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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [54,4,Mod(7,54)] 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("54.7"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(54, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([16])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 54.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [30] 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: \(3.18610314031\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(5\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 13.5
Character \(\chi\) \(=\) 54.13
Dual form 54.4.e.b.25.5

$q$-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+(-0.347296 - 1.96962i) q^{2} +(4.23116 - 3.01618i) q^{3} +(-3.75877 + 1.36808i) q^{4} +(-14.9069 - 12.5084i) q^{5} +(-7.41017 - 7.28624i) q^{6} +(11.7781 + 4.28689i) q^{7} +(4.00000 + 6.92820i) q^{8} +(8.80536 - 25.5238i) q^{9} +(-19.4596 + 33.7050i) q^{10} +(-9.63062 + 8.08105i) q^{11} +(-11.7776 + 17.1257i) q^{12} +(6.96648 - 39.5088i) q^{13} +(4.35302 - 24.6872i) q^{14} +(-100.801 - 7.96305i) q^{15} +(12.2567 - 10.2846i) q^{16} +(24.5383 - 42.5016i) q^{17} +(-53.3302 - 8.47885i) q^{18} +(67.8741 + 117.561i) q^{19} +(73.1441 + 26.6223i) q^{20} +(62.7651 - 17.3864i) q^{21} +(19.2612 + 16.1621i) q^{22} +(162.928 - 59.3010i) q^{23} +(37.8213 + 17.2496i) q^{24} +(44.0502 + 249.821i) q^{25} -80.2367 q^{26} +(-39.7275 - 134.554i) q^{27} -50.1361 q^{28} +(7.51379 + 42.6128i) q^{29} +(19.3236 + 201.304i) q^{30} +(-18.2216 + 6.63213i) q^{31} +(-24.5134 - 20.5692i) q^{32} +(-16.3748 + 63.2398i) q^{33} +(-92.2339 - 33.5704i) q^{34} +(-121.953 - 211.229i) q^{35} +(1.82130 + 107.985i) q^{36} +(124.441 - 215.538i) q^{37} +(207.978 - 174.514i) q^{38} +(-89.6894 - 188.180i) q^{39} +(27.0330 - 153.311i) q^{40} +(-68.5925 + 389.008i) q^{41} +(-56.0427 - 117.585i) q^{42} +(-164.521 + 138.049i) q^{43} +(25.1438 - 43.5503i) q^{44} +(-450.522 + 270.340i) q^{45} +(-173.385 - 300.311i) q^{46} +(267.360 + 97.3112i) q^{47} +(20.8399 - 80.4842i) q^{48} +(-142.406 - 119.493i) q^{49} +(476.753 - 173.524i) q^{50} +(-24.3669 - 253.843i) q^{51} +(27.8659 + 158.035i) q^{52} -695.130 q^{53} +(-251.222 + 124.978i) q^{54} +244.643 q^{55} +(17.4121 + 98.7488i) q^{56} +(641.771 + 292.700i) q^{57} +(81.3214 - 29.5986i) q^{58} +(-320.887 - 269.256i) q^{59} +(389.781 - 107.972i) q^{60} +(78.1126 + 28.4307i) q^{61} +(19.3911 + 33.5863i) q^{62} +(213.129 - 262.875i) q^{63} +(-32.0000 + 55.4256i) q^{64} +(-598.040 + 501.815i) q^{65} +(130.245 + 10.2891i) q^{66} +(104.917 - 595.014i) q^{67} +(-34.0883 + 193.324i) q^{68} +(510.512 - 742.332i) q^{69} +(-373.687 + 313.560i) q^{70} +(25.4078 - 44.0076i) q^{71} +(212.056 - 41.0899i) q^{72} +(444.921 + 770.626i) q^{73} +(-467.746 - 170.246i) q^{74} +(939.887 + 924.169i) q^{75} +(-415.956 - 349.029i) q^{76} +(-148.073 + 53.8943i) q^{77} +(-339.494 + 242.008i) q^{78} +(-29.3386 - 166.388i) q^{79} -311.353 q^{80} +(-573.931 - 449.493i) q^{81} +790.017 q^{82} +(257.027 + 1457.67i) q^{83} +(-212.134 + 151.219i) q^{84} +(-897.416 + 326.633i) q^{85} +(329.041 + 276.099i) q^{86} +(160.320 + 157.639i) q^{87} +(-94.5096 - 34.3987i) q^{88} +(401.246 + 694.978i) q^{89} +(688.931 + 793.467i) q^{90} +(251.422 - 435.476i) q^{91} +(-531.281 + 445.798i) q^{92} +(-57.0949 + 83.0212i) q^{93} +(98.8124 - 560.393i) q^{94} +(458.709 - 2601.47i) q^{95} +(-165.760 - 13.0947i) q^{96} +(1141.18 - 957.566i) q^{97} +(-185.898 + 321.985i) q^{98} +(121.458 + 316.967i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q - 12 q^{5} + 30 q^{6} + 33 q^{7} + 120 q^{8} + 42 q^{9} - 30 q^{10} - 39 q^{11} + 24 q^{12} - 60 q^{13} - 66 q^{14} - 153 q^{15} + 102 q^{17} + 168 q^{18} - 171 q^{19} + 96 q^{20} + 78 q^{21} + 78 q^{22}+ \cdots + 10296 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).

