LMFDB - Embedded newform 2151.4.a.c.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, 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("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.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:	$126.913108422$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.404541 q^{2} -7.83635 q^{4} +0.184451 q^{5} +12.9030 q^{7} +6.40644 q^{8} +O(q^{10})$	\(q-0.404541 q^{2} -7.83635 q^{4} +0.184451 q^{5} +12.9030 q^{7} +6.40644 q^{8} -0.0746179 q^{10} +9.52438 q^{11} -51.2689 q^{13} -5.21980 q^{14} +60.0991 q^{16} -40.8287 q^{17} +21.4706 q^{19} -1.44542 q^{20} -3.85300 q^{22} -16.4955 q^{23} -124.966 q^{25} +20.7403 q^{26} -101.113 q^{28} +181.639 q^{29} +56.6939 q^{31} -75.5641 q^{32} +16.5169 q^{34} +2.37998 q^{35} -3.43914 q^{37} -8.68574 q^{38} +1.18167 q^{40} +367.492 q^{41} +3.14636 q^{43} -74.6364 q^{44} +6.67310 q^{46} -423.535 q^{47} -176.511 q^{49} +50.5538 q^{50} +401.760 q^{52} +648.482 q^{53} +1.75678 q^{55} +82.6626 q^{56} -73.4802 q^{58} -328.782 q^{59} -285.831 q^{61} -22.9350 q^{62} -450.224 q^{64} -9.45659 q^{65} +206.277 q^{67} +319.948 q^{68} -0.962798 q^{70} +117.724 q^{71} -623.884 q^{73} +1.39127 q^{74} -168.251 q^{76} +122.894 q^{77} -1138.83 q^{79} +11.0853 q^{80} -148.665 q^{82} +737.634 q^{83} -7.53090 q^{85} -1.27283 q^{86} +61.0174 q^{88} +1347.30 q^{89} -661.524 q^{91} +129.265 q^{92} +171.337 q^{94} +3.96028 q^{95} -244.352 q^{97} +71.4060 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8}+O(q^{10}) $ 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8	$ 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8} - 68 q^{10} + 258 q^{11} - 134 q^{13} + 292 q^{14} + 327 q^{16} + 364 q^{17} - 278 q^{19} + 986 q^{20} - 179 q^{22} + 668 q^{23} + 490 q^{25} + 760 q^{26} - 802 q^{28} + 714 q^{29} - 608 q^{31} + 918 q^{32} - 228 q^{34} + 934 q^{35} - 1080 q^{37} + 1395 q^{38} - 563 q^{40} + 1796 q^{41} - 1934 q^{43} + 3157 q^{44} - 940 q^{46} + 2032 q^{47} + 762 q^{49} + 1754 q^{50} - 2328 q^{52} + 1790 q^{53} - 478 q^{55} + 3557 q^{56} - 2626 q^{58} + 3622 q^{59} + 324 q^{61} + 796 q^{62} + 2023 q^{64} + 2200 q^{65} - 2444 q^{67} - 357 q^{68} + 4305 q^{70} + 1298 q^{71} - 1368 q^{73} - 813 q^{74} + 1390 q^{76} + 1408 q^{77} - 1378 q^{79} + 7684 q^{80} + 9001 q^{82} + 3524 q^{83} + 60 q^{85} + 2543 q^{86} + 1749 q^{88} + 7854 q^{89} + 850 q^{91} + 496 q^{92} + 6634 q^{94} + 3696 q^{95} - 1746 q^{97} + 4632 q^{98}+O(q^{100}) $ 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8 - 68 * q^10 + 258 * q^11 - 134 * q^13 + 292 * q^14 + 327 * q^16 + 364 * q^17 - 278 * q^19 + 986 * q^20 - 179 * q^22 + 668 * q^23 + 490 * q^25 + 760 * q^26 - 802 * q^28 + 714 * q^29 - 608 * q^31 + 918 * q^32 - 228 * q^34 + 934 * q^35 - 1080 * q^37 + 1395 * q^38 - 563 * q^40 + 1796 * q^41 - 1934 * q^43 + 3157 * q^44 - 940 * q^46 + 2032 * q^47 + 762 * q^49 + 1754 * q^50 - 2328 * q^52 + 1790 * q^53 - 478 * q^55 + 3557 * q^56 - 2626 * q^58 + 3622 * q^59 + 324 * q^61 + 796 * q^62 + 2023 * q^64 + 2200 * q^65 - 2444 * q^67 - 357 * q^68 + 4305 * q^70 + 1298 * q^71 - 1368 * q^73 - 813 * q^74 + 1390 * q^76 + 1408 * q^77 - 1378 * q^79 + 7684 * q^80 + 9001 * q^82 + 3524 * q^83 + 60 * q^85 + 2543 * q^86 + 1749 * q^88 + 7854 * q^89 + 850 * q^91 + 496 * q^92 + 6634 * q^94 + 3696 * q^95 - 1746 * q^97 + 4632 * 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 $

$2$	−0.404541	−0.143027	−0.0715133	−	0.997440i	$-0.522783\pi$
$2$	−0.404541	−0.143027	−0.0715133	+	0.997440i	$0.522783\pi$
$3$	0	0
$3$	0	0
$4$	−7.83635	−0.979543
$5$	0.184451	0.0164978	0.00824890	−	0.999966i	$-0.497374\pi$
$5$	0.184451	0.0164978	0.00824890	+	0.999966i	$0.497374\pi$
$6$	0	0
$7$	12.9030	0.696699	0.348349	−	0.937365i	$-0.386742\pi$
$7$	12.9030	0.696699	0.348349	+	0.937365i	$0.386742\pi$
$8$	6.40644	0.283127
$9$	0	0
$10$	−0.0746179	−0.00235962
$11$	9.52438	0.261065	0.130532	−	0.991444i	$-0.458331\pi$
$11$	9.52438	0.261065	0.130532	+	0.991444i	$0.458331\pi$
$12$	0	0
$13$	−51.2689	−1.09380	−0.546901	−	0.837197i	$-0.684193\pi$
$13$	−51.2689	−1.09380	−0.546901	+	0.837197i	$0.684193\pi$
$14$	−5.21980	−0.0996465
$15$	0	0
$16$	60.0991	0.939049
$17$	−40.8287	−0.582495	−0.291248	−	0.956648i	$-0.594070\pi$
$17$	−40.8287	−0.582495	−0.291248	+	0.956648i	$0.594070\pi$
$18$	0	0
$19$	21.4706	0.259247	0.129624	−	0.991563i	$-0.458623\pi$
$19$	21.4706	0.259247	0.129624	+	0.991563i	$0.458623\pi$
$20$	−1.44542	−0.0161603
$21$	0	0
$22$	−3.85300	−0.0373392
$23$	−16.4955	−0.149546	−0.0747729	−	0.997201i	$-0.523823\pi$
$23$	−16.4955	−0.149546	−0.0747729	+	0.997201i	$0.523823\pi$
$24$	0	0
$25$	−124.966	−0.999728
$26$	20.7403	0.156443
$27$	0	0
$28$	−101.113	−0.682447
$29$	181.639	1.16308	0.581542	−	0.813516i	$-0.302450\pi$
$29$	181.639	1.16308	0.581542	+	0.813516i	$0.302450\pi$
$30$	0	0
$31$	56.6939	0.328469	0.164234	−	0.986421i	$-0.447485\pi$
$31$	56.6939	0.328469	0.164234	+	0.986421i	$0.447485\pi$
$32$	−75.5641	−0.417436
$33$	0	0
$34$	16.5169	0.0833124
$35$	2.37998	0.0114940
$36$	0	0
$37$	−3.43914	−0.0152808	−0.00764042	−	0.999971i	$-0.502432\pi$
$37$	−3.43914	−0.0152808	−0.00764042	+	0.999971i	$0.502432\pi$
$38$	−8.68574	−0.0370793
$39$	0	0
$40$	1.18167	0.00467098
$41$	367.492	1.39982	0.699909	−	0.714232i	$-0.253224\pi$
$41$	367.492	1.39982	0.699909	+	0.714232i	$0.253224\pi$
$42$	0	0
