LMFDB - Embedded newform 495.4.a.g.1.3

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.788.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x - 3 $ x^3 - x^2 - 7*x - 3
Coefficient ring:	$\Z[a_1, a_2]$
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			1.3
Root			$-0.476452$ 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)

$f(q)$	$=$	$q+2.82009 q^{2} -0.0470959 q^{4} -5.00000 q^{5} -7.12434 q^{7} -22.6935 q^{8} +O(q^{10})$	\(q+2.82009 q^{2} -0.0470959 q^{4} -5.00000 q^{5} -7.12434 q^{7} -22.6935 q^{8} -14.1004 q^{10} +11.0000 q^{11} +66.8837 q^{13} -20.0913 q^{14} -63.6210 q^{16} +40.9051 q^{17} +4.85136 q^{19} +0.235480 q^{20} +31.0210 q^{22} +128.183 q^{23} +25.0000 q^{25} +188.618 q^{26} +0.335527 q^{28} +224.689 q^{29} -267.207 q^{31} +2.13129 q^{32} +115.356 q^{34} +35.6217 q^{35} +418.418 q^{37} +13.6813 q^{38} +113.468 q^{40} +496.282 q^{41} -90.9549 q^{43} -0.518055 q^{44} +361.486 q^{46} +203.640 q^{47} -292.244 q^{49} +70.5022 q^{50} -3.14995 q^{52} -219.416 q^{53} -55.0000 q^{55} +161.676 q^{56} +633.644 q^{58} -585.745 q^{59} +156.973 q^{61} -753.547 q^{62} +514.979 q^{64} -334.419 q^{65} -638.333 q^{67} -1.92646 q^{68} +100.456 q^{70} +961.243 q^{71} +223.078 q^{73} +1179.98 q^{74} -0.228479 q^{76} -78.3678 q^{77} +415.648 q^{79} +318.105 q^{80} +1399.56 q^{82} -44.7029 q^{83} -204.526 q^{85} -256.501 q^{86} -249.629 q^{88} -809.758 q^{89} -476.503 q^{91} -6.03688 q^{92} +574.282 q^{94} -24.2568 q^{95} -429.786 q^{97} -824.154 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} - 5 q^{4} - 15 q^{5} - 16 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q - q^2 - 5 * q^4 - 15 * q^5 - 16 * q^7 + 3 * q^8	$ 3 q - q^{2} - 5 q^{4} - 15 q^{5} - 16 q^{7} + 3 q^{8} + 5 q^{10} + 33 q^{11} - 42 q^{13} + 76 q^{14} - 85 q^{16} + 34 q^{17} - 280 q^{19} + 25 q^{20} - 11 q^{22} + 112 q^{23} + 75 q^{25} + 314 q^{26} - 280 q^{28} + 290 q^{29} - 392 q^{31} + 23 q^{32} + 310 q^{34} + 80 q^{35} + 570 q^{37} + 512 q^{38} - 15 q^{40} + 662 q^{41} - 68 q^{43} - 55 q^{44} + 728 q^{46} - 264 q^{47} + 731 q^{49} - 25 q^{50} + 582 q^{52} + 94 q^{53} - 165 q^{55} - 204 q^{56} - 54 q^{58} + 612 q^{59} - 582 q^{61} - 320 q^{62} + 347 q^{64} + 210 q^{65} + 940 q^{67} - 678 q^{68} - 380 q^{70} + 1616 q^{71} + 738 q^{73} + 642 q^{74} + 880 q^{76} - 176 q^{77} + 124 q^{79} + 425 q^{80} + 2126 q^{82} + 1232 q^{83} - 170 q^{85} + 1000 q^{86} + 33 q^{88} - 838 q^{89} - 1736 q^{91} - 1248 q^{92} + 2432 q^{94} + 1400 q^{95} - 90 q^{97} - 3481 q^{98}+O(q^{100}) $ 3 * q - q^2 - 5 * q^4 - 15 * q^5 - 16 * q^7 + 3 * q^8 + 5 * q^10 + 33 * q^11 - 42 * q^13 + 76 * q^14 - 85 * q^16 + 34 * q^17 - 280 * q^19 + 25 * q^20 - 11 * q^22 + 112 * q^23 + 75 * q^25 + 314 * q^26 - 280 * q^28 + 290 * q^29 - 392 * q^31 + 23 * q^32 + 310 * q^34 + 80 * q^35 + 570 * q^37 + 512 * q^38 - 15 * q^40 + 662 * q^41 - 68 * q^43 - 55 * q^44 + 728 * q^46 - 264 * q^47 + 731 * q^49 - 25 * q^50 + 582 * q^52 + 94 * q^53 - 165 * q^55 - 204 * q^56 - 54 * q^58 + 612 * q^59 - 582 * q^61 - 320 * q^62 + 347 * q^64 + 210 * q^65 + 940 * q^67 - 678 * q^68 - 380 * q^70 + 1616 * q^71 + 738 * q^73 + 642 * q^74 + 880 * q^76 - 176 * q^77 + 124 * q^79 + 425 * q^80 + 2126 * q^82 + 1232 * q^83 - 170 * q^85 + 1000 * q^86 + 33 * q^88 - 838 * q^89 - 1736 * q^91 - 1248 * q^92 + 2432 * q^94 + 1400 * q^95 - 90 * q^97 - 3481 * 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 $

$2$	2.82009	0.997052	0.498526	−	0.866875i	$-0.333875\pi$
$2$	2.82009	0.997052	0.498526	+	0.866875i	$0.333875\pi$
$3$	0	0
$3$	0	0
$4$	−0.0470959	−0.00588699
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−7.12434	−0.384678	−0.192339	−	0.981329i	$-0.561607\pi$
$7$	−7.12434	−0.384678	−0.192339	+	0.981329i	$0.561607\pi$
$8$	−22.6935	−1.00292
$9$	0	0
$10$	−14.1004	−0.445895
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	66.8837	1.42694	0.713470	−	0.700686i	$-0.247123\pi$
$13$	66.8837	1.42694	0.713470	+	0.700686i	$0.247123\pi$
$14$	−20.0913	−0.383544
$15$	0	0
$16$	−63.6210	−0.994078
$17$	40.9051	0.583585	0.291793	−	0.956482i	$-0.405748\pi$
$17$	40.9051	0.583585	0.291793	+	0.956482i	$0.405748\pi$
$18$	0	0
$19$	4.85136	0.0585779	0.0292889	−	0.999571i	$-0.490676\pi$
$19$	4.85136	0.0585779	0.0292889	+	0.999571i	$0.490676\pi$
$20$	0.235480	0.00263274
$21$	0	0
$22$	31.0210	0.300623
$23$	128.183	1.16208	0.581042	−	0.813874i	$-0.302645\pi$
$23$	128.183	1.16208	0.581042	+	0.813874i	$0.302645\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	188.618	1.42273
$27$	0	0
$28$	0.335527	0.00226460
$29$	224.689	1.43875	0.719375	−	0.694622i	$-0.244428\pi$
$29$	224.689	1.43875	0.719375	+	0.694622i	$0.244428\pi$
$30$	0	0
$31$	−267.207	−1.54812	−0.774060	−	0.633112i	$-0.781777\pi$
$31$	−267.207	−1.54812	−0.774060	+	0.633112i	$0.781777\pi$
$32$	2.13129	0.0117738
$33$	0	0
$34$	115.356	0.581865
$35$	35.6217	0.172033
$36$	0	0
$37$	418.418	1.85912	0.929561	−	0.368669i	$-0.120187\pi$
$37$	418.418	1.85912	0.929561	+	0.368669i	$0.120187\pi$
$38$	13.6813	0.0584052
$39$	0	0
$40$	113.468	0.448520
$41$	496.282	1.89040	0.945198	−	0.326498i	$-0.105869\pi$
$41$	496.282	1.89040	0.945198	+	0.326498i	$0.105869\pi$
$42$	0	0
$43$	−90.9549	−0.322570	−0.161285	−	0.986908i	$-0.551564\pi$
$43$	−90.9549	−0.322570	−0.161285	+	0.986908i	$0.551564\pi$
$44$	−0.518055	−0.00177499
$45$	0	0
$46$	361.486	1.15866
