LMFDB - Embedded newform 693.4.a.s.1.5

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} - 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			1.5
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)

$f(q)$	$=$	\(q-0.315819 q^{2} -7.90026 q^{4} -5.90817 q^{5} -7.00000 q^{7} +5.02161 q^{8} +1.86591 q^{10} +11.0000 q^{11} +2.91171 q^{13} +2.21073 q^{14} +61.6161 q^{16} +60.0496 q^{17} +69.8907 q^{19} +46.6760 q^{20} -3.47401 q^{22} +120.487 q^{23} -90.0936 q^{25} -0.919575 q^{26} +55.3018 q^{28} -174.527 q^{29} +44.5860 q^{31} -59.6324 q^{32} -18.9648 q^{34} +41.3572 q^{35} -271.813 q^{37} -22.0728 q^{38} -29.6685 q^{40} -355.169 q^{41} +545.964 q^{43} -86.9028 q^{44} -38.0521 q^{46} +413.419 q^{47} +49.0000 q^{49} +28.4533 q^{50} -23.0033 q^{52} -709.617 q^{53} -64.9898 q^{55} -35.1512 q^{56} +55.1191 q^{58} -358.442 q^{59} -574.641 q^{61} -14.0811 q^{62} -474.096 q^{64} -17.2029 q^{65} +457.221 q^{67} -474.407 q^{68} -13.0614 q^{70} -87.8142 q^{71} +403.682 q^{73} +85.8438 q^{74} -552.154 q^{76} -77.0000 q^{77} -195.205 q^{79} -364.038 q^{80} +112.169 q^{82} -1407.01 q^{83} -354.783 q^{85} -172.426 q^{86} +55.2377 q^{88} -1250.19 q^{89} -20.3820 q^{91} -951.879 q^{92} -130.566 q^{94} -412.926 q^{95} +1773.32 q^{97} -15.4751 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 30 q^{4} - 10 q^{5} - 56 q^{7} - 15 q^{8} - 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}+ \cdots - 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

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	$-0.517780\pi$
$2$	−0.315819	−0.111659	−0.0558295	+	0.998440i	$0.517780\pi$
$3$	0	0
$3$	0	0
$4$	−7.90026	−0.987532
$5$	−5.90817	−0.528442	−0.264221	−	0.964462i	$-0.585115\pi$
$5$	−5.90817	−0.528442	−0.264221	+	0.964462i	$0.585115\pi$
$6$	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	$-0.359092\pi$
$17$	60.0496	0.856715	0.428358	+	0.903609i	$0.359092\pi$
$18$	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	$-0.316090\pi$
$23$	120.487	1.09232	0.546159	+	0.837682i	$0.316090\pi$
$24$	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	$-0.688728\pi$
$29$	−174.527	−1.11755	−0.558774	+	0.829320i	$0.688728\pi$
$30$	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	$-0.736479\pi$
$41$	−355.169	−1.35288	−0.676441	+	0.736497i	$0.736479\pi$
$42$	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	$-0.278302\pi$
$47$	413.419	1.28305	0.641524	+	0.767103i	$0.278302\pi$
$48$	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	$-0.871455\pi$
$53$	−709.617	−1.83912	−0.919560	+	0.392950i	$0.871455\pi$
$54$	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	$-0.629418\pi$
$59$	−358.442	−0.790936	−0.395468	+	0.918480i	$0.629418\pi$
$60$	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	$-0.523382\pi$
$71$	−87.8142	−0.146784	−0.0733918	+	0.997303i	$0.523382\pi$
$72$	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	$-0.880502\pi$
$83$	−1407.01	−1.86071	−0.930356	+	0.366657i	$0.880502\pi$
$84$	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	$-0.767309\pi$
$89$	−1250.19	−1.48899	−0.744494	+	0.667629i	$0.767309\pi$
$90$	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	$-0.197216\pi$
$101$	1652.74	1.62825	0.814127	+	0.580687i	$0.197216\pi$
$102$	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	$-0.621136\pi$
$107$	−822.234	−0.742882	−0.371441	+	0.928457i	$0.621136\pi$
$108$	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	$-0.707216\pi$
$113$	−1455.80	−1.21194	−0.605972	+	0.795486i	$0.707216\pi$
$114$	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	$-0.713844\pi$
$131$	−1866.42	−1.24481	−0.622405	+	0.782695i	$0.713844\pi$
$132$	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	$-0.877773\pi$
