LMFDB - Embedded newform 6003.2.a.h.1.5

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:	$5$
Coefficient field:	5.5.312617.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 11x^{2} - x - 3 $ x^5 - 2x^4 - 5x^3 + 11*x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
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			1.5
Root			$-2.28064$ 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)

$f(q)$	$=$	$q+2.50612 q^{2} +4.28064 q^{4} +2.73973 q^{5} -1.54091 q^{7} +5.71555 q^{8} +O(q^{10})$	\(q+2.50612 q^{2} +4.28064 q^{4} +2.73973 q^{5} -1.54091 q^{7} +5.71555 q^{8} +6.86609 q^{10} +1.30482 q^{11} +3.43491 q^{13} -3.86170 q^{14} +5.76257 q^{16} +2.28064 q^{17} -2.39036 q^{19} +11.7278 q^{20} +3.27003 q^{22} -1.00000 q^{23} +2.50612 q^{25} +8.60830 q^{26} -6.59606 q^{28} -1.00000 q^{29} +2.47133 q^{31} +3.01060 q^{32} +5.71555 q^{34} -4.22167 q^{35} +10.3035 q^{37} -5.99053 q^{38} +15.6591 q^{40} +3.81467 q^{41} +4.29040 q^{43} +5.58545 q^{44} -2.50612 q^{46} -10.4725 q^{47} -4.62561 q^{49} +6.28064 q^{50} +14.7036 q^{52} +0.703308 q^{53} +3.57485 q^{55} -8.80712 q^{56} -2.50612 q^{58} +14.9155 q^{59} -0.0657559 q^{61} +6.19346 q^{62} -3.98021 q^{64} +9.41073 q^{65} -0.500088 q^{67} +9.76257 q^{68} -10.5800 q^{70} +0.422671 q^{71} +11.1034 q^{73} +25.8218 q^{74} -10.2323 q^{76} -2.01060 q^{77} -2.63000 q^{79} +15.7879 q^{80} +9.56002 q^{82} -15.9522 q^{83} +6.24833 q^{85} +10.7523 q^{86} +7.45775 q^{88} -15.8300 q^{89} -5.29288 q^{91} -4.28064 q^{92} -26.2453 q^{94} -6.54895 q^{95} +3.71639 q^{97} -11.5923 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{2} + 8 q^{4} + 3 q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) $ 5 * q + 2 * q^2 + 8 * q^4 + 3 * q^5 - 5 * q^7 + 3 * q^8	$ 5 q + 2 q^{2} + 8 q^{4} + 3 q^{5} - 5 q^{7} + 3 q^{8} + 9 q^{10} + 8 q^{11} + 5 q^{13} + 2 q^{14} - 10 q^{16} - 2 q^{17} - 9 q^{19} + 12 q^{20} + 10 q^{22} - 5 q^{23} + 2 q^{25} + 3 q^{26} - 14 q^{28} - 5 q^{29} - 6 q^{31} + 8 q^{32} + 3 q^{34} + 15 q^{35} + 10 q^{37} + 10 q^{38} + 26 q^{40} + 11 q^{41} - 9 q^{43} + 16 q^{44} - 2 q^{46} + 13 q^{47} - 6 q^{49} + 18 q^{50} + 12 q^{52} - q^{53} + 13 q^{55} + 4 q^{56} - 2 q^{58} - 6 q^{59} + 23 q^{61} + 36 q^{62} - q^{64} + 20 q^{65} - 10 q^{67} + 10 q^{68} - 16 q^{70} + 11 q^{71} + 31 q^{73} + 18 q^{74} - 8 q^{76} - 3 q^{77} + 8 q^{79} + 8 q^{80} + 16 q^{82} - 7 q^{83} + 6 q^{85} + 36 q^{86} - 3 q^{88} - 3 q^{89} + 8 q^{91} - 8 q^{92} - 39 q^{94} + 11 q^{95} + 3 q^{97} - 38 q^{98}+O(q^{100}) $ 5 * q + 2 * q^2 + 8 * q^4 + 3 * q^5 - 5 * q^7 + 3 * q^8 + 9 * q^10 + 8 * q^11 + 5 * q^13 + 2 * q^14 - 10 * q^16 - 2 * q^17 - 9 * q^19 + 12 * q^20 + 10 * q^22 - 5 * q^23 + 2 * q^25 + 3 * q^26 - 14 * q^28 - 5 * q^29 - 6 * q^31 + 8 * q^32 + 3 * q^34 + 15 * q^35 + 10 * q^37 + 10 * q^38 + 26 * q^40 + 11 * q^41 - 9 * q^43 + 16 * q^44 - 2 * q^46 + 13 * q^47 - 6 * q^49 + 18 * q^50 + 12 * q^52 - q^53 + 13 * q^55 + 4 * q^56 - 2 * q^58 - 6 * q^59 + 23 * q^61 + 36 * q^62 - q^64 + 20 * q^65 - 10 * q^67 + 10 * q^68 - 16 * q^70 + 11 * q^71 + 31 * q^73 + 18 * q^74 - 8 * q^76 - 3 * q^77 + 8 * q^79 + 8 * q^80 + 16 * q^82 - 7 * q^83 + 6 * q^85 + 36 * q^86 - 3 * q^88 - 3 * q^89 + 8 * q^91 - 8 * q^92 - 39 * q^94 + 11 * q^95 + 3 * q^97 - 38 * 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 $

$2$	2.50612	1.77209	0.886047	−	0.463595i	$-0.153441\pi$
$2$	2.50612	1.77209	0.886047	+	0.463595i	$0.153441\pi$
$3$	0	0
$3$	0	0
$4$	4.28064	2.14032
$5$	2.73973	1.22524	0.612622	−	0.790376i	$-0.290115\pi$
$5$	2.73973	1.22524	0.612622	+	0.790376i	$0.290115\pi$
$6$	0	0
$7$	−1.54091	−0.582408	−0.291204	−	0.956661i	$-0.594056\pi$
$7$	−1.54091	−0.582408	−0.291204	+	0.956661i	$0.594056\pi$
$8$	5.71555	2.02075
$9$	0	0
$10$	6.86609	2.17125
$11$	1.30482	0.393418	0.196709	−	0.980462i	$-0.436975\pi$
$11$	1.30482	0.393418	0.196709	+	0.980462i	$0.436975\pi$
$12$	0	0
$13$	3.43491	0.952673	0.476336	−	0.879263i	$-0.341964\pi$
$13$	3.43491	0.952673	0.476336	+	0.879263i	$0.341964\pi$
$14$	−3.86170	−1.03208
$15$	0	0
$16$	5.76257	1.44064
$17$	2.28064	0.553135	0.276568	−	0.960994i	$-0.410803\pi$
$17$	2.28064	0.553135	0.276568	+	0.960994i	$0.410803\pi$
$18$	0	0
$19$	−2.39036	−0.548387	−0.274193	−	0.961675i	$-0.588411\pi$
$19$	−2.39036	−0.548387	−0.274193	+	0.961675i	$0.588411\pi$
$20$	11.7278	2.62241
$21$	0	0
$22$	3.27003	0.697173
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	2.50612	0.501224
$26$	8.60830	1.68823
$27$	0	0
$28$	−6.59606	−1.24654
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	2.47133	0.443865	0.221932	−	0.975062i	$-0.428764\pi$
$31$	2.47133	0.443865	0.221932	+	0.975062i	$0.428764\pi$
$32$	3.01060	0.532205
$33$	0	0
$34$	5.71555	0.980208
$35$	−4.22167	−0.713592
$36$	0	0
$37$	10.3035	1.69388	0.846941	−	0.531687i	$-0.178442\pi$
$37$	10.3035	1.69388	0.846941	+	0.531687i	$0.178442\pi$
$38$	−5.99053	−0.971793
$39$	0	0
$40$	15.6591	2.47591
$41$	3.81467	0.595751	0.297876	−	0.954605i	$-0.403722\pi$
$41$	3.81467	0.595751	0.297876	+	0.954605i	$0.403722\pi$
$42$	0	0
$43$	4.29040	0.654280	0.327140	−	0.944976i	$-0.393915\pi$
$43$	4.29040	0.654280	0.327140	+	0.944976i	$0.393915\pi$
$44$	5.58545	0.842039
$45$	0	0
$46$	−2.50612	−0.369507
$47$	−10.4725	−1.52757	−0.763783	−	0.645473i	$-0.776660\pi$
