LMFDB - Embedded newform 3344.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, 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("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.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:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.30278 q^{3} +1.30278 q^{5} +2.30278 q^{7} +7.90833 q^{9} +O(q^{10})$	$q+3.30278 q^{3} +1.30278 q^{5} +2.30278 q^{7} +7.90833 q^{9} -1.00000 q^{11} +0.302776 q^{13} +4.30278 q^{15} +2.60555 q^{17} -1.00000 q^{19} +7.60555 q^{21} +8.60555 q^{23} -3.30278 q^{25} +16.2111 q^{27} -4.69722 q^{29} -0.302776 q^{31} -3.30278 q^{33} +3.00000 q^{35} -9.21110 q^{37} +1.00000 q^{39} -6.90833 q^{41} -11.9083 q^{43} +10.3028 q^{45} +6.00000 q^{47} -1.69722 q^{49} +8.60555 q^{51} -3.39445 q^{53} -1.30278 q^{55} -3.30278 q^{57} -3.21110 q^{61} +18.2111 q^{63} +0.394449 q^{65} -11.5139 q^{67} +28.4222 q^{69} +13.3028 q^{71} +4.60555 q^{73} -10.9083 q^{75} -2.30278 q^{77} +5.81665 q^{79} +29.8167 q^{81} -10.6972 q^{83} +3.39445 q^{85} -15.5139 q^{87} -2.60555 q^{89} +0.697224 q^{91} -1.00000 q^{93} -1.30278 q^{95} +8.00000 q^{97} -7.90833 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} - q^{5} + q^{7} + 5 q^{9}+O(q^{10}) $ 2 * q + 3 * q^3 - q^5 + q^7 + 5 * q^9	$ 2 q + 3 q^{3} - q^{5} + q^{7} + 5 q^{9} - 2 q^{11} - 3 q^{13} + 5 q^{15} - 2 q^{17} - 2 q^{19} + 8 q^{21} + 10 q^{23} - 3 q^{25} + 18 q^{27} - 13 q^{29} + 3 q^{31} - 3 q^{33} + 6 q^{35} - 4 q^{37} + 2 q^{39} - 3 q^{41} - 13 q^{43} + 17 q^{45} + 12 q^{47} - 7 q^{49} + 10 q^{51} - 14 q^{53} + q^{55} - 3 q^{57} + 8 q^{61} + 22 q^{63} + 8 q^{65} - 5 q^{67} + 28 q^{69} + 23 q^{71} + 2 q^{73} - 11 q^{75} - q^{77} - 10 q^{79} + 38 q^{81} - 25 q^{83} + 14 q^{85} - 13 q^{87} + 2 q^{89} + 5 q^{91} - 2 q^{93} + q^{95} + 16 q^{97} - 5 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 - q^5 + q^7 + 5 * q^9 - 2 * q^11 - 3 * q^13 + 5 * q^15 - 2 * q^17 - 2 * q^19 + 8 * q^21 + 10 * q^23 - 3 * q^25 + 18 * q^27 - 13 * q^29 + 3 * q^31 - 3 * q^33 + 6 * q^35 - 4 * q^37 + 2 * q^39 - 3 * q^41 - 13 * q^43 + 17 * q^45 + 12 * q^47 - 7 * q^49 + 10 * q^51 - 14 * q^53 + q^55 - 3 * q^57 + 8 * q^61 + 22 * q^63 + 8 * q^65 - 5 * q^67 + 28 * q^69 + 23 * q^71 + 2 * q^73 - 11 * q^75 - q^77 - 10 * q^79 + 38 * q^81 - 25 * q^83 + 14 * q^85 - 13 * q^87 + 2 * q^89 + 5 * q^91 - 2 * q^93 + q^95 + 16 * q^97 - 5 * 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
$3$	3.30278	1.90686	0.953429	−	0.301617i	$-0.0975264\pi$
$3$	3.30278	1.90686	0.953429	+	0.301617i	$0.0975264\pi$
$4$	0	0
$5$	1.30278	0.582619	0.291309	−	0.956629i	$-0.405909\pi$
$5$	1.30278	0.582619	0.291309	+	0.956629i	$0.405909\pi$
$6$	0	0
$7$	2.30278	0.870367	0.435184	−	0.900342i	$-0.356683\pi$
$7$	2.30278	0.870367	0.435184	+	0.900342i	$0.356683\pi$
$8$	0	0
$9$	7.90833	2.63611
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.302776	0.0839749	0.0419874	−	0.999118i	$-0.486631\pi$
$13$	0.302776	0.0839749	0.0419874	+	0.999118i	$0.486631\pi$
$14$	0	0
$15$	4.30278	1.11097
$16$	0	0
$17$	2.60555	0.631939	0.315970	−	0.948769i	$-0.397670\pi$
$17$	2.60555	0.631939	0.315970	+	0.948769i	$0.397670\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	7.60555	1.65967
$22$	0	0
$23$	8.60555	1.79438	0.897191	−	0.441643i	$-0.145604\pi$
$23$	8.60555	1.79438	0.897191	+	0.441643i	$0.145604\pi$
$24$	0	0
$25$	−3.30278	−0.660555
$26$	0	0
$27$	16.2111	3.11983
$28$	0	0
$29$	−4.69722	−0.872253	−0.436126	−	0.899885i	$-0.643650\pi$
$29$	−4.69722	−0.872253	−0.436126	+	0.899885i	$0.643650\pi$
$30$	0	0
$31$	−0.302776	−0.0543801	−0.0271901	−	0.999630i	$-0.508656\pi$
$31$	−0.302776	−0.0543801	−0.0271901	+	0.999630i	$0.508656\pi$
$32$	0	0
$33$	−3.30278	−0.574939
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	−9.21110	−1.51430	−0.757148	−	0.653243i	$-0.773408\pi$
$37$	−9.21110	−1.51430	−0.757148	+	0.653243i	$0.773408\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−6.90833	−1.07890	−0.539450	−	0.842018i	$-0.681368\pi$
$41$	−6.90833	−1.07890	−0.539450	+	0.842018i	$0.681368\pi$
$42$	0	0
$43$	−11.9083	−1.81600	−0.908001	−	0.418967i	$-0.862392\pi$
$43$	−11.9083	−1.81600	−0.908001	+	0.418967i	$0.862392\pi$
$44$	0	0
$45$	10.3028	1.53585
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	−1.69722	−0.242461
$50$	0	0
$51$	8.60555	1.20502
$52$	0	0
$53$	−3.39445	−0.466263	−0.233132	−	0.972445i	$-0.574897\pi$
$53$	−3.39445	−0.466263	−0.233132	+	0.972445i	$0.574897\pi$
$54$	0	0
$55$	−1.30278	−0.175666
$56$	0	0
$57$	−3.30278	−0.437463
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−3.21110	−0.411140	−0.205570	−	0.978642i	$-0.565905\pi$
