LMFDB - Embedded newform 1088.4.a.y.1.2

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:	$3$
Coefficient field:	3.3.8396.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 20x + 8 $ x^3 - x^2 - 20*x + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 136)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.81129$ 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)

$f(q)$	$=$	$q-1.16862 q^{3} -1.28534 q^{5} -30.0364 q^{7} -25.6343 q^{9} +O(q^{10})$	$q-1.16862 q^{3} -1.28534 q^{5} -30.0364 q^{7} -25.6343 q^{9} +7.58621 q^{11} -15.2050 q^{13} +1.50208 q^{15} -17.0000 q^{17} +41.7431 q^{19} +35.1012 q^{21} -84.4540 q^{23} -123.348 q^{25} +61.5097 q^{27} -155.374 q^{29} -217.059 q^{31} -8.86541 q^{33} +38.6070 q^{35} -370.217 q^{37} +17.7689 q^{39} +97.9713 q^{41} +413.340 q^{43} +32.9488 q^{45} +393.446 q^{47} +559.184 q^{49} +19.8666 q^{51} +245.861 q^{53} -9.75086 q^{55} -48.7819 q^{57} +183.647 q^{59} -214.658 q^{61} +769.962 q^{63} +19.5436 q^{65} -727.621 q^{67} +98.6948 q^{69} -431.701 q^{71} +540.677 q^{73} +144.147 q^{75} -227.862 q^{77} +511.069 q^{79} +620.245 q^{81} +834.896 q^{83} +21.8508 q^{85} +181.574 q^{87} -495.264 q^{89} +456.703 q^{91} +253.660 q^{93} -53.6541 q^{95} -892.146 q^{97} -194.467 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{3} + 8 q^{5} - 2 q^{7} - 5 q^{9}+O(q^{10}) $ 3 * q + 4 * q^3 + 8 * q^5 - 2 * q^7 - 5 * q^9	$ 3 q + 4 q^{3} + 8 q^{5} - 2 q^{7} - 5 q^{9} + 84 q^{11} + 50 q^{13} - 148 q^{15} - 51 q^{17} + 224 q^{19} + 16 q^{21} - 234 q^{23} + 161 q^{25} + 292 q^{27} + 72 q^{29} - 2 q^{31} + 148 q^{33} + 428 q^{35} - 100 q^{37} + 156 q^{39} + 218 q^{41} - 44 q^{43} - 768 q^{45} - 16 q^{47} + 403 q^{49} - 68 q^{51} - 462 q^{53} + 460 q^{55} + 688 q^{57} + 68 q^{59} - 460 q^{61} + 586 q^{63} + 408 q^{65} + 1008 q^{67} - 400 q^{69} + 518 q^{71} + 838 q^{73} - 732 q^{75} + 904 q^{77} + 1238 q^{79} + 455 q^{81} + 1148 q^{83} - 136 q^{85} + 1108 q^{87} - 2506 q^{89} + 1416 q^{91} - 648 q^{93} + 40 q^{95} + 2098 q^{97} + 384 q^{99}+O(q^{100}) $ 3 * q + 4 * q^3 + 8 * q^5 - 2 * q^7 - 5 * q^9 + 84 * q^11 + 50 * q^13 - 148 * q^15 - 51 * q^17 + 224 * q^19 + 16 * q^21 - 234 * q^23 + 161 * q^25 + 292 * q^27 + 72 * q^29 - 2 * q^31 + 148 * q^33 + 428 * q^35 - 100 * q^37 + 156 * q^39 + 218 * q^41 - 44 * q^43 - 768 * q^45 - 16 * q^47 + 403 * q^49 - 68 * q^51 - 462 * q^53 + 460 * q^55 + 688 * q^57 + 68 * q^59 - 460 * q^61 + 586 * q^63 + 408 * q^65 + 1008 * q^67 - 400 * q^69 + 518 * q^71 + 838 * q^73 - 732 * q^75 + 904 * q^77 + 1238 * q^79 + 455 * q^81 + 1148 * q^83 - 136 * q^85 + 1108 * q^87 - 2506 * q^89 + 1416 * q^91 - 648 * q^93 + 40 * q^95 + 2098 * q^97 + 384 * 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
$3$	−1.16862	−0.224902	−0.112451	−	0.993657i	$-0.535870\pi$
$3$	−1.16862	−0.224902	−0.112451	+	0.993657i	$0.535870\pi$
$4$	0	0
$5$	−1.28534	−0.114964	−0.0574822	−	0.998347i	$-0.518307\pi$
$5$	−1.28534	−0.114964	−0.0574822	+	0.998347i	$0.518307\pi$
$6$	0	0
$7$	−30.0364	−1.62181	−0.810906	−	0.585176i	$-0.801025\pi$
$7$	−30.0364	−1.62181	−0.810906	+	0.585176i	$0.801025\pi$
$8$	0	0
$9$	−25.6343	−0.949419
$10$	0	0
$11$	7.58621	0.207939	0.103969	−	0.994580i	$-0.466846\pi$
$11$	7.58621	0.207939	0.103969	+	0.994580i	$0.466846\pi$
$12$	0	0
$13$	−15.2050	−0.324393	−0.162196	−	0.986758i	$-0.551858\pi$
$13$	−15.2050	−0.324393	−0.162196	+	0.986758i	$0.551858\pi$
$14$	0	0
$15$	1.50208	0.0258557
$16$	0	0
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	41.7431	0.504027	0.252014	−	0.967724i	$-0.418907\pi$
$19$	41.7431	0.504027	0.252014	+	0.967724i	$0.418907\pi$
$20$	0	0
$21$	35.1012	0.364748
$22$	0	0
$23$	−84.4540	−0.765646	−0.382823	−	0.923822i	$-0.625048\pi$
$23$	−84.4540	−0.765646	−0.382823	+	0.923822i	$0.625048\pi$
$24$	0	0
$25$	−123.348	−0.986783
$26$	0	0
$27$	61.5097	0.438427
$28$	0	0
$29$	−155.374	−0.994904	−0.497452	−	0.867491i	$-0.665731\pi$
$29$	−155.374	−0.994904	−0.497452	+	0.867491i	$0.665731\pi$
$30$	0	0
$31$	−217.059	−1.25758	−0.628789	−	0.777576i	$-0.716449\pi$
$31$	−217.059	−1.25758	−0.628789	+	0.777576i	$0.716449\pi$
$32$	0	0
$33$	−8.86541	−0.0467658
$34$	0	0
$35$	38.6070	0.186451
$36$	0	0
$37$	−370.217	−1.64496	−0.822478	−	0.568797i	$-0.807409\pi$
$37$	−370.217	−1.64496	−0.822478	+	0.568797i	$0.807409\pi$
$38$	0	0
$39$	17.7689	0.0729565
$40$	0	0
$41$	97.9713	0.373184	0.186592	−	0.982437i	$-0.440256\pi$
$41$	97.9713	0.373184	0.186592	+	0.982437i	$0.440256\pi$
$42$	0	0
$43$	413.340	1.46590	0.732950	−	0.680282i	$-0.238143\pi$
$43$	413.340	1.46590	0.732950	+	0.680282i	$0.238143\pi$
$44$	0	0
$45$	32.9488	0.109149
$46$	0	0
$47$	393.446	1.22106	0.610532	−	0.791992i	$-0.290956\pi$
$47$	393.446	1.22106	0.610532	+	0.791992i	$0.290956\pi$
$48$	0	0
$49$	559.184	1.63027
$50$	0	0
$51$	19.8666	0.0545466
$52$	0	0
$53$	245.861	0.637199	0.318600	−	0.947889i	$-0.396787\pi$
