LMFDB - Embedded newform 6003.2.a.v.1.6

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:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
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)

$f(q)$	$=$	$q-1.99322 q^{2} +1.97291 q^{4} +0.735105 q^{5} -3.83374 q^{7} +0.0540029 q^{8} +O(q^{10})$	\(q-1.99322 q^{2} +1.97291 q^{4} +0.735105 q^{5} -3.83374 q^{7} +0.0540029 q^{8} -1.46522 q^{10} +0.205441 q^{11} +6.99499 q^{13} +7.64148 q^{14} -4.05345 q^{16} +3.70930 q^{17} +4.87904 q^{19} +1.45029 q^{20} -0.409488 q^{22} +1.00000 q^{23} -4.45962 q^{25} -13.9425 q^{26} -7.56362 q^{28} -1.00000 q^{29} +5.25291 q^{31} +7.97140 q^{32} -7.39344 q^{34} -2.81821 q^{35} +4.20377 q^{37} -9.72497 q^{38} +0.0396978 q^{40} +11.5860 q^{41} +6.44126 q^{43} +0.405316 q^{44} -1.99322 q^{46} -0.322866 q^{47} +7.69760 q^{49} +8.88898 q^{50} +13.8005 q^{52} -12.4026 q^{53} +0.151021 q^{55} -0.207033 q^{56} +1.99322 q^{58} -8.24981 q^{59} +11.3425 q^{61} -10.4702 q^{62} -7.78180 q^{64} +5.14206 q^{65} +7.87929 q^{67} +7.31811 q^{68} +5.61729 q^{70} -8.34336 q^{71} +1.72520 q^{73} -8.37902 q^{74} +9.62589 q^{76} -0.787608 q^{77} -2.27322 q^{79} -2.97971 q^{80} -23.0933 q^{82} -2.31436 q^{83} +2.72673 q^{85} -12.8388 q^{86} +0.0110944 q^{88} -9.00709 q^{89} -26.8170 q^{91} +1.97291 q^{92} +0.643541 q^{94} +3.58661 q^{95} +15.7302 q^{97} -15.3430 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8}+O(q^{10}) $ 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8	$ 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8} + 8 q^{10} + 36 q^{13} - 7 q^{14} + 47 q^{16} - 18 q^{17} + 16 q^{19} + 25 q^{22} + 30 q^{23} + 56 q^{25} - 11 q^{26} + 27 q^{28} - 30 q^{29} + 14 q^{31} + 7 q^{32} + 3 q^{34} + 22 q^{35} + 40 q^{37} - 6 q^{38} + 30 q^{40} - 14 q^{41} + 34 q^{43} - 5 q^{44} - q^{46} + 2 q^{47} + 74 q^{49} + 21 q^{50} + 71 q^{52} - 16 q^{53} + 22 q^{55} - 14 q^{56} + q^{58} + 32 q^{59} + 46 q^{61} - 20 q^{62} + 68 q^{64} - 12 q^{65} + 14 q^{67} - 27 q^{68} + 32 q^{71} + 50 q^{73} + 26 q^{74} + 56 q^{76} - 34 q^{77} + 16 q^{79} - 2 q^{80} + 38 q^{82} + 14 q^{83} + 38 q^{85} - 10 q^{86} + 40 q^{88} + 2 q^{89} + 32 q^{91} + 37 q^{92} + 29 q^{94} + 28 q^{95} + 56 q^{97} - 8 q^{98}+O(q^{100}) $ 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8 + 8 * q^10 + 36 * q^13 - 7 * q^14 + 47 * q^16 - 18 * q^17 + 16 * q^19 + 25 * q^22 + 30 * q^23 + 56 * q^25 - 11 * q^26 + 27 * q^28 - 30 * q^29 + 14 * q^31 + 7 * q^32 + 3 * q^34 + 22 * q^35 + 40 * q^37 - 6 * q^38 + 30 * q^40 - 14 * q^41 + 34 * q^43 - 5 * q^44 - q^46 + 2 * q^47 + 74 * q^49 + 21 * q^50 + 71 * q^52 - 16 * q^53 + 22 * q^55 - 14 * q^56 + q^58 + 32 * q^59 + 46 * q^61 - 20 * q^62 + 68 * q^64 - 12 * q^65 + 14 * q^67 - 27 * q^68 + 32 * q^71 + 50 * q^73 + 26 * q^74 + 56 * q^76 - 34 * q^77 + 16 * q^79 - 2 * q^80 + 38 * q^82 + 14 * q^83 + 38 * q^85 - 10 * q^86 + 40 * q^88 + 2 * q^89 + 32 * q^91 + 37 * q^92 + 29 * q^94 + 28 * q^95 + 56 * q^97 - 8 * 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 $

$2$	−1.99322	−1.40942	−0.704708	−	0.709497i	$-0.748922\pi$
$2$	−1.99322	−1.40942	−0.704708	+	0.709497i	$0.748922\pi$
$3$	0	0
$3$	0	0
$4$	1.97291	0.986453
$5$	0.735105	0.328749	0.164375	−	0.986398i	$-0.447439\pi$
$5$	0.735105	0.328749	0.164375	+	0.986398i	$0.447439\pi$
$6$	0	0
$7$	−3.83374	−1.44902	−0.724510	−	0.689265i	$-0.757934\pi$
$7$	−3.83374	−1.44902	−0.724510	+	0.689265i	$0.757934\pi$
$8$	0.0540029	0.0190929
$9$	0	0
$10$	−1.46522	−0.463344
$11$	0.205441	0.0619428	0.0309714	−	0.999520i	$-0.490140\pi$
$11$	0.205441	0.0619428	0.0309714	+	0.999520i	$0.490140\pi$
$12$	0	0
$13$	6.99499	1.94006	0.970031	−	0.242981i	$-0.0781253\pi$
$13$	6.99499	1.94006	0.970031	+	0.242981i	$0.0781253\pi$
$14$	7.64148	2.04227
$15$	0	0
$16$	−4.05345	−1.01336
$17$	3.70930	0.899638	0.449819	−	0.893120i	$-0.351489\pi$
$17$	3.70930	0.899638	0.449819	+	0.893120i	$0.351489\pi$
$18$	0	0
$19$	4.87904	1.11933	0.559664	−	0.828720i	$-0.310930\pi$
$19$	4.87904	1.11933	0.559664	+	0.828720i	$0.310930\pi$
$20$	1.45029	0.324296
$21$	0	0
$22$	−0.409488	−0.0873032
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.45962	−0.891924
$26$	−13.9425	−2.73435
$27$	0	0
$28$	−7.56362	−1.42939
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	5.25291	0.943450	0.471725	−	0.881746i	$-0.343632\pi$
$31$	5.25291	0.943450	0.471725	+	0.881746i	$0.343632\pi$
$32$	7.97140	1.40916
$33$	0	0
$34$	−7.39344	−1.26796
$35$	−2.81821	−0.476364
$36$	0	0
$37$	4.20377	0.691096	0.345548	−	0.938401i	$-0.387693\pi$
$37$	4.20377	0.691096	0.345548	+	0.938401i	$0.387693\pi$
$38$	−9.72497	−1.57760
$39$	0	0
$40$	0.0396978	0.00627678
$41$	11.5860	1.80942	0.904712	−	0.426023i	$-0.140086\pi$
$41$	11.5860	1.80942	0.904712	+	0.426023i	$0.140086\pi$
$42$	0	0
$43$	6.44126	0.982283	0.491141	−	0.871080i	$-0.336580\pi$
$43$	6.44126	0.982283	0.491141	+	0.871080i	$0.336580\pi$
$44$	0.405316	0.0611037
$45$	0	0
$46$	−1.99322	−0.293884
$47$	−0.322866	−0.0470948	−0.0235474	−	0.999723i	$-0.507496\pi$
$47$	−0.322866	−0.0470948	−0.0235474	+	0.999723i	$0.507496\pi$
$48$	0	0
$49$	7.69760	1.09966
$50$	8.88898	1.25709
$51$	0	0
$52$	13.8005	1.91378
$53$	−12.4026	−1.70362	−0.851812	−	0.523847i	$-0.824496\pi$
