LMFDB - Embedded newform 1617.4.a.p.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1617,4,Mod(1,1617)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1617, 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("1617.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1617 = 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1617.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:	$95.4060884793$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 28x^{3} - 11x^{2} + 108x - 64 $ x^5 - 28x^3 - 11x^2 + 108*x - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-4.36278$ of defining polynomial
Character	$\chi$	$=$	1617.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.36278 q^{2} -3.00000 q^{3} +20.7594 q^{4} +1.66863 q^{5} -16.0883 q^{6} +68.4261 q^{8} +9.00000 q^{9} +8.94849 q^{10} +11.0000 q^{11} -62.2783 q^{12} -0.996604 q^{13} -5.00588 q^{15} +200.879 q^{16} +99.5115 q^{17} +48.2650 q^{18} -32.8695 q^{19} +34.6398 q^{20} +58.9906 q^{22} +72.6019 q^{23} -205.278 q^{24} -122.216 q^{25} -5.34457 q^{26} -27.0000 q^{27} +45.0027 q^{29} -26.8455 q^{30} +62.8102 q^{31} +529.860 q^{32} -33.0000 q^{33} +533.659 q^{34} +186.835 q^{36} -301.317 q^{37} -176.272 q^{38} +2.98981 q^{39} +114.178 q^{40} +307.353 q^{41} +214.347 q^{43} +228.354 q^{44} +15.0176 q^{45} +389.348 q^{46} +602.899 q^{47} -602.636 q^{48} -655.416 q^{50} -298.535 q^{51} -20.6889 q^{52} +592.072 q^{53} -144.795 q^{54} +18.3549 q^{55} +98.6084 q^{57} +241.339 q^{58} -695.030 q^{59} -103.919 q^{60} -442.815 q^{61} +336.837 q^{62} +1234.49 q^{64} -1.66296 q^{65} -176.972 q^{66} -555.143 q^{67} +2065.80 q^{68} -217.806 q^{69} -153.352 q^{71} +615.835 q^{72} +147.489 q^{73} -1615.90 q^{74} +366.647 q^{75} -682.352 q^{76} +16.0337 q^{78} +676.959 q^{79} +335.192 q^{80} +81.0000 q^{81} +1648.27 q^{82} +222.412 q^{83} +166.048 q^{85} +1149.50 q^{86} -135.008 q^{87} +752.687 q^{88} +1136.35 q^{89} +80.5364 q^{90} +1507.18 q^{92} -188.430 q^{93} +3233.22 q^{94} -54.8469 q^{95} -1589.58 q^{96} -1010.60 q^{97} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} - 15 q^{3} + 21 q^{4} - 7 q^{5} - 15 q^{6} + 60 q^{8} + 45 q^{9} - 55 q^{10} + 55 q^{11} - 63 q^{12} - 111 q^{13} + 21 q^{15} + 201 q^{16} - 136 q^{17} + 45 q^{18} - 111 q^{19} - 219 q^{20}+ \cdots + 495 q^{99}+O(q^{100}) $ 5 * q + 5 * q^2 - 15 * q^3 + 21 * q^4 - 7 * q^5 - 15 * q^6 + 60 * q^8 + 45 * q^9 - 55 * q^10 + 55 * q^11 - 63 * q^12 - 111 * q^13 + 21 * q^15 + 201 * q^16 - 136 * q^17 + 45 * q^18 - 111 * q^19 - 219 * q^20 + 55 * q^22 - 28 * q^23 - 180 * q^24 + 190 * q^25 + q^26 - 135 * q^27 + 61 * q^29 + 165 * q^30 + 280 * q^31 + 535 * q^32 - 165 * q^33 + 572 * q^34 + 189 * q^36 - 41 * q^37 - 267 * q^38 + 333 * q^39 + 336 * q^40 - 426 * q^41 + 424 * q^43 + 231 * q^44 - 63 * q^45 + 140 * q^46 - 75 * q^47 - 603 * q^48 + 490 * q^50 + 408 * q^51 + 269 * q^52 + 1500 * q^53 - 135 * q^54 - 77 * q^55 + 333 * q^57 - 1767 * q^58 - 757 * q^59 + 657 * q^60 - 658 * q^61 + 568 * q^62 - 748 * q^64 + 537 * q^65 - 165 * q^66 - 583 * q^67 + 1650 * q^68 + 84 * q^69 - 764 * q^71 + 540 * q^72 - 875 * q^73 - 825 * q^74 - 570 * q^75 - 213 * q^76 - 3 * q^78 - 244 * q^79 + 2577 * q^80 + 405 * q^81 + 2006 * q^82 - 924 * q^83 - 1402 * q^85 + 1272 * q^86 - 183 * q^87 + 660 * q^88 + 1110 * q^89 - 495 * q^90 - 2046 * q^92 - 840 * q^93 + 3349 * q^94 + 1923 * q^95 - 1605 * q^96 + 852 * q^97 + 495 * 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 $

$2$	5.36278	1.89603	0.948015	−	0.318226i	$-0.103087\pi$
$2$	5.36278	1.89603	0.948015	+	0.318226i	$0.103087\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	20.7594	2.59493
$5$	1.66863	0.149247	0.0746233	−	0.997212i	$-0.476225\pi$
$5$	1.66863	0.149247	0.0746233	+	0.997212i	$0.476225\pi$
$6$	−16.0883	−1.09467
$7$	0	0
$7$	0	0
$8$	68.4261	3.02403
$9$	9.00000	0.333333
$10$	8.94849	0.282976
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−62.2783	−1.49818
$13$	−0.996604	−0.0212622	−0.0106311	−	0.999943i	$-0.503384\pi$
$13$	−0.996604	−0.0212622	−0.0106311	+	0.999943i	$0.503384\pi$
$14$	0	0
$15$	−5.00588	−0.0861675
$16$	200.879	3.13873
$17$	99.5115	1.41971	0.709856	−	0.704347i	$-0.248760\pi$
$17$	99.5115	1.41971	0.709856	+	0.704347i	$0.248760\pi$
$18$	48.2650	0.632010
$19$	−32.8695	−0.396883	−0.198441	−	0.980113i	$-0.563588\pi$
$19$	−32.8695	−0.396883	−0.198441	+	0.980113i	$0.563588\pi$
$20$	34.6398	0.387284
$21$	0	0
$22$	58.9906	0.571675
$23$	72.6019	0.658198	0.329099	−	0.944295i	$-0.393255\pi$
$23$	72.6019	0.658198	0.329099	+	0.944295i	$0.393255\pi$
$24$	−205.278	−1.74593
$25$	−122.216	−0.977725
$26$	−5.34457	−0.0403137
$27$	−27.0000	−0.192450
$28$	0	0
$29$	45.0027	0.288165	0.144082	−	0.989566i	$-0.453977\pi$
$29$	45.0027	0.288165	0.144082	+	0.989566i	$0.453977\pi$
$30$	−26.8455	−0.163376
$31$	62.8102	0.363904	0.181952	−	0.983307i	$-0.441758\pi$
$31$	62.8102	0.363904	0.181952	+	0.983307i	$0.441758\pi$
$32$	529.860	2.92709
$33$	−33.0000	−0.174078
$34$	533.659	2.69182
$35$	0	0
$36$	186.835	0.864976
$37$	−301.317	−1.33882	−0.669408	−	0.742895i	$-0.733452\pi$
$37$	−301.317	−1.33882	−0.669408	+	0.742895i	$0.733452\pi$
$38$	−176.272	−0.752502
$39$	2.98981	0.0122757
$40$	114.178	0.451327
$41$	307.353	1.17074	0.585371	−	0.810765i	$-0.300949\pi$
$41$	307.353	1.17074	0.585371	+	0.810765i	$0.300949\pi$
$42$	0	0
$43$	214.347	0.760178	0.380089	−	0.924950i	$-0.375893\pi$
$43$	214.347	0.760178	0.380089	+	0.924950i	$0.375893\pi$
$44$	228.354	0.782401
$45$	15.0176	0.0497489
$46$	389.348	1.24796
$47$	602.899	1.87110	0.935552	−	0.353188i	$-0.114902\pi$
$47$	602.899	1.87110	0.935552	+	0.353188i	$0.114902\pi$
