LMFDB - Embedded newform 448.4.a.w.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,4,Mod(1,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, 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("448.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	448.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:	$26.4328556826$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 224)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.66908$ of defining polynomial
Character	$\chi$	$=$	448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.24797 q^{3} +18.9243 q^{5} +7.00000 q^{7} +12.0371 q^{9} +O(q^{10})$	$q+6.24797 q^{3} +18.9243 q^{5} +7.00000 q^{7} +12.0371 q^{9} +49.7423 q^{11} +38.6987 q^{13} +118.238 q^{15} -83.9969 q^{17} -136.741 q^{19} +43.7358 q^{21} +149.729 q^{23} +233.128 q^{25} -93.4878 q^{27} -63.8486 q^{29} +6.03564 q^{31} +310.789 q^{33} +132.470 q^{35} -37.1749 q^{37} +241.788 q^{39} -361.465 q^{41} +99.2433 q^{43} +227.793 q^{45} -343.368 q^{47} +49.0000 q^{49} -524.810 q^{51} -625.497 q^{53} +941.338 q^{55} -854.352 q^{57} +363.618 q^{59} -120.927 q^{61} +84.2596 q^{63} +732.345 q^{65} +173.794 q^{67} +935.503 q^{69} +468.385 q^{71} -710.075 q^{73} +1456.58 q^{75} +348.196 q^{77} +1079.70 q^{79} -909.110 q^{81} +1375.18 q^{83} -1589.58 q^{85} -398.924 q^{87} +462.708 q^{89} +270.891 q^{91} +37.7105 q^{93} -2587.72 q^{95} +456.619 q^{97} +598.753 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{5} + 21 q^{7} + 15 q^{9}+O(q^{10}) $ 3 * q + 6 * q^5 + 21 * q^7 + 15 * q^9	$ 3 q + 6 q^{5} + 21 q^{7} + 15 q^{9} + 6 q^{13} + 168 q^{15} - 66 q^{17} - 168 q^{19} + 336 q^{23} + 69 q^{25} - 168 q^{27} - 90 q^{29} + 504 q^{31} + 120 q^{33} + 42 q^{35} - 18 q^{37} + 840 q^{39} - 450 q^{41} + 150 q^{45} + 504 q^{47} + 147 q^{49} + 336 q^{51} + 78 q^{53} + 1176 q^{55} + 48 q^{57} + 504 q^{59} - 498 q^{61} + 105 q^{63} + 1068 q^{65} + 1008 q^{67} + 1224 q^{69} + 504 q^{71} - 234 q^{73} + 1848 q^{75} + 168 q^{79} - 981 q^{81} + 3024 q^{83} - 1476 q^{85} - 336 q^{87} + 246 q^{89} + 42 q^{91} - 1200 q^{93} - 2184 q^{95} - 2514 q^{97} + 2688 q^{99}+O(q^{100}) $ 3 * q + 6 * q^5 + 21 * q^7 + 15 * q^9 + 6 * q^13 + 168 * q^15 - 66 * q^17 - 168 * q^19 + 336 * q^23 + 69 * q^25 - 168 * q^27 - 90 * q^29 + 504 * q^31 + 120 * q^33 + 42 * q^35 - 18 * q^37 + 840 * q^39 - 450 * q^41 + 150 * q^45 + 504 * q^47 + 147 * q^49 + 336 * q^51 + 78 * q^53 + 1176 * q^55 + 48 * q^57 + 504 * q^59 - 498 * q^61 + 105 * q^63 + 1068 * q^65 + 1008 * q^67 + 1224 * q^69 + 504 * q^71 - 234 * q^73 + 1848 * q^75 + 168 * q^79 - 981 * q^81 + 3024 * q^83 - 1476 * q^85 - 336 * q^87 + 246 * q^89 + 42 * q^91 - 1200 * q^93 - 2184 * q^95 - 2514 * q^97 + 2688 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	6.24797	1.20242	0.601211	−	0.799090i	$-0.294685\pi$
$3$	6.24797	1.20242	0.601211	+	0.799090i	$0.294685\pi$
$4$	0	0
$5$	18.9243	1.69264	0.846320	−	0.532675i	$-0.178813\pi$
$5$	18.9243	1.69264	0.846320	+	0.532675i	$0.178813\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	12.0371	0.445818
$10$	0	0
$11$	49.7423	1.36344	0.681722	−	0.731611i	$-0.261231\pi$
$11$	49.7423	1.36344	0.681722	+	0.731611i	$0.261231\pi$
$12$	0	0
$13$	38.6987	0.825622	0.412811	−	0.910817i	$-0.364547\pi$
$13$	38.6987	0.825622	0.412811	+	0.910817i	$0.364547\pi$
$14$	0	0
$15$	118.238	2.03527
$16$	0	0
$17$	−83.9969	−1.19837	−0.599184	−	0.800612i	$-0.704508\pi$
$17$	−83.9969	−1.19837	−0.599184	+	0.800612i	$0.704508\pi$
$18$	0	0
$19$	−136.741	−1.65108	−0.825539	−	0.564345i	$-0.809129\pi$
$19$	−136.741	−1.65108	−0.825539	+	0.564345i	$0.809129\pi$
$20$	0	0
$21$	43.7358	0.454473
$22$	0	0
$23$	149.729	1.35742	0.678711	−	0.734406i	$-0.262539\pi$
$23$	149.729	1.35742	0.678711	+	0.734406i	$0.262539\pi$
$24$	0	0
$25$	233.128	1.86503
$26$	0	0
$27$	−93.4878	−0.666361
$28$	0	0
$29$	−63.8486	−0.408841	−0.204420	−	0.978883i	$-0.565531\pi$
$29$	−63.8486	−0.408841	−0.204420	+	0.978883i	$0.565531\pi$
$30$	0	0
$31$	6.03564	0.0349688	0.0174844	−	0.999847i	$-0.494434\pi$
$31$	6.03564	0.0349688	0.0174844	+	0.999847i	$0.494434\pi$
$32$	0	0
$33$	310.789	1.63943
$34$	0	0
$35$	132.470	0.639758
$36$	0	0
$37$	−37.1749	−0.165176	−0.0825880	−	0.996584i	$-0.526319\pi$
$37$	−37.1749	−0.165176	−0.0825880	+	0.996584i	$0.526319\pi$
$38$	0	0
$39$	241.788	0.992746
$40$	0	0
$41$	−361.465	−1.37686	−0.688432	−	0.725301i	$-0.741701\pi$
$41$	−361.465	−1.37686	−0.688432	+	0.725301i	$0.741701\pi$
$42$	0	0
$43$	99.2433	0.351964	0.175982	−	0.984393i	$-0.443690\pi$
$43$	99.2433	0.351964	0.175982	+	0.984393i	$0.443690\pi$
$44$	0	0
$45$	227.793	0.754609
$46$	0	0
$47$	−343.368	−1.06565	−0.532823	−	0.846227i	$-0.678869\pi$
$47$	−343.368	−1.06565	−0.532823	+	0.846227i	$0.678869\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−524.810	−1.44094
$52$	0	0
