LMFDB - Embedded newform 1143.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1143,4,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, 4, names="a")

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

chi := DirichletCharacter("1143.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1143 = 3^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 4 $
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:	$67.4391831366$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 5 x^{12} - 55 x^{11} + 264 x^{10} + 1126 x^{9} - 5085 x^{8} - 10823 x^{7} + 44242 x^{6} + \cdots - 130048 $ x^13 - 5x^12 - 55x^11 + 264x^10 + 1126x^9 - 5085x^8 - 10823x^7 + 44242x^6 + 52047x^5 - 178872x^4 - 113160x^3 + 298560x^2 + 65136x - 130048
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 127)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.48248$ 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)

$f(q)$	$=$	$q-4.48248 q^{2} +12.0927 q^{4} +5.99266 q^{5} -20.2899 q^{7} -18.3452 q^{8} +O(q^{10})$	\(q-4.48248 q^{2} +12.0927 q^{4} +5.99266 q^{5} -20.2899 q^{7} -18.3452 q^{8} -26.8620 q^{10} +8.70120 q^{11} -57.1649 q^{13} +90.9491 q^{14} -14.5090 q^{16} +34.0983 q^{17} +58.9447 q^{19} +72.4672 q^{20} -39.0030 q^{22} -147.409 q^{23} -89.0880 q^{25} +256.241 q^{26} -245.359 q^{28} -127.619 q^{29} -98.1327 q^{31} +211.798 q^{32} -152.845 q^{34} -121.591 q^{35} -72.2719 q^{37} -264.219 q^{38} -109.937 q^{40} +15.2752 q^{41} +427.027 q^{43} +105.221 q^{44} +660.756 q^{46} +356.718 q^{47} +68.6803 q^{49} +399.335 q^{50} -691.275 q^{52} +582.203 q^{53} +52.1434 q^{55} +372.223 q^{56} +572.049 q^{58} +79.1281 q^{59} -297.291 q^{61} +439.878 q^{62} -833.310 q^{64} -342.570 q^{65} -162.865 q^{67} +412.339 q^{68} +545.028 q^{70} -1137.70 q^{71} -756.758 q^{73} +323.957 q^{74} +712.798 q^{76} -176.547 q^{77} +55.7276 q^{79} -86.9475 q^{80} -68.4710 q^{82} +789.369 q^{83} +204.340 q^{85} -1914.14 q^{86} -159.626 q^{88} +1351.23 q^{89} +1159.87 q^{91} -1782.56 q^{92} -1598.98 q^{94} +353.236 q^{95} +1221.35 q^{97} -307.858 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 8 q^{2} + 34 q^{4} + 46 q^{5} - 26 q^{7} + 117 q^{8}+O(q^{10}) $ 13 * q + 8 * q^2 + 34 * q^4 + 46 * q^5 - 26 * q^7 + 117 * q^8	$ 13 q + 8 q^{2} + 34 q^{4} + 46 q^{5} - 26 q^{7} + 117 q^{8} - 25 q^{10} + 53 q^{11} - 75 q^{13} + 152 q^{14} + 82 q^{16} + 479 q^{17} - 209 q^{19} + 533 q^{20} - 407 q^{22} + 376 q^{23} + 9 q^{25} + 622 q^{26} - 258 q^{28} + 158 q^{29} - 307 q^{31} + 609 q^{32} + 948 q^{34} + 512 q^{35} - 171 q^{37} - 712 q^{38} + 1897 q^{40} + 641 q^{41} + 530 q^{43} - 1328 q^{44} + 2051 q^{46} + 555 q^{47} + 357 q^{49} - 808 q^{50} + 2473 q^{52} + 1640 q^{53} + 540 q^{55} - 551 q^{56} + 1328 q^{58} + 860 q^{59} + 191 q^{61} - 367 q^{62} + 1915 q^{64} + 1584 q^{65} + 912 q^{67} + 1873 q^{68} + 2329 q^{70} - 115 q^{71} - 2563 q^{73} - 470 q^{74} - 192 q^{76} + 3338 q^{77} + 169 q^{79} + 3194 q^{80} + 54 q^{82} + 1688 q^{83} + 480 q^{85} + 485 q^{86} - 1192 q^{88} + 2752 q^{89} - 398 q^{91} + 846 q^{92} + 531 q^{94} + 1766 q^{95} - 2124 q^{97} + 1479 q^{98}+O(q^{100}) $ 13 * q + 8 * q^2 + 34 * q^4 + 46 * q^5 - 26 * q^7 + 117 * q^8 - 25 * q^10 + 53 * q^11 - 75 * q^13 + 152 * q^14 + 82 * q^16 + 479 * q^17 - 209 * q^19 + 533 * q^20 - 407 * q^22 + 376 * q^23 + 9 * q^25 + 622 * q^26 - 258 * q^28 + 158 * q^29 - 307 * q^31 + 609 * q^32 + 948 * q^34 + 512 * q^35 - 171 * q^37 - 712 * q^38 + 1897 * q^40 + 641 * q^41 + 530 * q^43 - 1328 * q^44 + 2051 * q^46 + 555 * q^47 + 357 * q^49 - 808 * q^50 + 2473 * q^52 + 1640 * q^53 + 540 * q^55 - 551 * q^56 + 1328 * q^58 + 860 * q^59 + 191 * q^61 - 367 * q^62 + 1915 * q^64 + 1584 * q^65 + 912 * q^67 + 1873 * q^68 + 2329 * q^70 - 115 * q^71 - 2563 * q^73 - 470 * q^74 - 192 * q^76 + 3338 * q^77 + 169 * q^79 + 3194 * q^80 + 54 * q^82 + 1688 * q^83 + 480 * q^85 + 485 * q^86 - 1192 * q^88 + 2752 * q^89 - 398 * q^91 + 846 * q^92 + 531 * q^94 + 1766 * q^95 - 2124 * q^97 + 1479 * 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 $

$2$	−4.48248	−1.58480	−0.792398	−	0.610004i	$-0.791168\pi$
$2$	−4.48248	−1.58480	−0.792398	+	0.610004i	$0.791168\pi$
$3$	0	0
$3$	0	0
$4$	12.0927	1.51158
$5$	5.99266	0.536000	0.268000	−	0.963419i	$-0.413637\pi$
$5$	5.99266	0.536000	0.268000	+	0.963419i	$0.413637\pi$
$6$	0	0
$7$	−20.2899	−1.09555	−0.547776	−	0.836625i	$-0.684525\pi$
$7$	−20.2899	−1.09555	−0.547776	+	0.836625i	$0.684525\pi$
$8$	−18.3452	−0.810753
$9$	0	0
$10$	−26.8620	−0.849451
$11$	8.70120	0.238501	0.119251	−	0.992864i	$-0.461951\pi$
$11$	8.70120	0.238501	0.119251	+	0.992864i	$0.461951\pi$
$12$	0	0
$13$	−57.1649	−1.21959	−0.609796	−	0.792559i	$-0.708749\pi$
$13$	−57.1649	−1.21959	−0.609796	+	0.792559i	$0.708749\pi$
$14$	90.9491	1.73623
$15$	0	0
$16$	−14.5090	−0.226703
$17$	34.0983	0.486473	0.243237	−	0.969967i	$-0.421791\pi$
$17$	34.0983	0.486473	0.243237	+	0.969967i	$0.421791\pi$
$18$	0	0
$19$	58.9447	0.711729	0.355864	−	0.934538i	$-0.384186\pi$
$19$	58.9447	0.711729	0.355864	+	0.934538i	$0.384186\pi$
$20$	72.4672	0.810208
$21$	0	0
$22$	−39.0030	−0.377976
$23$	−147.409	−1.33638	−0.668192	−	0.743989i	$-0.732931\pi$
$23$	−147.409	−1.33638	−0.668192	+	0.743989i	$0.732931\pi$
$24$	0	0
$25$	−89.0880	−0.712704
$26$	256.241	1.93280
$27$	0	0
$28$	−245.359	−1.65602
$29$	−127.619	−0.817180	−0.408590	−	0.912718i	$-0.633979\pi$