$48$	−602.636	−1.81215
$49$	0	0
$50$	−655.416	−1.85380
$51$	−298.535	−0.819671
$52$	−20.6889	−0.0551738
$53$	592.072	1.53448	0.767239	−	0.641362i	$-0.221630\pi$
$53$	592.072	1.53448	0.767239	+	0.641362i	$0.221630\pi$
$54$	−144.795	−0.364891
$55$	18.3549	0.0449995
$56$	0	0
$57$	98.6084	0.229140
$58$	241.339	0.546369
$59$	−695.030	−1.53365	−0.766824	−	0.641857i	$-0.778164\pi$
$59$	−695.030	−1.53365	−0.766824	+	0.641857i	$0.778164\pi$
$60$	−103.919	−0.223599
$61$	−442.815	−0.929453	−0.464726	−	0.885454i	$-0.653847\pi$
$61$	−442.815	−0.929453	−0.464726	+	0.885454i	$0.653847\pi$
$62$	336.837	0.689974
$63$	0	0
$64$	1234.49	2.41112
$65$	−1.66296	−0.00317331
$66$	−176.972	−0.330056
$67$	−555.143	−1.01226	−0.506131	−	0.862457i	$-0.668925\pi$
$67$	−555.143	−1.01226	−0.506131	+	0.862457i	$0.668925\pi$
$68$	2065.80	3.68405
$69$	−217.806	−0.380011
$70$	0	0
$71$	−153.352	−0.256331	−0.128166	−	0.991753i	$-0.540909\pi$
$71$	−153.352	−0.256331	−0.128166	+	0.991753i	$0.540909\pi$
$72$	615.835	1.00801
$73$	147.489	0.236470	0.118235	−	0.992986i	$-0.462276\pi$
$73$	147.489	0.236470	0.118235	+	0.992986i	$0.462276\pi$
$74$	−1615.90	−2.53843
$75$	366.647	0.564490
$76$	−682.352	−1.02988
$77$	0	0
$78$	16.0337	0.0232751
$79$	676.959	0.964099	0.482050	−	0.876144i	$-0.339893\pi$
$79$	676.959	0.964099	0.482050	+	0.876144i	$0.339893\pi$
$80$	335.192	0.468445
$81$	81.0000	0.111111
$82$	1648.27	2.21976
$83$	222.412	0.294131	0.147065	−	0.989127i	$-0.453017\pi$
$83$	222.412	0.294131	0.147065	+	0.989127i	$0.453017\pi$
$84$	0	0
$85$	166.048	0.211887
$86$	1149.50	1.44132
$87$	−135.008	−0.166372
$88$	752.687	0.911781
$89$	1136.35	1.35340	0.676699	−	0.736260i	$-0.263410\pi$
$89$	1136.35	1.35340	0.676699	+	0.736260i	$0.263410\pi$
$90$	80.5364	0.0943253
$91$	0	0
$92$	1507.18	1.70798
$93$	−188.430	−0.210100
$94$	3233.22	3.54767
$95$	−54.8469	−0.0592334
$96$	−1589.58	−1.68996
$97$	−1010.60	−1.05785	−0.528923	−	0.848670i	$-0.677404\pi$
$97$	−1010.60	−1.05785	−0.528923	+	0.848670i	$0.677404\pi$
$98$	0	0
$99$	99.0000	0.100504
$100$	−2537.13	−2.53713
$101$	400.418	0.394486	0.197243	−	0.980355i	$-0.436801\pi$
$101$	400.418	0.394486	0.197243	+	0.980355i	$0.436801\pi$
$102$	−1600.98	−1.55412
$103$	419.845	0.401636	0.200818	−	0.979629i	$-0.435640\pi$
$103$	419.845	0.401636	0.200818	+	0.979629i	$0.435640\pi$
$104$	−68.1937	−0.0642975
$105$	0	0
$106$	3175.15	2.90941
$107$	2131.99	1.92623	0.963116	−	0.269085i	$-0.0867214\pi$
$107$	2131.99	1.92623	0.963116	+	0.269085i	$0.0867214\pi$
$108$	−560.505	−0.499394
$109$	−14.0057	−0.0123074	−0.00615369	−	0.999981i	$-0.501959\pi$
$109$	−14.0057	−0.0123074	−0.00615369	+	0.999981i	$0.501959\pi$
$110$	98.4333	0.0853205
$111$	903.950	0.772965
$112$	0	0
$113$	1322.98	1.10137	0.550686	−	0.834712i	$-0.314366\pi$
$113$	1322.98	1.10137	0.550686	+	0.834712i	$0.314366\pi$
$114$	528.816	0.434457
$115$	121.146	0.0982338
$116$	934.230	0.747768
$117$	−8.96944	−0.00708739
$118$	−3727.30	−2.90784
$119$	0	0
$120$	−342.533	−0.260574
$121$	121.000	0.0909091
$122$	−2374.72	−1.76227
$123$	−922.059	−0.675929
$124$	1303.90	0.944306
$125$	−412.511	−0.295169
$126$	0	0
$127$	−2199.01	−1.53646	−0.768232	−	0.640172i	$-0.778863\pi$
$127$	−2199.01	−1.53646	−0.768232	+	0.640172i	$0.778863\pi$
$128$	2381.45	1.64447
$129$	−643.042	−0.438889
$130$	−8.91810	−0.00601668
$131$	−1450.27	−0.967259	−0.483629	−	0.875273i	$-0.660682\pi$
$131$	−1450.27	−0.967259	−0.483629	+	0.875273i	$0.660682\pi$
$132$	−685.061	−0.451719
$133$	0	0
$134$	−2977.11	−1.91928
$135$	−45.0529	−0.0287225
$136$	6809.19	4.29326
$137$	−893.027	−0.556909	−0.278454	−	0.960449i	$-0.589822\pi$
$137$	−893.027	−0.556909	−0.278454	+	0.960449i	$0.589822\pi$
$138$	−1168.05	−0.720512
$139$	530.606	0.323780	0.161890	−	0.986809i	$-0.448241\pi$
$139$	530.606	0.323780	0.161890	+	0.986809i	$0.448241\pi$
$140$	0	0
$141$	−1808.70	−1.08028
$142$	−822.392	−0.486011
$143$	−10.9626	−0.00641079
$144$	1807.91	1.04624
$145$	75.0927	0.0430076
$146$	790.954	0.448355
$147$	0	0
$148$	−6255.17	−3.47413
$149$	2958.97	1.62690	0.813451	−	0.581633i	$-0.197586\pi$
$149$	2958.97	1.62690	0.813451	+	0.581633i	$0.197586\pi$
$150$	1966.25	1.07029
$151$	1668.28	0.899092	0.449546	−	0.893257i	$-0.351586\pi$
$151$	1668.28	0.899092	0.449546	+	0.893257i	$0.351586\pi$
$152$	−2249.13	−1.20019
$153$	895.604	0.473237
$154$	0	0
$155$	104.807	0.0543115
$156$	62.0668	0.0318546
$157$	1202.63	0.611342	0.305671	−	0.952137i	$-0.401119\pi$
$157$	1202.63	0.611342	0.305671	+	0.952137i	$0.401119\pi$
$158$	3630.38	1.82796
$159$	−1776.21	−0.885931
$160$	884.139	0.436858
$161$	0	0
$162$	434.385	0.210670
$163$	−2534.83	−1.21806	−0.609029	−	0.793148i	$-0.708441\pi$
$163$	−2534.83	−1.21806	−0.609029	+	0.793148i	$0.708441\pi$
$164$	6380.47	3.03799
$165$	−55.0647	−0.0259805
$166$	1192.75	0.557681
$167$	−2424.43	−1.12340	−0.561700	−	0.827341i	$-0.689853\pi$
$167$	−2424.43	−1.12340	−0.561700	+	0.827341i	$0.689853\pi$
$168$	0	0
$169$	−2196.01	−0.999548
$170$	890.478	0.401744
$171$	−295.825	−0.132294
$172$	4449.73	1.97261
$173$	−1977.78	−0.869180	−0.434590	−	0.900628i	$-0.643107\pi$
$173$	−1977.78	−0.869180	−0.434590	+	0.900628i	$0.643107\pi$
$174$	−724.018	−0.315447
$175$	0	0
$176$	2209.67	0.946363
$177$	2085.09	0.885452
$178$	6093.97	2.56608
$179$	1314.89	0.549049	0.274525	−	0.961580i	$-0.411480\pi$
$179$	1314.89	0.549049	0.274525	+	0.961580i	$0.411480\pi$
$180$	311.758	0.129095
$181$	−2118.94	−0.870163	−0.435081	−	0.900391i	$-0.643280\pi$
$181$	−2118.94	−0.870163	−0.435081	+	0.900391i	$0.643280\pi$
$182$	0	0
$183$	1328.44	0.536620
$184$	4967.87	1.99041
