LMFDB - Embedded newform 54.5.b.a.53.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [54,5,Mod(53,54)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(54, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("54.53");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 54 = 2 \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	54.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.58197800653$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			53.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	54.53
Dual form			54.5.b.a.53.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} -8.00000 q^{4} +33.9411i q^{5} -73.0000 q^{7} +22.6274i q^{8} +O(q^{10})$	\(q-2.82843i q^{2} -8.00000 q^{4} +33.9411i q^{5} -73.0000 q^{7} +22.6274i q^{8} +96.0000 q^{10} +169.706i q^{11} +95.0000 q^{13} +206.475i q^{14} +64.0000 q^{16} -101.823i q^{17} -313.000 q^{19} -271.529i q^{20} +480.000 q^{22} +644.881i q^{23} -527.000 q^{25} -268.701i q^{26} +584.000 q^{28} -1561.29i q^{29} -958.000 q^{31} -181.019i q^{32} -288.000 q^{34} -2477.70i q^{35} -385.000 q^{37} +885.298i q^{38} -768.000 q^{40} +2375.88i q^{41} +2546.00 q^{43} -1357.65i q^{44} +1824.00 q^{46} +169.706i q^{47} +2928.00 q^{49} +1490.58i q^{50} -760.000 q^{52} +2647.41i q^{53} -5760.00 q^{55} -1651.80i q^{56} -4416.00 q^{58} +2681.35i q^{59} +5615.00 q^{61} +2709.63i q^{62} -512.000 q^{64} +3224.41i q^{65} +23.0000 q^{67} +814.587i q^{68} -7008.00 q^{70} -407.294i q^{71} +6527.00 q^{73} +1088.94i q^{74} +2504.00 q^{76} -12388.5i q^{77} -6121.00 q^{79} +2172.23i q^{80} +6720.00 q^{82} +2511.64i q^{83} +3456.00 q^{85} -7201.18i q^{86} -3840.00 q^{88} +2749.23i q^{89} -6935.00 q^{91} -5159.05i q^{92} +480.000 q^{94} -10623.6i q^{95} +9935.00 q^{97} -8281.63i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 16 q^{4} - 146 q^{7}+O(q^{10}) $ 2 * q - 16 * q^4 - 146 * q^7	$ 2 q - 16 q^{4} - 146 q^{7} + 192 q^{10} + 190 q^{13} + 128 q^{16} - 626 q^{19} + 960 q^{22} - 1054 q^{25} + 1168 q^{28} - 1916 q^{31} - 576 q^{34} - 770 q^{37} - 1536 q^{40} + 5092 q^{43} + 3648 q^{46} + 5856 q^{49} - 1520 q^{52} - 11520 q^{55} - 8832 q^{58} + 11230 q^{61} - 1024 q^{64} + 46 q^{67} - 14016 q^{70} + 13054 q^{73} + 5008 q^{76} - 12242 q^{79} + 13440 q^{82} + 6912 q^{85} - 7680 q^{88} - 13870 q^{91} + 960 q^{94} + 19870 q^{97}+O(q^{100}) $ 2 * q - 16 * q^4 - 146 * q^7 + 192 * q^10 + 190 * q^13 + 128 * q^16 - 626 * q^19 + 960 * q^22 - 1054 * q^25 + 1168 * q^28 - 1916 * q^31 - 576 * q^34 - 770 * q^37 - 1536 * q^40 + 5092 * q^43 + 3648 * q^46 + 5856 * q^49 - 1520 * q^52 - 11520 * q^55 - 8832 * q^58 + 11230 * q^61 - 1024 * q^64 + 46 * q^67 - 14016 * q^70 + 13054 * q^73 + 5008 * q^76 - 12242 * q^79 + 13440 * q^82 + 6912 * q^85 - 7680 * q^88 - 13870 * q^91 + 960 * q^94 + 19870 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 2.82843i	− 0.707107i
$2$	− 2.82843i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−8.00000	−0.500000
$5$	33.9411i	1.35765i	0.734302	+	0.678823i	$0.237509\pi$
$5$	33.9411i	1.35765i	−0.734302	+	0.678823i	$0.762491\pi$
$6$	0	0
$7$	−73.0000	−1.48980	−0.744898	−	0.667178i	$-0.767502\pi$
$7$	−73.0000	−1.48980	−0.744898	+	0.667178i	$0.767502\pi$
$8$	22.6274i	0.353553i
$9$	0	0
$10$	96.0000	0.960000
$11$	169.706i	1.40253i	0.712903	+	0.701263i	$0.247380\pi$
$11$	169.706i	1.40253i	−0.712903	+	0.701263i	$0.752620\pi$
$12$	0	0
$13$	95.0000	0.562130	0.281065	−	0.959689i	$-0.409312\pi$
$13$	95.0000	0.562130	0.281065	+	0.959689i	$0.409312\pi$
$14$	206.475i	1.05344i
$15$	0	0
$16$	64.0000	0.250000
$17$	− 101.823i	− 0.352330i	−0.984361	−	0.176165i	$-0.943631\pi$
$17$	− 101.823i	− 0.352330i	0.984361	−	0.176165i	$-0.0563692\pi$
$18$	0	0
$19$	−313.000	−0.867036	−0.433518	−	0.901145i	$-0.642728\pi$
$19$	−313.000	−0.867036	−0.433518	+	0.901145i	$0.642728\pi$
$20$	− 271.529i	− 0.678823i
$21$	0	0
$22$	480.000	0.991736
$23$	644.881i	1.21906i	0.792764	+	0.609529i	$0.208641\pi$
$23$	644.881i	1.21906i	−0.792764	+	0.609529i	$0.791359\pi$
$24$	0	0
$25$	−527.000	−0.843200
$26$	− 268.701i	− 0.397486i
$27$	0	0
$28$	584.000	0.744898
$29$	− 1561.29i	− 1.85647i	−0.371994	−	0.928235i	$-0.621326\pi$
$29$	− 1561.29i	− 1.85647i	0.371994	−	0.928235i	$-0.378674\pi$
$30$	0	0
$31$	−958.000	−0.996878	−0.498439	−	0.866925i	$-0.666093\pi$
$31$	−958.000	−0.996878	−0.498439	+	0.866925i	$0.666093\pi$
$32$	− 181.019i	− 0.176777i
$33$	0	0
$34$	−288.000	−0.249135
$35$	− 2477.70i	− 2.02261i
$36$	0	0
$37$	−385.000	−0.281227	−0.140614	−	0.990065i	$-0.544908\pi$
$37$	−385.000	−0.281227	−0.140614	+	0.990065i	$0.544908\pi$
$38$	885.298i	0.613087i
$39$	0	0
$40$	−768.000	−0.480000
$41$	2375.88i	1.41337i	0.707527	+	0.706686i	$0.249811\pi$
$41$	2375.88i	1.41337i	−0.707527	+	0.706686i	$0.750189\pi$
$42$	0	0
$43$	2546.00	1.37696	0.688480	−	0.725255i	$-0.258278\pi$
$43$	2546.00	1.37696	0.688480	+	0.725255i	$0.258278\pi$
$44$	− 1357.65i	− 0.701263i
$45$	0	0
$46$	1824.00	0.862004
$47$	169.706i	0.0768246i	0.999262	+	0.0384123i	$0.0122300\pi$
$47$	169.706i	0.0768246i	−0.999262	+	0.0384123i	$0.987770\pi$
$48$	0	0
$49$	2928.00	1.21949
$50$	1490.58i	0.596232i
$51$	0	0
$52$	−760.000	−0.281065
$53$	2647.41i	0.942473i	0.882007	+	0.471237i	$0.156192\pi$
$53$	2647.41i	0.942473i	−0.882007	+	0.471237i	$0.843808\pi$
$54$	0	0
$55$	−5760.00	−1.90413
$56$	− 1651.80i	− 0.526722i
$57$	0	0
$58$	−4416.00	−1.31272
$59$	2681.35i	0.770281i	0.922858	+	0.385141i	$0.125847\pi$
