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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [162,4,Mod(19,162)] 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("162.19"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([16])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 162.e (of order \(9\), degree \(6\), not 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: \(9.55830942093\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(5\) over \(\Q(\zeta_{9})\)
Twist minimal: no (minimal twist has level 54)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 91.5
Character \(\chi\) \(=\) 162.91
Dual form 162.4.e.b.73.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.347296 + 1.96962i) q^{2} +(-3.75877 + 1.36808i) q^{4} +(14.9069 + 12.5084i) q^{5} +(11.7781 + 4.28689i) q^{7} +(-4.00000 - 6.92820i) q^{8} +(-19.4596 + 33.7050i) q^{10} +(9.63062 - 8.08105i) q^{11} +(6.96648 - 39.5088i) q^{13} +(-4.35302 + 24.6872i) q^{14} +(12.2567 - 10.2846i) q^{16} +(-24.5383 + 42.5016i) q^{17} +(67.8741 + 117.561i) q^{19} +(-73.1441 - 26.6223i) q^{20} +(19.2612 + 16.1621i) q^{22} +(-162.928 + 59.3010i) q^{23} +(44.0502 + 249.821i) q^{25} +80.2367 q^{26} -50.1361 q^{28} +(-7.51379 - 42.6128i) q^{29} +(-18.2216 + 6.63213i) q^{31} +(24.5134 + 20.5692i) q^{32} +(-92.2339 - 33.5704i) q^{34} +(121.953 + 211.229i) q^{35} +(124.441 - 215.538i) q^{37} +(-207.978 + 174.514i) q^{38} +(27.0330 - 153.311i) q^{40} +(68.5925 - 389.008i) q^{41} +(-164.521 + 138.049i) q^{43} +(-25.1438 + 43.5503i) q^{44} +(-173.385 - 300.311i) q^{46} +(-267.360 - 97.3112i) q^{47} +(-142.406 - 119.493i) q^{49} +(-476.753 + 173.524i) q^{50} +(27.8659 + 158.035i) q^{52} +695.130 q^{53} +244.643 q^{55} +(-17.4121 - 98.7488i) q^{56} +(81.3214 - 29.5986i) q^{58} +(320.887 + 269.256i) q^{59} +(78.1126 + 28.4307i) q^{61} +(-19.3911 - 33.5863i) q^{62} +(-32.0000 + 55.4256i) q^{64} +(598.040 - 501.815i) q^{65} +(104.917 - 595.014i) q^{67} +(34.0883 - 193.324i) q^{68} +(-373.687 + 313.560i) q^{70} +(-25.4078 + 44.0076i) q^{71} +(444.921 + 770.626i) q^{73} +(467.746 + 170.246i) q^{74} +(-415.956 - 349.029i) q^{76} +(148.073 - 53.8943i) q^{77} +(-29.3386 - 166.388i) q^{79} +311.353 q^{80} +790.017 q^{82} +(-257.027 - 1457.67i) q^{83} +(-897.416 + 326.633i) q^{85} +(-329.041 - 276.099i) q^{86} +(-94.5096 - 34.3987i) q^{88} +(-401.246 - 694.978i) q^{89} +(251.422 - 435.476i) q^{91} +(531.281 - 445.798i) q^{92} +(98.8124 - 560.393i) q^{94} +(-458.709 + 2601.47i) q^{95} +(1141.18 - 957.566i) q^{97} +(185.898 - 321.985i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q + 12 q^{5} + 33 q^{7} - 120 q^{8} - 30 q^{10} + 39 q^{11} - 60 q^{13} + 66 q^{14} - 102 q^{17} - 171 q^{19} - 96 q^{20} + 78 q^{22} - 48 q^{23} - 432 q^{25} + 468 q^{26} + 336 q^{28} + 381 q^{29} - 801 q^{31}+ \cdots - 4002 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(e\left(\frac{4}{9}\right)\)

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
\(3\) 0 0
\(4\) −3.75877 + 1.36808i −0.469846 + 0.171010i
\(5\) 14.9069 + 12.5084i 1.33331 + 1.11878i 0.983291 + 0.182041i \(0.0582705\pi\)
0.350022 + 0.936741i \(0.386174\pi\)
\(6\) 0 0
\(7\) 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
\(9\) 0 0
\(10\) −19.4596 + 33.7050i −0.615366 + 1.06584i
\(11\) 9.63062 − 8.08105i 0.263976 − 0.221503i −0.501186 − 0.865339i \(-0.667103\pi\)
0.765163 + 0.643837i \(0.222658\pi\)
\(12\) 0 0
\(13\) 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
\(15\) 0 0
\(16\) 12.2567 − 10.2846i 0.191511 − 0.160697i
\(17\) −24.5383 + 42.5016i −0.350083 + 0.606362i −0.986264 − 0.165178i \(-0.947180\pi\)
0.636181 + 0.771540i \(0.280513\pi\)
\(18\) 0 0
