LMFDB - Embedded newform 3726.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3726,2,Mod(1,3726)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3726, 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("3726.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3726 = 2 \cdot 3^{4} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3726.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:	$29.7522597931$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.1030257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 11x^{3} - 4x^{2} + 7x - 1 $ x^5 - x^4 - 11x^3 - 4x^2 + 7*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 414)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.165826$ of defining polynomial
Character	$\chi$	$=$	3726.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -4.26821 q^{5} -2.54195 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -4.26821 q^{5} -2.54195 q^{7} -1.00000 q^{8} +4.26821 q^{10} -2.16583 q^{11} -1.48845 q^{13} +2.54195 q^{14} +1.00000 q^{16} +4.14182 q^{17} +0.403627 q^{19} -4.26821 q^{20} +2.16583 q^{22} +1.00000 q^{23} +13.2176 q^{25} +1.48845 q^{26} -2.54195 q^{28} +3.47943 q^{29} +8.57236 q^{31} -1.00000 q^{32} -4.14182 q^{34} +10.8496 q^{35} +10.4100 q^{37} -0.403627 q^{38} +4.26821 q^{40} -7.68469 q^{41} +5.69876 q^{43} -2.16583 q^{44} -1.00000 q^{46} -7.22282 q^{47} -0.538465 q^{49} -13.2176 q^{50} -1.48845 q^{52} +2.62644 q^{53} +9.24420 q^{55} +2.54195 q^{56} -3.47943 q^{58} -11.5098 q^{59} -7.42501 q^{61} -8.57236 q^{62} +1.00000 q^{64} +6.35304 q^{65} +9.62586 q^{67} +4.14182 q^{68} -10.8496 q^{70} +4.30211 q^{71} -14.1757 q^{73} -10.4100 q^{74} +0.403627 q^{76} +5.50543 q^{77} +9.89961 q^{79} -4.26821 q^{80} +7.68469 q^{82} -4.52901 q^{83} -17.6782 q^{85} -5.69876 q^{86} +2.16583 q^{88} -5.78408 q^{89} +3.78358 q^{91} +1.00000 q^{92} +7.22282 q^{94} -1.72277 q^{95} -19.0646 q^{97} +0.538465 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{4} - 5 q^{5} + 3 q^{7} - 5 q^{8}+O(q^{10}) $ 5 * q - 5 * q^2 + 5 * q^4 - 5 * q^5 + 3 * q^7 - 5 * q^8	$ 5 q - 5 q^{2} + 5 q^{4} - 5 q^{5} + 3 q^{7} - 5 q^{8} + 5 q^{10} - 11 q^{11} - 3 q^{14} + 5 q^{16} - 11 q^{17} + 9 q^{19} - 5 q^{20} + 11 q^{22} + 5 q^{23} + 12 q^{25} + 3 q^{28} + 2 q^{29} + 4 q^{31} - 5 q^{32} + 11 q^{34} + 7 q^{35} + 4 q^{37} - 9 q^{38} + 5 q^{40} - 8 q^{41} + 5 q^{43} - 11 q^{44} - 5 q^{46} - 19 q^{47} + 4 q^{49} - 12 q^{50} + 3 q^{53} - 2 q^{55} - 3 q^{56} - 2 q^{58} - 19 q^{59} - 13 q^{61} - 4 q^{62} + 5 q^{64} + q^{65} + q^{67} - 11 q^{68} - 7 q^{70} - 27 q^{71} - 7 q^{73} - 4 q^{74} + 9 q^{76} - 19 q^{77} + 3 q^{79} - 5 q^{80} + 8 q^{82} - 9 q^{83} - 5 q^{85} - 5 q^{86} + 11 q^{88} - 9 q^{89} - 19 q^{91} + 5 q^{92} + 19 q^{94} - 17 q^{95} - 20 q^{97} - 4 q^{98}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^4 - 5 * q^5 + 3 * q^7 - 5 * q^8 + 5 * q^10 - 11 * q^11 - 3 * q^14 + 5 * q^16 - 11 * q^17 + 9 * q^19 - 5 * q^20 + 11 * q^22 + 5 * q^23 + 12 * q^25 + 3 * q^28 + 2 * q^29 + 4 * q^31 - 5 * q^32 + 11 * q^34 + 7 * q^35 + 4 * q^37 - 9 * q^38 + 5 * q^40 - 8 * q^41 + 5 * q^43 - 11 * q^44 - 5 * q^46 - 19 * q^47 + 4 * q^49 - 12 * q^50 + 3 * q^53 - 2 * q^55 - 3 * q^56 - 2 * q^58 - 19 * q^59 - 13 * q^61 - 4 * q^62 + 5 * q^64 + q^65 + q^67 - 11 * q^68 - 7 * q^70 - 27 * q^71 - 7 * q^73 - 4 * q^74 + 9 * q^76 - 19 * q^77 + 3 * q^79 - 5 * q^80 + 8 * q^82 - 9 * q^83 - 5 * q^85 - 5 * q^86 + 11 * q^88 - 9 * q^89 - 19 * q^91 + 5 * q^92 + 19 * q^94 - 17 * q^95 - 20 * q^97 - 4 * q^98

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−4.26821	−1.90880	−0.954401	−	0.298528i	$-0.903504\pi$
$5$	−4.26821	−1.90880	−0.954401	+	0.298528i	$0.903504\pi$
$6$	0	0
$7$	−2.54195	−0.960769	−0.480384	−	0.877058i	$-0.659503\pi$
$7$	−2.54195	−0.960769	−0.480384	+	0.877058i	$0.659503\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	4.26821	1.34973
$11$	−2.16583	−0.653021	−0.326511	−	0.945194i	$-0.605873\pi$
$11$	−2.16583	−0.653021	−0.326511	+	0.945194i	$0.605873\pi$
$12$	0	0
$13$	−1.48845	−0.412823	−0.206412	−	0.978465i	$-0.566179\pi$
$13$	−1.48845	−0.412823	−0.206412	+	0.978465i	$0.566179\pi$
$14$	2.54195	0.679366
$15$	0	0
$16$	1.00000	0.250000
$17$	4.14182	1.00454	0.502269	−	0.864711i	$-0.332499\pi$
$17$	4.14182	1.00454	0.502269	+	0.864711i	$0.332499\pi$
$18$	0	0
$19$	0.403627	0.0925984	0.0462992	−	0.998928i	$-0.485257\pi$
$19$	0.403627	0.0925984	0.0462992	+	0.998928i	$0.485257\pi$
$20$	−4.26821	−0.954401
$21$	0	0
$22$	2.16583	0.461756
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	13.2176	2.64352
$26$	1.48845	0.291910
$27$	0	0
$28$	−2.54195	−0.480384
$29$	3.47943	0.646114	0.323057	−	0.946379i	$-0.395289\pi$
$29$	3.47943	0.646114	0.323057	+	0.946379i	$0.395289\pi$
$30$	0	0
$31$	8.57236	1.53964	0.769821	−	0.638260i	$-0.220345\pi$
$31$	8.57236	1.53964	0.769821	+	0.638260i	$0.220345\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.14182	−0.710316
$35$	10.8496	1.83392
$36$	0	0
$37$	10.4100	1.71140	0.855699	−	0.517474i	$-0.173127\pi$
$37$	10.4100	1.71140	0.855699	+	0.517474i	$0.173127\pi$
$38$	−0.403627	−0.0654770
$39$	0	0
$40$	4.26821	0.674863
$41$	−7.68469	−1.20015	−0.600074	−	0.799945i	$-0.704862\pi$
$41$	−7.68469	−1.20015	−0.600074	+	0.799945i	$0.704862\pi$
$42$	0	0
$43$	5.69876	0.869052	0.434526	−	0.900659i	$-0.356916\pi$
