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Properties

Label 2646.2.h.o.361.1
Level $2646$
Weight $2$
Character 2646.361
Analytic conductor $21.128$
Analytic rank $0$
Dimension $6$
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] = [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 = [6,-3,0,-3,-2,0,0,6,0,1,-2,0,-8] 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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 - 1.41036i\) of defining polynomial
Character \(\chi\) \(=\) 2646.361
Dual form 2646.2.h.o.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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} -3.18194 q^{5} +1.00000 q^{8} +(1.59097 - 2.75564i) q^{10} -3.18194 q^{11} +(-2.85185 + 4.93955i) q^{13} +(-0.500000 + 0.866025i) q^{16} +(-0.760877 + 1.31788i) q^{17} +(0.641315 + 1.11079i) q^{19} +(1.59097 + 2.75564i) q^{20} +(1.59097 - 2.75564i) q^{22} -2.23912 q^{23} +5.12476 q^{25} +(-2.85185 - 4.93955i) q^{26} +(3.54063 + 6.13255i) q^{29} +(-4.71053 - 8.15888i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(-0.760877 - 1.31788i) q^{34} +(0.500000 + 0.866025i) q^{37} -1.28263 q^{38} -3.18194 q^{40} +(-2.80150 + 4.85235i) q^{41} +(3.41423 + 5.91362i) q^{43} +(1.59097 + 2.75564i) q^{44} +(1.11956 - 1.93914i) q^{46} +(2.91423 - 5.04759i) q^{47} +(-2.56238 + 4.43818i) q^{50} +5.70370 q^{52} +(-1.02859 + 1.78157i) q^{53} +10.1248 q^{55} -7.08126 q^{58} +(0.562382 + 0.974074i) q^{59} +(1.56238 - 2.70612i) q^{61} +9.42107 q^{62} +1.00000 q^{64} +(9.07442 - 15.7174i) q^{65} +(-5.48345 - 9.49761i) q^{67} +1.52175 q^{68} -8.69002 q^{71} +(2.48345 - 4.30146i) q^{73} -1.00000 q^{74} +(0.641315 - 1.11079i) q^{76} +(2.06922 - 3.58399i) q^{79} +(1.59097 - 2.75564i) q^{80} +(-2.80150 - 4.85235i) q^{82} +(-4.03379 - 6.98673i) q^{83} +(2.42107 - 4.19341i) q^{85} -6.82846 q^{86} -3.18194 q^{88} +(0.112725 + 0.195246i) q^{89} +(1.11956 + 1.93914i) q^{92} +(2.91423 + 5.04759i) q^{94} +(-2.04063 - 3.53447i) q^{95} +(-7.42107 - 12.8537i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - 3 q^{4} - 2 q^{5} + 6 q^{8} + q^{10} - 2 q^{11} - 8 q^{13} - 3 q^{16} - 4 q^{17} + 3 q^{19} + q^{20} + q^{22} - 14 q^{23} - 4 q^{25} - 8 q^{26} + 5 q^{29} - 20 q^{31} - 3 q^{32} - 4 q^{34}+ \cdots - 28 q^{97}+O(q^{100}) \) 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
\(5\) −3.18194 −1.42301 −0.711504 − 0.702682i \(-0.751986\pi\)
−0.711504 + 0.702682i \(0.751986\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.59097 − 2.75564i 0.503109 − 0.871411i
\(11\) −3.18194 −0.959392 −0.479696 − 0.877435i \(-0.659253\pi\)
−0.479696 + 0.877435i \(0.659253\pi\)
\(12\) 0 0
\(13\) −2.85185 + 4.93955i −0.790960 + 1.36998i 0.134412 + 0.990925i \(0.457085\pi\)
−0.925373 + 0.379058i \(0.876248\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −0.760877 + 1.31788i −0.184540 + 0.319632i −0.943421 − 0.331596i \(-0.892413\pi\)
0.758882 + 0.651229i \(0.225746\pi\)
\(18\) 0 0
\(19\) 0.641315 + 1.11079i 0.147128 + 0.254833i 0.930165 − 0.367142i \(-0.119664\pi\)
−0.783037 + 0.621975i \(0.786330\pi\)