$53$	−12.4026	−1.70362	−0.851812	+	0.523847i	$0.824496\pi$
$54$	0	0
$55$	0.151021	0.0203636
$56$	−0.207033	−0.0276660
$57$	0	0
$58$	1.99322	0.261722
$59$	−8.24981	−1.07403	−0.537017	−	0.843571i	$-0.680449\pi$
$59$	−8.24981	−1.07403	−0.537017	+	0.843571i	$0.680449\pi$
$60$	0	0
$61$	11.3425	1.45226	0.726131	−	0.687556i	$-0.241316\pi$
$61$	11.3425	1.45226	0.726131	+	0.687556i	$0.241316\pi$
$62$	−10.4702	−1.32971
$63$	0	0
$64$	−7.78180	−0.972726
$65$	5.14206	0.637794
$66$	0	0
$67$	7.87929	0.962609	0.481304	−	0.876554i	$-0.340163\pi$
$67$	7.87929	0.962609	0.481304	+	0.876554i	$0.340163\pi$
$68$	7.31811	0.887451
$69$	0	0
$70$	5.61729	0.671395
$71$	−8.34336	−0.990174	−0.495087	−	0.868843i	$-0.664864\pi$
$71$	−8.34336	−0.990174	−0.495087	+	0.868843i	$0.664864\pi$
$72$	0	0
$73$	1.72520	0.201919	0.100960	−	0.994891i	$-0.467809\pi$
$73$	1.72520	0.201919	0.100960	+	0.994891i	$0.467809\pi$
$74$	−8.37902	−0.974042
$75$	0	0
$76$	9.62589	1.10416
$77$	−0.787608	−0.0897563
$78$	0	0
$79$	−2.27322	−0.255758	−0.127879	−	0.991790i	$-0.540817\pi$
$79$	−2.27322	−0.255758	−0.127879	+	0.991790i	$0.540817\pi$
$80$	−2.97971	−0.333142
$81$	0	0
$82$	−23.0933	−2.55023
$83$	−2.31436	−0.254034	−0.127017	−	0.991901i	$-0.540540\pi$
$83$	−2.31436	−0.254034	−0.127017	+	0.991901i	$0.540540\pi$
$84$	0	0
$85$	2.72673	0.295755
$86$	−12.8388	−1.38444
$87$	0	0
$88$	0.0110944	0.00118267
$89$	−9.00709	−0.954749	−0.477375	−	0.878700i	$-0.658411\pi$
$89$	−9.00709	−0.954749	−0.477375	+	0.878700i	$0.658411\pi$
$90$	0	0
$91$	−26.8170	−2.81119
$92$	1.97291	0.205690
$93$	0	0
$94$	0.643541	0.0663761
$95$	3.58661	0.367978
$96$	0	0
$97$	15.7302	1.59716	0.798580	−	0.601889i	$-0.205585\pi$
$97$	15.7302	1.59716	0.798580	+	0.601889i	$0.205585\pi$
$98$	−15.3430	−1.54987
$99$	0	0
$100$	−8.79841	−0.879841
$101$	15.1163	1.50413	0.752066	−	0.659087i	$-0.229057\pi$
$101$	15.1163	1.50413	0.752066	+	0.659087i	$0.229057\pi$
$102$	0	0
$103$	−19.1194	−1.88389	−0.941944	−	0.335770i	$-0.891003\pi$
$103$	−19.1194	−1.88389	−0.941944	+	0.335770i	$0.891003\pi$
$104$	0.377750	0.0370414
$105$	0	0
$106$	24.7210	2.40112
$107$	−4.51704	−0.436679	−0.218339	−	0.975873i	$-0.570064\pi$
$107$	−4.51704	−0.436679	−0.218339	+	0.975873i	$0.570064\pi$
$108$	0	0
$109$	−1.24986	−0.119715	−0.0598573	−	0.998207i	$-0.519065\pi$
$109$	−1.24986	−0.119715	−0.0598573	+	0.998207i	$0.519065\pi$
$110$	−0.301017	−0.0287008
$111$	0	0
$112$	15.5399	1.46838
$113$	−1.73750	−0.163450	−0.0817252	−	0.996655i	$-0.526043\pi$
$113$	−1.73750	−0.163450	−0.0817252	+	0.996655i	$0.526043\pi$
$114$	0	0
$115$	0.735105	0.0685489
