LMFDB - Embedded newform 693.4.a.t.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:	$0$
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} - 45x^{6} + 77x^{5} + 540x^{4} - 915x^{3} - 1452x^{2} + 2660x - 672 $ x^8 - 2x^7 - 45x^6 + 77x^5 + 540x^4 - 915x^3 - 1452x^2 + 2660*x - 672
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.315819$ 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)

$f(q)$	$=$	$q+0.315819 q^{2} -7.90026 q^{4} +5.90817 q^{5} -7.00000 q^{7} -5.02161 q^{8} +O(q^{10})$	\(q+0.315819 q^{2} -7.90026 q^{4} +5.90817 q^{5} -7.00000 q^{7} -5.02161 q^{8} +1.86591 q^{10} -11.0000 q^{11} +2.91171 q^{13} -2.21073 q^{14} +61.6161 q^{16} -60.0496 q^{17} +69.8907 q^{19} -46.6760 q^{20} -3.47401 q^{22} -120.487 q^{23} -90.0936 q^{25} +0.919575 q^{26} +55.3018 q^{28} +174.527 q^{29} +44.5860 q^{31} +59.6324 q^{32} -18.9648 q^{34} -41.3572 q^{35} -271.813 q^{37} +22.0728 q^{38} -29.6685 q^{40} +355.169 q^{41} +545.964 q^{43} +86.9028 q^{44} -38.0521 q^{46} -413.419 q^{47} +49.0000 q^{49} -28.4533 q^{50} -23.0033 q^{52} +709.617 q^{53} -64.9898 q^{55} +35.1512 q^{56} +55.1191 q^{58} +358.442 q^{59} -574.641 q^{61} +14.0811 q^{62} -474.096 q^{64} +17.2029 q^{65} +457.221 q^{67} +474.407 q^{68} -13.0614 q^{70} +87.8142 q^{71} +403.682 q^{73} -85.8438 q^{74} -552.154 q^{76} +77.0000 q^{77} -195.205 q^{79} +364.038 q^{80} +112.169 q^{82} +1407.01 q^{83} -354.783 q^{85} +172.426 q^{86} +55.2377 q^{88} +1250.19 q^{89} -20.3820 q^{91} +951.879 q^{92} -130.566 q^{94} +412.926 q^{95} +1773.32 q^{97} +15.4751 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} + 30 q^{4} + 10 q^{5} - 56 q^{7} + 15 q^{8}+O(q^{10}) $ 8 * q + 2 * q^2 + 30 * q^4 + 10 * q^5 - 56 * q^7 + 15 * q^8	$ 8 q + 2 q^{2} + 30 q^{4} + 10 q^{5} - 56 q^{7} + 15 q^{8} - 13 q^{10} - 88 q^{11} - 148 q^{13} - 14 q^{14} + 266 q^{16} + 114 q^{17} + 58 q^{19} + 291 q^{20} - 22 q^{22} + 246 q^{23} + 244 q^{25} + 305 q^{26} - 210 q^{28} - 72 q^{29} + 252 q^{31} + 1272 q^{32} + 630 q^{34} - 70 q^{35} - 80 q^{37} + 1885 q^{38} - 342 q^{40} + 682 q^{41} - 106 q^{43} - 330 q^{44} + 120 q^{46} + 828 q^{47} + 392 q^{49} + 801 q^{50} - 1681 q^{52} + 462 q^{53} - 110 q^{55} - 105 q^{56} - 1087 q^{58} + 626 q^{59} - 854 q^{61} + 1350 q^{62} + 2997 q^{64} + 22 q^{65} + 130 q^{67} + 2202 q^{68} + 91 q^{70} + 326 q^{71} - 390 q^{73} - 359 q^{74} + 2041 q^{76} + 616 q^{77} - 508 q^{79} + 4391 q^{80} + 1528 q^{82} + 1596 q^{83} - 880 q^{85} - 414 q^{86} - 165 q^{88} + 4324 q^{89} + 1036 q^{91} + 2092 q^{92} - 1685 q^{94} + 1076 q^{95} - 964 q^{97} + 98 q^{98}+O(q^{100}) $ 8 * q + 2 * q^2 + 30 * q^4 + 10 * q^5 - 56 * q^7 + 15 * q^8 - 13 * q^10 - 88 * q^11 - 148 * q^13 - 14 * q^14 + 266 * q^16 + 114 * q^17 + 58 * q^19 + 291 * q^20 - 22 * q^22 + 246 * q^23 + 244 * q^25 + 305 * q^26 - 210 * q^28 - 72 * q^29 + 252 * q^31 + 1272 * q^32 + 630 * q^34 - 70 * q^35 - 80 * q^37 + 1885 * q^38 - 342 * q^40 + 682 * q^41 - 106 * q^43 - 330 * q^44 + 120 * q^46 + 828 * q^47 + 392 * q^49 + 801 * q^50 - 1681 * q^52 + 462 * q^53 - 110 * q^55 - 105 * q^56 - 1087 * q^58 + 626 * q^59 - 854 * q^61 + 1350 * q^62 + 2997 * q^64 + 22 * q^65 + 130 * q^67 + 2202 * q^68 + 91 * q^70 + 326 * q^71 - 390 * q^73 - 359 * q^74 + 2041 * q^76 + 616 * q^77 - 508 * q^79 + 4391 * q^80 + 1528 * q^82 + 1596 * q^83 - 880 * q^85 - 414 * q^86 - 165 * q^88 + 4324 * q^89 + 1036 * q^91 + 2092 * q^92 - 1685 * q^94 + 1076 * q^95 - 964 * q^97 + 98 * 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 $

$2$	0.315819	0.111659	0.0558295	−	0.998440i	$-0.482220\pi$
$2$	0.315819	0.111659	0.0558295	+	0.998440i	$0.482220\pi$
$3$	0	0
$3$	0	0
$4$	−7.90026	−0.987532
$5$	5.90817	0.528442	0.264221	−	0.964462i	$-0.414885\pi$
$5$	5.90817	0.528442	0.264221	+	0.964462i	$0.414885\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−5.02161	−0.221926
$9$	0	0
$10$	1.86591	0.0590053
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	2.91171	0.0621203	0.0310601	−	0.999518i	$-0.490112\pi$
$13$	2.91171	0.0621203	0.0310601	+	0.999518i	$0.490112\pi$
$14$	−2.21073	−0.0422031
$15$	0	0
$16$	61.6161	0.962752
$17$	−60.0496	−0.856715	−0.428358	−	0.903609i	$-0.640908\pi$
$17$	−60.0496	−0.856715	−0.428358	+	0.903609i	$0.640908\pi$
$18$	0	0
$19$	69.8907	0.843896	0.421948	−	0.906620i	$-0.361347\pi$
$19$	69.8907	0.843896	0.421948	+	0.906620i	$0.361347\pi$
$20$	−46.6760	−0.521854
$21$	0	0
$22$	−3.47401	−0.0336664
$23$	−120.487	−1.09232	−0.546159	−	0.837682i	$-0.683910\pi$
$23$	−120.487	−1.09232	−0.546159	+	0.837682i	$0.683910\pi$
$24$	0	0
$25$	−90.0936	−0.720749
$26$	0.919575	0.00693629
$27$	0	0
$28$	55.3018	0.373252
$29$	174.527	1.11755	0.558774	−	0.829320i	$-0.311272\pi$
$29$	174.527	1.11755	0.558774	+	0.829320i	$0.311272\pi$
$30$	0	0
$31$	44.5860	0.258319	0.129160	−	0.991624i	$-0.458772\pi$
$31$	44.5860	0.258319	0.129160	+	0.991624i	$0.458772\pi$
$32$	59.6324	0.329426
$33$	0	0
$34$	−18.9648	−0.0956599
$35$	−41.3572	−0.199732
$36$	0	0
$37$	−271.813	−1.20772	−0.603862	−	0.797089i	$-0.706372\pi$
$37$	−271.813	−1.20772	−0.603862	+	0.797089i	$0.706372\pi$
$38$	22.0728	0.0942285
$39$	0	0
$40$	−29.6685	−0.117275
$41$	355.169	1.35288	0.676441	−	0.736497i	$-0.263521\pi$
$41$	355.169	1.35288	0.676441	+	0.736497i	$0.263521\pi$
$42$	0	0
$43$	545.964	1.93625	0.968125	−	0.250468i	$-0.0805845\pi$
