LMFDB - Embedded newform 688.4.a.i.1.4

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			1.4
Root			$-0.299707$ 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)

$f(q)$	$=$	$q-1.43046 q^{3} +20.4116 q^{5} -29.9522 q^{7} -24.9538 q^{9} +O(q^{10})$	$q-1.43046 q^{3} +20.4116 q^{5} -29.9522 q^{7} -24.9538 q^{9} +22.8719 q^{11} -44.4397 q^{13} -29.1981 q^{15} -13.0970 q^{17} -5.41527 q^{19} +42.8456 q^{21} +175.226 q^{23} +291.634 q^{25} +74.3180 q^{27} +165.972 q^{29} +155.581 q^{31} -32.7174 q^{33} -611.374 q^{35} -95.3991 q^{37} +63.5693 q^{39} +189.928 q^{41} +43.0000 q^{43} -509.347 q^{45} +37.2349 q^{47} +554.137 q^{49} +18.7348 q^{51} +559.862 q^{53} +466.852 q^{55} +7.74635 q^{57} +82.3042 q^{59} -640.304 q^{61} +747.422 q^{63} -907.085 q^{65} +509.592 q^{67} -250.655 q^{69} -792.932 q^{71} +612.727 q^{73} -417.171 q^{75} -685.065 q^{77} -237.047 q^{79} +567.443 q^{81} -418.683 q^{83} -267.330 q^{85} -237.417 q^{87} +113.788 q^{89} +1331.07 q^{91} -222.553 q^{93} -110.534 q^{95} +1649.00 q^{97} -570.740 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.43046	−0.275293	−0.137646	−	0.990481i	$-0.543954\pi$
$3$	−1.43046	−0.275293	−0.137646	+	0.990481i	$0.543954\pi$
$4$	0	0
$5$	20.4116	1.82567	0.912835	−	0.408329i	$-0.133888\pi$
$5$	20.4116	1.82567	0.912835	+	0.408329i	$0.133888\pi$
$6$	0	0
$7$	−29.9522	−1.61727	−0.808635	−	0.588311i	$-0.799793\pi$
$7$	−29.9522	−1.61727	−0.808635	+	0.588311i	$0.799793\pi$
$8$	0	0
$9$	−24.9538	−0.924214
$10$	0	0
$11$	22.8719	0.626922	0.313461	−	0.949601i	$-0.398512\pi$
$11$	22.8719	0.626922	0.313461	+	0.949601i	$0.398512\pi$
$12$	0	0
$13$	−44.4397	−0.948104	−0.474052	−	0.880497i	$-0.657209\pi$
$13$	−44.4397	−0.948104	−0.474052	+	0.880497i	$0.657209\pi$
$14$	0	0
$15$	−29.1981	−0.502594
$16$	0	0
$17$	−13.0970	−0.186852	−0.0934260	−	0.995626i	$-0.529782\pi$
$17$	−13.0970	−0.186852	−0.0934260	+	0.995626i	$0.529782\pi$
$18$	0	0
$19$	−5.41527	−0.0653868	−0.0326934	−	0.999465i	$-0.510408\pi$
$19$	−5.41527	−0.0653868	−0.0326934	+	0.999465i	$0.510408\pi$
$20$	0	0
$21$	42.8456	0.445223
$22$	0	0
$23$	175.226	1.58857	0.794287	−	0.607542i	$-0.207845\pi$
$23$	175.226	1.58857	0.794287	+	0.607542i	$0.207845\pi$
$24$	0	0
$25$	291.634	2.33307
$26$	0	0
$27$	74.3180	0.529722
$28$	0	0
$29$	165.972	1.06277	0.531383	−	0.847132i	$-0.321672\pi$
$29$	165.972	1.06277	0.531383	+	0.847132i	$0.321672\pi$
$30$	0	0
$31$	155.581	0.901395	0.450697	−	0.892677i	$-0.351175\pi$
$31$	155.581	0.901395	0.450697	+	0.892677i	$0.351175\pi$
$32$	0	0
$33$	−32.7174	−0.172587
$34$	0	0
$35$	−611.374	−2.95260
$36$	0	0
$37$	−95.3991	−0.423879	−0.211939	−	0.977283i	$-0.567978\pi$
$37$	−95.3991	−0.423879	−0.211939	+	0.977283i	$0.567978\pi$
$38$	0	0
$39$	63.5693	0.261006
$40$	0	0
$41$	189.928	0.723457	0.361728	−	0.932284i	$-0.382187\pi$
$41$	189.928	0.723457	0.361728	+	0.932284i	$0.382187\pi$
$42$	0	0
$43$	43.0000	0.152499
$43$	43.0000	0.152499
$44$	0	0
$45$	−509.347	−1.68731
$46$	0	0
$47$	37.2349	0.115559	0.0577794	−	0.998329i	$-0.481598\pi$
$47$	37.2349	0.115559	0.0577794	+	0.998329i	$0.481598\pi$
$48$	0	0
$49$	554.137	1.61556
$50$	0	0
$51$	18.7348	0.0514390
$52$	0	0
$53$	559.862	1.45100	0.725500	−	0.688223i	$-0.241609\pi$
$53$	559.862	1.45100	0.725500	+	0.688223i	$0.241609\pi$
$54$	0	0
$55$	466.852	1.14455
$56$	0	0
$57$	7.74635	0.0180005
$58$	0	0
$59$	82.3042	0.181612	0.0908059	−	0.995869i	$-0.471056\pi$
$59$	82.3042	0.181612	0.0908059	+	0.995869i	$0.471056\pi$
$60$	0	0
$61$	−640.304	−1.34398	−0.671988	−	0.740562i	$-0.734559\pi$
$61$	−640.304	−1.34398	−0.671988	+	0.740562i	$0.734559\pi$
$62$	0	0
$63$	747.422	1.49470
$64$	0	0
$65$	−907.085	−1.73092
$66$	0	0
$67$	509.592	0.929203	0.464602	−	0.885520i	$-0.346198\pi$
$67$	509.592	0.929203	0.464602	+	0.885520i	$0.346198\pi$
$68$	0	0
$69$	−250.655	−0.437323
$70$	0	0
$71$	−792.932	−1.32540	−0.662702	−	0.748883i	$-0.730590\pi$
$71$	−792.932	−1.32540	−0.662702	+	0.748883i	$0.730590\pi$
$72$	0	0
$73$	612.727	0.982387	0.491194	−	0.871050i	$-0.336561\pi$
$73$	612.727	0.982387	0.491194	+	0.871050i	$0.336561\pi$
$74$	0	0
$75$	−417.171	−0.642277
$76$	0	0
$77$	−685.065	−1.01390
$78$	0	0
$79$	−237.047	−0.337594	−0.168797	−	0.985651i	$-0.553988\pi$
$79$	−237.047	−0.337594	−0.168797	+	0.985651i	$0.553988\pi$
$80$	0	0
$81$	567.443	0.778385
$82$	0	0
$83$	−418.683	−0.553691	−0.276846	−	0.960914i	$-0.589289\pi$
$83$	−418.683	−0.553691	−0.276846	+	0.960914i	$0.589289\pi$
$84$	0	0
$85$	−267.330	−0.341130
$86$	0	0
