LMFDB - Embedded newform 153.4.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [153,4,Mod(1,153)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(153, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("153.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 153 = 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	153.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:	$9.02729223088$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.2636.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 14x - 4 $ x^3 - 14*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 17)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.87707$ of defining polynomial
Character	$\chi$	$=$	153.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.36122 q^{2} -6.14708 q^{4} -3.03171 q^{5} -7.94049 q^{7} +19.2573 q^{8} +O(q^{10})$	\(q-1.36122 q^{2} -6.14708 q^{4} -3.03171 q^{5} -7.94049 q^{7} +19.2573 q^{8} +4.12682 q^{10} -27.6161 q^{11} +58.1117 q^{13} +10.8088 q^{14} +22.9632 q^{16} +17.0000 q^{17} +89.1688 q^{19} +18.6361 q^{20} +37.5916 q^{22} +115.269 q^{23} -115.809 q^{25} -79.1029 q^{26} +48.8108 q^{28} +128.558 q^{29} +273.460 q^{31} -185.316 q^{32} -23.1408 q^{34} +24.0732 q^{35} -132.351 q^{37} -121.379 q^{38} -58.3825 q^{40} +470.559 q^{41} +352.642 q^{43} +169.758 q^{44} -156.907 q^{46} -152.598 q^{47} -279.949 q^{49} +157.641 q^{50} -357.217 q^{52} -527.614 q^{53} +83.7239 q^{55} -152.912 q^{56} -174.995 q^{58} +292.020 q^{59} -53.8962 q^{61} -372.239 q^{62} +68.5514 q^{64} -176.178 q^{65} +52.9572 q^{67} -104.500 q^{68} -32.7690 q^{70} -788.400 q^{71} +295.780 q^{73} +180.159 q^{74} -548.127 q^{76} +219.285 q^{77} -720.325 q^{79} -69.6175 q^{80} -640.535 q^{82} +116.051 q^{83} -51.5390 q^{85} -480.024 q^{86} -531.812 q^{88} +813.329 q^{89} -461.435 q^{91} -708.569 q^{92} +207.720 q^{94} -270.334 q^{95} +794.693 q^{97} +381.072 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 25 q^{4} + 8 q^{5} + 22 q^{7} + 39 q^{8}+O(q^{10}) $ 3 * q - q^2 + 25 * q^4 + 8 * q^5 + 22 * q^7 + 39 * q^8	$ 3 q - q^{2} + 25 q^{4} + 8 q^{5} + 22 q^{7} + 39 q^{8} - 56 q^{10} + 28 q^{11} + 30 q^{13} - 92 q^{14} + 137 q^{16} + 51 q^{17} + 80 q^{19} + 168 q^{20} + 286 q^{22} - 142 q^{23} - 223 q^{25} - 26 q^{26} + 476 q^{28} + 456 q^{29} + 230 q^{31} + 71 q^{32} - 17 q^{34} + 332 q^{35} + 356 q^{37} - 724 q^{38} - 424 q^{40} + 294 q^{41} + 556 q^{43} + 1122 q^{44} - 704 q^{46} - 640 q^{47} - 269 q^{49} - 547 q^{50} - 774 q^{52} - 302 q^{53} + 76 q^{55} - 684 q^{56} - 1304 q^{58} - 636 q^{59} - 84 q^{61} - 508 q^{62} - 919 q^{64} - 408 q^{65} + 1008 q^{67} + 425 q^{68} - 1504 q^{70} + 402 q^{71} + 838 q^{73} - 836 q^{74} - 908 q^{76} + 504 q^{77} - 594 q^{79} + 40 q^{80} + 358 q^{82} + 2396 q^{83} + 136 q^{85} + 1264 q^{86} + 1838 q^{88} + 170 q^{89} - 1016 q^{91} - 4896 q^{92} - 2016 q^{94} + 472 q^{95} - 270 q^{97} - 2857 q^{98}+O(q^{100}) $ 3 * q - q^2 + 25 * q^4 + 8 * q^5 + 22 * q^7 + 39 * q^8 - 56 * q^10 + 28 * q^11 + 30 * q^13 - 92 * q^14 + 137 * q^16 + 51 * q^17 + 80 * q^19 + 168 * q^20 + 286 * q^22 - 142 * q^23 - 223 * q^25 - 26 * q^26 + 476 * q^28 + 456 * q^29 + 230 * q^31 + 71 * q^32 - 17 * q^34 + 332 * q^35 + 356 * q^37 - 724 * q^38 - 424 * q^40 + 294 * q^41 + 556 * q^43 + 1122 * q^44 - 704 * q^46 - 640 * q^47 - 269 * q^49 - 547 * q^50 - 774 * q^52 - 302 * q^53 + 76 * q^55 - 684 * q^56 - 1304 * q^58 - 636 * q^59 - 84 * q^61 - 508 * q^62 - 919 * q^64 - 408 * q^65 + 1008 * q^67 + 425 * q^68 - 1504 * q^70 + 402 * q^71 + 838 * q^73 - 836 * q^74 - 908 * q^76 + 504 * q^77 - 594 * q^79 + 40 * q^80 + 358 * q^82 + 2396 * q^83 + 136 * q^85 + 1264 * q^86 + 1838 * q^88 + 170 * q^89 - 1016 * q^91 - 4896 * q^92 - 2016 * q^94 + 472 * q^95 - 270 * q^97 - 2857 * 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 $

$2$	−1.36122	−0.481264	−0.240632	−	0.970616i	$-0.577355\pi$
$2$	−1.36122	−0.481264	−0.240632	+	0.970616i	$0.577355\pi$
$3$	0	0
$3$	0	0
$4$	−6.14708	−0.768385
$5$	−3.03171	−0.271164	−0.135582	−	0.990766i	$-0.543290\pi$
$5$	−3.03171	−0.271164	−0.135582	+	0.990766i	$0.543290\pi$
$6$	0	0
$7$	−7.94049	−0.428746	−0.214373	−	0.976752i	$-0.568771\pi$
$7$	−7.94049	−0.428746	−0.214373	+	0.976752i	$0.568771\pi$
$8$	19.2573	0.851061
$9$	0	0
$10$	4.12682	0.130502
$11$	−27.6161	−0.756961	−0.378481	−	0.925609i	$-0.623553\pi$
$11$	−27.6161	−0.756961	−0.378481	+	0.925609i	$0.623553\pi$
$12$	0	0
$13$	58.1117	1.23979	0.619896	−	0.784684i	$-0.287175\pi$
$13$	58.1117	1.23979	0.619896	+	0.784684i	$0.287175\pi$
$14$	10.8088	0.206340
$15$	0	0
$16$	22.9632	0.358799
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	89.1688	1.07667	0.538335	−	0.842731i	$-0.319054\pi$
$19$	89.1688	1.07667	0.538335	+	0.842731i	$0.319054\pi$
$20$	18.6361	0.208358
$21$	0	0
$22$	37.5916	0.364298
$23$	115.269	1.04501	0.522507	−	0.852635i	$-0.324997\pi$
$23$	115.269	1.04501	0.522507	+	0.852635i	$0.324997\pi$
$24$	0	0
$25$	−115.809	−0.926470
$26$	−79.1029	−0.596668
$27$	0	0
$28$	48.8108	0.329442
$29$	128.558	0.823191	0.411596	−	0.911367i	$-0.364972\pi$
$29$	128.558	0.823191	0.411596	+	0.911367i	$0.364972\pi$
$30$	0	0
$31$	273.460	1.58435	0.792174	−	0.610295i	$-0.208949\pi$
$31$	273.460	1.58435	0.792174	+	0.610295i	$0.208949\pi$
$32$	−185.316	−1.02374
$33$	0	0
$34$	−23.1408	−0.116724
$35$	24.0732	0.116260
$36$	0	0
$37$	−132.351	−0.588063	−0.294031	−	0.955796i	$-0.594997\pi$
$37$	−132.351	−0.588063	−0.294031	+	0.955796i	$0.594997\pi$
