LMFDB - Embedded newform 6003.2.a.u.1.11

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:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
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)

$f(q)$	$=$	$q-0.144466 q^{2} -1.97913 q^{4} +4.08805 q^{5} -0.828594 q^{7} +0.574849 q^{8} +O(q^{10})$	\(q-0.144466 q^{2} -1.97913 q^{4} +4.08805 q^{5} -0.828594 q^{7} +0.574849 q^{8} -0.590584 q^{10} -0.0328797 q^{11} -3.12976 q^{13} +0.119704 q^{14} +3.87521 q^{16} -1.27064 q^{17} -4.11917 q^{19} -8.09078 q^{20} +0.00474999 q^{22} -1.00000 q^{23} +11.7122 q^{25} +0.452143 q^{26} +1.63989 q^{28} -1.00000 q^{29} +6.85696 q^{31} -1.70953 q^{32} +0.183564 q^{34} -3.38733 q^{35} -3.30243 q^{37} +0.595079 q^{38} +2.35001 q^{40} -6.64755 q^{41} -1.68035 q^{43} +0.0650731 q^{44} +0.144466 q^{46} -5.69373 q^{47} -6.31343 q^{49} -1.69201 q^{50} +6.19419 q^{52} -7.40328 q^{53} -0.134414 q^{55} -0.476316 q^{56} +0.144466 q^{58} +2.70220 q^{59} +7.51934 q^{61} -0.990597 q^{62} -7.50346 q^{64} -12.7946 q^{65} -4.02039 q^{67} +2.51476 q^{68} +0.489354 q^{70} +3.73962 q^{71} +2.01339 q^{73} +0.477089 q^{74} +8.15236 q^{76} +0.0272439 q^{77} +15.8864 q^{79} +15.8421 q^{80} +0.960344 q^{82} -11.5703 q^{83} -5.19443 q^{85} +0.242754 q^{86} -0.0189008 q^{88} +7.36813 q^{89} +2.59330 q^{91} +1.97913 q^{92} +0.822550 q^{94} -16.8394 q^{95} -17.3350 q^{97} +0.912076 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8}+O(q^{10}) $ 22 * q + 3 * q^2 + 17 * q^4 - 6 * q^7 + 6 * q^8	$ 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8} - 12 q^{10} - 28 q^{13} + q^{14} + 3 q^{16} + 10 q^{17} - 8 q^{19} - 11 q^{22} - 22 q^{23} - 11 q^{26} - 21 q^{28} - 22 q^{29} - 18 q^{31} - 5 q^{32} - 33 q^{34} - 2 q^{35} - 28 q^{37} - 14 q^{38} - 30 q^{40} + 10 q^{41} - 14 q^{43} - 37 q^{44} - 3 q^{46} + 18 q^{47} + 2 q^{49} - 7 q^{50} - 57 q^{52} - 20 q^{53} - 42 q^{55} + 2 q^{56} - 3 q^{58} + 20 q^{59} - 38 q^{61} - 4 q^{62} - 24 q^{64} - 12 q^{65} - 50 q^{67} - 11 q^{68} - 48 q^{70} - 12 q^{71} - 46 q^{73} + 6 q^{74} - 16 q^{76} + 14 q^{77} - 20 q^{79} + 58 q^{80} - 42 q^{82} - 22 q^{83} - 66 q^{85} - 22 q^{86} - 68 q^{88} + 14 q^{89} - 16 q^{91} - 17 q^{92} - 27 q^{94} + 20 q^{95} - 48 q^{97} + 28 q^{98}+O(q^{100}) $ 22 * q + 3 * q^2 + 17 * q^4 - 6 * q^7 + 6 * q^8 - 12 * q^10 - 28 * q^13 + q^14 + 3 * q^16 + 10 * q^17 - 8 * q^19 - 11 * q^22 - 22 * q^23 - 11 * q^26 - 21 * q^28 - 22 * q^29 - 18 * q^31 - 5 * q^32 - 33 * q^34 - 2 * q^35 - 28 * q^37 - 14 * q^38 - 30 * q^40 + 10 * q^41 - 14 * q^43 - 37 * q^44 - 3 * q^46 + 18 * q^47 + 2 * q^49 - 7 * q^50 - 57 * q^52 - 20 * q^53 - 42 * q^55 + 2 * q^56 - 3 * q^58 + 20 * q^59 - 38 * q^61 - 4 * q^62 - 24 * q^64 - 12 * q^65 - 50 * q^67 - 11 * q^68 - 48 * q^70 - 12 * q^71 - 46 * q^73 + 6 * q^74 - 16 * q^76 + 14 * q^77 - 20 * q^79 + 58 * q^80 - 42 * q^82 - 22 * q^83 - 66 * q^85 - 22 * q^86 - 68 * q^88 + 14 * q^89 - 16 * q^91 - 17 * q^92 - 27 * q^94 + 20 * q^95 - 48 * q^97 + 28 * 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 $

$2$	−0.144466	−0.102153	−0.0510764	−	0.998695i	$-0.516265\pi$
$2$	−0.144466	−0.102153	−0.0510764	+	0.998695i	$0.516265\pi$
$3$	0	0
$3$	0	0
$4$	−1.97913	−0.989565
$5$	4.08805	1.82823	0.914116	−	0.405452i	$-0.132886\pi$
$5$	4.08805	1.82823	0.914116	+	0.405452i	$0.132886\pi$
$6$	0	0
$7$	−0.828594	−0.313179	−0.156589	−	0.987664i	$-0.550050\pi$
$7$	−0.828594	−0.313179	−0.156589	+	0.987664i	$0.550050\pi$
$8$	0.574849	0.203240
$9$	0	0
$10$	−0.590584	−0.186759
$11$	−0.0328797	−0.00991359	−0.00495680	−	0.999988i	$-0.501578\pi$
$11$	−0.0328797	−0.00991359	−0.00495680	+	0.999988i	$0.501578\pi$
$12$	0	0
$13$	−3.12976	−0.868038	−0.434019	−	0.900904i	$-0.642905\pi$
$13$	−3.12976	−0.868038	−0.434019	+	0.900904i	$0.642905\pi$
$14$	0.119704	0.0319921
$15$	0	0
$16$	3.87521	0.968803
$17$	−1.27064	−0.308175	−0.154087	−	0.988057i	$-0.549244\pi$
$17$	−1.27064	−0.308175	−0.154087	+	0.988057i	$0.549244\pi$
$18$	0	0
$19$	−4.11917	−0.945001	−0.472501	−	0.881330i	$-0.656649\pi$
$19$	−4.11917	−0.945001	−0.472501	+	0.881330i	$0.656649\pi$
$20$	−8.09078	−1.80915
$21$	0	0
$22$	0.00474999	0.00101270
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	11.7122	2.34243
$26$	0.452143	0.0886726
$27$	0	0
$28$	1.63989	0.309911
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	6.85696	1.23155	0.615773	−	0.787924i	$-0.288844\pi$
$31$	6.85696	1.23155	0.615773	+	0.787924i	$0.288844\pi$
$32$	−1.70953	−0.302206
$33$	0	0
$34$	0.183564	0.0314809
$35$	−3.38733	−0.572564
$36$	0	0
$37$	−3.30243	−0.542917	−0.271458	−	0.962450i	$-0.587506\pi$
$37$	−3.30243	−0.542917	−0.271458	+	0.962450i	$0.587506\pi$
$38$	0.595079	0.0965346
$39$	0	0
$40$	2.35001	0.371569
$41$	−6.64755	−1.03817	−0.519086	−	0.854722i	$-0.673728\pi$
$41$	−6.64755	−1.03817	−0.519086	+	0.854722i	$0.673728\pi$
$42$	0	0
$43$	−1.68035	−0.256251	−0.128126	−	0.991758i	$-0.540896\pi$
$43$	−1.68035	−0.256251	−0.128126	+	0.991758i	$0.540896\pi$
$44$	0.0650731	0.00981014
$45$	0	0
$46$	0.144466	0.0213003
$47$	−5.69373	−0.830515	−0.415258	−	0.909704i	$-0.636309\pi$
$47$	−5.69373	−0.830515	−0.415258	+	0.909704i	$0.636309\pi$
$48$	0	0
$49$	−6.31343	−0.901919
$50$	−1.69201	−0.239286
$51$	0	0
$52$	6.19419	0.858980
$53$	−7.40328	−1.01692	−0.508459	−	0.861086i	$-0.669785\pi$
