LMFDB - Embedded newform 495.4.a.h.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:	3.3.1772.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 12x + 8 $ x^3 - x^2 - 12*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.654334$ 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)

$f(q)$	$=$	$q-0.654334 q^{2} -7.57185 q^{4} -5.00000 q^{5} +3.49576 q^{7} +10.1892 q^{8} +O(q^{10})$	\(q-0.654334 q^{2} -7.57185 q^{4} -5.00000 q^{5} +3.49576 q^{7} +10.1892 q^{8} +3.27167 q^{10} +11.0000 q^{11} +23.2440 q^{13} -2.28739 q^{14} +53.9077 q^{16} -23.2270 q^{17} -28.9915 q^{19} +37.8592 q^{20} -7.19767 q^{22} +76.2048 q^{23} +25.0000 q^{25} -15.2093 q^{26} -26.4693 q^{28} +162.581 q^{29} -114.241 q^{31} -116.787 q^{32} +15.1982 q^{34} -17.4788 q^{35} -271.498 q^{37} +18.9701 q^{38} -50.9459 q^{40} +5.16904 q^{41} -98.0647 q^{43} -83.2903 q^{44} -49.8633 q^{46} -262.096 q^{47} -330.780 q^{49} -16.3583 q^{50} -176.000 q^{52} -7.88955 q^{53} -55.0000 q^{55} +35.6189 q^{56} -106.382 q^{58} +255.269 q^{59} -591.402 q^{61} +74.7515 q^{62} -354.844 q^{64} -116.220 q^{65} -331.340 q^{67} +175.871 q^{68} +11.4370 q^{70} -63.2225 q^{71} -937.901 q^{73} +177.650 q^{74} +219.519 q^{76} +38.4533 q^{77} -596.761 q^{79} -269.538 q^{80} -3.38227 q^{82} -117.623 q^{83} +116.135 q^{85} +64.1670 q^{86} +112.081 q^{88} +907.431 q^{89} +81.2554 q^{91} -577.011 q^{92} +171.498 q^{94} +144.958 q^{95} +40.7592 q^{97} +216.440 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + q^{4} - 15 q^{5} + 2 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q - q^2 + q^4 - 15 * q^5 + 2 * q^7 + 3 * q^8	$ 3 q - q^{2} + q^{4} - 15 q^{5} + 2 q^{7} + 3 q^{8} + 5 q^{10} + 33 q^{11} - 66 q^{13} + 88 q^{14} - 103 q^{16} + 100 q^{17} - 70 q^{19} - 5 q^{20} - 11 q^{22} + 10 q^{23} + 75 q^{25} - 88 q^{26} - 64 q^{28} - 34 q^{29} - 104 q^{31} - 37 q^{32} - 278 q^{34} - 10 q^{35} - 270 q^{37} - 154 q^{38} - 15 q^{40} - 82 q^{41} - 428 q^{43} + 11 q^{44} - 706 q^{46} + 246 q^{47} - 685 q^{49} - 25 q^{50} - 528 q^{52} + 298 q^{53} - 165 q^{55} - 288 q^{56} + 114 q^{58} - 204 q^{59} - 492 q^{61} - 584 q^{62} - 319 q^{64} + 330 q^{65} - 812 q^{67} + 798 q^{68} - 440 q^{70} - 442 q^{71} - 1044 q^{73} - 42 q^{74} + 106 q^{76} + 22 q^{77} - 1436 q^{79} + 515 q^{80} - 1246 q^{82} + 734 q^{83} - 500 q^{85} + 82 q^{86} + 33 q^{88} - 556 q^{89} - 176 q^{91} - 630 q^{92} + 38 q^{94} + 350 q^{95} - 1038 q^{97} + 143 q^{98}+O(q^{100}) $ 3 * q - q^2 + q^4 - 15 * q^5 + 2 * q^7 + 3 * q^8 + 5 * q^10 + 33 * q^11 - 66 * q^13 + 88 * q^14 - 103 * q^16 + 100 * q^17 - 70 * q^19 - 5 * q^20 - 11 * q^22 + 10 * q^23 + 75 * q^25 - 88 * q^26 - 64 * q^28 - 34 * q^29 - 104 * q^31 - 37 * q^32 - 278 * q^34 - 10 * q^35 - 270 * q^37 - 154 * q^38 - 15 * q^40 - 82 * q^41 - 428 * q^43 + 11 * q^44 - 706 * q^46 + 246 * q^47 - 685 * q^49 - 25 * q^50 - 528 * q^52 + 298 * q^53 - 165 * q^55 - 288 * q^56 + 114 * q^58 - 204 * q^59 - 492 * q^61 - 584 * q^62 - 319 * q^64 + 330 * q^65 - 812 * q^67 + 798 * q^68 - 440 * q^70 - 442 * q^71 - 1044 * q^73 - 42 * q^74 + 106 * q^76 + 22 * q^77 - 1436 * q^79 + 515 * q^80 - 1246 * q^82 + 734 * q^83 - 500 * q^85 + 82 * q^86 + 33 * q^88 - 556 * q^89 - 176 * q^91 - 630 * q^92 + 38 * q^94 + 350 * q^95 - 1038 * q^97 + 143 * 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 $

$2$	−0.654334	−0.231342	−0.115671	−	0.993288i	$-0.536902\pi$
$2$	−0.654334	−0.231342	−0.115671	+	0.993288i	$0.536902\pi$
$3$	0	0
$3$	0	0
$4$	−7.57185	−0.946481
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	3.49576	0.188753	0.0943766	−	0.995537i	$-0.469914\pi$
$7$	3.49576	0.188753	0.0943766	+	0.995537i	$0.469914\pi$
$8$	10.1892	0.450302
$9$	0	0
$10$	3.27167	0.103459
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	23.2440	0.495902	0.247951	−	0.968773i	$-0.420243\pi$
$13$	23.2440	0.495902	0.247951	+	0.968773i	$0.420243\pi$
$14$	−2.28739	−0.0436665
$15$	0	0
$16$	53.9077	0.842307
$17$	−23.2270	−0.331375	−0.165688	−	0.986178i	$-0.552984\pi$
$17$	−23.2270	−0.331375	−0.165688	+	0.986178i	$0.552984\pi$
$18$	0	0
$19$	−28.9915	−0.350058	−0.175029	−	0.984563i	$-0.556002\pi$
$19$	−28.9915	−0.350058	−0.175029	+	0.984563i	$0.556002\pi$
$20$	37.8592	0.423279
$21$	0	0
$22$	−7.19767	−0.0697522
$23$	76.2048	0.690861	0.345430	−	0.938444i	$-0.387733\pi$
$23$	76.2048	0.690861	0.345430	+	0.938444i	$0.387733\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−15.2093	−0.114723
$27$	0	0
$28$	−26.4693	−0.178651
$29$	162.581	1.04105	0.520525	−	0.853846i	$-0.325736\pi$
$29$	162.581	1.04105	0.520525	+	0.853846i	$0.325736\pi$
$30$	0	0
$31$	−114.241	−0.661878	−0.330939	−	0.943652i	$-0.607365\pi$
$31$	−114.241	−0.661878	−0.330939	+	0.943652i	$0.607365\pi$
$32$	−116.787	−0.645163
$33$	0	0
$34$	15.1982	0.0766610
$35$	−17.4788	−0.0844130
$36$	0	0
$37$	−271.498	−1.20632	−0.603162	−	0.797618i	$-0.706093\pi$
$37$	−271.498	−1.20632	−0.603162	+	0.797618i	$0.706093\pi$
$38$	18.9701	0.0809831
$39$	0	0
$40$	−50.9459	−0.201381
$41$	5.16904	0.0196895	0.00984473	−	0.999952i	$-0.496866\pi$
$41$	5.16904	0.0196895	0.00984473	+	0.999952i	$0.496866\pi$
$42$	0	0
$43$	−98.0647	−0.347784	−0.173892	−	0.984765i	$-0.555634\pi$
$43$	−98.0647	−0.347784	−0.173892	+	0.984765i	$0.555634\pi$
$44$	−83.2903	−0.285375
$45$	0	0
$46$	−49.8633	−0.159825