$38$	−121.379	−0.518163
$39$	0	0
$40$	−58.3825	−0.230777
$41$	470.559	1.79241	0.896207	−	0.443636i	$-0.146312\pi$
$41$	470.559	1.79241	0.896207	+	0.443636i	$0.146312\pi$
$42$	0	0
$43$	352.642	1.25064	0.625318	−	0.780370i	$-0.284969\pi$
$43$	352.642	1.25064	0.625318	+	0.780370i	$0.284969\pi$
$44$	169.758	0.581637
$45$	0	0
$46$	−156.907	−0.502928
$47$	−152.598	−0.473589	−0.236795	−	0.971560i	$-0.576097\pi$
$47$	−152.598	−0.473589	−0.236795	+	0.971560i	$0.576097\pi$
$48$	0	0
$49$	−279.949	−0.816177
$50$	157.641	0.445877
$51$	0	0
$52$	−357.217	−0.952637
$53$	−527.614	−1.36742	−0.683711	−	0.729753i	$-0.739635\pi$
$53$	−527.614	−1.36742	−0.683711	+	0.729753i	$0.739635\pi$
$54$	0	0
$55$	83.7239	0.205261
$56$	−152.912	−0.364889
$57$	0	0
$58$	−174.995	−0.396173
$59$	292.020	0.644368	0.322184	−	0.946677i	$-0.395583\pi$
$59$	292.020	0.644368	0.322184	+	0.946677i	$0.395583\pi$
$60$	0	0
$61$	−53.8962	−0.113126	−0.0565632	−	0.998399i	$-0.518014\pi$
$61$	−53.8962	−0.113126	−0.0565632	+	0.998399i	$0.518014\pi$
$62$	−372.239	−0.762490
$63$	0	0
$64$	68.5514	0.133889
$65$	−176.178	−0.336187
$66$	0	0
$67$	52.9572	0.0965635	0.0482817	−	0.998834i	$-0.484625\pi$
$67$	52.9572	0.0965635	0.0482817	+	0.998834i	$0.484625\pi$
$68$	−104.500	−0.186361
$69$	0	0
$70$	−32.7690	−0.0559520
$71$	−788.400	−1.31783	−0.658915	−	0.752218i	$-0.728984\pi$
$71$	−788.400	−1.31783	−0.658915	+	0.752218i	$0.728984\pi$
$72$	0	0
$73$	295.780	0.474224	0.237112	−	0.971482i	$-0.423799\pi$
$73$	295.780	0.474224	0.237112	+	0.971482i	$0.423799\pi$
$74$	180.159	0.283014
$75$	0	0
$76$	−548.127	−0.827296
$77$	219.285	0.324544
$78$	0	0
$79$	−720.325	−1.02586	−0.512930	−	0.858430i	$-0.671440\pi$
$79$	−720.325	−1.02586	−0.512930	+	0.858430i	$0.671440\pi$
$80$	−69.6175	−0.0972934
$81$	0	0
$82$	−640.535	−0.862625
$83$	116.051	0.153473	0.0767363	−	0.997051i	$-0.475550\pi$
$83$	116.051	0.153473	0.0767363	+	0.997051i	$0.475550\pi$
$84$	0	0
$85$	−51.5390	−0.0657669
$86$	−480.024	−0.601887
$87$	0	0
$88$	−531.812	−0.644220
$89$	813.329	0.968682	0.484341	−	0.874879i	$-0.339059\pi$
$89$	813.329	0.968682	0.484341	+	0.874879i	$0.339059\pi$
$90$	0	0
$91$	−461.435	−0.531556
$92$	−708.569	−0.802972
$93$	0	0
$94$	207.720	0.227922
$95$	−270.334	−0.291954
$96$	0	0
$97$	794.693	0.831844	0.415922	−	0.909400i	$-0.363459\pi$
$97$	794.693	0.831844	0.415922	+	0.909400i	$0.363459\pi$
$98$	381.072	0.392797
$99$	0	0
$100$	711.885	0.711885
$101$	−265.513	−0.261579	−0.130790	−	0.991410i	$-0.541751\pi$
$101$	−265.513	−0.261579	−0.130790	+	0.991410i	$0.541751\pi$
$102$	0	0
$103$	523.107	0.500420	0.250210	−	0.968192i	$-0.419500\pi$
$103$	523.107	0.500420	0.250210	+	0.968192i	$0.419500\pi$
$104$	1119.07	1.05514
$105$	0	0
$106$	718.199	0.658091
$107$	986.039	0.890878	0.445439	−	0.895312i	$-0.353048\pi$
$107$	986.039	0.890878	0.445439	+	0.895312i	$0.353048\pi$
$108$	0	0
$109$	1814.39	1.59438	0.797188	−	0.603732i	$-0.206320\pi$
$109$	1814.39	1.59438	0.797188	+	0.603732i	$0.206320\pi$
$110$	−113.967	−0.0987846
$111$	0	0
$112$	−182.339	−0.153834
$113$	707.339	0.588857	0.294429	−	0.955673i	$-0.404871\pi$
$113$	707.339	0.588857	0.294429	+	0.955673i	$0.404871\pi$
$114$	0	0
$115$	−349.463	−0.283370
$116$	−790.253	−0.632527
$117$	0	0
$118$	−397.503	−0.310112
$119$	−134.988	−0.103986
$120$	0	0
$121$	−568.350	−0.427010
$122$	73.3647	0.0544437
$123$	0	0
$124$	−1680.98	−1.21739
$125$	730.061	0.522389
$126$	0	0
$127$	2648.18	1.85030	0.925151	−	0.379600i	$-0.123938\pi$
$127$	2648.18	1.85030	0.925151	+	0.379600i	$0.123938\pi$
$128$	1389.22	0.959302
$129$	0	0
$130$	239.817	0.161795
$131$	1979.08	1.31995	0.659974	−	0.751289i	$-0.270567\pi$
$131$	1979.08	1.31995	0.659974	+	0.751289i	$0.270567\pi$
$132$	0	0
$133$	−708.044	−0.461618
$134$	−72.0865	−0.0464726
$135$	0	0
$136$	327.374	0.206413
$137$	−3141.92	−1.95936	−0.979679	−	0.200570i	$-0.935721\pi$
$137$	−3141.92	−1.95936	−0.979679	+	0.200570i	$0.935721\pi$
$138$	0	0
$139$	1468.07	0.895830	0.447915	−	0.894076i	$-0.352167\pi$
$139$	1468.07	0.895830	0.447915	+	0.894076i	$0.352167\pi$
$140$	−147.980	−0.0893327
$141$	0	0
$142$	1073.19	0.634224
$143$	−1604.82	−0.938474
$144$	0	0
$145$	−389.749	−0.223220
$146$	−402.621	−0.228227
$147$	0	0
$148$	813.570	0.451858
$149$	286.027	0.157263	0.0786316	−	0.996904i	$-0.474945\pi$
$149$	286.027	0.157263	0.0786316	+	0.996904i	$0.474945\pi$
$150$	0	0
$151$	−669.626	−0.360883	−0.180442	−	0.983586i	$-0.557753\pi$
$151$	−669.626	−0.360883	−0.180442	+	0.983586i	$0.557753\pi$
$152$	1717.15	0.916311
$153$	0	0
$154$	−298.496	−0.156191
$155$	−829.049	−0.429618
$156$	0	0
$157$	720.809	0.366413	0.183206	−	0.983074i	$-0.441352\pi$
$157$	720.809	0.366413	0.183206	+	0.983074i	$0.441352\pi$
$158$	980.522	0.493710
$159$	0	0
$160$	561.825	0.277601
$161$	−915.294	−0.448045
$162$	0	0
$163$	−676.599	−0.325125	−0.162562	−	0.986698i	$-0.551976\pi$
$163$	−676.599	−0.325125	−0.162562	+	0.986698i	$0.551976\pi$
$164$	−2892.56	−1.37726
$165$	0	0
$166$	−157.971	−0.0738609
$167$	2835.67	1.31396	0.656979	−	0.753909i	$-0.271834\pi$
$167$	2835.67	1.31396	0.656979	+	0.753909i	$0.271834\pi$
$168$	0	0
$169$	1179.97	0.537083
$170$	70.1560	0.0316513
$171$	0	0
$172$	−2167.72	−0.960970
$173$	177.314	0.0779243	0.0389621	−	0.999241i	$-0.487595\pi$
$173$	177.314	0.0779243	0.0389621	+	0.999241i	$0.487595\pi$
$174$	0	0
$175$	919.578	0.397220
$176$	−634.153	−0.271597
$177$	0	0
$178$	−1107.12	−0.466192
$179$	1023.76	0.427483	0.213741	−	0.976890i	$-0.431435\pi$
