LMFDB - Embedded newform 4416.2.a.bs.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4416,2,Mod(1,4416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4416, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4416.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4416 = 2^{6} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4416.a (trivial)

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:	yes
Analytic conductor:	$35.2619375326$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 552)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	4416.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.42864 q^{5} -0.622216 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +4.42864 q^{5} -0.622216 q^{7} +1.00000 q^{9} +5.80642 q^{11} -2.00000 q^{13} +4.42864 q^{15} +1.37778 q^{17} +6.42864 q^{19} -0.622216 q^{21} +1.00000 q^{23} +14.6128 q^{25} +1.00000 q^{27} +0.755569 q^{29} +1.24443 q^{31} +5.80642 q^{33} -2.75557 q^{35} -9.05086 q^{37} -2.00000 q^{39} -9.61285 q^{41} -5.18421 q^{43} +4.42864 q^{45} +5.24443 q^{47} -6.61285 q^{49} +1.37778 q^{51} -12.0415 q^{53} +25.7146 q^{55} +6.42864 q^{57} -12.8573 q^{59} -2.94914 q^{61} -0.622216 q^{63} -8.85728 q^{65} +5.18421 q^{67} +1.00000 q^{69} -6.10171 q^{71} +2.00000 q^{73} +14.6128 q^{75} -3.61285 q^{77} +8.62222 q^{79} +1.00000 q^{81} +11.9081 q^{83} +6.10171 q^{85} +0.755569 q^{87} +6.23506 q^{89} +1.24443 q^{91} +1.24443 q^{93} +28.4701 q^{95} -2.85728 q^{97} +5.80642 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 2 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 2 q^{7} + 3 q^{9} + 4 q^{11} - 6 q^{13} + 4 q^{17} + 6 q^{19} - 2 q^{21} + 3 q^{23} + 17 q^{25} + 3 q^{27} + 2 q^{29} + 4 q^{31} + 4 q^{33} - 8 q^{35} - 14 q^{37} - 6 q^{39} - 2 q^{41} - 2 q^{43} + 16 q^{47} + 7 q^{49} + 4 q^{51} + 4 q^{53} + 24 q^{55} + 6 q^{57} - 12 q^{59} - 22 q^{61} - 2 q^{63} + 2 q^{67} + 3 q^{69} + 8 q^{71} + 6 q^{73} + 17 q^{75} + 16 q^{77} + 26 q^{79} + 3 q^{81} - 4 q^{83} - 8 q^{85} + 2 q^{87} - 8 q^{89} + 4 q^{91} + 4 q^{93} + 32 q^{95} + 18 q^{97} + 4 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 2 * q^7 + 3 * q^9 + 4 * q^11 - 6 * q^13 + 4 * q^17 + 6 * q^19 - 2 * q^21 + 3 * q^23 + 17 * q^25 + 3 * q^27 + 2 * q^29 + 4 * q^31 + 4 * q^33 - 8 * q^35 - 14 * q^37 - 6 * q^39 - 2 * q^41 - 2 * q^43 + 16 * q^47 + 7 * q^49 + 4 * q^51 + 4 * q^53 + 24 * q^55 + 6 * q^57 - 12 * q^59 - 22 * q^61 - 2 * q^63 + 2 * q^67 + 3 * q^69 + 8 * q^71 + 6 * q^73 + 17 * q^75 + 16 * q^77 + 26 * q^79 + 3 * q^81 - 4 * q^83 - 8 * q^85 + 2 * q^87 - 8 * q^89 + 4 * q^91 + 4 * q^93 + 32 * q^95 + 18 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	4.42864	1.98055	0.990274	−	0.139132i	$-0.0444311\pi$
$5$	4.42864	1.98055	0.990274	+	0.139132i	$0.0444311\pi$
$6$	0	0
$7$	−0.622216	−0.235175	−0.117588	−	0.993063i	$-0.537516\pi$
$7$	−0.622216	−0.235175	−0.117588	+	0.993063i	$0.537516\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.80642	1.75070	0.875351	−	0.483487i	$-0.160630\pi$
$11$	5.80642	1.75070	0.875351	+	0.483487i	$0.160630\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	4.42864	1.14347
$16$	0	0
$17$	1.37778	0.334162	0.167081	−	0.985943i	$-0.446566\pi$
$17$	1.37778	0.334162	0.167081	+	0.985943i	$0.446566\pi$
$18$	0	0
$19$	6.42864	1.47483	0.737416	−	0.675439i	$-0.236046\pi$
$19$	6.42864	1.47483	0.737416	+	0.675439i	$0.236046\pi$
$20$	0	0
$21$	−0.622216	−0.135779
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	14.6128	2.92257
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.755569	0.140306	0.0701528	−	0.997536i	$-0.477651\pi$
$29$	0.755569	0.140306	0.0701528	+	0.997536i	$0.477651\pi$
$30$	0	0
$31$	1.24443	0.223506	0.111753	−	0.993736i	$-0.464353\pi$
$31$	1.24443	0.223506	0.111753	+	0.993736i	$0.464353\pi$
$32$	0	0
$33$	5.80642	1.01077
$34$	0	0
$35$	−2.75557	−0.465776
$36$	0	0
$37$	−9.05086	−1.48795	−0.743976	−	0.668207i	$-0.767062\pi$
$37$	−9.05086	−1.48795	−0.743976	+	0.668207i	$0.767062\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−9.61285	−1.50127	−0.750637	−	0.660715i	$-0.770253\pi$
$41$	−9.61285	−1.50127	−0.750637	+	0.660715i	$0.770253\pi$
$42$	0	0
$43$	−5.18421	−0.790584	−0.395292	−	0.918555i	$-0.629357\pi$
$43$	−5.18421	−0.790584	−0.395292	+	0.918555i	$0.629357\pi$
$44$	0	0
$45$	4.42864	0.660183
$46$	0	0
$47$	5.24443	0.764979	0.382489	−	0.923960i	$-0.375067\pi$
$47$	5.24443	0.764979	0.382489	+	0.923960i	$0.375067\pi$
$48$	0	0
$49$	−6.61285	−0.944693
$50$	0	0
$51$	1.37778	0.192928
$52$	0	0
$53$	−12.0415	−1.65403	−0.827013	−	0.562183i	$-0.809962\pi$
$53$	−12.0415	−1.65403	−0.827013	+	0.562183i	$0.809962\pi$
$54$	0	0
$55$	25.7146	3.46735
$56$	0	0
$57$	6.42864	0.851494
$58$	0	0
