LMFDB - Embedded newform 693.4.a.u.1.7

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:	$8$
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			1.7
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	$-0.182061\pi$
$2$	4.75652	1.68168	0.840841	+	0.541282i	$0.182061\pi$
$3$	0	0
$3$	0	0
$4$	14.6245	1.82806
$5$	20.4224	1.82664	0.913318	−	0.407248i	$-0.133511\pi$
$5$	20.4224	1.82664	0.913318	+	0.407248i	$0.133511\pi$
$6$	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	$-0.476046\pi$
$17$	10.5397	0.150367	0.0751837	+	0.997170i	$0.476046\pi$
$18$	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	$-0.573249\pi$
$23$	−50.3192	−0.456186	−0.228093	+	0.973639i	$0.573249\pi$
$24$	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	$-0.488216\pi$
$29$	11.5606	0.0740260	0.0370130	+	0.999315i	$0.488216\pi$
$30$	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	$-0.602371\pi$
$41$	−165.966	−0.632182	−0.316091	+	0.948729i	$0.602371\pi$
$42$	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	$-0.887444\pi$
$47$	−604.561	−1.87626	−0.938130	+	0.346282i	$0.887444\pi$
$48$	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	$-0.568324\pi$
$53$	−164.372	−0.426004	−0.213002	+	0.977052i	$0.568324\pi$
$54$	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	$-0.395317\pi$
$59$	292.736	0.645949	0.322974	+	0.946408i	$0.395317\pi$
$60$	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	$-0.503900\pi$
$71$	−14.6599	−0.0245044	−0.0122522	+	0.999925i	$0.503900\pi$
$72$	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	$-0.341371\pi$
$83$	722.856	0.955949	0.477974	+	0.878374i	$0.341371\pi$
$84$	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	$-0.596649\pi$
$89$	−502.073	−0.597973	−0.298986	+	0.954257i	$0.596649\pi$
$90$	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	$-0.564754\pi$
$101$	−410.136	−0.404060	−0.202030	+	0.979379i	$0.564754\pi$
$102$	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	$-0.115567\pi$
$107$	2069.33	1.86963	0.934813	+	0.355140i	$0.115567\pi$
$108$	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	$-0.491379\pi$
$113$	65.0582	0.0541607	0.0270804	+	0.999633i	$0.491379\pi$
$114$	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	$-0.730793\pi$
$131$	−1988.69	−1.32636	−0.663179	+	0.748461i	$0.730793\pi$
$132$	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	$-0.225477\pi$
$137$	2435.56	1.51886	0.759431	+	0.650587i	$0.225477\pi$
$138$	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	$-0.547481\pi$
$149$	−540.585	−0.297224	−0.148612	+	0.988896i	$0.547481\pi$
$150$	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	$-0.656551\pi$
$167$	−2038.25	−0.944459	−0.472229	+	0.881476i	$0.656551\pi$
$168$	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	$-0.355806\pi$
$173$	1991.77	0.875326	0.437663	+	0.899139i	$0.355806\pi$
$174$	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	$-0.313306\pi$
$179$	2650.93	1.10693	0.553463	+	0.832874i	$0.313306\pi$
$180$	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	$-0.517496\pi$
$191$	−290.033	−0.109874	−0.0549372	+	0.998490i	$0.517496\pi$
$192$	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	$-0.273587\pi$
$197$	3610.11	1.30563	0.652816	+	0.757516i	$0.273587\pi$
$198$	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	$-0.732849\pi$
$227$	−4569.24	−1.33600	−0.667999	+	0.744162i	$0.732849\pi$
$228$	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	$-0.273157\pi$
$233$	4650.88	1.30768	0.653840	+	0.756633i	$0.273157\pi$
$234$	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	$-0.782511\pi$
$239$	−5730.84	−1.55104	−0.775518	+	0.631326i	$0.782511\pi$
$240$	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	$-0.306198\pi$
$251$	4548.60	1.14385	0.571923	+	0.820308i	$0.306198\pi$
$252$	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	$-0.542515\pi$
$257$	−1097.31	−0.266336	−0.133168	+	0.991093i	$0.542515\pi$
$258$	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	$-0.486934\pi$