$53$	−7.40328	−1.01692	−0.508459	+	0.861086i	$0.669785\pi$
$54$	0	0
$55$	−0.134414	−0.0181243
$56$	−0.476316	−0.0636504
$57$	0	0
$58$	0.144466	0.0189693
$59$	2.70220	0.351796	0.175898	−	0.984408i	$-0.443717\pi$
$59$	2.70220	0.351796	0.175898	+	0.984408i	$0.443717\pi$
$60$	0	0
$61$	7.51934	0.962753	0.481376	−	0.876514i	$-0.340137\pi$
$61$	7.51934	0.962753	0.481376	+	0.876514i	$0.340137\pi$
$62$	−0.990597	−0.125806
$63$	0	0
$64$	−7.50346	−0.937932
$65$	−12.7946	−1.58698
$66$	0	0
$67$	−4.02039	−0.491169	−0.245585	−	0.969375i	$-0.578980\pi$
$67$	−4.02039	−0.491169	−0.245585	+	0.969375i	$0.578980\pi$
$68$	2.51476	0.304959
$69$	0	0
$70$	0.489354	0.0584890
$71$	3.73962	0.443812	0.221906	−	0.975068i	$-0.428772\pi$
$71$	3.73962	0.443812	0.221906	+	0.975068i	$0.428772\pi$
$72$	0	0
$73$	2.01339	0.235649	0.117825	−	0.993034i	$-0.462408\pi$
$73$	2.01339	0.235649	0.117825	+	0.993034i	$0.462408\pi$
$74$	0.477089	0.0554605
$75$	0	0
$76$	8.15236	0.935140
$77$	0.0272439	0.00310473
$78$	0	0
$79$	15.8864	1.78736	0.893682	−	0.448702i	$-0.148113\pi$
$79$	15.8864	1.78736	0.893682	+	0.448702i	$0.148113\pi$
$80$	15.8421	1.77120
$81$	0	0
$82$	0.960344	0.106052
$83$	−11.5703	−1.27000	−0.635002	−	0.772511i	$-0.719001\pi$
$83$	−11.5703	−1.27000	−0.635002	+	0.772511i	$0.719001\pi$
$84$	0	0
$85$	−5.19443	−0.563415
$86$	0.242754	0.0261768
$87$	0	0
$88$	−0.0189008	−0.00201484
$89$	7.36813	0.781020	0.390510	−	0.920599i	$-0.372299\pi$
$89$	7.36813	0.781020	0.390510	+	0.920599i	$0.372299\pi$
$90$	0	0
$91$	2.59330	0.271851
$92$	1.97913	0.206339
$93$	0	0
$94$	0.822550	0.0848395
$95$	−16.8394	−1.72768
$96$	0	0
$97$	−17.3350	−1.76010	−0.880051	−	0.474879i	$-0.842492\pi$
$97$	−17.3350	−1.76010	−0.880051	+	0.474879i	$0.842492\pi$
$98$	0.912076	0.0921336
$99$	0	0
$100$	−23.1799	−2.31799
$101$	2.51480	0.250232	0.125116	−	0.992142i	$-0.460070\pi$
$101$	2.51480	0.250232	0.125116	+	0.992142i	$0.460070\pi$
$102$	0	0
$103$	−7.00357	−0.690083	−0.345041	−	0.938588i	$-0.612135\pi$
$103$	−7.00357	−0.690083	−0.345041	+	0.938588i	$0.612135\pi$
$104$	−1.79914	−0.176420
$105$	0	0
$106$	1.06952	0.103881
$107$	−5.54507	−0.536062	−0.268031	−	0.963410i	$-0.586373\pi$
$107$	−5.54507	−0.536062	−0.268031	+	0.963410i	$0.586373\pi$
$108$	0	0
$109$	−5.88688	−0.563860	−0.281930	−	0.959435i	$-0.590975\pi$
$109$	−5.88688	−0.563860	−0.281930	+	0.959435i	$0.590975\pi$
$110$	0.0194182	0.00185145
$111$	0	0
$112$	−3.21098	−0.303409
$113$	−18.6370	−1.75322	−0.876611	−	0.481199i	$-0.840201\pi$
$113$	−18.6370	−1.75322	−0.876611	+	0.481199i	$0.840201\pi$
$114$	0	0
$115$	−4.08805	−0.381213
$116$	1.97913	0.183758
$117$	0	0
$118$	−0.390376	−0.0359370
$119$	1.05284	0.0965139
$120$	0	0
$121$	−10.9989	−0.999902
$122$	−1.08629	−0.0983480
$123$	0	0
$124$	−13.5708	−1.21869
$125$	27.4397	2.45428
$126$	0	0
$127$	−15.5960	−1.38393	−0.691963	−	0.721933i	$-0.743254\pi$
$127$	−15.5960	−1.38393	−0.691963	+	0.721933i	$0.743254\pi$
$128$	4.50306	0.398018
$129$	0	0
$130$	1.84838	0.162114
$131$	6.67906	0.583552	0.291776	−	0.956487i	$-0.405754\pi$
$131$	6.67906	0.583552	0.291776	+	0.956487i	$0.405754\pi$
$132$	0	0
$133$	3.41311	0.295955
$134$	0.580810	0.0501743
$135$	0	0
$136$	−0.730424	−0.0626334
$137$	−13.8703	−1.18502	−0.592508	−	0.805565i	$-0.701862\pi$
$137$	−13.8703	−1.18502	−0.592508	+	0.805565i	$0.701862\pi$
$138$	0	0
$139$	4.53966	0.385049	0.192524	−	0.981292i	$-0.438333\pi$
$139$	4.53966	0.385049	0.192524	+	0.981292i	$0.438333\pi$
$140$	6.70397	0.566589
$141$	0	0
$142$	−0.540248	−0.0453366
$143$	0.102905	0.00860538
$144$	0	0
$145$	−4.08805	−0.339494
$146$	−0.290866	−0.0240723
$147$	0	0
$148$	6.53594	0.537251
$149$	−15.0677	−1.23440	−0.617199	−	0.786807i	$-0.711733\pi$
$149$	−15.0677	−1.23440	−0.617199	+	0.786807i	$0.711733\pi$
$150$	0	0
$151$	4.02276	0.327368	0.163684	−	0.986513i	$-0.447662\pi$
$151$	4.02276	0.327368	0.163684	+	0.986513i	$0.447662\pi$
$152$	−2.36790	−0.192062
$153$	0	0
$154$	−0.00393581	−0.000317157 0
$155$	28.0316	2.25155
$156$	0	0
$157$	−0.424831	−0.0339052	−0.0169526	−	0.999856i	$-0.505396\pi$
$157$	−0.424831	−0.0339052	−0.0169526	+	0.999856i	$0.505396\pi$
$158$	−2.29505	−0.182584
$159$	0	0
$160$	−6.98866	−0.552502
$161$	0.828594	0.0653023
$162$	0	0
$163$	−10.2009	−0.798993	−0.399497	−	0.916735i	$-0.630815\pi$
$163$	−10.2009	−0.798993	−0.399497	+	0.916735i	$0.630815\pi$
$164$	13.1564	1.02734
$165$	0	0
$166$	1.67151	0.129735
$167$	19.4197	1.50274	0.751370	−	0.659882i	$-0.229394\pi$
$167$	19.4197	1.50274	0.751370	+	0.659882i	$0.229394\pi$
$168$	0	0
$169$	−3.20463	−0.246510
$170$	0.750419	0.0575545
$171$	0	0
$172$	3.32564	0.253577
$173$	22.8330	1.73596	0.867979	−	0.496601i	$-0.165419\pi$
$173$	22.8330	1.73596	0.867979	+	0.496601i	$0.165419\pi$
$174$	0	0
$175$	−9.70463	−0.733601
$176$	−0.127416	−0.00960432
$177$	0	0
$178$	−1.06444	−0.0797834
$179$	−9.79123	−0.731831	−0.365915	−	0.930648i	$-0.619244\pi$
$179$	−9.79123	−0.731831	−0.365915	+	0.930648i	$0.619244\pi$
$180$	0	0
$181$	20.8077	1.54662	0.773310	−	0.634028i	$-0.218599\pi$
$181$	20.8077	1.54662	0.773310	+	0.634028i	$0.218599\pi$
$182$	−0.374643	−0.0277704
$183$	0	0
$184$	−0.574849	−0.0423784
$185$	−13.5005	−0.992578
$186$	0	0
$187$	0.0417781	0.00305512
$188$	11.2686	0.821849
