LMFDB - Embedded newform 1071.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1071,2,Mod(1,1071)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1071, 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("1071.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1071 = 3^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1071.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:	$8.55197805648$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1224176.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 8x^{2} + 2x - 2 $ x^5 - 2x^4 - 5x^3 + 8x^2 + 2x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.03496$ of defining polynomial
Character	$\chi$	$=$	1071.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.03496 q^{2} +2.14106 q^{4} +2.37597 q^{5} -1.00000 q^{7} -0.287051 q^{8} +O(q^{10})$	\(q-2.03496 q^{2} +2.14106 q^{4} +2.37597 q^{5} -1.00000 q^{7} -0.287051 q^{8} -4.83501 q^{10} +6.22816 q^{11} -0.711126 q^{13} +2.03496 q^{14} -3.69798 q^{16} +1.00000 q^{17} +4.48025 q^{19} +5.08710 q^{20} -12.6740 q^{22} +0.858940 q^{23} +0.645243 q^{25} +1.44711 q^{26} -2.14106 q^{28} +2.23088 q^{29} -0.765088 q^{31} +8.09935 q^{32} -2.03496 q^{34} -2.37597 q^{35} +3.81783 q^{37} -9.11713 q^{38} -0.682025 q^{40} +1.00675 q^{41} -8.66362 q^{43} +13.3349 q^{44} -1.74791 q^{46} -3.55139 q^{47} +1.00000 q^{49} -1.31304 q^{50} -1.52256 q^{52} -0.356369 q^{53} +14.7979 q^{55} +0.287051 q^{56} -4.53974 q^{58} -13.1527 q^{59} +0.518528 q^{61} +1.55692 q^{62} -9.08588 q^{64} -1.68962 q^{65} +6.26644 q^{67} +2.14106 q^{68} +4.83501 q^{70} +11.2613 q^{71} -5.60009 q^{73} -7.76912 q^{74} +9.59248 q^{76} -6.22816 q^{77} +1.57007 q^{79} -8.78630 q^{80} -2.04870 q^{82} -5.82037 q^{83} +2.37597 q^{85} +17.6301 q^{86} -1.78780 q^{88} +10.0209 q^{89} +0.711126 q^{91} +1.83904 q^{92} +7.22694 q^{94} +10.6449 q^{95} -8.38640 q^{97} -2.03496 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{2} + 4 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8}+O(q^{10}) $ 5 * q + 2 * q^2 + 4 * q^4 + 3 * q^5 - 5 * q^7 + 6 * q^8	$ 5 q + 2 q^{2} + 4 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8} - 2 q^{10} + 7 q^{11} + 5 q^{13} - 2 q^{14} + 10 q^{16} + 5 q^{17} + 3 q^{19} + 8 q^{20} - 6 q^{22} + 11 q^{23} + 6 q^{25} + 16 q^{26} - 4 q^{28} + 22 q^{29} - 6 q^{31} + 24 q^{32} + 2 q^{34} - 3 q^{35} - 10 q^{37} - 10 q^{38} - 10 q^{40} + 27 q^{41} - q^{43} + 22 q^{44} - 4 q^{46} - 16 q^{47} + 5 q^{49} + 6 q^{50} + 28 q^{52} + 4 q^{53} + 3 q^{55} - 6 q^{56} + 6 q^{58} + 8 q^{59} - 20 q^{61} - 14 q^{62} + 14 q^{64} + 39 q^{65} + 4 q^{67} + 4 q^{68} + 2 q^{70} + 4 q^{71} - 8 q^{73} - 28 q^{74} - 12 q^{76} - 7 q^{77} + 6 q^{79} - 40 q^{80} + 8 q^{82} + 6 q^{83} + 3 q^{85} + 50 q^{86} + 2 q^{88} + 20 q^{89} - 5 q^{91} - 26 q^{92} - 30 q^{94} + 11 q^{95} - 18 q^{97} + 2 q^{98}+O(q^{100}) $ 5 * q + 2 * q^2 + 4 * q^4 + 3 * q^5 - 5 * q^7 + 6 * q^8 - 2 * q^10 + 7 * q^11 + 5 * q^13 - 2 * q^14 + 10 * q^16 + 5 * q^17 + 3 * q^19 + 8 * q^20 - 6 * q^22 + 11 * q^23 + 6 * q^25 + 16 * q^26 - 4 * q^28 + 22 * q^29 - 6 * q^31 + 24 * q^32 + 2 * q^34 - 3 * q^35 - 10 * q^37 - 10 * q^38 - 10 * q^40 + 27 * q^41 - q^43 + 22 * q^44 - 4 * q^46 - 16 * q^47 + 5 * q^49 + 6 * q^50 + 28 * q^52 + 4 * q^53 + 3 * q^55 - 6 * q^56 + 6 * q^58 + 8 * q^59 - 20 * q^61 - 14 * q^62 + 14 * q^64 + 39 * q^65 + 4 * q^67 + 4 * q^68 + 2 * q^70 + 4 * q^71 - 8 * q^73 - 28 * q^74 - 12 * q^76 - 7 * q^77 + 6 * q^79 - 40 * q^80 + 8 * q^82 + 6 * q^83 + 3 * q^85 + 50 * q^86 + 2 * q^88 + 20 * q^89 - 5 * q^91 - 26 * q^92 - 30 * q^94 + 11 * q^95 - 18 * q^97 + 2 * 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 $

$2$	−2.03496	−1.43893	−0.719467	−	0.694527i	$-0.755614\pi$
$2$	−2.03496	−1.43893	−0.719467	+	0.694527i	$0.755614\pi$
$3$	0	0
$3$	0	0
$4$	2.14106	1.07053
$5$	2.37597	1.06257	0.531283	−	0.847194i	$-0.321710\pi$
$5$	2.37597	1.06257	0.531283	+	0.847194i	$0.321710\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−0.287051	−0.101488
$9$	0	0
$10$	−4.83501	−1.52896
$11$	6.22816	1.87786	0.938930	−	0.344108i	$-0.111819\pi$
$11$	6.22816	1.87786	0.938930	+	0.344108i	$0.111819\pi$
$12$	0	0
$13$	−0.711126	−0.197231	−0.0986155	−	0.995126i	$-0.531441\pi$
$13$	−0.711126	−0.197231	−0.0986155	+	0.995126i	$0.531441\pi$
$14$	2.03496	0.543866
$15$	0	0
$16$	−3.69798	−0.924496
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	4.48025	1.02784	0.513920	−	0.857838i	$-0.328193\pi$
$19$	4.48025	1.02784	0.513920	+	0.857838i	$0.328193\pi$
$20$	5.08710	1.13751
$21$	0	0
$22$	−12.6740	−2.70212
$23$	0.858940	0.179101	0.0895507	−	0.995982i	$-0.471457\pi$
$23$	0.858940	0.179101	0.0895507	+	0.995982i	$0.471457\pi$
$24$	0	0
$25$	0.645243	0.129049
$26$	1.44711	0.283802
$27$	0	0
$28$	−2.14106	−0.404622
$29$	2.23088	0.414263	0.207132	−	0.978313i	$-0.433587\pi$
$29$	2.23088	0.414263	0.207132	+	0.978313i	$0.433587\pi$
$30$	0	0
$31$	−0.765088	−0.137414	−0.0687069	−	0.997637i	$-0.521887\pi$
$31$	−0.765088	−0.137414	−0.0687069	+	0.997637i	$0.521887\pi$
$32$	8.09935	1.43178
$33$	0	0
$34$	−2.03496	−0.348993
$35$	−2.37597	−0.401613
$36$	0	0
$37$	3.81783	0.627647	0.313823	−	0.949481i	$-0.398390\pi$
$37$	3.81783	0.627647	0.313823	+	0.949481i	$0.398390\pi$
$38$	−9.11713	−1.47899
$39$	0	0
$40$	−0.682025	−0.107838
$41$	1.00675	0.157229	0.0786143	−	0.996905i	$-0.474950\pi$
