LMFDB - Embedded newform 693.4.a.p.1.2

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} - 30x^{3} + 11x^{2} + 185x + 162 $ x^5 - x^4 - 30x^3 + 11x^2 + 185*x + 162
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.59998$ 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)

$f(q)$	$=$	$q-1.59998 q^{2} -5.44007 q^{4} +17.2762 q^{5} +7.00000 q^{7} +21.5038 q^{8} +O(q^{10})$	\(q-1.59998 q^{2} -5.44007 q^{4} +17.2762 q^{5} +7.00000 q^{7} +21.5038 q^{8} -27.6416 q^{10} -11.0000 q^{11} +46.1062 q^{13} -11.1999 q^{14} +9.11483 q^{16} -19.8752 q^{17} +76.5707 q^{19} -93.9838 q^{20} +17.5998 q^{22} +163.205 q^{23} +173.468 q^{25} -73.7690 q^{26} -38.0805 q^{28} -158.858 q^{29} +170.768 q^{31} -186.614 q^{32} +31.7999 q^{34} +120.934 q^{35} -245.971 q^{37} -122.511 q^{38} +371.505 q^{40} +3.33673 q^{41} -122.798 q^{43} +59.8407 q^{44} -261.125 q^{46} -390.972 q^{47} +49.0000 q^{49} -277.545 q^{50} -250.821 q^{52} +410.957 q^{53} -190.038 q^{55} +150.527 q^{56} +254.169 q^{58} -408.774 q^{59} -21.9747 q^{61} -273.225 q^{62} +225.660 q^{64} +796.541 q^{65} +618.424 q^{67} +108.122 q^{68} -193.491 q^{70} -929.041 q^{71} +868.090 q^{73} +393.549 q^{74} -416.549 q^{76} -77.0000 q^{77} +152.702 q^{79} +157.470 q^{80} -5.33871 q^{82} +100.924 q^{83} -343.368 q^{85} +196.475 q^{86} -236.542 q^{88} +1063.23 q^{89} +322.743 q^{91} -887.848 q^{92} +625.547 q^{94} +1322.85 q^{95} +1415.50 q^{97} -78.3990 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 21 q^{4} - 21 q^{5} + 35 q^{7} + 42 q^{8}+O(q^{10}) $ 5 * q + q^2 + 21 * q^4 - 21 * q^5 + 35 * q^7 + 42 * q^8	$ 5 q + q^{2} + 21 q^{4} - 21 q^{5} + 35 q^{7} + 42 q^{8} - 23 q^{10} - 55 q^{11} + 101 q^{13} + 7 q^{14} - 7 q^{16} + 20 q^{17} + 237 q^{19} - 85 q^{20} - 11 q^{22} + 80 q^{23} + 486 q^{25} - 165 q^{26} + 147 q^{28} + 11 q^{29} + 316 q^{31} - 453 q^{32} + 936 q^{34} - 147 q^{35} + 319 q^{37} - 89 q^{38} + 624 q^{40} - 1190 q^{41} + 88 q^{43} - 231 q^{44} + 1000 q^{46} - 377 q^{47} + 245 q^{49} + 644 q^{50} + 1001 q^{52} + 992 q^{53} + 231 q^{55} + 294 q^{56} + 721 q^{58} - 71 q^{59} - 574 q^{61} - 272 q^{62} - 1380 q^{64} - 589 q^{65} - 527 q^{67} + 2974 q^{68} - 161 q^{70} + 1156 q^{71} + 1061 q^{73} + 1609 q^{74} + 2399 q^{76} - 385 q^{77} + 588 q^{79} + 1643 q^{80} - 2602 q^{82} + 212 q^{83} + 1918 q^{85} + 4760 q^{86} - 462 q^{88} - 1030 q^{89} + 707 q^{91} + 1174 q^{92} - 1799 q^{94} + 3593 q^{95} + 2488 q^{97} + 49 q^{98}+O(q^{100}) $ 5 * q + q^2 + 21 * q^4 - 21 * q^5 + 35 * q^7 + 42 * q^8 - 23 * q^10 - 55 * q^11 + 101 * q^13 + 7 * q^14 - 7 * q^16 + 20 * q^17 + 237 * q^19 - 85 * q^20 - 11 * q^22 + 80 * q^23 + 486 * q^25 - 165 * q^26 + 147 * q^28 + 11 * q^29 + 316 * q^31 - 453 * q^32 + 936 * q^34 - 147 * q^35 + 319 * q^37 - 89 * q^38 + 624 * q^40 - 1190 * q^41 + 88 * q^43 - 231 * q^44 + 1000 * q^46 - 377 * q^47 + 245 * q^49 + 644 * q^50 + 1001 * q^52 + 992 * q^53 + 231 * q^55 + 294 * q^56 + 721 * q^58 - 71 * q^59 - 574 * q^61 - 272 * q^62 - 1380 * q^64 - 589 * q^65 - 527 * q^67 + 2974 * q^68 - 161 * q^70 + 1156 * q^71 + 1061 * q^73 + 1609 * q^74 + 2399 * q^76 - 385 * q^77 + 588 * q^79 + 1643 * q^80 - 2602 * q^82 + 212 * q^83 + 1918 * q^85 + 4760 * q^86 - 462 * q^88 - 1030 * q^89 + 707 * q^91 + 1174 * q^92 - 1799 * q^94 + 3593 * q^95 + 2488 * 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 $

$2$	−1.59998	−0.565678	−0.282839	−	0.959167i	$-0.591276\pi$
$2$	−1.59998	−0.565678	−0.282839	+	0.959167i	$0.591276\pi$
$3$	0	0
$3$	0	0
$4$	−5.44007	−0.680008
$5$	17.2762	1.54523	0.772616	−	0.634873i	$-0.218948\pi$
$5$	17.2762	1.54523	0.772616	+	0.634873i	$0.218948\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	21.5038	0.950344
$9$	0	0
$10$	−27.6416	−0.874104
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	46.1062	0.983658	0.491829	−	0.870692i	$-0.336328\pi$
$13$	46.1062	0.983658	0.491829	+	0.870692i	$0.336328\pi$
$14$	−11.1999	−0.213806
$15$	0	0
$16$	9.11483	0.142419
$17$	−19.8752	−0.283555	−0.141778	−	0.989899i	$-0.545282\pi$
$17$	−19.8752	−0.283555	−0.141778	+	0.989899i	$0.545282\pi$
$18$	0	0
$19$	76.5707	0.924553	0.462277	−	0.886736i	$-0.347033\pi$
$19$	76.5707	0.924553	0.462277	+	0.886736i	$0.347033\pi$
$20$	−93.9838	−1.05077
$21$	0	0
$22$	17.5998	0.170558
$23$	163.205	1.47960	0.739798	−	0.672829i	$-0.234921\pi$
$23$	163.205	1.47960	0.739798	+	0.672829i	$0.234921\pi$
$24$	0	0
$25$	173.468	1.38774
$26$	−73.7690	−0.556434
$27$	0	0
$28$	−38.0805	−0.257019
$29$	−158.858	−1.01721	−0.508605	−	0.861000i	$-0.669839\pi$
$29$	−158.858	−1.01721	−0.508605	+	0.861000i	$0.669839\pi$
$30$	0	0
$31$	170.768	0.989380	0.494690	−	0.869069i	$-0.335282\pi$
$31$	170.768	0.989380	0.494690	+	0.869069i	$0.335282\pi$
$32$	−186.614	−1.03091
$33$	0	0
$34$	31.7999	0.160401
$35$	120.934	0.584043
$36$	0	0
$37$	−245.971	−1.09290	−0.546452	−	0.837490i	$-0.684022\pi$
$37$	−245.971	−1.09290	−0.546452	+	0.837490i	$0.684022\pi$
$38$	−122.511	−0.523000
$39$	0	0
$40$	371.505	1.46850
$41$	3.33673	0.0127100	0.00635500	−	0.999980i	$-0.497977\pi$
$41$	3.33673	0.0127100	0.00635500	+	0.999980i	$0.497977\pi$
$42$	0	0
$43$	−122.798	−0.435502	−0.217751	−	0.976004i	$-0.569872\pi$
$43$	−122.798	−0.435502	−0.217751	+	0.976004i	$0.569872\pi$
$44$	59.8407	0.205030
$45$	0	0
$46$	−261.125	−0.836975
