LMFDB - Embedded newform 6003.2.a.p.1.4

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 18 x^{12} + 34 x^{11} + 124 x^{10} - 216 x^{9} - 420 x^{8} + 647 x^{7} + 750 x^{6} + \cdots - 64 $ x^14 - 2x^13 - 18x^12 + 34x^11 + 124x^10 - 216x^9 - 420x^8 + 647x^7 + 750x^6 - 939x^5 - 717x^4 + 604x^3 + 352x^2 - 128*x - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.37642$ 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)

$f(q)$	$=$	$q-1.37642 q^{2} -0.105481 q^{4} +1.48313 q^{5} +4.92788 q^{7} +2.89802 q^{8} +O(q^{10})$	\(q-1.37642 q^{2} -0.105481 q^{4} +1.48313 q^{5} +4.92788 q^{7} +2.89802 q^{8} -2.04140 q^{10} +4.78117 q^{11} +0.524666 q^{13} -6.78281 q^{14} -3.77791 q^{16} +2.51811 q^{17} +6.02409 q^{19} -0.156442 q^{20} -6.58088 q^{22} +1.00000 q^{23} -2.80033 q^{25} -0.722159 q^{26} -0.519799 q^{28} +1.00000 q^{29} -4.86009 q^{31} -0.596058 q^{32} -3.46596 q^{34} +7.30868 q^{35} -1.81490 q^{37} -8.29165 q^{38} +4.29813 q^{40} +6.88501 q^{41} +5.44082 q^{43} -0.504324 q^{44} -1.37642 q^{46} +8.73911 q^{47} +17.2840 q^{49} +3.85442 q^{50} -0.0553424 q^{52} +7.14027 q^{53} +7.09109 q^{55} +14.2811 q^{56} -1.37642 q^{58} +8.40962 q^{59} -11.8204 q^{61} +6.68951 q^{62} +8.37625 q^{64} +0.778147 q^{65} +8.80386 q^{67} -0.265613 q^{68} -10.0598 q^{70} +11.6325 q^{71} -13.0454 q^{73} +2.49805 q^{74} -0.635428 q^{76} +23.5611 q^{77} -5.81717 q^{79} -5.60313 q^{80} -9.47663 q^{82} -7.77527 q^{83} +3.73467 q^{85} -7.48882 q^{86} +13.8559 q^{88} -14.1010 q^{89} +2.58549 q^{91} -0.105481 q^{92} -12.0286 q^{94} +8.93450 q^{95} -8.12641 q^{97} -23.7900 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) $ 14 * q + 2 * q^2 + 12 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8	$ 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} - 5 q^{10} + 12 q^{11} + 13 q^{13} + 9 q^{14} + 14 q^{17} - 9 q^{19} + 2 q^{20} - 9 q^{22} + 14 q^{23} + 13 q^{25} + 16 q^{26} + 3 q^{28} + 14 q^{29} - 28 q^{31} + 4 q^{32} + 14 q^{34} + 9 q^{35} - 12 q^{37} - 2 q^{38} - 20 q^{40} + 25 q^{41} + 5 q^{43} + 37 q^{44} + 2 q^{46} + 17 q^{47} + 17 q^{49} + 44 q^{50} + 25 q^{52} + 17 q^{53} + q^{55} + 54 q^{56} + 2 q^{58} + 18 q^{59} - 13 q^{61} + 8 q^{62} + 20 q^{64} + 16 q^{65} + 2 q^{67} + 19 q^{68} + 14 q^{70} + 55 q^{71} + 19 q^{73} - 4 q^{74} - 32 q^{76} + 19 q^{77} - 68 q^{79} + 2 q^{80} - 12 q^{82} + 21 q^{83} + 16 q^{85} + 22 q^{86} - 25 q^{88} + 17 q^{89} - 30 q^{91} + 12 q^{92} + 16 q^{94} + 55 q^{95} + 25 q^{97} + 31 q^{98}+O(q^{100}) $ 14 * q + 2 * q^2 + 12 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8 - 5 * q^10 + 12 * q^11 + 13 * q^13 + 9 * q^14 + 14 * q^17 - 9 * q^19 + 2 * q^20 - 9 * q^22 + 14 * q^23 + 13 * q^25 + 16 * q^26 + 3 * q^28 + 14 * q^29 - 28 * q^31 + 4 * q^32 + 14 * q^34 + 9 * q^35 - 12 * q^37 - 2 * q^38 - 20 * q^40 + 25 * q^41 + 5 * q^43 + 37 * q^44 + 2 * q^46 + 17 * q^47 + 17 * q^49 + 44 * q^50 + 25 * q^52 + 17 * q^53 + q^55 + 54 * q^56 + 2 * q^58 + 18 * q^59 - 13 * q^61 + 8 * q^62 + 20 * q^64 + 16 * q^65 + 2 * q^67 + 19 * q^68 + 14 * q^70 + 55 * q^71 + 19 * q^73 - 4 * q^74 - 32 * q^76 + 19 * q^77 - 68 * q^79 + 2 * q^80 - 12 * q^82 + 21 * q^83 + 16 * q^85 + 22 * q^86 - 25 * q^88 + 17 * q^89 - 30 * q^91 + 12 * q^92 + 16 * q^94 + 55 * q^95 + 25 * 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 $

$2$	−1.37642	−0.973273	−0.486636	−	0.873605i	$-0.661776\pi$
$2$	−1.37642	−0.973273	−0.486636	+	0.873605i	$0.661776\pi$
$3$	0	0
$3$	0	0
$4$	−0.105481	−0.0527406
$5$	1.48313	0.663275	0.331638	−	0.943407i	$-0.392399\pi$
$5$	1.48313	0.663275	0.331638	+	0.943407i	$0.392399\pi$
$6$	0	0
$7$	4.92788	1.86256	0.931282	−	0.364298i	$-0.118691\pi$
$7$	4.92788	1.86256	0.931282	+	0.364298i	$0.118691\pi$
$8$	2.89802	1.02460
$9$	0	0
$10$	−2.04140	−0.645547
$11$	4.78117	1.44158	0.720789	−	0.693154i	$-0.243780\pi$
$11$	4.78117	1.44158	0.720789	+	0.693154i	$0.243780\pi$
$12$	0	0
$13$	0.524666	0.145516	0.0727581	−	0.997350i	$-0.476820\pi$
$13$	0.524666	0.145516	0.0727581	+	0.997350i	$0.476820\pi$
$14$	−6.78281	−1.81278
$15$	0	0
$16$	−3.77791	−0.944478
$17$	2.51811	0.610730	0.305365	−	0.952235i	$-0.401221\pi$
$17$	2.51811	0.610730	0.305365	+	0.952235i	$0.401221\pi$
$18$	0	0
$19$	6.02409	1.38202	0.691010	−	0.722845i	$-0.257166\pi$
$19$	6.02409	1.38202	0.691010	+	0.722845i	$0.257166\pi$
$20$	−0.156442	−0.0349815
$21$	0	0
$22$	−6.58088	−1.40305
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−2.80033	−0.560066
$26$	−0.722159	−0.141627
$27$	0	0
$28$	−0.519799	−0.0982328
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−4.86009	−0.872899	−0.436449	−	0.899729i	$-0.643764\pi$
$31$	−4.86009	−0.872899	−0.436449	+	0.899729i	$0.643764\pi$
$32$	−0.596058	−0.105369
$33$	0	0
$34$	−3.46596	−0.594407
$35$	7.30868	1.23539
$36$	0	0
$37$	−1.81490	−0.298367	−0.149183	−	0.988810i	$-0.547664\pi$
$37$	−1.81490	−0.298367	−0.149183	+	0.988810i	$0.547664\pi$
$38$	−8.29165	−1.34508
$39$	0	0
$40$	4.29813	0.679594
$41$	6.88501	1.07526	0.537629	−	0.843182i	$-0.319320\pi$
$41$	6.88501	1.07526	0.537629	+	0.843182i	$0.319320\pi$
$42$	0	0
$43$	5.44082	0.829717	0.414858	−	0.909886i	$-0.363831\pi$
$43$	5.44082	0.829717	0.414858	+	0.909886i	$0.363831\pi$
$44$	−0.504324	−0.0760297
$45$	0	0
$46$	−1.37642	−0.202941