$47$	−262.096	−0.813417	−0.406709	−	0.913558i	$-0.633324\pi$
$47$	−262.096	−0.813417	−0.406709	+	0.913558i	$0.633324\pi$
$48$	0	0
$49$	−330.780	−0.964372
$50$	−16.3583	−0.0462684
$51$	0	0
$52$	−176.000	−0.469362
$53$	−7.88955	−0.0204474	−0.0102237	−	0.999948i	$-0.503254\pi$
$53$	−7.88955	−0.0204474	−0.0102237	+	0.999948i	$0.503254\pi$
$54$	0	0
$55$	−55.0000	−0.134840
$56$	35.6189	0.0849960
$57$	0	0
$58$	−106.382	−0.240839
$59$	255.269	0.563275	0.281637	−	0.959521i	$-0.409122\pi$
$59$	255.269	0.563275	0.281637	+	0.959521i	$0.409122\pi$
$60$	0	0
$61$	−591.402	−1.24133	−0.620666	−	0.784075i	$-0.713138\pi$
$61$	−591.402	−1.24133	−0.620666	+	0.784075i	$0.713138\pi$
$62$	74.7515	0.153120
$63$	0	0
$64$	−354.844	−0.693054
$65$	−116.220	−0.221774
$66$	0	0
$67$	−331.340	−0.604174	−0.302087	−	0.953280i	$-0.597683\pi$
$67$	−331.340	−0.604174	−0.302087	+	0.953280i	$0.597683\pi$
$68$	175.871	0.313640
$69$	0	0
$70$	11.4370	0.0195282
$71$	−63.2225	−0.105678	−0.0528389	−	0.998603i	$-0.516827\pi$
$71$	−63.2225	−0.105678	−0.0528389	+	0.998603i	$0.516827\pi$
$72$	0	0
$73$	−937.901	−1.50374	−0.751870	−	0.659311i	$-0.770848\pi$
$73$	−937.901	−1.50374	−0.751870	+	0.659311i	$0.770848\pi$
$74$	177.650	0.279073
$75$	0	0
$76$	219.519	0.331324
$77$	38.4533	0.0569112
$78$	0	0
$79$	−596.761	−0.849884	−0.424942	−	0.905221i	$-0.639705\pi$
$79$	−596.761	−0.849884	−0.424942	+	0.905221i	$0.639705\pi$
$80$	−269.538	−0.376691
$81$	0	0
$82$	−3.38227	−0.00455500
$83$	−117.623	−0.155552	−0.0777761	−	0.996971i	$-0.524782\pi$
$83$	−117.623	−0.155552	−0.0777761	+	0.996971i	$0.524782\pi$
$84$	0	0
$85$	116.135	0.148196
$86$	64.1670	0.0804570
$87$	0	0
$88$	112.081	0.135771
$89$	907.431	1.08076	0.540379	−	0.841422i	$-0.318281\pi$
$89$	907.431	1.08076	0.540379	+	0.841422i	$0.318281\pi$
$90$	0	0
$91$	81.2554	0.0936030
$92$	−577.011	−0.653887
$93$	0	0
$94$	171.498	0.188177
$95$	144.958	0.156551
$96$	0	0
$97$	40.7592	0.0426646	0.0213323	−	0.999772i	$-0.493209\pi$
$97$	40.7592	0.0426646	0.0213323	+	0.999772i	$0.493209\pi$
$98$	216.440	0.223100
$99$	0	0
$100$	−189.296	−0.189296
$101$	−214.100	−0.210928	−0.105464	−	0.994423i	$-0.533633\pi$
$101$	−214.100	−0.210928	−0.105464	+	0.994423i	$0.533633\pi$
$102$	0	0
$103$	586.293	0.560865	0.280433	−	0.959874i	$-0.409522\pi$
$103$	586.293	0.560865	0.280433	+	0.959874i	$0.409522\pi$
$104$	236.837	0.223306
$105$	0	0
$106$	5.16240	0.00473034
$107$	−1697.61	−1.53377	−0.766887	−	0.641783i	$-0.778195\pi$
$107$	−1697.61	−1.53377	−0.766887	+	0.641783i	$0.778195\pi$
$108$	0	0
$109$	−260.945	−0.229303	−0.114651	−	0.993406i	$-0.536575\pi$
$109$	−260.945	−0.229303	−0.114651	+	0.993406i	$0.536575\pi$
$110$	35.9883	0.0311941
$111$	0	0
$112$	188.448	0.158988
$113$	1745.53	1.45315	0.726573	−	0.687089i	$-0.241112\pi$
$113$	1745.53	1.45315	0.726573	+	0.687089i	$0.241112\pi$
$114$	0	0
$115$	−381.024	−0.308962
$116$	−1231.04	−0.985335
$117$	0	0
$118$	−167.031	−0.130309
$119$	−81.1960	−0.0625481
$120$	0	0
$121$	121.000	0.0909091
$122$	386.974	0.287172
$123$	0	0
$124$	865.013	0.626455
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−744.282	−0.520034	−0.260017	−	0.965604i	$-0.583728\pi$
$127$	−744.282	−0.520034	−0.260017	+	0.965604i	$0.583728\pi$
$128$	1166.48	0.805496
$129$	0	0
$130$	76.0466	0.0513056
$131$	−1391.61	−0.928133	−0.464067	−	0.885800i	$-0.653610\pi$
$131$	−1391.61	−0.928133	−0.464067	+	0.885800i	$0.653610\pi$
$132$	0	0
$133$	−101.347	−0.0660746
$134$	216.807	0.139771
$135$	0	0
$136$	−236.664	−0.149219
$137$	−1946.61	−1.21394	−0.606972	−	0.794723i	$-0.707616\pi$
$137$	−1946.61	−1.21394	−0.606972	+	0.794723i	$0.707616\pi$
$138$	0	0
$139$	−488.758	−0.298244	−0.149122	−	0.988819i	$-0.547645\pi$
$139$	−488.758	−0.298244	−0.149122	+	0.988819i	$0.547645\pi$
$140$	132.347	0.0798953
$141$	0	0
$142$	41.3686	0.0244477
$143$	255.684	0.149520
$144$	0	0
$145$	−812.903	−0.465572
$146$	613.700	0.347878
$147$	0	0
$148$	2055.74	1.14176
$149$	−2455.03	−1.34982	−0.674912	−	0.737898i	$-0.735819\pi$
$149$	−2455.03	−1.34982	−0.674912	+	0.737898i	$0.735819\pi$
$150$	0	0
$151$	−649.725	−0.350158	−0.175079	−	0.984554i	$-0.556018\pi$
$151$	−649.725	−0.350158	−0.175079	+	0.984554i	$0.556018\pi$
$152$	−295.400	−0.157632
$153$	0	0
$154$	−25.1613	−0.0131659
$155$	571.203	0.296001
$156$	0	0
$157$	2389.98	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$157$	2389.98	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$158$	390.481	0.196614
$159$	0	0
$160$	583.935	0.288526
$161$	266.393	0.130402
$162$	0	0
$163$	−953.394	−0.458132	−0.229066	−	0.973411i	$-0.573567\pi$
$163$	−953.394	−0.458132	−0.229066	+	0.973411i	$0.573567\pi$
$164$	−39.1392	−0.0186357
$165$	0	0
$166$	76.9649	0.0359857
$167$	730.445	0.338464	0.169232	−	0.985576i	$-0.445871\pi$
$167$	730.445	0.338464	0.169232	+	0.985576i	$0.445871\pi$
$168$	0	0
$169$	−1656.72	−0.754081
$170$	−75.9911	−0.0342838
$171$	0	0
$172$	742.531	0.329171
$173$	594.321	0.261187	0.130594	−	0.991436i	$-0.458312\pi$
$173$	594.321	0.261187	0.130594	+	0.991436i	$0.458312\pi$
$174$	0	0
$175$	87.3939	0.0377506
$176$	592.984	0.253965
$177$	0	0
$178$	−593.762	−0.250025
$179$	1590.73	0.664228	0.332114	−	0.943239i	$-0.392238\pi$
$179$	1590.73	0.664228	0.332114	+	0.943239i	$0.392238\pi$
$180$	0	0
$181$	−1851.41	−0.760299	−0.380150	−	0.924925i	$-0.624128\pi$