$59$	−12.8573	−1.67388	−0.836938	−	0.547298i	$-0.815656\pi$
$59$	−12.8573	−1.67388	−0.836938	+	0.547298i	$0.815656\pi$
$60$	0	0
$61$	−2.94914	−0.377599	−0.188800	−	0.982016i	$-0.560460\pi$
$61$	−2.94914	−0.377599	−0.188800	+	0.982016i	$0.560460\pi$
$62$	0	0
$63$	−0.622216	−0.0783918
$64$	0	0
$65$	−8.85728	−1.09861
$66$	0	0
$67$	5.18421	0.633352	0.316676	−	0.948534i	$-0.397433\pi$
$67$	5.18421	0.633352	0.316676	+	0.948534i	$0.397433\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−6.10171	−0.724140	−0.362070	−	0.932151i	$-0.617930\pi$
$71$	−6.10171	−0.724140	−0.362070	+	0.932151i	$0.617930\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	14.6128	1.68735
$76$	0	0
$77$	−3.61285	−0.411722
$78$	0	0
$79$	8.62222	0.970075	0.485038	−	0.874493i	$-0.338806\pi$
$79$	8.62222	0.970075	0.485038	+	0.874493i	$0.338806\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.9081	1.30709	0.653544	−	0.756889i	$-0.273282\pi$
$83$	11.9081	1.30709	0.653544	+	0.756889i	$0.273282\pi$
$84$	0	0
$85$	6.10171	0.661823
$86$	0	0
$87$	0.755569	0.0810055
$88$	0	0
$89$	6.23506	0.660915	0.330458	−	0.943821i	$-0.392797\pi$
$89$	6.23506	0.660915	0.330458	+	0.943821i	$0.392797\pi$
$90$	0	0
$91$	1.24443	0.130452
$92$	0	0
$93$	1.24443	0.129042
$94$	0	0
$95$	28.4701	2.92097
$96$	0	0
$97$	−2.85728	−0.290113	−0.145056	−	0.989423i	$-0.546336\pi$
$97$	−2.85728	−0.290113	−0.145056	+	0.989423i	$0.546336\pi$
$98$	0	0
$99$	5.80642	0.583568
$100$	0	0
$101$	−15.7146	−1.56366	−0.781828	−	0.623494i	$-0.785713\pi$
$101$	−15.7146	−1.56366	−0.781828	+	0.623494i	$0.785713\pi$
$102$	0	0
$103$	−5.47949	−0.539911	−0.269955	−	0.962873i	$-0.587009\pi$
$103$	−5.47949	−0.539911	−0.269955	+	0.962873i	$0.587009\pi$
$104$	0	0
$105$	−2.75557	−0.268916
$106$	0	0
$107$	−0.295286	−0.0285464	−0.0142732	−	0.999898i	$-0.504543\pi$
$107$	−0.295286	−0.0285464	−0.0142732	+	0.999898i	$0.504543\pi$
$108$	0	0
$109$	9.31756	0.892461	0.446230	−	0.894918i	$-0.352766\pi$
$109$	9.31756	0.892461	0.446230	+	0.894918i	$0.352766\pi$
$110$	0	0
$111$	−9.05086	−0.859069
$112$	0	0
$113$	−11.4795	−1.07990	−0.539950	−	0.841697i	$-0.681557\pi$
$113$	−11.4795	−1.07990	−0.539950	+	0.841697i	$0.681557\pi$
$114$	0	0
$115$	4.42864	0.412973
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	−0.857279	−0.0785866
$120$	0	0
$121$	22.7146	2.06496
$122$	0	0
$123$	−9.61285	−0.866761
$124$	0	0
$125$	42.5718	3.80774
$126$	0	0
$127$	−8.47013	−0.751602	−0.375801	−	0.926700i	$-0.622632\pi$
$127$	−8.47013	−0.751602	−0.375801	+	0.926700i	$0.622632\pi$
$128$	0	0
$129$	−5.18421	−0.456444
$130$	0	0
$131$	−7.61285	−0.665138	−0.332569	−	0.943079i	$-0.607915\pi$
$131$	−7.61285	−0.665138	−0.332569	+	0.943079i	$0.607915\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	0	0
$135$	4.42864	0.381157
$136$	0	0
$137$	−5.37778	−0.459455	−0.229728	−	0.973255i	$-0.573784\pi$
$137$	−5.37778	−0.459455	−0.229728	+	0.973255i	$0.573784\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	5.24443	0.441661
$142$	0	0
$143$	−11.6128	−0.971115
$144$	0	0
$145$	3.34614	0.277882
$146$	0	0
$147$	−6.61285	−0.545418
$148$	0	0
$149$	−13.2859	−1.08842	−0.544212	−	0.838947i	$-0.683171\pi$
$149$	−13.2859	−1.08842	−0.544212	+	0.838947i	$0.683171\pi$
$150$	0	0
$151$	2.48886	0.202541	0.101270	−	0.994859i	$-0.467709\pi$
$151$	2.48886	0.202541	0.101270	+	0.994859i	$0.467709\pi$
$152$	0	0
$153$	1.37778	0.111387
$154$	0	0
$155$	5.51114	0.442665
$156$	0	0
$157$	11.1526	0.890072	0.445036	−	0.895513i	$-0.353191\pi$
$157$	11.1526	0.890072	0.445036	+	0.895513i	$0.353191\pi$
$158$	0	0
$159$	−12.0415	−0.954952
$160$	0	0
$161$	−0.622216	−0.0490375
$162$	0	0
$163$	−8.85728	−0.693756	−0.346878	−	0.937910i	$-0.612758\pi$
$163$	−8.85728	−0.693756	−0.346878	+	0.937910i	$0.612758\pi$
$164$	0	0
$165$	25.7146	2.00188
$166$	0	0
$167$	−1.89829	−0.146894	−0.0734470	−	0.997299i	$-0.523400\pi$
$167$	−1.89829	−0.146894	−0.0734470	+	0.997299i	$0.523400\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	6.42864	0.491610
$172$	0	0
$173$	16.7556	1.27390	0.636951	−	0.770904i	$-0.280195\pi$
$173$	16.7556	1.27390	0.636951	+	0.770904i	$0.280195\pi$
$174$	0	0
$175$	−9.09234	−0.687316
$176$	0	0
$177$	−12.8573	−0.966412
$178$	0	0
$179$	−18.9590	−1.41706	−0.708531	−	0.705680i	$-0.750642\pi$
$179$	−18.9590	−1.41706	−0.708531	+	0.705680i	$0.750642\pi$
$180$	0	0
$181$	−1.70471	−0.126710	−0.0633552	−	0.997991i	$-0.520180\pi$
$181$	−1.70471	−0.126710	−0.0633552	+	0.997991i	$0.520180\pi$
$182$	0	0
$183$	−2.94914	−0.218007
$184$	0	0
$185$	−40.0830	−2.94696
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	−0.622216	−0.0452595
$190$	0	0
$191$	10.3684	0.750232	0.375116	−	0.926978i	$-0.377603\pi$
$191$	10.3684	0.750232	0.375116	+	0.926978i	$0.377603\pi$
$192$	0	0
