LMFDB - Embedded newform 9800.2.a.cf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9800,2,Mod(1,9800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9800.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:	$78.2533939809$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1944.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 9x - 6 $ x^3 - 9*x - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.705720$ of defining polynomial
Character	$\chi$	$=$	9800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.705720 q^{3} -2.50196 q^{9} +O(q^{10})$	$q+0.705720 q^{3} -2.50196 q^{9} -4.50196 q^{11} +5.09052 q^{13} +2.00000 q^{17} +3.09052 q^{19} -5.79624 q^{23} -3.88284 q^{27} +9.50196 q^{29} -5.41144 q^{31} -3.17712 q^{33} -7.09052 q^{37} +3.59248 q^{39} +6.59248 q^{41} -4.70572 q^{43} +10.0944 q^{47} +1.41144 q^{51} -9.91340 q^{53} +2.18104 q^{57} -8.00000 q^{59} +8.91340 q^{61} +0.117159 q^{67} -4.09052 q^{69} +8.18104 q^{73} -14.1810 q^{79} +4.76568 q^{81} -10.7057 q^{83} +6.70572 q^{87} -4.09052 q^{89} -3.81896 q^{93} -2.00000 q^{97} +11.2637 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 9 q^{9}+O(q^{10}) $ 3 * q + 9 * q^9	$ 3 q + 9 q^{9} + 3 q^{11} + 3 q^{13} + 6 q^{17} - 3 q^{19} - 3 q^{23} - 18 q^{27} + 12 q^{29} - 12 q^{31} - 18 q^{33} - 9 q^{37} - 18 q^{39} - 9 q^{41} - 12 q^{43} - 15 q^{47} - 9 q^{53} - 18 q^{57} - 24 q^{59} + 6 q^{61} - 6 q^{67} - 18 q^{79} + 27 q^{81} - 30 q^{83} + 18 q^{87} - 36 q^{93} - 6 q^{97} + 63 q^{99}+O(q^{100}) $ 3 * q + 9 * q^9 + 3 * q^11 + 3 * q^13 + 6 * q^17 - 3 * q^19 - 3 * q^23 - 18 * q^27 + 12 * q^29 - 12 * q^31 - 18 * q^33 - 9 * q^37 - 18 * q^39 - 9 * q^41 - 12 * q^43 - 15 * q^47 - 9 * q^53 - 18 * q^57 - 24 * q^59 + 6 * q^61 - 6 * q^67 - 18 * q^79 + 27 * q^81 - 30 * q^83 + 18 * q^87 - 36 * q^93 - 6 * q^97 + 63 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.705720	0.407447	0.203724	−	0.979028i	$-0.434696\pi$
$3$	0.705720	0.407447	0.203724	+	0.979028i	$0.434696\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.50196	−0.833987
$10$	0	0
$11$	−4.50196	−1.35739	−0.678696	−	0.734419i	$-0.737455\pi$
$11$	−4.50196	−1.35739	−0.678696	+	0.734419i	$0.737455\pi$
$12$	0	0
$13$	5.09052	1.41186	0.705928	−	0.708283i	$-0.250530\pi$
$13$	5.09052	1.41186	0.705928	+	0.708283i	$0.250530\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	3.09052	0.709014	0.354507	−	0.935053i	$-0.384649\pi$
$19$	3.09052	0.709014	0.354507	+	0.935053i	$0.384649\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.79624	−1.20860	−0.604300	−	0.796757i	$-0.706547\pi$
$23$	−5.79624	−1.20860	−0.604300	+	0.796757i	$0.706547\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3.88284	−0.747253
$28$	0	0
$29$	9.50196	1.76447	0.882235	−	0.470810i	$-0.156038\pi$
$29$	9.50196	1.76447	0.882235	+	0.470810i	$0.156038\pi$
$30$	0	0
$31$	−5.41144	−0.971923	−0.485962	−	0.873980i	$-0.661531\pi$
$31$	−5.41144	−0.971923	−0.485962	+	0.873980i	$0.661531\pi$
$32$	0	0
$33$	−3.17712	−0.553066
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.09052	−1.16567	−0.582837	−	0.812589i	$-0.698057\pi$
$37$	−7.09052	−1.16567	−0.582837	+	0.812589i	$0.698057\pi$
$38$	0	0
$39$	3.59248	0.575257
$40$	0	0
$41$	6.59248	1.02957	0.514786	−	0.857319i	$-0.327871\pi$
$41$	6.59248	1.02957	0.514786	+	0.857319i	$0.327871\pi$
$42$	0	0
$43$	−4.70572	−0.717616	−0.358808	−	0.933411i	$-0.616817\pi$
$43$	−4.70572	−0.717616	−0.358808	+	0.933411i	$0.616817\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.0944	1.47243	0.736213	−	0.676750i	$-0.236612\pi$
$47$	10.0944	1.47243	0.736213	+	0.676750i	$0.236612\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1.41144	0.197641
$52$	0	0
$53$	−9.91340	−1.36171	−0.680855	−	0.732418i	$-0.738392\pi$
$53$	−9.91340	−1.36171	−0.680855	+	0.732418i	$0.738392\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.18104	0.288886
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	8.91340	1.14124	0.570622	−	0.821213i	$-0.306702\pi$
$61$	8.91340	1.14124	0.570622	+	0.821213i	$0.306702\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0.117159	0.0143132	0.00715662	−	0.999974i	$-0.497722\pi$
$67$	0.117159	0.0143132	0.00715662	+	0.999974i	$0.497722\pi$
$68$	0	0
$69$	−4.09052	−0.492441
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	8.18104	0.957518	0.478759	−	0.877946i	$-0.341087\pi$
$73$	8.18104	0.957518	0.478759	+	0.877946i	$0.341087\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−14.1810	−1.59549	−0.797746	−	0.602994i	$-0.793974\pi$