$41$	1.00675	0.157229	0.0786143	+	0.996905i	$0.474950\pi$
$42$	0	0
$43$	−8.66362	−1.32119	−0.660595	−	0.750742i	$-0.729696\pi$
$43$	−8.66362	−1.32119	−0.660595	+	0.750742i	$0.729696\pi$
$44$	13.3349	2.01031
$45$	0	0
$46$	−1.74791	−0.257715
$47$	−3.55139	−0.518024	−0.259012	−	0.965874i	$-0.583397\pi$
$47$	−3.55139	−0.518024	−0.259012	+	0.965874i	$0.583397\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.31304	−0.185692
$51$	0	0
$52$	−1.52256	−0.211142
$53$	−0.356369	−0.0489511	−0.0244755	−	0.999700i	$-0.507792\pi$
$53$	−0.356369	−0.0489511	−0.0244755	+	0.999700i	$0.507792\pi$
$54$	0	0
$55$	14.7979	1.99535
$56$	0.287051	0.0383588
$57$	0	0
$58$	−4.53974	−0.596097
$59$	−13.1527	−1.71233	−0.856167	−	0.516699i	$-0.827161\pi$
$59$	−13.1527	−1.71233	−0.856167	+	0.516699i	$0.827161\pi$
$60$	0	0
$61$	0.518528	0.0663907	0.0331954	−	0.999449i	$-0.489432\pi$
$61$	0.518528	0.0663907	0.0331954	+	0.999449i	$0.489432\pi$
$62$	1.55692	0.197729
$63$	0	0
$64$	−9.08588	−1.13573
$65$	−1.68962	−0.209571
$66$	0	0
$67$	6.26644	0.765567	0.382784	−	0.923838i	$-0.374966\pi$
$67$	6.26644	0.765567	0.382784	+	0.923838i	$0.374966\pi$
$68$	2.14106	0.259642
$69$	0	0
$70$	4.83501	0.577894
$71$	11.2613	1.33647	0.668235	−	0.743950i	$-0.267050\pi$
$71$	11.2613	1.33647	0.668235	+	0.743950i	$0.267050\pi$
$72$	0	0
$73$	−5.60009	−0.655441	−0.327721	−	0.944775i	$-0.606281\pi$
$73$	−5.60009	−0.655441	−0.327721	+	0.944775i	$0.606281\pi$
$74$	−7.76912	−0.903142
$75$	0	0
$76$	9.59248	1.10033
$77$	−6.22816	−0.709764
$78$	0	0
$79$	1.57007	0.176646	0.0883231	−	0.996092i	$-0.471849\pi$
$79$	1.57007	0.176646	0.0883231	+	0.996092i	$0.471849\pi$
$80$	−8.78630	−0.982339
$81$	0	0
$82$	−2.04870	−0.226241
$83$	−5.82037	−0.638868	−0.319434	−	0.947608i	$-0.603493\pi$
$83$	−5.82037	−0.638868	−0.319434	+	0.947608i	$0.603493\pi$
$84$	0	0
$85$	2.37597	0.257710
$86$	17.6301	1.90111
$87$	0	0
$88$	−1.78780	−0.190580
$89$	10.0209	1.06221	0.531104	−	0.847307i	$-0.321777\pi$
$89$	10.0209	1.06221	0.531104	+	0.847307i	$0.321777\pi$
$90$	0	0
$91$	0.711126	0.0745463
$92$	1.83904	0.191733
$93$	0	0
$94$	7.22694	0.745402
$95$	10.6449	1.09215
$96$	0	0
$97$	−8.38640	−0.851510	−0.425755	−	0.904839i	$-0.639991\pi$
$97$	−8.38640	−0.851510	−0.425755	+	0.904839i	$0.639991\pi$
$98$	−2.03496	−0.205562
$99$	0	0
$100$	1.38150	0.138150
$101$	17.6225	1.75351	0.876753	−	0.480942i	$-0.159705\pi$
$101$	17.6225	1.75351	0.876753	+	0.480942i	$0.159705\pi$
$102$	0	0
$103$	14.2425	1.40336	0.701678	−	0.712494i	$-0.252435\pi$
$103$	14.2425	1.40336	0.701678	+	0.712494i	$0.252435\pi$
$104$	0.204130	0.0200165
$105$	0	0
$106$	0.725197	0.0704374
$107$	14.0672	1.35993	0.679964	−	0.733246i	$-0.261995\pi$
$107$	14.0672	1.35993	0.679964	+	0.733246i	$0.261995\pi$
$108$	0	0
$109$	7.35607	0.704584	0.352292	−	0.935890i	$-0.385402\pi$
$109$	7.35607	0.704584	0.352292	+	0.935890i	$0.385402\pi$
$110$	−30.1132	−2.87118
$111$	0	0
$112$	3.69798	0.349426
$113$	8.31159	0.781888	0.390944	−	0.920414i	$-0.372149\pi$
$113$	8.31159	0.781888	0.390944	+	0.920414i	$0.372149\pi$
$114$	0	0
$115$	2.04082	0.190307
$116$	4.77644	0.443481
$117$	0	0
$118$	26.7652	2.46393
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	27.7900	2.52636
$122$	−1.05518	−0.0955318
$123$	0	0
$124$	−1.63810	−0.147106
$125$	−10.3468	−0.925444
$126$	0	0
$127$	−17.8451	−1.58350	−0.791750	−	0.610845i	$-0.790830\pi$
$127$	−17.8451	−1.58350	−0.791750	+	0.610845i	$0.790830\pi$
$128$	2.29070	0.202471
$129$	0	0
$130$	3.43830	0.301559
$131$	13.8583	1.21080	0.605402	−	0.795920i	$-0.293012\pi$
$131$	13.8583	1.21080	0.605402	+	0.795920i	$0.293012\pi$
$132$	0	0
$133$	−4.48025	−0.388487
$134$	−12.7519	−1.10160
$135$	0	0
$136$	−0.287051	−0.0246144
$137$	−6.92360	−0.591523	−0.295762	−	0.955262i	$-0.595573\pi$
$137$	−6.92360	−0.591523	−0.295762	+	0.955262i	$0.595573\pi$
$138$	0	0
$139$	16.9126	1.43451	0.717255	−	0.696810i	$-0.245398\pi$
$139$	16.9126	1.43451	0.717255	+	0.696810i	$0.245398\pi$
$140$	−5.08710	−0.429938
$141$	0	0
$142$	−22.9163	−1.92309
$143$	−4.42901	−0.370372
$144$	0	0
$145$	5.30050	0.440183
$146$	11.3960	0.943137
$147$	0	0
$148$	8.17420	0.671915
$149$	9.21370	0.754815	0.377408	−	0.926047i	$-0.376816\pi$
$149$	9.21370	0.754815	0.377408	+	0.926047i	$0.376816\pi$
$150$	0	0
$151$	−18.8731	−1.53587	−0.767936	−	0.640527i	$-0.778716\pi$
$151$	−18.8731	−1.53587	−0.767936	+	0.640527i	$0.778716\pi$
$152$	−1.28606	−0.104313
$153$	0	0
$154$	12.6740	1.02130
$155$	−1.81783	−0.146011
$156$	0	0
$157$	9.47875	0.756487	0.378243	−	0.925706i	$-0.376528\pi$
$157$	9.47875	0.756487	0.378243	+	0.925706i	$0.376528\pi$
$158$	−3.19502	−0.254182
$159$	0	0
$160$	19.2438	1.52136
$161$	−0.858940	−0.0676940
$162$	0	0
$163$	17.3744	1.36087	0.680433	−	0.732810i	$-0.261792\pi$
$163$	17.3744	1.36087	0.680433	+	0.732810i	$0.261792\pi$
$164$	2.15552	0.168318
$165$	0	0
$166$	11.8442	0.919289
$167$	−11.3551	−0.878688	−0.439344	−	0.898319i	$-0.644789\pi$
$167$	−11.3551	−0.878688	−0.439344	+	0.898319i	$0.644789\pi$
$168$	0	0
$169$	−12.4943	−0.961100
$170$	−4.83501	−0.370828
$171$	0	0
$172$	−18.5493	−1.41437
$173$	6.08034	0.462280	0.231140	−	0.972921i	$-0.425754\pi$
$173$	6.08034	0.462280	0.231140	+	0.972921i	$0.425754\pi$
$174$	0	0
$175$	−0.645243	−0.0487758
$176$	−23.0316	−1.73607
$177$	0	0
$178$	−20.3920	−1.52845
$179$	−6.81379	−0.509287	−0.254643	−	0.967035i	$-0.581958\pi$
