LMFDB - Embedded newform 688.4.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [688,4,Mod(1,688)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(688, 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("688.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 688 = 2^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	688.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:	$40.5933140839$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 32x^{4} - 16x^{3} + 251x^{2} + 276x + 60 $ x^6 - 32x^4 - 16x^3 + 251x^2 + 276x + 60
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.17112$ of defining polynomial
Character	$\chi$	$=$	688.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.46717 q^{3} -7.54340 q^{5} -4.58222 q^{7} -20.9131 q^{9} +O(q^{10})$	$q-2.46717 q^{3} -7.54340 q^{5} -4.58222 q^{7} -20.9131 q^{9} -26.9150 q^{11} -15.6529 q^{13} +18.6109 q^{15} +27.2420 q^{17} -38.3104 q^{19} +11.3051 q^{21} -82.5575 q^{23} -68.0971 q^{25} +118.210 q^{27} -34.2852 q^{29} -119.055 q^{31} +66.4040 q^{33} +34.5655 q^{35} +378.527 q^{37} +38.6185 q^{39} +385.478 q^{41} +43.0000 q^{43} +157.756 q^{45} -271.022 q^{47} -322.003 q^{49} -67.2108 q^{51} -329.363 q^{53} +203.031 q^{55} +94.5184 q^{57} +173.956 q^{59} +54.5012 q^{61} +95.8283 q^{63} +118.076 q^{65} +906.954 q^{67} +203.684 q^{69} +621.376 q^{71} -1025.87 q^{73} +168.007 q^{75} +123.331 q^{77} +737.945 q^{79} +273.009 q^{81} -558.465 q^{83} -205.498 q^{85} +84.5875 q^{87} +1631.31 q^{89} +71.7252 q^{91} +293.729 q^{93} +288.991 q^{95} -406.607 q^{97} +562.875 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 7 q^{3} + 43 q^{5} - 8 q^{7} + 81 q^{9}+O(q^{10}) $ 6 * q - 7 * q^3 + 43 * q^5 - 8 * q^7 + 81 * q^9	$ 6 q - 7 q^{3} + 43 q^{5} - 8 q^{7} + 81 q^{9} + 28 q^{11} + 56 q^{13} + 124 q^{15} + 19 q^{17} + 75 q^{19} - 18 q^{21} - 131 q^{23} + 105 q^{25} - 238 q^{27} + 515 q^{29} - 237 q^{31} + 540 q^{33} - 198 q^{35} + 269 q^{37} - 290 q^{39} + 471 q^{41} + 258 q^{43} + 334 q^{45} - 415 q^{47} + 350 q^{49} + 1241 q^{51} + 450 q^{53} + 1732 q^{55} - 1000 q^{57} - 356 q^{59} - 1328 q^{61} + 2290 q^{63} - 62 q^{65} + 632 q^{67} - 1130 q^{69} + 144 q^{71} + 864 q^{73} + 2494 q^{75} + 2660 q^{77} + 1613 q^{79} - 102 q^{81} + 682 q^{83} + 84 q^{85} - 449 q^{87} + 3378 q^{89} + 3900 q^{91} + 1879 q^{93} + 79 q^{95} - 55 q^{97} + 1612 q^{99}+O(q^{100}) $ 6 * q - 7 * q^3 + 43 * q^5 - 8 * q^7 + 81 * q^9 + 28 * q^11 + 56 * q^13 + 124 * q^15 + 19 * q^17 + 75 * q^19 - 18 * q^21 - 131 * q^23 + 105 * q^25 - 238 * q^27 + 515 * q^29 - 237 * q^31 + 540 * q^33 - 198 * q^35 + 269 * q^37 - 290 * q^39 + 471 * q^41 + 258 * q^43 + 334 * q^45 - 415 * q^47 + 350 * q^49 + 1241 * q^51 + 450 * q^53 + 1732 * q^55 - 1000 * q^57 - 356 * q^59 - 1328 * q^61 + 2290 * q^63 - 62 * q^65 + 632 * q^67 - 1130 * q^69 + 144 * q^71 + 864 * q^73 + 2494 * q^75 + 2660 * q^77 + 1613 * q^79 - 102 * q^81 + 682 * q^83 + 84 * q^85 - 449 * q^87 + 3378 * q^89 + 3900 * q^91 + 1879 * q^93 + 79 * q^95 - 55 * q^97 + 1612 * 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
$3$	−2.46717	−0.474808	−0.237404	−	0.971411i	$-0.576296\pi$
$3$	−2.46717	−0.474808	−0.237404	+	0.971411i	$0.576296\pi$
$4$	0	0
$5$	−7.54340	−0.674702	−0.337351	−	0.941379i	$-0.609531\pi$
$5$	−7.54340	−0.674702	−0.337351	+	0.941379i	$0.609531\pi$
$6$	0	0
$7$	−4.58222	−0.247417	−0.123708	−	0.992319i	$-0.539479\pi$
$7$	−4.58222	−0.247417	−0.123708	+	0.992319i	$0.539479\pi$
$8$	0	0
$9$	−20.9131	−0.774558
$10$	0	0
$11$	−26.9150	−0.737744	−0.368872	−	0.929480i	$-0.620256\pi$
$11$	−26.9150	−0.737744	−0.368872	+	0.929480i	$0.620256\pi$
$12$	0	0
$13$	−15.6529	−0.333949	−0.166975	−	0.985961i	$-0.553400\pi$
$13$	−15.6529	−0.333949	−0.166975	+	0.985961i	$0.553400\pi$
$14$	0	0
$15$	18.6109	0.320354
$16$	0	0
$17$	27.2420	0.388657	0.194328	−	0.980937i	$-0.437747\pi$
$17$	27.2420	0.388657	0.194328	+	0.980937i	$0.437747\pi$
$18$	0	0
$19$	−38.3104	−0.462580	−0.231290	−	0.972885i	$-0.574295\pi$
$19$	−38.3104	−0.462580	−0.231290	+	0.972885i	$0.574295\pi$
$20$	0	0
$21$	11.3051	0.117475
$22$	0	0
$23$	−82.5575	−0.748453	−0.374227	−	0.927337i	$-0.622092\pi$
$23$	−82.5575	−0.748453	−0.374227	+	0.927337i	$0.622092\pi$
$24$	0	0
$25$	−68.0971	−0.544777
$26$	0	0
$27$	118.210	0.842574
$28$	0	0
$29$	−34.2852	−0.219538	−0.109769	−	0.993957i	$-0.535011\pi$
$29$	−34.2852	−0.219538	−0.109769	+	0.993957i	$0.535011\pi$
$30$	0	0
$31$	−119.055	−0.689770	−0.344885	−	0.938645i	$-0.612082\pi$
$31$	−119.055	−0.689770	−0.344885	+	0.938645i	$0.612082\pi$
$32$	0	0
$33$	66.4040	0.350286
$34$	0	0
$35$	34.5655	0.166933
$36$	0	0
$37$	378.527	1.68188	0.840939	−	0.541129i	$-0.182003\pi$
$37$	378.527	1.68188	0.840939	+	0.541129i	$0.182003\pi$
$38$	0	0
$39$	38.6185	0.158562
$40$	0	0
$41$	385.478	1.46833	0.734166	−	0.678970i	$-0.237573\pi$
$41$	385.478	1.46833	0.734166	+	0.678970i	$0.237573\pi$
$42$	0	0
$43$	43.0000	0.152499
