LMFDB - Embedded newform 8450.2.a.br.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8450,2,Mod(1,8450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8450 = 2 \cdot 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8450.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:	$67.4735897080$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.2808.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 9x - 2 $ x^3 - 9*x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 650)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.88202$ of defining polynomial
Character	$\chi$	$=$	8450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +2.88202 q^{3} +1.00000 q^{4} -2.88202 q^{6} +1.88202 q^{7} -1.00000 q^{8} +5.30604 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +2.88202 q^{3} +1.00000 q^{4} -2.88202 q^{6} +1.88202 q^{7} -1.00000 q^{8} +5.30604 q^{9} +6.18806 q^{11} +2.88202 q^{12} -1.88202 q^{14} +1.00000 q^{16} -3.00000 q^{17} -5.30604 q^{18} +4.88202 q^{19} +5.42402 q^{21} -6.18806 q^{22} -0.575978 q^{23} -2.88202 q^{24} +6.64606 q^{27} +1.88202 q^{28} +5.07008 q^{29} +7.30604 q^{31} -1.00000 q^{32} +17.8341 q^{33} +3.00000 q^{34} +5.30604 q^{36} +9.18806 q^{37} -4.88202 q^{38} +5.45800 q^{41} -5.42402 q^{42} -4.00000 q^{43} +6.18806 q^{44} +0.575978 q^{46} -2.45800 q^{47} +2.88202 q^{48} -3.45800 q^{49} -8.64606 q^{51} -11.0701 q^{53} -6.64606 q^{54} -1.88202 q^{56} +14.0701 q^{57} -5.07008 q^{58} -8.83412 q^{59} +3.64606 q^{61} -7.30604 q^{62} +9.98608 q^{63} +1.00000 q^{64} -17.8341 q^{66} +3.57598 q^{67} -3.00000 q^{68} -1.65998 q^{69} -5.30604 q^{72} -11.8341 q^{73} -9.18806 q^{74} +4.88202 q^{76} +11.6461 q^{77} -11.1881 q^{79} +3.23596 q^{81} -5.45800 q^{82} +0.188063 q^{83} +5.42402 q^{84} +4.00000 q^{86} +14.6121 q^{87} -6.18806 q^{88} -11.4580 q^{89} -0.575978 q^{92} +21.0562 q^{93} +2.45800 q^{94} -2.88202 q^{96} -15.7640 q^{97} +3.45800 q^{98} +32.8341 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 3 q^{7} - 3 q^{8} + 9 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^7 - 3 * q^8 + 9 * q^9	$ 3 q - 3 q^{2} + 3 q^{4} - 3 q^{7} - 3 q^{8} + 9 q^{9} + 3 q^{11} + 3 q^{14} + 3 q^{16} - 9 q^{17} - 9 q^{18} + 6 q^{19} + 18 q^{21} - 3 q^{22} - 6 q^{27} - 3 q^{28} - 9 q^{29} + 15 q^{31} - 3 q^{32} + 12 q^{33} + 9 q^{34} + 9 q^{36} + 12 q^{37} - 6 q^{38} + 6 q^{41} - 18 q^{42} - 12 q^{43} + 3 q^{44} + 3 q^{47} - 9 q^{53} + 6 q^{54} + 3 q^{56} + 18 q^{57} + 9 q^{58} + 15 q^{59} - 15 q^{61} - 15 q^{62} - 15 q^{63} + 3 q^{64} - 12 q^{66} + 9 q^{67} - 9 q^{68} - 24 q^{69} - 9 q^{72} + 6 q^{73} - 12 q^{74} + 6 q^{76} + 9 q^{77} - 18 q^{79} + 27 q^{81} - 6 q^{82} - 15 q^{83} + 18 q^{84} + 12 q^{86} + 30 q^{87} - 3 q^{88} - 24 q^{89} - 6 q^{93} - 3 q^{94} - 30 q^{97} + 57 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^7 - 3 * q^8 + 9 * q^9 + 3 * q^11 + 3 * q^14 + 3 * q^16 - 9 * q^17 - 9 * q^18 + 6 * q^19 + 18 * q^21 - 3 * q^22 - 6 * q^27 - 3 * q^28 - 9 * q^29 + 15 * q^31 - 3 * q^32 + 12 * q^33 + 9 * q^34 + 9 * q^36 + 12 * q^37 - 6 * q^38 + 6 * q^41 - 18 * q^42 - 12 * q^43 + 3 * q^44 + 3 * q^47 - 9 * q^53 + 6 * q^54 + 3 * q^56 + 18 * q^57 + 9 * q^58 + 15 * q^59 - 15 * q^61 - 15 * q^62 - 15 * q^63 + 3 * q^64 - 12 * q^66 + 9 * q^67 - 9 * q^68 - 24 * q^69 - 9 * q^72 + 6 * q^73 - 12 * q^74 + 6 * q^76 + 9 * q^77 - 18 * q^79 + 27 * q^81 - 6 * q^82 - 15 * q^83 + 18 * q^84 + 12 * q^86 + 30 * q^87 - 3 * q^88 - 24 * q^89 - 6 * q^93 - 3 * q^94 - 30 * q^97 + 57 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	2.88202	1.66394	0.831968	−	0.554824i	$-0.187214\pi$
$3$	2.88202	1.66394	0.831968	+	0.554824i	$0.187214\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−2.88202	−1.17658
$7$	1.88202	0.711337	0.355668	−	0.934612i	$-0.384253\pi$
$7$	1.88202	0.711337	0.355668	+	0.934612i	$0.384253\pi$
$8$	−1.00000	−0.353553
$9$	5.30604	1.76868
$10$	0	0
$11$	6.18806	1.86577	0.932886	−	0.360173i	$-0.117282\pi$
$11$	6.18806	1.86577	0.932886	+	0.360173i	$0.117282\pi$
$12$	2.88202	0.831968
$13$	0	0
$13$	0	0
$14$	−1.88202	−0.502991
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	−5.30604	−1.25065
$19$	4.88202	1.12001	0.560006	−	0.828488i	$-0.310799\pi$
$19$	4.88202	1.12001	0.560006	+	0.828488i	$0.310799\pi$
$20$	0	0
$21$	5.42402	1.18362
$22$	−6.18806	−1.31930
$23$	−0.575978	−0.120100	−0.0600499	−	0.998195i	$-0.519126\pi$
$23$	−0.575978	−0.120100	−0.0600499	+	0.998195i	$0.519126\pi$
$24$	−2.88202	−0.588290
$25$	0	0
$26$	0	0
$27$	6.64606	1.27904
$28$	1.88202	0.355668
$29$	5.07008	0.941491	0.470745	−	0.882269i	$-0.343985\pi$
$29$	5.07008	0.941491	0.470745	+	0.882269i	$0.343985\pi$
$30$	0	0
$31$	7.30604	1.31220	0.656102	−	0.754672i	$-0.272204\pi$
$31$	7.30604	1.31220	0.656102	+	0.754672i	$0.272204\pi$
$32$	−1.00000	−0.176777
$33$	17.8341	3.10452
$34$	3.00000	0.514496
$35$	0	0
$36$	5.30604	0.884340
$37$	9.18806	1.51051	0.755254	−	0.655432i	$-0.227513\pi$
$37$	9.18806	1.51051	0.755254	+	0.655432i	$0.227513\pi$
$38$	−4.88202	−0.791968
$39$	0	0
$40$	0	0
$41$	5.45800	0.852396	0.426198	−	0.904630i	$-0.359853\pi$
$41$	5.45800	0.852396	0.426198	+	0.904630i	$0.359853\pi$
$42$	−5.42402	−0.836945
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	6.18806	0.932886
$45$	0	0
$46$	0.575978	0.0849233
$47$	−2.45800	−0.358536	−0.179268	−	0.983800i	$-0.557373\pi$
$47$	−2.45800	−0.358536	−0.179268	+	0.983800i	$0.557373\pi$
$48$	2.88202	0.415984