$179$	−6.81379	−0.509287	−0.254643	+	0.967035i	$0.581958\pi$
$180$	0	0
$181$	16.9880	1.26271	0.631354	−	0.775495i	$-0.282500\pi$
$181$	16.9880	1.26271	0.631354	+	0.775495i	$0.282500\pi$
$182$	−1.44711	−0.107267
$183$	0	0
$184$	−0.246560	−0.0181766
$185$	9.07105	0.666917
$186$	0	0
$187$	6.22816	0.455448
$188$	−7.60374	−0.554560
$189$	0	0
$190$	−21.6620	−1.57153
$191$	−17.5947	−1.27310	−0.636552	−	0.771233i	$-0.719640\pi$
$191$	−17.5947	−1.27310	−0.636552	+	0.771233i	$0.719640\pi$
$192$	0	0
$193$	−12.9550	−0.932519	−0.466259	−	0.884648i	$-0.654399\pi$
$193$	−12.9550	−0.932519	−0.466259	+	0.884648i	$0.654399\pi$
$194$	17.0660	1.22527
$195$	0	0
$196$	2.14106	0.152933
$197$	9.90182	0.705475	0.352738	−	0.935722i	$-0.385251\pi$
$197$	9.90182	0.705475	0.352738	+	0.935722i	$0.385251\pi$
$198$	0	0
$199$	−25.3350	−1.79595	−0.897974	−	0.440049i	$-0.854961\pi$
$199$	−25.3350	−1.79595	−0.897974	+	0.440049i	$0.854961\pi$
$200$	−0.185218	−0.0130969
$201$	0	0
$202$	−35.8611	−2.52318
$203$	−2.23088	−0.156577
$204$	0	0
$205$	2.39202	0.167066
$206$	−28.9829	−2.01934
$207$	0	0
$208$	2.62973	0.182339
$209$	27.9037	1.93014
$210$	0	0
$211$	−3.07939	−0.211994	−0.105997	−	0.994366i	$-0.533803\pi$
$211$	−3.07939	−0.211994	−0.105997	+	0.994366i	$0.533803\pi$
$212$	−0.763008	−0.0524036
$213$	0	0
$214$	−28.6262	−1.95685
$215$	−20.5845	−1.40385
$216$	0	0
$217$	0.765088	0.0519375
$218$	−14.9693	−1.01385
$219$	0	0
$220$	31.6833	2.13608
$221$	−0.711126	−0.0478355
$222$	0	0
$223$	−23.7888	−1.59302	−0.796509	−	0.604627i	$-0.793322\pi$
$223$	−23.7888	−1.59302	−0.796509	+	0.604627i	$0.793322\pi$
$224$	−8.09935	−0.541160
$225$	0	0
$226$	−16.9137	−1.12509
$227$	14.3094	0.949750	0.474875	−	0.880053i	$-0.342493\pi$
$227$	14.3094	0.949750	0.474875	+	0.880053i	$0.342493\pi$
$228$	0	0
$229$	−12.6854	−0.838275	−0.419137	−	0.907923i	$-0.637667\pi$
$229$	−12.6854	−0.838275	−0.419137	+	0.907923i	$0.637667\pi$
$230$	−4.15298	−0.273839
$231$	0	0
$232$	−0.640375	−0.0420427
$233$	6.03463	0.395342	0.197671	−	0.980268i	$-0.436662\pi$
$233$	6.03463	0.395342	0.197671	+	0.980268i	$0.436662\pi$
$234$	0	0
$235$	−8.43800	−0.550435
$236$	−28.1607	−1.83310
$237$	0	0
$238$	2.03496	0.131907
$239$	−24.6686	−1.59568	−0.797840	−	0.602869i	$-0.794024\pi$
$239$	−24.6686	−1.59568	−0.797840	+	0.602869i	$0.794024\pi$
$240$	0	0
$241$	−0.00986274	−0.000635315 0	−0.000317657	−	1.00000i	$-0.500101\pi$
$241$	−0.00986274	−0.000635315 0	−0.000317657		1.00000i	$0.500101\pi$
$242$	−56.5514	−3.63526
$243$	0	0
$244$	1.11020	0.0710733
$245$	2.37597	0.151795
$246$	0	0
$247$	−3.18602	−0.202722
$248$	0.219619	0.0139458
$249$	0	0
$250$	21.0553	1.33165
$251$	10.7253	0.676974	0.338487	−	0.940971i	$-0.390085\pi$
$251$	10.7253	0.676974	0.338487	+	0.940971i	$0.390085\pi$
$252$	0	0
$253$	5.34962	0.336327
$254$	36.3141	2.27855
$255$	0	0
$256$	13.5103	0.844392
$257$	2.94736	0.183851	0.0919255	−	0.995766i	$-0.470698\pi$
$257$	2.94736	0.183851	0.0919255	+	0.995766i	$0.470698\pi$
$258$	0	0
$259$	−3.81783	−0.237228
$260$	−3.61757	−0.224352
$261$	0	0
$262$	−28.2010	−1.74227
$263$	−9.36525	−0.577486	−0.288743	−	0.957407i	$-0.593237\pi$
$263$	−9.36525	−0.577486	−0.288743	+	0.957407i	$0.593237\pi$
$264$	0	0
$265$	−0.846723	−0.0520138
$266$	9.11713	0.559007
$267$	0	0
$268$	13.4168	0.819562
$269$	−29.7427	−1.81345	−0.906724	−	0.421725i	$-0.861425\pi$
$269$	−29.7427	−1.81345	−0.906724	+	0.421725i	$0.861425\pi$
$270$	0	0
$271$	0.211638	0.0128561	0.00642805	−	0.999979i	$-0.497954\pi$
$271$	0.211638	0.0128561	0.00642805	+	0.999979i	$0.497954\pi$
$272$	−3.69798	−0.224223
$273$	0	0
$274$	14.0892	0.851163
$275$	4.01868	0.242335
$276$	0	0
$277$	31.8731	1.91507	0.957535	−	0.288316i	$-0.0930952\pi$
$277$	31.8731	1.91507	0.957535	+	0.288316i	$0.0930952\pi$
$278$	−34.4165	−2.06417
$279$	0	0
$280$	0.682025	0.0407588
$281$	24.0209	1.43296	0.716482	−	0.697606i	$-0.245751\pi$
$281$	24.0209	1.43296	0.716482	+	0.697606i	$0.245751\pi$
$282$	0	0
$283$	−4.28176	−0.254524	−0.127262	−	0.991869i	$-0.540619\pi$
$283$	−4.28176	−0.254524	−0.127262	+	0.991869i	$0.540619\pi$
$284$	24.1111	1.43073
$285$	0	0
$286$	9.01285	0.532941
$287$	−1.00675	−0.0594268
$288$	0	0
$289$	1.00000	0.0588235
$290$	−10.7863	−0.633394
$291$	0	0
$292$	−11.9901	−0.701670
$293$	20.4252	1.19325	0.596626	−	0.802519i	$-0.296508\pi$
$293$	20.4252	1.19325	0.596626	+	0.802519i	$0.296508\pi$
$294$	0	0
$295$	−31.2504	−1.81947
$296$	−1.09591	−0.0636985
$297$	0	0
$298$	−18.7495	−1.08613
$299$	−0.610815	−0.0353243
$300$	0	0
$301$	8.66362	0.499363
$302$	38.4060	2.21002
$303$	0	0
$304$	−16.5679	−0.950233
$305$	1.23201	0.0705446
$306$	0	0
$307$	−26.1619	−1.49314	−0.746569	−	0.665308i	$-0.768300\pi$
$307$	−26.1619	−1.49314	−0.746569	+	0.665308i	$0.768300\pi$
$308$	−13.3349	−0.759824
$309$	0	0
$310$	3.69920	0.210101
$311$	−20.3060	−1.15145	−0.575723	−	0.817645i	$-0.695279\pi$
$311$	−20.3060	−1.15145	−0.575723	+	0.817645i	$0.695279\pi$
$312$	0	0
$313$	−6.59024	−0.372502	−0.186251	−	0.982502i	$-0.559634\pi$
$313$	−6.59024	−0.372502	−0.186251	+	0.982502i	$0.559634\pi$
$314$	−19.2889	−1.08853
$315$	0	0
$316$	3.36161	0.189105
$317$	−2.71394	−0.152430	−0.0762150	−	0.997091i	$-0.524284\pi$
$317$	−2.71394	−0.152430	−0.0762150	+	0.997091i	$0.524284\pi$
$318$	0	0
$319$	13.8943	0.777929
$320$	−21.5878	−1.20679