$29$	−127.619	−0.817180	−0.408590	+	0.912718i	$0.633979\pi$
$30$	0	0
$31$	−98.1327	−0.568553	−0.284277	−	0.958742i	$-0.591753\pi$
$31$	−98.1327	−0.568553	−0.284277	+	0.958742i	$0.591753\pi$
$32$	211.798	1.17003
$33$	0	0
$34$	−152.845	−0.770961
$35$	−121.591	−0.587216
$36$	0	0
$37$	−72.2719	−0.321120	−0.160560	−	0.987026i	$-0.551330\pi$
$37$	−72.2719	−0.321120	−0.160560	+	0.987026i	$0.551330\pi$
$38$	−264.219	−1.12795
$39$	0	0
$40$	−109.937	−0.434564
$41$	15.2752	0.0581852	0.0290926	−	0.999577i	$-0.490738\pi$
$41$	15.2752	0.0581852	0.0290926	+	0.999577i	$0.490738\pi$
$42$	0	0
$43$	427.027	1.51444	0.757221	−	0.653159i	$-0.226557\pi$
$43$	427.027	1.51444	0.757221	+	0.653159i	$0.226557\pi$
$44$	105.221	0.360514
$45$	0	0
$46$	660.756	2.11790
$47$	356.718	1.10708	0.553540	−	0.832823i	$-0.313277\pi$
$47$	356.718	1.10708	0.553540	+	0.832823i	$0.313277\pi$
$48$	0	0
$49$	68.6803	0.200234
$50$	399.335	1.12949
$51$	0	0
$52$	−691.275	−1.84351
$53$	582.203	1.50890	0.754450	−	0.656357i	$-0.227904\pi$
$53$	582.203	1.50890	0.754450	+	0.656357i	$0.227904\pi$
$54$	0	0
$55$	52.1434	0.127837
$56$	372.223	0.888222
$57$	0	0
$58$	572.049	1.29506
$59$	79.1281	0.174603	0.0873017	−	0.996182i	$-0.472176\pi$
$59$	79.1281	0.174603	0.0873017	+	0.996182i	$0.472176\pi$
$60$	0	0
$61$	−297.291	−0.624004	−0.312002	−	0.950081i	$-0.601000\pi$
$61$	−297.291	−0.624004	−0.312002	+	0.950081i	$0.601000\pi$
$62$	439.878	0.901041
$63$	0	0
$64$	−833.310	−1.62756
$65$	−342.570	−0.653701
$66$	0	0
$67$	−162.865	−0.296972	−0.148486	−	0.988915i	$-0.547440\pi$
$67$	−162.865	−0.296972	−0.148486	+	0.988915i	$0.547440\pi$
$68$	412.339	0.735344
$69$	0	0
$70$	545.028	0.930618
$71$	−1137.70	−1.90168	−0.950842	−	0.309675i	$-0.899780\pi$
$71$	−1137.70	−1.90168	−0.950842	+	0.309675i	$0.899780\pi$
$72$	0	0
$73$	−756.758	−1.21331	−0.606657	−	0.794964i	$-0.707490\pi$
$73$	−756.758	−1.21331	−0.606657	+	0.794964i	$0.707490\pi$
$74$	323.957	0.508909
$75$	0	0
$76$	712.798	1.07584
$77$	−176.547	−0.261290
$78$	0	0
$79$	55.7276	0.0793651	0.0396825	−	0.999212i	$-0.487365\pi$
$79$	55.7276	0.0793651	0.0396825	+	0.999212i	$0.487365\pi$
$80$	−86.9475	−0.121513
$81$	0	0
$82$	−68.4710	−0.0922117
$83$	789.369	1.04391	0.521955	−	0.852973i	$-0.325203\pi$
$83$	789.369	1.04391	0.521955	+	0.852973i	$0.325203\pi$
$84$	0	0
$85$	204.340	0.260750
$86$	−1914.14	−2.40008
$87$	0	0
$88$	−159.626	−0.193365
$89$	1351.23	1.60933	0.804664	−	0.593730i	$-0.202345\pi$
$89$	1351.23	1.60933	0.804664	+	0.593730i	$0.202345\pi$
$90$	0	0
$91$	1159.87	1.33613
$92$	−1782.56	−2.02005
$93$	0	0
$94$	−1598.98	−1.75450
$95$	353.236	0.381487
$96$	0	0
$97$	1221.35	1.27844	0.639222	−	0.769022i	$-0.279256\pi$
$97$	1221.35	1.27844	0.639222	+	0.769022i	$0.279256\pi$
$98$	−307.858	−0.317330
$99$	0	0
$100$	−1077.31	−1.07731
$101$	−1035.00	−1.01966	−0.509832	−	0.860274i	$-0.670292\pi$
$101$	−1035.00	−1.01966	−0.509832	+	0.860274i	$0.670292\pi$
$102$	0	0
$103$	−1436.04	−1.37376	−0.686878	−	0.726772i	$-0.741020\pi$
$103$	−1436.04	−1.37376	−0.686878	+	0.726772i	$0.741020\pi$
$104$	1048.70	0.988787
$105$	0	0
$106$	−2609.72	−2.39130
$107$	1344.95	1.21515	0.607574	−	0.794263i	$-0.292143\pi$
$107$	1344.95	1.21515	0.607574	+	0.794263i	$0.292143\pi$
$108$	0	0
$109$	491.906	0.432257	0.216129	−	0.976365i	$-0.430657\pi$
$109$	491.906	0.432257	0.216129	+	0.976365i	$0.430657\pi$
$110$	−233.732	−0.202595
$111$	0	0
$112$	294.386	0.248365
$113$	−866.204	−0.721111	−0.360556	−	0.932738i	$-0.617413\pi$
$113$	−866.204	−0.721111	−0.360556	+	0.932738i	$0.617413\pi$
$114$	0	0
$115$	−883.370	−0.716302
$116$	−1543.25	−1.23523
$117$	0	0
$118$	−354.690	−0.276711
$119$	−691.851	−0.532957
$120$	0	0
$121$	−1255.29	−0.943117
$122$	1332.60	0.988919
$123$	0	0
$124$	−1186.68	−0.859414
$125$	−1282.96	−0.918010
$126$	0	0
$127$	127.000	0.0887357
$127$	127.000	0.0887357
$128$	2040.91	1.40932
$129$	0	0
$130$	1535.56	1.03598
$131$	−1372.81	−0.915594	−0.457797	−	0.889057i	$-0.651361\pi$
$131$	−1372.81	−0.915594	−0.457797	+	0.889057i	$0.651361\pi$
$132$	0	0
$133$	−1195.98	−0.779736
$134$	730.039	0.470640
$135$	0	0
$136$	−625.541	−0.394410
$137$	1734.43	1.08163	0.540813	−	0.841143i	$-0.318117\pi$
$137$	1734.43	1.08163	0.540813	+	0.841143i	$0.318117\pi$
$138$	0	0
$139$	−1159.62	−0.707608	−0.353804	−	0.935320i	$-0.615112\pi$
$139$	−1159.62	−0.707608	−0.353804	+	0.935320i	$0.615112\pi$
$140$	−1470.35	−0.887625
$141$	0	0
$142$	5099.70	3.01378
$143$	−497.403	−0.290874
$144$	0	0
$145$	−764.777	−0.438009
$146$	3392.16	1.92285
$147$	0	0
$148$	−873.959	−0.485398
$149$	−3378.34	−1.85748	−0.928739	−	0.370735i	$-0.879106\pi$
$149$	−3378.34	−1.85748	−0.928739	+	0.370735i	$0.879106\pi$
$150$	0	0
$151$	−1035.75	−0.558199	−0.279099	−	0.960262i	$-0.590036\pi$
$151$	−1035.75	−0.558199	−0.279099	+	0.960262i	$0.590036\pi$
$152$	−1081.36	−0.577036
$153$	0	0
$154$	791.367	0.414092
$155$	−588.076	−0.304745
$156$	0	0
$157$	1974.36	1.00364	0.501819	−	0.864973i	$-0.332664\pi$
$157$	1974.36	1.00364	0.501819	+	0.864973i	$0.332664\pi$
$158$	−249.798	−0.125778
$159$	0	0
$160$	1269.24	0.627137
$161$	2990.91	1.46408
$162$	0	0
$163$	3758.38	1.80600	0.903002	−	0.429636i	$-0.141358\pi$
$163$	3758.38	1.80600	0.903002	+	0.429636i	$0.141358\pi$
$164$	184.718	0.0879517
$165$	0	0
$166$	−3538.33	−1.65438
$167$	−708.610	−0.328346	−0.164173	−	0.986432i	$-0.552496\pi$
$167$	−708.610	−0.328346	−0.164173	+	0.986432i	$0.552496\pi$
