LMFDB - Embedded newform 6003.2.a.k.1.10

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:	$1$
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} - x^{9} - 17x^{8} + 23x^{7} + 69x^{6} - 88x^{5} - 106x^{4} + 101x^{3} + 60x^{2} - 23x - 11 $ x^10 - x^9 - 17x^8 + 23x^7 + 69x^6 - 88x^5 - 106x^4 + 101x^3 + 60x^2 - 23x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$0.565113$ 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)

$f(q)$	$=$	$q+2.70414 q^{2} +5.31240 q^{4} -0.910949 q^{5} -2.94138 q^{7} +8.95720 q^{8} +O(q^{10})$	\(q+2.70414 q^{2} +5.31240 q^{4} -0.910949 q^{5} -2.94138 q^{7} +8.95720 q^{8} -2.46334 q^{10} -1.96757 q^{11} -6.90155 q^{13} -7.95393 q^{14} +13.5968 q^{16} +3.43588 q^{17} -4.84044 q^{19} -4.83932 q^{20} -5.32059 q^{22} +1.00000 q^{23} -4.17017 q^{25} -18.6628 q^{26} -15.6258 q^{28} -1.00000 q^{29} -6.10796 q^{31} +18.8532 q^{32} +9.29112 q^{34} +2.67945 q^{35} +10.7601 q^{37} -13.0893 q^{38} -8.15955 q^{40} -0.108786 q^{41} +4.09933 q^{43} -10.4525 q^{44} +2.70414 q^{46} +2.94339 q^{47} +1.65174 q^{49} -11.2767 q^{50} -36.6638 q^{52} -8.24232 q^{53} +1.79236 q^{55} -26.3466 q^{56} -2.70414 q^{58} -14.7490 q^{59} -4.18717 q^{61} -16.5168 q^{62} +23.7883 q^{64} +6.28696 q^{65} +7.67423 q^{67} +18.2528 q^{68} +7.24562 q^{70} -11.1823 q^{71} -15.2305 q^{73} +29.0969 q^{74} -25.7144 q^{76} +5.78738 q^{77} -10.2160 q^{79} -12.3860 q^{80} -0.294174 q^{82} -3.05803 q^{83} -3.12991 q^{85} +11.0852 q^{86} -17.6239 q^{88} +7.02861 q^{89} +20.3001 q^{91} +5.31240 q^{92} +7.95934 q^{94} +4.40940 q^{95} -12.0644 q^{97} +4.46653 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) $ 10 * q - 3 * q^2 + 17 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8	$ 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8} - 4 q^{10} - 9 q^{11} - 16 q^{13} - 16 q^{14} + 27 q^{16} + q^{19} - 21 q^{20} + 17 q^{22} + 10 q^{23} - 4 q^{25} - 28 q^{26} - 14 q^{28} - 10 q^{29} + 17 q^{31} - 21 q^{32} - 3 q^{34} - 29 q^{35} + q^{37} - 32 q^{38} + 13 q^{40} - 5 q^{43} - 33 q^{44} - 3 q^{46} - 15 q^{47} + 31 q^{49} + 22 q^{50} - 21 q^{52} - 35 q^{53} - 20 q^{55} - 18 q^{56} + 3 q^{58} - 49 q^{59} + 8 q^{61} - 15 q^{62} + 12 q^{64} + 3 q^{65} + 35 q^{67} + 18 q^{68} - 16 q^{70} - 30 q^{71} - 15 q^{73} - 23 q^{74} + 10 q^{76} - 23 q^{77} + 24 q^{79} - 23 q^{80} - 5 q^{82} - q^{83} + 10 q^{86} + 18 q^{88} - 15 q^{89} + 26 q^{91} + 17 q^{92} + 3 q^{94} - 7 q^{95} - 35 q^{97} - 3 q^{98}+O(q^{100}) $ 10 * q - 3 * q^2 + 17 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8 - 4 * q^10 - 9 * q^11 - 16 * q^13 - 16 * q^14 + 27 * q^16 + q^19 - 21 * q^20 + 17 * q^22 + 10 * q^23 - 4 * q^25 - 28 * q^26 - 14 * q^28 - 10 * q^29 + 17 * q^31 - 21 * q^32 - 3 * q^34 - 29 * q^35 + q^37 - 32 * q^38 + 13 * q^40 - 5 * q^43 - 33 * q^44 - 3 * q^46 - 15 * q^47 + 31 * q^49 + 22 * q^50 - 21 * q^52 - 35 * q^53 - 20 * q^55 - 18 * q^56 + 3 * q^58 - 49 * q^59 + 8 * q^61 - 15 * q^62 + 12 * q^64 + 3 * q^65 + 35 * q^67 + 18 * q^68 - 16 * q^70 - 30 * q^71 - 15 * q^73 - 23 * q^74 + 10 * q^76 - 23 * q^77 + 24 * q^79 - 23 * q^80 - 5 * q^82 - q^83 + 10 * q^86 + 18 * q^88 - 15 * q^89 + 26 * q^91 + 17 * q^92 + 3 * q^94 - 7 * q^95 - 35 * q^97 - 3 * 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 $

$2$	2.70414	1.91212	0.956059	−	0.293173i	$-0.0947112\pi$
$2$	2.70414	1.91212	0.956059	+	0.293173i	$0.0947112\pi$
$3$	0	0
$3$	0	0
$4$	5.31240	2.65620
$5$	−0.910949	−0.407389	−0.203694	−	0.979035i	$-0.565295\pi$
$5$	−0.910949	−0.407389	−0.203694	+	0.979035i	$0.565295\pi$
$6$	0	0
$7$	−2.94138	−1.11174	−0.555869	−	0.831270i	$-0.687614\pi$
$7$	−2.94138	−1.11174	−0.555869	+	0.831270i	$0.687614\pi$
$8$	8.95720	3.16685
$9$	0	0
$10$	−2.46334	−0.778976
$11$	−1.96757	−0.593245	−0.296622	−	0.954995i	$-0.595860\pi$
$11$	−1.96757	−0.593245	−0.296622	+	0.954995i	$0.595860\pi$
$12$	0	0
$13$	−6.90155	−1.91414	−0.957072	−	0.289849i	$-0.906395\pi$
$13$	−6.90155	−1.91414	−0.957072	+	0.289849i	$0.906395\pi$
$14$	−7.95393	−2.12578
$15$	0	0
$16$	13.5968	3.39919
$17$	3.43588	0.833323	0.416662	−	0.909062i	$-0.363200\pi$
$17$	3.43588	0.833323	0.416662	+	0.909062i	$0.363200\pi$
$18$	0	0
$19$	−4.84044	−1.11047	−0.555237	−	0.831692i	$-0.687372\pi$
$19$	−4.84044	−1.11047	−0.555237	+	0.831692i	$0.687372\pi$
$20$	−4.83932	−1.08211
$21$	0	0
$22$	−5.32059	−1.13435
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.17017	−0.834034
$26$	−18.6628	−3.66007
$27$	0	0
$28$	−15.6258	−2.95300
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−6.10796	−1.09702	−0.548511	−	0.836143i	$-0.684805\pi$
$31$	−6.10796	−1.09702	−0.548511	+	0.836143i	$0.684805\pi$
$32$	18.8532	3.33281
$33$	0	0
$34$	9.29112	1.59341
$35$	2.67945	0.452910
$36$	0	0
$37$	10.7601	1.76895	0.884476	−	0.466585i	$-0.154516\pi$
$37$	10.7601	1.76895	0.884476	+	0.466585i	$0.154516\pi$
$38$	−13.0893	−2.12336
$39$	0	0
$40$	−8.15955	−1.29014
$41$	−0.108786	−0.0169896	−0.00849479	−	0.999964i	$-0.502704\pi$
$41$	−0.108786	−0.0169896	−0.00849479	+	0.999964i	$0.502704\pi$
$42$	0	0
$43$	4.09933	0.625142	0.312571	−	0.949894i	$-0.398810\pi$
$43$	4.09933	0.625142	0.312571	+	0.949894i	$0.398810\pi$
$44$	−10.4525	−1.57578
$45$	0	0
$46$	2.70414	0.398704