$79$	−14.1810	−1.59549	−0.797746	+	0.602994i	$0.793974\pi$
$80$	0	0
$81$	4.76568	0.529520
$82$	0	0
$83$	−10.7057	−1.17511	−0.587553	−	0.809186i	$-0.699909\pi$
$83$	−10.7057	−1.17511	−0.587553	+	0.809186i	$0.699909\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.70572	0.718929
$88$	0	0
$89$	−4.09052	−0.433594	−0.216797	−	0.976217i	$-0.569561\pi$
$89$	−4.09052	−0.433594	−0.216797	+	0.976217i	$0.569561\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−3.81896	−0.396008
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	11.2637	1.13205
$100$	0	0
$101$	3.32092	0.330444	0.165222	−	0.986256i	$-0.447166\pi$
$101$	3.32092	0.330444	0.165222	+	0.986256i	$0.447166\pi$
$102$	0	0
$103$	−10.8868	−1.07270	−0.536352	−	0.843994i	$-0.680198\pi$
$103$	−10.8868	−1.07270	−0.536352	+	0.843994i	$0.680198\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.88284	−0.375368	−0.187684	−	0.982229i	$-0.560098\pi$
$107$	−3.88284	−0.375368	−0.187684	+	0.982229i	$0.560098\pi$
$108$	0	0
$109$	−10.2716	−0.983837	−0.491919	−	0.870641i	$-0.663704\pi$
$109$	−10.2716	−0.983837	−0.491919	+	0.870641i	$0.663704\pi$
$110$	0	0
$111$	−5.00392	−0.474951
$112$	0	0
$113$	−2.82288	−0.265554	−0.132777	−	0.991146i	$-0.542389\pi$
$113$	−2.82288	−0.265554	−0.132777	+	0.991146i	$0.542389\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−12.7363	−1.17747
$118$	0	0
$119$	0	0
$120$	0	0
$121$	9.26764	0.842513
$122$	0	0
$123$	4.65244	0.419497
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.73236	0.153722	0.0768610	−	0.997042i	$-0.475510\pi$
$127$	1.73236	0.153722	0.0768610	+	0.997042i	$0.475510\pi$
$128$	0	0
$129$	−3.32092	−0.292391
$130$	0	0
$131$	12.7363	1.11277	0.556387	−	0.830923i	$-0.312187\pi$
$131$	12.7363	1.11277	0.556387	+	0.830923i	$0.312187\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.00000	−0.341743	−0.170872	−	0.985293i	$-0.554658\pi$
$137$	−4.00000	−0.341743	−0.170872	+	0.985293i	$0.554658\pi$
$138$	0	0
$139$	1.41144	0.119717	0.0598584	−	0.998207i	$-0.480935\pi$
$139$	1.41144	0.119717	0.0598584	+	0.998207i	$0.480935\pi$
$140$	0	0
$141$	7.12384	0.599936
$142$	0	0
$143$	−22.9173	−1.91644
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	23.0944	1.89197	0.945985	−	0.324210i	$-0.105098\pi$
$149$	23.0944	1.89197	0.945985	+	0.324210i	$0.105098\pi$
$150$	0	0
$151$	−0.407520	−0.0331635	−0.0165817	−	0.999863i	$-0.505278\pi$
$151$	−0.407520	−0.0331635	−0.0165817	+	0.999863i	$0.505278\pi$
$152$	0	0
$153$	−5.00392	−0.404543
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−14.0944	−1.12486	−0.562429	−	0.826845i	$-0.690133\pi$
$157$	−14.0944	−1.12486	−0.562429	+	0.826845i	$0.690133\pi$
$158$	0	0
$159$	−6.99608	−0.554825
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.1810	1.26740	0.633698	−	0.773580i	$-0.281536\pi$
$163$	16.1810	1.26740	0.633698	+	0.773580i	$0.281536\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.79624	0.603291	0.301646	−	0.953420i	$-0.402464\pi$
$167$	7.79624	0.603291	0.301646	+	0.953420i	$0.402464\pi$
$168$	0	0
$169$	12.9134	0.993338
$170$	0	0
$171$	−7.73236	−0.591308
$172$	0	0
$173$	−8.90948	−0.677375	−0.338688	−	0.940899i	$-0.609983\pi$
$173$	−8.90948	−0.677375	−0.338688	+	0.940899i	$0.609983\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5.64576	−0.424361
$178$	0	0
$179$	1.67908	0.125500	0.0627502	−	0.998029i	$-0.480013\pi$
$179$	1.67908	0.125500	0.0627502	+	0.998029i	$0.480013\pi$
$180$	0	0
$181$	−12.5059	−0.929555	−0.464777	−	0.885428i	$-0.653866\pi$
$181$	−12.5059	−0.929555	−0.464777	+	0.885428i	$0.653866\pi$
$182$	0	0
$183$	6.29036	0.464997
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−9.00392	−0.658432
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.58856	0.476732	0.238366	−	0.971175i	$-0.423388\pi$
$191$	6.58856	0.476732	0.238366	+	0.971175i	$0.423388\pi$
$192$	0	0
$193$	−17.8268	−1.28320	−0.641601	−	0.767039i	$-0.721729\pi$
$193$	−17.8268	−1.28320	−0.641601	+	0.767039i	$0.721729\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5.09052	−0.362685	−0.181342	−	0.983420i	$-0.558044\pi$
$197$	−5.09052	−0.362685	−0.181342	+	0.983420i	$0.558044\pi$
$198$	0	0
$199$	−24.3621	−1.72698	−0.863491	−	0.504364i	$-0.831727\pi$
$199$	−24.3621	−1.72698	−0.863491	+	0.504364i	$0.831727\pi$
$200$	0	0
$201$	0.0826814	0.00583189
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	14.5020	1.00796
$208$	0	0
$209$	−13.9134	−0.962410
$210$	0	0
$211$	−8.26764	−0.569168	−0.284584	−	0.958651i	$-0.591855\pi$