$168$	0	0
$169$	1070.83	0.487403
$170$	−915.948	−0.413235
$171$	0	0
$172$	5163.89	2.28920
$173$	2153.41	0.946363	0.473181	−	0.880965i	$-0.343105\pi$
$173$	2153.41	0.946363	0.473181	+	0.880965i	$0.343105\pi$
$174$	0	0
$175$	1807.59	0.780804
$176$	−126.246	−0.0540689
$177$	0	0
$178$	−6056.87	−2.55046
$179$	791.823	0.330635	0.165317	−	0.986240i	$-0.447135\pi$
$179$	791.823	0.330635	0.165317	+	0.986240i	$0.447135\pi$
$180$	0	0
$181$	3423.95	1.40608	0.703040	−	0.711150i	$-0.251825\pi$
$181$	3423.95	1.40608	0.703040	+	0.711150i	$0.251825\pi$
$182$	−5199.10	−2.11749
$183$	0	0
$184$	2704.25	1.08348
$185$	−433.101	−0.172120
$186$	0	0
$187$	296.696	0.116024
$188$	4313.67	1.67344
$189$	0	0
$190$	−1583.37	−0.604579
$191$	1191.56	0.451404	0.225702	−	0.974196i	$-0.427532\pi$
$191$	1191.56	0.451404	0.225702	+	0.974196i	$0.427532\pi$
$192$	0	0
$193$	−3571.53	−1.33204	−0.666021	−	0.745933i	$-0.732004\pi$
$193$	−3571.53	−1.33204	−0.666021	+	0.745933i	$0.732004\pi$
$194$	−5474.67	−2.02607
$195$	0	0
$196$	830.526	0.302670
$197$	4386.46	1.58641	0.793204	−	0.608956i	$-0.208411\pi$
$197$	4386.46	1.58641	0.793204	+	0.608956i	$0.208411\pi$
$198$	0	0
$199$	−1350.45	−0.481061	−0.240531	−	0.970642i	$-0.577321\pi$
$199$	−1350.45	−0.481061	−0.240531	+	0.970642i	$0.577321\pi$
$200$	1634.34	0.577827
$201$	0	0
$202$	4639.35	1.61596
$203$	2589.37	0.895263
$204$	0	0
$205$	91.5394	0.0311873
$206$	6437.01	2.17713
$207$	0	0
$208$	829.405	0.276485
$209$	512.890	0.169748
$210$	0	0
$211$	4659.45	1.52024	0.760118	−	0.649785i	$-0.225141\pi$
$211$	4659.45	1.52024	0.760118	+	0.649785i	$0.225141\pi$
$212$	7040.38	2.28083
$213$	0	0
$214$	−6028.69	−1.92576
$215$	2559.03	0.811741
$216$	0	0
$217$	1991.10	0.622880
$218$	−2204.96	−0.685040
$219$	0	0
$220$	630.552	0.193235
$221$	−1949.22	−0.593299
$222$	0	0
$223$	3249.25	0.975722	0.487861	−	0.872921i	$-0.337777\pi$
$223$	3249.25	0.975722	0.487861	+	0.872921i	$0.337777\pi$
$224$	−4297.37	−1.28183
$225$	0	0
$226$	3882.74	1.14281
$227$	2213.55	0.647217	0.323608	−	0.946191i	$-0.395104\pi$
$227$	2213.55	0.647217	0.323608	+	0.946191i	$0.395104\pi$
$228$	0	0
$229$	5292.29	1.52718	0.763590	−	0.645701i	$-0.223435\pi$
$229$	5292.29	1.52718	0.763590	+	0.645701i	$0.223435\pi$
$230$	3959.69	1.13519
$231$	0	0
$232$	2341.20	0.662531
$233$	5572.29	1.56675	0.783375	−	0.621550i	$-0.213497\pi$
$233$	5572.29	1.56675	0.783375	+	0.621550i	$0.213497\pi$
$234$	0	0
$235$	2137.69	0.593395
$236$	956.868	0.263927
$237$	0	0
$238$	3101.21	0.844628
$239$	−294.219	−0.0796294	−0.0398147	−	0.999207i	$-0.512677\pi$
$239$	−294.219	−0.0796294	−0.0398147	+	0.999207i	$0.512677\pi$
$240$	0	0
$241$	2884.22	0.770908	0.385454	−	0.922727i	$-0.374045\pi$
$241$	2884.22	0.770908	0.385454	+	0.922727i	$0.374045\pi$
$242$	5626.81	1.49465
$243$	0	0
$244$	−3595.04	−0.943232
$245$	411.578	0.107325
$246$	0	0
$247$	−3369.57	−0.868019
$248$	1800.27	0.460956
$249$	0	0
$250$	5750.83	1.45486
$251$	−2422.46	−0.609181	−0.304590	−	0.952483i	$-0.598520\pi$
$251$	−2422.46	−0.609181	−0.304590	+	0.952483i	$0.598520\pi$
$252$	0	0
$253$	−1282.63	−0.318729
$254$	−569.275	−0.140628
$255$	0	0
$256$	−2481.87	−0.605926
$257$	−3089.95	−0.749983	−0.374991	−	0.927028i	$-0.622354\pi$
$257$	−3089.95	−0.749983	−0.374991	+	0.927028i	$0.622354\pi$
$258$	0	0
$259$	1466.39	0.351803
$260$	−4142.58	−0.988123
$261$	0	0
$262$	6153.59	1.45103
$263$	−2815.98	−0.660232	−0.330116	−	0.943940i	$-0.607088\pi$
$263$	−2815.98	−0.660232	−0.330116	+	0.943940i	$0.607088\pi$
$264$	0	0
$265$	3488.95	0.808771
$266$	5360.97	1.23572
$267$	0	0
$268$	−1969.47	−0.448897
$269$	3784.44	0.857774	0.428887	−	0.903358i	$-0.358906\pi$
$269$	3784.44	0.857774	0.428887	+	0.903358i	$0.358906\pi$
$270$	0	0
$271$	1369.04	0.306876	0.153438	−	0.988158i	$-0.450965\pi$
$271$	1369.04	0.306876	0.153438	+	0.988158i	$0.450965\pi$
$272$	−494.732	−0.110285
$273$	0	0
$274$	−7774.57	−1.71416
$275$	−775.172	−0.169981
$276$	0	0
$277$	5440.34	1.18007	0.590033	−	0.807379i	$-0.299115\pi$
$277$	5440.34	1.18007	0.590033	+	0.807379i	$0.299115\pi$
$278$	5197.97	1.12142
$279$	0	0
$280$	2230.61	0.476087
$281$	3506.57	0.744428	0.372214	−	0.928147i	$-0.378599\pi$
$281$	3506.57	0.744428	0.372214	+	0.928147i	$0.378599\pi$
$282$	0	0
$283$	−4260.58	−0.894930	−0.447465	−	0.894301i	$-0.647673\pi$
$283$	−4260.58	−0.894930	−0.447465	+	0.894301i	$0.647673\pi$
$284$	−13757.8	−2.87455
$285$	0	0
$286$	2229.60	0.460976
$287$	−309.933	−0.0637449
$288$	0	0
$289$	−3750.31	−0.763344
$290$	3428.10	0.694155
$291$	0	0
$292$	−9151.21	−1.83402
$293$	5874.40	1.17128	0.585642	−	0.810570i	$-0.300842\pi$
$293$	5874.40	1.17128	0.585642	+	0.810570i	$0.300842\pi$
$294$	0	0
$295$	474.188	0.0935874
$296$	1325.84	0.260349
$297$	0	0
$298$	15143.3	2.94372
$299$	8426.60	1.62984
$300$	0	0
$301$	−8664.34	−1.65915
$302$	4642.73	0.884632
$303$	0	0
$304$	−855.229	−0.161351
$305$	−1781.57	−0.334466
$306$	0	0
$307$	5101.93	0.948477	0.474239	−	0.880396i	$-0.342723\pi$
$307$	5101.93	0.948477	0.474239	+	0.880396i	$0.342723\pi$
$308$	−2134.92	−0.394961
$309$	0	0
$310$	2636.04	0.482958
$311$	7090.92	1.29289	0.646446	−	0.762960i	$-0.276255\pi$
$311$	7090.92	1.29289	0.646446	+	0.762960i	$0.276255\pi$
$312$	0	0
$313$	9643.89	1.74155	0.870775	−	0.491682i	$-0.163618\pi$
$313$	9643.89	1.74155	0.870775	+	0.491682i	$0.163618\pi$
$314$	−8850.04	−1.59056
$315$	0	0
$316$	673.894	0.119967
