LMFDB - Embedded newform 10000.2.a.be.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [10000,2,Mod(1,10000)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("10000.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(10000, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 10000 = 2^{4} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	10000.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,-5,0,0,0,-10,0,9,0,-1,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$79.8504020213$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.6152203125.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1 $ x^8 - 3x^7 - 8x^6 + 20x^5 + 26x^4 - 35x^3 - 27x^2 + 16*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 625)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.32675$ of defining polynomial
Character	$\chi$	$=$	10000.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.759083 q^{3} +2.04213 q^{7} -2.42379 q^{9} -1.34867 q^{11} +1.31964 q^{13} +4.08717 q^{17} +4.88680 q^{19} -1.55014 q^{21} +2.73618 q^{23} +4.11711 q^{27} -4.61914 q^{29} -7.15727 q^{31} +1.02375 q^{33} -8.64640 q^{37} -1.00171 q^{39} -10.0996 q^{41} -2.43460 q^{43} -7.57192 q^{47} -2.82971 q^{49} -3.10250 q^{51} -0.621245 q^{53} -3.70948 q^{57} +11.3412 q^{59} +0.647513 q^{61} -4.94970 q^{63} -10.9389 q^{67} -2.07698 q^{69} +2.95230 q^{71} +13.5390 q^{73} -2.75416 q^{77} -1.88929 q^{79} +4.14616 q^{81} -2.37481 q^{83} +3.50631 q^{87} +7.33882 q^{89} +2.69487 q^{91} +5.43296 q^{93} +5.79915 q^{97} +3.26890 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{3} - 10 q^{7} + 9 q^{9} - q^{11} + 10 q^{13} + 15 q^{17} + 10 q^{19} - 14 q^{21} - 30 q^{23} - 20 q^{27} + 10 q^{29} + 9 q^{31} + 5 q^{33} - 10 q^{37} - 8 q^{39} - 4 q^{41} - 30 q^{47} - 4 q^{49}+ \cdots + 22 q^{99}+O(q^{100}) $ 8 * q - 5 * q^3 - 10 * q^7 + 9 * q^9 - q^11 + 10 * q^13 + 15 * q^17 + 10 * q^19 - 14 * q^21 - 30 * q^23 - 20 * q^27 + 10 * q^29 + 9 * q^31 + 5 * q^33 - 10 * q^37 - 8 * q^39 - 4 * q^41 - 30 * q^47 - 4 * q^49 + 14 * q^51 + 10 * q^53 - 10 * q^57 + 5 * q^59 + 6 * q^61 - 10 * q^67 + 3 * q^69 + 9 * q^71 + 5 * q^77 + 20 * q^79 + 8 * q^81 - 40 * q^83 - 40 * q^87 - 5 * q^89 - 6 * q^91 - 40 * q^93 + 22 * 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
$3$	−0.759083	−0.438257	−0.219128	−	0.975696i	$-0.570321\pi$
$3$	−0.759083	−0.438257	−0.219128	+	0.975696i	$0.570321\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.04213	0.771852	0.385926	−	0.922530i	$-0.373882\pi$
$7$	2.04213	0.771852	0.385926	+	0.922530i	$0.373882\pi$
$8$	0	0
$9$	−2.42379	−0.807931
$10$	0	0
$11$	−1.34867	−0.406640	−0.203320	−	0.979112i	$-0.565173\pi$
$11$	−1.34867	−0.406640	−0.203320	+	0.979112i	$0.565173\pi$
$12$	0	0
$13$	1.31964	0.366002	0.183001	−	0.983113i	$-0.441419\pi$
$13$	1.31964	0.366002	0.183001	+	0.983113i	$0.441419\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.08717	0.991285	0.495643	−	0.868527i	$-0.334933\pi$
$17$	4.08717	0.991285	0.495643	+	0.868527i	$0.334933\pi$
$18$	0	0
$19$	4.88680	1.12111	0.560554	−	0.828118i	$-0.310588\pi$
$19$	4.88680	1.12111	0.560554	+	0.828118i	$0.310588\pi$
$20$	0	0
$21$	−1.55014	−0.338269
$22$	0	0
$23$	2.73618	0.570532	0.285266	−	0.958448i	$-0.407918\pi$
$23$	2.73618	0.570532	0.285266	+	0.958448i	$0.407918\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.11711	0.792338
$28$	0	0
$29$	−4.61914	−0.857753	−0.428876	−	0.903363i	$-0.641090\pi$
$29$	−4.61914	−0.857753	−0.428876	+	0.903363i	$0.641090\pi$
$30$	0	0
$31$	−7.15727	−1.28548	−0.642742	−	0.766083i	$-0.722203\pi$
$31$	−7.15727	−1.28548	−0.642742	+	0.766083i	$0.722203\pi$
$32$	0	0
$33$	1.02375	0.178213
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.64640	−1.42146	−0.710730	−	0.703465i	$-0.751635\pi$
$37$	−8.64640	−1.42146	−0.710730	+	0.703465i	$0.751635\pi$
$38$	0	0
$39$	−1.00171	−0.160403
$40$	0	0
$41$	−10.0996	−1.57729	−0.788645	−	0.614849i	$-0.789217\pi$
$41$	−10.0996	−1.57729	−0.788645	+	0.614849i	$0.789217\pi$
$42$	0	0
$43$	−2.43460	−0.371272	−0.185636	−	0.982619i	$-0.559435\pi$
$43$	−2.43460	−0.371272	−0.185636	+	0.982619i	$0.559435\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.57192	−1.10448	−0.552239	−	0.833686i	$-0.686226\pi$
$47$	−7.57192	−1.10448	−0.552239	+	0.833686i	$0.686226\pi$
$48$	0	0
$49$	−2.82971	−0.404244
$50$	0	0
$51$	−3.10250	−0.434437
$52$	0	0
$53$	−0.621245	−0.0853345	−0.0426673	−	0.999089i	$-0.513586\pi$
$53$	−0.621245	−0.0853345	−0.0426673	+	0.999089i	$0.513586\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.70948	−0.491333
$58$	0	0
$59$	11.3412	1.47650	0.738248	−	0.674530i	$-0.235653\pi$
$59$	11.3412	1.47650	0.738248	+	0.674530i	$0.235653\pi$
$60$	0	0
$61$	0.647513	0.0829056	0.0414528	−	0.999140i	$-0.486801\pi$
$61$	0.647513	0.0829056	0.0414528	+	0.999140i	$0.486801\pi$
$62$	0	0
$63$	−4.94970	−0.623603
