LMFDB - Embedded newform 6003.2.a.k.1.2

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			1.2
Root			$1.96386$ 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)

$f(q)$	$=$	$q-2.36221 q^{2} +3.58005 q^{4} -3.25906 q^{5} +3.14721 q^{7} -3.73241 q^{8} +O(q^{10})$	\(q-2.36221 q^{2} +3.58005 q^{4} -3.25906 q^{5} +3.14721 q^{7} -3.73241 q^{8} +7.69859 q^{10} -1.45931 q^{11} +1.99182 q^{13} -7.43438 q^{14} +1.65665 q^{16} +5.39964 q^{17} +1.55174 q^{19} -11.6676 q^{20} +3.44720 q^{22} +1.00000 q^{23} +5.62147 q^{25} -4.70511 q^{26} +11.2672 q^{28} -1.00000 q^{29} -7.97707 q^{31} +3.55146 q^{32} -12.7551 q^{34} -10.2569 q^{35} -10.1874 q^{37} -3.66555 q^{38} +12.1642 q^{40} +4.33929 q^{41} -4.79243 q^{43} -5.22439 q^{44} -2.36221 q^{46} +2.39075 q^{47} +2.90493 q^{49} -13.2791 q^{50} +7.13083 q^{52} -6.13279 q^{53} +4.75597 q^{55} -11.7467 q^{56} +2.36221 q^{58} -5.55214 q^{59} +6.34330 q^{61} +18.8435 q^{62} -11.7026 q^{64} -6.49147 q^{65} +8.62997 q^{67} +19.3310 q^{68} +24.2291 q^{70} -14.2522 q^{71} +0.308432 q^{73} +24.0649 q^{74} +5.55531 q^{76} -4.59275 q^{77} -2.60309 q^{79} -5.39913 q^{80} -10.2503 q^{82} +17.7781 q^{83} -17.5978 q^{85} +11.3207 q^{86} +5.44674 q^{88} +0.673720 q^{89} +6.26869 q^{91} +3.58005 q^{92} -5.64746 q^{94} -5.05722 q^{95} +5.30656 q^{97} -6.86206 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.36221	−1.67034	−0.835168	−	0.549994i	$-0.814630\pi$
$2$	−2.36221	−1.67034	−0.835168	+	0.549994i	$0.814630\pi$
$3$	0	0
$3$	0	0
$4$	3.58005	1.79002
$5$	−3.25906	−1.45750	−0.728748	−	0.684782i	$-0.759897\pi$
$5$	−3.25906	−1.45750	−0.728748	+	0.684782i	$0.759897\pi$
$6$	0	0
$7$	3.14721	1.18953	0.594767	−	0.803898i	$-0.297244\pi$
$7$	3.14721	1.18953	0.594767	+	0.803898i	$0.297244\pi$
$8$	−3.73241	−1.31961
$9$	0	0
$10$	7.69859	2.43451
$11$	−1.45931	−0.439998	−0.219999	−	0.975500i	$-0.570605\pi$
$11$	−1.45931	−0.439998	−0.219999	+	0.975500i	$0.570605\pi$
$12$	0	0
$13$	1.99182	0.552433	0.276216	−	0.961095i	$-0.410919\pi$
$13$	1.99182	0.552433	0.276216	+	0.961095i	$0.410919\pi$
$14$	−7.43438	−1.98692
$15$	0	0
$16$	1.65665	0.414163
$17$	5.39964	1.30961	0.654803	−	0.755800i	$-0.272752\pi$
$17$	5.39964	1.30961	0.654803	+	0.755800i	$0.272752\pi$
$18$	0	0
$19$	1.55174	0.355994	0.177997	−	0.984031i	$-0.443038\pi$
$19$	1.55174	0.355994	0.177997	+	0.984031i	$0.443038\pi$
$20$	−11.6676	−2.60895
$21$	0	0
$22$	3.44720	0.734945
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	5.62147	1.12429
$26$	−4.70511	−0.922749
$27$	0	0
$28$	11.2672	2.12929
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−7.97707	−1.43272	−0.716362	−	0.697729i	$-0.754194\pi$
$31$	−7.97707	−1.43272	−0.716362	+	0.697729i	$0.754194\pi$
$32$	3.55146	0.627815
$33$	0	0
$34$	−12.7551	−2.18748
$35$	−10.2569	−1.73374
$36$	0	0
$37$	−10.1874	−1.67481	−0.837403	−	0.546586i	$-0.815927\pi$
$37$	−10.1874	−1.67481	−0.837403	+	0.546586i	$0.815927\pi$
$38$	−3.66555	−0.594630
$39$	0	0
$40$	12.1642	1.92332
$41$	4.33929	0.677684	0.338842	−	0.940843i	$-0.389965\pi$
$41$	4.33929	0.677684	0.338842	+	0.940843i	$0.389965\pi$
$42$	0	0
$43$	−4.79243	−0.730839	−0.365420	−	0.930843i	$-0.619075\pi$
$43$	−4.79243	−0.730839	−0.365420	+	0.930843i	$0.619075\pi$
$44$	−5.22439	−0.787607
$45$	0	0
$46$	−2.36221	−0.348289
$47$	2.39075	0.348727	0.174363	−	0.984681i	$-0.444213\pi$
$47$	2.39075	0.348727	0.174363	+	0.984681i	$0.444213\pi$
$48$	0	0
$49$	2.90493	0.414990
$50$	−13.2791	−1.87795
$51$	0	0
$52$	7.13083	0.988868
$53$	−6.13279	−0.842404	−0.421202	−	0.906967i	$-0.638392\pi$
$53$	−6.13279	−0.842404	−0.421202	+	0.906967i	$0.638392\pi$
$54$	0	0
$55$	4.75597	0.641295
$56$	−11.7467	−1.56972
$57$	0	0
$58$	2.36221	0.310174
$59$	−5.55214	−0.722827	−0.361413	−	0.932406i	$-0.617706\pi$
$59$	−5.55214	−0.722827	−0.361413	+	0.932406i	$0.617706\pi$
$60$	0	0
$61$	6.34330	0.812176	0.406088	−	0.913834i	$-0.366893\pi$
$61$	6.34330	0.812176	0.406088	+	0.913834i	$0.366893\pi$
$62$	18.8435	2.39313
$63$	0	0
$64$	−11.7026	−1.46283
$65$	−6.49147	−0.805168
$66$	0	0
$67$	8.62997	1.05432	0.527159	−	0.849767i	$-0.323257\pi$
$67$	8.62997	1.05432	0.527159	+	0.849767i	$0.323257\pi$
$68$	19.3310	2.34423
$69$	0	0
$70$	24.2291	2.89593
$71$	−14.2522	−1.69142	−0.845710	−	0.533643i	$-0.820823\pi$
$71$	−14.2522	−1.69142	−0.845710	+	0.533643i	$0.820823\pi$
$72$	0	0
$73$	0.308432	0.0360993	0.0180496	−	0.999837i	$-0.494254\pi$
$73$	0.308432	0.0360993	0.0180496	+	0.999837i	$0.494254\pi$
$74$	24.0649	2.79749
$75$	0	0
$76$	5.55531	0.637238
$77$	−4.59275	−0.523392
$78$	0	0
$79$	−2.60309	−0.292871	−0.146436	−	0.989220i	$-0.546780\pi$
$79$	−2.60309	−0.292871	−0.146436	+	0.989220i	$0.546780\pi$
$80$	−5.39913	−0.603641
$81$	0	0
$82$	−10.2503	−1.13196
$83$	17.7781	1.95140	0.975702	−	0.219102i	$-0.0703129\pi$
