LMFDB - Embedded newform 10000.2.a.v.1.2

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:	$4$
Coefficient field:	4.4.18625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 14x^{2} + 9x + 41 $ x^4 - x^3 - 14x^2 + 9x + 41
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1250)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.26594$ 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)

$f(q)$	$=$	$q-1.64791 q^{3} +4.90244 q^{7} -0.284403 q^{9} +O(q^{10})$	$q-1.64791 q^{3} +4.90244 q^{7} -0.284403 q^{9} -4.63650 q^{11} +1.40043 q^{13} +0.217606 q^{17} +1.04834 q^{19} -8.07876 q^{21} -7.50201 q^{23} +5.41239 q^{27} +3.93231 q^{29} -8.69624 q^{31} +7.64051 q^{33} +8.90244 q^{37} -2.30778 q^{39} +2.64791 q^{41} +6.67778 q^{43} -4.63650 q^{47} +17.0339 q^{49} -0.358594 q^{51} -1.36350 q^{53} -1.72756 q^{57} -13.2844 q^{59} +3.83564 q^{61} -1.39427 q^{63} -3.17577 q^{67} +12.3626 q^{69} +4.22902 q^{71} -10.4159 q^{73} -22.7301 q^{77} +4.73317 q^{79} -8.06591 q^{81} +7.72611 q^{83} -6.48008 q^{87} -9.58022 q^{89} +6.86551 q^{91} +14.3306 q^{93} -5.34415 q^{97} +1.31863 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} - 3 q^{7} + 17 q^{9}+O(q^{10}) $ 4 * q + q^3 - 3 * q^7 + 17 * q^9	$ 4 q + q^{3} - 3 q^{7} + 17 q^{9} - 8 q^{11} + 4 q^{13} - 2 q^{17} - 5 q^{19} - 7 q^{21} - 9 q^{23} + 10 q^{27} - 10 q^{29} - 18 q^{31} - 22 q^{33} + 13 q^{37} + 16 q^{39} + 3 q^{41} + 16 q^{43} - 8 q^{47} + 23 q^{49} - 13 q^{51} - 16 q^{53} - 15 q^{57} - 35 q^{59} + 8 q^{61} - 54 q^{63} - 23 q^{67} + 19 q^{69} + 17 q^{71} - q^{73} - 29 q^{77} - 10 q^{79} + 24 q^{81} + 11 q^{83} - 40 q^{87} - 5 q^{89} + 17 q^{91} + 38 q^{93} + 3 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q + q^3 - 3 * q^7 + 17 * q^9 - 8 * q^11 + 4 * q^13 - 2 * q^17 - 5 * q^19 - 7 * q^21 - 9 * q^23 + 10 * q^27 - 10 * q^29 - 18 * q^31 - 22 * q^33 + 13 * q^37 + 16 * q^39 + 3 * q^41 + 16 * q^43 - 8 * q^47 + 23 * q^49 - 13 * q^51 - 16 * q^53 - 15 * q^57 - 35 * q^59 + 8 * q^61 - 54 * q^63 - 23 * q^67 + 19 * q^69 + 17 * q^71 - q^73 - 29 * q^77 - 10 * q^79 + 24 * q^81 + 11 * q^83 - 40 * q^87 - 5 * q^89 + 17 * q^91 + 38 * q^93 + 3 * q^97 - 4 * 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
$3$	−1.64791	−0.951419	−0.475710	−	0.879602i	$-0.657809\pi$
$3$	−1.64791	−0.951419	−0.475710	+	0.879602i	$0.657809\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.90244	1.85295	0.926474	−	0.376360i	$-0.122824\pi$
$7$	4.90244	1.85295	0.926474	+	0.376360i	$0.122824\pi$
$8$	0	0
$9$	−0.284403	−0.0948011
$10$	0	0
$11$	−4.63650	−1.39796	−0.698978	−	0.715143i	$-0.746362\pi$
$11$	−4.63650	−1.39796	−0.698978	+	0.715143i	$0.746362\pi$
$12$	0	0
$13$	1.40043	0.388409	0.194204	−	0.980961i	$-0.437787\pi$
$13$	1.40043	0.388409	0.194204	+	0.980961i	$0.437787\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.217606	0.0527771	0.0263886	−	0.999652i	$-0.491599\pi$
$17$	0.217606	0.0527771	0.0263886	+	0.999652i	$0.491599\pi$
$18$	0	0
$19$	1.04834	0.240505	0.120252	−	0.992743i	$-0.461630\pi$
$19$	1.04834	0.240505	0.120252	+	0.992743i	$0.461630\pi$
$20$	0	0
$21$	−8.07876	−1.76293
$22$	0	0
$23$	−7.50201	−1.56428	−0.782138	−	0.623105i	$-0.785871\pi$
$23$	−7.50201	−1.56428	−0.782138	+	0.623105i	$0.785871\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.41239	1.04162
$28$	0	0
$29$	3.93231	0.730212	0.365106	−	0.930966i	$-0.381033\pi$
$29$	3.93231	0.730212	0.365106	+	0.930966i	$0.381033\pi$
$30$	0	0
$31$	−8.69624	−1.56189	−0.780946	−	0.624599i	$-0.785262\pi$
$31$	−8.69624	−1.56189	−0.780946	+	0.624599i	$0.785262\pi$
$32$	0	0
$33$	7.64051	1.33004
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.90244	1.46355	0.731776	−	0.681545i	$-0.238692\pi$
$37$	8.90244	1.46355	0.731776	+	0.681545i	$0.238692\pi$
$38$	0	0
$39$	−2.30778	−0.369540
$40$	0	0
$41$	2.64791	0.413534	0.206767	−	0.978390i	$-0.433706\pi$
$41$	2.64791	0.413534	0.206767	+	0.978390i	$0.433706\pi$
$42$	0	0
$43$	6.67778	1.01835	0.509176	−	0.860662i	$-0.329950\pi$
$43$	6.67778	1.01835	0.509176	+	0.860662i	$0.329950\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.63650	−0.676303	−0.338151	−	0.941092i	$-0.609802\pi$
$47$	−4.63650	−0.676303	−0.338151	+	0.941092i	$0.609802\pi$
$48$	0	0
$49$	17.0339	2.43341
$50$	0	0
$51$	−0.358594	−0.0502132
$52$	0	0
$53$	−1.36350	−0.187292	−0.0936458	−	0.995606i	$-0.529852\pi$
$53$	−1.36350	−0.187292	−0.0936458	+	0.995606i	$0.529852\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.72756	−0.228821
$58$	0	0
$59$	−13.2844	−1.72948	−0.864741	−	0.502218i	$-0.832517\pi$
$59$	−13.2844	−1.72948	−0.864741	+	0.502218i	$0.832517\pi$
$60$	0	0
$61$	3.83564	0.491103	0.245552	−	0.969384i	$-0.421031\pi$
$61$	3.83564	0.491103	0.245552	+	0.969384i	$0.421031\pi$
