LMFDB - Embedded newform 693.4.a.h.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [693,4,Mod(1,693)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [2,-3,0,7,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$40.8883236340$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{37}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 9 $ x^2 - x - 9
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.54138$ 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)

$f(q)$	$=$	\(q+1.54138 q^{2} -5.62414 q^{4} +11.1655 q^{5} +7.00000 q^{7} -21.0000 q^{8} +17.2103 q^{10} +11.0000 q^{11} -79.1655 q^{13} +10.7897 q^{14} +12.6241 q^{16} -53.9104 q^{17} +130.745 q^{19} -62.7965 q^{20} +16.9552 q^{22} -58.5862 q^{23} -0.331050 q^{25} -122.024 q^{26} -39.3690 q^{28} -277.483 q^{29} +154.752 q^{31} +187.459 q^{32} -83.0965 q^{34} +78.1587 q^{35} -297.648 q^{37} +201.528 q^{38} -234.476 q^{40} -304.483 q^{41} +455.552 q^{43} -61.8656 q^{44} -90.3037 q^{46} -502.579 q^{47} +49.0000 q^{49} -0.510274 q^{50} +445.238 q^{52} +135.096 q^{53} +122.821 q^{55} -147.000 q^{56} -427.707 q^{58} +53.9309 q^{59} -263.297 q^{61} +238.531 q^{62} +187.952 q^{64} -883.925 q^{65} +123.572 q^{67} +303.200 q^{68} +120.472 q^{70} -139.793 q^{71} -1157.77 q^{73} -458.790 q^{74} -735.328 q^{76} +77.0000 q^{77} -62.4966 q^{79} +140.955 q^{80} -469.324 q^{82} -756.352 q^{83} -601.938 q^{85} +702.179 q^{86} -231.000 q^{88} -135.558 q^{89} -554.159 q^{91} +329.497 q^{92} -774.666 q^{94} +1459.84 q^{95} -571.248 q^{97} +75.5277 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} + 7 q^{4} - 2 q^{5} + 14 q^{7} - 42 q^{8} + 77 q^{10} + 22 q^{11} - 134 q^{13} - 21 q^{14} + 7 q^{16} + 26 q^{17} + 152 q^{19} - 229 q^{20} - 33 q^{22} - 178 q^{23} + 48 q^{25} + 127 q^{26}+ \cdots - 147 q^{98}+O(q^{100}) $ 2 * q - 3 * q^2 + 7 * q^4 - 2 * q^5 + 14 * q^7 - 42 * q^8 + 77 * q^10 + 22 * q^11 - 134 * q^13 - 21 * q^14 + 7 * q^16 + 26 * q^17 + 152 * q^19 - 229 * q^20 - 33 * q^22 - 178 * q^23 + 48 * q^25 + 127 * q^26 + 49 * q^28 - 190 * q^29 + 346 * q^31 + 381 * q^32 - 446 * q^34 - 14 * q^35 - 206 * q^37 + 105 * q^38 + 42 * q^40 - 244 * q^41 + 242 * q^43 + 77 * q^44 + 452 * q^46 - 920 * q^47 + 98 * q^49 - 220 * q^50 - 247 * q^52 + 550 * q^53 - 22 * q^55 - 294 * q^56 - 825 * q^58 + 412 * q^59 + 252 * q^61 - 630 * q^62 - 646 * q^64 - 162 * q^65 + 16 * q^67 + 1312 * q^68 + 539 * q^70 + 572 * q^71 - 1026 * q^73 - 875 * q^74 - 467 * q^76 + 154 * q^77 - 52 * q^79 + 215 * q^80 - 744 * q^82 - 138 * q^83 - 1654 * q^85 + 1672 * q^86 - 462 * q^88 - 1512 * q^89 - 938 * q^91 - 1178 * q^92 + 1121 * q^94 + 1180 * q^95 - 1106 * q^97 - 147 * q^98

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 $

$2$	1.54138	0.544961	0.272480	−	0.962161i	$-0.412156\pi$
$2$	1.54138	0.544961	0.272480	+	0.962161i	$0.412156\pi$
$3$	0	0
$3$	0	0
$4$	−5.62414	−0.703018
$5$	11.1655	0.998675	0.499337	−	0.866408i	$-0.333577\pi$
$5$	11.1655	0.998675	0.499337	+	0.866408i	$0.333577\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−21.0000	−0.928078
$9$	0	0
$10$	17.2103	0.544238
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−79.1655	−1.68897	−0.844483	−	0.535582i	$-0.820092\pi$
$13$	−79.1655	−1.68897	−0.844483	+	0.535582i	$0.820092\pi$
$14$	10.7897	0.205976
$15$	0	0
$16$	12.6241	0.197252
$17$	−53.9104	−0.769129	−0.384564	−	0.923098i	$-0.625648\pi$
$17$	−53.9104	−0.769129	−0.384564	+	0.923098i	$0.625648\pi$
$18$	0	0
$19$	130.745	1.57868	0.789340	−	0.613956i	$-0.210423\pi$
$19$	130.745	1.57868	0.789340	+	0.613956i	$0.210423\pi$
$20$	−62.7965	−0.702086
$21$	0	0
$22$	16.9552	0.164312
$23$	−58.5862	−0.531133	−0.265567	−	0.964093i	$-0.585559\pi$
$23$	−58.5862	−0.531133	−0.265567	+	0.964093i	$0.585559\pi$
$24$	0	0
$25$	−0.331050	−0.00264840
$26$	−122.024	−0.920420
$27$	0	0
$28$	−39.3690	−0.265716
$29$	−277.483	−1.77680	−0.888401	−	0.459068i	$-0.848184\pi$
$29$	−277.483	−1.77680	−0.888401	+	0.459068i	$0.848184\pi$
$30$	0	0
$31$	154.752	0.896588	0.448294	−	0.893886i	$-0.352032\pi$
$31$	154.752	0.896588	0.448294	+	0.893886i	$0.352032\pi$
$32$	187.459	1.03557
$33$	0	0
$34$	−83.0965	−0.419145
$35$	78.1587	0.377464
$36$	0	0
$37$	−297.648	−1.32252	−0.661258	−	0.750159i	$-0.729977\pi$
$37$	−297.648	−1.32252	−0.661258	+	0.750159i	$0.729977\pi$
$38$	201.528	0.860319
$39$	0	0
$40$	−234.476	−0.926848
$41$	−304.483	−1.15981	−0.579905	−	0.814684i	$-0.696910\pi$
$41$	−304.483	−1.15981	−0.579905	+	0.814684i	$0.696910\pi$
$42$	0	0
$43$	455.552	1.61561	0.807803	−	0.589453i	$-0.200657\pi$
$43$	455.552	1.61561	0.807803	+	0.589453i	$0.200657\pi$
$44$	−61.8656	−0.211968
$45$	0	0
$46$	−90.3037	−0.289447
$47$	−502.579	−1.55976	−0.779880	−	0.625929i	$-0.784720\pi$
$47$	−502.579	−1.55976	−0.779880	+	0.625929i	$0.784720\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−0.510274	−0.00144327
$51$	0	0
$52$	445.238	1.18737
$53$	135.096	0.350131	0.175065	−	0.984557i	$-0.443986\pi$
$53$	135.096	0.350131	0.175065	+	0.984557i	$0.443986\pi$
$54$	0	0
$55$	122.821	0.301112
$56$	−147.000	−0.350780
$57$	0	0
$58$	−427.707	−0.968287
