LMFDB - Embedded newform 6003.2.a.s.1.20

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 33 x^{18} + 64 x^{17} + 453 x^{16} - 846 x^{15} - 3353 x^{14} + 5985 x^{13} + \cdots - 256 $ x^20 - 2x^19 - 33x^18 + 64x^17 + 453x^16 - 846x^15 - 3353x^14 + 5985x^13 + 14484x^12 - 24566x^11 - 36791x^10 + 59410x^9 + 52109x^8 - 82362x^7 - 34967x^6 + 60661x^5 + 5201x^4 - 19624x^3 + 2768x^2 + 1408*x - 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Root			$-2.80084$ 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)

$f(q)$	$=$	$q+2.80084 q^{2} +5.84471 q^{4} +3.48103 q^{5} -1.41384 q^{7} +10.7684 q^{8} +O(q^{10})$	\(q+2.80084 q^{2} +5.84471 q^{4} +3.48103 q^{5} -1.41384 q^{7} +10.7684 q^{8} +9.74980 q^{10} +4.43439 q^{11} -2.37735 q^{13} -3.95993 q^{14} +18.4712 q^{16} +3.28679 q^{17} -3.46248 q^{19} +20.3456 q^{20} +12.4200 q^{22} -1.00000 q^{23} +7.11755 q^{25} -6.65859 q^{26} -8.26345 q^{28} -1.00000 q^{29} -4.18385 q^{31} +30.1980 q^{32} +9.20579 q^{34} -4.92160 q^{35} -10.7653 q^{37} -9.69784 q^{38} +37.4851 q^{40} -3.71341 q^{41} -7.08099 q^{43} +25.9177 q^{44} -2.80084 q^{46} -1.63781 q^{47} -5.00107 q^{49} +19.9351 q^{50} -13.8949 q^{52} -6.97182 q^{53} +15.4362 q^{55} -15.2248 q^{56} -2.80084 q^{58} -12.2928 q^{59} +5.25185 q^{61} -11.7183 q^{62} +47.6374 q^{64} -8.27563 q^{65} -11.9828 q^{67} +19.2103 q^{68} -13.7846 q^{70} -12.3124 q^{71} +8.68606 q^{73} -30.1520 q^{74} -20.2372 q^{76} -6.26950 q^{77} +9.51988 q^{79} +64.2986 q^{80} -10.4007 q^{82} -2.12581 q^{83} +11.4414 q^{85} -19.8327 q^{86} +47.7513 q^{88} +11.0119 q^{89} +3.36119 q^{91} -5.84471 q^{92} -4.58725 q^{94} -12.0530 q^{95} +12.3407 q^{97} -14.0072 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) $ 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8	$ 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8} + 7 q^{10} + 21 q^{13} + q^{14} + 58 q^{16} + 4 q^{17} + 7 q^{19} + 20 q^{20} + 7 q^{22} - 20 q^{23} + 47 q^{25} - 8 q^{26} + 11 q^{28} - 20 q^{29} + 28 q^{31} - 14 q^{32} + 16 q^{34} - 9 q^{35} + 14 q^{37} + 20 q^{38} + 34 q^{40} - 7 q^{41} + 3 q^{43} + q^{44} + 2 q^{46} - 3 q^{47} + 35 q^{49} + 24 q^{50} + 73 q^{52} + 19 q^{53} + 29 q^{55} + 30 q^{56} + 2 q^{58} - 20 q^{59} + 15 q^{61} - 12 q^{62} + 82 q^{64} + 28 q^{65} + 20 q^{67} + 23 q^{68} - 24 q^{70} - 63 q^{71} + 19 q^{73} - 16 q^{74} - 44 q^{76} + 7 q^{77} + 32 q^{79} + 56 q^{80} - 20 q^{82} + 21 q^{83} + 4 q^{85} + 6 q^{86} + 55 q^{88} + 13 q^{89} + 70 q^{91} - 30 q^{92} - 12 q^{94} - 9 q^{95} - 9 q^{97} - 31 q^{98}+O(q^{100}) $ 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8 + 7 * q^10 + 21 * q^13 + q^14 + 58 * q^16 + 4 * q^17 + 7 * q^19 + 20 * q^20 + 7 * q^22 - 20 * q^23 + 47 * q^25 - 8 * q^26 + 11 * q^28 - 20 * q^29 + 28 * q^31 - 14 * q^32 + 16 * q^34 - 9 * q^35 + 14 * q^37 + 20 * q^38 + 34 * q^40 - 7 * q^41 + 3 * q^43 + q^44 + 2 * q^46 - 3 * q^47 + 35 * q^49 + 24 * q^50 + 73 * q^52 + 19 * q^53 + 29 * q^55 + 30 * q^56 + 2 * q^58 - 20 * q^59 + 15 * q^61 - 12 * q^62 + 82 * q^64 + 28 * q^65 + 20 * q^67 + 23 * q^68 - 24 * q^70 - 63 * q^71 + 19 * q^73 - 16 * q^74 - 44 * q^76 + 7 * q^77 + 32 * q^79 + 56 * q^80 - 20 * q^82 + 21 * q^83 + 4 * q^85 + 6 * q^86 + 55 * q^88 + 13 * q^89 + 70 * q^91 - 30 * q^92 - 12 * q^94 - 9 * q^95 - 9 * q^97 - 31 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.80084	1.98049	0.990247	−	0.139326i	$-0.0444937\pi$
$2$	2.80084	1.98049	0.990247	+	0.139326i	$0.0444937\pi$
$3$	0	0
$3$	0	0
$4$	5.84471	2.92235
$5$	3.48103	1.55676	0.778381	−	0.627792i	$-0.216041\pi$
$5$	3.48103	1.55676	0.778381	+	0.627792i	$0.216041\pi$
$6$	0	0
$7$	−1.41384	−0.534380	−0.267190	−	0.963644i	$-0.586095\pi$
$7$	−1.41384	−0.534380	−0.267190	+	0.963644i	$0.586095\pi$
$8$	10.7684	3.80721
$9$	0	0
$10$	9.74980	3.08316
$11$	4.43439	1.33702	0.668509	−	0.743704i	$-0.266933\pi$
$11$	4.43439	1.33702	0.668509	+	0.743704i	$0.266933\pi$
$12$	0	0
$13$	−2.37735	−0.659359	−0.329680	−	0.944093i	$-0.606941\pi$
$13$	−2.37735	−0.659359	−0.329680	+	0.944093i	$0.606941\pi$
$14$	−3.95993	−1.05834
$15$	0	0
$16$	18.4712	4.61779
$17$	3.28679	0.797165	0.398582	−	0.917133i	$-0.369502\pi$
$17$	3.28679	0.797165	0.398582	+	0.917133i	$0.369502\pi$
$18$	0	0
$19$	−3.46248	−0.794347	−0.397173	−	0.917744i	$-0.630009\pi$
$19$	−3.46248	−0.794347	−0.397173	+	0.917744i	$0.630009\pi$
$20$	20.3456	4.54941
$21$	0	0
$22$	12.4200	2.64796
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	7.11755	1.42351
$26$	−6.65859	−1.30586
$27$	0	0
$28$	−8.26345	−1.56165
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−4.18385	−0.751443	−0.375721	−	0.926733i	$-0.622605\pi$
$31$	−4.18385	−0.751443	−0.375721	+	0.926733i	$0.622605\pi$
$32$	30.1980	5.33830
$33$	0	0
$34$	9.20579	1.57878
$35$	−4.92160	−0.831902
$36$	0	0
$37$	−10.7653	−1.76981	−0.884905	−	0.465771i	$-0.845777\pi$
$37$	−10.7653	−1.76981	−0.884905	+	0.465771i	$0.845777\pi$
$38$	−9.69784	−1.57320
$39$	0	0
$40$	37.4851	5.92691
$41$	−3.71341	−0.579937	−0.289968	−	0.957036i	$-0.593645\pi$
$41$	−3.71341	−0.579937	−0.289968	+	0.957036i	$0.593645\pi$
$42$	0	0
$43$	−7.08099	−1.07984	−0.539920	−	0.841716i	$-0.681546\pi$
$43$	−7.08099	−1.07984	−0.539920	+	0.841716i	$0.681546\pi$