$62$	0	0
$63$	−1.39427	−0.175661
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.17577	−0.387982	−0.193991	−	0.981003i	$-0.562143\pi$
$67$	−3.17577	−0.387982	−0.193991	+	0.981003i	$0.562143\pi$
$68$	0	0
$69$	12.3626	1.48828
$70$	0	0
$71$	4.22902	0.501892	0.250946	−	0.968001i	$-0.419258\pi$
$71$	4.22902	0.501892	0.250946	+	0.968001i	$0.419258\pi$
$72$	0	0
$73$	−10.4159	−1.21908	−0.609542	−	0.792754i	$-0.708647\pi$
$73$	−10.4159	−1.21908	−0.609542	+	0.792754i	$0.708647\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−22.7301	−2.59034
$78$	0	0
$79$	4.73317	0.532523	0.266261	−	0.963901i	$-0.414212\pi$
$79$	4.73317	0.532523	0.266261	+	0.963901i	$0.414212\pi$
$80$	0	0
$81$	−8.06591	−0.896212
$82$	0	0
$83$	7.72611	0.848051	0.424026	−	0.905650i	$-0.360617\pi$
$83$	7.72611	0.848051	0.424026	+	0.905650i	$0.360617\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.48008	−0.694738
$88$	0	0
$89$	−9.58022	−1.01550	−0.507750	−	0.861504i	$-0.669523\pi$
$89$	−9.58022	−1.01550	−0.507750	+	0.861504i	$0.669523\pi$
$90$	0	0
$91$	6.86551	0.719701
$92$	0	0
$93$	14.3306	1.48601
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−5.34415	−0.542616	−0.271308	−	0.962493i	$-0.587456\pi$
$97$	−5.34415	−0.542616	−0.271308	+	0.962493i	$0.587456\pi$
$98$	0	0
$99$	1.31863	0.132528
$100$	0	0
$101$	−13.6405	−1.35728	−0.678641	−	0.734470i	$-0.737431\pi$
$101$	−13.6405	−1.35728	−0.678641	+	0.734470i	$0.737431\pi$
$102$	0	0
$103$	−11.8128	−1.16395	−0.581976	−	0.813206i	$-0.697720\pi$
$103$	−11.8128	−1.16395	−0.581976	+	0.813206i	$0.697720\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.69679	−0.550730	−0.275365	−	0.961340i	$-0.588799\pi$
$107$	−5.69679	−0.550730	−0.275365	+	0.961340i	$0.588799\pi$
$108$	0	0
$109$	12.4642	1.19385	0.596927	−	0.802296i	$-0.296388\pi$
$109$	12.4642	1.19385	0.596927	+	0.802296i	$0.296388\pi$
$110$	0	0
$111$	−14.6704	−1.39245
$112$	0	0
$113$	0.00650049	0.000611515 0	0.000305757	−	1.00000i	$-0.499903\pi$
$113$	0.00650049	0.000611515 0	0.000305757		1.00000i	$0.499903\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.398286	−0.0368216
$118$	0	0
$119$	1.06680	0.0977932
$120$	0	0
$121$	10.4971	0.954282
$122$	0	0
$123$	−4.36350	−0.393444
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.3706	−0.920238	−0.460119	−	0.887857i	$-0.652193\pi$
$127$	−10.3706	−0.920238	−0.460119	+	0.887857i	$0.652193\pi$
$128$	0	0
$129$	−11.0044	−0.968880
$130$	0	0
$131$	−7.78150	−0.679873	−0.339936	−	0.940448i	$-0.610406\pi$
$131$	−7.78150	−0.679873	−0.339936	+	0.940448i	$0.610406\pi$
$132$	0	0
$133$	5.13940	0.445642
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.58761	0.135639	0.0678193	−	0.997698i	$-0.478396\pi$
$137$	1.58761	0.135639	0.0678193	+	0.997698i	$0.478396\pi$
$138$	0	0
$139$	4.31919	0.366349	0.183174	−	0.983080i	$-0.441363\pi$
$139$	4.31919	0.366349	0.183174	+	0.983080i	$0.441363\pi$
$140$	0	0
$141$	7.64051	0.643447
$142$	0	0
$143$	−6.49308	−0.542979
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−28.0703	−2.31520
$148$	0	0
$149$	−6.40748	−0.524921	−0.262461	−	0.964943i	$-0.584534\pi$
$149$	−6.40748	−0.524921	−0.262461	+	0.964943i	$0.584534\pi$
$150$	0	0
$151$	−1.40534	−0.114365	−0.0571824	−	0.998364i	$-0.518212\pi$
$151$	−1.40534	−0.114365	−0.0571824	+	0.998364i	$0.518212\pi$
$152$	0	0
$153$	−0.0618877	−0.00500333
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2.13851	−0.170671	−0.0853356	−	0.996352i	$-0.527196\pi$
$157$	−2.13851	−0.170671	−0.0853356	+	0.996352i	$0.527196\pi$
$158$	0	0
$159$	2.24693	0.178193
$160$	0	0
$161$	−36.7781	−2.89852
$162$	0	0
$163$	−5.64701	−0.442308	−0.221154	−	0.975239i	$-0.570982\pi$
$163$	−5.64701	−0.442308	−0.221154	+	0.975239i	$0.570982\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.224106	0.0173418	0.00867092	−	0.999962i	$-0.497240\pi$
$167$	0.224106	0.0173418	0.00867092	+	0.999962i	$0.497240\pi$
$168$	0	0
$169$	−11.0388	−0.849138
$170$	0	0
$171$	−0.298150	−0.0228001
$172$	0	0
$173$	6.50112	0.494271	0.247135	−	0.968981i	$-0.420511\pi$
$173$	6.50112	0.494271	0.247135	+	0.968981i	$0.420511\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	21.8915	1.64546
$178$	0	0
$179$	17.8353	1.33307	0.666536	−	0.745472i	$-0.267776\pi$
$179$	17.8353	1.33307	0.666536	+	0.745472i	$0.267776\pi$
$180$	0	0
$181$	19.5759	1.45506	0.727531	−	0.686075i	$-0.240668\pi$
$181$	19.5759	1.45506	0.727531	+	0.686075i	$0.240668\pi$
$182$	0	0
$183$	−6.32078	−0.467245
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.00893	−0.0737801
$188$	0	0
$189$	26.5339	1.93006
$190$	0	0
$191$	24.8637	1.79908	0.899538	−	0.436842i	$-0.143903\pi$
$191$	24.8637	1.79908	0.899538	+	0.436842i	$0.143903\pi$
$192$	0	0
$193$	6.92090	0.498177	0.249089	−	0.968481i	$-0.419869\pi$
