LMFDB - Embedded newform 693.4.a.t.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,4,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, 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("693.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	693.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:	$40.8883236340$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 45x^{6} + 77x^{5} + 540x^{4} - 915x^{3} - 1452x^{2} + 2660x - 672 $ x^8 - 2x^7 - 45x^6 + 77x^5 + 540x^4 - 915x^3 - 1452x^2 + 2660*x - 672
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.09348$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.09348 q^{2} -3.61736 q^{4} -6.99302 q^{5} -7.00000 q^{7} +24.3207 q^{8} +O(q^{10})$	\(q-2.09348 q^{2} -3.61736 q^{4} -6.99302 q^{5} -7.00000 q^{7} +24.3207 q^{8} +14.6397 q^{10} -11.0000 q^{11} +13.5929 q^{13} +14.6543 q^{14} -21.9758 q^{16} -93.3418 q^{17} -54.1929 q^{19} +25.2963 q^{20} +23.0282 q^{22} +108.854 q^{23} -76.0977 q^{25} -28.4564 q^{26} +25.3215 q^{28} -68.0108 q^{29} -203.765 q^{31} -148.559 q^{32} +195.409 q^{34} +48.9511 q^{35} -288.303 q^{37} +113.452 q^{38} -170.075 q^{40} +22.3285 q^{41} -285.795 q^{43} +39.7910 q^{44} -227.884 q^{46} +433.862 q^{47} +49.0000 q^{49} +159.309 q^{50} -49.1703 q^{52} +592.467 q^{53} +76.9232 q^{55} -170.245 q^{56} +142.379 q^{58} -271.371 q^{59} +819.457 q^{61} +426.576 q^{62} +486.812 q^{64} -95.0552 q^{65} -961.774 q^{67} +337.651 q^{68} -102.478 q^{70} +599.916 q^{71} +239.044 q^{73} +603.556 q^{74} +196.035 q^{76} +77.0000 q^{77} -185.996 q^{79} +153.677 q^{80} -46.7443 q^{82} -466.853 q^{83} +652.741 q^{85} +598.305 q^{86} -267.527 q^{88} -184.217 q^{89} -95.1502 q^{91} -393.765 q^{92} -908.280 q^{94} +378.972 q^{95} -1441.50 q^{97} -102.580 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} + 30 q^{4} + 10 q^{5} - 56 q^{7} + 15 q^{8}+O(q^{10}) $ 8 * q + 2 * q^2 + 30 * q^4 + 10 * q^5 - 56 * q^7 + 15 * q^8	$ 8 q + 2 q^{2} + 30 q^{4} + 10 q^{5} - 56 q^{7} + 15 q^{8} - 13 q^{10} - 88 q^{11} - 148 q^{13} - 14 q^{14} + 266 q^{16} + 114 q^{17} + 58 q^{19} + 291 q^{20} - 22 q^{22} + 246 q^{23} + 244 q^{25} + 305 q^{26} - 210 q^{28} - 72 q^{29} + 252 q^{31} + 1272 q^{32} + 630 q^{34} - 70 q^{35} - 80 q^{37} + 1885 q^{38} - 342 q^{40} + 682 q^{41} - 106 q^{43} - 330 q^{44} + 120 q^{46} + 828 q^{47} + 392 q^{49} + 801 q^{50} - 1681 q^{52} + 462 q^{53} - 110 q^{55} - 105 q^{56} - 1087 q^{58} + 626 q^{59} - 854 q^{61} + 1350 q^{62} + 2997 q^{64} + 22 q^{65} + 130 q^{67} + 2202 q^{68} + 91 q^{70} + 326 q^{71} - 390 q^{73} - 359 q^{74} + 2041 q^{76} + 616 q^{77} - 508 q^{79} + 4391 q^{80} + 1528 q^{82} + 1596 q^{83} - 880 q^{85} - 414 q^{86} - 165 q^{88} + 4324 q^{89} + 1036 q^{91} + 2092 q^{92} - 1685 q^{94} + 1076 q^{95} - 964 q^{97} + 98 q^{98}+O(q^{100}) $ 8 * q + 2 * q^2 + 30 * q^4 + 10 * q^5 - 56 * q^7 + 15 * q^8 - 13 * q^10 - 88 * q^11 - 148 * q^13 - 14 * q^14 + 266 * q^16 + 114 * q^17 + 58 * q^19 + 291 * q^20 - 22 * q^22 + 246 * q^23 + 244 * q^25 + 305 * q^26 - 210 * q^28 - 72 * q^29 + 252 * q^31 + 1272 * q^32 + 630 * q^34 - 70 * q^35 - 80 * q^37 + 1885 * q^38 - 342 * q^40 + 682 * q^41 - 106 * q^43 - 330 * q^44 + 120 * q^46 + 828 * q^47 + 392 * q^49 + 801 * q^50 - 1681 * q^52 + 462 * q^53 - 110 * q^55 - 105 * q^56 - 1087 * q^58 + 626 * q^59 - 854 * q^61 + 1350 * q^62 + 2997 * q^64 + 22 * q^65 + 130 * q^67 + 2202 * q^68 + 91 * q^70 + 326 * q^71 - 390 * q^73 - 359 * q^74 + 2041 * q^76 + 616 * q^77 - 508 * q^79 + 4391 * q^80 + 1528 * q^82 + 1596 * q^83 - 880 * q^85 - 414 * q^86 - 165 * q^88 + 4324 * q^89 + 1036 * q^91 + 2092 * q^92 - 1685 * q^94 + 1076 * q^95 - 964 * q^97 + 98 * q^98

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 $

$2$	−2.09348	−0.740155	−0.370078	−	0.929001i	$-0.620669\pi$
$2$	−2.09348	−0.740155	−0.370078	+	0.929001i	$0.620669\pi$
$3$	0	0
$3$	0	0
$4$	−3.61736	−0.452170
$5$	−6.99302	−0.625474	−0.312737	−	0.949840i	$-0.601246\pi$
$5$	−6.99302	−0.625474	−0.312737	+	0.949840i	$0.601246\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	24.3207	1.07483
$9$	0	0
$10$	14.6397	0.462948
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	13.5929	0.289999	0.145000	−	0.989432i	$-0.453682\pi$
$13$	13.5929	0.289999	0.145000	+	0.989432i	$0.453682\pi$
$14$	14.6543	0.279752
$15$	0	0
$16$	−21.9758	−0.343372
$17$	−93.3418	−1.33169	−0.665845	−	0.746090i	$-0.731929\pi$
$17$	−93.3418	−1.33169	−0.665845	+	0.746090i	$0.731929\pi$
$18$	0	0
$19$	−54.1929	−0.654353	−0.327176	−	0.944963i	$-0.606097\pi$
$19$	−54.1929	−0.654353	−0.327176	+	0.944963i	$0.606097\pi$
$20$	25.2963	0.282821
$21$	0	0
$22$	23.0282	0.223165
$23$	108.854	0.986855	0.493428	−	0.869787i	$-0.335744\pi$
$23$	108.854	0.986855	0.493428	+	0.869787i	$0.335744\pi$
$24$	0	0
$25$	−76.0977	−0.608782
$26$	−28.4564	−0.214644
$27$	0	0
$28$	25.3215	0.170904
$29$	−68.0108	−0.435493	−0.217746	−	0.976005i	$-0.569871\pi$
$29$	−68.0108	−0.435493	−0.217746	+	0.976005i	$0.569871\pi$
$30$	0	0
$31$	−203.765	−1.18055	−0.590277	−	0.807201i	$-0.700982\pi$
$31$	−203.765	−1.18055	−0.590277	+	0.807201i	$0.700982\pi$
$32$	−148.559	−0.820683
$33$	0	0
$34$	195.409	0.985657
$35$	48.9511	0.236407
$36$	0	0
$37$	−288.303	−1.28099	−0.640497	−	0.767960i	$-0.721272\pi$
$37$	−288.303	−1.28099	−0.640497	+	0.767960i	$0.721272\pi$
$38$	113.452	0.484323
