LMFDB - Embedded newform 2175.4.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,4,Mod(1,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2175.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2175.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:	$128.329154262$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 43x^{6} + 117x^{5} + 586x^{4} - 701x^{3} - 1792x^{2} + 924x + 1424 $ x^8 - 4x^7 - 43x^6 + 117x^5 + 586x^4 - 701x^3 - 1792x^2 + 924*x + 1424
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.81879$ of defining polynomial
Character	$\chi$	$=$	2175.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.818792 q^{2} +3.00000 q^{3} -7.32958 q^{4} -2.45638 q^{6} -13.8632 q^{7} +12.5517 q^{8} +9.00000 q^{9} +71.7303 q^{11} -21.9887 q^{12} +78.8785 q^{13} +11.3510 q^{14} +48.3594 q^{16} -93.8775 q^{17} -7.36913 q^{18} -15.4555 q^{19} -41.5895 q^{21} -58.7322 q^{22} +187.363 q^{23} +37.6552 q^{24} -64.5851 q^{26} +27.0000 q^{27} +101.611 q^{28} +29.0000 q^{29} -73.3710 q^{31} -140.010 q^{32} +215.191 q^{33} +76.8662 q^{34} -65.9662 q^{36} +127.123 q^{37} +12.6549 q^{38} +236.635 q^{39} -124.177 q^{41} +34.0531 q^{42} +41.9717 q^{43} -525.753 q^{44} -153.411 q^{46} +588.328 q^{47} +145.078 q^{48} -150.813 q^{49} -281.633 q^{51} -578.146 q^{52} +125.128 q^{53} -22.1074 q^{54} -174.007 q^{56} -46.3666 q^{57} -23.7450 q^{58} -55.2122 q^{59} +259.660 q^{61} +60.0755 q^{62} -124.768 q^{63} -272.236 q^{64} -176.196 q^{66} +664.277 q^{67} +688.083 q^{68} +562.089 q^{69} +265.186 q^{71} +112.966 q^{72} +5.27903 q^{73} -104.087 q^{74} +113.283 q^{76} -994.408 q^{77} -193.755 q^{78} -792.533 q^{79} +81.0000 q^{81} +101.675 q^{82} -1424.74 q^{83} +304.833 q^{84} -34.3661 q^{86} +87.0000 q^{87} +900.339 q^{88} -1499.69 q^{89} -1093.50 q^{91} -1373.29 q^{92} -220.113 q^{93} -481.718 q^{94} -420.030 q^{96} -764.065 q^{97} +123.484 q^{98} +645.572 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{2} + 24 q^{3} + 38 q^{4} + 12 q^{6} + q^{7} + 9 q^{8} + 72 q^{9} + 11 q^{11} + 114 q^{12} + 97 q^{13} - 69 q^{14} + 242 q^{16} + 119 q^{17} + 36 q^{18} + 220 q^{19} + 3 q^{21} + 129 q^{22} + 262 q^{23}+ \cdots + 99 q^{99}+O(q^{100}) $ 8 * q + 4 * q^2 + 24 * q^3 + 38 * q^4 + 12 * q^6 + q^7 + 9 * q^8 + 72 * q^9 + 11 * q^11 + 114 * q^12 + 97 * q^13 - 69 * q^14 + 242 * q^16 + 119 * q^17 + 36 * q^18 + 220 * q^19 + 3 * q^21 + 129 * q^22 + 262 * q^23 + 27 * q^24 + 43 * q^26 + 216 * q^27 + 13 * q^28 + 232 * q^29 + 108 * q^31 + 566 * q^32 + 33 * q^33 - 527 * q^34 + 342 * q^36 - 118 * q^37 + 1088 * q^38 + 291 * q^39 - 118 * q^41 - 207 * q^42 + 384 * q^43 - 1605 * q^44 + 154 * q^46 + 1309 * q^47 + 726 * q^48 + 427 * q^49 + 357 * q^51 + 747 * q^52 + 1912 * q^53 + 108 * q^54 - 2022 * q^56 + 660 * q^57 + 116 * q^58 - 1178 * q^59 + 572 * q^61 + 2572 * q^62 + 9 * q^63 - 1771 * q^64 + 387 * q^66 - 1147 * q^67 + 3963 * q^68 + 786 * q^69 + 84 * q^71 + 81 * q^72 + 222 * q^73 - 1758 * q^74 + 1390 * q^76 + 1753 * q^77 + 129 * q^78 + 390 * q^79 + 648 * q^81 + 2994 * q^82 + 1570 * q^83 + 39 * q^84 + 1668 * q^86 + 696 * q^87 + 1790 * q^88 - 119 * q^89 - 3529 * q^91 + 2934 * q^92 + 324 * q^93 - 2277 * q^94 + 1698 * q^96 + 2100 * q^97 + 7677 * q^98 + 99 * q^99

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 $

$2$	−0.818792	−0.289487	−0.144743	−	0.989469i	$-0.546236\pi$
$2$	−0.818792	−0.289487	−0.144743	+	0.989469i	$0.546236\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−7.32958	−0.916198
$5$	0	0
$5$	0	0
$6$	−2.45638	−0.167135
$7$	−13.8632	−0.748540	−0.374270	−	0.927320i	$-0.622107\pi$
$7$	−13.8632	−0.748540	−0.374270	+	0.927320i	$0.622107\pi$
$8$	12.5517	0.554714
$9$	9.00000	0.333333
$10$	0	0
$11$	71.7303	1.96614	0.983068	−	0.183243i	$-0.0586595\pi$
$11$	71.7303	1.96614	0.983068	+	0.183243i	$0.0586595\pi$
$12$	−21.9887	−0.528967
$13$	78.8785	1.68284	0.841421	−	0.540379i	$-0.181719\pi$
$13$	78.8785	1.68284	0.841421	+	0.540379i	$0.181719\pi$
$14$	11.3510	0.216692
$15$	0	0
$16$	48.3594	0.755615
$17$	−93.8775	−1.33933	−0.669666	−	0.742662i	$-0.733563\pi$
$17$	−93.8775	−1.33933	−0.669666	+	0.742662i	$0.733563\pi$
$18$	−7.36913	−0.0964955
$19$	−15.4555	−0.186618	−0.0933091	−	0.995637i	$-0.529744\pi$
$19$	−15.4555	−0.186618	−0.0933091	+	0.995637i	$0.529744\pi$
$20$	0	0
$21$	−41.5895	−0.432170
$22$	−58.7322	−0.569170
$23$	187.363	1.69860	0.849301	−	0.527909i	$-0.177024\pi$
$23$	187.363	1.69860	0.849301	+	0.527909i	$0.177024\pi$
$24$	37.6552	0.320264
$25$	0	0
$26$	−64.5851	−0.487161
$27$	27.0000	0.192450
$28$	101.611	0.685811
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−73.3710	−0.425091	−0.212545	−	0.977151i	$-0.568175\pi$
$31$	−73.3710	−0.425091	−0.212545	+	0.977151i	$0.568175\pi$
$32$	−140.010	−0.773454
$33$	215.191	1.13515
$34$	76.8662	0.387719
$35$	0	0
$36$	−65.9662	−0.305399
$37$	127.123	0.564836	0.282418	−	0.959291i	$-0.408863\pi$
$37$	127.123	0.564836	0.282418	+	0.959291i	$0.408863\pi$
$38$	12.6549	0.0540235
$39$	236.635	0.971590
$40$	0	0
$41$	−124.177	−0.473004	−0.236502	−	0.971631i	$-0.576001\pi$
$41$	−124.177	−0.473004	−0.236502	+	0.971631i	$0.576001\pi$
$42$	34.0531	0.125107
$43$	41.9717	0.148852	0.0744258	−	0.997227i	$-0.476288\pi$
$43$	41.9717	0.148852	0.0744258	+	0.997227i	$0.476288\pi$
$44$	−525.753	−1.80137
$45$	0	0
$46$	−153.411	−0.491723
$47$	588.328	1.82588	0.912941	−	0.408093i	$-0.133806\pi$
$47$	588.328	1.82588	0.912941	+	0.408093i	$0.133806\pi$
$48$	145.078	0.436255
$49$	−150.813	−0.439688