$47$	−10.4725	−1.52757	−0.763783	+	0.645473i	$0.776660\pi$
$48$	0	0
$49$	−4.62561	−0.660801
$50$	6.28064	0.888216
$51$	0	0
$52$	14.7036	2.03902
$53$	0.703308	0.0966067	0.0483034	−	0.998833i	$-0.484619\pi$
$53$	0.703308	0.0966067	0.0483034	+	0.998833i	$0.484619\pi$
$54$	0	0
$55$	3.57485	0.482033
$56$	−8.80712	−1.17690
$57$	0	0
$58$	−2.50612	−0.329070
$59$	14.9155	1.94183	0.970917	−	0.239417i	$-0.0769563\pi$
$59$	14.9155	1.94183	0.970917	+	0.239417i	$0.0769563\pi$
$60$	0	0
$61$	−0.0657559	−0.00841919	−0.00420959	−	0.999991i	$-0.501340\pi$
$61$	−0.0657559	−0.00841919	−0.00420959	+	0.999991i	$0.501340\pi$
$62$	6.19346	0.786570
$63$	0	0
$64$	−3.98021	−0.497527
$65$	9.41073	1.16726
$66$	0	0
$67$	−0.500088	−0.0610955	−0.0305477	−	0.999533i	$-0.509725\pi$
$67$	−0.500088	−0.0610955	−0.0305477	+	0.999533i	$0.509725\pi$
$68$	9.76257	1.18389
$69$	0	0
$70$	−10.5800	−1.26455
$71$	0.422671	0.0501619	0.0250809	−	0.999685i	$-0.492016\pi$
$71$	0.422671	0.0501619	0.0250809	+	0.999685i	$0.492016\pi$
$72$	0	0
$73$	11.1034	1.29956	0.649779	−	0.760123i	$-0.274861\pi$
$73$	11.1034	1.29956	0.649779	+	0.760123i	$0.274861\pi$
$74$	25.8218	3.00172
$75$	0	0
$76$	−10.2323	−1.17372
$77$	−2.01060	−0.229130
$78$	0	0
$79$	−2.63000	−0.295899	−0.147949	−	0.988995i	$-0.547267\pi$
$79$	−2.63000	−0.295899	−0.147949	+	0.988995i	$0.547267\pi$
$80$	15.7879	1.76514
$81$	0	0
$82$	9.56002	1.05573
$83$	−15.9522	−1.75098	−0.875491	−	0.483234i	$-0.839462\pi$
$83$	−15.9522	−1.75098	−0.875491	+	0.483234i	$0.839462\pi$
$84$	0	0
$85$	6.24833	0.677726
$86$	10.7523	1.15945
$87$	0	0
$88$	7.45775	0.794999
$89$	−15.8300	−1.67797	−0.838986	−	0.544152i	$-0.816851\pi$
$89$	−15.8300	−1.67797	−0.838986	+	0.544152i	$0.816851\pi$
$90$	0	0
$91$	−5.29288	−0.554844
$92$	−4.28064	−0.446287
$93$	0	0
$94$	−26.2453	−2.70699
$95$	−6.54895	−0.671908
$96$	0	0
$97$	3.71639	0.377342	0.188671	−	0.982040i	$-0.439582\pi$
$97$	3.71639	0.377342	0.188671	+	0.982040i	$0.439582\pi$
$98$	−11.5923	−1.17100
$99$	0	0
$100$	10.7278	1.07278
$101$	0.0279137	0.00277752	0.00138876	−	0.999999i	$-0.499558\pi$
$101$	0.0279137	0.00277752	0.00138876	+	0.999999i	$0.499558\pi$
$102$	0	0
$103$	−3.31676	−0.326810	−0.163405	−	0.986559i	$-0.552248\pi$
$103$	−3.31676	−0.326810	−0.163405	+	0.986559i	$0.552248\pi$
$104$	19.6324	1.92511
$105$	0	0
$106$	1.76257	0.171196
$107$	−0.970450	−0.0938170	−0.0469085	−	0.998899i	$-0.514937\pi$
$107$	−0.970450	−0.0938170	−0.0469085	+	0.998899i	$0.514937\pi$
$108$	0	0
$109$	5.57666	0.534148	0.267074	−	0.963676i	$-0.413943\pi$
$109$	5.57666	0.534148	0.267074	+	0.963676i	$0.413943\pi$
$110$	8.95901	0.854208
$111$	0	0
$112$	−8.87958	−0.839042
$113$	1.30663	0.122918	0.0614588	−	0.998110i	$-0.480425\pi$
$113$	1.30663	0.122918	0.0614588	+	0.998110i	$0.480425\pi$
$114$	0	0
$115$	−2.73973	−0.255481
$116$	−4.28064	−0.397447
$117$	0	0
$118$	37.3800	3.44111
$119$	−3.51425	−0.322150
$120$	0	0
$121$	−9.29745	−0.845223
$122$	−0.164792	−0.0149196
$123$	0	0
$124$	10.5789	0.950011
$125$	−6.83256	−0.611123
$126$	0	0
$127$	−4.65246	−0.412839	−0.206420	−	0.978464i	$-0.566181\pi$
$127$	−4.65246	−0.412839	−0.206420	+	0.978464i	$0.566181\pi$
$128$	−15.9961	−1.41387
$129$	0	0
$130$	23.5844	2.06849
$131$	13.3386	1.16540	0.582702	−	0.812686i	$-0.301996\pi$
$131$	13.3386	1.16540	0.582702	+	0.812686i	$0.301996\pi$
$132$	0	0
$133$	3.68332	0.319385
$134$	−1.25328	−0.108267
$135$	0	0
$136$	13.0351	1.11775
$137$	8.53689	0.729356	0.364678	−	0.931134i	$-0.381179\pi$
$137$	8.53689	0.729356	0.364678	+	0.931134i	$0.381179\pi$
$138$	0	0
$139$	−6.25654	−0.530673	−0.265336	−	0.964156i	$-0.585483\pi$
$139$	−6.25654	−0.530673	−0.265336	+	0.964156i	$0.585483\pi$
$140$	−18.0714	−1.52731
$141$	0	0
$142$	1.05927	0.0888916
$143$	4.48194	0.374798
$144$	0	0
$145$	−2.73973	−0.227522
$146$	27.8265	2.30294
$147$	0	0
$148$	44.1054	3.62545
$149$	3.96274	0.324640	0.162320	−	0.986738i	$-0.448102\pi$
$149$	3.96274	0.324640	0.162320	+	0.986738i	$0.448102\pi$
$150$	0	0
$151$	−6.48861	−0.528036	−0.264018	−	0.964518i	$-0.585048\pi$
$151$	−6.48861	−0.528036	−0.264018	+	0.964518i	$0.585048\pi$
$152$	−13.6622	−1.10815
$153$	0	0
$154$	−5.03881	−0.406039
$155$	6.77079	0.543843
$156$	0	0
$157$	−13.1793	−1.05183	−0.525913	−	0.850539i	$-0.676276\pi$
$157$	−13.1793	−1.05183	−0.525913	+	0.850539i	$0.676276\pi$
$158$	−6.59110	−0.524360
$159$	0	0
$160$	8.24824	0.652081
$161$	1.54091	0.121440
$162$	0	0
$163$	−25.3241	−1.98354	−0.991770	−	0.128033i	$-0.959134\pi$
$163$	−25.3241	−1.98354	−0.991770	+	0.128033i	$0.959134\pi$
$164$	16.3292	1.27510
$165$	0	0
$166$	−39.9782	−3.10291
$167$	6.97623	0.539837	0.269918	−	0.962883i	$-0.413003\pi$
$167$	6.97623	0.539837	0.269918	+	0.962883i	$0.413003\pi$
$168$	0	0
$169$	−1.20139	−0.0924144
$170$	15.6591	1.20099
$171$	0	0
$172$	18.3656	1.40037
$173$	−9.39066	−0.713959	−0.356979	−	0.934112i	$-0.616193\pi$
$173$	−9.39066	−0.713959	−0.356979	+	0.934112i	$0.616193\pi$
$174$	0	0
$175$	−3.86170	−0.291917
$176$	7.51911	0.566775
$177$	0	0
$178$	−39.6718	−2.97353
$179$	−14.0140	−1.04745	−0.523726	−	0.851887i	$-0.675458\pi$
$179$	−14.0140	−1.04745	−0.523726	+	0.851887i	$0.675458\pi$
$180$	0	0
$181$	11.7427	0.872828	0.436414	−	0.899746i	$-0.356248\pi$