$43$	43.0000	0.152499
$44$	0	0
$45$	157.756	0.522596
$46$	0	0
$47$	−271.022	−0.841120	−0.420560	−	0.907265i	$-0.638166\pi$
$47$	−271.022	−0.841120	−0.420560	+	0.907265i	$0.638166\pi$
$48$	0	0
$49$	−322.003	−0.938785
$50$	0	0
$51$	−67.2108	−0.184537
$52$	0	0
$53$	−329.363	−0.853612	−0.426806	−	0.904343i	$-0.640361\pi$
$53$	−329.363	−0.853612	−0.426806	+	0.904343i	$0.640361\pi$
$54$	0	0
$55$	203.031	0.497757
$56$	0	0
$57$	94.5184	0.219636
$58$	0	0
$59$	173.956	0.383849	0.191924	−	0.981410i	$-0.438527\pi$
$59$	173.956	0.383849	0.191924	+	0.981410i	$0.438527\pi$
$60$	0	0
$61$	54.5012	0.114396	0.0571981	−	0.998363i	$-0.481783\pi$
$61$	54.5012	0.114396	0.0571981	+	0.998363i	$0.481783\pi$
$62$	0	0
$63$	95.8283	0.191639
$64$	0	0
$65$	118.076	0.225316
$66$	0	0
$67$	906.954	1.65376	0.826881	−	0.562377i	$-0.190113\pi$
$67$	906.954	1.65376	0.826881	+	0.562377i	$0.190113\pi$
$68$	0	0
$69$	203.684	0.355371
$70$	0	0
$71$	621.376	1.03864	0.519322	−	0.854579i	$-0.326184\pi$
$71$	621.376	1.03864	0.519322	+	0.854579i	$0.326184\pi$
$72$	0	0
$73$	−1025.87	−1.64477	−0.822387	−	0.568928i	$-0.807358\pi$
$73$	−1025.87	−1.64477	−0.822387	+	0.568928i	$0.807358\pi$
$74$	0	0
$75$	168.007	0.258664
$76$	0	0
$77$	123.331	0.182530
$78$	0	0
$79$	737.945	1.05095	0.525476	−	0.850808i	$-0.323887\pi$
$79$	737.945	1.05095	0.525476	+	0.850808i	$0.323887\pi$
$80$	0	0
$81$	273.009	0.374497
$82$	0	0
$83$	−558.465	−0.738548	−0.369274	−	0.929321i	$-0.620394\pi$
$83$	−558.465	−0.738548	−0.369274	+	0.929321i	$0.620394\pi$
$84$	0	0
$85$	−205.498	−0.262228
$86$	0	0
$87$	84.5875	0.104238
$88$	0	0
$89$	1631.31	1.94291	0.971453	−	0.237231i	$-0.0762398\pi$
$89$	1631.31	1.94291	0.971453	+	0.237231i	$0.0762398\pi$
$90$	0	0
$91$	71.7252	0.0826247
$92$	0	0
$93$	293.729	0.327508
$94$	0	0
$95$	288.991	0.312104
$96$	0	0
$97$	−406.607	−0.425616	−0.212808	−	0.977094i	$-0.568261\pi$
$97$	−406.607	−0.425616	−0.212808	+	0.977094i	$0.568261\pi$
$98$	0	0
$99$	562.875	0.571425
$100$	0	0
$101$	1000.43	0.985606	0.492803	−	0.870141i	$-0.335972\pi$
$101$	1000.43	0.985606	0.492803	+	0.870141i	$0.335972\pi$
$102$	0	0
$103$	1659.81	1.58783	0.793913	−	0.608031i	$-0.208040\pi$
$103$	1659.81	1.58783	0.793913	+	0.608031i	$0.208040\pi$
$104$	0	0
$105$	−85.2792	−0.0792609
$106$	0	0
$107$	151.590	0.136961	0.0684803	−	0.997652i	$-0.478185\pi$
$107$	151.590	0.136961	0.0684803	+	0.997652i	$0.478185\pi$
$108$	0	0
$109$	1092.76	0.960254	0.480127	−	0.877199i	$-0.340591\pi$
$109$	1092.76	0.960254	0.480127	+	0.877199i	$0.340591\pi$
$110$	0	0
$111$	−933.892	−0.798569
$112$	0	0
$113$	−970.442	−0.807889	−0.403945	−	0.914783i	$-0.632361\pi$
$113$	−970.442	−0.807889	−0.403945	+	0.914783i	$0.632361\pi$
$114$	0	0
$115$	622.764	0.504983
$116$	0	0
$117$	327.351	0.258663
$118$	0	0
$119$	−124.829	−0.0961602
$120$	0	0
$121$	−606.582	−0.455734
$122$	0	0
$123$	−951.042	−0.697175
$124$	0	0
$125$	1456.61	1.04226
$126$	0	0
$127$	−2115.78	−1.47831	−0.739155	−	0.673535i	$-0.764775\pi$
$127$	−2115.78	−1.47831	−0.739155	+	0.673535i	$0.764775\pi$
$128$	0	0
$129$	−106.088	−0.0724075
$130$	0	0
$131$	1695.44	1.13077	0.565386	−	0.824826i	$-0.308727\pi$
$131$	1695.44	1.13077	0.565386	+	0.824826i	$0.308727\pi$
$132$	0	0
$133$	175.547	0.114450
$134$	0	0
$135$	−891.704	−0.568486
$136$	0	0
$137$	1613.51	1.00622	0.503108	−	0.864224i	$-0.332190\pi$
$137$	1613.51	1.00622	0.503108	+	0.864224i	$0.332190\pi$
$138$	0	0
$139$	−2072.45	−1.26463	−0.632313	−	0.774713i	$-0.717894\pi$
$139$	−2072.45	−1.26463	−0.632313	+	0.774713i	$0.717894\pi$
$140$	0	0
$141$	668.658	0.399370
$142$	0	0
$143$	421.299	0.246369
$144$	0	0
$145$	258.627	0.148123
$146$	0	0
$147$	794.438	0.445742
$148$	0	0
$149$	−811.396	−0.446122	−0.223061	−	0.974805i	$-0.571605\pi$
$149$	−811.396	−0.446122	−0.223061	+	0.974805i	$0.571605\pi$
$150$	0	0
$151$	944.326	0.508928	0.254464	−	0.967082i	$-0.418101\pi$
$151$	944.326	0.508928	0.254464	+	0.967082i	$0.418101\pi$
$152$	0	0
$153$	−569.714	−0.301037
$154$	0	0
$155$	898.078	0.465390
$156$	0	0
$157$	2138.58	1.08712	0.543558	−	0.839372i	$-0.317077\pi$
$157$	2138.58	1.08712	0.543558	+	0.839372i	$0.317077\pi$
$158$	0	0
$159$	812.595	0.405302
$160$	0	0
$161$	378.297	0.185180
$162$	0	0
$163$	2184.57	1.04975	0.524873	−	0.851180i	$-0.324113\pi$
$163$	2184.57	1.04975	0.524873	+	0.851180i	$0.324113\pi$
$164$	0	0
$165$	−500.912	−0.236339
$166$	0	0
$167$	−3334.42	−1.54506	−0.772531	−	0.634978i	$-0.781009\pi$
$167$	−3334.42	−1.54506	−0.772531	+	0.634978i	$0.781009\pi$
$168$	0	0
$169$	−1951.99	−0.888478
$170$	0	0
$171$	801.188	0.358295
$172$	0	0
$173$	1169.82	0.514105	0.257052	−	0.966397i	$-0.417249\pi$
$173$	1169.82	0.514105	0.257052	+	0.966397i	$0.417249\pi$
$174$	0	0
$175$	312.036	0.134787
$176$	0	0
$177$	−429.178	−0.182254
$178$	0	0
$179$	−206.597	−0.0862669	−0.0431334	−	0.999069i	$-0.513734\pi$
$179$	−206.597	−0.0862669	−0.0431334	+	0.999069i	$0.513734\pi$
$180$	0	0