$179$	1023.76	0.427483	0.213741	+	0.976890i	$0.431435\pi$
$180$	0	0
$181$	−3450.21	−1.41686	−0.708432	−	0.705779i	$-0.750597\pi$
$181$	−3450.21	−1.41686	−0.708432	+	0.705779i	$0.750597\pi$
$182$	628.116	0.255819
$183$	0	0
$184$	2219.78	0.889370
$185$	401.248	0.159461
$186$	0	0
$187$	−469.474	−0.183590
$188$	938.031	0.363899
$189$	0	0
$190$	367.984	0.140507
$191$	490.894	0.185968	0.0929839	−	0.995668i	$-0.470360\pi$
$191$	490.894	0.185968	0.0929839	+	0.995668i	$0.470360\pi$
$192$	0	0
$193$	−3548.80	−1.32357	−0.661783	−	0.749696i	$-0.730200\pi$
$193$	−3548.80	−1.32357	−0.661783	+	0.749696i	$0.730200\pi$
$194$	−1081.75	−0.400337
$195$	0	0
$196$	1720.87	0.627138
$197$	−1363.15	−0.492996	−0.246498	−	0.969143i	$-0.579280\pi$
$197$	−1363.15	−0.492996	−0.246498	+	0.969143i	$0.579280\pi$
$198$	0	0
$199$	3737.46	1.33137	0.665683	−	0.746235i	$-0.268140\pi$
$199$	3737.46	1.33137	0.665683	+	0.746235i	$0.268140\pi$
$200$	−2230.16	−0.788482
$201$	0	0
$202$	361.422	0.125889
$203$	−1020.81	−0.352940
$204$	0	0
$205$	−1426.60	−0.486038
$206$	−712.064	−0.240834
$207$	0	0
$208$	1334.43	0.444836
$209$	−2462.50	−0.814997
$210$	0	0
$211$	−5266.12	−1.71817	−0.859087	−	0.511829i	$-0.828968\pi$
$211$	−5266.12	−1.71817	−0.859087	+	0.511829i	$0.828968\pi$
$212$	3243.28	1.05071
$213$	0	0
$214$	−1342.22	−0.428748
$215$	−1069.11	−0.339128
$216$	0	0
$217$	−2171.40	−0.679283
$218$	−2469.78	−0.767316
$219$	0	0
$220$	−514.657	−0.157719
$221$	987.899	0.300694
$222$	0	0
$223$	704.546	0.211569	0.105785	−	0.994389i	$-0.466265\pi$
$223$	704.546	0.211569	0.105785	+	0.994389i	$0.466265\pi$
$224$	1471.50	0.438923
$225$	0	0
$226$	−962.845	−0.283396
$227$	2151.26	0.629006	0.314503	−	0.949256i	$-0.398162\pi$
$227$	2151.26	0.629006	0.314503	+	0.949256i	$0.398162\pi$
$228$	0	0
$229$	−3916.94	−1.13030	−0.565149	−	0.824989i	$-0.691181\pi$
$229$	−3916.94	−1.13030	−0.565149	+	0.824989i	$0.691181\pi$
$230$	475.696	0.136376
$231$	0	0
$232$	2475.67	0.700586
$233$	5192.74	1.46003	0.730017	−	0.683429i	$-0.239512\pi$
$233$	5192.74	1.46003	0.730017	+	0.683429i	$0.239512\pi$
$234$	0	0
$235$	462.632	0.128420
$236$	−1795.07	−0.495123
$237$	0	0
$238$	183.749	0.0500448
$239$	−334.305	−0.0904786	−0.0452393	−	0.998976i	$-0.514405\pi$
$239$	−334.305	−0.0904786	−0.0452393	+	0.998976i	$0.514405\pi$
$240$	0	0
$241$	−1918.45	−0.512773	−0.256386	−	0.966574i	$-0.582532\pi$
$241$	−1918.45	−0.512773	−0.256386	+	0.966574i	$0.582532\pi$
$242$	773.651	0.205505
$243$	0	0
$244$	331.304	0.0869245
$245$	848.722	0.221318
$246$	0	0
$247$	5181.75	1.33485
$248$	5266.09	1.34838
$249$	0	0
$250$	−993.775	−0.251407
$251$	−7695.71	−1.93525	−0.967627	−	0.252385i	$-0.918785\pi$
$251$	−7695.71	−1.93525	−0.967627	+	0.252385i	$0.918785\pi$
$252$	0	0
$253$	−3183.29	−0.791035
$254$	−3604.76	−0.890484
$255$	0	0
$256$	−2439.44	−0.595567
$257$	−5335.10	−1.29492	−0.647460	−	0.762099i	$-0.724169\pi$
$257$	−5335.10	−1.29492	−0.647460	+	0.762099i	$0.724169\pi$
$258$	0	0
$259$	1050.93	0.252130
$260$	1082.98	0.258321
$261$	0	0
$262$	−2693.97	−0.635244
$263$	−3934.15	−0.922396	−0.461198	−	0.887297i	$-0.652580\pi$
$263$	−3934.15	−0.922396	−0.461198	+	0.887297i	$0.652580\pi$
$264$	0	0
$265$	1599.57	0.370795
$266$	963.804	0.222160
$267$	0	0
$268$	−325.532	−0.0741979
$269$	−3424.04	−0.776088	−0.388044	−	0.921641i	$-0.626849\pi$
$269$	−3424.04	−0.776088	−0.388044	+	0.921641i	$0.626849\pi$
$270$	0	0
$271$	549.034	0.123068	0.0615340	−	0.998105i	$-0.480401\pi$
$271$	549.034	0.123068	0.0615340	+	0.998105i	$0.480401\pi$
$272$	390.374	0.0870216
$273$	0	0
$274$	4276.85	0.942970
$275$	3198.19	0.701302
$276$	0	0
$277$	5203.65	1.12873	0.564363	−	0.825527i	$-0.309122\pi$
$277$	5203.65	1.12873	0.564363	+	0.825527i	$0.309122\pi$
$278$	−1998.37	−0.431131
$279$	0	0
$280$	463.585	0.0989447
$281$	1986.73	0.421774	0.210887	−	0.977510i	$-0.432365\pi$
$281$	1986.73	0.421774	0.210887	+	0.977510i	$0.432365\pi$
$282$	0	0
$283$	753.696	0.158313	0.0791565	−	0.996862i	$-0.474777\pi$
$283$	753.696	0.158313	0.0791565	+	0.996862i	$0.474777\pi$
$284$	4846.36	1.01260
$285$	0	0
$286$	2184.51	0.451654
$287$	−3736.47	−0.768490
$288$	0	0
$289$	289.000	0.0588235
$290$	530.534	0.107428
$291$	0	0
$292$	−1818.18	−0.364387
$293$	7202.22	1.43603	0.718017	−	0.696025i	$-0.245050\pi$
$293$	7202.22	1.43603	0.718017	+	0.696025i	$0.245050\pi$
$294$	0	0
$295$	−885.318	−0.174729
$296$	−2548.72	−0.500477
$297$	0	0
$298$	−389.345	−0.0756852
$299$	6698.50	1.29560
$300$	0	0
$301$	−2800.15	−0.536205
$302$	911.509	0.173680
$303$	0	0
$304$	2047.60	0.386308
$305$	163.398	0.0306758
$306$	0	0
$307$	2425.71	0.450953	0.225477	−	0.974249i	$-0.427606\pi$
$307$	2425.71	0.450953	0.225477	+	0.974249i	$0.427606\pi$
$308$	−1347.96	−0.249375
$309$	0	0
$310$	1128.52	0.206760
$311$	9544.94	1.74033	0.870167	−	0.492757i	$-0.164011\pi$
$311$	9544.94	1.74033	0.870167	+	0.492757i	$0.164011\pi$
$312$	0	0
$313$	588.379	0.106253	0.0531264	−	0.998588i	$-0.483081\pi$
$313$	588.379	0.106253	0.0531264	+	0.998588i	$0.483081\pi$
$314$	−981.180	−0.176341
$315$	0	0
$316$	4427.89	0.788255
$317$	−7653.31	−1.35600	−0.678001	−	0.735061i	$-0.737154\pi$
$317$	−7653.31	−1.35600	−0.678001	+	0.735061i	$0.737154\pi$
$318$	0	0
$319$	−3550.26	−0.623124
$320$	−207.828	−0.0363060
$321$	0	0
$322$	1245.92	0.215628
$323$	1515.87	0.261131
$324$	0	0
$325$	−6729.85	−1.14863
$326$	921.001	0.156471
$327$	0	0
$328$	9061.70	1.52545
$329$	1211.70	0.203050
$330$	0	0