$50$	0	0
$51$	−281.633	−0.773264
$52$	−578.146	−1.54182
$53$	125.128	0.324294	0.162147	−	0.986767i	$-0.448158\pi$
$53$	125.128	0.324294	0.162147	+	0.986767i	$0.448158\pi$
$54$	−22.1074	−0.0557117
$55$	0	0
$56$	−174.007	−0.415225
$57$	−46.3666	−0.107744
$58$	−23.7450	−0.0537563
$59$	−55.2122	−0.121831	−0.0609154	−	0.998143i	$-0.519402\pi$
$59$	−55.2122	−0.121831	−0.0609154	+	0.998143i	$0.519402\pi$
$60$	0	0
$61$	259.660	0.545018	0.272509	−	0.962153i	$-0.412147\pi$
$61$	259.660	0.545018	0.272509	+	0.962153i	$0.412147\pi$
$62$	60.0755	0.123058
$63$	−124.768	−0.249513
$64$	−272.236	−0.531711
$65$	0	0
$66$	−176.196	−0.328610
$67$	664.277	1.21126	0.605630	−	0.795747i	$-0.292921\pi$
$67$	664.277	1.21126	0.605630	+	0.795747i	$0.292921\pi$
$68$	688.083	1.22709
$69$	562.089	0.980688
$70$	0	0
$71$	265.186	0.443264	0.221632	−	0.975130i	$-0.428862\pi$
$71$	265.186	0.443264	0.221632	+	0.975130i	$0.428862\pi$
$72$	112.966	0.184905
$73$	5.27903	0.00846388	0.00423194	−	0.999991i	$-0.498653\pi$
$73$	5.27903	0.00846388	0.00423194	+	0.999991i	$0.498653\pi$
$74$	−104.087	−0.163512
$75$	0	0
$76$	113.283	0.170979
$77$	−994.408	−1.47173
$78$	−193.755	−0.281262
$79$	−792.533	−1.12870	−0.564348	−	0.825537i	$-0.690872\pi$
$79$	−792.533	−1.12870	−0.564348	+	0.825537i	$0.690872\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	101.675	0.136928
$83$	−1424.74	−1.88416	−0.942081	−	0.335385i	$-0.891134\pi$
$83$	−1424.74	−1.88416	−0.942081	+	0.335385i	$0.891134\pi$
$84$	304.833	0.395953
$85$	0	0
$86$	−34.3661	−0.0430906
$87$	87.0000	0.107211
$88$	900.339	1.09064
$89$	−1499.69	−1.78614	−0.893071	−	0.449916i	$-0.851454\pi$
$89$	−1499.69	−1.78614	−0.893071	+	0.449916i	$0.851454\pi$
$90$	0	0
$91$	−1093.50	−1.25968
$92$	−1373.29	−1.55626
$93$	−220.113	−0.245426
$94$	−481.718	−0.528568
$95$	0	0
$96$	−420.030	−0.446554
$97$	−764.065	−0.799784	−0.399892	−	0.916562i	$-0.630952\pi$
$97$	−764.065	−0.799784	−0.399892	+	0.916562i	$0.630952\pi$
$98$	123.484	0.127284
$99$	645.572	0.655378
$100$	0	0
$101$	−1545.45	−1.52256	−0.761279	−	0.648425i	$-0.775428\pi$
$101$	−1545.45	−1.52256	−0.761279	+	0.648425i	$0.775428\pi$
$102$	230.598	0.223850
$103$	−204.337	−0.195475	−0.0977374	−	0.995212i	$-0.531161\pi$
$103$	−204.337	−0.195475	−0.0977374	+	0.995212i	$0.531161\pi$
$104$	990.062	0.933496
$105$	0	0
$106$	−102.454	−0.0938789
$107$	342.888	0.309797	0.154898	−	0.987930i	$-0.450495\pi$
$107$	342.888	0.309797	0.154898	+	0.987930i	$0.450495\pi$
$108$	−197.899	−0.176322
$109$	1312.12	1.15301	0.576504	−	0.817094i	$-0.304416\pi$
$109$	1312.12	1.15301	0.576504	+	0.817094i	$0.304416\pi$
$110$	0	0
$111$	381.370	0.326108
$112$	−670.414	−0.565608
$113$	−39.3605	−0.0327675	−0.0163837	−	0.999866i	$-0.505215\pi$
$113$	−39.3605	−0.0327675	−0.0163837	+	0.999866i	$0.505215\pi$
$114$	37.9646	0.0311905
$115$	0	0
$116$	−212.558	−0.170134
$117$	709.906	0.560948
$118$	45.2073	0.0352684
$119$	1301.44	1.00254
$120$	0	0
$121$	3814.23	2.86569
$122$	−212.608	−0.157775
$123$	−372.531	−0.273089
$124$	537.778	0.389467
$125$	0	0
$126$	102.159	0.0722308
$127$	1843.60	1.28813	0.644066	−	0.764970i	$-0.277246\pi$
$127$	1843.60	1.28813	0.644066	+	0.764970i	$0.277246\pi$
$128$	1342.99	0.927377
$129$	125.915	0.0859396
$130$	0	0
$131$	−2280.13	−1.52073	−0.760366	−	0.649494i	$-0.774981\pi$
$131$	−2280.13	−1.52073	−0.760366	+	0.649494i	$0.774981\pi$
$132$	−1577.26	−1.04002
$133$	214.263	0.139691
$134$	−543.905	−0.350643
$135$	0	0
$136$	−1178.33	−0.742946
$137$	−1201.01	−0.748975	−0.374487	−	0.927232i	$-0.622181\pi$
$137$	−1201.01	−0.748975	−0.374487	+	0.927232i	$0.622181\pi$
$138$	−460.233	−0.283896
$139$	−9.65588	−0.00589209	−0.00294605	−	0.999996i	$-0.500938\pi$
$139$	−9.65588	−0.00589209	−0.00294605	+	0.999996i	$0.500938\pi$
$140$	0	0
$141$	1764.98	1.05417
$142$	−217.132	−0.128319
$143$	5657.98	3.30870
$144$	435.234	0.251872
$145$	0	0
$146$	−4.32242	−0.00245018
$147$	−452.439	−0.253854
$148$	−931.760	−0.517501
$149$	−564.420	−0.310329	−0.155165	−	0.987889i	$-0.549591\pi$
$149$	−564.420	−0.310329	−0.155165	+	0.987889i	$0.549591\pi$
$150$	0	0
$151$	635.922	0.342719	0.171359	−	0.985209i	$-0.445184\pi$
$151$	635.922	0.342719	0.171359	+	0.985209i	$0.445184\pi$
$152$	−193.994	−0.103520
$153$	−844.898	−0.446444
$154$	814.213	0.426046
$155$	0	0
$156$	−1734.44	−0.890168
$157$	573.022	0.291288	0.145644	−	0.989337i	$-0.453475\pi$
$157$	573.022	0.291288	0.145644	+	0.989337i	$0.453475\pi$
$158$	648.920	0.326742
$159$	375.383	0.187232
$160$	0	0
$161$	−2597.44	−1.27147
$162$	−66.3221	−0.0321652
$163$	3278.02	1.57518	0.787591	−	0.616199i	$-0.211328\pi$
$163$	3278.02	1.57518	0.787591	+	0.616199i	$0.211328\pi$
$164$	910.164	0.433365
$165$	0	0
$166$	1166.57	0.545440
$167$	3924.81	1.81863	0.909314	−	0.416112i	$-0.136607\pi$
$167$	3924.81	1.81863	0.909314	+	0.416112i	$0.136607\pi$
$168$	−522.020	−0.239730
$169$	4024.82	1.83196
$170$	0	0
$171$	−139.100	−0.0622060
$172$	−307.635	−0.136378
$173$	2409.34	1.05884	0.529419	−	0.848361i	$-0.322410\pi$
$173$	2409.34	1.05884	0.529419	+	0.848361i	$0.322410\pi$
$174$	−71.2349	−0.0310362
$175$	0	0
$176$	3468.83	1.48564
$177$	−165.637	−0.0703390
$178$	1227.93	0.517064
$179$	−270.657	−0.113016	−0.0565081	−	0.998402i	$-0.517997\pi$
$179$	−270.657	−0.113016	−0.0565081	+	0.998402i	$0.517997\pi$
$180$	0	0
$181$	−176.505	−0.0724837	−0.0362418	−	0.999343i	$-0.511539\pi$
$181$	−176.505	−0.0724837	−0.0362418	+	0.999343i	$0.511539\pi$
$182$	895.353	0.364659
$183$	778.981	0.314666
