LMFDB - Embedded newform 693.4.a.m.1.4

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.509800.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 5x + 15 $ x^4 - x^3 - 10x^2 + 5x + 15
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.59222$ 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)

$f(q)$	$=$	$q+5.05706 q^{2} +17.5738 q^{4} -12.9427 q^{5} -7.00000 q^{7} +48.4155 q^{8} +O(q^{10})$	\(q+5.05706 q^{2} +17.5738 q^{4} -12.9427 q^{5} -7.00000 q^{7} +48.4155 q^{8} -65.4521 q^{10} -11.0000 q^{11} -55.5473 q^{13} -35.3994 q^{14} +104.249 q^{16} -59.2511 q^{17} -33.1777 q^{19} -227.453 q^{20} -55.6276 q^{22} -26.0532 q^{23} +42.5139 q^{25} -280.906 q^{26} -123.017 q^{28} +188.479 q^{29} -278.333 q^{31} +139.870 q^{32} -299.636 q^{34} +90.5990 q^{35} +201.567 q^{37} -167.781 q^{38} -626.628 q^{40} +126.784 q^{41} -454.826 q^{43} -193.312 q^{44} -131.753 q^{46} +129.710 q^{47} +49.0000 q^{49} +214.995 q^{50} -976.179 q^{52} -79.8855 q^{53} +142.370 q^{55} -338.908 q^{56} +953.150 q^{58} +593.670 q^{59} -49.8622 q^{61} -1407.55 q^{62} -126.661 q^{64} +718.932 q^{65} +295.997 q^{67} -1041.27 q^{68} +458.164 q^{70} -546.422 q^{71} -809.232 q^{73} +1019.34 q^{74} -583.059 q^{76} +77.0000 q^{77} -375.244 q^{79} -1349.27 q^{80} +641.154 q^{82} -85.9037 q^{83} +766.870 q^{85} -2300.08 q^{86} -532.570 q^{88} -750.306 q^{89} +388.831 q^{91} -457.856 q^{92} +655.953 q^{94} +429.409 q^{95} -451.876 q^{97} +247.796 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 22 q^{4} + 18 q^{5} - 28 q^{7} + 60 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 22 * q^4 + 18 * q^5 - 28 * q^7 + 60 * q^8	$ 4 q + 4 q^{2} + 22 q^{4} + 18 q^{5} - 28 q^{7} + 60 q^{8} - 92 q^{10} - 44 q^{11} - 134 q^{13} - 28 q^{14} - 6 q^{16} + 74 q^{17} - 164 q^{19} - 116 q^{20} - 44 q^{22} - 194 q^{23} + 38 q^{25} - 734 q^{26} - 154 q^{28} + 108 q^{29} - 412 q^{31} + 4 q^{32} - 346 q^{34} - 126 q^{35} + 286 q^{37} - 224 q^{38} - 540 q^{40} + 18 q^{41} - 496 q^{43} - 242 q^{44} - 284 q^{46} - 62 q^{47} + 196 q^{49} - 212 q^{50} - 822 q^{52} + 828 q^{53} - 198 q^{55} - 420 q^{56} + 1388 q^{58} + 1224 q^{59} - 350 q^{61} + 878 q^{62} - 718 q^{64} + 396 q^{65} - 1498 q^{67} - 1058 q^{68} + 644 q^{70} - 2326 q^{71} - 1630 q^{73} + 1156 q^{74} - 3152 q^{76} + 308 q^{77} - 1020 q^{79} - 3072 q^{80} + 2118 q^{82} + 1920 q^{83} + 2008 q^{85} - 1056 q^{86} - 660 q^{88} - 1550 q^{89} + 938 q^{91} - 2592 q^{92} - 1042 q^{94} - 2332 q^{95} - 2202 q^{97} + 196 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 22 * q^4 + 18 * q^5 - 28 * q^7 + 60 * q^8 - 92 * q^10 - 44 * q^11 - 134 * q^13 - 28 * q^14 - 6 * q^16 + 74 * q^17 - 164 * q^19 - 116 * q^20 - 44 * q^22 - 194 * q^23 + 38 * q^25 - 734 * q^26 - 154 * q^28 + 108 * q^29 - 412 * q^31 + 4 * q^32 - 346 * q^34 - 126 * q^35 + 286 * q^37 - 224 * q^38 - 540 * q^40 + 18 * q^41 - 496 * q^43 - 242 * q^44 - 284 * q^46 - 62 * q^47 + 196 * q^49 - 212 * q^50 - 822 * q^52 + 828 * q^53 - 198 * q^55 - 420 * q^56 + 1388 * q^58 + 1224 * q^59 - 350 * q^61 + 878 * q^62 - 718 * q^64 + 396 * q^65 - 1498 * q^67 - 1058 * q^68 + 644 * q^70 - 2326 * q^71 - 1630 * q^73 + 1156 * q^74 - 3152 * q^76 + 308 * q^77 - 1020 * q^79 - 3072 * q^80 + 2118 * q^82 + 1920 * q^83 + 2008 * q^85 - 1056 * q^86 - 660 * q^88 - 1550 * q^89 + 938 * q^91 - 2592 * q^92 - 1042 * q^94 - 2332 * q^95 - 2202 * q^97 + 196 * 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 $

$2$	5.05706	1.78794	0.893970	−	0.448127i	$-0.147909\pi$
$2$	5.05706	1.78794	0.893970	+	0.448127i	$0.147909\pi$
$3$	0	0
$3$	0	0
$4$	17.5738	2.19673
$5$	−12.9427	−1.15763	−0.578816	−	0.815458i	$-0.696485\pi$
$5$	−12.9427	−1.15763	−0.578816	+	0.815458i	$0.696485\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	48.4155	2.13968
$9$	0	0
$10$	−65.4521	−2.06978
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−55.5473	−1.18508	−0.592540	−	0.805541i	$-0.701875\pi$
$13$	−55.5473	−1.18508	−0.592540	+	0.805541i	$0.701875\pi$
$14$	−35.3994	−0.675778
$15$	0	0
$16$	104.249	1.62889
$17$	−59.2511	−0.845324	−0.422662	−	0.906287i	$-0.638904\pi$
$17$	−59.2511	−0.845324	−0.422662	+	0.906287i	$0.638904\pi$
$18$	0	0
$19$	−33.1777	−0.400604	−0.200302	−	0.979734i	$-0.564192\pi$
$19$	−33.1777	−0.400604	−0.200302	+	0.979734i	$0.564192\pi$
$20$	−227.453	−2.54300
$21$	0	0
$22$	−55.6276	−0.539084
$23$	−26.0532	−0.236195	−0.118097	−	0.993002i	$-0.537679\pi$
$23$	−26.0532	−0.236195	−0.118097	+	0.993002i	$0.537679\pi$
$24$	0	0
$25$	42.5139	0.340111
$26$	−280.906	−2.11885
$27$	0	0
$28$	−123.017	−0.830286
$29$	188.479	1.20689	0.603443	−	0.797406i	$-0.293795\pi$
$29$	188.479	1.20689	0.603443	+	0.797406i	$0.293795\pi$
$30$	0	0
$31$	−278.333	−1.61259	−0.806293	−	0.591516i	$-0.798529\pi$
$31$	−278.333	−1.61259	−0.806293	+	0.591516i	$0.798529\pi$
$32$	139.870	0.772682
$33$	0	0
$34$	−299.636	−1.51139
$35$	90.5990	0.437544
$36$	0	0
$37$	201.567	0.895608	0.447804	−	0.894132i	$-0.352206\pi$
$37$	201.567	0.895608	0.447804	+	0.894132i	$0.352206\pi$
$38$	−167.781	−0.716256
$39$	0	0
$40$	−626.628	−2.47696
$41$	126.784	0.482935	0.241467	−	0.970409i	$-0.422371\pi$
$41$	126.784	0.482935	0.241467	+	0.970409i	$0.422371\pi$
$42$	0	0
$43$	−454.826	−1.61303	−0.806514	−	0.591214i	$-0.798649\pi$
$43$	−454.826	−1.61303	−0.806514	+	0.591214i	$0.798649\pi$