$61$	−3.21110	−0.411140	−0.205570	+	0.978642i	$0.565905\pi$
$62$	0	0
$63$	18.2111	2.29438
$64$	0	0
$65$	0.394449	0.0489253
$66$	0	0
$67$	−11.5139	−1.40664	−0.703322	−	0.710871i	$-0.748301\pi$
$67$	−11.5139	−1.40664	−0.703322	+	0.710871i	$0.748301\pi$
$68$	0	0
$69$	28.4222	3.42163
$70$	0	0
$71$	13.3028	1.57875	0.789375	−	0.613912i	$-0.210405\pi$
$71$	13.3028	1.57875	0.789375	+	0.613912i	$0.210405\pi$
$72$	0	0
$73$	4.60555	0.539039	0.269520	−	0.962995i	$-0.413135\pi$
$73$	4.60555	0.539039	0.269520	+	0.962995i	$0.413135\pi$
$74$	0	0
$75$	−10.9083	−1.25959
$76$	0	0
$77$	−2.30278	−0.262426
$78$	0	0
$79$	5.81665	0.654425	0.327212	−	0.944951i	$-0.393891\pi$
$79$	5.81665	0.654425	0.327212	+	0.944951i	$0.393891\pi$
$80$	0	0
$81$	29.8167	3.31296
$82$	0	0
$83$	−10.6972	−1.17417	−0.587086	−	0.809524i	$-0.699725\pi$
$83$	−10.6972	−1.17417	−0.587086	+	0.809524i	$0.699725\pi$
$84$	0	0
$85$	3.39445	0.368180
$86$	0	0
$87$	−15.5139	−1.66326
$88$	0	0
$89$	−2.60555	−0.276188	−0.138094	−	0.990419i	$-0.544098\pi$
$89$	−2.60555	−0.276188	−0.138094	+	0.990419i	$0.544098\pi$
$90$	0	0
$91$	0.697224	0.0730890
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−1.30278	−0.133662
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	−7.90833	−0.794817
$100$	0	0
$101$	−13.8167	−1.37481	−0.687404	−	0.726275i	$-0.741250\pi$
$101$	−13.8167	−1.37481	−0.687404	+	0.726275i	$0.741250\pi$
$102$	0	0
$103$	5.30278	0.522498	0.261249	−	0.965271i	$-0.415866\pi$
$103$	5.30278	0.522498	0.261249	+	0.965271i	$0.415866\pi$
$104$	0	0
$105$	9.90833	0.966954
$106$	0	0
$107$	11.2111	1.08382	0.541909	−	0.840437i	$-0.317702\pi$
$107$	11.2111	1.08382	0.541909	+	0.840437i	$0.317702\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−30.4222	−2.88755
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	11.2111	1.04544
$116$	0	0
$117$	2.39445	0.221367
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−22.8167	−2.05731
$124$	0	0
$125$	−10.8167	−0.967471
$126$	0	0
$127$	12.6056	1.11856	0.559281	−	0.828978i	$-0.311077\pi$
$127$	12.6056	1.11856	0.559281	+	0.828978i	$0.311077\pi$
$128$	0	0
$129$	−39.3305	−3.46286
$130$	0	0
$131$	6.11943	0.534657	0.267329	−	0.963605i	$-0.413859\pi$
$131$	6.11943	0.534657	0.267329	+	0.963605i	$0.413859\pi$
$132$	0	0
$133$	−2.30278	−0.199676
$134$	0	0
$135$	21.1194	1.81767
$136$	0	0
$137$	−21.5139	−1.83805	−0.919027	−	0.394194i	$-0.871024\pi$
$137$	−21.5139	−1.83805	−0.919027	+	0.394194i	$0.871024\pi$
$138$	0	0
$139$	2.69722	0.228776	0.114388	−	0.993436i	$-0.463509\pi$
$139$	2.69722	0.228776	0.114388	+	0.993436i	$0.463509\pi$
$140$	0	0
$141$	19.8167	1.66886
$142$	0	0
$143$	−0.302776	−0.0253194
$144$	0	0
$145$	−6.11943	−0.508191
$146$	0	0
$147$	−5.60555	−0.462338
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−12.4222	−1.01090	−0.505452	−	0.862855i	$-0.668674\pi$
$151$	−12.4222	−1.01090	−0.505452	+	0.862855i	$0.668674\pi$
$152$	0	0
$153$	20.6056	1.66586
$154$	0	0
$155$	−0.394449	−0.0316829
$156$	0	0
$157$	11.9083	0.950388	0.475194	−	0.879881i	$-0.342378\pi$
$157$	11.9083	0.950388	0.475194	+	0.879881i	$0.342378\pi$
$158$	0	0
$159$	−11.2111	−0.889098
$160$	0	0
$161$	19.8167	1.56177
$162$	0	0
$163$	18.6056	1.45730	0.728650	−	0.684887i	$-0.240148\pi$
$163$	18.6056	1.45730	0.728650	+	0.684887i	$0.240148\pi$
$164$	0	0
$165$	−4.30278	−0.334971
$166$	0	0
$167$	−16.4222	−1.27079	−0.635394	−	0.772188i	$-0.719162\pi$
$167$	−16.4222	−1.27079	−0.635394	+	0.772188i	$0.719162\pi$
$168$	0	0
$169$	−12.9083	−0.992948
$170$	0	0
$171$	−7.90833	−0.604765
$172$	0	0
$173$	20.3305	1.54570	0.772851	−	0.634588i	$-0.218830\pi$
$173$	20.3305	1.54570	0.772851	+	0.634588i	$0.218830\pi$
$174$	0	0
$175$	−7.60555	−0.574926
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−12.1194	−0.905849	−0.452924	−	0.891549i	$-0.649619\pi$
$179$	−12.1194	−0.905849	−0.452924	+	0.891549i	$0.649619\pi$
$180$	0	0
$181$	21.0278	1.56298	0.781490	−	0.623917i	$-0.214460\pi$
$181$	21.0278	1.56298	0.781490	+	0.623917i	$0.214460\pi$
$182$	0	0
$183$	−10.6056	−0.783985
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	−2.60555	−0.190537
$188$	0	0
$189$	37.3305	2.71540
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	7.72498	0.556056	0.278028	−	0.960573i	$-0.410319\pi$
$193$	7.72498	0.556056	0.278028	+	0.960573i	$0.410319\pi$
$194$	0	0
$195$	1.30278	0.0932937
$196$	0	0
$197$	−21.6333	−1.54131	−0.770655	−	0.637253i	$-0.780071\pi$
$197$	−21.6333	−1.54131	−0.770655	+	0.637253i	$0.780071\pi$
$198$	0	0
$199$	23.8167	1.68832	0.844159	−	0.536093i	$-0.180100\pi$
$199$	23.8167	1.68832	0.844159	+	0.536093i	$0.180100\pi$