$193$	6.92090	0.498177	0.249089	+	0.968481i	$0.419869\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.69838	−0.121005	−0.0605024	−	0.998168i	$-0.519270\pi$
$197$	−1.69838	−0.121005	−0.0605024	+	0.998168i	$0.519270\pi$
$198$	0	0
$199$	9.23919	0.654949	0.327475	−	0.944860i	$-0.393802\pi$
$199$	9.23919	0.654949	0.327475	+	0.944860i	$0.393802\pi$
$200$	0	0
$201$	5.23337	0.369134
$202$	0	0
$203$	19.2779	1.35304
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.13360	0.148295
$208$	0	0
$209$	−4.86060	−0.336215
$210$	0	0
$211$	−8.13906	−0.560316	−0.280158	−	0.959954i	$-0.590387\pi$
$211$	−8.13906	−0.560316	−0.280158	+	0.959954i	$0.590387\pi$
$212$	0	0
$213$	−6.96902	−0.477510
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−42.6328	−2.89410
$218$	0	0
$219$	17.1644	1.15986
$220$	0	0
$221$	0.304741	0.0204991
$222$	0	0
$223$	−6.28876	−0.421127	−0.210563	−	0.977580i	$-0.567530\pi$
$223$	−6.28876	−0.421127	−0.210563	+	0.977580i	$0.567530\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−18.7676	−1.24565	−0.622825	−	0.782361i	$-0.714015\pi$
$227$	−18.7676	−1.24565	−0.622825	+	0.782361i	$0.714015\pi$
$228$	0	0
$229$	11.0071	0.727366	0.363683	−	0.931523i	$-0.381519\pi$
$229$	11.0071	0.727366	0.363683	+	0.931523i	$0.381519\pi$
$230$	0	0
$231$	37.4571	2.46450
$232$	0	0
$233$	−14.2656	−0.934571	−0.467285	−	0.884107i	$-0.654768\pi$
$233$	−14.2656	−0.934571	−0.467285	+	0.884107i	$0.654768\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−7.79982	−0.506653
$238$	0	0
$239$	17.7310	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$239$	17.7310	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$240$	0	0
$241$	20.3909	1.31349	0.656746	−	0.754111i	$-0.271932\pi$
$241$	20.3909	1.31349	0.656746	+	0.754111i	$0.271932\pi$
$242$	0	0
$243$	−2.94531	−0.188942
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.46812	0.0934141
$248$	0	0
$249$	−12.7319	−0.806852
$250$	0	0
$251$	−2.92090	−0.184366	−0.0921828	−	0.995742i	$-0.529384\pi$
$251$	−2.92090	−0.184366	−0.0921828	+	0.995742i	$0.529384\pi$
$252$	0	0
$253$	34.7830	2.18679
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.49149	0.0930366	0.0465183	−	0.998917i	$-0.485187\pi$
$257$	1.49149	0.0930366	0.0465183	+	0.998917i	$0.485187\pi$
$258$	0	0
$259$	43.6436	2.71188
$260$	0	0
$261$	−1.11836	−0.0692248
$262$	0	0
$263$	−26.7073	−1.64684	−0.823422	−	0.567430i	$-0.807938\pi$
$263$	−26.7073	−1.64684	−0.823422	+	0.567430i	$0.807938\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	15.7873	0.966167
$268$	0	0
$269$	−4.24202	−0.258640	−0.129320	−	0.991603i	$-0.541279\pi$
$269$	−4.24202	−0.258640	−0.129320	+	0.991603i	$0.541279\pi$
$270$	0	0
$271$	−19.0299	−1.15598	−0.577991	−	0.816043i	$-0.696163\pi$
$271$	−19.0299	−1.15598	−0.577991	+	0.816043i	$0.696163\pi$
$272$	0	0
$273$	−11.3137	−0.684738
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−20.7043	−1.24400	−0.622000	−	0.783017i	$-0.713680\pi$
$277$	−20.7043	−1.24400	−0.622000	+	0.783017i	$0.713680\pi$
$278$	0	0
$279$	2.47324	0.148069
$280$	0	0
$281$	−24.4220	−1.45690	−0.728448	−	0.685101i	$-0.759758\pi$
$281$	−24.4220	−1.45690	−0.728448	+	0.685101i	$0.759758\pi$
$282$	0	0
$283$	−12.1271	−0.720881	−0.360440	−	0.932782i	$-0.617374\pi$
$283$	−12.1271	−0.720881	−0.360440	+	0.932782i	$0.617374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.9812	0.766256
$288$	0	0
$289$	−16.9526	−0.997215
$290$	0	0
$291$	8.80666	0.516256
$292$	0	0
$293$	2.64141	0.154313	0.0771563	−	0.997019i	$-0.475416\pi$
$293$	2.64141	0.154313	0.0771563	+	0.997019i	$0.475416\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−25.0945	−1.45613
$298$	0	0
$299$	−10.5060	−0.607579
$300$	0	0
$301$	32.7374	1.88695
$302$	0	0
$303$	22.4783	1.29134
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−7.45983	−0.425755	−0.212878	−	0.977079i	$-0.568284\pi$
$307$	−7.45983	−0.425755	−0.212878	+	0.977079i	$0.568284\pi$
$308$	0	0
$309$	19.4664	1.10741
$310$	0	0
$311$	21.5777	1.22356	0.611781	−	0.791028i	$-0.290454\pi$
$311$	21.5777	1.22356	0.611781	+	0.791028i	$0.290454\pi$
$312$	0	0
$313$	−13.5463	−0.765684	−0.382842	−	0.923814i	$-0.625055\pi$
$313$	−13.5463	−0.765684	−0.382842	+	0.923814i	$0.625055\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.0031	0.786494	0.393247	−	0.919433i	$-0.371352\pi$
$317$	14.0031	0.786494	0.393247	+	0.919433i	$0.371352\pi$
$318$	0	0
$319$	−18.2321	−1.02080
$320$	0	0
$321$	9.38778	0.523975
$322$	0	0
$323$	0.228124	0.0126931
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−20.5398	−1.13585
$328$	0	0
$329$	−22.7301	−1.25315
$330$	0	0
$331$	23.8722	1.31214	0.656068	−	0.754702i	$-0.272218\pi$
$331$	23.8722	1.31214	0.656068	+	0.754702i	$0.272218\pi$
$332$	0	0
$333$	−2.53188	−0.138746