$43$	5.69876	0.869052	0.434526	+	0.900659i	$0.356916\pi$
$44$	−2.16583	−0.326511
$45$	0	0
$46$	−1.00000	−0.147442
$47$	−7.22282	−1.05356	−0.526778	−	0.850003i	$-0.676600\pi$
$47$	−7.22282	−1.05356	−0.526778	+	0.850003i	$0.676600\pi$
$48$	0	0
$49$	−0.538465	−0.0769236
$50$	−13.2176	−1.86925
$51$	0	0
$52$	−1.48845	−0.206412
$53$	2.62644	0.360770	0.180385	−	0.983596i	$-0.442266\pi$
$53$	2.62644	0.360770	0.180385	+	0.983596i	$0.442266\pi$
$54$	0	0
$55$	9.24420	1.24649
$56$	2.54195	0.339683
$57$	0	0
$58$	−3.47943	−0.456872
$59$	−11.5098	−1.49845	−0.749227	−	0.662314i	$-0.769575\pi$
$59$	−11.5098	−1.49845	−0.749227	+	0.662314i	$0.769575\pi$
$60$	0	0
$61$	−7.42501	−0.950675	−0.475338	−	0.879803i	$-0.657674\pi$
$61$	−7.42501	−0.950675	−0.475338	+	0.879803i	$0.657674\pi$
$62$	−8.57236	−1.08869
$63$	0	0
$64$	1.00000	0.125000
$65$	6.35304	0.787997
$66$	0	0
$67$	9.62586	1.17599	0.587993	−	0.808866i	$-0.299918\pi$
$67$	9.62586	1.17599	0.587993	+	0.808866i	$0.299918\pi$
$68$	4.14182	0.502269
$69$	0	0
$70$	−10.8496	−1.29678
$71$	4.30211	0.510567	0.255283	−	0.966866i	$-0.417831\pi$
$71$	4.30211	0.510567	0.255283	+	0.966866i	$0.417831\pi$
$72$	0	0
$73$	−14.1757	−1.65914	−0.829571	−	0.558401i	$-0.811415\pi$
$73$	−14.1757	−1.65914	−0.829571	+	0.558401i	$0.811415\pi$
$74$	−10.4100	−1.21014
$75$	0	0
$76$	0.403627	0.0462992
$77$	5.50543	0.627402
$78$	0	0
$79$	9.89961	1.11379	0.556897	−	0.830582i	$-0.311992\pi$
$79$	9.89961	1.11379	0.556897	+	0.830582i	$0.311992\pi$
$80$	−4.26821	−0.477200
$81$	0	0
$82$	7.68469	0.848632
$83$	−4.52901	−0.497124	−0.248562	−	0.968616i	$-0.579958\pi$
$83$	−4.52901	−0.497124	−0.248562	+	0.968616i	$0.579958\pi$
$84$	0	0
$85$	−17.6782	−1.91746
$86$	−5.69876	−0.614513
$87$	0	0
$88$	2.16583	0.230878
$89$	−5.78408	−0.613111	−0.306555	−	0.951853i	$-0.599176\pi$
$89$	−5.78408	−0.613111	−0.306555	+	0.951853i	$0.599176\pi$
$90$	0	0
$91$	3.78358	0.396627
$92$	1.00000	0.104257
$93$	0	0
$94$	7.22282	0.744977
$95$	−1.72277	−0.176752
$96$	0	0
$97$	−19.0646	−1.93572	−0.967861	−	0.251486i	$-0.919081\pi$
$97$	−19.0646	−1.93572	−0.967861	+	0.251486i	$0.919081\pi$
$98$	0.538465	0.0543932
$99$	0	0
$100$	13.2176	1.32176
$101$	6.74168	0.670822	0.335411	−	0.942072i	$-0.391125\pi$
$101$	6.74168	0.670822	0.335411	+	0.942072i	$0.391125\pi$
$102$	0	0
$103$	6.80668	0.670682	0.335341	−	0.942097i	$-0.391149\pi$
$103$	6.80668	0.670682	0.335341	+	0.942097i	$0.391149\pi$
$104$	1.48845	0.145955
$105$	0	0
$106$	−2.62644	−0.255103
$107$	−3.02658	−0.292591	−0.146295	−	0.989241i	$-0.546735\pi$
$107$	−3.02658	−0.292591	−0.146295	+	0.989241i	$0.546735\pi$
$108$	0	0
$109$	6.76695	0.648156	0.324078	−	0.946030i	$-0.394946\pi$
$109$	6.76695	0.648156	0.324078	+	0.946030i	$0.394946\pi$
$110$	−9.24420	−0.881400
$111$	0	0
$112$	−2.54195	−0.240192
$113$	3.29697	0.310152	0.155076	−	0.987903i	$-0.450438\pi$
$113$	3.29697	0.310152	0.155076	+	0.987903i	$0.450438\pi$
$114$	0	0
$115$	−4.26821	−0.398013
$116$	3.47943	0.323057
$117$	0	0
$118$	11.5098	1.05957
$119$	−10.5283	−0.965129
$120$	0	0
$121$	−6.30920	−0.573563
$122$	7.42501	0.672229
$123$	0	0
$124$	8.57236	0.769821
$125$	−35.0745	−3.13716
$126$	0	0
$127$	−16.7356	−1.48504	−0.742521	−	0.669823i	$-0.766370\pi$
$127$	−16.7356	−1.48504	−0.742521	+	0.669823i	$0.766370\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−6.35304	−0.557198
$131$	10.7245	0.937001	0.468500	−	0.883463i	$-0.344794\pi$
$131$	10.7245	0.937001	0.468500	+	0.883463i	$0.344794\pi$
$132$	0	0
$133$	−1.02600	−0.0889657
$134$	−9.62586	−0.831548
$135$	0	0
$136$	−4.14182	−0.355158
$137$	8.26608	0.706219	0.353109	−	0.935582i	$-0.385124\pi$
$137$	8.26608	0.706219	0.353109	+	0.935582i	$0.385124\pi$
$138$	0	0
$139$	4.56937	0.387569	0.193784	−	0.981044i	$-0.437924\pi$
$139$	4.56937	0.387569	0.193784	+	0.981044i	$0.437924\pi$
$140$	10.8496	0.916958
$141$	0	0
$142$	−4.30211	−0.361025
$143$	3.22373	0.269582
$144$	0	0
$145$	−14.8509	−1.23330
$146$	14.1757	1.17319
$147$	0	0
$148$	10.4100	0.855699
$149$	10.7181	0.878061	0.439030	−	0.898472i	$-0.355322\pi$
$149$	10.7181	0.878061	0.439030	+	0.898472i	$0.355322\pi$
$150$	0	0
$151$	−14.8276	−1.20666	−0.603328	−	0.797493i	$-0.706159\pi$
$151$	−14.8276	−1.20666	−0.603328	+	0.797493i	$0.706159\pi$
$152$	−0.403627	−0.0327385
$153$	0	0
$154$	−5.50543	−0.443640
$155$	−36.5887	−2.93887
$156$	0	0
$157$	1.50819	0.120366	0.0601832	−	0.998187i	$-0.480832\pi$
$157$	1.50819	0.120366	0.0601832	+	0.998187i	$0.480832\pi$
$158$	−9.89961	−0.787571
$159$	0	0
$160$	4.26821	0.337432
$161$	−2.54195	−0.200334
$162$	0	0
$163$	−13.3928	−1.04901	−0.524503	−	0.851409i	$-0.675749\pi$
$163$	−13.3928	−1.04901	−0.524503	+	0.851409i	$0.675749\pi$
$164$	−7.68469	−0.600074
$165$	0	0
$166$	4.52901	0.351520
$167$	−10.3753	−0.802868	−0.401434	−	0.915888i	$-0.631488\pi$
$167$	−10.3753	−0.802868	−0.401434	+	0.915888i	$0.631488\pi$
$168$	0	0
$169$	−10.7845	−0.829577
$170$	17.6782	1.35585
$171$	0	0
$172$	5.69876	0.434526
$173$	12.0190	0.913788	0.456894	−	0.889521i	$-0.348962\pi$
$173$	12.0190	0.913788	0.456894	+	0.889521i	$0.348962\pi$
$174$	0	0
$175$	−33.5986	−2.53982
$176$	−2.16583	−0.163255
$177$	0	0
$178$	5.78408	0.433535
$179$	−6.19100	−0.462737	−0.231368	−	0.972866i	$-0.574320\pi$
$179$	−6.19100	−0.462737	−0.231368	+	0.972866i	$0.574320\pi$
$180$	0	0
