LMFDB - Embedded newform 693.4.a.o.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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 42x^{3} + 18x^{2} + 368x + 352 $ x^5 - x^4 - 42x^3 + 18x^2 + 368*x + 352
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.22767$ 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)

$f(q)$	$=$	$q+1.22767 q^{2} -6.49284 q^{4} +2.21191 q^{5} +7.00000 q^{7} -17.7924 q^{8} +O(q^{10})$	\(q+1.22767 q^{2} -6.49284 q^{4} +2.21191 q^{5} +7.00000 q^{7} -17.7924 q^{8} +2.71549 q^{10} -11.0000 q^{11} +12.8361 q^{13} +8.59366 q^{14} +30.0996 q^{16} +45.5444 q^{17} +11.0493 q^{19} -14.3616 q^{20} -13.5043 q^{22} -177.525 q^{23} -120.107 q^{25} +15.7584 q^{26} -45.4499 q^{28} -58.5230 q^{29} +175.188 q^{31} +179.291 q^{32} +55.9132 q^{34} +15.4834 q^{35} +221.135 q^{37} +13.5648 q^{38} -39.3551 q^{40} +307.706 q^{41} +462.781 q^{43} +71.4212 q^{44} -217.942 q^{46} +293.151 q^{47} +49.0000 q^{49} -147.452 q^{50} -83.3427 q^{52} -400.608 q^{53} -24.3310 q^{55} -124.547 q^{56} -71.8467 q^{58} -16.3417 q^{59} +509.546 q^{61} +215.072 q^{62} -20.6874 q^{64} +28.3923 q^{65} +483.585 q^{67} -295.712 q^{68} +19.0084 q^{70} -202.883 q^{71} +885.910 q^{73} +271.479 q^{74} -71.7412 q^{76} -77.0000 q^{77} +289.526 q^{79} +66.5777 q^{80} +377.760 q^{82} -106.577 q^{83} +100.740 q^{85} +568.140 q^{86} +195.716 q^{88} +1586.10 q^{89} +89.8527 q^{91} +1152.64 q^{92} +359.891 q^{94} +24.4400 q^{95} -990.599 q^{97} +60.1556 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} + 45 q^{4} + 24 q^{5} + 35 q^{7} - 57 q^{8}+O(q^{10}) $ 5 * q - q^2 + 45 * q^4 + 24 * q^5 + 35 * q^7 - 57 * q^8	$ 5 q - q^{2} + 45 q^{4} + 24 q^{5} + 35 q^{7} - 57 q^{8} - 10 q^{10} - 55 q^{11} - 50 q^{13} - 7 q^{14} + 433 q^{16} - 222 q^{17} + 160 q^{19} + 430 q^{20} + 11 q^{22} - 54 q^{23} + 125 q^{25} + 1026 q^{26} + 315 q^{28} - 14 q^{29} - 34 q^{31} + 583 q^{32} - 750 q^{34} + 168 q^{35} + 1044 q^{37} + 156 q^{38} + 1158 q^{40} + 114 q^{41} + 672 q^{43} - 495 q^{44} - 1224 q^{46} + 292 q^{47} + 245 q^{49} - 1143 q^{50} - 914 q^{52} + 710 q^{53} - 264 q^{55} - 399 q^{56} - 810 q^{58} - 270 q^{59} + 138 q^{61} + 480 q^{62} + 3001 q^{64} + 196 q^{65} + 1942 q^{67} - 3130 q^{68} - 70 q^{70} + 278 q^{71} - 338 q^{73} - 462 q^{74} - 1332 q^{76} - 385 q^{77} + 576 q^{79} + 1602 q^{80} + 1386 q^{82} - 1644 q^{83} + 360 q^{85} - 2020 q^{86} + 627 q^{88} + 3656 q^{89} - 350 q^{91} + 748 q^{92} + 976 q^{94} + 1276 q^{95} + 692 q^{97} - 49 q^{98}+O(q^{100}) $ 5 * q - q^2 + 45 * q^4 + 24 * q^5 + 35 * q^7 - 57 * q^8 - 10 * q^10 - 55 * q^11 - 50 * q^13 - 7 * q^14 + 433 * q^16 - 222 * q^17 + 160 * q^19 + 430 * q^20 + 11 * q^22 - 54 * q^23 + 125 * q^25 + 1026 * q^26 + 315 * q^28 - 14 * q^29 - 34 * q^31 + 583 * q^32 - 750 * q^34 + 168 * q^35 + 1044 * q^37 + 156 * q^38 + 1158 * q^40 + 114 * q^41 + 672 * q^43 - 495 * q^44 - 1224 * q^46 + 292 * q^47 + 245 * q^49 - 1143 * q^50 - 914 * q^52 + 710 * q^53 - 264 * q^55 - 399 * q^56 - 810 * q^58 - 270 * q^59 + 138 * q^61 + 480 * q^62 + 3001 * q^64 + 196 * q^65 + 1942 * q^67 - 3130 * q^68 - 70 * q^70 + 278 * q^71 - 338 * q^73 - 462 * q^74 - 1332 * q^76 - 385 * q^77 + 576 * q^79 + 1602 * q^80 + 1386 * q^82 - 1644 * q^83 + 360 * q^85 - 2020 * q^86 + 627 * q^88 + 3656 * q^89 - 350 * q^91 + 748 * q^92 + 976 * q^94 + 1276 * q^95 + 692 * q^97 - 49 * 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 $

$2$	1.22767	0.434045	0.217023	−	0.976167i	$-0.430365\pi$
$2$	1.22767	0.434045	0.217023	+	0.976167i	$0.430365\pi$
$3$	0	0
$3$	0	0
$4$	−6.49284	−0.811605
$5$	2.21191	0.197839	0.0989196	−	0.995095i	$-0.468461\pi$
$5$	2.21191	0.197839	0.0989196	+	0.995095i	$0.468461\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−17.7924	−0.786319
$9$	0	0
$10$	2.71549	0.0858712
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	12.8361	0.273853	0.136927	−	0.990581i	$-0.456278\pi$
$13$	12.8361	0.273853	0.136927	+	0.990581i	$0.456278\pi$
$14$	8.59366	0.164054
$15$	0	0
$16$	30.0996	0.470307
$17$	45.5444	0.649772	0.324886	−	0.945753i	$-0.394674\pi$
$17$	45.5444	0.649772	0.324886	+	0.945753i	$0.394674\pi$
$18$	0	0
$19$	11.0493	0.133415	0.0667074	−	0.997773i	$-0.478751\pi$
$19$	11.0493	0.133415	0.0667074	+	0.997773i	$0.478751\pi$
$20$	−14.3616	−0.160567
$21$	0	0
$22$	−13.5043	−0.130870
$23$	−177.525	−1.60942	−0.804708	−	0.593671i	$-0.797678\pi$
$23$	−177.525	−1.60942	−0.804708	+	0.593671i	$0.797678\pi$
$24$	0	0
$25$	−120.107	−0.960860
$26$	15.7584	0.118865
$27$	0	0
$28$	−45.4499	−0.306758
$29$	−58.5230	−0.374740	−0.187370	−	0.982289i	$-0.559996\pi$
$29$	−58.5230	−0.374740	−0.187370	+	0.982289i	$0.559996\pi$
$30$	0	0
$31$	175.188	1.01499	0.507495	−	0.861654i	$-0.330571\pi$
$31$	175.188	1.01499	0.507495	+	0.861654i	$0.330571\pi$
$32$	179.291	0.990453
$33$	0	0
$34$	55.9132	0.282031
$35$	15.4834	0.0747762
$36$	0	0
$37$	221.135	0.982549	0.491274	−	0.871005i	$-0.336531\pi$
$37$	221.135	0.982549	0.491274	+	0.871005i	$0.336531\pi$
$38$	13.5648	0.0579080
$39$	0	0
$40$	−39.3551	−0.155565
$41$	307.706	1.17209	0.586043	−	0.810280i	$-0.300685\pi$
$41$	307.706	1.17209	0.586043	+	0.810280i	$0.300685\pi$
$42$	0	0
$43$	462.781	1.64124	0.820621	−	0.571473i	$-0.193628\pi$
$43$	462.781	1.64124	0.820621	+	0.571473i	$0.193628\pi$
$44$	71.4212	0.244708
$45$	0	0
$46$	−217.942	−0.698560