$44$	−193.312	−0.662339
$45$	0	0
$46$	−131.753	−0.422302
$47$	129.710	0.402558	0.201279	−	0.979534i	$-0.435490\pi$
$47$	129.710	0.402558	0.201279	+	0.979534i	$0.435490\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	214.995	0.608098
$51$	0	0
$52$	−976.179	−2.60330
$53$	−79.8855	−0.207040	−0.103520	−	0.994627i	$-0.533011\pi$
$53$	−79.8855	−0.207040	−0.103520	+	0.994627i	$0.533011\pi$
$54$	0	0
$55$	142.370	0.349039
$56$	−338.908	−0.808724
$57$	0	0
$58$	953.150	2.15784
$59$	593.670	1.30999	0.654994	−	0.755634i	$-0.272671\pi$
$59$	593.670	1.30999	0.654994	+	0.755634i	$0.272671\pi$
$60$	0	0
$61$	−49.8622	−0.104659	−0.0523295	−	0.998630i	$-0.516665\pi$
$61$	−49.8622	−0.104659	−0.0523295	+	0.998630i	$0.516665\pi$
$62$	−1407.55	−2.88321
$63$	0	0
$64$	−126.661	−0.247385
$65$	718.932	1.37189
$66$	0	0
$67$	295.997	0.539728	0.269864	−	0.962898i	$-0.413021\pi$
$67$	295.997	0.539728	0.269864	+	0.962898i	$0.413021\pi$
$68$	−1041.27	−1.85695
$69$	0	0
$70$	458.164	0.782302
$71$	−546.422	−0.913357	−0.456679	−	0.889632i	$-0.650961\pi$
$71$	−546.422	−0.913357	−0.456679	+	0.889632i	$0.650961\pi$
$72$	0	0
$73$	−809.232	−1.29744	−0.648722	−	0.761025i	$-0.724696\pi$
$73$	−809.232	−1.29744	−0.648722	+	0.761025i	$0.724696\pi$
$74$	1019.34	1.60129
$75$	0	0
$76$	−583.059	−0.880019
$77$	77.0000	0.113961
$78$	0	0
$79$	−375.244	−0.534408	−0.267204	−	0.963640i	$-0.586100\pi$
$79$	−375.244	−0.534408	−0.267204	+	0.963640i	$0.586100\pi$
$80$	−1349.27	−1.88566
$81$	0	0
$82$	641.154	0.863459
$83$	−85.9037	−0.113604	−0.0568022	−	0.998385i	$-0.518090\pi$
$83$	−85.9037	−0.113604	−0.0568022	+	0.998385i	$0.518090\pi$
$84$	0	0
$85$	766.870	0.978574
$86$	−2300.08	−2.88400
$87$	0	0
$88$	−532.570	−0.645138
$89$	−750.306	−0.893621	−0.446810	−	0.894629i	$-0.647440\pi$
$89$	−750.306	−0.893621	−0.446810	+	0.894629i	$0.647440\pi$
$90$	0	0
$91$	388.831	0.447918
$92$	−457.856	−0.518856
$93$	0	0
$94$	655.953	0.719749
$95$	429.409	0.463752
$96$	0	0
$97$	−451.876	−0.473001	−0.236500	−	0.971631i	$-0.576000\pi$
$97$	−451.876	−0.473001	−0.236500	+	0.971631i	$0.576000\pi$
$98$	247.796	0.255420
$99$	0	0
$100$	747.132	0.747132
$101$	1414.20	1.39325	0.696626	−	0.717434i	$-0.254684\pi$
$101$	1414.20	1.39325	0.696626	+	0.717434i	$0.254684\pi$
$102$	0	0
$103$	−27.8798	−0.0266707	−0.0133354	−	0.999911i	$-0.504245\pi$
$103$	−27.8798	−0.0266707	−0.0133354	+	0.999911i	$0.504245\pi$
$104$	−2689.35	−2.53569
$105$	0	0
$106$	−403.985	−0.370175
$107$	778.142	0.703045	0.351523	−	0.936179i	$-0.385664\pi$
$107$	778.142	0.703045	0.351523	+	0.936179i	$0.385664\pi$
$108$	0	0
$109$	−219.545	−0.192923	−0.0964615	−	0.995337i	$-0.530752\pi$
$109$	−219.545	−0.192923	−0.0964615	+	0.995337i	$0.530752\pi$
$110$	719.973	0.624061
$111$	0	0
$112$	−729.744	−0.615664
$113$	−607.402	−0.505660	−0.252830	−	0.967511i	$-0.581361\pi$
$113$	−607.402	−0.505660	−0.252830	+	0.967511i	$0.581361\pi$
$114$	0	0
$115$	337.200	0.273426
$116$	3312.30	2.65120
$117$	0	0
$118$	3002.22	2.34218
$119$	414.758	0.319502
$120$	0	0
$121$	121.000	0.0909091
$122$	−252.156	−0.187124
$123$	0	0
$124$	−4891.39	−3.54242
$125$	1067.59	0.763908
$126$	0	0
$127$	1949.05	1.36181	0.680905	−	0.732372i	$-0.261587\pi$
$127$	1949.05	1.36181	0.680905	+	0.732372i	$0.261587\pi$
$128$	−1759.49	−1.21499
$129$	0	0
$130$	3635.68	2.45285
$131$	1782.57	1.18888	0.594442	−	0.804138i	$-0.297373\pi$
$131$	1782.57	1.18888	0.594442	+	0.804138i	$0.297373\pi$
$132$	0	0
$133$	232.244	0.151414
$134$	1496.87	0.965001
$135$	0	0
$136$	−2868.67	−1.80872
$137$	3080.56	1.92109	0.960546	−	0.278120i	$-0.0897111\pi$
$137$	3080.56	1.92109	0.960546	+	0.278120i	$0.0897111\pi$
$138$	0	0
$139$	1171.35	0.714769	0.357384	−	0.933957i	$-0.383669\pi$
$139$	1171.35	0.714769	0.357384	+	0.933957i	$0.383669\pi$
$140$	1592.17	0.961165
$141$	0	0
$142$	−2763.29	−1.63303
$143$	611.020	0.357315
$144$	0	0
$145$	−2439.43	−1.39713
$146$	−4092.33	−2.31975
$147$	0	0
$148$	3542.31	1.96741
$149$	−1237.66	−0.680492	−0.340246	−	0.940337i	$-0.610510\pi$
$149$	−1237.66	−0.680492	−0.340246	+	0.940337i	$0.610510\pi$
$150$	0	0
$151$	1581.62	0.852388	0.426194	−	0.904632i	$-0.359854\pi$
$151$	1581.62	0.852388	0.426194	+	0.904632i	$0.359854\pi$
$152$	−1606.31	−0.857165
$153$	0	0
$154$	389.394	0.203755
$155$	3602.39	1.86678
$156$	0	0
$157$	−3209.77	−1.63164	−0.815821	−	0.578304i	$-0.803715\pi$
$157$	−3209.77	−1.63164	−0.815821	+	0.578304i	$0.803715\pi$
$158$	−1897.63	−0.955489
$159$	0	0
$160$	−1810.30	−0.894481
$161$	182.373	0.0892732
$162$	0	0
$163$	−139.967	−0.0672579	−0.0336290	−	0.999434i	$-0.510706\pi$
$163$	−139.967	−0.0672579	−0.0336290	+	0.999434i	$0.510706\pi$
$164$	2228.08	1.06088
$165$	0	0
$166$	−434.420	−0.203118
$167$	−1425.74	−0.660641	−0.330320	−	0.943869i	$-0.607157\pi$
$167$	−1425.74	−0.660641	−0.330320	+	0.943869i	$0.607157\pi$
$168$	0	0
$169$	888.497	0.404414
$170$	3878.11	1.74963
$171$	0	0
$172$	−7993.03	−3.54339
$173$	−1527.01	−0.671077	−0.335538	−	0.942027i	$-0.608918\pi$
$173$	−1527.01	−0.671077	−0.335538	+	0.942027i	$0.608918\pi$
$174$	0	0
$175$	−297.597	−0.128550
$176$	−1146.74	−0.491130
$177$	0	0
$178$	−3794.34	−1.59774
$179$	−4512.37	−1.88419	−0.942095	−	0.335346i	$-0.891147\pi$
$179$	−4512.37	−1.88419	−0.942095	+	0.335346i	$0.891147\pi$
$180$	0	0
$181$	−2973.55	−1.22112	−0.610559	−	0.791970i	$-0.709055\pi$