$47$	−390.972	−1.21338	−0.606692	−	0.794937i	$-0.707504\pi$
$47$	−390.972	−1.21338	−0.606692	+	0.794937i	$0.707504\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−277.545	−0.785016
$51$	0	0
$52$	−250.821	−0.668896
$53$	410.957	1.06508	0.532541	−	0.846404i	$-0.321237\pi$
$53$	410.957	1.06508	0.532541	+	0.846404i	$0.321237\pi$
$54$	0	0
$55$	−190.038	−0.465905
$56$	150.527	0.359196
$57$	0	0
$58$	254.169	0.575414
$59$	−408.774	−0.901998	−0.450999	−	0.892524i	$-0.648932\pi$
$59$	−408.774	−0.901998	−0.450999	+	0.892524i	$0.648932\pi$
$60$	0	0
$61$	−21.9747	−0.0461241	−0.0230621	−	0.999734i	$-0.507342\pi$
$61$	−21.9747	−0.0461241	−0.0230621	+	0.999734i	$0.507342\pi$
$62$	−273.225	−0.559671
$63$	0	0
$64$	225.660	0.440743
$65$	796.541	1.51998
$66$	0	0
$67$	618.424	1.12765	0.563824	−	0.825895i	$-0.309329\pi$
$67$	618.424	1.12765	0.563824	+	0.825895i	$0.309329\pi$
$68$	108.122	0.192820
$69$	0	0
$70$	−193.491	−0.330380
$71$	−929.041	−1.55291	−0.776457	−	0.630171i	$-0.782985\pi$
$71$	−929.041	−1.55291	−0.776457	+	0.630171i	$0.782985\pi$
$72$	0	0
$73$	868.090	1.39181	0.695906	−	0.718133i	$-0.255003\pi$
$73$	868.090	1.39181	0.695906	+	0.718133i	$0.255003\pi$
$74$	393.549	0.618232
$75$	0	0
$76$	−416.549	−0.628704
$77$	−77.0000	−0.113961
$78$	0	0
$79$	152.702	0.217472	0.108736	−	0.994071i	$-0.465320\pi$
$79$	152.702	0.217472	0.108736	+	0.994071i	$0.465320\pi$
$80$	157.470	0.220071
$81$	0	0
$82$	−5.33871	−0.00718977
$83$	100.924	0.133469	0.0667343	−	0.997771i	$-0.478742\pi$
$83$	100.924	0.133469	0.0667343	+	0.997771i	$0.478742\pi$
$84$	0	0
$85$	−343.368	−0.438159
$86$	196.475	0.246354
$87$	0	0
$88$	−236.542	−0.286540
$89$	1063.23	1.26631	0.633156	−	0.774024i	$-0.281759\pi$
$89$	1063.23	1.26631	0.633156	+	0.774024i	$0.281759\pi$
$90$	0	0
$91$	322.743	0.371788
$92$	−887.848	−1.00614
$93$	0	0
$94$	625.547	0.686385
$95$	1322.85	1.42865
$96$	0	0
$97$	1415.50	1.48167	0.740836	−	0.671686i	$-0.234430\pi$
$97$	1415.50	1.48167	0.740836	+	0.671686i	$0.234430\pi$
$98$	−78.3990	−0.0808112
$99$	0	0
$100$	−943.677	−0.943677
$101$	774.140	0.762671	0.381336	−	0.924437i	$-0.375464\pi$
$101$	774.140	0.762671	0.381336	+	0.924437i	$0.375464\pi$
$102$	0	0
$103$	759.159	0.726235	0.363118	−	0.931743i	$-0.381712\pi$
$103$	759.159	0.726235	0.363118	+	0.931743i	$0.381712\pi$
$104$	991.460	0.934814
$105$	0	0
$106$	−657.523	−0.602493
$107$	1181.22	1.06722	0.533612	−	0.845729i	$-0.320834\pi$
$107$	1181.22	1.06722	0.533612	+	0.845729i	$0.320834\pi$
$108$	0	0
$109$	738.091	0.648590	0.324295	−	0.945956i	$-0.394873\pi$
$109$	738.091	0.648590	0.324295	+	0.945956i	$0.394873\pi$
$110$	304.058	0.263552
$111$	0	0
$112$	63.8038	0.0538294
$113$	−620.181	−0.516298	−0.258149	−	0.966105i	$-0.583113\pi$
$113$	−620.181	−0.516298	−0.258149	+	0.966105i	$0.583113\pi$
$114$	0	0
$115$	2819.57	2.28632
$116$	864.195	0.691711
$117$	0	0
$118$	654.031	0.510241
$119$	−139.126	−0.107174
$120$	0	0
$121$	121.000	0.0909091
$122$	35.1591	0.0260914
$123$	0	0
$124$	−928.988	−0.672787
$125$	837.342	0.599153
$126$	0	0
$127$	−2291.79	−1.60129	−0.800645	−	0.599139i	$-0.795510\pi$
$127$	−2291.79	−1.60129	−0.800645	+	0.599139i	$0.795510\pi$
$128$	1131.86	0.781589
$129$	0	0
$130$	−1274.45	−0.859820
$131$	1491.77	0.994938	0.497469	−	0.867482i	$-0.334263\pi$
$131$	1491.77	0.994938	0.497469	+	0.867482i	$0.334263\pi$
$132$	0	0
$133$	535.995	0.349448
$134$	−989.465	−0.637886
$135$	0	0
$136$	−427.393	−0.269475
$137$	−2590.79	−1.61566	−0.807831	−	0.589414i	$-0.799359\pi$
$137$	−2590.79	−1.61566	−0.807831	+	0.589414i	$0.799359\pi$
$138$	0	0
$139$	2674.14	1.63178	0.815890	−	0.578207i	$-0.196248\pi$
$139$	2674.14	1.63178	0.815890	+	0.578207i	$0.196248\pi$
$140$	−657.886	−0.397154
$141$	0	0
$142$	1486.45	0.878449
$143$	−507.168	−0.296584
$144$	0	0
$145$	−2744.46	−1.57183
$146$	−1388.93	−0.787318
$147$	0	0
$148$	1338.10	0.743184
$149$	−822.355	−0.452147	−0.226074	−	0.974110i	$-0.572589\pi$
$149$	−822.355	−0.452147	−0.226074	+	0.974110i	$0.572589\pi$
$150$	0	0
$151$	−138.803	−0.0748053	−0.0374027	−	0.999300i	$-0.511908\pi$
$151$	−138.803	−0.0748053	−0.0374027	+	0.999300i	$0.511908\pi$
$152$	1646.56	0.878644
$153$	0	0
$154$	123.198	0.0644650
$155$	2950.22	1.52882
$156$	0	0
$157$	−634.746	−0.322664	−0.161332	−	0.986900i	$-0.551579\pi$
$157$	−634.746	−0.322664	−0.161332	+	0.986900i	$0.551579\pi$
$158$	−244.320	−0.123019
$159$	0	0
$160$	−3223.99	−1.59299
$161$	1142.44	0.559235
$162$	0	0
$163$	−608.367	−0.292337	−0.146169	−	0.989260i	$-0.546694\pi$
$163$	−608.367	−0.292337	−0.146169	+	0.989260i	$0.546694\pi$
$164$	−18.1520	−0.00864291
$165$	0	0
$166$	−161.477	−0.0755003
$167$	1545.50	0.716135	0.358067	−	0.933696i	$-0.383436\pi$
$167$	1545.50	0.716135	0.358067	+	0.933696i	$0.383436\pi$
$168$	0	0
$169$	−71.2185	−0.0324163
$170$	549.382	0.247857
$171$	0	0
$172$	668.031	0.296145
$173$	−126.598	−0.0556362	−0.0278181	−	0.999613i	$-0.508856\pi$
$173$	−126.598	−0.0556362	−0.0278181	+	0.999613i	$0.508856\pi$
$174$	0	0
$175$	1214.28	0.524518
$176$	−100.263	−0.0429410
$177$	0	0
$178$	−1701.14	−0.716325
$179$	144.263	0.0602387	0.0301194	−	0.999546i	$-0.490411\pi$
$179$	144.263	0.0602387	0.0301194	+	0.999546i	$0.490411\pi$
$180$	0	0
$181$	2925.01	1.20118	0.600591	−	0.799556i	$-0.294932\pi$
$181$	2925.01	1.20118	0.600591	+	0.799556i	$0.294932\pi$