$47$	2.94339	0.429337	0.214669	−	0.976687i	$-0.431133\pi$
$47$	2.94339	0.429337	0.214669	+	0.976687i	$0.431133\pi$
$48$	0	0
$49$	1.65174	0.235962
$50$	−11.2767	−1.59477
$51$	0	0
$52$	−36.6638	−5.08435
$53$	−8.24232	−1.13217	−0.566085	−	0.824347i	$-0.691543\pi$
$53$	−8.24232	−1.13217	−0.566085	+	0.824347i	$0.691543\pi$
$54$	0	0
$55$	1.79236	0.241681
$56$	−26.3466	−3.52071
$57$	0	0
$58$	−2.70414	−0.355072
$59$	−14.7490	−1.92016	−0.960078	−	0.279733i	$-0.909754\pi$
$59$	−14.7490	−1.92016	−0.960078	+	0.279733i	$0.909754\pi$
$60$	0	0
$61$	−4.18717	−0.536112	−0.268056	−	0.963403i	$-0.586381\pi$
$61$	−4.18717	−0.536112	−0.268056	+	0.963403i	$0.586381\pi$
$62$	−16.5168	−2.09764
$63$	0	0
$64$	23.7883	2.97354
$65$	6.28696	0.779801
$66$	0	0
$67$	7.67423	0.937556	0.468778	−	0.883316i	$-0.344694\pi$
$67$	7.67423	0.937556	0.468778	+	0.883316i	$0.344694\pi$
$68$	18.2528	2.21347
$69$	0	0
$70$	7.24562	0.866017
$71$	−11.1823	−1.32710	−0.663550	−	0.748132i	$-0.730951\pi$
$71$	−11.1823	−1.32710	−0.663550	+	0.748132i	$0.730951\pi$
$72$	0	0
$73$	−15.2305	−1.78260	−0.891299	−	0.453416i	$-0.850205\pi$
$73$	−15.2305	−1.78260	−0.891299	+	0.453416i	$0.850205\pi$
$74$	29.0969	3.38245
$75$	0	0
$76$	−25.7144	−2.94964
$77$	5.78738	0.659533
$78$	0	0
$79$	−10.2160	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$79$	−10.2160	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$80$	−12.3860	−1.38479
$81$	0	0
$82$	−0.294174	−0.0324861
$83$	−3.05803	−0.335663	−0.167831	−	0.985816i	$-0.553676\pi$
$83$	−3.05803	−0.335663	−0.167831	+	0.985816i	$0.553676\pi$
$84$	0	0
$85$	−3.12991	−0.339487
$86$	11.0852	1.19535
$87$	0	0
$88$	−17.6239	−1.87872
$89$	7.02861	0.745031	0.372515	−	0.928026i	$-0.378495\pi$
$89$	7.02861	0.745031	0.372515	+	0.928026i	$0.378495\pi$
$90$	0	0
$91$	20.3001	2.12803
$92$	5.31240	0.553856
$93$	0	0
$94$	7.95934	0.820943
$95$	4.40940	0.452395
$96$	0	0
$97$	−12.0644	−1.22496	−0.612478	−	0.790488i	$-0.709827\pi$
$97$	−12.0644	−1.22496	−0.612478	+	0.790488i	$0.709827\pi$
$98$	4.46653	0.451188
$99$	0	0
$100$	−22.1536	−2.21536
$101$	−6.01230	−0.598246	−0.299123	−	0.954215i	$-0.596694\pi$
$101$	−6.01230	−0.598246	−0.299123	+	0.954215i	$0.596694\pi$
$102$	0	0
$103$	8.13902	0.801962	0.400981	−	0.916086i	$-0.368669\pi$
$103$	8.13902	0.801962	0.400981	+	0.916086i	$0.368669\pi$
$104$	−61.8185	−6.06181
$105$	0	0
$106$	−22.2884	−2.16484
$107$	8.69377	0.840458	0.420229	−	0.907418i	$-0.361950\pi$
$107$	8.69377	0.840458	0.420229	+	0.907418i	$0.361950\pi$
$108$	0	0
$109$	4.84305	0.463880	0.231940	−	0.972730i	$-0.425493\pi$
$109$	4.84305	0.463880	0.231940	+	0.972730i	$0.425493\pi$
$110$	4.84679	0.462123
$111$	0	0
$112$	−39.9933	−3.77901
$113$	−5.49292	−0.516730	−0.258365	−	0.966047i	$-0.583184\pi$
$113$	−5.49292	−0.516730	−0.258365	+	0.966047i	$0.583184\pi$
$114$	0	0
$115$	−0.910949	−0.0849464
$116$	−5.31240	−0.493244
$117$	0	0
$118$	−39.8834	−3.67157
$119$	−10.1062	−0.926438
$120$	0	0
$121$	−7.12867	−0.648061
$122$	−11.3227	−1.02511
$123$	0	0
$124$	−32.4479	−2.91391
$125$	8.35356	0.747165
$126$	0	0
$127$	9.46429	0.839820	0.419910	−	0.907566i	$-0.362062\pi$
$127$	9.46429	0.839820	0.419910	+	0.907566i	$0.362062\pi$
$128$	26.6205	2.35295
$129$	0	0
$130$	17.0008	1.49107
$131$	0.702049	0.0613383	0.0306691	−	0.999530i	$-0.490236\pi$
$131$	0.702049	0.0613383	0.0306691	+	0.999530i	$0.490236\pi$
$132$	0	0
$133$	14.2376	1.23456
$134$	20.7522	1.79272
$135$	0	0
$136$	30.7759	2.63901
$137$	13.4489	1.14902	0.574509	−	0.818498i	$-0.305193\pi$
$137$	13.4489	1.14902	0.574509	+	0.818498i	$0.305193\pi$
$138$	0	0
$139$	21.6071	1.83269	0.916344	−	0.400391i	$-0.131126\pi$
$139$	21.6071	1.83269	0.916344	+	0.400391i	$0.131126\pi$
$140$	14.2343	1.20302
$141$	0	0
$142$	−30.2387	−2.53757
$143$	13.5793	1.13556
$144$	0	0
$145$	0.910949	0.0756502
$146$	−41.1855	−3.40854
$147$	0	0
$148$	57.1620	4.69869
$149$	4.15966	0.340773	0.170386	−	0.985377i	$-0.445498\pi$
$149$	4.15966	0.340773	0.170386	+	0.985377i	$0.445498\pi$
$150$	0	0
$151$	−10.9815	−0.893666	−0.446833	−	0.894617i	$-0.647448\pi$
$151$	−10.9815	−0.893666	−0.446833	+	0.894617i	$0.647448\pi$
$152$	−43.3568	−3.51670
$153$	0	0
$154$	15.6499	1.26111
$155$	5.56404	0.446914
$156$	0	0
$157$	13.0533	1.04177	0.520885	−	0.853627i	$-0.325602\pi$
$157$	13.0533	1.04177	0.520885	+	0.853627i	$0.325602\pi$
$158$	−27.6256	−2.19777
$159$	0	0
$160$	−17.1743	−1.35775
$161$	−2.94138	−0.231813
$162$	0	0
$163$	15.5057	1.21450	0.607250	−	0.794511i	$-0.292273\pi$
$163$	15.5057	1.21450	0.607250	+	0.794511i	$0.292273\pi$
$164$	−0.577916	−0.0451277
$165$	0	0
$166$	−8.26936	−0.641827
$167$	14.4844	1.12084	0.560419	−	0.828209i	$-0.310640\pi$
$167$	14.4844	1.12084	0.560419	+	0.828209i	$0.310640\pi$
$168$	0	0
$169$	34.6314	2.66395
$170$	−8.46373	−0.649139
$171$	0	0
$172$	21.7773	1.66050
$173$	−19.7531	−1.50180	−0.750900	−	0.660416i	$-0.770380\pi$
$173$	−19.7531	−1.50180	−0.750900	+	0.660416i	$0.770380\pi$
$174$	0	0
$175$	12.2661	0.927228
$176$	−26.7526	−2.01655
$177$	0	0
$178$	19.0064	1.42459
$179$	10.7033	0.800006	0.400003	−	0.916514i	$-0.369009\pi$
$179$	10.7033	0.800006	0.400003	+	0.916514i	$0.369009\pi$
$180$	0	0
$181$	−12.3302	−0.916499	−0.458250	−	0.888824i	$-0.651523\pi$