$181$	652.242	0.267849	0.133925	−	0.990992i	$-0.457242\pi$
$181$	652.242	0.267849	0.133925	+	0.990992i	$0.457242\pi$
$182$	0	0
$183$	−134.464	−0.0543162
$184$	0	0
$185$	−2855.38	−1.13477
$186$	0	0
$187$	−733.220	−0.286729
$188$	0	0
$189$	−541.664	−0.208467
$190$	0	0
$191$	1128.21	0.427404	0.213702	−	0.976899i	$-0.431448\pi$
$191$	1128.21	0.427404	0.213702	+	0.976899i	$0.431448\pi$
$192$	0	0
$193$	355.606	0.132627	0.0663136	−	0.997799i	$-0.478876\pi$
$193$	355.606	0.132627	0.0663136	+	0.997799i	$0.478876\pi$
$194$	0	0
$195$	−291.315	−0.106982
$196$	0	0
$197$	4347.00	1.57214	0.786068	−	0.618140i	$-0.212114\pi$
$197$	4347.00	1.57214	0.786068	+	0.618140i	$0.212114\pi$
$198$	0	0
$199$	−430.375	−0.153309	−0.0766545	−	0.997058i	$-0.524424\pi$
$199$	−430.375	−0.153309	−0.0766545	+	0.997058i	$0.524424\pi$
$200$	0	0
$201$	−2237.61	−0.785219
$202$	0	0
$203$	157.102	0.0543174
$204$	0	0
$205$	−2907.82	−0.990687
$206$	0	0
$207$	1726.53	0.579720
$208$	0	0
$209$	1031.13	0.341265
$210$	0	0
$211$	−103.702	−0.0338347	−0.0169174	−	0.999857i	$-0.505385\pi$
$211$	−103.702	−0.0338347	−0.0169174	+	0.999857i	$0.505385\pi$
$212$	0	0
$213$	−1533.04	−0.493156
$214$	0	0
$215$	−324.366	−0.102891
$216$	0	0
$217$	545.535	0.170661
$218$	0	0
$219$	2530.99	0.780951
$220$	0	0
$221$	−426.418	−0.129792
$222$	0	0
$223$	−4447.94	−1.33568	−0.667839	−	0.744306i	$-0.732781\pi$
$223$	−4447.94	−1.33568	−0.667839	+	0.744306i	$0.732781\pi$
$224$	0	0
$225$	1424.12	0.421961
$226$	0	0
$227$	−509.808	−0.149062	−0.0745311	−	0.997219i	$-0.523746\pi$
$227$	−509.808	−0.149062	−0.0745311	+	0.997219i	$0.523746\pi$
$228$	0	0
$229$	−4754.86	−1.37210	−0.686049	−	0.727556i	$-0.740656\pi$
$229$	−4754.86	−1.37210	−0.686049	+	0.727556i	$0.740656\pi$
$230$	0	0
$231$	−304.278	−0.0866667
$232$	0	0
$233$	893.223	0.251146	0.125573	−	0.992084i	$-0.459923\pi$
$233$	893.223	0.251146	0.125573	+	0.992084i	$0.459923\pi$
$234$	0	0
$235$	2044.43	0.567506
$236$	0	0
$237$	−1820.64	−0.499000
$238$	0	0
$239$	3883.75	1.05112	0.525562	−	0.850755i	$-0.323855\pi$
$239$	3883.75	1.05112	0.525562	+	0.850755i	$0.323855\pi$
$240$	0	0
$241$	−3118.83	−0.833616	−0.416808	−	0.908995i	$-0.636851\pi$
$241$	−3118.83	−0.833616	−0.416808	+	0.908995i	$0.636851\pi$
$242$	0	0
$243$	−3865.22	−1.02039
$244$	0	0
$245$	2429.00	0.633400
$246$	0	0
$247$	599.670	0.154478
$248$	0	0
$249$	1377.83	0.350668
$250$	0	0
$251$	−5196.19	−1.30669	−0.653347	−	0.757058i	$-0.726636\pi$
$251$	−5196.19	−1.30669	−0.653347	+	0.757058i	$0.726636\pi$
$252$	0	0
$253$	2222.04	0.552167
$254$	0	0
$255$	506.998	0.124508
$256$	0	0
$257$	−2441.32	−0.592550	−0.296275	−	0.955103i	$-0.595744\pi$
$257$	−2441.32	−0.592550	−0.296275	+	0.955103i	$0.595744\pi$
$258$	0	0
$259$	−1734.50	−0.416125
$260$	0	0
$261$	717.008	0.170045
$262$	0	0
$263$	−5831.01	−1.36713	−0.683566	−	0.729889i	$-0.739572\pi$
$263$	−5831.01	−1.36713	−0.683566	+	0.729889i	$0.739572\pi$
$264$	0	0
$265$	2484.52	0.575934
$266$	0	0
$267$	−4024.73	−0.922507
$268$	0	0
$269$	−2605.41	−0.590538	−0.295269	−	0.955414i	$-0.595409\pi$
$269$	−2605.41	−0.590538	−0.295269	+	0.955414i	$0.595409\pi$
$270$	0	0
$271$	−1793.84	−0.402097	−0.201049	−	0.979581i	$-0.564435\pi$
$271$	−1793.84	−0.402097	−0.201049	+	0.979581i	$0.564435\pi$
$272$	0	0
$273$	−176.958	−0.0392308
$274$	0	0
$275$	1832.83	0.401906
$276$	0	0
$277$	−825.071	−0.178967	−0.0894833	−	0.995988i	$-0.528522\pi$
$277$	−825.071	−0.178967	−0.0894833	+	0.995988i	$0.528522\pi$
$278$	0	0
$279$	2489.80	0.534267
$280$	0	0
$281$	−5114.69	−1.08582	−0.542912	−	0.839789i	$-0.682678\pi$
$281$	−5114.69	−1.08582	−0.542912	+	0.839789i	$0.682678\pi$
$282$	0	0
$283$	3703.40	0.777895	0.388947	−	0.921260i	$-0.372839\pi$
$283$	3703.40	0.777895	0.388947	+	0.921260i	$0.372839\pi$
$284$	0	0
$285$	−712.990	−0.148189
$286$	0	0
$287$	−1766.35	−0.363290
$288$	0	0
$289$	−4170.87	−0.848946
$290$	0	0
$291$	1003.17	0.202085
$292$	0	0
$293$	749.193	0.149380	0.0746899	−	0.997207i	$-0.476203\pi$
$293$	749.193	0.149380	0.0746899	+	0.997207i	$0.476203\pi$
$294$	0	0
$295$	−1312.22	−0.258984
$296$	0	0
$297$	−3181.62	−0.621603
$298$	0	0
$299$	1292.27	0.249946
$300$	0	0
$301$	−197.036	−0.0377307
$302$	0	0
$303$	−2468.23	−0.467973
$304$	0	0
$305$	−411.125	−0.0771834
$306$	0	0
$307$	4761.00	0.885097	0.442548	−	0.896745i	$-0.354075\pi$
$307$	4761.00	0.885097	0.442548	+	0.896745i	$0.354075\pi$
$308$	0	0
$309$	−4095.04	−0.753912
$310$	0	0
$311$	5995.99	1.09325	0.546626	−	0.837377i	$-0.315912\pi$
$311$	5995.99	1.09325	0.546626	+	0.837377i	$0.315912\pi$
$312$	0	0
$313$	398.261	0.0719203	0.0359602	−	0.999353i	$-0.488551\pi$
$313$	398.261	0.0719203	0.0359602	+	0.999353i	$0.488551\pi$
$314$	0	0
$315$	−722.871	−0.129299
$316$	0	0
$317$	4657.19	0.825154	0.412577	−	0.910923i	$-0.364629\pi$
$317$	4657.19	0.825154	0.412577	+	0.910923i	$0.364629\pi$
$318$	0	0
$319$	922.786	0.161963
$320$	0	0
$321$	−374.000	−0.0650300
$322$	0	0
$323$	−1043.65	−0.179785
