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Properties

Label 2646.2.e.q.1549.1
Level $2646$
Weight $2$
Character 2646.1549
Analytic conductor $21.128$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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] = [2646,2,Mod(1549,2646)] 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("2646.1549"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2646, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-8,0,8,0,0,0,-8,0,0,-8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.1284163748\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 882)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1549.1
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 2646.1549
Dual form 2646.2.e.q.2125.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +(-1.93185 - 3.34607i) q^{5} -1.00000 q^{8} +(1.93185 + 3.34607i) q^{10} +(-1.86603 + 3.23205i) q^{11} +(3.34607 - 5.79555i) q^{13} +1.00000 q^{16} +(-2.70831 - 4.69093i) q^{17} +(-1.48356 + 2.56961i) q^{19} +(-1.93185 - 3.34607i) q^{20} +(1.86603 - 3.23205i) q^{22} +(-0.732051 - 1.26795i) q^{23} +(-4.96410 + 8.59808i) q^{25} +(-3.34607 + 5.79555i) q^{26} +(-2.00000 - 3.46410i) q^{29} -1.79315 q^{31} -1.00000 q^{32} +(2.70831 + 4.69093i) q^{34} +(-0.267949 + 0.464102i) q^{37} +(1.48356 - 2.56961i) q^{38} +(1.93185 + 3.34607i) q^{40} +(0.637756 - 1.10463i) q^{41} +(-1.86603 - 3.23205i) q^{43} +(-1.86603 + 3.23205i) q^{44} +(0.732051 + 1.26795i) q^{46} +10.5558 q^{47} +(4.96410 - 8.59808i) q^{50} +(3.34607 - 5.79555i) q^{52} +(-1.46410 - 2.53590i) q^{53} +14.4195 q^{55} +(2.00000 + 3.46410i) q^{58} -8.62398 q^{59} -6.96953 q^{61} +1.79315 q^{62} +1.00000 q^{64} -25.8564 q^{65} +5.53590 q^{67} +(-2.70831 - 4.69093i) q^{68} -2.53590 q^{71} +(-3.41542 - 5.91567i) q^{73} +(0.267949 - 0.464102i) q^{74} +(-1.48356 + 2.56961i) q^{76} -4.92820 q^{79} +(-1.93185 - 3.34607i) q^{80} +(-0.637756 + 1.10463i) q^{82} +(8.95215 + 15.5056i) q^{83} +(-10.4641 + 18.1244i) q^{85} +(1.86603 + 3.23205i) q^{86} +(1.86603 - 3.23205i) q^{88} +(-3.53553 + 6.12372i) q^{89} +(-0.732051 - 1.26795i) q^{92} -10.5558 q^{94} +11.4641 q^{95} +(-2.94855 - 5.10703i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} - 8 q^{11} + 8 q^{16} + 8 q^{22} + 8 q^{23} - 12 q^{25} - 16 q^{29} - 8 q^{32} - 16 q^{37} - 8 q^{43} - 8 q^{44} - 8 q^{46} + 12 q^{50} + 16 q^{53} + 16 q^{58} + 8 q^{64}+ \cdots + 64 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.93185 − 3.34607i −0.863950 − 1.49641i −0.868086 − 0.496414i \(-0.834650\pi\)
0.00413535 − 0.999991i \(-0.498684\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.93185 + 3.34607i 0.610905 + 1.05812i
\(11\) −1.86603 + 3.23205i −0.562628 + 0.974500i 0.434638 + 0.900605i \(0.356876\pi\)
−0.997266 + 0.0738948i \(0.976457\pi\)
\(12\) 0 0
\(13\) 3.34607 − 5.79555i 0.928032 − 1.60740i 0.141420 − 0.989950i \(-0.454833\pi\)