\(19\) 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
\(21\) 0 0
\(22\) 19.2612 + 16.1621i 0.186660 + 0.156626i
\(23\) −162.928 + 59.3010i −1.47708 + 0.537614i −0.950013 − 0.312209i \(-0.898931\pi\)
−0.527068 + 0.849823i \(0.676709\pi\)
\(24\) 0 0
\(25\) 44.0502 + 249.821i 0.352401 + 1.99857i
\(26\) 80.2367 0.605219
\(27\) 0 0
\(28\) −50.1361 −0.338387
\(29\) −7.51379 − 42.6128i −0.0481130 − 0.272862i 0.951255 − 0.308405i \(-0.0997952\pi\)
−0.999368 + 0.0355426i \(0.988684\pi\)
\(30\) 0 0
\(31\) −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
\(33\) 0 0
\(34\) −92.2339 − 33.5704i −0.465235 − 0.169332i
\(35\) 121.953 + 211.229i 0.588968 + 1.02012i
\(36\) 0 0
\(37\) 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
\(39\) 0 0
\(40\) 27.0330 − 153.311i 0.106857 − 0.606017i
\(41\) 68.5925 − 389.008i 0.261277 − 1.48177i −0.518154 − 0.855287i \(-0.673381\pi\)
0.779431 − 0.626488i \(-0.215508\pi\)
\(42\) 0 0
\(43\) −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
\(45\) 0 0
\(46\) −173.385 − 300.311i −0.555742 − 0.962574i
\(47\) −267.360 − 97.3112i −0.829756 − 0.302006i −0.107997 − 0.994151i \(-0.534444\pi\)
−0.721759 + 0.692145i \(0.756666\pi\)
\(48\) 0 0
\(49\) −142.406 − 119.493i −0.415179 − 0.348376i
\(50\) −476.753 + 173.524i −1.34846 + 0.490800i
\(51\) 0 0
\(52\) 27.8659 + 158.035i 0.0743136 + 0.421453i
\(53\) 695.130 1.80157 0.900787 − 0.434262i \(-0.142991\pi\)
0.900787 + 0.434262i \(0.142991\pi\)
\(54\) 0 0
\(55\) 244.643 0.599777
\(56\) −17.4121 − 98.7488i −0.0415498 − 0.235641i
\(57\) 0 0
\(58\) 81.3214 − 29.5986i 0.184104 − 0.0670083i
\(59\) 320.887 + 269.256i 0.708066 + 0.594138i 0.924056 − 0.382258i \(-0.124853\pi\)
−0.215990 + 0.976396i \(0.569298\pi\)
\(60\) 0 0
\(61\) 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
\(63\) 0 0
\(64\) −32.0000 + 55.4256i −0.0625000 + 0.108253i
\(65\) 598.040 − 501.815i 1.14120 − 0.957577i
\(66\) 0 0
\(67\) 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
\(69\) 0 0
\(70\) −373.687 + 313.560i −0.638059 + 0.535395i
\(71\) −25.4078 + 44.0076i −0.0424697 + 0.0735597i −0.886479 − 0.462769i \(-0.846856\pi\)
0.844009 + 0.536329i \(0.180189\pi\)
\(72\) 0 0
\(73\) 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
\(75\) 0 0
\(76\) −415.956 − 349.029i −0.627809 − 0.526794i
\(77\) 148.073 − 53.8943i 0.219150 − 0.0797639i
\(78\) 0 0
\(79\) −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
\(81\) 0 0
\(82\) 790.017 1.06394
\(83\) −257.027 − 1457.67i −0.339909 − 1.92772i −0.372001 − 0.928232i \(-0.621328\pi\)
0.0320928 − 0.999485i \(-0.489783\pi\)
\(84\) 0 0
\(85\) −897.416 + 326.633i −1.14516 + 0.416803i
\(86\) −329.041 − 276.099i −0.412575 − 0.346192i
\(87\) 0 0
\(88\) −94.5096 − 34.3987i −0.114486 − 0.0416695i
\(89\) −401.246 − 694.978i −0.477887 − 0.827725i 0.521792 − 0.853073i \(-0.325264\pi\)
−0.999679 + 0.0253482i \(0.991931\pi\)
\(90\) 0 0
\(91\) 251.422 − 435.476i 0.289629 − 0.501651i
\(92\) 531.281 − 445.798i 0.602064 − 0.505192i
\(93\) 0 0
\(94\) 98.8124 − 560.393i 0.108423 − 0.614895i
\(95\) −458.709 + 2601.47i −0.495395 + 2.80953i
\(96\) 0 0
\(97\) 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
\(99\) 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 162.4.e.b.91.5 30
3.2 odd 2 54.4.e.b.13.5 ✓ 30
27.2 odd 18 54.4.e.b.25.5 yes 30
27.5 odd 18 1458.4.a.i.1.1 15
27.22 even 9 1458.4.a.j.1.15 15
27.25 even 9 inner 162.4.e.b.73.5 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
54.4.e.b.13.5 ✓ 30 3.2 odd 2
54.4.e.b.25.5 yes 30 27.2 odd 18
162.4.e.b.73.5 30 27.25 even 9 inner
162.4.e.b.91.5 30 1.1 even 1 trivial
1458.4.a.i.1.1 15 27.5 odd 18
1458.4.a.j.1.15 15 27.22 even 9