$263$	350.041	0.0820701	0.0410350	+	0.999158i	$0.486934\pi$
$264$	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	$-0.825752\pi$
$269$	−7534.44	−1.70774	−0.853872	+	0.520483i	$0.825752\pi$
$270$	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	$-0.248571\pi$
$281$	6691.39	1.42055	0.710275	+	0.703924i	$0.248571\pi$
$282$	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	$-0.685642\pi$
$293$	−5523.97	−1.10141	−0.550707	+	0.834699i	$0.685642\pi$
$294$	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	$-0.669081\pi$
$311$	−5556.42	−1.01311	−0.506553	+	0.862209i	$0.669081\pi$
$312$	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	$-0.133601\pi$
$317$	10308.3	1.82641	0.913203	+	0.407506i	$0.133601\pi$
$318$	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	$-0.528421\pi$
$347$	−1152.74	−0.178335	−0.0891677	+	0.996017i	$0.528421\pi$
$348$	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	$-0.387391\pi$
$353$	4595.35	0.692877	0.346438	+	0.938073i	$0.387391\pi$
$354$	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	$-0.391981\pi$
$359$	4528.50	0.665752	0.332876	+	0.942971i	$0.391981\pi$
$360$	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	$-0.398458\pi$
$383$	4701.46	0.627241	0.313620	+	0.949548i	$0.398458\pi$
$384$	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	$-0.777428\pi$
$389$	−11743.8	−1.53068	−0.765338	+	0.643629i	$0.777428\pi$
$390$	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	$-0.361338\pi$
$401$	6776.89	0.843945	0.421972	+	0.906609i	$0.361338\pi$
$402$	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	$-0.326333\pi$
$419$	8901.30	1.03784	0.518922	+	0.854821i	$0.326333\pi$
$420$	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	$-0.318294\pi$
$431$	9669.77	1.08069	0.540344	+	0.841444i	$0.318294\pi$
$432$	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	$-0.458869\pi$
$443$	2402.96	0.257716	0.128858	+	0.991663i	$0.458869\pi$
$444$	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	$-0.789863\pi$
$449$	−15030.3	−1.57978	−0.789890	+	0.613248i	$0.789863\pi$
$450$	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	$-0.584827\pi$
$461$	−5213.29	−0.526697	−0.263348	+	0.964701i	$0.584827\pi$
$462$	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	$-0.385581\pi$
$467$	7100.05	0.703535	0.351768	+	0.936087i	$0.385581\pi$
$468$	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	$-0.720627\pi$
$479$	−13396.6	−1.27788	−0.638941	+	0.769256i	$0.720627\pi$
$480$	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	$-0.518755\pi$
$491$	−1281.36	−0.117774	−0.0588868	+	0.998265i	$0.518755\pi$
$492$	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	$-0.122918\pi$
$503$	20900.8	1.85273	0.926363	+	0.376631i	$0.122918\pi$
$504$	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	$-0.757103\pi$
$509$	−16598.5	−1.44542	−0.722708	+	0.691153i	$0.757103\pi$
$510$	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	$-0.160120\pi$
$521$	20837.8	1.75225	0.876125	+	0.482085i	$0.160120\pi$
$522$	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	$-0.533713\pi$
$557$	−2779.41	−0.211431	−0.105716	+	0.994396i	$0.533713\pi$
$558$	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	$-0.255567\pi$
$563$	18558.7	1.38927	0.694633	+	0.719364i	$0.255567\pi$
$564$	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	$-0.387223\pi$
$569$	9417.67	0.693865	0.346932	+	0.937890i	$0.387223\pi$
$570$	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	$-0.722651\pi$
$587$	−18312.6	−1.28764	−0.643818	+	0.765179i	$0.722651\pi$
$588$	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	$-0.436961\pi$
$593$	5682.32	0.393499	0.196749	+	0.980454i	$0.436961\pi$
$594$	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	$-0.217319\pi$
$599$	22748.4	1.55171	0.775854	+	0.630912i	$0.217319\pi$
$600$	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	$-0.341125\pi$
$617$	14671.7	0.957308	0.478654	+	0.878004i	$0.341125\pi$