$43$	545.964	1.93625	0.968125	+	0.250468i	$0.0805845\pi$
$44$	86.9028	0.297752
$45$	0	0
$46$	−38.0521	−0.121967
$47$	−413.419	−1.28305	−0.641524	−	0.767103i	$-0.721698\pi$
$47$	−413.419	−1.28305	−0.641524	+	0.767103i	$0.721698\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−28.4533	−0.0804780
$51$	0	0
$52$	−23.0033	−0.0613458
$53$	709.617	1.83912	0.919560	−	0.392950i	$-0.128545\pi$
$53$	709.617	1.83912	0.919560	+	0.392950i	$0.128545\pi$
$54$	0	0
$55$	−64.9898	−0.159331
$56$	35.1512	0.0838800
$57$	0	0
$58$	55.1191	0.124784
$59$	358.442	0.790936	0.395468	−	0.918480i	$-0.370582\pi$
$59$	358.442	0.790936	0.395468	+	0.918480i	$0.370582\pi$
$60$	0	0
$61$	−574.641	−1.20615	−0.603076	−	0.797684i	$-0.706058\pi$
$61$	−574.641	−1.20615	−0.603076	+	0.797684i	$0.706058\pi$
$62$	14.0811	0.0288436
$63$	0	0
$64$	−474.096	−0.925969
$65$	17.2029	0.0328270
$66$	0	0
$67$	457.221	0.833709	0.416854	−	0.908973i	$-0.363132\pi$
$67$	457.221	0.833709	0.416854	+	0.908973i	$0.363132\pi$
$68$	474.407	0.846034
$69$	0	0
$70$	−13.0614	−0.0223019
$71$	87.8142	0.146784	0.0733918	−	0.997303i	$-0.476618\pi$
$71$	87.8142	0.146784	0.0733918	+	0.997303i	$0.476618\pi$
$72$	0	0
$73$	403.682	0.647225	0.323613	−	0.946190i	$-0.395103\pi$
$73$	403.682	0.647225	0.323613	+	0.946190i	$0.395103\pi$
$74$	−85.8438	−0.134853
$75$	0	0
$76$	−552.154	−0.833374
$77$	77.0000	0.113961
$78$	0	0
$79$	−195.205	−0.278003	−0.139001	−	0.990292i	$-0.544389\pi$
$79$	−195.205	−0.278003	−0.139001	+	0.990292i	$0.544389\pi$
$80$	364.038	0.508759
$81$	0	0
$82$	112.169	0.151061
$83$	1407.01	1.86071	0.930356	−	0.366657i	$-0.119498\pi$
$83$	1407.01	1.86071	0.930356	+	0.366657i	$0.119498\pi$
$84$	0	0
$85$	−354.783	−0.452725
$86$	172.426	0.216200
$87$	0	0
$88$	55.2377	0.0669131
$89$	1250.19	1.48899	0.744494	−	0.667629i	$-0.232691\pi$
$89$	1250.19	1.48899	0.744494	+	0.667629i	$0.232691\pi$
$90$	0	0
$91$	−20.3820	−0.0234793
$92$	951.879	1.07870
$93$	0	0
$94$	−130.566	−0.143264
$95$	412.926	0.445950
$96$	0	0
$97$	1773.32	1.85622	0.928110	−	0.372307i	$-0.121433\pi$
$97$	1773.32	1.85622	0.928110	+	0.372307i	$0.121433\pi$
$98$	15.4751	0.0159513
$99$	0	0
$100$	711.763	0.711763
$101$	−1652.74	−1.62825	−0.814127	−	0.580687i	$-0.802784\pi$
$101$	−1652.74	−1.62825	−0.814127	+	0.580687i	$0.802784\pi$
$102$	0	0
$103$	−851.256	−0.814338	−0.407169	−	0.913353i	$-0.633484\pi$
$103$	−851.256	−0.814338	−0.407169	+	0.913353i	$0.633484\pi$
$104$	−14.6215	−0.0137861
$105$	0	0
$106$	224.111	0.205354
$107$	822.234	0.742882	0.371441	−	0.928457i	$-0.378864\pi$
$107$	822.234	0.742882	0.371441	+	0.928457i	$0.378864\pi$
$108$	0	0
$109$	310.706	0.273030	0.136515	−	0.990638i	$-0.456410\pi$
$109$	310.706	0.273030	0.136515	+	0.990638i	$0.456410\pi$
$110$	−20.5250	−0.0177908
$111$	0	0
$112$	−431.313	−0.363886
$113$	1455.80	1.21194	0.605972	−	0.795486i	$-0.292784\pi$
$113$	1455.80	1.21194	0.605972	+	0.795486i	$0.292784\pi$
$114$	0	0
$115$	−711.858	−0.577227
$116$	−1378.81	−1.10362
$117$	0	0
$118$	113.203	0.0883151
$119$	420.347	0.323808
$120$	0	0
$121$	121.000	0.0909091
$122$	−181.483	−0.134678
$123$	0	0
$124$	−352.241	−0.255098
$125$	−1270.81	−0.909316
$126$	0	0
$127$	1182.69	0.826355	0.413177	−	0.910651i	$-0.364419\pi$
$127$	1182.69	0.826355	0.413177	+	0.910651i	$0.364419\pi$
$128$	−626.788	−0.432818
$129$	0	0
$130$	5.43300	0.00366543
$131$	1866.42	1.24481	0.622405	−	0.782695i	$-0.286156\pi$
$131$	1866.42	1.24481	0.622405	+	0.782695i	$0.286156\pi$
$132$	0	0
$133$	−489.235	−0.318963
$134$	144.399	0.0930910
$135$	0	0
$136$	301.545	0.190127
$137$	2973.54	1.85436	0.927179	−	0.374619i	$-0.122227\pi$
$137$	2973.54	1.85436	0.927179	+	0.374619i	$0.122227\pi$
$138$	0	0
$139$	−2904.10	−1.77211	−0.886054	−	0.463583i	$-0.846564\pi$
$139$	−2904.10	−1.77211	−0.886054	+	0.463583i	$0.846564\pi$
$140$	326.732	0.197242
$141$	0	0
$142$	27.7334	0.0163897
$143$	−32.0288	−0.0187300
$144$	0	0
$145$	1031.14	0.590560
$146$	127.491	0.0722685
$147$	0	0
$148$	2147.39	1.19267
$149$	−67.9990	−0.0373872	−0.0186936	−	0.999825i	$-0.505951\pi$
$149$	−67.9990	−0.0373872	−0.0186936	+	0.999825i	$0.505951\pi$
$150$	0	0
$151$	3128.90	1.68627	0.843133	−	0.537705i	$-0.180709\pi$
$151$	3128.90	1.68627	0.843133	+	0.537705i	$0.180709\pi$
$152$	−350.963	−0.187282
$153$	0	0
$154$	24.3181	0.0127247
$155$	263.422	0.136507
$156$	0	0
$157$	102.502	0.0521055	0.0260528	−	0.999661i	$-0.491706\pi$
$157$	102.502	0.0521055	0.0260528	+	0.999661i	$0.491706\pi$
$158$	−61.6493	−0.0310415
$159$	0	0
$160$	352.318	0.174082
$161$	843.410	0.412857
$162$	0	0
$163$	−3270.96	−1.57179	−0.785893	−	0.618363i	$-0.787796\pi$
$163$	−3270.96	−1.57179	−0.785893	+	0.618363i	$0.787796\pi$
$164$	−2805.93	−1.33601
$165$	0	0
$166$	444.360	0.207765
$167$	1018.68	0.472021	0.236011	−	0.971750i	$-0.424160\pi$
$167$	1018.68	0.472021	0.236011	+	0.971750i	$0.424160\pi$
$168$	0	0
$169$	−2188.52	−0.996141
$170$	−112.047	−0.0505507
$171$	0	0
$172$	−4313.26	−1.91211
$173$	−3510.34	−1.54270	−0.771348	−	0.636414i	$-0.780417\pi$
$173$	−3510.34	−1.54270	−0.771348	+	0.636414i	$0.780417\pi$
$174$	0	0
$175$	630.655	0.272417
$176$	−677.778	−0.290281
$177$	0	0
$178$	394.834	0.166259
$179$	−309.297	−0.129150	−0.0645752	−	0.997913i	$-0.520569\pi$
$179$	−309.297	−0.129150	−0.0645752	+	0.997913i	$0.520569\pi$
$180$	0	0
$181$	−1938.21	−0.795944	−0.397972	−	0.917398i	$-0.630286\pi$
