LMFDB - Embedded newform 1143.2.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1143,2,Mod(1,1143)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1143, 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("1143.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1143 = 3^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1143.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:	$9.12690095103$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 8x^{5} + 15x^{4} + 17x^{3} - 28x^{2} - 11x + 15 $ x^7 - 2x^6 - 8x^5 + 15x^4 + 17x^3 - 28x^2 - 11x + 15
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 127)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.24403$ of defining polynomial
Character	$\chi$	$=$	1143.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.24403 q^{2} -0.452382 q^{4} +2.65960 q^{5} -1.13710 q^{7} +3.05084 q^{8} +O(q^{10})$	\(q-1.24403 q^{2} -0.452382 q^{4} +2.65960 q^{5} -1.13710 q^{7} +3.05084 q^{8} -3.30863 q^{10} -0.387885 q^{11} -3.92836 q^{13} +1.41459 q^{14} -2.89059 q^{16} -5.40139 q^{17} +0.820438 q^{19} -1.20316 q^{20} +0.482542 q^{22} -8.63338 q^{23} +2.07348 q^{25} +4.88701 q^{26} +0.514402 q^{28} +8.20007 q^{29} +7.24550 q^{31} -2.50570 q^{32} +6.71951 q^{34} -3.02422 q^{35} -7.63586 q^{37} -1.02065 q^{38} +8.11403 q^{40} -1.27186 q^{41} -4.31624 q^{43} +0.175472 q^{44} +10.7402 q^{46} +7.17025 q^{47} -5.70701 q^{49} -2.57948 q^{50} +1.77712 q^{52} -12.0735 q^{53} -1.03162 q^{55} -3.46910 q^{56} -10.2012 q^{58} +7.80904 q^{59} +3.19459 q^{61} -9.01364 q^{62} +8.89835 q^{64} -10.4479 q^{65} -6.56628 q^{67} +2.44349 q^{68} +3.76223 q^{70} -14.2950 q^{71} -0.473248 q^{73} +9.49926 q^{74} -0.371151 q^{76} +0.441063 q^{77} -5.31508 q^{79} -7.68781 q^{80} +1.58224 q^{82} +5.39344 q^{83} -14.3655 q^{85} +5.36955 q^{86} -1.18338 q^{88} -10.1340 q^{89} +4.46692 q^{91} +3.90559 q^{92} -8.92003 q^{94} +2.18204 q^{95} +6.53946 q^{97} +7.09971 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{2} + 6 q^{4} - 8 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) $ 7 * q - 2 * q^2 + 6 * q^4 - 8 * q^5 - 3 * q^7 - 3 * q^8	$ 7 q - 2 q^{2} + 6 q^{4} - 8 q^{5} - 3 q^{7} - 3 q^{8} - 5 q^{10} - q^{13} + 4 q^{14} - 8 q^{16} - 24 q^{17} - 5 q^{19} - 11 q^{20} - 9 q^{22} + q^{23} + 7 q^{25} + 4 q^{26} - 26 q^{28} + 7 q^{29} - 8 q^{31} + 2 q^{32} - q^{34} - 4 q^{35} - 6 q^{37} - 29 q^{38} - 3 q^{40} - 14 q^{41} - q^{43} + 21 q^{44} - 3 q^{46} - 25 q^{47} - 10 q^{50} + 6 q^{52} - 29 q^{53} - 23 q^{55} - 9 q^{56} - 22 q^{58} + 12 q^{59} + 7 q^{61} - 4 q^{62} - 3 q^{64} - 3 q^{65} - 25 q^{67} - 53 q^{68} + 51 q^{70} - 7 q^{71} + 13 q^{73} - 11 q^{74} + 12 q^{76} - 19 q^{77} - 23 q^{79} + 14 q^{80} + 26 q^{82} - 26 q^{83} + 15 q^{85} - 5 q^{86} + 25 q^{88} - 13 q^{89} - 40 q^{91} + 32 q^{92} - 19 q^{94} + 40 q^{95} - 5 q^{97} + 11 q^{98}+O(q^{100}) $ 7 * q - 2 * q^2 + 6 * q^4 - 8 * q^5 - 3 * q^7 - 3 * q^8 - 5 * q^10 - q^13 + 4 * q^14 - 8 * q^16 - 24 * q^17 - 5 * q^19 - 11 * q^20 - 9 * q^22 + q^23 + 7 * q^25 + 4 * q^26 - 26 * q^28 + 7 * q^29 - 8 * q^31 + 2 * q^32 - q^34 - 4 * q^35 - 6 * q^37 - 29 * q^38 - 3 * q^40 - 14 * q^41 - q^43 + 21 * q^44 - 3 * q^46 - 25 * q^47 - 10 * q^50 + 6 * q^52 - 29 * q^53 - 23 * q^55 - 9 * q^56 - 22 * q^58 + 12 * q^59 + 7 * q^61 - 4 * q^62 - 3 * q^64 - 3 * q^65 - 25 * q^67 - 53 * q^68 + 51 * q^70 - 7 * q^71 + 13 * q^73 - 11 * q^74 + 12 * q^76 - 19 * q^77 - 23 * q^79 + 14 * q^80 + 26 * q^82 - 26 * q^83 + 15 * q^85 - 5 * q^86 + 25 * q^88 - 13 * q^89 - 40 * q^91 + 32 * q^92 - 19 * q^94 + 40 * q^95 - 5 * q^97 + 11 * 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 $

$2$	−1.24403	−0.879664	−0.439832	−	0.898080i	$-0.644962\pi$
$2$	−1.24403	−0.879664	−0.439832	+	0.898080i	$0.644962\pi$
$3$	0	0
$3$	0	0
$4$	−0.452382	−0.226191
$5$	2.65960	1.18941	0.594705	−	0.803944i	$-0.297269\pi$
$5$	2.65960	1.18941	0.594705	+	0.803944i	$0.297269\pi$
$6$	0	0
$7$	−1.13710	−0.429782	−0.214891	−	0.976638i	$-0.568940\pi$
$7$	−1.13710	−0.429782	−0.214891	+	0.976638i	$0.568940\pi$
$8$	3.05084	1.07864
$9$	0	0
$10$	−3.30863	−1.04628
$11$	−0.387885	−0.116952	−0.0584759	−	0.998289i	$-0.518624\pi$
$11$	−0.387885	−0.116952	−0.0584759	+	0.998289i	$0.518624\pi$
$12$	0	0
$13$	−3.92836	−1.08953	−0.544765	−	0.838589i	$-0.683381\pi$
$13$	−3.92836	−1.08953	−0.544765	+	0.838589i	$0.683381\pi$
$14$	1.41459	0.378064
$15$	0	0
$16$	−2.89059	−0.722647
$17$	−5.40139	−1.31003	−0.655015	−	0.755616i	$-0.727338\pi$
$17$	−5.40139	−1.31003	−0.655015	+	0.755616i	$0.727338\pi$
$18$	0	0
$19$	0.820438	0.188221	0.0941107	−	0.995562i	$-0.469999\pi$
$19$	0.820438	0.188221	0.0941107	+	0.995562i	$0.469999\pi$
$20$	−1.20316	−0.269034
$21$	0	0
$22$	0.482542	0.102878
$23$	−8.63338	−1.80019	−0.900093	−	0.435699i	$-0.856501\pi$
$23$	−8.63338	−1.80019	−0.900093	+	0.435699i	$0.856501\pi$
$24$	0	0
$25$	2.07348	0.414696
$26$	4.88701	0.958421
$27$	0	0
$28$	0.514402	0.0972128
$29$	8.20007	1.52272	0.761358	−	0.648332i	$-0.224533\pi$
$29$	8.20007	1.52272	0.761358	+	0.648332i	$0.224533\pi$
$30$	0	0
$31$	7.24550	1.30133	0.650665	−	0.759365i	$-0.274490\pi$
$31$	7.24550	1.30133	0.650665	+	0.759365i	$0.274490\pi$
$32$	−2.50570	−0.442950
$33$	0	0
$34$	6.71951	1.15239
$35$	−3.02422	−0.511187
$36$	0	0
$37$	−7.63586	−1.25533	−0.627664	−	0.778484i	$-0.715989\pi$
$37$	−7.63586	−1.25533	−0.627664	+	0.778484i	$0.715989\pi$
