LMFDB - Embedded newform 10000.2.a.be.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10000,2,Mod(1,10000)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$79.8504020213$
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			1.5
Root			$-1.47435$ 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)

$f(q)$	$=$	$q-0.710340 q^{3} -4.59110 q^{7} -2.49542 q^{9} +O(q^{10})$	$q-0.710340 q^{3} -4.59110 q^{7} -2.49542 q^{9} -3.91479 q^{11} +0.572939 q^{13} +0.232611 q^{17} +5.55010 q^{19} +3.26125 q^{21} -4.93267 q^{23} +3.90362 q^{27} +4.13062 q^{29} +3.49531 q^{31} +2.78083 q^{33} +5.41648 q^{37} -0.406982 q^{39} +10.4227 q^{41} +1.38833 q^{43} +0.920418 q^{47} +14.0782 q^{49} -0.165233 q^{51} -1.23118 q^{53} -3.94246 q^{57} -4.50780 q^{59} -11.6588 q^{61} +11.4567 q^{63} -2.95447 q^{67} +3.50387 q^{69} -3.20551 q^{71} +10.2922 q^{73} +17.9732 q^{77} +9.61509 q^{79} +4.71335 q^{81} -10.4834 q^{83} -2.93415 q^{87} +7.25828 q^{89} -2.63042 q^{91} -2.48286 q^{93} +8.31971 q^{97} +9.76903 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{3} - 10 q^{7} + 9 q^{9}+O(q^{10}) $ 8 * q - 5 * q^3 - 10 * q^7 + 9 * q^9	$ 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}+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

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
$3$	−0.710340	−0.410115	−0.205058	−	0.978750i	$-0.565738\pi$
$3$	−0.710340	−0.410115	−0.205058	+	0.978750i	$0.565738\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.59110	−1.73527	−0.867637	−	0.497198i	$-0.834362\pi$
$7$	−4.59110	−1.73527	−0.867637	+	0.497198i	$0.834362\pi$
$8$	0	0
$9$	−2.49542	−0.831806
$10$	0	0
$11$	−3.91479	−1.18035	−0.590177	−	0.807274i	$-0.700942\pi$
$11$	−3.91479	−1.18035	−0.590177	+	0.807274i	$0.700942\pi$
$12$	0	0
$13$	0.572939	0.158905	0.0794524	−	0.996839i	$-0.474683\pi$
$13$	0.572939	0.158905	0.0794524	+	0.996839i	$0.474683\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.232611	0.0564165	0.0282082	−	0.999602i	$-0.491020\pi$
$17$	0.232611	0.0564165	0.0282082	+	0.999602i	$0.491020\pi$
$18$	0	0
$19$	5.55010	1.27328	0.636640	−	0.771161i	$-0.280324\pi$
$19$	5.55010	1.27328	0.636640	+	0.771161i	$0.280324\pi$
$20$	0	0
$21$	3.26125	0.711662
$22$	0	0
$23$	−4.93267	−1.02853	−0.514266	−	0.857631i	$-0.671936\pi$
$23$	−4.93267	−1.02853	−0.514266	+	0.857631i	$0.671936\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.90362	0.751251
$28$	0	0
$29$	4.13062	0.767037	0.383519	−	0.923533i	$-0.374712\pi$
$29$	4.13062	0.767037	0.383519	+	0.923533i	$0.374712\pi$
$30$	0	0
$31$	3.49531	0.627777	0.313888	−	0.949460i	$-0.398368\pi$
$31$	3.49531	0.627777	0.313888	+	0.949460i	$0.398368\pi$
$32$	0	0
$33$	2.78083	0.484081
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.41648	0.890464	0.445232	−	0.895415i	$-0.353121\pi$
$37$	5.41648	0.890464	0.445232	+	0.895415i	$0.353121\pi$
$38$	0	0
$39$	−0.406982	−0.0651693
$40$	0	0
$41$	10.4227	1.62775	0.813876	−	0.581039i	$-0.197354\pi$
$41$	10.4227	1.62775	0.813876	+	0.581039i	$0.197354\pi$
$42$	0	0
$43$	1.38833	0.211718	0.105859	−	0.994381i	$-0.466241\pi$
$43$	1.38833	0.211718	0.105859	+	0.994381i	$0.466241\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.920418	0.134257	0.0671284	−	0.997744i	$-0.478616\pi$
$47$	0.920418	0.134257	0.0671284	+	0.997744i	$0.478616\pi$
$48$	0	0
$49$	14.0782	2.01118
$50$	0	0
$51$	−0.165233	−0.0231373
$52$	0	0
$53$	−1.23118	−0.169115	−0.0845576	−	0.996419i	$-0.526948\pi$
$53$	−1.23118	−0.169115	−0.0845576	+	0.996419i	$0.526948\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.94246	−0.522191
$58$	0	0
$59$	−4.50780	−0.586866	−0.293433	−	0.955980i	$-0.594798\pi$
$59$	−4.50780	−0.586866	−0.293433	+	0.955980i	$0.594798\pi$
$60$	0	0
$61$	−11.6588	−1.49275	−0.746376	−	0.665525i	$-0.768208\pi$
$61$	−11.6588	−1.49275	−0.746376	+	0.665525i	$0.768208\pi$
$62$	0	0
$63$	11.4567	1.44341
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.95447	−0.360946	−0.180473	−	0.983580i	$-0.557763\pi$
$67$	−2.95447	−0.360946	−0.180473	+	0.983580i	$0.557763\pi$
$68$	0	0
$69$	3.50387	0.421817
$70$	0	0
$71$	−3.20551	−0.380424	−0.190212	−	0.981743i	$-0.560918\pi$
$71$	−3.20551	−0.380424	−0.190212	+	0.981743i	$0.560918\pi$
$72$	0	0
$73$	10.2922	1.20461	0.602306	−	0.798266i	$-0.294249\pi$
$73$	10.2922	1.20461	0.602306	+	0.798266i	$0.294249\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	17.9732	2.04824
$78$	0	0
$79$	9.61509	1.08178	0.540891	−	0.841093i	$-0.318087\pi$
$79$	9.61509	1.08178	0.540891	+	0.841093i	$0.318087\pi$
$80$	0	0
$81$	4.71335	0.523706
$82$	0	0
$83$	−10.4834	−1.15070	−0.575351	−	0.817906i	$-0.695135\pi$
$83$	−10.4834	−1.15070	−0.575351	+	0.817906i	$0.695135\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.93415	−0.314574
