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(883,2646)] 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("2646.883"); 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([4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-3,0,-3,-1,0,0,6,0,2,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == 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 1765.1
Root \(0.500000 + 1.41036i\) of defining polynomial
Character \(\chi\) \(=\) 2646.1765
Dual form 2646.2.f.l.883.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-1.59097 + 2.75564i) q^{5} +1.00000 q^{8} +3.18194 q^{10} +(1.59097 + 2.75564i) q^{11} +(2.85185 - 4.93955i) q^{13} +(-0.500000 - 0.866025i) q^{16} -1.52175 q^{17} +1.28263 q^{19} +(-1.59097 - 2.75564i) q^{20} +(1.59097 - 2.75564i) q^{22} +(1.11956 - 1.93914i) q^{23} +(-2.56238 - 4.43818i) q^{25} -5.70370 q^{26} +(3.54063 + 6.13255i) q^{29} +(4.71053 - 8.15888i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(0.760877 + 1.31788i) q^{34} -1.00000 q^{37} +(-0.641315 - 1.11079i) q^{38} +(-1.59097 + 2.75564i) q^{40} +(2.80150 - 4.85235i) q^{41} +(3.41423 + 5.91362i) q^{43} -3.18194 q^{44} -2.23912 q^{46} +(-2.91423 - 5.04759i) q^{47} +(-2.56238 + 4.43818i) q^{50} +(2.85185 + 4.93955i) q^{52} +2.05718 q^{53} -10.1248 q^{55} +(3.54063 - 6.13255i) q^{58} +(-0.562382 + 0.974074i) q^{59} +(-1.56238 - 2.70612i) q^{61} -9.42107 q^{62} +1.00000 q^{64} +(9.07442 + 15.7174i) q^{65} +(-5.48345 + 9.49761i) q^{67} +(0.760877 - 1.31788i) q^{68} -8.69002 q^{71} +4.96690 q^{73} +(0.500000 + 0.866025i) q^{74} +(-0.641315 + 1.11079i) q^{76} +(2.06922 + 3.58399i) q^{79} +3.18194 q^{80} -5.60301 q^{82} +(4.03379 + 6.98673i) q^{83} +(2.42107 - 4.19341i) q^{85} +(3.41423 - 5.91362i) q^{86} +(1.59097 + 2.75564i) q^{88} +0.225450 q^{89} +(1.11956 + 1.93914i) q^{92} +(-2.91423 + 5.04759i) q^{94} +(-2.04063 + 3.53447i) q^{95} +(7.42107 + 12.8537i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - 3 q^{4} - q^{5} + 6 q^{8} + 2 q^{10} + q^{11} + 8 q^{13} - 3 q^{16} - 8 q^{17} + 6 q^{19} - q^{20} + q^{22} + 7 q^{23} + 2 q^{25} - 16 q^{26} + 5 q^{29} + 20 q^{31} - 3 q^{32} + 4 q^{34}+ \cdots + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) −1.59097 + 2.75564i −0.711504 + 1.23236i 0.252788 + 0.967522i \(0.418652\pi\)
−0.964292 + 0.264840i \(0.914681\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 3.18194 1.00622
\(11\) 1.59097 + 2.75564i 0.479696 + 0.830858i 0.999729 − 0.0232884i \(-0.00741361\pi\)
−0.520033 + 0.854146i \(0.674080\pi\)
\(12\) 0 0
\(13\) 2.85185 − 4.93955i 0.790960 − 1.36998i −0.134412 − 0.990925i \(-0.542915\pi\)
0.925373 − 0.379058i \(-0.123752\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −1.52175 −0.369079 −0.184540 − 0.982825i \(-0.559079\pi\)
−0.184540 + 0.982825i \(0.559079\pi\)
