LMFDB - Embedded newform 9984.2.a.bu.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9984,2,Mod(1,9984)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9984, 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("9984.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9984 = 2^{8} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9984.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:	$79.7226413780$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 14x^{6} + 24x^{5} + 65x^{4} - 82x^{3} - 126x^{2} + 84x + 92 $ x^8 - 2x^7 - 14x^6 + 24x^5 + 65x^4 - 82x^3 - 126x^2 + 84*x + 92
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 312)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.60416$ of defining polynomial
Character	$\chi$	$=$	9984.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.47174 q^{5} +2.93973 q^{7} +1.00000 q^{9} +4.32135 q^{11} -1.00000 q^{13} +1.47174 q^{15} +6.88672 q^{17} -6.56749 q^{19} +2.93973 q^{21} +3.54085 q^{23} -2.83399 q^{25} +1.00000 q^{27} -7.96637 q^{29} -4.92763 q^{31} +4.32135 q^{33} +4.32651 q^{35} -3.45073 q^{37} -1.00000 q^{39} +11.4381 q^{41} -1.43599 q^{43} +1.47174 q^{45} +6.41147 q^{47} +1.64203 q^{49} +6.88672 q^{51} +9.46174 q^{53} +6.35988 q^{55} -6.56749 q^{57} -0.602544 q^{59} +5.75217 q^{61} +2.93973 q^{63} -1.47174 q^{65} +8.57959 q^{67} +3.54085 q^{69} +3.26862 q^{71} -15.3835 q^{73} -2.83399 q^{75} +12.7036 q^{77} +8.85689 q^{79} +1.00000 q^{81} +9.95232 q^{83} +10.1354 q^{85} -7.96637 q^{87} +5.10329 q^{89} -2.93973 q^{91} -4.92763 q^{93} -9.66561 q^{95} +6.42874 q^{97} +4.32135 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} - 2 q^{5} - 2 q^{7} + 8 q^{9} - 8 q^{13} - 2 q^{15} + 8 q^{17} + 6 q^{19} - 2 q^{21} + 4 q^{23} + 16 q^{25} + 8 q^{27} - 4 q^{29} - 2 q^{31} - 4 q^{35} - 8 q^{37} - 8 q^{39} + 18 q^{41} + 16 q^{43}+ \cdots + 20 q^{97}+O(q^{100}) $ 8 * q + 8 * q^3 - 2 * q^5 - 2 * q^7 + 8 * q^9 - 8 * q^13 - 2 * q^15 + 8 * q^17 + 6 * q^19 - 2 * q^21 + 4 * q^23 + 16 * q^25 + 8 * q^27 - 4 * q^29 - 2 * q^31 - 4 * q^35 - 8 * q^37 - 8 * q^39 + 18 * q^41 + 16 * q^43 - 2 * q^45 + 12 * q^47 + 24 * q^49 + 8 * q^51 + 4 * q^53 - 12 * q^55 + 6 * q^57 - 4 * q^59 - 12 * q^61 - 2 * q^63 + 2 * q^65 + 22 * q^67 + 4 * q^69 + 16 * q^73 + 16 * q^75 + 20 * q^77 + 8 * q^81 + 16 * q^83 + 4 * q^85 - 4 * q^87 + 30 * q^89 + 2 * q^91 - 2 * q^93 - 12 * q^95 + 20 * q^97

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
$5$	1.47174	0.658181	0.329090	−	0.944298i	$-0.393258\pi$
$5$	1.47174	0.658181	0.329090	+	0.944298i	$0.393258\pi$
$6$	0	0
$7$	2.93973	1.11111	0.555557	−	0.831478i	$-0.312505\pi$
$7$	2.93973	1.11111	0.555557	+	0.831478i	$0.312505\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.32135	1.30293	0.651467	−	0.758677i	$-0.274154\pi$
$11$	4.32135	1.30293	0.651467	+	0.758677i	$0.274154\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	1.47174	0.380001
$16$	0	0
$17$	6.88672	1.67027	0.835137	−	0.550041i	$-0.185388\pi$
$17$	6.88672	1.67027	0.835137	+	0.550041i	$0.185388\pi$
$18$	0	0
$19$	−6.56749	−1.50668	−0.753342	−	0.657629i	$-0.771560\pi$
$19$	−6.56749	−1.50668	−0.753342	+	0.657629i	$0.771560\pi$
$20$	0	0
$21$	2.93973	0.641502
$22$	0	0
$23$	3.54085	0.738318	0.369159	−	0.929366i	$-0.379646\pi$
$23$	3.54085	0.738318	0.369159	+	0.929366i	$0.379646\pi$
$24$	0	0
$25$	−2.83399	−0.566798
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−7.96637	−1.47932	−0.739659	−	0.672982i	$-0.765013\pi$
$29$	−7.96637	−1.47932	−0.739659	+	0.672982i	$0.765013\pi$
$30$	0	0
$31$	−4.92763	−0.885028	−0.442514	−	0.896761i	$-0.645913\pi$
$31$	−4.92763	−0.885028	−0.442514	+	0.896761i	$0.645913\pi$
$32$	0	0
$33$	4.32135	0.752250
$34$	0	0
$35$	4.32651	0.731314
$36$	0	0
$37$	−3.45073	−0.567296	−0.283648	−	0.958928i	$-0.591545\pi$
$37$	−3.45073	−0.567296	−0.283648	+	0.958928i	$0.591545\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	11.4381	1.78633	0.893166	−	0.449727i	$-0.148479\pi$
$41$	11.4381	1.78633	0.893166	+	0.449727i	$0.148479\pi$
$42$	0	0
$43$	−1.43599	−0.218987	−0.109493	−	0.993988i	$-0.534923\pi$
$43$	−1.43599	−0.218987	−0.109493	+	0.993988i	$0.534923\pi$
$44$	0	0
$45$	1.47174	0.219394
$46$	0	0
$47$	6.41147	0.935209	0.467604	−	0.883938i	$-0.345117\pi$
$47$	6.41147	0.935209	0.467604	+	0.883938i	$0.345117\pi$
$48$	0	0
$49$	1.64203	0.234575
$50$	0	0
$51$	6.88672	0.964334
$52$	0	0
$53$	9.46174	1.29967	0.649835	−	0.760075i	$-0.274838\pi$
$53$	9.46174	1.29967	0.649835	+	0.760075i	$0.274838\pi$
$54$	0	0
$55$	6.35988	0.857566
$56$	0	0
$57$	−6.56749	−0.869885
$58$	0	0
$59$	−0.602544	−0.0784445	−0.0392223	−	0.999231i	$-0.512488\pi$
$59$	−0.602544	−0.0784445	−0.0392223	+	0.999231i	$0.512488\pi$
$60$	0	0
$61$	5.75217	0.736490	0.368245	−	0.929729i	$-0.379959\pi$
