LMFDB - Embedded newform 5265.2.a.bb.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5265,2,Mod(1,5265)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5265, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5265.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5265 = 3^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5265.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:	$42.0412366642$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 10x^{6} + 7x^{5} + 33x^{4} - 14x^{3} - 38x^{2} + 7x + 9 $ x^8 - x^7 - 10x^6 + 7x^5 + 33x^4 - 14x^3 - 38x^2 + 7x + 9
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 585)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.43149$ of defining polynomial
Character	$\chi$	$=$	5265.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.43149 q^{2} +0.0491573 q^{4} +1.00000 q^{5} -1.25071 q^{7} +2.79261 q^{8} +O(q^{10})$	\(q-1.43149 q^{2} +0.0491573 q^{4} +1.00000 q^{5} -1.25071 q^{7} +2.79261 q^{8} -1.43149 q^{10} -1.47395 q^{11} +1.00000 q^{13} +1.79038 q^{14} -4.09590 q^{16} -3.53506 q^{17} +4.76114 q^{19} +0.0491573 q^{20} +2.10994 q^{22} +6.94445 q^{23} +1.00000 q^{25} -1.43149 q^{26} -0.0614815 q^{28} -9.61166 q^{29} -0.103321 q^{31} +0.278013 q^{32} +5.06040 q^{34} -1.25071 q^{35} -9.80005 q^{37} -6.81551 q^{38} +2.79261 q^{40} +6.15088 q^{41} -5.67179 q^{43} -0.0724552 q^{44} -9.94090 q^{46} +2.11464 q^{47} -5.43573 q^{49} -1.43149 q^{50} +0.0491573 q^{52} +8.82702 q^{53} -1.47395 q^{55} -3.49274 q^{56} +13.7590 q^{58} -4.08373 q^{59} +13.4484 q^{61} +0.147903 q^{62} +7.79382 q^{64} +1.00000 q^{65} -11.1497 q^{67} -0.173774 q^{68} +1.79038 q^{70} +9.50636 q^{71} +8.24967 q^{73} +14.0286 q^{74} +0.234045 q^{76} +1.84348 q^{77} -1.61596 q^{79} -4.09590 q^{80} -8.80490 q^{82} -4.08130 q^{83} -3.53506 q^{85} +8.11910 q^{86} -4.11615 q^{88} -13.7502 q^{89} -1.25071 q^{91} +0.341370 q^{92} -3.02708 q^{94} +4.76114 q^{95} -11.2902 q^{97} +7.78118 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 5 q^{4} + 8 q^{5} - 6 q^{7} - 6 q^{8}+O(q^{10}) $ 8 * q - q^2 + 5 * q^4 + 8 * q^5 - 6 * q^7 - 6 * q^8	$ 8 q - q^{2} + 5 q^{4} + 8 q^{5} - 6 q^{7} - 6 q^{8} - q^{10} - 9 q^{11} + 8 q^{13} + 3 q^{14} - 13 q^{16} + 6 q^{17} - 11 q^{19} + 5 q^{20} - 4 q^{22} - 3 q^{23} + 8 q^{25} - q^{26} - 13 q^{28} - 8 q^{29} - 18 q^{31} - 3 q^{32} - 9 q^{34} - 6 q^{35} - 18 q^{37} + 8 q^{38} - 6 q^{40} + 17 q^{41} - 17 q^{43} + 5 q^{44} + 3 q^{46} - 11 q^{47} - 16 q^{49} - q^{50} + 5 q^{52} + 10 q^{53} - 9 q^{55} + q^{56} - 10 q^{58} - 7 q^{59} - 21 q^{61} - 29 q^{62} - 10 q^{64} + 8 q^{65} - 13 q^{67} + 16 q^{68} + 3 q^{70} - 34 q^{71} - 16 q^{73} + 4 q^{74} - 2 q^{76} - 18 q^{77} - 37 q^{79} - 13 q^{80} + q^{82} - 3 q^{83} + 6 q^{85} + 2 q^{86} - 19 q^{88} + 14 q^{89} - 6 q^{91} + 14 q^{92} - 44 q^{94} - 11 q^{95} - 17 q^{97} + 45 q^{98}+O(q^{100}) $ 8 * q - q^2 + 5 * q^4 + 8 * q^5 - 6 * q^7 - 6 * q^8 - q^10 - 9 * q^11 + 8 * q^13 + 3 * q^14 - 13 * q^16 + 6 * q^17 - 11 * q^19 + 5 * q^20 - 4 * q^22 - 3 * q^23 + 8 * q^25 - q^26 - 13 * q^28 - 8 * q^29 - 18 * q^31 - 3 * q^32 - 9 * q^34 - 6 * q^35 - 18 * q^37 + 8 * q^38 - 6 * q^40 + 17 * q^41 - 17 * q^43 + 5 * q^44 + 3 * q^46 - 11 * q^47 - 16 * q^49 - q^50 + 5 * q^52 + 10 * q^53 - 9 * q^55 + q^56 - 10 * q^58 - 7 * q^59 - 21 * q^61 - 29 * q^62 - 10 * q^64 + 8 * q^65 - 13 * q^67 + 16 * q^68 + 3 * q^70 - 34 * q^71 - 16 * q^73 + 4 * q^74 - 2 * q^76 - 18 * q^77 - 37 * q^79 - 13 * q^80 + q^82 - 3 * q^83 + 6 * q^85 + 2 * q^86 - 19 * q^88 + 14 * q^89 - 6 * q^91 + 14 * q^92 - 44 * q^94 - 11 * q^95 - 17 * q^97 + 45 * 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 $

$2$	−1.43149	−1.01221	−0.506107	−	0.862470i	$-0.668916\pi$
$2$	−1.43149	−1.01221	−0.506107	+	0.862470i	$0.668916\pi$
$3$	0	0
$3$	0	0
$4$	0.0491573	0.0245786
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.25071	−0.472724	−0.236362	−	0.971665i	$-0.575955\pi$
$7$	−1.25071	−0.472724	−0.236362	+	0.971665i	$0.575955\pi$
$8$	2.79261	0.987336
$9$	0	0
$10$	−1.43149	−0.452676
$11$	−1.47395	−0.444412	−0.222206	−	0.975000i	$-0.571326\pi$
$11$	−1.47395	−0.444412	−0.222206	+	0.975000i	$0.571326\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.79038	0.478498
$15$	0	0
$16$	−4.09590	−1.02397
$17$	−3.53506	−0.857378	−0.428689	−	0.903452i	$-0.641024\pi$
$17$	−3.53506	−0.857378	−0.428689	+	0.903452i	$0.641024\pi$
$18$	0	0
$19$	4.76114	1.09228	0.546140	−	0.837694i	$-0.316097\pi$
$19$	4.76114	1.09228	0.546140	+	0.837694i	$0.316097\pi$
$20$	0.0491573	0.0109919
$21$	0	0
$22$	2.10994	0.449840
$23$	6.94445	1.44802	0.724009	−	0.689790i	$-0.242297\pi$
$23$	6.94445	1.44802	0.724009	+	0.689790i	$0.242297\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.43149	−0.280738
$27$	0	0
$28$	−0.0614815	−0.0116189
$29$	−9.61166	−1.78484	−0.892420	−	0.451206i	$-0.850994\pi$
$29$	−9.61166	−1.78484	−0.892420	+	0.451206i	$0.850994\pi$
$30$	0	0
$31$	−0.103321	−0.0185570	−0.00927851	−	0.999957i	$-0.502953\pi$
$31$	−0.103321	−0.0185570	−0.00927851	+	0.999957i	$0.502953\pi$
$32$	0.278013	0.0491462
$33$	0	0
$34$	5.06040	0.867851
$35$	−1.25071	−0.211408
$36$	0	0
$37$	−9.80005	−1.61112	−0.805559	−	0.592516i	$-0.798135\pi$
$37$	−9.80005	−1.61112	−0.805559	+	0.592516i	$0.798135\pi$
$38$	−6.81551	−1.10562
$39$	0	0
$40$	2.79261	0.441550
$41$	6.15088	0.960605	0.480303	−	0.877103i	$-0.340527\pi$