$39$	0	0
$40$	−170.075	−0.672280
$41$	22.3285	0.0850520	0.0425260	−	0.999095i	$-0.486459\pi$
$41$	22.3285	0.0850520	0.0425260	+	0.999095i	$0.486459\pi$
$42$	0	0
$43$	−285.795	−1.01357	−0.506783	−	0.862074i	$-0.669165\pi$
$43$	−285.795	−1.01357	−0.506783	+	0.862074i	$0.669165\pi$
$44$	39.7910	0.136334
$45$	0	0
$46$	−227.884	−0.730426
$47$	433.862	1.34650	0.673248	−	0.739417i	$-0.264899\pi$
$47$	433.862	1.34650	0.673248	+	0.739417i	$0.264899\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	159.309	0.450593
$51$	0	0
$52$	−49.1703	−0.131129
$53$	592.467	1.53550	0.767751	−	0.640748i	$-0.221376\pi$
$53$	592.467	1.53550	0.767751	+	0.640748i	$0.221376\pi$
$54$	0	0
$55$	76.9232	0.188588
$56$	−170.245	−0.406248
$57$	0	0
$58$	142.379	0.322332
$59$	−271.371	−0.598806	−0.299403	−	0.954127i	$-0.596787\pi$
$59$	−271.371	−0.598806	−0.299403	+	0.954127i	$0.596787\pi$
$60$	0	0
$61$	819.457	1.72001	0.860006	−	0.510284i	$-0.170460\pi$
$61$	819.457	1.72001	0.860006	+	0.510284i	$0.170460\pi$
$62$	426.576	0.873794
$63$	0	0
$64$	486.812	0.950805
$65$	−95.0552	−0.181387
$66$	0	0
$67$	−961.774	−1.75372	−0.876861	−	0.480744i	$-0.840366\pi$
$67$	−961.774	−1.75372	−0.876861	+	0.480744i	$0.840366\pi$
$68$	337.651	0.602150
$69$	0	0
$70$	−102.478	−0.174978
$71$	599.916	1.00277	0.501387	−	0.865223i	$-0.332823\pi$
$71$	599.916	1.00277	0.501387	+	0.865223i	$0.332823\pi$
$72$	0	0
$73$	239.044	0.383260	0.191630	−	0.981467i	$-0.438623\pi$
$73$	239.044	0.383260	0.191630	+	0.981467i	$0.438623\pi$
$74$	603.556	0.948135
$75$	0	0
$76$	196.035	0.295879
$77$	77.0000	0.113961
$78$	0	0
$79$	−185.996	−0.264888	−0.132444	−	0.991190i	$-0.542283\pi$
$79$	−185.996	−0.264888	−0.132444	+	0.991190i	$0.542283\pi$
$80$	153.677	0.214771
$81$	0	0
$82$	−46.7443	−0.0629517
$83$	−466.853	−0.617395	−0.308697	−	0.951160i	$-0.599893\pi$
$83$	−466.853	−0.617395	−0.308697	+	0.951160i	$0.599893\pi$
$84$	0	0
$85$	652.741	0.832938
$86$	598.305	0.750196
$87$	0	0
$88$	−267.527	−0.324074
$89$	−184.217	−0.219404	−0.109702	−	0.993965i	$-0.534990\pi$
$89$	−184.217	−0.219404	−0.109702	+	0.993965i	$0.534990\pi$
$90$	0	0
$91$	−95.1502	−0.109609
$92$	−393.765	−0.446226
$93$	0	0
$94$	−908.280	−0.996616
$95$	378.972	0.409281
$96$	0	0
$97$	−1441.50	−1.50889	−0.754443	−	0.656366i	$-0.772093\pi$
$97$	−1441.50	−1.50889	−0.754443	+	0.656366i	$0.772093\pi$
$98$	−102.580	−0.105736
$99$	0	0
$100$	275.273	0.275273
$101$	1358.55	1.33843	0.669213	−	0.743070i	$-0.266631\pi$
$101$	1358.55	1.33843	0.669213	+	0.743070i	$0.266631\pi$
$102$	0	0
$103$	1805.19	1.72690	0.863451	−	0.504432i	$-0.168298\pi$
$103$	1805.19	1.72690	0.863451	+	0.504432i	$0.168298\pi$
$104$	330.588	0.311700
$105$	0	0
$106$	−1240.32	−1.13651
$107$	762.019	0.688479	0.344239	−	0.938882i	$-0.388137\pi$
$107$	762.019	0.688479	0.344239	+	0.938882i	$0.388137\pi$
$108$	0	0
$109$	1476.81	1.29773	0.648867	−	0.760902i	$-0.275243\pi$
$109$	1476.81	1.29773	0.648867	+	0.760902i	$0.275243\pi$
$110$	−161.037	−0.139584
$111$	0	0
$112$	153.831	0.129783
$113$	494.257	0.411467	0.205733	−	0.978608i	$-0.434042\pi$
$113$	494.257	0.411467	0.205733	+	0.978608i	$0.434042\pi$
$114$	0	0
$115$	−761.219	−0.617253
$116$	246.020	0.196917
$117$	0	0
$118$	568.109	0.443209
$119$	653.393	0.503331
$120$	0	0
$121$	121.000	0.0909091
$122$	−1715.51	−1.27308
$123$	0	0
$124$	737.090	0.533811
$125$	1406.28	1.00625
$126$	0	0
$127$	−129.976	−0.0908149	−0.0454075	−	0.998969i	$-0.514459\pi$
$127$	−129.976	−0.0908149	−0.0454075	+	0.998969i	$0.514459\pi$
$128$	169.346	0.116939
$129$	0	0
$130$	198.996	0.134255
$131$	2334.22	1.55680	0.778402	−	0.627766i	$-0.216030\pi$
$131$	2334.22	1.55680	0.778402	+	0.627766i	$0.216030\pi$
$132$	0	0
$133$	379.350	0.247322
$134$	2013.45	1.29803
$135$	0	0
$136$	−2270.13	−1.43134
$137$	−1111.32	−0.693042	−0.346521	−	0.938042i	$-0.612637\pi$
$137$	−1111.32	−0.693042	−0.346521	+	0.938042i	$0.612637\pi$
$138$	0	0
$139$	−1170.30	−0.714125	−0.357062	−	0.934081i	$-0.616222\pi$
$139$	−1170.30	−0.714125	−0.357062	+	0.934081i	$0.616222\pi$
$140$	−177.074	−0.106896
$141$	0	0
$142$	−1255.91	−0.742208
$143$	−149.522	−0.0874380
$144$	0	0
$145$	475.601	0.272390
$146$	−500.433	−0.283672
$147$	0	0
$148$	1042.90	0.579227
$149$	−1346.99	−0.740604	−0.370302	−	0.928911i	$-0.620746\pi$
$149$	−1346.99	−0.740604	−0.370302	+	0.928911i	$0.620746\pi$
$150$	0	0
$151$	−1591.91	−0.857933	−0.428967	−	0.903320i	$-0.641122\pi$
$151$	−1591.91	−0.857933	−0.428967	+	0.903320i	$0.641122\pi$
$152$	−1318.01	−0.703319
$153$	0	0
$154$	−161.198	−0.0843485
$155$	1424.93	0.738407
$156$	0	0
$157$	1080.97	0.549497	0.274749	−	0.961516i	$-0.411405\pi$
$157$	1080.97	0.549497	0.274749	+	0.961516i	$0.411405\pi$
$158$	389.378	0.196058
$159$	0	0
$160$	1038.88	0.513316
$161$	−761.979	−0.372996
$162$	0	0
$163$	4036.04	1.93943	0.969715	−	0.244237i	$-0.0785376\pi$
$163$	4036.04	1.93943	0.969715	+	0.244237i	$0.0785376\pi$
$164$	−80.7704	−0.0384580
$165$	0	0
$166$	977.345	0.456968
$167$	3173.23	1.47037	0.735186	−	0.677865i	$-0.237095\pi$
$167$	3173.23	1.47037	0.735186	+	0.677865i	$0.237095\pi$
$168$	0	0
$169$	−2012.23	−0.915901
$170$	−1366.50	−0.616503
$171$	0	0
$172$	1033.82	0.458304
$173$	−1460.26	−0.641743	−0.320872	−	0.947123i	$-0.603976\pi$
$173$	−1460.26	−0.641743	−0.320872	+	0.947123i	$0.603976\pi$
$174$	0	0
$175$	532.684	0.230098