$184$	2351.73	0.942238
$185$	0	0
$186$	180.227	0.0710476
$187$	−6733.86	−2.63331
$188$	−4312.19	−1.67287
$189$	−374.305	−0.144057
$190$	0	0
$191$	2520.29	0.954774	0.477387	−	0.878693i	$-0.341584\pi$
$191$	2520.29	0.954774	0.477387	+	0.878693i	$0.341584\pi$
$192$	−816.708	−0.306983
$193$	−1493.85	−0.557150	−0.278575	−	0.960414i	$-0.589862\pi$
$193$	−1493.85	−0.557150	−0.278575	+	0.960414i	$0.589862\pi$
$194$	625.610	0.231527
$195$	0	0
$196$	1105.40	0.402841
$197$	3155.05	1.14106	0.570528	−	0.821278i	$-0.306739\pi$
$197$	3155.05	1.14106	0.570528	+	0.821278i	$0.306739\pi$
$198$	−528.589	−0.189723
$199$	5460.52	1.94516	0.972578	−	0.232575i	$-0.0747152\pi$
$199$	5460.52	1.94516	0.972578	+	0.232575i	$0.0747152\pi$
$200$	0	0
$201$	1992.83	0.699321
$202$	1265.40	0.440760
$203$	−402.032	−0.139000
$204$	2064.25	0.708462
$205$	0	0
$206$	167.309	0.0565873
$207$	1686.27	0.566201
$208$	3814.52	1.27158
$209$	−1108.63	−0.366917
$210$	0	0
$211$	2596.75	0.847241	0.423621	−	0.905840i	$-0.360759\pi$
$211$	2596.75	0.847241	0.423621	+	0.905840i	$0.360759\pi$
$212$	−917.133	−0.297118
$213$	795.557	0.255919
$214$	−280.754	−0.0896820
$215$	0	0
$216$	338.897	0.106755
$217$	1017.15	0.318197
$218$	−1074.35	−0.333781
$219$	15.8371	0.00488662
$220$	0	0
$221$	−7404.92	−2.25389
$222$	−312.262	−0.0944040
$223$	1578.45	0.473995	0.236998	−	0.971510i	$-0.423837\pi$
$223$	1578.45	0.473995	0.236998	+	0.971510i	$0.423837\pi$
$224$	1940.98	0.578961
$225$	0	0
$226$	32.2281	0.00948575
$227$	2428.63	0.710106	0.355053	−	0.934846i	$-0.384463\pi$
$227$	2428.63	0.710106	0.355053	+	0.934846i	$0.384463\pi$
$228$	339.848	0.0987148
$229$	−4621.51	−1.33362	−0.666808	−	0.745229i	$-0.732340\pi$
$229$	−4621.51	−1.33362	−0.666808	+	0.745229i	$0.732340\pi$
$230$	0	0
$231$	−2983.22	−0.849704
$232$	364.000	0.103008
$233$	−3337.50	−0.938398	−0.469199	−	0.883092i	$-0.655457\pi$
$233$	−3337.50	−0.938398	−0.469199	+	0.883092i	$0.655457\pi$
$234$	−581.266	−0.162387
$235$	0	0
$236$	404.682	0.111621
$237$	−2377.60	−0.651653
$238$	−1065.61	−0.290223
$239$	−757.216	−0.204938	−0.102469	−	0.994736i	$-0.532674\pi$
$239$	−757.216	−0.204938	−0.102469	+	0.994736i	$0.532674\pi$
$240$	0	0
$241$	−480.856	−0.128525	−0.0642627	−	0.997933i	$-0.520470\pi$
$241$	−480.856	−0.128525	−0.0642627	+	0.997933i	$0.520470\pi$
$242$	−3123.06	−0.829578
$243$	243.000	0.0641500
$244$	−1903.20	−0.499344
$245$	0	0
$246$	305.025	0.0790556
$247$	−1219.11	−0.314049
$248$	−920.933	−0.235804
$249$	−4274.22	−1.08782
$250$	0	0
$251$	2236.26	0.562356	0.281178	−	0.959656i	$-0.409275\pi$
$251$	2236.26	0.562356	0.281178	+	0.959656i	$0.409275\pi$
$252$	914.500	0.228604
$253$	13439.6	3.33968
$254$	−1509.52	−0.372897
$255$	0	0
$256$	1078.26	0.263247
$257$	6940.70	1.68463	0.842314	−	0.538988i	$-0.181193\pi$
$257$	6940.70	1.68463	0.842314	+	0.538988i	$0.181193\pi$
$258$	−103.098	−0.0248784
$259$	−1762.33	−0.422802
$260$	0	0
$261$	261.000	0.0618984
$262$	1866.95	0.440232
$263$	−5980.13	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$263$	−5980.13	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$264$	2701.02	0.629682
$265$	0	0
$266$	−175.436	−0.0404387
$267$	−4499.06	−1.03123
$268$	−4868.87	−1.10975
$269$	1822.34	0.413049	0.206525	−	0.978441i	$-0.433785\pi$
$269$	1822.34	0.413049	0.206525	+	0.978441i	$0.433785\pi$
$270$	0	0
$271$	3201.60	0.717651	0.358826	−	0.933405i	$-0.383177\pi$
$271$	3201.60	0.717651	0.358826	+	0.933405i	$0.383177\pi$
$272$	−4539.86	−1.01202
$273$	−3280.51	−0.727274
$274$	983.380	0.216818
$275$	0	0
$276$	−4119.87	−0.898504
$277$	−924.254	−0.200480	−0.100240	−	0.994963i	$-0.531961\pi$
$277$	−924.254	−0.200480	−0.100240	+	0.994963i	$0.531961\pi$
$278$	7.90616	0.00170568
$279$	−660.339	−0.141697
$280$	0	0
$281$	3105.33	0.659247	0.329623	−	0.944113i	$-0.393078\pi$
$281$	3105.33	0.659247	0.329623	+	0.944113i	$0.393078\pi$
$282$	−1445.15	−0.305169
$283$	10.5568	0.00221745	0.00110873	−	0.999999i	$-0.499647\pi$
$283$	10.5568	0.00221745	0.00110873	+	0.999999i	$0.499647\pi$
$284$	−1943.70	−0.406117
$285$	0	0
$286$	−4632.70	−0.957823
$287$	1721.48	0.354063
$288$	−1260.09	−0.257818
$289$	3899.99	0.793811
$290$	0	0
$291$	−2292.20	−0.461756
$292$	−38.6930	−0.00775459
$293$	2892.07	0.576643	0.288322	−	0.957534i	$-0.406903\pi$
$293$	2892.07	0.576643	0.288322	+	0.957534i	$0.406903\pi$
$294$	370.453	0.0734873
$295$	0	0
$296$	1595.62	0.313322
$297$	1936.72	0.378383
$298$	462.142	0.0898362
$299$	14778.9	2.85848
$300$	0	0
$301$	−581.860	−0.111421
$302$	−520.687	−0.0992125
$303$	−4636.36	−0.879049
$304$	−747.420	−0.141012
$305$	0	0
$306$	691.795	0.129240
$307$	−6869.50	−1.27708	−0.638539	−	0.769590i	$-0.720461\pi$
$307$	−6869.50	−1.27708	−0.638539	+	0.769590i	$0.720461\pi$
$308$	7288.59	1.34840
$309$	−613.010	−0.112857
$310$	0	0
$311$	−3754.87	−0.684627	−0.342313	−	0.939586i	$-0.611210\pi$
$311$	−3754.87	−0.684627	−0.342313	+	0.939586i	$0.611210\pi$
$312$	2970.19	0.538954
$313$	−1004.33	−0.181368	−0.0906839	−	0.995880i	$-0.528905\pi$
$313$	−1004.33	−0.181368	−0.0906839	+	0.995880i	$0.528905\pi$
$314$	−469.186	−0.0843239
$315$	0	0
$316$	5808.94	1.03411
$317$	4555.66	0.807165	0.403582	−	0.914943i	$-0.367765\pi$
$317$	4555.66	0.807165	0.403582	+	0.914943i	$0.367765\pi$
$318$	−307.361	−0.0542010
$319$	2080.18	0.365102
$320$	0	0
$321$	1028.66	0.178861
$322$	2126.76	0.368074
$323$	1450.93	0.249944
$324$	−593.696	−0.101800
$325$	0	0
$326$	−2684.02	−0.455994
$327$	3936.35	0.665690
$328$	−1558.64	−0.262382
$329$	−8156.08	−1.36675