$189$	0	0
$190$	2.43271	0.176488
$191$	4.12720	0.298634	0.149317	−	0.988789i	$-0.452293\pi$
$191$	4.12720	0.298634	0.149317	+	0.988789i	$0.452293\pi$
$192$	0	0
$193$	23.4133	1.68533	0.842664	−	0.538440i	$-0.180986\pi$
$193$	23.4133	1.68533	0.842664	+	0.538440i	$0.180986\pi$
$194$	2.50432	0.179799
$195$	0	0
$196$	12.4951	0.892507
$197$	14.0371	1.00010	0.500051	−	0.865996i	$-0.333314\pi$
$197$	14.0371	1.00010	0.500051	+	0.865996i	$0.333314\pi$
$198$	0	0
$199$	0.624891	0.0442973	0.0221487	−	0.999755i	$-0.492949\pi$
$199$	0.624891	0.0442973	0.0221487	+	0.999755i	$0.492949\pi$
$200$	6.73273	0.476076
$201$	0	0
$202$	−0.363303	−0.0255619
$203$	0.828594	0.0581559
$204$	0	0
$205$	−27.1755	−1.89802
$206$	1.01178	0.0704939
$207$	0	0
$208$	−12.1285	−0.840958
$209$	0.135437	0.00936836
$210$	0	0
$211$	−13.4759	−0.927717	−0.463858	−	0.885909i	$-0.653535\pi$
$211$	−13.4759	−0.927717	−0.463858	+	0.885909i	$0.653535\pi$
$212$	14.6521	1.00631
$213$	0	0
$214$	0.801074	0.0547603
$215$	−6.86937	−0.468487
$216$	0	0
$217$	−5.68163	−0.385694
$218$	0.850453	0.0576000
$219$	0	0
$220$	0.266022	0.0179352
$221$	3.97679	0.267508
$222$	0	0
$223$	3.35902	0.224937	0.112468	−	0.993655i	$-0.464124\pi$
$223$	3.35902	0.224937	0.112468	+	0.993655i	$0.464124\pi$
$224$	1.41651	0.0946445
$225$	0	0
$226$	2.69241	0.179097
$227$	−15.9164	−1.05641	−0.528204	−	0.849118i	$-0.677134\pi$
$227$	−15.9164	−1.05641	−0.528204	+	0.849118i	$0.677134\pi$
$228$	0	0
$229$	−17.0653	−1.12771	−0.563853	−	0.825875i	$-0.690682\pi$
$229$	−17.0653	−1.12771	−0.563853	+	0.825875i	$0.690682\pi$
$230$	0.590584	0.0389420
$231$	0	0
$232$	−0.574849	−0.0377407
$233$	−17.5655	−1.15076	−0.575378	−	0.817887i	$-0.695145\pi$
$233$	−17.5655	−1.15076	−0.575378	+	0.817887i	$0.695145\pi$
$234$	0	0
$235$	−23.2763	−1.51838
$236$	−5.34800	−0.348125
$237$	0	0
$238$	−0.152100	−0.00985917
$239$	18.2475	1.18033	0.590166	−	0.807282i	$-0.299062\pi$
$239$	18.2475	1.18033	0.590166	+	0.807282i	$0.299062\pi$
$240$	0	0
$241$	−5.35400	−0.344882	−0.172441	−	0.985020i	$-0.555165\pi$
$241$	−5.35400	−0.344882	−0.172441	+	0.985020i	$0.555165\pi$
$242$	1.58897	0.102143
$243$	0	0
$244$	−14.8817	−0.952706
$245$	−25.8096	−1.64892
$246$	0	0
$247$	12.8920	0.820297
$248$	3.94171	0.250299
$249$	0	0
$250$	−3.96410	−0.250712
$251$	−14.0396	−0.886170	−0.443085	−	0.896480i	$-0.646116\pi$
$251$	−14.0396	−0.886170	−0.443085	+	0.896480i	$0.646116\pi$
$252$	0	0
$253$	0.0328797	0.00206713
$254$	2.25310	0.141372
$255$	0	0
$256$	14.3564	0.897273
$257$	9.51274	0.593388	0.296694	−	0.954973i	$-0.404116\pi$
$257$	9.51274	0.593388	0.296694	+	0.954973i	$0.404116\pi$
$258$	0	0
$259$	2.73637	0.170030
$260$	25.3222	1.57042
$261$	0	0
$262$	−0.964896	−0.0596115
$263$	4.44916	0.274347	0.137174	−	0.990547i	$-0.456198\pi$
$263$	4.44916	0.274347	0.137174	+	0.990547i	$0.456198\pi$
$264$	0	0
$265$	−30.2650	−1.85916
$266$	−0.493079	−0.0302326
$267$	0	0
$268$	7.95688	0.486044
$269$	25.7794	1.57180	0.785899	−	0.618355i	$-0.212201\pi$
$269$	25.7794	1.57180	0.785899	+	0.618355i	$0.212201\pi$
$270$	0	0
$271$	25.2632	1.53463	0.767314	−	0.641272i	$-0.221593\pi$
$271$	25.2632	1.53463	0.767314	+	0.641272i	$0.221593\pi$
$272$	−4.92399	−0.298561
$273$	0	0
$274$	2.00378	0.121053
$275$	−0.385092	−0.0232219
$276$	0	0
$277$	−23.9622	−1.43975	−0.719875	−	0.694104i	$-0.755801\pi$
$277$	−23.9622	−1.43975	−0.719875	+	0.694104i	$0.755801\pi$
$278$	−0.655826	−0.0393338
$279$	0	0
$280$	−1.94720	−0.116368
$281$	−15.0553	−0.898123	−0.449062	−	0.893501i	$-0.648242\pi$
$281$	−15.0553	−0.898123	−0.449062	+	0.893501i	$0.648242\pi$
$282$	0	0
$283$	23.6312	1.40473	0.702365	−	0.711817i	$-0.252128\pi$
$283$	23.6312	1.40473	0.702365	+	0.711817i	$0.252128\pi$
$284$	−7.40120	−0.439180
$285$	0	0
$286$	−0.0148663	−0.000879064 0
$287$	5.50811	0.325134
$288$	0	0
$289$	−15.3855	−0.905028
$290$	0.590584	0.0346803
$291$	0	0
$292$	−3.98476	−0.233190
$293$	−6.33215	−0.369928	−0.184964	−	0.982745i	$-0.559217\pi$
$293$	−6.33215	−0.369928	−0.184964	+	0.982745i	$0.559217\pi$
$294$	0	0
$295$	11.0467	0.643165
$296$	−1.89840	−0.110342
$297$	0	0
$298$	2.17678	0.126097
$299$	3.12976	0.180998
$300$	0	0
$301$	1.39233	0.0802525
$302$	−0.581152	−0.0334416
$303$	0	0
$304$	−15.9626	−0.915520
$305$	30.7395	1.76014
$306$	0	0
$307$	−13.6720	−0.780301	−0.390150	−	0.920751i	$-0.627577\pi$
$307$	−13.6720	−0.780301	−0.390150	+	0.920751i	$0.627577\pi$
$308$	−0.0539192	−0.00307233
$309$	0	0
$310$	−4.04961	−0.230002
$311$	21.5379	1.22130	0.610651	−	0.791900i	$-0.290908\pi$
$311$	21.5379	1.22130	0.610651	+	0.791900i	$0.290908\pi$
$312$	0	0
$313$	−17.1243	−0.967921	−0.483961	−	0.875090i	$-0.660802\pi$
$313$	−17.1243	−0.967921	−0.483961	+	0.875090i	$0.660802\pi$
$314$	0.0613736	0.00346351
$315$	0	0
$316$	−31.4413	−1.76871
$317$	−15.7386	−0.883971	−0.441985	−	0.897022i	$-0.645726\pi$
$317$	−15.7386	−0.883971	−0.441985	+	0.897022i	$0.645726\pi$
$318$	0	0
$319$	0.0328797	0.00184091
$320$	−30.6745	−1.71476
$321$	0	0
$322$	−0.119704	−0.00667082
$323$	5.23397	0.291226
$324$	0	0
$325$	−36.6562	−2.03332
$326$	1.47368	0.0816194
$327$	0	0
$328$	−3.82133	−0.210998
$329$	4.71779	0.260100
$330$	0	0
$331$	−3.18162	−0.174878	−0.0874389	−	0.996170i	$-0.527868\pi$
$331$	−3.18162	−0.174878	−0.0874389	+	0.996170i	$0.527868\pi$