$331$	752.266	0.124919	0.0624597	−	0.998047i	$-0.480106\pi$
$331$	752.266	0.124919	0.0624597	+	0.998047i	$0.480106\pi$
$332$	−713.373	−0.117926
$333$	0	0
$334$	−3859.98	−0.632361
$335$	−160.551	−0.0261845
$336$	0	0
$337$	−1968.57	−0.318204	−0.159102	−	0.987262i	$-0.550860\pi$
$337$	−1968.57	−0.318204	−0.159102	+	0.987262i	$0.550860\pi$
$338$	−1606.20	−0.258479
$339$	0	0
$340$	316.814	0.0505343
$341$	−7551.89	−1.19929
$342$	0	0
$343$	4946.52	0.778678
$344$	6790.93	1.06437
$345$	0	0
$346$	−241.363	−0.0375022
$347$	−3983.10	−0.616207	−0.308104	−	0.951353i	$-0.599694\pi$
$347$	−3983.10	−0.616207	−0.308104	+	0.951353i	$0.599694\pi$
$348$	0	0
$349$	1495.61	0.229393	0.114697	−	0.993401i	$-0.463410\pi$
$349$	1495.61	0.229393	0.114697	+	0.993401i	$0.463410\pi$
$350$	−1251.75	−0.191168
$351$	0	0
$352$	5117.72	0.774930
$353$	−6482.49	−0.977417	−0.488708	−	0.872447i	$-0.662532\pi$
$353$	−6482.49	−0.977417	−0.488708	+	0.872447i	$0.662532\pi$
$354$	0	0
$355$	2390.20	0.357348
$356$	−4999.59	−0.744320
$357$	0	0
$358$	−1393.56	−0.205732
$359$	4943.42	0.726751	0.363376	−	0.931643i	$-0.381624\pi$
$359$	4943.42	0.726751	0.363376	+	0.931643i	$0.381624\pi$
$360$	0	0
$361$	1092.08	0.159218
$362$	4696.50	0.681886
$363$	0	0
$364$	2836.48	0.408439
$365$	−896.717	−0.128593
$366$	0	0
$367$	−14.8871	−0.00211743	−0.00105872	−	0.999999i	$-0.500337\pi$
$367$	−14.8871	−0.00211743	−0.00105872	+	0.999999i	$0.500337\pi$
$368$	2646.95	0.374950
$369$	0	0
$370$	−546.188	−0.0767431
$371$	4189.51	0.586276
$372$	0	0
$373$	1923.18	0.266966	0.133483	−	0.991051i	$-0.457384\pi$
$373$	1923.18	0.266966	0.133483	+	0.991051i	$0.457384\pi$
$374$	639.058	0.0883554
$375$	0	0
$376$	−2938.63	−0.403053
$377$	7470.70	1.02059
$378$	0	0
$379$	−9592.87	−1.30014	−0.650069	−	0.759875i	$-0.725260\pi$
$379$	−9592.87	−1.30014	−0.650069	+	0.759875i	$0.725260\pi$
$380$	1661.76	0.224333
$381$	0	0
$382$	−668.215	−0.0894996
$383$	9083.77	1.21190	0.605951	−	0.795502i	$-0.292793\pi$
$383$	9083.77	1.21190	0.605951	+	0.795502i	$0.292793\pi$
$384$	0	0
$385$	−664.809	−0.0880046
$386$	4830.70	0.636985
$387$	0	0
$388$	−4885.04	−0.639176
$389$	1143.78	0.149079	0.0745396	−	0.997218i	$-0.476251\pi$
$389$	1143.78	0.149079	0.0745396	+	0.997218i	$0.476251\pi$
$390$	0	0
$391$	1959.58	0.253453
$392$	−5391.06	−0.694616
$393$	0	0
$394$	1855.55	0.237262
$395$	2183.81	0.278176
$396$	0	0
$397$	10604.5	1.34061	0.670307	−	0.742084i	$-0.266162\pi$
$397$	10604.5	1.34061	0.670307	+	0.742084i	$0.266162\pi$
$398$	−5087.51	−0.640739
$399$	0	0
$400$	−2659.33	−0.332417
$401$	−13785.4	−1.71674	−0.858368	−	0.513035i	$-0.828521\pi$
$401$	−13785.4	−1.71674	−0.858368	+	0.513035i	$0.828521\pi$
$402$	0	0
$403$	15891.2	1.96426
$404$	1632.13	0.200993
$405$	0	0
$406$	1389.55	0.169857
$407$	3655.01	0.445141
$408$	0	0
$409$	−9505.94	−1.14924	−0.574619	−	0.818421i	$-0.694850\pi$
$409$	−9505.94	−1.14924	−0.574619	+	0.818421i	$0.694850\pi$
$410$	1941.91	0.233913
$411$	0	0
$412$	−3215.58	−0.384515
$413$	−2318.78	−0.276270
$414$	0	0
$415$	−351.832	−0.0416162
$416$	−10769.1	−1.26922
$417$	0	0
$418$	3352.00	0.392229
$419$	−9680.86	−1.12874	−0.564369	−	0.825523i	$-0.690880\pi$
$419$	−9680.86	−1.12874	−0.564369	+	0.825523i	$0.690880\pi$
$420$	0	0
$421$	−12360.3	−1.43089	−0.715444	−	0.698671i	$-0.753775\pi$
$421$	−12360.3	−1.43089	−0.715444	+	0.698671i	$0.753775\pi$
$422$	7168.36	0.826897
$423$	0	0
$424$	−10160.4	−1.16376
$425$	−1968.75	−0.224702
$426$	0	0
$427$	427.962	0.0485025
$428$	−6061.25	−0.684537
$429$	0	0
$430$	1455.29	0.163210
$431$	−2970.58	−0.331990	−0.165995	−	0.986127i	$-0.553084\pi$
$431$	−2970.58	−0.331990	−0.165995	+	0.986127i	$0.553084\pi$
$432$	0	0
$433$	6131.50	0.680510	0.340255	−	0.940333i	$-0.389487\pi$
$433$	6131.50	0.680510	0.340255	+	0.940333i	$0.389487\pi$
$434$	2955.76	0.326915
$435$	0	0
$436$	−11153.2	−1.22509
$437$	10278.4	1.12513
$438$	0	0
$439$	−2544.91	−0.276679	−0.138339	−	0.990385i	$-0.544176\pi$
$439$	−2544.91	−0.276679	−0.138339	+	0.990385i	$0.544176\pi$
$440$	1612.30	0.174689
$441$	0	0
$442$	−1344.75	−0.144713
$443$	−8529.82	−0.914817	−0.457408	−	0.889257i	$-0.651222\pi$
$443$	−8529.82	−0.914817	−0.457408	+	0.889257i	$0.651222\pi$
$444$	0	0
$445$	−2465.77	−0.262672
$446$	−959.043	−0.101821
$447$	0	0
$448$	−544.331	−0.0574046
$449$	−8855.74	−0.930798	−0.465399	−	0.885101i	$-0.654089\pi$
$449$	−8855.74	−0.930798	−0.465399	+	0.885101i	$0.654089\pi$
$450$	0	0
$451$	−12995.0	−1.35679
$452$	−4348.07	−0.452469
$453$	0	0
$454$	−2928.35	−0.302718
$455$	1398.94	0.144139
$456$	0	0
$457$	−7154.78	−0.732356	−0.366178	−	0.930545i	$-0.619334\pi$
$457$	−7154.78	−0.732356	−0.366178	+	0.930545i	$0.619334\pi$
$458$	5331.82	0.543973
$459$	0	0
$460$	2148.17	0.217737
$461$	7263.06	0.733784	0.366892	−	0.930264i	$-0.380422\pi$
$461$	7263.06	0.733784	0.366892	+	0.930264i	$0.380422\pi$
$462$	0	0
$463$	352.898	0.0354224	0.0177112	−	0.999843i	$-0.494362\pi$
$463$	352.898	0.0354224	0.0177112	+	0.999843i	$0.494362\pi$
$464$	2952.09	0.295360
$465$	0	0
$466$	−7068.47	−0.702662
$467$	−1483.02	−0.146951	−0.0734753	−	0.997297i	$-0.523409\pi$
$467$	−1483.02	−0.146951	−0.0734753	+	0.997297i	$0.523409\pi$
$468$	0	0
$469$	−420.506	−0.0414012
$470$	−629.745	−0.0618042
$471$	0	0
$472$	5623.51	0.548396
$473$	−9738.60	−0.946683
$474$	0	0
$475$	−10326.5	−0.997502
$476$	829.783	0.0799014
$477$	0	0
$478$	455.063	0.0435441
$479$	9990.10	0.952942	0.476471	−	0.879190i	$-0.341916\pi$
