LMFDB - Embedded newform 1088.4.a.bg.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1088,4,Mod(1,1088)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1088, 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("1088.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1088 = 2^{6} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1088.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:	$64.1940780862$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 67x^{5} - 35x^{4} + 893x^{3} + 595x^{2} - 3064x - 2804 $ x^7 - x^6 - 67x^5 - 35x^4 + 893x^3 + 595x^2 - 3064*x - 2804
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 544)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.90461$ of defining polynomial
Character	$\chi$	$=$	1088.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.01688 q^{3} +14.1958 q^{5} -9.87960 q^{7} -10.8647 q^{9} +O(q^{10})$	$q+4.01688 q^{3} +14.1958 q^{5} -9.87960 q^{7} -10.8647 q^{9} +60.2213 q^{11} +51.2168 q^{13} +57.0229 q^{15} -17.0000 q^{17} +34.2361 q^{19} -39.6851 q^{21} +39.8677 q^{23} +76.5216 q^{25} -152.098 q^{27} +196.276 q^{29} -182.486 q^{31} +241.902 q^{33} -140.249 q^{35} +255.616 q^{37} +205.732 q^{39} -194.494 q^{41} -150.979 q^{43} -154.233 q^{45} +350.840 q^{47} -245.394 q^{49} -68.2869 q^{51} -71.9534 q^{53} +854.892 q^{55} +137.522 q^{57} -142.346 q^{59} +610.499 q^{61} +107.339 q^{63} +727.065 q^{65} -368.545 q^{67} +160.144 q^{69} +710.184 q^{71} +1015.72 q^{73} +307.378 q^{75} -594.962 q^{77} -374.294 q^{79} -317.611 q^{81} +773.711 q^{83} -241.329 q^{85} +788.418 q^{87} +1052.83 q^{89} -506.001 q^{91} -733.023 q^{93} +486.010 q^{95} -1596.33 q^{97} -654.287 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{7} + 67 q^{9}+O(q^{10}) $ 7 * q - 2 * q^7 + 67 * q^9	$ 7 q - 2 q^{7} + 67 q^{9} - 108 q^{11} + 34 q^{13} + 128 q^{15} - 119 q^{17} - 124 q^{19} + 296 q^{21} - 6 q^{23} + 197 q^{25} - 248 q^{29} + 50 q^{31} + 512 q^{33} - 640 q^{35} + 484 q^{37} + 1504 q^{39} - 366 q^{41} - 1412 q^{43} + 80 q^{45} + 1012 q^{47} + 1115 q^{49} + 146 q^{53} + 1024 q^{55} + 48 q^{57} - 2332 q^{59} + 548 q^{61} + 2838 q^{63} - 208 q^{65} - 924 q^{67} + 1672 q^{69} + 1286 q^{71} + 870 q^{73} - 3136 q^{75} + 1344 q^{77} + 1818 q^{79} + 3039 q^{81} - 1772 q^{83} + 384 q^{87} + 1706 q^{89} - 588 q^{91} + 5576 q^{93} + 2048 q^{95} + 1802 q^{97} - 5148 q^{99}+O(q^{100}) $ 7 * q - 2 * q^7 + 67 * q^9 - 108 * q^11 + 34 * q^13 + 128 * q^15 - 119 * q^17 - 124 * q^19 + 296 * q^21 - 6 * q^23 + 197 * q^25 - 248 * q^29 + 50 * q^31 + 512 * q^33 - 640 * q^35 + 484 * q^37 + 1504 * q^39 - 366 * q^41 - 1412 * q^43 + 80 * q^45 + 1012 * q^47 + 1115 * q^49 + 146 * q^53 + 1024 * q^55 + 48 * q^57 - 2332 * q^59 + 548 * q^61 + 2838 * q^63 - 208 * q^65 - 924 * q^67 + 1672 * q^69 + 1286 * q^71 + 870 * q^73 - 3136 * q^75 + 1344 * q^77 + 1818 * q^79 + 3039 * q^81 - 1772 * q^83 + 384 * q^87 + 1706 * q^89 - 588 * q^91 + 5576 * q^93 + 2048 * q^95 + 1802 * q^97 - 5148 * 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
$3$	4.01688	0.773048	0.386524	−	0.922279i	$-0.373676\pi$
$3$	4.01688	0.773048	0.386524	+	0.922279i	$0.373676\pi$
$4$	0	0
$5$	14.1958	1.26971	0.634857	−	0.772630i	$-0.281059\pi$
$5$	14.1958	1.26971	0.634857	+	0.772630i	$0.281059\pi$
$6$	0	0
$7$	−9.87960	−0.533448	−0.266724	−	0.963773i	$-0.585941\pi$
$7$	−9.87960	−0.533448	−0.266724	+	0.963773i	$0.585941\pi$
$8$	0	0
$9$	−10.8647	−0.402396
$10$	0	0
$11$	60.2213	1.65067	0.825337	−	0.564640i	$-0.190985\pi$
$11$	60.2213	1.65067	0.825337	+	0.564640i	$0.190985\pi$
$12$	0	0
$13$	51.2168	1.09269	0.546346	−	0.837560i	$-0.316019\pi$
$13$	51.2168	1.09269	0.546346	+	0.837560i	$0.316019\pi$
$14$	0	0
$15$	57.0229	0.981550
$16$	0	0
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	34.2361	0.413384	0.206692	−	0.978406i	$-0.433730\pi$
$19$	34.2361	0.413384	0.206692	+	0.978406i	$0.433730\pi$
$20$	0	0
$21$	−39.6851	−0.412381
$22$	0	0
$23$	39.8677	0.361435	0.180717	−	0.983535i	$-0.442158\pi$
$23$	39.8677	0.361435	0.180717	+	0.983535i	$0.442158\pi$
$24$	0	0
$25$	76.5216	0.612173
$26$	0	0
$27$	−152.098	−1.08412
$28$	0	0
$29$	196.276	1.25681	0.628407	−	0.777885i	$-0.283707\pi$
$29$	196.276	1.25681	0.628407	+	0.777885i	$0.283707\pi$
$30$	0	0
$31$	−182.486	−1.05727	−0.528636	−	0.848849i	$-0.677296\pi$
$31$	−182.486	−1.05727	−0.528636	+	0.848849i	$0.677296\pi$
$32$	0	0
$33$	241.902	1.27605
$34$	0	0
$35$	−140.249	−0.677326
$36$	0	0
$37$	255.616	1.13576	0.567878	−	0.823113i	$-0.307765\pi$
$37$	255.616	1.13576	0.567878	+	0.823113i	$0.307765\pi$
$38$	0	0
$39$	205.732	0.844703
$40$	0	0
$41$	−194.494	−0.740848	−0.370424	−	0.928863i	$-0.620788\pi$
$41$	−194.494	−0.740848	−0.370424	+	0.928863i	$0.620788\pi$
$42$	0	0
$43$	−150.979	−0.535443	−0.267721	−	0.963496i	$-0.586271\pi$
$43$	−150.979	−0.535443	−0.267721	+	0.963496i	$0.586271\pi$
$44$	0	0