$41$	6.15088	0.960605	0.480303	+	0.877103i	$0.340527\pi$
$42$	0	0
$43$	−5.67179	−0.864940	−0.432470	−	0.901648i	$-0.642358\pi$
$43$	−5.67179	−0.864940	−0.432470	+	0.901648i	$0.642358\pi$
$44$	−0.0724552	−0.0109230
$45$	0	0
$46$	−9.94090	−1.46571
$47$	2.11464	0.308452	0.154226	−	0.988036i	$-0.450712\pi$
$47$	2.11464	0.308452	0.154226	+	0.988036i	$0.450712\pi$
$48$	0	0
$49$	−5.43573	−0.776532
$50$	−1.43149	−0.202443
$51$	0	0
$52$	0.0491573	0.00681689
$53$	8.82702	1.21249	0.606243	−	0.795280i	$-0.292676\pi$
$53$	8.82702	1.21249	0.606243	+	0.795280i	$0.292676\pi$
$54$	0	0
$55$	−1.47395	−0.198747
$56$	−3.49274	−0.466737
$57$	0	0
$58$	13.7590	1.80664
$59$	−4.08373	−0.531656	−0.265828	−	0.964020i	$-0.585645\pi$
$59$	−4.08373	−0.531656	−0.265828	+	0.964020i	$0.585645\pi$
$60$	0	0
$61$	13.4484	1.72189	0.860944	−	0.508700i	$-0.169874\pi$
$61$	13.4484	1.72189	0.860944	+	0.508700i	$0.169874\pi$
$62$	0.147903	0.0187837
$63$	0	0
$64$	7.79382	0.974228
$65$	1.00000	0.124035
$66$	0	0
$67$	−11.1497	−1.36216	−0.681079	−	0.732210i	$-0.738489\pi$
$67$	−11.1497	−1.36216	−0.681079	+	0.732210i	$0.738489\pi$
$68$	−0.173774	−0.0210732
$69$	0	0
$70$	1.79038	0.213991
$71$	9.50636	1.12820	0.564098	−	0.825708i	$-0.309224\pi$
$71$	9.50636	1.12820	0.564098	+	0.825708i	$0.309224\pi$
$72$	0	0
$73$	8.24967	0.965550	0.482775	−	0.875744i	$-0.339629\pi$
$73$	8.24967	0.965550	0.482775	+	0.875744i	$0.339629\pi$
$74$	14.0286	1.63080
$75$	0	0
$76$	0.234045	0.0268467
$77$	1.84348	0.210084
$78$	0	0
$79$	−1.61596	−0.181809	−0.0909046	−	0.995860i	$-0.528976\pi$
$79$	−1.61596	−0.181809	−0.0909046	+	0.995860i	$0.528976\pi$
$80$	−4.09590	−0.457935
$81$	0	0
$82$	−8.80490	−0.972339
$83$	−4.08130	−0.447981	−0.223990	−	0.974591i	$-0.571908\pi$
$83$	−4.08130	−0.447981	−0.223990	+	0.974591i	$0.571908\pi$
$84$	0	0
$85$	−3.53506	−0.383431
$86$	8.11910	0.875505
$87$	0	0
$88$	−4.11615	−0.438783
$89$	−13.7502	−1.45752	−0.728761	−	0.684768i	$-0.759903\pi$
$89$	−13.7502	−1.45752	−0.728761	+	0.684768i	$0.759903\pi$
$90$	0	0
$91$	−1.25071	−0.131110
$92$	0.341370	0.0355903
$93$	0	0
$94$	−3.02708	−0.312220
$95$	4.76114	0.488482
$96$	0	0
$97$	−11.2902	−1.14635	−0.573173	−	0.819435i	$-0.694288\pi$
$97$	−11.2902	−1.14635	−0.573173	+	0.819435i	$0.694288\pi$
$98$	7.78118	0.786017
$99$	0	0
$100$	0.0491573	0.00491573
$101$	3.71394	0.369550	0.184775	−	0.982781i	$-0.440844\pi$
$101$	3.71394	0.369550	0.184775	+	0.982781i	$0.440844\pi$
$102$	0	0
$103$	−8.46371	−0.833954	−0.416977	−	0.908917i	$-0.636910\pi$
$103$	−8.46371	−0.833954	−0.416977	+	0.908917i	$0.636910\pi$
$104$	2.79261	0.273838
$105$	0	0
$106$	−12.6358	−1.22730
$107$	16.8261	1.62664	0.813321	−	0.581816i	$-0.197658\pi$
$107$	16.8261	1.62664	0.813321	+	0.581816i	$0.197658\pi$
$108$	0	0
$109$	−17.4145	−1.66801	−0.834003	−	0.551760i	$-0.813957\pi$
$109$	−17.4145	−1.66801	−0.834003	+	0.551760i	$0.813957\pi$
$110$	2.10994	0.201175
$111$	0	0
$112$	5.12278	0.484057
$113$	5.39829	0.507829	0.253914	−	0.967227i	$-0.418282\pi$
$113$	5.39829	0.507829	0.253914	+	0.967227i	$0.418282\pi$
$114$	0	0
$115$	6.94445	0.647574
$116$	−0.472483	−0.0438689
$117$	0	0
$118$	5.84581	0.538150
$119$	4.42133	0.405303
$120$	0	0
$121$	−8.82748	−0.802498
$122$	−19.2512	−1.74292
$123$	0	0
$124$	−0.00507899	−0.000456107 0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−16.9895	−1.50757	−0.753786	−	0.657119i	$-0.771775\pi$
$127$	−16.9895	−1.50757	−0.753786	+	0.657119i	$0.771775\pi$
$128$	−11.7128	−1.03527
$129$	0	0
$130$	−1.43149	−0.125550
$131$	0.373750	0.0326547	0.0163273	−	0.999867i	$-0.494803\pi$
$131$	0.373750	0.0326547	0.0163273	+	0.999867i	$0.494803\pi$
$132$	0	0
$133$	−5.95480	−0.516347
$134$	15.9607	1.37880
$135$	0	0
$136$	−9.87204	−0.846520
$137$	15.6740	1.33912	0.669560	−	0.742758i	$-0.266483\pi$
$137$	15.6740	1.33912	0.669560	+	0.742758i	$0.266483\pi$
$138$	0	0
$139$	−12.0331	−1.02064	−0.510318	−	0.859986i	$-0.670472\pi$
$139$	−12.0331	−1.02064	−0.510318	+	0.859986i	$0.670472\pi$
$140$	−0.0614815	−0.00519613
$141$	0	0
$142$	−13.6082	−1.14198
$143$	−1.47395	−0.123258
$144$	0	0
$145$	−9.61166	−0.798205
$146$	−11.8093	−0.977344
$147$	0	0
$148$	−0.481744	−0.0395991
$149$	−4.61382	−0.377979	−0.188990	−	0.981979i	$-0.560521\pi$
$149$	−4.61382	−0.377979	−0.188990	+	0.981979i	$0.560521\pi$
$150$	0	0
$151$	4.09191	0.332995	0.166498	−	0.986042i	$-0.446754\pi$
$151$	4.09191	0.332995	0.166498	+	0.986042i	$0.446754\pi$
$152$	13.2960	1.07845
$153$	0	0
$154$	−2.63892	−0.212650
$155$	−0.103321	−0.00829896
$156$	0	0
$157$	10.1548	0.810442	0.405221	−	0.914219i	$-0.367195\pi$
$157$	10.1548	0.810442	0.405221	+	0.914219i	$0.367195\pi$
$158$	2.31322	0.184030
$159$	0	0
$160$	0.278013	0.0219789
$161$	−8.68549	−0.684513
$162$	0	0
$163$	8.09508	0.634056	0.317028	−	0.948416i	$-0.397315\pi$
$163$	8.09508	0.634056	0.317028	+	0.948416i	$0.397315\pi$
$164$	0.302360	0.0236104
$165$	0	0
$166$	5.84233	0.453453
$167$	−8.52106	−0.659380	−0.329690	−	0.944089i	$-0.606944\pi$
$167$	−8.52106	−0.659380	−0.329690	+	0.944089i	$0.606944\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	5.06040	0.388115
$171$	0	0
$172$	−0.278810	−0.0212591
$173$	17.6337	1.34067	0.670334	−	0.742060i	$-0.266151\pi$
$173$	17.6337	1.34067	0.670334	+	0.742060i	$0.266151\pi$
$174$	0	0
$175$	−1.25071	−0.0945448
$176$	6.03713	0.455066
$177$	0	0
$178$	19.6833	1.47532
$179$	18.1466	1.35634	0.678171	−	0.734904i	$-0.262773\pi$
