LMFDB - Embedded newform 1216.4.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,4,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1216.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:	$71.7463225670$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{177}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 44 $ x^2 - x - 44
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.15207$ of defining polynomial
Character	$\chi$	$=$	1216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.15207 q^{3} -18.3041 q^{5} -21.8479 q^{7} +10.8479 q^{9} +O(q^{10})$	$q-6.15207 q^{3} -18.3041 q^{5} -21.8479 q^{7} +10.8479 q^{9} -8.30413 q^{11} -53.0645 q^{13} +112.608 q^{15} +74.2810 q^{17} -19.0000 q^{19} +134.410 q^{21} +163.977 q^{23} +210.041 q^{25} +99.3686 q^{27} +232.410 q^{29} -98.4331 q^{31} +51.0876 q^{33} +399.908 q^{35} -296.433 q^{37} +326.456 q^{39} -434.912 q^{41} -171.299 q^{43} -198.562 q^{45} +366.083 q^{47} +134.332 q^{49} -456.982 q^{51} -138.631 q^{53} +152.000 q^{55} +116.889 q^{57} +572.797 q^{59} +632.691 q^{61} -237.005 q^{63} +971.299 q^{65} -183.461 q^{67} -1008.80 q^{69} -56.6545 q^{71} +68.1521 q^{73} -1292.19 q^{75} +181.428 q^{77} +332.820 q^{79} -904.217 q^{81} +1152.91 q^{83} -1359.65 q^{85} -1429.80 q^{87} -368.479 q^{89} +1159.35 q^{91} +605.567 q^{93} +347.779 q^{95} -426.443 q^{97} -90.0827 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 10 q^{5} - 57 q^{7} + 35 q^{9}+O(q^{10}) $ 2 * q + q^3 - 10 * q^5 - 57 * q^7 + 35 * q^9	$ 2 q + q^{3} - 10 q^{5} - 57 q^{7} + 35 q^{9} + 10 q^{11} - 13 q^{13} + 172 q^{15} - 51 q^{17} - 38 q^{19} - 117 q^{21} + 155 q^{23} + 154 q^{25} + 79 q^{27} + 79 q^{29} + 16 q^{31} + 182 q^{33} + 108 q^{35} - 380 q^{37} + 613 q^{39} - 790 q^{41} + 296 q^{43} + 2 q^{45} + 200 q^{47} + 1027 q^{49} - 1353 q^{51} - 397 q^{53} + 304 q^{55} - 19 q^{57} + 201 q^{59} + 680 q^{61} - 1086 q^{63} + 1304 q^{65} - 939 q^{67} - 1073 q^{69} - 406 q^{71} + 123 q^{73} - 1693 q^{75} - 462 q^{77} - 106 q^{79} - 1702 q^{81} + 2226 q^{83} - 2400 q^{85} - 2527 q^{87} - 870 q^{89} - 249 q^{91} + 1424 q^{93} + 190 q^{95} - 1864 q^{97} + 352 q^{99}+O(q^{100}) $ 2 * q + q^3 - 10 * q^5 - 57 * q^7 + 35 * q^9 + 10 * q^11 - 13 * q^13 + 172 * q^15 - 51 * q^17 - 38 * q^19 - 117 * q^21 + 155 * q^23 + 154 * q^25 + 79 * q^27 + 79 * q^29 + 16 * q^31 + 182 * q^33 + 108 * q^35 - 380 * q^37 + 613 * q^39 - 790 * q^41 + 296 * q^43 + 2 * q^45 + 200 * q^47 + 1027 * q^49 - 1353 * q^51 - 397 * q^53 + 304 * q^55 - 19 * q^57 + 201 * q^59 + 680 * q^61 - 1086 * q^63 + 1304 * q^65 - 939 * q^67 - 1073 * q^69 - 406 * q^71 + 123 * q^73 - 1693 * q^75 - 462 * q^77 - 106 * q^79 - 1702 * q^81 + 2226 * q^83 - 2400 * q^85 - 2527 * q^87 - 870 * q^89 - 249 * q^91 + 1424 * q^93 + 190 * q^95 - 1864 * q^97 + 352 * 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
$3$	−6.15207	−1.18397	−0.591983	−	0.805950i	$-0.701655\pi$
$3$	−6.15207	−1.18397	−0.591983	+	0.805950i	$0.701655\pi$
$4$	0	0
$5$	−18.3041	−1.63717	−0.818586	−	0.574384i	$-0.805242\pi$
$5$	−18.3041	−1.63717	−0.818586	+	0.574384i	$0.805242\pi$
$6$	0	0
$7$	−21.8479	−1.17968	−0.589839	−	0.807521i	$-0.700809\pi$
$7$	−21.8479	−1.17968	−0.589839	+	0.807521i	$0.700809\pi$
$8$	0	0
$9$	10.8479	0.401775
$10$	0	0
$11$	−8.30413	−0.227617	−0.113809	−	0.993503i	$-0.536305\pi$
$11$	−8.30413	−0.227617	−0.113809	+	0.993503i	$0.536305\pi$
$12$	0	0
$13$	−53.0645	−1.13211	−0.566055	−	0.824367i	$-0.691531\pi$
$13$	−53.0645	−1.13211	−0.566055	+	0.824367i	$0.691531\pi$
$14$	0	0
$15$	112.608	1.93836
$16$	0	0
$17$	74.2810	1.05975	0.529876	−	0.848075i	$-0.322238\pi$
$17$	74.2810	1.05975	0.529876	+	0.848075i	$0.322238\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	134.410	1.39670
$22$	0	0
$23$	163.977	1.48659	0.743294	−	0.668964i	$-0.233262\pi$
$23$	163.977	1.48659	0.743294	+	0.668964i	$0.233262\pi$
$24$	0	0
$25$	210.041	1.68033
$26$	0	0
$27$	99.3686	0.708278
$28$	0	0
$29$	232.410	1.48819	0.744094	−	0.668075i	$-0.232882\pi$
$29$	232.410	1.48819	0.744094	+	0.668075i	$0.232882\pi$
$30$	0	0
$31$	−98.4331	−0.570294	−0.285147	−	0.958484i	$-0.592042\pi$
$31$	−98.4331	−0.570294	−0.285147	+	0.958484i	$0.592042\pi$
$32$	0	0
$33$	51.0876	0.269491
$34$	0	0
$35$	399.908	1.93133
$36$	0	0
$37$	−296.433	−1.31712	−0.658558	−	0.752530i	$-0.728833\pi$
$37$	−296.433	−1.31712	−0.658558	+	0.752530i	$0.728833\pi$
$38$	0	0
$39$	326.456	1.34038
$40$	0	0
$41$	−434.912	−1.65663	−0.828316	−	0.560261i	$-0.810701\pi$
$41$	−434.912	−1.65663	−0.828316	+	0.560261i	$0.810701\pi$
$42$	0	0
$43$	−171.299	−0.607509	−0.303755	−	0.952750i	$-0.598240\pi$
$43$	−171.299	−0.607509	−0.303755	+	0.952750i	$0.598240\pi$
$44$	0	0
$45$	−198.562	−0.657775
$46$	0	0
$47$	366.083	1.13614	0.568071	−	0.822980i	$-0.307690\pi$
$47$	366.083	1.13614	0.568071	+	0.822980i	$0.307690\pi$
$48$	0	0
$49$	134.332	0.391639
$50$	0	0
$51$	−456.982	−1.25471
