LMFDB - Embedded newform 7500.2.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7500,2,Mod(1,7500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7500, 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("7500.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7500 = 2^{2} \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7500.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:	$59.8878015160$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.5125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 6x^{2} + 7x + 11 $ x^4 - 2x^3 - 6x^2 + 7*x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 300)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.70636$ of defining polynomial
Character	$\chi$	$=$	7500.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.0883282 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.0883282 q^{7} +1.00000 q^{9} +2.26981 q^{11} -2.65177 q^{13} -2.08833 q^{17} +1.76095 q^{19} +0.0883282 q^{21} -4.74010 q^{23} +1.00000 q^{27} +3.70636 q^{29} -4.10620 q^{31} +2.26981 q^{33} -7.11909 q^{37} -2.65177 q^{39} -6.58938 q^{41} -1.79469 q^{43} -10.1110 q^{47} -6.99220 q^{49} -2.08833 q^{51} +0.961440 q^{53} +1.76095 q^{57} -8.97801 q^{59} +9.46618 q^{61} +0.0883282 q^{63} -13.9039 q^{67} -4.74010 q^{69} +6.14774 q^{71} +3.15765 q^{73} +0.200488 q^{77} +13.3522 q^{79} +1.00000 q^{81} +15.8195 q^{83} +3.70636 q^{87} -10.2508 q^{89} -0.234226 q^{91} -4.10620 q^{93} +12.8077 q^{97} +2.26981 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 4 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} - 4 q^{7} + 4 q^{9} - q^{11} - 5 q^{13} - 4 q^{17} - 5 q^{19} - 4 q^{21} - 9 q^{23} + 4 q^{27} + 6 q^{29} + 11 q^{31} - q^{33} - 2 q^{37} - 5 q^{39} + 6 q^{43} - 16 q^{47} - 4 q^{49} - 4 q^{51} + 2 q^{53} - 5 q^{57} + q^{59} - 22 q^{61} - 4 q^{63} - 36 q^{67} - 9 q^{69} + 20 q^{71} - 12 q^{73} + 11 q^{77} - 3 q^{79} + 4 q^{81} - 14 q^{83} + 6 q^{87} - 15 q^{89} - 10 q^{91} + 11 q^{93} + 12 q^{97} - q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^7 + 4 * q^9 - q^11 - 5 * q^13 - 4 * q^17 - 5 * q^19 - 4 * q^21 - 9 * q^23 + 4 * q^27 + 6 * q^29 + 11 * q^31 - q^33 - 2 * q^37 - 5 * q^39 + 6 * q^43 - 16 * q^47 - 4 * q^49 - 4 * q^51 + 2 * q^53 - 5 * q^57 + q^59 - 22 * q^61 - 4 * q^63 - 36 * q^67 - 9 * q^69 + 20 * q^71 - 12 * q^73 + 11 * q^77 - 3 * q^79 + 4 * q^81 - 14 * q^83 + 6 * q^87 - 15 * q^89 - 10 * q^91 + 11 * q^93 + 12 * q^97 - 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.0883282	0.0333849	0.0166925	−	0.999861i	$-0.494686\pi$
$7$	0.0883282	0.0333849	0.0166925	+	0.999861i	$0.494686\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.26981	0.684372	0.342186	−	0.939632i	$-0.388833\pi$
$11$	2.26981	0.684372	0.342186	+	0.939632i	$0.388833\pi$
$12$	0	0
$13$	−2.65177	−0.735469	−0.367735	−	0.929931i	$-0.619867\pi$
$13$	−2.65177	−0.735469	−0.367735	+	0.929931i	$0.619867\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.08833	−0.506494	−0.253247	−	0.967402i	$-0.581499\pi$
$17$	−2.08833	−0.506494	−0.253247	+	0.967402i	$0.581499\pi$
$18$	0	0
$19$	1.76095	0.403990	0.201995	−	0.979387i	$-0.435258\pi$
$19$	1.76095	0.403990	0.201995	+	0.979387i	$0.435258\pi$
$20$	0	0
$21$	0.0883282	0.0192748
$22$	0	0
$23$	−4.74010	−0.988379	−0.494190	−	0.869354i	$-0.664535\pi$
$23$	−4.74010	−0.988379	−0.494190	+	0.869354i	$0.664535\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.70636	0.688254	0.344127	−	0.938923i	$-0.388175\pi$
$29$	3.70636	0.688254	0.344127	+	0.938923i	$0.388175\pi$
$30$	0	0
$31$	−4.10620	−0.737495	−0.368748	−	0.929530i	$-0.620213\pi$
$31$	−4.10620	−0.737495	−0.368748	+	0.929530i	$0.620213\pi$
$32$	0	0
$33$	2.26981	0.395123
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.11909	−1.17037	−0.585185	−	0.810900i	$-0.698978\pi$
$37$	−7.11909	−1.17037	−0.585185	+	0.810900i	$0.698978\pi$
$38$	0	0
$39$	−2.65177	−0.424623
$40$	0	0
$41$	−6.58938	−1.02909	−0.514544	−	0.857464i	$-0.672039\pi$
$41$	−6.58938	−1.02909	−0.514544	+	0.857464i	$0.672039\pi$
$42$	0	0
$43$	−1.79469	−0.273688	−0.136844	−	0.990593i	$-0.543696\pi$
$43$	−1.79469	−0.273688	−0.136844	+	0.990593i	$0.543696\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.1110	−1.47484	−0.737422	−	0.675432i	$-0.763957\pi$
$47$	−10.1110	−1.47484	−0.737422	+	0.675432i	$0.763957\pi$
$48$	0	0
$49$	−6.99220	−0.998885
$50$	0	0
$51$	−2.08833	−0.292424
$52$	0	0
$53$	0.961440	0.132064	0.0660320	−	0.997818i	$-0.478966\pi$
$53$	0.961440	0.132064	0.0660320	+	0.997818i	$0.478966\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.76095	0.233244
$58$	0	0
$59$	−8.97801	−1.16884	−0.584419	−	0.811452i	$-0.698677\pi$
$59$	−8.97801	−1.16884	−0.584419	+	0.811452i	$0.698677\pi$
$60$	0	0
$61$	9.46618	1.21202	0.606010	−	0.795457i	$-0.292769\pi$
