LMFDB - Embedded newform 688.4.a.i.1.5

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			1.5
Root			$-4.15251$ 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)

$f(q)$	$=$	$q+6.49933 q^{3} +17.2665 q^{5} +23.3206 q^{7} +15.2413 q^{9} +O(q^{10})$	$q+6.49933 q^{3} +17.2665 q^{5} +23.3206 q^{7} +15.2413 q^{9} +60.5580 q^{11} +10.9419 q^{13} +112.221 q^{15} -3.57473 q^{17} -33.2403 q^{19} +151.568 q^{21} -63.7158 q^{23} +173.134 q^{25} -76.4239 q^{27} -89.3510 q^{29} -222.839 q^{31} +393.586 q^{33} +402.666 q^{35} -59.6535 q^{37} +71.1150 q^{39} -143.837 q^{41} +43.0000 q^{43} +263.164 q^{45} -379.013 q^{47} +200.850 q^{49} -23.2334 q^{51} -150.129 q^{53} +1045.63 q^{55} -216.040 q^{57} -207.310 q^{59} -486.557 q^{61} +355.435 q^{63} +188.929 q^{65} -1019.41 q^{67} -414.110 q^{69} -13.8437 q^{71} +411.158 q^{73} +1125.25 q^{75} +1412.25 q^{77} +1315.13 q^{79} -908.218 q^{81} -813.425 q^{83} -61.7233 q^{85} -580.721 q^{87} -350.573 q^{89} +255.171 q^{91} -1448.30 q^{93} -573.946 q^{95} -1187.03 q^{97} +922.981 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	6.49933	1.25080	0.625398	−	0.780306i	$-0.284937\pi$
$3$	6.49933	1.25080	0.625398	+	0.780306i	$0.284937\pi$
$4$	0	0
$5$	17.2665	1.54437	0.772183	−	0.635400i	$-0.219165\pi$
$5$	17.2665	1.54437	0.772183	+	0.635400i	$0.219165\pi$
$6$	0	0
$7$	23.3206	1.25919	0.629597	−	0.776922i	$-0.283220\pi$
$7$	23.3206	1.25919	0.629597	+	0.776922i	$0.283220\pi$
$8$	0	0
$9$	15.2413	0.564491
$10$	0	0
$11$	60.5580	1.65990	0.829951	−	0.557836i	$-0.188368\pi$
$11$	60.5580	1.65990	0.829951	+	0.557836i	$0.188368\pi$
$12$	0	0
$13$	10.9419	0.233441	0.116721	−	0.993165i	$-0.462762\pi$
$13$	10.9419	0.233441	0.116721	+	0.993165i	$0.462762\pi$
$14$	0	0
$15$	112.221	1.93169
$16$	0	0
$17$	−3.57473	−0.0510000	−0.0255000	−	0.999675i	$-0.508118\pi$
$17$	−3.57473	−0.0510000	−0.0255000	+	0.999675i	$0.508118\pi$
$18$	0	0
$19$	−33.2403	−0.401361	−0.200680	−	0.979657i	$-0.564315\pi$
$19$	−33.2403	−0.401361	−0.200680	+	0.979657i	$0.564315\pi$
$20$	0	0
$21$	151.568	1.57499
$22$	0	0
$23$	−63.7158	−0.577637	−0.288819	−	0.957384i	$-0.593262\pi$
$23$	−63.7158	−0.577637	−0.288819	+	0.957384i	$0.593262\pi$
$24$	0	0
$25$	173.134	1.38507
$26$	0	0
$27$	−76.4239	−0.544733
$28$	0	0
$29$	−89.3510	−0.572140	−0.286070	−	0.958209i	$-0.592349\pi$
$29$	−89.3510	−0.572140	−0.286070	+	0.958209i	$0.592349\pi$
$30$	0	0
$31$	−222.839	−1.29106	−0.645532	−	0.763733i	$-0.723364\pi$
$31$	−222.839	−1.29106	−0.645532	+	0.763733i	$0.723364\pi$
$32$	0	0
$33$	393.586	2.07620
$34$	0	0
$35$	402.666	1.94466
$36$	0	0
$37$	−59.6535	−0.265053	−0.132527	−	0.991179i	$-0.542309\pi$
$37$	−59.6535	−0.265053	−0.132527	+	0.991179i	$0.542309\pi$
$38$	0	0
$39$	71.1150	0.291987
$40$	0	0
$41$	−143.837	−0.547890	−0.273945	−	0.961745i	$-0.588329\pi$
$41$	−143.837	−0.547890	−0.273945	+	0.961745i	$0.588329\pi$
$42$	0	0
$43$	43.0000	0.152499
$43$	43.0000	0.152499
$44$	0	0
$45$	263.164	0.871782
$46$	0	0
$47$	−379.013	−1.17627	−0.588135	−	0.808763i	$-0.700138\pi$
$47$	−379.013	−1.17627	−0.588135	+	0.808763i	$0.700138\pi$
$48$	0	0
$49$	200.850	0.585569
$50$	0	0
$51$	−23.2334	−0.0637906
$52$	0	0
$53$	−150.129	−0.389090	−0.194545	−	0.980894i	$-0.562323\pi$
$53$	−150.129	−0.389090	−0.194545	+	0.980894i	$0.562323\pi$
$54$	0	0
$55$	1045.63	2.56350
$56$	0	0
$57$	−216.040	−0.502021
$58$	0	0
$59$	−207.310	−0.457448	−0.228724	−	0.973491i	$-0.573455\pi$
$59$	−207.310	−0.457448	−0.228724	+	0.973491i	$0.573455\pi$
$60$	0	0
$61$	−486.557	−1.02127	−0.510633	−	0.859799i	$-0.670589\pi$
$61$	−486.557	−1.02127	−0.510633	+	0.859799i	$0.670589\pi$
$62$	0	0
$63$	355.435	0.710804
$64$	0	0
$65$	188.929	0.360519
$66$	0	0
$67$	−1019.41	−1.85882	−0.929408	−	0.369054i	$-0.879682\pi$
$67$	−1019.41	−1.85882	−0.929408	+	0.369054i	$0.879682\pi$
$68$	0	0
$69$	−414.110	−0.722507
$70$	0	0
$71$	−13.8437	−0.0231400	−0.0115700	−	0.999933i	$-0.503683\pi$
$71$	−13.8437	−0.0231400	−0.0115700	+	0.999933i	$0.503683\pi$
$72$	0	0
$73$	411.158	0.659211	0.329605	−	0.944119i	$-0.393084\pi$
$73$	411.158	0.659211	0.329605	+	0.944119i	$0.393084\pi$
$74$	0	0
$75$	1125.25	1.73244
$76$	0	0
$77$	1412.25	2.09014
$78$	0	0
$79$	1315.13	1.87296	0.936481	−	0.350718i	$-0.114062\pi$
$79$	1315.13	1.87296	0.936481	+	0.350718i	$0.114062\pi$
$80$	0	0
$81$	−908.218	−1.24584
$82$	0	0
$83$	−813.425	−1.07572	−0.537861	−	0.843033i	$-0.680768\pi$
$83$	−813.425	−1.07572	−0.537861	+	0.843033i	$0.680768\pi$
$84$	0	0
$85$	−61.7233	−0.0787627