$53$	−625.497	−1.62111	−0.810553	−	0.585666i	$-0.800833\pi$
$53$	−625.497	−1.62111	−0.810553	+	0.585666i	$0.800833\pi$
$54$	0	0
$55$	941.338	2.30782
$56$	0	0
$57$	−854.352	−1.98529
$58$	0	0
$59$	363.618	0.802356	0.401178	−	0.916000i	$-0.368601\pi$
$59$	363.618	0.802356	0.401178	+	0.916000i	$0.368601\pi$
$60$	0	0
$61$	−120.927	−0.253822	−0.126911	−	0.991914i	$-0.540506\pi$
$61$	−120.927	−0.253822	−0.126911	+	0.991914i	$0.540506\pi$
$62$	0	0
$63$	84.2596	0.168503
$64$	0	0
$65$	732.345	1.39748
$66$	0	0
$67$	173.794	0.316901	0.158450	−	0.987367i	$-0.449350\pi$
$67$	173.794	0.316901	0.158450	+	0.987367i	$0.449350\pi$
$68$	0	0
$69$	935.503	1.63219
$70$	0	0
$71$	468.385	0.782917	0.391458	−	0.920196i	$-0.371971\pi$
$71$	468.385	0.782917	0.391458	+	0.920196i	$0.371971\pi$
$72$	0	0
$73$	−710.075	−1.13847	−0.569233	−	0.822176i	$-0.692760\pi$
$73$	−710.075	−1.13847	−0.569233	+	0.822176i	$0.692760\pi$
$74$	0	0
$75$	1456.58	2.24255
$76$	0	0
$77$	348.196	0.515333
$78$	0	0
$79$	1079.70	1.53767	0.768834	−	0.639448i	$-0.220837\pi$
$79$	1079.70	1.53767	0.768834	+	0.639448i	$0.220837\pi$
$80$	0	0
$81$	−909.110	−1.24706
$82$	0	0
$83$	1375.18	1.81862	0.909312	−	0.416115i	$-0.136609\pi$
$83$	1375.18	1.81862	0.909312	+	0.416115i	$0.136609\pi$
$84$	0	0
$85$	−1589.58	−2.02840
$86$	0	0
$87$	−398.924	−0.491599
$88$	0	0
$89$	462.708	0.551090	0.275545	−	0.961288i	$-0.411142\pi$
$89$	462.708	0.551090	0.275545	+	0.961288i	$0.411142\pi$
$90$	0	0
$91$	270.891	0.312056
$92$	0	0
$93$	37.7105	0.0420472
$94$	0	0
$95$	−2587.72	−2.79468
$96$	0	0
$97$	456.619	0.477966	0.238983	−	0.971024i	$-0.423186\pi$
$97$	456.619	0.477966	0.238983	+	0.971024i	$0.423186\pi$
$98$	0	0
$99$	598.753	0.607848
$100$	0	0
$101$	−1370.39	−1.35008	−0.675042	−	0.737780i	$-0.735874\pi$
$101$	−1370.39	−1.35008	−0.675042	+	0.737780i	$0.735874\pi$
$102$	0	0
$103$	543.120	0.519566	0.259783	−	0.965667i	$-0.416349\pi$
$103$	543.120	0.519566	0.259783	+	0.965667i	$0.416349\pi$
$104$	0	0
$105$	827.668	0.769258
$106$	0	0
$107$	−527.870	−0.476927	−0.238463	−	0.971152i	$-0.576644\pi$
$107$	−527.870	−0.476927	−0.238463	+	0.971152i	$0.576644\pi$
$108$	0	0
$109$	1460.10	1.28305	0.641524	−	0.767103i	$-0.278303\pi$
$109$	1460.10	1.28305	0.641524	+	0.767103i	$0.278303\pi$
$110$	0	0
$111$	−232.267	−0.198611
$112$	0	0
$113$	792.377	0.659651	0.329825	−	0.944042i	$-0.393010\pi$
$113$	792.377	0.659651	0.329825	+	0.944042i	$0.393010\pi$
$114$	0	0
$115$	2833.52	2.29762
$116$	0	0
$117$	465.819	0.368077
$118$	0	0
$119$	−587.978	−0.452940
$120$	0	0
$121$	1143.30	0.858979
$122$	0	0
$123$	−2258.42	−1.65557
$124$	0	0
$125$	2046.25	1.46418
$126$	0	0
$127$	488.510	0.341325	0.170662	−	0.985330i	$-0.445409\pi$
$127$	488.510	0.341325	0.170662	+	0.985330i	$0.445409\pi$
$128$	0	0
$129$	620.069	0.423209
$130$	0	0
$131$	−1812.31	−1.20872	−0.604359	−	0.796712i	$-0.706571\pi$
$131$	−1812.31	−1.20872	−0.604359	+	0.796712i	$0.706571\pi$
$132$	0	0
$133$	−957.186	−0.624049
$134$	0	0
$135$	−1769.19	−1.12791
$136$	0	0
$137$	−949.427	−0.592081	−0.296040	−	0.955175i	$-0.595666\pi$
$137$	−949.427	−0.592081	−0.296040	+	0.955175i	$0.595666\pi$
$138$	0	0
$139$	−2560.24	−1.56228	−0.781139	−	0.624357i	$-0.785361\pi$
$139$	−2560.24	−1.56228	−0.781139	+	0.624357i	$0.785361\pi$
$140$	0	0
$141$	−2145.35	−1.28136
$142$	0	0
$143$	1924.96	1.12569
$144$	0	0
$145$	−1208.29	−0.692020
$146$	0	0
$147$	306.150	0.171775
$148$	0	0
$149$	1562.56	0.859129	0.429565	−	0.903036i	$-0.358667\pi$
$149$	1562.56	0.859129	0.429565	+	0.903036i	$0.358667\pi$
$150$	0	0
$151$	−939.174	−0.506151	−0.253076	−	0.967446i	$-0.581442\pi$
$151$	−939.174	−0.506151	−0.253076	+	0.967446i	$0.581442\pi$
$152$	0	0
$153$	−1011.08	−0.534254
$154$	0	0
$155$	114.220	0.0591896
$156$	0	0
$157$	1614.93	0.820926	0.410463	−	0.911877i	$-0.365367\pi$
$157$	1614.93	0.820926	0.410463	+	0.911877i	$0.365367\pi$
$158$	0	0
$159$	−3908.08	−1.94925
$160$	0	0
$161$	1048.10	0.513057
$162$	0	0
$163$	1336.65	0.642297	0.321149	−	0.947029i	$-0.395931\pi$
$163$	1336.65	0.642297	0.321149	+	0.947029i	$0.395931\pi$
$164$	0	0
$165$	5881.45	2.77497
$166$	0	0
$167$	1657.29	0.767932	0.383966	−	0.923347i	$-0.374558\pi$
$167$	1657.29	0.767932	0.383966	+	0.923347i	$0.374558\pi$
$168$	0	0
$169$	−699.412	−0.318349
$170$	0	0
$171$	−1645.96	−0.736080
$172$	0	0
$173$	−2242.61	−0.985565	−0.492782	−	0.870153i	$-0.664020\pi$
$173$	−2242.61	−0.985565	−0.492782	+	0.870153i	$0.664020\pi$
$174$	0	0
$175$	1631.90	0.704914
$176$	0	0
$177$	2271.87	0.964771
$178$	0	0
$179$	−1242.65	−0.518882	−0.259441	−	0.965759i	$-0.583538\pi$
$179$	−1242.65	−0.518882	−0.259441	+	0.965759i	$0.583538\pi$
$180$	0	0
$181$	−1072.54	−0.440449	−0.220225	−	0.975449i	$-0.570679\pi$
$181$	−1072.54	−0.440449	−0.220225	+	0.975449i	$0.570679\pi$
$182$	0	0
$183$	−755.550	−0.305202
$184$	0	0
$185$	−703.508	−0.279583
$186$	0	0
$187$	−4178.20	−1.63391