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.9389	−1.33639	−0.668197	−	0.743984i	$-0.732934\pi$
$67$	−10.9389	−1.33639	−0.668197	+	0.743984i	$0.732934\pi$
$68$	0	0
$69$	−2.07698	−0.250039
$70$	0	0
$71$	2.95230	0.350374	0.175187	−	0.984535i	$-0.443947\pi$
$71$	2.95230	0.350374	0.175187	+	0.984535i	$0.443947\pi$
$72$	0	0
$73$	13.5390	1.58462	0.792309	−	0.610120i	$-0.208879\pi$
$73$	13.5390	1.58462	0.792309	+	0.610120i	$0.208879\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.75416	−0.313866
$78$	0	0
$79$	−1.88929	−0.212562	−0.106281	−	0.994336i	$-0.533894\pi$
$79$	−1.88929	−0.212562	−0.106281	+	0.994336i	$0.533894\pi$
$80$	0	0
$81$	4.14616	0.460684
$82$	0	0
$83$	−2.37481	−0.260669	−0.130335	−	0.991470i	$-0.541605\pi$
$83$	−2.37481	−0.260669	−0.130335	+	0.991470i	$0.541605\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.50631	0.375916
$88$	0	0
$89$	7.33882	0.777913	0.388957	−	0.921256i	$-0.372836\pi$
$89$	7.33882	0.777913	0.388957	+	0.921256i	$0.372836\pi$
$90$	0	0
$91$	2.69487	0.282499
$92$	0	0
$93$	5.43296	0.563371
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.79915	0.588814	0.294407	−	0.955680i	$-0.404878\pi$
$97$	5.79915	0.588814	0.294407	+	0.955680i	$0.404878\pi$
$98$	0	0
$99$	3.26890	0.328537
$100$	0	0
$101$	11.5536	1.14962	0.574812	−	0.818285i	$-0.305075\pi$
$101$	11.5536	1.14962	0.574812	+	0.818285i	$0.305075\pi$
$102$	0	0
$103$	−11.6071	−1.14368	−0.571842	−	0.820364i	$-0.693771\pi$
$103$	−11.6071	−1.14368	−0.571842	+	0.820364i	$0.693771\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.1703	−0.983204	−0.491602	−	0.870820i	$-0.663588\pi$
$107$	−10.1703	−0.983204	−0.491602	+	0.870820i	$0.663588\pi$
$108$	0	0
$109$	1.19475	0.114436	0.0572182	−	0.998362i	$-0.481777\pi$
$109$	1.19475	0.114436	0.0572182	+	0.998362i	$0.481777\pi$
$110$	0	0
$111$	6.56333	0.622964
$112$	0	0
$113$	4.48257	0.421685	0.210842	−	0.977520i	$-0.432379\pi$
$113$	4.48257	0.421685	0.210842	+	0.977520i	$0.432379\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.19853	−0.295704
$118$	0	0
$119$	8.34653	0.765125
$120$	0	0
$121$	−9.18108	−0.834644
$122$	0	0
$123$	7.66642	0.691258
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.17685	0.370635	0.185318	−	0.982679i	$-0.440669\pi$
$127$	4.17685	0.370635	0.185318	+	0.982679i	$0.440669\pi$
$128$	0	0
$129$	1.84806	0.162713
$130$	0	0
$131$	−3.38914	−0.296111	−0.148055	−	0.988979i	$-0.547301\pi$
$131$	−3.38914	−0.296111	−0.148055	+	0.988979i	$0.547301\pi$
$132$	0	0
$133$	9.97947	0.865330
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.33597	−0.712190	−0.356095	−	0.934450i	$-0.615892\pi$
$137$	−8.33597	−0.712190	−0.356095	+	0.934450i	$0.615892\pi$
$138$	0	0
$139$	3.43355	0.291230	0.145615	−	0.989341i	$-0.453484\pi$
$139$	3.43355	0.291230	0.145615	+	0.989341i	$0.453484\pi$
$140$	0	0
$141$	5.74771	0.484045
$142$	0	0
$143$	−1.77976	−0.148831
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.14798	0.177163
$148$	0	0
$149$	9.96023	0.815974	0.407987	−	0.912988i	$-0.366231\pi$
$149$	9.96023	0.815974	0.407987	+	0.912988i	$0.366231\pi$
$150$	0	0
$151$	21.0404	1.71225	0.856123	−	0.516772i	$-0.172866\pi$
$151$	21.0404	1.71225	0.856123	+	0.516772i	$0.172866\pi$
$152$	0	0
$153$	−9.90646	−0.800890
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.80843	0.623181	0.311590	−	0.950217i	$-0.399138\pi$
$157$	7.80843	0.623181	0.311590	+	0.950217i	$0.399138\pi$
$158$	0	0
$159$	0.471576	0.0373984
$160$	0	0
$161$	5.58762	0.440366
$162$	0	0
$163$	11.6785	0.914734	0.457367	−	0.889278i	$-0.348793\pi$
$163$	11.6785	0.914734	0.457367	+	0.889278i	$0.348793\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.51198	−0.581294	−0.290647	−	0.956830i	$-0.593871\pi$
$167$	−7.51198	−0.581294	−0.290647	+	0.956830i	$0.593871\pi$
$168$	0	0
$169$	−11.2586	−0.866043
$170$	0	0
$171$	−11.8446	−0.905779
$172$	0	0
$173$	5.31381	0.404002	0.202001	−	0.979385i	$-0.435256\pi$
$173$	5.31381	0.404002	0.202001	+	0.979385i	$0.435256\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−8.60889	−0.647084
$178$	0	0
$179$	−1.30674	−0.0976705	−0.0488352	−	0.998807i	$-0.515551\pi$
$179$	−1.30674	−0.0976705	−0.0488352	+	0.998807i	$0.515551\pi$
$180$	0	0
$181$	−20.1627	−1.49868	−0.749341	−	0.662184i	$-0.769630\pi$
$181$	−20.1627	−1.49868	−0.749341	+	0.662184i	$0.769630\pi$
$182$	0	0
$183$	−0.491516	−0.0363339
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−5.51225	−0.403096
$188$	0	0
$189$	8.40766	0.611568
$190$	0	0
$191$	−21.2957	−1.54090	−0.770452	−	0.637498i	$-0.779970\pi$
$191$	−21.2957	−1.54090	−0.770452	+	0.637498i	$0.779970\pi$
$192$	0	0
$193$	−7.99352	−0.575386	−0.287693	−	0.957723i	$-0.592888\pi$
$193$	−7.99352	−0.575386	−0.287693	+	0.957723i	$0.592888\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−21.7014	−1.54616	−0.773081	−	0.634307i	$-0.781285\pi$