$330$	0	0
$331$	−4344.37	−0.721414	−0.360707	−	0.932679i	$-0.617465\pi$
$331$	−4344.37	−0.721414	−0.360707	+	0.932679i	$0.617465\pi$
$332$	10442.7	1.72626
$333$	1144.11	0.188279
$334$	−3213.60	−0.526468
$335$	0	0
$336$	−2011.24	−0.326554
$337$	−8112.36	−1.31130	−0.655650	−	0.755065i	$-0.727606\pi$
$337$	−8112.36	−1.31130	−0.655650	+	0.755065i	$0.727606\pi$
$338$	−3295.49	−0.530328
$339$	−118.082	−0.0189183
$340$	0	0
$341$	−5262.92	−0.835786
$342$	113.894	0.0180078
$343$	6845.81	1.07766
$344$	526.817	0.0825700
$345$	0	0
$346$	−1972.75	−0.306519
$347$	10739.1	1.66139	0.830696	−	0.556726i	$-0.187943\pi$
$347$	10739.1	1.66139	0.830696	+	0.556726i	$0.187943\pi$
$348$	−637.673	−0.0982267
$349$	−7695.17	−1.18027	−0.590134	−	0.807306i	$-0.700925\pi$
$349$	−7695.17	−1.18027	−0.590134	+	0.807306i	$0.700925\pi$
$350$	0	0
$351$	2129.72	0.323863
$352$	−10043.0	−1.52072
$353$	6621.09	0.998315	0.499157	−	0.866511i	$-0.333643\pi$
$353$	6621.09	0.998315	0.499157	+	0.866511i	$0.333643\pi$
$354$	135.622	0.0203622
$355$	0	0
$356$	10992.1	1.63646
$357$	3904.32	0.578819
$358$	221.612	0.0327167
$359$	2419.99	0.355772	0.177886	−	0.984051i	$-0.443074\pi$
$359$	2419.99	0.355772	0.177886	+	0.984051i	$0.443074\pi$
$360$	0	0
$361$	−6620.13	−0.965174
$362$	144.521	0.0209831
$363$	11442.7	1.65451
$364$	8014.93	1.15411
$365$	0	0
$366$	−637.823	−0.0910917
$367$	−5254.54	−0.747369	−0.373685	−	0.927556i	$-0.621906\pi$
$367$	−5254.54	−0.747369	−0.373685	+	0.927556i	$0.621906\pi$
$368$	9060.75	1.28349
$369$	−1117.59	−0.157668
$370$	0	0
$371$	−1734.66	−0.242747
$372$	1613.33	0.224859
$373$	6000.18	0.832916	0.416458	−	0.909155i	$-0.363271\pi$
$373$	6000.18	0.832916	0.416458	+	0.909155i	$0.363271\pi$
$374$	5513.63	0.762308
$375$	0	0
$376$	7384.53	1.01284
$377$	2287.48	0.312496
$378$	306.478	0.0417025
$379$	9449.12	1.28066	0.640328	−	0.768101i	$-0.278798\pi$
$379$	9449.12	1.28066	0.640328	+	0.768101i	$0.278798\pi$
$380$	0	0
$381$	5530.79	0.743703
$382$	−2063.59	−0.276394
$383$	856.748	0.114302	0.0571511	−	0.998366i	$-0.481798\pi$
$383$	856.748	0.114302	0.0571511	+	0.998366i	$0.481798\pi$
$384$	4028.96	0.535421
$385$	0	0
$386$	1223.16	0.161288
$387$	377.745	0.0496172
$388$	5600.28	0.732760
$389$	−172.900	−0.0225356	−0.0112678	−	0.999937i	$-0.503587\pi$
$389$	−172.900	−0.0225356	−0.0112678	+	0.999937i	$0.503587\pi$
$390$	0	0
$391$	−17589.2	−2.27499
$392$	−1892.96	−0.243901
$393$	−6840.39	−0.877996
$394$	−2583.33	−0.330320
$395$	0	0
$396$	−4731.77	−0.600456
$397$	9250.58	1.16945	0.584727	−	0.811230i	$-0.301202\pi$
$397$	9250.58	1.16945	0.584727	+	0.811230i	$0.301202\pi$
$398$	−4471.03	−0.563097
$399$	642.788	0.0806507
$400$	0	0
$401$	9906.64	1.23370	0.616851	−	0.787080i	$-0.288408\pi$
$401$	9906.64	1.23370	0.616851	+	0.787080i	$0.288408\pi$
$402$	−1631.71	−0.202444
$403$	−5787.39	−0.715361
$404$	11327.5	1.39496
$405$	0	0
$406$	329.180	0.0402388
$407$	9118.58	1.11054
$408$	−3534.98	−0.428940
$409$	−9457.67	−1.14340	−0.571702	−	0.820462i	$-0.693716\pi$
$409$	−9457.67	−1.14340	−0.571702	+	0.820462i	$0.693716\pi$
$410$	0	0
$411$	−3603.04	−0.432421
$412$	1497.70	0.179094
$413$	765.415	0.0911952
$414$	−1380.70	−0.163908
$415$	0	0
$416$	−11043.8	−1.30160
$417$	−28.9676	−0.00340180
$418$	907.737	0.106217
$419$	4924.06	0.574119	0.287060	−	0.957913i	$-0.407322\pi$
$419$	4924.06	0.574119	0.287060	+	0.957913i	$0.407322\pi$
$420$	0	0
$421$	−13100.9	−1.51663	−0.758314	−	0.651889i	$-0.773977\pi$
$421$	−13100.9	−1.51663	−0.758314	+	0.651889i	$0.773977\pi$
$422$	−2126.20	−0.245265
$423$	5294.95	0.608627
$424$	1570.57	0.179891
$425$	0	0
$426$	−651.395	−0.0740850
$427$	−3599.71	−0.407968
$428$	−2513.23	−0.283835
$429$	16973.9	1.91028
$430$	0	0
$431$	−12506.6	−1.39774	−0.698868	−	0.715251i	$-0.746312\pi$
$431$	−12506.6	−1.39774	−0.698868	+	0.715251i	$0.746312\pi$
$432$	1305.70	0.145418
$433$	−3714.42	−0.412249	−0.206124	−	0.978526i	$-0.566085\pi$
$433$	−3714.42	−0.412249	−0.206124	+	0.978526i	$0.566085\pi$
$434$	−832.837	−0.0921139
$435$	0	0
$436$	−9617.26	−1.05638
$437$	−2895.79	−0.316990
$438$	−12.9673	−0.00141461
$439$	9298.90	1.01096	0.505481	−	0.862838i	$-0.331315\pi$
$439$	9298.90	1.01096	0.505481	+	0.862838i	$0.331315\pi$
$440$	0	0
$441$	−1357.32	−0.146563
$442$	6063.09	0.652470
$443$	10321.3	1.10696	0.553478	−	0.832864i	$-0.313300\pi$
$443$	10321.3	1.10696	0.553478	+	0.832864i	$0.313300\pi$
$444$	−2795.28	−0.298780
$445$	0	0
$446$	−1292.42	−0.137215
$447$	−1693.26	−0.179169
$448$	3774.05	0.398007
$449$	7340.29	0.771514	0.385757	−	0.922600i	$-0.373940\pi$
$449$	7340.29	0.771514	0.385757	+	0.922600i	$0.373940\pi$
$450$	0	0
$451$	−8907.24	−0.929990
$452$	288.496	0.0300215
$453$	1907.76	0.197869
$454$	−1988.54	−0.205566
$455$	0	0
$456$	−581.982	−0.0597671
$457$	−1423.35	−0.145692	−0.0728462	−	0.997343i	$-0.523208\pi$
$457$	−1423.35	−0.145692	−0.0728462	+	0.997343i	$0.523208\pi$
$458$	3784.06	0.386064
$459$	−2534.69	−0.257755
$460$	0	0
$461$	−7865.08	−0.794606	−0.397303	−	0.917687i	$-0.630054\pi$
$461$	−7865.08	−0.794606	−0.397303	+	0.917687i	$0.630054\pi$
$462$	2442.64	0.245978
$463$	−6096.09	−0.611899	−0.305950	−	0.952048i	$-0.598974\pi$
$463$	−6096.09	−0.611899	−0.305950	+	0.952048i	$0.598974\pi$
$464$	1402.42	0.140314
$465$	0	0
$466$	2732.72	0.271654
$467$	7535.45	0.746679	0.373339	−	0.927695i	$-0.378213\pi$
$467$	7535.45	0.746679	0.373339	+	0.927695i	$0.378213\pi$
$468$	−5203.32	−0.513939
$469$	−9208.98	−0.906676
$470$	0	0
$471$	1719.07	0.168175
