LMFDB - Embedded newform 6003.2.a.l.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.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:	$47.9341963334$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3x^{9} - 10x^{8} + 32x^{7} + 32x^{6} - 118x^{5} - 29x^{4} + 182x^{3} - 28x^{2} - 101x + 43 $ x^10 - 3x^9 - 10x^8 + 32x^7 + 32x^6 - 118x^5 - 29x^4 + 182x^3 - 28x^2 - 101*x + 43
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.685481$ of defining polynomial
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.685481 q^{2} -1.53012 q^{4} +0.380940 q^{5} -0.405330 q^{7} -2.41983 q^{8} +O(q^{10})$	\(q+0.685481 q^{2} -1.53012 q^{4} +0.380940 q^{5} -0.405330 q^{7} -2.41983 q^{8} +0.261127 q^{10} +2.04811 q^{11} +4.25903 q^{13} -0.277846 q^{14} +1.40149 q^{16} +7.27552 q^{17} -2.41258 q^{19} -0.582882 q^{20} +1.40394 q^{22} -1.00000 q^{23} -4.85488 q^{25} +2.91949 q^{26} +0.620202 q^{28} -1.00000 q^{29} -4.43663 q^{31} +5.80035 q^{32} +4.98723 q^{34} -0.154407 q^{35} +0.231841 q^{37} -1.65378 q^{38} -0.921809 q^{40} +7.25515 q^{41} -0.152414 q^{43} -3.13385 q^{44} -0.685481 q^{46} +6.36400 q^{47} -6.83571 q^{49} -3.32793 q^{50} -6.51681 q^{52} -10.1639 q^{53} +0.780208 q^{55} +0.980829 q^{56} -0.685481 q^{58} +2.00392 q^{59} +1.35879 q^{61} -3.04123 q^{62} +1.17306 q^{64} +1.62244 q^{65} +11.4500 q^{67} -11.1324 q^{68} -0.105843 q^{70} +2.73384 q^{71} -6.65340 q^{73} +0.158923 q^{74} +3.69153 q^{76} -0.830161 q^{77} -3.74935 q^{79} +0.533882 q^{80} +4.97327 q^{82} -7.16171 q^{83} +2.77154 q^{85} -0.104477 q^{86} -4.95608 q^{88} +10.1567 q^{89} -1.72631 q^{91} +1.53012 q^{92} +4.36240 q^{94} -0.919050 q^{95} +14.6780 q^{97} -4.68575 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 3 q^{2} + 9 q^{4} + 10 q^{5} + q^{7} + 9 q^{8}+O(q^{10}) $ 10 * q + 3 * q^2 + 9 * q^4 + 10 * q^5 + q^7 + 9 * q^8	$ 10 q + 3 q^{2} + 9 q^{4} + 10 q^{5} + q^{7} + 9 q^{8} - 6 q^{10} - 13 q^{13} + 12 q^{14} - 5 q^{16} + 22 q^{17} - 2 q^{19} - 3 q^{20} + 3 q^{22} - 10 q^{23} + 10 q^{25} + 25 q^{26} + 19 q^{28} - 10 q^{29} - 22 q^{31} + 31 q^{32} + 13 q^{34} + 15 q^{35} - 9 q^{37} + 10 q^{38} - 6 q^{40} + 25 q^{41} + 3 q^{43} + 27 q^{44} - 3 q^{46} + 17 q^{47} + 17 q^{49} - 2 q^{50} - 18 q^{52} + 43 q^{53} - 11 q^{55} + 7 q^{56} - 3 q^{58} + 7 q^{59} - 6 q^{61} - 3 q^{62} + 33 q^{64} - 11 q^{65} + 11 q^{67} + 51 q^{68} + 34 q^{70} + 17 q^{71} - 44 q^{73} - 9 q^{74} + 24 q^{76} + 71 q^{77} + 5 q^{79} - 38 q^{80} + 33 q^{82} + 32 q^{83} + 16 q^{85} + 9 q^{86} + 18 q^{88} + 10 q^{89} - 3 q^{91} - 9 q^{92} + 47 q^{94} + 8 q^{95} + 6 q^{97} + 73 q^{98}+O(q^{100}) $ 10 * q + 3 * q^2 + 9 * q^4 + 10 * q^5 + q^7 + 9 * q^8 - 6 * q^10 - 13 * q^13 + 12 * q^14 - 5 * q^16 + 22 * q^17 - 2 * q^19 - 3 * q^20 + 3 * q^22 - 10 * q^23 + 10 * q^25 + 25 * q^26 + 19 * q^28 - 10 * q^29 - 22 * q^31 + 31 * q^32 + 13 * q^34 + 15 * q^35 - 9 * q^37 + 10 * q^38 - 6 * q^40 + 25 * q^41 + 3 * q^43 + 27 * q^44 - 3 * q^46 + 17 * q^47 + 17 * q^49 - 2 * q^50 - 18 * q^52 + 43 * q^53 - 11 * q^55 + 7 * q^56 - 3 * q^58 + 7 * q^59 - 6 * q^61 - 3 * q^62 + 33 * q^64 - 11 * q^65 + 11 * q^67 + 51 * q^68 + 34 * q^70 + 17 * q^71 - 44 * q^73 - 9 * q^74 + 24 * q^76 + 71 * q^77 + 5 * q^79 - 38 * q^80 + 33 * q^82 + 32 * q^83 + 16 * q^85 + 9 * q^86 + 18 * q^88 + 10 * q^89 - 3 * q^91 - 9 * q^92 + 47 * q^94 + 8 * q^95 + 6 * q^97 + 73 * 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 $

$2$	0.685481	0.484708	0.242354	−	0.970188i	$-0.422080\pi$
$2$	0.685481	0.484708	0.242354	+	0.970188i	$0.422080\pi$
$3$	0	0
$3$	0	0
$4$	−1.53012	−0.765058
$5$	0.380940	0.170362	0.0851808	−	0.996366i	$-0.472853\pi$
$5$	0.380940	0.170362	0.0851808	+	0.996366i	$0.472853\pi$
$6$	0	0
$7$	−0.405330	−0.153200	−0.0766002	−	0.997062i	$-0.524407\pi$
$7$	−0.405330	−0.153200	−0.0766002	+	0.997062i	$0.524407\pi$
$8$	−2.41983	−0.855538
$9$	0	0
$10$	0.261127	0.0825757
$11$	2.04811	0.617529	0.308764	−	0.951139i	$-0.400085\pi$
$11$	2.04811	0.617529	0.308764	+	0.951139i	$0.400085\pi$
$12$	0	0
$13$	4.25903	1.18124	0.590621	−	0.806949i	$-0.298883\pi$
$13$	4.25903	1.18124	0.590621	+	0.806949i	$0.298883\pi$
$14$	−0.277846	−0.0742575
$15$	0	0
$16$	1.40149	0.350371
$17$	7.27552	1.76457	0.882286	−	0.470714i	$-0.156004\pi$
$17$	7.27552	1.76457	0.882286	+	0.470714i	$0.156004\pi$
$18$	0	0
$19$	−2.41258	−0.553485	−0.276742	−	0.960944i	$-0.589255\pi$
$19$	−2.41258	−0.553485	−0.276742	+	0.960944i	$0.589255\pi$
$20$	−0.582882	−0.130336
$21$	0	0
$22$	1.40394	0.299321
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.85488	−0.970977
$26$	2.91949	0.572558
$27$	0	0
$28$	0.620202	0.117207
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−4.43663	−0.796843	−0.398421	−	0.917203i	$-0.630442\pi$
$31$	−4.43663	−0.796843	−0.398421	+	0.917203i	$0.630442\pi$
$32$	5.80035	1.02537
$33$	0	0
$34$	4.98723	0.855303
$35$	−0.154407	−0.0260995
$36$	0	0
$37$	0.231841	0.0381145	0.0190572	−	0.999818i	$-0.493934\pi$
$37$	0.231841	0.0381145	0.0190572	+	0.999818i	$0.493934\pi$
$38$	−1.65378	−0.268279
$39$	0	0
$40$	−0.921809	−0.145751
$41$	7.25515	1.13306	0.566532	−	0.824040i	$-0.308285\pi$
$41$	7.25515	1.13306	0.566532	+	0.824040i	$0.308285\pi$
$42$	0	0
$43$	−0.152414	−0.0232430	−0.0116215	−	0.999932i	$-0.503699\pi$
$43$	−0.152414	−0.0232430	−0.0116215	+	0.999932i	$0.503699\pi$
