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Properties

Label 1638.2.bj.g.1135.4
Level $1638$
Weight $2$
Character 1638.1135
Analytic conductor $13.079$
Analytic rank $0$
Dimension $12$
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] = [1638,2,Mod(127,1638)] 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("1638.127"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1638, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.bj (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,6,0,0,0,0,0,-2,18,0,-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: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 39 x^{10} - 140 x^{9} + 460 x^{8} - 1066 x^{7} + 2127 x^{6} - 3172 x^{5} + \cdots + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1135.4
Root \(0.500000 - 2.47866i\) of defining polynomial
Character \(\chi\) \(=\) 1638.1135
Dual form 1638.2.bj.g.127.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} -1.56356i q^{5} +(-0.866025 + 0.500000i) q^{7} +1.00000i q^{8} +(0.781779 - 1.35408i) q^{10} +(2.48215 + 1.43307i) q^{11} +(2.99598 + 2.00602i) q^{13} -1.00000 q^{14} +(-0.500000 + 0.866025i) q^{16} +(1.11481 + 1.93090i) q^{17} +(-6.26657 + 3.61801i) q^{19} +(1.35408 - 0.781779i) q^{20} +(1.43307 + 2.48215i) q^{22} +(0.833676 - 1.44397i) q^{23} +2.55529 q^{25} +(1.59158 + 3.23525i) q^{26} +(-0.866025 - 0.500000i) q^{28} +(2.41379 - 4.18080i) q^{29} -0.597963i q^{31} +(-0.866025 + 0.500000i) q^{32} +2.22961i q^{34} +(0.781779 + 1.35408i) q^{35} +(0.0333971 + 0.0192818i) q^{37} -7.23602 q^{38} +1.56356 q^{40} +(6.88896 + 3.97734i) q^{41} +(5.04571 + 8.73942i) q^{43} +2.86614i q^{44} +(1.44397 - 0.833676i) q^{46} -7.02636i q^{47} +(0.500000 - 0.866025i) q^{49} +(2.21294 + 1.27764i) q^{50} +(-0.239275 + 3.59760i) q^{52} +5.98404 q^{53} +(2.24069 - 3.88098i) q^{55} +(-0.500000 - 0.866025i) q^{56} +(4.18080 - 2.41379i) q^{58} +(-0.776138 + 0.448103i) q^{59} +(7.12846 + 12.3469i) q^{61} +(0.298982 - 0.517851i) q^{62} -1.00000 q^{64} +(3.13653 - 4.68438i) q^{65} +(-1.42103 - 0.820432i) q^{67} +(-1.11481 + 1.93090i) q^{68} +1.56356i q^{70} +(1.98724 - 1.14733i) q^{71} +11.2277i q^{73} +(0.0192818 + 0.0333971i) q^{74} +(-6.26657 - 3.61801i) q^{76} -2.86614 q^{77} +4.26098 q^{79} +(1.35408 + 0.781779i) q^{80} +(3.97734 + 6.88896i) q^{82} +4.94829i q^{83} +(3.01907 - 1.74306i) q^{85} +10.0914i q^{86} +(-1.43307 + 2.48215i) q^{88} +(-2.09682 - 1.21060i) q^{89} +(-3.59760 - 0.239275i) q^{91} +1.66735 q^{92} +(3.51318 - 6.08501i) q^{94} +(5.65696 + 9.79815i) q^{95} +(-4.23338 + 2.44414i) q^{97} +(0.866025 - 0.500000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{4} - 2 q^{10} + 18 q^{11} - 8 q^{13} - 12 q^{14} - 6 q^{16} - 4 q^{17} + 12 q^{19} - 2 q^{22} + 6 q^{23} - 24 q^{25} + 14 q^{26} + 10 q^{29} - 2 q^{35} - 6 q^{37} - 8 q^{38} - 4 q^{40} + 24 q^{41}+ \cdots + 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 + 0.500000i 0.612372 + 0.353553i
\(3\) 0 0
\(4\) 0.500000 + 0.866025i 0.250000 + 0.433013i