$83$	17.7781	1.95140	0.975702	+	0.219102i	$0.0703129\pi$
$84$	0	0
$85$	−17.5978	−1.90874
$86$	11.3207	1.22075
$87$	0	0
$88$	5.44674	0.580624
$89$	0.673720	0.0714142	0.0357071	−	0.999362i	$-0.488632\pi$
$89$	0.673720	0.0714142	0.0357071	+	0.999362i	$0.488632\pi$
$90$	0	0
$91$	6.26869	0.657137
$92$	3.58005	0.373246
$93$	0	0
$94$	−5.64746	−0.582491
$95$	−5.05722	−0.518860
$96$	0	0
$97$	5.30656	0.538800	0.269400	−	0.963028i	$-0.413175\pi$
$97$	5.30656	0.538800	0.269400	+	0.963028i	$0.413175\pi$
$98$	−6.86206	−0.693173
$99$	0	0
$100$	20.1251	2.01251
$101$	−7.47895	−0.744184	−0.372092	−	0.928196i	$-0.621359\pi$
$101$	−7.47895	−0.744184	−0.372092	+	0.928196i	$0.621359\pi$
$102$	0	0
$103$	−5.92379	−0.583688	−0.291844	−	0.956466i	$-0.594269\pi$
$103$	−5.92379	−0.583688	−0.291844	+	0.956466i	$0.594269\pi$
$104$	−7.43431	−0.728994
$105$	0	0
$106$	14.4870	1.40710
$107$	−10.9806	−1.06154	−0.530768	−	0.847517i	$-0.678096\pi$
$107$	−10.9806	−1.06154	−0.530768	+	0.847517i	$0.678096\pi$
$108$	0	0
$109$	−12.4247	−1.19007	−0.595037	−	0.803698i	$-0.702863\pi$
$109$	−12.4247	−1.19007	−0.595037	+	0.803698i	$0.702863\pi$
$110$	−11.2346	−1.07118
$111$	0	0
$112$	5.21384	0.492661
$113$	6.26445	0.589310	0.294655	−	0.955604i	$-0.404795\pi$
$113$	6.26445	0.589310	0.294655	+	0.955604i	$0.404795\pi$
$114$	0	0
$115$	−3.25906	−0.303909
$116$	−3.58005	−0.332399
$117$	0	0
$118$	13.1153	1.20736
$119$	16.9938	1.55782
$120$	0	0
$121$	−8.87042	−0.806402
$122$	−14.9842	−1.35661
$123$	0	0
$124$	−28.5583	−2.56461
$125$	−2.02540	−0.181157
$126$	0	0
$127$	−5.48635	−0.486835	−0.243418	−	0.969922i	$-0.578269\pi$
$127$	−5.48635	−0.486835	−0.243418	+	0.969922i	$0.578269\pi$
$128$	20.5411	1.81560
$129$	0	0
$130$	15.3342	1.34490
$131$	−4.33605	−0.378843	−0.189421	−	0.981896i	$-0.560661\pi$
$131$	−4.33605	−0.378843	−0.189421	+	0.981896i	$0.560661\pi$
$132$	0	0
$133$	4.88366	0.423467
$134$	−20.3858	−1.76107
$135$	0	0
$136$	−20.1537	−1.72816
$137$	10.2133	0.872579	0.436289	−	0.899806i	$-0.356292\pi$
$137$	10.2133	0.872579	0.436289	+	0.899806i	$0.356292\pi$
$138$	0	0
$139$	18.4979	1.56897	0.784484	−	0.620150i	$-0.212928\pi$
$139$	18.4979	1.56897	0.784484	+	0.620150i	$0.212928\pi$
$140$	−36.7204	−3.10344
$141$	0	0
$142$	33.6666	2.82524
$143$	−2.90668	−0.243069
$144$	0	0
$145$	3.25906	0.270650
$146$	−0.728583	−0.0602979
$147$	0	0
$148$	−36.4716	−2.99794
$149$	−23.4402	−1.92030	−0.960148	−	0.279491i	$-0.909834\pi$
$149$	−23.4402	−1.92030	−0.960148	+	0.279491i	$0.909834\pi$
$150$	0	0
$151$	24.1988	1.96927	0.984637	−	0.174616i	$-0.0558684\pi$
$151$	24.1988	1.96927	0.984637	+	0.174616i	$0.0558684\pi$
$152$	−5.79174	−0.469772
$153$	0	0
$154$	10.8490	0.874241
$155$	25.9977	2.08819
$156$	0	0
$157$	6.75085	0.538777	0.269388	−	0.963032i	$-0.413178\pi$
$157$	6.75085	0.538777	0.269388	+	0.963032i	$0.413178\pi$
$158$	6.14906	0.489193
$159$	0	0
$160$	−11.5744	−0.915037
$161$	3.14721	0.248035
$162$	0	0
$163$	12.5645	0.984130	0.492065	−	0.870558i	$-0.336242\pi$
$163$	12.5645	0.984130	0.492065	+	0.870558i	$0.336242\pi$
$164$	15.5349	1.21307
$165$	0	0
$166$	−41.9957	−3.25950
$167$	−6.12109	−0.473664	−0.236832	−	0.971551i	$-0.576109\pi$
$167$	−6.12109	−0.473664	−0.236832	+	0.971551i	$0.576109\pi$
$168$	0	0
$169$	−9.03264	−0.694818
$170$	41.5696	3.18825
$171$	0	0
$172$	−17.1571	−1.30822
$173$	−10.1985	−0.775381	−0.387690	−	0.921790i	$-0.626727\pi$
$173$	−10.1985	−0.775381	−0.387690	+	0.921790i	$0.626727\pi$
$174$	0	0
$175$	17.6919	1.33738
$176$	−2.41757	−0.182231
$177$	0	0
$178$	−1.59147	−0.119286
$179$	20.4498	1.52849	0.764244	−	0.644927i	$-0.223112\pi$
$179$	20.4498	1.52849	0.764244	+	0.644927i	$0.223112\pi$
$180$	0	0
$181$	1.56194	0.116098	0.0580492	−	0.998314i	$-0.481512\pi$
$181$	1.56194	0.116098	0.0580492	+	0.998314i	$0.481512\pi$
$182$	−14.8080	−1.09764
$183$	0	0
$184$	−3.73241	−0.275157
$185$	33.2015	2.44102
$186$	0	0
$187$	−7.87974	−0.576224
$188$	8.55900	0.624230
$189$	0	0
$190$	11.9462	0.866671
$191$	−17.5488	−1.26979	−0.634895	−	0.772598i	$-0.718957\pi$
$191$	−17.5488	−1.26979	−0.634895	+	0.772598i	$0.718957\pi$
$192$	0	0
$193$	5.23453	0.376790	0.188395	−	0.982093i	$-0.439672\pi$
$193$	5.23453	0.376790	0.188395	+	0.982093i	$0.439672\pi$
$194$	−12.5352	−0.899977
$195$	0	0
$196$	10.3998	0.742842
$197$	2.58232	0.183983	0.0919913	−	0.995760i	$-0.470677\pi$
$197$	2.58232	0.183983	0.0919913	+	0.995760i	$0.470677\pi$
$198$	0	0
$199$	−16.8047	−1.19126	−0.595628	−	0.803260i	$-0.703097\pi$
$199$	−16.8047	−1.19126	−0.595628	+	0.803260i	$0.703097\pi$
$200$	−20.9816	−1.48363
$201$	0	0
$202$	17.6669	1.24304
$203$	−3.14721	−0.220891
$204$	0	0
$205$	−14.1420	−0.987721
$206$	13.9932	0.974955
$207$	0	0
$208$	3.29976	0.228797
$209$	−2.26447	−0.156637
$210$	0	0
$211$	8.97860	0.618112	0.309056	−	0.951044i	$-0.399987\pi$