$181$	11.7427	0.872828	0.436414	+	0.899746i	$0.356248\pi$
$182$	−13.2646	−0.983236
$183$	0	0
$184$	−5.71555	−0.421356
$185$	28.2287	2.07542
$186$	0	0
$187$	2.97582	0.217613
$188$	−44.8288	−3.26948
$189$	0	0
$190$	−16.4124	−1.19068
$191$	25.1512	1.81988	0.909939	−	0.414741i	$-0.136128\pi$
$191$	25.1512	1.81988	0.909939	+	0.414741i	$0.136128\pi$
$192$	0	0
$193$	3.76994	0.271367	0.135683	−	0.990752i	$-0.456677\pi$
$193$	3.76994	0.271367	0.135683	+	0.990752i	$0.456677\pi$
$194$	9.31371	0.668686
$195$	0	0
$196$	−19.8005	−1.41432
$197$	13.9870	0.996532	0.498266	−	0.867024i	$-0.333970\pi$
$197$	13.9870	0.996532	0.498266	+	0.867024i	$0.333970\pi$
$198$	0	0
$199$	3.46558	0.245669	0.122834	−	0.992427i	$-0.460802\pi$
$199$	3.46558	0.245669	0.122834	+	0.992427i	$0.460802\pi$
$200$	14.3238	1.01285
$201$	0	0
$202$	0.0699551	0.00492202
$203$	1.54091	0.108150
$204$	0	0
$205$	10.4512	0.729941
$206$	−8.31220	−0.579139
$207$	0	0
$208$	19.7939	1.37246
$209$	−3.11899	−0.215745
$210$	0	0
$211$	4.25674	0.293046	0.146523	−	0.989207i	$-0.453192\pi$
$211$	4.25674	0.293046	0.146523	+	0.989207i	$0.453192\pi$
$212$	3.01060	0.206769
$213$	0	0
$214$	−2.43206	−0.166253
$215$	11.7545	0.801652
$216$	0	0
$217$	−3.80809	−0.258510
$218$	13.9758	0.946560
$219$	0	0
$220$	15.3026	1.03170
$221$	7.83378	0.526957
$222$	0	0
$223$	−22.6777	−1.51861	−0.759306	−	0.650734i	$-0.774461\pi$
$223$	−22.6777	−1.51861	−0.759306	+	0.650734i	$0.774461\pi$
$224$	−4.63906	−0.309960
$225$	0	0
$226$	3.27457	0.217821
$227$	17.9337	1.19030	0.595149	−	0.803615i	$-0.297093\pi$
$227$	17.9337	1.19030	0.595149	+	0.803615i	$0.297093\pi$
$228$	0	0
$229$	0.181131	0.0119695	0.00598475	−	0.999982i	$-0.498095\pi$
$229$	0.181131	0.0119695	0.00598475	+	0.999982i	$0.498095\pi$
$230$	−6.86609	−0.452737
$231$	0	0
$232$	−5.71555	−0.375244
$233$	24.6936	1.61773	0.808865	−	0.587994i	$-0.200082\pi$
$233$	24.6936	1.61773	0.808865	+	0.587994i	$0.200082\pi$
$234$	0	0
$235$	−28.6917	−1.87164
$236$	63.8479	4.15614
$237$	0	0
$238$	−8.80712	−0.570881
$239$	−1.31609	−0.0851306	−0.0425653	−	0.999094i	$-0.513553\pi$
$239$	−1.31609	−0.0851306	−0.0425653	+	0.999094i	$0.513553\pi$
$240$	0	0
$241$	−2.66488	−0.171660	−0.0858299	−	0.996310i	$-0.527354\pi$
$241$	−2.66488	−0.171660	−0.0858299	+	0.996310i	$0.527354\pi$
$242$	−23.3005	−1.49781
$243$	0	0
$244$	−0.281477	−0.0180197
$245$	−12.6729	−0.809643
$246$	0	0
$247$	−8.21068	−0.522433
$248$	14.1250	0.896940
$249$	0	0
$250$	−17.1232	−1.08297
$251$	−13.6009	−0.858483	−0.429241	−	0.903190i	$-0.641219\pi$
$251$	−13.6009	−0.858483	−0.429241	+	0.903190i	$0.641219\pi$
$252$	0	0
$253$	−1.30482	−0.0820333
$254$	−11.6596	−0.731590
$255$	0	0
$256$	−32.1277	−2.00798
$257$	16.1780	1.00915	0.504577	−	0.863367i	$-0.331648\pi$
$257$	16.1780	1.00915	0.504577	+	0.863367i	$0.331648\pi$
$258$	0	0
$259$	−15.8767	−0.986530
$260$	40.2839	2.49830
$261$	0	0
$262$	33.4282	2.06520
$263$	−16.1452	−0.995557	−0.497778	−	0.867304i	$-0.665851\pi$
$263$	−16.1452	−0.995557	−0.497778	+	0.867304i	$0.665851\pi$
$264$	0	0
$265$	1.92687	0.118367
$266$	9.23085	0.565980
$267$	0	0
$268$	−2.14070	−0.130764
$269$	0.348955	0.0212762	0.0106381	−	0.999943i	$-0.496614\pi$
$269$	0.348955	0.0212762	0.0106381	+	0.999943i	$0.496614\pi$
$270$	0	0
$271$	−11.2443	−0.683043	−0.341522	−	0.939874i	$-0.610942\pi$
$271$	−11.2443	−0.683043	−0.341522	+	0.939874i	$0.610942\pi$
$272$	13.1423	0.796871
$273$	0	0
$274$	21.3945	1.29249
$275$	3.27003	0.197190
$276$	0	0
$277$	−32.0388	−1.92503	−0.962513	−	0.271234i	$-0.912568\pi$
$277$	−32.0388	−1.92503	−0.962513	+	0.271234i	$0.912568\pi$
$278$	−15.6796	−0.940402
$279$	0	0
$280$	−24.1291	−1.44199
$281$	−9.77595	−0.583184	−0.291592	−	0.956543i	$-0.594185\pi$
$281$	−9.77595	−0.583184	−0.291592	+	0.956543i	$0.594185\pi$
$282$	0	0
$283$	4.40306	0.261735	0.130867	−	0.991400i	$-0.458224\pi$
$283$	4.40306	0.261735	0.130867	+	0.991400i	$0.458224\pi$
$284$	1.80930	0.107362
$285$	0	0
$286$	11.2323	0.664178
$287$	−5.87805	−0.346970
$288$	0	0
$289$	−11.7987	−0.694041
$290$	−6.86609	−0.403191
$291$	0	0
$292$	47.5298	2.78147
$293$	−6.34380	−0.370609	−0.185305	−	0.982681i	$-0.559327\pi$
$293$	−6.34380	−0.370609	−0.185305	+	0.982681i	$0.559327\pi$
$294$	0	0
$295$	40.8645	2.37922
$296$	58.8900	3.42291
$297$	0	0
$298$	9.93109	0.575293
$299$	−3.43491	−0.198646
$300$	0	0
$301$	−6.61110	−0.381058
$302$	−16.2612	−0.935729
$303$	0	0
$304$	−13.7746	−0.790030
$305$	−0.180154	−0.0103156
$306$	0	0
$307$	−18.1678	−1.03689	−0.518447	−	0.855110i	$-0.673490\pi$
$307$	−18.1678	−1.03689	−0.518447	+	0.855110i	$0.673490\pi$
$308$	−8.60666	−0.490410
$309$	0	0
$310$	16.9684	0.963740
$311$	−15.5142	−0.879727	−0.439864	−	0.898065i	$-0.644973\pi$
$311$	−15.5142	−0.879727	−0.439864	+	0.898065i	$0.644973\pi$
$312$	0	0
$313$	−27.5360	−1.55642	−0.778212	−	0.628001i	$-0.783873\pi$
$313$	−27.5360	−1.55642	−0.778212	+	0.628001i	$0.783873\pi$
$314$	−33.0290	−1.86393
$315$	0	0
$316$	−11.2581	−0.633317
$317$	2.20692	0.123953	0.0619765	−	0.998078i	$-0.480260\pi$
$317$	2.20692	0.123953	0.0619765	+	0.998078i	$0.480260\pi$
$318$	0	0
$319$	−1.30482	−0.0730558
$320$	−10.9047	−0.609592
$321$	0	0
$322$	3.86170	0.215204
$323$	−5.45155	−0.303332
$324$	0	0
$325$	8.60830	0.477502
$326$	−63.4653	−3.51502
$327$	0	0