$181$	−1.84271	−0.136967	−0.0684837	−	0.997652i	$-0.521816\pi$
$181$	−1.84271	−0.136967	−0.0684837	+	0.997652i	$0.521816\pi$
$182$	−3.78358	−0.280458
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	−44.4322	−3.26672
$186$	0	0
$187$	−8.97046	−0.655985
$188$	−7.22282	−0.526778
$189$	0	0
$190$	1.72277	0.124983
$191$	23.5513	1.70411	0.852055	−	0.523452i	$-0.175356\pi$
$191$	23.5513	1.70411	0.852055	+	0.523452i	$0.175356\pi$
$192$	0	0
$193$	8.55772	0.615998	0.307999	−	0.951387i	$-0.400341\pi$
$193$	8.55772	0.615998	0.307999	+	0.951387i	$0.400341\pi$
$194$	19.0646	1.36876
$195$	0	0
$196$	−0.538465	−0.0384618
$197$	20.2436	1.44230	0.721149	−	0.692780i	$-0.243614\pi$
$197$	20.2436	1.44230	0.721149	+	0.692780i	$0.243614\pi$
$198$	0	0
$199$	−20.0901	−1.42415	−0.712075	−	0.702104i	$-0.752244\pi$
$199$	−20.0901	−1.42415	−0.712075	+	0.702104i	$0.752244\pi$
$200$	−13.2176	−0.934627
$201$	0	0
$202$	−6.74168	−0.474343
$203$	−8.84456	−0.620766
$204$	0	0
$205$	32.7999	2.29084
$206$	−6.80668	−0.474244
$207$	0	0
$208$	−1.48845	−0.103206
$209$	−0.874186	−0.0604687
$210$	0	0
$211$	9.45387	0.650831	0.325416	−	0.945571i	$-0.394496\pi$
$211$	9.45387	0.650831	0.325416	+	0.945571i	$0.394496\pi$
$212$	2.62644	0.180385
$213$	0	0
$214$	3.02658	0.206893
$215$	−24.3235	−1.65885
$216$	0	0
$217$	−21.7906	−1.47924
$218$	−6.76695	−0.458315
$219$	0	0
$220$	9.24420	0.623244
$221$	−6.16491	−0.414697
$222$	0	0
$223$	24.0625	1.61135	0.805673	−	0.592361i	$-0.201804\pi$
$223$	24.0625	1.61135	0.805673	+	0.592361i	$0.201804\pi$
$224$	2.54195	0.169842
$225$	0	0
$226$	−3.29697	−0.219311
$227$	17.1353	1.13731	0.568654	−	0.822577i	$-0.307464\pi$
$227$	17.1353	1.13731	0.568654	+	0.822577i	$0.307464\pi$
$228$	0	0
$229$	10.5172	0.694993	0.347497	−	0.937681i	$-0.387032\pi$
$229$	10.5172	0.694993	0.347497	+	0.937681i	$0.387032\pi$
$230$	4.26821	0.281437
$231$	0	0
$232$	−3.47943	−0.228436
$233$	−12.5529	−0.822365	−0.411182	−	0.911553i	$-0.634884\pi$
$233$	−12.5529	−0.822365	−0.411182	+	0.911553i	$0.634884\pi$
$234$	0	0
$235$	30.8285	2.01103
$236$	−11.5098	−0.749227
$237$	0	0
$238$	10.5283	0.682449
$239$	−8.36249	−0.540924	−0.270462	−	0.962731i	$-0.587177\pi$
$239$	−8.36249	−0.540924	−0.270462	+	0.962731i	$0.587177\pi$
$240$	0	0
$241$	1.22801	0.0791032	0.0395516	−	0.999218i	$-0.487407\pi$
$241$	1.22801	0.0791032	0.0395516	+	0.999218i	$0.487407\pi$
$242$	6.30920	0.405571
$243$	0	0
$244$	−7.42501	−0.475338
$245$	2.29828	0.146832
$246$	0	0
$247$	−0.600781	−0.0382268
$248$	−8.57236	−0.544346
$249$	0	0
$250$	35.0745	2.21831
$251$	4.50806	0.284546	0.142273	−	0.989827i	$-0.454559\pi$
$251$	4.50806	0.284546	0.142273	+	0.989827i	$0.454559\pi$
$252$	0	0
$253$	−2.16583	−0.136164
$254$	16.7356	1.05008
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.7546	0.795611	0.397806	−	0.917470i	$-0.369772\pi$
$257$	12.7546	0.795611	0.397806	+	0.917470i	$0.369772\pi$
$258$	0	0
$259$	−26.4618	−1.64426
$260$	6.35304	0.393999
$261$	0	0
$262$	−10.7245	−0.662560
$263$	−3.81151	−0.235028	−0.117514	−	0.993071i	$-0.537492\pi$
$263$	−3.81151	−0.235028	−0.117514	+	0.993071i	$0.537492\pi$
$264$	0	0
$265$	−11.2102	−0.688638
$266$	1.02600	0.0629082
$267$	0	0
$268$	9.62586	0.587993
$269$	−30.1954	−1.84105	−0.920525	−	0.390685i	$-0.872238\pi$
$269$	−30.1954	−1.84105	−0.920525	+	0.390685i	$0.872238\pi$
$270$	0	0
$271$	−6.55820	−0.398382	−0.199191	−	0.979961i	$-0.563831\pi$
$271$	−6.55820	−0.398382	−0.199191	+	0.979961i	$0.563831\pi$
$272$	4.14182	0.251135
$273$	0	0
$274$	−8.26608	−0.499372
$275$	−28.6271	−1.72628
$276$	0	0
$277$	1.52278	0.0914951	0.0457475	−	0.998953i	$-0.485433\pi$
$277$	1.52278	0.0914951	0.0457475	+	0.998953i	$0.485433\pi$
$278$	−4.56937	−0.274052
$279$	0	0
$280$	−10.8496	−0.648388
$281$	−6.92876	−0.413335	−0.206668	−	0.978411i	$-0.566262\pi$
$281$	−6.92876	−0.413335	−0.206668	+	0.978411i	$0.566262\pi$
$282$	0	0
$283$	−30.1809	−1.79407	−0.897033	−	0.441963i	$-0.854282\pi$
$283$	−30.1809	−1.79407	−0.897033	+	0.441963i	$0.854282\pi$
$284$	4.30211	0.255283
$285$	0	0
$286$	−3.22373	−0.190623
$287$	19.5341	1.15306
$288$	0	0
$289$	0.154656	0.00909744
$290$	14.8509	0.872077
$291$	0	0
$292$	−14.1757	−0.829571
$293$	27.0341	1.57935	0.789675	−	0.613526i	$-0.210249\pi$
$293$	27.0341	1.57935	0.789675	+	0.613526i	$0.210249\pi$
$294$	0	0
$295$	49.1264	2.86025
$296$	−10.4100	−0.605071
$297$	0	0
$298$	−10.7181	−0.620883
$299$	−1.48845	−0.0860796
$300$	0	0
$301$	−14.4860	−0.834958
$302$	14.8276	0.853234
$303$	0	0
$304$	0.403627	0.0231496
$305$	31.6915	1.81465
$306$	0	0
$307$	22.6804	1.29444	0.647218	−	0.762305i	$-0.275932\pi$
$307$	22.6804	1.29444	0.647218	+	0.762305i	$0.275932\pi$
$308$	5.50543	0.313701
$309$	0	0
$310$	36.5887	2.07810
$311$	−7.93521	−0.449965	−0.224982	−	0.974363i	$-0.572232\pi$
$311$	−7.93521	−0.449965	−0.224982	+	0.974363i	$0.572232\pi$
$312$	0	0
$313$	7.51420	0.424728	0.212364	−	0.977191i	$-0.431884\pi$
$313$	7.51420	0.424728	0.212364	+	0.977191i	$0.431884\pi$
$314$	−1.50819	−0.0851118
$315$	0	0
$316$	9.89961	0.556897
$317$	−9.67071	−0.543161	−0.271581	−	0.962416i	$-0.587546\pi$
$317$	−9.67071	−0.543161	−0.271581	+	0.962416i	$0.587546\pi$
$318$	0	0
$319$	−7.53584	−0.421926
$320$	−4.26821	−0.238600
$321$	0	0
$322$	2.54195	0.141658
$323$	1.67175	0.0930187
$324$	0	0
$325$	−19.6738	−1.09131