$44$	25.9177	3.90724
$45$	0	0
$46$	−2.80084	−0.412961
$47$	−1.63781	−0.238899	−0.119450	−	0.992840i	$-0.538113\pi$
$47$	−1.63781	−0.238899	−0.119450	+	0.992840i	$0.538113\pi$
$48$	0	0
$49$	−5.00107	−0.714438
$50$	19.9351	2.81925
$51$	0	0
$52$	−13.8949	−1.92688
$53$	−6.97182	−0.957653	−0.478827	−	0.877910i	$-0.658938\pi$
$53$	−6.97182	−0.957653	−0.478827	+	0.877910i	$0.658938\pi$
$54$	0	0
$55$	15.4362	2.08142
$56$	−15.2248	−2.03449
$57$	0	0
$58$	−2.80084	−0.367768
$59$	−12.2928	−1.60039	−0.800195	−	0.599740i	$-0.795271\pi$
$59$	−12.2928	−1.60039	−0.800195	+	0.599740i	$0.795271\pi$
$60$	0	0
$61$	5.25185	0.672431	0.336215	−	0.941785i	$-0.390853\pi$
$61$	5.25185	0.672431	0.336215	+	0.941785i	$0.390853\pi$
$62$	−11.7183	−1.48823
$63$	0	0
$64$	47.6374	5.95467
$65$	−8.27563	−1.02647
$66$	0	0
$67$	−11.9828	−1.46393	−0.731965	−	0.681342i	$-0.761397\pi$
$67$	−11.9828	−1.46393	−0.731965	+	0.681342i	$0.761397\pi$
$68$	19.2103	2.32960
$69$	0	0
$70$	−13.7846	−1.64758
$71$	−12.3124	−1.46121	−0.730604	−	0.682802i	$-0.760761\pi$
$71$	−12.3124	−1.46121	−0.730604	+	0.682802i	$0.760761\pi$
$72$	0	0
$73$	8.68606	1.01663	0.508313	−	0.861172i	$-0.330269\pi$
$73$	8.68606	1.01663	0.508313	+	0.861172i	$0.330269\pi$
$74$	−30.1520	−3.50510
$75$	0	0
$76$	−20.2372	−2.32136
$77$	−6.26950	−0.714476
$78$	0	0
$79$	9.51988	1.07107	0.535535	−	0.844513i	$-0.320110\pi$
$79$	9.51988	1.07107	0.535535	+	0.844513i	$0.320110\pi$
$80$	64.2986	7.18881
$81$	0	0
$82$	−10.4007	−1.14856
$83$	−2.12581	−0.233338	−0.116669	−	0.993171i	$-0.537222\pi$
$83$	−2.12581	−0.233338	−0.116669	+	0.993171i	$0.537222\pi$
$84$	0	0
$85$	11.4414	1.24100
$86$	−19.8327	−2.13862
$87$	0	0
$88$	47.7513	5.09031
$89$	11.0119	1.16726	0.583632	−	0.812018i	$-0.301631\pi$
$89$	11.0119	1.16726	0.583632	+	0.812018i	$0.301631\pi$
$90$	0	0
$91$	3.36119	0.352348
$92$	−5.84471	−0.609353
$93$	0	0
$94$	−4.58725	−0.473139
$95$	−12.0530	−1.23661
$96$	0	0
$97$	12.3407	1.25301	0.626505	−	0.779417i	$-0.284485\pi$
$97$	12.3407	1.25301	0.626505	+	0.779417i	$0.284485\pi$
$98$	−14.0072	−1.41494
$99$	0	0
$100$	41.6000	4.16000
$101$	17.4826	1.73959	0.869794	−	0.493416i	$-0.164252\pi$
$101$	17.4826	1.73959	0.869794	+	0.493416i	$0.164252\pi$
$102$	0	0
$103$	−10.0681	−0.992040	−0.496020	−	0.868311i	$-0.665206\pi$
$103$	−10.0681	−0.992040	−0.496020	+	0.868311i	$0.665206\pi$
$104$	−25.6003	−2.51032
$105$	0	0
$106$	−19.5270	−1.89663
$107$	8.64089	0.835346	0.417673	−	0.908597i	$-0.362846\pi$
$107$	8.64089	0.835346	0.417673	+	0.908597i	$0.362846\pi$
$108$	0	0
$109$	−7.68708	−0.736289	−0.368144	−	0.929769i	$-0.620007\pi$
$109$	−7.68708	−0.736289	−0.368144	+	0.929769i	$0.620007\pi$
$110$	43.2344	4.12224
$111$	0	0
$112$	−26.1152	−2.46765
$113$	5.51803	0.519092	0.259546	−	0.965731i	$-0.416427\pi$
$113$	5.51803	0.519092	0.259546	+	0.965731i	$0.416427\pi$
$114$	0	0
$115$	−3.48103	−0.324607
$116$	−5.84471	−0.542667
$117$	0	0
$118$	−34.4302	−3.16956
$119$	−4.64699	−0.425989
$120$	0	0
$121$	8.66381	0.787619
$122$	14.7096	1.33174
$123$	0	0
$124$	−24.4534	−2.19598
$125$	7.37123	0.659303
$126$	0	0
$127$	12.7499	1.13137	0.565684	−	0.824622i	$-0.308612\pi$
$127$	12.7499	1.13137	0.565684	+	0.824622i	$0.308612\pi$
$128$	73.0287	6.45489
$129$	0	0
$130$	−23.1787	−2.03291
$131$	14.4611	1.26347	0.631735	−	0.775185i	$-0.282343\pi$
$131$	14.4611	1.26347	0.631735	+	0.775185i	$0.282343\pi$
$132$	0	0
$133$	4.89537	0.424483
$134$	−33.5619	−2.89930
$135$	0	0
$136$	35.3935	3.03497
$137$	−9.36729	−0.800302	−0.400151	−	0.916449i	$-0.631042\pi$
$137$	−9.36729	−0.800302	−0.400151	+	0.916449i	$0.631042\pi$
$138$	0	0
$139$	−7.17859	−0.608880	−0.304440	−	0.952532i	$-0.598469\pi$
$139$	−7.17859	−0.608880	−0.304440	+	0.952532i	$0.598469\pi$
$140$	−28.7653	−2.43111
$141$	0	0
$142$	−34.4849	−2.89391
$143$	−10.5421	−0.881575
$144$	0	0
$145$	−3.48103	−0.289084
$146$	24.3283	2.01342
$147$	0	0
$148$	−62.9202	−5.17201
$149$	−12.0231	−0.984974	−0.492487	−	0.870320i	$-0.663912\pi$
$149$	−12.0231	−0.984974	−0.492487	+	0.870320i	$0.663912\pi$
$150$	0	0
$151$	14.1232	1.14933	0.574665	−	0.818389i	$-0.305132\pi$
$151$	14.1232	1.14933	0.574665	+	0.818389i	$0.305132\pi$
$152$	−37.2853	−3.02424
$153$	0	0
$154$	−17.5599	−1.41501
$155$	−14.5641	−1.16982
$156$	0	0
$157$	12.7785	1.01983	0.509917	−	0.860224i	$-0.329676\pi$
$157$	12.7785	1.01983	0.509917	+	0.860224i	$0.329676\pi$
$158$	26.6637	2.12125
$159$	0	0
$160$	105.120	8.31046
$161$	1.41384	0.111426
$162$	0	0
$163$	18.0242	1.41177	0.705883	−	0.708328i	$-0.250550\pi$
$163$	18.0242	1.41177	0.705883	+	0.708328i	$0.250550\pi$
$164$	−21.7038	−1.69478
$165$	0	0
$166$	−5.95405	−0.462124
$167$	−6.24398	−0.483174	−0.241587	−	0.970379i	$-0.577668\pi$
$167$	−6.24398	−0.483174	−0.241587	+	0.970379i	$0.577668\pi$
$168$	0	0
$169$	−7.34819	−0.565246
$170$	32.0456	2.45778
$171$	0	0
$172$	−41.3863	−3.15567
$173$	21.4424	1.63024	0.815119	−	0.579293i	$-0.196671\pi$
$173$	21.4424	1.63024	0.815119	+	0.579293i	$0.196671\pi$
$174$	0	0
$175$	−10.0630	−0.760694
$176$	81.9084	6.17407
$177$	0	0
$178$	30.8427	2.31176
$179$	−19.8711	−1.48524	−0.742618	−	0.669715i	$-0.766416\pi$
$179$	−19.8711	−1.48524	−0.742618	+	0.669715i	$0.766416\pi$
$180$	0	0