$179$	18.1466	1.35634	0.678171	+	0.734904i	$0.262773\pi$
$180$	0	0
$181$	0.832634	0.0618892	0.0309446	−	0.999521i	$-0.490148\pi$
$181$	0.832634	0.0618892	0.0309446	+	0.999521i	$0.490148\pi$
$182$	1.79038	0.132711
$183$	0	0
$184$	19.3931	1.42968
$185$	−9.80005	−0.720514
$186$	0	0
$187$	5.21049	0.381029
$188$	0.103950	0.00758133
$189$	0	0
$190$	−6.81551	−0.494449
$191$	−17.2903	−1.25108	−0.625541	−	0.780191i	$-0.715122\pi$
$191$	−17.2903	−1.25108	−0.625541	+	0.780191i	$0.715122\pi$
$192$	0	0
$193$	7.84624	0.564784	0.282392	−	0.959299i	$-0.408872\pi$
$193$	7.84624	0.564784	0.282392	+	0.959299i	$0.408872\pi$
$194$	16.1618	1.16035
$195$	0	0
$196$	−0.267205	−0.0190861
$197$	−3.70100	−0.263685	−0.131843	−	0.991271i	$-0.542089\pi$
$197$	−3.70100	−0.263685	−0.131843	+	0.991271i	$0.542089\pi$
$198$	0	0
$199$	5.29240	0.375168	0.187584	−	0.982249i	$-0.439934\pi$
$199$	5.29240	0.375168	0.187584	+	0.982249i	$0.439934\pi$
$200$	2.79261	0.197467
$201$	0	0
$202$	−5.31645	−0.374064
$203$	12.0214	0.843736
$204$	0	0
$205$	6.15088	0.429596
$206$	12.1157	0.844140
$207$	0	0
$208$	−4.09590	−0.283999
$209$	−7.01766	−0.485422
$210$	0	0
$211$	−19.9266	−1.37180	−0.685902	−	0.727694i	$-0.740592\pi$
$211$	−19.9266	−1.37180	−0.685902	+	0.727694i	$0.740592\pi$
$212$	0.433912	0.0298012
$213$	0	0
$214$	−24.0864	−1.64651
$215$	−5.67179	−0.386813
$216$	0	0
$217$	0.129225	0.00877235
$218$	24.9286	1.68838
$219$	0	0
$220$	−0.0724552	−0.00488493
$221$	−3.53506	−0.237794
$222$	0	0
$223$	9.46456	0.633794	0.316897	−	0.948460i	$-0.397359\pi$
$223$	9.46456	0.633794	0.316897	+	0.948460i	$0.397359\pi$
$224$	−0.347714	−0.0232326
$225$	0	0
$226$	−7.72759	−0.514032
$227$	11.9398	0.792470	0.396235	−	0.918149i	$-0.370316\pi$
$227$	11.9398	0.792470	0.396235	+	0.918149i	$0.370316\pi$
$228$	0	0
$229$	−19.0770	−1.26065	−0.630323	−	0.776333i	$-0.717077\pi$
$229$	−19.0770	−1.26065	−0.630323	+	0.776333i	$0.717077\pi$
$230$	−9.94090	−0.655483
$231$	0	0
$232$	−26.8416	−1.76224
$233$	−17.7574	−1.16333	−0.581664	−	0.813429i	$-0.697598\pi$
$233$	−17.7574	−1.16333	−0.581664	+	0.813429i	$0.697598\pi$
$234$	0	0
$235$	2.11464	0.137944
$236$	−0.200745	−0.0130674
$237$	0	0
$238$	−6.32909	−0.410254
$239$	12.2210	0.790513	0.395256	−	0.918571i	$-0.370656\pi$
$239$	12.2210	0.790513	0.395256	+	0.918571i	$0.370656\pi$
$240$	0	0
$241$	1.38487	0.0892076	0.0446038	−	0.999005i	$-0.485797\pi$
$241$	1.38487	0.0892076	0.0446038	+	0.999005i	$0.485797\pi$
$242$	12.6364	0.812301
$243$	0	0
$244$	0.661086	0.0423217
$245$	−5.43573	−0.347276
$246$	0	0
$247$	4.76114	0.302944
$248$	−0.288535	−0.0183220
$249$	0	0
$250$	−1.43149	−0.0905352
$251$	−22.7662	−1.43699	−0.718496	−	0.695531i	$-0.755169\pi$
$251$	−22.7662	−1.43699	−0.718496	+	0.695531i	$0.755169\pi$
$252$	0	0
$253$	−10.2358	−0.643516
$254$	24.3202	1.52599
$255$	0	0
$256$	1.17907	0.0736917
$257$	−5.16712	−0.322316	−0.161158	−	0.986929i	$-0.551523\pi$
$257$	−5.16712	−0.322316	−0.161158	+	0.986929i	$0.551523\pi$
$258$	0	0
$259$	12.2570	0.761614
$260$	0.0491573	0.00304861
$261$	0	0
$262$	−0.535018	−0.0330535
$263$	0.386768	0.0238491	0.0119246	−	0.999929i	$-0.496204\pi$
$263$	0.386768	0.0238491	0.0119246	+	0.999929i	$0.496204\pi$
$264$	0	0
$265$	8.82702	0.542240
$266$	8.52422	0.522654
$267$	0	0
$268$	−0.548091	−0.0334800
$269$	−1.52427	−0.0929363	−0.0464682	−	0.998920i	$-0.514797\pi$
$269$	−1.52427	−0.0929363	−0.0464682	+	0.998920i	$0.514797\pi$
$270$	0	0
$271$	−1.98110	−0.120343	−0.0601717	−	0.998188i	$-0.519165\pi$
$271$	−1.98110	−0.120343	−0.0601717	+	0.998188i	$0.519165\pi$
$272$	14.4793	0.877934
$273$	0	0
$274$	−22.4371	−1.35548
$275$	−1.47395	−0.0888823
$276$	0	0
$277$	5.88483	0.353585	0.176793	−	0.984248i	$-0.443428\pi$
$277$	5.88483	0.353585	0.176793	+	0.984248i	$0.443428\pi$
$278$	17.2253	1.03310
$279$	0	0
$280$	−3.49274	−0.208731
$281$	−14.2937	−0.852690	−0.426345	−	0.904561i	$-0.640199\pi$
$281$	−14.2937	−0.852690	−0.426345	+	0.904561i	$0.640199\pi$
$282$	0	0
$283$	−13.8013	−0.820402	−0.410201	−	0.911995i	$-0.634541\pi$
$283$	−13.8013	−0.820402	−0.410201	+	0.911995i	$0.634541\pi$
$284$	0.467307	0.0277295
$285$	0	0
$286$	2.10994	0.124763
$287$	−7.69296	−0.454101
$288$	0	0
$289$	−4.50334	−0.264902
$290$	13.7590	0.807954
$291$	0	0
$292$	0.405531	0.0237319
$293$	−17.1747	−1.00335	−0.501677	−	0.865055i	$-0.667283\pi$
$293$	−17.1747	−1.00335	−0.501677	+	0.865055i	$0.667283\pi$
$294$	0	0
$295$	−4.08373	−0.237764
$296$	−27.3677	−1.59071
$297$	0	0
$298$	6.60463	0.382596
$299$	6.94445	0.401608
$300$	0	0
$301$	7.09376	0.408878
$302$	−5.85753	−0.337063
$303$	0	0
$304$	−19.5011	−1.11847
$305$	13.4484	0.770052
$306$	0	0
$307$	−0.748107	−0.0426967	−0.0213484	−	0.999772i	$-0.506796\pi$
$307$	−0.748107	−0.0426967	−0.0213484	+	0.999772i	$0.506796\pi$
$308$	0.0906204	0.00516358
$309$	0	0
$310$	0.147903	0.00840032
$311$	1.41338	0.0801455	0.0400727	−	0.999197i	$-0.487241\pi$
$311$	1.41338	0.0801455	0.0400727	+	0.999197i	$0.487241\pi$
$312$	0	0
$313$	−9.30568	−0.525988	−0.262994	−	0.964797i	$-0.584710\pi$
$313$	−9.30568	−0.525988	−0.262994	+	0.964797i	$0.584710\pi$
$314$	−14.5365	−0.820341
$315$	0	0
$316$	−0.0794360	−0.00446862
$317$	26.2891	1.47654	0.738271	−	0.674504i	$-0.235643\pi$
$317$	26.2891	1.47654	0.738271	+	0.674504i	$0.235643\pi$
$318$	0	0
$319$	14.1671	0.793203
$320$	7.79382	0.435688
$321$	0	0
$322$	12.4332	0.692874