$52$	0	0
$53$	−138.631	−0.359292	−0.179646	−	0.983731i	$-0.557495\pi$
$53$	−138.631	−0.359292	−0.179646	+	0.983731i	$0.557495\pi$
$54$	0	0
$55$	152.000	0.372649
$56$	0	0
$57$	116.889	0.271620
$58$	0	0
$59$	572.797	1.26393	0.631964	−	0.774997i	$-0.282249\pi$
$59$	572.797	1.26393	0.631964	+	0.774997i	$0.282249\pi$
$60$	0	0
$61$	632.691	1.32800	0.663998	−	0.747734i	$-0.268858\pi$
$61$	632.691	1.32800	0.663998	+	0.747734i	$0.268858\pi$
$62$	0	0
$63$	−237.005	−0.473965
$64$	0	0
$65$	971.299	1.85346
$66$	0	0
$67$	−183.461	−0.334527	−0.167264	−	0.985912i	$-0.553493\pi$
$67$	−183.461	−0.334527	−0.167264	+	0.985912i	$0.553493\pi$
$68$	0	0
$69$	−1008.80	−1.76007
$70$	0	0
$71$	−56.6545	−0.0946994	−0.0473497	−	0.998878i	$-0.515078\pi$
$71$	−56.6545	−0.0946994	−0.0473497	+	0.998878i	$0.515078\pi$
$72$	0	0
$73$	68.1521	0.109268	0.0546342	−	0.998506i	$-0.482601\pi$
$73$	68.1521	0.109268	0.0546342	+	0.998506i	$0.482601\pi$
$74$	0	0
$75$	−1292.19	−1.98945
$76$	0	0
$77$	181.428	0.268515
$78$	0	0
$79$	332.820	0.473989	0.236995	−	0.971511i	$-0.423838\pi$
$79$	332.820	0.473989	0.236995	+	0.971511i	$0.423838\pi$
$80$	0	0
$81$	−904.217	−1.24035
$82$	0	0
$83$	1152.91	1.52468	0.762341	−	0.647176i	$-0.224050\pi$
$83$	1152.91	1.52468	0.762341	+	0.647176i	$0.224050\pi$
$84$	0	0
$85$	−1359.65	−1.73500
$86$	0	0
$87$	−1429.80	−1.76196
$88$	0	0
$89$	−368.479	−0.438862	−0.219431	−	0.975628i	$-0.570420\pi$
$89$	−368.479	−0.438862	−0.219431	+	0.975628i	$0.570420\pi$
$90$	0	0
$91$	1159.35	1.33553
$92$	0	0
$93$	605.567	0.675208
$94$	0	0
$95$	347.779	0.375593
$96$	0	0
$97$	−426.443	−0.446378	−0.223189	−	0.974775i	$-0.571647\pi$
$97$	−426.443	−0.446378	−0.223189	+	0.974775i	$0.571647\pi$
$98$	0	0
$99$	−90.0827	−0.0914510
$100$	0	0
$101$	403.124	0.397152	0.198576	−	0.980086i	$-0.436368\pi$
$101$	403.124	0.397152	0.198576	+	0.980086i	$0.436368\pi$
$102$	0	0
$103$	1135.68	1.08642	0.543211	−	0.839596i	$-0.317208\pi$
$103$	1135.68	1.08642	0.543211	+	0.839596i	$0.317208\pi$
$104$	0	0
$105$	−2460.26	−2.28663
$106$	0	0
$107$	−380.096	−0.343414	−0.171707	−	0.985148i	$-0.554928\pi$
$107$	−380.096	−0.343414	−0.171707	+	0.985148i	$0.554928\pi$
$108$	0	0
$109$	−1180.74	−1.03756	−0.518782	−	0.854907i	$-0.673614\pi$
$109$	−1180.74	−1.03756	−0.518782	+	0.854907i	$0.673614\pi$
$110$	0	0
$111$	1823.68	1.55942
$112$	0	0
$113$	−1132.51	−0.942807	−0.471404	−	0.881918i	$-0.656252\pi$
$113$	−1132.51	−0.942807	−0.471404	+	0.881918i	$0.656252\pi$
$114$	0	0
$115$	−3001.45	−2.43380
$116$	0	0
$117$	−575.640	−0.454854
$118$	0	0
$119$	−1622.89	−1.25017
$120$	0	0
$121$	−1262.04	−0.948190
$122$	0	0
$123$	2675.61	1.96140
$124$	0	0
$125$	−1556.61	−1.11382
$126$	0	0
$127$	40.0462	0.0279806	0.0139903	−	0.999902i	$-0.495547\pi$
$127$	40.0462	0.0279806	0.0139903	+	0.999902i	$0.495547\pi$
$128$	0	0
$129$	1053.84	0.719270
$130$	0	0
$131$	−177.898	−0.118649	−0.0593244	−	0.998239i	$-0.518895\pi$
$131$	−177.898	−0.118649	−0.0593244	+	0.998239i	$0.518895\pi$
$132$	0	0
$133$	415.111	0.270637
$134$	0	0
$135$	−1818.86	−1.15957
$136$	0	0
$137$	−24.7603	−0.0154410	−0.00772050	−	0.999970i	$-0.502458\pi$
$137$	−24.7603	−0.0154410	−0.00772050	+	0.999970i	$0.502458\pi$
$138$	0	0
$139$	2867.21	1.74959	0.874796	−	0.484491i	$-0.160995\pi$
$139$	2867.21	1.74959	0.874796	+	0.484491i	$0.160995\pi$
$140$	0	0
$141$	−2252.17	−1.34515
$142$	0	0
$143$	440.655	0.257688
$144$	0	0
$145$	−4254.06	−2.43642
$146$	0	0
$147$	−826.421	−0.463687
$148$	0	0
$149$	−1949.35	−1.07179	−0.535895	−	0.844285i	$-0.680026\pi$
$149$	−1949.35	−1.07179	−0.535895	+	0.844285i	$0.680026\pi$
$150$	0	0
$151$	−1120.99	−0.604136	−0.302068	−	0.953286i	$-0.597677\pi$
$151$	−1120.99	−0.604136	−0.302068	+	0.953286i	$0.597677\pi$
$152$	0	0
$153$	805.795	0.425782
$154$	0	0
$155$	1801.73	0.933669
$156$	0	0
$157$	2360.23	1.19979	0.599894	−	0.800079i	$-0.295209\pi$
$157$	2360.23	1.19979	0.599894	+	0.800079i	$0.295209\pi$
$158$	0	0
$159$	852.870	0.425390
$160$	0	0
$161$	−3582.56	−1.75370
$162$	0	0
$163$	861.825	0.414131	0.207065	−	0.978327i	$-0.433609\pi$
$163$	861.825	0.414131	0.207065	+	0.978327i	$0.433609\pi$
$164$	0	0
$165$	−935.114	−0.441203
$166$	0	0
$167$	−1686.51	−0.781472	−0.390736	−	0.920503i	$-0.627779\pi$
$167$	−1686.51	−0.781472	−0.390736	+	0.920503i	$0.627779\pi$
$168$	0	0
$169$	618.838	0.281674
$170$	0	0
$171$	−206.111	−0.0921736
$172$	0	0
$173$	3191.44	1.40255	0.701273	−	0.712893i	$-0.252616\pi$
$173$	3191.44	1.40255	0.701273	+	0.712893i	$0.252616\pi$
$174$	0	0
$175$	−4588.97	−1.98225
$176$	0	0
$177$	−3523.88	−1.49645
$178$	0	0
$179$	−1229.49	−0.513389	−0.256695	−	0.966493i	$-0.582633\pi$
$179$	−1229.49	−0.513389	−0.256695	+	0.966493i	$0.582633\pi$
$180$	0	0
$181$	3108.95	1.27672	0.638360	−	0.769738i	$-0.279613\pi$
$181$	3108.95	1.27672	0.638360	+	0.769738i	$0.279613\pi$
$182$	0	0
$183$	−3892.36	−1.57230
$184$	0	0
$185$	5425.95	2.15635
$186$	0	0