$38$	−1.02065	−0.165572
$39$	0	0
$40$	8.11403	1.28294
$41$	−1.27186	−0.198632	−0.0993158	−	0.995056i	$-0.531665\pi$
$41$	−1.27186	−0.198632	−0.0993158	+	0.995056i	$0.531665\pi$
$42$	0	0
$43$	−4.31624	−0.658221	−0.329110	−	0.944291i	$-0.606749\pi$
$43$	−4.31624	−0.658221	−0.329110	+	0.944291i	$0.606749\pi$
$44$	0.175472	0.0264534
$45$	0	0
$46$	10.7402	1.58356
$47$	7.17025	1.04589	0.522944	−	0.852367i	$-0.324834\pi$
$47$	7.17025	1.04589	0.522944	+	0.852367i	$0.324834\pi$
$48$	0	0
$49$	−5.70701	−0.815287
$50$	−2.57948	−0.364793
$51$	0	0
$52$	1.77712	0.246442
$53$	−12.0735	−1.65842	−0.829212	−	0.558935i	$-0.811210\pi$
$53$	−12.0735	−1.65842	−0.829212	+	0.558935i	$0.811210\pi$
$54$	0	0
$55$	−1.03162	−0.139104
$56$	−3.46910	−0.463579
$57$	0	0
$58$	−10.2012	−1.33948
$59$	7.80904	1.01665	0.508325	−	0.861165i	$-0.330265\pi$
$59$	7.80904	1.01665	0.508325	+	0.861165i	$0.330265\pi$
$60$	0	0
$61$	3.19459	0.409026	0.204513	−	0.978864i	$-0.434439\pi$
$61$	3.19459	0.409026	0.204513	+	0.978864i	$0.434439\pi$
$62$	−9.01364	−1.14473
$63$	0	0
$64$	8.89835	1.11229
$65$	−10.4479	−1.29590
$66$	0	0
$67$	−6.56628	−0.802198	−0.401099	−	0.916035i	$-0.631372\pi$
$67$	−6.56628	−0.802198	−0.401099	+	0.916035i	$0.631372\pi$
$68$	2.44349	0.296317
$69$	0	0
$70$	3.76223	0.449673
$71$	−14.2950	−1.69650	−0.848250	−	0.529595i	$-0.822344\pi$
$71$	−14.2950	−1.69650	−0.848250	+	0.529595i	$0.822344\pi$
$72$	0	0
$73$	−0.473248	−0.0553895	−0.0276947	−	0.999616i	$-0.508817\pi$
$73$	−0.473248	−0.0553895	−0.0276947	+	0.999616i	$0.508817\pi$
$74$	9.49926	1.10427
$75$	0	0
$76$	−0.371151	−0.0425740
$77$	0.441063	0.0502638
$78$	0	0
$79$	−5.31508	−0.597993	−0.298996	−	0.954254i	$-0.596652\pi$
$79$	−5.31508	−0.597993	−0.298996	+	0.954254i	$0.596652\pi$
$80$	−7.68781	−0.859523
$81$	0	0
$82$	1.58224	0.174729
$83$	5.39344	0.592007	0.296004	−	0.955187i	$-0.404346\pi$
$83$	5.39344	0.592007	0.296004	+	0.955187i	$0.404346\pi$
$84$	0	0
$85$	−14.3655	−1.55816
$86$	5.36955	0.579013
$87$	0	0
$88$	−1.18338	−0.126148
$89$	−10.1340	−1.07420	−0.537102	−	0.843517i	$-0.680481\pi$
$89$	−10.1340	−1.07420	−0.537102	+	0.843517i	$0.680481\pi$
$90$	0	0
$91$	4.46692	0.468261
$92$	3.90559	0.407186
$93$	0	0
$94$	−8.92003	−0.920030
$95$	2.18204	0.223872
$96$	0	0
$97$	6.53946	0.663982	0.331991	−	0.943283i	$-0.392280\pi$
$97$	6.53946	0.663982	0.331991	+	0.943283i	$0.392280\pi$
$98$	7.09971	0.717179
$99$	0	0
$100$	−0.938005	−0.0938005
$101$	−1.62625	−0.161818	−0.0809090	−	0.996721i	$-0.525782\pi$
$101$	−1.62625	−0.161818	−0.0809090	+	0.996721i	$0.525782\pi$
$102$	0	0
$103$	3.71247	0.365800	0.182900	−	0.983132i	$-0.441452\pi$
$103$	3.71247	0.365800	0.182900	+	0.983132i	$0.441452\pi$
$104$	−11.9848	−1.17521
$105$	0	0
$106$	15.0198	1.45886
$107$	−10.4934	−1.01444	−0.507218	−	0.861818i	$-0.669326\pi$
$107$	−10.4934	−1.01444	−0.507218	+	0.861818i	$0.669326\pi$
$108$	0	0
$109$	−15.1383	−1.44998	−0.724991	−	0.688759i	$-0.758156\pi$
$109$	−15.1383	−1.44998	−0.724991	+	0.688759i	$0.758156\pi$
$110$	1.28337	0.122364
$111$	0	0
$112$	3.28688	0.310581
$113$	1.26306	0.118819	0.0594093	−	0.998234i	$-0.481078\pi$
$113$	1.26306	0.118819	0.0594093	+	0.998234i	$0.481078\pi$
$114$	0	0
$115$	−22.9614	−2.14116
$116$	−3.70956	−0.344424
$117$	0	0
$118$	−9.71470	−0.894311
$119$	6.14190	0.563027
$120$	0	0
$121$	−10.8495	−0.986322
$122$	−3.97418	−0.359805
$123$	0	0
$124$	−3.27773	−0.294349
$125$	−7.78338	−0.696166
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−6.05844	−0.535495
$129$	0	0
$130$	12.9975	1.13996
$131$	5.76432	0.503631	0.251815	−	0.967775i	$-0.418972\pi$
$131$	5.76432	0.503631	0.251815	+	0.967775i	$0.418972\pi$
$132$	0	0
$133$	−0.932917	−0.0808942
$134$	8.16867	0.705665
$135$	0	0
$136$	−16.4788	−1.41305
$137$	7.76444	0.663361	0.331680	−	0.943392i	$-0.392384\pi$
$137$	7.76444	0.663361	0.331680	+	0.943392i	$0.392384\pi$
$138$	0	0
$139$	0.461442	0.0391390	0.0195695	−	0.999808i	$-0.493770\pi$
$139$	0.461442	0.0391390	0.0195695	+	0.999808i	$0.493770\pi$
$140$	1.36810	0.115626
$141$	0	0
$142$	17.7834	1.49235
$143$	1.52375	0.127423
$144$	0	0
$145$	21.8089	1.81113
$146$	0.588736	0.0487241
$147$	0	0
$148$	3.45433	0.283944
$149$	−22.2561	−1.82329	−0.911647	−	0.410975i	$-0.865188\pi$
$149$	−22.2561	−1.82329	−0.911647	+	0.410975i	$0.865188\pi$
$150$	0	0
$151$	−7.37200	−0.599925	−0.299963	−	0.953951i	$-0.596974\pi$
$151$	−7.37200	−0.599925	−0.299963	+	0.953951i	$0.596974\pi$
$152$	2.50303	0.203022
$153$	0	0
$154$	−0.548697	−0.0442152
$155$	19.2701	1.54782
$156$	0	0
$157$	15.2661	1.21836	0.609182	−	0.793030i	$-0.291498\pi$
$157$	15.2661	1.21836	0.609182	+	0.793030i	$0.291498\pi$
$158$	6.61213	0.526033
$159$	0	0
$160$	−6.66417	−0.526849
$161$	9.81699	0.773687
$162$	0	0
$163$	−18.4061	−1.44167	−0.720837	−	0.693104i	$-0.756243\pi$
$163$	−18.4061	−1.44167	−0.720837	+	0.693104i	$0.756243\pi$
$164$	0.575368	0.0449287
$165$	0	0
$166$	−6.70962	−0.520767
$167$	19.9318	1.54237	0.771184	−	0.636613i	$-0.219665\pi$
$167$	19.9318	1.54237	0.771184	+	0.636613i	$0.219665\pi$
$168$	0	0
$169$	2.43199	0.187076
$170$	17.8712	1.37066
$171$	0	0
$172$	1.95259	0.148884
$173$	−13.9274	−1.05888	−0.529441	−	0.848347i	$-0.677598\pi$
$173$	−13.9274	−1.05888	−0.529441	+	0.848347i	$0.677598\pi$
$174$	0	0
$175$	−2.35775	−0.178229
$176$	1.12122	0.0845148
$177$	0	0
$178$	12.6070	0.944938