$61$	9.46618	1.21202	0.606010	+	0.795457i	$0.292769\pi$
$62$	0	0
$63$	0.0883282	0.0111283
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−13.9039	−1.69863	−0.849314	−	0.527888i	$-0.822984\pi$
$67$	−13.9039	−1.69863	−0.849314	+	0.527888i	$0.822984\pi$
$68$	0	0
$69$	−4.74010	−0.570641
$70$	0	0
$71$	6.14774	0.729602	0.364801	−	0.931085i	$-0.381137\pi$
$71$	6.14774	0.729602	0.364801	+	0.931085i	$0.381137\pi$
$72$	0	0
$73$	3.15765	0.369575	0.184787	−	0.982779i	$-0.440840\pi$
$73$	3.15765	0.369575	0.184787	+	0.982779i	$0.440840\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.200488	0.0228477
$78$	0	0
$79$	13.3522	1.50224	0.751118	−	0.660167i	$-0.229515\pi$
$79$	13.3522	1.50224	0.751118	+	0.660167i	$0.229515\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	15.8195	1.73641	0.868207	−	0.496202i	$-0.165272\pi$
$83$	15.8195	1.73641	0.868207	+	0.496202i	$0.165272\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.70636	0.397364
$88$	0	0
$89$	−10.2508	−1.08658	−0.543291	−	0.839544i	$-0.682822\pi$
$89$	−10.2508	−1.08658	−0.543291	+	0.839544i	$0.682822\pi$
$90$	0	0
$91$	−0.234226	−0.0245536
$92$	0	0
$93$	−4.10620	−0.425793
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.8077	1.30043	0.650214	−	0.759751i	$-0.274679\pi$
$97$	12.8077	1.30043	0.650214	+	0.759751i	$0.274679\pi$
$98$	0	0
$99$	2.26981	0.228124
$100$	0	0
$101$	−2.72537	−0.271185	−0.135592	−	0.990765i	$-0.543294\pi$
$101$	−2.72537	−0.271185	−0.135592	+	0.990765i	$0.543294\pi$
$102$	0	0
$103$	0.359976	0.0354695	0.0177348	−	0.999843i	$-0.494355\pi$
$103$	0.359976	0.0354695	0.0177348	+	0.999843i	$0.494355\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.0286533	−0.00277002	−0.00138501	−	0.999999i	$-0.500441\pi$
$107$	−0.0286533	−0.00277002	−0.00138501	+	0.999999i	$0.500441\pi$
$108$	0	0
$109$	−18.9217	−1.81237	−0.906187	−	0.422877i	$-0.861020\pi$
$109$	−18.9217	−1.81237	−0.906187	+	0.422877i	$0.861020\pi$
$110$	0	0
$111$	−7.11909	−0.675714
$112$	0	0
$113$	−4.89380	−0.460370	−0.230185	−	0.973147i	$-0.573933\pi$
$113$	−4.89380	−0.460370	−0.230185	+	0.973147i	$0.573933\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.65177	−0.245156
$118$	0	0
$119$	−0.184458	−0.0169093
$120$	0	0
$121$	−5.84798	−0.531634
$122$	0	0
$123$	−6.58938	−0.594144
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.26997	0.290163	0.145081	−	0.989420i	$-0.453656\pi$
$127$	3.26997	0.290163	0.145081	+	0.989420i	$0.453656\pi$
$128$	0	0
$129$	−1.79469	−0.158014
$130$	0	0
$131$	3.59550	0.314141	0.157070	−	0.987587i	$-0.449795\pi$
$131$	3.59550	0.314141	0.157070	+	0.987587i	$0.449795\pi$
$132$	0	0
$133$	0.155542	0.0134872
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−17.0197	−1.45409	−0.727046	−	0.686589i	$-0.759107\pi$
$137$	−17.0197	−1.45409	−0.727046	+	0.686589i	$0.759107\pi$
$138$	0	0
$139$	−18.9860	−1.61037	−0.805185	−	0.593024i	$-0.797934\pi$
$139$	−18.9860	−1.61037	−0.805185	+	0.593024i	$0.797934\pi$
$140$	0	0
$141$	−10.1110	−0.851502
$142$	0	0
$143$	−6.01901	−0.503335
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−6.99220	−0.576707
$148$	0	0
$149$	20.3441	1.66665	0.833327	−	0.552780i	$-0.186433\pi$
$149$	20.3441	1.66665	0.833327	+	0.552780i	$0.186433\pi$
$150$	0	0
$151$	13.2609	1.07915	0.539577	−	0.841936i	$-0.318584\pi$
$151$	13.2609	1.07915	0.539577	+	0.841936i	$0.318584\pi$
$152$	0	0
$153$	−2.08833	−0.168831
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−12.8066	−1.02208	−0.511039	−	0.859558i	$-0.670739\pi$
$157$	−12.8066	−1.02208	−0.511039	+	0.859558i	$0.670739\pi$
$158$	0	0
$159$	0.961440	0.0762471
$160$	0	0
$161$	−0.418684	−0.0329970
$162$	0	0
$163$	−15.0647	−1.17996	−0.589978	−	0.807420i	$-0.700863\pi$
$163$	−15.0647	−1.17996	−0.589978	+	0.807420i	$0.700863\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.45128	0.267068	0.133534	−	0.991044i	$-0.457367\pi$
$167$	3.45128	0.267068	0.133534	+	0.991044i	$0.457367\pi$
$168$	0	0
$169$	−5.96810	−0.459085
$170$	0	0
$171$	1.76095	0.134663
$172$	0	0
$173$	2.41457	0.183576	0.0917880	−	0.995779i	$-0.470742\pi$
$173$	2.41457	0.183576	0.0917880	+	0.995779i	$0.470742\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−8.97801	−0.674829
$178$	0	0
$179$	−4.06948	−0.304167	−0.152084	−	0.988368i	$-0.548598\pi$
$179$	−4.06948	−0.304167	−0.152084	+	0.988368i	$0.548598\pi$
$180$	0	0
$181$	−13.3363	−0.991280	−0.495640	−	0.868528i	$-0.665066\pi$
$181$	−13.3363	−0.991280	−0.495640	+	0.868528i	$0.665066\pi$
$182$	0	0
$183$	9.46618	0.699760
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.74010	−0.346630
$188$	0	0
$189$	0.0883282	0.00642493
$190$	0	0
$191$	25.2529	1.82724	0.913618	−	0.406574i	$-0.133277\pi$
$191$	25.2529	1.82724	0.913618	+	0.406574i	$0.133277\pi$
$192$	0	0
$193$	24.6399	1.77362	0.886808	−	0.462139i	$-0.152918\pi$