$472$	−693.009	−0.0675812
$473$	3010.64	0.292663
$474$	1946.76	0.188645
$475$	0	0
$476$	−9539.00	−0.918528
$477$	1126.15	0.108098
$478$	620.002	0.0593269
$479$	16971.5	1.61889	0.809444	−	0.587197i	$-0.199768\pi$
$479$	16971.5	1.61889	0.809444	+	0.587197i	$0.199768\pi$
$480$	0	0
$481$	10027.3	0.950530
$482$	393.721	0.0372064
$483$	−7792.32	−0.734085
$484$	−27956.7	−2.62554
$485$	0	0
$486$	−198.966	−0.0185706
$487$	−17333.1	−1.61281	−0.806405	−	0.591364i	$-0.798590\pi$
$487$	−17333.1	−1.61281	−0.806405	+	0.591364i	$0.798590\pi$
$488$	3259.19	0.302329
$489$	9834.07	0.909431
$490$	0	0
$491$	2468.47	0.226885	0.113442	−	0.993545i	$-0.463812\pi$
$491$	2468.47	0.226885	0.113442	+	0.993545i	$0.463812\pi$
$492$	2730.49	0.250204
$493$	−2722.45	−0.248708
$494$	998.197	0.0909130
$495$	0	0
$496$	−3548.17	−0.321205
$497$	−3676.31	−0.331801
$498$	3499.70	0.314910
$499$	9085.35	0.815062	0.407531	−	0.913191i	$-0.366390\pi$
$499$	9085.35	0.815062	0.407531	+	0.913191i	$0.366390\pi$
$500$	0	0
$501$	11774.4	1.04998
$502$	−1831.03	−0.162795
$503$	−4534.24	−0.401932	−0.200966	−	0.979598i	$-0.564408\pi$
$503$	−4534.24	−0.401932	−0.200966	+	0.979598i	$0.564408\pi$
$504$	−1566.06	−0.138408
$505$	0	0
$506$	−11004.2	−0.966793
$507$	12074.5	1.05768
$508$	−13512.8	−1.18018
$509$	−13065.3	−1.13774	−0.568871	−	0.822426i	$-0.692620\pi$
$509$	−13065.3	−1.13774	−0.568871	+	0.822426i	$0.692620\pi$
$510$	0	0
$511$	−73.1840	−0.00633555
$512$	−11626.8	−1.00358
$513$	−417.300	−0.0359147
$514$	−5682.99	−0.487677
$515$	0	0
$516$	−922.904	−0.0787376
$517$	42200.9	3.58993
$518$	1442.98	0.122396
$519$	7228.02	0.611320
$520$	0	0
$521$	11144.0	0.937101	0.468550	−	0.883437i	$-0.344776\pi$
$521$	11144.0	0.937101	0.468550	+	0.883437i	$0.344776\pi$
$522$	−213.705	−0.0179188
$523$	4525.71	0.378385	0.189193	−	0.981940i	$-0.439413\pi$
$523$	4525.71	0.378385	0.189193	+	0.981940i	$0.439413\pi$
$524$	16712.4	1.39329
$525$	0	0
$526$	4896.49	0.405888
$527$	6887.89	0.569338
$528$	10406.5	0.857736
$529$	22937.8	1.88525
$530$	0	0
$531$	−496.910	−0.0406103
$532$	−1570.45	−0.127985
$533$	−9794.88	−0.795992
$534$	3683.80	0.298527
$535$	0	0
$536$	8337.83	0.671902
$537$	−811.972	−0.0652499
$538$	−1492.12	−0.119572
$539$	−10817.9	−0.864486
$540$	0	0
$541$	12028.0	0.955864	0.477932	−	0.878397i	$-0.341387\pi$
$541$	12028.0	0.955864	0.477932	+	0.878397i	$0.341387\pi$
$542$	−2621.45	−0.207750
$543$	−529.516	−0.0418485
$544$	13143.8	1.03591
$545$	0	0
$546$	2686.06	0.210536
$547$	−861.602	−0.0673482	−0.0336741	−	0.999433i	$-0.510721\pi$
$547$	−861.602	−0.0673482	−0.0336741	+	0.999433i	$0.510721\pi$
$548$	8802.93	0.686209
$549$	2336.94	0.181673
$550$	0	0
$551$	−448.211	−0.0346541
$552$	7055.19	0.544001
$553$	10987.0	0.844874
$554$	756.772	0.0580364
$555$	0	0
$556$	70.7736	0.00539832
$557$	−26268.4	−1.99826	−0.999129	−	0.0417358i	$-0.986711\pi$
$557$	−26268.4	−1.99826	−0.999129	+	0.0417358i	$0.986711\pi$
$558$	540.680	0.0410194
$559$	3310.66	0.250494
$560$	0	0
$561$	−20201.6	−1.52034
$562$	−2542.62	−0.190843
$563$	−24246.0	−1.81501	−0.907503	−	0.420045i	$-0.862014\pi$
$563$	−24246.0	−1.81501	−0.907503	+	0.420045i	$0.862014\pi$
$564$	−12936.6	−0.965831
$565$	0	0
$566$	−8.64386	−0.000641923 0
$567$	−1122.92	−0.0831711
$568$	3328.54	0.245885
$569$	−16401.5	−1.20841	−0.604206	−	0.796828i	$-0.706510\pi$
$569$	−16401.5	−1.20841	−0.604206	+	0.796828i	$0.706510\pi$
$570$	0	0
$571$	13490.3	0.988704	0.494352	−	0.869262i	$-0.335405\pi$
$571$	13490.3	0.988704	0.494352	+	0.869262i	$0.335405\pi$
$572$	−41470.6	−3.03142
$573$	7560.87	0.551239
$574$	−1409.54	−0.102496
$575$	0	0
$576$	−2450.12	−0.177237
$577$	−25614.9	−1.84811	−0.924057	−	0.382254i	$-0.875148\pi$
$577$	−25614.9	−1.84811	−0.924057	+	0.382254i	$0.875148\pi$
$578$	−3193.28	−0.229798
$579$	−4481.56	−0.321671
$580$	0	0
$581$	19751.4	1.41037
$582$	1876.83	0.133672
$583$	8975.44	0.637607
$584$	66.2609	0.00469503
$585$	0	0
$586$	−2368.00	−0.166930
$587$	389.674	0.0273996	0.0136998	−	0.999906i	$-0.495639\pi$
$587$	389.674	0.0273996	0.0136998	+	0.999906i	$0.495639\pi$
$588$	3316.19	0.232580
$589$	1133.99	0.0793296
$590$	0	0
$591$	9465.15	0.658789
$592$	6147.60	0.426799
$593$	−3592.08	−0.248750	−0.124375	−	0.992235i	$-0.539693\pi$
$593$	−3592.08	−0.248750	−0.124375	+	0.992235i	$0.539693\pi$
$594$	−1585.77	−0.109537
$595$	0	0
$596$	4136.96	0.284323
$597$	16381.6	1.12304
$598$	−12100.8	−0.827492
$599$	6355.87	0.433546	0.216773	−	0.976222i	$-0.430447\pi$
$599$	6355.87	0.433546	0.216773	+	0.976222i	$0.430447\pi$
$600$	0	0
$601$	−24779.9	−1.68185	−0.840927	−	0.541149i	$-0.817990\pi$
$601$	−24779.9	−1.68185	−0.840927	+	0.541149i	$0.817990\pi$
$602$	476.422	0.0322550
$603$	5978.50	0.403753
$604$	−4661.04	−0.313998
$605$	0	0
$606$	3796.21	0.254473
$607$	−14087.1	−0.941971	−0.470986	−	0.882141i	$-0.656102\pi$
$607$	−14087.1	−0.941971	−0.470986	+	0.882141i	$0.656102\pi$
$608$	2163.93	0.144341
$609$	−1206.09	−0.0802519
$610$	0	0
$611$	46406.4	3.07267
$612$	6192.75	0.409031
$613$	18561.9	1.22302	0.611508	−	0.791238i	$-0.290563\pi$
$613$	18561.9	1.22302	0.611508	+	0.791238i	$0.290563\pi$
$614$	5624.69	0.369697
$615$	0	0
$616$	−12481.5	−0.816389
$617$	21599.2	1.40932	0.704661	−	0.709544i	$-0.251099\pi$
$617$	21599.2	1.40932	0.704661	+	0.709544i	$0.251099\pi$
$618$	501.928	0.0326707
$619$	−14290.1	−0.927893	−0.463947	−	0.885863i	$-0.653567\pi$
$619$	−14290.1	−0.927893	−0.463947	+	0.885863i	$0.653567\pi$
$620$	0	0
$621$	5058.80	0.326896
$622$	3074.45	0.198190