$137$	−2973.54	−1.85436	−0.927179	+	0.374619i	$0.877773\pi$
$138$	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	$-0.494049\pi$
$149$	67.9990	0.0373872	0.0186936	+	0.999825i	$0.494049\pi$
$150$	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	$-0.575840\pi$
$167$	−1018.68	−0.472021	−0.236011	+	0.971750i	$0.575840\pi$
$168$	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	$-0.219583\pi$
$173$	3510.34	1.54270	0.771348	+	0.636414i	$0.219583\pi$
$174$	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	$-0.479431\pi$
$179$	309.297	0.129150	0.0645752	+	0.997913i	$0.479431\pi$
$180$	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	$-0.473609\pi$
$191$	437.216	0.165633	0.0828163	+	0.996565i	$0.473609\pi$
$192$	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	$-0.763183\pi$
$197$	−4068.89	−1.47156	−0.735778	+	0.677222i	$0.763183\pi$
$198$	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	$-0.387117\pi$
$227$	2375.23	0.694490	0.347245	+	0.937774i	$0.387117\pi$
$228$	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	$-0.297934\pi$
$233$	4218.29	1.18605	0.593024	+	0.805185i	$0.297934\pi$
$234$	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	$-0.854062\pi$
$239$	−6626.55	−1.79345	−0.896727	+	0.442584i	$0.854062\pi$
$240$	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	$-0.587381\pi$
$251$	−2155.94	−0.542158	−0.271079	+	0.962557i	$0.587381\pi$
$252$	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	$-0.802783\pi$
$257$	−6708.42	−1.62825	−0.814125	+	0.580690i	$0.802783\pi$
$258$	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	$-0.370959\pi$
$263$	3364.19	0.788763	0.394381	+	0.918947i	$0.370959\pi$
$264$	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	$-0.464543\pi$
$269$	980.866	0.222321	0.111161	+	0.993802i	$0.464543\pi$
$270$	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	$-0.192661\pi$
$281$	7747.25	1.64471	0.822353	+	0.568977i	$0.192661\pi$
$282$	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	$-0.602151\pi$
$293$	−3164.06	−0.630875	−0.315437	+	0.948946i	$0.602151\pi$
$294$	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	$-0.483996\pi$
$311$	551.287	0.100517	0.0502583	+	0.998736i	$0.483996\pi$
$312$	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	$-0.326291\pi$
$317$	5858.91	1.03807	0.519037	+	0.854752i	$0.326291\pi$
$318$	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	$-0.498065\pi$
$347$	78.5678	0.0121549	0.00607743	+	0.999982i	$0.498065\pi$
$348$	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	$-0.482711\pi$
$353$	720.110	0.108577	0.0542884	+	0.998525i	$0.482711\pi$
$354$	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	$-0.374800\pi$
$359$	5213.98	0.766527	0.383264	+	0.923639i	$0.374800\pi$
$360$	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	$-0.691042\pi$
$383$	−8466.69	−1.12958	−0.564788	+	0.825236i	$0.691042\pi$
$384$	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	$-0.306370\pi$
$389$	8769.09	1.14296	0.571479	+	0.820617i	$0.306370\pi$
$390$	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	$-0.337838\pi$
$401$	7832.40	0.975390	0.487695	+	0.873014i	$0.337838\pi$
$402$	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	$-0.547159\pi$
$419$	−2532.08	−0.295228	−0.147614	+	0.989045i	$0.547159\pi$
$420$	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	$-0.444293\pi$
$431$	3115.93	0.348234	0.174117	+	0.984725i	$0.444293\pi$
$432$	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	$-0.532680\pi$
$443$	−1911.22	−0.204977	−0.102488	+	0.994734i	$0.532680\pi$
$444$	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	$-0.817767\pi$
$449$	−15994.2	−1.68110	−0.840548	+	0.541737i	$0.817767\pi$
