LMFDB - Embedded newform 10000.2.a.bo.1.1

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 20 x^{10} + 11 x^{9} + 144 x^{8} - 29 x^{7} - 440 x^{6} + 4 x^{5} + 556 x^{4} + \cdots + 45 $ x^12 - x^11 - 20x^10 + 11x^9 + 144x^8 - 29x^7 - 440x^6 + 4x^5 + 556x^4 + 15x^3 - 285*x^2 + 45
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 5 $
Twist minimal:	no (minimal twist has level 5000)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.504162$ 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)

$f(q)$	$=$	$q-3.42065 q^{3} -3.76868 q^{7} +8.70082 q^{9} +O(q^{10})$	$q-3.42065 q^{3} -3.76868 q^{7} +8.70082 q^{9} +3.95163 q^{11} +3.68697 q^{13} -3.88348 q^{17} -0.0723216 q^{19} +12.8913 q^{21} -1.44731 q^{23} -19.5005 q^{27} -2.60731 q^{29} +1.00035 q^{31} -13.5171 q^{33} +11.8238 q^{37} -12.6118 q^{39} -6.61168 q^{41} -2.39793 q^{43} -0.145509 q^{47} +7.20298 q^{49} +13.2840 q^{51} -2.88987 q^{53} +0.247387 q^{57} +5.31862 q^{59} -9.06706 q^{61} -32.7907 q^{63} -10.8524 q^{67} +4.95073 q^{69} -9.71253 q^{71} +0.867634 q^{73} -14.8924 q^{77} +13.1995 q^{79} +40.6019 q^{81} +6.55497 q^{83} +8.91870 q^{87} -7.96887 q^{89} -13.8950 q^{91} -3.42184 q^{93} +11.1480 q^{97} +34.3825 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{3} + 26 q^{9}+O(q^{10}) $ 12 * q - 4 * q^3 + 26 * q^9	$ 12 q - 4 q^{3} + 26 q^{9} + q^{11} - 4 q^{13} - 8 q^{17} - 9 q^{19} + 12 q^{21} - 37 q^{27} + 8 q^{29} - 33 q^{31} - 26 q^{33} - 6 q^{37} - 14 q^{39} + 27 q^{41} - 50 q^{43} + 18 q^{47} + 12 q^{49} + 5 q^{51} - 22 q^{53} - 36 q^{57} - 33 q^{59} - 8 q^{61} - 26 q^{63} - 41 q^{67} + 3 q^{69} - 19 q^{71} + 5 q^{73} + 13 q^{77} - 58 q^{79} + 68 q^{81} - 18 q^{83} - 48 q^{87} + 44 q^{89} - 46 q^{91} - 10 q^{93} - 22 q^{97} - 5 q^{99}+O(q^{100}) $ 12 * q - 4 * q^3 + 26 * q^9 + q^11 - 4 * q^13 - 8 * q^17 - 9 * q^19 + 12 * q^21 - 37 * q^27 + 8 * q^29 - 33 * q^31 - 26 * q^33 - 6 * q^37 - 14 * q^39 + 27 * q^41 - 50 * q^43 + 18 * q^47 + 12 * q^49 + 5 * q^51 - 22 * q^53 - 36 * q^57 - 33 * q^59 - 8 * q^61 - 26 * q^63 - 41 * q^67 + 3 * q^69 - 19 * q^71 + 5 * q^73 + 13 * q^77 - 58 * q^79 + 68 * q^81 - 18 * q^83 - 48 * q^87 + 44 * q^89 - 46 * q^91 - 10 * q^93 - 22 * q^97 - 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.42065	−1.97491	−0.987456	−	0.157896i	$-0.949529\pi$
$3$	−3.42065	−1.97491	−0.987456	+	0.157896i	$0.949529\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.76868	−1.42443	−0.712214	−	0.701962i	$-0.752308\pi$
$7$	−3.76868	−1.42443	−0.712214	+	0.701962i	$0.752308\pi$
$8$	0	0
$9$	8.70082	2.90027
$10$	0	0
$11$	3.95163	1.19146	0.595731	−	0.803184i	$-0.296863\pi$
$11$	3.95163	1.19146	0.595731	+	0.803184i	$0.296863\pi$
$12$	0	0
$13$	3.68697	1.02258	0.511290	−	0.859408i	$-0.329168\pi$
$13$	3.68697	1.02258	0.511290	+	0.859408i	$0.329168\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.88348	−0.941882	−0.470941	−	0.882165i	$-0.656086\pi$
$17$	−3.88348	−0.941882	−0.470941	+	0.882165i	$0.656086\pi$
$18$	0	0
$19$	−0.0723216	−0.0165917	−0.00829586	−	0.999966i	$-0.502641\pi$
$19$	−0.0723216	−0.0165917	−0.00829586	+	0.999966i	$0.502641\pi$
$20$	0	0
$21$	12.8913	2.81312
$22$	0	0
$23$	−1.44731	−0.301785	−0.150892	−	0.988550i	$-0.548215\pi$
$23$	−1.44731	−0.301785	−0.150892	+	0.988550i	$0.548215\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−19.5005	−3.75287
$28$	0	0
$29$	−2.60731	−0.484166	−0.242083	−	0.970256i	$-0.577831\pi$
$29$	−2.60731	−0.484166	−0.242083	+	0.970256i	$0.577831\pi$
$30$	0	0
$31$	1.00035	0.179668	0.0898340	−	0.995957i	$-0.471366\pi$
$31$	1.00035	0.179668	0.0898340	+	0.995957i	$0.471366\pi$
$32$	0	0
$33$	−13.5171	−2.35303
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.8238	1.94382	0.971908	−	0.235359i	$-0.0756266\pi$
$37$	11.8238	1.94382	0.971908	+	0.235359i	$0.0756266\pi$
$38$	0	0
$39$	−12.6118	−2.01951
$40$	0	0
$41$	−6.61168	−1.03257	−0.516285	−	0.856417i	$-0.672686\pi$
$41$	−6.61168	−1.03257	−0.516285	+	0.856417i	$0.672686\pi$
$42$	0	0
$43$	−2.39793	−0.365680	−0.182840	−	0.983143i	$-0.558529\pi$
$43$	−2.39793	−0.365680	−0.182840	+	0.983143i	$0.558529\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.145509	−0.0212247	−0.0106124	−	0.999944i	$-0.503378\pi$
$47$	−0.145509	−0.0212247	−0.0106124	+	0.999944i	$0.503378\pi$
$48$	0	0
$49$	7.20298	1.02900
$50$	0	0
$51$	13.2840	1.86013
$52$	0	0
$53$	−2.88987	−0.396955	−0.198477	−	0.980105i	$-0.563600\pi$
$53$	−2.88987	−0.396955	−0.198477	+	0.980105i	$0.563600\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.247387	0.0327672
$58$	0	0
$59$	5.31862	0.692426	0.346213	−	0.938156i	$-0.387467\pi$
$59$	5.31862	0.692426	0.346213	+	0.938156i	$0.387467\pi$
$60$	0	0