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−20.6449	−1.12460	−0.562299	−	0.826934i	$-0.690083\pi$
$337$	−20.6449	−1.12460	−0.562299	+	0.826934i	$0.690083\pi$
$338$	0	0
$339$	−0.0107122	−0.000581807 0
$340$	0	0
$341$	40.3201	2.18346
$342$	0	0
$343$	49.1905	2.65604
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.01543	0.215559	0.107780	−	0.994175i	$-0.465626\pi$
$347$	4.01543	0.215559	0.107780	+	0.994175i	$0.465626\pi$
$348$	0	0
$349$	−4.19334	−0.224464	−0.112232	−	0.993682i	$-0.535800\pi$
$349$	−4.19334	−0.224464	−0.112232	+	0.993682i	$0.535800\pi$
$350$	0	0
$351$	7.57967	0.404573
$352$	0	0
$353$	−32.9526	−1.75389	−0.876946	−	0.480588i	$-0.840423\pi$
$353$	−32.9526	−1.75389	−0.876946	+	0.480588i	$0.840423\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.75798	−0.0930423
$358$	0	0
$359$	−19.2330	−1.01508	−0.507540	−	0.861628i	$-0.669445\pi$
$359$	−19.2330	−1.01508	−0.507540	+	0.861628i	$0.669445\pi$
$360$	0	0
$361$	−17.9010	−0.942158
$362$	0	0
$363$	−17.2982	−0.907922
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−25.6774	−1.34035	−0.670176	−	0.742202i	$-0.733781\pi$
$367$	−25.6774	−1.34035	−0.670176	+	0.742202i	$0.733781\pi$
$368$	0	0
$369$	−0.753073	−0.0392034
$370$	0	0
$371$	−6.68449	−0.347041
$372$	0	0
$373$	21.9354	1.13577	0.567887	−	0.823107i	$-0.307761\pi$
$373$	21.9354	1.13577	0.567887	+	0.823107i	$0.307761\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.50692	0.283621
$378$	0	0
$379$	−33.8910	−1.74087	−0.870433	−	0.492288i	$-0.836161\pi$
$379$	−33.8910	−1.74087	−0.870433	+	0.492288i	$0.836161\pi$
$380$	0	0
$381$	17.0897	0.875532
$382$	0	0
$383$	4.98611	0.254778	0.127389	−	0.991853i	$-0.459340\pi$
$383$	4.98611	0.254778	0.127389	+	0.991853i	$0.459340\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.89918	−0.0965408
$388$	0	0
$389$	1.92918	0.0978136	0.0489068	−	0.998803i	$-0.484426\pi$
$389$	1.92918	0.0978136	0.0489068	+	0.998803i	$0.484426\pi$
$390$	0	0
$391$	−1.63248	−0.0825580
$392$	0	0
$393$	12.8232	0.646844
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−8.25487	−0.414300	−0.207150	−	0.978309i	$-0.566419\pi$
$397$	−8.25487	−0.414300	−0.207150	+	0.978309i	$0.566419\pi$
$398$	0	0
$399$	−8.46925	−0.423993
$400$	0	0
$401$	−10.4229	−0.520495	−0.260248	−	0.965542i	$-0.583804\pi$
$401$	−10.4229	−0.520495	−0.260248	+	0.965542i	$0.583804\pi$
$402$	0	0
$403$	−12.1785	−0.606653
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−41.2761	−2.04598
$408$	0	0
$409$	−24.5399	−1.21342	−0.606710	−	0.794923i	$-0.707511\pi$
$409$	−24.5399	−1.21342	−0.606710	+	0.794923i	$0.707511\pi$
$410$	0	0
$411$	−2.61623	−0.129049
$412$	0	0
$413$	−65.1260	−3.20464
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−7.11762	−0.348551
$418$	0	0
$419$	−3.41800	−0.166980	−0.0834901	−	0.996509i	$-0.526607\pi$
$419$	−3.41800	−0.166980	−0.0834901	+	0.996509i	$0.526607\pi$
$420$	0	0
$421$	−7.07787	−0.344954	−0.172477	−	0.985014i	$-0.555177\pi$
$421$	−7.07787	−0.344954	−0.172477	+	0.985014i	$0.555177\pi$
$422$	0	0
$423$	1.31863	0.0641142
$424$	0	0
$425$	0	0
$426$	0	0
$427$	18.8040	0.909988
$428$	0	0
$429$	10.7000	0.516601
$430$	0	0
$431$	−33.2362	−1.60093	−0.800465	−	0.599380i	$-0.795414\pi$
$431$	−33.2362	−1.60093	−0.800465	+	0.599380i	$0.795414\pi$
$432$	0	0
$433$	30.3712	1.45955	0.729773	−	0.683689i	$-0.239626\pi$
$433$	30.3712	1.45955	0.729773	+	0.683689i	$0.239626\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.86462	−0.376216
$438$	0	0
$439$	30.4973	1.45556	0.727779	−	0.685812i	$-0.240553\pi$
$439$	30.4973	1.45556	0.727779	+	0.685812i	$0.240553\pi$
$440$	0	0
$441$	−4.84449	−0.230690
$442$	0	0
$443$	38.2866	1.81905	0.909526	−	0.415646i	$-0.136444\pi$
$443$	38.2866	1.81905	0.909526	+	0.415646i	$0.136444\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	10.5589	0.499420
$448$	0	0
$449$	−24.9258	−1.17632	−0.588161	−	0.808744i	$-0.700148\pi$
$449$	−24.9258	−1.17632	−0.588161	+	0.808744i	$0.700148\pi$
$450$	0	0
$451$	−12.2770	−0.578102
$452$	0	0
$453$	2.31587	0.108809
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.11022	0.192268	0.0961341	−	0.995368i	$-0.469352\pi$
$457$	4.11022	0.192268	0.0961341	+	0.995368i	$0.469352\pi$
$458$	0	0
$459$	1.17777	0.0549734
$460$	0	0
$461$	23.1783	1.07952	0.539762	−	0.841818i	$-0.318514\pi$
$461$	23.1783	1.07952	0.539762	+	0.841818i	$0.318514\pi$
$462$	0	0
$463$	−6.33363	−0.294349	−0.147174	−	0.989111i	$-0.547018\pi$
$463$	−6.33363	−0.294349	−0.147174	+	0.989111i	$0.547018\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−0.563198	−0.0260617	−0.0130309	−	0.999915i	$-0.504148\pi$
$467$	−0.563198	−0.0260617	−0.0130309	+	0.999915i	$0.504148\pi$
$468$	0	0
$469$	−15.5690	−0.718910
$470$	0	0
$471$	3.52406	0.162380