$45$	−154.233	−0.510928
$46$	0	0
$47$	350.840	1.08884	0.544418	−	0.838814i	$-0.316750\pi$
$47$	350.840	1.08884	0.544418	+	0.838814i	$0.316750\pi$
$48$	0	0
$49$	−245.394	−0.715433
$50$	0	0
$51$	−68.2869	−0.187492
$52$	0	0
$53$	−71.9534	−0.186482	−0.0932412	−	0.995644i	$-0.529723\pi$
$53$	−71.9534	−0.186482	−0.0932412	+	0.995644i	$0.529723\pi$
$54$	0	0
$55$	854.892	2.09588
$56$	0	0
$57$	137.522	0.319566
$58$	0	0
$59$	−142.346	−0.314099	−0.157049	−	0.987591i	$-0.550198\pi$
$59$	−142.346	−0.314099	−0.157049	+	0.987591i	$0.550198\pi$
$60$	0	0
$61$	610.499	1.28142	0.640708	−	0.767785i	$-0.278641\pi$
$61$	610.499	1.28142	0.640708	+	0.767785i	$0.278641\pi$
$62$	0	0
$63$	107.339	0.214658
$64$	0	0
$65$	727.065	1.38741
$66$	0	0
$67$	−368.545	−0.672013	−0.336007	−	0.941860i	$-0.609076\pi$
$67$	−368.545	−0.672013	−0.336007	+	0.941860i	$0.609076\pi$
$68$	0	0
$69$	160.144	0.279406
$70$	0	0
$71$	710.184	1.18709	0.593544	−	0.804801i	$-0.297728\pi$
$71$	710.184	1.18709	0.593544	+	0.804801i	$0.297728\pi$
$72$	0	0
$73$	1015.72	1.62851	0.814254	−	0.580509i	$-0.197146\pi$
$73$	1015.72	1.62851	0.814254	+	0.580509i	$0.197146\pi$
$74$	0	0
$75$	307.378	0.473239
$76$	0	0
$77$	−594.962	−0.880549
$78$	0	0
$79$	−374.294	−0.533055	−0.266528	−	0.963827i	$-0.585876\pi$
$79$	−374.294	−0.533055	−0.266528	+	0.963827i	$0.585876\pi$
$80$	0	0
$81$	−317.611	−0.435681
$82$	0	0
$83$	773.711	1.02320	0.511601	−	0.859223i	$-0.329052\pi$
$83$	773.711	1.02320	0.511601	+	0.859223i	$0.329052\pi$
$84$	0	0
$85$	−241.329	−0.307951
$86$	0	0
$87$	788.418	0.971578
$88$	0	0
$89$	1052.83	1.25393	0.626966	−	0.779046i	$-0.284296\pi$
$89$	1052.83	1.25393	0.626966	+	0.779046i	$0.284296\pi$
$90$	0	0
$91$	−506.001	−0.582894
$92$	0	0
$93$	−733.023	−0.817322
$94$	0	0
$95$	486.010	0.524880
$96$	0	0
$97$	−1596.33	−1.67096	−0.835479	−	0.549523i	$-0.814810\pi$
$97$	−1596.33	−1.67096	−0.835479	+	0.549523i	$0.814810\pi$
$98$	0	0
$99$	−654.287	−0.664225
$100$	0	0
$101$	1355.25	1.33517	0.667587	−	0.744532i	$-0.267327\pi$
$101$	1355.25	1.33517	0.667587	+	0.744532i	$0.267327\pi$
$102$	0	0
$103$	−1035.23	−0.990330	−0.495165	−	0.868799i	$-0.664892\pi$
$103$	−1035.23	−0.990330	−0.495165	+	0.868799i	$0.664892\pi$
$104$	0	0
$105$	−563.363	−0.523606
$106$	0	0
$107$	154.641	0.139717	0.0698585	−	0.997557i	$-0.477745\pi$
$107$	154.641	0.139717	0.0698585	+	0.997557i	$0.477745\pi$
$108$	0	0
$109$	427.905	0.376017	0.188009	−	0.982167i	$-0.439797\pi$
$109$	427.905	0.376017	0.188009	+	0.982167i	$0.439797\pi$
$110$	0	0
$111$	1026.78	0.877994
$112$	0	0
$113$	−772.563	−0.643156	−0.321578	−	0.946883i	$-0.604213\pi$
$113$	−772.563	−0.643156	−0.321578	+	0.946883i	$0.604213\pi$
$114$	0	0
$115$	565.956	0.458918
$116$	0	0
$117$	−556.455	−0.439695
$118$	0	0
$119$	167.953	0.129380
$120$	0	0
$121$	2295.61	1.72473
$122$	0	0
$123$	−781.257	−0.572712
$124$	0	0
$125$	−688.191	−0.492429
$126$	0	0
$127$	−1179.19	−0.823907	−0.411954	−	0.911205i	$-0.635153\pi$
$127$	−1179.19	−0.823907	−0.411954	+	0.911205i	$0.635153\pi$
$128$	0	0
$129$	−606.463	−0.413923
$130$	0	0
$131$	−2519.21	−1.68019	−0.840093	−	0.542442i	$-0.817500\pi$
$131$	−2519.21	−1.68019	−0.840093	+	0.542442i	$0.817500\pi$
$132$	0	0
$133$	−338.239	−0.220519
$134$	0	0
$135$	−2159.16	−1.37652
$136$	0	0
$137$	519.136	0.323743	0.161872	−	0.986812i	$-0.448247\pi$
$137$	519.136	0.323743	0.161872	+	0.986812i	$0.448247\pi$
$138$	0	0
$139$	−1381.62	−0.843075	−0.421537	−	0.906811i	$-0.638509\pi$
$139$	−1381.62	−0.843075	−0.421537	+	0.906811i	$0.638509\pi$
$140$	0	0
$141$	1409.28	0.841723
$142$	0	0
$143$	3084.34	1.80368
$144$	0	0
$145$	2786.31	1.59579
$146$	0	0
$147$	−985.716	−0.553064
$148$	0	0
$149$	1914.19	1.05246	0.526230	−	0.850343i	$-0.323605\pi$
$149$	1914.19	1.05246	0.526230	+	0.850343i	$0.323605\pi$
$150$	0	0
$151$	−1591.12	−0.857506	−0.428753	−	0.903422i	$-0.641047\pi$
$151$	−1591.12	−0.857506	−0.428753	+	0.903422i	$0.641047\pi$
$152$	0	0
$153$	184.700	0.0975954
$154$	0	0
$155$	−2590.54	−1.34243
$156$	0	0
$157$	1017.70	0.517331	0.258665	−	0.965967i	$-0.416717\pi$
$157$	1017.70	0.517331	0.258665	+	0.965967i	$0.416717\pi$
$158$	0	0
$159$	−289.028	−0.144160
$160$	0	0
$161$	−393.877	−0.192807
$162$	0	0
$163$	3366.72	1.61780	0.808900	−	0.587946i	$-0.200063\pi$
$163$	3366.72	1.61780	0.808900	+	0.587946i	$0.200063\pi$
$164$	0	0
$165$	3434.00	1.62022
$166$	0	0
$167$	−2328.20	−1.07881	−0.539405	−	0.842046i	$-0.681351\pi$
$167$	−2328.20	−1.07881	−0.539405	+	0.842046i	$0.681351\pi$
$168$	0	0
$169$	426.162	0.193974
$170$	0	0
$171$	−371.965	−0.166344
$172$	0	0
$173$	2292.81	1.00762	0.503812	−	0.863813i	$-0.331930\pi$
$173$	2292.81	1.00762	0.503812	+	0.863813i	$0.331930\pi$
$174$	0	0
$175$	−756.003	−0.326562
$176$	0	0
$177$	−571.785	−0.242814
$178$	0	0
$179$	1836.72	0.766943	0.383472	−	0.923553i	$-0.374728\pi$
$179$	1836.72	0.766943	0.383472	+	0.923553i	$0.374728\pi$
$180$	0	0