$187$	−616.840	−0.241218
$188$	0	0
$189$	−2171.00	−0.835539
$190$	0	0
$191$	1415.48	0.536233	0.268117	−	0.963386i	$-0.413599\pi$
$191$	1415.48	0.536233	0.268117	+	0.963386i	$0.413599\pi$
$192$	0	0
$193$	1443.40	0.538333	0.269167	−	0.963094i	$-0.413252\pi$
$193$	1443.40	0.538333	0.269167	+	0.963094i	$0.413252\pi$
$194$	0	0
$195$	−5975.50	−2.19443
$196$	0	0
$197$	−5271.92	−1.90664	−0.953322	−	0.301954i	$-0.902361\pi$
$197$	−5271.92	−1.90664	−0.953322	+	0.301954i	$0.902361\pi$
$198$	0	0
$199$	2510.19	0.894183	0.447091	−	0.894488i	$-0.352460\pi$
$199$	2510.19	0.894183	0.447091	+	0.894488i	$0.352460\pi$
$200$	0	0
$201$	1128.67	0.396069
$202$	0	0
$203$	−5077.68	−1.75558
$204$	0	0
$205$	7960.70	2.71219
$206$	0	0
$207$	1778.81	0.597275
$208$	0	0
$209$	157.779	0.0522190
$210$	0	0
$211$	1854.44	0.605046	0.302523	−	0.953142i	$-0.402171\pi$
$211$	1854.44	0.605046	0.302523	+	0.953142i	$0.402171\pi$
$212$	0	0
$213$	348.542	0.112121
$214$	0	0
$215$	3135.48	0.994597
$216$	0	0
$217$	2150.56	0.672763
$218$	0	0
$219$	−419.276	−0.129370
$220$	0	0
$221$	−3941.68	−1.19976
$222$	0	0
$223$	−1880.34	−0.564649	−0.282325	−	0.959319i	$-0.591105\pi$
$223$	−1880.34	−0.564649	−0.282325	+	0.959319i	$0.591105\pi$
$224$	0	0
$225$	2278.51	0.675115
$226$	0	0
$227$	1799.23	0.526075	0.263038	−	0.964786i	$-0.415276\pi$
$227$	1799.23	0.526075	0.263038	+	0.964786i	$0.415276\pi$
$228$	0	0
$229$	−4835.34	−1.39532	−0.697660	−	0.716429i	$-0.745776\pi$
$229$	−4835.34	−1.39532	−0.697660	+	0.716429i	$0.745776\pi$
$230$	0	0
$231$	−1116.16	−0.317913
$232$	0	0
$233$	865.299	0.243295	0.121647	−	0.992573i	$-0.461182\pi$
$233$	865.299	0.243295	0.121647	+	0.992573i	$0.461182\pi$
$234$	0	0
$235$	−6700.83	−1.86006
$236$	0	0
$237$	−2047.53	−0.561187
$238$	0	0
$239$	4764.27	1.28943	0.644717	−	0.764421i	$-0.276975\pi$
$239$	4764.27	1.28943	0.644717	+	0.764421i	$0.276975\pi$
$240$	0	0
$241$	615.336	0.164470	0.0822350	−	0.996613i	$-0.473794\pi$
$241$	615.336	0.164470	0.0822350	+	0.996613i	$0.473794\pi$
$242$	0	0
$243$	2879.85	0.760257
$244$	0	0
$245$	−2458.83	−0.641180
$246$	0	0
$247$	1008.22	0.259724
$248$	0	0
$249$	−7092.79	−1.80517
$250$	0	0
$251$	−1658.08	−0.416959	−0.208480	−	0.978027i	$-0.566852\pi$
$251$	−1658.08	−0.416959	−0.208480	+	0.978027i	$0.566852\pi$
$252$	0	0
$253$	−1361.69	−0.338373
$254$	0	0
$255$	8364.66	2.05418
$256$	0	0
$257$	3446.12	0.836432	0.418216	−	0.908348i	$-0.362656\pi$
$257$	3446.12	0.836432	0.418216	+	0.908348i	$0.362656\pi$
$258$	0	0
$259$	6476.45	1.55377
$260$	0	0
$261$	2521.17	0.597917
$262$	0	0
$263$	−5755.80	−1.34950	−0.674748	−	0.738048i	$-0.735748\pi$
$263$	−5755.80	−1.34950	−0.674748	+	0.738048i	$0.735748\pi$
$264$	0	0
$265$	2537.53	0.588223
$266$	0	0
$267$	2266.91	0.519598
$268$	0	0
$269$	−2257.28	−0.511631	−0.255816	−	0.966726i	$-0.582344\pi$
$269$	−2257.28	−0.511631	−0.255816	+	0.966726i	$0.582344\pi$
$270$	0	0
$271$	−7012.13	−1.57180	−0.785898	−	0.618357i	$-0.787799\pi$
$271$	−7012.13	−1.57180	−0.785898	+	0.618357i	$0.787799\pi$
$272$	0	0
$273$	−7132.39	−1.58122
$274$	0	0
$275$	−1744.21	−0.382472
$276$	0	0
$277$	372.810	0.0808664	0.0404332	−	0.999182i	$-0.487126\pi$
$277$	372.810	0.0808664	0.0404332	+	0.999182i	$0.487126\pi$
$278$	0	0
$279$	−1067.80	−0.229130
$280$	0	0
$281$	−1888.96	−0.401017	−0.200508	−	0.979692i	$-0.564259\pi$
$281$	−1888.96	−0.401017	−0.200508	+	0.979692i	$0.564259\pi$
$282$	0	0
$283$	−3884.43	−0.815920	−0.407960	−	0.913000i	$-0.633760\pi$
$283$	−3884.43	−0.815920	−0.407960	+	0.913000i	$0.633760\pi$
$284$	0	0
$285$	−2139.56	−0.444689
$286$	0	0
$287$	9501.94	1.95429
$288$	0	0
$289$	604.668	0.123075
$290$	0	0
$291$	2623.51	0.528497
$292$	0	0
$293$	1273.01	0.253823	0.126911	−	0.991914i	$-0.459494\pi$
$293$	1273.01	0.253823	0.126911	+	0.991914i	$0.459494\pi$
$294$	0	0
$295$	−10484.5	−2.06927
$296$	0	0
$297$	−825.170	−0.161216
$298$	0	0
$299$	−8701.35	−1.68298
$300$	0	0
$301$	3742.53	0.716665
$302$	0	0
$303$	−2480.05	−0.470214
$304$	0	0
$305$	−11580.9	−2.17416
$306$	0	0
$307$	819.153	0.152285	0.0761426	−	0.997097i	$-0.475740\pi$
$307$	819.153	0.152285	0.0761426	+	0.997097i	$0.475740\pi$
$308$	0	0
$309$	−6986.76	−1.28629
$310$	0	0
$311$	−2104.67	−0.383745	−0.191872	−	0.981420i	$-0.561456\pi$
$311$	−2104.67	−0.383745	−0.191872	+	0.981420i	$0.561456\pi$
$312$	0	0
$313$	2395.47	0.432587	0.216293	−	0.976328i	$-0.430603\pi$
$313$	2395.47	0.432587	0.216293	+	0.976328i	$0.430603\pi$
$314$	0	0
$315$	4338.17	0.775962
$316$	0	0
$317$	−2158.23	−0.382393	−0.191196	−	0.981552i	$-0.561237\pi$
$317$	−2158.23	−0.382393	−0.191196	+	0.981552i	$0.561237\pi$
$318$	0	0
$319$	−1929.96	−0.338737
$320$	0	0
$321$	2338.38	0.406590
$322$	0	0
$323$	−1411.34	−0.243124
$324$	0	0
$325$	−11145.7	−1.90232
$326$	0	0
$327$	7264.00	1.22844
$328$	0	0
$329$	−7998.15	−1.34028
$330$	0	0
$331$	517.782	0.0859815	0.0429908	−	0.999075i	$-0.486311\pi$