$59$	53.9309	0.119004	0.0595018	−	0.998228i	$-0.481049\pi$
$59$	53.9309	0.119004	0.0595018	+	0.998228i	$0.481049\pi$
$60$	0	0
$61$	−263.297	−0.552651	−0.276325	−	0.961064i	$-0.589117\pi$
$61$	−263.297	−0.552651	−0.276325	+	0.961064i	$0.589117\pi$
$62$	238.531	0.488605
$63$	0	0
$64$	187.952	0.367094
$65$	−883.925	−1.68673
$66$	0	0
$67$	123.572	0.225325	0.112663	−	0.993633i	$-0.464062\pi$
$67$	123.572	0.225325	0.112663	+	0.993633i	$0.464062\pi$
$68$	303.200	0.540711
$69$	0	0
$70$	120.472	0.205703
$71$	−139.793	−0.233668	−0.116834	−	0.993151i	$-0.537275\pi$
$71$	−139.793	−0.233668	−0.116834	+	0.993151i	$0.537275\pi$
$72$	0	0
$73$	−1157.77	−1.85626	−0.928131	−	0.372255i	$-0.878585\pi$
$73$	−1157.77	−1.85626	−0.928131	+	0.372255i	$0.878585\pi$
$74$	−458.790	−0.720719
$75$	0	0
$76$	−735.328	−1.10984
$77$	77.0000	0.113961
$78$	0	0
$79$	−62.4966	−0.0890052	−0.0445026	−	0.999009i	$-0.514170\pi$
$79$	−62.4966	−0.0890052	−0.0445026	+	0.999009i	$0.514170\pi$
$80$	140.955	0.196991
$81$	0	0
$82$	−469.324	−0.632051
$83$	−756.352	−1.00025	−0.500123	−	0.865954i	$-0.666712\pi$
$83$	−756.352	−1.00025	−0.500123	+	0.865954i	$0.666712\pi$
$84$	0	0
$85$	−601.938	−0.768110
$86$	702.179	0.880441
$87$	0	0
$88$	−231.000	−0.279826
$89$	−135.558	−0.161451	−0.0807255	−	0.996736i	$-0.525724\pi$
$89$	−135.558	−0.161451	−0.0807255	+	0.996736i	$0.525724\pi$
$90$	0	0
$91$	−554.159	−0.638369
$92$	329.497	0.373396
$93$	0	0
$94$	−774.666	−0.850008
$95$	1459.84	1.57659
$96$	0	0
$97$	−571.248	−0.597953	−0.298977	−	0.954260i	$-0.596645\pi$
$97$	−571.248	−0.597953	−0.298977	+	0.954260i	$0.596645\pi$
$98$	75.5277	0.0778515
$99$	0	0
$100$	1.86187	0.00186187
$101$	−1.99258	−0.00196306	−0.000981532	−	1.00000i	$-0.500312\pi$
$101$	−1.99258	−0.00196306	−0.000981532		1.00000i	$0.500312\pi$
$102$	0	0
$103$	409.186	0.391440	0.195720	−	0.980660i	$-0.437296\pi$
$103$	409.186	0.391440	0.195720	+	0.980660i	$0.437296\pi$
$104$	1662.48	1.56749
$105$	0	0
$106$	208.235	0.190807
$107$	−1157.78	−1.04605	−0.523023	−	0.852319i	$-0.675196\pi$
$107$	−1157.78	−1.04605	−0.523023	+	0.852319i	$0.675196\pi$
$108$	0	0
$109$	−1146.39	−1.00738	−0.503691	−	0.863884i	$-0.668025\pi$
$109$	−1146.39	−1.00738	−0.503691	+	0.863884i	$0.668025\pi$
$110$	189.314	0.164094
$111$	0	0
$112$	88.3690	0.0745543
$113$	−1224.50	−1.01939	−0.509694	−	0.860355i	$-0.670242\pi$
$113$	−1224.50	−1.01939	−0.509694	+	0.860355i	$0.670242\pi$
$114$	0	0
$115$	−654.146	−0.530429
$116$	1560.60	1.24912
$117$	0	0
$118$	83.1281	0.0648522
$119$	−377.373	−0.290703
$120$	0	0
$121$	121.000	0.0909091
$122$	−405.841	−0.301173
$123$	0	0
$124$	−870.346	−0.630317
$125$	−1399.39	−1.00132
$126$	0	0
$127$	−1948.15	−1.36118	−0.680590	−	0.732664i	$-0.738277\pi$
$127$	−1948.15	−1.36118	−0.680590	+	0.732664i	$0.738277\pi$
$128$	−1209.96	−0.835521
$129$	0	0
$130$	−1362.46	−0.919201
$131$	52.2215	0.0348291	0.0174145	−	0.999848i	$-0.494456\pi$
$131$	52.2215	0.0348291	0.0174145	+	0.999848i	$0.494456\pi$
$132$	0	0
$133$	915.214	0.596685
$134$	190.472	0.122793
$135$	0	0
$136$	1132.12	0.713811
$137$	1390.23	0.866973	0.433486	−	0.901160i	$-0.357283\pi$
$137$	1390.23	0.866973	0.433486	+	0.901160i	$0.357283\pi$
$138$	0	0
$139$	448.677	0.273786	0.136893	−	0.990586i	$-0.456288\pi$
$139$	448.677	0.273786	0.136893	+	0.990586i	$0.456288\pi$
$140$	−439.576	−0.265364
$141$	0	0
$142$	−215.475	−0.127340
$143$	−870.821	−0.509243
$144$	0	0
$145$	−3098.24	−1.77445
$146$	−1784.57	−1.01159
$147$	0	0
$148$	1674.02	0.929753
$149$	3343.54	1.83835	0.919173	−	0.393855i	$-0.128859\pi$
$149$	3343.54	1.83835	0.919173	+	0.393855i	$0.128859\pi$
$150$	0	0
$151$	1797.06	0.968493	0.484246	−	0.874932i	$-0.339094\pi$
$151$	1797.06	0.968493	0.484246	+	0.874932i	$0.339094\pi$
$152$	−2745.64	−1.46514
$153$	0	0
$154$	118.686	0.0621040
$155$	1727.88	0.895400
$156$	0	0
$157$	−2460.43	−1.25072	−0.625362	−	0.780335i	$-0.715049\pi$
$157$	−2460.43	−1.25072	−0.625362	+	0.780335i	$0.715049\pi$
$158$	−96.3311	−0.0485043
$159$	0	0
$160$	2093.07	1.03420
$161$	−410.103	−0.200749
$162$	0	0
$163$	2958.31	1.42155	0.710774	−	0.703420i	$-0.248345\pi$
$163$	2958.31	1.42155	0.710774	+	0.703420i	$0.248345\pi$
$164$	1712.46	0.815368
$165$	0	0
$166$	−1165.83	−0.545095
$167$	−2481.65	−1.14991	−0.574957	−	0.818184i	$-0.694981\pi$
$167$	−2481.65	−1.14991	−0.574957	+	0.818184i	$0.694981\pi$
$168$	0	0
$169$	4070.18	1.85261
$170$	−927.816	−0.418589
$171$	0	0
$172$	−2562.09	−1.13580
$173$	3064.48	1.34675	0.673377	−	0.739300i	$-0.264843\pi$
$173$	3064.48	1.34675	0.673377	+	0.739300i	$0.264843\pi$
$174$	0	0
$175$	−2.31735	−0.00100100
$176$	138.866	0.0594738
$177$	0	0
$178$	−208.947	−0.0879845
$179$	520.580	0.217374	0.108687	−	0.994076i	$-0.465335\pi$
$179$	520.580	0.217374	0.108687	+	0.994076i	$0.465335\pi$
$180$	0	0
$181$	4659.06	1.91329	0.956643	−	0.291261i	$-0.0940750\pi$
$181$	4659.06	1.91329	0.956643	+	0.291261i	$0.0940750\pi$
$182$	−854.170	−0.347886
$183$	0	0
$184$	1230.31	0.492933
$185$	−3323.40	−1.32076
$186$	0	0
$187$	−593.014	−0.231901
$188$	2826.58	1.09654
$189$	0	0
$190$	2250.16	0.859179
$191$	−1130.20	−0.428160	−0.214080	−	0.976816i	$-0.568675\pi$