$181$	2746.26	1.12778	0.563889	−	0.825850i	$-0.309304\pi$
$181$	2746.26	1.12778	0.563889	+	0.825850i	$0.309304\pi$
$182$	0	0
$183$	2452.30	0.990596
$184$	0	0
$185$	3628.68	1.44208
$186$	0	0
$187$	−1023.76	−0.400347
$188$	0	0
$189$	1502.67	0.578322
$190$	0	0
$191$	3701.09	1.40210	0.701051	−	0.713111i	$-0.252714\pi$
$191$	3701.09	1.40210	0.701051	+	0.713111i	$0.252714\pi$
$192$	0	0
$193$	1389.59	0.518262	0.259131	−	0.965842i	$-0.416564\pi$
$193$	1389.59	0.518262	0.259131	+	0.965842i	$0.416564\pi$
$194$	0	0
$195$	2920.53	1.07253
$196$	0	0
$197$	3404.70	1.23134	0.615672	−	0.788003i	$-0.288885\pi$
$197$	3404.70	1.23134	0.615672	+	0.788003i	$0.288885\pi$
$198$	0	0
$199$	−4282.21	−1.52542	−0.762708	−	0.646744i	$-0.776130\pi$
$199$	−4282.21	−1.52542	−0.762708	+	0.646744i	$0.776130\pi$
$200$	0	0
$201$	−1480.40	−0.519499
$202$	0	0
$203$	−1939.13	−0.670445
$204$	0	0
$205$	−2761.00	−0.940665
$206$	0	0
$207$	−433.151	−0.145440
$208$	0	0
$209$	2061.74	0.682363
$210$	0	0
$211$	18.1576	0.00592428	0.00296214	−	0.999996i	$-0.499057\pi$
$211$	18.1576	0.00592428	0.00296214	+	0.999996i	$0.499057\pi$
$212$	0	0
$213$	2852.72	0.917677
$214$	0	0
$215$	−2143.27	−0.679859
$216$	0	0
$217$	1802.89	0.563999
$218$	0	0
$219$	4080.02	1.25891
$220$	0	0
$221$	−870.686	−0.265017
$222$	0	0
$223$	−4945.40	−1.48506	−0.742531	−	0.669812i	$-0.766375\pi$
$223$	−4945.40	−1.48506	−0.742531	+	0.669812i	$0.766375\pi$
$224$	0	0
$225$	−831.384	−0.246336
$226$	0	0
$227$	6179.32	1.80677	0.903383	−	0.428835i	$-0.141076\pi$
$227$	6179.32	1.80677	0.903383	+	0.428835i	$0.141076\pi$
$228$	0	0
$229$	−6838.27	−1.97330	−0.986650	−	0.162856i	$-0.947929\pi$
$229$	−6838.27	−1.97330	−0.986650	+	0.162856i	$0.947929\pi$
$230$	0	0
$231$	−2389.89	−0.680707
$232$	0	0
$233$	6168.82	1.73447	0.867237	−	0.497895i	$-0.165893\pi$
$233$	6168.82	1.73447	0.867237	+	0.497895i	$0.165893\pi$
$234$	0	0
$235$	4980.47	1.38251
$236$	0	0
$237$	−1503.49	−0.412077
$238$	0	0
$239$	632.850	0.171279	0.0856394	−	0.996326i	$-0.472707\pi$
$239$	632.850	0.171279	0.0856394	+	0.996326i	$0.472707\pi$
$240$	0	0
$241$	410.736	0.109783	0.0548917	−	0.998492i	$-0.482519\pi$
$241$	410.736	0.109783	0.0548917	+	0.998492i	$0.482519\pi$
$242$	0	0
$243$	2830.84	0.747318
$244$	0	0
$245$	−3483.57	−0.908395
$246$	0	0
$247$	1753.46	0.451701
$248$	0	0
$249$	3107.90	0.790985
$250$	0	0
$251$	−3392.00	−0.852993	−0.426497	−	0.904489i	$-0.640252\pi$
$251$	−3392.00	−0.852993	−0.426497	+	0.904489i	$0.640252\pi$
$252$	0	0
$253$	2400.89	0.596611
$254$	0	0
$255$	−969.389	−0.238061
$256$	0	0
$257$	−4213.03	−1.02257	−0.511287	−	0.859410i	$-0.670831\pi$
$257$	−4213.03	−1.02257	−0.511287	+	0.859410i	$0.670831\pi$
$258$	0	0
$259$	−2525.38	−0.605866
$260$	0	0
$261$	−2132.48	−0.505738
$262$	0	0
$263$	−1282.19	−0.300620	−0.150310	−	0.988639i	$-0.548027\pi$
$263$	−1282.19	−0.300620	−0.150310	+	0.988639i	$0.548027\pi$
$264$	0	0
$265$	−1021.44	−0.236779
$266$	0	0
$267$	4229.10	0.969351
$268$	0	0
$269$	5951.38	1.34893	0.674465	−	0.738307i	$-0.264374\pi$
$269$	5951.38	1.34893	0.674465	+	0.738307i	$0.264374\pi$
$270$	0	0
$271$	−6709.65	−1.50399	−0.751997	−	0.659167i	$-0.770909\pi$
$271$	−6709.65	−1.50399	−0.751997	+	0.659167i	$0.770909\pi$
$272$	0	0
$273$	−2032.55	−0.450605
$274$	0	0
$275$	4608.23	1.01050
$276$	0	0
$277$	−3369.11	−0.730794	−0.365397	−	0.930852i	$-0.619067\pi$
$277$	−3369.11	−0.730794	−0.365397	+	0.930852i	$0.619067\pi$
$278$	0	0
$279$	1982.65	0.425442
$280$	0	0
$281$	−1964.14	−0.416978	−0.208489	−	0.978025i	$-0.566855\pi$
$281$	−1964.14	−0.416978	−0.208489	+	0.978025i	$0.566855\pi$
$282$	0	0
$283$	116.659	0.0245042	0.0122521	−	0.999925i	$-0.496100\pi$
$283$	116.659	0.0245042	0.0122521	+	0.999925i	$0.496100\pi$
$284$	0	0
$285$	1952.24	0.405757
$286$	0	0
$287$	1921.52	0.395204
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−6412.27	−1.29173
$292$	0	0
$293$	−3801.24	−0.757922	−0.378961	−	0.925413i	$-0.623718\pi$
$293$	−3801.24	−0.757922	−0.378961	+	0.925413i	$0.623718\pi$
$294$	0	0
$295$	−2020.72	−0.398816
$296$	0	0
$297$	−9159.53	−1.78953
$298$	0	0
$299$	2041.90	0.394936
$300$	0	0
$301$	1491.61	0.285631
$302$	0	0
$303$	5443.88	1.03215
$304$	0	0
$305$	8666.54	1.62703
$306$	0	0
$307$	−4487.26	−0.834207	−0.417103	−	0.908859i	$-0.636955\pi$
$307$	−4487.26	−0.834207	−0.417103	+	0.908859i	$0.636955\pi$
$308$	0	0
$309$	−4158.38	−0.765573
$310$	0	0
$311$	−4722.00	−0.860965	−0.430482	−	0.902599i	$-0.641657\pi$
$311$	−4722.00	−0.860965	−0.430482	+	0.902599i	$0.641657\pi$
$312$	0	0
$313$	−980.994	−0.177154	−0.0885768	−	0.996069i	$-0.528232\pi$
$313$	−980.994	−0.177154	−0.0885768	+	0.996069i	$0.528232\pi$
$314$	0	0
$315$	1523.76	0.272554
$316$	0	0
$317$	3171.02	0.561837	0.280919	−	0.959732i	$-0.409361\pi$
$317$	3171.02	0.561837	0.280919	+	0.959732i	$0.409361\pi$
$318$	0	0
$319$	11820.0	2.07459
$320$	0	0
$321$	621.174	0.108008
$322$	0	0