$181$	−0.142040	−0.0105578	−0.00527889	−	0.999986i	$-0.501680\pi$
$181$	−0.142040	−0.0105578	−0.00527889	+	0.999986i	$0.501680\pi$
$182$	9.41415	0.697823
$183$	0	0
$184$	−10.7684	−0.793857
$185$	−37.4744	−2.75517
$186$	0	0
$187$	14.5749	1.06582
$188$	−9.57253	−0.698148
$189$	0	0
$190$	−33.7584	−2.44910
$191$	14.5967	1.05618	0.528091	−	0.849188i	$-0.322908\pi$
$191$	14.5967	1.05618	0.528091	+	0.849188i	$0.322908\pi$
$192$	0	0
$193$	−17.3067	−1.24576	−0.622881	−	0.782317i	$-0.714038\pi$
$193$	−17.3067	−1.24576	−0.622881	+	0.782317i	$0.714038\pi$
$194$	34.5644	2.48158
$195$	0	0
$196$	−29.2298	−2.08784
$197$	−2.11009	−0.150338	−0.0751689	−	0.997171i	$-0.523950\pi$
$197$	−2.11009	−0.150338	−0.0751689	+	0.997171i	$0.523950\pi$
$198$	0	0
$199$	16.4313	1.16478	0.582391	−	0.812909i	$-0.302117\pi$
$199$	16.4313	1.16478	0.582391	+	0.812909i	$0.302117\pi$
$200$	76.6446	5.41959
$201$	0	0
$202$	48.9661	3.44524
$203$	1.41384	0.0992318
$204$	0	0
$205$	−12.9265	−0.902824
$206$	−28.1992	−1.96473
$207$	0	0
$208$	−43.9125	−3.04478
$209$	−15.3540	−1.06206
$210$	0	0
$211$	5.03335	0.346510	0.173255	−	0.984877i	$-0.444572\pi$
$211$	5.03335	0.346510	0.173255	+	0.984877i	$0.444572\pi$
$212$	−40.7482	−2.79860
$213$	0	0
$214$	24.2018	1.65440
$215$	−24.6491	−1.68105
$216$	0	0
$217$	5.91528	0.401556
$218$	−21.5303	−1.45821
$219$	0	0
$220$	90.2202	6.08264
$221$	−7.81387	−0.525618
$222$	0	0
$223$	−4.74485	−0.317739	−0.158869	−	0.987300i	$-0.550785\pi$
$223$	−4.74485	−0.317739	−0.158869	+	0.987300i	$0.550785\pi$
$224$	−42.6950	−2.85268
$225$	0	0
$226$	15.4551	1.02806
$227$	15.9414	1.05806	0.529032	−	0.848602i	$-0.322555\pi$
$227$	15.9414	1.05806	0.529032	+	0.848602i	$0.322555\pi$
$228$	0	0
$229$	24.6062	1.62602	0.813012	−	0.582247i	$-0.197826\pi$
$229$	24.6062	1.62602	0.813012	+	0.582247i	$0.197826\pi$
$230$	−9.74980	−0.642883
$231$	0	0
$232$	−10.7684	−0.706980
$233$	3.43090	0.224765	0.112383	−	0.993665i	$-0.464152\pi$
$233$	3.43090	0.224765	0.112383	+	0.993665i	$0.464152\pi$
$234$	0	0
$235$	−5.70127	−0.371910
$236$	−71.8479	−4.67690
$237$	0	0
$238$	−13.0155	−0.843668
$239$	−5.76650	−0.373004	−0.186502	−	0.982455i	$-0.559715\pi$
$239$	−5.76650	−0.373004	−0.186502	+	0.982455i	$0.559715\pi$
$240$	0	0
$241$	19.8383	1.27790	0.638948	−	0.769250i	$-0.279370\pi$
$241$	19.8383	1.27790	0.638948	+	0.769250i	$0.279370\pi$
$242$	24.2659	1.55987
$243$	0	0
$244$	30.6955	1.96508
$245$	−17.4089	−1.11221
$246$	0	0
$247$	8.23153	0.523760
$248$	−45.0534	−2.86090
$249$	0	0
$250$	20.6456	1.30574
$251$	1.97792	0.124845	0.0624226	−	0.998050i	$-0.480117\pi$
$251$	1.97792	0.124845	0.0624226	+	0.998050i	$0.480117\pi$
$252$	0	0
$253$	−4.43439	−0.278788
$254$	35.7103	2.24067
$255$	0	0
$256$	109.267	6.82919
$257$	0.846927	0.0528298	0.0264149	−	0.999651i	$-0.491591\pi$
$257$	0.846927	0.0528298	0.0264149	+	0.999651i	$0.491591\pi$
$258$	0	0
$259$	15.2204	0.945751
$260$	−48.3686	−2.99969
$261$	0	0
$262$	40.5031	2.50229
$263$	−0.347153	−0.0214064	−0.0107032	−	0.999943i	$-0.503407\pi$
$263$	−0.347153	−0.0214064	−0.0107032	+	0.999943i	$0.503407\pi$
$264$	0	0
$265$	−24.2691	−1.49084
$266$	13.7112	0.840685
$267$	0	0
$268$	−70.0359	−4.27812
$269$	−22.3131	−1.36045	−0.680227	−	0.733002i	$-0.738119\pi$
$269$	−22.3131	−1.36045	−0.680227	+	0.733002i	$0.738119\pi$
$270$	0	0
$271$	−21.0578	−1.27917	−0.639584	−	0.768721i	$-0.720893\pi$
$271$	−21.0578	−1.27917	−0.639584	+	0.768721i	$0.720893\pi$
$272$	60.7109	3.68114
$273$	0	0
$274$	−26.2363	−1.58499
$275$	31.5620	1.90326
$276$	0	0
$277$	2.42766	0.145864	0.0729320	−	0.997337i	$-0.476764\pi$
$277$	2.42766	0.145864	0.0729320	+	0.997337i	$0.476764\pi$
$278$	−20.1061	−1.20588
$279$	0	0
$280$	−52.9978	−3.16722
$281$	−11.6621	−0.695703	−0.347852	−	0.937550i	$-0.613089\pi$
$281$	−11.6621	−0.695703	−0.347852	+	0.937550i	$0.613089\pi$
$282$	0	0
$283$	−7.95812	−0.473061	−0.236530	−	0.971624i	$-0.576010\pi$
$283$	−7.95812	−0.473061	−0.236530	+	0.971624i	$0.576010\pi$
$284$	−71.9621	−4.27016
$285$	0	0
$286$	−29.5268	−1.74595
$287$	5.25015	0.309906
$288$	0	0
$289$	−6.19698	−0.364528
$290$	−9.74980	−0.572528
$291$	0	0
$292$	50.7675	2.97094
$293$	25.6929	1.50099	0.750497	−	0.660874i	$-0.229814\pi$
$293$	25.6929	1.50099	0.750497	+	0.660874i	$0.229814\pi$
$294$	0	0
$295$	−42.7917	−2.49143
$296$	−115.926	−6.73803
$297$	0	0
$298$	−33.6749	−1.95073
$299$	2.37735	0.137486
$300$	0	0
$301$	10.0114	0.577045
$302$	39.5568	2.27624
$303$	0	0
$304$	−63.9560	−3.66813
$305$	18.2818	1.04681
$306$	0	0
$307$	18.5614	1.05936	0.529679	−	0.848198i	$-0.322312\pi$
$307$	18.5614	1.05936	0.529679	+	0.848198i	$0.322312\pi$
$308$	−36.6434	−2.08795
$309$	0	0
$310$	−40.7917	−2.31682
$311$	−9.91454	−0.562202	−0.281101	−	0.959678i	$-0.590700\pi$
$311$	−9.91454	−0.562202	−0.281101	+	0.959678i	$0.590700\pi$
$312$	0	0
$313$	15.3040	0.865032	0.432516	−	0.901626i	$-0.357626\pi$
$313$	15.3040	0.865032	0.432516	+	0.901626i	$0.357626\pi$
$314$	35.7905	2.01977
$315$	0	0
$316$	55.6409	3.13004
$317$	−17.3563	−0.974829	−0.487414	−	0.873171i	$-0.662060\pi$
$317$	−17.3563	−0.974829	−0.487414	+	0.873171i	$0.662060\pi$
$318$	0	0
$319$	−4.43439	−0.248278
$320$	165.827	9.27001
$321$	0	0
$322$	3.95993	0.220678
$323$	−11.3804	−0.633225
