LMFDB - Embedded newform 693.4.a.r.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:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 43x^{6} + 57x^{5} + 560x^{4} - 439x^{3} - 2246x^{2} + 384x + 1056 $ x^8 - 2x^7 - 43x^6 + 57x^5 + 560x^4 - 439x^3 - 2246x^2 + 384*x + 1056
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.75652$ 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)

$f(q)$	$=$	\(q-4.75652 q^{2} +14.6245 q^{4} -20.4224 q^{5} +7.00000 q^{7} -31.5093 q^{8} +97.1395 q^{10} -11.0000 q^{11} -9.41327 q^{13} -33.2956 q^{14} +32.8790 q^{16} -10.5397 q^{17} -3.48558 q^{19} -298.667 q^{20} +52.3217 q^{22} +50.3192 q^{23} +292.075 q^{25} +44.7744 q^{26} +102.371 q^{28} -11.5606 q^{29} +169.249 q^{31} +95.6852 q^{32} +50.1321 q^{34} -142.957 q^{35} -283.695 q^{37} +16.5792 q^{38} +643.496 q^{40} +165.966 q^{41} -209.054 q^{43} -160.869 q^{44} -239.344 q^{46} +604.561 q^{47} +49.0000 q^{49} -1389.26 q^{50} -137.664 q^{52} +164.372 q^{53} +224.646 q^{55} -220.565 q^{56} +54.9883 q^{58} -292.736 q^{59} -202.920 q^{61} -805.036 q^{62} -718.160 q^{64} +192.242 q^{65} +750.145 q^{67} -154.137 q^{68} +679.977 q^{70} +14.6599 q^{71} +992.390 q^{73} +1349.40 q^{74} -50.9747 q^{76} -77.0000 q^{77} +1285.50 q^{79} -671.468 q^{80} -789.418 q^{82} -722.856 q^{83} +215.246 q^{85} +994.370 q^{86} +346.602 q^{88} +502.073 q^{89} -65.8929 q^{91} +735.891 q^{92} -2875.60 q^{94} +71.1840 q^{95} -532.376 q^{97} -233.069 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{2} + 30 q^{4} - 10 q^{5} + 56 q^{7} - 81 q^{8} + 9 q^{10} - 88 q^{11} + 16 q^{13} - 42 q^{14} + 122 q^{16} - 90 q^{17} - 42 q^{19} - 291 q^{20} + 66 q^{22} - 338 q^{23} + 244 q^{25} - 209 q^{26}+ \cdots - 294 q^{98}+O(q^{100}) $ 8 * q - 6 * q^2 + 30 * q^4 - 10 * q^5 + 56 * q^7 - 81 * q^8 + 9 * q^10 - 88 * q^11 + 16 * q^13 - 42 * q^14 + 122 * q^16 - 90 * q^17 - 42 * q^19 - 291 * q^20 + 66 * q^22 - 338 * q^23 + 244 * q^25 - 209 * q^26 + 210 * q^28 - 496 * q^29 - 8 * q^31 - 524 * q^32 - 302 * q^34 - 70 * q^35 - 360 * q^37 - 45 * q^38 - 6 * q^40 - 242 * q^41 - 66 * q^43 - 330 * q^44 + 344 * q^46 - 540 * q^47 + 392 * q^49 - 1171 * q^50 + 465 * q^52 - 906 * q^53 + 110 * q^55 - 567 * q^56 + 977 * q^58 - 1242 * q^59 - 318 * q^61 + 110 * q^62 + 525 * q^64 - 1258 * q^65 + 522 * q^67 - 678 * q^68 + 63 * q^70 - 858 * q^71 - 78 * q^73 - 1651 * q^74 + 1775 * q^76 - 616 * q^77 + 516 * q^79 - 567 * q^80 - 1212 * q^82 - 3192 * q^83 + 720 * q^85 - 1322 * q^86 + 891 * q^88 - 2356 * q^89 + 112 * q^91 - 4504 * q^92 - 423 * q^94 - 3308 * q^95 - 1556 * q^97 - 294 * 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 $

$2$	−4.75652	−1.68168	−0.840841	−	0.541282i	$-0.817939\pi$
$2$	−4.75652	−1.68168	−0.840841	+	0.541282i	$0.817939\pi$
$3$	0	0
$3$	0	0
$4$	14.6245	1.82806
$5$	−20.4224	−1.82664	−0.913318	−	0.407248i	$-0.866489\pi$
$5$	−20.4224	−1.82664	−0.913318	+	0.407248i	$0.866489\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−31.5093	−1.39253
$9$	0	0
$10$	97.1395	3.07182
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−9.41327	−0.200829	−0.100414	−	0.994946i	$-0.532017\pi$
$13$	−9.41327	−0.200829	−0.100414	+	0.994946i	$0.532017\pi$
$14$	−33.2956	−0.635616
$15$	0	0
$16$	32.8790	0.513734
$17$	−10.5397	−0.150367	−0.0751837	−	0.997170i	$-0.523954\pi$
$17$	−10.5397	−0.150367	−0.0751837	+	0.997170i	$0.523954\pi$
$18$	0	0
$19$	−3.48558	−0.0420867	−0.0210433	−	0.999779i	$-0.506699\pi$
$19$	−3.48558	−0.0420867	−0.0210433	+	0.999779i	$0.506699\pi$
$20$	−298.667	−3.33919
$21$	0	0
$22$	52.3217	0.507046
$23$	50.3192	0.456186	0.228093	−	0.973639i	$-0.426751\pi$
$23$	50.3192	0.456186	0.228093	+	0.973639i	$0.426751\pi$
$24$	0	0
$25$	292.075	2.33660
$26$	44.7744	0.337730
$27$	0	0
$28$	102.371	0.690940
$29$	−11.5606	−0.0740260	−0.0370130	−	0.999315i	$-0.511784\pi$
$29$	−11.5606	−0.0740260	−0.0370130	+	0.999315i	$0.511784\pi$
$30$	0	0
$31$	169.249	0.980582	0.490291	−	0.871559i	$-0.336891\pi$
$31$	169.249	0.980582	0.490291	+	0.871559i	$0.336891\pi$
$32$	95.6852	0.528591
$33$	0	0
$34$	50.1321	0.252870
$35$	−142.957	−0.690403
$36$	0	0
$37$	−283.695	−1.26052	−0.630260	−	0.776384i	$-0.717052\pi$
$37$	−283.695	−1.26052	−0.630260	+	0.776384i	$0.717052\pi$
$38$	16.5792	0.0707765
$39$	0	0
$40$	643.496	2.54364
$41$	165.966	0.632182	0.316091	−	0.948729i	$-0.397629\pi$
$41$	165.966	0.632182	0.316091	+	0.948729i	$0.397629\pi$
$42$	0	0
$43$	−209.054	−0.741406	−0.370703	−	0.928751i	$-0.620883\pi$
$43$	−209.054	−0.741406	−0.370703	+	0.928751i	$0.620883\pi$
$44$	−160.869	−0.551180
$45$	0	0
$46$	−239.344	−0.767160
$47$	604.561	1.87626	0.938130	−	0.346282i	$-0.112556\pi$
$47$	604.561	1.87626	0.938130	+	0.346282i	$0.112556\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−1389.26	−3.92942
$51$	0	0
$52$	−137.664	−0.367126
$53$	164.372	0.426004	0.213002	−	0.977052i	$-0.431676\pi$
$53$	164.372	0.426004	0.213002	+	0.977052i	$0.431676\pi$
$54$	0	0
$55$	224.646	0.550751
$56$	−220.565	−0.526326
$57$	0	0
$58$	54.9883	0.124488
$59$	−292.736	−0.645949	−0.322974	−	0.946408i	$-0.604683\pi$
