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Properties

Label 342.2.u.b.55.1
Level $342$
Weight $2$
Character 342.55
Analytic conductor $2.731$
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] = [342,2,Mod(55,342)] 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("342.55"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(342, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.u (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,-3,0,-3,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 55.1
Root \(-0.173648 + 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 342.55
Dual form 342.2.u.b.199.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.173648 + 0.984808i) q^{2} +(-0.939693 - 0.342020i) q^{4} +(-2.20574 + 0.802823i) q^{5} +(-1.78699 - 3.09516i) q^{7} +(0.500000 - 0.866025i) q^{8} +(-0.407604 - 2.31164i) q^{10} +(1.35844 - 2.35289i) q^{11} +(4.14543 - 3.47843i) q^{13} +(3.35844 - 1.22237i) q^{14} +(0.766044 + 0.642788i) q^{16} +(0.673648 - 3.82045i) q^{17} +(1.01114 + 4.24000i) q^{19} +2.34730 q^{20} +(2.08125 + 1.74638i) q^{22} +(-7.73783 - 2.81634i) q^{23} +(0.390530 - 0.327693i) q^{25} +(2.70574 + 4.68647i) q^{26} +(0.620615 + 3.51968i) q^{28} +(-0.613341 - 3.47843i) q^{29} +(-3.26604 - 5.65695i) q^{31} +(-0.766044 + 0.642788i) q^{32} +(3.64543 + 1.32683i) q^{34} +(6.42649 + 5.39246i) q^{35} +0.389185 q^{37} +(-4.35117 + 0.259515i) q^{38} +(-0.407604 + 2.31164i) q^{40} +(1.48886 + 1.24930i) q^{41} +(-4.71941 + 1.71772i) q^{43} +(-2.08125 + 1.74638i) q^{44} +(4.11721 - 7.13122i) q^{46} +(-0.518418 - 2.94010i) q^{47} +(-2.88666 + 4.99984i) q^{49} +(0.254900 + 0.441500i) q^{50} +(-5.08512 + 1.85083i) q^{52} +(7.80453 + 2.84062i) q^{53} +(-1.10741 + 6.28044i) q^{55} -3.57398 q^{56} +3.53209 q^{58} +(-0.474308 + 2.68993i) q^{59} +(-5.91147 - 2.15160i) q^{61} +(6.13816 - 2.23411i) q^{62} +(-0.500000 - 0.866025i) q^{64} +(-6.35117 + 11.0005i) q^{65} +(-2.59374 - 14.7098i) q^{67} +(-1.93969 + 3.35965i) q^{68} +(-6.42649 + 5.39246i) q^{70} +(8.47818 - 3.08580i) q^{71} +(7.88326 + 6.61484i) q^{73} +(-0.0675813 + 0.383273i) q^{74} +(0.500000 - 4.33013i) q^{76} -9.71007 q^{77} +(9.96451 + 8.36121i) q^{79} +(-2.20574 - 0.802823i) q^{80} +(-1.48886 + 1.24930i) q^{82} +(-4.08512 - 7.07564i) q^{83} +(1.58125 + 8.96773i) q^{85} +(-0.872111 - 4.94599i) q^{86} +(-1.35844 - 2.35289i) q^{88} +(-8.98158 + 7.53644i) q^{89} +(-18.1741 - 6.61484i) q^{91} +(6.30793 + 5.29298i) q^{92} +2.98545 q^{94} +(-5.63429 - 8.54055i) q^{95} +(-1.49407 + 8.47329i) q^{97} +(-4.42262 - 3.71102i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{5} - 3 q^{7} + 3 q^{8} - 6 q^{10} + 9 q^{13} + 12 q^{14} + 3 q^{17} + 12 q^{20} + 15 q^{22} - 27 q^{23} - 15 q^{25} + 6 q^{26} + 15 q^{28} + 3 q^{29} - 15 q^{31} + 6 q^{34} - 12 q^{35} - 6 q^{37}+ \cdots + 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.173648 + 0.984808i −0.122788 + 0.696364i
