LMFDB - Embedded newform 1856.2.a.bc.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,2,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1856.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1856.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:	$14.8202346151$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.68772992.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} + 17x^{2} - 8 $ x^6 - 9x^4 + 17x^2 - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.854088$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.15704 q^{3} -1.42586 q^{5} +1.70818 q^{7} +6.96693 q^{9} +O(q^{10})$	$q+3.15704 q^{3} -1.42586 q^{5} +1.70818 q^{7} +6.96693 q^{9} +3.15704 q^{11} -1.42586 q^{13} -4.50150 q^{15} -3.39279 q^{17} +0.363721 q^{19} +5.39279 q^{21} +7.29482 q^{23} -2.96693 q^{25} +12.5238 q^{27} -1.00000 q^{29} +1.44887 q^{31} +9.96693 q^{33} -2.43562 q^{35} +0.851718 q^{37} -4.50150 q^{39} +8.54107 q^{41} +6.57340 q^{43} -9.93385 q^{45} +11.1793 q^{47} -4.08214 q^{49} -10.7112 q^{51} -13.3597 q^{53} -4.50150 q^{55} +1.14828 q^{57} -10.7112 q^{59} -3.39279 q^{61} +11.9007 q^{63} +2.03307 q^{65} -12.6282 q^{67} +23.0301 q^{69} -5.58665 q^{71} +13.9339 q^{73} -9.36672 q^{75} +5.39279 q^{77} +4.86522 q^{79} +18.6373 q^{81} -5.33335 q^{83} +4.83763 q^{85} -3.15704 q^{87} -15.3266 q^{89} -2.43562 q^{91} +4.57414 q^{93} -0.518614 q^{95} +14.2445 q^{97} +21.9949 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{5} + 10 q^{9}+O(q^{10}) $ 6 * q - 4 * q^5 + 10 * q^9	$ 6 q - 4 q^{5} + 10 q^{9} - 4 q^{13} + 16 q^{17} - 4 q^{21} + 14 q^{25} - 6 q^{29} + 28 q^{33} - 4 q^{37} + 24 q^{41} + 4 q^{45} + 30 q^{49} - 12 q^{53} + 16 q^{57} + 16 q^{61} + 44 q^{65} + 20 q^{69} + 20 q^{73} - 4 q^{77} + 30 q^{81} + 20 q^{85} + 8 q^{89} + 32 q^{93} + 40 q^{97}+O(q^{100}) $ 6 * q - 4 * q^5 + 10 * q^9 - 4 * q^13 + 16 * q^17 - 4 * q^21 + 14 * q^25 - 6 * q^29 + 28 * q^33 - 4 * q^37 + 24 * q^41 + 4 * q^45 + 30 * q^49 - 12 * q^53 + 16 * q^57 + 16 * q^61 + 44 * q^65 + 20 * q^69 + 20 * q^73 - 4 * q^77 + 30 * q^81 + 20 * q^85 + 8 * q^89 + 32 * q^93 + 40 * q^97

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
$3$	3.15704	1.82272	0.911360	−	0.411610i	$-0.135033\pi$
$3$	3.15704	1.82272	0.911360	+	0.411610i	$0.135033\pi$
$4$	0	0
$5$	−1.42586	−0.637663	−0.318832	−	0.947811i	$-0.603290\pi$
$5$	−1.42586	−0.637663	−0.318832	+	0.947811i	$0.603290\pi$
$6$	0	0
$7$	1.70818	0.645630	0.322815	−	0.946462i	$-0.395371\pi$
$7$	1.70818	0.645630	0.322815	+	0.946462i	$0.395371\pi$
$8$	0	0
$9$	6.96693	2.32231
$10$	0	0
$11$	3.15704	0.951885	0.475942	−	0.879477i	$-0.342107\pi$
$11$	3.15704	0.951885	0.475942	+	0.879477i	$0.342107\pi$
$12$	0	0
$13$	−1.42586	−0.395462	−0.197731	−	0.980256i	$-0.563357\pi$
$13$	−1.42586	−0.395462	−0.197731	+	0.980256i	$0.563357\pi$
$14$	0	0
$15$	−4.50150	−1.16228
$16$	0	0
$17$	−3.39279	−0.822871	−0.411436	−	0.911439i	$-0.634973\pi$
$17$	−3.39279	−0.822871	−0.411436	+	0.911439i	$0.634973\pi$
$18$	0	0
$19$	0.363721	0.0834433	0.0417216	−	0.999129i	$-0.486716\pi$
$19$	0.363721	0.0834433	0.0417216	+	0.999129i	$0.486716\pi$
$20$	0	0
$21$	5.39279	1.17680
$22$	0	0
$23$	7.29482	1.52108	0.760538	−	0.649294i	$-0.224935\pi$
$23$	7.29482	1.52108	0.760538	+	0.649294i	$0.224935\pi$
$24$	0	0
$25$	−2.96693	−0.593385
$26$	0	0
$27$	12.5238	2.41020
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	1.44887	0.260224	0.130112	−	0.991499i	$-0.458466\pi$
$31$	1.44887	0.260224	0.130112	+	0.991499i	$0.458466\pi$
$32$	0	0
$33$	9.96693	1.73502
$34$	0	0
$35$	−2.43562	−0.411694
$36$	0	0
$37$	0.851718	0.140021	0.0700107	−	0.997546i	$-0.477697\pi$
$37$	0.851718	0.140021	0.0700107	+	0.997546i	$0.477697\pi$
$38$	0	0
$39$	−4.50150	−0.720817
$40$	0	0
$41$	8.54107	1.33389	0.666945	−	0.745107i	$-0.267601\pi$
$41$	8.54107	1.33389	0.666945	+	0.745107i	$0.267601\pi$
$42$	0	0
$43$	6.57340	1.00243	0.501217	−	0.865322i	$-0.332886\pi$
$43$	6.57340	1.00243	0.501217	+	0.865322i	$0.332886\pi$
$44$	0	0
$45$	−9.93385	−1.48085
$46$	0	0
$47$	11.1793	1.63067	0.815335	−	0.578990i	$-0.196553\pi$
$47$	11.1793	1.63067	0.815335	+	0.578990i	$0.196553\pi$
$48$	0	0
$49$	−4.08214	−0.583162
$50$	0	0
$51$	−10.7112	−1.49986
$52$	0	0
$53$	−13.3597	−1.83510	−0.917549	−	0.397623i	$-0.869835\pi$
$53$	−13.3597	−1.83510	−0.917549	+	0.397623i	$0.869835\pi$
$54$	0	0
$55$	−4.50150	−0.606982
$56$	0	0
$57$	1.14828	0.152094
$58$	0	0
$59$	−10.7112	−1.39448	−0.697238	−	0.716840i	$-0.745588\pi$
$59$	−10.7112	−1.39448	−0.697238	+	0.716840i	$0.745588\pi$
$60$	0	0
$61$	−3.39279	−0.434402	−0.217201	−	0.976127i	$-0.569693\pi$
$61$	−3.39279	−0.434402	−0.217201	+	0.976127i	$0.569693\pi$
$62$	0	0