$188$	0	0
$189$	−654.415	−0.251861
$190$	0	0
$191$	−29.1524	−0.0110439	−0.00552197	−	0.999985i	$-0.501758\pi$
$191$	−29.1524	−0.0110439	−0.00552197	+	0.999985i	$0.501758\pi$
$192$	0	0
$193$	−2202.82	−0.821567	−0.410783	−	0.911733i	$-0.634745\pi$
$193$	−2202.82	−0.821567	−0.410783	+	0.911733i	$0.634745\pi$
$194$	0	0
$195$	4575.67	1.68036
$196$	0	0
$197$	−4759.74	−1.72141	−0.860705	−	0.509105i	$-0.829977\pi$
$197$	−4759.74	−1.72141	−0.860705	+	0.509105i	$0.829977\pi$
$198$	0	0
$199$	4832.93	1.72160	0.860798	−	0.508947i	$-0.169965\pi$
$199$	4832.93	1.72160	0.860798	+	0.508947i	$0.169965\pi$
$200$	0	0
$201$	1085.86	0.381048
$202$	0	0
$203$	−446.940	−0.154527
$204$	0	0
$205$	−6840.47	−2.33053
$206$	0	0
$207$	1802.30	0.605163
$208$	0	0
$209$	−6801.81	−2.25115
$210$	0	0
$211$	−2520.60	−0.822395	−0.411197	−	0.911546i	$-0.634889\pi$
$211$	−2520.60	−0.822395	−0.411197	+	0.911546i	$0.634889\pi$
$212$	0	0
$213$	2926.45	0.941396
$214$	0	0
$215$	1878.11	0.595749
$216$	0	0
$217$	42.2495	0.0132170
$218$	0	0
$219$	−4436.53	−1.36892
$220$	0	0
$221$	−3250.57	−0.989398
$222$	0	0
$223$	−2524.34	−0.758036	−0.379018	−	0.925389i	$-0.623738\pi$
$223$	−2524.34	−0.758036	−0.379018	+	0.925389i	$0.623738\pi$
$224$	0	0
$225$	2806.19	0.831463
$226$	0	0
$227$	−4337.60	−1.26827	−0.634133	−	0.773224i	$-0.718643\pi$
$227$	−4337.60	−1.26827	−0.634133	+	0.773224i	$0.718643\pi$
$228$	0	0
$229$	−5538.22	−1.59815	−0.799074	−	0.601233i	$-0.794676\pi$
$229$	−5538.22	−1.59815	−0.799074	+	0.601233i	$0.794676\pi$
$230$	0	0
$231$	2175.52	0.619648
$232$	0	0
$233$	1457.62	0.409836	0.204918	−	0.978779i	$-0.434307\pi$
$233$	1457.62	0.409836	0.204918	+	0.978779i	$0.434307\pi$
$234$	0	0
$235$	−6497.99	−1.80375
$236$	0	0
$237$	6745.94	1.84893
$238$	0	0
$239$	−2795.76	−0.756664	−0.378332	−	0.925670i	$-0.623502\pi$
$239$	−2795.76	−0.756664	−0.378332	+	0.925670i	$0.623502\pi$
$240$	0	0
$241$	3601.50	0.962627	0.481314	−	0.876548i	$-0.340160\pi$
$241$	3601.50	0.962627	0.481314	+	0.876548i	$0.340160\pi$
$242$	0	0
$243$	−3155.92	−0.833137
$244$	0	0
$245$	927.290	0.241806
$246$	0	0
$247$	−5291.69	−1.36317
$248$	0	0
$249$	8592.09	2.18675
$250$	0	0
$251$	3040.21	0.764528	0.382264	−	0.924053i	$-0.375145\pi$
$251$	3040.21	0.764528	0.382264	+	0.924053i	$0.375145\pi$
$252$	0	0
$253$	7447.88	1.85077
$254$	0	0
$255$	−9931.65	−2.43900
$256$	0	0
$257$	−7273.23	−1.76534	−0.882668	−	0.469996i	$-0.844255\pi$
$257$	−7273.23	−1.76534	−0.882668	+	0.469996i	$0.844255\pi$
$258$	0	0
$259$	−260.224	−0.0624306
$260$	0	0
$261$	−768.551	−0.182269
$262$	0	0
$263$	−2853.28	−0.668975	−0.334488	−	0.942400i	$-0.608563\pi$
$263$	−2853.28	−0.668975	−0.334488	+	0.942400i	$0.608563\pi$
$264$	0	0
$265$	−11837.1	−2.74395
$266$	0	0
$267$	2890.98	0.662642
$268$	0	0
$269$	4249.78	0.963247	0.481623	−	0.876378i	$-0.340047\pi$
$269$	4249.78	0.963247	0.481623	+	0.876378i	$0.340047\pi$
$270$	0	0
$271$	1041.41	0.233437	0.116718	−	0.993165i	$-0.462763\pi$
$271$	1041.41	0.233437	0.116718	+	0.993165i	$0.462763\pi$
$272$	0	0
$273$	1692.52	0.375223
$274$	0	0
$275$	11596.4	2.54286
$276$	0	0
$277$	5252.63	1.13935	0.569675	−	0.821870i	$-0.307069\pi$
$277$	5252.63	1.13935	0.569675	+	0.821870i	$0.307069\pi$
$278$	0	0
$279$	72.6515	0.0155897
$280$	0	0
$281$	8888.67	1.88702	0.943512	−	0.331338i	$-0.107500\pi$
$281$	8888.67	1.88702	0.943512	+	0.331338i	$0.107500\pi$
$282$	0	0
$283$	−4040.87	−0.848781	−0.424390	−	0.905479i	$-0.639512\pi$
$283$	−4040.87	−0.848781	−0.424390	+	0.905479i	$0.639512\pi$
$284$	0	0
$285$	−16168.0	−3.36038
$286$	0	0
$287$	−2530.26	−0.520405
$288$	0	0
$289$	2142.48	0.436084
$290$	0	0
$291$	2852.94	0.574716
$292$	0	0
$293$	3004.41	0.599043	0.299521	−	0.954090i	$-0.403173\pi$
$293$	3004.41	0.599043	0.299521	+	0.954090i	$0.403173\pi$
$294$	0	0
$295$	6881.21	1.35810
$296$	0	0
$297$	−4650.30	−0.908545
$298$	0	0
$299$	5794.32	1.12072
$300$	0	0
$301$	694.703	0.133030
$302$	0	0
$303$	−8562.12	−1.62337
$304$	0	0
$305$	−2288.46	−0.429630
$306$	0	0
$307$	−7361.55	−1.36855	−0.684276	−	0.729223i	$-0.739882\pi$
$307$	−7361.55	−1.36855	−0.684276	+	0.729223i	$0.739882\pi$
$308$	0	0
$309$	3393.40	0.624737
$310$	0	0
$311$	5903.89	1.07646	0.538229	−	0.842798i	$-0.319093\pi$
$311$	5903.89	1.07646	0.538229	+	0.842798i	$0.319093\pi$
$312$	0	0
$313$	−364.298	−0.0657870	−0.0328935	−	0.999459i	$-0.510472\pi$
$313$	−364.298	−0.0657870	−0.0328935	+	0.999459i	$0.510472\pi$
$314$	0	0
$315$	1594.55	0.285215
$316$	0	0
$317$	3461.46	0.613296	0.306648	−	0.951823i	$-0.400793\pi$
$317$	3461.46	0.613296	0.306648	+	0.951823i	$0.400793\pi$
$318$	0	0
$319$	−3175.98	−0.557431
$320$	0	0
$321$	−3298.12	−0.573467
$322$	0	0
$323$	11485.8	1.97860
$324$	0	0
$325$	9021.76	1.53981
$326$	0	0
$327$	9122.65	1.54276
$328$	0	0
$329$	−2403.57	−0.402776
$330$	0	0
$331$	−5357.54	−0.889659	−0.444829	−	0.895615i	$-0.646736\pi$
