LMFDB - Embedded newform 9984.2.a.bn.1.2

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:	$6$
Coefficient field:	6.6.5163008.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 5x^{2} - 6x + 1 $ x^6 - 2x^5 - 5x^4 + 8x^3 + 5x^2 - 6*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 4992)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.218114$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.10575 q^{5} -0.685687 q^{7} +1.00000 q^{9} -2.76115 q^{11} +1.00000 q^{13} +1.10575 q^{15} -3.55207 q^{17} -0.578969 q^{19} +0.685687 q^{21} +2.35279 q^{23} -3.77731 q^{25} -1.00000 q^{27} -2.48990 q^{29} -6.82490 q^{31} +2.76115 q^{33} +0.758200 q^{35} -7.11584 q^{37} -1.00000 q^{39} +4.72320 q^{41} +11.4039 q^{43} -1.10575 q^{45} +4.66390 q^{47} -6.52983 q^{49} +3.55207 q^{51} -9.10892 q^{53} +3.05315 q^{55} +0.578969 q^{57} -3.01668 q^{59} -7.13869 q^{61} -0.685687 q^{63} -1.10575 q^{65} -7.03386 q^{67} -2.35279 q^{69} +7.53590 q^{71} +11.5846 q^{73} +3.77731 q^{75} +1.89328 q^{77} +17.0435 q^{79} +1.00000 q^{81} -11.1878 q^{83} +3.92771 q^{85} +2.48990 q^{87} +12.5299 q^{89} -0.685687 q^{91} +6.82490 q^{93} +0.640197 q^{95} -12.7727 q^{97} -2.76115 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 4 q^{7} + 6 q^{9} + 8 q^{11} + 6 q^{13} + 4 q^{19} - 4 q^{21} + 8 q^{23} + 2 q^{25} - 6 q^{27} + 4 q^{29} + 20 q^{31} - 8 q^{33} + 8 q^{35} - 20 q^{37} - 6 q^{39} + 16 q^{47} + 10 q^{49}+ \cdots + 8 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 + 4 * q^7 + 6 * q^9 + 8 * q^11 + 6 * q^13 + 4 * q^19 - 4 * q^21 + 8 * q^23 + 2 * q^25 - 6 * q^27 + 4 * q^29 + 20 * q^31 - 8 * q^33 + 8 * q^35 - 20 * q^37 - 6 * q^39 + 16 * q^47 + 10 * q^49 + 8 * q^53 + 16 * q^55 - 4 * q^57 - 8 * q^61 + 4 * q^63 + 4 * q^67 - 8 * q^69 + 24 * q^71 - 4 * q^73 - 2 * q^75 + 12 * q^77 + 40 * q^79 + 6 * q^81 + 16 * q^83 - 16 * q^85 - 4 * q^87 + 16 * q^89 + 4 * q^91 - 20 * q^93 - 8 * q^95 - 20 * q^97 + 8 * q^99

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
$5$	−1.10575	−0.494508	−0.247254	−	0.968951i	$-0.579528\pi$
$5$	−1.10575	−0.494508	−0.247254	+	0.968951i	$0.579528\pi$
$6$	0	0
$7$	−0.685687	−0.259165	−0.129583	−	0.991569i	$-0.541364\pi$
$7$	−0.685687	−0.259165	−0.129583	+	0.991569i	$0.541364\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.76115	−0.832517	−0.416259	−	0.909246i	$-0.636659\pi$
$11$	−2.76115	−0.832517	−0.416259	+	0.909246i	$0.636659\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.10575	0.285504
$16$	0	0
$17$	−3.55207	−0.861502	−0.430751	−	0.902471i	$-0.641751\pi$
$17$	−3.55207	−0.861502	−0.430751	+	0.902471i	$0.641751\pi$
$18$	0	0
$19$	−0.578969	−0.132825	−0.0664123	−	0.997792i	$-0.521155\pi$
$19$	−0.578969	−0.132825	−0.0664123	+	0.997792i	$0.521155\pi$
$20$	0	0
$21$	0.685687	0.149629
$22$	0	0
$23$	2.35279	0.490590	0.245295	−	0.969448i	$-0.421115\pi$
$23$	2.35279	0.490590	0.245295	+	0.969448i	$0.421115\pi$
$24$	0	0
$25$	−3.77731	−0.755462
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.48990	−0.462363	−0.231181	−	0.972911i	$-0.574259\pi$
$29$	−2.48990	−0.462363	−0.231181	+	0.972911i	$0.574259\pi$
$30$	0	0
$31$	−6.82490	−1.22579	−0.612894	−	0.790165i	$-0.709995\pi$
$31$	−6.82490	−1.22579	−0.612894	+	0.790165i	$0.709995\pi$
$32$	0	0
$33$	2.76115	0.480654
$34$	0	0
$35$	0.758200	0.128159
$36$	0	0
$37$	−7.11584	−1.16984	−0.584918	−	0.811092i	$-0.698873\pi$
$37$	−7.11584	−1.16984	−0.584918	+	0.811092i	$0.698873\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	4.72320	0.737639	0.368820	−	0.929501i	$-0.379762\pi$
$41$	4.72320	0.737639	0.368820	+	0.929501i	$0.379762\pi$
$42$	0	0
$43$	11.4039	1.73907	0.869537	−	0.493868i	$-0.164417\pi$
$43$	11.4039	1.73907	0.869537	+	0.493868i	$0.164417\pi$
$44$	0	0
$45$	−1.10575	−0.164836
$46$	0	0
$47$	4.66390	0.680299	0.340150	−	0.940371i	$-0.389522\pi$
$47$	4.66390	0.680299	0.340150	+	0.940371i	$0.389522\pi$
$48$	0	0
$49$	−6.52983	−0.932833
$50$	0	0
$51$	3.55207	0.497389
$52$	0	0
$53$	−9.10892	−1.25121	−0.625603	−	0.780141i	$-0.715147\pi$
$53$	−9.10892	−1.25121	−0.625603	+	0.780141i	$0.715147\pi$
$54$	0	0
$55$	3.05315	0.411686
$56$	0	0
$57$	0.578969	0.0766863
$58$	0	0
$59$	−3.01668	−0.392739	−0.196369	−	0.980530i	$-0.562915\pi$
$59$	−3.01668	−0.392739	−0.196369	+	0.980530i	$0.562915\pi$
$60$	0	0
$61$	−7.13869	−0.914016	−0.457008	−	0.889463i	$-0.651079\pi$
$61$	−7.13869	−0.914016	−0.457008	+	0.889463i	$0.651079\pi$
$62$	0	0
$63$	−0.685687	−0.0863884
$64$	0	0
$65$	−1.10575	−0.137152
$66$	0	0
$67$	−7.03386	−0.859322	−0.429661	−	0.902990i	$-0.641367\pi$
$67$	−7.03386	−0.859322	−0.429661	+	0.902990i	$0.641367\pi$
$68$	0	0
$69$	−2.35279	−0.283242
$70$	0	0
$71$	7.53590	0.894347	0.447174	−	0.894447i	$-0.352431\pi$