$63$	11.9007	1.49935
$64$	0	0
$65$	2.03307	0.252172
$66$	0	0
$67$	−12.6282	−1.54278	−0.771389	−	0.636364i	$-0.780438\pi$
$67$	−12.6282	−1.54278	−0.771389	+	0.636364i	$0.780438\pi$
$68$	0	0
$69$	23.0301	2.77250
$70$	0	0
$71$	−5.58665	−0.663013	−0.331506	−	0.943453i	$-0.607557\pi$
$71$	−5.58665	−0.663013	−0.331506	+	0.943453i	$0.607557\pi$
$72$	0	0
$73$	13.9339	1.63083	0.815417	−	0.578874i	$-0.196508\pi$
$73$	13.9339	1.63083	0.815417	+	0.578874i	$0.196508\pi$
$74$	0	0
$75$	−9.36672	−1.08158
$76$	0	0
$77$	5.39279	0.614565
$78$	0	0
$79$	4.86522	0.547380	0.273690	−	0.961818i	$-0.411756\pi$
$79$	4.86522	0.547380	0.273690	+	0.961818i	$0.411756\pi$
$80$	0	0
$81$	18.6373	2.07081
$82$	0	0
$83$	−5.33335	−0.585412	−0.292706	−	0.956203i	$-0.594556\pi$
$83$	−5.33335	−0.585412	−0.292706	+	0.956203i	$0.594556\pi$
$84$	0	0
$85$	4.83763	0.524715
$86$	0	0
$87$	−3.15704	−0.338471
$88$	0	0
$89$	−15.3266	−1.62462	−0.812310	−	0.583226i	$-0.801790\pi$
$89$	−15.3266	−1.62462	−0.812310	+	0.583226i	$0.801790\pi$
$90$	0	0
$91$	−2.43562	−0.255322
$92$	0	0
$93$	4.57414	0.474316
$94$	0	0
$95$	−0.518614	−0.0532087
$96$	0	0
$97$	14.2445	1.44631	0.723155	−	0.690686i	$-0.242691\pi$
$97$	14.2445	1.44631	0.723155	+	0.690686i	$0.242691\pi$
$98$	0	0
$99$	21.9949	2.21057
$100$	0	0
$101$	0.851718	0.0847491	0.0423745	−	0.999102i	$-0.486508\pi$
$101$	0.851718	0.0847491	0.0423745	+	0.999102i	$0.486508\pi$
$102$	0	0
$103$	−6.31409	−0.622146	−0.311073	−	0.950386i	$-0.600688\pi$
$103$	−6.31409	−0.622146	−0.311073	+	0.950386i	$0.600688\pi$
$104$	0	0
$105$	−7.68935	−0.750404
$106$	0	0
$107$	5.12453	0.495407	0.247703	−	0.968836i	$-0.420324\pi$
$107$	5.12453	0.495407	0.247703	+	0.968836i	$0.420324\pi$
$108$	0	0
$109$	19.6563	1.88273	0.941365	−	0.337390i	$-0.109544\pi$
$109$	19.6563	1.88273	0.941365	+	0.337390i	$0.109544\pi$
$110$	0	0
$111$	2.68891	0.255220
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	−10.4014	−0.969934
$116$	0	0
$117$	−9.93385	−0.918385
$118$	0	0
$119$	−5.79547	−0.531270
$120$	0	0
$121$	−1.03307	−0.0939157
$122$	0	0
$123$	26.9645	2.43131
$124$	0	0
$125$	11.3597	1.01604
$126$	0	0
$127$	6.46898	0.574029	0.287015	−	0.957926i	$-0.407337\pi$
$127$	6.46898	0.574029	0.287015	+	0.957926i	$0.407337\pi$
$128$	0	0
$129$	20.7525	1.82716
$130$	0	0
$131$	−14.9534	−1.30648	−0.653241	−	0.757150i	$-0.726591\pi$
$131$	−14.9534	−1.30648	−0.653241	+	0.757150i	$0.726591\pi$
$132$	0	0
$133$	0.621299	0.0538735
$134$	0	0
$135$	−17.8571	−1.53690
$136$	0	0
$137$	3.14828	0.268976	0.134488	−	0.990915i	$-0.457061\pi$
$137$	3.14828	0.268976	0.134488	+	0.990915i	$0.457061\pi$
$138$	0	0
$139$	−5.33335	−0.452369	−0.226185	−	0.974084i	$-0.572625\pi$
$139$	−5.33335	−0.452369	−0.226185	+	0.974084i	$0.572625\pi$
$140$	0	0
$141$	35.2936	2.97225
$142$	0	0
$143$	−4.50150	−0.376434
$144$	0	0
$145$	1.42586	0.118411
$146$	0	0
$147$	−12.8875	−1.06294
$148$	0	0
$149$	−20.2114	−1.65578	−0.827892	−	0.560887i	$-0.810460\pi$
$149$	−20.2114	−1.65578	−0.827892	+	0.560887i	$0.810460\pi$
$150$	0	0
$151$	−2.43562	−0.198208	−0.0991039	−	0.995077i	$-0.531598\pi$
$151$	−2.43562	−0.198208	−0.0991039	+	0.995077i	$0.531598\pi$
$152$	0	0
$153$	−23.6373	−1.91096
$154$	0	0
$155$	−2.06588	−0.165936
$156$	0	0
$157$	−7.32664	−0.584729	−0.292365	−	0.956307i	$-0.594442\pi$
$157$	−7.32664	−0.584729	−0.292365	+	0.956307i	$0.594442\pi$
$158$	0	0
$159$	−42.1772	−3.34487
$160$	0	0
$161$	12.4608	0.982052
$162$	0	0
$163$	12.8875	1.00943	0.504713	−	0.863287i	$-0.331598\pi$
$163$	12.8875	1.00943	0.504713	+	0.863287i	$0.331598\pi$
$164$	0	0
$165$	−14.2114	−1.10636
$166$	0	0
$167$	−21.6312	−1.67387	−0.836935	−	0.547302i	$-0.815655\pi$
$167$	−21.6312	−1.67387	−0.836935	+	0.547302i	$0.815655\pi$
$168$	0	0
$169$	−10.9669	−0.843610
$170$	0	0
$171$	2.53402	0.193781
$172$	0	0
$173$	−11.0821	−0.842559	−0.421280	−	0.906931i	$-0.638419\pi$
$173$	−11.0821	−0.842559	−0.421280	+	0.906931i	$0.638419\pi$
$174$	0	0
$175$	−5.06803	−0.383107
$176$	0	0
$177$	−33.8156	−2.54174
$178$	0	0
$179$	−7.04153	−0.526309	−0.263154	−	0.964754i	$-0.584763\pi$
$179$	−7.04153	−0.526309	−0.263154	+	0.964754i	$0.584763\pi$
$180$	0	0
$181$	16.2114	1.20499	0.602493	−	0.798124i	$-0.294174\pi$
$181$	16.2114	1.20499	0.602493	+	0.798124i	$0.294174\pi$
$182$	0	0
$183$	−10.7112	−0.791793
$184$	0	0
$185$	−1.21443	−0.0892866
$186$	0	0
$187$	−10.7112	−0.783279
$188$	0	0
$189$	21.3928	1.55610
$190$	0	0
$191$	13.7193	0.992696	0.496348	−	0.868124i	$-0.334674\pi$
$191$	13.7193	0.992696	0.496348	+	0.868124i	$0.334674\pi$
$192$	0	0
$193$	−9.62320	−0.692693	−0.346347	−	0.938107i	$-0.612578\pi$
$193$	−9.62320	−0.692693	−0.346347	+	0.938107i	$0.612578\pi$
$194$	0	0
$195$	6.41850	0.459638