$61$	−9.06706	−1.16092	−0.580459	−	0.814290i	$-0.697127\pi$
$61$	−9.06706	−1.16092	−0.580459	+	0.814290i	$0.697127\pi$
$62$	0	0
$63$	−32.7907	−4.13123
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.8524	−1.32584	−0.662918	−	0.748692i	$-0.730682\pi$
$67$	−10.8524	−1.32584	−0.662918	+	0.748692i	$0.730682\pi$
$68$	0	0
$69$	4.95073	0.595998
$70$	0	0
$71$	−9.71253	−1.15267	−0.576333	−	0.817215i	$-0.695517\pi$
$71$	−9.71253	−1.15267	−0.576333	+	0.817215i	$0.695517\pi$
$72$	0	0
$73$	0.867634	0.101549	0.0507745	−	0.998710i	$-0.483831\pi$
$73$	0.867634	0.101549	0.0507745	+	0.998710i	$0.483831\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−14.8924	−1.69715
$78$	0	0
$79$	13.1995	1.48506	0.742529	−	0.669814i	$-0.233626\pi$
$79$	13.1995	1.48506	0.742529	+	0.669814i	$0.233626\pi$
$80$	0	0
$81$	40.6019	4.51132
$82$	0	0
$83$	6.55497	0.719501	0.359750	−	0.933049i	$-0.382862\pi$
$83$	6.55497	0.719501	0.359750	+	0.933049i	$0.382862\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.91870	0.956185
$88$	0	0
$89$	−7.96887	−0.844699	−0.422349	−	0.906433i	$-0.638794\pi$
$89$	−7.96887	−0.844699	−0.422349	+	0.906433i	$0.638794\pi$
$90$	0	0
$91$	−13.8950	−1.45659
$92$	0	0
$93$	−3.42184	−0.354829
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.1480	1.13191	0.565955	−	0.824436i	$-0.308508\pi$
$97$	11.1480	1.13191	0.565955	+	0.824436i	$0.308508\pi$
$98$	0	0
$99$	34.3825	3.45557
$100$	0	0
$101$	−8.69306	−0.864992	−0.432496	−	0.901636i	$-0.642367\pi$
$101$	−8.69306	−0.864992	−0.432496	+	0.901636i	$0.642367\pi$
$102$	0	0
$103$	−4.32655	−0.426307	−0.213154	−	0.977019i	$-0.568373\pi$
$103$	−4.32655	−0.426307	−0.213154	+	0.977019i	$0.568373\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.42272	−0.910929	−0.455464	−	0.890254i	$-0.650527\pi$
$107$	−9.42272	−0.910929	−0.455464	+	0.890254i	$0.650527\pi$
$108$	0	0
$109$	6.18249	0.592175	0.296088	−	0.955161i	$-0.404318\pi$
$109$	6.18249	0.592175	0.296088	+	0.955161i	$0.404318\pi$
$110$	0	0
$111$	−40.4450	−3.83887
$112$	0	0
$113$	11.2323	1.05665	0.528325	−	0.849042i	$-0.322820\pi$
$113$	11.2323	1.05665	0.528325	+	0.849042i	$0.322820\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	32.0797	2.96576
$118$	0	0
$119$	14.6356	1.34164
$120$	0	0
$121$	4.61539	0.419581
$122$	0	0
$123$	22.6162	2.03924
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.29067	0.380736	0.190368	−	0.981713i	$-0.439032\pi$
$127$	4.29067	0.380736	0.190368	+	0.981713i	$0.439032\pi$
$128$	0	0
$129$	8.20246	0.722186
$130$	0	0
$131$	−6.71954	−0.587089	−0.293544	−	0.955945i	$-0.594835\pi$
$131$	−6.71954	−0.587089	−0.293544	+	0.955945i	$0.594835\pi$
$132$	0	0
$133$	0.272557	0.0236337
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.1950	1.12733	0.563663	−	0.826005i	$-0.309392\pi$
$137$	13.1950	1.12733	0.563663	+	0.826005i	$0.309392\pi$
$138$	0	0
$139$	6.82978	0.579295	0.289647	−	0.957133i	$-0.406462\pi$
$139$	6.82978	0.579295	0.289647	+	0.957133i	$0.406462\pi$
$140$	0	0
$141$	0.497737	0.0419170
$142$	0	0
$143$	14.5695	1.21837
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−24.6388	−2.03218
$148$	0	0
$149$	−17.2053	−1.40951	−0.704757	−	0.709448i	$-0.748944\pi$
$149$	−17.2053	−1.40951	−0.704757	+	0.709448i	$0.748944\pi$
$150$	0	0
$151$	8.18246	0.665879	0.332939	−	0.942948i	$-0.391960\pi$
$151$	8.18246	0.665879	0.332939	+	0.942948i	$0.391960\pi$
$152$	0	0
$153$	−33.7895	−2.73172
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−15.3674	−1.22645	−0.613227	−	0.789907i	$-0.710129\pi$
$157$	−15.3674	−1.22645	−0.613227	+	0.789907i	$0.710129\pi$
$158$	0	0
$159$	9.88524	0.783951
$160$	0	0
$161$	5.45445	0.429871
$162$	0	0
$163$	21.4103	1.67698	0.838491	−	0.544916i	$-0.183438\pi$
$163$	21.4103	1.67698	0.838491	+	0.544916i	$0.183438\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.3260	1.65025	0.825126	−	0.564949i	$-0.191104\pi$
$167$	21.3260	1.65025	0.825126	+	0.564949i	$0.191104\pi$
$168$	0	0
$169$	0.593724	0.0456710
$170$	0	0
$171$	−0.629258	−0.0481205
$172$	0	0
$173$	0.0299212	0.00227487	0.00113743	−	0.999999i	$-0.499638\pi$
$173$	0.0299212	0.00227487	0.00113743	+	0.999999i	$0.499638\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−18.1931	−1.36748
$178$	0	0
$179$	13.8591	1.03588	0.517938	−	0.855418i	$-0.326700\pi$
$179$	13.8591	1.03588	0.517938	+	0.855418i	$0.326700\pi$
$180$	0	0
$181$	7.18118	0.533773	0.266887	−	0.963728i	$-0.414005\pi$
$181$	7.18118	0.533773	0.266887	+	0.963728i	$0.414005\pi$
$182$	0	0
$183$	31.0152	2.29271
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−15.3461	−1.12222
$188$	0	0
$189$	73.4913	5.34570
$190$	0	0
$191$	−2.82214	−0.204203	−0.102102	−	0.994774i	$-0.532557\pi$