$59$	−292.736	−0.645949	−0.322974	+	0.946408i	$0.604683\pi$
$60$	0	0
$61$	−202.920	−0.425921	−0.212961	−	0.977061i	$-0.568311\pi$
$61$	−202.920	−0.425921	−0.212961	+	0.977061i	$0.568311\pi$
$62$	−805.036	−1.64903
$63$	0	0
$64$	−718.160	−1.40266
$65$	192.242	0.366841
$66$	0	0
$67$	750.145	1.36783	0.683917	−	0.729560i	$-0.260275\pi$
$67$	750.145	1.36783	0.683917	+	0.729560i	$0.260275\pi$
$68$	−154.137	−0.274880
$69$	0	0
$70$	679.977	1.16104
$71$	14.6599	0.0245044	0.0122522	−	0.999925i	$-0.496100\pi$
$71$	14.6599	0.0245044	0.0122522	+	0.999925i	$0.496100\pi$
$72$	0	0
$73$	992.390	1.59110	0.795551	−	0.605886i	$-0.207181\pi$
$73$	992.390	1.59110	0.795551	+	0.605886i	$0.207181\pi$
$74$	1349.40	2.11979
$75$	0	0
$76$	−50.9747	−0.0769369
$77$	−77.0000	−0.113961
$78$	0	0
$79$	1285.50	1.83077	0.915383	−	0.402585i	$-0.131888\pi$
$79$	1285.50	1.83077	0.915383	+	0.402585i	$0.131888\pi$
$80$	−671.468	−0.938405
$81$	0	0
$82$	−789.418	−1.06313
$83$	−722.856	−0.955949	−0.477974	−	0.878374i	$-0.658629\pi$
$83$	−722.856	−0.955949	−0.477974	+	0.878374i	$0.658629\pi$
$84$	0	0
$85$	215.246	0.274667
$86$	994.370	1.24681
$87$	0	0
$88$	346.602	0.419863
$89$	502.073	0.597973	0.298986	−	0.954257i	$-0.403351\pi$
$89$	502.073	0.597973	0.298986	+	0.954257i	$0.403351\pi$
$90$	0	0
$91$	−65.8929	−0.0759061
$92$	735.891	0.833934
$93$	0	0
$94$	−2875.60	−3.15528
$95$	71.1840	0.0768771
$96$	0	0
$97$	−532.376	−0.557264	−0.278632	−	0.960398i	$-0.589881\pi$
$97$	−532.376	−0.557264	−0.278632	+	0.960398i	$0.589881\pi$
$98$	−233.069	−0.240240
$99$	0	0
$100$	4271.43	4.27143
$101$	410.136	0.404060	0.202030	−	0.979379i	$-0.435246\pi$
$101$	410.136	0.404060	0.202030	+	0.979379i	$0.435246\pi$
$102$	0	0
$103$	−1443.12	−1.38053	−0.690267	−	0.723555i	$-0.742507\pi$
$103$	−1443.12	−1.38053	−0.690267	+	0.723555i	$0.742507\pi$
$104$	296.606	0.279659
$105$	0	0
$106$	−781.838	−0.716404
$107$	−2069.33	−1.86963	−0.934813	−	0.355140i	$-0.884433\pi$
$107$	−2069.33	−1.86963	−0.934813	+	0.355140i	$0.884433\pi$
$108$	0	0
$109$	−1944.14	−1.70840	−0.854198	−	0.519947i	$-0.825952\pi$
$109$	−1944.14	−1.70840	−0.854198	+	0.519947i	$0.825952\pi$
$110$	−1068.53	−0.926189
$111$	0	0
$112$	230.153	0.194173
$113$	−65.0582	−0.0541607	−0.0270804	−	0.999633i	$-0.508621\pi$
$113$	−65.0582	−0.0541607	−0.0270804	+	0.999633i	$0.508621\pi$
$114$	0	0
$115$	−1027.64	−0.833286
$116$	−169.068	−0.135324
$117$	0	0
$118$	1392.40	1.08628
$119$	−73.7777	−0.0568336
$120$	0	0
$121$	121.000	0.0909091
$122$	965.190	0.716264
$123$	0	0
$124$	2475.18	1.79256
$125$	−3412.07	−2.44148
$126$	0	0
$127$	−449.328	−0.313948	−0.156974	−	0.987603i	$-0.550174\pi$
$127$	−449.328	−0.313948	−0.156974	+	0.987603i	$0.550174\pi$
$128$	2650.46	1.83023
$129$	0	0
$130$	−914.401	−0.616910
$131$	1988.69	1.32636	0.663179	−	0.748461i	$-0.269207\pi$
$131$	1988.69	1.32636	0.663179	+	0.748461i	$0.269207\pi$
$132$	0	0
$133$	−24.3991	−0.0159073
$134$	−3568.08	−2.30026
$135$	0	0
$136$	332.098	0.209391
$137$	−2435.56	−1.51886	−0.759431	−	0.650587i	$-0.774523\pi$
$137$	−2435.56	−1.51886	−0.759431	+	0.650587i	$0.774523\pi$
$138$	0	0
$139$	−194.806	−0.118872	−0.0594360	−	0.998232i	$-0.518930\pi$
$139$	−194.806	−0.118872	−0.0594360	+	0.998232i	$0.518930\pi$
$140$	−2090.67	−1.26210
$141$	0	0
$142$	−69.7303	−0.0412087
$143$	103.546	0.0605521
$144$	0	0
$145$	236.096	0.135219
$146$	−4720.32	−2.67573
$147$	0	0
$148$	−4148.89	−2.30430
$149$	540.585	0.297224	0.148612	−	0.988896i	$-0.452519\pi$
$149$	540.585	0.297224	0.148612	+	0.988896i	$0.452519\pi$
$150$	0	0
$151$	−1585.07	−0.854244	−0.427122	−	0.904194i	$-0.640472\pi$
$151$	−1585.07	−0.854244	−0.427122	+	0.904194i	$0.640472\pi$
$152$	109.828	0.0586069
$153$	0	0
$154$	366.252	0.191646
$155$	−3456.47	−1.79117
$156$	0	0
$157$	−2488.18	−1.26483	−0.632414	−	0.774630i	$-0.717936\pi$
$157$	−2488.18	−1.26483	−0.632414	+	0.774630i	$0.717936\pi$
$158$	−6114.52	−3.07877
$159$	0	0
$160$	−1954.12	−0.965543
$161$	352.234	0.172422
$162$	0	0
$163$	739.943	0.355563	0.177782	−	0.984070i	$-0.443108\pi$
$163$	739.943	0.355563	0.177782	+	0.984070i	$0.443108\pi$
$164$	2427.16	1.15566
$165$	0	0
$166$	3438.28	1.60760
$167$	2038.25	0.944459	0.472229	−	0.881476i	$-0.343449\pi$
$167$	2038.25	0.944459	0.472229	+	0.881476i	$0.343449\pi$
$168$	0	0
$169$	−2108.39	−0.959668
$170$	−1023.82	−0.461902
$171$	0	0
$172$	−3057.30	−1.35533
$173$	−1991.77	−0.875326	−0.437663	−	0.899139i	$-0.644194\pi$
$173$	−1991.77	−0.875326	−0.437663	+	0.899139i	$0.644194\pi$
$174$	0	0
$175$	2044.52	0.883151
$176$	−361.669	−0.154897
$177$	0	0
$178$	−2388.12	−1.00560
$179$	−2650.93	−1.10693	−0.553463	−	0.832874i	$-0.686694\pi$
$179$	−2650.93	−1.10693	−0.553463	+	0.832874i	$0.686694\pi$
$180$	0	0
$181$	−4491.40	−1.84444	−0.922218	−	0.386671i	$-0.873625\pi$
$181$	−4491.40	−1.84444	−0.922218	+	0.386671i	$0.873625\pi$
$182$	313.421	0.127650
$183$	0	0
$184$	−1585.52	−0.635252
$185$	5793.74	2.30251
$186$	0	0
$187$	115.936	0.0453375
$188$	8841.37	3.42991
$189$	0	0
$190$	−338.588	−0.129283
$191$	290.033	0.109874	0.0549372	−	0.998490i	$-0.482504\pi$
$191$	290.033	0.109874	0.0549372	+	0.998490i	$0.482504\pi$