$326$	13.3928	0.741759
$327$	0	0
$328$	7.68469	0.424316
$329$	18.3601	1.01222
$330$	0	0
$331$	−28.5933	−1.57163	−0.785816	−	0.618461i	$-0.787757\pi$
$331$	−28.5933	−1.57163	−0.785816	+	0.618461i	$0.787757\pi$
$332$	−4.52901	−0.248562
$333$	0	0
$334$	10.3753	0.567713
$335$	−41.0852	−2.24473
$336$	0	0
$337$	−17.0049	−0.926318	−0.463159	−	0.886275i	$-0.653284\pi$
$337$	−17.0049	−0.926318	−0.463159	+	0.886275i	$0.653284\pi$
$338$	10.7845	0.586600
$339$	0	0
$340$	−17.6782	−0.958732
$341$	−18.5663	−1.00542
$342$	0	0
$343$	19.1624	1.03467
$344$	−5.69876	−0.307256
$345$	0	0
$346$	−12.0190	−0.646146
$347$	15.2315	0.817667	0.408834	−	0.912609i	$-0.365936\pi$
$347$	15.2315	0.817667	0.408834	+	0.912609i	$0.365936\pi$
$348$	0	0
$349$	−18.7597	−1.00418	−0.502092	−	0.864814i	$-0.667436\pi$
$349$	−18.7597	−1.00418	−0.502092	+	0.864814i	$0.667436\pi$
$350$	33.5986	1.79592
$351$	0	0
$352$	2.16583	0.115439
$353$	−16.0621	−0.854898	−0.427449	−	0.904039i	$-0.640588\pi$
$353$	−16.0621	−0.854898	−0.427449	+	0.904039i	$0.640588\pi$
$354$	0	0
$355$	−18.3623	−0.974570
$356$	−5.78408	−0.306555
$357$	0	0
$358$	6.19100	0.327204
$359$	3.70142	0.195353	0.0976767	−	0.995218i	$-0.468859\pi$
$359$	3.70142	0.195353	0.0976767	+	0.995218i	$0.468859\pi$
$360$	0	0
$361$	−18.8371	−0.991426
$362$	1.84271	0.0968505
$363$	0	0
$364$	3.78358	0.198314
$365$	60.5049	3.16697
$366$	0	0
$367$	3.27262	0.170829	0.0854146	−	0.996345i	$-0.472779\pi$
$367$	3.27262	0.170829	0.0854146	+	0.996345i	$0.472779\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	44.4322	2.30992
$371$	−6.67630	−0.346616
$372$	0	0
$373$	11.4557	0.593155	0.296577	−	0.955009i	$-0.404155\pi$
$373$	11.4557	0.593155	0.296577	+	0.955009i	$0.404155\pi$
$374$	8.97046	0.463851
$375$	0	0
$376$	7.22282	0.372488
$377$	−5.17898	−0.266731
$378$	0	0
$379$	5.68367	0.291951	0.145975	−	0.989288i	$-0.453368\pi$
$379$	5.68367	0.291951	0.145975	+	0.989288i	$0.453368\pi$
$380$	−1.72277	−0.0883760
$381$	0	0
$382$	−23.5513	−1.20499
$383$	8.77631	0.448448	0.224224	−	0.974538i	$-0.428015\pi$
$383$	8.77631	0.448448	0.224224	+	0.974538i	$0.428015\pi$
$384$	0	0
$385$	−23.4983	−1.19759
$386$	−8.55772	−0.435576
$387$	0	0
$388$	−19.0646	−0.967861
$389$	21.6724	1.09883	0.549416	−	0.835549i	$-0.314850\pi$
$389$	21.6724	1.09883	0.549416	+	0.835549i	$0.314850\pi$
$390$	0	0
$391$	4.14182	0.209461
$392$	0.538465	0.0271966
$393$	0	0
$394$	−20.2436	−1.01986
$395$	−42.2536	−2.12601
$396$	0	0
$397$	−12.8208	−0.643460	−0.321730	−	0.946832i	$-0.604264\pi$
$397$	−12.8208	−0.643460	−0.321730	+	0.946832i	$0.604264\pi$
$398$	20.0901	1.00703
$399$	0	0
$400$	13.2176	0.660881
$401$	−15.0561	−0.751864	−0.375932	−	0.926647i	$-0.622677\pi$
$401$	−15.0561	−0.751864	−0.375932	+	0.926647i	$0.622677\pi$
$402$	0	0
$403$	−12.7596	−0.635600
$404$	6.74168	0.335411
$405$	0	0
$406$	8.84456	0.438948
$407$	−22.5463	−1.11758
$408$	0	0
$409$	26.2503	1.29799	0.648997	−	0.760791i	$-0.275189\pi$
$409$	26.2503	1.29799	0.648997	+	0.760791i	$0.275189\pi$
$410$	−32.7999	−1.61987
$411$	0	0
$412$	6.80668	0.335341
$413$	29.2575	1.43967
$414$	0	0
$415$	19.3308	0.948911
$416$	1.48845	0.0729775
$417$	0	0
$418$	0.874186	0.0427578
$419$	−7.61740	−0.372134	−0.186067	−	0.982537i	$-0.559574\pi$
$419$	−7.61740	−0.372134	−0.186067	+	0.982537i	$0.559574\pi$
$420$	0	0
$421$	−14.7403	−0.718399	−0.359200	−	0.933261i	$-0.616950\pi$
$421$	−14.7403	−0.718399	−0.359200	+	0.933261i	$0.616950\pi$
$422$	−9.45387	−0.460207
$423$	0	0
$424$	−2.62644	−0.127551
$425$	54.7450	2.65552
$426$	0	0
$427$	18.8740	0.913379
$428$	−3.02658	−0.146295
$429$	0	0
$430$	24.3235	1.17298
$431$	−30.9125	−1.48900	−0.744500	−	0.667622i	$-0.767312\pi$
$431$	−30.9125	−1.48900	−0.744500	+	0.667622i	$0.767312\pi$
$432$	0	0
$433$	1.69567	0.0814885	0.0407442	−	0.999170i	$-0.487027\pi$
$433$	1.69567	0.0814885	0.0407442	+	0.999170i	$0.487027\pi$
$434$	21.7906	1.04598
$435$	0	0
$436$	6.76695	0.324078
$437$	0.403627	0.0193081
$438$	0	0
$439$	22.6381	1.08046	0.540230	−	0.841518i	$-0.318337\pi$
$439$	22.6381	1.08046	0.540230	+	0.841518i	$0.318337\pi$
$440$	−9.24420	−0.440700
$441$	0	0
$442$	6.16491	0.293235
$443$	13.5120	0.641973	0.320987	−	0.947084i	$-0.395986\pi$
$443$	13.5120	0.641973	0.320987	+	0.947084i	$0.395986\pi$
$444$	0	0
$445$	24.6877	1.17031
$446$	−24.0625	−1.13939
$447$	0	0
$448$	−2.54195	−0.120096
$449$	−35.9618	−1.69714	−0.848572	−	0.529080i	$-0.822537\pi$
$449$	−35.9618	−1.69714	−0.848572	+	0.529080i	$0.822537\pi$
$450$	0	0
$451$	16.6437	0.783721
$452$	3.29697	0.155076
$453$	0	0
$454$	−17.1353	−0.804198
$455$	−16.1491	−0.757083
$456$	0	0
$457$	−27.5883	−1.29053	−0.645264	−	0.763960i	$-0.723252\pi$
$457$	−27.5883	−1.29053	−0.645264	+	0.763960i	$0.723252\pi$
$458$	−10.5172	−0.491435
$459$	0	0
$460$	−4.26821	−0.199006
$461$	−32.3275	−1.50564	−0.752822	−	0.658225i	$-0.771308\pi$
$461$	−32.3275	−1.50564	−0.752822	+	0.658225i	$0.771308\pi$
$462$	0	0
$463$	−26.7669	−1.24396	−0.621982	−	0.783032i	$-0.713672\pi$
$463$	−26.7669	−1.24396	−0.621982	+	0.783032i	$0.713672\pi$
$464$	3.47943	0.161529
$465$	0	0
$466$	12.5529	0.581500
$467$	6.17053	0.285538	0.142769	−	0.989756i	$-0.454399\pi$
$467$	6.17053	0.285538	0.142769	+	0.989756i	$0.454399\pi$
$468$	0	0
$469$	−24.4685	−1.12985
$470$	−30.8285	−1.42201
$471$	0	0
$472$	11.5098	0.529783
$473$	−12.3425	−0.567510