$324$	0	0
$325$	−16.9209	−0.938604
$326$	50.4830	2.79599
$327$	0	0
$328$	−39.9875	−2.20794
$329$	2.31560	0.127663
$330$	0	0
$331$	−25.6184	−1.40811	−0.704057	−	0.710144i	$-0.748630\pi$
$331$	−25.6184	−1.40811	−0.704057	+	0.710144i	$0.748630\pi$
$332$	−12.4247	−0.681895
$333$	0	0
$334$	−17.4884	−0.956922
$335$	−41.7124	−2.27899
$336$	0	0
$337$	−16.3884	−0.892734	−0.446367	−	0.894850i	$-0.647282\pi$
$337$	−16.3884	−0.892734	−0.446367	+	0.894850i	$0.647282\pi$
$338$	−20.5811	−1.11947
$339$	0	0
$340$	66.8717	3.62663
$341$	−18.5528	−1.00469
$342$	0	0
$343$	16.9675	0.916161
$344$	−76.2509	−4.11117
$345$	0	0
$346$	60.0568	3.22868
$347$	25.0667	1.34565	0.672826	−	0.739800i	$-0.265080\pi$
$347$	25.0667	1.34565	0.672826	+	0.739800i	$0.265080\pi$
$348$	0	0
$349$	22.4766	1.20315	0.601573	−	0.798818i	$-0.294541\pi$
$349$	22.4766	1.20315	0.601573	+	0.798818i	$0.294541\pi$
$350$	−28.1850	−1.50655
$351$	0	0
$352$	133.910	7.13741
$353$	2.42803	0.129231	0.0646156	−	0.997910i	$-0.479418\pi$
$353$	2.42803	0.129231	0.0646156	+	0.997910i	$0.479418\pi$
$354$	0	0
$355$	−42.8596	−2.27475
$356$	64.3616	3.41116
$357$	0	0
$358$	−55.6558	−2.94150
$359$	−36.3433	−1.91812	−0.959062	−	0.283195i	$-0.908606\pi$
$359$	−36.3433	−1.91812	−0.959062	+	0.283195i	$0.908606\pi$
$360$	0	0
$361$	−7.01126	−0.369014
$362$	−0.397832	−0.0209096
$363$	0	0
$364$	19.6451	1.02969
$365$	30.2364	1.58265
$366$	0	0
$367$	12.2035	0.637016	0.318508	−	0.947920i	$-0.396818\pi$
$367$	12.2035	0.637016	0.318508	+	0.947920i	$0.396818\pi$
$368$	−18.4712	−0.962876
$369$	0	0
$370$	−104.960	−5.45660
$371$	9.85701	0.511750
$372$	0	0
$373$	4.00742	0.207497	0.103748	−	0.994604i	$-0.466916\pi$
$373$	4.00742	0.207497	0.103748	+	0.994604i	$0.466916\pi$
$374$	40.8220	2.11086
$375$	0	0
$376$	−17.6366	−0.909539
$377$	2.37735	0.122440
$378$	0	0
$379$	3.66384	0.188199	0.0940993	−	0.995563i	$-0.470003\pi$
$379$	3.66384	0.188199	0.0940993	+	0.995563i	$0.470003\pi$
$380$	−70.4461	−3.61381
$381$	0	0
$382$	40.8831	2.09176
$383$	12.5542	0.641488	0.320744	−	0.947166i	$-0.396067\pi$
$383$	12.5542	0.641488	0.320744	+	0.947166i	$0.396067\pi$
$384$	0	0
$385$	−21.8243	−1.11227
$386$	−48.4732	−2.46722
$387$	0	0
$388$	72.1279	3.66174
$389$	−18.2759	−0.926623	−0.463312	−	0.886195i	$-0.653339\pi$
$389$	−18.2759	−0.926623	−0.463312	+	0.886195i	$0.653339\pi$
$390$	0	0
$391$	−3.28679	−0.166220
$392$	−53.8535	−2.72001
$393$	0	0
$394$	−5.91003	−0.297743
$395$	33.1389	1.66740
$396$	0	0
$397$	9.01959	0.452680	0.226340	−	0.974048i	$-0.427324\pi$
$397$	9.01959	0.452680	0.226340	+	0.974048i	$0.427324\pi$
$398$	46.0213	2.30684
$399$	0	0
$400$	131.469	6.57347
$401$	−6.66076	−0.332622	−0.166311	−	0.986073i	$-0.553186\pi$
$401$	−6.66076	−0.332622	−0.166311	+	0.986073i	$0.553186\pi$
$402$	0	0
$403$	9.94650	0.495470
$404$	102.181	5.08369
$405$	0	0
$406$	3.95993	0.196528
$407$	−47.7377	−2.36627
$408$	0	0
$409$	3.95276	0.195451	0.0977257	−	0.995213i	$-0.468843\pi$
$409$	3.95276	0.195451	0.0977257	+	0.995213i	$0.468843\pi$
$410$	−36.2050	−1.78804
$411$	0	0
$412$	−58.8451	−2.89909
$413$	17.3800	0.855216
$414$	0	0
$415$	−7.39999	−0.363251
$416$	−71.7913	−3.51986
$417$	0	0
$418$	−43.0040	−2.10339
$419$	−4.41044	−0.215464	−0.107732	−	0.994180i	$-0.534359\pi$
$419$	−4.41044	−0.215464	−0.107732	+	0.994180i	$0.534359\pi$
$420$	0	0
$421$	14.8860	0.725497	0.362748	−	0.931887i	$-0.381838\pi$
$421$	14.8860	0.725497	0.362748	+	0.931887i	$0.381838\pi$
$422$	14.0976	0.686260
$423$	0	0
$424$	−75.0754	−3.64598
$425$	23.3939	1.13477
$426$	0	0
$427$	−7.42526	−0.359333
$428$	50.5035	2.44118
$429$	0	0
$430$	−69.0382	−3.32932
$431$	2.60324	0.125393	0.0626967	−	0.998033i	$-0.480030\pi$
$431$	2.60324	0.125393	0.0626967	+	0.998033i	$0.480030\pi$
$432$	0	0
$433$	5.93295	0.285119	0.142560	−	0.989786i	$-0.454467\pi$
$433$	5.93295	0.285119	0.142560	+	0.989786i	$0.454467\pi$
$434$	16.5678	0.795278
$435$	0	0
$436$	−44.9287	−2.15170
$437$	3.46248	0.165633
$438$	0	0
$439$	−8.48828	−0.405124	−0.202562	−	0.979269i	$-0.564927\pi$
$439$	−8.48828	−0.405124	−0.202562	+	0.979269i	$0.564927\pi$
$440$	166.224	7.92440
$441$	0	0
$442$	−21.8854	−1.04098
$443$	21.2540	1.00981	0.504904	−	0.863176i	$-0.331528\pi$
$443$	21.2540	1.00981	0.504904	+	0.863176i	$0.331528\pi$
$444$	0	0
$445$	38.3329	1.81715
$446$	−13.2896	−0.629279
$447$	0	0
$448$	−67.3514	−3.18206
$449$	10.8308	0.511139	0.255570	−	0.966791i	$-0.417737\pi$
$449$	10.8308	0.511139	0.255570	+	0.966791i	$0.417737\pi$
$450$	0	0
$451$	−16.4667	−0.775386
$452$	32.2512	1.51697
$453$	0	0
$454$	44.6492	2.09549
$455$	11.7004	0.548522
$456$	0	0
$457$	2.24941	0.105223	0.0526114	−	0.998615i	$-0.483246\pi$
$457$	2.24941	0.105223	0.0526114	+	0.998615i	$0.483246\pi$
$458$	68.9180	3.22033
$459$	0	0
$460$	−20.3456	−0.948617
$461$	−2.51632	−0.117197	−0.0585984	−	0.998282i	$-0.518663\pi$
$461$	−2.51632	−0.117197	−0.0585984	+	0.998282i	$0.518663\pi$
$462$	0	0
$463$	−15.2202	−0.707341	−0.353671	−	0.935370i	$-0.615067\pi$
$463$	−15.2202	−0.707341	−0.353671	+	0.935370i	$0.615067\pi$
$464$	−18.4712	−0.857502
$465$	0	0
$466$	9.60939	0.445146
$467$	6.08666	0.281657	0.140829	−	0.990034i	$-0.455023\pi$
$467$	6.08666	0.281657	0.140829	+	0.990034i	$0.455023\pi$