$317$	4662.64	0.826120	0.413060	−	0.910704i	$-0.364460\pi$
$317$	4662.64	0.826120	0.413060	+	0.910704i	$0.364460\pi$
$318$	0	0
$319$	−1110.44	−0.194898
$320$	−4993.75	−0.872372
$321$	0	0
$322$	−13406.7	−2.32027
$323$	2009.91	0.346237
$324$	0	0
$325$	5092.71	0.869208
$326$	−16846.9	−2.86215
$327$	0	0
$328$	−280.228	−0.0471738
$329$	−7237.78	−1.21286
$330$	0	0
$331$	−2759.66	−0.458261	−0.229131	−	0.973396i	$-0.573588\pi$
$331$	−2759.66	−0.458261	−0.229131	+	0.973396i	$0.573588\pi$
$332$	9545.56	1.57795
$333$	0	0
$334$	3176.33	0.520362
$335$	−975.994	−0.159177
$336$	0	0
$337$	−2462.01	−0.397965	−0.198982	−	0.980003i	$-0.563764\pi$
$337$	−2462.01	−0.397965	−0.198982	+	0.980003i	$0.563764\pi$
$338$	−4799.96	−0.772435
$339$	0	0
$340$	2471.01	0.394145
$341$	−853.872	−0.135601
$342$	0	0
$343$	5565.92	0.876185
$344$	−7833.91	−1.22784
$345$	0	0
$346$	−9652.63	−1.49979
$347$	5933.71	0.917977	0.458989	−	0.888442i	$-0.348212\pi$
$347$	5933.71	0.917977	0.458989	+	0.888442i	$0.348212\pi$
$348$	0	0
$349$	−8965.75	−1.37515	−0.687573	−	0.726116i	$-0.741324\pi$
$349$	−8965.75	−1.37515	−0.687573	+	0.726116i	$0.741324\pi$
$350$	−8102.48	−1.23742
$351$	0	0
$352$	1842.90	0.279054
$353$	−5153.48	−0.777031	−0.388515	−	0.921442i	$-0.627012\pi$
$353$	−5153.48	−0.777031	−0.388515	+	0.921442i	$0.627012\pi$
$354$	0	0
$355$	−6817.83	−1.01930
$356$	16340.0	2.43263
$357$	0	0
$358$	−3549.33	−0.523989
$359$	1783.87	0.262254	0.131127	−	0.991366i	$-0.458140\pi$
$359$	1783.87	0.262254	0.131127	+	0.991366i	$0.458140\pi$
$360$	0	0
$361$	−3384.52	−0.493442
$362$	−15347.8	−2.22835
$363$	0	0
$364$	14025.9	2.01966
$365$	−4535.00	−0.650336
$366$	0	0
$367$	−8063.33	−1.14687	−0.573436	−	0.819250i	$-0.694390\pi$
$367$	−8063.33	−1.14687	−0.573436	+	0.819250i	$0.694390\pi$
$368$	2138.75	0.302962
$369$	0	0
$370$	1941.37	0.272775
$371$	−11812.8	−1.65308
$372$	0	0
$373$	−3328.88	−0.462099	−0.231049	−	0.972942i	$-0.574216\pi$
$373$	−3328.88	−0.462099	−0.231049	+	0.972942i	$0.574216\pi$
$374$	−1329.93	−0.183875
$375$	0	0
$376$	−6544.08	−0.897568
$377$	7295.32	0.996626
$378$	0	0
$379$	−4980.68	−0.675040	−0.337520	−	0.941318i	$-0.609588\pi$
$379$	−4980.68	−0.675040	−0.337520	+	0.941318i	$0.609588\pi$
$380$	4271.56	0.576648
$381$	0	0
$382$	−5341.14	−0.715384
$383$	−3259.30	−0.434836	−0.217418	−	0.976079i	$-0.569764\pi$
$383$	−3259.30	−0.434836	−0.217418	+	0.976079i	$0.569764\pi$
$384$	0	0
$385$	−1057.98	−0.140052
$386$	16009.3	2.11102
$387$	0	0
$388$	14769.3	1.93247
$389$	13363.9	1.74184	0.870922	−	0.491422i	$-0.163523\pi$
$389$	13363.9	1.74184	0.870922	+	0.491422i	$0.163523\pi$
$390$	0	0
$391$	−5026.38	−0.650115
$392$	−1259.96	−0.162340
$393$	0	0
$394$	−19662.2	−2.51414
$395$	333.957	0.0425397
$396$	0	0
$397$	−6376.83	−0.806156	−0.403078	−	0.915166i	$-0.632060\pi$
$397$	−6376.83	−0.806156	−0.403078	+	0.915166i	$0.632060\pi$
$398$	6053.39	0.762384
$399$	0	0
$400$	1292.58	0.161572
$401$	15399.4	1.91772	0.958862	−	0.283874i	$-0.0916197\pi$
$401$	15399.4	1.91772	0.958862	+	0.283874i	$0.0916197\pi$
$402$	0	0
$403$	5609.74	0.693403
$404$	−12515.9	−1.54130
$405$	0	0
$406$	−11606.8	−1.41881
$407$	−628.852	−0.0765873
$408$	0	0
$409$	2084.12	0.251964	0.125982	−	0.992033i	$-0.459792\pi$
$409$	2084.12	0.251964	0.125982	+	0.992033i	$0.459792\pi$
$410$	−410.324	−0.0494255
$411$	0	0
$412$	−17365.5	−2.07655
$413$	−1605.50	−0.191287
$414$	0	0
$415$	4730.42	0.559536
$416$	−12107.4	−1.42696
$417$	0	0
$418$	−2299.02	−0.269016
$419$	16396.9	1.91179	0.955897	−	0.293702i	$-0.0948872\pi$
$419$	16396.9	1.91179	0.955897	+	0.293702i	$0.0948872\pi$
$420$	0	0
$421$	11887.3	1.37614	0.688069	−	0.725646i	$-0.258459\pi$
$421$	11887.3	1.37614	0.688069	+	0.725646i	$0.258459\pi$
$422$	−20885.9	−2.40926
$423$	0	0
$424$	−10680.7	−1.22335
$425$	−3037.75	−0.346711
$426$	0	0
$427$	6032.01	0.683628
$428$	16264.0	1.83679
$429$	0	0
$430$	−11470.8	−1.28644
$431$	−17178.6	−1.91987	−0.959937	−	0.280217i	$-0.909594\pi$
$431$	−17178.6	−1.91987	−0.959937	+	0.280217i	$0.909594\pi$
$432$	0	0
$433$	−3841.73	−0.426378	−0.213189	−	0.977011i	$-0.568385\pi$
$433$	−3841.73	−0.426378	−0.213189	+	0.977011i	$0.568385\pi$
$434$	−8925.08	−0.987138
$435$	0	0
$436$	5948.44	0.653392
$437$	−8688.96	−0.951143
$438$	0	0
$439$	−6038.50	−0.656496	−0.328248	−	0.944591i	$-0.606458\pi$
$439$	−6038.50	−0.656496	−0.328248	+	0.944591i	$0.606458\pi$
$440$	−956.583	−0.103644
$441$	0	0
$442$	8737.36	0.940258
$443$	−9137.40	−0.979980	−0.489990	−	0.871728i	$-0.663000\pi$
$443$	−9137.40	−0.979980	−0.489990	+	0.871728i	$0.663000\pi$
$444$	0	0
$445$	8097.48	0.862600
$446$	−14564.7	−1.54632
$447$	0	0
$448$	16907.8	1.78307
$449$	8984.62	0.944344	0.472172	−	0.881506i	$-0.343470\pi$
$449$	8984.62	0.944344	0.472172	+	0.881506i	$0.343470\pi$
$450$	0	0
$451$	132.913	0.0138772
$452$	−10474.7	−1.09002
$453$	0	0
$454$	−9922.18	−1.02571
$455$	6950.71	0.716164
$456$	0	0
$457$	10388.7	1.06337	0.531686	−	0.846942i	$-0.321559\pi$
$457$	10388.7	1.06337	0.531686	+	0.846942i	$0.321559\pi$
$458$	−23722.6	−2.42027
$459$	0	0
$460$	−10682.3	−1.08275
$461$	−4180.70	−0.422375	−0.211187	−	0.977446i	$-0.567733\pi$
$461$	−4180.70	−0.422375	−0.211187	+	0.977446i	$0.567733\pi$
$462$	0	0
$463$	−5078.19	−0.509727	−0.254864	−	0.966977i	$-0.582031\pi$
$463$	−5078.19	−0.509727	−0.254864	+	0.966977i	$0.582031\pi$
$464$	1851.62	0.185257
$465$	0	0
$466$	−24977.7	−2.48298