$197$	−21.7014	−1.54616	−0.773081	+	0.634307i	$0.781285\pi$
$198$	0	0
$199$	−9.34240	−0.662265	−0.331133	−	0.943584i	$-0.607431\pi$
$199$	−9.34240	−0.662265	−0.331133	+	0.943584i	$0.607431\pi$
$200$	0	0
$201$	8.30350	0.585684
$202$	0	0
$203$	−9.43288	−0.662058
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−6.63192	−0.460951
$208$	0	0
$209$	−6.59069	−0.455887
$210$	0	0
$211$	6.88147	0.473740	0.236870	−	0.971541i	$-0.423878\pi$
$211$	6.88147	0.473740	0.236870	+	0.971541i	$0.423878\pi$
$212$	0	0
$213$	−2.24104	−0.153554
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−14.6161	−0.992203
$218$	0	0
$219$	−10.2772	−0.694469
$220$	0	0
$221$	5.39359	0.362812
$222$	0	0
$223$	−8.51496	−0.570204	−0.285102	−	0.958497i	$-0.592028\pi$
$223$	−8.51496	−0.570204	−0.285102	+	0.958497i	$0.592028\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.9836	−1.39273	−0.696364	−	0.717689i	$-0.745200\pi$
$227$	−20.9836	−1.39273	−0.696364	+	0.717689i	$0.745200\pi$
$228$	0	0
$229$	29.7629	1.96679	0.983394	−	0.181485i	$-0.0580904\pi$
$229$	29.7629	1.96679	0.983394	+	0.181485i	$0.0580904\pi$
$230$	0	0
$231$	2.09064	0.137554
$232$	0	0
$233$	−2.27154	−0.148813	−0.0744066	−	0.997228i	$-0.523706\pi$
$233$	−2.27154	−0.148813	−0.0744066	+	0.997228i	$0.523706\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.43413	0.0931568
$238$	0	0
$239$	−15.4308	−0.998135	−0.499067	−	0.866563i	$-0.666324\pi$
$239$	−15.4308	−0.998135	−0.499067	+	0.866563i	$0.666324\pi$
$240$	0	0
$241$	−4.84184	−0.311890	−0.155945	−	0.987766i	$-0.549842\pi$
$241$	−4.84184	−0.311890	−0.155945	+	0.987766i	$0.549842\pi$
$242$	0	0
$243$	−15.4986	−0.994235
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.44880	0.410328
$248$	0	0
$249$	1.80268	0.114240
$250$	0	0
$251$	17.8293	1.12537	0.562687	−	0.826670i	$-0.309768\pi$
$251$	17.8293	1.12537	0.562687	+	0.826670i	$0.309768\pi$
$252$	0	0
$253$	−3.69020	−0.232001
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.88929	0.367364	0.183682	−	0.982986i	$-0.441198\pi$
$257$	5.88929	0.367364	0.183682	+	0.982986i	$0.441198\pi$
$258$	0	0
$259$	−17.6571	−1.09716
$260$	0	0
$261$	11.1958	0.693005
$262$	0	0
$263$	−23.6773	−1.46000	−0.730002	−	0.683445i	$-0.760481\pi$
$263$	−23.6773	−1.46000	−0.730002	+	0.683445i	$0.760481\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−5.57077	−0.340926
$268$	0	0
$269$	13.7460	0.838111	0.419056	−	0.907961i	$-0.362361\pi$
$269$	13.7460	0.838111	0.419056	+	0.907961i	$0.362361\pi$
$270$	0	0
$271$	−7.49882	−0.455521	−0.227760	−	0.973717i	$-0.573140\pi$
$271$	−7.49882	−0.455521	−0.227760	+	0.973717i	$0.573140\pi$
$272$	0	0
$273$	−2.04563	−0.123807
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.83798	0.170517	0.0852587	−	0.996359i	$-0.472828\pi$
$277$	2.83798	0.170517	0.0852587	+	0.996359i	$0.472828\pi$
$278$	0	0
$279$	17.3477	1.03858
$280$	0	0
$281$	−22.9609	−1.36974	−0.684868	−	0.728667i	$-0.740140\pi$
$281$	−22.9609	−1.36974	−0.684868	+	0.728667i	$0.740140\pi$
$282$	0	0
$283$	25.1247	1.49351	0.746753	−	0.665101i	$-0.231612\pi$
$283$	25.1247	1.49351	0.746753	+	0.665101i	$0.231612\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−20.6246	−1.21743
$288$	0	0
$289$	−0.295018	−0.0173540
$290$	0	0
$291$	−4.40203	−0.258052
$292$	0	0
$293$	−28.8755	−1.68692	−0.843461	−	0.537190i	$-0.819486\pi$
$293$	−28.8755	−1.68692	−0.843461	+	0.537190i	$0.819486\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.55263	−0.322196
$298$	0	0
$299$	3.61076	0.208816
$300$	0	0
$301$	−4.97176	−0.286567
$302$	0	0
$303$	−8.77012	−0.503830
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−23.9526	−1.36704	−0.683522	−	0.729930i	$-0.739553\pi$
$307$	−23.9526	−1.36704	−0.683522	+	0.729930i	$0.739553\pi$
$308$	0	0
$309$	8.81077	0.501227
$310$	0	0
$311$	9.88835	0.560717	0.280358	−	0.959895i	$-0.409547\pi$
$311$	9.88835	0.560717	0.280358	+	0.959895i	$0.409547\pi$
$312$	0	0
$313$	−18.7786	−1.06143	−0.530713	−	0.847551i	$-0.678076\pi$
$313$	−18.7786	−1.06143	−0.530713	+	0.847551i	$0.678076\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−25.8818	−1.45367	−0.726834	−	0.686813i	$-0.759009\pi$
$317$	−25.8818	−1.45367	−0.726834	+	0.686813i	$0.759009\pi$
$318$	0	0
$319$	6.22970	0.348796
$320$	0	0
$321$	7.72013	0.430895
$322$	0	0
$323$	19.9732	1.11134
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−0.906914	−0.0501525
$328$	0	0
$329$	−15.4628	−0.852494
$330$	0	0
$331$	−1.29743	−0.0713134	−0.0356567	−	0.999364i	$-0.511352\pi$
$331$	−1.29743	−0.0713134	−0.0356567	+	0.999364i	$0.511352\pi$
$332$	0	0
$333$	20.9571	1.14844
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−11.2769	−0.614293	−0.307147	−	0.951662i	$-0.599374\pi$
$337$	−11.2769	−0.614293	−0.307147	+	0.951662i	$0.599374\pi$
$338$	0	0
$339$	−3.40264	−0.184806
$340$	0	0
$341$	9.65281	0.522729
$342$	0	0
$343$	−20.0735	−1.08387