$176$	241.734	0.103531
$177$	0	0
$178$	385.654	0.162393
$179$	1949.59	0.814075	0.407037	−	0.913411i	$-0.366562\pi$
$179$	1949.59	0.814075	0.407037	+	0.913411i	$0.366562\pi$
$180$	0	0
$181$	3156.78	1.29636	0.648182	−	0.761486i	$-0.275530\pi$
$181$	3156.78	1.29636	0.648182	+	0.761486i	$0.275530\pi$
$182$	199.195	0.0811279
$183$	0	0
$184$	2647.41	1.06070
$185$	2016.11	0.801229
$186$	0	0
$187$	1026.76	0.401519
$188$	−1569.43	−0.608845
$189$	0	0
$190$	−793.368	−0.302932
$191$	−4298.46	−1.62841	−0.814203	−	0.580580i	$-0.802826\pi$
$191$	−4298.46	−1.62841	−0.814203	+	0.580580i	$0.802826\pi$
$192$	0	0
$193$	869.612	0.324332	0.162166	−	0.986764i	$-0.448152\pi$
$193$	869.612	0.324332	0.162166	+	0.986764i	$0.448152\pi$
$194$	3017.74	1.11681
$195$	0	0
$196$	−177.251	−0.0645957
$197$	578.333	0.209160	0.104580	−	0.994516i	$-0.466650\pi$
$197$	578.333	0.209160	0.104580	+	0.994516i	$0.466650\pi$
$198$	0	0
$199$	4791.42	1.70681	0.853404	−	0.521250i	$-0.174534\pi$
$199$	4791.42	1.70681	0.853404	+	0.521250i	$0.174534\pi$
$200$	−1850.75	−0.654338
$201$	0	0
$202$	−2844.10	−0.990644
$203$	476.076	0.164601
$204$	0	0
$205$	−156.144	−0.0531979
$206$	−3779.13	−1.27818
$207$	0	0
$208$	−298.715	−0.0995777
$209$	596.122	0.197295
$210$	0	0
$211$	3192.91	1.04175	0.520875	−	0.853633i	$-0.325606\pi$
$211$	3192.91	1.04175	0.520875	+	0.853633i	$0.325606\pi$
$212$	−2143.17	−0.694308
$213$	0	0
$214$	−1595.27	−0.509581
$215$	1998.57	0.633959
$216$	0	0
$217$	1426.35	0.446208
$218$	−3091.67	−0.960524
$219$	0	0
$220$	−278.259	−0.0852737
$221$	−1268.78	−0.386189
$222$	0	0
$223$	−644.701	−0.193598	−0.0967991	−	0.995304i	$-0.530860\pi$
$223$	−644.701	−0.193598	−0.0967991	+	0.995304i	$0.530860\pi$
$224$	1039.92	0.310189
$225$	0	0
$226$	−1034.71	−0.304549
$227$	−942.889	−0.275691	−0.137845	−	0.990454i	$-0.544018\pi$
$227$	−942.889	−0.275691	−0.137845	+	0.990454i	$0.544018\pi$
$228$	0	0
$229$	−2421.98	−0.698904	−0.349452	−	0.936954i	$-0.613632\pi$
$229$	−2421.98	−0.698904	−0.349452	+	0.936954i	$0.613632\pi$
$230$	1593.59	0.456863
$231$	0	0
$232$	−1654.07	−0.468081
$233$	5640.92	1.58605	0.793023	−	0.609192i	$-0.208506\pi$
$233$	5640.92	1.58605	0.793023	+	0.609192i	$0.208506\pi$
$234$	0	0
$235$	−3034.00	−0.842198
$236$	981.647	0.270762
$237$	0	0
$238$	−1367.86	−0.372543
$239$	−2184.21	−0.591149	−0.295575	−	0.955320i	$-0.595511\pi$
$239$	−2184.21	−0.591149	−0.295575	+	0.955320i	$0.595511\pi$
$240$	0	0
$241$	1519.98	0.406269	0.203134	−	0.979151i	$-0.434887\pi$
$241$	1519.98	0.406269	0.203134	+	0.979151i	$0.434887\pi$
$242$	−253.311	−0.0672869
$243$	0	0
$244$	−2964.27	−0.777738
$245$	−342.658	−0.0893535
$246$	0	0
$247$	−736.638	−0.189762
$248$	−4955.69	−1.26890
$249$	0	0
$250$	−2944.01	−0.744783
$251$	−3651.63	−0.918283	−0.459142	−	0.888363i	$-0.651843\pi$
$251$	−3651.63	−0.918283	−0.459142	+	0.888363i	$0.651843\pi$
$252$	0	0
$253$	−1197.40	−0.297548
$254$	272.101	0.0672172
$255$	0	0
$256$	−4249.02	−1.03736
$257$	−1510.10	−0.366528	−0.183264	−	0.983064i	$-0.558666\pi$
$257$	−1510.10	−0.366528	−0.183264	+	0.983064i	$0.558666\pi$
$258$	0	0
$259$	2018.12	0.484170
$260$	343.849	0.0820177
$261$	0	0
$262$	−4886.62	−1.15228
$263$	−1903.00	−0.446175	−0.223087	−	0.974798i	$-0.571614\pi$
$263$	−1903.00	−0.446175	−0.223087	+	0.974798i	$0.571614\pi$
$264$	0	0
$265$	−4143.13	−0.960417
$266$	−794.161	−0.183057
$267$	0	0
$268$	3479.08	0.792980
$269$	−149.202	−0.0338178	−0.0169089	−	0.999857i	$-0.505383\pi$
$269$	−149.202	−0.0338178	−0.0169089	+	0.999857i	$0.505383\pi$
$270$	0	0
$271$	−1389.42	−0.311444	−0.155722	−	0.987801i	$-0.549770\pi$
$271$	−1389.42	−0.311444	−0.155722	+	0.987801i	$0.549770\pi$
$272$	2051.26	0.457265
$273$	0	0
$274$	2326.53	0.512959
$275$	837.075	0.183555
$276$	0	0
$277$	−468.647	−0.101654	−0.0508272	−	0.998707i	$-0.516186\pi$
$277$	−468.647	−0.101654	−0.0508272	+	0.998707i	$0.516186\pi$
$278$	2449.99	0.528563
$279$	0	0
$280$	1190.52	0.254098
$281$	4609.75	0.978628	0.489314	−	0.872108i	$-0.337247\pi$
$281$	4609.75	0.978628	0.489314	+	0.872108i	$0.337247\pi$
$282$	0	0
$283$	−4188.07	−0.879700	−0.439850	−	0.898071i	$-0.644968\pi$
$283$	−4188.07	−0.879700	−0.439850	+	0.898071i	$0.644968\pi$
$284$	−2170.11	−0.453424
$285$	0	0
$286$	313.020	0.0647177
$287$	−156.300	−0.0321466
$288$	0	0
$289$	3799.70	0.773397
$290$	−995.658	−0.201611
$291$	0	0
$292$	−864.708	−0.173299
$293$	−4766.65	−0.950412	−0.475206	−	0.879874i	$-0.657627\pi$
$293$	−4766.65	−0.950412	−0.475206	+	0.879874i	$0.657627\pi$
$294$	0	0
$295$	1897.70	0.374538
$296$	−7011.73	−1.37685
$297$	0	0
$298$	2819.90	0.548162
$299$	1479.64	0.286187
$300$	0	0
$301$	2000.56	0.383092
$302$	3332.63	0.635004
$303$	0	0
$304$	1190.93	0.224687
$305$	−5730.48	−1.07582
$306$	0	0
$307$	−10210.5	−1.89819	−0.949094	−	0.314994i	$-0.897997\pi$
$307$	−10210.5	−1.89819	−0.949094	+	0.314994i	$0.897997\pi$
$308$	−278.537	−0.0515295
$309$	0	0
$310$	−2983.05	−0.546536
$311$	968.081	0.176511	0.0882554	−	0.996098i	$-0.471871\pi$
$311$	968.081	0.176511	0.0882554	+	0.996098i	$0.471871\pi$
$312$	0	0
$313$	1143.87	0.206567	0.103284	−	0.994652i	$-0.467065\pi$
$313$	1143.87	0.206567	0.103284	+	0.994652i	$0.467065\pi$
$314$	−2262.99	−0.406713
$315$	0	0
$316$	672.814	0.119774
$317$	−4477.92	−0.793391	−0.396696	−	0.917950i	$-0.629843\pi$
$317$	−4477.92	−0.793391	−0.396696	+	0.917950i	$0.629843\pi$