$331$	517.782	0.0859815	0.0429908	+	0.999075i	$0.486311\pi$
$332$	0	0
$333$	−3215.69	−0.529185
$334$	0	0
$335$	3358.10	0.547679
$336$	0	0
$337$	6503.63	1.05126	0.525631	−	0.850713i	$-0.323829\pi$
$337$	6503.63	1.05126	0.525631	+	0.850713i	$0.323829\pi$
$338$	0	0
$339$	6967.25	1.11625
$340$	0	0
$341$	817.402	0.129809
$342$	0	0
$343$	4558.96	0.717670
$344$	0	0
$345$	18465.2	2.88154
$346$	0	0
$347$	6058.33	0.937257	0.468628	−	0.883395i	$-0.344748\pi$
$347$	6058.33	0.937257	0.468628	+	0.883395i	$0.344748\pi$
$348$	0	0
$349$	10955.1	1.68027	0.840135	−	0.542377i	$-0.182476\pi$
$349$	10955.1	1.68027	0.840135	+	0.542377i	$0.182476\pi$
$350$	0	0
$351$	−5272.94	−0.801849
$352$	0	0
$353$	−1806.43	−0.272369	−0.136185	−	0.990683i	$-0.543484\pi$
$353$	−1806.43	−0.272369	−0.136185	+	0.990683i	$0.543484\pi$
$354$	0	0
$355$	1037.01	0.155039
$356$	0	0
$357$	9984.11	1.48015
$358$	0	0
$359$	7964.50	1.17089	0.585446	−	0.810711i	$-0.300919\pi$
$359$	7964.50	1.17089	0.585446	+	0.810711i	$0.300919\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	7764.16	1.12263
$364$	0	0
$365$	−1247.46	−0.178891
$366$	0	0
$367$	7311.58	1.03995	0.519974	−	0.854182i	$-0.325941\pi$
$367$	7311.58	1.03995	0.519974	+	0.854182i	$0.325941\pi$
$368$	0	0
$369$	−4717.90	−0.665594
$370$	0	0
$371$	3028.81	0.423849
$372$	0	0
$373$	−5518.38	−0.766035	−0.383017	−	0.923741i	$-0.625115\pi$
$373$	−5518.38	−0.766035	−0.383017	+	0.923741i	$0.625115\pi$
$374$	0	0
$375$	9576.36	1.31872
$376$	0	0
$377$	−12332.7	−1.68479
$378$	0	0
$379$	−1139.97	−0.154502	−0.0772512	−	0.997012i	$-0.524614\pi$
$379$	−1139.97	−0.154502	−0.0772512	+	0.997012i	$0.524614\pi$
$380$	0	0
$381$	−246.367	−0.0331280
$382$	0	0
$383$	10409.5	1.38877	0.694385	−	0.719604i	$-0.255677\pi$
$383$	10409.5	1.38877	0.694385	+	0.719604i	$0.255677\pi$
$384$	0	0
$385$	−3320.89	−0.439605
$386$	0	0
$387$	−1858.24	−0.244082
$388$	0	0
$389$	−10471.2	−1.36481	−0.682404	−	0.730975i	$-0.739066\pi$
$389$	−10471.2	−1.36481	−0.682404	+	0.730975i	$0.739066\pi$
$390$	0	0
$391$	12180.4	1.57542
$392$	0	0
$393$	1094.44	0.140476
$394$	0	0
$395$	−6091.98	−0.776002
$396$	0	0
$397$	9588.68	1.21220	0.606098	−	0.795390i	$-0.292734\pi$
$397$	9588.68	1.21220	0.606098	+	0.795390i	$0.292734\pi$
$398$	0	0
$399$	−2553.79	−0.320424
$400$	0	0
$401$	8549.30	1.06467	0.532334	−	0.846535i	$-0.321315\pi$
$401$	8549.30	1.06467	0.532334	+	0.846535i	$0.321315\pi$
$402$	0	0
$403$	5223.30	0.645635
$404$	0	0
$405$	16550.9	2.03067
$406$	0	0
$407$	2461.62	0.299798
$408$	0	0
$409$	266.960	0.0322746	0.0161373	−	0.999870i	$-0.494863\pi$
$409$	266.960	0.0322746	0.0161373	+	0.999870i	$0.494863\pi$
$410$	0	0
$411$	152.327	0.0182816
$412$	0	0
$413$	−12514.4	−1.49103
$414$	0	0
$415$	−21103.1	−2.49617
$416$	0	0
$417$	−17639.2	−2.07146
$418$	0	0
$419$	8643.83	1.00783	0.503913	−	0.863755i	$-0.331893\pi$
$419$	8643.83	1.00783	0.503913	+	0.863755i	$0.331893\pi$
$420$	0	0
$421$	16801.3	1.94500	0.972499	−	0.232905i	$-0.0748232\pi$
$421$	16801.3	1.94500	0.972499	+	0.232905i	$0.0748232\pi$
$422$	0	0
$423$	3971.24	0.456474
$424$	0	0
$425$	15602.1	1.78073
$426$	0	0
$427$	−13823.0	−1.56661
$428$	0	0
$429$	−2710.94	−0.305094
$430$	0	0
$431$	−12053.5	−1.34710	−0.673548	−	0.739144i	$-0.735231\pi$
$431$	−12053.5	−1.34710	−0.673548	+	0.739144i	$0.735231\pi$
$432$	0	0
$433$	9034.61	1.00271	0.501357	−	0.865240i	$-0.332834\pi$
$433$	9034.61	1.00271	0.501357	+	0.865240i	$0.332834\pi$
$434$	0	0
$435$	26171.3	2.88464
$436$	0	0
$437$	−3115.56	−0.341047
$438$	0	0
$439$	−3008.87	−0.327120	−0.163560	−	0.986533i	$-0.552298\pi$
$439$	−3008.87	−0.327120	−0.163560	+	0.986533i	$0.552298\pi$
$440$	0	0
$441$	1457.23	0.157351
$442$	0	0
$443$	−229.594	−0.0246237	−0.0123119	−	0.999924i	$-0.503919\pi$
$443$	−229.594	−0.0246237	−0.0123119	+	0.999924i	$0.503919\pi$
$444$	0	0
$445$	6744.70	0.718493
$446$	0	0
$447$	11992.5	1.26896
$448$	0	0
$449$	−7559.44	−0.794548	−0.397274	−	0.917700i	$-0.630044\pi$
$449$	−7559.44	−0.794548	−0.397274	+	0.917700i	$0.630044\pi$
$450$	0	0
$451$	3611.57	0.377078
$452$	0	0
$453$	6896.38	0.715276
$454$	0	0
$455$	−21220.9	−2.18648
$456$	0	0
$457$	11556.4	1.18290	0.591449	−	0.806343i	$-0.298556\pi$
$457$	11556.4	1.18290	0.591449	+	0.806343i	$0.298556\pi$
$458$	0	0
$459$	7381.20	0.750599
$460$	0	0
$461$	−9191.58	−0.928622	−0.464311	−	0.885672i	$-0.653698\pi$
$461$	−9191.58	−0.928622	−0.464311	+	0.885672i	$0.653698\pi$
$462$	0	0
$463$	−1356.03	−0.136113	−0.0680564	−	0.997681i	$-0.521680\pi$
$463$	−1356.03	−0.136113	−0.0680564	+	0.997681i	$0.521680\pi$
$464$	0	0
$465$	−11084.4	−1.10543
$466$	0	0
$467$	−14808.9	−1.46740	−0.733700	−	0.679473i	$-0.762208\pi$
$467$	−14808.9	−1.46740	−0.733700	+	0.679473i	$0.762208\pi$
$468$	0	0
$469$	4008.25	0.394635
$470$	0	0
$471$	−14520.3	−1.42051
$472$	0	0
$473$	1422.49	0.138280
$474$	0	0
$475$	−3990.79	−0.385494
$476$	0	0
$477$	−1503.86	−0.144355
$478$	0	0