$181$	−1851.41	−0.760299	−0.380150	+	0.924925i	$0.624128\pi$
$182$	−53.1681	−0.0216543
$183$	0	0
$184$	776.464	0.311096
$185$	1357.49	0.539485
$186$	0	0
$187$	−255.497	−0.0999134
$188$	1984.55	0.769884
$189$	0	0
$190$	−94.8506	−0.0362168
$191$	−2929.26	−1.10971	−0.554853	−	0.831948i	$-0.687226\pi$
$191$	−2929.26	−1.10971	−0.554853	+	0.831948i	$0.687226\pi$
$192$	0	0
$193$	2485.70	0.927072	0.463536	−	0.886078i	$-0.346580\pi$
$193$	2485.70	0.927072	0.463536	+	0.886078i	$0.346580\pi$
$194$	−26.6701	−0.00987011
$195$	0	0
$196$	2504.61	0.912760
$197$	−320.772	−0.116010	−0.0580052	−	0.998316i	$-0.518474\pi$
$197$	−320.772	−0.116010	−0.0580052	+	0.998316i	$0.518474\pi$
$198$	0	0
$199$	144.790	0.0515773	0.0257887	−	0.999667i	$-0.491790\pi$
$199$	144.790	0.0515773	0.0257887	+	0.999667i	$0.491790\pi$
$200$	254.730	0.0900605
$201$	0	0
$202$	140.093	0.0487965
$203$	568.342	0.196502
$204$	0	0
$205$	−25.8452	−0.00880540
$206$	−383.631	−0.129752
$207$	0	0
$208$	1253.03	0.417702
$209$	−318.907	−0.105547
$210$	0	0
$211$	3625.41	1.18286	0.591430	−	0.806356i	$-0.298564\pi$
$211$	3625.41	1.18286	0.591430	+	0.806356i	$0.298564\pi$
$212$	59.7385	0.0193531
$213$	0	0
$214$	1110.80	0.354826
$215$	490.323	0.155534
$216$	0	0
$217$	−399.358	−0.124932
$218$	170.745	0.0530474
$219$	0	0
$220$	416.452	0.127623
$221$	−539.889	−0.164330
$222$	0	0
$223$	1460.26	0.438502	0.219251	−	0.975668i	$-0.429639\pi$
$223$	1460.26	0.438502	0.219251	+	0.975668i	$0.429639\pi$
$224$	−408.259	−0.121777
$225$	0	0
$226$	−1142.16	−0.336173
$227$	4201.57	1.22849	0.614247	−	0.789114i	$-0.289460\pi$
$227$	4201.57	1.22849	0.614247	+	0.789114i	$0.289460\pi$
$228$	0	0
$229$	1227.39	0.354185	0.177093	−	0.984194i	$-0.443331\pi$
$229$	1227.39	0.354185	0.177093	+	0.984194i	$0.443331\pi$
$230$	249.317	0.0714759
$231$	0	0
$232$	1656.56	0.468788
$233$	4064.73	1.14287	0.571437	−	0.820646i	$-0.306386\pi$
$233$	4064.73	1.14287	0.571437	+	0.820646i	$0.306386\pi$
$234$	0	0
$235$	1310.48	0.363771
$236$	−1932.86	−0.533129
$237$	0	0
$238$	53.1293	0.0144700
$239$	−3883.64	−1.05109	−0.525547	−	0.850764i	$-0.676139\pi$
$239$	−3883.64	−1.05109	−0.525547	+	0.850764i	$0.676139\pi$
$240$	0	0
$241$	3714.51	0.992833	0.496416	−	0.868084i	$-0.334649\pi$
$241$	3714.51	0.992833	0.496416	+	0.868084i	$0.334649\pi$
$242$	−79.1744	−0.0210311
$243$	0	0
$244$	4478.01	1.17490
$245$	1653.90	0.431280
$246$	0	0
$247$	−673.879	−0.173595
$248$	−1164.02	−0.298045
$249$	0	0
$250$	81.7917	0.0206918
$251$	−1322.43	−0.332553	−0.166277	−	0.986079i	$-0.553174\pi$
$251$	−1322.43	−0.332553	−0.166277	+	0.986079i	$0.553174\pi$
$252$	0	0
$253$	838.253	0.208302
$254$	487.008	0.120306
$255$	0	0
$256$	2075.48	0.506709
$257$	−315.639	−0.0766109	−0.0383054	−	0.999266i	$-0.512196\pi$
$257$	−315.639	−0.0766109	−0.0383054	+	0.999266i	$0.512196\pi$
$258$	0	0
$259$	−949.091	−0.227698
$260$	880.000	0.209905
$261$	0	0
$262$	910.577	0.214716
$263$	1814.30	0.425378	0.212689	−	0.977120i	$-0.431778\pi$
$263$	1814.30	0.425378	0.212689	+	0.977120i	$0.431778\pi$
$264$	0	0
$265$	39.4478	0.00914436
$266$	66.3149	0.0152858
$267$	0	0
$268$	2508.86	0.571839
$269$	−4639.76	−1.05164	−0.525820	−	0.850596i	$-0.676241\pi$
$269$	−4639.76	−1.05164	−0.525820	+	0.850596i	$0.676241\pi$
$270$	0	0
$271$	1255.80	0.281493	0.140746	−	0.990046i	$-0.455050\pi$
$271$	1255.80	0.281493	0.140746	+	0.990046i	$0.455050\pi$
$272$	−1252.11	−0.279120
$273$	0	0
$274$	1273.73	0.280836
$275$	275.000	0.0603023
$276$	0	0
$277$	−1572.64	−0.341123	−0.170561	−	0.985347i	$-0.554558\pi$
$277$	−1572.64	−0.341123	−0.170561	+	0.985347i	$0.554558\pi$
$278$	319.811	0.0689963
$279$	0	0
$280$	−178.094	−0.0380114
$281$	−7579.80	−1.60916	−0.804578	−	0.593846i	$-0.797609\pi$
$281$	−7579.80	−1.60916	−0.804578	+	0.593846i	$0.797609\pi$
$282$	0	0
$283$	1868.74	0.392527	0.196263	−	0.980551i	$-0.437119\pi$
$283$	1868.74	0.392527	0.196263	+	0.980551i	$0.437119\pi$
$284$	478.711	0.100022
$285$	0	0
$286$	−167.303	−0.0345902
$287$	18.0697	0.00371645
$288$	0	0
$289$	−4373.51	−0.890190
$290$	531.910	0.107706
$291$	0	0
$292$	7101.64	1.42326
$293$	9079.48	1.81034	0.905168	−	0.425053i	$-0.139745\pi$
$293$	9079.48	1.81034	0.905168	+	0.425053i	$0.139745\pi$
$294$	0	0
$295$	−1276.35	−0.251904
$296$	−2766.34	−0.543211
$297$	0	0
$298$	1606.41	0.312271
$299$	1771.30	0.342599
$300$	0	0
$301$	−342.810	−0.0656454
$302$	425.137	0.0810062
$303$	0	0
$304$	−1562.86	−0.294857
$305$	2957.01	0.555141
$306$	0	0
$307$	−3296.83	−0.612899	−0.306450	−	0.951887i	$-0.599141\pi$
$307$	−3296.83	−0.612899	−0.306450	+	0.951887i	$0.599141\pi$
$308$	−291.163	−0.0538654
$309$	0	0
$310$	−373.757	−0.0684774
$311$	−6836.31	−1.24647	−0.623234	−	0.782036i	$-0.714181\pi$
$311$	−6836.31	−1.24647	−0.623234	+	0.782036i	$0.714181\pi$
$312$	0	0
$313$	−4011.25	−0.724375	−0.362187	−	0.932105i	$-0.617970\pi$
$313$	−4011.25	−0.724375	−0.362187	+	0.932105i	$0.617970\pi$
$314$	−1563.85	−0.281060
$315$	0	0
$316$	4518.58	0.804399
$317$	1733.31	0.307105	0.153552	−	0.988141i	$-0.450929\pi$
$317$	1733.31	0.307105	0.153552	+	0.988141i	$0.450929\pi$
$318$	0	0
$319$	1788.39	0.313889
$320$	1774.22	0.309943
$321$	0	0
$322$	−174.310	−0.0301675
$323$	673.387	0.116001
$324$	0	0
$325$	581.100	0.0991804
$326$	623.838	0.105985
$327$	0	0
$328$	52.6683	0.00886621
$329$	−916.223	−0.153535
$330$	0	0