$71$	7.53590	0.894347	0.447174	+	0.894447i	$0.352431\pi$
$72$	0	0
$73$	11.5846	1.35587	0.677935	−	0.735122i	$-0.262875\pi$
$73$	11.5846	1.35587	0.677935	+	0.735122i	$0.262875\pi$
$74$	0	0
$75$	3.77731	0.436166
$76$	0	0
$77$	1.89328	0.215760
$78$	0	0
$79$	17.0435	1.91755	0.958774	−	0.284169i	$-0.0917176\pi$
$79$	17.0435	1.91755	0.958774	+	0.284169i	$0.0917176\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.1878	−1.22802	−0.614009	−	0.789299i	$-0.710444\pi$
$83$	−11.1878	−1.22802	−0.614009	+	0.789299i	$0.710444\pi$
$84$	0	0
$85$	3.92771	0.426020
$86$	0	0
$87$	2.48990	0.266945
$88$	0	0
$89$	12.5299	1.32817	0.664085	−	0.747657i	$-0.268821\pi$
$89$	12.5299	1.32817	0.664085	+	0.747657i	$0.268821\pi$
$90$	0	0
$91$	−0.685687	−0.0718795
$92$	0	0
$93$	6.82490	0.707709
$94$	0	0
$95$	0.640197	0.0656828
$96$	0	0
$97$	−12.7727	−1.29687	−0.648435	−	0.761270i	$-0.724576\pi$
$97$	−12.7727	−1.29687	−0.648435	+	0.761270i	$0.724576\pi$
$98$	0	0
$99$	−2.76115	−0.277506
$100$	0	0
$101$	6.61486	0.658203	0.329101	−	0.944295i	$-0.393254\pi$
$101$	6.61486	0.658203	0.329101	+	0.944295i	$0.393254\pi$
$102$	0	0
$103$	−4.54312	−0.447647	−0.223823	−	0.974630i	$-0.571854\pi$
$103$	−4.54312	−0.447647	−0.223823	+	0.974630i	$0.571854\pi$
$104$	0	0
$105$	−0.758200	−0.0739928
$106$	0	0
$107$	−2.76305	−0.267114	−0.133557	−	0.991041i	$-0.542640\pi$
$107$	−2.76305	−0.267114	−0.133557	+	0.991041i	$0.542640\pi$
$108$	0	0
$109$	−13.7172	−1.31387	−0.656937	−	0.753945i	$-0.728148\pi$
$109$	−13.7172	−1.31387	−0.656937	+	0.753945i	$0.728148\pi$
$110$	0	0
$111$	7.11584	0.675405
$112$	0	0
$113$	−12.4874	−1.17471	−0.587357	−	0.809328i	$-0.699832\pi$
$113$	−12.4874	−1.17471	−0.587357	+	0.809328i	$0.699832\pi$
$114$	0	0
$115$	−2.60160	−0.242601
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	2.43560	0.223271
$120$	0	0
$121$	−3.37606	−0.306915
$122$	0	0
$123$	−4.72320	−0.425876
$124$	0	0
$125$	9.70554	0.868090
$126$	0	0
$127$	0.796554	0.0706828	0.0353414	−	0.999375i	$-0.488748\pi$
$127$	0.796554	0.0706828	0.0353414	+	0.999375i	$0.488748\pi$
$128$	0	0
$129$	−11.4039	−1.00405
$130$	0	0
$131$	−6.40167	−0.559316	−0.279658	−	0.960100i	$-0.590221\pi$
$131$	−6.40167	−0.559316	−0.279658	+	0.960100i	$0.590221\pi$
$132$	0	0
$133$	0.396991	0.0344235
$134$	0	0
$135$	1.10575	0.0951681
$136$	0	0
$137$	0.333293	0.0284752	0.0142376	−	0.999899i	$-0.495468\pi$
$137$	0.333293	0.0284752	0.0142376	+	0.999899i	$0.495468\pi$
$138$	0	0
$139$	−5.95367	−0.504984	−0.252492	−	0.967599i	$-0.581250\pi$
$139$	−5.95367	−0.504984	−0.252492	+	0.967599i	$0.581250\pi$
$140$	0	0
$141$	−4.66390	−0.392771
$142$	0	0
$143$	−2.76115	−0.230899
$144$	0	0
$145$	2.75322	0.228642
$146$	0	0
$147$	6.52983	0.538572
$148$	0	0
$149$	11.4682	0.939510	0.469755	−	0.882797i	$-0.344342\pi$
$149$	11.4682	0.939510	0.469755	+	0.882797i	$0.344342\pi$
$150$	0	0
$151$	6.82129	0.555109	0.277554	−	0.960710i	$-0.410476\pi$
$151$	6.82129	0.555109	0.277554	+	0.960710i	$0.410476\pi$
$152$	0	0
$153$	−3.55207	−0.287167
$154$	0	0
$155$	7.54666	0.606162
$156$	0	0
$157$	9.24604	0.737914	0.368957	−	0.929446i	$-0.379715\pi$
$157$	9.24604	0.737914	0.368957	+	0.929446i	$0.379715\pi$
$158$	0	0
$159$	9.10892	0.722385
$160$	0	0
$161$	−1.61327	−0.127144
$162$	0	0
$163$	−14.2252	−1.11420	−0.557100	−	0.830445i	$-0.688086\pi$
$163$	−14.2252	−1.11420	−0.557100	+	0.830445i	$0.688086\pi$
$164$	0	0
$165$	−3.05315	−0.237687
$166$	0	0
$167$	23.1974	1.79507	0.897535	−	0.440943i	$-0.145356\pi$
$167$	23.1974	1.79507	0.897535	+	0.440943i	$0.145356\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.578969	−0.0442749
$172$	0	0
$173$	−7.21309	−0.548401	−0.274200	−	0.961673i	$-0.588413\pi$
$173$	−7.21309	−0.548401	−0.274200	+	0.961673i	$0.588413\pi$
$174$	0	0
$175$	2.59005	0.195789
$176$	0	0
$177$	3.01668	0.226748
$178$	0	0
$179$	11.4129	0.853039	0.426520	−	0.904478i	$-0.359740\pi$
$179$	11.4129	0.853039	0.426520	+	0.904478i	$0.359740\pi$
$180$	0	0
$181$	−19.3259	−1.43648	−0.718242	−	0.695794i	$-0.755053\pi$
$181$	−19.3259	−1.43648	−0.718242	+	0.695794i	$0.755053\pi$
$182$	0	0
$183$	7.13869	0.527707
$184$	0	0
$185$	7.86836	0.578493
$186$	0	0
$187$	9.80778	0.717216
$188$	0	0
$189$	0.685687	0.0498764
$190$	0	0
$191$	3.74971	0.271319	0.135660	−	0.990756i	$-0.456685\pi$
$191$	3.74971	0.271319	0.135660	+	0.990756i	$0.456685\pi$
$192$	0	0
$193$	−6.51547	−0.468994	−0.234497	−	0.972117i	$-0.575344\pi$
$193$	−6.51547	−0.468994	−0.234497	+	0.972117i	$0.575344\pi$
$194$	0	0
$195$	1.10575	0.0791847
$196$	0	0
$197$	3.84850	0.274194	0.137097	−	0.990558i	$-0.456223\pi$
$197$	3.84850	0.274194	0.137097	+	0.990558i	$0.456223\pi$
$198$	0	0
$199$	3.11118	0.220546	0.110273	−	0.993901i	$-0.464828\pi$