$192$	0	0
$193$	−1593.17	−0.594193	−0.297096	−	0.954847i	$-0.596018\pi$
$193$	−1593.17	−0.594193	−0.297096	+	0.954847i	$0.596018\pi$
$194$	2532.26	0.937141
$195$	0	0
$196$	716.598	0.261151
$197$	−3610.11	−1.30563	−0.652816	−	0.757516i	$-0.726413\pi$
$197$	−3610.11	−1.30563	−0.652816	+	0.757516i	$0.726413\pi$
$198$	0	0
$199$	1880.64	0.669925	0.334962	−	0.942231i	$-0.391276\pi$
$199$	1880.64	0.669925	0.334962	+	0.942231i	$0.391276\pi$
$200$	−9203.07	−3.25378
$201$	0	0
$202$	−1950.82	−0.679500
$203$	−80.9244	−0.0279792
$204$	0	0
$205$	−3389.42	−1.15477
$206$	6864.23	2.32162
$207$	0	0
$208$	−309.499	−0.103172
$209$	38.3414	0.0126896
$210$	0	0
$211$	2239.49	0.730676	0.365338	−	0.930875i	$-0.380953\pi$
$211$	2239.49	0.730676	0.365338	+	0.930875i	$0.380953\pi$
$212$	2403.85	0.778760
$213$	0	0
$214$	9842.82	3.14412
$215$	4269.39	1.35428
$216$	0	0
$217$	1184.74	0.370625
$218$	9247.35	2.87298
$219$	0	0
$220$	3285.33	1.00680
$221$	99.2128	0.0301981
$222$	0	0
$223$	−3817.30	−1.14630	−0.573151	−	0.819450i	$-0.694279\pi$
$223$	−3817.30	−1.14630	−0.573151	+	0.819450i	$0.694279\pi$
$224$	669.796	0.199789
$225$	0	0
$226$	309.451	0.0910812
$227$	4569.24	1.33600	0.667999	−	0.744162i	$-0.267151\pi$
$227$	4569.24	1.33600	0.667999	+	0.744162i	$0.267151\pi$
$228$	0	0
$229$	−1857.55	−0.536029	−0.268014	−	0.963415i	$-0.586367\pi$
$229$	−1857.55	−0.536029	−0.268014	+	0.963415i	$0.586367\pi$
$230$	4887.98	1.40132
$231$	0	0
$232$	364.268	0.103083
$233$	−4650.88	−1.30768	−0.653840	−	0.756633i	$-0.726843\pi$
$233$	−4650.88	−1.30768	−0.653840	+	0.756633i	$0.726843\pi$
$234$	0	0
$235$	−12346.6	−3.42724
$236$	−4281.10	−1.18083
$237$	0	0
$238$	350.925	0.0955760
$239$	5730.84	1.55104	0.775518	−	0.631326i	$-0.217489\pi$
$239$	5730.84	1.55104	0.775518	+	0.631326i	$0.217489\pi$
$240$	0	0
$241$	−655.807	−0.175287	−0.0876437	−	0.996152i	$-0.527934\pi$
$241$	−655.807	−0.175287	−0.0876437	+	0.996152i	$0.527934\pi$
$242$	−575.539	−0.152880
$243$	0	0
$244$	−2967.59	−0.778608
$245$	−1000.70	−0.260948
$246$	0	0
$247$	32.8107	0.00845221
$248$	−5332.92	−1.36549
$249$	0	0
$250$	16229.6	4.10579
$251$	−4548.60	−1.14385	−0.571923	−	0.820308i	$-0.693802\pi$
$251$	−4548.60	−1.14385	−0.571923	+	0.820308i	$0.693802\pi$
$252$	0	0
$253$	−553.511	−0.137545
$254$	2137.24	0.527961
$255$	0	0
$256$	−6861.67	−1.67521
$257$	1097.31	0.266336	0.133168	−	0.991093i	$-0.457485\pi$
$257$	1097.31	0.266336	0.133168	+	0.991093i	$0.457485\pi$
$258$	0	0
$259$	−1985.87	−0.476432
$260$	2811.43	0.670606
$261$	0	0
$262$	−9459.25	−2.23051
$263$	−350.041	−0.0820701	−0.0410350	−	0.999158i	$-0.513066\pi$
$263$	−350.041	−0.0820701	−0.0410350	+	0.999158i	$0.513066\pi$
$264$	0	0
$265$	−3356.87	−0.778155
$266$	116.055	0.0267510
$267$	0	0
$268$	10970.5	2.50048
$269$	7534.44	1.70774	0.853872	−	0.520483i	$-0.174248\pi$
$269$	7534.44	1.70774	0.853872	+	0.520483i	$0.174248\pi$
$270$	0	0
$271$	3284.96	0.736337	0.368168	−	0.929759i	$-0.379985\pi$
$271$	3284.96	0.736337	0.368168	+	0.929759i	$0.379985\pi$
$272$	−346.534	−0.0772489
$273$	0	0
$274$	11584.8	2.55425
$275$	−3212.82	−0.704511
$276$	0	0
$277$	2905.52	0.630238	0.315119	−	0.949052i	$-0.397956\pi$
$277$	2905.52	0.630238	0.315119	+	0.949052i	$0.397956\pi$
$278$	926.596	0.199905
$279$	0	0
$280$	4504.47	0.961406
$281$	−6691.39	−1.42055	−0.710275	−	0.703924i	$-0.751429\pi$
$281$	−6691.39	−1.42055	−0.710275	+	0.703924i	$0.751429\pi$
$282$	0	0
$283$	4169.05	0.875704	0.437852	−	0.899047i	$-0.355739\pi$
$283$	4169.05	0.875704	0.437852	+	0.899047i	$0.355739\pi$
$284$	214.394	0.0447955
$285$	0	0
$286$	−492.518	−0.101829
$287$	1161.76	0.238942
$288$	0	0
$289$	−4801.92	−0.977390
$290$	−1122.99	−0.227395
$291$	0	0
$292$	14513.2	2.90863
$293$	5523.97	1.10141	0.550707	−	0.834699i	$-0.314358\pi$
$293$	5523.97	1.10141	0.550707	+	0.834699i	$0.314358\pi$
$294$	0	0
$295$	5978.37	1.17991
$296$	8939.04	1.75531
$297$	0	0
$298$	−2571.30	−0.499837
$299$	−473.668	−0.0916152
$300$	0	0
$301$	−1463.38	−0.280225
$302$	7539.39	1.43657
$303$	0	0
$304$	−114.602	−0.0216214
$305$	4144.11	0.778003
$306$	0	0
$307$	8309.31	1.54475	0.772374	−	0.635168i	$-0.219069\pi$
$307$	8309.31	1.54475	0.772374	+	0.635168i	$0.219069\pi$
$308$	−1126.08	−0.208326
$309$	0	0
$310$	16440.8	3.01217
$311$	5556.42	1.01311	0.506553	−	0.862209i	$-0.330919\pi$
$311$	5556.42	1.01311	0.506553	+	0.862209i	$0.330919\pi$
$312$	0	0
$313$	2102.94	0.379761	0.189881	−	0.981807i	$-0.439190\pi$
$313$	2102.94	0.379761	0.189881	+	0.981807i	$0.439190\pi$
$314$	11835.1	2.12704
$315$	0	0
$316$	18799.8	3.34674
$317$	−10308.3	−1.82641	−0.913203	−	0.407506i	$-0.866399\pi$
$317$	−10308.3	−1.82641	−0.913203	+	0.407506i	$0.866399\pi$
$318$	0	0
$319$	127.167	0.0223197
$320$	14666.6	2.56214
$321$	0	0
$322$	−1675.41	−0.289959
$323$	36.7369	0.00632847
$324$	0	0
$325$	−2749.38	−0.469256
$326$	−3519.55	−0.597944
$327$	0	0
$328$	−5229.46	−0.880331
$329$	4231.93	0.709160
$330$	0	0
$331$	4541.31	0.754118	0.377059	−	0.926189i	$-0.376935\pi$
$331$	4541.31	0.754118	0.377059	+	0.926189i	$0.376935\pi$
$332$	−10571.4	−1.74753
