LMFDB - Embedded newform 693.4.a.p.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [693,4,Mod(1,693)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	693.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,1,0,21,-21] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$40.8883236340$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 30x^{3} + 11x^{2} + 185x + 162 $ x^5 - x^4 - 30x^3 + 11x^2 + 185*x + 162
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.28053$ 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)

$f(q)$	$=$	\(q-1.28053 q^{2} -6.36023 q^{4} -16.8824 q^{5} +7.00000 q^{7} +18.3888 q^{8} +21.6185 q^{10} -11.0000 q^{11} -68.0397 q^{13} -8.96374 q^{14} +27.3344 q^{16} -119.045 q^{17} +16.8404 q^{19} +107.376 q^{20} +14.0859 q^{22} -199.722 q^{23} +160.016 q^{25} +87.1272 q^{26} -44.5216 q^{28} -181.053 q^{29} -31.5474 q^{31} -182.113 q^{32} +152.441 q^{34} -118.177 q^{35} +75.9047 q^{37} -21.5647 q^{38} -310.447 q^{40} -408.485 q^{41} -97.8817 q^{43} +69.9626 q^{44} +255.750 q^{46} -41.8437 q^{47} +49.0000 q^{49} -204.906 q^{50} +432.748 q^{52} +563.375 q^{53} +185.707 q^{55} +128.721 q^{56} +231.845 q^{58} +224.425 q^{59} -622.237 q^{61} +40.3975 q^{62} +14.5263 q^{64} +1148.67 q^{65} -280.572 q^{67} +757.155 q^{68} +151.329 q^{70} +807.229 q^{71} +1038.49 q^{73} -97.1985 q^{74} -107.109 q^{76} -77.0000 q^{77} +710.954 q^{79} -461.471 q^{80} +523.079 q^{82} +191.854 q^{83} +2009.77 q^{85} +125.341 q^{86} -202.276 q^{88} -1562.10 q^{89} -476.278 q^{91} +1270.28 q^{92} +53.5822 q^{94} -284.307 q^{95} +816.513 q^{97} -62.7462 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 21 q^{4} - 21 q^{5} + 35 q^{7} + 42 q^{8} - 23 q^{10} - 55 q^{11} + 101 q^{13} + 7 q^{14} - 7 q^{16} + 20 q^{17} + 237 q^{19} - 85 q^{20} - 11 q^{22} + 80 q^{23} + 486 q^{25} - 165 q^{26}+ \cdots + 49 q^{98}+O(q^{100}) $ 5 * q + q^2 + 21 * q^4 - 21 * q^5 + 35 * q^7 + 42 * q^8 - 23 * q^10 - 55 * q^11 + 101 * q^13 + 7 * q^14 - 7 * q^16 + 20 * q^17 + 237 * q^19 - 85 * q^20 - 11 * q^22 + 80 * q^23 + 486 * q^25 - 165 * q^26 + 147 * q^28 + 11 * q^29 + 316 * q^31 - 453 * q^32 + 936 * q^34 - 147 * q^35 + 319 * q^37 - 89 * q^38 + 624 * q^40 - 1190 * q^41 + 88 * q^43 - 231 * q^44 + 1000 * q^46 - 377 * q^47 + 245 * q^49 + 644 * q^50 + 1001 * q^52 + 992 * q^53 + 231 * q^55 + 294 * q^56 + 721 * q^58 - 71 * q^59 - 574 * q^61 - 272 * q^62 - 1380 * q^64 - 589 * q^65 - 527 * q^67 + 2974 * q^68 - 161 * q^70 + 1156 * q^71 + 1061 * q^73 + 1609 * q^74 + 2399 * q^76 - 385 * q^77 + 588 * q^79 + 1643 * q^80 - 2602 * q^82 + 212 * q^83 + 1918 * q^85 + 4760 * q^86 - 462 * q^88 - 1030 * q^89 + 707 * q^91 + 1174 * q^92 - 1799 * q^94 + 3593 * q^95 + 2488 * q^97 + 49 * 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 $

$2$	−1.28053	−0.452737	−0.226369	−	0.974042i	$-0.572685\pi$
$2$	−1.28053	−0.452737	−0.226369	+	0.974042i	$0.572685\pi$
$3$	0	0
$3$	0	0
$4$	−6.36023	−0.795029
$5$	−16.8824	−1.51001	−0.755004	−	0.655720i	$-0.772365\pi$
$5$	−16.8824	−1.51001	−0.755004	+	0.655720i	$0.772365\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	18.3888	0.812676
$9$	0	0
$10$	21.6185	0.683637
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−68.0397	−1.45160	−0.725801	−	0.687905i	$-0.758531\pi$
$13$	−68.0397	−1.45160	−0.725801	+	0.687905i	$0.758531\pi$
$14$	−8.96374	−0.171119
$15$	0	0
$16$	27.3344	0.427100
$17$	−119.045	−1.69839	−0.849197	−	0.528076i	$-0.822913\pi$
$17$	−119.045	−1.69839	−0.849197	+	0.528076i	$0.822913\pi$
$18$	0	0
$19$	16.8404	0.203340	0.101670	−	0.994818i	$-0.467581\pi$
$19$	16.8404	0.203340	0.101670	+	0.994818i	$0.467581\pi$
$20$	107.376	1.20050
$21$	0	0
$22$	14.0859	0.136505
$23$	−199.722	−1.81065	−0.905323	−	0.424724i	$-0.860371\pi$
$23$	−199.722	−1.81065	−0.905323	+	0.424724i	$0.860371\pi$
$24$	0	0
$25$	160.016	1.28013
$26$	87.1272	0.657194
$27$	0	0
$28$	−44.5216	−0.300493
$29$	−181.053	−1.15933	−0.579667	−	0.814853i	$-0.696817\pi$
$29$	−181.053	−1.15933	−0.579667	+	0.814853i	$0.696817\pi$
$30$	0	0
$31$	−31.5474	−0.182776	−0.0913882	−	0.995815i	$-0.529130\pi$
$31$	−31.5474	−0.182776	−0.0913882	+	0.995815i	$0.529130\pi$
$32$	−182.113	−1.00604
$33$	0	0
$34$	152.441	0.768926
$35$	−118.177	−0.570730
$36$	0	0
$37$	75.9047	0.337261	0.168630	−	0.985679i	$-0.446066\pi$
$37$	75.9047	0.337261	0.168630	+	0.985679i	$0.446066\pi$
$38$	−21.5647	−0.0920594
$39$	0	0
$40$	−310.447	−1.22715
$41$	−408.485	−1.55597	−0.777983	−	0.628285i	$-0.783757\pi$
$41$	−408.485	−1.55597	−0.777983	+	0.628285i	$0.783757\pi$
$42$	0	0
$43$	−97.8817	−0.347135	−0.173568	−	0.984822i	$-0.555530\pi$
$43$	−97.8817	−0.347135	−0.173568	+	0.984822i	$0.555530\pi$
$44$	69.9626	0.239710
$45$	0	0
$46$	255.750	0.819747
$47$	−41.8437	−0.129862	−0.0649311	−	0.997890i	$-0.520683\pi$
$47$	−41.8437	−0.129862	−0.0649311	+	0.997890i	$0.520683\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−204.906	−0.579561
$51$	0	0
$52$	432.748	1.15407
$53$	563.375	1.46010	0.730052	−	0.683392i	$-0.239496\pi$
$53$	563.375	1.46010	0.730052	+	0.683392i	$0.239496\pi$
$54$	0	0
$55$	185.707	0.455285
$56$	128.721	0.307163