$193$	24.6399	1.77362	0.886808	+	0.462139i	$0.152918\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−18.6201	−1.32663	−0.663315	−	0.748340i	$-0.730851\pi$
$197$	−18.6201	−1.32663	−0.663315	+	0.748340i	$0.730851\pi$
$198$	0	0
$199$	9.85708	0.698750	0.349375	−	0.936983i	$-0.386394\pi$
$199$	9.85708	0.698750	0.349375	+	0.936983i	$0.386394\pi$
$200$	0	0
$201$	−13.9039	−0.980703
$202$	0	0
$203$	0.327376	0.0229773
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.74010	−0.329460
$208$	0	0
$209$	3.99702	0.276480
$210$	0	0
$211$	−7.89878	−0.543775	−0.271887	−	0.962329i	$-0.587648\pi$
$211$	−7.89878	−0.543775	−0.271887	+	0.962329i	$0.587648\pi$
$212$	0	0
$213$	6.14774	0.421236
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−0.362693	−0.0246212
$218$	0	0
$219$	3.15765	0.213374
$220$	0	0
$221$	5.53777	0.372511
$222$	0	0
$223$	9.18174	0.614855	0.307427	−	0.951572i	$-0.400532\pi$
$223$	9.18174	0.614855	0.307427	+	0.951572i	$0.400532\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.5654	−0.767626	−0.383813	−	0.923411i	$-0.625389\pi$
$227$	−11.5654	−0.767626	−0.383813	+	0.923411i	$0.625389\pi$
$228$	0	0
$229$	24.9463	1.64850	0.824248	−	0.566229i	$-0.191598\pi$
$229$	24.9463	1.64850	0.824248	+	0.566229i	$0.191598\pi$
$230$	0	0
$231$	0.200488	0.0131911
$232$	0	0
$233$	−13.0200	−0.852967	−0.426484	−	0.904495i	$-0.640248\pi$
$233$	−13.0200	−0.852967	−0.426484	+	0.904495i	$0.640248\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	13.3522	0.867317
$238$	0	0
$239$	−26.4086	−1.70823	−0.854115	−	0.520084i	$-0.825901\pi$
$239$	−26.4086	−1.70823	−0.854115	+	0.520084i	$0.825901\pi$
$240$	0	0
$241$	−6.23591	−0.401690	−0.200845	−	0.979623i	$-0.564369\pi$
$241$	−6.23591	−0.401690	−0.200845	+	0.979623i	$0.564369\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.66964	−0.297122
$248$	0	0
$249$	15.8195	1.00252
$250$	0	0
$251$	7.46802	0.471377	0.235689	−	0.971829i	$-0.424265\pi$
$251$	7.46802	0.471377	0.235689	+	0.971829i	$0.424265\pi$
$252$	0	0
$253$	−10.7591	−0.676419
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−8.39880	−0.523903	−0.261951	−	0.965081i	$-0.584366\pi$
$257$	−8.39880	−0.523903	−0.261951	+	0.965081i	$0.584366\pi$
$258$	0	0
$259$	−0.628816	−0.0390727
$260$	0	0
$261$	3.70636	0.229418
$262$	0	0
$263$	10.0248	0.618156	0.309078	−	0.951037i	$-0.399980\pi$
$263$	10.0248	0.618156	0.309078	+	0.951037i	$0.399980\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−10.2508	−0.627339
$268$	0	0
$269$	−17.6192	−1.07426	−0.537130	−	0.843500i	$-0.680491\pi$
$269$	−17.6192	−1.07426	−0.537130	+	0.843500i	$0.680491\pi$
$270$	0	0
$271$	−4.85426	−0.294876	−0.147438	−	0.989071i	$-0.547103\pi$
$271$	−4.85426	−0.294876	−0.147438	+	0.989071i	$0.547103\pi$
$272$	0	0
$273$	−0.234226	−0.0141760
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−22.6120	−1.35862	−0.679311	−	0.733851i	$-0.737721\pi$
$277$	−22.6120	−1.35862	−0.679311	+	0.733851i	$0.737721\pi$
$278$	0	0
$279$	−4.10620	−0.245832
$280$	0	0
$281$	29.2542	1.74516	0.872580	−	0.488472i	$-0.162445\pi$
$281$	29.2542	1.74516	0.872580	+	0.488472i	$0.162445\pi$
$282$	0	0
$283$	−29.2529	−1.73890	−0.869452	−	0.494017i	$-0.835528\pi$
$283$	−29.2529	−1.73890	−0.869452	+	0.494017i	$0.835528\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.582028	−0.0343560
$288$	0	0
$289$	−12.6389	−0.743464
$290$	0	0
$291$	12.8077	0.750803
$292$	0	0
$293$	13.4104	0.783447	0.391723	−	0.920083i	$-0.371879\pi$
$293$	13.4104	0.783447	0.391723	+	0.920083i	$0.371879\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.26981	0.131708
$298$	0	0
$299$	12.5697	0.726923
$300$	0	0
$301$	−0.158522	−0.00913704
$302$	0	0
$303$	−2.72537	−0.156569
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−13.5444	−0.773022	−0.386511	−	0.922285i	$-0.626320\pi$
$307$	−13.5444	−0.773022	−0.386511	+	0.922285i	$0.626320\pi$
$308$	0	0
$309$	0.359976	0.0204783
$310$	0	0
$311$	−2.03882	−0.115611	−0.0578055	−	0.998328i	$-0.518410\pi$
$311$	−2.03882	−0.115611	−0.0578055	+	0.998328i	$0.518410\pi$
$312$	0	0
$313$	18.2786	1.03317	0.516583	−	0.856237i	$-0.327204\pi$
$313$	18.2786	1.03317	0.516583	+	0.856237i	$0.327204\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.7978	1.05579	0.527896	−	0.849309i	$-0.322981\pi$
$317$	18.7978	1.05579	0.527896	+	0.849309i	$0.322981\pi$
$318$	0	0
$319$	8.41272	0.471022
$320$	0	0
$321$	−0.0286533	−0.00159927
$322$	0	0
$323$	−3.67745	−0.204619
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−18.9217	−1.04637
$328$	0	0
$329$	−0.893088	−0.0492375
$330$	0	0
$331$	−8.32012	−0.457315	−0.228657	−	0.973507i	$-0.573434\pi$
$331$	−8.32012	−0.457315	−0.228657	+	0.973507i	$0.573434\pi$
$332$	0	0
$333$	−7.11909	−0.390124
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−17.5942	−0.958417	−0.479209	−	0.877701i	$-0.659076\pi$