$467$	−3968.97	−0.393280	−0.196640	−	0.980476i	$-0.563003\pi$
$467$	−3968.97	−0.393280	−0.196640	+	0.980476i	$0.563003\pi$
$468$	0	0
$469$	3304.51	0.325348
$470$	−9582.18	−0.940410
$471$	0	0
$472$	−1451.62	−0.141560
$473$	3715.65	0.361196
$474$	0	0
$475$	−5251.27	−0.507252
$476$	−8366.31	−0.805608
$477$	0	0
$478$	1318.83	0.126196
$479$	−843.501	−0.0804604	−0.0402302	−	0.999190i	$-0.512809\pi$
$479$	−843.501	−0.0804604	−0.0402302	+	0.999190i	$0.512809\pi$
$480$	0	0
$481$	4131.41	0.391635
$482$	−12928.5	−1.22173
$483$	0	0
$484$	−15179.8	−1.42560
$485$	7319.13	0.685246
$486$	0	0
$487$	4366.33	0.406277	0.203139	−	0.979150i	$-0.434886\pi$
$487$	4366.33	0.406277	0.203139	+	0.979150i	$0.434886\pi$
$488$	5453.88	0.505913
$489$	0	0
$490$	−1844.89	−0.170089
$491$	11403.8	1.04816	0.524078	−	0.851670i	$-0.324410\pi$
$491$	11403.8	1.04816	0.524078	+	0.851670i	$0.324410\pi$
$492$	0	0
$493$	−4351.58	−0.397536
$494$	15104.0	1.37563
$495$	0	0
$496$	1423.81	0.128893
$497$	23083.7	2.08339
$498$	0	0
$499$	13001.5	1.16639	0.583194	−	0.812333i	$-0.301803\pi$
$499$	13001.5	1.16639	0.583194	+	0.812333i	$0.301803\pi$
$500$	−15514.4	−1.38765
$501$	0	0
$502$	10858.6	0.965428
$503$	9223.90	0.817641	0.408821	−	0.912615i	$-0.365940\pi$
$503$	9223.90	0.817641	0.408821	+	0.912615i	$0.365940\pi$
$504$	0	0
$505$	−6202.39	−0.546540
$506$	5749.37	0.505120
$507$	0	0
$508$	1535.77	0.134131
$509$	−17452.8	−1.51981	−0.759903	−	0.650037i	$-0.774753\pi$
$509$	−17452.8	−1.51981	−0.759903	+	0.650037i	$0.774753\pi$
$510$	0	0
$511$	15354.6	1.32925
$512$	−5202.35	−0.449050
$513$	0	0
$514$	13850.6	1.18857
$515$	−8605.69	−0.736334
$516$	0	0
$517$	3103.88	0.264040
$518$	−6573.07	−0.557537
$519$	0	0
$520$	6284.53	0.529990
$521$	1764.28	0.148358	0.0741788	−	0.997245i	$-0.476366\pi$
$521$	1764.28	0.148358	0.0741788	+	0.997245i	$0.476366\pi$
$522$	0	0
$523$	−1343.12	−0.112296	−0.0561478	−	0.998422i	$-0.517882\pi$
$523$	−1343.12	−0.112296	−0.0561478	+	0.998422i	$0.517882\pi$
$524$	−16600.9	−1.38400
$525$	0	0
$526$	12622.6	1.04633
$527$	−3346.16	−0.276586
$528$	0	0
$529$	9562.29	0.785920
$530$	−15639.1	−1.28174
$531$	0	0
$532$	−14462.6	−1.17863
$533$	−873.208	−0.0709622
$534$	0	0
$535$	8059.81	0.651319
$536$	2987.79	0.240771
$537$	0	0
$538$	−16963.7	−1.35940
$539$	597.601	0.0477560
$540$	0	0
$541$	5800.90	0.460999	0.230499	−	0.973072i	$-0.425964\pi$
$541$	5800.90	0.460999	0.230499	+	0.973072i	$0.425964\pi$
$542$	−6136.72	−0.486337
$543$	0	0
$544$	7221.95	0.569189
$545$	2947.83	0.231690
$546$	0	0
$547$	−20172.9	−1.57684	−0.788420	−	0.615138i	$-0.789100\pi$
$547$	−20172.9	−1.57684	−0.788420	+	0.615138i	$0.789100\pi$
$548$	20973.9	1.63496
$549$	0	0
$550$	3474.70	0.269385
$551$	−7522.46	−0.581611
$552$	0	0
$553$	−1130.71	−0.0869486
$554$	−24386.2	−1.87016
$555$	0	0
$556$	−14022.9	−1.06961
$557$	7029.36	0.534728	0.267364	−	0.963596i	$-0.413847\pi$
$557$	7029.36	0.534728	0.267364	+	0.963596i	$0.413847\pi$
$558$	0	0
$559$	−24411.0	−1.84700
$560$	1764.16	0.133124
$561$	0	0
$562$	−15718.1	−1.17977
$563$	−11312.5	−0.846826	−0.423413	−	0.905937i	$-0.639168\pi$
$563$	−11312.5	−0.846826	−0.423413	+	0.905937i	$0.639168\pi$
$564$	0	0
$565$	−5190.87	−0.386516
$566$	19098.0	1.41828
$567$	0	0
$568$	20871.3	1.54180
$569$	927.916	0.0683660	0.0341830	−	0.999416i	$-0.489117\pi$
$569$	927.916	0.0683660	0.0341830	+	0.999416i	$0.489117\pi$
$570$	0	0
$571$	15832.0	1.16033	0.580164	−	0.814499i	$-0.302988\pi$
$571$	15832.0	1.16033	0.580164	+	0.814499i	$0.302988\pi$
$572$	−6014.92	−0.439679
$573$	0	0
$574$	1389.27	0.101023
$575$	13132.3	0.952445
$576$	0	0
$577$	−14721.1	−1.06212	−0.531062	−	0.847333i	$-0.678207\pi$
$577$	−14721.1	−1.06212	−0.531062	+	0.847333i	$0.678207\pi$
$578$	16810.7	1.20974
$579$	0	0
$580$	−9248.18	−0.662086
$581$	−16016.2	−1.14366
$582$	0	0
$583$	5065.87	0.359874
$584$	13882.9	0.983697
$585$	0	0
$586$	−26331.9	−1.85625
$587$	−8796.43	−0.618513	−0.309257	−	0.950979i	$-0.600080\pi$
$587$	−8796.43	−0.618513	−0.309257	+	0.950979i	$0.600080\pi$
$588$	0	0
$589$	−5784.40	−0.404656
$590$	−2125.54	−0.148317
$591$	0	0
$592$	1048.59	0.0727988
$593$	25527.4	1.76777	0.883883	−	0.467708i	$-0.154920\pi$
$593$	25527.4	1.76777	0.883883	+	0.467708i	$0.154920\pi$
$594$	0	0
$595$	−4146.03	−0.285665
$596$	−40853.0	−2.80773
$597$	0	0
$598$	−37772.1	−2.58297
$599$	−1545.78	−0.105441	−0.0527204	−	0.998609i	$-0.516789\pi$
$599$	−1545.78	−0.105441	−0.0527204	+	0.998609i	$0.516789\pi$
$600$	0	0
$601$	25564.6	1.73511	0.867556	−	0.497340i	$-0.165690\pi$
$601$	25564.6	1.73511	0.867556	+	0.497340i	$0.165690\pi$
$602$	38837.7	2.62942
$603$	0	0
$604$	−12524.9	−0.843763
$605$	−7522.53	−0.505511
$606$	0	0
$607$	−14665.6	−0.980658	−0.490329	−	0.871538i	$-0.663123\pi$
$607$	−14665.6	−0.980658	−0.490329	+	0.871538i	$0.663123\pi$
$608$	12484.4	0.832745
$609$	0	0
$610$	7985.84	0.530061
$611$	−20391.8	−1.35018
$612$	0	0
$613$	18431.9	1.21445	0.607225	−	0.794530i	$-0.292283\pi$
$613$	18431.9	1.21445	0.607225	+	0.794530i	$0.292283\pi$
$614$	−22869.3	−1.50314
$615$	0	0
$616$	3238.79	0.211842
$617$	17954.8	1.17153	0.585763	−	0.810482i	$-0.300795\pi$
$617$	17954.8	1.17153	0.585763	+	0.810482i	$0.300795\pi$
$618$	0	0
$619$	17919.5	1.16356	0.581781	−	0.813345i	$-0.302356\pi$
$619$	17919.5	1.16356	0.581781	+	0.813345i	$0.302356\pi$
$620$	−7111.40	−0.460646
$621$	0	0
$622$	−31784.9	−2.04897
$623$	−27416.4	−1.76310
$624$	0	0
$625$	3447.67	0.220651