$318$	0	0
$319$	748.119	0.131306
$320$	−3404.29	−0.594704
$321$	0	0
$322$	1595.19	0.276075
$323$	5058.47	0.871395
$324$	0	0
$325$	−1034.39	−0.176546
$326$	−8449.36	−1.43548
$327$	0	0
$328$	543.045	0.0914166
$329$	−3037.03	−0.508927
$330$	0	0
$331$	6910.13	1.14748	0.573739	−	0.819038i	$-0.305492\pi$
$331$	6910.13	1.14748	0.573739	+	0.819038i	$0.305492\pi$
$332$	1688.77	0.279167
$333$	0	0
$334$	−6643.09	−1.08830
$335$	6725.70	1.09691
$336$	0	0
$337$	2277.26	0.368101	0.184050	−	0.982917i	$-0.441079\pi$
$337$	2277.26	0.368101	0.184050	+	0.982917i	$0.441079\pi$
$338$	4212.56	0.677909
$339$	0	0
$340$	−2361.20	−0.376629
$341$	2241.41	0.355951
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−6950.72	−1.08941
$345$	0	0
$346$	3057.02	0.474990
$347$	−5231.70	−0.809372	−0.404686	−	0.914456i	$-0.632619\pi$
$347$	−5231.70	−0.809372	−0.404686	+	0.914456i	$0.632619\pi$
$348$	0	0
$349$	−1509.59	−0.231538	−0.115769	−	0.993276i	$-0.536933\pi$
$349$	−1509.59	−0.231538	−0.115769	+	0.993276i	$0.536933\pi$
$350$	−1115.16	−0.170308
$351$	0	0
$352$	1634.15	0.247445
$353$	−9194.02	−1.38626	−0.693128	−	0.720814i	$-0.743768\pi$
$353$	−9194.02	−1.38626	−0.693128	+	0.720814i	$0.743768\pi$
$354$	0	0
$355$	−4195.22	−0.627209
$356$	666.380	0.0992081
$357$	0	0
$358$	−4081.43	−0.602542
$359$	−10798.5	−1.58752	−0.793762	−	0.608228i	$-0.791881\pi$
$359$	−10798.5	−1.58752	−0.793762	+	0.608228i	$0.791881\pi$
$360$	0	0
$361$	−3922.13	−0.571822
$362$	−6608.64	−0.959510
$363$	0	0
$364$	344.192	0.0495620
$365$	−1671.64	−0.239719
$366$	0	0
$367$	−1315.27	−0.187076	−0.0935378	−	0.995616i	$-0.529818\pi$
$367$	−1315.27	−0.187076	−0.0935378	+	0.995616i	$0.529818\pi$
$368$	−2392.16	−0.338859
$369$	0	0
$370$	−4220.68	−0.593034
$371$	−4147.27	−0.580365
$372$	0	0
$373$	−6348.16	−0.881221	−0.440610	−	0.897698i	$-0.645238\pi$
$373$	−6348.16	−0.881221	−0.440610	+	0.897698i	$0.645238\pi$
$374$	−2149.50	−0.297187
$375$	0	0
$376$	10551.8	1.44726
$377$	−924.463	−0.126292
$378$	0	0
$379$	6776.47	0.918427	0.459213	−	0.888326i	$-0.348131\pi$
$379$	6776.47	0.918427	0.459213	+	0.888326i	$0.348131\pi$
$380$	−1370.88	−0.185065
$381$	0	0
$382$	8998.72	1.20527
$383$	14440.7	1.92659	0.963297	−	0.268437i	$-0.0865073\pi$
$383$	14440.7	1.92659	0.963297	+	0.268437i	$0.0865073\pi$
$384$	0	0
$385$	−538.462	−0.0712794
$386$	−1820.51	−0.240056
$387$	0	0
$388$	5214.42	0.682273
$389$	11087.0	1.44507	0.722534	−	0.691335i	$-0.242977\pi$
$389$	11087.0	1.44507	0.722534	+	0.691335i	$0.242977\pi$
$390$	0	0
$391$	−10160.6	−1.31418
$392$	1191.71	0.153547
$393$	0	0
$394$	−1210.73	−0.154811
$395$	1300.67	0.165681
$396$	0	0
$397$	6864.56	0.867814	0.433907	−	0.900958i	$-0.357135\pi$
$397$	6864.56	0.867814	0.433907	+	0.900958i	$0.357135\pi$
$398$	−10030.7	−1.26330
$399$	0	0
$400$	1672.31	0.209039
$401$	3464.90	0.431493	0.215747	−	0.976449i	$-0.430781\pi$
$401$	3464.90	0.431493	0.215747	+	0.976449i	$0.430781\pi$
$402$	0	0
$403$	−2769.75	−0.342360
$404$	−4914.38	−0.605196
$405$	0	0
$406$	−996.653	−0.121830
$407$	3171.34	0.386234
$408$	0	0
$409$	−4876.20	−0.589518	−0.294759	−	0.955572i	$-0.595239\pi$
$409$	−4876.20	−0.589518	−0.294759	+	0.955572i	$0.595239\pi$
$410$	326.883	0.0393747
$411$	0	0
$412$	−6530.03	−0.780854
$413$	1899.60	0.226327
$414$	0	0
$415$	3264.71	0.386165
$416$	−2019.35	−0.237997
$417$	0	0
$418$	−1247.97	−0.146029
$419$	8520.23	0.993414	0.496707	−	0.867918i	$-0.334542\pi$
$419$	8520.23	0.993414	0.496707	+	0.867918i	$0.334542\pi$
$420$	0	0
$421$	6164.41	0.713622	0.356811	−	0.934177i	$-0.383864\pi$
$421$	6164.41	0.713622	0.356811	+	0.934177i	$0.383864\pi$
$422$	−6684.29	−0.771057
$423$	0	0
$424$	14409.2	1.65041
$425$	7103.10	0.810708
$426$	0	0
$427$	−5736.20	−0.650103
$428$	−2756.50	−0.311309
$429$	0	0
$430$	−4183.95	−0.469228
$431$	639.993	0.0715252	0.0357626	−	0.999360i	$-0.488614\pi$
$431$	639.993	0.0715252	0.0357626	+	0.999360i	$0.488614\pi$
$432$	0	0
$433$	3938.71	0.437142	0.218571	−	0.975821i	$-0.429861\pi$
$433$	3938.71	0.437142	0.218571	+	0.975821i	$0.429861\pi$
$434$	−2986.03	−0.330263
$435$	0	0
$436$	−5342.16	−0.586796
$437$	−5899.12	−0.645751
$438$	0	0
$439$	110.576	0.0120216	0.00601082	−	0.999982i	$-0.498087\pi$
$439$	110.576	0.0120216	0.00601082	+	0.999982i	$0.498087\pi$
$440$	1870.82	0.202700
$441$	0	0
$442$	2656.17	0.285840
$443$	−16940.5	−1.81686	−0.908428	−	0.418041i	$-0.862717\pi$
$443$	−16940.5	−1.81686	−0.908428	+	0.418041i	$0.862717\pi$
$444$	0	0
$445$	1288.23	0.137232
$446$	1349.67	0.143293
$447$	0	0
$448$	−3407.69	−0.359371
$449$	5625.02	0.591228	0.295614	−	0.955308i	$-0.404476\pi$
$449$	5625.02	0.591228	0.295614	+	0.955308i	$0.404476\pi$
$450$	0	0
$451$	−245.614	−0.0256441
$452$	−1787.90	−0.186053
$453$	0	0
$454$	1973.92	0.204054
$455$	665.387	0.0685578
$456$	0	0
$457$	13399.8	1.37159	0.685793	−	0.727797i	$-0.259456\pi$
$457$	13399.8	1.37159	0.685793	+	0.727797i	$0.259456\pi$
$458$	5070.36	0.517298
$459$	0	0
$460$	2753.60	0.279103
$461$	−3548.83	−0.358537	−0.179269	−	0.983800i	$-0.557373\pi$
$461$	−3548.83	−0.358537	−0.179269	+	0.983800i	$0.557373\pi$
$462$	0	0
$463$	15722.0	1.57811	0.789053	−	0.614326i	$-0.210572\pi$
$463$	15722.0	1.57811	0.789053	+	0.614326i	$0.210572\pi$
$464$	1494.59	0.149536
$465$	0	0
$466$	−11809.1	−1.17392
$467$	19145.3	1.89708	0.948540	−	0.316657i	$-0.102560\pi$