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.15501	−0.115687	−0.0578434	−	0.998326i	$-0.518422\pi$
$347$	−2.15501	−0.115687	−0.0578434	+	0.998326i	$0.518422\pi$
$348$	0	0
$349$	5.60904	0.300245	0.150122	−	0.988667i	$-0.452033\pi$
$349$	5.60904	0.300245	0.150122	+	0.988667i	$0.452033\pi$
$350$	0	0
$351$	5.43309	0.289997
$352$	0	0
$353$	−18.4969	−0.984492	−0.492246	−	0.870456i	$-0.663824\pi$
$353$	−18.4969	−0.984492	−0.492246	+	0.870456i	$0.663824\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−6.33571	−0.335321
$358$	0	0
$359$	−11.2450	−0.593487	−0.296743	−	0.954957i	$-0.595901\pi$
$359$	−11.2450	−0.593487	−0.296743	+	0.954957i	$0.595901\pi$
$360$	0	0
$361$	4.88081	0.256885
$362$	0	0
$363$	6.96920	0.365788
$364$	0	0
$365$	0	0
$366$	0	0
$367$	28.5677	1.49122	0.745610	−	0.666383i	$-0.232158\pi$
$367$	28.5677	1.49122	0.745610	+	0.666383i	$0.232158\pi$
$368$	0	0
$369$	24.4793	1.27434
$370$	0	0
$371$	−1.26866	−0.0658656
$372$	0	0
$373$	14.0848	0.729284	0.364642	−	0.931148i	$-0.381191\pi$
$373$	14.0848	0.729284	0.364642	+	0.931148i	$0.381191\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.09559	−0.313939
$378$	0	0
$379$	−21.9903	−1.12957	−0.564783	−	0.825239i	$-0.691040\pi$
$379$	−21.9903	−1.12957	−0.564783	+	0.825239i	$0.691040\pi$
$380$	0	0
$381$	−3.17057	−0.162433
$382$	0	0
$383$	14.7436	0.753363	0.376682	−	0.926343i	$-0.377065\pi$
$383$	14.7436	0.753363	0.376682	+	0.926343i	$0.377065\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.90096	0.299963
$388$	0	0
$389$	−14.2568	−0.722847	−0.361423	−	0.932402i	$-0.617709\pi$
$389$	−14.2568	−0.722847	−0.361423	+	0.932402i	$0.617709\pi$
$390$	0	0
$391$	11.1832	0.565560
$392$	0	0
$393$	2.57264	0.129772
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−5.49065	−0.275568	−0.137784	−	0.990462i	$-0.543998\pi$
$397$	−5.49065	−0.275568	−0.137784	+	0.990462i	$0.543998\pi$
$398$	0	0
$399$	−7.57525	−0.379237
$400$	0	0
$401$	26.7528	1.33597	0.667985	−	0.744175i	$-0.267157\pi$
$401$	26.7528	1.33597	0.667985	+	0.744175i	$0.267157\pi$
$402$	0	0
$403$	−9.44500	−0.470489
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.6612	0.578022
$408$	0	0
$409$	5.13389	0.253854	0.126927	−	0.991912i	$-0.459489\pi$
$409$	5.13389	0.253854	0.126927	+	0.991912i	$0.459489\pi$
$410$	0	0
$411$	6.32769	0.312122
$412$	0	0
$413$	23.1601	1.13964
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.60635	−0.127633
$418$	0	0
$419$	−31.8516	−1.55605	−0.778026	−	0.628231i	$-0.783779\pi$
$419$	−31.8516	−1.55605	−0.778026	+	0.628231i	$0.783779\pi$
$420$	0	0
$421$	3.08044	0.150132	0.0750658	−	0.997179i	$-0.476083\pi$
$421$	3.08044	0.150132	0.0750658	+	0.997179i	$0.476083\pi$
$422$	0	0
$423$	18.3528	0.892343
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.32231	0.0639908
$428$	0	0
$429$	1.35098	0.0652261
$430$	0	0
$431$	−23.3471	−1.12459	−0.562295	−	0.826937i	$-0.690081\pi$
$431$	−23.3471	−1.12459	−0.562295	+	0.826937i	$0.690081\pi$
$432$	0	0
$433$	8.21415	0.394747	0.197374	−	0.980328i	$-0.436759\pi$
$433$	8.21415	0.394747	0.197374	+	0.980328i	$0.436759\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	13.3711	0.639628
$438$	0	0
$439$	24.3117	1.16034	0.580168	−	0.814497i	$-0.302987\pi$
$439$	24.3117	1.16034	0.580168	+	0.814497i	$0.302987\pi$
$440$	0	0
$441$	6.85863	0.326602
$442$	0	0
$443$	−0.0631363	−0.00299970	−0.00149985	−	0.999999i	$-0.500477\pi$
$443$	−0.0631363	−0.00299970	−0.00149985	+	0.999999i	$0.500477\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−7.56064	−0.357606
$448$	0	0
$449$	−11.5711	−0.546074	−0.273037	−	0.962004i	$-0.588028\pi$
$449$	−11.5711	−0.546074	−0.273037	+	0.962004i	$0.588028\pi$
$450$	0	0
$451$	13.6210	0.641389
$452$	0	0
$453$	−15.9714	−0.750403
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−41.3967	−1.93646	−0.968228	−	0.250071i	$-0.919546\pi$
$457$	−41.3967	−1.93646	−0.968228	+	0.250071i	$0.919546\pi$
$458$	0	0
$459$	16.8273	0.785432
$460$	0	0
$461$	29.3707	1.36793	0.683965	−	0.729515i	$-0.260254\pi$
$461$	29.3707	1.36793	0.683965	+	0.729515i	$0.260254\pi$
$462$	0	0
$463$	30.5501	1.41978	0.709891	−	0.704311i	$-0.248744\pi$
$463$	30.5501	1.41978	0.709891	+	0.704311i	$0.248744\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−32.7149	−1.51386	−0.756932	−	0.653494i	$-0.773303\pi$
$467$	−32.7149	−1.51386	−0.756932	+	0.653494i	$0.773303\pi$
$468$	0	0
$469$	−22.3386	−1.03150
$470$	0	0
$471$	−5.92724	−0.273113
$472$	0	0
$473$	3.28347	0.150974
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.50577	0.0689444
$478$	0	0
$479$	12.3353	0.563613	0.281807	−	0.959471i	$-0.409066\pi$
$479$	12.3353	0.563613	0.281807	+	0.959471i	$0.409066\pi$
$480$	0	0
$481$	−11.4101	−0.520256
$482$	0	0
$483$	−4.24147	−0.192993
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−35.2962	−1.59942	−0.799711	−	0.600385i	$-0.795014\pi$