$181$	−2973.55	−1.22112	−0.610559	+	0.791970i	$0.709055\pi$
$182$	1966.34	0.800851
$183$	0	0
$184$	−1261.38	−0.505381
$185$	−2608.83	−1.03678
$186$	0	0
$187$	651.762	0.254875
$188$	2279.51	0.884311
$189$	0	0
$190$	2171.55	0.829161
$191$	1275.56	0.483225	0.241613	−	0.970373i	$-0.422324\pi$
$191$	1275.56	0.483225	0.241613	+	0.970373i	$0.422324\pi$
$192$	0	0
$193$	−121.983	−0.0454950	−0.0227475	−	0.999741i	$-0.507241\pi$
$193$	−121.983	−0.0454950	−0.0227475	+	0.999741i	$0.507241\pi$
$194$	−2285.17	−0.845697
$195$	0	0
$196$	861.118	0.313819
$197$	3483.97	1.26001	0.630007	−	0.776589i	$-0.283052\pi$
$197$	3483.97	1.26001	0.630007	+	0.776589i	$0.283052\pi$
$198$	0	0
$199$	2655.65	0.946002	0.473001	−	0.881062i	$-0.343171\pi$
$199$	2655.65	0.946002	0.473001	+	0.881062i	$0.343171\pi$
$200$	2058.33	0.727730
$201$	0	0
$202$	7151.71	2.49105
$203$	−1319.35	−0.456160
$204$	0	0
$205$	−1640.93	−0.559061
$206$	−140.990	−0.0476856
$207$	0	0
$208$	−5790.75	−1.93037
$209$	364.954	0.120787
$210$	0	0
$211$	1909.91	0.623147	0.311573	−	0.950222i	$-0.399144\pi$
$211$	1909.91	0.623147	0.311573	+	0.950222i	$0.399144\pi$
$212$	−1403.89	−0.454811
$213$	0	0
$214$	3935.11	1.25700
$215$	5886.68	1.86729
$216$	0	0
$217$	1948.33	0.609500
$218$	−1110.25	−0.344935
$219$	0	0
$220$	2501.99	0.766745
$221$	3291.24	1.00178
$222$	0	0
$223$	−4381.22	−1.31564	−0.657821	−	0.753174i	$-0.728522\pi$
$223$	−4381.22	−1.31564	−0.657821	+	0.753174i	$0.728522\pi$
$224$	−979.092	−0.292046
$225$	0	0
$226$	−3071.67	−0.904090
$227$	−1711.28	−0.500359	−0.250180	−	0.968199i	$-0.580490\pi$
$227$	−1711.28	−0.500359	−0.250180	+	0.968199i	$0.580490\pi$
$228$	0	0
$229$	2617.84	0.755423	0.377712	−	0.925923i	$-0.376711\pi$
$229$	2617.84	0.755423	0.377712	+	0.925923i	$0.376711\pi$
$230$	1705.24	0.488870
$231$	0	0
$232$	9125.31	2.58235
$233$	−3593.09	−1.01026	−0.505131	−	0.863043i	$-0.668556\pi$
$233$	−3593.09	−1.01026	−0.505131	+	0.863043i	$0.668556\pi$
$234$	0	0
$235$	−1678.81	−0.466014
$236$	10433.1	2.87769
$237$	0	0
$238$	2097.45	0.571251
$239$	−1841.19	−0.498312	−0.249156	−	0.968463i	$-0.580153\pi$
$239$	−1841.19	−0.498312	−0.249156	+	0.968463i	$0.580153\pi$
$240$	0	0
$241$	2209.45	0.590552	0.295276	−	0.955412i	$-0.404588\pi$
$241$	2209.45	0.590552	0.295276	+	0.955412i	$0.404588\pi$
$242$	611.904	0.162540
$243$	0	0
$244$	−876.270	−0.229907
$245$	−634.193	−0.165376
$246$	0	0
$247$	1842.93	0.474748
$248$	−13475.6	−3.45042
$249$	0	0
$250$	5398.89	1.36582
$251$	−1066.68	−0.268240	−0.134120	−	0.990965i	$-0.542821\pi$
$251$	−1066.68	−0.268240	−0.134120	+	0.990965i	$0.542821\pi$
$252$	0	0
$253$	286.586	0.0712154
$254$	9856.44	2.43483
$255$	0	0
$256$	−7884.58	−1.92495
$257$	−2990.18	−0.725769	−0.362884	−	0.931834i	$-0.618208\pi$
$257$	−2990.18	−0.725769	−0.362884	+	0.931834i	$0.618208\pi$
$258$	0	0
$259$	−1410.97	−0.338508
$260$	12634.4	3.01366
$261$	0	0
$262$	9014.56	2.12565
$263$	−4240.32	−0.994181	−0.497090	−	0.867699i	$-0.665598\pi$
$263$	−4240.32	−0.994181	−0.497090	+	0.867699i	$0.665598\pi$
$264$	0	0
$265$	1033.93	0.239676
$266$	1174.47	0.270719
$267$	0	0
$268$	5201.80	1.18564
$269$	−5364.21	−1.21584	−0.607922	−	0.793997i	$-0.707997\pi$
$269$	−5364.21	−1.21584	−0.607922	+	0.793997i	$0.707997\pi$
$270$	0	0
$271$	1476.11	0.330876	0.165438	−	0.986220i	$-0.447096\pi$
$271$	1476.11	0.330876	0.165438	+	0.986220i	$0.447096\pi$
$272$	−6176.88	−1.37694
$273$	0	0
$274$	15578.6	3.43480
$275$	−467.653	−0.102547
$276$	0	0
$277$	2371.67	0.514439	0.257219	−	0.966353i	$-0.417194\pi$
$277$	2371.67	0.514439	0.257219	+	0.966353i	$0.417194\pi$
$278$	5923.60	1.27796
$279$	0	0
$280$	4386.39	0.936204
$281$	1619.54	0.343821	0.171911	−	0.985113i	$-0.445006\pi$
$281$	1619.54	0.343821	0.171911	+	0.985113i	$0.445006\pi$
$282$	0	0
$283$	−3779.82	−0.793947	−0.396973	−	0.917830i	$-0.629939\pi$
$283$	−3779.82	−0.793947	−0.396973	+	0.917830i	$0.629939\pi$
$284$	−9602.73	−2.00640
$285$	0	0
$286$	3089.96	0.638858
$287$	−887.488	−0.182532
$288$	0	0
$289$	−1402.31	−0.285427
$290$	−12336.4	−2.49798
$291$	0	0
$292$	−14221.3	−2.85014
$293$	−2884.18	−0.575070	−0.287535	−	0.957770i	$-0.592836\pi$
$293$	−2884.18	−0.575070	−0.287535	+	0.957770i	$0.592836\pi$
$294$	0	0
$295$	−7683.70	−1.51648
$296$	9758.98	1.91632
$297$	0	0
$298$	−6258.93	−1.21668
$299$	1447.19	0.279909
$300$	0	0
$301$	3183.78	0.609668
$302$	7998.35	1.52402
$303$	0	0
$304$	−3458.74	−0.652541
$305$	645.352	0.121156
$306$	0	0
$307$	−4242.88	−0.788774	−0.394387	−	0.918944i	$-0.629043\pi$
$307$	−4242.88	−0.788774	−0.394387	+	0.918944i	$0.629043\pi$
$308$	1353.19	0.250341
$309$	0	0
$310$	18217.5	3.33769
$311$	9658.23	1.76099	0.880495	−	0.474055i	$-0.157210\pi$
$311$	9658.23	1.76099	0.880495	+	0.474055i	$0.157210\pi$
$312$	0	0
$313$	10683.6	1.92930	0.964651	−	0.263529i	$-0.0848866\pi$
$313$	10683.6	1.92930	0.964651	+	0.263529i	$0.0848866\pi$
$314$	−16232.0	−2.91728
$315$	0	0
$316$	−6594.47	−1.17395
$317$	2077.90	0.368160	0.184080	−	0.982911i	$-0.441069\pi$
$317$	2077.90	0.368160	0.184080	+	0.982911i	$0.441069\pi$
$318$	0	0
$319$	−2073.27	−0.363890
$320$	1639.34	0.286380
$321$	0	0
$322$	922.269	0.159615
$323$	1965.81	0.338640
$324$	0	0
$325$	−2361.53	−0.403059
$326$	−707.820	−0.120253
$327$	0	0
$328$	6138.31	1.03333
$329$	−907.973	−0.152153
$330$	0	0