$59$	2681.35i	0.770281i	−0.922858	+	0.385141i	$0.874153\pi$
$60$	0	0
$61$	5615.00	1.50900	0.754501	−	0.656298i	$-0.227879\pi$
$61$	5615.00	1.50900	0.754501	+	0.656298i	$0.227879\pi$
$62$	2709.63i	0.704899i
$63$	0	0
$64$	−512.000	−0.125000
$65$	3224.41i	0.763173i
$66$	0	0
$67$	23.0000	0.00512364	0.00256182	−	0.999997i	$-0.499185\pi$
$67$	23.0000	0.00512364	0.00256182	+	0.999997i	$0.499185\pi$
$68$	814.587i	0.176165i
$69$	0	0
$70$	−7008.00	−1.43020
$71$	− 407.294i	− 0.0807962i	−0.999184	−	0.0403981i	$-0.987137\pi$
$71$	− 407.294i	− 0.0807962i	0.999184	−	0.0403981i	$-0.0128626\pi$
$72$	0	0
$73$	6527.00	1.22481	0.612404	−	0.790545i	$-0.290203\pi$
$73$	6527.00	1.22481	0.612404	+	0.790545i	$0.290203\pi$
$74$	1088.94i	0.198858i
$75$	0	0
$76$	2504.00	0.433518
$77$	− 12388.5i	− 2.08948i
$78$	0	0
$79$	−6121.00	−0.980772	−0.490386	−	0.871505i	$-0.663144\pi$
$79$	−6121.00	−0.980772	−0.490386	+	0.871505i	$0.663144\pi$
$80$	2172.23i	0.339411i
$81$	0	0
$82$	6720.00	0.999405
$83$	2511.64i	0.364587i	0.983244	+	0.182294i	$0.0583522\pi$
$83$	2511.64i	0.364587i	−0.983244	+	0.182294i	$0.941648\pi$
$84$	0	0
$85$	3456.00	0.478339
$86$	− 7201.18i	− 0.973658i
$87$	0	0
$88$	−3840.00	−0.495868
$89$	2749.23i	0.347081i	0.984827	+	0.173541i	$0.0555208\pi$
$89$	2749.23i	0.347081i	−0.984827	+	0.173541i	$0.944479\pi$
$90$	0	0
$91$	−6935.00	−0.837459
$92$	− 5159.05i	− 0.609529i
$93$	0	0
$94$	480.000	0.0543232
$95$	− 10623.6i	− 1.17713i
$96$	0	0
$97$	9935.00	1.05590	0.527952	−	0.849274i	$-0.322960\pi$
$97$	9935.00	1.05590	0.527952	+	0.849274i	$0.322960\pi$
$98$	− 8281.63i	− 0.862311i
$99$	0	0
$100$	4216.00	0.421600
$101$	− 9503.52i	− 0.931626i	−0.884883	−	0.465813i	$-0.845762\pi$
$101$	− 9503.52i	− 0.931626i	0.884883	−	0.465813i	$-0.154238\pi$
$102$	0	0
$103$	6215.00	0.585823	0.292912	−	0.956140i	$-0.405376\pi$
$103$	6215.00	0.585823	0.292912	+	0.956140i	$0.405376\pi$
$104$	2149.60i	0.198743i
$105$	0	0
$106$	7488.00	0.666429
$107$	7840.40i	0.684811i	0.939552	+	0.342405i	$0.111242\pi$
$107$	7840.40i	0.684811i	−0.939552	+	0.342405i	$0.888758\pi$
$108$	0	0
$109$	−15982.0	−1.34517	−0.672586	−	0.740019i	$-0.734817\pi$
$109$	−15982.0	−1.34517	−0.672586	+	0.740019i	$0.734817\pi$
$110$	16291.7i	1.34642i
$111$	0	0
$112$	−4672.00	−0.372449
$113$	2885.00i	0.225937i	0.993599	+	0.112969i	$0.0360360\pi$
$113$	2885.00i	0.225937i	−0.993599	+	0.112969i	$0.963964\pi$
$114$	0	0
$115$	−21888.0	−1.65505
$116$	12490.3i	0.928235i
$117$	0	0
$118$	7584.00	0.544671
$119$	7433.11i	0.524900i
$120$	0	0
$121$	−14159.0	−0.967079
$122$	− 15881.6i	− 1.06703i
$123$	0	0
$124$	7664.00	0.498439
$125$	3326.23i	0.212879i
$126$	0	0
$127$	−10942.0	−0.678405	−0.339203	−	0.940713i	$-0.610157\pi$
$127$	−10942.0	−0.678405	−0.339203	+	0.940713i	$0.610157\pi$
$128$	1448.15i	0.0883883i
$129$	0	0
$130$	9120.00	0.539645
$131$	746.705i	0.0435117i	0.999763	+	0.0217559i	$0.00692565\pi$
$131$	746.705i	0.0435117i	−0.999763	+	0.0217559i	$0.993074\pi$
$132$	0	0
$133$	22849.0	1.29171
$134$	− 65.0538i	− 0.00362296i
$135$	0	0
$136$	2304.00	0.124567
$137$	20738.0i	1.10491i	0.833543	+	0.552454i	$0.186309\pi$
$137$	20738.0i	1.10491i	−0.833543	+	0.552454i	$0.813691\pi$
$138$	0	0
$139$	−10777.0	−0.557787	−0.278893	−	0.960322i	$-0.589968\pi$
$139$	−10777.0	−0.557787	−0.278893	+	0.960322i	$0.589968\pi$
$140$	19821.6i	1.01131i
$141$	0	0
$142$	−1152.00	−0.0571315
$143$	16122.0i	0.788402i
$144$	0	0
$145$	52992.0	2.52043
$146$	− 18461.1i	− 0.866070i
$147$	0	0
$148$	3080.00	0.140614
$149$	36792.2i	1.65723i	0.559818	+	0.828615i	$0.310871\pi$
$149$	36792.2i	1.65723i	−0.559818	+	0.828615i	$0.689129\pi$
$150$	0	0
$151$	−505.000	−0.0221482	−0.0110741	−	0.999939i	$-0.503525\pi$
$151$	−505.000	−0.0221482	−0.0110741	+	0.999939i	$0.503525\pi$
$152$	− 7082.38i	− 0.306544i
$153$	0	0
$154$	−35040.0	−1.47748
$155$	− 32515.6i	− 1.35341i
$156$	0	0
$157$	−15790.0	−0.640594	−0.320297	−	0.947317i	$-0.603783\pi$
$157$	−15790.0	−0.640594	−0.320297	+	0.947317i	$0.603783\pi$
$158$	17312.8i	0.693511i
$159$	0	0
$160$	6144.00	0.240000
$161$	− 47076.3i	− 1.81615i
$162$	0	0
$163$	−12409.0	−0.467048	−0.233524	−	0.972351i	$-0.575026\pi$
$163$	−12409.0	−0.467048	−0.233524	+	0.972351i	$0.575026\pi$
$164$	− 19007.0i	− 0.706686i
$165$	0	0
$166$	7104.00	0.257802
$167$	− 14832.3i	− 0.531832i	−0.963996	−	0.265916i	$-0.914326\pi$
$167$	− 14832.3i	− 0.531832i	0.963996	−	0.265916i	$-0.0856744\pi$
$168$	0	0
$169$	−19536.0	−0.684010
$170$	− 9775.04i	− 0.338237i
$171$	0	0
$172$	−20368.0	−0.688480
$173$	6991.87i	0.233615i	0.993155	+	0.116808i	$0.0372661\pi$
$173$	6991.87i	0.233615i	−0.993155	+	0.116808i	$0.962734\pi$
$174$	0	0
$175$	38471.0	1.25620
$176$	10861.2i	0.350631i
$177$	0	0
$178$	7776.00	0.245424
$179$	− 52540.9i	− 1.63980i	−0.572506	−	0.819900i	$-0.694029\pi$
$179$	− 52540.9i	− 1.63980i	0.572506	−	0.819900i	$-0.305971\pi$
$180$	0	0
$181$	9263.00	0.282745	0.141372	−	0.989956i	$-0.454849\pi$
$181$	9263.00	0.282745	0.141372	+	0.989956i	$0.454849\pi$
$182$	19615.1i	0.592173i
$183$	0	0
$184$	−14592.0	−0.431002
$185$	− 13067.3i	− 0.381807i
$186$	0	0
$187$	17280.0	0.494152
$188$	− 1357.65i	− 0.0384123i
$189$	0	0
$190$	−30048.0	−0.832355
$191$	− 10216.3i	− 0.280044i	−0.990148	−	0.140022i	$-0.955283\pi$
$191$	− 10216.3i	− 0.280044i	0.990148	−	0.140022i	$-0.0447173\pi$
$192$	0	0
$193$	−5089.00	−0.136621	−0.0683106	−	0.997664i	$-0.521761\pi$