$61$	5.75217	0.736490	0.368245	+	0.929729i	$0.379959\pi$
$62$	0	0
$63$	2.93973	0.370371
$64$	0	0
$65$	−1.47174	−0.182546
$66$	0	0
$67$	8.57959	1.04816	0.524082	−	0.851668i	$-0.324409\pi$
$67$	8.57959	1.04816	0.524082	+	0.851668i	$0.324409\pi$
$68$	0	0
$69$	3.54085	0.426268
$70$	0	0
$71$	3.26862	0.387914	0.193957	−	0.981010i	$-0.437868\pi$
$71$	3.26862	0.387914	0.193957	+	0.981010i	$0.437868\pi$
$72$	0	0
$73$	−15.3835	−1.80050	−0.900249	−	0.435375i	$-0.856616\pi$
$73$	−15.3835	−1.80050	−0.900249	+	0.435375i	$0.856616\pi$
$74$	0	0
$75$	−2.83399	−0.327241
$76$	0	0
$77$	12.7036	1.44771
$78$	0	0
$79$	8.85689	0.996478	0.498239	−	0.867040i	$-0.333980\pi$
$79$	8.85689	0.996478	0.498239	+	0.867040i	$0.333980\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.95232	1.09241	0.546205	−	0.837652i	$-0.316072\pi$
$83$	9.95232	1.09241	0.546205	+	0.837652i	$0.316072\pi$
$84$	0	0
$85$	10.1354	1.09934
$86$	0	0
$87$	−7.96637	−0.854084
$88$	0	0
$89$	5.10329	0.540947	0.270474	−	0.962727i	$-0.412820\pi$
$89$	5.10329	0.540947	0.270474	+	0.962727i	$0.412820\pi$
$90$	0	0
$91$	−2.93973	−0.308168
$92$	0	0
$93$	−4.92763	−0.510971
$94$	0	0
$95$	−9.66561	−0.991670
$96$	0	0
$97$	6.42874	0.652739	0.326370	−	0.945242i	$-0.394175\pi$
$97$	6.42874	0.652739	0.326370	+	0.945242i	$0.394175\pi$
$98$	0	0
$99$	4.32135	0.434312
$100$	0	0
$101$	−2.54248	−0.252986	−0.126493	−	0.991968i	$-0.540372\pi$
$101$	−2.54248	−0.252986	−0.126493	+	0.991968i	$0.540372\pi$
$102$	0	0
$103$	10.5637	1.04088	0.520438	−	0.853899i	$-0.325769\pi$
$103$	10.5637	1.04088	0.520438	+	0.853899i	$0.325769\pi$
$104$	0	0
$105$	4.32651	0.422224
$106$	0	0
$107$	2.58527	0.249928	0.124964	−	0.992161i	$-0.460118\pi$
$107$	2.58527	0.249928	0.124964	+	0.992161i	$0.460118\pi$
$108$	0	0
$109$	2.47602	0.237160	0.118580	−	0.992945i	$-0.462166\pi$
$109$	2.47602	0.237160	0.118580	+	0.992945i	$0.462166\pi$
$110$	0	0
$111$	−3.45073	−0.327529
$112$	0	0
$113$	6.95821	0.654573	0.327287	−	0.944925i	$-0.393866\pi$
$113$	6.95821	0.654573	0.327287	+	0.944925i	$0.393866\pi$
$114$	0	0
$115$	5.21120	0.485947
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	20.2451	1.85587
$120$	0	0
$121$	7.67403	0.697639
$122$	0	0
$123$	11.4381	1.03134
$124$	0	0
$125$	−11.5296	−1.03124
$126$	0	0
$127$	−11.3823	−1.01002	−0.505008	−	0.863115i	$-0.668510\pi$
$127$	−11.3823	−1.01002	−0.505008	+	0.863115i	$0.668510\pi$
$128$	0	0
$129$	−1.43599	−0.126432
$130$	0	0
$131$	−1.41473	−0.123605	−0.0618026	−	0.998088i	$-0.519685\pi$
$131$	−1.41473	−0.123605	−0.0618026	+	0.998088i	$0.519685\pi$
$132$	0	0
$133$	−19.3067	−1.67410
$134$	0	0
$135$	1.47174	0.126667
$136$	0	0
$137$	17.8444	1.52455	0.762275	−	0.647253i	$-0.224082\pi$
$137$	17.8444	1.52455	0.762275	+	0.647253i	$0.224082\pi$
$138$	0	0
$139$	5.42851	0.460440	0.230220	−	0.973139i	$-0.426055\pi$
$139$	5.42851	0.460440	0.230220	+	0.973139i	$0.426055\pi$
$140$	0	0
$141$	6.41147	0.539943
$142$	0	0
$143$	−4.32135	−0.361369
$144$	0	0
$145$	−11.7244	−0.973658
$146$	0	0
$147$	1.64203	0.135432
$148$	0	0
$149$	−8.30643	−0.680489	−0.340245	−	0.940337i	$-0.610510\pi$
$149$	−8.30643	−0.680489	−0.340245	+	0.940337i	$0.610510\pi$
$150$	0	0
$151$	−23.0182	−1.87319	−0.936596	−	0.350410i	$-0.886042\pi$
$151$	−23.0182	−1.87319	−0.936596	+	0.350410i	$0.886042\pi$
$152$	0	0
$153$	6.88672	0.556758
$154$	0	0
$155$	−7.25217	−0.582508
$156$	0	0
$157$	13.7135	1.09445	0.547227	−	0.836985i	$-0.315684\pi$
$157$	13.7135	1.09445	0.547227	+	0.836985i	$0.315684\pi$
$158$	0	0
$159$	9.46174	0.750365
$160$	0	0
$161$	10.4092	0.820356
$162$	0	0
$163$	−13.1981	−1.03375	−0.516876	−	0.856060i	$-0.672905\pi$
$163$	−13.1981	−1.03375	−0.516876	+	0.856060i	$0.672905\pi$
$164$	0	0
$165$	6.35988	0.495116
$166$	0	0
$167$	−17.7828	−1.37608	−0.688038	−	0.725674i	$-0.741528\pi$
$167$	−17.7828	−1.37608	−0.688038	+	0.725674i	$0.741528\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.56749	−0.502228
$172$	0	0
$173$	−19.9839	−1.51935	−0.759674	−	0.650304i	$-0.774641\pi$
$173$	−19.9839	−1.51935	−0.759674	+	0.650304i	$0.774641\pi$
$174$	0	0
$175$	−8.33118	−0.629778
$176$	0	0
$177$	−0.602544	−0.0452900
$178$	0	0
$179$	−4.32651	−0.323379	−0.161689	−	0.986842i	$-0.551694\pi$
$179$	−4.32651	−0.323379	−0.161689	+	0.986842i	$0.551694\pi$
$180$	0	0
$181$	15.8189	1.17581	0.587905	−	0.808930i	$-0.299953\pi$
$181$	15.8189	1.17581	0.587905	+	0.808930i	$0.299953\pi$
$182$	0	0
$183$	5.75217	0.425213
$184$	0	0
$185$	−5.07856	−0.373383
$186$	0	0
$187$	29.7599	2.17626
$188$	0	0
$189$	2.93973	0.213834
$190$	0	0
$191$	6.76415	0.489437	0.244718	−	0.969594i	$-0.421304\pi$
$191$	6.76415	0.489437	0.244718	+	0.969594i	$0.421304\pi$
$192$	0	0
$193$	11.5595	0.832070	0.416035	−	0.909349i	$-0.363419\pi$
$193$	11.5595	0.832070	0.416035	+	0.909349i	$0.363419\pi$
$194$	0	0