$199$	3.11118	0.220546	0.110273	+	0.993901i	$0.464828\pi$
$200$	0	0
$201$	7.03386	0.496130
$202$	0	0
$203$	1.70729	0.119828
$204$	0	0
$205$	−5.22269	−0.364769
$206$	0	0
$207$	2.35279	0.163530
$208$	0	0
$209$	1.59862	0.110579
$210$	0	0
$211$	−1.12650	−0.0775513	−0.0387757	−	0.999248i	$-0.512346\pi$
$211$	−1.12650	−0.0775513	−0.0387757	+	0.999248i	$0.512346\pi$
$212$	0	0
$213$	−7.53590	−0.516352
$214$	0	0
$215$	−12.6099	−0.859986
$216$	0	0
$217$	4.67974	0.317682
$218$	0	0
$219$	−11.5846	−0.782812
$220$	0	0
$221$	−3.55207	−0.238938
$222$	0	0
$223$	−0.307748	−0.0206083	−0.0103042	−	0.999947i	$-0.503280\pi$
$223$	−0.307748	−0.0206083	−0.0103042	+	0.999947i	$0.503280\pi$
$224$	0	0
$225$	−3.77731	−0.251821
$226$	0	0
$227$	19.1157	1.26875	0.634376	−	0.773025i	$-0.281257\pi$
$227$	19.1157	1.26875	0.634376	+	0.773025i	$0.281257\pi$
$228$	0	0
$229$	4.65469	0.307590	0.153795	−	0.988103i	$-0.450850\pi$
$229$	4.65469	0.307590	0.153795	+	0.988103i	$0.450850\pi$
$230$	0	0
$231$	−1.89328	−0.124569
$232$	0	0
$233$	15.4230	1.01039	0.505196	−	0.863004i	$-0.331420\pi$
$233$	15.4230	1.01039	0.505196	+	0.863004i	$0.331420\pi$
$234$	0	0
$235$	−5.15712	−0.336413
$236$	0	0
$237$	−17.0435	−1.10710
$238$	0	0
$239$	13.6256	0.881370	0.440685	−	0.897662i	$-0.354736\pi$
$239$	13.6256	0.881370	0.440685	+	0.897662i	$0.354736\pi$
$240$	0	0
$241$	−25.5569	−1.64626	−0.823132	−	0.567850i	$-0.807775\pi$
$241$	−25.5569	−1.64626	−0.823132	+	0.567850i	$0.807775\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	7.22039	0.461294
$246$	0	0
$247$	−0.578969	−0.0368389
$248$	0	0
$249$	11.1878	0.708996
$250$	0	0
$251$	−20.7744	−1.31127	−0.655633	−	0.755079i	$-0.727598\pi$
$251$	−20.7744	−1.31127	−0.655633	+	0.755079i	$0.727598\pi$
$252$	0	0
$253$	−6.49639	−0.408425
$254$	0	0
$255$	−3.92771	−0.245963
$256$	0	0
$257$	0.897798	0.0560031	0.0280016	−	0.999608i	$-0.491086\pi$
$257$	0.897798	0.0560031	0.0280016	+	0.999608i	$0.491086\pi$
$258$	0	0
$259$	4.87923	0.303181
$260$	0	0
$261$	−2.48990	−0.154121
$262$	0	0
$263$	−14.8581	−0.916190	−0.458095	−	0.888903i	$-0.651468\pi$
$263$	−14.8581	−0.916190	−0.458095	+	0.888903i	$0.651468\pi$
$264$	0	0
$265$	10.0722	0.618732
$266$	0	0
$267$	−12.5299	−0.766819
$268$	0	0
$269$	18.8744	1.15079	0.575397	−	0.817874i	$-0.304848\pi$
$269$	18.8744	1.15079	0.575397	+	0.817874i	$0.304848\pi$
$270$	0	0
$271$	−5.89671	−0.358200	−0.179100	−	0.983831i	$-0.557318\pi$
$271$	−5.89671	−0.358200	−0.179100	+	0.983831i	$0.557318\pi$
$272$	0	0
$273$	0.685687	0.0414996
$274$	0	0
$275$	10.4297	0.628935
$276$	0	0
$277$	5.37429	0.322910	0.161455	−	0.986880i	$-0.448381\pi$
$277$	5.37429	0.322910	0.161455	+	0.986880i	$0.448381\pi$
$278$	0	0
$279$	−6.82490	−0.408596
$280$	0	0
$281$	0.0410347	0.00244792	0.00122396	−	0.999999i	$-0.499610\pi$
$281$	0.0410347	0.00244792	0.00122396	+	0.999999i	$0.499610\pi$
$282$	0	0
$283$	6.32265	0.375843	0.187921	−	0.982184i	$-0.439825\pi$
$283$	6.32265	0.375843	0.187921	+	0.982184i	$0.439825\pi$
$284$	0	0
$285$	−0.640197	−0.0379220
$286$	0	0
$287$	−3.23863	−0.191170
$288$	0	0
$289$	−4.38283	−0.257814
$290$	0	0
$291$	12.7727	0.748748
$292$	0	0
$293$	0.685714	0.0400598	0.0200299	−	0.999799i	$-0.493624\pi$
$293$	0.685714	0.0400598	0.0200299	+	0.999799i	$0.493624\pi$
$294$	0	0
$295$	3.33571	0.194212
$296$	0	0
$297$	2.76115	0.160218
$298$	0	0
$299$	2.35279	0.136065
$300$	0	0
$301$	−7.81948	−0.450707
$302$	0	0
$303$	−6.61486	−0.380014
$304$	0	0
$305$	7.89363	0.451988
$306$	0	0
$307$	−10.4663	−0.597341	−0.298670	−	0.954356i	$-0.596543\pi$
$307$	−10.4663	−0.597341	−0.298670	+	0.954356i	$0.596543\pi$
$308$	0	0
$309$	4.54312	0.258449
$310$	0	0
$311$	14.3985	0.816464	0.408232	−	0.912878i	$-0.366145\pi$
$311$	14.3985	0.816464	0.408232	+	0.912878i	$0.366145\pi$
$312$	0	0
$313$	31.5228	1.78178	0.890888	−	0.454224i	$-0.150083\pi$
$313$	31.5228	1.78178	0.890888	+	0.454224i	$0.150083\pi$
$314$	0	0
$315$	0.758200	0.0427198
$316$	0	0
$317$	13.7158	0.770357	0.385178	−	0.922842i	$-0.374140\pi$
$317$	13.7158	0.770357	0.385178	+	0.922842i	$0.374140\pi$
$318$	0	0
$319$	6.87498	0.384925
$320$	0	0
$321$	2.76305	0.154218
$322$	0	0
$323$	2.05654	0.114429
$324$	0	0
$325$	−3.77731	−0.209527
$326$	0	0
$327$	13.7172	0.758565
$328$	0	0
$329$	−3.19797	−0.176310
$330$	0	0
$331$	28.9849	1.59315	0.796577	−	0.604537i	$-0.206642\pi$
$331$	28.9849	1.59315	0.796577	+	0.604537i	$0.206642\pi$
$332$	0	0
$333$	−7.11584	−0.389945
$334$	0	0
$335$	7.77771	0.424942
$336$	0	0
$337$	−1.17449	−0.0639787	−0.0319894	−	0.999488i	$-0.510184\pi$
$337$	−1.17449	−0.0639787	−0.0319894	+	0.999488i	$0.510184\pi$
$338$	0	0
$339$	12.4874	0.678222
$340$	0	0
$341$	18.8446	1.02049
$342$	0	0
$343$	9.27723	0.500923