$47$	203.640	0.631998	0.315999	−	0.948760i	$-0.397660\pi$
$47$	203.640	0.631998	0.315999	+	0.948760i	$0.397660\pi$
$48$	0	0
$49$	−292.244	−0.852023
$50$	70.5022	0.199410
$51$	0	0
$52$	−3.14995	−0.00840038
$53$	−219.416	−0.568662	−0.284331	−	0.958726i	$-0.591771\pi$
$53$	−219.416	−0.568662	−0.284331	+	0.958726i	$0.591771\pi$
$54$	0	0
$55$	−55.0000	−0.134840
$56$	161.676	0.385802
$57$	0	0
$58$	633.644	1.43451
$59$	−585.745	−1.29250	−0.646250	−	0.763125i	$-0.723664\pi$
$59$	−585.745	−1.29250	−0.646250	+	0.763125i	$0.723664\pi$
$60$	0	0
$61$	156.973	0.329481	0.164740	−	0.986337i	$-0.447321\pi$
$61$	156.973	0.329481	0.164740	+	0.986337i	$0.447321\pi$
$62$	−753.547	−1.54356
$63$	0	0
$64$	514.979	1.00582
$65$	−334.419	−0.638147
$66$	0	0
$67$	−638.333	−1.16395	−0.581976	−	0.813206i	$-0.697720\pi$
$67$	−638.333	−1.16395	−0.581976	+	0.813206i	$0.697720\pi$
$68$	−1.92646	−0.00343556
$69$	0	0
$70$	100.456	0.171526
$71$	961.243	1.60674	0.803371	−	0.595479i	$-0.203038\pi$
$71$	961.243	1.60674	0.803371	+	0.595479i	$0.203038\pi$
$72$	0	0
$73$	223.078	0.357662	0.178831	−	0.983880i	$-0.442769\pi$
$73$	223.078	0.357662	0.178831	+	0.983880i	$0.442769\pi$
$74$	1179.98	1.85364
$75$	0	0
$76$	−0.228479	−0.000344847 0
$77$	−78.3678	−0.115985
$78$	0	0
$79$	415.648	0.591950	0.295975	−	0.955196i	$-0.404356\pi$
$79$	415.648	0.591950	0.295975	+	0.955196i	$0.404356\pi$
$80$	318.105	0.444565
$81$	0	0
$82$	1399.56	1.88482
$83$	−44.7029	−0.0591178	−0.0295589	−	0.999563i	$-0.509410\pi$
$83$	−44.7029	−0.0591178	−0.0295589	+	0.999563i	$0.509410\pi$
$84$	0	0
$85$	−204.526	−0.260987
$86$	−256.501	−0.321619
$87$	0	0
$88$	−249.629	−0.302392
$89$	−809.758	−0.964429	−0.482214	−	0.876053i	$-0.660167\pi$
$89$	−809.758	−0.964429	−0.482214	+	0.876053i	$0.660167\pi$
$90$	0	0
$91$	−476.503	−0.548913
$92$	−6.03688	−0.00684117
$93$	0	0
$94$	574.282	0.630135
$95$	−24.2568	−0.0261968
$96$	0	0
$97$	−429.786	−0.449878	−0.224939	−	0.974373i	$-0.572218\pi$
$97$	−429.786	−0.449878	−0.224939	+	0.974373i	$0.572218\pi$
$98$	−824.154	−0.849511
$99$	0	0
$100$	−1.17740	−0.00117740
$101$	1758.18	1.73213	0.866067	−	0.499928i	$-0.166640\pi$
$101$	1758.18	1.73213	0.866067	+	0.499928i	$0.166640\pi$
$102$	0	0
$103$	959.852	0.918223	0.459112	−	0.888379i	$-0.348168\pi$
$103$	959.852	0.918223	0.459112	+	0.888379i	$0.348168\pi$
$104$	−1517.83	−1.43111
$105$	0	0
$106$	−618.772	−0.566985
$107$	1120.32	1.01220	0.506098	−	0.862476i	$-0.331087\pi$
$107$	1120.32	1.01220	0.506098	+	0.862476i	$0.331087\pi$
$108$	0	0
$109$	−1811.63	−1.59195	−0.795973	−	0.605331i	$-0.793041\pi$
$109$	−1811.63	−1.59195	−0.795973	+	0.605331i	$0.793041\pi$
$110$	−155.105	−0.134442
$111$	0	0
$112$	453.258	0.382400
$113$	961.487	0.800435	0.400217	−	0.916420i	$-0.368935\pi$
$113$	961.487	0.800435	0.400217	+	0.916420i	$0.368935\pi$
$114$	0	0
$115$	−640.913	−0.519699
$116$	−10.5820	−0.00846991
$117$	0	0
$118$	−1651.85	−1.28869
$119$	−291.422	−0.224493
$120$	0	0
$121$	121.000	0.0909091
$122$	442.677	0.328509
$123$	0	0
$124$	12.5843	0.00911377
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1524.75	−1.06535	−0.532675	−	0.846320i	$-0.678813\pi$
$127$	−1524.75	−1.06535	−0.532675	+	0.846320i	$0.678813\pi$
$128$	1435.24	0.991079
$129$	0	0
$130$	−943.091	−0.636265
$131$	869.085	0.579636	0.289818	−	0.957082i	$-0.406405\pi$
$131$	869.085	0.579636	0.289818	+	0.957082i	$0.406405\pi$
$132$	0	0
$133$	−34.5628	−0.0225336
$134$	−1800.16	−1.16052
$135$	0	0
$136$	−928.281	−0.585290
$137$	−1050.15	−0.654893	−0.327447	−	0.944870i	$-0.606188\pi$
$137$	−1050.15	−0.654893	−0.327447	+	0.944870i	$0.606188\pi$
$138$	0	0
$139$	−272.869	−0.166507	−0.0832534	−	0.996528i	$-0.526531\pi$
$139$	−272.869	−0.166507	−0.0832534	+	0.996528i	$0.526531\pi$
$140$	−1.67764	−0.00101276
$141$	0	0
$142$	2710.79	1.60200
$143$	735.721	0.430238
$144$	0	0
$145$	−1123.45	−0.643429
$146$	629.100	0.356607
$147$	0	0
$148$	−19.7058	−0.0109446
$149$	199.929	0.109925	0.0549626	−	0.998488i	$-0.482496\pi$
$149$	199.929	0.109925	0.0549626	+	0.998488i	$0.482496\pi$
$150$	0	0
$151$	−2625.47	−1.41495	−0.707476	−	0.706737i	$-0.750166\pi$
$151$	−2625.47	−1.41495	−0.707476	+	0.706737i	$0.750166\pi$
$152$	−110.095	−0.0587490
$153$	0	0
$154$	−221.004	−0.115643
$155$	1336.03	0.692341
$156$	0	0
$157$	1581.68	0.804023	0.402012	−	0.915635i	$-0.368311\pi$
$157$	1581.68	0.804023	0.402012	+	0.915635i	$0.368311\pi$
$158$	1172.16	0.590205
$159$	0	0
$160$	−10.6565	−0.00526542
$161$	−913.216	−0.447028
$162$	0	0
$163$	1104.98	0.530974	0.265487	−	0.964114i	$-0.414467\pi$
$163$	1104.98	0.530974	0.265487	+	0.964114i	$0.414467\pi$
$164$	−23.3729	−0.0111287
$165$	0	0
$166$	−126.066	−0.0589436
$167$	200.333	0.0928276	0.0464138	−	0.998922i	$-0.485221\pi$
$167$	200.333	0.0928276	0.0464138	+	0.998922i	$0.485221\pi$
$168$	0	0
$169$	2276.43	1.03616
$170$	−576.780	−0.260218
$171$	0	0
$172$	4.28360	0.00189896
$173$	2070.79	0.910052	0.455026	−	0.890478i	$-0.349630\pi$
$173$	2070.79	0.910052	0.455026	+	0.890478i	$0.349630\pi$
$174$	0	0
$175$	−178.109	−0.0769357
$176$	−699.831	−0.299726
$177$	0	0
$178$	−2283.59	−0.961586
$179$	−2060.68	−0.860462	−0.430231	−	0.902719i	$-0.641568\pi$
$179$	−2060.68	−0.860462	−0.430231	+	0.902719i	$0.641568\pi$
$180$	0	0
$181$	4113.93	1.68943	0.844713	−	0.535220i	$-0.179771\pi$
$181$	4113.93	1.68943	0.844713	+	0.535220i	$0.179771\pi$