\(5\) − 1.56356i − 0.699244i −0.936891 − 0.349622i \(-0.886310\pi\)
0.936891 − 0.349622i \(-0.113690\pi\)
\(6\) 0 0
\(7\) −0.866025 + 0.500000i −0.327327 + 0.188982i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0.781779 − 1.35408i 0.247220 − 0.428198i
\(11\) 2.48215 + 1.43307i 0.748396 + 0.432087i 0.825114 − 0.564966i \(-0.191111\pi\)
−0.0767180 + 0.997053i \(0.524444\pi\)
\(12\) 0 0
\(13\) 2.99598 + 2.00602i 0.830935 + 0.556370i
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 1.11481 + 1.93090i 0.270380 + 0.468312i 0.968959 − 0.247221i \(-0.0795173\pi\)
−0.698579 + 0.715533i \(0.746184\pi\)
\(18\) 0 0
\(19\) −6.26657 + 3.61801i −1.43765 + 0.830028i −0.997686 − 0.0679872i \(-0.978342\pi\)
−0.439964 + 0.898015i \(0.645009\pi\)
\(20\) 1.35408 − 0.781779i 0.302782 − 0.174811i
\(21\) 0 0
\(22\) 1.43307 + 2.48215i 0.305531 + 0.529196i
\(23\) 0.833676 − 1.44397i 0.173833 − 0.301088i −0.765924 − 0.642932i \(-0.777718\pi\)
0.939757 + 0.341843i \(0.111051\pi\)
\(24\) 0 0
\(25\) 2.55529 0.511058
\(26\) 1.59158 + 3.23525i 0.312135 + 0.634485i
\(27\) 0 0
\(28\) −0.866025 − 0.500000i −0.163663 − 0.0944911i
\(29\) 2.41379 − 4.18080i 0.448229 − 0.776356i −0.550042 − 0.835137i \(-0.685388\pi\)
0.998271 + 0.0587816i \(0.0187216\pi\)
\(30\) 0 0
\(31\) − 0.597963i − 0.107397i −0.998557 − 0.0536987i \(-0.982899\pi\)
0.998557 − 0.0536987i \(-0.0171010\pi\)
\(32\) −0.866025 + 0.500000i −0.153093 + 0.0883883i
\(33\) 0 0
\(34\) 2.22961i 0.382375i
\(35\) 0.781779 + 1.35408i 0.132145 + 0.228881i
\(36\) 0 0
\(37\) 0.0333971 + 0.0192818i 0.00549045 + 0.00316991i 0.502743 − 0.864436i \(-0.332324\pi\)
−0.497252 + 0.867606i \(0.665658\pi\)
\(38\) −7.23602 −1.17384
\(39\) 0 0
\(40\) 1.56356 0.247220
\(41\) 6.88896 + 3.97734i 1.07588 + 0.621157i 0.929781 − 0.368114i \(-0.119996\pi\)
0.146095 + 0.989271i \(0.453330\pi\)
\(42\) 0 0
\(43\) 5.04571 + 8.73942i 0.769463 + 1.33275i 0.937854 + 0.347029i \(0.112809\pi\)
−0.168391 + 0.985720i \(0.553857\pi\)
\(44\) 2.86614i 0.432087i
\(45\) 0 0
\(46\) 1.44397 − 0.833676i 0.212902 − 0.122919i
\(47\) − 7.02636i − 1.02490i −0.858717 − 0.512450i \(-0.828738\pi\)
0.858717 − 0.512450i \(-0.171262\pi\)
\(48\) 0 0
\(49\) 0.500000 − 0.866025i 0.0714286 − 0.123718i
\(50\) 2.21294 + 1.27764i 0.312958 + 0.180686i
\(51\) 0 0
\(52\) −0.239275 + 3.59760i −0.0331814 + 0.498898i
\(53\) 5.98404 0.821971 0.410985 − 0.911642i \(-0.365185\pi\)
0.410985 + 0.911642i \(0.365185\pi\)
\(54\) 0 0
\(55\) 2.24069 − 3.88098i 0.302134 − 0.523312i
\(56\) −0.500000 − 0.866025i −0.0668153 − 0.115728i
\(57\) 0 0
\(58\) 4.18080 − 2.41379i 0.548966 − 0.316946i
\(59\) −0.776138 + 0.448103i −0.101044 + 0.0583381i −0.549671 − 0.835381i \(-0.685247\pi\)
0.448626 + 0.893719i \(0.351913\pi\)
\(60\) 0 0
\(61\) 7.12846 + 12.3469i 0.912706 + 1.58085i 0.810225 + 0.586119i \(0.199345\pi\)
0.102481 + 0.994735i \(0.467322\pi\)
\(62\) 0.298982 − 0.517851i 0.0379707 − 0.0657672i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 3.13653 − 4.68438i 0.389038 − 0.581026i
\(66\) 0 0