$116$	−1.97291	−0.183180
$117$	0	0
$118$	16.4437	1.51376
$119$	−14.2205	−1.30359
$120$	0	0
$121$	−10.9578	−0.996163
$122$	−22.6081	−2.04684
$123$	0	0
$124$	10.3635	0.930669
$125$	−6.95382	−0.621968
$126$	0	0
$127$	−7.77512	−0.689930	−0.344965	−	0.938615i	$-0.612109\pi$
$127$	−7.77512	−0.689930	−0.344965	+	0.938615i	$0.612109\pi$
$128$	−0.431984	−0.0381823
$129$	0	0
$130$	−10.2492	−0.898916
$131$	7.77889	0.679645	0.339822	−	0.940490i	$-0.389633\pi$
$131$	7.77889	0.679645	0.339822	+	0.940490i	$0.389633\pi$
$132$	0	0
$133$	−18.7050	−1.62193
$134$	−15.7051	−1.35672
$135$	0	0
$136$	0.200313	0.0171767
$137$	−21.8734	−1.86877	−0.934383	−	0.356269i	$-0.884049\pi$
$137$	−21.8734	−1.86877	−0.934383	+	0.356269i	$0.884049\pi$
$138$	0	0
$139$	2.10387	0.178448	0.0892239	−	0.996012i	$-0.471561\pi$
$139$	2.10387	0.178448	0.0892239	+	0.996012i	$0.471561\pi$
$140$	−5.56006	−0.469911
$141$	0	0
$142$	16.6301	1.39557
$143$	1.43706	0.120173
$144$	0	0
$145$	−0.735105	−0.0610472
$146$	−3.43870	−0.284589
$147$	0	0
$148$	8.29365	0.681734
$149$	2.90681	0.238135	0.119067	−	0.992886i	$-0.462010\pi$
$149$	2.90681	0.238135	0.119067	+	0.992886i	$0.462010\pi$
$150$	0	0
$151$	14.7679	1.20179	0.600896	−	0.799327i	$-0.294811\pi$
$151$	14.7679	1.20179	0.600896	+	0.799327i	$0.294811\pi$
$152$	0.263482	0.0213712
$153$	0	0
$154$	1.56987	0.126504
$155$	3.86144	0.310158
$156$	0	0
$157$	−3.71893	−0.296803	−0.148401	−	0.988927i	$-0.547413\pi$
$157$	−3.71893	−0.296803	−0.148401	+	0.988927i	$0.547413\pi$
$158$	4.53102	0.360469
$159$	0	0
$160$	5.85982	0.463259
$161$	−3.83374	−0.302141
$162$	0	0
$163$	22.3418	1.74994	0.874972	−	0.484173i	$-0.160880\pi$
$163$	22.3418	1.74994	0.874972	+	0.484173i	$0.160880\pi$
$164$	22.8580	1.78491
$165$	0	0
$166$	4.61302	0.358040
$167$	10.6214	0.821910	0.410955	−	0.911656i	$-0.365195\pi$
$167$	10.6214	0.821910	0.410955	+	0.911656i	$0.365195\pi$
$168$	0	0
$169$	35.9299	2.76384
$170$	−5.43495	−0.416842
$171$	0	0
$172$	12.7080	0.968976
$173$	−2.93433	−0.223093	−0.111546	−	0.993759i	$-0.535580\pi$
$173$	−2.93433	−0.223093	−0.111546	+	0.993759i	$0.535580\pi$
$174$	0	0
$175$	17.0970	1.29242
$176$	−0.832746	−0.0627706
$177$	0	0
$178$	17.9531	1.34564
$179$	−9.64171	−0.720655	−0.360328	−	0.932826i	$-0.617335\pi$
$179$	−9.64171	−0.720655	−0.360328	+	0.932826i	$0.617335\pi$
$180$	0	0
$181$	10.7004	0.795355	0.397678	−	0.917525i	$-0.369816\pi$
$181$	10.7004	0.795355	0.397678	+	0.917525i	$0.369816\pi$
$182$	53.4521	3.96213
$183$	0	0
$184$	0.0540029	0.00398115
$185$	3.09022	0.227197
$186$	0	0
$187$	0.762043	0.0557261
$188$	−0.636984	−0.0464568
$189$	0	0