$43$	3.14636	0.0111585	0.00557925	−	0.999984i	$-0.498224\pi$
$43$	3.14636	0.0111585	0.00557925	+	0.999984i	$0.498224\pi$
$44$	−74.6364	−0.255724
$45$	0	0
$46$	6.67310	0.0213890
$47$	−423.535	−1.31445	−0.657223	−	0.753696i	$-0.728269\pi$
$47$	−423.535	−1.31445	−0.657223	+	0.753696i	$0.728269\pi$
$48$	0	0
$49$	−176.511	−0.514611
$50$	50.5538	0.142988
$51$	0	0
$52$	401.760	1.07143
$53$	648.482	1.68068	0.840339	−	0.542062i	$-0.182356\pi$
$53$	648.482	1.68068	0.840339	+	0.542062i	$0.182356\pi$
$54$	0	0
$55$	1.75678	0.00430699
$56$	82.6626	0.197255
$57$	0	0
$58$	−73.4802	−0.166352
$59$	−328.782	−0.725487	−0.362743	−	0.931889i	$-0.618160\pi$
$59$	−328.782	−0.725487	−0.362743	+	0.931889i	$0.618160\pi$
$60$	0	0
$61$	−285.831	−0.599949	−0.299974	−	0.953947i	$-0.596978\pi$
$61$	−285.831	−0.599949	−0.299974	+	0.953947i	$0.596978\pi$
$62$	−22.9350	−0.0469798
$63$	0	0
$64$	−450.224	−0.879344
$65$	−9.45659	−0.0180453
$66$	0	0
$67$	206.277	0.376131	0.188065	−	0.982157i	$-0.439778\pi$
$67$	206.277	0.376131	0.188065	+	0.982157i	$0.439778\pi$
$68$	319.948	0.570579
$69$	0	0
$70$	−0.962798	−0.00164395
$71$	117.724	0.196778	0.0983891	−	0.995148i	$-0.468631\pi$
$71$	117.724	0.196778	0.0983891	+	0.995148i	$0.468631\pi$
$72$	0	0
$73$	−623.884	−1.00027	−0.500137	−	0.865946i	$-0.666717\pi$
$73$	−623.884	−1.00027	−0.500137	+	0.865946i	$0.666717\pi$
$74$	1.39127	0.00218557
$75$	0	0
$76$	−168.251	−0.253944
$77$	122.894	0.181883
$78$	0	0
$79$	−1138.83	−1.62188	−0.810941	−	0.585128i	$-0.801044\pi$
$79$	−1138.83	−1.62188	−0.810941	+	0.585128i	$0.801044\pi$
$80$	11.0853	0.0154922
$81$	0	0
$82$	−148.665	−0.200211
$83$	737.634	0.975492	0.487746	−	0.872986i	$-0.337819\pi$
$83$	737.634	0.975492	0.487746	+	0.872986i	$0.337819\pi$
$84$	0	0
$85$	−7.53090	−0.00960989
$86$	−1.27283	−0.00159596
$87$	0	0
$88$	61.0174	0.0739145
$89$	1347.30	1.60465	0.802323	−	0.596890i	$-0.203597\pi$
$89$	1347.30	1.60465	0.802323	+	0.596890i	$0.203597\pi$
$90$	0	0
$91$	−661.524	−0.762050
$92$	129.265	0.146487
$93$	0	0
$94$	171.337	0.188001
$95$	3.96028	0.00427701
$96$	0	0
$97$	−244.352	−0.255775	−0.127887	−	0.991789i	$-0.540820\pi$
$97$	−244.352	−0.255775	−0.127887	+	0.991789i	$0.540820\pi$
$98$	71.4060	0.0736030
$99$	0	0
$100$	979.277	0.979277
$101$	69.1289	0.0681048	0.0340524	−	0.999420i	$-0.489159\pi$
$101$	69.1289	0.0681048	0.0340524	+	0.999420i	$0.489159\pi$
$102$	0	0
$103$	−1654.01	−1.58227	−0.791137	−	0.611639i	$-0.790510\pi$
$103$	−1654.01	−1.58227	−0.791137	+	0.611639i	$0.790510\pi$
$104$	−328.451	−0.309685
$105$	0	0
$106$	−262.337	−0.240382
$107$	126.043	0.113879	0.0569396	−	0.998378i	$-0.481866\pi$
$107$	126.043	0.113879	0.0569396	+	0.998378i	$0.481866\pi$
$108$	0	0
$109$	1523.54	1.33879	0.669395	−	0.742906i	$-0.266553\pi$
$109$	1523.54	1.33879	0.669395	+	0.742906i	$0.266553\pi$
$110$	−0.710689	−0.000616014 0
$111$	0	0
$112$	775.462	0.654234
$113$	−2246.01	−1.86979	−0.934896	−	0.354923i	$-0.884507\pi$
$113$	−2246.01	−1.86979	−0.934896	+	0.354923i	$0.884507\pi$
$114$	0	0
$115$	−3.04261	−0.00246717
$116$	−1423.38	−1.13929
$117$	0	0
$118$	133.005	0.103764
$119$	−526.815	−0.405824
$120$	0	0
$121$	−1240.29	−0.931845
$122$	115.630	0.0858087
$123$	0	0
$124$	−444.273	−0.321749
$125$	−46.1065	−0.0329911
$126$	0	0
$127$	29.4297	0.0205627	0.0102813	−	0.999947i	$-0.496727\pi$
$127$	29.4297	0.0205627	0.0102813	+	0.999947i	$0.496727\pi$
$128$	786.647	0.543206
$129$	0	0
$130$	3.82557	0.00258096
$131$	995.070	0.663661	0.331831	−	0.943339i	$-0.392334\pi$
$131$	995.070	0.663661	0.331831	+	0.943339i	$0.392334\pi$
$132$	0	0
$133$	277.037	0.180617
$134$	−83.4475	−0.0537967
$135$	0	0
$136$	−261.567	−0.164920
$137$	1676.34	1.04540	0.522699	−	0.852518i	$-0.324925\pi$
$137$	1676.34	1.04540	0.522699	+	0.852518i	$0.324925\pi$
$138$	0	0
$139$	2714.78	1.65658	0.828289	−	0.560301i	$-0.189315\pi$
$139$	2714.78	1.65658	0.828289	+	0.560301i	$0.189315\pi$
$140$	−18.6503	−0.0112589
$141$	0	0
$142$	−47.6241	−0.0281445
$143$	−488.304	−0.285553
$144$	0	0
$145$	33.5034	0.0191883
$146$	252.386	0.143066
$147$	0	0
$148$	26.9503	0.0149682
$149$	377.824	0.207735	0.103868	−	0.994591i	$-0.466878\pi$
$149$	377.824	0.207735	0.103868	+	0.994591i	$0.466878\pi$
$150$	0	0
$151$	−311.215	−0.167724	−0.0838621	−	0.996477i	$-0.526726\pi$
$151$	−311.215	−0.167724	−0.0838621	+	0.996477i	$0.526726\pi$
$152$	137.550	0.0734001
$153$	0	0
$154$	−49.7154	−0.0260142
$155$	10.4572	0.00541901
$156$	0	0
$157$	−3418.87	−1.73793	−0.868967	−	0.494870i	$-0.835216\pi$
$157$	−3418.87	−1.73793	−0.868967	+	0.494870i	$0.835216\pi$
$158$	460.704	0.231972
$159$	0	0
$160$	−13.9379	−0.00688678
$161$	−212.842	−0.104188
$162$	0	0
$163$	3167.17	1.52191	0.760956	−	0.648803i	$-0.224730\pi$
$163$	3167.17	1.52191	0.760956	+	0.648803i	$0.224730\pi$
$164$	−2879.79	−1.37118
$165$	0	0
$166$	−298.403	−0.139521
$167$	−3870.93	−1.79366	−0.896830	−	0.442375i	$-0.854136\pi$
$167$	−3870.93	−1.79366	−0.896830	+	0.442375i	$0.854136\pi$
$168$	0	0
$169$	431.495	0.196402
$170$	3.04655	0.00137447
$171$	0	0
$172$	−24.6560	−0.0109302
$173$	4108.72	1.80567	0.902833	−	0.429991i	$-0.141483\pi$
$173$	4108.72	1.80567	0.902833	+	0.429991i	$0.141483\pi$
$174$	0	0
$175$	−1612.44	−0.696509
$176$	572.407	0.245152
$177$	0	0
$178$	−545.037	−0.229507
$179$	2608.77	1.08932	0.544661	−	0.838657i	$-0.316659\pi$