$191$	−1130.20	−0.428160	−0.214080	+	0.976816i	$0.568675\pi$
$192$	0	0
$193$	1748.02	0.651945	0.325973	−	0.945379i	$-0.394308\pi$
$193$	1748.02	0.651945	0.325973	+	0.945379i	$0.394308\pi$
$194$	−880.511	−0.325861
$195$	0	0
$196$	−275.583	−0.100431
$197$	2534.49	0.916622	0.458311	−	0.888792i	$-0.348455\pi$
$197$	2534.49	0.916622	0.458311	+	0.888792i	$0.348455\pi$
$198$	0	0
$199$	3210.26	1.14356	0.571781	−	0.820406i	$-0.306253\pi$
$199$	3210.26	1.14356	0.571781	+	0.820406i	$0.306253\pi$
$200$	6.95205	0.00245792
$201$	0	0
$202$	−3.07133	−0.00106979
$203$	−1942.38	−0.671568
$204$	0	0
$205$	−3399.71	−1.15827
$206$	630.712	0.213319
$207$	0	0
$208$	−999.397	−0.333152
$209$	1438.19	0.475990
$210$	0	0
$211$	−2574.51	−0.839985	−0.419992	−	0.907528i	$-0.637967\pi$
$211$	−2574.51	−0.839985	−0.419992	+	0.907528i	$0.637967\pi$
$212$	−759.802	−0.246148
$213$	0	0
$214$	−1784.58	−0.570053
$215$	5086.48	1.61346
$216$	0	0
$217$	1083.26	0.338878
$218$	−1767.03	−0.548983
$219$	0	0
$220$	−690.762	−0.211687
$221$	4267.84	1.29903
$222$	0	0
$223$	6269.84	1.88278	0.941388	−	0.337324i	$-0.109522\pi$
$223$	6269.84	1.88278	0.941388	+	0.337324i	$0.109522\pi$
$224$	1312.21	0.391410
$225$	0	0
$226$	−1887.42	−0.555527
$227$	−1762.28	−0.515270	−0.257635	−	0.966242i	$-0.582943\pi$
$227$	−1762.28	−0.515270	−0.257635	+	0.966242i	$0.582943\pi$
$228$	0	0
$229$	4596.52	1.32640	0.663202	−	0.748441i	$-0.269197\pi$
$229$	4596.52	1.32640	0.663202	+	0.748441i	$0.269197\pi$
$230$	−1008.29	−0.289063
$231$	0	0
$232$	5827.14	1.64901
$233$	−6296.24	−1.77030	−0.885150	−	0.465305i	$-0.845945\pi$
$233$	−6296.24	−1.77030	−0.885150	+	0.465305i	$0.845945\pi$
$234$	0	0
$235$	−5611.56	−1.55769
$236$	−303.315	−0.0836616
$237$	0	0
$238$	−581.675	−0.158422
$239$	5264.73	1.42488	0.712442	−	0.701731i	$-0.247589\pi$
$239$	5264.73	1.42488	0.712442	+	0.701731i	$0.247589\pi$
$240$	0	0
$241$	374.226	0.100025	0.0500125	−	0.998749i	$-0.484074\pi$
$241$	374.226	0.100025	0.0500125	+	0.998749i	$0.484074\pi$
$242$	186.507	0.0495419
$243$	0	0
$244$	1480.82	0.388523
$245$	547.111	0.142668
$246$	0	0
$247$	−10350.5	−2.66634
$248$	−3249.79	−0.832103
$249$	0	0
$250$	−2156.99	−0.545680
$251$	−4972.06	−1.25033	−0.625166	−	0.780492i	$-0.714969\pi$
$251$	−4972.06	−1.25033	−0.625166	+	0.780492i	$0.714969\pi$
$252$	0	0
$253$	−644.448	−0.160143
$254$	−3002.84	−0.741790
$255$	0	0
$256$	−3368.63	−0.822420
$257$	−4807.87	−1.16695	−0.583476	−	0.812130i	$-0.698308\pi$
$257$	−4807.87	−1.16695	−0.583476	+	0.812130i	$0.698308\pi$
$258$	0	0
$259$	−2083.54	−0.499864
$260$	4971.32	1.18580
$261$	0	0
$262$	80.4932	0.0189805
$263$	−2216.15	−0.519596	−0.259798	−	0.965663i	$-0.583656\pi$
$263$	−2216.15	−0.519596	−0.259798	+	0.965663i	$0.583656\pi$
$264$	0	0
$265$	1508.42	0.349667
$266$	1410.69	0.325170
$267$	0	0
$268$	−694.989	−0.158408
$269$	2296.59	0.520542	0.260271	−	0.965536i	$-0.416188\pi$
$269$	2296.59	0.520542	0.260271	+	0.965536i	$0.416188\pi$
$270$	0	0
$271$	5199.71	1.16553	0.582767	−	0.812639i	$-0.301970\pi$
$271$	5199.71	1.16553	0.582767	+	0.812639i	$0.301970\pi$
$272$	−680.572	−0.151712
$273$	0	0
$274$	2142.87	0.472466
$275$	−3.64155	−0.000798523 0
$276$	0	0
$277$	−3760.26	−0.815640	−0.407820	−	0.913062i	$-0.633711\pi$
$277$	−3760.26	−0.815640	−0.407820	+	0.913062i	$0.633711\pi$
$278$	691.582	0.149203
$279$	0	0
$280$	−1641.33	−0.350316
$281$	−6192.84	−1.31471	−0.657355	−	0.753581i	$-0.728325\pi$
$281$	−6192.84	−1.31471	−0.657355	+	0.753581i	$0.728325\pi$
$282$	0	0
$283$	4662.14	0.979278	0.489639	−	0.871925i	$-0.337129\pi$
$283$	4662.14	0.979278	0.489639	+	0.871925i	$0.337129\pi$
$284$	786.218	0.164273
$285$	0	0
$286$	−1342.27	−0.277517
$287$	−2131.38	−0.438367
$288$	0	0
$289$	−2006.67	−0.408441
$290$	−4775.57	−0.967004
$291$	0	0
$292$	6511.48	1.30498
$293$	8562.82	1.70732	0.853661	−	0.520830i	$-0.174377\pi$
$293$	8562.82	1.70732	0.853661	+	0.520830i	$0.174377\pi$
$294$	0	0
$295$	602.167	0.118846
$296$	6250.62	1.22740
$297$	0	0
$298$	5153.67	1.00183
$299$	4638.01	0.897066
$300$	0	0
$301$	3188.86	0.610641
$302$	2769.95	0.527790
$303$	0	0
$304$	1650.54	0.311398
$305$	−2939.85	−0.551919
$306$	0	0
$307$	−5923.67	−1.10124	−0.550622	−	0.834755i	$-0.685609\pi$
$307$	−5923.67	−1.10124	−0.550622	+	0.834755i	$0.685609\pi$
$308$	−433.059	−0.0801163
$309$	0	0
$310$	2663.33	0.487958
$311$	833.349	0.151945	0.0759725	−	0.997110i	$-0.475794\pi$
$311$	833.349	0.151945	0.0759725	+	0.997110i	$0.475794\pi$
$312$	0	0
$313$	1788.60	0.322996	0.161498	−	0.986873i	$-0.448367\pi$
$313$	1788.60	0.322996	0.161498	+	0.986873i	$0.448367\pi$
$314$	−3792.46	−0.681595
$315$	0	0
$316$	351.490	0.0625723
$317$	317.235	0.0562072	0.0281036	−	0.999605i	$-0.491053\pi$
$317$	317.235	0.0562072	0.0281036	+	0.999605i	$0.491053\pi$
$318$	0	0
$319$	−3052.31	−0.535726
$320$	2098.58	0.366607
$321$	0	0
$322$	−632.126	−0.109401
$323$	−7048.51	−1.21421
$324$	0	0
$325$	26.2078	0.00447306
$326$	4559.88	0.774688
$327$	0	0
$328$	6394.14	1.07639
$329$	−3518.06	−0.589534
$330$	0	0
$331$	−7119.71	−1.18228	−0.591140	−	0.806569i	$-0.701322\pi$
$331$	−7119.71	−1.18228	−0.591140	+	0.806569i	$0.701322\pi$