$49$	−3.45800	−0.494000
$50$	0	0
$51$	−8.64606	−1.21069
$52$	0	0
$53$	−11.0701	−1.52059	−0.760296	−	0.649576i	$-0.774946\pi$
$53$	−11.0701	−1.52059	−0.760296	+	0.649576i	$0.774946\pi$
$54$	−6.64606	−0.904414
$55$	0	0
$56$	−1.88202	−0.251496
$57$	14.0701	1.86363
$58$	−5.07008	−0.665735
$59$	−8.83412	−1.15011	−0.575053	−	0.818116i	$-0.695018\pi$
$59$	−8.83412	−1.15011	−0.575053	+	0.818116i	$0.695018\pi$
$60$	0	0
$61$	3.64606	0.466830	0.233415	−	0.972377i	$-0.425010\pi$
$61$	3.64606	0.466830	0.233415	+	0.972377i	$0.425010\pi$
$62$	−7.30604	−0.927868
$63$	9.98608	1.25813
$64$	1.00000	0.125000
$65$	0	0
$66$	−17.8341	−2.19523
$67$	3.57598	0.436875	0.218438	−	0.975851i	$-0.429904\pi$
$67$	3.57598	0.436875	0.218438	+	0.975851i	$0.429904\pi$
$68$	−3.00000	−0.363803
$69$	−1.65998	−0.199838
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−5.30604	−0.625323
$73$	−11.8341	−1.38508	−0.692540	−	0.721380i	$-0.743508\pi$
$73$	−11.8341	−1.38508	−0.692540	+	0.721380i	$0.743508\pi$
$74$	−9.18806	−1.06809
$75$	0	0
$76$	4.88202	0.560006
$77$	11.6461	1.32719
$78$	0	0
$79$	−11.1881	−1.25876	−0.629378	−	0.777100i	$-0.716690\pi$
$79$	−11.1881	−1.25876	−0.629378	+	0.777100i	$0.716690\pi$
$80$	0	0
$81$	3.23596	0.359551
$82$	−5.45800	−0.602735
$83$	0.188063	0.0206426	0.0103213	−	0.999947i	$-0.496715\pi$
$83$	0.188063	0.0206426	0.0103213	+	0.999947i	$0.496715\pi$
$84$	5.42402	0.591809
$85$	0	0
$86$	4.00000	0.431331
$87$	14.6121	1.56658
$88$	−6.18806	−0.659650
$89$	−11.4580	−1.21455	−0.607273	−	0.794493i	$-0.707736\pi$
$89$	−11.4580	−1.21455	−0.607273	+	0.794493i	$0.707736\pi$
$90$	0	0
$91$	0	0
$92$	−0.575978	−0.0600499
$93$	21.0562	2.18342
$94$	2.45800	0.253523
$95$	0	0
$96$	−2.88202	−0.294145
$97$	−15.7640	−1.60060	−0.800298	−	0.599603i	$-0.795325\pi$
$97$	−15.7640	−1.60060	−0.800298	+	0.599603i	$0.795325\pi$
$98$	3.45800	0.349311
$99$	32.8341	3.29995
$100$	0	0
$101$	−10.4941	−1.04420	−0.522101	−	0.852884i	$-0.674852\pi$
$101$	−10.4941	−1.04420	−0.522101	+	0.852884i	$0.674852\pi$
$102$	8.64606	0.856088
$103$	17.1881	1.69359	0.846795	−	0.531919i	$-0.178529\pi$
$103$	17.1881	1.69359	0.846795	+	0.531919i	$0.178529\pi$
$104$	0	0
$105$	0	0
$106$	11.0701	1.07522
$107$	1.49411	0.144441	0.0722203	−	0.997389i	$-0.476992\pi$
$107$	1.49411	0.144441	0.0722203	+	0.997389i	$0.476992\pi$
$108$	6.64606	0.639518
$109$	−12.9521	−1.24059	−0.620293	−	0.784370i	$-0.712986\pi$
$109$	−12.9521	−1.24059	−0.620293	+	0.784370i	$0.712986\pi$
$110$	0	0
$111$	26.4802	2.51339
$112$	1.88202	0.177834
$113$	−9.22204	−0.867537	−0.433768	−	0.901024i	$-0.642816\pi$
$113$	−9.22204	−0.867537	−0.433768	+	0.901024i	$0.642816\pi$
$114$	−14.0701	−1.31778
$115$	0	0
$116$	5.07008	0.470745
$117$	0	0
$118$	8.83412	0.813247
$119$	−5.64606	−0.517574
$120$	0	0
$121$	27.2921	2.48110
$122$	−3.64606	−0.330099
$123$	15.7301	1.41833
$124$	7.30604	0.656102
$125$	0	0
$126$	−9.98608	−0.889631
$127$	4.37613	0.388318	0.194159	−	0.980970i	$-0.437802\pi$
$127$	4.37613	0.388318	0.194159	+	0.980970i	$0.437802\pi$
$128$	−1.00000	−0.0883883
$129$	−11.5281	−1.01499
$130$	0	0
$131$	22.1402	1.93440	0.967198	−	0.254025i	$-0.0817545\pi$
$131$	22.1402	1.93440	0.967198	+	0.254025i	$0.0817545\pi$
$132$	17.8341	1.55226
$133$	9.18806	0.796706
$134$	−3.57598	−0.308917
$135$	0	0
$136$	3.00000	0.257248
$137$	5.83412	0.498443	0.249221	−	0.968447i	$-0.419825\pi$
$137$	5.83412	0.498443	0.249221	+	0.968447i	$0.419825\pi$
$138$	1.65998	0.141307
$139$	4.26994	0.362171	0.181086	−	0.983467i	$-0.442039\pi$
$139$	4.26994	0.362171	0.181086	+	0.983467i	$0.442039\pi$
$140$	0	0
$141$	−7.08400	−0.596581
$142$	0	0
$143$	0	0
$144$	5.30604	0.442170
$145$	0	0
$146$	11.8341	0.979399
$147$	−9.96602	−0.821984
$148$	9.18806	0.755254
$149$	16.9160	1.38581	0.692906	−	0.721028i	$-0.256330\pi$
$149$	16.9160	1.38581	0.692906	+	0.721028i	$0.256330\pi$
$150$	0	0
$151$	−17.4462	−1.41975	−0.709876	−	0.704326i	$-0.751249\pi$
$151$	−17.4462	−1.41975	−0.709876	+	0.704326i	$0.751249\pi$
$152$	−4.88202	−0.395984
$153$	−15.9181	−1.28690
$154$	−11.6461	−0.938466
$155$	0	0
$156$	0	0
$157$	−2.73006	−0.217883	−0.108941	−	0.994048i	$-0.534746\pi$
$157$	−2.73006	−0.217883	−0.108941	+	0.994048i	$0.534746\pi$
$158$	11.1881	0.890074
$159$	−31.9042	−2.53017
$160$	0	0
$161$	−1.08400	−0.0854314
$162$	−3.23596	−0.254241
$163$	15.7980	1.23740	0.618698	−	0.785629i	$-0.287660\pi$
$163$	15.7980	1.23740	0.618698	+	0.785629i	$0.287660\pi$
$164$	5.45800	0.426198
$165$	0	0
$166$	−0.188063	−0.0145965
$167$	−15.5642	−1.20439	−0.602197	−	0.798348i	$-0.705708\pi$
$167$	−15.5642	−1.20439	−0.602197	+	0.798348i	$0.705708\pi$
$168$	−5.42402	−0.418472
$169$	0	0
$170$	0	0
$171$	25.9042	1.98094
$172$	−4.00000	−0.304997
$173$	−5.64606	−0.429262	−0.214631	−	0.976695i	$-0.568855\pi$
$173$	−5.64606	−0.429262	−0.214631	+	0.976695i	$0.568855\pi$
$174$	−14.6121	−1.10774
$175$	0	0
$176$	6.18806	0.466443
$177$	−25.4601	−1.91370
$178$	11.4580	0.858813
$179$	−13.4941	−1.00860	−0.504298	−	0.863529i	$-0.668249\pi$
$179$	−13.4941	−1.00860	−0.504298	+	0.863529i	$0.668249\pi$
$180$	0	0
$181$	−2.73006	−0.202924	−0.101462	−	0.994839i	$-0.532352\pi$
$181$	−2.73006	−0.202924	−0.101462	+	0.994839i	$0.532352\pi$
$182$	0	0
$183$	10.5080	0.776776
$184$	0.575978	0.0424617
$185$	0	0
$186$	−21.0562	−1.54391
$187$	−18.5642	−1.35755
$188$	−2.45800	−0.179268
$189$	12.5080	0.909825
$190$	0	0
$191$	−4.84804	−0.350792	−0.175396	−	0.984498i	$-0.556121\pi$