$179$	2608.77	1.08932	0.544661	+	0.838657i	$0.316659\pi$
$180$	0	0
$181$	−942.424	−0.387016	−0.193508	−	0.981099i	$-0.561986\pi$
$181$	−942.424	−0.387016	−0.193508	+	0.981099i	$0.561986\pi$
$182$	267.613	0.108994
$183$	0	0
$184$	−105.678	−0.0423405
$185$	−0.634352	−0.000252100 0
$186$	0	0
$187$	−388.868	−0.152069
$188$	3318.97	1.28756
$189$	0	0
$190$	−1.60209	−0.000611727 0
$191$	1060.78	0.401859	0.200930	−	0.979606i	$-0.435604\pi$
$191$	1060.78	0.401859	0.200930	+	0.979606i	$0.435604\pi$
$192$	0	0
$193$	−236.424	−0.0881770	−0.0440885	−	0.999028i	$-0.514038\pi$
$193$	−236.424	−0.0881770	−0.0440885	+	0.999028i	$0.514038\pi$
$194$	98.8501	0.0365826
$195$	0	0
$196$	1383.20	0.504083
$197$	−2588.22	−0.936055	−0.468027	−	0.883714i	$-0.655035\pi$
$197$	−2588.22	−0.936055	−0.468027	+	0.883714i	$0.655035\pi$
$198$	0	0
$199$	5501.29	1.95968	0.979839	−	0.199788i	$-0.0640254\pi$
$199$	5501.29	1.95968	0.979839	+	0.199788i	$0.0640254\pi$
$200$	−800.588	−0.283050
$201$	0	0
$202$	−27.9654	−0.00974080
$203$	2343.69	0.810319
$204$	0	0
$205$	67.7842	0.0230939
$206$	669.113	0.226307
$207$	0	0
$208$	−3081.21	−1.02713
$209$	204.495	0.0676803
$210$	0	0
$211$	2554.52	0.833463	0.416731	−	0.909030i	$-0.363176\pi$
$211$	2554.52	0.833463	0.416731	+	0.909030i	$0.363176\pi$
$212$	−5081.73	−1.64630
$213$	0	0
$214$	−50.9897	−0.0162878
$215$	0.580349	0.000184091 0
$216$	0	0
$217$	731.524	0.228844
$218$	−616.332	−0.191483
$219$	0	0
$220$	−13.7667	−0.00421888
$221$	2093.24	0.637134
$222$	0	0
$223$	5489.52	1.64846	0.824228	−	0.566259i	$-0.191610\pi$
$223$	5489.52	1.64846	0.824228	+	0.566259i	$0.191610\pi$
$224$	−975.007	−0.290828
$225$	0	0
$226$	908.600	0.267430
$227$	3395.02	0.992667	0.496333	−	0.868132i	$-0.334679\pi$
$227$	3395.02	0.992667	0.496333	+	0.868132i	$0.334679\pi$
$228$	0	0
$229$	3633.53	1.04852	0.524258	−	0.851559i	$-0.324343\pi$
$229$	3633.53	1.04852	0.524258	+	0.851559i	$0.324343\pi$
$230$	1.23086	0.000352872 0
$231$	0	0
$232$	1163.66	0.329301
$233$	2145.84	0.603341	0.301671	−	0.953412i	$-0.402456\pi$
$233$	2145.84	0.603341	0.301671	+	0.953412i	$0.402456\pi$
$234$	0	0
$235$	−78.1215	−0.0216855
$236$	2576.45	0.710646
$237$	0	0
$238$	213.118	0.0580436
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	4630.83	1.23775	0.618876	−	0.785489i	$-0.287588\pi$
$241$	4630.83	1.23775	0.618876	+	0.785489i	$0.287588\pi$
$242$	501.746	0.133279
$243$	0	0
$244$	2239.87	0.587676
$245$	−32.5577	−0.00848994
$246$	0	0
$247$	−1100.77	−0.283565
$248$	363.206	0.0929985
$249$	0	0
$250$	18.6519	0.00471861
$251$	4227.27	1.06304	0.531519	−	0.847046i	$-0.321621\pi$
$251$	4227.27	1.06304	0.531519	+	0.847046i	$0.321621\pi$
$252$	0	0
$253$	−157.110	−0.0390411
$254$	−11.9055	−0.00294101
$255$	0	0
$256$	3283.56	0.801651
$257$	−1340.84	−0.325445	−0.162722	−	0.986672i	$-0.552028\pi$
$257$	−1340.84	−0.325445	−0.162722	+	0.986672i	$0.552028\pi$
$258$	0	0
$259$	−44.3754	−0.0106461
$260$	74.1051	0.0176762
$261$	0	0
$262$	−402.546	−0.0949213
$263$	2978.36	0.698303	0.349151	−	0.937066i	$-0.386470\pi$
$263$	2978.36	0.698303	0.349151	+	0.937066i	$0.386470\pi$
$264$	0	0
$265$	119.613	0.0277275
$266$	−112.073	−0.0258331
$267$	0	0
$268$	−1616.46	−0.368436
$269$	4987.59	1.13048	0.565239	−	0.824927i	$-0.308784\pi$
$269$	4987.59	1.13048	0.565239	+	0.824927i	$0.308784\pi$
$270$	0	0
$271$	4374.50	0.980561	0.490280	−	0.871565i	$-0.336894\pi$
$271$	4374.50	0.980561	0.490280	+	0.871565i	$0.336894\pi$
$272$	−2453.77	−0.546991
$273$	0	0
$274$	−678.147	−0.149520
$275$	−1190.22	−0.260993
$276$	0	0
$277$	4538.78	0.984508	0.492254	−	0.870452i	$-0.336173\pi$
$277$	4538.78	0.984508	0.492254	+	0.870452i	$0.336173\pi$
$278$	−1098.24	−0.236935
$279$	0	0
$280$	15.2472	0.00325427
$281$	559.302	0.118737	0.0593686	−	0.998236i	$-0.481091\pi$
$281$	559.302	0.118737	0.0593686	+	0.998236i	$0.481091\pi$
$282$	0	0
$283$	6170.65	1.29614	0.648069	−	0.761581i	$-0.275577\pi$
$283$	6170.65	1.29614	0.648069	+	0.761581i	$0.275577\pi$
$284$	−922.525	−0.192753
$285$	0	0
$286$	197.539	0.0408417
$287$	4741.76	0.975252
$288$	0	0
$289$	−3246.02	−0.660699
$290$	−13.5535	−0.00274444
$291$	0	0
$292$	4888.97	0.979812
$293$	−6784.64	−1.35277	−0.676387	−	0.736546i	$-0.736455\pi$
$293$	−6784.64	−1.35277	−0.676387	+	0.736546i	$0.736455\pi$
$294$	0	0
$295$	−60.6441	−0.0119689
$296$	−22.0326	−0.00432642
$297$	0	0
$298$	−152.845	−0.0297117
$299$	845.706	0.163573
$300$	0	0
$301$	40.5976	0.00777411
$302$	125.899	0.0239890
$303$	0	0
$304$	1290.37	0.243446
$305$	−52.7217	−0.00989783
$306$	0	0
$307$	−3206.20	−0.596051	−0.298025	−	0.954558i	$-0.596328\pi$
$307$	−3206.20	−0.596051	−0.298025	+	0.954558i	$0.596328\pi$
$308$	−963.037	−0.178163
$309$	0	0
$310$	−4.23038	−0.000775062 0
$311$	−1959.54	−0.357284	−0.178642	−	0.983914i	$-0.557170\pi$
$311$	−1959.54	−0.357284	−0.178642	+	0.983914i	$0.557170\pi$
$312$	0	0
$313$	−4963.00	−0.896246	−0.448123	−	0.893972i	$-0.647907\pi$
$313$	−4963.00	−0.896246	−0.448123	+	0.893972i	$0.647907\pi$
$314$	1383.07	0.248571
$315$	0	0
$316$	8924.29	1.58870
$317$	676.552	0.119871	0.0599353	−	0.998202i	$-0.480911\pi$
$317$	676.552	0.119871	0.0599353	+	0.998202i	$0.480911\pi$
$318$	0	0
$319$	1730.00	0.303640
$320$	−83.0443	−0.0145072
$321$	0	0
$322$	86.1033	0.0149017
$323$	−876.619	−0.151010
$324$	0	0
$325$	6406.86	1.09350
$326$	−1281.25	−0.217674
$327$	0	0
$328$	2354.31	0.396327