$185$	−502.785	−0.199814
$186$	−1010.51	−0.398357
$187$	1094.63	0.428059
$188$	12515.9	4.85538
$189$	0	0
$190$	−294.132	−0.112308
$191$	2136.61	0.809421	0.404711	−	0.914445i	$-0.367372\pi$
$191$	2136.61	0.809421	0.404711	+	0.914445i	$0.367372\pi$
$192$	−3703.48	−1.39206
$193$	−910.852	−0.339713	−0.169856	−	0.985469i	$-0.554330\pi$
$193$	−910.852	−0.339713	−0.169856	+	0.985469i	$0.554330\pi$
$194$	−5419.64	−2.00571
$195$	4.98888	0.00183211
$196$	0	0
$197$	−2485.86	−0.899038	−0.449519	−	0.893271i	$-0.648405\pi$
$197$	−2485.86	−0.899038	−0.449519	+	0.893271i	$0.648405\pi$
$198$	530.915	0.190558
$199$	384.145	0.136841	0.0684204	−	0.997657i	$-0.478204\pi$
$199$	384.145	0.136841	0.0684204	+	0.997657i	$0.478204\pi$
$200$	−8362.74	−2.95668
$201$	1665.43	0.584429
$202$	2147.36	0.747958
$203$	0	0
$204$	−6197.41	−2.12699
$205$	512.857	0.174729
$206$	2251.54	0.761515
$207$	653.417	0.219399
$208$	−200.196	−0.0667362
$209$	−361.564	−0.119665
$210$	0	0
$211$	3875.31	1.26440	0.632198	−	0.774807i	$-0.282153\pi$
$211$	3875.31	1.26440	0.632198	+	0.774807i	$0.282153\pi$
$212$	12291.1	3.98186
$213$	460.055	0.147993
$214$	11433.4	3.65219
$215$	357.666	0.113454
$216$	−1847.50	−0.581976
$217$	0	0
$218$	−75.1096	−0.0233352
$219$	−442.468	−0.136526
$220$	381.037	0.116771
$221$	−99.1736	−0.0301861
$222$	4847.69	1.46557
$223$	−449.799	−0.135071	−0.0675354	−	0.997717i	$-0.521514\pi$
$223$	−449.799	−0.135071	−0.0675354	+	0.997717i	$0.521514\pi$
$224$	0	0
$225$	−1099.94	−0.325908
$226$	7094.83	2.08823
$227$	−5873.32	−1.71729	−0.858647	−	0.512567i	$-0.828695\pi$
$227$	−5873.32	−1.71729	−0.858647	+	0.512567i	$0.828695\pi$
$228$	2047.06	0.594603
$229$	−4207.68	−1.21420	−0.607099	−	0.794626i	$-0.707667\pi$
$229$	−4207.68	−1.21420	−0.607099	+	0.794626i	$0.707667\pi$
$230$	649.677	0.186254
$231$	0	0
$232$	3079.36	0.871421
$233$	5405.81	1.51994	0.759970	−	0.649958i	$-0.225213\pi$
$233$	5405.81	1.51994	0.759970	+	0.649958i	$0.225213\pi$
$234$	−48.1011	−0.0134379
$235$	1006.01	0.279256
$236$	−14428.4	−3.97971
$237$	−2030.88	−0.556623
$238$	0	0
$239$	−5105.16	−1.38170	−0.690848	−	0.723000i	$-0.742763\pi$
$239$	−5105.16	−1.38170	−0.690848	+	0.723000i	$0.742763\pi$
$240$	−1005.58	−0.270457
$241$	−254.138	−0.0679273	−0.0339637	−	0.999423i	$-0.510813\pi$
$241$	−254.138	−0.0679273	−0.0339637	+	0.999423i	$0.510813\pi$
$242$	648.897	0.172366
$243$	−243.000	−0.0641500
$244$	−9192.58	−2.41186
$245$	0	0
$246$	−4944.80	−1.28158
$247$	32.7579	0.00843859
$248$	4297.85	1.10046
$249$	−667.235	−0.169816
$250$	−2212.21	−0.559649
$251$	−7030.70	−1.76802	−0.884012	−	0.467464i	$-0.845168\pi$
$251$	−7030.70	−1.76802	−0.884012	+	0.467464i	$0.845168\pi$
$252$	0	0
$253$	798.621	0.198454
$254$	−11792.8	−2.91318
$255$	−498.143	−0.122333
$256$	2895.22	0.706841
$257$	−6053.67	−1.46933	−0.734664	−	0.678431i	$-0.762660\pi$
$257$	−6053.67	−1.46933	−0.734664	+	0.678431i	$0.762660\pi$
$258$	−3448.49	−0.832146
$259$	0	0
$260$	−34.5221	−0.00823451
$261$	405.024	0.0960550
$262$	−7777.50	−1.83395
$263$	−6106.79	−1.43179	−0.715895	−	0.698208i	$-0.753981\pi$
$263$	−6106.79	−1.43179	−0.715895	+	0.698208i	$0.753981\pi$
$264$	−2258.06	−0.526417
$265$	987.947	0.229015
$266$	0	0
$267$	−3409.04	−0.781384
$268$	−11524.5	−2.62675
$269$	−4855.91	−1.10063	−0.550316	−	0.834956i	$-0.685493\pi$
$269$	−4855.91	−1.10063	−0.550316	+	0.834956i	$0.685493\pi$
$270$	−241.609	−0.0544588
$271$	8501.09	1.90555	0.952775	−	0.303676i	$-0.0982141\pi$
$271$	8501.09	1.90555	0.952775	+	0.303676i	$0.0982141\pi$
$272$	19989.7	4.45609
$273$	0	0
$274$	−4789.11	−1.05592
$275$	−1344.37	−0.294795
$276$	−4521.53	−0.986101
$277$	−6830.18	−1.48154	−0.740768	−	0.671761i	$-0.765538\pi$
$277$	−6830.18	−1.48154	−0.740768	+	0.671761i	$0.765538\pi$
$278$	2845.52	0.613896
$279$	565.291	0.121301
$280$	0	0
$281$	4987.88	1.05890	0.529452	−	0.848340i	$-0.322397\pi$
$281$	4987.88	1.05890	0.529452	+	0.848340i	$0.322397\pi$
$282$	−9699.66	−2.04825
$283$	1952.88	0.410200	0.205100	−	0.978741i	$-0.434248\pi$
$283$	1952.88	0.410200	0.205100	+	0.978741i	$0.434248\pi$
$284$	−3183.50	−0.665161
$285$	164.541	0.0341984
$286$	−58.7903	−0.0121550
$287$	0	0
$288$	4768.74	0.975697
$289$	4989.55	1.01558
$290$	402.706	0.0815438
$291$	3031.81	0.610748
$292$	3061.80	0.613624
$293$	3220.72	0.642172	0.321086	−	0.947050i	$-0.395952\pi$
$293$	3220.72	0.642172	0.321086	+	0.947050i	$0.395952\pi$
$294$	0	0
$295$	−1159.75	−0.228892
$296$	−20617.9	−4.04862
$297$	−297.000	−0.0580259
$298$	15868.3	3.08466
$299$	−72.3554	−0.0139947
$300$	7611.39	1.46481
$301$	0	0
$302$	8946.64	1.70471
$303$	−1201.25	−0.227757
$304$	−6602.78	−1.24571
$305$	−738.893	−0.138718
$306$	4802.93	0.897272
$307$	5325.08	0.989963	0.494981	−	0.868904i	$-0.335175\pi$
$307$	5325.08	0.989963	0.494981	+	0.868904i	$0.335175\pi$
$308$	0	0
$309$	−1259.53	−0.231885
$310$	562.056	0.102976
$311$	9625.36	1.75500	0.877498	−	0.479580i	$-0.159211\pi$
$311$	9625.36	1.75500	0.877498	+	0.479580i	$0.159211\pi$
$312$	204.581	0.0371222
$313$	4315.55	0.779327	0.389664	−	0.920957i	$-0.372591\pi$
$313$	4315.55	0.779327	0.389664	+	0.920957i	$0.372591\pi$
$314$	6449.46	1.15912
$315$	0	0
$316$	14053.3	2.50177
$317$	−7399.84	−1.31109	−0.655546	−	0.755155i	$-0.727562\pi$
$317$	−7399.84	−1.31109	−0.655546	+	0.755155i	$0.727562\pi$
$318$	−9525.45	−1.67975
$319$	495.029	0.0868850
$320$	2059.91	0.359852
$321$	−6395.96	−1.11211
$322$	0	0
$323$	−3270.89	−0.563459
$324$	1681.51	0.288325
$325$	121.801	0.0207886
$326$	−13593.8	−2.30947
$327$	42.0171	0.00710567
$328$	21031.0	3.54037
$329$	0	0
$330$	−295.300	−0.0492598