$47$	8.73911	1.27473	0.637365	−	0.770562i	$-0.280024\pi$
$47$	8.73911	1.27473	0.637365	+	0.770562i	$0.280024\pi$
$48$	0	0
$49$	17.2840	2.46915
$50$	3.85442	0.545097
$51$	0	0
$52$	−0.0553424	−0.00767461
$53$	7.14027	0.980791	0.490395	−	0.871500i	$-0.336852\pi$
$53$	7.14027	0.980791	0.490395	+	0.871500i	$0.336852\pi$
$54$	0	0
$55$	7.09109	0.956163
$56$	14.2811	1.90839
$57$	0	0
$58$	−1.37642	−0.180732
$59$	8.40962	1.09484	0.547420	−	0.836858i	$-0.315610\pi$
$59$	8.40962	1.09484	0.547420	+	0.836858i	$0.315610\pi$
$60$	0	0
$61$	−11.8204	−1.51345	−0.756724	−	0.653735i	$-0.773201\pi$
$61$	−11.8204	−1.51345	−0.756724	+	0.653735i	$0.773201\pi$
$62$	6.68951	0.849568
$63$	0	0
$64$	8.37625	1.04703
$65$	0.778147	0.0965173
$66$	0	0
$67$	8.80386	1.07556	0.537781	−	0.843084i	$-0.319263\pi$
$67$	8.80386	1.07556	0.537781	+	0.843084i	$0.319263\pi$
$68$	−0.265613	−0.0322103
$69$	0	0
$70$	−10.0598	−1.20237
$71$	11.6325	1.38053	0.690264	−	0.723557i	$-0.257494\pi$
$71$	11.6325	1.38053	0.690264	+	0.723557i	$0.257494\pi$
$72$	0	0
$73$	−13.0454	−1.52684	−0.763421	−	0.645901i	$-0.776482\pi$
$73$	−13.0454	−1.52684	−0.763421	+	0.645901i	$0.776482\pi$
$74$	2.49805	0.290392
$75$	0	0
$76$	−0.635428	−0.0728886
$77$	23.5611	2.68503
$78$	0	0
$79$	−5.81717	−0.654483	−0.327241	−	0.944941i	$-0.606119\pi$
$79$	−5.81717	−0.654483	−0.327241	+	0.944941i	$0.606119\pi$
$80$	−5.60313	−0.626449
$81$	0	0
$82$	−9.47663	−1.04652
$83$	−7.77527	−0.853447	−0.426723	−	0.904382i	$-0.640332\pi$
$83$	−7.77527	−0.853447	−0.426723	+	0.904382i	$0.640332\pi$
$84$	0	0
$85$	3.73467	0.405082
$86$	−7.48882	−0.807541
$87$	0	0
$88$	13.8559	1.47705
$89$	−14.1010	−1.49471	−0.747353	−	0.664427i	$-0.768676\pi$
$89$	−14.1010	−1.49471	−0.747353	+	0.664427i	$0.768676\pi$
$90$	0	0
$91$	2.58549	0.271033
$92$	−0.105481	−0.0109972
$93$	0	0
$94$	−12.0286	−1.24066
$95$	8.93450	0.916660
$96$	0	0
$97$	−8.12641	−0.825112	−0.412556	−	0.910932i	$-0.635364\pi$
$97$	−8.12641	−0.825112	−0.412556	+	0.910932i	$0.635364\pi$
$98$	−23.7900	−2.40315
$99$	0	0
$100$	0.295382	0.0295382
$101$	−6.34705	−0.631555	−0.315778	−	0.948833i	$-0.602265\pi$
$101$	−6.34705	−0.631555	−0.315778	+	0.948833i	$0.602265\pi$
$102$	0	0
$103$	−16.8362	−1.65892	−0.829459	−	0.558568i	$-0.811351\pi$
$103$	−16.8362	−1.65892	−0.829459	+	0.558568i	$0.811351\pi$
$104$	1.52049	0.149096
$105$	0	0
$106$	−9.82797	−0.954577
$107$	−6.76344	−0.653846	−0.326923	−	0.945051i	$-0.606012\pi$
$107$	−6.76344	−0.653846	−0.326923	+	0.945051i	$0.606012\pi$
$108$	0	0
$109$	−15.8184	−1.51512	−0.757562	−	0.652763i	$-0.773610\pi$
$109$	−15.8184	−1.51512	−0.757562	+	0.652763i	$0.773610\pi$
$110$	−9.76029	−0.930607
$111$	0	0
$112$	−18.6171	−1.75915
$113$	11.6740	1.09820	0.549098	−	0.835758i	$-0.314971\pi$
$113$	11.6740	1.09820	0.549098	+	0.835758i	$0.314971\pi$
$114$	0	0
$115$	1.48313	0.138302
$116$	−0.105481	−0.00979368
$117$	0	0
$118$	−11.5751	−1.06558
$119$	12.4089	1.13752
$120$	0	0
$121$	11.8596	1.07815
$122$	16.2698	1.47300
$123$	0	0
$124$	0.512649	0.0460372
$125$	−11.5689	−1.03475
$126$	0	0
$127$	16.1525	1.43330	0.716652	−	0.697431i	$-0.245673\pi$
$127$	16.1525	1.43330	0.716652	+	0.697431i	$0.245673\pi$
$128$	−10.3371	−0.913677
$129$	0	0
$130$	−1.07105	−0.0939376
$131$	5.52338	0.482580	0.241290	−	0.970453i	$-0.422430\pi$
$131$	5.52338	0.482580	0.241290	+	0.970453i	$0.422430\pi$
$132$	0	0
$133$	29.6860	2.57410
$134$	−12.1178	−1.04682
$135$	0	0
$136$	7.29751	0.625756
$137$	−2.67599	−0.228625	−0.114312	−	0.993445i	$-0.536467\pi$
$137$	−2.67599	−0.228625	−0.114312	+	0.993445i	$0.536467\pi$
$138$	0	0
$139$	20.2927	1.72121	0.860604	−	0.509275i	$-0.170086\pi$
$139$	20.2927	1.72121	0.860604	+	0.509275i	$0.170086\pi$
$140$	−0.770928	−0.0651553
$141$	0	0
$142$	−16.0112	−1.34363
$143$	2.50852	0.209773
$144$	0	0
$145$	1.48313	0.123167
$146$	17.9558	1.48603
$147$	0	0
$148$	0.191437	0.0157360
$149$	−10.7950	−0.884360	−0.442180	−	0.896926i	$-0.645795\pi$
$149$	−10.7950	−0.884360	−0.442180	+	0.896926i	$0.645795\pi$
$150$	0	0
$151$	−13.5193	−1.10019	−0.550094	−	0.835103i	$-0.685408\pi$
$151$	−13.5193	−1.10019	−0.550094	+	0.835103i	$0.685408\pi$
$152$	17.4579	1.41602
$153$	0	0
$154$	−32.4298	−2.61327
$155$	−7.20814	−0.578972
$156$	0	0
$157$	−6.39919	−0.510711	−0.255355	−	0.966847i	$-0.582192\pi$
$157$	−6.39919	−0.510711	−0.255355	+	0.966847i	$0.582192\pi$
$158$	8.00684	0.636990
$159$	0	0
$160$	−0.884030	−0.0698887
$161$	4.92788	0.388372
$162$	0	0
$163$	−8.82550	−0.691267	−0.345633	−	0.938370i	$-0.612336\pi$
$163$	−8.82550	−0.691267	−0.345633	+	0.938370i	$0.612336\pi$
$164$	−0.726239	−0.0567097
$165$	0	0
$166$	10.7020	0.830636
$167$	19.1613	1.48275	0.741373	−	0.671093i	$-0.234175\pi$
$167$	19.1613	1.48275	0.741373	+	0.671093i	$0.234175\pi$
$168$	0	0
$169$	−12.7247	−0.978825
$170$	−5.14046	−0.394255
$171$	0	0
$172$	−0.573904	−0.0437598
$173$	−10.0566	−0.764587	−0.382294	−	0.924041i	$-0.624866\pi$
$173$	−10.0566	−0.764587	−0.382294	+	0.924041i	$0.624866\pi$
$174$	0	0
$175$	−13.7997	−1.04316
$176$	−18.0629	−1.36154
$177$	0	0
$178$	19.4089	1.45476
$179$	−8.03079	−0.600249	−0.300125	−	0.953900i	$-0.597028\pi$
$179$	−8.03079	−0.600249	−0.300125	+	0.953900i	$0.597028\pi$
$180$	0	0
$181$	−13.1680	−0.978768	−0.489384	−	0.872068i	$-0.662778\pi$