$190$	−7.14888	−0.518634
$191$	−17.0137	−1.23107	−0.615536	−	0.788109i	$-0.711060\pi$
$191$	−17.0137	−1.23107	−0.615536	+	0.788109i	$0.711060\pi$
$192$	0	0
$193$	−0.184988	−0.0133157	−0.00665786	−	0.999978i	$-0.502119\pi$
$193$	−0.184988	−0.0133157	−0.00665786	+	0.999978i	$0.502119\pi$
$194$	−31.3537	−2.25106
$195$	0	0
$196$	15.1866	1.08476
$197$	27.5520	1.96300	0.981500	−	0.191461i	$-0.0613225\pi$
$197$	27.5520	1.96300	0.981500	+	0.191461i	$0.0613225\pi$
$198$	0	0
$199$	9.23085	0.654358	0.327179	−	0.944962i	$-0.393902\pi$
$199$	9.23085	0.654358	0.327179	+	0.944962i	$0.393902\pi$
$200$	−0.240832	−0.0170294
$201$	0	0
$202$	−30.1301	−2.11995
$203$	3.83374	0.269076
$204$	0	0
$205$	8.51691	0.594847
$206$	38.1090	2.65518
$207$	0	0
$208$	−28.3539	−1.96599
$209$	1.00235	0.0693343
$210$	0	0
$211$	−15.5991	−1.07388	−0.536941	−	0.843619i	$-0.680420\pi$
$211$	−15.5991	−1.07388	−0.536941	+	0.843619i	$0.680420\pi$
$212$	−24.4691	−1.68055
$213$	0	0
$214$	9.00344	0.615462
$215$	4.73500	0.322924
$216$	0	0
$217$	−20.1383	−1.36708
$218$	2.49123	0.168728
$219$	0	0
$220$	0.297950	0.0200878
$221$	25.9465	1.74535
$222$	0	0
$223$	7.52483	0.503900	0.251950	−	0.967740i	$-0.418928\pi$
$223$	7.52483	0.503900	0.251950	+	0.967740i	$0.418928\pi$
$224$	−30.5603	−2.04190
$225$	0	0
$226$	3.46321	0.230370
$227$	−11.5656	−0.767635	−0.383818	−	0.923409i	$-0.625391\pi$
$227$	−11.5656	−0.767635	−0.383818	+	0.923409i	$0.625391\pi$
$228$	0	0
$229$	−4.91669	−0.324904	−0.162452	−	0.986716i	$-0.551940\pi$
$229$	−4.91669	−0.324904	−0.162452	+	0.986716i	$0.551940\pi$
$230$	−1.46522	−0.0966139
$231$	0	0
$232$	−0.0540029	−0.00354546
$233$	4.86391	0.318645	0.159323	−	0.987227i	$-0.449069\pi$
$233$	4.86391	0.318645	0.159323	+	0.987227i	$0.449069\pi$
$234$	0	0
$235$	−0.237340	−0.0154824
$236$	−16.2761	−1.05948
$237$	0	0
$238$	28.3445	1.83730
$239$	−13.1237	−0.848899	−0.424450	−	0.905452i	$-0.639532\pi$
$239$	−13.1237	−0.848899	−0.424450	+	0.905452i	$0.639532\pi$
$240$	0	0
$241$	−7.35922	−0.474049	−0.237025	−	0.971504i	$-0.576172\pi$
$241$	−7.35922	−0.474049	−0.237025	+	0.971504i	$0.576172\pi$
$242$	21.8412	1.40401
$243$	0	0
$244$	22.3778	1.43259
$245$	5.65854	0.361511
$246$	0	0
$247$	34.1288	2.17157
$248$	0.283672	0.0180132
$249$	0	0
$250$	13.8605	0.876612
$251$	−7.85155	−0.495585	−0.247793	−	0.968813i	$-0.579705\pi$
$251$	−7.85155	−0.495585	−0.247793	+	0.968813i	$0.579705\pi$
$252$	0	0
$253$	0.205441	0.0129160
$254$	15.4975	0.972399
$255$	0	0
$256$	16.4246	1.02654
$257$	4.43790	0.276829	0.138414	−	0.990374i	$-0.455799\pi$
$257$	4.43790	0.276829	0.138414	+	0.990374i	$0.455799\pi$