$331$	−5357.54	−0.889659	−0.444829	+	0.895615i	$0.646736\pi$
$332$	0	0
$333$	−447.477	−0.0736384
$334$	0	0
$335$	3288.93	0.536399
$336$	0	0
$337$	−1379.32	−0.222957	−0.111478	−	0.993767i	$-0.535559\pi$
$337$	−1379.32	−0.222957	−0.111478	+	0.993767i	$0.535559\pi$
$338$	0	0
$339$	4950.74	0.793178
$340$	0	0
$341$	300.227	0.0476780
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	17703.7	2.76271
$346$	0	0
$347$	12271.4	1.89845	0.949226	−	0.314596i	$-0.101869\pi$
$347$	12271.4	1.89845	0.949226	+	0.314596i	$0.101869\pi$
$348$	0	0
$349$	−1538.02	−0.235898	−0.117949	−	0.993020i	$-0.537632\pi$
$349$	−1538.02	−0.235898	−0.117949	+	0.993020i	$0.537632\pi$
$350$	0	0
$351$	−3617.85	−0.550162
$352$	0	0
$353$	6690.58	1.00879	0.504396	−	0.863472i	$-0.331715\pi$
$353$	6690.58	1.00879	0.504396	+	0.863472i	$0.331715\pi$
$354$	0	0
$355$	8863.85	1.32520
$356$	0	0
$357$	−3673.67	−0.544625
$358$	0	0
$359$	8231.42	1.21013	0.605066	−	0.796175i	$-0.293147\pi$
$359$	8231.42	1.21013	0.605066	+	0.796175i	$0.293147\pi$
$360$	0	0
$361$	11839.0	1.72606
$362$	0	0
$363$	7143.31	1.03286
$364$	0	0
$365$	−13437.7	−1.92701
$366$	0	0
$367$	1986.86	0.282598	0.141299	−	0.989967i	$-0.454872\pi$
$367$	1986.86	0.282598	0.141299	+	0.989967i	$0.454872\pi$
$368$	0	0
$369$	−4350.99	−0.613830
$370$	0	0
$371$	−4378.48	−0.612720
$372$	0	0
$373$	−572.831	−0.0795177	−0.0397588	−	0.999209i	$-0.512659\pi$
$373$	−572.831	−0.0795177	−0.0397588	+	0.999209i	$0.512659\pi$
$374$	0	0
$375$	12784.9	1.76056
$376$	0	0
$377$	−2470.86	−0.337548
$378$	0	0
$379$	−12120.7	−1.64274	−0.821370	−	0.570395i	$-0.806790\pi$
$379$	−12120.7	−1.64274	−0.821370	+	0.570395i	$0.806790\pi$
$380$	0	0
$381$	3052.19	0.410416
$382$	0	0
$383$	−9576.11	−1.27759	−0.638794	−	0.769378i	$-0.720566\pi$
$383$	−9576.11	−1.27759	−0.638794	+	0.769378i	$0.720566\pi$
$384$	0	0
$385$	6589.37	0.872273
$386$	0	0
$387$	1194.60	0.156912
$388$	0	0
$389$	14432.7	1.88115	0.940573	−	0.339590i	$-0.110288\pi$
$389$	14432.7	1.88115	0.940573	+	0.339590i	$0.110288\pi$
$390$	0	0
$391$	−12576.8	−1.62669
$392$	0	0
$393$	−11323.2	−1.45339
$394$	0	0
$395$	20432.6	2.60272
$396$	0	0
$397$	−3975.33	−0.502559	−0.251279	−	0.967915i	$-0.580851\pi$
$397$	−3975.33	−0.502559	−0.251279	+	0.967915i	$0.580851\pi$
$398$	0	0
$399$	−5980.46	−0.750370
$400$	0	0
$401$	−4132.25	−0.514600	−0.257300	−	0.966332i	$-0.582833\pi$
$401$	−4132.25	−0.514600	−0.257300	+	0.966332i	$0.582833\pi$
$402$	0	0
$403$	233.571	0.0288710
$404$	0	0
$405$	−17204.3	−2.11083
$406$	0	0
$407$	−1849.16	−0.225208
$408$	0	0
$409$	1711.34	0.206895	0.103448	−	0.994635i	$-0.467013\pi$
$409$	1711.34	0.206895	0.103448	+	0.994635i	$0.467013\pi$
$410$	0	0
$411$	−5931.99	−0.711930
$412$	0	0
$413$	2545.33	0.303262
$414$	0	0
$415$	26024.3	3.07827
$416$	0	0
$417$	−15996.3	−1.87852
$418$	0	0
$419$	−180.837	−0.0210846	−0.0105423	−	0.999944i	$-0.503356\pi$
$419$	−180.837	−0.0210846	−0.0105423	+	0.999944i	$0.503356\pi$
$420$	0	0
$421$	−5955.14	−0.689396	−0.344698	−	0.938714i	$-0.612019\pi$
$421$	−5955.14	−0.689396	−0.344698	+	0.938714i	$0.612019\pi$
$422$	0	0
$423$	−4133.15	−0.475084
$424$	0	0
$425$	−19582.1	−2.23499
$426$	0	0
$427$	−846.492	−0.0959358
$428$	0	0
$429$	12027.1	1.35355
$430$	0	0
$431$	8079.15	0.902921	0.451461	−	0.892291i	$-0.350903\pi$
$431$	8079.15	0.902921	0.451461	+	0.892291i	$0.350903\pi$
$432$	0	0
$433$	4192.11	0.465266	0.232633	−	0.972565i	$-0.425266\pi$
$433$	4192.11	0.465266	0.232633	+	0.972565i	$0.425266\pi$
$434$	0	0
$435$	−7549.34	−0.832100
$436$	0	0
$437$	−20474.1	−2.24121
$438$	0	0
$439$	−4638.72	−0.504315	−0.252157	−	0.967686i	$-0.581140\pi$
$439$	−4638.72	−0.504315	−0.252157	+	0.967686i	$0.581140\pi$
$440$	0	0
$441$	589.817	0.0636883
$442$	0	0
$443$	−7905.06	−0.847812	−0.423906	−	0.905706i	$-0.639341\pi$
$443$	−7905.06	−0.847812	−0.423906	+	0.905706i	$0.639341\pi$
$444$	0	0
$445$	8756.42	0.932796
$446$	0	0
$447$	9762.85	1.03304
$448$	0	0
$449$	10245.9	1.07691	0.538455	−	0.842654i	$-0.319008\pi$
$449$	10245.9	1.07691	0.538455	+	0.842654i	$0.319008\pi$
$450$	0	0
$451$	−17980.1	−1.87728
$452$	0	0
$453$	−5867.93	−0.608608
$454$	0	0
$455$	5126.41	0.528198
$456$	0	0
$457$	−15100.5	−1.54567	−0.772835	−	0.634606i	$-0.781162\pi$
$457$	−15100.5	−1.54567	−0.772835	+	0.634606i	$0.781162\pi$
$458$	0	0
$459$	7852.69	0.798545
$460$	0	0
$461$	16221.4	1.63884	0.819422	−	0.573190i	$-0.194294\pi$
$461$	16221.4	1.63884	0.819422	+	0.573190i	$0.194294\pi$
$462$	0	0
$463$	−1409.06	−0.141435	−0.0707177	−	0.997496i	$-0.522529\pi$
$463$	−1409.06	−0.141435	−0.0707177	+	0.997496i	$0.522529\pi$
$464$	0	0
$465$	713.644	0.0711708
$466$	0	0
$467$	10791.4	1.06931	0.534653	−	0.845072i	$-0.320442\pi$
$467$	10791.4	1.06931	0.534653	+	0.845072i	$0.320442\pi$
$468$	0	0
$469$	1216.56	0.119777
$470$	0	0
$471$	10090.0	0.987099
$472$	0	0
$473$	4936.60	0.479884
$474$	0	0