$468$	0	0
$469$	16.9417	0.782295
$470$	−15.9683	−0.736564
$471$	0	0
$472$	−132.374	−6.09301
$473$	−31.3998	−1.44377
$474$	0	0
$475$	−24.6443	−1.13076
$476$	−27.1603	−1.24489
$477$	0	0
$478$	−16.1510	−0.738732
$479$	2.91970	0.133405	0.0667023	−	0.997773i	$-0.478752\pi$
$479$	2.91970	0.133405	0.0667023	+	0.997773i	$0.478752\pi$
$480$	0	0
$481$	25.5930	1.16694
$482$	55.5639	2.53087
$483$	0	0
$484$	50.6374	2.30170
$485$	42.9584	1.95064
$486$	0	0
$487$	35.3220	1.60059	0.800297	−	0.599604i	$-0.204675\pi$
$487$	35.3220	1.60059	0.800297	+	0.599604i	$0.204675\pi$
$488$	56.5541	2.56008
$489$	0	0
$490$	−48.7594	−2.20273
$491$	5.12284	0.231190	0.115595	−	0.993296i	$-0.463122\pi$
$491$	5.12284	0.231190	0.115595	+	0.993296i	$0.463122\pi$
$492$	0	0
$493$	−3.28679	−0.148030
$494$	23.0552	1.03730
$495$	0	0
$496$	−77.2807	−3.47001
$497$	17.4076	0.780840
$498$	0	0
$499$	8.30239	0.371666	0.185833	−	0.982581i	$-0.440502\pi$
$499$	8.30239	0.371666	0.185833	+	0.982581i	$0.440502\pi$
$500$	43.0827	1.92672
$501$	0	0
$502$	5.53984	0.247255
$503$	37.2320	1.66009	0.830045	−	0.557696i	$-0.188314\pi$
$503$	37.2320	1.66009	0.830045	+	0.557696i	$0.188314\pi$
$504$	0	0
$505$	60.8575	2.70812
$506$	−12.4200	−0.552137
$507$	0	0
$508$	74.5192	3.30626
$509$	41.3069	1.83090	0.915449	−	0.402434i	$-0.131836\pi$
$509$	41.3069	1.83090	0.915449	+	0.402434i	$0.131836\pi$
$510$	0	0
$511$	−12.2807	−0.543264
$512$	159.982	7.07027
$513$	0	0
$514$	2.37211	0.104629
$515$	−35.0474	−1.54437
$516$	0	0
$517$	−7.26270	−0.319413
$518$	42.6300	1.87305
$519$	0	0
$520$	−89.1153	−3.90796
$521$	−24.8969	−1.09075	−0.545377	−	0.838191i	$-0.683613\pi$
$521$	−24.8969	−1.09075	−0.545377	+	0.838191i	$0.683613\pi$
$522$	0	0
$523$	10.7859	0.471634	0.235817	−	0.971797i	$-0.424223\pi$
$523$	10.7859	0.471634	0.235817	+	0.971797i	$0.424223\pi$
$524$	84.5207	3.69230
$525$	0	0
$526$	−0.972319	−0.0423951
$527$	−13.7515	−0.599024
$528$	0	0
$529$	1.00000	0.0434783
$530$	−67.9738	−2.95260
$531$	0	0
$532$	28.6120	1.24049
$533$	8.82808	0.382387
$534$	0	0
$535$	30.0792	1.30044
$536$	−129.035	−5.57348
$537$	0	0
$538$	−62.4954	−2.69437
$539$	−22.1767	−0.955217
$540$	0	0
$541$	−42.3958	−1.82274	−0.911368	−	0.411592i	$-0.864973\pi$
$541$	−42.3958	−1.82274	−0.911368	+	0.411592i	$0.864973\pi$
$542$	−58.9794	−2.53338
$543$	0	0
$544$	99.2546	4.25550
$545$	−26.7589	−1.14623
$546$	0	0
$547$	−43.7711	−1.87151	−0.935757	−	0.352644i	$-0.885283\pi$
$547$	−43.7711	−1.87151	−0.935757	+	0.352644i	$0.885283\pi$
$548$	−54.7491	−2.33876
$549$	0	0
$550$	88.4000	3.76939
$551$	3.46248	0.147506
$552$	0	0
$553$	−13.4595	−0.572358
$554$	6.79949	0.288883
$555$	0	0
$556$	−41.9567	−1.77936
$557$	10.4656	0.443443	0.221722	−	0.975110i	$-0.428832\pi$
$557$	10.4656	0.443443	0.221722	+	0.975110i	$0.428832\pi$
$558$	0	0
$559$	16.8340	0.712002
$560$	−90.9077	−3.84155
$561$	0	0
$562$	−32.6637	−1.37784
$563$	−30.1027	−1.26868	−0.634338	−	0.773055i	$-0.718727\pi$
$563$	−30.1027	−1.26868	−0.634338	+	0.773055i	$0.718727\pi$
$564$	0	0
$565$	19.2084	0.808103
$566$	−22.2894	−0.936894
$567$	0	0
$568$	−132.584	−5.56312
$569$	7.93717	0.332744	0.166372	−	0.986063i	$-0.446795\pi$
$569$	7.93717	0.332744	0.166372	+	0.986063i	$0.446795\pi$
$570$	0	0
$571$	42.1256	1.76290	0.881450	−	0.472277i	$-0.156568\pi$
$571$	42.1256	1.76290	0.881450	+	0.472277i	$0.156568\pi$
$572$	−61.6155	−2.57627
$573$	0	0
$574$	14.7048	0.613768
$575$	−7.11755	−0.296822
$576$	0	0
$577$	−24.2008	−1.00749	−0.503745	−	0.863852i	$-0.668045\pi$
$577$	−24.2008	−1.00749	−0.503745	+	0.863852i	$0.668045\pi$
$578$	−17.3568	−0.721946
$579$	0	0
$580$	−20.3456	−0.844804
$581$	3.00554	0.124691
$582$	0	0
$583$	−30.9158	−1.28040
$584$	93.5350	3.87051
$585$	0	0
$586$	71.9616	2.97271
$587$	−1.25188	−0.0516706	−0.0258353	−	0.999666i	$-0.508225\pi$
$587$	−1.25188	−0.0516706	−0.0258353	+	0.999666i	$0.508225\pi$
$588$	0	0
$589$	14.4865	0.596906
$590$	−119.853	−4.93425
$591$	0	0
$592$	−198.848	−8.17262
$593$	−16.3746	−0.672426	−0.336213	−	0.941786i	$-0.609146\pi$
$593$	−16.3746	−0.672426	−0.336213	+	0.941786i	$0.609146\pi$
$594$	0	0
$595$	−16.1763	−0.663163
$596$	−70.2717	−2.87844
$597$	0	0
$598$	6.65859	0.272290
$599$	13.9988	0.571977	0.285989	−	0.958233i	$-0.407678\pi$
$599$	13.9988	0.571977	0.285989	+	0.958233i	$0.407678\pi$
$600$	0	0
$601$	−15.8577	−0.646850	−0.323425	−	0.946254i	$-0.604834\pi$
$601$	−15.8577	−0.646850	−0.323425	+	0.946254i	$0.604834\pi$
$602$	28.0402	1.14283
$603$	0	0
$604$	82.5460	3.35875
$605$	30.1589	1.22614
$606$	0	0
$607$	−44.6641	−1.81286	−0.906430	−	0.422356i	$-0.861203\pi$
$607$	−44.6641	−1.81286	−0.906430	+	0.422356i	$0.861203\pi$
$608$	−104.560	−4.24046
$609$	0	0
$610$	51.2045	2.07321
$611$	3.89366	0.157521
$612$	0	0
$613$	41.5875	1.67971	0.839853	−	0.542814i	$-0.182641\pi$
$613$	41.5875	1.67971	0.839853	+	0.542814i	$0.182641\pi$
$614$	51.9876	2.09805
$615$	0	0
$616$	−67.5125	−2.72016
$617$	38.3035	1.54204	0.771020	−	0.636811i	$-0.219747\pi$
$617$	38.3035	1.54204	0.771020	+	0.636811i	$0.219747\pi$
$618$	0	0
$619$	−35.0276	−1.40788	−0.703939	−	0.710260i	$-0.748577\pi$
$619$	−35.0276	−1.40788	−0.703939	+	0.710260i	$0.748577\pi$
$620$	−85.1229	−3.41862
$621$	0	0
$622$	−27.7690	−1.11344