$53$	245.861	0.637199	0.318600	+	0.947889i	$0.396787\pi$
$54$	0	0
$55$	−9.75086	−0.0239055
$56$	0	0
$57$	−48.7819	−0.113357
$58$	0	0
$59$	183.647	0.405234	0.202617	−	0.979258i	$-0.435055\pi$
$59$	183.647	0.405234	0.202617	+	0.979258i	$0.435055\pi$
$60$	0	0
$61$	−214.658	−0.450560	−0.225280	−	0.974294i	$-0.572330\pi$
$61$	−214.658	−0.450560	−0.225280	+	0.974294i	$0.572330\pi$
$62$	0	0
$63$	769.962	1.53978
$64$	0	0
$65$	19.5436	0.0372936
$66$	0	0
$67$	−727.621	−1.32676	−0.663381	−	0.748282i	$-0.730879\pi$
$67$	−727.621	−1.32676	−0.663381	+	0.748282i	$0.730879\pi$
$68$	0	0
$69$	98.6948	0.172195
$70$	0	0
$71$	−431.701	−0.721598	−0.360799	−	0.932644i	$-0.617496\pi$
$71$	−431.701	−0.721598	−0.360799	+	0.932644i	$0.617496\pi$
$72$	0	0
$73$	540.677	0.866869	0.433435	−	0.901185i	$-0.357302\pi$
$73$	540.677	0.866869	0.433435	+	0.901185i	$0.357302\pi$
$74$	0	0
$75$	144.147	0.221929
$76$	0	0
$77$	−227.862	−0.337238
$78$	0	0
$79$	511.069	0.727845	0.363922	−	0.931429i	$-0.381437\pi$
$79$	511.069	0.727845	0.363922	+	0.931429i	$0.381437\pi$
$80$	0	0
$81$	620.245	0.850816
$82$	0	0
$83$	834.896	1.10412	0.552059	−	0.833805i	$-0.313842\pi$
$83$	834.896	1.10412	0.552059	+	0.833805i	$0.313842\pi$
$84$	0	0
$85$	21.8508	0.0278829
$86$	0	0
$87$	181.574	0.223756
$88$	0	0
$89$	−495.264	−0.589864	−0.294932	−	0.955518i	$-0.595297\pi$
$89$	−495.264	−0.589864	−0.294932	+	0.955518i	$0.595297\pi$
$90$	0	0
$91$	456.703	0.526104
$92$	0	0
$93$	253.660	0.282831
$94$	0	0
$95$	−53.6541	−0.0579452
$96$	0	0
$97$	−892.146	−0.933853	−0.466926	−	0.884296i	$-0.654639\pi$
$97$	−892.146	−0.933853	−0.466926	+	0.884296i	$0.654639\pi$
$98$	0	0
$99$	−194.467	−0.197421
$100$	0	0
$101$	1855.55	1.82806	0.914028	−	0.405650i	$-0.132955\pi$
$101$	1855.55	1.82806	0.914028	+	0.405650i	$0.132955\pi$
$102$	0	0
$103$	182.909	0.174976	0.0874882	−	0.996166i	$-0.472116\pi$
$103$	182.909	0.174976	0.0874882	+	0.996166i	$0.472116\pi$
$104$	0	0
$105$	−45.1170	−0.0419330
$106$	0	0
$107$	−179.792	−0.162441	−0.0812203	−	0.996696i	$-0.525882\pi$
$107$	−179.792	−0.162441	−0.0812203	+	0.996696i	$0.525882\pi$
$108$	0	0
$109$	−372.690	−0.327497	−0.163749	−	0.986502i	$-0.552359\pi$
$109$	−372.690	−0.327497	−0.163749	+	0.986502i	$0.552359\pi$
$110$	0	0
$111$	432.645	0.369953
$112$	0	0
$113$	−1267.64	−1.05530	−0.527652	−	0.849461i	$-0.676928\pi$
$113$	−1267.64	−1.05530	−0.527652	+	0.849461i	$0.676928\pi$
$114$	0	0
$115$	108.552	0.0880220
$116$	0	0
$117$	389.770	0.307985
$118$	0	0
$119$	510.618	0.393347
$120$	0	0
$121$	−1273.45	−0.956761
$122$	0	0
$123$	−114.491	−0.0839297
$124$	0	0
$125$	319.212	0.228409
$126$	0	0
$127$	2173.49	1.51863	0.759316	−	0.650722i	$-0.225534\pi$
$127$	2173.49	1.51863	0.759316	+	0.650722i	$0.225534\pi$
$128$	0	0
$129$	−483.038	−0.329683
$130$	0	0
$131$	1329.37	0.886623	0.443311	−	0.896368i	$-0.353804\pi$
$131$	1329.37	0.886623	0.443311	+	0.896368i	$0.353804\pi$
$132$	0	0
$133$	−1253.81	−0.817438
$134$	0	0
$135$	−79.0608	−0.0504035
$136$	0	0
$137$	−1136.50	−0.708741	−0.354370	−	0.935105i	$-0.615305\pi$
$137$	−1136.50	−0.708741	−0.354370	+	0.935105i	$0.615305\pi$
$138$	0	0
$139$	2695.14	1.64459	0.822297	−	0.569059i	$-0.192692\pi$
$139$	2695.14	1.64459	0.822297	+	0.569059i	$0.192692\pi$
$140$	0	0
$141$	−459.790	−0.274619
$142$	0	0
$143$	−115.348	−0.0674539
$144$	0	0
$145$	199.708	0.114379
$146$	0	0
$147$	−653.475	−0.366651
$148$	0	0
$149$	522.901	0.287502	0.143751	−	0.989614i	$-0.454084\pi$
$149$	522.901	0.287502	0.143751	+	0.989614i	$0.454084\pi$
$150$	0	0
$151$	−642.332	−0.346174	−0.173087	−	0.984907i	$-0.555374\pi$
$151$	−642.332	−0.346174	−0.173087	+	0.984907i	$0.555374\pi$
$152$	0	0
$153$	435.783	0.230268
$154$	0	0
$155$	278.994	0.144577
$156$	0	0
$157$	898.206	0.456590	0.228295	−	0.973592i	$-0.426685\pi$
$157$	898.206	0.456590	0.228295	+	0.973592i	$0.426685\pi$
$158$	0	0
$159$	−287.318	−0.143307
$160$	0	0
$161$	2536.69	1.24173
$162$	0	0
$163$	277.508	0.133350	0.0666751	−	0.997775i	$-0.478761\pi$
$163$	277.508	0.133350	0.0666751	+	0.997775i	$0.478761\pi$
$164$	0	0
$165$	11.3951	0.00537639
$166$	0	0
$167$	2658.55	1.23189	0.615943	−	0.787791i	$-0.288775\pi$
$167$	2658.55	1.23189	0.615943	+	0.787791i	$0.288775\pi$
$168$	0	0
$169$	−1965.81	−0.894769
$170$	0	0
$171$	−1070.06	−0.478533
$172$	0	0
$173$	−432.801	−0.190204	−0.0951018	−	0.995468i	$-0.530318\pi$
$173$	−432.801	−0.190204	−0.0951018	+	0.995468i	$0.530318\pi$
$174$	0	0
$175$	3704.92	1.60038
$176$	0	0
$177$	−214.614	−0.0911378
$178$	0	0
$179$	577.672	0.241214	0.120607	−	0.992700i	$-0.461516\pi$
$179$	577.672	0.241214	0.120607	+	0.992700i	$0.461516\pi$
$180$	0	0
$181$	−3094.10	−1.27062	−0.635311	−	0.772257i	$-0.719128\pi$
$181$	−3094.10	−1.27062	−0.635311	+	0.772257i	$0.719128\pi$
$182$	0	0
$183$	250.855	0.101332
$184$	0	0
$185$	475.855	0.189111
$186$	0	0
$187$	−128.966	−0.0504326
$188$	0	0
$189$	−1847.53	−0.711047
$190$	0	0