$179$	6.39494	0.477980	0.238990	−	0.971022i	$-0.423184\pi$
$179$	6.39494	0.477980	0.238990	+	0.971022i	$0.423184\pi$
$180$	0	0
$181$	13.0180	0.967617	0.483808	−	0.875174i	$-0.339253\pi$
$181$	13.0180	0.967617	0.483808	+	0.875174i	$0.339253\pi$
$182$	−5.55700	−0.411912
$183$	0	0
$184$	−26.3391	−1.94174
$185$	−20.3083	−1.49310
$186$	0	0
$187$	2.09512	0.153210
$188$	−3.24369	−0.236570
$189$	0	0
$190$	−2.71453	−0.196933
$191$	14.6672	1.06128	0.530642	−	0.847596i	$-0.321951\pi$
$191$	14.6672	1.06128	0.530642	+	0.847596i	$0.321951\pi$
$192$	0	0
$193$	−0.759127	−0.0546432	−0.0273216	−	0.999627i	$-0.508698\pi$
$193$	−0.759127	−0.0546432	−0.0273216	+	0.999627i	$0.508698\pi$
$194$	−8.13530	−0.584081
$195$	0	0
$196$	2.58175	0.184411
$197$	0.299227	0.0213190	0.0106595	−	0.999943i	$-0.496607\pi$
$197$	0.299227	0.0213190	0.0106595	+	0.999943i	$0.496607\pi$
$198$	0	0
$199$	6.35587	0.450556	0.225278	−	0.974295i	$-0.427671\pi$
$199$	6.35587	0.450556	0.225278	+	0.974295i	$0.427671\pi$
$200$	6.32587	0.447306
$201$	0	0
$202$	2.02311	0.142346
$203$	−9.32428	−0.654436
$204$	0	0
$205$	−3.38265	−0.236254
$206$	−4.61843	−0.321781
$207$	0	0
$208$	11.3553	0.787346
$209$	−0.318236	−0.0220128
$210$	0	0
$211$	16.8192	1.15788	0.578941	−	0.815370i	$-0.303466\pi$
$211$	16.8192	1.15788	0.578941	+	0.815370i	$0.303466\pi$
$212$	5.46184	0.375120
$213$	0	0
$214$	13.0541	0.892362
$215$	−11.4795	−0.782894
$216$	0	0
$217$	−8.23883	−0.559288
$218$	18.8325	1.27550
$219$	0	0
$220$	0.466686	0.0314640
$221$	21.2186	1.42732
$222$	0	0
$223$	−10.1748	−0.681353	−0.340677	−	0.940181i	$-0.610656\pi$
$223$	−10.1748	−0.681353	−0.340677	+	0.940181i	$0.610656\pi$
$224$	2.84923	0.190372
$225$	0	0
$226$	−1.57129	−0.104521
$227$	−1.39003	−0.0922596	−0.0461298	−	0.998935i	$-0.514689\pi$
$227$	−1.39003	−0.0922596	−0.0461298	+	0.998935i	$0.514689\pi$
$228$	0	0
$229$	−7.58166	−0.501010	−0.250505	−	0.968115i	$-0.580597\pi$
$229$	−7.58166	−0.501010	−0.250505	+	0.968115i	$0.580597\pi$
$230$	28.5647	1.88350
$231$	0	0
$232$	25.0171	1.64246
$233$	2.70235	0.177037	0.0885184	−	0.996075i	$-0.471787\pi$
$233$	2.70235	0.177037	0.0885184	+	0.996075i	$0.471787\pi$
$234$	0	0
$235$	19.0700	1.24399
$236$	−3.53267	−0.229957
$237$	0	0
$238$	−7.64073	−0.495275
$239$	7.71939	0.499326	0.249663	−	0.968333i	$-0.419680\pi$
$239$	7.71939	0.499326	0.249663	+	0.968333i	$0.419680\pi$
$240$	0	0
$241$	7.14292	0.460116	0.230058	−	0.973177i	$-0.426108\pi$
$241$	7.14292	0.460116	0.230058	+	0.973177i	$0.426108\pi$
$242$	13.4972	0.867632
$243$	0	0
$244$	−1.44518	−0.0925179
$245$	−15.1784	−0.969711
$246$	0	0
$247$	−3.22297	−0.205073
$248$	22.1049	1.40366
$249$	0	0
$250$	9.68278	0.612393
$251$	−11.4785	−0.724519	−0.362259	−	0.932077i	$-0.617995\pi$
$251$	−11.4785	−0.724519	−0.362259	+	0.932077i	$0.617995\pi$
$252$	0	0
$253$	3.34876	0.210535
$254$	−1.24403	−0.0780576
$255$	0	0
$256$	−10.2598	−0.641238
$257$	−23.1729	−1.44549	−0.722743	−	0.691117i	$-0.757119\pi$
$257$	−23.1729	−1.44549	−0.722743	+	0.691117i	$0.757119\pi$
$258$	0	0
$259$	8.68271	0.539517
$260$	4.72643	0.293120
$261$	0	0
$262$	−7.17100	−0.443026
$263$	9.28819	0.572734	0.286367	−	0.958120i	$-0.407552\pi$
$263$	9.28819	0.572734	0.286367	+	0.958120i	$0.407552\pi$
$264$	0	0
$265$	−32.1107	−1.97255
$266$	1.16058	0.0711597
$267$	0	0
$268$	2.97046	0.181450
$269$	25.6009	1.56092	0.780458	−	0.625209i	$-0.214986\pi$
$269$	25.6009	1.56092	0.780458	+	0.625209i	$0.214986\pi$
$270$	0	0
$271$	10.2077	0.620075	0.310037	−	0.950724i	$-0.399658\pi$
$271$	10.2077	0.620075	0.310037	+	0.950724i	$0.399658\pi$
$272$	15.6132	0.946689
$273$	0	0
$274$	−9.65922	−0.583535
$275$	−0.804272	−0.0484994
$276$	0	0
$277$	−8.77510	−0.527245	−0.263622	−	0.964626i	$-0.584917\pi$
$277$	−8.77510	−0.527245	−0.263622	+	0.964626i	$0.584917\pi$
$278$	−0.574050	−0.0344292
$279$	0	0
$280$	−9.22643	−0.551385
$281$	16.7110	0.996894	0.498447	−	0.866920i	$-0.333904\pi$
$281$	16.7110	0.996894	0.498447	+	0.866920i	$0.333904\pi$
$282$	0	0
$283$	0.0861729	0.00512245	0.00256122	−	0.999997i	$-0.499185\pi$
$283$	0.0861729	0.00512245	0.00256122	+	0.999997i	$0.499185\pi$
$284$	6.46679	0.383733
$285$	0	0
$286$	−1.89560	−0.112089
$287$	1.44623	0.0853683
$288$	0	0
$289$	12.1750	0.716178
$290$	−27.1310	−1.59319
$291$	0	0
$292$	0.214089	0.0125286
$293$	−17.3794	−1.01532	−0.507658	−	0.861559i	$-0.669489\pi$
$293$	−17.3794	−1.01532	−0.507658	+	0.861559i	$0.669489\pi$
$294$	0	0
$295$	20.7689	1.20921
$296$	−23.2958	−1.35404
$297$	0	0
$298$	27.6874	1.60389
$299$	33.9150	1.96136
$300$	0	0
$301$	4.90798	0.282892
$302$	9.17102	0.527733
$303$	0	0
$304$	−2.37155	−0.136018
$305$	8.49635	0.486499
$306$	0	0
$307$	25.0765	1.43119	0.715595	−	0.698516i	$-0.246156\pi$
$307$	25.0765	1.43119	0.715595	+	0.698516i	$0.246156\pi$
$308$	−0.199529	−0.0113692
$309$	0	0
$310$	−23.9727	−1.36156
$311$	1.48709	0.0843253	0.0421627	−	0.999111i	$-0.486575\pi$
$311$	1.48709	0.0843253	0.0421627	+	0.999111i	$0.486575\pi$
$312$	0	0
$313$	11.1036	0.627610	0.313805	−	0.949487i	$-0.398396\pi$
$313$	11.1036	0.627610	0.313805	+	0.949487i	$0.398396\pi$
$314$	−18.9915	−1.07175
$315$	0	0
$316$	2.40444	0.135261
$317$	−3.10930	−0.174636	−0.0873178	−	0.996181i	$-0.527830\pi$
$317$	−3.10930	−0.174636	−0.0873178	+	0.996181i	$0.527830\pi$
$318$	0	0
$319$	−3.18069	−0.178084
$320$	23.6661	1.32297
$321$	0	0