$623$	−15.5691	−0.623762
$624$	0	0
$625$	−9.92828	−0.397131
$626$	42.8640	1.71319
$627$	0	0
$628$	74.6865	2.98031
$629$	−35.3835	−1.41083
$630$	0	0
$631$	0.0346178	0.00137811	0.000689056	−	1.00000i	$-0.499781\pi$
$631$	0.0346178	0.00137811	0.000689056		1.00000i	$0.499781\pi$
$632$	102.514	4.07778
$633$	0	0
$634$	−48.6123	−1.93064
$635$	44.3826	1.76127
$636$	0	0
$637$	11.8893	0.471071
$638$	−12.4200	−0.491713
$639$	0	0
$640$	254.215	10.0487
$641$	1.97337	0.0779435	0.0389718	−	0.999240i	$-0.487592\pi$
$641$	1.97337	0.0779435	0.0389718	+	0.999240i	$0.487592\pi$
$642$	0	0
$643$	−29.6819	−1.17054	−0.585270	−	0.810838i	$-0.699012\pi$
$643$	−29.6819	−1.17054	−0.585270	+	0.810838i	$0.699012\pi$
$644$	8.26345	0.325626
$645$	0	0
$646$	−31.8748	−1.25410
$647$	8.97043	0.352664	0.176332	−	0.984331i	$-0.443577\pi$
$647$	8.97043	0.352664	0.176332	+	0.984331i	$0.443577\pi$
$648$	0	0
$649$	−54.5112	−2.13975
$650$	−47.3928	−1.85890
$651$	0	0
$652$	105.346	4.12568
$653$	24.5006	0.958784	0.479392	−	0.877601i	$-0.340857\pi$
$653$	24.5006	0.958784	0.479392	+	0.877601i	$0.340857\pi$
$654$	0	0
$655$	50.3394	1.96692
$656$	−68.5910	−2.67803
$657$	0	0
$658$	6.48562	0.252836
$659$	21.1494	0.823863	0.411931	−	0.911215i	$-0.364854\pi$
$659$	21.1494	0.823863	0.411931	+	0.911215i	$0.364854\pi$
$660$	0	0
$661$	−29.9812	−1.16613	−0.583067	−	0.812424i	$-0.698147\pi$
$661$	−29.9812	−1.16613	−0.583067	+	0.812424i	$0.698147\pi$
$662$	−71.7530	−2.78876
$663$	0	0
$664$	−22.8915	−0.888364
$665$	17.0409	0.660819
$666$	0	0
$667$	1.00000	0.0387202
$668$	−36.4942	−1.41200
$669$	0	0
$670$	−116.830	−4.51353
$671$	23.2888	0.899052
$672$	0	0
$673$	−11.6843	−0.450398	−0.225199	−	0.974313i	$-0.572303\pi$
$673$	−11.6843	−0.450398	−0.225199	+	0.974313i	$0.572303\pi$
$674$	−45.9013	−1.76805
$675$	0	0
$676$	−42.9480	−1.65185
$677$	−47.4614	−1.82409	−0.912046	−	0.410088i	$-0.865498\pi$
$677$	−47.4614	−1.82409	−0.912046	+	0.410088i	$0.865498\pi$
$678$	0	0
$679$	−17.4478	−0.669583
$680$	123.206	4.72473
$681$	0	0
$682$	−51.9635	−1.98979
$683$	−48.2425	−1.84595	−0.922975	−	0.384860i	$-0.874250\pi$
$683$	−48.2425	−1.84595	−0.922975	+	0.384860i	$0.874250\pi$
$684$	0	0
$685$	−32.6078	−1.24588
$686$	47.5234	1.81445
$687$	0	0
$688$	−130.794	−4.98648
$689$	16.5745	0.631437
$690$	0	0
$691$	−17.1961	−0.654172	−0.327086	−	0.944995i	$-0.606067\pi$
$691$	−17.1961	−0.654172	−0.327086	+	0.944995i	$0.606067\pi$
$692$	125.325	4.76413
$693$	0	0
$694$	70.2079	2.66506
$695$	−24.9889	−0.947881
$696$	0	0
$697$	−12.2052	−0.462305
$698$	62.9534	2.38282
$699$	0	0
$700$	−58.8155	−2.22302
$701$	27.6256	1.04340	0.521702	−	0.853128i	$-0.325297\pi$
$701$	27.6256	1.04340	0.521702	+	0.853128i	$0.325297\pi$
$702$	0	0
$703$	37.2747	1.40584
$704$	211.243	7.96151
$705$	0	0
$706$	6.80053	0.255941
$707$	−24.7176	−0.929600
$708$	0	0
$709$	8.10255	0.304298	0.152149	−	0.988358i	$-0.451381\pi$
$709$	8.10255	0.304298	0.152149	+	0.988358i	$0.451381\pi$
$710$	−120.043	−4.50513
$711$	0	0
$712$	118.581	4.44402
$713$	4.18385	0.156687
$714$	0	0
$715$	−36.6974	−1.37240
$716$	−116.141	−4.34038
$717$	0	0
$718$	−101.792	−3.79883
$719$	−21.9351	−0.818041	−0.409020	−	0.912525i	$-0.634130\pi$
$719$	−21.9351	−0.818041	−0.409020	+	0.912525i	$0.634130\pi$
$720$	0	0
$721$	14.2347	0.530126
$722$	−19.6374	−0.730829
$723$	0	0
$724$	−0.830184	−0.0308536
$725$	−7.11755	−0.264339
$726$	0	0
$727$	49.7966	1.84685	0.923427	−	0.383773i	$-0.125376\pi$
$727$	49.7966	1.84685	0.923427	+	0.383773i	$0.125376\pi$
$728$	36.1946	1.34146
$729$	0	0
$730$	84.6873	3.13442
$731$	−23.2737	−0.860811
$732$	0	0
$733$	−5.55178	−0.205060	−0.102530	−	0.994730i	$-0.532694\pi$
$733$	−5.55178	−0.205060	−0.102530	+	0.994730i	$0.532694\pi$
$734$	34.1799	1.26160
$735$	0	0
$736$	−30.1980	−1.11311
$737$	−53.1363	−1.95730
$738$	0	0
$739$	−35.1640	−1.29353	−0.646765	−	0.762689i	$-0.723879\pi$
$739$	−35.1640	−1.29353	−0.646765	+	0.762689i	$0.723879\pi$
$740$	−219.027	−8.05159
$741$	0	0
$742$	27.6079	1.01352
$743$	−49.3711	−1.81125	−0.905625	−	0.424079i	$-0.860598\pi$
$743$	−49.3711	−1.81125	−0.905625	+	0.424079i	$0.860598\pi$
$744$	0	0
$745$	−41.8529	−1.53337
$746$	11.2242	0.410946
$747$	0	0
$748$	85.1862	3.11471
$749$	−12.2168	−0.446392
$750$	0	0
$751$	26.4065	0.963587	0.481793	−	0.876285i	$-0.339986\pi$
$751$	26.4065	0.963587	0.481793	+	0.876285i	$0.339986\pi$
$752$	−30.2523	−1.10319
$753$	0	0
$754$	6.65859	0.242491
$755$	49.1633	1.78923
$756$	0	0
$757$	41.0628	1.49245	0.746227	−	0.665692i	$-0.231863\pi$
$757$	41.0628	1.49245	0.746227	+	0.665692i	$0.231863\pi$
$758$	10.2618	0.372726
$759$	0	0
$760$	−129.791	−4.70802
$761$	−39.5526	−1.43378	−0.716890	−	0.697187i	$-0.754435\pi$
$761$	−39.5526	−1.43378	−0.716890	+	0.697187i	$0.754435\pi$
$762$	0	0
$763$	10.8683	0.393458
$764$	85.3135	3.08653
$765$	0	0
$766$	35.1622	1.27046
$767$	29.2244	1.05523
$768$	0	0
$769$	−9.25396	−0.333706	−0.166853	−	0.985982i	$-0.553361\pi$
$769$	−9.25396	−0.333706	−0.166853	+	0.985982i	$0.553361\pi$
$770$	−61.1263	−2.20284
$771$	0	0
$772$	−101.152	−3.64055
$773$	32.8456	1.18137	0.590687	−	0.806901i	$-0.298857\pi$
$773$	32.8456	1.18137	0.590687	+	0.806901i	$0.298857\pi$
$774$	0	0
$775$	−29.7788	−1.06969