$191$	502.111	0.190217	0.0951085	−	0.995467i	$-0.469680\pi$
$191$	502.111	0.190217	0.0951085	+	0.995467i	$0.469680\pi$
$192$	0	0
$193$	1013.05	0.377830	0.188915	−	0.981993i	$-0.439503\pi$
$193$	1013.05	0.377830	0.188915	+	0.981993i	$0.439503\pi$
$194$	0	0
$195$	−22.8391	−0.00838739
$196$	0	0
$197$	5431.69	1.96443	0.982213	−	0.187769i	$-0.0601258\pi$
$197$	5431.69	1.96443	0.982213	+	0.187769i	$0.0601258\pi$
$198$	0	0
$199$	2523.46	0.898911	0.449455	−	0.893303i	$-0.351618\pi$
$199$	2523.46	0.898911	0.449455	+	0.893303i	$0.351618\pi$
$200$	0	0
$201$	850.314	0.298391
$202$	0	0
$203$	4666.87	1.61355
$204$	0	0
$205$	−125.926	−0.0429029
$206$	0	0
$207$	2164.92	0.726920
$208$	0	0
$209$	316.672	0.104807
$210$	0	0
$211$	−257.746	−0.0840947	−0.0420473	−	0.999116i	$-0.513388\pi$
$211$	−257.746	−0.0840947	−0.0420473	+	0.999116i	$0.513388\pi$
$212$	0	0
$213$	504.495	0.162288
$214$	0	0
$215$	−531.282	−0.168526
$216$	0	0
$217$	6519.66	2.03955
$218$	0	0
$219$	−631.847	−0.194960
$220$	0	0
$221$	258.485	0.0786769
$222$	0	0
$223$	−3722.60	−1.11786	−0.558932	−	0.829214i	$-0.688789\pi$
$223$	−3722.60	−1.11786	−0.558932	+	0.829214i	$0.688789\pi$
$224$	0	0
$225$	3161.94	0.936871
$226$	0	0
$227$	4586.89	1.34116	0.670578	−	0.741839i	$-0.266046\pi$
$227$	4586.89	1.34116	0.670578	+	0.741839i	$0.266046\pi$
$228$	0	0
$229$	−4322.30	−1.24727	−0.623636	−	0.781715i	$-0.714345\pi$
$229$	−4322.30	−1.24727	−0.623636	+	0.781715i	$0.714345\pi$
$230$	0	0
$231$	266.285	0.0758453
$232$	0	0
$233$	5105.95	1.43563	0.717815	−	0.696234i	$-0.245142\pi$
$233$	5105.95	1.43563	0.717815	+	0.696234i	$0.245142\pi$
$234$	0	0
$235$	−505.712	−0.140379
$236$	0	0
$237$	−597.246	−0.163693
$238$	0	0
$239$	−2890.40	−0.782277	−0.391138	−	0.920332i	$-0.627919\pi$
$239$	−2890.40	−0.782277	−0.391138	+	0.920332i	$0.627919\pi$
$240$	0	0
$241$	−1741.00	−0.465344	−0.232672	−	0.972555i	$-0.574747\pi$
$241$	−1741.00	−0.465344	−0.232672	+	0.972555i	$0.574747\pi$
$242$	0	0
$243$	−2385.59	−0.629777
$244$	0	0
$245$	−718.742	−0.187423
$246$	0	0
$247$	−634.704	−0.163503
$248$	0	0
$249$	−975.679	−0.248318
$250$	0	0
$251$	−4314.41	−1.08495	−0.542477	−	0.840071i	$-0.682513\pi$
$251$	−4314.41	−1.08495	−0.542477	+	0.840071i	$0.682513\pi$
$252$	0	0
$253$	−640.685	−0.159208
$254$	0	0
$255$	−25.5353	−0.00627092
$256$	0	0
$257$	−3109.13	−0.754639	−0.377319	−	0.926083i	$-0.623154\pi$
$257$	−3109.13	−0.754639	−0.377319	+	0.926083i	$0.623154\pi$
$258$	0	0
$259$	11120.0	2.66781
$260$	0	0
$261$	3982.91	0.944581
$262$	0	0
$263$	−3840.90	−0.900533	−0.450266	−	0.892894i	$-0.648671\pi$
$263$	−3840.90	−0.900533	−0.450266	+	0.892894i	$0.648671\pi$
$264$	0	0
$265$	−316.015	−0.0732552
$266$	0	0
$267$	578.777	0.132661
$268$	0	0
$269$	7685.29	1.74193	0.870967	−	0.491341i	$-0.163493\pi$
$269$	7685.29	1.74193	0.870967	+	0.491341i	$0.163493\pi$
$270$	0	0
$271$	−4651.00	−1.04254	−0.521270	−	0.853392i	$-0.674541\pi$
$271$	−4651.00	−1.04254	−0.521270	+	0.853392i	$0.674541\pi$
$272$	0	0
$273$	−533.714	−0.118322
$274$	0	0
$275$	−935.743	−0.205191
$276$	0	0
$277$	−5774.67	−1.25259	−0.626293	−	0.779588i	$-0.715428\pi$
$277$	−5774.67	−1.25259	−0.626293	+	0.779588i	$0.715428\pi$
$278$	0	0
$279$	5564.15	1.19397
$280$	0	0
$281$	−8230.83	−1.74737	−0.873683	−	0.486495i	$-0.838275\pi$
$281$	−8230.83	−1.74737	−0.873683	+	0.486495i	$0.838275\pi$
$282$	0	0
$283$	−3229.82	−0.678420	−0.339210	−	0.940711i	$-0.610160\pi$
$283$	−3229.82	−0.678420	−0.339210	+	0.940711i	$0.610160\pi$
$284$	0	0
$285$	62.7014	0.0130320
$286$	0	0
$287$	−2942.70	−0.605234
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	1042.58	0.210025
$292$	0	0
$293$	5178.99	1.03263	0.516313	−	0.856400i	$-0.327304\pi$
$293$	5178.99	1.03263	0.516313	+	0.856400i	$0.327304\pi$
$294$	0	0
$295$	−236.049	−0.0465875
$296$	0	0
$297$	466.625	0.0911661
$298$	0	0
$299$	1284.12	0.248370
$300$	0	0
$301$	−12415.2	−2.37742
$302$	0	0
$303$	−2168.43	−0.411133
$304$	0	0
$305$	275.909	0.0517984
$306$	0	0
$307$	3937.41	0.731986	0.365993	−	0.930618i	$-0.380729\pi$
$307$	3937.41	0.731986	0.365993	+	0.930618i	$0.380729\pi$
$308$	0	0
$309$	−213.752	−0.0393525
$310$	0	0
$311$	8551.65	1.55923	0.779613	−	0.626261i	$-0.215416\pi$
$311$	8551.65	1.55923	0.779613	+	0.626261i	$0.215416\pi$
$312$	0	0
$313$	−7157.11	−1.29247	−0.646236	−	0.763138i	$-0.723658\pi$
$313$	−7157.11	−1.29247	−0.646236	+	0.763138i	$0.723658\pi$
$314$	0	0
$315$	−989.663	−0.177020
$316$	0	0
$317$	−8648.95	−1.53241	−0.766204	−	0.642598i	$-0.777857\pi$
$317$	−8648.95	−1.53241	−0.766204	+	0.642598i	$0.777857\pi$
$318$	0	0
$319$	−1178.70	−0.206879
$320$	0	0
$321$	210.109	0.0365332
$322$	0	0
$323$	−709.633	−0.122245
$324$	0	0
$325$	1875.51	0.320106
$326$	0	0
$327$	435.534	0.0736547
$328$	0	0
$329$	−11817.7	−1.98034
$330$	0	0
$331$	3974.03	0.659917	0.329959	−	0.943995i	$-0.392965\pi$
$331$	3974.03	0.659917	0.329959	+	0.943995i	$0.392965\pi$