$195$	−1.47174	−0.105393
$196$	0	0
$197$	−17.2065	−1.22591	−0.612955	−	0.790118i	$-0.710019\pi$
$197$	−17.2065	−1.22591	−0.612955	+	0.790118i	$0.710019\pi$
$198$	0	0
$199$	−22.7288	−1.61120	−0.805599	−	0.592461i	$-0.798157\pi$
$199$	−22.7288	−1.61120	−0.805599	+	0.592461i	$0.798157\pi$
$200$	0	0
$201$	8.57959	0.605157
$202$	0	0
$203$	−23.4190	−1.64369
$204$	0	0
$205$	16.8339	1.17573
$206$	0	0
$207$	3.54085	0.246106
$208$	0	0
$209$	−28.3804	−1.96311
$210$	0	0
$211$	−4.86274	−0.334765	−0.167382	−	0.985892i	$-0.553531\pi$
$211$	−4.86274	−0.334765	−0.167382	+	0.985892i	$0.553531\pi$
$212$	0	0
$213$	3.26862	0.223962
$214$	0	0
$215$	−2.11340	−0.144133
$216$	0	0
$217$	−14.4859	−0.983368
$218$	0	0
$219$	−15.3835	−1.03952
$220$	0	0
$221$	−6.88672	−0.463251
$222$	0	0
$223$	7.82908	0.524274	0.262137	−	0.965031i	$-0.415573\pi$
$223$	7.82908	0.524274	0.262137	+	0.965031i	$0.415573\pi$
$224$	0	0
$225$	−2.83399	−0.188933
$226$	0	0
$227$	−5.61608	−0.372752	−0.186376	−	0.982478i	$-0.559674\pi$
$227$	−5.61608	−0.372752	−0.186376	+	0.982478i	$0.559674\pi$
$228$	0	0
$229$	−6.99252	−0.462079	−0.231039	−	0.972944i	$-0.574213\pi$
$229$	−6.99252	−0.462079	−0.231039	+	0.972944i	$0.574213\pi$
$230$	0	0
$231$	12.7036	0.835836
$232$	0	0
$233$	−4.22318	−0.276670	−0.138335	−	0.990385i	$-0.544175\pi$
$233$	−4.22318	−0.276670	−0.138335	+	0.990385i	$0.544175\pi$
$234$	0	0
$235$	9.43599	0.615536
$236$	0	0
$237$	8.85689	0.575317
$238$	0	0
$239$	−5.20923	−0.336957	−0.168479	−	0.985705i	$-0.553885\pi$
$239$	−5.20923	−0.336957	−0.168479	+	0.985705i	$0.553885\pi$
$240$	0	0
$241$	11.1307	0.716995	0.358497	−	0.933531i	$-0.383289\pi$
$241$	11.1307	0.716995	0.358497	+	0.933531i	$0.383289\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.41663	0.154393
$246$	0	0
$247$	6.56749	0.417879
$248$	0	0
$249$	9.95232	0.630703
$250$	0	0
$251$	2.86165	0.180626	0.0903130	−	0.995913i	$-0.471213\pi$
$251$	2.86165	0.180626	0.0903130	+	0.995913i	$0.471213\pi$
$252$	0	0
$253$	15.3012	0.961981
$254$	0	0
$255$	10.1354	0.634706
$256$	0	0
$257$	10.6744	0.665849	0.332925	−	0.942953i	$-0.391965\pi$
$257$	10.6744	0.665849	0.332925	+	0.942953i	$0.391965\pi$
$258$	0	0
$259$	−10.1442	−0.630331
$260$	0	0
$261$	−7.96637	−0.493106
$262$	0	0
$263$	2.22929	0.137464	0.0687321	−	0.997635i	$-0.478105\pi$
$263$	2.22929	0.137464	0.0687321	+	0.997635i	$0.478105\pi$
$264$	0	0
$265$	13.9252	0.855418
$266$	0	0
$267$	5.10329	0.312316
$268$	0	0
$269$	−0.146614	−0.00893920	−0.00446960	−	0.999990i	$-0.501423\pi$
$269$	−0.146614	−0.00893920	−0.00446960	+	0.999990i	$0.501423\pi$
$270$	0	0
$271$	−11.2326	−0.682334	−0.341167	−	0.940003i	$-0.610822\pi$
$271$	−11.2326	−0.682334	−0.341167	+	0.940003i	$0.610822\pi$
$272$	0	0
$273$	−2.93973	−0.177921
$274$	0	0
$275$	−12.2467	−0.738501
$276$	0	0
$277$	−0.180247	−0.0108300	−0.00541498	−	0.999985i	$-0.501724\pi$
$277$	−0.180247	−0.0108300	−0.00541498	+	0.999985i	$0.501724\pi$
$278$	0	0
$279$	−4.92763	−0.295009
$280$	0	0
$281$	10.3531	0.617617	0.308809	−	0.951124i	$-0.400070\pi$
$281$	10.3531	0.617617	0.308809	+	0.951124i	$0.400070\pi$
$282$	0	0
$283$	−2.86450	−0.170277	−0.0851386	−	0.996369i	$-0.527133\pi$
$283$	−2.86450	−0.170277	−0.0851386	+	0.996369i	$0.527133\pi$
$284$	0	0
$285$	−9.66561	−0.572541
$286$	0	0
$287$	33.6250	1.98482
$288$	0	0
$289$	30.4269	1.78982
$290$	0	0
$291$	6.42874	0.376859
$292$	0	0
$293$	14.5083	0.847583	0.423791	−	0.905760i	$-0.360699\pi$
$293$	14.5083	0.847583	0.423791	+	0.905760i	$0.360699\pi$
$294$	0	0
$295$	−0.886785	−0.0516306
$296$	0	0
$297$	4.32135	0.250750
$298$	0	0
$299$	−3.54085	−0.204773
$300$	0	0
$301$	−4.22143	−0.243319
$302$	0	0
$303$	−2.54248	−0.146062
$304$	0	0
$305$	8.46568	0.484744
$306$	0	0
$307$	6.24005	0.356138	0.178069	−	0.984018i	$-0.443015\pi$
$307$	6.24005	0.356138	0.178069	+	0.984018i	$0.443015\pi$
$308$	0	0
$309$	10.5637	0.600950
$310$	0	0
$311$	−21.7669	−1.23429	−0.617143	−	0.786851i	$-0.711710\pi$
$311$	−21.7669	−1.23429	−0.617143	+	0.786851i	$0.711710\pi$
$312$	0	0
$313$	32.4755	1.83562	0.917812	−	0.397016i	$-0.129954\pi$
$313$	32.4755	1.83562	0.917812	+	0.397016i	$0.129954\pi$
$314$	0	0
$315$	4.32651	0.243771
$316$	0	0
$317$	23.3653	1.31232	0.656162	−	0.754620i	$-0.272179\pi$
$317$	23.3653	1.31232	0.656162	+	0.754620i	$0.272179\pi$
$318$	0	0
$319$	−34.4254	−1.92745
$320$	0	0
$321$	2.58527	0.144296
$322$	0	0
$323$	−45.2284	−2.51658
$324$	0	0
$325$	2.83399	0.157202
$326$	0	0
$327$	2.47602	0.136924
$328$	0	0
$329$	18.8480	1.03912
$330$	0	0
$331$	−13.6094	−0.748039	−0.374020	−	0.927421i	$-0.622021\pi$
$331$	−13.6094	−0.748039	−0.374020	+	0.927421i	$0.622021\pi$
$332$	0	0
$333$	−3.45073	−0.189099
$334$	0	0
$335$	12.6269	0.689881
$336$	0	0
$337$	−6.45391	−0.351567	−0.175783	−	0.984429i	$-0.556246\pi$