$331$	−11020.5	−1.83004	−0.915019	−	0.403410i	$-0.867825\pi$
$331$	−11020.5	−1.83004	−0.915019	+	0.403410i	$0.867825\pi$
$332$	−1509.66	−0.249558
$333$	0	0
$334$	−7210.05	−1.18119
$335$	−3831.00	−0.624806
$336$	0	0
$337$	−5932.54	−0.958950	−0.479475	−	0.877556i	$-0.659173\pi$
$337$	−5932.54	−0.958950	−0.479475	+	0.877556i	$0.659173\pi$
$338$	4493.18	0.723068
$339$	0	0
$340$	13476.9	2.14966
$341$	3061.67	0.486213
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−22020.6	−3.45137
$345$	0	0
$346$	−7722.17	−1.19985
$347$	4115.57	0.636701	0.318351	−	0.947973i	$-0.396871\pi$
$347$	4115.57	0.636701	0.318351	+	0.947973i	$0.396871\pi$
$348$	0	0
$349$	−3760.31	−0.576746	−0.288373	−	0.957518i	$-0.593114\pi$
$349$	−3760.31	−0.576746	−0.288373	+	0.957518i	$0.593114\pi$
$350$	−1504.97	−0.229840
$351$	0	0
$352$	−1538.57	−0.232972
$353$	888.701	0.133997	0.0669983	−	0.997753i	$-0.478658\pi$
$353$	888.701	0.133997	0.0669983	+	0.997753i	$0.478658\pi$
$354$	0	0
$355$	7072.18	1.05733
$356$	−13185.8	−1.96304
$357$	0	0
$358$	−22819.3	−3.36882
$359$	−10561.1	−1.55263	−0.776314	−	0.630347i	$-0.782913\pi$
$359$	−10561.1	−1.55263	−0.776314	+	0.630347i	$0.782913\pi$
$360$	0	0
$361$	−5758.24	−0.839516
$362$	−15037.4	−2.18329
$363$	0	0
$364$	6833.25	0.983955
$365$	10473.7	1.50196
$366$	0	0
$367$	2773.62	0.394500	0.197250	−	0.980353i	$-0.436799\pi$
$367$	2773.62	0.394500	0.197250	+	0.980353i	$0.436799\pi$
$368$	−2716.03	−0.384736
$369$	0	0
$370$	−13193.0	−1.85371
$371$	559.198	0.0782537
$372$	0	0
$373$	−11360.2	−1.57697	−0.788484	−	0.615056i	$-0.789134\pi$
$373$	−11360.2	−1.57697	−0.788484	+	0.615056i	$0.789134\pi$
$374$	3296.00	0.455701
$375$	0	0
$376$	6279.99	0.861346
$377$	−10469.5	−1.43026
$378$	0	0
$379$	−10948.5	−1.48387	−0.741936	−	0.670470i	$-0.766092\pi$
$379$	−10948.5	−1.48387	−0.741936	+	0.670470i	$0.766092\pi$
$380$	7546.37	1.01874
$381$	0	0
$382$	6450.56	0.863978
$383$	−11122.1	−1.48385	−0.741923	−	0.670485i	$-0.766086\pi$
$383$	−11122.1	−1.48385	−0.741923	+	0.670485i	$0.766086\pi$
$384$	0	0
$385$	−996.589	−0.131924
$386$	−616.875	−0.0813423
$387$	0	0
$388$	−7941.20	−1.03906
$389$	7938.70	1.03473	0.517363	−	0.855766i	$-0.326914\pi$
$389$	7938.70	1.03473	0.517363	+	0.855766i	$0.326914\pi$
$390$	0	0
$391$	1543.68	0.199661
$392$	2372.36	0.305669
$393$	0	0
$394$	17618.7	2.25283
$395$	4856.67	0.618647
$396$	0	0
$397$	1060.60	0.134080	0.0670402	−	0.997750i	$-0.478644\pi$
$397$	1060.60	0.134080	0.0670402	+	0.997750i	$0.478644\pi$
$398$	13429.8	1.69139
$399$	0	0
$400$	4432.04	0.554005
$401$	2776.24	0.345733	0.172866	−	0.984945i	$-0.444697\pi$
$401$	2776.24	0.345733	0.172866	+	0.984945i	$0.444697\pi$
$402$	0	0
$403$	15460.7	1.91104
$404$	24853.0	3.06060
$405$	0	0
$406$	−6672.05	−0.815587
$407$	−2217.24	−0.270036
$408$	0	0
$409$	8725.98	1.05494	0.527472	−	0.849573i	$-0.323140\pi$
$409$	8725.98	1.05494	0.527472	+	0.849573i	$0.323140\pi$
$410$	−8298.27	−0.999567
$411$	0	0
$412$	−489.956	−0.0585883
$413$	−4155.69	−0.495129
$414$	0	0
$415$	1111.83	0.131512
$416$	−7769.41	−0.915689
$417$	0	0
$418$	1845.59	0.215959
$419$	2390.49	0.278719	0.139359	−	0.990242i	$-0.455496\pi$
$419$	2390.49	0.278719	0.139359	+	0.990242i	$0.455496\pi$
$420$	0	0
$421$	−8626.42	−0.998636	−0.499318	−	0.866419i	$-0.666416\pi$
$421$	−8626.42	−0.998636	−0.499318	+	0.866419i	$0.666416\pi$
$422$	9658.55	1.11415
$423$	0	0
$424$	−3867.69	−0.442999
$425$	−2519.00	−0.287504
$426$	0	0
$427$	349.035	0.0395574
$428$	13674.9	1.54440
$429$	0	0
$430$	29769.3	3.33861
$431$	−4143.05	−0.463025	−0.231512	−	0.972832i	$-0.574367\pi$
$431$	−4143.05	−0.463025	−0.231512	+	0.972832i	$0.574367\pi$
$432$	0	0
$433$	161.696	0.0179460	0.00897302	−	0.999960i	$-0.497144\pi$
$433$	161.696	0.0179460	0.00897302	+	0.999960i	$0.497144\pi$
$434$	9852.84	1.08975
$435$	0	0
$436$	−3858.25	−0.423800
$437$	864.386	0.0946205
$438$	0	0
$439$	12921.0	1.40475	0.702376	−	0.711806i	$-0.252123\pi$
$439$	12921.0	1.40475	0.702376	+	0.711806i	$0.252123\pi$
$440$	6892.90	0.746833
$441$	0	0
$442$	16644.0	1.79112
$443$	10175.2	1.09128	0.545640	−	0.838020i	$-0.316287\pi$
$443$	10175.2	1.09128	0.545640	+	0.838020i	$0.316287\pi$
$444$	0	0
$445$	9711.00	1.03448
$446$	−22156.1	−2.35229
$447$	0	0
$448$	886.626	0.0935026
$449$	−3995.27	−0.419930	−0.209965	−	0.977709i	$-0.567335\pi$
$449$	−3995.27	−0.419930	−0.209965	+	0.977709i	$0.567335\pi$
$450$	0	0
$451$	−1394.62	−0.145610
$452$	−10674.4	−1.11080
$453$	0	0
$454$	−8654.04	−0.894613
$455$	−5032.53	−0.518524
$456$	0	0
$457$	12445.5	1.27391	0.636956	−	0.770900i	$-0.280193\pi$
$457$	12445.5	1.27391	0.636956	+	0.770900i	$0.280193\pi$
$458$	13238.6	1.35065
$459$	0	0
$460$	5925.89	0.600644
$461$	−11977.5	−1.21009	−0.605043	−	0.796193i	$-0.706844\pi$
$461$	−11977.5	−1.21009	−0.605043	+	0.796193i	$0.706844\pi$
$462$	0	0
$463$	17335.0	1.74002	0.870009	−	0.493037i	$-0.164113\pi$
$463$	17335.0	1.74002	0.870009	+	0.493037i	$0.164113\pi$
$464$	19648.8	1.96589
$465$	0	0
$466$	−18170.4	−1.80629
$467$	−4014.05	−0.397747	−0.198874	−	0.980025i	$-0.563728\pi$
$467$	−4014.05	−0.397747	−0.198874	+	0.980025i	$0.563728\pi$
$468$	0	0
$469$	−2071.98	−0.203998
$470$	−8489.82	−0.833204
$471$	0	0
$472$	28742.8	2.80296
$473$	5003.08	0.486346
$474$	0	0
$475$	−1410.51	−0.136250
$476$	7288.89	0.701861
$477$	0	0
$478$	−9310.99	−0.890952