0.786612 − 0.617448i \(-0.211833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.70831 − 4.69093i −0.656861 − 1.13772i −0.981424 − 0.191853i \(-0.938550\pi\)
0.324562 − 0.945864i \(-0.394783\pi\)
\(18\) 0 0
\(19\) −1.48356 + 2.56961i −0.340353 + 0.589509i −0.984498 − 0.175395i \(-0.943880\pi\)
0.644145 + 0.764903i \(0.277213\pi\)
\(20\) −1.93185 − 3.34607i −0.431975 − 0.748203i
\(21\) 0 0
\(22\) 1.86603 − 3.23205i 0.397838 − 0.689076i
\(23\) −0.732051 − 1.26795i −0.152643 − 0.264386i 0.779555 − 0.626334i \(-0.215445\pi\)
−0.932198 + 0.361948i \(0.882112\pi\)
\(24\) 0 0
\(25\) −4.96410 + 8.59808i −0.992820 + 1.71962i
\(26\) −3.34607 + 5.79555i −0.656217 + 1.13660i
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 − 3.46410i −0.371391 − 0.643268i 0.618389 − 0.785872i \(-0.287786\pi\)
−0.989780 + 0.142605i \(0.954452\pi\)
\(30\) 0 0
\(31\) −1.79315 −0.322059 −0.161030 − 0.986950i \(-0.551481\pi\)
−0.161030 + 0.986950i \(0.551481\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.70831 + 4.69093i 0.464471 + 0.804488i
\(35\) 0 0
\(36\) 0 0
\(37\) −0.267949 + 0.464102i −0.0440506 + 0.0762978i −0.887210 − 0.461366i \(-0.847360\pi\)
0.843159 + 0.537664i \(0.180693\pi\)
\(38\) 1.48356 − 2.56961i 0.240666 − 0.416845i
\(39\) 0 0
\(40\) 1.93185 + 3.34607i 0.305453 + 0.529059i
\(41\) 0.637756 − 1.10463i 0.0996008 − 0.172514i −0.811919 − 0.583771i \(-0.801577\pi\)
0.911519 + 0.411257i \(0.134910\pi\)
\(42\) 0 0
\(43\) −1.86603 − 3.23205i −0.284566 − 0.492883i 0.687938 − 0.725770i \(-0.258516\pi\)
−0.972504 + 0.232887i \(0.925183\pi\)
\(44\) −1.86603 + 3.23205i −0.281314 + 0.487250i
\(45\) 0 0
\(46\) 0.732051 + 1.26795i 0.107935 + 0.186949i
\(47\) 10.5558 1.53973 0.769863 − 0.638209i \(-0.220324\pi\)
0.769863 + 0.638209i \(0.220324\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 4.96410 − 8.59808i 0.702030 − 1.21595i
\(51\) 0 0
\(52\) 3.34607 − 5.79555i 0.464016 − 0.803699i
\(53\) −1.46410 − 2.53590i −0.201110 − 0.348332i 0.747776 − 0.663951i \(-0.231121\pi\)
−0.948886 + 0.315618i \(0.897788\pi\)
\(54\) 0 0
\(55\) 14.4195 1.94433
\(56\) 0 0
\(57\) 0 0
\(58\) 2.00000 + 3.46410i 0.262613 + 0.454859i
\(59\) −8.62398 −1.12275 −0.561373 − 0.827563i \(-0.689727\pi\)
−0.561373 + 0.827563i \(0.689727\pi\)
\(60\) 0 0
\(61\) −6.96953 −0.892357 −0.446179 − 0.894944i \(-0.647215\pi\)
−0.446179 + 0.894944i \(0.647215\pi\)
\(62\) 1.79315 0.227730
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −25.8564 −3.20709
\(66\) 0 0
\(67\) 5.53590 0.676318 0.338159 − 0.941089i \(-0.390196\pi\)
0.338159 + 0.941089i \(0.390196\pi\)
\(68\) −2.70831 − 4.69093i −0.328431 − 0.568859i
\(69\) 0 0
\(70\) 0 0
\(71\) −2.53590 −0.300956 −0.150478 − 0.988613i \(-0.548081\pi\)
−0.150478 + 0.988613i \(0.548081\pi\)
\(72\) 0 0
\(73\) −3.41542 − 5.91567i −0.399744 − 0.692377i 0.593950 − 0.804502i \(-0.297568\pi\)
−0.993694 + 0.112125i \(0.964234\pi\)