$181$	−13.1680	−0.978768	−0.489384	+	0.872068i	$0.662778\pi$
$182$	−3.55871	−0.263789
$183$	0	0
$184$	2.89802	0.213645
$185$	−2.69172	−0.197899
$186$	0	0
$187$	12.0395	0.880416
$188$	−0.921811	−0.0672300
$189$	0	0
$190$	−12.2976	−0.892160
$191$	−13.9071	−1.00628	−0.503142	−	0.864204i	$-0.667823\pi$
$191$	−13.9071	−1.00628	−0.503142	+	0.864204i	$0.667823\pi$
$192$	0	0
$193$	−12.5196	−0.901181	−0.450591	−	0.892731i	$-0.648787\pi$
$193$	−12.5196	−0.901181	−0.450591	+	0.892731i	$0.648787\pi$
$194$	11.1853	0.803058
$195$	0	0
$196$	−1.82314	−0.130224
$197$	−11.2711	−0.803031	−0.401516	−	0.915852i	$-0.631516\pi$
$197$	−11.2711	−0.803031	−0.401516	+	0.915852i	$0.631516\pi$
$198$	0	0
$199$	−3.41641	−0.242183	−0.121091	−	0.992641i	$-0.538639\pi$
$199$	−3.41641	−0.242183	−0.121091	+	0.992641i	$0.538639\pi$
$200$	−8.11540	−0.573846
$201$	0	0
$202$	8.73618	0.614675
$203$	4.92788	0.345870
$204$	0	0
$205$	10.2114	0.713192
$206$	23.1736	1.61458
$207$	0	0
$208$	−1.98214	−0.137437
$209$	28.8022	1.99229
$210$	0	0
$211$	6.89502	0.474673	0.237336	−	0.971428i	$-0.423726\pi$
$211$	6.89502	0.474673	0.237336	+	0.971428i	$0.423726\pi$
$212$	−0.753164	−0.0517275
$213$	0	0
$214$	9.30930	0.636371
$215$	8.06943	0.550331
$216$	0	0
$217$	−23.9500	−1.62583
$218$	21.7726	1.47463
$219$	0	0
$220$	−0.747977	−0.0504286
$221$	1.32116	0.0888712
$222$	0	0
$223$	−4.85054	−0.324816	−0.162408	−	0.986724i	$-0.551926\pi$
$223$	−4.85054	−0.324816	−0.162408	+	0.986724i	$0.551926\pi$
$224$	−2.93730	−0.196257
$225$	0	0
$226$	−16.0682	−1.06884
$227$	−18.5953	−1.23422	−0.617108	−	0.786879i	$-0.711696\pi$
$227$	−18.5953	−1.23422	−0.617108	+	0.786879i	$0.711696\pi$
$228$	0	0
$229$	17.0684	1.12791	0.563955	−	0.825805i	$-0.309279\pi$
$229$	17.0684	1.12791	0.563955	+	0.825805i	$0.309279\pi$
$230$	−2.04140	−0.134606
$231$	0	0
$232$	2.89802	0.190264
$233$	9.38867	0.615073	0.307536	−	0.951536i	$-0.400495\pi$
$233$	9.38867	0.615073	0.307536	+	0.951536i	$0.400495\pi$
$234$	0	0
$235$	12.9612	0.845496
$236$	−0.887057	−0.0577425
$237$	0	0
$238$	−17.0798	−1.10712
$239$	−14.9793	−0.968929	−0.484465	−	0.874811i	$-0.660986\pi$
$239$	−14.9793	−0.968929	−0.484465	+	0.874811i	$0.660986\pi$
$240$	0	0
$241$	10.6067	0.683239	0.341619	−	0.939838i	$-0.389025\pi$
$241$	10.6067	0.683239	0.341619	+	0.939838i	$0.389025\pi$
$242$	−16.3238	−1.04933
$243$	0	0
$244$	1.24683	0.0798201
$245$	25.6344	1.63772
$246$	0	0
$247$	3.16064	0.201106
$248$	−14.0846	−0.894375
$249$	0	0
$250$	15.9236	1.00710
$251$	−29.5202	−1.86330	−0.931650	−	0.363357i	$-0.881631\pi$
$251$	−29.5202	−1.86330	−0.931650	+	0.363357i	$0.881631\pi$
$252$	0	0
$253$	4.78117	0.300590
$254$	−22.2326	−1.39500
$255$	0	0
$256$	−2.52438	−0.157774
$257$	−23.5471	−1.46883	−0.734414	−	0.678701i	$-0.762543\pi$
$257$	−23.5471	−1.46883	−0.734414	+	0.678701i	$0.762543\pi$
$258$	0	0
$259$	−8.94359	−0.555728
$260$	−0.0820799	−0.00509038
$261$	0	0
$262$	−7.60246	−0.469682
$263$	4.17470	0.257423	0.128711	−	0.991682i	$-0.458916\pi$
$263$	4.17470	0.257423	0.128711	+	0.991682i	$0.458916\pi$
$264$	0	0
$265$	10.5899	0.650534
$266$	−40.8603	−2.50530
$267$	0	0
$268$	−0.928642	−0.0567258
$269$	23.3314	1.42254	0.711271	−	0.702918i	$-0.248120\pi$
$269$	23.3314	1.42254	0.711271	+	0.702918i	$0.248120\pi$
$270$	0	0
$271$	30.5545	1.85605	0.928027	−	0.372513i	$-0.121504\pi$
$271$	30.5545	1.85605	0.928027	+	0.372513i	$0.121504\pi$
$272$	−9.51318	−0.576821
$273$	0	0
$274$	3.68327	0.222514
$275$	−13.3889	−0.807379
$276$	0	0
$277$	4.64255	0.278944	0.139472	−	0.990226i	$-0.455459\pi$
$277$	4.64255	0.278944	0.139472	+	0.990226i	$0.455459\pi$
$278$	−27.9312	−1.67520
$279$	0	0
$280$	21.1807	1.26579
$281$	−23.1599	−1.38160	−0.690802	−	0.723044i	$-0.742742\pi$
$281$	−23.1599	−1.38160	−0.690802	+	0.723044i	$0.742742\pi$
$282$	0	0
$283$	6.01659	0.357649	0.178825	−	0.983881i	$-0.442770\pi$
$283$	6.01659	0.357649	0.178825	+	0.983881i	$0.442770\pi$
$284$	−1.22701	−0.0728099
$285$	0	0
$286$	−3.45277	−0.204166
$287$	33.9285	2.00274
$288$	0	0
$289$	−10.6591	−0.627008
$290$	−2.04140	−0.119875
$291$	0	0
$292$	1.37604	0.0805266
$293$	−19.9569	−1.16589	−0.582947	−	0.812510i	$-0.698101\pi$
$293$	−19.9569	−1.16589	−0.582947	+	0.812510i	$0.698101\pi$
$294$	0	0
$295$	12.4725	0.726180
$296$	−5.25960	−0.305708
$297$	0	0
$298$	14.8584	0.860723
$299$	0.524666	0.0303422
$300$	0	0
$301$	26.8117	1.54540
$302$	18.6082	1.07078
$303$	0	0
$304$	−22.7585	−1.30529
$305$	−17.5312	−1.00383
$306$	0	0
$307$	−30.5457	−1.74334	−0.871668	−	0.490097i	$-0.836961\pi$
$307$	−30.5457	−1.74334	−0.871668	+	0.490097i	$0.836961\pi$
$308$	−2.48525	−0.141610
$309$	0	0
$310$	9.92140	0.563497
$311$	20.4913	1.16196	0.580979	−	0.813919i	$-0.302670\pi$
$311$	20.4913	1.16196	0.580979	+	0.813919i	$0.302670\pi$
$312$	0	0
$313$	26.7843	1.51394	0.756969	−	0.653451i	$-0.226679\pi$
$313$	26.7843	1.51394	0.756969	+	0.653451i	$0.226679\pi$
$314$	8.80794	0.497061
$315$	0	0
$316$	0.613602	0.0345178
$317$	−29.1115	−1.63506	−0.817532	−	0.575883i	$-0.804658\pi$
$317$	−29.1115	−1.63506	−0.817532	+	0.575883i	$0.804658\pi$
$318$	0	0
$319$	4.78117	0.267694
$320$	12.4230	0.694469
$321$	0	0
$322$	−6.78281	−0.377991
$323$	15.1693	0.844042
$324$	0	0
$325$	−1.46924	−0.0814987
$326$	12.1476	0.672791
$327$	0	0
$328$	19.9529	1.10171