$193$	1.61285	0.116095	0.0580477	−	0.998314i	$-0.481512\pi$
$193$	1.61285	0.116095	0.0580477	+	0.998314i	$0.481512\pi$
$194$	0	0
$195$	−8.85728	−0.634283
$196$	0	0
$197$	−5.34614	−0.380897	−0.190448	−	0.981697i	$-0.560994\pi$
$197$	−5.34614	−0.380897	−0.190448	+	0.981697i	$0.560994\pi$
$198$	0	0
$199$	21.3590	1.51410	0.757051	−	0.653355i	$-0.226639\pi$
$199$	21.3590	1.51410	0.757051	+	0.653355i	$0.226639\pi$
$200$	0	0
$201$	5.18421	0.365666
$202$	0	0
$203$	−0.470127	−0.0329964
$204$	0	0
$205$	−42.5718	−2.97335
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	37.3274	2.58199
$210$	0	0
$211$	−6.36842	−0.438420	−0.219210	−	0.975678i	$-0.570348\pi$
$211$	−6.36842	−0.438420	−0.219210	+	0.975678i	$0.570348\pi$
$212$	0	0
$213$	−6.10171	−0.418082
$214$	0	0
$215$	−22.9590	−1.56579
$216$	0	0
$217$	−0.774305	−0.0525632
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	−2.75557	−0.185360
$222$	0	0
$223$	7.22570	0.483868	0.241934	−	0.970293i	$-0.422218\pi$
$223$	7.22570	0.483868	0.241934	+	0.970293i	$0.422218\pi$
$224$	0	0
$225$	14.6128	0.974190
$226$	0	0
$227$	15.0509	0.998960	0.499480	−	0.866325i	$-0.333524\pi$
$227$	15.0509	0.998960	0.499480	+	0.866325i	$0.333524\pi$
$228$	0	0
$229$	6.29529	0.416004	0.208002	−	0.978128i	$-0.433304\pi$
$229$	6.29529	0.416004	0.208002	+	0.978128i	$0.433304\pi$
$230$	0	0
$231$	−3.61285	−0.237708
$232$	0	0
$233$	27.7146	1.81564	0.907821	−	0.419359i	$-0.137745\pi$
$233$	27.7146	1.81564	0.907821	+	0.419359i	$0.137745\pi$
$234$	0	0
$235$	23.2257	1.51508
$236$	0	0
$237$	8.62222	0.560073
$238$	0	0
$239$	−15.2257	−0.984868	−0.492434	−	0.870350i	$-0.663893\pi$
$239$	−15.2257	−0.984868	−0.492434	+	0.870350i	$0.663893\pi$
$240$	0	0
$241$	5.14272	0.331272	0.165636	−	0.986187i	$-0.447032\pi$
$241$	5.14272	0.331272	0.165636	+	0.986187i	$0.447032\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−29.2859	−1.87101
$246$	0	0
$247$	−12.8573	−0.818089
$248$	0	0
$249$	11.9081	0.754647
$250$	0	0
$251$	11.4380	0.721961	0.360980	−	0.932573i	$-0.382442\pi$
$251$	11.4380	0.721961	0.360980	+	0.932573i	$0.382442\pi$
$252$	0	0
$253$	5.80642	0.365047
$254$	0	0
$255$	6.10171	0.382104
$256$	0	0
$257$	−4.10171	−0.255858	−0.127929	−	0.991783i	$-0.540833\pi$
$257$	−4.10171	−0.255858	−0.127929	+	0.991783i	$0.540833\pi$
$258$	0	0
$259$	5.63158	0.349930
$260$	0	0
$261$	0.755569	0.0467685
$262$	0	0
$263$	24.4701	1.50889	0.754446	−	0.656362i	$-0.227905\pi$
$263$	24.4701	1.50889	0.754446	+	0.656362i	$0.227905\pi$
$264$	0	0
$265$	−53.3274	−3.27588
$266$	0	0
$267$	6.23506	0.381580
$268$	0	0
$269$	14.8573	0.905864	0.452932	−	0.891545i	$-0.350378\pi$
$269$	14.8573	0.905864	0.452932	+	0.891545i	$0.350378\pi$
$270$	0	0
$271$	17.2444	1.04752	0.523762	−	0.851864i	$-0.324528\pi$
$271$	17.2444	1.04752	0.523762	+	0.851864i	$0.324528\pi$
$272$	0	0
$273$	1.24443	0.0753164
$274$	0	0
$275$	84.8484	5.11655
$276$	0	0
$277$	−14.5906	−0.876663	−0.438331	−	0.898813i	$-0.644430\pi$
$277$	−14.5906	−0.876663	−0.438331	+	0.898813i	$0.644430\pi$
$278$	0	0
$279$	1.24443	0.0745022
$280$	0	0
$281$	7.00937	0.418144	0.209072	−	0.977900i	$-0.432956\pi$
$281$	7.00937	0.418144	0.209072	+	0.977900i	$0.432956\pi$
$282$	0	0
$283$	7.08250	0.421011	0.210505	−	0.977593i	$-0.432489\pi$
$283$	7.08250	0.421011	0.210505	+	0.977593i	$0.432489\pi$
$284$	0	0
$285$	28.4701	1.68642
$286$	0	0
$287$	5.98126	0.353063
$288$	0	0
$289$	−15.1017	−0.888336
$290$	0	0
$291$	−2.85728	−0.167497
$292$	0	0
$293$	5.67307	0.331424	0.165712	−	0.986174i	$-0.447008\pi$
$293$	5.67307	0.331424	0.165712	+	0.986174i	$0.447008\pi$
$294$	0	0
$295$	−56.9403	−3.31519
$296$	0	0
$297$	5.80642	0.336923
$298$	0	0
$299$	−2.00000	−0.115663
$300$	0	0
$301$	3.22570	0.185926
$302$	0	0
$303$	−15.7146	−0.902778
$304$	0	0
$305$	−13.0607	−0.747853
$306$	0	0
$307$	−29.0607	−1.65858	−0.829291	−	0.558817i	$-0.811255\pi$
$307$	−29.0607	−1.65858	−0.829291	+	0.558817i	$0.811255\pi$
$308$	0	0
$309$	−5.47949	−0.311718
$310$	0	0
$311$	−14.3684	−0.814758	−0.407379	−	0.913259i	$-0.633557\pi$
$311$	−14.3684	−0.814758	−0.407379	+	0.913259i	$0.633557\pi$
$312$	0	0
$313$	−29.8163	−1.68532	−0.842658	−	0.538450i	$-0.819010\pi$
$313$	−29.8163	−1.68532	−0.842658	+	0.538450i	$0.819010\pi$
$314$	0	0
$315$	−2.75557	−0.155259
$316$	0	0
$317$	−20.7556	−1.16575	−0.582874	−	0.812562i	$-0.698072\pi$
$317$	−20.7556	−1.16575	−0.582874	+	0.812562i	$0.698072\pi$
$318$	0	0
$319$	4.38715	0.245633
$320$	0	0
$321$	−0.295286	−0.0164813
$322$	0	0
$323$	8.85728	0.492832
$324$	0	0
$325$	−29.2257	−1.62115
$326$	0	0
$327$	9.31756	0.515262
$328$	0	0
$329$	−3.26317	−0.179904
$330$	0	0
$331$	−8.85728	−0.486840	−0.243420	−	0.969921i	$-0.578269\pi$
$331$	−8.85728	−0.486840	−0.243420	+	0.969921i	$0.578269\pi$
$332$	0	0