$332$	22.8991	1.25675
$333$	0	0
$334$	−2.80548	−0.153509
$335$	−16.4356	−0.897971
$336$	0	0
$337$	−6.70392	−0.365186	−0.182593	−	0.983189i	$-0.558449\pi$
$337$	−6.70392	−0.365186	−0.182593	+	0.983189i	$0.558449\pi$
$338$	0.462959	0.0251817
$339$	0	0
$340$	10.2805	0.557536
$341$	−0.225454	−0.0122090
$342$	0	0
$343$	11.0314	0.595641
$344$	−0.965949	−0.0520805
$345$	0	0
$346$	−3.29859	−0.177333
$347$	11.8806	0.637785	0.318893	−	0.947791i	$-0.396689\pi$
$347$	11.8806	0.637785	0.318893	+	0.947791i	$0.396689\pi$
$348$	0	0
$349$	−21.8783	−1.17112	−0.585558	−	0.810630i	$-0.699125\pi$
$349$	−21.8783	−1.17112	−0.585558	+	0.810630i	$0.699125\pi$
$350$	1.40199	0.0749394
$351$	0	0
$352$	0.0562089	0.00299594
$353$	−10.4828	−0.557943	−0.278971	−	0.960299i	$-0.589993\pi$
$353$	−10.4828	−0.557943	−0.278971	+	0.960299i	$0.589993\pi$
$354$	0	0
$355$	15.2878	0.811391
$356$	−14.5825	−0.772870
$357$	0	0
$358$	1.41450	0.0747586
$359$	−30.0157	−1.58417	−0.792085	−	0.610411i	$-0.791004\pi$
$359$	−30.0157	−1.58417	−0.792085	+	0.610411i	$0.791004\pi$
$360$	0	0
$361$	−2.03248	−0.106972
$362$	−3.00600	−0.157992
$363$	0	0
$364$	−5.13247	−0.269014
$365$	8.23084	0.430822
$366$	0	0
$367$	−31.6753	−1.65344	−0.826719	−	0.562615i	$-0.809795\pi$
$367$	−31.6753	−1.65344	−0.826719	+	0.562615i	$0.809795\pi$
$368$	−3.87521	−0.202009
$369$	0	0
$370$	1.95037	0.101395
$371$	6.13431	0.318478
$372$	0	0
$373$	−27.6937	−1.43392	−0.716962	−	0.697113i	$-0.754468\pi$
$373$	−27.6937	−1.43392	−0.716962	+	0.697113i	$0.754468\pi$
$374$	−0.00603552	−0.000312089 0
$375$	0	0
$376$	−3.27303	−0.168794
$377$	3.12976	0.161191
$378$	0	0
$379$	−18.4048	−0.945393	−0.472696	−	0.881225i	$-0.656719\pi$
$379$	−18.4048	−0.945393	−0.472696	+	0.881225i	$0.656719\pi$
$380$	33.3273	1.70965
$381$	0	0
$382$	−0.596240	−0.0305063
$383$	−33.8734	−1.73085	−0.865425	−	0.501038i	$-0.832952\pi$
$383$	−33.8734	−1.73085	−0.865425	+	0.501038i	$0.832952\pi$
$384$	0	0
$385$	0.111374	0.00567616
$386$	−3.38243	−0.172161
$387$	0	0
$388$	34.3082	1.74173
$389$	−14.9510	−0.758045	−0.379023	−	0.925387i	$-0.623740\pi$
$389$	−14.9510	−0.758045	−0.379023	+	0.925387i	$0.623740\pi$
$390$	0	0
$391$	1.27064	0.0642589
$392$	−3.62927	−0.183306
$393$	0	0
$394$	−2.02788	−0.102163
$395$	64.9446	3.26772
$396$	0	0
$397$	−18.7899	−0.943036	−0.471518	−	0.881856i	$-0.656294\pi$
$397$	−18.7899	−0.943036	−0.471518	+	0.881856i	$0.656294\pi$
$398$	−0.0902754	−0.00452510
$399$	0	0
$400$	45.3871	2.26936
$401$	−18.9879	−0.948213	−0.474106	−	0.880468i	$-0.657229\pi$
$401$	−18.9879	−0.948213	−0.474106	+	0.880468i	$0.657229\pi$
$402$	0	0
$403$	−21.4606	−1.06903
$404$	−4.97712	−0.247621
$405$	0	0
$406$	−0.119704	−0.00594079
$407$	0.108583	0.00538225
$408$	0	0
$409$	6.58997	0.325853	0.162927	−	0.986638i	$-0.447907\pi$
$409$	6.58997	0.325853	0.162927	+	0.986638i	$0.447907\pi$
$410$	3.92594	0.193888
$411$	0	0
$412$	13.8610	0.682881
$413$	−2.23902	−0.110175
$414$	0	0
$415$	−47.2999	−2.32186
$416$	5.35042	0.262326
$417$	0	0
$418$	−0.0195660	−0.000957004 0
$419$	−22.2270	−1.08586	−0.542931	−	0.839778i	$-0.682685\pi$
$419$	−22.2270	−1.08586	−0.542931	+	0.839778i	$0.682685\pi$
$420$	0	0
$421$	8.80239	0.429002	0.214501	−	0.976724i	$-0.431187\pi$
$421$	8.80239	0.429002	0.214501	+	0.976724i	$0.431187\pi$
$422$	1.94680	0.0947689
$423$	0	0
$424$	−4.25577	−0.206678
$425$	−14.8819	−0.721879
$426$	0	0
$427$	−6.23048	−0.301514
$428$	10.9744	0.530468
$429$	0	0
$430$	0.992390	0.0478573
$431$	19.7085	0.949324	0.474662	−	0.880168i	$-0.342570\pi$
$431$	19.7085	0.949324	0.474662	+	0.880168i	$0.342570\pi$
$432$	0	0
$433$	17.9153	0.860953	0.430476	−	0.902602i	$-0.358346\pi$
$433$	17.9153	0.860953	0.430476	+	0.902602i	$0.358346\pi$
$434$	0.820802	0.0393998
$435$	0	0
$436$	11.6509	0.557976
$437$	4.11917	0.197046
$438$	0	0
$439$	−11.9184	−0.568834	−0.284417	−	0.958701i	$-0.591800\pi$
$439$	−11.9184	−0.568834	−0.284417	+	0.958701i	$0.591800\pi$
$440$	−0.0772676	−0.00368359
$441$	0	0
$442$	−0.574510	−0.0273267
$443$	19.4540	0.924286	0.462143	−	0.886805i	$-0.347081\pi$
$443$	19.4540	0.924286	0.462143	+	0.886805i	$0.347081\pi$
$444$	0	0
$445$	30.1213	1.42789
$446$	−0.485264	−0.0229779
$447$	0	0
$448$	6.21732	0.293741
$449$	3.01378	0.142229	0.0711146	−	0.997468i	$-0.477344\pi$
$449$	3.01378	0.142229	0.0711146	+	0.997468i	$0.477344\pi$
$450$	0	0
$451$	0.218569	0.0102920
$452$	36.8851	1.73493
$453$	0	0
$454$	2.29938	0.107915
$455$	10.6015	0.497007
$456$	0	0
$457$	4.97040	0.232506	0.116253	−	0.993220i	$-0.462912\pi$
$457$	4.97040	0.232506	0.116253	+	0.993220i	$0.462912\pi$
$458$	2.46535	0.115198
$459$	0	0
$460$	8.09078	0.377235
$461$	−9.83599	−0.458108	−0.229054	−	0.973414i	$-0.573563\pi$
$461$	−9.83599	−0.458108	−0.229054	+	0.973414i	$0.573563\pi$
$462$	0	0
$463$	5.42753	0.252239	0.126119	−	0.992015i	$-0.459748\pi$
$463$	5.42753	0.252239	0.126119	+	0.992015i	$0.459748\pi$
$464$	−3.87521	−0.179902
$465$	0	0
$466$	2.53762	0.117553
$467$	18.0781	0.836556	0.418278	−	0.908319i	$-0.362634\pi$
$467$	18.0781	0.836556	0.418278	+	0.908319i	$0.362634\pi$
$468$	0	0
$469$	3.33127	0.153824
$470$	3.36263	0.155106
$471$	0	0
$472$	1.55336	0.0714990
$473$	0.0552494	0.00254037
$474$	0	0
$475$	−48.2444	−2.21360
$476$	−2.08371	−0.0955067
$477$	0	0