$479$	−9834.66	−0.938115	−0.469058	−	0.883168i	$-0.655406\pi$
$479$	−9834.66	−0.938115	−0.469058	+	0.883168i	$0.655406\pi$
$480$	0	0
$481$	15730.1	1.49112
$482$	0	0
$483$	22040.1	2.07632
$484$	0	0
$485$	7805.67	0.730798
$486$	0	0
$487$	−3687.82	−0.343144	−0.171572	−	0.985172i	$-0.554885\pi$
$487$	−3687.82	−0.343144	−0.171572	+	0.985172i	$0.554885\pi$
$488$	0	0
$489$	−5302.00	−0.490317
$490$	0	0
$491$	−11197.4	−1.02919	−0.514593	−	0.857435i	$-0.672057\pi$
$491$	−11197.4	−1.02919	−0.514593	+	0.857435i	$0.672057\pi$
$492$	0	0
$493$	17263.6	1.57711
$494$	0	0
$495$	1648.89	0.149721
$496$	0	0
$497$	1237.78	0.111715
$498$	0	0
$499$	−12101.6	−1.08566	−0.542829	−	0.839843i	$-0.682647\pi$
$499$	−12101.6	−1.08566	−0.542829	+	0.839843i	$0.682647\pi$
$500$	0	0
$501$	10375.5	0.925236
$502$	0	0
$503$	7266.58	0.644136	0.322068	−	0.946716i	$-0.395622\pi$
$503$	7266.58	0.644136	0.322068	+	0.946716i	$0.395622\pi$
$504$	0	0
$505$	−7378.84	−0.650206
$506$	0	0
$507$	−3807.13	−0.333493
$508$	0	0
$509$	−2564.99	−0.223362	−0.111681	−	0.993744i	$-0.535624\pi$
$509$	−2564.99	−0.223362	−0.111681	+	0.993744i	$0.535624\pi$
$510$	0	0
$511$	−1488.98	−0.128902
$512$	0	0
$513$	−1888.00	−0.162490
$514$	0	0
$515$	−20787.6	−1.77866
$516$	0	0
$517$	−3040.00	−0.258606
$518$	0	0
$519$	−19633.9	−1.66057
$520$	0	0
$521$	11254.6	0.946401	0.473201	−	0.880955i	$-0.343099\pi$
$521$	11254.6	0.946401	0.473201	+	0.880955i	$0.343099\pi$
$522$	0	0
$523$	−18357.1	−1.53480	−0.767401	−	0.641168i	$-0.778450\pi$
$523$	−18357.1	−1.53480	−0.767401	+	0.641168i	$0.778450\pi$
$524$	0	0
$525$	28231.6	2.34691
$526$	0	0
$527$	−7311.71	−0.604370
$528$	0	0
$529$	14721.4	1.20995
$530$	0	0
$531$	6213.66	0.507815
$532$	0	0
$533$	23078.4	1.87549
$534$	0	0
$535$	6957.33	0.562227
$536$	0	0
$537$	7563.93	0.607836
$538$	0	0
$539$	−1115.51	−0.0891438
$540$	0	0
$541$	−11381.8	−0.904510	−0.452255	−	0.891889i	$-0.649380\pi$
$541$	−11381.8	−0.904510	−0.452255	+	0.891889i	$0.649380\pi$
$542$	0	0
$543$	−19126.5	−1.51159
$544$	0	0
$545$	21612.4	1.69867
$546$	0	0
$547$	−8996.84	−0.703248	−0.351624	−	0.936141i	$-0.614371\pi$
$547$	−8996.84	−0.703248	−0.351624	+	0.936141i	$0.614371\pi$
$548$	0	0
$549$	6863.39	0.533556
$550$	0	0
$551$	−4415.79	−0.341414
$552$	0	0
$553$	−7271.43	−0.559155
$554$	0	0
$555$	−33380.8	−2.55304
$556$	0	0
$557$	−10759.8	−0.818507	−0.409253	−	0.912421i	$-0.634211\pi$
$557$	−10759.8	−0.818507	−0.409253	+	0.912421i	$0.634211\pi$
$558$	0	0
$559$	9089.90	0.687767
$560$	0	0
$561$	3794.84	0.285594
$562$	0	0
$563$	25381.8	1.90003	0.950015	−	0.312205i	$-0.101068\pi$
$563$	25381.8	1.90003	0.950015	+	0.312205i	$0.101068\pi$
$564$	0	0
$565$	20729.5	1.54354
$566$	0	0
$567$	19755.3	1.46322
$568$	0	0
$569$	−5546.00	−0.408612	−0.204306	−	0.978907i	$-0.565494\pi$
$569$	−5546.00	−0.408612	−0.204306	+	0.978907i	$0.565494\pi$
$570$	0	0
$571$	7714.96	0.565431	0.282715	−	0.959204i	$-0.408765\pi$
$571$	7714.96	0.565431	0.282715	+	0.959204i	$0.408765\pi$
$572$	0	0
$573$	−8708.13	−0.634882
$574$	0	0
$575$	34441.9	2.49796
$576$	0	0
$577$	21335.1	1.53933	0.769665	−	0.638448i	$-0.220423\pi$
$577$	21335.1	1.53933	0.769665	+	0.638448i	$0.220423\pi$
$578$	0	0
$579$	−8879.90	−0.637368
$580$	0	0
$581$	−25188.8	−1.79863
$582$	0	0
$583$	1151.21	0.0817811
$584$	0	0
$585$	10536.6	0.744674
$586$	0	0
$587$	12370.9	0.869846	0.434923	−	0.900468i	$-0.356776\pi$
$587$	12370.9	0.869846	0.434923	+	0.900468i	$0.356776\pi$
$588$	0	0
$589$	1870.23	0.130834
$590$	0	0
$591$	32433.2	2.25740
$592$	0	0
$593$	12982.5	0.899034	0.449517	−	0.893272i	$-0.351596\pi$
$593$	12982.5	0.899034	0.449517	+	0.893272i	$0.351596\pi$
$594$	0	0
$595$	29705.5	2.04674
$596$	0	0
$597$	−15442.8	−1.05868
$598$	0	0
$599$	−13163.7	−0.897921	−0.448960	−	0.893552i	$-0.648206\pi$
$599$	−13163.7	−0.897921	−0.448960	+	0.893552i	$0.648206\pi$
$600$	0	0
$601$	−29325.4	−1.99036	−0.995180	−	0.0980640i	$-0.968735\pi$
$601$	−29325.4	−1.99036	−0.995180	+	0.0980640i	$0.968735\pi$
$602$	0	0
$603$	−1990.17	−0.134405
$604$	0	0
$605$	23100.6	1.55235
$606$	0	0
$607$	2170.11	0.145110	0.0725552	−	0.997364i	$-0.476885\pi$
$607$	2170.11	0.145110	0.0725552	+	0.997364i	$0.476885\pi$
$608$	0	0
$609$	31238.2	2.07855
$610$	0	0
$611$	−19426.0	−1.28624
$612$	0	0
$613$	4917.96	0.324036	0.162018	−	0.986788i	$-0.448200\pi$
$613$	4917.96	0.324036	0.162018	+	0.986788i	$0.448200\pi$
$614$	0	0
$615$	−48974.7	−3.21114
$616$	0	0
$617$	24763.5	1.61579	0.807894	−	0.589328i	$-0.200607\pi$
$617$	24763.5	1.61579	0.807894	+	0.589328i	$0.200607\pi$
$618$	0	0
$619$	−27552.0	−1.78902	−0.894512	−	0.447043i	$-0.852477\pi$
$619$	−27552.0	−1.78902	−0.894512	+	0.447043i	$0.852477\pi$
$620$	0	0
$621$	16294.2	1.05292
$622$	0	0
$623$	8050.51	0.517716
$624$	0	0
$625$	2237.20	0.143181
$626$	0	0
$627$	−970.664	−0.0618255
$628$	0	0
$629$	−22019.3	−1.39582
$630$	0	0
$631$	−377.839	−0.0238376	−0.0119188	−	0.999929i	$-0.503794\pi$