$332$	4253.83	0.703191
$333$	0	0
$334$	−3825.17	−0.626658
$335$	1379.75	0.225027
$336$	0	0
$337$	2230.78	0.360589	0.180294	−	0.983613i	$-0.442295\pi$
$337$	2230.78	0.360589	0.180294	+	0.983613i	$0.442295\pi$
$338$	6273.70	1.00960
$339$	0	0
$340$	3385.38	0.539995
$341$	1702.27	0.270331
$342$	0	0
$343$	343.000	0.0539949
$344$	−9566.59	−1.49941
$345$	0	0
$346$	4723.54	0.733927
$347$	11699.4	1.80997	0.904983	−	0.425448i	$-0.139883\pi$
$347$	11699.4	1.80997	0.904983	+	0.425448i	$0.139883\pi$
$348$	0	0
$349$	7095.65	1.08831	0.544157	−	0.838984i	$-0.316850\pi$
$349$	7095.65	1.08831	0.544157	+	0.838984i	$0.316850\pi$
$350$	−3.57192	−0.000545506 0
$351$	0	0
$352$	2062.04	0.312237
$353$	2798.12	0.421895	0.210947	−	0.977497i	$-0.432345\pi$
$353$	2798.12	0.421895	0.210947	+	0.977497i	$0.432345\pi$
$354$	0	0
$355$	−1560.87	−0.233358
$356$	762.399	0.113503
$357$	0	0
$358$	802.412	0.118460
$359$	6826.72	1.00362	0.501811	−	0.864977i	$-0.332667\pi$
$359$	6826.72	1.00362	0.501811	+	0.864977i	$0.332667\pi$
$360$	0	0
$361$	10235.2	1.49223
$362$	7181.38	1.04267
$363$	0	0
$364$	3116.67	0.448785
$365$	−12927.1	−1.85380
$366$	0	0
$367$	−8474.66	−1.20538	−0.602689	−	0.797976i	$-0.705904\pi$
$367$	−8474.66	−1.20538	−0.602689	+	0.797976i	$0.705904\pi$
$368$	−739.600	−0.104767
$369$	0	0
$370$	−5122.63	−0.719764
$371$	945.675	0.132337
$372$	0	0
$373$	−7278.84	−1.01041	−0.505206	−	0.862999i	$-0.668584\pi$
$373$	−7278.84	−1.01041	−0.505206	+	0.862999i	$0.668584\pi$
$374$	−914.061	−0.126377
$375$	0	0
$376$	10554.2	1.44758
$377$	21967.1	3.00096
$378$	0	0
$379$	−11047.8	−1.49733	−0.748664	−	0.662950i	$-0.769304\pi$
$379$	−11047.8	−1.49733	−0.748664	+	0.662950i	$0.769304\pi$
$380$	−8210.32	−1.10837
$381$	0	0
$382$	−1742.07	−0.233330
$383$	4700.44	0.627106	0.313553	−	0.949571i	$-0.398481\pi$
$383$	4700.44	0.627106	0.313553	+	0.949571i	$0.398481\pi$
$384$	0	0
$385$	859.745	0.113810
$386$	2694.37	0.355284
$387$	0	0
$388$	3212.78	0.420372
$389$	−5308.33	−0.691885	−0.345942	−	0.938256i	$-0.612441\pi$
$389$	−5308.33	−0.691885	−0.345942	+	0.938256i	$0.612441\pi$
$390$	0	0
$391$	3158.40	0.408510
$392$	−1029.00	−0.132583
$393$	0	0
$394$	3906.61	0.499523
$395$	−697.807	−0.0888873
$396$	0	0
$397$	7910.41	1.00003	0.500015	−	0.866017i	$-0.333328\pi$
$397$	7910.41	1.00003	0.500015	+	0.866017i	$0.333328\pi$
$398$	4948.23	0.623197
$399$	0	0
$400$	−4.17922	−0.000522403 0
$401$	9226.75	1.14903	0.574516	−	0.818493i	$-0.305190\pi$
$401$	9226.75	1.14903	0.574516	+	0.818493i	$0.305190\pi$
$402$	0	0
$403$	−12251.0	−1.51431
$404$	11.2066	0.00138007
$405$	0	0
$406$	−2993.95	−0.365978
$407$	−3274.13	−0.398754
$408$	0	0
$409$	−4012.26	−0.485070	−0.242535	−	0.970143i	$-0.577979\pi$
$409$	−4012.26	−0.485070	−0.242535	+	0.970143i	$0.577979\pi$
$410$	−5240.25	−0.631214
$411$	0	0
$412$	−2301.32	−0.275189
$413$	377.517	0.0449791
$414$	0	0
$415$	−8445.07	−0.998921
$416$	−14840.3	−1.74905
$417$	0	0
$418$	2216.80	0.259396
$419$	−2894.88	−0.337528	−0.168764	−	0.985657i	$-0.553978\pi$
$419$	−2894.88	−0.337528	−0.168764	+	0.985657i	$0.553978\pi$
$420$	0	0
$421$	−1379.07	−0.159648	−0.0798239	−	0.996809i	$-0.525436\pi$
$421$	−1379.07	−0.159648	−0.0798239	+	0.996809i	$0.525436\pi$
$422$	−3968.31	−0.457758
$423$	0	0
$424$	−2837.03	−0.324948
$425$	17.8470	0.00203696
$426$	0	0
$427$	−1843.08	−0.208882
$428$	6511.52	0.735389
$429$	0	0
$430$	7840.20	0.879274
$431$	−3372.16	−0.376870	−0.188435	−	0.982086i	$-0.560342\pi$
$431$	−3372.16	−0.376870	−0.188435	+	0.982086i	$0.560342\pi$
$432$	0	0
$433$	8134.86	0.902856	0.451428	−	0.892308i	$-0.350915\pi$
$433$	8134.86	0.902856	0.451428	+	0.892308i	$0.350915\pi$
$434$	1669.72	0.184675
$435$	0	0
$436$	6447.48	0.708207
$437$	−7659.84	−0.838490
$438$	0	0
$439$	1317.81	0.143270	0.0716352	−	0.997431i	$-0.477178\pi$
$439$	1317.81	0.143270	0.0716352	+	0.997431i	$0.477178\pi$
$440$	−2579.24	−0.279455
$441$	0	0
$442$	6578.38	0.707922
$443$	590.088	0.0632865	0.0316433	−	0.999499i	$-0.489926\pi$
$443$	590.088	0.0632865	0.0316433	+	0.999499i	$0.489926\pi$
$444$	0	0
$445$	−1513.58	−0.161237
$446$	9664.21	1.02604
$447$	0	0
$448$	1315.66	0.138748
$449$	2858.37	0.300434	0.150217	−	0.988653i	$-0.452003\pi$
$449$	2858.37	0.300434	0.150217	+	0.988653i	$0.452003\pi$
$450$	0	0
$451$	−3349.31	−0.349696
$452$	6886.74	0.716649
$453$	0	0
$454$	−2716.34	−0.280802
$455$	−6187.47	−0.637523
$456$	0	0
$457$	−3399.88	−0.348008	−0.174004	−	0.984745i	$-0.555671\pi$
$457$	−3399.88	−0.348008	−0.174004	+	0.984745i	$0.555671\pi$
$458$	7084.99	0.722838
$459$	0	0
$460$	3679.01	0.372901
$461$	−10462.2	−1.05700	−0.528498	−	0.848935i	$-0.677245\pi$
$461$	−10462.2	−1.05700	−0.528498	+	0.848935i	$0.677245\pi$
$462$	0	0
$463$	4820.84	0.483895	0.241947	−	0.970289i	$-0.422214\pi$
$463$	4820.84	0.483895	0.241947	+	0.970289i	$0.422214\pi$
$464$	−3502.98	−0.350478
$465$	0	0
$466$	−9704.90	−0.964744
$467$	1133.78	0.112345	0.0561723	−	0.998421i	$-0.482110\pi$
$467$	1133.78	0.112345	0.0561723	+	0.998421i	$0.482110\pi$
$468$	0	0
$469$	865.007	0.0851649
$470$	−8649.56	−0.848882
$471$	0	0
$472$	−1132.55	−0.110445