$193$	−5089.00	−0.136621	−0.0683106	+	0.997664i	$0.521761\pi$
$194$	− 28100.4i	− 0.746637i
$195$	0	0
$196$	−23424.0	−0.609746
$197$	− 11913.3i	− 0.306974i	−0.988151	−	0.153487i	$-0.950950\pi$
$197$	− 11913.3i	− 0.306974i	0.988151	−	0.153487i	$-0.0490502\pi$
$198$	0	0
$199$	44903.0	1.13389	0.566943	−	0.823757i	$-0.308126\pi$
$199$	44903.0	1.13389	0.566943	+	0.823757i	$0.308126\pi$
$200$	− 11924.6i	− 0.298116i
$201$	0	0
$202$	−26880.0	−0.658759
$203$	113974.i	2.76576i
$204$	0	0
$205$	−80640.0	−1.91886
$206$	− 17578.7i	− 0.414240i
$207$	0	0
$208$	6080.00	0.140533
$209$	− 53117.9i	− 1.21604i
$210$	0	0
$211$	66551.0	1.49482	0.747411	−	0.664362i	$-0.231296\pi$
$211$	66551.0	1.49482	0.747411	+	0.664362i	$0.231296\pi$
$212$	− 21179.3i	− 0.471237i
$213$	0	0
$214$	22176.0	0.484234
$215$	86414.1i	1.86942i
$216$	0	0
$217$	69934.0	1.48515
$218$	45203.9i	0.951181i
$219$	0	0
$220$	46080.0	0.952066
$221$	− 9673.22i	− 0.198055i
$222$	0	0
$223$	−7102.00	−0.142814	−0.0714070	−	0.997447i	$-0.522749\pi$
$223$	−7102.00	−0.142814	−0.0714070	+	0.997447i	$0.522749\pi$
$224$	13214.4i	0.263361i
$225$	0	0
$226$	8160.00	0.159762
$227$	5159.05i	0.100119i	0.998746	+	0.0500597i	$0.0159412\pi$
$227$	5159.05i	0.100119i	−0.998746	+	0.0500597i	$0.984059\pi$
$228$	0	0
$229$	−94990.0	−1.81137	−0.905684	−	0.423952i	$-0.860642\pi$
$229$	−94990.0	−1.81137	−0.905684	+	0.423952i	$0.860642\pi$
$230$	61908.6i	1.17030i
$231$	0	0
$232$	35328.0	0.656361
$233$	40322.1i	0.742730i	0.928487	+	0.371365i	$0.121110\pi$
$233$	40322.1i	0.742730i	−0.928487	+	0.371365i	$0.878890\pi$
$234$	0	0
$235$	−5760.00	−0.104301
$236$	− 21450.8i	− 0.385141i
$237$	0	0
$238$	21024.0	0.371160
$239$	− 95985.5i	− 1.68039i	−0.542285	−	0.840195i	$-0.682441\pi$
$239$	− 95985.5i	− 1.68039i	0.542285	−	0.840195i	$-0.317559\pi$
$240$	0	0
$241$	24191.0	0.416505	0.208252	−	0.978075i	$-0.433222\pi$
$241$	24191.0	0.416505	0.208252	+	0.978075i	$0.433222\pi$
$242$	40047.7i	0.683828i
$243$	0	0
$244$	−44920.0	−0.754501
$245$	99379.6i	1.65564i
$246$	0	0
$247$	−29735.0	−0.487387
$248$	− 21677.1i	− 0.352450i
$249$	0	0
$250$	9408.00	0.150528
$251$	− 19753.7i	− 0.313546i	−0.987635	−	0.156773i	$-0.949891\pi$
$251$	− 19753.7i	− 0.313546i	0.987635	−	0.156773i	$-0.0501092\pi$
$252$	0	0
$253$	−109440.	−1.70976
$254$	30948.6i	0.479705i
$255$	0	0
$256$	4096.00	0.0625000
$257$	− 16766.9i	− 0.253856i	−0.991912	−	0.126928i	$-0.959488\pi$
$257$	− 16766.9i	− 0.253856i	0.991912	−	0.126928i	$-0.0405117\pi$
$258$	0	0
$259$	28105.0	0.418971
$260$	− 25795.3i	− 0.381587i
$261$	0	0
$262$	2112.00	0.0307674
$263$	79693.8i	1.15216i	0.817393	+	0.576080i	$0.195418\pi$
$263$	79693.8i	1.15216i	−0.817393	+	0.576080i	$0.804582\pi$
$264$	0	0
$265$	−89856.0	−1.27954
$266$	− 64626.7i	− 0.913375i
$267$	0	0
$268$	−184.000	−0.00256182
$269$	62214.1i	0.859774i	0.902883	+	0.429887i	$0.141447\pi$
$269$	62214.1i	0.859774i	−0.902883	+	0.429887i	$0.858553\pi$
$270$	0	0
$271$	−81241.0	−1.10621	−0.553104	−	0.833112i	$-0.686557\pi$
$271$	−81241.0	−1.10621	−0.553104	+	0.833112i	$0.686557\pi$
$272$	− 6516.70i	− 0.0880825i
$273$	0	0
$274$	58656.0	0.781288
$275$	− 89434.9i	− 1.18261i
$276$	0	0
$277$	60722.0	0.791383	0.395691	−	0.918384i	$-0.370505\pi$
$277$	60722.0	0.791383	0.395691	+	0.918384i	$0.370505\pi$
$278$	30482.0i	0.394415i
$279$	0	0
$280$	56064.0	0.715102
$281$	− 103385.i	− 1.30931i	−0.755926	−	0.654657i	$-0.772813\pi$
$281$	− 103385.i	− 1.30931i	0.755926	−	0.654657i	$-0.227187\pi$
$282$	0	0
$283$	145202.	1.81301	0.906504	−	0.422197i	$-0.138741\pi$
$283$	145202.	1.81301	0.906504	+	0.422197i	$0.138741\pi$
$284$	3258.35i	0.0403981i
$285$	0	0
$286$	45600.0	0.557484
$287$	− 173439.i	− 2.10564i
$288$	0	0
$289$	73153.0	0.875864
$290$	− 149884.i	− 1.78221i
$291$	0	0
$292$	−52216.0	−0.612404
$293$	− 91607.1i	− 1.06707i	−0.845777	−	0.533536i	$-0.820863\pi$
$293$	− 91607.1i	− 1.06707i	0.845777	−	0.533536i	$-0.179137\pi$
$294$	0	0
$295$	−91008.0	−1.04577
$296$	− 8711.56i	− 0.0994288i
$297$	0	0
$298$	104064.	1.17184
$299$	61263.7i	0.685269i
$300$	0	0
$301$	−185858.	−2.05139
$302$	1428.36i	0.0156611i
$303$	0	0
$304$	−20032.0	−0.216759
$305$	190579.i	2.04869i
$306$	0	0
$307$	−8494.00	−0.0901230	−0.0450615	−	0.998984i	$-0.514348\pi$
$307$	−8494.00	−0.0901230	−0.0450615	+	0.998984i	$0.514348\pi$
$308$	99108.1i	1.04474i
$309$	0	0
$310$	−91968.0	−0.957003
$311$	− 46804.8i	− 0.483916i	−0.970287	−	0.241958i	$-0.922210\pi$
$311$	− 46804.8i	− 0.483916i	0.970287	−	0.241958i	$-0.0777896\pi$
$312$	0	0
$313$	62063.0	0.633496	0.316748	−	0.948510i	$-0.397409\pi$
$313$	62063.0	0.633496	0.316748	+	0.948510i	$0.397409\pi$
$314$	44660.9i	0.452968i
$315$	0	0
$316$	48968.0	0.490386
$317$	− 118862.i	− 1.18283i	−0.806366	−	0.591417i	$-0.798569\pi$
$317$	− 118862.i	− 1.18283i	0.806366	−	0.591417i	$-0.201431\pi$
$318$	0	0
$319$	264960.	2.60375
$320$	− 17377.9i	− 0.169706i
$321$	0	0
$322$	−133152.	−1.28421
$323$	31870.7i	0.305483i
$324$	0	0
$325$	−50065.0	−0.473988
$326$	35098.0i	0.330253i
$327$	0	0
$328$	−53760.0	−0.499703
$329$	− 12388.5i	− 0.114453i
$330$	0	0
$331$	38087.0	0.347633	0.173816	−	0.984778i	$-0.444390\pi$
$331$	38087.0	0.347633	0.173816	+	0.984778i	$0.444390\pi$
$332$	− 20093.1i	− 0.182294i
$333$	0	0
$334$	−41952.0	−0.376062
$335$	780.646i	0.00695608i
$336$	0	0
$337$	75119.0	0.661439	0.330720	−	0.943729i	$-0.392709\pi$
$337$	75119.0	0.661439	0.330720	+	0.943729i	$0.392709\pi$