$337$	−17.5942	−0.958417	−0.479209	+	0.877701i	$0.659076\pi$
$338$	0	0
$339$	−4.89380	−0.265795
$340$	0	0
$341$	−9.32028	−0.504721
$342$	0	0
$343$	−1.23591	−0.0667326
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−30.8192	−1.65446	−0.827231	−	0.561861i	$-0.810085\pi$
$347$	−30.8192	−1.65446	−0.827231	+	0.561861i	$0.810085\pi$
$348$	0	0
$349$	22.0376	1.17964	0.589822	−	0.807533i	$-0.299198\pi$
$349$	22.0376	1.17964	0.589822	+	0.807533i	$0.299198\pi$
$350$	0	0
$351$	−2.65177	−0.141541
$352$	0	0
$353$	−6.11611	−0.325527	−0.162764	−	0.986665i	$-0.552041\pi$
$353$	−6.11611	−0.325527	−0.162764	+	0.986665i	$0.552041\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.184458	−0.00976256
$358$	0	0
$359$	14.8107	0.781680	0.390840	−	0.920459i	$-0.372185\pi$
$359$	14.8107	0.781680	0.390840	+	0.920459i	$0.372185\pi$
$360$	0	0
$361$	−15.8990	−0.836792
$362$	0	0
$363$	−5.84798	−0.306939
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−13.3115	−0.694855	−0.347428	−	0.937707i	$-0.612945\pi$
$367$	−13.3115	−0.694855	−0.347428	+	0.937707i	$0.612945\pi$
$368$	0	0
$369$	−6.58938	−0.343029
$370$	0	0
$371$	0.0849222	0.00440894
$372$	0	0
$373$	17.0229	0.881410	0.440705	−	0.897652i	$-0.354728\pi$
$373$	17.0229	0.881410	0.440705	+	0.897652i	$0.354728\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9.82843	−0.506190
$378$	0	0
$379$	−2.83465	−0.145606	−0.0728031	−	0.997346i	$-0.523194\pi$
$379$	−2.83465	−0.145606	−0.0728031	+	0.997346i	$0.523194\pi$
$380$	0	0
$381$	3.26997	0.167526
$382$	0	0
$383$	−10.0485	−0.513453	−0.256726	−	0.966484i	$-0.582644\pi$
$383$	−10.0485	−0.513453	−0.256726	+	0.966484i	$0.582644\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.79469	−0.0912292
$388$	0	0
$389$	8.37128	0.424441	0.212220	−	0.977222i	$-0.431930\pi$
$389$	8.37128	0.424441	0.212220	+	0.977222i	$0.431930\pi$
$390$	0	0
$391$	9.89889	0.500608
$392$	0	0
$393$	3.59550	0.181369
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8.38397	0.420779	0.210390	−	0.977618i	$-0.432527\pi$
$397$	8.38397	0.420779	0.210390	+	0.977618i	$0.432527\pi$
$398$	0	0
$399$	0.155542	0.00778682
$400$	0	0
$401$	2.14450	0.107091	0.0535455	−	0.998565i	$-0.482948\pi$
$401$	2.14450	0.107091	0.0535455	+	0.998565i	$0.482948\pi$
$402$	0	0
$403$	10.8887	0.542405
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−16.1589	−0.800969
$408$	0	0
$409$	−3.58754	−0.177392	−0.0886962	−	0.996059i	$-0.528270\pi$
$409$	−3.58754	−0.177392	−0.0886962	+	0.996059i	$0.528270\pi$
$410$	0	0
$411$	−17.0197	−0.839521
$412$	0	0
$413$	−0.793011	−0.0390215
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−18.9860	−0.929747
$418$	0	0
$419$	−9.20715	−0.449799	−0.224899	−	0.974382i	$-0.572205\pi$
$419$	−9.20715	−0.449799	−0.224899	+	0.974382i	$0.572205\pi$
$420$	0	0
$421$	−40.9085	−1.99376	−0.996880	−	0.0789366i	$-0.974848\pi$
$421$	−40.9085	−1.99376	−0.996880	+	0.0789366i	$0.974848\pi$
$422$	0	0
$423$	−10.1110	−0.491615
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.836130	0.0404632
$428$	0	0
$429$	−6.01901	−0.290601
$430$	0	0
$431$	2.26272	0.108991	0.0544956	−	0.998514i	$-0.482645\pi$
$431$	2.26272	0.108991	0.0544956	+	0.998514i	$0.482645\pi$
$432$	0	0
$433$	4.57519	0.219870	0.109935	−	0.993939i	$-0.464936\pi$
$433$	4.57519	0.219870	0.109935	+	0.993939i	$0.464936\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.34709	−0.399295
$438$	0	0
$439$	−1.75299	−0.0836655	−0.0418328	−	0.999125i	$-0.513320\pi$
$439$	−1.75299	−0.0836655	−0.0418328	+	0.999125i	$0.513320\pi$
$440$	0	0
$441$	−6.99220	−0.332962
$442$	0	0
$443$	−20.8364	−0.989967	−0.494983	−	0.868902i	$-0.664826\pi$
$443$	−20.8364	−0.989967	−0.494983	+	0.868902i	$0.664826\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	20.3441	0.962243
$448$	0	0
$449$	25.1952	1.18904	0.594518	−	0.804082i	$-0.297343\pi$
$449$	25.1952	1.18904	0.594518	+	0.804082i	$0.297343\pi$
$450$	0	0
$451$	−14.9566	−0.704280
$452$	0	0
$453$	13.2609	0.623050
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−39.5166	−1.84851	−0.924255	−	0.381775i	$-0.875313\pi$
$457$	−39.5166	−1.84851	−0.924255	+	0.381775i	$0.875313\pi$
$458$	0	0
$459$	−2.08833	−0.0974748
$460$	0	0
$461$	14.5860	0.679337	0.339668	−	0.940545i	$-0.389685\pi$
$461$	14.5860	0.679337	0.339668	+	0.940545i	$0.389685\pi$
$462$	0	0
$463$	−9.87311	−0.458842	−0.229421	−	0.973327i	$-0.573683\pi$
$463$	−9.87311	−0.458842	−0.229421	+	0.973327i	$0.573683\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	21.1418	0.978325	0.489162	−	0.872193i	$-0.337303\pi$
$467$	21.1418	0.978325	0.489162	+	0.872193i	$0.337303\pi$
$468$	0	0
$469$	−1.22810	−0.0567085
$470$	0	0
$471$	−12.8066	−0.590097
$472$	0	0
$473$	−4.07360	−0.187304
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.961440	0.0440213
