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Properties

Label 648.2.i.j.433.2
Level $648$
Weight $2$
Character 648.433
Analytic conductor $5.174$
Analytic rank $0$
Dimension $4$
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] = [648,2,Mod(217,648)] 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("648.217"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(648, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 648.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,4,0,0] 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: \(5.17430605098\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 433.2
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 648.433
Dual form 648.2.i.j.217.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.86603 - 3.23205i) q^{5} +(-1.73205 - 3.00000i) q^{7} +(1.00000 + 1.73205i) q^{11} +(1.23205 - 2.13397i) q^{13} -2.26795 q^{17} -7.46410 q^{19} +(-2.46410 + 4.26795i) q^{23} +(-4.46410 - 7.73205i) q^{25} +(2.13397 + 3.69615i) q^{29} +(5.46410 - 9.46410i) q^{31} -12.9282 q^{35} -0.464102 q^{37} +(3.46410 - 6.00000i) q^{41} +(2.26795 + 3.92820i) q^{43} +(3.46410 + 6.00000i) q^{47} +(-2.50000 + 4.33013i) q^{49} +10.9282 q^{53} +7.46410 q^{55} +(4.00000 - 6.92820i) q^{59} +(-5.23205 - 9.06218i) q^{61} +(-4.59808 - 7.96410i) q^{65} +(0.267949 - 0.464102i) q^{67} +2.00000 q^{71} +1.00000 q^{73} +(3.46410 - 6.00000i) q^{77} +(-0.267949 - 0.464102i) q^{79} +(-1.46410 - 2.53590i) q^{83} +(-4.23205 + 7.33013i) q^{85} +5.19615 q^{89} -8.53590 q^{91} +(-13.9282 + 24.1244i) q^{95} +(5.92820 + 10.2679i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} + 4 q^{11} - 2 q^{13} - 16 q^{17} - 16 q^{19} + 4 q^{23} - 4 q^{25} + 12 q^{29} + 8 q^{31} - 24 q^{35} + 12 q^{37} + 16 q^{43} - 10 q^{49} + 16 q^{53} + 16 q^{55} + 16 q^{59} - 14 q^{61} - 8 q^{65}+ \cdots - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.86603 − 3.23205i 0.834512 − 1.44542i −0.0599153 − 0.998203i \(-0.519083\pi\)
0.894427 − 0.447214i \(-0.147584\pi\)
\(6\) 0 0
\(7\) −1.73205 − 3.00000i −0.654654 − 1.13389i −0.981981 − 0.188982i \(-0.939481\pi\)
0.327327 − 0.944911i \(-0.393852\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 + 1.73205i 0.301511 + 0.522233i 0.976478 − 0.215615i \(-0.0691756\pi\)
−0.674967 + 0.737848i \(0.735842\pi\)
\(12\) 0 0
\(13\) 1.23205 − 2.13397i 0.341709 − 0.591858i −0.643041 − 0.765832i \(-0.722327\pi\)
0.984750 + 0.173974i \(0.0556608\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.26795 −0.550058 −0.275029 − 0.961436i \(-0.588688\pi\)
−0.275029 + 0.961436i \(0.588688\pi\)
\(18\) 0 0
\(19\) −7.46410 −1.71238 −0.856191 − 0.516659i \(-0.827175\pi\)
−0.856191 + 0.516659i \(0.827175\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.46410 + 4.26795i −0.513801 + 0.889929i 0.486071 + 0.873919i \(0.338430\pi\)
−0.999872 + 0.0160097i \(0.994904\pi\)
\(24\) 0 0
\(25\) −4.46410 − 7.73205i −0.892820 − 1.54641i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.13397 + 3.69615i 0.396269 + 0.686358i 0.993262 − 0.115888i \(-0.0369714\pi\)
−0.596993 + 0.802246i \(0.703638\pi\)
\(30\) 0 0
\(31\) 5.46410 − 9.46410i 0.981382 − 1.69980i 0.324355 − 0.945935i \(-0.394853\pi\)
0.657027 − 0.753867i \(-0.271814\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −12.9282 −2.18527
\(36\) 0 0
\(37\) −0.464102 −0.0762978 −0.0381489 − 0.999272i \(-0.512146\pi\)