$332$	0	0
$333$	9490.27	1.56175
$334$	0	0
$335$	935.241	0.152530
$336$	0	0
$337$	6574.10	1.06265	0.531327	−	0.847167i	$-0.321694\pi$
$337$	6574.10	1.06265	0.531327	+	0.847167i	$0.321694\pi$
$338$	0	0
$339$	1481.39	0.237340
$340$	0	0
$341$	−1646.65	−0.261499
$342$	0	0
$343$	−6493.39	−1.02219
$344$	0	0
$345$	−126.856	−0.0197963
$346$	0	0
$347$	−10832.6	−1.67586	−0.837928	−	0.545781i	$-0.816233\pi$
$347$	−10832.6	−1.67586	−0.837928	+	0.545781i	$0.816233\pi$
$348$	0	0
$349$	−499.906	−0.0766743	−0.0383372	−	0.999265i	$-0.512206\pi$
$349$	−499.906	−0.0766743	−0.0383372	+	0.999265i	$0.512206\pi$
$350$	0	0
$351$	−935.254	−0.142223
$352$	0	0
$353$	11473.5	1.72995	0.864973	−	0.501818i	$-0.167335\pi$
$353$	11473.5	1.72995	0.864973	+	0.501818i	$0.167335\pi$
$354$	0	0
$355$	554.882	0.0829580
$356$	0	0
$357$	−596.720	−0.0884644
$358$	0	0
$359$	−2522.13	−0.370787	−0.185394	−	0.982664i	$-0.559356\pi$
$359$	−2522.13	−0.370787	−0.185394	+	0.982664i	$0.559356\pi$
$360$	0	0
$361$	−5116.51	−0.745956
$362$	0	0
$363$	1488.18	0.215177
$364$	0	0
$365$	−694.954	−0.0996590
$366$	0	0
$367$	8207.33	1.16735	0.583677	−	0.811986i	$-0.301613\pi$
$367$	8207.33	1.16735	0.583677	+	0.811986i	$0.301613\pi$
$368$	0	0
$369$	−2511.43	−0.354308
$370$	0	0
$371$	−7384.76	−1.03342
$372$	0	0
$373$	7506.69	1.04204	0.521021	−	0.853544i	$-0.325551\pi$
$373$	7506.69	1.04204	0.521021	+	0.853544i	$0.325551\pi$
$374$	0	0
$375$	−373.038	−0.0513696
$376$	0	0
$377$	2362.46	0.322740
$378$	0	0
$379$	−13096.7	−1.77501	−0.887507	−	0.460794i	$-0.847565\pi$
$379$	−13096.7	−1.77501	−0.887507	+	0.460794i	$0.847565\pi$
$380$	0	0
$381$	−2539.99	−0.341543
$382$	0	0
$383$	12694.2	1.69359	0.846795	−	0.531919i	$-0.178529\pi$
$383$	12694.2	1.69359	0.846795	+	0.531919i	$0.178529\pi$
$384$	0	0
$385$	292.880	0.0387703
$386$	0	0
$387$	−10595.7	−1.39175
$388$	0	0
$389$	5351.44	0.697503	0.348752	−	0.937215i	$-0.386606\pi$
$389$	5351.44	0.697503	0.348752	+	0.937215i	$0.386606\pi$
$390$	0	0
$391$	1435.72	0.185697
$392$	0	0
$393$	−1553.53	−0.199403
$394$	0	0
$395$	−656.897	−0.0836762
$396$	0	0
$397$	−7986.09	−1.00960	−0.504799	−	0.863237i	$-0.668433\pi$
$397$	−7986.09	−1.00960	−0.504799	+	0.863237i	$0.668433\pi$
$398$	0	0
$399$	1465.23	0.183843
$400$	0	0
$401$	−2930.11	−0.364894	−0.182447	−	0.983216i	$-0.558402\pi$
$401$	−2930.11	−0.364894	−0.182447	+	0.983216i	$0.558402\pi$
$402$	0	0
$403$	3300.38	0.407949
$404$	0	0
$405$	−797.226	−0.0978135
$406$	0	0
$407$	−2808.55	−0.342050
$408$	0	0
$409$	−1318.95	−0.159457	−0.0797285	−	0.996817i	$-0.525405\pi$
$409$	−1318.95	−0.159457	−0.0797285	+	0.996817i	$0.525405\pi$
$410$	0	0
$411$	1328.14	0.159397
$412$	0	0
$413$	−5516.09	−0.657213
$414$	0	0
$415$	−1073.13	−0.126934
$416$	0	0
$417$	−3149.60	−0.369872
$418$	0	0
$419$	−11191.2	−1.30483	−0.652416	−	0.757861i	$-0.726244\pi$
$419$	−11191.2	−1.30483	−0.652416	+	0.757861i	$0.726244\pi$
$420$	0	0
$421$	14340.3	1.66011	0.830054	−	0.557683i	$-0.188309\pi$
$421$	14340.3	1.66011	0.830054	+	0.557683i	$0.188309\pi$
$422$	0	0
$423$	−10085.7	−1.15930
$424$	0	0
$425$	2096.91	0.239330
$426$	0	0
$427$	6447.56	0.730724
$428$	0	0
$429$	134.799	0.0151705
$430$	0	0
$431$	−4327.51	−0.483640	−0.241820	−	0.970321i	$-0.577744\pi$
$431$	−4327.51	−0.483640	−0.241820	+	0.970321i	$0.577744\pi$
$432$	0	0
$433$	16771.0	1.86135	0.930674	−	0.365849i	$-0.119221\pi$
$433$	16771.0	1.86135	0.930674	+	0.365849i	$0.119221\pi$
$434$	0	0
$435$	−233.384	−0.0257239
$436$	0	0
$437$	−3525.37	−0.385907
$438$	0	0
$439$	3566.96	0.387794	0.193897	−	0.981022i	$-0.437887\pi$
$439$	3566.96	0.387794	0.193897	+	0.981022i	$0.437887\pi$
$440$	0	0
$441$	−14334.3	−1.54781
$442$	0	0
$443$	−5654.12	−0.606400	−0.303200	−	0.952927i	$-0.598055\pi$
$443$	−5654.12	−0.606400	−0.303200	+	0.952927i	$0.598055\pi$
$444$	0	0
$445$	636.583	0.0678133
$446$	0	0
$447$	−611.074	−0.0646595
$448$	0	0
$449$	−3179.47	−0.334184	−0.167092	−	0.985941i	$-0.553438\pi$
$449$	−3179.47	−0.334184	−0.167092	+	0.985941i	$0.553438\pi$
$450$	0	0
$451$	743.230	0.0775995
$452$	0	0
$453$	750.644	0.0778551
$454$	0	0
$455$	−587.019	−0.0604832
$456$	0	0
$457$	17359.1	1.77685	0.888427	−	0.459018i	$-0.151799\pi$
$457$	17359.1	1.77685	0.888427	+	0.459018i	$0.151799\pi$
$458$	0	0
$459$	−1045.66	−0.106334
$460$	0	0
$461$	−5090.46	−0.514287	−0.257144	−	0.966373i	$-0.582781\pi$
$461$	−5090.46	−0.514287	−0.257144	+	0.966373i	$0.582781\pi$
$462$	0	0
$463$	−12047.3	−1.20926	−0.604629	−	0.796507i	$-0.706679\pi$
$463$	−12047.3	−1.20926	−0.604629	+	0.796507i	$0.706679\pi$
$464$	0	0
$465$	−326.039	−0.0325155
$466$	0	0
$467$	−19507.9	−1.93301	−0.966507	−	0.256641i	$-0.917384\pi$
$467$	−19507.9	−1.93301	−0.966507	+	0.256641i	$0.917384\pi$
$468$	0	0
$469$	21855.1	2.15176
$470$	0	0
$471$	−1049.66	−0.102688
$472$	0	0
$473$	3135.68	0.304818
$474$	0	0
$475$	−5148.92	−0.497366
$476$	0	0
$477$	−6302.47	−0.604969
$478$	0	0