$324$	0	0
$325$	1065.92	0.181928
$326$	0	0
$327$	−2696.03	−0.455936
$328$	0	0
$329$	1241.88	0.208107
$330$	0	0
$331$	−10013.6	−1.66282	−0.831412	−	0.555656i	$-0.812467\pi$
$331$	−10013.6	−1.66282	−0.831412	+	0.555656i	$0.812467\pi$
$332$	0	0
$333$	−7916.16	−1.30271
$334$	0	0
$335$	−6841.52	−1.11580
$336$	0	0
$337$	−3872.11	−0.625897	−0.312949	−	0.949770i	$-0.601317\pi$
$337$	−3872.11	−0.625897	−0.312949	+	0.949770i	$0.601317\pi$
$338$	0	0
$339$	2394.25	0.383592
$340$	0	0
$341$	3204.36	0.508874
$342$	0	0
$343$	3047.19	0.479688
$344$	0	0
$345$	−1536.47	−0.239770
$346$	0	0
$347$	−10290.8	−1.59204	−0.796022	−	0.605267i	$-0.793066\pi$
$347$	−10290.8	−1.59204	−0.796022	+	0.605267i	$0.793066\pi$
$348$	0	0
$349$	700.572	0.107452	0.0537260	−	0.998556i	$-0.482890\pi$
$349$	700.572	0.107452	0.0537260	+	0.998556i	$0.482890\pi$
$350$	0	0
$351$	−1850.33	−0.281377
$352$	0	0
$353$	3608.33	0.544057	0.272028	−	0.962289i	$-0.412306\pi$
$353$	3608.33	0.544057	0.272028	+	0.962289i	$0.412306\pi$
$354$	0	0
$355$	−4687.29	−0.700776
$356$	0	0
$357$	307.975	0.0456576
$358$	0	0
$359$	12085.9	1.77680	0.888399	−	0.459072i	$-0.151818\pi$
$359$	12085.9	1.77680	0.888399	+	0.459072i	$0.151818\pi$
$360$	0	0
$361$	−5391.31	−0.786020
$362$	0	0
$363$	1496.54	0.216386
$364$	0	0
$365$	7738.52	1.10973
$366$	0	0
$367$	−968.974	−0.137820	−0.0689101	−	0.997623i	$-0.521952\pi$
$367$	−968.974	−0.137820	−0.0689101	+	0.997623i	$0.521952\pi$
$368$	0	0
$369$	−8061.53	−1.13731
$370$	0	0
$371$	1509.21	0.211198
$372$	0	0
$373$	13006.0	1.80543	0.902713	−	0.430243i	$-0.141572\pi$
$373$	13006.0	1.80543	0.902713	+	0.430243i	$0.141572\pi$
$374$	0	0
$375$	−3593.71	−0.494875
$376$	0	0
$377$	536.664	0.0733146
$378$	0	0
$379$	−1901.08	−0.257657	−0.128828	−	0.991667i	$-0.541122\pi$
$379$	−1901.08	−0.257657	−0.128828	+	0.991667i	$0.541122\pi$
$380$	0	0
$381$	5220.00	0.701913
$382$	0	0
$383$	9501.65	1.26765	0.633827	−	0.773475i	$-0.281483\pi$
$383$	9501.65	1.26765	0.633827	+	0.773475i	$0.281483\pi$
$384$	0	0
$385$	−930.332	−0.123154
$386$	0	0
$387$	−899.262	−0.118119
$388$	0	0
$389$	4640.26	0.604808	0.302404	−	0.953180i	$-0.402211\pi$
$389$	4640.26	0.604808	0.302404	+	0.953180i	$0.402211\pi$
$390$	0	0
$391$	−2249.03	−0.290891
$392$	0	0
$393$	−4182.94	−0.536899
$394$	0	0
$395$	−5566.61	−0.709080
$396$	0	0
$397$	4633.90	0.585815	0.292908	−	0.956141i	$-0.405377\pi$
$397$	4633.90	0.585815	0.292908	+	0.956141i	$0.405377\pi$
$398$	0	0
$399$	−433.104	−0.0543417
$400$	0	0
$401$	−11801.3	−1.46965	−0.734823	−	0.678258i	$-0.762735\pi$
$401$	−11801.3	−1.46965	−0.734823	+	0.678258i	$0.762735\pi$
$402$	0	0
$403$	1863.56	0.230348
$404$	0	0
$405$	−2059.41	−0.252674
$406$	0	0
$407$	−10188.1	−1.24080
$408$	0	0
$409$	5588.72	0.675658	0.337829	−	0.941207i	$-0.390307\pi$
$409$	5588.72	0.675658	0.337829	+	0.941207i	$0.390307\pi$
$410$	0	0
$411$	−3980.81	−0.477759
$412$	0	0
$413$	−797.103	−0.0949706
$414$	0	0
$415$	4212.72	0.498300
$416$	0	0
$417$	5113.09	0.600454
$418$	0	0
$419$	6908.94	0.805547	0.402773	−	0.915300i	$-0.368046\pi$
$419$	6908.94	0.805547	0.402773	+	0.915300i	$0.368046\pi$
$420$	0	0
$421$	16499.6	1.91007	0.955036	−	0.296491i	$-0.0958165\pi$
$421$	16499.6	1.91007	0.955036	+	0.296491i	$0.0958165\pi$
$422$	0	0
$423$	5667.90	0.651496
$424$	0	0
$425$	−1855.10	−0.211731
$426$	0	0
$427$	−249.737	−0.0283035
$428$	0	0
$429$	−1039.42	−0.116978
$430$	0	0
$431$	7101.67	0.793679	0.396839	−	0.917888i	$-0.370107\pi$
$431$	7101.67	0.793679	0.396839	+	0.917888i	$0.370107\pi$
$432$	0	0
$433$	−8935.72	−0.991739	−0.495870	−	0.868397i	$-0.665151\pi$
$433$	−8935.72	−0.991739	−0.495870	+	0.868397i	$0.665151\pi$
$434$	0	0
$435$	−638.077	−0.0703298
$436$	0	0
$437$	3162.81	0.346219
$438$	0	0
$439$	−17789.3	−1.93403	−0.967015	−	0.254720i	$-0.918017\pi$
$439$	−17789.3	−1.93403	−0.967015	+	0.254720i	$0.918017\pi$
$440$	0	0
$441$	6734.07	0.727143
$442$	0	0
$443$	−2744.62	−0.294359	−0.147180	−	0.989110i	$-0.547020\pi$
$443$	−2744.62	−0.294359	−0.147180	+	0.989110i	$0.547020\pi$
$444$	0	0
$445$	−12305.6	−1.31088
$446$	0	0
$447$	2001.85	0.211822
$448$	0	0
$449$	−12051.9	−1.26673	−0.633366	−	0.773853i	$-0.718327\pi$
$449$	−12051.9	−1.26673	−0.633366	+	0.773853i	$0.718327\pi$
$450$	0	0
$451$	−10375.2	−1.08325
$452$	0	0
$453$	−2329.81	−0.241643
$454$	0	0
$455$	−541.052	−0.0557471
$456$	0	0
$457$	130.425	0.0133502	0.00667509	−	0.999978i	$-0.497875\pi$
$457$	130.425	0.0133502	0.00667509	+	0.999978i	$0.497875\pi$
$458$	0	0
$459$	3220.28	0.327472
$460$	0	0
$461$	−981.122	−0.0991223	−0.0495612	−	0.998771i	$-0.515782\pi$
$461$	−981.122	−0.0991223	−0.0495612	+	0.998771i	$0.515782\pi$
$462$	0	0
$463$	−6359.89	−0.638378	−0.319189	−	0.947691i	$-0.603411\pi$
$463$	−6359.89	−0.638378	−0.319189	+	0.947691i	$0.603411\pi$
$464$	0	0
$465$	−2215.71	−0.220970
$466$	0	0
$467$	14159.9	1.40309	0.701546	−	0.712624i	$-0.252494\pi$
$467$	14159.9	1.40309	0.701546	+	0.712624i	$0.252494\pi$