$323$	−16.8309	−0.936497
$324$	0	0
$325$	1.00000	0.0554700
$326$	−11.5880	−0.641801
$327$	0	0
$328$	17.1770	0.948440
$329$	−2.64480	−0.145813
$330$	0	0
$331$	−12.9660	−0.712675	−0.356338	−	0.934357i	$-0.615975\pi$
$331$	−12.9660	−0.712675	−0.356338	+	0.934357i	$0.615975\pi$
$332$	−0.200625	−0.0110108
$333$	0	0
$334$	12.1978	0.667434
$335$	−11.1497	−0.609175
$336$	0	0
$337$	1.89298	0.103117	0.0515586	−	0.998670i	$-0.483581\pi$
$337$	1.89298	0.103117	0.0515586	+	0.998670i	$0.483581\pi$
$338$	−1.43149	−0.0778627
$339$	0	0
$340$	−0.173774	−0.00942422
$341$	0.152290	0.00824696
$342$	0	0
$343$	15.5535	0.839809
$344$	−15.8391	−0.853986
$345$	0	0
$346$	−25.2425	−1.35704
$347$	−29.0052	−1.55708	−0.778541	−	0.627594i	$-0.784040\pi$
$347$	−29.0052	−1.55708	−0.778541	+	0.627594i	$0.784040\pi$
$348$	0	0
$349$	−2.94640	−0.157717	−0.0788585	−	0.996886i	$-0.525128\pi$
$349$	−2.94640	−0.157717	−0.0788585	+	0.996886i	$0.525128\pi$
$350$	1.79038	0.0956996
$351$	0	0
$352$	−0.409776	−0.0218412
$353$	−29.4606	−1.56803	−0.784015	−	0.620742i	$-0.786831\pi$
$353$	−29.4606	−1.56803	−0.784015	+	0.620742i	$0.786831\pi$
$354$	0	0
$355$	9.50636	0.504545
$356$	−0.675924	−0.0358239
$357$	0	0
$358$	−25.9767	−1.37291
$359$	−24.7647	−1.30703	−0.653515	−	0.756913i	$-0.726707\pi$
$359$	−24.7647	−1.30703	−0.653515	+	0.756913i	$0.726707\pi$
$360$	0	0
$361$	3.66842	0.193075
$362$	−1.19191	−0.0626452
$363$	0	0
$364$	−0.0614815	−0.00322250
$365$	8.24967	0.431807
$366$	0	0
$367$	−9.97114	−0.520490	−0.260245	−	0.965543i	$-0.583803\pi$
$367$	−9.97114	−0.520490	−0.260245	+	0.965543i	$0.583803\pi$
$368$	−28.4438	−1.48273
$369$	0	0
$370$	14.0286	0.729315
$371$	−11.0400	−0.573170
$372$	0	0
$373$	−19.0913	−0.988509	−0.494255	−	0.869317i	$-0.664559\pi$
$373$	−19.0913	−0.988509	−0.494255	+	0.869317i	$0.664559\pi$
$374$	−7.45875	−0.385683
$375$	0	0
$376$	5.90536	0.304546
$377$	−9.61166	−0.495025
$378$	0	0
$379$	20.7186	1.06424	0.532122	−	0.846668i	$-0.321395\pi$
$379$	20.7186	1.06424	0.532122	+	0.846668i	$0.321395\pi$
$380$	0.234045	0.0120062
$381$	0	0
$382$	24.7509	1.26636
$383$	33.4247	1.70792	0.853961	−	0.520337i	$-0.174194\pi$
$383$	33.4247	1.70792	0.853961	+	0.520337i	$0.174194\pi$
$384$	0	0
$385$	1.84348	0.0939524
$386$	−11.2318	−0.571683
$387$	0	0
$388$	−0.554995	−0.0281756
$389$	−7.43237	−0.376836	−0.188418	−	0.982089i	$-0.560336\pi$
$389$	−7.43237	−0.376836	−0.188418	+	0.982089i	$0.560336\pi$
$390$	0	0
$391$	−24.5491	−1.24150
$392$	−15.1798	−0.766698
$393$	0	0
$394$	5.29794	0.266906
$395$	−1.61596	−0.0813075
$396$	0	0
$397$	−1.80255	−0.0904674	−0.0452337	−	0.998976i	$-0.514403\pi$
$397$	−1.80255	−0.0904674	−0.0452337	+	0.998976i	$0.514403\pi$
$398$	−7.57601	−0.379751
$399$	0	0
$400$	−4.09590	−0.204795
$401$	−25.6141	−1.27911	−0.639554	−	0.768746i	$-0.720881\pi$
$401$	−25.6141	−1.27911	−0.639554	+	0.768746i	$0.720881\pi$
$402$	0	0
$403$	−0.103321	−0.00514679
$404$	0.182567	0.00908305
$405$	0	0
$406$	−17.2085	−0.854042
$407$	14.4447	0.715999
$408$	0	0
$409$	12.5408	0.620101	0.310050	−	0.950720i	$-0.399654\pi$
$409$	12.5408	0.620101	0.310050	+	0.950720i	$0.399654\pi$
$410$	−8.80490	−0.434843
$411$	0	0
$412$	−0.416053	−0.0204975
$413$	5.10756	0.251327
$414$	0	0
$415$	−4.08130	−0.200343
$416$	0.278013	0.0136307
$417$	0	0
$418$	10.0457	0.491351
$419$	22.1070	1.08000	0.540000	−	0.841665i	$-0.318424\pi$
$419$	22.1070	1.08000	0.540000	+	0.841665i	$0.318424\pi$
$420$	0	0
$421$	−34.1718	−1.66543	−0.832717	−	0.553699i	$-0.813216\pi$
$421$	−34.1718	−1.66543	−0.832717	+	0.553699i	$0.813216\pi$
$422$	28.5247	1.38856
$423$	0	0
$424$	24.6504	1.19713
$425$	−3.53506	−0.171476
$426$	0	0
$427$	−16.8200	−0.813977
$428$	0.827126	0.0399806
$429$	0	0
$430$	8.11910	0.391538
$431$	−8.75050	−0.421497	−0.210748	−	0.977540i	$-0.567590\pi$
$431$	−8.75050	−0.421497	−0.210748	+	0.977540i	$0.567590\pi$
$432$	0	0
$433$	10.0228	0.481667	0.240834	−	0.970566i	$-0.422579\pi$
$433$	10.0228	0.481667	0.240834	+	0.970566i	$0.422579\pi$
$434$	−0.184984	−0.00887950
$435$	0	0
$436$	−0.856049	−0.0409973
$437$	33.0635	1.58164
$438$	0	0
$439$	−34.0383	−1.62456	−0.812279	−	0.583270i	$-0.801773\pi$
$439$	−34.0383	−1.62456	−0.812279	+	0.583270i	$0.801773\pi$
$440$	−4.11615	−0.196230
$441$	0	0
$442$	5.06040	0.240699
$443$	−27.0458	−1.28498	−0.642492	−	0.766292i	$-0.722099\pi$
$443$	−27.0458	−1.28498	−0.642492	+	0.766292i	$0.722099\pi$
$444$	0	0
$445$	−13.7502	−0.651823
$446$	−13.5484	−0.641536
$447$	0	0
$448$	−9.74781	−0.460541
$449$	−24.6140	−1.16161	−0.580804	−	0.814043i	$-0.697262\pi$
$449$	−24.6140	−1.16161	−0.580804	+	0.814043i	$0.697262\pi$
$450$	0	0
$451$	−9.06606	−0.426904
$452$	0.265365	0.0124817
$453$	0	0
$454$	−17.0916	−0.802150
$455$	−1.25071	−0.0586342
$456$	0	0
$457$	−17.4479	−0.816178	−0.408089	−	0.912942i	$-0.633805\pi$
$457$	−17.4479	−0.816178	−0.408089	+	0.912942i	$0.633805\pi$
$458$	27.3085	1.27604
$459$	0	0
$460$	0.341370	0.0159165
$461$	26.2226	1.22131	0.610653	−	0.791898i	$-0.290907\pi$
$461$	26.2226	1.22131	0.610653	+	0.791898i	$0.290907\pi$
$462$	0	0
$463$	−27.4477	−1.27560	−0.637801	−	0.770201i	$-0.720156\pi$
$463$	−27.4477	−1.27560	−0.637801	+	0.770201i	$0.720156\pi$
$464$	39.3684	1.82763
$465$	0	0
$466$	25.4195	1.17754
$467$	20.6056	0.953512	0.476756	−	0.879036i	$-0.341812\pi$
$467$	20.6056	0.953512	0.476756	+	0.879036i	$0.341812\pi$
$468$	0	0
$469$	13.9451	0.643924