$333$	0	0
$334$	−9694.98	−1.58828
$335$	−15319.8	−2.49853
$336$	0	0
$337$	−5462.54	−0.882978	−0.441489	−	0.897267i	$-0.645550\pi$
$337$	−5462.54	−0.882978	−0.441489	+	0.897267i	$0.645550\pi$
$338$	10028.6	1.61386
$339$	0	0
$340$	3147.85	0.502106
$341$	−1861.74	−0.295657
$342$	0	0
$343$	343.000	0.0539949
$344$	6587.15	1.03243
$345$	0	0
$346$	9473.88	1.47202
$347$	1152.74	0.178335	0.0891677	−	0.996017i	$-0.471579\pi$
$347$	1152.74	0.178335	0.0891677	+	0.996017i	$0.471579\pi$
$348$	0	0
$349$	228.903	0.0351086	0.0175543	−	0.999846i	$-0.494412\pi$
$349$	228.903	0.0351086	0.0175543	+	0.999846i	$0.494412\pi$
$350$	−9724.81	−1.48518
$351$	0	0
$352$	−1052.54	−0.159376
$353$	−4595.35	−0.692877	−0.346438	−	0.938073i	$-0.612609\pi$
$353$	−4595.35	−0.692877	−0.346438	+	0.938073i	$0.612609\pi$
$354$	0	0
$355$	−299.391	−0.0447607
$356$	7342.54	1.09313
$357$	0	0
$358$	12609.2	1.86150
$359$	−4528.50	−0.665752	−0.332876	−	0.942971i	$-0.608019\pi$
$359$	−4528.50	−0.665752	−0.332876	+	0.942971i	$0.608019\pi$
$360$	0	0
$361$	−6846.85	−0.998229
$362$	21363.4	3.10176
$363$	0	0
$364$	−963.648	−0.138761
$365$	−20267.0	−2.90636
$366$	0	0
$367$	−9717.48	−1.38215	−0.691074	−	0.722784i	$-0.742862\pi$
$367$	−9717.48	−1.38215	−0.691074	+	0.722784i	$0.742862\pi$
$368$	1654.44	0.234358
$369$	0	0
$370$	−27558.0	−3.87209
$371$	1150.60	0.161015
$372$	0	0
$373$	3647.49	0.506327	0.253164	−	0.967423i	$-0.418529\pi$
$373$	3647.49	0.506327	0.253164	+	0.967423i	$0.418529\pi$
$374$	−551.454	−0.0762433
$375$	0	0
$376$	−19049.3	−2.61275
$377$	108.823	0.0148665
$378$	0	0
$379$	4767.04	0.646085	0.323043	−	0.946384i	$-0.395294\pi$
$379$	4767.04	0.646085	0.323043	+	0.946384i	$0.395294\pi$
$380$	1041.03	0.140536
$381$	0	0
$382$	−1379.55	−0.184774
$383$	−4701.46	−0.627241	−0.313620	−	0.949548i	$-0.601542\pi$
$383$	−4701.46	−0.627241	−0.313620	+	0.949548i	$0.601542\pi$
$384$	0	0
$385$	1572.53	0.208164
$386$	7577.96	0.999244
$387$	0	0
$388$	−7785.71	−1.01871
$389$	11743.8	1.53068	0.765338	−	0.643629i	$-0.222572\pi$
$389$	11743.8	1.53068	0.765338	+	0.643629i	$0.222572\pi$
$390$	0	0
$391$	−530.348	−0.0685955
$392$	−1543.96	−0.198933
$393$	0	0
$394$	17171.5	2.19566
$395$	−26253.1	−3.34414
$396$	0	0
$397$	−11857.0	−1.49895	−0.749477	−	0.662030i	$-0.769695\pi$
$397$	−11857.0	−1.49895	−0.749477	+	0.662030i	$0.769695\pi$
$398$	−8945.30	−1.12660
$399$	0	0
$400$	9603.12	1.20039
$401$	−6776.89	−0.843945	−0.421972	−	0.906609i	$-0.638662\pi$
$401$	−6776.89	−0.843945	−0.421972	+	0.906609i	$0.638662\pi$
$402$	0	0
$403$	−1593.19	−0.196929
$404$	5998.01	0.738644
$405$	0	0
$406$	384.918	0.0470522
$407$	3120.65	0.380061
$408$	0	0
$409$	16441.2	1.98768	0.993842	−	0.110804i	$-0.0353425\pi$
$409$	16441.2	1.98768	0.993842	+	0.110804i	$0.0353425\pi$
$410$	16121.8	1.94195
$411$	0	0
$412$	−21104.9	−2.52369
$413$	−2049.15	−0.244146
$414$	0	0
$415$	14762.5	1.74617
$416$	−900.710	−0.106156
$417$	0	0
$418$	−182.371	−0.0213399
$419$	−8901.30	−1.03784	−0.518922	−	0.854821i	$-0.673667\pi$
$419$	−8901.30	−1.03784	−0.518922	+	0.854821i	$0.673667\pi$
$420$	0	0
$421$	6562.92	0.759755	0.379878	−	0.925037i	$-0.375966\pi$
$421$	6562.92	0.759755	0.379878	+	0.925037i	$0.375966\pi$
$422$	−10652.2	−1.22876
$423$	0	0
$424$	−5179.25	−0.593223
$425$	−3078.37	−0.351348
$426$	0	0
$427$	−1420.44	−0.160983
$428$	−30262.9	−3.41778
$429$	0	0
$430$	−20307.4	−2.27747
$431$	−9669.77	−1.08069	−0.540344	−	0.841444i	$-0.681706\pi$
$431$	−9669.77	−1.08069	−0.540344	+	0.841444i	$0.681706\pi$
$432$	0	0
$433$	−6658.03	−0.738948	−0.369474	−	0.929241i	$-0.620462\pi$
$433$	−6658.03	−0.738948	−0.369474	+	0.929241i	$0.620462\pi$
$434$	−5635.25	−0.623274
$435$	0	0
$436$	−28432.0	−3.12305
$437$	−175.392	−0.0191994
$438$	0	0
$439$	10345.5	1.12474	0.562372	−	0.826884i	$-0.309889\pi$
$439$	10345.5	1.12474	0.562372	+	0.826884i	$0.309889\pi$
$440$	−7078.46	−0.766937
$441$	0	0
$442$	−471.907	−0.0507836
$443$	−2402.96	−0.257716	−0.128858	−	0.991663i	$-0.541131\pi$
$443$	−2402.96	−0.257716	−0.128858	+	0.991663i	$0.541131\pi$
$444$	0	0
$445$	−10253.5	−1.09228
$446$	18157.1	1.92772
$447$	0	0
$448$	−5027.12	−0.530154
$449$	15030.3	1.57978	0.789890	−	0.613248i	$-0.210137\pi$
$449$	15030.3	1.57978	0.789890	+	0.613248i	$0.210137\pi$
$450$	0	0
$451$	−1825.62	−0.190610
$452$	−951.441	−0.0990089
$453$	0	0
$454$	−21733.7	−2.24672
$455$	1345.69	0.138653
$456$	0	0
$457$	−16018.3	−1.63961	−0.819807	−	0.572641i	$-0.805919\pi$
$457$	−16018.3	−1.63961	−0.819807	+	0.572641i	$0.805919\pi$
$458$	8835.48	0.901430
$459$	0	0
$460$	−15028.7	−1.52329
$461$	5213.29	0.526697	0.263348	−	0.964701i	$-0.415173\pi$
$461$	5213.29	0.526697	0.263348	+	0.964701i	$0.415173\pi$
$462$	0	0
$463$	9895.68	0.993285	0.496643	−	0.867955i	$-0.334566\pi$
$463$	9895.68	0.993285	0.496643	+	0.867955i	$0.334566\pi$
$464$	−380.102	−0.0380297
$465$	0	0
$466$	22122.0	2.19910
$467$	−7100.05	−0.703535	−0.351768	−	0.936087i	$-0.614419\pi$
$467$	−7100.05	−0.703535	−0.351768	+	0.936087i	$0.614419\pi$
$468$	0	0
$469$	5251.02	0.516993
$470$	58726.8	5.76354
$471$	0	0
$472$	9223.91	0.899502