$191$	−2.82214	−0.204203	−0.102102	+	0.994774i	$0.532557\pi$
$192$	0	0
$193$	17.9234	1.29015	0.645076	−	0.764118i	$-0.276826\pi$
$193$	17.9234	1.29015	0.645076	+	0.764118i	$0.276826\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−14.4954	−1.03275	−0.516377	−	0.856361i	$-0.672720\pi$
$197$	−14.4954	−1.03275	−0.516377	+	0.856361i	$0.672720\pi$
$198$	0	0
$199$	−0.880104	−0.0623889	−0.0311945	−	0.999513i	$-0.509931\pi$
$199$	−0.880104	−0.0623889	−0.0311945	+	0.999513i	$0.509931\pi$
$200$	0	0
$201$	37.1223	2.61841
$202$	0	0
$203$	9.82614	0.689660
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−12.5928	−0.875259
$208$	0	0
$209$	−0.285788	−0.0197684
$210$	0	0
$211$	12.9932	0.894486	0.447243	−	0.894412i	$-0.352406\pi$
$211$	12.9932	0.894486	0.447243	+	0.894412i	$0.352406\pi$
$212$	0	0
$213$	33.2231	2.27641
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−3.77000	−0.255924
$218$	0	0
$219$	−2.96787	−0.200550
$220$	0	0
$221$	−14.3183	−0.963150
$222$	0	0
$223$	20.6759	1.38456	0.692281	−	0.721628i	$-0.256606\pi$
$223$	20.6759	1.38456	0.692281	+	0.721628i	$0.256606\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.6103	−0.836973	−0.418487	−	0.908223i	$-0.637439\pi$
$227$	−12.6103	−0.836973	−0.418487	+	0.908223i	$0.637439\pi$
$228$	0	0
$229$	4.69149	0.310022	0.155011	−	0.987913i	$-0.450459\pi$
$229$	4.69149	0.310022	0.155011	+	0.987913i	$0.450459\pi$
$230$	0	0
$231$	50.9418	3.35173
$232$	0	0
$233$	−1.09367	−0.0716486	−0.0358243	−	0.999358i	$-0.511406\pi$
$233$	−1.09367	−0.0716486	−0.0358243	+	0.999358i	$0.511406\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−45.1508	−2.93286
$238$	0	0
$239$	21.1664	1.36914	0.684569	−	0.728948i	$-0.259990\pi$
$239$	21.1664	1.36914	0.684569	+	0.728948i	$0.259990\pi$
$240$	0	0
$241$	−6.18632	−0.398496	−0.199248	−	0.979949i	$-0.563850\pi$
$241$	−6.18632	−0.398496	−0.199248	+	0.979949i	$0.563850\pi$
$242$	0	0
$243$	−80.3832	−5.15658
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.266647	−0.0169664
$248$	0	0
$249$	−22.4222	−1.42095
$250$	0	0
$251$	−23.9447	−1.51138	−0.755689	−	0.654930i	$-0.772698\pi$
$251$	−23.9447	−1.51138	−0.755689	+	0.654930i	$0.772698\pi$
$252$	0	0
$253$	−5.71923	−0.359565
$254$	0	0
$255$	0	0
$256$	0	0
$257$	17.1167	1.06771	0.533854	−	0.845576i	$-0.320743\pi$
$257$	17.1167	1.06771	0.533854	+	0.845576i	$0.320743\pi$
$258$	0	0
$259$	−44.5601	−2.76883
$260$	0	0
$261$	−22.6858	−1.40421
$262$	0	0
$263$	17.6070	1.08569	0.542847	−	0.839832i	$-0.317346\pi$
$263$	17.6070	1.08569	0.542847	+	0.839832i	$0.317346\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	27.2587	1.66821
$268$	0	0
$269$	29.0019	1.76827	0.884137	−	0.467227i	$-0.154747\pi$
$269$	29.0019	1.76827	0.884137	+	0.467227i	$0.154747\pi$
$270$	0	0
$271$	−8.68477	−0.527562	−0.263781	−	0.964583i	$-0.584970\pi$
$271$	−8.68477	−0.527562	−0.263781	+	0.964583i	$0.584970\pi$
$272$	0	0
$273$	47.5299	2.87664
$274$	0	0
$275$	0	0
$276$	0	0
$277$	13.0902	0.786516	0.393258	−	0.919428i	$-0.371348\pi$
$277$	13.0902	0.786516	0.393258	+	0.919428i	$0.371348\pi$
$278$	0	0
$279$	8.70387	0.521087
$280$	0	0
$281$	−9.17892	−0.547569	−0.273784	−	0.961791i	$-0.588275\pi$
$281$	−9.17892	−0.547569	−0.273784	+	0.961791i	$0.588275\pi$
$282$	0	0
$283$	−17.9124	−1.06478	−0.532390	−	0.846499i	$-0.678706\pi$
$283$	−17.9124	−1.06478	−0.532390	+	0.846499i	$0.678706\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	24.9173	1.47082
$288$	0	0
$289$	−1.91859	−0.112858
$290$	0	0
$291$	−38.1334	−2.23542
$292$	0	0
$293$	−21.7875	−1.27284	−0.636420	−	0.771343i	$-0.719586\pi$
$293$	−21.7875	−1.27284	−0.636420	+	0.771343i	$0.719586\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−77.0588	−4.47141
$298$	0	0
$299$	−5.33618	−0.308599
$300$	0	0
$301$	9.03703	0.520886
$302$	0	0
$303$	29.7359	1.70828
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−9.30372	−0.530991	−0.265496	−	0.964112i	$-0.585536\pi$
$307$	−9.30372	−0.530991	−0.265496	+	0.964112i	$0.585536\pi$
$308$	0	0
$309$	14.7996	0.841919
$310$	0	0
$311$	−24.6490	−1.39771	−0.698857	−	0.715261i	$-0.746308\pi$
$311$	−24.6490	−1.39771	−0.698857	+	0.715261i	$0.746308\pi$
$312$	0	0
$313$	−27.8619	−1.57485	−0.787424	−	0.616411i	$-0.788586\pi$
$313$	−27.8619	−1.57485	−0.787424	+	0.616411i	$0.788586\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.4157	−1.53982	−0.769910	−	0.638153i	$-0.779699\pi$
$317$	−27.4157	−1.53982	−0.769910	+	0.638153i	$0.779699\pi$
$318$	0	0
$319$	−10.3031	−0.576865
$320$	0	0
$321$	32.2318	1.79900
$322$	0	0
$323$	0.280859	0.0156274
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−21.1481	−1.16949
$328$	0	0
$329$	0.548379	0.0302331
$330$	0	0
$331$	−11.3477	−0.623724	−0.311862	−	0.950127i	$-0.600953\pi$