$47$	293.151	0.909796	0.454898	−	0.890544i	$-0.349676\pi$
$47$	293.151	0.909796	0.454898	+	0.890544i	$0.349676\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−147.452	−0.417057
$51$	0	0
$52$	−83.3427	−0.222261
$53$	−400.608	−1.03826	−0.519130	−	0.854696i	$-0.673744\pi$
$53$	−400.608	−1.03826	−0.519130	+	0.854696i	$0.673744\pi$
$54$	0	0
$55$	−24.3310	−0.0596508
$56$	−124.547	−0.297201
$57$	0	0
$58$	−71.8467	−0.162654
$59$	−16.3417	−0.0360595	−0.0180298	−	0.999837i	$-0.505739\pi$
$59$	−16.3417	−0.0360595	−0.0180298	+	0.999837i	$0.505739\pi$
$60$	0	0
$61$	509.546	1.06952	0.534760	−	0.845004i	$-0.320402\pi$
$61$	509.546	1.06952	0.534760	+	0.845004i	$0.320402\pi$
$62$	215.072	0.440552
$63$	0	0
$64$	−20.6874	−0.0404051
$65$	28.3923	0.0541790
$66$	0	0
$67$	483.585	0.881781	0.440891	−	0.897561i	$-0.354663\pi$
$67$	483.585	0.881781	0.440891	+	0.897561i	$0.354663\pi$
$68$	−295.712	−0.527358
$69$	0	0
$70$	19.0084	0.0324563
$71$	−202.883	−0.339124	−0.169562	−	0.985519i	$-0.554235\pi$
$71$	−202.883	−0.339124	−0.169562	+	0.985519i	$0.554235\pi$
$72$	0	0
$73$	885.910	1.42038	0.710191	−	0.704009i	$-0.248608\pi$
$73$	885.910	1.42038	0.710191	+	0.704009i	$0.248608\pi$
$74$	271.479	0.426471
$75$	0	0
$76$	−71.7412	−0.108280
$77$	−77.0000	−0.113961
$78$	0	0
$79$	289.526	0.412331	0.206166	−	0.978517i	$-0.433901\pi$
$79$	289.526	0.412331	0.206166	+	0.978517i	$0.433901\pi$
$80$	66.5777	0.0930451
$81$	0	0
$82$	377.760	0.508739
$83$	−106.577	−0.140944	−0.0704722	−	0.997514i	$-0.522451\pi$
$83$	−106.577	−0.140944	−0.0704722	+	0.997514i	$0.522451\pi$
$84$	0	0
$85$	100.740	0.128550
$86$	568.140	0.712373
$87$	0	0
$88$	195.716	0.237084
$89$	1586.10	1.88906	0.944528	−	0.328432i	$-0.106520\pi$
$89$	1586.10	1.88906	0.944528	+	0.328432i	$0.106520\pi$
$90$	0	0
$91$	89.8527	0.103507
$92$	1152.64	1.30621
$93$	0	0
$94$	359.891	0.394893
$95$	24.4400	0.0263947
$96$	0	0
$97$	−990.599	−1.03691	−0.518454	−	0.855105i	$-0.673492\pi$
$97$	−990.599	−1.03691	−0.518454	+	0.855105i	$0.673492\pi$
$98$	60.1556	0.0620065
$99$	0	0
$100$	779.838	0.779838
$101$	496.401	0.489047	0.244523	−	0.969643i	$-0.421369\pi$
$101$	496.401	0.489047	0.244523	+	0.969643i	$0.421369\pi$
$102$	0	0
$103$	287.602	0.275128	0.137564	−	0.990493i	$-0.456073\pi$
$103$	287.602	0.275128	0.137564	+	0.990493i	$0.456073\pi$
$104$	−228.385	−0.215336
$105$	0	0
$106$	−491.813	−0.450652
$107$	1310.33	1.18388	0.591938	−	0.805984i	$-0.298363\pi$
$107$	1310.33	1.18388	0.591938	+	0.805984i	$0.298363\pi$
$108$	0	0
$109$	−1226.79	−1.07803	−0.539014	−	0.842297i	$-0.681203\pi$
$109$	−1226.79	−1.07803	−0.539014	+	0.842297i	$0.681203\pi$
$110$	−29.8703	−0.0258911
$111$	0	0
$112$	210.697	0.177759
$113$	−1717.53	−1.42984	−0.714921	−	0.699206i	$-0.753537\pi$
$113$	−1717.53	−1.42984	−0.714921	+	0.699206i	$0.753537\pi$
$114$	0	0
$115$	−392.670	−0.318406
$116$	379.980	0.304140
$117$	0	0
$118$	−20.0622	−0.0156515
$119$	318.810	0.245591
$120$	0	0
$121$	121.000	0.0909091
$122$	625.552	0.464220
$123$	0	0
$124$	−1137.47	−0.823771
$125$	−542.156	−0.387935
$126$	0	0
$127$	2764.27	1.93141	0.965707	−	0.259633i	$-0.0836015\pi$
$127$	2764.27	1.93141	0.965707	+	0.259633i	$0.0836015\pi$
$128$	−1459.73	−1.00799
$129$	0	0
$130$	34.8563	0.0235161
$131$	781.116	0.520965	0.260482	−	0.965479i	$-0.416118\pi$
$131$	781.116	0.520965	0.260482	+	0.965479i	$0.416118\pi$
$132$	0	0
$133$	77.3450	0.0504260
$134$	593.681	0.382733
$135$	0	0
$136$	−810.341	−0.510928
$137$	1562.95	0.974685	0.487343	−	0.873211i	$-0.337966\pi$
$137$	1562.95	0.974685	0.487343	+	0.873211i	$0.337966\pi$
$138$	0	0
$139$	1707.58	1.04198	0.520991	−	0.853562i	$-0.325563\pi$
$139$	1707.58	1.04198	0.520991	+	0.853562i	$0.325563\pi$
$140$	−100.531	−0.0606887
$141$	0	0
$142$	−249.073	−0.147195
$143$	−141.197	−0.0825699
$144$	0	0
$145$	−129.448	−0.0741382
$146$	1087.60	0.616511
$147$	0	0
$148$	−1435.79	−0.797441
$149$	−1825.21	−1.00353	−0.501767	−	0.865003i	$-0.667317\pi$
$149$	−1825.21	−1.00353	−0.501767	+	0.865003i	$0.667317\pi$
$150$	0	0
$151$	−1300.18	−0.700708	−0.350354	−	0.936617i	$-0.613939\pi$
$151$	−1300.18	−0.700708	−0.350354	+	0.936617i	$0.613939\pi$
$152$	−196.593	−0.104906
$153$	0	0
$154$	−94.5303	−0.0494641
$155$	387.500	0.200805
$156$	0	0
$157$	2068.27	1.05138	0.525688	−	0.850677i	$-0.323808\pi$
$157$	2068.27	1.05138	0.525688	+	0.850677i	$0.323808\pi$
$158$	355.441	0.178971
$159$	0	0
$160$	396.576	0.195950
$161$	−1242.68	−0.608302
$162$	0	0
$163$	1463.67	0.703336	0.351668	−	0.936125i	$-0.385615\pi$
$163$	1463.67	0.703336	0.351668	+	0.936125i	$0.385615\pi$
$164$	−1997.88	−0.951271
$165$	0	0
$166$	−130.841	−0.0611763
$167$	−3276.22	−1.51809	−0.759045	−	0.651038i	$-0.774334\pi$
$167$	−3276.22	−1.51809	−0.759045	+	0.651038i	$0.774334\pi$
$168$	0	0
$169$	−2032.23	−0.925004
$170$	123.675	0.0557967
$171$	0	0
$172$	−3004.76	−1.33204
$173$	795.944	0.349795	0.174897	−	0.984587i	$-0.444041\pi$
$173$	795.944	0.349795	0.174897	+	0.984587i	$0.444041\pi$
$174$	0	0
$175$	−840.752	−0.363171
$176$	−331.096	−0.141803
$177$	0	0
$178$	1947.20	0.819936
$179$	−1636.50	−0.683340	−0.341670	−	0.939820i	$-0.610992\pi$
$179$	−1636.50	−0.683340	−0.341670	+	0.939820i	$0.610992\pi$
$180$	0	0
$181$	3631.26	1.49121	0.745606	−	0.666387i	$-0.232160\pi$
$181$	3631.26	1.49121	0.745606	+	0.666387i	$0.232160\pi$
$182$	110.309	0.0449267