$87$	−237.417	−0.292572
$88$	0	0
$89$	113.788	0.135523	0.0677615	−	0.997702i	$-0.478414\pi$
$89$	113.788	0.135523	0.0677615	+	0.997702i	$0.478414\pi$
$90$	0	0
$91$	1331.07	1.53334
$92$	0	0
$93$	−222.553	−0.248147
$94$	0	0
$95$	−110.534	−0.119375
$96$	0	0
$97$	1649.00	1.72609	0.863045	−	0.505127i	$-0.168554\pi$
$97$	1649.00	1.72609	0.863045	+	0.505127i	$0.168554\pi$
$98$	0	0
$99$	−570.740	−0.579410
$100$	0	0
$101$	−165.065	−0.162619	−0.0813097	−	0.996689i	$-0.525910\pi$
$101$	−165.065	−0.162619	−0.0813097	+	0.996689i	$0.525910\pi$
$102$	0	0
$103$	1665.32	1.59309	0.796547	−	0.604576i	$-0.206658\pi$
$103$	1665.32	1.59309	0.796547	+	0.604576i	$0.206658\pi$
$104$	0	0
$105$	874.547	0.812829
$106$	0	0
$107$	631.683	0.570721	0.285360	−	0.958420i	$-0.407887\pi$
$107$	631.683	0.570721	0.285360	+	0.958420i	$0.407887\pi$
$108$	0	0
$109$	839.368	0.737586	0.368793	−	0.929512i	$-0.379771\pi$
$109$	839.368	0.737586	0.368793	+	0.929512i	$0.379771\pi$
$110$	0	0
$111$	136.465	0.116691
$112$	0	0
$113$	−1080.84	−0.899796	−0.449898	−	0.893080i	$-0.648540\pi$
$113$	−1080.84	−0.899796	−0.449898	+	0.893080i	$0.648540\pi$
$114$	0	0
$115$	3576.65	2.90021
$116$	0	0
$117$	1108.94	0.876251
$118$	0	0
$119$	392.284	0.302190
$120$	0	0
$121$	−807.876	−0.606969
$122$	0	0
$123$	−271.685	−0.199162
$124$	0	0
$125$	3401.26	2.43375
$126$	0	0
$127$	−594.837	−0.415616	−0.207808	−	0.978170i	$-0.566633\pi$
$127$	−594.837	−0.415616	−0.207808	+	0.978170i	$0.566633\pi$
$128$	0	0
$129$	−61.5099	−0.0419818
$130$	0	0
$131$	−1987.90	−1.32583	−0.662916	−	0.748694i	$-0.730681\pi$
$131$	−1987.90	−1.32583	−0.662916	+	0.748694i	$0.730681\pi$
$132$	0	0
$133$	162.200	0.105748
$134$	0	0
$135$	1516.95	0.967098
$136$	0	0
$137$	1717.01	1.07076	0.535379	−	0.844612i	$-0.320169\pi$
$137$	1717.01	1.07076	0.535379	+	0.844612i	$0.320169\pi$
$138$	0	0
$139$	1735.09	1.05876	0.529381	−	0.848384i	$-0.322424\pi$
$139$	1735.09	1.05876	0.529381	+	0.848384i	$0.322424\pi$
$140$	0	0
$141$	−53.2631	−0.0318125
$142$	0	0
$143$	−1016.42	−0.594387
$144$	0	0
$145$	3387.76	1.94026
$146$	0	0
$147$	−792.673	−0.444752
$148$	0	0
$149$	456.600	0.251048	0.125524	−	0.992091i	$-0.459939\pi$
$149$	456.600	0.251048	0.125524	+	0.992091i	$0.459939\pi$
$150$	0	0
$151$	483.618	0.260638	0.130319	−	0.991472i	$-0.458400\pi$
$151$	483.618	0.260638	0.130319	+	0.991472i	$0.458400\pi$
$152$	0	0
$153$	326.819	0.172691
$154$	0	0
$155$	3175.67	1.64565
$156$	0	0
$157$	2300.64	1.16950	0.584749	−	0.811214i	$-0.301193\pi$
$157$	2300.64	1.16950	0.584749	+	0.811214i	$0.301193\pi$
$158$	0	0
$159$	−800.862	−0.399450
$160$	0	0
$161$	−5248.42	−2.56915
$162$	0	0
$163$	−314.362	−0.151060	−0.0755299	−	0.997144i	$-0.524065\pi$
$163$	−314.362	−0.151060	−0.0755299	+	0.997144i	$0.524065\pi$
$164$	0	0
$165$	−667.815	−0.315087
$166$	0	0
$167$	282.441	0.130874	0.0654369	−	0.997857i	$-0.479156\pi$
$167$	282.441	0.130874	0.0654369	+	0.997857i	$0.479156\pi$
$168$	0	0
$169$	−222.114	−0.101099
$170$	0	0
$171$	135.131	0.0604314
$172$	0	0
$173$	1774.10	0.779667	0.389834	−	0.920885i	$-0.372532\pi$
$173$	1774.10	0.779667	0.389834	+	0.920885i	$0.372532\pi$
$174$	0	0
$175$	−8735.09	−3.77320
$176$	0	0
$177$	−117.733	−0.0499964
$178$	0	0
$179$	−890.113	−0.371677	−0.185839	−	0.982580i	$-0.559500\pi$
$179$	−890.113	−0.371677	−0.185839	+	0.982580i	$0.559500\pi$
$180$	0	0
$181$	−2883.82	−1.18427	−0.592135	−	0.805838i	$-0.701715\pi$
$181$	−2883.82	−1.18427	−0.592135	+	0.805838i	$0.701715\pi$
$182$	0	0
$183$	915.931	0.369987
$184$	0	0
$185$	−1947.25	−0.773863
$186$	0	0
$187$	−299.553	−0.117142
$188$	0	0
$189$	−2225.99	−0.856704
$190$	0	0
$191$	−1120.48	−0.424478	−0.212239	−	0.977218i	$-0.568076\pi$
$191$	−1120.48	−0.424478	−0.212239	+	0.977218i	$0.568076\pi$
$192$	0	0
$193$	−2777.72	−1.03598	−0.517992	−	0.855385i	$-0.673320\pi$
$193$	−2777.72	−1.03598	−0.517992	+	0.855385i	$0.673320\pi$
$194$	0	0
$195$	1297.55	0.476511
$196$	0	0
$197$	−1266.30	−0.457971	−0.228985	−	0.973430i	$-0.573541\pi$
$197$	−1266.30	−0.457971	−0.228985	+	0.973430i	$0.573541\pi$
$198$	0	0
$199$	5113.40	1.82150	0.910752	−	0.412955i	$-0.135503\pi$
$199$	5113.40	1.82150	0.910752	+	0.412955i	$0.135503\pi$
$200$	0	0
$201$	−728.953	−0.255803
$202$	0	0
$203$	−4971.23	−1.71878
$204$	0	0
$205$	3876.73	1.32079
$206$	0	0
$207$	−4372.56	−1.46818
$208$	0	0
$209$	−123.858	−0.0409924
$210$	0	0
$211$	1318.35	0.430138	0.215069	−	0.976599i	$-0.431002\pi$
$211$	1318.35	0.430138	0.215069	+	0.976599i	$0.431002\pi$
$212$	0	0
$213$	1134.26	0.364874