$331$	−5531.46	−0.918540	−0.459270	−	0.888297i	$-0.651889\pi$
$331$	−5531.46	−0.918540	−0.459270	+	0.888297i	$0.651889\pi$
$332$	890.626	0.147227
$333$	0	0
$334$	−477.954	−0.0783009
$335$	1656.70	0.270195
$336$	0	0
$337$	1905.85	0.308066	0.154033	−	0.988066i	$-0.450774\pi$
$337$	1905.85	0.308066	0.154033	+	0.988066i	$0.450774\pi$
$338$	1084.05	0.174451
$339$	0	0
$340$	−879.357	−0.140264
$341$	−1256.65	−0.199564
$342$	0	0
$343$	−2355.37	−0.370781
$344$	−999.199	−0.156608
$345$	0	0
$346$	−388.884	−0.0604235
$347$	−8642.29	−1.33701	−0.668505	−	0.743708i	$-0.733065\pi$
$347$	−8642.29	−1.33701	−0.668505	+	0.743708i	$0.733065\pi$
$348$	0	0
$349$	−9888.64	−1.51670	−0.758348	−	0.651850i	$-0.773993\pi$
$349$	−9888.64	−1.51670	−0.758348	+	0.651850i	$0.773993\pi$
$350$	−57.1848	−0.00873330
$351$	0	0
$352$	−1284.66	−0.194524
$353$	1043.79	0.157380	0.0786901	−	0.996899i	$-0.474926\pi$
$353$	1043.79	0.157380	0.0786901	+	0.996899i	$0.474926\pi$
$354$	0	0
$355$	316.112	0.0472606
$356$	−6870.93	−1.02292
$357$	0	0
$358$	−1040.87	−0.153664
$359$	1542.10	0.226710	0.113355	−	0.993555i	$-0.463840\pi$
$359$	1542.10	0.226710	0.113355	+	0.993555i	$0.463840\pi$
$360$	0	0
$361$	−6018.49	−0.877459
$362$	1211.44	0.175889
$363$	0	0
$364$	−615.253	−0.0885935
$365$	4689.51	0.672493
$366$	0	0
$367$	−7972.89	−1.13401	−0.567005	−	0.823714i	$-0.691898\pi$
$367$	−7972.89	−1.13401	−0.567005	+	0.823714i	$0.691898\pi$
$368$	4108.02	0.581917
$369$	0	0
$370$	−888.252	−0.124805
$371$	−27.5800	−0.00385951
$372$	0	0
$373$	−11051.0	−1.53405	−0.767026	−	0.641616i	$-0.778264\pi$
$373$	−11051.0	−1.53405	−0.767026	+	0.641616i	$0.778264\pi$
$374$	167.180	0.0231142
$375$	0	0
$376$	−2670.54	−0.366284
$377$	3779.02	0.516259
$378$	0	0
$379$	5725.91	0.776043	0.388022	−	0.921650i	$-0.373159\pi$
$379$	5725.91	0.776043	0.388022	+	0.921650i	$0.373159\pi$
$380$	−1097.60	−0.148172
$381$	0	0
$382$	1916.71	0.256722
$383$	10415.6	1.38959	0.694796	−	0.719206i	$-0.255494\pi$
$383$	10415.6	1.38959	0.694796	+	0.719206i	$0.255494\pi$
$384$	0	0
$385$	−192.267	−0.0254515
$386$	−1626.48	−0.214471
$387$	0	0
$388$	−308.622	−0.0403812
$389$	−9455.10	−1.23237	−0.616186	−	0.787601i	$-0.711323\pi$
$389$	−9455.10	−1.23237	−0.616186	+	0.787601i	$0.711323\pi$
$390$	0	0
$391$	−1770.01	−0.228934
$392$	−3370.37	−0.434259
$393$	0	0
$394$	209.892	0.0268380
$395$	2983.80	0.380080
$396$	0	0
$397$	−5439.87	−0.687706	−0.343853	−	0.939023i	$-0.611732\pi$
$397$	−5439.87	−0.687706	−0.343853	+	0.939023i	$0.611732\pi$
$398$	−94.7410	−0.0119320
$399$	0	0
$400$	1347.69	0.168461
$401$	−5383.45	−0.670416	−0.335208	−	0.942144i	$-0.608807\pi$
$401$	−5383.45	−0.670416	−0.335208	+	0.942144i	$0.608807\pi$
$402$	0	0
$403$	−2655.41	−0.328227
$404$	1621.13	0.199639
$405$	0	0
$406$	−371.885	−0.0454590
$407$	−2986.48	−0.363721
$408$	0	0
$409$	5302.44	0.641049	0.320524	−	0.947240i	$-0.396141\pi$
$409$	5302.44	0.641049	0.320524	+	0.947240i	$0.396141\pi$
$410$	16.9114	0.00203706
$411$	0	0
$412$	−4439.32	−0.530848
$413$	892.359	0.106320
$414$	0	0
$415$	588.117	0.0695651
$416$	−2714.60	−0.319938
$417$	0	0
$418$	208.671	0.0244173
$419$	−8647.52	−1.00826	−0.504128	−	0.863629i	$-0.668186\pi$
$419$	−8647.52	−1.00826	−0.504128	+	0.863629i	$0.668186\pi$
$420$	0	0
$421$	−5732.40	−0.663611	−0.331806	−	0.943348i	$-0.607658\pi$
$421$	−5732.40	−0.663611	−0.331806	+	0.943348i	$0.607658\pi$
$422$	−2372.23	−0.273645
$423$	0	0
$424$	−80.3881	−0.00920752
$425$	−580.676	−0.0662751
$426$	0	0
$427$	−2067.40	−0.234305
$428$	12854.0	1.45169
$429$	0	0
$430$	−320.835	−0.0359815
$431$	1315.46	0.147015	0.0735077	−	0.997295i	$-0.476581\pi$
$431$	1315.46	0.147015	0.0735077	+	0.997295i	$0.476581\pi$
$432$	0	0
$433$	1700.73	0.188757	0.0943784	−	0.995536i	$-0.469914\pi$
$433$	1700.73	0.188757	0.0943784	+	0.995536i	$0.469914\pi$
$434$	261.313	0.0289019
$435$	0	0
$436$	1975.84	0.217031
$437$	−2209.29	−0.241842
$438$	0	0
$439$	−8913.22	−0.969032	−0.484516	−	0.874783i	$-0.661004\pi$
$439$	−8913.22	−0.969032	−0.484516	+	0.874783i	$0.661004\pi$
$440$	−560.405	−0.0607188
$441$	0	0
$442$	353.267	0.0380163
$443$	4761.04	0.510618	0.255309	−	0.966860i	$-0.417823\pi$
$443$	4761.04	0.510618	0.255309	+	0.966860i	$0.417823\pi$
$444$	0	0
$445$	−4537.15	−0.483330
$446$	−955.495	−0.101444
$447$	0	0
$448$	−1240.45	−0.130816
$449$	11399.2	1.19814	0.599068	−	0.800698i	$-0.295538\pi$
$449$	11399.2	1.19814	0.599068	+	0.800698i	$0.295538\pi$
$450$	0	0
$451$	56.8594	0.00593660
$452$	−13216.9	−1.37537
$453$	0	0
$454$	−2749.23	−0.284202
$455$	−406.277	−0.0418605
$456$	0	0
$457$	−3617.29	−0.370262	−0.185131	−	0.982714i	$-0.559271\pi$
$457$	−3617.29	−0.370262	−0.185131	+	0.982714i	$0.559271\pi$
$458$	−803.124	−0.0819378
$459$	0	0
$460$	2885.06	0.292427
$461$	13221.5	1.33576	0.667881	−	0.744268i	$-0.267202\pi$
$461$	13221.5	1.33576	0.667881	+	0.744268i	$0.267202\pi$
$462$	0	0
$463$	1197.28	0.120177	0.0600887	−	0.998193i	$-0.480862\pi$
$463$	1197.28	0.120177	0.0600887	+	0.998193i	$0.480862\pi$
$464$	8764.34	0.876884
$465$	0	0
$466$	−2659.69	−0.264394
$467$	1558.44	0.154424	0.0772121	−	0.997015i	$-0.475398\pi$
$467$	1558.44	0.154424	0.0772121	+	0.997015i	$0.475398\pi$
$468$	0	0
$469$	−1158.28	−0.114040
$470$	−857.490	−0.0841555
$471$	0	0
$472$	2600.98	0.253644
$473$	−1078.71	−0.104861
$474$	0	0
$475$	−724.788	−0.0700117