$329$	−5464.89	−0.915773
$330$	0	0
$331$	10629.6	1.76512	0.882559	−	0.470202i	$-0.155819\pi$
$331$	10629.6	1.76512	0.882559	+	0.470202i	$0.155819\pi$
$332$	−5780.35	−0.955536
$333$	0	0
$334$	1565.95	0.256541
$335$	38.0480	0.00620533
$336$	0	0
$337$	2967.32	0.479645	0.239823	−	0.970817i	$-0.422911\pi$
$337$	2967.32	0.479645	0.239823	+	0.970817i	$0.422911\pi$
$338$	−174.557	−0.0280907
$339$	0	0
$340$	59.0147	0.00941330
$341$	539.974	0.0857515
$342$	0	0
$343$	−6703.28	−1.05523
$344$	20.1570	0.00315928
$345$	0	0
$346$	−1662.14	−0.258258
$347$	−5061.13	−0.782985	−0.391492	−	0.920181i	$-0.628041\pi$
$347$	−5061.13	−0.782985	−0.391492	+	0.920181i	$0.628041\pi$
$348$	0	0
$349$	−8.89325	−0.00136402	−0.000682012	−	1.00000i	$-0.500217\pi$
$349$	−8.89325	−0.00136402	−0.000682012		1.00000i	$0.500217\pi$
$350$	652.298	0.0996194
$351$	0	0
$352$	−719.701	−0.108978
$353$	−3516.80	−0.530256	−0.265128	−	0.964213i	$-0.585414\pi$
$353$	−3516.80	−0.530256	−0.265128	+	0.964213i	$0.585414\pi$
$354$	0	0
$355$	21.7143	0.00324641
$356$	−10557.9	−1.57182
$357$	0	0
$358$	−1055.35	−0.155802
$359$	11819.0	1.73755	0.868776	−	0.495206i	$-0.164907\pi$
$359$	11819.0	1.73755	0.868776	+	0.495206i	$0.164907\pi$
$360$	0	0
$361$	−6398.01	−0.932791
$362$	381.249	0.0553536
$363$	0	0
$364$	5183.93	0.746461
$365$	−115.076	−0.0165023
$366$	0	0
$367$	7770.52	1.10523	0.552613	−	0.833438i	$-0.313631\pi$
$367$	7770.52	1.10523	0.552613	+	0.833438i	$0.313631\pi$
$368$	−991.365	−0.140431
$369$	0	0
$370$	0.256621	3.60570e−5 0
$371$	8367.40	1.17093
$372$	0	0
$373$	11090.9	1.53959	0.769793	−	0.638293i	$-0.220359\pi$
$373$	11090.9	1.53959	0.769793	+	0.638293i	$0.220359\pi$
$374$	157.313	0.0217499
$375$	0	0
$376$	−2713.35	−0.372156
$377$	−9312.40	−1.27218
$378$	0	0
$379$	5723.79	0.775756	0.387878	−	0.921711i	$-0.373208\pi$
$379$	5723.79	0.775756	0.387878	+	0.921711i	$0.373208\pi$
$380$	−31.0341	−0.00418952
$381$	0	0
$382$	−429.127	−0.0574766
$383$	−8789.52	−1.17265	−0.586323	−	0.810077i	$-0.699425\pi$
$383$	−8789.52	−1.17265	−0.586323	+	0.810077i	$0.699425\pi$
$384$	0	0
$385$	22.6678	0.00300067
$386$	95.6430	0.0126117
$387$	0	0
$388$	1914.82	0.250542
$389$	11549.5	1.50536	0.752678	−	0.658389i	$-0.228762\pi$
$389$	11549.5	1.50536	0.752678	+	0.658389i	$0.228762\pi$
$390$	0	0
$391$	673.491	0.0871097
$392$	−1130.81	−0.145700
$393$	0	0
$394$	1047.04	0.133881
$395$	−210.059	−0.0267575
$396$	0	0
$397$	−9356.44	−1.18284	−0.591418	−	0.806365i	$-0.701432\pi$
$397$	−9356.44	−1.18284	−0.591418	+	0.806365i	$0.701432\pi$
$398$	−2225.49	−0.280286
$399$	0	0
$400$	−7510.34	−0.938793
$401$	−6189.90	−0.770845	−0.385423	−	0.922740i	$-0.625944\pi$
$401$	−6189.90	−0.770845	−0.385423	+	0.922740i	$0.625944\pi$
$402$	0	0
$403$	−2906.63	−0.359279
$404$	−541.718	−0.0667116
$405$	0	0
$406$	−948.118	−0.115897
$407$	−32.7557	−0.00398928
$408$	0	0
$409$	3555.45	0.429843	0.214921	−	0.976631i	$-0.431050\pi$
$409$	3555.45	0.429843	0.214921	+	0.976631i	$0.431050\pi$
$410$	−27.4215	−0.00330305
$411$	0	0
$412$	12961.4	1.54991
$413$	−4242.28	−0.505446
$414$	0	0
$415$	136.057	0.0160935
$416$	3874.08	0.456593
$417$	0	0
$418$	−82.7264	−0.00968009
$419$	1551.26	0.180869	0.0904343	−	0.995902i	$-0.471174\pi$
$419$	1551.26	0.180869	0.0904343	+	0.995902i	$0.471174\pi$
$420$	0	0
$421$	−13680.5	−1.58372	−0.791862	−	0.610699i	$-0.790888\pi$
$421$	−13680.5	−1.58372	−0.791862	+	0.610699i	$0.790888\pi$
$422$	−1033.41	−0.119207
$423$	0	0
$424$	4154.47	0.475846
$425$	5102.20	0.582337
$426$	0	0
$427$	−3688.09	−0.417984
$428$	−987.720	−0.111550
$429$	0	0
$430$	−0.234775	−2.63299e−5 0
$431$	962.547	0.107574	0.0537869	−	0.998552i	$-0.482871\pi$
$431$	962.547	0.107574	0.0537869	+	0.998552i	$0.482871\pi$
$432$	0	0
$433$	−13442.8	−1.49196	−0.745981	−	0.665967i	$-0.768019\pi$
$433$	−13442.8	−1.49196	−0.745981	+	0.665967i	$0.768019\pi$
$434$	−295.931	−0.0327308
$435$	0	0
$436$	−11938.9	−1.31140
$437$	−354.169	−0.0387694
$438$	0	0
$439$	9974.82	1.08445	0.542223	−	0.840234i	$-0.317583\pi$
$439$	9974.82	1.08445	0.542223	+	0.840234i	$0.317583\pi$
$440$	11.2547	0.00121943
$441$	0	0
$442$	−846.801	−0.0911272
$443$	−14323.2	−1.53616	−0.768079	−	0.640355i	$-0.778787\pi$
$443$	−14323.2	−1.53616	−0.768079	+	0.640355i	$0.778787\pi$
$444$	0	0
$445$	248.511	0.0264731
$446$	−2220.73	−0.235773
$447$	0	0
$448$	−5809.26	−0.612638
$449$	−1998.25	−0.210030	−0.105015	−	0.994471i	$-0.533489\pi$
$449$	−1998.25	−0.210030	−0.105015	+	0.994471i	$0.533489\pi$
$450$	0	0
$451$	3500.13	0.365443
$452$	17600.5	1.83154
$453$	0	0
$454$	−1373.42	−0.141978
$455$	−122.019	−0.0125722
$456$	0	0
$457$	10653.5	1.09048	0.545238	−	0.838281i	$-0.316439\pi$
$457$	10653.5	1.09048	0.545238	+	0.838281i	$0.316439\pi$
$458$	−1469.91	−0.149966
$459$	0	0
$460$	23.8430	0.00241670
$461$	5380.68	0.543608	0.271804	−	0.962353i	$-0.412380\pi$
$461$	5380.68	0.543608	0.271804	+	0.962353i	$0.412380\pi$
$462$	0	0
$463$	18451.5	1.85209	0.926043	−	0.377418i	$-0.123188\pi$
$463$	18451.5	1.85209	0.926043	+	0.377418i	$0.123188\pi$
$464$	10916.3	1.09219
$465$	0	0
$466$	−868.079	−0.0862939
$467$	10532.3	1.04363	0.521816	−	0.853058i	$-0.325255\pi$
$467$	10532.3	1.04363	0.521816	+	0.853058i	$0.325255\pi$
$468$	0	0
$469$	2661.60	0.262050
$470$	31.6033	0.00310160
$471$	0	0
$472$	−2106.32	−0.205405
$473$	29.9671	0.00291309
$474$	0	0
$475$	−2683.10	−0.259177