$329$	43.0653	2.37427
$330$	0	0
$331$	−22.7491	−1.25040	−0.625201	−	0.780464i	$-0.714983\pi$
$331$	−22.7491	−1.25040	−0.625201	+	0.780464i	$0.714983\pi$
$332$	0.820145	0.0450113
$333$	0	0
$334$	−26.3739	−1.44312
$335$	13.0573	0.713394
$336$	0	0
$337$	−21.1999	−1.15483	−0.577416	−	0.816450i	$-0.695939\pi$
$337$	−21.1999	−1.15483	−0.577416	+	0.816450i	$0.695939\pi$
$338$	17.5145	0.952664
$339$	0	0
$340$	−0.393938	−0.0213643
$341$	−23.2370	−1.25835
$342$	0	0
$343$	50.6785	2.73638
$344$	15.7676	0.850131
$345$	0	0
$346$	13.8420	0.744152
$347$	7.79477	0.418445	0.209223	−	0.977868i	$-0.432907\pi$
$347$	7.79477	0.418445	0.209223	+	0.977868i	$0.432907\pi$
$348$	0	0
$349$	17.0834	0.914451	0.457226	−	0.889351i	$-0.348843\pi$
$349$	17.0834	0.914451	0.457226	+	0.889351i	$0.348843\pi$
$350$	18.9941	1.01528
$351$	0	0
$352$	−2.84986	−0.151898
$353$	−13.2100	−0.703099	−0.351550	−	0.936169i	$-0.614345\pi$
$353$	−13.2100	−0.703099	−0.351550	+	0.936169i	$0.614345\pi$
$354$	0	0
$355$	17.2526	0.915670
$356$	1.48739	0.0788317
$357$	0	0
$358$	11.0537	0.584206
$359$	−9.87969	−0.521430	−0.260715	−	0.965416i	$-0.583958\pi$
$359$	−9.87969	−0.521430	−0.260715	+	0.965416i	$0.583958\pi$
$360$	0	0
$361$	17.2896	0.909981
$362$	18.1246	0.952608
$363$	0	0
$364$	−0.272721	−0.0142945
$365$	−19.3479	−1.01272
$366$	0	0
$367$	17.8042	0.929372	0.464686	−	0.885475i	$-0.346167\pi$
$367$	17.8042	0.929372	0.464686	+	0.885475i	$0.346167\pi$
$368$	−3.77791	−0.196937
$369$	0	0
$370$	3.70493	0.192610
$371$	35.1864	1.82679
$372$	0	0
$373$	−12.5979	−0.652297	−0.326148	−	0.945319i	$-0.605751\pi$
$373$	−12.5979	−0.652297	−0.326148	+	0.945319i	$0.605751\pi$
$374$	−16.5714	−0.856884
$375$	0	0
$376$	25.3261	1.30609
$377$	0.524666	0.0270217
$378$	0	0
$379$	−18.5593	−0.953328	−0.476664	−	0.879086i	$-0.658154\pi$
$379$	−18.5593	−0.953328	−0.476664	+	0.879086i	$0.658154\pi$
$380$	−0.942421	−0.0483452
$381$	0	0
$382$	19.1420	0.979388
$383$	−6.16130	−0.314828	−0.157414	−	0.987533i	$-0.550316\pi$
$383$	−6.16130	−0.314828	−0.157414	+	0.987533i	$0.550316\pi$
$384$	0	0
$385$	34.9441	1.78092
$386$	17.2322	0.877095
$387$	0	0
$388$	0.857183	0.0435169
$389$	17.4876	0.886656	0.443328	−	0.896359i	$-0.353798\pi$
$389$	17.4876	0.886656	0.443328	+	0.896359i	$0.353798\pi$
$390$	0	0
$391$	2.51811	0.127346
$392$	50.0894	2.52990
$393$	0	0
$394$	15.5137	0.781568
$395$	−8.62761	−0.434102
$396$	0	0
$397$	−16.6427	−0.835272	−0.417636	−	0.908614i	$-0.637141\pi$
$397$	−16.6427	−0.835272	−0.417636	+	0.908614i	$0.637141\pi$
$398$	4.70239	0.235710
$399$	0	0
$400$	10.5794	0.528970
$401$	−14.6527	−0.731722	−0.365861	−	0.930669i	$-0.619226\pi$
$401$	−14.6527	−0.731722	−0.365861	+	0.930669i	$0.619226\pi$
$402$	0	0
$403$	−2.54993	−0.127021
$404$	0.669495	0.0333086
$405$	0	0
$406$	−6.78281	−0.336625
$407$	−8.67733	−0.430119
$408$	0	0
$409$	−23.9553	−1.18451	−0.592257	−	0.805749i	$-0.701763\pi$
$409$	−23.9553	−1.18451	−0.592257	+	0.805749i	$0.701763\pi$
$410$	−14.0551	−0.694130
$411$	0	0
$412$	1.77590	0.0874923
$413$	41.4416	2.03921
$414$	0	0
$415$	−11.5317	−0.566070
$416$	−0.312732	−0.0153329
$417$	0	0
$418$	−39.6438	−1.93904
$419$	15.4521	0.754886	0.377443	−	0.926033i	$-0.376803\pi$
$419$	15.4521	0.754886	0.377443	+	0.926033i	$0.376803\pi$
$420$	0	0
$421$	4.79223	0.233559	0.116780	−	0.993158i	$-0.462743\pi$
$421$	4.79223	0.233559	0.116780	+	0.993158i	$0.462743\pi$
$422$	−9.49041	−0.461986
$423$	0	0
$424$	20.6926	1.00492
$425$	−7.05153	−0.342049
$426$	0	0
$427$	−58.2496	−2.81889
$428$	0.713416	0.0344842
$429$	0	0
$430$	−11.1069	−0.535622
$431$	24.6103	1.18544	0.592719	−	0.805410i	$-0.298055\pi$
$431$	24.6103	1.18544	0.592719	+	0.805410i	$0.298055\pi$
$432$	0	0
$433$	−12.2569	−0.589031	−0.294516	−	0.955647i	$-0.595158\pi$
$433$	−12.2569	−0.589031	−0.294516	+	0.955647i	$0.595158\pi$
$434$	32.9651	1.58238
$435$	0	0
$436$	1.66854	0.0799086
$437$	6.02409	0.288171
$438$	0	0
$439$	18.0437	0.861177	0.430588	−	0.902548i	$-0.358306\pi$
$439$	18.0437	0.861177	0.430588	+	0.902548i	$0.358306\pi$
$440$	20.5501	0.979688
$441$	0	0
$442$	−1.81847	−0.0864959
$443$	0.405869	0.0192834	0.00964171	−	0.999954i	$-0.496931\pi$
$443$	0.405869	0.0192834	0.00964171	+	0.999954i	$0.496931\pi$
$444$	0	0
$445$	−20.9136	−0.991401
$446$	6.67636	0.316135
$447$	0	0
$448$	41.2772	1.95016
$449$	−1.01695	−0.0479929	−0.0239965	−	0.999712i	$-0.507639\pi$
$449$	−1.01695	−0.0479929	−0.0239965	+	0.999712i	$0.507639\pi$
$450$	0	0
$451$	32.9184	1.55007
$452$	−1.23139	−0.0579195
$453$	0	0
$454$	25.5949	1.20123
$455$	3.83462	0.179770
$456$	0	0
$457$	−22.1164	−1.03456	−0.517281	−	0.855815i	$-0.673056\pi$
$457$	−22.1164	−1.03456	−0.517281	+	0.855815i	$0.673056\pi$
$458$	−23.4932	−1.09776
$459$	0	0
$460$	−0.156442	−0.00729415
$461$	−41.1018	−1.91430	−0.957150	−	0.289593i	$-0.906480\pi$
$461$	−41.1018	−1.91430	−0.957150	+	0.289593i	$0.906480\pi$
$462$	0	0
$463$	−14.1273	−0.656551	−0.328276	−	0.944582i	$-0.606467\pi$
$463$	−14.1273	−0.656551	−0.328276	+	0.944582i	$0.606467\pi$
$464$	−3.77791	−0.175385
$465$	0	0
$466$	−12.9227	−0.598633
$467$	26.8320	1.24164	0.620819	−	0.783954i	$-0.286800\pi$
$467$	26.8320	1.24164	0.620819	+	0.783954i	$0.286800\pi$
$468$	0	0
$469$	43.3844	2.00331
$470$	−17.8400	−0.822898
$471$	0	0
$472$	24.3712	1.12178