$478$	−2.63614	−0.120574
$479$	5.90145	0.269644	0.134822	−	0.990870i	$-0.456954\pi$
$479$	5.90145	0.269644	0.134822	+	0.990870i	$0.456954\pi$
$480$	0	0
$481$	10.3358	0.471272
$482$	0.773471	0.0352306
$483$	0	0
$484$	21.7683	0.989468
$485$	−70.8664	−3.21788
$486$	0	0
$487$	−8.42154	−0.381617	−0.190808	−	0.981627i	$-0.561111\pi$
$487$	−8.42154	−0.381617	−0.190808	+	0.981627i	$0.561111\pi$
$488$	4.32248	0.195670
$489$	0	0
$490$	3.72861	0.168442
$491$	20.5960	0.929485	0.464743	−	0.885446i	$-0.346147\pi$
$491$	20.5960	0.929485	0.464743	+	0.885446i	$0.346147\pi$
$492$	0	0
$493$	1.27064	0.0572266
$494$	−1.86245	−0.0837957
$495$	0	0
$496$	26.5722	1.19313
$497$	−3.09863	−0.138992
$498$	0	0
$499$	27.4511	1.22888	0.614439	−	0.788964i	$-0.289382\pi$
$499$	27.4511	1.22888	0.614439	+	0.788964i	$0.289382\pi$
$500$	−54.3067	−2.42867
$501$	0	0
$502$	2.02824	0.0905248
$503$	16.4801	0.734810	0.367405	−	0.930061i	$-0.380246\pi$
$503$	16.4801	0.734810	0.367405	+	0.930061i	$0.380246\pi$
$504$	0	0
$505$	10.2806	0.457483
$506$	−0.00474999	−0.000211163 0
$507$	0	0
$508$	30.8666	1.36948
$509$	−37.8732	−1.67870	−0.839351	−	0.543590i	$-0.817064\pi$
$509$	−37.8732	−1.67870	−0.839351	+	0.543590i	$0.817064\pi$
$510$	0	0
$511$	−1.66828	−0.0738004
$512$	−11.0801	−0.489677
$513$	0	0
$514$	−1.37427	−0.0606163
$515$	−28.6310	−1.26163
$516$	0	0
$517$	0.187208	0.00823339
$518$	−0.395313	−0.0173691
$519$	0	0
$520$	−7.35496	−0.322536
$521$	−3.02822	−0.132669	−0.0663343	−	0.997797i	$-0.521130\pi$
$521$	−3.02822	−0.132669	−0.0663343	+	0.997797i	$0.521130\pi$
$522$	0	0
$523$	−31.3330	−1.37010	−0.685049	−	0.728497i	$-0.740219\pi$
$523$	−31.3330	−1.37010	−0.685049	+	0.728497i	$0.740219\pi$
$524$	−13.2187	−0.577462
$525$	0	0
$526$	−0.642753	−0.0280253
$527$	−8.71271	−0.379531
$528$	0	0
$529$	1.00000	0.0434783
$530$	4.37226	0.189919
$531$	0	0
$532$	−6.75500	−0.292866
$533$	20.8052	0.901173
$534$	0	0
$535$	−22.6685	−0.980046
$536$	−2.31112	−0.0998251
$537$	0	0
$538$	−3.72425	−0.160564
$539$	0.207584	0.00894126
$540$	0	0
$541$	−6.56857	−0.282405	−0.141202	−	0.989981i	$-0.545097\pi$
$541$	−6.56857	−0.282405	−0.141202	+	0.989981i	$0.545097\pi$
$542$	−3.64967	−0.156767
$543$	0	0
$544$	2.17220	0.0931322
$545$	−24.0659	−1.03087
$546$	0	0
$547$	3.77035	0.161209	0.0806043	−	0.996746i	$-0.474315\pi$
$547$	3.77035	0.161209	0.0806043	+	0.996746i	$0.474315\pi$
$548$	27.4510	1.17265
$549$	0	0
$550$	0.0556327	0.00237219
$551$	4.11917	0.175482
$552$	0	0
$553$	−13.1634	−0.559765
$554$	3.46172	0.147075
$555$	0	0
$556$	−8.98457	−0.381031
$557$	16.1536	0.684452	0.342226	−	0.939618i	$-0.388819\pi$
$557$	16.1536	0.684452	0.342226	+	0.939618i	$0.388819\pi$
$558$	0	0
$559$	5.25909	0.222436
$560$	−13.1266	−0.554702
$561$	0	0
$562$	2.17498	0.0917458
$563$	12.6580	0.533469	0.266735	−	0.963770i	$-0.414055\pi$
$563$	12.6580	0.533469	0.266735	+	0.963770i	$0.414055\pi$
$564$	0	0
$565$	−76.1891	−3.20530
$566$	−3.41390	−0.143497
$567$	0	0
$568$	2.14972	0.0902002
$569$	10.4132	0.436543	0.218272	−	0.975888i	$-0.429958\pi$
$569$	10.4132	0.436543	0.218272	+	0.975888i	$0.429958\pi$
$570$	0	0
$571$	−23.0090	−0.962897	−0.481449	−	0.876474i	$-0.659889\pi$
$571$	−23.0090	−0.962897	−0.481449	+	0.876474i	$0.659889\pi$
$572$	−0.203663	−0.00851558
$573$	0	0
$574$	−0.795735	−0.0332133
$575$	−11.7122	−0.488431
$576$	0	0
$577$	24.7624	1.03087	0.515435	−	0.856928i	$-0.327630\pi$
$577$	24.7624	1.03087	0.515435	+	0.856928i	$0.327630\pi$
$578$	2.22268	0.0924512
$579$	0	0
$580$	8.09078	0.335952
$581$	9.58706	0.397738
$582$	0	0
$583$	0.243417	0.0100813
$584$	1.15739	0.0478933
$585$	0	0
$586$	0.914780	0.0377892
$587$	−10.1827	−0.420283	−0.210142	−	0.977671i	$-0.567392\pi$
$587$	−10.1827	−0.420283	−0.210142	+	0.977671i	$0.567392\pi$
$588$	0	0
$589$	−28.2449	−1.16381
$590$	−1.59588	−0.0657012
$591$	0	0
$592$	−12.7976	−0.525979
$593$	−15.1467	−0.621998	−0.310999	−	0.950410i	$-0.600664\pi$
$593$	−15.1467	−0.621998	−0.310999	+	0.950410i	$0.600664\pi$
$594$	0	0
$595$	4.30407	0.176450
$596$	29.8210	1.22152
$597$	0	0
$598$	−0.452143	−0.0184895
$599$	−7.06037	−0.288479	−0.144239	−	0.989543i	$-0.546074\pi$
$599$	−7.06037	−0.288479	−0.144239	+	0.989543i	$0.546074\pi$
$600$	0	0
$601$	14.9345	0.609193	0.304596	−	0.952482i	$-0.401478\pi$
$601$	14.9345	0.609193	0.304596	+	0.952482i	$0.401478\pi$
$602$	−0.201144	−0.00819802
$603$	0	0
$604$	−7.96157	−0.323952
$605$	−44.9642	−1.82805
$606$	0	0
$607$	4.17525	0.169468	0.0847341	−	0.996404i	$-0.472996\pi$
$607$	4.17525	0.169468	0.0847341	+	0.996404i	$0.472996\pi$
$608$	7.04185	0.285585
$609$	0	0
$610$	−4.44080	−0.179803
$611$	17.8200	0.720919
$612$	0	0
$613$	−2.61552	−0.105640	−0.0528198	−	0.998604i	$-0.516821\pi$
$613$	−2.61552	−0.105640	−0.0528198	+	0.998604i	$0.516821\pi$
$614$	1.97513	0.0797099
$615$	0	0
$616$	0.0156611	0.000631004 0
$617$	−29.8600	−1.20212	−0.601060	−	0.799204i	$-0.705255\pi$
$617$	−29.8600	−1.20212	−0.601060	+	0.799204i	$0.705255\pi$
$618$	0	0
$619$	3.08187	0.123871	0.0619354	−	0.998080i	$-0.480273\pi$
$619$	3.08187	0.123871	0.0619354	+	0.998080i	$0.480273\pi$
$620$	−55.4782	−2.22806
$621$	0	0
$622$	−3.11149	−0.124760
$623$	−6.10518	−0.244599
$624$	0	0
$625$	53.6140	2.14456
$626$	2.47388	0.0988759
$627$	0	0
$628$	0.840795	0.0335514