$472$	0	0
$473$	−30.9615	−1.42361
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.387785	0.0177554
$478$	0	0
$479$	0.911154	0.0416317	0.0208158	−	0.999783i	$-0.493374\pi$
$479$	0.911154	0.0416317	0.0208158	+	0.999783i	$0.493374\pi$
$480$	0	0
$481$	12.4672	0.568457
$482$	0	0
$483$	60.6069	2.75771
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0.135380	0.00613465	0.00306733	−	0.999995i	$-0.499024\pi$
$487$	0.135380	0.00613465	0.00306733	+	0.999995i	$0.499024\pi$
$488$	0	0
$489$	9.30575	0.420821
$490$	0	0
$491$	25.1222	1.13375	0.566874	−	0.823804i	$-0.308153\pi$
$491$	25.1222	1.13375	0.566874	+	0.823804i	$0.308153\pi$
$492$	0	0
$493$	0.855693	0.0385385
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20.7325	0.929979
$498$	0	0
$499$	−16.7375	−0.749275	−0.374637	−	0.927171i	$-0.622233\pi$
$499$	−16.7375	−0.749275	−0.374637	+	0.927171i	$0.622233\pi$
$500$	0	0
$501$	−0.369306	−0.0164994
$502$	0	0
$503$	−15.7489	−0.702210	−0.351105	−	0.936336i	$-0.614194\pi$
$503$	−15.7489	−0.702210	−0.351105	+	0.936336i	$0.614194\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	18.1909	0.807887
$508$	0	0
$509$	−2.93142	−0.129933	−0.0649664	−	0.997887i	$-0.520694\pi$
$509$	−2.93142	−0.129933	−0.0649664	+	0.997887i	$0.520694\pi$
$510$	0	0
$511$	−51.0631	−2.25890
$512$	0	0
$513$	5.67400	0.250513
$514$	0	0
$515$	0	0
$516$	0	0
$517$	21.4971	0.945441
$518$	0	0
$519$	−10.7132	−0.470259
$520$	0	0
$521$	−33.2059	−1.45478	−0.727389	−	0.686225i	$-0.759267\pi$
$521$	−33.2059	−1.45478	−0.727389	+	0.686225i	$0.759267\pi$
$522$	0	0
$523$	14.8347	0.648678	0.324339	−	0.945941i	$-0.394858\pi$
$523$	14.8347	0.648678	0.324339	+	0.945941i	$0.394858\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.89235	−0.0824321
$528$	0	0
$529$	33.2801	1.44696
$530$	0	0
$531$	3.77813	0.163957
$532$	0	0
$533$	3.70820	0.160620
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−29.3909	−1.26831
$538$	0	0
$539$	−78.9776	−3.40180
$540$	0	0
$541$	−18.6902	−0.803553	−0.401777	−	0.915738i	$-0.631607\pi$
$541$	−18.6902	−0.803553	−0.401777	+	0.915738i	$0.631607\pi$
$542$	0	0
$543$	−32.2592	−1.38437
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−7.11845	−0.304363	−0.152182	−	0.988353i	$-0.548630\pi$
$547$	−7.11845	−0.304363	−0.152182	+	0.988353i	$0.548630\pi$
$548$	0	0
$549$	−1.09087	−0.0465571
$550$	0	0
$551$	4.12238	0.175619
$552$	0	0
$553$	23.2041	0.986737
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−43.4953	−1.84296	−0.921478	−	0.388431i	$-0.873017\pi$
$557$	−43.4953	−1.84296	−0.921478	+	0.388431i	$0.873017\pi$
$558$	0	0
$559$	9.35175	0.395537
$560$	0	0
$561$	1.66262	0.0701958
$562$	0	0
$563$	−1.59825	−0.0673581	−0.0336791	−	0.999433i	$-0.510722\pi$
$563$	−1.59825	−0.0673581	−0.0336791	+	0.999433i	$0.510722\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−39.5426	−1.66063
$568$	0	0
$569$	16.3333	0.684727	0.342364	−	0.939568i	$-0.388773\pi$
$569$	16.3333	0.684727	0.342364	+	0.939568i	$0.388773\pi$
$570$	0	0
$571$	−26.4867	−1.10843	−0.554216	−	0.832373i	$-0.686982\pi$
$571$	−26.4867	−1.10843	−0.554216	+	0.832373i	$0.686982\pi$
$572$	0	0
$573$	−40.9731	−1.71168
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−5.48713	−0.228432	−0.114216	−	0.993456i	$-0.536436\pi$
$577$	−5.48713	−0.228432	−0.114216	+	0.993456i	$0.536436\pi$
$578$	0	0
$579$	−11.4050	−0.473976
$580$	0	0
$581$	37.8768	1.57139
$582$	0	0
$583$	6.32188	0.261825
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−44.8139	−1.84967	−0.924835	−	0.380369i	$-0.875797\pi$
$587$	−44.8139	−1.84967	−0.924835	+	0.380369i	$0.875797\pi$
$588$	0	0
$589$	−9.11658	−0.375642
$590$	0	0
$591$	2.79878	0.115126
$592$	0	0
$593$	28.8514	1.18479	0.592393	−	0.805649i	$-0.298183\pi$
$593$	28.8514	1.18479	0.592393	+	0.805649i	$0.298183\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−15.2253	−0.623131
$598$	0	0
$599$	28.0259	1.14511	0.572555	−	0.819867i	$-0.305952\pi$
$599$	28.0259	1.14511	0.572555	+	0.819867i	$0.305952\pi$
$600$	0	0
$601$	−7.93479	−0.323667	−0.161833	−	0.986818i	$-0.551741\pi$
$601$	−7.93479	−0.323667	−0.161833	+	0.986818i	$0.551741\pi$
$602$	0	0
$603$	0.903199	0.0367811
$604$	0	0
$605$	0	0
$606$	0	0
$607$	42.3848	1.72034	0.860172	−	0.510004i	$-0.170356\pi$
$607$	42.3848	1.72034	0.860172	+	0.510004i	$0.170356\pi$
$608$	0	0
$609$	−31.7682	−1.28731
$610$	0	0
$611$	−6.49308	−0.262682
$612$	0	0
$613$	−18.1485	−0.733010	−0.366505	−	0.930416i	$-0.619446\pi$
$613$	−18.1485	−0.733010	−0.366505	+	0.930416i	$0.619446\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	48.7227	1.96150	0.980752	−	0.195258i	$-0.0625546\pi$
$617$	48.7227	1.96150	0.980752	+	0.195258i	$0.0625546\pi$
$618$	0	0
$619$	29.3106	1.17809	0.589045	−	0.808100i	$-0.299504\pi$