\(18\) 0 0
\(19\) 1.28263 0.294256 0.147128 − 0.989117i \(-0.452997\pi\)
0.147128 + 0.989117i \(0.452997\pi\)
\(20\) −1.59097 − 2.75564i −0.355752 − 0.616181i
\(21\) 0 0
\(22\) 1.59097 − 2.75564i 0.339196 − 0.587505i
\(23\) 1.11956 − 1.93914i 0.233445 − 0.404338i −0.725375 − 0.688354i \(-0.758334\pi\)
0.958820 + 0.284016i \(0.0916669\pi\)
\(24\) 0 0
\(25\) −2.56238 − 4.43818i −0.512476 − 0.887635i
\(26\) −5.70370 −1.11859
\(27\) 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 −0.0386810 − 0.999252i \(-0.512316\pi\)
0.884718 − 0.466127i \(-0.154351\pi\)
\(32\) −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
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) −0.641315 − 1.11079i −0.104035 − 0.180194i
\(39\) 0 0
\(40\) −1.59097 + 2.75564i −0.251555 + 0.435706i
\(41\) 2.80150 − 4.85235i 0.437522 − 0.757810i −0.559976 − 0.828509i \(-0.689190\pi\)
0.997498 + 0.0706992i \(0.0225230\pi\)
\(42\) 0 0
\(43\) 3.41423 + 5.91362i 0.520665 + 0.901819i 0.999711 + 0.0240288i \(0.00764935\pi\)
−0.479046 + 0.877790i \(0.659017\pi\)
\(44\) −3.18194 −0.479696
\(45\) 0 0
\(46\) −2.23912 −0.330141
\(47\) −2.91423 − 5.04759i −0.425084 − 0.736267i 0.571344 − 0.820711i \(-0.306422\pi\)
−0.996428 + 0.0844432i \(0.973089\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −2.56238 + 4.43818i −0.362375 + 0.627653i
\(51\) 0 0
\(52\) 2.85185 + 4.93955i 0.395480 + 0.684992i
\(53\) 2.05718 0.282575 0.141288 − 0.989969i \(-0.454876\pi\)
0.141288 + 0.989969i \(0.454876\pi\)
\(54\) 0 0
\(55\) −10.1248 −1.36522
\(56\) 0 0
\(57\) 0 0
\(58\) 3.54063 − 6.13255i 0.464907 − 0.805243i
\(59\) −0.562382 + 0.974074i −0.0732159 + 0.126814i −0.900309 − 0.435251i \(-0.856660\pi\)
0.827093 + 0.562065i \(0.189993\pi\)
\(60\) 0 0
\(61\) −1.56238 − 2.70612i −0.200042 − 0.346484i 0.748499 − 0.663135i \(-0.230775\pi\)
−0.948542 + 0.316652i \(0.897441\pi\)
\(62\) −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 0.308019 + 0.951380i \(0.400334\pi\)
−0.977929 + 0.208938i \(0.932999\pi\)
\(68\) 0.760877 − 1.31788i 0.0922699 − 0.159816i
\(69\) 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
\(73\) 4.96690 0.581331 0.290666 − 0.956825i \(-0.406123\pi\)
0.290666 + 0.956825i \(0.406123\pi\)
\(74\) 0.500000 + 0.866025i 0.0581238 + 0.100673i
\(75\) 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 0.958633 − 0.284646i \(-0.0918762\pi\)
−0.725827 + 0.687877i \(0.758543\pi\)
\(80\) 3.18194 0.355752
\(81\) 0 0
\(82\) −5.60301 −0.618749
\(83\) 4.03379 + 6.98673i 0.442766 + 0.766893i 0.997894 − 0.0648718i \(-0.0206639\pi\)
−0.555127 + 0.831765i \(0.687331\pi\)
\(84\) 0 0
\(85\) 2.42107 − 4.19341i 0.262602 − 0.454839i
\(86\) 3.41423 − 5.91362i 0.368166 − 0.637682i
\(87\) 0 0
\(88\) 1.59097 + 2.75564i 0.169598 + 0.293753i