$468$	0	0
$469$	−4155.86	−0.409168
$470$	0	0
$471$	−5276.24	−0.516170
$472$	0	0
$473$	−1157.35	−0.112505
$474$	0	0
$475$	2608.83	0.252003
$476$	0	0
$477$	6887.98	0.661172
$478$	0	0
$479$	20583.3	1.96342	0.981709	−	0.190390i	$-0.0609751\pi$
$479$	20583.3	1.96342	0.981709	+	0.190390i	$0.0609751\pi$
$480$	0	0
$481$	−5925.06	−0.561662
$482$	0	0
$483$	−933.324	−0.0879248
$484$	0	0
$485$	3067.20	0.287164
$486$	0	0
$487$	17672.3	1.64437	0.822186	−	0.569218i	$-0.192754\pi$
$487$	17672.3	1.64437	0.822186	+	0.569218i	$0.192754\pi$
$488$	0	0
$489$	−5389.71	−0.498428
$490$	0	0
$491$	3484.50	0.320271	0.160136	−	0.987095i	$-0.448807\pi$
$491$	3484.50	0.320271	0.160136	+	0.987095i	$0.448807\pi$
$492$	0	0
$493$	−933.998	−0.0853249
$494$	0	0
$495$	−4245.99	−0.385542
$496$	0	0
$497$	−2847.28	−0.256978
$498$	0	0
$499$	−1393.45	−0.125009	−0.0625046	−	0.998045i	$-0.519909\pi$
$499$	−1393.45	−0.125009	−0.0625046	+	0.998045i	$0.519909\pi$
$500$	0	0
$501$	8226.59	0.733607
$502$	0	0
$503$	−3191.15	−0.282875	−0.141438	−	0.989947i	$-0.545172\pi$
$503$	−3191.15	−0.282875	−0.141438	+	0.989947i	$0.545172\pi$
$504$	0	0
$505$	−7546.63	−0.664991
$506$	0	0
$507$	4815.89	0.421856
$508$	0	0
$509$	10831.8	0.943247	0.471624	−	0.881800i	$-0.343668\pi$
$509$	10831.8	0.943247	0.471624	+	0.881800i	$0.343668\pi$
$510$	0	0
$511$	4700.75	0.406945
$512$	0	0
$513$	−4528.67	−0.389757
$514$	0	0
$515$	−12520.6	−1.07131
$516$	0	0
$517$	7294.56	0.620531
$518$	0	0
$519$	−2886.16	−0.244101
$520$	0	0
$521$	14277.5	1.20059	0.600296	−	0.799778i	$-0.295049\pi$
$521$	14277.5	1.20059	0.600296	+	0.799778i	$0.295049\pi$
$522$	0	0
$523$	−4181.28	−0.349588	−0.174794	−	0.984605i	$-0.555926\pi$
$523$	−4181.28	−0.349588	−0.174794	+	0.984605i	$0.555926\pi$
$524$	0	0
$525$	−769.847	−0.0639978
$526$	0	0
$527$	−3243.29	−0.268084
$528$	0	0
$529$	−5351.26	−0.439817
$530$	0	0
$531$	−3637.94	−0.297313
$532$	0	0
$533$	−6033.87	−0.490348
$534$	0	0
$535$	−1143.51	−0.0924077
$536$	0	0
$537$	509.710	0.0409602
$538$	0	0
$539$	8666.72	0.692583
$540$	0	0
$541$	5123.26	0.407146	0.203573	−	0.979060i	$-0.434745\pi$
$541$	5123.26	0.407146	0.203573	+	0.979060i	$0.434745\pi$
$542$	0	0
$543$	−1609.19	−0.127177
$544$	0	0
$545$	−8243.14	−0.647885
$546$	0	0
$547$	15435.6	1.20654	0.603271	−	0.797537i	$-0.293864\pi$
$547$	15435.6	1.20654	0.603271	+	0.797537i	$0.293864\pi$
$548$	0	0
$549$	−1139.79	−0.0886064
$550$	0	0
$551$	1313.48	0.101554
$552$	0	0
$553$	−3381.43	−0.260023
$554$	0	0
$555$	7044.73	0.538796
$556$	0	0
$557$	−978.643	−0.0744460	−0.0372230	−	0.999307i	$-0.511851\pi$
$557$	−978.643	−0.0744460	−0.0372230	+	0.999307i	$0.511851\pi$
$558$	0	0
$559$	−673.076	−0.0509268
$560$	0	0
$561$	1808.98	0.136141
$562$	0	0
$563$	24362.4	1.82371	0.911857	−	0.410507i	$-0.134648\pi$
$563$	24362.4	1.82371	0.911857	+	0.410507i	$0.134648\pi$
$564$	0	0
$565$	7320.43	0.545085
$566$	0	0
$567$	−1250.99	−0.0926569
$568$	0	0
$569$	−12733.1	−0.938135	−0.469067	−	0.883162i	$-0.655410\pi$
$569$	−12733.1	−0.938135	−0.469067	+	0.883162i	$0.655410\pi$
$570$	0	0
$571$	6363.95	0.466415	0.233208	−	0.972427i	$-0.425078\pi$
$571$	6363.95	0.466415	0.233208	+	0.972427i	$0.425078\pi$
$572$	0	0
$573$	−2783.48	−0.202935
$574$	0	0
$575$	5621.93	0.407740
$576$	0	0
$577$	−11981.3	−0.864449	−0.432225	−	0.901766i	$-0.642271\pi$
$577$	−11981.3	−0.864449	−0.432225	+	0.901766i	$0.642271\pi$
$578$	0	0
$579$	−877.340	−0.0629724
$580$	0	0
$581$	2559.01	0.182729
$582$	0	0
$583$	8864.80	0.629747
$584$	0	0
$585$	−2469.34	−0.174521
$586$	0	0
$587$	−17927.4	−1.26055	−0.630276	−	0.776371i	$-0.717058\pi$
$587$	−17927.4	−1.26055	−0.630276	+	0.776371i	$0.717058\pi$
$588$	0	0
$589$	4561.04	0.319074
$590$	0	0
$591$	−10724.8	−0.746462
$592$	0	0
$593$	19927.7	1.37999	0.689993	−	0.723816i	$-0.257613\pi$
$593$	19927.7	1.37999	0.689993	+	0.723816i	$0.257613\pi$
$594$	0	0
$595$	941.636	0.0648795
$596$	0	0
$597$	1061.81	0.0727923
$598$	0	0
$599$	4155.72	0.283469	0.141735	−	0.989905i	$-0.454732\pi$
$599$	4155.72	0.283469	0.141735	+	0.989905i	$0.454732\pi$
$600$	0	0
$601$	−4104.20	−0.278559	−0.139279	−	0.990253i	$-0.544479\pi$
$601$	−4104.20	−0.278559	−0.139279	+	0.990253i	$0.544479\pi$
$602$	0	0
$603$	−18967.2	−1.28093
$604$	0	0
$605$	4575.69	0.307485
$606$	0	0
$607$	−5870.16	−0.392525	−0.196262	−	0.980551i	$-0.562880\pi$
$607$	−5870.16	−0.392525	−0.196262	+	0.980551i	$0.562880\pi$
$608$	0	0
$609$	−387.599	−0.0257903
$610$	0	0
$611$	4242.29	0.280891
$612$	0	0
$613$	29802.4	1.96363	0.981817	−	0.189830i	$-0.0607936\pi$
$613$	29802.4	1.96363	0.981817	+	0.189830i	$0.0607936\pi$
$614$	0	0
$615$	7174.09	0.470386
$616$	0	0
$617$	27394.8	1.78748	0.893738	−	0.448588i	$-0.148073\pi$
$617$	27394.8	1.78748	0.893738	+	0.448588i	$0.148073\pi$
$618$	0	0
$619$	25129.8	1.63174	0.815872	−	0.578232i	$-0.196257\pi$
$619$	25129.8	1.63174	0.815872	+	0.578232i	$0.196257\pi$
$620$	0	0
$621$	−9759.11	−0.630627
$622$	0	0
