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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [342,2,Mod(29,342)] 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("342.29"); 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([3, 17])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.x (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 281.1
Character \(\chi\) \(=\) 342.281
Dual form 342.2.x.b.185.1

$q$-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+(-0.766044 + 0.642788i) q^{2} +(-1.62796 + 0.591401i) q^{3} +(0.173648 - 0.984808i) q^{4} +(0.0346602 - 0.0952281i) q^{5} +(0.866942 - 1.49947i) q^{6} +(1.30506 + 2.26042i) q^{7} +(0.500000 + 0.866025i) q^{8} +(2.30049 - 1.92555i) q^{9} +(0.0346602 + 0.0952281i) q^{10} -3.50077i q^{11} +(0.299725 + 1.70592i) q^{12} +(1.48386 + 4.07686i) q^{13} +(-2.45270 - 0.892710i) q^{14} +(-0.000107279 + 0.175525i) q^{15} +(-0.939693 - 0.342020i) q^{16} +(-2.30657 + 6.33725i) q^{17} +(-0.524556 + 2.95378i) q^{18} +(-4.07978 + 1.53473i) q^{19} +(-0.0877627 - 0.0506698i) q^{20} +(-3.46139 - 2.90806i) q^{21} +(2.25025 + 2.68175i) q^{22} +(-3.08818 - 0.544530i) q^{23} +(-1.32615 - 1.11415i) q^{24} +(3.82236 + 3.20734i) q^{25} +(-3.75725 - 2.16925i) q^{26} +(-2.60632 + 4.49523i) q^{27} +(2.45270 - 0.892710i) q^{28} +(0.579160 - 3.28458i) q^{29} +(-0.112743 - 0.134529i) q^{30} +9.69994i q^{31} +(0.939693 - 0.342020i) q^{32} +(2.07036 + 5.69910i) q^{33} +(-2.30657 - 6.33725i) q^{34} +(0.260489 - 0.0459313i) q^{35} +(-1.49682 - 2.59991i) q^{36} +3.15406i q^{37} +(2.13879 - 3.79810i) q^{38} +(-4.82671 - 5.75940i) q^{39} +(0.0998001 - 0.0175974i) q^{40} +(-6.18649 + 5.19108i) q^{41} +(4.52084 + 0.00276307i) q^{42} +(-1.56634 - 8.88314i) q^{43} +(-3.44759 - 0.607902i) q^{44} +(-0.103631 - 0.285811i) q^{45} +(2.71570 - 1.56791i) q^{46} +(-3.56978 - 0.629448i) q^{47} +(1.73205 + 0.00105860i) q^{48} +(0.0936613 - 0.162226i) q^{49} -4.98973 q^{50} +(0.00713918 - 11.6809i) q^{51} +(4.27259 - 0.753373i) q^{52} +(8.61956 + 7.23267i) q^{53} +(-0.892918 - 5.11886i) q^{54} +(-0.333372 - 0.121337i) q^{55} +(-1.30506 + 2.26042i) q^{56} +(5.73407 - 4.91126i) q^{57} +(1.66762 + 2.88841i) q^{58} +(0.963679 + 5.46529i) q^{59} +(0.172840 + 0.0305853i) q^{60} +(11.5303 - 4.19670i) q^{61} +(-6.23500 - 7.43058i) q^{62} +(7.35483 + 2.68712i) q^{63} +(-0.500000 + 0.866025i) q^{64} +0.439662 q^{65} +(-5.24930 - 3.03496i) q^{66} +(4.12214 - 4.91257i) q^{67} +(5.84044 + 3.37198i) q^{68} +(5.34946 - 0.939884i) q^{69} +(-0.170022 + 0.202625i) q^{70} +(1.27070 - 1.06625i) q^{71} +(2.81782 + 1.02951i) q^{72} +(0.643609 + 3.65009i) q^{73} +(-2.02739 - 