$331$	−11.3477	−0.623724	−0.311862	+	0.950127i	$0.600953\pi$
$332$	0	0
$333$	102.877	5.63760
$334$	0	0
$335$	0	0
$336$	0	0
$337$	12.8690	0.701017	0.350508	−	0.936560i	$-0.386009\pi$
$337$	12.8690	0.701017	0.350508	+	0.936560i	$0.386009\pi$
$338$	0	0
$339$	−38.4219	−2.08679
$340$	0	0
$341$	3.95301	0.214068
$342$	0	0
$343$	−0.764965	−0.0413042
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−15.4978	−0.831966	−0.415983	−	0.909372i	$-0.636563\pi$
$347$	−15.4978	−0.831966	−0.415983	+	0.909372i	$0.636563\pi$
$348$	0	0
$349$	13.0522	0.698668	0.349334	−	0.936998i	$-0.386408\pi$
$349$	13.0522	0.698668	0.349334	+	0.936998i	$0.386408\pi$
$350$	0	0
$351$	−71.8977	−3.83762
$352$	0	0
$353$	−15.9684	−0.849914	−0.424957	−	0.905214i	$-0.639711\pi$
$353$	−15.9684	−0.849914	−0.424957	+	0.905214i	$0.639711\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−50.0632	−2.64963
$358$	0	0
$359$	−34.3026	−1.81042	−0.905212	−	0.424961i	$-0.860288\pi$
$359$	−34.3026	−1.81042	−0.905212	+	0.424961i	$0.860288\pi$
$360$	0	0
$361$	−18.9948	−0.999725
$362$	0	0
$363$	−15.7876	−0.828635
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0.938953	0.0490130	0.0245065	−	0.999700i	$-0.492199\pi$
$367$	0.938953	0.0490130	0.0245065	+	0.999700i	$0.492199\pi$
$368$	0	0
$369$	−57.5271	−2.99474
$370$	0	0
$371$	10.8910	0.565434
$372$	0	0
$373$	−33.9345	−1.75706	−0.878530	−	0.477688i	$-0.841475\pi$
$373$	−33.9345	−1.75706	−0.878530	+	0.477688i	$0.841475\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9.61308	−0.495099
$378$	0	0
$379$	−7.13651	−0.366578	−0.183289	−	0.983059i	$-0.558674\pi$
$379$	−7.13651	−0.366578	−0.183289	+	0.983059i	$0.558674\pi$
$380$	0	0
$381$	−14.6769	−0.751919
$382$	0	0
$383$	11.2628	0.575505	0.287752	−	0.957705i	$-0.407092\pi$
$383$	11.2628	0.575505	0.287752	+	0.957705i	$0.407092\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−20.8639	−1.06057
$388$	0	0
$389$	11.0850	0.562033	0.281017	−	0.959703i	$-0.409328\pi$
$389$	11.0850	0.562033	0.281017	+	0.959703i	$0.409328\pi$
$390$	0	0
$391$	5.62060	0.284246
$392$	0	0
$393$	22.9852	1.15945
$394$	0	0
$395$	0	0
$396$	0	0
$397$	24.6762	1.23846	0.619231	−	0.785209i	$-0.287444\pi$
$397$	24.6762	1.23846	0.619231	+	0.785209i	$0.287444\pi$
$398$	0	0
$399$	−0.932322	−0.0466745
$400$	0	0
$401$	−5.07442	−0.253404	−0.126702	−	0.991941i	$-0.540439\pi$
$401$	−5.07442	−0.253404	−0.126702	+	0.991941i	$0.540439\pi$
$402$	0	0
$403$	3.68826	0.183725
$404$	0	0
$405$	0	0
$406$	0	0
$407$	46.7232	2.31598
$408$	0	0
$409$	−1.39890	−0.0691713	−0.0345856	−	0.999402i	$-0.511011\pi$
$409$	−1.39890	−0.0691713	−0.0345856	+	0.999402i	$0.511011\pi$
$410$	0	0
$411$	−45.1355	−2.22637
$412$	0	0
$413$	−20.0442	−0.986311
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−23.3623	−1.14406
$418$	0	0
$419$	12.0710	0.589709	0.294854	−	0.955542i	$-0.404729\pi$
$419$	12.0710	0.589709	0.294854	+	0.955542i	$0.404729\pi$
$420$	0	0
$421$	−29.5353	−1.43946	−0.719731	−	0.694253i	$-0.755735\pi$
$421$	−29.5353	−1.43946	−0.719731	+	0.694253i	$0.755735\pi$
$422$	0	0
$423$	−1.26605	−0.0615576
$424$	0	0
$425$	0	0
$426$	0	0
$427$	34.1709	1.65364
$428$	0	0
$429$	−49.8372	−2.40616
$430$	0	0
$431$	−0.267267	−0.0128738	−0.00643689	−	0.999979i	$-0.502049\pi$
$431$	−0.267267	−0.0128738	−0.00643689	+	0.999979i	$0.502049\pi$
$432$	0	0
$433$	−13.1641	−0.632628	−0.316314	−	0.948655i	$-0.602445\pi$
$433$	−13.1641	−0.632628	−0.316314	+	0.948655i	$0.602445\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.104672	0.00500713
$438$	0	0
$439$	6.21345	0.296552	0.148276	−	0.988946i	$-0.452628\pi$
$439$	6.21345	0.296552	0.148276	+	0.988946i	$0.452628\pi$
$440$	0	0
$441$	62.6719	2.98437
$442$	0	0
$443$	3.60370	0.171217	0.0856085	−	0.996329i	$-0.472717\pi$
$443$	3.60370	0.171217	0.0856085	+	0.996329i	$0.472717\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	58.8533	2.78367
$448$	0	0
$449$	21.4541	1.01248	0.506241	−	0.862392i	$-0.331035\pi$
$449$	21.4541	1.01248	0.506241	+	0.862392i	$0.331035\pi$
$450$	0	0
$451$	−26.1269	−1.23027
$452$	0	0
$453$	−27.9893	−1.31505
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−27.5872	−1.29048	−0.645238	−	0.763981i	$-0.723242\pi$
$457$	−27.5872	−1.29048	−0.645238	+	0.763981i	$0.723242\pi$
$458$	0	0
$459$	75.7298	3.53477
$460$	0	0
$461$	11.4464	0.533113	0.266557	−	0.963819i	$-0.414114\pi$
$461$	11.4464	0.533113	0.266557	+	0.963819i	$0.414114\pi$
$462$	0	0
$463$	−0.536092	−0.0249143	−0.0124571	−	0.999922i	$-0.503965\pi$
$463$	−0.536092	−0.0249143	−0.0124571	+	0.999922i	$0.503965\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.96429	−0.322269	−0.161134	−	0.986932i	$-0.551515\pi$