$183$	0	0
$184$	3158.59	1.26551
$185$	489.130	0.194387
$186$	0	0
$187$	−500.988	−0.195914
$188$	−1903.38	−0.738394
$189$	0	0
$190$	30.0042	0.0114565
$191$	−208.684	−0.0790568	−0.0395284	−	0.999218i	$-0.512586\pi$
$191$	−208.684	−0.0790568	−0.0395284	+	0.999218i	$0.512586\pi$
$192$	0	0
$193$	808.914	0.301694	0.150847	−	0.988557i	$-0.451800\pi$
$193$	808.914	0.301694	0.150847	+	0.988557i	$0.451800\pi$
$194$	−1216.12	−0.450065
$195$	0	0
$196$	−318.149	−0.115944
$197$	−4224.83	−1.52795	−0.763976	−	0.645245i	$-0.776755\pi$
$197$	−4224.83	−1.52795	−0.763976	+	0.645245i	$0.776755\pi$
$198$	0	0
$199$	344.463	0.122705	0.0613526	−	0.998116i	$-0.480459\pi$
$199$	344.463	0.122705	0.0613526	+	0.998116i	$0.480459\pi$
$200$	2136.99	0.755542
$201$	0	0
$202$	609.414	0.212268
$203$	−409.661	−0.141638
$204$	0	0
$205$	680.617	0.231885
$206$	353.079	0.119418
$207$	0	0
$208$	386.362	0.128795
$209$	−121.542	−0.0402261
$210$	0	0
$211$	−5991.08	−1.95470	−0.977352	−	0.211619i	$-0.932126\pi$
$211$	−5991.08	−1.95470	−0.977352	+	0.211619i	$0.932126\pi$
$212$	2601.08	0.842656
$213$	0	0
$214$	1608.65	0.513856
$215$	1023.63	0.324702
$216$	0	0
$217$	1226.32	0.383631
$218$	−1506.09	−0.467913
$219$	0	0
$220$	157.977	0.0484128
$221$	584.612	0.177942
$222$	0	0
$223$	−2148.98	−0.645320	−0.322660	−	0.946515i	$-0.604577\pi$
$223$	−2148.98	−0.645320	−0.322660	+	0.946515i	$0.604577\pi$
$224$	1255.04	0.374356
$225$	0	0
$226$	−2108.56	−0.620616
$227$	2067.49	0.604513	0.302256	−	0.953227i	$-0.402260\pi$
$227$	2067.49	0.604513	0.302256	+	0.953227i	$0.402260\pi$
$228$	0	0
$229$	−6394.72	−1.84530	−0.922652	−	0.385633i	$-0.873983\pi$
$229$	−6394.72	−1.84530	−0.922652	+	0.385633i	$0.873983\pi$
$230$	−482.067	−0.138203
$231$	0	0
$232$	1041.26	0.294665
$233$	5341.34	1.50181	0.750907	−	0.660408i	$-0.229617\pi$
$233$	5341.34	1.50181	0.750907	+	0.660408i	$0.229617\pi$
$234$	0	0
$235$	648.423	0.179993
$236$	106.104	0.0292661
$237$	0	0
$238$	391.393	0.106598
$239$	4325.48	1.17068	0.585339	−	0.810789i	$-0.300962\pi$
$239$	4325.48	1.17068	0.585339	+	0.810789i	$0.300962\pi$
$240$	0	0
$241$	6620.57	1.76958	0.884790	−	0.465991i	$-0.154302\pi$
$241$	6620.57	1.76958	0.884790	+	0.465991i	$0.154302\pi$
$242$	148.548	0.0394587
$243$	0	0
$244$	−3308.40	−0.868027
$245$	108.384	0.0282628
$246$	0	0
$247$	141.830	0.0365361
$248$	−3117.01	−0.798106
$249$	0	0
$250$	−665.586	−0.168381
$251$	−4621.48	−1.16217	−0.581086	−	0.813842i	$-0.697372\pi$
$251$	−4621.48	−1.16217	−0.581086	+	0.813842i	$0.697372\pi$
$252$	0	0
$253$	1952.78	0.485257
$254$	3393.61	0.838322
$255$	0	0
$256$	−1626.56	−0.397109
$257$	−6233.98	−1.51309	−0.756547	−	0.653939i	$-0.773115\pi$
$257$	−6233.98	−1.51309	−0.756547	+	0.653939i	$0.773115\pi$
$258$	0	0
$259$	1547.94	0.371368
$260$	−184.347	−0.0439719
$261$	0	0
$262$	958.949	0.226122
$263$	−1289.39	−0.302308	−0.151154	−	0.988510i	$-0.548299\pi$
$263$	−1289.39	−0.302308	−0.151154	+	0.988510i	$0.548299\pi$
$264$	0	0
$265$	−886.109	−0.205408
$266$	94.9538	0.0218872
$267$	0	0
$268$	−3139.84	−0.715658
$269$	−2310.16	−0.523616	−0.261808	−	0.965120i	$-0.584319\pi$
$269$	−2310.16	−0.523616	−0.261808	+	0.965120i	$0.584319\pi$
$270$	0	0
$271$	7584.99	1.70020	0.850102	−	0.526619i	$-0.176540\pi$
$271$	7584.99	1.70020	0.850102	+	0.526619i	$0.176540\pi$
$272$	1370.87	0.305592
$273$	0	0
$274$	1918.78	0.423058
$275$	1321.18	0.289710
$276$	0	0
$277$	−339.509	−0.0736431	−0.0368215	−	0.999322i	$-0.511723\pi$
$277$	−339.509	−0.0736431	−0.0368215	+	0.999322i	$0.511723\pi$
$278$	2096.34	0.452267
$279$	0	0
$280$	−275.486	−0.0587979
$281$	4829.29	1.02524	0.512618	−	0.858617i	$-0.328676\pi$
$281$	4829.29	1.02524	0.512618	+	0.858617i	$0.328676\pi$
$282$	0	0
$283$	−2858.99	−0.600527	−0.300264	−	0.953856i	$-0.597075\pi$
$283$	−2858.99	−0.600527	−0.300264	+	0.953856i	$0.597075\pi$
$284$	1317.29	0.275235
$285$	0	0
$286$	−173.343	−0.0358391
$287$	2153.94	0.443007
$288$	0	0
$289$	−2838.71	−0.577796
$290$	−158.918	−0.0321793
$291$	0	0
$292$	−5752.07	−1.15279
$293$	−8749.66	−1.74458	−0.872288	−	0.488993i	$-0.837364\pi$
$293$	−8749.66	−1.74458	−0.872288	+	0.488993i	$0.837364\pi$
$294$	0	0
$295$	−36.1465	−0.00713399
$296$	−3934.51	−0.772596
$297$	0	0
$298$	−2240.74	−0.435580
$299$	−2278.73	−0.440744
$300$	0	0
$301$	3239.46	0.620331
$302$	−1596.18	−0.304139
$303$	0	0
$304$	332.579	0.0627458
$305$	1127.07	0.211593
$306$	0	0
$307$	−3970.00	−0.738045	−0.369023	−	0.929420i	$-0.620308\pi$
$307$	−3970.00	−0.738045	−0.369023	+	0.929420i	$0.620308\pi$
$308$	499.948	0.0924909
$309$	0	0
$310$	475.721	0.0871585
$311$	−2412.36	−0.439847	−0.219923	−	0.975517i	$-0.570581\pi$
$311$	−2412.36	−0.439847	−0.219923	+	0.975517i	$0.570581\pi$
$312$	0	0
$313$	−6809.58	−1.22971	−0.614857	−	0.788639i	$-0.710786\pi$
$313$	−6809.58	−1.22971	−0.614857	+	0.788639i	$0.710786\pi$
$314$	2539.15	0.456345
$315$	0	0
$316$	−1879.84	−0.334650
$317$	10686.8	1.89347	0.946733	−	0.322021i	$-0.104362\pi$
$317$	10686.8	1.89347	0.946733	+	0.322021i	$0.104362\pi$
$318$	0	0
$319$	643.753	0.112988
$320$	−45.7587	−0.00799371
$321$	0	0
$322$	−1525.59	−0.264031
$323$	503.232	0.0866892
$324$	0	0
$325$	−1541.71	−0.263135
$326$	1796.90	0.305280
$327$	0	0
$328$	−5474.81	−0.921634
$329$	2052.05	0.343870
$330$	0	0
$331$	3924.27	0.651654	0.325827	−	0.945429i	$-0.394357\pi$