$211$	−8.26764	−0.569168	−0.284584	+	0.958651i	$0.591855\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	5.77352	0.390138
$220$	0	0
$221$	10.1810	0.684851
$222$	0	0
$223$	−17.8268	−1.19377	−0.596885	−	0.802327i	$-0.703595\pi$
$223$	−17.8268	−1.19377	−0.596885	+	0.802327i	$0.703595\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.35816	0.355634	0.177817	−	0.984064i	$-0.443097\pi$
$227$	5.35816	0.355634	0.177817	+	0.984064i	$0.443097\pi$
$228$	0	0
$229$	−24.1810	−1.59793	−0.798964	−	0.601379i	$-0.794618\pi$
$229$	−24.1810	−1.59793	−0.798964	+	0.601379i	$0.794618\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−10.0078	−0.650079
$238$	0	0
$239$	−12.7696	−0.825997	−0.412998	−	0.910732i	$-0.635519\pi$
$239$	−12.7696	−0.825997	−0.412998	+	0.910732i	$0.635519\pi$
$240$	0	0
$241$	22.0944	1.42323	0.711614	−	0.702571i	$-0.247965\pi$
$241$	22.0944	1.42323	0.711614	+	0.702571i	$0.247965\pi$
$242$	0	0
$243$	15.0118	0.963005
$244$	0	0
$245$	0	0
$246$	0	0
$247$	15.7324	1.00103
$248$	0	0
$249$	−7.55524	−0.478794
$250$	0	0
$251$	13.5059	0.852484	0.426242	−	0.904609i	$-0.359837\pi$
$251$	13.5059	0.852484	0.426242	+	0.904609i	$0.359837\pi$
$252$	0	0
$253$	26.0944	1.64054
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−16.3621	−1.02064	−0.510319	−	0.859985i	$-0.670473\pi$
$257$	−16.3621	−1.02064	−0.510319	+	0.859985i	$0.670473\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−23.7735	−1.47154
$262$	0	0
$263$	−9.11324	−0.561946	−0.280973	−	0.959716i	$-0.590657\pi$
$263$	−9.11324	−0.561946	−0.280973	+	0.959716i	$0.590657\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.88676	−0.176667
$268$	0	0
$269$	−8.91340	−0.543460	−0.271730	−	0.962374i	$-0.587596\pi$
$269$	−8.91340	−0.543460	−0.271730	+	0.962374i	$0.587596\pi$
$270$	0	0
$271$	−16.3621	−0.993926	−0.496963	−	0.867772i	$-0.665551\pi$
$271$	−16.3621	−0.993926	−0.496963	+	0.867772i	$0.665551\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−11.1771	−0.671568	−0.335784	−	0.941939i	$-0.609001\pi$
$277$	−11.1771	−0.671568	−0.335784	+	0.941939i	$0.609001\pi$
$278$	0	0
$279$	13.5392	0.810571
$280$	0	0
$281$	−28.2755	−1.68677	−0.843387	−	0.537307i	$-0.819442\pi$
$281$	−28.2755	−1.68677	−0.843387	+	0.537307i	$0.819442\pi$
$282$	0	0
$283$	−12.8229	−0.762241	−0.381121	−	0.924525i	$-0.624462\pi$
$283$	−12.8229	−0.762241	−0.381121	+	0.924525i	$0.624462\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−1.41144	−0.0827400
$292$	0	0
$293$	8.90948	0.520497	0.260249	−	0.965542i	$-0.416195\pi$
$293$	8.90948	0.520497	0.260249	+	0.965542i	$0.416195\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	17.4804	1.01432
$298$	0	0
$299$	−29.5059	−1.70637
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2.34364	0.134638
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−9.12108	−0.520567	−0.260284	−	0.965532i	$-0.583816\pi$
$307$	−9.12108	−0.520567	−0.260284	+	0.965532i	$0.583816\pi$
$308$	0	0
$309$	−7.68300	−0.437071
$310$	0	0
$311$	13.1771	0.747206	0.373603	−	0.927589i	$-0.378122\pi$
$311$	13.1771	0.747206	0.373603	+	0.927589i	$0.378122\pi$
$312$	0	0
$313$	−17.5392	−0.991374	−0.495687	−	0.868501i	$-0.665084\pi$
$313$	−17.5392	−0.991374	−0.495687	+	0.868501i	$0.665084\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.1850	0.740541	0.370271	−	0.928924i	$-0.379265\pi$
$317$	13.1850	0.740541	0.370271	+	0.928924i	$0.379265\pi$
$318$	0	0
$319$	−42.7774	−2.39508
$320$	0	0
$321$	−2.74020	−0.152943
$322$	0	0
$323$	6.18104	0.343922
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−7.24884	−0.400862
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8.50196	0.467310	0.233655	−	0.972320i	$-0.424931\pi$
$331$	8.50196	0.467310	0.233655	+	0.972320i	$0.424931\pi$
$332$	0	0
$333$	17.7402	0.972157
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−28.1889	−1.53555	−0.767773	−	0.640722i	$-0.778635\pi$
$337$	−28.1889	−1.53555	−0.767773	+	0.640722i	$0.778635\pi$
$338$	0	0
$339$	−1.99216	−0.108199
$340$	0	0
$341$	24.3621	1.31928
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.70572	−0.145251	−0.0726253	−	0.997359i	$-0.523138\pi$
$347$	−2.70572	−0.145251	−0.0726253	+	0.997359i	$0.523138\pi$
$348$	0	0
$349$	13.6830	0.732434	0.366217	−	0.930529i	$-0.380653\pi$
$349$	13.6830	0.732434	0.366217	+	0.930529i	$0.380653\pi$
$350$	0	0
$351$	−19.7657	−1.05501
$352$	0	0
$353$	−20.3621	−1.08376	−0.541882	−	0.840454i	$-0.682288\pi$