$344$	0	0
$345$	2.60160	0.140066
$346$	0	0
$347$	5.19324	0.278787	0.139394	−	0.990237i	$-0.455485\pi$
$347$	5.19324	0.278787	0.139394	+	0.990237i	$0.455485\pi$
$348$	0	0
$349$	8.87053	0.474828	0.237414	−	0.971409i	$-0.423700\pi$
$349$	8.87053	0.474828	0.237414	+	0.971409i	$0.423700\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−5.11101	−0.272032	−0.136016	−	0.990707i	$-0.543430\pi$
$353$	−5.11101	−0.272032	−0.136016	+	0.990707i	$0.543430\pi$
$354$	0	0
$355$	−8.33285	−0.442262
$356$	0	0
$357$	−2.43560	−0.128906
$358$	0	0
$359$	36.1704	1.90900	0.954499	−	0.298214i	$-0.0963907\pi$
$359$	36.1704	1.90900	0.954499	+	0.298214i	$0.0963907\pi$
$360$	0	0
$361$	−18.6648	−0.982358
$362$	0	0
$363$	3.37606	0.177197
$364$	0	0
$365$	−12.8097	−0.670489
$366$	0	0
$367$	28.0824	1.46589	0.732945	−	0.680288i	$-0.238145\pi$
$367$	28.0824	1.46589	0.732945	+	0.680288i	$0.238145\pi$
$368$	0	0
$369$	4.72320	0.245880
$370$	0	0
$371$	6.24587	0.324269
$372$	0	0
$373$	18.7204	0.969304	0.484652	−	0.874707i	$-0.338946\pi$
$373$	18.7204	0.969304	0.484652	+	0.874707i	$0.338946\pi$
$374$	0	0
$375$	−9.70554	−0.501192
$376$	0	0
$377$	−2.48990	−0.128236
$378$	0	0
$379$	13.8256	0.710171	0.355086	−	0.934834i	$-0.384452\pi$
$379$	13.8256	0.710171	0.355086	+	0.934834i	$0.384452\pi$
$380$	0	0
$381$	−0.796554	−0.0408087
$382$	0	0
$383$	8.59161	0.439011	0.219505	−	0.975611i	$-0.429556\pi$
$383$	8.59161	0.439011	0.219505	+	0.975611i	$0.429556\pi$
$384$	0	0
$385$	−2.09350	−0.106695
$386$	0	0
$387$	11.4039	0.579691
$388$	0	0
$389$	−27.2446	−1.38136	−0.690679	−	0.723162i	$-0.742688\pi$
$389$	−27.2446	−1.38136	−0.690679	+	0.723162i	$0.742688\pi$
$390$	0	0
$391$	−8.35725	−0.422644
$392$	0	0
$393$	6.40167	0.322921
$394$	0	0
$395$	−18.8460	−0.948243
$396$	0	0
$397$	21.5335	1.08073	0.540367	−	0.841429i	$-0.318285\pi$
$397$	21.5335	1.08073	0.540367	+	0.841429i	$0.318285\pi$
$398$	0	0
$399$	−0.396991	−0.0198744
$400$	0	0
$401$	0.890079	0.0444484	0.0222242	−	0.999753i	$-0.492925\pi$
$401$	0.890079	0.0444484	0.0222242	+	0.999753i	$0.492925\pi$
$402$	0	0
$403$	−6.82490	−0.339973
$404$	0	0
$405$	−1.10575	−0.0549453
$406$	0	0
$407$	19.6479	0.973909
$408$	0	0
$409$	6.28608	0.310826	0.155413	−	0.987850i	$-0.450329\pi$
$409$	6.28608	0.310826	0.155413	+	0.987850i	$0.450329\pi$
$410$	0	0
$411$	−0.333293	−0.0164401
$412$	0	0
$413$	2.06850	0.101784
$414$	0	0
$415$	12.3709	0.607265
$416$	0	0
$417$	5.95367	0.291553
$418$	0	0
$419$	−5.63655	−0.275364	−0.137682	−	0.990477i	$-0.543965\pi$
$419$	−5.63655	−0.275364	−0.137682	+	0.990477i	$0.543965\pi$
$420$	0	0
$421$	−23.9587	−1.16768	−0.583838	−	0.811870i	$-0.698450\pi$
$421$	−23.9587	−1.16768	−0.583838	+	0.811870i	$0.698450\pi$
$422$	0	0
$423$	4.66390	0.226766
$424$	0	0
$425$	13.4172	0.650832
$426$	0	0
$427$	4.89491	0.236881
$428$	0	0
$429$	2.76115	0.133309
$430$	0	0
$431$	29.2799	1.41036	0.705182	−	0.709026i	$-0.250865\pi$
$431$	29.2799	1.41036	0.705182	+	0.709026i	$0.250865\pi$
$432$	0	0
$433$	11.2630	0.541266	0.270633	−	0.962683i	$-0.412767\pi$
$433$	11.2630	0.541266	0.270633	+	0.962683i	$0.412767\pi$
$434$	0	0
$435$	−2.75322	−0.132007
$436$	0	0
$437$	−1.36219	−0.0651624
$438$	0	0
$439$	30.4516	1.45338	0.726688	−	0.686967i	$-0.241058\pi$
$439$	30.4516	1.45338	0.726688	+	0.686967i	$0.241058\pi$
$440$	0	0
$441$	−6.52983	−0.310944
$442$	0	0
$443$	−26.1510	−1.24247	−0.621236	−	0.783624i	$-0.713369\pi$
$443$	−26.1510	−1.24247	−0.621236	+	0.783624i	$0.713369\pi$
$444$	0	0
$445$	−13.8550	−0.656790
$446$	0	0
$447$	−11.4682	−0.542426
$448$	0	0
$449$	−31.2462	−1.47460	−0.737301	−	0.675565i	$-0.763900\pi$
$449$	−31.2462	−1.47460	−0.737301	+	0.675565i	$0.763900\pi$
$450$	0	0
$451$	−13.0414	−0.614098
$452$	0	0
$453$	−6.82129	−0.320492
$454$	0	0
$455$	0.758200	0.0355450
$456$	0	0
$457$	−10.9309	−0.511327	−0.255663	−	0.966766i	$-0.582294\pi$
$457$	−10.9309	−0.511327	−0.255663	+	0.966766i	$0.582294\pi$
$458$	0	0
$459$	3.55207	0.165796
$460$	0	0
$461$	−3.40242	−0.158467	−0.0792333	−	0.996856i	$-0.525247\pi$
$461$	−3.40242	−0.158467	−0.0792333	+	0.996856i	$0.525247\pi$
$462$	0	0
$463$	9.31244	0.432786	0.216393	−	0.976306i	$-0.430571\pi$
$463$	9.31244	0.432786	0.216393	+	0.976306i	$0.430571\pi$
$464$	0	0
$465$	−7.54666	−0.349968
$466$	0	0
$467$	17.8530	0.826140	0.413070	−	0.910699i	$-0.364456\pi$
$467$	17.8530	0.826140	0.413070	+	0.910699i	$0.364456\pi$
$468$	0	0
$469$	4.82302	0.222706
$470$	0	0
$471$	−9.24604	−0.426035
$472$	0	0
$473$	−31.4878	−1.44781
$474$	0	0
$475$	2.18694	0.100344
$476$	0	0
$477$	−9.10892	−0.417069
$478$	0	0
$479$	−9.89685	−0.452199	−0.226099	−	0.974104i	$-0.572597\pi$
$479$	−9.89685	−0.452199	−0.226099	+	0.974104i	$0.572597\pi$
$480$	0	0