$321$	0	0
$322$	1.74791	0.0974071
$323$	4.48025	0.249288
$324$	0	0
$325$	−0.458849	−0.0254524
$326$	−35.3562	−1.95820
$327$	0	0
$328$	−0.288990	−0.0159568
$329$	3.55139	0.195795
$330$	0	0
$331$	29.6948	1.63217	0.816086	−	0.577930i	$-0.196139\pi$
$331$	29.6948	1.63217	0.816086	+	0.577930i	$0.196139\pi$
$332$	−12.4618	−0.683928
$333$	0	0
$334$	23.1073	1.26437
$335$	14.8889	0.813466
$336$	0	0
$337$	−16.2270	−0.883941	−0.441971	−	0.897030i	$-0.645720\pi$
$337$	−16.2270	−0.883941	−0.441971	+	0.897030i	$0.645720\pi$
$338$	25.4254	1.38296
$339$	0	0
$340$	5.08710	0.275887
$341$	−4.76509	−0.258044
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	2.48690	0.134085
$345$	0	0
$346$	−12.3733	−0.665190
$347$	−9.55393	−0.512882	−0.256441	−	0.966560i	$-0.582550\pi$
$347$	−9.55393	−0.512882	−0.256441	+	0.966560i	$0.582550\pi$
$348$	0	0
$349$	−30.5686	−1.63630	−0.818149	−	0.575006i	$-0.805000\pi$
$349$	−30.5686	−1.63630	−0.818149	+	0.575006i	$0.805000\pi$
$350$	1.31304	0.0701851
$351$	0	0
$352$	50.4440	2.68867
$353$	−7.15269	−0.380699	−0.190350	−	0.981716i	$-0.560962\pi$
$353$	−7.15269	−0.380699	−0.190350	+	0.981716i	$0.560962\pi$
$354$	0	0
$355$	26.7565	1.42009
$356$	21.4552	1.13713
$357$	0	0
$358$	13.8658	0.732830
$359$	22.6821	1.19712	0.598558	−	0.801079i	$-0.295740\pi$
$359$	22.6821	1.19712	0.598558	+	0.801079i	$0.295740\pi$
$360$	0	0
$361$	1.07264	0.0564546
$362$	−34.5699	−1.81695
$363$	0	0
$364$	1.52256	0.0798040
$365$	−13.3057	−0.696450
$366$	0	0
$367$	−37.1236	−1.93784	−0.968918	−	0.247383i	$-0.920429\pi$
$367$	−37.1236	−1.93784	−0.968918	+	0.247383i	$0.920429\pi$
$368$	−3.17635	−0.165578
$369$	0	0
$370$	−18.4592	−0.959649
$371$	0.356369	0.0185018
$372$	0	0
$373$	−24.4480	−1.26587	−0.632934	−	0.774206i	$-0.718150\pi$
$373$	−24.4480	−1.26587	−0.632934	+	0.774206i	$0.718150\pi$
$374$	−12.6740	−0.655359
$375$	0	0
$376$	1.01943	0.0525731
$377$	−1.58643	−0.0817056
$378$	0	0
$379$	−30.3773	−1.56038	−0.780189	−	0.625544i	$-0.784877\pi$
$379$	−30.3773	−1.56038	−0.780189	+	0.625544i	$0.784877\pi$
$380$	22.7915	1.16918
$381$	0	0
$382$	35.8044	1.83191
$383$	−15.8789	−0.811376	−0.405688	−	0.914012i	$-0.632968\pi$
$383$	−15.8789	−0.811376	−0.405688	+	0.914012i	$0.632968\pi$
$384$	0	0
$385$	−14.7979	−0.754172
$386$	26.3628	1.34183
$387$	0	0
$388$	−17.9558	−0.911567
$389$	12.1136	0.614181	0.307091	−	0.951680i	$-0.400645\pi$
$389$	12.1136	0.614181	0.307091	+	0.951680i	$0.400645\pi$
$390$	0	0
$391$	0.858940	0.0434385
$392$	−0.287051	−0.0144983
$393$	0	0
$394$	−20.1498	−1.01513
$395$	3.73043	0.187698
$396$	0	0
$397$	14.5259	0.729033	0.364516	−	0.931197i	$-0.381234\pi$
$397$	14.5259	0.729033	0.364516	+	0.931197i	$0.381234\pi$
$398$	51.5556	2.58425
$399$	0	0
$400$	−2.38610	−0.119305
$401$	17.7338	0.885586	0.442793	−	0.896624i	$-0.353988\pi$
$401$	17.7338	0.885586	0.442793	+	0.896624i	$0.353988\pi$
$402$	0	0
$403$	0.544074	0.0271023
$404$	37.7308	1.87718
$405$	0	0
$406$	4.53974	0.225304
$407$	23.7780	1.17863
$408$	0	0
$409$	33.5126	1.65709	0.828546	−	0.559921i	$-0.189169\pi$
$409$	33.5126	1.65709	0.828546	+	0.559921i	$0.189169\pi$
$410$	−4.86766	−0.240397
$411$	0	0
$412$	30.4941	1.50233
$413$	13.1527	0.647201
$414$	0	0
$415$	−13.8290	−0.678840
$416$	−5.75966	−0.282390
$417$	0	0
$418$	−56.7829	−2.77734
$419$	11.5650	0.564987	0.282493	−	0.959269i	$-0.408838\pi$
$419$	11.5650	0.564987	0.282493	+	0.959269i	$0.408838\pi$
$420$	0	0
$421$	4.39396	0.214149	0.107074	−	0.994251i	$-0.465852\pi$
$421$	4.39396	0.214149	0.107074	+	0.994251i	$0.465852\pi$
$422$	6.26644	0.305045
$423$	0	0
$424$	0.102296	0.00496794
$425$	0.645243	0.0312989
$426$	0	0
$427$	−0.518528	−0.0250933
$428$	30.1187	1.45584
$429$	0	0
$430$	41.8887	2.02005
$431$	−1.11062	−0.0534967	−0.0267484	−	0.999642i	$-0.508515\pi$
$431$	−1.11062	−0.0534967	−0.0267484	+	0.999642i	$0.508515\pi$
$432$	0	0
$433$	−9.09356	−0.437008	−0.218504	−	0.975836i	$-0.570118\pi$
$433$	−9.09356	−0.437008	−0.218504	+	0.975836i	$0.570118\pi$
$434$	−1.55692	−0.0747347
$435$	0	0
$436$	15.7498	0.754278
$437$	3.84827	0.184088
$438$	0	0
$439$	22.6990	1.08336	0.541682	−	0.840583i	$-0.317788\pi$
$439$	22.6990	1.08336	0.541682	+	0.840583i	$0.317788\pi$
$440$	−4.24776	−0.202504
$441$	0	0
$442$	1.44711	0.0688321
$443$	−30.1883	−1.43429	−0.717144	−	0.696925i	$-0.754551\pi$
$443$	−30.1883	−1.43429	−0.717144	+	0.696925i	$0.754551\pi$
$444$	0	0
$445$	23.8093	1.12867
$446$	48.4093	2.29225
$447$	0	0
$448$	9.08588	0.429267
$449$	27.4391	1.29493	0.647466	−	0.762094i	$-0.275829\pi$
$449$	27.4391	1.29493	0.647466	+	0.762094i	$0.275829\pi$
$450$	0	0
$451$	6.27022	0.295253
$452$	17.7956	0.837035
$453$	0	0
$454$	−29.1191	−1.36663
$455$	1.68962	0.0792104
$456$	0	0
$457$	−12.7424	−0.596062	−0.298031	−	0.954556i	$-0.596330\pi$
$457$	−12.7424	−0.596062	−0.298031	+	0.954556i	$0.596330\pi$
$458$	25.8143	1.20622
$459$	0	0
$460$	4.36951	0.203730
$461$	−10.6909	−0.497926	−0.248963	−	0.968513i	$-0.580090\pi$
$461$	−10.6909	−0.497926	−0.248963	+	0.968513i	$0.580090\pi$
$462$	0	0
$463$	14.3352	0.666211	0.333106	−	0.942890i	$-0.391903\pi$
$463$	14.3352	0.666211	0.333106	+	0.942890i	$0.391903\pi$
$464$	−8.24974	−0.382985
$465$	0	0
$466$	−12.2802	−0.568871
$467$	−41.3800	−1.91484	−0.957419	−	0.288701i	$-0.906777\pi$
$467$	−41.3800	−1.91484	−0.957419	+	0.288701i	$0.906777\pi$
$468$	0	0
$469$	−6.26644	−0.289357