\(20\) 1.59097 + 2.75564i 0.355752 + 0.616181i
\(21\) 0 0
\(22\) 1.59097 − 2.75564i 0.339196 − 0.587505i
\(23\) −2.23912 −0.466889 −0.233445 − 0.972370i \(-0.575000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(24\) 0 0
\(25\) 5.12476 1.02495
\(26\) −2.85185 − 4.93955i −0.559293 − 0.968725i
\(27\) 0 0
\(28\) 0 0
\(29\) 3.54063 + 6.13255i 0.657478 + 1.13879i 0.981266 + 0.192656i \(0.0617101\pi\)
−0.323788 + 0.946130i \(0.604957\pi\)
\(30\) 0 0
\(31\) −4.71053 − 8.15888i −0.846037 − 1.46538i −0.884718 − 0.466127i \(-0.845649\pi\)
0.0386810 − 0.999252i \(-0.487684\pi\)
\(32\) −0.500000 − 0.866025i −0.0883883 − 0.153093i
\(33\) 0 0
\(34\) −0.760877 − 1.31788i −0.130489 − 0.226014i
\(35\) 0 0
\(36\) 0 0
\(37\) 0.500000 + 0.866025i 0.0821995 + 0.142374i 0.904194 − 0.427121i \(-0.140472\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −1.28263 −0.208070
\(39\) 0 0
\(40\) −3.18194 −0.503109
\(41\) −2.80150 + 4.85235i −0.437522 + 0.757810i −0.997498 − 0.0706992i \(-0.977477\pi\)
0.559976 + 0.828509i \(0.310810\pi\)
\(42\) 0 0
\(43\) 3.41423 + 5.91362i 0.520665 + 0.901819i 0.999711 + 0.0240288i \(0.00764935\pi\)
−0.479046 + 0.877790i \(0.659017\pi\)
\(44\) 1.59097 + 2.75564i 0.239848 + 0.415429i
\(45\) 0 0
\(46\) 1.11956 − 1.93914i 0.165070 − 0.285910i
\(47\) 2.91423 − 5.04759i 0.425084 − 0.736267i −0.571344 − 0.820711i \(-0.693578\pi\)
0.996428 + 0.0844432i \(0.0269112\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −2.56238 + 4.43818i −0.362375 + 0.627653i
\(51\) 0 0
\(52\) 5.70370 0.790960
\(53\) −1.02859 + 1.78157i −0.141288 + 0.244717i −0.927982 − 0.372626i \(-0.878458\pi\)
0.786694 + 0.617343i \(0.211791\pi\)
\(54\) 0 0
\(55\) 10.1248 1.36522
\(56\) 0 0
\(57\) 0 0
\(58\) −7.08126 −0.929815
\(59\) 0.562382 + 0.974074i 0.0732159 + 0.126814i 0.900309 − 0.435251i \(-0.143340\pi\)
−0.827093 + 0.562065i \(0.810007\pi\)
\(60\) 0 0
\(61\) 1.56238 − 2.70612i 0.200042 − 0.346484i −0.748499 − 0.663135i \(-0.769225\pi\)
0.948542 + 0.316652i \(0.102559\pi\)
\(62\) 9.42107 1.19648
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 9.07442 − 15.7174i 1.12554 − 1.94950i
\(66\) 0 0
\(67\) −5.48345 − 9.49761i −0.669910 − 1.16032i −0.977929 − 0.208938i \(-0.932999\pi\)
0.308019 − 0.951380i \(-0.400334\pi\)
\(68\) 1.52175 0.184540
\(69\) 0 0
\(70\) 0 0
\(71\) −8.69002 −1.03132 −0.515658 − 0.856794i \(-0.672452\pi\)
−0.515658 + 0.856794i \(0.672452\pi\)
\(72\) 0 0
\(73\) 2.48345 − 4.30146i 0.290666 − 0.503448i −0.683302 − 0.730136i \(-0.739457\pi\)
0.973967 + 0.226689i \(0.0727899\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) 0.641315 − 1.11079i 0.0735639 − 0.127416i
\(77\) 0 0
\(78\) 0 0
\(79\) 2.06922 − 3.58399i 0.232805 − 0.403231i −0.725827 − 0.687877i \(-0.758543\pi\)
0.958633 + 0.284646i \(0.0918762\pi\)
\(80\) 1.59097 − 2.75564i 0.177876 − 0.308090i
\(81\) 0 0
\(82\) −2.80150 − 4.85235i −0.309374 − 0.535852i
\(83\) −4.03379 − 6.98673i −0.442766 − 0.766893i 0.555127 − 0.831765i \(-0.312669\pi\)
−0.997894 + 0.0648718i \(0.979336\pi\)
\(84\) 0 0
\(85\) 2.42107 − 4.19341i 0.262602 − 0.454839i
\(86\) −6.82846 −0.736332
\(87\) 0 0