$476$	4128.30	0.397522
$477$	0	0
$478$	−96.6852	−0.00925163
$479$	−1778.89	−0.169686	−0.0848432	−	0.996394i	$-0.527039\pi$
$479$	−1778.89	−0.169686	−0.0848432	+	0.996394i	$0.527039\pi$
$480$	0	0
$481$	176.321	0.0167142
$482$	−1873.36	−0.177032
$483$	0	0
$484$	9719.31	0.912783
$485$	−45.0709	−0.00421972
$486$	0	0
$487$	2452.57	0.228206	0.114103	−	0.993469i	$-0.463601\pi$
$487$	2452.57	0.228206	0.114103	+	0.993469i	$0.463601\pi$
$488$	−1831.16	−0.169862
$489$	0	0
$490$	13.1709	0.00121429
$491$	5289.02	0.486131	0.243065	−	0.970010i	$-0.421847\pi$
$491$	5289.02	0.486131	0.243065	+	0.970010i	$0.421847\pi$
$492$	0	0
$493$	−7416.07	−0.677491
$494$	445.308	0.0405574
$495$	0	0
$496$	3407.25	0.308448
$497$	1519.00	0.137095
$498$	0	0
$499$	263.856	0.0236709	0.0118355	−	0.999930i	$-0.496233\pi$
$499$	263.856	0.0236709	0.0118355	+	0.999930i	$0.496233\pi$
$500$	361.306	0.0323162
$501$	0	0
$502$	−1710.10	−0.152043
$503$	11985.0	1.06239	0.531197	−	0.847249i	$-0.321743\pi$
$503$	11985.0	1.06239	0.531197	+	0.847249i	$0.321743\pi$
$504$	0	0
$505$	12.7509	0.00112358
$506$	63.5572	0.00558392
$507$	0	0
$508$	−230.621	−0.0201420
$509$	13047.5	1.13619	0.568094	−	0.822964i	$-0.307681\pi$
$509$	13047.5	1.13619	0.568094	+	0.822964i	$0.307681\pi$
$510$	0	0
$511$	−8050.00	−0.696890
$512$	−7621.51	−0.657864
$513$	0	0
$514$	542.424	0.0465473
$515$	−305.083	−0.0261040
$516$	0	0
$517$	−4033.91	−0.343155
$518$	17.9516	0.00152268
$519$	0	0
$520$	−60.5831	−0.00510912
$521$	19081.9	1.60459	0.802297	−	0.596925i	$-0.203611\pi$
$521$	19081.9	1.60459	0.802297	+	0.596925i	$0.203611\pi$
$522$	0	0
$523$	−5187.50	−0.433716	−0.216858	−	0.976203i	$-0.569581\pi$
$523$	−5187.50	−0.433716	−0.216858	+	0.976203i	$0.569581\pi$
$524$	−7797.71	−0.650085
$525$	0	0
$526$	−1204.87	−0.0998759
$527$	−2314.74	−0.191331
$528$	0	0
$529$	−11894.9	−0.977636
$530$	−48.3884	−0.00396577
$531$	0	0
$532$	−2170.96	−0.176923
$533$	−18840.9	−1.53112
$534$	0	0
$535$	23.2488	0.00187876
$536$	1321.50	0.106493
$537$	0	0
$538$	−2017.68	−0.161689
$539$	−1681.16	−0.134347
$540$	0	0
$541$	4545.96	0.361268	0.180634	−	0.983550i	$-0.442185\pi$
$541$	4545.96	0.361268	0.180634	+	0.983550i	$0.442185\pi$
$542$	−1769.66	−0.140246
$543$	0	0
$544$	3085.18	0.243155
$545$	281.017	0.0220871
$546$	0	0
$547$	−12750.0	−0.996617	−0.498309	−	0.867000i	$-0.666045\pi$
$547$	−12750.0	−0.996617	−0.498309	+	0.867000i	$0.666045\pi$
$548$	−13136.4	−1.02401
$549$	0	0
$550$	481.494	0.0373290
$551$	3899.90	0.301527
$552$	0	0
$553$	−14694.4	−1.12996
$554$	−1836.12	−0.140811
$555$	0	0
$556$	−21273.9	−1.62269
$557$	−24131.1	−1.83567	−0.917833	−	0.396968i	$-0.870063\pi$
$557$	−24131.1	−1.83567	−0.917833	+	0.396968i	$0.870063\pi$
$558$	0	0
$559$	−161.310	−0.0122052
$560$	143.035	0.0107934
$561$	0	0
$562$	−226.260	−0.0169826
$563$	20514.4	1.53566	0.767830	−	0.640653i	$-0.221336\pi$
$563$	20514.4	1.53566	0.767830	+	0.640653i	$0.221336\pi$
$564$	0	0
$565$	−414.278	−0.0308474
$566$	−2496.28	−0.185382
$567$	0	0
$568$	754.192	0.0557133
$569$	−3103.28	−0.228640	−0.114320	−	0.993444i	$-0.536469\pi$
$569$	−3103.28	−0.228640	−0.114320	+	0.993444i	$0.536469\pi$
$570$	0	0
$571$	−20969.7	−1.53687	−0.768435	−	0.639928i	$-0.778964\pi$
$571$	−20969.7	−1.53687	−0.768435	+	0.639928i	$0.778964\pi$
$572$	3826.52	0.279711
$573$	0	0
$574$	−1918.23	−0.139487
$575$	2061.38	0.149505
$576$	0	0
$577$	12527.8	0.903883	0.451941	−	0.892048i	$-0.350732\pi$
$577$	12527.8	0.903883	0.451941	+	0.892048i	$0.350732\pi$
$578$	1313.14	0.0944976
$579$	0	0
$580$	−262.544	−0.0187958
$581$	9517.72	0.679624
$582$	0	0
$583$	6176.40	0.438765
$584$	−3996.88	−0.283205
$585$	0	0
$586$	2744.66	0.193483
$587$	6325.45	0.444769	0.222384	−	0.974959i	$-0.428616\pi$
$587$	6325.45	0.444769	0.222384	+	0.974959i	$0.428616\pi$
$588$	0	0
$589$	1217.25	0.0851547
$590$	24.5330	0.00171188
$591$	0	0
$592$	−206.689	−0.0143494
$593$	11526.0	0.798175	0.399087	−	0.916913i	$-0.369327\pi$
$593$	11526.0	0.798175	0.399087	+	0.916913i	$0.369327\pi$
$594$	0	0
$595$	−97.1715	−0.00669520
$596$	−2960.76	−0.203486
$597$	0	0
$598$	−342.122	−0.0233953
$599$	−6404.00	−0.436829	−0.218414	−	0.975856i	$-0.570088\pi$
$599$	−6404.00	−0.436829	−0.218414	+	0.975856i	$0.570088\pi$
$600$	0	0
$601$	12541.7	0.851224	0.425612	−	0.904906i	$-0.360059\pi$
$601$	12541.7	0.851224	0.425612	+	0.904906i	$0.360059\pi$
$602$	−16.4234	−0.00111191
$603$	0	0
$604$	2438.79	0.164293
$605$	−228.772	−0.0153734
$606$	0	0
$607$	3939.71	0.263440	0.131720	−	0.991287i	$-0.457950\pi$
$607$	3939.71	0.263440	0.131720	+	0.991287i	$0.457950\pi$
$608$	−1622.41	−0.108219
$609$	0	0
$610$	21.3281	0.00141565
$611$	21714.2	1.43774
$612$	0	0
$613$	−12338.8	−0.812987	−0.406494	−	0.913654i	$-0.633249\pi$
$613$	−12338.8	−0.812987	−0.406494	+	0.913654i	$0.633249\pi$
$614$	1297.04	0.0852512
$615$	0	0
$616$	787.311	0.0514962
$617$	25307.0	1.65125	0.825625	−	0.564219i	$-0.190823\pi$
$617$	25307.0	1.65125	0.825625	+	0.564219i	$0.190823\pi$
$618$	0	0
$619$	−6563.41	−0.426181	−0.213090	−	0.977032i	$-0.568353\pi$
$619$	−6563.41	−0.426181	−0.213090	+	0.977032i	$0.568353\pi$
$620$	−81.9466	−0.00530815
$621$	0	0
$622$	792.713	0.0511011
$623$	17384.3	1.11796
$624$	0	0
$625$	15612.2	0.999184
$626$	2007.73	0.128187
$627$	0	0
$628$	26791.5	1.70238
$629$	140.416	0.00890101
$630$	0	0
$631$	2421.92	0.152797	0.0763985	−	0.997077i	$-0.475658\pi$