$470$	17.1710	0.792039
$471$	0	0
$472$	3.77549	0.173781
$473$	−53.9584	−2.48101
$474$	0	0
$475$	2.89085	0.132641
$476$	−2.14106	−0.0981353
$477$	0	0
$478$	50.1996	2.29608
$479$	18.0800	0.826097	0.413049	−	0.910709i	$-0.364464\pi$
$479$	18.0800	0.826097	0.413049	+	0.910709i	$0.364464\pi$
$480$	0	0
$481$	−2.71496	−0.123791
$482$	0.0200703	0.000914176 0
$483$	0	0
$484$	59.5000	2.70454
$485$	−19.9258	−0.904786
$486$	0	0
$487$	−6.49253	−0.294205	−0.147102	−	0.989121i	$-0.546995\pi$
$487$	−6.49253	−0.294205	−0.147102	+	0.989121i	$0.546995\pi$
$488$	−0.148844	−0.00673785
$489$	0	0
$490$	−4.83501	−0.218423
$491$	5.97550	0.269671	0.134835	−	0.990868i	$-0.456949\pi$
$491$	5.97550	0.269671	0.134835	+	0.990868i	$0.456949\pi$
$492$	0	0
$493$	2.23088	0.100474
$494$	6.48343	0.291703
$495$	0	0
$496$	2.82928	0.127038
$497$	−11.2613	−0.505138
$498$	0	0
$499$	−44.2238	−1.97973	−0.989865	−	0.142009i	$-0.954644\pi$
$499$	−44.2238	−1.97973	−0.989865	+	0.142009i	$0.954644\pi$
$500$	−22.1531	−0.990716
$501$	0	0
$502$	−21.8255	−0.974121
$503$	−26.2355	−1.16979	−0.584893	−	0.811111i	$-0.698863\pi$
$503$	−26.2355	−1.16979	−0.584893	+	0.811111i	$0.698863\pi$
$504$	0	0
$505$	41.8706	1.86322
$506$	−10.8862	−0.483953
$507$	0	0
$508$	−38.2075	−1.69518
$509$	−21.3451	−0.946105	−0.473053	−	0.881034i	$-0.656848\pi$
$509$	−21.3451	−0.946105	−0.473053	+	0.881034i	$0.656848\pi$
$510$	0	0
$511$	5.60009	0.247734
$512$	−32.0743	−1.41750
$513$	0	0
$514$	−5.99775	−0.264549
$515$	33.8398	1.49116
$516$	0	0
$517$	−22.1186	−0.972776
$518$	7.76912	0.341356
$519$	0	0
$520$	0.485006	0.0212689
$521$	7.53663	0.330186	0.165093	−	0.986278i	$-0.447208\pi$
$521$	7.53663	0.330186	0.165093	+	0.986278i	$0.447208\pi$
$522$	0	0
$523$	6.46618	0.282746	0.141373	−	0.989956i	$-0.454848\pi$
$523$	6.46618	0.282746	0.141373	+	0.989956i	$0.454848\pi$
$524$	29.6714	1.29620
$525$	0	0
$526$	19.0579	0.830964
$527$	−0.765088	−0.0333277
$528$	0	0
$529$	−22.2622	−0.967923
$530$	1.72305	0.0748444
$531$	0	0
$532$	−9.59248	−0.415887
$533$	−0.715929	−0.0310103
$534$	0	0
$535$	33.4233	1.44501
$536$	−1.79879	−0.0776958
$537$	0	0
$538$	60.5253	2.60943
$539$	6.22816	0.268266
$540$	0	0
$541$	28.4060	1.22127	0.610635	−	0.791912i	$-0.290914\pi$
$541$	28.4060	1.22127	0.610635	+	0.791912i	$0.290914\pi$
$542$	−0.430675	−0.0184991
$543$	0	0
$544$	8.09935	0.347257
$545$	17.4778	0.748668
$546$	0	0
$547$	12.7440	0.544893	0.272446	−	0.962171i	$-0.412167\pi$
$547$	12.7440	0.544893	0.272446	+	0.962171i	$0.412167\pi$
$548$	−14.8238	−0.633243
$549$	0	0
$550$	−8.17784	−0.348704
$551$	9.99488	0.425796
$552$	0	0
$553$	−1.57007	−0.0667660
$554$	−64.8605	−2.75566
$555$	0	0
$556$	36.2110	1.53569
$557$	24.1926	1.02508	0.512538	−	0.858665i	$-0.328705\pi$
$557$	24.1926	1.02508	0.512538	+	0.858665i	$0.328705\pi$
$558$	0	0
$559$	6.16093	0.260580
$560$	8.78630	0.371289
$561$	0	0
$562$	−48.8815	−2.06194
$563$	27.8074	1.17194	0.585971	−	0.810332i	$-0.300713\pi$
$563$	27.8074	1.17194	0.585971	+	0.810332i	$0.300713\pi$
$564$	0	0
$565$	19.7481	0.830809
$566$	8.71320	0.366243
$567$	0	0
$568$	−3.23257	−0.135635
$569$	−11.3609	−0.476273	−0.238136	−	0.971232i	$-0.576537\pi$
$569$	−11.3609	−0.476273	−0.238136	+	0.971232i	$0.576537\pi$
$570$	0	0
$571$	32.9047	1.37702	0.688509	−	0.725227i	$-0.258265\pi$
$571$	32.9047	1.37702	0.688509	+	0.725227i	$0.258265\pi$
$572$	−9.48277	−0.396494
$573$	0	0
$574$	2.04870	0.0855112
$575$	0.554225	0.0231128
$576$	0	0
$577$	10.1812	0.423851	0.211925	−	0.977286i	$-0.432027\pi$
$577$	10.1812	0.423851	0.211925	+	0.977286i	$0.432027\pi$
$578$	−2.03496	−0.0846432
$579$	0	0
$580$	11.3487	0.471229
$581$	5.82037	0.241469
$582$	0	0
$583$	−2.21952	−0.0919233
$584$	1.60751	0.0665194
$585$	0	0
$586$	−41.5644	−1.71701
$587$	−41.2839	−1.70397	−0.851984	−	0.523568i	$-0.824600\pi$
$587$	−41.2839	−1.70397	−0.851984	+	0.523568i	$0.824600\pi$
$588$	0	0
$589$	−3.42778	−0.141239
$590$	63.5933	2.61810
$591$	0	0
$592$	−14.1183	−0.580257
$593$	13.3060	0.546413	0.273207	−	0.961955i	$-0.411916\pi$
$593$	13.3060	0.546413	0.273207	+	0.961955i	$0.411916\pi$
$594$	0	0
$595$	−2.37597	−0.0974054
$596$	19.7271	0.808053
$597$	0	0
$598$	1.24298	0.0508294
$599$	−13.3174	−0.544134	−0.272067	−	0.962278i	$-0.587707\pi$
$599$	−13.3174	−0.544134	−0.272067	+	0.962278i	$0.587707\pi$
$600$	0	0
$601$	−19.9280	−0.812880	−0.406440	−	0.913677i	$-0.633230\pi$
$601$	−19.9280	−0.812880	−0.406440	+	0.913677i	$0.633230\pi$
$602$	−17.6301	−0.718550
$603$	0	0
$604$	−40.4084	−1.64420
$605$	66.0282	2.68443
$606$	0	0
$607$	9.95250	0.403959	0.201980	−	0.979390i	$-0.435262\pi$
$607$	9.95250	0.403959	0.201980	+	0.979390i	$0.435262\pi$
$608$	36.2871	1.47164
$609$	0	0
$610$	−2.50709	−0.101509
$611$	2.52549	0.102170
$612$	0	0
$613$	28.1459	1.13680	0.568402	−	0.822751i	$-0.307562\pi$
$613$	28.1459	1.13680	0.568402	+	0.822751i	$0.307562\pi$
$614$	53.2384	2.14853
$615$	0	0
$616$	1.78780	0.0720325
$617$	32.4338	1.30573	0.652867	−	0.757473i	$-0.273566\pi$
$617$	32.4338	1.30573	0.652867	+	0.757473i	$0.273566\pi$
$618$	0	0
$619$	−25.1743	−1.01184	−0.505920	−	0.862581i	$-0.668847\pi$
$619$	−25.1743	−1.01184	−0.505920	+	0.862581i	$0.668847\pi$
$620$	−3.89208	−0.156310
$621$	0	0
$622$	41.3218	1.65685
$623$	−10.0209	−0.401477
$624$	0	0
$625$	−27.8099	−1.11240
$626$	13.4109	0.536006
$627$	0	0
$628$	20.2946	0.809842