$473$	2299.60	0.223542
$474$	0	0
$475$	−1018.05	−0.0983397
$476$	−1078.96	−0.103895
$477$	0	0
$478$	−27258.9	−2.60835
$479$	13396.6	1.27788	0.638941	−	0.769256i	$-0.279373\pi$
$479$	13396.6	1.27788	0.638941	+	0.769256i	$0.279373\pi$
$480$	0	0
$481$	2670.50	0.253148
$482$	3119.36	0.294778
$483$	0	0
$484$	1769.56	0.166187
$485$	10872.4	1.01792
$486$	0	0
$487$	−691.582	−0.0643503	−0.0321751	−	0.999482i	$-0.510243\pi$
$487$	−691.582	−0.0643503	−0.0321751	+	0.999482i	$0.510243\pi$
$488$	6393.85	0.593107
$489$	0	0
$490$	4759.84	0.438832
$491$	1281.36	0.117774	0.0588868	−	0.998265i	$-0.481245\pi$
$491$	1281.36	0.117774	0.0588868	+	0.998265i	$0.481245\pi$
$492$	0	0
$493$	121.845	0.0111311
$494$	−156.065	−0.0142139
$495$	0	0
$496$	5564.74	0.503758
$497$	102.620	0.00926181
$498$	0	0
$499$	5185.81	0.465228	0.232614	−	0.972569i	$-0.425272\pi$
$499$	5185.81	0.465228	0.232614	+	0.972569i	$0.425272\pi$
$500$	−49899.6	−4.46316
$501$	0	0
$502$	21635.5	1.92358
$503$	−20900.8	−1.85273	−0.926363	−	0.376631i	$-0.877082\pi$
$503$	−20900.8	−1.85273	−0.926363	+	0.376631i	$0.877082\pi$
$504$	0	0
$505$	−8375.96	−0.738070
$506$	2632.79	0.231307
$507$	0	0
$508$	−6571.18	−0.573915
$509$	16598.5	1.44542	0.722708	−	0.691153i	$-0.242897\pi$
$509$	16598.5	1.44542	0.722708	+	0.691153i	$0.242897\pi$
$510$	0	0
$511$	6946.73	0.601380
$512$	11434.0	0.986944
$513$	0	0
$514$	−5219.38	−0.447893
$515$	29472.0	2.52173
$516$	0	0
$517$	−6650.17	−0.565714
$518$	9445.81	0.801207
$519$	0	0
$520$	−6057.40	−0.510836
$521$	−20837.8	−1.75225	−0.876125	−	0.482085i	$-0.839880\pi$
$521$	−20837.8	−1.75225	−0.876125	+	0.482085i	$0.839880\pi$
$522$	0	0
$523$	−8513.29	−0.711779	−0.355889	−	0.934528i	$-0.615822\pi$
$523$	−8513.29	−0.711779	−0.355889	+	0.934528i	$0.615822\pi$
$524$	29083.5	2.42466
$525$	0	0
$526$	1664.97	0.138016
$527$	−1783.83	−0.147448
$528$	0	0
$529$	−9634.98	−0.791894
$530$	15967.0	1.30861
$531$	0	0
$532$	−356.823	−0.0290794
$533$	−1562.28	−0.126960
$534$	0	0
$535$	42260.8	3.41513
$536$	−23636.6	−1.90475
$537$	0	0
$538$	−35837.7	−2.87188
$539$	−539.000	−0.0430730
$540$	0	0
$541$	23541.5	1.87085	0.935423	−	0.353532i	$-0.115019\pi$
$541$	23541.5	1.87085	0.935423	+	0.353532i	$0.115019\pi$
$542$	−15625.0	−1.23828
$543$	0	0
$544$	−1008.49	−0.0794829
$545$	39704.1	3.12062
$546$	0	0
$547$	2249.62	0.175844	0.0879220	−	0.996127i	$-0.471977\pi$
$547$	2249.62	0.175844	0.0879220	+	0.996127i	$0.471977\pi$
$548$	−35618.8	−2.77657
$549$	0	0
$550$	15281.8	1.18476
$551$	40.2955	0.00311551
$552$	0	0
$553$	8998.53	0.691964
$554$	−13820.2	−1.05986
$555$	0	0
$556$	−2848.93	−0.217305
$557$	2779.41	0.211431	0.105716	−	0.994396i	$-0.466287\pi$
$557$	2779.41	0.211431	0.105716	+	0.994396i	$0.466287\pi$
$558$	0	0
$559$	1967.88	0.148896
$560$	−4700.27	−0.354684
$561$	0	0
$562$	31827.7	2.38891
$563$	−18558.7	−1.38927	−0.694633	−	0.719364i	$-0.744433\pi$
$563$	−18558.7	−1.38927	−0.694633	+	0.719364i	$0.744433\pi$
$564$	0	0
$565$	1328.65	0.0989320
$566$	−19830.1	−1.47266
$567$	0	0
$568$	−461.925	−0.0341231
$569$	−9417.67	−0.693865	−0.346932	−	0.937890i	$-0.612777\pi$
$569$	−9417.67	−0.693865	−0.346932	+	0.937890i	$0.612777\pi$
$570$	0	0
$571$	19252.5	1.41102	0.705509	−	0.708701i	$-0.250718\pi$
$571$	19252.5	1.41102	0.705509	+	0.708701i	$0.250718\pi$
$572$	1514.30	0.110693
$573$	0	0
$574$	−5525.93	−0.401825
$575$	14697.0	1.06592
$576$	0	0
$577$	17970.2	1.29655	0.648274	−	0.761407i	$-0.275491\pi$
$577$	17970.2	1.29655	0.648274	+	0.761407i	$0.275491\pi$
$578$	22840.4	1.64366
$579$	0	0
$580$	3452.77	0.247187
$581$	−5059.99	−0.361315
$582$	0	0
$583$	−1808.09	−0.128445
$584$	−31269.5	−2.21565
$585$	0	0
$586$	−26274.9	−1.85223
$587$	18312.6	1.28764	0.643818	−	0.765179i	$-0.277349\pi$
$587$	18312.6	1.28764	0.643818	+	0.765179i	$0.277349\pi$
$588$	0	0
$589$	−589.932	−0.0412694
$590$	−28436.2	−1.98424
$591$	0	0
$592$	−9327.61	−0.647572
$593$	−5682.32	−0.393499	−0.196749	−	0.980454i	$-0.563039\pi$
$593$	−5682.32	−0.393499	−0.196749	+	0.980454i	$0.563039\pi$
$594$	0	0
$595$	1506.72	0.103814
$596$	7905.75	0.543343
$597$	0	0
$598$	2253.01	0.154068
$599$	−22748.4	−1.55171	−0.775854	−	0.630912i	$-0.782681\pi$
$599$	−22748.4	−1.55171	−0.775854	+	0.630912i	$0.782681\pi$
$600$	0	0
$601$	−4805.47	−0.326155	−0.163077	−	0.986613i	$-0.552142\pi$
$601$	−4805.47	−0.326155	−0.163077	+	0.986613i	$0.552142\pi$
$602$	6960.59	0.471250
$603$	0	0
$604$	−23180.7	−1.56161
$605$	−2471.11	−0.166058
$606$	0	0
$607$	−15730.2	−1.05184	−0.525922	−	0.850533i	$-0.676280\pi$
$607$	−15730.2	−1.05184	−0.525922	+	0.850533i	$0.676280\pi$
$608$	−333.518	−0.0222466
$609$	0	0
$610$	−19711.5	−1.30835
$611$	−5690.90	−0.376807
$612$	0	0
$613$	−17586.7	−1.15876	−0.579380	−	0.815058i	$-0.696705\pi$
$613$	−17586.7	−1.15876	−0.579380	+	0.815058i	$0.696705\pi$
$614$	−39523.4	−2.59777
$615$	0	0
$616$	2426.22	0.158693
$617$	−14671.7	−0.957308	−0.478654	−	0.878004i	$-0.658875\pi$
$617$	−14671.7	−0.957308	−0.478654	+	0.878004i	$0.658875\pi$
$618$	0	0
$619$	−1935.25	−0.125661	−0.0628305	−	0.998024i	$-0.520013\pi$
$619$	−1935.25	−0.125661	−0.0628305	+	0.998024i	$0.520013\pi$