$191$	−4.84804	−0.350792	−0.175396	+	0.984498i	$0.556121\pi$
$192$	2.88202	0.207992
$193$	10.6822	0.768919	0.384460	−	0.923142i	$-0.374388\pi$
$193$	10.6822	0.768919	0.384460	+	0.923142i	$0.374388\pi$
$194$	15.7640	1.13179
$195$	0	0
$196$	−3.45800	−0.247000
$197$	15.5642	1.10890	0.554451	−	0.832216i	$-0.312928\pi$
$197$	15.5642	1.10890	0.554451	+	0.832216i	$0.312928\pi$
$198$	−32.8341	−2.33342
$199$	0.272066	0.0192862	0.00964311	−	0.999954i	$-0.496930\pi$
$199$	0.272066	0.0192862	0.00964311	+	0.999954i	$0.496930\pi$
$200$	0	0
$201$	10.3060	0.726932
$202$	10.4941	0.738363
$203$	9.54200	0.669717
$204$	−8.64606	−0.605345
$205$	0	0
$206$	−17.1881	−1.19755
$207$	−3.05616	−0.212418
$208$	0	0
$209$	30.2103	2.08969
$210$	0	0
$211$	−17.0222	−1.17186	−0.585928	−	0.810363i	$-0.699270\pi$
$211$	−17.0222	−1.17186	−0.585928	+	0.810363i	$0.699270\pi$
$212$	−11.0701	−0.760296
$213$	0	0
$214$	−1.49411	−0.102135
$215$	0	0
$216$	−6.64606	−0.452207
$217$	13.7501	0.933419
$218$	12.9521	0.877227
$219$	−34.1062	−2.30468
$220$	0	0
$221$	0	0
$222$	−26.4802	−1.77723
$223$	−1.65998	−0.111161	−0.0555803	−	0.998454i	$-0.517701\pi$
$223$	−1.65998	−0.111161	−0.0555803	+	0.998454i	$0.517701\pi$
$224$	−1.88202	−0.125748
$225$	0	0
$226$	9.22204	0.613441
$227$	−8.45800	−0.561377	−0.280689	−	0.959799i	$-0.590563\pi$
$227$	−8.45800	−0.561377	−0.280689	+	0.959799i	$0.590563\pi$
$228$	14.0701	0.931814
$229$	9.38792	0.620371	0.310185	−	0.950676i	$-0.399609\pi$
$229$	9.38792	0.620371	0.310185	+	0.950676i	$0.399609\pi$
$230$	0	0
$231$	33.5642	2.20836
$232$	−5.07008	−0.332867
$233$	−20.9882	−1.37498	−0.687492	−	0.726192i	$-0.741288\pi$
$233$	−20.9882	−1.37498	−0.687492	+	0.726192i	$0.741288\pi$
$234$	0	0
$235$	0	0
$236$	−8.83412	−0.575053
$237$	−32.2442	−2.09449
$238$	5.64606	0.365980
$239$	−14.4580	−0.935210	−0.467605	−	0.883937i	$-0.654883\pi$
$239$	−14.4580	−0.935210	−0.467605	+	0.883937i	$0.654883\pi$
$240$	0	0
$241$	9.92992	0.639642	0.319821	−	0.947478i	$-0.396377\pi$
$241$	9.92992	0.639642	0.319821	+	0.947478i	$0.396377\pi$
$242$	−27.2921	−1.75440
$243$	−10.6121	−0.680766
$244$	3.64606	0.233415
$245$	0	0
$246$	−15.7301	−1.00291
$247$	0	0
$248$	−7.30604	−0.463934
$249$	0.542001	0.0343479
$250$	0	0
$251$	14.6461	0.924451	0.462226	−	0.886762i	$-0.347051\pi$
$251$	14.6461	0.924451	0.462226	+	0.886762i	$0.347051\pi$
$252$	9.98608	0.629064
$253$	−3.56419	−0.224079
$254$	−4.37613	−0.274583
$255$	0	0
$256$	1.00000	0.0625000
$257$	−23.0701	−1.43907	−0.719536	−	0.694455i	$-0.755646\pi$
$257$	−23.0701	−1.43907	−0.719536	+	0.694455i	$0.755646\pi$
$258$	11.5281	0.717707
$259$	17.2921	1.07448
$260$	0	0
$261$	26.9021	1.66520
$262$	−22.1402	−1.36782
$263$	−9.56419	−0.589753	−0.294877	−	0.955535i	$-0.595279\pi$
$263$	−9.56419	−0.589753	−0.294877	+	0.955535i	$0.595279\pi$
$264$	−17.8341	−1.09761
$265$	0	0
$266$	−9.18806	−0.563356
$267$	−33.0222	−2.02093
$268$	3.57598	0.218438
$269$	5.07008	0.309128	0.154564	−	0.987983i	$-0.450603\pi$
$269$	5.07008	0.309128	0.154564	+	0.987983i	$0.450603\pi$
$270$	0	0
$271$	−0.730064	−0.0443482	−0.0221741	−	0.999754i	$-0.507059\pi$
$271$	−0.730064	−0.0443482	−0.0221741	+	0.999754i	$0.507059\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	−5.83412	−0.352452
$275$	0	0
$276$	−1.65998	−0.0999191
$277$	5.56419	0.334320	0.167160	−	0.985930i	$-0.446540\pi$
$277$	5.56419	0.334320	0.167160	+	0.985930i	$0.446540\pi$
$278$	−4.26994	−0.256094
$279$	38.7662	2.32087
$280$	0	0
$281$	23.2921	1.38949	0.694746	−	0.719255i	$-0.255517\pi$
$281$	23.2921	1.38949	0.694746	+	0.719255i	$0.255517\pi$
$282$	7.08400	0.421846
$283$	14.1062	0.838526	0.419263	−	0.907865i	$-0.362289\pi$
$283$	14.1062	0.838526	0.419263	+	0.907865i	$0.362289\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.2721	0.606341
$288$	−5.30604	−0.312662
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−45.4323	−2.66329
$292$	−11.8341	−0.692540
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	9.96602	0.581230
$295$	0	0
$296$	−9.18806	−0.534045
$297$	41.1262	2.38639
$298$	−16.9160	−0.979917
$299$	0	0
$300$	0	0
$301$	−7.52808	−0.433911
$302$	17.4462	1.00392
$303$	−30.2442	−1.73749
$304$	4.88202	0.280003
$305$	0	0
$306$	15.9181	0.909979
$307$	−9.42189	−0.537736	−0.268868	−	0.963177i	$-0.586650\pi$
$307$	−9.42189	−0.537736	−0.268868	+	0.963177i	$0.586650\pi$
$308$	11.6461	0.663596
$309$	49.5364	2.81802
$310$	0	0
$311$	31.7044	1.79779	0.898895	−	0.438165i	$-0.144371\pi$
$311$	31.7044	1.79779	0.898895	+	0.438165i	$0.144371\pi$
$312$	0	0
$313$	15.3740	0.868990	0.434495	−	0.900674i	$-0.356927\pi$
$313$	15.3740	0.868990	0.434495	+	0.900674i	$0.356927\pi$
$314$	2.73006	0.154066
$315$	0	0
$316$	−11.1881	−0.629378
$317$	3.18806	0.179059	0.0895297	−	0.995984i	$-0.471464\pi$
$317$	3.18806	0.179059	0.0895297	+	0.995984i	$0.471464\pi$
$318$	31.9042	1.78910
$319$	31.3740	1.75661
$320$	0	0
$321$	4.30604	0.240340
$322$	1.08400	0.0604091
$323$	−14.6461	−0.814929
$324$	3.23596	0.179775
$325$	0	0
$326$	−15.7980	−0.874971
$327$	−37.3282	−2.06426
$328$	−5.45800	−0.301368
$329$	−4.62600	−0.255040
$330$	0	0
$331$	−9.02219	−0.495904	−0.247952	−	0.968772i	$-0.579758\pi$
$331$	−9.02219	−0.495904	−0.247952	+	0.968772i	$0.579758\pi$
$332$	0.188063	0.0103213
$333$	48.7523	2.67161
$334$	15.5642	0.851635
$335$	0	0
$336$	5.42402	0.295905
$337$	−16.2921	−0.887489	−0.443744	−	0.896153i	$-0.646350\pi$