2.41615i) q^{74} +(-8.11945 - 2.96086i) q^{75} +(0.802963 + 4.28430i) q^{76} +(7.91322 - 4.56870i) q^{77} +(7.39955 + 1.30940i) q^{78} +(-2.75149 + 7.55967i) q^{79} +(-0.0651399 + 0.0776307i) q^{80} +(1.58450 - 8.85942i) q^{81} +(1.40236 - 7.95320i) q^{82} +(5.57871 - 3.22087i) q^{83} +(-3.46494 + 2.90383i) q^{84} +(0.523538 + 0.439301i) q^{85} +(6.90986 + 5.79806i) q^{86} +(0.999657 + 5.68967i) q^{87} +(3.03176 - 1.75038i) q^{88} +(1.00556 - 5.70280i) q^{89} +(0.263102 + 0.152331i) q^{90} +(-7.27891 + 8.67467i) q^{91} +(-1.07251 + 2.94671i) q^{92} +(-5.73656 - 15.7911i) q^{93} +(3.13921 - 1.81242i) q^{94} +(0.00474295 + 0.441704i) q^{95} +(-1.32751 + 1.11253i) q^{96} +(-8.34594 - 9.94630i) q^{97} +(0.0325282 + 0.184477i) q^{98} +(-6.74092 - 8.05348i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 6 q^{3} - 9 q^{5} - 9 q^{6} + 9 q^{7} + 24 q^{8} - 9 q^{10} - 6 q^{12} + 15 q^{13} - 6 q^{14} + 12 q^{15} - 27 q^{17} - 9 q^{18} - 12 q^{19} - 9 q^{20} - 15 q^{21} + 18 q^{22} - 3 q^{24} - 9 q^{25}+ \cdots + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(325\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.766044 + 0.642788i −0.541675 + 0.454519i
\(3\) −1.62796 + 0.591401i −0.939901 + 0.341446i
\(4\) 0.173648 − 0.984808i 0.0868241 − 0.492404i
\(5\) 0.0346602 − 0.0952281i 0.0155005 − 0.0425873i −0.931701 − 0.363226i \(-0.881675\pi\)
0.947202 + 0.320639i \(0.103898\pi\)
\(6\) 0.866942 − 1.49947i 0.353928 − 0.612156i
\(7\) 1.30506 + 2.26042i 0.493265 + 0.854359i 0.999970 − 0.00776006i \(-0.00247013\pi\)
−0.506705 + 0.862119i \(0.669137\pi\)
\(8\) 0.500000 + 0.866025i 0.176777 + 0.306186i
\(9\) 2.30049 − 1.92555i 0.766830 − 0.641851i
\(10\) 0.0346602 + 0.0952281i 0.0109605 + 0.0301138i
\(11\) − 3.50077i − 1.05552i −0.849393 − 0.527761i \(-0.823032\pi\)
0.849393 − 0.527761i \(-0.176968\pi\)
\(12\) 0.299725 + 1.70592i 0.0865231 + 0.492457i
\(13\) 1.48386 + 4.07686i 0.411548 + 1.13072i 0.956368 + 0.292165i \(0.0943756\pi\)
−0.544820 + 0.838553i \(0.683402\pi\)
\(14\) −2.45270 − 0.892710i −0.655512 − 0.238587i
\(15\) −0.000107279 0.175525i −2.76992e−5 0.0453205i
\(16\) −0.939693 − 0.342020i −0.234923 − 0.0855050i
\(17\) −2.30657 + 6.33725i −0.559426 + 1.53701i 0.261049 + 0.965326i \(0.415932\pi\)
−0.820474 + 0.571684i \(0.806290\pi\)
\(18\) −0.524556 + 2.95378i −0.123639 + 0.696214i
\(19\) −4.07978 + 1.53473i −0.935966 + 0.352090i
\(20\) −0.0877627 − 0.0506698i −0.0196243 − 0.0113301i
\(21\) −3.46139 − 2.90806i −0.755337 − 0.634590i
\(22\) 2.25025 + 2.68175i 0.479755 + 0.571750i
\(23\) −3.08818 − 0.544530i −0.643930 − 0.113542i −0.157859 − 0.987462i \(-0.550459\pi\)
−0.486071 + 0.873919i \(0.661570\pi\)