$44$	−3.13385	−0.472445
$45$	0	0
$46$	−0.685481	−0.101069
$47$	6.36400	0.928285	0.464143	−	0.885760i	$-0.346362\pi$
$47$	6.36400	0.928285	0.464143	+	0.885760i	$0.346362\pi$
$48$	0	0
$49$	−6.83571	−0.976530
$50$	−3.32793	−0.470641
$51$	0	0
$52$	−6.51681	−0.903719
$53$	−10.1639	−1.39611	−0.698056	−	0.716043i	$-0.745951\pi$
$53$	−10.1639	−1.39611	−0.698056	+	0.716043i	$0.745951\pi$
$54$	0	0
$55$	0.780208	0.105203
$56$	0.980829	0.131069
$57$	0	0
$58$	−0.685481	−0.0900081
$59$	2.00392	0.260888	0.130444	−	0.991456i	$-0.458360\pi$
$59$	2.00392	0.260888	0.130444	+	0.991456i	$0.458360\pi$
$60$	0	0
$61$	1.35879	0.173975	0.0869876	−	0.996209i	$-0.472276\pi$
$61$	1.35879	0.173975	0.0869876	+	0.996209i	$0.472276\pi$
$62$	−3.04123	−0.386236
$63$	0	0
$64$	1.17306	0.146632
$65$	1.62244	0.201238
$66$	0	0
$67$	11.4500	1.39885	0.699423	−	0.714708i	$-0.253441\pi$
$67$	11.4500	1.39885	0.699423	+	0.714708i	$0.253441\pi$
$68$	−11.1324	−1.35000
$69$	0	0
$70$	−0.105843	−0.0126506
$71$	2.73384	0.324447	0.162223	−	0.986754i	$-0.448133\pi$
$71$	2.73384	0.324447	0.162223	+	0.986754i	$0.448133\pi$
$72$	0	0
$73$	−6.65340	−0.778721	−0.389361	−	0.921085i	$-0.627304\pi$
$73$	−6.65340	−0.778721	−0.389361	+	0.921085i	$0.627304\pi$
$74$	0.158923	0.0184744
$75$	0	0
$76$	3.69153	0.423448
$77$	−0.830161	−0.0946057
$78$	0	0
$79$	−3.74935	−0.421835	−0.210918	−	0.977504i	$-0.567645\pi$
$79$	−3.74935	−0.421835	−0.210918	+	0.977504i	$0.567645\pi$
$80$	0.533882	0.0596898
$81$	0	0
$82$	4.97327	0.549206
$83$	−7.16171	−0.786100	−0.393050	−	0.919517i	$-0.628580\pi$
$83$	−7.16171	−0.786100	−0.393050	+	0.919517i	$0.628580\pi$
$84$	0	0
$85$	2.77154	0.300615
$86$	−0.104477	−0.0112661
$87$	0	0
$88$	−4.95608	−0.528319
$89$	10.1567	1.07661	0.538305	−	0.842750i	$-0.319065\pi$
$89$	10.1567	1.07661	0.538305	+	0.842750i	$0.319065\pi$
$90$	0	0
$91$	−1.72631	−0.180967
$92$	1.53012	0.159526
$93$	0	0
$94$	4.36240	0.449948
$95$	−0.919050	−0.0942925
$96$	0	0
$97$	14.6780	1.49033	0.745164	−	0.666881i	$-0.232371\pi$
$97$	14.6780	1.49033	0.745164	+	0.666881i	$0.232371\pi$
$98$	−4.68575	−0.473332
$99$	0	0
$100$	7.42854	0.742854
$101$	9.09294	0.904782	0.452391	−	0.891820i	$-0.350571\pi$
$101$	9.09294	0.904782	0.452391	+	0.891820i	$0.350571\pi$
$102$	0	0
$103$	−7.02304	−0.692000	−0.346000	−	0.938234i	$-0.612460\pi$
$103$	−7.02304	−0.692000	−0.346000	+	0.938234i	$0.612460\pi$
$104$	−10.3061	−1.01060
$105$	0	0
$106$	−6.96713	−0.676707
$107$	−1.08274	−0.104673	−0.0523364	−	0.998630i	$-0.516667\pi$
$107$	−1.08274	−0.104673	−0.0523364	+	0.998630i	$0.516667\pi$
$108$	0	0
$109$	10.8562	1.03984	0.519918	−	0.854216i	$-0.325963\pi$
$109$	10.8562	1.03984	0.519918	+	0.854216i	$0.325963\pi$
$110$	0.534817	0.0509928
$111$	0	0
$112$	−0.568064	−0.0536770
$113$	−3.45717	−0.325223	−0.162612	−	0.986690i	$-0.551992\pi$
$113$	−3.45717	−0.325223	−0.162612	+	0.986690i	$0.551992\pi$
$114$	0	0
$115$	−0.380940	−0.0355228
$116$	1.53012	0.142068
$117$	0	0
$118$	1.37365	0.126455
$119$	−2.94899	−0.270333
$120$	0	0
$121$	−6.80524	−0.618658
$122$	0.931425	0.0843273
$123$	0	0
$124$	6.78856	0.609631
$125$	−3.75412	−0.335779
$126$	0	0
$127$	19.4040	1.72182	0.860911	−	0.508755i	$-0.169894\pi$
$127$	19.4040	1.72182	0.860911	+	0.508755i	$0.169894\pi$
$128$	−10.7966	−0.954292
$129$	0	0
$130$	1.11215	0.0975419
$131$	8.65276	0.755995	0.377998	−	0.925807i	$-0.376613\pi$
$131$	8.65276	0.755995	0.377998	+	0.925807i	$0.376613\pi$
$132$	0	0
$133$	0.977893	0.0847941
$134$	7.84879	0.678032
$135$	0	0
$136$	−17.6055	−1.50966
$137$	11.8377	1.01136	0.505681	−	0.862720i	$-0.331241\pi$
$137$	11.8377	1.01136	0.505681	+	0.862720i	$0.331241\pi$
$138$	0	0
$139$	−2.08959	−0.177237	−0.0886183	−	0.996066i	$-0.528245\pi$
$139$	−2.08959	−0.177237	−0.0886183	+	0.996066i	$0.528245\pi$
$140$	0.236260	0.0199676
$141$	0	0
$142$	1.87399	0.157262
$143$	8.72297	0.729451
$144$	0	0
$145$	−0.380940	−0.0316353
$146$	−4.56078	−0.377453
$147$	0	0
$148$	−0.354744	−0.0291598
$149$	−1.89443	−0.155198	−0.0775990	−	0.996985i	$-0.524725\pi$
$149$	−1.89443	−0.155198	−0.0775990	+	0.996985i	$0.524725\pi$
$150$	0	0
$151$	9.08912	0.739662	0.369831	−	0.929099i	$-0.379416\pi$
$151$	9.08912	0.739662	0.369831	+	0.929099i	$0.379416\pi$
$152$	5.83804	0.473527
$153$	0	0
$154$	−0.569060	−0.0458561
$155$	−1.69009	−0.135751
$156$	0	0
$157$	11.2110	0.894736	0.447368	−	0.894350i	$-0.352361\pi$
$157$	11.2110	0.894736	0.447368	+	0.894350i	$0.352361\pi$
$158$	−2.57011	−0.204467
$159$	0	0
$160$	2.20958	0.174683
$161$	0.405330	0.0319445
$162$	0	0
$163$	1.75880	0.137760	0.0688800	−	0.997625i	$-0.478057\pi$
$163$	1.75880	0.137760	0.0688800	+	0.997625i	$0.478057\pi$
$164$	−11.1012	−0.866860
$165$	0	0
$166$	−4.90922	−0.381029
$167$	−2.58551	−0.200073	−0.100036	−	0.994984i	$-0.531896\pi$
$167$	−2.58551	−0.200073	−0.100036	+	0.994984i	$0.531896\pi$
$168$	0	0
$169$	5.13935	0.395335
$170$	1.89984	0.145711
$171$	0	0
$172$	0.233211	0.0177822
$173$	−1.81880	−0.138281	−0.0691405	−	0.997607i	$-0.522026\pi$
$173$	−1.81880	−0.138281	−0.0691405	+	0.997607i	$0.522026\pi$
$174$	0	0
$175$	1.96783	0.148754
$176$	2.87040	0.216364
$177$	0	0
$178$	6.96224	0.521842
$179$	1.55527	0.116246	0.0581230	−	0.998309i	$-0.481488\pi$
$179$	1.55527	0.116246	0.0581230	+	0.998309i	$0.481488\pi$
$180$	0	0