$196$	0	0
$197$	−11.7034	−0.833835	−0.416918	−	0.908944i	$-0.636890\pi$
$197$	−11.7034	−0.833835	−0.416918	+	0.908944i	$0.636890\pi$
$198$	0	0
$199$	19.7142	1.39750	0.698750	−	0.715366i	$-0.253740\pi$
$199$	19.7142	1.39750	0.698750	+	0.715366i	$0.253740\pi$
$200$	0	0
$201$	−39.8677	−2.81205
$202$	0	0
$203$	−1.70818	−0.119890
$204$	0	0
$205$	−12.1784	−0.850573
$206$	0	0
$207$	50.8225	3.53241
$208$	0	0
$209$	1.14828	0.0794284
$210$	0	0
$211$	−19.2016	−1.32189	−0.660945	−	0.750434i	$-0.729844\pi$
$211$	−19.2016	−1.32189	−0.660945	+	0.750434i	$0.729844\pi$
$212$	0	0
$213$	−17.6373	−1.20849
$214$	0	0
$215$	−9.37273	−0.639215
$216$	0	0
$217$	2.47492	0.168009
$218$	0	0
$219$	43.9898	2.97255
$220$	0	0
$221$	4.83763	0.325414
$222$	0	0
$223$	−13.1468	−0.880374	−0.440187	−	0.897906i	$-0.645088\pi$
$223$	−13.1468	−0.880374	−0.440187	+	0.897906i	$0.645088\pi$
$224$	0	0
$225$	−20.6704	−1.37802
$226$	0	0
$227$	−15.5259	−1.03049	−0.515246	−	0.857043i	$-0.672299\pi$
$227$	−15.5259	−1.03049	−0.515246	+	0.857043i	$0.672299\pi$
$228$	0	0
$229$	9.16237	0.605466	0.302733	−	0.953075i	$-0.402101\pi$
$229$	9.16237	0.605466	0.302733	+	0.953075i	$0.402101\pi$
$230$	0	0
$231$	17.0253	1.12018
$232$	0	0
$233$	−7.96693	−0.521931	−0.260965	−	0.965348i	$-0.584041\pi$
$233$	−7.96693	−0.521931	−0.260965	+	0.965348i	$0.584041\pi$
$234$	0	0
$235$	−15.9401	−1.03982
$236$	0	0
$237$	15.3597	0.997721
$238$	0	0
$239$	−21.3779	−1.38282	−0.691410	−	0.722463i	$-0.743010\pi$
$239$	−21.3779	−1.38282	−0.691410	+	0.722463i	$0.743010\pi$
$240$	0	0
$241$	3.96693	0.255532	0.127766	−	0.991804i	$-0.459219\pi$
$241$	3.96693	0.255532	0.127766	+	0.991804i	$0.459219\pi$
$242$	0	0
$243$	21.2675	1.36431
$244$	0	0
$245$	5.82055	0.371861
$246$	0	0
$247$	−0.518614	−0.0329986
$248$	0	0
$249$	−16.8376	−1.06704
$250$	0	0
$251$	26.9585	1.70161	0.850803	−	0.525485i	$-0.176116\pi$
$251$	26.9585	1.70161	0.850803	+	0.525485i	$0.176116\pi$
$252$	0	0
$253$	23.0301	1.44789
$254$	0	0
$255$	15.2726	0.956409
$256$	0	0
$257$	27.3077	1.70340	0.851702	−	0.524026i	$-0.175571\pi$
$257$	27.3077	1.70340	0.851702	+	0.524026i	$0.175571\pi$
$258$	0	0
$259$	1.45488	0.0904020
$260$	0	0
$261$	−6.96693	−0.431242
$262$	0	0
$263$	−5.38383	−0.331981	−0.165991	−	0.986127i	$-0.553082\pi$
$263$	−5.38383	−0.331981	−0.165991	+	0.986127i	$0.553082\pi$
$264$	0	0
$265$	19.0491	1.17017
$266$	0	0
$267$	−48.3869	−2.96123
$268$	0	0
$269$	−7.98592	−0.486910	−0.243455	−	0.969912i	$-0.578281\pi$
$269$	−7.98592	−0.486910	−0.243455	+	0.969912i	$0.578281\pi$
$270$	0	0
$271$	20.3911	1.23867	0.619337	−	0.785126i	$-0.287402\pi$
$271$	20.3911	1.23867	0.619337	+	0.785126i	$0.287402\pi$
$272$	0	0
$273$	−7.68935	−0.465381
$274$	0	0
$275$	−9.36672	−0.564834
$276$	0	0
$277$	17.8677	1.07357	0.536783	−	0.843720i	$-0.319639\pi$
$277$	17.8677	1.07357	0.536783	+	0.843720i	$0.319639\pi$
$278$	0	0
$279$	10.0942	0.604322
$280$	0	0
$281$	0.522079	0.0311446	0.0155723	−	0.999879i	$-0.495043\pi$
$281$	0.522079	0.0311446	0.0155723	+	0.999879i	$0.495043\pi$
$282$	0	0
$283$	6.56738	0.390390	0.195195	−	0.980764i	$-0.437466\pi$
$283$	6.56738	0.390390	0.195195	+	0.980764i	$0.437466\pi$
$284$	0	0
$285$	−1.63729	−0.0969846
$286$	0	0
$287$	14.5896	0.861199
$288$	0	0
$289$	−5.48901	−0.322883
$290$	0	0
$291$	44.9705	2.63622
$292$	0	0
$293$	−32.7856	−1.91535	−0.957677	−	0.287846i	$-0.907061\pi$
$293$	−32.7856	−1.91535	−0.957677	+	0.287846i	$0.907061\pi$
$294$	0	0
$295$	15.2726	0.889206
$296$	0	0
$297$	39.5381	2.29423
$298$	0	0
$299$	−10.4014	−0.601528
$300$	0	0
$301$	11.2285	0.647201
$302$	0	0
$303$	2.68891	0.154474
$304$	0	0
$305$	4.83763	0.277002
$306$	0	0
$307$	−6.05478	−0.345565	−0.172782	−	0.984960i	$-0.555276\pi$
$307$	−6.05478	−0.345565	−0.172782	+	0.984960i	$0.555276\pi$
$308$	0	0
$309$	−19.9339	−1.13400
$310$	0	0
$311$	6.88664	0.390505	0.195253	−	0.980753i	$-0.437447\pi$
$311$	6.88664	0.390505	0.195253	+	0.980753i	$0.437447\pi$
$312$	0	0
$313$	−2.26349	−0.127940	−0.0639701	−	0.997952i	$-0.520376\pi$
$313$	−2.26349	−0.127940	−0.0639701	+	0.997952i	$0.520376\pi$
$314$	0	0
$315$	−16.9688	−0.956082
$316$	0	0
$317$	−19.3928	−1.08921	−0.544604	−	0.838694i	$-0.683320\pi$
$317$	−19.3928	−1.08921	−0.544604	+	0.838694i	$0.683320\pi$
$318$	0	0
$319$	−3.15704	−0.176761
$320$	0	0
$321$	16.1784	0.902988
$322$	0	0
$323$	−1.23403	−0.0686631
$324$	0	0
$325$	4.23042	0.234661
$326$	0	0
$327$	62.0557	3.43169
$328$	0	0
$329$	19.0962	1.05281
$330$	0	0
$331$	27.8948	1.53324	0.766618	−	0.642104i	$-0.221938\pi$
$331$	27.8948	1.53324	0.766618	+	0.642104i	$0.221938\pi$
$332$	0	0
$333$	5.93385	0.325173
$334$	0	0
$335$	18.0060	0.983773
$336$	0	0
$337$	−17.0962	−0.931290	−0.465645	−	0.884971i	$-0.654178\pi$