$620$	−50549.0	−3.27435
$621$	0	0
$622$	−26429.2	−1.70372
$623$	3514.51	0.226012
$624$	0	0
$625$	33173.3	2.12309
$626$	−10002.7	−0.638638
$627$	0	0
$628$	−36388.2	−2.31218
$629$	2990.06	0.189541
$630$	0	0
$631$	−1.75233	−0.000110554 0	−5.52768e−5	−	1.00000i	$-0.500018\pi$
$631$	−1.75233	−0.000110554 0	−5.52768e−5		1.00000i	$0.500018\pi$
$632$	−40505.3	−2.54939
$633$	0	0
$634$	49031.5	3.07143
$635$	9176.36	0.573469
$636$	0	0
$637$	−461.250	−0.0286898
$638$	−604.872	−0.0375346
$639$	0	0
$640$	−54128.7	−3.34317
$641$	−9110.30	−0.561366	−0.280683	−	0.959801i	$-0.590561\pi$
$641$	−9110.30	−0.561366	−0.280683	+	0.959801i	$0.590561\pi$
$642$	0	0
$643$	−9073.67	−0.556502	−0.278251	−	0.960508i	$-0.589755\pi$
$643$	−9073.67	−0.556502	−0.278251	+	0.960508i	$0.589755\pi$
$644$	5151.23	0.315197
$645$	0	0
$646$	−174.740	−0.0106425
$647$	−4814.56	−0.292550	−0.146275	−	0.989244i	$-0.546728\pi$
$647$	−4814.56	−0.292550	−0.146275	+	0.989244i	$0.546728\pi$
$648$	0	0
$649$	3220.10	0.194761
$650$	13077.5	0.789139
$651$	0	0
$652$	10821.3	0.649990
$653$	−10772.3	−0.645564	−0.322782	−	0.946473i	$-0.604618\pi$
$653$	−10772.3	−0.645564	−0.322782	+	0.946473i	$0.604618\pi$
$654$	0	0
$655$	−40613.9	−2.42277
$656$	5456.78	0.324773
$657$	0	0
$658$	−20129.2	−1.19258
$659$	17455.5	1.03182	0.515910	−	0.856643i	$-0.327454\pi$
$659$	17455.5	1.03182	0.515910	+	0.856643i	$0.327454\pi$
$660$	0	0
$661$	15398.9	0.906124	0.453062	−	0.891479i	$-0.350332\pi$
$661$	15398.9	0.906124	0.453062	+	0.891479i	$0.350332\pi$
$662$	−21600.8	−1.26819
$663$	0	0
$664$	22776.7	1.33119
$665$	498.288	0.0290568
$666$	0	0
$667$	−581.722	−0.0337696
$668$	29808.3	1.72652
$669$	0	0
$670$	72868.8	4.20174
$671$	2232.11	0.128420
$672$	0	0
$673$	14982.1	0.858126	0.429063	−	0.903275i	$-0.358844\pi$
$673$	14982.1	0.858126	0.429063	+	0.903275i	$0.358844\pi$
$674$	25982.7	1.48489
$675$	0	0
$676$	−30834.1	−1.75433
$677$	−27825.6	−1.57965	−0.789827	−	0.613330i	$-0.789830\pi$
$677$	−27825.6	−1.57965	−0.789827	+	0.613330i	$0.789830\pi$
$678$	0	0
$679$	−3726.63	−0.210626
$680$	−6782.24	−0.382481
$681$	0	0
$682$	8855.40	0.497200
$683$	−32920.0	−1.84429	−0.922145	−	0.386845i	$-0.873565\pi$
$683$	−32920.0	−1.84429	−0.922145	+	0.386845i	$0.873565\pi$
$684$	0	0
$685$	49740.1	2.77441
$686$	−1631.49	−0.0908023
$687$	0	0
$688$	−6873.49	−0.380885
$689$	−1547.28	−0.0855539
$690$	0	0
$691$	−3157.37	−0.173823	−0.0869116	−	0.996216i	$-0.527700\pi$
$691$	−3157.37	−0.173823	−0.0869116	+	0.996216i	$0.527700\pi$
$692$	−29128.5	−1.60014
$693$	0	0
$694$	−5483.03	−0.299904
$695$	3978.40	0.217136
$696$	0	0
$697$	−1749.22	−0.0950596
$698$	−1088.78	−0.0590416
$699$	0	0
$700$	29900.0	1.61445
$701$	15251.7	0.821754	0.410877	−	0.911691i	$-0.365223\pi$
$701$	15251.7	0.821754	0.410877	+	0.911691i	$0.365223\pi$
$702$	0	0
$703$	988.843	0.0530511
$704$	7899.76	0.422917
$705$	0	0
$706$	21857.8	1.16520
$707$	2870.95	0.152720
$708$	0	0
$709$	−30795.6	−1.63124	−0.815622	−	0.578585i	$-0.803605\pi$
$709$	−30795.6	−1.63124	−0.815622	+	0.578585i	$0.803605\pi$
$710$	1424.06	0.0752733
$711$	0	0
$712$	−15820.0	−0.832694
$713$	8516.48	0.447328
$714$	0	0
$715$	−2114.66	−0.110607
$716$	−38768.4	−2.02352
$717$	0	0
$718$	21539.9	1.11958
$719$	−12226.1	−0.634151	−0.317076	−	0.948400i	$-0.602701\pi$
$719$	−12226.1	−0.634151	−0.317076	+	0.948400i	$0.602701\pi$
$720$	0	0
$721$	−10101.9	−0.521793
$722$	32567.2	1.67870
$723$	0	0
$724$	−65684.2	−3.37173
$725$	−3376.57	−0.172969
$726$	0	0
$727$	18279.6	0.932534	0.466267	−	0.884644i	$-0.345599\pi$
$727$	18279.6	0.932534	0.466267	+	0.884644i	$0.345599\pi$
$728$	2076.24	0.105701
$729$	0	0
$730$	96400.3	4.88758
$731$	2203.36	0.111483
$732$	0	0
$733$	7407.39	0.373258	0.186629	−	0.982430i	$-0.440244\pi$
$733$	7407.39	0.373258	0.186629	+	0.982430i	$0.440244\pi$
$734$	46221.4	2.32433
$735$	0	0
$736$	4814.80	0.241136
$737$	−8251.60	−0.412417
$738$	0	0
$739$	−19188.4	−0.955151	−0.477576	−	0.878591i	$-0.658484\pi$
$739$	−19188.4	−0.955151	−0.477576	+	0.878591i	$0.658484\pi$
$740$	84730.3	4.20912
$741$	0	0
$742$	−5472.87	−0.270775
$743$	−24324.6	−1.20105	−0.600527	−	0.799604i	$-0.705043\pi$
$743$	−24324.6	−1.20105	−0.600527	+	0.799604i	$0.705043\pi$
$744$	0	0
$745$	−11040.0	−0.542920
$746$	−17349.4	−0.851481
$747$	0	0
$748$	1695.51	0.0828795
$749$	−14485.3	−0.706652
$750$	0	0
$751$	18846.7	0.915748	0.457874	−	0.889017i	$-0.348611\pi$
$751$	18846.7	0.915748	0.457874	+	0.889017i	$0.348611\pi$
$752$	19877.3	0.963899
$753$	0	0
$754$	−517.620	−0.0250008
$755$	32370.9	1.56039
$756$	0	0
$757$	22771.4	1.09332	0.546659	−	0.837356i	$-0.315900\pi$
$757$	22771.4	1.09332	0.546659	+	0.837356i	$0.315900\pi$
$758$	−22674.5	−1.08651
$759$	0	0
$760$	−2242.96	−0.107053
$761$	−24040.1	−1.14514	−0.572570	−	0.819856i	$-0.694053\pi$
$761$	−24040.1	−1.14514	−0.572570	+	0.819856i	$0.694053\pi$
$762$	0	0
$763$	−13609.0	−0.645713
$764$	4241.57	0.200857
$765$	0	0
$766$	22362.6	1.05482
$767$	2755.60	0.129725
$768$	0	0
$769$	6179.39	0.289772	0.144886	−	0.989448i	$-0.453718\pi$
$769$	6179.39	0.289772	0.144886	+	0.989448i	$0.453718\pi$