$182$	−516.383	−0.210312
$183$	0	0
$184$	3509.54	1.40612
$185$	−4249.46	−1.68879
$186$	0	0
$187$	218.627	0.0854951
$188$	2126.91	0.825111
$189$	0	0
$190$	−2116.54	−0.808156
$191$	3771.70	1.42885	0.714426	−	0.699711i	$-0.246688\pi$
$191$	3771.70	1.42885	0.714426	+	0.699711i	$0.246688\pi$
$192$	0	0
$193$	−1282.92	−0.478480	−0.239240	−	0.970960i	$-0.576898\pi$
$193$	−1282.92	−0.478480	−0.239240	+	0.970960i	$0.576898\pi$
$194$	−2264.77	−0.838149
$195$	0	0
$196$	−266.563	−0.0971440
$197$	−3052.31	−1.10390	−0.551949	−	0.833878i	$-0.686116\pi$
$197$	−3052.31	−1.10390	−0.551949	+	0.833878i	$0.686116\pi$
$198$	0	0
$199$	5040.28	1.79546	0.897729	−	0.440547i	$-0.145216\pi$
$199$	5040.28	1.79546	0.897729	+	0.440547i	$0.145216\pi$
$200$	3730.22	1.31883
$201$	0	0
$202$	−1238.61	−0.431426
$203$	−1112.00	−0.384469
$204$	0	0
$205$	57.6462	0.0196399
$206$	−1214.64	−0.410815
$207$	0	0
$208$	420.250	0.140092
$209$	−842.277	−0.278763
$210$	0	0
$211$	3556.28	1.16031	0.580153	−	0.814508i	$-0.302993\pi$
$211$	3556.28	1.16031	0.580153	+	0.814508i	$0.302993\pi$
$212$	−2235.63	−0.724264
$213$	0	0
$214$	−1889.93	−0.603706
$215$	−2121.49	−0.672952
$216$	0	0
$217$	1195.37	0.373951
$218$	−1180.93	−0.366893
$219$	0	0
$220$	1033.82	0.316819
$221$	−916.369	−0.278922
$222$	0	0
$223$	1767.85	0.530870	0.265435	−	0.964129i	$-0.414484\pi$
$223$	1767.85	0.530870	0.265435	+	0.964129i	$0.414484\pi$
$224$	−1306.30	−0.389646
$225$	0	0
$226$	992.277	0.292059
$227$	−4836.74	−1.41421	−0.707105	−	0.707109i	$-0.749999\pi$
$227$	−4836.74	−1.41421	−0.707105	+	0.707109i	$0.749999\pi$
$228$	0	0
$229$	4941.30	1.42590	0.712949	−	0.701216i	$-0.247359\pi$
$229$	4941.30	1.42590	0.712949	+	0.701216i	$0.247359\pi$
$230$	−4511.26	−1.29332
$231$	0	0
$232$	−3416.04	−0.966700
$233$	−4822.14	−1.35583	−0.677916	−	0.735140i	$-0.737117\pi$
$233$	−4822.14	−1.35583	−0.677916	+	0.735140i	$0.737117\pi$
$234$	0	0
$235$	−6754.51	−1.87496
$236$	2223.76	0.613366
$237$	0	0
$238$	222.599	0.0606259
$239$	−4821.80	−1.30501	−0.652503	−	0.757786i	$-0.726281\pi$
$239$	−4821.80	−1.30501	−0.652503	+	0.757786i	$0.726281\pi$
$240$	0	0
$241$	1544.41	0.412798	0.206399	−	0.978468i	$-0.433825\pi$
$241$	1544.41	0.412798	0.206399	+	0.978468i	$0.433825\pi$
$242$	−193.598	−0.0514253
$243$	0	0
$244$	119.544	0.0313648
$245$	846.535	0.220747
$246$	0	0
$247$	3530.38	0.909445
$248$	3672.16	0.940252
$249$	0	0
$250$	−1339.73	−0.338928
$251$	7568.20	1.90319	0.951595	−	0.307354i	$-0.0994436\pi$
$251$	7568.20	1.90319	0.951595	+	0.307354i	$0.0994436\pi$
$252$	0	0
$253$	−1795.26	−0.446115
$254$	3666.82	0.905815
$255$	0	0
$256$	−3616.24	−0.882871
$257$	−1415.19	−0.343490	−0.171745	−	0.985141i	$-0.554941\pi$
$257$	−1415.19	−0.343490	−0.171745	+	0.985141i	$0.554941\pi$
$258$	0	0
$259$	−1721.80	−0.413079
$260$	−4333.23	−1.03360
$261$	0	0
$262$	−2386.81	−0.562815
$263$	7875.57	1.84650	0.923248	−	0.384204i	$-0.125524\pi$
$263$	7875.57	1.84650	0.923248	+	0.384204i	$0.125524\pi$
$264$	0	0
$265$	7099.79	1.64580
$266$	−857.580	−0.197675
$267$	0	0
$268$	−3364.27	−0.766810
$269$	2062.01	0.467371	0.233685	−	0.972312i	$-0.424921\pi$
$269$	2062.01	0.467371	0.233685	+	0.972312i	$0.424921\pi$
$270$	0	0
$271$	−8247.07	−1.84861	−0.924306	−	0.381651i	$-0.875355\pi$
$271$	−8247.07	−1.84861	−0.924306	+	0.381651i	$0.875355\pi$
$272$	−181.159	−0.0403837
$273$	0	0
$274$	4145.21	0.913945
$275$	−1908.15	−0.418420
$276$	0	0
$277$	−6637.33	−1.43971	−0.719853	−	0.694126i	$-0.755791\pi$
$277$	−6637.33	−1.43971	−0.719853	+	0.694126i	$0.755791\pi$
$278$	−4278.57	−0.923062
$279$	0	0
$280$	2600.53	0.555042
$281$	−5600.82	−1.18903	−0.594515	−	0.804085i	$-0.702656\pi$
$281$	−5600.82	−1.18903	−0.594515	+	0.804085i	$0.702656\pi$
$282$	0	0
$283$	−934.745	−0.196342	−0.0981711	−	0.995170i	$-0.531299\pi$
$283$	−934.745	−0.196342	−0.0981711	+	0.995170i	$0.531299\pi$
$284$	5054.04	1.05599
$285$	0	0
$286$	811.459	0.167771
$287$	23.3571	0.00480393
$288$	0	0
$289$	−4517.98	−0.919596
$290$	4391.08	0.889148
$291$	0	0
$292$	−4722.47	−0.946444
$293$	−9169.86	−1.82836	−0.914179	−	0.405311i	$-0.867163\pi$
$293$	−9169.86	−1.82836	−0.914179	+	0.405311i	$0.867163\pi$
$294$	0	0
$295$	−7062.08	−1.39380
$296$	−5289.33	−1.03864
$297$	0	0
$298$	1315.75	0.255770
$299$	7524.78	1.45542
$300$	0	0
$301$	−859.589	−0.164604
$302$	222.081	0.0423157
$303$	0	0
$304$	697.929	0.131674
$305$	−379.640	−0.0712725
$306$	0	0
$307$	−6193.71	−1.15144	−0.575722	−	0.817645i	$-0.695279\pi$
$307$	−6193.71	−1.15144	−0.575722	+	0.817645i	$0.695279\pi$
$308$	418.885	0.0774941
$309$	0	0
$310$	−4720.29	−0.864822
$311$	3373.58	0.615106	0.307553	−	0.951531i	$-0.400490\pi$
$311$	3373.58	0.615106	0.307553	+	0.951531i	$0.400490\pi$
$312$	0	0
$313$	7175.09	1.29572	0.647860	−	0.761760i	$-0.275664\pi$
$313$	7175.09	1.29572	0.647860	+	0.761760i	$0.275664\pi$
$314$	1015.58	0.182524
$315$	0	0
$316$	−830.709	−0.147883
$317$	6302.06	1.11659	0.558295	−	0.829643i	$-0.311456\pi$
$317$	6302.06	1.11659	0.558295	+	0.829643i	$0.311456\pi$
$318$	0	0
$319$	1747.43	0.306700
$320$	3898.56	0.681050
$321$	0	0
$322$	−1827.88	−0.316347
$323$	−1521.86	−0.262162
$324$	0	0
$325$	7997.94	1.36506
$326$	973.375	0.165369
$327$	0	0
$328$	71.7526	0.0120789
$329$	−2736.80	−0.458616
$330$	0	0
$331$	−7404.03	−1.22949	−0.614747	−	0.788725i	$-0.710742\pi$