$337$	−16.2921	−0.887489	−0.443744	+	0.896153i	$0.646350\pi$
$338$	0	0
$339$	−26.5781	−1.44352
$340$	0	0
$341$	45.2103	2.44827
$342$	−25.9042	−1.40074
$343$	−19.6822	−1.06274
$344$	4.00000	0.215666
$345$	0	0
$346$	5.64606	0.303534
$347$	−12.3421	−0.662561	−0.331281	−	0.943532i	$-0.607481\pi$
$347$	−12.3421	−0.662561	−0.331281	+	0.943532i	$0.607481\pi$
$348$	14.6121	0.783290
$349$	18.7523	1.00379	0.501893	−	0.864930i	$-0.332637\pi$
$349$	18.7523	1.00379	0.501893	+	0.864930i	$0.332637\pi$
$350$	0	0
$351$	0	0
$352$	−6.18806	−0.329825
$353$	0.707877	0.0376765	0.0188382	−	0.999823i	$-0.494003\pi$
$353$	0.707877	0.0376765	0.0188382	+	0.999823i	$0.494003\pi$
$354$	25.4601	1.35319
$355$	0	0
$356$	−11.4580	−0.607273
$357$	−16.2721	−0.861209
$358$	13.4941	0.713186
$359$	−0.0221876	−0.00117101	−0.000585507	−	1.00000i	$-0.500186\pi$
$359$	−0.0221876	−0.00117101	−0.000585507		1.00000i	$0.500186\pi$
$360$	0	0
$361$	4.83412	0.254428
$362$	2.73006	0.143489
$363$	78.6565	4.12839
$364$	0	0
$365$	0	0
$366$	−10.5080	−0.549263
$367$	−15.8363	−0.826646	−0.413323	−	0.910585i	$-0.635632\pi$
$367$	−15.8363	−0.826646	−0.413323	+	0.910585i	$0.635632\pi$
$368$	−0.575978	−0.0300249
$369$	28.9604	1.50762
$370$	0	0
$371$	−20.8341	−1.08165
$372$	21.0562	1.09171
$373$	−4.35394	−0.225438	−0.112719	−	0.993627i	$-0.535956\pi$
$373$	−4.35394	−0.225438	−0.112719	+	0.993627i	$0.535956\pi$
$374$	18.5642	0.959931
$375$	0	0
$376$	2.45800	0.126762
$377$	0	0
$378$	−12.5080	−0.643343
$379$	−2.42402	−0.124514	−0.0622568	−	0.998060i	$-0.519830\pi$
$379$	−2.42402	−0.124514	−0.0622568	+	0.998060i	$0.519830\pi$
$380$	0	0
$381$	12.6121	0.646137
$382$	4.84804	0.248047
$383$	−7.46013	−0.381195	−0.190597	−	0.981668i	$-0.561042\pi$
$383$	−7.46013	−0.381195	−0.190597	+	0.981668i	$0.561042\pi$
$384$	−2.88202	−0.147072
$385$	0	0
$386$	−10.6822	−0.543708
$387$	−21.2242	−1.07889
$388$	−15.7640	−0.800298
$389$	−6.57598	−0.333415	−0.166708	−	0.986006i	$-0.553314\pi$
$389$	−6.57598	−0.333415	−0.166708	+	0.986006i	$0.553314\pi$
$390$	0	0
$391$	1.72793	0.0873854
$392$	3.45800	0.174655
$393$	63.8084	3.21871
$394$	−15.5642	−0.784113
$395$	0	0
$396$	32.8341	1.64998
$397$	6.95210	0.348916	0.174458	−	0.984665i	$-0.444183\pi$
$397$	6.95210	0.348916	0.174458	+	0.984665i	$0.444183\pi$
$398$	−0.272066	−0.0136374
$399$	26.4802	1.32567
$400$	0	0
$401$	6.91813	0.345475	0.172737	−	0.984968i	$-0.444739\pi$
$401$	6.91813	0.345475	0.172737	+	0.984968i	$0.444739\pi$
$402$	−10.3060	−0.514019
$403$	0	0
$404$	−10.4941	−0.522101
$405$	0	0
$406$	−9.54200	−0.473562
$407$	56.8563	2.81826
$408$	8.64606	0.428044
$409$	15.9299	0.787684	0.393842	−	0.919178i	$-0.371146\pi$
$409$	15.9299	0.787684	0.393842	+	0.919178i	$0.371146\pi$
$410$	0	0
$411$	16.8141	0.829377
$412$	17.1881	0.846795
$413$	−16.6260	−0.818112
$414$	3.05616	0.150202
$415$	0	0
$416$	0	0
$417$	12.3060	0.602629
$418$	−30.2103	−1.47763
$419$	−22.9500	−1.12118	−0.560590	−	0.828094i	$-0.689426\pi$
$419$	−22.9500	−1.12118	−0.560590	+	0.828094i	$0.689426\pi$
$420$	0	0
$421$	24.7523	1.20635	0.603175	−	0.797609i	$-0.293902\pi$
$421$	24.7523	1.20635	0.603175	+	0.797609i	$0.293902\pi$
$422$	17.0222	0.828627
$423$	−13.0422	−0.634136
$424$	11.0701	0.537611
$425$	0	0
$426$	0	0
$427$	6.86196	0.332074
$428$	1.49411	0.0722203
$429$	0	0
$430$	0	0
$431$	−32.1041	−1.54640	−0.773199	−	0.634163i	$-0.781345\pi$
$431$	−32.1041	−1.54640	−0.773199	+	0.634163i	$0.781345\pi$
$432$	6.64606	0.319759
$433$	16.2103	0.779015	0.389507	−	0.921023i	$-0.372645\pi$
$433$	16.2103	0.779015	0.389507	+	0.921023i	$0.372645\pi$
$434$	−13.7501	−0.660027
$435$	0	0
$436$	−12.9521	−0.620293
$437$	−2.81194	−0.134513
$438$	34.1062	1.62966
$439$	−22.1041	−1.05497	−0.527485	−	0.849565i	$-0.676865\pi$
$439$	−22.1041	−1.05497	−0.527485	+	0.849565i	$0.676865\pi$
$440$	0	0
$441$	−18.3483	−0.873728
$442$	0	0
$443$	−2.64606	−0.125718	−0.0628591	−	0.998022i	$-0.520022\pi$
$443$	−2.64606	−0.125718	−0.0628591	+	0.998022i	$0.520022\pi$
$444$	26.4802	1.25669
$445$	0	0
$446$	1.65998	0.0786024
$447$	48.7523	2.30590
$448$	1.88202	0.0889171
$449$	35.1262	1.65771	0.828855	−	0.559463i	$-0.188993\pi$
$449$	35.1262	1.65771	0.828855	+	0.559463i	$0.188993\pi$
$450$	0	0
$451$	33.7744	1.59038
$452$	−9.22204	−0.433768
$453$	−50.2803	−2.36238
$454$	8.45800	0.396954
$455$	0	0
$456$	−14.0701	−0.658892
$457$	18.9181	0.884953	0.442476	−	0.896780i	$-0.354100\pi$
$457$	18.9181	0.884953	0.442476	+	0.896780i	$0.354100\pi$
$458$	−9.38792	−0.438668
$459$	−19.9382	−0.930635
$460$	0	0
$461$	8.81194	0.410413	0.205206	−	0.978719i	$-0.434213\pi$
$461$	8.81194	0.410413	0.205206	+	0.978719i	$0.434213\pi$
$462$	−33.5642	−1.56155
$463$	28.5621	1.32739	0.663696	−	0.748003i	$-0.268987\pi$
$463$	28.5621	1.32739	0.663696	+	0.748003i	$0.268987\pi$
$464$	5.07008	0.235373
$465$	0	0
$466$	20.9882	0.972260
$467$	22.1402	1.02452	0.512262	−	0.858829i	$-0.328808\pi$
$467$	22.1402	1.02452	0.512262	+	0.858829i	$0.328808\pi$
$468$	0	0
$469$	6.73006	0.310765
$470$	0	0
$471$	−7.86810	−0.362543
$472$	8.83412	0.406624
$473$	−24.7523	−1.13811
$474$	32.2442	1.48103
$475$	0	0
$476$	−5.64606	−0.258787
$477$	−58.7383	−2.68944
$478$	14.4580	0.661293
$479$	−19.7501	−0.902406	−0.451203	−	0.892421i	$-0.649005\pi$
$479$	−19.7501	−0.902406	−0.451203	+	0.892421i	$0.649005\pi$
$480$	0	0
$481$	0	0
$482$	−9.92992	−0.452295
$483$	−3.12412	−0.142152
$484$	27.2921	1.24055
$485$	0	0