$473$	26.0135	1.19610
$474$	0	0
$475$	−16.8694	−0.774023
$476$	−1.30891	−0.0599937
$477$	0	0
$478$	20.6177	0.943032
$479$	14.9429	0.682759	0.341379	−	0.939926i	$-0.389106\pi$
$479$	14.9429	0.682759	0.341379	+	0.939926i	$0.389106\pi$
$480$	0	0
$481$	−0.952214	−0.0434172
$482$	−14.5992	−0.664978
$483$	0	0
$484$	−1.25097	−0.0568622
$485$	−12.0525	−0.547276
$486$	0	0
$487$	7.07224	0.320474	0.160237	−	0.987079i	$-0.448774\pi$
$487$	7.07224	0.320474	0.160237	+	0.987079i	$0.448774\pi$
$488$	−34.2557	−1.55068
$489$	0	0
$490$	−35.2836	−1.59395
$491$	23.1087	1.04288	0.521440	−	0.853288i	$-0.325395\pi$
$491$	23.1087	1.04288	0.521440	+	0.853288i	$0.325395\pi$
$492$	0	0
$493$	2.51811	0.113410
$494$	−4.35035	−0.195731
$495$	0	0
$496$	18.3610	0.824433
$497$	57.3238	2.57132
$498$	0	0
$499$	34.2464	1.53308	0.766541	−	0.642196i	$-0.221976\pi$
$499$	34.2464	1.53308	0.766541	+	0.642196i	$0.221976\pi$
$500$	1.22030	0.0545735
$501$	0	0
$502$	40.6321	1.81350
$503$	30.4574	1.35803	0.679015	−	0.734124i	$-0.262407\pi$
$503$	30.4574	1.35803	0.679015	+	0.734124i	$0.262407\pi$
$504$	0	0
$505$	−9.41349	−0.418895
$506$	−6.58088	−0.292556
$507$	0	0
$508$	−1.70379	−0.0755933
$509$	23.0616	1.02219	0.511095	−	0.859524i	$-0.329240\pi$
$509$	23.0616	1.02219	0.511095	+	0.859524i	$0.329240\pi$
$510$	0	0
$511$	−64.2860	−2.84384
$512$	24.1488	1.06723
$513$	0	0
$514$	32.4106	1.42957
$515$	−24.9702	−1.10032
$516$	0	0
$517$	41.7832	1.83762
$518$	12.3101	0.540874
$519$	0	0
$520$	2.25508	0.0988919
$521$	−7.60212	−0.333055	−0.166527	−	0.986037i	$-0.553255\pi$
$521$	−7.60212	−0.333055	−0.166527	+	0.986037i	$0.553255\pi$
$522$	0	0
$523$	−9.05401	−0.395904	−0.197952	−	0.980212i	$-0.563429\pi$
$523$	−9.05401	−0.395904	−0.197952	+	0.980212i	$0.563429\pi$
$524$	−0.582613	−0.0254515
$525$	0	0
$526$	−5.74612	−0.250543
$527$	−12.2382	−0.533106
$528$	0	0
$529$	1.00000	0.0434783
$530$	−14.5761	−0.633147
$531$	0	0
$532$	−3.13132	−0.135760
$533$	3.61233	0.156467
$534$	0	0
$535$	−10.0310	−0.433680
$536$	25.5137	1.10203
$537$	0	0
$538$	−32.1137	−1.38452
$539$	82.6380	3.55947
$540$	0	0
$541$	20.6458	0.887632	0.443816	−	0.896118i	$-0.353624\pi$
$541$	20.6458	0.887632	0.443816	+	0.896118i	$0.353624\pi$
$542$	−42.0557	−1.80645
$543$	0	0
$544$	−1.50094	−0.0643522
$545$	−23.4607	−1.00494
$546$	0	0
$547$	−7.07622	−0.302557	−0.151279	−	0.988491i	$-0.548339\pi$
$547$	−7.07622	−0.302557	−0.151279	+	0.988491i	$0.548339\pi$
$548$	0.282266	0.0120578
$549$	0	0
$550$	18.4286	0.785800
$551$	6.02409	0.256635
$552$	0	0
$553$	−28.6663	−1.21902
$554$	−6.39008	−0.271488
$555$	0	0
$556$	−2.14050	−0.0907775
$557$	−10.9839	−0.465404	−0.232702	−	0.972548i	$-0.574757\pi$
$557$	−10.9839	−0.465404	−0.232702	+	0.972548i	$0.574757\pi$
$558$	0	0
$559$	2.85461	0.120737
$560$	−27.6116	−1.16680
$561$	0	0
$562$	31.8776	1.34468
$563$	−4.36253	−0.183859	−0.0919293	−	0.995766i	$-0.529303\pi$
$563$	−4.36253	−0.183859	−0.0919293	+	0.995766i	$0.529303\pi$
$564$	0	0
$565$	17.3140	0.728406
$566$	−8.28133	−0.348090
$567$	0	0
$568$	33.7113	1.41449
$569$	−12.5393	−0.525676	−0.262838	−	0.964840i	$-0.584659\pi$
$569$	−12.5393	−0.525676	−0.262838	+	0.964840i	$0.584659\pi$
$570$	0	0
$571$	−26.7780	−1.12062	−0.560312	−	0.828282i	$-0.689319\pi$
$571$	−26.7780	−1.12062	−0.560312	+	0.828282i	$0.689319\pi$
$572$	−0.264602	−0.0110636
$573$	0	0
$574$	−46.6997	−1.94921
$575$	−2.80033	−0.116782
$576$	0	0
$577$	−30.5483	−1.27174	−0.635871	−	0.771795i	$-0.719359\pi$
$577$	−30.5483	−1.27174	−0.635871	+	0.771795i	$0.719359\pi$
$578$	14.6714	0.610250
$579$	0	0
$580$	−0.156442	−0.00649591
$581$	−38.3156	−1.58960
$582$	0	0
$583$	34.1389	1.41389
$584$	−37.8056	−1.56441
$585$	0	0
$586$	27.4690	1.13473
$587$	−17.8790	−0.737945	−0.368973	−	0.929440i	$-0.620290\pi$
$587$	−17.8790	−0.737945	−0.368973	+	0.929440i	$0.620290\pi$
$588$	0	0
$589$	−29.2776	−1.20636
$590$	−17.1674	−0.706771
$591$	0	0
$592$	6.85651	0.281801
$593$	24.1803	0.992966	0.496483	−	0.868047i	$-0.334625\pi$
$593$	24.1803	0.992966	0.496483	+	0.868047i	$0.334625\pi$
$594$	0	0
$595$	18.4040	0.754492
$596$	1.13867	0.0466417
$597$	0	0
$598$	−0.722159	−0.0295313
$599$	33.7750	1.38001	0.690004	−	0.723805i	$-0.257609\pi$
$599$	33.7750	1.38001	0.690004	+	0.723805i	$0.257609\pi$
$600$	0	0
$601$	47.8919	1.95355	0.976776	−	0.214263i	$-0.0687350\pi$
$601$	47.8919	1.95355	0.976776	+	0.214263i	$0.0687350\pi$
$602$	−36.9040	−1.50410
$603$	0	0
$604$	1.42604	0.0580246
$605$	17.5893	0.715108
$606$	0	0
$607$	1.35443	0.0549746	0.0274873	−	0.999622i	$-0.491249\pi$
$607$	1.35443	0.0549746	0.0274873	+	0.999622i	$0.491249\pi$
$608$	−3.59071	−0.145622
$609$	0	0
$610$	24.1302	0.977002
$611$	4.58511	0.185494
$612$	0	0
$613$	−22.4155	−0.905353	−0.452677	−	0.891675i	$-0.649531\pi$
$613$	−22.4155	−0.905353	−0.452677	+	0.891675i	$0.649531\pi$
$614$	42.0436	1.69674
$615$	0	0
$616$	68.2804	2.75109
$617$	3.75650	0.151231	0.0756155	−	0.997137i	$-0.475908\pi$
$617$	3.75650	0.151231	0.0756155	+	0.997137i	$0.475908\pi$
$618$	0	0
$619$	−4.17326	−0.167738	−0.0838688	−	0.996477i	$-0.526728\pi$
$619$	−4.17326	−0.167738	−0.0838688	+	0.996477i	$0.526728\pi$
$620$	0.760324	0.0305353
$621$	0	0
$622$	−28.2046	−1.13090
$623$	−69.4882	−2.78399
$624$	0	0
$625$	−3.15649	−0.126260
$626$	−36.8663	−1.47347