$479$	−13941.9	−1.32990	−0.664951	−	0.746887i	$-0.731548\pi$
$479$	−13941.9	−1.32990	−0.664951	+	0.746887i	$0.731548\pi$
$480$	0	0
$481$	5629.16	0.533612
$482$	0	0
$483$	−2964.43	−0.279268
$484$	0	0
$485$	1146.71	0.107360
$486$	0	0
$487$	−964.221	−0.0897187	−0.0448594	−	0.998993i	$-0.514284\pi$
$487$	−964.221	−0.0897187	−0.0448594	+	0.998993i	$0.514284\pi$
$488$	0	0
$489$	−324.302	−0.0299907
$490$	0	0
$491$	16193.2	1.48837	0.744183	−	0.667976i	$-0.232839\pi$
$491$	16193.2	1.48837	0.744183	+	0.667976i	$0.232839\pi$
$492$	0	0
$493$	2641.36	0.241300
$494$	0	0
$495$	249.957	0.0226964
$496$	0	0
$497$	12966.7	1.17030
$498$	0	0
$499$	−1317.09	−0.118158	−0.0590790	−	0.998253i	$-0.518816\pi$
$499$	−1317.09	−0.118158	−0.0590790	+	0.998253i	$0.518816\pi$
$500$	0	0
$501$	−3106.85	−0.277053
$502$	0	0
$503$	−14479.9	−1.28356	−0.641778	−	0.766890i	$-0.721803\pi$
$503$	−14479.9	−1.28356	−0.641778	+	0.766890i	$0.721803\pi$
$504$	0	0
$505$	−2385.01	−0.210161
$506$	0	0
$507$	2297.29	0.201235
$508$	0	0
$509$	18916.3	1.64725	0.823624	−	0.567136i	$-0.191948\pi$
$509$	18916.3	1.64725	0.823624	+	0.567136i	$0.191948\pi$
$510$	0	0
$511$	−16240.0	−1.40590
$512$	0	0
$513$	2567.60	0.220979
$514$	0	0
$515$	−235.101	−0.0201161
$516$	0	0
$517$	2984.76	0.253907
$518$	0	0
$519$	505.781	0.0427771
$520$	0	0
$521$	−18078.6	−1.52022	−0.760112	−	0.649792i	$-0.774856\pi$
$521$	−18078.6	−1.52022	−0.760112	+	0.649792i	$0.774856\pi$
$522$	0	0
$523$	1976.16	0.165223	0.0826114	−	0.996582i	$-0.473674\pi$
$523$	1976.16	0.165223	0.0826114	+	0.996582i	$0.473674\pi$
$524$	0	0
$525$	−4329.66	−0.359927
$526$	0	0
$527$	3690.00	0.305007
$528$	0	0
$529$	−5034.53	−0.413785
$530$	0	0
$531$	−4707.67	−0.384737
$532$	0	0
$533$	−1489.65	−0.121058
$534$	0	0
$535$	231.094	0.0186749
$536$	0	0
$537$	−675.081	−0.0542493
$538$	0	0
$539$	4242.09	0.338997
$540$	0	0
$541$	−3955.27	−0.314326	−0.157163	−	0.987573i	$-0.550235\pi$
$541$	−3955.27	−0.314326	−0.157163	+	0.987573i	$0.550235\pi$
$542$	0	0
$543$	3615.83	0.285765
$544$	0	0
$545$	479.033	0.0376505
$546$	0	0
$547$	15036.5	1.17535	0.587675	−	0.809097i	$-0.300043\pi$
$547$	15036.5	1.17535	0.587675	+	0.809097i	$0.300043\pi$
$548$	0	0
$549$	5502.62	0.427771
$550$	0	0
$551$	−6485.79	−0.501459
$552$	0	0
$553$	−15350.7	−1.18043
$554$	0	0
$555$	−556.095	−0.0425314
$556$	0	0
$557$	−23145.5	−1.76070	−0.880348	−	0.474328i	$-0.842691\pi$
$557$	−23145.5	−1.76070	−0.880348	+	0.474328i	$0.842691\pi$
$558$	0	0
$559$	−6284.83	−0.475528
$560$	0	0
$561$	150.712	0.0113424
$562$	0	0
$563$	−7004.72	−0.524358	−0.262179	−	0.965019i	$-0.584441\pi$
$563$	−7004.72	−0.524358	−0.262179	+	0.965019i	$0.584441\pi$
$564$	0	0
$565$	1629.35	0.121322
$566$	0	0
$567$	−18629.9	−1.37986
$568$	0	0
$569$	−7000.62	−0.515785	−0.257892	−	0.966174i	$-0.583028\pi$
$569$	−7000.62	−0.515785	−0.257892	+	0.966174i	$0.583028\pi$
$570$	0	0
$571$	−8499.45	−0.622927	−0.311463	−	0.950258i	$-0.600819\pi$
$571$	−8499.45	−0.622927	−0.311463	+	0.950258i	$0.600819\pi$
$572$	0	0
$573$	−586.778	−0.0427801
$574$	0	0
$575$	10417.2	0.755527
$576$	0	0
$577$	−9055.00	−0.653318	−0.326659	−	0.945142i	$-0.605923\pi$
$577$	−9055.00	−0.653318	−0.326659	+	0.945142i	$0.605923\pi$
$578$	0	0
$579$	−1183.88	−0.0849746
$580$	0	0
$581$	−25077.3	−1.79067
$582$	0	0
$583$	1865.15	0.132498
$584$	0	0
$585$	−500.987	−0.0354073
$586$	0	0
$587$	−2733.75	−0.192221	−0.0961105	−	0.995371i	$-0.530640\pi$
$587$	−2733.75	−0.192221	−0.0961105	+	0.995371i	$0.530640\pi$
$588$	0	0
$589$	−9060.70	−0.633854
$590$	0	0
$591$	−6347.60	−0.441802
$592$	0	0
$593$	19803.0	1.37135	0.685675	−	0.727908i	$-0.259507\pi$
$593$	19803.0	1.37135	0.685675	+	0.727908i	$0.259507\pi$
$594$	0	0
$595$	−656.318	−0.0452209
$596$	0	0
$597$	−2948.97	−0.202166
$598$	0	0
$599$	−5177.02	−0.353134	−0.176567	−	0.984289i	$-0.556499\pi$
$599$	−5177.02	−0.353134	−0.176567	+	0.984289i	$0.556499\pi$
$600$	0	0
$601$	15728.5	1.06752	0.533761	−	0.845635i	$-0.320778\pi$
$601$	15728.5	1.06752	0.533761	+	0.845635i	$0.320778\pi$
$602$	0	0
$603$	18652.1	1.25965
$604$	0	0
$605$	1636.82	0.109993
$606$	0	0
$607$	4410.43	0.294916	0.147458	−	0.989068i	$-0.452891\pi$
$607$	4410.43	0.294916	0.147458	+	0.989068i	$0.452891\pi$
$608$	0	0
$609$	−5453.81	−0.362889
$610$	0	0
$611$	−5982.35	−0.396104
$612$	0	0
$613$	6957.62	0.458427	0.229214	−	0.973376i	$-0.426385\pi$
$613$	6957.62	0.458427	0.229214	+	0.973376i	$0.426385\pi$
$614$	0	0
$615$	147.160	0.00964892
$616$	0	0
$617$	−7389.00	−0.482123	−0.241061	−	0.970510i	$-0.577496\pi$
$617$	−7389.00	−0.482123	−0.241061	+	0.970510i	$0.577496\pi$
$618$	0	0
$619$	3923.27	0.254749	0.127374	−	0.991855i	$-0.459345\pi$
$619$	3923.27	0.254749	0.127374	+	0.991855i	$0.459345\pi$
$620$	0	0
$621$	−5194.73	−0.335680
$622$	0	0
$623$	14875.9	0.956648
$624$	0	0
$625$	15008.2	0.960524
$626$	0	0
$627$	−370.070	−0.0235712
$628$	0	0
$629$	6293.70	0.398960