$214$	0	0
$215$	877.699	0.278412
$216$	0	0
$217$	−4660.01	−1.45780
$218$	0	0
$219$	−876.483	−0.270444
$220$	0	0
$221$	582.026	0.177155
$222$	0	0
$223$	812.504	0.243988	0.121994	−	0.992531i	$-0.461071\pi$
$223$	812.504	0.243988	0.121994	+	0.992531i	$0.461071\pi$
$224$	0	0
$225$	−7277.36	−2.15626
$226$	0	0
$227$	−1890.51	−0.552763	−0.276382	−	0.961048i	$-0.589135\pi$
$227$	−1890.51	−0.552763	−0.276382	+	0.961048i	$0.589135\pi$
$228$	0	0
$229$	1368.43	0.394884	0.197442	−	0.980315i	$-0.436737\pi$
$229$	1368.43	0.394884	0.197442	+	0.980315i	$0.436737\pi$
$230$	0	0
$231$	979.960	0.279120
$232$	0	0
$233$	−3535.33	−0.994023	−0.497012	−	0.867744i	$-0.665569\pi$
$233$	−3535.33	−0.994023	−0.497012	+	0.867744i	$0.665569\pi$
$234$	0	0
$235$	760.023	0.210972
$236$	0	0
$237$	339.088	0.0929371
$238$	0	0
$239$	−1515.24	−0.410095	−0.205047	−	0.978752i	$-0.565735\pi$
$239$	−1515.24	−0.410095	−0.205047	+	0.978752i	$0.565735\pi$
$240$	0	0
$241$	−6717.46	−1.79548	−0.897738	−	0.440530i	$-0.854791\pi$
$241$	−6717.46	−1.79548	−0.897738	+	0.440530i	$0.854791\pi$
$242$	0	0
$243$	−2818.29	−0.744006
$244$	0	0
$245$	11310.8	2.94948
$246$	0	0
$247$	240.653	0.0619934
$248$	0	0
$249$	598.910	0.152427
$250$	0	0
$251$	2291.38	0.576217	0.288109	−	0.957598i	$-0.406974\pi$
$251$	2291.38	0.576217	0.288109	+	0.957598i	$0.406974\pi$
$252$	0	0
$253$	4007.76	0.995912
$254$	0	0
$255$	382.406	0.0939107
$256$	0	0
$257$	−2236.18	−0.542760	−0.271380	−	0.962472i	$-0.587480\pi$
$257$	−2236.18	−0.542760	−0.271380	+	0.962472i	$0.587480\pi$
$258$	0	0
$259$	2857.42	0.685526
$260$	0	0
$261$	−4141.63	−0.982224
$262$	0	0
$263$	6393.87	1.49910	0.749550	−	0.661948i	$-0.230270\pi$
$263$	6393.87	1.49910	0.749550	+	0.661948i	$0.230270\pi$
$264$	0	0
$265$	11427.7	2.64905
$266$	0	0
$267$	−162.770	−0.0373085
$268$	0	0
$269$	−2688.95	−0.609473	−0.304736	−	0.952437i	$-0.598568\pi$
$269$	−2688.95	−0.609473	−0.304736	+	0.952437i	$0.598568\pi$
$270$	0	0
$271$	1057.95	0.237143	0.118572	−	0.992945i	$-0.462168\pi$
$271$	1057.95	0.237143	0.118572	+	0.992945i	$0.462168\pi$
$272$	0	0
$273$	−1904.04	−0.422117
$274$	0	0
$275$	6670.22	1.46265
$276$	0	0
$277$	1904.17	0.413035	0.206518	−	0.978443i	$-0.433787\pi$
$277$	1904.17	0.413035	0.206518	+	0.978443i	$0.433787\pi$
$278$	0	0
$279$	−3882.34	−0.833081
$280$	0	0
$281$	6520.37	1.38424	0.692122	−	0.721781i	$-0.256676\pi$
$281$	6520.37	1.38424	0.692122	+	0.721781i	$0.256676\pi$
$282$	0	0
$283$	−130.376	−0.0273854	−0.0136927	−	0.999906i	$-0.504359\pi$
$283$	−130.376	−0.0273854	−0.0136927	+	0.999906i	$0.504359\pi$
$284$	0	0
$285$	158.115	0.0328630
$286$	0	0
$287$	−5688.76	−1.17002
$288$	0	0
$289$	−4741.47	−0.965086
$290$	0	0
$291$	−2358.84	−0.475180
$292$	0	0
$293$	−321.150	−0.0640333	−0.0320166	−	0.999487i	$-0.510193\pi$
$293$	−321.150	−0.0640333	−0.0320166	+	0.999487i	$0.510193\pi$
$294$	0	0
$295$	1679.96	0.331563
$296$	0	0
$297$	1699.79	0.332094
$298$	0	0
$299$	−7787.00	−1.50613
$300$	0	0
$301$	−1287.95	−0.246631
$302$	0	0
$303$	236.119	0.0447679
$304$	0	0
$305$	−13069.6	−2.45366
$306$	0	0
$307$	−3944.87	−0.733374	−0.366687	−	0.930344i	$-0.619508\pi$
$307$	−3944.87	−0.733374	−0.366687	+	0.930344i	$0.619508\pi$
$308$	0	0
$309$	−2382.18	−0.438567
$310$	0	0
$311$	−7046.04	−1.28471	−0.642354	−	0.766408i	$-0.722042\pi$
$311$	−7046.04	−1.28471	−0.642354	+	0.766408i	$0.722042\pi$
$312$	0	0
$313$	−230.283	−0.0415858	−0.0207929	−	0.999784i	$-0.506619\pi$
$313$	−230.283	−0.0415858	−0.0207929	+	0.999784i	$0.506619\pi$
$314$	0	0
$315$	15256.1	2.72883
$316$	0	0
$317$	6574.02	1.16477	0.582387	−	0.812911i	$-0.302119\pi$
$317$	6574.02	1.16477	0.582387	+	0.812911i	$0.302119\pi$
$318$	0	0
$319$	3796.10	0.666271
$320$	0	0
$321$	−903.600	−0.157115
$322$	0	0
$323$	70.9237	0.0122177
$324$	0	0
$325$	−12960.1	−2.21199
$326$	0	0
$327$	−1200.68	−0.203052
$328$	0	0
$329$	−1115.27	−0.186890
$330$	0	0
$331$	−8326.42	−1.38266	−0.691331	−	0.722538i	$-0.742975\pi$
$331$	−8326.42	−1.38266	−0.691331	+	0.722538i	$0.742975\pi$
$332$	0	0
$333$	2380.57	0.391755
$334$	0	0
$335$	10401.6	1.69642
$336$	0	0
$337$	9021.52	1.45826	0.729130	−	0.684375i	$-0.239925\pi$
$337$	9021.52	1.45826	0.729130	+	0.684375i	$0.239925\pi$
$338$	0	0
$339$	1546.10	0.247707
$340$	0	0
$341$	3558.44	0.565104
$342$	0	0
$343$	−6324.03	−0.995526
$344$	0	0
$345$	−5116.27	−0.798408
$346$	0	0
$347$	−133.107	−0.0205925	−0.0102962	−	0.999947i	$-0.503277\pi$
$347$	−133.107	−0.0205925	−0.0102962	+	0.999947i	$0.503277\pi$
$348$	0	0