$467$	19145.3	1.89708	0.948540	+	0.316657i	$0.102560\pi$
$468$	0	0
$469$	6732.41	0.662844
$470$	6351.61	0.623358
$471$	0	0
$472$	−6599.93	−0.643615
$473$	3143.74	0.305601
$474$	0	0
$475$	4123.96	0.398358
$476$	−2363.56	−0.227591
$477$	0	0
$478$	4572.59	0.437543
$479$	−4243.61	−0.404793	−0.202396	−	0.979304i	$-0.564873\pi$
$479$	−4243.61	−0.404793	−0.202396	+	0.979304i	$0.564873\pi$
$480$	0	0
$481$	−3918.87	−0.371487
$482$	−3182.05	−0.300702
$483$	0	0
$484$	−437.700	−0.0411064
$485$	10080.4	0.943769
$486$	0	0
$487$	−5633.82	−0.524215	−0.262107	−	0.965039i	$-0.584417\pi$
$487$	−5633.82	−0.524215	−0.262107	+	0.965039i	$0.584417\pi$
$488$	19929.7	1.84872
$489$	0	0
$490$	717.346	0.0661355
$491$	−2089.90	−0.192089	−0.0960445	−	0.995377i	$-0.530619\pi$
$491$	−2089.90	−0.192089	−0.0960445	+	0.995377i	$0.530619\pi$
$492$	0	0
$493$	6348.25	0.579941
$494$	1542.13	0.140453
$495$	0	0
$496$	4477.90	0.405370
$497$	−4199.41	−0.379013
$498$	0	0
$499$	−15107.3	−1.35530	−0.677651	−	0.735383i	$-0.737002\pi$
$499$	−15107.3	−1.35530	−0.677651	+	0.735383i	$0.737002\pi$
$500$	−5087.02	−0.454997
$501$	0	0
$502$	7644.61	0.679672
$503$	20468.9	1.81444	0.907220	−	0.420656i	$-0.138200\pi$
$503$	20468.9	1.81444	0.907220	+	0.420656i	$0.138200\pi$
$504$	0	0
$505$	−9500.39	−0.837152
$506$	2506.72	0.220232
$507$	0	0
$508$	470.170	0.0410638
$509$	9681.40	0.843066	0.421533	−	0.906813i	$-0.361492\pi$
$509$	9681.40	0.843066	0.421533	+	0.906813i	$0.361492\pi$
$510$	0	0
$511$	−1673.31	−0.144859
$512$	7540.45	0.650867
$513$	0	0
$514$	3161.37	0.271288
$515$	−12623.7	−1.08013
$516$	0	0
$517$	−4772.48	−0.405984
$518$	−4224.89	−0.358361
$519$	0	0
$520$	−2311.81	−0.194960
$521$	14360.2	1.20755	0.603775	−	0.797155i	$-0.293663\pi$
$521$	14360.2	1.20755	0.603775	+	0.797155i	$0.293663\pi$
$522$	0	0
$523$	9440.79	0.789325	0.394663	−	0.918826i	$-0.370861\pi$
$523$	9440.79	0.789325	0.394663	+	0.918826i	$0.370861\pi$
$524$	−8443.70	−0.703940
$525$	0	0
$526$	3983.88	0.330239
$527$	19019.8	1.57213
$528$	0	0
$529$	−317.766	−0.0261171
$530$	8673.54	0.710858
$531$	0	0
$532$	−1372.25	−0.111832
$533$	303.509	0.0246650
$534$	0	0
$535$	−5328.81	−0.430626
$536$	−23391.0	−1.88496
$537$	0	0
$538$	312.350	0.0250304
$539$	−539.000	−0.0430730
$540$	0	0
$541$	17757.4	1.41118	0.705590	−	0.708620i	$-0.250682\pi$
$541$	17757.4	1.41118	0.705590	+	0.708620i	$0.250682\pi$
$542$	2908.72	0.230517
$543$	0	0
$544$	13866.8	1.09289
$545$	−10327.4	−0.811699
$546$	0	0
$547$	3081.08	0.240836	0.120418	−	0.992723i	$-0.461576\pi$
$547$	3081.08	0.240836	0.120418	+	0.992723i	$0.461576\pi$
$548$	4020.06	0.313373
$549$	0	0
$550$	−1752.40	−0.135859
$551$	3685.70	0.284966
$552$	0	0
$553$	1301.97	0.100118
$554$	981.102	0.0752401
$555$	0	0
$556$	4233.39	0.322906
$557$	−9681.04	−0.736443	−0.368222	−	0.929738i	$-0.620033\pi$
$557$	−9681.04	−0.736443	−0.368222	+	0.929738i	$0.620033\pi$
$558$	0	0
$559$	−3884.78	−0.293933
$560$	−1075.74	−0.0811757
$561$	0	0
$562$	−9650.40	−0.724337
$563$	9538.12	0.714003	0.357002	−	0.934104i	$-0.383799\pi$
$563$	9538.12	0.714003	0.357002	+	0.934104i	$0.383799\pi$
$564$	0	0
$565$	−3456.34	−0.257362
$566$	8767.63	0.651115
$567$	0	0
$568$	14590.3	1.07781
$569$	−17921.0	−1.32036	−0.660181	−	0.751107i	$-0.729520\pi$
$569$	−17921.0	−1.32036	−0.660181	+	0.751107i	$0.729520\pi$
$570$	0	0
$571$	−22573.5	−1.65441	−0.827207	−	0.561897i	$-0.810072\pi$
$571$	−22573.5	−1.65441	−0.827207	+	0.561897i	$0.810072\pi$
$572$	540.874	0.0395368
$573$	0	0
$574$	327.210	0.0237935
$575$	−8283.56	−0.600779
$576$	0	0
$577$	9454.60	0.682149	0.341075	−	0.940036i	$-0.389209\pi$
$577$	9454.60	0.682149	0.341075	+	0.940036i	$0.389209\pi$
$578$	−7954.58	−0.572434
$579$	0	0
$580$	−1720.42	−0.123166
$581$	3267.97	0.233353
$582$	0	0
$583$	−6517.14	−0.462971
$584$	5813.71	0.411940
$585$	0	0
$586$	9978.87	0.703453
$587$	11553.1	0.812345	0.406173	−	0.913796i	$-0.366863\pi$
$587$	11553.1	0.812345	0.406173	+	0.913796i	$0.366863\pi$
$588$	0	0
$589$	11042.6	0.772499
$590$	−3972.80	−0.277216
$591$	0	0
$592$	6335.71	0.439858
$593$	12795.9	0.886115	0.443057	−	0.896493i	$-0.353894\pi$
$593$	12795.9	0.886115	0.443057	+	0.896493i	$0.353894\pi$
$594$	0	0
$595$	−4569.19	−0.314821
$596$	4872.56	0.334879
$597$	0	0
$598$	−3097.59	−0.211823
$599$	26094.7	1.77997	0.889983	−	0.455993i	$-0.150716\pi$
$599$	26094.7	1.77997	0.889983	+	0.455993i	$0.150716\pi$
$600$	0	0
$601$	−6283.94	−0.426501	−0.213251	−	0.976998i	$-0.568405\pi$
$601$	−6283.94	−0.426501	−0.213251	+	0.976998i	$0.568405\pi$
$602$	−4188.13	−0.283547
$603$	0	0
$604$	5758.52	0.387932
$605$	−846.155	−0.0568613
$606$	0	0
$607$	−16754.4	−1.12033	−0.560165	−	0.828381i	$-0.689262\pi$
$607$	−16754.4	−1.12033	−0.560165	+	0.828381i	$0.689262\pi$
$608$	8050.87	0.537016
$609$	0	0
$610$	11996.6	0.796277
$611$	5897.44	0.390482
$612$	0	0
$613$	−25906.0	−1.70690	−0.853452	−	0.521171i	$-0.825495\pi$
$613$	−25906.0	−1.70690	−0.853452	+	0.521171i	$0.825495\pi$
$614$	21375.4	1.40495
$615$	0	0
$616$	1872.69	0.122488
$617$	27420.4	1.78915	0.894573	−	0.446922i	$-0.147480\pi$
$617$	27420.4	1.78915	0.894573	+	0.446922i	$0.147480\pi$
$618$	0	0
$619$	3612.48	0.234568	0.117284	−	0.993098i	$-0.462581\pi$
$619$	3612.48	0.234568	0.117284	+	0.993098i	$0.462581\pi$
$620$	−5154.48	−0.333885
$621$	0	0
$622$	−2026.65	−0.130645
$623$	1289.52	0.0829271