$629$	4.19619	0.167313
$630$	0	0
$631$	29.1425	1.16014	0.580072	−	0.814565i	$-0.303024\pi$
$631$	29.1425	1.16014	0.580072	+	0.814565i	$0.303024\pi$
$632$	9.13230	0.363263
$633$	0	0
$634$	2.27370	0.0903001
$635$	−63.7575	−2.53014
$636$	0	0
$637$	19.7595	0.782900
$638$	−0.00474999	−0.000188054 0
$639$	0	0
$640$	18.4088	0.727670
$641$	−7.05943	−0.278831	−0.139415	−	0.990234i	$-0.544522\pi$
$641$	−7.05943	−0.278831	−0.139415	+	0.990234i	$0.544522\pi$
$642$	0	0
$643$	−13.9954	−0.551923	−0.275962	−	0.961169i	$-0.588996\pi$
$643$	−13.9954	−0.551923	−0.275962	+	0.961169i	$0.588996\pi$
$644$	−1.63989	−0.0646209
$645$	0	0
$646$	−0.756130	−0.0297495
$647$	9.22726	0.362761	0.181381	−	0.983413i	$-0.441943\pi$
$647$	9.22726	0.362761	0.181381	+	0.983413i	$0.441943\pi$
$648$	0	0
$649$	−0.0888474	−0.00348756
$650$	5.29558	0.207710
$651$	0	0
$652$	20.1888	0.790656
$653$	−18.2056	−0.712440	−0.356220	−	0.934402i	$-0.615935\pi$
$653$	−18.2056	−0.712440	−0.356220	+	0.934402i	$0.615935\pi$
$654$	0	0
$655$	27.3043	1.06687
$656$	−25.7607	−1.00578
$657$	0	0
$658$	−0.681559	−0.0265700
$659$	13.3730	0.520937	0.260469	−	0.965482i	$-0.416123\pi$
$659$	13.3730	0.520937	0.260469	+	0.965482i	$0.416123\pi$
$660$	0	0
$661$	28.4406	1.10621	0.553106	−	0.833111i	$-0.313443\pi$
$661$	28.4406	1.10621	0.553106	+	0.833111i	$0.313443\pi$
$662$	0.459636	0.0178643
$663$	0	0
$664$	−6.65116	−0.258115
$665$	13.9530	0.541074
$666$	0	0
$667$	1.00000	0.0387202
$668$	−38.4340	−1.48706
$669$	0	0
$670$	2.37438	0.0917303
$671$	−0.247233	−0.00954434
$672$	0	0
$673$	−8.48921	−0.327235	−0.163618	−	0.986524i	$-0.552316\pi$
$673$	−8.48921	−0.327235	−0.163618	+	0.986524i	$0.552316\pi$
$674$	0.968489	0.0373048
$675$	0	0
$676$	6.34237	0.243937
$677$	−25.5811	−0.983163	−0.491582	−	0.870832i	$-0.663581\pi$
$677$	−25.5811	−0.983163	−0.491582	+	0.870832i	$0.663581\pi$
$678$	0	0
$679$	14.3637	0.551227
$680$	−2.98601	−0.114508
$681$	0	0
$682$	0.0325705	0.00124719
$683$	−17.8005	−0.681116	−0.340558	−	0.940224i	$-0.610616\pi$
$683$	−17.8005	−0.681116	−0.340558	+	0.940224i	$0.610616\pi$
$684$	0	0
$685$	−56.7023	−2.16648
$686$	−1.59367	−0.0608464
$687$	0	0
$688$	−6.51172	−0.248257
$689$	23.1705	0.882724
$690$	0	0
$691$	−4.62167	−0.175817	−0.0879084	−	0.996129i	$-0.528018\pi$
$691$	−4.62167	−0.175817	−0.0879084	+	0.996129i	$0.528018\pi$
$692$	−45.1894	−1.71784
$693$	0	0
$694$	−1.71635	−0.0651516
$695$	18.5584	0.703958
$696$	0	0
$697$	8.44662	0.319939
$698$	3.16066	0.119633
$699$	0	0
$700$	19.2067	0.725946
$701$	40.4095	1.52625	0.763123	−	0.646254i	$-0.223665\pi$
$701$	40.4095	1.52625	0.763123	+	0.646254i	$0.223665\pi$
$702$	0	0
$703$	13.6033	0.513057
$704$	0.246711	0.00929828
$705$	0	0
$706$	1.51441	0.0569955
$707$	−2.08375	−0.0783674
$708$	0	0
$709$	−21.3793	−0.802917	−0.401459	−	0.915877i	$-0.631497\pi$
$709$	−21.3793	−0.802917	−0.401459	+	0.915877i	$0.631497\pi$
$710$	−2.20856	−0.0828859
$711$	0	0
$712$	4.23556	0.158734
$713$	−6.85696	−0.256795
$714$	0	0
$715$	0.420682	0.0157326
$716$	19.3781	0.724194
$717$	0	0
$718$	4.33625	0.161827
$719$	3.84124	0.143254	0.0716270	−	0.997431i	$-0.477181\pi$
$719$	3.84124	0.143254	0.0716270	+	0.997431i	$0.477181\pi$
$720$	0	0
$721$	5.80312	0.216119
$722$	0.293624	0.0109275
$723$	0	0
$724$	−41.1810	−1.53048
$725$	−11.7122	−0.434979
$726$	0	0
$727$	−23.2952	−0.863972	−0.431986	−	0.901880i	$-0.642187\pi$
$727$	−23.2952	−0.863972	−0.431986	+	0.901880i	$0.642187\pi$
$728$	1.49075	0.0552510
$729$	0	0
$730$	−1.18908	−0.0440097
$731$	2.13512	0.0789702
$732$	0	0
$733$	26.1377	0.965419	0.482709	−	0.875781i	$-0.339653\pi$
$733$	26.1377	0.965419	0.482709	+	0.875781i	$0.339653\pi$
$734$	4.57601	0.168903
$735$	0	0
$736$	1.70953	0.0630143
$737$	0.132189	0.00486925
$738$	0	0
$739$	48.0772	1.76855	0.884274	−	0.466969i	$-0.154654\pi$
$739$	48.0772	1.76855	0.884274	+	0.466969i	$0.154654\pi$
$740$	26.7193	0.982220
$741$	0	0
$742$	−0.886199	−0.0325334
$743$	28.8522	1.05849	0.529243	−	0.848470i	$-0.322476\pi$
$743$	28.8522	1.05849	0.529243	+	0.848470i	$0.322476\pi$
$744$	0	0
$745$	−61.5977	−2.25677
$746$	4.00079	0.146479
$747$	0	0
$748$	−0.0826843	−0.00302324
$749$	4.59461	0.167883
$750$	0	0
$751$	30.4943	1.11275	0.556376	−	0.830930i	$-0.312191\pi$
$751$	30.4943	1.11275	0.556376	+	0.830930i	$0.312191\pi$
$752$	−22.0644	−0.804606
$753$	0	0
$754$	−0.452143	−0.0164661
$755$	16.4453	0.598504
$756$	0	0
$757$	1.88355	0.0684587	0.0342294	−	0.999414i	$-0.489102\pi$
$757$	1.88355	0.0684587	0.0342294	+	0.999414i	$0.489102\pi$
$758$	2.65887	0.0965746
$759$	0	0
$760$	−9.68009	−0.351134
$761$	−37.0313	−1.34238	−0.671192	−	0.741283i	$-0.734218\pi$
$761$	−37.0313	−1.34238	−0.671192	+	0.741283i	$0.734218\pi$
$762$	0	0
$763$	4.87783	0.176589
$764$	−8.16826	−0.295517
$765$	0	0
$766$	4.89356	0.176811
$767$	−8.45722	−0.305373
$768$	0	0
$769$	45.3148	1.63409	0.817046	−	0.576572i	$-0.195610\pi$
$769$	45.3148	1.63409	0.817046	+	0.576572i	$0.195610\pi$
$770$	−0.0160898	−0.000579836 0
$771$	0	0
$772$	−46.3380	−1.66774
$773$	−22.0003	−0.791297	−0.395648	−	0.918402i	$-0.629480\pi$
$773$	−22.0003	−0.791297	−0.395648	+	0.918402i	$0.629480\pi$
$774$	0	0
$775$	80.3098	2.88481
$776$	−9.96500	−0.357723
$777$	0	0
$778$	2.15991	0.0774365
$779$	27.3823	0.981074
$780$	0	0