$88$	0	0
$89$	7.25828	0.769376	0.384688	−	0.923047i	$-0.374309\pi$
$89$	7.25828	0.769376	0.384688	+	0.923047i	$0.374309\pi$
$90$	0	0
$91$	−2.63042	−0.275743
$92$	0	0
$93$	−2.48286	−0.257461
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.31971	0.844739	0.422369	−	0.906424i	$-0.361199\pi$
$97$	8.31971	0.844739	0.422369	+	0.906424i	$0.361199\pi$
$98$	0	0
$99$	9.76903	0.981824
$100$	0	0
$101$	−3.56513	−0.354744	−0.177372	−	0.984144i	$-0.556760\pi$
$101$	−3.56513	−0.354744	−0.177372	+	0.984144i	$0.556760\pi$
$102$	0	0
$103$	−0.399323	−0.0393465	−0.0196732	−	0.999806i	$-0.506263\pi$
$103$	−0.399323	−0.0393465	−0.0196732	+	0.999806i	$0.506263\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.64372	−0.158904	−0.0794522	−	0.996839i	$-0.525317\pi$
$107$	−1.64372	−0.158904	−0.0794522	+	0.996839i	$0.525317\pi$
$108$	0	0
$109$	0.0749154	0.00717560	0.00358780	−	0.999994i	$-0.498858\pi$
$109$	0.0749154	0.00717560	0.00358780	+	0.999994i	$0.498858\pi$
$110$	0	0
$111$	−3.84755	−0.365193
$112$	0	0
$113$	−14.1328	−1.32951	−0.664753	−	0.747063i	$-0.731463\pi$
$113$	−14.1328	−1.32951	−0.664753	+	0.747063i	$0.731463\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.42972	−0.132178
$118$	0	0
$119$	−1.06794	−0.0978981
$120$	0	0
$121$	4.32557	0.393234
$122$	0	0
$123$	−7.40366	−0.667565
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.8124	1.04819	0.524093	−	0.851661i	$-0.324405\pi$
$127$	11.8124	1.04819	0.524093	+	0.851661i	$0.324405\pi$
$128$	0	0
$129$	−0.986184	−0.0868287
$130$	0	0
$131$	16.7373	1.46234	0.731170	−	0.682195i	$-0.238974\pi$
$131$	16.7373	1.46234	0.731170	+	0.682195i	$0.238974\pi$
$132$	0	0
$133$	−25.4811	−2.20949
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.4221	−0.890423	−0.445211	−	0.895425i	$-0.646872\pi$
$137$	−10.4221	−0.890423	−0.445211	+	0.895425i	$0.646872\pi$
$138$	0	0
$139$	−6.65993	−0.564888	−0.282444	−	0.959284i	$-0.591145\pi$
$139$	−6.65993	−0.564888	−0.282444	+	0.959284i	$0.591145\pi$
$140$	0	0
$141$	−0.653810	−0.0550608
$142$	0	0
$143$	−2.24294	−0.187564
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−10.0003	−0.824814
$148$	0	0
$149$	−12.0316	−0.985667	−0.492834	−	0.870124i	$-0.664039\pi$
$149$	−12.0316	−0.985667	−0.492834	+	0.870124i	$0.664039\pi$
$150$	0	0
$151$	1.54218	0.125501	0.0627505	−	0.998029i	$-0.480013\pi$
$151$	1.54218	0.125501	0.0627505	+	0.998029i	$0.480013\pi$
$152$	0	0
$153$	−0.580462	−0.0469276
$154$	0	0
$155$	0	0
$156$	0	0
$157$	9.82482	0.784106	0.392053	−	0.919943i	$-0.371765\pi$
$157$	9.82482	0.784106	0.392053	+	0.919943i	$0.371765\pi$
$158$	0	0
$159$	0.874555	0.0693567
$160$	0	0
$161$	22.6464	1.78478
$162$	0	0
$163$	−5.58107	−0.437143	−0.218572	−	0.975821i	$-0.570140\pi$
$163$	−5.58107	−0.437143	−0.218572	+	0.975821i	$0.570140\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.8607	−1.69163	−0.845817	−	0.533473i	$-0.820887\pi$
$167$	−21.8607	−1.69163	−0.845817	+	0.533473i	$0.820887\pi$
$168$	0	0
$169$	−12.6717	−0.974749
$170$	0	0
$171$	−13.8498	−1.05912
$172$	0	0
$173$	−23.4385	−1.78200	−0.891000	−	0.454004i	$-0.849995\pi$
$173$	−23.4385	−1.78200	−0.891000	+	0.454004i	$0.849995\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.20207	0.240683
$178$	0	0
$179$	6.31873	0.472284	0.236142	−	0.971719i	$-0.424117\pi$
$179$	6.31873	0.472284	0.236142	+	0.971719i	$0.424117\pi$
$180$	0	0
$181$	−13.3377	−0.991385	−0.495693	−	0.868498i	$-0.665086\pi$
$181$	−13.3377	−0.991385	−0.495693	+	0.868498i	$0.665086\pi$
$182$	0	0
$183$	8.28169	0.612200
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−0.910624	−0.0665914
$188$	0	0
$189$	−17.9219	−1.30363
$190$	0	0
$191$	2.78083	0.201214	0.100607	−	0.994926i	$-0.467922\pi$
$191$	2.78083	0.201214	0.100607	+	0.994926i	$0.467922\pi$
$192$	0	0
$193$	−22.5667	−1.62438	−0.812192	−	0.583391i	$-0.801726\pi$
$193$	−22.5667	−1.62438	−0.812192	+	0.583391i	$0.801726\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.27182	−0.0906135	−0.0453068	−	0.998973i	$-0.514427\pi$
$197$	−1.27182	−0.0906135	−0.0453068	+	0.998973i	$0.514427\pi$
$198$	0	0
$199$	8.62648	0.611515	0.305757	−	0.952109i	$-0.401090\pi$
$199$	8.62648	0.611515	0.305757	+	0.952109i	$0.401090\pi$
$200$	0	0
$201$	2.09868	0.148030
$202$	0	0
$203$	−18.9641	−1.33102
$204$	0	0
$205$	0	0
$206$	0	0
$207$	12.3091	0.855538
$208$	0	0
$209$	−21.7275	−1.50292
$210$	0	0
$211$	21.0316	1.44787	0.723937	−	0.689866i	$-0.242330\pi$
$211$	21.0316	1.44787	0.723937	+	0.689866i	$0.242330\pi$
$212$	0	0
$213$	2.27700	0.156018
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−16.0473	−1.08936
$218$	0	0
$219$	−7.31097	−0.494029
$220$	0	0
$221$	0.133272	0.00896485
$222$	0	0