$181$	−1938.21	−0.795944	−0.397972	+	0.917398i	$0.630286\pi$
$182$	−6.43702	−0.00262167
$183$	0	0
$184$	605.039	0.242413
$185$	−1605.92	−0.638213
$186$	0	0
$187$	660.545	0.258309
$188$	3266.11	1.26705
$189$	0	0
$190$	130.410	0.0497943
$191$	−437.216	−0.165633	−0.0828163	−	0.996565i	$-0.526391\pi$
$191$	−437.216	−0.165633	−0.0828163	+	0.996565i	$0.526391\pi$
$192$	0	0
$193$	17.5821	0.00655746	0.00327873	−	0.999995i	$-0.498956\pi$
$193$	17.5821	0.00655746	0.00327873	+	0.999995i	$0.498956\pi$
$194$	560.048	0.207263
$195$	0	0
$196$	−387.113	−0.141076
$197$	4068.89	1.47156	0.735778	−	0.677222i	$-0.236817\pi$
$197$	4068.89	1.47156	0.735778	+	0.677222i	$0.236817\pi$
$198$	0	0
$199$	3833.34	1.36552	0.682759	−	0.730643i	$-0.260780\pi$
$199$	3833.34	1.36552	0.682759	+	0.730643i	$0.260780\pi$
$200$	452.414	0.159953
$201$	0	0
$202$	−521.967	−0.181809
$203$	−1221.69	−0.422394
$204$	0	0
$205$	2098.40	0.714920
$206$	−268.843	−0.0909281
$207$	0	0
$208$	179.409	0.0598065
$209$	−768.797	−0.254444
$210$	0	0
$211$	−57.2507	−0.0186792	−0.00933958	−	0.999956i	$-0.502973\pi$
$211$	−57.2507	−0.0186792	−0.00933958	+	0.999956i	$0.502973\pi$
$212$	−5606.16	−1.81619
$213$	0	0
$214$	259.677	0.0829494
$215$	3225.65	1.02320
$216$	0	0
$217$	−312.102	−0.0976354
$218$	98.1270	0.0304862
$219$	0	0
$220$	513.436	0.157345
$221$	−174.847	−0.0532194
$222$	0	0
$223$	−796.167	−0.239082	−0.119541	−	0.992829i	$-0.538142\pi$
$223$	−796.167	−0.239082	−0.119541	+	0.992829i	$0.538142\pi$
$224$	−417.427	−0.124511
$225$	0	0
$226$	459.768	0.135324
$227$	−2375.23	−0.694490	−0.347245	−	0.937774i	$-0.612883\pi$
$227$	−2375.23	−0.694490	−0.347245	+	0.937774i	$0.612883\pi$
$228$	0	0
$229$	−4005.14	−1.15575	−0.577876	−	0.816125i	$-0.696118\pi$
$229$	−4005.14	−1.15575	−0.577876	+	0.816125i	$0.696118\pi$
$230$	−224.818	−0.0644525
$231$	0	0
$232$	−876.407	−0.248013
$233$	−4218.29	−1.18605	−0.593024	−	0.805185i	$-0.702066\pi$
$233$	−4218.29	−1.18605	−0.593024	+	0.805185i	$0.702066\pi$
$234$	0	0
$235$	−2442.55	−0.678017
$236$	−2831.79	−0.781075
$237$	0	0
$238$	132.754	0.0361560
$239$	6626.55	1.79345	0.896727	−	0.442584i	$-0.145938\pi$
$239$	6626.55	1.79345	0.896727	+	0.442584i	$0.145938\pi$
$240$	0	0
$241$	−6040.48	−1.61453	−0.807265	−	0.590189i	$-0.799053\pi$
$241$	−6040.48	−1.61453	−0.807265	+	0.590189i	$0.799053\pi$
$242$	38.2141	0.0101508
$243$	0	0
$244$	4539.81	1.19111
$245$	289.500	0.0754918
$246$	0	0
$247$	203.502	0.0524231
$248$	−223.894	−0.0573276
$249$	0	0
$250$	−401.346	−0.101533
$251$	2155.94	0.542158	0.271079	−	0.962557i	$-0.412619\pi$
$251$	2155.94	0.542158	0.271079	+	0.962557i	$0.412619\pi$
$252$	0	0
$253$	1325.36	0.329346
$254$	373.517	0.0922699
$255$	0	0
$256$	3594.82	0.877641
$257$	6708.42	1.62825	0.814125	−	0.580690i	$-0.197217\pi$
$257$	6708.42	1.62825	0.814125	+	0.580690i	$0.197217\pi$
$258$	0	0
$259$	1902.69	0.456477
$260$	−135.907	−0.0324177
$261$	0	0
$262$	589.452	0.138994
$263$	−3364.19	−0.788763	−0.394381	−	0.918947i	$-0.629041\pi$
$263$	−3364.19	−0.788763	−0.394381	+	0.918947i	$0.629041\pi$
$264$	0	0
$265$	4192.53	0.971869
$266$	−154.510	−0.0356150
$267$	0	0
$268$	−3612.17	−0.823314
$269$	−980.866	−0.222321	−0.111161	−	0.993802i	$-0.535457\pi$
$269$	−980.866	−0.222321	−0.111161	+	0.993802i	$0.535457\pi$
$270$	0	0
$271$	−384.406	−0.0861660	−0.0430830	−	0.999071i	$-0.513718\pi$
$271$	−384.406	−0.0861660	−0.0430830	+	0.999071i	$0.513718\pi$
$272$	−3700.02	−0.824804
$273$	0	0
$274$	939.102	0.207056
$275$	991.029	0.217314
$276$	0	0
$277$	−188.648	−0.0409197	−0.0204598	−	0.999791i	$-0.506513\pi$
$277$	−188.648	−0.0409197	−0.0204598	+	0.999791i	$0.506513\pi$
$278$	−917.172	−0.197872
$279$	0	0
$280$	207.679	0.0443258
$281$	−7747.25	−1.64471	−0.822353	−	0.568977i	$-0.807339\pi$
$281$	−7747.25	−1.64471	−0.822353	+	0.568977i	$0.807339\pi$
$282$	0	0
$283$	3829.43	0.804368	0.402184	−	0.915559i	$-0.368251\pi$
$283$	3829.43	0.804368	0.402184	+	0.915559i	$0.368251\pi$
$284$	−693.755	−0.144953
$285$	0	0
$286$	−10.1153	−0.00209137
$287$	−2486.19	−0.511341
$288$	0	0
$289$	−1307.05	−0.266039
$290$	325.653	0.0659413
$291$	0	0
$292$	−3189.19	−0.639156
$293$	3164.06	0.630875	0.315437	−	0.948946i	$-0.397849\pi$
$293$	3164.06	0.630875	0.315437	+	0.948946i	$0.397849\pi$
$294$	0	0
$295$	2117.74	0.417964
$296$	1364.94	0.268025
$297$	0	0
$298$	−21.4754	−0.00417462
$299$	−350.824	−0.0678551
$300$	0	0
$301$	−3821.75	−0.731834
$302$	988.166	0.188287
$303$	0	0
$304$	4306.39	0.812463
$305$	−3395.07	−0.637381
$306$	0	0
$307$	1737.84	0.323075	0.161538	−	0.986867i	$-0.448355\pi$
$307$	1737.84	0.323075	0.161538	+	0.986867i	$0.448355\pi$
$308$	−608.320	−0.112540
$309$	0	0
$310$	83.1936	0.0152422
$311$	−551.287	−0.100517	−0.0502583	−	0.998736i	$-0.516004\pi$
$311$	−551.287	−0.100517	−0.0502583	+	0.998736i	$0.516004\pi$
$312$	0	0
$313$	8193.33	1.47960	0.739800	−	0.672827i	$-0.234920\pi$
$313$	8193.33	1.47960	0.739800	+	0.672827i	$0.234920\pi$
$314$	32.3722	0.00581805
$315$	0	0
$316$	1542.17	0.274537
$317$	−5858.91	−1.03807	−0.519037	−	0.854752i	$-0.673709\pi$
$317$	−5858.91	−1.03807	−0.519037	+	0.854752i	$0.673709\pi$
$318$	0	0
$319$	−1919.80	−0.336954
$320$	−2801.04	−0.489321
$321$	0	0
$322$	266.365	0.0460992
$323$	−4196.90	−0.722978
$324$	0	0
$325$	−262.327	−0.0447731
$326$	−1033.03	−0.175504
$327$	0	0