$338$	55256.2i	0.483668i
$339$	0	0
$340$	−27648.0	−0.239170
$341$	− 162578.i	− 1.39815i
$342$	0	0
$343$	−38471.0	−0.326998
$344$	57609.4i	0.486829i
$345$	0	0
$346$	19776.0	0.165191
$347$	199710.i	1.65859i	0.558808	+	0.829297i	$0.311259\pi$
$347$	199710.i	1.65859i	−0.558808	+	0.829297i	$0.688741\pi$
$348$	0	0
$349$	−63073.0	−0.517836	−0.258918	−	0.965899i	$-0.583366\pi$
$349$	−63073.0	−0.517836	−0.258918	+	0.965899i	$0.583366\pi$
$350$	− 108812.i	− 0.888265i
$351$	0	0
$352$	30720.0	0.247934
$353$	216544.i	1.73779i	0.494996	+	0.868895i	$0.335170\pi$
$353$	216544.i	1.73779i	−0.494996	+	0.868895i	$0.664830\pi$
$354$	0	0
$355$	13824.0	0.109693
$356$	− 21993.8i	− 0.173541i
$357$	0	0
$358$	−148608.	−1.15951
$359$	243052.i	1.88587i	0.332981	+	0.942933i	$0.391945\pi$
$359$	243052.i	1.88587i	−0.332981	+	0.942933i	$0.608055\pi$
$360$	0	0
$361$	−32352.0	−0.248249
$362$	− 26199.7i	− 0.199931i
$363$	0	0
$364$	55480.0	0.418730
$365$	221534.i	1.66285i
$366$	0	0
$367$	19367.0	0.143791	0.0718953	−	0.997412i	$-0.477095\pi$
$367$	19367.0	0.143791	0.0718953	+	0.997412i	$0.477095\pi$
$368$	41272.4i	0.304764i
$369$	0	0
$370$	−36960.0	−0.269978
$371$	− 193261.i	− 1.40409i
$372$	0	0
$373$	−130657.	−0.939107	−0.469553	−	0.882904i	$-0.655585\pi$
$373$	−130657.	−0.939107	−0.469553	+	0.882904i	$0.655585\pi$
$374$	− 48875.2i	− 0.349418i
$375$	0	0
$376$	−3840.00	−0.0271616
$377$	− 148323.i	− 1.04358i
$378$	0	0
$379$	−123433.	−0.859316	−0.429658	−	0.902992i	$-0.641366\pi$
$379$	−123433.	−0.859316	−0.429658	+	0.902992i	$0.641366\pi$
$380$	84988.6i	0.588564i
$381$	0	0
$382$	−28896.0	−0.198021
$383$	224554.i	1.53082i	0.643543	+	0.765410i	$0.277464\pi$
$383$	224554.i	1.53082i	−0.643543	+	0.765410i	$0.722536\pi$
$384$	0	0
$385$	420480.	2.83677
$386$	14393.9i	0.0966057i
$387$	0	0
$388$	−79480.0	−0.527952
$389$	− 83936.4i	− 0.554691i	−0.960770	−	0.277346i	$-0.910545\pi$
$389$	− 83936.4i	− 0.554691i	0.960770	−	0.277346i	$-0.0894547\pi$
$390$	0	0
$391$	65664.0	0.429511
$392$	66253.1i	0.431155i
$393$	0	0
$394$	−33696.0	−0.217063
$395$	− 207754.i	− 1.33154i
$396$	0	0
$397$	261650.	1.66012	0.830060	−	0.557673i	$-0.188306\pi$
$397$	261650.	1.66012	0.830060	+	0.557673i	$0.188306\pi$
$398$	− 127005.i	− 0.801778i
$399$	0	0
$400$	−33728.0	−0.210800
$401$	6245.17i	0.0388379i	0.999811	+	0.0194189i	$0.00618163\pi$
$401$	6245.17i	0.0388379i	−0.999811	+	0.0194189i	$0.993818\pi$
$402$	0	0
$403$	−91010.0	−0.560375
$404$	76028.1i	0.465813i
$405$	0	0
$406$	322368.	1.95569
$407$	− 65336.7i	− 0.394428i
$408$	0	0
$409$	296159.	1.77043	0.885214	−	0.465184i	$-0.154012\pi$
$409$	296159.	1.77043	0.885214	+	0.465184i	$0.154012\pi$
$410$	228084.i	1.35684i
$411$	0	0
$412$	−49720.0	−0.292912
$413$	− 195738.i	− 1.14756i
$414$	0	0
$415$	−85248.0	−0.494980
$416$	− 17196.8i	− 0.0993715i
$417$	0	0
$418$	−150240.	−0.859870
$419$	− 145981.i	− 0.831510i	−0.909477	−	0.415755i	$-0.863517\pi$
$419$	− 145981.i	− 0.831510i	0.909477	−	0.415755i	$-0.136483\pi$
$420$	0	0
$421$	−315937.	−1.78253	−0.891264	−	0.453485i	$-0.850181\pi$
$421$	−315937.	−1.78253	−0.891264	+	0.453485i	$0.850181\pi$
$422$	− 188235.i	− 1.05700i
$423$	0	0
$424$	−59904.0	−0.333215
$425$	53660.9i	0.297085i
$426$	0	0
$427$	−409895.	−2.24811
$428$	− 62723.2i	− 0.342405i
$429$	0	0
$430$	244416.	1.32188
$431$	712.764i	0.00383699i	0.999998	+	0.00191850i	$0.000610677\pi$
$431$	712.764i	0.00383699i	−0.999998	+	0.00191850i	$0.999389\pi$
$432$	0	0
$433$	−29374.0	−0.156671	−0.0783353	−	0.996927i	$-0.524960\pi$
$433$	−29374.0	−0.156671	−0.0783353	+	0.996927i	$0.524960\pi$
$434$	− 197803.i	− 1.05016i
$435$	0	0
$436$	127856.	0.672586
$437$	− 201848.i	− 1.05697i
$438$	0	0
$439$	−11422.0	−0.0592670	−0.0296335	−	0.999561i	$-0.509434\pi$
$439$	−11422.0	−0.0592670	−0.0296335	+	0.999561i	$0.509434\pi$
$440$	− 130334.i	− 0.673212i
$441$	0	0
$442$	−27360.0	−0.140046
$443$	104742.i	0.533722i	0.963735	+	0.266861i	$0.0859864\pi$
$443$	104742.i	0.533722i	−0.963735	+	0.266861i	$0.914014\pi$
$444$	0	0
$445$	−93312.0	−0.471213
$446$	20087.5i	0.100985i
$447$	0	0
$448$	37376.0	0.186224
$449$	− 83597.0i	− 0.414666i	−0.978270	−	0.207333i	$-0.933522\pi$
$449$	− 83597.0i	− 0.414666i	0.978270	−	0.207333i	$-0.0664783\pi$
$450$	0	0
$451$	−403200.	−1.98229
$452$	− 23080.0i	− 0.112969i
$453$	0	0
$454$	14592.0	0.0707951
$455$	− 235382.i	− 1.13697i
$456$	0	0
$457$	174626.	0.836135	0.418068	−	0.908416i	$-0.362708\pi$
$457$	174626.	0.836135	0.418068	+	0.908416i	$0.362708\pi$
$458$	268672.i	1.28083i
$459$	0	0
$460$	175104.	0.827524
$461$	65947.6i	0.310311i	0.987890	+	0.155156i	$0.0495879\pi$
$461$	65947.6i	0.310311i	−0.987890	+	0.155156i	$0.950412\pi$
$462$	0	0
$463$	6647.00	0.0310073	0.0155036	−	0.999880i	$-0.495065\pi$
$463$	6647.00	0.0310073	0.0155036	+	0.999880i	$0.495065\pi$
$464$	− 99922.7i	− 0.464118i
$465$	0	0
$466$	114048.	0.525189
$467$	232259.i	1.06497i	0.846438	+	0.532487i	$0.178742\pi$
$467$	232259.i	1.06497i	−0.846438	+	0.532487i	$0.821258\pi$
$468$	0	0
$469$	−1679.00	−0.00763317
$470$	16291.7i	0.0737517i
$471$	0	0
$472$	−60672.0	−0.272336
$473$	432071.i	1.93122i
$474$	0	0
$475$	164951.	0.731085
$476$	− 59464.9i	− 0.262450i
$477$	0	0
$478$	−271488.	−1.18821
$479$	− 292776.i	− 1.27604i	−0.770019	−	0.638021i	$-0.779753\pi$
$479$	− 292776.i	− 1.27604i	0.770019	−	0.638021i	$-0.220247\pi$
$480$	0	0
$481$	−36575.0	−0.158086
$482$	− 68422.5i	− 0.294513i
$483$	0	0