$467$	−6.96429	−0.322269	−0.161134	+	0.986932i	$0.551515\pi$
$468$	0	0
$469$	40.8994	1.88856
$470$	0	0
$471$	52.5665	2.42214
$472$	0	0
$473$	−9.47572	−0.435694
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−25.1443	−1.15128
$478$	0	0
$479$	14.0542	0.642155	0.321077	−	0.947053i	$-0.395955\pi$
$479$	14.0542	0.642155	0.321077	+	0.947053i	$0.395955\pi$
$480$	0	0
$481$	43.5939	1.98771
$482$	0	0
$483$	−18.6578	−0.848957
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.44305	0.0653910	0.0326955	−	0.999465i	$-0.489591\pi$
$487$	1.44305	0.0653910	0.0326955	+	0.999465i	$0.489591\pi$
$488$	0	0
$489$	−73.2370	−3.31189
$490$	0	0
$491$	30.7195	1.38635	0.693175	−	0.720769i	$-0.256211\pi$
$491$	30.7195	1.38635	0.693175	+	0.720769i	$0.256211\pi$
$492$	0	0
$493$	10.1255	0.456027
$494$	0	0
$495$	0	0
$496$	0	0
$497$	36.6035	1.64189
$498$	0	0
$499$	−10.8449	−0.485484	−0.242742	−	0.970091i	$-0.578047\pi$
$499$	−10.8449	−0.485484	−0.242742	+	0.970091i	$0.578047\pi$
$500$	0	0
$501$	−72.9486	−3.25910
$502$	0	0
$503$	−17.6255	−0.785882	−0.392941	−	0.919564i	$-0.628542\pi$
$503$	−17.6255	−0.785882	−0.392941	+	0.919564i	$0.628542\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2.03092	−0.0901963
$508$	0	0
$509$	6.01417	0.266573	0.133287	−	0.991078i	$-0.457447\pi$
$509$	6.01417	0.266573	0.133287	+	0.991078i	$0.457447\pi$
$510$	0	0
$511$	−3.26984	−0.144649
$512$	0	0
$513$	1.41031	0.0622666
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−0.575000	−0.0252885
$518$	0	0
$519$	−0.102350	−0.00449266
$520$	0	0
$521$	−14.5378	−0.636913	−0.318457	−	0.947937i	$-0.603165\pi$
$521$	−14.5378	−0.636913	−0.318457	+	0.947937i	$0.603165\pi$
$522$	0	0
$523$	−14.7324	−0.644204	−0.322102	−	0.946705i	$-0.604389\pi$
$523$	−14.7324	−0.644204	−0.322102	+	0.946705i	$0.604389\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.88484	−0.169226
$528$	0	0
$529$	−20.9053	−0.908926
$530$	0	0
$531$	46.2764	2.00823
$532$	0	0
$533$	−24.3770	−1.05589
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−47.4070	−2.04576
$538$	0	0
$539$	28.4635	1.22601
$540$	0	0
$541$	−12.4953	−0.537216	−0.268608	−	0.963250i	$-0.586564\pi$
$541$	−12.4953	−0.537216	−0.268608	+	0.963250i	$0.586564\pi$
$542$	0	0
$543$	−24.5643	−1.05415
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−18.1367	−0.775471	−0.387736	−	0.921771i	$-0.626743\pi$
$547$	−18.1367	−0.775471	−0.387736	+	0.921771i	$0.626743\pi$
$548$	0	0
$549$	−78.8909	−3.36698
$550$	0	0
$551$	0.188565	0.00803315
$552$	0	0
$553$	−49.7447	−2.11536
$554$	0	0
$555$	0	0
$556$	0	0
$557$	36.0780	1.52867	0.764336	−	0.644818i	$-0.223067\pi$
$557$	36.0780	1.52867	0.764336	+	0.644818i	$0.223067\pi$
$558$	0	0
$559$	−8.84108	−0.373938
$560$	0	0
$561$	52.4935	2.21628
$562$	0	0
$563$	−25.7068	−1.08341	−0.541705	−	0.840568i	$-0.682221\pi$
$563$	−25.7068	−1.08341	−0.541705	+	0.840568i	$0.682221\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−153.016	−6.42605
$568$	0	0
$569$	14.8438	0.622286	0.311143	−	0.950363i	$-0.399288\pi$
$569$	14.8438	0.622286	0.311143	+	0.950363i	$0.399288\pi$
$570$	0	0
$571$	36.5775	1.53072	0.765361	−	0.643601i	$-0.222561\pi$
$571$	36.5775	1.53072	0.765361	+	0.643601i	$0.222561\pi$
$572$	0	0
$573$	9.65356	0.403283
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−28.8315	−1.20027	−0.600135	−	0.799899i	$-0.704886\pi$
$577$	−28.8315	−1.20027	−0.600135	+	0.799899i	$0.704886\pi$
$578$	0	0
$579$	−61.3095	−2.54794
$580$	0	0
$581$	−24.7036	−1.02488
$582$	0	0
$583$	−11.4197	−0.472956
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−25.5572	−1.05486	−0.527430	−	0.849598i	$-0.676844\pi$
$587$	−25.5572	−1.05486	−0.527430	+	0.849598i	$0.676844\pi$
$588$	0	0
$589$	−0.0723469	−0.00298100
$590$	0	0
$591$	49.5836	2.03960
$592$	0	0
$593$	27.7017	1.13757	0.568786	−	0.822486i	$-0.307413\pi$
$593$	27.7017	1.13757	0.568786	+	0.822486i	$0.307413\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.01053	0.123213
$598$	0	0
$599$	10.7768	0.440328	0.220164	−	0.975463i	$-0.429341\pi$
$599$	10.7768	0.440328	0.220164	+	0.975463i	$0.429341\pi$
$600$	0	0
$601$	9.56300	0.390083	0.195041	−	0.980795i	$-0.437516\pi$
$601$	9.56300	0.390083	0.195041	+	0.980795i	$0.437516\pi$
$602$	0	0
$603$	−94.4251	−3.84529
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−35.6290	−1.44614	−0.723069	−	0.690776i	$-0.757269\pi$
$607$	−35.6290	−1.44614	−0.723069	+	0.690776i	$0.757269\pi$
$608$	0	0
$609$	−33.6118	−1.36202
$610$	0	0
$611$	−0.536489	−0.0217040
$612$	0	0
$613$	−19.7107	−0.796108	−0.398054	−	0.917362i	$-0.630314\pi$
$613$	−19.7107	−0.796108	−0.398054	+	0.917362i	$0.630314\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−31.1501	−1.25406	−0.627028	−	0.778997i	$-0.715729\pi$