$328$	−1783.52	−0.300239
$329$	2893.93	0.484947
$330$	0	0
$331$	1952.05	0.324153	0.162076	−	0.986778i	$-0.448181\pi$
$331$	1952.05	0.324153	0.162076	+	0.986778i	$0.448181\pi$
$332$	−11115.7	−1.83751
$333$	0	0
$334$	321.718	0.0527054
$335$	2701.34	0.440567
$336$	0	0
$337$	−8151.87	−1.31769	−0.658844	−	0.752280i	$-0.728954\pi$
$337$	−8151.87	−1.31769	−0.658844	+	0.752280i	$0.728954\pi$
$338$	−691.177	−0.111228
$339$	0	0
$340$	2802.88	0.447080
$341$	−490.447	−0.0778861
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−2741.62	−0.429704
$345$	0	0
$346$	−1108.63	−0.172256
$347$	−78.5678	−0.0121549	−0.00607743	−	0.999982i	$-0.501935\pi$
$347$	−78.5678	−0.0121549	−0.00607743	+	0.999982i	$0.501935\pi$
$348$	0	0
$349$	−942.516	−0.144561	−0.0722804	−	0.997384i	$-0.523028\pi$
$349$	−942.516	−0.144561	−0.0722804	+	0.997384i	$0.523028\pi$
$350$	199.173	0.0304178
$351$	0	0
$352$	−655.956	−0.0993256
$353$	−720.110	−0.108577	−0.0542884	−	0.998525i	$-0.517289\pi$
$353$	−720.110	−0.108577	−0.0542884	+	0.998525i	$0.517289\pi$
$354$	0	0
$355$	518.821	0.0775666
$356$	−9876.83	−1.47042
$357$	0	0
$358$	−97.6818	−0.0144208
$359$	−5213.98	−0.766527	−0.383264	−	0.923639i	$-0.625200\pi$
$359$	−5213.98	−0.766527	−0.383264	+	0.923639i	$0.625200\pi$
$360$	0	0
$361$	−1974.29	−0.287840
$362$	−612.123	−0.0888743
$363$	0	0
$364$	161.023	0.0231865
$365$	2385.02	0.342021
$366$	0	0
$367$	−6256.57	−0.889892	−0.444946	−	0.895557i	$-0.646777\pi$
$367$	−6256.57	−0.889892	−0.444946	+	0.895557i	$0.646777\pi$
$368$	−7423.95	−1.05163
$369$	0	0
$370$	−507.179	−0.0712622
$371$	−4967.32	−0.695122
$372$	0	0
$373$	−2406.21	−0.334019	−0.167009	−	0.985955i	$-0.553411\pi$
$373$	−2406.21	−0.334019	−0.167009	+	0.985955i	$0.553411\pi$
$374$	208.613	0.0288425
$375$	0	0
$376$	2076.03	0.284742
$377$	508.173	0.0694225
$378$	0	0
$379$	−1843.79	−0.249892	−0.124946	−	0.992164i	$-0.539876\pi$
$379$	−1843.79	−0.249892	−0.124946	+	0.992164i	$0.539876\pi$
$380$	−3262.22	−0.440390
$381$	0	0
$382$	−138.081	−0.0184944
$383$	8466.69	1.12958	0.564788	−	0.825236i	$-0.308958\pi$
$383$	8466.69	1.12958	0.564788	+	0.825236i	$0.308958\pi$
$384$	0	0
$385$	454.929	0.0602216
$386$	5.55278	0.000732199 0
$387$	0	0
$388$	−14009.7	−1.83308
$389$	−8769.09	−1.14296	−0.571479	−	0.820617i	$-0.693630\pi$
$389$	−8769.09	−1.14296	−0.571479	+	0.820617i	$0.693630\pi$
$390$	0	0
$391$	7235.20	0.935805
$392$	−246.059	−0.0317037
$393$	0	0
$394$	1285.03	0.164312
$395$	−1153.30	−0.146909
$396$	0	0
$397$	7946.64	1.00461	0.502306	−	0.864690i	$-0.332485\pi$
$397$	7946.64	1.00461	0.502306	+	0.864690i	$0.332485\pi$
$398$	1210.64	0.152472
$399$	0	0
$400$	−5551.22	−0.693902
$401$	−7832.40	−0.975390	−0.487695	−	0.873014i	$-0.662162\pi$
$401$	−7832.40	−0.975390	−0.487695	+	0.873014i	$0.662162\pi$
$402$	0	0
$403$	129.822	0.0160469
$404$	13057.1	1.60795
$405$	0	0
$406$	−385.834	−0.0471640
$407$	2989.95	0.364143
$408$	0	0
$409$	7353.60	0.889027	0.444514	−	0.895772i	$-0.353377\pi$
$409$	7353.60	0.889027	0.444514	+	0.895772i	$0.353377\pi$
$410$	662.715	0.0798272
$411$	0	0
$412$	6725.15	0.804185
$413$	−2509.10	−0.298946
$414$	0	0
$415$	8312.83	0.983279
$416$	173.632	0.0204640
$417$	0	0
$418$	−242.801	−0.0284110
$419$	2532.08	0.295228	0.147614	−	0.989045i	$-0.452841\pi$
$419$	2532.08	0.295228	0.147614	+	0.989045i	$0.452841\pi$
$420$	0	0
$421$	−3330.80	−0.385589	−0.192795	−	0.981239i	$-0.561755\pi$
$421$	−3330.80	−0.385589	−0.192795	+	0.981239i	$0.561755\pi$
$422$	−18.0809	−0.00208570
$423$	0	0
$424$	−3563.42	−0.408148
$425$	5410.08	0.617476
$426$	0	0
$427$	4022.49	0.455882
$428$	−6495.86	−0.733620
$429$	0	0
$430$	1018.72	0.114249
$431$	−3115.93	−0.348234	−0.174117	−	0.984725i	$-0.555707\pi$
$431$	−3115.93	−0.348234	−0.174117	+	0.984725i	$0.555707\pi$
$432$	0	0
$433$	−12771.7	−1.41748	−0.708741	−	0.705469i	$-0.750736\pi$
$433$	−12771.7	−1.41748	−0.708741	+	0.705469i	$0.750736\pi$
$434$	−98.5679	−0.0109019
$435$	0	0
$436$	−2454.66	−0.269626
$437$	−8420.93	−0.921802
$438$	0	0
$439$	−15313.8	−1.66490	−0.832449	−	0.554102i	$-0.813062\pi$
$439$	−15313.8	−1.66490	−0.832449	+	0.554102i	$0.813062\pi$
$440$	326.353	0.0353597
$441$	0	0
$442$	−55.2201	−0.00594242
$443$	1911.22	0.204977	0.102488	−	0.994734i	$-0.467320\pi$
$443$	1911.22	0.204977	0.102488	+	0.994734i	$0.467320\pi$
$444$	0	0
$445$	7386.33	0.786844
$446$	−251.445	−0.0266956
$447$	0	0
$448$	3318.67	0.349983
$449$	15994.2	1.68110	0.840548	−	0.541737i	$-0.182233\pi$
$449$	15994.2	1.68110	0.840548	+	0.541737i	$0.182233\pi$
$450$	0	0
$451$	−3906.86	−0.407909
$452$	−11501.2	−1.19683
$453$	0	0
$454$	−750.142	−0.0775460
$455$	−120.420	−0.0124074
$456$	0	0
$457$	−2724.26	−0.278852	−0.139426	−	0.990232i	$-0.544526\pi$
$457$	−2724.26	−0.278852	−0.139426	+	0.990232i	$0.544526\pi$
$458$	−1264.90	−0.129050
$459$	0	0
$460$	5623.86	0.570030
$461$	−1935.41	−0.195533	−0.0977666	−	0.995209i	$-0.531170\pi$
$461$	−1935.41	−0.195533	−0.0977666	+	0.995209i	$0.531170\pi$
$462$	0	0
$463$	16266.1	1.63272	0.816361	−	0.577542i	$-0.195988\pi$
$463$	16266.1	1.63272	0.816361	+	0.577542i	$0.195988\pi$
$464$	10753.7	1.07592
$465$	0	0
$466$	−1332.22	−0.132433
$467$	14971.5	1.48351	0.741754	−	0.670672i	$-0.233994\pi$
$467$	14971.5	1.48351	0.741754	+	0.670672i	$0.233994\pi$
$468$	0	0
$469$	−3200.55	−0.315112
$470$	−771.403	−0.0757067
$471$	0	0