$473$	5011.07	0.487123
$474$	0	0
$475$	−43.2831	−0.00418098
$476$	2122.40	0.204370
$477$	0	0
$478$	8114.96	0.776506
$479$	−9278.95	−0.885106	−0.442553	−	0.896742i	$-0.645927\pi$
$479$	−9278.95	−0.885106	−0.442553	+	0.896742i	$0.645927\pi$
$480$	0	0
$481$	23563.5	2.23369
$482$	576.825	0.0545097
$483$	0	0
$484$	−680.521	−0.0639107
$485$	−6378.29	−0.597161
$486$	0	0
$487$	−3853.58	−0.358567	−0.179284	−	0.983797i	$-0.557378\pi$
$487$	−3853.58	−0.358567	−0.179284	+	0.983797i	$0.557378\pi$
$488$	5529.23	0.512903
$489$	0	0
$490$	843.306	0.0777484
$491$	−6237.37	−0.573296	−0.286648	−	0.958036i	$-0.592541\pi$
$491$	−6237.37	−0.573296	−0.286648	+	0.958036i	$0.592541\pi$
$492$	0	0
$493$	14959.2	1.36659
$494$	−15954.0	−1.45305
$495$	0	0
$496$	1953.61	0.176854
$497$	−978.554	−0.0883182
$498$	0	0
$499$	2171.63	0.194821	0.0974105	−	0.995244i	$-0.468944\pi$
$499$	2171.63	0.194821	0.0974105	+	0.995244i	$0.468944\pi$
$500$	7870.35	0.703946
$501$	0	0
$502$	−7663.83	−0.681382
$503$	13952.7	1.23682	0.618411	−	0.785855i	$-0.287777\pi$
$503$	13952.7	1.23682	0.618411	+	0.785855i	$0.287777\pi$
$504$	0	0
$505$	−22.2482	−0.00196046
$506$	−993.340	−0.0872715
$507$	0	0
$508$	10956.7	0.956935
$509$	5585.84	0.486420	0.243210	−	0.969974i	$-0.421800\pi$
$509$	5585.84	0.486420	0.243210	+	0.969974i	$0.421800\pi$
$510$	0	0
$511$	−8104.41	−0.701601
$512$	4487.36	0.387334
$513$	0	0
$514$	−7410.76	−0.635943
$515$	4568.78	0.390921
$516$	0	0
$517$	−5528.37	−0.470285
$518$	−3211.53	−0.272406
$519$	0	0
$520$	18562.4	1.56542
$521$	−4169.32	−0.350598	−0.175299	−	0.984515i	$-0.556089\pi$
$521$	−4169.32	−0.350598	−0.175299	+	0.984515i	$0.556089\pi$
$522$	0	0
$523$	−9157.35	−0.765627	−0.382814	−	0.923826i	$-0.625045\pi$
$523$	−9157.35	−0.765627	−0.382814	+	0.923826i	$0.625045\pi$
$524$	−293.701	−0.0244855
$525$	0	0
$526$	−3415.94	−0.283159
$527$	−8342.72	−0.689592
$528$	0	0
$529$	−8734.66	−0.717897
$530$	2325.05	0.190555
$531$	0	0
$532$	−5147.30	−0.419480
$533$	24104.5	1.95888
$534$	0	0
$535$	−12927.2	−1.04466
$536$	−2595.02	−0.209119
$537$	0	0
$538$	3539.93	0.283675
$539$	539.000	0.0430730
$540$	0	0
$541$	5427.01	0.431285	0.215643	−	0.976472i	$-0.430815\pi$
$541$	5427.01	0.431285	0.215643	+	0.976472i	$0.430815\pi$
$542$	8014.74	0.635171
$543$	0	0
$544$	−10106.0	−0.796489
$545$	−12800.1	−1.00605
$546$	0	0
$547$	−18288.3	−1.42953	−0.714764	−	0.699366i	$-0.753466\pi$
$547$	−18288.3	−1.42953	−0.714764	+	0.699366i	$0.753466\pi$
$548$	−7818.85	−0.609497
$549$	0	0
$550$	−5.61302	−0.000435164 0
$551$	−36279.5	−2.80500
$552$	0	0
$553$	−437.476	−0.0336408
$554$	−5796.00	−0.444492
$555$	0	0
$556$	−2523.42	−0.192477
$557$	−16404.9	−1.24793	−0.623965	−	0.781452i	$-0.714479\pi$
$557$	−16404.9	−1.24793	−0.623965	+	0.781452i	$0.714479\pi$
$558$	0	0
$559$	−36064.0	−2.72870
$560$	986.686	0.0744556
$561$	0	0
$562$	−9545.52	−0.716466
$563$	−2143.26	−0.160440	−0.0802199	−	0.996777i	$-0.525562\pi$
$563$	−2143.26	−0.160440	−0.0802199	+	0.996777i	$0.525562\pi$
$564$	0	0
$565$	−13672.1	−1.01804
$566$	7186.14	0.533668
$567$	0	0
$568$	2935.66	0.216862
$569$	−22176.2	−1.63388	−0.816938	−	0.576726i	$-0.804330\pi$
$569$	−22176.2	−1.63388	−0.816938	+	0.576726i	$0.804330\pi$
$570$	0	0
$571$	14873.1	1.09005	0.545026	−	0.838419i	$-0.316520\pi$
$571$	14873.1	1.09005	0.545026	+	0.838419i	$0.316520\pi$
$572$	4897.62	0.358007
$573$	0	0
$574$	−3285.27	−0.238893
$575$	19.3950	0.00140665
$576$	0	0
$577$	−19737.8	−1.42408	−0.712040	−	0.702139i	$-0.752229\pi$
$577$	−19737.8	−1.42408	−0.712040	+	0.702139i	$0.752229\pi$
$578$	−3093.04	−0.222584
$579$	0	0
$580$	17425.0	1.24747
$581$	−5294.47	−0.378058
$582$	0	0
$583$	1486.06	0.105568
$584$	24313.2	1.72275
$585$	0	0
$586$	13198.6	0.930423
$587$	1295.83	0.0911150	0.0455575	−	0.998962i	$-0.485494\pi$
$587$	1295.83	0.0911150	0.0455575	+	0.998962i	$0.485494\pi$
$588$	0	0
$589$	20233.0	1.41543
$590$	928.169	0.0647663
$591$	0	0
$592$	−3757.56	−0.260869
$593$	−17061.1	−1.18147	−0.590737	−	0.806864i	$-0.701163\pi$
$593$	−17061.1	−1.18147	−0.590737	+	0.806864i	$0.701163\pi$
$594$	0	0
$595$	−4213.56	−0.290318
$596$	−18804.5	−1.29239
$597$	0	0
$598$	7148.94	0.488866
$599$	2275.93	0.155245	0.0776226	−	0.996983i	$-0.475267\pi$
$599$	2275.93	0.155245	0.0776226	+	0.996983i	$0.475267\pi$
$600$	0	0
$601$	−19259.9	−1.30720	−0.653600	−	0.756840i	$-0.726742\pi$
$601$	−19259.9	−1.30720	−0.653600	+	0.756840i	$0.726742\pi$
$602$	4915.25	0.332775
$603$	0	0
$604$	−10106.9	−0.680868
$605$	1351.03	0.0907886
$606$	0	0
$607$	22972.5	1.53612	0.768061	−	0.640376i	$-0.221222\pi$
$607$	22972.5	1.53612	0.768061	+	0.640376i	$0.221222\pi$
$608$	24509.3	1.63484
$609$	0	0
$610$	−4531.43	−0.300774
$611$	39787.0	2.63438
$612$	0	0
$613$	−23830.6	−1.57016	−0.785082	−	0.619392i	$-0.787379\pi$
$613$	−23830.6	−1.57016	−0.785082	+	0.619392i	$0.787379\pi$
$614$	−9130.63	−0.600134
$615$	0	0
$616$	−1617.00	−0.105764
$617$	3176.27	0.207248	0.103624	−	0.994617i	$-0.466956\pi$
$617$	3176.27	0.207248	0.103624	+	0.994617i	$0.466956\pi$
$618$	0	0
$619$	3767.38	0.244627	0.122313	−	0.992492i	$-0.460969\pi$
$619$	3767.38	0.244627	0.122313	+	0.992492i	$0.460969\pi$