$776$	132.890	4.77047
$777$	0	0
$778$	−51.1878	−1.83517
$779$	12.8576	0.460671
$780$	0	0
$781$	−54.5978	−1.95366
$782$	−9.20579	−0.329198
$783$	0	0
$784$	−92.3756	−3.29913
$785$	44.4822	1.58764
$786$	0	0
$787$	−25.5920	−0.912257	−0.456129	−	0.889914i	$-0.650764\pi$
$787$	−25.5920	−0.912257	−0.456129	+	0.889914i	$0.650764\pi$
$788$	−12.3329	−0.439340
$789$	0	0
$790$	92.8169	3.30228
$791$	−7.80158	−0.277392
$792$	0	0
$793$	−12.4855	−0.443373
$794$	25.2624	0.896530
$795$	0	0
$796$	96.0359	3.40390
$797$	11.2097	0.397069	0.198534	−	0.980094i	$-0.436382\pi$
$797$	11.2097	0.397069	0.198534	+	0.980094i	$0.436382\pi$
$798$	0	0
$799$	−5.38315	−0.190442
$800$	214.935	7.59912
$801$	0	0
$802$	−18.6557	−0.658756
$803$	38.5174	1.35925
$804$	0	0
$805$	4.92160	0.173464
$806$	27.8586	0.981276
$807$	0	0
$808$	188.260	6.62297
$809$	−8.56734	−0.301212	−0.150606	−	0.988594i	$-0.548122\pi$
$809$	−8.56734	−0.301212	−0.150606	+	0.988594i	$0.548122\pi$
$810$	0	0
$811$	21.1612	0.743069	0.371535	−	0.928419i	$-0.378832\pi$
$811$	21.1612	0.743069	0.371535	+	0.928419i	$0.378832\pi$
$812$	8.26345	0.289990
$813$	0	0
$814$	−133.706	−4.68638
$815$	62.7428	2.19778
$816$	0	0
$817$	24.5177	0.857767
$818$	11.0710	0.387090
$819$	0	0
$820$	−75.5514	−2.63837
$821$	11.7177	0.408950	0.204475	−	0.978872i	$-0.434451\pi$
$821$	11.7177	0.408950	0.204475	+	0.978872i	$0.434451\pi$
$822$	0	0
$823$	−3.38360	−0.117945	−0.0589724	−	0.998260i	$-0.518782\pi$
$823$	−3.38360	−0.117945	−0.0589724	+	0.998260i	$0.518782\pi$
$824$	−108.417	−3.77690
$825$	0	0
$826$	48.6787	1.69375
$827$	−10.4103	−0.362003	−0.181001	−	0.983483i	$-0.557934\pi$
$827$	−10.4103	−0.362003	−0.181001	+	0.983483i	$0.557934\pi$
$828$	0	0
$829$	−14.8931	−0.517259	−0.258630	−	0.965977i	$-0.583271\pi$
$829$	−14.8931	−0.517259	−0.258630	+	0.965977i	$0.583271\pi$
$830$	−20.7262	−0.719417
$831$	0	0
$832$	−113.251	−3.92627
$833$	−16.4375	−0.569525
$834$	0	0
$835$	−21.7355	−0.752187
$836$	−89.7394	−3.10370
$837$	0	0
$838$	−12.3529	−0.426725
$839$	35.8545	1.23784	0.618918	−	0.785456i	$-0.287571\pi$
$839$	35.8545	1.23784	0.618918	+	0.785456i	$0.287571\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	41.6932	1.43684
$843$	0	0
$844$	29.4184	1.01262
$845$	−25.5793	−0.879953
$846$	0	0
$847$	−12.2492	−0.420888
$848$	−128.778	−4.42224
$849$	0	0
$850$	65.5226	2.24741
$851$	10.7653	0.369031
$852$	0	0
$853$	11.4424	0.391779	0.195890	−	0.980626i	$-0.437241\pi$
$853$	11.4424	0.391779	0.195890	+	0.980626i	$0.437241\pi$
$854$	−20.7970	−0.711657
$855$	0	0
$856$	93.0486	3.18034
$857$	34.4455	1.17664	0.588319	−	0.808629i	$-0.299790\pi$
$857$	34.4455	1.17664	0.588319	+	0.808629i	$0.299790\pi$
$858$	0	0
$859$	−2.68062	−0.0914617	−0.0457308	−	0.998954i	$-0.514562\pi$
$859$	−2.68062	−0.0914617	−0.0457308	+	0.998954i	$0.514562\pi$
$860$	−144.067	−4.91263
$861$	0	0
$862$	7.29125	0.248341
$863$	−40.5322	−1.37973	−0.689866	−	0.723937i	$-0.742330\pi$
$863$	−40.5322	−1.37973	−0.689866	+	0.723937i	$0.742330\pi$
$864$	0	0
$865$	74.6417	2.53789
$866$	16.6172	0.564677
$867$	0	0
$868$	34.5731	1.17349
$869$	42.2148	1.43204
$870$	0	0
$871$	28.4873	0.965256
$872$	−82.7776	−2.80320
$873$	0	0
$874$	9.69784	0.328034
$875$	−10.4217	−0.352318
$876$	0	0
$877$	26.5521	0.896600	0.448300	−	0.893883i	$-0.352030\pi$
$877$	26.5521	0.896600	0.448300	+	0.893883i	$0.352030\pi$
$878$	−23.7743	−0.802344
$879$	0	0
$880$	285.125	9.61157
$881$	−2.87095	−0.0967248	−0.0483624	−	0.998830i	$-0.515400\pi$
$881$	−2.87095	−0.0967248	−0.0483624	+	0.998830i	$0.515400\pi$
$882$	0	0
$883$	28.0387	0.943578	0.471789	−	0.881711i	$-0.343608\pi$
$883$	28.0387	0.943578	0.471789	+	0.881711i	$0.343608\pi$
$884$	−45.6698	−1.53604
$885$	0	0
$886$	59.5290	1.99992
$887$	−17.7793	−0.596972	−0.298486	−	0.954414i	$-0.596482\pi$
$887$	−17.7793	−0.596972	−0.298486	+	0.954414i	$0.596482\pi$
$888$	0	0
$889$	−18.0262	−0.604580
$890$	107.364	3.59886
$891$	0	0
$892$	−27.7323	−0.928544
$893$	5.67089	0.189769
$894$	0	0
$895$	−69.1718	−2.31216
$896$	−103.251	−3.44936
$897$	0	0
$898$	30.3355	1.01231
$899$	4.18385	0.139539
$900$	0	0
$901$	−22.9149	−0.763407
$902$	−46.1206	−1.53565
$903$	0	0
$904$	59.4203	1.97629
$905$	−0.494446	−0.0164360
$906$	0	0
$907$	48.0328	1.59490	0.797451	−	0.603383i	$-0.206181\pi$
$907$	48.0328	1.59490	0.797451	+	0.603383i	$0.206181\pi$
$908$	93.1725	3.09204
$909$	0	0
$910$	32.7709	1.08634
$911$	−36.9017	−1.22261	−0.611304	−	0.791396i	$-0.709355\pi$
$911$	−36.9017	−1.22261	−0.611304	+	0.791396i	$0.709355\pi$
$912$	0	0
$913$	−9.42666	−0.311977
$914$	6.30023	0.208393
$915$	0	0
$916$	143.816	4.75181
$917$	−20.4456	−0.675172
$918$	0	0
$919$	−9.31252	−0.307192	−0.153596	−	0.988134i	$-0.549085\pi$
$919$	−9.31252	−0.307192	−0.153596	+	0.988134i	$0.549085\pi$
$920$	−37.4851	−1.23585
$921$	0	0
$922$	−7.04781	−0.232107
$923$	29.2708	0.963460
$924$	0	0
$925$	−76.6228	−2.51934
$926$	−42.6293	−1.40088
$927$	0	0
$928$	−30.1980	−0.991297
$929$	30.7813	1.00990	0.504951	−	0.863148i	$-0.331511\pi$
$929$	30.7813	1.00990	0.504951	+	0.863148i	$0.331511\pi$
$930$	0	0
$931$	17.3161	0.567512
$932$	20.0526	0.656844
$933$	0	0
$934$	17.0478	0.557820
$935$	50.7357	1.65924
$936$	0	0
$937$	40.1120	1.31040	0.655202	−	0.755454i	$-0.272584\pi$
