LMFDB - Embedded newform 495.4.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,4,Mod(1,495)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, 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("495.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	495.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:	$29.2059454528$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1957.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 10 $ x^3 - x^2 - 9*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.04096$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.793499 q^{2} -7.37036 q^{4} -5.00000 q^{5} -2.90793 q^{7} +12.1964 q^{8} +O(q^{10})$	\(q-0.793499 q^{2} -7.37036 q^{4} -5.00000 q^{5} -2.90793 q^{7} +12.1964 q^{8} +3.96749 q^{10} -11.0000 q^{11} +68.4882 q^{13} +2.30744 q^{14} +49.2851 q^{16} +31.0515 q^{17} +54.9249 q^{19} +36.8518 q^{20} +8.72849 q^{22} -180.615 q^{23} +25.0000 q^{25} -54.3453 q^{26} +21.4325 q^{28} -67.3690 q^{29} +153.367 q^{31} -136.679 q^{32} -24.6393 q^{34} +14.5397 q^{35} -324.485 q^{37} -43.5828 q^{38} -60.9818 q^{40} +25.4570 q^{41} +133.864 q^{43} +81.0740 q^{44} +143.318 q^{46} -113.784 q^{47} -334.544 q^{49} -19.8375 q^{50} -504.783 q^{52} -91.6741 q^{53} +55.0000 q^{55} -35.4662 q^{56} +53.4573 q^{58} -434.698 q^{59} -60.2384 q^{61} -121.696 q^{62} -285.826 q^{64} -342.441 q^{65} -439.825 q^{67} -228.861 q^{68} -11.5372 q^{70} -436.620 q^{71} -91.5134 q^{73} +257.478 q^{74} -404.816 q^{76} +31.9873 q^{77} +947.462 q^{79} -246.425 q^{80} -20.2001 q^{82} +944.385 q^{83} -155.258 q^{85} -106.221 q^{86} -134.160 q^{88} -413.702 q^{89} -199.159 q^{91} +1331.20 q^{92} +90.2872 q^{94} -274.624 q^{95} -1463.39 q^{97} +265.460 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 17 q^{4} - 15 q^{5} + 6 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q - q^2 + 17 * q^4 - 15 * q^5 + 6 * q^7 + 3 * q^8	$ 3 q - q^{2} + 17 q^{4} - 15 q^{5} + 6 q^{7} + 3 q^{8} + 5 q^{10} - 33 q^{11} - 20 q^{13} - 144 q^{14} + 25 q^{16} - 32 q^{17} + 116 q^{19} - 85 q^{20} + 11 q^{22} - 240 q^{23} + 75 q^{25} - 302 q^{26} + 160 q^{28} - 238 q^{29} + 92 q^{31} - 197 q^{32} + 354 q^{34} - 30 q^{35} - 90 q^{37} - 324 q^{38} - 15 q^{40} + 46 q^{41} - 134 q^{43} - 187 q^{44} - 240 q^{46} + 220 q^{47} - 457 q^{49} - 25 q^{50} - 1530 q^{52} + 798 q^{53} + 165 q^{55} - 688 q^{56} - 978 q^{58} - 1236 q^{59} + 342 q^{61} + 1792 q^{62} - 1919 q^{64} + 100 q^{65} + 764 q^{67} - 1074 q^{68} + 720 q^{70} - 1816 q^{71} + 100 q^{73} + 1874 q^{74} + 396 q^{76} - 66 q^{77} - 96 q^{79} - 125 q^{80} - 910 q^{82} - 858 q^{83} + 160 q^{85} - 188 q^{86} - 33 q^{88} - 838 q^{89} + 332 q^{91} + 688 q^{92} - 3112 q^{94} - 580 q^{95} - 1322 q^{97} - 1017 q^{98}+O(q^{100}) $ 3 * q - q^2 + 17 * q^4 - 15 * q^5 + 6 * q^7 + 3 * q^8 + 5 * q^10 - 33 * q^11 - 20 * q^13 - 144 * q^14 + 25 * q^16 - 32 * q^17 + 116 * q^19 - 85 * q^20 + 11 * q^22 - 240 * q^23 + 75 * q^25 - 302 * q^26 + 160 * q^28 - 238 * q^29 + 92 * q^31 - 197 * q^32 + 354 * q^34 - 30 * q^35 - 90 * q^37 - 324 * q^38 - 15 * q^40 + 46 * q^41 - 134 * q^43 - 187 * q^44 - 240 * q^46 + 220 * q^47 - 457 * q^49 - 25 * q^50 - 1530 * q^52 + 798 * q^53 + 165 * q^55 - 688 * q^56 - 978 * q^58 - 1236 * q^59 + 342 * q^61 + 1792 * q^62 - 1919 * q^64 + 100 * q^65 + 764 * q^67 - 1074 * q^68 + 720 * q^70 - 1816 * q^71 + 100 * q^73 + 1874 * q^74 + 396 * q^76 - 66 * q^77 - 96 * q^79 - 125 * q^80 - 910 * q^82 - 858 * q^83 + 160 * q^85 - 188 * q^86 - 33 * q^88 - 838 * q^89 + 332 * q^91 + 688 * q^92 - 3112 * q^94 - 580 * q^95 - 1322 * q^97 - 1017 * 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 $

$2$	−0.793499	−0.280544	−0.140272	−	0.990113i	$-0.544798\pi$
$2$	−0.793499	−0.280544	−0.140272	+	0.990113i	$0.544798\pi$
$3$	0	0
$3$	0	0
$4$	−7.37036	−0.921295
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−2.90793	−0.157014	−0.0785068	−	0.996914i	$-0.525015\pi$
$7$	−2.90793	−0.157014	−0.0785068	+	0.996914i	$0.525015\pi$
$8$	12.1964	0.539008
$9$	0	0
$10$	3.96749	0.125463
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	68.4882	1.46117	0.730585	−	0.682822i	$-0.239247\pi$
$13$	68.4882	1.46117	0.730585	+	0.682822i	$0.239247\pi$
$14$	2.30744	0.0440493
$15$	0	0
$16$	49.2851	0.770079
$17$	31.0515	0.443006	0.221503	−	0.975160i	$-0.428904\pi$
$17$	31.0515	0.443006	0.221503	+	0.975160i	$0.428904\pi$
$18$	0	0
$19$	54.9249	0.663191	0.331595	−	0.943422i	$-0.392413\pi$
$19$	54.9249	0.663191	0.331595	+	0.943422i	$0.392413\pi$
$20$	36.8518	0.412016
$21$	0	0
$22$	8.72849	0.0845873
$23$	−180.615	−1.63743	−0.818713	−	0.574203i	$-0.805312\pi$
$23$	−180.615	−1.63743	−0.818713	+	0.574203i	$0.805312\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−54.3453	−0.409923
$27$	0	0
$28$	21.4325	0.144656
$29$	−67.3690	−0.431383	−0.215692	−	0.976462i	$-0.569201\pi$
$29$	−67.3690	−0.431383	−0.215692	+	0.976462i	$0.569201\pi$
$30$	0	0
$31$	153.367	0.888562	0.444281	−	0.895887i	$-0.353459\pi$
$31$	153.367	0.888562	0.444281	+	0.895887i	$0.353459\pi$
$32$	−136.679	−0.755050
$33$	0	0
$34$	−24.6393	−0.124283
$35$	14.5397	0.0702186
$36$	0	0
$37$	−324.485	−1.44176	−0.720878	−	0.693062i	$-0.756261\pi$
$37$	−324.485	−1.44176	−0.720878	+	0.693062i	$0.756261\pi$
$38$	−43.5828	−0.186054
$39$	0	0
$40$	−60.9818	−0.241052
$41$	25.4570	0.0969686	0.0484843	−	0.998824i	$-0.484561\pi$