$86$	0	0
$87$	−580.721	−0.715630
$88$	0	0
$89$	−350.573	−0.417535	−0.208768	−	0.977965i	$-0.566945\pi$
$89$	−350.573	−0.417535	−0.208768	+	0.977965i	$0.566945\pi$
$90$	0	0
$91$	255.171	0.293948
$92$	0	0
$93$	−1448.30	−1.61486
$94$	0	0
$95$	−573.946	−0.619848
$96$	0	0
$97$	−1187.03	−1.24252	−0.621262	−	0.783603i	$-0.713380\pi$
$97$	−1187.03	−1.24252	−0.621262	+	0.783603i	$0.713380\pi$
$98$	0	0
$99$	922.981	0.937001
$100$	0	0
$101$	469.014	0.462066	0.231033	−	0.972946i	$-0.425790\pi$
$101$	469.014	0.462066	0.231033	+	0.972946i	$0.425790\pi$
$102$	0	0
$103$	1299.82	1.24345	0.621725	−	0.783235i	$-0.286432\pi$
$103$	1299.82	1.24345	0.621725	+	0.783235i	$0.286432\pi$
$104$	0	0
$105$	2617.06	2.43237
$106$	0	0
$107$	1480.92	1.33800	0.669000	−	0.743263i	$-0.266723\pi$
$107$	1480.92	1.33800	0.669000	+	0.743263i	$0.266723\pi$
$108$	0	0
$109$	350.586	0.308073	0.154037	−	0.988065i	$-0.450773\pi$
$109$	350.586	0.308073	0.154037	+	0.988065i	$0.450773\pi$
$110$	0	0
$111$	−387.707	−0.331528
$112$	0	0
$113$	785.106	0.653598	0.326799	−	0.945094i	$-0.394030\pi$
$113$	785.106	0.653598	0.326799	+	0.945094i	$0.394030\pi$
$114$	0	0
$115$	−1100.15	−0.892084
$116$	0	0
$117$	166.768	0.131776
$118$	0	0
$119$	−83.3649	−0.0642189
$120$	0	0
$121$	2336.27	1.75528
$122$	0	0
$123$	−934.841	−0.685299
$124$	0	0
$125$	831.100	0.594687
$126$	0	0
$127$	731.562	0.511147	0.255573	−	0.966790i	$-0.417736\pi$
$127$	731.562	0.511147	0.255573	+	0.966790i	$0.417736\pi$
$128$	0	0
$129$	279.471	0.190745
$130$	0	0
$131$	−2462.56	−1.64240	−0.821202	−	0.570638i	$-0.806696\pi$
$131$	−2462.56	−1.64240	−0.821202	+	0.570638i	$0.806696\pi$
$132$	0	0
$133$	−775.184	−0.505391
$134$	0	0
$135$	−1319.58	−0.841267
$136$	0	0
$137$	2384.64	1.48710	0.743552	−	0.668678i	$-0.233139\pi$
$137$	2384.64	1.48710	0.743552	+	0.668678i	$0.233139\pi$
$138$	0	0
$139$	3086.87	1.88363	0.941816	−	0.336129i	$-0.109118\pi$
$139$	3086.87	1.88363	0.941816	+	0.336129i	$0.109118\pi$
$140$	0	0
$141$	−2463.33	−1.47127
$142$	0	0
$143$	662.619	0.387490
$144$	0	0
$145$	−1542.78	−0.883594
$146$	0	0
$147$	1305.39	0.732427
$148$	0	0
$149$	−1445.58	−0.794808	−0.397404	−	0.917644i	$-0.630089\pi$
$149$	−1445.58	−0.794808	−0.397404	+	0.917644i	$0.630089\pi$
$150$	0	0
$151$	825.878	0.445093	0.222546	−	0.974922i	$-0.428563\pi$
$151$	825.878	0.445093	0.222546	+	0.974922i	$0.428563\pi$
$152$	0	0
$153$	−54.4834	−0.0287891
$154$	0	0
$155$	−3847.65	−1.99388
$156$	0	0
$157$	1187.82	0.603809	0.301904	−	0.953338i	$-0.402378\pi$
$157$	1187.82	0.603809	0.301904	+	0.953338i	$0.402378\pi$
$158$	0	0
$159$	−975.737	−0.486673
$160$	0	0
$161$	−1485.89	−0.727357
$162$	0	0
$163$	−1514.60	−0.727809	−0.363904	−	0.931436i	$-0.618557\pi$
$163$	−1514.60	−0.727809	−0.363904	+	0.931436i	$0.618557\pi$
$164$	0	0
$165$	6795.88	3.20641
$166$	0	0
$167$	3354.48	1.55435	0.777177	−	0.629282i	$-0.216651\pi$
$167$	3354.48	1.55435	0.777177	+	0.629282i	$0.216651\pi$
$168$	0	0
$169$	−2077.27	−0.945505
$170$	0	0
$171$	−506.625	−0.226565
$172$	0	0
$173$	1654.64	0.727166	0.363583	−	0.931562i	$-0.381553\pi$
$173$	1654.64	0.727166	0.363583	+	0.931562i	$0.381553\pi$
$174$	0	0
$175$	4037.58	1.74407
$176$	0	0
$177$	−1347.38	−0.572175
$178$	0	0
$179$	1334.05	0.557049	0.278524	−	0.960429i	$-0.410155\pi$
$179$	1334.05	0.557049	0.278524	+	0.960429i	$0.410155\pi$
$180$	0	0
$181$	2771.58	1.13817	0.569087	−	0.822277i	$-0.307297\pi$
$181$	2771.58	1.13817	0.569087	+	0.822277i	$0.307297\pi$
$182$	0	0
$183$	−3162.29	−1.27740
$184$	0	0
$185$	−1030.01	−0.409339
$186$	0	0
$187$	−216.479	−0.0846550
$188$	0	0
$189$	−1782.25	−0.685924
$190$	0	0
$191$	−81.5135	−0.0308802	−0.0154401	−	0.999881i	$-0.504915\pi$
$191$	−81.5135	−0.0308802	−0.0154401	+	0.999881i	$0.504915\pi$
$192$	0	0
$193$	3305.29	1.23275	0.616373	−	0.787454i	$-0.288601\pi$
$193$	3305.29	1.23275	0.616373	+	0.787454i	$0.288601\pi$
$194$	0	0
$195$	1227.91	0.450936
$196$	0	0
$197$	2241.99	0.810840	0.405420	−	0.914131i	$-0.367125\pi$
$197$	2241.99	0.810840	0.405420	+	0.914131i	$0.367125\pi$
$198$	0	0
$199$	4074.29	1.45135	0.725675	−	0.688037i	$-0.241528\pi$
$199$	4074.29	1.45135	0.725675	+	0.688037i	$0.241528\pi$
$200$	0	0
$201$	−6625.48	−2.32500
$202$	0	0
$203$	−2083.72	−0.720435
$204$	0	0
$205$	−2483.56	−0.846143
$206$	0	0
$207$	−971.109	−0.326071
$208$	0	0
$209$	−2012.97	−0.666220
$210$	0	0
$211$	−4267.62	−1.39239	−0.696196	−	0.717851i	$-0.745126\pi$
$211$	−4267.62	−1.39239	−0.696196	+	0.717851i	$0.745126\pi$
$212$	0	0