$328$	21.8029	1.20387
$329$	16.1371	0.889667
$330$	0	0
$331$	−12.6766	−0.696770	−0.348385	−	0.937352i	$-0.613270\pi$
$331$	−12.6766	−0.696770	−0.348385	+	0.937352i	$0.613270\pi$
$332$	−68.2856	−3.74766
$333$	0	0
$334$	17.4833	0.956642
$335$	−1.37011	−0.0748569
$336$	0	0
$337$	18.5875	1.01253	0.506264	−	0.862379i	$-0.331026\pi$
$337$	18.5875	1.01253	0.506264	+	0.862379i	$0.331026\pi$
$338$	−3.01082	−0.163767
$339$	0	0
$340$	26.7468	1.45055
$341$	3.22464	0.174624
$342$	0	0
$343$	17.9140	0.967264
$344$	24.5220	1.32214
$345$	0	0
$346$	−23.5341	−1.26520
$347$	5.50028	0.295271	0.147635	−	0.989042i	$-0.452834\pi$
$347$	5.50028	0.295271	0.147635	+	0.989042i	$0.452834\pi$
$348$	0	0
$349$	18.0812	0.967863	0.483931	−	0.875106i	$-0.339208\pi$
$349$	18.0812	0.967863	0.483931	+	0.875106i	$0.339208\pi$
$350$	−9.67787	−0.517304
$351$	0	0
$352$	3.92829	0.209379
$353$	0.598193	0.0318386	0.0159193	−	0.999873i	$-0.494933\pi$
$353$	0.598193	0.0318386	0.0159193	+	0.999873i	$0.494933\pi$
$354$	0	0
$355$	1.15801	0.0614606
$356$	−67.7623	−3.59140
$357$	0	0
$358$	−35.1206	−1.85618
$359$	5.43779	0.286996	0.143498	−	0.989651i	$-0.454165\pi$
$359$	5.43779	0.286996	0.143498	+	0.989651i	$0.454165\pi$
$360$	0	0
$361$	−13.2862	−0.699272
$362$	29.4286	1.54673
$363$	0	0
$364$	−22.6569	−1.18754
$365$	30.4204	1.59228
$366$	0	0
$367$	−12.9691	−0.676984	−0.338492	−	0.940969i	$-0.609917\pi$
$367$	−12.9691	−0.676984	−0.338492	+	0.940969i	$0.609917\pi$
$368$	−5.76257	−0.300395
$369$	0	0
$370$	70.7446	3.67784
$371$	−1.08373	−0.0562645
$372$	0	0
$373$	−3.85806	−0.199763	−0.0998815	−	0.994999i	$-0.531846\pi$
$373$	−3.85806	−0.199763	−0.0998815	+	0.994999i	$0.531846\pi$
$374$	7.45775	0.385631
$375$	0	0
$376$	−59.8559	−3.08683
$377$	−3.43491	−0.176907
$378$	0	0
$379$	12.5992	0.647180	0.323590	−	0.946197i	$-0.395110\pi$
$379$	12.5992	0.647180	0.323590	+	0.946197i	$0.395110\pi$
$380$	−28.0337	−1.43810
$381$	0	0
$382$	63.0320	3.22500
$383$	3.18743	0.162870	0.0814349	−	0.996679i	$-0.474050\pi$
$383$	3.18743	0.162870	0.0814349	+	0.996679i	$0.474050\pi$
$384$	0	0
$385$	−5.50851	−0.280740
$386$	9.44793	0.480887
$387$	0	0
$388$	15.9085	0.807632
$389$	37.9270	1.92298	0.961488	−	0.274849i	$-0.0886277\pi$
$389$	37.9270	1.92298	0.961488	+	0.274849i	$0.0886277\pi$
$390$	0	0
$391$	−2.28064	−0.115337
$392$	−26.4379	−1.33531
$393$	0	0
$394$	35.0531	1.76595
$395$	−7.20550	−0.362548
$396$	0	0
$397$	10.4782	0.525886	0.262943	−	0.964811i	$-0.415307\pi$
$397$	10.4782	0.525886	0.262943	+	0.964811i	$0.415307\pi$
$398$	8.68517	0.435348
$399$	0	0
$400$	14.4417	0.722085
$401$	−15.9163	−0.794822	−0.397411	−	0.917641i	$-0.630091\pi$
$401$	−15.9163	−0.794822	−0.397411	+	0.917641i	$0.630091\pi$
$402$	0	0
$403$	8.48881	0.422858
$404$	0.119488	0.00594477
$405$	0	0
$406$	3.86170	0.191653
$407$	13.4442	0.666403
$408$	0	0
$409$	8.90182	0.440167	0.220083	−	0.975481i	$-0.429367\pi$
$409$	8.90182	0.440167	0.220083	+	0.975481i	$0.429367\pi$
$410$	26.1919	1.29352
$411$	0	0
$412$	−14.1979	−0.699478
$413$	−22.9834	−1.13094
$414$	0	0
$415$	−43.7048	−2.14538
$416$	10.3412	0.507017
$417$	0	0
$418$	−7.81656	−0.382321
$419$	−12.9512	−0.632706	−0.316353	−	0.948641i	$-0.602458\pi$
$419$	−12.9512	−0.632706	−0.316353	+	0.948641i	$0.602458\pi$
$420$	0	0
$421$	2.12359	0.103497	0.0517487	−	0.998660i	$-0.483521\pi$
$421$	2.12359	0.103497	0.0517487	+	0.998660i	$0.483521\pi$
$422$	10.6679	0.519305
$423$	0	0
$424$	4.01979	0.195218
$425$	5.71555	0.277245
$426$	0	0
$427$	0.101324	0.00490340
$428$	−4.15415	−0.200798
$429$	0	0
$430$	29.4583	1.42060
$431$	26.2219	1.26306	0.631532	−	0.775349i	$-0.282426\pi$
$431$	26.2219	1.26306	0.631532	+	0.775349i	$0.282426\pi$
$432$	0	0
$433$	17.6826	0.849771	0.424885	−	0.905247i	$-0.360314\pi$
$433$	17.6826	0.849771	0.424885	+	0.905247i	$0.360314\pi$
$434$	−9.54354	−0.458104
$435$	0	0
$436$	23.8717	1.14325
$437$	2.39036	0.114347
$438$	0	0
$439$	−12.0888	−0.576967	−0.288484	−	0.957485i	$-0.593151\pi$
$439$	−12.0888	−0.576967	−0.288484	+	0.957485i	$0.593151\pi$
$440$	20.4322	0.974068
$441$	0	0
$442$	19.6324	0.933818
$443$	−21.3112	−1.01253	−0.506264	−	0.862379i	$-0.668974\pi$
$443$	−21.3112	−1.01253	−0.506264	+	0.862379i	$0.668974\pi$
$444$	0	0
$445$	−43.3698	−2.05593
$446$	−56.8330	−2.69112
$447$	0	0
$448$	6.13313	0.289763
$449$	−6.48212	−0.305910	−0.152955	−	0.988233i	$-0.548879\pi$
$449$	−6.48212	−0.305910	−0.152955	+	0.988233i	$0.548879\pi$
$450$	0	0
$451$	4.97745	0.234379
$452$	5.59321	0.263083
$453$	0	0
$454$	44.9439	2.10932
$455$	−14.5010	−0.679820
$456$	0	0
$457$	−11.5228	−0.539012	−0.269506	−	0.962999i	$-0.586860\pi$
$457$	−11.5228	−0.539012	−0.269506	+	0.962999i	$0.586860\pi$
$458$	0.453937	0.0212111
$459$	0	0
$460$	−11.7278	−0.546811
$461$	−41.6112	−1.93803	−0.969013	−	0.247010i	$-0.920552\pi$
$461$	−41.6112	−1.93803	−0.969013	+	0.247010i	$0.920552\pi$
$462$	0	0
$463$	3.04216	0.141381	0.0706906	−	0.997498i	$-0.477480\pi$
$463$	3.04216	0.141381	0.0706906	+	0.997498i	$0.477480\pi$
$464$	−5.76257	−0.267521
$465$	0	0
$466$	61.8851	2.86677
$467$	16.7837	0.776657	0.388328	−	0.921521i	$-0.373053\pi$
$467$	16.7837	0.776657	0.388328	+	0.921521i	$0.373053\pi$
$468$	0	0
$469$	0.770589	0.0355825
$470$	−71.9049	−3.31673
$471$	0	0
$472$	85.2503	3.92396