$481$	−7.11584	−0.324454
$482$	0	0
$483$	1.61327	0.0734066
$484$	0	0
$485$	14.1234	0.641313
$486$	0	0
$487$	20.8666	0.945554	0.472777	−	0.881182i	$-0.343252\pi$
$487$	20.8666	0.945554	0.472777	+	0.881182i	$0.343252\pi$
$488$	0	0
$489$	14.2252	0.643284
$490$	0	0
$491$	−8.99319	−0.405857	−0.202928	−	0.979194i	$-0.565046\pi$
$491$	−8.99319	−0.405857	−0.202928	+	0.979194i	$0.565046\pi$
$492$	0	0
$493$	8.84429	0.398327
$494$	0	0
$495$	3.05315	0.137229
$496$	0	0
$497$	−5.16727	−0.231784
$498$	0	0
$499$	14.9129	0.667595	0.333797	−	0.942645i	$-0.391670\pi$
$499$	14.9129	0.667595	0.333797	+	0.942645i	$0.391670\pi$
$500$	0	0
$501$	−23.1974	−1.03638
$502$	0	0
$503$	−7.56905	−0.337487	−0.168744	−	0.985660i	$-0.553971\pi$
$503$	−7.56905	−0.337487	−0.168744	+	0.985660i	$0.553971\pi$
$504$	0	0
$505$	−7.31440	−0.325486
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	5.16738	0.229040	0.114520	−	0.993421i	$-0.463467\pi$
$509$	5.16738	0.229040	0.114520	+	0.993421i	$0.463467\pi$
$510$	0	0
$511$	−7.94338	−0.351394
$512$	0	0
$513$	0.578969	0.0255621
$514$	0	0
$515$	5.02357	0.221365
$516$	0	0
$517$	−12.8777	−0.566361
$518$	0	0
$519$	7.21309	0.316619
$520$	0	0
$521$	−14.5480	−0.637358	−0.318679	−	0.947863i	$-0.603239\pi$
$521$	−14.5480	−0.637358	−0.318679	+	0.947863i	$0.603239\pi$
$522$	0	0
$523$	11.6629	0.509984	0.254992	−	0.966943i	$-0.417927\pi$
$523$	11.6629	0.509984	0.254992	+	0.966943i	$0.417927\pi$
$524$	0	0
$525$	−2.59005	−0.113039
$526$	0	0
$527$	24.2425	1.05602
$528$	0	0
$529$	−17.4644	−0.759321
$530$	0	0
$531$	−3.01668	−0.130913
$532$	0	0
$533$	4.72320	0.204584
$534$	0	0
$535$	3.05525	0.132090
$536$	0	0
$537$	−11.4129	−0.492502
$538$	0	0
$539$	18.0298	0.776600
$540$	0	0
$541$	−15.2582	−0.656003	−0.328001	−	0.944677i	$-0.606375\pi$
$541$	−15.2582	−0.656003	−0.328001	+	0.944677i	$0.606375\pi$
$542$	0	0
$543$	19.3259	0.829354
$544$	0	0
$545$	15.1679	0.649721
$546$	0	0
$547$	−28.2101	−1.20618	−0.603089	−	0.797674i	$-0.706064\pi$
$547$	−28.2101	−1.20618	−0.603089	+	0.797674i	$0.706064\pi$
$548$	0	0
$549$	−7.13869	−0.304672
$550$	0	0
$551$	1.44157	0.0614131
$552$	0	0
$553$	−11.6865	−0.496962
$554$	0	0
$555$	−7.86836	−0.333993
$556$	0	0
$557$	18.9541	0.803112	0.401556	−	0.915834i	$-0.368469\pi$
$557$	18.9541	0.803112	0.401556	+	0.915834i	$0.368469\pi$
$558$	0	0
$559$	11.4039	0.482332
$560$	0	0
$561$	−9.80778	−0.414085
$562$	0	0
$563$	3.54239	0.149294	0.0746471	−	0.997210i	$-0.476217\pi$
$563$	3.54239	0.149294	0.0746471	+	0.997210i	$0.476217\pi$
$564$	0	0
$565$	13.8080	0.580906
$566$	0	0
$567$	−0.685687	−0.0287961
$568$	0	0
$569$	23.7895	0.997306	0.498653	−	0.866802i	$-0.333828\pi$
$569$	23.7895	0.997306	0.498653	+	0.866802i	$0.333828\pi$
$570$	0	0
$571$	34.7499	1.45424	0.727118	−	0.686512i	$-0.240859\pi$
$571$	34.7499	1.45424	0.727118	+	0.686512i	$0.240859\pi$
$572$	0	0
$573$	−3.74971	−0.156646
$574$	0	0
$575$	−8.88721	−0.370622
$576$	0	0
$577$	19.5537	0.814032	0.407016	−	0.913421i	$-0.366569\pi$
$577$	19.5537	0.814032	0.407016	+	0.913421i	$0.366569\pi$
$578$	0	0
$579$	6.51547	0.270774
$580$	0	0
$581$	7.67131	0.318259
$582$	0	0
$583$	25.1511	1.04165
$584$	0	0
$585$	−1.10575	−0.0457173
$586$	0	0
$587$	−20.7215	−0.855269	−0.427634	−	0.903952i	$-0.640653\pi$
$587$	−20.7215	−0.855269	−0.427634	+	0.903952i	$0.640653\pi$
$588$	0	0
$589$	3.95141	0.162815
$590$	0	0
$591$	−3.84850	−0.158306
$592$	0	0
$593$	11.8943	0.488441	0.244221	−	0.969720i	$-0.421468\pi$
$593$	11.8943	0.488441	0.244221	+	0.969720i	$0.421468\pi$
$594$	0	0
$595$	−2.69318	−0.110410
$596$	0	0
$597$	−3.11118	−0.127332
$598$	0	0
$599$	−28.2654	−1.15489	−0.577446	−	0.816429i	$-0.695951\pi$
$599$	−28.2654	−1.15489	−0.577446	+	0.816429i	$0.695951\pi$
$600$	0	0
$601$	9.99876	0.407858	0.203929	−	0.978986i	$-0.434629\pi$
$601$	9.99876	0.407858	0.203929	+	0.978986i	$0.434629\pi$
$602$	0	0
$603$	−7.03386	−0.286441
$604$	0	0
$605$	3.73309	0.151772
$606$	0	0
$607$	−0.899468	−0.0365083	−0.0182541	−	0.999833i	$-0.505811\pi$
$607$	−0.899468	−0.0365083	−0.0182541	+	0.999833i	$0.505811\pi$
$608$	0	0
$609$	−1.70729	−0.0691829
$610$	0	0
$611$	4.66390	0.188681
$612$	0	0
$613$	−7.32954	−0.296038	−0.148019	−	0.988985i	$-0.547290\pi$
$613$	−7.32954	−0.296038	−0.148019	+	0.988985i	$0.547290\pi$
$614$	0	0
$615$	5.22269	0.210599
$616$	0	0
$617$	46.5169	1.87270	0.936349	−	0.351070i	$-0.114182\pi$
$617$	46.5169	1.87270	0.936349	+	0.351070i	$0.114182\pi$
$618$	0	0
$619$	−0.661456	−0.0265862	−0.0132931	−	0.999912i	$-0.504231\pi$
$619$	−0.661456	−0.0265862	−0.0132931	+	0.999912i	$0.504231\pi$
$620$	0	0
$621$	−2.35279	−0.0944141
$622$	0	0
$623$	−8.59160	−0.344215
$624$	0	0
$625$	8.15461	0.326184
$626$	0	0
$627$	−1.59862	−0.0638427
$628$	0	0
