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Properties

Label 2646.2.h.t.361.1
Level $2646$
Weight $2$
Character 2646.361
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(361,2646)] 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("2646.361"); 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, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,4,0,-4,0,0,0,-8,0,0,16,0,0] 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 361.1
Root \(0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 2646.361
Dual form 2646.2.h.t.667.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} -3.86370 q^{5} -1.00000 q^{8} +(-1.93185 + 3.34607i) q^{10} +3.73205 q^{11} +(-3.34607 + 5.79555i) q^{13} +(-0.500000 + 0.866025i) q^{16} +(2.70831 - 4.69093i) q^{17} +(1.48356 + 2.56961i) q^{19} +(1.93185 + 3.34607i) q^{20} +(1.86603 - 3.23205i) q^{22} +1.46410 q^{23} +9.92820 q^{25} +(3.34607 + 5.79555i) q^{26} +(-2.00000 - 3.46410i) q^{29} +(-0.896575 - 1.55291i) q^{31} +(0.500000 + 0.866025i) q^{32} +(-2.70831 - 4.69093i) q^{34} +(-0.267949 - 0.464102i) q^{37} +2.96713 q^{38} +3.86370 q^{40} +(-0.637756 + 1.10463i) q^{41} +(-1.86603 - 3.23205i) q^{43} +(-1.86603 - 3.23205i) q^{44} +(0.732051 - 1.26795i) q^{46} +(5.27792 - 9.14162i) q^{47} +(4.96410 - 8.59808i) q^{50} +6.69213 q^{52} +(-1.46410 + 2.53590i) q^{53} -14.4195 q^{55} -4.00000 q^{58} +(-4.31199 - 7.46859i) q^{59} +(-3.48477 + 6.03579i) q^{61} -1.79315 q^{62} +1.00000 q^{64} +(12.9282 - 22.3923i) q^{65} +(-2.76795 - 4.79423i) q^{67} -5.41662 q^{68} -2.53590 q^{71} +(3.41542 - 5.91567i) q^{73} -0.535898 q^{74} +(1.48356 - 2.56961i) q^{76} +(2.46410 - 4.26795i) q^{79} +(1.93185 - 3.34607i) q^{80} +(0.637756 + 1.10463i) q^{82} +(-8.95215 - 15.5056i) q^{83} +(-10.4641 + 18.1244i) q^{85} -3.73205 q^{86} -3.73205 q^{88} +(3.53553 + 6.12372i) q^{89} +(-0.732051 - 1.26795i) q^{92} +(-5.27792 - 9.14162i) q^{94} +(-5.73205 - 9.92820i) q^{95} +(2.94855 + 5.10703i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{4} - 8 q^{8} + 16 q^{11} - 4 q^{16} + 8 q^{22} - 16 q^{23} + 24 q^{25} - 16 q^{29} + 4 q^{32} - 16 q^{37} - 8 q^{43} - 8 q^{44} - 8 q^{46} + 12 q^{50} + 16 q^{53} - 32 q^{58} + 8 q^{64}+ \cdots - 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 − 0.866025i 0.353553 − 0.612372i
\(3\) 0 0
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) −3.86370 −1.72790 −0.863950 − 0.503577i \(-0.832017\pi\)
−0.863950 + 0.503577i \(0.832017\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.93185 + 3.34607i −0.610905 + 1.05812i
\(11\) 3.73205 1.12526 0.562628 − 0.826710i \(-0.309790\pi\)
0.562628 + 0.826710i \(0.309790\pi\)