$182$	−1343.78	−0.547294
$183$	0	0
$184$	−2908.91	−1.16548
$185$	−2092.09	−0.831424
$186$	0	0
$187$	449.956	0.175958
$188$	−9.59060	−0.00372056
$189$	0	0
$190$	−68.4064	−0.0261196
$191$	−2771.96	−1.05012	−0.525058	−	0.851066i	$-0.675956\pi$
$191$	−2771.96	−1.05012	−0.525058	+	0.851066i	$0.675956\pi$
$192$	0	0
$193$	219.884	0.0820082	0.0410041	−	0.999159i	$-0.486944\pi$
$193$	219.884	0.0820082	0.0410041	+	0.999159i	$0.486944\pi$
$194$	−1212.03	−0.448552
$195$	0	0
$196$	13.7635	0.00501585
$197$	−2211.07	−0.799655	−0.399827	−	0.916590i	$-0.630930\pi$
$197$	−2211.07	−0.799655	−0.399827	+	0.916590i	$0.630930\pi$
$198$	0	0
$199$	−471.758	−0.168050	−0.0840252	−	0.996464i	$-0.526778\pi$
$199$	−471.758	−0.168050	−0.0840252	+	0.996464i	$0.526778\pi$
$200$	−567.338	−0.200584
$201$	0	0
$202$	4958.23	1.72703
$203$	−1600.76	−0.553456
$204$	0	0
$205$	−2481.41	−0.845411
$206$	2706.87	0.915517
$207$	0	0
$208$	−4255.21	−1.41849
$209$	53.3650	0.0176619
$210$	0	0
$211$	5623.68	1.83483	0.917417	−	0.397928i	$-0.130271\pi$
$211$	5623.68	1.83483	0.917417	+	0.397928i	$0.130271\pi$
$212$	10.3336	0.00334771
$213$	0	0
$214$	3159.39	1.00921
$215$	454.774	0.144257
$216$	0	0
$217$	1903.67	0.595529
$218$	−5108.94	−1.58725
$219$	0	0
$220$	2.59028	0.000793802 0
$221$	2735.89	0.832741
$222$	0	0
$223$	−4421.93	−1.32787	−0.663933	−	0.747792i	$-0.731114\pi$
$223$	−4421.93	−1.32787	−0.663933	+	0.747792i	$0.731114\pi$
$224$	−15.1840	−0.00452914
$225$	0	0
$226$	2711.48	0.798075
$227$	3489.83	1.02039	0.510194	−	0.860060i	$-0.329574\pi$
$227$	3489.83	1.02039	0.510194	+	0.860060i	$0.329574\pi$
$228$	0	0
$229$	−84.8948	−0.0244978	−0.0122489	−	0.999925i	$-0.503899\pi$
$229$	−84.8948	−0.0244978	−0.0122489	+	0.999925i	$0.503899\pi$
$230$	−1807.43	−0.518167
$231$	0	0
$232$	−5099.00	−1.44295
$233$	−4075.86	−1.14600	−0.573001	−	0.819555i	$-0.694221\pi$
$233$	−4075.86	−1.14600	−0.573001	+	0.819555i	$0.694221\pi$
$234$	0	0
$235$	−1018.20	−0.282638
$236$	27.5862	0.00760894
$237$	0	0
$238$	−821.836	−0.223831
$239$	−763.901	−0.206747	−0.103374	−	0.994643i	$-0.532964\pi$
$239$	−763.901	−0.206747	−0.103374	+	0.994643i	$0.532964\pi$
$240$	0	0
$241$	2526.32	0.675248	0.337624	−	0.941281i	$-0.390377\pi$
$241$	2526.32	0.675248	0.337624	+	0.941281i	$0.390377\pi$
$242$	341.231	0.0906411
$243$	0	0
$244$	−7.39278	−0.00193965
$245$	1461.22	0.381036
$246$	0	0
$247$	324.477	0.0835870
$248$	6063.86	1.55264
$249$	0	0
$250$	−352.511	−0.0891791
$251$	−963.988	−0.242416	−0.121208	−	0.992627i	$-0.538677\pi$
$251$	−963.988	−0.242416	−0.121208	+	0.992627i	$0.538677\pi$
$252$	0	0
$253$	1410.01	0.350381
$254$	−4299.92	−1.06221
$255$	0	0
$256$	−72.3368	−0.0176604
$257$	−1517.07	−0.368218	−0.184109	−	0.982906i	$-0.558940\pi$
$257$	−1517.07	−0.368218	−0.184109	+	0.982906i	$0.558940\pi$
$258$	0	0
$259$	−2980.95	−0.715164
$260$	15.7498	0.00375676
$261$	0	0
$262$	2450.90	0.577927
$263$	1420.58	0.333068	0.166534	−	0.986036i	$-0.446742\pi$
$263$	1420.58	0.333068	0.166534	+	0.986036i	$0.446742\pi$
$264$	0	0
$265$	1097.08	0.254313
$266$	−97.4701	−0.0224672
$267$	0	0
$268$	30.0629	0.00685217
$269$	−6870.16	−1.55718	−0.778590	−	0.627533i	$-0.784065\pi$
$269$	−6870.16	−1.55718	−0.778590	+	0.627533i	$0.784065\pi$
$270$	0	0
$271$	2499.86	0.560353	0.280177	−	0.959948i	$-0.409607\pi$
$271$	2499.86	0.560353	0.280177	+	0.959948i	$0.409607\pi$
$272$	−2602.42	−0.580129
$273$	0	0
$274$	−2961.52	−0.652963
$275$	275.000	0.0603023
$276$	0	0
$277$	−1562.11	−0.338837	−0.169419	−	0.985544i	$-0.554189\pi$
$277$	−1562.11	−0.338837	−0.169419	+	0.985544i	$0.554189\pi$
$278$	−769.515	−0.166016
$279$	0	0
$280$	−808.382	−0.172536
$281$	7476.58	1.58724	0.793621	−	0.608412i	$-0.208193\pi$
$281$	7476.58	1.58724	0.793621	+	0.608412i	$0.208193\pi$
$282$	0	0
$283$	2680.33	0.563001	0.281501	−	0.959561i	$-0.409168\pi$
$283$	2680.33	0.563001	0.281501	+	0.959561i	$0.409168\pi$
$284$	−45.2706	−0.00945887
$285$	0	0
$286$	2074.80	0.428970
$287$	−3535.68	−0.727194
$288$	0	0
$289$	−3239.77	−0.659428
$290$	−3168.22	−0.641532
$291$	0	0
$292$	−10.5061	−0.00210555
$293$	−1320.86	−0.263363	−0.131682	−	0.991292i	$-0.542038\pi$
$293$	−1320.86	−0.263363	−0.131682	+	0.991292i	$0.542038\pi$
$294$	0	0
$295$	2928.73	0.578024
$296$	−9495.38	−1.86455
$297$	0	0
$298$	563.819	0.109601
$299$	8573.33	1.65822
$300$	0	0
$301$	647.994	0.124085
$302$	−7404.06	−1.41078
$303$	0	0
$304$	−308.649	−0.0582310
$305$	−784.864	−0.147348
$306$	0	0
$307$	−6644.48	−1.23525	−0.617623	−	0.786475i	$-0.711904\pi$
$307$	−6644.48	−1.23525	−0.617623	+	0.786475i	$0.711904\pi$
$308$	3.69080	0.000682802 0
$309$	0	0
$310$	3767.73	0.690300
$311$	6911.90	1.26025	0.630125	−	0.776493i	$-0.283004\pi$
$311$	6911.90	1.26025	0.630125	+	0.776493i	$0.283004\pi$
$312$	0	0
$313$	−3967.78	−0.716524	−0.358262	−	0.933621i	$-0.616630\pi$
$313$	−3967.78	−0.716524	−0.358262	+	0.933621i	$0.616630\pi$
$314$	4460.47	0.801653
$315$	0	0
$316$	−19.5753	−0.00348480
$317$	3402.43	0.602838	0.301419	−	0.953492i	$-0.402540\pi$
$317$	3402.43	0.602838	0.301419	+	0.953492i	$0.402540\pi$
$318$	0	0
$319$	2471.58	0.433800
$320$	−2574.89	−0.449815
$321$	0	0
$322$	−2575.35	−0.445710
$323$	198.446	0.0341852
$324$	0	0
$325$	1672.09	0.285388
$326$	3116.15	0.529409
$327$	0	0
$328$	−11262.4	−1.89592
$329$	−1450.80	−0.243116
$330$	0	0
$331$	4518.53	0.750335	0.375167	−	0.926957i	$-0.377585\pi$