$337$	−6.45391	−0.351567	−0.175783	+	0.984429i	$0.556246\pi$
$338$	0	0
$339$	6.95821	0.377918
$340$	0	0
$341$	−21.2940	−1.15313
$342$	0	0
$343$	−15.7510	−0.850474
$344$	0	0
$345$	5.21120	0.280562
$346$	0	0
$347$	−6.01451	−0.322876	−0.161438	−	0.986883i	$-0.551613\pi$
$347$	−6.01451	−0.322876	−0.161438	+	0.986883i	$0.551613\pi$
$348$	0	0
$349$	−20.5964	−1.10250	−0.551249	−	0.834340i	$-0.685849\pi$
$349$	−20.5964	−1.10250	−0.551249	+	0.834340i	$0.685849\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	2.96820	0.157981	0.0789907	−	0.996875i	$-0.474830\pi$
$353$	2.96820	0.157981	0.0789907	+	0.996875i	$0.474830\pi$
$354$	0	0
$355$	4.81054	0.255317
$356$	0	0
$357$	20.2451	1.07149
$358$	0	0
$359$	−8.05700	−0.425232	−0.212616	−	0.977136i	$-0.568198\pi$
$359$	−8.05700	−0.425232	−0.212616	+	0.977136i	$0.568198\pi$
$360$	0	0
$361$	24.1319	1.27010
$362$	0	0
$363$	7.67403	0.402782
$364$	0	0
$365$	−22.6404	−1.18505
$366$	0	0
$367$	−0.102750	−0.00536350	−0.00268175	−	0.999996i	$-0.500854\pi$
$367$	−0.102750	−0.00536350	−0.00268175	+	0.999996i	$0.500854\pi$
$368$	0	0
$369$	11.4381	0.595444
$370$	0	0
$371$	27.8150	1.44408
$372$	0	0
$373$	−2.65302	−0.137368	−0.0686842	−	0.997638i	$-0.521880\pi$
$373$	−2.65302	−0.137368	−0.0686842	+	0.997638i	$0.521880\pi$
$374$	0	0
$375$	−11.5296	−0.595385
$376$	0	0
$377$	7.96637	0.410289
$378$	0	0
$379$	35.9425	1.84624	0.923121	−	0.384510i	$-0.125629\pi$
$379$	35.9425	1.84624	0.923121	+	0.384510i	$0.125629\pi$
$380$	0	0
$381$	−11.3823	−0.583133
$382$	0	0
$383$	10.8548	0.554653	0.277327	−	0.960776i	$-0.410552\pi$
$383$	10.8548	0.554653	0.277327	+	0.960776i	$0.410552\pi$
$384$	0	0
$385$	18.6964	0.952854
$386$	0	0
$387$	−1.43599	−0.0729956
$388$	0	0
$389$	0.412406	0.0209098	0.0104549	−	0.999945i	$-0.496672\pi$
$389$	0.412406	0.0209098	0.0104549	+	0.999945i	$0.496672\pi$
$390$	0	0
$391$	24.3848	1.23319
$392$	0	0
$393$	−1.41473	−0.0713634
$394$	0	0
$395$	13.0350	0.655862
$396$	0	0
$397$	−2.06389	−0.103584	−0.0517918	−	0.998658i	$-0.516493\pi$
$397$	−2.06389	−0.103584	−0.0517918	+	0.998658i	$0.516493\pi$
$398$	0	0
$399$	−19.3067	−0.966541
$400$	0	0
$401$	20.7908	1.03824	0.519120	−	0.854701i	$-0.326260\pi$
$401$	20.7908	1.03824	0.519120	+	0.854701i	$0.326260\pi$
$402$	0	0
$403$	4.92763	0.245463
$404$	0	0
$405$	1.47174	0.0731312
$406$	0	0
$407$	−14.9118	−0.739150
$408$	0	0
$409$	9.26345	0.458048	0.229024	−	0.973421i	$-0.426447\pi$
$409$	9.26345	0.458048	0.229024	+	0.973421i	$0.426447\pi$
$410$	0	0
$411$	17.8444	0.880200
$412$	0	0
$413$	−1.77132	−0.0871608
$414$	0	0
$415$	14.6472	0.719002
$416$	0	0
$417$	5.42851	0.265835
$418$	0	0
$419$	−35.1941	−1.71934	−0.859672	−	0.510847i	$-0.829332\pi$
$419$	−35.1941	−1.71934	−0.859672	+	0.510847i	$0.829332\pi$
$420$	0	0
$421$	−38.5231	−1.87750	−0.938751	−	0.344596i	$-0.888016\pi$
$421$	−38.5231	−1.87750	−0.938751	+	0.344596i	$0.888016\pi$
$422$	0	0
$423$	6.41147	0.311736
$424$	0	0
$425$	−19.5169	−0.946709
$426$	0	0
$427$	16.9098	0.818325
$428$	0	0
$429$	−4.32135	−0.208637
$430$	0	0
$431$	−1.89823	−0.0914344	−0.0457172	−	0.998954i	$-0.514557\pi$
$431$	−1.89823	−0.0914344	−0.0457172	+	0.998954i	$0.514557\pi$
$432$	0	0
$433$	−11.4500	−0.550251	−0.275126	−	0.961408i	$-0.588719\pi$
$433$	−11.4500	−0.550251	−0.275126	+	0.961408i	$0.588719\pi$
$434$	0	0
$435$	−11.7244	−0.562142
$436$	0	0
$437$	−23.2545	−1.11241
$438$	0	0
$439$	−38.8346	−1.85348	−0.926738	−	0.375709i	$-0.877399\pi$
$439$	−38.8346	−1.85348	−0.926738	+	0.375709i	$0.877399\pi$
$440$	0	0
$441$	1.64203	0.0781918
$442$	0	0
$443$	−31.9374	−1.51739	−0.758696	−	0.651445i	$-0.774163\pi$
$443$	−31.9374	−1.51739	−0.758696	+	0.651445i	$0.774163\pi$
$444$	0	0
$445$	7.51069	0.356041
$446$	0	0
$447$	−8.30643	−0.392881
$448$	0	0
$449$	8.33864	0.393525	0.196762	−	0.980451i	$-0.436957\pi$
$449$	8.33864	0.393525	0.196762	+	0.980451i	$0.436957\pi$
$450$	0	0
$451$	49.4280	2.32747
$452$	0	0
$453$	−23.0182	−1.08149
$454$	0	0
$455$	−4.32651	−0.202830
$456$	0	0
$457$	23.6899	1.10817	0.554084	−	0.832461i	$-0.313069\pi$
$457$	23.6899	1.10817	0.554084	+	0.832461i	$0.313069\pi$
$458$	0	0
$459$	6.88672	0.321445
$460$	0	0
$461$	20.5605	0.957599	0.478800	−	0.877924i	$-0.341072\pi$
$461$	20.5605	0.957599	0.478800	+	0.877924i	$0.341072\pi$
$462$	0	0
$463$	24.3069	1.12964	0.564820	−	0.825214i	$-0.308946\pi$
$463$	24.3069	1.12964	0.564820	+	0.825214i	$0.308946\pi$
$464$	0	0
$465$	−7.25217	−0.336311
$466$	0	0
$467$	29.7179	1.37518	0.687590	−	0.726099i	$-0.258669\pi$
$467$	29.7179	1.37518	0.687590	+	0.726099i	$0.258669\pi$
$468$	0	0
$469$	25.2217	1.16463
$470$	0	0
$471$	13.7135	0.631883
$472$	0	0
$473$	−6.20542	−0.285326
$474$	0	0
$475$	18.6122	0.853986
$476$	0	0
$477$	9.46174	0.433224
$478$	0	0
$479$	−21.9556	−1.00318	−0.501588	−	0.865106i	$-0.667251\pi$