$484$	113272.	0.483539
$485$	337205.i	1.43354i
$486$	0	0
$487$	53399.0	0.225152	0.112576	−	0.993643i	$-0.464090\pi$
$487$	53399.0	0.225152	0.112576	+	0.993643i	$0.464090\pi$
$488$	127053.i	0.533513i
$489$	0	0
$490$	281088.	1.17071
$491$	− 79795.6i	− 0.330991i	−0.986211	−	0.165495i	$-0.947078\pi$
$491$	− 79795.6i	− 0.330991i	0.986211	−	0.165495i	$-0.0529223\pi$
$492$	0	0
$493$	−158976.	−0.654090
$494$	84103.3i	0.344635i
$495$	0	0
$496$	−61312.0	−0.249220
$497$	29732.4i	0.120370i
$498$	0	0
$499$	51026.0	0.204923	0.102461	−	0.994737i	$-0.467328\pi$
$499$	51026.0	0.204923	0.102461	+	0.994737i	$0.467328\pi$
$500$	− 26609.8i	− 0.106439i
$501$	0	0
$502$	−55872.0	−0.221711
$503$	− 136138.i	− 0.538075i	−0.963130	−	0.269038i	$-0.913294\pi$
$503$	− 136138.i	− 0.538075i	0.963130	−	0.269038i	$-0.0867056\pi$
$504$	0	0
$505$	322560.	1.26482
$506$	309543.i	1.20898i
$507$	0	0
$508$	87536.0	0.339203
$509$	− 52710.6i	− 0.203452i	−0.994812	−	0.101726i	$-0.967563\pi$
$509$	− 52710.6i	− 0.203452i	0.994812	−	0.101726i	$-0.0324365\pi$
$510$	0	0
$511$	−476471.	−1.82471
$512$	− 11585.2i	− 0.0441942i
$513$	0	0
$514$	−47424.0	−0.179503
$515$	210944.i	0.795340i
$516$	0	0
$517$	−28800.0	−0.107749
$518$	− 79492.9i	− 0.296257i
$519$	0	0
$520$	−72960.0	−0.269822
$521$	− 73618.3i	− 0.271213i	−0.990763	−	0.135606i	$-0.956702\pi$
$521$	− 73618.3i	− 0.271213i	0.990763	−	0.135606i	$-0.0432983\pi$
$522$	0	0
$523$	477047.	1.74405	0.872023	−	0.489465i	$-0.162808\pi$
$523$	477047.	1.74405	0.872023	+	0.489465i	$0.162808\pi$
$524$	− 5973.64i	− 0.0217559i
$525$	0	0
$526$	225408.	0.814700
$527$	97546.8i	0.351230i
$528$	0	0
$529$	−136031.	−0.486101
$530$	254151.i	0.904774i
$531$	0	0
$532$	−182792.	−0.645853
$533$	225708.i	0.794499i
$534$	0	0
$535$	−266112.	−0.929730
$536$	520.431i	0.00181148i
$537$	0	0
$538$	175968.	0.607952
$539$	496898.i	1.71037i
$540$	0	0
$541$	159743.	0.545792	0.272896	−	0.962044i	$-0.412018\pi$
$541$	159743.	0.545792	0.272896	+	0.962044i	$0.412018\pi$
$542$	229784.i	0.782207i
$543$	0	0
$544$	−18432.0	−0.0622837
$545$	− 542447.i	− 1.82627i
$546$	0	0
$547$	82727.0	0.276486	0.138243	−	0.990398i	$-0.455855\pi$
$547$	82727.0	0.276486	0.138243	+	0.990398i	$0.455855\pi$
$548$	− 165904.i	− 0.552454i
$549$	0	0
$550$	−252960.	−0.836231
$551$	488684.i	1.60963i
$552$	0	0
$553$	446833.	1.46115
$554$	− 171748.i	− 0.559592i
$555$	0	0
$556$	86216.0	0.278893
$557$	558298.i	1.79951i	0.436391	+	0.899757i	$0.356257\pi$
$557$	558298.i	1.79951i	−0.436391	+	0.899757i	$0.643743\pi$
$558$	0	0
$559$	241870.	0.774031
$560$	− 158573.i	− 0.505654i
$561$	0	0
$562$	−292416.	−0.925824
$563$	− 426504.i	− 1.34557i	−0.739838	−	0.672785i	$-0.765098\pi$
$563$	− 426504.i	− 1.34557i	0.739838	−	0.672785i	$-0.234902\pi$
$564$	0	0
$565$	−97920.0	−0.306743
$566$	− 410693.i	− 1.28199i
$567$	0	0
$568$	9216.00	0.0285658
$569$	− 480640.i	− 1.48455i	−0.670094	−	0.742276i	$-0.733746\pi$
$569$	− 480640.i	− 1.48455i	0.670094	−	0.742276i	$-0.266254\pi$
$570$	0	0
$571$	−284665.	−0.873096	−0.436548	−	0.899681i	$-0.643799\pi$
$571$	−284665.	−0.873096	−0.436548	+	0.899681i	$0.643799\pi$
$572$	− 128976.i	− 0.394201i
$573$	0	0
$574$	−490560.	−1.48891
$575$	− 339852.i	− 1.02791i
$576$	0	0
$577$	85871.0	0.257926	0.128963	−	0.991649i	$-0.458835\pi$
$577$	85871.0	0.257926	0.128963	+	0.991649i	$0.458835\pi$
$578$	− 206908.i	− 0.619329i
$579$	0	0
$580$	−423936.	−1.26021
$581$	− 183350.i	− 0.543161i
$582$	0	0
$583$	−449280.	−1.32184
$584$	147689.i	0.433035i
$585$	0	0
$586$	−259104.	−0.754534
$587$	− 76401.5i	− 0.221731i	−0.993835	−	0.110865i	$-0.964638\pi$
$587$	− 76401.5i	− 0.221731i	0.993835	−	0.110865i	$-0.0353622\pi$
$588$	0	0
$589$	299854.	0.864329
$590$	257409.i	0.739470i
$591$	0	0
$592$	−24640.0	−0.0703068
$593$	− 612366.i	− 1.74141i	−0.491804	−	0.870706i	$-0.663662\pi$
$593$	− 612366.i	− 1.74141i	0.491804	−	0.870706i	$-0.336338\pi$
$594$	0	0
$595$	−252288.	−0.712628
$596$	− 294337.i	− 0.828615i
$597$	0	0
$598$	173280.	0.484558
$599$	488684.i	1.36199i	0.732287	+	0.680996i	$0.238453\pi$
$599$	488684.i	1.36199i	−0.732287	+	0.680996i	$0.761547\pi$
$600$	0	0
$601$	472418.	1.30791	0.653954	−	0.756534i	$-0.273109\pi$
$601$	472418.	1.30791	0.653954	+	0.756534i	$0.273109\pi$
$602$	525686.i	1.45055i
$603$	0	0
$604$	4040.00	0.0110741
$605$	− 480572.i	− 1.31295i
$606$	0	0
$607$	477527.	1.29605	0.648023	−	0.761621i	$-0.275596\pi$
$607$	477527.	1.29605	0.648023	+	0.761621i	$0.275596\pi$
$608$	56659.1i	0.153272i
$609$	0	0
$610$	539040.	1.44864
$611$	16122.0i	0.0431854i
$612$	0	0
$613$	−242737.	−0.645974	−0.322987	−	0.946403i	$-0.604687\pi$
$613$	−242737.	−0.645974	−0.322987	+	0.946403i	$0.604687\pi$
$614$	24024.7i	0.0637266i
$615$	0	0
$616$	280320.	0.738742
$617$	− 158403.i	− 0.416096i	−0.978119	−	0.208048i	$-0.933289\pi$
$617$	− 158403.i	− 0.416096i	0.978119	−	0.208048i	$-0.0667111\pi$
$618$	0	0
$619$	520199.	1.35765	0.678826	−	0.734300i	$-0.262489\pi$
$619$	520199.	1.35765	0.678826	+	0.734300i	$0.262489\pi$
$620$	260125.i	0.676703i
$621$	0	0
$622$	−132384.	−0.342180
$623$	− 200694.i	− 0.517080i
$624$	0	0
$625$	−442271.	−1.13221
$626$	− 175541.i	− 0.447950i
$627$	0	0
$628$	126320.	0.320297
$629$	39202.0i	0.0990848i
$630$	0	0
$631$	313895.	0.788362	0.394181	−	0.919033i	$-0.371028\pi$
$631$	313895.	0.788362	0.394181	+	0.919033i	$0.371028\pi$
$632$	− 138502.i	− 0.346755i
$633$	0	0
$634$	−336192.	−0.836390
$635$	− 371384.i	− 0.921034i