$211$	8.97860	0.618112	0.309056	+	0.951044i	$0.399987\pi$
$212$	−21.9557	−1.50792
$213$	0	0
$214$	25.9386	1.77312
$215$	15.6188	1.06520
$216$	0	0
$217$	−25.1055	−1.70427
$218$	29.3499	1.98783
$219$	0	0
$220$	17.0266	1.14793
$221$	10.7551	0.723469
$222$	0	0
$223$	7.08920	0.474728	0.237364	−	0.971421i	$-0.423717\pi$
$223$	7.08920	0.474728	0.237364	+	0.971421i	$0.423717\pi$
$224$	11.1772	0.746806
$225$	0	0
$226$	−14.7980	−0.984346
$227$	−6.89075	−0.457355	−0.228678	−	0.973502i	$-0.573440\pi$
$227$	−6.89075	−0.457355	−0.228678	+	0.973502i	$0.573440\pi$
$228$	0	0
$229$	−14.4803	−0.956883	−0.478442	−	0.878119i	$-0.658798\pi$
$229$	−14.4803	−0.956883	−0.478442	+	0.878119i	$0.658798\pi$
$230$	7.69859	0.507630
$231$	0	0
$232$	3.73241	0.245045
$233$	−6.05994	−0.397000	−0.198500	−	0.980101i	$-0.563607\pi$
$233$	−6.05994	−0.397000	−0.198500	+	0.980101i	$0.563607\pi$
$234$	0	0
$235$	−7.79160	−0.508268
$236$	−19.8769	−1.29388
$237$	0	0
$238$	−40.1430	−2.60208
$239$	11.5519	0.747233	0.373616	−	0.927583i	$-0.378118\pi$
$239$	11.5519	0.747233	0.373616	+	0.927583i	$0.378118\pi$
$240$	0	0
$241$	−10.0097	−0.644780	−0.322390	−	0.946607i	$-0.604486\pi$
$241$	−10.0097	−0.644780	−0.322390	+	0.946607i	$0.604486\pi$
$242$	20.9538	1.34696
$243$	0	0
$244$	22.7093	1.45381
$245$	−9.46733	−0.604846
$246$	0	0
$247$	3.09080	0.196663
$248$	29.7737	1.89063
$249$	0	0
$250$	4.78443	0.302594
$251$	−5.01221	−0.316368	−0.158184	−	0.987410i	$-0.550564\pi$
$251$	−5.01221	−0.316368	−0.158184	+	0.987410i	$0.550564\pi$
$252$	0	0
$253$	−1.45931	−0.0917459
$254$	12.9599	0.813179
$255$	0	0
$256$	−25.1173	−1.56983
$257$	7.41284	0.462400	0.231200	−	0.972906i	$-0.425735\pi$
$257$	7.41284	0.462400	0.231200	+	0.972906i	$0.425735\pi$
$258$	0	0
$259$	−32.0620	−1.99224
$260$	−23.2398	−1.44127
$261$	0	0
$262$	10.2427	0.632795
$263$	−9.73785	−0.600462	−0.300231	−	0.953867i	$-0.597064\pi$
$263$	−9.73785	−0.600462	−0.300231	+	0.953867i	$0.597064\pi$
$264$	0	0
$265$	19.9871	1.22780
$266$	−11.5362	−0.707332
$267$	0	0
$268$	30.8957	1.88726
$269$	−0.499903	−0.0304796	−0.0152398	−	0.999884i	$-0.504851\pi$
$269$	−0.499903	−0.0304796	−0.0152398	+	0.999884i	$0.504851\pi$
$270$	0	0
$271$	−9.54186	−0.579626	−0.289813	−	0.957083i	$-0.593593\pi$
$271$	−9.54186	−0.579626	−0.289813	+	0.957083i	$0.593593\pi$
$272$	8.94534	0.542391
$273$	0	0
$274$	−24.1259	−1.45750
$275$	−8.20345	−0.494687
$276$	0	0
$277$	−19.7349	−1.18576	−0.592878	−	0.805292i	$-0.702008\pi$
$277$	−19.7349	−1.18576	−0.592878	+	0.805292i	$0.702008\pi$
$278$	−43.6959	−2.62070
$279$	0	0
$280$	38.2831	2.28786
$281$	23.4853	1.40102	0.700509	−	0.713644i	$-0.252957\pi$
$281$	23.4853	1.40102	0.700509	+	0.713644i	$0.252957\pi$
$282$	0	0
$283$	−16.2076	−0.963440	−0.481720	−	0.876325i	$-0.659988\pi$
$283$	−16.2076	−0.963440	−0.481720	+	0.876325i	$0.659988\pi$
$284$	−51.0234	−3.02768
$285$	0	0
$286$	6.86621	0.406007
$287$	13.6567	0.806128
$288$	0	0
$289$	12.1561	0.715067
$290$	−7.69859	−0.452077
$291$	0	0
$292$	1.10420	0.0646186
$293$	18.5400	1.08312	0.541559	−	0.840662i	$-0.317834\pi$
$293$	18.5400	1.08312	0.541559	+	0.840662i	$0.317834\pi$
$294$	0	0
$295$	18.0948	1.05352
$296$	38.0237	2.21009
$297$	0	0
$298$	55.3708	3.20754
$299$	1.99182	0.115190
$300$	0	0
$301$	−15.0828	−0.869358
$302$	−57.1628	−3.28935
$303$	0	0
$304$	2.57070	0.147440
$305$	−20.6732	−1.18374
$306$	0	0
$307$	−12.5888	−0.718482	−0.359241	−	0.933245i	$-0.616964\pi$
$307$	−12.5888	−0.718482	−0.359241	+	0.933245i	$0.616964\pi$
$308$	−16.4423	−0.936885
$309$	0	0
$310$	−61.4122	−3.48798
$311$	16.3135	0.925053	0.462527	−	0.886605i	$-0.346943\pi$
$311$	16.3135	0.925053	0.462527	+	0.886605i	$0.346943\pi$
$312$	0	0
$313$	12.7002	0.717858	0.358929	−	0.933365i	$-0.383142\pi$
$313$	12.7002	0.717858	0.358929	+	0.933365i	$0.383142\pi$
$314$	−15.9469	−0.899938
$315$	0	0
$316$	−9.31921	−0.524246
$317$	15.8264	0.888900	0.444450	−	0.895804i	$-0.353399\pi$
$317$	15.8264	0.888900	0.444450	+	0.895804i	$0.353399\pi$
$318$	0	0
$319$	1.45931	0.0817056
$320$	38.1395	2.13206
$321$	0	0
$322$	−7.43438	−0.414302
$323$	8.37885	0.466212
$324$	0	0
$325$	11.1970	0.621097
$326$	−29.6801	−1.64383
$327$	0	0
$328$	−16.1960	−0.894276
$329$	7.52419	0.414822
$330$	0	0
$331$	7.32656	0.402704	0.201352	−	0.979519i	$-0.435466\pi$
$331$	7.32656	0.402704	0.201352	+	0.979519i	$0.435466\pi$
$332$	63.6466	3.49306
$333$	0	0
$334$	14.4593	0.791179
$335$	−28.1256	−1.53666
$336$	0	0
$337$	−0.348221	−0.0189688	−0.00948441	−	0.999955i	$-0.503019\pi$
$337$	−0.348221	−0.0189688	−0.00948441	+	0.999955i	$0.503019\pi$
$338$	21.3370	1.16058
$339$	0	0
$340$	−63.0008	−3.41670
$341$	11.6410	0.630395
$342$	0	0
$343$	−12.8880	−0.695889
$344$	17.8873	0.964421
$345$	0	0
$346$	24.0911	1.29515
$347$	−23.0544	−1.23763	−0.618814	−	0.785538i	$-0.712386\pi$