$200$	0	0
$201$	−38.0278	−2.68227
$202$	0	0
$203$	−10.8167	−0.759180
$204$	0	0
$205$	−9.00000	−0.628587
$206$	0	0
$207$	68.0555	4.73019
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−17.3944	−1.19748	−0.598742	−	0.800942i	$-0.704332\pi$
$211$	−17.3944	−1.19748	−0.598742	+	0.800942i	$0.704332\pi$
$212$	0	0
$213$	43.9361	3.01045
$214$	0	0
$215$	−15.5139	−1.05804
$216$	0	0
$217$	−0.697224	−0.0473307
$218$	0	0
$219$	15.2111	1.02787
$220$	0	0
$221$	0.788897	0.0530670
$222$	0	0
$223$	10.7889	0.722478	0.361239	−	0.932473i	$-0.382354\pi$
$223$	10.7889	0.722478	0.361239	+	0.932473i	$0.382354\pi$
$224$	0	0
$225$	−26.1194	−1.74130
$226$	0	0
$227$	19.8167	1.31528	0.657639	−	0.753333i	$-0.271555\pi$
$227$	19.8167	1.31528	0.657639	+	0.753333i	$0.271555\pi$
$228$	0	0
$229$	1.09167	0.0721398	0.0360699	−	0.999349i	$-0.488516\pi$
$229$	1.09167	0.0721398	0.0360699	+	0.999349i	$0.488516\pi$
$230$	0	0
$231$	−7.60555	−0.500409
$232$	0	0
$233$	−5.21110	−0.341391	−0.170695	−	0.985324i	$-0.554601\pi$
$233$	−5.21110	−0.341391	−0.170695	+	0.985324i	$0.554601\pi$
$234$	0	0
$235$	7.81665	0.509902
$236$	0	0
$237$	19.2111	1.24790
$238$	0	0
$239$	−22.9361	−1.48361	−0.741806	−	0.670615i	$-0.766030\pi$
$239$	−22.9361	−1.48361	−0.741806	+	0.670615i	$0.766030\pi$
$240$	0	0
$241$	23.9083	1.54007	0.770035	−	0.638001i	$-0.220239\pi$
$241$	23.9083	1.54007	0.770035	+	0.638001i	$0.220239\pi$
$242$	0	0
$243$	49.8444	3.19752
$244$	0	0
$245$	−2.21110	−0.141262
$246$	0	0
$247$	−0.302776	−0.0192652
$248$	0	0
$249$	−35.3305	−2.23898
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	0	0
$253$	−8.60555	−0.541026
$254$	0	0
$255$	11.2111	0.702066
$256$	0	0
$257$	2.60555	0.162530	0.0812649	−	0.996693i	$-0.474104\pi$
$257$	2.60555	0.162530	0.0812649	+	0.996693i	$0.474104\pi$
$258$	0	0
$259$	−21.2111	−1.31799
$260$	0	0
$261$	−37.1472	−2.29935
$262$	0	0
$263$	16.6972	1.02959	0.514797	−	0.857312i	$-0.327867\pi$
$263$	16.6972	1.02959	0.514797	+	0.857312i	$0.327867\pi$
$264$	0	0
$265$	−4.42221	−0.271654
$266$	0	0
$267$	−8.60555	−0.526651
$268$	0	0
$269$	1.81665	0.110763	0.0553817	−	0.998465i	$-0.482362\pi$
$269$	1.81665	0.110763	0.0553817	+	0.998465i	$0.482362\pi$
$270$	0	0
$271$	−11.5139	−0.699418	−0.349709	−	0.936858i	$-0.613720\pi$
$271$	−11.5139	−0.699418	−0.349709	+	0.936858i	$0.613720\pi$
$272$	0	0
$273$	2.30278	0.139370
$274$	0	0
$275$	3.30278	0.199165
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	−2.39445	−0.143352
$280$	0	0
$281$	−18.5139	−1.10445	−0.552223	−	0.833697i	$-0.686220\pi$
$281$	−18.5139	−1.10445	−0.552223	+	0.833697i	$0.686220\pi$
$282$	0	0
$283$	−27.9361	−1.66063	−0.830314	−	0.557296i	$-0.811839\pi$
$283$	−27.9361	−1.66063	−0.830314	+	0.557296i	$0.811839\pi$
$284$	0	0
$285$	−4.30278	−0.254874
$286$	0	0
$287$	−15.9083	−0.939039
$288$	0	0
$289$	−10.2111	−0.600653
$290$	0	0
$291$	26.4222	1.54890
$292$	0	0
$293$	28.5416	1.66742	0.833710	−	0.552202i	$-0.186212\pi$
$293$	28.5416	1.66742	0.833710	+	0.552202i	$0.186212\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−16.2111	−0.940664
$298$	0	0
$299$	2.60555	0.150683
$300$	0	0
$301$	−27.4222	−1.58059
$302$	0	0
$303$	−45.6333	−2.62157
$304$	0	0
$305$	−4.18335	−0.239538
$306$	0	0
$307$	23.8167	1.35929	0.679644	−	0.733542i	$-0.262134\pi$
$307$	23.8167	1.35929	0.679644	+	0.733542i	$0.262134\pi$
$308$	0	0
$309$	17.5139	0.996330
$310$	0	0
$311$	−4.18335	−0.237216	−0.118608	−	0.992941i	$-0.537843\pi$
$311$	−4.18335	−0.237216	−0.118608	+	0.992941i	$0.537843\pi$
$312$	0	0
$313$	−7.11943	−0.402414	−0.201207	−	0.979549i	$-0.564486\pi$
$313$	−7.11943	−0.402414	−0.201207	+	0.979549i	$0.564486\pi$
$314$	0	0
$315$	23.7250	1.33675
$316$	0	0
$317$	7.81665	0.439027	0.219514	−	0.975609i	$-0.429553\pi$
$317$	7.81665	0.439027	0.219514	+	0.975609i	$0.429553\pi$
$318$	0	0
$319$	4.69722	0.262994
$320$	0	0
$321$	37.0278	2.06669
$322$	0	0
$323$	−2.60555	−0.144977
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	6.60555	0.365288
$328$	0	0
$329$	13.8167	0.761737
$330$	0	0
$331$	9.72498	0.534533	0.267267	−	0.963623i	$-0.413880\pi$
$331$	9.72498	0.534533	0.267267	+	0.963623i	$0.413880\pi$
$332$	0	0
$333$	−72.8444	−3.99185
$334$	0	0
$335$	−15.0000	−0.819538
$336$	0	0
$337$	−5.69722	−0.310348	−0.155174	−	0.987887i	$-0.549594\pi$
$337$	−5.69722	−0.310348	−0.155174	+	0.987887i	$0.549594\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.302776	0.0163962
$342$	0	0
$343$	−20.0278	−1.08140
$344$	0	0
$345$	37.0278	1.99351
$346$	0	0
$347$	−17.2111	−0.923940	−0.461970	−	0.886895i	$-0.652857\pi$
$347$	−17.2111	−0.923940	−0.461970	+	0.886895i	$0.652857\pi$