$181$	12.3318	0.916613	0.458306	−	0.888794i	$-0.348456\pi$
$181$	12.3318	0.916613	0.458306	+	0.888794i	$0.348456\pi$
$182$	−1.18336	−0.0877162
$183$	0	0
$184$	2.41983	0.178392
$185$	0.0883176	0.00649324
$186$	0	0
$187$	14.9011	1.08967
$188$	−9.73766	−0.710192
$189$	0	0
$190$	−0.629991	−0.0457044
$191$	−14.5240	−1.05092	−0.525459	−	0.850819i	$-0.676106\pi$
$191$	−14.5240	−1.05092	−0.525459	+	0.850819i	$0.676106\pi$
$192$	0	0
$193$	18.8358	1.35583	0.677916	−	0.735139i	$-0.262883\pi$
$193$	18.8358	1.35583	0.677916	+	0.735139i	$0.262883\pi$
$194$	10.0615	0.722374
$195$	0	0
$196$	10.4594	0.747102
$197$	−0.504424	−0.0359387	−0.0179694	−	0.999839i	$-0.505720\pi$
$197$	−0.504424	−0.0359387	−0.0179694	+	0.999839i	$0.505720\pi$
$198$	0	0
$199$	−9.59737	−0.680339	−0.340170	−	0.940364i	$-0.610485\pi$
$199$	−9.59737	−0.680339	−0.340170	+	0.940364i	$0.610485\pi$
$200$	11.7480	0.830708
$201$	0	0
$202$	6.23304	0.438555
$203$	0.405330	0.0284486
$204$	0	0
$205$	2.76378	0.193031
$206$	−4.81416	−0.335418
$207$	0	0
$208$	5.96897	0.413874
$209$	−4.94124	−0.341793
$210$	0	0
$211$	21.5393	1.48283	0.741414	−	0.671047i	$-0.234155\pi$
$211$	21.5393	1.48283	0.741414	+	0.671047i	$0.234155\pi$
$212$	15.5519	1.06811
$213$	0	0
$214$	−0.742200	−0.0507358
$215$	−0.0580607	−0.00395971
$216$	0	0
$217$	1.79830	0.122077
$218$	7.44172	0.504017
$219$	0	0
$220$	−1.19381	−0.0804865
$221$	30.9867	2.08439
$222$	0	0
$223$	−2.63204	−0.176254	−0.0881272	−	0.996109i	$-0.528088\pi$
$223$	−2.63204	−0.176254	−0.0881272	+	0.996109i	$0.528088\pi$
$224$	−2.35106	−0.157087
$225$	0	0
$226$	−2.36983	−0.157639
$227$	13.1447	0.872445	0.436222	−	0.899839i	$-0.356316\pi$
$227$	13.1447	0.872445	0.436222	+	0.899839i	$0.356316\pi$
$228$	0	0
$229$	18.4205	1.21726	0.608629	−	0.793455i	$-0.291720\pi$
$229$	18.4205	1.21726	0.608629	+	0.793455i	$0.291720\pi$
$230$	−0.261127	−0.0172182
$231$	0	0
$232$	2.41983	0.158869
$233$	24.7411	1.62085	0.810423	−	0.585846i	$-0.199238\pi$
$233$	24.7411	1.62085	0.810423	+	0.585846i	$0.199238\pi$
$234$	0	0
$235$	2.42430	0.158144
$236$	−3.06623	−0.199594
$237$	0	0
$238$	−2.02147	−0.131033
$239$	24.7224	1.59916	0.799581	−	0.600558i	$-0.205055\pi$
$239$	24.7224	1.59916	0.799581	+	0.600558i	$0.205055\pi$
$240$	0	0
$241$	−7.40517	−0.477009	−0.238504	−	0.971141i	$-0.576657\pi$
$241$	−7.40517	−0.477009	−0.238504	+	0.971141i	$0.576657\pi$
$242$	−4.66486	−0.299869
$243$	0	0
$244$	−2.07911	−0.133101
$245$	−2.60399	−0.166363
$246$	0	0
$247$	−10.2753	−0.653800
$248$	10.7359	0.681729
$249$	0	0
$250$	−2.57338	−0.162755
$251$	−25.6232	−1.61732	−0.808662	−	0.588274i	$-0.799808\pi$
$251$	−25.6232	−1.61732	−0.808662	+	0.588274i	$0.799808\pi$
$252$	0	0
$253$	−2.04811	−0.128764
$254$	13.3010	0.834582
$255$	0	0
$256$	−9.74697	−0.609186
$257$	−17.6329	−1.09991	−0.549956	−	0.835193i	$-0.685356\pi$
$257$	−17.6329	−1.09991	−0.549956	+	0.835193i	$0.685356\pi$
$258$	0	0
$259$	−0.0939722	−0.00583915
$260$	−2.48251	−0.153959
$261$	0	0
$262$	5.93131	0.366437
$263$	25.1637	1.55166	0.775829	−	0.630944i	$-0.217332\pi$
$263$	25.1637	1.55166	0.775829	+	0.630944i	$0.217332\pi$
$264$	0	0
$265$	−3.87182	−0.237844
$266$	0.670327	0.0411004
$267$	0	0
$268$	−17.5199	−1.07020
$269$	21.8356	1.33134	0.665671	−	0.746245i	$-0.268145\pi$
$269$	21.8356	1.33134	0.665671	+	0.746245i	$0.268145\pi$
$270$	0	0
$271$	5.93344	0.360431	0.180215	−	0.983627i	$-0.442321\pi$
$271$	5.93344	0.360431	0.180215	+	0.983627i	$0.442321\pi$
$272$	10.1965	0.618255
$273$	0	0
$274$	8.11452	0.490216
$275$	−9.94334	−0.599606
$276$	0	0
$277$	−10.8109	−0.649564	−0.324782	−	0.945789i	$-0.605291\pi$
$277$	−10.8109	−0.649564	−0.324782	+	0.945789i	$0.605291\pi$
$278$	−1.43237	−0.0859080
$279$	0	0
$280$	0.373637	0.0223291
$281$	20.1779	1.20371	0.601857	−	0.798604i	$-0.294428\pi$
$281$	20.1779	1.20371	0.601857	+	0.798604i	$0.294428\pi$
$282$	0	0
$283$	9.56629	0.568657	0.284328	−	0.958727i	$-0.408229\pi$
$283$	9.56629	0.568657	0.284328	+	0.958727i	$0.408229\pi$
$284$	−4.18309	−0.248220
$285$	0	0
$286$	5.97943	0.353571
$287$	−2.94073	−0.173586
$288$	0	0
$289$	35.9331	2.11371
$290$	−0.261127	−0.0153339
$291$	0	0
$292$	10.1805	0.595767
$293$	30.6285	1.78934	0.894668	−	0.446731i	$-0.147412\pi$
$293$	30.6285	1.78934	0.894668	+	0.446731i	$0.147412\pi$
$294$	0	0
$295$	0.763373	0.0444453
$296$	−0.561016	−0.0326084
$297$	0	0
$298$	−1.29860	−0.0752257
$299$	−4.25903	−0.246306
$300$	0	0
$301$	0.0617781	0.00356083
$302$	6.23042	0.358520
$303$	0	0
$304$	−3.38120	−0.193925
$305$	0.517618	0.0296387
$306$	0	0
$307$	−30.5073	−1.74114	−0.870572	−	0.492040i	$-0.836251\pi$
$307$	−30.5073	−1.74114	−0.870572	+	0.492040i	$0.836251\pi$
$308$	1.27024	0.0723788
$309$	0	0
$310$	−1.15853	−0.0657998
$311$	19.3976	1.09994	0.549969	−	0.835185i	$-0.314639\pi$
$311$	19.3976	1.09994	0.549969	+	0.835185i	$0.314639\pi$
$312$	0	0
$313$	−29.2764	−1.65480	−0.827399	−	0.561615i	$-0.810180\pi$
$313$	−29.2764	−1.65480	−0.827399	+	0.561615i	$0.810180\pi$
$314$	7.68494	0.433686
$315$	0	0
$316$	5.73694	0.322728
$317$	3.86042	0.216823	0.108411	−	0.994106i	$-0.465424\pi$
$317$	3.86042	0.216823	0.108411	+	0.994106i	$0.465424\pi$
$318$	0	0
$319$	−2.04811	−0.114672
$320$	0.446864	0.0249805
$321$	0	0
$322$	0.277846	0.0154838
$323$	−17.5528	−0.976664
$324$	0	0
$325$	−20.6771	−1.14696
$326$	1.20563	0.0667734