$331$	4518.53	0.750335	0.375167	+	0.926957i	$0.377585\pi$
$332$	2.10532	0.000348026 0
$333$	0	0
$334$	564.956	0.0925540
$335$	3191.66	0.520535
$336$	0	0
$337$	−615.717	−0.0995260	−0.0497630	−	0.998761i	$-0.515847\pi$
$337$	−615.717	−0.0995260	−0.0497630	+	0.998761i	$0.515847\pi$
$338$	6419.75	1.03310
$339$	0	0
$340$	9.63232	0.00153643
$341$	−2939.27	−0.466776
$342$	0	0
$343$	4525.69	0.712433
$344$	2064.09	0.323512
$345$	0	0
$346$	5839.80	0.907369
$347$	8001.58	1.23789	0.618944	−	0.785435i	$-0.287561\pi$
$347$	8001.58	1.23789	0.618944	+	0.785435i	$0.287561\pi$
$348$	0	0
$349$	934.074	0.143266	0.0716330	−	0.997431i	$-0.477179\pi$
$349$	934.074	0.143266	0.0716330	+	0.997431i	$0.477179\pi$
$350$	−502.282	−0.0767089
$351$	0	0
$352$	23.4442	0.00354994
$353$	−11951.7	−1.80205	−0.901026	−	0.433764i	$-0.857185\pi$
$353$	−11951.7	−1.80205	−0.901026	+	0.433764i	$0.857185\pi$
$354$	0	0
$355$	−4806.22	−0.718556
$356$	38.1363	0.00567758
$357$	0	0
$358$	−5811.31	−0.857926
$359$	−2014.59	−0.296173	−0.148086	−	0.988974i	$-0.547311\pi$
$359$	−2014.59	−0.296173	−0.148086	+	0.988974i	$0.547311\pi$
$360$	0	0
$361$	−6835.46	−0.996569
$362$	11601.7	1.68445
$363$	0	0
$364$	22.4413	0.00323144
$365$	−1115.39	−0.159951
$366$	0	0
$367$	4133.31	0.587893	0.293947	−	0.955822i	$-0.405031\pi$
$367$	4133.31	0.587893	0.293947	+	0.955822i	$0.405031\pi$
$368$	−8155.10	−1.15520
$369$	0	0
$370$	−5899.88	−0.828973
$371$	1563.19	0.218752
$372$	0	0
$373$	3255.81	0.451955	0.225978	−	0.974133i	$-0.427442\pi$
$373$	3255.81	0.451955	0.225978	+	0.974133i	$0.427442\pi$
$374$	1268.92	0.175439
$375$	0	0
$376$	−4621.30	−0.633844
$377$	15028.1	2.05301
$378$	0	0
$379$	−3279.44	−0.444469	−0.222234	−	0.974993i	$-0.571335\pi$
$379$	−3279.44	−0.444469	−0.222234	+	0.974993i	$0.571335\pi$
$380$	1.14240	0.000154220 0
$381$	0	0
$382$	−7817.18	−1.04702
$383$	−10824.2	−1.44410	−0.722050	−	0.691841i	$-0.756800\pi$
$383$	−10824.2	−1.44410	−0.722050	+	0.691841i	$0.756800\pi$
$384$	0	0
$385$	391.839	0.0518700
$386$	620.092	0.0817664
$387$	0	0
$388$	20.2412	0.00264843
$389$	8828.59	1.15071	0.575356	−	0.817903i	$-0.304863\pi$
$389$	8828.59	1.15071	0.575356	+	0.817903i	$0.304863\pi$
$390$	0	0
$391$	5243.32	0.678174
$392$	6632.04	0.854512
$393$	0	0
$394$	−6235.41	−0.797297
$395$	−2078.24	−0.264728
$396$	0	0
$397$	−8975.60	−1.13469	−0.567346	−	0.823480i	$-0.692030\pi$
$397$	−8975.60	−1.13469	−0.567346	+	0.823480i	$0.692030\pi$
$398$	−1330.40	−0.167555
$399$	0	0
$400$	−1590.53	−0.198816
$401$	−10084.8	−1.25589	−0.627947	−	0.778257i	$-0.716104\pi$
$401$	−10084.8	−1.25589	−0.627947	+	0.778257i	$0.716104\pi$
$402$	0	0
$403$	−17871.8	−2.20907
$404$	−82.8031	−0.0101971
$405$	0	0
$406$	−4514.30	−0.551825
$407$	4602.60	0.560546
$408$	0	0
$409$	7533.27	0.910749	0.455374	−	0.890300i	$-0.349505\pi$
$409$	7533.27	0.910749	0.455374	+	0.890300i	$0.349505\pi$
$410$	−6997.80	−0.842919
$411$	0	0
$412$	−45.2051	−0.00540557
$413$	4173.05	0.497197
$414$	0	0
$415$	223.514	0.0264383
$416$	142.549	0.0168005
$417$	0	0
$418$	150.494	0.0176098
$419$	1987.86	0.231774	0.115887	−	0.993262i	$-0.463029\pi$
$419$	1987.86	0.231774	0.115887	+	0.993262i	$0.463029\pi$
$420$	0	0
$421$	13280.0	1.53736	0.768678	−	0.639635i	$-0.220915\pi$
$421$	13280.0	1.53736	0.768678	+	0.639635i	$0.220915\pi$
$422$	15859.3	1.82942
$423$	0	0
$424$	4979.32	0.570323
$425$	1022.63	0.116717
$426$	0	0
$427$	−1118.33	−0.126744
$428$	−52.7623	−0.00595879
$429$	0	0
$430$	1282.50	0.143832
$431$	11935.3	1.33388	0.666939	−	0.745112i	$-0.267604\pi$
$431$	11935.3	1.33388	0.666939	+	0.745112i	$0.267604\pi$
$432$	0	0
$433$	12937.5	1.43588	0.717942	−	0.696102i	$-0.245084\pi$
$433$	12937.5	1.43588	0.717942	+	0.696102i	$0.245084\pi$
$434$	5368.53	0.593773
$435$	0	0
$436$	85.3202	0.00937178
$437$	621.860	0.0680723
$438$	0	0
$439$	−2181.67	−0.237187	−0.118594	−	0.992943i	$-0.537839\pi$
$439$	−2181.67	−0.237187	−0.118594	+	0.992943i	$0.537839\pi$
$440$	1248.14	0.135234
$441$	0	0
$442$	7715.45	0.830286
$443$	8733.61	0.936673	0.468337	−	0.883550i	$-0.344853\pi$
$443$	8733.61	0.936673	0.468337	+	0.883550i	$0.344853\pi$
$444$	0	0
$445$	4048.79	0.431306
$446$	−12470.2	−1.32395
$447$	0	0
$448$	−3668.88	−0.386916
$449$	10113.3	1.06298	0.531490	−	0.847065i	$-0.321632\pi$
$449$	10113.3	1.06298	0.531490	+	0.847065i	$0.321632\pi$
$450$	0	0
$451$	5459.10	0.569976
$452$	−45.2821	−0.00471215
$453$	0	0
$454$	9841.62	1.01738
$455$	2382.51	0.245481
$456$	0	0
$457$	18924.7	1.93711	0.968557	−	0.248793i	$-0.0800338\pi$
$457$	18924.7	1.93711	0.968557	+	0.248793i	$0.0800338\pi$
$458$	−239.411	−0.0244256
$459$	0	0
$460$	30.1844	0.00305946
$461$	−12279.2	−1.24056	−0.620280	−	0.784380i	$-0.712981\pi$
$461$	−12279.2	−1.24056	−0.620280	+	0.784380i	$0.712981\pi$
$462$	0	0
$463$	−2512.53	−0.252197	−0.126098	−	0.992018i	$-0.540245\pi$
$463$	−2512.53	−0.252197	−0.126098	+	0.992018i	$0.540245\pi$
$464$	−14295.0	−1.43023
$465$	0	0
$466$	−11494.3	−1.14262
$467$	−13924.8	−1.37979	−0.689894	−	0.723910i	$-0.742343\pi$
$467$	−13924.8	−1.37979	−0.689894	+	0.723910i	$0.742343\pi$
$468$	0	0
$469$	4547.70	0.447747
$470$	−2871.41	−0.281805
$471$	0	0
$472$	13292.6	1.29628
$473$	−1000.50	−0.0972584
$474$	0	0
$475$	121.284	0.0117156
$476$	13.7248	0.00132159
$477$	0	0
$478$	−2154.27	−0.206138
$479$	17622.0	1.68094	0.840468	−	0.541862i	$-0.182280\pi$