$450$	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	$-0.468830\pi$
$461$	1935.41	0.195533	0.0977666	+	0.995209i	$0.468830\pi$
$462$	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	$-0.766006\pi$
$467$	−14971.5	−1.48351	−0.741754	+	0.670672i	$0.766006\pi$
$468$	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	$-0.683119\pi$
$479$	−11407.5	−1.08815	−0.544075	+	0.839037i	$0.683119\pi$
$480$	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	$-0.598005\pi$
$491$	−6594.27	−0.606100	−0.303050	+	0.952975i	$0.598005\pi$
$492$	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	$-0.638797\pi$
$503$	−9529.33	−0.844715	−0.422358	+	0.906429i	$0.638797\pi$
$504$	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	$-0.826211\pi$
$509$	−19628.2	−1.70924	−0.854622	+	0.519250i	$0.826211\pi$
$510$	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	$-0.809507\pi$
$521$	−19650.6	−1.65242	−0.826209	+	0.563364i	$0.809507\pi$
$522$	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	$-0.727013\pi$
$557$	−17200.9	−1.30849	−0.654244	+	0.756284i	$0.727013\pi$
$558$	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	$-0.615837\pi$
$563$	−9509.54	−0.711864	−0.355932	+	0.934512i	$0.615837\pi$
$564$	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	$-0.712689\pi$
$569$	−16818.3	−1.23912	−0.619561	+	0.784948i	$0.712689\pi$
$570$	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	$-0.632456\pi$
$587$	−11497.5	−0.808434	−0.404217	+	0.914663i	$0.632456\pi$
$588$	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	$-0.624095\pi$
$593$	−10976.4	−0.760112	−0.380056	+	0.924963i	$0.624095\pi$
$594$	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	$-0.315348\pi$
$599$	16070.8	1.09622	0.548108	+	0.836408i	$0.315348\pi$
$600$	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	$-0.376723\pi$
$617$	11576.5	0.755355	0.377678	+	0.925937i	$0.376723\pi$
$618$	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	$-0.263024\pi$
$641$	21993.0	1.35518	0.677590	+	0.735440i	$0.263024\pi$
$642$	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	$-0.618139\pi$
$647$	−11937.5	−0.725365	−0.362682	+	0.931913i	$0.618139\pi$
$648$	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	$-0.287489\pi$
$653$	20662.1	1.23824	0.619120	+	0.785296i	$0.287489\pi$
$654$	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	$-0.562687\pi$
$659$	−6620.23	−0.391331	−0.195666	+	0.980671i	$0.562687\pi$
$660$	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	$-0.641138\pi$
$677$	−15114.1	−0.858024	−0.429012	+	0.903299i	$0.641138\pi$
$678$	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	$-0.467216\pi$
$683$	3670.31	0.205623	0.102812	+	0.994701i	$0.467216\pi$
$684$	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	$-0.439450\pi$
$701$	7018.53	0.378154	0.189077	+	0.981962i	$0.439450\pi$
$702$	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	$-0.741449\pi$
$719$	−26523.0	−1.37572	−0.687859	+	0.725845i	$0.741449\pi$
$720$	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	$-0.717665\pi$
$743$	−25589.5	−1.26351	−0.631754	+	0.775169i	$0.717665\pi$
$744$	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	$-0.746364\pi$
$761$	−29347.7	−1.39797	−0.698984	+	0.715137i	$0.746364\pi$
$762$	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	$-0.602973\pi$
$773$	−13663.7	−0.635770	−0.317885	+	0.948129i	$0.602973\pi$
$774$	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	$-0.412147\pi$
$797$	12263.0	0.545015	0.272507	+	0.962154i	$0.412147\pi$
$798$	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	$-0.410841\pi$
$809$	12722.6	0.552909	0.276454	+	0.961027i	$0.410841\pi$