$474$	0	0
$475$	5.33499	0.244786
$476$	−10.5283	−0.482565
$477$	0	0
$478$	8.36249	0.382491
$479$	−13.5302	−0.618211	−0.309106	−	0.951028i	$-0.600030\pi$
$479$	−13.5302	−0.618211	−0.309106	+	0.951028i	$0.600030\pi$
$480$	0	0
$481$	−15.4949	−0.706505
$482$	−1.22801	−0.0559344
$483$	0	0
$484$	−6.30920	−0.286782
$485$	81.3719	3.69491
$486$	0	0
$487$	7.57109	0.343079	0.171539	−	0.985177i	$-0.445126\pi$
$487$	7.57109	0.343079	0.171539	+	0.985177i	$0.445126\pi$
$488$	7.42501	0.336115
$489$	0	0
$490$	−2.29828	−0.103826
$491$	1.71106	0.0772192	0.0386096	−	0.999254i	$-0.487707\pi$
$491$	1.71106	0.0772192	0.0386096	+	0.999254i	$0.487707\pi$
$492$	0	0
$493$	14.4112	0.649046
$494$	0.600781	0.0270304
$495$	0	0
$496$	8.57236	0.384911
$497$	−10.9358	−0.490536
$498$	0	0
$499$	3.99670	0.178917	0.0894584	−	0.995991i	$-0.471486\pi$
$499$	3.99670	0.178917	0.0894584	+	0.995991i	$0.471486\pi$
$500$	−35.0745	−1.56858
$501$	0	0
$502$	−4.50806	−0.201204
$503$	−17.3156	−0.772064	−0.386032	−	0.922485i	$-0.626155\pi$
$503$	−17.3156	−0.772064	−0.386032	+	0.922485i	$0.626155\pi$
$504$	0	0
$505$	−28.7749	−1.28047
$506$	2.16583	0.0962827
$507$	0	0
$508$	−16.7356	−0.742521
$509$	−11.0572	−0.490102	−0.245051	−	0.969510i	$-0.578805\pi$
$509$	−11.0572	−0.490102	−0.245051	+	0.969510i	$0.578805\pi$
$510$	0	0
$511$	36.0340	1.59405
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−12.7546	−0.562582
$515$	−29.0523	−1.28020
$516$	0	0
$517$	15.6434	0.687994
$518$	26.4618	1.16267
$519$	0	0
$520$	−6.35304	−0.278599
$521$	−36.6628	−1.60623	−0.803114	−	0.595825i	$-0.796825\pi$
$521$	−36.6628	−1.60623	−0.803114	+	0.595825i	$0.796825\pi$
$522$	0	0
$523$	−24.3763	−1.06590	−0.532950	−	0.846147i	$-0.678917\pi$
$523$	−24.3763	−1.06590	−0.532950	+	0.846147i	$0.678917\pi$
$524$	10.7245	0.468500
$525$	0	0
$526$	3.81151	0.166190
$527$	35.5052	1.54663
$528$	0	0
$529$	1.00000	0.0434783
$530$	11.2102	0.486941
$531$	0	0
$532$	−1.02600	−0.0444828
$533$	11.4383	0.495448
$534$	0	0
$535$	12.9181	0.558497
$536$	−9.62586	−0.415774
$537$	0	0
$538$	30.1954	1.30182
$539$	1.16622	0.0502327
$540$	0	0
$541$	21.5000	0.924356	0.462178	−	0.886787i	$-0.347068\pi$
$541$	21.5000	0.924356	0.462178	+	0.886787i	$0.347068\pi$
$542$	6.55820	0.281699
$543$	0	0
$544$	−4.14182	−0.177579
$545$	−28.8827	−1.23720
$546$	0	0
$547$	−22.3915	−0.957392	−0.478696	−	0.877981i	$-0.658890\pi$
$547$	−22.3915	−0.957392	−0.478696	+	0.877981i	$0.658890\pi$
$548$	8.26608	0.353109
$549$	0	0
$550$	28.6271	1.22066
$551$	1.40439	0.0598291
$552$	0	0
$553$	−25.1644	−1.07010
$554$	−1.52278	−0.0646968
$555$	0	0
$556$	4.56937	0.193784
$557$	31.9418	1.35342	0.676709	−	0.736250i	$-0.263405\pi$
$557$	31.9418	1.35342	0.676709	+	0.736250i	$0.263405\pi$
$558$	0	0
$559$	−8.48234	−0.358765
$560$	10.8496	0.458479
$561$	0	0
$562$	6.92876	0.292272
$563$	−3.46698	−0.146116	−0.0730579	−	0.997328i	$-0.523276\pi$
$563$	−3.46698	−0.146116	−0.0730579	+	0.997328i	$0.523276\pi$
$564$	0	0
$565$	−14.0721	−0.592020
$566$	30.1809	1.26860
$567$	0	0
$568$	−4.30211	−0.180513
$569$	−18.2215	−0.763884	−0.381942	−	0.924186i	$-0.624745\pi$
$569$	−18.2215	−0.763884	−0.381942	+	0.924186i	$0.624745\pi$
$570$	0	0
$571$	25.3837	1.06227	0.531136	−	0.847286i	$-0.321765\pi$
$571$	25.3837	1.06227	0.531136	+	0.847286i	$0.321765\pi$
$572$	3.22373	0.134791
$573$	0	0
$574$	−19.5341	−0.815339
$575$	13.2176	0.551213
$576$	0	0
$577$	−27.8419	−1.15907	−0.579537	−	0.814946i	$-0.696767\pi$
$577$	−27.8419	−1.15907	−0.579537	+	0.814946i	$0.696767\pi$
$578$	−0.154656	−0.00643286
$579$	0	0
$580$	−14.8509	−0.616652
$581$	11.5126	0.477621
$582$	0	0
$583$	−5.68842	−0.235590
$584$	14.1757	0.586595
$585$	0	0
$586$	−27.0341	−1.11677
$587$	25.2009	1.04015	0.520075	−	0.854120i	$-0.325904\pi$
$587$	25.2009	1.04015	0.520075	+	0.854120i	$0.325904\pi$
$588$	0	0
$589$	3.46004	0.142568
$590$	−49.1264	−2.02250
$591$	0	0
$592$	10.4100	0.427850
$593$	10.9848	0.451091	0.225545	−	0.974233i	$-0.427584\pi$
$593$	10.9848	0.451091	0.225545	+	0.974233i	$0.427584\pi$
$594$	0	0
$595$	44.9371	1.84224
$596$	10.7181	0.439030
$597$	0	0
$598$	1.48845	0.0608674
$599$	−17.3657	−0.709543	−0.354771	−	0.934953i	$-0.615441\pi$
$599$	−17.3657	−0.709543	−0.354771	+	0.934953i	$0.615441\pi$
$600$	0	0
$601$	−33.0965	−1.35003	−0.675017	−	0.737802i	$-0.735864\pi$
$601$	−33.0965	−1.35003	−0.675017	+	0.737802i	$0.735864\pi$
$602$	14.4860	0.590405
$603$	0	0
$604$	−14.8276	−0.603328
$605$	26.9290	1.09482
$606$	0	0
$607$	−33.2549	−1.34977	−0.674887	−	0.737921i	$-0.735807\pi$
$607$	−33.2549	−1.34977	−0.674887	+	0.737921i	$0.735807\pi$
$608$	−0.403627	−0.0163692
$609$	0	0
$610$	−31.6915	−1.28315
$611$	10.7508	0.434932
$612$	0	0
$613$	30.1349	1.21714	0.608569	−	0.793501i	$-0.291744\pi$
$613$	30.1349	1.21714	0.608569	+	0.793501i	$0.291744\pi$
$614$	−22.6804	−0.915305
$615$	0	0
$616$	−5.50543	−0.221820
$617$	−33.6053	−1.35290	−0.676450	−	0.736489i	$-0.736482\pi$
$617$	−33.6053	−1.35290	−0.676450	+	0.736489i	$0.736482\pi$
$618$	0	0
$619$	−0.830425	−0.0333776	−0.0166888	−	0.999861i	$-0.505312\pi$
$619$	−0.830425	−0.0333776	−0.0166888	+	0.999861i	$0.505312\pi$
$620$	−36.5887	−1.46944
$621$	0	0
$622$	7.93521	0.318173
$623$	14.7029	0.589058
$624$	0	0
$625$	83.6174	3.34470
$626$	−7.51420	−0.300328
$627$	0	0
$628$	1.50819	0.0601832
$629$	43.1164	1.71917