$331$	3924.27	0.651654	0.325827	+	0.945429i	$0.394357\pi$
$332$	691.989	0.114391
$333$	0	0
$334$	−4022.10	−0.658920
$335$	1069.65	0.174451
$336$	0	0
$337$	−4292.08	−0.693783	−0.346891	−	0.937905i	$-0.612763\pi$
$337$	−4292.08	−0.693783	−0.346891	+	0.937905i	$0.612763\pi$
$338$	−2494.90	−0.401494
$339$	0	0
$340$	−654.088	−0.104332
$341$	−1927.07	−0.306031
$342$	0	0
$343$	343.000	0.0539949
$344$	−8233.96	−1.29054
$345$	0	0
$346$	977.153	0.151827
$347$	1481.08	0.229131	0.114565	−	0.993416i	$-0.463452\pi$
$347$	1481.08	0.229131	0.114565	+	0.993416i	$0.463452\pi$
$348$	0	0
$349$	8449.08	1.29590	0.647949	−	0.761683i	$-0.275627\pi$
$349$	8449.08	1.29590	0.647949	+	0.761683i	$0.275627\pi$
$350$	−1032.16	−0.157633
$351$	0	0
$352$	−1972.20	−0.298633
$353$	4430.47	0.668017	0.334008	−	0.942570i	$-0.391599\pi$
$353$	4430.47	0.668017	0.334008	+	0.942570i	$0.391599\pi$
$354$	0	0
$355$	−448.760	−0.0670921
$356$	−10298.3	−1.53317
$357$	0	0
$358$	−2009.08	−0.296601
$359$	−1340.69	−0.197101	−0.0985503	−	0.995132i	$-0.531421\pi$
$359$	−1340.69	−0.197101	−0.0985503	+	0.995132i	$0.531421\pi$
$360$	0	0
$361$	−6736.91	−0.982201
$362$	4457.97	0.647254
$363$	0	0
$364$	−583.399	−0.0840067
$365$	1959.55	0.281007
$366$	0	0
$367$	−10543.8	−1.49967	−0.749837	−	0.661623i	$-0.769868\pi$
$367$	−10543.8	−1.49967	−0.749837	+	0.661623i	$0.769868\pi$
$368$	−5343.44	−0.756919
$369$	0	0
$370$	600.488	0.0843726
$371$	−2804.26	−0.392425
$372$	0	0
$373$	2702.46	0.375142	0.187571	−	0.982251i	$-0.439938\pi$
$373$	2702.46	0.375142	0.187571	+	0.982251i	$0.439938\pi$
$374$	−615.046	−0.0850354
$375$	0	0
$376$	−5215.84	−0.715389
$377$	−751.207	−0.102624
$378$	0	0
$379$	2784.94	0.377447	0.188724	−	0.982030i	$-0.439565\pi$
$379$	2784.94	0.377447	0.188724	+	0.982030i	$0.439565\pi$
$380$	−158.685	−0.0214220
$381$	0	0
$382$	−256.194	−0.0343142
$383$	9176.46	1.22427	0.612135	−	0.790753i	$-0.290311\pi$
$383$	9176.46	1.22427	0.612135	+	0.790753i	$0.290311\pi$
$384$	0	0
$385$	−170.317	−0.0225459
$386$	993.077	0.130949
$387$	0	0
$388$	6431.80	0.841560
$389$	3142.13	0.409544	0.204772	−	0.978810i	$-0.434355\pi$
$389$	3142.13	0.409544	0.204772	+	0.978810i	$0.434355\pi$
$390$	0	0
$391$	−8085.27	−1.04575
$392$	−871.826	−0.112331
$393$	0	0
$394$	−5186.67	−0.663200
$395$	640.405	0.0815753
$396$	0	0
$397$	−11192.0	−1.41489	−0.707444	−	0.706769i	$-0.750152\pi$
$397$	−11192.0	−1.41489	−0.707444	+	0.706769i	$0.750152\pi$
$398$	422.885	0.0532596
$399$	0	0
$400$	−3615.19	−0.451899
$401$	3429.07	0.427031	0.213515	−	0.976940i	$-0.431509\pi$
$401$	3429.07	0.427031	0.213515	+	0.976940i	$0.431509\pi$
$402$	0	0
$403$	2248.73	0.277959
$404$	−3223.05	−0.396912
$405$	0	0
$406$	−502.927	−0.0614774
$407$	−2432.48	−0.296250
$408$	0	0
$409$	12313.7	1.48869	0.744347	−	0.667793i	$-0.232761\pi$
$409$	12313.7	1.48869	0.744347	+	0.667793i	$0.232761\pi$
$410$	835.571	0.100649
$411$	0	0
$412$	−1867.35	−0.223295
$413$	−114.392	−0.0136292
$414$	0	0
$415$	−235.740	−0.0278843
$416$	2301.40	0.271239
$417$	0	0
$418$	−149.213	−0.0174599
$419$	11674.8	1.36122	0.680609	−	0.732647i	$-0.261715\pi$
$419$	11674.8	1.36122	0.680609	+	0.732647i	$0.261715\pi$
$420$	0	0
$421$	6294.54	0.728687	0.364344	−	0.931265i	$-0.381293\pi$
$421$	6294.54	0.728687	0.364344	+	0.931265i	$0.381293\pi$
$422$	−7355.04	−0.848430
$423$	0	0
$424$	7127.76	0.816403
$425$	−5470.22	−0.624340
$426$	0	0
$427$	3566.82	0.404240
$428$	−8507.78	−0.960838
$429$	0	0
$430$	1256.67	0.140935
$431$	5110.96	0.571198	0.285599	−	0.958349i	$-0.407807\pi$
$431$	5110.96	0.571198	0.285599	+	0.958349i	$0.407807\pi$
$432$	0	0
$433$	−6754.23	−0.749624	−0.374812	−	0.927101i	$-0.622293\pi$
$433$	−6754.23	−0.749624	−0.374812	+	0.927101i	$0.622293\pi$
$434$	1505.51	0.166513
$435$	0	0
$436$	7965.33	0.874932
$437$	−1961.53	−0.214720
$438$	0	0
$439$	−6529.02	−0.709825	−0.354912	−	0.934900i	$-0.615489\pi$
$439$	−6529.02	−0.709825	−0.354912	+	0.934900i	$0.615489\pi$
$440$	432.906	0.0469045
$441$	0	0
$442$	717.708	0.0772351
$443$	1529.54	0.164042	0.0820211	−	0.996631i	$-0.473863\pi$
$443$	1529.54	0.164042	0.0820211	+	0.996631i	$0.473863\pi$
$444$	0	0
$445$	3508.30	0.373729
$446$	−2638.23	−0.280098
$447$	0	0
$448$	−144.812	−0.0152717
$449$	−9778.96	−1.02783	−0.513917	−	0.857840i	$-0.671806\pi$
$449$	−9778.96	−1.02783	−0.513917	+	0.857840i	$0.671806\pi$
$450$	0	0
$451$	−3384.76	−0.353397
$452$	11151.7	1.16047
$453$	0	0
$454$	2538.19	0.262386
$455$	198.746	0.0204777
$456$	0	0
$457$	10614.0	1.08643	0.543217	−	0.839592i	$-0.317206\pi$
$457$	10614.0	1.08643	0.543217	+	0.839592i	$0.317206\pi$
$458$	−7850.58	−0.800946
$459$	0	0
$460$	2549.54	0.258420
$461$	−7536.27	−0.761386	−0.380693	−	0.924701i	$-0.624315\pi$
$461$	−7536.27	−0.761386	−0.380693	+	0.924701i	$0.624315\pi$
$462$	0	0
$463$	−3945.44	−0.396026	−0.198013	−	0.980199i	$-0.563449\pi$
$463$	−3945.44	−0.396026	−0.198013	+	0.980199i	$0.563449\pi$
$464$	−1761.52	−0.176242
$465$	0	0
$466$	6557.37	0.651855
$467$	−9456.18	−0.937002	−0.468501	−	0.883463i	$-0.655206\pi$
$467$	−9456.18	−0.937002	−0.468501	+	0.883463i	$0.655206\pi$
$468$	0	0
$469$	3385.10	0.333282
$470$	796.046	0.0781253
$471$	0	0
$472$	290.758	0.0283543
$473$	−5090.59	−0.494853
$474$	0	0
$475$	−1327.10	−0.128193
$476$	−2069.98	−0.199323
$477$	0	0
$478$	5310.24	0.508127
$479$	2775.63	0.264764	0.132382	−	0.991199i	$-0.457737\pi$