$629$	25.2759	1.00782
$630$	0	0
$631$	31.5950	1.25778	0.628889	−	0.777495i	$-0.283510\pi$
$631$	31.5950	1.25778	0.628889	+	0.777495i	$0.283510\pi$
$632$	0	0
$633$	1.12650	0.0447743
$634$	0	0
$635$	−0.880793	−0.0349532
$636$	0	0
$637$	−6.52983	−0.258721
$638$	0	0
$639$	7.53590	0.298116
$640$	0	0
$641$	−10.8982	−0.430451	−0.215226	−	0.976564i	$-0.569049\pi$
$641$	−10.8982	−0.430451	−0.215226	+	0.976564i	$0.569049\pi$
$642$	0	0
$643$	5.08026	0.200346	0.100173	−	0.994970i	$-0.468060\pi$
$643$	5.08026	0.200346	0.100173	+	0.994970i	$0.468060\pi$
$644$	0	0
$645$	12.6099	0.496513
$646$	0	0
$647$	−12.3338	−0.484893	−0.242447	−	0.970165i	$-0.577950\pi$
$647$	−12.3338	−0.484893	−0.242447	+	0.970165i	$0.577950\pi$
$648$	0	0
$649$	8.32951	0.326962
$650$	0	0
$651$	−4.67974	−0.183414
$652$	0	0
$653$	48.7543	1.90790	0.953952	−	0.299961i	$-0.0969736\pi$
$653$	48.7543	1.90790	0.953952	+	0.299961i	$0.0969736\pi$
$654$	0	0
$655$	7.07866	0.276586
$656$	0	0
$657$	11.5846	0.451957
$658$	0	0
$659$	30.5060	1.18834	0.594172	−	0.804338i	$-0.297480\pi$
$659$	30.5060	1.18834	0.594172	+	0.804338i	$0.297480\pi$
$660$	0	0
$661$	−13.0033	−0.505770	−0.252885	−	0.967496i	$-0.581380\pi$
$661$	−13.0033	−0.505770	−0.252885	+	0.967496i	$0.581380\pi$
$662$	0	0
$663$	3.55207	0.137951
$664$	0	0
$665$	−0.438974	−0.0170227
$666$	0	0
$667$	−5.85820	−0.226831
$668$	0	0
$669$	0.307748	0.0118982
$670$	0	0
$671$	19.7110	0.760934
$672$	0	0
$673$	24.3174	0.937367	0.468683	−	0.883366i	$-0.344729\pi$
$673$	24.3174	0.937367	0.468683	+	0.883366i	$0.344729\pi$
$674$	0	0
$675$	3.77731	0.145389
$676$	0	0
$677$	23.3502	0.897420	0.448710	−	0.893677i	$-0.351884\pi$
$677$	23.3502	0.897420	0.448710	+	0.893677i	$0.351884\pi$
$678$	0	0
$679$	8.75806	0.336104
$680$	0	0
$681$	−19.1157	−0.732514
$682$	0	0
$683$	35.6322	1.36343	0.681714	−	0.731619i	$-0.261235\pi$
$683$	35.6322	1.36343	0.681714	+	0.731619i	$0.261235\pi$
$684$	0	0
$685$	−0.368540	−0.0140812
$686$	0	0
$687$	−4.65469	−0.177587
$688$	0	0
$689$	−9.10892	−0.347022
$690$	0	0
$691$	−15.6695	−0.596095	−0.298047	−	0.954551i	$-0.596335\pi$
$691$	−15.6695	−0.596095	−0.298047	+	0.954551i	$0.596335\pi$
$692$	0	0
$693$	1.89328	0.0719198
$694$	0	0
$695$	6.58330	0.249719
$696$	0	0
$697$	−16.7771	−0.635478
$698$	0	0
$699$	−15.4230	−0.583351
$700$	0	0
$701$	47.9494	1.81102	0.905511	−	0.424323i	$-0.139488\pi$
$701$	47.9494	1.81102	0.905511	+	0.424323i	$0.139488\pi$
$702$	0	0
$703$	4.11985	0.155383
$704$	0	0
$705$	5.15712	0.194228
$706$	0	0
$707$	−4.53572	−0.170583
$708$	0	0
$709$	−38.1397	−1.43237	−0.716183	−	0.697913i	$-0.754112\pi$
$709$	−38.1397	−1.43237	−0.716183	+	0.697913i	$0.754112\pi$
$710$	0	0
$711$	17.0435	0.639183
$712$	0	0
$713$	−16.0575	−0.601360
$714$	0	0
$715$	3.05315	0.114181
$716$	0	0
$717$	−13.6256	−0.508859
$718$	0	0
$719$	15.4056	0.574531	0.287265	−	0.957851i	$-0.407254\pi$
$719$	15.4056	0.574531	0.287265	+	0.957851i	$0.407254\pi$
$720$	0	0
$721$	3.11516	0.116015
$722$	0	0
$723$	25.5569	0.950471
$724$	0	0
$725$	9.40512	0.349297
$726$	0	0
$727$	−30.6861	−1.13809	−0.569043	−	0.822308i	$-0.692686\pi$
$727$	−30.6861	−1.13809	−0.569043	+	0.822308i	$0.692686\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−40.5073	−1.49822
$732$	0	0
$733$	39.3294	1.45266	0.726332	−	0.687344i	$-0.241224\pi$
$733$	39.3294	1.45266	0.726332	+	0.687344i	$0.241224\pi$
$734$	0	0
$735$	−7.22039	−0.266328
$736$	0	0
$737$	19.4215	0.715401
$738$	0	0
$739$	−22.8826	−0.841750	−0.420875	−	0.907119i	$-0.638277\pi$
$739$	−22.8826	−0.841750	−0.420875	+	0.907119i	$0.638277\pi$
$740$	0	0
$741$	0.578969	0.0212690
$742$	0	0
$743$	−34.1646	−1.25338	−0.626690	−	0.779269i	$-0.715591\pi$
$743$	−34.1646	−1.25338	−0.626690	+	0.779269i	$0.715591\pi$
$744$	0	0
$745$	−12.6810	−0.464595
$746$	0	0
$747$	−11.1878	−0.409339
$748$	0	0
$749$	1.89459	0.0692267
$750$	0	0
$751$	48.5164	1.77039	0.885195	−	0.465221i	$-0.154025\pi$
$751$	48.5164	1.77039	0.885195	+	0.465221i	$0.154025\pi$
$752$	0	0
$753$	20.7744	0.757060
$754$	0	0
$755$	−7.54267	−0.274506
$756$	0	0
$757$	−1.04163	−0.0378588	−0.0189294	−	0.999821i	$-0.506026\pi$
$757$	−1.04163	−0.0378588	−0.0189294	+	0.999821i	$0.506026\pi$
$758$	0	0
$759$	6.49639	0.235804
$760$	0	0
$761$	−33.0167	−1.19685	−0.598427	−	0.801177i	$-0.704207\pi$
$761$	−33.0167	−1.19685	−0.598427	+	0.801177i	$0.704207\pi$
$762$	0	0
$763$	9.40573	0.340510
$764$	0	0
$765$	3.92771	0.142007
$766$	0	0
$767$	−3.01668	−0.108926
$768$	0	0
$769$	−7.43777	−0.268213	−0.134106	−	0.990967i	$-0.542816\pi$
$769$	−7.43777	−0.268213	−0.134106	+	0.990967i	$0.542816\pi$
$770$	0	0
$771$	−0.897798	−0.0323334
$772$	0	0
$773$	51.5396	1.85375	0.926876	−	0.375368i	$-0.122484\pi$
$773$	51.5396	1.85375	0.926876	+	0.375368i	$0.122484\pi$