$630$	0	0
$631$	−4.57382	−0.182081	−0.0910404	−	0.995847i	$-0.529019\pi$
$631$	−4.57382	−0.182081	−0.0910404	+	0.995847i	$0.529019\pi$
$632$	−9.89961	−0.393785
$633$	0	0
$634$	9.67071	0.384073
$635$	71.4309	2.83465
$636$	0	0
$637$	0.801481	0.0317558
$638$	7.53584	0.298347
$639$	0	0
$640$	4.26821	0.168716
$641$	−21.1220	−0.834268	−0.417134	−	0.908845i	$-0.636965\pi$
$641$	−21.1220	−0.834268	−0.417134	+	0.908845i	$0.636965\pi$
$642$	0	0
$643$	−8.78671	−0.346514	−0.173257	−	0.984877i	$-0.555429\pi$
$643$	−8.78671	−0.346514	−0.173257	+	0.984877i	$0.555429\pi$
$644$	−2.54195	−0.100167
$645$	0	0
$646$	−1.67175	−0.0657741
$647$	−26.9104	−1.05796	−0.528979	−	0.848635i	$-0.677425\pi$
$647$	−26.9104	−1.05796	−0.528979	+	0.848635i	$0.677425\pi$
$648$	0	0
$649$	24.9283	0.978522
$650$	19.6738	0.771671
$651$	0	0
$652$	−13.3928	−0.524503
$653$	−0.828552	−0.0324238	−0.0162119	−	0.999869i	$-0.505161\pi$
$653$	−0.828552	−0.0324238	−0.0162119	+	0.999869i	$0.505161\pi$
$654$	0	0
$655$	−45.7743	−1.78855
$656$	−7.68469	−0.300037
$657$	0	0
$658$	−18.3601	−0.715750
$659$	−44.0223	−1.71487	−0.857433	−	0.514595i	$-0.827942\pi$
$659$	−44.0223	−1.71487	−0.857433	+	0.514595i	$0.827942\pi$
$660$	0	0
$661$	−45.3532	−1.76403	−0.882017	−	0.471218i	$-0.843815\pi$
$661$	−45.3532	−1.76403	−0.882017	+	0.471218i	$0.843815\pi$
$662$	28.5933	1.11131
$663$	0	0
$664$	4.52901	0.175760
$665$	4.37919	0.169818
$666$	0	0
$667$	3.47943	0.134724
$668$	−10.3753	−0.401434
$669$	0	0
$670$	41.0852	1.58726
$671$	16.0813	0.620811
$672$	0	0
$673$	−11.4297	−0.440581	−0.220291	−	0.975434i	$-0.570701\pi$
$673$	−11.4297	−0.440581	−0.220291	+	0.975434i	$0.570701\pi$
$674$	17.0049	0.655006
$675$	0	0
$676$	−10.7845	−0.414789
$677$	21.7327	0.835254	0.417627	−	0.908619i	$-0.362862\pi$
$677$	21.7327	0.835254	0.417627	+	0.908619i	$0.362862\pi$
$678$	0	0
$679$	48.4615	1.85978
$680$	17.6782	0.677926
$681$	0	0
$682$	18.5663	0.710939
$683$	26.4259	1.01116	0.505580	−	0.862780i	$-0.331279\pi$
$683$	26.4259	1.01116	0.505580	+	0.862780i	$0.331279\pi$
$684$	0	0
$685$	−35.2814	−1.34803
$686$	−19.1624	−0.731625
$687$	0	0
$688$	5.69876	0.217263
$689$	−3.90934	−0.148934
$690$	0	0
$691$	−16.5931	−0.631230	−0.315615	−	0.948887i	$-0.602211\pi$
$691$	−16.5931	−0.631230	−0.315615	+	0.948887i	$0.602211\pi$
$692$	12.0190	0.456894
$693$	0	0
$694$	−15.2315	−0.578178
$695$	−19.5030	−0.739792
$696$	0	0
$697$	−31.8286	−1.20559
$698$	18.7597	0.710065
$699$	0	0
$700$	−33.5986	−1.26991
$701$	−45.7538	−1.72810	−0.864049	−	0.503408i	$-0.832079\pi$
$701$	−45.7538	−1.72810	−0.864049	+	0.503408i	$0.832079\pi$
$702$	0	0
$703$	4.20177	0.158473
$704$	−2.16583	−0.0816276
$705$	0	0
$706$	16.0621	0.604504
$707$	−17.1370	−0.644505
$708$	0	0
$709$	−31.1197	−1.16873	−0.584363	−	0.811493i	$-0.698655\pi$
$709$	−31.1197	−1.16873	−0.584363	+	0.811493i	$0.698655\pi$
$710$	18.3623	0.689125
$711$	0	0
$712$	5.78408	0.216767
$713$	8.57236	0.321038
$714$	0	0
$715$	−13.7596	−0.514579
$716$	−6.19100	−0.231368
$717$	0	0
$718$	−3.70142	−0.138136
$719$	36.3532	1.35574	0.677872	−	0.735180i	$-0.262902\pi$
$719$	36.3532	1.35574	0.677872	+	0.735180i	$0.262902\pi$
$720$	0	0
$721$	−17.3023	−0.644370
$722$	18.8371	0.701044
$723$	0	0
$724$	−1.84271	−0.0684837
$725$	45.9898	1.70802
$726$	0	0
$727$	2.73345	0.101378	0.0506891	−	0.998714i	$-0.483858\pi$
$727$	2.73345	0.101378	0.0506891	+	0.998714i	$0.483858\pi$
$728$	−3.78358	−0.140229
$729$	0	0
$730$	−60.5049	−2.23939
$731$	23.6032	0.872996
$732$	0	0
$733$	−14.0489	−0.518907	−0.259454	−	0.965756i	$-0.583542\pi$
$733$	−14.0489	−0.518907	−0.259454	+	0.965756i	$0.583542\pi$
$734$	−3.27262	−0.120795
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−20.8480	−0.767944
$738$	0	0
$739$	−52.8938	−1.94573	−0.972865	−	0.231375i	$-0.925678\pi$
$739$	−52.8938	−1.94573	−0.972865	+	0.231375i	$0.925678\pi$
$740$	−44.4322	−1.63336
$741$	0	0
$742$	6.67630	0.245095
$743$	7.15211	0.262385	0.131193	−	0.991357i	$-0.458119\pi$
$743$	7.15211	0.262385	0.131193	+	0.991357i	$0.458119\pi$
$744$	0	0
$745$	−45.7471	−1.67604
$746$	−11.4557	−0.419424
$747$	0	0
$748$	−8.97046	−0.327992
$749$	7.69343	0.281112
$750$	0	0
$751$	−28.6960	−1.04713	−0.523566	−	0.851985i	$-0.675399\pi$
$751$	−28.6960	−1.04713	−0.523566	+	0.851985i	$0.675399\pi$
$752$	−7.22282	−0.263389
$753$	0	0
$754$	5.17898	0.188607
$755$	63.2875	2.30327
$756$	0	0
$757$	25.1275	0.913273	0.456637	−	0.889653i	$-0.349054\pi$
$757$	25.1275	0.913273	0.456637	+	0.889653i	$0.349054\pi$
$758$	−5.68367	−0.206440
$759$	0	0
$760$	1.72277	0.0624913
$761$	14.6081	0.529544	0.264772	−	0.964311i	$-0.414703\pi$
$761$	14.6081	0.529544	0.264772	+	0.964311i	$0.414703\pi$
$762$	0	0
$763$	−17.2013	−0.622728
$764$	23.5513	0.852055
$765$	0	0
$766$	−8.77631	−0.317101
$767$	17.1319	0.618596
$768$	0	0
$769$	16.1350	0.581844	0.290922	−	0.956747i	$-0.406038\pi$
$769$	16.1350	0.581844	0.290922	+	0.956747i	$0.406038\pi$
$770$	23.4983	0.846822
$771$	0	0
$772$	8.55772	0.307999
$773$	−3.45901	−0.124412	−0.0622059	−	0.998063i	$-0.519814\pi$
$773$	−3.45901	−0.124412	−0.0622059	+	0.998063i	$0.519814\pi$
$774$	0	0
$775$	113.306	4.07008
$776$	19.0646	0.684381
$777$	0	0
$778$	−21.6724	−0.776992
$779$	−3.10175	−0.111132
$780$	0	0
$781$	−9.31762	−0.333411
$782$	−4.14182	−0.148111
$783$	0	0
$784$	−0.538465	−0.0192309