$347$	−23.0544	−1.23763	−0.618814	+	0.785538i	$0.712386\pi$
$348$	0	0
$349$	2.29304	0.122744	0.0613719	−	0.998115i	$-0.480452\pi$
$349$	2.29304	0.122744	0.0613719	+	0.998115i	$0.480452\pi$
$350$	−41.7921	−2.23388
$351$	0	0
$352$	−5.18267	−0.276237
$353$	29.5910	1.57497	0.787486	−	0.616333i	$-0.211382\pi$
$353$	29.5910	1.57497	0.787486	+	0.616333i	$0.211382\pi$
$354$	0	0
$355$	46.4486	2.46524
$356$	2.41195	0.127833
$357$	0	0
$358$	−48.3067	−2.55309
$359$	−26.7858	−1.41370	−0.706850	−	0.707363i	$-0.749884\pi$
$359$	−26.7858	−1.41370	−0.706850	+	0.707363i	$0.749884\pi$
$360$	0	0
$361$	−16.5921	−0.873268
$362$	−3.68964	−0.193923
$363$	0	0
$364$	22.4422	1.17629
$365$	−1.00520	−0.0526145
$366$	0	0
$367$	5.84694	0.305208	0.152604	−	0.988287i	$-0.451234\pi$
$367$	5.84694	0.305208	0.152604	+	0.988287i	$0.451234\pi$
$368$	1.65665	0.0863591
$369$	0	0
$370$	−78.4290	−4.07733
$371$	−19.3012	−1.00207
$372$	0	0
$373$	2.69778	0.139686	0.0698430	−	0.997558i	$-0.477750\pi$
$373$	2.69778	0.139686	0.0698430	+	0.997558i	$0.477750\pi$
$374$	18.6136	0.962487
$375$	0	0
$376$	−8.92327	−0.460182
$377$	−1.99182	−0.102584
$378$	0	0
$379$	17.2760	0.887409	0.443704	−	0.896173i	$-0.353664\pi$
$379$	17.2760	0.887409	0.443704	+	0.896173i	$0.353664\pi$
$380$	−18.1051	−0.928772
$381$	0	0
$382$	41.4541	2.12098
$383$	−20.4954	−1.04726	−0.523632	−	0.851945i	$-0.675423\pi$
$383$	−20.4954	−1.04726	−0.523632	+	0.851945i	$0.675423\pi$
$384$	0	0
$385$	14.9680	0.762842
$386$	−12.3651	−0.629365
$387$	0	0
$388$	18.9977	0.964464
$389$	−38.8139	−1.96794	−0.983972	−	0.178321i	$-0.942934\pi$
$389$	−38.8139	−1.96794	−0.983972	+	0.178321i	$0.942934\pi$
$390$	0	0
$391$	5.39964	0.273072
$392$	−10.8424	−0.547623
$393$	0	0
$394$	−6.09999	−0.307313
$395$	8.48364	0.426858
$396$	0	0
$397$	−36.1654	−1.81509	−0.907544	−	0.419957i	$-0.862045\pi$
$397$	−36.1654	−1.81509	−0.907544	+	0.419957i	$0.862045\pi$
$398$	39.6964	1.98980
$399$	0	0
$400$	9.31283	0.465641
$401$	−31.4018	−1.56813	−0.784066	−	0.620677i	$-0.786858\pi$
$401$	−31.4018	−1.56813	−0.784066	+	0.620677i	$0.786858\pi$
$402$	0	0
$403$	−15.8889	−0.791483
$404$	−26.7750	−1.33211
$405$	0	0
$406$	7.43438	0.368962
$407$	14.8666	0.736911
$408$	0	0
$409$	9.87827	0.488449	0.244224	−	0.969719i	$-0.421467\pi$
$409$	9.87827	0.488449	0.244224	+	0.969719i	$0.421467\pi$
$410$	33.4064	1.64983
$411$	0	0
$412$	−21.2074	−1.04482
$413$	−17.4737	−0.859827
$414$	0	0
$415$	−57.9400	−2.84416
$416$	7.07388	0.346825
$417$	0	0
$418$	5.34916	0.261636
$419$	−12.3077	−0.601273	−0.300636	−	0.953739i	$-0.597199\pi$
$419$	−12.3077	−0.601273	−0.300636	+	0.953739i	$0.597199\pi$
$420$	0	0
$421$	26.2999	1.28178	0.640890	−	0.767633i	$-0.278566\pi$
$421$	26.2999	1.28178	0.640890	+	0.767633i	$0.278566\pi$
$422$	−21.2094	−1.03246
$423$	0	0
$424$	22.8901	1.11164
$425$	30.3539	1.47238
$426$	0	0
$427$	19.9637	0.966110
$428$	−39.3111	−1.90018
$429$	0	0
$430$	−36.8950	−1.77923
$431$	30.2923	1.45913	0.729564	−	0.683913i	$-0.239723\pi$
$431$	30.2923	1.45913	0.729564	+	0.683913i	$0.239723\pi$
$432$	0	0
$433$	−18.9515	−0.910750	−0.455375	−	0.890300i	$-0.650495\pi$
$433$	−18.9515	−0.910750	−0.455375	+	0.890300i	$0.650495\pi$
$434$	59.3045	2.84671
$435$	0	0
$436$	−44.4812	−2.13026
$437$	1.55174	0.0742299
$438$	0	0
$439$	6.83911	0.326413	0.163206	−	0.986592i	$-0.447816\pi$
$439$	6.83911	0.326413	0.163206	+	0.986592i	$0.447816\pi$
$440$	−17.7512	−0.846257
$441$	0	0
$442$	−25.4059	−1.20844
$443$	0.0392883	0.00186664	0.000933321	−	1.00000i	$-0.499703\pi$
$443$	0.0392883	0.00186664	0.000933321		1.00000i	$0.499703\pi$
$444$	0	0
$445$	−2.19569	−0.104086
$446$	−16.7462	−0.792955
$447$	0	0
$448$	−36.8305	−1.74008
$449$	25.4521	1.20116	0.600580	−	0.799564i	$-0.294936\pi$
$449$	25.4521	1.20116	0.600580	+	0.799564i	$0.294936\pi$
$450$	0	0
$451$	−6.33237	−0.298179
$452$	22.4270	1.05488
$453$	0	0
$454$	16.2774	0.763937
$455$	−20.4300	−0.957775
$456$	0	0
$457$	11.6847	0.546588	0.273294	−	0.961931i	$-0.411887\pi$
$457$	11.6847	0.546588	0.273294	+	0.961931i	$0.411887\pi$
$458$	34.2055	1.59832
$459$	0	0
$460$	−11.6676	−0.544004
$461$	1.88294	0.0876973	0.0438486	−	0.999038i	$-0.486038\pi$
$461$	1.88294	0.0876973	0.0438486	+	0.999038i	$0.486038\pi$
$462$	0	0
$463$	−11.5245	−0.535588	−0.267794	−	0.963476i	$-0.586295\pi$
$463$	−11.5245	−0.535588	−0.267794	+	0.963476i	$0.586295\pi$
$464$	−1.65665	−0.0769082
$465$	0	0
$466$	14.3149	0.663124
$467$	−18.6749	−0.864169	−0.432085	−	0.901833i	$-0.642222\pi$
$467$	−18.6749	−0.864169	−0.432085	+	0.901833i	$0.642222\pi$
$468$	0	0
$469$	27.1603	1.25415
$470$	18.4054	0.848978
$471$	0	0
$472$	20.7229	0.953848
$473$	6.99364	0.321568
$474$	0	0
$475$	8.72307	0.400242
$476$	60.8387	2.78854
$477$	0	0
$478$	−27.2881	−1.24813
$479$	26.1025	1.19265	0.596327	−	0.802742i	$-0.296626\pi$