$478$	0	0
$479$	41.4475	1.89378	0.946892	−	0.321551i	$-0.104204\pi$
$479$	41.4475	1.89378	0.946892	+	0.321551i	$0.104204\pi$
$480$	0	0
$481$	18.8782	0.860772
$482$	0	0
$483$	−0.418684	−0.0190508
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.03970	0.183056	0.0915281	−	0.995802i	$-0.470825\pi$
$487$	4.03970	0.183056	0.0915281	+	0.995802i	$0.470825\pi$
$488$	0	0
$489$	−15.0647	−0.681247
$490$	0	0
$491$	28.9752	1.30763	0.653816	−	0.756654i	$-0.273167\pi$
$491$	28.9752	1.30763	0.653816	+	0.756654i	$0.273167\pi$
$492$	0	0
$493$	−7.74010	−0.348597
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.543019	0.0243577
$498$	0	0
$499$	23.6824	1.06017	0.530086	−	0.847944i	$-0.322160\pi$
$499$	23.6824	1.06017	0.530086	+	0.847944i	$0.322160\pi$
$500$	0	0
$501$	3.45128	0.154192
$502$	0	0
$503$	−13.3422	−0.594898	−0.297449	−	0.954738i	$-0.596136\pi$
$503$	−13.3422	−0.594898	−0.297449	+	0.954738i	$0.596136\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−5.96810	−0.265053
$508$	0	0
$509$	−10.3709	−0.459683	−0.229841	−	0.973228i	$-0.573821\pi$
$509$	−10.3709	−0.459683	−0.229841	+	0.973228i	$0.573821\pi$
$510$	0	0
$511$	0.278909	0.0123382
$512$	0	0
$513$	1.76095	0.0777479
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−22.9501	−1.00934
$518$	0	0
$519$	2.41457	0.105988
$520$	0	0
$521$	29.5976	1.29669	0.648347	−	0.761345i	$-0.275460\pi$
$521$	29.5976	1.29669	0.648347	+	0.761345i	$0.275460\pi$
$522$	0	0
$523$	−2.13978	−0.0935658	−0.0467829	−	0.998905i	$-0.514897\pi$
$523$	−2.13978	−0.0935658	−0.0467829	+	0.998905i	$0.514897\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.57509	0.373537
$528$	0	0
$529$	−0.531447	−0.0231064
$530$	0	0
$531$	−8.97801	−0.389612
$532$	0	0
$533$	17.4735	0.756863
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4.06948	−0.175611
$538$	0	0
$539$	−15.8709	−0.683610
$540$	0	0
$541$	−8.61207	−0.370262	−0.185131	−	0.982714i	$-0.559271\pi$
$541$	−8.61207	−0.370262	−0.185131	+	0.982714i	$0.559271\pi$
$542$	0	0
$543$	−13.3363	−0.572316
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−23.3085	−0.996601	−0.498300	−	0.867004i	$-0.666042\pi$
$547$	−23.3085	−0.996601	−0.498300	+	0.867004i	$0.666042\pi$
$548$	0	0
$549$	9.46618	0.404007
$550$	0	0
$551$	6.52673	0.278048
$552$	0	0
$553$	1.17937	0.0501520
$554$	0	0
$555$	0	0
$556$	0	0
$557$	37.2790	1.57956	0.789781	−	0.613389i	$-0.210194\pi$
$557$	37.2790	1.57956	0.789781	+	0.613389i	$0.210194\pi$
$558$	0	0
$559$	4.75911	0.201289
$560$	0	0
$561$	−4.74010	−0.200127
$562$	0	0
$563$	10.7486	0.453000	0.226500	−	0.974011i	$-0.427272\pi$
$563$	10.7486	0.453000	0.226500	+	0.974011i	$0.427272\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.0883282	0.00370943
$568$	0	0
$569$	7.54687	0.316381	0.158191	−	0.987409i	$-0.449434\pi$
$569$	7.54687	0.316381	0.158191	+	0.987409i	$0.449434\pi$
$570$	0	0
$571$	29.6221	1.23965	0.619824	−	0.784741i	$-0.287204\pi$
$571$	29.6221	1.23965	0.619824	+	0.784741i	$0.287204\pi$
$572$	0	0
$573$	25.2529	1.05496
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−15.5747	−0.648381	−0.324191	−	0.945992i	$-0.605092\pi$
$577$	−15.5747	−0.648381	−0.324191	+	0.945992i	$0.605092\pi$
$578$	0	0
$579$	24.6399	1.02400
$580$	0	0
$581$	1.39731	0.0579700
$582$	0	0
$583$	2.18228	0.0903809
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−39.0913	−1.61347	−0.806735	−	0.590913i	$-0.798768\pi$
$587$	−39.0913	−1.61347	−0.806735	+	0.590913i	$0.798768\pi$
$588$	0	0
$589$	−7.23082	−0.297941
$590$	0	0
$591$	−18.6201	−0.765930
$592$	0	0
$593$	18.6722	0.766775	0.383387	−	0.923588i	$-0.374757\pi$
$593$	18.6722	0.766775	0.383387	+	0.923588i	$0.374757\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.85708	0.403424
$598$	0	0
$599$	−14.0186	−0.572783	−0.286392	−	0.958113i	$-0.592456\pi$
$599$	−14.0186	−0.572783	−0.286392	+	0.958113i	$0.592456\pi$
$600$	0	0
$601$	−9.89791	−0.403744	−0.201872	−	0.979412i	$-0.564703\pi$
$601$	−9.89791	−0.403744	−0.201872	+	0.979412i	$0.564703\pi$
$602$	0	0
$603$	−13.9039	−0.566209
$604$	0	0
$605$	0	0
$606$	0	0
$607$	22.2953	0.904939	0.452469	−	0.891780i	$-0.350543\pi$
$607$	22.2953	0.904939	0.452469	+	0.891780i	$0.350543\pi$
$608$	0	0
$609$	0.327376	0.0132660
$610$	0	0
$611$	26.8121	1.08470
$612$	0	0
$613$	16.4288	0.663551	0.331776	−	0.943358i	$-0.392352\pi$
$613$	16.4288	0.663551	0.331776	+	0.943358i	$0.392352\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.78516	0.152385	0.0761924	−	0.997093i	$-0.475724\pi$
$617$	3.78516	0.152385	0.0761924	+	0.997093i	$0.475724\pi$
$618$	0	0
$619$	−0.422090	−0.0169652	−0.00848262	−	0.999964i	$-0.502700\pi$
$619$	−0.422090	−0.0169652	−0.00848262	+	0.999964i	$0.502700\pi$
$620$	0	0
$621$	−4.74010	−0.190214
$622$	0	0
$623$	−0.905434	−0.0362755
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.99702	0.159626