$630$	0	0
$631$	−18684.3	−1.17878	−0.589390	−	0.807849i	$-0.700632\pi$
$631$	−18684.3	−1.17878	−0.589390	+	0.807849i	$0.700632\pi$
$632$	0	0
$633$	301.208	0.0189130
$634$	0	0
$635$	−2793.68	−0.174588
$636$	0	0
$637$	−8502.39	−0.528850
$638$	0	0
$639$	11066.4	0.685099
$640$	0	0
$641$	26642.0	1.64165	0.820823	−	0.571182i	$-0.193515\pi$
$641$	26642.0	1.64165	0.820823	+	0.571182i	$0.193515\pi$
$642$	0	0
$643$	8875.90	0.544372	0.272186	−	0.962245i	$-0.412253\pi$
$643$	8875.90	0.544372	0.272186	+	0.962245i	$0.412253\pi$
$644$	0	0
$645$	620.868	0.0379018
$646$	0	0
$647$	16201.7	0.984473	0.492237	−	0.870461i	$-0.336180\pi$
$647$	16201.7	0.984473	0.492237	+	0.870461i	$0.336180\pi$
$648$	0	0
$649$	1393.18	0.0842639
$650$	0	0
$651$	−7619.02	−0.458699
$652$	0	0
$653$	−13913.3	−0.833798	−0.416899	−	0.908953i	$-0.636883\pi$
$653$	−13913.3	−0.833798	−0.416899	+	0.908953i	$0.636883\pi$
$654$	0	0
$655$	−1708.69	−0.101930
$656$	0	0
$657$	−13859.9	−0.823022
$658$	0	0
$659$	11346.8	0.670729	0.335365	−	0.942088i	$-0.391141\pi$
$659$	11346.8	0.670729	0.335365	+	0.942088i	$0.391141\pi$
$660$	0	0
$661$	−10485.2	−0.616986	−0.308493	−	0.951227i	$-0.599825\pi$
$661$	−10485.2	−0.616986	−0.308493	+	0.951227i	$0.599825\pi$
$662$	0	0
$663$	−302.071	−0.0176945
$664$	0	0
$665$	1611.57	0.0939762
$666$	0	0
$667$	13121.9	0.761745
$668$	0	0
$669$	4350.31	0.251409
$670$	0	0
$671$	−1628.44	−0.0936890
$672$	0	0
$673$	−277.557	−0.0158975	−0.00794876	−	0.999968i	$-0.502530\pi$
$673$	−277.557	−0.0158975	−0.00794876	+	0.999968i	$0.502530\pi$
$674$	0	0
$675$	−7587.09	−0.432633
$676$	0	0
$677$	8151.49	0.462758	0.231379	−	0.972864i	$-0.425676\pi$
$677$	8151.49	0.462758	0.231379	+	0.972864i	$0.425676\pi$
$678$	0	0
$679$	26796.8	1.51453
$680$	0	0
$681$	−5360.34	−0.301628
$682$	0	0
$683$	−9376.93	−0.525327	−0.262663	−	0.964888i	$-0.584601\pi$
$683$	−9376.93	−0.525327	−0.262663	+	0.964888i	$0.584601\pi$
$684$	0	0
$685$	1460.78	0.0814799
$686$	0	0
$687$	5051.14	0.280514
$688$	0	0
$689$	−3738.31	−0.206703
$690$	0	0
$691$	−15608.9	−0.859320	−0.429660	−	0.902991i	$-0.641367\pi$
$691$	−15608.9	−0.859320	−0.429660	+	0.902991i	$0.641367\pi$
$692$	0	0
$693$	5841.09	0.320180
$694$	0	0
$695$	−3464.17	−0.189070
$696$	0	0
$697$	−1665.51	−0.0905104
$698$	0	0
$699$	−5966.93	−0.322875
$700$	0	0
$701$	−10219.3	−0.550610	−0.275305	−	0.961357i	$-0.588779\pi$
$701$	−10219.3	−0.550610	−0.275305	+	0.961357i	$0.588779\pi$
$702$	0	0
$703$	−15454.0	−0.829103
$704$	0	0
$705$	590.986	0.0315714
$706$	0	0
$707$	−55733.9	−2.96476
$708$	0	0
$709$	−5044.05	−0.267184	−0.133592	−	0.991036i	$-0.542651\pi$
$709$	−5044.05	−0.267184	−0.133592	+	0.991036i	$0.542651\pi$
$710$	0	0
$711$	−13100.9	−0.691030
$712$	0	0
$713$	18331.5	0.962860
$714$	0	0
$715$	148.262	0.00775479
$716$	0	0
$717$	3377.78	0.175935
$718$	0	0
$719$	2995.14	0.155354	0.0776772	−	0.996979i	$-0.475250\pi$
$719$	2995.14	0.155354	0.0776772	+	0.996979i	$0.475250\pi$
$720$	0	0
$721$	−5493.93	−0.283779
$722$	0	0
$723$	2034.58	0.104657
$724$	0	0
$725$	19165.1	0.981755
$726$	0	0
$727$	19956.8	1.01809	0.509047	−	0.860738i	$-0.329998\pi$
$727$	19956.8	1.01809	0.509047	+	0.860738i	$0.329998\pi$
$728$	0	0
$729$	−13958.8	−0.709178
$730$	0	0
$731$	−7026.78	−0.355533
$732$	0	0
$733$	1975.36	0.0995386	0.0497693	−	0.998761i	$-0.484151\pi$
$733$	1975.36	0.0995386	0.0497693	+	0.998761i	$0.484151\pi$
$734$	0	0
$735$	839.938	0.0421518
$736$	0	0
$737$	−5519.88	−0.275885
$738$	0	0
$739$	37767.3	1.87996	0.939982	−	0.341225i	$-0.110842\pi$
$739$	37767.3	1.87996	0.939982	+	0.341225i	$0.110842\pi$
$740$	0	0
$741$	741.729	0.0367721
$742$	0	0
$743$	−35262.1	−1.74111	−0.870553	−	0.492074i	$-0.836239\pi$
$743$	−35262.1	−1.74111	−0.870553	+	0.492074i	$0.836239\pi$
$744$	0	0
$745$	−672.106	−0.0330524
$746$	0	0
$747$	−21402.0	−1.04827
$748$	0	0
$749$	5400.30	0.263448
$750$	0	0
$751$	5472.38	0.265899	0.132949	−	0.991123i	$-0.457555\pi$
$751$	5472.38	0.265899	0.132949	+	0.991123i	$0.457555\pi$
$752$	0	0
$753$	5041.92	0.244008
$754$	0	0
$755$	825.616	0.0397977
$756$	0	0
$757$	32341.0	1.55278	0.776389	−	0.630254i	$-0.217049\pi$
$757$	32341.0	1.55278	0.776389	+	0.630254i	$0.217049\pi$
$758$	0	0
$759$	748.719	0.0358060
$760$	0	0
$761$	−5874.34	−0.279822	−0.139911	−	0.990164i	$-0.544682\pi$
$761$	−5874.34	−0.279822	−0.139911	+	0.990164i	$0.544682\pi$
$762$	0	0
$763$	11194.3	0.531139
$764$	0	0
$765$	−560.130	−0.0264726
$766$	0	0
$767$	−2792.35	−0.131455
$768$	0	0
$769$	4730.89	0.221847	0.110923	−	0.993829i	$-0.464619\pi$
$769$	4730.89	0.221847	0.110923	+	0.993829i	$0.464619\pi$
$770$	0	0
$771$	3633.40	0.169719
$772$	0	0
$773$	22031.8	1.02513	0.512567	−	0.858647i	$-0.328695\pi$
$773$	22031.8	1.02513	0.512567	+	0.858647i	$0.328695\pi$
$774$	0	0
$775$	26773.7	1.24096
$776$	0	0
$777$	−12995.1	−0.599994
$778$	0	0
$779$	4089.62	0.188095
$780$	0	0
$781$	−3274.97	−0.150048
$782$	0	0