$349$	−1322.67	−0.202868	−0.101434	−	0.994842i	$-0.532343\pi$
$349$	−1322.67	−0.202868	−0.101434	+	0.994842i	$0.532343\pi$
$350$	0	0
$351$	−3302.67	−0.502232
$352$	0	0
$353$	12515.7	1.88709	0.943543	−	0.331251i	$-0.107471\pi$
$353$	12515.7	1.88709	0.943543	+	0.331251i	$0.107471\pi$
$354$	0	0
$355$	−16185.0	−2.41975
$356$	0	0
$357$	−561.148	−0.0831907
$358$	0	0
$359$	12654.2	1.86035	0.930174	−	0.367118i	$-0.119656\pi$
$359$	12654.2	1.86035	0.930174	+	0.367118i	$0.119656\pi$
$360$	0	0
$361$	−6829.67	−0.995725
$362$	0	0
$363$	1155.64	0.167094
$364$	0	0
$365$	12506.7	1.79351
$366$	0	0
$367$	13307.8	1.89281	0.946407	−	0.322976i	$-0.104683\pi$
$367$	13307.8	1.89281	0.946407	+	0.322976i	$0.104683\pi$
$368$	0	0
$369$	−4739.41	−0.668629
$370$	0	0
$371$	−16769.1	−2.34666
$372$	0	0
$373$	−6552.61	−0.909602	−0.454801	−	0.890593i	$-0.650290\pi$
$373$	−6552.61	−0.909602	−0.454801	+	0.890593i	$0.650290\pi$
$374$	0	0
$375$	−4865.38	−0.669993
$376$	0	0
$377$	−7375.74	−1.00761
$378$	0	0
$379$	−11236.0	−1.52284	−0.761418	−	0.648262i	$-0.775496\pi$
$379$	−11236.0	−1.52284	−0.761418	+	0.648262i	$0.775496\pi$
$380$	0	0
$381$	850.893	0.114416
$382$	0	0
$383$	−6353.36	−0.847628	−0.423814	−	0.905749i	$-0.639309\pi$
$383$	−6353.36	−0.847628	−0.423814	+	0.905749i	$0.639309\pi$
$384$	0	0
$385$	−13983.3	−1.85105
$386$	0	0
$387$	−1073.01	−0.140941
$388$	0	0
$389$	3788.91	0.493844	0.246922	−	0.969035i	$-0.420581\pi$
$389$	3788.91	0.493844	0.246922	+	0.969035i	$0.420581\pi$
$390$	0	0
$391$	−2294.94	−0.296828
$392$	0	0
$393$	2843.62	0.364992
$394$	0	0
$395$	−4838.52	−0.616335
$396$	0	0
$397$	−2601.66	−0.328901	−0.164451	−	0.986385i	$-0.552585\pi$
$397$	−2601.66	−0.328901	−0.164451	+	0.986385i	$0.552585\pi$
$398$	0	0
$399$	−232.021	−0.0291117
$400$	0	0
$401$	−1698.36	−0.211501	−0.105750	−	0.994393i	$-0.533724\pi$
$401$	−1698.36	−0.211501	−0.105750	+	0.994393i	$0.533724\pi$
$402$	0	0
$403$	−6913.99	−0.854616
$404$	0	0
$405$	11582.4	1.42107
$406$	0	0
$407$	−2181.96	−0.265739
$408$	0	0
$409$	−5511.51	−0.666324	−0.333162	−	0.942870i	$-0.608116\pi$
$409$	−5511.51	−0.666324	−0.333162	+	0.942870i	$0.608116\pi$
$410$	0	0
$411$	−2456.12	−0.294772
$412$	0	0
$413$	−2465.20	−0.293715
$414$	0	0
$415$	−8545.99	−1.01086
$416$	0	0
$417$	−2481.98	−0.291470
$418$	0	0
$419$	6137.26	0.715573	0.357786	−	0.933803i	$-0.383532\pi$
$419$	6137.26	0.715573	0.357786	+	0.933803i	$0.383532\pi$
$420$	0	0
$421$	9168.60	1.06140	0.530701	−	0.847559i	$-0.321929\pi$
$421$	9168.60	1.06140	0.530701	+	0.847559i	$0.321929\pi$
$422$	0	0
$423$	−929.150	−0.106801
$424$	0	0
$425$	−3819.52	−0.435939
$426$	0	0
$427$	19178.5	2.17357
$428$	0	0
$429$	1453.95	0.163630
$430$	0	0
$431$	−12535.8	−1.40099	−0.700494	−	0.713658i	$-0.747037\pi$
$431$	−12535.8	−1.40099	−0.700494	+	0.713658i	$0.747037\pi$
$432$	0	0
$433$	7163.93	0.795095	0.397548	−	0.917582i	$-0.369861\pi$
$433$	7163.93	0.795095	0.397548	+	0.917582i	$0.369861\pi$
$434$	0	0
$435$	−4846.06	−0.534140
$436$	0	0
$437$	−948.898	−0.103872
$438$	0	0
$439$	2988.31	0.324884	0.162442	−	0.986718i	$-0.448063\pi$
$439$	2988.31	0.324884	0.162442	+	0.986718i	$0.448063\pi$
$440$	0	0
$441$	−13827.8	−1.49312
$442$	0	0
$443$	11257.1	1.20732	0.603660	−	0.797242i	$-0.293708\pi$
$443$	11257.1	1.20732	0.603660	+	0.797242i	$0.293708\pi$
$444$	0	0
$445$	2322.60	0.247420
$446$	0	0
$447$	−653.150	−0.0691117
$448$	0	0
$449$	−4960.39	−0.521371	−0.260685	−	0.965424i	$-0.583948\pi$
$449$	−4960.39	−0.521371	−0.260685	+	0.965424i	$0.583948\pi$
$450$	0	0
$451$	4344.01	0.453551
$452$	0	0
$453$	−691.798	−0.0717517
$454$	0	0
$455$	27169.2	2.79937
$456$	0	0
$457$	−4911.00	−0.502685	−0.251343	−	0.967898i	$-0.580872\pi$
$457$	−4911.00	−0.502685	−0.251343	+	0.967898i	$0.580872\pi$
$458$	0	0
$459$	−973.341	−0.0989797
$460$	0	0
$461$	−3089.54	−0.312135	−0.156068	−	0.987746i	$-0.549882\pi$
$461$	−3089.54	−0.312135	−0.156068	+	0.987746i	$0.549882\pi$
$462$	0	0
$463$	−2821.14	−0.283174	−0.141587	−	0.989926i	$-0.545221\pi$
$463$	−2821.14	−0.283174	−0.141587	+	0.989926i	$0.545221\pi$
$464$	0	0
$465$	−4542.67	−0.453035
$466$	0	0
$467$	−483.354	−0.0478950	−0.0239475	−	0.999713i	$-0.507623\pi$
$467$	−483.354	−0.0478950	−0.0239475	+	0.999713i	$0.507623\pi$
$468$	0	0
$469$	−15263.4	−1.50277
$470$	0	0
$471$	−3290.99	−0.321955
$472$	0	0
$473$	983.492	0.0956047
$474$	0	0
$475$	−1579.28	−0.152552
$476$	0	0
$477$	−13970.7	−1.34103
$478$	0	0
$479$	5570.85	0.531396	0.265698	−	0.964056i	$-0.414398\pi$