$337$	−17.0962	−0.931290	−0.465645	+	0.884971i	$0.654178\pi$
$338$	0	0
$339$	6.31409	0.342934
$340$	0	0
$341$	4.57414	0.247704
$342$	0	0
$343$	−18.9302	−1.02214
$344$	0	0
$345$	−32.8376	−1.76792
$346$	0	0
$347$	−12.8370	−0.689126	−0.344563	−	0.938763i	$-0.611973\pi$
$347$	−12.8370	−0.689126	−0.344563	+	0.938763i	$0.611973\pi$
$348$	0	0
$349$	3.12929	0.167507	0.0837536	−	0.996486i	$-0.473309\pi$
$349$	3.12929	0.167507	0.0837536	+	0.996486i	$0.473309\pi$
$350$	0	0
$351$	−17.8571	−0.953142
$352$	0	0
$353$	−25.2746	−1.34523	−0.672615	−	0.739993i	$-0.734829\pi$
$353$	−25.2746	−1.34523	−0.672615	+	0.739993i	$0.734829\pi$
$354$	0	0
$355$	7.96577	0.422779
$356$	0	0
$357$	−18.2966	−0.968357
$358$	0	0
$359$	−29.6030	−1.56238	−0.781192	−	0.624291i	$-0.785388\pi$
$359$	−29.6030	−1.56238	−0.781192	+	0.624291i	$0.785388\pi$
$360$	0	0
$361$	−18.8677	−0.993037
$362$	0	0
$363$	−3.26146	−0.171182
$364$	0	0
$365$	−19.8677	−1.03992
$366$	0	0
$367$	−34.4142	−1.79641	−0.898204	−	0.439578i	$-0.855128\pi$
$367$	−34.4142	−1.79641	−0.898204	+	0.439578i	$0.855128\pi$
$368$	0	0
$369$	59.5050	3.09771
$370$	0	0
$371$	−22.8207	−1.18479
$372$	0	0
$373$	−5.35971	−0.277515	−0.138758	−	0.990326i	$-0.544311\pi$
$373$	−5.35971	−0.277515	−0.138758	+	0.990326i	$0.544311\pi$
$374$	0	0
$375$	35.8631	1.85196
$376$	0	0
$377$	1.42586	0.0734355
$378$	0	0
$379$	−19.0972	−0.980955	−0.490478	−	0.871454i	$-0.663178\pi$
$379$	−19.0972	−0.980955	−0.490478	+	0.871454i	$0.663178\pi$
$380$	0	0
$381$	20.4229	1.04629
$382$	0	0
$383$	15.0638	0.769724	0.384862	−	0.922974i	$-0.374249\pi$
$383$	15.0638	0.769724	0.384862	+	0.922974i	$0.374249\pi$
$384$	0	0
$385$	−7.68935	−0.391886
$386$	0	0
$387$	45.7964	2.32796
$388$	0	0
$389$	−2.06615	−0.104758	−0.0523789	−	0.998627i	$-0.516680\pi$
$389$	−2.06615	−0.104758	−0.0523789	+	0.998627i	$0.516680\pi$
$390$	0	0
$391$	−24.7498	−1.25165
$392$	0	0
$393$	−47.2084	−2.38135
$394$	0	0
$395$	−6.93712	−0.349044
$396$	0	0
$397$	−4.87071	−0.244454	−0.122227	−	0.992502i	$-0.539004\pi$
$397$	−4.87071	−0.244454	−0.122227	+	0.992502i	$0.539004\pi$
$398$	0	0
$399$	1.96147	0.0981962
$400$	0	0
$401$	−4.56006	−0.227718	−0.113859	−	0.993497i	$-0.536321\pi$
$401$	−4.56006	−0.227718	−0.113859	+	0.993497i	$0.536321\pi$
$402$	0	0
$403$	−2.06588	−0.102909
$404$	0	0
$405$	−26.5741	−1.32048
$406$	0	0
$407$	2.68891	0.133284
$408$	0	0
$409$	−1.40687	−0.0695652	−0.0347826	−	0.999395i	$-0.511074\pi$
$409$	−1.40687	−0.0695652	−0.0347826	+	0.999395i	$0.511074\pi$
$410$	0	0
$411$	9.93927	0.490268
$412$	0	0
$413$	−18.2966	−0.900315
$414$	0	0
$415$	7.60461	0.373296
$416$	0	0
$417$	−16.8376	−0.824542
$418$	0	0
$419$	27.4266	1.33988	0.669940	−	0.742416i	$-0.266320\pi$
$419$	27.4266	1.33988	0.669940	+	0.742416i	$0.266320\pi$
$420$	0	0
$421$	29.0962	1.41806	0.709032	−	0.705177i	$-0.249132\pi$
$421$	29.0962	1.41806	0.709032	+	0.705177i	$0.249132\pi$
$422$	0	0
$423$	77.8854	3.78692
$424$	0	0
$425$	10.0661	0.488280
$426$	0	0
$427$	−5.79547	−0.280463
$428$	0	0
$429$	−14.2114	−0.686134
$430$	0	0
$431$	−21.8845	−1.05414	−0.527069	−	0.849823i	$-0.676709\pi$
$431$	−21.8845	−1.05414	−0.527069	+	0.849823i	$0.676709\pi$
$432$	0	0
$433$	8.23042	0.395529	0.197764	−	0.980250i	$-0.436632\pi$
$433$	8.23042	0.395529	0.197764	+	0.980250i	$0.436632\pi$
$434$	0	0
$435$	4.50150	0.215830
$436$	0	0
$437$	2.65328	0.126924
$438$	0	0
$439$	−32.8610	−1.56837	−0.784184	−	0.620528i	$-0.786918\pi$
$439$	−32.8610	−1.56837	−0.784184	+	0.620528i	$0.786918\pi$
$440$	0	0
$441$	−28.4399	−1.35428
$442$	0	0
$443$	21.4763	1.02037	0.510184	−	0.860065i	$-0.329577\pi$
$443$	21.4763	1.02037	0.510184	+	0.860065i	$0.329577\pi$
$444$	0	0
$445$	21.8536	1.03596
$446$	0	0
$447$	−63.8084	−3.01803
$448$	0	0
$449$	−22.4890	−1.06132	−0.530661	−	0.847584i	$-0.678056\pi$
$449$	−22.4890	−1.06132	−0.530661	+	0.847584i	$0.678056\pi$
$450$	0	0
$451$	26.9645	1.26971
$452$	0	0
$453$	−7.68935	−0.361277
$454$	0	0
$455$	3.47285	0.162810
$456$	0	0
$457$	25.8677	1.21004	0.605020	−	0.796210i	$-0.293165\pi$
$457$	25.8677	1.21004	0.605020	+	0.796210i	$0.293165\pi$
$458$	0	0
$459$	−42.4904	−1.98328
$460$	0	0
$461$	−0.296565	−0.0138124	−0.00690620	−	0.999976i	$-0.502198\pi$
$461$	−0.296565	−0.0138124	−0.00690620	+	0.999976i	$0.502198\pi$
$462$	0	0
$463$	−19.1956	−0.892093	−0.446047	−	0.895010i	$-0.647168\pi$
$463$	−19.1956	−0.892093	−0.446047	+	0.895010i	$0.647168\pi$
$464$	0	0
$465$	−6.52208	−0.302454
$466$	0	0
$467$	11.9512	0.553036	0.276518	−	0.961009i	$-0.410819\pi$
$467$	11.9512	0.553036	0.276518	+	0.961009i	$0.410819\pi$
$468$	0	0
$469$	−21.5711	−0.996063
$470$	0	0
$471$	−23.1305	−1.06580
$472$	0	0
$473$	20.7525	0.954201