$629$	3.81783	0.152227
$630$	0	0
$631$	−1.11754	−0.0444885	−0.0222442	−	0.999753i	$-0.507081\pi$
$631$	−1.11754	−0.0444885	−0.0222442	+	0.999753i	$0.507081\pi$
$632$	−0.450689	−0.0179275
$633$	0	0
$634$	5.52276	0.219337
$635$	−42.3995	−1.68257
$636$	0	0
$637$	−0.711126	−0.0281758
$638$	−28.2742	−1.11939
$639$	0	0
$640$	5.44263	0.215139
$641$	37.7746	1.49201	0.746004	−	0.665941i	$-0.231970\pi$
$641$	37.7746	1.49201	0.746004	+	0.665941i	$0.231970\pi$
$642$	0	0
$643$	7.32969	0.289055	0.144527	−	0.989501i	$-0.453834\pi$
$643$	7.32969	0.289055	0.144527	+	0.989501i	$0.453834\pi$
$644$	−1.83904	−0.0724684
$645$	0	0
$646$	−9.11713	−0.358709
$647$	−34.4612	−1.35481	−0.677404	−	0.735611i	$-0.736895\pi$
$647$	−34.4612	−1.35481	−0.677404	+	0.735611i	$0.736895\pi$
$648$	0	0
$649$	−81.9170	−3.21552
$650$	0.933740	0.0366243
$651$	0	0
$652$	37.1996	1.45685
$653$	8.24803	0.322771	0.161385	−	0.986891i	$-0.448404\pi$
$653$	8.24803	0.322771	0.161385	+	0.986891i	$0.448404\pi$
$654$	0	0
$655$	32.9269	1.28656
$656$	−3.72296	−0.145357
$657$	0	0
$658$	−7.22694	−0.281735
$659$	21.7000	0.845314	0.422657	−	0.906290i	$-0.361098\pi$
$659$	21.7000	0.845314	0.422657	+	0.906290i	$0.361098\pi$
$660$	0	0
$661$	−22.5845	−0.878436	−0.439218	−	0.898380i	$-0.644744\pi$
$661$	−22.5845	−0.878436	−0.439218	+	0.898380i	$0.644744\pi$
$662$	−60.4277	−2.34859
$663$	0	0
$664$	1.67074	0.0648374
$665$	−10.6449	−0.412793
$666$	0	0
$667$	1.91619	0.0741952
$668$	−24.3120	−0.940661
$669$	0	0
$670$	−30.2983	−1.17052
$671$	3.22948	0.124673
$672$	0	0
$673$	14.1344	0.544841	0.272421	−	0.962178i	$-0.412176\pi$
$673$	14.1344	0.544841	0.272421	+	0.962178i	$0.412176\pi$
$674$	33.0213	1.27193
$675$	0	0
$676$	−26.7510	−1.02889
$677$	−38.8982	−1.49498	−0.747490	−	0.664273i	$-0.768741\pi$
$677$	−38.8982	−1.49498	−0.747490	+	0.664273i	$0.768741\pi$
$678$	0	0
$679$	8.38640	0.321840
$680$	−0.682025	−0.0261545
$681$	0	0
$682$	9.69676	0.371308
$683$	−16.2099	−0.620256	−0.310128	−	0.950695i	$-0.600372\pi$
$683$	−16.2099	−0.620256	−0.310128	+	0.950695i	$0.600372\pi$
$684$	0	0
$685$	−16.4503	−0.628533
$686$	2.03496	0.0776951
$687$	0	0
$688$	32.0379	1.22143
$689$	0.253424	0.00965467
$690$	0	0
$691$	45.6349	1.73603	0.868017	−	0.496534i	$-0.165394\pi$
$691$	45.6349	1.73603	0.868017	+	0.496534i	$0.165394\pi$
$692$	13.0184	0.494885
$693$	0	0
$694$	19.4419	0.738003
$695$	40.1839	1.52426
$696$	0	0
$697$	1.00675	0.0381335
$698$	62.2058	2.35452
$699$	0	0
$700$	−1.38150	−0.0522159
$701$	−15.3825	−0.580987	−0.290494	−	0.956877i	$-0.593820\pi$
$701$	−15.3825	−0.580987	−0.290494	+	0.956877i	$0.593820\pi$
$702$	0	0
$703$	17.1048	0.645120
$704$	−56.5883	−2.13275
$705$	0	0
$706$	14.5554	0.547801
$707$	−17.6225	−0.662763
$708$	0	0
$709$	−33.1206	−1.24387	−0.621935	−	0.783069i	$-0.713653\pi$
$709$	−33.1206	−1.24387	−0.621935	+	0.783069i	$0.713653\pi$
$710$	−54.4484	−2.04341
$711$	0	0
$712$	−2.87650	−0.107801
$713$	−0.657165	−0.0246110
$714$	0	0
$715$	−10.5232	−0.393545
$716$	−14.5887	−0.545207
$717$	0	0
$718$	−46.1572	−1.72257
$719$	−6.33055	−0.236090	−0.118045	−	0.993008i	$-0.537663\pi$
$719$	−6.33055	−0.236090	−0.118045	+	0.993008i	$0.537663\pi$
$720$	0	0
$721$	−14.2425	−0.530419
$722$	−2.18277	−0.0812344
$723$	0	0
$724$	36.3723	1.35177
$725$	1.43946	0.0534601
$726$	0	0
$727$	−43.7632	−1.62309	−0.811543	−	0.584292i	$-0.801372\pi$
$727$	−43.7632	−1.62309	−0.811543	+	0.584292i	$0.801372\pi$
$728$	−0.204130	−0.00756554
$729$	0	0
$730$	27.0765	1.00215
$731$	−8.66362	−0.320436
$732$	0	0
$733$	−49.8143	−1.83993	−0.919967	−	0.391995i	$-0.871785\pi$
$733$	−49.8143	−1.83993	−0.919967	+	0.391995i	$0.871785\pi$
$734$	75.5450	2.78842
$735$	0	0
$736$	6.95685	0.256433
$737$	39.0284	1.43763
$738$	0	0
$739$	−9.85389	−0.362481	−0.181241	−	0.983439i	$-0.558011\pi$
$739$	−9.85389	−0.362481	−0.181241	+	0.983439i	$0.558011\pi$
$740$	19.4217	0.713955
$741$	0	0
$742$	−0.725197	−0.0266228
$743$	−23.5796	−0.865053	−0.432526	−	0.901621i	$-0.642378\pi$
$743$	−23.5796	−0.865053	−0.432526	+	0.901621i	$0.642378\pi$
$744$	0	0
$745$	21.8915	0.802042
$746$	49.7506	1.82150
$747$	0	0
$748$	13.3349	0.487571
$749$	−14.0672	−0.514004
$750$	0	0
$751$	15.2642	0.556997	0.278498	−	0.960437i	$-0.410163\pi$
$751$	15.2642	0.556997	0.278498	+	0.960437i	$0.410163\pi$
$752$	13.1330	0.478911
$753$	0	0
$754$	3.22833	0.117569
$755$	−44.8420	−1.63197
$756$	0	0
$757$	−14.4107	−0.523766	−0.261883	−	0.965100i	$-0.584343\pi$
$757$	−14.4107	−0.523766	−0.261883	+	0.965100i	$0.584343\pi$
$758$	61.8166	2.24528
$759$	0	0
$760$	−3.05564	−0.110840
$761$	6.52689	0.236600	0.118300	−	0.992978i	$-0.462256\pi$
$761$	6.52689	0.236600	0.118300	+	0.992978i	$0.462256\pi$
$762$	0	0
$763$	−7.35607	−0.266308
$764$	−37.6712	−1.36290
$765$	0	0
$766$	32.3130	1.16752
$767$	9.35322	0.337725
$768$	0	0
$769$	−47.7238	−1.72096	−0.860482	−	0.509481i	$-0.829838\pi$
$769$	−47.7238	−1.72096	−0.860482	+	0.509481i	$0.829838\pi$
$770$	30.1132	1.08520
$771$	0	0
$772$	−27.7374	−0.998289
$773$	6.87624	0.247321	0.123661	−	0.992325i	$-0.460537\pi$
$773$	6.87624	0.247321	0.123661	+	0.992325i	$0.460537\pi$
$774$	0	0
$775$	−0.493668	−0.0177331
$776$	2.40732	0.0864179
$777$	0	0
$778$	−24.6506	−0.883766
$779$	4.51051	0.161606
$780$	0	0
$781$	70.1371	2.50970
$782$	−1.74791	−0.0625051
$783$	0	0
$784$	−3.69798	−0.132071
$785$	22.5213	0.803818
$786$	0	0