$57$	0	0
$58$	231.845	0.524874
$59$	224.425	0.495213	0.247607	−	0.968861i	$-0.420356\pi$
$59$	224.425	0.495213	0.247607	+	0.968861i	$0.420356\pi$
$60$	0	0
$61$	−622.237	−1.30605	−0.653027	−	0.757335i	$-0.726501\pi$
$61$	−622.237	−1.30605	−0.653027	+	0.757335i	$0.726501\pi$
$62$	40.3975	0.0827497
$63$	0	0
$64$	14.5263	0.0283716
$65$	1148.67	2.19193
$66$	0	0
$67$	−280.572	−0.511603	−0.255801	−	0.966729i	$-0.582339\pi$
$67$	−280.572	−0.511603	−0.255801	+	0.966729i	$0.582339\pi$
$68$	757.155	1.35027
$69$	0	0
$70$	151.329	0.258390
$71$	807.229	1.34930	0.674651	−	0.738137i	$-0.264294\pi$
$71$	807.229	1.34930	0.674651	+	0.738137i	$0.264294\pi$
$72$	0	0
$73$	1038.49	1.66501	0.832505	−	0.554017i	$-0.186906\pi$
$73$	1038.49	1.66501	0.832505	+	0.554017i	$0.186906\pi$
$74$	−97.1985	−0.152691
$75$	0	0
$76$	−107.109	−0.161661
$77$	−77.0000	−0.113961
$78$	0	0
$79$	710.954	1.01251	0.506257	−	0.862383i	$-0.331029\pi$
$79$	710.954	1.01251	0.506257	+	0.862383i	$0.331029\pi$
$80$	−461.471	−0.644925
$81$	0	0
$82$	523.079	0.704444
$83$	191.854	0.253719	0.126860	−	0.991921i	$-0.459510\pi$
$83$	191.854	0.253719	0.126860	+	0.991921i	$0.459510\pi$
$84$	0	0
$85$	2009.77	2.56459
$86$	125.341	0.157161
$87$	0	0
$88$	−202.276	−0.245031
$89$	−1562.10	−1.86047	−0.930236	−	0.366962i	$-0.880398\pi$
$89$	−1562.10	−1.86047	−0.930236	+	0.366962i	$0.880398\pi$
$90$	0	0
$91$	−476.278	−0.548654
$92$	1270.28	1.43952
$93$	0	0
$94$	53.5822	0.0587935
$95$	−284.307	−0.307045
$96$	0	0
$97$	816.513	0.854684	0.427342	−	0.904090i	$-0.359450\pi$
$97$	816.513	0.854684	0.427342	+	0.904090i	$0.359450\pi$
$98$	−62.7462	−0.0646767
$99$	0	0
$100$	−1017.74	−1.01774
$101$	990.633	0.975957	0.487979	−	0.872856i	$-0.337734\pi$
$101$	990.633	0.975957	0.487979	+	0.872856i	$0.337734\pi$
$102$	0	0
$103$	−1947.55	−1.86308	−0.931542	−	0.363634i	$-0.881536\pi$
$103$	−1947.55	−1.86308	−0.931542	+	0.363634i	$0.881536\pi$
$104$	−1251.17	−1.17968
$105$	0	0
$106$	−721.421	−0.661043
$107$	479.820	0.433513	0.216757	−	0.976226i	$-0.430452\pi$
$107$	479.820	0.433513	0.216757	+	0.976226i	$0.430452\pi$
$108$	0	0
$109$	−474.762	−0.417192	−0.208596	−	0.978002i	$-0.566889\pi$
$109$	−474.762	−0.417192	−0.208596	+	0.978002i	$0.566889\pi$
$110$	−237.803	−0.206124
$111$	0	0
$112$	191.341	0.161429
$113$	−486.828	−0.405283	−0.202641	−	0.979253i	$-0.564953\pi$
$113$	−486.828	−0.405283	−0.202641	+	0.979253i	$0.564953\pi$
$114$	0	0
$115$	3371.78	2.73409
$116$	1151.54	0.921704
$117$	0	0
$118$	−287.383	−0.224201
$119$	−833.316	−0.641933
$120$	0	0
$121$	121.000	0.0909091
$122$	796.795	0.591299
$123$	0	0
$124$	200.648	0.145313
$125$	−591.150	−0.422993
$126$	0	0
$127$	−1060.45	−0.740943	−0.370471	−	0.928844i	$-0.620804\pi$
$127$	−1060.45	−0.740943	−0.370471	+	0.928844i	$0.620804\pi$
$128$	1438.30	0.993196
$129$	0	0
$130$	−1470.92	−0.992369
$131$	−1988.88	−1.32648	−0.663242	−	0.748405i	$-0.730820\pi$
$131$	−1988.88	−1.32648	−0.663242	+	0.748405i	$0.730820\pi$
$132$	0	0
$133$	117.883	0.0768552
$134$	359.283	0.231622
$135$	0	0
$136$	−2189.09	−1.38024
$137$	1270.95	0.792590	0.396295	−	0.918123i	$-0.370296\pi$
$137$	1270.95	0.792590	0.396295	+	0.918123i	$0.370296\pi$
$138$	0	0
$139$	−1086.86	−0.663210	−0.331605	−	0.943418i	$-0.607590\pi$
$139$	−1086.86	−0.663210	−0.331605	+	0.943418i	$0.607590\pi$
$140$	751.632	0.453747
$141$	0	0
$142$	−1033.68	−0.610879
$143$	748.437	0.437674
$144$	0	0
$145$	3056.61	1.75060
$146$	−1329.82	−0.753812
$147$	0	0
$148$	−482.772	−0.268132
$149$	811.871	0.446383	0.223192	−	0.974775i	$-0.428352\pi$
$149$	811.871	0.446383	0.223192	+	0.974775i	$0.428352\pi$
$150$	0	0
$151$	−719.203	−0.387602	−0.193801	−	0.981041i	$-0.562082\pi$
$151$	−719.203	−0.387602	−0.193801	+	0.981041i	$0.562082\pi$
$152$	309.674	0.165249
$153$	0	0
$154$	98.6011	0.0515942
$155$	532.595	0.275994
$156$	0	0
$157$	3411.37	1.73412	0.867061	−	0.498201i	$-0.166006\pi$
$157$	3411.37	1.73412	0.867061	+	0.498201i	$0.166006\pi$
$158$	−910.401	−0.458402
$159$	0	0
$160$	3074.50	1.51913
$161$	−1398.05	−0.684360
$162$	0	0
$163$	−2691.36	−1.29327	−0.646637	−	0.762798i	$-0.723825\pi$
$163$	−2691.36	−1.29327	−0.646637	+	0.762798i	$0.723825\pi$
$164$	2598.06	1.23704
$165$	0	0
$166$	−245.675	−0.114868
$167$	2449.16	1.13486	0.567429	−	0.823422i	$-0.307938\pi$
$167$	2449.16	1.13486	0.567429	+	0.823422i	$0.307938\pi$
$168$	0	0
$169$	2432.40	1.10715
$170$	−2573.58	−1.16108
$171$	0	0
$172$	622.550	0.275983
$173$	−914.239	−0.401782	−0.200891	−	0.979614i	$-0.564384\pi$
$173$	−914.239	−0.401782	−0.200891	+	0.979614i	$0.564384\pi$
$174$	0	0
$175$	1120.11	0.483842
$176$	−300.679	−0.128776
$177$	0	0
$178$	2000.32	0.842305
$179$	3651.00	1.52452	0.762258	−	0.647273i	$-0.224091\pi$
$179$	3651.00	1.52452	0.762258	+	0.647273i	$0.224091\pi$
$180$	0	0
$181$	−193.523	−0.0794722	−0.0397361	−	0.999210i	$-0.512652\pi$
$181$	−193.523	−0.0794722	−0.0397361	+	0.999210i	$0.512652\pi$
$182$	609.890	0.248396
$183$	0	0
$184$	−3672.64	−1.47147
$185$	−1281.45	−0.509267
$186$	0	0
$187$	1309.50	0.512085
$188$	266.135	0.103244
$189$	0	0
$190$	364.064	0.139011