$631$	−377.839	−0.0238376	−0.0119188	+	0.999929i	$0.503794\pi$
$632$	0	0
$633$	−11408.6	−0.716354
$634$	0	0
$635$	−733.012	−0.0458090
$636$	0	0
$637$	−7128.27	−0.443379
$638$	0	0
$639$	−614.584	−0.0380479
$640$	0	0
$641$	−14047.3	−0.865576	−0.432788	−	0.901496i	$-0.642470\pi$
$641$	−14047.3	−0.865576	−0.432788	+	0.901496i	$0.642470\pi$
$642$	0	0
$643$	5768.94	0.353818	0.176909	−	0.984227i	$-0.443390\pi$
$643$	5768.94	0.353818	0.176909	+	0.984227i	$0.443390\pi$
$644$	0	0
$645$	−19289.7	−1.17757
$646$	0	0
$647$	−4568.59	−0.277604	−0.138802	−	0.990320i	$-0.544325\pi$
$647$	−4568.59	−0.277604	−0.138802	+	0.990320i	$0.544325\pi$
$648$	0	0
$649$	−4756.58	−0.287692
$650$	0	0
$651$	−13230.4	−0.796528
$652$	0	0
$653$	16532.7	0.990774	0.495387	−	0.868672i	$-0.335026\pi$
$653$	16532.7	0.990774	0.495387	+	0.868672i	$0.335026\pi$
$654$	0	0
$655$	3256.26	0.194248
$656$	0	0
$657$	739.309	0.0439014
$658$	0	0
$659$	−18630.0	−1.10125	−0.550624	−	0.834753i	$-0.685610\pi$
$659$	−18630.0	−1.10125	−0.550624	+	0.834753i	$0.685610\pi$
$660$	0	0
$661$	−28851.3	−1.69771	−0.848854	−	0.528627i	$-0.822707\pi$
$661$	−28851.3	−1.69771	−0.848854	+	0.528627i	$0.822707\pi$
$662$	0	0
$663$	24249.5	1.42047
$664$	0	0
$665$	−7598.24	−0.443079
$666$	0	0
$667$	38109.9	2.21232
$668$	0	0
$669$	11568.0	0.668525
$670$	0	0
$671$	−5253.95	−0.302275
$672$	0	0
$673$	−3873.41	−0.221856	−0.110928	−	0.993828i	$-0.535382\pi$
$673$	−3873.41	−0.221856	−0.110928	+	0.993828i	$0.535382\pi$
$674$	0	0
$675$	20871.5	1.19014
$676$	0	0
$677$	−5025.24	−0.285282	−0.142641	−	0.989775i	$-0.545559\pi$
$677$	−5025.24	−0.285282	−0.142641	+	0.989775i	$0.545559\pi$
$678$	0	0
$679$	9316.90	0.526583
$680$	0	0
$681$	−11069.0	−0.622855
$682$	0	0
$683$	−14157.0	−0.793122	−0.396561	−	0.918008i	$-0.629796\pi$
$683$	−14157.0	−0.793122	−0.396561	+	0.918008i	$0.629796\pi$
$684$	0	0
$685$	453.217	0.0252796
$686$	0	0
$687$	29747.4	1.65201
$688$	0	0
$689$	7356.40	0.406758
$690$	0	0
$691$	−2437.65	−0.134200	−0.0671002	−	0.997746i	$-0.521375\pi$
$691$	−2437.65	−0.134200	−0.0671002	+	0.997746i	$0.521375\pi$
$692$	0	0
$693$	1968.12	0.107883
$694$	0	0
$695$	−52481.7	−2.86438
$696$	0	0
$697$	−32305.7	−1.75562
$698$	0	0
$699$	−5323.38	−0.288052
$700$	0	0
$701$	−9496.96	−0.511691	−0.255845	−	0.966718i	$-0.582354\pi$
$701$	−9496.96	−0.511691	−0.255845	+	0.966718i	$0.582354\pi$
$702$	0	0
$703$	5632.23	0.302167
$704$	0	0
$705$	41223.9	2.20225
$706$	0	0
$707$	−8807.43	−0.468511
$708$	0	0
$709$	−6350.33	−0.336377	−0.168189	−	0.985755i	$-0.553792\pi$
$709$	−6350.33	−0.336377	−0.168189	+	0.985755i	$0.553792\pi$
$710$	0	0
$711$	3610.41	0.190437
$712$	0	0
$713$	−16140.7	−0.847792
$714$	0	0
$715$	−8065.80	−0.421879
$716$	0	0
$717$	−29310.1	−1.52665
$718$	0	0
$719$	19722.6	1.02299	0.511496	−	0.859286i	$-0.329092\pi$
$719$	19722.6	1.02299	0.511496	+	0.859286i	$0.329092\pi$
$720$	0	0
$721$	−24812.2	−1.28163
$722$	0	0
$723$	−3785.59	−0.194727
$724$	0	0
$725$	48815.7	2.50065
$726$	0	0
$727$	−28325.8	−1.44504	−0.722520	−	0.691350i	$-0.757016\pi$
$727$	−28325.8	−1.44504	−0.722520	+	0.691350i	$0.757016\pi$
$728$	0	0
$729$	6696.82	0.340234
$730$	0	0
$731$	−12724.3	−0.643809
$732$	0	0
$733$	−33981.8	−1.71234	−0.856170	−	0.516694i	$-0.827163\pi$
$733$	−33981.8	−1.71234	−0.856170	+	0.516694i	$0.827163\pi$
$734$	0	0
$735$	15126.9	0.759135
$736$	0	0
$737$	1523.49	0.0761443
$738$	0	0
$739$	9147.10	0.455320	0.227660	−	0.973741i	$-0.426893\pi$
$739$	9147.10	0.455320	0.227660	+	0.973741i	$0.426893\pi$
$740$	0	0
$741$	−6202.67	−0.307504
$742$	0	0
$743$	34159.2	1.68665	0.843324	−	0.537405i	$-0.180596\pi$
$743$	34159.2	1.68665	0.843324	+	0.537405i	$0.180596\pi$
$744$	0	0
$745$	35681.1	1.75470
$746$	0	0
$747$	12506.7	0.612579
$748$	0	0
$749$	8304.31	0.405117
$750$	0	0
$751$	−28361.8	−1.37808	−0.689038	−	0.724725i	$-0.741967\pi$
$751$	−28361.8	−1.37808	−0.689038	+	0.724725i	$0.741967\pi$
$752$	0	0
$753$	10200.6	0.493666
$754$	0	0
$755$	20518.7	0.989074
$756$	0	0
$757$	−11464.9	−0.550462	−0.275231	−	0.961378i	$-0.588754\pi$
$757$	−11464.9	−0.550462	−0.275231	+	0.961378i	$0.588754\pi$
$758$	0	0
$759$	8377.18	0.400623
$760$	0	0
$761$	14289.3	0.680666	0.340333	−	0.940305i	$-0.389460\pi$
$761$	14289.3	0.680666	0.340333	+	0.940305i	$0.389460\pi$
$762$	0	0
$763$	25796.7	1.22399
$764$	0	0
$765$	−14749.4	−0.697079
$766$	0	0
$767$	−30395.2	−1.43091
$768$	0	0
$769$	100.811	0.00472734	0.00236367	−	0.999997i	$-0.499248\pi$
$769$	100.811	0.00472734	0.00236367	+	0.999997i	$0.499248\pi$
$770$	0	0
$771$	−21200.8	−0.990307
$772$	0	0
$773$	−52.1073	−0.00242454	−0.00121227	−	0.999999i	$-0.500386\pi$
$773$	−52.1073	−0.00242454	−0.00121227	+	0.999999i	$0.500386\pi$
$774$	0	0
$775$	−20675.0	−0.958282
$776$	0	0
$777$	−39843.6	−1.83961
$778$	0	0
$779$	8263.34	0.380057
$780$	0	0
$781$	470.467	0.0215552
$782$	0	0
$783$	23094.3	1.05405
$784$	0	0
$785$	−43201.9	−1.96426
$786$	0	0