$181$	−12.3302	−0.916499	−0.458250	+	0.888824i	$0.651523\pi$
$182$	54.8944	4.06904
$183$	0	0
$184$	8.95720	0.660333
$185$	−9.80192	−0.720652
$186$	0	0
$187$	−6.76034	−0.494365
$188$	15.6364	1.14040
$189$	0	0
$190$	11.9236	0.865032
$191$	2.23872	0.161988	0.0809940	−	0.996715i	$-0.474191\pi$
$191$	2.23872	0.161988	0.0809940	+	0.996715i	$0.474191\pi$
$192$	0	0
$193$	20.6216	1.48437	0.742187	−	0.670193i	$-0.233789\pi$
$193$	20.6216	1.48437	0.742187	+	0.670193i	$0.233789\pi$
$194$	−32.6239	−2.34226
$195$	0	0
$196$	8.77468	0.626763
$197$	−1.49237	−0.106327	−0.0531635	−	0.998586i	$-0.516930\pi$
$197$	−1.49237	−0.106327	−0.0531635	+	0.998586i	$0.516930\pi$
$198$	0	0
$199$	17.4876	1.23966	0.619832	−	0.784735i	$-0.287201\pi$
$199$	17.4876	1.23966	0.619832	+	0.784735i	$0.287201\pi$
$200$	−37.3531	−2.64126
$201$	0	0
$202$	−16.2581	−1.14392
$203$	2.94138	0.206445
$204$	0	0
$205$	0.0990988	0.00692136
$206$	22.0091	1.53345
$207$	0	0
$208$	−93.8387	−6.50654
$209$	9.52391	0.658783
$210$	0	0
$211$	12.5135	0.861465	0.430733	−	0.902480i	$-0.358255\pi$
$211$	12.5135	0.861465	0.430733	+	0.902480i	$0.358255\pi$
$212$	−43.7865	−3.00727
$213$	0	0
$214$	23.5092	1.60706
$215$	−3.73428	−0.254676
$216$	0	0
$217$	17.9658	1.21960
$218$	13.0963	0.886995
$219$	0	0
$220$	9.52171	0.641954
$221$	−23.7129	−1.59510
$222$	0	0
$223$	−12.8067	−0.857601	−0.428800	−	0.903399i	$-0.641064\pi$
$223$	−12.8067	−0.857601	−0.428800	+	0.903399i	$0.641064\pi$
$224$	−55.4545	−3.70521
$225$	0	0
$226$	−14.8536	−0.988049
$227$	−8.12890	−0.539534	−0.269767	−	0.962926i	$-0.586947\pi$
$227$	−8.12890	−0.539534	−0.269767	+	0.962926i	$0.586947\pi$
$228$	0	0
$229$	−22.7523	−1.50352	−0.751758	−	0.659439i	$-0.770794\pi$
$229$	−22.7523	−1.50352	−0.751758	+	0.659439i	$0.770794\pi$
$230$	−2.46334	−0.162428
$231$	0	0
$232$	−8.95720	−0.588069
$233$	−17.6542	−1.15657	−0.578284	−	0.815835i	$-0.696277\pi$
$233$	−17.6542	−1.15657	−0.578284	+	0.815835i	$0.696277\pi$
$234$	0	0
$235$	−2.68128	−0.174907
$236$	−78.3525	−5.10031
$237$	0	0
$238$	−27.3287	−1.77146
$239$	−19.0506	−1.23228	−0.616142	−	0.787635i	$-0.711305\pi$
$239$	−19.0506	−1.23228	−0.616142	+	0.787635i	$0.711305\pi$
$240$	0	0
$241$	28.7347	1.85096	0.925481	−	0.378793i	$-0.123661\pi$
$241$	28.7347	1.85096	0.925481	+	0.378793i	$0.123661\pi$
$242$	−19.2769	−1.23917
$243$	0	0
$244$	−22.2439	−1.42402
$245$	−1.50465	−0.0961284
$246$	0	0
$247$	33.4065	2.12561
$248$	−54.7102	−3.47410
$249$	0	0
$250$	22.5892	1.42867
$251$	21.0070	1.32595	0.662975	−	0.748642i	$-0.269294\pi$
$251$	21.0070	1.32595	0.662975	+	0.748642i	$0.269294\pi$
$252$	0	0
$253$	−1.96757	−0.123700
$254$	25.5928	1.60583
$255$	0	0
$256$	24.4092	1.52557
$257$	−21.0350	−1.31213	−0.656065	−	0.754705i	$-0.727780\pi$
$257$	−21.0350	−1.31213	−0.656065	+	0.754705i	$0.727780\pi$
$258$	0	0
$259$	−31.6496	−1.96661
$260$	33.3988	2.07131
$261$	0	0
$262$	1.89844	0.117286
$263$	−23.7998	−1.46756	−0.733778	−	0.679389i	$-0.762245\pi$
$263$	−23.7998	−1.46756	−0.733778	+	0.679389i	$0.762245\pi$
$264$	0	0
$265$	7.50834	0.461233
$266$	38.5005	2.36062
$267$	0	0
$268$	40.7686	2.49034
$269$	15.6853	0.956349	0.478175	−	0.878265i	$-0.341299\pi$
$269$	15.6853	0.956349	0.478175	+	0.878265i	$0.341299\pi$
$270$	0	0
$271$	3.24376	0.197044	0.0985222	−	0.995135i	$-0.468588\pi$
$271$	3.24376	0.197044	0.0985222	+	0.995135i	$0.468588\pi$
$272$	46.7169	2.83263
$273$	0	0
$274$	36.3678	2.19706
$275$	8.20511	0.494787
$276$	0	0
$277$	−9.00248	−0.540907	−0.270453	−	0.962733i	$-0.587174\pi$
$277$	−9.00248	−0.540907	−0.270453	+	0.962733i	$0.587174\pi$
$278$	58.4287	3.50432
$279$	0	0
$280$	24.0004	1.43430
$281$	−18.2368	−1.08792	−0.543958	−	0.839112i	$-0.683075\pi$
$281$	−18.2368	−1.08792	−0.543958	+	0.839112i	$0.683075\pi$
$282$	0	0
$283$	1.44497	0.0858944	0.0429472	−	0.999077i	$-0.486325\pi$
$283$	1.44497	0.0858944	0.0429472	+	0.999077i	$0.486325\pi$
$284$	−59.4050	−3.52504
$285$	0	0
$286$	36.7203	2.17132
$287$	0.319982	0.0188880
$288$	0	0
$289$	−5.19473	−0.305572
$290$	2.46334	0.144652
$291$	0	0
$292$	−80.9106	−4.73493
$293$	23.8208	1.39162	0.695812	−	0.718224i	$-0.255045\pi$
$293$	23.8208	1.39162	0.695812	+	0.718224i	$0.255045\pi$
$294$	0	0
$295$	13.4356	0.782250
$296$	96.3805	5.60200
$297$	0	0
$298$	11.2483	0.651598
$299$	−6.90155	−0.399127
$300$	0	0
$301$	−12.0577	−0.694995
$302$	−29.6957	−1.70879
$303$	0	0
$304$	−65.8144	−3.77471
$305$	3.81430	0.218406
$306$	0	0
$307$	4.63178	0.264350	0.132175	−	0.991226i	$-0.457804\pi$
$307$	4.63178	0.264350	0.132175	+	0.991226i	$0.457804\pi$
$308$	30.7449	1.75185
$309$	0	0
$310$	15.0460	0.854553
$311$	−22.9214	−1.29975	−0.649877	−	0.760039i	$-0.725180\pi$
$311$	−22.9214	−1.29975	−0.649877	+	0.760039i	$0.725180\pi$
$312$	0	0
$313$	−30.8433	−1.74337	−0.871684	−	0.490069i	$-0.836972\pi$
$313$	−30.8433	−1.74337	−0.871684	+	0.490069i	$0.836972\pi$
$314$	35.2981	1.99199
$315$	0	0
$316$	−54.2715	−3.05301
$317$	−6.49887	−0.365013	−0.182507	−	0.983205i	$-0.558421\pi$
$317$	−6.49887	−0.365013	−0.182507	+	0.983205i	$0.558421\pi$
$318$	0	0
$319$	1.96757	0.110163
$320$	−21.6699	−1.21139
$321$	0	0
$322$	−7.95393	−0.443255
$323$	−16.6312	−0.925384
$324$	0	0
$325$	28.7806	1.59646
$326$	41.9296	2.32227
$327$	0	0