$476$	614.804	0.0592006
$477$	0	0
$478$	2541.19	0.243162
$479$	2142.72	0.204392	0.102196	−	0.994764i	$-0.467413\pi$
$479$	2142.72	0.204392	0.102196	+	0.994764i	$0.467413\pi$
$480$	0	0
$481$	−6310.70	−0.598219
$482$	−2430.53	−0.229684
$483$	0	0
$484$	−916.194	−0.0860437
$485$	−203.796	−0.0190802
$486$	0	0
$487$	7073.49	0.658174	0.329087	−	0.944300i	$-0.393259\pi$
$487$	7073.49	0.658174	0.329087	+	0.944300i	$0.393259\pi$
$488$	−6025.90	−0.558975
$489$	0	0
$490$	−1082.20	−0.0997732
$491$	−8583.19	−0.788909	−0.394454	−	0.918916i	$-0.629066\pi$
$491$	−8583.19	−0.788909	−0.394454	+	0.918916i	$0.629066\pi$
$492$	0	0
$493$	−3776.26	−0.344978
$494$	440.941	0.0401597
$495$	0	0
$496$	−6158.45	−0.557505
$497$	−221.010	−0.0199470
$498$	0	0
$499$	11167.4	1.00185	0.500923	−	0.865492i	$-0.332994\pi$
$499$	11167.4	1.00185	0.500923	+	0.865492i	$0.332994\pi$
$500$	946.481	0.0846558
$501$	0	0
$502$	865.308	0.0769334
$503$	8272.55	0.733310	0.366655	−	0.930357i	$-0.380503\pi$
$503$	8272.55	0.733310	0.366655	+	0.930357i	$0.380503\pi$
$504$	0	0
$505$	1070.50	0.0943299
$506$	−548.497	−0.0481890
$507$	0	0
$508$	5635.59	0.492202
$509$	439.792	0.0382975	0.0191488	−	0.999817i	$-0.493904\pi$
$509$	439.792	0.0382975	0.0191488	+	0.999817i	$0.493904\pi$
$510$	0	0
$511$	−3278.67	−0.283836
$512$	−10689.9	−0.922719
$513$	0	0
$514$	206.533	0.0177233
$515$	−2931.46	−0.250827
$516$	0	0
$517$	−2883.05	−0.245254
$518$	621.022	0.0526760
$519$	0	0
$520$	−1184.19	−0.0998654
$521$	−23071.7	−1.94009	−0.970047	−	0.242916i	$-0.921896\pi$
$521$	−23071.7	−1.94009	−0.970047	+	0.242916i	$0.921896\pi$
$522$	0	0
$523$	−12046.4	−1.00718	−0.503588	−	0.863944i	$-0.667987\pi$
$523$	−12046.4	−1.00718	−0.503588	+	0.863944i	$0.667987\pi$
$524$	10537.1	0.878461
$525$	0	0
$526$	−1187.16	−0.0984077
$527$	2653.47	0.219330
$528$	0	0
$529$	−6359.83	−0.522711
$530$	−25.8120	−0.00211547
$531$	0	0
$532$	767.386	0.0625384
$533$	120.149	0.00976404
$534$	0	0
$535$	8488.03	0.685924
$536$	−3376.08	−0.272061
$537$	0	0
$538$	3035.95	0.243288
$539$	−3638.58	−0.290769
$540$	0	0
$541$	7103.33	0.564503	0.282252	−	0.959340i	$-0.408919\pi$
$541$	7103.33	0.564503	0.282252	+	0.959340i	$0.408919\pi$
$542$	−821.713	−0.0651211
$543$	0	0
$544$	2712.62	0.213791
$545$	1304.73	0.102547
$546$	0	0
$547$	17746.5	1.38717	0.693586	−	0.720373i	$-0.256030\pi$
$547$	17746.5	1.38717	0.693586	+	0.720373i	$0.256030\pi$
$548$	14739.5	1.14897
$549$	0	0
$550$	−179.942	−0.0139504
$551$	−4713.46	−0.364429
$552$	0	0
$553$	−2086.13	−0.160418
$554$	1029.03	0.0789159
$555$	0	0
$556$	3700.80	0.282282
$557$	2996.58	0.227952	0.113976	−	0.993484i	$-0.463641\pi$
$557$	2996.58	0.227952	0.113976	+	0.993484i	$0.463641\pi$
$558$	0	0
$559$	−2279.41	−0.172467
$560$	−942.240	−0.0711016
$561$	0	0
$562$	4959.72	0.372265
$563$	25636.3	1.91908	0.959540	−	0.281573i	$-0.0908563\pi$
$563$	25636.3	1.91908	0.959540	+	0.281573i	$0.0908563\pi$
$564$	0	0
$565$	−8727.64	−0.649867
$566$	−1222.78	−0.0908078
$567$	0	0
$568$	−644.185	−0.0475870
$569$	−555.677	−0.0409406	−0.0204703	−	0.999790i	$-0.506516\pi$
$569$	−555.677	−0.0409406	−0.0204703	+	0.999790i	$0.506516\pi$
$570$	0	0
$571$	2142.63	0.157034	0.0785169	−	0.996913i	$-0.474982\pi$
$571$	2142.63	0.157034	0.0785169	+	0.996913i	$0.474982\pi$
$572$	−1936.00	−0.141518
$573$	0	0
$574$	−11.8236	−0.000859770 0
$575$	1905.12	0.138172
$576$	0	0
$577$	−19107.8	−1.37863	−0.689314	−	0.724463i	$-0.742088\pi$
$577$	−19107.8	−1.37863	−0.689314	+	0.724463i	$0.742088\pi$
$578$	2861.73	0.205938
$579$	0	0
$580$	6155.18	0.440655
$581$	−411.183	−0.0293610
$582$	0	0
$583$	−86.7851	−0.00616513
$584$	−9556.45	−0.677138
$585$	0	0
$586$	−5941.00	−0.418807
$587$	−16167.2	−1.13678	−0.568391	−	0.822758i	$-0.692434\pi$
$587$	−16167.2	−1.13678	−0.568391	+	0.822758i	$0.692434\pi$
$588$	0	0
$589$	3312.01	0.231696
$590$	835.156	0.0582760
$591$	0	0
$592$	−14635.8	−1.01610
$593$	8646.53	0.598770	0.299385	−	0.954132i	$-0.403219\pi$
$593$	8646.53	0.598770	0.299385	+	0.954132i	$0.403219\pi$
$594$	0	0
$595$	405.980	0.0279724
$596$	18589.1	1.27758
$597$	0	0
$598$	−1159.02	−0.0792575
$599$	7314.60	0.498942	0.249471	−	0.968382i	$-0.419743\pi$
$599$	7314.60	0.498942	0.249471	+	0.968382i	$0.419743\pi$
$600$	0	0
$601$	−8609.65	−0.584351	−0.292175	−	0.956365i	$-0.594379\pi$
$601$	−8609.65	−0.584351	−0.292175	+	0.956365i	$0.594379\pi$
$602$	224.312	0.0151865
$603$	0	0
$604$	4919.62	0.331418
$605$	−605.000	−0.0406558
$606$	0	0
$607$	−5635.76	−0.376851	−0.188426	−	0.982087i	$-0.560338\pi$
$607$	−5635.76	−0.376851	−0.188426	+	0.982087i	$0.560338\pi$
$608$	3385.83	0.225845
$609$	0	0
$610$	−1934.87	−0.128427
$611$	−6092.15	−0.403375
$612$	0	0
$613$	−7072.43	−0.465992	−0.232996	−	0.972478i	$-0.574853\pi$
$613$	−7072.43	−0.465992	−0.232996	+	0.972478i	$0.574853\pi$
$614$	2157.23	0.141789
$615$	0	0
$616$	391.808	0.0256273
$617$	−22030.7	−1.43747	−0.718737	−	0.695282i	$-0.755279\pi$
$617$	−22030.7	−1.43747	−0.718737	+	0.695282i	$0.755279\pi$
$618$	0	0
$619$	4607.86	0.299201	0.149600	−	0.988747i	$-0.452201\pi$
$619$	4607.86	0.299201	0.149600	+	0.988747i	$0.452201\pi$
$620$	−4325.06	−0.280159
$621$	0	0
$622$	4473.22	0.288360
$623$	3172.16	0.203996
$624$	0	0
$625$	625.000	0.0400000
$626$	2624.70	0.167578
$627$	0	0
$628$	−18096.6	−1.14989
$629$	6306.09	0.399746
$630$	0	0