$631$	2421.92	0.152797	0.0763985	+	0.997077i	$0.475658\pi$
$632$	−7295.87	−0.459200
$633$	0	0
$634$	−273.693	−0.0171447
$635$	5.42833	0.000339239 0
$636$	0	0
$637$	9049.54	0.562882
$638$	−699.853	−0.0434286
$639$	0	0
$640$	145.098	0.00896170
$641$	−23523.6	−1.44949	−0.724747	−	0.689015i	$-0.758043\pi$
$641$	−23523.6	−1.44949	−0.724747	+	0.689015i	$0.758043\pi$
$642$	0	0
$643$	−3877.61	−0.237820	−0.118910	−	0.992905i	$-0.537940\pi$
$643$	−3877.61	−0.237820	−0.118910	+	0.992905i	$0.537940\pi$
$644$	1667.91	0.102057
$645$	0	0
$646$	354.628	0.0215985
$647$	−19418.1	−1.17991	−0.589956	−	0.807436i	$-0.700855\pi$
$647$	−19418.1	−1.17991	−0.589956	+	0.807436i	$0.700855\pi$
$648$	0	0
$649$	−3131.44	−0.189399
$650$	−2591.84	−0.156400
$651$	0	0
$652$	−24819.0	−1.49078
$653$	8925.79	0.534905	0.267453	−	0.963571i	$-0.413818\pi$
$653$	8925.79	0.534905	0.267453	+	0.963571i	$0.413818\pi$
$654$	0	0
$655$	183.542	0.0109489
$656$	22085.9	1.31450
$657$	0	0
$658$	2210.77	0.130980
$659$	17122.9	1.01216	0.506080	−	0.862486i	$-0.331094\pi$
$659$	17122.9	1.01216	0.506080	+	0.862486i	$0.331094\pi$
$660$	0	0
$661$	−22027.7	−1.29618	−0.648092	−	0.761562i	$-0.724433\pi$
$661$	−22027.7	−1.29618	−0.648092	+	0.761562i	$0.724433\pi$
$662$	−4300.09	−0.252459
$663$	0	0
$664$	4725.61	0.276189
$665$	51.0997	0.00297979
$666$	0	0
$667$	−2996.22	−0.173934
$668$	30333.9	1.75697
$669$	0	0
$670$	−15.3920	−0.000887527 0
$671$	−2722.36	−0.156625
$672$	0	0
$673$	18061.4	1.03450	0.517248	−	0.855835i	$-0.326956\pi$
$673$	18061.4	1.03450	0.517248	+	0.855835i	$0.326956\pi$
$674$	−1200.40	−0.0686021
$675$	0	0
$676$	−3381.34	−0.192384
$677$	31127.6	1.76710	0.883552	−	0.468332i	$-0.155145\pi$
$677$	31127.6	1.76710	0.883552	+	0.468332i	$0.155145\pi$
$678$	0	0
$679$	−3152.88	−0.178198
$680$	−48.2463	−0.00272082
$681$	0	0
$682$	−218.442	−0.0122647
$683$	−8837.55	−0.495109	−0.247554	−	0.968874i	$-0.579627\pi$
$683$	−8837.55	−0.495109	−0.247554	+	0.968874i	$0.579627\pi$
$684$	0	0
$685$	309.203	0.0172467
$686$	2711.75	0.150926
$687$	0	0
$688$	189.093	0.0104784
$689$	−33246.9	−1.83833
$690$	0	0
$691$	−25935.3	−1.42782	−0.713910	−	0.700238i	$-0.753077\pi$
$691$	−25935.3	−1.42782	−0.713910	+	0.700238i	$0.753077\pi$
$692$	−32197.4	−1.76873
$693$	0	0
$694$	2047.43	0.111988
$695$	500.743	0.0273299
$696$	0	0
$697$	−15004.2	−0.815388
$698$	3.59768	0.000195092 0
$699$	0	0
$700$	12635.7	0.682261
$701$	11698.6	0.630313	0.315156	−	0.949040i	$-0.397943\pi$
$701$	11698.6	0.630313	0.315156	+	0.949040i	$0.397943\pi$
$702$	0	0
$703$	−73.8405	−0.00396152
$704$	−4288.11	−0.229566
$705$	0	0
$706$	1422.69	0.0758407
$707$	891.973	0.0474485
$708$	0	0
$709$	−1708.24	−0.0904856	−0.0452428	−	0.998976i	$-0.514406\pi$
$709$	−1708.24	−0.0904856	−0.0452428	+	0.998976i	$0.514406\pi$
$710$	−8.78431	−0.000464323 0
$711$	0	0
$712$	8631.40	0.454319
$713$	−935.195	−0.0491211
$714$	0	0
$715$	−90.0682	−0.00471099
$716$	−20443.2	−1.06704
$717$	0	0
$718$	−4781.25	−0.248516
$719$	−13629.2	−0.706932	−0.353466	−	0.935447i	$-0.614997\pi$
$719$	−13629.2	−0.706932	−0.353466	+	0.935447i	$0.614997\pi$
$720$	0	0
$721$	−21341.7	−1.10237
$722$	2588.25	0.133414
$723$	0	0
$724$	7385.16	0.379099
$725$	−22698.6	−1.16277
$726$	0	0
$727$	1061.74	0.0541646	0.0270823	−	0.999633i	$-0.491378\pi$
$727$	1061.74	0.0541646	0.0270823	+	0.999633i	$0.491378\pi$
$728$	−4238.02	−0.215757
$729$	0	0
$730$	46.5529	0.00236027
$731$	−128.462	−0.00649977
$732$	0	0
$733$	30272.1	1.52541	0.762706	−	0.646746i	$-0.223871\pi$
$733$	30272.1	1.52541	0.762706	+	0.646746i	$0.223871\pi$
$734$	−3143.49	−0.158077
$735$	0	0
$736$	1246.47	0.0624258
$737$	1964.66	0.0981944
$738$	0	0
$739$	−19487.2	−0.970027	−0.485013	−	0.874507i	$-0.661185\pi$
$739$	−19487.2	−0.970027	−0.485013	+	0.874507i	$0.661185\pi$
$740$	4.97100	0.000246943 0
$741$	0	0
$742$	−3384.95	−0.167474
$743$	−12497.7	−0.617085	−0.308543	−	0.951211i	$-0.599841\pi$
$743$	−12497.7	−0.617085	−0.308543	+	0.951211i	$0.599841\pi$
$744$	0	0
$745$	69.6900	0.00342717
$746$	−4486.72	−0.220202
$747$	0	0
$748$	3047.31	0.148958
$749$	1626.34	0.0793395
$750$	0	0
$751$	−24201.2	−1.17592	−0.587959	−	0.808891i	$-0.700068\pi$
$751$	−24201.2	−1.17592	−0.587959	+	0.808891i	$0.700068\pi$
$752$	−25454.1	−1.23433
$753$	0	0
$754$	3767.24	0.181956
$755$	−57.4040	−0.00276708
$756$	0	0
$757$	1771.59	0.0850586	0.0425293	−	0.999095i	$-0.486458\pi$
$757$	1771.59	0.0850586	0.0425293	+	0.999095i	$0.486458\pi$
$758$	−2315.51	−0.110954
$759$	0	0
$760$	25.3713	0.00121094
$761$	10966.9	0.522403	0.261202	−	0.965284i	$-0.415881\pi$
$761$	10966.9	0.522403	0.261202	+	0.965284i	$0.415881\pi$
$762$	0	0
$763$	19658.2	0.932734
$764$	−8312.61	−0.393638
$765$	0	0
$766$	3555.72	0.167720
$767$	16856.2	0.793538
$768$	0	0
$769$	22166.3	1.03945	0.519726	−	0.854333i	$-0.326034\pi$
$769$	22166.3	1.03945	0.519726	+	0.854333i	$0.326034\pi$
$770$	−9.17006	−0.000429176 0
$771$	0	0
$772$	1852.70	0.0863731
$773$	32142.8	1.49560	0.747798	−	0.663927i	$-0.231111\pi$
$773$	32142.8	1.49560	0.747798	+	0.663927i	$0.231111\pi$
$774$	0	0
$775$	−7084.81	−0.328379
$776$	−1565.42	−0.0724168
$777$	0	0
$778$	−4672.25	−0.215306
$779$	7890.28	0.362899
$780$	0	0
$781$	1121.25	0.0513718
$782$	−272.454	−0.0124590
$783$	0	0
$784$	−10608.2	−0.483244
$785$	−630.614	−0.0286721
$786$	0	0
$787$	32823.5	1.48670	0.743349	−	0.668904i	$-0.233236\pi$