$223$	−6.33537	−0.424248	−0.212124	−	0.977243i	$-0.568038\pi$
$223$	−6.33537	−0.424248	−0.212124	+	0.977243i	$0.568038\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.2624	−0.747512	−0.373756	−	0.927527i	$-0.621930\pi$
$227$	−11.2624	−0.747512	−0.373756	+	0.927527i	$0.621930\pi$
$228$	0	0
$229$	15.0408	0.993927	0.496964	−	0.867771i	$-0.334448\pi$
$229$	15.0408	0.993927	0.496964	+	0.867771i	$0.334448\pi$
$230$	0	0
$231$	−12.7671	−0.840013
$232$	0	0
$233$	7.82749	0.512796	0.256398	−	0.966571i	$-0.417464\pi$
$233$	7.82749	0.512796	0.256398	+	0.966571i	$0.417464\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−6.82999	−0.443655
$238$	0	0
$239$	−8.73886	−0.565270	−0.282635	−	0.959228i	$-0.591208\pi$
$239$	−8.73886	−0.565270	−0.282635	+	0.959228i	$0.591208\pi$
$240$	0	0
$241$	−0.600892	−0.0387068	−0.0193534	−	0.999813i	$-0.506161\pi$
$241$	−0.600892	−0.0387068	−0.0193534	+	0.999813i	$0.506161\pi$
$242$	0	0
$243$	−15.0589	−0.966031
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.17987	0.202330
$248$	0	0
$249$	7.44678	0.471921
$250$	0	0
$251$	14.1908	0.895712	0.447856	−	0.894106i	$-0.352188\pi$
$251$	14.1908	0.895712	0.447856	+	0.894106i	$0.352188\pi$
$252$	0	0
$253$	19.3103	1.21403
$254$	0	0
$255$	0	0
$256$	0	0
$257$	17.6859	1.10322	0.551609	−	0.834103i	$-0.314014\pi$
$257$	17.6859	1.10322	0.551609	+	0.834103i	$0.314014\pi$
$258$	0	0
$259$	−24.8676	−1.54520
$260$	0	0
$261$	−10.3076	−0.638026
$262$	0	0
$263$	−24.3348	−1.50055	−0.750274	−	0.661127i	$-0.770078\pi$
$263$	−24.3348	−1.50055	−0.750274	+	0.661127i	$0.770078\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−5.15585	−0.315533
$268$	0	0
$269$	29.9819	1.82803	0.914013	−	0.405685i	$-0.132967\pi$
$269$	29.9819	1.82803	0.914013	+	0.405685i	$0.132967\pi$
$270$	0	0
$271$	27.6981	1.68254	0.841271	−	0.540613i	$-0.181808\pi$
$271$	27.6981	1.68254	0.841271	+	0.540613i	$0.181808\pi$
$272$	0	0
$273$	1.86850	0.113087
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2.29407	−0.137838	−0.0689188	−	0.997622i	$-0.521955\pi$
$277$	−2.29407	−0.137838	−0.0689188	+	0.997622i	$0.521955\pi$
$278$	0	0
$279$	−8.72226	−0.522188
$280$	0	0
$281$	−1.61829	−0.0965388	−0.0482694	−	0.998834i	$-0.515371\pi$
$281$	−1.61829	−0.0965388	−0.0482694	+	0.998834i	$0.515371\pi$
$282$	0	0
$283$	−12.9922	−0.772306	−0.386153	−	0.922435i	$-0.626196\pi$
$283$	−12.9922	−0.772306	−0.386153	+	0.922435i	$0.626196\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−47.8517	−2.82459
$288$	0	0
$289$	−16.9459	−0.996817
$290$	0	0
$291$	−5.90983	−0.346440
$292$	0	0
$293$	11.7009	0.683572	0.341786	−	0.939778i	$-0.388968\pi$
$293$	11.7009	0.683572	0.341786	+	0.939778i	$0.388968\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−15.2818	−0.886742
$298$	0	0
$299$	−2.82612	−0.163439
$300$	0	0
$301$	−6.37395	−0.367388
$302$	0	0
$303$	2.53246	0.145486
$304$	0	0
$305$	0	0
$306$	0	0
$307$	26.5673	1.51628	0.758138	−	0.652094i	$-0.226109\pi$
$307$	26.5673	1.51628	0.758138	+	0.652094i	$0.226109\pi$
$308$	0	0
$309$	0.283655	0.0161366
$310$	0	0
$311$	13.4910	0.765005	0.382502	−	0.923955i	$-0.375062\pi$
$311$	13.4910	0.765005	0.382502	+	0.923955i	$0.375062\pi$
$312$	0	0
$313$	−16.9944	−0.960578	−0.480289	−	0.877110i	$-0.659468\pi$
$313$	−16.9944	−0.960578	−0.480289	+	0.877110i	$0.659468\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−16.7959	−0.943351	−0.471675	−	0.881772i	$-0.656351\pi$
$317$	−16.7959	−0.943351	−0.471675	+	0.881772i	$0.656351\pi$
$318$	0	0
$319$	−16.1705	−0.905375
$320$	0	0
$321$	1.16760	0.0651691
$322$	0	0
$323$	1.29101	0.0718340
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−0.0532154	−0.00294282
$328$	0	0
$329$	−4.22574	−0.232972
$330$	0	0
$331$	−12.8344	−0.705442	−0.352721	−	0.935729i	$-0.614743\pi$
$331$	−12.8344	−0.705442	−0.352721	+	0.935729i	$0.614743\pi$
$332$	0	0
$333$	−13.5164	−0.740693
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−21.2375	−1.15688	−0.578440	−	0.815725i	$-0.696338\pi$
$337$	−21.2375	−1.15688	−0.578440	+	0.815725i	$0.696338\pi$
$338$	0	0
$339$	10.0391	0.545251
$340$	0	0
$341$	−13.6834	−0.740998
$342$	0	0
$343$	−32.4969	−1.75467
$344$	0	0
$345$	0	0
$346$	0	0
$347$	28.6281	1.53684	0.768419	−	0.639947i	$-0.221044\pi$
$347$	28.6281	1.53684	0.768419	+	0.639947i	$0.221044\pi$
$348$	0	0
$349$	19.9124	1.06588	0.532942	−	0.846152i	$-0.321086\pi$
$349$	19.9124	1.06588	0.532942	+	0.846152i	$0.321086\pi$
$350$	0	0
$351$	2.23654	0.119377
$352$	0	0
$353$	−24.6916	−1.31420	−0.657099	−	0.753804i	$-0.728217\pi$
$353$	−24.6916	−1.31420	−0.657099	+	0.753804i	$0.728217\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.758602	0.0401495
$358$	0	0
$359$	−21.5011	−1.13478	−0.567391	−	0.823448i	$-0.692047\pi$