$213$	−89.9746	−0.0289435
$214$	0	0
$215$	742.461	0.235514
$216$	0	0
$217$	−5196.73	−1.62570
$218$	0	0
$219$	2672.25	0.824538
$220$	0	0
$221$	−39.1143	−0.0119055
$222$	0	0
$223$	889.370	0.267070	0.133535	−	0.991044i	$-0.457367\pi$
$223$	889.370	0.267070	0.133535	+	0.991044i	$0.457367\pi$
$224$	0	0
$225$	2638.78	0.781859
$226$	0	0
$227$	−232.389	−0.0679479	−0.0339739	−	0.999423i	$-0.510816\pi$
$227$	−232.389	−0.0679479	−0.0339739	+	0.999423i	$0.510816\pi$
$228$	0	0
$229$	851.806	0.245803	0.122902	−	0.992419i	$-0.460780\pi$
$229$	851.806	0.245803	0.122902	+	0.992419i	$0.460780\pi$
$230$	0	0
$231$	9178.67	2.61434
$232$	0	0
$233$	−2083.62	−0.585848	−0.292924	−	0.956136i	$-0.594628\pi$
$233$	−2083.62	−0.585848	−0.292924	+	0.956136i	$0.594628\pi$
$234$	0	0
$235$	−6544.24	−1.81659
$236$	0	0
$237$	8547.48	2.34269
$238$	0	0
$239$	−986.983	−0.267124	−0.133562	−	0.991040i	$-0.542642\pi$
$239$	−986.983	−0.267124	−0.133562	+	0.991040i	$0.542642\pi$
$240$	0	0
$241$	14.8772	0.00397644	0.00198822	−	0.999998i	$-0.499367\pi$
$241$	14.8772	0.00397644	0.00198822	+	0.999998i	$0.499367\pi$
$242$	0	0
$243$	−3839.36	−1.01356
$244$	0	0
$245$	3467.99	0.904333
$246$	0	0
$247$	−363.712	−0.0936942
$248$	0	0
$249$	−5286.72	−1.34551
$250$	0	0
$251$	3027.27	0.761274	0.380637	−	0.924725i	$-0.375705\pi$
$251$	3027.27	0.761274	0.380637	+	0.924725i	$0.375705\pi$
$252$	0	0
$253$	−3858.50	−0.958822
$254$	0	0
$255$	−401.160	−0.0985161
$256$	0	0
$257$	5759.92	1.39803	0.699016	−	0.715106i	$-0.253622\pi$
$257$	5759.92	1.39803	0.699016	+	0.715106i	$0.253622\pi$
$258$	0	0
$259$	−1391.15	−0.333753
$260$	0	0
$261$	−1361.82	−0.322968
$262$	0	0
$263$	−2545.09	−0.596719	−0.298359	−	0.954454i	$-0.596439\pi$
$263$	−2545.09	−0.596719	−0.298359	+	0.954454i	$0.596439\pi$
$264$	0	0
$265$	−2592.21	−0.600898
$266$	0	0
$267$	−2278.49	−0.522252
$268$	0	0
$269$	6757.89	1.53173	0.765866	−	0.643001i	$-0.222311\pi$
$269$	6757.89	1.53173	0.765866	+	0.643001i	$0.222311\pi$
$270$	0	0
$271$	−2485.54	−0.557143	−0.278571	−	0.960415i	$-0.589861\pi$
$271$	−2485.54	−0.557143	−0.278571	+	0.960415i	$0.589861\pi$
$272$	0	0
$273$	1658.44	0.367669
$274$	0	0
$275$	10484.6	2.29908
$276$	0	0
$277$	−2090.19	−0.453385	−0.226692	−	0.973966i	$-0.572791\pi$
$277$	−2090.19	−0.453385	−0.226692	+	0.973966i	$0.572791\pi$
$278$	0	0
$279$	−3396.34	−0.728795
$280$	0	0
$281$	−6049.22	−1.28422	−0.642111	−	0.766612i	$-0.721941\pi$
$281$	−6049.22	−1.28422	−0.642111	+	0.766612i	$0.721941\pi$
$282$	0	0
$283$	650.183	0.136570	0.0682851	−	0.997666i	$-0.478247\pi$
$283$	650.183	0.136570	0.0682851	+	0.997666i	$0.478247\pi$
$284$	0	0
$285$	−3730.26	−0.775304
$286$	0	0
$287$	−3354.35	−0.689900
$288$	0	0
$289$	−4900.22	−0.997399
$290$	0	0
$291$	−7714.92	−1.55415
$292$	0	0
$293$	−2172.20	−0.433110	−0.216555	−	0.976270i	$-0.569482\pi$
$293$	−2172.20	−0.433110	−0.216555	+	0.976270i	$0.569482\pi$
$294$	0	0
$295$	−3579.53	−0.706468
$296$	0	0
$297$	−4628.08	−0.904203
$298$	0	0
$299$	−697.171	−0.134844
$300$	0	0
$301$	1002.79	0.192025
$302$	0	0
$303$	3048.28	0.577950
$304$	0	0
$305$	−8401.16	−1.57721
$306$	0	0
$307$	−1257.47	−0.233771	−0.116885	−	0.993145i	$-0.537291\pi$
$307$	−1257.47	−0.233771	−0.116885	+	0.993145i	$0.537291\pi$
$308$	0	0
$309$	8447.98	1.55530
$310$	0	0
$311$	1316.02	0.239950	0.119975	−	0.992777i	$-0.461719\pi$
$311$	1316.02	0.239950	0.119975	+	0.992777i	$0.461719\pi$
$312$	0	0
$313$	3137.09	0.566514	0.283257	−	0.959044i	$-0.408585\pi$
$313$	3137.09	0.566514	0.283257	+	0.959044i	$0.408585\pi$
$314$	0	0
$315$	6137.14	1.09774
$316$	0	0
$317$	571.518	0.101261	0.0506303	−	0.998717i	$-0.483877\pi$
$317$	571.518	0.101261	0.0506303	+	0.998717i	$0.483877\pi$
$318$	0	0
$319$	−5410.92	−0.949696
$320$	0	0
$321$	9624.99	1.67356
$322$	0	0
$323$	118.825	0.0204694
$324$	0	0
$325$	1894.41	0.323332
$326$	0	0
$327$	2278.57	0.385337
$328$	0	0
$329$	−8838.80	−1.48115
$330$	0	0
$331$	2257.30	0.374841	0.187420	−	0.982280i	$-0.439987\pi$
$331$	2257.30	0.374841	0.187420	+	0.982280i	$0.439987\pi$
$332$	0	0
$333$	−909.194	−0.149620
$334$	0	0
$335$	−17601.7	−2.87069
$336$	0	0
$337$	−9222.66	−1.49077	−0.745386	−	0.666633i	$-0.767735\pi$
$337$	−9222.66	−1.49077	−0.745386	+	0.666633i	$0.767735\pi$
$338$	0	0
$339$	5102.66	0.817517
$340$	0	0
$341$	−13494.7	−2.14304
$342$	0	0
$343$	−3315.02	−0.521849
$344$	0	0
$345$	−7150.24	−1.11582
$346$	0	0
$347$	−8840.48	−1.36767	−0.683835	−	0.729636i	$-0.739689\pi$
$347$	−8840.48	−1.36767	−0.683835	+	0.729636i	$0.739689\pi$