$323$	−582.014	−0.100260
$324$	0	0
$325$	3919.19	0.668916
$326$	0	0
$327$	1718.84	0.290680
$328$	0	0
$329$	−3466.16	−0.580838
$330$	0	0
$331$	−1643.10	−0.272850	−0.136425	−	0.990650i	$-0.543561\pi$
$331$	−1643.10	−0.272850	−0.136425	+	0.990650i	$0.543561\pi$
$332$	0	0
$333$	−2777.19	−0.457024
$334$	0	0
$335$	−5231.80	−0.853264
$336$	0	0
$337$	−9936.95	−1.60623	−0.803116	−	0.595822i	$-0.796826\pi$
$337$	−9936.95	−1.60623	−0.803116	+	0.595822i	$0.796826\pi$
$338$	0	0
$339$	−3103.29	−0.497191
$340$	0	0
$341$	−10989.5	−1.74521
$342$	0	0
$343$	5813.09	0.915094
$344$	0	0
$345$	2273.37	0.354766
$346$	0	0
$347$	−11337.7	−1.75401	−0.877004	−	0.480484i	$-0.840461\pi$
$347$	−11337.7	−1.75401	−0.877004	+	0.480484i	$0.840461\pi$
$348$	0	0
$349$	627.001	0.0961678	0.0480839	−	0.998843i	$-0.484689\pi$
$349$	627.001	0.0961678	0.0480839	+	0.998843i	$0.484689\pi$
$350$	0	0
$351$	−7789.97	−1.18461
$352$	0	0
$353$	5362.12	0.808490	0.404245	−	0.914651i	$-0.367534\pi$
$353$	5362.12	0.808490	0.404245	+	0.914651i	$0.367534\pi$
$354$	0	0
$355$	10081.6	1.50726
$356$	0	0
$357$	674.647	0.100017
$358$	0	0
$359$	5063.48	0.744402	0.372201	−	0.928152i	$-0.378603\pi$
$359$	5063.48	0.744402	0.372201	+	0.928152i	$0.378603\pi$
$360$	0	0
$361$	−5686.89	−0.829113
$362$	0	0
$363$	9221.18	1.33330
$364$	0	0
$365$	14419.0	2.06774
$366$	0	0
$367$	−77.7642	−0.0110607	−0.00553033	−	0.999985i	$-0.501760\pi$
$367$	−77.7642	−0.0110607	−0.00553033	+	0.999985i	$0.501760\pi$
$368$	0	0
$369$	2113.11	0.298115
$370$	0	0
$371$	710.871	0.0994786
$372$	0	0
$373$	−2917.41	−0.404981	−0.202491	−	0.979284i	$-0.564904\pi$
$373$	−2917.41	−0.404981	−0.202491	+	0.979284i	$0.564904\pi$
$374$	0	0
$375$	−2764.38	−0.380672
$376$	0	0
$377$	10052.7	1.37331
$378$	0	0
$379$	13918.1	1.88634	0.943169	−	0.332313i	$-0.107829\pi$
$379$	13918.1	1.88634	0.943169	+	0.332313i	$0.107829\pi$
$380$	0	0
$381$	−4736.66	−0.636920
$382$	0	0
$383$	−2374.14	−0.316744	−0.158372	−	0.987380i	$-0.550624\pi$
$383$	−2374.14	−0.316744	−0.158372	+	0.987380i	$0.550624\pi$
$384$	0	0
$385$	−8445.99	−1.11804
$386$	0	0
$387$	1640.34	0.215460
$388$	0	0
$389$	−3839.71	−0.500465	−0.250232	−	0.968186i	$-0.580507\pi$
$389$	−3839.71	−0.500465	−0.250232	+	0.968186i	$0.580507\pi$
$390$	0	0
$391$	−677.751	−0.0876608
$392$	0	0
$393$	−10119.4	−1.29887
$394$	0	0
$395$	−5313.41	−0.676827
$396$	0	0
$397$	−9335.29	−1.18016	−0.590082	−	0.807344i	$-0.700904\pi$
$397$	−9335.29	−1.18016	−0.590082	+	0.807344i	$0.700904\pi$
$398$	0	0
$399$	−1358.66	−0.170472
$400$	0	0
$401$	−3648.62	−0.454373	−0.227187	−	0.973851i	$-0.572953\pi$
$401$	−3648.62	−0.454373	−0.227187	+	0.973851i	$0.572953\pi$
$402$	0	0
$403$	−9346.34	−1.15527
$404$	0	0
$405$	−4508.76	−0.553190
$406$	0	0
$407$	15393.5	1.87476
$408$	0	0
$409$	3796.95	0.459040	0.229520	−	0.973304i	$-0.426284\pi$
$409$	3796.95	0.459040	0.229520	+	0.973304i	$0.426284\pi$
$410$	0	0
$411$	2085.31	0.250269
$412$	0	0
$413$	1406.32	0.167555
$414$	0	0
$415$	10983.5	1.29917
$416$	0	0
$417$	−5549.79	−0.651737
$418$	0	0
$419$	−7116.97	−0.829801	−0.414901	−	0.909867i	$-0.636184\pi$
$419$	−7116.97	−0.829801	−0.414901	+	0.909867i	$0.636184\pi$
$420$	0	0
$421$	−4495.19	−0.520385	−0.260192	−	0.965557i	$-0.583786\pi$
$421$	−4495.19	−0.520385	−0.260192	+	0.965557i	$0.583786\pi$
$422$	0	0
$423$	−3811.77	−0.438144
$424$	0	0
$425$	−1300.87	−0.148474
$426$	0	0
$427$	−6031.48	−0.683569
$428$	0	0
$429$	12389.4	1.39433
$430$	0	0
$431$	2891.80	0.323186	0.161593	−	0.986857i	$-0.448337\pi$
$431$	2891.80	0.323186	0.161593	+	0.986857i	$0.448337\pi$
$432$	0	0
$433$	−14298.6	−1.58694	−0.793471	−	0.608608i	$-0.791728\pi$
$433$	−14298.6	−1.58694	−0.793471	+	0.608608i	$0.791728\pi$
$434$	0	0
$435$	11192.3	1.23363
$436$	0	0
$437$	1364.92	0.149411
$438$	0	0
$439$	−4699.43	−0.510914	−0.255457	−	0.966820i	$-0.582226\pi$
$439$	−4699.43	−0.510914	−0.255457	+	0.966820i	$0.582226\pi$
$440$	0	0
$441$	2666.13	0.287888
$442$	0	0
$443$	−2195.37	−0.235451	−0.117726	−	0.993046i	$-0.537560\pi$
$443$	−2195.37	−0.235451	−0.117726	+	0.993046i	$0.537560\pi$
$444$	0	0
$445$	14945.8	1.59214
$446$	0	0
$447$	7689.06	0.813602
$448$	0	0
$449$	1843.65	0.193780	0.0968902	−	0.995295i	$-0.469110\pi$
$449$	1843.65	0.193780	0.0968902	+	0.995295i	$0.469110\pi$
$450$	0	0
$451$	−11712.7	−1.22290
$452$	0	0
$453$	−6391.33	−0.662894
$454$	0	0
$455$	−7183.11	−0.740109
$456$	0	0
$457$	5769.52	0.590562	0.295281	−	0.955410i	$-0.404587\pi$
$457$	5769.52	0.590562	0.295281	+	0.955410i	$0.404587\pi$
$458$	0	0
$459$	2585.66	0.262938
$460$	0	0
$461$	−7335.80	−0.741133	−0.370567	−	0.928806i	$-0.620836\pi$
$461$	−7335.80	−0.741133	−0.370567	+	0.928806i	$0.620836\pi$
$462$	0	0
$463$	2659.54	0.266953	0.133477	−	0.991052i	$-0.457386\pi$
$463$	2659.54	0.266953	0.133477	+	0.991052i	$0.457386\pi$
$464$	0	0
$465$	−10405.9	−1.03776
$466$	0	0
$467$	−5657.03	−0.560549	−0.280274	−	0.959920i	$-0.590425\pi$