$479$	2775.63	0.264764	0.132382	+	0.991199i	$0.457737\pi$
$480$	0	0
$481$	2838.51	0.269074
$482$	8127.85	0.768078
$483$	0	0
$484$	−785.633	−0.0737822
$485$	−2191.12	−0.205141
$486$	0	0
$487$	20902.4	1.94492	0.972461	−	0.233066i	$-0.0748759\pi$
$487$	20902.4	1.94492	0.972461	+	0.233066i	$0.0748759\pi$
$488$	−9066.02	−0.840983
$489$	0	0
$490$	133.059	0.0122673
$491$	7832.44	0.719904	0.359952	−	0.932971i	$-0.382793\pi$
$491$	7832.44	0.719904	0.359952	+	0.932971i	$0.382793\pi$
$492$	0	0
$493$	−2665.39	−0.243495
$494$	174.120	0.0158583
$495$	0	0
$496$	5273.10	0.477357
$497$	−1420.18	−0.128177
$498$	0	0
$499$	−11585.6	−1.03936	−0.519681	−	0.854360i	$-0.673949\pi$
$499$	−11585.6	−1.03936	−0.519681	+	0.854360i	$0.673949\pi$
$500$	3520.13	0.314850
$501$	0	0
$502$	−5673.63	−0.504435
$503$	168.525	0.0149387	0.00746934	−	0.999972i	$-0.497622\pi$
$503$	168.525	0.0149387	0.00746934	+	0.999972i	$0.497622\pi$
$504$	0	0
$505$	1097.99	0.0967526
$506$	2397.36	0.210624
$507$	0	0
$508$	−17948.0	−1.56755
$509$	5404.59	0.470637	0.235319	−	0.971918i	$-0.424387\pi$
$509$	5404.59	0.470637	0.235319	+	0.971918i	$0.424387\pi$
$510$	0	0
$511$	6201.37	0.536854
$512$	9680.94	0.835628
$513$	0	0
$514$	−7653.25	−0.656752
$515$	636.149	0.0544312
$516$	0	0
$517$	−3224.66	−0.274314
$518$	1900.36	0.161191
$519$	0	0
$520$	−505.166	−0.0426019
$521$	−5372.28	−0.451754	−0.225877	−	0.974156i	$-0.572525\pi$
$521$	−5372.28	−0.451754	−0.225877	+	0.974156i	$0.572525\pi$
$522$	0	0
$523$	18418.3	1.53991	0.769957	−	0.638096i	$-0.220278\pi$
$523$	18418.3	1.53991	0.769957	+	0.638096i	$0.220278\pi$
$524$	−5071.66	−0.422818
$525$	0	0
$526$	−1582.94	−0.131215
$527$	7978.83	0.659513
$528$	0	0
$529$	19348.2	1.59022
$530$	−1087.85	−0.0891566
$531$	0	0
$532$	−502.188	−0.0409260
$533$	3949.74	0.320980
$534$	0	0
$535$	2898.34	0.234217
$536$	−8604.12	−0.693361
$537$	0	0
$538$	−2836.10	−0.227273
$539$	−539.000	−0.0430730
$540$	0	0
$541$	2953.93	0.234749	0.117375	−	0.993088i	$-0.462552\pi$
$541$	2953.93	0.234749	0.117375	+	0.993088i	$0.462552\pi$
$542$	9311.83	0.737965
$543$	0	0
$544$	8165.70	0.643569
$545$	−2713.54	−0.213276
$546$	0	0
$547$	13476.6	1.05341	0.526706	−	0.850048i	$-0.323427\pi$
$547$	13476.6	1.05341	0.526706	+	0.850048i	$0.323427\pi$
$548$	−10148.0	−0.791059
$549$	0	0
$550$	1621.97	0.125747
$551$	−646.637	−0.0499958
$552$	0	0
$553$	2026.68	0.155847
$554$	−416.804	−0.0319644
$555$	0	0
$556$	−11087.1	−0.845677
$557$	−8327.77	−0.633499	−0.316749	−	0.948509i	$-0.602591\pi$
$557$	−8327.77	−0.633499	−0.316749	+	0.948509i	$0.602591\pi$
$558$	0	0
$559$	5940.30	0.449460
$560$	466.044	0.0351677
$561$	0	0
$562$	5928.75	0.444999
$563$	−4910.23	−0.367569	−0.183785	−	0.982967i	$-0.558835\pi$
$563$	−4910.23	−0.367569	−0.183785	+	0.982967i	$0.558835\pi$
$564$	0	0
$565$	−3799.03	−0.282879
$566$	−3509.88	−0.260656
$567$	0	0
$568$	3609.77	0.266660
$569$	9451.76	0.696377	0.348188	−	0.937425i	$-0.386797\pi$
$569$	9451.76	0.696377	0.348188	+	0.937425i	$0.386797\pi$
$570$	0	0
$571$	−9705.19	−0.711295	−0.355648	−	0.934620i	$-0.615740\pi$
$571$	−9705.19	−0.711295	−0.355648	+	0.934620i	$0.615740\pi$
$572$	916.770	0.0670141
$573$	0	0
$574$	2644.32	0.192285
$575$	21322.1	1.54642
$576$	0	0
$577$	2933.73	0.211669	0.105834	−	0.994384i	$-0.466249\pi$
$577$	2933.73	0.211669	0.105834	+	0.994384i	$0.466249\pi$
$578$	−3484.99	−0.250790
$579$	0	0
$580$	840.482	0.0601709
$581$	−746.041	−0.0532720
$582$	0	0
$583$	4406.69	0.313047
$584$	−15762.4	−1.11687
$585$	0	0
$586$	−10741.7	−0.757225
$587$	23344.2	1.64143	0.820714	−	0.571339i	$-0.193576\pi$
$587$	23344.2	1.64143	0.820714	+	0.571339i	$0.193576\pi$
$588$	0	0
$589$	1935.70	0.135415
$590$	−44.3758	−0.00309648
$591$	0	0
$592$	6656.07	0.462099
$593$	−26704.7	−1.84929	−0.924645	−	0.380830i	$-0.875638\pi$
$593$	−26704.7	−1.84929	−0.924645	+	0.380830i	$0.875638\pi$
$594$	0	0
$595$	705.180	0.0485875
$596$	11850.8	0.814473
$597$	0	0
$598$	−2797.52	−0.191303
$599$	1767.91	0.120593	0.0602963	−	0.998181i	$-0.480795\pi$
$599$	1767.91	0.120593	0.0602963	+	0.998181i	$0.480795\pi$
$600$	0	0
$601$	−3390.95	−0.230149	−0.115075	−	0.993357i	$-0.536711\pi$
$601$	−3390.95	−0.230149	−0.115075	+	0.993357i	$0.536711\pi$
$602$	3976.98	0.269252
$603$	0	0
$604$	8441.83	0.568697
$605$	267.641	0.0179854
$606$	0	0
$607$	−12925.9	−0.864323	−0.432162	−	0.901796i	$-0.642249\pi$
$607$	−12925.9	−0.864323	−0.432162	+	0.901796i	$0.642249\pi$
$608$	1981.04	0.132141
$609$	0	0
$610$	1383.66	0.0918409
$611$	3762.91	0.249151
$612$	0	0
$613$	12339.8	0.813049	0.406525	−	0.913640i	$-0.366741\pi$
$613$	12339.8	0.813049	0.406525	+	0.913640i	$0.366741\pi$
$614$	−4873.83	−0.320345
$615$	0	0
$616$	1370.01	0.0896093
$617$	6235.47	0.406857	0.203428	−	0.979090i	$-0.434792\pi$
$617$	6235.47	0.406857	0.203428	+	0.979090i	$0.434792\pi$
$618$	0	0
$619$	9830.39	0.638314	0.319157	−	0.947702i	$-0.396600\pi$
$619$	9830.39	0.638314	0.319157	+	0.947702i	$0.396600\pi$
$620$	−2515.98	−0.162974
$621$	0	0
$622$	−2961.57	−0.190913
$623$	11102.7	0.713996
$624$	0	0
$625$	13814.2	0.884111
$626$	−8359.89	−0.533752
$627$	0	0
$628$	−13428.9	−0.853302
$629$	10071.4	0.638433
$630$	0	0
$631$	−13040.9	−0.822742	−0.411371	−	0.911468i	$-0.634950\pi$
$631$	−13040.9	−0.822742	−0.411371	+	0.911468i	$0.634950\pi$
$632$	−5151.34	−0.324224
$633$	0	0
$634$	13119.8	0.821850