\(12\) 0 0
\(13\) −3.34607 + 5.79555i −0.928032 + 1.60740i −0.141420 + 0.989950i \(0.545167\pi\)
−0.786612 + 0.617448i \(0.788167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 2.70831 − 4.69093i 0.656861 − 1.13772i −0.324562 − 0.945864i \(-0.605217\pi\)
0.981424 − 0.191853i \(-0.0614497\pi\)
\(18\) 0 0
\(19\) 1.48356 + 2.56961i 0.340353 + 0.589509i 0.984498 − 0.175395i \(-0.0561201\pi\)
−0.644145 + 0.764903i \(0.722787\pi\)
\(20\) 1.93185 + 3.34607i 0.431975 + 0.748203i
\(21\) 0 0
\(22\) 1.86603 − 3.23205i 0.397838 − 0.689076i
\(23\) 1.46410 0.305286 0.152643 − 0.988281i \(-0.451221\pi\)
0.152643 + 0.988281i \(0.451221\pi\)
\(24\) 0 0
\(25\) 9.92820 1.98564
\(26\) 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
\(31\) −0.896575 − 1.55291i −0.161030 − 0.278912i 0.774209 − 0.632931i \(-0.218148\pi\)
−0.935238 + 0.354019i \(0.884815\pi\)
\(32\) 0.500000 + 0.866025i 0.0883883 + 0.153093i
\(33\) 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 0.843159 − 0.537664i \(-0.180693\pi\)
−0.887210 + 0.461366i \(0.847360\pi\)
\(38\) 2.96713 0.481332
\(39\) 0 0
\(40\) 3.86370 0.610905
\(41\) −0.637756 + 1.10463i −0.0996008 + 0.172514i −0.911519 − 0.411257i \(-0.865090\pi\)
0.811919 + 0.583771i \(0.198423\pi\)
\(42\) 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
\(47\) 5.27792 − 9.14162i 0.769863 − 1.33344i −0.167773 − 0.985826i \(-0.553658\pi\)
0.937637 − 0.347617i \(-0.113009\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 4.96410 − 8.59808i 0.702030 − 1.21595i
\(51\) 0 0
\(52\) 6.69213 0.928032
\(53\) −1.46410 + 2.53590i −0.201110 + 0.348332i −0.948886 − 0.315618i \(-0.897788\pi\)
0.747776 + 0.663951i \(0.231121\pi\)
\(54\) 0 0
\(55\) −14.4195 −1.94433
\(56\) 0 0
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) −4.31199 − 7.46859i −0.561373 − 0.972327i −0.997377 − 0.0723823i \(-0.976940\pi\)
0.436004 − 0.899945i \(-0.356394\pi\)
\(60\) 0 0
\(61\) −3.48477 + 6.03579i −0.446179 + 0.772804i −0.998133 − 0.0610700i \(-0.980549\pi\)
0.551955 + 0.833874i \(0.313882\pi\)
\(62\) −1.79315 −0.227730
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 12.9282 − 22.3923i 1.60355 − 2.77742i
\(66\) 0 0
\(67\) −2.76795 − 4.79423i −0.338159 − 0.585708i 0.645928 − 0.763399i \(-0.276471\pi\)
−0.984086 + 0.177690i \(0.943137\pi\)
\(68\) −5.41662 −0.656861
\(69\) 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 \(-0.702432\pi\)
0.993694 + 0.112125i \(0.0357656\pi\)
\(74\) −0.535898 −0.0622969
\(75\) 0 0
\(76\) 1.48356 − 2.56961i 0.170176 − 0.294754i
\(77\) 0 0
\(78\) 0 0
\(79\) 2.46410 − 4.26795i 0.277233 − 0.480182i −0.693463 − 0.720492i \(-0.743916\pi\)