$328$	−0.974421	−0.0538034
$329$	−8.65763	−0.477310
$330$	0	0
$331$	2.03510	0.111859	0.0559295	−	0.998435i	$-0.482188\pi$
$331$	2.03510	0.111859	0.0559295	+	0.998435i	$0.482188\pi$
$332$	−16.2455	−0.891586
$333$	0	0
$334$	39.1680	2.14318
$335$	−6.99083	−0.381950
$336$	0	0
$337$	5.19917	0.283217	0.141608	−	0.989923i	$-0.454773\pi$
$337$	5.19917	0.283217	0.141608	+	0.989923i	$0.454773\pi$
$338$	93.6482	5.09379
$339$	0	0
$340$	−16.6273	−0.901744
$341$	12.0178	0.650802
$342$	0	0
$343$	15.7313	0.849410
$344$	36.7185	1.97973
$345$	0	0
$346$	−53.4152	−2.87162
$347$	−6.09729	−0.327320	−0.163660	−	0.986517i	$-0.552330\pi$
$347$	−6.09729	−0.327320	−0.163660	+	0.986517i	$0.552330\pi$
$348$	0	0
$349$	12.8479	0.687735	0.343867	−	0.939018i	$-0.388263\pi$
$349$	12.8479	0.687735	0.343867	+	0.939018i	$0.388263\pi$
$350$	33.1692	1.77297
$351$	0	0
$352$	−37.0950	−1.97717
$353$	27.2447	1.45009	0.725045	−	0.688702i	$-0.241819\pi$
$353$	27.2447	1.45009	0.725045	+	0.688702i	$0.241819\pi$
$354$	0	0
$355$	10.1865	0.540646
$356$	37.3388	1.97895
$357$	0	0
$358$	28.9434	1.52971
$359$	−8.52864	−0.450124	−0.225062	−	0.974344i	$-0.572259\pi$
$359$	−8.52864	−0.450124	−0.225062	+	0.974344i	$0.572259\pi$
$360$	0	0
$361$	4.42990	0.233152
$362$	−33.3427	−1.75246
$363$	0	0
$364$	107.842	5.65246
$365$	13.8742	0.726210
$366$	0	0
$367$	4.20088	0.219284	0.109642	−	0.993971i	$-0.465030\pi$
$367$	4.20088	0.219284	0.109642	+	0.993971i	$0.465030\pi$
$368$	13.5968	0.708780
$369$	0	0
$370$	−26.5058	−1.37797
$371$	24.2438	1.25868
$372$	0	0
$373$	−10.2967	−0.533143	−0.266572	−	0.963815i	$-0.585891\pi$
$373$	−10.2967	−0.533143	−0.266572	+	0.963815i	$0.585891\pi$
$374$	−18.2809	−0.945284
$375$	0	0
$376$	26.3645	1.35965
$377$	6.90155	0.355448
$378$	0	0
$379$	−22.9237	−1.17751	−0.588755	−	0.808311i	$-0.700382\pi$
$379$	−22.9237	−1.17751	−0.588755	+	0.808311i	$0.700382\pi$
$380$	23.4245	1.20165
$381$	0	0
$382$	6.05382	0.309740
$383$	21.1104	1.07869	0.539345	−	0.842085i	$-0.318672\pi$
$383$	21.1104	1.07869	0.539345	+	0.842085i	$0.318672\pi$
$384$	0	0
$385$	−5.27201	−0.268686
$386$	55.7637	2.83830
$387$	0	0
$388$	−64.0910	−3.25373
$389$	4.07634	0.206678	0.103339	−	0.994646i	$-0.467047\pi$
$389$	4.07634	0.206678	0.103339	+	0.994646i	$0.467047\pi$
$390$	0	0
$391$	3.43588	0.173760
$392$	14.7949	0.747257
$393$	0	0
$394$	−4.03559	−0.203310
$395$	9.30627	0.468249
$396$	0	0
$397$	24.0777	1.20843	0.604214	−	0.796822i	$-0.293487\pi$
$397$	24.0777	1.20843	0.604214	+	0.796822i	$0.293487\pi$
$398$	47.2890	2.37038
$399$	0	0
$400$	−56.7008	−2.83504
$401$	−12.3329	−0.615874	−0.307937	−	0.951407i	$-0.599639\pi$
$401$	−12.3329	−0.615874	−0.307937	+	0.951407i	$0.599639\pi$
$402$	0	0
$403$	42.1544	2.09986
$404$	−31.9397	−1.58906
$405$	0	0
$406$	7.95393	0.394747
$407$	−21.1713	−1.04942
$408$	0	0
$409$	−20.0144	−0.989649	−0.494825	−	0.868993i	$-0.664768\pi$
$409$	−20.0144	−0.989649	−0.494825	+	0.868993i	$0.664768\pi$
$410$	0.267978	0.0132345
$411$	0	0
$412$	43.2377	2.13017
$413$	43.3825	2.13471
$414$	0	0
$415$	2.78571	0.136745
$416$	−130.116	−6.37948
$417$	0	0
$418$	25.7540	1.25967
$419$	5.01137	0.244822	0.122411	−	0.992480i	$-0.460937\pi$
$419$	5.01137	0.244822	0.122411	+	0.992480i	$0.460937\pi$
$420$	0	0
$421$	−14.9922	−0.730675	−0.365337	−	0.930875i	$-0.619046\pi$
$421$	−14.9922	−0.730675	−0.365337	+	0.930875i	$0.619046\pi$
$422$	33.8383	1.64722
$423$	0	0
$424$	−73.8281	−3.58541
$425$	−14.3282	−0.695020
$426$	0	0
$427$	12.3161	0.596016
$428$	46.1847	2.23242
$429$	0	0
$430$	−10.0980	−0.486971
$431$	−11.8290	−0.569782	−0.284891	−	0.958560i	$-0.591957\pi$
$431$	−11.8290	−0.569782	−0.284891	+	0.958560i	$0.591957\pi$
$432$	0	0
$433$	1.83235	0.0880573	0.0440286	−	0.999030i	$-0.485981\pi$
$433$	1.83235	0.0880573	0.0440286	+	0.999030i	$0.485981\pi$
$434$	48.5822	2.33202
$435$	0	0
$436$	25.7282	1.23216
$437$	−4.84044	−0.231550
$438$	0	0
$439$	12.3719	0.590479	0.295240	−	0.955423i	$-0.404600\pi$
$439$	12.3719	0.590479	0.295240	+	0.955423i	$0.404600\pi$
$440$	16.0545	0.765368
$441$	0	0
$442$	−64.1231	−3.05002
$443$	−17.8116	−0.846256	−0.423128	−	0.906070i	$-0.639068\pi$
$443$	−17.8116	−0.846256	−0.423128	+	0.906070i	$0.639068\pi$
$444$	0	0
$445$	−6.40270	−0.303517
$446$	−34.6312	−1.63983
$447$	0	0
$448$	−69.9705	−3.30579
$449$	31.3888	1.48133	0.740665	−	0.671874i	$-0.234511\pi$
$449$	31.3888	1.48133	0.740665	+	0.671874i	$0.234511\pi$
$450$	0	0
$451$	0.214045	0.0100790
$452$	−29.1805	−1.37254
$453$	0	0
$454$	−21.9817	−1.03165
$455$	−18.4924	−0.866935
$456$	0	0
$457$	13.0475	0.610338	0.305169	−	0.952298i	$-0.401287\pi$
$457$	13.0475	0.610338	0.305169	+	0.952298i	$0.401287\pi$
$458$	−61.5256	−2.87490
$459$	0	0
$460$	−4.83932	−0.225635
$461$	−13.2091	−0.615210	−0.307605	−	0.951514i	$-0.599527\pi$
$461$	−13.2091	−0.615210	−0.307605	+	0.951514i	$0.599527\pi$
$462$	0	0
$463$	9.26754	0.430699	0.215350	−	0.976537i	$-0.430911\pi$
$463$	9.26754	0.430699	0.215350	+	0.976537i	$0.430911\pi$
$464$	−13.5968	−0.631214
$465$	0	0
$466$	−47.7396	−2.21150
$467$	−29.5387	−1.36689	−0.683444	−	0.730003i	$-0.739519\pi$
$467$	−29.5387	−1.36689	−0.683444	+	0.730003i	$0.739519\pi$
$468$	0	0
$469$	−22.5729	−1.04232
$470$	−7.25056	−0.334443
$471$	0	0