$628$	0	0
$629$	14.8670	0.592786
$630$	0	0
$631$	−34.9223	−1.39023	−0.695117	−	0.718897i	$-0.744647\pi$
$631$	−34.9223	−1.39023	−0.695117	+	0.718897i	$0.744647\pi$
$632$	0	0
$633$	−7.89878	−0.313949
$634$	0	0
$635$	0	0
$636$	0	0
$637$	18.5417	0.734650
$638$	0	0
$639$	6.14774	0.243201
$640$	0	0
$641$	3.65274	0.144275	0.0721373	−	0.997395i	$-0.477018\pi$
$641$	3.65274	0.144275	0.0721373	+	0.997395i	$0.477018\pi$
$642$	0	0
$643$	−34.3967	−1.35647	−0.678236	−	0.734844i	$-0.737255\pi$
$643$	−34.3967	−1.35647	−0.678236	+	0.734844i	$0.737255\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−31.4429	−1.23615	−0.618074	−	0.786120i	$-0.712087\pi$
$647$	−31.4429	−1.23615	−0.618074	+	0.786120i	$0.712087\pi$
$648$	0	0
$649$	−20.3783	−0.799920
$650$	0	0
$651$	−0.362693	−0.0142151
$652$	0	0
$653$	34.8800	1.36496	0.682481	−	0.730904i	$-0.260901\pi$
$653$	34.8800	1.36496	0.682481	+	0.730904i	$0.260901\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.15765	0.123192
$658$	0	0
$659$	−43.2173	−1.68351	−0.841754	−	0.539862i	$-0.818477\pi$
$659$	−43.2173	−1.68351	−0.841754	+	0.539862i	$0.818477\pi$
$660$	0	0
$661$	−10.0354	−0.390332	−0.195166	−	0.980770i	$-0.562525\pi$
$661$	−10.0354	−0.390332	−0.195166	+	0.980770i	$0.562525\pi$
$662$	0	0
$663$	5.53777	0.215069
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−17.5685	−0.680256
$668$	0	0
$669$	9.18174	0.354987
$670$	0	0
$671$	21.4864	0.829473
$672$	0	0
$673$	−29.8864	−1.15204	−0.576019	−	0.817437i	$-0.695394\pi$
$673$	−29.8864	−1.15204	−0.576019	+	0.817437i	$0.695394\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.6279	0.869662	0.434831	−	0.900512i	$-0.356808\pi$
$677$	22.6279	0.869662	0.434831	+	0.900512i	$0.356808\pi$
$678$	0	0
$679$	1.13128	0.0434147
$680$	0	0
$681$	−11.5654	−0.443189
$682$	0	0
$683$	−25.6607	−0.981880	−0.490940	−	0.871194i	$-0.663346\pi$
$683$	−25.6607	−0.981880	−0.490940	+	0.871194i	$0.663346\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	24.9463	0.951760
$688$	0	0
$689$	−2.54952	−0.0971290
$690$	0	0
$691$	−34.2631	−1.30343	−0.651716	−	0.758463i	$-0.725950\pi$
$691$	−34.2631	−1.30343	−0.651716	+	0.758463i	$0.725950\pi$
$692$	0	0
$693$	0.200488	0.00761590
$694$	0	0
$695$	0	0
$696$	0	0
$697$	13.7608	0.521227
$698$	0	0
$699$	−13.0200	−0.492461
$700$	0	0
$701$	42.0813	1.58939	0.794695	−	0.607009i	$-0.207631\pi$
$701$	42.0813	1.58939	0.794695	+	0.607009i	$0.207631\pi$
$702$	0	0
$703$	−12.5364	−0.472818
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−0.240727	−0.00905347
$708$	0	0
$709$	−20.7728	−0.780139	−0.390069	−	0.920785i	$-0.627549\pi$
$709$	−20.7728	−0.780139	−0.390069	+	0.920785i	$0.627549\pi$
$710$	0	0
$711$	13.3522	0.500746
$712$	0	0
$713$	19.4638	0.728925
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−26.4086	−0.986247
$718$	0	0
$719$	25.7878	0.961721	0.480860	−	0.876797i	$-0.340324\pi$
$719$	25.7878	0.961721	0.480860	+	0.876797i	$0.340324\pi$
$720$	0	0
$721$	0.0317960	0.00118415
$722$	0	0
$723$	−6.23591	−0.231916
$724$	0	0
$725$	0	0
$726$	0	0
$727$	9.26839	0.343746	0.171873	−	0.985119i	$-0.445018\pi$
$727$	9.26839	0.343746	0.171873	+	0.985119i	$0.445018\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	3.74790	0.138621
$732$	0	0
$733$	21.0388	0.777086	0.388543	−	0.921431i	$-0.372978\pi$
$733$	21.0388	0.777086	0.388543	+	0.921431i	$0.372978\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−31.5591	−1.16249
$738$	0	0
$739$	−8.85805	−0.325849	−0.162924	−	0.986639i	$-0.552093\pi$
$739$	−8.85805	−0.325849	−0.162924	+	0.986639i	$0.552093\pi$
$740$	0	0
$741$	−4.66964	−0.171544
$742$	0	0
$743$	−13.9773	−0.512778	−0.256389	−	0.966574i	$-0.582533\pi$
$743$	−13.9773	−0.512778	−0.256389	+	0.966574i	$0.582533\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	15.8195	0.578805
$748$	0	0
$749$	−0.00253090	−9.24769e−5 0
$750$	0	0
$751$	−32.8762	−1.19967	−0.599834	−	0.800124i	$-0.704767\pi$
$751$	−32.8762	−1.19967	−0.599834	+	0.800124i	$0.704767\pi$
$752$	0	0
$753$	7.46802	0.272150
$754$	0	0
$755$	0	0
$756$	0	0
$757$	31.4556	1.14327	0.571636	−	0.820507i	$-0.306309\pi$
$757$	31.4556	1.14327	0.571636	+	0.820507i	$0.306309\pi$
$758$	0	0
$759$	−10.7591	−0.390531
$760$	0	0
$761$	37.8792	1.37312	0.686559	−	0.727074i	$-0.259120\pi$
$761$	37.8792	1.37312	0.686559	+	0.727074i	$0.259120\pi$
$762$	0	0
$763$	−1.67132	−0.0605059
$764$	0	0
$765$	0	0
$766$	0	0
$767$	23.8076	0.859644
$768$	0	0
$769$	14.1944	0.511862	0.255931	−	0.966695i	$-0.417618\pi$
$769$	14.1944	0.511862	0.255931	+	0.966695i	$0.417618\pi$
$770$	0	0
$771$	−8.39880	−0.302475
$772$	0	0
$773$	36.3783	1.30844	0.654218	−	0.756306i	$-0.272998\pi$
$773$	36.3783	1.30844	0.654218	+	0.756306i	$0.272998\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−0.628816	−0.0225586