$470$	−3.02708	−0.139629
$471$	0	0
$472$	−11.4043	−0.524923
$473$	8.35992	0.384389
$474$	0	0
$475$	4.76114	0.218456
$476$	0.217341	0.00996180
$477$	0	0
$478$	−17.4943	−0.800169
$479$	−16.9016	−0.772255	−0.386127	−	0.922445i	$-0.626187\pi$
$479$	−16.9016	−0.772255	−0.386127	+	0.922445i	$0.626187\pi$
$480$	0	0
$481$	−9.80005	−0.446844
$482$	−1.98243	−0.0902973
$483$	0	0
$484$	−0.433935	−0.0197243
$485$	−11.2902	−0.512661
$486$	0	0
$487$	3.72906	0.168980	0.0844899	−	0.996424i	$-0.473074\pi$
$487$	3.72906	0.168980	0.0844899	+	0.996424i	$0.473074\pi$
$488$	37.5560	1.70008
$489$	0	0
$490$	7.78118	0.351518
$491$	−20.9624	−0.946019	−0.473010	−	0.881057i	$-0.656832\pi$
$491$	−20.9624	−0.946019	−0.473010	+	0.881057i	$0.656832\pi$
$492$	0	0
$493$	33.9778	1.53028
$494$	−6.81551	−0.306644
$495$	0	0
$496$	0.423193	0.0190019
$497$	−11.8897	−0.533325
$498$	0	0
$499$	−10.3235	−0.462145	−0.231072	−	0.972937i	$-0.574223\pi$
$499$	−10.3235	−0.462145	−0.231072	+	0.972937i	$0.574223\pi$
$500$	0.0491573	0.00219838
$501$	0	0
$502$	32.5896	1.45454
$503$	−15.9668	−0.711923	−0.355962	−	0.934501i	$-0.615847\pi$
$503$	−15.9668	−0.711923	−0.355962	+	0.934501i	$0.615847\pi$
$504$	0	0
$505$	3.71394	0.165268
$506$	14.6524	0.651377
$507$	0	0
$508$	−0.835157	−0.0370541
$509$	−19.3864	−0.859287	−0.429644	−	0.902999i	$-0.641361\pi$
$509$	−19.3864	−0.859287	−0.429644	+	0.902999i	$0.641361\pi$
$510$	0	0
$511$	−10.3179	−0.456439
$512$	21.7378	0.960682
$513$	0	0
$514$	7.39668	0.326253
$515$	−8.46371	−0.372956
$516$	0	0
$517$	−3.11687	−0.137080
$518$	−17.5458	−0.770917
$519$	0	0
$520$	2.79261	0.122464
$521$	−43.0972	−1.88812	−0.944060	−	0.329773i	$-0.893028\pi$
$521$	−43.0972	−1.88812	−0.944060	+	0.329773i	$0.893028\pi$
$522$	0	0
$523$	30.2809	1.32409	0.662045	−	0.749464i	$-0.269689\pi$
$523$	30.2809	1.32409	0.662045	+	0.749464i	$0.269689\pi$
$524$	0.0183725	0.000802607 0
$525$	0	0
$526$	−0.553654	−0.0241405
$527$	0.365247	0.0159104
$528$	0	0
$529$	25.2254	1.09676
$530$	−12.6358	−0.548863
$531$	0	0
$532$	−0.292722	−0.0126911
$533$	6.15088	0.266424
$534$	0	0
$535$	16.8261	0.727456
$536$	−31.1368	−1.34491
$537$	0	0
$538$	2.18197	0.0940715
$539$	8.01197	0.345100
$540$	0	0
$541$	18.2422	0.784292	0.392146	−	0.919903i	$-0.371733\pi$
$541$	18.2422	0.784292	0.392146	+	0.919903i	$0.371733\pi$
$542$	2.83592	0.121813
$543$	0	0
$544$	−0.982793	−0.0421369
$545$	−17.4145	−0.745955
$546$	0	0
$547$	−7.23354	−0.309284	−0.154642	−	0.987971i	$-0.549422\pi$
$547$	−7.23354	−0.309284	−0.154642	+	0.987971i	$0.549422\pi$
$548$	0.770491	0.0329138
$549$	0	0
$550$	2.10994	0.0899680
$551$	−45.7624	−1.94954
$552$	0	0
$553$	2.02109	0.0859455
$554$	−8.42406	−0.357904
$555$	0	0
$556$	−0.591516	−0.0250858
$557$	−0.960304	−0.0406894	−0.0203447	−	0.999793i	$-0.506476\pi$
$557$	−0.960304	−0.0406894	−0.0203447	+	0.999793i	$0.506476\pi$
$558$	0	0
$559$	−5.67179	−0.239891
$560$	5.12278	0.216477
$561$	0	0
$562$	20.4612	0.863105
$563$	−8.88373	−0.374405	−0.187202	−	0.982321i	$-0.559942\pi$
$563$	−8.88373	−0.374405	−0.187202	+	0.982321i	$0.559942\pi$
$564$	0	0
$565$	5.39829	0.227108
$566$	19.7564	0.830423
$567$	0	0
$568$	26.5475	1.11391
$569$	−11.8371	−0.496238	−0.248119	−	0.968730i	$-0.579812\pi$
$569$	−11.8371	−0.496238	−0.248119	+	0.968730i	$0.579812\pi$
$570$	0	0
$571$	−34.9871	−1.46416	−0.732082	−	0.681216i	$-0.761451\pi$
$571$	−34.9871	−1.46416	−0.732082	+	0.681216i	$0.761451\pi$
$572$	−0.0724552	−0.00302950
$573$	0	0
$574$	11.0124	0.459648
$575$	6.94445	0.289604
$576$	0	0
$577$	−11.2588	−0.468709	−0.234355	−	0.972151i	$-0.575298\pi$
$577$	−11.2588	−0.468709	−0.234355	+	0.972151i	$0.575298\pi$
$578$	6.44648	0.268138
$579$	0	0
$580$	−0.472483	−0.0196188
$581$	5.10452	0.211771
$582$	0	0
$583$	−13.0106	−0.538842
$584$	23.0381	0.953323
$585$	0	0
$586$	24.5853	1.01561
$587$	15.2220	0.628280	0.314140	−	0.949377i	$-0.398284\pi$
$587$	15.2220	0.628280	0.314140	+	0.949377i	$0.398284\pi$
$588$	0	0
$589$	−0.491926	−0.0202695
$590$	5.84581	0.240668
$591$	0	0
$592$	40.1400	1.64974
$593$	12.6917	0.521184	0.260592	−	0.965449i	$-0.416082\pi$
$593$	12.6917	0.521184	0.260592	+	0.965449i	$0.416082\pi$
$594$	0	0
$595$	4.42133	0.181257
$596$	−0.226803	−0.00929021
$597$	0	0
$598$	−9.94090	−0.406514
$599$	22.4838	0.918662	0.459331	−	0.888265i	$-0.348089\pi$
$599$	22.4838	0.918662	0.459331	+	0.888265i	$0.348089\pi$
$600$	0	0
$601$	−23.5083	−0.958923	−0.479462	−	0.877563i	$-0.659168\pi$
$601$	−23.5083	−0.958923	−0.479462	+	0.877563i	$0.659168\pi$
$602$	−10.1546	−0.413872
$603$	0	0
$604$	0.201147	0.00818457
$605$	−8.82748	−0.358888
$606$	0	0
$607$	−37.9335	−1.53967	−0.769836	−	0.638242i	$-0.779662\pi$
$607$	−37.9335	−1.53967	−0.769836	+	0.638242i	$0.779662\pi$
$608$	1.32366	0.0536814
$609$	0	0
$610$	−19.2512	−0.779458
$611$	2.11464	0.0855492
$612$	0	0
$613$	24.5881	0.993103	0.496551	−	0.868007i	$-0.334599\pi$
$613$	24.5881	0.993103	0.496551	+	0.868007i	$0.334599\pi$
$614$	1.07091	0.0432182
$615$	0	0
$616$	5.14811	0.207423
$617$	−9.05098	−0.364379	−0.182189	−	0.983263i	$-0.558318\pi$
$617$	−9.05098	−0.364379	−0.182189	+	0.983263i	$0.558318\pi$
$618$	0	0
$619$	−12.1436	−0.488093	−0.244047	−	0.969764i	$-0.578475\pi$
$619$	−12.1436	−0.488093	−0.244047	+	0.969764i	$0.578475\pi$
$620$	−0.00507899	−0.000203977 0
$621$	0	0
$622$	−2.02324	−0.0811244
$623$	17.1975	0.689005
$624$	0	0
$625$	1.00000	0.0400000