−0.0381489 + 0.999272i \(0.512146\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.46410 − 6.00000i 0.541002 − 0.937043i −0.457845 − 0.889032i \(-0.651379\pi\)
0.998847 − 0.0480106i \(-0.0152881\pi\)
\(42\) 0 0
\(43\) 2.26795 + 3.92820i 0.345859 + 0.599045i 0.985509 − 0.169621i \(-0.0542542\pi\)
−0.639650 + 0.768666i \(0.720921\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.46410 + 6.00000i 0.505291 + 0.875190i 0.999981 + 0.00612051i \(0.00194823\pi\)
−0.494690 + 0.869069i \(0.664718\pi\)
\(48\) 0 0
\(49\) −2.50000 + 4.33013i −0.357143 + 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.9282 1.50110 0.750552 − 0.660811i \(-0.229788\pi\)
0.750552 + 0.660811i \(0.229788\pi\)
\(54\) 0 0
\(55\) 7.46410 1.00646
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000 − 6.92820i 0.520756 − 0.901975i −0.478953 − 0.877841i \(-0.658984\pi\)
0.999709 − 0.0241347i \(-0.00768307\pi\)
\(60\) 0 0
\(61\) −5.23205 − 9.06218i −0.669895 − 1.16029i −0.977933 − 0.208919i \(-0.933006\pi\)
0.308038 − 0.951374i \(-0.400328\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.59808 − 7.96410i −0.570321 − 0.987825i
\(66\) 0 0
\(67\) 0.267949 − 0.464102i 0.0327352 − 0.0566990i −0.849194 − 0.528082i \(-0.822912\pi\)
0.881929 + 0.471383i \(0.156245\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 − 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 − 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.46410 − 6.00000i 0.394771 − 0.683763i
\(78\) 0 0
\(79\) −0.267949 − 0.464102i −0.0301466 − 0.0522155i 0.850558 − 0.525880i \(-0.176264\pi\)
−0.880705 + 0.473665i \(0.842931\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1.46410 − 2.53590i −0.160706 − 0.278351i 0.774416 − 0.632677i \(-0.218044\pi\)
−0.935122 + 0.354326i \(0.884710\pi\)
\(84\) 0 0
\(85\) −4.23205 + 7.33013i −0.459030 + 0.795064i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.19615 0.550791 0.275396 − 0.961331i \(-0.411191\pi\)
0.275396 + 0.961331i \(0.411191\pi\)
\(90\) 0 0
\(91\) −8.53590 −0.894805
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −13.9282 + 24.1244i −1.42900 + 2.47511i
\(96\) 0 0
\(97\) 5.92820 + 10.2679i 0.601918 + 1.04255i 0.992530 + 0.121997i \(0.0389299\pi\)
−0.390613 + 0.920555i \(0.627737\pi\)
\(98\) 0 0
\(99\) 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 648.2.i.j.433.2 4
3.2 odd 2 648.2.i.i.433.1 4
4.3 odd 2 1296.2.i.t.433.2 4
9.2 odd 6 648.2.i.i.217.1 4
9.4 even 3 648.2.a.e.1.1 ✓ 2
9.5 odd 6 648.2.a.h.1.2 yes 2
9.7 even 3 inner 648.2.i.j.217.2 4
12.11 even 2 1296.2.i.r.433.1 4
36.7 odd 6 1296.2.i.t.865.2 4
36.11 even 6 1296.2.i.r.865.1 4
36.23 even 6 1296.2.a.q.1.2 2
36.31 odd 6 1296.2.a.m.1.1 2
72.5 odd 6 5184.2.a.bg.1.1 2
72.13 even 6 5184.2.a.cb.1.2 2
72.59 even 6 5184.2.a.bi.1.1 2
72.67 odd 6 5184.2.a.bz.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
648.2.a.e.1.1 ✓ 2 9.4 even 3
648.2.a.h.1.2 yes 2 9.5 odd 6
648.2.i.i.217.1 4 9.2 odd 6
648.2.i.i.433.1 4 3.2 odd 2
648.2.i.j.217.2 4 9.7 even 3 inner
648.2.i.j.433.2 4 1.1 even 1 trivial
1296.2.a.m.1.1 2 36.31 odd 6
1296.2.a.q.1.2 2 36.23 even 6
1296.2.i.r.433.1 4 12.11 even 2
1296.2.i.r.865.1 4 36.11 even 6
1296.2.i.t.433.2 4 4.3 odd 2
1296.2.i.t.865.2 4 36.7 odd 6
5184.2.a.bg.1.1 2 72.5 odd 6
5184.2.a.bi.1.1 2 72.59 even 6
5184.2.a.bz.1.2 2 72.67 odd 6
5184.2.a.cb.1.2 2 72.13 even 6