$348$	0	0
$349$	1431.64	0.219581	0.109791	−	0.993955i	$-0.464982\pi$
$349$	1431.64	0.219581	0.109791	+	0.993955i	$0.464982\pi$
$350$	0	0
$351$	−836.222	−0.127163
$352$	0	0
$353$	−8308.06	−1.25267	−0.626336	−	0.779553i	$-0.715446\pi$
$353$	−8308.06	−1.25267	−0.626336	+	0.779553i	$0.715446\pi$
$354$	0	0
$355$	−239.032	−0.0357367
$356$	0	0
$357$	−541.816	−0.0803247
$358$	0	0
$359$	−11642.2	−1.71157	−0.855786	−	0.517330i	$-0.826926\pi$
$359$	−11642.2	−1.71157	−0.855786	+	0.517330i	$0.826926\pi$
$360$	0	0
$361$	−5754.08	−0.838909
$362$	0	0
$363$	15184.2	2.19549
$364$	0	0
$365$	7099.27	1.01806
$366$	0	0
$367$	−4373.93	−0.622119	−0.311059	−	0.950391i	$-0.600684\pi$
$367$	−4373.93	−0.622119	−0.311059	+	0.950391i	$0.600684\pi$
$368$	0	0
$369$	−2192.25	−0.309279
$370$	0	0
$371$	−3501.09	−0.489940
$372$	0	0
$373$	7970.63	1.10644	0.553222	−	0.833034i	$-0.313398\pi$
$373$	7970.63	1.10644	0.553222	+	0.833034i	$0.313398\pi$
$374$	0	0
$375$	5401.59	0.743832
$376$	0	0
$377$	−977.669	−0.133561
$378$	0	0
$379$	2020.56	0.273850	0.136925	−	0.990581i	$-0.456278\pi$
$379$	2020.56	0.273850	0.136925	+	0.990581i	$0.456278\pi$
$380$	0	0
$381$	4754.66	0.639340
$382$	0	0
$383$	−11625.0	−1.55094	−0.775468	−	0.631386i	$-0.782486\pi$
$383$	−11625.0	−1.55094	−0.775468	+	0.631386i	$0.782486\pi$
$384$	0	0
$385$	24384.7	3.22794
$386$	0	0
$387$	655.375	0.0860841
$388$	0	0
$389$	6212.26	0.809702	0.404851	−	0.914383i	$-0.367323\pi$
$389$	6212.26	0.809702	0.404851	+	0.914383i	$0.367323\pi$
$390$	0	0
$391$	227.767	0.0294595
$392$	0	0
$393$	−16005.0	−2.05431
$394$	0	0
$395$	22707.8	2.89254
$396$	0	0
$397$	−1647.75	−0.208307	−0.104154	−	0.994561i	$-0.533213\pi$
$397$	−1647.75	−0.208307	−0.104154	+	0.994561i	$0.533213\pi$
$398$	0	0
$399$	−5038.18	−0.632141
$400$	0	0
$401$	6893.37	0.858450	0.429225	−	0.903198i	$-0.358787\pi$
$401$	6893.37	0.858450	0.429225	+	0.903198i	$0.358787\pi$
$402$	0	0
$403$	−2438.28	−0.301388
$404$	0	0
$405$	−15681.8	−1.92404
$406$	0	0
$407$	−3612.49	−0.439962
$408$	0	0
$409$	−2962.23	−0.358125	−0.179062	−	0.983838i	$-0.557306\pi$
$409$	−2962.23	−0.358125	−0.179062	+	0.983838i	$0.557306\pi$
$410$	0	0
$411$	15498.5	1.86006
$412$	0	0
$413$	−4834.59	−0.576016
$414$	0	0
$415$	−14045.0	−1.66131
$416$	0	0
$417$	20062.6	2.35604
$418$	0	0
$419$	5520.25	0.643633	0.321816	−	0.946802i	$-0.395707\pi$
$419$	5520.25	0.643633	0.321816	+	0.946802i	$0.395707\pi$
$420$	0	0
$421$	11017.9	1.27548	0.637742	−	0.770250i	$-0.279868\pi$
$421$	11017.9	1.27548	0.637742	+	0.770250i	$0.279868\pi$
$422$	0	0
$423$	−5776.63	−0.663994
$424$	0	0
$425$	−618.906	−0.0706385
$426$	0	0
$427$	−11346.8	−1.28597
$428$	0	0
$429$	4306.58	0.484671
$430$	0	0
$431$	−8825.55	−0.986338	−0.493169	−	0.869934i	$-0.664162\pi$
$431$	−8825.55	−0.986338	−0.493169	+	0.869934i	$0.664162\pi$
$432$	0	0
$433$	4570.88	0.507303	0.253652	−	0.967296i	$-0.418368\pi$
$433$	4570.88	0.507303	0.253652	+	0.967296i	$0.418368\pi$
$434$	0	0
$435$	−10027.0	−1.10520
$436$	0	0
$437$	2117.93	0.231841
$438$	0	0
$439$	3059.73	0.332649	0.166325	−	0.986071i	$-0.446810\pi$
$439$	3059.73	0.332649	0.166325	+	0.986071i	$0.446810\pi$
$440$	0	0
$441$	3061.21	0.330549
$442$	0	0
$443$	2764.55	0.296496	0.148248	−	0.988950i	$-0.452637\pi$
$443$	2764.55	0.296496	0.148248	+	0.988950i	$0.452637\pi$
$444$	0	0
$445$	−6053.18	−0.644828
$446$	0	0
$447$	−9395.29	−0.994143
$448$	0	0
$449$	7567.33	0.795377	0.397688	−	0.917521i	$-0.369812\pi$
$449$	7567.33	0.795377	0.397688	+	0.917521i	$0.369812\pi$
$450$	0	0
$451$	−8710.46	−0.909444
$452$	0	0
$453$	5367.65	0.556720
$454$	0	0
$455$	4405.93	0.453963
$456$	0	0
$457$	−8079.05	−0.826963	−0.413482	−	0.910512i	$-0.635687\pi$
$457$	−8079.05	−0.826963	−0.413482	+	0.910512i	$0.635687\pi$
$458$	0	0
$459$	273.195	0.0277813
$460$	0	0
$461$	14466.4	1.46154	0.730770	−	0.682624i	$-0.239161\pi$
$461$	14466.4	1.46154	0.730770	+	0.682624i	$0.239161\pi$
$462$	0	0
$463$	7966.79	0.799672	0.399836	−	0.916587i	$-0.369067\pi$
$463$	7966.79	0.799672	0.399836	+	0.916587i	$0.369067\pi$
$464$	0	0
$465$	−25007.2	−2.49393
$466$	0	0
$467$	−6737.29	−0.667590	−0.333795	−	0.942646i	$-0.608329\pi$
$467$	−6737.29	−0.667590	−0.333795	+	0.942646i	$0.608329\pi$
$468$	0	0
$469$	−23773.2	−2.34061
$470$	0	0
$471$	7720.00	0.755242
$472$	0	0
$473$	2603.99	0.253133
$474$	0	0
$475$	−5755.02	−0.555912
$476$	0	0
$477$	−2288.15	−0.219638
$478$	0	0
$479$	18115.7	1.72803	0.864014	−	0.503468i	$-0.167943\pi$
$479$	18115.7	1.72803	0.864014	+	0.503468i	$0.167943\pi$