$623$	20790.4	1.33700
$624$	11443.5	0.734148
$625$	0	0
$626$	822.338	0.0525036
$627$	−3325.89	−0.211839
$628$	−4200.01	−0.266877
$629$	−11934.0	−0.756503
$630$	0	0
$631$	21344.9	1.34664	0.673318	−	0.739353i	$-0.264869\pi$
$631$	21344.9	1.34664	0.673318	+	0.739353i	$0.264869\pi$
$632$	−9947.67	−0.626103
$633$	7790.26	0.489155
$634$	−3730.14	−0.233663
$635$	0	0
$636$	−2751.40	−0.171541
$637$	−11895.9	−0.739926
$638$	−1703.23	−0.105692
$639$	2386.67	0.147755
$640$	0	0
$641$	24178.6	1.48985	0.744927	−	0.667146i	$-0.232484\pi$
$641$	24178.6	1.48985	0.744927	+	0.667146i	$0.232484\pi$
$642$	−842.262	−0.0517779
$643$	−11994.8	−0.735658	−0.367829	−	0.929893i	$-0.619899\pi$
$643$	−11994.8	−0.735658	−0.367829	+	0.929893i	$0.619899\pi$
$644$	19038.1	1.16492
$645$	0	0
$646$	−1188.01	−0.0723553
$647$	−5641.51	−0.342798	−0.171399	−	0.985202i	$-0.554829\pi$
$647$	−5641.51	−0.342798	−0.171399	+	0.985202i	$0.554829\pi$
$648$	1016.69	0.0616348
$649$	−3960.39	−0.239536
$650$	0	0
$651$	3051.46	0.183711
$652$	−24026.5	−1.44318
$653$	−21090.4	−1.26390	−0.631952	−	0.775008i	$-0.717746\pi$
$653$	−21090.4	−1.26390	−0.631952	+	0.775008i	$0.717746\pi$
$654$	−3223.05	−0.192708
$655$	0	0
$656$	−6005.12	−0.357409
$657$	47.5112	0.00282129
$658$	6678.13	0.395654
$659$	−13160.3	−0.777926	−0.388963	−	0.921253i	$-0.627167\pi$
$659$	−13160.3	−0.777926	−0.388963	+	0.921253i	$0.627167\pi$
$660$	0	0
$661$	32282.2	1.89959	0.949796	−	0.312869i	$-0.101290\pi$
$661$	32282.2	1.89959	0.949796	+	0.312869i	$0.101290\pi$
$662$	3557.13	0.208840
$663$	−22214.8	−1.30128
$664$	−17883.0	−1.04517
$665$	0	0
$666$	−936.787	−0.0545042
$667$	5433.52	0.315423
$668$	−28767.2	−1.66622
$669$	4735.35	0.273661
$670$	0	0
$671$	18625.5	1.07158
$672$	5822.95	0.334263
$673$	2556.97	0.146455	0.0732273	−	0.997315i	$-0.476670\pi$
$673$	2556.97	0.146455	0.0732273	+	0.997315i	$0.476670\pi$
$674$	6642.33	0.379604
$675$	0	0
$676$	−29500.2	−1.67844
$677$	−24565.3	−1.39457	−0.697284	−	0.716795i	$-0.745608\pi$
$677$	−24565.3	−1.39457	−0.697284	+	0.716795i	$0.745608\pi$
$678$	96.6842	0.00547660
$679$	10592.4	0.598670
$680$	0	0
$681$	7285.90	0.409980
$682$	4309.23	0.241949
$683$	2925.25	0.163882	0.0819411	−	0.996637i	$-0.473888\pi$
$683$	2925.25	0.163882	0.0819411	+	0.996637i	$0.473888\pi$
$684$	1019.54	0.0569930
$685$	0	0
$686$	−5605.29	−0.311969
$687$	−13864.5	−0.769964
$688$	2029.72	0.112475
$689$	9869.88	0.545737
$690$	0	0
$691$	−3667.88	−0.201929	−0.100964	−	0.994890i	$-0.532193\pi$
$691$	−3667.88	−0.201929	−0.100964	+	0.994890i	$0.532193\pi$
$692$	−17659.5	−0.970104
$693$	−8949.67	−0.490577
$694$	−8793.06	−0.480951
$695$	0	0
$696$	1092.00	0.0594715
$697$	11657.4	0.633510
$698$	6300.74	0.341672
$699$	−10012.5	−0.541784
$700$	0	0
$701$	13843.7	0.745891	0.372946	−	0.927853i	$-0.378348\pi$
$701$	13843.7	0.745891	0.372946	+	0.927853i	$0.378348\pi$
$702$	−1743.80	−0.0937541
$703$	−1964.76	−0.105409
$704$	−19527.6	−1.04542
$705$	0	0
$706$	−5421.30	−0.288999
$707$	21424.9	1.13970
$708$	1214.05	0.0644444
$709$	23902.3	1.26610	0.633052	−	0.774109i	$-0.281802\pi$
$709$	23902.3	1.26610	0.633052	+	0.774109i	$0.281802\pi$
$710$	0	0
$711$	−7132.80	−0.376232
$712$	−18823.7	−0.990797
$713$	−13747.0	−0.722060
$714$	−3196.82	−0.167560
$715$	0	0
$716$	1983.81	0.103545
$717$	−2271.65	−0.118321
$718$	−1981.47	−0.102991
$719$	16629.2	0.862537	0.431269	−	0.902224i	$-0.358066\pi$
$719$	16629.2	0.862537	0.431269	+	0.902224i	$0.358066\pi$
$720$	0	0
$721$	2832.75	0.146321
$722$	5420.51	0.279405
$723$	−1442.57	−0.0742042
$724$	1293.71	0.0664094
$725$	0	0
$726$	−9369.18	−0.478957
$727$	8889.10	0.453478	0.226739	−	0.973956i	$-0.427194\pi$
$727$	8889.10	0.453478	0.226739	+	0.973956i	$0.427194\pi$
$728$	−13725.4	−0.698759
$729$	729.000	0.0370370
$730$	0	0
$731$	−3940.20	−0.199362
$732$	−5709.60	−0.288296
$733$	17622.0	0.887973	0.443987	−	0.896033i	$-0.353564\pi$
$733$	17622.0	0.887973	0.443987	+	0.896033i	$0.353564\pi$
$734$	4302.37	0.216353
$735$	0	0
$736$	−26232.7	−1.31379
$737$	47648.8	2.38150
$738$	915.075	0.0456428
$739$	−25697.3	−1.27915	−0.639573	−	0.768730i	$-0.720889\pi$
$739$	−25697.3	−1.27915	−0.639573	+	0.768730i	$0.720889\pi$
$740$	0	0
$741$	−3657.33	−0.181316
$742$	1420.33	0.0702721
$743$	35603.8	1.75798	0.878989	−	0.476843i	$-0.158219\pi$
$743$	35603.8	1.75798	0.878989	+	0.476843i	$0.158219\pi$
$744$	−2762.80	−0.136141
$745$	0	0
$746$	−4912.90	−0.241118
$747$	−12822.7	−0.628054
$748$	49356.4	2.41263
$749$	−4753.51	−0.231895
$750$	0	0
$751$	−890.657	−0.0432763	−0.0216382	−	0.999766i	$-0.506888\pi$
$751$	−890.657	−0.0432763	−0.0216382	+	0.999766i	$0.506888\pi$
$752$	28451.2	1.37966
$753$	6708.77	0.324676
$754$	−1872.97	−0.0904634
$755$	0	0
$756$	2743.50	0.131984
$757$	27316.1	1.31152	0.655761	−	0.754969i	$-0.272348\pi$
$757$	27316.1	1.31152	0.655761	+	0.754969i	$0.272348\pi$
$758$	−7736.86	−0.370733
$759$	40318.8	1.92817
$760$	0	0
$761$	−19155.6	−0.912472	−0.456236	−	0.889859i	$-0.650803\pi$
$761$	−19155.6	−0.912472	−0.456236	+	0.889859i	$0.650803\pi$
$762$	−4528.56	−0.215292
$763$	−18190.1	−0.863073
$764$	−18472.7	−0.874762
$765$	0	0
$766$	−701.498	−0.0330890
$767$	−4355.06	−0.205022
$768$	3234.78	0.151986
$769$	−6601.93	−0.309586	−0.154793	−	0.987947i	$-0.549471\pi$
$769$	−6601.93	−0.309586	−0.154793	+	0.987947i	$0.549471\pi$
$770$	0	0
$771$	20822.1	0.972620
$772$	10949.3	0.510460
$773$	−10501.3	−0.488622	−0.244311	−	0.969697i	$-0.578562\pi$
$773$	−10501.3	−0.488622	−0.244311	+	0.969697i	$0.578562\pi$