$467$	−5657.03	−0.560549	−0.280274	+	0.959920i	$0.590425\pi$
$468$	0	0
$469$	3641.07	0.358484
$470$	0	0
$471$	4087.96	0.399922
$472$	0	0
$473$	−9092.14	−0.883842
$474$	0	0
$475$	2619.80	0.253063
$476$	0	0
$477$	781.753	0.0750398
$478$	0	0
$479$	−1924.95	−0.183618	−0.0918090	−	0.995777i	$-0.529265\pi$
$479$	−1924.95	−0.183618	−0.0918090	+	0.995777i	$0.529265\pi$
$480$	0	0
$481$	13091.8	1.24103
$482$	0	0
$483$	−1582.16	−0.149049
$484$	0	0
$485$	−22661.2	−2.12164
$486$	0	0
$487$	9457.20	0.879973	0.439986	−	0.898004i	$-0.354983\pi$
$487$	9457.20	0.879973	0.439986	+	0.898004i	$0.354983\pi$
$488$	0	0
$489$	13523.7	1.25064
$490$	0	0
$491$	−17148.2	−1.57615	−0.788074	−	0.615580i	$-0.788922\pi$
$491$	−17148.2	−1.57615	−0.788074	+	0.615580i	$0.788922\pi$
$492$	0	0
$493$	−3336.70	−0.304822
$494$	0	0
$495$	−9288.14	−0.843376
$496$	0	0
$497$	−7016.33	−0.633250
$498$	0	0
$499$	−19203.5	−1.72278	−0.861389	−	0.507946i	$-0.830405\pi$
$499$	−19203.5	−1.72278	−0.861389	+	0.507946i	$0.830405\pi$
$500$	0	0
$501$	−9352.08	−0.833973
$502$	0	0
$503$	−7853.63	−0.696175	−0.348087	−	0.937462i	$-0.613169\pi$
$503$	−7853.63	−0.696175	−0.348087	+	0.937462i	$0.613169\pi$
$504$	0	0
$505$	19238.9	1.69529
$506$	0	0
$507$	1711.84	0.149952
$508$	0	0
$509$	6822.18	0.594082	0.297041	−	0.954865i	$-0.404000\pi$
$509$	6822.18	0.594082	0.297041	+	0.954865i	$0.404000\pi$
$510$	0	0
$511$	−10034.9	−0.868724
$512$	0	0
$513$	−5207.24	−0.448158
$514$	0	0
$515$	−14695.9	−1.25744
$516$	0	0
$517$	21128.1	1.79731
$518$	0	0
$519$	9209.93	0.778942
$520$	0	0
$521$	−6072.25	−0.510614	−0.255307	−	0.966860i	$-0.582177\pi$
$521$	−6072.25	−0.510614	−0.255307	+	0.966860i	$0.582177\pi$
$522$	0	0
$523$	−13825.4	−1.15591	−0.577955	−	0.816069i	$-0.696149\pi$
$523$	−13825.4	−1.15591	−0.577955	+	0.816069i	$0.696149\pi$
$524$	0	0
$525$	−3036.77	−0.252449
$526$	0	0
$527$	3102.26	0.256426
$528$	0	0
$529$	−10577.6	−0.869365
$530$	0	0
$531$	1546.54	0.126392
$532$	0	0
$533$	−9961.34	−0.809519
$534$	0	0
$535$	2195.26	0.177401
$536$	0	0
$537$	7377.87	0.592884
$538$	0	0
$539$	−14777.9	−1.18095
$540$	0	0
$541$	−17020.5	−1.35262	−0.676312	−	0.736615i	$-0.736423\pi$
$541$	−17020.5	−1.35262	−0.676312	+	0.736615i	$0.736423\pi$
$542$	0	0
$543$	11031.4	0.871827
$544$	0	0
$545$	6074.47	0.477435
$546$	0	0
$547$	21180.6	1.65560	0.827802	−	0.561020i	$-0.189591\pi$
$547$	21180.6	1.65560	0.827802	+	0.561020i	$0.189591\pi$
$548$	0	0
$549$	−6632.89	−0.515637
$550$	0	0
$551$	6719.74	0.519547
$552$	0	0
$553$	3697.87	0.284357
$554$	0	0
$555$	14575.9	1.11480
$556$	0	0
$557$	−12123.8	−0.922263	−0.461131	−	0.887332i	$-0.652556\pi$
$557$	−12123.8	−0.922263	−0.461131	+	0.887332i	$0.652556\pi$
$558$	0	0
$559$	−7732.65	−0.585074
$560$	0	0
$561$	−4112.33	−0.309488
$562$	0	0
$563$	−3995.69	−0.299108	−0.149554	−	0.988754i	$-0.547784\pi$
$563$	−3995.69	−0.299108	−0.149554	+	0.988754i	$0.547784\pi$
$564$	0	0
$565$	−10967.2	−0.816624
$566$	0	0
$567$	3137.87	0.232413
$568$	0	0
$569$	−19168.8	−1.41230	−0.706148	−	0.708064i	$-0.749569\pi$
$569$	−19168.8	−1.41230	−0.706148	+	0.708064i	$0.749569\pi$
$570$	0	0
$571$	−19127.7	−1.40187	−0.700937	−	0.713223i	$-0.747235\pi$
$571$	−19127.7	−1.40187	−0.700937	+	0.713223i	$0.747235\pi$
$572$	0	0
$573$	14866.8	1.08389
$574$	0	0
$575$	3050.74	0.221260
$576$	0	0
$577$	11272.7	0.813328	0.406664	−	0.913578i	$-0.366692\pi$
$577$	11272.7	0.813328	0.406664	+	0.913578i	$0.366692\pi$
$578$	0	0
$579$	5581.80	0.400642
$580$	0	0
$581$	−7643.95	−0.545825
$582$	0	0
$583$	−4333.13	−0.307822
$584$	0	0
$585$	−7899.34	−0.558287
$586$	0	0
$587$	−22029.2	−1.54896	−0.774482	−	0.632596i	$-0.781989\pi$
$587$	−22029.2	−1.54896	−0.774482	+	0.632596i	$0.781989\pi$
$588$	0	0
$589$	−6247.60	−0.437059
$590$	0	0
$591$	13676.3	0.951888
$592$	0	0
$593$	−20575.4	−1.42484	−0.712421	−	0.701753i	$-0.752401\pi$
$593$	−20575.4	−1.42484	−0.712421	+	0.701753i	$0.752401\pi$
$594$	0	0
$595$	2384.23	0.164276
$596$	0	0
$597$	−17201.1	−1.17922
$598$	0	0
$599$	21007.3	1.43295	0.716473	−	0.697615i	$-0.245755\pi$
$599$	21007.3	1.43295	0.716473	+	0.697615i	$0.245755\pi$
$600$	0	0
$601$	20499.1	1.39131	0.695653	−	0.718378i	$-0.255115\pi$
$601$	20499.1	1.39131	0.695653	+	0.718378i	$0.255115\pi$
$602$	0	0
$603$	4004.13	0.270416
$604$	0	0
$605$	32588.1	2.18991
$606$	0	0
$607$	−2031.26	−0.135826	−0.0679131	−	0.997691i	$-0.521634\pi$
$607$	−2031.26	−0.135826	−0.0679131	+	0.997691i	$0.521634\pi$
$608$	0	0
$609$	−7789.25	−0.518287
$610$	0	0
$611$	17968.9	1.18976
$612$	0	0
$613$	−7817.91	−0.515110	−0.257555	−	0.966264i	$-0.582917\pi$
$613$	−7817.91	−0.515110	−0.257555	+	0.966264i	$0.582917\pi$
$614$	0	0
$615$	−11090.6	−0.727180
$616$	0	0
$617$	−18880.6	−1.23194	−0.615968	−	0.787771i	$-0.711235\pi$
$617$	−18880.6	−1.23194	−0.615968	+	0.787771i	$0.711235\pi$
$618$	0	0
$619$	16860.2	1.09478	0.547389	−	0.836879i	$-0.315622\pi$