$479$	26.1025	1.19265	0.596327	+	0.802742i	$0.296626\pi$
$480$	0	0
$481$	−20.2916	−0.925217
$482$	23.6450	1.07700
$483$	0	0
$484$	−31.7565	−1.44348
$485$	−17.2944	−0.785298
$486$	0	0
$487$	15.2270	0.690000	0.345000	−	0.938603i	$-0.387879\pi$
$487$	15.2270	0.690000	0.345000	+	0.938603i	$0.387879\pi$
$488$	−23.6758	−1.07175
$489$	0	0
$490$	22.3639	1.01030
$491$	−34.5319	−1.55840	−0.779201	−	0.626774i	$-0.784375\pi$
$491$	−34.5319	−1.55840	−0.779201	+	0.626774i	$0.784375\pi$
$492$	0	0
$493$	−5.39964	−0.243188
$494$	−7.30112	−0.328493
$495$	0	0
$496$	−13.2152	−0.593382
$497$	−44.8545	−2.01200
$498$	0	0
$499$	−29.8743	−1.33736	−0.668678	−	0.743552i	$-0.733140\pi$
$499$	−29.8743	−1.33736	−0.668678	+	0.743552i	$0.733140\pi$
$500$	−7.25104	−0.324276
$501$	0	0
$502$	11.8399	0.528441
$503$	18.6093	0.829748	0.414874	−	0.909879i	$-0.363826\pi$
$503$	18.6093	0.829748	0.414874	+	0.909879i	$0.363826\pi$
$504$	0	0
$505$	24.3744	1.08464
$506$	3.44720	0.153247
$507$	0	0
$508$	−19.6414	−0.871447
$509$	25.5195	1.13113	0.565567	−	0.824702i	$-0.308657\pi$
$509$	25.5195	1.13113	0.565567	+	0.824702i	$0.308657\pi$
$510$	0	0
$511$	0.970701	0.0429413
$512$	18.2502	0.806551
$513$	0	0
$514$	−17.5107	−0.772364
$515$	19.3060	0.850723
$516$	0	0
$517$	−3.48884	−0.153439
$518$	75.7373	3.32771
$519$	0	0
$520$	24.2289	1.06251
$521$	21.9999	0.963833	0.481917	−	0.876217i	$-0.339941\pi$
$521$	21.9999	0.963833	0.481917	+	0.876217i	$0.339941\pi$
$522$	0	0
$523$	21.1377	0.924286	0.462143	−	0.886805i	$-0.347081\pi$
$523$	21.1377	0.924286	0.462143	+	0.886805i	$0.347081\pi$
$524$	−15.5233	−0.678138
$525$	0	0
$526$	23.0029	1.00297
$527$	−43.0733	−1.87630
$528$	0	0
$529$	1.00000	0.0434783
$530$	−47.2139	−2.05084
$531$	0	0
$532$	17.4837	0.758016
$533$	8.64311	0.374375
$534$	0	0
$535$	35.7865	1.54718
$536$	−32.2106	−1.39129
$537$	0	0
$538$	1.18088	0.0509113
$539$	−4.23918	−0.182595
$540$	0	0
$541$	−10.5851	−0.455087	−0.227544	−	0.973768i	$-0.573069\pi$
$541$	−10.5851	−0.455087	−0.227544	+	0.973768i	$0.573069\pi$
$542$	22.5399	0.968171
$543$	0	0
$544$	19.1766	0.822189
$545$	40.4930	1.73453
$546$	0	0
$547$	−0.756387	−0.0323408	−0.0161704	−	0.999869i	$-0.505147\pi$
$547$	−0.756387	−0.0323408	−0.0161704	+	0.999869i	$0.505147\pi$
$548$	36.5640	1.56194
$549$	0	0
$550$	19.3783	0.826293
$551$	−1.55174	−0.0661064
$552$	0	0
$553$	−8.19248	−0.348380
$554$	46.6181	1.98061
$555$	0	0
$556$	66.2232	2.80849
$557$	−43.0236	−1.82297	−0.911483	−	0.411337i	$-0.865062\pi$
$557$	−43.0236	−1.82297	−0.911483	+	0.411337i	$0.865062\pi$
$558$	0	0
$559$	−9.54569	−0.403740
$560$	−16.9922	−0.718052
$561$	0	0
$562$	−55.4773	−2.34017
$563$	−39.1328	−1.64925	−0.824626	−	0.565678i	$-0.808614\pi$
$563$	−39.1328	−1.64925	−0.824626	+	0.565678i	$0.808614\pi$
$564$	0	0
$565$	−20.4162	−0.858917
$566$	38.2857	1.60927
$567$	0	0
$568$	53.1949	2.23201
$569$	−17.9418	−0.752158	−0.376079	−	0.926588i	$-0.622728\pi$
$569$	−17.9418	−0.752158	−0.376079	+	0.926588i	$0.622728\pi$
$570$	0	0
$571$	16.5548	0.692797	0.346399	−	0.938087i	$-0.387404\pi$
$571$	16.5548	0.692797	0.346399	+	0.938087i	$0.387404\pi$
$572$	−10.4061	−0.435100
$573$	0	0
$574$	−32.2600	−1.34650
$575$	5.62147	0.234431
$576$	0	0
$577$	23.6425	0.984251	0.492125	−	0.870524i	$-0.336220\pi$
$577$	23.6425	0.984251	0.492125	+	0.870524i	$0.336220\pi$
$578$	−28.7154	−1.19440
$579$	0	0
$580$	11.6676	0.484470
$581$	55.9515	2.32126
$582$	0	0
$583$	8.94963	0.370656
$584$	−1.15120	−0.0476368
$585$	0	0
$586$	−43.7954	−1.80917
$587$	−40.7585	−1.68228	−0.841142	−	0.540814i	$-0.818116\pi$
$587$	−40.7585	−1.68228	−0.841142	+	0.540814i	$0.818116\pi$
$588$	0	0
$589$	−12.3784	−0.510041
$590$	−42.7437	−1.75973
$591$	0	0
$592$	−16.8771	−0.693643
$593$	−9.13372	−0.375077	−0.187539	−	0.982257i	$-0.560051\pi$
$593$	−9.13372	−0.375077	−0.187539	+	0.982257i	$0.560051\pi$
$594$	0	0
$595$	−55.3838	−2.27052
$596$	−83.9171	−3.43738
$597$	0	0
$598$	−4.70511	−0.192406
$599$	−43.6351	−1.78288	−0.891442	−	0.453136i	$-0.850305\pi$
$599$	−43.6351	−1.78288	−0.891442	+	0.453136i	$0.850305\pi$
$600$	0	0
$601$	4.90324	0.200007	0.100004	−	0.994987i	$-0.468115\pi$
$601$	4.90324	0.200007	0.100004	+	0.994987i	$0.468115\pi$
$602$	35.6288	1.45212
$603$	0	0
$604$	86.6330	3.52505
$605$	28.9092	1.17533
$606$	0	0
$607$	−2.74891	−0.111575	−0.0557875	−	0.998443i	$-0.517767\pi$
$607$	−2.74891	−0.111575	−0.0557875	+	0.998443i	$0.517767\pi$
$608$	5.51094	0.223498
$609$	0	0
$610$	48.8344	1.97725
$611$	4.76196	0.192648
$612$	0	0
$613$	−43.8181	−1.76980	−0.884899	−	0.465783i	$-0.845773\pi$
$613$	−43.8181	−1.76980	−0.884899	+	0.465783i	$0.845773\pi$
$614$	29.7375	1.20011
$615$	0	0
$616$	17.1420	0.690672
$617$	−10.9918	−0.442511	−0.221256	−	0.975216i	$-0.571016\pi$
$617$	−10.9918	−0.442511	−0.221256	+	0.975216i	$0.571016\pi$