$41$	25.4570	0.0969686	0.0484843	+	0.998824i	$0.484561\pi$
$42$	0	0
$43$	133.864	0.474746	0.237373	−	0.971419i	$-0.423714\pi$
$43$	133.864	0.474746	0.237373	+	0.971419i	$0.423714\pi$
$44$	81.0740	0.277781
$45$	0	0
$46$	143.318	0.459371
$47$	−113.784	−0.353129	−0.176564	−	0.984289i	$-0.556498\pi$
$47$	−113.784	−0.353129	−0.176564	+	0.984289i	$0.556498\pi$
$48$	0	0
$49$	−334.544	−0.975347
$50$	−19.8375	−0.0561088
$51$	0	0
$52$	−504.783	−1.34617
$53$	−91.6741	−0.237593	−0.118796	−	0.992919i	$-0.537904\pi$
$53$	−91.6741	−0.237593	−0.118796	+	0.992919i	$0.537904\pi$
$54$	0	0
$55$	55.0000	0.134840
$56$	−35.4662	−0.0846316
$57$	0	0
$58$	53.4573	0.121022
$59$	−434.698	−0.959200	−0.479600	−	0.877487i	$-0.659218\pi$
$59$	−434.698	−0.959200	−0.479600	+	0.877487i	$0.659218\pi$
$60$	0	0
$61$	−60.2384	−0.126438	−0.0632191	−	0.998000i	$-0.520137\pi$
$61$	−60.2384	−0.126438	−0.0632191	+	0.998000i	$0.520137\pi$
$62$	−121.696	−0.249281
$63$	0	0
$64$	−285.826	−0.558254
$65$	−342.441	−0.653455
$66$	0	0
$67$	−439.825	−0.801987	−0.400994	−	0.916081i	$-0.631335\pi$
$67$	−439.825	−0.801987	−0.400994	+	0.916081i	$0.631335\pi$
$68$	−228.861	−0.408139
$69$	0	0
$70$	−11.5372	−0.0196994
$71$	−436.620	−0.729820	−0.364910	−	0.931043i	$-0.618900\pi$
$71$	−436.620	−0.729820	−0.364910	+	0.931043i	$0.618900\pi$
$72$	0	0
$73$	−91.5134	−0.146724	−0.0733619	−	0.997305i	$-0.523373\pi$
$73$	−91.5134	−0.146724	−0.0733619	+	0.997305i	$0.523373\pi$
$74$	257.478	0.404477
$75$	0	0
$76$	−404.816	−0.610994
$77$	31.9873	0.0473414
$78$	0	0
$79$	947.462	1.34934	0.674670	−	0.738120i	$-0.264286\pi$
$79$	947.462	1.34934	0.674670	+	0.738120i	$0.264286\pi$
$80$	−246.425	−0.344390
$81$	0	0
$82$	−20.2001	−0.0272040
$83$	944.385	1.24891	0.624456	−	0.781060i	$-0.285321\pi$
$83$	944.385	1.24891	0.624456	+	0.781060i	$0.285321\pi$
$84$	0	0
$85$	−155.258	−0.198118
$86$	−106.221	−0.133187
$87$	0	0
$88$	−134.160	−0.162517
$89$	−413.702	−0.492722	−0.246361	−	0.969178i	$-0.579235\pi$
$89$	−413.702	−0.492722	−0.246361	+	0.969178i	$0.579235\pi$
$90$	0	0
$91$	−199.159	−0.229424
$92$	1331.20	1.50855
$93$	0	0
$94$	90.2872	0.0990683
$95$	−274.624	−0.296588
$96$	0	0
$97$	−1463.39	−1.53180	−0.765899	−	0.642960i	$-0.777706\pi$
$97$	−1463.39	−1.53180	−0.765899	+	0.642960i	$0.777706\pi$
$98$	265.460	0.273628
$99$	0	0
$100$	−184.259	−0.184259
$101$	−1959.24	−1.93022	−0.965109	−	0.261847i	$-0.915668\pi$
$101$	−1959.24	−1.93022	−0.965109	+	0.261847i	$0.915668\pi$
$102$	0	0
$103$	−151.029	−0.144479	−0.0722397	−	0.997387i	$-0.523015\pi$
$103$	−151.029	−0.144479	−0.0722397	+	0.997387i	$0.523015\pi$
$104$	835.307	0.787583
$105$	0	0
$106$	72.7433	0.0666552
$107$	−54.4430	−0.0491888	−0.0245944	−	0.999698i	$-0.507829\pi$
$107$	−54.4430	−0.0491888	−0.0245944	+	0.999698i	$0.507829\pi$
$108$	0	0
$109$	619.673	0.544531	0.272266	−	0.962222i	$-0.412227\pi$
$109$	619.673	0.544531	0.272266	+	0.962222i	$0.412227\pi$
$110$	−43.6424	−0.0378286
$111$	0	0
$112$	−143.318	−0.120913
$113$	−171.324	−0.142627	−0.0713133	−	0.997454i	$-0.522719\pi$
$113$	−171.324	−0.142627	−0.0713133	+	0.997454i	$0.522719\pi$
$114$	0	0
$115$	903.074	0.732279
$116$	496.534	0.397431
$117$	0	0
$118$	344.932	0.269098
$119$	−90.2957	−0.0695580
$120$	0	0
$121$	121.000	0.0909091
$122$	47.7991	0.0354715
$123$	0	0
$124$	−1130.37	−0.818628
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1085.23	0.758257	0.379128	−	0.925344i	$-0.376224\pi$
$127$	1085.23	0.758257	0.379128	+	0.925344i	$0.376224\pi$
$128$	1320.23	0.911665
$129$	0	0
$130$	271.727	0.183323
$131$	−2354.89	−1.57059	−0.785296	−	0.619120i	$-0.787489\pi$
$131$	−2354.89	−1.57059	−0.785296	+	0.619120i	$0.787489\pi$
$132$	0	0
$133$	−159.718	−0.104130
$134$	349.000	0.224993
$135$	0	0
$136$	378.716	0.238784
$137$	−2408.70	−1.50211	−0.751056	−	0.660239i	$-0.770455\pi$
$137$	−2408.70	−1.50211	−0.751056	+	0.660239i	$0.770455\pi$
$138$	0	0
$139$	−2604.89	−1.58952	−0.794762	−	0.606921i	$-0.792405\pi$
$139$	−2604.89	−1.58952	−0.794762	+	0.606921i	$0.792405\pi$
$140$	−107.163	−0.0646921
$141$	0	0
$142$	346.457	0.204747
$143$	−753.370	−0.440559
$144$	0	0
$145$	336.845	0.192920
$146$	72.6158	0.0411625
$147$	0	0
$148$	2391.57	1.32828
$149$	304.932	0.167658	0.0838288	−	0.996480i	$-0.473285\pi$
$149$	304.932	0.167658	0.0838288	+	0.996480i	$0.473285\pi$
$150$	0	0
$151$	1263.74	0.681071	0.340536	−	0.940232i	$-0.389392\pi$
$151$	1263.74	0.681071	0.340536	+	0.940232i	$0.389392\pi$
$152$	669.883	0.357465
$153$	0	0
$154$	−25.3819	−0.0132814
$155$	−766.833	−0.397377
$156$	0	0
$157$	−3714.09	−1.88801	−0.944003	−	0.329937i	$-0.892972\pi$
$157$	−3714.09	−1.88801	−0.944003	+	0.329937i	$0.892972\pi$
$158$	−751.810	−0.378549
$159$	0	0
$160$	683.393	0.337668
$161$	525.216	0.257098
$162$	0	0
$163$	1862.92	0.895186	0.447593	−	0.894238i	$-0.352281\pi$
$163$	1862.92	0.895186	0.447593	+	0.894238i	$0.352281\pi$
$164$	−187.627	−0.0893367
$165$	0	0
$166$	−749.369	−0.350375
$167$	−1092.64	−0.506295	−0.253147	−	0.967428i	$-0.581466\pi$
$167$	−1092.64	−0.506295	−0.253147	+	0.967428i	$0.581466\pi$
$168$	0	0
$169$	2493.64	1.13502
$170$	123.197	0.0555809
$171$	0	0
$172$	−986.627	−0.437381
$173$	2645.18	1.16248	0.581241	−	0.813732i	$-0.302567\pi$
$173$	2645.18	1.16248	0.581241	+	0.813732i	$0.302567\pi$
$174$	0	0
$175$	−72.6983	−0.0314027
$176$	−542.136	−0.232188
$177$	0	0
$178$	328.272	0.138230