$322$	−12.2127	−0.680585
$323$	−4.43151	−0.246576
$324$	0	0
$325$	−8.14537	−0.451824
$326$	22.8978	1.26819
$327$	0	0
$328$	−3.88025	−0.214251
$329$	−8.15327	−0.449504
$330$	0	0
$331$	27.7366	1.52454	0.762272	−	0.647257i	$-0.224084\pi$
$331$	27.7366	1.52454	0.762272	+	0.647257i	$0.224084\pi$
$332$	−2.43990	−0.133907
$333$	0	0
$334$	−24.7958	−1.35677
$335$	−17.4637	−0.954143
$336$	0	0
$337$	13.8646	0.755253	0.377627	−	0.925958i	$-0.376740\pi$
$337$	13.8646	0.755253	0.377627	+	0.925958i	$0.376740\pi$
$338$	−3.02548	−0.164564
$339$	0	0
$340$	6.49871	0.352442
$341$	−2.81042	−0.152193
$342$	0	0
$343$	14.4491	0.780178
$344$	−13.1682	−0.709981
$345$	0	0
$346$	17.3262	0.931461
$347$	19.3863	1.04071	0.520355	−	0.853950i	$-0.325800\pi$
$347$	19.3863	1.04071	0.520355	+	0.853950i	$0.325800\pi$
$348$	0	0
$349$	17.7032	0.947631	0.473815	−	0.880624i	$-0.342876\pi$
$349$	17.7032	0.947631	0.473815	+	0.880624i	$0.342876\pi$
$350$	2.93312	0.156782
$351$	0	0
$352$	0.971925	0.0518038
$353$	−30.9262	−1.64603	−0.823017	−	0.568017i	$-0.807711\pi$
$353$	−30.9262	−1.64603	−0.823017	+	0.568017i	$0.807711\pi$
$354$	0	0
$355$	−38.0189	−2.01784
$356$	4.58444	0.242975
$357$	0	0
$358$	−7.95552	−0.420462
$359$	20.9608	1.10627	0.553134	−	0.833092i	$-0.313432\pi$
$359$	20.9608	1.10627	0.553134	+	0.833092i	$0.313432\pi$
$360$	0	0
$361$	−18.3269	−0.964573
$362$	−16.1948	−0.851178
$363$	0	0
$364$	−2.02075	−0.105916
$365$	−1.25865	−0.0658808
$366$	0	0
$367$	28.5541	1.49051	0.745255	−	0.666780i	$-0.232328\pi$
$367$	28.5541	1.49051	0.745255	+	0.666780i	$0.232328\pi$
$368$	24.9555	1.30090
$369$	0	0
$370$	25.2643	1.31343
$371$	13.7287	0.712761
$372$	0	0
$373$	4.17300	0.216070	0.108035	−	0.994147i	$-0.465544\pi$
$373$	4.17300	0.216070	0.108035	+	0.994147i	$0.465544\pi$
$374$	−2.60640	−0.134774
$375$	0	0
$376$	21.8753	1.12813
$377$	−32.2128	−1.65904
$378$	0	0
$379$	−9.38931	−0.482296	−0.241148	−	0.970488i	$-0.577524\pi$
$379$	−9.38931	−0.482296	−0.241148	+	0.970488i	$0.577524\pi$
$380$	−0.987115	−0.0506379
$381$	0	0
$382$	−18.2465	−0.933573
$383$	−11.7549	−0.600648	−0.300324	−	0.953837i	$-0.597095\pi$
$383$	−11.7549	−0.600648	−0.300324	+	0.953837i	$0.597095\pi$
$384$	0	0
$385$	1.17305	0.0597842
$386$	0.944379	0.0480676
$387$	0	0
$388$	−2.95833	−0.150187
$389$	−18.6401	−0.945090	−0.472545	−	0.881307i	$-0.656665\pi$
$389$	−18.6401	−0.945090	−0.472545	+	0.881307i	$0.656665\pi$
$390$	0	0
$391$	46.6323	2.35830
$392$	−17.4112	−0.879398
$393$	0	0
$394$	−0.372248	−0.0187536
$395$	−14.1360	−0.711259
$396$	0	0
$397$	−7.02368	−0.352508	−0.176254	−	0.984345i	$-0.556398\pi$
$397$	−7.02368	−0.352508	−0.176254	+	0.984345i	$0.556398\pi$
$398$	−7.90691	−0.396338
$399$	0	0
$400$	−5.99358	−0.299679
$401$	−6.31172	−0.315192	−0.157596	−	0.987504i	$-0.550374\pi$
$401$	−6.31172	−0.315192	−0.157596	+	0.987504i	$0.550374\pi$
$402$	0	0
$403$	−28.4629	−1.41784
$404$	0.735687	0.0366018
$405$	0	0
$406$	11.5997	0.575684
$407$	2.96184	0.146813
$408$	0	0
$409$	−12.7328	−0.629594	−0.314797	−	0.949159i	$-0.601936\pi$
$409$	−12.7328	−0.629594	−0.314797	+	0.949159i	$0.601936\pi$
$410$	4.20813	0.207825
$411$	0	0
$412$	−1.67945	−0.0827407
$413$	−8.87963	−0.436938
$414$	0	0
$415$	14.3444	0.704139
$416$	9.84330	0.482607
$417$	0	0
$418$	0.395896	0.0193639
$419$	−14.1641	−0.691960	−0.345980	−	0.938242i	$-0.612453\pi$
$419$	−14.1641	−0.691960	−0.345980	+	0.938242i	$0.612453\pi$
$420$	0	0
$421$	−20.9206	−1.01961	−0.509804	−	0.860291i	$-0.670282\pi$
$421$	−20.9206	−1.01961	−0.509804	+	0.860291i	$0.670282\pi$
$422$	−20.9236	−1.01855
$423$	0	0
$424$	−36.8344	−1.78884
$425$	−11.1997	−0.543264
$426$	0	0
$427$	−3.63256	−0.175792
$428$	4.74702	0.229456
$429$	0	0
$430$	14.2809	0.688684
$431$	8.11460	0.390866	0.195433	−	0.980717i	$-0.437389\pi$
$431$	8.11460	0.390866	0.195433	+	0.980717i	$0.437389\pi$
$432$	0	0
$433$	9.52965	0.457966	0.228983	−	0.973430i	$-0.426460\pi$
$433$	9.52965	0.457966	0.228983	+	0.973430i	$0.426460\pi$
$434$	10.2494	0.491986
$435$	0	0
$436$	6.84827	0.327973
$437$	−7.08316	−0.338833
$438$	0	0
$439$	−36.2854	−1.73181	−0.865904	−	0.500210i	$-0.833256\pi$
$439$	−36.2854	−1.73181	−0.865904	+	0.500210i	$0.833256\pi$
$440$	−3.14731	−0.150042
$441$	0	0
$442$	−26.3966	−1.25556
$443$	6.41377	0.304727	0.152364	−	0.988324i	$-0.451311\pi$
$443$	6.41377	0.304727	0.152364	+	0.988324i	$0.451311\pi$
$444$	0	0
$445$	−26.9524	−1.27767
$446$	12.6578	0.599362
$447$	0	0
$448$	−10.1183	−0.478044
$449$	−17.1079	−0.807369	−0.403685	−	0.914898i	$-0.632271\pi$
$449$	−17.1079	−0.807369	−0.403685	+	0.914898i	$0.632271\pi$
$450$	0	0
$451$	0.493337	0.0232303
$452$	−0.571385	−0.0268757
$453$	0	0
$454$	1.72924	0.0811575
$455$	11.8802	0.556954
$456$	0	0
$457$	22.4370	1.04956	0.524780	−	0.851238i	$-0.324148\pi$
$457$	22.4370	1.04956	0.524780	+	0.851238i	$0.324148\pi$
$458$	9.43184	0.440721
$459$	0	0
$460$	10.3873	0.484311
$461$	30.5419	1.42248	0.711238	−	0.702951i	$-0.248135\pi$
$461$	30.5419	1.42248	0.711238	+	0.702951i	$0.248135\pi$
$462$	0	0
$463$	6.12658	0.284726	0.142363	−	0.989815i	$-0.454530\pi$
$463$	6.12658	0.284726	0.142363	+	0.989815i	$0.454530\pi$
$464$	−23.7030	−1.10039
$465$	0	0
$466$	−3.36181	−0.155733
$467$	−26.6943	−1.23526	−0.617632	−	0.786467i	$-0.711908\pi$
$467$	−26.6943	−1.23526	−0.617632	+	0.786467i	$0.711908\pi$
$468$	0	0
$469$	7.46649	0.344771
$470$	−23.7237	−1.09429