$327$	0	0
$328$	−17.5562	−0.969380
$329$	−2.57952	−0.142214
$330$	0	0
$331$	−25.0203	−1.37524	−0.687621	−	0.726070i	$-0.741345\pi$
$331$	−25.0203	−1.37524	−0.687621	+	0.726070i	$0.741345\pi$
$332$	10.9582	0.601412
$333$	0	0
$334$	−1.77232	−0.0969769
$335$	4.36178	0.238310
$336$	0	0
$337$	−33.8767	−1.84538	−0.922692	−	0.385539i	$-0.874015\pi$
$337$	−33.8767	−1.84538	−0.922692	+	0.385539i	$0.874015\pi$
$338$	3.52293	0.191622
$339$	0	0
$340$	−4.24077	−0.229988
$341$	−9.08671	−0.492073
$342$	0	0
$343$	5.60803	0.302805
$344$	0.368816	0.0198852
$345$	0	0
$346$	−1.24675	−0.0670260
$347$	−24.6846	−1.32514	−0.662569	−	0.749001i	$-0.730534\pi$
$347$	−24.6846	−1.32514	−0.662569	+	0.749001i	$0.730534\pi$
$348$	0	0
$349$	12.5166	0.670000	0.335000	−	0.942218i	$-0.391264\pi$
$349$	12.5166	0.670000	0.335000	+	0.942218i	$0.391264\pi$
$350$	1.34891	0.0721023
$351$	0	0
$352$	11.8798	0.633193
$353$	33.8638	1.80239	0.901195	−	0.433415i	$-0.142691\pi$
$353$	33.8638	1.80239	0.901195	+	0.433415i	$0.142691\pi$
$354$	0	0
$355$	1.04143	0.0552732
$356$	−15.5410	−0.823669
$357$	0	0
$358$	1.06610	0.0563454
$359$	18.0635	0.953357	0.476678	−	0.879078i	$-0.341841\pi$
$359$	18.0635	0.953357	0.476678	+	0.879078i	$0.341841\pi$
$360$	0	0
$361$	−13.1794	−0.693655
$362$	8.45319	0.444290
$363$	0	0
$364$	2.64146	0.138450
$365$	−2.53455	−0.132664
$366$	0	0
$367$	−9.57309	−0.499711	−0.249856	−	0.968283i	$-0.580383\pi$
$367$	−9.57309	−0.499711	−0.249856	+	0.968283i	$0.580383\pi$
$368$	−1.40149	−0.0730575
$369$	0	0
$370$	0.0605400	0.00314733
$371$	4.11972	0.213885
$372$	0	0
$373$	9.57257	0.495649	0.247824	−	0.968805i	$-0.420284\pi$
$373$	9.57257	0.495649	0.247824	+	0.968805i	$0.420284\pi$
$374$	10.2144	0.528174
$375$	0	0
$376$	−15.3998	−0.794184
$377$	−4.25903	−0.219351
$378$	0	0
$379$	2.22469	0.114274	0.0571372	−	0.998366i	$-0.481803\pi$
$379$	2.22469	0.114274	0.0571372	+	0.998366i	$0.481803\pi$
$380$	1.40625	0.0721392
$381$	0	0
$382$	−9.95591	−0.509388
$383$	−36.0845	−1.84383	−0.921915	−	0.387392i	$-0.873376\pi$
$383$	−36.0845	−1.84383	−0.921915	+	0.387392i	$0.873376\pi$
$384$	0	0
$385$	−0.316242	−0.0161172
$386$	12.9116	0.657183
$387$	0	0
$388$	−22.4591	−1.14019
$389$	10.6296	0.538944	0.269472	−	0.963008i	$-0.413151\pi$
$389$	10.6296	0.538944	0.269472	+	0.963008i	$0.413151\pi$
$390$	0	0
$391$	−7.27552	−0.367939
$392$	16.5412	0.835458
$393$	0	0
$394$	−0.345773	−0.0174198
$395$	−1.42828	−0.0718645
$396$	0	0
$397$	−31.1017	−1.56095	−0.780475	−	0.625187i	$-0.785023\pi$
$397$	−31.1017	−1.56095	−0.780475	+	0.625187i	$0.785023\pi$
$398$	−6.57881	−0.329766
$399$	0	0
$400$	−6.80405	−0.340202
$401$	2.46629	0.123160	0.0615802	−	0.998102i	$-0.480386\pi$
$401$	2.46629	0.123160	0.0615802	+	0.998102i	$0.480386\pi$
$402$	0	0
$403$	−18.8958	−0.941265
$404$	−13.9133	−0.692210
$405$	0	0
$406$	0.277846	0.0137893
$407$	0.474836	0.0235368
$408$	0	0
$409$	5.00550	0.247506	0.123753	−	0.992313i	$-0.460507\pi$
$409$	5.00550	0.247506	0.123753	+	0.992313i	$0.460507\pi$
$410$	1.89452	0.0935636
$411$	0	0
$412$	10.7461	0.529420
$413$	−0.812249	−0.0399682
$414$	0	0
$415$	−2.72818	−0.133921
$416$	24.7039	1.21121
$417$	0	0
$418$	−3.38713	−0.165670
$419$	−19.3411	−0.944874	−0.472437	−	0.881364i	$-0.656626\pi$
$419$	−19.3411	−0.944874	−0.472437	+	0.881364i	$0.656626\pi$
$420$	0	0
$421$	3.05325	0.148806	0.0744032	−	0.997228i	$-0.476295\pi$
$421$	3.05325	0.148806	0.0744032	+	0.997228i	$0.476295\pi$
$422$	14.7648	0.718739
$423$	0	0
$424$	24.5948	1.19443
$425$	−35.3218	−1.71336
$426$	0	0
$427$	−0.550759	−0.0266531
$428$	1.65672	0.0800807
$429$	0	0
$430$	−0.0397995	−0.00191930
$431$	−30.1870	−1.45406	−0.727028	−	0.686607i	$-0.759099\pi$
$431$	−30.1870	−1.45406	−0.727028	+	0.686607i	$0.759099\pi$
$432$	0	0
$433$	6.16536	0.296288	0.148144	−	0.988966i	$-0.452670\pi$
$433$	6.16536	0.296288	0.148144	+	0.988966i	$0.452670\pi$
$434$	1.23270	0.0591715
$435$	0	0
$436$	−16.6112	−0.795534
$437$	2.41258	0.115410
$438$	0	0
$439$	−1.07730	−0.0514166	−0.0257083	−	0.999669i	$-0.508184\pi$
$439$	−1.07730	−0.0514166	−0.0257083	+	0.999669i	$0.508184\pi$
$440$	−1.88797	−0.0900053
$441$	0	0
$442$	21.2408	1.01032
$443$	−3.82763	−0.181856	−0.0909281	−	0.995857i	$-0.528983\pi$
$443$	−3.82763	−0.181856	−0.0909281	+	0.995857i	$0.528983\pi$
$444$	0	0
$445$	3.86910	0.183413
$446$	−1.80421	−0.0854320
$447$	0	0
$448$	−0.475476	−0.0224641
$449$	−2.55571	−0.120611	−0.0603056	−	0.998180i	$-0.519208\pi$
$449$	−2.55571	−0.120611	−0.0603056	+	0.998180i	$0.519208\pi$
$450$	0	0
$451$	14.8594	0.699700
$452$	5.28987	0.248815
$453$	0	0
$454$	9.01045	0.422881
$455$	−0.657622	−0.0308298
$456$	0	0
$457$	27.9350	1.30675	0.653373	−	0.757036i	$-0.273353\pi$
$457$	27.9350	1.30675	0.653373	+	0.757036i	$0.273353\pi$
$458$	12.6269	0.590015
$459$	0	0
$460$	0.582882	0.0271770
$461$	21.0902	0.982269	0.491135	−	0.871084i	$-0.336582\pi$
$461$	21.0902	0.982269	0.491135	+	0.871084i	$0.336582\pi$
$462$	0	0
$463$	12.8599	0.597650	0.298825	−	0.954308i	$-0.403405\pi$
$463$	12.8599	0.597650	0.298825	+	0.954308i	$0.403405\pi$
$464$	−1.40149	−0.0650623
$465$	0	0
$466$	16.9596	0.785637
$467$	2.52024	0.116623	0.0583114	−	0.998298i	$-0.481428\pi$
$467$	2.52024	0.116623	0.0583114	+	0.998298i	$0.481428\pi$
$468$	0	0
$469$	−4.64105	−0.214304
$470$	1.66181	0.0766538
$471$	0	0