\(74\) 0.267949 − 0.464102i 0.0311485 − 0.0539507i
\(75\) 0 0
\(76\) −1.48356 + 2.56961i −0.170176 + 0.294754i
\(77\) 0 0
\(78\) 0 0
\(79\) −4.92820 −0.554466 −0.277233 − 0.960803i \(-0.589417\pi\)
−0.277233 + 0.960803i \(0.589417\pi\)
\(80\) −1.93185 − 3.34607i −0.215988 − 0.374101i
\(81\) 0 0
\(82\) −0.637756 + 1.10463i −0.0704284 + 0.121986i
\(83\) 8.95215 + 15.5056i 0.982626 + 1.70196i 0.652043 + 0.758182i \(0.273912\pi\)
0.330583 + 0.943777i \(0.392755\pi\)
\(84\) 0 0
\(85\) −10.4641 + 18.1244i −1.13499 + 1.96586i
\(86\) 1.86603 + 3.23205i 0.201219 + 0.348521i
\(87\) 0 0
\(88\) 1.86603 − 3.23205i 0.198919 − 0.344538i
\(89\) −3.53553 + 6.12372i −0.374766 + 0.649113i −0.990292 − 0.139003i \(-0.955610\pi\)
0.615526 + 0.788116i \(0.288944\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.732051 − 1.26795i −0.0763216 − 0.132193i
\(93\) 0 0
\(94\) −10.5558 −1.08875
\(95\) 11.4641 1.17619
\(96\) 0 0
\(97\) −2.94855 − 5.10703i −0.299379 − 0.518540i 0.676615 − 0.736337i \(-0.263446\pi\)
−0.975994 + 0.217797i \(0.930113\pi\)
\(98\) 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 2646.2.e.q.1549.1 8
3.2 odd 2 882.2.e.s.373.1 8
7.2 even 3 2646.2.f.r.1765.1 8
7.3 odd 6 2646.2.h.t.361.1 8
7.4 even 3 2646.2.h.t.361.4 8
7.5 odd 6 2646.2.f.r.1765.4 8
7.6 odd 2 inner 2646.2.e.q.1549.4 8
9.2 odd 6 882.2.h.q.79.3 8
9.7 even 3 2646.2.h.t.667.4 8
21.2 odd 6 882.2.f.q.589.3 yes 8
21.5 even 6 882.2.f.q.589.2 yes 8
21.11 odd 6 882.2.h.q.67.4 8
21.17 even 6 882.2.h.q.67.1 8
21.20 even 2 882.2.e.s.373.4 8
63.2 odd 6 882.2.f.q.295.3 yes 8
63.5 even 6 7938.2.a.cp.1.4 4
63.11 odd 6 882.2.e.s.655.1 8
63.16 even 3 2646.2.f.r.883.1 8
63.20 even 6 882.2.h.q.79.2 8
63.23 odd 6 7938.2.a.cp.1.1 4
63.25 even 3 inner 2646.2.e.q.2125.1 8
63.34 odd 6 2646.2.h.t.667.1 8
63.38 even 6 882.2.e.s.655.4 8
63.40 odd 6 7938.2.a.ci.1.1 4
63.47 even 6 882.2.f.q.295.2 ✓ 8
63.52 odd 6 inner 2646.2.e.q.2125.4 8
63.58 even 3 7938.2.a.ci.1.4 4
63.61 odd 6 2646.2.f.r.883.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
882.2.e.s.373.1 8 3.2 odd 2
882.2.e.s.373.4 8 21.20 even 2
882.2.e.s.655.1 8 63.11 odd 6
882.2.e.s.655.4 8 63.38 even 6
882.2.f.q.295.2 ✓ 8 63.47 even 6
882.2.f.q.295.3 yes 8 63.2 odd 6
882.2.f.q.589.2 yes 8 21.5 even 6
882.2.f.q.589.3 yes 8 21.2 odd 6
882.2.h.q.67.1 8 21.17 even 6
882.2.h.q.67.4 8 21.11 odd 6
882.2.h.q.79.2 8 63.20 even 6
882.2.h.q.79.3 8 9.2 odd 6
2646.2.e.q.1549.1 8 1.1 even 1 trivial
2646.2.e.q.1549.4 8 7.6 odd 2 inner
2646.2.e.q.2125.1 8 63.25 even 3 inner
2646.2.e.q.2125.4 8 63.52 odd 6 inner
2646.2.f.r.883.1 8 63.16 even 3
2646.2.f.r.883.4 8 63.61 odd 6
2646.2.f.r.1765.1 8 7.2 even 3
2646.2.f.r.1765.4 8 7.5 odd 6
2646.2.h.t.361.1 8 7.3 odd 6
2646.2.h.t.361.4 8 7.4 even 3
2646.2.h.t.667.1 8 63.34 odd 6
2646.2.h.t.667.4 8 9.7 even 3
7938.2.a.ci.1.1 4 63.40 odd 6
7938.2.a.ci.1.4 4 63.58 even 3
7938.2.a.cp.1.1 4 63.23 odd 6
7938.2.a.cp.1.4 4 63.5 even 6