$487$	−35.2962	−1.59942	−0.799711	+	0.600385i	$0.795014\pi$
$488$	0	0
$489$	−8.86498	−0.400888
$490$	0	0
$491$	−15.3156	−0.691183	−0.345591	−	0.938385i	$-0.612322\pi$
$491$	−15.3156	−0.691183	−0.345591	+	0.938385i	$0.612322\pi$
$492$	0	0
$493$	−18.8792	−0.850278
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.02898	0.270437
$498$	0	0
$499$	35.1777	1.57477	0.787386	−	0.616461i	$-0.211434\pi$
$499$	35.1777	1.57477	0.787386	+	0.616461i	$0.211434\pi$
$500$	0	0
$501$	5.70221	0.254756
$502$	0	0
$503$	−7.33494	−0.327049	−0.163524	−	0.986539i	$-0.552286\pi$
$503$	−7.33494	−0.327049	−0.163524	+	0.986539i	$0.552286\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8.54617	0.379549
$508$	0	0
$509$	5.70697	0.252957	0.126479	−	0.991969i	$-0.459633\pi$
$509$	5.70697	0.252957	0.126479	+	0.991969i	$0.459633\pi$
$510$	0	0
$511$	27.6484	1.22309
$512$	0	0
$513$	20.1195	0.888297
$514$	0	0
$515$	0	0
$516$	0	0
$517$	10.2120	0.449125
$518$	0	0
$519$	−4.03362	−0.177056
$520$	0	0
$521$	−33.0901	−1.44970	−0.724852	−	0.688905i	$-0.758092\pi$
$521$	−33.0901	−1.44970	−0.724852	+	0.688905i	$0.758092\pi$
$522$	0	0
$523$	31.4719	1.37617	0.688086	−	0.725629i	$-0.258451\pi$
$523$	31.4719	1.37617	0.688086	+	0.725629i	$0.258451\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−29.2530	−1.27428
$528$	0	0
$529$	−15.5133	−0.674493
$530$	0	0
$531$	−27.4887	−1.19291
$532$	0	0
$533$	−13.3278	−0.577291
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0.991925	0.0428047
$538$	0	0
$539$	3.81635	0.164382
$540$	0	0
$541$	−22.8759	−0.983512	−0.491756	−	0.870733i	$-0.663645\pi$
$541$	−22.8759	−0.983512	−0.491756	+	0.870733i	$0.663645\pi$
$542$	0	0
$543$	15.3052	0.656808
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−17.7548	−0.759139	−0.379570	−	0.925163i	$-0.623928\pi$
$547$	−17.7548	−0.759139	−0.379570	+	0.925163i	$0.623928\pi$
$548$	0	0
$549$	−1.56944	−0.0669820
$550$	0	0
$551$	−22.5728	−0.961634
$552$	0	0
$553$	−3.85818	−0.164067
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.27448	−0.0963727	−0.0481864	−	0.998838i	$-0.515344\pi$
$557$	−2.27448	−0.0963727	−0.0481864	+	0.998838i	$0.515344\pi$
$558$	0	0
$559$	−3.21278	−0.135886
$560$	0	0
$561$	4.18426	0.176659
$562$	0	0
$563$	0.397699	0.0167610	0.00838050	−	0.999965i	$-0.497332\pi$
$563$	0.397699	0.0167610	0.00838050	+	0.999965i	$0.497332\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	8.46699	0.355580
$568$	0	0
$569$	−11.1876	−0.469010	−0.234505	−	0.972115i	$-0.575347\pi$
$569$	−11.1876	−0.469010	−0.234505	+	0.972115i	$0.575347\pi$
$570$	0	0
$571$	18.9124	0.791458	0.395729	−	0.918367i	$-0.370492\pi$
$571$	18.9124	0.791458	0.395729	+	0.918367i	$0.370492\pi$
$572$	0	0
$573$	16.1652	0.675312
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−6.01809	−0.250537	−0.125268	−	0.992123i	$-0.539979\pi$
$577$	−6.01809	−0.250537	−0.125268	+	0.992123i	$0.539979\pi$
$578$	0	0
$579$	6.06774	0.252167
$580$	0	0
$581$	−4.84967	−0.201198
$582$	0	0
$583$	0.837855	0.0347004
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.3326	0.674118	0.337059	−	0.941483i	$-0.390568\pi$
$587$	16.3326	0.674118	0.337059	+	0.941483i	$0.390568\pi$
$588$	0	0
$589$	−34.9761	−1.44117
$590$	0	0
$591$	16.4732	0.677615
$592$	0	0
$593$	8.40604	0.345195	0.172597	−	0.984992i	$-0.444784\pi$
$593$	8.40604	0.345195	0.172597	+	0.984992i	$0.444784\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.09165	0.290242
$598$	0	0
$599$	−13.1918	−0.539001	−0.269501	−	0.963000i	$-0.586859\pi$
$599$	−13.1918	−0.539001	−0.269501	+	0.963000i	$0.586859\pi$
$600$	0	0
$601$	−6.06690	−0.247474	−0.123737	−	0.992315i	$-0.539488\pi$
$601$	−6.06690	−0.247474	−0.123737	+	0.992315i	$0.539488\pi$
$602$	0	0
$603$	26.5135	1.07971
$604$	0	0
$605$	0	0
$606$	0	0
$607$	14.9141	0.605345	0.302672	−	0.953095i	$-0.402121\pi$
$607$	14.9141	0.605345	0.302672	+	0.953095i	$0.402121\pi$
$608$	0	0
$609$	7.16034	0.290151
$610$	0	0
$611$	−9.99219	−0.404241
$612$	0	0
$613$	−36.4772	−1.47330	−0.736651	−	0.676273i	$-0.763594\pi$
$613$	−36.4772	−1.47330	−0.736651	+	0.676273i	$0.763594\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	16.2341	0.653559	0.326780	−	0.945101i	$-0.394036\pi$
$617$	16.2341	0.653559	0.326780	+	0.945101i	$0.394036\pi$
$618$	0	0
$619$	10.0360	0.403382	0.201691	−	0.979449i	$-0.435356\pi$
$619$	10.0360	0.403382	0.201691	+	0.979449i	$0.435356\pi$
$620$	0	0
$621$	11.2651	0.452054
$622$	0	0
$623$	14.9868	0.600434
$624$	0	0
$625$	0	0
$626$	0	0
$627$	5.00288	0.199796
$628$	0	0
$629$	−35.3393	−1.40907
$630$	0	0
$631$	−28.7580	−1.14484	−0.572420	−	0.819961i	$-0.693995\pi$
$631$	−28.7580	−1.14484	−0.572420	+	0.819961i	$0.693995\pi$
$632$	0	0
$633$	−5.22361	−0.207620
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−3.73419	−0.147954
$638$	0	0
$639$	−7.15577	−0.283078
$640$	0	0
$641$	−18.2437	−0.720582	−0.360291	−	0.932840i	$-0.617323\pi$