$473$	5.59819	0.257405
$474$	0	0
$475$	−5.99053	−0.274865
$476$	−15.0432	−0.689504
$477$	0	0
$478$	−3.29827	−0.150859
$479$	−8.59016	−0.392494	−0.196247	−	0.980554i	$-0.562875\pi$
$479$	−8.59016	−0.392494	−0.196247	+	0.980554i	$0.562875\pi$
$480$	0	0
$481$	35.3915	1.61371
$482$	−6.67850	−0.304197
$483$	0	0
$484$	−39.7990	−1.80904
$485$	10.1819	0.462336
$486$	0	0
$487$	17.5263	0.794192	0.397096	−	0.917777i	$-0.370018\pi$
$487$	17.5263	0.794192	0.397096	+	0.917777i	$0.370018\pi$
$488$	−0.375831	−0.0170131
$489$	0	0
$490$	−31.7598	−1.43476
$491$	2.78790	0.125816	0.0629081	−	0.998019i	$-0.479963\pi$
$491$	2.78790	0.125816	0.0629081	+	0.998019i	$0.479963\pi$
$492$	0	0
$493$	−2.28064	−0.102715
$494$	−20.5769	−0.925801
$495$	0	0
$496$	14.2412	0.639450
$497$	−0.651297	−0.0292147
$498$	0	0
$499$	−20.1875	−0.903715	−0.451857	−	0.892090i	$-0.649238\pi$
$499$	−20.1875	−0.903715	−0.451857	+	0.892090i	$0.649238\pi$
$500$	−29.2477	−1.30800
$501$	0	0
$502$	−34.0856	−1.52131
$503$	−18.5368	−0.826516	−0.413258	−	0.910614i	$-0.635609\pi$
$503$	−18.5368	−0.826516	−0.413258	+	0.910614i	$0.635609\pi$
$504$	0	0
$505$	0.0764760	0.00340314
$506$	−3.27003	−0.145371
$507$	0	0
$508$	−19.9155	−0.883608
$509$	−4.94638	−0.219245	−0.109622	−	0.993973i	$-0.534964\pi$
$509$	−4.94638	−0.219245	−0.109622	+	0.993973i	$0.534964\pi$
$510$	0	0
$511$	−17.1093	−0.756873
$512$	−48.5237	−2.14446
$513$	0	0
$514$	40.5439	1.78832
$515$	−9.08703	−0.400423
$516$	0	0
$517$	−13.6647	−0.600972
$518$	−39.7889	−1.74822
$519$	0	0
$520$	53.7875	2.35874
$521$	25.3305	1.10975	0.554876	−	0.831933i	$-0.312766\pi$
$521$	25.3305	1.10975	0.554876	+	0.831933i	$0.312766\pi$
$522$	0	0
$523$	27.7478	1.21333	0.606664	−	0.794959i	$-0.292507\pi$
$523$	27.7478	1.21333	0.606664	+	0.794959i	$0.292507\pi$
$524$	57.0979	2.49433
$525$	0	0
$526$	−40.4619	−1.76422
$527$	5.63621	0.245517
$528$	0	0
$529$	1.00000	0.0434783
$530$	4.82897	0.209757
$531$	0	0
$532$	15.7670	0.683585
$533$	13.1030	0.567556
$534$	0	0
$535$	−2.65877	−0.114949
$536$	−2.85828	−0.123459
$537$	0	0
$538$	0.874524	0.0377034
$539$	−6.03558	−0.259971
$540$	0	0
$541$	19.2053	0.825702	0.412851	−	0.910799i	$-0.364533\pi$
$541$	19.2053	0.825702	0.412851	+	0.910799i	$0.364533\pi$
$542$	−28.1796	−1.21042
$543$	0	0
$544$	6.86609	0.294381
$545$	15.2786	0.654461
$546$	0	0
$547$	−20.6006	−0.880820	−0.440410	−	0.897797i	$-0.645167\pi$
$547$	−20.6006	−0.880820	−0.440410	+	0.897797i	$0.645167\pi$
$548$	36.5433	1.56105
$549$	0	0
$550$	8.19509	0.349440
$551$	2.39036	0.101833
$552$	0	0
$553$	4.05259	0.172334
$554$	−80.2932	−3.41133
$555$	0	0
$556$	−26.7820	−1.13581
$557$	−35.2426	−1.49328	−0.746639	−	0.665229i	$-0.768334\pi$
$557$	−35.2426	−1.49328	−0.746639	+	0.665229i	$0.768334\pi$
$558$	0	0
$559$	14.7371	0.623314
$560$	−24.3277	−1.02803
$561$	0	0
$562$	−24.4997	−1.03346
$563$	20.8449	0.878507	0.439253	−	0.898363i	$-0.355243\pi$
$563$	20.8449	0.878507	0.439253	+	0.898363i	$0.355243\pi$
$564$	0	0
$565$	3.57982	0.150604
$566$	11.0346	0.463819
$567$	0	0
$568$	2.41580	0.101365
$569$	−28.9690	−1.21444	−0.607221	−	0.794533i	$-0.707716\pi$
$569$	−28.9690	−1.21444	−0.607221	+	0.794533i	$0.707716\pi$
$570$	0	0
$571$	−34.4758	−1.44277	−0.721384	−	0.692535i	$-0.756494\pi$
$571$	−34.4758	−1.44277	−0.721384	+	0.692535i	$0.756494\pi$
$572$	19.1855	0.802188
$573$	0	0
$574$	−14.7311	−0.614864
$575$	−2.50612	−0.104512
$576$	0	0
$577$	15.4028	0.641227	0.320613	−	0.947210i	$-0.396111\pi$
$577$	15.4028	0.641227	0.320613	+	0.947210i	$0.396111\pi$
$578$	−29.5690	−1.22991
$579$	0	0
$580$	−11.7278	−0.486970
$581$	24.5809	1.01979
$582$	0	0
$583$	0.917689	0.0380068
$584$	63.4622	2.62608
$585$	0	0
$586$	−15.8983	−0.656754
$587$	−20.3345	−0.839294	−0.419647	−	0.907687i	$-0.637846\pi$
$587$	−20.3345	−0.839294	−0.419647	+	0.907687i	$0.637846\pi$
$588$	0	0
$589$	−5.90738	−0.243409
$590$	102.411	4.21620
$591$	0	0
$592$	59.3745	2.44028
$593$	0.213068	0.00874965	0.00437482	−	0.999990i	$-0.498607\pi$
$593$	0.213068	0.00874965	0.00437482	+	0.999990i	$0.498607\pi$
$594$	0	0
$595$	−9.62809	−0.394713
$596$	16.9630	0.694833
$597$	0	0
$598$	−8.60830	−0.352019
$599$	25.4984	1.04184	0.520918	−	0.853607i	$-0.325590\pi$
$599$	25.4984	1.04184	0.520918	+	0.853607i	$0.325590\pi$
$600$	0	0
$601$	12.9760	0.529301	0.264651	−	0.964344i	$-0.414743\pi$
$601$	12.9760	0.529301	0.264651	+	0.964344i	$0.414743\pi$
$602$	−16.5682	−0.675270
$603$	0	0
$604$	−27.7754	−1.13016
$605$	−25.4725	−1.03560
$606$	0	0
$607$	−12.1310	−0.492382	−0.246191	−	0.969221i	$-0.579179\pi$
$607$	−12.1310	−0.492382	−0.246191	+	0.969221i	$0.579179\pi$
$608$	−7.19643	−0.291854
$609$	0	0
$610$	−0.451486	−0.0182801
$611$	−35.9720	−1.45527
$612$	0	0
$613$	10.5478	0.426022	0.213011	−	0.977050i	$-0.431673\pi$
$613$	10.5478	0.426022	0.213011	+	0.977050i	$0.431673\pi$
$614$	−45.5308	−1.83747
$615$	0	0
$616$	−11.4917	−0.463014
$617$	−9.77913	−0.393693	−0.196846	−	0.980434i	$-0.563070\pi$
$617$	−9.77913	−0.393693	−0.196846	+	0.980434i	$0.563070\pi$
$618$	0	0
$619$	−9.84498	−0.395703	−0.197852	−	0.980232i	$-0.563396\pi$
$619$	−9.84498	−0.395703	−0.197852	+	0.980232i	$0.563396\pi$
$620$	28.9833	1.16400
$621$	0	0
$622$	−38.8803	−1.55896
$623$	24.3925	0.977264
$624$	0	0
$625$	−31.2500	−1.25000