$479$	5570.85	0.531396	0.265698	+	0.964056i	$0.414398\pi$
$480$	0	0
$481$	4239.51	0.401881
$482$	0	0
$483$	7507.68	0.707269
$484$	0	0
$485$	33658.8	3.15127
$486$	0	0
$487$	−453.859	−0.0422306	−0.0211153	−	0.999777i	$-0.506722\pi$
$487$	−453.859	−0.0422306	−0.0211153	+	0.999777i	$0.506722\pi$
$488$	0	0
$489$	449.684	0.0415857
$490$	0	0
$491$	−2787.17	−0.256178	−0.128089	−	0.991763i	$-0.540884\pi$
$491$	−2787.17	−0.256178	−0.128089	+	0.991763i	$0.540884\pi$
$492$	0	0
$493$	−2173.73	−0.198580
$494$	0	0
$495$	−11649.7	−1.05781
$496$	0	0
$497$	23750.1	2.14354
$498$	0	0
$499$	325.118	0.0291669	0.0145834	−	0.999894i	$-0.495358\pi$
$499$	325.118	0.0291669	0.0145834	+	0.999894i	$0.495358\pi$
$500$	0	0
$501$	−404.022	−0.0360286
$502$	0	0
$503$	11441.8	1.01424	0.507120	−	0.861875i	$-0.330710\pi$
$503$	11441.8	1.01424	0.507120	+	0.861875i	$0.330710\pi$
$504$	0	0
$505$	−3369.24	−0.296889
$506$	0	0
$507$	317.727	0.0278318
$508$	0	0
$509$	−8398.29	−0.731332	−0.365666	−	0.930746i	$-0.619159\pi$
$509$	−8398.29	−0.731332	−0.365666	+	0.930746i	$0.619159\pi$
$510$	0	0
$511$	−18352.5	−1.58878
$512$	0	0
$513$	−402.452	−0.0346368
$514$	0	0
$515$	33991.8	2.90846
$516$	0	0
$517$	851.632	0.0724463
$518$	0	0
$519$	−2537.79	−0.214637
$520$	0	0
$521$	9762.74	0.820947	0.410474	−	0.911873i	$-0.365363\pi$
$521$	9762.74	0.820947	0.410474	+	0.911873i	$0.365363\pi$
$522$	0	0
$523$	14406.0	1.20445	0.602227	−	0.798325i	$-0.294280\pi$
$523$	14406.0	1.20445	0.602227	+	0.798325i	$0.294280\pi$
$524$	0	0
$525$	12495.2	1.03874
$526$	0	0
$527$	−2037.65	−0.168427
$528$	0	0
$529$	18537.3	1.52357
$530$	0	0
$531$	−2053.80	−0.167848
$532$	0	0
$533$	−8440.33	−0.685912
$534$	0	0
$535$	12893.7	1.04195
$536$	0	0
$537$	1273.27	0.102320
$538$	0	0
$539$	12674.2	1.01283
$540$	0	0
$541$	22048.2	1.75217	0.876087	−	0.482153i	$-0.160145\pi$
$541$	22048.2	1.75217	0.876087	+	0.482153i	$0.160145\pi$
$542$	0	0
$543$	4125.21	0.326021
$544$	0	0
$545$	17132.8	1.34659
$546$	0	0
$547$	−6165.00	−0.481895	−0.240947	−	0.970538i	$-0.577458\pi$
$547$	−6165.00	−0.481895	−0.240947	+	0.970538i	$0.577458\pi$
$548$	0	0
$549$	15978.0	1.24212
$550$	0	0
$551$	−898.784	−0.0694909
$552$	0	0
$553$	7100.10	0.545980
$554$	0	0
$555$	2785.47	0.213039
$556$	0	0
$557$	−12519.9	−0.952400	−0.476200	−	0.879337i	$-0.657986\pi$
$557$	−12519.9	−0.952400	−0.476200	+	0.879337i	$0.657986\pi$
$558$	0	0
$559$	−1910.91	−0.144584
$560$	0	0
$561$	428.499	0.0322482
$562$	0	0
$563$	−21124.7	−1.58135	−0.790674	−	0.612237i	$-0.790270\pi$
$563$	−21124.7	−1.58135	−0.790674	+	0.612237i	$0.790270\pi$
$564$	0	0
$565$	−22061.7	−1.64273
$566$	0	0
$567$	−16996.2	−1.25886
$568$	0	0
$569$	−22782.7	−1.67856	−0.839279	−	0.543700i	$-0.817023\pi$
$569$	−22782.7	−1.67856	−0.839279	+	0.543700i	$0.817023\pi$
$570$	0	0
$571$	12990.7	0.952092	0.476046	−	0.879420i	$-0.342070\pi$
$571$	12990.7	0.952092	0.476046	+	0.879420i	$0.342070\pi$
$572$	0	0
$573$	1602.81	0.116856
$574$	0	0
$575$	51101.9	3.70626
$576$	0	0
$577$	−25759.9	−1.85858	−0.929289	−	0.369353i	$-0.879579\pi$
$577$	−25759.9	−1.85858	−0.929289	+	0.369353i	$0.879579\pi$
$578$	0	0
$579$	3973.43	0.285199
$580$	0	0
$581$	12540.5	0.895468
$582$	0	0
$583$	12805.1	0.909663
$584$	0	0
$585$	22635.2	1.59974
$586$	0	0
$587$	4369.06	0.307207	0.153604	−	0.988133i	$-0.450912\pi$
$587$	4369.06	0.307207	0.153604	+	0.988133i	$0.450912\pi$
$588$	0	0
$589$	−842.515	−0.0589393
$590$	0	0
$591$	1811.40	0.126076
$592$	0	0
$593$	3241.12	0.224447	0.112223	−	0.993683i	$-0.464203\pi$
$593$	3241.12	0.224447	0.112223	+	0.993683i	$0.464203\pi$
$594$	0	0
$595$	8007.15	0.551699
$596$	0	0
$597$	−7314.53	−0.501447
$598$	0	0
$599$	4294.37	0.292927	0.146463	−	0.989216i	$-0.453211\pi$
$599$	4294.37	0.292927	0.146463	+	0.989216i	$0.453211\pi$
$600$	0	0
$601$	1277.55	0.0867093	0.0433546	−	0.999060i	$-0.486195\pi$
$601$	1277.55	0.0867093	0.0433546	+	0.999060i	$0.486195\pi$
$602$	0	0
$603$	−12716.3	−0.858782
$604$	0	0
$605$	−16490.0	−1.10813
$606$	0	0
$607$	−17305.3	−1.15717	−0.578584	−	0.815623i	$-0.696394\pi$
$607$	−17305.3	−1.15717	−0.578584	+	0.815623i	$0.696394\pi$
$608$	0	0
$609$	7111.17	0.473168
$610$	0	0
$611$	−1654.71	−0.109562
$612$	0	0
$613$	−12812.9	−0.844225	−0.422112	−	0.906544i	$-0.638711\pi$
$613$	−12812.9	−0.844225	−0.422112	+	0.906544i	$0.638711\pi$
$614$	0	0
$615$	−5545.52	−0.363605
$616$	0	0
$617$	−3963.52	−0.258615	−0.129307	−	0.991605i	$-0.541275\pi$
$617$	−3963.52	−0.258615	−0.129307	+	0.991605i	$0.541275\pi$
$618$	0	0
$619$	−26937.7	−1.74914	−0.874570	−	0.484899i	$-0.838856\pi$