$623$	−7475.03	−0.480708
$624$	0	0
$625$	−2475.65	−0.158442
$626$	0	0
$627$	−2543.96	−0.162035
$628$	0	0
$629$	10311.9	0.653673
$630$	0	0
$631$	1190.63	0.0751160	0.0375580	−	0.999294i	$-0.488042\pi$
$631$	1190.63	0.0751160	0.0375580	+	0.999294i	$0.488042\pi$
$632$	0	0
$633$	255.850	0.0160650
$634$	0	0
$635$	15960.2	0.997420
$636$	0	0
$637$	5040.29	0.313507
$638$	0	0
$639$	−12994.9	−0.804490
$640$	0	0
$641$	28828.0	1.77635	0.888174	−	0.459508i	$-0.151974\pi$
$641$	28828.0	1.77635	0.888174	+	0.459508i	$0.151974\pi$
$642$	0	0
$643$	−16853.2	−1.03363	−0.516816	−	0.856096i	$-0.672883\pi$
$643$	−16853.2	−1.03363	−0.516816	+	0.856096i	$0.672883\pi$
$644$	0	0
$645$	800.268	0.0488535
$646$	0	0
$647$	20234.3	1.22951	0.614754	−	0.788719i	$-0.289255\pi$
$647$	20234.3	1.22951	0.614754	+	0.788719i	$0.289255\pi$
$648$	0	0
$649$	−4682.02	−0.283182
$650$	0	0
$651$	−1345.93	−0.0810310
$652$	0	0
$653$	24965.3	1.49612	0.748062	−	0.663629i	$-0.230985\pi$
$653$	24965.3	1.49612	0.748062	+	0.663629i	$0.230985\pi$
$654$	0	0
$655$	−12789.4	−0.762935
$656$	0	0
$657$	21454.0	1.27397
$658$	0	0
$659$	−9033.00	−0.533954	−0.266977	−	0.963703i	$-0.586025\pi$
$659$	−9033.00	−0.533954	−0.266977	+	0.963703i	$0.586025\pi$
$660$	0	0
$661$	−19710.5	−1.15983	−0.579915	−	0.814677i	$-0.696914\pi$
$661$	−19710.5	−1.15983	−0.579915	+	0.814677i	$0.696914\pi$
$662$	0	0
$663$	1052.05	0.0616261
$664$	0	0
$665$	−1324.22	−0.0772196
$666$	0	0
$667$	2830.50	0.164314
$668$	0	0
$669$	10973.8	0.634190
$670$	0	0
$671$	−1466.90	−0.0843951
$672$	0	0
$673$	19293.1	1.10504	0.552521	−	0.833499i	$-0.313666\pi$
$673$	19293.1	1.10504	0.552521	+	0.833499i	$0.313666\pi$
$674$	0	0
$675$	−8049.74	−0.459014
$676$	0	0
$677$	−16029.1	−0.909968	−0.454984	−	0.890500i	$-0.650355\pi$
$677$	−16029.1	−0.909968	−0.454984	+	0.890500i	$0.650355\pi$
$678$	0	0
$679$	1863.16	0.105304
$680$	0	0
$681$	1257.78	0.0707759
$682$	0	0
$683$	4293.48	0.240535	0.120267	−	0.992742i	$-0.461625\pi$
$683$	4293.48	0.240535	0.120267	+	0.992742i	$0.461625\pi$
$684$	0	0
$685$	−12171.4	−0.678896
$686$	0	0
$687$	11731.1	0.651482
$688$	0	0
$689$	5155.49	0.285063
$690$	0	0
$691$	19631.7	1.08079	0.540393	−	0.841413i	$-0.318276\pi$
$691$	19631.7	1.08079	0.540393	+	0.841413i	$0.318276\pi$
$692$	0	0
$693$	−2579.22	−0.141380
$694$	0	0
$695$	15633.3	0.853246
$696$	0	0
$697$	10501.2	0.570677
$698$	0	0
$699$	−2203.74	−0.119246
$700$	0	0
$701$	−13399.7	−0.721967	−0.360983	−	0.932572i	$-0.617559\pi$
$701$	−13399.7	−0.721967	−0.360983	+	0.932572i	$0.617559\pi$
$702$	0	0
$703$	−14501.5	−0.778003
$704$	0	0
$705$	−5043.96	−0.269456
$706$	0	0
$707$	−4584.18	−0.243855
$708$	0	0
$709$	−510.833	−0.0270589	−0.0135294	−	0.999908i	$-0.504307\pi$
$709$	−510.833	−0.0270589	−0.0135294	+	0.999908i	$0.504307\pi$
$710$	0	0
$711$	−15432.7	−0.814023
$712$	0	0
$713$	9828.87	0.516261
$714$	0	0
$715$	−3178.03	−0.166226
$716$	0	0
$717$	−9581.88	−0.499082
$718$	0	0
$719$	29685.4	1.53975	0.769875	−	0.638195i	$-0.220319\pi$
$719$	29685.4	1.53975	0.769875	+	0.638195i	$0.220319\pi$
$720$	0	0
$721$	−7605.63	−0.392855
$722$	0	0
$723$	7694.69	0.395807
$724$	0	0
$725$	2334.72	0.119599
$726$	0	0
$727$	10575.9	0.539529	0.269765	−	0.962926i	$-0.413054\pi$
$727$	10575.9	0.539529	0.269765	+	0.962926i	$0.413054\pi$
$728$	0	0
$729$	2164.94	0.109990
$730$	0	0
$731$	1171.41	0.0592696
$732$	0	0
$733$	−4428.93	−0.223173	−0.111587	−	0.993755i	$-0.535593\pi$
$733$	−4428.93	−0.223173	−0.111587	+	0.993755i	$0.535593\pi$
$734$	0	0
$735$	−5992.76	−0.300743
$736$	0	0
$737$	−24410.7	−1.22005
$738$	0	0
$739$	−20131.6	−1.00210	−0.501052	−	0.865417i	$-0.667053\pi$
$739$	−20131.6	−1.00210	−0.501052	+	0.865417i	$0.667053\pi$
$740$	0	0
$741$	−1479.49	−0.0733474
$742$	0	0
$743$	7362.63	0.363538	0.181769	−	0.983341i	$-0.441818\pi$
$743$	7362.63	0.363538	0.181769	+	0.983341i	$0.441818\pi$
$744$	0	0
$745$	6120.69	0.300999
$746$	0	0
$747$	11679.2	0.572048
$748$	0	0
$749$	−694.621	−0.0338864
$750$	0	0
$751$	7494.93	0.364173	0.182087	−	0.983283i	$-0.441715\pi$
$751$	7494.93	0.364173	0.182087	+	0.983283i	$0.441715\pi$
$752$	0	0
$753$	12819.9	0.620429
$754$	0	0
$755$	−7123.43	−0.343375
$756$	0	0
$757$	26060.4	1.25123	0.625615	−	0.780132i	$-0.284848\pi$
$757$	26060.4	1.25123	0.625615	+	0.780132i	$0.284848\pi$
$758$	0	0
$759$	−5482.15	−0.262173
$760$	0	0
$761$	−23064.0	−1.09865	−0.549323	−	0.835610i	$-0.685114\pi$
$761$	−23064.0	−1.09865	−0.549323	+	0.835610i	$0.685114\pi$
$762$	0	0
$763$	−5007.28	−0.237583
$764$	0	0
$765$	4297.58	0.203110
$766$	0	0
$767$	−2722.91	−0.128186
$768$	0	0
$769$	19455.7	0.912341	0.456171	−	0.889892i	$-0.349221\pi$
$769$	19455.7	0.912341	0.456171	+	0.889892i	$0.349221\pi$
$770$	0	0
$771$	6023.16	0.281347
$772$	0	0
$773$	−14025.4	−0.652599	−0.326300	−	0.945266i	$-0.605802\pi$
$773$	−14025.4	−0.652599	−0.326300	+	0.945266i	$0.605802\pi$
$774$	0	0
$775$	8107.28	0.375771
$776$	0	0
$777$	4279.30	0.197579
$778$	0	0