$783$	−9557.00	−0.436193
$784$	0	0
$785$	−1154.50	−0.0524916
$786$	0	0
$787$	19869.3	0.899955	0.449978	−	0.893040i	$-0.351432\pi$
$787$	19869.3	0.899955	0.449978	+	0.893040i	$0.351432\pi$
$788$	0	0
$789$	4488.56	0.202531
$790$	0	0
$791$	38075.3	1.71151
$792$	0	0
$793$	3263.88	0.146159
$794$	0	0
$795$	369.302	0.0164752
$796$	0	0
$797$	−15708.3	−0.698140	−0.349070	−	0.937097i	$-0.613502\pi$
$797$	−15708.3	−0.698140	−0.349070	+	0.937097i	$0.613502\pi$
$798$	0	0
$799$	−6688.58	−0.296151
$800$	0	0
$801$	12695.8	0.560028
$802$	0	0
$803$	4101.69	0.180256
$804$	0	0
$805$	−3260.51	−0.142755
$806$	0	0
$807$	−8981.20	−0.391764
$808$	0	0
$809$	−33319.6	−1.44803	−0.724015	−	0.689785i	$-0.757705\pi$
$809$	−33319.6	−1.44803	−0.724015	+	0.689785i	$0.757705\pi$
$810$	0	0
$811$	9038.44	0.391347	0.195673	−	0.980669i	$-0.437311\pi$
$811$	9038.44	0.391347	0.195673	+	0.980669i	$0.437311\pi$
$812$	0	0
$813$	5435.26	0.234469
$814$	0	0
$815$	−356.692	−0.0153305
$816$	0	0
$817$	17254.1	0.738854
$818$	0	0
$819$	−11707.3	−0.499494
$820$	0	0
$821$	21590.8	0.917815	0.458907	−	0.888484i	$-0.348241\pi$
$821$	21590.8	0.917815	0.458907	+	0.888484i	$0.348241\pi$
$822$	0	0
$823$	18740.9	0.793764	0.396882	−	0.917870i	$-0.370092\pi$
$823$	18740.9	0.793764	0.396882	+	0.917870i	$0.370092\pi$
$824$	0	0
$825$	1093.53	0.0461477
$826$	0	0
$827$	36493.1	1.53445	0.767224	−	0.641379i	$-0.221637\pi$
$827$	36493.1	1.53445	0.767224	+	0.641379i	$0.221637\pi$
$828$	0	0
$829$	29244.1	1.22520	0.612599	−	0.790394i	$-0.290124\pi$
$829$	29244.1	1.22520	0.612599	+	0.790394i	$0.290124\pi$
$830$	0	0
$831$	6748.41	0.281708
$832$	0	0
$833$	−9506.13	−0.395400
$834$	0	0
$835$	−3417.15	−0.141623
$836$	0	0
$837$	−13351.2	−0.551356
$838$	0	0
$839$	18879.0	0.776846	0.388423	−	0.921481i	$-0.373020\pi$
$839$	18879.0	0.776846	0.388423	+	0.921481i	$0.373020\pi$
$840$	0	0
$841$	−247.923	−0.0101654
$842$	0	0
$843$	9618.73	0.392985
$844$	0	0
$845$	2526.73	0.102867
$846$	0	0
$847$	38249.8	1.55169
$848$	0	0
$849$	3774.44	0.152578
$850$	0	0
$851$	31266.3	1.25945
$852$	0	0
$853$	4398.33	0.176549	0.0882744	−	0.996096i	$-0.471865\pi$
$853$	4398.33	0.176549	0.0882744	+	0.996096i	$0.471865\pi$
$854$	0	0
$855$	1375.39	0.0550143
$856$	0	0
$857$	−25903.0	−1.03247	−0.516236	−	0.856446i	$-0.672667\pi$
$857$	−25903.0	−1.03247	−0.516236	+	0.856446i	$0.672667\pi$
$858$	0	0
$859$	−29283.3	−1.16313	−0.581567	−	0.813499i	$-0.697560\pi$
$859$	−29283.3	−1.16313	−0.581567	+	0.813499i	$0.697560\pi$
$860$	0	0
$861$	3438.91	0.136118
$862$	0	0
$863$	−33015.8	−1.30228	−0.651141	−	0.758957i	$-0.725709\pi$
$863$	−33015.8	−1.30228	−0.651141	+	0.758957i	$0.725709\pi$
$864$	0	0
$865$	556.296	0.0218666
$866$	0	0
$867$	−337.732	−0.0132295
$868$	0	0
$869$	3877.07	0.151347
$870$	0	0
$871$	11063.5	0.430392
$872$	0	0
$873$	22869.6	0.886618
$874$	0	0
$875$	−9587.96	−0.370437
$876$	0	0
$877$	471.610	0.0181587	0.00907933	−	0.999959i	$-0.497110\pi$
$877$	471.610	0.0181587	0.00907933	+	0.999959i	$0.497110\pi$
$878$	0	0
$879$	−6052.28	−0.232239
$880$	0	0
$881$	−35583.2	−1.36076	−0.680380	−	0.732859i	$-0.738185\pi$
$881$	−35583.2	−1.36076	−0.680380	+	0.732859i	$0.738185\pi$
$882$	0	0
$883$	21916.2	0.835267	0.417633	−	0.908616i	$-0.362860\pi$
$883$	21916.2	0.835267	0.417633	+	0.908616i	$0.362860\pi$
$884$	0	0
$885$	275.852	0.0104776
$886$	0	0
$887$	−25302.2	−0.957797	−0.478899	−	0.877870i	$-0.658964\pi$
$887$	−25302.2	−0.957797	−0.478899	+	0.877870i	$0.658964\pi$
$888$	0	0
$889$	−65283.8	−2.46293
$890$	0	0
$891$	4705.31	0.176918
$892$	0	0
$893$	16423.7	0.615450
$894$	0	0
$895$	−742.506	−0.0277310
$896$	0	0
$897$	−1500.65	−0.0558589
$898$	0	0
$899$	33725.3	1.25117
$900$	0	0
$901$	−4179.63	−0.154543
$902$	0	0
$903$	14508.7	0.534684
$904$	0	0
$905$	3976.97	0.146076
$906$	0	0
$907$	−44853.4	−1.64204	−0.821021	−	0.570899i	$-0.806595\pi$
$907$	−44853.4	−1.64204	−0.821021	+	0.570899i	$0.806595\pi$
$908$	0	0
$909$	−47565.7	−1.73559
$910$	0	0
$911$	7513.57	0.273256	0.136628	−	0.990622i	$-0.456374\pi$
$911$	7513.57	0.273256	0.136628	+	0.990622i	$0.456374\pi$
$912$	0	0
$913$	6333.70	0.229589
$914$	0	0
$915$	−322.434	−0.0116495
$916$	0	0
$917$	−39929.4	−1.43794
$918$	0	0
$919$	47162.6	1.69287	0.846436	−	0.532490i	$-0.178744\pi$
$919$	47162.6	1.69287	0.846436	+	0.532490i	$0.178744\pi$
$920$	0	0
$921$	−4601.34	−0.164625
$922$	0	0
$923$	6564.01	0.234081
$924$	0	0
$925$	45665.5	1.62322
$926$	0	0
$927$	−4688.75	−0.166126
$928$	0	0
$929$	29757.0	1.05091	0.525456	−	0.850821i	$-0.323895\pi$
$929$	29757.0	1.05091	0.525456	+	0.850821i	$0.323895\pi$
$930$	0	0
$931$	23342.1	0.821703
$932$	0	0
$933$	−9993.65	−0.350672
$934$	0	0
$935$	165.765	0.00579795
$936$	0	0
$937$	28802.1	1.00419	0.502093	−	0.864814i	$-0.332564\pi$
$937$	28802.1	1.00419	0.502093	+	0.864814i	$0.332564\pi$
$938$	0	0
$939$	8363.96	0.290679
$940$	0	0
$941$	−30380.2	−1.05246	−0.526230	−	0.850342i	$-0.676395\pi$