$619$	29.3106	1.17809	0.589045	+	0.808100i	$0.299504\pi$
$620$	0	0
$621$	−40.6038	−1.62937
$622$	0	0
$623$	−46.9664	−1.88167
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8.00982	0.319881
$628$	0	0
$629$	1.93722	0.0772420
$630$	0	0
$631$	6.49317	0.258489	0.129245	−	0.991613i	$-0.458745\pi$
$631$	6.49317	0.258489	0.129245	+	0.991613i	$0.458745\pi$
$632$	0	0
$633$	13.4124	0.533095
$634$	0	0
$635$	0	0
$636$	0	0
$637$	23.8547	0.945159
$638$	0	0
$639$	−1.20275	−0.0475799
$640$	0	0
$641$	−13.0813	−0.516682	−0.258341	−	0.966054i	$-0.583176\pi$
$641$	−13.0813	−0.516682	−0.258341	+	0.966054i	$0.583176\pi$
$642$	0	0
$643$	20.7411	0.817950	0.408975	−	0.912546i	$-0.365886\pi$
$643$	20.7411	0.817950	0.408975	+	0.912546i	$0.365886\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26.0560	1.02437	0.512184	−	0.858876i	$-0.328837\pi$
$647$	26.0560	1.02437	0.512184	+	0.858876i	$0.328837\pi$
$648$	0	0
$649$	61.5931	2.41774
$650$	0	0
$651$	70.2548	2.75350
$652$	0	0
$653$	−29.4792	−1.15361	−0.576805	−	0.816882i	$-0.695701\pi$
$653$	−29.4792	−1.15361	−0.576805	+	0.816882i	$0.695701\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.96230	0.115570
$658$	0	0
$659$	4.71961	0.183850	0.0919250	−	0.995766i	$-0.470698\pi$
$659$	4.71961	0.183850	0.0919250	+	0.995766i	$0.470698\pi$
$660$	0	0
$661$	−3.45526	−0.134394	−0.0671971	−	0.997740i	$-0.521406\pi$
$661$	−3.45526	−0.134394	−0.0671971	+	0.997740i	$0.521406\pi$
$662$	0	0
$663$	−0.502185	−0.0195032
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−29.5002	−1.14225
$668$	0	0
$669$	10.3633	0.400668
$670$	0	0
$671$	−17.7839	−0.686541
$672$	0	0
$673$	−22.6691	−0.873828	−0.436914	−	0.899503i	$-0.643929\pi$
$673$	−22.6691	−0.873828	−0.436914	+	0.899503i	$0.643929\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21.6755	0.833057	0.416529	−	0.909123i	$-0.363247\pi$
$677$	21.6755	0.833057	0.416529	+	0.909123i	$0.363247\pi$
$678$	0	0
$679$	−26.1994	−1.00544
$680$	0	0
$681$	30.9273	1.18514
$682$	0	0
$683$	−47.4569	−1.81589	−0.907944	−	0.419092i	$-0.862348\pi$
$683$	−47.4569	−1.81589	−0.907944	+	0.419092i	$0.862348\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−18.1386	−0.692031
$688$	0	0
$689$	−1.90949	−0.0727457
$690$	0	0
$691$	−30.9333	−1.17676	−0.588379	−	0.808585i	$-0.700234\pi$
$691$	−30.9333	−1.17676	−0.588379	+	0.808585i	$0.700234\pi$
$692$	0	0
$693$	6.46452	0.245567
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0.576199	0.0218251
$698$	0	0
$699$	23.5084	0.889169
$700$	0	0
$701$	−22.1963	−0.838343	−0.419172	−	0.907907i	$-0.637679\pi$
$701$	−22.1963	−0.838343	−0.419172	+	0.907907i	$0.637679\pi$
$702$	0	0
$703$	9.33274	0.351991
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−66.8718	−2.51497
$708$	0	0
$709$	7.03984	0.264387	0.132193	−	0.991224i	$-0.457798\pi$
$709$	7.03984	0.264387	0.132193	+	0.991224i	$0.457798\pi$
$710$	0	0
$711$	−1.34613	−0.0504837
$712$	0	0
$713$	65.2393	2.44323
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−29.2191	−1.09121
$718$	0	0
$719$	−43.4455	−1.62024	−0.810121	−	0.586262i	$-0.800599\pi$
$719$	−43.4455	−1.62024	−0.810121	+	0.586262i	$0.800599\pi$
$720$	0	0
$721$	−57.9116	−2.15674
$722$	0	0
$723$	−33.6023	−1.24968
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−14.5677	−0.540286	−0.270143	−	0.962820i	$-0.587071\pi$
$727$	−14.5677	−0.540286	−0.270143	+	0.962820i	$0.587071\pi$
$728$	0	0
$729$	29.0513	1.07597
$730$	0	0
$731$	1.45312	0.0537457
$732$	0	0
$733$	−29.7741	−1.09973	−0.549866	−	0.835253i	$-0.685321\pi$
$733$	−29.7741	−1.09973	−0.549866	+	0.835253i	$0.685321\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14.7245	0.542382
$738$	0	0
$739$	−13.7526	−0.505898	−0.252949	−	0.967480i	$-0.581400\pi$
$739$	−13.7526	−0.505898	−0.252949	+	0.967480i	$0.581400\pi$
$740$	0	0
$741$	−2.41932	−0.0888760
$742$	0	0
$743$	−13.3186	−0.488613	−0.244307	−	0.969698i	$-0.578560\pi$
$743$	−13.3186	−0.488613	−0.244307	+	0.969698i	$0.578560\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−2.19733	−0.0803962
$748$	0	0
$749$	−27.9282	−1.02047
$750$	0	0
$751$	−15.1992	−0.554628	−0.277314	−	0.960779i	$-0.589444\pi$
$751$	−15.1992	−0.554628	−0.277314	+	0.960779i	$0.589444\pi$
$752$	0	0
$753$	4.81337	0.175409
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−12.8568	−0.467288	−0.233644	−	0.972322i	$-0.575065\pi$
$757$	−12.8568	−0.467288	−0.233644	+	0.972322i	$0.575065\pi$
$758$	0	0
$759$	−57.3192	−2.08056
$760$	0	0
$761$	−36.7187	−1.33105	−0.665526	−	0.746374i	$-0.731793\pi$
$761$	−36.7187	−1.33105	−0.665526	+	0.746374i	$0.731793\pi$
$762$	0	0
$763$	61.1049	2.21215
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−18.6039	−0.671746
$768$	0	0
$769$	29.0027	1.04586	0.522932	−	0.852374i	$-0.324838\pi$