\(67\) −1.42103 − 0.820432i −0.173606 − 0.100232i 0.410679 − 0.911780i \(-0.365292\pi\)
−0.584285 + 0.811548i \(0.698625\pi\)
\(68\) −1.11481 + 1.93090i −0.135190 + 0.234156i
\(69\) 0 0
\(70\) 1.56356i 0.186881i
\(71\) 1.98724 − 1.14733i 0.235841 − 0.136163i −0.377422 − 0.926041i \(-0.623190\pi\)
0.613264 + 0.789878i \(0.289856\pi\)
\(72\) 0 0
\(73\) 11.2277i 1.31411i 0.753844 + 0.657054i \(0.228198\pi\)
−0.753844 + 0.657054i \(0.771802\pi\)
\(74\) 0.0192818 + 0.0333971i 0.00224147 + 0.00388233i
\(75\) 0 0
\(76\) −6.26657 − 3.61801i −0.718825 − 0.415014i
\(77\) −2.86614 −0.326627
\(78\) 0 0
\(79\) 4.26098 0.479397 0.239699 − 0.970847i \(-0.422951\pi\)
0.239699 + 0.970847i \(0.422951\pi\)
\(80\) 1.35408 + 0.781779i 0.151391 + 0.0874055i
\(81\) 0 0
\(82\) 3.97734 + 6.88896i 0.439224 + 0.760759i
\(83\) 4.94829i 0.543145i 0.962418 + 0.271572i \(0.0875437\pi\)
−0.962418 + 0.271572i \(0.912456\pi\)
\(84\) 0 0
\(85\) 3.01907 − 1.74306i 0.327465 − 0.189062i
\(86\) 10.0914i 1.08818i
\(87\) 0 0
\(88\) −1.43307 + 2.48215i −0.152766 + 0.264598i
\(89\) −2.09682 − 1.21060i −0.222263 − 0.128323i 0.384735 − 0.923027i \(-0.374293\pi\)
−0.606997 + 0.794704i \(0.707626\pi\)
\(90\) 0 0
\(91\) −3.59760 − 0.239275i −0.377131 − 0.0250828i
\(92\) 1.66735 0.173833
\(93\) 0 0
\(94\) 3.51318 − 6.08501i 0.362357 − 0.627621i
\(95\) 5.65696 + 9.79815i 0.580392 + 1.00527i
\(96\) 0 0
\(97\) −4.23338 + 2.44414i −0.429835 + 0.248165i −0.699276 − 0.714852i \(-0.746494\pi\)
0.269442 + 0.963017i \(0.413161\pi\)
\(98\) 0.866025 − 0.500000i 0.0874818 − 0.0505076i
\(99\) 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 1638.2.bj.g.1135.4 12
3.2 odd 2 182.2.m.b.43.1 ✓ 12
12.11 even 2 1456.2.cc.d.225.6 12
13.10 even 6 inner 1638.2.bj.g.127.6 12
21.2 odd 6 1274.2.o.d.459.4 12
21.5 even 6 1274.2.o.e.459.6 12
21.11 odd 6 1274.2.v.e.667.6 12
21.17 even 6 1274.2.v.d.667.4 12
21.20 even 2 1274.2.m.c.589.3 12
39.17 odd 6 2366.2.d.r.337.6 12
39.20 even 12 2366.2.a.bf.1.6 6
39.23 odd 6 182.2.m.b.127.1 yes 12
39.32 even 12 2366.2.a.bh.1.6 6
39.35 odd 6 2366.2.d.r.337.12 12
156.23 even 6 1456.2.cc.d.673.6 12
273.23 odd 6 1274.2.v.e.361.6 12
273.62 even 6 1274.2.m.c.491.3 12
273.101 even 6 1274.2.o.e.569.3 12
273.179 odd 6 1274.2.o.d.569.1 12
273.257 even 6 1274.2.v.d.361.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
182.2.m.b.43.1 ✓ 12 3.2 odd 2
182.2.m.b.127.1 yes 12 39.23 odd 6
1274.2.m.c.491.3 12 273.62 even 6
1274.2.m.c.589.3 12 21.20 even 2
1274.2.o.d.459.4 12 21.2 odd 6
1274.2.o.d.569.1 12 273.179 odd 6
1274.2.o.e.459.6 12 21.5 even 6
1274.2.o.e.569.3 12 273.101 even 6
1274.2.v.d.361.4 12 273.257 even 6
1274.2.v.d.667.4 12 21.17 even 6
1274.2.v.e.361.6 12 273.23 odd 6
1274.2.v.e.667.6 12 21.11 odd 6
1456.2.cc.d.225.6 12 12.11 even 2
1456.2.cc.d.673.6 12 156.23 even 6
1638.2.bj.g.127.6 12 13.10 even 6 inner
1638.2.bj.g.1135.4 12 1.1 even 1 trivial
2366.2.a.bf.1.6 6 39.20 even 12
2366.2.a.bh.1.6 6 39.32 even 12
2366.2.d.r.337.6 12 39.17 odd 6
2366.2.d.r.337.12 12 39.35 odd 6