$619$	−26937.7	−1.74914	−0.874570	+	0.484899i	$0.838856\pi$
$620$	0	0
$621$	13022.5	0.841503
$622$	0	0
$623$	−3408.22	−0.219177
$624$	0	0
$625$	32971.0	2.11015
$626$	0	0
$627$	177.174	0.0112849
$628$	0	0
$629$	1249.44	0.0792026
$630$	0	0
$631$	2401.37	0.151501	0.0757504	−	0.997127i	$-0.475865\pi$
$631$	2401.37	0.151501	0.0757504	+	0.997127i	$0.475865\pi$
$632$	0	0
$633$	−1885.85	−0.118414
$634$	0	0
$635$	−12141.6	−0.758778
$636$	0	0
$637$	−24625.7	−1.53172
$638$	0	0
$639$	19786.6	1.22496
$640$	0	0
$641$	−22872.8	−1.40940	−0.704698	−	0.709508i	$-0.748917\pi$
$641$	−22872.8	−1.40940	−0.704698	+	0.709508i	$0.748917\pi$
$642$	0	0
$643$	−27051.9	−1.65913	−0.829566	−	0.558409i	$-0.811412\pi$
$643$	−27051.9	−1.65913	−0.829566	+	0.558409i	$0.811412\pi$
$644$	0	0
$645$	−1255.52	−0.0766448
$646$	0	0
$647$	−12947.9	−0.786760	−0.393380	−	0.919376i	$-0.628694\pi$
$647$	−12947.9	−0.786760	−0.393380	+	0.919376i	$0.628694\pi$
$648$	0	0
$649$	1882.45	0.113856
$650$	0	0
$651$	6665.97	0.401321
$652$	0	0
$653$	16218.4	0.971934	0.485967	−	0.873977i	$-0.338467\pi$
$653$	16218.4	0.971934	0.485967	+	0.873977i	$0.338467\pi$
$654$	0	0
$655$	−40576.3	−2.42053
$656$	0	0
$657$	−15289.8	−0.907936
$658$	0	0
$659$	26252.1	1.55180	0.775901	−	0.630855i	$-0.217296\pi$
$659$	26252.1	1.55180	0.775901	+	0.630855i	$0.217296\pi$
$660$	0	0
$661$	24649.9	1.45049	0.725243	−	0.688493i	$-0.241727\pi$
$661$	24649.9	1.45049	0.725243	+	0.688493i	$0.241727\pi$
$662$	0	0
$663$	−832.566	−0.0487695
$664$	0	0
$665$	3310.75	0.193061
$666$	0	0
$667$	29082.7	1.68828
$668$	0	0
$669$	−1162.26	−0.0671681
$670$	0	0
$671$	−14645.0	−0.842567
$672$	0	0
$673$	4424.70	0.253432	0.126716	−	0.991939i	$-0.459556\pi$
$673$	4424.70	0.253432	0.126716	+	0.991939i	$0.459556\pi$
$674$	0	0
$675$	21673.6	1.23588
$676$	0	0
$677$	18153.6	1.03057	0.515287	−	0.857018i	$-0.327685\pi$
$677$	18153.6	1.03057	0.515287	+	0.857018i	$0.327685\pi$
$678$	0	0
$679$	−49391.3	−2.79155
$680$	0	0
$681$	2704.30	0.152172
$682$	0	0
$683$	−19887.7	−1.11417	−0.557087	−	0.830454i	$-0.688081\pi$
$683$	−19887.7	−1.11417	−0.557087	+	0.830454i	$0.688081\pi$
$684$	0	0
$685$	35046.9	1.95485
$686$	0	0
$687$	−1957.49	−0.108709
$688$	0	0
$689$	−24880.1	−1.37570
$690$	0	0
$691$	−8550.10	−0.470711	−0.235355	−	0.971909i	$-0.575625\pi$
$691$	−8550.10	−0.470711	−0.235355	+	0.971909i	$0.575625\pi$
$692$	0	0
$693$	17095.0	0.937062
$694$	0	0
$695$	35415.9	1.93295
$696$	0	0
$697$	−2487.48	−0.135179
$698$	0	0
$699$	5057.17	0.273648
$700$	0	0
$701$	−19280.9	−1.03884	−0.519421	−	0.854518i	$-0.673852\pi$
$701$	−19280.9	−1.03884	−0.519421	+	0.854518i	$0.673852\pi$
$702$	0	0
$703$	516.612	0.0277161
$704$	0	0
$705$	−1087.19	−0.0580791
$706$	0	0
$707$	4944.06	0.262999
$708$	0	0
$709$	19384.3	1.02679	0.513394	−	0.858153i	$-0.328388\pi$
$709$	19384.3	1.02679	0.513394	+	0.858153i	$0.328388\pi$
$710$	0	0
$711$	5915.23	0.312009
$712$	0	0
$713$	27261.9	1.43193
$714$	0	0
$715$	−20746.8	−1.08515
$716$	0	0
$717$	2167.49	0.112896
$718$	0	0
$719$	11746.0	0.609253	0.304627	−	0.952472i	$-0.401468\pi$
$719$	11746.0	0.609253	0.304627	+	0.952472i	$0.401468\pi$
$720$	0	0
$721$	−49880.0	−2.57646
$722$	0	0
$723$	9609.08	0.494282
$724$	0	0
$725$	48403.0	2.47951
$726$	0	0
$727$	−10042.8	−0.512333	−0.256167	−	0.966633i	$-0.582460\pi$
$727$	−10042.8	−0.512333	−0.256167	+	0.966633i	$0.582460\pi$
$728$	0	0
$729$	−11289.5	−0.573566
$730$	0	0
$731$	−563.170	−0.0284947
$732$	0	0
$733$	−23541.9	−1.18627	−0.593136	−	0.805102i	$-0.702111\pi$
$733$	−23541.9	−1.18627	−0.593136	+	0.805102i	$0.702111\pi$
$734$	0	0
$735$	−16179.7	−0.811970
$736$	0	0
$737$	11655.3	0.582538
$738$	0	0
$739$	39871.6	1.98471	0.992354	−	0.123422i	$-0.0393868\pi$
$739$	39871.6	1.98471	0.992354	+	0.123422i	$0.0393868\pi$
$740$	0	0
$741$	−344.245	−0.0170664
$742$	0	0
$743$	11387.7	0.562281	0.281141	−	0.959667i	$-0.409287\pi$
$743$	11387.7	0.562281	0.281141	+	0.959667i	$0.409287\pi$
$744$	0	0
$745$	9319.94	0.458331
$746$	0	0
$747$	10447.7	0.511729
$748$	0	0
$749$	−18920.3	−0.923009
$750$	0	0
$751$	26156.9	1.27094	0.635471	−	0.772125i	$-0.280806\pi$
$751$	26156.9	1.27094	0.635471	+	0.772125i	$0.280806\pi$
$752$	0	0
$753$	−3277.73	−0.158629
$754$	0	0
$755$	9871.43	0.475839
$756$	0	0
$757$	9147.11	0.439178	0.219589	−	0.975593i	$-0.429528\pi$
$757$	9147.11	0.439178	0.219589	+	0.975593i	$0.429528\pi$
$758$	0	0
$759$	−5732.95	−0.274167
$760$	0	0
$761$	−41329.2	−1.96870	−0.984350	−	0.176225i	$-0.943611\pi$