$258$	0	0
$259$	−16.1162	−1.00141
$260$	10.1448	0.629154
$261$	0	0
$262$	−15.5050	−0.957902
$263$	23.1392	1.42682	0.713411	−	0.700746i	$-0.247149\pi$
$263$	23.1392	1.42682	0.713411	+	0.700746i	$0.247149\pi$
$264$	0	0
$265$	−9.11720	−0.560065
$266$	37.2831	2.28597
$267$	0	0
$268$	15.5451	0.949568
$269$	−19.1142	−1.16541	−0.582706	−	0.812683i	$-0.698006\pi$
$269$	−19.1142	−1.16541	−0.582706	+	0.812683i	$0.698006\pi$
$270$	0	0
$271$	6.26721	0.380706	0.190353	−	0.981716i	$-0.439037\pi$
$271$	6.26721	0.380706	0.190353	+	0.981716i	$0.439037\pi$
$272$	−15.0355	−0.911660
$273$	0	0
$274$	43.5983	2.63387
$275$	−0.916189	−0.0552483
$276$	0	0
$277$	−7.29983	−0.438604	−0.219302	−	0.975657i	$-0.570378\pi$
$277$	−7.29983	−0.438604	−0.219302	+	0.975657i	$0.570378\pi$
$278$	−4.19346	−0.251507
$279$	0	0
$280$	−0.152191	−0.00909517
$281$	−8.88233	−0.529875	−0.264938	−	0.964266i	$-0.585351\pi$
$281$	−8.88233	−0.529875	−0.264938	+	0.964266i	$0.585351\pi$
$282$	0	0
$283$	−7.70543	−0.458040	−0.229020	−	0.973422i	$-0.573552\pi$
$283$	−7.70543	−0.458040	−0.229020	+	0.973422i	$0.573552\pi$
$284$	−16.4607	−0.976761
$285$	0	0
$286$	−2.86437	−0.169374
$287$	−44.4177	−2.62189
$288$	0	0
$289$	−3.24108	−0.190652
$290$	1.46522	0.0860409
$291$	0	0
$292$	3.40366	0.199184
$293$	−0.218722	−0.0127779	−0.00638894	−	0.999980i	$-0.502034\pi$
$293$	−0.218722	−0.0127779	−0.00638894	+	0.999980i	$0.502034\pi$
$294$	0	0
$295$	−6.06448	−0.353088
$296$	0.227016	0.0131950
$297$	0	0
$298$	−5.79389	−0.335631
$299$	6.99499	0.404531
$300$	0	0
$301$	−24.6941	−1.42335
$302$	−29.4355	−1.69382
$303$	0	0
$304$	−19.7769	−1.13429
$305$	8.33796	0.477430
$306$	0	0
$307$	−29.1831	−1.66557	−0.832784	−	0.553598i	$-0.813255\pi$
$307$	−29.1831	−1.66557	−0.832784	+	0.553598i	$0.813255\pi$
$308$	−1.55388	−0.0885404
$309$	0	0
$310$	−7.69668	−0.437142
$311$	18.9236	1.07306	0.536528	−	0.843882i	$-0.319735\pi$
$311$	18.9236	1.07306	0.536528	+	0.843882i	$0.319735\pi$
$312$	0	0
$313$	6.13022	0.346500	0.173250	−	0.984878i	$-0.444573\pi$
$313$	6.13022	0.346500	0.173250	+	0.984878i	$0.444573\pi$
$314$	7.41262	0.418319
$315$	0	0
$316$	−4.48486	−0.252293
$317$	16.3437	0.917953	0.458976	−	0.888448i	$-0.348216\pi$
$317$	16.3437	0.917953	0.458976	+	0.888448i	$0.348216\pi$
$318$	0	0
$319$	−0.205441	−0.0115025
$320$	−5.72045	−0.319783
$321$	0	0
$322$	7.64148	0.425843
$323$	18.0978	1.00699
$324$	0	0
$325$	−31.1950	−1.73039
$326$	−44.5320	−2.46640
$327$	0	0
$328$	0.625676	0.0345472
$329$	1.23778	0.0682412
$330$	0	0
$331$	−15.5579	−0.855139	−0.427569	−	0.903983i	$-0.640630\pi$
$331$	−15.5579	−0.855139	−0.427569	+	0.903983i	$0.640630\pi$