$191$	−3300.22	−1.25024	−0.625119	−	0.780529i	$-0.714950\pi$
$191$	−3300.22	−1.25024	−0.625119	+	0.780529i	$0.714950\pi$
$192$	0	0
$193$	2659.32	0.991824	0.495912	−	0.868373i	$-0.334834\pi$
$193$	2659.32	0.991824	0.495912	+	0.868373i	$0.334834\pi$
$194$	−1045.57	−0.386947
$195$	0	0
$196$	−311.651	−0.113576
$197$	−1318.61	−0.476890	−0.238445	−	0.971156i	$-0.576638\pi$
$197$	−1318.61	−0.476890	−0.238445	+	0.971156i	$0.576638\pi$
$198$	0	0
$199$	−1213.60	−0.432309	−0.216155	−	0.976359i	$-0.569352\pi$
$199$	−1213.60	−0.432309	−0.216155	+	0.976359i	$0.569352\pi$
$200$	2942.49	1.04033
$201$	0	0
$202$	−1268.54	−0.441852
$203$	−1267.37	−0.438187
$204$	0	0
$205$	6896.21	2.34952
$206$	2493.90	0.843487
$207$	0	0
$208$	−1859.83	−0.619979
$209$	−185.244	−0.0613092
$210$	0	0
$211$	−999.365	−0.326062	−0.163031	−	0.986621i	$-0.552127\pi$
$211$	−999.365	−0.326062	−0.163031	+	0.986621i	$0.552127\pi$
$212$	−3583.20	−1.16083
$213$	0	0
$214$	−614.425	−0.196268
$215$	1652.48	0.524177
$216$	0	0
$217$	−220.831	−0.0690830
$218$	607.949	0.188878
$219$	0	0
$220$	−1181.14	−0.361965
$221$	8099.80	2.46539
$222$	0	0
$223$	594.639	0.178565	0.0892825	−	0.996006i	$-0.471543\pi$
$223$	594.639	0.178565	0.0892825	+	0.996006i	$0.471543\pi$
$224$	−1274.79	−0.380248
$225$	0	0
$226$	623.400	0.183487
$227$	−3854.66	−1.12706	−0.563531	−	0.826095i	$-0.690557\pi$
$227$	−3854.66	−1.12706	−0.563531	+	0.826095i	$0.690557\pi$
$228$	0	0
$229$	−1878.04	−0.541942	−0.270971	−	0.962588i	$-0.587345\pi$
$229$	−1878.04	−0.541942	−0.270971	+	0.962588i	$0.587345\pi$
$230$	−4317.68	−1.23782
$231$	0	0
$232$	−3329.34	−0.942164
$233$	2960.57	0.832418	0.416209	−	0.909269i	$-0.363359\pi$
$233$	2960.57	0.832418	0.416209	+	0.909269i	$0.363359\pi$
$234$	0	0
$235$	706.422	0.196093
$236$	−1427.39	−0.393709
$237$	0	0
$238$	1067.09	0.290627
$239$	1857.27	0.502664	0.251332	−	0.967901i	$-0.419131\pi$
$239$	1857.27	0.502664	0.251332	+	0.967901i	$0.419131\pi$
$240$	0	0
$241$	3597.60	0.961583	0.480792	−	0.876835i	$-0.340349\pi$
$241$	3597.60	0.961583	0.480792	+	0.876835i	$0.340349\pi$
$242$	−154.945	−0.0411579
$243$	0	0
$244$	3957.57	1.03835
$245$	−827.238	−0.215716
$246$	0	0
$247$	−1145.82	−0.295168
$248$	−580.117	−0.148538
$249$	0	0
$250$	756.988	0.191504
$251$	−910.320	−0.228920	−0.114460	−	0.993428i	$-0.536514\pi$
$251$	−910.320	−0.228920	−0.114460	+	0.993428i	$0.536514\pi$
$252$	0	0
$253$	2196.94	0.545930
$254$	1357.94	0.335452
$255$	0	0
$256$	−1958.00	−0.478028
$257$	669.516	0.162503	0.0812515	−	0.996694i	$-0.474108\pi$
$257$	669.516	0.162503	0.0812515	+	0.996694i	$0.474108\pi$
$258$	0	0
$259$	531.333	0.127473
$260$	−7305.84	−1.74265
$261$	0	0
$262$	2546.83	0.600549
$263$	−5046.95	−1.18330	−0.591651	−	0.806195i	$-0.701523\pi$
$263$	−5046.95	−1.18330	−0.591651	+	0.806195i	$0.701523\pi$
$264$	0	0
$265$	−9511.13	−2.20477
$266$	−150.953	−0.0347952
$267$	0	0
$268$	1784.51	0.406739
$269$	−7226.13	−1.63786	−0.818931	−	0.573892i	$-0.805433\pi$
$269$	−7226.13	−1.63786	−0.818931	+	0.573892i	$0.805433\pi$
$270$	0	0
$271$	−4741.05	−1.06272	−0.531362	−	0.847145i	$-0.678320\pi$
$271$	−4741.05	−1.06272	−0.531362	+	0.847145i	$0.678320\pi$
$272$	−3254.03	−0.725384
$273$	0	0
$274$	−1627.50	−0.358835
$275$	−1760.17	−0.385972
$276$	0	0
$277$	−7997.89	−1.73482	−0.867412	−	0.497590i	$-0.834218\pi$
$277$	−7997.89	−1.73482	−0.867412	+	0.497590i	$0.834218\pi$
$278$	1391.76	0.300260
$279$	0	0
$280$	−2173.13	−0.463818
$281$	304.490	0.0646418	0.0323209	−	0.999478i	$-0.489710\pi$
$281$	304.490	0.0646418	0.0323209	+	0.999478i	$0.489710\pi$
$282$	0	0
$283$	6994.49	1.46919	0.734593	−	0.678508i	$-0.237373\pi$
$283$	6994.49	1.46919	0.734593	+	0.678508i	$0.237373\pi$
$284$	−5134.16	−1.07273
$285$	0	0
$286$	−958.399	−0.198151
$287$	−2859.39	−0.588100
$288$	0	0
$289$	9258.75	1.88454
$290$	−3914.09	−0.792564
$291$	0	0
$292$	−6605.02	−1.32373
$293$	−3090.16	−0.616141	−0.308070	−	0.951364i	$-0.599683\pi$
$293$	−3090.16	−0.616141	−0.308070	+	0.951364i	$0.599683\pi$
$294$	0	0
$295$	−3788.83	−0.747776
$296$	1395.79	0.274084
$297$	0	0
$298$	−1039.63	−0.202094
$299$	13589.0	2.62834
$300$	0	0
$301$	−685.172	−0.131205
$302$	920.964	0.175482
$303$	0	0
$304$	460.323	0.0868465
$305$	10504.9	1.97215
$306$	0	0
$307$	2816.61	0.523624	0.261812	−	0.965119i	$-0.415680\pi$
$307$	2816.61	0.523624	0.261812	+	0.965119i	$0.415680\pi$
$308$	489.738	0.0906020
$309$	0	0
$310$	−682.006	−0.124953
$311$	1962.61	0.357844	0.178922	−	0.983863i	$-0.442739\pi$
$311$	1962.61	0.357844	0.178922	+	0.983863i	$0.442739\pi$
$312$	0	0
$313$	−9270.94	−1.67420	−0.837100	−	0.547050i	$-0.815751\pi$
$313$	−9270.94	−1.67420	−0.837100	+	0.547050i	$0.815751\pi$
$314$	−4368.38	−0.785102
$315$	0	0
$316$	−4521.83	−0.804978
$317$	2929.73	0.519084	0.259542	−	0.965732i	$-0.416428\pi$
$317$	2929.73	0.519084	0.259542	+	0.965732i	$0.416428\pi$
$318$	0	0
$319$	1991.58	0.349552
$320$	−245.238	−0.0428414
$321$	0	0
$322$	1790.25	0.309835
$323$	−2004.77	−0.345351
$324$	0	0
$325$	−10887.4	−1.85823
$326$	3446.38	0.585513
$327$	0	0
$328$	−7511.53	−1.26450
$329$	−292.906	−0.0490833
$330$	0	0
$331$	−6024.72	−1.00045	−0.500224	−	0.865896i	$-0.666749\pi$