$486$	10.6121	0.481374
$487$	3.78623	0.171570	0.0857852	−	0.996314i	$-0.472660\pi$
$487$	3.78623	0.171570	0.0857852	+	0.996314i	$0.472660\pi$
$488$	−3.64606	−0.165049
$489$	45.5302	2.05895
$490$	0	0
$491$	−18.4441	−0.832370	−0.416185	−	0.909280i	$-0.636633\pi$
$491$	−18.4441	−0.832370	−0.416185	+	0.909280i	$0.636633\pi$
$492$	15.7301	0.709166
$493$	−15.2103	−0.685035
$494$	0	0
$495$	0	0
$496$	7.30604	0.328051
$497$	0	0
$498$	−0.542001	−0.0242877
$499$	−9.98608	−0.447038	−0.223519	−	0.974700i	$-0.571755\pi$
$499$	−9.98608	−0.447038	−0.223519	+	0.974700i	$0.571755\pi$
$500$	0	0
$501$	−44.8563	−2.00403
$502$	−14.6461	−0.653686
$503$	−2.30391	−0.102726	−0.0513632	−	0.998680i	$-0.516357\pi$
$503$	−2.30391	−0.102726	−0.0513632	+	0.998680i	$0.516357\pi$
$504$	−9.98608	−0.444815
$505$	0	0
$506$	3.56419	0.158448
$507$	0	0
$508$	4.37613	0.194159
$509$	19.0201	0.843049	0.421525	−	0.906817i	$-0.361495\pi$
$509$	19.0201	0.843049	0.421525	+	0.906817i	$0.361495\pi$
$510$	0	0
$511$	−22.2721	−0.985258
$512$	−1.00000	−0.0441942
$513$	32.4462	1.43254
$514$	23.0701	1.01758
$515$	0	0
$516$	−11.5281	−0.507496
$517$	−15.2103	−0.668946
$518$	−17.2921	−0.759772
$519$	−16.2721	−0.714264
$520$	0	0
$521$	−21.2220	−0.929754	−0.464877	−	0.885375i	$-0.653902\pi$
$521$	−21.2220	−0.929754	−0.464877	+	0.885375i	$0.653902\pi$
$522$	−26.9021	−1.17747
$523$	−28.3143	−1.23810	−0.619049	−	0.785352i	$-0.712482\pi$
$523$	−28.3143	−1.23810	−0.619049	+	0.785352i	$0.712482\pi$
$524$	22.1402	0.967198
$525$	0	0
$526$	9.56419	0.417018
$527$	−21.9181	−0.954769
$528$	17.8341	0.776131
$529$	−22.6682	−0.985576
$530$	0	0
$531$	−46.8742	−2.03417
$532$	9.18806	0.398353
$533$	0	0
$534$	33.0222	1.42901
$535$	0	0
$536$	−3.57598	−0.154459
$537$	−38.8903	−1.67824
$538$	−5.07008	−0.218587
$539$	−21.3983	−0.921691
$540$	0	0
$541$	24.6204	1.05851	0.529256	−	0.848462i	$-0.322471\pi$
$541$	24.6204	1.05851	0.529256	+	0.848462i	$0.322471\pi$
$542$	0.730064	0.0313589
$543$	−7.86810	−0.337653
$544$	3.00000	0.128624
$545$	0	0
$546$	0	0
$547$	17.9382	0.766981	0.383491	−	0.923545i	$-0.374722\pi$
$547$	17.9382	0.766981	0.383491	+	0.923545i	$0.374722\pi$
$548$	5.83412	0.249221
$549$	19.3462	0.825674
$550$	0	0
$551$	24.7523	1.05448
$552$	1.65998	0.0706535
$553$	−21.0562	−0.895399
$554$	−5.56419	−0.236400
$555$	0	0
$556$	4.26994	0.181086
$557$	−26.4802	−1.12200	−0.561001	−	0.827815i	$-0.689584\pi$
$557$	−26.4802	−1.12200	−0.561001	+	0.827815i	$0.689584\pi$
$558$	−38.7662	−1.64110
$559$	0	0
$560$	0	0
$561$	−53.5024	−2.25887
$562$	−23.2921	−0.982519
$563$	−14.2803	−0.601844	−0.300922	−	0.953649i	$-0.597294\pi$
$563$	−14.2803	−0.601844	−0.300922	+	0.953649i	$0.597294\pi$
$564$	−7.08400	−0.298290
$565$	0	0
$566$	−14.1062	−0.592927
$567$	6.09014	0.255762
$568$	0	0
$569$	4.85983	0.203735	0.101867	−	0.994798i	$-0.467518\pi$
$569$	4.85983	0.203735	0.101867	+	0.994798i	$0.467518\pi$
$570$	0	0
$571$	6.53987	0.273685	0.136843	−	0.990593i	$-0.456305\pi$
$571$	6.53987	0.273685	0.136843	+	0.990593i	$0.456305\pi$
$572$	0	0
$573$	−13.9722	−0.583695
$574$	−10.2721	−0.428748
$575$	0	0
$576$	5.30604	0.221085
$577$	−43.7383	−1.82085	−0.910425	−	0.413673i	$-0.864246\pi$
$577$	−43.7383	−1.82085	−0.910425	+	0.413673i	$0.864246\pi$
$578$	8.00000	0.332756
$579$	30.7862	1.27943
$580$	0	0
$581$	0.353938	0.0146838
$582$	45.4323	1.88323
$583$	−68.5024	−2.83708
$584$	11.8341	0.489700
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−13.2721	−0.547797	−0.273898	−	0.961759i	$-0.588313\pi$
$587$	−13.2721	−0.547797	−0.273898	+	0.961759i	$0.588313\pi$
$588$	−9.96602	−0.410992
$589$	35.6682	1.46968
$590$	0	0
$591$	44.8563	1.84514
$592$	9.18806	0.377627
$593$	−25.6704	−1.05416	−0.527078	−	0.849817i	$-0.676712\pi$
$593$	−25.6704	−1.05416	−0.527078	+	0.849817i	$0.676712\pi$
$594$	−41.1262	−1.68743
$595$	0	0
$596$	16.9160	0.692906
$597$	0.784099	0.0320910
$598$	0	0
$599$	16.8480	0.688392	0.344196	−	0.938898i	$-0.388151\pi$
$599$	16.8480	0.688392	0.344196	+	0.938898i	$0.388151\pi$
$600$	0	0
$601$	−16.2921	−0.664570	−0.332285	−	0.943179i	$-0.607820\pi$
$601$	−16.2921	−0.664570	−0.332285	+	0.943179i	$0.607820\pi$
$602$	7.52808	0.306822
$603$	18.9743	0.772693
$604$	−17.4462	−0.709876
$605$	0	0
$606$	30.2442	1.22859
$607$	31.5642	1.28115	0.640575	−	0.767895i	$-0.278696\pi$
$607$	31.5642	1.28115	0.640575	+	0.767895i	$0.278696\pi$
$608$	−4.88202	−0.197992
$609$	27.5002	1.11437
$610$	0	0
$611$	0	0
$612$	−15.9181	−0.643452
$613$	−27.8320	−1.12412	−0.562062	−	0.827095i	$-0.689992\pi$
$613$	−27.8320	−1.12412	−0.562062	+	0.827095i	$0.689992\pi$
$614$	9.42189	0.380237
$615$	0	0
$616$	−11.6461	−0.469233
$617$	1.08400	0.0436403	0.0218202	−	0.999762i	$-0.493054\pi$
$617$	1.08400	0.0436403	0.0218202	+	0.999762i	$0.493054\pi$
$618$	−49.5364	−1.99264
$619$	−27.0562	−1.08748	−0.543740	−	0.839254i	$-0.682992\pi$
$619$	−27.0562	−1.08748	−0.543740	+	0.839254i	$0.682992\pi$
$620$	0	0
$621$	−3.82799	−0.153612
$622$	−31.7044	−1.27123
$623$	−21.5642	−0.863951
$624$	0	0
$625$	0	0
$626$	−15.3740	−0.614468
$627$	87.0666	3.47710
$628$	−2.73006	−0.108941
$629$	−27.5642	−1.09906
$630$	0	0
$631$	23.4240	0.932496	0.466248	−	0.884654i	$-0.345606\pi$
$631$	23.4240	0.932496	0.466248	+	0.884654i	$0.345606\pi$
$632$	11.1881	0.445037
$633$	−49.0583	−1.94989
$634$	−3.18806	−0.126614
$635$	0	0
$636$	−31.9042	−1.26508
$637$	0	0
$638$	−31.3740	−1.24211
$639$	0	0
$640$	0	0