$475$	−31878.2	−3.07931
$476$	0	0
$477$	−7529.16	−0.722718
$478$	0	0
$479$	−38.6870	−0.00369030	−0.00184515	−	0.999998i	$-0.500587\pi$
$479$	−38.6870	−0.00369030	−0.00184515	+	0.999998i	$0.500587\pi$
$480$	0	0
$481$	−1438.62	−0.136373
$482$	0	0
$483$	6548.52	0.616911
$484$	0	0
$485$	8641.20	0.809024
$486$	0	0
$487$	3827.61	0.356151	0.178076	−	0.984017i	$-0.443013\pi$
$487$	3827.61	0.356151	0.178076	+	0.984017i	$0.443013\pi$
$488$	0	0
$489$	8351.34	0.772312
$490$	0	0
$491$	−17847.2	−1.64039	−0.820195	−	0.572084i	$-0.806135\pi$
$491$	−17847.2	−1.64039	−0.820195	+	0.572084i	$0.806135\pi$
$492$	0	0
$493$	5363.08	0.489941
$494$	0	0
$495$	11331.0	1.02887
$496$	0	0
$497$	3278.70	0.295915
$498$	0	0
$499$	6245.41	0.560286	0.280143	−	0.959958i	$-0.409618\pi$
$499$	6245.41	0.560286	0.280143	+	0.959958i	$0.409618\pi$
$500$	0	0
$501$	10354.7	0.923378
$502$	0	0
$503$	7715.52	0.683932	0.341966	−	0.939712i	$-0.388907\pi$
$503$	7715.52	0.683932	0.341966	+	0.939712i	$0.388907\pi$
$504$	0	0
$505$	−25933.6	−2.28520
$506$	0	0
$507$	−4369.90	−0.382789
$508$	0	0
$509$	−1316.33	−0.114627	−0.0573135	−	0.998356i	$-0.518253\pi$
$509$	−1316.33	−0.114627	−0.0573135	+	0.998356i	$0.518253\pi$
$510$	0	0
$511$	−4970.53	−0.430300
$512$	0	0
$513$	12783.6	1.10021
$514$	0	0
$515$	10278.2	0.879437
$516$	0	0
$517$	−17079.9	−1.45295
$518$	0	0
$519$	−14011.8	−1.18506
$520$	0	0
$521$	13517.7	1.13670	0.568352	−	0.822786i	$-0.307581\pi$
$521$	13517.7	1.13670	0.568352	+	0.822786i	$0.307581\pi$
$522$	0	0
$523$	9646.55	0.806528	0.403264	−	0.915084i	$-0.367876\pi$
$523$	9646.55	0.806528	0.403264	+	0.915084i	$0.367876\pi$
$524$	0	0
$525$	10196.1	0.847604
$526$	0	0
$527$	−506.975	−0.0419055
$528$	0	0
$529$	10251.8	0.842593
$530$	0	0
$531$	4376.90	0.357705
$532$	0	0
$533$	−13988.2	−1.13677
$534$	0	0
$535$	−9989.57	−0.807265
$536$	0	0
$537$	−7764.03	−0.623915
$538$	0	0
$539$	2437.38	0.194778
$540$	0	0
$541$	−10000.7	−0.794759	−0.397379	−	0.917654i	$-0.630080\pi$
$541$	−10000.7	−0.794759	−0.397379	+	0.917654i	$0.630080\pi$
$542$	0	0
$543$	−6701.20	−0.529606
$544$	0	0
$545$	27631.3	2.17174
$546$	0	0
$547$	20263.2	1.58389	0.791947	−	0.610590i	$-0.209068\pi$
$547$	20263.2	1.58389	0.791947	+	0.610590i	$0.209068\pi$
$548$	0	0
$549$	−1455.61	−0.113159
$550$	0	0
$551$	8730.70	0.675028
$552$	0	0
$553$	7557.91	0.581184
$554$	0	0
$555$	−4395.49	−0.336177
$556$	0	0
$557$	−13723.9	−1.04399	−0.521994	−	0.852949i	$-0.674812\pi$
$557$	−13723.9	−1.04399	−0.521994	+	0.852949i	$0.674812\pi$
$558$	0	0
$559$	3840.59	0.290589
$560$	0	0
$561$	−26105.3	−1.96464
$562$	0	0
$563$	−883.647	−0.0661479	−0.0330739	−	0.999453i	$-0.510530\pi$
$563$	−883.647	−0.0661479	−0.0330739	+	0.999453i	$0.510530\pi$
$564$	0	0
$565$	14995.2	1.11655
$566$	0	0
$567$	−6363.77	−0.471346
$568$	0	0
$569$	9710.34	0.715428	0.357714	−	0.933831i	$-0.383556\pi$
$569$	9710.34	0.715428	0.357714	+	0.933831i	$0.383556\pi$
$570$	0	0
$571$	16209.9	1.18803	0.594015	−	0.804454i	$-0.297542\pi$
$571$	16209.9	1.18803	0.594015	+	0.804454i	$0.297542\pi$
$572$	0	0
$573$	−182.143	−0.0132795
$574$	0	0
$575$	34906.1	2.53163
$576$	0	0
$577$	5860.69	0.422848	0.211424	−	0.977394i	$-0.432190\pi$
$577$	5860.69	0.422848	0.211424	+	0.977394i	$0.432190\pi$
$578$	0	0
$579$	−13763.1	−0.987870
$580$	0	0
$581$	9626.27	0.687375
$582$	0	0
$583$	−31113.7	−2.21029
$584$	0	0
$585$	8815.30	0.623022
$586$	0	0
$587$	−5685.62	−0.399780	−0.199890	−	0.979818i	$-0.564058\pi$
$587$	−5685.62	−0.399780	−0.199890	+	0.979818i	$0.564058\pi$
$588$	0	0
$589$	−825.318	−0.0577362
$590$	0	0
$591$	−29738.7	−2.06986
$592$	0	0
$593$	102.429	0.00709321	0.00354660	−	0.999994i	$-0.498871\pi$
$593$	102.429	0.00709321	0.00354660	+	0.999994i	$0.498871\pi$
$594$	0	0
$595$	−11127.1	−0.766664
$596$	0	0
$597$	30196.0	2.07008
$598$	0	0
$599$	17357.5	1.18399	0.591995	−	0.805942i	$-0.298340\pi$
$599$	17357.5	1.18399	0.591995	+	0.805942i	$0.298340\pi$
$600$	0	0
$601$	5417.26	0.367678	0.183839	−	0.982956i	$-0.441147\pi$
$601$	5417.26	0.367678	0.183839	+	0.982956i	$0.441147\pi$
$602$	0	0
$603$	2091.98	0.141280
$604$	0	0
$605$	21636.2	1.45394
$606$	0	0
$607$	1543.98	0.103242	0.0516211	−	0.998667i	$-0.483561\pi$
$607$	1543.98	0.103242	0.0516211	+	0.998667i	$0.483561\pi$
$608$	0	0
$609$	−2792.47	−0.185807
$610$	0	0
$611$	−13287.9	−0.879820
$612$	0	0
$613$	−17476.4	−1.15149	−0.575747	−	0.817628i	$-0.695289\pi$
$613$	−17476.4	−1.15149	−0.575747	+	0.817628i	$0.695289\pi$
$614$	0	0
$615$	−42739.0	−2.80228
$616$	0	0
$617$	−14451.3	−0.942929	−0.471465	−	0.881885i	$-0.656274\pi$
$617$	−14451.3	−0.942929	−0.471465	+	0.881885i	$0.656274\pi$
$618$	0	0
$619$	−3082.20	−0.200136	−0.100068	−	0.994981i	$-0.531906\pi$
$619$	−3082.20	−0.200136	−0.100068	+	0.994981i	$0.531906\pi$
$620$	0	0
$621$	−13997.9	−0.904532
$622$	0	0
$623$	3238.96	0.208292
$624$	0	0
$625$	9582.83	0.613301
$626$	0	0