\(n\) \(29\)
\(\chi(n)\) \(e\left(\frac{4}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.347296 − 1.96962i −0.122788 − 0.696364i
\(3\) 4.23116 − 3.01618i 0.814286 − 0.580463i
\(4\) −3.75877 + 1.36808i −0.469846 + 0.171010i
\(5\) −14.9069 − 12.5084i −1.33331 − 1.11878i −0.983291 − 0.182041i \(-0.941729\pi\)
−0.350022 − 0.936741i \(-0.613826\pi\)
\(6\) −7.41017 − 7.28624i −0.504198 − 0.495766i
\(7\) 11.7781 + 4.28689i 0.635959 + 0.231470i 0.639823 − 0.768522i \(-0.279008\pi\)
−0.00386374 + 0.999993i \(0.501230\pi\)
\(8\) 4.00000 + 6.92820i 0.176777 + 0.306186i
\(9\) 8.80536 − 25.5238i 0.326125 − 0.945327i
\(10\) −19.4596 + 33.7050i −0.615366 + 1.06584i
\(11\) −9.63062 + 8.08105i −0.263976 + 0.221503i −0.765163 − 0.643837i \(-0.777342\pi\)
0.501186 + 0.865339i \(0.332897\pi\)
\(12\) −11.7776 + 17.1257i −0.283324 + 0.411980i
\(13\) 6.96648 − 39.5088i 0.148627 − 0.842906i −0.815756 − 0.578396i \(-0.803679\pi\)
0.964383 − 0.264510i \(-0.0852102\pi\)
\(14\) 4.35302 − 24.6872i 0.0830996 − 0.471281i
\(15\) −100.801 − 7.96305i −1.73511 − 0.137070i
\(16\) 12.2567 − 10.2846i 0.191511 − 0.160697i
\(17\) 24.5383 − 42.5016i 0.350083 − 0.606362i −0.636181 − 0.771540i \(-0.719487\pi\)
0.986264 + 0.165178i \(0.0528200\pi\)
\(18\) −53.3302 − 8.47885i −0.698336 − 0.111027i
\(19\) 67.8741 + 117.561i 0.819546 + 1.41950i 0.906017 + 0.423241i \(0.139108\pi\)
−0.0864706 + 0.996254i \(0.527559\pi\)
\(20\) 73.1441 + 26.6223i 0.817775 + 0.297646i
\(21\) 62.7651 − 17.3864i 0.652213 − 0.180668i
\(22\) 19.2612 + 16.1621i 0.186660 + 0.156626i
\(23\) 162.928 − 59.3010i 1.47708 − 0.537614i 0.527068 − 0.849823i \(-0.323291\pi\)
0.950013 + 0.312209i \(0.101069\pi\)
\(24\) 37.8213 + 17.2496i 0.321677 + 0.146711i
\(25\) 44.0502 + 249.821i 0.352401 + 1.99857i
\(26\) −80.2367 −0.605219
\(27\) −39.7275 − 134.554i −0.283169 − 0.959070i
\(28\) −50.1361 −0.338387
\(29\) 7.51379 + 42.6128i 0.0481130 + 0.272862i 0.999368 − 0.0355426i \(-0.0113159\pi\)
−0.951255 + 0.308405i \(0.900205\pi\)
\(30\) 19.3236 + 201.304i 0.117600 + 1.22510i
\(31\) −18.2216 + 6.63213i −0.105571 + 0.0384247i −0.394266 − 0.918997i \(-0.629001\pi\)
0.288695 + 0.957421i \(0.406779\pi\)
\(32\) −24.5134 − 20.5692i −0.135419 − 0.113630i
\(33\) −16.3748 + 63.2398i −0.0863783 + 0.333595i
\(34\) −92.2339 − 33.5704i −0.465235 − 0.169332i