$331$	−7404.03	−1.22949	−0.614747	+	0.788725i	$0.710742\pi$
$332$	−549.035	−0.0907597
$333$	0	0
$334$	−2472.77	−0.405102
$335$	10684.0	1.74248
$336$	0	0
$337$	7008.68	1.13290	0.566450	−	0.824096i	$-0.308316\pi$
$337$	7008.68	1.13290	0.566450	+	0.824096i	$0.308316\pi$
$338$	113.948	0.0183372
$339$	0	0
$340$	1867.94	0.297952
$341$	−1878.45	−0.298309
$342$	0	0
$343$	343.000	0.0539949
$344$	−2640.64	−0.413877
$345$	0	0
$346$	202.554	0.0314722
$347$	10025.8	1.55104	0.775521	−	0.631322i	$-0.217487\pi$
$347$	10025.8	1.55104	0.775521	+	0.631322i	$0.217487\pi$
$348$	0	0
$349$	−10746.5	−1.64827	−0.824136	−	0.566391i	$-0.808339\pi$
$349$	−10746.5	−1.64827	−0.824136	+	0.566391i	$0.808339\pi$
$350$	−1942.82	−0.296708
$351$	0	0
$352$	2052.76	0.310830
$353$	606.136	0.0913919	0.0456960	−	0.998955i	$-0.485449\pi$
$353$	606.136	0.0913919	0.0456960	+	0.998955i	$0.485449\pi$
$354$	0	0
$355$	−16050.3	−2.39961
$356$	−5784.02	−0.861102
$357$	0	0
$358$	−230.818	−0.0340757
$359$	−8527.32	−1.25363	−0.626817	−	0.779166i	$-0.715643\pi$
$359$	−8527.32	−1.25363	−0.626817	+	0.779166i	$0.715643\pi$
$360$	0	0
$361$	−995.934	−0.145201
$362$	−4679.95	−0.679483
$363$	0	0
$364$	−1755.75	−0.252819
$365$	14997.3	2.15067
$366$	0	0
$367$	5504.16	0.782874	0.391437	−	0.920205i	$-0.371978\pi$
$367$	5504.16	0.782874	0.391437	+	0.920205i	$0.371978\pi$
$368$	1487.59	0.210723
$369$	0	0
$370$	6799.05	0.955312
$371$	2876.70	0.402563
$372$	0	0
$373$	−7091.36	−0.984387	−0.492194	−	0.870486i	$-0.663805\pi$
$373$	−7091.36	−0.984387	−0.492194	+	0.870486i	$0.663805\pi$
$374$	−349.799	−0.0483627
$375$	0	0
$376$	−8407.39	−1.15313
$377$	−7324.32	−1.00059
$378$	0	0
$379$	−77.3370	−0.0104816	−0.00524081	−	0.999986i	$-0.501668\pi$
$379$	−77.3370	−0.0104816	−0.00524081	+	0.999986i	$0.501668\pi$
$380$	−7196.40	−0.971493
$381$	0	0
$382$	−6034.65	−0.808271
$383$	−4843.75	−0.646224	−0.323112	−	0.946361i	$-0.604729\pi$
$383$	−4843.75	−0.646224	−0.323112	+	0.946361i	$0.604729\pi$
$384$	0	0
$385$	−1330.27	−0.176096
$386$	2052.65	0.270666
$387$	0	0
$388$	−7700.41	−1.00755
$389$	−9882.00	−1.28801	−0.644007	−	0.765020i	$-0.722729\pi$
$389$	−9882.00	−1.28801	−0.644007	+	0.765020i	$0.722729\pi$
$390$	0	0
$391$	−3243.74	−0.419547
$392$	1053.69	0.135763
$393$	0	0
$394$	4883.63	0.624451
$395$	2638.11	0.336045
$396$	0	0
$397$	−10135.5	−1.28133	−0.640663	−	0.767823i	$-0.721340\pi$
$397$	−10135.5	−1.28133	−0.640663	+	0.767823i	$0.721340\pi$
$398$	−8064.35	−1.01565
$399$	0	0
$400$	1581.13	0.197641
$401$	−9220.96	−1.14831	−0.574155	−	0.818746i	$-0.694669\pi$
$401$	−9220.96	−1.14831	−0.574155	+	0.818746i	$0.694669\pi$
$402$	0	0
$403$	7873.45	0.973212
$404$	−4211.37	−0.518623
$405$	0	0
$406$	1779.18	0.217486
$407$	2705.69	0.329523
$408$	0	0
$409$	−5710.89	−0.690429	−0.345214	−	0.938524i	$-0.612194\pi$
$409$	−5710.89	−0.690429	−0.345214	+	0.938524i	$0.612194\pi$
$410$	−92.2327	−0.0111099
$411$	0	0
$412$	−4129.88	−0.493846
$413$	−2861.42	−0.340923
$414$	0	0
$415$	1743.59	0.206240
$416$	−8604.07	−1.01406
$417$	0	0
$418$	1347.63	0.157690
$419$	−4520.73	−0.527093	−0.263547	−	0.964647i	$-0.584892\pi$
$419$	−4520.73	−0.527093	−0.263547	+	0.964647i	$0.584892\pi$
$420$	0	0
$421$	3798.26	0.439705	0.219853	−	0.975533i	$-0.429442\pi$
$421$	3798.26	0.439705	0.219853	+	0.975533i	$0.429442\pi$
$422$	−5689.98	−0.656359
$423$	0	0
$424$	8837.15	1.01219
$425$	−3447.71	−0.393502
$426$	0	0
$427$	−153.823	−0.0174333
$428$	−6425.92	−0.725721
$429$	0	0
$430$	3394.35	0.380674
$431$	3468.51	0.387638	0.193819	−	0.981037i	$-0.437912\pi$
$431$	3468.51	0.387638	0.193819	+	0.981037i	$0.437912\pi$
$432$	0	0
$433$	−9859.98	−1.09432	−0.547160	−	0.837028i	$-0.684291\pi$
$433$	−9859.98	−1.09432	−0.547160	+	0.837028i	$0.684291\pi$
$434$	−1912.57	−0.211536
$435$	0	0
$436$	−4015.26	−0.441046
$437$	12496.8	1.36796
$438$	0	0
$439$	8464.33	0.920228	0.460114	−	0.887860i	$-0.347808\pi$
$439$	8464.33	0.920228	0.460114	+	0.887860i	$0.347808\pi$
$440$	−4086.55	−0.442770
$441$	0	0
$442$	1466.17	0.157780
$443$	7825.41	0.839270	0.419635	−	0.907693i	$-0.362158\pi$
$443$	7825.41	0.839270	0.419635	+	0.907693i	$0.362158\pi$
$444$	0	0
$445$	18368.5	1.95675
$446$	−2828.52	−0.300302
$447$	0	0
$448$	1579.62	0.166585
$449$	−1050.94	−0.110461	−0.0552303	−	0.998474i	$-0.517589\pi$
$449$	−1050.94	−0.110461	−0.0552303	+	0.998474i	$0.517589\pi$
$450$	0	0
$451$	−36.7041	−0.00383221
$452$	3373.83	0.351087
$453$	0	0
$454$	7738.68	0.799988
$455$	5575.79	0.574499
$456$	0	0
$457$	14397.2	1.47369	0.736843	−	0.676064i	$-0.236316\pi$
$457$	14397.2	1.47369	0.736843	+	0.676064i	$0.236316\pi$
$458$	−7905.98	−0.806599
$459$	0	0
$460$	−15338.7	−1.55472
$461$	5755.83	0.581509	0.290755	−	0.956798i	$-0.406094\pi$
$461$	5755.83	0.581509	0.290755	+	0.956798i	$0.406094\pi$
$462$	0	0
$463$	−10056.0	−1.00938	−0.504689	−	0.863302i	$-0.668393\pi$
$463$	−10056.0	−1.00938	−0.504689	+	0.863302i	$0.668393\pi$
$464$	−1447.96	−0.144870
$465$	0	0
$466$	7715.32	0.766964
$467$	−2607.32	−0.258356	−0.129178	−	0.991621i	$-0.541234\pi$
$467$	−2607.32	−0.258356	−0.129178	+	0.991621i	$0.541234\pi$
$468$	0	0
$469$	4328.97	0.426211
$470$	10807.1	1.06062
$471$	0	0
$472$	−8790.21	−0.857208
$473$	1350.78	0.131309
$474$	0	0
$475$	13282.5	1.28304
$476$	756.856	0.0728791
$477$	0	0