$787$	4.15484	0.148104	0.0740519	−	0.997254i	$-0.476407\pi$
$787$	4.15484	0.148104	0.0740519	+	0.997254i	$0.476407\pi$
$788$	21.2004	0.755232
$789$	0	0
$790$	−7.59128	−0.270086
$791$	−8.31159	−0.295526
$792$	0	0
$793$	−0.368739	−0.0130943
$794$	−29.5596	−1.04903
$795$	0	0
$796$	−54.2437	−1.92262
$797$	3.31075	0.117273	0.0586363	−	0.998279i	$-0.481325\pi$
$797$	3.31075	0.117273	0.0586363	+	0.998279i	$0.481325\pi$
$798$	0	0
$799$	−3.55139	−0.125639
$800$	5.22605	0.184769
$801$	0	0
$802$	−36.0876	−1.27430
$803$	−34.8783	−1.23083
$804$	0	0
$805$	−2.04082	−0.0719294
$806$	−1.10717	−0.0389983
$807$	0	0
$808$	−5.05856	−0.177960
$809$	−2.64325	−0.0929316	−0.0464658	−	0.998920i	$-0.514796\pi$
$809$	−2.64325	−0.0929316	−0.0464658	+	0.998920i	$0.514796\pi$
$810$	0	0
$811$	46.1370	1.62009	0.810045	−	0.586367i	$-0.199443\pi$
$811$	46.1370	1.62009	0.810045	+	0.586367i	$0.199443\pi$
$812$	−4.77644	−0.167620
$813$	0	0
$814$	−48.3873	−1.69598
$815$	41.2811	1.44601
$816$	0	0
$817$	−38.8152	−1.35797
$818$	−68.1968	−2.38445
$819$	0	0
$820$	5.12146	0.178849
$821$	−44.2923	−1.54581	−0.772906	−	0.634520i	$-0.781198\pi$
$821$	−44.2923	−1.54581	−0.772906	+	0.634520i	$0.781198\pi$
$822$	0	0
$823$	−19.3480	−0.674429	−0.337214	−	0.941428i	$-0.609485\pi$
$823$	−19.3480	−0.674429	−0.337214	+	0.941428i	$0.609485\pi$
$824$	−4.08833	−0.142424
$825$	0	0
$826$	−26.7652	−0.931280
$827$	−9.90983	−0.344598	−0.172299	−	0.985045i	$-0.555120\pi$
$827$	−9.90983	−0.344598	−0.172299	+	0.985045i	$0.555120\pi$
$828$	0	0
$829$	1.92888	0.0669928	0.0334964	−	0.999439i	$-0.489336\pi$
$829$	1.92888	0.0669928	0.0334964	+	0.999439i	$0.489336\pi$
$830$	28.1415	0.976806
$831$	0	0
$832$	6.46120	0.224002
$833$	1.00000	0.0346479
$834$	0	0
$835$	−26.9795	−0.933665
$836$	59.7435	2.06627
$837$	0	0
$838$	−23.5343	−0.812979
$839$	−23.5048	−0.811475	−0.405737	−	0.913990i	$-0.632985\pi$
$839$	−23.5048	−0.811475	−0.405737	+	0.913990i	$0.632985\pi$
$840$	0	0
$841$	−24.0232	−0.828386
$842$	−8.94154	−0.308146
$843$	0	0
$844$	−6.59316	−0.226946
$845$	−29.6861	−1.02123
$846$	0	0
$847$	−27.7900	−0.954874
$848$	1.31785	0.0452551
$849$	0	0
$850$	−1.31304	−0.0450370
$851$	3.27929	0.112412
$852$	0	0
$853$	−49.8274	−1.70606	−0.853029	−	0.521863i	$-0.825237\pi$
$853$	−49.8274	−1.70606	−0.853029	+	0.521863i	$0.825237\pi$
$854$	1.05518	0.0361076
$855$	0	0
$856$	−4.03800	−0.138016
$857$	−11.3370	−0.387266	−0.193633	−	0.981074i	$-0.562027\pi$
$857$	−11.3370	−0.387266	−0.193633	+	0.981074i	$0.562027\pi$
$858$	0	0
$859$	−25.0407	−0.854378	−0.427189	−	0.904162i	$-0.640496\pi$
$859$	−25.0407	−0.854378	−0.427189	+	0.904162i	$0.640496\pi$
$860$	−44.0727	−1.50287
$861$	0	0
$862$	2.26007	0.0769782
$863$	−47.6479	−1.62195	−0.810976	−	0.585079i	$-0.801063\pi$
$863$	−47.6479	−1.62195	−0.810976	+	0.585079i	$0.801063\pi$
$864$	0	0
$865$	14.4467	0.491203
$866$	18.5050	0.628826
$867$	0	0
$868$	1.63810	0.0556007
$869$	9.77862	0.331717
$870$	0	0
$871$	−4.45623	−0.150994
$872$	−2.11157	−0.0715067
$873$	0	0
$874$	−7.83107	−0.264890
$875$	10.3468	0.349785
$876$	0	0
$877$	27.8933	0.941890	0.470945	−	0.882163i	$-0.343913\pi$
$877$	27.8933	0.941890	0.470945	+	0.882163i	$0.343913\pi$
$878$	−46.1916	−1.55889
$879$	0	0
$880$	−54.7225	−1.84469
$881$	−53.1557	−1.79086	−0.895431	−	0.445200i	$-0.853133\pi$
$881$	−53.1557	−1.79086	−0.895431	+	0.445200i	$0.853133\pi$
$882$	0	0
$883$	42.1837	1.41960	0.709798	−	0.704405i	$-0.248786\pi$
$883$	42.1837	1.41960	0.709798	+	0.704405i	$0.248786\pi$
$884$	−1.52256	−0.0512094
$885$	0	0
$886$	61.4319	2.06385
$887$	47.2859	1.58770	0.793852	−	0.608110i	$-0.208072\pi$
$887$	47.2859	1.58770	0.793852	+	0.608110i	$0.208072\pi$
$888$	0	0
$889$	17.8451	0.598507
$890$	−48.4509	−1.62408
$891$	0	0
$892$	−50.9333	−1.70537
$893$	−15.9111	−0.532445
$894$	0	0
$895$	−16.1894	−0.541151
$896$	−2.29070	−0.0765268
$897$	0	0
$898$	−55.8375	−1.86332
$899$	−1.70682	−0.0569255
$900$	0	0
$901$	−0.356369	−0.0118724
$902$	−12.7597	−0.424850
$903$	0	0
$904$	−2.38585	−0.0793522
$905$	40.3630	1.34171
$906$	0	0
$907$	−1.76575	−0.0586307	−0.0293153	−	0.999570i	$-0.509333\pi$
$907$	−1.76575	−0.0586307	−0.0293153	+	0.999570i	$0.509333\pi$
$908$	30.6373	1.01674
$909$	0	0
$910$	−3.43830	−0.113979
$911$	−14.2835	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$911$	−14.2835	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$912$	0	0
$913$	−36.2502	−1.19971
$914$	25.9302	0.857694
$915$	0	0
$916$	−27.1602	−0.897398
$917$	−13.8583	−0.457641
$918$	0	0
$919$	7.68006	0.253342	0.126671	−	0.991945i	$-0.459571\pi$
$919$	7.68006	0.253342	0.126671	+	0.991945i	$0.459571\pi$
$920$	−0.585819	−0.0193139
$921$	0	0
$922$	21.7556	0.716483
$923$	−8.00820	−0.263593
$924$	0	0
$925$	2.46343	0.0809970
$926$	−29.1715	−0.958634
$927$	0	0
$928$	18.0686	0.593132
$929$	3.01953	0.0990677	0.0495338	−	0.998772i	$-0.484226\pi$
$929$	3.01953	0.0990677	0.0495338	+	0.998772i	$0.484226\pi$
$930$	0	0
$931$	4.48025	0.146834
$932$	12.9205	0.423225
$933$	0	0
$934$	84.2066	2.75533
$935$	14.7979	0.483944
$936$	0	0
$937$	6.44249	0.210467	0.105233	−	0.994448i	$-0.466441\pi$
$937$	6.44249	0.210467	0.105233	+	0.994448i	$0.466441\pi$
$938$	12.7519	0.416366
$939$	0	0
$940$	−18.0663	−0.589257
$941$	−5.37139	−0.175102	−0.0875512	−	0.996160i	$-0.527904\pi$
$941$	−5.37139	−0.175102	−0.0875512	+	0.996160i	$0.527904\pi$