\(35\) −121.953 − 211.229i −0.588968 − 1.02012i
\(36\) 1.82130 + 107.985i 0.00843196 + 0.499929i
\(37\) 124.441 − 215.538i 0.552919 − 0.957684i −0.445143 − 0.895459i \(-0.646847\pi\)
0.998062 − 0.0622244i \(-0.0198195\pi\)
\(38\) 207.978 − 174.514i 0.887856 − 0.745000i
\(39\) −89.6894 − 188.180i −0.368251 − 0.772640i
\(40\) 27.0330 − 153.311i 0.106857 − 0.606017i
\(41\) −68.5925 + 389.008i −0.261277 + 1.48177i 0.518154 + 0.855287i \(0.326619\pi\)
−0.779431 + 0.626488i \(0.784492\pi\)
\(42\) −56.0427 − 117.585i −0.205895 − 0.431994i
\(43\) −164.521 + 138.049i −0.583469 + 0.489589i −0.886084 − 0.463524i \(-0.846585\pi\)
0.302615 + 0.953113i \(0.402140\pi\)
\(44\) 25.1438 − 43.5503i 0.0861492 − 0.149215i
\(45\) −450.522 + 270.340i −1.49244 + 0.895554i
\(46\) −173.385 − 300.311i −0.555742 − 0.962574i
\(47\) 267.360 + 97.3112i 0.829756 + 0.302006i 0.721759 − 0.692145i \(-0.243334\pi\)
0.107997 + 0.994151i \(0.465556\pi\)
\(48\) 20.8399 − 80.4842i 0.0626662 − 0.242018i
\(49\) −142.406 − 119.493i −0.415179 − 0.348376i
\(50\) 476.753 − 173.524i 1.34846 − 0.490800i
\(51\) −24.3669 − 253.843i −0.0669029 − 0.696963i
\(52\) 27.8659 + 158.035i 0.0743136 + 0.421453i
\(53\) −695.130 −1.80157 −0.900787 − 0.434262i \(-0.857009\pi\)
−0.900787 + 0.434262i \(0.857009\pi\)
\(54\) −251.222 + 124.978i −0.633092 + 0.314951i
\(55\) 244.643 0.599777
\(56\) 17.4121 + 98.7488i 0.0415498 + 0.235641i
\(57\) 641.771 + 292.700i 1.49131 + 0.680160i
\(58\) 81.3214 − 29.5986i 0.184104 − 0.0670083i
\(59\) −320.887 − 269.256i −0.708066 − 0.594138i 0.215990 − 0.976396i \(-0.430702\pi\)
−0.924056 + 0.382258i \(0.875147\pi\)
\(60\) 389.781 − 107.972i 0.838676 − 0.232320i
\(61\) 78.1126 + 28.4307i 0.163956 + 0.0596750i 0.422694 − 0.906272i \(-0.361085\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(62\) 19.3911 + 33.5863i 0.0397204 + 0.0687978i
\(63\) 213.129 − 262.875i 0.426217 − 0.525701i
\(64\) −32.0000 + 55.4256i −0.0625000 + 0.108253i
\(65\) −598.040 + 501.815i −1.14120 + 0.957577i
\(66\) 130.245 + 10.2891i 0.242910 + 0.0191894i
\(67\) 104.917 − 595.014i 0.191308 − 1.08496i −0.726270 − 0.687409i \(-0.758748\pi\)
0.917579 − 0.397554i \(-0.130141\pi\)
\(68\) −34.0883 + 193.324i −0.0607913 + 0.344765i
\(69\) 510.512 − 742.332i 0.890702 − 1.29516i
\(70\) −373.687 + 313.560i −0.638059 + 0.535395i