$787$	15261.8	0.691266	0.345633	−	0.938370i	$-0.387664\pi$
$787$	15261.8	0.691266	0.345633	+	0.938370i	$0.387664\pi$
$788$	0	0
$789$	35410.0	1.59776
$790$	0	0
$791$	24742.9	1.11221
$792$	0	0
$793$	−33573.4	−1.50344
$794$	0	0
$795$	−15611.0	−0.696436
$796$	0	0
$797$	−24432.3	−1.08587	−0.542934	−	0.839775i	$-0.682687\pi$
$797$	−24432.3	−1.08587	−0.542934	+	0.839775i	$0.682687\pi$
$798$	0	0
$799$	27193.0	1.20403
$800$	0	0
$801$	−3997.24	−0.176324
$802$	0	0
$803$	−565.944	−0.0248714
$804$	0	0
$805$	65575.6	2.87110
$806$	0	0
$807$	13886.9	0.605754
$808$	0	0
$809$	3635.86	0.158010	0.0790050	−	0.996874i	$-0.474826\pi$
$809$	3635.86	0.158010	0.0790050	+	0.996874i	$0.474826\pi$
$810$	0	0
$811$	1087.93	0.0471051	0.0235526	−	0.999723i	$-0.492502\pi$
$811$	1087.93	0.0471051	0.0235526	+	0.999723i	$0.492502\pi$
$812$	0	0
$813$	43139.1	1.86095
$814$	0	0
$815$	−15775.0	−0.678003
$816$	0	0
$817$	3254.69	0.139372
$818$	0	0
$819$	12576.5	0.536581
$820$	0	0
$821$	−10953.7	−0.465636	−0.232818	−	0.972520i	$-0.574795\pi$
$821$	−10953.7	−0.465636	−0.232818	+	0.972520i	$0.574795\pi$
$822$	0	0
$823$	−13502.5	−0.571893	−0.285947	−	0.958246i	$-0.592308\pi$
$823$	−13502.5	−0.571893	−0.285947	+	0.958246i	$0.592308\pi$
$824$	0	0
$825$	10730.5	0.452834
$826$	0	0
$827$	26812.4	1.12740	0.563699	−	0.825980i	$-0.309378\pi$
$827$	26812.4	1.12740	0.563699	+	0.825980i	$0.309378\pi$
$828$	0	0
$829$	4497.94	0.188444	0.0942218	−	0.995551i	$-0.469964\pi$
$829$	4497.94	0.188444	0.0942218	+	0.995551i	$0.469964\pi$
$830$	0	0
$831$	−2293.55	−0.0957430
$832$	0	0
$833$	9978.33	0.415040
$834$	0	0
$835$	30870.0	1.27940
$836$	0	0
$837$	−9781.16	−0.403926
$838$	0	0
$839$	−30324.9	−1.24783	−0.623917	−	0.781491i	$-0.714460\pi$
$839$	−30324.9	−1.24783	−0.623917	+	0.781491i	$0.714460\pi$
$840$	0	0
$841$	29625.4	1.21470
$842$	0	0
$843$	11621.0	0.474790
$844$	0	0
$845$	−11327.3	−0.461149
$846$	0	0
$847$	27573.0	1.11856
$848$	0	0
$849$	23897.3	0.966022
$850$	0	0
$851$	−48608.2	−1.95801
$852$	0	0
$853$	−17230.8	−0.691642	−0.345821	−	0.938300i	$-0.612400\pi$
$853$	−17230.8	−0.691642	−0.345821	+	0.938300i	$0.612400\pi$
$854$	0	0
$855$	3772.68	0.150904
$856$	0	0
$857$	−6724.12	−0.268018	−0.134009	−	0.990980i	$-0.542785\pi$
$857$	−6724.12	−0.268018	−0.134009	+	0.990980i	$0.542785\pi$
$858$	0	0
$859$	−36183.5	−1.43721	−0.718606	−	0.695417i	$-0.755219\pi$
$859$	−36183.5	−1.43721	−0.718606	+	0.695417i	$0.755219\pi$
$860$	0	0
$861$	−58456.6	−2.31381
$862$	0	0
$863$	−14350.8	−0.566056	−0.283028	−	0.959112i	$-0.591339\pi$
$863$	−14350.8	−0.566056	−0.283028	+	0.959112i	$0.591339\pi$
$864$	0	0
$865$	−58416.5	−2.29621
$866$	0	0
$867$	−3719.96	−0.145717
$868$	0	0
$869$	−2763.78	−0.107888
$870$	0	0
$871$	9735.27	0.378722
$872$	0	0
$873$	−4626.02	−0.179344
$874$	0	0
$875$	34008.7	1.31395
$876$	0	0
$877$	42978.6	1.65483	0.827414	−	0.561592i	$-0.189811\pi$
$877$	42978.6	1.65483	0.827414	+	0.561592i	$0.189811\pi$
$878$	0	0
$879$	−7831.65	−0.300518
$880$	0	0
$881$	−7298.75	−0.279116	−0.139558	−	0.990214i	$-0.544568\pi$
$881$	−7298.75	−0.279116	−0.139558	+	0.990214i	$0.544568\pi$
$882$	0	0
$883$	46722.1	1.78066	0.890331	−	0.455314i	$-0.150473\pi$
$883$	46722.1	1.78066	0.890331	+	0.455314i	$0.150473\pi$
$884$	0	0
$885$	64501.7	2.44994
$886$	0	0
$887$	−35268.7	−1.33507	−0.667535	−	0.744578i	$-0.732651\pi$
$887$	−35268.7	−1.33507	−0.667535	+	0.744578i	$0.732651\pi$
$888$	0	0
$889$	−874.928	−0.0330080
$890$	0	0
$891$	7508.74	0.282326
$892$	0	0
$893$	−6955.57	−0.260649
$894$	0	0
$895$	22504.8	0.840506
$896$	0	0
$897$	53531.3	1.99259
$898$	0	0
$899$	−22876.8	−0.848704
$900$	0	0
$901$	−10297.7	−0.380761
$902$	0	0
$903$	−23024.3	−0.848507
$904$	0	0
$905$	−56906.6	−2.09021
$906$	0	0
$907$	−46292.7	−1.69474	−0.847368	−	0.531007i	$-0.821814\pi$
$907$	−46292.7	−1.69474	−0.847368	+	0.531007i	$0.821814\pi$
$908$	0	0
$909$	4373.06	0.159566
$910$	0	0
$911$	42085.5	1.53058	0.765288	−	0.643688i	$-0.222597\pi$
$911$	42085.5	1.53058	0.765288	+	0.643688i	$0.222597\pi$
$912$	0	0
$913$	−9573.94	−0.347044
$914$	0	0
$915$	71246.2	2.57413
$916$	0	0
$917$	3886.70	0.139967
$918$	0	0
$919$	−21331.8	−0.765691	−0.382845	−	0.923812i	$-0.625056\pi$
$919$	−21331.8	−0.765691	−0.382845	+	0.923812i	$0.625056\pi$
$920$	0	0
$921$	−5039.49	−0.180300
$922$	0	0
$923$	3006.34	0.107210
$924$	0	0
$925$	−62263.2	−2.21319
$926$	0	0
$927$	12319.7	0.436498
$928$	0	0
$929$	−34331.5	−1.21247	−0.606233	−	0.795287i	$-0.707320\pi$
$929$	−34331.5	−1.21247	−0.606233	+	0.795287i	$0.707320\pi$
$930$	0	0
$931$	−2552.31	−0.0898481
$932$	0	0
$933$	12948.0	0.454341
$934$	0	0
$935$	11290.7	0.394915
$936$	0	0
$937$	13625.1	0.475041	0.237521	−	0.971383i	$-0.423665\pi$
$937$	13625.1	0.475041	0.237521	+	0.971383i	$0.423665\pi$
$938$	0	0
$939$	−14737.1	−0.512168
$940$	0	0
$941$	−7086.48	−0.245497	−0.122748	−	0.992438i	$-0.539171\pi$
$941$	−7086.48	−0.245497	−0.122748	+	0.992438i	$0.539171\pi$