$353$	−20.3621	−1.08376	−0.541882	+	0.840454i	$0.682288\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−18.4154	−0.971925	−0.485963	−	0.873980i	$-0.661531\pi$
$359$	−18.4154	−0.971925	−0.485963	+	0.873980i	$0.661531\pi$
$360$	0	0
$361$	−9.44868	−0.497299
$362$	0	0
$363$	6.54036	0.343280
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−25.6230	−1.33751	−0.668756	−	0.743482i	$-0.733173\pi$
$367$	−25.6230	−1.33751	−0.668756	+	0.743482i	$0.733173\pi$
$368$	0	0
$369$	−16.4941	−0.858650
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−28.1889	−1.45956	−0.729782	−	0.683679i	$-0.760379\pi$
$373$	−28.1889	−1.45956	−0.729782	+	0.683679i	$0.760379\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	48.3699	2.49118
$378$	0	0
$379$	21.7402	1.11672	0.558359	−	0.829599i	$-0.311431\pi$
$379$	21.7402	1.11672	0.558359	+	0.829599i	$0.311431\pi$
$380$	0	0
$381$	1.22256	0.0626336
$382$	0	0
$383$	−20.2116	−1.03276	−0.516382	−	0.856358i	$-0.672722\pi$
$383$	−20.2116	−1.03276	−0.516382	+	0.856358i	$0.672722\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	11.7735	0.598482
$388$	0	0
$389$	13.8268	0.701046	0.350523	−	0.936554i	$-0.386004\pi$
$389$	13.8268	0.701046	0.350523	+	0.936554i	$0.386004\pi$
$390$	0	0
$391$	−11.5925	−0.586257
$392$	0	0
$393$	8.98824	0.453397
$394$	0	0
$395$	0	0
$396$	0	0
$397$	36.3699	1.82535	0.912677	−	0.408682i	$-0.134011\pi$
$397$	36.3699	1.82535	0.912677	+	0.408682i	$0.134011\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.81896	−0.340523	−0.170261	−	0.985399i	$-0.554461\pi$
$401$	−6.81896	−0.340523	−0.170261	+	0.985399i	$0.554461\pi$
$402$	0	0
$403$	−27.5470	−1.37222
$404$	0	0
$405$	0	0
$406$	0	0
$407$	31.9212	1.58228
$408$	0	0
$409$	17.8640	0.883320	0.441660	−	0.897182i	$-0.354390\pi$
$409$	17.8640	0.883320	0.441660	+	0.897182i	$0.354390\pi$
$410$	0	0
$411$	−2.82288	−0.139242
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0.996080	0.0487783
$418$	0	0
$419$	−18.4487	−0.901277	−0.450639	−	0.892706i	$-0.648804\pi$
$419$	−18.4487	−0.901277	−0.450639	+	0.892706i	$0.648804\pi$
$420$	0	0
$421$	−10.7324	−0.523063	−0.261532	−	0.965195i	$-0.584228\pi$
$421$	−10.7324	−0.523063	−0.261532	+	0.965195i	$0.584228\pi$
$422$	0	0
$423$	−25.2559	−1.22798
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−16.1732	−0.780850
$430$	0	0
$431$	6.58856	0.317360	0.158680	−	0.987330i	$-0.449276\pi$
$431$	6.58856	0.317360	0.158680	+	0.987330i	$0.449276\pi$
$432$	0	0
$433$	5.17712	0.248797	0.124398	−	0.992232i	$-0.460300\pi$
$433$	5.17712	0.248797	0.124398	+	0.992232i	$0.460300\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−17.9134	−0.856914
$438$	0	0
$439$	−31.0118	−1.48011	−0.740055	−	0.672546i	$-0.765201\pi$
$439$	−31.0118	−1.48011	−0.740055	+	0.672546i	$0.765201\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.87892	−0.231805	−0.115902	−	0.993261i	$-0.536976\pi$
$443$	−4.87892	−0.231805	−0.115902	+	0.993261i	$0.536976\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	16.2982	0.770878
$448$	0	0
$449$	27.7696	1.31053	0.655264	−	0.755400i	$-0.272557\pi$
$449$	27.7696	1.31053	0.655264	+	0.755400i	$0.272557\pi$
$450$	0	0
$451$	−29.6791	−1.39753
$452$	0	0
$453$	−0.287595	−0.0135124
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.0039	1.16963	0.584817	−	0.811165i	$-0.301166\pi$
$457$	25.0039	1.16963	0.584817	+	0.811165i	$0.301166\pi$
$458$	0	0
$459$	−7.76568	−0.362471
$460$	0	0
$461$	4.00784	0.186664	0.0933318	−	0.995635i	$-0.470248\pi$
$461$	4.00784	0.186664	0.0933318	+	0.995635i	$0.470248\pi$
$462$	0	0
$463$	36.8002	1.71025	0.855124	−	0.518423i	$-0.173481\pi$
$463$	36.8002	1.71025	0.855124	+	0.518423i	$0.173481\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−23.4753	−1.08631	−0.543154	−	0.839633i	$-0.682770\pi$
$467$	−23.4753	−1.08631	−0.543154	+	0.839633i	$0.682770\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−9.94672	−0.458321
$472$	0	0
$473$	21.1850	0.974086
$474$	0	0
$475$	0	0
$476$	0	0
$477$	24.8029	1.13565
$478$	0	0
$479$	37.7735	1.72592	0.862958	−	0.505275i	$-0.168609\pi$
$479$	37.7735	1.72592	0.862958	+	0.505275i	$0.168609\pi$
$480$	0	0
$481$	−36.0944	−1.64576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	6.17320	0.279734	0.139867	−	0.990170i	$-0.455332\pi$
$487$	6.17320	0.279734	0.139867	+	0.990170i	$0.455332\pi$
$488$	0	0
$489$	11.4193	0.516398
$490$	0	0
$491$	39.7814	1.79531	0.897654	−	0.440701i	$-0.145270\pi$
$491$	39.7814	1.79531	0.897654	+	0.440701i	$0.145270\pi$
$492$	0	0
$493$	19.0039	0.855893
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	3.76568	0.168575	0.0842875	−	0.996441i	$-0.473139\pi$