$620$	−9717.87	−0.629482
$621$	0	0
$622$	1284.51	0.0828041
$623$	−948.908	−0.0610228
$624$	0	0
$625$	−15583.5	−0.997345
$626$	2756.92	0.176020
$627$	0	0
$628$	13837.8	0.879281
$629$	16046.3	1.01719
$630$	0	0
$631$	7347.85	0.463571	0.231786	−	0.972767i	$-0.425543\pi$
$631$	7347.85	0.463571	0.231786	+	0.972767i	$0.425543\pi$
$632$	1312.43	0.0826038
$633$	0	0
$634$	488.979	0.0306307
$635$	−21752.1	−1.35938
$636$	0	0
$637$	−3879.11	−0.241281
$638$	−4704.78	−0.291950
$639$	0	0
$640$	−13509.9	−0.834414
$641$	−13732.5	−0.846179	−0.423089	−	0.906088i	$-0.639054\pi$
$641$	−13732.5	−0.846179	−0.423089	+	0.906088i	$0.639054\pi$
$642$	0	0
$643$	21566.9	1.32273	0.661364	−	0.750065i	$-0.269978\pi$
$643$	21566.9	1.32273	0.661364	+	0.750065i	$0.269978\pi$
$644$	2306.48	0.141131
$645$	0	0
$646$	−10864.4	−0.661696
$647$	24177.3	1.46910	0.734550	−	0.678554i	$-0.237393\pi$
$647$	24177.3	1.46910	0.734550	+	0.678554i	$0.237393\pi$
$648$	0	0
$649$	593.240	0.0358809
$650$	40.3961	0.00243764
$651$	0	0
$652$	−16637.9	−0.999374
$653$	−27365.2	−1.63994	−0.819972	−	0.572404i	$-0.806011\pi$
$653$	−27365.2	−1.63994	−0.819972	+	0.572404i	$0.806011\pi$
$654$	0	0
$655$	583.080	0.0347829
$656$	−3843.84	−0.228775
$657$	0	0
$658$	−5422.66	−0.321273
$659$	−3190.05	−0.188569	−0.0942843	−	0.995545i	$-0.530056\pi$
$659$	−3190.05	−0.188569	−0.0942843	+	0.995545i	$0.530056\pi$
$660$	0	0
$661$	−9967.01	−0.586493	−0.293246	−	0.956037i	$-0.594736\pi$
$661$	−9967.01	−0.586493	−0.293246	+	0.956037i	$0.594736\pi$
$662$	−10974.2	−0.644296
$663$	0	0
$664$	15883.4	0.928306
$665$	10218.8	0.595894
$666$	0	0
$667$	16256.7	0.943719
$668$	13957.1	0.808410
$669$	0	0
$670$	2126.72	0.122631
$671$	−2896.26	−0.166630
$672$	0	0
$673$	−25568.9	−1.46450	−0.732249	−	0.681037i	$-0.761529\pi$
$673$	−25568.9	−1.46450	−0.732249	+	0.681037i	$0.761529\pi$
$674$	3438.48	0.196507
$675$	0	0
$676$	−22891.3	−1.30242
$677$	25104.8	1.42520	0.712598	−	0.701573i	$-0.247519\pi$
$677$	25104.8	1.42520	0.712598	+	0.701573i	$0.247519\pi$
$678$	0	0
$679$	−3998.74	−0.226005
$680$	12640.7	0.712865
$681$	0	0
$682$	2623.85	0.147320
$683$	10322.5	0.578301	0.289150	−	0.957284i	$-0.406627\pi$
$683$	10322.5	0.578301	0.289150	+	0.957284i	$0.406627\pi$
$684$	0	0
$685$	15522.6	0.865824
$686$	528.694	0.0294251
$687$	0	0
$688$	5750.95	0.318682
$689$	−10695.0	−0.591359
$690$	0	0
$691$	18143.4	0.998853	0.499427	−	0.866356i	$-0.333544\pi$
$691$	18143.4	0.998853	0.499427	+	0.866356i	$0.333544\pi$
$692$	−17235.1	−0.946792
$693$	0	0
$694$	18033.3	0.986360
$695$	5009.71	0.273423
$696$	0	0
$697$	16414.8	0.892044
$698$	10937.1	0.593088
$699$	0	0
$700$	13.0331	0.000703722 0
$701$	20743.7	1.11766	0.558830	−	0.829282i	$-0.311251\pi$
$701$	20743.7	1.11766	0.558830	+	0.829282i	$0.311251\pi$
$702$	0	0
$703$	−38916.0	−2.08783
$704$	2067.47	0.110683
$705$	0	0
$706$	4312.97	0.229916
$707$	−13.9481	−0.000741968 0
$708$	0	0
$709$	−15741.1	−0.833806	−0.416903	−	0.908951i	$-0.636884\pi$
$709$	−15741.1	−0.833806	−0.416903	+	0.908951i	$0.636884\pi$
$710$	−2405.89	−0.127171
$711$	0	0
$712$	2846.72	0.149839
$713$	−9066.31	−0.476208
$714$	0	0
$715$	−9723.17	−0.508568
$716$	−2927.82	−0.152818
$717$	0	0
$718$	10522.6	0.546934
$719$	17939.6	0.930506	0.465253	−	0.885178i	$-0.345963\pi$
$719$	17939.6	0.930506	0.465253	+	0.885178i	$0.345963\pi$
$720$	0	0
$721$	2864.30	0.147950
$722$	15776.4	0.813208
$723$	0	0
$724$	−26203.2	−1.34508
$725$	91.8607	0.00470569
$726$	0	0
$727$	−32942.6	−1.68057	−0.840285	−	0.542144i	$-0.817613\pi$
$727$	−32942.6	−1.68057	−0.840285	+	0.542144i	$0.817613\pi$
$728$	11637.3	0.592456
$729$	0	0
$730$	−19925.7	−1.01025
$731$	−24559.0	−1.24261
$732$	0	0
$733$	−4868.78	−0.245337	−0.122669	−	0.992448i	$-0.539145\pi$
$733$	−4868.78	−0.245337	−0.122669	+	0.992448i	$0.539145\pi$
$734$	−13062.7	−0.656884
$735$	0	0
$736$	−10982.5	−0.550027
$737$	1359.30	0.0679381
$738$	0	0
$739$	−15026.5	−0.747984	−0.373992	−	0.927432i	$-0.622011\pi$
$739$	−15026.5	−0.747984	−0.373992	+	0.927432i	$0.622011\pi$
$740$	18691.3	0.928521
$741$	0	0
$742$	1457.65	0.0721184
$743$	27390.8	1.35245	0.676226	−	0.736694i	$-0.263614\pi$
$743$	27390.8	1.35245	0.676226	+	0.736694i	$0.263614\pi$
$744$	0	0
$745$	37332.4	1.83591
$746$	−11219.5	−0.550635
$747$	0	0
$748$	3335.20	0.163031
$749$	−8104.46	−0.395368
$750$	0	0
$751$	−15310.0	−0.743903	−0.371951	−	0.928252i	$-0.621311\pi$
$751$	−15310.0	−0.743903	−0.371951	+	0.928252i	$0.621311\pi$
$752$	−6344.63	−0.307666
$753$	0	0
$754$	33859.6	1.63541
$755$	20065.1	0.967209
$756$	0	0
$757$	−314.895	−0.0151190	−0.00755948	−	0.999971i	$-0.502406\pi$
$757$	−314.895	−0.0151190	−0.00755948	+	0.999971i	$0.502406\pi$
$758$	−17028.9	−0.815985
$759$	0	0
$760$	−30656.5	−1.46320
$761$	7111.08	0.338734	0.169367	−	0.985553i	$-0.445828\pi$
$761$	7111.08	0.338734	0.169367	+	0.985553i	$0.445828\pi$
$762$	0	0
$763$	−8024.75	−0.380754
$764$	6356.42	0.301004
$765$	0	0
$766$	7245.17	0.341748
$767$	−4269.47	−0.200993
$768$	0	0
$769$	−6715.66	−0.314919	−0.157460	−	0.987525i	$-0.550330\pi$
$769$	−6715.66	−0.314919	−0.157460	+	0.987525i	$0.550330\pi$