$774$	0	0
$775$	25.7798	0.926037
$776$	0	0
$777$	−4.87923	−0.175042
$778$	0	0
$779$	−2.73458	−0.0979766
$780$	0	0
$781$	−20.8077	−0.744560
$782$	0	0
$783$	2.48990	0.0889818
$784$	0	0
$785$	−10.2238	−0.364904
$786$	0	0
$787$	3.30283	0.117733	0.0588667	−	0.998266i	$-0.481251\pi$
$787$	3.30283	0.117733	0.0588667	+	0.998266i	$0.481251\pi$
$788$	0	0
$789$	14.8581	0.528962
$790$	0	0
$791$	8.56243	0.304445
$792$	0	0
$793$	−7.13869	−0.253502
$794$	0	0
$795$	−10.0722	−0.357225
$796$	0	0
$797$	−50.1597	−1.77675	−0.888374	−	0.459121i	$-0.848165\pi$
$797$	−50.1597	−1.77675	−0.888374	+	0.459121i	$0.848165\pi$
$798$	0	0
$799$	−16.5665	−0.586079
$800$	0	0
$801$	12.5299	0.442723
$802$	0	0
$803$	−31.9867	−1.12879
$804$	0	0
$805$	1.78388	0.0628737
$806$	0	0
$807$	−18.8744	−0.664411
$808$	0	0
$809$	−40.9805	−1.44080	−0.720399	−	0.693560i	$-0.756041\pi$
$809$	−40.9805	−1.44080	−0.720399	+	0.693560i	$0.756041\pi$
$810$	0	0
$811$	43.0183	1.51058	0.755288	−	0.655393i	$-0.227497\pi$
$811$	43.0183	1.51058	0.755288	+	0.655393i	$0.227497\pi$
$812$	0	0
$813$	5.89671	0.206807
$814$	0	0
$815$	15.7295	0.550981
$816$	0	0
$817$	−6.60249	−0.230992
$818$	0	0
$819$	−0.685687	−0.0239598
$820$	0	0
$821$	39.9408	1.39394	0.696971	−	0.717099i	$-0.254531\pi$
$821$	39.9408	1.39394	0.696971	+	0.717099i	$0.254531\pi$
$822$	0	0
$823$	−0.699056	−0.0243675	−0.0121838	−	0.999926i	$-0.503878\pi$
$823$	−0.699056	−0.0243675	−0.0121838	+	0.999926i	$0.503878\pi$
$824$	0	0
$825$	−10.4297	−0.363116
$826$	0	0
$827$	37.1391	1.29145	0.645726	−	0.763569i	$-0.276555\pi$
$827$	37.1391	1.29145	0.645726	+	0.763569i	$0.276555\pi$
$828$	0	0
$829$	−37.6179	−1.30652	−0.653262	−	0.757132i	$-0.726600\pi$
$829$	−37.6179	−1.30652	−0.653262	+	0.757132i	$0.726600\pi$
$830$	0	0
$831$	−5.37429	−0.186432
$832$	0	0
$833$	23.1944	0.803638
$834$	0	0
$835$	−25.6506	−0.887676
$836$	0	0
$837$	6.82490	0.235903
$838$	0	0
$839$	27.7951	0.959592	0.479796	−	0.877380i	$-0.340711\pi$
$839$	27.7951	0.959592	0.479796	+	0.877380i	$0.340711\pi$
$840$	0	0
$841$	−22.8004	−0.786221
$842$	0	0
$843$	−0.0410347	−0.00141331
$844$	0	0
$845$	−1.10575	−0.0380391
$846$	0	0
$847$	2.31492	0.0795417
$848$	0	0
$849$	−6.32265	−0.216993
$850$	0	0
$851$	−16.7420	−0.573910
$852$	0	0
$853$	8.03933	0.275261	0.137631	−	0.990484i	$-0.456051\pi$
$853$	8.03933	0.275261	0.137631	+	0.990484i	$0.456051\pi$
$854$	0	0
$855$	0.640197	0.0218943
$856$	0	0
$857$	−1.86836	−0.0638220	−0.0319110	−	0.999491i	$-0.510159\pi$
$857$	−1.86836	−0.0638220	−0.0319110	+	0.999491i	$0.510159\pi$
$858$	0	0
$859$	−14.4919	−0.494458	−0.247229	−	0.968957i	$-0.579520\pi$
$859$	−14.4919	−0.494458	−0.247229	+	0.968957i	$0.579520\pi$
$860$	0	0
$861$	3.23863	0.110372
$862$	0	0
$863$	−30.2816	−1.03080	−0.515398	−	0.856951i	$-0.672356\pi$
$863$	−30.2816	−1.03080	−0.515398	+	0.856951i	$0.672356\pi$
$864$	0	0
$865$	7.97590	0.271189
$866$	0	0
$867$	4.38283	0.148849
$868$	0	0
$869$	−47.0597	−1.59639
$870$	0	0
$871$	−7.03386	−0.238333
$872$	0	0
$873$	−12.7727	−0.432290
$874$	0	0
$875$	−6.65496	−0.224979
$876$	0	0
$877$	−32.6612	−1.10289	−0.551445	−	0.834211i	$-0.685924\pi$
$877$	−32.6612	−1.10289	−0.551445	+	0.834211i	$0.685924\pi$
$878$	0	0
$879$	−0.685714	−0.0231285
$880$	0	0
$881$	19.6921	0.663444	0.331722	−	0.943377i	$-0.392370\pi$
$881$	19.6921	0.663444	0.331722	+	0.943377i	$0.392370\pi$
$882$	0	0
$883$	−45.0848	−1.51722	−0.758611	−	0.651543i	$-0.774122\pi$
$883$	−45.0848	−1.51722	−0.758611	+	0.651543i	$0.774122\pi$
$884$	0	0
$885$	−3.33571	−0.112129
$886$	0	0
$887$	35.2426	1.18333	0.591665	−	0.806184i	$-0.298471\pi$
$887$	35.2426	1.18333	0.591665	+	0.806184i	$0.298471\pi$
$888$	0	0
$889$	−0.546187	−0.0183185
$890$	0	0
$891$	−2.76115	−0.0925019
$892$	0	0
$893$	−2.70025	−0.0903605
$894$	0	0
$895$	−12.6198	−0.421835
$896$	0	0
$897$	−2.35279	−0.0785573
$898$	0	0
$899$	16.9933	0.566759
$900$	0	0
$901$	32.3555	1.07792
$902$	0	0
$903$	7.81948	0.260216
$904$	0	0
$905$	21.3697	0.710352
$906$	0	0
$907$	29.3483	0.974495	0.487247	−	0.873264i	$-0.338001\pi$
$907$	29.3483	0.974495	0.487247	+	0.873264i	$0.338001\pi$
$908$	0	0
$909$	6.61486	0.219401
$910$	0	0
$911$	−3.06426	−0.101523	−0.0507617	−	0.998711i	$-0.516165\pi$
$911$	−3.06426	−0.101523	−0.0507617	+	0.998711i	$0.516165\pi$
$912$	0	0
$913$	30.8911	1.02235
$914$	0	0
$915$	−7.89363	−0.260956
$916$	0	0
$917$	4.38954	0.144955
$918$	0	0
$919$	−11.7866	−0.388803	−0.194401	−	0.980922i	$-0.562276\pi$
$919$	−11.7866	−0.388803	−0.194401	+	0.980922i	$0.562276\pi$
$920$	0	0
$921$	10.4663	0.344875
$922$	0	0
$923$	7.53590	0.248047
$924$	0	0
$925$	26.8787	0.883767
$926$	0	0
$927$	−4.54312	−0.149216
$928$	0	0
$929$	26.2340	0.860708	0.430354	−	0.902660i	$-0.358389\pi$