$617$	−31.1501	−1.25406	−0.627028	+	0.778997i	$0.715729\pi$
$618$	0	0
$619$	−13.1321	−0.527824	−0.263912	−	0.964547i	$-0.585013\pi$
$619$	−13.1321	−0.527824	−0.263912	+	0.964547i	$0.585013\pi$
$620$	0	0
$621$	28.2233	1.13256
$622$	0	0
$623$	30.0322	1.20321
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.977581	0.0390408
$628$	0	0
$629$	−45.9174	−1.83085
$630$	0	0
$631$	−35.4286	−1.41039	−0.705196	−	0.709012i	$-0.749141\pi$
$631$	−35.4286	−1.41039	−0.705196	+	0.709012i	$0.749141\pi$
$632$	0	0
$633$	−44.4450	−1.76653
$634$	0	0
$635$	0	0
$636$	0	0
$637$	26.5571	1.05223
$638$	0	0
$639$	−84.5070	−3.34305
$640$	0	0
$641$	12.2806	0.485053	0.242527	−	0.970145i	$-0.422024\pi$
$641$	12.2806	0.485053	0.242527	+	0.970145i	$0.422024\pi$
$642$	0	0
$643$	43.6669	1.72205	0.861027	−	0.508560i	$-0.169822\pi$
$643$	43.6669	1.72205	0.861027	+	0.508560i	$0.169822\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−43.1816	−1.69765	−0.848823	−	0.528678i	$-0.822688\pi$
$647$	−43.1816	−1.69765	−0.848823	+	0.528678i	$0.822688\pi$
$648$	0	0
$649$	21.0172	0.824999
$650$	0	0
$651$	12.8958	0.505428
$652$	0	0
$653$	5.02161	0.196511	0.0982554	−	0.995161i	$-0.468674\pi$
$653$	5.02161	0.196511	0.0982554	+	0.995161i	$0.468674\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	7.54914	0.294520
$658$	0	0
$659$	1.17693	0.0458466	0.0229233	−	0.999737i	$-0.492703\pi$
$659$	1.17693	0.0458466	0.0229233	+	0.999737i	$0.492703\pi$
$660$	0	0
$661$	−19.6608	−0.764717	−0.382359	−	0.924014i	$-0.624888\pi$
$661$	−19.6608	−0.764717	−0.382359	+	0.924014i	$0.624888\pi$
$662$	0	0
$663$	48.9777	1.90214
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.77359	0.146114
$668$	0	0
$669$	−70.7251	−2.73439
$670$	0	0
$671$	−35.8297	−1.38319
$672$	0	0
$673$	47.8685	1.84520	0.922598	−	0.385762i	$-0.126061\pi$
$673$	47.8685	1.84520	0.922598	+	0.385762i	$0.126061\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	12.0777	0.464183	0.232092	−	0.972694i	$-0.425443\pi$
$677$	12.0777	0.464183	0.232092	+	0.972694i	$0.425443\pi$
$678$	0	0
$679$	−42.0133	−1.61232
$680$	0	0
$681$	43.1353	1.65295
$682$	0	0
$683$	−3.39442	−0.129884	−0.0649420	−	0.997889i	$-0.520686\pi$
$683$	−3.39442	−0.129884	−0.0649420	+	0.997889i	$0.520686\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−16.0479	−0.612267
$688$	0	0
$689$	−10.6549	−0.405918
$690$	0	0
$691$	13.5857	0.516824	0.258412	−	0.966035i	$-0.416801\pi$
$691$	13.5857	0.516824	0.258412	+	0.966035i	$0.416801\pi$
$692$	0	0
$693$	−129.577	−4.92221
$694$	0	0
$695$	0	0
$696$	0	0
$697$	25.6763	0.972560
$698$	0	0
$699$	3.74105	0.141500
$700$	0	0
$701$	−46.8052	−1.76781	−0.883903	−	0.467670i	$-0.845094\pi$
$701$	−46.8052	−1.76781	−0.883903	+	0.467670i	$0.845094\pi$
$702$	0	0
$703$	−0.855114	−0.0322513
$704$	0	0
$705$	0	0
$706$	0	0
$707$	32.7614	1.23212
$708$	0	0
$709$	9.09300	0.341495	0.170747	−	0.985315i	$-0.445382\pi$
$709$	9.09300	0.341495	0.170747	+	0.985315i	$0.445382\pi$
$710$	0	0
$711$	114.846	4.30708
$712$	0	0
$713$	−1.44782	−0.0542211
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−72.4026	−2.70393
$718$	0	0
$719$	42.5875	1.58825	0.794123	−	0.607757i	$-0.207931\pi$
$719$	42.5875	1.58825	0.794123	+	0.607757i	$0.207931\pi$
$720$	0	0
$721$	16.3054	0.607244
$722$	0	0
$723$	21.1612	0.786995
$724$	0	0
$725$	0	0
$726$	0	0
$727$	38.2124	1.41722	0.708610	−	0.705600i	$-0.249323\pi$
$727$	38.2124	1.41722	0.708610	+	0.705600i	$0.249323\pi$
$728$	0	0
$729$	153.157	5.67247
$730$	0	0
$731$	9.31230	0.344428
$732$	0	0
$733$	−31.3999	−1.15978	−0.579891	−	0.814694i	$-0.696905\pi$
$733$	−31.3999	−1.15978	−0.579891	+	0.814694i	$0.696905\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−42.8848	−1.57968
$738$	0	0
$739$	−31.2745	−1.15045	−0.575226	−	0.817994i	$-0.695086\pi$
$739$	−31.2745	−1.15045	−0.575226	+	0.817994i	$0.695086\pi$
$740$	0	0
$741$	0.912106	0.0335071
$742$	0	0
$743$	−7.79596	−0.286006	−0.143003	−	0.989722i	$-0.545676\pi$
$743$	−7.79596	−0.286006	−0.143003	+	0.989722i	$0.545676\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	57.0336	2.08675
$748$	0	0
$749$	35.5113	1.29755
$750$	0	0
$751$	3.39100	0.123739	0.0618696	−	0.998084i	$-0.480294\pi$
$751$	3.39100	0.123739	0.0618696	+	0.998084i	$0.480294\pi$
$752$	0	0
$753$	81.9065	2.98484
$754$	0	0
$755$	0	0
$756$	0	0
$757$	34.6809	1.26050	0.630250	−	0.776392i	$-0.282952\pi$
$757$	34.6809	1.26050	0.630250	+	0.776392i	$0.282952\pi$
$758$	0	0
$759$	19.5635	0.710109
$760$	0	0
$761$	13.5014	0.489426	0.244713	−	0.969596i	$-0.421306\pi$
$761$	13.5014	0.489426	0.244713	+	0.969596i	$0.421306\pi$
$762$	0	0
$763$	−23.2999	−0.843511
$764$	0	0
$765$	0	0
$766$	0	0
$767$	19.6096	0.708061