\(88\) −3.18194 −0.339196
\(89\) 0.112725 + 0.195246i 0.0119488 + 0.0206960i 0.871938 − 0.489616i \(-0.162863\pi\)
−0.859989 + 0.510312i \(0.829530\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.11956 + 1.93914i 0.116722 + 0.202169i
\(93\) 0 0
\(94\) 2.91423 + 5.04759i 0.300580 + 0.520620i
\(95\) −2.04063 − 3.53447i −0.209364 − 0.362629i
\(96\) 0 0
\(97\) −7.42107 − 12.8537i −0.753495 − 1.30509i −0.946119 − 0.323819i \(-0.895033\pi\)
0.192624 − 0.981273i \(-0.438300\pi\)
\(98\) 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.o.361.1 6
3.2 odd 2 882.2.h.p.67.3 6
7.2 even 3 2646.2.e.p.1549.3 6
7.3 odd 6 2646.2.f.l.1765.1 6
7.4 even 3 2646.2.f.m.1765.3 6
7.5 odd 6 378.2.e.d.37.1 6
7.6 odd 2 378.2.h.c.361.3 6
9.2 odd 6 882.2.e.o.655.1 6
9.7 even 3 2646.2.e.p.2125.3 6
21.2 odd 6 882.2.e.o.373.1 6
21.5 even 6 126.2.e.c.121.3 yes 6
21.11 odd 6 882.2.f.o.589.2 6
21.17 even 6 882.2.f.n.589.2 6
21.20 even 2 126.2.h.d.67.1 yes 6
28.19 even 6 3024.2.q.g.2305.1 6
28.27 even 2 3024.2.t.h.1873.3 6
63.2 odd 6 882.2.h.p.79.3 6
63.4 even 3 7938.2.a.bz.1.1 3
63.5 even 6 1134.2.g.m.163.3 6
63.11 odd 6 882.2.f.o.295.2 6
63.13 odd 6 1134.2.g.l.487.1 6
63.16 even 3 inner 2646.2.h.o.667.1 6
63.20 even 6 126.2.e.c.25.3 ✓ 6
63.25 even 3 2646.2.f.m.883.3 6
63.31 odd 6 7938.2.a.ca.1.3 3
63.32 odd 6 7938.2.a.bw.1.3 3
63.34 odd 6 378.2.e.d.235.1 6
63.38 even 6 882.2.f.n.295.2 6
63.40 odd 6 1134.2.g.l.163.1 6
63.41 even 6 1134.2.g.m.487.3 6
63.47 even 6 126.2.h.d.79.1 yes 6
63.52 odd 6 2646.2.f.l.883.1 6
63.59 even 6 7938.2.a.bv.1.1 3
63.61 odd 6 378.2.h.c.289.3 6
84.47 odd 6 1008.2.q.g.625.1 6
84.83 odd 2 1008.2.t.h.193.3 6
252.47 odd 6 1008.2.t.h.961.3 6
252.83 odd 6 1008.2.q.g.529.1 6
252.187 even 6 3024.2.t.h.289.3 6
252.223 even 6 3024.2.q.g.2881.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.e.c.25.3 ✓ 6 63.20 even 6
126.2.e.c.121.3 yes 6 21.5 even 6
126.2.h.d.67.1 yes 6 21.20 even 2
126.2.h.d.79.1 yes 6 63.47 even 6
378.2.e.d.37.1 6 7.5 odd 6
378.2.e.d.235.1 6 63.34 odd 6
378.2.h.c.289.3 6 63.61 odd 6
378.2.h.c.361.3 6 7.6 odd 2
882.2.e.o.373.1 6 21.2 odd 6
882.2.e.o.655.1 6 9.2 odd 6
882.2.f.n.295.2 6 63.38 even 6
882.2.f.n.589.2 6 21.17 even 6
882.2.f.o.295.2 6 63.11 odd 6
882.2.f.o.589.2 6 21.11 odd 6
882.2.h.p.67.3 6 3.2 odd 2
882.2.h.p.79.3 6 63.2 odd 6
1008.2.q.g.529.1 6 252.83 odd 6
1008.2.q.g.625.1 6 84.47 odd 6
1008.2.t.h.193.3 6 84.83 odd 2
1008.2.t.h.961.3 6 252.47 odd 6
1134.2.g.l.163.1 6 63.40 odd 6
1134.2.g.l.487.1 6 63.13 odd 6
1134.2.g.m.163.3 6 63.5 even 6
1134.2.g.m.487.3 6 63.41 even 6
2646.2.e.p.1549.3 6 7.2 even 3
2646.2.e.p.2125.3 6 9.7 even 3
2646.2.f.l.883.1 6 63.52 odd 6
2646.2.f.l.1765.1 6 7.3 odd 6
2646.2.f.m.883.3 6 63.25 even 3
2646.2.f.m.1765.3 6 7.4 even 3
2646.2.h.o.361.1 6 1.1 even 1 trivial
2646.2.h.o.667.1 6 63.16 even 3 inner
3024.2.q.g.2305.1 6 28.19 even 6
3024.2.q.g.2881.1 6 252.223 even 6
3024.2.t.h.289.3 6 252.187 even 6
3024.2.t.h.1873.3 6 28.27 even 2
7938.2.a.bv.1.1 3 63.59 even 6
7938.2.a.bw.1.3 3 63.32 odd 6
7938.2.a.bz.1.1 3 63.4 even 3
7938.2.a.ca.1.3 3 63.31 odd 6