$479$	17622.0	1.68094	0.840468	+	0.541862i	$0.182280\pi$
$480$	0	0
$481$	27985.4	2.65285
$482$	7124.46	0.673257
$483$	0	0
$484$	−5.69861	−0.000535181 0
$485$	2148.93	0.201191
$486$	0	0
$487$	−6109.87	−0.568510	−0.284255	−	0.958749i	$-0.591746\pi$
$487$	−6109.87	−0.568510	−0.284255	+	0.958749i	$0.591746\pi$
$488$	−3562.27	−0.330443
$489$	0	0
$490$	4120.77	0.379913
$491$	−4134.23	−0.379990	−0.189995	−	0.981785i	$-0.560847\pi$
$491$	−4134.23	−0.379990	−0.189995	+	0.981785i	$0.560847\pi$
$492$	0	0
$493$	9190.95	0.839634
$494$	915.055	0.0833406
$495$	0	0
$496$	17000.0	1.53895
$497$	−6848.23	−0.618078
$498$	0	0
$499$	16556.0	1.48527	0.742633	−	0.669698i	$-0.233577\pi$
$499$	16556.0	1.48527	0.742633	+	0.669698i	$0.233577\pi$
$500$	5.88699	0.000526548 0
$501$	0	0
$502$	−2718.53	−0.241701
$503$	−22181.6	−1.96626	−0.983132	−	0.182899i	$-0.941452\pi$
$503$	−22181.6	−1.96626	−0.983132	+	0.182899i	$0.941452\pi$
$504$	0	0
$505$	−8790.90	−0.774634
$506$	3976.35	0.349348
$507$	0	0
$508$	71.8094	0.00627171
$509$	19760.4	1.72076	0.860380	−	0.509654i	$-0.170226\pi$
$509$	19760.4	1.72076	0.860380	+	0.509654i	$0.170226\pi$
$510$	0	0
$511$	−1589.28	−0.137585
$512$	−11685.9	−1.00869
$513$	0	0
$514$	−4278.26	−0.367132
$515$	−4799.26	−0.410642
$516$	0	0
$517$	2240.04	0.190555
$518$	−8406.55	−0.713055
$519$	0	0
$520$	7589.14	0.640011
$521$	10283.0	0.864697	0.432349	−	0.901707i	$-0.357685\pi$
$521$	10283.0	0.864697	0.432349	+	0.901707i	$0.357685\pi$
$522$	0	0
$523$	−1923.24	−0.160798	−0.0803992	−	0.996763i	$-0.525620\pi$
$523$	−1923.24	−0.160798	−0.0803992	+	0.996763i	$0.525620\pi$
$524$	−40.9304	−0.00341231
$525$	0	0
$526$	4006.17	0.332086
$527$	−10930.1	−0.903460
$528$	0	0
$529$	4263.77	0.350437
$530$	3093.86	0.253564
$531$	0	0
$532$	1.62777	0.000132655 0
$533$	33193.2	2.69748
$534$	0	0
$535$	−5601.58	−0.452668
$536$	14486.0	1.16735
$537$	0	0
$538$	−19374.5	−1.55259
$539$	−3214.68	−0.256894
$540$	0	0
$541$	−10243.2	−0.814027	−0.407014	−	0.913422i	$-0.633430\pi$
$541$	−10243.2	−0.814027	−0.407014	+	0.913422i	$0.633430\pi$
$542$	7049.83	0.558701
$543$	0	0
$544$	87.1807	0.00687103
$545$	9058.13	0.711940
$546$	0	0
$547$	−483.940	−0.0378277	−0.0189139	−	0.999821i	$-0.506021\pi$
$547$	−483.940	−0.0378277	−0.0189139	+	0.999821i	$0.506021\pi$
$548$	49.4578	0.00385535
$549$	0	0
$550$	775.525	0.0601245
$551$	1090.05	0.0842789
$552$	0	0
$553$	−2961.22	−0.227710
$554$	−4405.28	−0.337839
$555$	0	0
$556$	12.8510	0.000980223 0
$557$	−23789.2	−1.80966	−0.904831	−	0.425771i	$-0.860003\pi$
$557$	−23789.2	−1.80966	−0.904831	+	0.425771i	$0.860003\pi$
$558$	0	0
$559$	−6083.40	−0.460287
$560$	−2266.29	−0.171015
$561$	0	0
$562$	21084.6	1.58256
$563$	−11331.9	−0.848280	−0.424140	−	0.905597i	$-0.639424\pi$
$563$	−11331.9	−0.848280	−0.424140	+	0.905597i	$0.639424\pi$
$564$	0	0
$565$	−4807.44	−0.357965
$566$	7558.78	0.561342
$567$	0	0
$568$	−21814.0	−1.61144
$569$	−2432.39	−0.179211	−0.0896056	−	0.995977i	$-0.528561\pi$
$569$	−2432.39	−0.179211	−0.0896056	+	0.995977i	$0.528561\pi$
$570$	0	0
$571$	20528.2	1.50451	0.752257	−	0.658870i	$-0.228965\pi$
$571$	20528.2	1.50451	0.752257	+	0.658870i	$0.228965\pi$
$572$	−34.6495	−0.00253281
$573$	0	0
$574$	−9970.94	−0.725051
$575$	3204.56	0.232417
$576$	0	0
$577$	13961.9	1.00735	0.503675	−	0.863893i	$-0.331981\pi$
$577$	13961.9	1.00735	0.503675	+	0.863893i	$0.331981\pi$
$578$	−9136.45	−0.657484
$579$	0	0
$580$	52.9098	0.00378786
$581$	318.479	0.0227413
$582$	0	0
$583$	−2413.57	−0.171458
$584$	−5062.43	−0.358707
$585$	0	0
$586$	−3724.94	−0.262587
$587$	10785.8	0.758393	0.379196	−	0.925316i	$-0.376200\pi$
$587$	10785.8	0.758393	0.379196	+	0.925316i	$0.376200\pi$
$588$	0	0
$589$	−1296.32	−0.0906856
$590$	8259.27	0.576320
$591$	0	0
$592$	−26620.2	−1.84811
$593$	8493.60	0.588179	0.294090	−	0.955778i	$-0.404984\pi$
$593$	8493.60	0.588179	0.294090	+	0.955778i	$0.404984\pi$
$594$	0	0
$595$	1457.11	0.100396
$596$	−9.41586	−0.000647128 0
$597$	0	0
$598$	24177.6	1.65333
$599$	12058.7	0.822549	0.411274	−	0.911512i	$-0.365084\pi$
$599$	12058.7	0.822549	0.411274	+	0.911512i	$0.365084\pi$
$600$	0	0
$601$	−12006.7	−0.814914	−0.407457	−	0.913224i	$-0.633584\pi$
$601$	−12006.7	−0.814914	−0.407457	+	0.913224i	$0.633584\pi$
$602$	1827.40	0.123720
$603$	0	0
$604$	123.649	0.00832981
$605$	−605.000	−0.0406558
$606$	0	0
$607$	−27404.7	−1.83249	−0.916245	−	0.400619i	$-0.868795\pi$
$607$	−27404.7	−1.83249	−0.916245	+	0.400619i	$0.868795\pi$
$608$	10.3397	0.000689686 0
$609$	0	0
$610$	−2213.39	−0.146914
$611$	13620.2	0.901822
$612$	0	0
$613$	−19837.8	−1.30708	−0.653541	−	0.756891i	$-0.726717\pi$
$613$	−19837.8	−1.30708	−0.653541	+	0.756891i	$0.726717\pi$
$614$	−18738.0	−1.23160
$615$	0	0
$616$	1778.44	0.116324
$617$	6884.10	0.449179	0.224589	−	0.974453i	$-0.427896\pi$
$617$	6884.10	0.449179	0.224589	+	0.974453i	$0.427896\pi$
$618$	0	0
$619$	−11903.8	−0.772944	−0.386472	−	0.922301i	$-0.626306\pi$
$619$	−11903.8	−0.772944	−0.386472	+	0.922301i	$0.626306\pi$
$620$	−62.9217	−0.00407580
$621$	0	0
$622$	19492.2	1.25654
$623$	5768.99	0.370995
$624$	0	0
$625$	625.000	0.0400000
$626$	−11189.5	−0.714411
$627$	0	0
$628$	−74.4906	−0.00473328
$629$	17115.4	1.08496
$630$	0	0
$631$	−912.730	−0.0575836	−0.0287918	−	0.999585i	$-0.509166\pi$
$631$	−912.730	−0.0575836	−0.0287918	+	0.999585i	$0.509166\pi$
$632$	−9432.51	−0.593679
$633$	0	0
$634$	9595.16	0.601061
