000i) q^{56} +(9.58258 + 16.5975i) q^{58} +(1.73205 + 3.00000i) q^{59} +(-5.79129 + 10.0308i) q^{61} +20.0616 q^{62} -12.5826 q^{64} +(1.82740 - 3.16515i) q^{65} +(-4.29129 - 7.43273i) q^{67} +(4.83465 + 8.37386i) q^{68} +(1.00000 - 1.73205i) q^{70} +4.47315 q^{71} +15.1652 q^{73} +(3.28335 - 5.68693i) q^{74} +(-7.79129 - 13.4949i) q^{76} +(-1.32288 - 2.29129i) q^{77} +(-0.291288 + 0.504525i) q^{79} +1.63670 q^{80} -2.00000 q^{82} +(-4.83465 + 8.37386i) q^{83} +(-1.58258 - 2.74110i) q^{85} +(0.637600 + 1.10436i) q^{86} +(-2.29129 + 3.96863i) q^{88} +1.82740 q^{89} -4.00000 q^{91} +(-4.83465 + 8.37386i) q^{92} +(14.3739 + 24.8963i) q^{94} +(2.55040 + 4.41742i) q^{95} +(0.791288 - 1.37055i) q^{97} -2.18890 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{4} + 4 q^{7} + 16 q^{10} - 16 q^{13} - 2 q^{16} + 8 q^{19} - 14 q^{22} - 20 q^{25} - 4 q^{28} - 12 q^{34} + 24 q^{37} + 12 q^{40} + 16 q^{43} - 24 q^{46} - 4 q^{49} - 8 q^{52} + 56 q^{55} + 40 q^{58}+ \cdots - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.09445 − 1.89564i 0.773893 − 1.34042i −0.161521 − 0.986869i \(-0.551640\pi\)
0.935414 − 0.353553i \(-0.115027\pi\)
\(3\) 0 0
\(4\) −1.39564 − 2.41733i −0.697822 − 1.20866i
\(5\) 0.456850 + 0.791288i 0.204310 + 0.353875i 0.949913 − 0.312516i \(-0.101172\pi\)
−0.745603 + 0.666390i \(0.767838\pi\)
\(6\) 0 0
\(7\) 0.500000 − 0.866025i 0.188982 − 0.327327i
\(8\) −1.73205 −0.612372
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) 1.32288 − 2.29129i 0.398862 − 0.690849i −0.594724 − 0.803930i \(-0.702739\pi\)
0.993586 + 0.113081i \(0.0360719\pi\)
\(12\) 0 0
\(13\) −2.00000 − 3.46410i −0.554700 − 0.960769i −0.997927 − 0.0643593i \(-0.979500\pi\)
0.443227 − 0.896410i \(-0.353834\pi\)
\(14\) −1.09445 − 1.89564i −0.292504 − 0.506632i
\(15\) 0 0
\(16\) 0.895644 − 1.55130i 0.223911 − 0.387825i
\(17\) −3.46410 −0.840168 −0.420084 − 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 0 0
\(19\) 5.58258 1.28073 0.640365 − 0.768070i \(-0.278783\pi\)
0.640365 + 0.768070i \(0.278783\pi\)
\(20\) 1.27520 − 2.20871i 0.285144 − 0.493883i
\(21\) 0 0
\(22\) −2.89564 − 5.01540i −0.617353 − 1.06929i
\(23\) −1.73205 − 3.00000i −0.361158 − 0.625543i 0.626994 − 0.779024i \(-0.284285\pi\)
−0.988152 + 0.153481i \(0.950952\pi\)
\(24\) 0 0
\(25\) 2.08258 − 3.60713i 0.416515 − 0.721425i
\(26\) −8.75560 −1.71712
\(27\) 0 0
\(28\) −2.79129 −0.527504
\(29\) −4.37780 + 7.58258i −0.812937 + 1.40805i 0.0978621 + 0.995200i \(0.468800\pi\)
−0.910799 + 0.412849i \(0.864534\pi\)
\(30\) 0 0
\(31\) 4.58258 + 7.93725i 0.823055 + 1.42557i 0.903397 + 0.428806i \(0.141065\pi\)
−0.0803419 + 0.996767i \(0.525601\pi\)
\(32\) −3.69253 − 6.39564i −0.652753 − 1.13060i
\(33\) 0 0
\(34\) −3.79129 + 6.56670i −0.650201 + 1.12618i
\(35\) 0.913701 0.154444
\(36\) 0 0
\(37\) 3.00000 0.493197 0.246598 − 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 6.10985 − 10.5826i 0.991149 − 1.71672i
\(39\) 0 0
\(40\) −0.791288 − 1.37055i −0.125114 − 0.216703i
\(41\) −0.456850 − 0.791288i −0.0713480 − 0.123578i 0.828144 − 0.560515i \(-0.189397\pi\)
−0.899492 + 0.436937i \(0.856063\pi\)
\(42\) 0 0
\(43\) −0.291288 + 0.504525i −0.0444210 + 0.0769394i −0.887381 − 0.461037i \(-0.847478\pi\)
0.842960 + 0.537976i \(0.180811\pi\)