$179$	−658.228	−0.274851	−0.137425	−	0.990512i	$-0.543883\pi$
$179$	−658.228	−0.274851	−0.137425	+	0.990512i	$0.543883\pi$
$180$	0	0
$181$	3986.11	1.63694	0.818468	−	0.574552i	$-0.194824\pi$
$181$	3986.11	1.63694	0.818468	+	0.574552i	$0.194824\pi$
$182$	158.033	0.0643635
$183$	0	0
$184$	−2202.84	−0.882586
$185$	1622.43	0.644773
$186$	0	0
$187$	−341.567	−0.133571
$188$	838.627	0.325336
$189$	0	0
$190$	217.914	0.0832060
$191$	−2104.50	−0.797260	−0.398630	−	0.917112i	$-0.630514\pi$
$191$	−2104.50	−0.797260	−0.398630	+	0.917112i	$0.630514\pi$
$192$	0	0
$193$	−433.649	−0.161734	−0.0808672	−	0.996725i	$-0.525769\pi$
$193$	−433.649	−0.161734	−0.0808672	+	0.996725i	$0.525769\pi$
$194$	1161.20	0.429737
$195$	0	0
$196$	2465.71	0.898582
$197$	4898.20	1.77148	0.885742	−	0.464177i	$-0.153650\pi$
$197$	4898.20	1.77148	0.885742	+	0.464177i	$0.153650\pi$
$198$	0	0
$199$	1866.38	0.664844	0.332422	−	0.943131i	$-0.392134\pi$
$199$	1866.38	0.664844	0.332422	+	0.943131i	$0.392134\pi$
$200$	304.909	0.107802
$201$	0	0
$202$	1554.66	0.541512
$203$	195.905	0.0677331
$204$	0	0
$205$	−127.285	−0.0433657
$206$	119.842	0.0405328
$207$	0	0
$208$	3375.45	1.12522
$209$	−604.173	−0.199960
$210$	0	0
$211$	−4165.49	−1.35907	−0.679537	−	0.733642i	$-0.737819\pi$
$211$	−4165.49	−1.35907	−0.679537	+	0.733642i	$0.737819\pi$
$212$	675.671	0.218893
$213$	0	0
$214$	43.2004	0.0137996
$215$	−669.320	−0.212313
$216$	0	0
$217$	−445.980	−0.139516
$218$	−491.710	−0.152765
$219$	0	0
$220$	−405.370	−0.124227
$221$	2126.66	0.647307
$222$	0	0
$223$	−3151.18	−0.946271	−0.473136	−	0.880990i	$-0.656878\pi$
$223$	−3151.18	−0.946271	−0.473136	+	0.880990i	$0.656878\pi$
$224$	397.452	0.118553
$225$	0	0
$226$	135.945	0.0400131
$227$	3430.20	1.00295	0.501477	−	0.865171i	$-0.332790\pi$
$227$	3430.20	1.00295	0.501477	+	0.865171i	$0.332790\pi$
$228$	0	0
$229$	−4041.79	−1.16633	−0.583164	−	0.812354i	$-0.698186\pi$
$229$	−4041.79	−1.16633	−0.583164	+	0.812354i	$0.698186\pi$
$230$	−716.589	−0.205437
$231$	0	0
$232$	−821.657	−0.232519
$233$	2565.83	0.721431	0.360715	−	0.932676i	$-0.382533\pi$
$233$	2565.83	0.721431	0.360715	+	0.932676i	$0.382533\pi$
$234$	0	0
$235$	568.918	0.157924
$236$	3203.88	0.883706
$237$	0	0
$238$	71.6496	0.0195141
$239$	5540.15	1.49942	0.749712	−	0.661764i	$-0.230192\pi$
$239$	5540.15	1.49942	0.749712	+	0.661764i	$0.230192\pi$
$240$	0	0
$241$	−5980.22	−1.59842	−0.799211	−	0.601050i	$-0.794749\pi$
$241$	−5980.22	−1.59842	−0.799211	+	0.601050i	$0.794749\pi$
$242$	−96.0134	−0.0255040
$243$	0	0
$244$	443.979	0.116487
$245$	1672.72	0.436188
$246$	0	0
$247$	3761.71	0.969035
$248$	1870.51	0.478942
$249$	0	0
$250$	99.1874	0.0250926
$251$	−1527.19	−0.384044	−0.192022	−	0.981391i	$-0.561505\pi$
$251$	−1527.19	−0.384044	−0.192022	+	0.981391i	$0.561505\pi$
$252$	0	0
$253$	1986.76	0.493703
$254$	−861.129	−0.212725
$255$	0	0
$256$	1239.01	0.302492
$257$	1193.28	0.289629	0.144814	−	0.989459i	$-0.453741\pi$
$257$	1193.28	0.289629	0.144814	+	0.989459i	$0.453741\pi$
$258$	0	0
$259$	943.581	0.226376
$260$	2523.91	0.602025
$261$	0	0
$262$	1868.60	0.440621
$263$	6793.71	1.59285	0.796423	−	0.604740i	$-0.206723\pi$
$263$	6793.71	1.59285	0.796423	+	0.604740i	$0.206723\pi$
$264$	0	0
$265$	458.370	0.106255
$266$	126.736	0.0292131
$267$	0	0
$268$	3241.67	0.738867
$269$	−7443.13	−1.68705	−0.843524	−	0.537091i	$-0.819523\pi$
$269$	−7443.13	−1.68705	−0.843524	+	0.537091i	$0.819523\pi$
$270$	0	0
$271$	−7912.17	−1.77354	−0.886771	−	0.462209i	$-0.847057\pi$
$271$	−7912.17	−1.77354	−0.886771	+	0.462209i	$0.847057\pi$
$272$	1530.38	0.341150
$273$	0	0
$274$	1911.30	0.421409
$275$	−275.000	−0.0603023
$276$	0	0
$277$	3462.64	0.751083	0.375541	−	0.926806i	$-0.377457\pi$
$277$	3462.64	0.751083	0.375541	+	0.926806i	$0.377457\pi$
$278$	2066.98	0.445932
$279$	0	0
$280$	177.331	0.0378484
$281$	7546.11	1.60200	0.801002	−	0.598662i	$-0.204301\pi$
$281$	7546.11	1.60200	0.801002	+	0.598662i	$0.204301\pi$
$282$	0	0
$283$	3070.04	0.644859	0.322430	−	0.946593i	$-0.395500\pi$
$283$	3070.04	0.644859	0.322430	+	0.946593i	$0.395500\pi$
$284$	3218.05	0.672380
$285$	0	0
$286$	597.799	0.123596
$287$	−74.0272	−0.0152254
$288$	0	0
$289$	−3948.80	−0.803746
$290$	−267.286	−0.0541227
$291$	0	0
$292$	674.487	0.135176
$293$	−3760.43	−0.749785	−0.374892	−	0.927068i	$-0.622320\pi$
$293$	−3760.43	−0.749785	−0.374892	+	0.927068i	$0.622320\pi$
$294$	0	0
$295$	2173.49	0.428967
$296$	−3957.54	−0.777119
$297$	0	0
$298$	−241.963	−0.0470354
$299$	−12370.0	−2.39256
$300$	0	0
$301$	−389.268	−0.0745416
$302$	−1002.78	−0.191071
$303$	0	0
$304$	2706.98	0.510709
$305$	301.192	0.0565449
$306$	0	0
$307$	2132.91	0.396521	0.198260	−	0.980149i	$-0.436471\pi$
$307$	2132.91	0.396521	0.198260	+	0.980149i	$0.436471\pi$
$308$	−235.758	−0.0436154
$309$	0	0
$310$	608.481	0.111482
$311$	−354.565	−0.0646481	−0.0323241	−	0.999477i	$-0.510291\pi$
$311$	−354.565	−0.0646481	−0.0323241	+	0.999477i	$0.510291\pi$
$312$	0	0
$313$	8700.26	1.57114	0.785572	−	0.618771i	$-0.212369\pi$
$313$	8700.26	1.57114	0.785572	+	0.618771i	$0.212369\pi$
$314$	2947.13	0.529669
$315$	0	0
$316$	−6983.14	−1.24314
$317$	−8115.01	−1.43781	−0.718903	−	0.695111i	$-0.755355\pi$
$317$	−8115.01	−1.43781	−0.718903	+	0.695111i	$0.755355\pi$
$318$	0	0
$319$	741.059	0.130067
$320$	1429.13	0.249659
$321$	0	0
$322$	−416.758	−0.0721274
$323$	1705.50	0.293797
$324$	0	0
$325$	1712.21	0.292234