$331$	−4899.29	−0.813562	−0.406781	−	0.913526i	$-0.633349\pi$
$331$	−4899.29	−0.813562	−0.406781	+	0.913526i	$0.633349\pi$
$332$	4617.14	0.763248
$333$	−2711.85	−0.446272
$334$	−13001.7	−2.13000
$335$	−926.327	−0.151077
$336$	0	0
$337$	−1445.02	−0.233577	−0.116788	−	0.993157i	$-0.537260\pi$
$337$	−1445.02	−0.233577	−0.116788	+	0.993157i	$0.537260\pi$
$338$	−11776.7	−1.89517
$339$	−3968.93	−0.635878
$340$	3447.06	0.549832
$341$	690.912	0.109721
$342$	−1586.45	−0.250834
$343$	0	0
$344$	14666.9	2.29880
$345$	−363.437	−0.0567153
$346$	−10606.4	−1.64799
$347$	4351.93	0.673268	0.336634	−	0.941636i	$-0.390711\pi$
$347$	4351.93	0.673268	0.336634	+	0.941636i	$0.390711\pi$
$348$	−2802.69	−0.431724
$349$	−7971.30	−1.22262	−0.611309	−	0.791392i	$-0.709357\pi$
$349$	−7971.30	−1.22262	−0.611309	+	0.791392i	$0.709357\pi$
$350$	0	0
$351$	26.9083	0.00409191
$352$	5828.46	0.882551
$353$	1169.88	0.176393	0.0881963	−	0.996103i	$-0.471890\pi$
$353$	1169.88	0.176393	0.0881963	+	0.996103i	$0.471890\pi$
$354$	11181.9	1.67884
$355$	−255.887	−0.0382565
$356$	23589.9	3.51197
$357$	0	0
$358$	7051.49	1.04101
$359$	11998.8	1.76400	0.881998	−	0.471253i	$-0.156198\pi$
$359$	11998.8	1.76400	0.881998	+	0.471253i	$0.156198\pi$
$360$	1027.60	0.150442
$361$	−5778.60	−0.842484
$362$	−11363.4	−1.64985
$363$	−363.000	−0.0524864
$364$	0	0
$365$	246.105	0.0352924
$366$	7124.16	1.01745
$367$	8489.94	1.20755	0.603776	−	0.797154i	$-0.293662\pi$
$367$	8489.94	1.20755	0.603776	+	0.797154i	$0.293662\pi$
$368$	14584.2	2.06590
$369$	2766.18	0.390248
$370$	−2696.33	−0.378853
$371$	0	0
$372$	−3911.71	−0.545196
$373$	−13306.1	−1.84708	−0.923542	−	0.383498i	$-0.874719\pi$
$373$	−13306.1	−1.84708	−0.923542	+	0.383498i	$0.874719\pi$
$374$	5870.25	0.811613
$375$	1237.53	0.170416
$376$	41254.0	5.65828
$377$	−44.8498	−0.00612701
$378$	0	0
$379$	−10645.0	−1.44274	−0.721368	−	0.692552i	$-0.756486\pi$
$379$	−10645.0	−1.44274	−0.721368	+	0.692552i	$0.756486\pi$
$380$	−1138.59	−0.153707
$381$	6597.04	0.887078
$382$	11458.2	1.53469
$383$	−2299.39	−0.306771	−0.153385	−	0.988166i	$-0.549018\pi$
$383$	−2299.39	−0.306771	−0.153385	+	0.988166i	$0.549018\pi$
$384$	−7144.34	−0.949435
$385$	0	0
$386$	−4884.70	−0.644105
$387$	1929.12	0.253393
$388$	−20979.5	−2.74504
$389$	−3243.77	−0.422791	−0.211395	−	0.977401i	$-0.567801\pi$
$389$	−3243.77	−0.422791	−0.211395	+	0.977401i	$0.567801\pi$
$390$	26.7543	0.00347373
$391$	7224.73	0.934451
$392$	0	0
$393$	4350.82	0.558447
$394$	−13331.1	−1.70460
$395$	1129.59	0.143888
$396$	2055.18	0.260800
$397$	−13850.8	−1.75102	−0.875508	−	0.483203i	$-0.839473\pi$
$397$	−13850.8	−1.75102	−0.875508	+	0.483203i	$0.839473\pi$
$398$	2060.09	0.259454
$399$	0	0
$400$	−24550.5	−3.06882
$401$	1238.44	0.154227	0.0771133	−	0.997022i	$-0.475430\pi$
$401$	1238.44	0.154227	0.0771133	+	0.997022i	$0.475430\pi$
$402$	8931.33	1.10810
$403$	−62.5969	−0.00773740
$404$	8312.46	1.02366
$405$	135.159	0.0165830
$406$	0	0
$407$	−3314.48	−0.403668
$408$	−20427.6	−2.47871
$409$	3951.51	0.477726	0.238863	−	0.971053i	$-0.423225\pi$
$409$	3951.51	0.477726	0.238863	+	0.971053i	$0.423225\pi$
$410$	2750.34	0.331292
$411$	2679.08	0.321531
$412$	8715.74	1.04222
$413$	0	0
$414$	3504.14	0.415988
$415$	371.122	0.0438980
$416$	−528.061	−0.0622363
$417$	−1591.82	−0.186934
$418$	−1938.99	−0.226888
$419$	11603.6	1.35292	0.676462	−	0.736477i	$-0.263512\pi$
$419$	11603.6	1.35292	0.676462	+	0.736477i	$0.263512\pi$
$420$	0	0
$421$	−10498.9	−1.21541	−0.607703	−	0.794165i	$-0.707909\pi$
$421$	−10498.9	−1.21541	−0.607703	+	0.794165i	$0.707909\pi$
$422$	20782.5	2.39733
$423$	5426.09	0.623702
$424$	40513.1	4.64031
$425$	−12161.9	−1.38809
$426$	2467.18	0.280599
$427$	0	0
$428$	44258.8	4.99844
$429$	32.8879	0.00370127
$430$	1918.08	0.215112
$431$	−11109.2	−1.24156	−0.620778	−	0.783986i	$-0.713183\pi$
$431$	−11109.2	−1.24156	−0.620778	+	0.783986i	$0.713183\pi$
$432$	−5423.72	−0.604049
$433$	−12535.7	−1.39129	−0.695644	−	0.718387i	$-0.744881\pi$
$433$	−12535.7	−1.39129	−0.695644	+	0.718387i	$0.744881\pi$
$434$	0	0
$435$	−225.278	−0.0248305
$436$	−290.751	−0.0319368
$437$	−2386.39	−0.261227
$438$	−2372.86	−0.258858
$439$	−2227.80	−0.242203	−0.121102	−	0.992640i	$-0.538643\pi$
$439$	−2227.80	−0.242203	−0.121102	+	0.992640i	$0.538643\pi$
$440$	1255.95	0.136080
$441$	0	0
$442$	−531.846	−0.0572338
$443$	−833.252	−0.0893657	−0.0446828	−	0.999001i	$-0.514228\pi$
$443$	−833.252	−0.0893657	−0.0446828	+	0.999001i	$0.514228\pi$
$444$	18765.5	2.00579
$445$	1896.14	0.201990
$446$	−2412.17	−0.256098
$447$	−8876.92	−0.939293
$448$	0	0
$449$	−4903.89	−0.515432	−0.257716	−	0.966221i	$-0.582970\pi$
$449$	−4903.89	−0.515432	−0.257716	+	0.966221i	$0.582970\pi$
$450$	−5898.75	−0.617932
$451$	3380.88	0.352992
$452$	27464.2	2.85798
$453$	−5004.85	−0.519091
$454$	−31497.3	−3.25604
$455$	0	0
$456$	6747.39	0.692929
$457$	12100.7	1.23861	0.619307	−	0.785149i	$-0.287414\pi$
$457$	12100.7	1.23861	0.619307	+	0.785149i	$0.287414\pi$
$458$	−22564.9	−2.30216
$459$	−2686.81	−0.273224
$460$	2514.91	0.254910
$461$	1271.16	0.128425	0.0642125	−	0.997936i	$-0.479546\pi$
$461$	1271.16	0.128425	0.0642125	+	0.997936i	$0.479546\pi$
$462$	0	0
$463$	−16020.3	−1.60805	−0.804024	−	0.594596i	$-0.797312\pi$
$463$	−16020.3	−1.60805	−0.804024	+	0.594596i	$0.797312\pi$
$464$	9040.08	0.904472
$465$	−314.420	−0.0313568
$466$	28990.2	2.88185
$467$	10696.1	1.05986	0.529930	−	0.848042i	$-0.322218\pi$
$467$	10696.1	1.05986	0.529930	+	0.848042i	$0.322218\pi$
$468$	−186.200	−0.0183913
$469$	0	0
$470$	5395.04	0.529478
$471$	−3607.90	−0.352958
$472$	−47558.2	−4.63780
$473$	2357.82	0.229202