\(44\) −7.38505 −1.11334
\(45\) 0 0
\(46\) −7.58258 −1.11799
\(47\) −6.56670 + 11.3739i −0.957852 + 1.65905i −0.230150 + 0.973155i \(0.573922\pi\)
−0.727702 + 0.685893i \(0.759412\pi\)
\(48\) 0 0
\(49\) −0.500000 − 0.866025i −0.0714286 − 0.123718i
\(50\) −4.55855 − 7.89564i −0.644677 − 1.11661i
\(51\) 0 0
\(52\) −5.58258 + 9.66930i −0.774164 + 1.34089i
\(53\) 8.66025 1.18958 0.594789 − 0.803882i \(-0.297236\pi\)
0.594789 + 0.803882i \(0.297236\pi\)
\(54\) 0 0
\(55\) 2.41742 0.325965
\(56\) −0.866025 + 1.50000i −0.115728 + 0.200446i
\(57\) 0 0
\(58\) 9.58258 + 16.5975i 1.25825 + 2.17936i
\(59\) 1.73205 + 3.00000i 0.225494 + 0.390567i 0.956467 − 0.291839i \(-0.0942671\pi\)
−0.730974 + 0.682406i \(0.760934\pi\)
\(60\) 0 0
\(61\) −5.79129 + 10.0308i −0.741498 + 1.28431i 0.210315 + 0.977634i \(0.432551\pi\)
−0.951813 + 0.306679i \(0.900782\pi\)
\(62\) 20.0616 2.54783
\(63\) 0 0
\(64\) −12.5826 −1.57282
\(65\) 1.82740 − 3.16515i 0.226661 − 0.392589i
\(66\) 0 0
\(67\) −4.29129 − 7.43273i −0.524264 − 0.908052i −0.999601 − 0.0282483i \(-0.991007\pi\)
0.475337 − 0.879804i \(-0.342326\pi\)
\(68\) 4.83465 + 8.37386i 0.586288 + 1.01548i
\(69\) 0 0
\(70\) 1.00000 − 1.73205i 0.119523 − 0.207020i
\(71\) 4.47315 0.530866 0.265433 − 0.964129i \(-0.414485\pi\)
0.265433 + 0.964129i \(0.414485\pi\)
\(72\) 0 0
\(73\) 15.1652 1.77495 0.887473 − 0.460859i \(-0.152459\pi\)
0.887473 + 0.460859i \(0.152459\pi\)
\(74\) 3.28335 − 5.68693i 0.381682 − 0.661092i
\(75\) 0 0
\(76\) −7.79129 − 13.4949i −0.893722 − 1.54797i
\(77\) −1.32288 − 2.29129i −0.150756 − 0.261116i
\(78\) 0 0
\(79\) −0.291288 + 0.504525i −0.0327724 + 0.0567635i −0.881946 − 0.471350i \(-0.843767\pi\)
0.849174 + 0.528113i \(0.177100\pi\)
\(80\) 1.63670 0.182989
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) −4.83465 + 8.37386i −0.530672 + 0.919151i 0.468687 + 0.883364i \(0.344727\pi\)
−0.999359 + 0.0357868i \(0.988606\pi\)
\(84\) 0 0
\(85\) −1.58258 − 2.74110i −0.171654 − 0.297314i
\(86\) 0.637600 + 1.10436i 0.0687542 + 0.119086i
\(87\) 0 0
\(88\) −2.29129 + 3.96863i −0.244252 + 0.423057i
\(89\) 1.82740 0.193704 0.0968521 − 0.995299i \(-0.469123\pi\)
0.0968521 + 0.995299i \(0.469123\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) −4.83465 + 8.37386i −0.504047 + 0.873036i
\(93\) 0 0
\(94\) 14.3739 + 24.8963i 1.48255 + 2.56785i
\(95\) 2.55040 + 4.41742i 0.261666 + 0.453218i
\(96\) 0 0
\(97\) 0.791288 − 1.37055i 0.0803431 − 0.139158i −0.823054 − 0.567963i \(-0.807732\pi\)
0.903397 + 0.428804i \(0.141065\pi\)
\(98\) −2.18890 −0.221112
\(99\) 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 567.2.f.n.379.4 8
3.2 odd 2 inner 567.2.f.n.379.1 8
9.2 odd 6 567.2.a.i.1.4 yes 4
9.4 even 3 inner 567.2.f.n.190.4 8
9.5 odd 6 inner 567.2.f.n.190.1 8
9.7 even 3 567.2.a.i.1.1 ✓ 4
36.7 odd 6 9072.2.a.ci.1.2 4
36.11 even 6 9072.2.a.ci.1.3 4
63.20 even 6 3969.2.a.u.1.4 4
63.34 odd 6 3969.2.a.u.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.a.i.1.1 ✓ 4 9.7 even 3
567.2.a.i.1.4 yes 4 9.2 odd 6
567.2.f.n.190.1 8 9.5 odd 6 inner
567.2.f.n.190.4 8 9.4 even 3 inner
567.2.f.n.379.1 8 3.2 odd 2 inner
567.2.f.n.379.4 8 1.1 even 1 trivial
3969.2.a.u.1.1 4 63.34 odd 6
3969.2.a.u.1.4 4 63.20 even 6
9072.2.a.ci.1.2 4 36.7 odd 6
9072.2.a.ci.1.3 4 36.11 even 6