$472$	−132.110	−6.08084
$473$	−8.06573	−0.370862
$474$	0	0
$475$	20.1855	0.926173
$476$	−53.6884	−2.46080
$477$	0	0
$478$	−51.5157	−2.35627
$479$	32.2746	1.47467	0.737333	−	0.675529i	$-0.236085\pi$
$479$	32.2746	1.47467	0.737333	+	0.675529i	$0.236085\pi$
$480$	0	0
$481$	−74.2615	−3.38603
$482$	77.7027	3.53926
$483$	0	0
$484$	−37.8703	−1.72138
$485$	10.9901	0.499033
$486$	0	0
$487$	6.35116	0.287798	0.143899	−	0.989592i	$-0.454036\pi$
$487$	6.35116	0.287798	0.143899	+	0.989592i	$0.454036\pi$
$488$	−37.5053	−1.69778
$489$	0	0
$490$	−4.06878	−0.183809
$491$	13.1332	0.592695	0.296347	−	0.955080i	$-0.404231\pi$
$491$	13.1332	0.592695	0.296347	+	0.955080i	$0.404231\pi$
$492$	0	0
$493$	−3.43588	−0.154744
$494$	90.3361	4.06441
$495$	0	0
$496$	−83.0485	−3.72899
$497$	32.8916	1.47539
$498$	0	0
$499$	−17.3213	−0.775410	−0.387705	−	0.921784i	$-0.626732\pi$
$499$	−17.3213	−0.775410	−0.387705	+	0.921784i	$0.626732\pi$
$500$	44.3774	1.98462
$501$	0	0
$502$	56.8059	2.53537
$503$	−21.7050	−0.967778	−0.483889	−	0.875129i	$-0.660776\pi$
$503$	−21.7050	−0.967778	−0.483889	+	0.875129i	$0.660776\pi$
$504$	0	0
$505$	5.47690	0.243719
$506$	−5.32059	−0.236529
$507$	0	0
$508$	50.2780	2.23073
$509$	−42.5936	−1.88793	−0.943963	−	0.330050i	$-0.892934\pi$
$509$	−42.5936	−1.88793	−0.943963	+	0.330050i	$0.892934\pi$
$510$	0	0
$511$	44.7988	1.98178
$512$	12.7649	0.564135
$513$	0	0
$514$	−56.8818	−2.50895
$515$	−7.41424	−0.326710
$516$	0	0
$517$	−5.79132	−0.254702
$518$	−85.5852	−3.76040
$519$	0	0
$520$	56.3135	2.46951
$521$	−22.0512	−0.966083	−0.483041	−	0.875598i	$-0.660468\pi$
$521$	−22.0512	−0.966083	−0.483041	+	0.875598i	$0.660468\pi$
$522$	0	0
$523$	18.4842	0.808258	0.404129	−	0.914702i	$-0.367575\pi$
$523$	18.4842	0.808258	0.404129	+	0.914702i	$0.367575\pi$
$524$	3.72956	0.162927
$525$	0	0
$526$	−64.3580	−2.80614
$527$	−20.9862	−0.914174
$528$	0	0
$529$	1.00000	0.0434783
$530$	20.3036	0.881933
$531$	0	0
$532$	75.6358	3.27923
$533$	0.750794	0.0325205
$534$	0	0
$535$	−7.91958	−0.342393
$536$	68.7396	2.96910
$537$	0	0
$538$	42.4153	1.82865
$539$	−3.24991	−0.139983
$540$	0	0
$541$	0.400817	0.0172325	0.00861623	−	0.999963i	$-0.497257\pi$
$541$	0.400817	0.0172325	0.00861623	+	0.999963i	$0.497257\pi$
$542$	8.77160	0.376772
$543$	0	0
$544$	64.7774	2.77731
$545$	−4.41178	−0.188980
$546$	0	0
$547$	9.57942	0.409587	0.204793	−	0.978805i	$-0.434348\pi$
$547$	9.57942	0.409587	0.204793	+	0.978805i	$0.434348\pi$
$548$	71.4459	3.05202
$549$	0	0
$550$	22.1878	0.946091
$551$	4.84044	0.206210
$552$	0	0
$553$	30.0492	1.27782
$554$	−24.3440	−1.03428
$555$	0	0
$556$	114.785	4.86798
$557$	26.0683	1.10455	0.552275	−	0.833662i	$-0.313760\pi$
$557$	26.0683	1.10455	0.552275	+	0.833662i	$0.313760\pi$
$558$	0	0
$559$	−28.2917	−1.19661
$560$	36.4319	1.53953
$561$	0	0
$562$	−49.3150	−2.08023
$563$	−28.8780	−1.21706	−0.608532	−	0.793529i	$-0.708241\pi$
$563$	−28.8780	−1.21706	−0.608532	+	0.793529i	$0.708241\pi$
$564$	0	0
$565$	5.00377	0.210510
$566$	3.90740	0.164240
$567$	0	0
$568$	−100.162	−4.20272
$569$	5.33108	0.223491	0.111745	−	0.993737i	$-0.464356\pi$
$569$	5.33108	0.223491	0.111745	+	0.993737i	$0.464356\pi$
$570$	0	0
$571$	18.6732	0.781448	0.390724	−	0.920508i	$-0.372225\pi$
$571$	18.6732	0.781448	0.390724	+	0.920508i	$0.372225\pi$
$572$	72.1385	3.01626
$573$	0	0
$574$	0.865279	0.0361160
$575$	−4.17017	−0.173908
$576$	0	0
$577$	−21.9351	−0.913170	−0.456585	−	0.889680i	$-0.650927\pi$
$577$	−21.9351	−0.913170	−0.456585	+	0.889680i	$0.650927\pi$
$578$	−14.0473	−0.584290
$579$	0	0
$580$	4.83932	0.200942
$581$	8.99484	0.373169
$582$	0	0
$583$	16.2174	0.671654
$584$	−136.423	−5.64522
$585$	0	0
$586$	64.4148	2.66095
$587$	−6.55171	−0.270418	−0.135209	−	0.990817i	$-0.543171\pi$
$587$	−6.55171	−0.270418	−0.135209	+	0.990817i	$0.543171\pi$
$588$	0	0
$589$	29.5652	1.21821
$590$	36.3318	1.49575
$591$	0	0
$592$	146.303	6.01301
$593$	−2.79927	−0.114952	−0.0574762	−	0.998347i	$-0.518305\pi$
$593$	−2.79927	−0.114952	−0.0574762	+	0.998347i	$0.518305\pi$
$594$	0	0
$595$	9.20627	0.377420
$596$	22.0978	0.905159
$597$	0	0
$598$	−18.6628	−0.763178
$599$	−37.4352	−1.52956	−0.764781	−	0.644290i	$-0.777153\pi$
$599$	−37.4352	−1.52956	−0.764781	+	0.644290i	$0.777153\pi$
$600$	0	0
$601$	14.0859	0.574575	0.287287	−	0.957844i	$-0.407247\pi$
$601$	14.0859	0.574575	0.287287	+	0.957844i	$0.407247\pi$
$602$	−32.6058	−1.32891
$603$	0	0
$604$	−58.3383	−2.37375
$605$	6.49385	0.264013
$606$	0	0
$607$	7.59659	0.308336	0.154168	−	0.988045i	$-0.450730\pi$
$607$	7.59659	0.308336	0.154168	+	0.988045i	$0.450730\pi$
$608$	−91.2579	−3.70100
$609$	0	0
$610$	10.3144	0.417618
$611$	−20.3139	−0.821813
$612$	0	0
$613$	−41.3566	−1.67038	−0.835189	−	0.549963i	$-0.814642\pi$
$613$	−41.3566	−1.67038	−0.835189	+	0.549963i	$0.814642\pi$
$614$	12.5250	0.505468
$615$	0	0
$616$	51.8387	2.08864
$617$	28.3537	1.14148	0.570738	−	0.821132i	$-0.306657\pi$
$617$	28.3537	1.14148	0.570738	+	0.821132i	$0.306657\pi$
$618$	0	0
$619$	−21.0951	−0.847883	−0.423941	−	0.905690i	$-0.639354\pi$
$619$	−21.0951	−0.847883	−0.423941	+	0.905690i	$0.639354\pi$
$620$	29.5584	1.18709
$621$	0	0
$622$	−61.9828	−2.48528
$623$	−20.6738	−0.828280
$624$	0	0
$625$	13.2412	0.529648