\(89\) 0.225450 0.0238977 0.0119488 − 0.999929i \(-0.496196\pi\)
0.0119488 + 0.999929i \(0.496196\pi\)
\(90\) 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 \(0.104967\pi\)
−0.192624 + 0.981273i \(0.561700\pi\)
\(98\) 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.f.l.1765.1 6
3.2 odd 2 882.2.f.n.589.2 6
7.2 even 3 378.2.h.c.361.3 6
7.3 odd 6 2646.2.e.p.1549.3 6
7.4 even 3 378.2.e.d.37.1 6
7.5 odd 6 2646.2.h.o.361.1 6
7.6 odd 2 2646.2.f.m.1765.3 6
9.2 odd 6 882.2.f.n.295.2 6
9.4 even 3 7938.2.a.ca.1.3 3
9.5 odd 6 7938.2.a.bv.1.1 3
9.7 even 3 inner 2646.2.f.l.883.1 6
21.2 odd 6 126.2.h.d.67.1 yes 6
21.5 even 6 882.2.h.p.67.3 6
21.11 odd 6 126.2.e.c.121.3 yes 6
21.17 even 6 882.2.e.o.373.1 6
21.20 even 2 882.2.f.o.589.2 6
28.11 odd 6 3024.2.q.g.2305.1 6
28.23 odd 6 3024.2.t.h.1873.3 6
63.2 odd 6 126.2.e.c.25.3 ✓ 6
63.4 even 3 1134.2.g.l.163.1 6
63.11 odd 6 126.2.h.d.79.1 yes 6
63.13 odd 6 7938.2.a.bz.1.1 3
63.16 even 3 378.2.e.d.235.1 6
63.20 even 6 882.2.f.o.295.2 6
63.23 odd 6 1134.2.g.m.487.3 6
63.25 even 3 378.2.h.c.289.3 6
63.32 odd 6 1134.2.g.m.163.3 6
63.34 odd 6 2646.2.f.m.883.3 6
63.38 even 6 882.2.h.p.79.3 6
63.41 even 6 7938.2.a.bw.1.3 3
63.47 even 6 882.2.e.o.655.1 6
63.52 odd 6 2646.2.h.o.667.1 6
63.58 even 3 1134.2.g.l.487.1 6
63.61 odd 6 2646.2.e.p.2125.3 6
84.11 even 6 1008.2.q.g.625.1 6
84.23 even 6 1008.2.t.h.193.3 6
252.11 even 6 1008.2.t.h.961.3 6
252.79 odd 6 3024.2.q.g.2881.1 6
252.151 odd 6 3024.2.t.h.289.3 6
252.191 even 6 1008.2.q.g.529.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.e.c.25.3 ✓ 6 63.2 odd 6
126.2.e.c.121.3 yes 6 21.11 odd 6
126.2.h.d.67.1 yes 6 21.2 odd 6
126.2.h.d.79.1 yes 6 63.11 odd 6
378.2.e.d.37.1 6 7.4 even 3
378.2.e.d.235.1 6 63.16 even 3
378.2.h.c.289.3 6 63.25 even 3
378.2.h.c.361.3 6 7.2 even 3
882.2.e.o.373.1 6 21.17 even 6
882.2.e.o.655.1 6 63.47 even 6
882.2.f.n.295.2 6 9.2 odd 6
882.2.f.n.589.2 6 3.2 odd 2
882.2.f.o.295.2 6 63.20 even 6
882.2.f.o.589.2 6 21.20 even 2
882.2.h.p.67.3 6 21.5 even 6
882.2.h.p.79.3 6 63.38 even 6
1008.2.q.g.529.1 6 252.191 even 6
1008.2.q.g.625.1 6 84.11 even 6
1008.2.t.h.193.3 6 84.23 even 6
1008.2.t.h.961.3 6 252.11 even 6
1134.2.g.l.163.1 6 63.4 even 3
1134.2.g.l.487.1 6 63.58 even 3
1134.2.g.m.163.3 6 63.32 odd 6
1134.2.g.m.487.3 6 63.23 odd 6
2646.2.e.p.1549.3 6 7.3 odd 6
2646.2.e.p.2125.3 6 63.61 odd 6
2646.2.f.l.883.1 6 9.7 even 3 inner
2646.2.f.l.1765.1 6 1.1 even 1 trivial
2646.2.f.m.883.3 6 63.34 odd 6
2646.2.f.m.1765.3 6 7.6 odd 2
2646.2.h.o.361.1 6 7.5 odd 6
2646.2.h.o.667.1 6 63.52 odd 6
3024.2.q.g.2305.1 6 28.11 odd 6
3024.2.q.g.2881.1 6 252.79 odd 6
3024.2.t.h.289.3 6 252.151 odd 6
3024.2.t.h.1873.3 6 28.23 odd 6
7938.2.a.bv.1.1 3 9.5 odd 6
7938.2.a.bw.1.3 3 63.41 even 6
7938.2.a.bz.1.1 3 63.13 odd 6
7938.2.a.ca.1.3 3 9.4 even 3