$474$	0	0
$475$	−1.07913	−0.0495140
$476$	0	0
$477$	−93.0761	−4.26166
$478$	0	0
$479$	−26.7052	−1.22019	−0.610096	−	0.792327i	$-0.708869\pi$
$479$	−26.7052	−1.22019	−0.610096	+	0.792327i	$0.708869\pi$
$480$	0	0
$481$	−1.21443	−0.0553732
$482$	0	0
$483$	39.3394	1.79001
$484$	0	0
$485$	−20.3106	−0.922259
$486$	0	0
$487$	10.9200	0.494832	0.247416	−	0.968909i	$-0.420418\pi$
$487$	10.9200	0.494832	0.247416	+	0.968909i	$0.420418\pi$
$488$	0	0
$489$	40.6864	1.83990
$490$	0	0
$491$	43.5217	1.96410	0.982052	−	0.188608i	$-0.0603977\pi$
$491$	43.5217	1.96410	0.982052	+	0.188608i	$0.0603977\pi$
$492$	0	0
$493$	3.39279	0.152803
$494$	0	0
$495$	−31.3616	−1.40960
$496$	0	0
$497$	−9.54297	−0.428061
$498$	0	0
$499$	−4.60591	−0.206189	−0.103094	−	0.994672i	$-0.532874\pi$
$499$	−4.60591	−0.206189	−0.103094	+	0.994672i	$0.532874\pi$
$500$	0	0
$501$	−68.2906	−3.05100
$502$	0	0
$503$	29.1853	1.30131	0.650654	−	0.759374i	$-0.274495\pi$
$503$	29.1853	1.30131	0.650654	+	0.759374i	$0.274495\pi$
$504$	0	0
$505$	−1.21443	−0.0540414
$506$	0	0
$507$	−34.6231	−1.53766
$508$	0	0
$509$	4.80456	0.212958	0.106479	−	0.994315i	$-0.466042\pi$
$509$	4.80456	0.212958	0.106479	+	0.994315i	$0.466042\pi$
$510$	0	0
$511$	23.8015	1.05291
$512$	0	0
$513$	4.55515	0.201115
$514$	0	0
$515$	9.00300	0.396719
$516$	0	0
$517$	35.2936	1.55221
$518$	0	0
$519$	−34.9868	−1.53575
$520$	0	0
$521$	9.18136	0.402242	0.201121	−	0.979566i	$-0.435541\pi$
$521$	9.18136	0.402242	0.201121	+	0.979566i	$0.435541\pi$
$522$	0	0
$523$	9.93927	0.434614	0.217307	−	0.976103i	$-0.430273\pi$
$523$	9.93927	0.434614	0.217307	+	0.976103i	$0.430273\pi$
$524$	0	0
$525$	−16.0000	−0.698297
$526$	0	0
$527$	−4.91570	−0.214131
$528$	0	0
$529$	30.2144	1.31367
$530$	0	0
$531$	−74.6240	−3.23840
$532$	0	0
$533$	−12.1784	−0.527503
$534$	0	0
$535$	−7.30685	−0.315903
$536$	0	0
$537$	−22.2304	−0.959313
$538$	0	0
$539$	−12.8875	−0.555103
$540$	0	0
$541$	−36.5871	−1.57300	−0.786502	−	0.617588i	$-0.788110\pi$
$541$	−36.5871	−1.57300	−0.786502	+	0.617588i	$0.788110\pi$
$542$	0	0
$543$	51.1802	2.19635
$544$	0	0
$545$	−28.0271	−1.20055
$546$	0	0
$547$	0.253293	0.0108300	0.00541500	−	0.999985i	$-0.498276\pi$
$547$	0.253293	0.0108300	0.00541500	+	0.999985i	$0.498276\pi$
$548$	0	0
$549$	−23.6373	−1.00881
$550$	0	0
$551$	−0.363721	−0.0154950
$552$	0	0
$553$	8.31065	0.353405
$554$	0	0
$555$	−3.83401	−0.162744
$556$	0	0
$557$	−11.2144	−0.475171	−0.237585	−	0.971367i	$-0.576356\pi$
$557$	−11.2144	−0.475171	−0.237585	+	0.971367i	$0.576356\pi$
$558$	0	0
$559$	−9.37273	−0.396424
$560$	0	0
$561$	−33.8156	−1.42770
$562$	0	0
$563$	−15.3676	−0.647666	−0.323833	−	0.946114i	$-0.604972\pi$
$563$	−15.3676	−0.647666	−0.323833	+	0.946114i	$0.604972\pi$
$564$	0	0
$565$	−2.85172	−0.119973
$566$	0	0
$567$	31.8358	1.33698
$568$	0	0
$569$	37.1423	1.55709	0.778543	−	0.627592i	$-0.215959\pi$
$569$	37.1423	1.55709	0.778543	+	0.627592i	$0.215959\pi$
$570$	0	0
$571$	−2.68891	−0.112527	−0.0562637	−	0.998416i	$-0.517919\pi$
$571$	−2.68891	−0.112527	−0.0562637	+	0.998416i	$0.517919\pi$
$572$	0	0
$573$	43.3126	1.80941
$574$	0	0
$575$	−21.6432	−0.902584
$576$	0	0
$577$	15.2605	0.635303	0.317651	−	0.948208i	$-0.397106\pi$
$577$	15.2605	0.635303	0.317651	+	0.948208i	$0.397106\pi$
$578$	0	0
$579$	−30.3809	−1.26259
$580$	0	0
$581$	−9.11031	−0.377959
$582$	0	0
$583$	−42.1772	−1.74680
$584$	0	0
$585$	14.1643	0.585621
$586$	0	0
$587$	24.7942	1.02337	0.511684	−	0.859174i	$-0.329022\pi$
$587$	24.7942	1.02337	0.511684	+	0.859174i	$0.329022\pi$
$588$	0	0
$589$	0.526984	0.0217140
$590$	0	0
$591$	−36.9483	−1.51985
$592$	0	0
$593$	25.0491	1.02864	0.514321	−	0.857598i	$-0.328044\pi$
$593$	25.0491	1.02864	0.514321	+	0.857598i	$0.328044\pi$
$594$	0	0
$595$	8.26353	0.338772
$596$	0	0
$597$	62.2385	2.54725
$598$	0	0
$599$	2.48610	0.101579	0.0507896	−	0.998709i	$-0.483826\pi$
$599$	2.48610	0.101579	0.0507896	+	0.998709i	$0.483826\pi$
$600$	0	0
$601$	13.7555	0.561098	0.280549	−	0.959840i	$-0.409483\pi$
$601$	13.7555	0.561098	0.280549	+	0.959840i	$0.409483\pi$
$602$	0	0
$603$	−87.9796	−3.58281
$604$	0	0
$605$	1.47302	0.0598866
$606$	0	0
$607$	37.4729	1.52098	0.760489	−	0.649351i	$-0.224959\pi$
$607$	37.4729	1.52098	0.760489	+	0.649351i	$0.224959\pi$
$608$	0	0
$609$	−5.39279	−0.218527
$610$	0	0
$611$	−15.9401	−0.644868
$612$	0	0
$613$	6.44185	0.260184	0.130092	−	0.991502i	$-0.458473\pi$
$613$	6.44185	0.260184	0.130092	+	0.991502i	$0.458473\pi$
$614$	0	0
$615$	−38.4476	−1.55036
$616$	0	0
$617$	−18.4229	−0.741676	−0.370838	−	0.928697i	$-0.620930\pi$
$617$	−18.4229	−0.741676	−0.370838	+	0.928697i	$0.620930\pi$
$618$	0	0
$619$	9.05348	0.363890	0.181945	−	0.983309i	$-0.441761\pi$