$641$	6.22204	0.245756	0.122878	−	0.992422i	$-0.460788\pi$
$641$	6.22204	0.245756	0.122878	+	0.992422i	$0.460788\pi$
$642$	−4.30604	−0.169946
$643$	37.8363	1.49212	0.746058	−	0.665881i	$-0.231944\pi$
$643$	37.8363	1.49212	0.746058	+	0.665881i	$0.231944\pi$
$644$	−1.08400	−0.0427157
$645$	0	0
$646$	14.6461	0.576242
$647$	38.8563	1.52760	0.763800	−	0.645453i	$-0.223332\pi$
$647$	38.8563	1.52760	0.763800	+	0.645453i	$0.223332\pi$
$648$	−3.23596	−0.127120
$649$	−54.6661	−2.14583
$650$	0	0
$651$	39.6281	1.55315
$652$	15.7980	0.618698
$653$	25.3740	0.992961	0.496481	−	0.868048i	$-0.334625\pi$
$653$	25.3740	0.992961	0.496481	+	0.868048i	$0.334625\pi$
$654$	37.3282	1.45965
$655$	0	0
$656$	5.45800	0.213099
$657$	−62.7924	−2.44976
$658$	4.62600	0.180340
$659$	16.5059	0.642978	0.321489	−	0.946913i	$-0.395817\pi$
$659$	16.5059	0.642978	0.321489	+	0.946913i	$0.395817\pi$
$660$	0	0
$661$	8.81194	0.342745	0.171372	−	0.985206i	$-0.445180\pi$
$661$	8.81194	0.342745	0.171372	+	0.985206i	$0.445180\pi$
$662$	9.02219	0.350657
$663$	0	0
$664$	−0.188063	−0.00729826
$665$	0	0
$666$	−48.7523	−1.88911
$667$	−2.92026	−0.113073
$668$	−15.5642	−0.602197
$669$	−4.78410	−0.184964
$670$	0	0
$671$	22.5621	0.870999
$672$	−5.42402	−0.209236
$673$	4.08187	0.157345	0.0786723	−	0.996901i	$-0.474932\pi$
$673$	4.08187	0.157345	0.0786723	+	0.996901i	$0.474932\pi$
$674$	16.2921	0.627549
$675$	0	0
$676$	0	0
$677$	24.4441	0.939462	0.469731	−	0.882810i	$-0.344351\pi$
$677$	24.4441	0.939462	0.469731	+	0.882810i	$0.344351\pi$
$678$	26.5781	1.02073
$679$	−29.6682	−1.13856
$680$	0	0
$681$	−24.3761	−0.934095
$682$	−45.2103	−1.73119
$683$	−23.4802	−0.898444	−0.449222	−	0.893420i	$-0.648299\pi$
$683$	−23.4802	−0.898444	−0.449222	+	0.893420i	$0.648299\pi$
$684$	25.9042	0.990472
$685$	0	0
$686$	19.6822	0.751469
$687$	27.0562	1.03226
$688$	−4.00000	−0.152499
$689$	0	0
$690$	0	0
$691$	11.4358	0.435039	0.217519	−	0.976056i	$-0.430203\pi$
$691$	11.4358	0.435039	0.217519	+	0.976056i	$0.430203\pi$
$692$	−5.64606	−0.214631
$693$	61.7945	2.34738
$694$	12.3421	0.468502
$695$	0	0
$696$	−14.6121	−0.553870
$697$	−16.3740	−0.620209
$698$	−18.7523	−0.709783
$699$	−60.4885	−2.28788
$700$	0	0
$701$	9.23383	0.348757	0.174378	−	0.984679i	$-0.444208\pi$
$701$	9.23383	0.348757	0.174378	+	0.984679i	$0.444208\pi$
$702$	0	0
$703$	44.8563	1.69179
$704$	6.18806	0.233221
$705$	0	0
$706$	−0.707877	−0.0266413
$707$	−19.7501	−0.742780
$708$	−25.4601	−0.956850
$709$	18.1999	0.683510	0.341755	−	0.939789i	$-0.388979\pi$
$709$	18.1999	0.683510	0.341755	+	0.939789i	$0.388979\pi$
$710$	0	0
$711$	−59.3643	−2.22634
$712$	11.4580	0.429407
$713$	−4.20812	−0.157595
$714$	16.2721	0.608967
$715$	0	0
$716$	−13.4941	−0.504298
$717$	−41.6682	−1.55613
$718$	0.0221876	0.000828033 0
$719$	16.8480	0.628326	0.314163	−	0.949369i	$-0.398276\pi$
$719$	16.8480	0.628326	0.314163	+	0.949369i	$0.398276\pi$
$720$	0	0
$721$	32.3483	1.20471
$722$	−4.83412	−0.179907
$723$	28.6182	1.06432
$724$	−2.73006	−0.101462
$725$	0	0
$726$	−78.6565	−2.91922
$727$	35.0201	1.29882	0.649411	−	0.760438i	$-0.275015\pi$
$727$	35.0201	1.29882	0.649411	+	0.760438i	$0.275015\pi$
$728$	0	0
$729$	−40.2921	−1.49230
$730$	0	0
$731$	12.0000	0.443836
$732$	10.5080	0.388388
$733$	18.1083	0.668846	0.334423	−	0.942423i	$-0.391459\pi$
$733$	18.1083	0.668846	0.334423	+	0.942423i	$0.391459\pi$
$734$	15.8363	0.584527
$735$	0	0
$736$	0.575978	0.0212308
$737$	22.1284	0.815109
$738$	−28.9604	−1.06605
$739$	15.9181	0.585558	0.292779	−	0.956180i	$-0.405420\pi$
$739$	15.9181	0.585558	0.292779	+	0.956180i	$0.405420\pi$
$740$	0	0
$741$	0	0
$742$	20.8341	0.764845
$743$	−43.8584	−1.60901	−0.804505	−	0.593946i	$-0.797569\pi$
$743$	−43.8584	−1.60901	−0.804505	+	0.593946i	$0.797569\pi$
$744$	−21.0562	−0.771956
$745$	0	0
$746$	4.35394	0.159409
$747$	0.997870	0.0365102
$748$	−18.5642	−0.678774
$749$	2.81194	0.102746
$750$	0	0
$751$	−39.7723	−1.45131	−0.725656	−	0.688058i	$-0.758464\pi$
$751$	−39.7723	−1.45131	−0.725656	+	0.688058i	$0.758464\pi$
$752$	−2.45800	−0.0896340
$753$	42.2103	1.53823
$754$	0	0
$755$	0	0
$756$	12.5080	0.454912
$757$	42.2943	1.53721	0.768605	−	0.639723i	$-0.220951\pi$
$757$	42.2943	1.53721	0.768605	+	0.639723i	$0.220951\pi$
$758$	2.42402	0.0880444
$759$	−10.2721	−0.372852
$760$	0	0
$761$	−3.99787	−0.144923	−0.0724613	−	0.997371i	$-0.523085\pi$
$761$	−3.99787	−0.144923	−0.0724613	+	0.997371i	$0.523085\pi$
$762$	−12.6121	−0.456888
$763$	−24.3761	−0.882475
$764$	−4.84804	−0.175396
$765$	0	0
$766$	7.46013	0.269545
$767$	0	0
$768$	2.88202	0.103996
$769$	39.9063	1.43906	0.719530	−	0.694462i	$-0.244357\pi$
$769$	39.9063	1.43906	0.719530	+	0.694462i	$0.244357\pi$
$770$	0	0
$771$	−66.4885	−2.39452
$772$	10.6822	0.384460
$773$	15.1881	0.546277	0.273138	−	0.961975i	$-0.411938\pi$
$773$	15.1881	0.546277	0.273138	+	0.961975i	$0.411938\pi$
$774$	21.2242	0.762887
$775$	0	0
$776$	15.7640	0.565896
$777$	49.8363	1.78787
$778$	6.57598	0.235760
$779$	26.6461	0.954694
$780$	0	0
$781$	0	0
$782$	−1.72793	−0.0617908
$783$	33.6961	1.20420
$784$	−3.45800	−0.123500
$785$	0	0
$786$	−63.8084	−2.27597
$787$	7.68217	0.273840	0.136920	−	0.990582i	$-0.456280\pi$
$787$	7.68217	0.273840	0.136920	+	0.990582i	$0.456280\pi$
$788$	15.5642	0.554451
$789$	−27.5642	−0.981311
$790$	0	0
$791$	−17.3561	−0.617111
$792$	−32.8341	−1.16671
$793$	0	0
$794$	−6.95210	−0.246721
$795$	0	0
$796$	0.272066	0.00964311