$499$	3.76568	0.168575	0.0842875	+	0.996441i	$0.473139\pi$
$500$	0	0
$501$	5.50196	0.245809
$502$	0	0
$503$	18.7057	0.834047	0.417023	−	0.908896i	$-0.363073\pi$
$503$	18.7057	0.834047	0.417023	+	0.908896i	$0.363073\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9.11324	0.404733
$508$	0	0
$509$	−12.3248	−0.546289	−0.273144	−	0.961973i	$-0.588064\pi$
$509$	−12.3248	−0.546289	−0.273144	+	0.961973i	$0.588064\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−12.0000	−0.529813
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−45.4448	−1.99866
$518$	0	0
$519$	−6.28759	−0.275995
$520$	0	0
$521$	29.5592	1.29501	0.647505	−	0.762061i	$-0.275812\pi$
$521$	29.5592	1.29501	0.647505	+	0.762061i	$0.275812\pi$
$522$	0	0
$523$	−43.0039	−1.88043	−0.940215	−	0.340581i	$-0.889376\pi$
$523$	−43.0039	−1.88043	−0.940215	+	0.340581i	$0.889376\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−10.8229	−0.471452
$528$	0	0
$529$	10.5964	0.460713
$530$	0	0
$531$	20.0157	0.868606
$532$	0	0
$533$	33.5592	1.45361
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.18496	0.0511348
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−4.96668	−0.213534	−0.106767	−	0.994284i	$-0.534050\pi$
$541$	−4.96668	−0.213534	−0.106767	+	0.994284i	$0.534050\pi$
$542$	0	0
$543$	−8.82564	−0.378745
$544$	0	0
$545$	0	0
$546$	0	0
$547$	22.1250	0.945997	0.472998	−	0.881063i	$-0.343172\pi$
$547$	22.1250	0.945997	0.472998	+	0.881063i	$0.343172\pi$
$548$	0	0
$549$	−22.3010	−0.951782
$550$	0	0
$551$	29.3660	1.25103
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.55916	−0.235549	−0.117775	−	0.993040i	$-0.537576\pi$
$557$	−5.55916	−0.235549	−0.117775	+	0.993040i	$0.537576\pi$
$558$	0	0
$559$	−23.9546	−1.01317
$560$	0	0
$561$	−6.35424	−0.268276
$562$	0	0
$563$	9.48316	0.399668	0.199834	−	0.979830i	$-0.435960\pi$
$563$	9.48316	0.399668	0.199834	+	0.979830i	$0.435960\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−34.2833	−1.43723	−0.718616	−	0.695407i	$-0.755224\pi$
$569$	−34.2833	−1.43723	−0.718616	+	0.695407i	$0.755224\pi$
$570$	0	0
$571$	−31.5470	−1.32020	−0.660101	−	0.751177i	$-0.729487\pi$
$571$	−31.5470	−1.32020	−0.660101	+	0.751177i	$0.729487\pi$
$572$	0	0
$573$	4.64968	0.194243
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−10.6418	−0.443025	−0.221513	−	0.975157i	$-0.571099\pi$
$577$	−10.6418	−0.443025	−0.221513	+	0.975157i	$0.571099\pi$
$578$	0	0
$579$	−12.5807	−0.522837
$580$	0	0
$581$	0	0
$582$	0	0
$583$	44.6297	1.84837
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.64184	0.274138	0.137069	−	0.990562i	$-0.456232\pi$
$587$	6.64184	0.274138	0.137069	+	0.990562i	$0.456232\pi$
$588$	0	0
$589$	−16.7242	−0.689107
$590$	0	0
$591$	−3.59248	−0.147775
$592$	0	0
$593$	13.5392	0.555988	0.277994	−	0.960583i	$-0.410330\pi$
$593$	13.5392	0.555988	0.277994	+	0.960583i	$0.410330\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−17.1928	−0.703654
$598$	0	0
$599$	26.1810	1.06973	0.534864	−	0.844938i	$-0.320363\pi$
$599$	26.1810	1.06973	0.534864	+	0.844938i	$0.320363\pi$
$600$	0	0
$601$	36.0078	1.46879	0.734395	−	0.678722i	$-0.237466\pi$
$601$	36.0078	1.46879	0.734395	+	0.678722i	$0.237466\pi$
$602$	0	0
$603$	−0.293127	−0.0119371
$604$	0	0
$605$	0	0
$606$	0	0
$607$	23.7430	0.963697	0.481849	−	0.876255i	$-0.339966\pi$
$607$	23.7430	0.963697	0.481849	+	0.876255i	$0.339966\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	51.3860	2.07885
$612$	0	0
$613$	9.55916	0.386091	0.193045	−	0.981190i	$-0.438164\pi$
$613$	9.55916	0.386091	0.193045	+	0.981190i	$0.438164\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19.8268	0.798197	0.399098	−	0.916908i	$-0.369323\pi$
$617$	19.8268	0.798197	0.399098	+	0.916908i	$0.369323\pi$
$618$	0	0
$619$	−27.1516	−1.09132	−0.545658	−	0.838008i	$-0.683720\pi$
$619$	−27.1516	−1.09132	−0.545658	+	0.838008i	$0.683720\pi$
$620$	0	0
$621$	22.5059	0.903130
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−9.81896	−0.392131
$628$	0	0
$629$	−14.1810	−0.565435
$630$	0	0
$631$	−49.7735	−1.98145	−0.990726	−	0.135873i	$-0.956616\pi$
$631$	−49.7735	−1.98145	−0.990726	+	0.135873i	$0.956616\pi$
$632$	0	0
$633$	−5.83464	−0.231906
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	20.2304	0.799053	0.399526	−	0.916722i	$-0.369175\pi$
$641$	20.2304	0.799053	0.399526	+	0.916722i	$0.369175\pi$
$642$	0	0
$643$	−25.1850	−0.993198	−0.496599	−	0.867980i	$-0.665418\pi$