$479$	9990.10	0.952942	0.476471	+	0.879190i	$0.341916\pi$
$480$	0	0
$481$	−7691.13	−0.729075
$482$	2611.44	0.246779
$483$	0	0
$484$	3493.69	0.328108
$485$	−2409.27	−0.225566
$486$	0	0
$487$	−1129.88	−0.105133	−0.0525663	−	0.998617i	$-0.516740\pi$
$487$	−1129.88	−0.105133	−0.0525663	+	0.998617i	$0.516740\pi$
$488$	−1037.90	−0.0962774
$489$	0	0
$490$	−1155.30	−0.106512
$491$	−18774.9	−1.72566	−0.862832	−	0.505491i	$-0.831311\pi$
$491$	−18774.9	−1.72566	−0.862832	+	0.505491i	$0.831311\pi$
$492$	0	0
$493$	2185.48	0.199653
$494$	−7053.51	−0.642414
$495$	0	0
$496$	6279.49	0.568463
$497$	6260.28	0.565014
$498$	0	0
$499$	17329.1	1.55462	0.777310	−	0.629118i	$-0.216584\pi$
$499$	17329.1	1.55462	0.777310	+	0.629118i	$0.216584\pi$
$500$	−4487.74	−0.401396
$501$	0	0
$502$	10475.6	0.931369
$503$	20837.0	1.84707	0.923533	−	0.383518i	$-0.125288\pi$
$503$	20837.0	1.84707	0.923533	+	0.383518i	$0.125288\pi$
$504$	0	0
$505$	804.957	0.0709309
$506$	4333.16	0.380697
$507$	0	0
$508$	−16278.6	−1.42174
$509$	−11835.0	−1.03060	−0.515301	−	0.857009i	$-0.672320\pi$
$509$	−11835.0	−1.03060	−0.515301	+	0.857009i	$0.672320\pi$
$510$	0	0
$511$	−2348.63	−0.203322
$512$	−7793.12	−0.672676
$513$	0	0
$514$	7262.26	0.623199
$515$	−1585.91	−0.135696
$516$	0	0
$517$	4214.16	0.358489
$518$	−1430.55	−0.121341
$519$	0	0
$520$	−3392.71	−0.286115
$521$	−7686.37	−0.646346	−0.323173	−	0.946340i	$-0.604750\pi$
$521$	−7686.37	−0.646346	−0.323173	+	0.946340i	$0.604750\pi$
$522$	0	0
$523$	11476.4	0.959518	0.479759	−	0.877400i	$-0.340724\pi$
$523$	11476.4	0.959518	0.479759	+	0.877400i	$0.340724\pi$
$524$	−12165.6	−1.01423
$525$	0	0
$526$	5355.25	0.443916
$527$	4648.81	0.384261
$528$	0	0
$529$	1120.01	0.0920535
$530$	−2177.37	−0.178451
$531$	0	0
$532$	4352.40	0.354700
$533$	27345.0	2.22222
$534$	0	0
$535$	−2989.38	−0.241574
$536$	1019.81	0.0821814
$537$	0	0
$538$	4660.88	0.373504
$539$	7731.09	0.617814
$540$	0	0
$541$	−546.481	−0.0434289	−0.0217145	−	0.999764i	$-0.506912\pi$
$541$	−546.481	−0.0434289	−0.0217145	+	0.999764i	$0.506912\pi$
$542$	−747.357	−0.0592283
$543$	0	0
$544$	−3150.38	−0.248293
$545$	−5500.69	−0.432337
$546$	0	0
$547$	8397.33	0.656388	0.328194	−	0.944610i	$-0.393560\pi$
$547$	8397.33	0.656388	0.328194	+	0.944610i	$0.393560\pi$
$548$	19313.6	1.50554
$549$	0	0
$550$	−4353.44	−0.337512
$551$	11463.3	0.886305
$552$	0	0
$553$	5719.73	0.439833
$554$	−7083.32	−0.543215
$555$	0	0
$556$	−9024.36	−0.688342
$557$	4881.65	0.371350	0.185675	−	0.982611i	$-0.440553\pi$
$557$	4881.65	0.371350	0.185675	+	0.982611i	$0.440553\pi$
$558$	0	0
$559$	20492.6	1.55053
$560$	552.797	0.0417142
$561$	0	0
$562$	−2704.38	−0.202985
$563$	−7198.57	−0.538870	−0.269435	−	0.963019i	$-0.586837\pi$
$563$	−7198.57	−0.538870	−0.269435	+	0.963019i	$0.586837\pi$
$564$	0	0
$565$	−2144.44	−0.159677
$566$	−1025.95	−0.0761904
$567$	0	0
$568$	−15182.5	−1.12155
$569$	23946.9	1.76433	0.882167	−	0.470937i	$-0.156084\pi$
$569$	23946.9	1.76433	0.882167	+	0.470937i	$0.156084\pi$
$570$	0	0
$571$	1593.15	0.116763	0.0583813	−	0.998294i	$-0.481406\pi$
$571$	1593.15	0.116763	0.0583813	+	0.998294i	$0.481406\pi$
$572$	9864.95	0.721109
$573$	0	0
$574$	5086.16	0.369847
$575$	−13349.2	−0.968174
$576$	0	0
$577$	12937.4	0.933435	0.466717	−	0.884406i	$-0.345436\pi$
$577$	12937.4	0.933435	0.466717	+	0.884406i	$0.345436\pi$
$578$	−393.393	−0.0283097
$579$	0	0
$580$	2395.82	0.171519
$581$	−921.499	−0.0658007
$582$	0	0
$583$	14570.6	1.03508
$584$	5695.92	0.403594
$585$	0	0
$586$	−9803.82	−0.691112
$587$	12899.2	0.906998	0.453499	−	0.891257i	$-0.350176\pi$
$587$	12899.2	0.906998	0.453499	+	0.891257i	$0.350176\pi$
$588$	0	0
$589$	24384.1	1.70582
$590$	1205.11	0.0840911
$591$	0	0
$592$	−3039.19	−0.210997
$593$	−4357.13	−0.301730	−0.150865	−	0.988554i	$-0.548206\pi$
$593$	−4357.13	−0.301730	−0.150865	+	0.988554i	$0.548206\pi$
$594$	0	0
$595$	409.245	0.0281973
$596$	−1758.23	−0.120839
$597$	0	0
$598$	−9118.14	−0.623526
$599$	−13726.8	−0.936328	−0.468164	−	0.883642i	$-0.655084\pi$
$599$	−13726.8	−0.936328	−0.468164	+	0.883642i	$0.655084\pi$
$600$	0	0
$601$	2531.41	0.171811	0.0859056	−	0.996303i	$-0.472622\pi$
$601$	2531.41	0.171811	0.0859056	+	0.996303i	$0.472622\pi$
$602$	3811.62	0.258057
$603$	0	0
$604$	4116.24	0.277297
$605$	1723.07	0.115790
$606$	0	0
$607$	185.004	0.0123708	0.00618540	−	0.999981i	$-0.498031\pi$
$607$	185.004	0.0123708	0.00618540	+	0.999981i	$0.498031\pi$
$608$	−16524.4	−1.10223
$609$	0	0
$610$	−222.420	−0.0147632
$611$	−8867.73	−0.587152
$612$	0	0
$613$	−17706.9	−1.16668	−0.583339	−	0.812228i	$-0.698254\pi$
$613$	−17706.9	−1.16668	−0.583339	+	0.812228i	$0.698254\pi$
$614$	−3301.93	−0.217028
$615$	0	0
$616$	4222.84	0.276207
$617$	6183.89	0.403491	0.201746	−	0.979438i	$-0.435339\pi$
$617$	6183.89	0.403491	0.201746	+	0.979438i	$0.435339\pi$
$618$	0	0
$619$	−1247.51	−0.0810046	−0.0405023	−	0.999179i	$-0.512896\pi$
$619$	−1247.51	−0.0810046	−0.0405023	+	0.999179i	$0.512896\pi$
$620$	5096.23	0.330112
$621$	0	0
$622$	−12992.8	−0.837561
$623$	−6458.23	−0.415318
$624$	0	0
$625$	12262.8	0.784817
$626$	−800.914	−0.0511357
$627$	0	0
$628$	−4430.87	−0.281546
$629$	−2249.96	−0.142626
$630$	0	0
$631$	24053.3	1.51750	0.758752	−	0.651379i	$-0.225809\pi$
$631$	24053.3	1.51750	0.758752	+	0.651379i	$0.225809\pi$
$632$	−13871.5	−0.873069
$633$	0	0
$634$	10417.8	0.652596
$635$	−8028.51	−0.501735
$636$	0	0
$637$	−16268.3	−1.01189
$638$	4832.69	0.299887
$639$	0	0
$640$	−4211.70	−0.260128