$627$	0	0
$628$	0.674994	0.0269352
$629$	−4.57010	−0.182222
$630$	0	0
$631$	−33.2279	−1.32278	−0.661390	−	0.750042i	$-0.730033\pi$
$631$	−33.2279	−1.32278	−0.661390	+	0.750042i	$0.730033\pi$
$632$	−16.8582	−0.670585
$633$	0	0
$634$	40.0695	1.59136
$635$	23.9563	0.950675
$636$	0	0
$637$	9.06835	0.359301
$638$	−6.58088	−0.260540
$639$	0	0
$640$	−15.3312	−0.606019
$641$	−12.6802	−0.500838	−0.250419	−	0.968138i	$-0.580568\pi$
$641$	−12.6802	−0.500838	−0.250419	+	0.968138i	$0.580568\pi$
$642$	0	0
$643$	−9.73223	−0.383802	−0.191901	−	0.981414i	$-0.561465\pi$
$643$	−9.73223	−0.383802	−0.191901	+	0.981414i	$0.561465\pi$
$644$	−0.519799	−0.0204829
$645$	0	0
$646$	−20.8792	−0.821483
$647$	−0.0196233	−0.000771470 0	−0.000385735	−	1.00000i	$-0.500123\pi$
$647$	−0.0196233	−0.000771470 0	−0.000385735		1.00000i	$0.500123\pi$
$648$	0	0
$649$	40.2079	1.57830
$650$	2.02228	0.0793205
$651$	0	0
$652$	0.930924	0.0364578
$653$	26.8683	1.05144	0.525719	−	0.850658i	$-0.323796\pi$
$653$	26.8683	1.05144	0.525719	+	0.850658i	$0.323796\pi$
$654$	0	0
$655$	8.19188	0.320083
$656$	−26.0110	−1.01556
$657$	0	0
$658$	−59.2757	−2.31081
$659$	29.6280	1.15414	0.577071	−	0.816694i	$-0.304196\pi$
$659$	29.6280	1.15414	0.577071	+	0.816694i	$0.304196\pi$
$660$	0	0
$661$	12.3242	0.479356	0.239678	−	0.970852i	$-0.422958\pi$
$661$	12.3242	0.479356	0.239678	+	0.970852i	$0.422958\pi$
$662$	31.3121	1.21698
$663$	0	0
$664$	−22.5329	−0.874445
$665$	44.0281	1.70734
$666$	0	0
$667$	1.00000	0.0387202
$668$	−2.02116	−0.0782010
$669$	0	0
$670$	−17.9722	−0.694327
$671$	−56.5154	−2.18175
$672$	0	0
$673$	−5.47404	−0.211009	−0.105504	−	0.994419i	$-0.533646\pi$
$673$	−5.47404	−0.211009	−0.105504	+	0.994419i	$0.533646\pi$
$674$	29.1798	1.12397
$675$	0	0
$676$	1.34222	0.0516238
$677$	41.7308	1.60385	0.801923	−	0.597427i	$-0.203810\pi$
$677$	41.7308	1.60385	0.801923	+	0.597427i	$0.203810\pi$
$678$	0	0
$679$	−40.0460	−1.53682
$680$	10.8231	0.415049
$681$	0	0
$682$	31.9837	1.22472
$683$	1.83472	0.0702037	0.0351019	−	0.999384i	$-0.488824\pi$
$683$	1.83472	0.0702037	0.0351019	+	0.999384i	$0.488824\pi$
$684$	0	0
$685$	−3.96883	−0.151641
$686$	−69.7547	−2.66324
$687$	0	0
$688$	−20.5549	−0.783649
$689$	3.74626	0.142721
$690$	0	0
$691$	33.0805	1.25844	0.629220	−	0.777227i	$-0.283374\pi$
$691$	33.0805	1.25844	0.629220	+	0.777227i	$0.283374\pi$
$692$	1.06078	0.0403248
$693$	0	0
$694$	−10.7288	−0.407261
$695$	30.0967	1.14163
$696$	0	0
$697$	17.3372	0.656693
$698$	−23.5138	−0.890010
$699$	0	0
$700$	1.45561	0.0550169
$701$	−16.2570	−0.614020	−0.307010	−	0.951706i	$-0.599328\pi$
$701$	−16.2570	−0.614020	−0.307010	+	0.951706i	$0.599328\pi$
$702$	0	0
$703$	−10.9331	−0.412349
$704$	40.0483	1.50938
$705$	0	0
$706$	18.1825	0.684307
$707$	−31.2775	−1.17631
$708$	0	0
$709$	0.0198177	0.000744270 0	0.000372135	−	1.00000i	$-0.499882\pi$
$709$	0.0198177	0.000744270 0	0.000372135		1.00000i	$0.499882\pi$
$710$	−23.7467	−0.891197
$711$	0	0
$712$	−40.8650	−1.53148
$713$	−4.86009	−0.182012
$714$	0	0
$715$	3.72046	0.139137
$716$	0.847097	0.0316575
$717$	0	0
$718$	13.5986	0.507493
$719$	26.1632	0.975723	0.487862	−	0.872921i	$-0.337777\pi$
$719$	26.1632	0.975723	0.487862	+	0.872921i	$0.337777\pi$
$720$	0	0
$721$	−82.9667	−3.08984
$722$	−23.7977	−0.885660
$723$	0	0
$724$	1.38897	0.0516208
$725$	−2.80033	−0.104002
$726$	0	0
$727$	−17.3000	−0.641623	−0.320811	−	0.947143i	$-0.603956\pi$
$727$	−17.3000	−0.641623	−0.320811	+	0.947143i	$0.603956\pi$
$728$	7.49280	0.277702
$729$	0	0
$730$	26.6308	0.985649
$731$	13.7006	0.506733
$732$	0	0
$733$	−33.2459	−1.22797	−0.613983	−	0.789319i	$-0.710434\pi$
$733$	−33.2459	−1.22797	−0.613983	+	0.789319i	$0.710434\pi$
$734$	−24.5060	−0.904533
$735$	0	0
$736$	−0.596058	−0.0219710
$737$	42.0928	1.55051
$738$	0	0
$739$	22.0846	0.812394	0.406197	−	0.913786i	$-0.366855\pi$
$739$	22.0846	0.812394	0.406197	+	0.913786i	$0.366855\pi$
$740$	0.283926	0.0104373
$741$	0	0
$742$	−48.4311	−1.77796
$743$	21.7767	0.798909	0.399455	−	0.916753i	$-0.369200\pi$
$743$	21.7767	0.798909	0.399455	+	0.916753i	$0.369200\pi$
$744$	0	0
$745$	−16.0104	−0.586574
$746$	17.3400	0.634862
$747$	0	0
$748$	−1.26994	−0.0464336
$749$	−33.3294	−1.21783
$750$	0	0
$751$	−17.9180	−0.653836	−0.326918	−	0.945053i	$-0.606010\pi$
$751$	−17.9180	−0.653836	−0.326918	+	0.945053i	$0.606010\pi$
$752$	−33.0156	−1.20395
$753$	0	0
$754$	−0.722159	−0.0262995
$755$	−20.0509	−0.729728
$756$	0	0
$757$	−8.33584	−0.302971	−0.151486	−	0.988459i	$-0.548406\pi$
$757$	−8.33584	−0.302971	−0.151486	+	0.988459i	$0.548406\pi$
$758$	25.5453	0.927848
$759$	0	0
$760$	25.8923	0.939213
$761$	−30.3649	−1.10073	−0.550363	−	0.834925i	$-0.685511\pi$
$761$	−30.3649	−1.10073	−0.550363	+	0.834925i	$0.685511\pi$
$762$	0	0
$763$	−77.9510	−2.82202
$764$	1.46694	0.0530720
$765$	0	0
$766$	8.48051	0.306413
$767$	4.41224	0.159317
$768$	0	0
$769$	13.4279	0.484222	0.242111	−	0.970249i	$-0.422160\pi$
$769$	13.4279	0.484222	0.242111	+	0.970249i	$0.422160\pi$
$770$	−48.0976	−1.73332
$771$	0	0
$772$	1.32058	0.0475288
$773$	38.3206	1.37830	0.689149	−	0.724620i	$-0.257985\pi$
$773$	38.3206	1.37830	0.689149	+	0.724620i	$0.257985\pi$
$774$	0	0
$775$	13.6099	0.488881
$776$	−23.5505	−0.845412
$777$	0	0
$778$	−24.0702	−0.862958