$781$	−0.122958	−0.00439977
$782$	−0.183564	−0.00656423
$783$	0	0
$784$	−24.4659	−0.873782
$785$	−1.73673	−0.0619866
$786$	0	0
$787$	−11.0240	−0.392962	−0.196481	−	0.980508i	$-0.562951\pi$
$787$	−11.0240	−0.392962	−0.196481	+	0.980508i	$0.562951\pi$
$788$	−27.7813	−0.989666
$789$	0	0
$790$	−9.38228	−0.333806
$791$	15.4425	0.549072
$792$	0	0
$793$	−23.5337	−0.835706
$794$	2.71450	0.0963339
$795$	0	0
$796$	−1.23674	−0.0438351
$797$	43.6770	1.54712	0.773559	−	0.633724i	$-0.218475\pi$
$797$	43.6770	1.54712	0.773559	+	0.633724i	$0.218475\pi$
$798$	0	0
$799$	7.23466	0.255944
$800$	−20.0223	−0.707897
$801$	0	0
$802$	2.74311	0.0968626
$803$	−0.0661995	−0.00233613
$804$	0	0
$805$	3.38733	0.119388
$806$	3.10033	0.109204
$807$	0	0
$808$	1.44563	0.0508571
$809$	−40.8283	−1.43544	−0.717722	−	0.696329i	$-0.754815\pi$
$809$	−40.8283	−1.43544	−0.717722	+	0.696329i	$0.754815\pi$
$810$	0	0
$811$	−46.1439	−1.62033	−0.810165	−	0.586202i	$-0.800622\pi$
$811$	−46.1439	−1.62033	−0.810165	+	0.586202i	$0.800622\pi$
$812$	−1.63989	−0.0575490
$813$	0	0
$814$	−0.0156865	−0.000549813 0
$815$	−41.7017	−1.46075
$816$	0	0
$817$	6.92165	0.242158
$818$	−0.952027	−0.0332868
$819$	0	0
$820$	53.7839	1.87821
$821$	4.16325	0.145298	0.0726491	−	0.997358i	$-0.476855\pi$
$821$	4.16325	0.145298	0.0726491	+	0.997358i	$0.476855\pi$
$822$	0	0
$823$	23.8760	0.832265	0.416132	−	0.909304i	$-0.363385\pi$
$823$	23.8760	0.832265	0.416132	+	0.909304i	$0.363385\pi$
$824$	−4.02600	−0.140252
$825$	0	0
$826$	0.323463	0.0112547
$827$	0.634486	0.0220633	0.0110316	−	0.999939i	$-0.496488\pi$
$827$	0.634486	0.0220633	0.0110316	+	0.999939i	$0.496488\pi$
$828$	0	0
$829$	−1.12420	−0.0390449	−0.0195225	−	0.999809i	$-0.506215\pi$
$829$	−1.12420	−0.0390449	−0.0195225	+	0.999809i	$0.506215\pi$
$830$	6.83323	0.237185
$831$	0	0
$832$	23.4840	0.814161
$833$	8.02208	0.277949
$834$	0	0
$835$	79.3886	2.74736
$836$	−0.268047	−0.00927060
$837$	0	0
$838$	3.21105	0.110924
$839$	53.5715	1.84949	0.924747	−	0.380583i	$-0.124277\pi$
$839$	53.5715	1.84949	0.924747	+	0.380583i	$0.124277\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−1.27165	−0.0438238
$843$	0	0
$844$	26.6705	0.918036
$845$	−13.1007	−0.450677
$846$	0	0
$847$	9.11363	0.313148
$848$	−28.6893	−0.985194
$849$	0	0
$850$	2.14993	0.0737420
$851$	3.30243	0.113206
$852$	0	0
$853$	7.24339	0.248009	0.124005	−	0.992282i	$-0.460426\pi$
$853$	7.24339	0.248009	0.124005	+	0.992282i	$0.460426\pi$
$854$	0.900092	0.0308005
$855$	0	0
$856$	−3.18758	−0.108949
$857$	40.9608	1.39919	0.699597	−	0.714537i	$-0.253363\pi$
$857$	40.9608	1.39919	0.699597	+	0.714537i	$0.253363\pi$
$858$	0	0
$859$	−5.68806	−0.194074	−0.0970370	−	0.995281i	$-0.530937\pi$
$859$	−5.68806	−0.194074	−0.0970370	+	0.995281i	$0.530937\pi$
$860$	13.5954	0.463598
$861$	0	0
$862$	−2.84720	−0.0969761
$863$	28.9336	0.984911	0.492455	−	0.870338i	$-0.336099\pi$
$863$	28.9336	0.984911	0.492455	+	0.870338i	$0.336099\pi$
$864$	0	0
$865$	93.3423	3.17373
$866$	−2.58815	−0.0879488
$867$	0	0
$868$	11.2447	0.381669
$869$	−0.522341	−0.0177192
$870$	0	0
$871$	12.5828	0.426354
$872$	−3.38406	−0.114599
$873$	0	0
$874$	−0.595079	−0.0201289
$875$	−22.7364	−0.768629
$876$	0	0
$877$	40.0680	1.35300	0.676500	−	0.736443i	$-0.263496\pi$
$877$	40.0680	1.35300	0.676500	+	0.736443i	$0.263496\pi$
$878$	1.72180	0.0581081
$879$	0	0
$880$	−0.520882	−0.0175589
$881$	−15.0697	−0.507709	−0.253855	−	0.967242i	$-0.581699\pi$
$881$	−15.0697	−0.507709	−0.253855	+	0.967242i	$0.581699\pi$
$882$	0	0
$883$	−40.9333	−1.37751	−0.688757	−	0.724992i	$-0.741843\pi$
$883$	−40.9333	−1.37751	−0.688757	+	0.724992i	$0.741843\pi$
$884$	−7.87057	−0.264716
$885$	0	0
$886$	−2.81044	−0.0944185
$887$	4.17043	0.140029	0.0700147	−	0.997546i	$-0.477695\pi$
$887$	4.17043	0.140029	0.0700147	+	0.997546i	$0.477695\pi$
$888$	0	0
$889$	12.9228	0.433416
$890$	−4.35150	−0.145863
$891$	0	0
$892$	−6.64794	−0.222589
$893$	23.4534	0.784838
$894$	0	0
$895$	−40.0270	−1.33796
$896$	−3.73121	−0.124651
$897$	0	0
$898$	−0.435389	−0.0145291
$899$	−6.85696	−0.228692
$900$	0	0
$901$	9.40689	0.313389
$902$	−0.0315758	−0.00105136
$903$	0	0
$904$	−10.7135	−0.356325
$905$	85.0628	2.82758
$906$	0	0
$907$	−23.2656	−0.772523	−0.386262	−	0.922389i	$-0.626234\pi$
$907$	−23.2656	−0.772523	−0.386262	+	0.922389i	$0.626234\pi$
$908$	31.5006	1.04538
$909$	0	0
$910$	−1.53156	−0.0507707
$911$	2.66516	0.0883007	0.0441504	−	0.999025i	$-0.485942\pi$
$911$	2.66516	0.0883007	0.0441504	+	0.999025i	$0.485942\pi$
$912$	0	0
$913$	0.380427	0.0125903
$914$	−0.718054	−0.0237511
$915$	0	0
$916$	33.7744	1.11594
$917$	−5.53422	−0.182756
$918$	0	0
$919$	16.2554	0.536218	0.268109	−	0.963389i	$-0.413601\pi$
$919$	16.2554	0.536218	0.268109	+	0.963389i	$0.413601\pi$
$920$	−2.35001	−0.0774776
$921$	0	0
$922$	1.42097	0.0467970
$923$	−11.7041	−0.385246
$924$	0	0
$925$	−38.6786	−1.27175
$926$	−0.784093	−0.0257669
$927$	0	0
$928$	1.70953	0.0561182
$929$	38.1670	1.25222	0.626109	−	0.779735i	$-0.284646\pi$
$929$	38.1670	1.25222	0.626109	+	0.779735i	$0.284646\pi$
$930$	0	0
$931$	26.0061	0.852315
$932$	34.7645	1.13875
$933$	0	0
$934$	−2.61168	−0.0854566
$935$	0.170791	0.00558547
$936$	0	0
$937$	19.8536	0.648588	0.324294	−	0.945956i	$-0.394873\pi$
$937$	19.8536	0.648588	0.324294	+	0.945956i	$0.394873\pi$