$618$	0	0
$619$	22.9060	0.920670	0.460335	−	0.887745i	$-0.347729\pi$
$619$	22.9060	0.920670	0.460335	+	0.887745i	$0.347729\pi$
$620$	93.0732	3.73791
$621$	0	0
$622$	−38.5359	−1.54515
$623$	2.12034	0.0849496
$624$	0	0
$625$	−21.5064	−0.860257
$626$	−30.0006	−1.19906
$627$	0	0
$628$	24.1684	0.964423
$629$	−55.0086	−2.19333
$630$	0	0
$631$	−23.3247	−0.928540	−0.464270	−	0.885694i	$-0.653683\pi$
$631$	−23.3247	−0.928540	−0.464270	+	0.885694i	$0.653683\pi$
$632$	9.71582	0.386475
$633$	0	0
$634$	−37.3854	−1.48476
$635$	17.8804	0.709560
$636$	0	0
$637$	5.78611	0.229254
$638$	−3.44720	−0.136476
$639$	0	0
$640$	−66.9447	−2.64622
$641$	−47.3515	−1.87027	−0.935136	−	0.354290i	$-0.884722\pi$
$641$	−47.3515	−1.87027	−0.935136	+	0.354290i	$0.884722\pi$
$642$	0	0
$643$	−8.49217	−0.334898	−0.167449	−	0.985881i	$-0.553553\pi$
$643$	−8.49217	−0.334898	−0.167449	+	0.985881i	$0.553553\pi$
$644$	11.2672	0.443989
$645$	0	0
$646$	−19.7926	−0.778731
$647$	−29.7442	−1.16936	−0.584682	−	0.811263i	$-0.698781\pi$
$647$	−29.7442	−1.16936	−0.584682	+	0.811263i	$0.698781\pi$
$648$	0	0
$649$	8.10228	0.318042
$650$	−26.4496	−1.03744
$651$	0	0
$652$	44.9816	1.76162
$653$	−27.5308	−1.07736	−0.538681	−	0.842510i	$-0.681077\pi$
$653$	−27.5308	−1.07736	−0.538681	+	0.842510i	$0.681077\pi$
$654$	0	0
$655$	14.1315	0.552162
$656$	7.18871	0.280672
$657$	0	0
$658$	−17.7737	−0.692893
$659$	−46.5492	−1.81330	−0.906651	−	0.421882i	$-0.861370\pi$
$659$	−46.5492	−1.81330	−0.906651	+	0.421882i	$0.861370\pi$
$660$	0	0
$661$	39.8206	1.54884	0.774421	−	0.632671i	$-0.218041\pi$
$661$	39.8206	1.54884	0.774421	+	0.632671i	$0.218041\pi$
$662$	−17.3069	−0.672651
$663$	0	0
$664$	−66.3553	−2.57509
$665$	−15.9161	−0.617201
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−21.9138	−0.847871
$669$	0	0
$670$	66.4386	2.56675
$671$	−9.25682	−0.357356
$672$	0	0
$673$	−26.3989	−1.01760	−0.508801	−	0.860884i	$-0.669911\pi$
$673$	−26.3989	−1.01760	−0.508801	+	0.860884i	$0.669911\pi$
$674$	0.822573	0.0316843
$675$	0	0
$676$	−32.3373	−1.24374
$677$	−17.7173	−0.680933	−0.340466	−	0.940257i	$-0.610585\pi$
$677$	−17.7173	−0.680933	−0.340466	+	0.940257i	$0.610585\pi$
$678$	0	0
$679$	16.7009	0.640920
$680$	65.6821	2.51879
$681$	0	0
$682$	−27.4985	−1.05297
$683$	−26.2424	−1.00414	−0.502068	−	0.864828i	$-0.667427\pi$
$683$	−26.2424	−1.00414	−0.502068	+	0.864828i	$0.667427\pi$
$684$	0	0
$685$	−33.2857	−1.27178
$686$	30.4443	1.16237
$687$	0	0
$688$	−7.93940	−0.302687
$689$	−12.2154	−0.465371
$690$	0	0
$691$	−48.7391	−1.85412	−0.927061	−	0.374910i	$-0.877674\pi$
$691$	−48.7391	−1.85412	−0.927061	+	0.374910i	$0.877674\pi$
$692$	−36.5113	−1.38795
$693$	0	0
$694$	54.4595	2.06725
$695$	−60.2856	−2.28676
$696$	0	0
$697$	23.4306	0.887498
$698$	−5.41666	−0.205024
$699$	0	0
$700$	63.3380	2.39395
$701$	1.46943	0.0554995	0.0277497	−	0.999615i	$-0.491166\pi$
$701$	1.46943	0.0554995	0.0277497	+	0.999615i	$0.491166\pi$
$702$	0	0
$703$	−15.8083	−0.596221
$704$	17.0777	0.643640
$705$	0	0
$706$	−69.9003	−2.63073
$707$	−23.5378	−0.885232
$708$	0	0
$709$	14.1235	0.530418	0.265209	−	0.964191i	$-0.414559\pi$
$709$	14.1235	0.530418	0.265209	+	0.964191i	$0.414559\pi$
$710$	−109.722	−4.11778
$711$	0	0
$712$	−2.51460	−0.0942387
$713$	−7.97707	−0.298743
$714$	0	0
$715$	9.47306	0.354272
$716$	73.2112	2.73603
$717$	0	0
$718$	63.2737	2.36136
$719$	−11.4231	−0.426011	−0.213005	−	0.977051i	$-0.568325\pi$
$719$	−11.4231	−0.426011	−0.213005	+	0.977051i	$0.568325\pi$
$720$	0	0
$721$	−18.6434	−0.694316
$722$	39.1941	1.45865
$723$	0	0
$724$	5.59184	0.207819
$725$	−5.62147	−0.208776
$726$	0	0
$727$	15.9489	0.591513	0.295757	−	0.955263i	$-0.404428\pi$
$727$	15.9489	0.591513	0.295757	+	0.955263i	$0.404428\pi$
$728$	−23.3973	−0.867163
$729$	0	0
$730$	2.37449	0.0878840
$731$	−25.8774	−0.957111
$732$	0	0
$733$	−28.1439	−1.03952	−0.519760	−	0.854312i	$-0.673979\pi$
$733$	−28.1439	−1.03952	−0.519760	+	0.854312i	$0.673979\pi$
$734$	−13.8117	−0.509800
$735$	0	0
$736$	3.55146	0.130908
$737$	−12.5938	−0.463898
$738$	0	0
$739$	−45.8284	−1.68582	−0.842912	−	0.538051i	$-0.819161\pi$
$739$	−45.8284	−1.68582	−0.842912	+	0.538051i	$0.819161\pi$
$740$	118.863	4.36949
$741$	0	0
$742$	45.5935	1.67379
$743$	−15.6780	−0.575172	−0.287586	−	0.957755i	$-0.592853\pi$
$743$	−15.6780	−0.575172	−0.287586	+	0.957755i	$0.592853\pi$
$744$	0	0
$745$	76.3930	2.79882
$746$	−6.37274	−0.233323
$747$	0	0
$748$	−28.2099	−1.03145
$749$	−34.5583	−1.26273
$750$	0	0
$751$	48.5188	1.77047	0.885237	−	0.465140i	$-0.153996\pi$
$751$	48.5188	1.77047	0.885237	+	0.465140i	$0.153996\pi$
$752$	3.96065	0.144430
$753$	0	0
$754$	4.70511	0.171350
$755$	−78.8654	−2.87021
$756$	0	0
$757$	−29.3527	−1.06684	−0.533421	−	0.845850i	$-0.679094\pi$
$757$	−29.3527	−1.06684	−0.533421	+	0.845850i	$0.679094\pi$