$626$	−43228.6	−2.76000
$627$	0	0
$628$	23875.3	1.51708
$629$	−2464.35	−0.156216
$630$	0	0
$631$	−23214.4	−1.46458	−0.732290	−	0.680993i	$-0.761548\pi$
$631$	−23214.4	−1.46458	−0.732290	+	0.680993i	$0.761548\pi$
$632$	−1022.34	−0.0643455
$633$	0	0
$634$	−20900.2	−1.30923
$635$	761.068	0.0475623
$636$	0	0
$637$	−3926.10	−0.244204
$638$	4977.52	0.308874
$639$	0	0
$640$	12230.5	0.755395
$641$	1102.69	0.0679463	0.0339731	−	0.999423i	$-0.489184\pi$
$641$	1102.69	0.0679463	0.0339731	+	0.999423i	$0.489184\pi$
$642$	0	0
$643$	26164.4	1.60470	0.802350	−	0.596854i	$-0.203583\pi$
$643$	26164.4	1.60470	0.802350	+	0.596854i	$0.203583\pi$
$644$	36168.0	2.21307
$645$	0	0
$646$	−9009.40	−0.548716
$647$	−22462.7	−1.36491	−0.682456	−	0.730927i	$-0.739088\pi$
$647$	−22462.7	−1.36491	−0.682456	+	0.730927i	$0.739088\pi$
$648$	0	0
$649$	688.509	0.0416431
$650$	−22828.0	−1.37752
$651$	0	0
$652$	45448.7	2.72992
$653$	−22228.2	−1.33209	−0.666047	−	0.745910i	$-0.732015\pi$
$653$	−22228.2	−1.33209	−0.666047	+	0.745910i	$0.732015\pi$
$654$	0	0
$655$	−8226.78	−0.490759
$656$	−221.629	−0.0131908
$657$	0	0
$658$	32443.2	1.92214
$659$	−95.8262	−0.00566443	−0.00283221	−	0.999996i	$-0.500902\pi$
$659$	−95.8262	−0.00566443	−0.00283221	+	0.999996i	$0.500902\pi$
$660$	0	0
$661$	8655.48	0.509318	0.254659	−	0.967031i	$-0.418037\pi$
$661$	8655.48	0.509318	0.254659	+	0.967031i	$0.418037\pi$
$662$	12370.1	0.726251
$663$	0	0
$664$	−14481.2	−0.846352
$665$	−7167.12	−0.417939
$666$	0	0
$667$	18812.1	1.09207
$668$	−8568.97	−0.496322
$669$	0	0
$670$	4374.88	0.252263
$671$	−2586.79	−0.148826
$672$	0	0
$673$	16241.9	0.930282	0.465141	−	0.885237i	$-0.346004\pi$
$673$	16241.9	0.930282	0.465141	+	0.885237i	$0.346004\pi$
$674$	11035.9	0.630693
$675$	0	0
$676$	12949.1	0.736750
$677$	−11.9360	−0.000677604 0	−0.000338802	−	1.00000i	$-0.500108\pi$
$677$	−11.9360	−0.000677604 0	−0.000338802		1.00000i	$0.500108\pi$
$678$	0	0
$679$	−24781.0	−1.40060
$680$	−3748.66	−0.211404
$681$	0	0
$682$	3827.47	0.214899
$683$	25012.6	1.40129	0.700644	−	0.713511i	$-0.252896\pi$
$683$	25012.6	1.40129	0.700644	+	0.713511i	$0.252896\pi$
$684$	0	0
$685$	10393.9	0.579751
$686$	−24949.1	−1.38858
$687$	0	0
$688$	−6195.73	−0.343329
$689$	−33281.6	−1.84024
$690$	0	0
$691$	12198.3	0.671555	0.335777	−	0.941941i	$-0.391001\pi$
$691$	12198.3	0.671555	0.335777	+	0.941941i	$0.391001\pi$
$692$	26040.4	1.43050
$693$	0	0
$694$	−26597.7	−1.45481
$695$	−6949.20	−0.379278
$696$	0	0
$697$	520.860	0.0283055
$698$	40188.8	2.17933
$699$	0	0
$700$	21858.5	1.18025
$701$	26331.6	1.41873	0.709367	−	0.704840i	$-0.248981\pi$
$701$	26331.6	1.41873	0.709367	+	0.704840i	$0.248981\pi$
$702$	0	0
$703$	−4260.05	−0.228550
$704$	−7250.80	−0.388174
$705$	0	0
$706$	23100.4	1.23144
$707$	21000.0	1.11709
$708$	0	0
$709$	29196.0	1.54651	0.773256	−	0.634094i	$-0.218627\pi$
$709$	29196.0	1.54651	0.773256	+	0.634094i	$0.218627\pi$
$710$	30560.8	1.61539
$711$	0	0
$712$	−24788.7	−1.30477
$713$	14465.6	0.759805
$714$	0	0
$715$	−2980.77	−0.155908
$716$	9575.23	0.499781
$717$	0	0
$718$	−7996.17	−0.415619
$719$	12729.8	0.660282	0.330141	−	0.943932i	$-0.392904\pi$
$719$	12729.8	0.660282	0.330141	+	0.943932i	$0.392904\pi$
$720$	0	0
$721$	29137.1	1.50502
$722$	15171.0	0.782005
$723$	0	0
$724$	41404.7	2.12541
$725$	11369.3	0.582407
$726$	0	0
$727$	−31504.9	−1.60722	−0.803611	−	0.595155i	$-0.797091\pi$
$727$	−31504.9	−1.60722	−0.803611	+	0.595155i	$0.797091\pi$
$728$	−21278.1	−1.08327
$729$	0	0
$730$	20328.1	1.03065
$731$	14560.9	0.736736
$732$	0	0
$733$	−24769.0	−1.24811	−0.624054	−	0.781381i	$-0.714516\pi$
$733$	−24769.0	−1.24811	−0.624054	+	0.781381i	$0.714516\pi$
$734$	36143.7	1.81756
$735$	0	0
$736$	−31220.9	−1.56361
$737$	−1417.12	−0.0708281
$738$	0	0
$739$	−22258.6	−1.10798	−0.553989	−	0.832524i	$-0.686895\pi$
$739$	−22258.6	−1.10798	−0.553989	+	0.832524i	$0.686895\pi$
$740$	−5237.34	−0.260174
$741$	0	0
$742$	52950.9	2.61979
$743$	−12747.8	−0.629436	−0.314718	−	0.949185i	$-0.601910\pi$
$743$	−12747.8	−0.629436	−0.314718	+	0.949185i	$0.601910\pi$
$744$	0	0
$745$	−20245.2	−0.995608
$746$	14921.6	0.732333
$747$	0	0
$748$	3587.84	0.175380
$749$	−27288.8	−1.33126
$750$	0	0
$751$	−2079.69	−0.101051	−0.0505253	−	0.998723i	$-0.516090\pi$
$751$	−2079.69	−0.101051	−0.0505253	+	0.998723i	$0.516090\pi$
$752$	−5175.63	−0.250978
$753$	0	0
$754$	−32701.1	−1.57945
$755$	−6206.89	−0.299195
$756$	0	0
$757$	1497.14	0.0718817	0.0359409	−	0.999354i	$-0.488557\pi$
$757$	1497.14	0.0718817	0.0359409	+	0.999354i	$0.488557\pi$
$758$	22325.8	1.06980
$759$	0	0
$760$	−6480.20	−0.309291
$761$	−15622.5	−0.744173	−0.372087	−	0.928198i	$-0.621358\pi$
$761$	−15622.5	−0.744173	−0.372087	+	0.928198i	$0.621358\pi$
$762$	0	0
$763$	−9980.72	−0.473560
$764$	14409.1	0.682334
$765$	0	0
$766$	14609.7	0.689127
$767$	−4523.35	−0.212945
$768$	0	0
$769$	−195.022	−0.00914520	−0.00457260	−	0.999990i	$-0.501456\pi$
$769$	−195.022	−0.00914520	−0.00457260	+	0.999990i	$0.501456\pi$
$770$	4742.40	0.221953
$771$	0	0
$772$	−43189.2	−2.01349
$773$	−15353.1	−0.714377	−0.357188	−	0.934032i	$-0.616265\pi$
$773$	−15353.1	−0.714377	−0.357188	+	0.934032i	$0.616265\pi$
$774$	0	0
$775$	8742.44	0.405210
$776$	−22405.9	−1.03650
$777$	0	0
$778$	−59903.5	−2.76047
$779$	900.395	0.0414121
$780$	0	0
$781$	−9899.32	−0.453554
$782$	22530.7	1.03030
$783$	0	0
$784$	−996.482	−0.0453937
$785$	11831.7	0.537950