$761$	−41329.2	−1.96870	−0.984350	+	0.176225i	$0.943611\pi$
$762$	0	0
$763$	−25140.9	−1.19287
$764$	0	0
$765$	6670.90	0.315277
$766$	0	0
$767$	−3657.57	−0.172187
$768$	0	0
$769$	−3722.69	−0.174569	−0.0872846	−	0.996183i	$-0.527819\pi$
$769$	−3722.69	−0.174569	−0.0872846	+	0.996183i	$0.527819\pi$
$770$	0	0
$771$	3198.78	0.149418
$772$	0	0
$773$	34945.5	1.62600	0.813002	−	0.582261i	$-0.197832\pi$
$773$	34945.5	1.62600	0.813002	+	0.582261i	$0.197832\pi$
$774$	0	0
$775$	45372.8	2.10302
$776$	0	0
$777$	−4087.43	−0.188721
$778$	0	0
$779$	−1028.51	−0.0473045
$780$	0	0
$781$	−18135.9	−0.830925
$782$	0	0
$783$	12334.7	0.562971
$784$	0	0
$785$	46959.8	2.13512
$786$	0	0
$787$	26135.3	1.18377	0.591883	−	0.806024i	$-0.298385\pi$
$787$	26135.3	1.18377	0.591883	+	0.806024i	$0.298385\pi$
$788$	0	0
$789$	−9146.20	−0.412691
$790$	0	0
$791$	32373.6	1.45521
$792$	0	0
$793$	28454.9	1.27423
$794$	0	0
$795$	−16346.9	−0.729263
$796$	0	0
$797$	20345.9	0.904250	0.452125	−	0.891955i	$-0.350666\pi$
$797$	20345.9	0.904250	0.452125	+	0.891955i	$0.350666\pi$
$798$	0	0
$799$	−487.664	−0.0215924
$800$	0	0
$801$	−2839.45	−0.125252
$802$	0	0
$803$	14014.2	0.615880
$804$	0	0
$805$	−107129.	−4.69042
$806$	0	0
$807$	3846.44	0.167783
$808$	0	0
$809$	−21515.4	−0.935034	−0.467517	−	0.883984i	$-0.654851\pi$
$809$	−21515.4	−0.935034	−0.467517	+	0.883984i	$0.654851\pi$
$810$	0	0
$811$	19685.9	0.852363	0.426181	−	0.904638i	$-0.359858\pi$
$811$	19685.9	0.852363	0.426181	+	0.904638i	$0.359858\pi$
$812$	0	0
$813$	−1513.36	−0.0652838
$814$	0	0
$815$	−6416.64	−0.275785
$816$	0	0
$817$	−232.857	−0.00997139
$818$	0	0
$819$	−33215.2	−1.41713
$820$	0	0
$821$	23501.4	0.999029	0.499515	−	0.866305i	$-0.333512\pi$
$821$	23501.4	0.999029	0.499515	+	0.866305i	$0.333512\pi$
$822$	0	0
$823$	−25153.6	−1.06537	−0.532684	−	0.846314i	$-0.678817\pi$
$823$	−25153.6	−1.06537	−0.532684	+	0.846314i	$0.678817\pi$
$824$	0	0
$825$	−9541.51	−0.402658
$826$	0	0
$827$	−31043.6	−1.30531	−0.652655	−	0.757655i	$-0.726345\pi$
$827$	−31043.6	−1.30531	−0.652655	+	0.757655i	$0.726345\pi$
$828$	0	0
$829$	16479.2	0.690404	0.345202	−	0.938528i	$-0.387810\pi$
$829$	16479.2	0.690404	0.345202	+	0.938528i	$0.387810\pi$
$830$	0	0
$831$	−2723.85	−0.113706
$832$	0	0
$833$	−7257.52	−0.301871
$834$	0	0
$835$	5765.08	0.238933
$836$	0	0
$837$	11562.5	0.477489
$838$	0	0
$839$	−10409.3	−0.428329	−0.214164	−	0.976798i	$-0.568703\pi$
$839$	−10409.3	−0.428329	−0.214164	+	0.976798i	$0.568703\pi$
$840$	0	0
$841$	3157.71	0.129473
$842$	0	0
$843$	−9327.15	−0.381072
$844$	0	0
$845$	−4533.71	−0.184573
$846$	0	0
$847$	24197.7	0.981633
$848$	0	0
$849$	186.499	0.00753901
$850$	0	0
$851$	−16716.4	−0.673363
$852$	0	0
$853$	1701.91	0.0683147	0.0341573	−	0.999416i	$-0.489125\pi$
$853$	1701.91	0.0683147	0.0341573	+	0.999416i	$0.489125\pi$
$854$	0	0
$855$	2758.25	0.110328
$856$	0	0
$857$	−13987.3	−0.557524	−0.278762	−	0.960360i	$-0.589924\pi$
$857$	−13987.3	−0.557524	−0.278762	+	0.960360i	$0.589924\pi$
$858$	0	0
$859$	845.375	0.0335784	0.0167892	−	0.999859i	$-0.494656\pi$
$859$	845.375	0.0335784	0.0167892	+	0.999859i	$0.494656\pi$
$860$	0	0
$861$	8137.56	0.322099
$862$	0	0
$863$	16970.3	0.669381	0.334691	−	0.942328i	$-0.391368\pi$
$863$	16970.3	0.669381	0.334691	+	0.942328i	$0.391368\pi$
$864$	0	0
$865$	36212.3	1.42342
$866$	0	0
$867$	6782.50	0.265681
$868$	0	0
$869$	−5421.72	−0.211645
$870$	0	0
$871$	−22646.1	−0.880981
$872$	0	0
$873$	−41148.8	−1.59528
$874$	0	0
$875$	−101876.	−3.93602
$876$	0	0
$877$	−3402.23	−0.130998	−0.0654989	−	0.997853i	$-0.520864\pi$
$877$	−3402.23	−0.130998	−0.0654989	+	0.997853i	$0.520864\pi$
$878$	0	0
$879$	459.393	0.0176279
$880$	0	0
$881$	−9753.40	−0.372986	−0.186493	−	0.982456i	$-0.559712\pi$
$881$	−9753.40	−0.372986	−0.186493	+	0.982456i	$0.559712\pi$
$882$	0	0
$883$	−42672.3	−1.62632	−0.813159	−	0.582042i	$-0.802254\pi$
$883$	−42672.3	−1.62632	−0.813159	+	0.582042i	$0.802254\pi$
$884$	0	0
$885$	−2403.12	−0.0912769
$886$	0	0
$887$	−13027.3	−0.493139	−0.246570	−	0.969125i	$-0.579303\pi$
$887$	−13027.3	−0.493139	−0.246570	+	0.969125i	$0.579303\pi$
$888$	0	0
$889$	17816.7	0.672164
$890$	0	0
$891$	12978.5	0.487987
$892$	0	0
$893$	−201.637	−0.00755601
$894$	0	0
$895$	−18168.6	−0.678560
$896$	0	0
$897$	11139.0	0.414628
$898$	0	0
$899$	25822.1	0.957972
$900$	0	0
$901$	−7332.50	−0.271122
$902$	0	0
$903$	1842.36	0.0678958
$904$	0	0
$905$	−58863.5	−2.16209
$906$	0	0
$907$	2184.13	0.0799591	0.0399796	−	0.999200i	$-0.487271\pi$