$635$	6114.33	0.382110
$636$	0	0
$637$	628.969	0.0391219
$638$	790.313	0.0490420
$639$	0	0
$640$	−3228.78	−0.199420
$641$	8274.75	0.509880	0.254940	−	0.966957i	$-0.417944\pi$
$641$	8274.75	0.509880	0.254940	+	0.966957i	$0.417944\pi$
$642$	0	0
$643$	−2407.55	−0.147659	−0.0738294	−	0.997271i	$-0.523522\pi$
$643$	−2407.55	−0.147659	−0.0738294	+	0.997271i	$0.523522\pi$
$644$	8068.50	0.493701
$645$	0	0
$646$	617.801	0.0376270
$647$	24597.2	1.49462	0.747309	−	0.664477i	$-0.231346\pi$
$647$	24597.2	1.49462	0.747309	+	0.664477i	$0.231346\pi$
$648$	0	0
$649$	179.759	0.0108724
$650$	−1892.71	−0.114212
$651$	0	0
$652$	−9503.39	−0.570831
$653$	−6681.60	−0.400415	−0.200208	−	0.979753i	$-0.564162\pi$
$653$	−6681.60	−0.400415	−0.200208	+	0.979753i	$0.564162\pi$
$654$	0	0
$655$	1727.76	0.103067
$656$	9261.83	0.551240
$657$	0	0
$658$	2519.24	0.149255
$659$	8185.27	0.483844	0.241922	−	0.970296i	$-0.422222\pi$
$659$	8185.27	0.483844	0.241922	+	0.970296i	$0.422222\pi$
$660$	0	0
$661$	−5645.12	−0.332178	−0.166089	−	0.986111i	$-0.553114\pi$
$661$	−5645.12	−0.332178	−0.166089	+	0.986111i	$0.553114\pi$
$662$	4817.70	0.282848
$663$	0	0
$664$	1896.26	0.110827
$665$	171.080	0.00997625
$666$	0	0
$667$	10389.3	0.603112
$668$	21271.9	1.23209
$669$	0	0
$670$	1313.17	0.0757196
$671$	−5605.00	−0.322472
$672$	0	0
$673$	−5422.21	−0.310566	−0.155283	−	0.987870i	$-0.549629\pi$
$673$	−5422.21	−0.310566	−0.155283	+	0.987870i	$0.549629\pi$
$674$	−5269.24	−0.301133
$675$	0	0
$676$	13195.0	0.750738
$677$	21717.0	1.23287	0.616435	−	0.787406i	$-0.288576\pi$
$677$	21717.0	1.23287	0.616435	+	0.787406i	$0.288576\pi$
$678$	0	0
$679$	−6934.19	−0.391915
$680$	−1792.40	−0.101082
$681$	0	0
$682$	−2365.80	−0.132831
$683$	−5011.91	−0.280784	−0.140392	−	0.990096i	$-0.544836\pi$
$683$	−5011.91	−0.280784	−0.140392	+	0.990096i	$0.544836\pi$
$684$	0	0
$685$	3457.11	0.192831
$686$	421.089	0.0234362
$687$	0	0
$688$	13929.5	0.771887
$689$	−5142.25	−0.284331
$690$	0	0
$691$	8199.88	0.451430	0.225715	−	0.974193i	$-0.427528\pi$
$691$	8199.88	0.451430	0.225715	+	0.974193i	$0.427528\pi$
$692$	−5167.93	−0.283895
$693$	0	0
$694$	1818.27	0.0994532
$695$	3777.02	0.206145
$696$	0	0
$697$	14014.3	0.761589
$698$	10372.6	0.562479
$699$	0	0
$700$	5458.87	0.294751
$701$	−33113.7	−1.78415	−0.892073	−	0.451891i	$-0.850749\pi$
$701$	−33113.7	−1.78415	−0.892073	+	0.451891i	$0.850749\pi$
$702$	0	0
$703$	2443.38	0.131086
$704$	227.561	0.0121826
$705$	0	0
$706$	5439.13	0.289950
$707$	3474.80	0.184842
$708$	0	0
$709$	−8338.06	−0.441668	−0.220834	−	0.975311i	$-0.570878\pi$
$709$	−8338.06	−0.441668	−0.220834	+	0.975311i	$0.570878\pi$
$710$	−550.927	−0.0291210
$711$	0	0
$712$	−28220.4	−1.48540
$713$	−31100.3	−1.63354
$714$	0	0
$715$	−312.315	−0.0163356
$716$	10625.5	0.554602
$717$	0	0
$718$	−1645.92	−0.0855506
$719$	−13316.7	−0.690724	−0.345362	−	0.938470i	$-0.612244\pi$
$719$	−13316.7	−0.690724	−0.345362	+	0.938470i	$0.612244\pi$
$720$	0	0
$721$	2013.21	0.103989
$722$	−8270.68	−0.426320
$723$	0	0
$724$	−23577.2	−1.21027
$725$	7029.05	0.360072
$726$	0	0
$727$	−14439.2	−0.736616	−0.368308	−	0.929704i	$-0.620063\pi$
$727$	−14439.2	−0.736616	−0.368308	+	0.929704i	$0.620063\pi$
$728$	−1598.69	−0.0813894
$729$	0	0
$730$	2405.68	0.121970
$731$	21077.0	1.06643
$732$	0	0
$733$	−26816.8	−1.35130	−0.675650	−	0.737223i	$-0.736137\pi$
$733$	−26816.8	−1.35130	−0.675650	+	0.737223i	$0.736137\pi$
$734$	−12944.2	−0.650926
$735$	0	0
$736$	−31828.7	−1.59405
$737$	−5319.44	−0.265867
$738$	0	0
$739$	15313.3	0.762257	0.381129	−	0.924522i	$-0.375536\pi$
$739$	15313.3	0.762257	0.381129	+	0.924522i	$0.375536\pi$
$740$	−3175.84	−0.157765
$741$	0	0
$742$	−3442.69	−0.170330
$743$	−4586.65	−0.226471	−0.113235	−	0.993568i	$-0.536121\pi$
$743$	−4586.65	−0.226471	−0.113235	+	0.993568i	$0.536121\pi$
$744$	0	0
$745$	−4037.19	−0.198539
$746$	3317.72	0.162829
$747$	0	0
$748$	3252.83	0.159004
$749$	9172.33	0.447463
$750$	0	0
$751$	20980.9	1.01945	0.509724	−	0.860338i	$-0.329748\pi$
$751$	20980.9	1.01945	0.509724	+	0.860338i	$0.329748\pi$
$752$	8823.72	0.427883
$753$	0	0
$754$	−922.231	−0.0445434
$755$	−2875.87	−0.138627
$756$	0	0
$757$	2320.42	0.111410	0.0557049	−	0.998447i	$-0.482259\pi$
$757$	2320.42	0.111410	0.0557049	+	0.998447i	$0.482259\pi$
$758$	3418.97	0.163829
$759$	0	0
$760$	−434.846	−0.0207546
$761$	−15977.5	−0.761084	−0.380542	−	0.924764i	$-0.624263\pi$
$761$	−15977.5	−0.761084	−0.380542	+	0.924764i	$0.624263\pi$
$762$	0	0
$763$	−8587.52	−0.407456
$764$	1354.95	0.0641629
$765$	0	0
$766$	11265.6	0.531389
$767$	−209.764	−0.00987503
$768$	0	0
$769$	−5354.46	−0.251088	−0.125544	−	0.992088i	$-0.540068\pi$
$769$	−5354.46	−0.251088	−0.125544	+	0.992088i	$0.540068\pi$
$770$	−209.092	−0.00978593
$771$	0	0
$772$	−5252.15	−0.244856
$773$	36773.3	1.71105	0.855527	−	0.517758i	$-0.173233\pi$
$773$	36773.3	1.71105	0.855527	+	0.517758i	$0.173233\pi$
$774$	0	0
$775$	−21041.4	−0.975264
$776$	17625.1	0.815340
$777$	0	0
$778$	3857.49	0.177761
$779$	3399.93	0.156374
$780$	0	0
$781$	2231.72	0.102250
$782$	−9926.01	−0.453905
$783$	0	0
$784$	1474.88	0.0671867
$785$	4574.83	0.208003
$786$	0	0
$787$	38254.9	1.73271	0.866353	−	0.499432i	$-0.166458\pi$
$787$	38254.9	1.73271	0.866353	+	0.499432i	$0.166458\pi$
$788$	27431.1	1.24009
$789$	0	0
$790$	786.203	0.0354074
$791$	−12022.7	−0.540429