$636$	0	0
$637$	278160.	0.685513
$638$	− 749420.i	− 1.84113i
$639$	0	0
$640$	−49152.0	−0.120000
$641$	677601.i	1.64914i	0.565759	+	0.824570i	$0.308583\pi$
$641$	677601.i	1.64914i	−0.565759	+	0.824570i	$0.691417\pi$
$642$	0	0
$643$	155282.	0.375577	0.187789	−	0.982209i	$-0.439868\pi$
$643$	155282.	0.375577	0.187789	+	0.982209i	$0.439868\pi$
$644$	376611.i	0.908073i
$645$	0	0
$646$	90144.0	0.216009
$647$	− 672645.i	− 1.60686i	−0.595401	−	0.803429i	$-0.703007\pi$
$647$	− 672645.i	− 1.60686i	0.595401	−	0.803429i	$-0.296993\pi$
$648$	0	0
$649$	−455040.	−1.08034
$650$	141605.i	0.335160i
$651$	0	0
$652$	99272.0	0.233524
$653$	384824.i	0.902477i	0.892403	+	0.451239i	$0.149018\pi$
$653$	384824.i	0.902477i	−0.892403	+	0.451239i	$0.850982\pi$
$654$	0	0
$655$	−25344.0	−0.0590735
$656$	152056.i	0.353343i
$657$	0	0
$658$	−35040.0	−0.0809305
$659$	108815.i	0.250564i	0.992121	+	0.125282i	$0.0399836\pi$
$659$	108815.i	0.250564i	−0.992121	+	0.125282i	$0.960016\pi$
$660$	0	0
$661$	−411889.	−0.942708	−0.471354	−	0.881944i	$-0.656235\pi$
$661$	−411889.	−0.942708	−0.471354	+	0.881944i	$0.656235\pi$
$662$	− 107726.i	− 0.245814i
$663$	0	0
$664$	−56832.0	−0.128901
$665$	775521.i	1.75368i
$666$	0	0
$667$	1.00685e6	2.26314
$668$	118658.i	0.265916i
$669$	0	0
$670$	2208.00	0.00491869
$671$	952897.i	2.11642i
$672$	0	0
$673$	−678097.	−1.49714	−0.748569	−	0.663057i	$-0.769259\pi$
$673$	−678097.	−1.49714	−0.748569	+	0.663057i	$0.769259\pi$
$674$	− 212469.i	− 0.467708i
$675$	0	0
$676$	156288.	0.342005
$677$	467539.i	1.02009i	0.860146	+	0.510047i	$0.170372\pi$
$677$	467539.i	1.02009i	−0.860146	+	0.510047i	$0.829628\pi$
$678$	0	0
$679$	−725255.	−1.57308
$680$	78200.4i	0.169118i
$681$	0	0
$682$	−459840.	−0.988640
$683$	− 626010.i	− 1.34196i	−0.741475	−	0.670981i	$-0.765873\pi$
$683$	− 626010.i	− 1.34196i	0.741475	−	0.670981i	$-0.234127\pi$
$684$	0	0
$685$	−703872.	−1.50007
$686$	108812.i	0.231223i
$687$	0	0
$688$	162944.	0.344240
$689$	251504.i	0.529793i
$690$	0	0
$691$	214226.	0.448659	0.224329	−	0.974513i	$-0.427981\pi$
$691$	214226.	0.448659	0.224329	+	0.974513i	$0.427981\pi$
$692$	− 55935.0i	− 0.116808i
$693$	0	0
$694$	564864.	1.17280
$695$	− 365784.i	− 0.757277i
$696$	0	0
$697$	241920.	0.497973
$698$	178397.i	0.366166i
$699$	0	0
$700$	−307768.	−0.628098
$701$	− 282356.i	− 0.574594i	−0.957842	−	0.287297i	$-0.907243\pi$
$701$	− 282356.i	− 0.574594i	0.957842	−	0.287297i	$-0.0927567\pi$
$702$	0	0
$703$	120505.	0.243834
$704$	− 86889.3i	− 0.175316i
$705$	0	0
$706$	612480.	1.22880
$707$	693757.i	1.38793i
$708$	0	0
$709$	−67969.0	−0.135213	−0.0676065	−	0.997712i	$-0.521536\pi$
$709$	−67969.0	−0.135213	−0.0676065	+	0.997712i	$0.521536\pi$
$710$	− 39100.2i	− 0.0775643i
$711$	0	0
$712$	−62208.0	−0.122712
$713$	− 617796.i	− 1.21525i
$714$	0	0
$715$	−547200.	−1.07037
$716$	420327.i	0.819900i
$717$	0	0
$718$	687456.	1.33351
$719$	728037.i	1.40830i	0.710050	+	0.704151i	$0.248672\pi$
$719$	728037.i	1.40830i	−0.710050	+	0.704151i	$0.751328\pi$
$720$	0	0
$721$	−453695.	−0.872757
$722$	91505.3i	0.175538i
$723$	0	0
$724$	−74104.0	−0.141372
$725$	822801.i	1.56538i
$726$	0	0
$727$	369890.	0.699848	0.349924	−	0.936778i	$-0.386207\pi$
$727$	369890.	0.699848	0.349924	+	0.936778i	$0.386207\pi$
$728$	− 156921.i	− 0.296087i
$729$	0	0
$730$	626592.	1.17582
$731$	− 259242.i	− 0.485145i
$732$	0	0
$733$	242258.	0.450890	0.225445	−	0.974256i	$-0.427616\pi$
$733$	242258.	0.450890	0.225445	+	0.974256i	$0.427616\pi$
$734$	− 54778.1i	− 0.101675i
$735$	0	0
$736$	116736.	0.215501
$737$	3903.23i	0.00718603i
$738$	0	0
$739$	−798190.	−1.46156	−0.730781	−	0.682612i	$-0.760844\pi$
$739$	−798190.	−1.46156	−0.730781	+	0.682612i	$0.760844\pi$
$740$	104539.i	0.190903i
$741$	0	0
$742$	−546624.	−0.992844
$743$	− 569362.i	− 1.03136i	−0.856781	−	0.515681i	$-0.827539\pi$
$743$	− 569362.i	− 1.03136i	0.856781	−	0.515681i	$-0.172461\pi$
$744$	0	0
$745$	−1.24877e6	−2.24993
$746$	369554.i	0.664049i
$747$	0	0
$748$	−138240.	−0.247076
$749$	− 572349.i	− 1.02023i
$750$	0	0
$751$	831479.	1.47425	0.737125	−	0.675756i	$-0.236183\pi$
$751$	831479.	1.47425	0.737125	+	0.675756i	$0.236183\pi$
$752$	10861.2i	0.0192062i
$753$	0	0
$754$	−419520.	−0.737921
$755$	− 17140.3i	− 0.0300693i
$756$	0	0
$757$	−954529.	−1.66570	−0.832851	−	0.553497i	$-0.813293\pi$
$757$	−954529.	−1.66570	−0.832851	+	0.553497i	$0.813293\pi$
$758$	349121.i	0.607628i
$759$	0	0
$760$	240384.	0.416177
$761$	139736.i	0.241289i	0.992696	+	0.120645i	$0.0384961\pi$
$761$	139736.i	0.241289i	−0.992696	+	0.120645i	$0.961504\pi$
$762$	0	0
$763$	1.16669e6	2.00403
$764$	81730.2i	0.140022i
$765$	0	0
$766$	635136.	1.08245
$767$	254728.i	0.432998i
$768$	0	0
$769$	1.08614e6	1.83668	0.918342	−	0.395788i	$-0.129529\pi$
$769$	1.08614e6	1.83668	0.918342	+	0.395788i	$0.129529\pi$
$770$	− 1.18930e6i	− 2.00590i
$771$	0	0
$772$	40712.0	0.0683106
$773$	− 1.11822e6i	− 1.87141i	−0.352778	−	0.935707i	$-0.614763\pi$
$773$	− 1.11822e6i	− 1.87141i	0.352778	−	0.935707i	$-0.385237\pi$
$774$	0	0
$775$	504866.	0.840568
$776$	224803.i	0.373318i
$777$	0	0
$778$	−237408.	−0.392226
$779$	− 743650.i	− 1.22544i
$780$	0	0
$781$	69120.0	0.113319
$782$	− 185726.i	− 0.303710i
$783$	0	0
$784$	187392.	0.304873
$785$	− 535930.i	− 0.869699i
$786$	0	0
$787$	−676633.	−1.09246	−0.546228	−	0.837637i	$-0.683937\pi$
$787$	−676633.	−1.09246	−0.546228	+	0.837637i	$0.683937\pi$
$788$	95306.7i	0.153487i