\(3\) 0 0
\(4\) −0.939693 − 0.342020i −0.469846 − 0.171010i
\(5\) −2.20574 + 0.802823i −0.986436 + 0.359033i −0.784339 − 0.620332i \(-0.786998\pi\)
−0.202097 + 0.979366i \(0.564775\pi\)
\(6\) 0 0
\(7\) −1.78699 − 3.09516i −0.675418 − 1.16986i −0.976346 − 0.216212i \(-0.930630\pi\)
0.300928 − 0.953647i \(-0.402704\pi\)
\(8\) 0.500000 − 0.866025i 0.176777 − 0.306186i
\(9\) 0 0
\(10\) −0.407604 − 2.31164i −0.128896 − 0.731003i
\(11\) 1.35844 − 2.35289i 0.409585 − 0.709423i −0.585258 − 0.810847i \(-0.699007\pi\)
0.994843 + 0.101425i \(0.0323401\pi\)
\(12\) 0 0
\(13\) 4.14543 − 3.47843i 1.14974 − 0.964742i 0.150022 − 0.988683i \(-0.452066\pi\)
0.999713 + 0.0239402i \(0.00762112\pi\)
\(14\) 3.35844 − 1.22237i 0.897581 − 0.326693i
\(15\) 0 0
\(16\) 0.766044 + 0.642788i 0.191511 + 0.160697i
\(17\) 0.673648 − 3.82045i 0.163384 − 0.926595i −0.787332 − 0.616530i \(-0.788538\pi\)
0.950715 − 0.310065i \(-0.100351\pi\)
\(18\) 0 0
\(19\) 1.01114 + 4.24000i 0.231972 + 0.972722i
\(20\) 2.34730 0.524871
\(21\) 0 0
\(22\) 2.08125 + 1.74638i 0.443724 + 0.372329i
\(23\) −7.73783 − 2.81634i −1.61345 − 0.587247i −0.631330 − 0.775514i \(-0.717491\pi\)
−0.982118 + 0.188267i \(0.939713\pi\)
\(24\) 0 0
\(25\) 0.390530 − 0.327693i 0.0781059 − 0.0655386i
\(26\) 2.70574 + 4.68647i 0.530639 + 0.919093i
\(27\) 0 0
\(28\) 0.620615 + 3.51968i 0.117285 + 0.665157i
\(29\) −0.613341 − 3.47843i −0.113895 − 0.645928i −0.987292 − 0.158919i \(-0.949199\pi\)
0.873397 − 0.487009i \(-0.161912\pi\)
\(30\) 0 0
\(31\) −3.26604 − 5.65695i −0.586599 − 1.01602i −0.994674 − 0.103071i \(-0.967133\pi\)
0.408075 − 0.912948i \(-0.366200\pi\)
\(32\) −0.766044 + 0.642788i −0.135419 + 0.113630i
\(33\) 0 0
\(34\) 3.64543 + 1.32683i 0.625186 + 0.227549i
\(35\) 6.42649 + 5.39246i 1.08627 + 0.911493i
\(36\) 0 0
\(37\) 0.389185 0.0639817 0.0319908 − 0.999488i \(-0.489815\pi\)
0.0319908 + 0.999488i \(0.489815\pi\)
\(38\) −4.35117 + 0.259515i −0.705852 + 0.0420989i
\(39\) 0 0
\(40\) −0.407604 + 2.31164i −0.0644478 + 0.365502i
\(41\) 1.48886 + 1.24930i 0.232520 + 0.195108i 0.751602 − 0.659617i \(-0.229282\pi\)
−0.519082 + 0.854725i \(0.673726\pi\)
\(42\) 0 0
\(43\) −4.71941 + 1.71772i −0.719703 + 0.261950i −0.675800 − 0.737085i \(-0.736202\pi\)
−0.0439033 + 0.999036i \(0.513979\pi\)
\(44\) −2.08125 + 1.74638i −0.313761 + 0.263276i
\(45\) 0 0
\(46\) 4.11721 − 7.13122i 0.607050 − 1.05144i
\(47\) −0.518418 − 2.94010i −0.0756191 − 0.428857i −0.998989 − 0.0449466i \(-0.985688\pi\)
0.923370 − 0.383911i \(-0.125423\pi\)
\(48\) 0 0
\(49\) −2.88666 + 4.99984i −0.412380 + 0.714263i
\(50\) 0.254900 + 0.441500i 0.0360483 + 0.0624375i
\(51\) 0 0
\(52\) −5.08512 + 1.85083i −0.705180 + 0.256664i
\(53\) 7.80453 + 2.84062i 1.07203 + 0.390189i 0.816937 − 0.576727i \(-0.195670\pi\)
0.255097 + 0.966915i \(0.417892\pi\)
\(54\) 0 0
\(55\) −1.10741 + 6.28044i −0.149323 + 0.846854i
\(56\) −3.57398 −0.477593