$479$	−21.9556	−1.00318	−0.501588	+	0.865106i	$0.667251\pi$
$480$	0	0
$481$	3.45073	0.157340
$482$	0	0
$483$	10.4092	0.473633
$484$	0	0
$485$	9.46141	0.429620
$486$	0	0
$487$	41.3497	1.87374	0.936868	−	0.349683i	$-0.113711\pi$
$487$	41.3497	1.87374	0.936868	+	0.349683i	$0.113711\pi$
$488$	0	0
$489$	−13.1981	−0.596838
$490$	0	0
$491$	−20.8229	−0.939726	−0.469863	−	0.882739i	$-0.655697\pi$
$491$	−20.8229	−0.939726	−0.469863	+	0.882739i	$0.655697\pi$
$492$	0	0
$493$	−54.8621	−2.47087
$494$	0	0
$495$	6.35988	0.285855
$496$	0	0
$497$	9.60886	0.431016
$498$	0	0
$499$	−25.0654	−1.12208	−0.561042	−	0.827788i	$-0.689599\pi$
$499$	−25.0654	−1.12208	−0.561042	+	0.827788i	$0.689599\pi$
$500$	0	0
$501$	−17.7828	−0.794478
$502$	0	0
$503$	−24.4343	−1.08947	−0.544737	−	0.838607i	$-0.683370\pi$
$503$	−24.4343	−1.08947	−0.544737	+	0.838607i	$0.683370\pi$
$504$	0	0
$505$	−3.74186	−0.166510
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−34.3045	−1.52052	−0.760260	−	0.649619i	$-0.774929\pi$
$509$	−34.3045	−1.52052	−0.760260	+	0.649619i	$0.774929\pi$
$510$	0	0
$511$	−45.2233	−2.00056
$512$	0	0
$513$	−6.56749	−0.289962
$514$	0	0
$515$	15.5470	0.685085
$516$	0	0
$517$	27.7062	1.21852
$518$	0	0
$519$	−19.9839	−0.877196
$520$	0	0
$521$	−24.5126	−1.07392	−0.536959	−	0.843609i	$-0.680427\pi$
$521$	−24.5126	−1.07392	−0.536959	+	0.843609i	$0.680427\pi$
$522$	0	0
$523$	−21.8796	−0.956730	−0.478365	−	0.878161i	$-0.658770\pi$
$523$	−21.8796	−0.956730	−0.478365	+	0.878161i	$0.658770\pi$
$524$	0	0
$525$	−8.33118	−0.363602
$526$	0	0
$527$	−33.9352	−1.47824
$528$	0	0
$529$	−10.4624	−0.454886
$530$	0	0
$531$	−0.602544	−0.0261482
$532$	0	0
$533$	−11.4381	−0.495439
$534$	0	0
$535$	3.80484	0.164498
$536$	0	0
$537$	−4.32651	−0.186703
$538$	0	0
$539$	7.09577	0.305636
$540$	0	0
$541$	−25.8833	−1.11281	−0.556405	−	0.830911i	$-0.687820\pi$
$541$	−25.8833	−1.11281	−0.556405	+	0.830911i	$0.687820\pi$
$542$	0	0
$543$	15.8189	0.678855
$544$	0	0
$545$	3.64405	0.156094
$546$	0	0
$547$	42.9380	1.83590	0.917949	−	0.396698i	$-0.129844\pi$
$547$	42.9380	1.83590	0.917949	+	0.396698i	$0.129844\pi$
$548$	0	0
$549$	5.75217	0.245497
$550$	0	0
$551$	52.3190	2.22886
$552$	0	0
$553$	26.0369	1.10720
$554$	0	0
$555$	−5.07856	−0.215573
$556$	0	0
$557$	15.9949	0.677726	0.338863	−	0.940836i	$-0.389958\pi$
$557$	15.9949	0.677726	0.338863	+	0.940836i	$0.389958\pi$
$558$	0	0
$559$	1.43599	0.0607360
$560$	0	0
$561$	29.7599	1.25646
$562$	0	0
$563$	1.36950	0.0577174	0.0288587	−	0.999584i	$-0.490813\pi$
$563$	1.36950	0.0577174	0.0288587	+	0.999584i	$0.490813\pi$
$564$	0	0
$565$	10.2406	0.430827
$566$	0	0
$567$	2.93973	0.123457
$568$	0	0
$569$	1.23722	0.0518672	0.0259336	−	0.999664i	$-0.491744\pi$
$569$	1.23722	0.0518672	0.0259336	+	0.999664i	$0.491744\pi$
$570$	0	0
$571$	−8.08889	−0.338510	−0.169255	−	0.985572i	$-0.554136\pi$
$571$	−8.08889	−0.338510	−0.169255	+	0.985572i	$0.554136\pi$
$572$	0	0
$573$	6.76415	0.282577
$574$	0	0
$575$	−10.0347	−0.418478
$576$	0	0
$577$	−4.86588	−0.202569	−0.101285	−	0.994857i	$-0.532295\pi$
$577$	−4.86588	−0.202569	−0.101285	+	0.994857i	$0.532295\pi$
$578$	0	0
$579$	11.5595	0.480396
$580$	0	0
$581$	29.2572	1.21379
$582$	0	0
$583$	40.8875	1.69339
$584$	0	0
$585$	−1.47174	−0.0608488
$586$	0	0
$587$	16.5324	0.682364	0.341182	−	0.939997i	$-0.389173\pi$
$587$	16.5324	0.682364	0.341182	+	0.939997i	$0.389173\pi$
$588$	0	0
$589$	32.3621	1.33346
$590$	0	0
$591$	−17.2065	−0.707779
$592$	0	0
$593$	21.3723	0.877656	0.438828	−	0.898571i	$-0.355394\pi$
$593$	21.3723	0.877656	0.438828	+	0.898571i	$0.355394\pi$
$594$	0	0
$595$	29.7955	1.22150
$596$	0	0
$597$	−22.7288	−0.930226
$598$	0	0
$599$	1.02621	0.0419299	0.0209649	−	0.999780i	$-0.493326\pi$
$599$	1.02621	0.0419299	0.0209649	+	0.999780i	$0.493326\pi$
$600$	0	0
$601$	−37.0228	−1.51019	−0.755096	−	0.655614i	$-0.772410\pi$
$601$	−37.0228	−1.51019	−0.755096	+	0.655614i	$0.772410\pi$
$602$	0	0
$603$	8.57959	0.349388
$604$	0	0
$605$	11.2941	0.459172
$606$	0	0
$607$	43.5674	1.76835	0.884174	−	0.467158i	$-0.154722\pi$
$607$	43.5674	1.76835	0.884174	+	0.467158i	$0.154722\pi$
$608$	0	0
$609$	−23.4190	−0.948985
$610$	0	0
$611$	−6.41147	−0.259380
$612$	0	0
$613$	−18.5030	−0.747327	−0.373664	−	0.927564i	$-0.621899\pi$
$613$	−18.5030	−0.747327	−0.373664	+	0.927564i	$0.621899\pi$
$614$	0	0
$615$	16.8339	0.678807
$616$	0	0
$617$	28.9669	1.16617	0.583083	−	0.812413i	$-0.301846\pi$
$617$	28.9669	1.16617	0.583083	+	0.812413i	$0.301846\pi$
$618$	0	0
$619$	−34.5680	−1.38941	−0.694703	−	0.719297i	$-0.744464\pi$
$619$	−34.5680	−1.38941	−0.694703	+	0.719297i	$0.744464\pi$
$620$	0	0
$621$	3.54085	0.142089
$622$	0	0
$623$	15.0023	0.601055
$624$	0	0
$625$	−2.79853	−0.111941
$626$	0	0
$627$	−28.3804	−1.13340
$628$	0	0
$629$	−23.7642	−0.947540
$630$	0	0