$789$	0	0
$790$	−587616.	−0.941541
$791$	− 210605.i	− 0.336601i
$792$	0	0
$793$	533425.	0.848256
$794$	− 740058.i	− 1.17388i
$795$	0	0
$796$	−359224.	−0.566943
$797$	− 834816.i	− 1.31424i	−0.753787	−	0.657119i	$-0.771775\pi$
$797$	− 834816.i	− 1.31424i	0.753787	−	0.657119i	$-0.228225\pi$
$798$	0	0
$799$	17280.0	0.0270676
$800$	95397.2i	0.149058i
$801$	0	0
$802$	17664.0	0.0274625
$803$	1.10767e6i	1.71782i
$804$	0	0
$805$	1.59782e6	2.46568
$806$	257415.i	0.396245i
$807$	0	0
$808$	215040.	0.329379
$809$	726204.i	1.10959i	0.831988	+	0.554794i	$0.187203\pi$
$809$	726204.i	1.10959i	−0.831988	+	0.554794i	$0.812797\pi$
$810$	0	0
$811$	−664846.	−1.01083	−0.505416	−	0.862876i	$-0.668661\pi$
$811$	−664846.	−1.01083	−0.505416	+	0.862876i	$0.668661\pi$
$812$	− 911794.i	− 1.38288i
$813$	0	0
$814$	−184800.	−0.278903
$815$	− 421175.i	− 0.634085i
$816$	0	0
$817$	−796898.	−1.19387
$818$	− 837664.i	− 1.25188i
$819$	0	0
$820$	645120.	0.959429
$821$	947942.i	1.40636i	0.711014	+	0.703178i	$0.248236\pi$
$821$	947942.i	1.40636i	−0.711014	+	0.703178i	$0.751764\pi$
$822$	0	0
$823$	−293017.	−0.432607	−0.216303	−	0.976326i	$-0.569400\pi$
$823$	−293017.	−0.432607	−0.216303	+	0.976326i	$0.569400\pi$
$824$	140629.i	0.207120i
$825$	0	0
$826$	−553632.	−0.811449
$827$	643218.i	0.940475i	0.882540	+	0.470238i	$0.155832\pi$
$827$	643218.i	0.940475i	−0.882540	+	0.470238i	$0.844168\pi$
$828$	0	0
$829$	−227761.	−0.331414	−0.165707	−	0.986175i	$-0.552991\pi$
$829$	−227761.	−0.331414	−0.165707	+	0.986175i	$0.552991\pi$
$830$	241118.i	0.350004i
$831$	0	0
$832$	−48640.0	−0.0702663
$833$	− 298139.i	− 0.429664i
$834$	0	0
$835$	503424.	0.722040
$836$	424943.i	0.608020i
$837$	0	0
$838$	−412896.	−0.587967
$839$	− 943224.i	− 1.33996i	−0.742380	−	0.669979i	$-0.766303\pi$
$839$	− 943224.i	− 1.33996i	0.742380	−	0.669979i	$-0.233697\pi$
$840$	0	0
$841$	−1.73035e6	−2.44648
$842$	893605.i	1.26044i
$843$	0	0
$844$	−532408.	−0.747411
$845$	− 663074.i	− 0.928642i
$846$	0	0
$847$	1.03361e6	1.44075
$848$	169434.i	0.235618i
$849$	0	0
$850$	151776.	0.210071
$851$	− 248279.i	− 0.342832i
$852$	0	0
$853$	766943.	1.05406	0.527030	−	0.849847i	$-0.323306\pi$
$853$	766943.	1.05406	0.527030	+	0.849847i	$0.323306\pi$
$854$	1.15936e6i	1.58965i
$855$	0	0
$856$	−177408.	−0.242117
$857$	839296.i	1.14276i	0.820687	+	0.571378i	$0.193591\pi$
$857$	839296.i	1.14276i	−0.820687	+	0.571378i	$0.806409\pi$
$858$	0	0
$859$	−1.18423e6	−1.60491	−0.802455	−	0.596712i	$-0.796473\pi$
$859$	−1.18423e6	−1.60491	−0.802455	+	0.596712i	$0.796473\pi$
$860$	− 691313.i	− 0.934712i
$861$	0	0
$862$	2016.00	0.00271316
$863$	385605.i	0.517751i	0.965911	+	0.258876i	$0.0833520\pi$
$863$	385605.i	0.517751i	−0.965911	+	0.258876i	$0.916648\pi$
$864$	0	0
$865$	−237312.	−0.317167
$866$	83082.2i	0.110783i
$867$	0	0
$868$	−559472.	−0.742573
$869$	− 1.03877e6i	− 1.37556i
$870$	0	0
$871$	2185.00	0.00288015
$872$	− 361631.i	− 0.475590i
$873$	0	0
$874$	−570912.	−0.747388
$875$	− 242815.i	− 0.317146i
$876$	0	0
$877$	1.06570e6	1.38559	0.692793	−	0.721136i	$-0.256380\pi$
$877$	1.06570e6	1.38559	0.692793	+	0.721136i	$0.256380\pi$
$878$	32306.3i	0.0419081i
$879$	0	0
$880$	−368640.	−0.476033
$881$	505553.i	0.651351i	0.945482	+	0.325675i	$0.105592\pi$
$881$	505553.i	0.651351i	−0.945482	+	0.325675i	$0.894408\pi$
$882$	0	0
$883$	211367.	0.271091	0.135546	−	0.990771i	$-0.456721\pi$
$883$	211367.	0.271091	0.135546	+	0.990771i	$0.456721\pi$
$884$	77385.8i	0.0990277i
$885$	0	0
$886$	296256.	0.377398
$887$	341651.i	0.434246i	0.976144	+	0.217123i	$0.0696673\pi$
$887$	341651.i	0.434246i	−0.976144	+	0.217123i	$0.930333\pi$
$888$	0	0
$889$	798766.	1.01069
$890$	263926.i	0.333198i
$891$	0	0
$892$	56816.0	0.0714070
$893$	− 53117.9i	− 0.0666097i
$894$	0	0
$895$	1.78330e6	2.22627
$896$	− 105715.i	− 0.131681i
$897$	0	0
$898$	−236448.	−0.293213
$899$	1.49572e6i	1.85068i
$900$	0	0
$901$	269568.	0.332062
$902$	1.14042e6i	1.40169i
$903$	0	0
$904$	−65280.0	−0.0798810
$905$	314397.i	0.383867i
$906$	0	0
$907$	1.48630e6	1.80672	0.903359	−	0.428885i	$-0.141093\pi$
$907$	1.48630e6	1.80672	0.903359	+	0.428885i	$0.141093\pi$
$908$	− 41272.4i	− 0.0500597i
$909$	0	0
$910$	−665760.	−0.803961
$911$	1.15637e6i	1.39335i	0.717385	+	0.696677i	$0.245339\pi$
$911$	1.15637e6i	1.39335i	−0.717385	+	0.696677i	$0.754661\pi$
$912$	0	0
$913$	−426240.	−0.511343
$914$	− 493917.i	− 0.591237i
$915$	0	0
$916$	759920.	0.905684
$917$	− 54509.4i	− 0.0648236i
$918$	0	0
$919$	−203038.	−0.240407	−0.120203	−	0.992749i	$-0.538355\pi$
$919$	−203038.	−0.240407	−0.120203	+	0.992749i	$0.538355\pi$
$920$	− 495269.i	− 0.585148i
$921$	0	0
$922$	186528.	0.219423
$923$	− 38692.9i	− 0.0454180i
$924$	0	0
$925$	202895.	0.237131
$926$	− 18800.6i	− 0.0219255i
$927$	0	0
$928$	−282624.	−0.328181
$929$	1.17328e6i	1.35947i	0.733459	+	0.679734i	$0.237905\pi$
$929$	1.17328e6i	1.35947i	−0.733459	+	0.679734i	$0.762095\pi$
$930$	0	0
$931$	−916464.	−1.05734
$932$	− 322576.i	− 0.371365i
$933$	0	0
$934$	656928.	0.753050
$935$	586503.i	0.670883i
$936$	0	0
$937$	−1.69454e6	−1.93007	−0.965037	−	0.262114i	$-0.915580\pi$
$937$	−1.69454e6	−1.93007	−0.965037	+	0.262114i	$0.915580\pi$
$938$	4748.93i	0.00539747i
$939$	0	0
$940$	46080.0	0.0521503
$941$	1.23474e6i	1.39443i	0.716860	+	0.697217i	$0.245578\pi$
$941$	1.23474e6i	1.39443i	−0.716860	+	0.697217i	$0.754422\pi$
$942$	0	0
$943$	−1.53216e6	−1.72298
$944$	171606.i	0.192570i
$945$	0	0