$474$	−10891.2	−1.05537
$475$	4017.17	0.388043
$476$	0	0
$477$	5328.64	0.511492
$478$	−27377.9	−2.61974
$479$	12055.2	1.14993	0.574963	−	0.818179i	$-0.305016\pi$
$479$	12055.2	1.14993	0.574963	+	0.818179i	$0.305016\pi$
$480$	−2652.42	−0.252220
$481$	300.293	0.0284661
$482$	−1362.89	−0.128792
$483$	0	0
$484$	2511.89	0.235903
$485$	−1686.32	−0.157880
$486$	−1303.16	−0.121630
$487$	−14273.2	−1.32809	−0.664045	−	0.747693i	$-0.731162\pi$
$487$	−14273.2	−1.32809	−0.664045	+	0.747693i	$0.731162\pi$
$488$	−30300.1	−2.81070
$489$	7604.50	0.703246
$490$	0	0
$491$	−15616.7	−1.43538	−0.717691	−	0.696362i	$-0.754801\pi$
$491$	−15616.7	−1.43538	−0.717691	+	0.696362i	$0.754801\pi$
$492$	−19141.4	−1.75399
$493$	4478.28	0.409111
$494$	175.673	0.0159998
$495$	165.194	0.0149998
$496$	12617.2	1.14220
$497$	0	0
$498$	−3578.24	−0.321977
$499$	5820.03	0.522125	0.261062	−	0.965322i	$-0.415927\pi$
$499$	5820.03	0.522125	0.261062	+	0.965322i	$0.415927\pi$
$500$	−8563.49	−0.765942
$501$	7273.28	0.648596
$502$	−37704.1	−3.35223
$503$	1744.38	0.154629	0.0773143	−	0.997007i	$-0.475366\pi$
$503$	1744.38	0.154629	0.0773143	+	0.997007i	$0.475366\pi$
$504$	0	0
$505$	668.149	0.0588757
$506$	4282.83	0.376275
$507$	6588.02	0.577089
$508$	−45650.3	−3.98701
$509$	4561.13	0.397187	0.198594	−	0.980082i	$-0.436363\pi$
$509$	4561.13	0.397187	0.198594	+	0.980082i	$0.436363\pi$
$510$	−2671.43	−0.231947
$511$	0	0
$512$	−3525.13	−0.304278
$513$	887.476	0.0763802
$514$	−32464.5	−2.78589
$515$	700.565	0.0599429
$516$	−13349.2	−1.13889
$517$	6631.89	0.564159
$518$	0	0
$519$	5933.35	0.501822
$520$	−113.790	−0.00959619
$521$	311.151	0.0261646	0.0130823	−	0.999914i	$-0.495836\pi$
$521$	311.151	0.0261646	0.0130823	+	0.999914i	$0.495836\pi$
$522$	2172.06	0.182123
$523$	−18159.9	−1.51831	−0.759154	−	0.650911i	$-0.774387\pi$
$523$	−18159.9	−1.51831	−0.759154	+	0.650911i	$0.774387\pi$
$524$	−30106.8	−2.50997
$525$	0	0
$526$	−32749.4	−2.71472
$527$	6250.34	0.516639
$528$	−6629.00	−0.546383
$529$	−6895.96	−0.566776
$530$	5298.14	0.434220
$531$	−6255.27	−0.511216
$532$	0	0
$533$	−306.309	−0.0248925
$534$	−18281.9	−1.48153
$535$	3557.49	0.287484
$536$	−37986.3	−3.06111
$537$	−3944.68	−0.316994
$538$	−26041.2	−2.08683
$539$	0	0
$540$	−935.274	−0.0745329
$541$	10858.8	0.862951	0.431476	−	0.902125i	$-0.357993\pi$
$541$	10858.8	0.862951	0.431476	+	0.902125i	$0.357993\pi$
$542$	45589.5	3.61298
$543$	6356.81	0.502389
$544$	52727.2	4.15562
$545$	−23.3703	−0.00183683
$546$	0	0
$547$	9560.10	0.747277	0.373638	−	0.927574i	$-0.378110\pi$
$547$	9560.10	0.747277	0.373638	+	0.927574i	$0.378110\pi$
$548$	−18538.7	−1.44514
$549$	−3985.33	−0.309818
$550$	−7209.58	−0.558941
$551$	−1479.21	−0.114368
$552$	−14903.6	−1.14917
$553$	0	0
$554$	−36628.8	−2.80904
$555$	1508.36	0.115362
$556$	11015.1	0.840186
$557$	−961.471	−0.0731397	−0.0365699	−	0.999331i	$-0.511643\pi$
$557$	−961.471	−0.0731397	−0.0365699	+	0.999331i	$0.511643\pi$
$558$	3031.53	0.229991
$559$	−213.619	−0.0161630
$560$	0	0
$561$	−3283.88	−0.247140
$562$	26748.9	2.00772
$563$	−26376.5	−1.97449	−0.987245	−	0.159206i	$-0.949106\pi$
$563$	−26376.5	−1.97449	−0.987245	+	0.159206i	$0.949106\pi$
$564$	−37547.6	−2.80326
$565$	2207.55	0.164376
$566$	10472.9	0.777751
$567$	0	0
$568$	−10493.3	−0.775154
$569$	8256.23	0.608294	0.304147	−	0.952625i	$-0.401629\pi$
$569$	8256.23	0.608294	0.304147	+	0.952625i	$0.401629\pi$
$570$	882.396	0.0648412
$571$	−25003.0	−1.83248	−0.916239	−	0.400632i	$-0.868791\pi$
$571$	−25003.0	−1.83248	−0.916239	+	0.400632i	$0.868791\pi$
$572$	−227.578	−0.0166355
$573$	−6409.82	−0.467320
$574$	0	0
$575$	−8873.09	−0.643537
$576$	11110.5	0.803708
$577$	5560.17	0.401166	0.200583	−	0.979677i	$-0.435716\pi$
$577$	5560.17	0.401166	0.200583	+	0.979677i	$0.435716\pi$
$578$	26757.9	1.92557
$579$	2732.56	0.196133
$580$	1558.88	0.111602
$581$	0	0
$582$	16258.9	1.15800
$583$	6512.79	0.462662
$584$	10092.1	0.715094
$585$	−14.9666	−0.00105777
$586$	17272.0	1.21758
$587$	−3926.82	−0.276111	−0.138055	−	0.990425i	$-0.544085\pi$
$587$	−3926.82	−0.276111	−0.138055	+	0.990425i	$0.544085\pi$
$588$	0	0
$589$	−2064.54	−0.144427
$590$	−6219.47	−0.433986
$591$	7457.59	0.519060
$592$	−60528.1	−4.20218
$593$	14397.6	0.997027	0.498514	−	0.866882i	$-0.333879\pi$
$593$	14397.6	0.997027	0.498514	+	0.866882i	$0.333879\pi$
$594$	−1592.75	−0.110019
$595$	0	0
$596$	61426.6	4.22170
$597$	−1152.44	−0.0790051
$598$	−388.026	−0.0265344
$599$	−14711.0	−1.00347	−0.501734	−	0.865022i	$-0.667304\pi$
$599$	−14711.0	−1.00347	−0.501734	+	0.865022i	$0.667304\pi$
$600$	25088.2	1.70704
$601$	−27436.8	−1.86218	−0.931090	−	0.364790i	$-0.881141\pi$
$601$	−27436.8	−1.86218	−0.931090	+	0.364790i	$0.881141\pi$
$602$	0	0
$603$	−4996.29	−0.337420
$604$	34632.6	2.33308
$605$	201.904	0.0135679
$606$	−6442.07	−0.431834
$607$	−6090.69	−0.407271	−0.203636	−	0.979047i	$-0.565276\pi$
$607$	−6090.69	−0.407271	−0.203636	+	0.979047i	$0.565276\pi$
$608$	−17416.2	−1.16171
$609$	0	0
$610$	−3962.52	−0.263013
$611$	−600.852	−0.0397837
$612$	18592.2	1.22802
$613$	5175.61	0.341013	0.170506	−	0.985357i	$-0.445460\pi$
$613$	5175.61	0.341013	0.170506	+	0.985357i	$0.445460\pi$
$614$	28557.3	1.87700
$615$	−1538.57	−0.100880
$616$	0	0
$617$	−10544.9	−0.688044	−0.344022	−	0.938962i	$-0.611789\pi$
$617$	−10544.9	−0.688044	−0.344022	+	0.938962i	$0.611789\pi$
$618$	−6754.61	−0.439661
$619$	8378.59	0.544045	0.272023	−	0.962291i	$-0.412307\pi$
$619$	8378.59	0.544045	0.272023	+	0.962291i	$0.412307\pi$
$620$	2175.73	0.140934
$621$	−1960.25	−0.126670
$622$	51618.7	3.32753
$623$	0	0
$624$	600.589	0.0385302
$625$	14588.6	0.933673