$326$	−1478.23	−0.251139
$327$	0	0
$328$	310.483	0.0522669
$329$	330.875	0.0554461
$330$	0	0
$331$	−6738.63	−1.11900	−0.559499	−	0.828831i	$-0.689006\pi$
$331$	−6738.63	−1.11900	−0.559499	+	0.828831i	$0.689006\pi$
$332$	−6960.46	−1.15062
$333$	0	0
$334$	867.011	0.142038
$335$	2199.12	0.358660
$336$	0	0
$337$	−10836.8	−1.75169	−0.875843	−	0.482595i	$-0.839694\pi$
$337$	−10836.8	−1.75169	−0.875843	+	0.482595i	$0.839694\pi$
$338$	−1978.70	−0.318423
$339$	0	0
$340$	1144.30	0.182525
$341$	−1687.03	−0.267912
$342$	0	0
$343$	1970.25	0.310156
$344$	1632.66	0.255892
$345$	0	0
$346$	−2098.95	−0.326128
$347$	−4745.54	−0.734161	−0.367081	−	0.930189i	$-0.619643\pi$
$347$	−4745.54	−0.734161	−0.367081	+	0.930189i	$0.619643\pi$
$348$	0	0
$349$	−744.830	−0.114240	−0.0571201	−	0.998367i	$-0.518192\pi$
$349$	−744.830	−0.114240	−0.0571201	+	0.998367i	$0.518192\pi$
$350$	57.6861	0.00880985
$351$	0	0
$352$	1503.46	0.227656
$353$	5863.06	0.884021	0.442011	−	0.897010i	$-0.354265\pi$
$353$	5863.06	0.884021	0.442011	+	0.897010i	$0.354265\pi$
$354$	0	0
$355$	2183.10	0.326386
$356$	3049.13	0.453943
$357$	0	0
$358$	522.303	0.0771078
$359$	−12398.9	−1.82281	−0.911403	−	0.411515i	$-0.865000\pi$
$359$	−12398.9	−1.82281	−0.911403	+	0.411515i	$0.865000\pi$
$360$	0	0
$361$	−3842.26	−0.560178
$362$	−3162.98	−0.459233
$363$	0	0
$364$	1467.87	0.211367
$365$	457.567	0.0656169
$366$	0	0
$367$	10059.5	1.43080	0.715399	−	0.698717i	$-0.246245\pi$
$367$	10059.5	1.43080	0.715399	+	0.698717i	$0.246245\pi$
$368$	−8901.62	−1.26095
$369$	0	0
$370$	−1287.39	−0.180887
$371$	266.582	0.0373053
$372$	0	0
$373$	−10527.6	−1.46139	−0.730694	−	0.682705i	$-0.760803\pi$
$373$	−10527.6	−1.46139	−0.730694	+	0.682705i	$0.760803\pi$
$374$	271.033	0.0374727
$375$	0	0
$376$	−1387.75	−0.190339
$377$	−4613.98	−0.630324
$378$	0	0
$379$	−3772.28	−0.511264	−0.255632	−	0.966774i	$-0.582284\pi$
$379$	−3772.28	−0.511264	−0.255632	+	0.966774i	$0.582284\pi$
$380$	2024.08	0.273245
$381$	0	0
$382$	1669.92	0.223667
$383$	−6667.97	−0.889602	−0.444801	−	0.895629i	$-0.646726\pi$
$383$	−6667.97	−0.889602	−0.444801	+	0.895629i	$0.646726\pi$
$384$	0	0
$385$	−159.936	−0.0211717
$386$	344.100	0.0453736
$387$	0	0
$388$	10785.7	1.41124
$389$	−3460.19	−0.450999	−0.225500	−	0.974243i	$-0.572401\pi$
$389$	−3460.19	−0.450999	−0.225500	+	0.974243i	$0.572401\pi$
$390$	0	0
$391$	−5608.37	−0.725389
$392$	−4080.22	−0.525720
$393$	0	0
$394$	−3886.72	−0.496980
$395$	−4737.31	−0.603443
$396$	0	0
$397$	−3250.11	−0.410878	−0.205439	−	0.978670i	$-0.565862\pi$
$397$	−3250.11	−0.410878	−0.205439	+	0.978670i	$0.565862\pi$
$398$	−1480.97	−0.186518
$399$	0	0
$400$	1232.13	0.154016
$401$	2870.79	0.357507	0.178753	−	0.983894i	$-0.442794\pi$
$401$	2870.79	0.357507	0.178753	+	0.983894i	$0.442794\pi$
$402$	0	0
$403$	10503.8	1.29834
$404$	14440.3	1.77830
$405$	0	0
$406$	−155.450	−0.0190021
$407$	3569.34	0.434706
$408$	0	0
$409$	−3166.04	−0.382765	−0.191382	−	0.981516i	$-0.561297\pi$
$409$	−3166.04	−0.382765	−0.191382	+	0.981516i	$0.561297\pi$
$410$	101.000	0.0121660
$411$	0	0
$412$	1113.14	0.133108
$413$	1264.07	0.150607
$414$	0	0
$415$	−4721.93	−0.558531
$416$	−9360.87	−1.10326
$417$	0	0
$418$	479.411	0.0560975
$419$	−2984.75	−0.348007	−0.174003	−	0.984745i	$-0.555670\pi$
$419$	−2984.75	−0.348007	−0.174003	+	0.984745i	$0.555670\pi$
$420$	0	0
$421$	1280.15	0.148197	0.0740985	−	0.997251i	$-0.476392\pi$
$421$	1280.15	0.148197	0.0740985	+	0.997251i	$0.476392\pi$
$422$	3305.32	0.381280
$423$	0	0
$424$	−1118.09	−0.128064
$425$	776.288	0.0886012
$426$	0	0
$427$	175.169	0.0198525
$428$	401.264	0.0453174
$429$	0	0
$430$	531.105	0.0595632
$431$	−5952.92	−0.665294	−0.332647	−	0.943051i	$-0.607942\pi$
$431$	−5952.92	−0.665294	−0.332647	+	0.943051i	$0.607942\pi$
$432$	0	0
$433$	4178.14	0.463715	0.231858	−	0.972750i	$-0.425520\pi$
$433$	4178.14	0.463715	0.231858	+	0.972750i	$0.425520\pi$
$434$	353.884	0.0391405
$435$	0	0
$436$	−4567.21	−0.501674
$437$	−9920.25	−1.08593
$438$	0	0
$439$	389.487	0.0423445	0.0211722	−	0.999776i	$-0.493260\pi$
$439$	389.487	0.0423445	0.0211722	+	0.999776i	$0.493260\pi$
$440$	670.800	0.0726799
$441$	0	0
$442$	−1687.50	−0.181598
$443$	4395.08	0.471369	0.235685	−	0.971830i	$-0.424267\pi$
$443$	4395.08	0.471369	0.235685	+	0.971830i	$0.424267\pi$
$444$	0	0
$445$	2068.51	0.220352
$446$	2500.46	0.265471
$447$	0	0
$448$	831.164	0.0876536
$449$	5354.13	0.562755	0.281378	−	0.959597i	$-0.409209\pi$
$449$	5354.13	0.562755	0.281378	+	0.959597i	$0.409209\pi$
$450$	0	0
$451$	−280.027	−0.0292371
$452$	1262.72	0.131401
$453$	0	0
$454$	−2721.86	−0.281373
$455$	995.796	0.102601
$456$	0	0
$457$	−3780.82	−0.387001	−0.193501	−	0.981100i	$-0.561984\pi$
$457$	−3780.82	−0.387001	−0.193501	+	0.981100i	$0.561984\pi$
$458$	3207.16	0.327207
$459$	0	0
$460$	−6655.98	−0.674645
$461$	−3549.14	−0.358568	−0.179284	−	0.983797i	$-0.557378\pi$
$461$	−3549.14	−0.358568	−0.179284	+	0.983797i	$0.557378\pi$
$462$	0	0
$463$	3119.32	0.313104	0.156552	−	0.987670i	$-0.449962\pi$
$463$	3119.32	0.313104	0.156552	+	0.987670i	$0.449962\pi$
$464$	−3320.29	−0.332199
$465$	0	0
$466$	−2035.99	−0.202393
$467$	−524.225	−0.0519448	−0.0259724	−	0.999663i	$-0.508268\pi$
$467$	−524.225	−0.0519448	−0.0259724	+	0.999663i	$0.508268\pi$
$468$	0	0
$469$	1278.98	0.125923
$470$	−451.436	−0.0443047
$471$	0	0
$472$	−5301.73	−0.517017