$626$	−83.4048	−3.33353
$627$	0	0
$628$	69.3445	2.76715
$629$	36.9705	1.47411
$630$	0	0
$631$	16.2747	0.647884	0.323942	−	0.946077i	$-0.394992\pi$
$631$	16.2747	0.647884	0.323942	+	0.946077i	$0.394992\pi$
$632$	−91.5068	−3.63995
$633$	0	0
$634$	−17.5739	−0.697949
$635$	−8.62148	−0.342133
$636$	0	0
$637$	−11.3995	−0.451666
$638$	5.32059	0.210644
$639$	0	0
$640$	−24.2500	−0.958564
$641$	−7.18264	−0.283697	−0.141849	−	0.989888i	$-0.545305\pi$
$641$	−7.18264	−0.283697	−0.141849	+	0.989888i	$0.545305\pi$
$642$	0	0
$643$	−32.4045	−1.27791	−0.638955	−	0.769244i	$-0.720633\pi$
$643$	−32.4045	−1.27791	−0.638955	+	0.769244i	$0.720633\pi$
$644$	−15.6258	−0.615743
$645$	0	0
$646$	−44.9731	−1.76944
$647$	19.2642	0.757354	0.378677	−	0.925529i	$-0.376379\pi$
$647$	19.2642	0.757354	0.378677	+	0.925529i	$0.376379\pi$
$648$	0	0
$649$	29.0197	1.13912
$650$	77.8270	3.05263
$651$	0	0
$652$	82.3724	3.22595
$653$	−14.5351	−0.568803	−0.284401	−	0.958705i	$-0.591795\pi$
$653$	−14.5351	−0.568803	−0.284401	+	0.958705i	$0.591795\pi$
$654$	0	0
$655$	−0.639531	−0.0249885
$656$	−1.47914	−0.0577508
$657$	0	0
$658$	−23.4115	−0.912674
$659$	−2.80123	−0.109120	−0.0545601	−	0.998510i	$-0.517376\pi$
$659$	−2.80123	−0.109120	−0.0545601	+	0.998510i	$0.517376\pi$
$660$	0	0
$661$	33.1626	1.28988	0.644939	−	0.764234i	$-0.276883\pi$
$661$	33.1626	1.28988	0.644939	+	0.764234i	$0.276883\pi$
$662$	5.50319	0.213888
$663$	0	0
$664$	−27.3914	−1.06299
$665$	−12.9697	−0.502945
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	76.9470	2.97717
$669$	0	0
$670$	−18.9042	−0.730334
$671$	8.23854	0.318045
$672$	0	0
$673$	−16.5269	−0.637064	−0.318532	−	0.947912i	$-0.603190\pi$
$673$	−16.5269	−0.637064	−0.318532	+	0.947912i	$0.603190\pi$
$674$	14.0593	0.541544
$675$	0	0
$676$	183.975	7.07598
$677$	−42.2959	−1.62556	−0.812782	−	0.582568i	$-0.802048\pi$
$677$	−42.2959	−1.62556	−0.812782	+	0.582568i	$0.802048\pi$
$678$	0	0
$679$	35.4861	1.36183
$680$	−28.0352	−1.07510
$681$	0	0
$682$	32.4980	1.24441
$683$	9.71545	0.371751	0.185876	−	0.982573i	$-0.440488\pi$
$683$	9.71545	0.371751	0.185876	+	0.982573i	$0.440488\pi$
$684$	0	0
$685$	−12.2513	−0.468097
$686$	42.5397	1.62417
$687$	0	0
$688$	55.7377	2.12498
$689$	56.8848	2.16714
$690$	0	0
$691$	−34.5907	−1.31589	−0.657946	−	0.753065i	$-0.728575\pi$
$691$	−34.5907	−1.31589	−0.657946	+	0.753065i	$0.728575\pi$
$692$	−104.936	−3.98908
$693$	0	0
$694$	−16.4879	−0.625874
$695$	−19.6829	−0.746617
$696$	0	0
$697$	−0.373777	−0.0141578
$698$	34.7427	1.31503
$699$	0	0
$700$	65.1622	2.46290
$701$	−17.2076	−0.649924	−0.324962	−	0.945727i	$-0.605351\pi$
$701$	−17.2076	−0.649924	−0.324962	+	0.945727i	$0.605351\pi$
$702$	0	0
$703$	−52.0837	−1.96438
$704$	−46.8051	−1.76404
$705$	0	0
$706$	73.6736	2.77274
$707$	17.6845	0.665093
$708$	0	0
$709$	−49.6732	−1.86552	−0.932758	−	0.360504i	$-0.882605\pi$
$709$	−49.6732	−1.86552	−0.932758	+	0.360504i	$0.882605\pi$
$710$	27.5459	1.03378
$711$	0	0
$712$	62.9566	2.35940
$713$	−6.10796	−0.228745
$714$	0	0
$715$	−12.3700	−0.462613
$716$	56.8604	2.12497
$717$	0	0
$718$	−23.0627	−0.860691
$719$	22.9217	0.854837	0.427418	−	0.904054i	$-0.359423\pi$
$719$	22.9217	0.854837	0.427418	+	0.904054i	$0.359423\pi$
$720$	0	0
$721$	−23.9400	−0.891572
$722$	11.9791	0.445815
$723$	0	0
$724$	−65.5031	−2.43440
$725$	4.17017	0.154876
$726$	0	0
$727$	−9.59833	−0.355982	−0.177991	−	0.984032i	$-0.556960\pi$
$727$	−9.59833	−0.355982	−0.177991	+	0.984032i	$0.556960\pi$
$728$	181.832	6.73914
$729$	0	0
$730$	37.5179	1.38860
$731$	14.0848	0.520946
$732$	0	0
$733$	39.7734	1.46906	0.734532	−	0.678574i	$-0.237402\pi$
$733$	39.7734	1.46906	0.734532	+	0.678574i	$0.237402\pi$
$734$	11.3598	0.419297
$735$	0	0
$736$	18.8532	0.694939
$737$	−15.0996	−0.556201
$738$	0	0
$739$	4.06890	0.149677	0.0748384	−	0.997196i	$-0.476156\pi$
$739$	4.06890	0.149677	0.0748384	+	0.997196i	$0.476156\pi$
$740$	−52.0717	−1.91419
$741$	0	0
$742$	65.5588	2.40674
$743$	−22.9187	−0.840805	−0.420403	−	0.907338i	$-0.638111\pi$
$743$	−22.9187	−0.840805	−0.420403	+	0.907338i	$0.638111\pi$
$744$	0	0
$745$	−3.78924	−0.138827
$746$	−27.8438	−1.01943
$747$	0	0
$748$	−35.9136	−1.31313
$749$	−25.5717	−0.934370
$750$	0	0
$751$	22.4424	0.818934	0.409467	−	0.912325i	$-0.365715\pi$
$751$	22.4424	0.818934	0.409467	+	0.912325i	$0.365715\pi$
$752$	40.0205	1.45940
$753$	0	0
$754$	18.6628	0.679658
$755$	10.0036	0.364069
$756$	0	0
$757$	−29.2186	−1.06197	−0.530984	−	0.847382i	$-0.678178\pi$
$757$	−29.2186	−1.06197	−0.530984	+	0.847382i	$0.678178\pi$
$758$	−61.9889	−2.25154
$759$	0	0
$760$	39.4959	1.43267
$761$	1.02230	0.0370584	0.0185292	−	0.999828i	$-0.494102\pi$
$761$	1.02230	0.0370584	0.0185292	+	0.999828i	$0.494102\pi$
$762$	0	0
$763$	−14.2453	−0.515714
$764$	11.8930	0.430272
$765$	0	0
$766$	57.0855	2.06258
$767$	101.791	3.67546
$768$	0	0
$769$	−28.6115	−1.03176	−0.515879	−	0.856662i	$-0.672534\pi$
$769$	−28.6115	−1.03176	−0.515879	+	0.856662i	$0.672534\pi$
$770$	−14.2563	−0.513760
$771$	0	0
$772$	109.550	3.94279
$773$	−14.2293	−0.511792	−0.255896	−	0.966704i	$-0.582370\pi$
$773$	−14.2293	−0.511792	−0.255896	+	0.966704i	$0.582370\pi$
$774$	0	0
$775$	25.4712	0.914954
$776$	−108.063	−3.87925
$777$	0	0