$626$	−69.0084	−2.75813
$627$	0	0
$628$	−56.4159	−2.25124
$629$	23.4985	0.936946
$630$	0	0
$631$	−45.3495	−1.80533	−0.902667	−	0.430340i	$-0.858394\pi$
$631$	−45.3495	−1.80533	−0.902667	+	0.430340i	$0.858394\pi$
$632$	−15.0319	−0.597937
$633$	0	0
$634$	5.53081	0.219656
$635$	−12.7465	−0.505829
$636$	0	0
$637$	−15.8886	−0.629527
$638$	−3.27003	−0.129462
$639$	0	0
$640$	−43.8250	−1.73233
$641$	19.4724	0.769113	0.384556	−	0.923101i	$-0.374354\pi$
$641$	19.4724	0.769113	0.384556	+	0.923101i	$0.374354\pi$
$642$	0	0
$643$	26.3465	1.03900	0.519502	−	0.854469i	$-0.326117\pi$
$643$	26.3465	1.03900	0.519502	+	0.854469i	$0.326117\pi$
$644$	6.59606	0.259921
$645$	0	0
$646$	−13.6622	−0.537533
$647$	31.5187	1.23913	0.619564	−	0.784946i	$-0.287309\pi$
$647$	31.5187	1.23913	0.619564	+	0.784946i	$0.287309\pi$
$648$	0	0
$649$	19.4620	0.763952
$650$	21.5734	0.846179
$651$	0	0
$652$	−108.403	−4.24541
$653$	−15.0575	−0.589244	−0.294622	−	0.955614i	$-0.595194\pi$
$653$	−15.0575	−0.589244	−0.294622	+	0.955614i	$0.595194\pi$
$654$	0	0
$655$	36.5443	1.42790
$656$	21.9823	0.858265
$657$	0	0
$658$	40.4415	1.57657
$659$	−17.5364	−0.683122	−0.341561	−	0.939860i	$-0.610956\pi$
$659$	−17.5364	−0.683122	−0.341561	+	0.939860i	$0.610956\pi$
$660$	0	0
$661$	29.0640	1.13046	0.565230	−	0.824933i	$-0.308787\pi$
$661$	29.0640	1.13046	0.565230	+	0.824933i	$0.308787\pi$
$662$	−31.7691	−1.23474
$663$	0	0
$664$	−91.1756	−3.53830
$665$	10.0913	0.391324
$666$	0	0
$667$	1.00000	0.0387202
$668$	29.8627	1.15542
$669$	0	0
$670$	−3.43365	−0.132654
$671$	−0.0857996	−0.00331226
$672$	0	0
$673$	−24.3551	−0.938820	−0.469410	−	0.882980i	$-0.655533\pi$
$673$	−24.3551	−0.938820	−0.469410	+	0.882980i	$0.655533\pi$
$674$	46.5826	1.79429
$675$	0	0
$676$	−5.14270	−0.197796
$677$	−23.4269	−0.900369	−0.450185	−	0.892936i	$-0.648642\pi$
$677$	−23.4269	−0.900369	−0.450185	+	0.892936i	$0.648642\pi$
$678$	0	0
$679$	−5.72661	−0.219767
$680$	35.7126	1.36952
$681$	0	0
$682$	8.08134	0.309451
$683$	−3.73192	−0.142798	−0.0713990	−	0.997448i	$-0.522746\pi$
$683$	−3.73192	−0.142798	−0.0713990	+	0.997448i	$0.522746\pi$
$684$	0	0
$685$	23.3888	0.893639
$686$	44.8946	1.71408
$687$	0	0
$688$	24.7237	0.942583
$689$	2.41580	0.0920346
$690$	0	0
$691$	−40.6745	−1.54733	−0.773665	−	0.633595i	$-0.781579\pi$
$691$	−40.6745	−1.54733	−0.773665	+	0.633595i	$0.781579\pi$
$692$	−40.1980	−1.52810
$693$	0	0
$694$	13.7844	0.523248
$695$	−17.1412	−0.650204
$696$	0	0
$697$	8.69987	0.329531
$698$	45.3136	1.71514
$699$	0	0
$700$	−16.5305	−0.624795
$701$	14.6991	0.555177	0.277588	−	0.960700i	$-0.410465\pi$
$701$	14.6991	0.555177	0.277588	+	0.960700i	$0.410465\pi$
$702$	0	0
$703$	−24.6290	−0.928902
$704$	−5.19346	−0.195736
$705$	0	0
$706$	1.49914	0.0564210
$707$	−0.0430124	−0.00161765
$708$	0	0
$709$	22.3017	0.837558	0.418779	−	0.908088i	$-0.362458\pi$
$709$	22.3017	0.837558	0.418779	+	0.908088i	$0.362458\pi$
$710$	2.90210	0.108914
$711$	0	0
$712$	−90.4769	−3.39077
$713$	−2.47133	−0.0925522
$714$	0	0
$715$	12.2793	0.459220
$716$	−59.9886	−2.24188
$717$	0	0
$718$	13.6278	0.508583
$719$	−6.13956	−0.228967	−0.114484	−	0.993425i	$-0.536521\pi$
$719$	−6.13956	−0.228967	−0.114484	+	0.993425i	$0.536521\pi$
$720$	0	0
$721$	5.11082	0.190337
$722$	−33.2967	−1.23918
$723$	0	0
$724$	50.2662	1.86813
$725$	−2.50612	−0.0930749
$726$	0	0
$727$	8.21938	0.304840	0.152420	−	0.988316i	$-0.451293\pi$
$727$	8.21938	0.304840	0.152420	+	0.988316i	$0.451293\pi$
$728$	−30.2517	−1.12120
$729$	0	0
$730$	76.2372	2.82166
$731$	9.78484	0.361905
$732$	0	0
$733$	−30.0927	−1.11150	−0.555750	−	0.831349i	$-0.687569\pi$
$733$	−30.0927	−1.11150	−0.555750	+	0.831349i	$0.687569\pi$
$734$	−32.5022	−1.19968
$735$	0	0
$736$	−3.01060	−0.110972
$737$	−0.652525	−0.0240361
$738$	0	0
$739$	−8.66756	−0.318841	−0.159421	−	0.987211i	$-0.550963\pi$
$739$	−8.66756	−0.318841	−0.159421	+	0.987211i	$0.550963\pi$
$740$	120.837	4.44206
$741$	0	0
$742$	−2.71596	−0.0997060
$743$	−23.5733	−0.864821	−0.432411	−	0.901677i	$-0.642337\pi$
$743$	−23.5733	−0.864821	−0.432411	+	0.901677i	$0.642337\pi$
$744$	0	0
$745$	10.8568	0.397763
$746$	−9.66877	−0.353999
$747$	0	0
$748$	12.7384	0.465762
$749$	1.49537	0.0546397
$750$	0	0
$751$	−7.07810	−0.258284	−0.129142	−	0.991626i	$-0.541222\pi$
$751$	−7.07810	−0.258284	−0.129142	+	0.991626i	$0.541222\pi$
$752$	−60.3484	−2.20068
$753$	0	0
$754$	−8.60830	−0.313496
$755$	−17.7770	−0.646973
$756$	0	0
$757$	40.0268	1.45480	0.727400	−	0.686214i	$-0.240729\pi$
$757$	40.0268	1.45480	0.727400	+	0.686214i	$0.240729\pi$
$758$	31.5752	1.14686
$759$	0	0
$760$	−37.4308	−1.35776
$761$	52.4927	1.90286	0.951430	−	0.307866i	$-0.0996148\pi$
$761$	52.4927	1.90286	0.951430	+	0.307866i	$0.0996148\pi$
$762$	0	0
$763$	−8.59312	−0.311092
$764$	107.663	3.89512
$765$	0	0
$766$	7.98807	0.288621
$767$	51.2334	1.84993
$768$	0	0
$769$	4.45410	0.160619	0.0803095	−	0.996770i	$-0.474409\pi$
$769$	4.45410	0.160619	0.0803095	+	0.996770i	$0.474409\pi$
$770$	−13.8050	−0.497497
$771$	0	0
$772$	16.1378	0.580811
$773$	−18.6124	−0.669443	−0.334721	−	0.942317i	$-0.608642\pi$
$773$	−18.6124	−0.669443	−0.334721	+	0.942317i	$0.608642\pi$
$774$	0	0
$775$	6.19346	0.222476
$776$	21.2412	0.762514
$777$	0	0
$778$	95.0496	3.40769
$779$	−9.11844	−0.326702