$797$	−33.0784	−1.17170	−0.585848	−	0.810421i	$-0.699238\pi$
$797$	−33.0784	−1.17170	−0.585848	+	0.810421i	$0.699238\pi$
$798$	−26.4802	−0.937388
$799$	7.37400	0.260873
$800$	0	0
$801$	−60.7966	−2.14814
$802$	−6.91813	−0.244288
$803$	−73.2303	−2.58424
$804$	10.3060	0.363466
$805$	0	0
$806$	0	0
$807$	14.6121	0.514370
$808$	10.4941	0.369181
$809$	23.2921	0.818907	0.409454	−	0.912331i	$-0.365719\pi$
$809$	23.2921	0.818907	0.409454	+	0.912331i	$0.365719\pi$
$810$	0	0
$811$	2.14983	0.0754906	0.0377453	−	0.999287i	$-0.487982\pi$
$811$	2.14983	0.0754906	0.0377453	+	0.999287i	$0.487982\pi$
$812$	9.54200	0.334859
$813$	−2.10406	−0.0737926
$814$	−56.8563	−1.99281
$815$	0	0
$816$	−8.64606	−0.302673
$817$	−19.5281	−0.683201
$818$	−15.9299	−0.556976
$819$	0	0
$820$	0	0
$821$	26.8563	0.937292	0.468646	−	0.883386i	$-0.344742\pi$
$821$	26.8563	0.937292	0.468646	+	0.883386i	$0.344742\pi$
$822$	−16.8141	−0.586458
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	−17.1881	−0.598775
$825$	0	0
$826$	16.6260	0.578493
$827$	−32.0243	−1.11359	−0.556797	−	0.830648i	$-0.687970\pi$
$827$	−32.0243	−1.11359	−0.556797	+	0.830648i	$0.687970\pi$
$828$	−3.05616	−0.106209
$829$	19.2699	0.669273	0.334636	−	0.942347i	$-0.391387\pi$
$829$	19.2699	0.669273	0.334636	+	0.942347i	$0.391387\pi$
$830$	0	0
$831$	16.0361	0.556286
$832$	0	0
$833$	10.3740	0.359438
$834$	−12.3060	−0.426123
$835$	0	0
$836$	30.2103	1.04484
$837$	48.5564	1.67835
$838$	22.9500	0.792794
$839$	−14.8119	−0.511365	−0.255682	−	0.966761i	$-0.582300\pi$
$839$	−14.8119	−0.511365	−0.255682	+	0.966761i	$0.582300\pi$
$840$	0	0
$841$	−3.29425	−0.113595
$842$	−24.7523	−0.853019
$843$	67.1284	2.31202
$844$	−17.0222	−0.585928
$845$	0	0
$846$	13.0422	0.448402
$847$	51.3643	1.76490
$848$	−11.0701	−0.380148
$849$	40.6543	1.39525
$850$	0	0
$851$	−5.29212	−0.181412
$852$	0	0
$853$	36.5524	1.25153	0.625765	−	0.780012i	$-0.284787\pi$
$853$	36.5524	1.25153	0.625765	+	0.780012i	$0.284787\pi$
$854$	−6.86196	−0.234812
$855$	0	0
$856$	−1.49411	−0.0510675
$857$	−27.9299	−0.954068	−0.477034	−	0.878885i	$-0.658288\pi$
$857$	−27.9299	−0.954068	−0.477034	+	0.878885i	$0.658288\pi$
$858$	0	0
$859$	−46.6461	−1.59154	−0.795772	−	0.605597i	$-0.792934\pi$
$859$	−46.6461	−1.59154	−0.795772	+	0.605597i	$0.792934\pi$
$860$	0	0
$861$	29.6043	1.00891
$862$	32.1041	1.09347
$863$	37.1062	1.26311	0.631555	−	0.775331i	$-0.282417\pi$
$863$	37.1062	1.26311	0.631555	+	0.775331i	$0.282417\pi$
$864$	−6.64606	−0.226104
$865$	0	0
$866$	−16.2103	−0.550847
$867$	−23.0562	−0.783028
$868$	13.7501	0.466710
$869$	−69.2324	−2.34855
$870$	0	0
$871$	0	0
$872$	12.9521	0.438614
$873$	−83.6447	−2.83094
$874$	2.81194	0.0951152
$875$	0	0
$876$	−34.1062	−1.15234
$877$	46.8924	1.58344	0.791722	−	0.610881i	$-0.209185\pi$
$877$	46.8924	1.58344	0.791722	+	0.610881i	$0.209185\pi$
$878$	22.1041	0.745976
$879$	17.2921	0.583249
$880$	0	0
$881$	−30.2220	−1.01821	−0.509103	−	0.860705i	$-0.670023\pi$
$881$	−30.2220	−1.01821	−0.509103	+	0.860705i	$0.670023\pi$
$882$	18.3483	0.617819
$883$	−3.93818	−0.132530	−0.0662652	−	0.997802i	$-0.521108\pi$
$883$	−3.93818	−0.132530	−0.0662652	+	0.997802i	$0.521108\pi$
$884$	0	0
$885$	0	0
$886$	2.64606	0.0888962
$887$	4.16375	0.139805	0.0699024	−	0.997554i	$-0.477731\pi$
$887$	4.16375	0.139805	0.0699024	+	0.997554i	$0.477731\pi$
$888$	−26.4802	−0.888617
$889$	8.23596	0.276225
$890$	0	0
$891$	20.0243	0.670840
$892$	−1.65998	−0.0555803
$893$	−12.0000	−0.401565
$894$	−48.7523	−1.63052
$895$	0	0
$896$	−1.88202	−0.0628739
$897$	0	0
$898$	−35.1262	−1.17218
$899$	37.0422	1.23543
$900$	0	0
$901$	33.2103	1.10639
$902$	−33.7744	−1.12457
$903$	−21.6961	−0.722001
$904$	9.22204	0.306720
$905$	0	0
$906$	50.2803	1.67045
$907$	27.6682	0.918709	0.459355	−	0.888253i	$-0.348081\pi$
$907$	27.6682	0.918709	0.459355	+	0.888253i	$0.348081\pi$
$908$	−8.45800	−0.280689
$909$	−55.6822	−1.84686
$910$	0	0
$911$	−48.5524	−1.60861	−0.804306	−	0.594215i	$-0.797463\pi$
$911$	−48.5524	−1.60861	−0.804306	+	0.594215i	$0.797463\pi$
$912$	14.0701	0.465907
$913$	1.16375	0.0385144
$914$	−18.9181	−0.625756
$915$	0	0
$916$	9.38792	0.310185
$917$	41.6682	1.37601
$918$	19.9382	0.658058
$919$	−8.26781	−0.272730	−0.136365	−	0.990659i	$-0.543542\pi$
$919$	−8.26781	−0.272730	−0.136365	+	0.990659i	$0.543542\pi$
$920$	0	0
$921$	−27.1541	−0.894758
$922$	−8.81194	−0.290206
$923$	0	0
$924$	33.5642	1.10418
$925$	0	0
$926$	−28.5621	−0.938607
$927$	91.2006	2.99542
$928$	−5.07008	−0.166434
$929$	−24.0000	−0.787414	−0.393707	−	0.919236i	$-0.628808\pi$
$929$	−24.0000	−0.787414	−0.393707	+	0.919236i	$0.628808\pi$
$930$	0	0
$931$	−16.8820	−0.553286
$932$	−20.9882	−0.687492
$933$	91.3726	2.99140
$934$	−22.1402	−0.724448
$935$	0	0
$936$	0	0
$937$	−46.8764	−1.53138	−0.765692	−	0.643207i	$-0.777603\pi$
$937$	−46.8764	−1.53138	−0.765692	+	0.643207i	$0.777603\pi$
$938$	−6.73006	−0.219744
$939$	44.3082	1.44594
$940$	0	0
$941$	−10.2721	−0.334860	−0.167430	−	0.985884i	$-0.553547\pi$
$941$	−10.2721	−0.334860	−0.167430	+	0.985884i	$0.553547\pi$
$942$	7.86810	0.256357
$943$	−3.14369	−0.102373
$944$	−8.83412	−0.287526
$945$	0	0
$946$	24.7523	0.804765
$947$	−38.1262	−1.23894	−0.619468	−	0.785022i	$-0.712652\pi$
$947$	−38.1262	−1.23894	−0.619468	+	0.785022i	$0.712652\pi$
$948$	−32.2442	−1.04724
$949$	0	0
$950$	0	0
$951$	9.18806	0.297943
$952$	5.64606	0.182990
$953$	44.2921	1.43476	0.717381	−	0.696681i	$-0.245341\pi$