$348$	0	0
$349$	3.57779	0.191515	0.0957575	−	0.995405i	$-0.469473\pi$
$349$	3.57779	0.191515	0.0957575	+	0.995405i	$0.469473\pi$
$350$	0	0
$351$	4.90833	0.261987
$352$	0	0
$353$	−0.788897	−0.0419888	−0.0209944	−	0.999780i	$-0.506683\pi$
$353$	−0.788897	−0.0419888	−0.0209944	+	0.999780i	$0.506683\pi$
$354$	0	0
$355$	17.3305	0.919809
$356$	0	0
$357$	19.8167	1.04881
$358$	0	0
$359$	27.5139	1.45213	0.726063	−	0.687628i	$-0.241348\pi$
$359$	27.5139	1.45213	0.726063	+	0.687628i	$0.241348\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	3.30278	0.173351
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	−7.21110	−0.376416	−0.188208	−	0.982129i	$-0.560268\pi$
$367$	−7.21110	−0.376416	−0.188208	+	0.982129i	$0.560268\pi$
$368$	0	0
$369$	−54.6333	−2.84410
$370$	0	0
$371$	−7.81665	−0.405820
$372$	0	0
$373$	17.9083	0.927258	0.463629	−	0.886029i	$-0.346547\pi$
$373$	17.9083	0.927258	0.463629	+	0.886029i	$0.346547\pi$
$374$	0	0
$375$	−35.7250	−1.84483
$376$	0	0
$377$	−1.42221	−0.0732473
$378$	0	0
$379$	31.5139	1.61876	0.809380	−	0.587286i	$-0.199804\pi$
$379$	31.5139	1.61876	0.809380	+	0.587286i	$0.199804\pi$
$380$	0	0
$381$	41.6333	2.13294
$382$	0	0
$383$	−21.5139	−1.09931	−0.549654	−	0.835392i	$-0.685240\pi$
$383$	−21.5139	−1.09931	−0.549654	+	0.835392i	$0.685240\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	0	0
$387$	−94.1749	−4.78718
$388$	0	0
$389$	−19.5416	−0.990800	−0.495400	−	0.868665i	$-0.664979\pi$
$389$	−19.5416	−0.990800	−0.495400	+	0.868665i	$0.664979\pi$
$390$	0	0
$391$	22.4222	1.13394
$392$	0	0
$393$	20.2111	1.01952
$394$	0	0
$395$	7.57779	0.381280
$396$	0	0
$397$	35.7527	1.79438	0.897189	−	0.441646i	$-0.145605\pi$
$397$	35.7527	1.79438	0.897189	+	0.441646i	$0.145605\pi$
$398$	0	0
$399$	−7.60555	−0.380754
$400$	0	0
$401$	15.3944	0.768762	0.384381	−	0.923175i	$-0.374415\pi$
$401$	15.3944	0.768762	0.384381	+	0.923175i	$0.374415\pi$
$402$	0	0
$403$	−0.0916731	−0.00456656
$404$	0	0
$405$	38.8444	1.93019
$406$	0	0
$407$	9.21110	0.456577
$408$	0	0
$409$	−38.1472	−1.88626	−0.943128	−	0.332428i	$-0.892132\pi$
$409$	−38.1472	−1.88626	−0.943128	+	0.332428i	$0.892132\pi$
$410$	0	0
$411$	−71.0555	−3.50491
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−13.9361	−0.684095
$416$	0	0
$417$	8.90833	0.436243
$418$	0	0
$419$	−23.2111	−1.13394	−0.566968	−	0.823740i	$-0.691884\pi$
$419$	−23.2111	−1.13394	−0.566968	+	0.823740i	$0.691884\pi$
$420$	0	0
$421$	20.2389	0.986382	0.493191	−	0.869921i	$-0.335830\pi$
$421$	20.2389	0.986382	0.493191	+	0.869921i	$0.335830\pi$
$422$	0	0
$423$	47.4500	2.30710
$424$	0	0
$425$	−8.60555	−0.417431
$426$	0	0
$427$	−7.39445	−0.357842
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−17.2111	−0.829030	−0.414515	−	0.910043i	$-0.636049\pi$
$431$	−17.2111	−0.829030	−0.414515	+	0.910043i	$0.636049\pi$
$432$	0	0
$433$	26.2389	1.26096	0.630480	−	0.776206i	$-0.282858\pi$
$433$	26.2389	1.26096	0.630480	+	0.776206i	$0.282858\pi$
$434$	0	0
$435$	−20.2111	−0.969048
$436$	0	0
$437$	−8.60555	−0.411659
$438$	0	0
$439$	10.7889	0.514926	0.257463	−	0.966288i	$-0.417113\pi$
$439$	10.7889	0.514926	0.257463	+	0.966288i	$0.417113\pi$
$440$	0	0
$441$	−13.4222	−0.639153
$442$	0	0
$443$	3.63331	0.172624	0.0863118	−	0.996268i	$-0.472492\pi$
$443$	3.63331	0.172624	0.0863118	+	0.996268i	$0.472492\pi$
$444$	0	0
$445$	−3.39445	−0.160912
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−10.1833	−0.480582	−0.240291	−	0.970701i	$-0.577243\pi$
$449$	−10.1833	−0.480582	−0.240291	+	0.970701i	$0.577243\pi$
$450$	0	0
$451$	6.90833	0.325300
$452$	0	0
$453$	−41.0278	−1.92765
$454$	0	0
$455$	0.908327	0.0425830
$456$	0	0
$457$	0.183346	0.00857657	0.00428829	−	0.999991i	$-0.498635\pi$
$457$	0.183346	0.00857657	0.00428829	+	0.999991i	$0.498635\pi$
$458$	0	0
$459$	42.2389	1.97154
$460$	0	0
$461$	−22.4222	−1.04431	−0.522153	−	0.852852i	$-0.674871\pi$
$461$	−22.4222	−1.04431	−0.522153	+	0.852852i	$0.674871\pi$
$462$	0	0
$463$	−13.2111	−0.613972	−0.306986	−	0.951714i	$-0.599320\pi$
$463$	−13.2111	−0.613972	−0.306986	+	0.951714i	$0.599320\pi$
$464$	0	0
$465$	−1.30278	−0.0604148
$466$	0	0
$467$	15.3944	0.712370	0.356185	−	0.934415i	$-0.384077\pi$
$467$	15.3944	0.712370	0.356185	+	0.934415i	$0.384077\pi$
$468$	0	0
$469$	−26.5139	−1.22430
$470$	0	0
$471$	39.3305	1.81226
$472$	0	0
$473$	11.9083	0.547545
$474$	0	0
$475$	3.30278	0.151542
$476$	0	0
$477$	−26.8444	−1.22912
$478$	0	0
$479$	−3.27502	−0.149639	−0.0748197	−	0.997197i	$-0.523838\pi$
$479$	−3.27502	−0.149639	−0.0748197	+	0.997197i	$0.523838\pi$
$480$	0	0
$481$	−2.78890	−0.127163
$482$	0	0
$483$	65.4500	2.97808
$484$	0	0