$787$	32823.5	1.48670	0.743349	+	0.668904i	$0.233236\pi$
$788$	20282.2	0.916906
$789$	0	0
$790$	84.9773	0.00382703
$791$	−28980.3	−1.30268
$792$	0	0
$793$	14654.2	0.656225
$794$	3785.06	0.169177
$795$	0	0
$796$	−43110.0	−1.91959
$797$	23237.5	1.03277	0.516383	−	0.856358i	$-0.327278\pi$
$797$	23237.5	1.03277	0.516383	+	0.856358i	$0.327278\pi$
$798$	0	0
$799$	17292.4	0.765658
$800$	9442.94	0.417323
$801$	0	0
$802$	2504.07	0.110251
$803$	−5942.11	−0.261136
$804$	0	0
$805$	−39.2590	−0.00171888
$806$	1175.85	0.0513865
$807$	0	0
$808$	442.870	0.0192823
$809$	9173.83	0.398683	0.199341	−	0.979930i	$-0.436120\pi$
$809$	9173.83	0.398683	0.199341	+	0.979930i	$0.436120\pi$
$810$	0	0
$811$	−16200.8	−0.701463	−0.350732	−	0.936476i	$-0.614067\pi$
$811$	−16200.8	−0.701463	−0.350732	+	0.936476i	$0.614067\pi$
$812$	−18366.0	−0.793743
$813$	0	0
$814$	13.2510	0.000570574 0
$815$	584.187	0.0251082
$816$	0	0
$817$	67.5544	0.00289281
$818$	−1438.32	−0.0614790
$819$	0	0
$820$	−531.180	−0.0226215
$821$	17784.6	0.756012	0.378006	−	0.925803i	$-0.376610\pi$
$821$	17784.6	0.756012	0.378006	+	0.925803i	$0.376610\pi$
$822$	0	0
$823$	9027.80	0.382369	0.191184	−	0.981554i	$-0.438767\pi$
$823$	9027.80	0.382369	0.191184	+	0.981554i	$0.438767\pi$
$824$	−10596.3	−0.447985
$825$	0	0
$826$	1716.18	0.0722922
$827$	−3216.13	−0.135231	−0.0676154	−	0.997711i	$-0.521539\pi$
$827$	−3216.13	−0.135231	−0.0676154	+	0.997711i	$0.521539\pi$
$828$	0	0
$829$	3257.41	0.136471	0.0682356	−	0.997669i	$-0.478263\pi$
$829$	3257.41	0.136471	0.0682356	+	0.997669i	$0.478263\pi$
$830$	−55.0407	−0.00230179
$831$	0	0
$832$	23082.5	0.961828
$833$	7206.74	0.299758
$834$	0	0
$835$	−713.996	−0.0295914
$836$	−1602.49	−0.0662958
$837$	0	0
$838$	−627.547	−0.0258690
$839$	−5708.72	−0.234907	−0.117454	−	0.993078i	$-0.537473\pi$
$839$	−5708.72	−0.234907	−0.117454	+	0.993078i	$0.537473\pi$
$840$	0	0
$841$	8603.57	0.352764
$842$	5534.33	0.226515
$843$	0	0
$844$	−20018.1	−0.816413
$845$	79.5897	0.00324020
$846$	0	0
$847$	−16003.5	−0.649216
$848$	38973.2	1.57824
$849$	0	0
$850$	−2064.05	−0.0832897
$851$	56.7303	0.00228518
$852$	0	0
$853$	−33403.4	−1.34081	−0.670405	−	0.741995i	$-0.733880\pi$
$853$	−33403.4	−1.34081	−0.670405	+	0.741995i	$0.733880\pi$
$854$	1491.98	0.0597828
$855$	0	0
$856$	807.490	0.0322423
$857$	−14391.3	−0.573626	−0.286813	−	0.957987i	$-0.592596\pi$
$857$	−14391.3	−0.573626	−0.286813	+	0.957987i	$0.592596\pi$
$858$	0	0
$859$	1941.48	0.0771156	0.0385578	−	0.999256i	$-0.487724\pi$
$859$	1941.48	0.0771156	0.0385578	+	0.999256i	$0.487724\pi$
$860$	−4.54782	−0.000180325 0
$861$	0	0
$862$	−389.389	−0.0153859
$863$	2702.99	0.106618	0.0533088	−	0.998578i	$-0.483023\pi$
$863$	2702.99	0.106618	0.0533088	+	0.998578i	$0.483023\pi$
$864$	0	0
$865$	757.858	0.0297895
$866$	5438.16	0.213390
$867$	0	0
$868$	−5732.48	−0.224162
$869$	−10846.7	−0.423416
$870$	0	0
$871$	−10575.6	−0.411412
$872$	9760.44	0.379048
$873$	0	0
$874$	143.276	0.00554505
$875$	−594.914	−0.0229849
$876$	0	0
$877$	27667.3	1.06529	0.532644	−	0.846340i	$-0.321199\pi$
$877$	27667.3	1.06529	0.532644	+	0.846340i	$0.321199\pi$
$878$	−4035.22	−0.155105
$879$	0	0
$880$	105.581	0.00404447
$881$	−10539.1	−0.403031	−0.201516	−	0.979485i	$-0.564587\pi$
$881$	−10539.1	−0.403031	−0.201516	+	0.979485i	$0.564587\pi$
$882$	0	0
$883$	3391.56	0.129258	0.0646291	−	0.997909i	$-0.479414\pi$
$883$	3391.56	0.129258	0.0646291	+	0.997909i	$0.479414\pi$
$884$	−16403.4	−0.624101
$885$	0	0
$886$	5794.33	0.219712
$887$	−44990.7	−1.70309	−0.851545	−	0.524281i	$-0.824334\pi$
$887$	−44990.7	−1.70309	−0.851545	+	0.524281i	$0.824334\pi$
$888$	0	0
$889$	379.732	0.0143260
$890$	−100.533	−0.00378636
$891$	0	0
$892$	−43017.8	−1.61473
$893$	−9093.57	−0.340767
$894$	0	0
$895$	481.190	0.0179714
$896$	10150.1	0.378451
$897$	0	0
$898$	808.375	0.0300399
$899$	10297.8	0.382037
$900$	0	0
$901$	−26476.7	−0.978987
$902$	−1415.95	−0.0522681
$903$	0	0
$904$	−14388.9	−0.529389
$905$	−173.831	−0.00638490
$906$	0	0
$907$	−19107.6	−0.699511	−0.349755	−	0.936841i	$-0.613735\pi$
$907$	−19107.6	−0.699511	−0.349755	+	0.936841i	$0.613735\pi$
$908$	−26604.5	−0.972360
$909$	0	0
$910$	49.3615	0.00179815
$911$	16317.9	0.593453	0.296726	−	0.954963i	$-0.404105\pi$
$911$	16317.9	0.593453	0.296726	+	0.954963i	$0.404105\pi$
$912$	0	0
$913$	7025.51	0.254666
$914$	−4309.75	−0.155967
$915$	0	0
$916$	−28473.6	−1.02707
$917$	12839.4	0.462372
$918$	0	0
$919$	4182.05	0.150112	0.0750562	−	0.997179i	$-0.476086\pi$
$919$	4182.05	0.150112	0.0750562	+	0.997179i	$0.476086\pi$
$920$	−19.4923	−0.000698525 0
$921$	0	0
$922$	−2176.70	−0.0777505
$923$	−6035.57	−0.215236
$924$	0	0
$925$	429.775	0.0152767
$926$	−7464.40	−0.264898
$927$	0	0
$928$	−13725.4	−0.485514
$929$	15712.7	0.554917	0.277458	−	0.960738i	$-0.410508\pi$
$929$	15712.7	0.554917	0.277458	+	0.960738i	$0.410508\pi$
$930$	0	0
$931$	−3789.81	−0.133411
$932$	−16815.5	−0.590999
$933$	0	0
$934$	−4260.74	−0.149267
$935$	−71.7271	−0.00250880
$936$	0	0
$937$	−3278.53	−0.114306	−0.0571531	−	0.998365i	$-0.518202\pi$
$937$	−3278.53	−0.114306	−0.0571531	+	0.998365i	$0.518202\pi$
$938$	−1076.73	−0.0374801
$939$	0	0
$940$	612.187	0.0212418
$941$	−1103.22	−0.0382190	−0.0191095	−	0.999817i	$-0.506083\pi$
$941$	−1103.22	−0.0382190	−0.0191095	+	0.999817i	$0.506083\pi$
$942$	0	0
$943$	−6061.96	−0.209337