$641$	−18.2437	−0.720582	−0.360291	+	0.932840i	$0.617323\pi$
$642$	0	0
$643$	33.9734	1.33978	0.669890	−	0.742461i	$-0.266341\pi$
$643$	33.9734	1.33978	0.669890	+	0.742461i	$0.266341\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−47.3728	−1.86242	−0.931208	−	0.364488i	$-0.881244\pi$
$647$	−47.3728	−1.86242	−0.931208	+	0.364488i	$0.881244\pi$
$648$	0	0
$649$	−15.2955	−0.600402
$650$	0	0
$651$	11.0948	0.434840
$652$	0	0
$653$	−37.3959	−1.46342	−0.731708	−	0.681618i	$-0.761277\pi$
$653$	−37.3959	−1.46342	−0.731708	+	0.681618i	$0.761277\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−32.8157	−1.28026
$658$	0	0
$659$	9.48154	0.369348	0.184674	−	0.982800i	$-0.440877\pi$
$659$	9.48154	0.369348	0.184674	+	0.982800i	$0.440877\pi$
$660$	0	0
$661$	−8.78089	−0.341537	−0.170768	−	0.985311i	$-0.554625\pi$
$661$	−8.78089	−0.341537	−0.170768	+	0.985311i	$0.554625\pi$
$662$	0	0
$663$	−4.09418	−0.159005
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−12.6388	−0.489375
$668$	0	0
$669$	6.46356	0.249896
$670$	0	0
$671$	−0.873283	−0.0337127
$672$	0	0
$673$	48.0259	1.85126	0.925631	−	0.378427i	$-0.123535\pi$
$673$	48.0259	1.85126	0.925631	+	0.378427i	$0.123535\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−15.4151	−0.592451	−0.296226	−	0.955118i	$-0.595728\pi$
$677$	−15.4151	−0.592451	−0.296226	+	0.955118i	$0.595728\pi$
$678$	0	0
$679$	11.8426	0.454478
$680$	0	0
$681$	15.9283	0.610372
$682$	0	0
$683$	27.8221	1.06458	0.532292	−	0.846561i	$-0.321331\pi$
$683$	27.8221	1.06458	0.532292	+	0.846561i	$0.321331\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−22.5925	−0.861957
$688$	0	0
$689$	−0.819818	−0.0312326
$690$	0	0
$691$	20.1261	0.765634	0.382817	−	0.923824i	$-0.374954\pi$
$691$	20.1261	0.765634	0.382817	+	0.923824i	$0.374954\pi$
$692$	0	0
$693$	6.67552	0.253582
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−41.2787	−1.56354
$698$	0	0
$699$	1.72428	0.0652184
$700$	0	0
$701$	31.6216	1.19433	0.597166	−	0.802118i	$-0.296293\pi$
$701$	31.6216	1.19433	0.597166	+	0.802118i	$0.296293\pi$
$702$	0	0
$703$	−42.2532	−1.59361
$704$	0	0
$705$	0	0
$706$	0	0
$707$	23.5939	0.887340
$708$	0	0
$709$	−4.76008	−0.178768	−0.0893842	−	0.995997i	$-0.528490\pi$
$709$	−4.76008	−0.178768	−0.0893842	+	0.995997i	$0.528490\pi$
$710$	0	0
$711$	4.57926	0.171736
$712$	0	0
$713$	−19.5835	−0.733409
$714$	0	0
$715$	0	0
$716$	0	0
$717$	11.7132	0.437439
$718$	0	0
$719$	23.0727	0.860467	0.430234	−	0.902718i	$-0.358431\pi$
$719$	23.0727	0.860467	0.430234	+	0.902718i	$0.358431\pi$
$720$	0	0
$721$	−23.7033	−0.882755
$722$	0	0
$723$	3.67535	0.136688
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−15.5807	−0.577856	−0.288928	−	0.957351i	$-0.593299\pi$
$727$	−15.5807	−0.577856	−0.288928	+	0.957351i	$0.593299\pi$
$728$	0	0
$729$	−0.673755	−0.0249539
$730$	0	0
$731$	−9.95061	−0.368037
$732$	0	0
$733$	17.2215	0.636092	0.318046	−	0.948075i	$-0.396973\pi$
$733$	17.2215	0.636092	0.318046	+	0.948075i	$0.396973\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14.7529	0.543431
$738$	0	0
$739$	27.4878	1.01116	0.505578	−	0.862781i	$-0.331279\pi$
$739$	27.4878	1.01116	0.505578	+	0.862781i	$0.331279\pi$
$740$	0	0
$741$	−4.89517	−0.179829
$742$	0	0
$743$	−48.4801	−1.77856	−0.889280	−	0.457362i	$-0.848794\pi$
$743$	−48.4801	−1.77856	−0.889280	+	0.457362i	$0.848794\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	5.75605	0.210603
$748$	0	0
$749$	−20.7691	−0.758888
$750$	0	0
$751$	−3.29720	−0.120316	−0.0601582	−	0.998189i	$-0.519161\pi$
$751$	−3.29720	−0.120316	−0.0601582	+	0.998189i	$0.519161\pi$
$752$	0	0
$753$	−13.5339	−0.493202
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−35.7934	−1.30093	−0.650466	−	0.759535i	$-0.725426\pi$
$757$	−35.7934	−1.30093	−0.650466	+	0.759535i	$0.725426\pi$
$758$	0	0
$759$	2.80117	0.101676
$760$	0	0
$761$	−0.664110	−0.0240740	−0.0120370	−	0.999928i	$-0.503832\pi$
$761$	−0.664110	−0.0240740	−0.0120370	+	0.999928i	$0.503832\pi$
$762$	0	0
$763$	2.43983	0.0883279
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14.9662	0.540400
$768$	0	0
$769$	25.7090	0.927091	0.463545	−	0.886073i	$-0.346577\pi$
$769$	25.7090	0.927091	0.463545	+	0.886073i	$0.346577\pi$
$770$	0	0
$771$	−4.47046	−0.161000
$772$	0	0
$773$	−38.5458	−1.38640	−0.693199	−	0.720746i	$-0.743799\pi$
$773$	−38.5458	−1.38640	−0.693199	+	0.720746i	$0.743799\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	13.4032	0.480836
$778$	0	0
$779$	−49.3546	−1.76831
$780$	0	0
$781$	−3.98169	−0.142476
$782$	0	0
$783$	−19.0175	−0.679630
$784$	0	0
$785$	0	0
$786$	0	0
$787$	36.2312	1.29150	0.645751	−	0.763548i	$-0.276544\pi$
$787$	36.2312	1.29150	0.645751	+	0.763548i	$0.276544\pi$
$788$	0	0
$789$	17.9730	0.639856
$790$	0	0
$791$	9.15398	0.325478
$792$	0	0
$793$	0.854482	0.0303436
$794$	0	0
$795$	0	0
$796$	0	0