$479$	6809.91	0.649588	0.324794	−	0.945785i	$-0.394705\pi$
$479$	6809.91	0.649588	0.324794	+	0.945785i	$0.394705\pi$
$480$	0	0
$481$	−11196.5	−1.06137
$482$	11173.3	1.05587
$483$	0	0
$484$	2126.43	0.199703
$485$	5848.51	0.547561
$486$	0	0
$487$	−16342.2	−1.52060	−0.760302	−	0.649570i	$-0.774949\pi$
$487$	−16342.2	−1.52060	−0.760302	+	0.649570i	$0.774949\pi$
$488$	−2414.10	−0.223937
$489$	0	0
$490$	−3207.15	−0.295682
$491$	−1686.53	−0.155014	−0.0775070	−	0.996992i	$-0.524696\pi$
$491$	−1686.53	−0.155014	−0.0775070	+	0.996992i	$0.524696\pi$
$492$	0	0
$493$	−11167.6	−1.02021
$494$	9319.79	0.848821
$495$	0	0
$496$	−29016.0	−2.62673
$497$	3824.95	0.345217
$498$	0	0
$499$	1992.19	0.178723	0.0893616	−	0.995999i	$-0.471517\pi$
$499$	1992.19	0.178723	0.0893616	+	0.995999i	$0.471517\pi$
$500$	18761.7	1.67810
$501$	0	0
$502$	−5394.27	−0.479597
$503$	−11525.3	−1.02165	−0.510824	−	0.859685i	$-0.670660\pi$
$503$	−11525.3	−1.02165	−0.510824	+	0.859685i	$0.670660\pi$
$504$	0	0
$505$	−18303.6	−1.61287
$506$	1449.28	0.127329
$507$	0	0
$508$	34252.2	2.99153
$509$	12191.9	1.06168	0.530842	−	0.847471i	$-0.321876\pi$
$509$	12191.9	1.06168	0.530842	+	0.847471i	$0.321876\pi$
$510$	0	0
$511$	5664.63	0.490388
$512$	−25796.8	−2.22670
$513$	0	0
$514$	−15121.5	−1.29763
$515$	360.841	0.0308748
$516$	0	0
$517$	−1426.81	−0.121376
$518$	−7135.37	−0.605232
$519$	0	0
$520$	34807.4	2.93540
$521$	−11196.1	−0.941482	−0.470741	−	0.882271i	$-0.656013\pi$
$521$	−11196.1	−0.941482	−0.470741	+	0.882271i	$0.656013\pi$
$522$	0	0
$523$	−11969.2	−1.00072	−0.500361	−	0.865817i	$-0.666799\pi$
$523$	−11969.2	−1.00072	−0.500361	+	0.865817i	$0.666799\pi$
$524$	31326.6	2.61166
$525$	0	0
$526$	−21443.6	−1.77754
$527$	16491.6	1.36316
$528$	0	0
$529$	−11488.2	−0.944212
$530$	5228.67	0.428526
$531$	0	0
$532$	4081.41	0.332616
$533$	−7042.50	−0.572316
$534$	0	0
$535$	−10071.3	−0.813867
$536$	14330.8	1.15485
$537$	0	0
$538$	−27127.1	−2.17385
$539$	−539.000	−0.0430730
$540$	0	0
$541$	10468.1	0.831905	0.415953	−	0.909386i	$-0.363448\pi$
$541$	10468.1	0.831905	0.415953	+	0.909386i	$0.363448\pi$
$542$	7464.78	0.591587
$543$	0	0
$544$	−8287.47	−0.653166
$545$	2841.51	0.223334
$546$	0	0
$547$	−18969.5	−1.48277	−0.741386	−	0.671078i	$-0.765831\pi$
$547$	−18969.5	−1.48277	−0.741386	+	0.671078i	$0.765831\pi$
$548$	54137.2	4.22012
$549$	0	0
$550$	−2364.95	−0.183349
$551$	−6253.30	−0.483484
$552$	0	0
$553$	2626.71	0.201987
$554$	11993.6	0.919786
$555$	0	0
$556$	20585.2	1.57015
$557$	−11524.3	−0.876658	−0.438329	−	0.898815i	$-0.644430\pi$
$557$	−11524.3	−0.876658	−0.438329	+	0.898815i	$0.644430\pi$
$558$	0	0
$559$	25264.3	1.91157
$560$	9444.87	0.712712
$561$	0	0
$562$	8190.12	0.614732
$563$	−14117.8	−1.05683	−0.528415	−	0.848986i	$-0.677213\pi$
$563$	−14117.8	−1.05683	−0.528415	+	0.848986i	$0.677213\pi$
$564$	0	0
$565$	7861.43	0.585368
$566$	−19114.8	−1.41953
$567$	0	0
$568$	−26455.3	−1.95429
$569$	16291.9	1.20034	0.600168	−	0.799874i	$-0.295101\pi$
$569$	16291.9	1.20034	0.600168	+	0.799874i	$0.295101\pi$
$570$	0	0
$571$	4581.10	0.335750	0.167875	−	0.985808i	$-0.446310\pi$
$571$	4581.10	0.335750	0.167875	+	0.985808i	$0.446310\pi$
$572$	10738.0	0.784925
$573$	0	0
$574$	−4488.08	−0.326357
$575$	−1107.62	−0.0803324
$576$	0	0
$577$	20971.6	1.51310	0.756550	−	0.653935i	$-0.226883\pi$
$577$	20971.6	1.51310	0.756550	+	0.653935i	$0.226883\pi$
$578$	−7091.54	−0.510327
$579$	0	0
$580$	−42870.2	−3.06912
$581$	601.326	0.0429384
$582$	0	0
$583$	878.740	0.0624249
$584$	−39179.4	−2.77612
$585$	0	0
$586$	−14585.5	−1.02819
$587$	−26603.0	−1.87057	−0.935283	−	0.353900i	$-0.884855\pi$
$587$	−26603.0	−1.87057	−0.935283	+	0.353900i	$0.884855\pi$
$588$	0	0
$589$	9234.45	0.646008
$590$	−38856.9	−2.71138
$591$	0	0
$592$	21013.2	1.45885
$593$	13431.6	0.930135	0.465068	−	0.885275i	$-0.346030\pi$
$593$	13431.6	0.930135	0.465068	+	0.885275i	$0.346030\pi$
$594$	0	0
$595$	−5368.09	−0.369866
$596$	−21750.5	−1.49486
$597$	0	0
$598$	7318.50	0.500461
$599$	7329.80	0.499979	0.249990	−	0.968249i	$-0.419573\pi$
$599$	7329.80	0.499979	0.249990	+	0.968249i	$0.419573\pi$
$600$	0	0
$601$	−9241.26	−0.627219	−0.313610	−	0.949552i	$-0.601538\pi$
$601$	−9241.26	−0.627219	−0.313610	+	0.949552i	$0.601538\pi$
$602$	16100.6	1.09005
$603$	0	0
$604$	27795.2	1.87247
$605$	−1566.07	−0.105239
$606$	0	0
$607$	11068.2	0.740109	0.370054	−	0.929010i	$-0.379339\pi$
$607$	11068.2	0.740109	0.370054	+	0.929010i	$0.379339\pi$
$608$	−4640.57	−0.309539
$609$	0	0
$610$	3263.58	0.216621
$611$	−7205.06	−0.477063
$612$	0	0
$613$	−22528.7	−1.48438	−0.742191	−	0.670189i	$-0.766213\pi$
$613$	−22528.7	−1.48438	−0.742191	+	0.670189i	$0.766213\pi$
$614$	−21456.5	−1.41028
$615$	0	0
$616$	3727.99	0.243839
$617$	−6859.59	−0.447579	−0.223790	−	0.974637i	$-0.571843\pi$
$617$	−6859.59	−0.447579	−0.223790	+	0.974637i	$0.571843\pi$
$618$	0	0
$619$	14905.2	0.967833	0.483917	−	0.875114i	$-0.339214\pi$
$619$	14905.2	0.967833	0.483917	+	0.875114i	$0.339214\pi$
$620$	63307.8	4.10081
$621$	0	0
$622$	48842.2	3.14854
$623$	5252.14	0.337757
$624$	0	0
$625$	−19131.8	−1.22444
$626$	54027.5	3.44948
$627$	0	0
$628$	−56408.1	−3.58428
$629$	−11943.1	−0.757079
$630$	0	0
$631$	20709.3	1.30653	0.653267	−	0.757128i	$-0.273398\pi$
$631$	20709.3	1.30653	0.653267	+	0.757128i	$0.273398\pi$
$632$	−18167.6	−1.14346