$480$	0	0
$481$	−652.722	−0.0618743
$482$	0	0
$483$	−9657.28	−0.909776
$484$	0	0
$485$	−20496.0	−1.91891
$486$	0	0
$487$	14966.4	1.39260	0.696298	−	0.717753i	$-0.254829\pi$
$487$	14966.4	1.39260	0.696298	+	0.717753i	$0.254829\pi$
$488$	0	0
$489$	−9843.90	−0.910341
$490$	0	0
$491$	15713.8	1.44430	0.722151	−	0.691736i	$-0.243154\pi$
$491$	15713.8	1.44430	0.722151	+	0.691736i	$0.243154\pi$
$492$	0	0
$493$	319.406	0.0291791
$494$	0	0
$495$	15936.7	1.44707
$496$	0	0
$497$	−322.843	−0.0291378
$498$	0	0
$499$	15391.5	1.38080	0.690398	−	0.723430i	$-0.257435\pi$
$499$	15391.5	1.38080	0.690398	+	0.723430i	$0.257435\pi$
$500$	0	0
$501$	21801.8	1.94418
$502$	0	0
$503$	12150.9	1.07710	0.538548	−	0.842595i	$-0.318973\pi$
$503$	12150.9	1.07710	0.538548	+	0.842595i	$0.318973\pi$
$504$	0	0
$505$	8098.25	0.713599
$506$	0	0
$507$	−13500.9	−1.18263
$508$	0	0
$509$	16646.6	1.44960	0.724799	−	0.688960i	$-0.241933\pi$
$509$	16646.6	1.44960	0.724799	+	0.688960i	$0.241933\pi$
$510$	0	0
$511$	9588.44	0.830074
$512$	0	0
$513$	2540.35	0.218634
$514$	0	0
$515$	22443.5	1.92034
$516$	0	0
$517$	−22952.3	−1.95249
$518$	0	0
$519$	10754.0	0.909536
$520$	0	0
$521$	16675.9	1.40227	0.701135	−	0.713028i	$-0.252677\pi$
$521$	16675.9	1.40227	0.701135	+	0.713028i	$0.252677\pi$
$522$	0	0
$523$	−14326.5	−1.19781	−0.598903	−	0.800821i	$-0.704397\pi$
$523$	−14326.5	−1.19781	−0.598903	+	0.800821i	$0.704397\pi$
$524$	0	0
$525$	26241.5	2.18148
$526$	0	0
$527$	796.588	0.0658443
$528$	0	0
$529$	−8107.30	−0.666335
$530$	0	0
$531$	−3159.67	−0.258226
$532$	0	0
$533$	−1573.84	−0.127900
$534$	0	0
$535$	25570.4	2.06636
$536$	0	0
$537$	8670.44	0.696754
$538$	0	0
$539$	12163.1	0.971987
$540$	0	0
$541$	−8159.73	−0.648455	−0.324228	−	0.945979i	$-0.605104\pi$
$541$	−8159.73	−0.648455	−0.324228	+	0.945979i	$0.605104\pi$
$542$	0	0
$543$	18013.4	1.42362
$544$	0	0
$545$	6053.40	0.475778
$546$	0	0
$547$	6089.67	0.476006	0.238003	−	0.971264i	$-0.423507\pi$
$547$	6089.67	0.476006	0.238003	+	0.971264i	$0.423507\pi$
$548$	0	0
$549$	−7415.74	−0.576496
$550$	0	0
$551$	2970.06	0.229635
$552$	0	0
$553$	30669.7	2.35842
$554$	0	0
$555$	−6694.37	−0.512000
$556$	0	0
$557$	4160.66	0.316504	0.158252	−	0.987399i	$-0.449414\pi$
$557$	4160.66	0.316504	0.158252	+	0.987399i	$0.449414\pi$
$558$	0	0
$559$	470.501	0.0355995
$560$	0	0
$561$	−1406.97	−0.105886
$562$	0	0
$563$	−7731.24	−0.578744	−0.289372	−	0.957217i	$-0.593446\pi$
$563$	−7731.24	−0.578744	−0.289372	+	0.957217i	$0.593446\pi$
$564$	0	0
$565$	13556.1	1.00939
$566$	0	0
$567$	−21180.2	−1.56875
$568$	0	0
$569$	9134.32	0.672989	0.336494	−	0.941685i	$-0.390759\pi$
$569$	9134.32	0.672989	0.336494	+	0.941685i	$0.390759\pi$
$570$	0	0
$571$	−10421.5	−0.763797	−0.381898	−	0.924204i	$-0.624730\pi$
$571$	−10421.5	−0.763797	−0.381898	+	0.924204i	$0.624730\pi$
$572$	0	0
$573$	−529.783	−0.0386248
$574$	0	0
$575$	−11031.3	−0.800067
$576$	0	0
$577$	806.571	0.0581941	0.0290971	−	0.999577i	$-0.490737\pi$
$577$	806.571	0.0581941	0.0290971	+	0.999577i	$0.490737\pi$
$578$	0	0
$579$	21482.2	1.54191
$580$	0	0
$581$	−18969.6	−1.35454
$582$	0	0
$583$	−9091.50	−0.645852
$584$	0	0
$585$	2879.51	0.203510
$586$	0	0
$587$	−11514.2	−0.809614	−0.404807	−	0.914402i	$-0.632661\pi$
$587$	−11514.2	−0.809614	−0.404807	+	0.914402i	$0.632661\pi$
$588$	0	0
$589$	7407.23	0.518183
$590$	0	0
$591$	14571.5	1.01420
$592$	0	0
$593$	2702.41	0.187141	0.0935706	−	0.995613i	$-0.470172\pi$
$593$	2702.41	0.187141	0.0935706	+	0.995613i	$0.470172\pi$
$594$	0	0
$595$	−1439.42	−0.0991775
$596$	0	0
$597$	26480.1	1.81534
$598$	0	0
$599$	27360.1	1.86628	0.933142	−	0.359507i	$-0.117055\pi$
$599$	27360.1	1.86628	0.933142	+	0.359507i	$0.117055\pi$
$600$	0	0
$601$	−11506.8	−0.780985	−0.390492	−	0.920606i	$-0.627695\pi$
$601$	−11506.8	−0.780985	−0.390492	+	0.920606i	$0.627695\pi$
$602$	0	0
$603$	−15537.1	−1.04929
$604$	0	0
$605$	40339.4	2.71079
$606$	0	0
$607$	−8189.12	−0.547588	−0.273794	−	0.961788i	$-0.588279\pi$
$607$	−8189.12	−0.547588	−0.273794	+	0.961788i	$0.588279\pi$
$608$	0	0
$609$	−13542.8	−0.901117
$610$	0	0
$611$	−4147.12	−0.274590
$612$	0	0
$613$	−24210.1	−1.59517	−0.797583	−	0.603209i	$-0.793889\pi$
$613$	−24210.1	−1.59517	−0.797583	+	0.603209i	$0.793889\pi$
$614$	0	0
$615$	−16141.5	−1.05835
$616$	0	0
$617$	−8779.45	−0.572848	−0.286424	−	0.958103i	$-0.592467\pi$
$617$	−8779.45	−0.572848	−0.286424	+	0.958103i	$0.592467\pi$
$618$	0	0
$619$	6089.56	0.395412	0.197706	−	0.980261i	$-0.436651\pi$