$331$	−6024.72	−1.00045	−0.500224	+	0.865896i	$0.666749\pi$
$332$	−1220.24	−0.201714
$333$	0	0
$334$	−3136.23	−0.513793
$335$	4736.74	0.772524
$336$	0	0
$337$	1851.77	0.299324	0.149662	−	0.988737i	$-0.452181\pi$
$337$	1851.77	0.299324	0.149662	+	0.988737i	$0.452181\pi$
$338$	−3114.77	−0.501247
$339$	0	0
$340$	−12782.6	−2.03892
$341$	347.021	0.0551092
$342$	0	0
$343$	343.000	0.0539949
$344$	−1799.92	−0.282109
$345$	0	0
$346$	1170.71	0.181902
$347$	−3931.79	−0.608269	−0.304134	−	0.952629i	$-0.598367\pi$
$347$	−3931.79	−0.608269	−0.304134	+	0.952629i	$0.598367\pi$
$348$	0	0
$349$	−746.311	−0.114467	−0.0572337	−	0.998361i	$-0.518228\pi$
$349$	−746.311	−0.114467	−0.0572337	+	0.998361i	$0.518228\pi$
$350$	−1434.34	−0.219053
$351$	0	0
$352$	2003.24	0.303333
$353$	8623.77	1.30027	0.650137	−	0.759817i	$-0.274711\pi$
$353$	8623.77	1.30027	0.650137	+	0.759817i	$0.274711\pi$
$354$	0	0
$355$	−13628.0	−2.03746
$356$	9935.30	1.47913
$357$	0	0
$358$	−4675.23	−0.690205
$359$	−4195.46	−0.616791	−0.308395	−	0.951258i	$-0.599792\pi$
$359$	−4195.46	−0.616791	−0.308395	+	0.951258i	$0.599792\pi$
$360$	0	0
$361$	−6575.40	−0.958653
$362$	247.813	0.0359800
$363$	0	0
$364$	3029.24	0.436196
$365$	−17532.2	−2.51418
$366$	0	0
$367$	−11009.4	−1.56590	−0.782948	−	0.622087i	$-0.786285\pi$
$367$	−11009.4	−1.56590	−0.782948	+	0.622087i	$0.786285\pi$
$368$	−5459.28	−0.773328
$369$	0	0
$370$	1640.95	0.230564
$371$	3943.62	0.551867
$372$	0	0
$373$	−647.517	−0.0898852	−0.0449426	−	0.998990i	$-0.514310\pi$
$373$	−647.517	−0.0898852	−0.0449426	+	0.998990i	$0.514310\pi$
$374$	−1676.86	−0.231840
$375$	0	0
$376$	−769.453	−0.105536
$377$	12318.8	1.68289
$378$	0	0
$379$	10934.0	1.48191	0.740955	−	0.671555i	$-0.234373\pi$
$379$	10934.0	1.48191	0.740955	+	0.671555i	$0.234373\pi$
$380$	1808.26	0.244109
$381$	0	0
$382$	4226.04	0.566029
$383$	662.006	0.0883210	0.0441605	−	0.999024i	$-0.485939\pi$
$383$	662.006	0.0883210	0.0441605	+	0.999024i	$0.485939\pi$
$384$	0	0
$385$	1299.95	0.172081
$386$	−3405.35	−0.449036
$387$	0	0
$388$	−5193.21	−0.679499
$389$	−46.1336	−0.00601302	−0.00300651	−	0.999995i	$-0.500957\pi$
$389$	−46.1336	−0.00601302	−0.00300651	+	0.999995i	$0.500957\pi$
$390$	0	0
$391$	23775.9	3.07519
$392$	901.050	0.116097
$393$	0	0
$394$	1688.53	0.215906
$395$	−12002.6	−1.52890
$396$	0	0
$397$	11516.2	1.45587	0.727936	−	0.685645i	$-0.240480\pi$
$397$	11516.2	1.45587	0.727936	+	0.685645i	$0.240480\pi$
$398$	1554.05	0.195723
$399$	0	0
$400$	4373.94	0.546742
$401$	9201.87	1.14593	0.572967	−	0.819578i	$-0.305792\pi$
$401$	9201.87	1.14593	0.572967	+	0.819578i	$0.305792\pi$
$402$	0	0
$403$	2146.47	0.265319
$404$	−6300.66	−0.775914
$405$	0	0
$406$	1622.91	0.198384
$407$	−834.952	−0.101688
$408$	0	0
$409$	5529.75	0.668529	0.334265	−	0.942479i	$-0.391512\pi$
$409$	5529.75	0.668529	0.334265	+	0.942479i	$0.391512\pi$
$410$	−8830.83	−1.06372
$411$	0	0
$412$	12386.9	1.48121
$413$	1570.97	0.187173
$414$	0	0
$415$	−3238.96	−0.383118
$416$	12390.9	1.46037
$417$	0	0
$418$	237.212	0.0277570
$419$	13372.9	1.55921	0.779604	−	0.626273i	$-0.215420\pi$
$419$	13372.9	1.55921	0.779604	+	0.626273i	$0.215420\pi$
$420$	0	0
$421$	−6100.40	−0.706213	−0.353106	−	0.935583i	$-0.614875\pi$
$421$	−6100.40	−0.706213	−0.353106	+	0.935583i	$0.614875\pi$
$422$	1279.72	0.147620
$423$	0	0
$424$	10359.8	1.18659
$425$	−19049.1	−2.17416
$426$	0	0
$427$	−4355.66	−0.493642
$428$	−3051.76	−0.344656
$429$	0	0
$430$	−2116.05	−0.237314
$431$	4465.34	0.499043	0.249522	−	0.968369i	$-0.419727\pi$
$431$	4465.34	0.499043	0.249522	+	0.968369i	$0.419727\pi$
$432$	0	0
$433$	−9331.67	−1.03568	−0.517842	−	0.855476i	$-0.673265\pi$
$433$	−9331.67	−1.03568	−0.517842	+	0.855476i	$0.673265\pi$
$434$	282.782	0.0312765
$435$	0	0
$436$	3019.60	0.331680
$437$	−3363.40	−0.368176
$438$	0	0
$439$	2725.68	0.296332	0.148166	−	0.988963i	$-0.452663\pi$
$439$	2725.68	0.296332	0.148166	+	0.988963i	$0.452663\pi$
$440$	3414.91	0.369999
$441$	0	0
$442$	−10372.1	−1.11617
$443$	14731.7	1.57996	0.789980	−	0.613132i	$-0.210091\pi$
$443$	14731.7	1.57996	0.789980	+	0.613132i	$0.210091\pi$
$444$	0	0
$445$	26372.0	2.80933
$446$	−761.456	−0.0808430
$447$	0	0
$448$	101.684	0.0107235
$449$	1591.42	0.167269	0.0836345	−	0.996497i	$-0.473347\pi$
$449$	1591.42	0.167269	0.0836345	+	0.996497i	$0.473347\pi$
$450$	0	0
$451$	4493.33	0.469141
$452$	3096.34	0.322212
$453$	0	0
$454$	4936.03	0.510263
$455$	8040.72	0.828472
$456$	0	0
$457$	−9627.76	−0.985487	−0.492744	−	0.870175i	$-0.664006\pi$
$457$	−9627.76	−0.985487	−0.492744	+	0.870175i	$0.664006\pi$
$458$	2404.90	0.245357
$459$	0	0
$460$	−21445.3	−2.17368
$461$	−15234.5	−1.53913	−0.769567	−	0.638566i	$-0.779528\pi$
$461$	−15234.5	−1.53913	−0.769567	+	0.638566i	$0.779528\pi$
$462$	0	0
$463$	5534.70	0.555549	0.277775	−	0.960646i	$-0.410403\pi$
$463$	5534.70	0.555549	0.277775	+	0.960646i	$0.410403\pi$
$464$	−4948.98	−0.495152
$465$	0	0
$466$	−3791.11	−0.376866
$467$	2265.02	0.224438	0.112219	−	0.993683i	$-0.464204\pi$
$467$	2265.02	0.224438	0.112219	+	0.993683i	$0.464204\pi$
$468$	0	0
$469$	−1964.01	−0.193368