$626$	13.3210	0.532413
$627$	0	0
$628$	0.499183	0.0199196
$629$	34.6438	1.38134
$630$	0	0
$631$	−39.4322	−1.56977	−0.784885	−	0.619641i	$-0.787278\pi$
$631$	−39.4322	−1.56977	−0.784885	+	0.619641i	$0.787278\pi$
$632$	−4.51273	−0.179507
$633$	0	0
$634$	−37.6325	−1.49458
$635$	−16.9895	−0.674207
$636$	0	0
$637$	−5.43573	−0.215371
$638$	−20.2800	−0.802892
$639$	0	0
$640$	−11.7128	−0.462989
$641$	38.2297	1.50998	0.754990	−	0.655736i	$-0.227641\pi$
$641$	38.2297	1.50998	0.754990	+	0.655736i	$0.227641\pi$
$642$	0	0
$643$	31.4125	1.23879	0.619393	−	0.785081i	$-0.287379\pi$
$643$	31.4125	1.23879	0.619393	+	0.785081i	$0.287379\pi$
$644$	−0.426955	−0.0168244
$645$	0	0
$646$	24.0932	0.947936
$647$	−45.8883	−1.80406	−0.902028	−	0.431677i	$-0.857922\pi$
$647$	−45.8883	−1.80406	−0.902028	+	0.431677i	$0.857922\pi$
$648$	0	0
$649$	6.01920	0.236274
$650$	−1.43149	−0.0561476
$651$	0	0
$652$	0.397932	0.0155842
$653$	−30.9169	−1.20987	−0.604935	−	0.796275i	$-0.706801\pi$
$653$	−30.9169	−1.20987	−0.604935	+	0.796275i	$0.706801\pi$
$654$	0	0
$655$	0.373750	0.0146036
$656$	−25.1934	−0.983636
$657$	0	0
$658$	3.78600	0.147594
$659$	−29.3819	−1.14456	−0.572278	−	0.820060i	$-0.693940\pi$
$659$	−29.3819	−1.14456	−0.572278	+	0.820060i	$0.693940\pi$
$660$	0	0
$661$	12.8322	0.499115	0.249558	−	0.968360i	$-0.419715\pi$
$661$	12.8322	0.499115	0.249558	+	0.968360i	$0.419715\pi$
$662$	18.5607	0.721380
$663$	0	0
$664$	−11.3975	−0.442307
$665$	−5.95480	−0.230917
$666$	0	0
$667$	−66.7477	−2.58448
$668$	−0.418872	−0.0162067
$669$	0	0
$670$	15.9607	0.616616
$671$	−19.8222	−0.765227
$672$	0	0
$673$	22.9034	0.882862	0.441431	−	0.897295i	$-0.354471\pi$
$673$	22.9034	0.882862	0.441431	+	0.897295i	$0.354471\pi$
$674$	−2.70978	−0.104377
$675$	0	0
$676$	0.0491573	0.00189066
$677$	28.7535	1.10509	0.552543	−	0.833484i	$-0.313657\pi$
$677$	28.7535	1.10509	0.552543	+	0.833484i	$0.313657\pi$
$678$	0	0
$679$	14.1208	0.541905
$680$	−9.87204	−0.378575
$681$	0	0
$682$	−0.218001	−0.00834769
$683$	−20.8274	−0.796938	−0.398469	−	0.917182i	$-0.630458\pi$
$683$	−20.8274	−0.796938	−0.398469	+	0.917182i	$0.630458\pi$
$684$	0	0
$685$	15.6740	0.598873
$686$	−22.2646	−0.850067
$687$	0	0
$688$	23.2311	0.885677
$689$	8.82702	0.336283
$690$	0	0
$691$	−37.8415	−1.43956	−0.719779	−	0.694203i	$-0.755757\pi$
$691$	−37.8415	−1.43956	−0.719779	+	0.694203i	$0.755757\pi$
$692$	0.866826	0.0329518
$693$	0	0
$694$	41.5206	1.57610
$695$	−12.0331	−0.456442
$696$	0	0
$697$	−21.7437	−0.823602
$698$	4.21773	0.159644
$699$	0	0
$700$	−0.0614815	−0.00232378
$701$	8.41612	0.317872	0.158936	−	0.987289i	$-0.449194\pi$
$701$	8.41612	0.317872	0.158936	+	0.987289i	$0.449194\pi$
$702$	0	0
$703$	−46.6594	−1.75979
$704$	−11.4877	−0.432958
$705$	0	0
$706$	42.1725	1.58718
$707$	−4.64506	−0.174695
$708$	0	0
$709$	−20.7398	−0.778900	−0.389450	−	0.921048i	$-0.627335\pi$
$709$	−20.7398	−0.778900	−0.389450	+	0.921048i	$0.627335\pi$
$710$	−13.6082	−0.510708
$711$	0	0
$712$	−38.3990	−1.43906
$713$	−0.717509	−0.0268709
$714$	0	0
$715$	−1.47395	−0.0551225
$716$	0.892038	0.0333370
$717$	0	0
$718$	35.4504	1.32300
$719$	−20.7856	−0.775173	−0.387587	−	0.921833i	$-0.626691\pi$
$719$	−20.7856	−0.775173	−0.387587	+	0.921833i	$0.626691\pi$
$720$	0	0
$721$	10.5856	0.394230
$722$	−5.25130	−0.195433
$723$	0	0
$724$	0.0409300	0.00152115
$725$	−9.61166	−0.356968
$726$	0	0
$727$	21.9926	0.815662	0.407831	−	0.913057i	$-0.366285\pi$
$727$	21.9926	0.815662	0.407831	+	0.913057i	$0.366285\pi$
$728$	−3.49274	−0.129450
$729$	0	0
$730$	−11.8093	−0.437082
$731$	20.0501	0.741581
$732$	0	0
$733$	36.4104	1.34485	0.672425	−	0.740165i	$-0.265253\pi$
$733$	36.4104	1.34485	0.672425	+	0.740165i	$0.265253\pi$
$734$	14.2736	0.526847
$735$	0	0
$736$	1.93065	0.0711647
$737$	16.4341	0.605359
$738$	0	0
$739$	−51.0462	−1.87776	−0.938881	−	0.344241i	$-0.888136\pi$
$739$	−51.0462	−1.87776	−0.938881	+	0.344241i	$0.888136\pi$
$740$	−0.481744	−0.0177092
$741$	0	0
$742$	15.8037	0.580172
$743$	−46.9278	−1.72162	−0.860808	−	0.508930i	$-0.830041\pi$
$743$	−46.9278	−1.72162	−0.860808	+	0.508930i	$0.830041\pi$
$744$	0	0
$745$	−4.61382	−0.169037
$746$	27.3289	1.00058
$747$	0	0
$748$	0.256134	0.00936517
$749$	−21.0446	−0.768952
$750$	0	0
$751$	−22.7572	−0.830421	−0.415210	−	0.909725i	$-0.636292\pi$
$751$	−22.7572	−0.830421	−0.415210	+	0.909725i	$0.636292\pi$
$752$	−8.66135	−0.315847
$753$	0	0
$754$	13.7590	0.501072
$755$	4.09191	0.148920
$756$	0	0
$757$	−51.9029	−1.88644	−0.943222	−	0.332162i	$-0.892222\pi$
$757$	−51.9029	−1.88644	−0.943222	+	0.332162i	$0.892222\pi$
$758$	−29.6584	−1.07724
$759$	0	0
$760$	13.2960	0.482296
$761$	20.1364	0.729943	0.364972	−	0.931019i	$-0.381079\pi$
$761$	20.1364	0.729943	0.364972	+	0.931019i	$0.381079\pi$
$762$	0	0
$763$	21.7805	0.788506
$764$	−0.849944	−0.0307499
$765$	0	0
$766$	−47.8470	−1.72878
$767$	−4.08373	−0.147455
$768$	0	0
$769$	18.2535	0.658240	0.329120	−	0.944288i	$-0.393248\pi$
$769$	18.2535	0.658240	0.329120	+	0.944288i	$0.393248\pi$
$770$	−2.63892	−0.0951000
$771$	0	0
$772$	0.385700	0.0138816
$773$	24.9928	0.898929	0.449465	−	0.893298i	$-0.351615\pi$
$773$	24.9928	0.898929	0.449465	+	0.893298i	$0.351615\pi$
$774$	0	0
$775$	−0.103321	−0.00371141
$776$	−31.5291	−1.13183
$777$	0	0
$778$	10.6393	0.381439
$779$	29.2852	1.04925
$780$	0	0
$781$	−14.0119	−0.501384
$782$	35.1417	1.25666
$783$	0	0
$784$	22.2642	0.795149