$929$	26.2340	0.860708	0.430354	+	0.902660i	$0.358389\pi$
$930$	0	0
$931$	3.78057	0.123903
$932$	0	0
$933$	−14.3985	−0.471386
$934$	0	0
$935$	−10.8450	−0.354669
$936$	0	0
$937$	−28.4449	−0.929255	−0.464627	−	0.885506i	$-0.653812\pi$
$937$	−28.4449	−0.929255	−0.464627	+	0.885506i	$0.653812\pi$
$938$	0	0
$939$	−31.5228	−1.02871
$940$	0	0
$941$	54.6424	1.78129	0.890646	−	0.454698i	$-0.150253\pi$
$941$	54.6424	1.78129	0.890646	+	0.454698i	$0.150253\pi$
$942$	0	0
$943$	11.1127	0.361879
$944$	0	0
$945$	−0.758200	−0.0246643
$946$	0	0
$947$	2.41371	0.0784350	0.0392175	−	0.999231i	$-0.487513\pi$
$947$	2.41371	0.0784350	0.0392175	+	0.999231i	$0.487513\pi$
$948$	0	0
$949$	11.5846	0.376051
$950$	0	0
$951$	−13.7158	−0.444766
$952$	0	0
$953$	−55.6943	−1.80411	−0.902057	−	0.431617i	$-0.857943\pi$
$953$	−55.6943	−1.80411	−0.902057	+	0.431617i	$0.857943\pi$
$954$	0	0
$955$	−4.14625	−0.134170
$956$	0	0
$957$	−6.87498	−0.222237
$958$	0	0
$959$	−0.228535	−0.00737977
$960$	0	0
$961$	15.5793	0.502558
$962$	0	0
$963$	−2.76305	−0.0890380
$964$	0	0
$965$	7.20451	0.231921
$966$	0	0
$967$	−13.6118	−0.437725	−0.218862	−	0.975756i	$-0.570235\pi$
$967$	−13.6118	−0.437725	−0.218862	+	0.975756i	$0.570235\pi$
$968$	0	0
$969$	−2.05654	−0.0660654
$970$	0	0
$971$	−38.1923	−1.22565	−0.612825	−	0.790219i	$-0.709967\pi$
$971$	−38.1923	−1.22565	−0.612825	+	0.790219i	$0.709967\pi$
$972$	0	0
$973$	4.08236	0.130874
$974$	0	0
$975$	3.77731	0.120971
$976$	0	0
$977$	15.2697	0.488521	0.244260	−	0.969710i	$-0.421455\pi$
$977$	15.2697	0.488521	0.244260	+	0.969710i	$0.421455\pi$
$978$	0	0
$979$	−34.5970	−1.10572
$980$	0	0
$981$	−13.7172	−0.437958
$982$	0	0
$983$	−35.5633	−1.13429	−0.567147	−	0.823617i	$-0.691953\pi$
$983$	−35.5633	−1.13429	−0.567147	+	0.823617i	$0.691953\pi$
$984$	0	0
$985$	−4.25549	−0.135591
$986$	0	0
$987$	3.19797	0.101793
$988$	0	0
$989$	26.8309	0.853172
$990$	0	0
$991$	−31.8367	−1.01133	−0.505663	−	0.862731i	$-0.668752\pi$
$991$	−31.8367	−1.01133	−0.505663	+	0.862731i	$0.668752\pi$
$992$	0	0
$993$	−28.9849	−0.919808
$994$	0	0
$995$	−3.44020	−0.109062
$996$	0	0
$997$	8.18204	0.259128	0.129564	−	0.991571i	$-0.458642\pi$
$997$	8.18204	0.259128	0.129564	+	0.991571i	$0.458642\pi$
$998$	0	0
$999$	7.11584	0.225135

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9984.2.a.bn.1.2		6
4.3	odd	2		9984.2.a.bo.1.2		6
8.3	odd	2		9984.2.a.bl.1.5		6
8.5	even	2		9984.2.a.bq.1.5		6
16.3	odd	4		4992.2.g.k.2497.11	yes	12
16.5	even	4		4992.2.g.i.2497.8	yes	12
16.11	odd	4		4992.2.g.k.2497.2	yes	12
16.13	even	4		4992.2.g.i.2497.5	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4992.2.g.i.2497.5	✓	12	16.13	even	4
4992.2.g.i.2497.8	yes	12	16.5	even	4
4992.2.g.k.2497.2	yes	12	16.11	odd	4
4992.2.g.k.2497.11	yes	12	16.3	odd	4
9984.2.a.bl.1.5		6	8.3	odd	2
9984.2.a.bn.1.2		6	1.1	even	1	trivial
9984.2.a.bo.1.2		6	4.3	odd	2
9984.2.a.bq.1.5		6	8.5	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:	\( N \)	\(=\)	\( 9984 = 2^{8} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9984.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.10575 q^{5} -0.685687 q^{7} +1.00000 q^{9} -2.76115 q^{11} +1.00000 q^{13} +1.10575 q^{15} -3.55207 q^{17} -0.578969 q^{19} +0.685687 q^{21} +2.35279 q^{23} -3.77731 q^{25} -1.00000 q^{27} -2.48990 q^{29} -6.82490 q^{31} +2.76115 q^{33} +0.758200 q^{35} -7.11584 q^{37} -1.00000 q^{39} +4.72320 q^{41} +11.4039 q^{43} -1.10575 q^{45} +4.66390 q^{47} -6.52983 q^{49} +3.55207 q^{51} -9.10892 q^{53} +3.05315 q^{55} +0.578969 q^{57} -3.01668 q^{59} -7.13869 q^{61} -0.685687 q^{63} -1.10575 q^{65} -7.03386 q^{67} -2.35279 q^{69} +7.53590 q^{71} +11.5846 q^{73} +3.77731 q^{75} +1.89328 q^{77} +17.0435 q^{79} +1.00000 q^{81} -11.1878 q^{83} +3.92771 q^{85} +2.48990 q^{87} +12.5299 q^{89} -0.685687 q^{91} +6.82490 q^{93} +0.640197 q^{95} -12.7727 q^{97} -2.76115 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 4 q^{7} + 6 q^{9} + 8 q^{11} + 6 q^{13} + 4 q^{19} - 4 q^{21} + 8 q^{23} + 2 q^{25} - 6 q^{27} + 4 q^{29} + 20 q^{31} - 8 q^{33} + 8 q^{35} - 20 q^{37} - 6 q^{39} + 16 q^{47} + 10 q^{49}+ \cdots + 8 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 + 4 * q^7 + 6 * q^9 + 8 * q^11 + 6 * q^13 + 4 * q^19 - 4 * q^21 + 8 * q^23 + 2 * q^25 - 6 * q^27 + 4 * q^29 + 20 * q^31 - 8 * q^33 + 8 * q^35 - 20 * q^37 - 6 * q^39 + 16 * q^47 + 10 * q^49 + 8 * q^53 + 16 * q^55 - 4 * q^57 - 8 * q^61 + 4 * q^63 + 4 * q^67 - 8 * q^69 + 24 * q^71 - 4 * q^73 - 2 * q^75 + 12 * q^77 + 40 * q^79 + 6 * q^81 + 16 * q^83 - 16 * q^85 - 4 * q^87 + 16 * q^89 + 4 * q^91 - 20 * q^93 - 8 * q^95 - 20 * q^97 + 8 * q^99

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