$778$	0	0
$779$	−11.6036	−0.415742
$780$	0	0
$781$	13.9542	0.499320
$782$	0	0
$783$	3.70636	0.132455
$784$	0	0
$785$	0	0
$786$	0	0
$787$	34.4472	1.22791	0.613954	−	0.789341i	$-0.289578\pi$
$787$	34.4472	1.22791	0.613954	+	0.789341i	$0.289578\pi$
$788$	0	0
$789$	10.0248	0.356892
$790$	0	0
$791$	−0.432260	−0.0153694
$792$	0	0
$793$	−25.1021	−0.891403
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−36.9018	−1.30713	−0.653564	−	0.756872i	$-0.726727\pi$
$797$	−36.9018	−1.30713	−0.653564	+	0.756872i	$0.726727\pi$
$798$	0	0
$799$	21.1151	0.747000
$800$	0	0
$801$	−10.2508	−0.362194
$802$	0	0
$803$	7.16725	0.252927
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−17.6192	−0.620224
$808$	0	0
$809$	36.1037	1.26934	0.634670	−	0.772783i	$-0.281136\pi$
$809$	36.1037	1.26934	0.634670	+	0.772783i	$0.281136\pi$
$810$	0	0
$811$	13.0666	0.458830	0.229415	−	0.973329i	$-0.426319\pi$
$811$	13.0666	0.458830	0.229415	+	0.973329i	$0.426319\pi$
$812$	0	0
$813$	−4.85426	−0.170246
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−3.16036	−0.110567
$818$	0	0
$819$	−0.234226	−0.00818453
$820$	0	0
$821$	−25.8859	−0.903424	−0.451712	−	0.892164i	$-0.649187\pi$
$821$	−25.8859	−0.903424	−0.451712	+	0.892164i	$0.649187\pi$
$822$	0	0
$823$	−39.1757	−1.36558	−0.682789	−	0.730615i	$-0.739233\pi$
$823$	−39.1757	−1.36558	−0.682789	+	0.730615i	$0.739233\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	45.4245	1.57956	0.789782	−	0.613388i	$-0.210194\pi$
$827$	45.4245	1.57956	0.789782	+	0.613388i	$0.210194\pi$
$828$	0	0
$829$	16.1544	0.561065	0.280533	−	0.959845i	$-0.409489\pi$
$829$	16.1544	0.561065	0.280533	+	0.959845i	$0.409489\pi$
$830$	0	0
$831$	−22.6120	−0.784401
$832$	0	0
$833$	14.6020	0.505929
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−4.10620	−0.141931
$838$	0	0
$839$	10.4813	0.361856	0.180928	−	0.983496i	$-0.442090\pi$
$839$	10.4813	0.361856	0.180928	+	0.983496i	$0.442090\pi$
$840$	0	0
$841$	−15.2629	−0.526306
$842$	0	0
$843$	29.2542	1.00757
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−0.516541	−0.0177486
$848$	0	0
$849$	−29.2529	−1.00396
$850$	0	0
$851$	33.7452	1.15677
$852$	0	0
$853$	38.2211	1.30867	0.654333	−	0.756207i	$-0.272950\pi$
$853$	38.2211	1.30867	0.654333	+	0.756207i	$0.272950\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−25.3637	−0.866408	−0.433204	−	0.901296i	$-0.642617\pi$
$857$	−25.3637	−0.866408	−0.433204	+	0.901296i	$0.642617\pi$
$858$	0	0
$859$	35.7717	1.22051	0.610257	−	0.792204i	$-0.291066\pi$
$859$	35.7717	1.22051	0.610257	+	0.792204i	$0.291066\pi$
$860$	0	0
$861$	−0.582028	−0.0198355
$862$	0	0
$863$	−44.2905	−1.50767	−0.753833	−	0.657066i	$-0.771797\pi$
$863$	−44.2905	−1.50767	−0.753833	+	0.657066i	$0.771797\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−12.6389	−0.429239
$868$	0	0
$869$	30.3069	1.02809
$870$	0	0
$871$	36.8699	1.24929
$872$	0	0
$873$	12.8077	0.433476
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−9.80714	−0.331164	−0.165582	−	0.986196i	$-0.552950\pi$
$877$	−9.80714	−0.331164	−0.165582	+	0.986196i	$0.552950\pi$
$878$	0	0
$879$	13.4104	0.452323
$880$	0	0
$881$	−0.982291	−0.0330942	−0.0165471	−	0.999863i	$-0.505267\pi$
$881$	−0.982291	−0.0330942	−0.0165471	+	0.999863i	$0.505267\pi$
$882$	0	0
$883$	16.1321	0.542890	0.271445	−	0.962454i	$-0.412499\pi$
$883$	16.1321	0.542890	0.271445	+	0.962454i	$0.412499\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−50.4066	−1.69249	−0.846244	−	0.532796i	$-0.821141\pi$
$887$	−50.4066	−1.69249	−0.846244	+	0.532796i	$0.821141\pi$
$888$	0	0
$889$	0.288830	0.00968706
$890$	0	0
$891$	2.26981	0.0760414
$892$	0	0
$893$	−17.8050	−0.595822
$894$	0	0
$895$	0	0
$896$	0	0
$897$	12.5697	0.419689
$898$	0	0
$899$	−15.2191	−0.507584
$900$	0	0
$901$	−2.00780	−0.0668896
$902$	0	0
$903$	−0.158522	−0.00527527
$904$	0	0
$905$	0	0
$906$	0	0
$907$	26.1281	0.867570	0.433785	−	0.901016i	$-0.357178\pi$
$907$	26.1281	0.867570	0.433785	+	0.901016i	$0.357178\pi$
$908$	0	0
$909$	−2.72537	−0.0903949
$910$	0	0
$911$	−31.4435	−1.04177	−0.520886	−	0.853627i	$-0.674398\pi$
$911$	−31.4435	−1.04177	−0.520886	+	0.853627i	$0.674398\pi$
$912$	0	0
$913$	35.9072	1.18835
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.317584	0.0104876
$918$	0	0
$919$	−49.1184	−1.62027	−0.810133	−	0.586246i	$-0.800605\pi$
$919$	−49.1184	−1.62027	−0.810133	+	0.586246i	$0.800605\pi$
$920$	0	0
$921$	−13.5444	−0.446304
$922$	0	0
$923$	−16.3024	−0.536600
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.359976	0.0118232
$928$	0	0
$929$	16.0095	0.525256	0.262628	−	0.964897i	$-0.415411\pi$
$929$	16.0095	0.525256	0.262628	+	0.964897i	$0.415411\pi$
$930$	0	0
$931$	−12.3129	−0.403540
$932$	0	0
$933$	−2.03882	−0.0667481
$934$	0	0
$935$	0	0
$936$	0	0
$937$	30.2670	0.988779	0.494390	−	0.869240i	$-0.335392\pi$