$785$	−6.43725	−0.229755
$786$	0	0
$787$	7.55969	0.269474	0.134737	−	0.990881i	$-0.456981\pi$
$787$	7.55969	0.269474	0.134737	+	0.990881i	$0.456981\pi$
$788$	20.2436	0.721149
$789$	0	0
$790$	42.2536	1.50332
$791$	−8.38074	−0.297985
$792$	0	0
$793$	11.0518	0.392461
$794$	12.8208	0.454995
$795$	0	0
$796$	−20.0901	−0.712075
$797$	−23.5661	−0.834756	−0.417378	−	0.908733i	$-0.637051\pi$
$797$	−23.5661	−0.834756	−0.417378	+	0.908733i	$0.637051\pi$
$798$	0	0
$799$	−29.9156	−1.05834
$800$	−13.2176	−0.467313
$801$	0	0
$802$	15.0561	0.531648
$803$	30.7021	1.08346
$804$	0	0
$805$	10.8496	0.382398
$806$	12.7596	0.449437
$807$	0	0
$808$	−6.74168	−0.237172
$809$	10.6692	0.375108	0.187554	−	0.982254i	$-0.439944\pi$
$809$	10.6692	0.375108	0.187554	+	0.982254i	$0.439944\pi$
$810$	0	0
$811$	33.5190	1.17701	0.588506	−	0.808493i	$-0.299716\pi$
$811$	33.5190	1.17701	0.588506	+	0.808493i	$0.299716\pi$
$812$	−8.84456	−0.310383
$813$	0	0
$814$	22.5463	0.790248
$815$	57.1633	2.00234
$816$	0	0
$817$	2.30017	0.0804729
$818$	−26.2503	−0.917820
$819$	0	0
$820$	32.7999	1.14542
$821$	43.1499	1.50594	0.752972	−	0.658053i	$-0.228620\pi$
$821$	43.1499	1.50594	0.752972	+	0.658053i	$0.228620\pi$
$822$	0	0
$823$	−11.7746	−0.410437	−0.205219	−	0.978716i	$-0.565791\pi$
$823$	−11.7746	−0.410437	−0.205219	+	0.978716i	$0.565791\pi$
$824$	−6.80668	−0.237122
$825$	0	0
$826$	−29.2575	−1.01800
$827$	32.2654	1.12198	0.560989	−	0.827824i	$-0.310421\pi$
$827$	32.2654	1.12198	0.560989	+	0.827824i	$0.310421\pi$
$828$	0	0
$829$	−34.1835	−1.18724	−0.593620	−	0.804745i	$-0.702302\pi$
$829$	−34.1835	−1.18724	−0.593620	+	0.804745i	$0.702302\pi$
$830$	−19.3308	−0.670981
$831$	0	0
$832$	−1.48845	−0.0516029
$833$	−2.23022	−0.0772727
$834$	0	0
$835$	44.2841	1.53252
$836$	−0.874186	−0.0302344
$837$	0	0
$838$	7.61740	0.263139
$839$	18.5033	0.638804	0.319402	−	0.947619i	$-0.396518\pi$
$839$	18.5033	0.638804	0.319402	+	0.947619i	$0.396518\pi$
$840$	0	0
$841$	−16.8936	−0.582537
$842$	14.7403	0.507985
$843$	0	0
$844$	9.45387	0.325416
$845$	46.0305	1.58350
$846$	0	0
$847$	16.0377	0.551062
$848$	2.62644	0.0901924
$849$	0	0
$850$	−54.7450	−1.87774
$851$	10.4100	0.356851
$852$	0	0
$853$	−17.2310	−0.589980	−0.294990	−	0.955500i	$-0.595316\pi$
$853$	−17.2310	−0.589980	−0.294990	+	0.955500i	$0.595316\pi$
$854$	−18.8740	−0.645857
$855$	0	0
$856$	3.02658	0.103446
$857$	32.9790	1.12654	0.563271	−	0.826272i	$-0.309543\pi$
$857$	32.9790	1.12654	0.563271	+	0.826272i	$0.309543\pi$
$858$	0	0
$859$	23.8450	0.813582	0.406791	−	0.913521i	$-0.366648\pi$
$859$	23.8450	0.813582	0.406791	+	0.913521i	$0.366648\pi$
$860$	−24.3235	−0.829424
$861$	0	0
$862$	30.9125	1.05288
$863$	22.0815	0.751664	0.375832	−	0.926688i	$-0.377357\pi$
$863$	22.0815	0.751664	0.375832	+	0.926688i	$0.377357\pi$
$864$	0	0
$865$	−51.2996	−1.74424
$866$	−1.69567	−0.0576211
$867$	0	0
$868$	−21.7906	−0.739620
$869$	−21.4408	−0.727330
$870$	0	0
$871$	−14.3277	−0.485474
$872$	−6.76695	−0.229158
$873$	0	0
$874$	−0.403627	−0.0136529
$875$	89.1579	3.01409
$876$	0	0
$877$	−7.26472	−0.245312	−0.122656	−	0.992449i	$-0.539141\pi$
$877$	−7.26472	−0.245312	−0.122656	+	0.992449i	$0.539141\pi$
$878$	−22.6381	−0.764000
$879$	0	0
$880$	9.24420	0.311622
$881$	52.7936	1.77866	0.889331	−	0.457263i	$-0.151170\pi$
$881$	52.7936	1.77866	0.889331	+	0.457263i	$0.151170\pi$
$882$	0	0
$883$	11.5394	0.388332	0.194166	−	0.980969i	$-0.437800\pi$
$883$	11.5394	0.388332	0.194166	+	0.980969i	$0.437800\pi$
$884$	−6.16491	−0.207348
$885$	0	0
$886$	−13.5120	−0.453944
$887$	2.46345	0.0827145	0.0413572	−	0.999144i	$-0.486832\pi$
$887$	2.46345	0.0827145	0.0413572	+	0.999144i	$0.486832\pi$
$888$	0	0
$889$	42.5411	1.42678
$890$	−24.6877	−0.827532
$891$	0	0
$892$	24.0625	0.805673
$893$	−2.91532	−0.0975576
$894$	0	0
$895$	26.4245	0.883273
$896$	2.54195	0.0849208
$897$	0	0
$898$	35.9618	1.20006
$899$	29.8269	0.994784
$900$	0	0
$901$	10.8783	0.362407
$902$	−16.6437	−0.554175
$903$	0	0
$904$	−3.29697	−0.109655
$905$	7.86506	0.261443
$906$	0	0
$907$	−24.0331	−0.798006	−0.399003	−	0.916950i	$-0.630644\pi$
$907$	−24.0331	−0.798006	−0.399003	+	0.916950i	$0.630644\pi$
$908$	17.1353	0.568654
$909$	0	0
$910$	16.1491	0.535339
$911$	50.4063	1.67004	0.835018	−	0.550223i	$-0.185457\pi$
$911$	50.4063	1.67004	0.835018	+	0.550223i	$0.185457\pi$
$912$	0	0
$913$	9.80906	0.324632
$914$	27.5883	0.912540
$915$	0	0
$916$	10.5172	0.347497
$917$	−27.2611	−0.900241
$918$	0	0
$919$	44.6135	1.47166	0.735832	−	0.677165i	$-0.236791\pi$
$919$	44.6135	1.47166	0.735832	+	0.677165i	$0.236791\pi$
$920$	4.26821	0.140719
$921$	0	0
$922$	32.3275	1.06465
$923$	−6.40350	−0.210774
$924$	0	0
$925$	137.596	4.52412
$926$	26.7669	0.879615
$927$	0	0
$928$	−3.47943	−0.114218
$929$	−3.57453	−0.117277	−0.0586383	−	0.998279i	$-0.518676\pi$
$929$	−3.57453	−0.117277	−0.0586383	+	0.998279i	$0.518676\pi$
$930$	0	0
$931$	−0.217339	−0.00712300
$932$	−12.5529	−0.411182
$933$	0	0
$934$	−6.17053	−0.201906
$935$	38.2878	1.25215
$936$	0	0
$937$	56.8302	1.85656	0.928280	−	0.371882i	$-0.121287\pi$
$937$	56.8302	1.85656	0.928280	+	0.371882i	$0.121287\pi$
$938$	24.4685	0.798925
$939$	0	0
$940$	30.8285	1.00551
$941$	−15.8198	−0.515710	−0.257855	−	0.966184i	$-0.583016\pi$
$941$	−15.8198	−0.515710	−0.257855	+	0.966184i	$0.583016\pi$
$942$	0	0
$943$	−7.68469	−0.250248