$478$	7714.79	0.738214
$479$	5944.97	0.567083	0.283541	−	0.958960i	$-0.408491\pi$
$479$	5944.97	0.567083	0.283541	+	0.958960i	$0.408491\pi$
$480$	0	0
$481$	−11340.8	−1.07504
$482$	−2471.03	−0.233511
$483$	0	0
$484$	−658.248	−0.0618189
$485$	24454.5	2.28953
$486$	0	0
$487$	15895.1	1.47901	0.739504	−	0.673152i	$-0.235060\pi$
$487$	15895.1	1.47901	0.739504	+	0.673152i	$0.235060\pi$
$488$	−472.540	−0.0438338
$489$	0	0
$490$	−1354.44	−0.124872
$491$	−3655.59	−0.335997	−0.167999	−	0.985787i	$-0.553730\pi$
$491$	−3655.59	−0.335997	−0.167999	+	0.985787i	$0.553730\pi$
$492$	0	0
$493$	3157.32	0.288435
$494$	−5648.54	−0.514453
$495$	0	0
$496$	1556.52	0.140907
$497$	−6503.28	−0.586946
$498$	0	0
$499$	21171.9	1.89937	0.949684	−	0.313208i	$-0.101404\pi$
$499$	21171.9	1.89937	0.949684	+	0.313208i	$0.101404\pi$
$500$	−4555.19	−0.407429
$501$	0	0
$502$	−12109.0	−1.07659
$503$	12270.3	1.08768	0.543842	−	0.839188i	$-0.316969\pi$
$503$	12270.3	1.08768	0.543842	+	0.839188i	$0.316969\pi$
$504$	0	0
$505$	13374.2	1.17850
$506$	2872.38	0.252357
$507$	0	0
$508$	12467.5	1.08889
$509$	3168.41	0.275908	0.137954	−	0.990439i	$-0.455947\pi$
$509$	3168.41	0.275908	0.137954	+	0.990439i	$0.455947\pi$
$510$	0	0
$511$	6076.63	0.526056
$512$	−3268.99	−0.282168
$513$	0	0
$514$	2264.27	0.194305
$515$	13115.4	1.12220
$516$	0	0
$517$	4300.69	0.365849
$518$	2754.85	0.233670
$519$	0	0
$520$	17128.7	1.44450
$521$	1305.16	0.109751	0.0548755	−	0.998493i	$-0.482524\pi$
$521$	1305.16	0.109751	0.0548755	+	0.998493i	$0.482524\pi$
$522$	0	0
$523$	707.998	0.0591943	0.0295971	−	0.999562i	$-0.490578\pi$
$523$	707.998	0.0591943	0.0295971	+	0.999562i	$0.490578\pi$
$524$	−8115.35	−0.676566
$525$	0	0
$526$	−12600.8	−1.04452
$527$	−3394.04	−0.280544
$528$	0	0
$529$	14469.0	1.18920
$530$	−11359.5	−0.930992
$531$	0	0
$532$	−2915.85	−0.237628
$533$	153.844	0.0125023
$534$	0	0
$535$	20407.0	1.64911
$536$	13298.5	1.07165
$537$	0	0
$538$	−3299.17	−0.264382
$539$	−539.000	−0.0430730
$540$	0	0
$541$	−5424.05	−0.431050	−0.215525	−	0.976498i	$-0.569146\pi$
$541$	−5424.05	−0.431050	−0.215525	+	0.976498i	$0.569146\pi$
$542$	13195.1	1.04572
$543$	0	0
$544$	3708.99	0.292319
$545$	12751.4	1.00222
$546$	0	0
$547$	−15353.7	−1.20014	−0.600071	−	0.799947i	$-0.704861\pi$
$547$	−15353.7	−1.20014	−0.600071	+	0.799947i	$0.704861\pi$
$548$	14094.0	1.09866
$549$	0	0
$550$	3053.00	0.236691
$551$	−12163.8	−0.940465
$552$	0	0
$553$	1068.91	0.0821968
$554$	10619.6	0.814410
$555$	0	0
$556$	−14547.5	−1.10962
$557$	−13685.2	−1.04104	−0.520519	−	0.853850i	$-0.674262\pi$
$557$	−13685.2	−1.04104	−0.520519	+	0.853850i	$0.674262\pi$
$558$	0	0
$559$	−5661.77	−0.428385
$560$	1102.29	0.0831789
$561$	0	0
$562$	8961.21	0.672608
$563$	1565.67	0.117203	0.0586015	−	0.998281i	$-0.481336\pi$
$563$	1565.67	0.117203	0.0586015	+	0.998281i	$0.481336\pi$
$564$	0	0
$565$	−10714.4	−0.797801
$566$	1495.57	0.111066
$567$	0	0
$568$	−19977.9	−1.47580
$569$	−20508.1	−1.51097	−0.755487	−	0.655164i	$-0.772600\pi$
$569$	−20508.1	−1.51097	−0.755487	+	0.655164i	$0.772600\pi$
$570$	0	0
$571$	971.535	0.0712040	0.0356020	−	0.999366i	$-0.488665\pi$
$571$	971.535	0.0712040	0.0356020	+	0.999366i	$0.488665\pi$
$572$	2759.03	0.201680
$573$	0	0
$574$	−37.3709	−0.00271748
$575$	28310.9	2.05330
$576$	0	0
$577$	13961.1	1.00729	0.503647	−	0.863910i	$-0.331991\pi$
$577$	13961.1	1.00729	0.503647	+	0.863910i	$0.331991\pi$
$578$	7228.67	0.520196
$579$	0	0
$580$	14930.0	1.06885
$581$	706.471	0.0504464
$582$	0	0
$583$	−4520.53	−0.321134
$584$	18667.3	1.32270
$585$	0	0
$586$	14671.6	1.03426
$587$	23834.3	1.67589	0.837946	−	0.545753i	$-0.183756\pi$
$587$	23834.3	1.67589	0.837946	+	0.545753i	$0.183756\pi$
$588$	0	0
$589$	13075.8	0.914735
$590$	11299.2	0.788440
$591$	0	0
$592$	−2241.99	−0.155651
$593$	−25102.9	−1.73837	−0.869184	−	0.494489i	$-0.835355\pi$
$593$	−25102.9	−1.73837	−0.869184	+	0.494489i	$0.835355\pi$
$594$	0	0
$595$	−2403.58	−0.165608
$596$	4473.66	0.307464
$597$	0	0
$598$	−12039.5	−0.823297
$599$	12122.2	0.826875	0.413437	−	0.910533i	$-0.364328\pi$
$599$	12122.2	0.826875	0.413437	+	0.910533i	$0.364328\pi$
$600$	0	0
$601$	20672.9	1.40310	0.701552	−	0.712619i	$-0.252491\pi$
$601$	20672.9	1.40310	0.701552	+	0.712619i	$0.252491\pi$
$602$	1375.32	0.0931130
$603$	0	0
$604$	755.096	0.0508682
$605$	2090.42	0.140476
$606$	0	0
$607$	−3571.58	−0.238824	−0.119412	−	0.992845i	$-0.538101\pi$
$607$	−3571.58	−0.238824	−0.119412	+	0.992845i	$0.538101\pi$
$608$	−14289.2	−0.953129
$609$	0	0
$610$	607.416	0.0403173
$611$	−18026.2	−1.19356
$612$	0	0
$613$	−25996.1	−1.71284	−0.856420	−	0.516280i	$-0.827317\pi$
$613$	−25996.1	−1.71284	−0.856420	+	0.516280i	$0.827317\pi$
$614$	9909.81	0.651347
$615$	0	0
$616$	−1655.79	−0.108302
$617$	−7266.40	−0.474123	−0.237062	−	0.971495i	$-0.576184\pi$
$617$	−7266.40	−0.474123	−0.237062	+	0.971495i	$0.576184\pi$
$618$	0	0
$619$	16227.6	1.05371	0.526853	−	0.849957i	$-0.323372\pi$
$619$	16227.6	1.05371	0.526853	+	0.849957i	$0.323372\pi$
$620$	−16049.4	−1.03961
$621$	0	0
$622$	−5397.65	−0.347952
$623$	7442.58	0.478621
$624$	0	0
$625$	−7217.38	−0.461912
$626$	−11480.0	−0.732960
$627$	0	0
$628$	3453.06	0.219414
$629$	4888.73	0.309899
$630$	0	0
$631$	−3162.02	−0.199490	−0.0997450	−	0.995013i	$-0.531803\pi$
$631$	−3162.02	−0.199490	−0.0997450	+	0.995013i	$0.531803\pi$