$473$	−1472.51	−0.143141
$474$	0	0
$475$	1373.12	0.132638
$476$	665.512	0.0640834
$477$	0	0
$478$	−4396.10	−0.420655
$479$	4648.94	0.443456	0.221728	−	0.975109i	$-0.428830\pi$
$479$	4648.94	0.443456	0.221728	+	0.975109i	$0.428830\pi$
$480$	0	0
$481$	−22223.4	−2.10665
$482$	4745.30	0.448428
$483$	0	0
$484$	−891.813	−0.0837541
$485$	7316.94	0.685041
$486$	0	0
$487$	−17883.3	−1.66400	−0.832002	−	0.554773i	$-0.812805\pi$
$487$	−17883.3	−1.66400	−0.832002	+	0.554773i	$0.812805\pi$
$488$	−734.689	−0.0681513
$489$	0	0
$490$	−1327.30	−0.122370
$491$	10702.7	0.983715	0.491857	−	0.870676i	$-0.336318\pi$
$491$	10702.7	0.983715	0.491857	+	0.870676i	$0.336318\pi$
$492$	0	0
$493$	−2091.91	−0.191105
$494$	−2984.91	−0.271857
$495$	0	0
$496$	7558.68	0.684264
$497$	1269.66	0.114592
$498$	0	0
$499$	4092.39	0.367135	0.183567	−	0.983007i	$-0.441235\pi$
$499$	4092.39	0.367135	0.183567	+	0.983007i	$0.441235\pi$
$500$	921.295	0.0824031
$501$	0	0
$502$	1211.82	0.107741
$503$	−945.979	−0.0838551	−0.0419275	−	0.999121i	$-0.513350\pi$
$503$	−945.979	−0.0838551	−0.0419275	+	0.999121i	$0.513350\pi$
$504$	0	0
$505$	9796.22	0.863220
$506$	−1576.49	−0.138505
$507$	0	0
$508$	−7998.54	−0.698578
$509$	−3674.38	−0.319968	−0.159984	−	0.987120i	$-0.551144\pi$
$509$	−3674.38	−0.319968	−0.159984	+	0.987120i	$0.551144\pi$
$510$	0	0
$511$	266.115	0.0230376
$512$	−11545.0	−0.996527
$513$	0	0
$514$	−946.864	−0.0812537
$515$	755.147	0.0646131
$516$	0	0
$517$	1251.62	0.106472
$518$	−748.730	−0.0635083
$519$	0	0
$520$	−4176.54	−0.352218
$521$	545.436	0.0458656	0.0229328	−	0.999737i	$-0.492700\pi$
$521$	545.436	0.0458656	0.0229328	+	0.999737i	$0.492700\pi$
$522$	0	0
$523$	9060.79	0.757554	0.378777	−	0.925488i	$-0.376345\pi$
$523$	9060.79	0.757554	0.378777	+	0.925488i	$0.376345\pi$
$524$	17356.4	1.44698
$525$	0	0
$526$	−5390.80	−0.446864
$527$	4762.26	0.393638
$528$	0	0
$529$	20454.7	1.68117
$530$	−363.716	−0.0298091
$531$	0	0
$532$	1177.18	0.0959344
$533$	1743.50	0.141688
$534$	0	0
$535$	272.215	0.0219979
$536$	−5364.26	−0.432278
$537$	0	0
$538$	5906.12	0.473292
$539$	3679.98	0.294078
$540$	0	0
$541$	−15842.3	−1.25899	−0.629495	−	0.777005i	$-0.716738\pi$
$541$	−15842.3	−1.25899	−0.629495	+	0.777005i	$0.716738\pi$
$542$	6278.30	0.497557
$543$	0	0
$544$	−4244.08	−0.334491
$545$	−3098.36	−0.243522
$546$	0	0
$547$	19567.5	1.52952	0.764759	−	0.644316i	$-0.222858\pi$
$547$	19567.5	1.52952	0.764759	+	0.644316i	$0.222858\pi$
$548$	17753.0	1.38389
$549$	0	0
$550$	218.212	0.0169175
$551$	−3700.23	−0.286089
$552$	0	0
$553$	−2755.16	−0.211865
$554$	−2747.60	−0.210712
$555$	0	0
$556$	19199.0	1.46442
$557$	−24262.5	−1.84567	−0.922833	−	0.385200i	$-0.874133\pi$
$557$	−24262.5	−1.84567	−0.922833	+	0.385200i	$0.874133\pi$
$558$	0	0
$559$	9168.11	0.693685
$560$	716.589	0.0540739
$561$	0	0
$562$	−5987.83	−0.449433
$563$	−9322.42	−0.697857	−0.348928	−	0.937149i	$-0.613454\pi$
$563$	−9322.42	−0.697857	−0.348928	+	0.937149i	$0.613454\pi$
$564$	0	0
$565$	856.620	0.0637845
$566$	−2436.08	−0.180912
$567$	0	0
$568$	−5325.18	−0.393379
$569$	−12441.6	−0.916662	−0.458331	−	0.888782i	$-0.651553\pi$
$569$	−12441.6	−0.916662	−0.458331	+	0.888782i	$0.651553\pi$
$570$	0	0
$571$	−2994.01	−0.219432	−0.109716	−	0.993963i	$-0.534994\pi$
$571$	−2994.01	−0.219432	−0.109716	+	0.993963i	$0.534994\pi$
$572$	5552.61	0.405885
$573$	0	0
$574$	58.7405	0.00427140
$575$	−4515.37	−0.327485
$576$	0	0
$577$	5436.07	0.392212	0.196106	−	0.980583i	$-0.437170\pi$
$577$	5436.07	0.392212	0.196106	+	0.980583i	$0.437170\pi$
$578$	3133.37	0.225486
$579$	0	0
$580$	−2482.67	−0.177737
$581$	−2746.21	−0.196096
$582$	0	0
$583$	1008.41	0.0716368
$584$	−1116.13	−0.0790853
$585$	0	0
$586$	2983.90	0.210348
$587$	−7342.27	−0.516265	−0.258133	−	0.966109i	$-0.583107\pi$
$587$	−7342.27	−0.516265	−0.258133	+	0.966109i	$0.583107\pi$
$588$	0	0
$589$	8423.63	0.589286
$590$	−1724.66	−0.120344
$591$	0	0
$592$	−15992.3	−1.11027
$593$	−3362.02	−0.232819	−0.116409	−	0.993201i	$-0.537138\pi$
$593$	−3362.02	−0.232819	−0.116409	+	0.993201i	$0.537138\pi$
$594$	0	0
$595$	451.479	0.0311073
$596$	−2247.46	−0.154462
$597$	0	0
$598$	9815.58	0.671219
$599$	2197.77	0.149914	0.0749569	−	0.997187i	$-0.476118\pi$
$599$	2197.77	0.149914	0.0749569	+	0.997187i	$0.476118\pi$
$600$	0	0
$601$	−10659.0	−0.723442	−0.361721	−	0.932286i	$-0.617811\pi$
$601$	−10659.0	−0.723442	−0.361721	+	0.932286i	$0.617811\pi$
$602$	308.884	0.0209122
$603$	0	0
$604$	−9314.22	−0.627467
$605$	−605.000	−0.0406558
$606$	0	0
$607$	−18305.0	−1.22402	−0.612008	−	0.790851i	$-0.709638\pi$
$607$	−18305.0	−1.22402	−0.612008	+	0.790851i	$0.709638\pi$
$608$	−7507.05	−0.500742
$609$	0	0
$610$	−238.995	−0.0158633
$611$	−7792.84	−0.515981
$612$	0	0
$613$	2721.13	0.179291	0.0896455	−	0.995974i	$-0.471427\pi$
$613$	2721.13	0.179291	0.0896455	+	0.995974i	$0.471427\pi$
$614$	−1692.47	−0.111242
$615$	0	0
$616$	390.128	0.0255174
$617$	21160.0	1.38066	0.690331	−	0.723493i	$-0.257465\pi$
$617$	21160.0	1.38066	0.690331	+	0.723493i	$0.257465\pi$
$618$	0	0
$619$	15215.3	0.987971	0.493986	−	0.869470i	$-0.335539\pi$
$619$	15215.3	0.987971	0.493986	+	0.869470i	$0.335539\pi$
$620$	5651.83	0.366102
$621$	0	0
$622$	281.347	0.0181367
$623$	1203.02	0.0773641
$624$	0	0
$625$	625.000	0.0400000
$626$	−6903.65	−0.440775
$627$	0	0
$628$	27374.2	1.73941
$629$	−10075.8	−0.638707
$630$	0	0