$758$	−40.8096	−1.48227
$759$	0	0
$760$	18.8756	0.684691
$761$	42.6826	1.54724	0.773621	−	0.633648i	$-0.218443\pi$
$761$	42.6826	1.54724	0.773621	+	0.633648i	$0.218443\pi$
$762$	0	0
$763$	−39.1033	−1.41563
$764$	−62.8257	−2.27296
$765$	0	0
$766$	48.4144	1.74928
$767$	−11.0589	−0.399313
$768$	0	0
$769$	−6.84739	−0.246923	−0.123462	−	0.992349i	$-0.539400\pi$
$769$	−6.84739	−0.246923	−0.123462	+	0.992349i	$0.539400\pi$
$770$	−35.3577	−1.27420
$771$	0	0
$772$	18.7399	0.674463
$773$	21.5447	0.774909	0.387454	−	0.921889i	$-0.373354\pi$
$773$	21.5447	0.774909	0.387454	+	0.921889i	$0.373354\pi$
$774$	0	0
$775$	−44.8428	−1.61080
$776$	−19.8063	−0.711004
$777$	0	0
$778$	91.6868	3.28713
$779$	6.73347	0.241251
$780$	0	0
$781$	20.7983	0.744221
$782$	−12.7551	−0.456122
$783$	0	0
$784$	4.81246	0.171874
$785$	−22.0014	−0.785265
$786$	0	0
$787$	−30.5152	−1.08775	−0.543875	−	0.839166i	$-0.683043\pi$
$787$	−30.5152	−1.08775	−0.543875	+	0.839166i	$0.683043\pi$
$788$	9.24483	0.329333
$789$	0	0
$790$	−20.0402	−0.712997
$791$	19.7155	0.701004
$792$	0	0
$793$	12.6347	0.448672
$794$	85.4303	3.03181
$795$	0	0
$796$	−60.1618	−2.13238
$797$	33.2045	1.17616	0.588082	−	0.808801i	$-0.299883\pi$
$797$	33.2045	1.17616	0.588082	+	0.808801i	$0.299883\pi$
$798$	0	0
$799$	12.9092	0.456695
$800$	19.9644	0.705848
$801$	0	0
$802$	74.1778	2.61931
$803$	−0.450098	−0.0158836
$804$	0	0
$805$	−10.2569	−0.361510
$806$	37.5330	1.32204
$807$	0	0
$808$	27.9145	0.982030
$809$	−25.3486	−0.891210	−0.445605	−	0.895230i	$-0.647011\pi$
$809$	−25.3486	−0.891210	−0.445605	+	0.895230i	$0.647011\pi$
$810$	0	0
$811$	−29.7398	−1.04431	−0.522153	−	0.852852i	$-0.674871\pi$
$811$	−29.7398	−1.04431	−0.522153	+	0.852852i	$0.674871\pi$
$812$	−11.2672	−0.395400
$813$	0	0
$814$	−35.1181	−1.23089
$815$	−40.9486	−1.43437
$816$	0	0
$817$	−7.43662	−0.260174
$818$	−23.3346	−0.815874
$819$	0	0
$820$	−50.6291	−1.76805
$821$	−8.37546	−0.292306	−0.146153	−	0.989262i	$-0.546689\pi$
$821$	−8.37546	−0.292306	−0.146153	+	0.989262i	$0.546689\pi$
$822$	0	0
$823$	42.0013	1.46407	0.732037	−	0.681265i	$-0.238570\pi$
$823$	42.0013	1.46407	0.732037	+	0.681265i	$0.238570\pi$
$824$	22.1100	0.770239
$825$	0	0
$826$	41.2767	1.43620
$827$	−12.4090	−0.431504	−0.215752	−	0.976448i	$-0.569220\pi$
$827$	−12.4090	−0.431504	−0.215752	+	0.976448i	$0.569220\pi$
$828$	0	0
$829$	−48.7584	−1.69345	−0.846725	−	0.532031i	$-0.821429\pi$
$829$	−48.7584	−1.69345	−0.846725	+	0.532031i	$0.821429\pi$
$830$	136.867	4.75071
$831$	0	0
$832$	−23.3095	−0.808113
$833$	15.6856	0.543473
$834$	0	0
$835$	19.9490	0.690364
$836$	−8.10691	−0.280383
$837$	0	0
$838$	29.0735	1.00433
$839$	23.4433	0.809352	0.404676	−	0.914460i	$-0.367384\pi$
$839$	23.4433	0.809352	0.404676	+	0.914460i	$0.367384\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−62.1260	−2.14100
$843$	0	0
$844$	32.1438	1.10644
$845$	29.4379	1.01269
$846$	0	0
$847$	−27.9171	−0.959242
$848$	−10.1599	−0.348893
$849$	0	0
$850$	−71.7024	−2.45937
$851$	−10.1874	−0.349221
$852$	0	0
$853$	8.35330	0.286011	0.143006	−	0.989722i	$-0.454323\pi$
$853$	8.35330	0.286011	0.143006	+	0.989722i	$0.454323\pi$
$854$	−47.1585	−1.61373
$855$	0	0
$856$	40.9842	1.40081
$857$	−18.4877	−0.631528	−0.315764	−	0.948838i	$-0.602261\pi$
$857$	−18.4877	−0.631528	−0.315764	+	0.948838i	$0.602261\pi$
$858$	0	0
$859$	31.3092	1.06826	0.534128	−	0.845404i	$-0.320640\pi$
$859$	31.3092	1.06826	0.534128	+	0.845404i	$0.320640\pi$
$860$	55.9162	1.90673
$861$	0	0
$862$	−71.5568	−2.43723
$863$	33.8041	1.15070	0.575352	−	0.817906i	$-0.304865\pi$
$863$	33.8041	1.15070	0.575352	+	0.817906i	$0.304865\pi$
$864$	0	0
$865$	33.2377	1.13011
$866$	44.7674	1.52126
$867$	0	0
$868$	−89.8789	−3.05069
$869$	3.79872	0.128863
$870$	0	0
$871$	17.1894	0.582440
$872$	46.3743	1.57043
$873$	0	0
$874$	−3.66555	−0.123989
$875$	−6.37436	−0.215493
$876$	0	0
$877$	−4.54543	−0.153488	−0.0767441	−	0.997051i	$-0.524452\pi$
$877$	−4.54543	−0.153488	−0.0767441	+	0.997051i	$0.524452\pi$
$878$	−16.1554	−0.545219
$879$	0	0
$880$	7.87900	0.265601
$881$	14.0971	0.474944	0.237472	−	0.971394i	$-0.423681\pi$
$881$	14.0971	0.474944	0.237472	+	0.971394i	$0.423681\pi$
$882$	0	0
$883$	−38.9958	−1.31231	−0.656156	−	0.754625i	$-0.727819\pi$
$883$	−38.9958	−1.31231	−0.656156	+	0.754625i	$0.727819\pi$
$884$	38.5039	1.29503
$885$	0	0
$886$	−0.0928072	−0.00311792
$887$	30.6389	1.02875	0.514377	−	0.857564i	$-0.328023\pi$
$887$	30.6389	1.02875	0.514377	+	0.857564i	$0.328023\pi$
$888$	0	0
$889$	−17.2667	−0.579107
$890$	5.18670	0.173858
$891$	0	0
$892$	25.3797	0.849775
$893$	3.70983	0.124145
$894$	0	0
$895$	−66.6470	−2.22776
$896$	64.6472	2.15971
$897$	0	0
$898$	−60.1234	−2.00634
$899$	7.97707	0.266050
$900$	0	0
$901$	−33.1149	−1.10322
$902$	14.9584	0.498060
$903$	0	0