$631$	22201.4	1.40067	0.700337	−	0.713813i	$-0.253033\pi$
$631$	22201.4	1.40067	0.700337	+	0.713813i	$0.253033\pi$
$632$	−6080.50	−0.382705
$633$	0	0
$634$	−1134.16	−0.0710462
$635$	3721.41	0.232566
$636$	0	0
$637$	−7688.64	−0.478234
$638$	−1170.20	−0.0726156
$639$	0	0
$640$	−5832.41	−0.360229
$641$	−8757.92	−0.539652	−0.269826	−	0.962909i	$-0.586966\pi$
$641$	−8757.92	−0.539652	−0.269826	+	0.962909i	$0.586966\pi$
$642$	0	0
$643$	28494.2	1.74759	0.873796	−	0.486292i	$-0.161651\pi$
$643$	28494.2	1.74759	0.873796	+	0.486292i	$0.161651\pi$
$644$	−2017.09	−0.123423
$645$	0	0
$646$	−440.619	−0.0268358
$647$	30050.5	1.82598	0.912989	−	0.407984i	$-0.133768\pi$
$647$	30050.5	1.82598	0.912989	+	0.407984i	$0.133768\pi$
$648$	0	0
$649$	2807.96	0.169834
$650$	−380.233	−0.0229446
$651$	0	0
$652$	7218.95	0.433614
$653$	−10864.3	−0.651074	−0.325537	−	0.945529i	$-0.605545\pi$
$653$	−10864.3	−0.651074	−0.325537	+	0.945529i	$0.605545\pi$
$654$	0	0
$655$	6958.05	0.415074
$656$	278.651	0.0165846
$657$	0	0
$658$	599.515	0.0355191
$659$	16448.7	0.972305	0.486152	−	0.873874i	$-0.338400\pi$
$659$	16448.7	0.972305	0.486152	+	0.873874i	$0.338400\pi$
$660$	0	0
$661$	26238.0	1.54394	0.771968	−	0.635661i	$-0.219273\pi$
$661$	26238.0	1.54394	0.771968	+	0.635661i	$0.219273\pi$
$662$	3619.42	0.212497
$663$	0	0
$664$	−1198.49	−0.0700456
$665$	506.736	0.0295495
$666$	0	0
$667$	12389.4	0.719221
$668$	−5530.82	−0.320350
$669$	0	0
$670$	−1084.03	−0.0625073
$671$	−6505.42	−0.374276
$672$	0	0
$673$	14935.1	0.855432	0.427716	−	0.903913i	$-0.359318\pi$
$673$	14935.1	0.855432	0.427716	+	0.903913i	$0.359318\pi$
$674$	−1247.06	−0.0712685
$675$	0	0
$676$	12544.4	0.713724
$677$	5111.59	0.290184	0.145092	−	0.989418i	$-0.453652\pi$
$677$	5111.59	0.290184	0.145092	+	0.989418i	$0.453652\pi$
$678$	0	0
$679$	142.484	0.00805308
$680$	1183.32	0.0667328
$681$	0	0
$682$	822.266	0.0461675
$683$	−30728.9	−1.72154	−0.860769	−	0.508997i	$-0.830017\pi$
$683$	−30728.9	−1.72154	−0.860769	+	0.508997i	$0.830017\pi$
$684$	0	0
$685$	9733.06	0.542892
$686$	1541.20	0.0857772
$687$	0	0
$688$	−5286.44	−0.292941
$689$	−183.385	−0.0101399
$690$	0	0
$691$	−4423.29	−0.243516	−0.121758	−	0.992560i	$-0.538853\pi$
$691$	−4423.29	−0.243516	−0.121758	+	0.992560i	$0.538853\pi$
$692$	−4500.11	−0.247209
$693$	0	0
$694$	5654.94	0.309306
$695$	2443.79	0.133379
$696$	0	0
$697$	−120.061	−0.00652460
$698$	6470.47	0.350875
$699$	0	0
$700$	−661.733	−0.0357302
$701$	17635.6	0.950194	0.475097	−	0.879933i	$-0.342413\pi$
$701$	17635.6	0.950194	0.475097	+	0.879933i	$0.342413\pi$
$702$	0	0
$703$	7871.14	0.422284
$704$	−3903.28	−0.208964
$705$	0	0
$706$	−682.985	−0.0364086
$707$	−748.441	−0.0398133
$708$	0	0
$709$	9791.11	0.518636	0.259318	−	0.965792i	$-0.416502\pi$
$709$	9791.11	0.518636	0.259318	+	0.965792i	$0.416502\pi$
$710$	−206.843	−0.0109333
$711$	0	0
$712$	9245.98	0.486668
$713$	−8705.68	−0.457266
$714$	0	0
$715$	−1278.42	−0.0668674
$716$	−12044.8	−0.628679
$717$	0	0
$718$	−1009.05	−0.0524476
$719$	−37165.5	−1.92773	−0.963867	−	0.266383i	$-0.914171\pi$
$719$	−37165.5	−1.92773	−0.963867	+	0.266383i	$0.914171\pi$
$720$	0	0
$721$	2049.54	0.105865
$722$	3938.10	0.202993
$723$	0	0
$724$	14018.6	0.719609
$725$	4064.52	0.208210
$726$	0	0
$727$	20491.6	1.04538	0.522690	−	0.852523i	$-0.324928\pi$
$727$	20491.6	1.04538	0.522690	+	0.852523i	$0.324928\pi$
$728$	827.926	0.0421497
$729$	0	0
$730$	−3068.50	−0.155576
$731$	2277.75	0.115247
$732$	0	0
$733$	3771.76	0.190059	0.0950294	−	0.995474i	$-0.469706\pi$
$733$	3771.76	0.190059	0.0950294	+	0.995474i	$0.469706\pi$
$734$	5216.93	0.262344
$735$	0	0
$736$	−8899.73	−0.445718
$737$	−3644.74	−0.182165
$738$	0	0
$739$	17388.4	0.865554	0.432777	−	0.901501i	$-0.357534\pi$
$739$	17388.4	0.865554	0.432777	+	0.901501i	$0.357534\pi$
$740$	−10278.7	−0.510612
$741$	0	0
$742$	18.0465	0.000892867 0
$743$	−19870.1	−0.981110	−0.490555	−	0.871410i	$-0.663206\pi$
$743$	−19870.1	−0.981110	−0.490555	+	0.871410i	$0.663206\pi$
$744$	0	0
$745$	12275.2	0.603660
$746$	7231.07	0.354890
$747$	0	0
$748$	1934.59	0.0945661
$749$	−5934.42	−0.289504
$750$	0	0
$751$	25552.4	1.24157	0.620785	−	0.783981i	$-0.286814\pi$
$751$	25552.4	1.24157	0.620785	+	0.783981i	$0.286814\pi$
$752$	−14129.0	−0.685147
$753$	0	0
$754$	−2472.74	−0.119432
$755$	3248.62	0.156595
$756$	0	0
$757$	6907.53	0.331649	0.165825	−	0.986155i	$-0.446971\pi$
$757$	6907.53	0.331649	0.165825	+	0.986155i	$0.446971\pi$
$758$	−3746.66	−0.179531
$759$	0	0
$760$	1477.00	0.0704952
$761$	20341.1	0.968940	0.484470	−	0.874808i	$-0.339013\pi$
$761$	20341.1	0.968940	0.484470	+	0.874808i	$0.339013\pi$
$762$	0	0
$763$	−912.201	−0.0432816
$764$	22179.9	1.05032
$765$	0	0
$766$	−6815.30	−0.321471
$767$	5933.48	0.279329
$768$	0	0
$769$	12240.6	0.574000	0.287000	−	0.957931i	$-0.407342\pi$
$769$	12240.6	0.574000	0.287000	+	0.957931i	$0.407342\pi$
$770$	125.806	0.00588799
$771$	0	0
$772$	−18821.4	−0.877456
$773$	−10294.7	−0.479008	−0.239504	−	0.970895i	$-0.576985\pi$
$773$	−10294.7	−0.479008	−0.239504	+	0.970895i	$0.576985\pi$
$774$	0	0
$775$	−2856.02	−0.132376
$776$	415.303	0.0192120
$777$	0	0
$778$	6186.79	0.285099
$779$	−149.858	−0.00689246
$780$	0	0
$781$	−695.447	−0.0318631
$782$	1158.18	0.0529621
$783$	0	0
$784$	−17831.6	−0.812298
$785$	−11949.9	−0.543326
$786$	0	0