$778$	11.0230	0.395194
$779$	0.526574	0.0188665
$780$	0	0
$781$	22.0021	0.787295
$782$	9.29112	0.332250
$783$	0	0
$784$	22.4583	0.802081
$785$	−11.8909	−0.424405
$786$	0	0
$787$	39.2969	1.40078	0.700392	−	0.713758i	$-0.253009\pi$
$787$	39.2969	1.40078	0.700392	+	0.713758i	$0.253009\pi$
$788$	−7.92807	−0.282426
$789$	0	0
$790$	25.1655	0.895348
$791$	16.1568	0.574469
$792$	0	0
$793$	28.8979	1.02620
$794$	65.1097	2.31066
$795$	0	0
$796$	92.9011	3.29279
$797$	−43.1769	−1.52940	−0.764702	−	0.644384i	$-0.777114\pi$
$797$	−43.1769	−1.52940	−0.764702	+	0.644384i	$0.777114\pi$
$798$	0	0
$799$	10.1131	0.357777
$800$	−78.6212	−2.77968
$801$	0	0
$802$	−33.3498	−1.17762
$803$	29.9671	1.05752
$804$	0	0
$805$	2.67945	0.0944382
$806$	113.991	4.01518
$807$	0	0
$808$	−53.8533	−1.89455
$809$	14.6184	0.513954	0.256977	−	0.966417i	$-0.417274\pi$
$809$	14.6184	0.513954	0.256977	+	0.966417i	$0.417274\pi$
$810$	0	0
$811$	−13.1087	−0.460308	−0.230154	−	0.973154i	$-0.573923\pi$
$811$	−13.1087	−0.460308	−0.230154	+	0.973154i	$0.573923\pi$
$812$	15.6258	0.548358
$813$	0	0
$814$	−57.2502	−2.00662
$815$	−14.1249	−0.494774
$816$	0	0
$817$	−19.8426	−0.694204
$818$	−54.1219	−1.89233
$819$	0	0
$820$	0.526452	0.0183845
$821$	28.8859	1.00812	0.504062	−	0.863667i	$-0.331838\pi$
$821$	28.8859	1.00812	0.504062	+	0.863667i	$0.331838\pi$
$822$	0	0
$823$	0.170861	0.00595583	0.00297792	−	0.999996i	$-0.499052\pi$
$823$	0.170861	0.00595583	0.00297792	+	0.999996i	$0.499052\pi$
$824$	72.9029	2.53969
$825$	0	0
$826$	117.312	4.08182
$827$	11.2743	0.392046	0.196023	−	0.980599i	$-0.437197\pi$
$827$	11.2743	0.392046	0.196023	+	0.980599i	$0.437197\pi$
$828$	0	0
$829$	−55.2534	−1.91903	−0.959515	−	0.281659i	$-0.909115\pi$
$829$	−55.2534	−1.91903	−0.959515	+	0.281659i	$0.909115\pi$
$830$	7.53296	0.261473
$831$	0	0
$832$	−164.176	−5.69178
$833$	5.67517	0.196633
$834$	0	0
$835$	−13.1946	−0.456617
$836$	50.5948	1.74986
$837$	0	0
$838$	13.5515	0.468128
$839$	−29.0767	−1.00384	−0.501919	−	0.864914i	$-0.667373\pi$
$839$	−29.0767	−1.00384	−0.501919	+	0.864914i	$0.667373\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−40.5411	−1.39714
$843$	0	0
$844$	66.4767	2.28822
$845$	−31.5474	−1.08526
$846$	0	0
$847$	20.9681	0.720474
$848$	−112.069	−3.84846
$849$	0	0
$850$	−38.7455	−1.32896
$851$	10.7601	0.368852
$852$	0	0
$853$	−35.0242	−1.19920	−0.599602	−	0.800298i	$-0.704675\pi$
$853$	−35.0242	−1.19920	−0.599602	+	0.800298i	$0.704675\pi$
$854$	33.3044	1.13965
$855$	0	0
$856$	77.8718	2.66160
$857$	44.7294	1.52793	0.763964	−	0.645259i	$-0.223250\pi$
$857$	44.7294	1.52793	0.763964	+	0.645259i	$0.223250\pi$
$858$	0	0
$859$	−40.7912	−1.39178	−0.695889	−	0.718149i	$-0.744989\pi$
$859$	−40.7912	−1.39178	−0.695889	+	0.718149i	$0.744989\pi$
$860$	−19.8380	−0.676470
$861$	0	0
$862$	−31.9873	−1.08949
$863$	34.3990	1.17095	0.585477	−	0.810689i	$-0.300907\pi$
$863$	34.3990	1.17095	0.585477	+	0.810689i	$0.300907\pi$
$864$	0	0
$865$	17.9941	0.611817
$866$	4.95495	0.168376
$867$	0	0
$868$	95.4417	3.23950
$869$	20.1007	0.681870
$870$	0	0
$871$	−52.9641	−1.79462
$872$	43.3802	1.46904
$873$	0	0
$874$	−13.0893	−0.442751
$875$	−24.5710	−0.830652
$876$	0	0
$877$	3.38095	0.114166	0.0570832	−	0.998369i	$-0.481820\pi$
$877$	3.38095	0.114166	0.0570832	+	0.998369i	$0.481820\pi$
$878$	33.4554	1.12907
$879$	0	0
$880$	24.3703	0.821521
$881$	41.7646	1.40708	0.703542	−	0.710653i	$-0.251601\pi$
$881$	41.7646	1.40708	0.703542	+	0.710653i	$0.251601\pi$
$882$	0	0
$883$	−32.6000	−1.09708	−0.548539	−	0.836125i	$-0.684816\pi$
$883$	−32.6000	−1.09708	−0.548539	+	0.836125i	$0.684816\pi$
$884$	−125.972	−4.23691
$885$	0	0
$886$	−48.1652	−1.61814
$887$	−29.8570	−1.00250	−0.501250	−	0.865302i	$-0.667126\pi$
$887$	−29.8570	−1.00250	−0.501250	+	0.865302i	$0.667126\pi$
$888$	0	0
$889$	−27.8381	−0.933660
$890$	−17.3138	−0.580361
$891$	0	0
$892$	−68.0343	−2.27796
$893$	−14.2473	−0.476768
$894$	0	0
$895$	−9.75020	−0.325913
$896$	−78.3012	−2.61586
$897$	0	0
$898$	84.8799	2.83248
$899$	6.10796	0.203712
$900$	0	0
$901$	−28.3196	−0.943464
$902$	0.578808	0.0192722
$903$	0	0
$904$	−49.2011	−1.63641
$905$	11.2322	0.373372
$906$	0	0
$907$	45.3545	1.50597	0.752986	−	0.658037i	$-0.228613\pi$
$907$	45.3545	1.50597	0.752986	+	0.658037i	$0.228613\pi$
$908$	−43.1839	−1.43311
$909$	0	0
$910$	−50.0060	−1.65768
$911$	−47.6815	−1.57976	−0.789879	−	0.613262i	$-0.789857\pi$
$911$	−47.6815	−1.57976	−0.789879	+	0.613262i	$0.789857\pi$
$912$	0	0
$913$	6.01689	0.199130
$914$	35.2824	1.16704
$915$	0	0
$916$	−120.869	−3.99364
$917$	−2.06500	−0.0681921
$918$	0	0
$919$	18.1949	0.600193	0.300097	−	0.953909i	$-0.402981\pi$
$919$	18.1949	0.600193	0.300097	+	0.953909i	$0.402981\pi$
$920$	−8.15955	−0.269012
$921$	0	0
$922$	−35.7193	−1.17635
$923$	77.1755	2.54026
$924$	0	0
$925$	−44.8715	−1.47537
$926$	25.0608	0.823548
$927$	0	0
$928$	−18.8532	−0.618887
$929$	45.9593	1.50787	0.753937	−	0.656946i	$-0.228152\pi$
$929$	45.9593	1.50787	0.753937	+	0.656946i	$0.228152\pi$
$930$	0	0
$931$	−7.99513	−0.262030
$932$	−93.7864	−3.07207
$933$	0	0
$934$	−79.8769	−2.61365
$935$	6.15832	0.201399
$936$	0	0
$937$	−10.9410	−0.357428	−0.178714	−	0.983901i	$-0.557194\pi$
$937$	−10.9410	−0.357428	−0.178714	+	0.983901i	$0.557194\pi$