$953$	44.2921	1.43476	0.717381	+	0.696681i	$0.245341\pi$
$954$	58.7383	1.90172
$955$	0	0
$956$	−14.4580	−0.467605
$957$	90.4205	2.92288
$958$	19.7501	0.638097
$959$	10.9799	0.354561
$960$	0	0
$961$	22.3783	0.721879
$962$	0	0
$963$	7.92779	0.255469
$964$	9.92992	0.319821
$965$	0	0
$966$	3.12412	0.100517
$967$	−44.1262	−1.41900	−0.709502	−	0.704703i	$-0.751080\pi$
$967$	−44.1262	−1.41900	−0.709502	+	0.704703i	$0.751080\pi$
$968$	−27.2921	−0.877202
$969$	−42.2103	−1.35599
$970$	0	0
$971$	−0.809807	−0.0259879	−0.0129940	−	0.999916i	$-0.504136\pi$
$971$	−0.809807	−0.0259879	−0.0129940	+	0.999916i	$0.504136\pi$
$972$	−10.6121	−0.340383
$973$	8.03611	0.257626
$974$	−3.78623	−0.121319
$975$	0	0
$976$	3.64606	0.116708
$977$	−62.0465	−1.98504	−0.992522	−	0.122068i	$-0.961048\pi$
$977$	−62.0465	−1.98504	−0.992522	+	0.122068i	$0.961048\pi$
$978$	−45.5302	−1.45590
$979$	−70.9028	−2.26606
$980$	0	0
$981$	−68.7244	−2.19420
$982$	18.4441	0.588574
$983$	−2.72580	−0.0869397	−0.0434698	−	0.999055i	$-0.513841\pi$
$983$	−2.72580	−0.0869397	−0.0434698	+	0.999055i	$0.513841\pi$
$984$	−15.7301	−0.501456
$985$	0	0
$986$	15.2103	0.484393
$987$	−13.3322	−0.424370
$988$	0	0
$989$	2.30391	0.0732602
$990$	0	0
$991$	25.2921	0.803431	0.401715	−	0.915765i	$-0.368414\pi$
$991$	25.2921	0.803431	0.401715	+	0.915765i	$0.368414\pi$
$992$	−7.30604	−0.231967
$993$	−26.0021	−0.825153
$994$	0	0
$995$	0	0
$996$	0.542001	0.0171740
$997$	2.29425	0.0726597	0.0363299	−	0.999340i	$-0.488433\pi$
$997$	2.29425	0.0726597	0.0363299	+	0.999340i	$0.488433\pi$
$998$	9.98608	0.316104
$999$	61.0644	1.93199

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.br.1.3		3
5.4	even	2		8450.2.a.ce.1.1		3
13.5	odd	4		650.2.d.c.51.6	yes	6
13.8	odd	4		650.2.d.c.51.3	✓	6
13.12	even	2		8450.2.a.cd.1.3		3
65.8	even	4		650.2.c.e.649.5		6
65.18	even	4		650.2.c.f.649.5		6
65.34	odd	4		650.2.d.d.51.4	yes	6
65.44	odd	4		650.2.d.d.51.1	yes	6
65.47	even	4		650.2.c.f.649.2		6
65.57	even	4		650.2.c.e.649.2		6
65.64	even	2		8450.2.a.bq.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
650.2.c.e.649.2		6	65.57	even	4
650.2.c.e.649.5		6	65.8	even	4
650.2.c.f.649.2		6	65.47	even	4
650.2.c.f.649.5		6	65.18	even	4
650.2.d.c.51.3	✓	6	13.8	odd	4
650.2.d.c.51.6	yes	6	13.5	odd	4
650.2.d.d.51.1	yes	6	65.44	odd	4
650.2.d.d.51.4	yes	6	65.34	odd	4
8450.2.a.bq.1.1		3	65.64	even	2
8450.2.a.br.1.3		3	1.1	even	1	trivial
8450.2.a.cd.1.3		3	13.12	even	2
8450.2.a.ce.1.1		3	5.4	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.88202 q^{3} +1.00000 q^{4} -2.88202 q^{6} +1.88202 q^{7} -1.00000 q^{8} +5.30604 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +2.88202 q^{3} +1.00000 q^{4} -2.88202 q^{6} +1.88202 q^{7} -1.00000 q^{8} +5.30604 q^{9} +6.18806 q^{11} +2.88202 q^{12} -1.88202 q^{14} +1.00000 q^{16} -3.00000 q^{17} -5.30604 q^{18} +4.88202 q^{19} +5.42402 q^{21} -6.18806 q^{22} -0.575978 q^{23} -2.88202 q^{24} +6.64606 q^{27} +1.88202 q^{28} +5.07008 q^{29} +7.30604 q^{31} -1.00000 q^{32} +17.8341 q^{33} +3.00000 q^{34} +5.30604 q^{36} +9.18806 q^{37} -4.88202 q^{38} +5.45800 q^{41} -5.42402 q^{42} -4.00000 q^{43} +6.18806 q^{44} +0.575978 q^{46} -2.45800 q^{47} +2.88202 q^{48} -3.45800 q^{49} -8.64606 q^{51} -11.0701 q^{53} -6.64606 q^{54} -1.88202 q^{56} +14.0701 q^{57} -5.07008 q^{58} -8.83412 q^{59} +3.64606 q^{61} -7.30604 q^{62} +9.98608 q^{63} +1.00000 q^{64} -17.8341 q^{66} +3.57598 q^{67} -3.00000 q^{68} -1.65998 q^{69} -5.30604 q^{72} -11.8341 q^{73} -9.18806 q^{74} +4.88202 q^{76} +11.6461 q^{77} -11.1881 q^{79} +3.23596 q^{81} -5.45800 q^{82} +0.188063 q^{83} +5.42402 q^{84} +4.00000 q^{86} +14.6121 q^{87} -6.18806 q^{88} -11.4580 q^{89} -0.575978 q^{92} +21.0562 q^{93} +2.45800 q^{94} -2.88202 q^{96} -15.7640 q^{97} +3.45800 q^{98} +32.8341 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{7} - 3 q^{8} + 9 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^7 - 3 * q^8 + 9 * q^9	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{7} - 3 q^{8} + 9 q^{9} + 3 q^{11} + 3 q^{14} + 3 q^{16} - 9 q^{17} - 9 q^{18} + 6 q^{19} + 18 q^{21} - 3 q^{22} - 6 q^{27} - 3 q^{28} - 9 q^{29} + 15 q^{31} - 3 q^{32} + 12 q^{33} + 9 q^{34} + 9 q^{36} + 12 q^{37} - 6 q^{38} + 6 q^{41} - 18 q^{42} - 12 q^{43} + 3 q^{44} + 3 q^{47} - 9 q^{53} + 6 q^{54} + 3 q^{56} + 18 q^{57} + 9 q^{58} + 15 q^{59} - 15 q^{61} - 15 q^{62} - 15 q^{63} + 3 q^{64} - 12 q^{66} + 9 q^{67} - 9 q^{68} - 24 q^{69} - 9 q^{72} + 6 q^{73} - 12 q^{74} + 6 q^{76} + 9 q^{77} - 18 q^{79} + 27 q^{81} - 6 q^{82} - 15 q^{83} + 18 q^{84} + 12 q^{86} + 30 q^{87} - 3 q^{88} - 24 q^{89} - 6 q^{93} - 3 q^{94} - 30 q^{97} + 57 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^7 - 3 * q^8 + 9 * q^9 + 3 * q^11 + 3 * q^14 + 3 * q^16 - 9 * q^17 - 9 * q^18 + 6 * q^19 + 18 * q^21 - 3 * q^22 - 6 * q^27 - 3 * q^28 - 9 * q^29 + 15 * q^31 - 3 * q^32 + 12 * q^33 + 9 * q^34 + 9 * q^36 + 12 * q^37 - 6 * q^38 + 6 * q^41 - 18 * q^42 - 12 * q^43 + 3 * q^44 + 3 * q^47 - 9 * q^53 + 6 * q^54 + 3 * q^56 + 18 * q^57 + 9 * q^58 + 15 * q^59 - 15 * q^61 - 15 * q^62 - 15 * q^63 + 3 * q^64 - 12 * q^66 + 9 * q^67 - 9 * q^68 - 24 * q^69 - 9 * q^72 + 6 * q^73 - 12 * q^74 + 6 * q^76 + 9 * q^77 - 18 * q^79 + 27 * q^81 - 6 * q^82 - 15 * q^83 + 18 * q^84 + 12 * q^86 + 30 * q^87 - 3 * q^88 - 24 * q^89 - 6 * q^93 - 3 * q^94 - 30 * q^97 + 57 * q^99

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