0.970696 + 0.240310i \(0.0772492\pi\)
\(80\) 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 \(-0.726088\pi\)
−0.330583 − 0.943777i \(-0.607245\pi\)
\(84\) 0 0
\(85\) −10.4641 + 18.1244i −1.13499 + 1.96586i
\(86\) −3.73205 −0.402437
\(87\) 0 0
\(88\) −3.73205 −0.397838
\(89\) 3.53553 + 6.12372i 0.374766 + 0.649113i 0.990292 − 0.139003i \(-0.0443898\pi\)
−0.615526 + 0.788116i \(0.711056\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.732051 − 1.26795i −0.0763216 − 0.132193i
\(93\) 0 0
\(94\) −5.27792 − 9.14162i −0.544376 − 0.942886i
\(95\) −5.73205 − 9.92820i −0.588096 − 1.01861i
\(96\) 0 0
\(97\) 2.94855 + 5.10703i 0.299379 + 0.518540i 0.975994 − 0.217797i \(-0.0698870\pi\)
−0.676615 + 0.736337i \(0.736554\pi\)
\(98\) 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.h.t.361.1 8
3.2 odd 2 882.2.h.q.67.1 8
7.2 even 3 2646.2.e.q.1549.4 8
7.3 odd 6 2646.2.f.r.1765.1 8
7.4 even 3 2646.2.f.r.1765.4 8
7.5 odd 6 2646.2.e.q.1549.1 8
7.6 odd 2 inner 2646.2.h.t.361.4 8
9.2 odd 6 882.2.e.s.655.4 8
9.7 even 3 2646.2.e.q.2125.4 8
21.2 odd 6 882.2.e.s.373.4 8
21.5 even 6 882.2.e.s.373.1 8
21.11 odd 6 882.2.f.q.589.2 yes 8
21.17 even 6 882.2.f.q.589.3 yes 8
21.20 even 2 882.2.h.q.67.4 8
63.2 odd 6 882.2.h.q.79.2 8
63.4 even 3 7938.2.a.ci.1.1 4
63.11 odd 6 882.2.f.q.295.2 ✓ 8
63.16 even 3 inner 2646.2.h.t.667.1 8
63.20 even 6 882.2.e.s.655.1 8
63.25 even 3 2646.2.f.r.883.4 8
63.31 odd 6 7938.2.a.ci.1.4 4
63.32 odd 6 7938.2.a.cp.1.4 4
63.34 odd 6 2646.2.e.q.2125.1 8
63.38 even 6 882.2.f.q.295.3 yes 8
63.47 even 6 882.2.h.q.79.3 8
63.52 odd 6 2646.2.f.r.883.1 8
63.59 even 6 7938.2.a.cp.1.1 4
63.61 odd 6 inner 2646.2.h.t.667.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
882.2.e.s.373.1 8 21.5 even 6
882.2.e.s.373.4 8 21.2 odd 6
882.2.e.s.655.1 8 63.20 even 6
882.2.e.s.655.4 8 9.2 odd 6
882.2.f.q.295.2 ✓ 8 63.11 odd 6
882.2.f.q.295.3 yes 8 63.38 even 6
882.2.f.q.589.2 yes 8 21.11 odd 6
882.2.f.q.589.3 yes 8 21.17 even 6
882.2.h.q.67.1 8 3.2 odd 2
882.2.h.q.67.4 8 21.20 even 2
882.2.h.q.79.2 8 63.2 odd 6
882.2.h.q.79.3 8 63.47 even 6
2646.2.e.q.1549.1 8 7.5 odd 6
2646.2.e.q.1549.4 8 7.2 even 3
2646.2.e.q.2125.1 8 63.34 odd 6
2646.2.e.q.2125.4 8 9.7 even 3
2646.2.f.r.883.1 8 63.52 odd 6
2646.2.f.r.883.4 8 63.25 even 3
2646.2.f.r.1765.1 8 7.3 odd 6
2646.2.f.r.1765.4 8 7.4 even 3
2646.2.h.t.361.1 8 1.1 even 1 trivial
2646.2.h.t.361.4 8 7.6 odd 2 inner
2646.2.h.t.667.1 8 63.16 even 3 inner
2646.2.h.t.667.4 8 63.61 odd 6 inner
7938.2.a.ci.1.1 4 63.4 even 3
7938.2.a.ci.1.4 4 63.31 odd 6
7938.2.a.cp.1.1 4 63.59 even 6
7938.2.a.cp.1.4 4 63.32 odd 6