$359$	−21.5011	−1.13478	−0.567391	+	0.823448i	$0.692047\pi$
$360$	0	0
$361$	11.8036	0.621241
$362$	0	0
$363$	−3.07263	−0.161271
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.14383	−0.0597075	−0.0298538	−	0.999554i	$-0.509504\pi$
$367$	−1.14383	−0.0597075	−0.0298538	+	0.999554i	$0.509504\pi$
$368$	0	0
$369$	−26.0090	−1.35397
$370$	0	0
$371$	5.65246	0.293461
$372$	0	0
$373$	−11.5395	−0.597493	−0.298747	−	0.954332i	$-0.596569\pi$
$373$	−11.5395	−0.597493	−0.298747	+	0.954332i	$0.596569\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.36660	0.121886
$378$	0	0
$379$	−21.1053	−1.08410	−0.542052	−	0.840345i	$-0.682352\pi$
$379$	−21.1053	−1.08410	−0.542052	+	0.840345i	$0.682352\pi$
$380$	0	0
$381$	−8.39086	−0.429877
$382$	0	0
$383$	−0.858129	−0.0438483	−0.0219242	−	0.999760i	$-0.506979\pi$
$383$	−0.858129	−0.0438483	−0.0219242	+	0.999760i	$0.506979\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.46445	−0.176108
$388$	0	0
$389$	−33.9346	−1.72055	−0.860277	−	0.509827i	$-0.829710\pi$
$389$	−33.9346	−1.72055	−0.860277	+	0.509827i	$0.829710\pi$
$390$	0	0
$391$	−1.14739	−0.0580262
$392$	0	0
$393$	−11.8891	−0.599728
$394$	0	0
$395$	0	0
$396$	0	0
$397$	26.8286	1.34649	0.673245	−	0.739420i	$-0.264900\pi$
$397$	26.8286	1.34649	0.673245	+	0.739420i	$0.264900\pi$
$398$	0	0
$399$	18.1002	0.906145
$400$	0	0
$401$	3.79757	0.189642	0.0948208	−	0.995494i	$-0.469772\pi$
$401$	3.79757	0.189642	0.0948208	+	0.995494i	$0.469772\pi$
$402$	0	0
$403$	2.00260	0.0997567
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−21.2044	−1.05106
$408$	0	0
$409$	−14.1754	−0.700927	−0.350463	−	0.936576i	$-0.613976\pi$
$409$	−14.1754	−0.700927	−0.350463	+	0.936576i	$0.613976\pi$
$410$	0	0
$411$	7.40326	0.365176
$412$	0	0
$413$	20.6958	1.01837
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.73082	0.231669
$418$	0	0
$419$	19.5969	0.957369	0.478685	−	0.877987i	$-0.341114\pi$
$419$	19.5969	0.957369	0.478685	+	0.877987i	$0.341114\pi$
$420$	0	0
$421$	−25.7840	−1.25664	−0.628318	−	0.777957i	$-0.716256\pi$
$421$	−25.7840	−1.25664	−0.628318	+	0.777957i	$0.716256\pi$
$422$	0	0
$423$	−2.29683	−0.111676
$424$	0	0
$425$	0	0
$426$	0	0
$427$	53.5266	2.59033
$428$	0	0
$429$	1.59325	0.0769227
$430$	0	0
$431$	−9.42533	−0.454002	−0.227001	−	0.973894i	$-0.572892\pi$
$431$	−9.42533	−0.454002	−0.227001	+	0.973894i	$0.572892\pi$
$432$	0	0
$433$	1.75161	0.0841770	0.0420885	−	0.999114i	$-0.486599\pi$
$433$	1.75161	0.0841770	0.0420885	+	0.999114i	$0.486599\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−27.3768	−1.30961
$438$	0	0
$439$	−28.0830	−1.34033	−0.670165	−	0.742212i	$-0.733777\pi$
$439$	−28.0830	−1.34033	−0.670165	+	0.742212i	$0.733777\pi$
$440$	0	0
$441$	−35.1311	−1.67291
$442$	0	0
$443$	29.9110	1.42111	0.710557	−	0.703640i	$-0.248443\pi$
$443$	29.9110	1.42111	0.710557	+	0.703640i	$0.248443\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	8.54653	0.404237
$448$	0	0
$449$	6.29974	0.297303	0.148652	−	0.988890i	$-0.452507\pi$
$449$	6.29974	0.297303	0.148652	+	0.988890i	$0.452507\pi$
$450$	0	0
$451$	−40.8026	−1.92132
$452$	0	0
$453$	−1.09547	−0.0514699
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.1809	0.897243	0.448622	−	0.893722i	$-0.351915\pi$
$457$	19.1809	0.897243	0.448622	+	0.893722i	$0.351915\pi$
$458$	0	0
$459$	0.908025	0.0423830
$460$	0	0
$461$	7.07110	0.329334	0.164667	−	0.986349i	$-0.447345\pi$
$461$	7.07110	0.329334	0.164667	+	0.986349i	$0.447345\pi$
$462$	0	0
$463$	9.61842	0.447006	0.223503	−	0.974703i	$-0.428251\pi$
$463$	9.61842	0.447006	0.223503	+	0.974703i	$0.428251\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.2220	0.889488	0.444744	−	0.895658i	$-0.353295\pi$
$467$	19.2220	0.889488	0.444744	+	0.895658i	$0.353295\pi$
$468$	0	0
$469$	13.5643	0.626341
$470$	0	0
$471$	−6.97896	−0.321574
$472$	0	0
$473$	−5.43500	−0.249902
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.07230	0.140671
$478$	0	0
$479$	37.9996	1.73625	0.868123	−	0.496350i	$-0.165327\pi$
$479$	37.9996	1.73625	0.868123	+	0.496350i	$0.165327\pi$
$480$	0	0
$481$	3.10332	0.141499
$482$	0	0
$483$	−16.0866	−0.731967
$484$	0	0
$485$	0	0
$486$	0	0
$487$	16.7490	0.758969	0.379485	−	0.925198i	$-0.376101\pi$
$487$	16.7490	0.758969	0.379485	+	0.925198i	$0.376101\pi$
$488$	0	0
$489$	3.96446	0.179279
$490$	0	0
$491$	8.95055	0.403933	0.201966	−	0.979392i	$-0.435267\pi$
$491$	8.95055	0.403933	0.201966	+	0.979392i	$0.435267\pi$
$492$	0	0
$493$	0.960829	0.0432736
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14.7168	0.660140
$498$	0	0
$499$	36.3310	1.62640	0.813200	−	0.581985i	$-0.197724\pi$
$499$	36.3310	1.62640	0.813200	+	0.581985i	$0.197724\pi$
$500$	0	0