$942$	0	0
$943$	0.864742	0.0281599
$944$	48.6384	1.58305
$945$	0	0
$946$	109.803	3.57001
$947$	−49.6361	−1.61296	−0.806478	−	0.591263i	$-0.798629\pi$
$947$	−49.6361	−1.61296	−0.806478	+	0.591263i	$0.798629\pi$
$948$	0	0
$949$	3.98237	0.129273
$950$	−5.88276	−0.190862
$951$	0	0
$952$	0.287051	0.00930338
$953$	38.4453	1.24537	0.622683	−	0.782474i	$-0.286042\pi$
$953$	38.4453	1.24537	0.622683	+	0.782474i	$0.286042\pi$
$954$	0	0
$955$	−41.8044	−1.35276
$956$	−52.8170	−1.70822
$957$	0	0
$958$	−36.7921	−1.18870
$959$	6.92360	0.223575
$960$	0	0
$961$	−30.4146	−0.981117
$962$	5.52483	0.178128
$963$	0	0
$964$	−0.0211167	−0.000680124 0
$965$	−30.7806	−0.990864
$966$	0	0
$967$	26.1458	0.840794	0.420397	−	0.907340i	$-0.361891\pi$
$967$	26.1458	0.840794	0.420397	+	0.907340i	$0.361891\pi$
$968$	−7.97713	−0.256395
$969$	0	0
$970$	40.5483	1.30193
$971$	31.0678	0.997014	0.498507	−	0.866886i	$-0.333882\pi$
$971$	31.0678	0.997014	0.498507	+	0.866886i	$0.333882\pi$
$972$	0	0
$973$	−16.9126	−0.542194
$974$	13.2120	0.423341
$975$	0	0
$976$	−1.91751	−0.0613779
$977$	46.1920	1.47781	0.738907	−	0.673808i	$-0.235342\pi$
$977$	46.1920	1.47781	0.738907	+	0.673808i	$0.235342\pi$
$978$	0	0
$979$	62.4114	1.99468
$980$	5.08710	0.162501
$981$	0	0
$982$	−12.1599	−0.388038
$983$	−36.2383	−1.15582	−0.577911	−	0.816100i	$-0.696132\pi$
$983$	−36.2383	−1.15582	−0.577911	+	0.816100i	$0.696132\pi$
$984$	0	0
$985$	23.5264	0.749615
$986$	−4.53974	−0.144575
$987$	0	0
$988$	−6.82147	−0.217020
$989$	−7.44153	−0.236627
$990$	0	0
$991$	5.86012	0.186153	0.0930764	−	0.995659i	$-0.470330\pi$
$991$	5.86012	0.186153	0.0930764	+	0.995659i	$0.470330\pi$
$992$	−6.19671	−0.196746
$993$	0	0
$994$	22.9163	0.726860
$995$	−60.1951	−1.90831
$996$	0	0
$997$	−53.5733	−1.69668	−0.848341	−	0.529450i	$-0.822398\pi$
$997$	−53.5733	−1.69668	−0.848341	+	0.529450i	$0.822398\pi$
$998$	89.9937	2.84870
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1071.2.a.n.1.1	yes	5
3.2	odd	2		1071.2.a.l.1.5	✓	5
7.6	odd	2		7497.2.a.bu.1.1		5
21.20	even	2		7497.2.a.bq.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1071.2.a.l.1.5	✓	5	3.2	odd	2
1071.2.a.n.1.1	yes	5	1.1	even	1	trivial
7497.2.a.bq.1.5		5	21.20	even	2
7497.2.a.bu.1.1		5	7.6	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:	\( N \)	\(=\)	\( 1071 = 3^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1071.a (trivial)

\(f(q)\)	\(=\)	\(q-2.03496 q^{2} +2.14106 q^{4} +2.37597 q^{5} -1.00000 q^{7} -0.287051 q^{8} +O(q^{10})\)	\(q-2.03496 q^{2} +2.14106 q^{4} +2.37597 q^{5} -1.00000 q^{7} -0.287051 q^{8} -4.83501 q^{10} +6.22816 q^{11} -0.711126 q^{13} +2.03496 q^{14} -3.69798 q^{16} +1.00000 q^{17} +4.48025 q^{19} +5.08710 q^{20} -12.6740 q^{22} +0.858940 q^{23} +0.645243 q^{25} +1.44711 q^{26} -2.14106 q^{28} +2.23088 q^{29} -0.765088 q^{31} +8.09935 q^{32} -2.03496 q^{34} -2.37597 q^{35} +3.81783 q^{37} -9.11713 q^{38} -0.682025 q^{40} +1.00675 q^{41} -8.66362 q^{43} +13.3349 q^{44} -1.74791 q^{46} -3.55139 q^{47} +1.00000 q^{49} -1.31304 q^{50} -1.52256 q^{52} -0.356369 q^{53} +14.7979 q^{55} +0.287051 q^{56} -4.53974 q^{58} -13.1527 q^{59} +0.518528 q^{61} +1.55692 q^{62} -9.08588 q^{64} -1.68962 q^{65} +6.26644 q^{67} +2.14106 q^{68} +4.83501 q^{70} +11.2613 q^{71} -5.60009 q^{73} -7.76912 q^{74} +9.59248 q^{76} -6.22816 q^{77} +1.57007 q^{79} -8.78630 q^{80} -2.04870 q^{82} -5.82037 q^{83} +2.37597 q^{85} +17.6301 q^{86} -1.78780 q^{88} +10.0209 q^{89} +0.711126 q^{91} +1.83904 q^{92} +7.22694 q^{94} +10.6449 q^{95} -8.38640 q^{97} -2.03496 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} + 4 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8}+O(q^{10}) \) 5 * q + 2 * q^2 + 4 * q^4 + 3 * q^5 - 5 * q^7 + 6 * q^8	\( 5 q + 2 q^{2} + 4 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{8} - 2 q^{10} + 7 q^{11} + 5 q^{13} - 2 q^{14} + 10 q^{16} + 5 q^{17} + 3 q^{19} + 8 q^{20} - 6 q^{22} + 11 q^{23} + 6 q^{25} + 16 q^{26} - 4 q^{28} + 22 q^{29} - 6 q^{31} + 24 q^{32} + 2 q^{34} - 3 q^{35} - 10 q^{37} - 10 q^{38} - 10 q^{40} + 27 q^{41} - q^{43} + 22 q^{44} - 4 q^{46} - 16 q^{47} + 5 q^{49} + 6 q^{50} + 28 q^{52} + 4 q^{53} + 3 q^{55} - 6 q^{56} + 6 q^{58} + 8 q^{59} - 20 q^{61} - 14 q^{62} + 14 q^{64} + 39 q^{65} + 4 q^{67} + 4 q^{68} + 2 q^{70} + 4 q^{71} - 8 q^{73} - 28 q^{74} - 12 q^{76} - 7 q^{77} + 6 q^{79} - 40 q^{80} + 8 q^{82} + 6 q^{83} + 3 q^{85} + 50 q^{86} + 2 q^{88} + 20 q^{89} - 5 q^{91} - 26 q^{92} - 30 q^{94} + 11 q^{95} - 18 q^{97} + 2 q^{98}+O(q^{100}) \) 5 * q + 2 * q^2 + 4 * q^4 + 3 * q^5 - 5 * q^7 + 6 * q^8 - 2 * q^10 + 7 * q^11 + 5 * q^13 - 2 * q^14 + 10 * q^16 + 5 * q^17 + 3 * q^19 + 8 * q^20 - 6 * q^22 + 11 * q^23 + 6 * q^25 + 16 * q^26 - 4 * q^28 + 22 * q^29 - 6 * q^31 + 24 * q^32 + 2 * q^34 - 3 * q^35 - 10 * q^37 - 10 * q^38 - 10 * q^40 + 27 * q^41 - q^43 + 22 * q^44 - 4 * q^46 - 16 * q^47 + 5 * q^49 + 6 * q^50 + 28 * q^52 + 4 * q^53 + 3 * q^55 - 6 * q^56 + 6 * q^58 + 8 * q^59 - 20 * q^61 - 14 * q^62 + 14 * q^64 + 39 * q^65 + 4 * q^67 + 4 * q^68 + 2 * q^70 + 4 * q^71 - 8 * q^73 - 28 * q^74 - 12 * q^76 - 7 * q^77 + 6 * q^79 - 40 * q^80 + 8 * q^82 + 6 * q^83 + 3 * q^85 + 50 * q^86 + 2 * q^88 + 20 * q^89 - 5 * q^91 - 26 * q^92 - 30 * q^94 + 11 * q^95 - 18 * q^97 + 2 * q^98

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