$938$	−61.0403	−1.99303
$939$	0	0
$940$	−14.2440	−0.464588
$941$	−14.1704	−0.461941	−0.230971	−	0.972961i	$-0.574190\pi$
$941$	−14.1704	−0.461941	−0.230971	+	0.972961i	$0.574190\pi$
$942$	0	0
$943$	−0.108786	−0.00354257
$944$	−200.539	−6.52698
$945$	0	0
$946$	−21.8109	−0.709133
$947$	−0.302202	−0.00982025	−0.00491012	−	0.999988i	$-0.501563\pi$
$947$	−0.302202	−0.00982025	−0.00491012	+	0.999988i	$0.501563\pi$
$948$	0	0
$949$	105.114	3.41215
$950$	54.5845	1.77095
$951$	0	0
$952$	−90.5236	−2.93389
$953$	2.28603	0.0740517	0.0370258	−	0.999314i	$-0.488212\pi$
$953$	2.28603	0.0740517	0.0370258	+	0.999314i	$0.488212\pi$
$954$	0	0
$955$	−2.03936	−0.0659921
$956$	−101.205	−3.27319
$957$	0	0
$958$	87.2753	2.81974
$959$	−39.5584	−1.27741
$960$	0	0
$961$	6.30715	0.203456
$962$	−200.814	−6.47449
$963$	0	0
$964$	152.650	4.91652
$965$	−18.7852	−0.604717
$966$	0	0
$967$	22.0246	0.708264	0.354132	−	0.935196i	$-0.384776\pi$
$967$	22.0246	0.708264	0.354132	+	0.935196i	$0.384776\pi$
$968$	−63.8529	−2.05231
$969$	0	0
$970$	29.7187	0.954211
$971$	−46.4904	−1.49195	−0.745974	−	0.665975i	$-0.768016\pi$
$971$	−46.4904	−1.49195	−0.745974	+	0.665975i	$0.768016\pi$
$972$	0	0
$973$	−63.5547	−2.03747
$974$	17.1744	0.550305
$975$	0	0
$976$	−56.9319	−1.82235
$977$	25.2675	0.808378	0.404189	−	0.914675i	$-0.367554\pi$
$977$	25.2675	0.808378	0.404189	+	0.914675i	$0.367554\pi$
$978$	0	0
$979$	−13.8293	−0.441986
$980$	−7.99328	−0.255336
$981$	0	0
$982$	35.5142	1.13330
$983$	49.1117	1.56642	0.783209	−	0.621758i	$-0.213581\pi$
$983$	49.1117	1.56642	0.783209	+	0.621758i	$0.213581\pi$
$984$	0	0
$985$	1.35947	0.0433165
$986$	−9.29112	−0.295889
$987$	0	0
$988$	177.469	5.64604
$989$	4.09933	0.130351
$990$	0	0
$991$	45.7654	1.45379	0.726894	−	0.686750i	$-0.240963\pi$
$991$	45.7654	1.45379	0.726894	+	0.686750i	$0.240963\pi$
$992$	−115.155	−3.65616
$993$	0	0
$994$	88.9435	2.82112
$995$	−15.9303	−0.505025
$996$	0	0
$997$	14.5550	0.460961	0.230480	−	0.973077i	$-0.425970\pi$
$997$	14.5550	0.460961	0.230480	+	0.973077i	$0.425970\pi$
$998$	−46.8394	−1.48268
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.k.1.10		10
3.2	odd	2		2001.2.a.k.1.1	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.k.1.1	✓	10	3.2	odd	2
6003.2.a.k.1.10		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)

\(f(q)\)	\(=\)	\(q+2.70414 q^{2} +5.31240 q^{4} -0.910949 q^{5} -2.94138 q^{7} +8.95720 q^{8} +O(q^{10})\)	\(q+2.70414 q^{2} +5.31240 q^{4} -0.910949 q^{5} -2.94138 q^{7} +8.95720 q^{8} -2.46334 q^{10} -1.96757 q^{11} -6.90155 q^{13} -7.95393 q^{14} +13.5968 q^{16} +3.43588 q^{17} -4.84044 q^{19} -4.83932 q^{20} -5.32059 q^{22} +1.00000 q^{23} -4.17017 q^{25} -18.6628 q^{26} -15.6258 q^{28} -1.00000 q^{29} -6.10796 q^{31} +18.8532 q^{32} +9.29112 q^{34} +2.67945 q^{35} +10.7601 q^{37} -13.0893 q^{38} -8.15955 q^{40} -0.108786 q^{41} +4.09933 q^{43} -10.4525 q^{44} +2.70414 q^{46} +2.94339 q^{47} +1.65174 q^{49} -11.2767 q^{50} -36.6638 q^{52} -8.24232 q^{53} +1.79236 q^{55} -26.3466 q^{56} -2.70414 q^{58} -14.7490 q^{59} -4.18717 q^{61} -16.5168 q^{62} +23.7883 q^{64} +6.28696 q^{65} +7.67423 q^{67} +18.2528 q^{68} +7.24562 q^{70} -11.1823 q^{71} -15.2305 q^{73} +29.0969 q^{74} -25.7144 q^{76} +5.78738 q^{77} -10.2160 q^{79} -12.3860 q^{80} -0.294174 q^{82} -3.05803 q^{83} -3.12991 q^{85} +11.0852 q^{86} -17.6239 q^{88} +7.02861 q^{89} +20.3001 q^{91} +5.31240 q^{92} +7.95934 q^{94} +4.40940 q^{95} -12.0644 q^{97} +4.46653 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) \) 10 * q - 3 * q^2 + 17 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8	\( 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8} - 4 q^{10} - 9 q^{11} - 16 q^{13} - 16 q^{14} + 27 q^{16} + q^{19} - 21 q^{20} + 17 q^{22} + 10 q^{23} - 4 q^{25} - 28 q^{26} - 14 q^{28} - 10 q^{29} + 17 q^{31} - 21 q^{32} - 3 q^{34} - 29 q^{35} + q^{37} - 32 q^{38} + 13 q^{40} - 5 q^{43} - 33 q^{44} - 3 q^{46} - 15 q^{47} + 31 q^{49} + 22 q^{50} - 21 q^{52} - 35 q^{53} - 20 q^{55} - 18 q^{56} + 3 q^{58} - 49 q^{59} + 8 q^{61} - 15 q^{62} + 12 q^{64} + 3 q^{65} + 35 q^{67} + 18 q^{68} - 16 q^{70} - 30 q^{71} - 15 q^{73} - 23 q^{74} + 10 q^{76} - 23 q^{77} + 24 q^{79} - 23 q^{80} - 5 q^{82} - q^{83} + 10 q^{86} + 18 q^{88} - 15 q^{89} + 26 q^{91} + 17 q^{92} + 3 q^{94} - 7 q^{95} - 35 q^{97} - 3 q^{98}+O(q^{100}) \) 10 * q - 3 * q^2 + 17 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8 - 4 * q^10 - 9 * q^11 - 16 * q^13 - 16 * q^14 + 27 * q^16 + q^19 - 21 * q^20 + 17 * q^22 + 10 * q^23 - 4 * q^25 - 28 * q^26 - 14 * q^28 - 10 * q^29 + 17 * q^31 - 21 * q^32 - 3 * q^34 - 29 * q^35 + q^37 - 32 * q^38 + 13 * q^40 - 5 * q^43 - 33 * q^44 - 3 * q^46 - 15 * q^47 + 31 * q^49 + 22 * q^50 - 21 * q^52 - 35 * q^53 - 20 * q^55 - 18 * q^56 + 3 * q^58 - 49 * q^59 + 8 * q^61 - 15 * q^62 + 12 * q^64 + 3 * q^65 + 35 * q^67 + 18 * q^68 - 16 * q^70 - 30 * q^71 - 15 * q^73 - 23 * q^74 + 10 * q^76 - 23 * q^77 + 24 * q^79 - 23 * q^80 - 5 * q^82 - q^83 + 10 * q^86 + 18 * q^88 - 15 * q^89 + 26 * q^91 + 17 * q^92 + 3 * q^94 - 7 * q^95 - 35 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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