$792$	0	0
$793$	6540.58	0.292892
$794$	−13740.0	−0.614126
$795$	0	0
$796$	−2236.54	−0.0995881
$797$	26969.6	1.19864	0.599318	−	0.800511i	$-0.295438\pi$
$797$	26969.6	1.19864	0.599318	+	0.800511i	$0.295438\pi$
$798$	0	0
$799$	13351.4	0.591160
$800$	−21534.2	−0.951686
$801$	0	0
$802$	4209.75	0.185351
$803$	−9745.01	−0.428262
$804$	0	0
$805$	−2748.69	−0.120346
$806$	2760.69	0.120647
$807$	0	0
$808$	−8832.14	−0.384546
$809$	−7996.42	−0.347514	−0.173757	−	0.984789i	$-0.555591\pi$
$809$	−7996.42	−0.347514	−0.173757	+	0.984789i	$0.555591\pi$
$810$	0	0
$811$	2150.99	0.0931337	0.0465669	−	0.998915i	$-0.485172\pi$
$811$	2150.99	0.0931337	0.0465669	+	0.998915i	$0.485172\pi$
$812$	2659.86	0.114954
$813$	0	0
$814$	−2986.27	−0.128586
$815$	3237.51	0.139147
$816$	0	0
$817$	5113.39	0.218966
$818$	15117.2	0.646160
$819$	0	0
$820$	−4419.14	−0.188199
$821$	27882.7	1.18528	0.592638	−	0.805469i	$-0.298086\pi$
$821$	27882.7	1.18528	0.592638	+	0.805469i	$0.298086\pi$
$822$	0	0
$823$	18462.7	0.781978	0.390989	−	0.920395i	$-0.372133\pi$
$823$	18462.7	0.781978	0.390989	+	0.920395i	$0.372133\pi$
$824$	−5117.11	−0.216339
$825$	0	0
$826$	−140.435	−0.00591570
$827$	−28769.2	−1.20968	−0.604838	−	0.796348i	$-0.706762\pi$
$827$	−28769.2	−1.20968	−0.604838	+	0.796348i	$0.706762\pi$
$828$	0	0
$829$	31112.2	1.30346	0.651732	−	0.758449i	$-0.274043\pi$
$829$	31112.2	1.30346	0.651732	+	0.758449i	$0.274043\pi$
$830$	−289.409	−0.0121031
$831$	0	0
$832$	−265.546	−0.0110651
$833$	2231.67	0.0928246
$834$	0	0
$835$	−7246.69	−0.300338
$836$	789.153	0.0326476
$837$	0	0
$838$	14332.7	0.590830
$839$	−13494.6	−0.555287	−0.277643	−	0.960684i	$-0.589553\pi$
$839$	−13494.6	−0.555287	−0.277643	+	0.960684i	$0.589553\pi$
$840$	0	0
$841$	−20964.1	−0.859570
$842$	7727.60	0.316283
$843$	0	0
$844$	38899.1	1.58645
$845$	−4495.12	−0.183002
$846$	0	0
$847$	847.000	0.0343604
$848$	−12058.2	−0.488300
$849$	0	0
$850$	−6715.60	−0.270992
$851$	−39257.0	−1.58133
$852$	0	0
$853$	850.969	0.0341578	0.0170789	−	0.999854i	$-0.494563\pi$
$853$	850.969	0.0341578	0.0170789	+	0.999854i	$0.494563\pi$
$854$	4378.86	0.175459
$855$	0	0
$856$	−23313.9	−0.930903
$857$	−47221.3	−1.88220	−0.941102	−	0.338122i	$-0.890208\pi$
$857$	−47221.3	−1.88220	−0.941102	+	0.338122i	$0.890208\pi$
$858$	0	0
$859$	32876.5	1.30586	0.652929	−	0.757419i	$-0.273540\pi$
$859$	32876.5	1.30586	0.652929	+	0.757419i	$0.273540\pi$
$860$	−6646.26	−0.263530
$861$	0	0
$862$	6274.55	0.247926
$863$	33117.4	1.30629	0.653145	−	0.757232i	$-0.273449\pi$
$863$	33117.4	1.30629	0.653145	+	0.757232i	$0.273449\pi$
$864$	0	0
$865$	1760.56	0.0692031
$866$	−8291.93	−0.325371
$867$	0	0
$868$	−7962.28	−0.311356
$869$	−3184.78	−0.124323
$870$	0	0
$871$	6207.35	0.241479
$872$	21827.5	0.847673
$873$	0	0
$874$	−2408.10	−0.0931981
$875$	−3795.09	−0.146626
$876$	0	0
$877$	−16817.3	−0.647526	−0.323763	−	0.946138i	$-0.604948\pi$
$877$	−16817.3	−0.647526	−0.323763	+	0.946138i	$0.604948\pi$
$878$	−8015.45	−0.308096
$879$	0	0
$880$	−732.354	−0.0280542
$881$	23372.2	0.893792	0.446896	−	0.894586i	$-0.352529\pi$
$881$	23372.2	0.893792	0.446896	+	0.894586i	$0.352529\pi$
$882$	0	0
$883$	−46228.1	−1.76183	−0.880916	−	0.473272i	$-0.843073\pi$
$883$	−46228.1	−1.76183	−0.880916	+	0.473272i	$0.843073\pi$
$884$	−3795.79	−0.144419
$885$	0	0
$886$	1877.76	0.0712017
$887$	−36945.6	−1.39855	−0.699275	−	0.714853i	$-0.746494\pi$
$887$	−36945.6	−1.39855	−0.699275	+	0.714853i	$0.746494\pi$
$888$	0	0
$889$	19349.9	0.730006
$890$	4307.02	0.162215
$891$	0	0
$892$	13953.0	0.523744
$893$	3239.10	0.121380
$894$	0	0
$895$	−3619.79	−0.135191
$896$	−10218.1	−0.380985
$897$	0	0
$898$	−12005.3	−0.446127
$899$	−10252.5	−0.380357
$900$	0	0
$901$	−18245.4	−0.674632
$902$	−4155.36	−0.153391
$903$	0	0
$904$	30559.0	1.12431
$905$	8032.02	0.295020
$906$	0	0
$907$	16331.0	0.597864	0.298932	−	0.954274i	$-0.403370\pi$
$907$	16331.0	0.597864	0.298932	+	0.954274i	$0.403370\pi$
$908$	−13423.9	−0.490625
$909$	0	0
$910$	243.994	0.00888826
$911$	20518.3	0.746213	0.373107	−	0.927788i	$-0.378293\pi$
$911$	20518.3	0.746213	0.373107	+	0.927788i	$0.378293\pi$
$912$	0	0
$913$	1172.35	0.0424963
$914$	13030.4	0.471562
$915$	0	0
$916$	41519.9	1.49766
$917$	5467.81	0.196906
$918$	0	0
$919$	47280.4	1.69710	0.848550	−	0.529115i	$-0.177476\pi$
$919$	47280.4	1.69710	0.848550	+	0.529115i	$0.177476\pi$
$920$	6986.52	0.250368
$921$	0	0
$922$	−9252.02	−0.330476
$923$	−2604.23	−0.0928704
$924$	0	0
$925$	−26559.9	−0.944091
$926$	−4843.68	−0.171893
$927$	0	0
$928$	−10492.7	−0.371162
$929$	35214.0	1.24363	0.621816	−	0.783163i	$-0.286395\pi$
$929$	35214.0	1.24363	0.621816	+	0.783163i	$0.286395\pi$
$930$	0	0
$931$	541.415	0.0190592
$932$	−34680.4	−1.21888
$933$	0	0
$934$	−11609.0	−0.406701
$935$	−1108.14	−0.0387594
$936$	0	0
$937$	−53330.3	−1.85937	−0.929683	−	0.368360i	$-0.879919\pi$
$937$	−53330.3	−1.85937	−0.929683	+	0.368360i	$0.879919\pi$
$938$	4155.77	0.144659
$939$	0	0
$940$	−4210.10	−0.146083
$941$	20304.7	0.703416	0.351708	−	0.936110i	$-0.385601\pi$
$941$	20304.7	0.703416	0.351708	+	0.936110i	$0.385601\pi$
$942$	0	0
$943$	−54625.5	−1.88638
$944$	−491.880	−0.0169590
$945$	0	0
$946$	−6249.54	−0.214789
$947$	20902.4	0.717252	0.358626	−	0.933481i	$-0.383245\pi$
$947$	20902.4	0.717252	0.358626	+	0.933481i	$0.383245\pi$
$948$	0	0
$949$	11371.6	0.388977