$907$	2184.13	0.0799591	0.0399796	+	0.999200i	$0.487271\pi$
$908$	0	0
$909$	4118.99	0.150295
$910$	0	0
$911$	−15349.5	−0.558236	−0.279118	−	0.960257i	$-0.590042\pi$
$911$	−15349.5	−0.558236	−0.279118	+	0.960257i	$0.590042\pi$
$912$	0	0
$913$	−9576.07	−0.347121
$914$	0	0
$915$	18695.6	0.675474
$916$	0	0
$917$	59542.2	2.14423
$918$	0	0
$919$	50672.7	1.81887	0.909433	−	0.415850i	$-0.136516\pi$
$919$	50672.7	1.81887	0.909433	+	0.415850i	$0.136516\pi$
$920$	0	0
$921$	5642.99	0.201892
$922$	0	0
$923$	35237.6	1.25662
$924$	0	0
$925$	−27821.6	−0.988940
$926$	0	0
$927$	−41556.0	−1.47236
$928$	0	0
$929$	−40538.7	−1.43168	−0.715841	−	0.698263i	$-0.753957\pi$
$929$	−40538.7	−1.43168	−0.715841	+	0.698263i	$0.753957\pi$
$930$	0	0
$931$	−3000.80	−0.105636
$932$	0	0
$933$	10079.1	0.353671
$934$	0	0
$935$	−6114.36	−0.213862
$936$	0	0
$937$	13492.4	0.470413	0.235206	−	0.971945i	$-0.424423\pi$
$937$	13492.4	0.470413	0.235206	+	0.971945i	$0.424423\pi$
$938$	0	0
$939$	329.411	0.0114483
$940$	0	0
$941$	22597.2	0.782835	0.391417	−	0.920213i	$-0.371985\pi$
$941$	22597.2	0.782835	0.391417	+	0.920213i	$0.371985\pi$
$942$	0	0
$943$	33280.3	1.14926
$944$	0	0
$945$	−45436.0	−1.56406
$946$	0	0
$947$	−21796.6	−0.747933	−0.373967	−	0.927442i	$-0.622003\pi$
$947$	−21796.6	−0.747933	−0.373967	+	0.927442i	$0.622003\pi$
$948$	0	0
$949$	−27229.4	−0.931405
$950$	0	0
$951$	−9403.89	−0.320654
$952$	0	0
$953$	−40073.5	−1.36213	−0.681064	−	0.732224i	$-0.738482\pi$
$953$	−40073.5	−1.36213	−0.681064	+	0.732224i	$0.738482\pi$
$954$	0	0
$955$	−22870.9	−0.774957
$956$	0	0
$957$	−5430.18	−0.183420
$958$	0	0
$959$	−51428.2	−1.73170
$960$	0	0
$961$	−5585.45	−0.187488
$962$	0	0
$963$	−15762.9	−0.527468
$964$	0	0
$965$	−56697.8	−1.89137
$966$	0	0
$967$	20836.9	0.692935	0.346468	−	0.938062i	$-0.387381\pi$
$967$	20836.9	0.692935	0.346468	+	0.938062i	$0.387381\pi$
$968$	0	0
$969$	−101.454	−0.00336343
$970$	0	0
$971$	5902.12	0.195065	0.0975325	−	0.995232i	$-0.468905\pi$
$971$	5902.12	0.195065	0.0975325	+	0.995232i	$0.468905\pi$
$972$	0	0
$973$	−51969.7	−1.71230
$974$	0	0
$975$	18539.0	0.608946
$976$	0	0
$977$	17199.6	0.563218	0.281609	−	0.959529i	$-0.409132\pi$
$977$	17199.6	0.563218	0.281609	+	0.959529i	$0.409132\pi$
$978$	0	0
$979$	2602.56	0.0849623
$980$	0	0
$981$	−20945.4	−0.681687
$982$	0	0
$983$	−8184.53	−0.265561	−0.132780	−	0.991145i	$-0.542390\pi$
$983$	−8184.53	−0.265561	−0.132780	+	0.991145i	$0.542390\pi$
$984$	0	0
$985$	−25847.3	−0.836104
$986$	0	0
$987$	1595.35	0.0514494
$988$	0	0
$989$	7534.73	0.242255
$990$	0	0
$991$	43473.7	1.39353	0.696765	−	0.717300i	$-0.254622\pi$
$991$	43473.7	1.39353	0.696765	+	0.717300i	$0.254622\pi$
$992$	0	0
$993$	11910.6	0.380637
$994$	0	0
$995$	104373.	3.32546
$996$	0	0
$997$	42253.7	1.34221	0.671107	−	0.741361i	$-0.265819\pi$
$997$	42253.7	1.34221	0.671107	+	0.741361i	$0.265819\pi$
$998$	0	0
$999$	−7089.87	−0.224538

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.4.a.i.1.4		6
4.3	odd	2		43.4.a.b.1.3	✓	6
12.11	even	2		387.4.a.h.1.4		6
20.19	odd	2		1075.4.a.b.1.4		6
28.27	even	2		2107.4.a.c.1.3		6
172.171	even	2		1849.4.a.c.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.a.b.1.3	✓	6	4.3	odd	2
387.4.a.h.1.4		6	12.11	even	2
688.4.a.i.1.4		6	1.1	even	1	trivial
1075.4.a.b.1.4		6	20.19	odd	2
1849.4.a.c.1.4		6	172.171	even	2
2107.4.a.c.1.3		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)

\(f(q)\)	\(=\)	\(q-1.43046 q^{3} +20.4116 q^{5} -29.9522 q^{7} -24.9538 q^{9} +O(q^{10})\)	\(q-1.43046 q^{3} +20.4116 q^{5} -29.9522 q^{7} -24.9538 q^{9} +22.8719 q^{11} -44.4397 q^{13} -29.1981 q^{15} -13.0970 q^{17} -5.41527 q^{19} +42.8456 q^{21} +175.226 q^{23} +291.634 q^{25} +74.3180 q^{27} +165.972 q^{29} +155.581 q^{31} -32.7174 q^{33} -611.374 q^{35} -95.3991 q^{37} +63.5693 q^{39} +189.928 q^{41} +43.0000 q^{43} -509.347 q^{45} +37.2349 q^{47} +554.137 q^{49} +18.7348 q^{51} +559.862 q^{53} +466.852 q^{55} +7.74635 q^{57} +82.3042 q^{59} -640.304 q^{61} +747.422 q^{63} -907.085 q^{65} +509.592 q^{67} -250.655 q^{69} -792.932 q^{71} +612.727 q^{73} -417.171 q^{75} -685.065 q^{77} -237.047 q^{79} +567.443 q^{81} -418.683 q^{83} -267.330 q^{85} -237.417 q^{87} +113.788 q^{89} +1331.07 q^{91} -222.553 q^{93} -110.534 q^{95} +1649.00 q^{97} -570.740 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 7 q^{3} + 43 q^{5} - 8 q^{7} + 81 q^{9}+O(q^{10}) \) 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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