$633$	0	0
$634$	10508.1	0.658248
$635$	−25225.9	−1.57647
$636$	0	0
$637$	−2721.82	−0.169297
$638$	−10484.7	−0.650613
$639$	0	0
$640$	22772.6	1.40651
$641$	−30663.6	−1.88945	−0.944727	−	0.327859i	$-0.893673\pi$
$641$	−30663.6	−1.88945	−0.944727	+	0.327859i	$0.893673\pi$
$642$	0	0
$643$	−12820.9	−0.786326	−0.393163	−	0.919469i	$-0.628619\pi$
$643$	−12820.9	−0.786326	−0.393163	+	0.919469i	$0.628619\pi$
$644$	3204.99	0.196109
$645$	0	0
$646$	9941.23	0.605468
$647$	21345.2	1.29701	0.648506	−	0.761209i	$-0.275394\pi$
$647$	21345.2	1.29701	0.648506	+	0.761209i	$0.275394\pi$
$648$	0	0
$649$	−6530.37	−0.394976
$650$	−11942.4	−0.720645
$651$	0	0
$652$	−2459.75	−0.147748
$653$	17194.7	1.03044	0.515222	−	0.857057i	$-0.327709\pi$
$653$	17194.7	1.03044	0.515222	+	0.857057i	$0.327709\pi$
$654$	0	0
$655$	−23071.3	−1.37629
$656$	13217.1	0.786649
$657$	0	0
$658$	−4591.67	−0.272040
$659$	−27892.6	−1.64877	−0.824386	−	0.566027i	$-0.808480\pi$
$659$	−27892.6	−1.64877	−0.824386	+	0.566027i	$0.808480\pi$
$660$	0	0
$661$	−17082.0	−1.00516	−0.502582	−	0.864529i	$-0.667617\pi$
$661$	−17082.0	−1.00516	−0.502582	+	0.864529i	$0.667617\pi$
$662$	−55731.4	−3.27200
$663$	0	0
$664$	−4159.07	−0.243077
$665$	−3005.86	−0.175282
$666$	0	0
$667$	−4910.49	−0.285060
$668$	−25055.7	−1.45125
$669$	0	0
$670$	−19373.6	−1.11712
$671$	548.484	0.0315559
$672$	0	0
$673$	−27727.6	−1.58814	−0.794071	−	0.607825i	$-0.792042\pi$
$673$	−27727.6	−1.58814	−0.794071	+	0.607825i	$0.792042\pi$
$674$	−30001.2	−1.71454
$675$	0	0
$676$	15614.3	0.888388
$677$	3018.10	0.171337	0.0856685	−	0.996324i	$-0.472697\pi$
$677$	3018.10	0.171337	0.0856685	+	0.996324i	$0.472697\pi$
$678$	0	0
$679$	3163.13	0.178778
$680$	37128.4	2.09384
$681$	0	0
$682$	15483.0	0.869320
$683$	−33614.8	−1.88321	−0.941607	−	0.336714i	$-0.890684\pi$
$683$	−33614.8	−1.88321	−0.941607	+	0.336714i	$0.890684\pi$
$684$	0	0
$685$	−39870.8	−2.22392
$686$	−1734.57	−0.0965397
$687$	0	0
$688$	−47415.2	−2.62745
$689$	4437.42	0.245359
$690$	0	0
$691$	−14923.1	−0.821565	−0.410783	−	0.911733i	$-0.634744\pi$
$691$	−14923.1	−0.821565	−0.410783	+	0.911733i	$0.634744\pi$
$692$	−26835.4	−1.47417
$693$	0	0
$694$	20812.7	1.13838
$695$	−15160.5	−0.827439
$696$	0	0
$697$	−7512.09	−0.408236
$698$	−19016.1	−1.03119
$699$	0	0
$700$	−5229.93	−0.282389
$701$	−15440.5	−0.831928	−0.415964	−	0.909381i	$-0.636556\pi$
$701$	−15440.5	−0.831928	−0.415964	+	0.909381i	$0.636556\pi$
$702$	0	0
$703$	−6687.54	−0.358784
$704$	1393.27	0.0745893
$705$	0	0
$706$	4494.21	0.239578
$707$	−9899.43	−0.526600
$708$	0	0
$709$	25441.7	1.34765	0.673825	−	0.738891i	$-0.264650\pi$
$709$	25441.7	1.34765	0.673825	+	0.738891i	$0.264650\pi$
$710$	35764.5	1.89044
$711$	0	0
$712$	−36326.4	−1.91206
$713$	7251.49	0.380884
$714$	0	0
$715$	−7908.26	−0.413639
$716$	−79299.6	−4.13906
$717$	0	0
$718$	−53408.1	−2.77601
$719$	11335.3	0.587951	0.293976	−	0.955813i	$-0.405022\pi$
$719$	11335.3	0.587951	0.293976	+	0.955813i	$0.405022\pi$
$720$	0	0
$721$	195.159	0.0100806
$722$	−29119.8	−1.50101
$723$	0	0
$724$	−52256.8	−2.68247
$725$	8012.98	0.410475
$726$	0	0
$727$	−11169.8	−0.569830	−0.284915	−	0.958553i	$-0.591965\pi$
$727$	−11169.8	−0.569830	−0.284915	+	0.958553i	$0.591965\pi$
$728$	18825.4	0.958402
$729$	0	0
$730$	52965.9	2.68542
$731$	26948.9	1.36353
$732$	0	0
$733$	25635.5	1.29177	0.645885	−	0.763435i	$-0.276488\pi$
$733$	25635.5	1.29177	0.645885	+	0.763435i	$0.276488\pi$
$734$	14026.3	0.705343
$735$	0	0
$736$	−3644.08	−0.182503
$737$	−3255.97	−0.162734
$738$	0	0
$739$	20947.2	1.04270	0.521350	−	0.853343i	$-0.325429\pi$
$739$	20947.2	1.04270	0.521350	+	0.853343i	$0.325429\pi$
$740$	−45847.2	−2.27753
$741$	0	0
$742$	2827.90	0.139913
$743$	4063.24	0.200627	0.100314	−	0.994956i	$-0.468015\pi$
$743$	4063.24	0.200627	0.100314	+	0.994956i	$0.468015\pi$
$744$	0	0
$745$	16018.7	0.787759
$746$	−57449.2	−2.81952
$747$	0	0
$748$	11454.0	0.559891
$749$	−5446.99	−0.265726
$750$	0	0
$751$	24644.9	1.19747	0.598737	−	0.800945i	$-0.295669\pi$
$751$	24644.9	1.19747	0.598737	+	0.800945i	$0.295669\pi$
$752$	13522.2	0.655724
$753$	0	0
$754$	−52944.9	−2.55721
$755$	−20470.5	−0.986751
$756$	0	0
$757$	−17258.3	−0.828618	−0.414309	−	0.910136i	$-0.635977\pi$
$757$	−17258.3	−0.828618	−0.414309	+	0.910136i	$0.635977\pi$
$758$	−55367.3	−2.65308
$759$	0	0
$760$	20790.0	0.992282
$761$	−12931.6	−0.615990	−0.307995	−	0.951388i	$-0.599658\pi$
$761$	−12931.6	−0.615990	−0.307995	+	0.951388i	$0.599658\pi$
$762$	0	0
$763$	1536.82	0.0729180
$764$	22416.4	1.06152
$765$	0	0
$766$	−56245.1	−2.65303
$767$	−32976.7	−1.55244
$768$	0	0
$769$	13584.8	0.637034	0.318517	−	0.947917i	$-0.396815\pi$
$769$	13584.8	0.637034	0.318517	+	0.947917i	$0.396815\pi$
$770$	−5039.81	−0.235873
$771$	0	0
$772$	−2143.71	−0.0999402
$773$	−2239.95	−0.104224	−0.0521121	−	0.998641i	$-0.516595\pi$
$773$	−2239.95	−0.104224	−0.0521121	+	0.998641i	$0.516595\pi$
$774$	0	0
$775$	−11833.0	−0.548458
$776$	−21877.8	−1.01207
$777$	0	0
$778$	40146.5	1.85003
$779$	−4206.40	−0.193466
$780$	0	0
$781$	6010.64	0.275388
$782$	7806.50	0.356982
$783$	0	0
$784$	5108.21	0.232699
$785$	41543.2	1.88884
$786$	0	0
$787$	10614.2	0.480758	0.240379	−	0.970679i	$-0.422728\pi$
$787$	10614.2	0.480758	0.240379	+	0.970679i	$0.422728\pi$
$788$	61226.8	2.76791
$789$	0	0
$790$	24560.5	1.10610