$950$	−1629.24	−0.0556415
$951$	0	0
$952$	−5672.39	−0.193113
$953$	−44662.5	−1.51811	−0.759055	−	0.651027i	$-0.774339\pi$
$953$	−44662.5	−1.51811	−0.759055	+	0.651027i	$0.774339\pi$
$954$	0	0
$955$	−461.590	−0.0156405
$956$	−28084.6	−0.950127
$957$	0	0
$958$	3407.55	0.114920
$959$	10940.7	0.368396
$960$	0	0
$961$	899.884	0.0302066
$962$	3484.74	0.116790
$963$	0	0
$964$	−42986.3	−1.43620
$965$	1789.25	0.0596869
$966$	0	0
$967$	−40483.6	−1.34629	−0.673146	−	0.739509i	$-0.735058\pi$
$967$	−40483.6	−1.34629	−0.673146	+	0.739509i	$0.735058\pi$
$968$	−2152.88	−0.0714835
$969$	0	0
$970$	−2689.96	−0.0890406
$971$	−55314.7	−1.82815	−0.914075	−	0.405544i	$-0.867082\pi$
$971$	−55314.7	−1.82815	−0.914075	+	0.405544i	$0.867082\pi$
$972$	0	0
$973$	11953.1	0.393832
$974$	25661.1	0.844184
$975$	0	0
$976$	15337.1	0.503002
$977$	−37534.5	−1.22910	−0.614552	−	0.788876i	$-0.710663\pi$
$977$	−37534.5	−1.22910	−0.614552	+	0.788876i	$0.710663\pi$
$978$	0	0
$979$	−17447.1	−0.569572
$980$	−703.717	−0.0229382
$981$	0	0
$982$	9615.61	0.312471
$983$	−985.287	−0.0319692	−0.0159846	−	0.999872i	$-0.505088\pi$
$983$	−985.287	−0.0319692	−0.0159846	+	0.999872i	$0.505088\pi$
$984$	0	0
$985$	−9344.93	−0.302289
$986$	−3272.21	−0.105688
$987$	0	0
$988$	−920.878	−0.0296529
$989$	−82155.2	−2.64144
$990$	0	0
$991$	−9167.83	−0.293870	−0.146935	−	0.989146i	$-0.546941\pi$
$991$	−9167.83	−0.293870	−0.146935	+	0.989146i	$0.546941\pi$
$992$	31409.7	1.00530
$993$	0	0
$994$	−1743.51	−0.0556346
$995$	761.921	0.0242759
$996$	0	0
$997$	−21433.7	−0.680854	−0.340427	−	0.940271i	$-0.610572\pi$
$997$	−21433.7	−0.680854	−0.340427	+	0.940271i	$0.610572\pi$
$998$	−14223.2	−0.451131
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.o.1.3		5
3.2	odd	2		77.4.a.e.1.3	✓	5
12.11	even	2		1232.4.a.y.1.5		5
15.14	odd	2		1925.4.a.r.1.3		5
21.20	even	2		539.4.a.h.1.3		5
33.32	even	2		847.4.a.f.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.e.1.3	✓	5	3.2	odd	2
539.4.a.h.1.3		5	21.20	even	2
693.4.a.o.1.3		5	1.1	even	1	trivial
847.4.a.f.1.3		5	33.32	even	2
1232.4.a.y.1.5		5	12.11	even	2
1925.4.a.r.1.3		5	15.14	odd	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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q+1.22767 q^{2} -6.49284 q^{4} +2.21191 q^{5} +7.00000 q^{7} -17.7924 q^{8} +O(q^{10})\)	\(q+1.22767 q^{2} -6.49284 q^{4} +2.21191 q^{5} +7.00000 q^{7} -17.7924 q^{8} +2.71549 q^{10} -11.0000 q^{11} +12.8361 q^{13} +8.59366 q^{14} +30.0996 q^{16} +45.5444 q^{17} +11.0493 q^{19} -14.3616 q^{20} -13.5043 q^{22} -177.525 q^{23} -120.107 q^{25} +15.7584 q^{26} -45.4499 q^{28} -58.5230 q^{29} +175.188 q^{31} +179.291 q^{32} +55.9132 q^{34} +15.4834 q^{35} +221.135 q^{37} +13.5648 q^{38} -39.3551 q^{40} +307.706 q^{41} +462.781 q^{43} +71.4212 q^{44} -217.942 q^{46} +293.151 q^{47} +49.0000 q^{49} -147.452 q^{50} -83.3427 q^{52} -400.608 q^{53} -24.3310 q^{55} -124.547 q^{56} -71.8467 q^{58} -16.3417 q^{59} +509.546 q^{61} +215.072 q^{62} -20.6874 q^{64} +28.3923 q^{65} +483.585 q^{67} -295.712 q^{68} +19.0084 q^{70} -202.883 q^{71} +885.910 q^{73} +271.479 q^{74} -71.7412 q^{76} -77.0000 q^{77} +289.526 q^{79} +66.5777 q^{80} +377.760 q^{82} -106.577 q^{83} +100.740 q^{85} +568.140 q^{86} +195.716 q^{88} +1586.10 q^{89} +89.8527 q^{91} +1152.64 q^{92} +359.891 q^{94} +24.4400 q^{95} -990.599 q^{97} +60.1556 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + 45 q^{4} + 24 q^{5} + 35 q^{7} - 57 q^{8}+O(q^{10}) \) 5 * q - q^2 + 45 * q^4 + 24 * q^5 + 35 * q^7 - 57 * q^8	\( 5 q - q^{2} + 45 q^{4} + 24 q^{5} + 35 q^{7} - 57 q^{8} - 10 q^{10} - 55 q^{11} - 50 q^{13} - 7 q^{14} + 433 q^{16} - 222 q^{17} + 160 q^{19} + 430 q^{20} + 11 q^{22} - 54 q^{23} + 125 q^{25} + 1026 q^{26} + 315 q^{28} - 14 q^{29} - 34 q^{31} + 583 q^{32} - 750 q^{34} + 168 q^{35} + 1044 q^{37} + 156 q^{38} + 1158 q^{40} + 114 q^{41} + 672 q^{43} - 495 q^{44} - 1224 q^{46} + 292 q^{47} + 245 q^{49} - 1143 q^{50} - 914 q^{52} + 710 q^{53} - 264 q^{55} - 399 q^{56} - 810 q^{58} - 270 q^{59} + 138 q^{61} + 480 q^{62} + 3001 q^{64} + 196 q^{65} + 1942 q^{67} - 3130 q^{68} - 70 q^{70} + 278 q^{71} - 338 q^{73} - 462 q^{74} - 1332 q^{76} - 385 q^{77} + 576 q^{79} + 1602 q^{80} + 1386 q^{82} - 1644 q^{83} + 360 q^{85} - 2020 q^{86} + 627 q^{88} + 3656 q^{89} - 350 q^{91} + 748 q^{92} + 976 q^{94} + 1276 q^{95} + 692 q^{97} - 49 q^{98}+O(q^{100}) \) 5 * q - q^2 + 45 * q^4 + 24 * q^5 + 35 * q^7 - 57 * q^8 - 10 * q^10 - 55 * q^11 - 50 * q^13 - 7 * q^14 + 433 * q^16 - 222 * q^17 + 160 * q^19 + 430 * q^20 + 11 * q^22 - 54 * q^23 + 125 * q^25 + 1026 * q^26 + 315 * q^28 - 14 * q^29 - 34 * q^31 + 583 * q^32 - 750 * q^34 + 168 * q^35 + 1044 * q^37 + 156 * q^38 + 1158 * q^40 + 114 * q^41 + 672 * q^43 - 495 * q^44 - 1224 * q^46 + 292 * q^47 + 245 * q^49 - 1143 * q^50 - 914 * q^52 + 710 * q^53 - 264 * q^55 - 399 * q^56 - 810 * q^58 - 270 * q^59 + 138 * q^61 + 480 * q^62 + 3001 * q^64 + 196 * q^65 + 1942 * q^67 - 3130 * q^68 - 70 * q^70 + 278 * q^71 - 338 * q^73 - 462 * q^74 - 1332 * q^76 - 385 * q^77 + 576 * q^79 + 1602 * q^80 + 1386 * q^82 - 1644 * q^83 + 360 * q^85 - 2020 * q^86 + 627 * q^88 + 3656 * q^89 - 350 * q^91 + 748 * q^92 + 976 * q^94 + 1276 * q^95 + 692 * q^97 - 49 * q^98

Introduction

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