$774$	−309.295	−0.0143635
$775$	0	0
$776$	−9590.34	−0.443651
$777$	−5286.99	−0.244105
$778$	141.569	0.00652376
$779$	1919.22	0.0882711
$780$	0	0
$781$	19021.8	0.871517
$782$	14401.9	0.658580
$783$	783.000	0.0357371
$784$	−7293.22	−0.332235
$785$	0	0
$786$	5600.86	0.254168
$787$	−27169.2	−1.23059	−0.615297	−	0.788296i	$-0.710964\pi$
$787$	−27169.2	−1.23059	−0.615297	+	0.788296i	$0.710964\pi$
$788$	−23125.2	−1.04543
$789$	−17940.4	−0.809500
$790$	0	0
$791$	545.661	0.0245278
$792$	8103.05	0.363547
$793$	20481.6	0.917180
$794$	−7574.30	−0.338541
$795$	0	0
$796$	−40023.3	−1.78215
$797$	−3072.45	−0.136552	−0.0682759	−	0.997666i	$-0.521750\pi$
$797$	−3072.45	−0.136552	−0.0682759	+	0.997666i	$0.521750\pi$
$798$	−526.309	−0.0233473
$799$	−55230.8	−2.44546
$800$	0	0
$801$	−13497.2	−0.595380
$802$	−8111.48	−0.357140
$803$	378.666	0.0166411
$804$	−14606.6	−0.640716
$805$	0	0
$806$	4738.67	0.207087
$807$	5467.03	0.238474
$808$	−19398.1	−0.844583
$809$	−149.629	−0.00650269	−0.00325134	−	0.999995i	$-0.501035\pi$
$809$	−149.629	−0.00650269	−0.00325134	+	0.999995i	$0.501035\pi$
$810$	0	0
$811$	−10550.4	−0.456813	−0.228406	−	0.973566i	$-0.573351\pi$
$811$	−10550.4	−0.456813	−0.228406	+	0.973566i	$0.573351\pi$
$812$	2946.72	0.127352
$813$	9604.81	0.414336
$814$	−7466.22	−0.321488
$815$	0	0
$816$	−13619.6	−0.584290
$817$	−648.695	−0.0277784
$818$	7743.86	0.331000
$819$	−9841.54	−0.419892
$820$	0	0
$821$	29361.9	1.24816	0.624078	−	0.781362i	$-0.285475\pi$
$821$	29361.9	1.24816	0.624078	+	0.781362i	$0.285475\pi$
$822$	2950.14	0.125180
$823$	−23398.7	−0.991042	−0.495521	−	0.868596i	$-0.665023\pi$
$823$	−23398.7	−0.991042	−0.495521	+	0.868596i	$0.665023\pi$
$824$	−2564.78	−0.108433
$825$	0	0
$826$	−626.716	−0.0263998
$827$	−15323.5	−0.644317	−0.322158	−	0.946686i	$-0.604408\pi$
$827$	−15323.5	−0.644317	−0.322158	+	0.946686i	$0.604408\pi$
$828$	−12359.6	−0.518752
$829$	10782.3	0.451730	0.225865	−	0.974159i	$-0.427479\pi$
$829$	10782.3	0.451730	0.225865	+	0.974159i	$0.427479\pi$
$830$	0	0
$831$	−2772.76	−0.115747
$832$	−21473.6	−0.894786
$833$	14157.9	0.588888
$834$	23.7185	0.000984776 0
$835$	0	0
$836$	8125.79	0.336168
$837$	−1981.02	−0.0818088
$838$	−4031.78	−0.166200
$839$	41081.5	1.69045	0.845227	−	0.534408i	$-0.179465\pi$
$839$	41081.5	1.69045	0.845227	+	0.534408i	$0.179465\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	10726.9	0.439044
$843$	9315.98	0.380616
$844$	−19033.1	−0.776240
$845$	0	0
$846$	−4335.46	−0.176189
$847$	−52877.3	−2.14508
$848$	6051.10	0.245042
$849$	31.6705	0.00128025
$850$	0	0
$851$	23818.2	0.959432
$852$	−5831.10	−0.234472
$853$	23051.2	0.925274	0.462637	−	0.886548i	$-0.346903\pi$
$853$	23051.2	0.925274	0.462637	+	0.886548i	$0.346903\pi$
$854$	2947.41	0.118101
$855$	0	0
$856$	4303.84	0.171848
$857$	−1000.28	−0.0398704	−0.0199352	−	0.999801i	$-0.506346\pi$
$857$	−1000.28	−0.0398704	−0.0199352	+	0.999801i	$0.506346\pi$
$858$	−13898.1	−0.553000
$859$	−16370.6	−0.650242	−0.325121	−	0.945672i	$-0.605405\pi$
$859$	−16370.6	−0.650242	−0.325121	+	0.945672i	$0.605405\pi$
$860$	0	0
$861$	5164.45	0.204418
$862$	10240.3	0.404626
$863$	−16312.7	−0.643441	−0.321721	−	0.946835i	$-0.604261\pi$
$863$	−16312.7	−0.643441	−0.321721	+	0.946835i	$0.604261\pi$
$864$	−3780.27	−0.148851
$865$	0	0
$866$	3041.34	0.119340
$867$	11700.0	0.458307
$868$	−7455.30	−0.291532
$869$	−56848.6	−2.21917
$870$	0	0
$871$	52397.2	2.03836
$872$	16469.3	0.639589
$873$	−6876.59	−0.266595
$874$	2371.05	0.0917644
$875$	0	0
$876$	−116.079	−0.00447711
$877$	15674.8	0.603534	0.301767	−	0.953382i	$-0.402424\pi$
$877$	15674.8	0.603534	0.301767	+	0.953382i	$0.402424\pi$
$878$	−7613.86	−0.292660
$879$	8676.21	0.332925
$880$	0	0
$881$	−14789.4	−0.565569	−0.282784	−	0.959184i	$-0.591258\pi$
$881$	−14789.4	−0.565569	−0.282784	+	0.959184i	$0.591258\pi$
$882$	1111.36	0.0424279
$883$	−5049.14	−0.192432	−0.0962158	−	0.995360i	$-0.530674\pi$
$883$	−5049.14	−0.192432	−0.0962158	+	0.995360i	$0.530674\pi$
$884$	54274.9	2.06500
$885$	0	0
$886$	−8451.03	−0.320449
$887$	37670.6	1.42599	0.712997	−	0.701167i	$-0.247337\pi$
$887$	37670.6	1.42599	0.712997	+	0.701167i	$0.247337\pi$
$888$	4786.85	0.180897
$889$	−25558.1	−0.964218
$890$	0	0
$891$	5810.15	0.218459
$892$	−11569.4	−0.434273
$893$	−9092.92	−0.340743
$894$	1386.43	0.0518670
$895$	0	0
$896$	−18618.0	−0.694179
$897$	44336.7	1.65034
$898$	−6010.17	−0.223343
$899$	−2127.76	−0.0789374
$900$	0	0
$901$	−11746.7	−0.434338
$902$	7293.17	0.269220
$903$	−1745.58	−0.0643292
$904$	−494.043	−0.0181766
$905$	0	0
$906$	−1562.06	−0.0572804
$907$	29554.5	1.08196	0.540981	−	0.841035i	$-0.318053\pi$
$907$	29554.5	1.08196	0.540981	+	0.841035i	$0.318053\pi$
$908$	−17800.9	−0.650597
$909$	−13909.1	−0.507519
$910$	0	0
$911$	898.948	0.0326932	0.0163466	−	0.999866i	$-0.494796\pi$
$911$	898.948	0.0326932	0.0163466	+	0.999866i	$0.494796\pi$
$912$	−2242.26	−0.0814130
$913$	−102197.	−3.70452
$914$	1165.43	0.0421760
$915$	0	0
$916$	33873.7	1.22186
$917$	31609.8	1.13833
$918$	2075.39	0.0746165
$919$	−36963.4	−1.32678	−0.663390	−	0.748274i	$-0.730883\pi$
$919$	−36963.4	−1.32678	−0.663390	+	0.748274i	$0.730883\pi$
$920$	0	0
$921$	−20608.5	−0.737321
$922$	6439.87	0.230028
$923$	20917.4	0.745944
$924$	21865.8	0.778497
$925$	0	0
$926$	4991.43	0.177137
$927$	−1839.03	−0.0651583
$928$	−4060.29	−0.143627
$929$	−35541.5	−1.25520	−0.627600	−	0.778536i	$-0.715963\pi$
$929$	−35541.5	−1.25520	−0.627600	+	0.778536i	$0.715963\pi$
$930$	0	0
$931$	2330.90	0.0820537