$946$	1.22208e6	1.36558
$947$	− 1.52331e6i	− 1.69859i	−0.527918	−	0.849295i	$-0.677027\pi$
$947$	− 1.52331e6i	− 1.69859i	0.527918	−	0.849295i	$-0.322973\pi$
$948$	0	0
$949$	620065.	0.688501
$950$	− 466552.i	− 0.516955i
$951$	0	0
$952$	−168192.	−0.185580
$953$	− 618373.i	− 0.680871i	−0.940268	−	0.340436i	$-0.889425\pi$
$953$	− 618373.i	− 0.680871i	0.940268	−	0.340436i	$-0.110575\pi$
$954$	0	0
$955$	346752.	0.380200
$956$	767884.i	0.840195i
$957$	0	0
$958$	−828096.	−0.902297
$959$	− 1.51388e6i	− 1.64609i
$960$	0	0
$961$	−5757.00	−0.00623375
$962$	103450.i	0.111784i
$963$	0	0
$964$	−193528.	−0.208252
$965$	− 172726.i	− 0.185483i
$966$	0	0
$967$	−1.17444e6	−1.25597	−0.627984	−	0.778227i	$-0.716119\pi$
$967$	−1.17444e6	−1.25597	−0.627984	+	0.778227i	$0.716119\pi$
$968$	− 320382.i	− 0.341914i
$969$	0	0
$970$	953760.	1.01367
$971$	− 1.02607e6i	− 1.08828i	−0.838995	−	0.544140i	$-0.816856\pi$
$971$	− 1.02607e6i	− 1.08828i	0.838995	−	0.544140i	$-0.183144\pi$
$972$	0	0
$973$	786721.	0.830989
$974$	− 151035.i	− 0.159206i
$975$	0	0
$976$	359360.	0.377251
$977$	296849.i	0.310990i	0.987837	+	0.155495i	$0.0496973\pi$
$977$	296849.i	0.310990i	−0.987837	+	0.155495i	$0.950303\pi$
$978$	0	0
$979$	−466560.	−0.486791
$980$	− 795037.i	− 0.827819i
$981$	0	0
$982$	−225696.	−0.234046
$983$	− 1.08832e6i	− 1.12629i	−0.826358	−	0.563145i	$-0.809591\pi$
$983$	− 1.08832e6i	− 1.12629i	0.826358	−	0.563145i	$-0.190409\pi$
$984$	0	0
$985$	404352.	0.416761
$986$	449652.i	0.462512i
$987$	0	0
$988$	237880.	0.243694
$989$	1.64187e6i	1.67859i
$990$	0	0
$991$	149735.	0.152467	0.0762335	−	0.997090i	$-0.475711\pi$
$991$	149735.	0.152467	0.0762335	+	0.997090i	$0.475711\pi$
$992$	173417.i	0.176225i
$993$	0	0
$994$	84096.0	0.0851143
$995$	1.52406e6i	1.53941i
$996$	0	0
$997$	402290.	0.404715	0.202357	−	0.979312i	$-0.435140\pi$
$997$	402290.	0.404715	0.202357	+	0.979312i	$0.435140\pi$
$998$	− 144323.i	− 0.144902i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.5.b.a.53.1	✓	2
3.2	odd	2	inner	54.5.b.a.53.2	yes	2
4.3	odd	2		432.5.e.g.161.2		2
5.2	odd	4		1350.5.b.b.1349.3		4
5.3	odd	4		1350.5.b.b.1349.2		4
5.4	even	2		1350.5.d.a.701.2		2
9.2	odd	6		162.5.d.b.53.2		4
9.4	even	3		162.5.d.b.107.2		4
9.5	odd	6		162.5.d.b.107.1		4
9.7	even	3		162.5.d.b.53.1		4
12.11	even	2		432.5.e.g.161.1		2
15.2	even	4		1350.5.b.b.1349.1		4
15.8	even	4		1350.5.b.b.1349.4		4
15.14	odd	2		1350.5.d.a.701.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.5.b.a.53.1	✓	2	1.1	even	1	trivial
54.5.b.a.53.2	yes	2	3.2	odd	2	inner
162.5.d.b.53.1		4	9.7	even	3
162.5.d.b.53.2		4	9.2	odd	6
162.5.d.b.107.1		4	9.5	odd	6
162.5.d.b.107.2		4	9.4	even	3
432.5.e.g.161.1		2	12.11	even	2
432.5.e.g.161.2		2	4.3	odd	2
1350.5.b.b.1349.1		4	15.2	even	4
1350.5.b.b.1349.2		4	5.3	odd	4
1350.5.b.b.1349.3		4	5.2	odd	4
1350.5.b.b.1349.4		4	15.8	even	4
1350.5.d.a.701.1		2	15.14	odd	2
1350.5.d.a.701.2		2	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	54.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -8.00000 q^{4} +33.9411i q^{5} -73.0000 q^{7} +22.6274i q^{8} +O(q^{10})\)	\(q-2.82843i q^{2} -8.00000 q^{4} +33.9411i q^{5} -73.0000 q^{7} +22.6274i q^{8} +96.0000 q^{10} +169.706i q^{11} +95.0000 q^{13} +206.475i q^{14} +64.0000 q^{16} -101.823i q^{17} -313.000 q^{19} -271.529i q^{20} +480.000 q^{22} +644.881i q^{23} -527.000 q^{25} -268.701i q^{26} +584.000 q^{28} -1561.29i q^{29} -958.000 q^{31} -181.019i q^{32} -288.000 q^{34} -2477.70i q^{35} -385.000 q^{37} +885.298i q^{38} -768.000 q^{40} +2375.88i q^{41} +2546.00 q^{43} -1357.65i q^{44} +1824.00 q^{46} +169.706i q^{47} +2928.00 q^{49} +1490.58i q^{50} -760.000 q^{52} +2647.41i q^{53} -5760.00 q^{55} -1651.80i q^{56} -4416.00 q^{58} +2681.35i q^{59} +5615.00 q^{61} +2709.63i q^{62} -512.000 q^{64} +3224.41i q^{65} +23.0000 q^{67} +814.587i q^{68} -7008.00 q^{70} -407.294i q^{71} +6527.00 q^{73} +1088.94i q^{74} +2504.00 q^{76} -12388.5i q^{77} -6121.00 q^{79} +2172.23i q^{80} +6720.00 q^{82} +2511.64i q^{83} +3456.00 q^{85} -7201.18i q^{86} -3840.00 q^{88} +2749.23i q^{89} -6935.00 q^{91} -5159.05i q^{92} +480.000 q^{94} -10623.6i q^{95} +9935.00 q^{97} -8281.63i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 16 q^{4} - 146 q^{7}+O(q^{10}) \) 2 * q - 16 * q^4 - 146 * q^7	\( 2 q - 16 q^{4} - 146 q^{7} + 192 q^{10} + 190 q^{13} + 128 q^{16} - 626 q^{19} + 960 q^{22} - 1054 q^{25} + 1168 q^{28} - 1916 q^{31} - 576 q^{34} - 770 q^{37} - 1536 q^{40} + 5092 q^{43} + 3648 q^{46} + 5856 q^{49} - 1520 q^{52} - 11520 q^{55} - 8832 q^{58} + 11230 q^{61} - 1024 q^{64} + 46 q^{67} - 14016 q^{70} + 13054 q^{73} + 5008 q^{76} - 12242 q^{79} + 13440 q^{82} + 6912 q^{85} - 7680 q^{88} - 13870 q^{91} + 960 q^{94} + 19870 q^{97}+O(q^{100}) \) 2 * q - 16 * q^4 - 146 * q^7 + 192 * q^10 + 190 * q^13 + 128 * q^16 - 626 * q^19 + 960 * q^22 - 1054 * q^25 + 1168 * q^28 - 1916 * q^31 - 576 * q^34 - 770 * q^37 - 1536 * q^40 + 5092 * q^43 + 3648 * q^46 + 5856 * q^49 - 1520 * q^52 - 11520 * q^55 - 8832 * q^58 + 11230 * q^61 - 1024 * q^64 + 46 * q^67 - 14016 * q^70 + 13054 * q^73 + 5008 * q^76 - 12242 * q^79 + 13440 * q^82 + 6912 * q^85 - 7680 * q^88 - 13870 * q^91 + 960 * q^94 + 19870 * q^97

Introduction

L-functions

Modular forms

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Representations

Groups

Database

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