$501$	15.5286	0.693765
$502$	0	0
$503$	−12.2945	−0.548184	−0.274092	−	0.961703i	$-0.588377\pi$
$503$	−12.2945	−0.548184	−0.274092	+	0.961703i	$0.588377\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9.00125	0.399759
$508$	0	0
$509$	−31.1292	−1.37978	−0.689889	−	0.723915i	$-0.742341\pi$
$509$	−31.1292	−1.37978	−0.689889	+	0.723915i	$0.742341\pi$
$510$	0	0
$511$	−47.2526	−2.09033
$512$	0	0
$513$	21.6654	0.956553
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−3.60324	−0.158470
$518$	0	0
$519$	16.6493	0.730825
$520$	0	0
$521$	16.2169	0.710476	0.355238	−	0.934776i	$-0.384400\pi$
$521$	16.2169	0.710476	0.355238	+	0.934776i	$0.384400\pi$
$522$	0	0
$523$	30.5932	1.33775	0.668873	−	0.743377i	$-0.266777\pi$
$523$	30.5932	1.33775	0.668873	+	0.743377i	$0.266777\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.813049	0.0354170
$528$	0	0
$529$	1.33119	0.0578776
$530$	0	0
$531$	11.2488	0.488158
$532$	0	0
$533$	5.97157	0.258657
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4.48845	−0.193691
$538$	0	0
$539$	−55.1133	−2.37390
$540$	0	0
$541$	18.3014	0.786840	0.393420	−	0.919359i	$-0.371292\pi$
$541$	18.3014	0.786840	0.393420	+	0.919359i	$0.371292\pi$
$542$	0	0
$543$	9.47432	0.406582
$544$	0	0
$545$	0	0
$546$	0	0
$547$	6.55603	0.280316	0.140158	−	0.990129i	$-0.455239\pi$
$547$	6.55603	0.280316	0.140158	+	0.990129i	$0.455239\pi$
$548$	0	0
$549$	29.0935	1.24168
$550$	0	0
$551$	22.9254	0.976653
$552$	0	0
$553$	−44.1439	−1.87719
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−35.5383	−1.50581	−0.752904	−	0.658131i	$-0.771347\pi$
$557$	−35.5383	−1.50581	−0.752904	+	0.658131i	$0.771347\pi$
$558$	0	0
$559$	0.795427	0.0336430
$560$	0	0
$561$	0.646853	0.0273101
$562$	0	0
$563$	−38.2479	−1.61196	−0.805979	−	0.591944i	$-0.798361\pi$
$563$	−38.2479	−1.61196	−0.805979	+	0.591944i	$0.798361\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−21.6395	−0.908773
$568$	0	0
$569$	7.55897	0.316889	0.158444	−	0.987368i	$-0.449352\pi$
$569$	7.55897	0.316889	0.158444	+	0.987368i	$0.449352\pi$
$570$	0	0
$571$	−23.0262	−0.963617	−0.481809	−	0.876277i	$-0.660020\pi$
$571$	−23.0262	−0.963617	−0.481809	+	0.876277i	$0.660020\pi$
$572$	0	0
$573$	−1.97533	−0.0825207
$574$	0	0
$575$	0	0
$576$	0	0
$577$	22.3168	0.929059	0.464529	−	0.885558i	$-0.346223\pi$
$577$	22.3168	0.929059	0.464529	+	0.885558i	$0.346223\pi$
$578$	0	0
$579$	16.0300	0.666184
$580$	0	0
$581$	48.1304	1.99678
$582$	0	0
$583$	4.81980	0.199616
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21.0786	0.870007	0.435004	−	0.900429i	$-0.356747\pi$
$587$	21.0786	0.870007	0.435004	+	0.900429i	$0.356747\pi$
$588$	0	0
$589$	19.3993	0.799335
$590$	0	0
$591$	0.903426	0.0371620
$592$	0	0
$593$	34.3547	1.41078	0.705390	−	0.708819i	$-0.250772\pi$
$593$	34.3547	1.41078	0.705390	+	0.708819i	$0.250772\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.12774	−0.250792
$598$	0	0
$599$	−0.498231	−0.0203572	−0.0101786	−	0.999948i	$-0.503240\pi$
$599$	−0.498231	−0.0203572	−0.0101786	+	0.999948i	$0.503240\pi$
$600$	0	0
$601$	27.8635	1.13657	0.568287	−	0.822830i	$-0.307606\pi$
$601$	27.8635	1.13657	0.568287	+	0.822830i	$0.307606\pi$
$602$	0	0
$603$	7.37264	0.300237
$604$	0	0
$605$	0	0
$606$	0	0
$607$	14.1000	0.572303	0.286152	−	0.958184i	$-0.407624\pi$
$607$	14.1000	0.572303	0.286152	+	0.958184i	$0.407624\pi$
$608$	0	0
$609$	13.4710	0.545871
$610$	0	0
$611$	0.527344	0.0213340
$612$	0	0
$613$	−31.8257	−1.28543	−0.642714	−	0.766106i	$-0.722192\pi$
$613$	−31.8257	−1.28543	−0.642714	+	0.766106i	$0.722192\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−33.9662	−1.36743	−0.683715	−	0.729749i	$-0.739637\pi$
$617$	−33.9662	−1.36743	−0.683715	+	0.729749i	$0.739637\pi$
$618$	0	0
$619$	−1.12842	−0.0453550	−0.0226775	−	0.999743i	$-0.507219\pi$
$619$	−1.12842	−0.0453550	−0.0226775	+	0.999743i	$0.507219\pi$
$620$	0	0
$621$	−19.2552	−0.772686
$622$	0	0
$623$	−33.3235	−1.33508
$624$	0	0
$625$	0	0
$626$	0	0
$627$	15.4339	0.616370
$628$	0	0
$629$	1.25993	0.0502369
$630$	0	0
$631$	−32.7801	−1.30496	−0.652478	−	0.757807i	$-0.726271\pi$
$631$	−32.7801	−1.30496	−0.652478	+	0.757807i	$0.726271\pi$
$632$	0	0
$633$	−14.9396	−0.593795
$634$	0	0
$635$	0	0
$636$	0	0
$637$	8.06597	0.319586
$638$	0	0
$639$	7.99909	0.316439
$640$	0	0
$641$	30.4126	1.20123	0.600613	−	0.799540i	$-0.294923\pi$
$641$	30.4126	1.20123	0.600613	+	0.799540i	$0.294923\pi$
$642$	0	0
$643$	−1.06932	−0.0421700	−0.0210850	−	0.999778i	$-0.506712\pi$
$643$	−1.06932	−0.0421700	−0.0210850	+	0.999778i	$0.506712\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−30.5680	−1.20175	−0.600876	−	0.799343i	$-0.705181\pi$
$647$	−30.5680	−1.20175	−0.600876	+	0.799343i	$0.705181\pi$
$648$	0	0