$769$	29.0027	1.04586	0.522932	+	0.852374i	$0.324838\pi$
$770$	0	0
$771$	−2.45784	−0.0885169
$772$	0	0
$773$	−4.26719	−0.153480	−0.0767401	−	0.997051i	$-0.524451\pi$
$773$	−4.26719	−0.153480	−0.0767401	+	0.997051i	$0.524451\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−71.9206	−2.58014
$778$	0	0
$779$	2.77589	0.0994567
$780$	0	0
$781$	−19.6078	−0.701623
$782$	0	0
$783$	21.2832	0.760599
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−15.2379	−0.543174	−0.271587	−	0.962414i	$-0.587548\pi$
$787$	−15.2379	−0.543174	−0.271587	+	0.962414i	$0.587548\pi$
$788$	0	0
$789$	44.0112	1.56684
$790$	0	0
$791$	0.0318682	0.00113310
$792$	0	0
$793$	5.37154	0.190749
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−4.29102	−0.151996	−0.0759979	−	0.997108i	$-0.524214\pi$
$797$	−4.29102	−0.151996	−0.0759979	+	0.997108i	$0.524214\pi$
$798$	0	0
$799$	−1.00893	−0.0356933
$800$	0	0
$801$	2.72464	0.0962706
$802$	0	0
$803$	48.2931	1.70423
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.99045	0.246075
$808$	0	0
$809$	−18.8191	−0.661643	−0.330821	−	0.943693i	$-0.607326\pi$
$809$	−18.8191	−0.661643	−0.330821	+	0.943693i	$0.607326\pi$
$810$	0	0
$811$	−40.4529	−1.42049	−0.710246	−	0.703953i	$-0.751416\pi$
$811$	−40.4529	−1.42049	−0.710246	+	0.703953i	$0.751416\pi$
$812$	0	0
$813$	31.3595	1.09982
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7.00055	0.244918
$818$	0	0
$819$	−1.95257	−0.0682284
$820$	0	0
$821$	14.6674	0.511896	0.255948	−	0.966691i	$-0.417612\pi$
$821$	14.6674	0.511896	0.255948	+	0.966691i	$0.417612\pi$
$822$	0	0
$823$	30.8675	1.07597	0.537987	−	0.842953i	$-0.319185\pi$
$823$	30.8675	1.07597	0.537987	+	0.842953i	$0.319185\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	41.7259	1.45095	0.725475	−	0.688248i	$-0.241620\pi$
$827$	41.7259	1.45095	0.725475	+	0.688248i	$0.241620\pi$
$828$	0	0
$829$	−55.1198	−1.91439	−0.957195	−	0.289445i	$-0.906529\pi$
$829$	−55.1198	−1.91439	−0.957195	+	0.289445i	$0.906529\pi$
$830$	0	0
$831$	34.1187	1.18357
$832$	0	0
$833$	3.70667	0.128428
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−47.0675	−1.62689
$838$	0	0
$839$	−40.3227	−1.39209	−0.696046	−	0.717997i	$-0.745059\pi$
$839$	−40.3227	−1.39209	−0.696046	+	0.717997i	$0.745059\pi$
$840$	0	0
$841$	−13.5369	−0.466791
$842$	0	0
$843$	40.2452	1.38612
$844$	0	0
$845$	0	0
$846$	0	0
$847$	51.4614	1.76823
$848$	0	0
$849$	19.9843	0.685860
$850$	0	0
$851$	−66.7862	−2.28940
$852$	0	0
$853$	−4.92347	−0.168577	−0.0842883	−	0.996441i	$-0.526862\pi$
$853$	−4.92347	−0.168577	−0.0842883	+	0.996441i	$0.526862\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−31.8663	−1.08853	−0.544266	−	0.838913i	$-0.683192\pi$
$857$	−31.8663	−1.08853	−0.544266	+	0.838913i	$0.683192\pi$
$858$	0	0
$859$	−47.9469	−1.63593	−0.817964	−	0.575269i	$-0.804897\pi$
$859$	−47.9469	−1.63593	−0.817964	+	0.575269i	$0.804897\pi$
$860$	0	0
$861$	−21.3918	−0.729031
$862$	0	0
$863$	−11.4172	−0.388645	−0.194323	−	0.980938i	$-0.562251\pi$
$863$	−11.4172	−0.388645	−0.194323	+	0.980938i	$0.562251\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	27.9364	0.948769
$868$	0	0
$869$	−21.9453	−0.744444
$870$	0	0
$871$	−4.44744	−0.150696
$872$	0	0
$873$	1.51989	0.0514406
$874$	0	0
$875$	0	0
$876$	0	0
$877$	34.6526	1.17014	0.585068	−	0.810984i	$-0.301068\pi$
$877$	34.6526	1.17014	0.585068	+	0.810984i	$0.301068\pi$
$878$	0	0
$879$	−4.35279	−0.146816
$880$	0	0
$881$	−7.94774	−0.267766	−0.133883	−	0.990997i	$-0.542745\pi$
$881$	−7.94774	−0.267766	−0.133883	+	0.990997i	$0.542745\pi$
$882$	0	0
$883$	47.2816	1.59115	0.795576	−	0.605854i	$-0.207168\pi$
$883$	47.2816	1.59115	0.795576	+	0.605854i	$0.207168\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3.20709	−0.107683	−0.0538417	−	0.998549i	$-0.517147\pi$
$887$	−3.20709	−0.107683	−0.0538417	+	0.998549i	$0.517147\pi$
$888$	0	0
$889$	−50.8410	−1.70515
$890$	0	0
$891$	37.3975	1.25286
$892$	0	0
$893$	−4.86060	−0.162654
$894$	0	0
$895$	0	0
$896$	0	0
$897$	17.3130	0.578063
$898$	0	0
$899$	−34.1963	−1.14051
$900$	0	0
$901$	−0.296706	−0.00988471
$902$	0	0
$903$	−53.9482	−1.79528
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−27.1200	−0.900506	−0.450253	−	0.892901i	$-0.648666\pi$
$907$	−27.1200	−0.900506	−0.450253	+	0.892901i	$0.648666\pi$
$908$	0	0
$909$	3.87941	0.128672
$910$	0	0
$911$	−11.8377	−0.392199	−0.196100	−	0.980584i	$-0.562828\pi$
$911$	−11.8377	−0.392199	−0.196100	+	0.980584i	$0.562828\pi$
$912$	0	0
$913$	−35.8221	−1.18554
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−38.1483	−1.25977
$918$	0	0
$919$	17.6313	0.581604	0.290802	−	0.956783i	$-0.406078\pi$
$919$	17.6313	0.581604	0.290802	+	0.956783i	$0.406078\pi$
$920$	0	0
$921$	12.2931	0.405072
$922$	0	0