$624$	0	0
$625$	−321.920	−0.0206029
$626$	−2394.67	−0.152892
$627$	0	0
$628$	−3910.27	−0.248466
$629$	26910.8	1.70589
$630$	0	0
$631$	7508.72	0.473720	0.236860	−	0.971544i	$-0.423882\pi$
$631$	7508.72	0.473720	0.236860	+	0.971544i	$0.423882\pi$
$632$	−4523.54	−0.284710
$633$	0	0
$634$	9374.42	0.587233
$635$	908.923	0.0568024
$636$	0	0
$637$	666.051	0.0414284
$638$	−1566.17	−0.0971869
$639$	0	0
$640$	−1184.24	−0.0731423
$641$	−15144.1	−0.933159	−0.466580	−	0.884479i	$-0.654514\pi$
$641$	−15144.1	−0.933159	−0.466580	+	0.884479i	$0.654514\pi$
$642$	0	0
$643$	1896.99	0.116345	0.0581726	−	0.998307i	$-0.481473\pi$
$643$	1896.99	0.116345	0.0581726	+	0.998307i	$0.481473\pi$
$644$	2756.35	0.168658
$645$	0	0
$646$	−10589.8	−0.644968
$647$	−8478.01	−0.515154	−0.257577	−	0.966258i	$-0.582924\pi$
$647$	−8478.01	−0.515154	−0.257577	+	0.966258i	$0.582924\pi$
$648$	0	0
$649$	2985.08	0.180547
$650$	2165.46	0.130672
$651$	0	0
$652$	−14599.8	−0.876952
$653$	−28619.2	−1.71509	−0.857547	−	0.514405i	$-0.828013\pi$
$653$	−28619.2	−1.71509	−0.857547	+	0.514405i	$0.828013\pi$
$654$	0	0
$655$	−16323.2	−0.973741
$656$	−490.688	−0.0292045
$657$	0	0
$658$	6357.96	0.376685
$659$	13750.0	0.812780	0.406390	−	0.913700i	$-0.366787\pi$
$659$	13750.0	0.812780	0.406390	+	0.913700i	$0.366787\pi$
$660$	0	0
$661$	−11608.8	−0.683100	−0.341550	−	0.939864i	$-0.610952\pi$
$661$	−11608.8	−0.683100	−0.341550	+	0.939864i	$0.610952\pi$
$662$	−14466.2	−0.849312
$663$	0	0
$664$	−11354.2	−0.663595
$665$	−2652.80	−0.154694
$666$	0	0
$667$	−7403.26	−0.429768
$668$	−11478.7	−0.664858
$669$	0	0
$670$	−14080.1	−0.811882
$671$	−9014.03	−0.518603
$672$	0	0
$673$	−28286.3	−1.62015	−0.810073	−	0.586329i	$-0.800572\pi$
$673$	−28286.3	−1.62015	−0.810073	+	0.586329i	$0.800572\pi$
$674$	−4767.38	−0.272452
$675$	0	0
$676$	7278.97	0.414143
$677$	14535.0	0.825148	0.412574	−	0.910924i	$-0.364630\pi$
$677$	14535.0	0.825148	0.412574	+	0.910924i	$0.364630\pi$
$678$	0	0
$679$	10090.5	0.570305
$680$	15875.1	0.895268
$681$	0	0
$682$	−4692.34	−0.263459
$683$	−2583.97	−0.144762	−0.0723812	−	0.997377i	$-0.523060\pi$
$683$	−2583.97	−0.144762	−0.0723812	+	0.997377i	$0.523060\pi$
$684$	0	0
$685$	7771.50	0.433480
$686$	718.062	0.0399646
$687$	0	0
$688$	6280.58	0.348030
$689$	8053.33	0.445294
$690$	0	0
$691$	−2187.71	−0.120440	−0.0602202	−	0.998185i	$-0.519180\pi$
$691$	−2187.71	−0.120440	−0.0602202	+	0.998185i	$0.519180\pi$
$692$	5282.29	0.290177
$693$	0	0
$694$	10952.4	0.599061
$695$	8183.91	0.446667
$696$	0	0
$697$	−2084.19	−0.113263
$698$	3160.30	0.171374
$699$	0	0
$700$	−1926.91	−0.104043
$701$	−10636.4	−0.573082	−0.286541	−	0.958068i	$-0.592506\pi$
$701$	−10636.4	−0.573082	−0.286541	+	0.958068i	$0.592506\pi$
$702$	0	0
$703$	15624.0	0.838222
$704$	−5354.93	−0.286679
$705$	0	0
$706$	19247.5	1.02605
$707$	−9509.87	−0.505878
$708$	0	0
$709$	2296.86	0.121665	0.0608324	−	0.998148i	$-0.480624\pi$
$709$	2296.86	0.121665	0.0608324	+	0.998148i	$0.480624\pi$
$710$	8782.59	0.464232
$711$	0	0
$712$	−4480.28	−0.235823
$713$	−22180.6	−1.16504
$714$	0	0
$715$	1045.61	0.0546902
$716$	−7052.38	−0.368100
$717$	0	0
$718$	22606.3	1.17501
$719$	13507.6	0.700624	0.350312	−	0.936633i	$-0.386076\pi$
$719$	13507.6	0.700624	0.350312	+	0.936633i	$0.386076\pi$
$720$	0	0
$721$	−12636.4	−0.652708
$722$	8210.88	0.423237
$723$	0	0
$724$	−11419.2	−0.586176
$725$	5175.47	0.265120
$726$	0	0
$727$	19678.8	1.00392	0.501959	−	0.864892i	$-0.332613\pi$
$727$	19678.8	1.00392	0.501959	+	0.864892i	$0.332613\pi$
$728$	−2314.12	−0.117812
$729$	0	0
$730$	3499.53	0.177430
$731$	26676.6	1.34975
$732$	0	0
$733$	12254.2	0.617486	0.308743	−	0.951145i	$-0.400092\pi$
$733$	12254.2	0.617486	0.308743	+	0.951145i	$0.400092\pi$
$734$	2753.49	0.138465
$735$	0	0
$736$	−16171.3	−0.809895
$737$	10579.5	0.528767
$738$	0	0
$739$	9407.00	0.468257	0.234129	−	0.972206i	$-0.424776\pi$
$739$	9407.00	0.468257	0.234129	+	0.972206i	$0.424776\pi$
$740$	−7293.00	−0.362292
$741$	0	0
$742$	8682.21	0.429560
$743$	−1201.12	−0.0593067	−0.0296534	−	0.999560i	$-0.509440\pi$
$743$	−1201.12	−0.0593067	−0.0296534	+	0.999560i	$0.509440\pi$
$744$	0	0
$745$	9419.55	0.463229
$746$	13289.7	0.652240
$747$	0	0
$748$	−3714.16	−0.181555
$749$	−5334.14	−0.260220
$750$	0	0
$751$	15855.7	0.770418	0.385209	−	0.922829i	$-0.374129\pi$
$751$	15855.7	0.770418	0.385209	+	0.922829i	$0.374129\pi$
$752$	−9534.48	−0.462349
$753$	0	0
$754$	1935.34	0.0934761
$755$	11132.3	0.536615
$756$	0	0
$757$	−27304.0	−1.31094	−0.655469	−	0.755222i	$-0.727529\pi$
$757$	−27304.0	−1.31094	−0.655469	+	0.755222i	$0.727529\pi$
$758$	−14186.4	−0.679779
$759$	0	0
$760$	9216.85	0.439908
$761$	14086.4	0.671002	0.335501	−	0.942040i	$-0.391094\pi$
$761$	14086.4	0.671002	0.335501	+	0.942040i	$0.391094\pi$
$762$	0	0
$763$	−10337.7	−0.490497
$764$	15549.1	0.736316
$765$	0	0
$766$	−30231.3	−1.42598
$767$	−3688.72	−0.173653
$768$	0	0
$769$	−22751.7	−1.06690	−0.533452	−	0.845831i	$-0.679105\pi$
$769$	−22751.7	−1.06690	−0.533452	+	0.845831i	$0.679105\pi$
$770$	1127.26	0.0527579
$771$	0	0
$772$	−3145.70	−0.146653
$773$	−20276.2	−0.943445	−0.471722	−	0.881747i	$-0.656368\pi$
$773$	−20276.2	−0.943445	−0.471722	+	0.881747i	$0.656368\pi$
$774$	0	0
$775$	15506.0	0.718700
$776$	−35058.2	−1.62180
$777$	0	0
$778$	−23210.3	−1.06958
$779$	−1210.05	−0.0556540
$780$	0	0