$770$	1325.20	0.0620217
$771$	0	0
$772$	−9831.13	−0.458329
$773$	−21178.0	−0.985405	−0.492702	−	0.870198i	$-0.663991\pi$
$773$	−21178.0	−0.985405	−0.492702	+	0.870198i	$0.663991\pi$
$774$	0	0
$775$	−51.2306	−0.00237452
$776$	11996.2	0.554947
$777$	0	0
$778$	−8182.17	−0.377050
$779$	−39809.6	−1.83097
$780$	0	0
$781$	−1537.73	−0.0704535
$782$	4868.30	0.222622
$783$	0	0
$784$	618.583	0.0281789
$785$	−27472.0	−1.24907
$786$	0	0
$787$	−4152.15	−0.188066	−0.0940331	−	0.995569i	$-0.529976\pi$
$787$	−4152.15	−0.188066	−0.0940331	+	0.995569i	$0.529976\pi$
$788$	−14254.3	−0.644402
$789$	0	0
$790$	−1075.59	−0.0484401
$791$	−8571.48	−0.385293
$792$	0	0
$793$	20844.0	0.933409
$794$	12193.0	0.544977
$795$	0	0
$796$	−18054.9	−0.803945
$797$	−19804.0	−0.880168	−0.440084	−	0.897957i	$-0.645051\pi$
$797$	−19804.0	−0.880168	−0.440084	+	0.897957i	$0.645051\pi$
$798$	0	0
$799$	27094.2	1.19966
$800$	−62.0582	−0.00274261
$801$	0	0
$802$	14221.9	0.626177
$803$	−12735.5	−0.559684
$804$	0	0
$805$	−4579.02	−0.200483
$806$	−18883.5	−0.825238
$807$	0	0
$808$	41.8443	0.00182188
$809$	−30562.6	−1.32821	−0.664106	−	0.747638i	$-0.731188\pi$
$809$	−30562.6	−1.32821	−0.664106	+	0.747638i	$0.731188\pi$
$810$	0	0
$811$	4221.09	0.182765	0.0913825	−	0.995816i	$-0.470871\pi$
$811$	4221.09	0.182765	0.0913825	+	0.995816i	$0.470871\pi$
$812$	10924.2	0.472125
$813$	0	0
$814$	−5046.69	−0.217305
$815$	33031.0	1.41966
$816$	0	0
$817$	59561.1	2.55052
$818$	−6184.43	−0.264344
$819$	0	0
$820$	19120.5	0.814287
$821$	18569.2	0.789365	0.394682	−	0.918818i	$-0.370855\pi$
$821$	18569.2	0.789365	0.394682	+	0.918818i	$0.370855\pi$
$822$	0	0
$823$	−5263.94	−0.222952	−0.111476	−	0.993767i	$-0.535558\pi$
$823$	−5263.94	−0.222952	−0.111476	+	0.993767i	$0.535558\pi$
$824$	−8592.91	−0.363287
$825$	0	0
$826$	581.897	0.0245118
$827$	30511.9	1.28295	0.641477	−	0.767142i	$-0.278322\pi$
$827$	30511.9	1.28295	0.641477	+	0.767142i	$0.278322\pi$
$828$	0	0
$829$	26979.3	1.13031	0.565157	−	0.824983i	$-0.308816\pi$
$829$	26979.3	1.13031	0.565157	+	0.824983i	$0.308816\pi$
$830$	−13017.1	−0.544372
$831$	0	0
$832$	−14879.3	−0.620009
$833$	−2641.61	−0.109876
$834$	0	0
$835$	−27708.9	−1.14839
$836$	−8088.61	−0.334630
$837$	0	0
$838$	−4462.11	−0.183939
$839$	−13394.2	−0.551155	−0.275578	−	0.961279i	$-0.588869\pi$
$839$	−13394.2	−0.551155	−0.275578	+	0.961279i	$0.588869\pi$
$840$	0	0
$841$	52607.7	2.15703
$842$	−2125.67	−0.0870017
$843$	0	0
$844$	14479.4	0.590524
$845$	45445.7	1.85015
$846$	0	0
$847$	847.000	0.0343604
$848$	1705.48	0.0690641
$849$	0	0
$850$	27.5091	0.00111006
$851$	17438.1	0.702432
$852$	0	0
$853$	8004.81	0.321312	0.160656	−	0.987010i	$-0.448639\pi$
$853$	8004.81	0.321312	0.160656	+	0.987010i	$0.448639\pi$
$854$	−2840.89	−0.113833
$855$	0	0
$856$	24313.4	0.970811
$857$	−15387.1	−0.613317	−0.306659	−	0.951820i	$-0.599211\pi$
$857$	−15387.1	−0.613317	−0.306659	+	0.951820i	$0.599211\pi$
$858$	0	0
$859$	19182.6	0.761934	0.380967	−	0.924589i	$-0.375591\pi$
$859$	19182.6	0.761934	0.380967	+	0.924589i	$0.375591\pi$
$860$	−28607.1	−1.13429
$861$	0	0
$862$	−5197.78	−0.205379
$863$	−44743.6	−1.76488	−0.882439	−	0.470428i	$-0.844100\pi$
$863$	−44743.6	−1.76488	−0.882439	+	0.470428i	$0.844100\pi$
$864$	0	0
$865$	34216.6	1.34497
$866$	12538.9	0.492021
$867$	0	0
$868$	−6092.42	−0.238238
$869$	−687.462	−0.0268361
$870$	0	0
$871$	−9782.68	−0.380567
$872$	24074.3	0.934928
$873$	0	0
$874$	−11806.7	−0.456944
$875$	−9795.71	−0.378463
$876$	0	0
$877$	−40200.1	−1.54785	−0.773923	−	0.633280i	$-0.781708\pi$
$877$	−40200.1	−1.54785	−0.773923	+	0.633280i	$0.781708\pi$
$878$	2031.25	0.0780767
$879$	0	0
$880$	1550.51	0.0593950
$881$	−22326.7	−0.853808	−0.426904	−	0.904297i	$-0.640396\pi$
$881$	−22326.7	−0.853808	−0.426904	+	0.904297i	$0.640396\pi$
$882$	0	0
$883$	3823.30	0.145713	0.0728563	−	0.997342i	$-0.476789\pi$
$883$	3823.30	0.145713	0.0728563	+	0.997342i	$0.476789\pi$
$884$	−24003.0	−0.913243
$885$	0	0
$886$	909.550	0.0344886
$887$	−12613.8	−0.477487	−0.238743	−	0.971083i	$-0.576735\pi$
$887$	−12613.8	−0.477487	−0.238743	+	0.971083i	$0.576735\pi$
$888$	0	0
$889$	−13637.0	−0.514478
$890$	−2333.00	−0.0878679
$891$	0	0
$892$	−35262.5	−1.32363
$893$	−65709.7	−2.46236
$894$	0	0
$895$	5812.55	0.217086
$896$	−8469.74	−0.315797
$897$	0	0
$898$	4405.84	0.163725
$899$	−42941.0	−1.59306
$900$	0	0
$901$	−7283.10	−0.269296
$902$	−5162.57	−0.190571
$903$	0	0
$904$	25714.4	0.946072
$905$	52020.8	1.91075
$906$	0	0
$907$	22803.0	0.834799	0.417399	−	0.908723i	$-0.362942\pi$
$907$	22803.0	0.834799	0.417399	+	0.908723i	$0.362942\pi$
$908$	9911.30	0.362244
$909$	0	0
$910$	−9537.25	−0.347425
$911$	26549.0	0.965542	0.482771	−	0.875747i	$-0.339630\pi$
$911$	26549.0	0.965542	0.482771	+	0.875747i	$0.339630\pi$
$912$	0	0
$913$	−8319.87	−0.301586
$914$	−5240.52	−0.189651
$915$	0	0
$916$	−25851.5	−0.932486
$917$	365.550	0.0131642
$918$	0	0
$919$	−4076.31	−0.146317	−0.0731583	−	0.997320i	$-0.523308\pi$
$919$	−4076.31	−0.146317	−0.0731583	+	0.997320i	$0.523308\pi$
$920$	13737.1	0.492280
$921$	0	0
$922$	−16126.3	−0.576021
$923$	11066.8	0.394657