$471$	0	0
$472$	23.8242	1.09660
$473$	1.67421	0.0769801
$474$	0	0
$475$	1.70116	0.0780547
$476$	−2.77849	−0.127352
$477$	0	0
$478$	−9.60317	−0.439239
$479$	−8.21828	−0.375503	−0.187751	−	0.982217i	$-0.560120\pi$
$479$	−8.21828	−0.375503	−0.187751	+	0.982217i	$0.560120\pi$
$480$	0	0
$481$	29.9964	1.36772
$482$	−8.88603	−0.404747
$483$	0	0
$484$	4.90814	0.223097
$485$	17.3924	0.789746
$486$	0	0
$487$	8.66668	0.392725	0.196362	−	0.980531i	$-0.437087\pi$
$487$	8.66668	0.392725	0.196362	+	0.980531i	$0.437087\pi$
$488$	9.74621	0.441190
$489$	0	0
$490$	18.8824	0.853020
$491$	25.2052	1.13749	0.568747	−	0.822513i	$-0.307428\pi$
$491$	25.2052	1.13749	0.568747	+	0.822513i	$0.307428\pi$
$492$	0	0
$493$	−44.2918	−1.99480
$494$	4.00949	0.180395
$495$	0	0
$496$	−20.9437	−0.940402
$497$	16.2548	0.729126
$498$	0	0
$499$	−36.4130	−1.63007	−0.815036	−	0.579411i	$-0.803283\pi$
$499$	−36.4130	−1.63007	−0.815036	+	0.579411i	$0.803283\pi$
$500$	3.52106	0.157467
$501$	0	0
$502$	14.2797	0.637333
$503$	2.84431	0.126822	0.0634108	−	0.997988i	$-0.479802\pi$
$503$	2.84431	0.126822	0.0634108	+	0.997988i	$0.479802\pi$
$504$	0	0
$505$	−4.32518	−0.192468
$506$	−4.16597	−0.185200
$507$	0	0
$508$	−0.452382	−0.0200712
$509$	−9.76983	−0.433040	−0.216520	−	0.976278i	$-0.569471\pi$
$509$	−9.76983	−0.433040	−0.216520	+	0.976278i	$0.569471\pi$
$510$	0	0
$511$	0.538128	0.0238054
$512$	24.8804	1.09957
$513$	0	0
$514$	28.8279	1.27154
$515$	9.87368	0.435086
$516$	0	0
$517$	−2.78123	−0.122319
$518$	−10.8016	−0.474594
$519$	0	0
$520$	−31.8748	−1.39780
$521$	12.1384	0.531793	0.265897	−	0.964002i	$-0.414332\pi$
$521$	12.1384	0.531793	0.265897	+	0.964002i	$0.414332\pi$
$522$	0	0
$523$	10.2942	0.450133	0.225067	−	0.974343i	$-0.427740\pi$
$523$	10.2942	0.450133	0.225067	+	0.974343i	$0.427740\pi$
$524$	−2.60767	−0.113917
$525$	0	0
$526$	−11.5548	−0.503814
$527$	−39.1358	−1.70478
$528$	0	0
$529$	51.5353	2.24067
$530$	39.9468	1.73518
$531$	0	0
$532$	0.422035	0.0182975
$533$	4.99633	0.216415
$534$	0	0
$535$	−27.9083	−1.20658
$536$	−20.0327	−0.865280
$537$	0	0
$538$	−31.8484	−1.37308
$539$	2.21367	0.0953493
$540$	0	0
$541$	14.9256	0.641702	0.320851	−	0.947130i	$-0.396031\pi$
$541$	14.9256	0.641702	0.320851	+	0.947130i	$0.396031\pi$
$542$	−12.6987	−0.545457
$543$	0	0
$544$	13.5343	0.580278
$545$	−40.2617	−1.72462
$546$	0	0
$547$	−24.2654	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$547$	−24.2654	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$548$	−3.51249	−0.150046
$549$	0	0
$550$	1.00054	0.0426632
$551$	6.72765	0.286608
$552$	0	0
$553$	6.04376	0.257007
$554$	10.9165	0.463798
$555$	0	0
$556$	−0.208748	−0.00885289
$557$	25.3184	1.07277	0.536387	−	0.843972i	$-0.319789\pi$
$557$	25.3184	1.07277	0.536387	+	0.843972i	$0.319789\pi$
$558$	0	0
$559$	16.9557	0.717152
$560$	8.74178	0.369408
$561$	0	0
$562$	−20.7890	−0.876932
$563$	17.6753	0.744925	0.372463	−	0.928047i	$-0.378513\pi$
$563$	17.6753	0.744925	0.372463	+	0.928047i	$0.378513\pi$
$564$	0	0
$565$	3.35924	0.141324
$566$	−0.107202	−0.00450603
$567$	0	0
$568$	−43.6117	−1.82991
$569$	15.2031	0.637346	0.318673	−	0.947865i	$-0.396763\pi$
$569$	15.2031	0.637346	0.318673	+	0.947865i	$0.396763\pi$
$570$	0	0
$571$	−37.0756	−1.55156	−0.775782	−	0.631001i	$-0.782645\pi$
$571$	−37.0756	−1.55156	−0.775782	+	0.631001i	$0.782645\pi$
$572$	−0.689318	−0.0288218
$573$	0	0
$574$	−1.79916	−0.0750954
$575$	−17.9012	−0.746530
$576$	0	0
$577$	0.568616	0.0236718	0.0118359	−	0.999930i	$-0.496232\pi$
$577$	0.568616	0.0236718	0.0118359	+	0.999930i	$0.496232\pi$
$578$	−15.1461	−0.629996
$579$	0	0
$580$	−9.86596	−0.409662
$581$	−6.13286	−0.254434
$582$	0	0
$583$	4.68313	0.193956
$584$	−1.44381	−0.0597451
$585$	0	0
$586$	21.6206	0.893138
$587$	7.84556	0.323821	0.161910	−	0.986805i	$-0.448234\pi$
$587$	7.84556	0.323821	0.161910	+	0.986805i	$0.448234\pi$
$588$	0	0
$589$	5.94448	0.244938
$590$	−25.8372	−1.06370
$591$	0	0
$592$	22.0721	0.907159
$593$	15.8166	0.649508	0.324754	−	0.945798i	$-0.394718\pi$
$593$	15.8166	0.649508	0.324754	+	0.945798i	$0.394718\pi$
$594$	0	0
$595$	16.3350	0.669670
$596$	10.0683	0.412412
$597$	0	0
$598$	−42.1914	−1.72533
$599$	−2.64225	−0.107960	−0.0539798	−	0.998542i	$-0.517191\pi$
$599$	−2.64225	−0.107960	−0.0539798	+	0.998542i	$0.517191\pi$
$600$	0	0
$601$	−6.01002	−0.245154	−0.122577	−	0.992459i	$-0.539116\pi$
$601$	−6.01002	−0.245154	−0.122577	+	0.992459i	$0.539116\pi$
$602$	−6.10569	−0.248850
$603$	0	0
$604$	3.33496	0.135698
$605$	−28.8555	−1.17314
$606$	0	0
$607$	−14.5942	−0.592360	−0.296180	−	0.955132i	$-0.595713\pi$
$607$	−14.5942	−0.592360	−0.296180	+	0.955132i	$0.595713\pi$
$608$	−2.05577	−0.0833726
$609$	0	0
$610$	−10.5697	−0.427956
$611$	−28.1673	−1.13953
$612$	0	0
$613$	−35.3496	−1.42776	−0.713879	−	0.700269i	$-0.753063\pi$
$613$	−35.3496	−1.42776	−0.713879	+	0.700269i	$0.753063\pi$
$614$	−31.1960	−1.25897
$615$	0	0
$616$	1.34561	0.0542163
$617$	−21.5619	−0.868049	−0.434024	−	0.900901i	$-0.642907\pi$
$617$	−21.5619	−0.868049	−0.434024	+	0.900901i	$0.642907\pi$
$618$	0	0
$619$	−18.8180	−0.756359	−0.378180	−	0.925732i	$-0.623450\pi$
$619$	−18.8180	−0.756359	−0.378180	+	0.925732i	$0.623450\pi$
$620$	−8.71746	−0.350102
$621$	0	0
$622$	−1.84999	−0.0741780
$623$	11.5234	0.461673
$624$	0	0
$625$	−31.0681	−1.24272
$626$	−13.8132	−0.552086
$627$	0	0
$628$	−6.90609	−0.275583