$619$	9.05348	0.363890	0.181945	+	0.983309i	$0.441761\pi$
$620$	0	0
$621$	91.3586	3.66610
$622$	0	0
$623$	−26.1806	−1.04890
$624$	0	0
$625$	−1.36271	−0.0545085
$626$	0	0
$627$	3.62518	0.144776
$628$	0	0
$629$	−2.88969	−0.115220
$630$	0	0
$631$	−13.8742	−0.552324	−0.276162	−	0.961111i	$-0.589063\pi$
$631$	−13.8742	−0.552324	−0.276162	+	0.961111i	$0.589063\pi$
$632$	0	0
$633$	−60.6202	−2.40944
$634$	0	0
$635$	−9.22385	−0.366037
$636$	0	0
$637$	5.82055	0.230619
$638$	0	0
$639$	−38.9218	−1.53972
$640$	0	0
$641$	31.8818	1.25926	0.629628	−	0.776897i	$-0.283208\pi$
$641$	31.8818	1.25926	0.629628	+	0.776897i	$0.283208\pi$
$642$	0	0
$643$	−6.35855	−0.250757	−0.125378	−	0.992109i	$-0.540015\pi$
$643$	−6.35855	−0.250757	−0.125378	+	0.992109i	$0.540015\pi$
$644$	0	0
$645$	−29.5901	−1.16511
$646$	0	0
$647$	20.6504	0.811853	0.405926	−	0.913906i	$-0.366949\pi$
$647$	20.6504	0.811853	0.405926	+	0.913906i	$0.366949\pi$
$648$	0	0
$649$	−33.8156	−1.32738
$650$	0	0
$651$	7.81344	0.306233
$652$	0	0
$653$	30.2445	1.18356	0.591779	−	0.806100i	$-0.298426\pi$
$653$	30.2445	1.18356	0.591779	+	0.806100i	$0.298426\pi$
$654$	0	0
$655$	21.3214	0.833096
$656$	0	0
$657$	97.0761	3.78730
$658$	0	0
$659$	−19.7202	−0.768189	−0.384095	−	0.923294i	$-0.625486\pi$
$659$	−19.7202	−0.768189	−0.384095	+	0.923294i	$0.625486\pi$
$660$	0	0
$661$	39.0821	1.52012	0.760059	−	0.649854i	$-0.225170\pi$
$661$	39.0821	1.52012	0.760059	+	0.649854i	$0.225170\pi$
$662$	0	0
$663$	15.2726	0.593139
$664$	0	0
$665$	−0.885885	−0.0343531
$666$	0	0
$667$	−7.29482	−0.282457
$668$	0	0
$669$	−41.5050	−1.60468
$670$	0	0
$671$	−10.7112	−0.413500
$672$	0	0
$673$	23.2415	0.895894	0.447947	−	0.894060i	$-0.352155\pi$
$673$	23.2415	0.895894	0.447947	+	0.894060i	$0.352155\pi$
$674$	0	0
$675$	−37.1571	−1.43018
$676$	0	0
$677$	−7.67527	−0.294984	−0.147492	−	0.989063i	$-0.547120\pi$
$677$	−7.67527	−0.294984	−0.147492	+	0.989063i	$0.547120\pi$
$678$	0	0
$679$	24.3321	0.933781
$680$	0	0
$681$	−49.0160	−1.87830
$682$	0	0
$683$	7.97780	0.305262	0.152631	−	0.988283i	$-0.451225\pi$
$683$	7.97780	0.305262	0.152631	+	0.988283i	$0.451225\pi$
$684$	0	0
$685$	−4.48901	−0.171516
$686$	0	0
$687$	28.9260	1.10360
$688$	0	0
$689$	19.0491	0.725711
$690$	0	0
$691$	−30.3244	−1.15359	−0.576797	−	0.816888i	$-0.695698\pi$
$691$	−30.3244	−1.15359	−0.576797	+	0.816888i	$0.695698\pi$
$692$	0	0
$693$	37.5711	1.42721
$694$	0	0
$695$	7.60461	0.288459
$696$	0	0
$697$	−28.9780	−1.09762
$698$	0	0
$699$	−25.1519	−0.951334
$700$	0	0
$701$	17.4920	0.660664	0.330332	−	0.943865i	$-0.392839\pi$
$701$	17.4920	0.660664	0.330332	+	0.943865i	$0.392839\pi$
$702$	0	0
$703$	0.309787	0.0116839
$704$	0	0
$705$	−50.3236	−1.89530
$706$	0	0
$707$	1.45488	0.0547165
$708$	0	0
$709$	23.5522	0.884520	0.442260	−	0.896887i	$-0.354177\pi$
$709$	23.5522	0.884520	0.442260	+	0.896887i	$0.354177\pi$
$710$	0	0
$711$	33.8956	1.27119
$712$	0	0
$713$	10.5692	0.395821
$714$	0	0
$715$	6.41850	0.240038
$716$	0	0
$717$	−67.4909	−2.52049
$718$	0	0
$719$	36.9483	1.37794	0.688969	−	0.724791i	$-0.258064\pi$
$719$	36.9483	1.37794	0.688969	+	0.724791i	$0.258064\pi$
$720$	0	0
$721$	−10.7856	−0.401676
$722$	0	0
$723$	12.5238	0.465764
$724$	0	0
$725$	2.96693	0.110189
$726$	0	0
$727$	0.673508	0.0249790	0.0124895	−	0.999922i	$-0.496024\pi$
$727$	0.673508	0.0249790	0.0124895	+	0.999922i	$0.496024\pi$
$728$	0	0
$729$	11.2304	0.415941
$730$	0	0
$731$	−22.3021	−0.824874
$732$	0	0
$733$	21.9339	0.810145	0.405073	−	0.914284i	$-0.367246\pi$
$733$	21.9339	0.810145	0.405073	+	0.914284i	$0.367246\pi$
$734$	0	0
$735$	18.3757	0.677799
$736$	0	0
$737$	−39.8677	−1.46855
$738$	0	0
$739$	14.3303	0.527150	0.263575	−	0.964639i	$-0.415098\pi$
$739$	14.3303	0.527150	0.263575	+	0.964639i	$0.415098\pi$
$740$	0	0
$741$	−1.63729	−0.0601473
$742$	0	0
$743$	−22.5135	−0.825941	−0.412970	−	0.910745i	$-0.635509\pi$
$743$	−22.5135	−0.825941	−0.412970	+	0.910745i	$0.635509\pi$
$744$	0	0
$745$	28.8186	1.05583
$746$	0	0
$747$	−37.1571	−1.35951
$748$	0	0
$749$	8.75359	0.319849
$750$	0	0
$751$	−8.43045	−0.307631	−0.153816	−	0.988100i	$-0.549156\pi$
$751$	−8.43045	−0.307631	−0.153816	+	0.988100i	$0.549156\pi$
$752$	0	0
$753$	85.1092	3.10155
$754$	0	0
$755$	3.47285	0.126390
$756$	0	0
$757$	−32.9639	−1.19809	−0.599047	−	0.800714i	$-0.704454\pi$
$757$	−32.9639	−1.19809	−0.599047	+	0.800714i	$0.704454\pi$
$758$	0	0
$759$	72.7070	2.63910
$760$	0	0
$761$	−8.16427	−0.295955	−0.147977	−	0.988991i	$-0.547276\pi$
$761$	−8.16427	−0.295955	−0.147977	+	0.988991i	$0.547276\pi$
$762$	0	0
$763$	33.5764	1.21555
$764$	0	0
$765$	33.7034	1.21855
$766$	0	0
$767$	15.2726	0.551462
$768$	0	0
$769$	−2.77149	−0.0999424	−0.0499712	−	0.998751i	$-0.515913\pi$