$631$	27192.9	1.71558	0.857790	−	0.514001i	$-0.171837\pi$
$631$	27192.9	1.71558	0.857790	+	0.514001i	$0.171837\pi$
$632$	11555.6	0.727305
$633$	0	0
$634$	6439.25	0.403368
$635$	−5426.15	−0.339103
$636$	0	0
$637$	−22912.3	−1.42515
$638$	−588.030	−0.0364895
$639$	0	0
$640$	−6601.16	−0.407709
$641$	−19887.7	−1.22546	−0.612728	−	0.790294i	$-0.709928\pi$
$641$	−19887.7	−1.22546	−0.612728	+	0.790294i	$0.709928\pi$
$642$	0	0
$643$	−920.348	−0.0564463	−0.0282232	−	0.999602i	$-0.508985\pi$
$643$	−920.348	−0.0564463	−0.0282232	+	0.999602i	$0.508985\pi$
$644$	−3871.03	−0.236863
$645$	0	0
$646$	−1353.31	−0.0824232
$647$	21378.9	1.29906	0.649528	−	0.760337i	$-0.274966\pi$
$647$	21378.9	1.29906	0.649528	+	0.760337i	$0.274966\pi$
$648$	0	0
$649$	4781.67	0.289210
$650$	−1358.63	−0.0819846
$651$	0	0
$652$	−13730.4	−0.824730
$653$	16874.7	1.01127	0.505633	−	0.862749i	$-0.331259\pi$
$653$	16874.7	1.01127	0.505633	+	0.862749i	$0.331259\pi$
$654$	0	0
$655$	11774.4	0.702390
$656$	1254.65	0.0746735
$657$	0	0
$658$	−262.549	−0.0155551
$659$	19891.2	1.17580	0.587898	−	0.808935i	$-0.299956\pi$
$659$	19891.2	1.17580	0.587898	+	0.808935i	$0.299956\pi$
$660$	0	0
$661$	29758.4	1.75109	0.875543	−	0.483140i	$-0.160504\pi$
$661$	29758.4	1.75109	0.875543	+	0.483140i	$0.160504\pi$
$662$	5347.09	0.313929
$663$	0	0
$664$	11518.1	0.673174
$665$	798.589	0.0465684
$666$	0	0
$667$	12167.8	0.706358
$668$	8053.17	0.466447
$669$	0	0
$670$	−1745.00	−0.100620
$671$	662.622	0.0381226
$672$	0	0
$673$	−26748.0	−1.53203	−0.766017	−	0.642820i	$-0.777764\pi$
$673$	−26748.0	−1.53203	−0.766017	+	0.642820i	$0.777764\pi$
$674$	8599.00	0.491426
$675$	0	0
$676$	−18379.0	−1.04569
$677$	2938.53	0.166820	0.0834098	−	0.996515i	$-0.473419\pi$
$677$	2938.53	0.166820	0.0834098	+	0.996515i	$0.473419\pi$
$678$	0	0
$679$	4255.43	0.240513
$680$	−1893.58	−0.106787
$681$	0	0
$682$	1338.66	0.0751611
$683$	17208.6	0.964083	0.482042	−	0.876148i	$-0.339895\pi$
$683$	17208.6	0.964083	0.482042	+	0.876148i	$0.339895\pi$
$684$	0	0
$685$	12043.5	0.671765
$686$	−1563.39	−0.0870126
$687$	0	0
$688$	6597.50	0.365592
$689$	−6278.59	−0.347163
$690$	0	0
$691$	19023.6	1.04731	0.523655	−	0.851930i	$-0.324568\pi$
$691$	19023.6	1.04731	0.523655	+	0.851930i	$0.324568\pi$
$692$	−19495.9	−1.07099
$693$	0	0
$694$	3765.58	0.205965
$695$	13024.5	0.710857
$696$	0	0
$697$	790.478	0.0429576
$698$	591.022	0.0320494
$699$	0	0
$700$	535.813	0.0289312
$701$	8531.68	0.459682	0.229841	−	0.973228i	$-0.426179\pi$
$701$	8531.68	0.459682	0.229841	+	0.973228i	$0.426179\pi$
$702$	0	0
$703$	−17822.3	−0.956160
$704$	3144.09	0.168320
$705$	0	0
$706$	−4652.34	−0.248007
$707$	5697.35	0.303071
$708$	0	0
$709$	−11511.0	−0.609736	−0.304868	−	0.952395i	$-0.598612\pi$
$709$	−11511.0	−0.609736	−0.304868	+	0.952395i	$0.598612\pi$
$710$	−1732.29	−0.0915656
$711$	0	0
$712$	−5045.66	−0.265581
$713$	−27700.3	−1.45496
$714$	0	0
$715$	3766.85	0.197024
$716$	4851.38	0.253218
$717$	0	0
$718$	9838.49	0.511378
$719$	23498.9	1.21886	0.609430	−	0.792840i	$-0.291398\pi$
$719$	23498.9	1.21886	0.609430	+	0.792840i	$0.291398\pi$
$720$	0	0
$721$	439.184	0.0226852
$722$	3048.83	0.157155
$723$	0	0
$724$	−29379.1	−1.50810
$725$	−1684.23	−0.0862767
$726$	0	0
$727$	19718.4	1.00593	0.502967	−	0.864306i	$-0.332242\pi$
$727$	19718.4	1.00593	0.502967	+	0.864306i	$0.332242\pi$
$728$	−2429.02	−0.123661
$729$	0	0
$730$	−363.079	−0.0184084
$731$	4156.68	0.210315
$732$	0	0
$733$	−3618.29	−0.182325	−0.0911627	−	0.995836i	$-0.529058\pi$
$733$	−3618.29	−0.182325	−0.0911627	+	0.995836i	$0.529058\pi$
$734$	−7982.22	−0.401402
$735$	0	0
$736$	24686.2	1.23634
$737$	4838.07	0.241808
$738$	0	0
$739$	−29651.3	−1.47597	−0.737985	−	0.674817i	$-0.764223\pi$
$739$	−29651.3	−1.47597	−0.737985	+	0.674817i	$0.764223\pi$
$740$	−11957.9	−0.594026
$741$	0	0
$742$	−211.533	−0.0104658
$743$	28501.9	1.40731	0.703657	−	0.710540i	$-0.251549\pi$
$743$	28501.9	1.40731	0.703657	+	0.710540i	$0.251549\pi$
$744$	0	0
$745$	−1524.66	−0.0749787
$746$	8353.63	0.409984
$747$	0	0
$748$	2517.47	0.123059
$749$	158.317	0.00772331
$750$	0	0
$751$	33121.8	1.60936	0.804681	−	0.593707i	$-0.202336\pi$
$751$	33121.8	1.60936	0.804681	+	0.593707i	$0.202336\pi$
$752$	−5607.84	−0.271937
$753$	0	0
$754$	3661.19	0.176834
$755$	−6318.70	−0.304584
$756$	0	0
$757$	−7189.40	−0.345182	−0.172591	−	0.984994i	$-0.555214\pi$
$757$	−7189.40	−0.345182	−0.172591	+	0.984994i	$0.555214\pi$
$758$	2993.30	0.143432
$759$	0	0
$760$	−3349.42	−0.159863
$761$	12289.2	0.585392	0.292696	−	0.956205i	$-0.405448\pi$
$761$	12289.2	0.585392	0.292696	+	0.956205i	$0.405448\pi$
$762$	0	0
$763$	−1801.97	−0.0854988
$764$	15511.0	0.734511
$765$	0	0
$766$	5291.03	0.249573
$767$	−29771.7	−1.40155
$768$	0	0
$769$	−38979.9	−1.82790	−0.913948	−	0.405832i	$-0.866982\pi$
$769$	−38979.9	−1.82790	−0.913948	+	0.405832i	$0.866982\pi$
$770$	126.909	0.00593960
$771$	0	0
$772$	3196.15	0.149005
$773$	6846.25	0.318554	0.159277	−	0.987234i	$-0.449084\pi$
$773$	6846.25	0.318554	0.159277	+	0.987234i	$0.449084\pi$
$774$	0	0
$775$	3834.16	0.177712
$776$	−17848.0	−0.825652
$777$	0	0
$778$	2745.66	0.126525
$779$	1398.22	0.0643087
$780$	0	0
$781$	4802.82	0.220049
$782$	4450.23	0.203504
$783$	0	0
$784$	−16488.0	−0.751094
$785$	18570.5	0.844342
$786$	0	0
$787$	17561.3	0.795416	0.397708	−	0.917512i	$-0.369806\pi$
$787$	17561.3	0.795416	0.397708	+	0.917512i	$0.369806\pi$