$629$	41.2443	1.64452
$630$	0	0
$631$	−33.8509	−1.34758	−0.673791	−	0.738922i	$-0.735335\pi$
$631$	−33.8509	−1.34758	−0.673791	+	0.738922i	$0.735335\pi$
$632$	−16.2155	−0.645017
$633$	0	0
$634$	3.86807	0.153621
$635$	2.65960	0.105543
$636$	0	0
$637$	22.4192	0.888280
$638$	3.95688	0.156654
$639$	0	0
$640$	−16.1130	−0.636923
$641$	31.9022	1.26006	0.630030	−	0.776571i	$-0.283043\pi$
$641$	31.9022	1.26006	0.630030	+	0.776571i	$0.283043\pi$
$642$	0	0
$643$	1.32854	0.0523925	0.0261963	−	0.999657i	$-0.491661\pi$
$643$	1.32854	0.0523925	0.0261963	+	0.999657i	$0.491661\pi$
$644$	−4.44103	−0.175001
$645$	0	0
$646$	5.51294	0.216904
$647$	−1.19511	−0.0469845	−0.0234922	−	0.999724i	$-0.507478\pi$
$647$	−1.19511	−0.0469845	−0.0234922	+	0.999724i	$0.507478\pi$
$648$	0	0
$649$	−3.02901	−0.118899
$650$	10.1331	0.397453
$651$	0	0
$652$	8.32657	0.326094
$653$	−18.0941	−0.708076	−0.354038	−	0.935231i	$-0.615192\pi$
$653$	−18.0941	−0.708076	−0.354038	+	0.935231i	$0.615192\pi$
$654$	0	0
$655$	15.3308	0.599024
$656$	3.67643	0.143540
$657$	0	0
$658$	10.1429	0.395413
$659$	−10.5226	−0.409901	−0.204951	−	0.978772i	$-0.565703\pi$
$659$	−10.5226	−0.409901	−0.204951	+	0.978772i	$0.565703\pi$
$660$	0	0
$661$	−11.2974	−0.439417	−0.219709	−	0.975566i	$-0.570511\pi$
$661$	−11.2974	−0.439417	−0.219709	+	0.975566i	$0.570511\pi$
$662$	−34.5053	−1.34109
$663$	0	0
$664$	16.4545	0.638560
$665$	−2.48119	−0.0962164
$666$	0	0
$667$	−70.7944	−2.74117
$668$	−9.01678	−0.348870
$669$	0	0
$670$	21.7254	0.839325
$671$	−1.23914	−0.0478363
$672$	0	0
$673$	17.4645	0.673207	0.336603	−	0.941647i	$-0.390722\pi$
$673$	17.4645	0.673207	0.336603	+	0.941647i	$0.390722\pi$
$674$	−17.2480	−0.664369
$675$	0	0
$676$	−1.10019	−0.0423150
$677$	−25.7431	−0.989388	−0.494694	−	0.869067i	$-0.664720\pi$
$677$	−25.7431	−0.989388	−0.494694	+	0.869067i	$0.664720\pi$
$678$	0	0
$679$	−7.43600	−0.285367
$680$	−43.8271	−1.68069
$681$	0	0
$682$	3.49626	0.133879
$683$	−2.08134	−0.0796404	−0.0398202	−	0.999207i	$-0.512679\pi$
$683$	−2.08134	−0.0796404	−0.0398202	+	0.999207i	$0.512679\pi$
$684$	0	0
$685$	20.6503	0.789008
$686$	−17.9752	−0.686295
$687$	0	0
$688$	12.4765	0.475661
$689$	47.4290	1.80690
$690$	0	0
$691$	−6.75347	−0.256914	−0.128457	−	0.991715i	$-0.541002\pi$
$691$	−6.75347	−0.256914	−0.128457	+	0.991715i	$0.541002\pi$
$692$	6.30051	0.239510
$693$	0	0
$694$	−24.1172	−0.915475
$695$	1.22725	0.0465524
$696$	0	0
$697$	6.86983	0.260213
$698$	−22.0234	−0.833597
$699$	0	0
$700$	1.06660	0.0403138
$701$	30.5737	1.15475	0.577376	−	0.816479i	$-0.304077\pi$
$701$	30.5737	1.15475	0.577376	+	0.816479i	$0.304077\pi$
$702$	0	0
$703$	−6.26475	−0.236280
$704$	−3.45154	−0.130085
$705$	0	0
$706$	38.4732	1.44796
$707$	1.84920	0.0695465
$708$	0	0
$709$	45.2251	1.69846	0.849232	−	0.528019i	$-0.177065\pi$
$709$	45.2251	1.69846	0.849232	+	0.528019i	$0.177065\pi$
$710$	47.2968	1.77502
$711$	0	0
$712$	−30.9173	−1.15867
$713$	−62.5532	−2.34264
$714$	0	0
$715$	4.05257	0.151558
$716$	−2.89296	−0.108115
$717$	0	0
$718$	−26.0759	−0.973145
$719$	19.9625	0.744475	0.372238	−	0.928137i	$-0.378591\pi$
$719$	19.9625	0.744475	0.372238	+	0.928137i	$0.378591\pi$
$720$	0	0
$721$	−4.22143	−0.157214
$722$	22.7992	0.848500
$723$	0	0
$724$	−5.88909	−0.218866
$725$	17.0027	0.631464
$726$	0	0
$727$	7.36788	0.273260	0.136630	−	0.990622i	$-0.456373\pi$
$727$	7.36788	0.273260	0.136630	+	0.990622i	$0.456373\pi$
$728$	13.6279	0.505083
$729$	0	0
$730$	1.56580	0.0579530
$731$	23.3137	0.862289
$732$	0	0
$733$	−5.91824	−0.218595	−0.109298	−	0.994009i	$-0.534860\pi$
$733$	−5.91824	−0.218595	−0.109298	+	0.994009i	$0.534860\pi$
$734$	−35.5222	−1.31115
$735$	0	0
$736$	21.6327	0.797392
$737$	2.54696	0.0938185
$738$	0	0
$739$	46.4192	1.70756	0.853780	−	0.520635i	$-0.174305\pi$
$739$	46.4192	1.70756	0.853780	+	0.520635i	$0.174305\pi$
$740$	9.18713	0.337726
$741$	0	0
$742$	−17.0790	−0.626990
$743$	−9.30932	−0.341526	−0.170763	−	0.985312i	$-0.554623\pi$
$743$	−9.30932	−0.341526	−0.170763	+	0.985312i	$0.554623\pi$
$744$	0	0
$745$	−59.1924	−2.16864
$746$	−5.19135	−0.190069
$747$	0	0
$748$	−0.947794	−0.0346548
$749$	11.9320	0.435986
$750$	0	0
$751$	−11.7635	−0.429257	−0.214628	−	0.976696i	$-0.568854\pi$
$751$	−11.7635	−0.429257	−0.214628	+	0.976696i	$0.568854\pi$
$752$	−20.7262	−0.755808
$753$	0	0
$754$	40.0738	1.45940
$755$	−19.6066	−0.713557
$756$	0	0
$757$	9.06800	0.329582	0.164791	−	0.986329i	$-0.447305\pi$
$757$	9.06800	0.329582	0.164791	+	0.986329i	$0.447305\pi$
$758$	11.6806	0.424259
$759$	0	0
$760$	6.65706	0.241477
$761$	−10.2029	−0.369854	−0.184927	−	0.982752i	$-0.559205\pi$
$761$	−10.2029	−0.369854	−0.184927	+	0.982752i	$0.559205\pi$
$762$	0	0
$763$	17.2137	0.623176
$764$	−6.63519	−0.240053
$765$	0	0
$766$	14.6235	0.528368
$767$	−30.6767	−1.10767
$768$	0	0
$769$	−37.9643	−1.36903	−0.684515	−	0.728999i	$-0.739986\pi$
$769$	−37.9643	−1.36903	−0.684515	+	0.728999i	$0.739986\pi$
$770$	−1.45931	−0.0525901
$771$	0	0
$772$	0.343415	0.0123598
$773$	−36.0374	−1.29618	−0.648088	−	0.761565i	$-0.724431\pi$
$773$	−36.0374	−1.29618	−0.648088	+	0.761565i	$0.724431\pi$
$774$	0	0
$775$	15.0234	0.539657
$776$	19.9509	0.716195
$777$	0	0
$778$	23.1889	0.831362
$779$	−1.04348	−0.0373867
$780$	0	0
$781$	5.54481	0.198409
$782$	−58.0121	−2.07451
$783$	0	0
$784$	16.4966	0.589165
$785$	40.6016	1.44913
$786$	0	0