$618$	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	$-0.409439\pi$
$641$	9110.30	0.561366	0.280683	+	0.959801i	$0.409439\pi$
$642$	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	$-0.453272\pi$
$647$	4814.56	0.292550	0.146275	+	0.989244i	$0.453272\pi$
$648$	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	$-0.395382\pi$
$653$	10772.3	0.645564	0.322782	+	0.946473i	$0.395382\pi$
$654$	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	$-0.672546\pi$
$659$	−17455.5	−1.03182	−0.515910	+	0.856643i	$0.672546\pi$
$660$	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	$-0.210170\pi$
$677$	27825.6	1.57965	0.789827	+	0.613330i	$0.210170\pi$
$678$	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	$-0.126435\pi$
$683$	32920.0	1.84429	0.922145	+	0.386845i	$0.126435\pi$
$684$	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	$-0.634777\pi$
$701$	−15251.7	−0.821754	−0.410877	+	0.911691i	$0.634777\pi$
$702$	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	$-0.397299\pi$
$719$	12226.1	0.634151	0.317076	+	0.948400i	$0.397299\pi$
$720$	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	$-0.294957\pi$
$743$	24324.6	1.20105	0.600527	+	0.799604i	$0.294957\pi$
$744$	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	$-0.305947\pi$
$761$	24040.1	1.14514	0.572570	+	0.819856i	$0.305947\pi$
$762$	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	$-0.679565\pi$
$773$	−22981.9	−1.06934	−0.534672	+	0.845060i	$0.679565\pi$
$774$	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	$-0.420872\pi$
$797$	11071.7	0.492072	0.246036	+	0.969261i	$0.420872\pi$
$798$	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	$-0.421993\pi$
$809$	11165.5	0.485241	0.242620	+	0.970121i	$0.421993\pi$
$810$	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	$-0.631949\pi$
$821$	−18949.2	−0.805521	−0.402761	+	0.915305i	$0.631949\pi$
$822$	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	$-0.475934\pi$
$827$	3592.80	0.151069	0.0755344	+	0.997143i	$0.475934\pi$
$828$	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	$-0.240008\pi$
$839$	35430.1	1.45790	0.728952	+	0.684565i	$0.240008\pi$
$840$	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	$-0.269858\pi$
$857$	33199.2	1.32329	0.661647	+	0.749816i	$0.269858\pi$
$858$	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	$-0.695730\pi$
$863$	−29250.4	−1.15376	−0.576880	+	0.816829i	$0.695730\pi$
$864$	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	$-0.520124\pi$
$881$	−3304.16	−0.126356	−0.0631782	+	0.998002i	$0.520124\pi$
$882$	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	$-0.557184\pi$
$887$	−9440.57	−0.357366	−0.178683	+	0.983907i	$0.557184\pi$
$888$	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	$-0.148188\pi$
$911$	49140.5	1.78715	0.893576	+	0.448912i	$0.148188\pi$
$912$	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	$-0.689399\pi$
$929$	−31742.8	−1.12104	−0.560520	+	0.828141i	$0.689399\pi$
$930$	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	$-0.646510\pi$
$941$	−25644.1	−0.888389	−0.444195	+	0.895930i	$0.646510\pi$
$942$	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	$-0.441818\pi$
$947$	10594.3	0.363535	0.181767	+	0.983342i	$0.441818\pi$
$948$	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	$-0.650931\pi$
$953$	−26865.9	−0.913190	−0.456595	+	0.889675i	$0.650931\pi$
$954$	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	$-0.559104\pi$
$971$	−11171.9	−0.369231	−0.184615	+	0.982811i	$0.559104\pi$
$972$	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	$-0.651513\pi$
$977$	−27986.4	−0.916441	−0.458221	+	0.888839i	$0.651513\pi$
$978$	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	$-0.460753\pi$
$983$	7580.83	0.245972	0.122986	+	0.992408i	$0.460753\pi$
$984$	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.u.1.7	yes	8
3.2	odd	2		693.4.a.r.1.2	✓	8

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

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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