$781$	−6599.07	−0.302347
$782$	21271.1	0.972701
$783$	0	0
$784$	−1076.82	−0.0490532
$785$	−7559.26	−0.343696
$786$	0	0
$787$	30782.4	1.39425	0.697124	−	0.716950i	$-0.254463\pi$
$787$	30782.4	1.39425	0.697124	+	0.716950i	$0.254463\pi$
$788$	−2092.04	−0.0945758
$789$	0	0
$790$	−2722.93	−0.122630
$791$	−3459.80	−0.155520
$792$	0	0
$793$	11138.8	0.498802
$794$	−14370.8	−0.642317
$795$	0	0
$796$	−17332.3	−0.771767
$797$	34592.5	1.53743	0.768714	−	0.639593i	$-0.220897\pi$
$797$	34592.5	1.53743	0.768714	+	0.639593i	$0.220897\pi$
$798$	0	0
$799$	−40497.5	−1.79311
$800$	11305.0	0.499617
$801$	0	0
$802$	−7253.68	−0.319372
$803$	−2629.48	−0.115557
$804$	0	0
$805$	5328.53	0.233300
$806$	5798.40	0.253399
$807$	0	0
$808$	33040.9	1.43858
$809$	−39785.7	−1.72904	−0.864518	−	0.502601i	$-0.832376\pi$
$809$	−39785.7	−1.72904	−0.864518	+	0.502601i	$0.832376\pi$
$810$	0	0
$811$	−8711.79	−0.377204	−0.188602	−	0.982054i	$-0.560396\pi$
$811$	−8711.79	−0.377204	−0.188602	+	0.982054i	$0.560396\pi$
$812$	−1722.14	−0.0744275
$813$	0	0
$814$	−6639.12	−0.285873
$815$	−28224.1	−1.21306
$816$	0	0
$817$	15488.1	0.663229
$818$	10208.2	0.436335
$819$	0	0
$820$	564.828	0.0240545
$821$	−1148.86	−0.0488374	−0.0244187	−	0.999702i	$-0.507773\pi$
$821$	−1148.86	−0.0488374	−0.0244187	+	0.999702i	$0.507773\pi$
$822$	0	0
$823$	20030.0	0.848360	0.424180	−	0.905578i	$-0.360562\pi$
$823$	20030.0	0.848360	0.424180	+	0.905578i	$0.360562\pi$
$824$	43903.5	1.85613
$825$	0	0
$826$	−3976.76	−0.167517
$827$	34601.5	1.45491	0.727456	−	0.686154i	$-0.240703\pi$
$827$	34601.5	1.45491	0.727456	+	0.686154i	$0.240703\pi$
$828$	0	0
$829$	−36147.7	−1.51443	−0.757214	−	0.653166i	$-0.773440\pi$
$829$	−36147.7	−1.51443	−0.757214	+	0.653166i	$0.773440\pi$
$830$	−6834.59	−0.285822
$831$	0	0
$832$	6617.18	0.275733
$833$	−4573.75	−0.190241
$834$	0	0
$835$	−22190.5	−0.919680
$836$	−2156.39	−0.0892108
$837$	0	0
$838$	−17836.9	−0.735281
$839$	7564.84	0.311284	0.155642	−	0.987814i	$-0.450255\pi$
$839$	7564.84	0.311284	0.155642	+	0.987814i	$0.450255\pi$
$840$	0	0
$841$	−19763.5	−0.810346
$842$	−12905.0	−0.528191
$843$	0	0
$844$	−11549.9	−0.471048
$845$	14071.6	0.572872
$846$	0	0
$847$	−847.000	−0.0343604
$848$	−13020.0	−0.527249
$849$	0	0
$850$	−14870.2	−0.600050
$851$	−31383.0	−1.26416
$852$	0	0
$853$	33128.9	1.32979	0.664895	−	0.746937i	$-0.268476\pi$
$853$	33128.9	1.32979	0.664895	+	0.746937i	$0.268476\pi$
$854$	12008.6	0.481178
$855$	0	0
$856$	18532.8	0.739998
$857$	−29744.5	−1.18559	−0.592797	−	0.805352i	$-0.701976\pi$
$857$	−29744.5	−1.18559	−0.592797	+	0.805352i	$0.701976\pi$
$858$	0	0
$859$	−3418.95	−0.135801	−0.0679005	−	0.997692i	$-0.521630\pi$
$859$	−3418.95	−0.135801	−0.0679005	+	0.997692i	$0.521630\pi$
$860$	−7229.54	−0.286657
$861$	0	0
$862$	−1339.81	−0.0529398
$863$	−18222.3	−0.718764	−0.359382	−	0.933191i	$-0.617012\pi$
$863$	−18222.3	−0.718764	−0.359382	+	0.933191i	$0.617012\pi$
$864$	0	0
$865$	10211.6	0.401394
$866$	−8245.59	−0.323553
$867$	0	0
$868$	−5159.63	−0.201762
$869$	2045.95	0.0798668
$870$	0	0
$871$	−13073.3	−0.508578
$872$	35917.0	1.39484
$873$	0	0
$874$	12349.7	0.477956
$875$	−9843.96	−0.380327
$876$	0	0
$877$	−33406.0	−1.28625	−0.643124	−	0.765762i	$-0.722362\pi$
$877$	−33406.0	−1.28625	−0.643124	+	0.765762i	$0.722362\pi$
$878$	−231.488	−0.00889788
$879$	0	0
$880$	−1690.45	−0.0647558
$881$	14129.7	0.540341	0.270170	−	0.962813i	$-0.412920\pi$
$881$	14129.7	0.540341	0.270170	+	0.962813i	$0.412920\pi$
$882$	0	0
$883$	22044.4	0.840150	0.420075	−	0.907489i	$-0.362004\pi$
$883$	22044.4	0.840150	0.420075	+	0.907489i	$0.362004\pi$
$884$	4589.65	0.174623
$885$	0	0
$886$	35464.5	1.34476
$887$	−597.301	−0.0226104	−0.0113052	−	0.999936i	$-0.503599\pi$
$887$	−597.301	−0.0226104	−0.0113052	+	0.999936i	$0.503599\pi$
$888$	0	0
$889$	909.831	0.0343248
$890$	−2696.89	−0.101573
$891$	0	0
$892$	2332.12	0.0875393
$893$	−23512.2	−0.881083
$894$	0	0
$895$	−13633.5	−0.509183
$896$	−1185.42	−0.0441988
$897$	0	0
$898$	−11775.8	−0.437600
$899$	13858.2	0.514123
$900$	0	0
$901$	−55302.0	−2.04481
$902$	514.187	0.0189807
$903$	0	0
$904$	12020.6	0.442258
$905$	−22075.4	−0.810842
$906$	0	0
$907$	48604.6	1.77937	0.889686	−	0.456573i	$-0.150923\pi$
$907$	48604.6	1.77937	0.889686	+	0.456573i	$0.150923\pi$
$908$	3410.77	0.124659
$909$	0	0
$910$	−1392.97	−0.0507434
$911$	16134.7	0.586791	0.293395	−	0.955991i	$-0.405215\pi$
$911$	16134.7	0.586791	0.293395	+	0.955991i	$0.405215\pi$
$912$	0	0
$913$	5135.38	0.186152
$914$	−28052.1	−1.01519
$915$	0	0
$916$	8761.18	0.316023
$917$	−16339.5	−0.588417
$918$	0	0
$919$	8686.36	0.311792	0.155896	−	0.987774i	$-0.450174\pi$
$919$	8686.36	0.311792	0.155896	+	0.987774i	$0.450174\pi$
$920$	−18513.3	−0.663442
$921$	0	0
$922$	7429.40	0.265373
$923$	8154.58	0.290803
$924$	0	0
$925$	21939.2	0.779846
$926$	−32913.6	−1.16804
$927$	0	0
$928$	10103.6	0.357401
$929$	−35449.0	−1.25193	−0.625966	−	0.779850i	$-0.715295\pi$
$929$	−35449.0	−1.25193	−0.625966	+	0.779850i	$0.715295\pi$
$930$	0	0
$931$	−2655.45	−0.0934790
$932$	−20405.2	−0.717162
$933$	0	0
$934$	−40080.1	−1.40413
$935$	−7180.15	−0.251140
$936$	0	0
$937$	20315.3	0.708293	0.354147	−	0.935190i	$-0.384771\pi$
$937$	20315.3	0.708293	0.354147	+	0.935190i	$0.384771\pi$
$938$	−14094.1	−0.490608
$939$	0	0
$940$	10975.1	0.380817
$941$	−45109.4	−1.56273	−0.781363	−	0.624077i	$-0.785475\pi$