$632$	3283.68	0.206674
$633$	0	0
$634$	−10083.2	−0.631630
$635$	−39593.5	−2.47436
$636$	0	0
$637$	2259.20	0.140523
$638$	−2795.86	−0.173494
$639$	0	0
$640$	19554.3	1.20774
$641$	8679.48	0.534819	0.267409	−	0.963583i	$-0.413832\pi$
$641$	8679.48	0.534819	0.267409	+	0.963583i	$0.413832\pi$
$642$	0	0
$643$	−19226.2	−1.17917	−0.589586	−	0.807705i	$-0.700709\pi$
$643$	−19226.2	−1.17917	−0.589586	+	0.807705i	$0.700709\pi$
$644$	−6214.94	−0.380284
$645$	0	0
$646$	2434.94	0.148299
$647$	−11211.6	−0.681258	−0.340629	−	0.940198i	$-0.610640\pi$
$647$	−11211.6	−0.681258	−0.340629	+	0.940198i	$0.610640\pi$
$648$	0	0
$649$	4496.52	0.271963
$650$	−12796.5	−0.772188
$651$	0	0
$652$	3309.56	0.198792
$653$	12404.5	0.743378	0.371689	−	0.928357i	$-0.378779\pi$
$653$	12404.5	0.743378	0.371689	+	0.928357i	$0.378779\pi$
$654$	0	0
$655$	25772.2	1.53741
$656$	30.4138	0.00181015
$657$	0	0
$658$	4378.83	0.259429
$659$	−20954.9	−1.23868	−0.619339	−	0.785124i	$-0.712599\pi$
$659$	−20954.9	−1.23868	−0.619339	+	0.785124i	$0.712599\pi$
$660$	0	0
$661$	−20349.0	−1.19741	−0.598703	−	0.800971i	$-0.704317\pi$
$661$	−20349.0	−1.19741	−0.598703	+	0.800971i	$0.704317\pi$
$662$	11846.3	0.695498
$663$	0	0
$664$	2170.26	0.126841
$665$	9259.96	0.539979
$666$	0	0
$667$	−25926.4	−1.50506
$668$	−8407.63	−0.486977
$669$	0	0
$670$	−17094.2	−0.985683
$671$	241.722	0.0139070
$672$	0	0
$673$	31712.5	1.81639	0.908193	−	0.418553i	$-0.137462\pi$
$673$	31712.5	1.81639	0.908193	+	0.418553i	$0.137462\pi$
$674$	−11213.7	−0.640857
$675$	0	0
$676$	387.433	0.0220433
$677$	−1195.61	−0.0678744	−0.0339372	−	0.999424i	$-0.510805\pi$
$677$	−1195.61	−0.0678744	−0.0339372	+	0.999424i	$0.510805\pi$
$678$	0	0
$679$	9908.49	0.560019
$680$	−7383.73	−0.416402
$681$	0	0
$682$	3005.47	0.168747
$683$	−16709.8	−0.936140	−0.468070	−	0.883691i	$-0.655051\pi$
$683$	−16709.8	−0.936140	−0.468070	+	0.883691i	$0.655051\pi$
$684$	0	0
$685$	−44759.0	−2.49657
$686$	−548.793	−0.0305438
$687$	0	0
$688$	−1119.29	−0.0620238
$689$	18947.7	1.04768
$690$	0	0
$691$	−11154.4	−0.614088	−0.307044	−	0.951695i	$-0.599340\pi$
$691$	−11154.4	−0.614088	−0.307044	+	0.951695i	$0.599340\pi$
$692$	688.701	0.0378331
$693$	0	0
$694$	−16041.0	−0.877391
$695$	46199.0	2.52148
$696$	0	0
$697$	−66.3182	−0.00360399
$698$	17194.2	0.932392
$699$	0	0
$700$	−6605.74	−0.356676
$701$	−10388.9	−0.559749	−0.279874	−	0.960037i	$-0.590293\pi$
$701$	−10388.9	−0.559749	−0.279874	+	0.960037i	$0.590293\pi$
$702$	0	0
$703$	−18834.2	−1.01045
$704$	−2482.26	−0.132889
$705$	0	0
$706$	−969.805	−0.0516984
$707$	5418.98	0.288263
$708$	0	0
$709$	−4288.90	−0.227183	−0.113592	−	0.993528i	$-0.536236\pi$
$709$	−4288.90	−0.227183	−0.113592	+	0.993528i	$0.536236\pi$
$710$	25680.2	1.35741
$711$	0	0
$712$	22863.4	1.20343
$713$	27870.2	1.46388
$714$	0	0
$715$	−8761.95	−0.458291
$716$	−784.800	−0.0409628
$717$	0	0
$718$	13643.5	0.709153
$719$	−31876.4	−1.65339	−0.826697	−	0.562647i	$-0.809783\pi$
$719$	−31876.4	−1.65339	−0.826697	+	0.562647i	$0.809783\pi$
$720$	0	0
$721$	5314.12	0.274491
$722$	1593.47	0.0821371
$723$	0	0
$724$	−15912.2	−0.816814
$725$	−27556.7	−1.41163
$726$	0	0
$727$	14030.2	0.715754	0.357877	−	0.933769i	$-0.383501\pi$
$727$	14030.2	0.715754	0.357877	+	0.933769i	$0.383501\pi$
$728$	6940.22	0.353326
$729$	0	0
$730$	−23995.4	−1.21659
$731$	2440.64	0.123489
$732$	0	0
$733$	−5466.09	−0.275436	−0.137718	−	0.990471i	$-0.543977\pi$
$733$	−5466.09	−0.275436	−0.137718	+	0.990471i	$0.543977\pi$
$734$	−8806.54	−0.442855
$735$	0	0
$736$	−30456.5	−1.52533
$737$	−6802.66	−0.339999
$738$	0	0
$739$	28957.0	1.44141	0.720704	−	0.693243i	$-0.243819\pi$
$739$	28957.0	1.44141	0.720704	+	0.693243i	$0.243819\pi$
$740$	23117.3	1.14839
$741$	0	0
$742$	−4602.66	−0.227721
$743$	17742.1	0.876037	0.438019	−	0.898966i	$-0.355680\pi$
$743$	17742.1	0.876037	0.438019	+	0.898966i	$0.355680\pi$
$744$	0	0
$745$	−14207.2	−0.698672
$746$	11346.0	0.556846
$747$	0	0
$748$	−1189.35	−0.0581374
$749$	8268.55	0.403373
$750$	0	0
$751$	30053.7	1.46029	0.730143	−	0.683294i	$-0.239453\pi$
$751$	30053.7	1.46029	0.730143	+	0.683294i	$0.239453\pi$
$752$	−3563.64	−0.172809
$753$	0	0
$754$	11718.8	0.566011
$755$	−2397.99	−0.115592
$756$	0	0
$757$	−8581.32	−0.412012	−0.206006	−	0.978551i	$-0.566047\pi$
$757$	−8581.32	−0.412012	−0.206006	+	0.978551i	$0.566047\pi$
$758$	123.738	0.00592923
$759$	0	0
$760$	28446.4	1.35771
$761$	−30413.3	−1.44873	−0.724363	−	0.689419i	$-0.757866\pi$
$761$	−30413.3	−1.44873	−0.724363	+	0.689419i	$0.757866\pi$
$762$	0	0
$763$	5166.63	0.245144
$764$	−20518.3	−0.971632
$765$	0	0
$766$	7749.90	0.365555
$767$	−18847.0	−0.887258
$768$	0	0
$769$	−24735.2	−1.15992	−0.579958	−	0.814646i	$-0.696931\pi$
$769$	−24735.2	−1.15992	−0.579958	+	0.814646i	$0.696931\pi$
$770$	2128.40	0.0996134
$771$	0	0
$772$	6979.18	0.325371
$773$	−25108.1	−1.16827	−0.584137	−	0.811655i	$-0.698567\pi$
$773$	−25108.1	−1.16827	−0.584137	+	0.811655i	$0.698567\pi$
$774$	0	0
$775$	29622.7	1.37301
$776$	30438.7	1.40810
$777$	0	0
$778$	15811.0	0.728601
$779$	255.496	0.0117511
$780$	0	0
$781$	10219.4	0.468221
$782$	5189.92	0.237329
$783$	0	0
$784$	446.627	0.0203456
$785$	−10966.0	−0.498591
$786$	0	0
$787$	−36364.0	−1.64706	−0.823531	−	0.567271i	$-0.807999\pi$