$470$	−904.597	−0.0887786
$471$	0	0
$472$	4126.89	0.402448
$473$	1076.70	0.104665
$474$	0	0
$475$	2694.73	0.260300
$476$	5300.08	0.510355
$477$	0	0
$478$	−2378.29	−0.227575
$479$	2150.36	0.205120	0.102560	−	0.994727i	$-0.467297\pi$
$479$	2150.36	0.205120	0.102560	+	0.994727i	$0.467297\pi$
$480$	0	0
$481$	−5164.53	−0.489569
$482$	−4606.84	−0.435344
$483$	0	0
$484$	−769.588	−0.0722754
$485$	−13784.7	−1.29058
$486$	0	0
$487$	−19739.0	−1.83667	−0.918336	−	0.395801i	$-0.870467\pi$
$487$	−19739.0	−1.83667	−0.918336	+	0.395801i	$0.870467\pi$
$488$	−11442.2	−1.06140
$489$	0	0
$490$	1059.31	0.0976624
$491$	−18494.8	−1.69992	−0.849958	−	0.526850i	$-0.823373\pi$
$491$	−18494.8	−1.69992	−0.849958	+	0.526850i	$0.823373\pi$
$492$	0	0
$493$	21553.5	1.96901
$494$	1467.26	0.133634
$495$	0	0
$496$	−862.329	−0.0780639
$497$	5650.60	0.509988
$498$	0	0
$499$	1228.36	0.110199	0.0550993	−	0.998481i	$-0.482452\pi$
$499$	1228.36	0.110199	0.0550993	+	0.998481i	$0.482452\pi$
$500$	3759.85	0.336291
$501$	0	0
$502$	1165.70	0.103641
$503$	−17887.6	−1.58562	−0.792812	−	0.609466i	$-0.791384\pi$
$503$	−17887.6	−1.58562	−0.792812	+	0.609466i	$0.791384\pi$
$504$	0	0
$505$	−16724.3	−1.47370
$506$	−2813.26	−0.247163
$507$	0	0
$508$	6744.71	0.589071
$509$	−6755.97	−0.588317	−0.294158	−	0.955757i	$-0.595039\pi$
$509$	−6755.97	−0.588317	−0.294158	+	0.955757i	$0.595039\pi$
$510$	0	0
$511$	7269.41	0.629315
$512$	−8999.12	−0.776775
$513$	0	0
$514$	−857.339	−0.0735712
$515$	32879.3	2.81327
$516$	0	0
$517$	460.280	0.0391549
$518$	−680.390	−0.0577116
$519$	0	0
$520$	21122.7	1.78133
$521$	−10300.4	−0.866158	−0.433079	−	0.901356i	$-0.642573\pi$
$521$	−10300.4	−0.866158	−0.433079	+	0.901356i	$0.642573\pi$
$522$	0	0
$523$	−1205.75	−0.100811	−0.0504053	−	0.998729i	$-0.516051\pi$
$523$	−1205.75	−0.100811	−0.0504053	+	0.998729i	$0.516051\pi$
$524$	12649.7	1.05459
$525$	0	0
$526$	6462.79	0.535724
$527$	3755.56	0.310426
$528$	0	0
$529$	27721.8	2.27844
$530$	12179.3	0.998181
$531$	0	0
$532$	−749.762	−0.0611021
$533$	27793.2	2.25864
$534$	0	0
$535$	−8100.51	−0.654609
$536$	−5159.38	−0.415767
$537$	0	0
$538$	9253.31	0.741521
$539$	−539.000	−0.0430730
$540$	0	0
$541$	3970.16	0.315509	0.157755	−	0.987478i	$-0.449575\pi$
$541$	3970.16	0.315509	0.157755	+	0.987478i	$0.449575\pi$
$542$	6071.08	0.481135
$543$	0	0
$544$	21679.6	1.70865
$545$	8015.13	0.629964
$546$	0	0
$547$	−24218.7	−1.89308	−0.946541	−	0.322584i	$-0.895449\pi$
$547$	−24218.7	−1.89308	−0.946541	+	0.322584i	$0.895449\pi$
$548$	−8083.56	−0.630132
$549$	0	0
$550$	2253.96	0.174744
$551$	−3049.01	−0.235739
$552$	0	0
$553$	4976.68	0.382694
$554$	10241.6	0.785420
$555$	0	0
$556$	6912.68	0.527271
$557$	15753.5	1.19838	0.599190	−	0.800607i	$-0.295489\pi$
$557$	15753.5	1.19838	0.599190	+	0.800607i	$0.295489\pi$
$558$	0	0
$559$	6659.84	0.503902
$560$	−3230.30	−0.243759
$561$	0	0
$562$	−389.910	−0.0292658
$563$	7797.63	0.583714	0.291857	−	0.956462i	$-0.405727\pi$
$563$	7797.63	0.583714	0.291857	+	0.956462i	$0.405727\pi$
$564$	0	0
$565$	8218.83	0.611980
$566$	−8956.69	−0.665155
$567$	0	0
$568$	14843.9	1.09655
$569$	−5074.55	−0.373877	−0.186939	−	0.982372i	$-0.559857\pi$
$569$	−5074.55	−0.373877	−0.186939	+	0.982372i	$0.559857\pi$
$570$	0	0
$571$	19247.4	1.41065	0.705323	−	0.708886i	$-0.250802\pi$
$571$	19247.4	1.41065	0.705323	+	0.708886i	$0.250802\pi$
$572$	−4760.23	−0.347964
$573$	0	0
$574$	3661.55	0.266255
$575$	−31958.6	−2.31786
$576$	0	0
$577$	13981.0	1.00873	0.504364	−	0.863491i	$-0.331727\pi$
$577$	13981.0	1.00873	0.504364	+	0.863491i	$0.331727\pi$
$578$	−11856.1	−0.853202
$579$	0	0
$580$	−19440.8	−1.39178
$581$	1342.98	0.0958969
$582$	0	0
$583$	−6197.12	−0.440238
$584$	19096.5	1.35311
$585$	0	0
$586$	3957.06	0.278950
$587$	864.831	0.0608099	0.0304049	−	0.999538i	$-0.490320\pi$
$587$	864.831	0.0608099	0.0304049	+	0.999538i	$0.490320\pi$
$588$	0	0
$589$	−531.270	−0.0371657
$590$	4851.72	0.338546
$591$	0	0
$592$	2074.81	0.144044
$593$	−3157.59	−0.218662	−0.109331	−	0.994005i	$-0.534871\pi$
$593$	−3157.59	−0.218662	−0.109331	+	0.994005i	$0.534871\pi$
$594$	0	0
$595$	14068.4	0.969324
$596$	−5163.69	−0.354888
$597$	0	0
$598$	−17401.2	−1.18995
$599$	−11270.2	−0.768758	−0.384379	−	0.923175i	$-0.625584\pi$
$599$	−11270.2	−0.768758	−0.384379	+	0.923175i	$0.625584\pi$
$600$	0	0
$601$	−22222.4	−1.50827	−0.754135	−	0.656719i	$-0.771944\pi$
$601$	−22222.4	−1.50827	−0.754135	+	0.656719i	$0.771944\pi$
$602$	877.386	0.0594013
$603$	0	0
$604$	4574.30	0.308155
$605$	−2042.77	−0.137274
$606$	0	0
$607$	1829.17	0.122312	0.0611562	−	0.998128i	$-0.480521\pi$
$607$	1829.17	0.122312	0.0611562	+	0.998128i	$0.480521\pi$
$608$	−3066.85	−0.204568
$609$	0	0
$610$	−13451.8	−0.892866
$611$	2847.03	0.188508
$612$	0	0
$613$	11172.5	0.736136	0.368068	−	0.929799i	$-0.380019\pi$
$613$	11172.5	0.736136	0.368068	+	0.929799i	$0.380019\pi$
$614$	−3606.77	−0.237064
$615$	0	0
$616$	−1415.94	−0.0926131
$617$	14278.2	0.931634	0.465817	−	0.884881i	$-0.345760\pi$
$617$	14278.2	0.931634	0.465817	+	0.884881i	$0.345760\pi$
$618$	0	0
$619$	−12918.1	−0.838809	−0.419405	−	0.907799i	$-0.637761\pi$