$787$	35428.6	1.60469	0.802346	−	0.596860i	$-0.203585\pi$
$787$	35428.6	1.60469	0.802346	+	0.596860i	$0.203585\pi$
$788$	2428.83	0.109802
$789$	0	0
$790$	−1952.40	−0.0879283
$791$	6101.94	0.274286
$792$	0	0
$793$	−13746.5	−0.615579
$794$	3559.49	0.159095
$795$	0	0
$796$	−1096.33	−0.0488170
$797$	3502.01	0.155643	0.0778215	−	0.996967i	$-0.475204\pi$
$797$	3502.01	0.155643	0.0778215	+	0.996967i	$0.475204\pi$
$798$	0	0
$799$	6087.70	0.269546
$800$	−2919.68	−0.129033
$801$	0	0
$802$	3522.57	0.155095
$803$	−10316.9	−0.453395
$804$	0	0
$805$	−1331.97	−0.0583176
$806$	1737.52	0.0759326
$807$	0	0
$808$	−2181.50	−0.0949814
$809$	−36538.1	−1.58790	−0.793949	−	0.607984i	$-0.791979\pi$
$809$	−36538.1	−1.58790	−0.793949	+	0.607984i	$0.791979\pi$
$810$	0	0
$811$	−9400.19	−0.407010	−0.203505	−	0.979074i	$-0.565233\pi$
$811$	−9400.19	−0.407010	−0.203505	+	0.979074i	$0.565233\pi$
$812$	−4303.40	−0.185985
$813$	0	0
$814$	1954.15	0.0841438
$815$	4766.97	0.204883
$816$	0	0
$817$	2843.04	0.121745
$818$	−3469.57	−0.148301
$819$	0	0
$820$	195.696	0.00833414
$821$	22281.7	0.947182	0.473591	−	0.880745i	$-0.342958\pi$
$821$	22281.7	0.947182	0.473591	+	0.880745i	$0.342958\pi$
$822$	0	0
$823$	44896.3	1.90156	0.950782	−	0.309861i	$-0.100282\pi$
$823$	44896.3	1.90156	0.950782	+	0.309861i	$0.100282\pi$
$824$	5973.84	0.252559
$825$	0	0
$826$	−583.900	−0.0245962
$827$	38313.8	1.61100	0.805502	−	0.592594i	$-0.201896\pi$
$827$	38313.8	1.61100	0.805502	+	0.592594i	$0.201896\pi$
$828$	0	0
$829$	8129.31	0.340582	0.170291	−	0.985394i	$-0.445529\pi$
$829$	8129.31	0.340582	0.170291	+	0.985394i	$0.445529\pi$
$830$	−384.824	−0.0160933
$831$	0	0
$832$	−8247.98	−0.343687
$833$	7683.03	0.319569
$834$	0	0
$835$	−3652.22	−0.151366
$836$	2414.71	0.0998978
$837$	0	0
$838$	5658.36	0.233252
$839$	15401.3	0.633744	0.316872	−	0.948468i	$-0.397367\pi$
$839$	15401.3	0.633744	0.316872	+	0.948468i	$0.397367\pi$
$840$	0	0
$841$	2043.47	0.0837864
$842$	3750.90	0.153521
$843$	0	0
$844$	−27451.1	−1.11955
$845$	8283.58	0.337235
$846$	0	0
$847$	422.987	0.0171594
$848$	−425.307	−0.0172230
$849$	0	0
$850$	379.955	0.0153322
$851$	−20689.5	−0.833403
$852$	0	0
$853$	−24884.8	−0.998872	−0.499436	−	0.866351i	$-0.666459\pi$
$853$	−24884.8	−0.998872	−0.499436	+	0.866351i	$0.666459\pi$
$854$	1352.77	0.0542046
$855$	0	0
$856$	−17297.2	−0.690662
$857$	41081.0	1.63746	0.818728	−	0.574182i	$-0.194680\pi$
$857$	41081.0	1.63746	0.818728	+	0.574182i	$0.194680\pi$
$858$	0	0
$859$	−1304.61	−0.0518192	−0.0259096	−	0.999664i	$-0.508248\pi$
$859$	−1304.61	−0.0518192	−0.0259096	+	0.999664i	$0.508248\pi$
$860$	−3712.65	−0.147210
$861$	0	0
$862$	−860.752	−0.0340108
$863$	−40341.8	−1.59125	−0.795625	−	0.605789i	$-0.792858\pi$
$863$	−40341.8	−1.59125	−0.795625	+	0.605789i	$0.792858\pi$
$864$	0	0
$865$	−2971.61	−0.116806
$866$	−1112.84	−0.0436673
$867$	0	0
$868$	3023.87	0.118245
$869$	−6564.37	−0.256250
$870$	0	0
$871$	−7701.67	−0.299611
$872$	−2658.82	−0.103256
$873$	0	0
$874$	1445.61	0.0559481
$875$	−436.970	−0.0168826
$876$	0	0
$877$	15848.0	0.610204	0.305102	−	0.952320i	$-0.401310\pi$
$877$	15848.0	0.610204	0.305102	+	0.952320i	$0.401310\pi$
$878$	5832.22	0.224178
$879$	0	0
$880$	−2964.92	−0.113577
$881$	3126.33	0.119556	0.0597779	−	0.998212i	$-0.480961\pi$
$881$	3126.33	0.119556	0.0597779	+	0.998212i	$0.480961\pi$
$882$	0	0
$883$	6162.62	0.234868	0.117434	−	0.993081i	$-0.462533\pi$
$883$	6162.62	0.234868	0.117434	+	0.993081i	$0.462533\pi$
$884$	4087.96	0.155535
$885$	0	0
$886$	−3115.31	−0.118127
$887$	42742.4	1.61798	0.808991	−	0.587821i	$-0.200014\pi$
$887$	42742.4	1.61798	0.808991	+	0.587821i	$0.200014\pi$
$888$	0	0
$889$	−2601.83	−0.0981580
$890$	2968.81	0.111814
$891$	0	0
$892$	−11056.8	−0.415034
$893$	7598.55	0.284743
$894$	0	0
$895$	−7953.66	−0.297052
$896$	4077.74	0.152040
$897$	0	0
$898$	−7458.90	−0.277179
$899$	−18573.3	−0.689049
$900$	0	0
$901$	183.251	0.00677577
$902$	−37.2050	−0.00137338
$903$	0	0
$904$	17785.5	0.654355
$905$	9257.05	0.340016
$906$	0	0
$907$	−53596.9	−1.96214	−0.981068	−	0.193665i	$-0.937962\pi$
$907$	−53596.9	−1.96214	−0.981068	+	0.193665i	$0.937962\pi$
$908$	−31813.6	−1.16275
$909$	0	0
$910$	265.841	0.00968410
$911$	27222.7	0.990041	0.495021	−	0.868881i	$-0.335160\pi$
$911$	27222.7	0.990041	0.495021	+	0.868881i	$0.335160\pi$
$912$	0	0
$913$	−1293.86	−0.0469008
$914$	2366.91	0.0856570
$915$	0	0
$916$	−9293.63	−0.335230
$917$	−4864.73	−0.175188
$918$	0	0
$919$	−28731.0	−1.03128	−0.515642	−	0.856804i	$-0.672446\pi$
$919$	−28731.0	−1.03128	−0.515642	+	0.856804i	$0.672446\pi$
$920$	−3882.32	−0.139126
$921$	0	0
$922$	−8651.26	−0.309018
$923$	−1469.54	−0.0524058
$924$	0	0
$925$	−6787.45	−0.241265
$926$	−783.418	−0.0278021
$927$	0	0
$928$	−18987.3	−0.671648
$929$	−16550.7	−0.584511	−0.292255	−	0.956340i	$-0.594406\pi$
$929$	−16550.7	−0.584511	−0.292255	+	0.956340i	$0.594406\pi$
$930$	0	0
$931$	9589.80	0.337587
$932$	−30777.5	−1.08171
$933$	0	0
$934$	−1019.74	−0.0357248
$935$	1277.49	0.0446826
$936$	0	0
$937$	−50653.4	−1.76604	−0.883018	−	0.469340i	$-0.844492\pi$
$937$	−50653.4	−1.76604	−0.883018	+	0.469340i	$0.844492\pi$
$938$	757.904	0.0263821
$939$	0	0
$940$	−9922.75	−0.344302
$941$	−39223.0	−1.35880	−0.679401	−	0.733767i	$-0.737760\pi$
$941$	−39223.0	−1.35880	−0.679401	+	0.733767i	$0.737760\pi$
$942$	0	0