$787$	−10.6592	−0.379959	−0.189979	−	0.981788i	$-0.560842\pi$
$787$	−10.6592	−0.379959	−0.189979	+	0.981788i	$0.560842\pi$
$788$	−0.135365	−0.00482217
$789$	0	0
$790$	17.5856	0.625669
$791$	−1.43622	−0.0510661
$792$	0	0
$793$	−12.5495	−0.445646
$794$	8.73769	0.310089
$795$	0	0
$796$	−2.87528	−0.101912
$797$	35.8524	1.26996	0.634979	−	0.772529i	$-0.281009\pi$
$797$	35.8524	1.26996	0.634979	+	0.772529i	$0.281009\pi$
$798$	0	0
$799$	−38.7293	−1.37014
$800$	−5.19553	−0.183690
$801$	0	0
$802$	7.85198	0.277263
$803$	0.183566	0.00647790
$804$	0	0
$805$	26.1093	0.920231
$806$	35.4088	1.24722
$807$	0	0
$808$	−4.96144	−0.174543
$809$	−43.9379	−1.54478	−0.772388	−	0.635151i	$-0.780938\pi$
$809$	−43.9379	−1.54478	−0.772388	+	0.635151i	$0.780938\pi$
$810$	0	0
$811$	27.6010	0.969204	0.484602	−	0.874735i	$-0.338964\pi$
$811$	27.6010	0.969204	0.484602	+	0.874735i	$0.338964\pi$
$812$	4.21813	0.148027
$813$	0	0
$814$	−3.68462	−0.129146
$815$	−48.9528	−1.71474
$816$	0	0
$817$	−3.54121	−0.123891
$818$	15.8400	0.553832
$819$	0	0
$820$	1.53025	0.0534386
$821$	−29.7144	−1.03704	−0.518521	−	0.855065i	$-0.673517\pi$
$821$	−29.7144	−1.03704	−0.518521	+	0.855065i	$0.673517\pi$
$822$	0	0
$823$	48.2092	1.68047	0.840233	−	0.542225i	$-0.182418\pi$
$823$	48.2092	1.68047	0.840233	+	0.542225i	$0.182418\pi$
$824$	11.3262	0.394565
$825$	0	0
$826$	11.0466	0.384359
$827$	−43.6202	−1.51682	−0.758412	−	0.651775i	$-0.774024\pi$
$827$	−43.6202	−1.51682	−0.758412	+	0.651775i	$0.774024\pi$
$828$	0	0
$829$	−24.8866	−0.864346	−0.432173	−	0.901791i	$-0.642253\pi$
$829$	−24.8866	−0.864346	−0.432173	+	0.901791i	$0.642253\pi$
$830$	−17.8449	−0.619406
$831$	0	0
$832$	−34.9559	−1.21188
$833$	30.8258	1.06805
$834$	0	0
$835$	53.0106	1.83451
$836$	0.143964	0.00497910
$837$	0	0
$838$	17.6206	0.608692
$839$	13.3581	0.461172	0.230586	−	0.973052i	$-0.425936\pi$
$839$	13.3581	0.461172	0.230586	+	0.973052i	$0.425936\pi$
$840$	0	0
$841$	38.2412	1.31866
$842$	26.0259	0.896913
$843$	0	0
$844$	−7.60870	−0.261902
$845$	6.46813	0.222511
$846$	0	0
$847$	12.3370	0.423904
$848$	34.8995	1.19845
$849$	0	0
$850$	13.9328	0.477890
$851$	65.9233	2.25982
$852$	0	0
$853$	37.6996	1.29081	0.645406	−	0.763840i	$-0.276688\pi$
$853$	37.6996	1.29081	0.645406	+	0.763840i	$0.276688\pi$
$854$	4.51903	0.154638
$855$	0	0
$856$	−32.0137	−1.09421
$857$	43.5517	1.48770	0.743849	−	0.668348i	$-0.232998\pi$
$857$	43.5517	1.48770	0.743849	+	0.668348i	$0.232998\pi$
$858$	0	0
$859$	34.8547	1.18923	0.594614	−	0.804011i	$-0.297305\pi$
$859$	34.8547	1.18923	0.594614	+	0.804011i	$0.297305\pi$
$860$	5.19311	0.177084
$861$	0	0
$862$	−10.0948	−0.343831
$863$	21.4982	0.731806	0.365903	−	0.930653i	$-0.380760\pi$
$863$	21.4982	0.731806	0.365903	+	0.930653i	$0.380760\pi$
$864$	0	0
$865$	−37.0414	−1.25945
$866$	−11.8552	−0.402856
$867$	0	0
$868$	3.72710	0.126506
$869$	2.06164	0.0699363
$870$	0	0
$871$	25.7947	0.874020
$872$	−46.1844	−1.56400
$873$	0	0
$874$	8.81168	0.298060
$875$	8.85045	0.299200
$876$	0	0
$877$	−3.92412	−0.132508	−0.0662541	−	0.997803i	$-0.521105\pi$
$877$	−3.92412	−0.132508	−0.0662541	+	0.997803i	$0.521105\pi$
$878$	45.1403	1.52341
$879$	0	0
$880$	2.98199	0.100523
$881$	13.1295	0.442343	0.221171	−	0.975235i	$-0.429012\pi$
$881$	13.1295	0.442343	0.221171	+	0.975235i	$0.429012\pi$
$882$	0	0
$883$	45.5718	1.53361	0.766807	−	0.641877i	$-0.221844\pi$
$883$	45.5718	1.53361	0.766807	+	0.641877i	$0.221844\pi$
$884$	−9.59891	−0.322846
$885$	0	0
$886$	−7.97894	−0.268058
$887$	50.9532	1.71084	0.855421	−	0.517934i	$-0.173299\pi$
$887$	50.9532	1.71084	0.855421	+	0.517934i	$0.173299\pi$
$888$	0	0
$889$	−1.13710	−0.0381370
$890$	33.5297	1.12392
$891$	0	0
$892$	4.60288	0.154116
$893$	5.88275	0.196859
$894$	0	0
$895$	17.0080	0.568515
$896$	6.88903	0.230146
$897$	0	0
$898$	21.2827	0.710214
$899$	59.4136	1.98156
$900$	0	0
$901$	65.2137	2.17258
$902$	−0.613727	−0.0204349
$903$	0	0
$904$	3.85340	0.128162
$905$	34.6226	1.15089
$906$	0	0
$907$	16.4438	0.546007	0.273004	−	0.962013i	$-0.411983\pi$
$907$	16.4438	0.546007	0.273004	+	0.962013i	$0.411983\pi$
$908$	0.628825	0.0208683
$909$	0	0
$910$	−14.7794	−0.489932
$911$	−44.9911	−1.49062	−0.745310	−	0.666718i	$-0.767699\pi$
$911$	−44.9911	−1.49062	−0.745310	+	0.666718i	$0.767699\pi$
$912$	0	0
$913$	−2.09204	−0.0692363
$914$	−27.9124	−0.923261
$915$	0	0
$916$	3.42981	0.113324
$917$	−6.55459	−0.216452
$918$	0	0
$919$	−50.1663	−1.65483	−0.827417	−	0.561587i	$-0.810191\pi$
$919$	−50.1663	−1.65483	−0.827417	+	0.561587i	$0.810191\pi$
$920$	−70.0515	−2.30953
$921$	0	0
$922$	−37.9951	−1.25130
$923$	56.1558	1.84839
$924$	0	0
$925$	−15.8328	−0.520580
$926$	−7.62166	−0.250463
$927$	0	0
$928$	−20.5469	−0.674487
$929$	−32.3580	−1.06163	−0.530816	−	0.847487i	$-0.678114\pi$
$929$	−32.3580	−1.06163	−0.530816	+	0.847487i	$0.678114\pi$
$930$	0	0
$931$	−4.68225	−0.153455
$932$	−1.22249	−0.0400441
$933$	0	0
$934$	33.2086	1.08662
$935$	5.57218	0.182230
$936$	0	0
$937$	57.9477	1.89307	0.946535	−	0.322602i	$-0.104557\pi$
$937$	57.9477	1.89307	0.946535	+	0.322602i	$0.104557\pi$
$938$	−9.28856	−0.303282
$939$	0	0
$940$	−8.62692	−0.281379
$941$	56.9973	1.85806	0.929029	−	0.370007i	$-0.120645\pi$
$941$	56.9973	1.85806	0.929029	+	0.370007i	$0.120645\pi$
$942$	0	0
$943$	10.9805	0.357574
$944$	−22.5727	−0.734679
$945$	0	0
$946$	−2.08277	−0.0677166