$791$	4251.82	0.191122
$792$	0	0
$793$	2769.71	0.124029
$794$	5363.51	0.239728
$795$	0	0
$796$	46670.1	2.07811
$797$	−1020.65	−0.0453616	−0.0226808	−	0.999743i	$-0.507220\pi$
$797$	−1020.65	−0.0453616	−0.0226808	+	0.999743i	$0.507220\pi$
$798$	0	0
$799$	−7685.49	−0.340292
$800$	5946.43	0.262798
$801$	0	0
$802$	14039.6	0.618150
$803$	8901.55	0.391194
$804$	0	0
$805$	−2360.40	−0.103345
$806$	78185.5	3.41683
$807$	0	0
$808$	68469.4	2.98112
$809$	44041.1	1.91397	0.956985	−	0.290139i	$-0.0937016\pi$
$809$	44041.1	1.91397	0.956985	+	0.290139i	$0.0937016\pi$
$810$	0	0
$811$	−16722.4	−0.724048	−0.362024	−	0.932169i	$-0.617914\pi$
$811$	−16722.4	−0.724048	−0.362024	+	0.932169i	$0.617914\pi$
$812$	−23186.1	−1.00206
$813$	0	0
$814$	−11212.7	−0.482808
$815$	1811.55	0.0778599
$816$	0	0
$817$	15090.0	0.646186
$818$	44127.8	1.88618
$819$	0	0
$820$	−28837.4	−1.22811
$821$	−30420.1	−1.29314	−0.646570	−	0.762855i	$-0.723797\pi$
$821$	−30420.1	−1.29314	−0.646570	+	0.762855i	$0.723797\pi$
$822$	0	0
$823$	−330.412	−0.0139945	−0.00699724	−	0.999976i	$-0.502227\pi$
$823$	−330.412	−0.0139945	−0.00699724	+	0.999976i	$0.502227\pi$
$824$	−1349.82	−0.0570668
$825$	0	0
$826$	−21015.6	−0.885261
$827$	−4475.72	−0.188194	−0.0940968	−	0.995563i	$-0.529996\pi$
$827$	−4475.72	−0.188194	−0.0940968	+	0.995563i	$0.529996\pi$
$828$	0	0
$829$	−10923.6	−0.457651	−0.228825	−	0.973467i	$-0.573489\pi$
$829$	−10923.6	−0.457651	−0.228825	+	0.973467i	$0.573489\pi$
$830$	5622.58	0.235136
$831$	0	0
$832$	7035.67	0.293170
$833$	−2903.30	−0.120761
$834$	0	0
$835$	18452.9	0.764779
$836$	6413.65	0.265336
$837$	0	0
$838$	12088.8	0.498332
$839$	28354.8	1.16677	0.583383	−	0.812197i	$-0.301729\pi$
$839$	28354.8	1.16677	0.583383	+	0.812197i	$0.301729\pi$
$840$	0	0
$841$	11135.4	0.456574
$842$	−43624.3	−1.78550
$843$	0	0
$844$	33564.5	1.36888
$845$	−11499.6	−0.468162
$846$	0	0
$847$	−847.000	−0.0343604
$848$	−8327.99	−0.337246
$849$	0	0
$850$	−12738.7	−0.514040
$851$	−5251.49	−0.211538
$852$	0	0
$853$	39699.8	1.59355	0.796774	−	0.604278i	$-0.206538\pi$
$853$	39699.8	1.59355	0.796774	+	0.604278i	$0.206538\pi$
$854$	1765.09	0.0707262
$855$	0	0
$856$	37674.1	1.50429
$857$	−6093.09	−0.242866	−0.121433	−	0.992600i	$-0.538749\pi$
$857$	−6093.09	−0.242866	−0.121433	+	0.992600i	$0.538749\pi$
$858$	0	0
$859$	20688.1	0.821733	0.410867	−	0.911695i	$-0.365226\pi$
$859$	20688.1	0.821733	0.410867	+	0.911695i	$0.365226\pi$
$860$	103452.	4.10194
$861$	0	0
$862$	−20951.6	−0.827860
$863$	38835.1	1.53182	0.765910	−	0.642948i	$-0.222289\pi$
$863$	38835.1	1.53182	0.765910	+	0.642948i	$0.222289\pi$
$864$	0	0
$865$	19763.6	0.776860
$866$	817.708	0.0320864
$867$	0	0
$868$	34239.7	1.33891
$869$	4127.68	0.161130
$870$	0	0
$871$	−16441.8	−0.639621
$872$	−10629.4	−0.412794
$873$	0	0
$874$	4371.25	0.169176
$875$	−7473.16	−0.288730
$876$	0	0
$877$	−14454.3	−0.556543	−0.278272	−	0.960502i	$-0.589762\pi$
$877$	−14454.3	−0.556543	−0.278272	+	0.960502i	$0.589762\pi$
$878$	65342.3	2.51161
$879$	0	0
$880$	14841.9	0.568547
$881$	21890.5	0.837130	0.418565	−	0.908187i	$-0.362533\pi$
$881$	21890.5	0.837130	0.418565	+	0.908187i	$0.362533\pi$
$882$	0	0
$883$	−34415.2	−1.31162	−0.655812	−	0.754924i	$-0.727674\pi$
$883$	−34415.2	−1.31162	−0.655812	+	0.754924i	$0.727674\pi$
$884$	57839.7	2.20063
$885$	0	0
$886$	51456.4	1.95114
$887$	288.789	0.0109319	0.00546594	−	0.999985i	$-0.498260\pi$
$887$	288.789	0.0109319	0.00546594	+	0.999985i	$0.498260\pi$
$888$	0	0
$889$	−13643.3	−0.514716
$890$	49109.1	1.84960
$891$	0	0
$892$	−76994.9	−2.89011
$893$	−4303.49	−0.161266
$894$	0	0
$895$	58402.3	2.18120
$896$	12316.5	0.459223
$897$	0	0
$898$	−20204.3	−0.750809
$899$	−52460.1	−1.94621
$900$	0	0
$901$	4733.30	0.175016
$902$	−7052.69	−0.260343
$903$	0	0
$904$	−29407.7	−1.08195
$905$	38485.9	1.41361
$906$	0	0
$907$	−36248.0	−1.32701	−0.663503	−	0.748174i	$-0.730931\pi$
$907$	−36248.0	−1.32701	−0.663503	+	0.748174i	$0.730931\pi$
$908$	−30073.7	−1.09915
$909$	0	0
$910$	−25449.8	−0.927090
$911$	4600.52	0.167313	0.0836564	−	0.996495i	$-0.473340\pi$
$911$	4600.52	0.167313	0.0836564	+	0.996495i	$0.473340\pi$
$912$	0	0
$913$	944.941	0.0342530
$914$	62937.8	2.27768
$915$	0	0
$916$	46005.6	1.65946
$917$	−12478.0	−0.449356
$918$	0	0
$919$	−46257.9	−1.66040	−0.830200	−	0.557466i	$-0.811774\pi$
$919$	−46257.9	−1.66040	−0.830200	+	0.557466i	$0.811774\pi$
$920$	16325.7	0.585046
$921$	0	0
$922$	−60571.1	−2.16356
$923$	30352.2	1.08240
$924$	0	0
$925$	8569.42	0.304606
$926$	87664.3	3.11105
$927$	0	0
$928$	26362.6	0.932539
$929$	3303.31	0.116661	0.0583305	−	0.998297i	$-0.481422\pi$
$929$	3303.31	0.116661	0.0583305	+	0.998297i	$0.481422\pi$
$930$	0	0
$931$	−1625.71	−0.0572291
$932$	−63144.3	−2.21927
$933$	0	0
$934$	−20299.3	−0.711148
$935$	−8435.57	−0.295051
$936$	0	0
$937$	−44736.9	−1.55975	−0.779877	−	0.625932i	$-0.784719\pi$
$937$	−44736.9	−1.55975	−0.779877	+	0.625932i	$0.784719\pi$
$938$	−10478.1	−0.364736
$939$	0	0
$940$	−29503.1	−1.02371
$941$	21009.4	0.727829	0.363914	−	0.931432i	$-0.381440\pi$
$941$	21009.4	0.727829	0.363914	+	0.931432i	$0.381440\pi$
$942$	0	0
$943$	−3303.13	−0.114067
$944$	61889.6	2.13383
$945$	0	0
$946$	25300.9	0.869558
$947$	−11343.0	−0.389227	−0.194613	−	0.980880i	$-0.562345\pi$
$947$	−11343.0	−0.389227	−0.194613	+	0.980880i	$0.562345\pi$