\(71\) 25.4078 − 44.0076i 0.0424697 − 0.0735597i −0.844009 − 0.536329i \(-0.819811\pi\)
0.886479 + 0.462769i \(0.153144\pi\)
\(72\) 212.056 − 41.0899i 0.347097 − 0.0672569i
\(73\) 444.921 + 770.626i 0.713343 + 1.23555i 0.963595 + 0.267366i \(0.0861533\pi\)
−0.250252 + 0.968181i \(0.580513\pi\)
\(74\) −467.746 − 170.246i −0.734789 − 0.267441i
\(75\) 939.887 + 924.169i 1.44705 + 1.42285i
\(76\) −415.956 − 349.029i −0.627809 − 0.526794i
\(77\) −148.073 + 53.8943i −0.219150 + 0.0797639i
\(78\) −339.494 + 242.008i −0.492822 + 0.351308i
\(79\) −29.3386 − 166.388i −0.0417830 − 0.236963i 0.956763 − 0.290868i \(-0.0939442\pi\)
−0.998546 + 0.0539054i \(0.982833\pi\)
\(80\) −311.353 −0.435129
\(81\) −573.931 − 449.493i −0.787285 − 0.616589i
\(82\) 790.017 1.06394
\(83\) 257.027 + 1457.67i 0.339909 + 1.92772i 0.372001 + 0.928232i \(0.378672\pi\)
−0.0320928 + 0.999485i \(0.510217\pi\)
\(84\) −212.134 + 151.219i −0.275544 + 0.196421i
\(85\) −897.416 + 326.633i −1.14516 + 0.416803i
\(86\) 329.041 + 276.099i 0.412575 + 0.346192i
\(87\) 160.320 + 157.639i 0.197564 + 0.194260i
\(88\) −94.5096 − 34.3987i −0.114486 − 0.0416695i
\(89\) 401.246 + 694.978i 0.477887 + 0.827725i 0.999679 − 0.0253482i \(-0.00806946\pi\)
−0.521792 + 0.853073i \(0.674736\pi\)
\(90\) 688.931 + 793.467i 0.806886 + 0.929320i
\(91\) 251.422 − 435.476i 0.289629 − 0.501651i
\(92\) −531.281 + 445.798i −0.602064 + 0.505192i
\(93\) −57.0949 + 83.0212i −0.0636609 + 0.0925688i
\(94\) 98.8124 − 560.393i 0.108423 − 0.614895i
\(95\) 458.709 − 2601.47i 0.495395 − 2.80953i
\(96\) −165.760 − 13.0947i −0.176228 − 0.0139216i
\(97\) 1141.18 − 957.566i 1.19453 − 1.00233i 0.194763 − 0.980850i \(-0.437606\pi\)
0.999769 − 0.0214809i \(-0.00683810\pi\)
\(98\) −185.898 + 321.985i −0.191618 + 0.331892i
\(99\) 121.458 + 316.967i 0.123303 + 0.321781i
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 54.4.e.b.13.5 ✓ 30
3.2 odd 2 162.4.e.b.91.5 30
27.2 odd 18 162.4.e.b.73.5 30
27.5 odd 18 1458.4.a.j.1.15 15
27.22 even 9 1458.4.a.i.1.1 15
27.25 even 9 inner 54.4.e.b.25.5 yes 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
54.4.e.b.13.5 ✓ 30 1.1 even 1 trivial
54.4.e.b.25.5 yes 30 27.25 even 9 inner
162.4.e.b.73.5 30 27.2 odd 18
162.4.e.b.91.5 30 3.2 odd 2
1458.4.a.i.1.1 15 27.22 even 9
1458.4.a.j.1.15 15 27.5 odd 18