$937$	40.1120	1.31040	0.655202	+	0.755454i	$0.272584\pi$
$938$	47.4510	1.54933
$939$	0	0
$940$	−33.3222	−1.08685
$941$	47.6356	1.55287	0.776437	−	0.630194i	$-0.217025\pi$
$941$	47.6356	1.55287	0.776437	+	0.630194i	$0.217025\pi$
$942$	0	0
$943$	3.71341	0.120925
$944$	−227.063	−7.39027
$945$	0	0
$946$	−87.9460	−2.85937
$947$	−36.9283	−1.20001	−0.600004	−	0.799997i	$-0.704834\pi$
$947$	−36.9283	−1.20001	−0.600004	+	0.799997i	$0.704834\pi$
$948$	0	0
$949$	−20.6498	−0.670322
$950$	−69.0248	−2.23946
$951$	0	0
$952$	−50.0406	−1.62183
$953$	−9.59892	−0.310939	−0.155470	−	0.987841i	$-0.549689\pi$
$953$	−9.59892	−0.310939	−0.155470	+	0.987841i	$0.549689\pi$
$954$	0	0
$955$	50.8115	1.64422
$956$	−33.7035	−1.09005
$957$	0	0
$958$	8.17762	0.264207
$959$	13.2438	0.427665
$960$	0	0
$961$	−13.4954	−0.435334
$962$	71.6819	2.31112
$963$	0	0
$964$	115.949	3.73446
$965$	−60.2450	−1.93935
$966$	0	0
$967$	−31.4665	−1.01190	−0.505948	−	0.862564i	$-0.668857\pi$
$967$	−31.4665	−1.01190	−0.505948	+	0.862564i	$0.668857\pi$
$968$	93.2954	2.99863
$969$	0	0
$970$	120.320	3.86323
$971$	−31.9428	−1.02509	−0.512547	−	0.858659i	$-0.671298\pi$
$971$	−31.9428	−1.02509	−0.512547	+	0.858659i	$0.671298\pi$
$972$	0	0
$973$	10.1493	0.325373
$974$	98.9314	3.16997
$975$	0	0
$976$	97.0078	3.10515
$977$	18.8977	0.604592	0.302296	−	0.953214i	$-0.402247\pi$
$977$	18.8977	0.604592	0.302296	+	0.953214i	$0.402247\pi$
$978$	0	0
$979$	48.8313	1.56065
$980$	−101.750	−3.25027
$981$	0	0
$982$	14.3482	0.457871
$983$	−9.51432	−0.303460	−0.151730	−	0.988422i	$-0.548484\pi$
$983$	−9.51432	−0.303460	−0.151730	+	0.988422i	$0.548484\pi$
$984$	0	0
$985$	−7.34528	−0.234040
$986$	−9.20579	−0.293172
$987$	0	0
$988$	48.1109	1.53061
$989$	7.08099	0.225162
$990$	0	0
$991$	37.2984	1.18482	0.592412	−	0.805635i	$-0.298176\pi$
$991$	37.2984	1.18482	0.592412	+	0.805635i	$0.298176\pi$
$992$	−126.344	−4.01143
$993$	0	0
$994$	48.7560	1.54645
$995$	57.1977	1.81329
$996$	0	0
$997$	−5.35625	−0.169634	−0.0848171	−	0.996397i	$-0.527031\pi$
$997$	−5.35625	−0.169634	−0.0848171	+	0.996397i	$0.527031\pi$
$998$	23.2537	0.736082
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.s.1.20		20
3.2	odd	2		2001.2.a.o.1.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.o.1.1	✓	20	3.2	odd	2
6003.2.a.s.1.20		20	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)

\(f(q)\)	\(=\)	\(q+2.80084 q^{2} +5.84471 q^{4} +3.48103 q^{5} -1.41384 q^{7} +10.7684 q^{8} +O(q^{10})\)	\(q+2.80084 q^{2} +5.84471 q^{4} +3.48103 q^{5} -1.41384 q^{7} +10.7684 q^{8} +9.74980 q^{10} +4.43439 q^{11} -2.37735 q^{13} -3.95993 q^{14} +18.4712 q^{16} +3.28679 q^{17} -3.46248 q^{19} +20.3456 q^{20} +12.4200 q^{22} -1.00000 q^{23} +7.11755 q^{25} -6.65859 q^{26} -8.26345 q^{28} -1.00000 q^{29} -4.18385 q^{31} +30.1980 q^{32} +9.20579 q^{34} -4.92160 q^{35} -10.7653 q^{37} -9.69784 q^{38} +37.4851 q^{40} -3.71341 q^{41} -7.08099 q^{43} +25.9177 q^{44} -2.80084 q^{46} -1.63781 q^{47} -5.00107 q^{49} +19.9351 q^{50} -13.8949 q^{52} -6.97182 q^{53} +15.4362 q^{55} -15.2248 q^{56} -2.80084 q^{58} -12.2928 q^{59} +5.25185 q^{61} -11.7183 q^{62} +47.6374 q^{64} -8.27563 q^{65} -11.9828 q^{67} +19.2103 q^{68} -13.7846 q^{70} -12.3124 q^{71} +8.68606 q^{73} -30.1520 q^{74} -20.2372 q^{76} -6.26950 q^{77} +9.51988 q^{79} +64.2986 q^{80} -10.4007 q^{82} -2.12581 q^{83} +11.4414 q^{85} -19.8327 q^{86} +47.7513 q^{88} +11.0119 q^{89} +3.36119 q^{91} -5.84471 q^{92} -4.58725 q^{94} -12.0530 q^{95} +12.3407 q^{97} -14.0072 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) \) 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8} + 7 q^{10} + 21 q^{13} + q^{14} + 58 q^{16} + 4 q^{17} + 7 q^{19} + 20 q^{20} + 7 q^{22} - 20 q^{23} + 47 q^{25} - 8 q^{26} + 11 q^{28} - 20 q^{29} + 28 q^{31} - 14 q^{32} + 16 q^{34} - 9 q^{35} + 14 q^{37} + 20 q^{38} + 34 q^{40} - 7 q^{41} + 3 q^{43} + q^{44} + 2 q^{46} - 3 q^{47} + 35 q^{49} + 24 q^{50} + 73 q^{52} + 19 q^{53} + 29 q^{55} + 30 q^{56} + 2 q^{58} - 20 q^{59} + 15 q^{61} - 12 q^{62} + 82 q^{64} + 28 q^{65} + 20 q^{67} + 23 q^{68} - 24 q^{70} - 63 q^{71} + 19 q^{73} - 16 q^{74} - 44 q^{76} + 7 q^{77} + 32 q^{79} + 56 q^{80} - 20 q^{82} + 21 q^{83} + 4 q^{85} + 6 q^{86} + 55 q^{88} + 13 q^{89} + 70 q^{91} - 30 q^{92} - 12 q^{94} - 9 q^{95} - 9 q^{97} - 31 q^{98}+O(q^{100}) \) 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8 + 7 * q^10 + 21 * q^13 + q^14 + 58 * q^16 + 4 * q^17 + 7 * q^19 + 20 * q^20 + 7 * q^22 - 20 * q^23 + 47 * q^25 - 8 * q^26 + 11 * q^28 - 20 * q^29 + 28 * q^31 - 14 * q^32 + 16 * q^34 - 9 * q^35 + 14 * q^37 + 20 * q^38 + 34 * q^40 - 7 * q^41 + 3 * q^43 + q^44 + 2 * q^46 - 3 * q^47 + 35 * q^49 + 24 * q^50 + 73 * q^52 + 19 * q^53 + 29 * q^55 + 30 * q^56 + 2 * q^58 - 20 * q^59 + 15 * q^61 - 12 * q^62 + 82 * q^64 + 28 * q^65 + 20 * q^67 + 23 * q^68 - 24 * q^70 - 63 * q^71 + 19 * q^73 - 16 * q^74 - 44 * q^76 + 7 * q^77 + 32 * q^79 + 56 * q^80 - 20 * q^82 + 21 * q^83 + 4 * q^85 + 6 * q^86 + 55 * q^88 + 13 * q^89 + 70 * q^91 - 30 * q^92 - 12 * q^94 - 9 * q^95 - 9 * q^97 - 31 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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