$788$	−36101.5	−1.63206
$789$	0	0
$790$	3759.05	0.169292
$791$	498.199	0.0223943
$792$	0	0
$793$	−4125.62	−0.184748
$794$	2578.96	0.115269
$795$	0	0
$796$	−13755.9	−0.612517
$797$	40618.8	1.80526	0.902629	−	0.430419i	$-0.141634\pi$
$797$	40618.8	1.80526	0.902629	+	0.430419i	$0.141634\pi$
$798$	0	0
$799$	−3533.16	−0.156438
$800$	−3416.96	−0.151010
$801$	0	0
$802$	−2277.97	−0.100296
$803$	1006.65	0.0442389
$804$	0	0
$805$	−2626.08	−0.114978
$806$	−8334.75	−0.364242
$807$	0	0
$808$	−23895.7	−1.04040
$809$	33200.7	1.44286	0.721430	−	0.692488i	$-0.243485\pi$
$809$	33200.7	1.44286	0.721430	+	0.692488i	$0.243485\pi$
$810$	0	0
$811$	37275.9	1.61397	0.806987	−	0.590570i	$-0.201097\pi$
$811$	37275.9	1.61397	0.806987	+	0.590570i	$0.201097\pi$
$812$	−1443.89	−0.0624021
$813$	0	0
$814$	−2832.26	−0.121954
$815$	−9314.61	−0.400339
$816$	0	0
$817$	7352.47	0.314847
$818$	2512.25	0.107382
$819$	0	0
$820$	938.135	0.0399526
$821$	26000.2	1.10525	0.552626	−	0.833429i	$-0.313626\pi$
$821$	26000.2	1.10525	0.552626	+	0.833429i	$0.313626\pi$
$822$	0	0
$823$	−43390.1	−1.83777	−0.918884	−	0.394527i	$-0.870908\pi$
$823$	−43390.1	−1.83777	−0.918884	+	0.394527i	$0.870908\pi$
$824$	−1842.01	−0.0778756
$825$	0	0
$826$	−1003.04	−0.0422521
$827$	33496.3	1.40844	0.704221	−	0.709981i	$-0.251297\pi$
$827$	33496.3	1.40844	0.704221	+	0.709981i	$0.251297\pi$
$828$	0	0
$829$	14895.4	0.624051	0.312025	−	0.950074i	$-0.398993\pi$
$829$	14895.4	0.624051	0.312025	+	0.950074i	$0.398993\pi$
$830$	3746.84	0.156693
$831$	0	0
$832$	−19575.7	−0.815705
$833$	−10388.1	−0.432084
$834$	0	0
$835$	5463.21	0.226422
$836$	4452.97	0.184222
$837$	0	0
$838$	2368.40	0.0976313
$839$	20519.9	0.844370	0.422185	−	0.906510i	$-0.361263\pi$
$839$	20519.9	0.844370	0.422185	+	0.906510i	$0.361263\pi$
$840$	0	0
$841$	−19850.4	−0.813908
$842$	−1015.80	−0.0415758
$843$	0	0
$844$	30701.2	1.25211
$845$	−12468.2	−0.507596
$846$	0	0
$847$	−351.860	−0.0142740
$848$	−4518.16	−0.182965
$849$	0	0
$850$	−615.984	−0.0248565
$851$	58606.8	2.36077
$852$	0	0
$853$	27624.2	1.10883	0.554416	−	0.832240i	$-0.312942\pi$
$853$	27624.2	1.10883	0.554416	+	0.832240i	$0.312942\pi$
$854$	−138.997	−0.00556951
$855$	0	0
$856$	−664.006	−0.0265132
$857$	31086.1	1.23907	0.619533	−	0.784970i	$-0.287322\pi$
$857$	31086.1	1.23907	0.619533	+	0.784970i	$0.287322\pi$
$858$	0	0
$859$	−32657.7	−1.29717	−0.648583	−	0.761144i	$-0.724638\pi$
$859$	−32657.7	−1.29717	−0.648583	+	0.761144i	$0.724638\pi$
$860$	4933.13	0.195603
$861$	0	0
$862$	4723.63	0.186644
$863$	10499.4	0.414141	0.207070	−	0.978326i	$-0.433607\pi$
$863$	10499.4	0.414141	0.207070	+	0.978326i	$0.433607\pi$
$864$	0	0
$865$	−13225.9	−0.519878
$866$	−3315.35	−0.130093
$867$	0	0
$868$	3287.03	0.128536
$869$	−10422.1	−0.406841
$870$	0	0
$871$	−30122.8	−1.17184
$872$	7557.76	0.293507
$873$	0	0
$874$	7871.70	0.304650
$875$	363.492	0.0140437
$876$	0	0
$877$	6736.30	0.259371	0.129686	−	0.991555i	$-0.458603\pi$
$877$	6736.30	0.259371	0.129686	+	0.991555i	$0.458603\pi$
$878$	−309.058	−0.0118795
$879$	0	0
$880$	2710.68	0.103837
$881$	−38860.1	−1.48607	−0.743037	−	0.669251i	$-0.766615\pi$
$881$	−38860.1	−1.48607	−0.743037	+	0.669251i	$0.766615\pi$
$882$	0	0
$883$	−13526.6	−0.515524	−0.257762	−	0.966208i	$-0.582985\pi$
$883$	−13526.6	−0.515524	−0.257762	+	0.966208i	$0.582985\pi$
$884$	−15674.3	−0.596361
$885$	0	0
$886$	−3487.49	−0.132240
$887$	−2089.69	−0.0791036	−0.0395518	−	0.999218i	$-0.512593\pi$
$887$	−2089.69	−0.0791036	−0.0395518	+	0.999218i	$0.512593\pi$
$888$	0	0
$889$	−3155.78	−0.119057
$890$	−1641.36	−0.0618185
$891$	0	0
$892$	23225.3	0.871795
$893$	−6249.55	−0.234192
$894$	0	0
$895$	3291.14	0.122917
$896$	−3839.14	−0.143144
$897$	0	0
$898$	−4248.50	−0.157878
$899$	−10332.2	−0.383311
$900$	0	0
$901$	−2846.62	−0.105255
$902$	222.201	0.00820231
$903$	0	0
$904$	−2089.53	−0.0768769
$905$	−19930.6	−0.732060
$906$	0	0
$907$	20510.3	0.750862	0.375431	−	0.926850i	$-0.377495\pi$
$907$	20510.3	0.750862	0.375431	+	0.926850i	$0.377495\pi$
$908$	−25281.8	−0.924016
$909$	0	0
$910$	−790.163	−0.0287842
$911$	−34898.0	−1.26918	−0.634589	−	0.772850i	$-0.718831\pi$
$911$	−34898.0	−1.26918	−0.634589	+	0.772850i	$0.718831\pi$
$912$	0	0
$913$	−10388.2	−0.376561
$914$	3000.08	0.108571
$915$	0	0
$916$	29789.5	1.07453
$917$	6847.86	0.246604
$918$	0	0
$919$	−680.620	−0.0244304	−0.0122152	−	0.999925i	$-0.503888\pi$
$919$	−680.620	−0.0244304	−0.0122152	+	0.999925i	$0.503888\pi$
$920$	11014.2	0.394705
$921$	0	0
$922$	2816.24	0.100594
$923$	−29903.3	−1.06639
$924$	0	0
$925$	−8112.13	−0.288351
$926$	−2475.18	−0.0878396
$927$	0	0
$928$	9207.90	0.325716
$929$	−46344.7	−1.63673	−0.818364	−	0.574700i	$-0.805119\pi$
$929$	−46344.7	−1.63673	−0.818364	+	0.574700i	$0.805119\pi$
$930$	0	0
$931$	−18374.8	−0.646841
$932$	−18911.1	−0.664651
$933$	0	0
$934$	415.972	0.0145728
$935$	1707.83	0.0597349
$936$	0	0
$937$	50003.7	1.74338	0.871691	−	0.490056i	$-0.163024\pi$
$937$	50003.7	1.74338	0.871691	+	0.490056i	$0.163024\pi$
$938$	−1014.87	−0.0353269
$939$	0	0
$940$	−4193.13	−0.145495
$941$	−10202.4	−0.353442	−0.176721	−	0.984261i	$-0.556549\pi$
$941$	−10202.4	−0.353442	−0.176721	+	0.984261i	$0.556549\pi$
$942$	0	0
$943$	−4597.91	−0.158779
$944$	−21424.1	−0.738660
$945$	0	0
$946$	1168.43	0.0401575
$947$	−1386.09	−0.0475625	−0.0237813	−	0.999717i	$-0.507571\pi$