$485$	10.4222	0.473248
$486$	0	0
$487$	−10.3305	−0.468121	−0.234061	−	0.972222i	$-0.575201\pi$
$487$	−10.3305	−0.468121	−0.234061	+	0.972222i	$0.575201\pi$
$488$	0	0
$489$	61.4500	2.77886
$490$	0	0
$491$	−31.9361	−1.44126	−0.720628	−	0.693322i	$-0.756146\pi$
$491$	−31.9361	−1.44126	−0.720628	+	0.693322i	$0.756146\pi$
$492$	0	0
$493$	−12.2389	−0.551210
$494$	0	0
$495$	−10.3028	−0.463075
$496$	0	0
$497$	30.6333	1.37409
$498$	0	0
$499$	34.0000	1.52205	0.761025	−	0.648723i	$-0.224697\pi$
$499$	34.0000	1.52205	0.761025	+	0.648723i	$0.224697\pi$
$500$	0	0
$501$	−54.2389	−2.42321
$502$	0	0
$503$	3.11943	0.139088	0.0695442	−	0.997579i	$-0.477845\pi$
$503$	3.11943	0.139088	0.0695442	+	0.997579i	$0.477845\pi$
$504$	0	0
$505$	−18.0000	−0.800989
$506$	0	0
$507$	−42.6333	−1.89341
$508$	0	0
$509$	10.4222	0.461956	0.230978	−	0.972959i	$-0.425807\pi$
$509$	10.4222	0.461956	0.230978	+	0.972959i	$0.425807\pi$
$510$	0	0
$511$	10.6056	0.469162
$512$	0	0
$513$	−16.2111	−0.715738
$514$	0	0
$515$	6.90833	0.304417
$516$	0	0
$517$	−6.00000	−0.263880
$518$	0	0
$519$	67.1472	2.94743
$520$	0	0
$521$	7.57779	0.331989	0.165995	−	0.986127i	$-0.446917\pi$
$521$	7.57779	0.331989	0.165995	+	0.986127i	$0.446917\pi$
$522$	0	0
$523$	17.0278	0.744572	0.372286	−	0.928118i	$-0.378574\pi$
$523$	17.0278	0.744572	0.372286	+	0.928118i	$0.378574\pi$
$524$	0	0
$525$	−25.1194	−1.09630
$526$	0	0
$527$	−0.788897	−0.0343649
$528$	0	0
$529$	51.0555	2.21980
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.09167	−0.0906004
$534$	0	0
$535$	14.6056	0.631453
$536$	0	0
$537$	−40.0278	−1.72733
$538$	0	0
$539$	1.69722	0.0731046
$540$	0	0
$541$	−17.8167	−0.765998	−0.382999	−	0.923749i	$-0.625109\pi$
$541$	−17.8167	−0.765998	−0.382999	+	0.923749i	$0.625109\pi$
$542$	0	0
$543$	69.4500	2.98038
$544$	0	0
$545$	2.60555	0.111610
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	0	0
$549$	−25.3944	−1.08381
$550$	0	0
$551$	4.69722	0.200108
$552$	0	0
$553$	13.3944	0.569590
$554$	0	0
$555$	−39.6333	−1.68234
$556$	0	0
$557$	15.6333	0.662405	0.331202	−	0.943560i	$-0.392546\pi$
$557$	15.6333	0.662405	0.331202	+	0.943560i	$0.392546\pi$
$558$	0	0
$559$	−3.60555	−0.152499
$560$	0	0
$561$	−8.60555	−0.363327
$562$	0	0
$563$	21.6333	0.911735	0.455868	−	0.890048i	$-0.349329\pi$
$563$	21.6333	0.911735	0.455868	+	0.890048i	$0.349329\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	68.6611	2.88349
$568$	0	0
$569$	44.7250	1.87497	0.937484	−	0.348027i	$-0.113148\pi$
$569$	44.7250	1.87497	0.937484	+	0.348027i	$0.113148\pi$
$570$	0	0
$571$	−39.9361	−1.67127	−0.835637	−	0.549283i	$-0.814901\pi$
$571$	−39.9361	−1.67127	−0.835637	+	0.549283i	$0.814901\pi$
$572$	0	0
$573$	19.8167	0.827853
$574$	0	0
$575$	−28.4222	−1.18529
$576$	0	0
$577$	−28.9083	−1.20347	−0.601735	−	0.798696i	$-0.705524\pi$
$577$	−28.9083	−1.20347	−0.601735	+	0.798696i	$0.705524\pi$
$578$	0	0
$579$	25.5139	1.06032
$580$	0	0
$581$	−24.6333	−1.02196
$582$	0	0
$583$	3.39445	0.140584
$584$	0	0
$585$	3.11943	0.128973
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	0.302776	0.0124757
$590$	0	0
$591$	−71.4500	−2.93906
$592$	0	0
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	0	0
$595$	7.81665	0.320452
$596$	0	0
$597$	78.6611	3.21938
$598$	0	0
$599$	45.1194	1.84353	0.921765	−	0.387749i	$-0.126747\pi$
$599$	45.1194	1.84353	0.921765	+	0.387749i	$0.126747\pi$
$600$	0	0
$601$	14.9083	0.608123	0.304062	−	0.952652i	$-0.401657\pi$
$601$	14.9083	0.608123	0.304062	+	0.952652i	$0.401657\pi$
$602$	0	0
$603$	−91.0555	−3.70807
$604$	0	0
$605$	1.30278	0.0529654
$606$	0	0
$607$	17.8167	0.723156	0.361578	−	0.932342i	$-0.382238\pi$
$607$	17.8167	0.723156	0.361578	+	0.932342i	$0.382238\pi$
$608$	0	0
$609$	−35.7250	−1.44765
$610$	0	0
$611$	1.81665	0.0734939
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	−29.7250	−1.19863
$616$	0	0
$617$	12.9083	0.519670	0.259835	−	0.965653i	$-0.416332\pi$
$617$	12.9083	0.519670	0.259835	+	0.965653i	$0.416332\pi$
$618$	0	0
$619$	−22.8444	−0.918194	−0.459097	−	0.888386i	$-0.651827\pi$
$619$	−22.8444	−0.918194	−0.459097	+	0.888386i	$0.651827\pi$
$620$	0	0
$621$	139.505	5.59816
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	2.42221	0.0968882
$626$	0	0
$627$	3.30278	0.131900
$628$	0	0
$629$	−24.0000	−0.956943
$630$	0	0
$631$	8.97224	0.357179	0.178590	−	0.983924i	$-0.442847\pi$
$631$	8.97224	0.357179	0.178590	+	0.983924i	$0.442847\pi$
$632$	0	0
$633$	−57.4500	−2.28343
$634$	0	0
$635$	16.4222	0.651695
$636$	0	0
$637$	−0.513878	−0.0203606
$638$	0	0
$639$	105.203	4.16175
$640$	0	0
$641$	−30.2389	−1.19436	−0.597182	−	0.802106i	$-0.703713\pi$