$780$	0	0
$781$	0.551510	0.0197346
$782$	−5.71555	−0.204388
$783$	0	0
$784$	−26.6554	−0.951979
$785$	−36.1078	−1.28874
$786$	0	0
$787$	40.6671	1.44963	0.724813	−	0.688946i	$-0.241926\pi$
$787$	40.6671	1.44963	0.724813	+	0.688946i	$0.241926\pi$
$788$	59.8733	2.13290
$789$	0	0
$790$	−18.0578	−0.642469
$791$	−2.01340	−0.0715881
$792$	0	0
$793$	−0.225866	−0.00802073
$794$	26.2596	0.931920
$795$	0	0
$796$	14.8349	0.525809
$797$	−3.35214	−0.118739	−0.0593695	−	0.998236i	$-0.518909\pi$
$797$	−3.35214	−0.118739	−0.0593695	+	0.998236i	$0.518909\pi$
$798$	0	0
$799$	−23.8839	−0.844951
$800$	7.54493	0.266754
$801$	0	0
$802$	−39.8882	−1.40850
$803$	14.4880	0.511269
$804$	0	0
$805$	4.22167	0.148794
$806$	21.2740	0.749344
$807$	0	0
$808$	0.159542	0.00561267
$809$	43.8960	1.54330	0.771651	−	0.636046i	$-0.219431\pi$
$809$	43.8960	1.54330	0.771651	+	0.636046i	$0.219431\pi$
$810$	0	0
$811$	3.65506	0.128346	0.0641732	−	0.997939i	$-0.479559\pi$
$811$	3.65506	0.128346	0.0641732	+	0.997939i	$0.479559\pi$
$812$	6.59606	0.231476
$813$	0	0
$814$	33.6927	1.18093
$815$	−69.3813	−2.43032
$816$	0	0
$817$	−10.2556	−0.358798
$818$	22.3090	0.780017
$819$	0	0
$820$	44.7376	1.56231
$821$	6.66144	0.232486	0.116243	−	0.993221i	$-0.462915\pi$
$821$	6.66144	0.232486	0.116243	+	0.993221i	$0.462915\pi$
$822$	0	0
$823$	−42.8887	−1.49501	−0.747503	−	0.664258i	$-0.768747\pi$
$823$	−42.8887	−1.49501	−0.747503	+	0.664258i	$0.768747\pi$
$824$	−18.9571	−0.660402
$825$	0	0
$826$	−57.5991	−2.00413
$827$	53.6110	1.86424	0.932119	−	0.362152i	$-0.117958\pi$
$827$	53.6110	1.86424	0.932119	+	0.362152i	$0.117958\pi$
$828$	0	0
$829$	24.5007	0.850945	0.425473	−	0.904971i	$-0.360108\pi$
$829$	24.5007	0.850945	0.425473	+	0.904971i	$0.360108\pi$
$830$	−109.529	−3.80182
$831$	0	0
$832$	−13.6717	−0.473980
$833$	−10.5493	−0.365513
$834$	0	0
$835$	19.1130	0.661432
$836$	−13.3513	−0.461763
$837$	0	0
$838$	−32.4572	−1.12122
$839$	13.9703	0.482310	0.241155	−	0.970487i	$-0.422474\pi$
$839$	13.9703	0.482310	0.241155	+	0.970487i	$0.422474\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	5.32197	0.183407
$843$	0	0
$844$	18.2215	0.627211
$845$	−3.29148	−0.113230
$846$	0	0
$847$	14.3265	0.492264
$848$	4.05286	0.139176
$849$	0	0
$850$	14.3238	0.491304
$851$	−10.3035	−0.353199
$852$	0	0
$853$	−52.9583	−1.81326	−0.906628	−	0.421930i	$-0.861353\pi$
$853$	−52.9583	−1.81326	−0.906628	+	0.421930i	$0.861353\pi$
$854$	0.253929	0.00868929
$855$	0	0
$856$	−5.54666	−0.189581
$857$	−5.62978	−0.192310	−0.0961548	−	0.995366i	$-0.530654\pi$
$857$	−5.62978	−0.192310	−0.0961548	+	0.995366i	$0.530654\pi$
$858$	0	0
$859$	−14.5237	−0.495543	−0.247772	−	0.968818i	$-0.579698\pi$
$859$	−14.5237	−0.495543	−0.247772	+	0.968818i	$0.579698\pi$
$860$	50.3169	1.71579
$861$	0	0
$862$	65.7152	2.23827
$863$	34.0906	1.16046	0.580228	−	0.814454i	$-0.302963\pi$
$863$	34.0906	1.16046	0.580228	+	0.814454i	$0.302963\pi$
$864$	0	0
$865$	−25.7279	−0.874774
$866$	44.3147	1.50587
$867$	0	0
$868$	−16.3011	−0.553294
$869$	−3.43168	−0.116412
$870$	0	0
$871$	−1.71776	−0.0582040
$872$	31.8737	1.07938
$873$	0	0
$874$	5.99053	0.202633
$875$	10.5283	0.355923
$876$	0	0
$877$	−38.1227	−1.28731	−0.643657	−	0.765314i	$-0.722584\pi$
$877$	−38.1227	−1.28731	−0.643657	+	0.765314i	$0.722584\pi$
$878$	−30.2960	−1.02244
$879$	0	0
$880$	20.6003	0.694437
$881$	−15.4602	−0.520867	−0.260433	−	0.965492i	$-0.583865\pi$
$881$	−15.4602	−0.520867	−0.260433	+	0.965492i	$0.583865\pi$
$882$	0	0
$883$	48.4095	1.62911	0.814554	−	0.580088i	$-0.196982\pi$
$883$	48.4095	1.62911	0.814554	+	0.580088i	$0.196982\pi$
$884$	33.5336	1.12786
$885$	0	0
$886$	−53.4085	−1.79429
$887$	14.6449	0.491727	0.245864	−	0.969304i	$-0.420928\pi$
$887$	14.6449	0.491727	0.245864	+	0.969304i	$0.420928\pi$
$888$	0	0
$889$	7.16901	0.240441
$890$	−108.690	−3.64330
$891$	0	0
$892$	−97.0750	−3.25031
$893$	25.0330	0.837697
$894$	0	0
$895$	−38.3945	−1.28338
$896$	24.6485	0.823448
$897$	0	0
$898$	−16.2450	−0.542102
$899$	−2.47133	−0.0824236
$900$	0	0
$901$	1.60399	0.0534366
$902$	12.4741	0.415342
$903$	0	0
$904$	7.46811	0.248386
$905$	32.1718	1.06943
$906$	0	0
$907$	−7.41188	−0.246107	−0.123054	−	0.992400i	$-0.539269\pi$
$907$	−7.41188	−0.246107	−0.123054	+	0.992400i	$0.539269\pi$
$908$	76.7675	2.54762
$909$	0	0
$910$	−36.3414	−1.20470
$911$	48.5281	1.60781	0.803904	−	0.594759i	$-0.202752\pi$
$911$	48.5281	1.60781	0.803904	+	0.594759i	$0.202752\pi$
$912$	0	0
$913$	−20.8147	−0.688868
$914$	−28.8774	−0.955180
$915$	0	0
$916$	0.775358	0.0256185
$917$	−20.5536	−0.678740
$918$	0	0
$919$	−19.1819	−0.632753	−0.316376	−	0.948634i	$-0.602466\pi$
$919$	−19.1819	−0.632753	−0.316376	+	0.948634i	$0.602466\pi$
$920$	−15.6591	−0.516264
$921$	0	0
$922$	−104.283	−3.43436
$923$	1.45184	0.0477879
$924$	0	0
$925$	25.8218	0.849014
$926$	7.62402	0.250541
$927$	0	0
$928$	−3.01060	−0.0988279
$929$	−38.9302	−1.27726	−0.638629	−	0.769515i	$-0.720498\pi$
$929$	−38.9302	−1.27726	−0.638629	+	0.769515i	$0.720498\pi$
$930$	0	0
$931$	11.0569	0.362375
$932$	105.704	3.46246
$933$	0	0
$934$	42.0619	1.37631
$935$	8.15293	0.266629
$936$	0	0
$937$	46.2552	1.51109	0.755545	−	0.655097i	$-0.227372\pi$
$937$	46.2552	1.51109	0.755545	+	0.655097i	$0.227372\pi$
$938$	1.93119	0.0630555
$939$	0	0