$787$	−36364.0	−1.64706	−0.823531	+	0.567271i	$0.807999\pi$
$788$	16604.8	0.750660
$789$	0	0
$790$	−4220.93	−0.190094
$791$	−4341.27	−0.195142
$792$	0	0
$793$	−1013.17	−0.0453704
$794$	16216.6	0.724818
$795$	0	0
$796$	−27419.5	−1.22093
$797$	18605.4	0.826897	0.413448	−	0.910528i	$-0.364324\pi$
$797$	18605.4	0.826897	0.413448	+	0.910528i	$0.364324\pi$
$798$	0	0
$799$	7770.63	0.344062
$800$	−32371.6	−1.43063
$801$	0	0
$802$	14753.3	0.649574
$803$	−9548.99	−0.419647
$804$	0	0
$805$	19737.0	0.864147
$806$	−12597.4	−0.550525
$807$	0	0
$808$	16647.0	0.724800
$809$	−26846.8	−1.16673	−0.583364	−	0.812211i	$-0.698264\pi$
$809$	−26846.8	−1.16673	−0.583364	+	0.812211i	$0.698264\pi$
$810$	0	0
$811$	−32344.3	−1.40045	−0.700224	−	0.713924i	$-0.746916\pi$
$811$	−32344.3	−1.40045	−0.700224	+	0.713924i	$0.746916\pi$
$812$	6049.37	0.261442
$813$	0	0
$814$	−4329.04	−0.186404
$815$	−10510.3	−0.451729
$816$	0	0
$817$	−9402.76	−0.402645
$818$	9137.31	0.390561
$819$	0	0
$820$	−313.599	−0.0133553
$821$	−16953.3	−0.720677	−0.360338	−	0.932822i	$-0.617339\pi$
$821$	−16953.3	−0.720677	−0.360338	+	0.932822i	$0.617339\pi$
$822$	0	0
$823$	37814.0	1.60159	0.800797	−	0.598936i	$-0.204410\pi$
$823$	37814.0	1.60159	0.800797	+	0.598936i	$0.204410\pi$
$824$	16324.8	0.690173
$825$	0	0
$826$	4578.21	0.192853
$827$	11894.6	0.500141	0.250071	−	0.968228i	$-0.419546\pi$
$827$	11894.6	0.500141	0.250071	+	0.968228i	$0.419546\pi$
$828$	0	0
$829$	30744.7	1.28807	0.644033	−	0.764998i	$-0.277260\pi$
$829$	30744.7	1.28807	0.644033	+	0.764998i	$0.277260\pi$
$830$	−2789.71	−0.116665
$831$	0	0
$832$	10404.3	0.433540
$833$	−973.884	−0.0405079
$834$	0	0
$835$	26700.4	1.10659
$836$	4582.04	0.189561
$837$	0	0
$838$	7233.08	0.298165
$839$	24677.8	1.01546	0.507731	−	0.861516i	$-0.330484\pi$
$839$	24677.8	1.01546	0.507731	+	0.861516i	$0.330484\pi$
$840$	0	0
$841$	846.708	0.0347168
$842$	−6077.14	−0.248732
$843$	0	0
$844$	−19346.4	−0.789017
$845$	−1230.39	−0.0500907
$846$	0	0
$847$	847.000	0.0343604
$848$	3745.80	0.151688
$849$	0	0
$850$	5516.26	0.222595
$851$	−40143.9	−1.61706
$852$	0	0
$853$	−1855.69	−0.0744873	−0.0372437	−	0.999306i	$-0.511858\pi$
$853$	−1855.69	−0.0744873	−0.0372437	+	0.999306i	$0.511858\pi$
$854$	246.114	0.00986163
$855$	0	0
$856$	25400.8	1.01423
$857$	−3984.84	−0.158832	−0.0794162	−	0.996842i	$-0.525306\pi$
$857$	−3984.84	−0.158832	−0.0794162	+	0.996842i	$0.525306\pi$
$858$	0	0
$859$	−12526.1	−0.497538	−0.248769	−	0.968563i	$-0.580026\pi$
$859$	−12526.1	−0.497538	−0.248769	+	0.968563i	$0.580026\pi$
$860$	11541.1	0.457613
$861$	0	0
$862$	−5549.54	−0.219279
$863$	22040.9	0.869387	0.434693	−	0.900579i	$-0.356857\pi$
$863$	22040.9	0.869387	0.434693	+	0.900579i	$0.356857\pi$
$864$	0	0
$865$	−2187.14	−0.0859709
$866$	15775.8	0.619033
$867$	0	0
$868$	−6502.91	−0.254290
$869$	−1679.72	−0.0655704
$870$	0	0
$871$	28513.2	1.10922
$872$	15871.8	0.616383
$873$	0	0
$874$	−19994.5	−0.773828
$875$	5861.39	0.226459
$876$	0	0
$877$	21039.5	0.810096	0.405048	−	0.914295i	$-0.367255\pi$
$877$	21039.5	0.810096	0.405048	+	0.914295i	$0.367255\pi$
$878$	−13542.7	−0.520553
$879$	0	0
$880$	−1732.17	−0.0663538
$881$	−33610.9	−1.28533	−0.642667	−	0.766146i	$-0.722172\pi$
$881$	−33610.9	−1.28533	−0.642667	+	0.766146i	$0.722172\pi$
$882$	0	0
$883$	−4618.52	−0.176020	−0.0880099	−	0.996120i	$-0.528051\pi$
$883$	−4618.52	−0.176020	−0.0880099	+	0.996120i	$0.528051\pi$
$884$	4985.11	0.189669
$885$	0	0
$886$	−12520.5	−0.474757
$887$	40095.6	1.51779	0.758895	−	0.651213i	$-0.225740\pi$
$887$	40095.6	1.51779	0.758895	+	0.651213i	$0.225740\pi$
$888$	0	0
$889$	−16042.6	−0.605231
$890$	−29389.3	−1.10689
$891$	0	0
$892$	−9617.22	−0.360996
$893$	−29937.0	−1.12184
$894$	0	0
$895$	2492.32	0.0930828
$896$	7923.03	0.295413
$897$	0	0
$898$	1681.48	0.0624852
$899$	−27127.7	−1.00641
$900$	0	0
$901$	−8167.85	−0.302009
$902$	58.7258	0.00216780
$903$	0	0
$904$	−13336.3	−0.490661
$905$	50533.1	1.85611
$906$	0	0
$907$	−52281.2	−1.91397	−0.956984	−	0.290141i	$-0.906298\pi$
$907$	−52281.2	−1.91397	−0.956984	+	0.290141i	$0.906298\pi$
$908$	26312.2	0.961674
$909$	0	0
$910$	−8921.15	−0.324981
$911$	−30635.6	−1.11416	−0.557081	−	0.830458i	$-0.688079\pi$
$911$	−30635.6	−1.11416	−0.557081	+	0.830458i	$0.688079\pi$
$912$	0	0
$913$	−1110.17	−0.0402423
$914$	−23035.3	−0.833632
$915$	0	0
$916$	−26881.0	−0.969622
$917$	10442.4	0.376051
$918$	0	0
$919$	−14574.9	−0.523159	−0.261579	−	0.965182i	$-0.584243\pi$
$919$	−14574.9	−0.523159	−0.261579	+	0.965182i	$0.584243\pi$
$920$	60631.6	2.17279
$921$	0	0
$922$	−9209.21	−0.328947
$923$	−42834.5	−1.52754
$924$	0	0
$925$	−42668.1	−1.51667
$926$	16089.4	0.570983
$927$	0	0
$928$	29645.1	1.04865
$929$	18955.8	0.669451	0.334726	−	0.942316i	$-0.391356\pi$
$929$	18955.8	0.669451	0.334726	+	0.942316i	$0.391356\pi$
$930$	0	0
$931$	3751.96	0.132079
$932$	26232.7	0.921976
$933$	0	0
$934$	4171.66	0.146147
$935$	3777.05	0.132110
$936$	0	0
$937$	6520.68	0.227344	0.113672	−	0.993518i	$-0.463739\pi$
$937$	6520.68	0.227344	0.113672	+	0.993518i	$0.463739\pi$
$938$	−6926.26	−0.241098
$939$	0	0
$940$	36745.0	1.27499
$941$	−12631.4	−0.437589	−0.218795	−	0.975771i	$-0.570212\pi$
$941$	−12631.4	−0.437589	−0.218795	+	0.975771i	$0.570212\pi$
$942$	0	0
$943$	544.573	0.0188057
$944$	−3725.91	−0.128462
$945$	0	0
$946$	−2161.22	−0.0742785