$641$	21286.8	1.31167	0.655834	−	0.754905i	$-0.272317\pi$
$641$	21286.8	1.31167	0.655834	+	0.754905i	$0.272317\pi$
$642$	0	0
$643$	−1789.41	−0.109747	−0.0548736	−	0.998493i	$-0.517476\pi$
$643$	−1789.41	−0.109747	−0.0548736	+	0.998493i	$0.517476\pi$
$644$	5626.38	0.344271
$645$	0	0
$646$	−2063.43	−0.125673
$647$	4378.61	0.266060	0.133030	−	0.991112i	$-0.457529\pi$
$647$	4378.61	0.266060	0.133030	+	0.991112i	$0.457529\pi$
$648$	0	0
$649$	−8064.45	−0.487762
$650$	9160.81	0.552795
$651$	0	0
$652$	4159.11	0.249821
$653$	7665.15	0.459358	0.229679	−	0.973266i	$-0.426232\pi$
$653$	7665.15	0.459358	0.229679	+	0.973266i	$0.426232\pi$
$654$	0	0
$655$	−5999.99	−0.357922
$656$	10805.5	0.643117
$657$	0	0
$658$	−1649.39	−0.0977205
$659$	4710.22	0.278428	0.139214	−	0.990262i	$-0.455542\pi$
$659$	4710.22	0.278428	0.139214	+	0.990262i	$0.455542\pi$
$660$	0	0
$661$	−31266.6	−1.83983	−0.919916	−	0.392116i	$-0.871743\pi$
$661$	−31266.6	−1.83983	−0.919916	+	0.392116i	$0.871743\pi$
$662$	−1024.00	−0.0601192
$663$	0	0
$664$	2234.82	0.130614
$665$	2146.58	0.125174
$666$	0	0
$667$	14818.7	0.860246
$668$	−17431.1	−1.00962
$669$	0	0
$670$	218.545	0.0126017
$671$	1488.40	0.0856322
$672$	0	0
$673$	11723.0	0.671454	0.335727	−	0.941959i	$-0.391018\pi$
$673$	11723.0	0.671454	0.335727	+	0.941959i	$0.391018\pi$
$674$	2679.65	0.153140
$675$	0	0
$676$	−7253.37	−0.412686
$677$	289.531	0.0164366	0.00821829	−	0.999966i	$-0.497384\pi$
$677$	289.531	0.0164366	0.00821829	+	0.999966i	$0.497384\pi$
$678$	0	0
$679$	−6310.25	−0.356650
$680$	−992.502	−0.0559716
$681$	0	0
$682$	10279.8	0.577175
$683$	−1720.10	−0.0963660	−0.0481830	−	0.998839i	$-0.515343\pi$
$683$	−1720.10	−0.0963660	−0.0481830	+	0.998839i	$0.515343\pi$
$684$	0	0
$685$	9525.37	0.531308
$686$	−6733.30	−0.374750
$687$	0	0
$688$	8097.77	0.448728
$689$	−30660.5	−1.69532
$690$	0	0
$691$	−16777.7	−0.923665	−0.461832	−	0.886967i	$-0.652808\pi$
$691$	−16777.7	−0.923665	−0.461832	+	0.886967i	$0.652808\pi$
$692$	−1089.96	−0.0598758
$693$	0	0
$694$	5421.88	0.296559
$695$	−4450.77	−0.242917
$696$	0	0
$697$	7999.50	0.434724
$698$	−2035.86	−0.110399
$699$	0	0
$700$	−5652.71	−0.305218
$701$	−23981.1	−1.29209	−0.646043	−	0.763301i	$-0.723577\pi$
$701$	−23981.1	−1.29209	−0.646043	+	0.763301i	$0.723577\pi$
$702$	0	0
$703$	−11801.6	−0.633150
$704$	−1893.12	−0.101349
$705$	0	0
$706$	8824.10	0.470396
$707$	2108.30	0.112151
$708$	0	0
$709$	−7709.28	−0.408361	−0.204181	−	0.978933i	$-0.565453\pi$
$709$	−7709.28	−0.408361	−0.204181	+	0.978933i	$0.565453\pi$
$710$	−3253.59	−0.171979
$711$	0	0
$712$	15662.5	0.824407
$713$	31521.5	1.65567
$714$	0	0
$715$	4865.34	0.254480
$716$	−6293.13	−0.328471
$717$	0	0
$718$	−6729.09	−0.349760
$719$	11976.5	0.621209	0.310605	−	0.950539i	$-0.399468\pi$
$719$	11976.5	0.621209	0.310605	+	0.950539i	$0.399468\pi$
$720$	0	0
$721$	−4153.72	−0.214553
$722$	−1486.56	−0.0766260
$723$	0	0
$724$	21208.7	1.08870
$725$	−14888.1	−0.762662
$726$	0	0
$727$	−18597.3	−0.948745	−0.474372	−	0.880324i	$-0.657325\pi$
$727$	−18597.3	−0.948745	−0.474372	+	0.880324i	$0.657325\pi$
$728$	−8886.00	−0.452386
$729$	0	0
$730$	1220.63	0.0618870
$731$	5994.91	0.303324
$732$	0	0
$733$	−23569.5	−1.18767	−0.593833	−	0.804588i	$-0.702386\pi$
$733$	−23569.5	−1.18767	−0.593833	+	0.804588i	$0.702386\pi$
$734$	20.2646	0.00101905
$735$	0	0
$736$	−21361.3	−1.06982
$737$	−1462.47	−0.0730948
$738$	0	0
$739$	10149.1	0.505199	0.252599	−	0.967571i	$-0.418715\pi$
$739$	10149.1	0.505199	0.252599	+	0.967571i	$0.418715\pi$
$740$	−2466.50	−0.122528
$741$	0	0
$742$	−5702.85	−0.282154
$743$	−27758.0	−1.37058	−0.685291	−	0.728269i	$-0.740325\pi$
$743$	−27758.0	−1.37058	−0.685291	+	0.728269i	$0.740325\pi$
$744$	0	0
$745$	−867.148	−0.0426441
$746$	−2617.87	−0.128481
$747$	0	0
$748$	2885.89	0.141068
$749$	−7829.63	−0.381960
$750$	0	0
$751$	−815.225	−0.0396112	−0.0198056	−	0.999804i	$-0.506305\pi$
$751$	−815.225	−0.0396112	−0.0198056	+	0.999804i	$0.506305\pi$
$752$	−3504.13	−0.169924
$753$	0	0
$754$	−10169.3	−0.491172
$755$	2030.11	0.0978585
$756$	0	0
$757$	−13239.4	−0.635659	−0.317829	−	0.948148i	$-0.602954\pi$
$757$	−13239.4	−0.635659	−0.317829	+	0.948148i	$0.602954\pi$
$758$	13058.0	0.625710
$759$	0	0
$760$	−5205.90	−0.248471
$761$	11028.2	0.525324	0.262662	−	0.964888i	$-0.415400\pi$
$761$	11028.2	0.525324	0.262662	+	0.964888i	$0.415400\pi$
$762$	0	0
$763$	−14407.1	−0.683582
$764$	−3017.56	−0.142895
$765$	0	0
$766$	−12365.0	−0.583246
$767$	16969.8	0.798882
$768$	0	0
$769$	−18921.2	−0.887277	−0.443639	−	0.896206i	$-0.646313\pi$
$769$	−18921.2	−0.887277	−0.443639	+	0.896206i	$0.646313\pi$
$770$	904.952	0.0423535
$771$	0	0
$772$	21814.7	1.01701
$773$	−38728.6	−1.80203	−0.901016	−	0.433786i	$-0.857177\pi$
$773$	−38728.6	−1.80203	−0.901016	+	0.433786i	$0.857177\pi$
$774$	0	0
$775$	−31669.0	−1.46785
$776$	15303.6	0.707949
$777$	0	0
$778$	−1556.93	−0.0717465
$779$	41959.2	1.92984
$780$	0	0
$781$	21772.5	0.997545
$782$	−2667.42	−0.121978
$783$	0	0
$784$	−6428.50	−0.292844
$785$	−2185.28	−0.0993579
$786$	0	0
$787$	−20587.3	−0.932477	−0.466239	−	0.884659i	$-0.654391\pi$
$787$	−20587.3	−0.932477	−0.466239	+	0.884659i	$0.654391\pi$
$788$	8379.37	0.378811
$789$	0	0
$790$	−2972.66	−0.133876
$791$	−5616.62	−0.252470
$792$	0	0
$793$	−3132.00	−0.140253
$794$	−14435.0	−0.645190
$795$	0	0
$796$	−22974.5	−1.02300
$797$	15871.4	0.705385	0.352693	−	0.935739i	$-0.385266\pi$
$797$	15871.4	0.705385	0.352693	+	0.935739i	$0.385266\pi$