$626$	23143.4	1.47763
$627$	1084.69	0.0690885
$628$	24966.0	1.58639
$629$	−29984.5	−1.90073
$630$	0	0
$631$	−2651.03	−0.167251	−0.0836257	−	0.996497i	$-0.526650\pi$
$631$	−2651.03	−0.167251	−0.0836257	+	0.996497i	$0.526650\pi$
$632$	46321.6	2.91547
$633$	−11625.9	−0.730000
$634$	−39683.7	−2.48587
$635$	−3669.33	−0.229312
$636$	−36873.2	−2.29893
$637$	0	0
$638$	2654.73	0.164737
$639$	−1380.17	−0.0854437
$640$	3973.75	0.245431
$641$	−6716.00	−0.413831	−0.206916	−	0.978359i	$-0.566343\pi$
$641$	−6716.00	−0.413831	−0.206916	+	0.978359i	$0.566343\pi$
$642$	−34300.1	−2.10860
$643$	−752.048	−0.0461242	−0.0230621	−	0.999734i	$-0.507342\pi$
$643$	−752.048	−0.0461242	−0.0230621	+	0.999734i	$0.507342\pi$
$644$	0	0
$645$	−1073.00	−0.0655026
$646$	−17541.1	−1.06834
$647$	10412.8	0.632717	0.316358	−	0.948640i	$-0.397540\pi$
$647$	10412.8	0.632717	0.316358	+	0.948640i	$0.397540\pi$
$648$	5542.51	0.336004
$649$	−7645.33	−0.462412
$650$	653.190	0.0394157
$651$	0	0
$652$	−52621.7	−3.16077
$653$	15442.9	0.925462	0.462731	−	0.886499i	$-0.346870\pi$
$653$	15442.9	0.925462	0.462731	+	0.886499i	$0.346870\pi$
$654$	225.329	0.0134726
$655$	−2419.96	−0.144360
$656$	61740.6	3.67464
$657$	1327.40	0.0788234
$658$	0	0
$659$	−5333.13	−0.315249	−0.157625	−	0.987499i	$-0.550384\pi$
$659$	−5333.13	−0.315249	−0.157625	+	0.987499i	$0.550384\pi$
$660$	−1143.11	−0.0674175
$661$	−15118.3	−0.889614	−0.444807	−	0.895626i	$-0.646728\pi$
$661$	−15118.3	−0.889614	−0.444807	+	0.895626i	$0.646728\pi$
$662$	−26273.8	−1.54254
$663$	297.521	0.0174280
$664$	15218.8	0.889461
$665$	0	0
$666$	−14543.1	−0.846145
$667$	3267.28	0.189670
$668$	−50329.8	−2.91514
$669$	1349.40	0.0779831
$670$	−4967.69	−0.286446
$671$	−4870.96	−0.280241
$672$	0	0
$673$	−813.476	−0.0465931	−0.0232966	−	0.999729i	$-0.507416\pi$
$673$	−813.476	−0.0465931	−0.0232966	+	0.999729i	$0.507416\pi$
$674$	−7749.33	−0.442868
$675$	3299.82	0.188163
$676$	−45587.9	−2.59376
$677$	−10390.3	−0.589854	−0.294927	−	0.955520i	$-0.595295\pi$
$677$	−10390.3	−0.589854	−0.294927	+	0.955520i	$0.595295\pi$
$678$	−21284.5	−1.20564
$679$	0	0
$680$	11362.0	0.640754
$681$	17620.0	0.991480
$682$	3705.21	0.208035
$683$	24315.7	1.36225	0.681124	−	0.732168i	$-0.261491\pi$
$683$	24315.7	1.36225	0.681124	+	0.732168i	$0.261491\pi$
$684$	−6141.17	−0.343294
$685$	−1490.13	−0.0831167
$686$	0	0
$687$	12623.0	0.701018
$688$	43057.8	2.38599
$689$	−590.061	−0.0326263
$690$	−1949.03	−0.107534
$691$	15379.7	0.846701	0.423351	−	0.905966i	$-0.360854\pi$
$691$	15379.7	0.846701	0.423351	+	0.905966i	$0.360854\pi$
$692$	−41057.7	−2.25546
$693$	0	0
$694$	23338.5	1.27654
$695$	885.384	0.0483230
$696$	−9238.07	−0.503115
$697$	30585.2	1.66212
$698$	−42748.3	−2.31812
$699$	−16217.4	−0.877538
$700$	0	0
$701$	−2397.08	−0.129153	−0.0645766	−	0.997913i	$-0.520570\pi$
$701$	−2397.08	−0.129153	−0.0645766	+	0.997913i	$0.520570\pi$
$702$	144.303	0.00775838
$703$	9904.13	0.531353
$704$	13579.4	0.726981
$705$	−3018.04	−0.161228
$706$	6273.83	0.334446
$707$	0	0
$708$	43285.3	2.29769
$709$	17121.2	0.906912	0.453456	−	0.891279i	$-0.350191\pi$
$709$	17121.2	0.906912	0.453456	+	0.891279i	$0.350191\pi$
$710$	−1372.27	−0.0725355
$711$	6092.63	0.321366
$712$	77755.6	4.09272
$713$	4560.14	0.239521
$714$	0	0
$715$	−18.2926	−0.000956788 0
$716$	27296.5	1.42474
$717$	15315.5	0.797723
$718$	64347.2	3.34459
$719$	−34131.4	−1.77036	−0.885179	−	0.465251i	$-0.845964\pi$
$719$	−34131.4	−1.77036	−0.885179	+	0.465251i	$0.845964\pi$
$720$	3016.73	0.156148
$721$	0	0
$722$	−30989.4	−1.59737
$723$	762.415	0.0392179
$724$	−43988.0	−2.25801
$725$	−5500.03	−0.281746
$726$	−1946.69	−0.0995158
$727$	1143.65	0.0583434	0.0291717	−	0.999574i	$-0.490713\pi$
$727$	1143.65	0.0583434	0.0291717	+	0.999574i	$0.490713\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	1319.81	0.0669154
$731$	21330.0	1.07923
$732$	27577.7	1.39249
$733$	−24224.7	−1.22068	−0.610342	−	0.792138i	$-0.708968\pi$
$733$	−24224.7	−1.22068	−0.610342	+	0.792138i	$0.708968\pi$
$734$	45529.7	2.28955
$735$	0	0
$736$	38468.9	1.92660
$737$	−6106.57	−0.305208
$738$	14834.4	0.739921
$739$	25815.9	1.28505	0.642525	−	0.766265i	$-0.277887\pi$
$739$	25815.9	1.28505	0.642525	+	0.766265i	$0.277887\pi$
$740$	−10437.5	−0.518502
$741$	−98.2736	−0.00487202
$742$	0	0
$743$	−17221.1	−0.850312	−0.425156	−	0.905120i	$-0.639781\pi$
$743$	−17221.1	−0.850312	−0.425156	+	0.905120i	$0.639781\pi$
$744$	−12893.6	−0.635351
$745$	4937.42	0.242810
$746$	−71357.5	−3.50212
$747$	2001.70	0.0980436
$748$	22723.8	1.11078
$749$	0	0
$750$	6636.62	0.323113
$751$	−15631.7	−0.759531	−0.379766	−	0.925083i	$-0.623995\pi$
$751$	−15631.7	−0.759531	−0.379766	+	0.925083i	$0.623995\pi$
$752$	121110.	5.87289
$753$	21092.1	1.02077
$754$	−240.520	−0.0116170
$755$	2783.74	0.134186
$756$	0	0
$757$	24756.7	1.18864	0.594319	−	0.804230i	$-0.297422\pi$
$757$	24756.7	1.18864	0.594319	+	0.804230i	$0.297422\pi$
$758$	−57086.8	−2.73547
$759$	−2395.86	−0.114578
$760$	−3752.96	−0.179124
$761$	9087.20	0.432866	0.216433	−	0.976297i	$-0.430558\pi$
$761$	9087.20	0.432866	0.216433	+	0.976297i	$0.430558\pi$
$762$	35378.5	1.68193
$763$	0	0
$764$	44354.8	2.10039
$765$	1494.43	0.0706290
$766$	−12331.1	−0.581647
$767$	692.670	0.0326087
$768$	−8685.66	−0.408095
$769$	−10699.9	−0.501753	−0.250876	−	0.968019i	$-0.580719\pi$
$769$	−10699.9	−0.501753	−0.250876	+	0.968019i	$0.580719\pi$
$770$	0	0
$771$	18161.0	0.848317
$772$	−18908.8	−0.881531
$773$	6586.45	0.306466	0.153233	−	0.988190i	$-0.451032\pi$
$773$	6586.45	0.306466	0.153233	+	0.988190i	$0.451032\pi$
$774$	10345.5	0.480440
$775$	−7676.39	−0.355799