$779$	41.4759	1.48603
$780$	0	0
$781$	55.6172	1.99014
$782$	−3.46596	−0.123942
$783$	0	0
$784$	−65.2975	−2.33205
$785$	−9.49081	−0.338742
$786$	0	0
$787$	39.6436	1.41314	0.706570	−	0.707643i	$-0.250241\pi$
$787$	39.6436	1.41314	0.706570	+	0.707643i	$0.250241\pi$
$788$	1.18889	0.0423523
$789$	0	0
$790$	11.8752	0.422499
$791$	57.5280	2.04546
$792$	0	0
$793$	−6.20177	−0.220231
$794$	22.9072	0.812947
$795$	0	0
$796$	0.360367	0.0127729
$797$	−40.1420	−1.42190	−0.710951	−	0.703242i	$-0.751735\pi$
$797$	−40.1420	−1.42190	−0.710951	+	0.703242i	$0.751735\pi$
$798$	0	0
$799$	22.0060	0.778516
$800$	1.66916	0.0590137
$801$	0	0
$802$	20.1682	0.712165
$803$	−62.3721	−2.20106
$804$	0	0
$805$	7.30868	0.257597
$806$	3.50976	0.123626
$807$	0	0
$808$	−18.3939	−0.647094
$809$	−0.173541	−0.00610138	−0.00305069	−	0.999995i	$-0.500971\pi$
$809$	−0.173541	−0.00610138	−0.00305069	+	0.999995i	$0.500971\pi$
$810$	0	0
$811$	−0.873050	−0.0306569	−0.0153285	−	0.999883i	$-0.504879\pi$
$811$	−0.873050	−0.0306569	−0.0153285	+	0.999883i	$0.504879\pi$
$812$	−0.519799	−0.0182414
$813$	0	0
$814$	11.9436	0.418623
$815$	−13.0893	−0.458500
$816$	0	0
$817$	32.7760	1.14669
$818$	32.9724	1.15285
$819$	0	0
$820$	−1.07711	−0.0376142
$821$	5.99107	0.209090	0.104545	−	0.994520i	$-0.466661\pi$
$821$	5.99107	0.209090	0.104545	+	0.994520i	$0.466661\pi$
$822$	0	0
$823$	18.7788	0.654589	0.327294	−	0.944922i	$-0.393863\pi$
$823$	18.7788	0.654589	0.327294	+	0.944922i	$0.393863\pi$
$824$	−48.7915	−1.69973
$825$	0	0
$826$	−57.0409	−1.98471
$827$	53.3963	1.85677	0.928387	−	0.371616i	$-0.121196\pi$
$827$	53.3963	1.85677	0.928387	+	0.371616i	$0.121196\pi$
$828$	0	0
$829$	−31.4472	−1.09221	−0.546103	−	0.837718i	$-0.683889\pi$
$829$	−31.4472	−1.09221	−0.546103	+	0.837718i	$0.683889\pi$
$830$	15.8724	0.550940
$831$	0	0
$832$	4.39473	0.152360
$833$	43.5230	1.50798
$834$	0	0
$835$	28.4187	0.983469
$836$	−3.03809	−0.105075
$837$	0	0
$838$	−21.2685	−0.734710
$839$	−43.6126	−1.50568	−0.752838	−	0.658206i	$-0.771316\pi$
$839$	−43.6126	−1.50568	−0.752838	+	0.658206i	$0.771316\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−6.59610	−0.227317
$843$	0	0
$844$	−0.727295	−0.0250345
$845$	−18.8724	−0.649230
$846$	0	0
$847$	58.4428	2.00812
$848$	−26.9753	−0.926335
$849$	0	0
$850$	9.70583	0.332907
$851$	−1.81490	−0.0622138
$852$	0	0
$853$	24.3777	0.834678	0.417339	−	0.908751i	$-0.362963\pi$
$853$	24.3777	0.834678	0.417339	+	0.908751i	$0.362963\pi$
$854$	80.1756	2.74355
$855$	0	0
$856$	−19.6006	−0.669933
$857$	−42.7652	−1.46083	−0.730416	−	0.683003i	$-0.760674\pi$
$857$	−42.7652	−1.46083	−0.730416	+	0.683003i	$0.760674\pi$
$858$	0	0
$859$	3.97878	0.135754	0.0678771	−	0.997694i	$-0.478377\pi$
$859$	3.97878	0.135754	0.0678771	+	0.997694i	$0.478377\pi$
$860$	−0.851173	−0.0290248
$861$	0	0
$862$	−33.8740	−1.15375
$863$	31.5448	1.07380	0.536899	−	0.843646i	$-0.319595\pi$
$863$	31.5448	1.07380	0.536899	+	0.843646i	$0.319595\pi$
$864$	0	0
$865$	−14.9152	−0.507132
$866$	16.8706	0.573288
$867$	0	0
$868$	2.52627	0.0857473
$869$	−27.8129	−0.943488
$870$	0	0
$871$	4.61909	0.156512
$872$	−45.8419	−1.55240
$873$	0	0
$874$	−8.29165	−0.280469
$875$	−57.0101	−1.92729
$876$	0	0
$877$	−30.3834	−1.02598	−0.512988	−	0.858396i	$-0.671461\pi$
$877$	−30.3834	−1.02598	−0.512988	+	0.858396i	$0.671461\pi$
$878$	−24.8356	−0.838160
$879$	0	0
$880$	−26.7895	−0.903075
$881$	6.35686	0.214168	0.107084	−	0.994250i	$-0.465849\pi$
$881$	6.35686	0.214168	0.107084	+	0.994250i	$0.465849\pi$
$882$	0	0
$883$	4.78642	0.161076	0.0805380	−	0.996752i	$-0.474336\pi$
$883$	4.78642	0.161076	0.0805380	+	0.996752i	$0.474336\pi$
$884$	−0.139358	−0.00468712
$885$	0	0
$886$	−0.558645	−0.0187680
$887$	17.9828	0.603804	0.301902	−	0.953339i	$-0.402378\pi$
$887$	17.9828	0.603804	0.301902	+	0.953339i	$0.402378\pi$
$888$	0	0
$889$	79.5977	2.66962
$890$	28.7858	0.964904
$891$	0	0
$892$	0.511641	0.0171310
$893$	52.6452	1.76170
$894$	0	0
$895$	−11.9107	−0.398130
$896$	−50.9399	−1.70178
$897$	0	0
$898$	1.39975	0.0467102
$899$	−4.86009	−0.162093
$900$	0	0
$901$	17.9799	0.598999
$902$	−45.3094	−1.50864
$903$	0	0
$904$	33.8314	1.12522
$905$	−19.5298	−0.649192
$906$	0	0
$907$	6.79373	0.225582	0.112791	−	0.993619i	$-0.464021\pi$
$907$	6.79373	0.225582	0.112791	+	0.993619i	$0.464021\pi$
$908$	1.96146	0.0650933
$909$	0	0
$910$	−5.27803	−0.174965
$911$	30.4854	1.01003	0.505013	−	0.863112i	$-0.331488\pi$
$911$	30.4854	1.01003	0.505013	+	0.863112i	$0.331488\pi$
$912$	0	0
$913$	−37.1749	−1.23031
$914$	30.4414	1.00691
$915$	0	0
$916$	−1.80039	−0.0594867
$917$	27.2186	0.898836
$918$	0	0
$919$	31.7293	1.04665	0.523326	−	0.852132i	$-0.324691\pi$
$919$	31.7293	1.04665	0.523326	+	0.852132i	$0.324691\pi$
$920$	4.29813	0.141705
$921$	0	0
$922$	56.5731	1.86314
$923$	6.10320	0.200889
$924$	0	0
$925$	5.08231	0.167105
$926$	19.4450	0.639003
$927$	0	0
$928$	−0.596058	−0.0195666
$929$	−20.9650	−0.687838	−0.343919	−	0.938999i	$-0.611755\pi$
$929$	−20.9650	−0.687838	−0.343919	+	0.938999i	$0.611755\pi$
$930$	0	0
$931$	104.121	3.41241
$932$	−0.990329	−0.0324393
$933$	0	0
$934$	−36.9320	−1.20845
$935$	17.8561	0.583958
$936$	0	0
$937$	12.3540	0.403588	0.201794	−	0.979428i	$-0.435323\pi$
$937$	12.3540	0.403588	0.201794	+	0.979428i	$0.435323\pi$