$472$	−1799.96	−0.175529
$473$	−6005.60	−0.583801
$474$	0	0
$475$	−6296.70	−0.608237
$476$	−3320.85	−0.319771
$477$	0	0
$478$	2092.79	0.200255
$479$	11407.5	1.08815	0.544075	−	0.839037i	$-0.316881\pi$
$479$	11407.5	1.08815	0.544075	+	0.839037i	$0.316881\pi$
$480$	0	0
$481$	−791.442	−0.0750242
$482$	−1907.70	−0.180277
$483$	0	0
$484$	−955.931	−0.0897757
$485$	10477.1	0.980905
$486$	0	0
$487$	15644.5	1.45568	0.727842	−	0.685744i	$-0.240523\pi$
$487$	15644.5	1.45568	0.727842	+	0.685744i	$0.240523\pi$
$488$	2885.62	0.267676
$489$	0	0
$490$	91.4297	0.00842933
$491$	6594.27	0.606100	0.303050	−	0.952975i	$-0.401995\pi$
$491$	6594.27	0.606100	0.303050	+	0.952975i	$0.401995\pi$
$492$	0	0
$493$	−10480.3	−0.957421
$494$	64.2697	0.00585350
$495$	0	0
$496$	2747.22	0.248697
$497$	−614.699	−0.0554790
$498$	0	0
$499$	5492.47	0.492739	0.246369	−	0.969176i	$-0.420762\pi$
$499$	5492.47	0.492739	0.246369	+	0.969176i	$0.420762\pi$
$500$	10039.7	0.897979
$501$	0	0
$502$	680.887	0.0605368
$503$	9529.33	0.844715	0.422358	−	0.906429i	$-0.361203\pi$
$503$	9529.33	0.844715	0.422358	+	0.906429i	$0.361203\pi$
$504$	0	0
$505$	−9764.65	−0.860438
$506$	418.573	0.0367744
$507$	0	0
$508$	−9343.58	−0.816052
$509$	19628.2	1.70924	0.854622	−	0.519250i	$-0.173789\pi$
$509$	19628.2	1.70924	0.854622	+	0.519250i	$0.173789\pi$
$510$	0	0
$511$	−2825.78	−0.244628
$512$	6149.62	0.530815
$513$	0	0
$514$	2118.65	0.181809
$515$	−5029.36	−0.430331
$516$	0	0
$517$	4547.60	0.386854
$518$	600.907	0.0509697
$519$	0	0
$520$	−86.3861	−0.00728515
$521$	19650.6	1.65242	0.826209	−	0.563364i	$-0.190493\pi$
$521$	19650.6	1.65242	0.826209	+	0.563364i	$0.190493\pi$
$522$	0	0
$523$	−4045.28	−0.338218	−0.169109	−	0.985597i	$-0.554089\pi$
$523$	−4045.28	−0.338218	−0.169109	+	0.985597i	$0.554089\pi$
$524$	−14745.2	−1.22929
$525$	0	0
$526$	−1062.47	−0.0880724
$527$	−2677.37	−0.221306
$528$	0	0
$529$	2350.14	0.193157
$530$	1324.08	0.108518
$531$	0	0
$532$	3865.08	0.314986
$533$	1034.15	0.0840414
$534$	0	0
$535$	4857.89	0.392570
$536$	−2295.99	−0.185021
$537$	0	0
$538$	−309.776	−0.0248242
$539$	−539.000	−0.0430730
$540$	0	0
$541$	8478.65	0.673800	0.336900	−	0.941540i	$-0.390622\pi$
$541$	8478.65	0.673800	0.336900	+	0.941540i	$0.390622\pi$
$542$	−121.403	−0.00962120
$543$	0	0
$544$	−3580.90	−0.282224
$545$	1835.70	0.144281
$546$	0	0
$547$	−20754.0	−1.62226	−0.811130	−	0.584866i	$-0.801147\pi$
$547$	−20754.0	−1.62226	−0.811130	+	0.584866i	$0.801147\pi$
$548$	−23491.8	−1.83124
$549$	0	0
$550$	312.986	0.0242650
$551$	12197.8	0.943095
$552$	0	0
$553$	1366.43	0.105075
$554$	−59.5786	−0.00456905
$555$	0	0
$556$	22943.2	1.75001
$557$	17200.9	1.30849	0.654244	−	0.756284i	$-0.272987\pi$
$557$	17200.9	1.30849	0.654244	+	0.756284i	$0.272987\pi$
$558$	0	0
$559$	1589.69	0.120280
$560$	−2548.27	−0.192293
$561$	0	0
$562$	−2446.73	−0.183646
$563$	9509.54	0.711864	0.355932	−	0.934512i	$-0.384163\pi$
$563$	9509.54	0.711864	0.355932	+	0.934512i	$0.384163\pi$
$564$	0	0
$565$	8601.08	0.640443
$566$	1209.41	0.0898149
$567$	0	0
$568$	−440.968	−0.0325750
$569$	16818.3	1.23912	0.619561	−	0.784948i	$-0.287311\pi$
$569$	16818.3	1.23912	0.619561	+	0.784948i	$0.287311\pi$
$570$	0	0
$571$	−5473.88	−0.401182	−0.200591	−	0.979675i	$-0.564286\pi$
$571$	−5473.88	−0.401182	−0.200591	+	0.979675i	$0.564286\pi$
$572$	253.036	0.0184965
$573$	0	0
$574$	−785.185	−0.0570958
$575$	10855.1	0.787286
$576$	0	0
$577$	20076.7	1.44854	0.724268	−	0.689519i	$-0.242178\pi$
$577$	20076.7	1.44854	0.724268	+	0.689519i	$0.242178\pi$
$578$	−412.792	−0.0297057
$579$	0	0
$580$	−8146.24	−0.583197
$581$	−9849.05	−0.703283
$582$	0	0
$583$	−7805.78	−0.554516
$584$	−2027.13	−0.143636
$585$	0	0
$586$	999.271	0.0704428
$587$	11497.5	0.808434	0.404217	−	0.914663i	$-0.367544\pi$
$587$	11497.5	0.808434	0.404217	+	0.914663i	$0.367544\pi$
$588$	0	0
$589$	3116.15	0.217994
$590$	668.822	0.0466694
$591$	0	0
$592$	−16748.1	−1.16274
$593$	10976.4	0.760112	0.380056	−	0.924963i	$-0.375905\pi$
$593$	10976.4	0.760112	0.380056	+	0.924963i	$0.375905\pi$
$594$	0	0
$595$	2483.48	0.171114
$596$	537.210	0.0369211
$597$	0	0
$598$	−110.797	−0.00757663
$599$	−16070.8	−1.09622	−0.548108	−	0.836408i	$-0.684652\pi$
$599$	−16070.8	−1.09622	−0.548108	+	0.836408i	$0.684652\pi$
$600$	0	0
$601$	21112.5	1.43294	0.716469	−	0.697619i	$-0.245757\pi$
$601$	21112.5	1.43294	0.716469	+	0.697619i	$0.245757\pi$
$602$	−1206.98	−0.0817158
$603$	0	0
$604$	−24719.1	−1.66524
$605$	714.888	0.0480402
$606$	0	0
$607$	5734.02	0.383421	0.191711	−	0.981452i	$-0.438597\pi$
$607$	5734.02	0.383421	0.191711	+	0.981452i	$0.438597\pi$
$608$	4167.75	0.278001
$609$	0	0
$610$	−1072.23	−0.0711693
$611$	−1203.76	−0.0797034
$612$	0	0
$613$	−6140.30	−0.404575	−0.202288	−	0.979326i	$-0.564838\pi$
$613$	−6140.30	−0.404575	−0.202288	+	0.979326i	$0.564838\pi$
$614$	548.845	0.0360742
$615$	0	0
$616$	−386.664	−0.0252908
$617$	−11576.5	−0.755355	−0.377678	−	0.925937i	$-0.623277\pi$
$617$	−11576.5	−0.755355	−0.377678	+	0.925937i	$0.623277\pi$
$618$	0	0
$619$	−1734.81	−0.112646	−0.0563229	−	0.998413i	$-0.517938\pi$
$619$	−1734.81	−0.112646	−0.0563229	+	0.998413i	$0.517938\pi$
$620$	−2081.10	−0.134805
$621$	0	0
$622$	−174.107	−0.0112236
$623$	−8751.34	−0.562785
$624$	0	0
$625$	3753.55	0.240227
$626$	2587.61	0.165210
$627$	0	0
$628$	−809.794	−0.0514559