$769$	−2.77149	−0.0999424	−0.0499712	+	0.998751i	$0.515913\pi$
$770$	0	0
$771$	86.2115	3.10483
$772$	0	0
$773$	14.7335	0.529927	0.264964	−	0.964258i	$-0.414640\pi$
$773$	14.7335	0.529927	0.264964	+	0.964258i	$0.414640\pi$
$774$	0	0
$775$	−4.29869	−0.154413
$776$	0	0
$777$	4.59313	0.164778
$778$	0	0
$779$	3.10656	0.111304
$780$	0	0
$781$	−17.6373	−0.631112
$782$	0	0
$783$	−12.5238	−0.447563
$784$	0	0
$785$	10.4468	0.372861
$786$	0	0
$787$	5.58665	0.199142	0.0995712	−	0.995030i	$-0.468253\pi$
$787$	5.58665	0.199142	0.0995712	+	0.995030i	$0.468253\pi$
$788$	0	0
$789$	−16.9970	−0.605109
$790$	0	0
$791$	3.41635	0.121471
$792$	0	0
$793$	4.83763	0.171789
$794$	0	0
$795$	60.1387	2.13290
$796$	0	0
$797$	−19.3928	−0.686928	−0.343464	−	0.939166i	$-0.611600\pi$
$797$	−19.3928	−0.686928	−0.343464	+	0.939166i	$0.611600\pi$
$798$	0	0
$799$	−37.9290	−1.34183
$800$	0	0
$801$	−106.780	−3.77287
$802$	0	0
$803$	43.9898	1.55237
$804$	0	0
$805$	−17.7674	−0.626218
$806$	0	0
$807$	−25.2119	−0.887500
$808$	0	0
$809$	−38.3947	−1.34989	−0.674943	−	0.737870i	$-0.735832\pi$
$809$	−38.3947	−1.34989	−0.674943	+	0.737870i	$0.735832\pi$
$810$	0	0
$811$	−51.0758	−1.79351	−0.896757	−	0.442524i	$-0.854083\pi$
$811$	−51.0758	−1.79351	−0.896757	+	0.442524i	$0.854083\pi$
$812$	0	0
$813$	64.3757	2.25775
$814$	0	0
$815$	−18.3757	−0.643674
$816$	0	0
$817$	2.39088	0.0836463
$818$	0	0
$819$	−16.9688	−0.592937
$820$	0	0
$821$	−43.4578	−1.51669	−0.758345	−	0.651854i	$-0.773991\pi$
$821$	−43.4578	−1.51669	−0.758345	+	0.651854i	$0.773991\pi$
$822$	0	0
$823$	11.2272	0.391357	0.195678	−	0.980668i	$-0.437309\pi$
$823$	11.2272	0.391357	0.195678	+	0.980668i	$0.437309\pi$
$824$	0	0
$825$	−29.5711	−1.02954
$826$	0	0
$827$	14.3424	0.498733	0.249366	−	0.968409i	$-0.419778\pi$
$827$	14.3424	0.498733	0.249366	+	0.968409i	$0.419778\pi$
$828$	0	0
$829$	35.8536	1.24525	0.622624	−	0.782521i	$-0.286067\pi$
$829$	35.8536	1.24525	0.622624	+	0.782521i	$0.286067\pi$
$830$	0	0
$831$	56.4091	1.95681
$832$	0	0
$833$	13.8498	0.479868
$834$	0	0
$835$	30.8430	1.06737
$836$	0	0
$837$	18.1453	0.627193
$838$	0	0
$839$	−12.2045	−0.421346	−0.210673	−	0.977557i	$-0.567566\pi$
$839$	−12.2045	−0.421346	−0.210673	+	0.977557i	$0.567566\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	1.64823	0.0567680
$844$	0	0
$845$	15.6373	0.537939
$846$	0	0
$847$	−1.76467	−0.0606348
$848$	0	0
$849$	20.7335	0.711572
$850$	0	0
$851$	6.21313	0.212983
$852$	0	0
$853$	12.9179	0.442299	0.221150	−	0.975240i	$-0.429019\pi$
$853$	12.9179	0.442299	0.221150	+	0.975240i	$0.429019\pi$
$854$	0	0
$855$	−3.61315	−0.123567
$856$	0	0
$857$	−31.3456	−1.07075	−0.535373	−	0.844616i	$-0.679829\pi$
$857$	−31.3456	−1.07075	−0.535373	+	0.844616i	$0.679829\pi$
$858$	0	0
$859$	44.9765	1.53458	0.767290	−	0.641300i	$-0.221605\pi$
$859$	44.9765	1.53458	0.767290	+	0.641300i	$0.221605\pi$
$860$	0	0
$861$	46.0601	1.56973
$862$	0	0
$863$	−14.7985	−0.503746	−0.251873	−	0.967760i	$-0.581046\pi$
$863$	−14.7985	−0.503746	−0.251873	+	0.967760i	$0.581046\pi$
$864$	0	0
$865$	15.8016	0.537269
$866$	0	0
$867$	−17.3290	−0.588525
$868$	0	0
$869$	15.3597	0.521043
$870$	0	0
$871$	18.0060	0.610110
$872$	0	0
$873$	99.2404	3.35878
$874$	0	0
$875$	19.4044	0.655988
$876$	0	0
$877$	−11.1954	−0.378043	−0.189022	−	0.981973i	$-0.560532\pi$
$877$	−11.1954	−0.378043	−0.189022	+	0.981973i	$0.560532\pi$
$878$	0	0
$879$	−103.505	−3.49115
$880$	0	0
$881$	18.3567	0.618453	0.309227	−	0.950988i	$-0.399930\pi$
$881$	18.3567	0.618453	0.309227	+	0.950988i	$0.399930\pi$
$882$	0	0
$883$	47.6594	1.60387	0.801934	−	0.597413i	$-0.203805\pi$
$883$	47.6594	1.60387	0.801934	+	0.597413i	$0.203805\pi$
$884$	0	0
$885$	48.2163	1.62077
$886$	0	0
$887$	34.4622	1.15713	0.578563	−	0.815638i	$-0.303614\pi$
$887$	34.4622	1.15713	0.578563	+	0.815638i	$0.303614\pi$
$888$	0	0
$889$	11.0502	0.370610
$890$	0	0
$891$	58.8387	1.97117
$892$	0	0
$893$	4.06615	0.136068
$894$	0	0
$895$	10.0402	0.335608
$896$	0	0
$897$	−32.8376	−1.09642
$898$	0	0
$899$	−1.44887	−0.0483225
$900$	0	0
$901$	45.3266	1.51005
$902$	0	0
$903$	35.4489	1.17967
$904$	0	0
$905$	−23.1152	−0.768376
$906$	0	0
$907$	−22.8233	−0.757835	−0.378918	−	0.925430i	$-0.623704\pi$
$907$	−22.8233	−0.757835	−0.378918	+	0.925430i	$0.623704\pi$
$908$	0	0
$909$	5.93385	0.196814
$910$	0	0
$911$	57.8580	1.91692	0.958461	−	0.285225i	$-0.0920683\pi$
$911$	57.8580	1.91692	0.958461	+	0.285225i	$0.0920683\pi$
$912$	0	0
$913$	−16.8376	−0.557244
$914$	0	0
$915$	15.2726	0.504897
$916$	0	0
$917$	−25.5430	−0.843503
$918$	0	0
$919$	−54.4477	−1.79606	−0.898031	−	0.439931i	$-0.855003\pi$
$919$	−54.4477	−1.79606	−0.898031	+	0.439931i	$0.855003\pi$