$923$	5.92243	0.194939
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3.35960	0.110344
$928$	0	0
$929$	15.4866	0.508098	0.254049	−	0.967191i	$-0.418238\pi$
$929$	15.4866	0.508098	0.254049	+	0.967191i	$0.418238\pi$
$930$	0	0
$931$	17.8572	0.585247
$932$	0	0
$933$	−35.5581	−1.16412
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−52.5603	−1.71707	−0.858534	−	0.512756i	$-0.828625\pi$
$937$	−52.5603	−1.71707	−0.858534	+	0.512756i	$0.828625\pi$
$938$	0	0
$939$	22.3231	0.728486
$940$	0	0
$941$	42.6648	1.39083	0.695416	−	0.718607i	$-0.255220\pi$
$941$	42.6648	1.39083	0.695416	+	0.718607i	$0.255220\pi$
$942$	0	0
$943$	−19.8646	−0.646881
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−52.3731	−1.70190	−0.850948	−	0.525250i	$-0.823972\pi$
$947$	−52.3731	−1.70190	−0.850948	+	0.525250i	$0.823972\pi$
$948$	0	0
$949$	−14.5867	−0.473503
$950$	0	0
$951$	−23.0758	−0.748286
$952$	0	0
$953$	−36.7756	−1.19128	−0.595640	−	0.803252i	$-0.703101\pi$
$953$	−36.7756	−1.19128	−0.595640	+	0.803252i	$0.703101\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	30.0449	0.971213
$958$	0	0
$959$	7.78316	0.251331
$960$	0	0
$961$	44.6246	1.43950
$962$	0	0
$963$	1.62019	0.0522098
$964$	0	0
$965$	0	0
$966$	0	0
$967$	25.7549	0.828223	0.414112	−	0.910226i	$-0.364092\pi$
$967$	25.7549	0.828223	0.414112	+	0.910226i	$0.364092\pi$
$968$	0	0
$969$	−0.375926	−0.0120765
$970$	0	0
$971$	42.7716	1.37261	0.686303	−	0.727315i	$-0.259232\pi$
$971$	42.7716	1.37261	0.686303	+	0.727315i	$0.259232\pi$
$972$	0	0
$973$	21.1745	0.678825
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−21.2176	−0.678811	−0.339406	−	0.940640i	$-0.610226\pi$
$977$	−21.2176	−0.678811	−0.339406	+	0.940640i	$0.610226\pi$
$978$	0	0
$979$	44.4186	1.41963
$980$	0	0
$981$	−3.54486	−0.113179
$982$	0	0
$983$	−52.8489	−1.68562	−0.842810	−	0.538212i	$-0.819100\pi$
$983$	−52.8489	−1.68562	−0.842810	+	0.538212i	$0.819100\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	37.4571	1.19227
$988$	0	0
$989$	−50.0968	−1.59298
$990$	0	0
$991$	−29.3077	−0.930988	−0.465494	−	0.885051i	$-0.654123\pi$
$991$	−29.3077	−0.930988	−0.465494	+	0.885051i	$0.654123\pi$
$992$	0	0
$993$	−39.3392	−1.24839
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0.0369248	0.00116942	0.000584710	−	1.00000i	$-0.499814\pi$
$997$	0.0369248	0.00116942	0.000584710		1.00000i	$0.499814\pi$
$998$	0	0
$999$	48.1835	1.52446

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.v.1.2		4
4.3	odd	2		1250.2.a.g.1.3	✓	4
5.4	even	2		10000.2.a.u.1.3		4
20.3	even	4		1250.2.b.d.1249.7		8
20.7	even	4		1250.2.b.d.1249.2		8
20.19	odd	2		1250.2.a.j.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1250.2.a.g.1.3	✓	4	4.3	odd	2
1250.2.a.j.1.2	yes	4	20.19	odd	2
1250.2.b.d.1249.2		8	20.7	even	4
1250.2.b.d.1249.7		8	20.3	even	4
10000.2.a.u.1.3		4	5.4	even	2
10000.2.a.v.1.2		4	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-1.64791 q^{3} +4.90244 q^{7} -0.284403 q^{9} +O(q^{10})\)	\(q-1.64791 q^{3} +4.90244 q^{7} -0.284403 q^{9} -4.63650 q^{11} +1.40043 q^{13} +0.217606 q^{17} +1.04834 q^{19} -8.07876 q^{21} -7.50201 q^{23} +5.41239 q^{27} +3.93231 q^{29} -8.69624 q^{31} +7.64051 q^{33} +8.90244 q^{37} -2.30778 q^{39} +2.64791 q^{41} +6.67778 q^{43} -4.63650 q^{47} +17.0339 q^{49} -0.358594 q^{51} -1.36350 q^{53} -1.72756 q^{57} -13.2844 q^{59} +3.83564 q^{61} -1.39427 q^{63} -3.17577 q^{67} +12.3626 q^{69} +4.22902 q^{71} -10.4159 q^{73} -22.7301 q^{77} +4.73317 q^{79} -8.06591 q^{81} +7.72611 q^{83} -6.48008 q^{87} -9.58022 q^{89} +6.86551 q^{91} +14.3306 q^{93} -5.34415 q^{97} +1.31863 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} - 3 q^{7} + 17 q^{9}+O(q^{10}) \) 4 * q + q^3 - 3 * q^7 + 17 * q^9	\( 4 q + q^{3} - 3 q^{7} + 17 q^{9} - 8 q^{11} + 4 q^{13} - 2 q^{17} - 5 q^{19} - 7 q^{21} - 9 q^{23} + 10 q^{27} - 10 q^{29} - 18 q^{31} - 22 q^{33} + 13 q^{37} + 16 q^{39} + 3 q^{41} + 16 q^{43} - 8 q^{47} + 23 q^{49} - 13 q^{51} - 16 q^{53} - 15 q^{57} - 35 q^{59} + 8 q^{61} - 54 q^{63} - 23 q^{67} + 19 q^{69} + 17 q^{71} - q^{73} - 29 q^{77} - 10 q^{79} + 24 q^{81} + 11 q^{83} - 40 q^{87} - 5 q^{89} + 17 q^{91} + 38 q^{93} + 3 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q + q^3 - 3 * q^7 + 17 * q^9 - 8 * q^11 + 4 * q^13 - 2 * q^17 - 5 * q^19 - 7 * q^21 - 9 * q^23 + 10 * q^27 - 10 * q^29 - 18 * q^31 - 22 * q^33 + 13 * q^37 + 16 * q^39 + 3 * q^41 + 16 * q^43 - 8 * q^47 + 23 * q^49 - 13 * q^51 - 16 * q^53 - 15 * q^57 - 35 * q^59 + 8 * q^61 - 54 * q^63 - 23 * q^67 + 19 * q^69 + 17 * q^71 - q^73 - 29 * q^77 - 10 * q^79 + 24 * q^81 + 11 * q^83 - 40 * q^87 - 5 * q^89 + 17 * q^91 + 38 * q^93 + 3 * q^97 - 4 * q^99

Introduction

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