$940$	−122.819	−4.00591
$941$	43.0101	1.40209	0.701045	−	0.713117i	$-0.252717\pi$
$941$	43.0101	1.40209	0.701045	+	0.713117i	$0.252717\pi$
$942$	0	0
$943$	−3.81467	−0.124223
$944$	85.9517	2.79749
$945$	0	0
$946$	14.0297	0.456146
$947$	−22.5579	−0.733034	−0.366517	−	0.930411i	$-0.619450\pi$
$947$	−22.5579	−0.733034	−0.366517	+	0.930411i	$0.619450\pi$
$948$	0	0
$949$	38.1393	1.23805
$950$	−15.0130	−0.487086
$951$	0	0
$952$	−20.0858	−0.650986
$953$	−40.6257	−1.31600	−0.657998	−	0.753020i	$-0.728597\pi$
$953$	−40.6257	−1.31600	−0.657998	+	0.753020i	$0.728597\pi$
$954$	0	0
$955$	68.9076	2.22980
$956$	−5.63369	−0.182207
$957$	0	0
$958$	−21.5280	−0.695537
$959$	−13.1546	−0.424783
$960$	0	0
$961$	−24.8925	−0.802984
$962$	88.6954	2.85966
$963$	0	0
$964$	−11.4074	−0.367407
$965$	10.3286	0.332490
$966$	0	0
$967$	−1.11416	−0.0358290	−0.0179145	−	0.999840i	$-0.505703\pi$
$967$	−1.11416	−0.0358290	−0.0179145	+	0.999840i	$0.505703\pi$
$968$	−53.1400	−1.70798
$969$	0	0
$970$	25.5171	0.819303
$971$	−14.4416	−0.463454	−0.231727	−	0.972781i	$-0.574438\pi$
$971$	−14.4416	−0.463454	−0.231727	+	0.972781i	$0.574438\pi$
$972$	0	0
$973$	9.64074	0.309068
$974$	43.9230	1.40738
$975$	0	0
$976$	−0.378923	−0.0121290
$977$	4.51331	0.144394	0.0721968	−	0.997390i	$-0.476999\pi$
$977$	4.51331	0.144394	0.0721968	+	0.997390i	$0.476999\pi$
$978$	0	0
$979$	−20.6552	−0.660144
$980$	−54.2481	−1.73289
$981$	0	0
$982$	6.98681	0.222958
$983$	8.42418	0.268690	0.134345	−	0.990935i	$-0.457107\pi$
$983$	8.42418	0.268690	0.134345	+	0.990935i	$0.457107\pi$
$984$	0	0
$985$	38.3206	1.22100
$986$	−5.71555	−0.182020
$987$	0	0
$988$	−35.1469	−1.11817
$989$	−4.29040	−0.136427
$990$	0	0
$991$	−40.3559	−1.28195	−0.640974	−	0.767563i	$-0.721469\pi$
$991$	−40.3559	−1.28195	−0.640974	+	0.767563i	$0.721469\pi$
$992$	7.44020	0.236227
$993$	0	0
$994$	−1.63223	−0.0517711
$995$	9.49477	0.301004
$996$	0	0
$997$	−21.8586	−0.692270	−0.346135	−	0.938185i	$-0.612506\pi$
$997$	−21.8586	−0.692270	−0.346135	+	0.938185i	$0.612506\pi$
$998$	−50.5922	−1.60147
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.h.1.5		5
3.2	odd	2		2001.2.a.h.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.h.1.1	✓	5	3.2	odd	2
6003.2.a.h.1.5		5	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)

\(f(q)\)	\(=\)	\(q+2.50612 q^{2} +4.28064 q^{4} +2.73973 q^{5} -1.54091 q^{7} +5.71555 q^{8} +O(q^{10})\)	\(q+2.50612 q^{2} +4.28064 q^{4} +2.73973 q^{5} -1.54091 q^{7} +5.71555 q^{8} +6.86609 q^{10} +1.30482 q^{11} +3.43491 q^{13} -3.86170 q^{14} +5.76257 q^{16} +2.28064 q^{17} -2.39036 q^{19} +11.7278 q^{20} +3.27003 q^{22} -1.00000 q^{23} +2.50612 q^{25} +8.60830 q^{26} -6.59606 q^{28} -1.00000 q^{29} +2.47133 q^{31} +3.01060 q^{32} +5.71555 q^{34} -4.22167 q^{35} +10.3035 q^{37} -5.99053 q^{38} +15.6591 q^{40} +3.81467 q^{41} +4.29040 q^{43} +5.58545 q^{44} -2.50612 q^{46} -10.4725 q^{47} -4.62561 q^{49} +6.28064 q^{50} +14.7036 q^{52} +0.703308 q^{53} +3.57485 q^{55} -8.80712 q^{56} -2.50612 q^{58} +14.9155 q^{59} -0.0657559 q^{61} +6.19346 q^{62} -3.98021 q^{64} +9.41073 q^{65} -0.500088 q^{67} +9.76257 q^{68} -10.5800 q^{70} +0.422671 q^{71} +11.1034 q^{73} +25.8218 q^{74} -10.2323 q^{76} -2.01060 q^{77} -2.63000 q^{79} +15.7879 q^{80} +9.56002 q^{82} -15.9522 q^{83} +6.24833 q^{85} +10.7523 q^{86} +7.45775 q^{88} -15.8300 q^{89} -5.29288 q^{91} -4.28064 q^{92} -26.2453 q^{94} -6.54895 q^{95} +3.71639 q^{97} -11.5923 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} + 8 q^{4} + 3 q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) \) 5 * q + 2 * q^2 + 8 * q^4 + 3 * q^5 - 5 * q^7 + 3 * q^8	\( 5 q + 2 q^{2} + 8 q^{4} + 3 q^{5} - 5 q^{7} + 3 q^{8} + 9 q^{10} + 8 q^{11} + 5 q^{13} + 2 q^{14} - 10 q^{16} - 2 q^{17} - 9 q^{19} + 12 q^{20} + 10 q^{22} - 5 q^{23} + 2 q^{25} + 3 q^{26} - 14 q^{28} - 5 q^{29} - 6 q^{31} + 8 q^{32} + 3 q^{34} + 15 q^{35} + 10 q^{37} + 10 q^{38} + 26 q^{40} + 11 q^{41} - 9 q^{43} + 16 q^{44} - 2 q^{46} + 13 q^{47} - 6 q^{49} + 18 q^{50} + 12 q^{52} - q^{53} + 13 q^{55} + 4 q^{56} - 2 q^{58} - 6 q^{59} + 23 q^{61} + 36 q^{62} - q^{64} + 20 q^{65} - 10 q^{67} + 10 q^{68} - 16 q^{70} + 11 q^{71} + 31 q^{73} + 18 q^{74} - 8 q^{76} - 3 q^{77} + 8 q^{79} + 8 q^{80} + 16 q^{82} - 7 q^{83} + 6 q^{85} + 36 q^{86} - 3 q^{88} - 3 q^{89} + 8 q^{91} - 8 q^{92} - 39 q^{94} + 11 q^{95} + 3 q^{97} - 38 q^{98}+O(q^{100}) \) 5 * q + 2 * q^2 + 8 * q^4 + 3 * q^5 - 5 * q^7 + 3 * q^8 + 9 * q^10 + 8 * q^11 + 5 * q^13 + 2 * q^14 - 10 * q^16 - 2 * q^17 - 9 * q^19 + 12 * q^20 + 10 * q^22 - 5 * q^23 + 2 * q^25 + 3 * q^26 - 14 * q^28 - 5 * q^29 - 6 * q^31 + 8 * q^32 + 3 * q^34 + 15 * q^35 + 10 * q^37 + 10 * q^38 + 26 * q^40 + 11 * q^41 - 9 * q^43 + 16 * q^44 - 2 * q^46 + 13 * q^47 - 6 * q^49 + 18 * q^50 + 12 * q^52 - q^53 + 13 * q^55 + 4 * q^56 - 2 * q^58 - 6 * q^59 + 23 * q^61 + 36 * q^62 - q^64 + 20 * q^65 - 10 * q^67 + 10 * q^68 - 16 * q^70 + 11 * q^71 + 31 * q^73 + 18 * q^74 - 8 * q^76 - 3 * q^77 + 8 * q^79 + 8 * q^80 + 16 * q^82 - 7 * q^83 + 6 * q^85 + 36 * q^86 - 3 * q^88 - 3 * q^89 + 8 * q^91 - 8 * q^92 - 39 * q^94 + 11 * q^95 + 3 * q^97 - 38 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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