\(57\) 0 0
\(58\) 3.53209 0.463786
\(59\) −0.474308 + 2.68993i −0.0617496 + 0.350199i 0.938241 + 0.345981i \(0.112454\pi\)
−0.999991 + 0.00421836i \(0.998657\pi\)
\(60\) 0 0
\(61\) −5.91147 − 2.15160i −0.756887 − 0.275484i −0.0653860 − 0.997860i \(-0.520828\pi\)
−0.691501 + 0.722376i \(0.743050\pi\)
\(62\) 6.13816 − 2.23411i 0.779547 − 0.283732i
\(63\) 0 0
\(64\) −0.500000 − 0.866025i −0.0625000 − 0.108253i
\(65\) −6.35117 + 11.0005i −0.787765 + 1.36445i
\(66\) 0 0
\(67\) −2.59374 − 14.7098i −0.316876 − 1.79709i −0.561503 − 0.827475i \(-0.689777\pi\)
0.244627 − 0.969617i \(-0.421335\pi\)
\(68\) −1.93969 + 3.35965i −0.235222 + 0.407417i
\(69\) 0 0
\(70\) −6.42649 + 5.39246i −0.768112 + 0.644523i
\(71\) 8.47818 − 3.08580i 1.00617 − 0.366218i 0.214212 − 0.976787i \(-0.431282\pi\)
0.791963 + 0.610570i \(0.209059\pi\)
\(72\) 0 0
\(73\) 7.88326 + 6.61484i 0.922665 + 0.774208i 0.974486 − 0.224448i \(-0.0720580\pi\)
−0.0518207 + 0.998656i \(0.516502\pi\)
\(74\) −0.0675813 + 0.383273i −0.00785617 + 0.0445546i
\(75\) 0 0
\(76\) 0.500000 − 4.33013i 0.0573539 − 0.496700i
\(77\) −9.71007 −1.10657
\(78\) 0 0
\(79\) 9.96451 + 8.36121i 1.12109 + 0.940710i 0.998659 − 0.0517663i \(-0.0164851\pi\)
0.122435 + 0.992476i \(0.460930\pi\)
\(80\) −2.20574 − 0.802823i −0.246609 − 0.0897583i
\(81\) 0 0
\(82\) −1.48886 + 1.24930i −0.164417 + 0.137962i
\(83\) −4.08512 − 7.07564i −0.448400 − 0.776652i 0.549882 − 0.835243i \(-0.314673\pi\)
−0.998282 + 0.0585902i \(0.981339\pi\)
\(84\) 0 0
\(85\) 1.58125 + 8.96773i 0.171511 + 0.972686i
\(86\) −0.872111 − 4.94599i −0.0940422 − 0.533340i
\(87\) 0 0
\(88\) −1.35844 − 2.35289i −0.144810 − 0.250819i
\(89\) −8.98158 + 7.53644i −0.952046 + 0.798861i −0.979641 − 0.200758i \(-0.935660\pi\)
0.0275951 + 0.999619i \(0.491215\pi\)
\(90\) 0 0
\(91\) −18.1741 − 6.61484i −1.90516 − 0.693423i
\(92\) 6.30793 + 5.29298i 0.657648 + 0.551832i
\(93\) 0 0
\(94\) 2.98545 0.307926
\(95\) −5.63429 − 8.54055i −0.578065 − 0.876242i
\(96\) 0 0
\(97\) −1.49407 + 8.47329i −0.151700 + 0.860333i 0.810041 + 0.586373i \(0.199445\pi\)
−0.961741 + 0.273960i \(0.911666\pi\)
\(98\) −4.42262 − 3.71102i −0.446752 − 0.374869i
\(99\) 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 342.2.u.b.55.1 6
3.2 odd 2 114.2.i.c.55.1 ✓ 6
12.11 even 2 912.2.bo.d.625.1 6
19.3 odd 18 6498.2.a.bu.1.2 3
19.9 even 9 inner 342.2.u.b.199.1 6
19.16 even 9 6498.2.a.bp.1.2 3
57.35 odd 18 2166.2.a.r.1.2 3
57.41 even 18 2166.2.a.p.1.2 3
57.47 odd 18 114.2.i.c.85.1 yes 6
228.47 even 18 912.2.bo.d.769.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.c.55.1 ✓ 6 3.2 odd 2
114.2.i.c.85.1 yes 6 57.47 odd 18
342.2.u.b.55.1 6 1.1 even 1 trivial
342.2.u.b.199.1 6 19.9 even 9 inner
912.2.bo.d.625.1 6 12.11 even 2
912.2.bo.d.769.1 6 228.47 even 18
2166.2.a.p.1.2 3 57.41 even 18
2166.2.a.r.1.2 3 57.35 odd 18
6498.2.a.bp.1.2 3 19.16 even 9
6498.2.a.bu.1.2 3 19.3 odd 18