$932$	24462.5	0.859758
$933$	−11264.6	−0.395270
$934$	−6169.96	−0.216154
$935$	0	0
$936$	8910.56	0.311165
$937$	−17380.4	−0.605968	−0.302984	−	0.952996i	$-0.597983\pi$
$937$	−17380.4	−0.605968	−0.302984	+	0.952996i	$0.597983\pi$
$938$	7540.24	0.262471
$939$	−3012.99	−0.104713
$940$	0	0
$941$	12456.3	0.431525	0.215763	−	0.976446i	$-0.430776\pi$
$941$	12456.3	0.431525	0.215763	+	0.976446i	$0.430776\pi$
$942$	−1407.56	−0.0486844
$943$	−23266.1	−0.803446
$944$	−2670.03	−0.0920572
$945$	0	0
$946$	−2465.09	−0.0847219
$947$	36639.9	1.25727	0.628636	−	0.777700i	$-0.283614\pi$
$947$	36639.9	1.25727	0.628636	+	0.777700i	$0.283614\pi$
$948$	17426.8	0.597043
$949$	416.402	0.0142434
$950$	0	0
$951$	13667.0	0.466017
$952$	16335.3	0.556125
$953$	−4554.66	−0.154816	−0.0774082	−	0.996999i	$-0.524664\pi$
$953$	−4554.66	−0.154816	−0.0774082	+	0.996999i	$0.524664\pi$
$954$	−922.082	−0.0312930
$955$	0	0
$956$	5550.07	0.187764
$957$	6240.53	0.210792
$958$	−13896.1	−0.468647
$959$	16649.8	0.560638
$960$	0	0
$961$	−24407.7	−0.819298
$962$	−8210.26	−0.275166
$963$	3085.99	0.103266
$964$	3524.47	0.117755
$965$	0	0
$966$	6380.29	0.212508
$967$	−13804.0	−0.459056	−0.229528	−	0.973302i	$-0.573718\pi$
$967$	−13804.0	−0.459056	−0.229528	+	0.973302i	$0.573718\pi$
$968$	47875.2	1.58964
$969$	4352.78	0.144305
$970$	0	0
$971$	−24931.6	−0.823988	−0.411994	−	0.911187i	$-0.635168\pi$
$971$	−24931.6	−0.823988	−0.411994	+	0.911187i	$0.635168\pi$
$972$	−1781.09	−0.0587741
$973$	133.861	0.00441047
$974$	14192.2	0.466887
$975$	0	0
$976$	12557.0	0.411824
$977$	24423.4	0.799770	0.399885	−	0.916565i	$-0.369050\pi$
$977$	24423.4	0.799770	0.399885	+	0.916565i	$0.369050\pi$
$978$	−8052.05	−0.263268
$979$	−107573.	−3.51180
$980$	0	0
$981$	11809.0	0.384336
$982$	−2021.16	−0.0656801
$983$	−52340.5	−1.69827	−0.849136	−	0.528174i	$-0.822877\pi$
$983$	−52340.5	−1.69827	−0.849136	+	0.528174i	$0.822877\pi$
$984$	−4675.91	−0.151486
$985$	0	0
$986$	2229.12	0.0719976
$987$	−24468.2	−0.789091
$988$	8935.56	0.287731
$989$	7863.93	0.252840
$990$	0	0
$991$	6746.36	0.216251	0.108126	−	0.994137i	$-0.465515\pi$
$991$	6746.36	0.216251	0.108126	+	0.994137i	$0.465515\pi$
$992$	10272.7	0.328788
$993$	−13033.1	−0.416509
$994$	3010.13	0.0960519
$995$	0	0
$996$	31328.2	0.996659
$997$	−15842.8	−0.503257	−0.251629	−	0.967824i	$-0.580966\pi$
$997$	−15842.8	−0.503257	−0.251629	+	0.967824i	$0.580966\pi$
$998$	−7439.01	−0.235950
$999$	3432.33	0.108703

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.4.a.o.1.3		8
5.4	even	2		435.4.a.k.1.6	✓	8
15.14	odd	2		1305.4.a.p.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.k.1.6	✓	8	5.4	even	2
1305.4.a.p.1.3		8	15.14	odd	2
2175.4.a.o.1.3		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2175.a (trivial)

\(f(q)\)	\(=\)	\(q-0.818792 q^{2} +3.00000 q^{3} -7.32958 q^{4} -2.45638 q^{6} -13.8632 q^{7} +12.5517 q^{8} +9.00000 q^{9} +71.7303 q^{11} -21.9887 q^{12} +78.8785 q^{13} +11.3510 q^{14} +48.3594 q^{16} -93.8775 q^{17} -7.36913 q^{18} -15.4555 q^{19} -41.5895 q^{21} -58.7322 q^{22} +187.363 q^{23} +37.6552 q^{24} -64.5851 q^{26} +27.0000 q^{27} +101.611 q^{28} +29.0000 q^{29} -73.3710 q^{31} -140.010 q^{32} +215.191 q^{33} +76.8662 q^{34} -65.9662 q^{36} +127.123 q^{37} +12.6549 q^{38} +236.635 q^{39} -124.177 q^{41} +34.0531 q^{42} +41.9717 q^{43} -525.753 q^{44} -153.411 q^{46} +588.328 q^{47} +145.078 q^{48} -150.813 q^{49} -281.633 q^{51} -578.146 q^{52} +125.128 q^{53} -22.1074 q^{54} -174.007 q^{56} -46.3666 q^{57} -23.7450 q^{58} -55.2122 q^{59} +259.660 q^{61} +60.0755 q^{62} -124.768 q^{63} -272.236 q^{64} -176.196 q^{66} +664.277 q^{67} +688.083 q^{68} +562.089 q^{69} +265.186 q^{71} +112.966 q^{72} +5.27903 q^{73} -104.087 q^{74} +113.283 q^{76} -994.408 q^{77} -193.755 q^{78} -792.533 q^{79} +81.0000 q^{81} +101.675 q^{82} -1424.74 q^{83} +304.833 q^{84} -34.3661 q^{86} +87.0000 q^{87} +900.339 q^{88} -1499.69 q^{89} -1093.50 q^{91} -1373.29 q^{92} -220.113 q^{93} -481.718 q^{94} -420.030 q^{96} -764.065 q^{97} +123.484 q^{98} +645.572 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 24 q^{3} + 38 q^{4} + 12 q^{6} + q^{7} + 9 q^{8} + 72 q^{9} + 11 q^{11} + 114 q^{12} + 97 q^{13} - 69 q^{14} + 242 q^{16} + 119 q^{17} + 36 q^{18} + 220 q^{19} + 3 q^{21} + 129 q^{22} + 262 q^{23}+ \cdots + 99 q^{99}+O(q^{100}) \) 8 * q + 4 * q^2 + 24 * q^3 + 38 * q^4 + 12 * q^6 + q^7 + 9 * q^8 + 72 * q^9 + 11 * q^11 + 114 * q^12 + 97 * q^13 - 69 * q^14 + 242 * q^16 + 119 * q^17 + 36 * q^18 + 220 * q^19 + 3 * q^21 + 129 * q^22 + 262 * q^23 + 27 * q^24 + 43 * q^26 + 216 * q^27 + 13 * q^28 + 232 * q^29 + 108 * q^31 + 566 * q^32 + 33 * q^33 - 527 * q^34 + 342 * q^36 - 118 * q^37 + 1088 * q^38 + 291 * q^39 - 118 * q^41 - 207 * q^42 + 384 * q^43 - 1605 * q^44 + 154 * q^46 + 1309 * q^47 + 726 * q^48 + 427 * q^49 + 357 * q^51 + 747 * q^52 + 1912 * q^53 + 108 * q^54 - 2022 * q^56 + 660 * q^57 + 116 * q^58 - 1178 * q^59 + 572 * q^61 + 2572 * q^62 + 9 * q^63 - 1771 * q^64 + 387 * q^66 - 1147 * q^67 + 3963 * q^68 + 786 * q^69 + 84 * q^71 + 81 * q^72 + 222 * q^73 - 1758 * q^74 + 1390 * q^76 + 1753 * q^77 + 129 * q^78 + 390 * q^79 + 648 * q^81 + 2994 * q^82 + 1570 * q^83 + 39 * q^84 + 1668 * q^86 + 696 * q^87 + 1790 * q^88 - 119 * q^89 - 3529 * q^91 + 2934 * q^92 + 324 * q^93 - 2277 * q^94 + 1698 * q^96 + 2100 * q^97 + 7677 * q^98 + 99 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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