$779$	−14767.8	−0.679220
$780$	0	0
$781$	−16724.3	−0.766253
$782$	0	0
$783$	−4052.85	−0.184977
$784$	0	0
$785$	−16132.1	−0.733479
$786$	0	0
$787$	41219.6	1.86699	0.933495	−	0.358589i	$-0.116742\pi$
$787$	41219.6	1.86699	0.933495	+	0.358589i	$0.116742\pi$
$788$	0	0
$789$	14386.1	0.649124
$790$	0	0
$791$	4446.78	0.199885
$792$	0	0
$793$	−853.104	−0.0382025
$794$	0	0
$795$	−6129.73	−0.273458
$796$	0	0
$797$	10289.5	0.457305	0.228653	−	0.973508i	$-0.426568\pi$
$797$	10289.5	0.457305	0.228653	+	0.973508i	$0.426568\pi$
$798$	0	0
$799$	−7383.19	−0.326907
$800$	0	0
$801$	−34115.7	−1.50489
$802$	0	0
$803$	27611.2	1.21342
$804$	0	0
$805$	−2853.65	−0.124941
$806$	0	0
$807$	6428.00	0.280392
$808$	0	0
$809$	−20063.0	−0.871912	−0.435956	−	0.899968i	$-0.643590\pi$
$809$	−20063.0	−0.871912	−0.435956	+	0.899968i	$0.643590\pi$
$810$	0	0
$811$	−26523.6	−1.14842	−0.574210	−	0.818708i	$-0.694691\pi$
$811$	−26523.6	−1.14842	−0.574210	+	0.818708i	$0.694691\pi$
$812$	0	0
$813$	4425.73	0.190919
$814$	0	0
$815$	−16479.1	−0.708267
$816$	0	0
$817$	−1647.35	−0.0705427
$818$	0	0
$819$	−1499.99	−0.0639976
$820$	0	0
$821$	−9204.45	−0.391276	−0.195638	−	0.980676i	$-0.562678\pi$
$821$	−9204.45	−0.391276	−0.195638	+	0.980676i	$0.562678\pi$
$822$	0	0
$823$	−15569.2	−0.659427	−0.329713	−	0.944081i	$-0.606952\pi$
$823$	−15569.2	−0.659427	−0.329713	+	0.944081i	$0.606952\pi$
$824$	0	0
$825$	−4521.92	−0.190828
$826$	0	0
$827$	11786.2	0.495582	0.247791	−	0.968814i	$-0.420295\pi$
$827$	11786.2	0.495582	0.247791	+	0.968814i	$0.420295\pi$
$828$	0	0
$829$	−33661.5	−1.41027	−0.705133	−	0.709075i	$-0.749113\pi$
$829$	−33661.5	−1.41027	−0.705133	+	0.709075i	$0.749113\pi$
$830$	0	0
$831$	2035.59	0.0849747
$832$	0	0
$833$	−8772.02	−0.364865
$834$	0	0
$835$	25152.9	1.04246
$836$	0	0
$837$	−14073.4	−0.581182
$838$	0	0
$839$	2686.31	0.110538	0.0552691	−	0.998471i	$-0.482398\pi$
$839$	2686.31	0.110538	0.0552691	+	0.998471i	$0.482398\pi$
$840$	0	0
$841$	−23213.5	−0.951803
$842$	0	0
$843$	12618.8	0.515558
$844$	0	0
$845$	14724.6	0.599458
$846$	0	0
$847$	2779.49	0.112756
$848$	0	0
$849$	−9136.93	−0.369350
$850$	0	0
$851$	−31250.3	−1.25881
$852$	0	0
$853$	7920.78	0.317940	0.158970	−	0.987283i	$-0.449183\pi$
$853$	7920.78	0.317940	0.158970	+	0.987283i	$0.449183\pi$
$854$	0	0
$855$	−6043.68	−0.241742
$856$	0	0
$857$	36678.1	1.46196	0.730980	−	0.682398i	$-0.239063\pi$
$857$	36678.1	1.46196	0.730980	+	0.682398i	$0.239063\pi$
$858$	0	0
$859$	−31205.9	−1.23950	−0.619750	−	0.784799i	$-0.712766\pi$
$859$	−31205.9	−1.23950	−0.619750	+	0.784799i	$0.712766\pi$
$860$	0	0
$861$	4357.88	0.172493
$862$	0	0
$863$	−38340.4	−1.51231	−0.756155	−	0.654392i	$-0.772924\pi$
$863$	−38340.4	−1.51231	−0.756155	+	0.654392i	$0.772924\pi$
$864$	0	0
$865$	−8824.46	−0.346868
$866$	0	0
$867$	10290.3	0.403086
$868$	0	0
$869$	−19861.8	−0.775334
$870$	0	0
$871$	−14196.5	−0.552273
$872$	0	0
$873$	8503.40	0.329664
$874$	0	0
$875$	−6674.51	−0.257874
$876$	0	0
$877$	−38409.6	−1.47890	−0.739452	−	0.673210i	$-0.764915\pi$
$877$	−38409.6	−1.47890	−0.739452	+	0.673210i	$0.764915\pi$
$878$	0	0
$879$	−1848.39	−0.0709267
$880$	0	0
$881$	41203.4	1.57568	0.787842	−	0.615877i	$-0.211198\pi$
$881$	41203.4	1.57568	0.787842	+	0.615877i	$0.211198\pi$
$882$	0	0
$883$	−30194.6	−1.15077	−0.575384	−	0.817883i	$-0.695148\pi$
$883$	−30194.6	−1.15077	−0.575384	+	0.817883i	$0.695148\pi$
$884$	0	0
$885$	3237.47	0.122967
$886$	0	0
$887$	13570.8	0.513712	0.256856	−	0.966450i	$-0.417313\pi$
$887$	13570.8	0.513712	0.256856	+	0.966450i	$0.417313\pi$
$888$	0	0
$889$	9694.99	0.365759
$890$	0	0
$891$	−7348.03	−0.276283
$892$	0	0
$893$	10383.0	0.389085
$894$	0	0
$895$	1558.44	0.0582045
$896$	0	0
$897$	−3188.25	−0.118676
$898$	0	0
$899$	4081.82	0.151431
$900$	0	0
$901$	−8972.51	−0.331762
$902$	0	0
$903$	486.121	0.0179148
$904$	0	0
$905$	−4920.12	−0.180719
$906$	0	0
$907$	−21966.6	−0.804176	−0.402088	−	0.915601i	$-0.631715\pi$
$907$	−21966.6	−0.804176	−0.402088	+	0.915601i	$0.631715\pi$
$908$	0	0
$909$	−20922.0	−0.763409
$910$	0	0
$911$	−27091.0	−0.985252	−0.492626	−	0.870241i	$-0.663963\pi$
$911$	−27091.0	−0.985252	−0.492626	+	0.870241i	$0.663963\pi$
$912$	0	0
$913$	15031.1	0.544859
$914$	0	0
$915$	1014.32	0.0366473
$916$	0	0
$917$	−7768.88	−0.279772
$918$	0	0
$919$	24533.7	0.880622	0.440311	−	0.897845i	$-0.354868\pi$
$919$	24533.7	0.880622	0.440311	+	0.897845i	$0.354868\pi$
$920$	0	0
$921$	−11746.2	−0.420251
$922$	0	0
$923$	−9726.35	−0.346855
$924$	0	0
$925$	−25776.6	−0.916248
$926$	0	0
$927$	−34711.8	−1.22986
$928$	0	0
$929$	9220.40	0.325631	0.162816	−	0.986657i	$-0.447942\pi$
$929$	9220.40	0.325631	0.162816	+	0.986657i	$0.447942\pi$
$930$	0	0
$931$	12336.1	0.434263
$932$	0	0
$933$	−14793.1	−0.519084
$934$	0	0
$935$	5530.97	0.193457
$936$	0	0
$937$	22526.6	0.785392	0.392696	−	0.919668i	$-0.371542\pi$
$937$	22526.6	0.785392	0.392696	+	0.919668i	$0.371542\pi$
$938$	0	0