$770$	−7479.74	−0.350067
$771$	0	0
$772$	−23299.3	−1.08622
$773$	22981.9	1.06934	0.534672	−	0.845060i	$-0.320435\pi$
$773$	22981.9	1.06934	0.534672	+	0.845060i	$0.320435\pi$
$774$	0	0
$775$	49433.4	2.29123
$776$	16774.8	0.776006
$777$	0	0
$778$	−55859.4	−2.57411
$779$	−578.487	−0.0266065
$780$	0	0
$781$	−161.259	−0.00738837
$782$	2522.61	0.115356
$783$	0	0
$784$	1611.07	0.0733906
$785$	50814.6	2.31038
$786$	0	0
$787$	32606.7	1.47688	0.738439	−	0.674320i	$-0.235563\pi$
$787$	32606.7	1.47688	0.738439	+	0.674320i	$0.235563\pi$
$788$	−52795.9	−2.38677
$789$	0	0
$790$	124873.	5.62378
$791$	−455.408	−0.0204708
$792$	0	0
$793$	1910.14	0.0855371
$794$	56397.9	2.52077
$795$	0	0
$796$	27503.3	1.22466
$797$	−11071.7	−0.492072	−0.246036	−	0.969261i	$-0.579128\pi$
$797$	−11071.7	−0.492072	−0.246036	+	0.969261i	$0.579128\pi$
$798$	0	0
$799$	−6371.87	−0.282129
$800$	27947.2	1.23510
$801$	0	0
$802$	32234.4	1.41925
$803$	−10916.3	−0.479735
$804$	0	0
$805$	−7193.47	−0.314952
$806$	7578.02	0.331172
$807$	0	0
$808$	−12923.1	−0.562665
$809$	−11165.5	−0.485241	−0.242620	−	0.970121i	$-0.578007\pi$
$809$	−11165.5	−0.485241	−0.242620	+	0.970121i	$0.578007\pi$
$810$	0	0
$811$	−29993.4	−1.29866	−0.649329	−	0.760508i	$-0.724950\pi$
$811$	−29993.4	−1.29866	−0.649329	+	0.760508i	$0.724950\pi$
$812$	−1183.48	−0.0511476
$813$	0	0
$814$	−14843.4	−0.639142
$815$	−15111.4	−0.649484
$816$	0	0
$817$	728.676	0.0312033
$818$	−78202.6	−3.34265
$819$	0	0
$820$	−49568.4	−2.11098
$821$	18949.2	0.805521	0.402761	−	0.915305i	$-0.368051\pi$
$821$	18949.2	0.805521	0.402761	+	0.915305i	$0.368051\pi$
$822$	0	0
$823$	−23956.5	−1.01467	−0.507333	−	0.861750i	$-0.669368\pi$
$823$	−23956.5	−1.01467	−0.507333	+	0.861750i	$0.669368\pi$
$824$	45471.8	1.92243
$825$	0	0
$826$	9746.83	0.410576
$827$	−3592.80	−0.151069	−0.0755344	−	0.997143i	$-0.524066\pi$
$827$	−3592.80	−0.151069	−0.0755344	+	0.997143i	$0.524066\pi$
$828$	0	0
$829$	21053.0	0.882029	0.441015	−	0.897500i	$-0.354619\pi$
$829$	21053.0	0.882029	0.441015	+	0.897500i	$0.354619\pi$
$830$	−70217.9	−2.93650
$831$	0	0
$832$	6760.23	0.281693
$833$	−516.444	−0.0214811
$834$	0	0
$835$	−41626.0	−1.72518
$836$	560.722	0.0231973
$837$	0	0
$838$	42339.2	1.74532
$839$	−35430.1	−1.45790	−0.728952	−	0.684565i	$-0.759992\pi$
$839$	−35430.1	−1.45790	−0.728952	+	0.684565i	$0.759992\pi$
$840$	0	0
$841$	−24255.4	−0.994520
$842$	−31216.6	−1.27767
$843$	0	0
$844$	32751.3	1.33572
$845$	43058.4	1.75296
$846$	0	0
$847$	847.000	0.0343604
$848$	5404.38	0.218853
$849$	0	0
$850$	14642.3	0.590856
$851$	−14275.3	−0.575031
$852$	0	0
$853$	7079.82	0.284183	0.142092	−	0.989854i	$-0.454617\pi$
$853$	7079.82	0.284183	0.142092	+	0.989854i	$0.454617\pi$
$854$	6756.33	0.270722
$855$	0	0
$856$	65203.3	2.60351
$857$	−33199.2	−1.32329	−0.661647	−	0.749816i	$-0.730142\pi$
$857$	−33199.2	−1.32329	−0.661647	+	0.749816i	$0.730142\pi$
$858$	0	0
$859$	−34495.1	−1.37015	−0.685074	−	0.728474i	$-0.740230\pi$
$859$	−34495.1	−1.37015	−0.685074	+	0.728474i	$0.740230\pi$
$860$	62437.5	2.47570
$861$	0	0
$862$	45994.4	1.81737
$863$	29250.4	1.15376	0.576880	−	0.816829i	$-0.304270\pi$
$863$	29250.4	1.15376	0.576880	+	0.816829i	$0.304270\pi$
$864$	0	0
$865$	40676.7	1.59890
$866$	31669.0	1.24268
$867$	0	0
$868$	17326.2	0.677524
$869$	−14140.5	−0.551997
$870$	0	0
$871$	−7061.32	−0.274700
$872$	61258.6	2.37899
$873$	0	0
$874$	834.253	0.0322872
$875$	−23884.5	−0.922792
$876$	0	0
$877$	−18933.8	−0.729017	−0.364509	−	0.931200i	$-0.618763\pi$
$877$	−18933.8	−0.729017	−0.364509	+	0.931200i	$0.618763\pi$
$878$	−49208.5	−1.89146
$879$	0	0
$880$	7386.14	0.282940
$881$	3304.16	0.126356	0.0631782	−	0.998002i	$-0.479876\pi$
$881$	3304.16	0.126356	0.0631782	+	0.998002i	$0.479876\pi$
$882$	0	0
$883$	−49863.9	−1.90040	−0.950200	−	0.311642i	$-0.899121\pi$
$883$	−49863.9	−1.90040	−0.950200	+	0.311642i	$0.899121\pi$
$884$	1450.93	0.0552038
$885$	0	0
$886$	11429.7	0.433396
$887$	9440.57	0.357366	0.178683	−	0.983907i	$-0.442816\pi$
$887$	9440.57	0.357366	0.178683	+	0.983907i	$0.442816\pi$
$888$	0	0
$889$	−3145.30	−0.118661
$890$	48771.1	1.83687
$891$	0	0
$892$	−55826.0	−2.09551
$893$	−2107.25	−0.0789656
$894$	0	0
$895$	54138.4	2.02195
$896$	18553.2	0.691762
$897$	0	0
$898$	−71491.6	−2.65669
$899$	−1956.63	−0.0725886
$900$	0	0
$901$	−1732.43	−0.0640572
$902$	8683.60	0.320546
$903$	0	0
$904$	2049.94	0.0754204
$905$	91725.1	3.36911
$906$	0	0
$907$	−15019.6	−0.549852	−0.274926	−	0.961465i	$-0.588653\pi$
$907$	−15019.6	−0.549852	−0.274926	+	0.961465i	$0.588653\pi$
$908$	66822.7	2.44228
$909$	0	0
$910$	−6400.81	−0.233170
$911$	−49140.5	−1.78715	−0.893576	−	0.448912i	$-0.851812\pi$
$911$	−49140.5	−1.78715	−0.893576	+	0.448912i	$0.851812\pi$
$912$	0	0
$913$	7951.41	0.288229
$914$	76191.2	2.75731
$915$	0	0
$916$	−27165.7	−0.979891
$917$	13920.8	0.501316
$918$	0	0
$919$	−25127.9	−0.901953	−0.450976	−	0.892536i	$-0.648924\pi$
$919$	−25127.9	−0.901953	−0.450976	+	0.892536i	$0.648924\pi$
$920$	32380.2	1.16037
$921$	0	0
$922$	−24797.1	−0.885737
$923$	−137.998	−0.00492119
$924$	0	0