$937$	30.2670	0.988779	0.494390	+	0.869240i	$0.335392\pi$
$938$	0	0
$939$	18.2786	0.596499
$940$	0	0
$941$	−12.3112	−0.401333	−0.200666	−	0.979660i	$-0.564311\pi$
$941$	−12.3112	−0.401333	−0.200666	+	0.979660i	$0.564311\pi$
$942$	0	0
$943$	31.2343	1.01713
$944$	0	0
$945$	0	0
$946$	0	0
$947$	40.5844	1.31882	0.659409	−	0.751785i	$-0.270807\pi$
$947$	40.5844	1.31882	0.659409	+	0.751785i	$0.270807\pi$
$948$	0	0
$949$	−8.37336	−0.271811
$950$	0	0
$951$	18.7978	0.609562
$952$	0	0
$953$	8.03153	0.260167	0.130083	−	0.991503i	$-0.458475\pi$
$953$	8.03153	0.260167	0.130083	+	0.991503i	$0.458475\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	8.41272	0.271945
$958$	0	0
$959$	−1.50332	−0.0485447
$960$	0	0
$961$	−14.1391	−0.456101
$962$	0	0
$963$	−0.0286533	−0.000923340 0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−26.9707	−0.867320	−0.433660	−	0.901076i	$-0.642778\pi$
$967$	−26.9707	−0.867320	−0.433660	+	0.901076i	$0.642778\pi$
$968$	0	0
$969$	−3.67745	−0.118137
$970$	0	0
$971$	29.7540	0.954850	0.477425	−	0.878673i	$-0.341570\pi$
$971$	29.7540	0.954850	0.477425	+	0.878673i	$0.341570\pi$
$972$	0	0
$973$	−1.67700	−0.0537620
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.42199	0.0454935	0.0227467	−	0.999741i	$-0.492759\pi$
$977$	1.42199	0.0454935	0.0227467	+	0.999741i	$0.492759\pi$
$978$	0	0
$979$	−23.2673	−0.743627
$980$	0	0
$981$	−18.9217	−0.604125
$982$	0	0
$983$	−2.02962	−0.0647348	−0.0323674	−	0.999476i	$-0.510305\pi$
$983$	−2.02962	−0.0647348	−0.0323674	+	0.999476i	$0.510305\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.893088	−0.0284273
$988$	0	0
$989$	8.50701	0.270507
$990$	0	0
$991$	20.2314	0.642672	0.321336	−	0.946965i	$-0.395868\pi$
$991$	20.2314	0.642672	0.321336	+	0.946965i	$0.395868\pi$
$992$	0	0
$993$	−8.32012	−0.264031
$994$	0	0
$995$	0	0
$996$	0	0
$997$	18.3001	0.579571	0.289786	−	0.957092i	$-0.406416\pi$
$997$	18.3001	0.579571	0.289786	+	0.957092i	$0.406416\pi$
$998$	0	0
$999$	−7.11909	−0.225238

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.a.f.1.3		4
5.2	odd	4		7500.2.d.c.1249.3		8
5.3	odd	4		7500.2.d.c.1249.6		8
5.4	even	2		7500.2.a.e.1.2		4
25.2	odd	20		1500.2.o.b.649.2		16
25.9	even	10		300.2.m.b.181.2	yes	8
25.11	even	5		1500.2.m.a.601.2		8
25.12	odd	20		1500.2.o.b.349.4		16
25.13	odd	20		1500.2.o.b.349.1		16
25.14	even	10		300.2.m.b.121.2	✓	8
25.16	even	5		1500.2.m.a.901.2		8
25.23	odd	20		1500.2.o.b.649.3		16
75.14	odd	10		900.2.n.b.721.1		8
75.59	odd	10		900.2.n.b.181.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.m.b.121.2	✓	8	25.14	even	10
300.2.m.b.181.2	yes	8	25.9	even	10
900.2.n.b.181.1		8	75.59	odd	10
900.2.n.b.721.1		8	75.14	odd	10
1500.2.m.a.601.2		8	25.11	even	5
1500.2.m.a.901.2		8	25.16	even	5
1500.2.o.b.349.1		16	25.13	odd	20
1500.2.o.b.349.4		16	25.12	odd	20
1500.2.o.b.649.2		16	25.2	odd	20
1500.2.o.b.649.3		16	25.23	odd	20
7500.2.a.e.1.2		4	5.4	even	2
7500.2.a.f.1.3		4	1.1	even	1	trivial
7500.2.d.c.1249.3		8	5.2	odd	4
7500.2.d.c.1249.6		8	5.3	odd	4

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:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.0883282 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.0883282 q^{7} +1.00000 q^{9} +2.26981 q^{11} -2.65177 q^{13} -2.08833 q^{17} +1.76095 q^{19} +0.0883282 q^{21} -4.74010 q^{23} +1.00000 q^{27} +3.70636 q^{29} -4.10620 q^{31} +2.26981 q^{33} -7.11909 q^{37} -2.65177 q^{39} -6.58938 q^{41} -1.79469 q^{43} -10.1110 q^{47} -6.99220 q^{49} -2.08833 q^{51} +0.961440 q^{53} +1.76095 q^{57} -8.97801 q^{59} +9.46618 q^{61} +0.0883282 q^{63} -13.9039 q^{67} -4.74010 q^{69} +6.14774 q^{71} +3.15765 q^{73} +0.200488 q^{77} +13.3522 q^{79} +1.00000 q^{81} +15.8195 q^{83} +3.70636 q^{87} -10.2508 q^{89} -0.234226 q^{91} -4.10620 q^{93} +12.8077 q^{97} +2.26981 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} - 4 q^{7} + 4 q^{9} - q^{11} - 5 q^{13} - 4 q^{17} - 5 q^{19} - 4 q^{21} - 9 q^{23} + 4 q^{27} + 6 q^{29} + 11 q^{31} - q^{33} - 2 q^{37} - 5 q^{39} + 6 q^{43} - 16 q^{47} - 4 q^{49} - 4 q^{51} + 2 q^{53} - 5 q^{57} + q^{59} - 22 q^{61} - 4 q^{63} - 36 q^{67} - 9 q^{69} + 20 q^{71} - 12 q^{73} + 11 q^{77} - 3 q^{79} + 4 q^{81} - 14 q^{83} + 6 q^{87} - 15 q^{89} - 10 q^{91} + 11 q^{93} + 12 q^{97} - q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^7 + 4 * q^9 - q^11 - 5 * q^13 - 4 * q^17 - 5 * q^19 - 4 * q^21 - 9 * q^23 + 4 * q^27 + 6 * q^29 + 11 * q^31 - q^33 - 2 * q^37 - 5 * q^39 + 6 * q^43 - 16 * q^47 - 4 * q^49 - 4 * q^51 + 2 * q^53 - 5 * q^57 + q^59 - 22 * q^61 - 4 * q^63 - 36 * q^67 - 9 * q^69 + 20 * q^71 - 12 * q^73 + 11 * q^77 - 3 * q^79 + 4 * q^81 - 14 * q^83 + 6 * q^87 - 15 * q^89 - 10 * q^91 + 11 * q^93 + 12 * q^97 - q^99

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