$333$	−9.05086	−0.495984
$334$	0	0
$335$	22.9590	1.25438
$336$	0	0
$337$	−19.9813	−1.08845	−0.544224	−	0.838940i	$-0.683176\pi$
$337$	−19.9813	−1.08845	−0.544224	+	0.838940i	$0.683176\pi$
$338$	0	0
$339$	−11.4795	−0.623481
$340$	0	0
$341$	7.22570	0.391293
$342$	0	0
$343$	8.47013	0.457344
$344$	0	0
$345$	4.42864	0.238430
$346$	0	0
$347$	3.73329	0.200414	0.100207	−	0.994967i	$-0.468050\pi$
$347$	3.73329	0.200414	0.100207	+	0.994967i	$0.468050\pi$
$348$	0	0
$349$	4.10171	0.219560	0.109780	−	0.993956i	$-0.464985\pi$
$349$	4.10171	0.219560	0.109780	+	0.993956i	$0.464985\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	5.73329	0.305152	0.152576	−	0.988292i	$-0.451243\pi$
$353$	5.73329	0.305152	0.152576	+	0.988292i	$0.451243\pi$
$354$	0	0
$355$	−27.0223	−1.43419
$356$	0	0
$357$	−0.857279	−0.0453720
$358$	0	0
$359$	20.8573	1.10081	0.550403	−	0.834899i	$-0.314474\pi$
$359$	20.8573	1.10081	0.550403	+	0.834899i	$0.314474\pi$
$360$	0	0
$361$	22.3274	1.17513
$362$	0	0
$363$	22.7146	1.19221
$364$	0	0
$365$	8.85728	0.463611
$366$	0	0
$367$	15.8479	0.827254	0.413627	−	0.910446i	$-0.364262\pi$
$367$	15.8479	0.827254	0.413627	+	0.910446i	$0.364262\pi$
$368$	0	0
$369$	−9.61285	−0.500425
$370$	0	0
$371$	7.49240	0.388986
$372$	0	0
$373$	−25.0509	−1.29708	−0.648542	−	0.761179i	$-0.724621\pi$
$373$	−25.0509	−1.29708	−0.648542	+	0.761179i	$0.724621\pi$
$374$	0	0
$375$	42.5718	2.19840
$376$	0	0
$377$	−1.51114	−0.0778275
$378$	0	0
$379$	9.45091	0.485461	0.242730	−	0.970094i	$-0.421957\pi$
$379$	9.45091	0.485461	0.242730	+	0.970094i	$0.421957\pi$
$380$	0	0
$381$	−8.47013	−0.433938
$382$	0	0
$383$	−37.9813	−1.94075	−0.970376	−	0.241600i	$-0.922328\pi$
$383$	−37.9813	−1.94075	−0.970376	+	0.241600i	$0.922328\pi$
$384$	0	0
$385$	−16.0000	−0.815436
$386$	0	0
$387$	−5.18421	−0.263528
$388$	0	0
$389$	35.0005	1.77459	0.887297	−	0.461198i	$-0.152580\pi$
$389$	35.0005	1.77459	0.887297	+	0.461198i	$0.152580\pi$
$390$	0	0
$391$	1.37778	0.0696776
$392$	0	0
$393$	−7.61285	−0.384017
$394$	0	0
$395$	38.1847	1.92128
$396$	0	0
$397$	−33.4291	−1.67776	−0.838880	−	0.544317i	$-0.816789\pi$
$397$	−33.4291	−1.67776	−0.838880	+	0.544317i	$0.816789\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	0	0
$401$	32.0701	1.60150	0.800751	−	0.598997i	$-0.204434\pi$
$401$	32.0701	1.60150	0.800751	+	0.598997i	$0.204434\pi$
$402$	0	0
$403$	−2.48886	−0.123979
$404$	0	0
$405$	4.42864	0.220061
$406$	0	0
$407$	−52.5531	−2.60496
$408$	0	0
$409$	11.3274	0.560104	0.280052	−	0.959985i	$-0.409648\pi$
$409$	11.3274	0.560104	0.280052	+	0.959985i	$0.409648\pi$
$410$	0	0
$411$	−5.37778	−0.265267
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	52.7368	2.58875
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	−11.3176	−0.552899	−0.276450	−	0.961028i	$-0.589158\pi$
$419$	−11.3176	−0.552899	−0.276450	+	0.961028i	$0.589158\pi$
$420$	0	0
$421$	−0.580728	−0.0283030	−0.0141515	−	0.999900i	$-0.504505\pi$
$421$	−0.580728	−0.0283030	−0.0141515	+	0.999900i	$0.504505\pi$
$422$	0	0
$423$	5.24443	0.254993
$424$	0	0
$425$	20.1334	0.976611
$426$	0	0
$427$	1.83500	0.0888021
$428$	0	0
$429$	−11.6128	−0.560674
$430$	0	0
$431$	32.3497	1.55823	0.779115	−	0.626881i	$-0.215669\pi$
$431$	32.3497	1.55823	0.779115	+	0.626881i	$0.215669\pi$
$432$	0	0
$433$	5.61285	0.269736	0.134868	−	0.990864i	$-0.456939\pi$
$433$	5.61285	0.269736	0.134868	+	0.990864i	$0.456939\pi$
$434$	0	0
$435$	3.34614	0.160435
$436$	0	0
$437$	6.42864	0.307524
$438$	0	0
$439$	10.4889	0.500606	0.250303	−	0.968168i	$-0.419470\pi$
$439$	10.4889	0.500606	0.250303	+	0.968168i	$0.419470\pi$
$440$	0	0
$441$	−6.61285	−0.314898
$442$	0	0
$443$	−20.9403	−0.994901	−0.497451	−	0.867492i	$-0.665730\pi$
$443$	−20.9403	−0.994901	−0.497451	+	0.867492i	$0.665730\pi$
$444$	0	0
$445$	27.6128	1.30897
$446$	0	0
$447$	−13.2859	−0.628402
$448$	0	0
$449$	−4.87601	−0.230113	−0.115057	−	0.993359i	$-0.536705\pi$
$449$	−4.87601	−0.230113	−0.115057	+	0.993359i	$0.536705\pi$
$450$	0	0
$451$	−55.8163	−2.62829
$452$	0	0
$453$	2.48886	0.116937
$454$	0	0
$455$	5.51114	0.258366
$456$	0	0
$457$	22.8573	1.06922	0.534609	−	0.845099i	$-0.320459\pi$
$457$	22.8573	1.06922	0.534609	+	0.845099i	$0.320459\pi$
$458$	0	0
$459$	1.37778	0.0643095
$460$	0	0
$461$	−23.5941	−1.09889	−0.549444	−	0.835531i	$-0.685160\pi$
$461$	−23.5941	−1.09889	−0.549444	+	0.835531i	$0.685160\pi$
$462$	0	0
$463$	−10.4889	−0.487459	−0.243729	−	0.969843i	$-0.578371\pi$
$463$	−10.4889	−0.487459	−0.243729	+	0.969843i	$0.578371\pi$
$464$	0	0
$465$	5.51114	0.255573
$466$	0	0
$467$	−15.0509	−0.696471	−0.348235	−	0.937407i	$-0.613219\pi$
$467$	−15.0509	−0.696471	−0.348235	+	0.937407i	$0.613219\pi$
$468$	0	0
$469$	−3.22570	−0.148949
$470$	0	0
$471$	11.1526	0.513883
$472$	0	0