$947$	−1386.09	−0.0475625	−0.0237813	+	0.999717i	$0.507571\pi$
$948$	0	0
$949$	−6267.59	−0.214388
$950$	−1089.57	−0.0372109
$951$	0	0
$952$	−1101.28	−0.0374923
$953$	−16825.6	−0.571915	−0.285958	−	0.958242i	$-0.592312\pi$
$953$	−16825.6	−0.571915	−0.285958	+	0.958242i	$0.592312\pi$
$954$	0	0
$955$	10522.5	0.356545
$956$	−40832.9	−1.38141
$957$	0	0
$958$	−3688.93	−0.124409
$959$	7004.35	0.235852
$960$	0	0
$961$	−6269.72	−0.210457
$962$	17634.2	0.591009
$963$	0	0
$964$	44076.4	1.47262
$965$	2168.24	0.0723298
$966$	0	0
$967$	−39600.8	−1.31694	−0.658468	−	0.752609i	$-0.728795\pi$
$967$	−39600.8	−1.31694	−0.658468	+	0.752609i	$0.728795\pi$
$968$	1475.76	0.0490007
$969$	0	0
$970$	−5805.98	−0.192184
$971$	−36947.4	−1.22111	−0.610555	−	0.791974i	$-0.709054\pi$
$971$	−36947.4	−1.22111	−0.610555	+	0.791974i	$0.709054\pi$
$972$	0	0
$973$	7574.85	0.249577
$974$	14190.4	0.466827
$975$	0	0
$976$	−2968.85	−0.0973675
$977$	−20284.0	−0.664220	−0.332110	−	0.943241i	$-0.607760\pi$
$977$	−20284.0	−0.664220	−0.332110	+	0.943241i	$0.607760\pi$
$978$	0	0
$979$	4550.72	0.148561
$980$	−12328.5	−0.401858
$981$	0	0
$982$	−8492.54	−0.275976
$983$	30081.3	0.976038	0.488019	−	0.872833i	$-0.337720\pi$
$983$	30081.3	0.976038	0.488019	+	0.872833i	$0.337720\pi$
$984$	0	0
$985$	−24491.0	−0.792232
$986$	1659.93	0.0536135
$987$	0	0
$988$	−27725.1	−0.892767
$989$	−24177.8	−0.777362
$990$	0	0
$991$	−41404.2	−1.32719	−0.663596	−	0.748091i	$-0.730971\pi$
$991$	−41404.2	−1.32719	−0.663596	+	0.748091i	$0.730971\pi$
$992$	−20961.9	−0.670909
$993$	0	0
$994$	−1007.48	−0.0321481
$995$	−9331.88	−0.297327
$996$	0	0
$997$	23882.5	0.758642	0.379321	−	0.925265i	$-0.376158\pi$
$997$	23882.5	0.758642	0.379321	+	0.925265i	$0.376158\pi$
$998$	−3247.30	−0.102998
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.4.a.i.1.2		3
3.2	odd	2		165.4.a.g.1.2	✓	3
5.4	even	2		2475.4.a.z.1.2		3
15.2	even	4		825.4.c.m.199.4		6
15.8	even	4		825.4.c.m.199.3		6
15.14	odd	2		825.4.a.p.1.2		3
33.32	even	2		1815.4.a.q.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.g.1.2	✓	3	3.2	odd	2
495.4.a.i.1.2		3	1.1	even	1	trivial
825.4.a.p.1.2		3	15.14	odd	2
825.4.c.m.199.3		6	15.8	even	4
825.4.c.m.199.4		6	15.2	even	4
1815.4.a.q.1.2		3	33.32	even	2
2475.4.a.z.1.2		3	5.4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q-0.793499 q^{2} -7.37036 q^{4} -5.00000 q^{5} -2.90793 q^{7} +12.1964 q^{8} +O(q^{10})\)	\(q-0.793499 q^{2} -7.37036 q^{4} -5.00000 q^{5} -2.90793 q^{7} +12.1964 q^{8} +3.96749 q^{10} -11.0000 q^{11} +68.4882 q^{13} +2.30744 q^{14} +49.2851 q^{16} +31.0515 q^{17} +54.9249 q^{19} +36.8518 q^{20} +8.72849 q^{22} -180.615 q^{23} +25.0000 q^{25} -54.3453 q^{26} +21.4325 q^{28} -67.3690 q^{29} +153.367 q^{31} -136.679 q^{32} -24.6393 q^{34} +14.5397 q^{35} -324.485 q^{37} -43.5828 q^{38} -60.9818 q^{40} +25.4570 q^{41} +133.864 q^{43} +81.0740 q^{44} +143.318 q^{46} -113.784 q^{47} -334.544 q^{49} -19.8375 q^{50} -504.783 q^{52} -91.6741 q^{53} +55.0000 q^{55} -35.4662 q^{56} +53.4573 q^{58} -434.698 q^{59} -60.2384 q^{61} -121.696 q^{62} -285.826 q^{64} -342.441 q^{65} -439.825 q^{67} -228.861 q^{68} -11.5372 q^{70} -436.620 q^{71} -91.5134 q^{73} +257.478 q^{74} -404.816 q^{76} +31.9873 q^{77} +947.462 q^{79} -246.425 q^{80} -20.2001 q^{82} +944.385 q^{83} -155.258 q^{85} -106.221 q^{86} -134.160 q^{88} -413.702 q^{89} -199.159 q^{91} +1331.20 q^{92} +90.2872 q^{94} -274.624 q^{95} -1463.39 q^{97} +265.460 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 17 q^{4} - 15 q^{5} + 6 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q - q^2 + 17 * q^4 - 15 * q^5 + 6 * q^7 + 3 * q^8	\( 3 q - q^{2} + 17 q^{4} - 15 q^{5} + 6 q^{7} + 3 q^{8} + 5 q^{10} - 33 q^{11} - 20 q^{13} - 144 q^{14} + 25 q^{16} - 32 q^{17} + 116 q^{19} - 85 q^{20} + 11 q^{22} - 240 q^{23} + 75 q^{25} - 302 q^{26} + 160 q^{28} - 238 q^{29} + 92 q^{31} - 197 q^{32} + 354 q^{34} - 30 q^{35} - 90 q^{37} - 324 q^{38} - 15 q^{40} + 46 q^{41} - 134 q^{43} - 187 q^{44} - 240 q^{46} + 220 q^{47} - 457 q^{49} - 25 q^{50} - 1530 q^{52} + 798 q^{53} + 165 q^{55} - 688 q^{56} - 978 q^{58} - 1236 q^{59} + 342 q^{61} + 1792 q^{62} - 1919 q^{64} + 100 q^{65} + 764 q^{67} - 1074 q^{68} + 720 q^{70} - 1816 q^{71} + 100 q^{73} + 1874 q^{74} + 396 q^{76} - 66 q^{77} - 96 q^{79} - 125 q^{80} - 910 q^{82} - 858 q^{83} + 160 q^{85} - 188 q^{86} - 33 q^{88} - 838 q^{89} + 332 q^{91} + 688 q^{92} - 3112 q^{94} - 580 q^{95} - 1322 q^{97} - 1017 q^{98}+O(q^{100}) \) 3 * q - q^2 + 17 * q^4 - 15 * q^5 + 6 * q^7 + 3 * q^8 + 5 * q^10 - 33 * q^11 - 20 * q^13 - 144 * q^14 + 25 * q^16 - 32 * q^17 + 116 * q^19 - 85 * q^20 + 11 * q^22 - 240 * q^23 + 75 * q^25 - 302 * q^26 + 160 * q^28 - 238 * q^29 + 92 * q^31 - 197 * q^32 + 354 * q^34 - 30 * q^35 - 90 * q^37 - 324 * q^38 - 15 * q^40 + 46 * q^41 - 134 * q^43 - 187 * q^44 - 240 * q^46 + 220 * q^47 - 457 * q^49 - 25 * q^50 - 1530 * q^52 + 798 * q^53 + 165 * q^55 - 688 * q^56 - 978 * q^58 - 1236 * q^59 + 342 * q^61 + 1792 * q^62 - 1919 * q^64 + 100 * q^65 + 764 * q^67 - 1074 * q^68 + 720 * q^70 - 1816 * q^71 + 100 * q^73 + 1874 * q^74 + 396 * q^76 - 66 * q^77 - 96 * q^79 - 125 * q^80 - 910 * q^82 - 858 * q^83 + 160 * q^85 - 188 * q^86 - 33 * q^88 - 838 * q^89 + 332 * q^91 + 688 * q^92 - 3112 * q^94 - 580 * q^95 - 1322 * q^97 - 1017 * q^98

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