$942$	0	0
$943$	−71315.6	−2.46273
$944$	0	0
$945$	39738.3	1.36792
$946$	0	0
$947$	38735.9	1.32920	0.664598	−	0.747202i	$-0.268603\pi$
$947$	38735.9	1.32920	0.664598	+	0.747202i	$0.268603\pi$
$948$	0	0
$949$	−3616.45	−0.123704
$950$	0	0
$951$	13277.6	0.452740
$952$	0	0
$953$	30964.3	1.05250	0.526249	−	0.850330i	$-0.323598\pi$
$953$	30964.3	1.05250	0.526249	+	0.850330i	$0.323598\pi$
$954$	0	0
$955$	−25909.1	−0.877906
$956$	0	0
$957$	11873.3	0.401053
$958$	0	0
$959$	540.962	0.0182154
$960$	0	0
$961$	−20101.9	−0.674765
$962$	0	0
$963$	−4123.26	−0.137975
$964$	0	0
$965$	−26420.2	−0.881344
$966$	0	0
$967$	−10968.0	−0.364743	−0.182371	−	0.983230i	$-0.558377\pi$
$967$	−10968.0	−0.364743	−0.182371	+	0.983230i	$0.558377\pi$
$968$	0	0
$969$	8682.65	0.287850
$970$	0	0
$971$	−30081.2	−0.994183	−0.497091	−	0.867698i	$-0.665599\pi$
$971$	−30081.2	−0.994183	−0.497091	+	0.867698i	$0.665599\pi$
$972$	0	0
$973$	−62642.5	−2.06395
$974$	0	0
$975$	68569.3	2.25228
$976$	0	0
$977$	26628.1	0.871962	0.435981	−	0.899956i	$-0.356401\pi$
$977$	26628.1	0.871962	0.435981	+	0.899956i	$0.356401\pi$
$978$	0	0
$979$	3059.90	0.0998926
$980$	0	0
$981$	−12808.6	−0.416867
$982$	0	0
$983$	−25495.3	−0.827236	−0.413618	−	0.910450i	$-0.635735\pi$
$983$	−25495.3	−0.827236	−0.413618	+	0.910450i	$0.635735\pi$
$984$	0	0
$985$	96498.0	3.12150
$986$	0	0
$987$	49205.2	1.58685
$988$	0	0
$989$	−28089.1	−0.903116
$990$	0	0
$991$	54385.4	1.74330	0.871649	−	0.490130i	$-0.163051\pi$
$991$	54385.4	1.74330	0.871649	+	0.490130i	$0.163051\pi$
$992$	0	0
$993$	−3185.43	−0.101799
$994$	0	0
$995$	−45946.8	−1.46393
$996$	0	0
$997$	36846.5	1.17045	0.585227	−	0.810870i	$-0.301006\pi$
$997$	36846.5	1.17045	0.585227	+	0.810870i	$0.301006\pi$
$998$	0	0
$999$	−29456.1	−0.932884

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.l.1.1		2
4.3	odd	2		1216.4.a.j.1.2		2
8.3	odd	2		38.4.a.b.1.1	✓	2
8.5	even	2		304.4.a.d.1.2		2
24.11	even	2		342.4.a.k.1.1		2
40.3	even	4		950.4.b.g.799.3		4
40.19	odd	2		950.4.a.h.1.2		2
40.27	even	4		950.4.b.g.799.2		4
56.27	even	2		1862.4.a.b.1.2		2
152.75	even	2		722.4.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.4.a.b.1.1	✓	2	8.3	odd	2
304.4.a.d.1.2		2	8.5	even	2
342.4.a.k.1.1		2	24.11	even	2
722.4.a.i.1.2		2	152.75	even	2
950.4.a.h.1.2		2	40.19	odd	2
950.4.b.g.799.2		4	40.27	even	4
950.4.b.g.799.3		4	40.3	even	4
1216.4.a.j.1.2		2	4.3	odd	2
1216.4.a.l.1.1		2	1.1	even	1	trivial
1862.4.a.b.1.2		2	56.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

\(f(q)\)	\(=\)	\(q-6.15207 q^{3} -18.3041 q^{5} -21.8479 q^{7} +10.8479 q^{9} +O(q^{10})\)	\(q-6.15207 q^{3} -18.3041 q^{5} -21.8479 q^{7} +10.8479 q^{9} -8.30413 q^{11} -53.0645 q^{13} +112.608 q^{15} +74.2810 q^{17} -19.0000 q^{19} +134.410 q^{21} +163.977 q^{23} +210.041 q^{25} +99.3686 q^{27} +232.410 q^{29} -98.4331 q^{31} +51.0876 q^{33} +399.908 q^{35} -296.433 q^{37} +326.456 q^{39} -434.912 q^{41} -171.299 q^{43} -198.562 q^{45} +366.083 q^{47} +134.332 q^{49} -456.982 q^{51} -138.631 q^{53} +152.000 q^{55} +116.889 q^{57} +572.797 q^{59} +632.691 q^{61} -237.005 q^{63} +971.299 q^{65} -183.461 q^{67} -1008.80 q^{69} -56.6545 q^{71} +68.1521 q^{73} -1292.19 q^{75} +181.428 q^{77} +332.820 q^{79} -904.217 q^{81} +1152.91 q^{83} -1359.65 q^{85} -1429.80 q^{87} -368.479 q^{89} +1159.35 q^{91} +605.567 q^{93} +347.779 q^{95} -426.443 q^{97} -90.0827 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 10 q^{5} - 57 q^{7} + 35 q^{9}+O(q^{10}) \) 2 * q + q^3 - 10 * q^5 - 57 * q^7 + 35 * q^9	\( 2 q + q^{3} - 10 q^{5} - 57 q^{7} + 35 q^{9} + 10 q^{11} - 13 q^{13} + 172 q^{15} - 51 q^{17} - 38 q^{19} - 117 q^{21} + 155 q^{23} + 154 q^{25} + 79 q^{27} + 79 q^{29} + 16 q^{31} + 182 q^{33} + 108 q^{35} - 380 q^{37} + 613 q^{39} - 790 q^{41} + 296 q^{43} + 2 q^{45} + 200 q^{47} + 1027 q^{49} - 1353 q^{51} - 397 q^{53} + 304 q^{55} - 19 q^{57} + 201 q^{59} + 680 q^{61} - 1086 q^{63} + 1304 q^{65} - 939 q^{67} - 1073 q^{69} - 406 q^{71} + 123 q^{73} - 1693 q^{75} - 462 q^{77} - 106 q^{79} - 1702 q^{81} + 2226 q^{83} - 2400 q^{85} - 2527 q^{87} - 870 q^{89} - 249 q^{91} + 1424 q^{93} + 190 q^{95} - 1864 q^{97} + 352 q^{99}+O(q^{100}) \) 2 * q + q^3 - 10 * q^5 - 57 * q^7 + 35 * q^9 + 10 * q^11 - 13 * q^13 + 172 * q^15 - 51 * q^17 - 38 * q^19 - 117 * q^21 + 155 * q^23 + 154 * q^25 + 79 * q^27 + 79 * q^29 + 16 * q^31 + 182 * q^33 + 108 * q^35 - 380 * q^37 + 613 * q^39 - 790 * q^41 + 296 * q^43 + 2 * q^45 + 200 * q^47 + 1027 * q^49 - 1353 * q^51 - 397 * q^53 + 304 * q^55 - 19 * q^57 + 201 * q^59 + 680 * q^61 - 1086 * q^63 + 1304 * q^65 - 939 * q^67 - 1073 * q^69 - 406 * q^71 + 123 * q^73 - 1693 * q^75 - 462 * q^77 - 106 * q^79 - 1702 * q^81 + 2226 * q^83 - 2400 * q^85 - 2527 * q^87 - 870 * q^89 - 249 * q^91 + 1424 * q^93 + 190 * q^95 - 1864 * q^97 + 352 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more