$947$	35170.9	1.20686	0.603432	−	0.797415i	$-0.293800\pi$
$947$	35170.9	1.20686	0.603432	+	0.797415i	$0.293800\pi$
$948$	0	0
$949$	40024.3	1.36907
$950$	−21251.8	−0.725789
$951$	0	0
$952$	−2991.75	−0.101852
$953$	−39539.0	−1.34396	−0.671980	−	0.740570i	$-0.734556\pi$
$953$	−39539.0	−1.34396	−0.671980	+	0.740570i	$0.734556\pi$
$954$	0	0
$955$	65160.8	2.20791
$956$	26230.9	0.887415
$957$	0	0
$958$	−9511.83	−0.320786
$959$	−18135.5	−0.610663
$960$	0	0
$961$	−629.373	−0.0211263
$962$	18145.1	0.608129
$963$	0	0
$964$	−8401.70	−0.280706
$965$	−22164.0	−0.739363
$966$	0	0
$967$	−29986.0	−0.997194	−0.498597	−	0.866834i	$-0.666151\pi$
$967$	−29986.0	−0.997194	−0.498597	+	0.866834i	$0.666151\pi$
$968$	2601.96	0.0863949
$969$	0	0
$970$	−39126.7	−1.29514
$971$	14879.8	0.491777	0.245889	−	0.969298i	$-0.420920\pi$
$971$	14879.8	0.491777	0.245889	+	0.969298i	$0.420920\pi$
$972$	0	0
$973$	18719.0	0.616755
$974$	−25431.9	−0.836643
$975$	0	0
$976$	−200.296	−0.00656896
$977$	−40864.1	−1.33813	−0.669067	−	0.743202i	$-0.733306\pi$
$977$	−40864.1	−1.33813	−0.669067	+	0.743202i	$0.733306\pi$
$978$	0	0
$979$	−11695.5	−0.381807
$980$	−4605.21	−0.150110
$981$	0	0
$982$	5848.88	0.190066
$983$	−48180.6	−1.56330	−0.781650	−	0.623718i	$-0.785621\pi$
$983$	−48180.6	−1.56330	−0.781650	+	0.623718i	$0.785621\pi$
$984$	0	0
$985$	−52732.4	−1.70578
$986$	−5051.65	−0.163162
$987$	0	0
$988$	−19205.5	−0.618430
$989$	−20041.4	−0.644367
$990$	0	0
$991$	−22380.4	−0.717392	−0.358696	−	0.933454i	$-0.616779\pi$
$991$	−22380.4	−0.717392	−0.358696	+	0.933454i	$0.616779\pi$
$992$	−31867.7	−1.01996
$993$	0	0
$994$	10405.1	0.332023
$995$	87077.1	2.77440
$996$	0	0
$997$	−12425.1	−0.394691	−0.197346	−	0.980334i	$-0.563232\pi$
$997$	−12425.1	−0.394691	−0.197346	+	0.980334i	$0.563232\pi$
$998$	−33874.6	−1.07443
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.p.1.2		5
3.2	odd	2		231.4.a.k.1.4	✓	5
21.20	even	2		1617.4.a.n.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.k.1.4	✓	5	3.2	odd	2
693.4.a.p.1.2		5	1.1	even	1	trivial
1617.4.a.n.1.4		5	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.59998 q^{2} -5.44007 q^{4} +17.2762 q^{5} +7.00000 q^{7} +21.5038 q^{8} +O(q^{10})\)	\(q-1.59998 q^{2} -5.44007 q^{4} +17.2762 q^{5} +7.00000 q^{7} +21.5038 q^{8} -27.6416 q^{10} -11.0000 q^{11} +46.1062 q^{13} -11.1999 q^{14} +9.11483 q^{16} -19.8752 q^{17} +76.5707 q^{19} -93.9838 q^{20} +17.5998 q^{22} +163.205 q^{23} +173.468 q^{25} -73.7690 q^{26} -38.0805 q^{28} -158.858 q^{29} +170.768 q^{31} -186.614 q^{32} +31.7999 q^{34} +120.934 q^{35} -245.971 q^{37} -122.511 q^{38} +371.505 q^{40} +3.33673 q^{41} -122.798 q^{43} +59.8407 q^{44} -261.125 q^{46} -390.972 q^{47} +49.0000 q^{49} -277.545 q^{50} -250.821 q^{52} +410.957 q^{53} -190.038 q^{55} +150.527 q^{56} +254.169 q^{58} -408.774 q^{59} -21.9747 q^{61} -273.225 q^{62} +225.660 q^{64} +796.541 q^{65} +618.424 q^{67} +108.122 q^{68} -193.491 q^{70} -929.041 q^{71} +868.090 q^{73} +393.549 q^{74} -416.549 q^{76} -77.0000 q^{77} +152.702 q^{79} +157.470 q^{80} -5.33871 q^{82} +100.924 q^{83} -343.368 q^{85} +196.475 q^{86} -236.542 q^{88} +1063.23 q^{89} +322.743 q^{91} -887.848 q^{92} +625.547 q^{94} +1322.85 q^{95} +1415.50 q^{97} -78.3990 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 21 q^{4} - 21 q^{5} + 35 q^{7} + 42 q^{8}+O(q^{10}) \) 5 * q + q^2 + 21 * q^4 - 21 * q^5 + 35 * q^7 + 42 * q^8	\( 5 q + q^{2} + 21 q^{4} - 21 q^{5} + 35 q^{7} + 42 q^{8} - 23 q^{10} - 55 q^{11} + 101 q^{13} + 7 q^{14} - 7 q^{16} + 20 q^{17} + 237 q^{19} - 85 q^{20} - 11 q^{22} + 80 q^{23} + 486 q^{25} - 165 q^{26} + 147 q^{28} + 11 q^{29} + 316 q^{31} - 453 q^{32} + 936 q^{34} - 147 q^{35} + 319 q^{37} - 89 q^{38} + 624 q^{40} - 1190 q^{41} + 88 q^{43} - 231 q^{44} + 1000 q^{46} - 377 q^{47} + 245 q^{49} + 644 q^{50} + 1001 q^{52} + 992 q^{53} + 231 q^{55} + 294 q^{56} + 721 q^{58} - 71 q^{59} - 574 q^{61} - 272 q^{62} - 1380 q^{64} - 589 q^{65} - 527 q^{67} + 2974 q^{68} - 161 q^{70} + 1156 q^{71} + 1061 q^{73} + 1609 q^{74} + 2399 q^{76} - 385 q^{77} + 588 q^{79} + 1643 q^{80} - 2602 q^{82} + 212 q^{83} + 1918 q^{85} + 4760 q^{86} - 462 q^{88} - 1030 q^{89} + 707 q^{91} + 1174 q^{92} - 1799 q^{94} + 3593 q^{95} + 2488 q^{97} + 49 q^{98}+O(q^{100}) \) 5 * q + q^2 + 21 * q^4 - 21 * q^5 + 35 * q^7 + 42 * q^8 - 23 * q^10 - 55 * q^11 + 101 * q^13 + 7 * q^14 - 7 * q^16 + 20 * q^17 + 237 * q^19 - 85 * q^20 - 11 * q^22 + 80 * q^23 + 486 * q^25 - 165 * q^26 + 147 * q^28 + 11 * q^29 + 316 * q^31 - 453 * q^32 + 936 * q^34 - 147 * q^35 + 319 * q^37 - 89 * q^38 + 624 * q^40 - 1190 * q^41 + 88 * q^43 - 231 * q^44 + 1000 * q^46 - 377 * q^47 + 245 * q^49 + 644 * q^50 + 1001 * q^52 + 992 * q^53 + 231 * q^55 + 294 * q^56 + 721 * q^58 - 71 * q^59 - 574 * q^61 - 272 * q^62 - 1380 * q^64 - 589 * q^65 - 527 * q^67 + 2974 * q^68 - 161 * q^70 + 1156 * q^71 + 1061 * q^73 + 1609 * q^74 + 2399 * q^76 - 385 * q^77 + 588 * q^79 + 1643 * q^80 - 2602 * q^82 + 212 * q^83 + 1918 * q^85 + 4760 * q^86 - 462 * q^88 - 1030 * q^89 + 707 * q^91 + 1174 * q^92 - 1799 * q^94 + 3593 * q^95 + 2488 * q^97 + 49 * q^98

Introduction

L-functions

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Database

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