$641$	−30.2389	−1.19436	−0.597182	+	0.802106i	$0.703713\pi$
$642$	0	0
$643$	2.42221	0.0955224	0.0477612	−	0.998859i	$-0.484791\pi$
$643$	2.42221	0.0955224	0.0477612	+	0.998859i	$0.484791\pi$
$644$	0	0
$645$	−51.2389	−2.01753
$646$	0	0
$647$	−4.18335	−0.164464	−0.0822322	−	0.996613i	$-0.526205\pi$
$647$	−4.18335	−0.164464	−0.0822322	+	0.996613i	$0.526205\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−2.30278	−0.0902529
$652$	0	0
$653$	−17.3305	−0.678196	−0.339098	−	0.940751i	$-0.610122\pi$
$653$	−17.3305	−0.678196	−0.339098	+	0.940751i	$0.610122\pi$
$654$	0	0
$655$	7.97224	0.311501
$656$	0	0
$657$	36.4222	1.42097
$658$	0	0
$659$	7.02776	0.273763	0.136881	−	0.990587i	$-0.456292\pi$
$659$	7.02776	0.273763	0.136881	+	0.990587i	$0.456292\pi$
$660$	0	0
$661$	−19.6333	−0.763647	−0.381824	−	0.924235i	$-0.624704\pi$
$661$	−19.6333	−0.763647	−0.381824	+	0.924235i	$0.624704\pi$
$662$	0	0
$663$	2.60555	0.101191
$664$	0	0
$665$	−3.00000	−0.116335
$666$	0	0
$667$	−40.4222	−1.56515
$668$	0	0
$669$	35.6333	1.37766
$670$	0	0
$671$	3.21110	0.123963
$672$	0	0
$673$	−2.30278	−0.0887655	−0.0443827	−	0.999015i	$-0.514132\pi$
$673$	−2.30278	−0.0887655	−0.0443827	+	0.999015i	$0.514132\pi$
$674$	0	0
$675$	−53.5416	−2.06082
$676$	0	0
$677$	−4.69722	−0.180529	−0.0902645	−	0.995918i	$-0.528771\pi$
$677$	−4.69722	−0.180529	−0.0902645	+	0.995918i	$0.528771\pi$
$678$	0	0
$679$	18.4222	0.706979
$680$	0	0
$681$	65.4500	2.50805
$682$	0	0
$683$	46.4222	1.77630	0.888148	−	0.459557i	$-0.151992\pi$
$683$	46.4222	1.77630	0.888148	+	0.459557i	$0.151992\pi$
$684$	0	0
$685$	−28.0278	−1.07089
$686$	0	0
$687$	3.60555	0.137560
$688$	0	0
$689$	−1.02776	−0.0391544
$690$	0	0
$691$	−3.57779	−0.136106	−0.0680529	−	0.997682i	$-0.521679\pi$
$691$	−3.57779	−0.136106	−0.0680529	+	0.997682i	$0.521679\pi$
$692$	0	0
$693$	−18.2111	−0.691783
$694$	0	0
$695$	3.51388	0.133289
$696$	0	0
$697$	−18.0000	−0.681799
$698$	0	0
$699$	−17.2111	−0.650984
$700$	0	0
$701$	15.6333	0.590462	0.295231	−	0.955426i	$-0.404603\pi$
$701$	15.6333	0.590462	0.295231	+	0.955426i	$0.404603\pi$
$702$	0	0
$703$	9.21110	0.347403
$704$	0	0
$705$	25.8167	0.972311
$706$	0	0
$707$	−31.8167	−1.19659
$708$	0	0
$709$	18.3028	0.687375	0.343688	−	0.939084i	$-0.388324\pi$
$709$	18.3028	0.687375	0.343688	+	0.939084i	$0.388324\pi$
$710$	0	0
$711$	46.0000	1.72513
$712$	0	0
$713$	−2.60555	−0.0975787
$714$	0	0
$715$	−0.394449	−0.0147515
$716$	0	0
$717$	−75.7527	−2.82904
$718$	0	0
$719$	4.18335	0.156012	0.0780062	−	0.996953i	$-0.475145\pi$
$719$	4.18335	0.156012	0.0780062	+	0.996953i	$0.475145\pi$
$720$	0	0
$721$	12.2111	0.454765
$722$	0	0
$723$	78.9638	2.93670
$724$	0	0
$725$	15.5139	0.576171
$726$	0	0
$727$	43.6333	1.61827	0.809135	−	0.587623i	$-0.199936\pi$
$727$	43.6333	1.61827	0.809135	+	0.587623i	$0.199936\pi$
$728$	0	0
$729$	75.1749	2.78426
$730$	0	0
$731$	−31.0278	−1.14760
$732$	0	0
$733$	7.21110	0.266348	0.133174	−	0.991093i	$-0.457483\pi$
$733$	7.21110	0.266348	0.133174	+	0.991093i	$0.457483\pi$
$734$	0	0
$735$	−7.30278	−0.269367
$736$	0	0
$737$	11.5139	0.424119
$738$	0	0
$739$	−35.3583	−1.30068	−0.650338	−	0.759645i	$-0.725373\pi$
$739$	−35.3583	−1.30068	−0.650338	+	0.759645i	$0.725373\pi$
$740$	0	0
$741$	−1.00000	−0.0367359
$742$	0	0
$743$	−45.6333	−1.67412	−0.837062	−	0.547108i	$-0.815729\pi$
$743$	−45.6333	−1.67412	−0.837062	+	0.547108i	$0.815729\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−84.5971	−3.09525
$748$	0	0
$749$	25.8167	0.943320
$750$	0	0
$751$	−16.8444	−0.614661	−0.307331	−	0.951603i	$-0.599436\pi$
$751$	−16.8444	−0.614661	−0.307331	+	0.951603i	$0.599436\pi$
$752$	0	0
$753$	−79.2666	−2.88864
$754$	0	0
$755$	−16.1833	−0.588972
$756$	0	0
$757$	−23.5416	−0.855635	−0.427818	−	0.903865i	$-0.640717\pi$
$757$	−23.5416	−0.855635	−0.427818	+	0.903865i	$0.640717\pi$
$758$	0	0
$759$	−28.4222	−1.03166
$760$	0	0
$761$	−30.2389	−1.09616	−0.548079	−	0.836427i	$-0.684641\pi$
$761$	−30.2389	−1.09616	−0.548079	+	0.836427i	$0.684641\pi$
$762$	0	0
$763$	4.60555	0.166732
$764$	0	0
$765$	26.8444	0.970562
$766$	0	0
$767$	0	0
$768$	0	0
$769$	5.63331	0.203142	0.101571	−	0.994828i	$-0.467613\pi$
$769$	5.63331	0.203142	0.101571	+	0.994828i	$0.467613\pi$
$770$	0	0
$771$	8.60555	0.309921
$772$	0	0
$773$	19.5778	0.704164	0.352082	−	0.935969i	$-0.385474\pi$
$773$	19.5778	0.704164	0.352082	+	0.935969i	$0.385474\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	−70.0555	−2.51323
$778$	0	0
$779$	6.90833	0.247516
$780$	0	0
$781$	−13.3028	−0.476011
$782$	0	0
$783$	−76.1472	−2.72128
$784$	0	0
$785$	15.5139	0.553714
$786$	0	0