$619$	16860.2	1.09478	0.547389	+	0.836879i	$0.315622\pi$
$620$	0	0
$621$	−6063.80	−0.391839
$622$	0	0
$623$	−10401.6	−0.668908
$624$	0	0
$625$	−19334.6	−1.23742
$626$	0	0
$627$	8281.77	0.527499
$628$	0	0
$629$	−4345.47	−0.275461
$630$	0	0
$631$	−5789.26	−0.365241	−0.182620	−	0.983184i	$-0.558458\pi$
$631$	−5789.26	−0.365241	−0.182620	+	0.983184i	$0.558458\pi$
$632$	0	0
$633$	72.9370	0.00457976
$634$	0	0
$635$	−16739.6	−1.04613
$636$	0	0
$637$	−12568.3	−0.781748
$638$	0	0
$639$	−7715.93	−0.477680
$640$	0	0
$641$	−4747.96	−0.292563	−0.146282	−	0.989243i	$-0.546731\pi$
$641$	−4747.96	−0.292563	−0.146282	+	0.989243i	$0.546731\pi$
$642$	0	0
$643$	24608.1	1.50925	0.754625	−	0.656157i	$-0.227819\pi$
$643$	24608.1	1.50925	0.754625	+	0.656157i	$0.227819\pi$
$644$	0	0
$645$	−8609.25	−0.525564
$646$	0	0
$647$	−15126.4	−0.919134	−0.459567	−	0.888143i	$-0.651995\pi$
$647$	−15126.4	−0.919134	−0.459567	+	0.888143i	$0.651995\pi$
$648$	0	0
$649$	−8572.25	−0.518475
$650$	0	0
$651$	7241.97	0.435999
$652$	0	0
$653$	17608.3	1.05523	0.527615	−	0.849484i	$-0.323086\pi$
$653$	17608.3	1.05523	0.527615	+	0.849484i	$0.323086\pi$
$654$	0	0
$655$	−35762.3	−2.13336
$656$	0	0
$657$	−11035.5	−0.655305
$658$	0	0
$659$	−4983.53	−0.294584	−0.147292	−	0.989093i	$-0.547056\pi$
$659$	−4983.53	−0.294584	−0.147292	+	0.989093i	$0.547056\pi$
$660$	0	0
$661$	12178.7	0.716636	0.358318	−	0.933600i	$-0.383350\pi$
$661$	12178.7	0.716636	0.358318	+	0.933600i	$0.383350\pi$
$662$	0	0
$663$	−3497.44	−0.204871
$664$	0	0
$665$	−4801.58	−0.279996
$666$	0	0
$667$	7825.10	0.454256
$668$	0	0
$669$	−19865.1	−1.14802
$670$	0	0
$671$	36765.1	2.11520
$672$	0	0
$673$	−13407.2	−0.767917	−0.383959	−	0.923350i	$-0.625439\pi$
$673$	−13407.2	−0.767917	−0.383959	+	0.923350i	$0.625439\pi$
$674$	0	0
$675$	−11638.8	−0.663669
$676$	0	0
$677$	−2154.14	−0.122290	−0.0611450	−	0.998129i	$-0.519475\pi$
$677$	−2154.14	−0.122290	−0.0611450	+	0.998129i	$0.519475\pi$
$678$	0	0
$679$	15771.1	0.891369
$680$	0	0
$681$	24821.6	1.39672
$682$	0	0
$683$	27829.8	1.55912	0.779558	−	0.626330i	$-0.215444\pi$
$683$	27829.8	1.55912	0.779558	+	0.626330i	$0.215444\pi$
$684$	0	0
$685$	7369.57	0.411061
$686$	0	0
$687$	−27468.5	−1.52546
$688$	0	0
$689$	−3685.23	−0.203768
$690$	0	0
$691$	2940.02	0.161857	0.0809287	−	0.996720i	$-0.474211\pi$
$691$	2940.02	0.161857	0.0809287	+	0.996720i	$0.474211\pi$
$692$	0	0
$693$	6464.09	0.354330
$694$	0	0
$695$	−19613.2	−1.07046
$696$	0	0
$697$	3306.39	0.179682
$698$	0	0
$699$	24779.4	1.34083
$700$	0	0
$701$	−31363.2	−1.68983	−0.844915	−	0.534900i	$-0.820349\pi$
$701$	−31363.2	−1.68983	−0.844915	+	0.534900i	$0.820349\pi$
$702$	0	0
$703$	8751.28	0.469503
$704$	0	0
$705$	20005.9	1.06875
$706$	0	0
$707$	−13389.3	−0.712246
$708$	0	0
$709$	29484.9	1.56182	0.780910	−	0.624644i	$-0.214756\pi$
$709$	29484.9	1.56182	0.780910	+	0.624644i	$0.214756\pi$
$710$	0	0
$711$	4066.59	0.214499
$712$	0	0
$713$	−7275.29	−0.382134
$714$	0	0
$715$	43784.8	2.29015
$716$	0	0
$717$	2542.08	0.132407
$718$	0	0
$719$	2019.45	0.104746	0.0523732	−	0.998628i	$-0.483321\pi$
$719$	2019.45	0.104746	0.0523732	+	0.998628i	$0.483321\pi$
$720$	0	0
$721$	10227.6	0.528290
$722$	0	0
$723$	1649.87	0.0848679
$724$	0	0
$725$	15019.4	0.769388
$726$	0	0
$727$	−15239.9	−0.777465	−0.388733	−	0.921351i	$-0.627087\pi$
$727$	−15239.9	−0.777465	−0.388733	+	0.921351i	$0.627087\pi$
$728$	0	0
$729$	19946.6	1.01339
$730$	0	0
$731$	2566.64	0.129864
$732$	0	0
$733$	21343.6	1.07550	0.537752	−	0.843103i	$-0.319274\pi$
$733$	21343.6	1.07550	0.537752	+	0.843103i	$0.319274\pi$
$734$	0	0
$735$	−13993.1	−0.702233
$736$	0	0
$737$	−22194.2	−1.10927
$738$	0	0
$739$	−10706.6	−0.532949	−0.266475	−	0.963842i	$-0.585859\pi$
$739$	−10706.6	−0.532949	−0.266475	+	0.963842i	$0.585859\pi$
$740$	0	0
$741$	7043.45	0.349187
$742$	0	0
$743$	29819.2	1.47235	0.736177	−	0.676789i	$-0.236629\pi$
$743$	29819.2	1.47235	0.736177	+	0.676789i	$0.236629\pi$
$744$	0	0
$745$	27173.5	1.33632
$746$	0	0
$747$	−8406.14	−0.411733
$748$	0	0
$749$	−1527.79	−0.0745318
$750$	0	0
$751$	−17903.8	−0.869934	−0.434967	−	0.900447i	$-0.643240\pi$
$751$	−17903.8	−0.869934	−0.434967	+	0.900447i	$0.643240\pi$
$752$	0	0
$753$	−13625.3	−0.659405
$754$	0	0
$755$	−22587.3	−1.08879
$756$	0	0
$757$	3664.20	0.175928	0.0879641	−	0.996124i	$-0.471964\pi$
$757$	3664.20	0.175928	0.0879641	+	0.996124i	$0.471964\pi$
$758$	0	0
$759$	9644.07	0.461209
$760$	0	0
$761$	−603.032	−0.0287252	−0.0143626	−	0.999897i	$-0.504572\pi$
$761$	−603.032	−0.0287252	−0.0143626	+	0.999897i	$0.504572\pi$
$762$	0	0
$763$	−4227.53	−0.200586
$764$	0	0
$765$	2621.97	0.123918
$766$	0	0
$767$	−7290.49	−0.343213
$768$	0	0
$769$	−12780.9	−0.599338	−0.299669	−	0.954043i	$-0.596876\pi$
$769$	−12780.9	−0.599338	−0.299669	+	0.954043i	$0.596876\pi$
$770$	0	0
$771$	−16923.2	−0.790499
$772$	0	0