$472$	−4.84914	−0.223200
$473$	−0.312161	−0.0143532
$474$	0	0
$475$	11.7128	0.537421
$476$	4.51229	0.206820
$477$	0	0
$478$	16.9468	0.775127
$479$	16.0009	0.731102	0.365551	−	0.930791i	$-0.380881\pi$
$479$	16.0009	0.731102	0.365551	+	0.930791i	$0.380881\pi$
$480$	0	0
$481$	0.987419	0.0450224
$482$	−5.07610	−0.231210
$483$	0	0
$484$	10.4128	0.473309
$485$	5.59145	0.253895
$486$	0	0
$487$	−6.01214	−0.272436	−0.136218	−	0.990679i	$-0.543495\pi$
$487$	−6.01214	−0.272436	−0.136218	+	0.990679i	$0.543495\pi$
$488$	−3.28804	−0.148842
$489$	0	0
$490$	−1.78499	−0.0806376
$491$	−3.60095	−0.162508	−0.0812542	−	0.996693i	$-0.525893\pi$
$491$	−3.60095	−0.162508	−0.0812542	+	0.996693i	$0.525893\pi$
$492$	0	0
$493$	−7.27552	−0.327673
$494$	−7.04350	−0.316902
$495$	0	0
$496$	−6.21787	−0.279191
$497$	−1.10811	−0.0497054
$498$	0	0
$499$	29.8906	1.33809	0.669043	−	0.743224i	$-0.266704\pi$
$499$	29.8906	1.33809	0.669043	+	0.743224i	$0.266704\pi$
$500$	5.74424	0.256890
$501$	0	0
$502$	−17.5642	−0.783930
$503$	−42.3227	−1.88707	−0.943537	−	0.331267i	$-0.892524\pi$
$503$	−42.3227	−1.88707	−0.943537	+	0.331267i	$0.892524\pi$
$504$	0	0
$505$	3.46387	0.154140
$506$	−1.40394	−0.0624128
$507$	0	0
$508$	−29.6903	−1.31729
$509$	6.95265	0.308171	0.154085	−	0.988058i	$-0.450757\pi$
$509$	6.95265	0.308171	0.154085	+	0.988058i	$0.450757\pi$
$510$	0	0
$511$	2.69682	0.119300
$512$	14.9118	0.659015
$513$	0	0
$514$	−12.0870	−0.533137
$515$	−2.67536	−0.117890
$516$	0	0
$517$	13.0342	0.573243
$518$	−0.0644162	−0.00283028
$519$	0	0
$520$	−3.92602	−0.172167
$521$	13.7784	0.603641	0.301821	−	0.953365i	$-0.402406\pi$
$521$	13.7784	0.603641	0.301821	+	0.953365i	$0.402406\pi$
$522$	0	0
$523$	−16.4050	−0.717342	−0.358671	−	0.933464i	$-0.616770\pi$
$523$	−16.4050	−0.717342	−0.358671	+	0.933464i	$0.616770\pi$
$524$	−13.2397	−0.578380
$525$	0	0
$526$	17.2492	0.752101
$527$	−32.2788	−1.40609
$528$	0	0
$529$	1.00000	0.0434783
$530$	−2.65406	−0.115285
$531$	0	0
$532$	−1.49629	−0.0648724
$533$	30.8999	1.33842
$534$	0	0
$535$	−0.412461	−0.0178322
$536$	−27.7071	−1.19677
$537$	0	0
$538$	14.9679	0.645313
$539$	−14.0003	−0.603035
$540$	0	0
$541$	−16.0303	−0.689195	−0.344598	−	0.938750i	$-0.611985\pi$
$541$	−16.0303	−0.689195	−0.344598	+	0.938750i	$0.611985\pi$
$542$	4.06726	0.174704
$543$	0	0
$544$	42.2005	1.80933
$545$	4.13556	0.177148
$546$	0	0
$547$	−22.4551	−0.960112	−0.480056	−	0.877238i	$-0.659384\pi$
$547$	−22.4551	−0.960112	−0.480056	+	0.877238i	$0.659384\pi$
$548$	−18.1130	−0.773751
$549$	0	0
$550$	−6.81597	−0.290634
$551$	2.41258	0.102780
$552$	0	0
$553$	1.51973	0.0646253
$554$	−7.41066	−0.314849
$555$	0	0
$556$	3.19731	0.135596
$557$	4.16780	0.176595	0.0882977	−	0.996094i	$-0.471857\pi$
$557$	4.16780	0.176595	0.0882977	+	0.996094i	$0.471857\pi$
$558$	0	0
$559$	−0.649137	−0.0274556
$560$	−0.216398	−0.00914450
$561$	0	0
$562$	13.8316	0.583450
$563$	−36.6909	−1.54634	−0.773169	−	0.634200i	$-0.781330\pi$
$563$	−36.6909	−1.54634	−0.773169	+	0.634200i	$0.781330\pi$
$564$	0	0
$565$	−1.31698	−0.0554056
$566$	6.55751	0.275633
$567$	0	0
$568$	−6.61541	−0.277577
$569$	16.9007	0.708513	0.354257	−	0.935148i	$-0.384734\pi$
$569$	16.9007	0.708513	0.354257	+	0.935148i	$0.384734\pi$
$570$	0	0
$571$	36.2803	1.51828	0.759141	−	0.650926i	$-0.225619\pi$
$571$	36.2803	1.51828	0.759141	+	0.650926i	$0.225619\pi$
$572$	−13.3472	−0.558073
$573$	0	0
$574$	−2.01582	−0.0841386
$575$	4.85488	0.202463
$576$	0	0
$577$	−25.5514	−1.06372	−0.531860	−	0.846833i	$-0.678507\pi$
$577$	−25.5514	−1.06372	−0.531860	+	0.846833i	$0.678507\pi$
$578$	24.6315	1.02453
$579$	0	0
$580$	0.582882	0.0242029
$581$	2.90286	0.120431
$582$	0	0
$583$	−20.8167	−0.862140
$584$	16.1001	0.666226
$585$	0	0
$586$	20.9953	0.867306
$587$	−13.2110	−0.545278	−0.272639	−	0.962116i	$-0.587897\pi$
$587$	−13.2110	−0.545278	−0.272639	+	0.962116i	$0.587897\pi$
$588$	0	0
$589$	10.7037	0.441040
$590$	0.523278	0.0215430
$591$	0	0
$592$	0.324922	0.0133542
$593$	−6.11339	−0.251047	−0.125523	−	0.992091i	$-0.540061\pi$
$593$	−6.11339	−0.251047	−0.125523	+	0.992091i	$0.540061\pi$
$594$	0	0
$595$	−1.12339	−0.0460544
$596$	2.89870	0.118735
$597$	0	0
$598$	−2.91949	−0.119387
$599$	−15.9128	−0.650181	−0.325090	−	0.945683i	$-0.605395\pi$
$599$	−15.9128	−0.650181	−0.325090	+	0.945683i	$0.605395\pi$
$600$	0	0
$601$	−10.5788	−0.431517	−0.215758	−	0.976447i	$-0.569222\pi$
$601$	−10.5788	−0.431517	−0.215758	+	0.976447i	$0.569222\pi$
$602$	0.0423477	0.00172596
$603$	0	0
$604$	−13.9074	−0.565884
$605$	−2.59239	−0.105396
$606$	0	0
$607$	22.6096	0.917694	0.458847	−	0.888515i	$-0.348263\pi$
$607$	22.6096	0.917694	0.458847	+	0.888515i	$0.348263\pi$
$608$	−13.9938	−0.567524
$609$	0	0
$610$	0.354817	0.0143661
$611$	27.1045	1.09653
$612$	0	0
$613$	−38.7632	−1.56563	−0.782815	−	0.622254i	$-0.786217\pi$
$613$	−38.7632	−1.56563	−0.782815	+	0.622254i	$0.786217\pi$
$614$	−20.9122	−0.843947
$615$	0	0
$616$	2.00885	0.0809388
$617$	−2.04217	−0.0822148	−0.0411074	−	0.999155i	$-0.513089\pi$
$617$	−2.04217	−0.0822148	−0.0411074	+	0.999155i	$0.513089\pi$
$618$	0	0
$619$	26.9437	1.08296	0.541479	−	0.840714i	$-0.317865\pi$
$619$	26.9437	1.08296	0.541479	+	0.840714i	$0.317865\pi$
$620$	2.58603	0.103858
$621$	0	0
$622$	13.2967	0.533149
$623$	−4.11683	−0.164937
$624$	0	0
$625$	22.8443	0.913773