$798$	0	0
$799$	−2594.17	−0.114862
$800$	21461.3	0.948463
$801$	0	0
$802$	18765.0	0.826204
$803$	−8168.28	−0.358969
$804$	0	0
$805$	2774.90	0.121494
$806$	−21631.4	−0.945329
$807$	0	0
$808$	−5113.06	−0.222620
$809$	−39667.1	−1.72388	−0.861942	−	0.507007i	$-0.830752\pi$
$809$	−39667.1	−1.72388	−0.861942	+	0.507007i	$0.830752\pi$
$810$	0	0
$811$	8003.87	0.346552	0.173276	−	0.984873i	$-0.444565\pi$
$811$	8003.87	0.346552	0.173276	+	0.984873i	$0.444565\pi$
$812$	6275.00	0.271194
$813$	0	0
$814$	−4975.28	−0.214230
$815$	2051.25	0.0881621
$816$	0	0
$817$	31444.7	1.34652
$818$	12939.7	0.553088
$819$	0	0
$820$	8769.40	0.373464
$821$	13279.1	0.564489	0.282244	−	0.959343i	$-0.408921\pi$
$821$	13279.1	0.564489	0.282244	+	0.959343i	$0.408921\pi$
$822$	0	0
$823$	28934.0	1.22549	0.612745	−	0.790281i	$-0.290065\pi$
$823$	28934.0	1.22549	0.612745	+	0.790281i	$0.290065\pi$
$824$	10073.6	0.425887
$825$	0	0
$826$	3156.37	0.132959
$827$	13679.6	0.575193	0.287597	−	0.957752i	$-0.407144\pi$
$827$	13679.6	0.575193	0.287597	+	0.957752i	$0.407144\pi$
$828$	0	0
$829$	16514.5	0.691886	0.345943	−	0.938256i	$-0.387559\pi$
$829$	16514.5	0.691886	0.345943	+	0.938256i	$0.387559\pi$
$830$	478.921	0.0200284
$831$	0	0
$832$	3983.64	0.165995
$833$	−4759.13	−0.197952
$834$	0	0
$835$	−8596.93	−0.356298
$836$	15137.1	0.626231
$837$	0	0
$838$	13177.8	0.543221
$839$	87.9839	0.00362043	0.00181022	−	0.999998i	$-0.499424\pi$
$839$	87.9839	0.00362043	0.00181022	+	0.999998i	$0.499424\pi$
$840$	0	0
$841$	−7861.94	−0.322356
$842$	16825.1	0.688635
$843$	0	0
$844$	32371.3	1.32022
$845$	−3577.33	−0.145638
$846$	0	0
$847$	4512.98	0.183079
$848$	−12115.7	−0.490630
$849$	0	0
$850$	2679.90	0.108141
$851$	−15256.0	−0.614534
$852$	0	0
$853$	8162.96	0.327660	0.163830	−	0.986489i	$-0.447615\pi$
$853$	8162.96	0.327660	0.163830	+	0.986489i	$0.447615\pi$
$854$	−582.551	−0.0233425
$855$	0	0
$856$	18988.4	0.758191
$857$	−18724.9	−0.746361	−0.373181	−	0.927759i	$-0.621733\pi$
$857$	−18724.9	−0.746361	−0.373181	+	0.927759i	$0.621733\pi$
$858$	0	0
$859$	−46422.5	−1.84391	−0.921953	−	0.387301i	$-0.873407\pi$
$859$	−46422.5	−1.84391	−0.921953	+	0.387301i	$0.873407\pi$
$860$	6571.88	0.260580
$861$	0	0
$862$	4043.61	0.159775
$863$	−29112.3	−1.14831	−0.574157	−	0.818746i	$-0.694670\pi$
$863$	−29112.3	−1.14831	−0.574157	+	0.818746i	$0.694670\pi$
$864$	0	0
$865$	−537.563	−0.0211303
$866$	−8346.33	−0.327505
$867$	0	0
$868$	13347.8	0.521950
$869$	19892.6	0.776536
$870$	0	0
$871$	3077.43	0.119719
$872$	34940.2	1.35691
$873$	0	0
$874$	−13991.2	−0.541487
$875$	−5797.04	−0.223972
$876$	0	0
$877$	39163.0	1.50791	0.753957	−	0.656924i	$-0.228143\pi$
$877$	39163.0	1.50791	0.753957	+	0.656924i	$0.228143\pi$
$878$	3464.19	0.133156
$879$	0	0
$880$	1922.57	0.0736473
$881$	35073.2	1.34125	0.670627	−	0.741795i	$-0.266025\pi$
$881$	35073.2	1.34125	0.670627	+	0.741795i	$0.266025\pi$
$882$	0	0
$883$	−48775.7	−1.85893	−0.929463	−	0.368915i	$-0.879729\pi$
$883$	−48775.7	−1.85893	−0.929463	+	0.368915i	$0.879729\pi$
$884$	−6072.69	−0.231048
$885$	0	0
$886$	11611.0	0.440269
$887$	−13296.0	−0.503309	−0.251654	−	0.967817i	$-0.580975\pi$
$887$	−13296.0	−0.503309	−0.251654	+	0.967817i	$0.580975\pi$
$888$	0	0
$889$	−21027.9	−0.793309
$890$	3356.46	0.126415
$891$	0	0
$892$	−4330.90	−0.162566
$893$	−13607.0	−0.509899
$894$	0	0
$895$	−3103.74	−0.115918
$896$	−11031.1	−0.411297
$897$	0	0
$898$	12054.6	0.447960
$899$	35155.3	1.30422
$900$	0	0
$901$	−8969.43	−0.331648
$902$	17689.1	0.652974
$903$	0	0
$904$	13621.4	0.501153
$905$	10460.0	0.384202
$906$	0	0
$907$	−11675.0	−0.427410	−0.213705	−	0.976898i	$-0.568553\pi$
$907$	−11675.0	−0.427410	−0.213705	+	0.976898i	$0.568553\pi$
$908$	−13224.0	−0.483319
$909$	0	0
$910$	−1904.26	−0.0693689
$911$	−18552.9	−0.674738	−0.337369	−	0.941372i	$-0.609537\pi$
$911$	−18552.9	−0.674738	−0.337369	+	0.941372i	$0.609537\pi$
$912$	0	0
$913$	−3204.87	−0.116173
$914$	9739.24	0.352457
$915$	0	0
$916$	24077.7	0.868504
$917$	−15714.9	−0.565922
$918$	0	0
$919$	33956.8	1.21886	0.609429	−	0.792841i	$-0.291399\pi$
$919$	33956.8	1.21886	0.609429	+	0.792841i	$0.291399\pi$
$920$	−6729.71	−0.241165
$921$	0	0
$922$	−9886.63	−0.353144
$923$	−45815.3	−1.63383
$924$	0	0
$925$	15327.4	0.544823
$926$	−480.372	−0.0170475
$927$	0	0
$928$	−23823.8	−0.842732
$929$	23695.3	0.836832	0.418416	−	0.908256i	$-0.362585\pi$
$929$	23695.3	0.836832	0.418416	+	0.908256i	$0.362585\pi$
$930$	0	0
$931$	−24962.7	−0.878753
$932$	−31920.2	−1.12187
$933$	0	0
$934$	2018.72	0.0707221
$935$	1423.31	0.0497830
$936$	0	0
$937$	7990.62	0.278593	0.139297	−	0.990251i	$-0.455516\pi$
$937$	7990.62	0.278593	0.139297	+	0.990251i	$0.455516\pi$
$938$	572.402	0.0199249
$939$	0	0
$940$	−2843.83	−0.0986762
$941$	−24385.9	−0.844799	−0.422400	−	0.906410i	$-0.638812\pi$
$941$	−24385.9	−0.844799	−0.422400	+	0.906410i	$0.638812\pi$
$942$	0	0
$943$	54241.0	1.87310
$944$	6705.69	0.231199
$945$	0	0
$946$	13256.4	0.455605
$947$	1174.62	0.0403064	0.0201532	−	0.999797i	$-0.493585\pi$
$947$	1174.62	0.0403064	0.0201532	+	0.999797i	$0.493585\pi$
$948$	0	0
$949$	17188.3	0.587939
$950$	14056.7	0.480063
$951$	0	0
$952$	−2599.51	−0.0884985
$953$	33546.9	1.14029	0.570143	−	0.821546i	$-0.306888\pi$
$953$	33546.9	1.14029	0.570143	+	0.821546i	$0.306888\pi$
$954$	0	0
$955$	−1488.25	−0.0504277
$956$	2055.00	0.0695224
$957$	0	0
$958$	−13598.7	−0.458617
$959$	24948.4	0.840067
$960$	0	0
$961$	44989.1	1.51016
$962$	10469.3	0.350878