$939$	−982.580	−0.0341483
$940$	0	0
$941$	−776.782	−0.0269100	−0.0134550	−	0.999909i	$-0.504283\pi$
$941$	−776.782	−0.0269100	−0.0134550	+	0.999909i	$0.504283\pi$
$942$	0	0
$943$	−31824.1	−1.09898
$944$	0	0
$945$	4085.99	0.140653
$946$	0	0
$947$	45677.6	1.56739	0.783697	−	0.621143i	$-0.213331\pi$
$947$	45677.6	1.56739	0.783697	+	0.621143i	$0.213331\pi$
$948$	0	0
$949$	16057.8	0.549271
$950$	0	0
$951$	−11490.1	−0.391790
$952$	0	0
$953$	−41584.5	−1.41349	−0.706743	−	0.707470i	$-0.749836\pi$
$953$	−41584.5	−1.41349	−0.706743	+	0.707470i	$0.749836\pi$
$954$	0	0
$955$	−8510.52	−0.288371
$956$	0	0
$957$	−2276.67	−0.0769012
$958$	0	0
$959$	−7393.47	−0.248955
$960$	0	0
$961$	−15617.0	−0.524217
$962$	0	0
$963$	−3170.22	−0.106084
$964$	0	0
$965$	−2682.48	−0.0894839
$966$	0	0
$967$	−5961.61	−0.198255	−0.0991274	−	0.995075i	$-0.531605\pi$
$967$	−5961.61	−0.198255	−0.0991274	+	0.995075i	$0.531605\pi$
$968$	0	0
$969$	2574.87	0.0853631
$970$	0	0
$971$	25246.9	0.834410	0.417205	−	0.908812i	$-0.363010\pi$
$971$	25246.9	0.834410	0.417205	+	0.908812i	$0.363010\pi$
$972$	0	0
$973$	9496.43	0.312890
$974$	0	0
$975$	−2629.81	−0.0863807
$976$	0	0
$977$	44505.1	1.45736	0.728682	−	0.684852i	$-0.240133\pi$
$977$	44505.1	1.45736	0.728682	+	0.684852i	$0.240133\pi$
$978$	0	0
$979$	−43906.8	−1.43337
$980$	0	0
$981$	−22853.0	−0.743772
$982$	0	0
$983$	−14970.7	−0.485750	−0.242875	−	0.970058i	$-0.578090\pi$
$983$	−14970.7	−0.485750	−0.242875	+	0.970058i	$0.578090\pi$
$984$	0	0
$985$	−32791.1	−1.06072
$986$	0	0
$987$	−3063.94	−0.0988109
$988$	0	0
$989$	−3549.97	−0.114138
$990$	0	0
$991$	10815.1	0.346674	0.173337	−	0.984863i	$-0.444545\pi$
$991$	10815.1	0.346674	0.173337	+	0.984863i	$0.444545\pi$
$992$	0	0
$993$	24705.2	0.789522
$994$	0	0
$995$	3246.49	0.103438
$996$	0	0
$997$	−39556.1	−1.25652	−0.628262	−	0.778002i	$-0.716233\pi$
$997$	−39556.1	−1.25652	−0.628262	+	0.778002i	$0.716233\pi$
$998$	0	0
$999$	44745.6	1.41711

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.4.a.i.1.3		6
4.3	odd	2		43.4.a.b.1.5	✓	6
12.11	even	2		387.4.a.h.1.2		6
20.19	odd	2		1075.4.a.b.1.2		6
28.27	even	2		2107.4.a.c.1.5		6
172.171	even	2		1849.4.a.c.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.a.b.1.5	✓	6	4.3	odd	2
387.4.a.h.1.2		6	12.11	even	2
688.4.a.i.1.3		6	1.1	even	1	trivial
1075.4.a.b.1.2		6	20.19	odd	2
1849.4.a.c.1.2		6	172.171	even	2
2107.4.a.c.1.5		6	28.27	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:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	688.a (trivial)

\(f(q)\)	\(=\)	\(q-2.46717 q^{3} -7.54340 q^{5} -4.58222 q^{7} -20.9131 q^{9} +O(q^{10})\)	\(q-2.46717 q^{3} -7.54340 q^{5} -4.58222 q^{7} -20.9131 q^{9} -26.9150 q^{11} -15.6529 q^{13} +18.6109 q^{15} +27.2420 q^{17} -38.3104 q^{19} +11.3051 q^{21} -82.5575 q^{23} -68.0971 q^{25} +118.210 q^{27} -34.2852 q^{29} -119.055 q^{31} +66.4040 q^{33} +34.5655 q^{35} +378.527 q^{37} +38.6185 q^{39} +385.478 q^{41} +43.0000 q^{43} +157.756 q^{45} -271.022 q^{47} -322.003 q^{49} -67.2108 q^{51} -329.363 q^{53} +203.031 q^{55} +94.5184 q^{57} +173.956 q^{59} +54.5012 q^{61} +95.8283 q^{63} +118.076 q^{65} +906.954 q^{67} +203.684 q^{69} +621.376 q^{71} -1025.87 q^{73} +168.007 q^{75} +123.331 q^{77} +737.945 q^{79} +273.009 q^{81} -558.465 q^{83} -205.498 q^{85} +84.5875 q^{87} +1631.31 q^{89} +71.7252 q^{91} +293.729 q^{93} +288.991 q^{95} -406.607 q^{97} +562.875 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 7 q^{3} + 43 q^{5} - 8 q^{7} + 81 q^{9}+O(q^{10}) \) 6 * q - 7 * q^3 + 43 * q^5 - 8 * q^7 + 81 * q^9	\( 6 q - 7 q^{3} + 43 q^{5} - 8 q^{7} + 81 q^{9} + 28 q^{11} + 56 q^{13} + 124 q^{15} + 19 q^{17} + 75 q^{19} - 18 q^{21} - 131 q^{23} + 105 q^{25} - 238 q^{27} + 515 q^{29} - 237 q^{31} + 540 q^{33} - 198 q^{35} + 269 q^{37} - 290 q^{39} + 471 q^{41} + 258 q^{43} + 334 q^{45} - 415 q^{47} + 350 q^{49} + 1241 q^{51} + 450 q^{53} + 1732 q^{55} - 1000 q^{57} - 356 q^{59} - 1328 q^{61} + 2290 q^{63} - 62 q^{65} + 632 q^{67} - 1130 q^{69} + 144 q^{71} + 864 q^{73} + 2494 q^{75} + 2660 q^{77} + 1613 q^{79} - 102 q^{81} + 682 q^{83} + 84 q^{85} - 449 q^{87} + 3378 q^{89} + 3900 q^{91} + 1879 q^{93} + 79 q^{95} - 55 q^{97} + 1612 q^{99}+O(q^{100}) \) 6 * q - 7 * q^3 + 43 * q^5 - 8 * q^7 + 81 * q^9 + 28 * q^11 + 56 * q^13 + 124 * q^15 + 19 * q^17 + 75 * q^19 - 18 * q^21 - 131 * q^23 + 105 * q^25 - 238 * q^27 + 515 * q^29 - 237 * q^31 + 540 * q^33 - 198 * q^35 + 269 * q^37 - 290 * q^39 + 471 * q^41 + 258 * q^43 + 334 * q^45 - 415 * q^47 + 350 * q^49 + 1241 * q^51 + 450 * q^53 + 1732 * q^55 - 1000 * q^57 - 356 * q^59 - 1328 * q^61 + 2290 * q^63 - 62 * q^65 + 632 * q^67 - 1130 * q^69 + 144 * q^71 + 864 * q^73 + 2494 * q^75 + 2660 * q^77 + 1613 * q^79 - 102 * q^81 + 682 * q^83 + 84 * q^85 - 449 * q^87 + 3378 * q^89 + 3900 * q^91 + 1879 * q^93 + 79 * q^95 - 55 * q^97 + 1612 * q^99

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