$768$	0	0
$769$	−18.8933	−0.681310	−0.340655	−	0.940188i	$-0.610649\pi$
$769$	−18.8933	−0.681310	−0.340655	+	0.940188i	$0.610649\pi$
$770$	0	0
$771$	−58.5501	−2.10863
$772$	0	0
$773$	−10.7892	−0.388060	−0.194030	−	0.980996i	$-0.562156\pi$
$773$	−10.7892	−0.388060	−0.194030	+	0.980996i	$0.562156\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	152.424	5.46819
$778$	0	0
$779$	0.478167	0.0171321
$780$	0	0
$781$	−38.3803	−1.37336
$782$	0	0
$783$	50.8440	1.81701
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−34.5898	−1.23299	−0.616496	−	0.787358i	$-0.711448\pi$
$787$	−34.5898	−1.23299	−0.616496	+	0.787358i	$0.711448\pi$
$788$	0	0
$789$	−60.2273	−2.14415
$790$	0	0
$791$	−42.3312	−1.50512
$792$	0	0
$793$	−33.4299	−1.18713
$794$	0	0
$795$	0	0
$796$	0	0
$797$	6.51613	0.230813	0.115407	−	0.993318i	$-0.463183\pi$
$797$	6.51613	0.230813	0.115407	+	0.993318i	$0.463183\pi$
$798$	0	0
$799$	0.565083	0.0199912
$800$	0	0
$801$	−69.3358	−2.44986
$802$	0	0
$803$	3.42857	0.120992
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−99.2051	−3.49219
$808$	0	0
$809$	17.6137	0.619264	0.309632	−	0.950856i	$-0.399794\pi$
$809$	17.6137	0.619264	0.309632	+	0.950856i	$0.399794\pi$
$810$	0	0
$811$	9.33225	0.327700	0.163850	−	0.986485i	$-0.447609\pi$
$811$	9.33225	0.327700	0.163850	+	0.986485i	$0.447609\pi$
$812$	0	0
$813$	29.7075	1.04189
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0.173422	0.00606726
$818$	0	0
$819$	−120.898	−4.22452
$820$	0	0
$821$	13.2652	0.462959	0.231480	−	0.972840i	$-0.425643\pi$
$821$	13.2652	0.462959	0.231480	+	0.972840i	$0.425643\pi$
$822$	0	0
$823$	40.8504	1.42396	0.711978	−	0.702202i	$-0.247800\pi$
$823$	40.8504	1.42396	0.711978	+	0.702202i	$0.247800\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−8.10663	−0.281895	−0.140948	−	0.990017i	$-0.545015\pi$
$827$	−8.10663	−0.281895	−0.140948	+	0.990017i	$0.545015\pi$
$828$	0	0
$829$	−37.9061	−1.31653	−0.658267	−	0.752784i	$-0.728710\pi$
$829$	−37.9061	−1.31653	−0.658267	+	0.752784i	$0.728710\pi$
$830$	0	0
$831$	−44.7771	−1.55330
$832$	0	0
$833$	−27.9726	−0.969194
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−19.5073	−0.674272
$838$	0	0
$839$	0.714266	0.0246592	0.0123296	−	0.999924i	$-0.496075\pi$
$839$	0.714266	0.0246592	0.0123296	+	0.999924i	$0.496075\pi$
$840$	0	0
$841$	−22.2019	−0.765583
$842$	0	0
$843$	31.3979	1.08140
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−17.3939	−0.597663
$848$	0	0
$849$	61.2718	2.10284
$850$	0	0
$851$	−17.1127	−0.586615
$852$	0	0
$853$	7.11289	0.243541	0.121770	−	0.992558i	$-0.461143\pi$
$853$	7.11289	0.243541	0.121770	+	0.992558i	$0.461143\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19.7346	0.674120	0.337060	−	0.941483i	$-0.390567\pi$
$857$	19.7346	0.674120	0.337060	+	0.941483i	$0.390567\pi$
$858$	0	0
$859$	−44.4365	−1.51615	−0.758077	−	0.652165i	$-0.773861\pi$
$859$	−44.4365	−1.51615	−0.758077	+	0.652165i	$0.773861\pi$
$860$	0	0
$861$	−85.2334	−2.90475
$862$	0	0
$863$	10.6482	0.362469	0.181235	−	0.983440i	$-0.441991\pi$
$863$	10.6482	0.362469	0.181235	+	0.983440i	$0.441991\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6.56282	0.222885
$868$	0	0
$869$	52.1595	1.76939
$870$	0	0
$871$	−40.0125	−1.35577
$872$	0	0
$873$	96.9969	3.28285
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−31.2559	−1.05544	−0.527718	−	0.849420i	$-0.676952\pi$
$877$	−31.2559	−1.05544	−0.527718	+	0.849420i	$0.676952\pi$
$878$	0	0
$879$	74.5274	2.51375
$880$	0	0
$881$	−29.0482	−0.978660	−0.489330	−	0.872099i	$-0.662759\pi$
$881$	−29.0482	−0.978660	−0.489330	+	0.872099i	$0.662759\pi$
$882$	0	0
$883$	41.5789	1.39924	0.699621	−	0.714514i	$-0.253352\pi$
$883$	41.5789	1.39924	0.699621	+	0.714514i	$0.253352\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24.9565	−0.837956	−0.418978	−	0.907996i	$-0.637612\pi$
$887$	−24.9565	−0.837956	−0.418978	+	0.907996i	$0.637612\pi$
$888$	0	0
$889$	−16.1702	−0.542331
$890$	0	0
$891$	160.444	5.37506
$892$	0	0
$893$	0.0105235	0.000352155 0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	18.2532	0.609456
$898$	0	0
$899$	−2.60823	−0.0869892
$900$	0	0
$901$	11.2228	0.373885
$902$	0	0
$903$	−30.9125	−1.02870
$904$	0	0
$905$	0	0
$906$	0	0
$907$	9.59802	0.318697	0.159349	−	0.987222i	$-0.449061\pi$
$907$	9.59802	0.318697	0.159349	+	0.987222i	$0.449061\pi$
$908$	0	0
$909$	−75.6368	−2.50871
$910$	0	0
$911$	38.9442	1.29028	0.645140	−	0.764064i	$-0.276799\pi$
$911$	38.9442	1.29028	0.645140	+	0.764064i	$0.276799\pi$
$912$	0	0
$913$	25.9028	0.857258
$914$	0	0
$915$	0	0
$916$	0	0
$917$	25.3238	0.836266
$918$	0	0
$919$	−19.1466	−0.631589	−0.315795	−	0.948828i	$-0.602271\pi$