$773$	−15919.8	−0.740743	−0.370372	−	0.928884i	$-0.620770\pi$
$773$	−15919.8	−0.740743	−0.370372	+	0.928884i	$0.620770\pi$
$774$	0	0
$775$	−13964.1	−0.647233
$776$	0	0
$777$	−10144.1	−0.468364
$778$	0	0
$779$	−6658.70	−0.306255
$780$	0	0
$781$	42768.2	1.95950
$782$	0	0
$783$	−29853.2	−1.36254
$784$	0	0
$785$	14447.0	0.656862
$786$	0	0
$787$	−14004.2	−0.634300	−0.317150	−	0.948375i	$-0.602726\pi$
$787$	−14004.2	−0.634300	−0.317150	+	0.948375i	$0.602726\pi$
$788$	0	0
$789$	−5150.38	−0.232393
$790$	0	0
$791$	7632.61	0.343090
$792$	0	0
$793$	31267.8	1.40019
$794$	0	0
$795$	−4102.99	−0.183042
$796$	0	0
$797$	35890.2	1.59510	0.797551	−	0.603252i	$-0.206129\pi$
$797$	35890.2	1.59510	0.797551	+	0.603252i	$0.206129\pi$
$798$	0	0
$799$	−5964.28	−0.264082
$800$	0	0
$801$	−11438.7	−0.504578
$802$	0	0
$803$	61168.0	2.68814
$804$	0	0
$805$	−5591.41	−0.244809
$806$	0	0
$807$	23906.0	1.04279
$808$	0	0
$809$	3780.80	0.164309	0.0821544	−	0.996620i	$-0.473820\pi$
$809$	3780.80	0.164309	0.0821544	+	0.996620i	$0.473820\pi$
$810$	0	0
$811$	31627.6	1.36941	0.684707	−	0.728818i	$-0.259930\pi$
$811$	31627.6	1.36941	0.684707	+	0.728818i	$0.259930\pi$
$812$	0	0
$813$	−26951.8	−1.16266
$814$	0	0
$815$	47793.3	2.05414
$816$	0	0
$817$	−5168.92	−0.221344
$818$	0	0
$819$	5497.55	0.234554
$820$	0	0
$821$	−36041.9	−1.53212	−0.766060	−	0.642769i	$-0.777786\pi$
$821$	−36041.9	−1.53212	−0.766060	+	0.642769i	$0.777786\pi$
$822$	0	0
$823$	−5132.22	−0.217373	−0.108686	−	0.994076i	$-0.534664\pi$
$823$	−5132.22	−0.217373	−0.108686	+	0.994076i	$0.534664\pi$
$824$	0	0
$825$	18510.7	0.781164
$826$	0	0
$827$	43411.1	1.82533	0.912667	−	0.408703i	$-0.134019\pi$
$827$	43411.1	1.82533	0.912667	+	0.408703i	$0.134019\pi$
$828$	0	0
$829$	19608.0	0.821488	0.410744	−	0.911751i	$-0.365269\pi$
$829$	19608.0	0.821488	0.410744	+	0.911751i	$0.365269\pi$
$830$	0	0
$831$	−13533.3	−0.564939
$832$	0	0
$833$	4171.69	0.173518
$834$	0	0
$835$	−33050.7	−1.36978
$836$	0	0
$837$	27755.7	1.14621
$838$	0	0
$839$	39957.8	1.64422	0.822109	−	0.569331i	$-0.192798\pi$
$839$	39957.8	1.64422	0.822109	+	0.569331i	$0.192798\pi$
$840$	0	0
$841$	14135.4	0.579583
$842$	0	0
$843$	−7889.72	−0.322344
$844$	0	0
$845$	6049.72	0.246292
$846$	0	0
$847$	−22679.7	−0.920051
$848$	0	0
$849$	468.607	0.0189429
$850$	0	0
$851$	10190.8	0.410501
$852$	0	0
$853$	15291.9	0.613814	0.306907	−	0.951739i	$-0.400706\pi$
$853$	15291.9	0.613814	0.306907	+	0.951739i	$0.400706\pi$
$854$	0	0
$855$	−5280.35	−0.211210
$856$	0	0
$857$	46263.3	1.84402	0.922010	−	0.387166i	$-0.126546\pi$
$857$	46263.3	1.84402	0.922010	+	0.387166i	$0.126546\pi$
$858$	0	0
$859$	38492.2	1.52891	0.764457	−	0.644675i	$-0.223007\pi$
$859$	38492.2	1.52891	0.764457	+	0.644675i	$0.223007\pi$
$860$	0	0
$861$	7718.50	0.305512
$862$	0	0
$863$	33188.8	1.30911	0.654553	−	0.756016i	$-0.272857\pi$
$863$	33188.8	1.30911	0.654553	+	0.756016i	$0.272857\pi$
$864$	0	0
$865$	32548.3	1.27939
$866$	0	0
$867$	1160.88	0.0454734
$868$	0	0
$869$	−22540.5	−0.879900
$870$	0	0
$871$	−18875.7	−0.734303
$872$	0	0
$873$	17343.7	0.672387
$874$	0	0
$875$	6799.05	0.262686
$876$	0	0
$877$	−8448.30	−0.325289	−0.162645	−	0.986685i	$-0.552002\pi$
$877$	−8448.30	−0.325289	−0.162645	+	0.986685i	$0.552002\pi$
$878$	0	0
$879$	−15269.1	−0.585910
$880$	0	0
$881$	−11721.8	−0.448262	−0.224131	−	0.974559i	$-0.571954\pi$
$881$	−11721.8	−0.448262	−0.224131	+	0.974559i	$0.571954\pi$
$882$	0	0
$883$	20348.6	0.775519	0.387760	−	0.921761i	$-0.373249\pi$
$883$	20348.6	0.775519	0.387760	+	0.921761i	$0.373249\pi$
$884$	0	0
$885$	−8116.96	−0.308304
$886$	0	0
$887$	−41522.0	−1.57178	−0.785892	−	0.618363i	$-0.787796\pi$
$887$	−41522.0	−1.57178	−0.785892	+	0.618363i	$0.787796\pi$
$888$	0	0
$889$	11649.9	0.439512
$890$	0	0
$891$	−19127.0	−0.719167
$892$	0	0
$893$	12011.4	0.450108
$894$	0	0
$895$	26073.8	0.973798
$896$	0	0
$897$	8202.05	0.305305
$898$	0	0
$899$	−35817.7	−1.32879
$900$	0	0
$901$	1223.21	0.0452286
$902$	0	0
$903$	5991.61	0.220806
$904$	0	0
$905$	38985.5	1.43196
$906$	0	0
$907$	−41036.0	−1.50229	−0.751145	−	0.660138i	$-0.770498\pi$
$907$	−41036.0	−1.50229	−0.751145	+	0.660138i	$0.770498\pi$
$908$	0	0
$909$	−14724.4	−0.537269
$910$	0	0
$911$	−6771.68	−0.246274	−0.123137	−	0.992390i	$-0.539295\pi$
$911$	−6771.68	−0.246274	−0.123137	+	0.992390i	$0.539295\pi$
$912$	0	0
$913$	46593.9	1.68897
$914$	0	0
$915$	34812.4	1.25777
$916$	0	0
$917$	24888.8	0.896292
$918$	0	0
$919$	−3589.93	−0.128858	−0.0644292	−	0.997922i	$-0.520523\pi$
$919$	−3589.93	−0.128858	−0.0644292	+	0.997922i	$0.520523\pi$
$920$	0	0
$921$	−18024.8	−0.644882
$922$	0	0
$923$	36373.3	1.29712
$924$	0	0
$925$	19560.1	0.695279
$926$	0	0
$927$	11247.4	0.398505
$928$	0	0
$929$	−38282.3	−1.35199	−0.675996	−	0.736906i	$-0.736286\pi$
$929$	−38282.3	−1.35199	−0.675996	+	0.736906i	$0.736286\pi$
$930$	0	0
$931$	−8401.32	−0.295749
$932$	0	0
$933$	−18967.7	−0.665568
$934$	0	0