$776$	−69151.5	−3.19896
$777$	0	0
$778$	−17395.6	−0.801623
$779$	−10102.5	−0.464648
$780$	103.566	0.00475419
$781$	−1686.87	−0.0772867
$782$	38744.7	1.77175
$783$	−1215.07	−0.0554574
$784$	0	0
$785$	2006.75	0.0912406
$786$	23332.5	1.05883
$787$	5581.26	0.252796	0.126398	−	0.991980i	$-0.459658\pi$
$787$	5581.26	0.252796	0.126398	+	0.991980i	$0.459658\pi$
$788$	−51605.1	−2.33294
$789$	18320.4	0.826644
$790$	6057.76	0.272817
$791$	0	0
$792$	6774.18	0.303927
$793$	441.311	0.0197622
$794$	−74279.0	−3.31998
$795$	−2963.84	−0.132222
$796$	7974.64	0.355092
$797$	20336.2	0.903821	0.451911	−	0.892063i	$-0.350743\pi$
$797$	20336.2	0.903821	0.451911	+	0.892063i	$0.350743\pi$
$798$	0	0
$799$	59995.5	2.65643
$800$	−64757.2	−2.86189
$801$	10227.1	0.451132
$802$	6641.49	0.292418
$803$	1622.38	0.0712985
$804$	34573.4	1.51655
$805$	0	0
$806$	−335.693	−0.0146703
$807$	14567.7	0.635450
$808$	27399.1	1.19294
$809$	5494.43	0.238781	0.119391	−	0.992847i	$-0.461906\pi$
$809$	5494.43	0.238781	0.119391	+	0.992847i	$0.461906\pi$
$810$	724.827	0.0314418
$811$	−33839.3	−1.46518	−0.732590	−	0.680671i	$-0.761688\pi$
$811$	−33839.3	−1.46518	−0.732590	+	0.680671i	$0.761688\pi$
$812$	0	0
$813$	−25503.3	−1.10017
$814$	−17774.9	−0.765367
$815$	−4229.69	−0.181791
$816$	−59969.2	−2.57272
$817$	−7045.48	−0.301702
$818$	21191.1	0.905782
$819$	0	0
$820$	10646.6	0.453410
$821$	35849.2	1.52393	0.761965	−	0.647618i	$-0.224235\pi$
$821$	35849.2	1.52393	0.761965	+	0.647618i	$0.224235\pi$
$822$	14367.3	0.609633
$823$	−14873.5	−0.629961	−0.314981	−	0.949098i	$-0.601998\pi$
$823$	−14873.5	−0.629961	−0.314981	+	0.949098i	$0.601998\pi$
$824$	28728.3	1.21456
$825$	4033.12	0.170200
$826$	0	0
$827$	5757.86	0.242104	0.121052	−	0.992646i	$-0.461373\pi$
$827$	5757.86	0.242104	0.121052	+	0.992646i	$0.461373\pi$
$828$	13564.6	0.569326
$829$	38718.4	1.62213	0.811064	−	0.584957i	$-0.198889\pi$
$829$	38718.4	1.62213	0.811064	+	0.584957i	$0.198889\pi$
$830$	1990.25	0.0832319
$831$	20490.5	0.855366
$832$	−1230.30	−0.0512657
$833$	0	0
$834$	−8536.57	−0.354433
$835$	−4045.47	−0.167664
$836$	−7505.87	−0.310521
$837$	−1695.87	−0.0700334
$838$	62227.8	2.56518
$839$	36011.2	1.48182	0.740908	−	0.671606i	$-0.234395\pi$
$839$	36011.2	1.48182	0.740908	+	0.671606i	$0.234395\pi$
$840$	0	0
$841$	−22363.8	−0.916961
$842$	−56303.4	−2.30445
$843$	−14963.7	−0.611359
$844$	80449.3	3.28102
$845$	−3664.32	−0.149179
$846$	29099.0	1.18256
$847$	0	0
$848$	118935.	4.81631
$849$	−5858.63	−0.236829
$850$	−65221.5	−2.63186
$851$	−21876.2	−0.881205
$852$	9550.49	0.384031
$853$	12086.6	0.485157	0.242578	−	0.970132i	$-0.422007\pi$
$853$	12086.6	0.485157	0.242578	+	0.970132i	$0.422007\pi$
$854$	0	0
$855$	−493.622	−0.0197445
$856$	145883.	5.82499
$857$	8653.97	0.344941	0.172470	−	0.985015i	$-0.444825\pi$
$857$	8653.97	0.344941	0.172470	+	0.985015i	$0.444825\pi$
$858$	176.371	0.00701772
$859$	46141.6	1.83275	0.916374	−	0.400324i	$-0.131102\pi$
$859$	46141.6	1.83275	0.916374	+	0.400324i	$0.131102\pi$
$860$	7424.94	0.294405
$861$	0	0
$862$	−59576.1	−2.35403
$863$	16324.1	0.643893	0.321946	−	0.946758i	$-0.395663\pi$
$863$	16324.1	0.643893	0.321946	+	0.946758i	$0.395663\pi$
$864$	−14306.2	−0.563319
$865$	−3300.19	−0.129722
$866$	−67226.2	−2.63792
$867$	−14968.6	−0.586346
$868$	0	0
$869$	7446.55	0.290687
$870$	−1208.12	−0.0470793
$871$	553.258	0.0215229
$872$	−958.356	−0.0372179
$873$	−9095.42	−0.352615
$874$	−12797.7	−0.495295
$875$	0	0
$876$	−9185.39	−0.354276
$877$	17297.6	0.666017	0.333008	−	0.942924i	$-0.391936\pi$
$877$	17297.6	0.666017	0.333008	+	0.942924i	$0.391936\pi$
$878$	−11947.2	−0.459225
$879$	−9662.15	−0.370758
$880$	3687.11	0.141241
$881$	−45762.2	−1.75002	−0.875009	−	0.484106i	$-0.839145\pi$
$881$	−45762.2	−1.75002	−0.875009	+	0.484106i	$0.839145\pi$
$882$	0	0
$883$	−1535.96	−0.0585380	−0.0292690	−	0.999572i	$-0.509318\pi$
$883$	−1535.96	−0.0585380	−0.0292690	+	0.999572i	$0.509318\pi$
$884$	−2058.79	−0.0783309
$885$	3479.24	0.132151
$886$	−4468.55	−0.169440
$887$	47975.2	1.81607	0.908033	−	0.418899i	$-0.137584\pi$
$887$	47975.2	1.81607	0.908033	+	0.418899i	$0.137584\pi$
$888$	61853.8	2.33747
$889$	0	0
$890$	10168.6	0.382979
$891$	891.000	0.0335013
$892$	−9337.57	−0.350499
$893$	−19817.0	−0.742610
$894$	−47605.0	−1.78093
$895$	2194.07	0.0819437
$896$	0	0
$897$	217.066	0.00807985
$898$	−26298.5	−0.977274
$899$	2826.62	0.104865
$900$	−22834.2	−0.845710
$901$	58918.0	2.17851
$902$	18130.9	0.669284
$903$	0	0
$904$	90526.0	3.33059
$905$	−3535.72	−0.129869
$906$	−26839.9	−0.984212
$907$	16461.0	0.602622	0.301311	−	0.953526i	$-0.402576\pi$
$907$	16461.0	0.602622	0.301311	+	0.953526i	$0.402576\pi$
$908$	−121927.	−4.45626
$909$	3603.76	0.131495
$910$	0	0
$911$	−4712.18	−0.171374	−0.0856869	−	0.996322i	$-0.527308\pi$
$911$	−4712.18	−0.171374	−0.0856869	+	0.996322i	$0.527308\pi$
$912$	19808.3	0.719210
$913$	2446.53	0.0886837
$914$	64893.4	2.34845
$915$	2216.68	0.0800887
$916$	−87349.1	−3.15076
$917$	0	0
$918$	−14408.8	−0.518040
$919$	16055.3	0.576294	0.288147	−	0.957586i	$-0.406961\pi$
$919$	16055.3	0.576294	0.288147	+	0.957586i	$0.406961\pi$
$920$	8289.52	0.297062
$921$	−15975.2	−0.571555
$922$	6816.97	0.243498
$923$	152.831	0.00545016
$924$	0	0
$925$	36825.6	1.30899
$926$	−85913.4	−3.04891
$927$	3778.60	0.133879
$928$	23845.1	0.843485
$929$	−9378.49	−0.331215	−0.165607	−	0.986192i	$-0.552958\pi$
$929$	−9378.49	−0.331215	−0.165607	+	0.986192i	$0.552958\pi$
$930$	−1686.17	−0.0594533
$931$	0	0
$932$	112222.	3.94414
$933$	−28876.1	−1.01325
$934$	57360.6	2.00953
$935$	1826.52	0.0638864
$936$	−613.743	−0.0214325