$810$	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	$-0.238741\pi$
$821$	34423.9	1.46334	0.731671	+	0.681658i	$0.238741\pi$
$822$	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	$-0.573398\pi$
$827$	−10870.9	−0.457095	−0.228547	+	0.973533i	$0.573398\pi$
$828$	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	$-0.496015\pi$
$839$	608.545	0.0250409	0.0125204	+	0.999922i	$0.496015\pi$
$840$	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	$-0.427805\pi$
$857$	11283.0	0.449733	0.224866	+	0.974390i	$0.427805\pi$
$858$	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	$-0.616538\pi$
$863$	−18151.7	−0.715979	−0.357989	+	0.933726i	$0.616538\pi$
$864$	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	$-0.417275\pi$
$881$	13439.4	0.513943	0.256971	+	0.966419i	$0.417275\pi$
$882$	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	$-0.550460\pi$
$887$	−8340.53	−0.315724	−0.157862	+	0.987461i	$0.550460\pi$
$888$	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	$-0.229961\pi$
$911$	41255.4	1.50039	0.750193	+	0.661219i	$0.229961\pi$
$912$	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	$-0.657264\pi$
$929$	−26854.6	−0.948408	−0.474204	+	0.880415i	$0.657264\pi$
$930$	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	$-0.132761\pi$
$941$	52782.7	1.82855	0.914276	+	0.405093i	$0.132761\pi$
$942$	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	$-0.0847762\pi$
$947$	56229.8	1.92949	0.964743	+	0.263195i	$0.0847762\pi$
$948$	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	$-0.415649\pi$
$953$	15410.4	0.523812	0.261906	+	0.965093i	$0.415649\pi$
$954$	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	$-0.372503\pi$
$971$	23595.7	0.779836	0.389918	+	0.920850i	$0.372503\pi$
$972$	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	$-0.408250\pi$
$977$	17361.9	0.568533	0.284267	+	0.958745i	$0.408250\pi$
$978$	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	$-0.143356\pi$
$983$	55493.4	1.80057	0.900287	+	0.435296i	$0.143356\pi$
$984$	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.s.1.5	✓	8
3.2	odd	2		693.4.a.t.1.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.4.a.s.1.5	✓	8	1.1	even	1	trivial
693.4.a.t.1.4	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)

\(f(q)\)	\(=\)	\(q-0.315819 q^{2} -7.90026 q^{4} -5.90817 q^{5} -7.00000 q^{7} +5.02161 q^{8} +1.86591 q^{10} +11.0000 q^{11} +2.91171 q^{13} +2.21073 q^{14} +61.6161 q^{16} +60.0496 q^{17} +69.8907 q^{19} +46.6760 q^{20} -3.47401 q^{22} +120.487 q^{23} -90.0936 q^{25} -0.919575 q^{26} +55.3018 q^{28} -174.527 q^{29} +44.5860 q^{31} -59.6324 q^{32} -18.9648 q^{34} +41.3572 q^{35} -271.813 q^{37} -22.0728 q^{38} -29.6685 q^{40} -355.169 q^{41} +545.964 q^{43} -86.9028 q^{44} -38.0521 q^{46} +413.419 q^{47} +49.0000 q^{49} +28.4533 q^{50} -23.0033 q^{52} -709.617 q^{53} -64.9898 q^{55} -35.1512 q^{56} +55.1191 q^{58} -358.442 q^{59} -574.641 q^{61} -14.0811 q^{62} -474.096 q^{64} -17.2029 q^{65} +457.221 q^{67} -474.407 q^{68} -13.0614 q^{70} -87.8142 q^{71} +403.682 q^{73} +85.8438 q^{74} -552.154 q^{76} -77.0000 q^{77} -195.205 q^{79} -364.038 q^{80} +112.169 q^{82} -1407.01 q^{83} -354.783 q^{85} -172.426 q^{86} +55.2377 q^{88} -1250.19 q^{89} -20.3820 q^{91} -951.879 q^{92} -130.566 q^{94} -412.926 q^{95} +1773.32 q^{97} -15.4751 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 30 q^{4} - 10 q^{5} - 56 q^{7} - 15 q^{8} - 13 q^{10} + 88 q^{11} - 148 q^{13} + 14 q^{14} + 266 q^{16} - 114 q^{17} + 58 q^{19} - 291 q^{20} - 22 q^{22} - 246 q^{23} + 244 q^{25} - 305 q^{26}+ \cdots - 98 q^{98}+O(q^{100}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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