$920$	0	0
$921$	−19.1152	−0.629868
$922$	0	0
$923$	7.96577	0.262196
$924$	0	0
$925$	−2.52698	−0.0830867
$926$	0	0
$927$	−43.9898	−1.44481
$928$	0	0
$929$	40.6814	1.33471	0.667357	−	0.744738i	$-0.267425\pi$
$929$	40.6814	1.33471	0.667357	+	0.744738i	$0.267425\pi$
$930$	0	0
$931$	−1.48476	−0.0486610
$932$	0	0
$933$	21.7414	0.711782
$934$	0	0
$935$	15.2726	0.499468
$936$	0	0
$937$	18.6213	0.608331	0.304166	−	0.952619i	$-0.401622\pi$
$937$	18.6213	0.608331	0.304166	+	0.952619i	$0.401622\pi$
$938$	0	0
$939$	−7.14594	−0.233199
$940$	0	0
$941$	−12.3437	−0.402394	−0.201197	−	0.979551i	$-0.564483\pi$
$941$	−12.3437	−0.402394	−0.201197	+	0.979551i	$0.564483\pi$
$942$	0	0
$943$	62.3056	2.02895
$944$	0	0
$945$	−30.5031	−0.992266
$946$	0	0
$947$	35.2461	1.14534	0.572672	−	0.819784i	$-0.305907\pi$
$947$	35.2461	1.14534	0.572672	+	0.819784i	$0.305907\pi$
$948$	0	0
$949$	−19.8677	−0.644933
$950$	0	0
$951$	−61.2239	−1.98532
$952$	0	0
$953$	−32.4559	−1.05135	−0.525675	−	0.850685i	$-0.676187\pi$
$953$	−32.4559	−1.05135	−0.525675	+	0.850685i	$0.676187\pi$
$954$	0	0
$955$	−19.5618	−0.633006
$956$	0	0
$957$	−9.96693	−0.322185
$958$	0	0
$959$	5.37782	0.173659
$960$	0	0
$961$	−28.9008	−0.932283
$962$	0	0
$963$	35.7022	1.15049
$964$	0	0
$965$	13.7213	0.441705
$966$	0	0
$967$	−20.9097	−0.672412	−0.336206	−	0.941788i	$-0.609144\pi$
$967$	−20.9097	−0.672412	−0.336206	+	0.941788i	$0.609144\pi$
$968$	0	0
$969$	−3.89588	−0.125154
$970$	0	0
$971$	−9.88533	−0.317235	−0.158618	−	0.987340i	$-0.550704\pi$
$971$	−9.88533	−0.317235	−0.158618	+	0.987340i	$0.550704\pi$
$972$	0	0
$973$	−9.11031	−0.292063
$974$	0	0
$975$	13.3556	0.427722
$976$	0	0
$977$	15.2795	0.488834	0.244417	−	0.969670i	$-0.421403\pi$
$977$	15.2795	0.488834	0.244417	+	0.969670i	$0.421403\pi$
$978$	0	0
$979$	−48.3869	−1.54645
$980$	0	0
$981$	136.944	4.37228
$982$	0	0
$983$	−38.9157	−1.24122	−0.620610	−	0.784120i	$-0.713115\pi$
$983$	−38.9157	−1.24122	−0.620610	+	0.784120i	$0.713115\pi$
$984$	0	0
$985$	16.6874	0.531706
$986$	0	0
$987$	60.2876	1.91898
$988$	0	0
$989$	47.9517	1.52478
$990$	0	0
$991$	49.7973	1.58186	0.790932	−	0.611905i	$-0.209596\pi$
$991$	49.7973	1.58186	0.790932	+	0.611905i	$0.209596\pi$
$992$	0	0
$993$	88.0651	2.79466
$994$	0	0
$995$	−28.1096	−0.891135
$996$	0	0
$997$	20.8235	0.659488	0.329744	−	0.944070i	$-0.393037\pi$
$997$	20.8235	0.659488	0.329744	+	0.944070i	$0.393037\pi$
$998$	0	0
$999$	10.6667	0.337480

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.2.a.bc.1.6		6
4.3	odd	2	inner	1856.2.a.bc.1.1		6
8.3	odd	2		928.2.a.i.1.6	yes	6
8.5	even	2		928.2.a.i.1.1	✓	6
24.5	odd	2		8352.2.a.bi.1.4		6
24.11	even	2		8352.2.a.bi.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.2.a.i.1.1	✓	6	8.5	even	2
928.2.a.i.1.6	yes	6	8.3	odd	2
1856.2.a.bc.1.1		6	4.3	odd	2	inner
1856.2.a.bc.1.6		6	1.1	even	1	trivial
8352.2.a.bi.1.3		6	24.11	even	2
8352.2.a.bi.1.4		6	24.5	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:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q+3.15704 q^{3} -1.42586 q^{5} +1.70818 q^{7} +6.96693 q^{9} +O(q^{10})\)	\(q+3.15704 q^{3} -1.42586 q^{5} +1.70818 q^{7} +6.96693 q^{9} +3.15704 q^{11} -1.42586 q^{13} -4.50150 q^{15} -3.39279 q^{17} +0.363721 q^{19} +5.39279 q^{21} +7.29482 q^{23} -2.96693 q^{25} +12.5238 q^{27} -1.00000 q^{29} +1.44887 q^{31} +9.96693 q^{33} -2.43562 q^{35} +0.851718 q^{37} -4.50150 q^{39} +8.54107 q^{41} +6.57340 q^{43} -9.93385 q^{45} +11.1793 q^{47} -4.08214 q^{49} -10.7112 q^{51} -13.3597 q^{53} -4.50150 q^{55} +1.14828 q^{57} -10.7112 q^{59} -3.39279 q^{61} +11.9007 q^{63} +2.03307 q^{65} -12.6282 q^{67} +23.0301 q^{69} -5.58665 q^{71} +13.9339 q^{73} -9.36672 q^{75} +5.39279 q^{77} +4.86522 q^{79} +18.6373 q^{81} -5.33335 q^{83} +4.83763 q^{85} -3.15704 q^{87} -15.3266 q^{89} -2.43562 q^{91} +4.57414 q^{93} -0.518614 q^{95} +14.2445 q^{97} +21.9949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{5} + 10 q^{9}+O(q^{10}) \) 6 * q - 4 * q^5 + 10 * q^9	\( 6 q - 4 q^{5} + 10 q^{9} - 4 q^{13} + 16 q^{17} - 4 q^{21} + 14 q^{25} - 6 q^{29} + 28 q^{33} - 4 q^{37} + 24 q^{41} + 4 q^{45} + 30 q^{49} - 12 q^{53} + 16 q^{57} + 16 q^{61} + 44 q^{65} + 20 q^{69} + 20 q^{73} - 4 q^{77} + 30 q^{81} + 20 q^{85} + 8 q^{89} + 32 q^{93} + 40 q^{97}+O(q^{100}) \) 6 * q - 4 * q^5 + 10 * q^9 - 4 * q^13 + 16 * q^17 - 4 * q^21 + 14 * q^25 - 6 * q^29 + 28 * q^33 - 4 * q^37 + 24 * q^41 + 4 * q^45 + 30 * q^49 - 12 * q^53 + 16 * q^57 + 16 * q^61 + 44 * q^65 + 20 * q^69 + 20 * q^73 - 4 * q^77 + 30 * q^81 + 20 * q^85 + 8 * q^89 + 32 * q^93 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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