$937$	−9100.65	−0.317295	−0.158647	−	0.987335i	$-0.550713\pi$
$937$	−9100.65	−0.317295	−0.158647	+	0.987335i	$0.550713\pi$
$938$	0	0
$939$	−12946.7	−0.449945
$940$	20884.3	0.724650
$941$	−9738.84	−0.337383	−0.168691	−	0.985669i	$-0.553954\pi$
$941$	−9738.84	−0.337383	−0.168691	+	0.985669i	$0.553954\pi$
$942$	−19348.4	−0.669219
$943$	22314.4	0.770580
$944$	−139617.	−4.81371
$945$	0	0
$946$	12644.5	0.434574
$947$	47225.8	1.62052	0.810260	−	0.586071i	$-0.199326\pi$
$947$	47225.8	1.62052	0.810260	+	0.586071i	$0.199326\pi$
$948$	−42159.9	−1.44440
$949$	−146.989	−0.00502787
$950$	21543.2	0.735740
$951$	22199.5	0.756960
$952$	0	0
$953$	972.630	0.0330604	0.0165302	−	0.999863i	$-0.494738\pi$
$953$	972.630	0.0330604	0.0165302	+	0.999863i	$0.494738\pi$
$954$	28576.4	0.969805
$955$	3565.20	0.120803
$956$	−105980.	−3.58541
$957$	−1485.09	−0.0501631
$958$	64649.3	2.18030
$959$	0	0
$960$	−6179.74	−0.207761
$961$	−25845.9	−0.867574
$962$	1610.41	0.0539726
$963$	19187.9	0.642078
$964$	−5275.77	−0.176267
$965$	−1519.87	−0.0507010
$966$	0	0
$967$	−21987.5	−0.731199	−0.365599	−	0.930772i	$-0.619136\pi$
$967$	−21987.5	−0.731199	−0.365599	+	0.930772i	$0.619136\pi$
$968$	8279.56	0.274912
$969$	9812.68	0.325313
$970$	−9043.36	−0.299345
$971$	18194.5	0.601326	0.300663	−	0.953730i	$-0.402792\pi$
$971$	18194.5	0.601326	0.300663	+	0.953730i	$0.402792\pi$
$972$	−5044.54	−0.166465
$973$	0	0
$974$	−76544.0	−2.51810
$975$	−365.402	−0.0120023
$976$	−88952.0	−2.91730
$977$	−14356.0	−0.470103	−0.235051	−	0.971983i	$-0.575526\pi$
$977$	−14356.0	−0.470103	−0.235051	+	0.971983i	$0.575526\pi$
$978$	40781.3	1.33338
$979$	12499.8	0.408065
$980$	0	0
$981$	−126.051	−0.00410246
$982$	−83749.1	−2.72153
$983$	−25765.5	−0.836005	−0.418002	−	0.908446i	$-0.637270\pi$
$983$	−25765.5	−0.836005	−0.418002	+	0.908446i	$0.637270\pi$
$984$	−63092.9	−2.04403
$985$	−4147.98	−0.134178
$986$	24016.1	0.775687
$987$	0	0
$988$	680.035	0.0218976
$989$	15562.0	0.500347
$990$	885.900	0.0284402
$991$	7750.45	0.248437	0.124219	−	0.992255i	$-0.460358\pi$
$991$	7750.45	0.248437	0.124219	+	0.992255i	$0.460358\pi$
$992$	33280.6	1.06518
$993$	14697.9	0.469711
$994$	0	0
$995$	640.995	0.0204230
$996$	−13851.4	−0.440662
$997$	−12105.1	−0.384528	−0.192264	−	0.981343i	$-0.561583\pi$
$997$	−12105.1	−0.384528	−0.192264	+	0.981343i	$0.561583\pi$
$998$	31211.5	0.989964
$999$	8135.55	0.257655

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1617.4.a.p.1.5		5
7.6	odd	2		231.4.a.l.1.5	✓	5
21.20	even	2		693.4.a.n.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.l.1.5	✓	5	7.6	odd	2
693.4.a.n.1.1		5	21.20	even	2
1617.4.a.p.1.5		5	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:	\( N \)	\(=\)	\( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1617.a (trivial)

\(f(q)\)	\(=\)	\(q+5.36278 q^{2} -3.00000 q^{3} +20.7594 q^{4} +1.66863 q^{5} -16.0883 q^{6} +68.4261 q^{8} +9.00000 q^{9} +8.94849 q^{10} +11.0000 q^{11} -62.2783 q^{12} -0.996604 q^{13} -5.00588 q^{15} +200.879 q^{16} +99.5115 q^{17} +48.2650 q^{18} -32.8695 q^{19} +34.6398 q^{20} +58.9906 q^{22} +72.6019 q^{23} -205.278 q^{24} -122.216 q^{25} -5.34457 q^{26} -27.0000 q^{27} +45.0027 q^{29} -26.8455 q^{30} +62.8102 q^{31} +529.860 q^{32} -33.0000 q^{33} +533.659 q^{34} +186.835 q^{36} -301.317 q^{37} -176.272 q^{38} +2.98981 q^{39} +114.178 q^{40} +307.353 q^{41} +214.347 q^{43} +228.354 q^{44} +15.0176 q^{45} +389.348 q^{46} +602.899 q^{47} -602.636 q^{48} -655.416 q^{50} -298.535 q^{51} -20.6889 q^{52} +592.072 q^{53} -144.795 q^{54} +18.3549 q^{55} +98.6084 q^{57} +241.339 q^{58} -695.030 q^{59} -103.919 q^{60} -442.815 q^{61} +336.837 q^{62} +1234.49 q^{64} -1.66296 q^{65} -176.972 q^{66} -555.143 q^{67} +2065.80 q^{68} -217.806 q^{69} -153.352 q^{71} +615.835 q^{72} +147.489 q^{73} -1615.90 q^{74} +366.647 q^{75} -682.352 q^{76} +16.0337 q^{78} +676.959 q^{79} +335.192 q^{80} +81.0000 q^{81} +1648.27 q^{82} +222.412 q^{83} +166.048 q^{85} +1149.50 q^{86} -135.008 q^{87} +752.687 q^{88} +1136.35 q^{89} +80.5364 q^{90} +1507.18 q^{92} -188.430 q^{93} +3233.22 q^{94} -54.8469 q^{95} -1589.58 q^{96} -1010.60 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} - 15 q^{3} + 21 q^{4} - 7 q^{5} - 15 q^{6} + 60 q^{8} + 45 q^{9} - 55 q^{10} + 55 q^{11} - 63 q^{12} - 111 q^{13} + 21 q^{15} + 201 q^{16} - 136 q^{17} + 45 q^{18} - 111 q^{19} - 219 q^{20}+ \cdots + 495 q^{99}+O(q^{100}) \) 5 * q + 5 * q^2 - 15 * q^3 + 21 * q^4 - 7 * q^5 - 15 * q^6 + 60 * q^8 + 45 * q^9 - 55 * q^10 + 55 * q^11 - 63 * q^12 - 111 * q^13 + 21 * q^15 + 201 * q^16 - 136 * q^17 + 45 * q^18 - 111 * q^19 - 219 * q^20 + 55 * q^22 - 28 * q^23 - 180 * q^24 + 190 * q^25 + q^26 - 135 * q^27 + 61 * q^29 + 165 * q^30 + 280 * q^31 + 535 * q^32 - 165 * q^33 + 572 * q^34 + 189 * q^36 - 41 * q^37 - 267 * q^38 + 333 * q^39 + 336 * q^40 - 426 * q^41 + 424 * q^43 + 231 * q^44 - 63 * q^45 + 140 * q^46 - 75 * q^47 - 603 * q^48 + 490 * q^50 + 408 * q^51 + 269 * q^52 + 1500 * q^53 - 135 * q^54 - 77 * q^55 + 333 * q^57 - 1767 * q^58 - 757 * q^59 + 657 * q^60 - 658 * q^61 + 568 * q^62 - 748 * q^64 + 537 * q^65 - 165 * q^66 - 583 * q^67 + 1650 * q^68 + 84 * q^69 - 764 * q^71 + 540 * q^72 - 875 * q^73 - 825 * q^74 - 570 * q^75 - 213 * q^76 - 3 * q^78 - 244 * q^79 + 2577 * q^80 + 405 * q^81 + 2006 * q^82 - 924 * q^83 - 1402 * q^85 + 1272 * q^86 - 183 * q^87 + 660 * q^88 + 1110 * q^89 - 495 * q^90 - 2046 * q^92 - 840 * q^93 + 3349 * q^94 + 1923 * q^95 - 1605 * q^96 + 852 * q^97 + 495 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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