$619$	6089.56	0.395412	0.197706	+	0.980261i	$0.436651\pi$
$620$	0	0
$621$	4869.40	0.314658
$622$	0	0
$623$	−8175.57	−0.525758
$624$	0	0
$625$	−7291.47	−0.466654
$626$	0	0
$627$	−13082.9	−0.833305
$628$	0	0
$629$	213.245	0.0135177
$630$	0	0
$631$	−13768.3	−0.868634	−0.434317	−	0.900760i	$-0.643010\pi$
$631$	−13768.3	−0.868634	−0.434317	+	0.900760i	$0.643010\pi$
$632$	0	0
$633$	−27736.6	−1.74160
$634$	0	0
$635$	12631.5	0.789398
$636$	0	0
$637$	2197.68	0.136696
$638$	0	0
$639$	−210.995	−0.0130623
$640$	0	0
$641$	393.612	0.0242539	0.0121269	−	0.999926i	$-0.496140\pi$
$641$	393.612	0.0242539	0.0121269	+	0.999926i	$0.496140\pi$
$642$	0	0
$643$	29944.9	1.83656	0.918281	−	0.395929i	$-0.129577\pi$
$643$	29944.9	1.83656	0.918281	+	0.395929i	$0.129577\pi$
$644$	0	0
$645$	4825.50	0.294580
$646$	0	0
$647$	8372.05	0.508716	0.254358	−	0.967110i	$-0.418136\pi$
$647$	8372.05	0.508716	0.254358	+	0.967110i	$0.418136\pi$
$648$	0	0
$649$	−12554.3	−0.759320
$650$	0	0
$651$	−33775.3	−2.03342
$652$	0	0
$653$	7203.34	0.431682	0.215841	−	0.976429i	$-0.430751\pi$
$653$	7203.34	0.431682	0.215841	+	0.976429i	$0.430751\pi$
$654$	0	0
$655$	−42519.9	−2.53647
$656$	0	0
$657$	6266.56	0.372119
$658$	0	0
$659$	28054.8	1.65836	0.829182	−	0.558979i	$-0.188807\pi$
$659$	28054.8	1.65836	0.829182	+	0.558979i	$0.188807\pi$
$660$	0	0
$661$	−11582.1	−0.681531	−0.340766	−	0.940148i	$-0.610686\pi$
$661$	−11582.1	−0.681531	−0.340766	+	0.940148i	$0.610686\pi$
$662$	0	0
$663$	−254.217	−0.0148914
$664$	0	0
$665$	−13384.8	−0.780509
$666$	0	0
$667$	5693.07	0.330489
$668$	0	0
$669$	5780.31	0.334050
$670$	0	0
$671$	−29464.9	−1.69520
$672$	0	0
$673$	9582.65	0.548862	0.274431	−	0.961607i	$-0.411511\pi$
$673$	9582.65	0.548862	0.274431	+	0.961607i	$0.411511\pi$
$674$	0	0
$675$	−13231.5	−0.754492
$676$	0	0
$677$	8247.63	0.468216	0.234108	−	0.972211i	$-0.424783\pi$
$677$	8247.63	0.468216	0.234108	+	0.972211i	$0.424783\pi$
$678$	0	0
$679$	−27682.3	−1.56458
$680$	0	0
$681$	−1510.37	−0.0849890
$682$	0	0
$683$	−23171.1	−1.29812	−0.649061	−	0.760736i	$-0.724838\pi$
$683$	−23171.1	−1.29812	−0.649061	+	0.760736i	$0.724838\pi$
$684$	0	0
$685$	41174.4	2.29663
$686$	0	0
$687$	5536.16	0.307450
$688$	0	0
$689$	−1642.69	−0.0908297
$690$	0	0
$691$	−24019.8	−1.32237	−0.661185	−	0.750223i	$-0.729946\pi$
$691$	−24019.8	−1.32237	−0.661185	+	0.750223i	$0.729946\pi$
$692$	0	0
$693$	21524.5	1.17987
$694$	0	0
$695$	53299.6	2.90902
$696$	0	0
$697$	514.177	0.0279424
$698$	0	0
$699$	−13542.1	−0.732776
$700$	0	0
$701$	−756.339	−0.0407511	−0.0203755	−	0.999792i	$-0.506486\pi$
$701$	−756.339	−0.0407511	−0.0203755	+	0.999792i	$0.506486\pi$
$702$	0	0
$703$	1982.90	0.106382
$704$	0	0
$705$	−42533.2	−2.27219
$706$	0	0
$707$	10937.7	0.581830
$708$	0	0
$709$	−26580.1	−1.40795	−0.703975	−	0.710224i	$-0.748594\pi$
$709$	−26580.1	−1.40795	−0.703975	+	0.710224i	$0.748594\pi$
$710$	0	0
$711$	20044.3	1.05727
$712$	0	0
$713$	14198.3	0.745767
$714$	0	0
$715$	11441.1	0.598426
$716$	0	0
$717$	−6414.72	−0.334118
$718$	0	0
$719$	−14125.2	−0.732656	−0.366328	−	0.930486i	$-0.619385\pi$
$719$	−14125.2	−0.732656	−0.366328	+	0.930486i	$0.619385\pi$
$720$	0	0
$721$	30312.7	1.56575
$722$	0	0
$723$	96.6916	0.00497372
$724$	0	0
$725$	−15469.6	−0.792453
$726$	0	0
$727$	19206.2	0.979805	0.489903	−	0.871777i	$-0.337032\pi$
$727$	19206.2	0.979805	0.489903	+	0.871777i	$0.337032\pi$
$728$	0	0
$729$	−431.393	−0.0219170
$730$	0	0
$731$	−153.713	−0.00777742
$732$	0	0
$733$	−29831.4	−1.50320	−0.751602	−	0.659617i	$-0.770718\pi$
$733$	−29831.4	−1.50320	−0.751602	+	0.659617i	$0.770718\pi$
$734$	0	0
$735$	22539.6	1.13114
$736$	0	0
$737$	−61733.4	−3.08545
$738$	0	0
$739$	−11720.8	−0.583431	−0.291715	−	0.956505i	$-0.594226\pi$
$739$	−11720.8	−0.583431	−0.291715	+	0.956505i	$0.594226\pi$
$740$	0	0
$741$	−2363.89	−0.117192
$742$	0	0
$743$	−4257.57	−0.210222	−0.105111	−	0.994460i	$-0.533520\pi$
$743$	−4257.57	−0.210222	−0.105111	+	0.994460i	$0.533520\pi$
$744$	0	0
$745$	−24960.1	−1.22747
$746$	0	0
$747$	−12397.6	−0.607236
$748$	0	0
$749$	34535.9	1.68480
$750$	0	0
$751$	16414.4	0.797564	0.398782	−	0.917046i	$-0.369433\pi$
$751$	16414.4	0.797564	0.398782	+	0.917046i	$0.369433\pi$
$752$	0	0
$753$	19675.2	0.952199
$754$	0	0
$755$	14260.1	0.687386
$756$	0	0
$757$	−22813.5	−1.09534	−0.547668	−	0.836696i	$-0.684484\pi$
$757$	−22813.5	−1.09534	−0.547668	+	0.836696i	$0.684484\pi$
$758$	0	0
$759$	−25077.7	−1.19929
$760$	0	0
$761$	1695.44	0.0807617	0.0403808	−	0.999184i	$-0.487143\pi$