$473$	−30.1017	−1.38408
$474$	0	0
$475$	93.9407	4.31030
$476$	0	0
$477$	−12.0415	−0.551342
$478$	0	0
$479$	33.7146	1.54046	0.770229	−	0.637768i	$-0.220142\pi$
$479$	33.7146	1.54046	0.770229	+	0.637768i	$0.220142\pi$
$480$	0	0
$481$	18.1017	0.825367
$482$	0	0
$483$	−0.622216	−0.0283118
$484$	0	0
$485$	−12.6539	−0.574582
$486$	0	0
$487$	12.2034	0.552990	0.276495	−	0.961015i	$-0.410827\pi$
$487$	12.2034	0.552990	0.276495	+	0.961015i	$0.410827\pi$
$488$	0	0
$489$	−8.85728	−0.400540
$490$	0	0
$491$	−6.75557	−0.304875	−0.152437	−	0.988313i	$-0.548712\pi$
$491$	−6.75557	−0.304875	−0.152437	+	0.988313i	$0.548712\pi$
$492$	0	0
$493$	1.04101	0.0468848
$494$	0	0
$495$	25.7146	1.15578
$496$	0	0
$497$	3.79658	0.170300
$498$	0	0
$499$	37.5941	1.68294	0.841472	−	0.540301i	$-0.181690\pi$
$499$	37.5941	1.68294	0.841472	+	0.540301i	$0.181690\pi$
$500$	0	0
$501$	−1.89829	−0.0848093
$502$	0	0
$503$	8.59057	0.383035	0.191517	−	0.981489i	$-0.438659\pi$
$503$	8.59057	0.383035	0.191517	+	0.981489i	$0.438659\pi$
$504$	0	0
$505$	−69.5941	−3.09690
$506$	0	0
$507$	−9.00000	−0.399704
$508$	0	0
$509$	3.77784	0.167450	0.0837250	−	0.996489i	$-0.473318\pi$
$509$	3.77784	0.167450	0.0837250	+	0.996489i	$0.473318\pi$
$510$	0	0
$511$	−1.24443	−0.0550504
$512$	0	0
$513$	6.42864	0.283831
$514$	0	0
$515$	−24.2667	−1.06932
$516$	0	0
$517$	30.4514	1.33925
$518$	0	0
$519$	16.7556	0.735488
$520$	0	0
$521$	30.8256	1.35050	0.675248	−	0.737591i	$-0.264037\pi$
$521$	30.8256	1.35050	0.675248	+	0.737591i	$0.264037\pi$
$522$	0	0
$523$	6.89877	0.301662	0.150831	−	0.988560i	$-0.451805\pi$
$523$	6.89877	0.301662	0.150831	+	0.988560i	$0.451805\pi$
$524$	0	0
$525$	−9.09234	−0.396822
$526$	0	0
$527$	1.71456	0.0746873
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−12.8573	−0.557958
$532$	0	0
$533$	19.2257	0.832757
$534$	0	0
$535$	−1.30772	−0.0565375
$536$	0	0
$537$	−18.9590	−0.818141
$538$	0	0
$539$	−38.3970	−1.65388
$540$	0	0
$541$	30.5906	1.31519	0.657596	−	0.753371i	$-0.271574\pi$
$541$	30.5906	1.31519	0.657596	+	0.753371i	$0.271574\pi$
$542$	0	0
$543$	−1.70471	−0.0731563
$544$	0	0
$545$	41.2641	1.76756
$546$	0	0
$547$	−27.3461	−1.16924	−0.584618	−	0.811308i	$-0.698756\pi$
$547$	−27.3461	−1.16924	−0.584618	+	0.811308i	$0.698756\pi$
$548$	0	0
$549$	−2.94914	−0.125866
$550$	0	0
$551$	4.85728	0.206927
$552$	0	0
$553$	−5.36488	−0.228138
$554$	0	0
$555$	−40.0830	−1.70143
$556$	0	0
$557$	−34.1432	−1.44669	−0.723347	−	0.690485i	$-0.757397\pi$
$557$	−34.1432	−1.44669	−0.723347	+	0.690485i	$0.757397\pi$
$558$	0	0
$559$	10.3684	0.438537
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	31.0509	1.30864	0.654319	−	0.756219i	$-0.272955\pi$
$563$	31.0509	1.30864	0.654319	+	0.756219i	$0.272955\pi$
$564$	0	0
$565$	−50.8385	−2.13879
$566$	0	0
$567$	−0.622216	−0.0261306
$568$	0	0
$569$	−21.8479	−0.915912	−0.457956	−	0.888975i	$-0.651418\pi$
$569$	−21.8479	−0.915912	−0.457956	+	0.888975i	$0.651418\pi$
$570$	0	0
$571$	18.1619	0.760053	0.380027	−	0.924976i	$-0.375915\pi$
$571$	18.1619	0.760053	0.380027	+	0.924976i	$0.375915\pi$
$572$	0	0
$573$	10.3684	0.433147
$574$	0	0
$575$	14.6128	0.609398
$576$	0	0
$577$	−39.3274	−1.63722	−0.818611	−	0.574349i	$-0.805255\pi$
$577$	−39.3274	−1.63722	−0.818611	+	0.574349i	$0.805255\pi$
$578$	0	0
$579$	1.61285	0.0670277
$580$	0	0
$581$	−7.40943	−0.307395
$582$	0	0
$583$	−69.9180	−2.89571
$584$	0	0
$585$	−8.85728	−0.366203
$586$	0	0
$587$	−38.8385	−1.60304	−0.801519	−	0.597969i	$-0.795975\pi$
$587$	−38.8385	−1.60304	−0.801519	+	0.597969i	$0.795975\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	−5.34614	−0.219911
$592$	0	0
$593$	29.6128	1.21605	0.608027	−	0.793916i	$-0.291961\pi$
$593$	29.6128	1.21605	0.608027	+	0.793916i	$0.291961\pi$
$594$	0	0
$595$	−3.79658	−0.155645
$596$	0	0
$597$	21.3590	0.874168
$598$	0	0
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	0	0
$601$	0.672594	0.0274357	0.0137178	−	0.999906i	$-0.495633\pi$
$601$	0.672594	0.0274357	0.0137178	+	0.999906i	$0.495633\pi$
$602$	0	0
$603$	5.18421	0.211117
$604$	0	0
$605$	100.595	4.08975
$606$	0	0
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	0	0
$609$	−0.470127	−0.0190505
$610$	0	0
$611$	−10.4889	−0.424334
$612$	0	0
$613$	−2.94914	−0.119115	−0.0595574	−	0.998225i	$-0.518969\pi$
$613$	−2.94914	−0.119115	−0.0595574	+	0.998225i	$0.518969\pi$
$614$	0	0
$615$	−42.5718	−1.71666
$616$	0	0
$617$	39.1753	1.57714	0.788569	−	0.614946i	$-0.210822\pi$
$617$	39.1753	1.57714	0.788569	+	0.614946i	$0.210822\pi$
$618$	0	0
$619$	−23.3689	−0.939275	−0.469638	−	0.882859i	$-0.655615\pi$
$619$	−23.3689	−0.939275	−0.469638	+	0.882859i	$0.655615\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−3.87955	−0.155431
$624$	0	0
$625$	115.471	4.61884
$626$	0	0