$627$	−42497.5	−2.70683
$628$	0	0
$629$	3122.57	0.197941
$630$	0	0
$631$	−8498.63	−0.536173	−0.268087	−	0.963395i	$-0.586391\pi$
$631$	−8498.63	−0.536173	−0.268087	+	0.963395i	$0.586391\pi$
$632$	0	0
$633$	−15748.6	−0.988865
$634$	0	0
$635$	9244.69	0.577739
$636$	0	0
$637$	1896.24	0.117946
$638$	0	0
$639$	5637.99	0.349038
$640$	0	0
$641$	20070.2	1.23670	0.618349	−	0.785904i	$-0.287802\pi$
$641$	20070.2	1.23670	0.618349	+	0.785904i	$0.287802\pi$
$642$	0	0
$643$	29430.7	1.80503	0.902514	−	0.430661i	$-0.141719\pi$
$643$	29430.7	1.80503	0.902514	+	0.430661i	$0.141719\pi$
$644$	0	0
$645$	11734.4	0.716341
$646$	0	0
$647$	4380.00	0.266144	0.133072	−	0.991106i	$-0.457516\pi$
$647$	4380.00	0.266144	0.133072	+	0.991106i	$0.457516\pi$
$648$	0	0
$649$	18087.2	1.09397
$650$	0	0
$651$	263.973	0.0158924
$652$	0	0
$653$	25596.2	1.53393	0.766964	−	0.641690i	$-0.221766\pi$
$653$	25596.2	1.53393	0.766964	+	0.641690i	$0.221766\pi$
$654$	0	0
$655$	−34296.6	−2.04592
$656$	0	0
$657$	−8547.23	−0.507548
$658$	0	0
$659$	−5179.96	−0.306195	−0.153097	−	0.988211i	$-0.548925\pi$
$659$	−5179.96	−0.306195	−0.153097	+	0.988211i	$0.548925\pi$
$660$	0	0
$661$	−3520.21	−0.207141	−0.103571	−	0.994622i	$-0.533027\pi$
$661$	−3520.21	−0.207141	−0.103571	+	0.994622i	$0.533027\pi$
$662$	0	0
$663$	−20309.4	−1.18967
$664$	0	0
$665$	−18114.1	−1.05629
$666$	0	0
$667$	−9559.99	−0.554969
$668$	0	0
$669$	−15772.0	−0.911479
$670$	0	0
$671$	−6015.21	−0.346073
$672$	0	0
$673$	−18724.3	−1.07247	−0.536233	−	0.844070i	$-0.680153\pi$
$673$	−18724.3	−1.07247	−0.536233	+	0.844070i	$0.680153\pi$
$674$	0	0
$675$	−21794.7	−1.24278
$676$	0	0
$677$	−24545.2	−1.39342	−0.696712	−	0.717351i	$-0.745355\pi$
$677$	−24545.2	−1.39342	−0.696712	+	0.717351i	$0.745355\pi$
$678$	0	0
$679$	3196.34	0.180654
$680$	0	0
$681$	−27101.2	−1.52499
$682$	0	0
$683$	26422.5	1.48028	0.740138	−	0.672455i	$-0.234760\pi$
$683$	26422.5	1.48028	0.740138	+	0.672455i	$0.234760\pi$
$684$	0	0
$685$	−17967.2	−1.00218
$686$	0	0
$687$	−34602.6	−1.92165
$688$	0	0
$689$	−24205.9	−1.33842
$690$	0	0
$691$	−23366.4	−1.28640	−0.643198	−	0.765700i	$-0.722393\pi$
$691$	−23366.4	−1.28640	−0.643198	+	0.765700i	$0.722393\pi$
$692$	0	0
$693$	4191.27	0.229745
$694$	0	0
$695$	−48450.7	−2.64437
$696$	0	0
$697$	30362.0	1.64999
$698$	0	0
$699$	9107.16	0.492796
$700$	0	0
$701$	7694.54	0.414577	0.207289	−	0.978280i	$-0.433536\pi$
$701$	7694.54	0.414577	0.207289	+	0.978280i	$0.433536\pi$
$702$	0	0
$703$	5083.32	0.272718
$704$	0	0
$705$	−40599.2	−2.16887
$706$	0	0
$707$	−9592.70	−0.510284
$708$	0	0
$709$	10450.4	0.553558	0.276779	−	0.960934i	$-0.410733\pi$
$709$	10450.4	0.553558	0.276779	+	0.960934i	$0.410733\pi$
$710$	0	0
$711$	12996.5	0.685520
$712$	0	0
$713$	903.711	0.0474674
$714$	0	0
$715$	36428.5	1.90539
$716$	0	0
$717$	−17467.8	−0.909829
$718$	0	0
$719$	−13749.5	−0.713171	−0.356586	−	0.934263i	$-0.616059\pi$
$719$	−13749.5	−0.713171	−0.356586	+	0.934263i	$0.616059\pi$
$720$	0	0
$721$	3801.84	0.196377
$722$	0	0
$723$	22502.1	1.15748
$724$	0	0
$725$	−14884.9	−0.762499
$726$	0	0
$727$	24646.6	1.25735	0.628675	−	0.777668i	$-0.283598\pi$
$727$	24646.6	1.25735	0.628675	+	0.777668i	$0.283598\pi$
$728$	0	0
$729$	4827.90	0.245283
$730$	0	0
$731$	−8336.13	−0.421782
$732$	0	0
$733$	25033.2	1.26142	0.630712	−	0.776017i	$-0.282763\pi$
$733$	25033.2	1.26142	0.630712	+	0.776017i	$0.282763\pi$
$734$	0	0
$735$	5793.68	0.290752
$736$	0	0
$737$	8644.93	0.432076
$738$	0	0
$739$	1987.30	0.0989228	0.0494614	−	0.998776i	$-0.484250\pi$
$739$	1987.30	0.0989228	0.0494614	+	0.998776i	$0.484250\pi$
$740$	0	0
$741$	−33062.3	−1.63910
$742$	0	0
$743$	−9220.44	−0.455269	−0.227635	−	0.973747i	$-0.573099\pi$
$743$	−9220.44	−0.455269	−0.227635	+	0.973747i	$0.573099\pi$
$744$	0	0
$745$	29570.4	1.45420
$746$	0	0
$747$	16553.2	0.810775
$748$	0	0
$749$	−3695.09	−0.180261
$750$	0	0
$751$	8210.43	0.398939	0.199469	−	0.979904i	$-0.436078\pi$
$751$	8210.43	0.398939	0.199469	+	0.979904i	$0.436078\pi$
$752$	0	0
$753$	18995.1	0.919285
$754$	0	0
$755$	−17773.2	−0.856732
$756$	0	0
$757$	13005.5	0.624431	0.312215	−	0.950011i	$-0.398929\pi$
$757$	13005.5	0.624431	0.312215	+	0.950011i	$0.398929\pi$
$758$	0	0
$759$	46534.1	2.22540
$760$	0	0
$761$	−18970.7	−0.903663	−0.451831	−	0.892103i	$-0.649229\pi$
$761$	−18970.7	−0.903663	−0.451831	+	0.892103i	$0.649229\pi$
$762$	0	0
$763$	10220.7	0.484946
$764$	0	0
$765$	−19133.9	−0.904298
$766$	0	0
$767$	14071.5	0.662443
$768$	0	0
$769$	−14087.4	−0.660606	−0.330303	−	0.943875i	$-0.607151\pi$
$769$	−14087.4	−0.660606	−0.330303	+	0.943875i	$0.607151\pi$
$770$	0	0
$771$	−45442.9	−2.12268
$772$	0	0
$773$	−19146.2	−0.890870	−0.445435	−	0.895314i	$-0.646951\pi$
$773$	−19146.2	−0.890870	−0.445435	+	0.895314i	$0.646951\pi$
$774$	0	0
$775$	1407.08	0.0652178
$776$	0	0
$777$	−1625.87	−0.0750679
$778$	0	0
$779$	49427.1	2.27331