$761$	1695.44	0.0807617	0.0403808	+	0.999184i	$0.487143\pi$
$762$	0	0
$763$	8175.86	0.387924
$764$	0	0
$765$	−940.741	−0.0444609
$766$	0	0
$767$	−2268.36	−0.106787
$768$	0	0
$769$	21604.9	1.01313	0.506563	−	0.862203i	$-0.330916\pi$
$769$	21604.9	1.01313	0.506563	+	0.862203i	$0.330916\pi$
$770$	0	0
$771$	37435.6	1.74865
$772$	0	0
$773$	34435.4	1.60227	0.801136	−	0.598482i	$-0.204229\pi$
$773$	34435.4	1.60227	0.801136	+	0.598482i	$0.204229\pi$
$774$	0	0
$775$	−38580.9	−1.78821
$776$	0	0
$777$	−9041.57	−0.417457
$778$	0	0
$779$	4781.18	0.219902
$780$	0	0
$781$	−838.345	−0.0384102
$782$	0	0
$783$	6828.54	0.311663
$784$	0	0
$785$	20509.5	0.932502
$786$	0	0
$787$	12352.5	0.559490	0.279745	−	0.960074i	$-0.409750\pi$
$787$	12352.5	0.559490	0.279745	+	0.960074i	$0.409750\pi$
$788$	0	0
$789$	−16541.4	−0.746374
$790$	0	0
$791$	18309.1	0.823006
$792$	0	0
$793$	−5323.85	−0.238406
$794$	0	0
$795$	−16847.6	−0.751601
$796$	0	0
$797$	29811.8	1.32495	0.662477	−	0.749083i	$-0.269505\pi$
$797$	29811.8	1.32495	0.662477	+	0.749083i	$0.269505\pi$
$798$	0	0
$799$	1354.87	0.0599897
$800$	0	0
$801$	−5343.17	−0.235695
$802$	0	0
$803$	24898.9	1.09423
$804$	0	0
$805$	−25656.2	−1.12331
$806$	0	0
$807$	43921.7	1.91588
$808$	0	0
$809$	−1273.45	−0.0553424	−0.0276712	−	0.999617i	$-0.508809\pi$
$809$	−1273.45	−0.0553424	−0.0276712	+	0.999617i	$0.508809\pi$
$810$	0	0
$811$	−37825.5	−1.63777	−0.818886	−	0.573956i	$-0.805408\pi$
$811$	−37825.5	−1.63777	−0.818886	+	0.573956i	$0.805408\pi$
$812$	0	0
$813$	−16154.3	−0.696872
$814$	0	0
$815$	−26152.0	−1.12400
$816$	0	0
$817$	−1429.33	−0.0612070
$818$	0	0
$819$	3889.14	0.165931
$820$	0	0
$821$	189.193	0.00804247	0.00402124	−	0.999992i	$-0.498720\pi$
$821$	189.193	0.00804247	0.00402124	+	0.999992i	$0.498720\pi$
$822$	0	0
$823$	−26388.5	−1.11767	−0.558837	−	0.829278i	$-0.688752\pi$
$823$	−26388.5	−1.11767	−0.558837	+	0.829278i	$0.688752\pi$
$824$	0	0
$825$	68143.0	2.87568
$826$	0	0
$827$	37247.2	1.56616	0.783079	−	0.621923i	$-0.213648\pi$
$827$	37247.2	1.56616	0.783079	+	0.621923i	$0.213648\pi$
$828$	0	0
$829$	−18429.8	−0.772128	−0.386064	−	0.922472i	$-0.626166\pi$
$829$	−18429.8	−0.772128	−0.386064	+	0.922472i	$0.626166\pi$
$830$	0	0
$831$	−13584.9	−0.567092
$832$	0	0
$833$	−717.985	−0.0298640
$834$	0	0
$835$	57920.2	2.40049
$836$	0	0
$837$	17030.2	0.703285
$838$	0	0
$839$	−24344.3	−1.00174	−0.500870	−	0.865523i	$-0.666986\pi$
$839$	−24344.3	−1.00174	−0.500870	+	0.865523i	$0.666986\pi$
$840$	0	0
$841$	−16405.4	−0.672656
$842$	0	0
$843$	−39315.9	−1.60630
$844$	0	0
$845$	−35867.4	−1.46021
$846$	0	0
$847$	54483.3	2.21023
$848$	0	0
$849$	4225.75	0.170822
$850$	0	0
$851$	3800.87	0.153105
$852$	0	0
$853$	14146.8	0.567852	0.283926	−	0.958846i	$-0.408363\pi$
$853$	14146.8	0.567852	0.283926	+	0.958846i	$0.408363\pi$
$854$	0	0
$855$	−8747.66	−0.349899
$856$	0	0
$857$	8682.05	0.346060	0.173030	−	0.984917i	$-0.444644\pi$
$857$	8682.05	0.346060	0.173030	+	0.984917i	$0.444644\pi$
$858$	0	0
$859$	25428.8	1.01003	0.505017	−	0.863110i	$-0.331486\pi$
$859$	25428.8	1.01003	0.505017	+	0.863110i	$0.331486\pi$
$860$	0	0
$861$	−21801.0	−0.862924
$862$	0	0
$863$	39552.6	1.56012	0.780062	−	0.625703i	$-0.215188\pi$
$863$	39552.6	1.56012	0.780062	+	0.625703i	$0.215188\pi$
$864$	0	0
$865$	28569.9	1.12301
$866$	0	0
$867$	−31848.1	−1.24754
$868$	0	0
$869$	79641.9	3.10894
$870$	0	0
$871$	−11154.3	−0.433924
$872$	0	0
$873$	−18091.9	−0.701395
$874$	0	0
$875$	19381.8	0.748826
$876$	0	0
$877$	1955.60	0.0752975	0.0376487	−	0.999291i	$-0.488013\pi$
$877$	1955.60	0.0752975	0.0376487	+	0.999291i	$0.488013\pi$
$878$	0	0
$879$	−14117.8	−0.541733
$880$	0	0
$881$	3037.09	0.116143	0.0580716	−	0.998312i	$-0.481505\pi$
$881$	3037.09	0.116143	0.0580716	+	0.998312i	$0.481505\pi$
$882$	0	0
$883$	−15702.3	−0.598443	−0.299222	−	0.954184i	$-0.596727\pi$
$883$	−15702.3	−0.598443	−0.299222	+	0.954184i	$0.596727\pi$
$884$	0	0
$885$	−23264.5	−0.883648
$886$	0	0
$887$	−2964.83	−0.112231	−0.0561157	−	0.998424i	$-0.517872\pi$
$887$	−2964.83	−0.112231	−0.0561157	+	0.998424i	$0.517872\pi$
$888$	0	0
$889$	17060.5	0.643633
$890$	0	0
$891$	−54999.9	−2.06797
$892$	0	0
$893$	12598.5	0.472109
$894$	0	0
$895$	23034.5	0.860287
$896$	0	0
$897$	−4531.14	−0.168663
$898$	0	0
$899$	19910.9	0.738670
$900$	0	0
$901$	536.670	0.0198436
$902$	0	0
$903$	6517.43	0.240184
$904$	0	0
$905$	47855.5	1.75776
$906$	0	0
$907$	−47065.4	−1.72302	−0.861511	−	0.507738i	$-0.830482\pi$