$627$	37.3274	1.49071
$628$	0	0
$629$	−12.4701	−0.497217
$630$	0	0
$631$	22.7239	0.904625	0.452313	−	0.891859i	$-0.350599\pi$
$631$	22.7239	0.904625	0.452313	+	0.891859i	$0.350599\pi$
$632$	0	0
$633$	−6.36842	−0.253122
$634$	0	0
$635$	−37.5111	−1.48858
$636$	0	0
$637$	13.2257	0.524021
$638$	0	0
$639$	−6.10171	−0.241380
$640$	0	0
$641$	45.5812	1.80035	0.900175	−	0.435529i	$-0.143439\pi$
$641$	45.5812	1.80035	0.900175	+	0.435529i	$0.143439\pi$
$642$	0	0
$643$	13.8381	0.545720	0.272860	−	0.962054i	$-0.412030\pi$
$643$	13.8381	0.545720	0.272860	+	0.962054i	$0.412030\pi$
$644$	0	0
$645$	−22.9590	−0.904009
$646$	0	0
$647$	−18.5718	−0.730134	−0.365067	−	0.930981i	$-0.618954\pi$
$647$	−18.5718	−0.730134	−0.365067	+	0.930981i	$0.618954\pi$
$648$	0	0
$649$	−74.6548	−2.93046
$650$	0	0
$651$	−0.774305	−0.0303474
$652$	0	0
$653$	10.0000	0.391330	0.195665	−	0.980671i	$-0.437313\pi$
$653$	10.0000	0.391330	0.195665	+	0.980671i	$0.437313\pi$
$654$	0	0
$655$	−33.7146	−1.31734
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	−33.2988	−1.29714	−0.648569	−	0.761156i	$-0.724632\pi$
$659$	−33.2988	−1.29714	−0.648569	+	0.761156i	$0.724632\pi$
$660$	0	0
$661$	−7.68598	−0.298950	−0.149475	−	0.988766i	$-0.547758\pi$
$661$	−7.68598	−0.298950	−0.149475	+	0.988766i	$0.547758\pi$
$662$	0	0
$663$	−2.75557	−0.107017
$664$	0	0
$665$	−17.7146	−0.686941
$666$	0	0
$667$	0.755569	0.0292557
$668$	0	0
$669$	7.22570	0.279362
$670$	0	0
$671$	−17.1240	−0.661064
$672$	0	0
$673$	−31.3274	−1.20758	−0.603791	−	0.797142i	$-0.706344\pi$
$673$	−31.3274	−1.20758	−0.603791	+	0.797142i	$0.706344\pi$
$674$	0	0
$675$	14.6128	0.562449
$676$	0	0
$677$	3.89523	0.149706	0.0748529	−	0.997195i	$-0.476151\pi$
$677$	3.89523	0.149706	0.0748529	+	0.997195i	$0.476151\pi$
$678$	0	0
$679$	1.77784	0.0682274
$680$	0	0
$681$	15.0509	0.576750
$682$	0	0
$683$	−10.8760	−0.416159	−0.208080	−	0.978112i	$-0.566721\pi$
$683$	−10.8760	−0.416159	−0.208080	+	0.978112i	$0.566721\pi$
$684$	0	0
$685$	−23.8163	−0.909973
$686$	0	0
$687$	6.29529	0.240180
$688$	0	0
$689$	24.0830	0.917488
$690$	0	0
$691$	10.5718	0.402172	0.201086	−	0.979574i	$-0.435553\pi$
$691$	10.5718	0.402172	0.201086	+	0.979574i	$0.435553\pi$
$692$	0	0
$693$	−3.61285	−0.137241
$694$	0	0
$695$	−17.7146	−0.671951
$696$	0	0
$697$	−13.2444	−0.501669
$698$	0	0
$699$	27.7146	1.04826
$700$	0	0
$701$	−18.2065	−0.687649	−0.343825	−	0.939034i	$-0.611723\pi$
$701$	−18.2065	−0.687649	−0.343825	+	0.939034i	$0.611723\pi$
$702$	0	0
$703$	−58.1847	−2.19448
$704$	0	0
$705$	23.2257	0.874730
$706$	0	0
$707$	9.77784	0.367734
$708$	0	0
$709$	17.1338	0.643474	0.321737	−	0.946829i	$-0.395733\pi$
$709$	17.1338	0.643474	0.321737	+	0.946829i	$0.395733\pi$
$710$	0	0
$711$	8.62222	0.323358
$712$	0	0
$713$	1.24443	0.0466043
$714$	0	0
$715$	−51.4291	−1.92334
$716$	0	0
$717$	−15.2257	−0.568614
$718$	0	0
$719$	−3.34614	−0.124790	−0.0623950	−	0.998052i	$-0.519874\pi$
$719$	−3.34614	−0.124790	−0.0623950	+	0.998052i	$0.519874\pi$
$720$	0	0
$721$	3.40943	0.126974
$722$	0	0
$723$	5.14272	0.191260
$724$	0	0
$725$	11.0410	0.410053
$726$	0	0
$727$	9.09234	0.337216	0.168608	−	0.985683i	$-0.446073\pi$
$727$	9.09234	0.337216	0.168608	+	0.985683i	$0.446073\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−7.14272	−0.264183
$732$	0	0
$733$	40.3595	1.49071	0.745357	−	0.666666i	$-0.232279\pi$
$733$	40.3595	1.49071	0.745357	+	0.666666i	$0.232279\pi$
$734$	0	0
$735$	−29.2859	−1.08023
$736$	0	0
$737$	30.1017	1.10881
$738$	0	0
$739$	41.9180	1.54198	0.770989	−	0.636849i	$-0.219762\pi$
$739$	41.9180	1.54198	0.770989	+	0.636849i	$0.219762\pi$
$740$	0	0
$741$	−12.8573	−0.472324
$742$	0	0
$743$	−34.9590	−1.28252	−0.641260	−	0.767323i	$-0.721588\pi$
$743$	−34.9590	−1.28252	−0.641260	+	0.767323i	$0.721588\pi$
$744$	0	0
$745$	−58.8385	−2.15568
$746$	0	0
$747$	11.9081	0.435696
$748$	0	0
$749$	0.183732	0.00671341
$750$	0	0
$751$	−25.7462	−0.939492	−0.469746	−	0.882802i	$-0.655655\pi$
$751$	−25.7462	−0.939492	−0.469746	+	0.882802i	$0.655655\pi$
$752$	0	0
$753$	11.4380	0.416824
$754$	0	0
$755$	11.0223	0.401142
$756$	0	0
$757$	−40.9304	−1.48764	−0.743821	−	0.668379i	$-0.766988\pi$
$757$	−40.9304	−1.48764	−0.743821	+	0.668379i	$0.766988\pi$
$758$	0	0
$759$	5.80642	0.210760
$760$	0	0
$761$	−2.26671	−0.0821680	−0.0410840	−	0.999156i	$-0.513081\pi$
$761$	−2.26671	−0.0821680	−0.0410840	+	0.999156i	$0.513081\pi$
$762$	0	0
$763$	−5.79753	−0.209885
$764$	0	0
$765$	6.10171	0.220608
$766$	0	0
$767$	25.7146	0.928499
$768$	0	0
$769$	−41.6128	−1.50060	−0.750299	−	0.661099i	$-0.770090\pi$
$769$	−41.6128	−1.50060	−0.750299	+	0.661099i	$0.770090\pi$
$770$	0	0
$771$	−4.10171	−0.147719
$772$	0	0
$773$	−2.44738	−0.0880260	−0.0440130	−	0.999031i	$-0.514014\pi$