$626$	−20.0684	−0.802094
$627$	0	0
$628$	−17.1541	−0.684525
$629$	1.68676	0.0672557
$630$	0	0
$631$	−40.1935	−1.60008	−0.800039	−	0.599948i	$-0.795188\pi$
$631$	−40.1935	−1.60008	−0.800039	+	0.599948i	$0.795188\pi$
$632$	9.07279	0.360896
$633$	0	0
$634$	2.64625	0.105096
$635$	7.39174	0.293332
$636$	0	0
$637$	−29.1135	−1.15352
$638$	−1.40394	−0.0555826
$639$	0	0
$640$	−4.11285	−0.162575
$641$	27.8057	1.09826	0.549129	−	0.835737i	$-0.314959\pi$
$641$	27.8057	1.09826	0.549129	+	0.835737i	$0.314959\pi$
$642$	0	0
$643$	4.49178	0.177139	0.0885693	−	0.996070i	$-0.471771\pi$
$643$	4.49178	0.177139	0.0885693	+	0.996070i	$0.471771\pi$
$644$	−0.620202	−0.0244394
$645$	0	0
$646$	−12.0321	−0.473397
$647$	12.7241	0.500237	0.250119	−	0.968215i	$-0.419530\pi$
$647$	12.7241	0.500237	0.250119	+	0.968215i	$0.419530\pi$
$648$	0	0
$649$	4.10425	0.161106
$650$	−14.1738	−0.555941
$651$	0	0
$652$	−2.69117	−0.105394
$653$	44.7671	1.75187	0.875936	−	0.482427i	$-0.160245\pi$
$653$	44.7671	1.75187	0.875936	+	0.482427i	$0.160245\pi$
$654$	0	0
$655$	3.29618	0.128793
$656$	10.1680	0.396993
$657$	0	0
$658$	−1.76821	−0.0689322
$659$	17.0118	0.662685	0.331342	−	0.943511i	$-0.392498\pi$
$659$	17.0118	0.662685	0.331342	+	0.943511i	$0.392498\pi$
$660$	0	0
$661$	−9.60435	−0.373566	−0.186783	−	0.982401i	$-0.559806\pi$
$661$	−9.60435	−0.373566	−0.186783	+	0.982401i	$0.559806\pi$
$662$	−17.1510	−0.666591
$663$	0	0
$664$	17.3301	0.672539
$665$	0.372519	0.0144457
$666$	0	0
$667$	1.00000	0.0387202
$668$	3.95613	0.153067
$669$	0	0
$670$	2.98992	0.115511
$671$	2.78295	0.107435
$672$	0	0
$673$	4.07544	0.157097	0.0785483	−	0.996910i	$-0.474972\pi$
$673$	4.07544	0.157097	0.0785483	+	0.996910i	$0.474972\pi$
$674$	−23.2219	−0.894473
$675$	0	0
$676$	−7.86381	−0.302454
$677$	41.3737	1.59012	0.795061	−	0.606529i	$-0.207439\pi$
$677$	41.3737	1.59012	0.795061	+	0.606529i	$0.207439\pi$
$678$	0	0
$679$	−5.94945	−0.228319
$680$	−6.70664	−0.257188
$681$	0	0
$682$	−6.22877	−0.238512
$683$	13.2590	0.507342	0.253671	−	0.967291i	$-0.418362\pi$
$683$	13.2590	0.507342	0.253671	+	0.967291i	$0.418362\pi$
$684$	0	0
$685$	4.50945	0.172297
$686$	3.84420	0.146772
$687$	0	0
$688$	−0.213606	−0.00814367
$689$	−43.2882	−1.64915
$690$	0	0
$691$	48.7952	1.85626	0.928129	−	0.372260i	$-0.121417\pi$
$691$	48.7952	1.85626	0.928129	+	0.372260i	$0.121417\pi$
$692$	2.78298	0.105793
$693$	0	0
$694$	−16.9208	−0.642305
$695$	−0.796008	−0.0301943
$696$	0	0
$697$	52.7850	1.99937
$698$	8.57991	0.324755
$699$	0	0
$700$	−3.01101	−0.113805
$701$	−38.1143	−1.43956	−0.719779	−	0.694203i	$-0.755757\pi$
$701$	−38.1143	−1.43956	−0.719779	+	0.694203i	$0.755757\pi$
$702$	0	0
$703$	−0.559336	−0.0210958
$704$	2.40255	0.0905496
$705$	0	0
$706$	23.2130	0.873633
$707$	−3.68564	−0.138613
$708$	0	0
$709$	−27.0213	−1.01481	−0.507403	−	0.861709i	$-0.669394\pi$
$709$	−27.0213	−1.01481	−0.507403	+	0.861709i	$0.669394\pi$
$710$	0.713879	0.0267914
$711$	0	0
$712$	−24.5775	−0.921082
$713$	4.43663	0.166153
$714$	0	0
$715$	3.32293	0.124270
$716$	−2.37974	−0.0889349
$717$	0	0
$718$	12.3822	0.462100
$719$	−14.5423	−0.542336	−0.271168	−	0.962532i	$-0.587410\pi$
$719$	−14.5423	−0.542336	−0.271168	+	0.962532i	$0.587410\pi$
$720$	0	0
$721$	2.84665	0.106015
$722$	−9.03426	−0.336220
$723$	0	0
$724$	−18.8690	−0.701262
$725$	4.85488	0.180306
$726$	0	0
$727$	−36.2248	−1.34350	−0.671751	−	0.740777i	$-0.734457\pi$
$727$	−36.2248	−1.34350	−0.671751	+	0.740777i	$0.734457\pi$
$728$	4.17738	0.154824
$729$	0	0
$730$	−1.73738	−0.0643034
$731$	−1.10889	−0.0410139
$732$	0	0
$733$	37.5280	1.38613	0.693065	−	0.720875i	$-0.256260\pi$
$733$	37.5280	1.38613	0.693065	+	0.720875i	$0.256260\pi$
$734$	−6.56217	−0.242214
$735$	0	0
$736$	−5.80035	−0.213804
$737$	23.4510	0.863827
$738$	0	0
$739$	26.3674	0.969940	0.484970	−	0.874531i	$-0.338830\pi$
$739$	26.3674	0.969940	0.484970	+	0.874531i	$0.338830\pi$
$740$	−0.135136	−0.00496770
$741$	0	0
$742$	2.82399	0.103672
$743$	−2.01265	−0.0738371	−0.0369185	−	0.999318i	$-0.511754\pi$
$743$	−2.01265	−0.0738371	−0.0369185	+	0.999318i	$0.511754\pi$
$744$	0	0
$745$	−0.721665	−0.0264398
$746$	6.56181	0.240245
$747$	0	0
$748$	−22.8004	−0.833663
$749$	0.438869	0.0160359
$750$	0	0
$751$	−5.58774	−0.203899	−0.101950	−	0.994790i	$-0.532508\pi$
$751$	−5.58774	−0.203899	−0.101950	+	0.994790i	$0.532508\pi$
$752$	8.91906	0.325245
$753$	0	0
$754$	−2.91949	−0.106321
$755$	3.46241	0.126010
$756$	0	0
$757$	−1.47747	−0.0536994	−0.0268497	−	0.999639i	$-0.508548\pi$
$757$	−1.47747	−0.0536994	−0.0268497	+	0.999639i	$0.508548\pi$
$758$	1.52498	0.0553898
$759$	0	0
$760$	2.22394	0.0806709
$761$	−52.4805	−1.90242	−0.951208	−	0.308551i	$-0.900156\pi$
$761$	−52.4805	−1.90242	−0.951208	+	0.308551i	$0.900156\pi$
$762$	0	0
$763$	−4.40034	−0.159303
$764$	22.2234	0.804013
$765$	0	0
$766$	−24.7352	−0.893720
$767$	8.53476	0.308172
$768$	0	0
$769$	−16.8920	−0.609140	−0.304570	−	0.952490i	$-0.598513\pi$
$769$	−16.8920	−0.609140	−0.304570	+	0.952490i	$0.598513\pi$
$770$	−0.216778	−0.00781213
$771$	0	0
$772$	−28.8210	−1.03729
$773$	16.1080	0.579363	0.289682	−	0.957123i	$-0.406451\pi$
$773$	16.1080	0.579363	0.289682	+	0.957123i	$0.406451\pi$
$774$	0	0
$775$	21.5393	0.773716
$776$	−35.5183	−1.27503
$777$	0	0
$778$	7.28642	0.261231
$779$	−17.5037	−0.627134
$780$	0	0