$786$	0	0
$787$	−27041.7	−1.22482	−0.612410	−	0.790540i	$-0.709800\pi$
$787$	−27041.7	−1.22482	−0.612410	+	0.790540i	$0.709800\pi$
$788$	53044.0	2.39799
$789$	0	0
$790$	−1496.95	−0.0674168
$791$	17575.2	0.790015
$792$	0	0
$793$	16994.6	0.761030
$794$	28584.0	1.27759
$795$	0	0
$796$	−16330.6	−0.727163
$797$	16608.2	0.738134	0.369067	−	0.929403i	$-0.379677\pi$
$797$	16608.2	0.738134	0.369067	+	0.929403i	$0.379677\pi$
$798$	0	0
$799$	12163.5	0.538565
$800$	−18868.7	−0.833886
$801$	0	0
$802$	−69027.3	−3.03920
$803$	−6584.71	−0.289376
$804$	0	0
$805$	17923.5	0.784746
$806$	−25145.6	−1.09890
$807$	0	0
$808$	18987.3	0.826695
$809$	−3737.98	−0.162448	−0.0812239	−	0.996696i	$-0.525883\pi$
$809$	−3737.98	−0.162448	−0.0812239	+	0.996696i	$0.525883\pi$
$810$	0	0
$811$	−14001.4	−0.606233	−0.303117	−	0.952954i	$-0.598027\pi$
$811$	−14001.4	−0.606233	−0.303117	+	0.952954i	$0.598027\pi$
$812$	31312.4	1.35326
$813$	0	0
$814$	2818.82	0.121375
$815$	22522.7	0.968019
$816$	0	0
$817$	25171.0	1.07787
$818$	−9342.05	−0.399312
$819$	0	0
$820$	1106.95	0.0471421
$821$	−29196.5	−1.24113	−0.620564	−	0.784156i	$-0.713096\pi$
$821$	−29196.5	−1.24113	−0.620564	+	0.784156i	$0.713096\pi$
$822$	0	0
$823$	−20622.6	−0.873463	−0.436731	−	0.899592i	$-0.643864\pi$
$823$	−20622.6	−0.873463	−0.436731	+	0.899592i	$0.643864\pi$
$824$	26344.4	1.11378
$825$	0	0
$826$	7196.63	0.303151
$827$	4618.12	0.194181	0.0970906	−	0.995276i	$-0.469046\pi$
$827$	4618.12	0.194181	0.0970906	+	0.995276i	$0.469046\pi$
$828$	0	0
$829$	17840.5	0.747437	0.373718	−	0.927542i	$-0.378083\pi$
$829$	17840.5	0.747437	0.373718	+	0.927542i	$0.378083\pi$
$830$	−21204.0	−0.886750
$831$	0	0
$832$	47636.1	1.98496
$833$	2341.88	0.0974085
$834$	0	0
$835$	−4246.46	−0.175994
$836$	6202.20	0.256588
$837$	0	0
$838$	−73498.9	−3.02981
$839$	−3178.71	−0.130800	−0.0654001	−	0.997859i	$-0.520832\pi$
$839$	−3178.71	−0.130800	−0.0654001	+	0.997859i	$0.520832\pi$
$840$	0	0
$841$	−8102.43	−0.332217
$842$	−53284.8	−2.18090
$843$	0	0
$844$	56345.1	2.29796
$845$	6417.10	0.261248
$846$	0	0
$847$	25469.7	1.03323
$848$	−8447.18	−0.342072
$849$	0	0
$850$	13616.6	0.549467
$851$	10653.5	0.429139
$852$	0	0
$853$	20694.2	0.830661	0.415331	−	0.909670i	$-0.363666\pi$
$853$	20694.2	0.830661	0.415331	+	0.909670i	$0.363666\pi$
$854$	−27038.4	−1.08341
$855$	0	0
$856$	−24673.3	−0.985184
$857$	−19147.6	−0.763209	−0.381605	−	0.924326i	$-0.624628\pi$
$857$	−19147.6	−0.763209	−0.381605	+	0.924326i	$0.624628\pi$
$858$	0	0
$859$	24033.8	0.954624	0.477312	−	0.878734i	$-0.341611\pi$
$859$	24033.8	0.954624	0.477312	+	0.878734i	$0.341611\pi$
$860$	30945.4	1.22701
$861$	0	0
$862$	77002.9	3.04261
$863$	23599.9	0.930882	0.465441	−	0.885079i	$-0.345896\pi$
$863$	23599.9	0.930882	0.465441	+	0.885079i	$0.345896\pi$
$864$	0	0
$865$	12904.7	0.507251
$866$	17220.5	0.675723
$867$	0	0
$868$	24077.7	0.941533
$869$	484.897	0.0189287
$870$	0	0
$871$	9310.15	0.362184
$872$	−9024.13	−0.350454
$873$	0	0
$874$	38948.1	1.50737
$875$	26031.1	1.00573
$876$	0	0
$877$	−11519.0	−0.443521	−0.221760	−	0.975101i	$-0.571180\pi$
$877$	−11519.0	−0.443521	−0.221760	+	0.975101i	$0.571180\pi$
$878$	27067.5	1.04041
$879$	0	0
$880$	−756.548	−0.0289809
$881$	−34539.7	−1.32085	−0.660427	−	0.750890i	$-0.729625\pi$
$881$	−34539.7	−1.32085	−0.660427	+	0.750890i	$0.729625\pi$
$882$	0	0
$883$	3728.90	0.142115	0.0710575	−	0.997472i	$-0.477363\pi$
$883$	3728.90	0.142115	0.0710575	+	0.997472i	$0.477363\pi$
$884$	−23571.3	−0.896819
$885$	0	0
$886$	40958.3	1.55307
$887$	−19658.2	−0.744147	−0.372074	−	0.928203i	$-0.621353\pi$
$887$	−19658.2	−0.744147	−0.372074	+	0.928203i	$0.621353\pi$
$888$	0	0
$889$	−2576.82	−0.0972145
$890$	−36296.8	−1.36705
$891$	0	0
$892$	39292.1	1.47488
$893$	21026.7	0.787940
$894$	0	0
$895$	4745.13	0.177220
$896$	−41409.9	−1.54398
$897$	0	0
$898$	−40273.4	−1.49659
$899$	12523.6	0.464610
$900$	0	0
$901$	19852.1	0.734040
$902$	−595.780	−0.0219926
$903$	0	0
$904$	15890.7	0.584643
$905$	20518.6	0.753659
$906$	0	0
$907$	−18626.3	−0.681892	−0.340946	−	0.940083i	$-0.610747\pi$
$907$	−18626.3	−0.681892	−0.340946	+	0.940083i	$0.610747\pi$
$908$	26767.6	0.978321
$909$	0	0
$910$	−31156.5	−1.13497
$911$	3653.75	0.132881	0.0664403	−	0.997790i	$-0.478836\pi$
$911$	3653.75	0.132881	0.0664403	+	0.997790i	$0.478836\pi$
$912$	0	0
$913$	6868.46	0.248973
$914$	−46567.0	−1.68523
$915$	0	0
$916$	63997.8	2.30846
$917$	27854.2	1.00308
$918$	0	0
$919$	−44548.5	−1.59904	−0.799521	−	0.600638i	$-0.794913\pi$
$919$	−44548.5	−1.59904	−0.799521	+	0.600638i	$0.794913\pi$
$920$	16205.6	0.580743
$921$	0	0
$922$	18739.9	0.669378
$923$	65036.2	2.31928
$924$	0	0
$925$	6438.56	0.228863
$926$	22762.9	0.807814
$927$	0	0
$928$	−27029.4	−0.956126
$929$	−12777.8	−0.451267	−0.225633	−	0.974212i	$-0.572445\pi$
$929$	−12777.8	−0.451267	−0.225633	+	0.974212i	$0.572445\pi$
$930$	0	0
$931$	4048.34	0.142512
$932$	67383.7	2.36827
$933$	0	0
$934$	17790.8	0.623269
$935$	1778.00	0.0621891
$936$	0	0
$937$	−34627.4	−1.20729	−0.603643	−	0.797255i	$-0.706285\pi$
$937$	−34627.4	−1.20729	−0.603643	+	0.797255i	$0.706285\pi$
$938$	−14812.4	−0.515611
$939$	0	0
$940$	25850.4	0.896964
$941$	−21761.8	−0.753893	−0.376946	−	0.926235i	$-0.623026\pi$
$941$	−21761.8	−0.753893	−0.376946	+	0.926235i	$0.623026\pi$
$942$	0	0
$943$	−2251.70	−0.0777577
$944$	−1148.07	−0.0395831
$945$	0	0
$946$	−16655.3	−0.572422