$938$	−59.7149	−1.94976
$939$	0	0
$940$	−1.36716	−0.0445920
$941$	−4.48801	−0.146305	−0.0731524	−	0.997321i	$-0.523306\pi$
$941$	−4.48801	−0.146305	−0.0731524	+	0.997321i	$0.523306\pi$
$942$	0	0
$943$	6.88501	0.224207
$944$	−31.7708	−1.03405
$945$	0	0
$946$	−35.8054	−1.16413
$947$	−24.5673	−0.798330	−0.399165	−	0.916879i	$-0.630700\pi$
$947$	−24.5673	−0.798330	−0.399165	+	0.916879i	$0.630700\pi$
$948$	0	0
$949$	−6.84445	−0.222180
$950$	23.2194	0.753335
$951$	0	0
$952$	35.9613	1.16551
$953$	17.9457	0.581319	0.290660	−	0.956826i	$-0.406125\pi$
$953$	17.9457	0.581319	0.290660	+	0.956826i	$0.406125\pi$
$954$	0	0
$955$	−20.6260	−0.667443
$956$	1.58003	0.0511019
$957$	0	0
$958$	−20.5676	−0.664511
$959$	−13.1869	−0.425829
$960$	0	0
$961$	−7.37948	−0.238048
$962$	1.31064	0.0422568
$963$	0	0
$964$	−1.11881	−0.0360344
$965$	−18.5682	−0.597731
$966$	0	0
$967$	47.2541	1.51959	0.759794	−	0.650164i	$-0.225300\pi$
$967$	47.2541	1.51959	0.759794	+	0.650164i	$0.225300\pi$
$968$	34.3694	1.10467
$969$	0	0
$970$	16.5892	0.532649
$971$	−11.7649	−0.377553	−0.188776	−	0.982020i	$-0.560452\pi$
$971$	−11.7649	−0.377553	−0.188776	+	0.982020i	$0.560452\pi$
$972$	0	0
$973$	100.000	3.20586
$974$	−9.73433	−0.311908
$975$	0	0
$976$	44.6564	1.42942
$977$	4.23462	0.135478	0.0677388	−	0.997703i	$-0.478422\pi$
$977$	4.23462	0.135478	0.0677388	+	0.997703i	$0.478422\pi$
$978$	0	0
$979$	−67.4195	−2.15474
$980$	−2.70395	−0.0863745
$981$	0	0
$982$	−31.8071	−1.01501
$983$	−19.8985	−0.634664	−0.317332	−	0.948315i	$-0.602787\pi$
$983$	−19.8985	−0.634664	−0.317332	+	0.948315i	$0.602787\pi$
$984$	0	0
$985$	−16.7165	−0.532631
$986$	−3.46596	−0.110379
$987$	0	0
$988$	−0.333388	−0.0106065
$989$	5.44082	0.173008
$990$	0	0
$991$	−39.7115	−1.26148	−0.630739	−	0.775995i	$-0.717248\pi$
$991$	−39.7115	−1.26148	−0.630739	+	0.775995i	$0.717248\pi$
$992$	2.89690	0.0919766
$993$	0	0
$994$	−78.9014	−2.50260
$995$	−5.06697	−0.160634
$996$	0	0
$997$	28.2172	0.893647	0.446824	−	0.894622i	$-0.352555\pi$
$997$	28.2172	0.893647	0.446824	+	0.894622i	$0.352555\pi$
$998$	−47.1373	−1.49211
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.p.1.4		14
3.2	odd	2		2001.2.a.m.1.11	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.m.1.11	✓	14	3.2	odd	2
6003.2.a.p.1.4		14	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)

\(f(q)\)	\(=\)	\(q-1.37642 q^{2} -0.105481 q^{4} +1.48313 q^{5} +4.92788 q^{7} +2.89802 q^{8} +O(q^{10})\)	\(q-1.37642 q^{2} -0.105481 q^{4} +1.48313 q^{5} +4.92788 q^{7} +2.89802 q^{8} -2.04140 q^{10} +4.78117 q^{11} +0.524666 q^{13} -6.78281 q^{14} -3.77791 q^{16} +2.51811 q^{17} +6.02409 q^{19} -0.156442 q^{20} -6.58088 q^{22} +1.00000 q^{23} -2.80033 q^{25} -0.722159 q^{26} -0.519799 q^{28} +1.00000 q^{29} -4.86009 q^{31} -0.596058 q^{32} -3.46596 q^{34} +7.30868 q^{35} -1.81490 q^{37} -8.29165 q^{38} +4.29813 q^{40} +6.88501 q^{41} +5.44082 q^{43} -0.504324 q^{44} -1.37642 q^{46} +8.73911 q^{47} +17.2840 q^{49} +3.85442 q^{50} -0.0553424 q^{52} +7.14027 q^{53} +7.09109 q^{55} +14.2811 q^{56} -1.37642 q^{58} +8.40962 q^{59} -11.8204 q^{61} +6.68951 q^{62} +8.37625 q^{64} +0.778147 q^{65} +8.80386 q^{67} -0.265613 q^{68} -10.0598 q^{70} +11.6325 q^{71} -13.0454 q^{73} +2.49805 q^{74} -0.635428 q^{76} +23.5611 q^{77} -5.81717 q^{79} -5.60313 q^{80} -9.47663 q^{82} -7.77527 q^{83} +3.73467 q^{85} -7.48882 q^{86} +13.8559 q^{88} -14.1010 q^{89} +2.58549 q^{91} -0.105481 q^{92} -12.0286 q^{94} +8.93450 q^{95} -8.12641 q^{97} -23.7900 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) 14 * q + 2 * q^2 + 12 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8	\( 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} - 5 q^{10} + 12 q^{11} + 13 q^{13} + 9 q^{14} + 14 q^{17} - 9 q^{19} + 2 q^{20} - 9 q^{22} + 14 q^{23} + 13 q^{25} + 16 q^{26} + 3 q^{28} + 14 q^{29} - 28 q^{31} + 4 q^{32} + 14 q^{34} + 9 q^{35} - 12 q^{37} - 2 q^{38} - 20 q^{40} + 25 q^{41} + 5 q^{43} + 37 q^{44} + 2 q^{46} + 17 q^{47} + 17 q^{49} + 44 q^{50} + 25 q^{52} + 17 q^{53} + q^{55} + 54 q^{56} + 2 q^{58} + 18 q^{59} - 13 q^{61} + 8 q^{62} + 20 q^{64} + 16 q^{65} + 2 q^{67} + 19 q^{68} + 14 q^{70} + 55 q^{71} + 19 q^{73} - 4 q^{74} - 32 q^{76} + 19 q^{77} - 68 q^{79} + 2 q^{80} - 12 q^{82} + 21 q^{83} + 16 q^{85} + 22 q^{86} - 25 q^{88} + 17 q^{89} - 30 q^{91} + 12 q^{92} + 16 q^{94} + 55 q^{95} + 25 q^{97} + 31 q^{98}+O(q^{100}) \) 14 * q + 2 * q^2 + 12 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8 - 5 * q^10 + 12 * q^11 + 13 * q^13 + 9 * q^14 + 14 * q^17 - 9 * q^19 + 2 * q^20 - 9 * q^22 + 14 * q^23 + 13 * q^25 + 16 * q^26 + 3 * q^28 + 14 * q^29 - 28 * q^31 + 4 * q^32 + 14 * q^34 + 9 * q^35 - 12 * q^37 - 2 * q^38 - 20 * q^40 + 25 * q^41 + 5 * q^43 + 37 * q^44 + 2 * q^46 + 17 * q^47 + 17 * q^49 + 44 * q^50 + 25 * q^52 + 17 * q^53 + q^55 + 54 * q^56 + 2 * q^58 + 18 * q^59 - 13 * q^61 + 8 * q^62 + 20 * q^64 + 16 * q^65 + 2 * q^67 + 19 * q^68 + 14 * q^70 + 55 * q^71 + 19 * q^73 - 4 * q^74 - 32 * q^76 + 19 * q^77 - 68 * q^79 + 2 * q^80 - 12 * q^82 + 21 * q^83 + 16 * q^85 + 22 * q^86 - 25 * q^88 + 17 * q^89 - 30 * q^91 + 12 * q^92 + 16 * q^94 + 55 * q^95 + 25 * q^97 + 31 * q^98

Introduction

L-functions

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