\(24\) −1.32615 − 1.11415i −0.270699 − 0.227425i
\(25\) 3.82236 + 3.20734i 0.764471 + 0.641467i
\(26\) −3.75725 − 2.16925i −0.736858 − 0.425425i
\(27\) −2.60632 + 4.49523i −0.501587 + 0.865107i
\(28\) 2.45270 − 0.892710i 0.463517 − 0.168706i
\(29\) 0.579160 − 3.28458i 0.107547 − 0.609931i −0.882625 − 0.470078i \(-0.844226\pi\)
0.990172 − 0.139853i \(-0.0446630\pi\)
\(30\) −0.112743 − 0.134529i −0.0205840 − 0.0245616i
\(31\) 9.69994i 1.74216i 0.491141 + 0.871080i \(0.336580\pi\)
−0.491141 + 0.871080i \(0.663420\pi\)
\(32\) 0.939693 − 0.342020i 0.166116 − 0.0604612i
\(33\) 2.07036 + 5.69910i 0.360403 + 0.992087i
\(34\) −2.30657 − 6.33725i −0.395574 − 1.08683i
\(35\) 0.260489 − 0.0459313i 0.0440307 − 0.00776380i
\(36\) −1.49682 − 2.59991i −0.249471 − 0.433318i
\(37\) 3.15406i 0.518524i 0.965807 + 0.259262i \(0.0834793\pi\)
−0.965807 + 0.259262i \(0.916521\pi\)
\(38\) 2.13879 − 3.79810i 0.346958 − 0.616133i
\(39\) −4.82671 − 5.75940i −0.772893 − 0.922242i
\(40\) 0.0998001 − 0.0175974i 0.0157798 − 0.00278240i
\(41\) −6.18649 + 5.19108i −0.966167 + 0.810711i −0.981945 − 0.189165i \(-0.939422\pi\)
0.0157780 + 0.999876i \(0.494978\pi\)
\(42\) 4.52084 + 0.00276307i 0.697581 + 0.000426351i
\(43\) −1.56634 − 8.88314i −0.238864 − 1.35467i −0.834321 − 0.551279i \(-0.814140\pi\)
0.595457 − 0.803388i \(-0.296971\pi\)
\(44\) −3.44759 − 0.607902i −0.519743 − 0.0916447i
\(45\) −0.103631 − 0.285811i −0.0154484 − 0.0426062i
\(46\) 2.71570 − 1.56791i 0.400408 − 0.231176i
\(47\) −3.56978 − 0.629448i −0.520705 − 0.0918144i −0.0928827 − 0.995677i \(-0.529608\pi\)
−0.427823 + 0.903863i \(0.640719\pi\)
\(48\) 1.73205 + 0.00105860i 0.250000 + 0.000152796i
\(49\) 0.0936613 − 0.162226i 0.0133802 − 0.0231752i
\(50\) −4.98973 −0.705654
\(51\) 0.00713918 − 11.6809i 0.000999686 − 1.63565i
\(52\) 4.27259 − 0.753373i 0.592502 − 0.104474i
\(53\) 8.61956 + 7.23267i 1.18399 + 0.993484i 0.999944 + 0.0105654i \(0.00336314\pi\)
0.184043 + 0.982918i \(0.441081\pi\)
\(54\) −0.892918 − 5.11886i −0.121511 − 0.696588i
\(55\) −0.333372 − 0.121337i −0.0449518 − 0.0163611i
\(56\) −1.30506 + 2.26042i −0.174395 + 0.302062i
\(57\) 5.73407 − 4.91126i 0.759496 − 0.650512i
\(58\) 1.66762 + 2.88841i 0.218970 + 0.379267i
\(59\) 0.963679 + 5.46529i 0.125460 + 0.711521i 0.981033 + 0.193839i \(0.0620938\pi\)
−0.855573 + 0.517682i \(0.826795\pi\)
\(60\) 0.172840 + 0.0305853i 0.0223136 + 0.00394855i
\(61\) 11.5303 − 4.19670i 1.47631 − 0.537332i 0.526502 − 0.850174i \(-0.323503\pi\)
0.949805 + 0.312842i \(0.101281\pi\)
\(62\) −6.23500 − 7.43058i −0.791846 − 0.943685i
\(63\) 7.35483 + 2.68712i 0.926621 + 0.338546i
\(64\) −0.500000 + 0.866025i −0.0625000 + 0.108253i
