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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 73.5
Character \(\chi\) \(=\) 162.73
Dual form 162.4.e.b.91.5

$q$-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 \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(e\left(\frac{5}{9}\right)\)

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
\(3\) 0 0
\(4\) −3.75877 − 1.36808i −0.469846 − 0.171010i
\(5\) 14.9069 − 12.5084i 1.33331 − 1.11878i 0.350022 − 0.936741i \(-0.386174\pi\)
0.983291 − 0.182041i \(-0.0582705\pi\)
\(6\) 0 0
\(7\) 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
\(9\) 0 0
\(10\) −19.4596 − 33.7050i −0.615366 − 1.06584i
\(11\) 9.63062 + 8.08105i 0.263976 + 0.221503i 0.765163 − 0.643837i \(-0.222658\pi\)
−0.501186 + 0.865339i \(0.667103\pi\)
\(12\) 0 0
\(13\) 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
\(15\) 0 0
\(16\) 12.2567 + 10.2846i 0.191511 + 0.160697i
\(17\) −24.5383 − 42.5016i −0.350083 − 0.606362i 0.636181 − 0.771540i \(-0.280513\pi\)
−0.986264 + 0.165178i \(0.947180\pi\)
\(18\) 0 0
\(19\) 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
\(21\) 0 0
\(22\) 19.2612 − 16.1621i 0.186660 − 0.156626i
\(23\) −162.928 − 59.3010i −1.47708 − 0.537614i −0.527068 − 0.849823i \(-0.676709\pi\)
−0.950013 + 0.312209i \(0.898931\pi\)
\(24\) 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 −0.999368 − 0.0355426i \(-0.988684\pi\)
0.951255 + 0.308405i \(0.0997952\pi\)
\(30\) 0 0
\(31\) −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
\(33\) 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 0.998062 + 0.0622244i \(0.0198195\pi\)
−0.445143 + 0.895459i \(0.646847\pi\)
\(38\) −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 0.779431 + 0.626488i \(0.215508\pi\)
−0.518154 + 0.855287i \(0.673381\pi\)
\(42\) 0 0
\(43\) −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
\(45\) 0 0
\(46\) −173.385 + 300.311i −0.555742 + 0.962574i
\(47\) −267.360 + 97.3112i −0.829756 + 0.302006i −0.721759 − 0.692145i \(-0.756666\pi\)
−0.107997 + 0.994151i \(0.534444\pi\)
\(48\) 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 −0.215990 − 0.976396i \(-0.569298\pi\)
0.924056 + 0.382258i \(0.124853\pi\)
\(60\) 0 0
\(61\) 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
\(63\) 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 0.917579 + 0.397554i \(0.130141\pi\)
−0.726270 + 0.687409i \(0.758748\pi\)
\(68\) 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 0.844009 − 0.536329i \(-0.180189\pi\)
−0.886479 + 0.462769i \(0.846856\pi\)
\(72\) 0 0
\(73\) 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
\(75\) 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 −0.998546 − 0.0539054i \(-0.982833\pi\)
0.956763 + 0.290868i \(0.0939442\pi\)
\(80\) 311.353 0.435129
\(81\) 0 0
\(82\) 790.017 1.06394
\(83\) −257.027 + 1457.67i −0.339909 + 1.92772i 0.0320928 + 0.999485i \(0.489783\pi\)
−0.372001 + 0.928232i \(0.621328\pi\)
\(84\) 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 −0.999679 − 0.0253482i \(-0.991931\pi\)
0.521792 + 0.853073i \(0.325264\pi\)
\(90\) 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 0.999769 + 0.0214809i \(0.00683810\pi\)
0.194763 + 0.980850i \(0.437606\pi\)
\(98\) 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.73.5 30
3.2 odd 2 54.4.e.b.25.5 yes 30
27.11 odd 18 1458.4.a.i.1.1 15
27.13 even 9 inner 162.4.e.b.91.5 30
27.14 odd 18 54.4.e.b.13.5 ✓ 30
27.16 even 9 1458.4.a.j.1.15 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
54.4.e.b.13.5 ✓ 30 27.14 odd 18
54.4.e.b.25.5 yes 30 3.2 odd 2
162.4.e.b.73.5 30 1.1 even 1 trivial
162.4.e.b.91.5 30 27.13 even 9 inner
1458.4.a.i.1.1 15 27.11 odd 18
1458.4.a.j.1.15 15 27.16 even 9