$787$	−21.0278	−0.749559	−0.374779	−	0.927114i	$-0.622282\pi$
$787$	−21.0278	−0.749559	−0.374779	+	0.927114i	$0.622282\pi$
$788$	0	0
$789$	55.1472	1.96329
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−0.972244	−0.0345254
$794$	0	0
$795$	−14.6056	−0.518006
$796$	0	0
$797$	9.39445	0.332768	0.166384	−	0.986061i	$-0.446791\pi$
$797$	9.39445	0.332768	0.166384	+	0.986061i	$0.446791\pi$
$798$	0	0
$799$	15.6333	0.553067
$800$	0	0
$801$	−20.6056	−0.728061
$802$	0	0
$803$	−4.60555	−0.162526
$804$	0	0
$805$	25.8167	0.909917
$806$	0	0
$807$	6.00000	0.211210
$808$	0	0
$809$	18.7889	0.660582	0.330291	−	0.943879i	$-0.392853\pi$
$809$	18.7889	0.660582	0.330291	+	0.943879i	$0.392853\pi$
$810$	0	0
$811$	13.6333	0.478730	0.239365	−	0.970930i	$-0.423061\pi$
$811$	13.6333	0.478730	0.239365	+	0.970930i	$0.423061\pi$
$812$	0	0
$813$	−38.0278	−1.33369
$814$	0	0
$815$	24.2389	0.849050
$816$	0	0
$817$	11.9083	0.416620
$818$	0	0
$819$	5.51388	0.192670
$820$	0	0
$821$	−27.3944	−0.956073	−0.478036	−	0.878340i	$-0.658651\pi$
$821$	−27.3944	−0.956073	−0.478036	+	0.878340i	$0.658651\pi$
$822$	0	0
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	0	0
$825$	10.9083	0.379779
$826$	0	0
$827$	50.0555	1.74060	0.870300	−	0.492521i	$-0.163925\pi$
$827$	50.0555	1.74060	0.870300	+	0.492521i	$0.163925\pi$
$828$	0	0
$829$	−56.4222	−1.95962	−0.979812	−	0.199921i	$-0.935932\pi$
$829$	−56.4222	−1.95962	−0.979812	+	0.199921i	$0.935932\pi$
$830$	0	0
$831$	−33.0278	−1.14572
$832$	0	0
$833$	−4.42221	−0.153220
$834$	0	0
$835$	−21.3944	−0.740385
$836$	0	0
$837$	−4.90833	−0.169657
$838$	0	0
$839$	−45.1194	−1.55770	−0.778848	−	0.627213i	$-0.784196\pi$
$839$	−45.1194	−1.55770	−0.778848	+	0.627213i	$0.784196\pi$
$840$	0	0
$841$	−6.93608	−0.239175
$842$	0	0
$843$	−61.1472	−2.10602
$844$	0	0
$845$	−16.8167	−0.578510
$846$	0	0
$847$	2.30278	0.0791243
$848$	0	0
$849$	−92.2666	−3.16658
$850$	0	0
$851$	−79.2666	−2.71722
$852$	0	0
$853$	−17.5778	−0.601852	−0.300926	−	0.953647i	$-0.597296\pi$
$853$	−17.5778	−0.601852	−0.300926	+	0.953647i	$0.597296\pi$
$854$	0	0
$855$	−10.3028	−0.352347
$856$	0	0
$857$	10.5416	0.360095	0.180048	−	0.983658i	$-0.442375\pi$
$857$	10.5416	0.360095	0.180048	+	0.983658i	$0.442375\pi$
$858$	0	0
$859$	48.8444	1.66655	0.833275	−	0.552859i	$-0.186463\pi$
$859$	48.8444	1.66655	0.833275	+	0.552859i	$0.186463\pi$
$860$	0	0
$861$	−52.5416	−1.79061
$862$	0	0
$863$	−14.7250	−0.501244	−0.250622	−	0.968085i	$-0.580635\pi$
$863$	−14.7250	−0.501244	−0.250622	+	0.968085i	$0.580635\pi$
$864$	0	0
$865$	26.4861	0.900555
$866$	0	0
$867$	−33.7250	−1.14536
$868$	0	0
$869$	−5.81665	−0.197316
$870$	0	0
$871$	−3.48612	−0.118123
$872$	0	0
$873$	63.2666	2.14125
$874$	0	0
$875$	−24.9083	−0.842055
$876$	0	0
$877$	−42.5694	−1.43747	−0.718733	−	0.695286i	$-0.755278\pi$
$877$	−42.5694	−1.43747	−0.718733	+	0.695286i	$0.755278\pi$
$878$	0	0
$879$	94.2666	3.17953
$880$	0	0
$881$	−23.0917	−0.777978	−0.388989	−	0.921242i	$-0.627176\pi$
$881$	−23.0917	−0.777978	−0.388989	+	0.921242i	$0.627176\pi$
$882$	0	0
$883$	2.97224	0.100024	0.0500120	−	0.998749i	$-0.484074\pi$
$883$	2.97224	0.100024	0.0500120	+	0.998749i	$0.484074\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.8444	1.10281	0.551404	−	0.834239i	$-0.314092\pi$
$887$	32.8444	1.10281	0.551404	+	0.834239i	$0.314092\pi$
$888$	0	0
$889$	29.0278	0.973560
$890$	0	0
$891$	−29.8167	−0.998895
$892$	0	0
$893$	−6.00000	−0.200782
$894$	0	0
$895$	−15.7889	−0.527765
$896$	0	0
$897$	8.60555	0.287331
$898$	0	0
$899$	1.42221	0.0474332
$900$	0	0
$901$	−8.84441	−0.294650
$902$	0	0
$903$	−90.5694	−3.01396
$904$	0	0
$905$	27.3944	0.910622
$906$	0	0
$907$	−52.8444	−1.75467	−0.877335	−	0.479879i	$-0.840681\pi$
$907$	−52.8444	−1.75467	−0.877335	+	0.479879i	$0.840681\pi$
$908$	0	0
$909$	−109.267	−3.62414
$910$	0	0
$911$	15.6333	0.517955	0.258977	−	0.965883i	$-0.416615\pi$
$911$	15.6333	0.517955	0.258977	+	0.965883i	$0.416615\pi$
$912$	0	0
$913$	10.6972	0.354026
$914$	0	0
$915$	−13.8167	−0.456764
$916$	0	0
$917$	14.0917	0.465348
$918$	0	0
$919$	19.1194	0.630692	0.315346	−	0.948977i	$-0.397879\pi$
$919$	19.1194	0.630692	0.315346	+	0.948977i	$0.397879\pi$
$920$	0	0
$921$	78.6611	2.59197
$922$	0	0
$923$	4.02776	0.132575
$924$	0	0
$925$	30.4222	1.00028
$926$	0	0
$927$	41.9361	1.37736
$928$	0	0
$929$	9.51388	0.312140	0.156070	−	0.987746i	$-0.450117\pi$
$929$	9.51388	0.312140	0.156070	+	0.987746i	$0.450117\pi$
$930$	0	0
$931$	1.69722	0.0556243
$932$	0	0
$933$	−13.8167	−0.452337
$934$	0	0
$935$	−3.39445	−0.111010
$936$	0	0
$937$	58.8444	1.92236	0.961182	−	0.275917i	$-0.0889814\pi$
$937$	58.8444	1.92236	0.961182	+	0.275917i	$0.0889814\pi$