$643$	−25.1850	−0.993198	−0.496599	+	0.867980i	$0.665418\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19.5698	0.769367	0.384683	−	0.923049i	$-0.374311\pi$
$647$	19.5698	0.769367	0.384683	+	0.923049i	$0.374311\pi$
$648$	0	0
$649$	36.0157	1.41374
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−18.2676	−0.714868	−0.357434	−	0.933938i	$-0.616348\pi$
$653$	−18.2676	−0.714868	−0.357434	+	0.933938i	$0.616348\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−20.4686	−0.798558
$658$	0	0
$659$	46.0078	1.79221	0.896105	−	0.443841i	$-0.146385\pi$
$659$	46.0078	1.79221	0.896105	+	0.443841i	$0.146385\pi$
$660$	0	0
$661$	48.4604	1.88489	0.942446	−	0.334357i	$-0.108519\pi$
$661$	48.4604	1.88489	0.942446	+	0.334357i	$0.108519\pi$
$662$	0	0
$663$	7.18496	0.279041
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−55.0756	−2.13254
$668$	0	0
$669$	−12.5807	−0.486399
$670$	0	0
$671$	−40.1278	−1.54912
$672$	0	0
$673$	15.0118	0.578661	0.289330	−	0.957229i	$-0.406567\pi$
$673$	15.0118	0.578661	0.289330	+	0.957229i	$0.406567\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.91340	0.304137	0.152068	−	0.988370i	$-0.451407\pi$
$677$	7.91340	0.304137	0.152068	+	0.988370i	$0.451407\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	3.78136	0.144902
$682$	0	0
$683$	−29.7096	−1.13681	−0.568404	−	0.822750i	$-0.692439\pi$
$683$	−29.7096	−1.13681	−0.568404	+	0.822750i	$0.692439\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−17.0650	−0.651072
$688$	0	0
$689$	−50.4644	−1.92254
$690$	0	0
$691$	21.7735	0.828304	0.414152	−	0.910208i	$-0.364078\pi$
$691$	21.7735	0.828304	0.414152	+	0.910208i	$0.364078\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	13.1850	0.499416
$698$	0	0
$699$	−12.7030	−0.480470
$700$	0	0
$701$	24.0905	0.909886	0.454943	−	0.890520i	$-0.349660\pi$
$701$	24.0905	0.909886	0.454943	+	0.890520i	$0.349660\pi$
$702$	0	0
$703$	−21.9134	−0.826479
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−24.0372	−0.902738	−0.451369	−	0.892337i	$-0.649064\pi$
$709$	−24.0372	−0.902738	−0.451369	+	0.892337i	$0.649064\pi$
$710$	0	0
$711$	35.4804	1.33062
$712$	0	0
$713$	31.3660	1.17467
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−9.01176	−0.336550
$718$	0	0
$719$	0.234318	0.00873858	0.00436929	−	0.999990i	$-0.498609\pi$
$719$	0.234318	0.00873858	0.00436929	+	0.999990i	$0.498609\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	15.5925	0.579891
$724$	0	0
$725$	0	0
$726$	0	0
$727$	9.44200	0.350184	0.175092	−	0.984552i	$-0.443978\pi$
$727$	9.44200	0.350184	0.175092	+	0.984552i	$0.443978\pi$
$728$	0	0
$729$	−3.70295	−0.137146
$730$	0	0
$731$	−9.41144	−0.348095
$732$	0	0
$733$	−28.7441	−1.06169	−0.530844	−	0.847469i	$-0.678125\pi$
$733$	−28.7441	−1.06169	−0.530844	+	0.847469i	$0.678125\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.527445	−0.0194287
$738$	0	0
$739$	−26.1477	−0.961859	−0.480930	−	0.876759i	$-0.659701\pi$
$739$	−26.1477	−0.961859	−0.480930	+	0.876759i	$0.659701\pi$
$740$	0	0
$741$	11.1026	0.407865
$742$	0	0
$743$	−32.7547	−1.20165	−0.600827	−	0.799379i	$-0.705162\pi$
$743$	−32.7547	−1.20165	−0.600827	+	0.799379i	$0.705162\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	26.7853	0.980022
$748$	0	0
$749$	0	0
$750$	0	0
$751$	26.1810	0.955360	0.477680	−	0.878534i	$-0.341478\pi$
$751$	26.1810	0.955360	0.477680	+	0.878534i	$0.341478\pi$
$752$	0	0
$753$	9.53136	0.347342
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−16.8229	−0.611438	−0.305719	−	0.952122i	$-0.598897\pi$
$757$	−16.8229	−0.611438	−0.305719	+	0.952122i	$0.598897\pi$
$758$	0	0
$759$	18.4154	0.668435
$760$	0	0
$761$	5.90556	0.214076	0.107038	−	0.994255i	$-0.465863\pi$
$761$	5.90556	0.214076	0.107038	+	0.994255i	$0.465863\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−40.7242	−1.47046
$768$	0	0
$769$	−0.275481	−0.00993410	−0.00496705	−	0.999988i	$-0.501581\pi$
$769$	−0.275481	−0.00993410	−0.00496705	+	0.999988i	$0.501581\pi$
$770$	0	0
$771$	−11.5470	−0.415857
$772$	0	0
$773$	7.37812	0.265372	0.132686	−	0.991158i	$-0.457640\pi$
$773$	7.37812	0.265372	0.132686	+	0.991158i	$0.457640\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	20.3742	0.729981
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−36.8946	−1.31851
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−18.6603	−0.665167	−0.332584	−	0.943074i	$-0.607920\pi$
$787$	−18.6603	−0.665167	−0.332584	+	0.943074i	$0.607920\pi$
$788$	0	0
$789$	−6.43139	−0.228964
$790$	0	0
$791$	0	0
$792$	0	0
$793$	45.3738	1.61127
$794$	0	0