$780$	0	0
$781$	23298.6	1.06746
$782$	0	0
$783$	5969.06	0.272435
$784$	0	0
$785$	30561.4	1.38953
$786$	0	0
$787$	8743.30	0.396016	0.198008	−	0.980200i	$-0.436553\pi$
$787$	8743.30	0.396016	0.198008	+	0.980200i	$0.436553\pi$
$788$	0	0
$789$	−17827.2	−0.804391
$790$	0	0
$791$	5546.64	0.249325
$792$	0	0
$793$	−4679.73	−0.209561
$794$	0	0
$795$	−73957.7	−3.29938
$796$	0	0
$797$	−27427.9	−1.21900	−0.609501	−	0.792785i	$-0.708630\pi$
$797$	−27427.9	−1.21900	−0.609501	+	0.792785i	$0.708630\pi$
$798$	0	0
$799$	28841.8	1.27703
$800$	0	0
$801$	5569.66	0.245686
$802$	0	0
$803$	−35320.8	−1.55223
$804$	0	0
$805$	19834.6	0.868421
$806$	0	0
$807$	26552.5	1.15823
$808$	0	0
$809$	−3197.69	−0.138967	−0.0694837	−	0.997583i	$-0.522135\pi$
$809$	−3197.69	−0.138967	−0.0694837	+	0.997583i	$0.522135\pi$
$810$	0	0
$811$	−212.937	−0.00921977	−0.00460989	−	0.999989i	$-0.501467\pi$
$811$	−212.937	−0.00921977	−0.00460989	+	0.999989i	$0.501467\pi$
$812$	0	0
$813$	6506.71	0.280689
$814$	0	0
$815$	25295.1	1.08718
$816$	0	0
$817$	−13570.6	−0.581121
$818$	0	0
$819$	3260.73	0.139120
$820$	0	0
$821$	15964.7	0.678652	0.339326	−	0.940669i	$-0.389801\pi$
$821$	15964.7	0.678652	0.339326	+	0.940669i	$0.389801\pi$
$822$	0	0
$823$	−23155.8	−0.980755	−0.490378	−	0.871510i	$-0.663141\pi$
$823$	−23155.8	−0.980755	−0.490378	+	0.871510i	$0.663141\pi$
$824$	0	0
$825$	72453.7	3.05759
$826$	0	0
$827$	14573.7	0.612788	0.306394	−	0.951905i	$-0.400877\pi$
$827$	14573.7	0.612788	0.306394	+	0.951905i	$0.400877\pi$
$828$	0	0
$829$	8738.62	0.366110	0.183055	−	0.983103i	$-0.441401\pi$
$829$	8738.62	0.366110	0.183055	+	0.983103i	$0.441401\pi$
$830$	0	0
$831$	32818.2	1.36998
$832$	0	0
$833$	−4115.85	−0.171195
$834$	0	0
$835$	31363.0	1.29983
$836$	0	0
$837$	−564.259	−0.0233018
$838$	0	0
$839$	7481.12	0.307839	0.153920	−	0.988083i	$-0.450810\pi$
$839$	7481.12	0.307839	0.153920	+	0.988083i	$0.450810\pi$
$840$	0	0
$841$	−20312.4	−0.832849
$842$	0	0
$843$	55536.1	2.26900
$844$	0	0
$845$	−13235.9	−0.538850
$846$	0	0
$847$	8003.11	0.324664
$848$	0	0
$849$	−25247.2	−1.02059
$850$	0	0
$851$	−5566.16	−0.224213
$852$	0	0
$853$	−3731.71	−0.149791	−0.0748954	−	0.997191i	$-0.523862\pi$
$853$	−3731.71	−0.149791	−0.0748954	+	0.997191i	$0.523862\pi$
$854$	0	0
$855$	−31148.6	−1.24592
$856$	0	0
$857$	−1485.43	−0.0592079	−0.0296039	−	0.999562i	$-0.509425\pi$
$857$	−1485.43	−0.0592079	−0.0296039	+	0.999562i	$0.509425\pi$
$858$	0	0
$859$	−22522.2	−0.894582	−0.447291	−	0.894388i	$-0.647611\pi$
$859$	−22522.2	−0.894582	−0.447291	+	0.894388i	$0.647611\pi$
$860$	0	0
$861$	−15809.0	−0.625747
$862$	0	0
$863$	21775.6	0.858921	0.429461	−	0.903086i	$-0.358704\pi$
$863$	21775.6	0.858921	0.429461	+	0.903086i	$0.358704\pi$
$864$	0	0
$865$	−42439.8	−1.66821
$866$	0	0
$867$	13386.1	0.524356
$868$	0	0
$869$	53706.9	2.09653
$870$	0	0
$871$	6725.61	0.261640
$872$	0	0
$873$	5496.37	0.213086
$874$	0	0
$875$	14323.8	0.553408
$876$	0	0
$877$	−27599.1	−1.06266	−0.531331	−	0.847164i	$-0.678308\pi$
$877$	−27599.1	−1.06266	−0.531331	+	0.847164i	$0.678308\pi$
$878$	0	0
$879$	18771.5	0.720302
$880$	0	0
$881$	−26664.6	−1.01970	−0.509849	−	0.860264i	$-0.670299\pi$
$881$	−26664.6	−1.01970	−0.509849	+	0.860264i	$0.670299\pi$
$882$	0	0
$883$	−25546.5	−0.973624	−0.486812	−	0.873507i	$-0.661840\pi$
$883$	−25546.5	−0.973624	−0.486812	+	0.873507i	$0.661840\pi$
$884$	0	0
$885$	42993.6	1.63301
$886$	0	0
$887$	42457.6	1.60720	0.803600	−	0.595170i	$-0.202915\pi$
$887$	42457.6	1.60720	0.803600	+	0.595170i	$0.202915\pi$
$888$	0	0
$889$	3419.57	0.129009
$890$	0	0
$891$	−45221.3	−1.70030
$892$	0	0
$893$	46952.4	1.75946
$894$	0	0
$895$	−23516.2	−0.878281
$896$	0	0
$897$	36202.7	1.34757
$898$	0	0
$899$	−385.367	−0.0142967
$900$	0	0
$901$	52539.8	1.94268
$902$	0	0
$903$	4340.48	0.159958
$904$	0	0
$905$	−20297.1	−0.745522
$906$	0	0
$907$	28864.4	1.05670	0.528350	−	0.849026i	$-0.322811\pi$
$907$	28864.4	1.05670	0.528350	+	0.849026i	$0.322811\pi$
$908$	0	0
$909$	−16495.4	−0.601891
$910$	0	0
$911$	41320.8	1.50276	0.751382	−	0.659867i	$-0.229387\pi$
$911$	41320.8	1.50276	0.751382	+	0.659867i	$0.229387\pi$
$912$	0	0
$913$	68404.8	2.47959
$914$	0	0
$915$	−14298.2	−0.516596
$916$	0	0
$917$	−12686.1	−0.456852
$918$	0	0
$919$	−54504.6	−1.95641	−0.978205	−	0.207641i	$-0.933421\pi$
$919$	−54504.6	−1.95641	−0.978205	+	0.207641i	$0.933421\pi$
$920$	0	0
$921$	−45994.7	−1.64558
$922$	0	0
$923$	18125.9	0.646393
$924$	0	0
$925$	−8666.52	−0.308058
$926$	0	0
$927$	6537.59	0.231632
$928$	0	0
$929$	19632.9	0.693363	0.346682	−	0.937983i	$-0.387308\pi$
$929$	19632.9	0.693363	0.346682	+	0.937983i	$0.387308\pi$
$930$	0	0
$931$	−6700.30	−0.235868
$932$	0	0
$933$	36887.3	1.29436
$934$	0	0
$935$	−79069.5	−2.76561
$936$	0	0
$937$	24262.2	0.845902	0.422951	−	0.906153i	$-0.360994\pi$
$937$	24262.2	0.845902	0.422951	+	0.906153i	$0.360994\pi$