$963$	0	0
$964$	11792.9	0.394007
$965$	10758.9	0.358903
$966$	0	0
$967$	24766.8	0.823625	0.411813	−	0.911269i	$-0.364896\pi$
$967$	24766.8	0.823625	0.411813	+	0.911269i	$0.364896\pi$
$968$	−10944.9	−0.363411
$969$	0	0
$970$	3279.56	0.108557
$971$	−42324.3	−1.39882	−0.699409	−	0.714721i	$-0.746553\pi$
$971$	−42324.3	−1.39882	−0.699409	+	0.714721i	$0.746553\pi$
$972$	0	0
$973$	−11657.2	−0.384083
$974$	1538.01	0.0505966
$975$	0	0
$976$	−1237.63	−0.0405896
$977$	11320.4	0.370698	0.185349	−	0.982673i	$-0.440658\pi$
$977$	11320.4	0.370698	0.185349	+	0.982673i	$0.440658\pi$
$978$	0	0
$979$	−22461.0	−0.733254
$980$	−5217.16	−0.170057
$981$	0	0
$982$	25556.8	0.830500
$983$	11311.9	0.367032	0.183516	−	0.983017i	$-0.441252\pi$
$983$	11311.9	0.367032	0.183516	+	0.983017i	$0.441252\pi$
$984$	0	0
$985$	4132.66	0.133683
$986$	−2974.92	−0.0960860
$987$	0	0
$988$	−31852.6	−1.02568
$989$	40648.8	1.30693
$990$	0	0
$991$	−29405.5	−0.942580	−0.471290	−	0.881978i	$-0.656212\pi$
$991$	−29405.5	−0.942580	−0.471290	+	0.881978i	$0.656212\pi$
$992$	−50676.5	−1.62196
$993$	0	0
$994$	−8521.63	−0.271921
$995$	−11330.9	−0.361018
$996$	0	0
$997$	−54905.9	−1.74412	−0.872060	−	0.489398i	$-0.837216\pi$
$997$	−54905.9	−1.74412	−0.872060	+	0.489398i	$0.837216\pi$
$998$	−23588.7	−0.748183
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.a.g.1.2		3
3.2	odd	2		17.4.a.b.1.2	✓	3
4.3	odd	2		2448.4.a.bi.1.1		3
12.11	even	2		272.4.a.h.1.2		3
15.2	even	4		425.4.b.f.324.4		6
15.8	even	4		425.4.b.f.324.3		6
15.14	odd	2		425.4.a.g.1.2		3
21.20	even	2		833.4.a.d.1.2		3
24.5	odd	2		1088.4.a.v.1.2		3
24.11	even	2		1088.4.a.x.1.2		3
33.32	even	2		2057.4.a.e.1.2		3
51.38	odd	4		289.4.b.b.288.3		6
51.47	odd	4		289.4.b.b.288.4		6
51.50	odd	2		289.4.a.b.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.4.a.b.1.2	✓	3	3.2	odd	2
153.4.a.g.1.2		3	1.1	even	1	trivial
272.4.a.h.1.2		3	12.11	even	2
289.4.a.b.1.2		3	51.50	odd	2
289.4.b.b.288.3		6	51.38	odd	4
289.4.b.b.288.4		6	51.47	odd	4
425.4.a.g.1.2		3	15.14	odd	2
425.4.b.f.324.3		6	15.8	even	4
425.4.b.f.324.4		6	15.2	even	4
833.4.a.d.1.2		3	21.20	even	2
1088.4.a.v.1.2		3	24.5	odd	2
1088.4.a.x.1.2		3	24.11	even	2
2057.4.a.e.1.2		3	33.32	even	2
2448.4.a.bi.1.1		3	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 153 = 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	153.a (trivial)

\(f(q)\)	\(=\)	\(q-1.36122 q^{2} -6.14708 q^{4} -3.03171 q^{5} -7.94049 q^{7} +19.2573 q^{8} +O(q^{10})\)	\(q-1.36122 q^{2} -6.14708 q^{4} -3.03171 q^{5} -7.94049 q^{7} +19.2573 q^{8} +4.12682 q^{10} -27.6161 q^{11} +58.1117 q^{13} +10.8088 q^{14} +22.9632 q^{16} +17.0000 q^{17} +89.1688 q^{19} +18.6361 q^{20} +37.5916 q^{22} +115.269 q^{23} -115.809 q^{25} -79.1029 q^{26} +48.8108 q^{28} +128.558 q^{29} +273.460 q^{31} -185.316 q^{32} -23.1408 q^{34} +24.0732 q^{35} -132.351 q^{37} -121.379 q^{38} -58.3825 q^{40} +470.559 q^{41} +352.642 q^{43} +169.758 q^{44} -156.907 q^{46} -152.598 q^{47} -279.949 q^{49} +157.641 q^{50} -357.217 q^{52} -527.614 q^{53} +83.7239 q^{55} -152.912 q^{56} -174.995 q^{58} +292.020 q^{59} -53.8962 q^{61} -372.239 q^{62} +68.5514 q^{64} -176.178 q^{65} +52.9572 q^{67} -104.500 q^{68} -32.7690 q^{70} -788.400 q^{71} +295.780 q^{73} +180.159 q^{74} -548.127 q^{76} +219.285 q^{77} -720.325 q^{79} -69.6175 q^{80} -640.535 q^{82} +116.051 q^{83} -51.5390 q^{85} -480.024 q^{86} -531.812 q^{88} +813.329 q^{89} -461.435 q^{91} -708.569 q^{92} +207.720 q^{94} -270.334 q^{95} +794.693 q^{97} +381.072 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 25 q^{4} + 8 q^{5} + 22 q^{7} + 39 q^{8}+O(q^{10}) \) 3 * q - q^2 + 25 * q^4 + 8 * q^5 + 22 * q^7 + 39 * q^8	\( 3 q - q^{2} + 25 q^{4} + 8 q^{5} + 22 q^{7} + 39 q^{8} - 56 q^{10} + 28 q^{11} + 30 q^{13} - 92 q^{14} + 137 q^{16} + 51 q^{17} + 80 q^{19} + 168 q^{20} + 286 q^{22} - 142 q^{23} - 223 q^{25} - 26 q^{26} + 476 q^{28} + 456 q^{29} + 230 q^{31} + 71 q^{32} - 17 q^{34} + 332 q^{35} + 356 q^{37} - 724 q^{38} - 424 q^{40} + 294 q^{41} + 556 q^{43} + 1122 q^{44} - 704 q^{46} - 640 q^{47} - 269 q^{49} - 547 q^{50} - 774 q^{52} - 302 q^{53} + 76 q^{55} - 684 q^{56} - 1304 q^{58} - 636 q^{59} - 84 q^{61} - 508 q^{62} - 919 q^{64} - 408 q^{65} + 1008 q^{67} + 425 q^{68} - 1504 q^{70} + 402 q^{71} + 838 q^{73} - 836 q^{74} - 908 q^{76} + 504 q^{77} - 594 q^{79} + 40 q^{80} + 358 q^{82} + 2396 q^{83} + 136 q^{85} + 1264 q^{86} + 1838 q^{88} + 170 q^{89} - 1016 q^{91} - 4896 q^{92} - 2016 q^{94} + 472 q^{95} - 270 q^{97} - 2857 q^{98}+O(q^{100}) \) 3 * q - q^2 + 25 * q^4 + 8 * q^5 + 22 * q^7 + 39 * q^8 - 56 * q^10 + 28 * q^11 + 30 * q^13 - 92 * q^14 + 137 * q^16 + 51 * q^17 + 80 * q^19 + 168 * q^20 + 286 * q^22 - 142 * q^23 - 223 * q^25 - 26 * q^26 + 476 * q^28 + 456 * q^29 + 230 * q^31 + 71 * q^32 - 17 * q^34 + 332 * q^35 + 356 * q^37 - 724 * q^38 - 424 * q^40 + 294 * q^41 + 556 * q^43 + 1122 * q^44 - 704 * q^46 - 640 * q^47 - 269 * q^49 - 547 * q^50 - 774 * q^52 - 302 * q^53 + 76 * q^55 - 684 * q^56 - 1304 * q^58 - 636 * q^59 - 84 * q^61 - 508 * q^62 - 919 * q^64 - 408 * q^65 + 1008 * q^67 + 425 * q^68 - 1504 * q^70 + 402 * q^71 + 838 * q^73 - 836 * q^74 - 908 * q^76 + 504 * q^77 - 594 * q^79 + 40 * q^80 + 358 * q^82 + 2396 * q^83 + 136 * q^85 + 1264 * q^86 + 1838 * q^88 + 170 * q^89 - 1016 * q^91 - 4896 * q^92 - 2016 * q^94 + 472 * q^95 - 270 * q^97 - 2857 * q^98

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