$629$	16322.3	1.03468
$630$	0	0
$631$	−4137.48	−0.261031	−0.130516	−	0.991446i	$-0.541663\pi$
$631$	−4137.48	−0.261031	−0.130516	+	0.991446i	$0.541663\pi$
$632$	980.241	0.0616960
$633$	0	0
$634$	−1850.36	−0.115910
$635$	6987.55	0.436681
$636$	0	0
$637$	142.674	0.00887433
$638$	−606.310	−0.0376239
$639$	0	0
$640$	−3703.17	−0.228720
$641$	−21993.0	−1.35518	−0.677590	−	0.735440i	$-0.736976\pi$
$641$	−21993.0	−1.35518	−0.677590	+	0.735440i	$0.736976\pi$
$642$	0	0
$643$	8688.04	0.532850	0.266425	−	0.963856i	$-0.414157\pi$
$643$	8688.04	0.532850	0.266425	+	0.963856i	$0.414157\pi$
$644$	−6663.15	−0.407710
$645$	0	0
$646$	−1325.46	−0.0807270
$647$	11937.5	0.725365	0.362682	−	0.931913i	$-0.381861\pi$
$647$	11937.5	0.725365	0.362682	+	0.931913i	$0.381861\pi$
$648$	0	0
$649$	−3942.87	−0.238476
$650$	−82.8478	−0.00499932
$651$	0	0
$652$	25841.4	1.55219
$653$	−20662.1	−1.23824	−0.619120	−	0.785296i	$-0.712511\pi$
$653$	−20662.1	−1.23824	−0.619120	+	0.785296i	$0.712511\pi$
$654$	0	0
$655$	11027.1	0.657811
$656$	21884.2	1.30249
$657$	0	0
$658$	913.959	0.0541487
$659$	6620.23	0.391331	0.195666	−	0.980671i	$-0.437313\pi$
$659$	6620.23	0.391331	0.195666	+	0.980671i	$0.437313\pi$
$660$	0	0
$661$	−26958.9	−1.58635	−0.793176	−	0.608993i	$-0.791574\pi$
$661$	−26958.9	−1.58635	−0.793176	+	0.608993i	$0.791574\pi$
$662$	616.496	0.0361945
$663$	0	0
$664$	−7065.44	−0.412940
$665$	−2890.48	−0.168553
$666$	0	0
$667$	−21028.3	−1.22072
$668$	−8047.81	−0.466136
$669$	0	0
$670$	853.135	0.0491932
$671$	6321.05	0.363668
$672$	0	0
$673$	−1054.58	−0.0604029	−0.0302014	−	0.999544i	$-0.509615\pi$
$673$	−1054.58	−0.0604029	−0.0302014	+	0.999544i	$0.509615\pi$
$674$	−2574.52	−0.147132
$675$	0	0
$676$	17289.9	0.983721
$677$	15114.1	0.858024	0.429012	−	0.903299i	$-0.358862\pi$
$677$	15114.1	0.858024	0.429012	+	0.903299i	$0.358862\pi$
$678$	0	0
$679$	−12413.2	−0.701585
$680$	1781.58	0.100471
$681$	0	0
$682$	−154.892	−0.00869668
$683$	−3670.31	−0.205623	−0.102812	−	0.994701i	$-0.532784\pi$
$683$	−3670.31	−0.205623	−0.102812	+	0.994701i	$0.532784\pi$
$684$	0	0
$685$	17568.2	0.979921
$686$	−108.326	−0.00602902
$687$	0	0
$688$	33640.2	1.86413
$689$	2066.20	0.114247
$690$	0	0
$691$	−13438.6	−0.739836	−0.369918	−	0.929064i	$-0.620614\pi$
$691$	−13438.6	−0.739836	−0.369918	+	0.929064i	$0.620614\pi$
$692$	27732.6	1.52346
$693$	0	0
$694$	−24.8132	−0.00135720
$695$	−17157.9	−0.936457
$696$	0	0
$697$	−21327.8	−1.15903
$698$	−297.665	−0.0161415
$699$	0	0
$700$	−4982.34	−0.269021
$701$	−7018.53	−0.378154	−0.189077	−	0.981962i	$-0.560550\pi$
$701$	−7018.53	−0.378154	−0.189077	+	0.981962i	$0.560550\pi$
$702$	0	0
$703$	−18997.2	−1.01919
$704$	5215.06	0.279190
$705$	0	0
$706$	−227.425	−0.0121236
$707$	11569.2	0.615422
$708$	0	0
$709$	19329.5	1.02389	0.511944	−	0.859019i	$-0.328926\pi$
$709$	19329.5	1.02389	0.511944	+	0.859019i	$0.328926\pi$
$710$	163.854	0.00866101
$711$	0	0
$712$	−6277.97	−0.330445
$713$	−5372.04	−0.282166
$714$	0	0
$715$	−189.232	−0.00989771
$716$	2443.52	0.127540
$717$	0	0
$718$	−1646.67	−0.0855896
$719$	26523.0	1.37572	0.687859	−	0.725845i	$-0.258551\pi$
$719$	26523.0	1.37572	0.687859	+	0.725845i	$0.258551\pi$
$720$	0	0
$721$	5958.80	0.307791
$722$	−623.520	−0.0321399
$723$	0	0
$724$	15312.3	0.786021
$725$	−15723.8	−0.805472
$726$	0	0
$727$	−3371.33	−0.171989	−0.0859944	−	0.996296i	$-0.527407\pi$
$727$	−3371.33	−0.171989	−0.0859944	+	0.996296i	$0.527407\pi$
$728$	102.350	0.00521065
$729$	0	0
$730$	753.236	0.0381897
$731$	−32784.9	−1.65881
$732$	0	0
$733$	3582.65	0.180530	0.0902649	−	0.995918i	$-0.471229\pi$
$733$	3582.65	0.180530	0.0902649	+	0.995918i	$0.471229\pi$
$734$	−1975.94	−0.0993644
$735$	0	0
$736$	−7184.94	−0.359837
$737$	−5029.44	−0.251373
$738$	0	0
$739$	9324.88	0.464169	0.232085	−	0.972696i	$-0.425445\pi$
$739$	9324.88	0.464169	0.232085	+	0.972696i	$0.425445\pi$
$740$	12687.2	0.630256
$741$	0	0
$742$	−1568.77	−0.0776166
$743$	25589.5	1.26351	0.631754	−	0.775169i	$-0.282335\pi$
$743$	25589.5	1.26351	0.631754	+	0.775169i	$0.282335\pi$
$744$	0	0
$745$	−401.749	−0.0197570
$746$	−759.928	−0.0372962
$747$	0	0
$748$	−5218.48	−0.255089
$749$	−5755.64	−0.280783
$750$	0	0
$751$	−410.886	−0.0199647	−0.00998233	−	0.999950i	$-0.503178\pi$
$751$	−410.886	−0.0199647	−0.00998233	+	0.999950i	$0.503178\pi$
$752$	−25473.3	−1.23526
$753$	0	0
$754$	160.491	0.00775164
$755$	18486.0	0.891094
$756$	0	0
$757$	19538.4	0.938093	0.469047	−	0.883173i	$-0.344598\pi$
$757$	19538.4	0.938093	0.469047	+	0.883173i	$0.344598\pi$
$758$	−582.304	−0.0279027
$759$	0	0
$760$	−2073.55	−0.0989678
$761$	29347.7	1.39797	0.698984	−	0.715137i	$-0.253636\pi$
$761$	29347.7	1.39797	0.698984	+	0.715137i	$0.253636\pi$
$762$	0	0
$763$	−2174.95	−0.103196
$764$	3454.12	0.163568
$765$	0	0
$766$	2673.94	0.126127
$767$	1043.68	0.0491332
$768$	0	0
$769$	−36589.3	−1.71579	−0.857897	−	0.513822i	$-0.828229\pi$
$769$	−36589.3	−1.71579	−0.857897	+	0.513822i	$0.828229\pi$
$770$	143.675	0.00672428
$771$	0	0
$772$	−138.903	−0.00647571
$773$	13663.7	0.635770	0.317885	−	0.948129i	$-0.397027\pi$
$773$	13663.7	0.635770	0.317885	+	0.948129i	$0.397027\pi$
$774$	0	0
$775$	−4016.92	−0.186183
$776$	−8904.91	−0.411943
$777$	0	0
$778$	−2769.45	−0.127621
$779$	24823.0	1.14169
$780$	0	0
$781$	−965.956	−0.0442569
$782$	2285.01	0.104491
$783$	0	0
$784$	3019.19	0.137536