$795$	0	0
$796$	0	0
$797$	9.81896	0.347805	0.173903	−	0.984763i	$-0.444362\pi$
$797$	9.81896	0.347805	0.173903	+	0.984763i	$0.444362\pi$
$798$	0	0
$799$	20.1889	0.714231
$800$	0	0
$801$	10.2343	0.361612
$802$	0	0
$803$	−36.8307	−1.29973
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−6.29036	−0.221431
$808$	0	0
$809$	−32.0039	−1.12520	−0.562599	−	0.826730i	$-0.690198\pi$
$809$	−32.0039	−1.12520	−0.562599	+	0.826730i	$0.690198\pi$
$810$	0	0
$811$	−43.5137	−1.52797	−0.763987	−	0.645232i	$-0.776761\pi$
$811$	−43.5137	−1.52797	−0.763987	+	0.645232i	$0.776761\pi$
$812$	0	0
$813$	−11.5470	−0.404972
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−14.5431	−0.508799
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−34.7242	−1.21188	−0.605941	−	0.795510i	$-0.707203\pi$
$821$	−34.7242	−1.21188	−0.605941	+	0.795510i	$0.707203\pi$
$822$	0	0
$823$	23.3021	0.812261	0.406130	−	0.913815i	$-0.366878\pi$
$823$	23.3021	0.812261	0.406130	+	0.913815i	$0.366878\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−11.2943	−0.392741	−0.196370	−	0.980530i	$-0.562915\pi$
$827$	−11.2943	−0.392741	−0.196370	+	0.980530i	$0.562915\pi$
$828$	0	0
$829$	−20.0078	−0.694901	−0.347450	−	0.937698i	$-0.612953\pi$
$829$	−20.0078	−0.694901	−0.347450	+	0.937698i	$0.612953\pi$
$830$	0	0
$831$	−7.88791	−0.273629
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	21.0118	0.726273
$838$	0	0
$839$	−13.5846	−0.468994	−0.234497	−	0.972117i	$-0.575344\pi$
$839$	−13.5846	−0.468994	−0.234497	+	0.972117i	$0.575344\pi$
$840$	0	0
$841$	61.2872	2.11335
$842$	0	0
$843$	−19.9546	−0.687272
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−9.04936	−0.310573
$850$	0	0
$851$	41.0984	1.40883
$852$	0	0
$853$	−29.6336	−1.01464	−0.507318	−	0.861759i	$-0.669363\pi$
$853$	−29.6336	−1.01464	−0.507318	+	0.861759i	$0.669363\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.8307	−0.643245	−0.321623	−	0.946868i	$-0.604228\pi$
$857$	−18.8307	−0.643245	−0.321623	+	0.946868i	$0.604228\pi$
$858$	0	0
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−19.8574	−0.675952	−0.337976	−	0.941155i	$-0.609742\pi$
$863$	−19.8574	−0.675952	−0.337976	+	0.941155i	$0.609742\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−9.17436	−0.311577
$868$	0	0
$869$	63.8425	2.16571
$870$	0	0
$871$	0.596400	0.0202082
$872$	0	0
$873$	5.00392	0.169357
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13.5513	0.457595	0.228798	−	0.973474i	$-0.426521\pi$
$877$	13.5513	0.457595	0.228798	+	0.973474i	$0.426521\pi$
$878$	0	0
$879$	6.28759	0.212075
$880$	0	0
$881$	47.5964	1.60356	0.801782	−	0.597617i	$-0.203886\pi$
$881$	47.5964	1.60356	0.801782	+	0.597617i	$0.203886\pi$
$882$	0	0
$883$	−36.6497	−1.23336	−0.616680	−	0.787214i	$-0.711523\pi$
$883$	−36.6497	−1.23336	−0.616680	+	0.787214i	$0.711523\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	46.1250	1.54873	0.774363	−	0.632742i	$-0.218071\pi$
$887$	46.1250	1.54873	0.774363	+	0.632742i	$0.218071\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−21.4549	−0.718767
$892$	0	0
$893$	31.1971	1.04397
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−20.8229	−0.695256
$898$	0	0
$899$	−51.4193	−1.71493
$900$	0	0
$901$	−19.8268	−0.660526
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	5.06780	0.168274	0.0841368	−	0.996454i	$-0.473187\pi$
$907$	5.06780	0.168274	0.0841368	+	0.996454i	$0.473187\pi$
$908$	0	0
$909$	−8.30881	−0.275586
$910$	0	0
$911$	−2.41536	−0.0800244	−0.0400122	−	0.999199i	$-0.512740\pi$
$911$	−2.41536	−0.0800244	−0.0400122	+	0.999199i	$0.512740\pi$
$912$	0	0
$913$	48.1967	1.59508
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	36.5510	1.20570	0.602852	−	0.797853i	$-0.294031\pi$
$919$	36.5510	1.20570	0.602852	+	0.797853i	$0.294031\pi$
$920$	0	0
$921$	−6.43692	−0.212104
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	27.2382	0.894621
$928$	0	0
$929$	16.2304	0.532502	0.266251	−	0.963904i	$-0.414215\pi$
$929$	16.2304	0.532502	0.266251	+	0.963904i	$0.414215\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	9.29935	0.304447
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−44.1889	−1.44359	−0.721794	−	0.692108i	$-0.756682\pi$
$937$	−44.1889	−1.44359	−0.721794	+	0.692108i	$0.756682\pi$
$938$	0	0
$939$	−12.3778	−0.403933
$940$	0	0
$941$	−23.4726	−0.765183	−0.382592	−	0.923918i	$-0.624968\pi$
$941$	−23.4726	−0.765183	−0.382592	+	0.923918i	$0.624968\pi$
$942$	0	0
$943$	−38.2116	−1.24434