$773$	−2.44738	−0.0880260	−0.0440130	+	0.999031i	$0.514014\pi$
$774$	0	0
$775$	18.1847	0.653213
$776$	0	0
$777$	5.63158	0.202032
$778$	0	0
$779$	−61.7975	−2.21413
$780$	0	0
$781$	−35.4291	−1.26775
$782$	0	0
$783$	0.755569	0.0270018
$784$	0	0
$785$	49.3907	1.76283
$786$	0	0
$787$	−36.4099	−1.29787	−0.648936	−	0.760843i	$-0.724786\pi$
$787$	−36.4099	−1.29787	−0.648936	+	0.760843i	$0.724786\pi$
$788$	0	0
$789$	24.4701	0.871160
$790$	0	0
$791$	7.14272	0.253966
$792$	0	0
$793$	5.89829	0.209454
$794$	0	0
$795$	−53.3274	−1.89133
$796$	0	0
$797$	−0.428639	−0.0151832	−0.00759159	−	0.999971i	$-0.502417\pi$
$797$	−0.428639	−0.0151832	−0.00759159	+	0.999971i	$0.502417\pi$
$798$	0	0
$799$	7.22570	0.255627
$800$	0	0
$801$	6.23506	0.220305
$802$	0	0
$803$	11.6128	0.409808
$804$	0	0
$805$	−2.75557	−0.0971210
$806$	0	0
$807$	14.8573	0.523001
$808$	0	0
$809$	−35.9813	−1.26503	−0.632517	−	0.774547i	$-0.717978\pi$
$809$	−35.9813	−1.26503	−0.632517	+	0.774547i	$0.717978\pi$
$810$	0	0
$811$	4.12045	0.144688	0.0723442	−	0.997380i	$-0.476952\pi$
$811$	4.12045	0.144688	0.0723442	+	0.997380i	$0.476952\pi$
$812$	0	0
$813$	17.2444	0.604789
$814$	0	0
$815$	−39.2257	−1.37402
$816$	0	0
$817$	−33.3274	−1.16598
$818$	0	0
$819$	1.24443	0.0434839
$820$	0	0
$821$	24.4514	0.853359	0.426680	−	0.904403i	$-0.359683\pi$
$821$	24.4514	0.853359	0.426680	+	0.904403i	$0.359683\pi$
$822$	0	0
$823$	12.2667	0.427590	0.213795	−	0.976878i	$-0.431417\pi$
$823$	12.2667	0.427590	0.213795	+	0.976878i	$0.431417\pi$
$824$	0	0
$825$	84.8484	2.95404
$826$	0	0
$827$	−21.5022	−0.747706	−0.373853	−	0.927488i	$-0.621964\pi$
$827$	−21.5022	−0.747706	−0.373853	+	0.927488i	$0.621964\pi$
$828$	0	0
$829$	2.38715	0.0829092	0.0414546	−	0.999140i	$-0.486801\pi$
$829$	2.38715	0.0829092	0.0414546	+	0.999140i	$0.486801\pi$
$830$	0	0
$831$	−14.5906	−0.506141
$832$	0	0
$833$	−9.11108	−0.315680
$834$	0	0
$835$	−8.40684	−0.290931
$836$	0	0
$837$	1.24443	0.0430138
$838$	0	0
$839$	−1.24443	−0.0429625	−0.0214813	−	0.999769i	$-0.506838\pi$
$839$	−1.24443	−0.0429625	−0.0214813	+	0.999769i	$0.506838\pi$
$840$	0	0
$841$	−28.4291	−0.980314
$842$	0	0
$843$	7.00937	0.241415
$844$	0	0
$845$	−39.8578	−1.37115
$846$	0	0
$847$	−14.1334	−0.485628
$848$	0	0
$849$	7.08250	0.243071
$850$	0	0
$851$	−9.05086	−0.310259
$852$	0	0
$853$	−38.0000	−1.30110	−0.650548	−	0.759465i	$-0.725461\pi$
$853$	−38.0000	−1.30110	−0.650548	+	0.759465i	$0.725461\pi$
$854$	0	0
$855$	28.4701	0.973658
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	5.98126	0.203841
$862$	0	0
$863$	−40.8573	−1.39080	−0.695399	−	0.718624i	$-0.744772\pi$
$863$	−40.8573	−1.39080	−0.695399	+	0.718624i	$0.744772\pi$
$864$	0	0
$865$	74.2044	2.52302
$866$	0	0
$867$	−15.1017	−0.512881
$868$	0	0
$869$	50.0642	1.69831
$870$	0	0
$871$	−10.3684	−0.351320
$872$	0	0
$873$	−2.85728	−0.0967042
$874$	0	0
$875$	−26.4889	−0.895487
$876$	0	0
$877$	−7.89829	−0.266706	−0.133353	−	0.991069i	$-0.542574\pi$
$877$	−7.89829	−0.266706	−0.133353	+	0.991069i	$0.542574\pi$
$878$	0	0
$879$	5.67307	0.191348
$880$	0	0
$881$	−48.9906	−1.65054	−0.825268	−	0.564741i	$-0.808976\pi$
$881$	−48.9906	−1.65054	−0.825268	+	0.564741i	$0.808976\pi$
$882$	0	0
$883$	8.85728	0.298071	0.149036	−	0.988832i	$-0.452383\pi$
$883$	8.85728	0.298071	0.149036	+	0.988832i	$0.452383\pi$
$884$	0	0
$885$	−56.9403	−1.91403
$886$	0	0
$887$	−51.1624	−1.71787	−0.858933	−	0.512088i	$-0.828872\pi$
$887$	−51.1624	−1.71787	−0.858933	+	0.512088i	$0.828872\pi$
$888$	0	0
$889$	5.27025	0.176758
$890$	0	0
$891$	5.80642	0.194523
$892$	0	0
$893$	33.7146	1.12821
$894$	0	0
$895$	−83.9625	−2.80656
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	0	0
$899$	0.940253	0.0313592
$900$	0	0
$901$	−16.5906	−0.552712
$902$	0	0
$903$	3.22570	0.107344
$904$	0	0
$905$	−7.54956	−0.250956
$906$	0	0
$907$	24.0228	0.797662	0.398831	−	0.917024i	$-0.369416\pi$
$907$	24.0228	0.797662	0.398831	+	0.917024i	$0.369416\pi$
$908$	0	0
$909$	−15.7146	−0.521219
$910$	0	0
$911$	34.8385	1.15425	0.577126	−	0.816655i	$-0.304174\pi$
$911$	34.8385	1.15425	0.577126	+	0.816655i	$0.304174\pi$
$912$	0	0
$913$	69.1437	2.28832
$914$	0	0
$915$	−13.0607	−0.431773
$916$	0	0
$917$	4.73683	0.156424
$918$	0	0
$919$	−38.6035	−1.27341	−0.636706	−	0.771107i	$-0.719703\pi$
$919$	−38.6035	−1.27341	−0.636706	+	0.771107i	$0.719703\pi$
$920$	0	0
$921$	−29.0607	−0.957583
$922$	0	0
$923$	12.2034	0.401680
$924$	0	0
$925$	−132.259	−4.34864
$926$	0	0
$927$	−5.47949	−0.179970
$928$	0	0
$929$	−29.8163	−0.978240	−0.489120	−	0.872216i	$-0.662682\pi$
$929$	−29.8163	−0.978240	−0.489120	+	0.872216i	$0.662682\pi$
$930$	0	0
$931$	−42.5116	−1.39326
$932$	0	0
$933$	−14.3684	−0.470401
$934$	0	0
$935$	35.4291	1.15866
$936$	0	0
