Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [378,2,Mod(289,378)] 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("378.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(378, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.01834519640\)
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 361.3
Root \(0.500000 + 1.41036i\) of defining polynomial
Character \(\chi\) \(=\) 378.361
Dual form 378.2.h.c.289.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +3.18194 q^{5} +(0.710533 - 2.54856i) q^{7} +1.00000 q^{8} +(-1.59097 + 2.75564i) q^{10} -3.18194 q^{11} +(2.85185 - 4.93955i) q^{13} +(1.85185 + 1.88962i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(0.760877 - 1.31788i) q^{17} +(-0.641315 - 1.11079i) q^{19} +(-1.59097 - 2.75564i) q^{20} +(1.59097 - 2.75564i) q^{22} -2.23912 q^{23} +5.12476 q^{25} +(2.85185 + 4.93955i) q^{26} +(-2.56238 + 0.658939i) q^{28} +(3.54063 + 6.13255i) q^{29} +(4.71053 + 8.15888i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(0.760877 + 1.31788i) q^{34} +(2.26088 - 8.10936i) q^{35} +(0.500000 + 0.866025i) q^{37} +1.28263 q^{38} +3.18194 q^{40} +(2.80150 - 4.85235i) q^{41} +(3.41423 + 5.91362i) q^{43} +(1.59097 + 2.75564i) q^{44} +(1.11956 - 1.93914i) q^{46} +(-2.91423 + 5.04759i) q^{47} +(-5.99028 - 3.62167i) q^{49} +(-2.56238 + 4.43818i) q^{50} -5.70370 q^{52} +(-1.02859 + 1.78157i) q^{53} -10.1248 q^{55} +(0.710533 - 2.54856i) q^{56} -7.08126 q^{58} +(-0.562382 - 0.974074i) q^{59} +(-1.56238 + 2.70612i) q^{61} -9.42107 q^{62} +1.00000 q^{64} +(9.07442 - 15.7174i) q^{65} +(-5.48345 - 9.49761i) q^{67} -1.52175 q^{68} +(5.89248 + 6.01266i) q^{70} -8.69002 q^{71} +(-2.48345 + 4.30146i) q^{73} -1.00000 q^{74} +(-0.641315 + 1.11079i) q^{76} +(-2.26088 + 8.10936i) q^{77} +(2.06922 - 3.58399i) q^{79} +(-1.59097 + 2.75564i) q^{80} +(2.80150 + 4.85235i) q^{82} +(4.03379 + 6.98673i) q^{83} +(2.42107 - 4.19341i) q^{85} -6.82846 q^{86} -3.18194 q^{88} +(-0.112725 - 0.195246i) q^{89} +(-10.5624 - 10.7778i) q^{91} +(1.11956 + 1.93914i) q^{92} +(-2.91423 - 5.04759i) q^{94} +(-2.04063 - 3.53447i) q^{95} +(7.42107 + 12.8537i) q^{97} +(6.13160 - 3.37690i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - 3 q^{4} + 2 q^{5} - 4 q^{7} + 6 q^{8} - q^{10} - 2 q^{11} + 8 q^{13} + 2 q^{14} - 3 q^{16} + 4 q^{17} - 3 q^{19} - q^{20} + q^{22} - 14 q^{23} - 4 q^{25} + 8 q^{26} + 2 q^{28} + 5 q^{29}+ \cdots + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

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 \(-0.248014\pi\)
0.711504 + 0.702682i \(0.248014\pi\)
\(6\) 0 0
\(7\) 0.710533 − 2.54856i 0.268556 − 0.963264i
\(8\) 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 \(-0.542915\pi\)
0.925373 − 0.379058i \(-0.123752\pi\)
\(14\) 1.85185 + 1.88962i 0.494927 + 0.505022i
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 0.760877 − 1.31788i 0.184540 − 0.319632i −0.758882 − 0.651229i \(-0.774254\pi\)
0.943421 + 0.331596i \(0.107587\pi\)
\(18\) 0 0
\(19\) −0.641315 − 1.11079i −0.147128 − 0.254833i 0.783037 − 0.621975i \(-0.213670\pi\)
−0.930165 + 0.367142i \(0.880336\pi\)
\(20\) −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
\(28\) −2.56238 + 0.658939i −0.484245 + 0.124528i
\(29\) 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 \(0.154351\pi\)
−0.0386810 + 0.999252i \(0.512316\pi\)
\(32\) −0.500000 − 0.866025i −0.0883883 − 0.153093i
\(33\) 0 0
\(34\) 0.760877 + 1.31788i 0.130489 + 0.226014i
\(35\) 2.26088 − 8.10936i 0.382158 − 1.37073i
\(36\) 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 −0.559976 − 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 \(0.659017\pi\)
\(44\) 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 −0.996428 − 0.0844432i \(-0.973089\pi\)
0.571344 + 0.820711i \(0.306422\pi\)
\(48\) 0 0
\(49\) −5.99028 − 3.62167i −0.855755 − 0.517381i
\(50\) −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