$944$	−11.5098	−0.374613
$945$	0	0
$946$	12.3425	0.401290
$947$	13.2161	0.429467	0.214734	−	0.976673i	$-0.431112\pi$
$947$	13.2161	0.429467	0.214734	+	0.976673i	$0.431112\pi$
$948$	0	0
$949$	21.0999	0.684932
$950$	−5.33499	−0.173090
$951$	0	0
$952$	10.5283	0.341225
$953$	−8.19166	−0.265354	−0.132677	−	0.991159i	$-0.542357\pi$
$953$	−8.19166	−0.265354	−0.132677	+	0.991159i	$0.542357\pi$
$954$	0	0
$955$	−100.522	−3.25281
$956$	−8.36249	−0.270462
$957$	0	0
$958$	13.5302	0.437141
$959$	−21.0120	−0.678513
$960$	0	0
$961$	42.4854	1.37050
$962$	15.4949	0.499574
$963$	0	0
$964$	1.22801	0.0395516
$965$	−36.5261	−1.17582
$966$	0	0
$967$	−28.0819	−0.903052	−0.451526	−	0.892258i	$-0.649120\pi$
$967$	−28.0819	−0.903052	−0.451526	+	0.892258i	$0.649120\pi$
$968$	6.30920	0.202785
$969$	0	0
$970$	−81.3719	−2.61270
$971$	−25.2831	−0.811373	−0.405687	−	0.914012i	$-0.632968\pi$
$971$	−25.2831	−0.811373	−0.405687	+	0.914012i	$0.632968\pi$
$972$	0	0
$973$	−11.6151	−0.372364
$974$	−7.57109	−0.242593
$975$	0	0
$976$	−7.42501	−0.237669
$977$	45.0405	1.44097	0.720487	−	0.693469i	$-0.243918\pi$
$977$	45.0405	1.44097	0.720487	+	0.693469i	$0.243918\pi$
$978$	0	0
$979$	12.5273	0.400374
$980$	2.29828	0.0734159
$981$	0	0
$982$	−1.71106	−0.0546022
$983$	−22.2720	−0.710368	−0.355184	−	0.934796i	$-0.615582\pi$
$983$	−22.2720	−0.710368	−0.355184	+	0.934796i	$0.615582\pi$
$984$	0	0
$985$	−86.4040	−2.75306
$986$	−14.4112	−0.458945
$987$	0	0
$988$	−0.600781	−0.0191134
$989$	5.69876	0.181210
$990$	0	0
$991$	26.0392	0.827164	0.413582	−	0.910467i	$-0.364278\pi$
$991$	26.0392	0.827164	0.413582	+	0.910467i	$0.364278\pi$
$992$	−8.57236	−0.272173
$993$	0	0
$994$	10.9358	0.346862
$995$	85.7488	2.71842
$996$	0	0
$997$	−7.83000	−0.247979	−0.123989	−	0.992284i	$-0.539569\pi$
$997$	−7.83000	−0.247979	−0.123989	+	0.992284i	$0.539569\pi$
$998$	−3.99670	−0.126513
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3726.2.a.q.1.1		5
3.2	odd	2		3726.2.a.v.1.5		5
9.2	odd	6		414.2.e.b.139.2	✓	10
9.4	even	3		1242.2.e.d.829.5		10
9.5	odd	6		414.2.e.b.277.2	yes	10
9.7	even	3		1242.2.e.d.415.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.2.e.b.139.2	✓	10	9.2	odd	6
414.2.e.b.277.2	yes	10	9.5	odd	6
1242.2.e.d.415.5		10	9.7	even	3
1242.2.e.d.829.5		10	9.4	even	3
3726.2.a.q.1.1		5	1.1	even	1	trivial
3726.2.a.v.1.5		5	3.2	odd	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:	\( N \)	\(=\)	\( 3726 = 2 \cdot 3^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3726.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.26821 q^{5} -2.54195 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.26821 q^{5} -2.54195 q^{7} -1.00000 q^{8} +4.26821 q^{10} -2.16583 q^{11} -1.48845 q^{13} +2.54195 q^{14} +1.00000 q^{16} +4.14182 q^{17} +0.403627 q^{19} -4.26821 q^{20} +2.16583 q^{22} +1.00000 q^{23} +13.2176 q^{25} +1.48845 q^{26} -2.54195 q^{28} +3.47943 q^{29} +8.57236 q^{31} -1.00000 q^{32} -4.14182 q^{34} +10.8496 q^{35} +10.4100 q^{37} -0.403627 q^{38} +4.26821 q^{40} -7.68469 q^{41} +5.69876 q^{43} -2.16583 q^{44} -1.00000 q^{46} -7.22282 q^{47} -0.538465 q^{49} -13.2176 q^{50} -1.48845 q^{52} +2.62644 q^{53} +9.24420 q^{55} +2.54195 q^{56} -3.47943 q^{58} -11.5098 q^{59} -7.42501 q^{61} -8.57236 q^{62} +1.00000 q^{64} +6.35304 q^{65} +9.62586 q^{67} +4.14182 q^{68} -10.8496 q^{70} +4.30211 q^{71} -14.1757 q^{73} -10.4100 q^{74} +0.403627 q^{76} +5.50543 q^{77} +9.89961 q^{79} -4.26821 q^{80} +7.68469 q^{82} -4.52901 q^{83} -17.6782 q^{85} -5.69876 q^{86} +2.16583 q^{88} -5.78408 q^{89} +3.78358 q^{91} +1.00000 q^{92} +7.22282 q^{94} -1.72277 q^{95} -19.0646 q^{97} +0.538465 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{4} - 5 q^{5} + 3 q^{7} - 5 q^{8}+O(q^{10}) \) 5 * q - 5 * q^2 + 5 * q^4 - 5 * q^5 + 3 * q^7 - 5 * q^8	\( 5 q - 5 q^{2} + 5 q^{4} - 5 q^{5} + 3 q^{7} - 5 q^{8} + 5 q^{10} - 11 q^{11} - 3 q^{14} + 5 q^{16} - 11 q^{17} + 9 q^{19} - 5 q^{20} + 11 q^{22} + 5 q^{23} + 12 q^{25} + 3 q^{28} + 2 q^{29} + 4 q^{31} - 5 q^{32} + 11 q^{34} + 7 q^{35} + 4 q^{37} - 9 q^{38} + 5 q^{40} - 8 q^{41} + 5 q^{43} - 11 q^{44} - 5 q^{46} - 19 q^{47} + 4 q^{49} - 12 q^{50} + 3 q^{53} - 2 q^{55} - 3 q^{56} - 2 q^{58} - 19 q^{59} - 13 q^{61} - 4 q^{62} + 5 q^{64} + q^{65} + q^{67} - 11 q^{68} - 7 q^{70} - 27 q^{71} - 7 q^{73} - 4 q^{74} + 9 q^{76} - 19 q^{77} + 3 q^{79} - 5 q^{80} + 8 q^{82} - 9 q^{83} - 5 q^{85} - 5 q^{86} + 11 q^{88} - 9 q^{89} - 19 q^{91} + 5 q^{92} + 19 q^{94} - 17 q^{95} - 20 q^{97} - 4 q^{98}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^4 - 5 * q^5 + 3 * q^7 - 5 * q^8 + 5 * q^10 - 11 * q^11 - 3 * q^14 + 5 * q^16 - 11 * q^17 + 9 * q^19 - 5 * q^20 + 11 * q^22 + 5 * q^23 + 12 * q^25 + 3 * q^28 + 2 * q^29 + 4 * q^31 - 5 * q^32 + 11 * q^34 + 7 * q^35 + 4 * q^37 - 9 * q^38 + 5 * q^40 - 8 * q^41 + 5 * q^43 - 11 * q^44 - 5 * q^46 - 19 * q^47 + 4 * q^49 - 12 * q^50 + 3 * q^53 - 2 * q^55 - 3 * q^56 - 2 * q^58 - 19 * q^59 - 13 * q^61 - 4 * q^62 + 5 * q^64 + q^65 + q^67 - 11 * q^68 - 7 * q^70 - 27 * q^71 - 7 * q^73 - 4 * q^74 + 9 * q^76 - 19 * q^77 + 3 * q^79 - 5 * q^80 + 8 * q^82 - 9 * q^83 - 5 * q^85 - 5 * q^86 + 11 * q^88 - 9 * q^89 - 19 * q^91 + 5 * q^92 + 19 * q^94 - 17 * q^95 - 20 * q^97 - 4 * q^98

Introduction

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