$619$	−12918.1	−0.838809	−0.419405	+	0.907799i	$0.637761\pi$
$620$	−3387.43	−0.219423
$621$	0	0
$622$	−2513.19	−0.162009
$623$	−10934.7	−0.703192
$624$	0	0
$625$	−10021.9	−0.641404
$626$	11871.8	0.757973
$627$	0	0
$628$	−21697.1	−1.37868
$629$	−9036.09	−0.572802
$630$	0	0
$631$	−17365.6	−1.09558	−0.547792	−	0.836614i	$-0.684532\pi$
$631$	−17365.6	−1.09558	−0.547792	+	0.836614i	$0.684532\pi$
$632$	13073.6	0.822846
$633$	0	0
$634$	−3751.61	−0.235009
$635$	17902.9	1.11883
$636$	0	0
$637$	−3333.95	−0.207372
$638$	−2550.29	−0.158255
$639$	0	0
$640$	−24282.0	−1.49973
$641$	3234.39	0.199299	0.0996494	−	0.995023i	$-0.468228\pi$
$641$	3234.39	0.199299	0.0996494	+	0.995023i	$0.468228\pi$
$642$	0	0
$643$	−4372.53	−0.268174	−0.134087	−	0.990970i	$-0.542810\pi$
$643$	−4372.53	−0.268174	−0.134087	+	0.990970i	$0.542810\pi$
$644$	8891.94	0.544086
$645$	0	0
$646$	2567.18	0.156353
$647$	−9149.07	−0.555930	−0.277965	−	0.960591i	$-0.589660\pi$
$647$	−9149.07	−0.555930	−0.277965	+	0.960591i	$0.589660\pi$
$648$	0	0
$649$	−2468.67	−0.149312
$650$	13941.7	0.841291
$651$	0	0
$652$	17117.7	1.02819
$653$	33267.8	1.99367	0.996837	−	0.0794734i	$-0.0253239\pi$
$653$	33267.8	1.99367	0.996837	+	0.0794734i	$0.0253239\pi$
$654$	0	0
$655$	33577.1	2.00300
$656$	−11165.7	−0.664554
$657$	0	0
$658$	375.076	0.0222218
$659$	15125.2	0.894073	0.447037	−	0.894516i	$-0.352479\pi$
$659$	15125.2	0.894073	0.447037	+	0.894516i	$0.352479\pi$
$660$	0	0
$661$	12789.3	0.752568	0.376284	−	0.926504i	$-0.377202\pi$
$661$	12789.3	0.752568	0.376284	+	0.926504i	$0.377202\pi$
$662$	7714.86	0.452940
$663$	0	0
$664$	3527.96	0.206192
$665$	−1990.15	−0.116052
$666$	0	0
$667$	36160.2	2.09914
$668$	−15577.2	−0.902246
$669$	0	0
$670$	−6065.55	−0.349750
$671$	6844.60	0.393790
$672$	0	0
$673$	−18730.3	−1.07281	−0.536404	−	0.843962i	$-0.680217\pi$
$673$	−18730.3	−1.07281	−0.536404	+	0.843962i	$0.680217\pi$
$674$	−2371.25	−0.135515
$675$	0	0
$676$	−15470.6	−0.880214
$677$	6160.79	0.349747	0.174873	−	0.984591i	$-0.444048\pi$
$677$	6160.79	0.349747	0.174873	+	0.984591i	$0.444048\pi$
$678$	0	0
$679$	5715.59	0.323040
$680$	36957.2	2.08418
$681$	0	0
$682$	−444.372	−0.0249500
$683$	−3050.21	−0.170883	−0.0854415	−	0.996343i	$-0.527230\pi$
$683$	−3050.21	−0.170883	−0.0854415	+	0.996343i	$0.527230\pi$
$684$	0	0
$685$	−21456.7	−1.19682
$686$	−439.223	−0.0244455
$687$	0	0
$688$	−2675.54	−0.148262
$689$	−38331.9	−2.11949
$690$	0	0
$691$	−12418.3	−0.683668	−0.341834	−	0.939760i	$-0.611048\pi$
$691$	−12418.3	−0.683668	−0.341834	+	0.939760i	$0.611048\pi$
$692$	5814.77	0.319428
$693$	0	0
$694$	5034.79	0.275386
$695$	18348.8	1.00145
$696$	0	0
$697$	48628.1	2.64264
$698$	955.676	0.0518236
$699$	0	0
$700$	−7124.16	−0.384669
$701$	15271.1	0.822796	0.411398	−	0.911456i	$-0.365041\pi$
$701$	15271.1	0.822796	0.411398	+	0.911456i	$0.365041\pi$
$702$	0	0
$703$	1278.27	0.0685785
$704$	−159.789	−0.00855436
$705$	0	0
$706$	−11043.0	−0.588682
$707$	6934.43	0.368877
$708$	0	0
$709$	24174.5	1.28052	0.640262	−	0.768157i	$-0.278826\pi$
$709$	24174.5	1.28052	0.640262	+	0.768157i	$0.278826\pi$
$710$	17451.1	0.922433
$711$	0	0
$712$	−28725.0	−1.51196
$713$	6300.69	0.330944
$714$	0	0
$715$	−12635.4	−0.660892
$716$	−23221.2	−1.21203
$717$	0	0
$718$	5372.43	0.279244
$719$	−21724.7	−1.12683	−0.563416	−	0.826173i	$-0.690513\pi$
$719$	−21724.7	−1.12683	−0.563416	+	0.826173i	$0.690513\pi$
$720$	0	0
$721$	−13632.8	−0.704179
$722$	8420.02	0.434018
$723$	0	0
$724$	1230.85	0.0631827
$725$	−28971.3	−1.48409
$726$	0	0
$727$	−32751.2	−1.67081	−0.835403	−	0.549638i	$-0.814766\pi$
$727$	−32751.2	−1.67081	−0.835403	+	0.549638i	$0.814766\pi$
$728$	−8758.16	−0.445878
$729$	0	0
$730$	22450.5	1.13826
$731$	11652.3	0.589572
$732$	0	0
$733$	−5646.65	−0.284535	−0.142267	−	0.989828i	$-0.545439\pi$
$733$	−5646.65	−0.284535	−0.142267	+	0.989828i	$0.545439\pi$
$734$	14097.9	0.708940
$735$	0	0
$736$	36371.9	1.82158
$737$	3086.30	0.154254
$738$	0	0
$739$	34189.8	1.70188	0.850942	−	0.525260i	$-0.176032\pi$
$739$	34189.8	1.70188	0.850942	+	0.525260i	$0.176032\pi$
$740$	8150.35	0.404882
$741$	0	0
$742$	−5049.95	−0.249851
$743$	15998.8	0.789960	0.394980	−	0.918690i	$-0.370751\pi$
$743$	15998.8	0.789960	0.394980	+	0.918690i	$0.370751\pi$
$744$	0	0
$745$	−13706.3	−0.674042
$746$	829.168	0.0406944
$747$	0	0
$748$	−8328.70	−0.407122
$749$	3358.74	0.163853
$750$	0	0
$751$	−18097.7	−0.879354	−0.439677	−	0.898156i	$-0.644907\pi$
$751$	−18097.7	−0.879354	−0.439677	+	0.898156i	$0.644907\pi$
$752$	−1143.77	−0.0554642
$753$	0	0
$754$	−15774.6	−0.761908
$755$	12141.9	0.585282
$756$	0	0
$757$	8597.91	0.412809	0.206404	−	0.978467i	$-0.433824\pi$
$757$	8597.91	0.412809	0.206404	+	0.978467i	$0.433824\pi$
$758$	−14001.4	−0.670915
$759$	0	0
$760$	−5228.05	−0.249528
$761$	−31479.9	−1.49953	−0.749767	−	0.661702i	$-0.769835\pi$
$761$	−31479.9	−1.49953	−0.749767	+	0.661702i	$0.769835\pi$
$762$	0	0
$763$	−3323.33	−0.157684
$764$	20990.2	0.993976
$765$	0	0
$766$	−847.721	−0.0399862
$767$	−15269.8	−0.718853
$768$	0	0
$769$	31854.0	1.49374	0.746868	−	0.664972i	$-0.231557\pi$
$769$	31854.0	1.49374	0.746868	+	0.664972i	$0.231557\pi$