$924$	0	0
$925$	98.5365	0.00350255
$926$	7430.75	0.263704
$927$	0	0
$928$	−52016.6	−1.84001
$929$	−16336.6	−0.576949	−0.288474	−	0.957488i	$-0.593148\pi$
$929$	−16336.6	−0.576949	−0.288474	+	0.957488i	$0.593148\pi$
$930$	0	0
$931$	6406.50	0.225526
$932$	35410.9	1.24455
$933$	0	0
$934$	1747.58	0.0612234
$935$	−6621.32	−0.231594
$936$	0	0
$937$	24654.3	0.859575	0.429788	−	0.902930i	$-0.358588\pi$
$937$	24654.3	0.859575	0.429788	+	0.902930i	$0.358588\pi$
$938$	1333.31	0.0464115
$939$	0	0
$940$	31560.2	1.09509
$941$	33651.2	1.16578	0.582890	−	0.812551i	$-0.301922\pi$
$941$	33651.2	1.16578	0.582890	+	0.812551i	$0.301922\pi$
$942$	0	0
$943$	17838.5	0.616014
$944$	680.832	0.0234737
$945$	0	0
$946$	7723.97	0.265463
$947$	−7228.64	−0.248046	−0.124023	−	0.992279i	$-0.539580\pi$
$947$	−7228.64	−0.248046	−0.124023	+	0.992279i	$0.539580\pi$
$948$	0	0
$949$	91655.7	3.13516
$950$	−66.7158	−0.00227847
$951$	0	0
$952$	7924.83	0.269795
$953$	−11299.0	−0.384061	−0.192031	−	0.981389i	$-0.561507\pi$
$953$	−11299.0	−0.384061	−0.192031	+	0.981389i	$0.561507\pi$
$954$	0	0
$955$	−12619.3	−0.427592
$956$	−29609.6	−1.00172
$957$	0	0
$958$	−14302.4	−0.482348
$959$	9731.60	0.327685
$960$	0	0
$961$	−5842.91	−0.196130
$962$	36320.3	1.21727
$963$	0	0
$964$	−2104.70	−0.0703194
$965$	19517.6	0.651081
$966$	0	0
$967$	−4363.41	−0.145106	−0.0725531	−	0.997365i	$-0.523115\pi$
$967$	−4363.41	−0.145106	−0.0725531	+	0.997365i	$0.523115\pi$
$968$	−2541.00	−0.0843707
$969$	0	0
$970$	−9831.37	−0.325429
$971$	16418.2	0.542622	0.271311	−	0.962492i	$-0.412543\pi$
$971$	16418.2	0.542622	0.271311	+	0.962492i	$0.412543\pi$
$972$	0	0
$973$	3140.74	0.103481
$974$	−5939.83	−0.195405
$975$	0	0
$976$	−3323.90	−0.109012
$977$	13884.6	0.454664	0.227332	−	0.973817i	$-0.427000\pi$
$977$	13884.6	0.454664	0.227332	+	0.973817i	$0.427000\pi$
$978$	0	0
$979$	−1491.14	−0.0486793
$980$	−3077.03	−0.100298
$981$	0	0
$982$	−9614.16	−0.312424
$983$	−7127.22	−0.231254	−0.115627	−	0.993293i	$-0.536888\pi$
$983$	−7127.22	−0.231254	−0.115627	+	0.993293i	$0.536888\pi$
$984$	0	0
$985$	28298.9	0.915408
$986$	23057.8	0.744738
$987$	0	0
$988$	58212.6	1.87448
$989$	−26689.1	−0.858102
$990$	0	0
$991$	52353.5	1.67817	0.839084	−	0.544002i	$-0.183091\pi$
$991$	52353.5	1.67817	0.839084	+	0.544002i	$0.183091\pi$
$992$	29009.5	0.928482
$993$	0	0
$994$	−1508.32	−0.0481299
$995$	35844.2	1.14205
$996$	0	0
$997$	12262.9	0.389539	0.194770	−	0.980849i	$-0.437604\pi$
$997$	12262.9	0.389539	0.194770	+	0.980849i	$0.437604\pi$
$998$	3347.31	0.106170
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.h.1.2		2
3.2	odd	2		231.4.a.i.1.1	✓	2
21.20	even	2		1617.4.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.i.1.1	✓	2	3.2	odd	2
693.4.a.h.1.2		2	1.1	even	1	trivial
1617.4.a.l.1.1		2	21.20	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.54138 q^{2} -5.62414 q^{4} +11.1655 q^{5} +7.00000 q^{7} -21.0000 q^{8} +17.2103 q^{10} +11.0000 q^{11} -79.1655 q^{13} +10.7897 q^{14} +12.6241 q^{16} -53.9104 q^{17} +130.745 q^{19} -62.7965 q^{20} +16.9552 q^{22} -58.5862 q^{23} -0.331050 q^{25} -122.024 q^{26} -39.3690 q^{28} -277.483 q^{29} +154.752 q^{31} +187.459 q^{32} -83.0965 q^{34} +78.1587 q^{35} -297.648 q^{37} +201.528 q^{38} -234.476 q^{40} -304.483 q^{41} +455.552 q^{43} -61.8656 q^{44} -90.3037 q^{46} -502.579 q^{47} +49.0000 q^{49} -0.510274 q^{50} +445.238 q^{52} +135.096 q^{53} +122.821 q^{55} -147.000 q^{56} -427.707 q^{58} +53.9309 q^{59} -263.297 q^{61} +238.531 q^{62} +187.952 q^{64} -883.925 q^{65} +123.572 q^{67} +303.200 q^{68} +120.472 q^{70} -139.793 q^{71} -1157.77 q^{73} -458.790 q^{74} -735.328 q^{76} +77.0000 q^{77} -62.4966 q^{79} +140.955 q^{80} -469.324 q^{82} -756.352 q^{83} -601.938 q^{85} +702.179 q^{86} -231.000 q^{88} -135.558 q^{89} -554.159 q^{91} +329.497 q^{92} -774.666 q^{94} +1459.84 q^{95} -571.248 q^{97} +75.5277 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 7 q^{4} - 2 q^{5} + 14 q^{7} - 42 q^{8} + 77 q^{10} + 22 q^{11} - 134 q^{13} - 21 q^{14} + 7 q^{16} + 26 q^{17} + 152 q^{19} - 229 q^{20} - 33 q^{22} - 178 q^{23} + 48 q^{25} + 127 q^{26}+ \cdots - 147 q^{98}+O(q^{100}) \) 2 * q - 3 * q^2 + 7 * q^4 - 2 * q^5 + 14 * q^7 - 42 * q^8 + 77 * q^10 + 22 * q^11 - 134 * q^13 - 21 * q^14 + 7 * q^16 + 26 * q^17 + 152 * q^19 - 229 * q^20 - 33 * q^22 - 178 * q^23 + 48 * q^25 + 127 * q^26 + 49 * q^28 - 190 * q^29 + 346 * q^31 + 381 * q^32 - 446 * q^34 - 14 * q^35 - 206 * q^37 + 105 * q^38 + 42 * q^40 - 244 * q^41 + 242 * q^43 + 77 * q^44 + 452 * q^46 - 920 * q^47 + 98 * q^49 - 220 * q^50 - 247 * q^52 + 550 * q^53 - 22 * q^55 - 294 * q^56 - 825 * q^58 + 412 * q^59 + 252 * q^61 - 630 * q^62 - 646 * q^64 - 162 * q^65 + 16 * q^67 + 1312 * q^68 + 539 * q^70 + 572 * q^71 - 1026 * q^73 - 875 * q^74 - 467 * q^76 + 154 * q^77 - 52 * q^79 + 215 * q^80 - 744 * q^82 - 138 * q^83 - 1654 * q^85 + 1672 * q^86 - 462 * q^88 - 1512 * q^89 - 938 * q^91 - 1178 * q^92 + 1121 * q^94 + 1180 * q^95 - 1106 * q^97 - 147 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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