$781$	5.59920	0.200355
$782$	−4.98723	−0.178343
$783$	0	0
$784$	−9.58014	−0.342148
$785$	4.27072	0.152429
$786$	0	0
$787$	33.4200	1.19129	0.595647	−	0.803246i	$-0.296896\pi$
$787$	33.4200	1.19129	0.595647	+	0.803246i	$0.296896\pi$
$788$	0.771827	0.0274952
$789$	0	0
$790$	−0.979058	−0.0348333
$791$	1.40130	0.0498244
$792$	0	0
$793$	5.78713	0.205507
$794$	−21.3196	−0.756605
$795$	0	0
$796$	14.6851	0.520499
$797$	−15.4354	−0.546751	−0.273376	−	0.961907i	$-0.588140\pi$
$797$	−15.4354	−0.546751	−0.273376	+	0.961907i	$0.588140\pi$
$798$	0	0
$799$	46.3014	1.63803
$800$	−28.1600	−0.995607
$801$	0	0
$802$	1.69059	0.0596969
$803$	−13.6269	−0.480883
$804$	0	0
$805$	0.154407	0.00544211
$806$	−12.9527	−0.456239
$807$	0	0
$808$	−22.0034	−0.774075
$809$	−35.4508	−1.24638	−0.623191	−	0.782070i	$-0.714164\pi$
$809$	−35.4508	−1.24638	−0.623191	+	0.782070i	$0.714164\pi$
$810$	0	0
$811$	−47.1404	−1.65532	−0.827662	−	0.561227i	$-0.810329\pi$
$811$	−47.1404	−1.65532	−0.827662	+	0.561227i	$0.810329\pi$
$812$	−0.620202	−0.0217648
$813$	0	0
$814$	0.325491	0.0114085
$815$	0.669998	0.0234690
$816$	0	0
$817$	0.367712	0.0128646
$818$	3.43117	0.119968
$819$	0	0
$820$	−4.22890	−0.147680
$821$	−45.2666	−1.57982	−0.789908	−	0.613225i	$-0.789872\pi$
$821$	−45.2666	−1.57982	−0.789908	+	0.613225i	$0.789872\pi$
$822$	0	0
$823$	5.85033	0.203930	0.101965	−	0.994788i	$-0.467487\pi$
$823$	5.85033	0.203930	0.101965	+	0.994788i	$0.467487\pi$
$824$	16.9945	0.592033
$825$	0	0
$826$	−0.556781	−0.0193729
$827$	7.19163	0.250077	0.125039	−	0.992152i	$-0.460095\pi$
$827$	7.19163	0.250077	0.125039	+	0.992152i	$0.460095\pi$
$828$	0	0
$829$	−4.12571	−0.143292	−0.0716460	−	0.997430i	$-0.522825\pi$
$829$	−4.12571	−0.143292	−0.0716460	+	0.997430i	$0.522825\pi$
$830$	−1.87012	−0.0649127
$831$	0	0
$832$	4.99609	0.173208
$833$	−49.7333	−1.72316
$834$	0	0
$835$	−0.984924	−0.0340847
$836$	7.56067	0.261491
$837$	0	0
$838$	−13.2580	−0.457988
$839$	13.2562	0.457656	0.228828	−	0.973467i	$-0.426511\pi$
$839$	13.2562	0.457656	0.228828	+	0.973467i	$0.426511\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	2.09295	0.0721277
$843$	0	0
$844$	−32.9577	−1.13445
$845$	1.95779	0.0673499
$846$	0	0
$847$	2.75837	0.0947787
$848$	−14.2445	−0.489158
$849$	0	0
$850$	−24.2124	−0.830479
$851$	−0.231841	−0.00794741
$852$	0	0
$853$	37.0406	1.26824	0.634122	−	0.773233i	$-0.281362\pi$
$853$	37.0406	1.26824	0.634122	+	0.773233i	$0.281362\pi$
$854$	−0.377535	−0.0129190
$855$	0	0
$856$	2.62005	0.0895516
$857$	45.7412	1.56249	0.781245	−	0.624224i	$-0.214585\pi$
$857$	45.7412	1.56249	0.781245	+	0.624224i	$0.214585\pi$
$858$	0	0
$859$	−4.79929	−0.163750	−0.0818748	−	0.996643i	$-0.526091\pi$
$859$	−4.79929	−0.163750	−0.0818748	+	0.996643i	$0.526091\pi$
$860$	0.0888396	0.00302941
$861$	0	0
$862$	−20.6926	−0.704793
$863$	46.5349	1.58407	0.792033	−	0.610478i	$-0.209023\pi$
$863$	46.5349	1.58407	0.792033	+	0.610478i	$0.209023\pi$
$864$	0	0
$865$	−0.692855	−0.0235578
$866$	4.22624	0.143613
$867$	0	0
$868$	−2.75161	−0.0933957
$869$	−7.67909	−0.260495
$870$	0	0
$871$	48.7661	1.65238
$872$	−26.2701	−0.889619
$873$	0	0
$874$	1.65378	0.0559400
$875$	1.52166	0.0514414
$876$	0	0
$877$	32.9850	1.11383	0.556913	−	0.830571i	$-0.311986\pi$
$877$	32.9850	1.11383	0.556913	+	0.830571i	$0.311986\pi$
$878$	−0.738467	−0.0249220
$879$	0	0
$880$	1.09345	0.0368602
$881$	23.0207	0.775587	0.387793	−	0.921746i	$-0.373237\pi$
$881$	23.0207	0.775587	0.387793	+	0.921746i	$0.373237\pi$
$882$	0	0
$883$	−14.1135	−0.474956	−0.237478	−	0.971393i	$-0.576321\pi$
$883$	−14.1135	−0.474956	−0.237478	+	0.971393i	$0.576321\pi$
$884$	−47.4132	−1.59468
$885$	0	0
$886$	−2.62377	−0.0881472
$887$	21.9389	0.736635	0.368317	−	0.929700i	$-0.379934\pi$
$887$	21.9389	0.736635	0.368317	+	0.929700i	$0.379934\pi$
$888$	0	0
$889$	−7.86501	−0.263784
$890$	2.65220	0.0889018
$891$	0	0
$892$	4.02732	0.134845
$893$	−15.3537	−0.513792
$894$	0	0
$895$	0.592463	0.0198038
$896$	4.37618	0.146198
$897$	0	0
$898$	−1.75189	−0.0584613
$899$	4.43663	0.147970
$900$	0	0
$901$	−73.9473	−2.46354
$902$	10.1858	0.339150
$903$	0	0
$904$	8.36576	0.278241
$905$	4.69766	0.156156
$906$	0	0
$907$	−59.8533	−1.98740	−0.993699	−	0.112084i	$-0.964247\pi$
$907$	−59.8533	−1.98740	−0.993699	+	0.112084i	$0.964247\pi$
$908$	−20.1129	−0.667471
$909$	0	0
$910$	−0.450788	−0.0149435
$911$	29.5116	0.977764	0.488882	−	0.872350i	$-0.337405\pi$
$911$	29.5116	0.977764	0.488882	+	0.872350i	$0.337405\pi$
$912$	0	0
$913$	−14.6680	−0.485439
$914$	19.1489	0.633391
$915$	0	0
$916$	−28.1854	−0.931273
$917$	−3.50723	−0.115819
$918$	0	0
$919$	−7.57158	−0.249763	−0.124882	−	0.992172i	$-0.539855\pi$
$919$	−7.57158	−0.249763	−0.124882	+	0.992172i	$0.539855\pi$
$920$	0.921809	0.0303912
$921$	0	0
$922$	14.4569	0.476114
$923$	11.6435	0.383250
$924$	0	0
$925$	−1.12556	−0.0370083
$926$	8.81522	0.289686
$927$	0	0
$928$	−5.80035	−0.190406
$929$	−20.9609	−0.687704	−0.343852	−	0.939024i	$-0.611732\pi$
$929$	−20.9609	−0.687704	−0.343852	+	0.939024i	$0.611732\pi$
$930$	0	0
$931$	16.4917	0.540494
$932$	−37.8568	−1.24004
$933$	0	0
$934$	1.72758	0.0565281
$935$	5.67641	0.185639
$936$	0	0
$937$	4.10207	0.134009	0.0670044	−	0.997753i	$-0.478656\pi$
$937$	4.10207	0.134009	0.0670044	+	0.997753i	$0.478656\pi$
$938$	−3.18135	−0.103875
$939$	0	0