$943$	393.905	0.0136027
$944$	13761.0	0.474451
$945$	0	0
$946$	705.837	0.0242587
$947$	27300.4	0.936794	0.468397	−	0.883518i	$-0.344832\pi$
$947$	27300.4	0.936794	0.468397	+	0.883518i	$0.344832\pi$
$948$	0	0
$949$	−21800.6	−0.745708
$950$	474.253	0.0161966
$951$	0	0
$952$	−827.321	−0.0281656
$953$	−2420.26	−0.0822664	−0.0411332	−	0.999154i	$-0.513097\pi$
$953$	−2420.26	−0.0822664	−0.0411332	+	0.999154i	$0.513097\pi$
$954$	0	0
$955$	14646.3	0.496276
$956$	29406.3	0.994841
$957$	0	0
$958$	−1402.06	−0.0472843
$959$	−6804.88	−0.229136
$960$	0	0
$961$	−16740.1	−0.561917
$962$	4129.30	0.138393
$963$	0	0
$964$	−28125.7	−0.939698
$965$	−12428.5	−0.414599
$966$	0	0
$967$	43229.4	1.43760	0.718802	−	0.695215i	$-0.244691\pi$
$967$	43229.4	1.43760	0.718802	+	0.695215i	$0.244691\pi$
$968$	1232.89	0.0409366
$969$	0	0
$970$	133.350	0.00441405
$971$	33895.2	1.12024	0.560118	−	0.828413i	$-0.310756\pi$
$971$	33895.2	1.12024	0.560118	+	0.828413i	$0.310756\pi$
$972$	0	0
$973$	−1708.58	−0.0562944
$974$	−4628.42	−0.152263
$975$	0	0
$976$	−31881.1	−1.04558
$977$	29081.2	0.952293	0.476146	−	0.879366i	$-0.342033\pi$
$977$	29081.2	0.952293	0.476146	+	0.879366i	$0.342033\pi$
$978$	0	0
$979$	9981.74	0.325861
$980$	−12523.1	−0.408199
$981$	0	0
$982$	5616.27	0.182508
$983$	−52689.5	−1.70960	−0.854798	−	0.518961i	$-0.826319\pi$
$983$	−52689.5	−1.70960	−0.854798	+	0.518961i	$0.826319\pi$
$984$	0	0
$985$	1603.86	0.0518814
$986$	2470.94	0.0798080
$987$	0	0
$988$	5102.51	0.164304
$989$	−7473.00	−0.240270
$990$	0	0
$991$	2394.92	0.0767679	0.0383840	−	0.999263i	$-0.487779\pi$
$991$	2394.92	0.0767679	0.0383840	+	0.999263i	$0.487779\pi$
$992$	13341.8	0.427020
$993$	0	0
$994$	144.614	0.00461458
$995$	−723.950	−0.0230661
$996$	0	0
$997$	33352.5	1.05946	0.529731	−	0.848166i	$-0.322293\pi$
$997$	33352.5	1.05946	0.529731	+	0.848166i	$0.322293\pi$
$998$	−7307.20	−0.231769
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.4.a.h.1.2	✓	3
3.2	odd	2		495.4.a.j.1.2	yes	3
5.4	even	2		2475.4.a.x.1.2		3
15.14	odd	2		2475.4.a.u.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.4.a.h.1.2	✓	3	1.1	even	1	trivial
495.4.a.j.1.2	yes	3	3.2	odd	2
2475.4.a.u.1.2		3	15.14	odd	2
2475.4.a.x.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)

\(f(q)\)	\(=\)	\(q-0.654334 q^{2} -7.57185 q^{4} -5.00000 q^{5} +3.49576 q^{7} +10.1892 q^{8} +O(q^{10})\)	\(q-0.654334 q^{2} -7.57185 q^{4} -5.00000 q^{5} +3.49576 q^{7} +10.1892 q^{8} +3.27167 q^{10} +11.0000 q^{11} +23.2440 q^{13} -2.28739 q^{14} +53.9077 q^{16} -23.2270 q^{17} -28.9915 q^{19} +37.8592 q^{20} -7.19767 q^{22} +76.2048 q^{23} +25.0000 q^{25} -15.2093 q^{26} -26.4693 q^{28} +162.581 q^{29} -114.241 q^{31} -116.787 q^{32} +15.1982 q^{34} -17.4788 q^{35} -271.498 q^{37} +18.9701 q^{38} -50.9459 q^{40} +5.16904 q^{41} -98.0647 q^{43} -83.2903 q^{44} -49.8633 q^{46} -262.096 q^{47} -330.780 q^{49} -16.3583 q^{50} -176.000 q^{52} -7.88955 q^{53} -55.0000 q^{55} +35.6189 q^{56} -106.382 q^{58} +255.269 q^{59} -591.402 q^{61} +74.7515 q^{62} -354.844 q^{64} -116.220 q^{65} -331.340 q^{67} +175.871 q^{68} +11.4370 q^{70} -63.2225 q^{71} -937.901 q^{73} +177.650 q^{74} +219.519 q^{76} +38.4533 q^{77} -596.761 q^{79} -269.538 q^{80} -3.38227 q^{82} -117.623 q^{83} +116.135 q^{85} +64.1670 q^{86} +112.081 q^{88} +907.431 q^{89} +81.2554 q^{91} -577.011 q^{92} +171.498 q^{94} +144.958 q^{95} +40.7592 q^{97} +216.440 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + q^{4} - 15 q^{5} + 2 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q - q^2 + q^4 - 15 * q^5 + 2 * q^7 + 3 * q^8	\( 3 q - q^{2} + q^{4} - 15 q^{5} + 2 q^{7} + 3 q^{8} + 5 q^{10} + 33 q^{11} - 66 q^{13} + 88 q^{14} - 103 q^{16} + 100 q^{17} - 70 q^{19} - 5 q^{20} - 11 q^{22} + 10 q^{23} + 75 q^{25} - 88 q^{26} - 64 q^{28} - 34 q^{29} - 104 q^{31} - 37 q^{32} - 278 q^{34} - 10 q^{35} - 270 q^{37} - 154 q^{38} - 15 q^{40} - 82 q^{41} - 428 q^{43} + 11 q^{44} - 706 q^{46} + 246 q^{47} - 685 q^{49} - 25 q^{50} - 528 q^{52} + 298 q^{53} - 165 q^{55} - 288 q^{56} + 114 q^{58} - 204 q^{59} - 492 q^{61} - 584 q^{62} - 319 q^{64} + 330 q^{65} - 812 q^{67} + 798 q^{68} - 440 q^{70} - 442 q^{71} - 1044 q^{73} - 42 q^{74} + 106 q^{76} + 22 q^{77} - 1436 q^{79} + 515 q^{80} - 1246 q^{82} + 734 q^{83} - 500 q^{85} + 82 q^{86} + 33 q^{88} - 556 q^{89} - 176 q^{91} - 630 q^{92} + 38 q^{94} + 350 q^{95} - 1038 q^{97} + 143 q^{98}+O(q^{100}) \) 3 * q - q^2 + q^4 - 15 * q^5 + 2 * q^7 + 3 * q^8 + 5 * q^10 + 33 * q^11 - 66 * q^13 + 88 * q^14 - 103 * q^16 + 100 * q^17 - 70 * q^19 - 5 * q^20 - 11 * q^22 + 10 * q^23 + 75 * q^25 - 88 * q^26 - 64 * q^28 - 34 * q^29 - 104 * q^31 - 37 * q^32 - 278 * q^34 - 10 * q^35 - 270 * q^37 - 154 * q^38 - 15 * q^40 - 82 * q^41 - 428 * q^43 + 11 * q^44 - 706 * q^46 + 246 * q^47 - 685 * q^49 - 25 * q^50 - 528 * q^52 + 298 * q^53 - 165 * q^55 - 288 * q^56 + 114 * q^58 - 204 * q^59 - 492 * q^61 - 584 * q^62 - 319 * q^64 + 330 * q^65 - 812 * q^67 + 798 * q^68 - 440 * q^70 - 442 * q^71 - 1044 * q^73 - 42 * q^74 + 106 * q^76 + 22 * q^77 - 1436 * q^79 + 515 * q^80 - 1246 * q^82 + 734 * q^83 - 500 * q^85 + 82 * q^86 + 33 * q^88 - 556 * q^89 - 176 * q^91 - 630 * q^92 + 38 * q^94 + 350 * q^95 - 1038 * q^97 + 143 * q^98

Introduction

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