$941$	−30380.2	−1.05246	−0.526230	+	0.850342i	$0.676395\pi$
$942$	0	0
$943$	−8274.06	−0.285727
$944$	0	0
$945$	2374.70	0.0817450
$946$	0	0
$947$	57677.1	1.97915	0.989574	−	0.144026i	$-0.0460048\pi$
$947$	57677.1	1.97915	0.989574	+	0.144026i	$0.0460048\pi$
$948$	0	0
$949$	−8221.00	−0.281206
$950$	0	0
$951$	10107.4	0.344641
$952$	0	0
$953$	30081.6	1.02250	0.511248	−	0.859433i	$-0.329184\pi$
$953$	30081.6	1.02250	0.511248	+	0.859433i	$0.329184\pi$
$954$	0	0
$955$	−645.383	−0.0218682
$956$	0	0
$957$	1377.45	0.0465275
$958$	0	0
$959$	34136.2	1.14944
$960$	0	0
$961$	17323.5	0.581501
$962$	0	0
$963$	4608.85	0.154224
$964$	0	0
$965$	−1302.12	−0.0434370
$966$	0	0
$967$	44350.8	1.47490	0.737448	−	0.675404i	$-0.236031\pi$
$967$	44350.8	1.47490	0.737448	+	0.675404i	$0.236031\pi$
$968$	0	0
$969$	829.293	0.0274930
$970$	0	0
$971$	646.546	0.0213683	0.0106842	−	0.999943i	$-0.496599\pi$
$971$	646.546	0.0213683	0.0106842	+	0.999943i	$0.496599\pi$
$972$	0	0
$973$	−80952.2	−2.66722
$974$	0	0
$975$	−2191.76	−0.0719922
$976$	0	0
$977$	2673.22	0.0875372	0.0437686	−	0.999042i	$-0.486064\pi$
$977$	2673.22	0.0875372	0.0437686	+	0.999042i	$0.486064\pi$
$978$	0	0
$979$	−3757.18	−0.122656
$980$	0	0
$981$	9553.65	0.310932
$982$	0	0
$983$	5637.62	0.182922	0.0914609	−	0.995809i	$-0.470846\pi$
$983$	5637.62	0.182922	0.0914609	+	0.995809i	$0.470846\pi$
$984$	0	0
$985$	−6981.57	−0.225839
$986$	0	0
$987$	13810.4	0.445381
$988$	0	0
$989$	−34908.2	−1.12236
$990$	0	0
$991$	−43188.7	−1.38439	−0.692197	−	0.721708i	$-0.743357\pi$
$991$	−43188.7	−1.38439	−0.692197	+	0.721708i	$0.743357\pi$
$992$	0	0
$993$	−4644.14	−0.148416
$994$	0	0
$995$	−3243.50	−0.103343
$996$	0	0
$997$	39179.4	1.24456	0.622279	−	0.782795i	$-0.286207\pi$
$997$	39179.4	1.24456	0.622279	+	0.782795i	$0.286207\pi$
$998$	0	0
$999$	−22771.9	−0.721194

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1088.4.a.y.1.2		3
4.3	odd	2		1088.4.a.u.1.2		3
8.3	odd	2		272.4.a.j.1.2		3
8.5	even	2		136.4.a.b.1.2	✓	3
24.5	odd	2		1224.4.a.i.1.2		3
24.11	even	2		2448.4.a.bj.1.2		3
136.101	even	2		2312.4.a.d.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.a.b.1.2	✓	3	8.5	even	2
272.4.a.j.1.2		3	8.3	odd	2
1088.4.a.u.1.2		3	4.3	odd	2
1088.4.a.y.1.2		3	1.1	even	1	trivial
1224.4.a.i.1.2		3	24.5	odd	2
2312.4.a.d.1.2		3	136.101	even	2
2448.4.a.bj.1.2		3	24.11	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:	\( N \)	\(=\)	\( 1088 = 2^{6} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1088.a (trivial)

\(f(q)\)	\(=\)	\(q-1.16862 q^{3} -1.28534 q^{5} -30.0364 q^{7} -25.6343 q^{9} +O(q^{10})\)	\(q-1.16862 q^{3} -1.28534 q^{5} -30.0364 q^{7} -25.6343 q^{9} +7.58621 q^{11} -15.2050 q^{13} +1.50208 q^{15} -17.0000 q^{17} +41.7431 q^{19} +35.1012 q^{21} -84.4540 q^{23} -123.348 q^{25} +61.5097 q^{27} -155.374 q^{29} -217.059 q^{31} -8.86541 q^{33} +38.6070 q^{35} -370.217 q^{37} +17.7689 q^{39} +97.9713 q^{41} +413.340 q^{43} +32.9488 q^{45} +393.446 q^{47} +559.184 q^{49} +19.8666 q^{51} +245.861 q^{53} -9.75086 q^{55} -48.7819 q^{57} +183.647 q^{59} -214.658 q^{61} +769.962 q^{63} +19.5436 q^{65} -727.621 q^{67} +98.6948 q^{69} -431.701 q^{71} +540.677 q^{73} +144.147 q^{75} -227.862 q^{77} +511.069 q^{79} +620.245 q^{81} +834.896 q^{83} +21.8508 q^{85} +181.574 q^{87} -495.264 q^{89} +456.703 q^{91} +253.660 q^{93} -53.6541 q^{95} -892.146 q^{97} -194.467 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{3} + 8 q^{5} - 2 q^{7} - 5 q^{9}+O(q^{10}) \) 3 * q + 4 * q^3 + 8 * q^5 - 2 * q^7 - 5 * q^9	\( 3 q + 4 q^{3} + 8 q^{5} - 2 q^{7} - 5 q^{9} + 84 q^{11} + 50 q^{13} - 148 q^{15} - 51 q^{17} + 224 q^{19} + 16 q^{21} - 234 q^{23} + 161 q^{25} + 292 q^{27} + 72 q^{29} - 2 q^{31} + 148 q^{33} + 428 q^{35} - 100 q^{37} + 156 q^{39} + 218 q^{41} - 44 q^{43} - 768 q^{45} - 16 q^{47} + 403 q^{49} - 68 q^{51} - 462 q^{53} + 460 q^{55} + 688 q^{57} + 68 q^{59} - 460 q^{61} + 586 q^{63} + 408 q^{65} + 1008 q^{67} - 400 q^{69} + 518 q^{71} + 838 q^{73} - 732 q^{75} + 904 q^{77} + 1238 q^{79} + 455 q^{81} + 1148 q^{83} - 136 q^{85} + 1108 q^{87} - 2506 q^{89} + 1416 q^{91} - 648 q^{93} + 40 q^{95} + 2098 q^{97} + 384 q^{99}+O(q^{100}) \) 3 * q + 4 * q^3 + 8 * q^5 - 2 * q^7 - 5 * q^9 + 84 * q^11 + 50 * q^13 - 148 * q^15 - 51 * q^17 + 224 * q^19 + 16 * q^21 - 234 * q^23 + 161 * q^25 + 292 * q^27 + 72 * q^29 - 2 * q^31 + 148 * q^33 + 428 * q^35 - 100 * q^37 + 156 * q^39 + 218 * q^41 - 44 * q^43 - 768 * q^45 - 16 * q^47 + 403 * q^49 - 68 * q^51 - 462 * q^53 + 460 * q^55 + 688 * q^57 + 68 * q^59 - 460 * q^61 + 586 * q^63 + 408 * q^65 + 1008 * q^67 - 400 * q^69 + 518 * q^71 + 838 * q^73 - 732 * q^75 + 904 * q^77 + 1238 * q^79 + 455 * q^81 + 1148 * q^83 - 136 * q^85 + 1108 * q^87 - 2506 * q^89 + 1416 * q^91 - 648 * q^93 + 40 * q^95 + 2098 * q^97 + 384 * q^99

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