$785$	605.600	0.0275348
$786$	0	0
$787$	4847.67	0.219569	0.109784	−	0.993955i	$-0.464984\pi$
$787$	4847.67	0.219569	0.109784	+	0.993955i	$0.464984\pi$
$788$	−32145.3	−1.45321
$789$	0	0
$790$	−364.235	−0.0164036
$791$	−10190.6	−0.458072
$792$	0	0
$793$	−1673.19	−0.0749265
$794$	2509.70	0.112174
$795$	0	0
$796$	−30284.4	−1.34849
$797$	−12263.0	−0.545015	−0.272507	−	0.962154i	$-0.587853\pi$
$797$	−12263.0	−0.545015	−0.272507	+	0.962154i	$0.587853\pi$
$798$	0	0
$799$	24825.6	1.09921
$800$	−5372.50	−0.237433
$801$	0	0
$802$	−2473.62	−0.108911
$803$	−4440.51	−0.195146
$804$	0	0
$805$	4983.00	0.218171
$806$	41.0002	0.00179177
$807$	0	0
$808$	8299.40	0.361351
$809$	−12722.6	−0.552909	−0.276454	−	0.961027i	$-0.589159\pi$
$809$	−12722.6	−0.552909	−0.276454	+	0.961027i	$0.589159\pi$
$810$	0	0
$811$	−29124.5	−1.26104	−0.630518	−	0.776175i	$-0.717157\pi$
$811$	−29124.5	−1.26104	−0.630518	+	0.776175i	$0.717157\pi$
$812$	9651.68	0.417127
$813$	0	0
$814$	944.282	0.0406598
$815$	−19325.4	−0.830598
$816$	0	0
$817$	38157.8	1.63399
$818$	2322.41	0.0992678
$819$	0	0
$820$	−16577.9	−0.706007
$821$	−34423.9	−1.46334	−0.731671	−	0.681658i	$-0.761259\pi$
$821$	−34423.9	−1.46334	−0.731671	+	0.681658i	$0.761259\pi$
$822$	0	0
$823$	36006.1	1.52502	0.762511	−	0.646976i	$-0.223967\pi$
$823$	36006.1	1.52502	0.762511	+	0.646976i	$0.223967\pi$
$824$	4274.67	0.180722
$825$	0	0
$826$	−792.421	−0.0333800
$827$	10870.9	0.457095	0.228547	−	0.973533i	$-0.426602\pi$
$827$	10870.9	0.457095	0.228547	+	0.973533i	$0.426602\pi$
$828$	0	0
$829$	11156.1	0.467389	0.233695	−	0.972310i	$-0.424918\pi$
$829$	11156.1	0.467389	0.233695	+	0.972310i	$0.424918\pi$
$830$	2625.35	0.109792
$831$	0	0
$832$	−1380.43	−0.0575215
$833$	−2942.43	−0.122388
$834$	0	0
$835$	6018.51	0.249436
$836$	6073.70	0.251272
$837$	0	0
$838$	799.681	0.0329648
$839$	−608.545	−0.0250409	−0.0125204	−	0.999922i	$-0.503985\pi$
$839$	−608.545	−0.0250409	−0.0125204	+	0.999922i	$0.503985\pi$
$840$	0	0
$841$	6070.79	0.248915
$842$	−1051.93	−0.0430545
$843$	0	0
$844$	452.296	0.0184463
$845$	−12930.1	−0.526403
$846$	0	0
$847$	−847.000	−0.0343604
$848$	43723.9	1.77062
$849$	0	0
$850$	1708.61	0.0689467
$851$	32750.0	1.31922
$852$	0	0
$853$	−8347.05	−0.335050	−0.167525	−	0.985868i	$-0.553577\pi$
$853$	−8347.05	−0.335050	−0.167525	+	0.985868i	$0.553577\pi$
$854$	1270.38	0.0509033
$855$	0	0
$856$	−4128.93	−0.164865
$857$	−11283.0	−0.449733	−0.224866	−	0.974390i	$-0.572195\pi$
$857$	−11283.0	−0.449733	−0.224866	+	0.974390i	$0.572195\pi$
$858$	0	0
$859$	47077.4	1.86992	0.934960	−	0.354752i	$-0.115435\pi$
$859$	47077.4	1.86992	0.934960	+	0.354752i	$0.115435\pi$
$860$	−25483.4	−1.01044
$861$	0	0
$862$	−984.070	−0.0388835
$863$	18151.7	0.715979	0.357989	−	0.933726i	$-0.383462\pi$
$863$	18151.7	0.715979	0.357989	+	0.933726i	$0.383462\pi$
$864$	0	0
$865$	−20739.7	−0.815226
$866$	−4033.55	−0.158274
$867$	0	0
$868$	2465.69	0.0964181
$869$	2147.25	0.0838210
$870$	0	0
$871$	1331.30	0.0517902
$872$	−1560.25	−0.0605924
$873$	0	0
$874$	−2659.49	−0.102927
$875$	8895.66	0.343689
$876$	0	0
$877$	−29337.8	−1.12961	−0.564804	−	0.825225i	$-0.691048\pi$
$877$	−29337.8	−1.12961	−0.564804	+	0.825225i	$0.691048\pi$
$878$	−4836.41	−0.185901
$879$	0	0
$880$	−4004.42	−0.153397
$881$	−13439.4	−0.513943	−0.256971	−	0.966419i	$-0.582725\pi$
$881$	−13439.4	−0.513943	−0.256971	+	0.966419i	$0.582725\pi$
$882$	0	0
$883$	20543.7	0.782957	0.391478	−	0.920187i	$-0.371964\pi$
$883$	20543.7	0.782957	0.391478	+	0.920187i	$0.371964\pi$
$884$	1381.34	0.0525559
$885$	0	0
$886$	603.599	0.0228875
$887$	8340.53	0.315724	0.157862	−	0.987461i	$-0.449540\pi$
$887$	8340.53	0.315724	0.157862	+	0.987461i	$0.449540\pi$
$888$	0	0
$889$	−8278.85	−0.312333
$890$	2332.75	0.0878582
$891$	0	0
$892$	6289.93	0.236101
$893$	−28894.1	−1.08276
$894$	0	0
$895$	−1827.38	−0.0682486
$896$	4387.52	0.163590
$897$	0	0
$898$	5051.27	0.187709
$899$	7781.48	0.288684
$900$	0	0
$901$	−42612.2	−1.57560
$902$	−1233.86	−0.0455467
$903$	0	0
$904$	−7310.43	−0.268962
$905$	−11451.3	−0.420611
$906$	0	0
$907$	20125.1	0.736763	0.368381	−	0.929675i	$-0.379912\pi$
$907$	20125.1	0.736763	0.368381	+	0.929675i	$0.379912\pi$
$908$	18764.9	0.685831
$909$	0	0
$910$	−38.0310	−0.00138540
$911$	−41255.4	−1.50039	−0.750193	−	0.661219i	$-0.770039\pi$
$911$	−41255.4	−1.50039	−0.750193	+	0.661219i	$0.770039\pi$
$912$	0	0
$913$	−15477.1	−0.561026
$914$	−860.373	−0.0311363
$915$	0	0
$916$	31641.7	1.14134
$917$	−13065.0	−0.470494
$918$	0	0
$919$	1034.04	0.0371164	0.0185582	−	0.999828i	$-0.494092\pi$
$919$	1034.04	0.0371164	0.0185582	+	0.999828i	$0.494092\pi$
$920$	3574.67	0.128101
$921$	0	0
$922$	−611.238	−0.0218330
$923$	255.690	0.00911824
$924$	0	0
$925$	24488.6	0.870466
$926$	5137.15	0.182308
$927$	0	0
$928$	10407.5	0.368149
$929$	26854.6	0.948408	0.474204	−	0.880415i	$-0.342736\pi$
$929$	26854.6	0.948408	0.474204	+	0.880415i	$0.342736\pi$
$930$	0	0
$931$	3424.64	0.120557
$932$	33325.6	1.17126
$933$	0	0
$934$	4728.29	0.165647
$935$	3902.61	0.136502
$936$	0	0
$937$	−2310.22	−0.0805461	−0.0402730	−	0.999189i	$-0.512823\pi$
$937$	−2310.22	−0.0805461	−0.0402730	+	0.999189i	$0.512823\pi$
$938$	−1010.80	−0.0351851
$939$	0	0
$940$	19296.7	0.669564
$941$	−52782.7	−1.82855	−0.914276	−	0.405093i	$-0.867239\pi$
$941$	−52782.7	−1.82855	−0.914276	+	0.405093i	$0.867239\pi$