$938$	0	0
$939$	−23.5139	−0.767346
$940$	0	0
$941$	−14.3667	−0.468341	−0.234170	−	0.972196i	$-0.575237\pi$
$941$	−14.3667	−0.468341	−0.234170	+	0.972196i	$0.575237\pi$
$942$	0	0
$943$	−59.4500	−1.93596
$944$	0	0
$945$	48.6333	1.58204
$946$	0	0
$947$	−4.18335	−0.135940	−0.0679702	−	0.997687i	$-0.521652\pi$
$947$	−4.18335	−0.135940	−0.0679702	+	0.997687i	$0.521652\pi$
$948$	0	0
$949$	1.39445	0.0452657
$950$	0	0
$951$	25.8167	0.837162
$952$	0	0
$953$	21.6333	0.700772	0.350386	−	0.936605i	$-0.386050\pi$
$953$	21.6333	0.700772	0.350386	+	0.936605i	$0.386050\pi$
$954$	0	0
$955$	7.81665	0.252941
$956$	0	0
$957$	15.5139	0.501492
$958$	0	0
$959$	−49.5416	−1.59978
$960$	0	0
$961$	−30.9083	−0.997043
$962$	0	0
$963$	88.6611	2.85706
$964$	0	0
$965$	10.0639	0.323969
$966$	0	0
$967$	19.6333	0.631365	0.315682	−	0.948865i	$-0.397767\pi$
$967$	19.6333	0.631365	0.315682	+	0.948865i	$0.397767\pi$
$968$	0	0
$969$	−8.60555	−0.276450
$970$	0	0
$971$	−22.9361	−0.736054	−0.368027	−	0.929815i	$-0.619967\pi$
$971$	−22.9361	−0.736054	−0.368027	+	0.929815i	$0.619967\pi$
$972$	0	0
$973$	6.21110	0.199119
$974$	0	0
$975$	−3.30278	−0.105773
$976$	0	0
$977$	−15.6333	−0.500154	−0.250077	−	0.968226i	$-0.580456\pi$
$977$	−15.6333	−0.500154	−0.250077	+	0.968226i	$0.580456\pi$
$978$	0	0
$979$	2.60555	0.0832738
$980$	0	0
$981$	15.8167	0.504987
$982$	0	0
$983$	−19.6972	−0.628244	−0.314122	−	0.949383i	$-0.601710\pi$
$983$	−19.6972	−0.628244	−0.314122	+	0.949383i	$0.601710\pi$
$984$	0	0
$985$	−28.1833	−0.897996
$986$	0	0
$987$	45.6333	1.45252
$988$	0	0
$989$	−102.478	−3.25860
$990$	0	0
$991$	15.0917	0.479403	0.239701	−	0.970847i	$-0.422950\pi$
$991$	15.0917	0.479403	0.239701	+	0.970847i	$0.422950\pi$
$992$	0	0
$993$	32.1194	1.01928
$994$	0	0
$995$	31.0278	0.983646
$996$	0	0
$997$	21.8167	0.690940	0.345470	−	0.938430i	$-0.387719\pi$
$997$	21.8167	0.690940	0.345470	+	0.938430i	$0.387719\pi$
$998$	0	0
$999$	−149.322	−4.72434

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.o.1.2		2
4.3	odd	2		418.2.a.d.1.1	✓	2
12.11	even	2		3762.2.a.bb.1.1		2
44.43	even	2		4598.2.a.bc.1.1		2
76.75	even	2		7942.2.a.bb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.d.1.1	✓	2	4.3	odd	2
3344.2.a.o.1.2		2	1.1	even	1	trivial
3762.2.a.bb.1.1		2	12.11	even	2
4598.2.a.bc.1.1		2	44.43	even	2
7942.2.a.bb.1.2		2	76.75	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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q+3.30278 q^{3} +1.30278 q^{5} +2.30278 q^{7} +7.90833 q^{9} +O(q^{10})\)	\(q+3.30278 q^{3} +1.30278 q^{5} +2.30278 q^{7} +7.90833 q^{9} -1.00000 q^{11} +0.302776 q^{13} +4.30278 q^{15} +2.60555 q^{17} -1.00000 q^{19} +7.60555 q^{21} +8.60555 q^{23} -3.30278 q^{25} +16.2111 q^{27} -4.69722 q^{29} -0.302776 q^{31} -3.30278 q^{33} +3.00000 q^{35} -9.21110 q^{37} +1.00000 q^{39} -6.90833 q^{41} -11.9083 q^{43} +10.3028 q^{45} +6.00000 q^{47} -1.69722 q^{49} +8.60555 q^{51} -3.39445 q^{53} -1.30278 q^{55} -3.30278 q^{57} -3.21110 q^{61} +18.2111 q^{63} +0.394449 q^{65} -11.5139 q^{67} +28.4222 q^{69} +13.3028 q^{71} +4.60555 q^{73} -10.9083 q^{75} -2.30278 q^{77} +5.81665 q^{79} +29.8167 q^{81} -10.6972 q^{83} +3.39445 q^{85} -15.5139 q^{87} -2.60555 q^{89} +0.697224 q^{91} -1.00000 q^{93} -1.30278 q^{95} +8.00000 q^{97} -7.90833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} - q^{5} + q^{7} + 5 q^{9}+O(q^{10}) \) 2 * q + 3 * q^3 - q^5 + q^7 + 5 * q^9	\( 2 q + 3 q^{3} - q^{5} + q^{7} + 5 q^{9} - 2 q^{11} - 3 q^{13} + 5 q^{15} - 2 q^{17} - 2 q^{19} + 8 q^{21} + 10 q^{23} - 3 q^{25} + 18 q^{27} - 13 q^{29} + 3 q^{31} - 3 q^{33} + 6 q^{35} - 4 q^{37} + 2 q^{39} - 3 q^{41} - 13 q^{43} + 17 q^{45} + 12 q^{47} - 7 q^{49} + 10 q^{51} - 14 q^{53} + q^{55} - 3 q^{57} + 8 q^{61} + 22 q^{63} + 8 q^{65} - 5 q^{67} + 28 q^{69} + 23 q^{71} + 2 q^{73} - 11 q^{75} - q^{77} - 10 q^{79} + 38 q^{81} - 25 q^{83} + 14 q^{85} - 13 q^{87} + 2 q^{89} + 5 q^{91} - 2 q^{93} + q^{95} + 16 q^{97} - 5 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 - q^5 + q^7 + 5 * q^9 - 2 * q^11 - 3 * q^13 + 5 * q^15 - 2 * q^17 - 2 * q^19 + 8 * q^21 + 10 * q^23 - 3 * q^25 + 18 * q^27 - 13 * q^29 + 3 * q^31 - 3 * q^33 + 6 * q^35 - 4 * q^37 + 2 * q^39 - 3 * q^41 - 13 * q^43 + 17 * q^45 + 12 * q^47 - 7 * q^49 + 10 * q^51 - 14 * q^53 + q^55 - 3 * q^57 + 8 * q^61 + 22 * q^63 + 8 * q^65 - 5 * q^67 + 28 * q^69 + 23 * q^71 + 2 * q^73 - 11 * q^75 - q^77 - 10 * q^79 + 38 * q^81 - 25 * q^83 + 14 * q^85 - 13 * q^87 + 2 * q^89 + 5 * q^91 - 2 * q^93 + q^95 + 16 * q^97 - 5 * q^99

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