$635$	7623.74	0.476439
$636$	0	0
$637$	−19546.4	−1.21578
$638$	6970.09	0.432521
$639$	0	0
$640$	−7176.18	−0.443224
$641$	−17066.1	−1.05159	−0.525796	−	0.850610i	$-0.676233\pi$
$641$	−17066.1	−1.05159	−0.525796	+	0.850610i	$0.676233\pi$
$642$	0	0
$643$	−27311.7	−1.67507	−0.837535	−	0.546384i	$-0.816004\pi$
$643$	−27311.7	−1.67507	−0.837535	+	0.546384i	$0.816004\pi$
$644$	43.0088	0.00263165
$645$	0	0
$646$	559.634	0.0340844
$647$	−15190.8	−0.923050	−0.461525	−	0.887127i	$-0.652697\pi$
$647$	−15190.8	−0.923050	−0.461525	+	0.887127i	$0.652697\pi$
$648$	0	0
$649$	−6443.20	−0.389704
$650$	4715.45	0.284547
$651$	0	0
$652$	−52.0401	−0.00312584
$653$	−9474.30	−0.567776	−0.283888	−	0.958857i	$-0.591624\pi$
$653$	−9474.30	−0.567776	−0.283888	+	0.958857i	$0.591624\pi$
$654$	0	0
$655$	−4345.42	−0.259221
$656$	−31574.0	−1.87920
$657$	0	0
$658$	−4091.38	−0.242399
$659$	1220.71	0.0721582	0.0360791	−	0.999349i	$-0.488513\pi$
$659$	1220.71	0.0721582	0.0360791	+	0.999349i	$0.488513\pi$
$660$	0	0
$661$	−22686.2	−1.33494	−0.667468	−	0.744639i	$-0.732622\pi$
$661$	−22686.2	−1.33494	−0.667468	+	0.744639i	$0.732622\pi$
$662$	12742.7	0.748123
$663$	0	0
$664$	1014.47	0.0592906
$665$	172.814	0.0100773
$666$	0	0
$667$	28801.3	1.67195
$668$	−9.43486	−0.000546475 0
$669$	0	0
$670$	9000.78	0.519001
$671$	1726.70	0.0993421
$672$	0	0
$673$	−2761.82	−0.158188	−0.0790938	−	0.996867i	$-0.525203\pi$
$673$	−2761.82	−0.158188	−0.0790938	+	0.996867i	$0.525203\pi$
$674$	−1736.38	−0.0992326
$675$	0	0
$676$	−107.211	−0.00609984
$677$	−16975.2	−0.963680	−0.481840	−	0.876259i	$-0.660031\pi$
$677$	−16975.2	−0.963680	−0.481840	+	0.876259i	$0.660031\pi$
$678$	0	0
$679$	3061.94	0.173058
$680$	4641.41	0.261750
$681$	0	0
$682$	−8289.02	−0.465400
$683$	−2447.72	−0.137130	−0.0685649	−	0.997647i	$-0.521842\pi$
$683$	−2447.72	−0.137130	−0.0685649	+	0.997647i	$0.521842\pi$
$684$	0	0
$685$	5250.75	0.292877
$686$	12762.9	0.710333
$687$	0	0
$688$	5786.64	0.320659
$689$	−14675.3	−0.811446
$690$	0	0
$691$	−27460.5	−1.51179	−0.755894	−	0.654694i	$-0.772797\pi$
$691$	−27460.5	−1.51179	−0.755894	+	0.654694i	$0.772797\pi$
$692$	−97.5256	−0.00535747
$693$	0	0
$694$	22565.2	1.23424
$695$	1364.34	0.0744641
$696$	0	0
$697$	20300.5	1.10321
$698$	2634.17	0.142844
$699$	0	0
$700$	8.38819	0.000452919 0
$701$	7980.51	0.429986	0.214993	−	0.976616i	$-0.431027\pi$
$701$	7980.51	0.429986	0.214993	+	0.976616i	$0.431027\pi$
$702$	0	0
$703$	2029.90	0.108903
$704$	5664.76	0.303265
$705$	0	0
$706$	−33704.9	−1.79674
$707$	−12525.9	−0.666314
$708$	0	0
$709$	4867.53	0.257834	0.128917	−	0.991655i	$-0.458850\pi$
$709$	4867.53	0.257834	0.128917	+	0.991655i	$0.458850\pi$
$710$	−13554.0	−0.716438
$711$	0	0
$712$	18376.3	0.967247
$713$	−34251.2	−1.79905
$714$	0	0
$715$	−3678.61	−0.192408
$716$	97.0498	0.00506553
$717$	0	0
$718$	−5681.33	−0.295300
$719$	−17353.3	−0.900094	−0.450047	−	0.893005i	$-0.648593\pi$
$719$	−17353.3	−0.900094	−0.450047	+	0.893005i	$0.648593\pi$
$720$	0	0
$721$	−6838.31	−0.353221
$722$	−19276.6	−0.993631
$723$	0	0
$724$	−193.749	−0.00994563
$725$	5617.23	0.287750
$726$	0	0
$727$	−37570.5	−1.91666	−0.958330	−	0.285663i	$-0.907786\pi$
$727$	−37570.5	−1.91666	−0.958330	+	0.285663i	$0.907786\pi$
$728$	10813.5	0.550516
$729$	0	0
$730$	−3145.50	−0.159480
$731$	−3720.52	−0.188247
$732$	0	0
$733$	10035.5	0.505689	0.252845	−	0.967507i	$-0.418634\pi$
$733$	10035.5	0.505689	0.252845	+	0.967507i	$0.418634\pi$
$734$	11656.3	0.586160
$735$	0	0
$736$	273.194	0.0136822
$737$	−7021.66	−0.350945
$738$	0	0
$739$	−20918.8	−1.04129	−0.520643	−	0.853775i	$-0.674308\pi$
$739$	−20918.8	−1.04129	−0.520643	+	0.853775i	$0.674308\pi$
$740$	98.5289	0.00489459
$741$	0	0
$742$	4408.34	0.218107
$743$	−25389.4	−1.25363	−0.626815	−	0.779168i	$-0.715642\pi$
$743$	−25389.4	−1.25363	−0.626815	+	0.779168i	$0.715642\pi$
$744$	0	0
$745$	−999.647	−0.0491600
$746$	9181.66	0.450623
$747$	0	0
$748$	−21.1911	−0.00103586
$749$	−7981.51	−0.389370
$750$	0	0
$751$	14657.0	0.712170	0.356085	−	0.934454i	$-0.384111\pi$
$751$	14657.0	0.712170	0.356085	+	0.934454i	$0.384111\pi$
$752$	−12955.8	−0.628255
$753$	0	0
$754$	42380.5	2.04696
$755$	13127.4	0.632786
$756$	0	0
$757$	7874.42	0.378072	0.189036	−	0.981970i	$-0.439464\pi$
$757$	7874.42	0.378072	0.189036	+	0.981970i	$0.439464\pi$
$758$	−9248.33	−0.443159
$759$	0	0
$760$	550.473	0.0262734
$761$	20257.8	0.964975	0.482487	−	0.875903i	$-0.339733\pi$
$761$	20257.8	0.964975	0.482487	+	0.875903i	$0.339733\pi$
$762$	0	0
$763$	12906.6	0.612387
$764$	130.548	0.00618202
$765$	0	0
$766$	−30525.2	−1.43984
$767$	−39176.8	−1.84432
$768$	0	0
$769$	−39900.7	−1.87107	−0.935536	−	0.353231i	$-0.885083\pi$
$769$	−39900.7	−1.87107	−0.935536	+	0.353231i	$0.885083\pi$
$770$	1105.02	0.0517171
$771$	0	0
$772$	−10.3556	−0.000482781 0
$773$	−22980.3	−1.06927	−0.534635	−	0.845083i	$-0.679551\pi$
$773$	−22980.3	−1.06927	−0.534635	+	0.845083i	$0.679551\pi$
$774$	0	0
$775$	−6680.17	−0.309624
$776$	9753.36	0.451192
$777$	0	0
$778$	24897.4	1.14732
$779$	2407.65	0.110735
$780$	0	0
$781$	10573.7	0.484451
$782$	14786.6	0.676175
$783$	0	0
$784$	18592.8	0.846977
$785$	−7908.39	−0.359570
$786$	0	0
$787$	−2717.00	−0.123063	−0.0615316	−	0.998105i	$-0.519598\pi$
$787$	−2717.00	−0.123063	−0.0615316	+	0.998105i	$0.519598\pi$
$788$	104.132	0.00470756
$789$	0	0
$790$	−5860.82	−0.263948