$785$	10.1548	0.362441
$786$	0	0
$787$	54.8848	1.95643	0.978216	−	0.207591i	$-0.0665623\pi$
$787$	54.8848	1.95643	0.978216	+	0.207591i	$0.0665623\pi$
$788$	−0.181931	−0.00648103
$789$	0	0
$790$	2.31322	0.0823007
$791$	−6.75170	−0.240063
$792$	0	0
$793$	13.4484	0.477566
$794$	2.58033	0.0915725
$795$	0	0
$796$	0.260160	0.00922113
$797$	9.52820	0.337506	0.168753	−	0.985658i	$-0.446026\pi$
$797$	9.52820	0.337506	0.168753	+	0.985658i	$0.446026\pi$
$798$	0	0
$799$	−7.47538	−0.264460
$800$	0.278013	0.00982925
$801$	0	0
$802$	36.6663	1.29473
$803$	−12.1596	−0.429102
$804$	0	0
$805$	−8.68549	−0.306123
$806$	0.147903	0.00520966
$807$	0	0
$808$	10.3716	0.364870
$809$	35.3596	1.24318	0.621589	−	0.783344i	$-0.286488\pi$
$809$	35.3596	1.24318	0.621589	+	0.783344i	$0.286488\pi$
$810$	0	0
$811$	−0.112318	−0.00394404	−0.00197202	−	0.999998i	$-0.500628\pi$
$811$	−0.112318	−0.00394404	−0.00197202	+	0.999998i	$0.500628\pi$
$812$	0.590939	0.0207379
$813$	0	0
$814$	−20.6775	−0.724745
$815$	8.09508	0.283558
$816$	0	0
$817$	−27.0042	−0.944756
$818$	−17.9519	−0.627675
$819$	0	0
$820$	0.302360	0.0105589
$821$	43.1941	1.50748	0.753742	−	0.657171i	$-0.228247\pi$
$821$	43.1941	1.50748	0.753742	+	0.657171i	$0.228247\pi$
$822$	0	0
$823$	−28.5766	−0.996119	−0.498060	−	0.867143i	$-0.665954\pi$
$823$	−28.5766	−0.996119	−0.498060	+	0.867143i	$0.665954\pi$
$824$	−23.6358	−0.823393
$825$	0	0
$826$	−7.31141	−0.254396
$827$	−6.91464	−0.240445	−0.120223	−	0.992747i	$-0.538361\pi$
$827$	−6.91464	−0.240445	−0.120223	+	0.992747i	$0.538361\pi$
$828$	0	0
$829$	44.7141	1.55299	0.776493	−	0.630126i	$-0.216997\pi$
$829$	44.7141	1.55299	0.776493	+	0.630126i	$0.216997\pi$
$830$	5.84233	0.202790
$831$	0	0
$832$	7.79382	0.270202
$833$	19.2156	0.665782
$834$	0	0
$835$	−8.52106	−0.294883
$836$	−0.344969	−0.0119310
$837$	0	0
$838$	−31.6459	−1.09319
$839$	−23.3180	−0.805025	−0.402513	−	0.915414i	$-0.631863\pi$
$839$	−23.3180	−0.805025	−0.402513	+	0.915414i	$0.631863\pi$
$840$	0	0
$841$	63.3839	2.18565
$842$	48.9166	1.68578
$843$	0	0
$844$	−0.979538	−0.0337171
$845$	1.00000	0.0344010
$846$	0	0
$847$	11.0406	0.379360
$848$	−36.1546	−1.24155
$849$	0	0
$850$	5.06040	0.173570
$851$	−68.0560	−2.33293
$852$	0	0
$853$	−8.01906	−0.274567	−0.137284	−	0.990532i	$-0.543837\pi$
$853$	−8.01906	−0.274567	−0.137284	+	0.990532i	$0.543837\pi$
$854$	24.0776	0.823920
$855$	0	0
$856$	46.9887	1.60604
$857$	6.57766	0.224689	0.112344	−	0.993669i	$-0.464164\pi$
$857$	6.57766	0.224689	0.112344	+	0.993669i	$0.464164\pi$
$858$	0	0
$859$	5.77553	0.197058	0.0985292	−	0.995134i	$-0.468586\pi$
$859$	5.77553	0.197058	0.0985292	+	0.995134i	$0.468586\pi$
$860$	−0.278810	−0.00950734
$861$	0	0
$862$	12.5262	0.426645
$863$	14.4279	0.491131	0.245565	−	0.969380i	$-0.421026\pi$
$863$	14.4279	0.491131	0.245565	+	0.969380i	$0.421026\pi$
$864$	0	0
$865$	17.6337	0.599565
$866$	−14.3476	−0.487551
$867$	0	0
$868$	0.00635234	0.000215612 0
$869$	2.38183	0.0807981
$870$	0	0
$871$	−11.1497	−0.377795
$872$	−48.6319	−1.64688
$873$	0	0
$874$	−47.3300	−1.60096
$875$	−1.25071	−0.0422817
$876$	0	0
$877$	53.6259	1.81082	0.905410	−	0.424539i	$-0.139564\pi$
$877$	53.6259	1.81082	0.905410	+	0.424539i	$0.139564\pi$
$878$	48.7253	1.64440
$879$	0	0
$880$	6.03713	0.203512
$881$	54.2756	1.82859	0.914297	−	0.405046i	$-0.132744\pi$
$881$	54.2756	1.82859	0.914297	+	0.405046i	$0.132744\pi$
$882$	0	0
$883$	−8.60336	−0.289526	−0.144763	−	0.989466i	$-0.546242\pi$
$883$	−8.60336	−0.289526	−0.144763	+	0.989466i	$0.546242\pi$
$884$	−0.173774	−0.00584465
$885$	0	0
$886$	38.7157	1.30068
$887$	14.5614	0.488924	0.244462	−	0.969659i	$-0.421389\pi$
$887$	14.5614	0.488924	0.244462	+	0.969659i	$0.421389\pi$
$888$	0	0
$889$	21.2489	0.712666
$890$	19.6833	0.659785
$891$	0	0
$892$	0.465252	0.0155778
$893$	10.0681	0.336916
$894$	0	0
$895$	18.1466	0.606574
$896$	14.6493	0.489399
$897$	0	0
$898$	35.2347	1.17580
$899$	0.993088	0.0331213
$900$	0	0
$901$	−31.2041	−1.03956
$902$	12.9780	0.432119
$903$	0	0
$904$	15.0753	0.501397
$905$	0.832634	0.0276777
$906$	0	0
$907$	29.0033	0.963040	0.481520	−	0.876435i	$-0.340085\pi$
$907$	29.0033	0.963040	0.481520	+	0.876435i	$0.340085\pi$
$908$	0.586927	0.0194778
$909$	0	0
$910$	1.79038	0.0593504
$911$	−4.31280	−0.142890	−0.0714448	−	0.997445i	$-0.522761\pi$
$911$	−4.31280	−0.142890	−0.0714448	+	0.997445i	$0.522761\pi$
$912$	0	0
$913$	6.01561	0.199088
$914$	24.9764	0.826147
$915$	0	0
$916$	−0.937775	−0.0309850
$917$	−0.467452	−0.0154366
$918$	0	0
$919$	−58.1294	−1.91751	−0.958757	−	0.284228i	$-0.908263\pi$
$919$	−58.1294	−1.91751	−0.958757	+	0.284228i	$0.908263\pi$
$920$	19.3931	0.639373
$921$	0	0
$922$	−37.5373	−1.23622
$923$	9.50636	0.312906
$924$	0	0
$925$	−9.80005	−0.322224
$926$	39.2910	1.29118
$927$	0	0
$928$	−2.67217	−0.0877182
$929$	4.72816	0.155126	0.0775630	−	0.996987i	$-0.475286\pi$
$929$	4.72816	0.155126	0.0775630	+	0.996987i	$0.475286\pi$
$930$	0	0
$931$	−25.8802	−0.848190
$932$	−0.872906	−0.0285930
$933$	0	0
$934$	−29.4966	−0.965159
$935$	5.21049	0.170401
$936$	0	0
$937$	18.4570	0.602965	0.301482	−	0.953472i	$-0.402519\pi$
$937$	18.4570	0.602965	0.301482	+	0.953472i	$0.402519\pi$
$938$	−19.9622	−0.651790
$939$	0	0
$940$	0.103950	0.00339047
$941$	0.196255	0.00639774	0.00319887	−	0.999995i	$-0.498982\pi$
$941$	0.196255	0.00639774	0.00319887	+	0.999995i	$0.498982\pi$
$942$	0	0
$943$	42.7145	1.39097