$770$	−1664.62	−0.0779077
$771$	0	0
$772$	−16913.9	−0.788529
$773$	27.0929	0.00126062	0.000630312	−	1.00000i	$-0.499799\pi$
$773$	27.0929	0.00126062	0.000630312		1.00000i	$0.499799\pi$
$774$	0	0
$775$	−5048.07	−0.233977
$776$	15014.7	0.694582
$777$	0	0
$778$	59.0756	0.00272232
$779$	−6879.05	−0.316390
$780$	0	0
$781$	−8879.52	−0.406830
$782$	−30445.9	−1.39225
$783$	0	0
$784$	1339.39	0.0610143
$785$	−57592.2	−2.61854
$786$	0	0
$787$	−6357.30	−0.287946	−0.143973	−	0.989582i	$-0.545988\pi$
$787$	−6357.30	−0.287946	−0.143973	+	0.989582i	$0.545988\pi$
$788$	8386.69	0.379142
$789$	0	0
$790$	15369.8	0.692192
$791$	−3407.80	−0.153182
$792$	0	0
$793$	42336.8	1.89587
$794$	−14746.9	−0.659127
$795$	0	0
$796$	7718.76	0.343699
$797$	9946.90	0.442079	0.221040	−	0.975265i	$-0.429055\pi$
$797$	9946.90	0.442079	0.221040	+	0.975265i	$0.429055\pi$
$798$	0	0
$799$	4981.28	0.220557
$800$	−29140.9	−1.28786
$801$	0	0
$802$	−11783.3	−0.518807
$803$	−11423.4	−0.502019
$804$	0	0
$805$	23602.5	1.03339
$806$	−2748.63	−0.120120
$807$	0	0
$808$	18216.5	0.793137
$809$	−21882.2	−0.950972	−0.475486	−	0.879723i	$-0.657728\pi$
$809$	−21882.2	−0.950972	−0.475486	+	0.879723i	$0.657728\pi$
$810$	0	0
$811$	−5407.69	−0.234143	−0.117071	−	0.993124i	$-0.537351\pi$
$811$	−5407.69	−0.234143	−0.117071	+	0.993124i	$0.537351\pi$
$812$	8060.77	0.348372
$813$	0	0
$814$	1069.18	0.0460379
$815$	45436.7	1.95285
$816$	0	0
$817$	−1648.37	−0.0705864
$818$	−7081.03	−0.302668
$819$	0	0
$820$	−43861.5	−1.86794
$821$	−5294.69	−0.225074	−0.112537	−	0.993648i	$-0.535898\pi$
$821$	−5294.69	−0.225074	−0.112537	+	0.993648i	$0.535898\pi$
$822$	0	0
$823$	7539.03	0.319312	0.159656	−	0.987173i	$-0.448961\pi$
$823$	7539.03	0.319312	0.159656	+	0.987173i	$0.448961\pi$
$824$	−35813.0	−1.51408
$825$	0	0
$826$	−2011.68	−0.0847402
$827$	18827.2	0.791640	0.395820	−	0.918328i	$-0.370460\pi$
$827$	18827.2	0.791640	0.395820	+	0.918328i	$0.370460\pi$
$828$	0	0
$829$	−5480.45	−0.229607	−0.114803	−	0.993388i	$-0.536624\pi$
$829$	−5480.45	−0.229607	−0.114803	+	0.993388i	$0.536624\pi$
$830$	4147.59	0.173452
$831$	0	0
$832$	−988.363	−0.0411843
$833$	−5833.21	−0.242628
$834$	0	0
$835$	−41347.7	−1.71365
$836$	1178.20	0.0487426
$837$	0	0
$838$	−17124.4	−0.705911
$839$	32885.7	1.35321	0.676603	−	0.736348i	$-0.263451\pi$
$839$	32885.7	1.35321	0.676603	+	0.736348i	$0.263451\pi$
$840$	0	0
$841$	8391.18	0.344056
$842$	7811.78	0.319729
$843$	0	0
$844$	6356.19	0.259229
$845$	−41064.8	−1.67180
$846$	0	0
$847$	847.000	0.0343604
$848$	15399.5	0.623611
$849$	0	0
$850$	24393.0	0.984322
$851$	−15159.8	−0.610660
$852$	0	0
$853$	−24852.9	−0.997593	−0.498797	−	0.866719i	$-0.666225\pi$
$853$	−24852.9	−0.997593	−0.498797	+	0.866719i	$0.666225\pi$
$854$	5577.57	0.223490
$855$	0	0
$856$	8823.29	0.352306
$857$	26129.7	1.04151	0.520755	−	0.853706i	$-0.325651\pi$
$857$	26129.7	1.04151	0.520755	+	0.853706i	$0.325651\pi$
$858$	0	0
$859$	3354.66	0.133248	0.0666238	−	0.997778i	$-0.478777\pi$
$859$	3354.66	0.133248	0.0666238	+	0.997778i	$0.478777\pi$
$860$	−10510.1	−0.416736
$861$	0	0
$862$	−5718.02	−0.225936
$863$	37414.5	1.47579	0.737893	−	0.674918i	$-0.235821\pi$
$863$	37414.5	1.47579	0.737893	+	0.674918i	$0.235821\pi$
$864$	0	0
$865$	15434.6	0.606694
$866$	11949.5	0.468893
$867$	0	0
$868$	1404.54	0.0549230
$869$	−7820.49	−0.305284
$870$	0	0
$871$	19090.1	0.742643
$872$	−8730.29	−0.339042
$873$	0	0
$874$	4306.94	0.166687
$875$	−4138.05	−0.159876
$876$	0	0
$877$	5518.99	0.212501	0.106250	−	0.994339i	$-0.466115\pi$
$877$	5518.99	0.212501	0.106250	+	0.994339i	$0.466115\pi$
$878$	−3490.33	−0.134160
$879$	0	0
$880$	5076.18	0.194452
$881$	−35273.5	−1.34892	−0.674458	−	0.738313i	$-0.735623\pi$
$881$	−35273.5	−1.34892	−0.674458	+	0.738313i	$0.735623\pi$
$882$	0	0
$883$	34154.4	1.30168	0.650842	−	0.759213i	$-0.274416\pi$
$883$	34154.4	1.30168	0.650842	+	0.759213i	$0.274416\pi$
$884$	−51516.6	−1.96006
$885$	0	0
$886$	−18864.4	−0.715307
$887$	31925.4	1.20851	0.604257	−	0.796790i	$-0.293470\pi$
$887$	31925.4	1.20851	0.604257	+	0.796790i	$0.293470\pi$
$888$	0	0
$889$	−7423.15	−0.280050
$890$	−33770.2	−1.27189
$891$	0	0
$892$	−3782.04	−0.141964
$893$	−704.664	−0.0264061
$894$	0	0
$895$	−61637.7	−2.30203
$896$	10068.1	0.375393
$897$	0	0
$898$	−2037.87	−0.0757288
$899$	5711.74	0.211899
$900$	0	0
$901$	−67067.1	−2.47983
$902$	−5753.87	−0.212398
$903$	0	0
$904$	−8952.17	−0.329364
$905$	3267.14	0.120004
$906$	0	0
$907$	21299.4	0.779751	0.389875	−	0.920868i	$-0.372518\pi$
$907$	21299.4	0.779751	0.389875	+	0.920868i	$0.372518\pi$
$908$	24516.5	0.896046
$909$	0	0
$910$	−10296.4	−0.375080
$911$	−25231.7	−0.917631	−0.458816	−	0.888531i	$-0.651726\pi$
$911$	−25231.7	−0.917631	−0.458816	+	0.888531i	$0.651726\pi$
$912$	0	0
$913$	−2110.39	−0.0764992
$914$	12328.7	0.446167
$915$	0	0
$916$	11944.8	0.430859
$917$	−13922.2	−0.501364
$918$	0	0
$919$	17335.9	0.622263	0.311131	−	0.950367i	$-0.399292\pi$
$919$	17335.9	0.622263	0.311131	+	0.950367i	$0.399292\pi$
$920$	62003.0	2.22193
$921$	0	0
$922$	19508.3	0.696824
$923$	−54923.6	−1.95865
$924$	0	0
$925$	12145.9	0.431736