$940$	−3.70947	−0.120989
$941$	26.3583	0.859254	0.429627	−	0.903006i	$-0.358645\pi$
$941$	26.3583	0.859254	0.429627	+	0.903006i	$0.358645\pi$
$942$	0	0
$943$	−7.25515	−0.236260
$944$	2.80846	0.0914077
$945$	0	0
$946$	−0.213981	−0.00695711
$947$	−42.4089	−1.37810	−0.689052	−	0.724712i	$-0.741973\pi$
$947$	−42.4089	−1.37810	−0.689052	+	0.724712i	$0.741973\pi$
$948$	0	0
$949$	−28.3370	−0.919859
$950$	8.02891	0.260492
$951$	0	0
$952$	7.13604	0.231280
$953$	−11.9842	−0.388206	−0.194103	−	0.980981i	$-0.562180\pi$
$953$	−11.9842	−0.388206	−0.194103	+	0.980981i	$0.562180\pi$
$954$	0	0
$955$	−5.53276	−0.179036
$956$	−37.8282	−1.22345
$957$	0	0
$958$	10.9683	0.354371
$959$	−4.79818	−0.154941
$960$	0	0
$961$	−11.3163	−0.365042
$962$	0.676857	0.0218227
$963$	0	0
$964$	11.3308	0.364939
$965$	7.17532	0.230982
$966$	0	0
$967$	43.6153	1.40257	0.701287	−	0.712879i	$-0.252609\pi$
$967$	43.6153	1.40257	0.701287	+	0.712879i	$0.252609\pi$
$968$	16.4675	0.529286
$969$	0	0
$970$	3.83283	0.123065
$971$	−51.1912	−1.64280	−0.821402	−	0.570349i	$-0.806808\pi$
$971$	−51.1912	−1.64280	−0.821402	+	0.570349i	$0.806808\pi$
$972$	0	0
$973$	0.846973	0.0271527
$974$	−4.12121	−0.132052
$975$	0	0
$976$	1.90432	0.0609559
$977$	40.7254	1.30292	0.651460	−	0.758683i	$-0.274157\pi$
$977$	40.7254	1.30292	0.651460	+	0.758683i	$0.274157\pi$
$978$	0	0
$979$	20.8021	0.664838
$980$	3.98441	0.127277
$981$	0	0
$982$	−2.46838	−0.0787692
$983$	−62.2725	−1.98618	−0.993092	−	0.117342i	$-0.962563\pi$
$983$	−62.2725	−1.98618	−0.993092	+	0.117342i	$0.962563\pi$
$984$	0	0
$985$	−0.192155	−0.00612257
$986$	−4.98723	−0.158826
$987$	0	0
$988$	15.7224	0.500195
$989$	0.152414	0.00484649
$990$	0	0
$991$	51.7484	1.64384	0.821921	−	0.569601i	$-0.192902\pi$
$991$	51.7484	1.64384	0.821921	+	0.569601i	$0.192902\pi$
$992$	−25.7340	−0.817055
$993$	0	0
$994$	−0.759586	−0.0240926
$995$	−3.65602	−0.115904
$996$	0	0
$997$	−40.0808	−1.26937	−0.634686	−	0.772770i	$-0.718871\pi$
$997$	−40.0808	−1.26937	−0.634686	+	0.772770i	$0.718871\pi$
$998$	20.4894	0.648581
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.l.1.5		10
3.2	odd	2		667.2.a.a.1.6	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.a.1.6	✓	10	3.2	odd	2
6003.2.a.l.1.5		10	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q+0.685481 q^{2} -1.53012 q^{4} +0.380940 q^{5} -0.405330 q^{7} -2.41983 q^{8} +O(q^{10})\)	\(q+0.685481 q^{2} -1.53012 q^{4} +0.380940 q^{5} -0.405330 q^{7} -2.41983 q^{8} +0.261127 q^{10} +2.04811 q^{11} +4.25903 q^{13} -0.277846 q^{14} +1.40149 q^{16} +7.27552 q^{17} -2.41258 q^{19} -0.582882 q^{20} +1.40394 q^{22} -1.00000 q^{23} -4.85488 q^{25} +2.91949 q^{26} +0.620202 q^{28} -1.00000 q^{29} -4.43663 q^{31} +5.80035 q^{32} +4.98723 q^{34} -0.154407 q^{35} +0.231841 q^{37} -1.65378 q^{38} -0.921809 q^{40} +7.25515 q^{41} -0.152414 q^{43} -3.13385 q^{44} -0.685481 q^{46} +6.36400 q^{47} -6.83571 q^{49} -3.32793 q^{50} -6.51681 q^{52} -10.1639 q^{53} +0.780208 q^{55} +0.980829 q^{56} -0.685481 q^{58} +2.00392 q^{59} +1.35879 q^{61} -3.04123 q^{62} +1.17306 q^{64} +1.62244 q^{65} +11.4500 q^{67} -11.1324 q^{68} -0.105843 q^{70} +2.73384 q^{71} -6.65340 q^{73} +0.158923 q^{74} +3.69153 q^{76} -0.830161 q^{77} -3.74935 q^{79} +0.533882 q^{80} +4.97327 q^{82} -7.16171 q^{83} +2.77154 q^{85} -0.104477 q^{86} -4.95608 q^{88} +10.1567 q^{89} -1.72631 q^{91} +1.53012 q^{92} +4.36240 q^{94} -0.919050 q^{95} +14.6780 q^{97} -4.68575 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 3 q^{2} + 9 q^{4} + 10 q^{5} + q^{7} + 9 q^{8}+O(q^{10}) \) 10 * q + 3 * q^2 + 9 * q^4 + 10 * q^5 + q^7 + 9 * q^8	\( 10 q + 3 q^{2} + 9 q^{4} + 10 q^{5} + q^{7} + 9 q^{8} - 6 q^{10} - 13 q^{13} + 12 q^{14} - 5 q^{16} + 22 q^{17} - 2 q^{19} - 3 q^{20} + 3 q^{22} - 10 q^{23} + 10 q^{25} + 25 q^{26} + 19 q^{28} - 10 q^{29} - 22 q^{31} + 31 q^{32} + 13 q^{34} + 15 q^{35} - 9 q^{37} + 10 q^{38} - 6 q^{40} + 25 q^{41} + 3 q^{43} + 27 q^{44} - 3 q^{46} + 17 q^{47} + 17 q^{49} - 2 q^{50} - 18 q^{52} + 43 q^{53} - 11 q^{55} + 7 q^{56} - 3 q^{58} + 7 q^{59} - 6 q^{61} - 3 q^{62} + 33 q^{64} - 11 q^{65} + 11 q^{67} + 51 q^{68} + 34 q^{70} + 17 q^{71} - 44 q^{73} - 9 q^{74} + 24 q^{76} + 71 q^{77} + 5 q^{79} - 38 q^{80} + 33 q^{82} + 32 q^{83} + 16 q^{85} + 9 q^{86} + 18 q^{88} + 10 q^{89} - 3 q^{91} - 9 q^{92} + 47 q^{94} + 8 q^{95} + 6 q^{97} + 73 q^{98}+O(q^{100}) \) 10 * q + 3 * q^2 + 9 * q^4 + 10 * q^5 + q^7 + 9 * q^8 - 6 * q^10 - 13 * q^13 + 12 * q^14 - 5 * q^16 + 22 * q^17 - 2 * q^19 - 3 * q^20 + 3 * q^22 - 10 * q^23 + 10 * q^25 + 25 * q^26 + 19 * q^28 - 10 * q^29 - 22 * q^31 + 31 * q^32 + 13 * q^34 + 15 * q^35 - 9 * q^37 + 10 * q^38 - 6 * q^40 + 25 * q^41 + 3 * q^43 + 27 * q^44 - 3 * q^46 + 17 * q^47 + 17 * q^49 - 2 * q^50 - 18 * q^52 + 43 * q^53 - 11 * q^55 + 7 * q^56 - 3 * q^58 + 7 * q^59 - 6 * q^61 - 3 * q^62 + 33 * q^64 - 11 * q^65 + 11 * q^67 + 51 * q^68 + 34 * q^70 + 17 * q^71 - 44 * q^73 - 9 * q^74 + 24 * q^76 + 71 * q^77 + 5 * q^79 - 38 * q^80 + 33 * q^82 + 32 * q^83 + 16 * q^85 + 9 * q^86 + 18 * q^88 + 10 * q^89 - 3 * q^91 - 9 * q^92 + 47 * q^94 + 8 * q^95 + 6 * q^97 + 73 * q^98

Introduction

L-functions

Modular forms

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Database

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