$791$	−6849.96	−0.307910
$792$	0	0
$793$	10498.9	0.470149
$794$	−25312.0	−1.13135
$795$	0	0
$796$	22.2179	0.000989311 0
$797$	29346.0	1.30425	0.652125	−	0.758111i	$-0.273878\pi$
$797$	29346.0	1.30425	0.652125	+	0.758111i	$0.273878\pi$
$798$	0	0
$799$	8329.90	0.368825
$800$	53.2823	0.00235477
$801$	0	0
$802$	−28440.2	−1.25219
$803$	2453.86	0.107839
$804$	0	0
$805$	4566.08	0.199917
$806$	−50400.0	−2.20256
$807$	0	0
$808$	−39899.3	−1.73719
$809$	19651.7	0.854037	0.427019	−	0.904243i	$-0.359564\pi$
$809$	19651.7	0.854037	0.427019	+	0.904243i	$0.359564\pi$
$810$	0	0
$811$	−33226.2	−1.43863	−0.719316	−	0.694683i	$-0.755544\pi$
$811$	−33226.2	−1.43863	−0.719316	+	0.694683i	$0.755544\pi$
$812$	75.3895	0.00325819
$813$	0	0
$814$	12979.7	0.558894
$815$	−5524.91	−0.237459
$816$	0	0
$817$	−441.255	−0.0188954
$818$	21244.5	0.908064
$819$	0	0
$820$	116.864	0.00497692
$821$	6761.60	0.287432	0.143716	−	0.989619i	$-0.454095\pi$
$821$	6761.60	0.287432	0.143716	+	0.989619i	$0.454095\pi$
$822$	0	0
$823$	30624.0	1.29707	0.648534	−	0.761186i	$-0.275383\pi$
$823$	30624.0	1.29707	0.648534	+	0.761186i	$0.275383\pi$
$824$	−21782.4	−0.920906
$825$	0	0
$826$	11768.4	0.495731
$827$	−18267.5	−0.768106	−0.384053	−	0.923311i	$-0.625472\pi$
$827$	−18267.5	−0.768106	−0.384053	+	0.923311i	$0.625472\pi$
$828$	0	0
$829$	−15153.2	−0.634853	−0.317427	−	0.948283i	$-0.602819\pi$
$829$	−15153.2	−0.634853	−0.317427	+	0.948283i	$0.602819\pi$
$830$	630.331	0.0263604
$831$	0	0
$832$	34443.7	1.43524
$833$	−11954.3	−0.497228
$834$	0	0
$835$	−1001.66	−0.0415138
$836$	−2.51327	−0.000103975 0
$837$	0	0
$838$	5605.93	0.231090
$839$	−22328.6	−0.918794	−0.459397	−	0.888231i	$-0.651934\pi$
$839$	−22328.6	−0.918794	−0.459397	+	0.888231i	$0.651934\pi$
$840$	0	0
$841$	26096.3	1.07000
$842$	37450.8	1.53283
$843$	0	0
$844$	−264.852	−0.0108016
$845$	−11382.2	−0.463383
$846$	0	0
$847$	−862.045	−0.0349708
$848$	13959.5	0.565294
$849$	0	0
$850$	2883.90	0.116373
$851$	53633.9	2.16045
$852$	0	0
$853$	−888.939	−0.0356819	−0.0178410	−	0.999841i	$-0.505679\pi$
$853$	−888.939	−0.0356819	−0.0178410	+	0.999841i	$0.505679\pi$
$854$	−3153.79	−0.126370
$855$	0	0
$856$	−25423.9	−1.01515
$857$	−11712.7	−0.466861	−0.233430	−	0.972374i	$-0.574995\pi$
$857$	−11712.7	−0.466861	−0.233430	+	0.972374i	$0.574995\pi$
$858$	0	0
$859$	−9943.60	−0.394961	−0.197480	−	0.980307i	$-0.563276\pi$
$859$	−9943.60	−0.394961	−0.197480	+	0.980307i	$0.563276\pi$
$860$	−21.4180	−0.000849242 0
$861$	0	0
$862$	33658.5	1.32995
$863$	−21912.2	−0.864309	−0.432154	−	0.901800i	$-0.642246\pi$
$863$	−21912.2	−0.864309	−0.432154	+	0.901800i	$0.642246\pi$
$864$	0	0
$865$	−10353.9	−0.406987
$866$	36485.0	1.43165
$867$	0	0
$868$	−89.6552	−0.00350587
$869$	4572.13	0.178480
$870$	0	0
$871$	−42694.1	−1.66089
$872$	41112.2	1.59660
$873$	0	0
$874$	1753.70	0.0678717
$875$	890.543	0.0344067
$876$	0	0
$877$	−3185.48	−0.122652	−0.0613261	−	0.998118i	$-0.519533\pi$
$877$	−3185.48	−0.122652	−0.0613261	+	0.998118i	$0.519533\pi$
$878$	−6152.50	−0.236488
$879$	0	0
$880$	3499.16	0.134041
$881$	2812.86	0.107568	0.0537842	−	0.998553i	$-0.482872\pi$
$881$	2812.86	0.107568	0.0537842	+	0.998553i	$0.482872\pi$
$882$	0	0
$883$	32216.0	1.22781	0.613905	−	0.789380i	$-0.289598\pi$
$883$	32216.0	1.22781	0.613905	+	0.789380i	$0.289598\pi$
$884$	−128.849	−0.00490234
$885$	0	0
$886$	24629.6	0.933912
$887$	−44471.4	−1.68343	−0.841715	−	0.539921i	$-0.818454\pi$
$887$	−44471.4	−1.68343	−0.841715	+	0.539921i	$0.818454\pi$
$888$	0	0
$889$	10862.8	0.409817
$890$	11417.9	0.430034
$891$	0	0
$892$	208.255	0.00781713
$893$	987.930	0.0370211
$894$	0	0
$895$	10303.4	0.384810
$896$	−10225.1	−0.381246
$897$	0	0
$898$	28520.5	1.05985
$899$	−60038.5	−2.22736
$900$	0	0
$901$	−8975.23	−0.331862
$902$	15395.2	0.568296
$903$	0	0
$904$	−21819.5	−0.802773
$905$	−20569.7	−0.755534
$906$	0	0
$907$	−21224.4	−0.777005	−0.388502	−	0.921448i	$-0.627008\pi$
$907$	−21224.4	−0.777005	−0.388502	+	0.921448i	$0.627008\pi$
$908$	−164.357	−0.00600701
$909$	0	0
$910$	6718.90	0.244758
$911$	−9599.56	−0.349119	−0.174560	−	0.984647i	$-0.555850\pi$
$911$	−9599.56	−0.349119	−0.174560	+	0.984647i	$0.555850\pi$
$912$	0	0
$913$	−491.732	−0.0178247
$914$	53369.4	1.93140
$915$	0	0
$916$	3.99820	0.000144219 0
$917$	−6191.66	−0.222973
$918$	0	0
$919$	51795.7	1.85918	0.929588	−	0.368600i	$-0.120163\pi$
$919$	51795.7	1.85918	0.929588	+	0.368600i	$0.120163\pi$
$920$	14544.6	0.521218
$921$	0	0
$922$	−34628.4	−1.23690
$923$	64291.6	2.29272
$924$	0	0
$925$	10460.4	0.371824
$926$	−7085.55	−0.251453
$927$	0	0
$928$	478.878	0.0169396
$929$	18821.8	0.664719	0.332360	−	0.943153i	$-0.392155\pi$
$929$	18821.8	0.664719	0.332360	+	0.943153i	$0.392155\pi$
$930$	0	0
$931$	−1417.78	−0.0499097
$932$	191.956	0.00674650
$933$	0	0
$934$	−39269.1	−1.37572
$935$	−2249.78	−0.0786906
$936$	0	0
$937$	−13493.1	−0.470437	−0.235218	−	0.971943i	$-0.575581\pi$
$937$	−13493.1	−0.470437	−0.235218	+	0.971943i	$0.575581\pi$
$938$	12824.9	0.446427
$939$	0	0
$940$	47.9530	0.00166389
$941$	−19364.0	−0.670826	−0.335413	−	0.942071i	$-0.608876\pi$
$941$	−19364.0	−0.670826	−0.335413	+	0.942071i	$0.608876\pi$
$942$	0	0
$943$	63614.7	2.19680
$944$	37265.7	1.28485
$945$	0	0
$946$	−2821.51	−0.0969717
$947$	−3304.64	−0.113396	−0.0566982	−	0.998391i	$-0.518057\pi$
$947$	−3304.64	−0.113396	−0.0566982	+	0.998391i	$0.518057\pi$