$937$	−35.5111	−1.16010	−0.580049	−	0.814581i	$-0.696967\pi$
$937$	−35.5111	−1.16010	−0.580049	+	0.814581i	$0.696967\pi$
$938$	0	0
$939$	−29.8163	−0.973017
$940$	0	0
$941$	21.2029	0.691196	0.345598	−	0.938383i	$-0.387676\pi$
$941$	21.2029	0.691196	0.345598	+	0.938383i	$0.387676\pi$
$942$	0	0
$943$	−9.61285	−0.313037
$944$	0	0
$945$	−2.75557	−0.0896387
$946$	0	0
$947$	−37.2070	−1.20906	−0.604532	−	0.796581i	$-0.706640\pi$
$947$	−37.2070	−1.20906	−0.604532	+	0.796581i	$0.706640\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	0	0
$951$	−20.7556	−0.673045
$952$	0	0
$953$	39.2958	1.27291	0.636457	−	0.771312i	$-0.280399\pi$
$953$	39.2958	1.27291	0.636457	+	0.771312i	$0.280399\pi$
$954$	0	0
$955$	45.9180	1.48587
$956$	0	0
$957$	4.38715	0.141816
$958$	0	0
$959$	3.34614	0.108053
$960$	0	0
$961$	−29.4514	−0.950045
$962$	0	0
$963$	−0.295286	−0.00951547
$964$	0	0
$965$	7.14272	0.229932
$966$	0	0
$967$	23.9367	0.769753	0.384876	−	0.922968i	$-0.374244\pi$
$967$	23.9367	0.769753	0.384876	+	0.922968i	$0.374244\pi$
$968$	0	0
$969$	8.85728	0.284537
$970$	0	0
$971$	−51.8448	−1.66378	−0.831890	−	0.554940i	$-0.812741\pi$
$971$	−51.8448	−1.66378	−0.831890	+	0.554940i	$0.812741\pi$
$972$	0	0
$973$	2.48886	0.0797893
$974$	0	0
$975$	−29.2257	−0.935971
$976$	0	0
$977$	29.1111	0.931346	0.465673	−	0.884957i	$-0.345812\pi$
$977$	29.1111	0.931346	0.465673	+	0.884957i	$0.345812\pi$
$978$	0	0
$979$	36.2034	1.15707
$980$	0	0
$981$	9.31756	0.297487
$982$	0	0
$983$	−1.36488	−0.0435328	−0.0217664	−	0.999763i	$-0.506929\pi$
$983$	−1.36488	−0.0435328	−0.0217664	+	0.999763i	$0.506929\pi$
$984$	0	0
$985$	−23.6761	−0.754384
$986$	0	0
$987$	−3.26317	−0.103868
$988$	0	0
$989$	−5.18421	−0.164848
$990$	0	0
$991$	−38.3881	−1.21944	−0.609719	−	0.792618i	$-0.708718\pi$
$991$	−38.3881	−1.21944	−0.609719	+	0.792618i	$0.708718\pi$
$992$	0	0
$993$	−8.85728	−0.281077
$994$	0	0
$995$	94.5915	2.99875
$996$	0	0
$997$	−12.3051	−0.389707	−0.194854	−	0.980832i	$-0.562423\pi$
$997$	−12.3051	−0.389707	−0.194854	+	0.980832i	$0.562423\pi$
$998$	0	0
$999$	−9.05086	−0.286356

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4416.2.a.bs.1.3		3
4.3	odd	2		4416.2.a.bp.1.3		3
8.3	odd	2		552.2.a.g.1.1	✓	3
8.5	even	2		1104.2.a.o.1.1		3
24.5	odd	2		3312.2.a.bf.1.3		3
24.11	even	2		1656.2.a.n.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.a.g.1.1	✓	3	8.3	odd	2
1104.2.a.o.1.1		3	8.5	even	2
1656.2.a.n.1.3		3	24.11	even	2
3312.2.a.bf.1.3		3	24.5	odd	2
4416.2.a.bp.1.3		3	4.3	odd	2
4416.2.a.bs.1.3		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4416 = 2^{6} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4416.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.42864 q^{5} -0.622216 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +4.42864 q^{5} -0.622216 q^{7} +1.00000 q^{9} +5.80642 q^{11} -2.00000 q^{13} +4.42864 q^{15} +1.37778 q^{17} +6.42864 q^{19} -0.622216 q^{21} +1.00000 q^{23} +14.6128 q^{25} +1.00000 q^{27} +0.755569 q^{29} +1.24443 q^{31} +5.80642 q^{33} -2.75557 q^{35} -9.05086 q^{37} -2.00000 q^{39} -9.61285 q^{41} -5.18421 q^{43} +4.42864 q^{45} +5.24443 q^{47} -6.61285 q^{49} +1.37778 q^{51} -12.0415 q^{53} +25.7146 q^{55} +6.42864 q^{57} -12.8573 q^{59} -2.94914 q^{61} -0.622216 q^{63} -8.85728 q^{65} +5.18421 q^{67} +1.00000 q^{69} -6.10171 q^{71} +2.00000 q^{73} +14.6128 q^{75} -3.61285 q^{77} +8.62222 q^{79} +1.00000 q^{81} +11.9081 q^{83} +6.10171 q^{85} +0.755569 q^{87} +6.23506 q^{89} +1.24443 q^{91} +1.24443 q^{93} +28.4701 q^{95} -2.85728 q^{97} +5.80642 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 2 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 2 q^{7} + 3 q^{9} + 4 q^{11} - 6 q^{13} + 4 q^{17} + 6 q^{19} - 2 q^{21} + 3 q^{23} + 17 q^{25} + 3 q^{27} + 2 q^{29} + 4 q^{31} + 4 q^{33} - 8 q^{35} - 14 q^{37} - 6 q^{39} - 2 q^{41} - 2 q^{43} + 16 q^{47} + 7 q^{49} + 4 q^{51} + 4 q^{53} + 24 q^{55} + 6 q^{57} - 12 q^{59} - 22 q^{61} - 2 q^{63} + 2 q^{67} + 3 q^{69} + 8 q^{71} + 6 q^{73} + 17 q^{75} + 16 q^{77} + 26 q^{79} + 3 q^{81} - 4 q^{83} - 8 q^{85} + 2 q^{87} - 8 q^{89} + 4 q^{91} + 4 q^{93} + 32 q^{95} + 18 q^{97} + 4 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 2 * q^7 + 3 * q^9 + 4 * q^11 - 6 * q^13 + 4 * q^17 + 6 * q^19 - 2 * q^21 + 3 * q^23 + 17 * q^25 + 3 * q^27 + 2 * q^29 + 4 * q^31 + 4 * q^33 - 8 * q^35 - 14 * q^37 - 6 * q^39 - 2 * q^41 - 2 * q^43 + 16 * q^47 + 7 * q^49 + 4 * q^51 + 4 * q^53 + 24 * q^55 + 6 * q^57 - 12 * q^59 - 22 * q^61 - 2 * q^63 + 2 * q^67 + 3 * q^69 + 8 * q^71 + 6 * q^73 + 17 * q^75 + 16 * q^77 + 26 * q^79 + 3 * q^81 - 4 * q^83 - 8 * q^85 + 2 * q^87 - 8 * q^89 + 4 * q^91 + 4 * q^93 + 32 * q^95 + 18 * q^97 + 4 * q^99

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