$904$	−23.3815	−0.777658
$905$	−5.09047	−0.169213
$906$	0	0
$907$	34.1387	1.13356	0.566779	−	0.823870i	$-0.308189\pi$
$907$	34.1387	1.13356	0.566779	+	0.823870i	$0.308189\pi$
$908$	−24.6692	−0.818677
$909$	0	0
$910$	48.2601	1.59981
$911$	−22.4448	−0.743630	−0.371815	−	0.928307i	$-0.621264\pi$
$911$	−22.4448	−0.743630	−0.371815	+	0.928307i	$0.621264\pi$
$912$	0	0
$913$	−25.9438	−0.858613
$914$	−27.6018	−0.912985
$915$	0	0
$916$	−51.8401	−1.71284
$917$	−13.6465	−0.450646
$918$	0	0
$919$	−22.5490	−0.743824	−0.371912	−	0.928268i	$-0.621298\pi$
$919$	−22.5490	−0.743824	−0.371912	+	0.928268i	$0.621298\pi$
$920$	12.1642	0.401040
$921$	0	0
$922$	−4.44790	−0.146484
$923$	−28.3878	−0.934396
$924$	0	0
$925$	−57.2684	−1.88297
$926$	27.2233	0.894612
$927$	0	0
$928$	−3.55146	−0.116582
$929$	50.6080	1.66039	0.830197	−	0.557470i	$-0.188228\pi$
$929$	50.6080	1.66039	0.830197	+	0.557470i	$0.188228\pi$
$930$	0	0
$931$	4.50770	0.147734
$932$	−21.6949	−0.710640
$933$	0	0
$934$	44.1140	1.44345
$935$	25.6805	0.839843
$936$	0	0
$937$	−3.29556	−0.107661	−0.0538306	−	0.998550i	$-0.517143\pi$
$937$	−3.29556	−0.107661	−0.0538306	+	0.998550i	$0.517143\pi$
$938$	−64.1585	−2.09485
$939$	0	0
$940$	−27.8943	−0.909812
$941$	30.5639	0.996353	0.498177	−	0.867076i	$-0.334003\pi$
$941$	30.5639	0.996353	0.498177	+	0.867076i	$0.334003\pi$
$942$	0	0
$943$	4.33929	0.141307
$944$	−9.19797	−0.299369
$945$	0	0
$946$	−16.5205	−0.537126
$947$	−12.8774	−0.418458	−0.209229	−	0.977867i	$-0.567095\pi$
$947$	−12.8774	−0.418458	−0.209229	+	0.977867i	$0.567095\pi$
$948$	0	0
$949$	0.614343	0.0199424
$950$	−20.6057	−0.668539
$951$	0	0
$952$	−63.4279	−2.05571
$953$	−20.7913	−0.673496	−0.336748	−	0.941595i	$-0.609327\pi$
$953$	−20.7913	−0.673496	−0.336748	+	0.941595i	$0.609327\pi$
$954$	0	0
$955$	57.1927	1.85071
$956$	41.3565	1.33757
$957$	0	0
$958$	−61.6597	−1.99213
$959$	32.1433	1.03796
$960$	0	0
$961$	32.6336	1.05270
$962$	47.9331	1.54542
$963$	0	0
$964$	−35.8351	−1.15417
$965$	−17.0596	−0.549169
$966$	0	0
$967$	−49.7242	−1.59902	−0.799511	−	0.600652i	$-0.794908\pi$
$967$	−49.7242	−1.59902	−0.799511	+	0.600652i	$0.794908\pi$
$968$	33.1081	1.06413
$969$	0	0
$970$	40.8530	1.31171
$971$	−36.5618	−1.17333	−0.586663	−	0.809831i	$-0.699559\pi$
$971$	−36.5618	−1.17333	−0.586663	+	0.809831i	$0.699559\pi$
$972$	0	0
$973$	58.2166	1.86634
$974$	−35.9694	−1.15253
$975$	0	0
$976$	10.5086	0.336374
$977$	−30.8588	−0.987260	−0.493630	−	0.869672i	$-0.664330\pi$
$977$	−30.8588	−0.987260	−0.493630	+	0.869672i	$0.664330\pi$
$978$	0	0
$979$	−0.983165	−0.0314221
$980$	−33.8935	−1.08269
$981$	0	0
$982$	81.5716	2.60306
$983$	31.4282	1.00240	0.501202	−	0.865330i	$-0.332891\pi$
$983$	31.4282	1.00240	0.501202	+	0.865330i	$0.332891\pi$
$984$	0	0
$985$	−8.41593	−0.268154
$986$	12.7551	0.406205
$987$	0	0
$988$	11.0652	0.352031
$989$	−4.79243	−0.152391
$990$	0	0
$991$	−10.6121	−0.337106	−0.168553	−	0.985693i	$-0.553909\pi$
$991$	−10.6121	−0.337106	−0.168553	+	0.985693i	$0.553909\pi$
$992$	−28.3302	−0.899485
$993$	0	0
$994$	105.956	3.36072
$995$	54.7676	1.73625
$996$	0	0
$997$	3.37475	0.106879	0.0534397	−	0.998571i	$-0.482982\pi$
$997$	3.37475	0.106879	0.0534397	+	0.998571i	$0.482982\pi$
$998$	70.5694	2.23384
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.k.1.2		10
3.2	odd	2		2001.2.a.k.1.9	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.k.1.9	✓	10	3.2	odd	2
6003.2.a.k.1.2		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)

\(f(q)\)	\(=\)	\(q-2.36221 q^{2} +3.58005 q^{4} -3.25906 q^{5} +3.14721 q^{7} -3.73241 q^{8} +O(q^{10})\)	\(q-2.36221 q^{2} +3.58005 q^{4} -3.25906 q^{5} +3.14721 q^{7} -3.73241 q^{8} +7.69859 q^{10} -1.45931 q^{11} +1.99182 q^{13} -7.43438 q^{14} +1.65665 q^{16} +5.39964 q^{17} +1.55174 q^{19} -11.6676 q^{20} +3.44720 q^{22} +1.00000 q^{23} +5.62147 q^{25} -4.70511 q^{26} +11.2672 q^{28} -1.00000 q^{29} -7.97707 q^{31} +3.55146 q^{32} -12.7551 q^{34} -10.2569 q^{35} -10.1874 q^{37} -3.66555 q^{38} +12.1642 q^{40} +4.33929 q^{41} -4.79243 q^{43} -5.22439 q^{44} -2.36221 q^{46} +2.39075 q^{47} +2.90493 q^{49} -13.2791 q^{50} +7.13083 q^{52} -6.13279 q^{53} +4.75597 q^{55} -11.7467 q^{56} +2.36221 q^{58} -5.55214 q^{59} +6.34330 q^{61} +18.8435 q^{62} -11.7026 q^{64} -6.49147 q^{65} +8.62997 q^{67} +19.3310 q^{68} +24.2291 q^{70} -14.2522 q^{71} +0.308432 q^{73} +24.0649 q^{74} +5.55531 q^{76} -4.59275 q^{77} -2.60309 q^{79} -5.39913 q^{80} -10.2503 q^{82} +17.7781 q^{83} -17.5978 q^{85} +11.3207 q^{86} +5.44674 q^{88} +0.673720 q^{89} +6.26869 q^{91} +3.58005 q^{92} -5.64746 q^{94} -5.05722 q^{95} +5.30656 q^{97} -6.86206 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) \) 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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