\(65\) 0.439662 0.0545334
\(66\) −5.24930 − 3.03496i −0.646144 − 0.373578i
\(67\) 4.12214 − 4.91257i 0.503599 − 0.600166i −0.453022 − 0.891499i \(-0.649654\pi\)
0.956622 + 0.291333i \(0.0940987\pi\)
\(68\) 5.84044 + 3.37198i 0.708258 + 0.408913i
\(69\) 5.34946 − 0.939884i 0.644000 − 0.113149i
\(70\) −0.170022 + 0.202625i −0.0203215 + 0.0242183i
\(71\) 1.27070 − 1.06625i 0.150805 − 0.126540i −0.564264 − 0.825595i \(-0.690840\pi\)
0.715069 + 0.699054i \(0.246395\pi\)
\(72\) 2.81782 + 1.02951i 0.332083 + 0.121328i
\(73\) 0.643609 + 3.65009i 0.0753288 + 0.427211i 0.999028 + 0.0440910i \(0.0140392\pi\)
−0.923699 + 0.383120i \(0.874850\pi\)
\(74\) −2.02739 − 2.41615i −0.235679 − 0.280872i
\(75\) −8.11945 − 2.96086i −0.937554 − 0.341891i
\(76\) 0.802963 + 4.28430i 0.0921061 + 0.491443i
\(77\) 7.91322 − 4.56870i 0.901795 − 0.520651i
\(78\) 7.39955 + 1.30940i 0.837834 + 0.148261i
\(79\) −2.75149 + 7.55967i −0.309567 + 0.850529i 0.683173 + 0.730256i \(0.260599\pi\)
−0.992741 + 0.120273i \(0.961623\pi\)
\(80\) −0.0651399 + 0.0776307i −0.00728286 + 0.00867937i
\(81\) 1.58450 − 8.85942i 0.176055 − 0.984380i
\(82\) 1.40236 − 7.95320i 0.154865 − 0.878284i
\(83\) 5.57871 − 3.22087i 0.612343 − 0.353536i −0.161539 − 0.986866i \(-0.551646\pi\)
0.773882 + 0.633330i \(0.218312\pi\)
\(84\) −3.46494 + 2.90383i −0.378056 + 0.316833i
\(85\) 0.523538 + 0.439301i 0.0567857 + 0.0476489i
\(86\) 6.90986 + 5.79806i 0.745109 + 0.625221i
\(87\) 0.999657 + 5.68967i 0.107174 + 0.609996i
\(88\) 3.03176 − 1.75038i 0.323186 − 0.186592i
\(89\) 1.00556 − 5.70280i 0.106589 − 0.604496i −0.883985 − 0.467516i \(-0.845149\pi\)
0.990574 − 0.136980i \(-0.0437397\pi\)
\(90\) 0.263102 + 0.152331i 0.0277334 + 0.0160571i
\(91\) −7.27891 + 8.67467i −0.763037 + 0.909352i
\(92\) −1.07251 + 2.94671i −0.111817 + 0.307216i
\(93\) −5.73656 − 15.7911i −0.594853 − 1.63746i
\(94\) 3.13921 − 1.81242i 0.323785 − 0.186937i
\(95\) 0.00474295 + 0.441704i 0.000486617 + 0.0453179i
\(96\) −1.32751 + 1.11253i −0.135488 + 0.113547i
\(97\) −8.34594 − 9.94630i −0.847402 − 1.00989i −0.999767 − 0.0215700i \(-0.993134\pi\)
0.152365 − 0.988324i \(-0.451311\pi\)
\(98\) 0.0325282 + 0.184477i 0.00328585 + 0.0186350i
\(99\) −6.74092 − 8.05348i −0.677487 − 0.809405i
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.x.b.281.1 yes 48
9.5 odd 6 342.2.bf.b.167.1 yes 48
19.14 odd 18 342.2.bf.b.299.1 yes 48
171.14 even 18 inner 342.2.x.b.185.1 ✓ 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
342.2.x.b.185.1 ✓ 48 171.14 even 18 inner
342.2.x.b.281.1 yes 48 1.1 even 1 trivial
342.2.bf.b.167.1 yes 48 9.5 odd 6
342.2.bf.b.299.1 yes 48 19.14 odd 18