$797$	33.8276	1.19823	0.599117	−	0.800662i	$-0.295518\pi$
$797$	33.8276	1.19823	0.599117	+	0.800662i	$0.295518\pi$
$798$	0	0
$799$	−30.9478	−1.09485
$800$	0	0
$801$	−17.7878	−0.628501
$802$	0	0
$803$	−18.2596	−0.644369
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−10.4344	−0.367308
$808$	0	0
$809$	−1.46409	−0.0514747	−0.0257374	−	0.999669i	$-0.508193\pi$
$809$	−1.46409	−0.0514747	−0.0257374	+	0.999669i	$0.508193\pi$
$810$	0	0
$811$	40.6885	1.42877	0.714383	−	0.699755i	$-0.246707\pi$
$811$	40.6885	1.42877	0.714383	+	0.699755i	$0.246707\pi$
$812$	0	0
$813$	5.69222	0.199635
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−11.8974	−0.416237
$818$	0	0
$819$	−6.53181	−0.228240
$820$	0	0
$821$	−29.9533	−1.04538	−0.522689	−	0.852523i	$-0.675071\pi$
$821$	−29.9533	−1.04538	−0.522689	+	0.852523i	$0.675071\pi$
$822$	0	0
$823$	−18.9369	−0.660098	−0.330049	−	0.943964i	$-0.607065\pi$
$823$	−18.9369	−0.660098	−0.330049	+	0.943964i	$0.607065\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−20.2639	−0.704644	−0.352322	−	0.935879i	$-0.614608\pi$
$827$	−20.2639	−0.704644	−0.352322	+	0.935879i	$0.614608\pi$
$828$	0	0
$829$	16.6400	0.577932	0.288966	−	0.957339i	$-0.406689\pi$
$829$	16.6400	0.577932	0.288966	+	0.957339i	$0.406689\pi$
$830$	0	0
$831$	−2.15426	−0.0747304
$832$	0	0
$833$	−11.5655	−0.400721
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−29.4672	−1.01854
$838$	0	0
$839$	7.13666	0.246385	0.123192	−	0.992383i	$-0.460687\pi$
$839$	7.13666	0.246385	0.123192	+	0.992383i	$0.460687\pi$
$840$	0	0
$841$	−7.66354	−0.264260
$842$	0	0
$843$	17.4293	0.600295
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−18.7490	−0.644222
$848$	0	0
$849$	−19.0717	−0.654539
$850$	0	0
$851$	−23.6581	−0.810988
$852$	0	0
$853$	31.8252	1.08968	0.544838	−	0.838541i	$-0.316591\pi$
$853$	31.8252	1.08968	0.544838	+	0.838541i	$0.316591\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	43.1535	1.47409	0.737047	−	0.675842i	$-0.236220\pi$
$857$	43.1535	1.47409	0.737047	+	0.675842i	$0.236220\pi$
$858$	0	0
$859$	−30.8915	−1.05400	−0.527002	−	0.849864i	$-0.676684\pi$
$859$	−30.8915	−1.05400	−0.527002	+	0.849864i	$0.676684\pi$
$860$	0	0
$861$	15.6558	0.533549
$862$	0	0
$863$	−38.1910	−1.30004	−0.650019	−	0.759918i	$-0.725239\pi$
$863$	−38.1910	−1.30004	−0.650019	+	0.759918i	$0.725239\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0.223943	0.00760551
$868$	0	0
$869$	2.54804	0.0864362
$870$	0	0
$871$	−14.4353	−0.489123
$872$	0	0
$873$	−14.0559	−0.475722
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−31.7247	−1.07127	−0.535634	−	0.844450i	$-0.679927\pi$
$877$	−31.7247	−1.07127	−0.535634	+	0.844450i	$0.679927\pi$
$878$	0	0
$879$	21.9189	0.739305
$880$	0	0
$881$	−38.4409	−1.29511	−0.647554	−	0.762020i	$-0.724208\pi$
$881$	−38.4409	−1.29511	−0.647554	+	0.762020i	$0.724208\pi$
$882$	0	0
$883$	12.1497	0.408871	0.204435	−	0.978880i	$-0.434464\pi$
$883$	12.1497	0.408871	0.204435	+	0.978880i	$0.434464\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24.2431	−0.814005	−0.407002	−	0.913427i	$-0.633426\pi$
$887$	−24.2431	−0.814005	−0.407002	+	0.913427i	$0.633426\pi$
$888$	0	0
$889$	8.52966	0.286075
$890$	0	0
$891$	−5.59180	−0.187332
$892$	0	0
$893$	−37.0025	−1.23824
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.74086	−0.0915148
$898$	0	0
$899$	33.0604	1.10263
$900$	0	0
$901$	−2.53913	−0.0845908
$902$	0	0
$903$	3.77397	0.125590
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−32.5046	−1.07930	−0.539649	−	0.841890i	$-0.681443\pi$
$907$	−32.5046	−1.07930	−0.539649	+	0.841890i	$0.681443\pi$
$908$	0	0
$909$	−28.0035	−0.928818
$910$	0	0
$911$	−44.2258	−1.46527	−0.732633	−	0.680624i	$-0.761709\pi$
$911$	−44.2258	−1.46527	−0.732633	+	0.680624i	$0.761709\pi$
$912$	0	0
$913$	3.20284	0.105999
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6.92106	−0.228554
$918$	0	0
$919$	39.7936	1.31267	0.656335	−	0.754469i	$-0.272106\pi$
$919$	39.7936	1.31267	0.656335	+	0.754469i	$0.272106\pi$
$920$	0	0
$921$	18.1820	0.599116
$922$	0	0
$923$	3.89597	0.128237
$924$	0	0
$925$	0	0
$926$	0	0
$927$	28.1333	0.924018
$928$	0	0
$929$	−6.64622	−0.218055	−0.109028	−	0.994039i	$-0.534774\pi$
$929$	−6.64622	−0.218055	−0.109028	+	0.994039i	$0.534774\pi$
$930$	0	0
$931$	−13.8282	−0.453202
$932$	0	0
$933$	−7.50607	−0.245738
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−0.229585	−0.00750023	−0.00375011	−	0.999993i	$-0.501194\pi$
$937$	−0.229585	−0.00750023	−0.00375011	+	0.999993i	$0.501194\pi$
$938$	0	0
$939$	14.2545	0.465177
$940$	0	0
$941$	28.3476	0.924106	0.462053	−	0.886852i	$-0.347113\pi$
$941$	28.3476	0.924106	0.462053	+	0.886852i	$0.347113\pi$
$942$	0	0
$943$	−27.6342	−0.899894
$944$	0	0
$945$	0	0
$946$	0	0
$947$	48.0036	1.55991	0.779954	−	0.625837i	$-0.215242\pi$
$947$	48.0036	1.55991	0.779954	+	0.625837i	$0.215242\pi$
$948$	0	0
