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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 25.5
Character \(\chi\) \(=\) 54.25
Dual form 54.4.e.b.13.5

$q$-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 \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).

\(n\) \(29\)
\(\chi(n)\) \(e\left(\frac{5}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.347296 + 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 −0.350022 + 0.936741i \(0.613826\pi\)
−0.983291 + 0.182041i \(0.941729\pi\)
\(6\) −7.41017 + 7.28624i −0.504198 + 0.495766i
\(7\) 11.7781 − 4.28689i 0.635959 − 0.231470i −0.00386374 − 0.999993i \(-0.501230\pi\)
0.639823 + 0.768522i \(0.279008\pi\)
\(8\) 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 0.501186 − 0.865339i \(-0.332897\pi\)
−0.765163 + 0.643837i \(0.777342\pi\)
\(12\) −11.7776 − 17.1257i −0.283324 − 0.411980i
\(13\) 6.96648 + 39.5088i 0.148627 + 0.842906i 0.964383 + 0.264510i \(0.0852102\pi\)
−0.815756 + 0.578396i \(0.803679\pi\)
\(14\) 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 0.986264 − 0.165178i \(-0.0528200\pi\)
−0.636181 + 0.771540i \(0.719487\pi\)
\(18\) −53.3302 + 8.47885i −0.698336 + 0.111027i
\(19\) 67.8741 − 117.561i 0.819546 − 1.41950i −0.0864706 − 0.996254i \(-0.527559\pi\)
0.906017 − 0.423241i \(-0.139108\pi\)
\(20\) 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 0.950013 − 0.312209i \(-0.101069\pi\)
0.527068 + 0.849823i \(0.323291\pi\)
\(24\) 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 −0.951255 − 0.308405i \(-0.900205\pi\)
0.999368 + 0.0355426i \(0.0113159\pi\)
\(30\) 19.3236 − 201.304i 0.117600 − 1.22510i
\(31\) −18.2216 − 6.63213i −0.105571 − 0.0384247i 0.288695 − 0.957421i \(-0.406779\pi\)
−0.394266 + 0.918997i \(0.629001\pi\)
\(32\) −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 0.998062 + 0.0622244i \(0.0198195\pi\)
−0.445143 + 0.895459i \(0.646847\pi\)
\(38\) 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 −0.779431 − 0.626488i \(-0.784492\pi\)
0.518154 − 0.855287i \(-0.326619\pi\)
\(42\) −56.0427 + 117.585i −0.205895 + 0.431994i
\(43\) −164.521 − 138.049i −0.583469 − 0.489589i 0.302615 − 0.953113i \(-0.402140\pi\)
−0.886084 + 0.463524i \(0.846585\pi\)
\(44\) 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 0.107997 − 0.994151i \(-0.465556\pi\)
0.721759 + 0.692145i \(0.243334\pi\)
\(48\) 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 −0.924056 − 0.382258i \(-0.875147\pi\)
0.215990 + 0.976396i \(0.430702\pi\)
\(60\) 389.781 + 107.972i 0.838676 + 0.232320i
\(61\) 78.1126 − 28.4307i 0.163956 − 0.0596750i −0.258738 − 0.965947i \(-0.583307\pi\)
0.422694 + 0.906272i \(0.361085\pi\)
\(62\) 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 0.917579 + 0.397554i \(0.130141\pi\)
−0.726270 + 0.687409i \(0.758748\pi\)
\(68\) −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 0.886479 − 0.462769i \(-0.153144\pi\)
−0.844009 + 0.536329i \(0.819811\pi\)
\(72\) 212.056 + 41.0899i 0.347097 + 0.0672569i
\(73\) 444.921 − 770.626i 0.713343 − 1.23555i −0.250252 − 0.968181i \(-0.580513\pi\)
0.963595 − 0.267366i \(-0.0861533\pi\)
\(74\) −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 −0.998546 − 0.0539054i \(-0.982833\pi\)
0.956763 + 0.290868i \(0.0939442\pi\)
\(80\) −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 −0.0320928 − 0.999485i \(-0.510217\pi\)
0.372001 − 0.928232i \(-0.378672\pi\)
\(84\) −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 −0.521792 − 0.853073i \(-0.674736\pi\)
0.999679 + 0.0253482i \(0.00806946\pi\)
\(90\) 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 0.999769 + 0.0214809i \(0.00683810\pi\)
0.194763 + 0.980850i \(0.437606\pi\)
\(98\) −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.25.5 yes 30
3.2 odd 2 162.4.e.b.73.5 30
27.11 odd 18 1458.4.a.j.1.15 15
27.13 even 9 inner 54.4.e.b.13.5 ✓ 30
27.14 odd 18 162.4.e.b.91.5 30
27.16 even 9 1458.4.a.i.1.1 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
54.4.e.b.13.5 ✓ 30 27.13 even 9 inner
54.4.e.b.25.5 yes 30 1.1 even 1 trivial
162.4.e.b.73.5 30 3.2 odd 2
162.4.e.b.91.5 30 27.14 odd 18
1458.4.a.i.1.1 15 27.16 even 9
1458.4.a.j.1.15 15 27.11 odd 18