$942$	0	0
$943$	−42793.3	−1.47778
$944$	22085.8	0.761476
$945$	0	0
$946$	−1896.68	−0.0651866
$947$	−56229.8	−1.92949	−0.964743	−	0.263195i	$-0.915224\pi$
$947$	−56229.8	−1.92949	−0.964743	+	0.263195i	$0.915224\pi$
$948$	0	0
$949$	1175.41	0.0402058
$950$	−1988.62	−0.0679151
$951$	0	0
$952$	−2110.82	−0.0718613
$953$	−15410.4	−0.523812	−0.261906	−	0.965093i	$-0.584351\pi$
$953$	−15410.4	−0.523812	−0.261906	+	0.965093i	$0.584351\pi$
$954$	0	0
$955$	−2583.14	−0.0875273
$956$	−52351.4	−1.77109
$957$	0	0
$958$	3602.72	0.121502
$959$	−20814.8	−0.700881
$960$	0	0
$961$	−27803.1	−0.933271
$962$	−249.953	−0.00837712
$963$	0	0
$964$	47721.4	1.59440
$965$	103.878	0.00346524
$966$	0	0
$967$	−35125.4	−1.16811	−0.584053	−	0.811716i	$-0.698534\pi$
$967$	−35125.4	−1.16811	−0.584053	+	0.811716i	$0.698534\pi$
$968$	−607.614	−0.0201751
$969$	0	0
$970$	3308.86	0.109527
$971$	−23595.7	−0.779836	−0.389918	−	0.920850i	$-0.627497\pi$
$971$	−23595.7	−0.779836	−0.389918	+	0.920850i	$0.627497\pi$
$972$	0	0
$973$	20328.7	0.669794
$974$	4940.82	0.162540
$975$	0	0
$976$	−35407.2	−1.16122
$977$	−17361.9	−0.568533	−0.284267	−	0.958745i	$-0.591750\pi$
$977$	−17361.9	−0.568533	−0.284267	+	0.958745i	$0.591750\pi$
$978$	0	0
$979$	−13752.1	−0.448947
$980$	−2287.13	−0.0745506
$981$	0	0
$982$	2082.60	0.0676765
$983$	−55493.4	−1.80057	−0.900287	−	0.435296i	$-0.856644\pi$
$983$	−55493.4	−1.80057	−0.900287	+	0.435296i	$0.856644\pi$
$984$	0	0
$985$	24039.7	0.777633
$986$	−3309.88	−0.106905
$987$	0	0
$988$	−1607.72	−0.0517695
$989$	−65781.6	−2.11500
$990$	0	0
$991$	−12348.7	−0.395831	−0.197915	−	0.980219i	$-0.563417\pi$
$991$	−12348.7	−0.395831	−0.197915	+	0.980219i	$0.563417\pi$
$992$	2658.77	0.0850969
$993$	0	0
$994$	−194.134	−0.00619472
$995$	22648.0	0.721598
$996$	0	0
$997$	−30574.2	−0.971208	−0.485604	−	0.874179i	$-0.661400\pi$
$997$	−30574.2	−0.971208	−0.485604	+	0.874179i	$0.661400\pi$
$998$	1734.63	0.0550187
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.t.1.4	yes	8
3.2	odd	2		693.4.a.s.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.4.a.s.1.5	✓	8	3.2	odd	2
693.4.a.t.1.4	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)

\(f(q)\)	\(=\)	\(q+0.315819 q^{2} -7.90026 q^{4} +5.90817 q^{5} -7.00000 q^{7} -5.02161 q^{8} +O(q^{10})\)	\(q+0.315819 q^{2} -7.90026 q^{4} +5.90817 q^{5} -7.00000 q^{7} -5.02161 q^{8} +1.86591 q^{10} -11.0000 q^{11} +2.91171 q^{13} -2.21073 q^{14} +61.6161 q^{16} -60.0496 q^{17} +69.8907 q^{19} -46.6760 q^{20} -3.47401 q^{22} -120.487 q^{23} -90.0936 q^{25} +0.919575 q^{26} +55.3018 q^{28} +174.527 q^{29} +44.5860 q^{31} +59.6324 q^{32} -18.9648 q^{34} -41.3572 q^{35} -271.813 q^{37} +22.0728 q^{38} -29.6685 q^{40} +355.169 q^{41} +545.964 q^{43} +86.9028 q^{44} -38.0521 q^{46} -413.419 q^{47} +49.0000 q^{49} -28.4533 q^{50} -23.0033 q^{52} +709.617 q^{53} -64.9898 q^{55} +35.1512 q^{56} +55.1191 q^{58} +358.442 q^{59} -574.641 q^{61} +14.0811 q^{62} -474.096 q^{64} +17.2029 q^{65} +457.221 q^{67} +474.407 q^{68} -13.0614 q^{70} +87.8142 q^{71} +403.682 q^{73} -85.8438 q^{74} -552.154 q^{76} +77.0000 q^{77} -195.205 q^{79} +364.038 q^{80} +112.169 q^{82} +1407.01 q^{83} -354.783 q^{85} +172.426 q^{86} +55.2377 q^{88} +1250.19 q^{89} -20.3820 q^{91} +951.879 q^{92} -130.566 q^{94} +412.926 q^{95} +1773.32 q^{97} +15.4751 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} + 30 q^{4} + 10 q^{5} - 56 q^{7} + 15 q^{8}+O(q^{10}) \) 8 * q + 2 * q^2 + 30 * q^4 + 10 * q^5 - 56 * q^7 + 15 * q^8	\( 8 q + 2 q^{2} + 30 q^{4} + 10 q^{5} - 56 q^{7} + 15 q^{8} - 13 q^{10} - 88 q^{11} - 148 q^{13} - 14 q^{14} + 266 q^{16} + 114 q^{17} + 58 q^{19} + 291 q^{20} - 22 q^{22} + 246 q^{23} + 244 q^{25} + 305 q^{26} - 210 q^{28} - 72 q^{29} + 252 q^{31} + 1272 q^{32} + 630 q^{34} - 70 q^{35} - 80 q^{37} + 1885 q^{38} - 342 q^{40} + 682 q^{41} - 106 q^{43} - 330 q^{44} + 120 q^{46} + 828 q^{47} + 392 q^{49} + 801 q^{50} - 1681 q^{52} + 462 q^{53} - 110 q^{55} - 105 q^{56} - 1087 q^{58} + 626 q^{59} - 854 q^{61} + 1350 q^{62} + 2997 q^{64} + 22 q^{65} + 130 q^{67} + 2202 q^{68} + 91 q^{70} + 326 q^{71} - 390 q^{73} - 359 q^{74} + 2041 q^{76} + 616 q^{77} - 508 q^{79} + 4391 q^{80} + 1528 q^{82} + 1596 q^{83} - 880 q^{85} - 414 q^{86} - 165 q^{88} + 4324 q^{89} + 1036 q^{91} + 2092 q^{92} - 1685 q^{94} + 1076 q^{95} - 964 q^{97} + 98 q^{98}+O(q^{100}) \) 8 * q + 2 * q^2 + 30 * q^4 + 10 * q^5 - 56 * q^7 + 15 * q^8 - 13 * q^10 - 88 * q^11 - 148 * q^13 - 14 * q^14 + 266 * q^16 + 114 * q^17 + 58 * q^19 + 291 * q^20 - 22 * q^22 + 246 * q^23 + 244 * q^25 + 305 * q^26 - 210 * q^28 - 72 * q^29 + 252 * q^31 + 1272 * q^32 + 630 * q^34 - 70 * q^35 - 80 * q^37 + 1885 * q^38 - 342 * q^40 + 682 * q^41 - 106 * q^43 - 330 * q^44 + 120 * q^46 + 828 * q^47 + 392 * q^49 + 801 * q^50 - 1681 * q^52 + 462 * q^53 - 110 * q^55 - 105 * q^56 - 1087 * q^58 + 626 * q^59 - 854 * q^61 + 1350 * q^62 + 2997 * q^64 + 22 * q^65 + 130 * q^67 + 2202 * q^68 + 91 * q^70 + 326 * q^71 - 390 * q^73 - 359 * q^74 + 2041 * q^76 + 616 * q^77 - 508 * q^79 + 4391 * q^80 + 1528 * q^82 + 1596 * q^83 - 880 * q^85 - 414 * q^86 - 165 * q^88 + 4324 * q^89 + 1036 * q^91 + 2092 * q^92 - 1685 * q^94 + 1076 * q^95 - 964 * q^97 + 98 * q^98

Introduction

L-functions

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