$919$	−19.1466	−0.631589	−0.315795	+	0.948828i	$0.602271\pi$
$920$	0	0
$921$	31.8247	1.04866
$922$	0	0
$923$	−35.8098	−1.17869
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−37.6445	−1.23641
$928$	0	0
$929$	23.8092	0.781153	0.390577	−	0.920570i	$-0.372276\pi$
$929$	23.8092	0.781153	0.390577	+	0.920570i	$0.372276\pi$
$930$	0	0
$931$	−0.520931	−0.0170728
$932$	0	0
$933$	84.3154	2.76036
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−6.05650	−0.197857	−0.0989286	−	0.995095i	$-0.531542\pi$
$937$	−6.05650	−0.197857	−0.0989286	+	0.995095i	$0.531542\pi$
$938$	0	0
$939$	95.3058	3.11019
$940$	0	0
$941$	47.8615	1.56024	0.780120	−	0.625630i	$-0.215158\pi$
$941$	47.8615	1.56024	0.780120	+	0.625630i	$0.215158\pi$
$942$	0	0
$943$	9.56914	0.311614
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−27.9826	−0.909314	−0.454657	−	0.890667i	$-0.650238\pi$
$947$	−27.9826	−0.909314	−0.454657	+	0.890667i	$0.650238\pi$
$948$	0	0
$949$	3.19894	0.103842
$950$	0	0
$951$	93.7795	3.04101
$952$	0	0
$953$	−37.4714	−1.21382	−0.606908	−	0.794772i	$-0.707590\pi$
$953$	−37.4714	−1.21382	−0.606908	+	0.794772i	$0.707590\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	35.2434	1.13926
$958$	0	0
$959$	−49.7278	−1.60579
$960$	0	0
$961$	−29.9993	−0.967719
$962$	0	0
$963$	−81.9854	−2.64194
$964$	0	0
$965$	0	0
$966$	0	0
$967$	18.3063	0.588692	0.294346	−	0.955699i	$-0.404898\pi$
$967$	18.3063	0.588692	0.294346	+	0.955699i	$0.404898\pi$
$968$	0	0
$969$	−0.960721	−0.0308628
$970$	0	0
$971$	−13.8309	−0.443855	−0.221928	−	0.975063i	$-0.571235\pi$
$971$	−13.8309	−0.443855	−0.221928	+	0.975063i	$0.571235\pi$
$972$	0	0
$973$	−25.7393	−0.825164
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−6.13957	−0.196422	−0.0982111	−	0.995166i	$-0.531312\pi$
$977$	−6.13957	−0.196422	−0.0982111	+	0.995166i	$0.531312\pi$
$978$	0	0
$979$	−31.4900	−1.00643
$980$	0	0
$981$	53.7928	1.71747
$982$	0	0
$983$	−53.7555	−1.71453	−0.857267	−	0.514872i	$-0.827840\pi$
$983$	−53.7555	−1.71453	−0.857267	+	0.514872i	$0.827840\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.87581	−0.0597077
$988$	0	0
$989$	3.47054	0.110357
$990$	0	0
$991$	51.0570	1.62188	0.810939	−	0.585130i	$-0.198957\pi$
$991$	51.0570	1.62188	0.810939	+	0.585130i	$0.198957\pi$
$992$	0	0
$993$	38.8163	1.23180
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−55.4203	−1.75518	−0.877589	−	0.479413i	$-0.840850\pi$
$997$	−55.4203	−1.75518	−0.877589	+	0.479413i	$0.840850\pi$
$998$	0	0
$999$	−230.570	−7.29490

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.bo.1.1		12
4.3	odd	2		5000.2.a.p.1.12	yes	12
5.4	even	2		10000.2.a.bp.1.12		12
20.19	odd	2		5000.2.a.o.1.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5000.2.a.o.1.1	✓	12	20.19	odd	2
5000.2.a.p.1.12	yes	12	4.3	odd	2
10000.2.a.bo.1.1		12	1.1	even	1	trivial
10000.2.a.bp.1.12		12	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)

\(f(q)\)	\(=\)	\(q-3.42065 q^{3} -3.76868 q^{7} +8.70082 q^{9} +O(q^{10})\)	\(q-3.42065 q^{3} -3.76868 q^{7} +8.70082 q^{9} +3.95163 q^{11} +3.68697 q^{13} -3.88348 q^{17} -0.0723216 q^{19} +12.8913 q^{21} -1.44731 q^{23} -19.5005 q^{27} -2.60731 q^{29} +1.00035 q^{31} -13.5171 q^{33} +11.8238 q^{37} -12.6118 q^{39} -6.61168 q^{41} -2.39793 q^{43} -0.145509 q^{47} +7.20298 q^{49} +13.2840 q^{51} -2.88987 q^{53} +0.247387 q^{57} +5.31862 q^{59} -9.06706 q^{61} -32.7907 q^{63} -10.8524 q^{67} +4.95073 q^{69} -9.71253 q^{71} +0.867634 q^{73} -14.8924 q^{77} +13.1995 q^{79} +40.6019 q^{81} +6.55497 q^{83} +8.91870 q^{87} -7.96887 q^{89} -13.8950 q^{91} -3.42184 q^{93} +11.1480 q^{97} +34.3825 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} + 26 q^{9}+O(q^{10}) \) 12 * q - 4 * q^3 + 26 * q^9	\( 12 q - 4 q^{3} + 26 q^{9} + q^{11} - 4 q^{13} - 8 q^{17} - 9 q^{19} + 12 q^{21} - 37 q^{27} + 8 q^{29} - 33 q^{31} - 26 q^{33} - 6 q^{37} - 14 q^{39} + 27 q^{41} - 50 q^{43} + 18 q^{47} + 12 q^{49} + 5 q^{51} - 22 q^{53} - 36 q^{57} - 33 q^{59} - 8 q^{61} - 26 q^{63} - 41 q^{67} + 3 q^{69} - 19 q^{71} + 5 q^{73} + 13 q^{77} - 58 q^{79} + 68 q^{81} - 18 q^{83} - 48 q^{87} + 44 q^{89} - 46 q^{91} - 10 q^{93} - 22 q^{97} - 5 q^{99}+O(q^{100}) \) 12 * q - 4 * q^3 + 26 * q^9 + q^11 - 4 * q^13 - 8 * q^17 - 9 * q^19 + 12 * q^21 - 37 * q^27 + 8 * q^29 - 33 * q^31 - 26 * q^33 - 6 * q^37 - 14 * q^39 + 27 * q^41 - 50 * q^43 + 18 * q^47 + 12 * q^49 + 5 * q^51 - 22 * q^53 - 36 * q^57 - 33 * q^59 - 8 * q^61 - 26 * q^63 - 41 * q^67 + 3 * q^69 - 19 * q^71 + 5 * q^73 + 13 * q^77 - 58 * q^79 + 68 * q^81 - 18 * q^83 - 48 * q^87 + 44 * q^89 - 46 * q^91 - 10 * q^93 - 22 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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