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−56.0717	−1.82209	−0.911043	−	0.412311i	$-0.864722\pi$
$947$	−56.0717	−1.82209	−0.911043	+	0.412311i	$0.864722\pi$
$948$	0	0
$949$	41.6458	1.35188
$950$	0	0
$951$	9.30489	0.301732
$952$	0	0
$953$	22.7163	0.735854	0.367927	−	0.929855i	$-0.380068\pi$
$953$	22.7163	0.735854	0.367927	+	0.929855i	$0.380068\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−30.1889	−0.975868
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.71632	−0.0553653
$962$	0	0
$963$	9.71471	0.313052
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−10.9400	−0.351808	−0.175904	−	0.984407i	$-0.556285\pi$
$967$	−10.9400	−0.351808	−0.175904	+	0.984407i	$0.556285\pi$
$968$	0	0
$969$	4.36208	0.140130
$970$	0	0
$971$	−25.7402	−0.826042	−0.413021	−	0.910721i	$-0.635526\pi$
$971$	−25.7402	−0.826042	−0.413021	+	0.910721i	$0.635526\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−16.8229	−0.538212	−0.269106	−	0.963111i	$-0.586728\pi$
$977$	−16.8229	−0.538212	−0.269106	+	0.963111i	$0.586728\pi$
$978$	0	0
$979$	18.4154	0.588557
$980$	0	0
$981$	25.6990	0.820507
$982$	0	0
$983$	−43.9851	−1.40291	−0.701454	−	0.712715i	$-0.747465\pi$
$983$	−43.9851	−1.40291	−0.701454	+	0.712715i	$0.747465\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	27.2755	0.867310
$990$	0	0
$991$	−44.2343	−1.40515	−0.702575	−	0.711610i	$-0.747966\pi$
$991$	−44.2343	−1.40515	−0.702575	+	0.711610i	$0.747966\pi$
$992$	0	0
$993$	6.00000	0.190404
$994$	0	0
$995$	0	0
$996$	0	0
$997$	34.1967	1.08302	0.541510	−	0.840694i	$-0.317853\pi$
$997$	34.1967	1.08302	0.541510	+	0.840694i	$0.317853\pi$
$998$	0	0
$999$	27.5314	0.871054

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9800.2.a.cf.1.2		3
5.4	even	2		1960.2.a.v.1.2		3
7.3	odd	6		1400.2.q.j.401.2		6
7.5	odd	6		1400.2.q.j.1201.2		6
7.6	odd	2		9800.2.a.ce.1.2		3
20.19	odd	2		3920.2.a.cb.1.2		3
35.3	even	12		1400.2.bh.i.849.3		12
35.4	even	6		1960.2.q.w.961.2		6
35.9	even	6		1960.2.q.w.361.2		6
35.12	even	12		1400.2.bh.i.249.3		12
35.17	even	12		1400.2.bh.i.849.4		12
35.19	odd	6		280.2.q.e.81.2	✓	6
35.24	odd	6		280.2.q.e.121.2	yes	6
35.33	even	12		1400.2.bh.i.249.4		12
35.34	odd	2		1960.2.a.w.1.2		3
105.59	even	6		2520.2.bi.q.1801.2		6
105.89	even	6		2520.2.bi.q.361.2		6
140.19	even	6		560.2.q.l.81.2		6
140.59	even	6		560.2.q.l.401.2		6
140.139	even	2		3920.2.a.cc.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.q.e.81.2	✓	6	35.19	odd	6
280.2.q.e.121.2	yes	6	35.24	odd	6
560.2.q.l.81.2		6	140.19	even	6
560.2.q.l.401.2		6	140.59	even	6
1400.2.q.j.401.2		6	7.3	odd	6
1400.2.q.j.1201.2		6	7.5	odd	6
1400.2.bh.i.249.3		12	35.12	even	12
1400.2.bh.i.249.4		12	35.33	even	12
1400.2.bh.i.849.3		12	35.3	even	12
1400.2.bh.i.849.4		12	35.17	even	12
1960.2.a.v.1.2		3	5.4	even	2
1960.2.a.w.1.2		3	35.34	odd	2
1960.2.q.w.361.2		6	35.9	even	6
1960.2.q.w.961.2		6	35.4	even	6
2520.2.bi.q.361.2		6	105.89	even	6
2520.2.bi.q.1801.2		6	105.59	even	6
3920.2.a.cb.1.2		3	20.19	odd	2
3920.2.a.cc.1.2		3	140.139	even	2
9800.2.a.ce.1.2		3	7.6	odd	2
9800.2.a.cf.1.2		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9800.a (trivial)

\(f(q)\)	\(=\)	\(q+0.705720 q^{3} -2.50196 q^{9} +O(q^{10})\)	\(q+0.705720 q^{3} -2.50196 q^{9} -4.50196 q^{11} +5.09052 q^{13} +2.00000 q^{17} +3.09052 q^{19} -5.79624 q^{23} -3.88284 q^{27} +9.50196 q^{29} -5.41144 q^{31} -3.17712 q^{33} -7.09052 q^{37} +3.59248 q^{39} +6.59248 q^{41} -4.70572 q^{43} +10.0944 q^{47} +1.41144 q^{51} -9.91340 q^{53} +2.18104 q^{57} -8.00000 q^{59} +8.91340 q^{61} +0.117159 q^{67} -4.09052 q^{69} +8.18104 q^{73} -14.1810 q^{79} +4.76568 q^{81} -10.7057 q^{83} +6.70572 q^{87} -4.09052 q^{89} -3.81896 q^{93} -2.00000 q^{97} +11.2637 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 9 q^{9}+O(q^{10}) \) 3 * q + 9 * q^9	\( 3 q + 9 q^{9} + 3 q^{11} + 3 q^{13} + 6 q^{17} - 3 q^{19} - 3 q^{23} - 18 q^{27} + 12 q^{29} - 12 q^{31} - 18 q^{33} - 9 q^{37} - 18 q^{39} - 9 q^{41} - 12 q^{43} - 15 q^{47} - 9 q^{53} - 18 q^{57} - 24 q^{59} + 6 q^{61} - 6 q^{67} - 18 q^{79} + 27 q^{81} - 30 q^{83} + 18 q^{87} - 36 q^{93} - 6 q^{97} + 63 q^{99}+O(q^{100}) \) 3 * q + 9 * q^9 + 3 * q^11 + 3 * q^13 + 6 * q^17 - 3 * q^19 - 3 * q^23 - 18 * q^27 + 12 * q^29 - 12 * q^31 - 18 * q^33 - 9 * q^37 - 18 * q^39 - 9 * q^41 - 12 * q^43 - 15 * q^47 - 9 * q^53 - 18 * q^57 - 24 * q^59 + 6 * q^61 - 6 * q^67 - 18 * q^79 + 27 * q^81 - 30 * q^83 + 18 * q^87 - 36 * q^93 - 6 * q^97 + 63 * q^99

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