$941$	−45109.4	−1.56273	−0.781363	+	0.624077i	$0.785475\pi$
$942$	0	0
$943$	2430.56	0.0839340
$944$	5963.61	0.205613
$945$	0	0
$946$	−6581.35	−0.226193
$947$	−19141.9	−0.656842	−0.328421	−	0.944531i	$-0.606516\pi$
$947$	−19141.9	−0.656842	−0.328421	+	0.944531i	$0.606516\pi$
$948$	0	0
$949$	3249.30	0.111145
$950$	−8633.40	−0.294847
$951$	0	0
$952$	15890.9	0.540996
$953$	21864.2	0.743180	0.371590	−	0.928397i	$-0.378813\pi$
$953$	21864.2	0.743180	0.371590	+	0.928397i	$0.378813\pi$
$954$	0	0
$955$	30059.2	1.01853
$956$	7901.07	0.267300
$957$	0	0
$958$	8883.90	0.299610
$959$	7779.27	0.261945
$960$	0	0
$961$	11729.0	0.393709
$962$	8204.07	0.274958
$963$	0	0
$964$	−5498.33	−0.183702
$965$	−6081.21	−0.202861
$966$	0	0
$967$	−27394.8	−0.911021	−0.455510	−	0.890230i	$-0.650543\pi$
$967$	−27394.8	−0.911021	−0.455510	+	0.890230i	$0.650543\pi$
$968$	2942.80	0.0977120
$969$	0	0
$970$	−21103.1	−0.698536
$971$	6029.69	0.199281	0.0996405	−	0.995024i	$-0.468231\pi$
$971$	6029.69	0.199281	0.0996405	+	0.995024i	$0.468231\pi$
$972$	0	0
$973$	8192.08	0.269914
$974$	11794.3	0.388000
$975$	0	0
$976$	−18008.3	−0.590605
$977$	2257.94	0.0739384	0.0369692	−	0.999316i	$-0.488230\pi$
$977$	2257.94	0.0739384	0.0369692	+	0.999316i	$0.488230\pi$
$978$	0	0
$979$	2026.39	0.0661529
$980$	1239.52	0.0404030
$981$	0	0
$982$	4375.15	0.142176
$983$	18653.3	0.605237	0.302619	−	0.953112i	$-0.402139\pi$
$983$	18653.3	0.605237	0.302619	+	0.953112i	$0.402139\pi$
$984$	0	0
$985$	−4044.29	−0.130824
$986$	−13289.9	−0.429247
$987$	0	0
$988$	2664.68	0.0858045
$989$	−31110.0	−1.00024
$990$	0	0
$991$	−18262.4	−0.585394	−0.292697	−	0.956205i	$-0.594553\pi$
$991$	−18262.4	−0.585394	−0.292697	+	0.956205i	$0.594553\pi$
$992$	30271.1	0.968860
$993$	0	0
$994$	8791.36	0.280528
$995$	−33506.5	−1.06756
$996$	0	0
$997$	51536.4	1.63708	0.818542	−	0.574446i	$-0.194783\pi$
$997$	51536.4	1.63708	0.818542	+	0.574446i	$0.194783\pi$
$998$	31626.8	1.00313
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.t.1.3	yes	8
3.2	odd	2		693.4.a.s.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.4.a.s.1.6	✓	8	3.2	odd	2
693.4.a.t.1.3	yes	8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q-2.09348 q^{2} -3.61736 q^{4} -6.99302 q^{5} -7.00000 q^{7} +24.3207 q^{8} +O(q^{10})\)	\(q-2.09348 q^{2} -3.61736 q^{4} -6.99302 q^{5} -7.00000 q^{7} +24.3207 q^{8} +14.6397 q^{10} -11.0000 q^{11} +13.5929 q^{13} +14.6543 q^{14} -21.9758 q^{16} -93.3418 q^{17} -54.1929 q^{19} +25.2963 q^{20} +23.0282 q^{22} +108.854 q^{23} -76.0977 q^{25} -28.4564 q^{26} +25.3215 q^{28} -68.0108 q^{29} -203.765 q^{31} -148.559 q^{32} +195.409 q^{34} +48.9511 q^{35} -288.303 q^{37} +113.452 q^{38} -170.075 q^{40} +22.3285 q^{41} -285.795 q^{43} +39.7910 q^{44} -227.884 q^{46} +433.862 q^{47} +49.0000 q^{49} +159.309 q^{50} -49.1703 q^{52} +592.467 q^{53} +76.9232 q^{55} -170.245 q^{56} +142.379 q^{58} -271.371 q^{59} +819.457 q^{61} +426.576 q^{62} +486.812 q^{64} -95.0552 q^{65} -961.774 q^{67} +337.651 q^{68} -102.478 q^{70} +599.916 q^{71} +239.044 q^{73} +603.556 q^{74} +196.035 q^{76} +77.0000 q^{77} -185.996 q^{79} +153.677 q^{80} -46.7443 q^{82} -466.853 q^{83} +652.741 q^{85} +598.305 q^{86} -267.527 q^{88} -184.217 q^{89} -95.1502 q^{91} -393.765 q^{92} -908.280 q^{94} +378.972 q^{95} -1441.50 q^{97} -102.580 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} + 30 q^{4} + 10 q^{5} - 56 q^{7} + 15 q^{8}+O(q^{10}) \) 8 * q + 2 * q^2 + 30 * q^4 + 10 * q^5 - 56 * q^7 + 15 * q^8	\( 8 q + 2 q^{2} + 30 q^{4} + 10 q^{5} - 56 q^{7} + 15 q^{8} - 13 q^{10} - 88 q^{11} - 148 q^{13} - 14 q^{14} + 266 q^{16} + 114 q^{17} + 58 q^{19} + 291 q^{20} - 22 q^{22} + 246 q^{23} + 244 q^{25} + 305 q^{26} - 210 q^{28} - 72 q^{29} + 252 q^{31} + 1272 q^{32} + 630 q^{34} - 70 q^{35} - 80 q^{37} + 1885 q^{38} - 342 q^{40} + 682 q^{41} - 106 q^{43} - 330 q^{44} + 120 q^{46} + 828 q^{47} + 392 q^{49} + 801 q^{50} - 1681 q^{52} + 462 q^{53} - 110 q^{55} - 105 q^{56} - 1087 q^{58} + 626 q^{59} - 854 q^{61} + 1350 q^{62} + 2997 q^{64} + 22 q^{65} + 130 q^{67} + 2202 q^{68} + 91 q^{70} + 326 q^{71} - 390 q^{73} - 359 q^{74} + 2041 q^{76} + 616 q^{77} - 508 q^{79} + 4391 q^{80} + 1528 q^{82} + 1596 q^{83} - 880 q^{85} - 414 q^{86} - 165 q^{88} + 4324 q^{89} + 1036 q^{91} + 2092 q^{92} - 1685 q^{94} + 1076 q^{95} - 964 q^{97} + 98 q^{98}+O(q^{100}) \) 8 * q + 2 * q^2 + 30 * q^4 + 10 * q^5 - 56 * q^7 + 15 * q^8 - 13 * q^10 - 88 * q^11 - 148 * q^13 - 14 * q^14 + 266 * q^16 + 114 * q^17 + 58 * q^19 + 291 * q^20 - 22 * q^22 + 246 * q^23 + 244 * q^25 + 305 * q^26 - 210 * q^28 - 72 * q^29 + 252 * q^31 + 1272 * q^32 + 630 * q^34 - 70 * q^35 - 80 * q^37 + 1885 * q^38 - 342 * q^40 + 682 * q^41 - 106 * q^43 - 330 * q^44 + 120 * q^46 + 828 * q^47 + 392 * q^49 + 801 * q^50 - 1681 * q^52 + 462 * q^53 - 110 * q^55 - 105 * q^56 - 1087 * q^58 + 626 * q^59 - 854 * q^61 + 1350 * q^62 + 2997 * q^64 + 22 * q^65 + 130 * q^67 + 2202 * q^68 + 91 * q^70 + 326 * q^71 - 390 * q^73 - 359 * q^74 + 2041 * q^76 + 616 * q^77 - 508 * q^79 + 4391 * q^80 + 1528 * q^82 + 1596 * q^83 - 880 * q^85 - 414 * q^86 - 165 * q^88 + 4324 * q^89 + 1036 * q^91 + 2092 * q^92 - 1685 * q^94 + 1076 * q^95 - 964 * q^97 + 98 * q^98

Introduction

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