$649$	17.6471	0.692709
$650$	0	0
$651$	11.3991	0.446765
$652$	0	0
$653$	−33.3502	−1.30509	−0.652546	−	0.757749i	$-0.726299\pi$
$653$	−33.3502	−1.30509	−0.652546	+	0.757749i	$0.726299\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−25.6833	−1.00200
$658$	0	0
$659$	10.1150	0.394026	0.197013	−	0.980401i	$-0.436876\pi$
$659$	10.1150	0.394026	0.197013	+	0.980401i	$0.436876\pi$
$660$	0	0
$661$	−30.8383	−1.19947	−0.599735	−	0.800199i	$-0.704727\pi$
$661$	−30.8383	−1.19947	−0.599735	+	0.800199i	$0.704727\pi$
$662$	0	0
$663$	−0.0946685	−0.00367662
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−20.3750	−0.788922
$668$	0	0
$669$	4.50027	0.173990
$670$	0	0
$671$	45.6416	1.76197
$672$	0	0
$673$	−30.8250	−1.18821	−0.594107	−	0.804386i	$-0.702495\pi$
$673$	−30.8250	−1.18821	−0.594107	+	0.804386i	$0.702495\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	41.2521	1.58545	0.792723	−	0.609582i	$-0.208663\pi$
$677$	41.2521	1.58545	0.792723	+	0.609582i	$0.208663\pi$
$678$	0	0
$679$	−38.1967	−1.46585
$680$	0	0
$681$	8.00014	0.306566
$682$	0	0
$683$	−22.3514	−0.855254	−0.427627	−	0.903955i	$-0.640650\pi$
$683$	−22.3514	−0.855254	−0.427627	+	0.903955i	$0.640650\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−10.6841	−0.407625
$688$	0	0
$689$	−0.705390	−0.0268732
$690$	0	0
$691$	−9.18901	−0.349566	−0.174783	−	0.984607i	$-0.555922\pi$
$691$	−9.18901	−0.349566	−0.174783	+	0.984607i	$0.555922\pi$
$692$	0	0
$693$	−44.8506	−1.70373
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.42443	0.0918320
$698$	0	0
$699$	−5.56018	−0.210305
$700$	0	0
$701$	−35.9929	−1.35943	−0.679717	−	0.733475i	$-0.737897\pi$
$701$	−35.9929	−1.35943	−0.679717	+	0.733475i	$0.737897\pi$
$702$	0	0
$703$	30.0620	1.13381
$704$	0	0
$705$	0	0
$706$	0	0
$707$	16.3679	0.615578
$708$	0	0
$709$	20.2461	0.760359	0.380180	−	0.924913i	$-0.375862\pi$
$709$	20.2461	0.760359	0.380180	+	0.924913i	$0.375862\pi$
$710$	0	0
$711$	−23.9937	−0.899832
$712$	0	0
$713$	−17.2412	−0.645688
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.20756	0.231826
$718$	0	0
$719$	37.0265	1.38086	0.690428	−	0.723401i	$-0.257422\pi$
$719$	37.0265	1.38086	0.690428	+	0.723401i	$0.257422\pi$
$720$	0	0
$721$	1.83333	0.0682769
$722$	0	0
$723$	0.426838	0.0158743
$724$	0	0
$725$	0	0
$726$	0	0
$727$	12.0790	0.447985	0.223993	−	0.974591i	$-0.428091\pi$
$727$	12.0790	0.447985	0.223993	+	0.974591i	$0.428091\pi$
$728$	0	0
$729$	−3.44309	−0.127522
$730$	0	0
$731$	0.322940	0.0119444
$732$	0	0
$733$	44.6116	1.64777	0.823884	−	0.566759i	$-0.191803\pi$
$733$	44.6116	1.64777	0.823884	+	0.566759i	$0.191803\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.5661	0.426044
$738$	0	0
$739$	14.6463	0.538774	0.269387	−	0.963032i	$-0.413179\pi$
$739$	14.6463	0.538774	0.269387	+	0.963032i	$0.413179\pi$
$740$	0	0
$741$	−2.25879	−0.0829787
$742$	0	0
$743$	−24.4397	−0.896604	−0.448302	−	0.893882i	$-0.647971\pi$
$743$	−24.4397	−0.896604	−0.448302	+	0.893882i	$0.647971\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	26.1604	0.957161
$748$	0	0
$749$	7.54648	0.275743
$750$	0	0
$751$	−31.2863	−1.14165	−0.570826	−	0.821071i	$-0.693377\pi$
$751$	−31.2863	−1.14165	−0.570826	+	0.821071i	$0.693377\pi$
$752$	0	0
$753$	−10.0803	−0.367345
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−11.3251	−0.411617	−0.205808	−	0.978592i	$-0.565982\pi$
$757$	−11.3251	−0.411617	−0.205808	+	0.978592i	$0.565982\pi$
$758$	0	0
$759$	−13.7169	−0.497892
$760$	0	0
$761$	−41.3338	−1.49835	−0.749174	−	0.662373i	$-0.769549\pi$
$761$	−41.3338	−1.49835	−0.749174	+	0.662373i	$0.769549\pi$
$762$	0	0
$763$	−0.343944	−0.0124516
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.58270	−0.0932558
$768$	0	0
$769$	22.6491	0.816747	0.408374	−	0.912815i	$-0.366096\pi$
$769$	22.6491	0.816747	0.408374	+	0.912815i	$0.366096\pi$
$770$	0	0
$771$	−12.5630	−0.452446
$772$	0	0
$773$	16.8541	0.606200	0.303100	−	0.952959i	$-0.401978\pi$
$773$	16.8541	0.606200	0.303100	+	0.952959i	$0.401978\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	17.6645	0.633710
$778$	0	0
$779$	57.8470	2.07258
$780$	0	0
$781$	12.5489	0.449035
$782$	0	0
$783$	16.1244	0.576238
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−20.1087	−0.716798	−0.358399	−	0.933568i	$-0.616677\pi$
$787$	−20.1087	−0.716798	−0.358399	+	0.933568i	$0.616677\pi$
$788$	0	0
$789$	17.2860	0.615398
$790$	0	0
$791$	64.8854	2.30706
$792$	0	0
$793$	−6.67976	−0.237205
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−54.1727	−1.91890	−0.959448	−	0.281887i	$-0.909040\pi$
$797$	−54.1727	−1.91890	−0.959448	+	0.281887i	$0.909040\pi$
$798$	0	0
$799$	0.214100	0.00757430
$800$	0	0
$801$	−18.1124	−0.639971
$802$	0	0