$938$	−0.481255	−0.0157135
$939$	0	0
$940$	46.0667	1.50253
$941$	53.3118	1.73791	0.868957	−	0.494887i	$-0.164791\pi$
$941$	53.3118	1.73791	0.868957	+	0.494887i	$0.164791\pi$
$942$	0	0
$943$	6.64755	0.216474
$944$	10.4716	0.340821
$945$	0	0
$946$	−0.00798166	−0.000259506 0
$947$	−13.6427	−0.443330	−0.221665	−	0.975123i	$-0.571149\pi$
$947$	−13.6427	−0.443330	−0.221665	+	0.975123i	$0.571149\pi$
$948$	0	0
$949$	−6.30142	−0.204553
$950$	6.96967	0.226126
$951$	0	0
$952$	0.605225	0.0196155
$953$	11.6267	0.376625	0.188312	−	0.982109i	$-0.439698\pi$
$953$	11.6267	0.376625	0.188312	+	0.982109i	$0.439698\pi$
$954$	0	0
$955$	16.8722	0.545972
$956$	−36.1141	−1.16801
$957$	0	0
$958$	−0.852558	−0.0275449
$959$	11.4928	0.371122
$960$	0	0
$961$	16.0179	0.516705
$962$	−1.49317	−0.0481418
$963$	0	0
$964$	10.5963	0.341283
$965$	95.7149	3.08117
$966$	0	0
$967$	24.1651	0.777099	0.388549	−	0.921428i	$-0.372976\pi$
$967$	24.1651	0.777099	0.388549	+	0.921428i	$0.372976\pi$
$968$	−6.32272	−0.203220
$969$	0	0
$970$	10.2378	0.328715
$971$	−13.0260	−0.418023	−0.209011	−	0.977913i	$-0.567025\pi$
$971$	−13.0260	−0.418023	−0.209011	+	0.977913i	$0.567025\pi$
$972$	0	0
$973$	−3.76153	−0.120589
$974$	1.21663	0.0389832
$975$	0	0
$976$	29.1390	0.932718
$977$	−57.2221	−1.83070	−0.915348	−	0.402664i	$-0.868084\pi$
$977$	−57.2221	−1.83070	−0.915348	+	0.402664i	$0.868084\pi$
$978$	0	0
$979$	−0.242262	−0.00774271
$980$	51.0806	1.63171
$981$	0	0
$982$	−2.97542	−0.0949496
$983$	1.19935	0.0382533	0.0191267	−	0.999817i	$-0.493911\pi$
$983$	1.19935	0.0382533	0.0191267	+	0.999817i	$0.493911\pi$
$984$	0	0
$985$	57.3844	1.82842
$986$	−0.183564	−0.00584586
$987$	0	0
$988$	−25.5149	−0.811737
$989$	1.68035	0.0534321
$990$	0	0
$991$	−8.08001	−0.256670	−0.128335	−	0.991731i	$-0.540963\pi$
$991$	−8.08001	−0.256670	−0.128335	+	0.991731i	$0.540963\pi$
$992$	−11.7222	−0.372180
$993$	0	0
$994$	0.447646	0.0141985
$995$	2.55459	0.0809858
$996$	0	0
$997$	−45.2942	−1.43448	−0.717241	−	0.696825i	$-0.754595\pi$
$997$	−45.2942	−1.43448	−0.717241	+	0.696825i	$0.754595\pi$
$998$	−3.96574	−0.125533
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.u.1.11	yes	22
3.2	odd	2		6003.2.a.t.1.12	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.t.1.12	✓	22	3.2	odd	2
6003.2.a.u.1.11	yes	22	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)

\(f(q)\)	\(=\)	\(q-0.144466 q^{2} -1.97913 q^{4} +4.08805 q^{5} -0.828594 q^{7} +0.574849 q^{8} +O(q^{10})\)	\(q-0.144466 q^{2} -1.97913 q^{4} +4.08805 q^{5} -0.828594 q^{7} +0.574849 q^{8} -0.590584 q^{10} -0.0328797 q^{11} -3.12976 q^{13} +0.119704 q^{14} +3.87521 q^{16} -1.27064 q^{17} -4.11917 q^{19} -8.09078 q^{20} +0.00474999 q^{22} -1.00000 q^{23} +11.7122 q^{25} +0.452143 q^{26} +1.63989 q^{28} -1.00000 q^{29} +6.85696 q^{31} -1.70953 q^{32} +0.183564 q^{34} -3.38733 q^{35} -3.30243 q^{37} +0.595079 q^{38} +2.35001 q^{40} -6.64755 q^{41} -1.68035 q^{43} +0.0650731 q^{44} +0.144466 q^{46} -5.69373 q^{47} -6.31343 q^{49} -1.69201 q^{50} +6.19419 q^{52} -7.40328 q^{53} -0.134414 q^{55} -0.476316 q^{56} +0.144466 q^{58} +2.70220 q^{59} +7.51934 q^{61} -0.990597 q^{62} -7.50346 q^{64} -12.7946 q^{65} -4.02039 q^{67} +2.51476 q^{68} +0.489354 q^{70} +3.73962 q^{71} +2.01339 q^{73} +0.477089 q^{74} +8.15236 q^{76} +0.0272439 q^{77} +15.8864 q^{79} +15.8421 q^{80} +0.960344 q^{82} -11.5703 q^{83} -5.19443 q^{85} +0.242754 q^{86} -0.0189008 q^{88} +7.36813 q^{89} +2.59330 q^{91} +1.97913 q^{92} +0.822550 q^{94} -16.8394 q^{95} -17.3350 q^{97} +0.912076 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8}+O(q^{10}) \) 22 * q + 3 * q^2 + 17 * q^4 - 6 * q^7 + 6 * q^8	\( 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8} - 12 q^{10} - 28 q^{13} + q^{14} + 3 q^{16} + 10 q^{17} - 8 q^{19} - 11 q^{22} - 22 q^{23} - 11 q^{26} - 21 q^{28} - 22 q^{29} - 18 q^{31} - 5 q^{32} - 33 q^{34} - 2 q^{35} - 28 q^{37} - 14 q^{38} - 30 q^{40} + 10 q^{41} - 14 q^{43} - 37 q^{44} - 3 q^{46} + 18 q^{47} + 2 q^{49} - 7 q^{50} - 57 q^{52} - 20 q^{53} - 42 q^{55} + 2 q^{56} - 3 q^{58} + 20 q^{59} - 38 q^{61} - 4 q^{62} - 24 q^{64} - 12 q^{65} - 50 q^{67} - 11 q^{68} - 48 q^{70} - 12 q^{71} - 46 q^{73} + 6 q^{74} - 16 q^{76} + 14 q^{77} - 20 q^{79} + 58 q^{80} - 42 q^{82} - 22 q^{83} - 66 q^{85} - 22 q^{86} - 68 q^{88} + 14 q^{89} - 16 q^{91} - 17 q^{92} - 27 q^{94} + 20 q^{95} - 48 q^{97} + 28 q^{98}+O(q^{100}) \) 22 * q + 3 * q^2 + 17 * q^4 - 6 * q^7 + 6 * q^8 - 12 * q^10 - 28 * q^13 + q^14 + 3 * q^16 + 10 * q^17 - 8 * q^19 - 11 * q^22 - 22 * q^23 - 11 * q^26 - 21 * q^28 - 22 * q^29 - 18 * q^31 - 5 * q^32 - 33 * q^34 - 2 * q^35 - 28 * q^37 - 14 * q^38 - 30 * q^40 + 10 * q^41 - 14 * q^43 - 37 * q^44 - 3 * q^46 + 18 * q^47 + 2 * q^49 - 7 * q^50 - 57 * q^52 - 20 * q^53 - 42 * q^55 + 2 * q^56 - 3 * q^58 + 20 * q^59 - 38 * q^61 - 4 * q^62 - 24 * q^64 - 12 * q^65 - 50 * q^67 - 11 * q^68 - 48 * q^70 - 12 * q^71 - 46 * q^73 + 6 * q^74 - 16 * q^76 + 14 * q^77 - 20 * q^79 + 58 * q^80 - 42 * q^82 - 22 * q^83 - 66 * q^85 - 22 * q^86 - 68 * q^88 + 14 * q^89 - 16 * q^91 - 17 * q^92 - 27 * q^94 + 20 * q^95 - 48 * q^97 + 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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