$948$	0	0
$949$	44950.6	1.53758
$950$	−7133.04	−0.243607
$951$	0	0
$952$	20080.7	0.683634
$953$	45452.4	1.54496	0.772481	−	0.635038i	$-0.219015\pi$
$953$	45452.4	1.54496	0.772481	+	0.635038i	$0.219015\pi$
$954$	0	0
$955$	−16509.2	−0.559397
$956$	−32356.7	−1.09466
$957$	0	0
$958$	34438.1	1.16142
$959$	−21563.9	−0.726105
$960$	0	0
$961$	47678.5	1.60043
$962$	−56621.5	−1.89766
$963$	0	0
$964$	38828.5	1.29728
$965$	1578.79	0.0526664
$966$	0	0
$967$	4435.17	0.147493	0.0737464	−	0.997277i	$-0.476504\pi$
$967$	4435.17	0.147493	0.0737464	+	0.997277i	$0.476504\pi$
$968$	5858.27	0.194517
$969$	0	0
$970$	29576.2	0.979006
$971$	33349.3	1.10219	0.551097	−	0.834441i	$-0.314209\pi$
$971$	33349.3	1.10219	0.551097	+	0.834441i	$0.314209\pi$
$972$	0	0
$973$	−8199.47	−0.270157
$974$	−82643.3	−2.71875
$975$	0	0
$976$	−5198.09	−0.170478
$977$	39719.6	1.30066	0.650328	−	0.759653i	$-0.274631\pi$
$977$	39719.6	1.30066	0.650328	+	0.759653i	$0.274631\pi$
$978$	0	0
$979$	8253.36	0.269437
$980$	−11145.2	−0.363286
$981$	0	0
$982$	−8528.86	−0.277156
$983$	−16117.6	−0.522963	−0.261482	−	0.965208i	$-0.584211\pi$
$983$	−16117.6	−0.522963	−0.261482	+	0.965208i	$0.584211\pi$
$984$	0	0
$985$	−45092.1	−1.45863
$986$	−56475.2	−1.82407
$987$	0	0
$988$	32387.3	1.04289
$989$	11849.7	0.380989
$990$	0	0
$991$	−14926.8	−0.478471	−0.239236	−	0.970962i	$-0.576897\pi$
$991$	−14926.8	−0.478471	−0.239236	+	0.970962i	$0.576897\pi$
$992$	−38930.6	−1.24602
$993$	0	0
$994$	19343.0	0.617227
$995$	−34371.4	−1.09512
$996$	0	0
$997$	51206.2	1.62660	0.813298	−	0.581847i	$-0.197670\pi$
$997$	51206.2	1.62660	0.813298	+	0.581847i	$0.197670\pi$
$998$	10074.6	0.319546
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.m.1.4		4
3.2	odd	2		77.4.a.c.1.1	✓	4
12.11	even	2		1232.4.a.w.1.4		4
15.14	odd	2		1925.4.a.q.1.4		4
21.20	even	2		539.4.a.f.1.1		4
33.32	even	2		847.4.a.e.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.c.1.1	✓	4	3.2	odd	2
539.4.a.f.1.1		4	21.20	even	2
693.4.a.m.1.4		4	1.1	even	1	trivial
847.4.a.e.1.4		4	33.32	even	2
1232.4.a.w.1.4		4	12.11	even	2
1925.4.a.q.1.4		4	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)

\(f(q)\)	\(=\)	\(q+5.05706 q^{2} +17.5738 q^{4} -12.9427 q^{5} -7.00000 q^{7} +48.4155 q^{8} +O(q^{10})\)	\(q+5.05706 q^{2} +17.5738 q^{4} -12.9427 q^{5} -7.00000 q^{7} +48.4155 q^{8} -65.4521 q^{10} -11.0000 q^{11} -55.5473 q^{13} -35.3994 q^{14} +104.249 q^{16} -59.2511 q^{17} -33.1777 q^{19} -227.453 q^{20} -55.6276 q^{22} -26.0532 q^{23} +42.5139 q^{25} -280.906 q^{26} -123.017 q^{28} +188.479 q^{29} -278.333 q^{31} +139.870 q^{32} -299.636 q^{34} +90.5990 q^{35} +201.567 q^{37} -167.781 q^{38} -626.628 q^{40} +126.784 q^{41} -454.826 q^{43} -193.312 q^{44} -131.753 q^{46} +129.710 q^{47} +49.0000 q^{49} +214.995 q^{50} -976.179 q^{52} -79.8855 q^{53} +142.370 q^{55} -338.908 q^{56} +953.150 q^{58} +593.670 q^{59} -49.8622 q^{61} -1407.55 q^{62} -126.661 q^{64} +718.932 q^{65} +295.997 q^{67} -1041.27 q^{68} +458.164 q^{70} -546.422 q^{71} -809.232 q^{73} +1019.34 q^{74} -583.059 q^{76} +77.0000 q^{77} -375.244 q^{79} -1349.27 q^{80} +641.154 q^{82} -85.9037 q^{83} +766.870 q^{85} -2300.08 q^{86} -532.570 q^{88} -750.306 q^{89} +388.831 q^{91} -457.856 q^{92} +655.953 q^{94} +429.409 q^{95} -451.876 q^{97} +247.796 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 22 q^{4} + 18 q^{5} - 28 q^{7} + 60 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 22 * q^4 + 18 * q^5 - 28 * q^7 + 60 * q^8	\( 4 q + 4 q^{2} + 22 q^{4} + 18 q^{5} - 28 q^{7} + 60 q^{8} - 92 q^{10} - 44 q^{11} - 134 q^{13} - 28 q^{14} - 6 q^{16} + 74 q^{17} - 164 q^{19} - 116 q^{20} - 44 q^{22} - 194 q^{23} + 38 q^{25} - 734 q^{26} - 154 q^{28} + 108 q^{29} - 412 q^{31} + 4 q^{32} - 346 q^{34} - 126 q^{35} + 286 q^{37} - 224 q^{38} - 540 q^{40} + 18 q^{41} - 496 q^{43} - 242 q^{44} - 284 q^{46} - 62 q^{47} + 196 q^{49} - 212 q^{50} - 822 q^{52} + 828 q^{53} - 198 q^{55} - 420 q^{56} + 1388 q^{58} + 1224 q^{59} - 350 q^{61} + 878 q^{62} - 718 q^{64} + 396 q^{65} - 1498 q^{67} - 1058 q^{68} + 644 q^{70} - 2326 q^{71} - 1630 q^{73} + 1156 q^{74} - 3152 q^{76} + 308 q^{77} - 1020 q^{79} - 3072 q^{80} + 2118 q^{82} + 1920 q^{83} + 2008 q^{85} - 1056 q^{86} - 660 q^{88} - 1550 q^{89} + 938 q^{91} - 2592 q^{92} - 1042 q^{94} - 2332 q^{95} - 2202 q^{97} + 196 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 22 * q^4 + 18 * q^5 - 28 * q^7 + 60 * q^8 - 92 * q^10 - 44 * q^11 - 134 * q^13 - 28 * q^14 - 6 * q^16 + 74 * q^17 - 164 * q^19 - 116 * q^20 - 44 * q^22 - 194 * q^23 + 38 * q^25 - 734 * q^26 - 154 * q^28 + 108 * q^29 - 412 * q^31 + 4 * q^32 - 346 * q^34 - 126 * q^35 + 286 * q^37 - 224 * q^38 - 540 * q^40 + 18 * q^41 - 496 * q^43 - 242 * q^44 - 284 * q^46 - 62 * q^47 + 196 * q^49 - 212 * q^50 - 822 * q^52 + 828 * q^53 - 198 * q^55 - 420 * q^56 + 1388 * q^58 + 1224 * q^59 - 350 * q^61 + 878 * q^62 - 718 * q^64 + 396 * q^65 - 1498 * q^67 - 1058 * q^68 + 644 * q^70 - 2326 * q^71 - 1630 * q^73 + 1156 * q^74 - 3152 * q^76 + 308 * q^77 - 1020 * q^79 - 3072 * q^80 + 2118 * q^82 + 1920 * q^83 + 2008 * q^85 - 1056 * q^86 - 660 * q^88 - 1550 * q^89 + 938 * q^91 - 2592 * q^92 - 1042 * q^94 - 2332 * q^95 - 2202 * q^97 + 196 * q^98

Introduction

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