$947$	−8871.49	−0.304419	−0.152209	−	0.988348i	$-0.548639\pi$
$947$	−8871.49	−0.304419	−0.152209	+	0.988348i	$0.548639\pi$
$948$	0	0
$949$	43260.0	1.47975
$950$	23538.7	0.803891
$951$	0	0
$952$	12692.2	0.432096
$953$	21681.9	0.736985	0.368493	−	0.929631i	$-0.379874\pi$
$953$	21681.9	0.736985	0.368493	+	0.929631i	$0.379874\pi$
$954$	0	0
$955$	7140.61	0.241953
$956$	−3557.89	−0.120366
$957$	0	0
$958$	3780.98	0.127513
$959$	−35191.5	−1.18498
$960$	0	0
$961$	−20161.0	−0.676747
$962$	−18519.0	−0.620661
$963$	0	0
$964$	34877.9	1.16529
$965$	−21403.0	−0.713974
$966$	0	0
$967$	34370.9	1.14301	0.571506	−	0.820598i	$-0.306360\pi$
$967$	34370.9	1.14301	0.571506	+	0.820598i	$0.306360\pi$
$968$	23028.6	0.764635
$969$	0	0
$970$	−32807.9	−1.08598
$971$	28443.6	0.940060	0.470030	−	0.882650i	$-0.344243\pi$
$971$	28443.6	0.940060	0.470030	+	0.882650i	$0.344243\pi$
$972$	0	0
$973$	23528.5	0.775221
$974$	−19572.0	−0.643867
$975$	0	0
$976$	4313.40	0.141464
$977$	13505.7	0.442258	0.221129	−	0.975245i	$-0.429026\pi$
$977$	13505.7	0.442258	0.221129	+	0.975245i	$0.429026\pi$
$978$	0	0
$979$	11757.3	0.383827
$980$	4977.07	0.162231
$981$	0	0
$982$	−51117.2	−1.66111
$983$	−8992.66	−0.291782	−0.145891	−	0.989301i	$-0.546605\pi$
$983$	−8992.66	−0.291782	−0.145891	+	0.989301i	$0.546605\pi$
$984$	0	0
$985$	26286.6	0.850315
$986$	19505.9	0.630014
$987$	0	0
$988$	−40747.0	−1.31208
$989$	−62947.4	−2.02388
$990$	0	0
$991$	−42022.2	−1.34700	−0.673502	−	0.739186i	$-0.735211\pi$
$991$	−42022.2	−1.34700	−0.673502	+	0.739186i	$0.735211\pi$
$992$	−20784.3	−0.665225
$993$	0	0
$994$	−103472.	−3.30176
$995$	−8092.82	−0.257849
$996$	0	0
$997$	−14379.8	−0.456784	−0.228392	−	0.973569i	$-0.573347\pi$
$997$	−14379.8	−0.456784	−0.228392	+	0.973569i	$0.573347\pi$
$998$	−58279.1	−1.84849
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.4.a.d.1.1		13
3.2	odd	2		127.4.a.b.1.13	✓	13
12.11	even	2		2032.4.a.g.1.11		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
127.4.a.b.1.13	✓	13	3.2	odd	2
1143.4.a.d.1.1		13	1.1	even	1	trivial
2032.4.a.g.1.11		13	12.11	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1143.a (trivial)

\(f(q)\)	\(=\)	\(q-4.48248 q^{2} +12.0927 q^{4} +5.99266 q^{5} -20.2899 q^{7} -18.3452 q^{8} +O(q^{10})\)	\(q-4.48248 q^{2} +12.0927 q^{4} +5.99266 q^{5} -20.2899 q^{7} -18.3452 q^{8} -26.8620 q^{10} +8.70120 q^{11} -57.1649 q^{13} +90.9491 q^{14} -14.5090 q^{16} +34.0983 q^{17} +58.9447 q^{19} +72.4672 q^{20} -39.0030 q^{22} -147.409 q^{23} -89.0880 q^{25} +256.241 q^{26} -245.359 q^{28} -127.619 q^{29} -98.1327 q^{31} +211.798 q^{32} -152.845 q^{34} -121.591 q^{35} -72.2719 q^{37} -264.219 q^{38} -109.937 q^{40} +15.2752 q^{41} +427.027 q^{43} +105.221 q^{44} +660.756 q^{46} +356.718 q^{47} +68.6803 q^{49} +399.335 q^{50} -691.275 q^{52} +582.203 q^{53} +52.1434 q^{55} +372.223 q^{56} +572.049 q^{58} +79.1281 q^{59} -297.291 q^{61} +439.878 q^{62} -833.310 q^{64} -342.570 q^{65} -162.865 q^{67} +412.339 q^{68} +545.028 q^{70} -1137.70 q^{71} -756.758 q^{73} +323.957 q^{74} +712.798 q^{76} -176.547 q^{77} +55.7276 q^{79} -86.9475 q^{80} -68.4710 q^{82} +789.369 q^{83} +204.340 q^{85} -1914.14 q^{86} -159.626 q^{88} +1351.23 q^{89} +1159.87 q^{91} -1782.56 q^{92} -1598.98 q^{94} +353.236 q^{95} +1221.35 q^{97} -307.858 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 8 q^{2} + 34 q^{4} + 46 q^{5} - 26 q^{7} + 117 q^{8}+O(q^{10}) \) 13 * q + 8 * q^2 + 34 * q^4 + 46 * q^5 - 26 * q^7 + 117 * q^8	\( 13 q + 8 q^{2} + 34 q^{4} + 46 q^{5} - 26 q^{7} + 117 q^{8} - 25 q^{10} + 53 q^{11} - 75 q^{13} + 152 q^{14} + 82 q^{16} + 479 q^{17} - 209 q^{19} + 533 q^{20} - 407 q^{22} + 376 q^{23} + 9 q^{25} + 622 q^{26} - 258 q^{28} + 158 q^{29} - 307 q^{31} + 609 q^{32} + 948 q^{34} + 512 q^{35} - 171 q^{37} - 712 q^{38} + 1897 q^{40} + 641 q^{41} + 530 q^{43} - 1328 q^{44} + 2051 q^{46} + 555 q^{47} + 357 q^{49} - 808 q^{50} + 2473 q^{52} + 1640 q^{53} + 540 q^{55} - 551 q^{56} + 1328 q^{58} + 860 q^{59} + 191 q^{61} - 367 q^{62} + 1915 q^{64} + 1584 q^{65} + 912 q^{67} + 1873 q^{68} + 2329 q^{70} - 115 q^{71} - 2563 q^{73} - 470 q^{74} - 192 q^{76} + 3338 q^{77} + 169 q^{79} + 3194 q^{80} + 54 q^{82} + 1688 q^{83} + 480 q^{85} + 485 q^{86} - 1192 q^{88} + 2752 q^{89} - 398 q^{91} + 846 q^{92} + 531 q^{94} + 1766 q^{95} - 2124 q^{97} + 1479 q^{98}+O(q^{100}) \) 13 * q + 8 * q^2 + 34 * q^4 + 46 * q^5 - 26 * q^7 + 117 * q^8 - 25 * q^10 + 53 * q^11 - 75 * q^13 + 152 * q^14 + 82 * q^16 + 479 * q^17 - 209 * q^19 + 533 * q^20 - 407 * q^22 + 376 * q^23 + 9 * q^25 + 622 * q^26 - 258 * q^28 + 158 * q^29 - 307 * q^31 + 609 * q^32 + 948 * q^34 + 512 * q^35 - 171 * q^37 - 712 * q^38 + 1897 * q^40 + 641 * q^41 + 530 * q^43 - 1328 * q^44 + 2051 * q^46 + 555 * q^47 + 357 * q^49 - 808 * q^50 + 2473 * q^52 + 1640 * q^53 + 540 * q^55 - 551 * q^56 + 1328 * q^58 + 860 * q^59 + 191 * q^61 - 367 * q^62 + 1915 * q^64 + 1584 * q^65 + 912 * q^67 + 1873 * q^68 + 2329 * q^70 - 115 * q^71 - 2563 * q^73 - 470 * q^74 - 192 * q^76 + 3338 * q^77 + 169 * q^79 + 3194 * q^80 + 54 * q^82 + 1688 * q^83 + 480 * q^85 + 485 * q^86 - 1192 * q^88 + 2752 * q^89 - 398 * q^91 + 846 * q^92 + 531 * q^94 + 1766 * q^95 - 2124 * q^97 + 1479 * q^98

Introduction

L-functions

Modular forms

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Database

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