$938$	0	0
$939$	−2276.12	−0.0791037
$940$	0	0
$941$	−21809.3	−0.755540	−0.377770	−	0.925900i	$-0.623309\pi$
$941$	−21809.3	−0.755540	−0.377770	+	0.925900i	$0.623309\pi$
$942$	0	0
$943$	−54121.9	−1.86898
$944$	0	0
$945$	−12384.3	−0.426309
$946$	0	0
$947$	−44809.7	−1.53761	−0.768806	−	0.639482i	$-0.779149\pi$
$947$	−44809.7	−1.53761	−0.768806	+	0.639482i	$0.779149\pi$
$948$	0	0
$949$	−27479.0	−0.939942
$950$	0	0
$951$	21627.1	0.737440
$952$	0	0
$953$	14114.0	0.479747	0.239873	−	0.970804i	$-0.422894\pi$
$953$	14114.0	0.479747	0.239873	+	0.970804i	$0.422894\pi$
$954$	0	0
$955$	−551.688	−0.0186934
$956$	0	0
$957$	−19843.4	−0.670268
$958$	0	0
$959$	−6645.99	−0.223785
$960$	0	0
$961$	−29754.6	−0.998777
$962$	0	0
$963$	−6354.02	−0.212622
$964$	0	0
$965$	−41686.8	−1.39062
$966$	0	0
$967$	44247.4	1.47146	0.735729	−	0.677276i	$-0.236840\pi$
$967$	44247.4	1.47146	0.735729	+	0.677276i	$0.236840\pi$
$968$	0	0
$969$	71762.9	2.37911
$970$	0	0
$971$	28073.3	0.927820	0.463910	−	0.885882i	$-0.346446\pi$
$971$	28073.3	0.927820	0.463910	+	0.885882i	$0.346446\pi$
$972$	0	0
$973$	−17921.7	−0.590486
$974$	0	0
$975$	56367.7	1.85150
$976$	0	0
$977$	21636.8	0.708519	0.354260	−	0.935147i	$-0.384733\pi$
$977$	21636.8	0.708519	0.354260	+	0.935147i	$0.384733\pi$
$978$	0	0
$979$	23016.2	0.751380
$980$	0	0
$981$	17575.3	0.572005
$982$	0	0
$983$	10355.5	0.336000	0.168000	−	0.985787i	$-0.446269\pi$
$983$	10355.5	0.336000	0.168000	+	0.985787i	$0.446269\pi$
$984$	0	0
$985$	−90074.7	−2.91372
$986$	0	0
$987$	−15017.4	−0.484307
$988$	0	0
$989$	14859.6	0.477764
$990$	0	0
$991$	22285.2	0.714342	0.357171	−	0.934039i	$-0.383741\pi$
$991$	22285.2	0.714342	0.357171	+	0.934039i	$0.383741\pi$
$992$	0	0
$993$	−33473.7	−1.06975
$994$	0	0
$995$	91459.8	2.91404
$996$	0	0
$997$	22234.4	0.706290	0.353145	−	0.935569i	$-0.385112\pi$
$997$	22234.4	0.706290	0.353145	+	0.935569i	$0.385112\pi$
$998$	0	0
$999$	3475.40	0.110067

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.a.w.1.3		3
4.3	odd	2		448.4.a.v.1.1		3
8.3	odd	2		224.4.a.f.1.3	✓	3
8.5	even	2		224.4.a.g.1.1	yes	3
24.5	odd	2		2016.4.a.x.1.3		3
24.11	even	2		2016.4.a.w.1.3		3
56.13	odd	2		1568.4.a.x.1.3		3
56.27	even	2		1568.4.a.w.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.4.a.f.1.3	✓	3	8.3	odd	2
224.4.a.g.1.1	yes	3	8.5	even	2
448.4.a.v.1.1		3	4.3	odd	2
448.4.a.w.1.3		3	1.1	even	1	trivial
1568.4.a.w.1.1		3	56.27	even	2
1568.4.a.x.1.3		3	56.13	odd	2
2016.4.a.w.1.3		3	24.11	even	2
2016.4.a.x.1.3		3	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

\(f(q)\)	\(=\)	\(q+6.24797 q^{3} +18.9243 q^{5} +7.00000 q^{7} +12.0371 q^{9} +O(q^{10})\)	\(q+6.24797 q^{3} +18.9243 q^{5} +7.00000 q^{7} +12.0371 q^{9} +49.7423 q^{11} +38.6987 q^{13} +118.238 q^{15} -83.9969 q^{17} -136.741 q^{19} +43.7358 q^{21} +149.729 q^{23} +233.128 q^{25} -93.4878 q^{27} -63.8486 q^{29} +6.03564 q^{31} +310.789 q^{33} +132.470 q^{35} -37.1749 q^{37} +241.788 q^{39} -361.465 q^{41} +99.2433 q^{43} +227.793 q^{45} -343.368 q^{47} +49.0000 q^{49} -524.810 q^{51} -625.497 q^{53} +941.338 q^{55} -854.352 q^{57} +363.618 q^{59} -120.927 q^{61} +84.2596 q^{63} +732.345 q^{65} +173.794 q^{67} +935.503 q^{69} +468.385 q^{71} -710.075 q^{73} +1456.58 q^{75} +348.196 q^{77} +1079.70 q^{79} -909.110 q^{81} +1375.18 q^{83} -1589.58 q^{85} -398.924 q^{87} +462.708 q^{89} +270.891 q^{91} +37.7105 q^{93} -2587.72 q^{95} +456.619 q^{97} +598.753 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{5} + 21 q^{7} + 15 q^{9}+O(q^{10}) \) 3 * q + 6 * q^5 + 21 * q^7 + 15 * q^9	\( 3 q + 6 q^{5} + 21 q^{7} + 15 q^{9} + 6 q^{13} + 168 q^{15} - 66 q^{17} - 168 q^{19} + 336 q^{23} + 69 q^{25} - 168 q^{27} - 90 q^{29} + 504 q^{31} + 120 q^{33} + 42 q^{35} - 18 q^{37} + 840 q^{39} - 450 q^{41} + 150 q^{45} + 504 q^{47} + 147 q^{49} + 336 q^{51} + 78 q^{53} + 1176 q^{55} + 48 q^{57} + 504 q^{59} - 498 q^{61} + 105 q^{63} + 1068 q^{65} + 1008 q^{67} + 1224 q^{69} + 504 q^{71} - 234 q^{73} + 1848 q^{75} + 168 q^{79} - 981 q^{81} + 3024 q^{83} - 1476 q^{85} - 336 q^{87} + 246 q^{89} + 42 q^{91} - 1200 q^{93} - 2184 q^{95} - 2514 q^{97} + 2688 q^{99}+O(q^{100}) \) 3 * q + 6 * q^5 + 21 * q^7 + 15 * q^9 + 6 * q^13 + 168 * q^15 - 66 * q^17 - 168 * q^19 + 336 * q^23 + 69 * q^25 - 168 * q^27 - 90 * q^29 + 504 * q^31 + 120 * q^33 + 42 * q^35 - 18 * q^37 + 840 * q^39 - 450 * q^41 + 150 * q^45 + 504 * q^47 + 147 * q^49 + 336 * q^51 + 78 * q^53 + 1176 * q^55 + 48 * q^57 + 504 * q^59 - 498 * q^61 + 105 * q^63 + 1068 * q^65 + 1008 * q^67 + 1224 * q^69 + 504 * q^71 - 234 * q^73 + 1848 * q^75 + 168 * q^79 - 981 * q^81 + 3024 * q^83 - 1476 * q^85 - 336 * q^87 + 246 * q^89 + 42 * q^91 - 1200 * q^93 - 2184 * q^95 - 2514 * q^97 + 2688 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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