$631$	−17.5139	−0.697218	−0.348609	−	0.937268i	$-0.613346\pi$
$631$	−17.5139	−0.697218	−0.348609	+	0.937268i	$0.613346\pi$
$632$	0	0
$633$	−4.86274	−0.193277
$634$	0	0
$635$	−16.7517	−0.664773
$636$	0	0
$637$	−1.64203	−0.0650595
$638$	0	0
$639$	3.26862	0.129305
$640$	0	0
$641$	−11.0016	−0.434535	−0.217268	−	0.976112i	$-0.569714\pi$
$641$	−11.0016	−0.434535	−0.217268	+	0.976112i	$0.569714\pi$
$642$	0	0
$643$	−24.8557	−0.980214	−0.490107	−	0.871662i	$-0.663042\pi$
$643$	−24.8557	−0.980214	−0.490107	+	0.871662i	$0.663042\pi$
$644$	0	0
$645$	−2.11340	−0.0832152
$646$	0	0
$647$	32.7660	1.28816	0.644082	−	0.764956i	$-0.277239\pi$
$647$	32.7660	1.28816	0.644082	+	0.764956i	$0.277239\pi$
$648$	0	0
$649$	−2.60380	−0.102208
$650$	0	0
$651$	−14.4859	−0.567748
$652$	0	0
$653$	37.6400	1.47297	0.736484	−	0.676455i	$-0.236485\pi$
$653$	37.6400	1.47297	0.736484	+	0.676455i	$0.236485\pi$
$654$	0	0
$655$	−2.08210	−0.0813545
$656$	0	0
$657$	−15.3835	−0.600166
$658$	0	0
$659$	−36.8040	−1.43368	−0.716840	−	0.697238i	$-0.754412\pi$
$659$	−36.8040	−1.43368	−0.716840	+	0.697238i	$0.754412\pi$
$660$	0	0
$661$	2.08080	0.0809336	0.0404668	−	0.999181i	$-0.487115\pi$
$661$	2.08080	0.0809336	0.0404668	+	0.999181i	$0.487115\pi$
$662$	0	0
$663$	−6.88672	−0.267458
$664$	0	0
$665$	−28.4143	−1.10186
$666$	0	0
$667$	−28.2077	−1.09221
$668$	0	0
$669$	7.82908	0.302690
$670$	0	0
$671$	24.8571	0.959599
$672$	0	0
$673$	44.3467	1.70944	0.854720	−	0.519089i	$-0.173729\pi$
$673$	44.3467	1.70944	0.854720	+	0.519089i	$0.173729\pi$
$674$	0	0
$675$	−2.83399	−0.109080
$676$	0	0
$677$	36.1463	1.38922	0.694608	−	0.719388i	$-0.255578\pi$
$677$	36.1463	1.38922	0.694608	+	0.719388i	$0.255578\pi$
$678$	0	0
$679$	18.8988	0.725268
$680$	0	0
$681$	−5.61608	−0.215209
$682$	0	0
$683$	−8.12516	−0.310900	−0.155450	−	0.987844i	$-0.549683\pi$
$683$	−8.12516	−0.310900	−0.155450	+	0.987844i	$0.549683\pi$
$684$	0	0
$685$	26.2623	1.00343
$686$	0	0
$687$	−6.99252	−0.266781
$688$	0	0
$689$	−9.46174	−0.360464
$690$	0	0
$691$	8.04956	0.306220	0.153110	−	0.988209i	$-0.451071\pi$
$691$	8.04956	0.306220	0.153110	+	0.988209i	$0.451071\pi$
$692$	0	0
$693$	12.7036	0.482570
$694$	0	0
$695$	7.98934	0.303053
$696$	0	0
$697$	78.7710	2.98367
$698$	0	0
$699$	−4.22318	−0.159735
$700$	0	0
$701$	32.5847	1.23071	0.615353	−	0.788252i	$-0.289014\pi$
$701$	32.5847	1.23071	0.615353	+	0.788252i	$0.289014\pi$
$702$	0	0
$703$	22.6626	0.854736
$704$	0	0
$705$	9.43599	0.355380
$706$	0	0
$707$	−7.47420	−0.281096
$708$	0	0
$709$	−19.1070	−0.717578	−0.358789	−	0.933419i	$-0.616810\pi$
$709$	−19.1070	−0.717578	−0.358789	+	0.933419i	$0.616810\pi$
$710$	0	0
$711$	8.85689	0.332159
$712$	0	0
$713$	−17.4480	−0.653433
$714$	0	0
$715$	−6.35988	−0.237846
$716$	0	0
$717$	−5.20923	−0.194542
$718$	0	0
$719$	−16.2181	−0.604832	−0.302416	−	0.953176i	$-0.597793\pi$
$719$	−16.2181	−0.604832	−0.302416	+	0.953176i	$0.597793\pi$
$720$	0	0
$721$	31.0546	1.15653
$722$	0	0
$723$	11.1307	0.413957
$724$	0	0
$725$	22.5766	0.838475
$726$	0	0
$727$	−7.74395	−0.287207	−0.143604	−	0.989635i	$-0.545869\pi$
$727$	−7.74395	−0.287207	−0.143604	+	0.989635i	$0.545869\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−9.88928	−0.365768
$732$	0	0
$733$	−40.9216	−1.51147	−0.755737	−	0.654875i	$-0.772721\pi$
$733$	−40.9216	−1.51147	−0.755737	+	0.654875i	$0.772721\pi$
$734$	0	0
$735$	2.41663	0.0891388
$736$	0	0
$737$	37.0754	1.36569
$738$	0	0
$739$	22.5269	0.828667	0.414334	−	0.910125i	$-0.364015\pi$
$739$	22.5269	0.828667	0.414334	+	0.910125i	$0.364015\pi$
$740$	0	0
$741$	6.56749	0.241263
$742$	0	0
$743$	−4.15224	−0.152331	−0.0761655	−	0.997095i	$-0.524268\pi$
$743$	−4.15224	−0.152331	−0.0761655	+	0.997095i	$0.524268\pi$
$744$	0	0
$745$	−12.2249	−0.447885
$746$	0	0
$747$	9.95232	0.364136
$748$	0	0
$749$	7.60002	0.277699
$750$	0	0
$751$	−40.2084	−1.46723	−0.733613	−	0.679567i	$-0.762168\pi$
$751$	−40.2084	−1.46723	−0.733613	+	0.679567i	$0.762168\pi$
$752$	0	0
$753$	2.86165	0.104284
$754$	0	0
$755$	−33.8767	−1.23290
$756$	0	0
$757$	−37.6737	−1.36928	−0.684638	−	0.728884i	$-0.740040\pi$
$757$	−37.6737	−1.36928	−0.684638	+	0.728884i	$0.740040\pi$
$758$	0	0
$759$	15.3012	0.555400
$760$	0	0
$761$	5.75751	0.208710	0.104355	−	0.994540i	$-0.466722\pi$
$761$	5.75751	0.208710	0.104355	+	0.994540i	$0.466722\pi$
$762$	0	0
$763$	7.27884	0.263512
$764$	0	0
$765$	10.1354	0.366447
$766$	0	0
$767$	0.602544	0.0217566
$768$	0	0
$769$	17.6221	0.635469	0.317734	−	0.948180i	$-0.397078\pi$
$769$	17.6221	0.635469	0.317734	+	0.948180i	$0.397078\pi$
$770$	0	0
$771$	10.6744	0.384428
$772$	0	0
$773$	−26.8263	−0.964873	−0.482437	−	0.875931i	$-0.660248\pi$
$773$	−26.8263	−0.964873	−0.482437	+	0.875931i	$0.660248\pi$
$774$	0	0
$775$	13.9649	0.501633
$776$	0	0
$777$	−10.1442	−0.363922
$778$	0	0