$949$	17.8666	0.579973
$950$	0	0
$951$	19.6464	0.637080
$952$	0	0
$953$	26.8885	0.871004	0.435502	−	0.900188i	$-0.356571\pi$
$953$	26.8885	0.871004	0.435502	+	0.900188i	$0.356571\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−4.72886	−0.152862
$958$	0	0
$959$	−17.0231	−0.549705
$960$	0	0
$961$	20.2265	0.652468
$962$	0	0
$963$	24.6508	0.794361
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−3.78016	−0.121562	−0.0607809	−	0.998151i	$-0.519359\pi$
$967$	−3.78016	−0.121562	−0.0607809	+	0.998151i	$0.519359\pi$
$968$	0	0
$969$	−15.1613	−0.487051
$970$	0	0
$971$	−11.2875	−0.362232	−0.181116	−	0.983462i	$-0.557971\pi$
$971$	−11.2875	−0.362232	−0.181116	+	0.983462i	$0.557971\pi$
$972$	0	0
$973$	7.01175	0.224786
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−20.2798	−0.648807	−0.324403	−	0.945919i	$-0.605164\pi$
$977$	−20.2798	−0.648807	−0.324403	+	0.945919i	$0.605164\pi$
$978$	0	0
$979$	−9.89766	−0.316331
$980$	0	0
$981$	−2.89583	−0.0924567
$982$	0	0
$983$	−28.3127	−0.903035	−0.451517	−	0.892262i	$-0.649117\pi$
$983$	−28.3127	−0.903035	−0.451517	+	0.892262i	$0.649117\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	11.7376	0.373611
$988$	0	0
$989$	−6.66148	−0.211823
$990$	0	0
$991$	−23.8610	−0.757970	−0.378985	−	0.925403i	$-0.623727\pi$
$991$	−23.8610	−0.757970	−0.378985	+	0.925403i	$0.623727\pi$
$992$	0	0
$993$	0.984859	0.0312536
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−13.6720	−0.432996	−0.216498	−	0.976283i	$-0.569464\pi$
$997$	−13.6720	−0.432996	−0.216498	+	0.976283i	$0.569464\pi$
$998$	0	0
$999$	−35.5982	−1.12628

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.be.1.4		8
4.3	odd	2		625.2.a.g.1.8	yes	8
5.4	even	2		10000.2.a.bn.1.5		8
12.11	even	2		5625.2.a.s.1.1		8
20.3	even	4		625.2.b.d.624.1		16
20.7	even	4		625.2.b.d.624.16		16
20.19	odd	2		625.2.a.e.1.1	✓	8
60.59	even	2		5625.2.a.be.1.8		8
100.3	even	20		625.2.e.k.249.1		32
100.11	odd	10		625.2.d.m.501.4		16
100.19	odd	10		625.2.d.p.251.4		16
100.23	even	20		625.2.e.j.124.1		32
100.27	even	20		625.2.e.j.124.8		32
100.31	odd	10		625.2.d.n.251.1		16
100.39	odd	10		625.2.d.q.501.1		16
100.47	even	20		625.2.e.k.249.8		32
100.59	odd	10		625.2.d.q.126.1		16
100.63	even	20		625.2.e.j.499.8		32
100.67	even	20		625.2.e.k.374.1		32
100.71	odd	10		625.2.d.n.376.1		16
100.79	odd	10		625.2.d.p.376.4		16
100.83	even	20		625.2.e.k.374.8		32
100.87	even	20		625.2.e.j.499.1		32
100.91	odd	10		625.2.d.m.126.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.2.a.e.1.1	✓	8	20.19	odd	2
625.2.a.g.1.8	yes	8	4.3	odd	2
625.2.b.d.624.1		16	20.3	even	4
625.2.b.d.624.16		16	20.7	even	4
625.2.d.m.126.4		16	100.91	odd	10
625.2.d.m.501.4		16	100.11	odd	10
625.2.d.n.251.1		16	100.31	odd	10
625.2.d.n.376.1		16	100.71	odd	10
625.2.d.p.251.4		16	100.19	odd	10
625.2.d.p.376.4		16	100.79	odd	10
625.2.d.q.126.1		16	100.59	odd	10
625.2.d.q.501.1		16	100.39	odd	10
625.2.e.j.124.1		32	100.23	even	20
625.2.e.j.124.8		32	100.27	even	20
625.2.e.j.499.1		32	100.87	even	20
625.2.e.j.499.8		32	100.63	even	20
625.2.e.k.249.1		32	100.3	even	20
625.2.e.k.249.8		32	100.47	even	20
625.2.e.k.374.1		32	100.67	even	20
625.2.e.k.374.8		32	100.83	even	20
5625.2.a.s.1.1		8	12.11	even	2
5625.2.a.be.1.8		8	60.59	even	2
10000.2.a.be.1.4		8	1.1	even	1	trivial
10000.2.a.bn.1.5		8	5.4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 10000 = 2^{4} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	10000.a (trivial)

\(f(q)\)	\(=\)	\(q-0.759083 q^{3} +2.04213 q^{7} -2.42379 q^{9} -1.34867 q^{11} +1.31964 q^{13} +4.08717 q^{17} +4.88680 q^{19} -1.55014 q^{21} +2.73618 q^{23} +4.11711 q^{27} -4.61914 q^{29} -7.15727 q^{31} +1.02375 q^{33} -8.64640 q^{37} -1.00171 q^{39} -10.0996 q^{41} -2.43460 q^{43} -7.57192 q^{47} -2.82971 q^{49} -3.10250 q^{51} -0.621245 q^{53} -3.70948 q^{57} +11.3412 q^{59} +0.647513 q^{61} -4.94970 q^{63} -10.9389 q^{67} -2.07698 q^{69} +2.95230 q^{71} +13.5390 q^{73} -2.75416 q^{77} -1.88929 q^{79} +4.14616 q^{81} -2.37481 q^{83} +3.50631 q^{87} +7.33882 q^{89} +2.69487 q^{91} +5.43296 q^{93} +5.79915 q^{97} +3.26890 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{3} - 10 q^{7} + 9 q^{9} - q^{11} + 10 q^{13} + 15 q^{17} + 10 q^{19} - 14 q^{21} - 30 q^{23} - 20 q^{27} + 10 q^{29} + 9 q^{31} + 5 q^{33} - 10 q^{37} - 8 q^{39} - 4 q^{41} - 30 q^{47} - 4 q^{49}+ \cdots + 22 q^{99}+O(q^{100}) \) 8 * q - 5 * q^3 - 10 * q^7 + 9 * q^9 - q^11 + 10 * q^13 + 15 * q^17 + 10 * q^19 - 14 * q^21 - 30 * q^23 - 20 * q^27 + 10 * q^29 + 9 * q^31 + 5 * q^33 - 10 * q^37 - 8 * q^39 - 4 * q^41 - 30 * q^47 - 4 * q^49 + 14 * q^51 + 10 * q^53 - 10 * q^57 + 5 * q^59 + 6 * q^61 - 10 * q^67 + 3 * q^69 + 9 * q^71 + 5 * q^77 + 20 * q^79 + 8 * q^81 - 40 * q^83 - 40 * q^87 - 5 * q^89 - 6 * q^91 - 40 * q^93 + 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more