$948$	0	0
$949$	14920.3	0.510361
$950$	342.032	0.0116810
$951$	0	0
$952$	6613.39	0.225148
$953$	38598.0	1.31197	0.655987	−	0.754772i	$-0.272252\pi$
$953$	38598.0	1.31197	0.655987	+	0.754772i	$0.272252\pi$
$954$	0	0
$955$	13859.8	0.469626
$956$	35.9766	0.00121712
$957$	0	0
$958$	49695.5	1.67598
$959$	7481.63	0.251923
$960$	0	0
$961$	41608.4	1.39668
$962$	78921.2	2.64503
$963$	0	0
$964$	−118.979	−0.00397518
$965$	−1099.42	−0.0366752
$966$	0	0
$967$	−23257.1	−0.773421	−0.386710	−	0.922201i	$-0.626389\pi$
$967$	−23257.1	−0.773421	−0.386710	+	0.922201i	$0.626389\pi$
$968$	−2745.92	−0.0911747
$969$	0	0
$970$	6060.17	0.200598
$971$	−32050.7	−1.05928	−0.529638	−	0.848224i	$-0.677672\pi$
$971$	−32050.7	−1.05928	−0.529638	+	0.848224i	$0.677672\pi$
$972$	0	0
$973$	1944.01	0.0640515
$974$	−17230.4	−0.566835
$975$	0	0
$976$	−9986.77	−0.327529
$977$	19808.8	0.648660	0.324330	−	0.945944i	$-0.394861\pi$
$977$	19808.8	0.648660	0.324330	+	0.945944i	$0.394861\pi$
$978$	0	0
$979$	−8907.34	−0.290786
$980$	−68.8174	−0.00224316
$981$	0	0
$982$	−11658.9	−0.378870
$983$	−10806.7	−0.350640	−0.175320	−	0.984511i	$-0.556096\pi$
$983$	−10806.7	−0.350640	−0.175320	+	0.984511i	$0.556096\pi$
$984$	0	0
$985$	11055.3	0.357616
$986$	25919.3	0.837158
$987$	0	0
$988$	−15.2816	−0.000492076 0
$989$	−11658.8	−0.374853
$990$	0	0
$991$	−38387.8	−1.23050	−0.615251	−	0.788331i	$-0.710945\pi$
$991$	−38387.8	−1.23050	−0.615251	+	0.788331i	$0.710945\pi$
$992$	−569.495	−0.0182273
$993$	0	0
$994$	−19312.6	−0.616256
$995$	2358.79	0.0751544
$996$	0	0
$997$	−45526.2	−1.44617	−0.723083	−	0.690761i	$-0.757276\pi$
$997$	−45526.2	−1.44617	−0.723083	+	0.690761i	$0.757276\pi$
$998$	46689.4	1.48089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.4.a.g.1.3		3
3.2	odd	2		165.4.a.f.1.1	✓	3
5.4	even	2		2475.4.a.w.1.1		3
15.2	even	4		825.4.c.o.199.2		6
15.8	even	4		825.4.c.o.199.5		6
15.14	odd	2		825.4.a.n.1.3		3
33.32	even	2		1815.4.a.p.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.f.1.1	✓	3	3.2	odd	2
495.4.a.g.1.3		3	1.1	even	1	trivial
825.4.a.n.1.3		3	15.14	odd	2
825.4.c.o.199.2		6	15.2	even	4
825.4.c.o.199.5		6	15.8	even	4
1815.4.a.p.1.3		3	33.32	even	2
2475.4.a.w.1.1		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)

\(f(q)\)	\(=\)	\(q+2.82009 q^{2} -0.0470959 q^{4} -5.00000 q^{5} -7.12434 q^{7} -22.6935 q^{8} +O(q^{10})\)	\(q+2.82009 q^{2} -0.0470959 q^{4} -5.00000 q^{5} -7.12434 q^{7} -22.6935 q^{8} -14.1004 q^{10} +11.0000 q^{11} +66.8837 q^{13} -20.0913 q^{14} -63.6210 q^{16} +40.9051 q^{17} +4.85136 q^{19} +0.235480 q^{20} +31.0210 q^{22} +128.183 q^{23} +25.0000 q^{25} +188.618 q^{26} +0.335527 q^{28} +224.689 q^{29} -267.207 q^{31} +2.13129 q^{32} +115.356 q^{34} +35.6217 q^{35} +418.418 q^{37} +13.6813 q^{38} +113.468 q^{40} +496.282 q^{41} -90.9549 q^{43} -0.518055 q^{44} +361.486 q^{46} +203.640 q^{47} -292.244 q^{49} +70.5022 q^{50} -3.14995 q^{52} -219.416 q^{53} -55.0000 q^{55} +161.676 q^{56} +633.644 q^{58} -585.745 q^{59} +156.973 q^{61} -753.547 q^{62} +514.979 q^{64} -334.419 q^{65} -638.333 q^{67} -1.92646 q^{68} +100.456 q^{70} +961.243 q^{71} +223.078 q^{73} +1179.98 q^{74} -0.228479 q^{76} -78.3678 q^{77} +415.648 q^{79} +318.105 q^{80} +1399.56 q^{82} -44.7029 q^{83} -204.526 q^{85} -256.501 q^{86} -249.629 q^{88} -809.758 q^{89} -476.503 q^{91} -6.03688 q^{92} +574.282 q^{94} -24.2568 q^{95} -429.786 q^{97} -824.154 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} - 5 q^{4} - 15 q^{5} - 16 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q - q^2 - 5 * q^4 - 15 * q^5 - 16 * q^7 + 3 * q^8	\( 3 q - q^{2} - 5 q^{4} - 15 q^{5} - 16 q^{7} + 3 q^{8} + 5 q^{10} + 33 q^{11} - 42 q^{13} + 76 q^{14} - 85 q^{16} + 34 q^{17} - 280 q^{19} + 25 q^{20} - 11 q^{22} + 112 q^{23} + 75 q^{25} + 314 q^{26} - 280 q^{28} + 290 q^{29} - 392 q^{31} + 23 q^{32} + 310 q^{34} + 80 q^{35} + 570 q^{37} + 512 q^{38} - 15 q^{40} + 662 q^{41} - 68 q^{43} - 55 q^{44} + 728 q^{46} - 264 q^{47} + 731 q^{49} - 25 q^{50} + 582 q^{52} + 94 q^{53} - 165 q^{55} - 204 q^{56} - 54 q^{58} + 612 q^{59} - 582 q^{61} - 320 q^{62} + 347 q^{64} + 210 q^{65} + 940 q^{67} - 678 q^{68} - 380 q^{70} + 1616 q^{71} + 738 q^{73} + 642 q^{74} + 880 q^{76} - 176 q^{77} + 124 q^{79} + 425 q^{80} + 2126 q^{82} + 1232 q^{83} - 170 q^{85} + 1000 q^{86} + 33 q^{88} - 838 q^{89} - 1736 q^{91} - 1248 q^{92} + 2432 q^{94} + 1400 q^{95} - 90 q^{97} - 3481 q^{98}+O(q^{100}) \) 3 * q - q^2 - 5 * q^4 - 15 * q^5 - 16 * q^7 + 3 * q^8 + 5 * q^10 + 33 * q^11 - 42 * q^13 + 76 * q^14 - 85 * q^16 + 34 * q^17 - 280 * q^19 + 25 * q^20 - 11 * q^22 + 112 * q^23 + 75 * q^25 + 314 * q^26 - 280 * q^28 + 290 * q^29 - 392 * q^31 + 23 * q^32 + 310 * q^34 + 80 * q^35 + 570 * q^37 + 512 * q^38 - 15 * q^40 + 662 * q^41 - 68 * q^43 - 55 * q^44 + 728 * q^46 - 264 * q^47 + 731 * q^49 - 25 * q^50 + 582 * q^52 + 94 * q^53 - 165 * q^55 - 204 * q^56 - 54 * q^58 + 612 * q^59 - 582 * q^61 - 320 * q^62 + 347 * q^64 + 210 * q^65 + 940 * q^67 - 678 * q^68 - 380 * q^70 + 1616 * q^71 + 738 * q^73 + 642 * q^74 + 880 * q^76 - 176 * q^77 + 124 * q^79 + 425 * q^80 + 2126 * q^82 + 1232 * q^83 - 170 * q^85 + 1000 * q^86 + 33 * q^88 - 838 * q^89 - 1736 * q^91 - 1248 * q^92 + 2432 * q^94 + 1400 * q^95 - 90 * q^97 - 3481 * q^98

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