$779$	−75.1196	−2.69144
$780$	0	0
$781$	14.1248	0.505426
$782$	0	0
$783$	−7.96637	−0.284695
$784$	0	0
$785$	20.1826	0.720348
$786$	0	0
$787$	−13.2119	−0.470955	−0.235477	−	0.971880i	$-0.575665\pi$
$787$	−13.2119	−0.470955	−0.235477	+	0.971880i	$0.575665\pi$
$788$	0	0
$789$	2.22929	0.0793650
$790$	0	0
$791$	20.4553	0.727306
$792$	0	0
$793$	−5.75217	−0.204266
$794$	0	0
$795$	13.9252	0.493876
$796$	0	0
$797$	−30.2374	−1.07106	−0.535531	−	0.844516i	$-0.679888\pi$
$797$	−30.2374	−1.07106	−0.535531	+	0.844516i	$0.679888\pi$
$798$	0	0
$799$	44.1540	1.56206
$800$	0	0
$801$	5.10329	0.180316
$802$	0	0
$803$	−66.4773	−2.34593
$804$	0	0
$805$	15.3195	0.539943
$806$	0	0
$807$	−0.146614	−0.00516105
$808$	0	0
$809$	20.8919	0.734519	0.367259	−	0.930119i	$-0.380296\pi$
$809$	20.8919	0.734519	0.367259	+	0.930119i	$0.380296\pi$
$810$	0	0
$811$	−2.19807	−0.0771846	−0.0385923	−	0.999255i	$-0.512287\pi$
$811$	−2.19807	−0.0771846	−0.0385923	+	0.999255i	$0.512287\pi$
$812$	0	0
$813$	−11.2326	−0.393946
$814$	0	0
$815$	−19.4241	−0.680396
$816$	0	0
$817$	9.43086	0.329944
$818$	0	0
$819$	−2.93973	−0.102723
$820$	0	0
$821$	28.6736	1.00071	0.500357	−	0.865819i	$-0.333202\pi$
$821$	28.6736	1.00071	0.500357	+	0.865819i	$0.333202\pi$
$822$	0	0
$823$	−32.9907	−1.14998	−0.574992	−	0.818159i	$-0.694995\pi$
$823$	−32.9907	−1.14998	−0.574992	+	0.818159i	$0.694995\pi$
$824$	0	0
$825$	−12.2467	−0.426374
$826$	0	0
$827$	3.70664	0.128893	0.0644463	−	0.997921i	$-0.479472\pi$
$827$	3.70664	0.128893	0.0644463	+	0.997921i	$0.479472\pi$
$828$	0	0
$829$	−38.6208	−1.34136	−0.670678	−	0.741748i	$-0.733997\pi$
$829$	−38.6208	−1.34136	−0.670678	+	0.741748i	$0.733997\pi$
$830$	0	0
$831$	−0.180247	−0.00625268
$832$	0	0
$833$	11.3082	0.391805
$834$	0	0
$835$	−26.1716	−0.905707
$836$	0	0
$837$	−4.92763	−0.170324
$838$	0	0
$839$	−24.7828	−0.855599	−0.427799	−	0.903874i	$-0.640711\pi$
$839$	−24.7828	−0.855599	−0.427799	+	0.903874i	$0.640711\pi$
$840$	0	0
$841$	34.4630	1.18838
$842$	0	0
$843$	10.3531	0.356581
$844$	0	0
$845$	1.47174	0.0506293
$846$	0	0
$847$	22.5596	0.775157
$848$	0	0
$849$	−2.86450	−0.0983095
$850$	0	0
$851$	−12.2185	−0.418845
$852$	0	0
$853$	48.8930	1.67406	0.837032	−	0.547154i	$-0.184289\pi$
$853$	48.8930	1.67406	0.837032	+	0.547154i	$0.184289\pi$
$854$	0	0
$855$	−9.66561	−0.330557
$856$	0	0
$857$	40.4927	1.38320	0.691602	−	0.722279i	$-0.256905\pi$
$857$	40.4927	1.38320	0.691602	+	0.722279i	$0.256905\pi$
$858$	0	0
$859$	−36.2116	−1.23552	−0.617762	−	0.786365i	$-0.711961\pi$
$859$	−36.2116	−1.23552	−0.617762	+	0.786365i	$0.711961\pi$
$860$	0	0
$861$	33.6250	1.14594
$862$	0	0
$863$	−18.0925	−0.615877	−0.307938	−	0.951406i	$-0.599639\pi$
$863$	−18.0925	−0.615877	−0.307938	+	0.951406i	$0.599639\pi$
$864$	0	0
$865$	−29.4110	−1.00001
$866$	0	0
$867$	30.4269	1.03335
$868$	0	0
$869$	38.2737	1.29835
$870$	0	0
$871$	−8.57959	−0.290708
$872$	0	0
$873$	6.42874	0.217580
$874$	0	0
$875$	−33.8939	−1.14582
$876$	0	0
$877$	−4.27060	−0.144208	−0.0721039	−	0.997397i	$-0.522971\pi$
$877$	−4.27060	−0.144208	−0.0721039	+	0.997397i	$0.522971\pi$
$878$	0	0
$879$	14.5083	0.489352
$880$	0	0
$881$	24.9886	0.841886	0.420943	−	0.907087i	$-0.361699\pi$
$881$	24.9886	0.841886	0.420943	+	0.907087i	$0.361699\pi$
$882$	0	0
$883$	−34.7898	−1.17077	−0.585385	−	0.810756i	$-0.699056\pi$
$883$	−34.7898	−1.17077	−0.585385	+	0.810756i	$0.699056\pi$
$884$	0	0
$885$	−0.886785	−0.0298090
$886$	0	0
$887$	−21.6922	−0.728354	−0.364177	−	0.931330i	$-0.618650\pi$
$887$	−21.6922	−0.728354	−0.364177	+	0.931330i	$0.618650\pi$
$888$	0	0
$889$	−33.4609	−1.12224
$890$	0	0
$891$	4.32135	0.144771
$892$	0	0
$893$	−42.1072	−1.40906
$894$	0	0
$895$	−6.36748	−0.212842
$896$	0	0
$897$	−3.54085	−0.118226
$898$	0	0
$899$	39.2553	1.30924
$900$	0	0
$901$	65.1604	2.17081
$902$	0	0
$903$	−4.22143	−0.140481
$904$	0	0
$905$	23.2813	0.773896
$906$	0	0
$907$	19.1200	0.634870	0.317435	−	0.948280i	$-0.397179\pi$
$907$	19.1200	0.634870	0.317435	+	0.948280i	$0.397179\pi$
$908$	0	0
$909$	−2.54248	−0.0843287
$910$	0	0
$911$	−12.1149	−0.401386	−0.200693	−	0.979654i	$-0.564319\pi$
$911$	−12.1149	−0.401386	−0.200693	+	0.979654i	$0.564319\pi$
$912$	0	0
$913$	43.0074	1.42334
$914$	0	0
$915$	8.46568	0.279867
$916$	0	0
$917$	−4.15891	−0.137339
$918$	0	0
$919$	−7.22863	−0.238451	−0.119225	−	0.992867i	$-0.538041\pi$
$919$	−7.22863	−0.238451	−0.119225	+	0.992867i	$0.538041\pi$
$920$	0	0
$921$	6.24005	0.205617
$922$	0	0
$923$	−3.26862	−0.107588
$924$	0	0
$925$	9.77933	0.321543
$926$	0	0
$927$	10.5637	0.346959
$928$	0	0
$929$	−41.9587	−1.37662	−0.688310	−	0.725416i	$-0.741647\pi$
$929$	−41.9587	−1.37662	−0.688310	+	0.725416i	$0.741647\pi$
$930$	0	0
$931$	−10.7840	−0.353431
$932$	0	0
$933$	−21.7669	−0.712616
$934$	0	0
$935$	43.7987	1.43237
$936$	0	0