$944$	16.7265	0.544403
$945$	0	0
$946$	−11.9671	−0.389085
$947$	−15.3290	−0.498126	−0.249063	−	0.968487i	$-0.580123\pi$
$947$	−15.3290	−0.498126	−0.249063	+	0.968487i	$0.580123\pi$
$948$	0	0
$949$	8.24967	0.267795
$950$	−6.81551	−0.221124
$951$	0	0
$952$	12.3471	0.400170
$953$	−3.12675	−0.101285	−0.0506427	−	0.998717i	$-0.516127\pi$
$953$	−3.12675	−0.101285	−0.0506427	+	0.998717i	$0.516127\pi$
$954$	0	0
$955$	−17.2903	−0.559501
$956$	0.600753	0.0194297
$957$	0	0
$958$	24.1945	0.781688
$959$	−19.6036	−0.633034
$960$	0	0
$961$	−30.9893	−0.999656
$962$	14.0286	0.452302
$963$	0	0
$964$	0.0680767	0.00219260
$965$	7.84624	0.252579
$966$	0	0
$967$	33.9373	1.09135	0.545675	−	0.837997i	$-0.316274\pi$
$967$	33.9373	1.09135	0.545675	+	0.837997i	$0.316274\pi$
$968$	−24.6517	−0.792335
$969$	0	0
$970$	16.1618	0.518923
$971$	−28.0546	−0.900314	−0.450157	−	0.892949i	$-0.648632\pi$
$971$	−28.0546	−0.900314	−0.450157	+	0.892949i	$0.648632\pi$
$972$	0	0
$973$	15.0499	0.482479
$974$	−5.33811	−0.171044
$975$	0	0
$976$	−55.0832	−1.76317
$977$	36.4095	1.16484	0.582422	−	0.812886i	$-0.302105\pi$
$977$	36.4095	1.16484	0.582422	+	0.812886i	$0.302105\pi$
$978$	0	0
$979$	20.2671	0.647739
$980$	−0.267205	−0.00853557
$981$	0	0
$982$	30.0074	0.957574
$983$	7.42511	0.236824	0.118412	−	0.992965i	$-0.462220\pi$
$983$	7.42511	0.236824	0.118412	+	0.992965i	$0.462220\pi$
$984$	0	0
$985$	−3.70100	−0.117924
$986$	−48.6388	−1.54898
$987$	0	0
$988$	0.234045	0.00744595
$989$	−39.3875	−1.25245
$990$	0	0
$991$	15.5263	0.493209	0.246605	−	0.969116i	$-0.420685\pi$
$991$	15.5263	0.493209	0.246605	+	0.969116i	$0.420685\pi$
$992$	−0.0287246	−0.000912008 0
$993$	0	0
$994$	17.0199	0.539840
$995$	5.29240	0.167780
$996$	0	0
$997$	20.4753	0.648459	0.324229	−	0.945979i	$-0.394895\pi$
$997$	20.4753	0.648459	0.324229	+	0.945979i	$0.394895\pi$
$998$	14.7780	0.467790
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.bb.1.3		8
3.2	odd	2		5265.2.a.be.1.6		8
9.2	odd	6		1755.2.i.e.1171.3		16
9.4	even	3		585.2.i.f.196.6	✓	16
9.5	odd	6		1755.2.i.e.586.3		16
9.7	even	3		585.2.i.f.391.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.i.f.196.6	✓	16	9.4	even	3
585.2.i.f.391.6	yes	16	9.7	even	3
1755.2.i.e.586.3		16	9.5	odd	6
1755.2.i.e.1171.3		16	9.2	odd	6
5265.2.a.bb.1.3		8	1.1	even	1	trivial
5265.2.a.be.1.6		8	3.2	odd	2

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:	\( N \)	\(=\)	\( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5265.a (trivial)

\(f(q)\)	\(=\)	\(q-1.43149 q^{2} +0.0491573 q^{4} +1.00000 q^{5} -1.25071 q^{7} +2.79261 q^{8} +O(q^{10})\)	\(q-1.43149 q^{2} +0.0491573 q^{4} +1.00000 q^{5} -1.25071 q^{7} +2.79261 q^{8} -1.43149 q^{10} -1.47395 q^{11} +1.00000 q^{13} +1.79038 q^{14} -4.09590 q^{16} -3.53506 q^{17} +4.76114 q^{19} +0.0491573 q^{20} +2.10994 q^{22} +6.94445 q^{23} +1.00000 q^{25} -1.43149 q^{26} -0.0614815 q^{28} -9.61166 q^{29} -0.103321 q^{31} +0.278013 q^{32} +5.06040 q^{34} -1.25071 q^{35} -9.80005 q^{37} -6.81551 q^{38} +2.79261 q^{40} +6.15088 q^{41} -5.67179 q^{43} -0.0724552 q^{44} -9.94090 q^{46} +2.11464 q^{47} -5.43573 q^{49} -1.43149 q^{50} +0.0491573 q^{52} +8.82702 q^{53} -1.47395 q^{55} -3.49274 q^{56} +13.7590 q^{58} -4.08373 q^{59} +13.4484 q^{61} +0.147903 q^{62} +7.79382 q^{64} +1.00000 q^{65} -11.1497 q^{67} -0.173774 q^{68} +1.79038 q^{70} +9.50636 q^{71} +8.24967 q^{73} +14.0286 q^{74} +0.234045 q^{76} +1.84348 q^{77} -1.61596 q^{79} -4.09590 q^{80} -8.80490 q^{82} -4.08130 q^{83} -3.53506 q^{85} +8.11910 q^{86} -4.11615 q^{88} -13.7502 q^{89} -1.25071 q^{91} +0.341370 q^{92} -3.02708 q^{94} +4.76114 q^{95} -11.2902 q^{97} +7.78118 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 5 q^{4} + 8 q^{5} - 6 q^{7} - 6 q^{8}+O(q^{10}) \) 8 * q - q^2 + 5 * q^4 + 8 * q^5 - 6 * q^7 - 6 * q^8	\( 8 q - q^{2} + 5 q^{4} + 8 q^{5} - 6 q^{7} - 6 q^{8} - q^{10} - 9 q^{11} + 8 q^{13} + 3 q^{14} - 13 q^{16} + 6 q^{17} - 11 q^{19} + 5 q^{20} - 4 q^{22} - 3 q^{23} + 8 q^{25} - q^{26} - 13 q^{28} - 8 q^{29} - 18 q^{31} - 3 q^{32} - 9 q^{34} - 6 q^{35} - 18 q^{37} + 8 q^{38} - 6 q^{40} + 17 q^{41} - 17 q^{43} + 5 q^{44} + 3 q^{46} - 11 q^{47} - 16 q^{49} - q^{50} + 5 q^{52} + 10 q^{53} - 9 q^{55} + q^{56} - 10 q^{58} - 7 q^{59} - 21 q^{61} - 29 q^{62} - 10 q^{64} + 8 q^{65} - 13 q^{67} + 16 q^{68} + 3 q^{70} - 34 q^{71} - 16 q^{73} + 4 q^{74} - 2 q^{76} - 18 q^{77} - 37 q^{79} - 13 q^{80} + q^{82} - 3 q^{83} + 6 q^{85} + 2 q^{86} - 19 q^{88} + 14 q^{89} - 6 q^{91} + 14 q^{92} - 44 q^{94} - 11 q^{95} - 17 q^{97} + 45 q^{98}+O(q^{100}) \) 8 * q - q^2 + 5 * q^4 + 8 * q^5 - 6 * q^7 - 6 * q^8 - q^10 - 9 * q^11 + 8 * q^13 + 3 * q^14 - 13 * q^16 + 6 * q^17 - 11 * q^19 + 5 * q^20 - 4 * q^22 - 3 * q^23 + 8 * q^25 - q^26 - 13 * q^28 - 8 * q^29 - 18 * q^31 - 3 * q^32 - 9 * q^34 - 6 * q^35 - 18 * q^37 + 8 * q^38 - 6 * q^40 + 17 * q^41 - 17 * q^43 + 5 * q^44 + 3 * q^46 - 11 * q^47 - 16 * q^49 - q^50 + 5 * q^52 + 10 * q^53 - 9 * q^55 + q^56 - 10 * q^58 - 7 * q^59 - 21 * q^61 - 29 * q^62 - 10 * q^64 + 8 * q^65 - 13 * q^67 + 16 * q^68 + 3 * q^70 - 34 * q^71 - 16 * q^73 + 4 * q^74 - 2 * q^76 - 18 * q^77 - 37 * q^79 - 13 * q^80 + q^82 - 3 * q^83 + 6 * q^85 + 2 * q^86 - 19 * q^88 + 14 * q^89 - 6 * q^91 + 14 * q^92 - 44 * q^94 - 11 * q^95 - 17 * q^97 + 45 * q^98

Introduction

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