$926$	−7087.37	−0.251518
$927$	0	0
$928$	32972.1	1.16634
$929$	−22981.2	−0.811614	−0.405807	−	0.913959i	$-0.633009\pi$
$929$	−22981.2	−0.811614	−0.405807	+	0.913959i	$0.633009\pi$
$930$	0	0
$931$	825.180	0.0290485
$932$	−18829.9	−0.661796
$933$	0	0
$934$	−2900.43	−0.101611
$935$	−22107.5	−0.773253
$936$	0	0
$937$	−15884.3	−0.553809	−0.276904	−	0.960897i	$-0.589309\pi$
$937$	−15884.3	−0.553809	−0.276904	+	0.960897i	$0.589309\pi$
$938$	2514.98	0.0875447
$939$	0	0
$940$	−4493.01	−0.155900
$941$	−22366.2	−0.774832	−0.387416	−	0.921905i	$-0.626632\pi$
$941$	−22366.2	−0.774832	−0.387416	+	0.921905i	$0.626632\pi$
$942$	0	0
$943$	81583.3	2.81730
$944$	6134.51	0.211506
$945$	0	0
$946$	−1378.75	−0.0473858
$947$	26760.3	0.918261	0.459130	−	0.888369i	$-0.348161\pi$
$947$	26760.3	0.918261	0.459130	+	0.888369i	$0.348161\pi$
$948$	0	0
$949$	−70658.4	−2.41693
$950$	−3450.69	−0.117848
$951$	0	0
$952$	−15323.7	−0.521683
$953$	33171.7	1.12753	0.563766	−	0.825935i	$-0.309352\pi$
$953$	33171.7	1.12753	0.563766	+	0.825935i	$0.309352\pi$
$954$	0	0
$955$	55715.7	1.88787
$956$	−11812.7	−0.399632
$957$	0	0
$958$	−2753.60	−0.0928652
$959$	8896.67	0.299571
$960$	0	0
$961$	−28795.8	−0.966593
$962$	6613.36	0.221646
$963$	0	0
$964$	−22881.6	−0.764487
$965$	−44895.7	−1.49766
$966$	0	0
$967$	7857.98	0.261319	0.130660	−	0.991427i	$-0.458291\pi$
$967$	7857.98	0.261319	0.130660	+	0.991427i	$0.458291\pi$
$968$	2225.04	0.0738797
$969$	0	0
$970$	17651.8	0.584294
$971$	434.865	0.0143723	0.00718614	−	0.999974i	$-0.497713\pi$
$971$	434.865	0.0143723	0.00718614	+	0.999974i	$0.497713\pi$
$972$	0	0
$973$	−7608.02	−0.250670
$974$	25276.5	0.831530
$975$	0	0
$976$	−17008.5	−0.557816
$977$	−19558.5	−0.640463	−0.320232	−	0.947339i	$-0.603761\pi$
$977$	−19558.5	−0.640463	−0.320232	+	0.947339i	$0.603761\pi$
$978$	0	0
$979$	17183.1	0.560953
$980$	5261.43	0.171500
$981$	0	0
$982$	23683.2	0.769615
$983$	59419.8	1.92797	0.963987	−	0.265950i	$-0.0856854\pi$
$983$	59419.8	1.92797	0.963987	+	0.265950i	$0.0856854\pi$
$984$	0	0
$985$	22261.4	0.720108
$986$	−27600.0	−0.891442
$987$	0	0
$988$	7287.66	0.234667
$989$	19549.1	0.628539
$990$	0	0
$991$	−25376.7	−0.813439	−0.406719	−	0.913553i	$-0.633327\pi$
$991$	−25376.7	−0.813439	−0.406719	+	0.913553i	$0.633327\pi$
$992$	5745.18	0.183881
$993$	0	0
$994$	−7235.79	−0.230891
$995$	20488.4	0.652791
$996$	0	0
$997$	−38621.4	−1.22683	−0.613416	−	0.789760i	$-0.710205\pi$
$997$	−38621.4	−1.22683	−0.613416	+	0.789760i	$0.710205\pi$
$998$	−1572.96	−0.0498910
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.p.1.3		5
3.2	odd	2		231.4.a.k.1.3	✓	5
21.20	even	2		1617.4.a.n.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.k.1.3	✓	5	3.2	odd	2
693.4.a.p.1.3		5	1.1	even	1	trivial
1617.4.a.n.1.3		5	21.20	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q-1.28053 q^{2} -6.36023 q^{4} -16.8824 q^{5} +7.00000 q^{7} +18.3888 q^{8} +21.6185 q^{10} -11.0000 q^{11} -68.0397 q^{13} -8.96374 q^{14} +27.3344 q^{16} -119.045 q^{17} +16.8404 q^{19} +107.376 q^{20} +14.0859 q^{22} -199.722 q^{23} +160.016 q^{25} +87.1272 q^{26} -44.5216 q^{28} -181.053 q^{29} -31.5474 q^{31} -182.113 q^{32} +152.441 q^{34} -118.177 q^{35} +75.9047 q^{37} -21.5647 q^{38} -310.447 q^{40} -408.485 q^{41} -97.8817 q^{43} +69.9626 q^{44} +255.750 q^{46} -41.8437 q^{47} +49.0000 q^{49} -204.906 q^{50} +432.748 q^{52} +563.375 q^{53} +185.707 q^{55} +128.721 q^{56} +231.845 q^{58} +224.425 q^{59} -622.237 q^{61} +40.3975 q^{62} +14.5263 q^{64} +1148.67 q^{65} -280.572 q^{67} +757.155 q^{68} +151.329 q^{70} +807.229 q^{71} +1038.49 q^{73} -97.1985 q^{74} -107.109 q^{76} -77.0000 q^{77} +710.954 q^{79} -461.471 q^{80} +523.079 q^{82} +191.854 q^{83} +2009.77 q^{85} +125.341 q^{86} -202.276 q^{88} -1562.10 q^{89} -476.278 q^{91} +1270.28 q^{92} +53.5822 q^{94} -284.307 q^{95} +816.513 q^{97} -62.7462 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 21 q^{4} - 21 q^{5} + 35 q^{7} + 42 q^{8} - 23 q^{10} - 55 q^{11} + 101 q^{13} + 7 q^{14} - 7 q^{16} + 20 q^{17} + 237 q^{19} - 85 q^{20} - 11 q^{22} + 80 q^{23} + 486 q^{25} - 165 q^{26}+ \cdots + 49 q^{98}+O(q^{100}) \) 5 * q + q^2 + 21 * q^4 - 21 * q^5 + 35 * q^7 + 42 * q^8 - 23 * q^10 - 55 * q^11 + 101 * q^13 + 7 * q^14 - 7 * q^16 + 20 * q^17 + 237 * q^19 - 85 * q^20 - 11 * q^22 + 80 * q^23 + 486 * q^25 - 165 * q^26 + 147 * q^28 + 11 * q^29 + 316 * q^31 - 453 * q^32 + 936 * q^34 - 147 * q^35 + 319 * q^37 - 89 * q^38 + 624 * q^40 - 1190 * q^41 + 88 * q^43 - 231 * q^44 + 1000 * q^46 - 377 * q^47 + 245 * q^49 + 644 * q^50 + 1001 * q^52 + 992 * q^53 + 231 * q^55 + 294 * q^56 + 721 * q^58 - 71 * q^59 - 574 * q^61 - 272 * q^62 - 1380 * q^64 - 589 * q^65 - 527 * q^67 + 2974 * q^68 - 161 * q^70 + 1156 * q^71 + 1061 * q^73 + 1609 * q^74 + 2399 * q^76 - 385 * q^77 + 588 * q^79 + 1643 * q^80 - 2602 * q^82 + 212 * q^83 + 1918 * q^85 + 4760 * q^86 - 462 * q^88 - 1030 * q^89 + 707 * q^91 + 1174 * q^92 - 1799 * q^94 + 3593 * q^95 + 2488 * q^97 + 49 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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