$944$	−19759.5	−0.681267
$945$	0	0
$946$	−12.1229	−0.000416649 0
$947$	6263.88	0.214941	0.107470	−	0.994208i	$-0.465725\pi$
$947$	6263.88	0.214941	0.107470	+	0.994208i	$0.465725\pi$
$948$	0	0
$949$	31985.8	1.09410
$950$	1085.42	0.0370692
$951$	0	0
$952$	−3375.01	−0.114900
$953$	−35492.7	−1.20642	−0.603212	−	0.797581i	$-0.706113\pi$
$953$	−35492.7	−1.20642	−0.603212	+	0.797581i	$0.706113\pi$
$954$	0	0
$955$	195.661	0.00662979
$956$	−1872.89	−0.0633614
$957$	0	0
$958$	719.635	0.0242697
$959$	21629.9	0.728327
$960$	0	0
$961$	−26576.8	−0.892108
$962$	−71.3289	−0.00239058
$963$	0	0
$964$	−36288.8	−1.21243
$965$	−43.6086	−0.00145473
$966$	0	0
$967$	−26252.5	−0.873033	−0.436517	−	0.899696i	$-0.643788\pi$
$967$	−26252.5	−0.873033	−0.436517	+	0.899696i	$0.643788\pi$
$968$	−7945.82	−0.263831
$969$	0	0
$970$	18.2330	0.000603532 0
$971$	3795.71	0.125448	0.0627241	−	0.998031i	$-0.480021\pi$
$971$	3795.71	0.125448	0.0627241	+	0.998031i	$0.480021\pi$
$972$	0	0
$973$	35028.9	1.15414
$974$	−992.162	−0.0326396
$975$	0	0
$976$	−17178.2	−0.563381
$977$	−25750.2	−0.843215	−0.421607	−	0.906779i	$-0.638534\pi$
$977$	−25750.2	−0.843215	−0.421607	+	0.906779i	$0.638534\pi$
$978$	0	0
$979$	12832.2	0.418916
$980$	255.133	0.00831626
$981$	0	0
$982$	−2139.62	−0.0695296
$983$	43871.3	1.42347	0.711737	−	0.702446i	$-0.247909\pi$
$983$	43871.3	1.42347	0.711737	+	0.702446i	$0.247909\pi$
$984$	0	0
$985$	−477.399	−0.0154428
$986$	3000.10	0.0968993
$987$	0	0
$988$	8626.05	0.277765
$989$	−51.9008	−0.00166871
$990$	0	0
$991$	−15752.8	−0.504948	−0.252474	−	0.967604i	$-0.581244\pi$
$991$	−15752.8	−0.504948	−0.252474	+	0.967604i	$0.581244\pi$
$992$	−4284.02	−0.137115
$993$	0	0
$994$	−614.496	−0.0196083
$995$	1014.72	0.0323304
$996$	0	0
$997$	−49307.5	−1.56628	−0.783142	−	0.621843i	$-0.786384\pi$
$997$	−49307.5	−1.56628	−0.783142	+	0.621843i	$0.786384\pi$
$998$	−106.740	−0.00338558
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.c.1.13		28
3.2	odd	2		717.4.a.a.1.16	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.a.1.16	✓	28	3.2	odd	2
2151.4.a.c.1.13		28	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-0.404541 q^{2} -7.83635 q^{4} +0.184451 q^{5} +12.9030 q^{7} +6.40644 q^{8} +O(q^{10})\)	\(q-0.404541 q^{2} -7.83635 q^{4} +0.184451 q^{5} +12.9030 q^{7} +6.40644 q^{8} -0.0746179 q^{10} +9.52438 q^{11} -51.2689 q^{13} -5.21980 q^{14} +60.0991 q^{16} -40.8287 q^{17} +21.4706 q^{19} -1.44542 q^{20} -3.85300 q^{22} -16.4955 q^{23} -124.966 q^{25} +20.7403 q^{26} -101.113 q^{28} +181.639 q^{29} +56.6939 q^{31} -75.5641 q^{32} +16.5169 q^{34} +2.37998 q^{35} -3.43914 q^{37} -8.68574 q^{38} +1.18167 q^{40} +367.492 q^{41} +3.14636 q^{43} -74.6364 q^{44} +6.67310 q^{46} -423.535 q^{47} -176.511 q^{49} +50.5538 q^{50} +401.760 q^{52} +648.482 q^{53} +1.75678 q^{55} +82.6626 q^{56} -73.4802 q^{58} -328.782 q^{59} -285.831 q^{61} -22.9350 q^{62} -450.224 q^{64} -9.45659 q^{65} +206.277 q^{67} +319.948 q^{68} -0.962798 q^{70} +117.724 q^{71} -623.884 q^{73} +1.39127 q^{74} -168.251 q^{76} +122.894 q^{77} -1138.83 q^{79} +11.0853 q^{80} -148.665 q^{82} +737.634 q^{83} -7.53090 q^{85} -1.27283 q^{86} +61.0174 q^{88} +1347.30 q^{89} -661.524 q^{91} +129.265 q^{92} +171.337 q^{94} +3.96028 q^{95} -244.352 q^{97} +71.4060 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8}+O(q^{10}) \) 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8	\( 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8} - 68 q^{10} + 258 q^{11} - 134 q^{13} + 292 q^{14} + 327 q^{16} + 364 q^{17} - 278 q^{19} + 986 q^{20} - 179 q^{22} + 668 q^{23} + 490 q^{25} + 760 q^{26} - 802 q^{28} + 714 q^{29} - 608 q^{31} + 918 q^{32} - 228 q^{34} + 934 q^{35} - 1080 q^{37} + 1395 q^{38} - 563 q^{40} + 1796 q^{41} - 1934 q^{43} + 3157 q^{44} - 940 q^{46} + 2032 q^{47} + 762 q^{49} + 1754 q^{50} - 2328 q^{52} + 1790 q^{53} - 478 q^{55} + 3557 q^{56} - 2626 q^{58} + 3622 q^{59} + 324 q^{61} + 796 q^{62} + 2023 q^{64} + 2200 q^{65} - 2444 q^{67} - 357 q^{68} + 4305 q^{70} + 1298 q^{71} - 1368 q^{73} - 813 q^{74} + 1390 q^{76} + 1408 q^{77} - 1378 q^{79} + 7684 q^{80} + 9001 q^{82} + 3524 q^{83} + 60 q^{85} + 2543 q^{86} + 1749 q^{88} + 7854 q^{89} + 850 q^{91} + 496 q^{92} + 6634 q^{94} + 3696 q^{95} - 1746 q^{97} + 4632 q^{98}+O(q^{100}) \) 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8 - 68 * q^10 + 258 * q^11 - 134 * q^13 + 292 * q^14 + 327 * q^16 + 364 * q^17 - 278 * q^19 + 986 * q^20 - 179 * q^22 + 668 * q^23 + 490 * q^25 + 760 * q^26 - 802 * q^28 + 714 * q^29 - 608 * q^31 + 918 * q^32 - 228 * q^34 + 934 * q^35 - 1080 * q^37 + 1395 * q^38 - 563 * q^40 + 1796 * q^41 - 1934 * q^43 + 3157 * q^44 - 940 * q^46 + 2032 * q^47 + 762 * q^49 + 1754 * q^50 - 2328 * q^52 + 1790 * q^53 - 478 * q^55 + 3557 * q^56 - 2626 * q^58 + 3622 * q^59 + 324 * q^61 + 796 * q^62 + 2023 * q^64 + 2200 * q^65 - 2444 * q^67 - 357 * q^68 + 4305 * q^70 + 1298 * q^71 - 1368 * q^73 - 813 * q^74 + 1390 * q^76 + 1408 * q^77 - 1378 * q^79 + 7684 * q^80 + 9001 * q^82 + 3524 * q^83 + 60 * q^85 + 2543 * q^86 + 1749 * q^88 + 7854 * q^89 + 850 * q^91 + 496 * q^92 + 6634 * q^94 + 3696 * q^95 - 1746 * q^97 + 4632 * q^98

Introduction

L-functions

Modular forms

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Database

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