$937$	−19.5281	−0.637954	−0.318977	−	0.947763i	$-0.603339\pi$
$937$	−19.5281	−0.637954	−0.318977	+	0.947763i	$0.603339\pi$
$938$	0	0
$939$	32.4755	1.05980
$940$	0	0
$941$	11.7669	0.383591	0.191795	−	0.981435i	$-0.438569\pi$
$941$	11.7669	0.383591	0.191795	+	0.981435i	$0.438569\pi$
$942$	0	0
$943$	40.5006	1.31888
$944$	0	0
$945$	4.32651	0.140741
$946$	0	0
$947$	−34.6707	−1.12665	−0.563323	−	0.826236i	$-0.690477\pi$
$947$	−34.6707	−1.12665	−0.563323	+	0.826236i	$0.690477\pi$
$948$	0	0
$949$	15.3835	0.499368
$950$	0	0
$951$	23.3653	0.757670
$952$	0	0
$953$	60.1507	1.94847	0.974236	−	0.225533i	$-0.0724123\pi$
$953$	60.1507	1.94847	0.974236	+	0.225533i	$0.0724123\pi$
$954$	0	0
$955$	9.95505	0.322138
$956$	0	0
$957$	−34.4254	−1.11282
$958$	0	0
$959$	52.4578	1.69395
$960$	0	0
$961$	−6.71847	−0.216725
$962$	0	0
$963$	2.58527	0.0833093
$964$	0	0
$965$	17.0125	0.547652
$966$	0	0
$967$	−39.2013	−1.26063	−0.630315	−	0.776340i	$-0.717074\pi$
$967$	−39.2013	−1.26063	−0.630315	+	0.776340i	$0.717074\pi$
$968$	0	0
$969$	−45.2284	−1.45295
$970$	0	0
$971$	42.7280	1.37121	0.685603	−	0.727976i	$-0.259539\pi$
$971$	42.7280	1.37121	0.685603	+	0.727976i	$0.259539\pi$
$972$	0	0
$973$	15.9584	0.511602
$974$	0	0
$975$	2.83399	0.0907604
$976$	0	0
$977$	−15.5858	−0.498633	−0.249316	−	0.968422i	$-0.580206\pi$
$977$	−15.5858	−0.498633	−0.249316	+	0.968422i	$0.580206\pi$
$978$	0	0
$979$	22.0531	0.704819
$980$	0	0
$981$	2.47602	0.0790532
$982$	0	0
$983$	43.4450	1.38568	0.692840	−	0.721091i	$-0.256359\pi$
$983$	43.4450	1.38568	0.692840	+	0.721091i	$0.256359\pi$
$984$	0	0
$985$	−25.3234	−0.806870
$986$	0	0
$987$	18.8480	0.599939
$988$	0	0
$989$	−5.08464	−0.161682
$990$	0	0
$991$	−10.3905	−0.330064	−0.165032	−	0.986288i	$-0.552773\pi$
$991$	−10.3905	−0.330064	−0.165032	+	0.986288i	$0.552773\pi$
$992$	0	0
$993$	−13.6094	−0.431881
$994$	0	0
$995$	−33.4507	−1.06046
$996$	0	0
$997$	−16.4960	−0.522433	−0.261217	−	0.965280i	$-0.584124\pi$
$997$	−16.4960	−0.522433	−0.261217	+	0.965280i	$0.584124\pi$
$998$	0	0
$999$	−3.45073	−0.109176

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9984.2.a.bu.1.6		8
4.3	odd	2		9984.2.a.bs.1.6		8
8.3	odd	2		9984.2.a.bv.1.3		8
8.5	even	2		9984.2.a.bt.1.3		8
16.3	odd	4		1248.2.g.b.625.3		16
16.5	even	4		312.2.g.b.157.4	yes	16
16.11	odd	4		1248.2.g.b.625.14		16
16.13	even	4		312.2.g.b.157.3	✓	16
48.5	odd	4		936.2.g.e.469.13		16
48.11	even	4		3744.2.g.e.1873.6		16
48.29	odd	4		936.2.g.e.469.14		16
48.35	even	4		3744.2.g.e.1873.11		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.g.b.157.3	✓	16	16.13	even	4
312.2.g.b.157.4	yes	16	16.5	even	4
936.2.g.e.469.13		16	48.5	odd	4
936.2.g.e.469.14		16	48.29	odd	4
1248.2.g.b.625.3		16	16.3	odd	4
1248.2.g.b.625.14		16	16.11	odd	4
3744.2.g.e.1873.6		16	48.11	even	4
3744.2.g.e.1873.11		16	48.35	even	4
9984.2.a.bs.1.6		8	4.3	odd	2
9984.2.a.bt.1.3		8	8.5	even	2
9984.2.a.bu.1.6		8	1.1	even	1	trivial
9984.2.a.bv.1.3		8	8.3	odd	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:	\( N \)	\(=\)	\( 9984 = 2^{8} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9984.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.47174 q^{5} +2.93973 q^{7} +1.00000 q^{9} +4.32135 q^{11} -1.00000 q^{13} +1.47174 q^{15} +6.88672 q^{17} -6.56749 q^{19} +2.93973 q^{21} +3.54085 q^{23} -2.83399 q^{25} +1.00000 q^{27} -7.96637 q^{29} -4.92763 q^{31} +4.32135 q^{33} +4.32651 q^{35} -3.45073 q^{37} -1.00000 q^{39} +11.4381 q^{41} -1.43599 q^{43} +1.47174 q^{45} +6.41147 q^{47} +1.64203 q^{49} +6.88672 q^{51} +9.46174 q^{53} +6.35988 q^{55} -6.56749 q^{57} -0.602544 q^{59} +5.75217 q^{61} +2.93973 q^{63} -1.47174 q^{65} +8.57959 q^{67} +3.54085 q^{69} +3.26862 q^{71} -15.3835 q^{73} -2.83399 q^{75} +12.7036 q^{77} +8.85689 q^{79} +1.00000 q^{81} +9.95232 q^{83} +10.1354 q^{85} -7.96637 q^{87} +5.10329 q^{89} -2.93973 q^{91} -4.92763 q^{93} -9.66561 q^{95} +6.42874 q^{97} +4.32135 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} - 2 q^{5} - 2 q^{7} + 8 q^{9} - 8 q^{13} - 2 q^{15} + 8 q^{17} + 6 q^{19} - 2 q^{21} + 4 q^{23} + 16 q^{25} + 8 q^{27} - 4 q^{29} - 2 q^{31} - 4 q^{35} - 8 q^{37} - 8 q^{39} + 18 q^{41} + 16 q^{43}+ \cdots + 20 q^{97}+O(q^{100}) \) 8 * q + 8 * q^3 - 2 * q^5 - 2 * q^7 + 8 * q^9 - 8 * q^13 - 2 * q^15 + 8 * q^17 + 6 * q^19 - 2 * q^21 + 4 * q^23 + 16 * q^25 + 8 * q^27 - 4 * q^29 - 2 * q^31 - 4 * q^35 - 8 * q^37 - 8 * q^39 + 18 * q^41 + 16 * q^43 - 2 * q^45 + 12 * q^47 + 24 * q^49 + 8 * q^51 + 4 * q^53 - 12 * q^55 + 6 * q^57 - 4 * q^59 - 12 * q^61 - 2 * q^63 + 2 * q^65 + 22 * q^67 + 4 * q^69 + 16 * q^73 + 16 * q^75 + 20 * q^77 + 8 * q^81 + 16 * q^83 + 4 * q^85 - 4 * q^87 + 30 * q^89 + 2 * q^91 - 2 * q^93 - 12 * q^95 + 20 * q^97

Introduction

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