$907$	−47065.4	−1.72302	−0.861511	+	0.507738i	$0.830482\pi$
$908$	0	0
$909$	7148.37	0.260832
$910$	0	0
$911$	40218.3	1.46267	0.731335	−	0.682018i	$-0.238898\pi$
$911$	40218.3	1.46267	0.731335	+	0.682018i	$0.238898\pi$
$912$	0	0
$913$	−49259.4	−1.78559
$914$	0	0
$915$	−54601.9	−1.97277
$916$	0	0
$917$	−57428.4	−2.06810
$918$	0	0
$919$	13015.3	0.467175	0.233587	−	0.972336i	$-0.424953\pi$
$919$	13015.3	0.467175	0.233587	+	0.972336i	$0.424953\pi$
$920$	0	0
$921$	−8172.72	−0.292400
$922$	0	0
$923$	−151.476	−0.00540184
$924$	0	0
$925$	−10328.0	−0.367117
$926$	0	0
$927$	19811.0	0.701917
$928$	0	0
$929$	−17365.8	−0.613298	−0.306649	−	0.951823i	$-0.599208\pi$
$929$	−17365.8	−0.613298	−0.306649	+	0.951823i	$0.599208\pi$
$930$	0	0
$931$	−6676.32	−0.235024
$932$	0	0
$933$	8553.22	0.300128
$934$	0	0
$935$	−3737.84	−0.130738
$936$	0	0
$937$	−41655.6	−1.45232	−0.726162	−	0.687523i	$-0.758698\pi$
$937$	−41655.6	−1.45232	−0.726162	+	0.687523i	$0.758698\pi$
$938$	0	0
$939$	20389.0	0.708593
$940$	0	0
$941$	−11377.5	−0.394149	−0.197075	−	0.980388i	$-0.563144\pi$
$941$	−11377.5	−0.394149	−0.197075	+	0.980388i	$0.563144\pi$
$942$	0	0
$943$	9164.66	0.316482
$944$	0	0
$945$	−30773.3	−1.05932
$946$	0	0
$947$	−39835.0	−1.36691	−0.683455	−	0.729992i	$-0.739524\pi$
$947$	−39835.0	−1.36691	−0.683455	+	0.729992i	$0.739524\pi$
$948$	0	0
$949$	4498.84	0.153887
$950$	0	0
$951$	3714.48	0.126656
$952$	0	0
$953$	16468.3	0.559769	0.279885	−	0.960034i	$-0.409704\pi$
$953$	16468.3	0.559769	0.279885	+	0.960034i	$0.409704\pi$
$954$	0	0
$955$	−1407.46	−0.0476903
$956$	0	0
$957$	−35167.3	−1.18788
$958$	0	0
$959$	55611.2	1.87255
$960$	0	0
$961$	19866.1	0.666848
$962$	0	0
$963$	22571.1	0.755289
$964$	0	0
$965$	57070.9	1.90381
$966$	0	0
$967$	−55135.3	−1.83354	−0.916770	−	0.399416i	$-0.869213\pi$
$967$	−55135.3	−1.83354	−0.916770	+	0.399416i	$0.869213\pi$
$968$	0	0
$969$	772.284	0.0256030
$970$	0	0
$971$	13040.1	0.430975	0.215488	−	0.976507i	$-0.430866\pi$
$971$	13040.1	0.430975	0.215488	+	0.976507i	$0.430866\pi$
$972$	0	0
$973$	71987.6	2.37186
$974$	0	0
$975$	12312.4	0.404423
$976$	0	0
$977$	−45144.9	−1.47831	−0.739157	−	0.673534i	$-0.764776\pi$
$977$	−45144.9	−1.47831	−0.739157	+	0.673534i	$0.764776\pi$
$978$	0	0
$979$	−21230.0	−0.693068
$980$	0	0
$981$	5343.37	0.173905
$982$	0	0
$983$	26573.6	0.862225	0.431113	−	0.902298i	$-0.358121\pi$
$983$	26573.6	0.862225	0.431113	+	0.902298i	$0.358121\pi$
$984$	0	0
$985$	38711.5	1.25223
$986$	0	0
$987$	−57446.3	−1.85262
$988$	0	0
$989$	−2739.78	−0.0880889
$990$	0	0
$991$	−20658.0	−0.662183	−0.331092	−	0.943599i	$-0.607417\pi$
$991$	−20658.0	−0.662183	−0.331092	+	0.943599i	$0.607417\pi$
$992$	0	0
$993$	14670.9	0.468850
$994$	0	0
$995$	70348.9	2.24142
$996$	0	0
$997$	−27284.2	−0.866701	−0.433350	−	0.901226i	$-0.642669\pi$
$997$	−27284.2	−0.866701	−0.433350	+	0.901226i	$0.642669\pi$
$998$	0	0
$999$	4558.95	0.144383

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.4.a.i.1.5		6
4.3	odd	2		43.4.a.b.1.6	✓	6
12.11	even	2		387.4.a.h.1.1		6
20.19	odd	2		1075.4.a.b.1.1		6
28.27	even	2		2107.4.a.c.1.6		6
172.171	even	2		1849.4.a.c.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.a.b.1.6	✓	6	4.3	odd	2
387.4.a.h.1.1		6	12.11	even	2
688.4.a.i.1.5		6	1.1	even	1	trivial
1075.4.a.b.1.1		6	20.19	odd	2
1849.4.a.c.1.1		6	172.171	even	2
2107.4.a.c.1.6		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)

\(f(q)\)	\(=\)	\(q+6.49933 q^{3} +17.2665 q^{5} +23.3206 q^{7} +15.2413 q^{9} +O(q^{10})\)	\(q+6.49933 q^{3} +17.2665 q^{5} +23.3206 q^{7} +15.2413 q^{9} +60.5580 q^{11} +10.9419 q^{13} +112.221 q^{15} -3.57473 q^{17} -33.2403 q^{19} +151.568 q^{21} -63.7158 q^{23} +173.134 q^{25} -76.4239 q^{27} -89.3510 q^{29} -222.839 q^{31} +393.586 q^{33} +402.666 q^{35} -59.6535 q^{37} +71.1150 q^{39} -143.837 q^{41} +43.0000 q^{43} +263.164 q^{45} -379.013 q^{47} +200.850 q^{49} -23.2334 q^{51} -150.129 q^{53} +1045.63 q^{55} -216.040 q^{57} -207.310 q^{59} -486.557 q^{61} +355.435 q^{63} +188.929 q^{65} -1019.41 q^{67} -414.110 q^{69} -13.8437 q^{71} +411.158 q^{73} +1125.25 q^{75} +1412.25 q^{77} +1315.13 q^{79} -908.218 q^{81} -813.425 q^{83} -61.7233 q^{85} -580.721 q^{87} -350.573 q^{89} +255.171 q^{91} -1448.30 q^{93} -573.946 q^{95} -1187.03 q^{97} +922.981 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 7 q^{3} + 43 q^{5} - 8 q^{7} + 81 q^{9}+O(q^{10}) \) 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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