$925$	−82860.2	−2.94533
$926$	−47069.0	−1.67039
$927$	0	0
$928$	−1106.18	−0.0391295
$929$	31742.8	1.12104	0.560520	−	0.828141i	$-0.310601\pi$
$929$	31742.8	1.12104	0.560520	+	0.828141i	$0.310601\pi$
$930$	0	0
$931$	−170.793	−0.00601239
$932$	−68016.6	−2.39051
$933$	0	0
$934$	33771.5	1.18312
$935$	−2367.70	−0.0828151
$936$	0	0
$937$	20656.6	0.720193	0.360096	−	0.932915i	$-0.382744\pi$
$937$	20656.6	0.720193	0.360096	+	0.932915i	$0.382744\pi$
$938$	−24976.6	−0.869417
$939$	0	0
$940$	−180562.	−6.26520
$941$	25644.1	0.888389	0.444195	−	0.895930i	$-0.353490\pi$
$941$	25644.1	0.888389	0.444195	+	0.895930i	$0.353490\pi$
$942$	0	0
$943$	8351.25	0.288393
$944$	−9624.86	−0.331846
$945$	0	0
$946$	−10938.1	−0.375927
$947$	−10594.3	−0.363535	−0.181767	−	0.983342i	$-0.558182\pi$
$947$	−10594.3	−0.363535	−0.181767	+	0.983342i	$0.558182\pi$
$948$	0	0
$949$	−9341.64	−0.319539
$950$	4842.37	0.165376
$951$	0	0
$952$	2324.69	0.0791423
$953$	26865.9	0.913190	0.456595	−	0.889675i	$-0.349069\pi$
$953$	26865.9	0.913190	0.456595	+	0.889675i	$0.349069\pi$
$954$	0	0
$955$	−5923.16	−0.200701
$956$	83810.5	2.83538
$957$	0	0
$958$	−63721.1	−2.14899
$959$	−17048.9	−0.574076
$960$	0	0
$961$	−1145.75	−0.0384595
$962$	−12702.3	−0.425715
$963$	0	0
$964$	−9590.82	−0.320435
$965$	32536.5	1.08537
$966$	0	0
$967$	−1932.19	−0.0642555	−0.0321278	−	0.999484i	$-0.510228\pi$
$967$	−1932.19	−0.0642555	−0.0321278	+	0.999484i	$0.510228\pi$
$968$	−3812.63	−0.126593
$969$	0	0
$970$	−51714.7	−1.71181
$971$	11171.9	0.369231	0.184615	−	0.982811i	$-0.440896\pi$
$971$	11171.9	0.369231	0.184615	+	0.982811i	$0.440896\pi$
$972$	0	0
$973$	−1363.64	−0.0449294
$974$	3289.52	0.108217
$975$	0	0
$976$	−6671.78	−0.218810
$977$	27986.4	0.916441	0.458221	−	0.888839i	$-0.348487\pi$
$977$	27986.4	0.916441	0.458221	+	0.888839i	$0.348487\pi$
$978$	0	0
$979$	−5522.80	−0.180296
$980$	−14634.7	−0.477028
$981$	0	0
$982$	−6094.79	−0.198058
$983$	−7580.83	−0.245972	−0.122986	−	0.992408i	$-0.539247\pi$
$983$	−7580.83	−0.245972	−0.122986	+	0.992408i	$0.539247\pi$
$984$	0	0
$985$	73727.1	2.38492
$986$	−579.559	−0.0187190
$987$	0	0
$988$	479.839	0.0154511
$989$	−10519.4	−0.338219
$990$	0	0
$991$	19489.4	0.624723	0.312361	−	0.949963i	$-0.398880\pi$
$991$	19489.4	0.624723	0.312361	+	0.949963i	$0.398880\pi$
$992$	16194.6	0.518326
$993$	0	0
$994$	−488.112	−0.0155754
$995$	−38407.2	−1.22371
$996$	0	0
$997$	56728.1	1.80200	0.901001	−	0.433816i	$-0.142833\pi$
$997$	56728.1	1.80200	0.901001	+	0.433816i	$0.142833\pi$
$998$	−24666.4	−0.782365
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.r.1.2	✓	8
3.2	odd	2		693.4.a.u.1.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.4.a.r.1.2	✓	8	1.1	even	1	trivial
693.4.a.u.1.7	yes	8	3.2	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)

\(f(q)\)	\(=\)	\(q-4.75652 q^{2} +14.6245 q^{4} -20.4224 q^{5} +7.00000 q^{7} -31.5093 q^{8} +97.1395 q^{10} -11.0000 q^{11} -9.41327 q^{13} -33.2956 q^{14} +32.8790 q^{16} -10.5397 q^{17} -3.48558 q^{19} -298.667 q^{20} +52.3217 q^{22} +50.3192 q^{23} +292.075 q^{25} +44.7744 q^{26} +102.371 q^{28} -11.5606 q^{29} +169.249 q^{31} +95.6852 q^{32} +50.1321 q^{34} -142.957 q^{35} -283.695 q^{37} +16.5792 q^{38} +643.496 q^{40} +165.966 q^{41} -209.054 q^{43} -160.869 q^{44} -239.344 q^{46} +604.561 q^{47} +49.0000 q^{49} -1389.26 q^{50} -137.664 q^{52} +164.372 q^{53} +224.646 q^{55} -220.565 q^{56} +54.9883 q^{58} -292.736 q^{59} -202.920 q^{61} -805.036 q^{62} -718.160 q^{64} +192.242 q^{65} +750.145 q^{67} -154.137 q^{68} +679.977 q^{70} +14.6599 q^{71} +992.390 q^{73} +1349.40 q^{74} -50.9747 q^{76} -77.0000 q^{77} +1285.50 q^{79} -671.468 q^{80} -789.418 q^{82} -722.856 q^{83} +215.246 q^{85} +994.370 q^{86} +346.602 q^{88} +502.073 q^{89} -65.8929 q^{91} +735.891 q^{92} -2875.60 q^{94} +71.1840 q^{95} -532.376 q^{97} -233.069 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{2} + 30 q^{4} - 10 q^{5} + 56 q^{7} - 81 q^{8} + 9 q^{10} - 88 q^{11} + 16 q^{13} - 42 q^{14} + 122 q^{16} - 90 q^{17} - 42 q^{19} - 291 q^{20} + 66 q^{22} - 338 q^{23} + 244 q^{25} - 209 q^{26}+ \cdots - 294 q^{98}+O(q^{100}) \) 8 * q - 6 * q^2 + 30 * q^4 - 10 * q^5 + 56 * q^7 - 81 * q^8 + 9 * q^10 - 88 * q^11 + 16 * q^13 - 42 * q^14 + 122 * q^16 - 90 * q^17 - 42 * q^19 - 291 * q^20 + 66 * q^22 - 338 * q^23 + 244 * q^25 - 209 * q^26 + 210 * q^28 - 496 * q^29 - 8 * q^31 - 524 * q^32 - 302 * q^34 - 70 * q^35 - 360 * q^37 - 45 * q^38 - 6 * q^40 - 242 * q^41 - 66 * q^43 - 330 * q^44 + 344 * q^46 - 540 * q^47 + 392 * q^49 - 1171 * q^50 + 465 * q^52 - 906 * q^53 + 110 * q^55 - 567 * q^56 + 977 * q^58 - 1242 * q^59 - 318 * q^61 + 110 * q^62 + 525 * q^64 - 1258 * q^65 + 522 * q^67 - 678 * q^68 + 63 * q^70 - 858 * q^71 - 78 * q^73 - 1651 * q^74 + 1775 * q^76 - 616 * q^77 + 516 * q^79 - 567 * q^80 - 1212 * q^82 - 3192 * q^83 + 720 * q^85 - 1322 * q^86 + 891 * q^88 - 2356 * q^89 + 112 * q^91 - 4504 * q^92 - 423 * q^94 - 3308 * q^95 - 1556 * q^97 - 294 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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