$947$	43.1879	1.40342	0.701709	−	0.712464i	$-0.252421\pi$
$947$	43.1879	1.40342	0.701709	+	0.712464i	$0.252421\pi$
$948$	0	0
$949$	1.85909	0.0603485
$950$	−2.11630	−0.0686619
$951$	0	0
$952$	18.7380	0.607302
$953$	27.6331	0.895124	0.447562	−	0.894253i	$-0.352292\pi$
$953$	27.6331	0.895124	0.447562	+	0.894253i	$0.352292\pi$
$954$	0	0
$955$	39.0090	1.26230
$956$	−3.49211	−0.112943
$957$	0	0
$958$	10.2238	0.330316
$959$	−8.82892	−0.285101
$960$	0	0
$961$	21.4973	0.693460
$962$	−37.3165	−1.20313
$963$	0	0
$964$	−3.23133	−0.104074
$965$	−2.01898	−0.0649931
$966$	0	0
$967$	48.4511	1.55808	0.779041	−	0.626973i	$-0.215706\pi$
$967$	48.4511	1.55808	0.779041	+	0.626973i	$0.215706\pi$
$968$	−33.1003	−1.06388
$969$	0	0
$970$	−21.6367	−0.694711
$971$	−5.66511	−0.181802	−0.0909010	−	0.995860i	$-0.528975\pi$
$971$	−5.66511	−0.181802	−0.0909010	+	0.995860i	$0.528975\pi$
$972$	0	0
$973$	−0.524705	−0.0168213
$974$	−10.7816	−0.345466
$975$	0	0
$976$	−9.23425	−0.295581
$977$	−19.6202	−0.627706	−0.313853	−	0.949472i	$-0.601620\pi$
$977$	−19.6202	−0.627706	−0.313853	+	0.949472i	$0.601620\pi$
$978$	0	0
$979$	3.93083	0.125630
$980$	6.86642	0.219340
$981$	0	0
$982$	−31.3561	−1.00061
$983$	−19.4545	−0.620501	−0.310250	−	0.950655i	$-0.600413\pi$
$983$	−19.4545	−0.620501	−0.310250	+	0.950655i	$0.600413\pi$
$984$	0	0
$985$	0.795823	0.0253570
$986$	55.1005	1.75476
$987$	0	0
$988$	1.45802	0.0463856
$989$	37.2638	1.18492
$990$	0	0
$991$	−35.6062	−1.13107	−0.565534	−	0.824725i	$-0.691330\pi$
$991$	−35.6062	−1.13107	−0.565534	+	0.824725i	$0.691330\pi$
$992$	−18.1551	−0.576424
$993$	0	0
$994$	−20.2215	−0.641386
$995$	16.9041	0.535896
$996$	0	0
$997$	−14.0161	−0.443893	−0.221946	−	0.975059i	$-0.571241\pi$
$997$	−14.0161	−0.443893	−0.221946	+	0.975059i	$0.571241\pi$
$998$	45.2990	1.43392
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.2.a.i.1.3		7
3.2	odd	2		127.2.a.b.1.5	✓	7
12.11	even	2		2032.2.a.p.1.1		7
15.14	odd	2		3175.2.a.j.1.3		7
21.20	even	2		6223.2.a.h.1.5		7
24.5	odd	2		8128.2.a.bi.1.1		7
24.11	even	2		8128.2.a.bj.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
127.2.a.b.1.5	✓	7	3.2	odd	2
1143.2.a.i.1.3		7	1.1	even	1	trivial
2032.2.a.p.1.1		7	12.11	even	2
3175.2.a.j.1.3		7	15.14	odd	2
6223.2.a.h.1.5		7	21.20	even	2
8128.2.a.bi.1.1		7	24.5	odd	2
8128.2.a.bj.1.7		7	24.11	even	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:	\( N \)	\(=\)	\( 1143 = 3^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1143.a (trivial)

\(f(q)\)	\(=\)	\(q-1.24403 q^{2} -0.452382 q^{4} +2.65960 q^{5} -1.13710 q^{7} +3.05084 q^{8} +O(q^{10})\)	\(q-1.24403 q^{2} -0.452382 q^{4} +2.65960 q^{5} -1.13710 q^{7} +3.05084 q^{8} -3.30863 q^{10} -0.387885 q^{11} -3.92836 q^{13} +1.41459 q^{14} -2.89059 q^{16} -5.40139 q^{17} +0.820438 q^{19} -1.20316 q^{20} +0.482542 q^{22} -8.63338 q^{23} +2.07348 q^{25} +4.88701 q^{26} +0.514402 q^{28} +8.20007 q^{29} +7.24550 q^{31} -2.50570 q^{32} +6.71951 q^{34} -3.02422 q^{35} -7.63586 q^{37} -1.02065 q^{38} +8.11403 q^{40} -1.27186 q^{41} -4.31624 q^{43} +0.175472 q^{44} +10.7402 q^{46} +7.17025 q^{47} -5.70701 q^{49} -2.57948 q^{50} +1.77712 q^{52} -12.0735 q^{53} -1.03162 q^{55} -3.46910 q^{56} -10.2012 q^{58} +7.80904 q^{59} +3.19459 q^{61} -9.01364 q^{62} +8.89835 q^{64} -10.4479 q^{65} -6.56628 q^{67} +2.44349 q^{68} +3.76223 q^{70} -14.2950 q^{71} -0.473248 q^{73} +9.49926 q^{74} -0.371151 q^{76} +0.441063 q^{77} -5.31508 q^{79} -7.68781 q^{80} +1.58224 q^{82} +5.39344 q^{83} -14.3655 q^{85} +5.36955 q^{86} -1.18338 q^{88} -10.1340 q^{89} +4.46692 q^{91} +3.90559 q^{92} -8.92003 q^{94} +2.18204 q^{95} +6.53946 q^{97} +7.09971 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{2} + 6 q^{4} - 8 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 7 * q - 2 * q^2 + 6 * q^4 - 8 * q^5 - 3 * q^7 - 3 * q^8	\( 7 q - 2 q^{2} + 6 q^{4} - 8 q^{5} - 3 q^{7} - 3 q^{8} - 5 q^{10} - q^{13} + 4 q^{14} - 8 q^{16} - 24 q^{17} - 5 q^{19} - 11 q^{20} - 9 q^{22} + q^{23} + 7 q^{25} + 4 q^{26} - 26 q^{28} + 7 q^{29} - 8 q^{31} + 2 q^{32} - q^{34} - 4 q^{35} - 6 q^{37} - 29 q^{38} - 3 q^{40} - 14 q^{41} - q^{43} + 21 q^{44} - 3 q^{46} - 25 q^{47} - 10 q^{50} + 6 q^{52} - 29 q^{53} - 23 q^{55} - 9 q^{56} - 22 q^{58} + 12 q^{59} + 7 q^{61} - 4 q^{62} - 3 q^{64} - 3 q^{65} - 25 q^{67} - 53 q^{68} + 51 q^{70} - 7 q^{71} + 13 q^{73} - 11 q^{74} + 12 q^{76} - 19 q^{77} - 23 q^{79} + 14 q^{80} + 26 q^{82} - 26 q^{83} + 15 q^{85} - 5 q^{86} + 25 q^{88} - 13 q^{89} - 40 q^{91} + 32 q^{92} - 19 q^{94} + 40 q^{95} - 5 q^{97} + 11 q^{98}+O(q^{100}) \) 7 * q - 2 * q^2 + 6 * q^4 - 8 * q^5 - 3 * q^7 - 3 * q^8 - 5 * q^10 - q^13 + 4 * q^14 - 8 * q^16 - 24 * q^17 - 5 * q^19 - 11 * q^20 - 9 * q^22 + q^23 + 7 * q^25 + 4 * q^26 - 26 * q^28 + 7 * q^29 - 8 * q^31 + 2 * q^32 - q^34 - 4 * q^35 - 6 * q^37 - 29 * q^38 - 3 * q^40 - 14 * q^41 - q^43 + 21 * q^44 - 3 * q^46 - 25 * q^47 - 10 * q^50 + 6 * q^52 - 29 * q^53 - 23 * q^55 - 9 * q^56 - 22 * q^58 + 12 * q^59 + 7 * q^61 - 4 * q^62 - 3 * q^64 - 3 * q^65 - 25 * q^67 - 53 * q^68 + 51 * q^70 - 7 * q^71 + 13 * q^73 - 11 * q^74 + 12 * q^76 - 19 * q^77 - 23 * q^79 + 14 * q^80 + 26 * q^82 - 26 * q^83 + 15 * q^85 - 5 * q^86 + 25 * q^88 - 13 * q^89 - 40 * q^91 + 32 * q^92 - 19 * q^94 + 40 * q^95 - 5 * q^97 + 11 * q^98

Introduction

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