$803$	−40.2918	−1.42187
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−21.2973	−0.749701
$808$	0	0
$809$	15.3366	0.539206	0.269603	−	0.962972i	$-0.413108\pi$
$809$	15.3366	0.539206	0.269603	+	0.962972i	$0.413108\pi$
$810$	0	0
$811$	−11.5666	−0.406158	−0.203079	−	0.979162i	$-0.565095\pi$
$811$	−11.5666	−0.406158	−0.203079	+	0.979162i	$0.565095\pi$
$812$	0	0
$813$	−19.6751	−0.690036
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7.70535	0.269576
$818$	0	0
$819$	6.56400	0.229365
$820$	0	0
$821$	33.3675	1.16453	0.582267	−	0.812997i	$-0.302166\pi$
$821$	33.3675	1.16453	0.582267	+	0.812997i	$0.302166\pi$
$822$	0	0
$823$	−29.9821	−1.04511	−0.522554	−	0.852606i	$-0.675021\pi$
$823$	−29.9821	−1.04511	−0.522554	+	0.852606i	$0.675021\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24.7230	0.859702	0.429851	−	0.902900i	$-0.358566\pi$
$827$	24.7230	0.859702	0.429851	+	0.902900i	$0.358566\pi$
$828$	0	0
$829$	−0.211406	−0.00734242	−0.00367121	−	0.999993i	$-0.501169\pi$
$829$	−0.211406	−0.00734242	−0.00367121	+	0.999993i	$0.501169\pi$
$830$	0	0
$831$	1.62957	0.0565293
$832$	0	0
$833$	3.27475	0.113464
$834$	0	0
$835$	0	0
$836$	0	0
$837$	13.6444	0.471618
$838$	0	0
$839$	5.51714	0.190473	0.0952365	−	0.995455i	$-0.469639\pi$
$839$	5.51714	0.190473	0.0952365	+	0.995455i	$0.469639\pi$
$840$	0	0
$841$	−11.9380	−0.411654
$842$	0	0
$843$	1.14953	0.0395920
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−19.8591	−0.682368
$848$	0	0
$849$	9.22888	0.316734
$850$	0	0
$851$	−26.7177	−0.915871
$852$	0	0
$853$	1.53946	0.0527100	0.0263550	−	0.999653i	$-0.491610\pi$
$853$	1.53946	0.0527100	0.0263550	+	0.999653i	$0.491610\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	45.3407	1.54881	0.774404	−	0.632691i	$-0.218050\pi$
$857$	45.3407	1.54881	0.774404	+	0.632691i	$0.218050\pi$
$858$	0	0
$859$	21.7964	0.743685	0.371842	−	0.928296i	$-0.378726\pi$
$859$	21.7964	0.743685	0.371842	+	0.928296i	$0.378726\pi$
$860$	0	0
$861$	33.9910	1.15841
$862$	0	0
$863$	−12.7882	−0.435314	−0.217657	−	0.976025i	$-0.569841\pi$
$863$	−12.7882	−0.435314	−0.217657	+	0.976025i	$0.569841\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	12.0374	0.408810
$868$	0	0
$869$	−37.6410	−1.27688
$870$	0	0
$871$	−1.69273	−0.0573561
$872$	0	0
$873$	−20.7611	−0.702658
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−9.55340	−0.322595	−0.161298	−	0.986906i	$-0.551568\pi$
$877$	−9.55340	−0.322595	−0.161298	+	0.986906i	$0.551568\pi$
$878$	0	0
$879$	−8.31160	−0.280343
$880$	0	0
$881$	−39.1333	−1.31843	−0.659217	−	0.751953i	$-0.729112\pi$
$881$	−39.1333	−1.31843	−0.659217	+	0.751953i	$0.729112\pi$
$882$	0	0
$883$	−35.0254	−1.17870	−0.589349	−	0.807879i	$-0.700616\pi$
$883$	−35.0254	−1.17870	−0.589349	+	0.807879i	$0.700616\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	36.2438	1.21695	0.608474	−	0.793574i	$-0.291782\pi$
$887$	36.2438	1.21695	0.608474	+	0.793574i	$0.291782\pi$
$888$	0	0
$889$	−54.2322	−1.81889
$890$	0	0
$891$	−18.4518	−0.618158
$892$	0	0
$893$	5.10841	0.170946
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.00751	0.0670287
$898$	0	0
$899$	14.4378	0.481528
$900$	0	0
$901$	−0.286386	−0.00954089
$902$	0	0
$903$	4.52768	0.150672
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−40.8532	−1.35651	−0.678255	−	0.734827i	$-0.737263\pi$
$907$	−40.8532	−1.35651	−0.678255	+	0.734827i	$0.737263\pi$
$908$	0	0
$909$	8.89649	0.295078
$910$	0	0
$911$	−12.7468	−0.422320	−0.211160	−	0.977452i	$-0.567724\pi$
$911$	−12.7468	−0.422320	−0.211160	+	0.977452i	$0.567724\pi$
$912$	0	0
$913$	41.0403	1.35824
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−76.8425	−2.53756
$918$	0	0
$919$	−9.67206	−0.319052	−0.159526	−	0.987194i	$-0.550997\pi$
$919$	−9.67206	−0.319052	−0.159526	+	0.987194i	$0.550997\pi$
$920$	0	0
$921$	−18.8718	−0.621848
$922$	0	0
$923$	−1.83656	−0.0604512
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.996477	0.0327286
$928$	0	0
$929$	10.9195	0.358257	0.179128	−	0.983826i	$-0.442672\pi$
$929$	10.9195	0.358257	0.179128	+	0.983826i	$0.442672\pi$
$930$	0	0
$931$	78.1356	2.56079
$932$	0	0
$933$	−9.58320	−0.313740
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−15.9466	−0.520954	−0.260477	−	0.965480i	$-0.583880\pi$
$937$	−15.9466	−0.520954	−0.260477	+	0.965480i	$0.583880\pi$
$938$	0	0
$939$	12.0718	0.393948
$940$	0	0
$941$	5.92701	0.193215	0.0966074	−	0.995323i	$-0.469201\pi$
$941$	5.92701	0.193215	0.0966074	+	0.995323i	$0.469201\pi$
$942$	0	0
$943$	−51.4117	−1.67419
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−0.990382	−0.0321831	−0.0160916	−	0.999871i	$-0.505122\pi$
$947$	−0.990382	−0.0321831	−0.0160916	+	0.999871i	$0.505122\pi$
$948$	0	0
$949$	5.89681	0.191418