$935$	−14533.2	−0.508326
$936$	0	0
$937$	−26099.5	−0.909961	−0.454980	−	0.890501i	$-0.650354\pi$
$937$	−26099.5	−0.909961	−0.454980	+	0.890501i	$0.650354\pi$
$938$	0	0
$939$	−3940.53	−0.136948
$940$	0	0
$941$	−33481.9	−1.15991	−0.579956	−	0.814648i	$-0.696930\pi$
$941$	−33481.9	−1.15991	−0.579956	+	0.814648i	$0.696930\pi$
$942$	0	0
$943$	−7754.02	−0.267768
$944$	0	0
$945$	21331.6	0.734303
$946$	0	0
$947$	−25045.8	−0.859430	−0.429715	−	0.902964i	$-0.641386\pi$
$947$	−25045.8	−0.859430	−0.429715	+	0.902964i	$0.641386\pi$
$948$	0	0
$949$	52021.9	1.77946
$950$	0	0
$951$	12737.6	0.434327
$952$	0	0
$953$	−3465.20	−0.117785	−0.0588924	−	0.998264i	$-0.518757\pi$
$953$	−3465.20	−0.117785	−0.0588924	+	0.998264i	$0.518757\pi$
$954$	0	0
$955$	52540.1	1.78027
$956$	0	0
$957$	47479.6	1.60376
$958$	0	0
$959$	−5128.86	−0.172700
$960$	0	0
$961$	3510.05	0.117822
$962$	0	0
$963$	−1680.13	−0.0562216
$964$	0	0
$965$	19726.3	0.658045
$966$	0	0
$967$	−45361.5	−1.50851	−0.754254	−	0.656583i	$-0.772001\pi$
$967$	−45361.5	−1.50851	−0.754254	+	0.656583i	$0.772001\pi$
$968$	0	0
$969$	−2337.88	−0.0775061
$970$	0	0
$971$	−50036.4	−1.65370	−0.826851	−	0.562421i	$-0.809870\pi$
$971$	−50036.4	−1.65370	−0.826851	+	0.562421i	$0.809870\pi$
$972$	0	0
$973$	13649.8	0.449737
$974$	0	0
$975$	15742.9	0.517104
$976$	0	0
$977$	−6689.64	−0.219059	−0.109530	−	0.993984i	$-0.534934\pi$
$977$	−6689.64	−0.219059	−0.109530	+	0.993984i	$0.534934\pi$
$978$	0	0
$979$	63403.0	2.06983
$980$	0	0
$981$	−4649.06	−0.151308
$982$	0	0
$983$	−31585.2	−1.02483	−0.512416	−	0.858737i	$-0.671249\pi$
$983$	−31585.2	−1.02483	−0.512416	+	0.858737i	$0.671249\pi$
$984$	0	0
$985$	48332.5	1.56345
$986$	0	0
$987$	−13923.1	−0.449016
$988$	0	0
$989$	−6019.18	−0.193528
$990$	0	0
$991$	30182.2	0.967476	0.483738	−	0.875213i	$-0.339279\pi$
$991$	30182.2	0.967476	0.483738	+	0.875213i	$0.339279\pi$
$992$	0	0
$993$	−6600.15	−0.210926
$994$	0	0
$995$	−60789.5	−1.93684
$996$	0	0
$997$	24670.2	0.783663	0.391832	−	0.920037i	$-0.371842\pi$
$997$	24670.2	0.783663	0.391832	+	0.920037i	$0.371842\pi$
$998$	0	0
$999$	−38878.6	−1.23130

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1088.4.a.bg.1.6		7
4.3	odd	2		1088.4.a.bh.1.2		7
8.3	odd	2		544.4.a.l.1.6	yes	7
8.5	even	2		544.4.a.k.1.2	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.4.a.k.1.2	✓	7	8.5	even	2
544.4.a.l.1.6	yes	7	8.3	odd	2
1088.4.a.bg.1.6		7	1.1	even	1	trivial
1088.4.a.bh.1.2		7	4.3	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:	\( N \)	\(=\)	\( 1088 = 2^{6} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1088.a (trivial)

\(f(q)\)	\(=\)	\(q+4.01688 q^{3} +14.1958 q^{5} -9.87960 q^{7} -10.8647 q^{9} +O(q^{10})\)	\(q+4.01688 q^{3} +14.1958 q^{5} -9.87960 q^{7} -10.8647 q^{9} +60.2213 q^{11} +51.2168 q^{13} +57.0229 q^{15} -17.0000 q^{17} +34.2361 q^{19} -39.6851 q^{21} +39.8677 q^{23} +76.5216 q^{25} -152.098 q^{27} +196.276 q^{29} -182.486 q^{31} +241.902 q^{33} -140.249 q^{35} +255.616 q^{37} +205.732 q^{39} -194.494 q^{41} -150.979 q^{43} -154.233 q^{45} +350.840 q^{47} -245.394 q^{49} -68.2869 q^{51} -71.9534 q^{53} +854.892 q^{55} +137.522 q^{57} -142.346 q^{59} +610.499 q^{61} +107.339 q^{63} +727.065 q^{65} -368.545 q^{67} +160.144 q^{69} +710.184 q^{71} +1015.72 q^{73} +307.378 q^{75} -594.962 q^{77} -374.294 q^{79} -317.611 q^{81} +773.711 q^{83} -241.329 q^{85} +788.418 q^{87} +1052.83 q^{89} -506.001 q^{91} -733.023 q^{93} +486.010 q^{95} -1596.33 q^{97} -654.287 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{7} + 67 q^{9}+O(q^{10}) \) 7 * q - 2 * q^7 + 67 * q^9	\( 7 q - 2 q^{7} + 67 q^{9} - 108 q^{11} + 34 q^{13} + 128 q^{15} - 119 q^{17} - 124 q^{19} + 296 q^{21} - 6 q^{23} + 197 q^{25} - 248 q^{29} + 50 q^{31} + 512 q^{33} - 640 q^{35} + 484 q^{37} + 1504 q^{39} - 366 q^{41} - 1412 q^{43} + 80 q^{45} + 1012 q^{47} + 1115 q^{49} + 146 q^{53} + 1024 q^{55} + 48 q^{57} - 2332 q^{59} + 548 q^{61} + 2838 q^{63} - 208 q^{65} - 924 q^{67} + 1672 q^{69} + 1286 q^{71} + 870 q^{73} - 3136 q^{75} + 1344 q^{77} + 1818 q^{79} + 3039 q^{81} - 1772 q^{83} + 384 q^{87} + 1706 q^{89} - 588 q^{91} + 5576 q^{93} + 2048 q^{95} + 1802 q^{97} - 5148 q^{99}+O(q^{100}) \) 7 * q - 2 * q^7 + 67 * q^9 - 108 * q^11 + 34 * q^13 + 128 * q^15 - 119 * q^17 - 124 * q^19 + 296 * q^21 - 6 * q^23 + 197 * q^25 - 248 * q^29 + 50 * q^31 + 512 * q^33 - 640 * q^35 + 484 * q^37 + 1504 * q^39 - 366 * q^41 - 1412 * q^43 + 80 * q^45 + 1012 * q^47 + 1115 * q^49 + 146 * q^53 + 1024 * q^55 + 48 * q^57 - 2332 * q^59 + 548 * q^61 + 2838 * q^63 - 208 * q^65 - 924 * q^67 + 1672 * q^69 + 1286 * q^71 + 870 * q^73 - 3136 * q^75 + 1344 * q^77 + 1818 * q^79 + 3039 * q^81 - 1772 * q^83 + 384 * q^87 + 1706 * q^89 - 588 * q^91 + 5576 * q^93 + 2048 * q^95 + 1802 * q^97 - 5148 * q^99

Introduction

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