\(56\) 0.710533 − 2.54856i 0.0949490 − 0.340565i
\(57\) 0 0
\(58\) −7.08126 −0.929815
\(59\) −0.562382 − 0.974074i −0.0732159 − 0.126814i 0.827093 − 0.562065i \(-0.189993\pi\)
−0.900309 + 0.435251i \(0.856660\pi\)
\(60\) 0 0
\(61\) −1.56238 + 2.70612i −0.200042 + 0.346484i −0.948542 − 0.316652i \(-0.897441\pi\)
0.748499 + 0.663135i \(0.230775\pi\)
\(62\) −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
\(70\) 5.89248 + 6.01266i 0.704286 + 0.718650i
\(71\) −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 −0.973967 − 0.226689i \(-0.927210\pi\)
0.683302 + 0.730136i \(0.260543\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −0.641315 + 1.11079i −0.0735639 + 0.127416i
\(77\) −2.26088 + 8.10936i −0.257651 + 0.924148i
\(78\) 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 0.997894 − 0.0648718i \(-0.0206639\pi\)
−0.555127 + 0.831765i \(0.687331\pi\)
\(84\) 0 0
\(85\) 2.42107 − 4.19341i 0.262602 − 0.454839i
\(86\) −6.82846 −0.736332
\(87\) 0 0
\(88\) −3.18194 −0.339196
\(89\) −0.112725 − 0.195246i −0.0119488 − 0.0206960i 0.859989 − 0.510312i \(-0.170470\pi\)
−0.871938 + 0.489616i \(0.837137\pi\)
\(90\) 0 0
\(91\) −10.5624 − 10.7778i −1.10724 − 1.12982i
\(92\) 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 \(0.561700\pi\)
\(98\) 6.13160 − 3.37690i 0.619385 − 0.341119i
\(99\) 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 378.2.h.c.361.3 6
3.2 odd 2 126.2.h.d.67.1 yes 6
4.3 odd 2 3024.2.t.h.1873.3 6
7.2 even 3 378.2.e.d.37.1 6
7.3 odd 6 2646.2.f.m.1765.3 6
7.4 even 3 2646.2.f.l.1765.1 6
7.5 odd 6 2646.2.e.p.1549.3 6
7.6 odd 2 2646.2.h.o.361.1 6
9.2 odd 6 126.2.e.c.25.3 ✓ 6
9.4 even 3 1134.2.g.l.487.1 6
9.5 odd 6 1134.2.g.m.487.3 6
9.7 even 3 378.2.e.d.235.1 6
12.11 even 2 1008.2.t.h.193.3 6
21.2 odd 6 126.2.e.c.121.3 yes 6
21.5 even 6 882.2.e.o.373.1 6
21.11 odd 6 882.2.f.n.589.2 6
21.17 even 6 882.2.f.o.589.2 6
21.20 even 2 882.2.h.p.67.3 6
28.23 odd 6 3024.2.q.g.2305.1 6
36.7 odd 6 3024.2.q.g.2881.1 6
36.11 even 6 1008.2.q.g.529.1 6
63.2 odd 6 126.2.h.d.79.1 yes 6
63.4 even 3 7938.2.a.ca.1.3 3
63.11 odd 6 882.2.f.n.295.2 6
63.16 even 3 inner 378.2.h.c.289.3 6
63.20 even 6 882.2.e.o.655.1 6
63.23 odd 6 1134.2.g.m.163.3 6
63.25 even 3 2646.2.f.l.883.1 6
63.31 odd 6 7938.2.a.bz.1.1 3
63.32 odd 6 7938.2.a.bv.1.1 3
63.34 odd 6 2646.2.e.p.2125.3 6
63.38 even 6 882.2.f.o.295.2 6
63.47 even 6 882.2.h.p.79.3 6
63.52 odd 6 2646.2.f.m.883.3 6
63.58 even 3 1134.2.g.l.163.1 6
63.59 even 6 7938.2.a.bw.1.3 3
63.61 odd 6 2646.2.h.o.667.1 6
84.23 even 6 1008.2.q.g.625.1 6
252.79 odd 6 3024.2.t.h.289.3 6
252.191 even 6 1008.2.t.h.961.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.e.c.25.3 ✓ 6 9.2 odd 6
126.2.e.c.121.3 yes 6 21.2 odd 6
126.2.h.d.67.1 yes 6 3.2 odd 2
126.2.h.d.79.1 yes 6 63.2 odd 6
378.2.e.d.37.1 6 7.2 even 3
378.2.e.d.235.1 6 9.7 even 3
378.2.h.c.289.3 6 63.16 even 3 inner
378.2.h.c.361.3 6 1.1 even 1 trivial
882.2.e.o.373.1 6 21.5 even 6
882.2.e.o.655.1 6 63.20 even 6
882.2.f.n.295.2 6 63.11 odd 6
882.2.f.n.589.2 6 21.11 odd 6
882.2.f.o.295.2 6 63.38 even 6
882.2.f.o.589.2 6 21.17 even 6
882.2.h.p.67.3 6 21.20 even 2
882.2.h.p.79.3 6 63.47 even 6
1008.2.q.g.529.1 6 36.11 even 6
1008.2.q.g.625.1 6 84.23 even 6
1008.2.t.h.193.3 6 12.11 even 2
1008.2.t.h.961.3 6 252.191 even 6
1134.2.g.l.163.1 6 63.58 even 3
1134.2.g.l.487.1 6 9.4 even 3
1134.2.g.m.163.3 6 63.23 odd 6
1134.2.g.m.487.3 6 9.5 odd 6
2646.2.e.p.1549.3 6 7.5 odd 6
2646.2.e.p.2125.3 6 63.34 odd 6
2646.2.f.l.883.1 6 63.25 even 3
2646.2.f.l.1765.1 6 7.4 even 3
2646.2.f.m.883.3 6 63.52 odd 6
2646.2.f.m.1765.3 6 7.3 odd 6
2646.2.h.o.361.1 6 7.6 odd 2
2646.2.h.o.667.1 6 63.61 odd 6
3024.2.q.g.2305.1 6 28.23 odd 6
3024.2.q.g.2881.1 6 36.7 odd 6
3024.2.t.h.289.3 6 252.79 odd 6
3024.2.t.h.1873.3 6 4.3 odd 2
7938.2.a.bv.1.1 3 63.32 odd 6
7938.2.a.bw.1.3 3 63.59 even 6
7938.2.a.bz.1.1 3 63.31 odd 6
7938.2.a.ca.1.3 3 63.4 even 3