$950$	0	0
$951$	11.9308	0.386882
$952$	0	0
$953$	53.5993	1.73625	0.868125	−	0.496345i	$-0.165325\pi$
$953$	53.5993	1.73625	0.868125	+	0.496345i	$0.165325\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	11.4866	0.371308
$958$	0	0
$959$	47.8491	1.54513
$960$	0	0
$961$	−18.7828	−0.605896
$962$	0	0
$963$	4.10176	0.132177
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−29.7350	−0.956212	−0.478106	−	0.878302i	$-0.658677\pi$
$967$	−29.7350	−0.956212	−0.478106	+	0.878302i	$0.658677\pi$
$968$	0	0
$969$	−0.917060	−0.0294602
$970$	0	0
$971$	−60.6166	−1.94528	−0.972640	−	0.232317i	$-0.925369\pi$
$971$	−60.6166	−1.94528	−0.972640	+	0.232317i	$0.925369\pi$
$972$	0	0
$973$	30.5764	0.980236
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−44.5130	−1.42410	−0.712048	−	0.702131i	$-0.752232\pi$
$977$	−44.5130	−1.42410	−0.712048	+	0.702131i	$0.752232\pi$
$978$	0	0
$979$	−28.4146	−0.908135
$980$	0	0
$981$	−0.186945	−0.00596870
$982$	0	0
$983$	38.2401	1.21967	0.609834	−	0.792529i	$-0.291236\pi$
$983$	38.2401	1.21967	0.609834	+	0.792529i	$0.291236\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3.00171	0.0955455
$988$	0	0
$989$	−6.84815	−0.217759
$990$	0	0
$991$	−22.7082	−0.721349	−0.360675	−	0.932692i	$-0.617453\pi$
$991$	−22.7082	−0.721349	−0.360675	+	0.932692i	$0.617453\pi$
$992$	0	0
$993$	9.11678	0.289312
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−20.6284	−0.653309	−0.326654	−	0.945144i	$-0.605921\pi$
$997$	−20.6284	−0.653309	−0.326654	+	0.945144i	$0.605921\pi$
$998$	0	0
$999$	21.1439	0.668962

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.be.1.5		8
4.3	odd	2		625.2.a.g.1.2	yes	8
5.4	even	2		10000.2.a.bn.1.4		8
12.11	even	2		5625.2.a.s.1.7		8
20.3	even	4		625.2.b.d.624.12		16
20.7	even	4		625.2.b.d.624.5		16
20.19	odd	2		625.2.a.e.1.7	✓	8
60.59	even	2		5625.2.a.be.1.2		8
100.3	even	20		625.2.e.j.249.6		32
100.11	odd	10		625.2.d.n.501.1		16
100.19	odd	10		625.2.d.q.251.1		16
100.23	even	20		625.2.e.k.124.6		32
100.27	even	20		625.2.e.k.124.3		32
100.31	odd	10		625.2.d.m.251.4		16
100.39	odd	10		625.2.d.p.501.4		16
100.47	even	20		625.2.e.j.249.3		32
100.59	odd	10		625.2.d.p.126.4		16
100.63	even	20		625.2.e.k.499.3		32
100.67	even	20		625.2.e.j.374.6		32
100.71	odd	10		625.2.d.m.376.4		16
100.79	odd	10		625.2.d.q.376.1		16
100.83	even	20		625.2.e.j.374.3		32
100.87	even	20		625.2.e.k.499.6		32
100.91	odd	10		625.2.d.n.126.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.2.a.e.1.7	✓	8	20.19	odd	2
625.2.a.g.1.2	yes	8	4.3	odd	2
625.2.b.d.624.5		16	20.7	even	4
625.2.b.d.624.12		16	20.3	even	4
625.2.d.m.251.4		16	100.31	odd	10
625.2.d.m.376.4		16	100.71	odd	10
625.2.d.n.126.1		16	100.91	odd	10
625.2.d.n.501.1		16	100.11	odd	10
625.2.d.p.126.4		16	100.59	odd	10
625.2.d.p.501.4		16	100.39	odd	10
625.2.d.q.251.1		16	100.19	odd	10
625.2.d.q.376.1		16	100.79	odd	10
625.2.e.j.249.3		32	100.47	even	20
625.2.e.j.249.6		32	100.3	even	20
625.2.e.j.374.3		32	100.83	even	20
625.2.e.j.374.6		32	100.67	even	20
625.2.e.k.124.3		32	100.27	even	20
625.2.e.k.124.6		32	100.23	even	20
625.2.e.k.499.3		32	100.63	even	20
625.2.e.k.499.6		32	100.87	even	20
5625.2.a.s.1.7		8	12.11	even	2
5625.2.a.be.1.2		8	60.59	even	2
10000.2.a.be.1.5		8	1.1	even	1	trivial
10000.2.a.bn.1.4		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-0.710340 q^{3} -4.59110 q^{7} -2.49542 q^{9} +O(q^{10})\)	\(q-0.710340 q^{3} -4.59110 q^{7} -2.49542 q^{9} -3.91479 q^{11} +0.572939 q^{13} +0.232611 q^{17} +5.55010 q^{19} +3.26125 q^{21} -4.93267 q^{23} +3.90362 q^{27} +4.13062 q^{29} +3.49531 q^{31} +2.78083 q^{33} +5.41648 q^{37} -0.406982 q^{39} +10.4227 q^{41} +1.38833 q^{43} +0.920418 q^{47} +14.0782 q^{49} -0.165233 q^{51} -1.23118 q^{53} -3.94246 q^{57} -4.50780 q^{59} -11.6588 q^{61} +11.4567 q^{63} -2.95447 q^{67} +3.50387 q^{69} -3.20551 q^{71} +10.2922 q^{73} +17.9732 q^{77} +9.61509 q^{79} +4.71335 q^{81} -10.4834 q^{83} -2.93415 q^{87} +7.25828 q^{89} -2.63042 q^{91} -2.48286 q^{93} +8.31971 q^{97} +9.76903 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{3} - 10 q^{7} + 9 q^{9}+O(q^{10}) \) 8 * q - 5 * q^3 - 10 * q^7 + 9 * q^9	\( 8 q - 5 q^{3} - 10 q^{7} + 9 q^{9} - q^{11} + 10 q^{13} + 15 q^{17} + 10 q^{19} - 14 q^{21} - 30 q^{23} - 20 q^{27} + 10 q^{29} + 9 q^{31} + 5 q^{33} - 10 q^{37} - 8 q^{39} - 4 q^{41} - 30 q^{47} - 4 q^{49} + 14 q^{51} + 10 q^{53} - 10 q^{57} + 5 q^{59} + 6 q^{61} - 10 q^{67} + 3 q^{69} + 9 q^{71} + 5 q^{77} + 20 q^{79} + 8 q^{81} - 40 q^{83} - 40 q^{87} - 5 q^{89} - 6 q^{91} - 40 q^{93} + 22 q^{99}+O(q^{100}) \) 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

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