$332$	−4.56602	−0.250593
$333$	0	0
$334$	−21.1708	−1.15841
$335$	5.79211	0.316457
$336$	0	0
$337$	29.7973	1.62316	0.811580	−	0.584241i	$-0.198608\pi$
$337$	29.7973	1.62316	0.811580	+	0.584241i	$0.198608\pi$
$338$	−71.6161	−3.89540
$339$	0	0
$340$	5.37958	0.291749
$341$	1.07916	0.0584399
$342$	0	0
$343$	−2.67440	−0.144404
$344$	0.347847	0.0187546
$345$	0	0
$346$	5.84875	0.314431
$347$	22.5165	1.20875	0.604374	−	0.796701i	$-0.293423\pi$
$347$	22.5165	1.20875	0.604374	+	0.796701i	$0.293423\pi$
$348$	0	0
$349$	1.53805	0.0823298	0.0411649	−	0.999152i	$-0.486893\pi$
$349$	1.53805	0.0823298	0.0411649	+	0.999152i	$0.486893\pi$
$350$	−34.0781	−1.82155
$351$	0	0
$352$	1.63765	0.0872872
$353$	−15.1412	−0.805883	−0.402942	−	0.915226i	$-0.632012\pi$
$353$	−15.1412	−0.805883	−0.402942	+	0.915226i	$0.632012\pi$
$354$	0	0
$355$	−6.13325	−0.325519
$356$	−17.7701	−0.941815
$357$	0	0
$358$	19.2180	1.01570
$359$	7.91870	0.417933	0.208966	−	0.977923i	$-0.432990\pi$
$359$	7.91870	0.417933	0.208966	+	0.977923i	$0.432990\pi$
$360$	0	0
$361$	4.80501	0.252895
$362$	−21.3282	−1.12099
$363$	0	0
$364$	−52.9075	−2.77310
$365$	1.26820	0.0663808
$366$	0	0
$367$	−32.7861	−1.71142	−0.855709	−	0.517457i	$-0.826879\pi$
$367$	−32.7861	−1.71142	−0.855709	+	0.517457i	$0.826879\pi$
$368$	−4.05345	−0.211301
$369$	0	0
$370$	−6.15946	−0.320215
$371$	47.5483	2.46858
$372$	0	0
$373$	19.5365	1.01156	0.505780	−	0.862662i	$-0.331205\pi$
$373$	19.5365	1.01156	0.505780	+	0.862662i	$0.331205\pi$
$374$	−1.51892	−0.0785412
$375$	0	0
$376$	−0.0174357	−0.000899177 0
$377$	−6.99499	−0.360260
$378$	0	0
$379$	−18.2511	−0.937494	−0.468747	−	0.883332i	$-0.655294\pi$
$379$	−18.2511	−0.937494	−0.468747	+	0.883332i	$0.655294\pi$
$380$	7.07604	0.362993
$381$	0	0
$382$	33.9121	1.73509
$383$	15.1886	0.776103	0.388052	−	0.921638i	$-0.373148\pi$
$383$	15.1886	0.776103	0.388052	+	0.921638i	$0.373148\pi$
$384$	0	0
$385$	−0.578975	−0.0295073
$386$	0.368721	0.0187674
$387$	0	0
$388$	31.0342	1.57552
$389$	−24.2781	−1.23095	−0.615474	−	0.788158i	$-0.711035\pi$
$389$	−24.2781	−1.23095	−0.615474	+	0.788158i	$0.711035\pi$
$390$	0	0
$391$	3.70930	0.187587
$392$	0.415693	0.0209956
$393$	0	0
$394$	−54.9171	−2.76668
$395$	−1.67106	−0.0840801
$396$	0	0
$397$	25.3709	1.27333	0.636665	−	0.771141i	$-0.280314\pi$
$397$	25.3709	1.27333	0.636665	+	0.771141i	$0.280314\pi$
$398$	−18.3991	−0.922262
$399$	0	0
$400$	18.0769	0.903843
$401$	−16.2577	−0.811873	−0.405936	−	0.913901i	$-0.633055\pi$
$401$	−16.2577	−0.811873	−0.405936	+	0.913901i	$0.633055\pi$
$402$	0	0
$403$	36.7440	1.83035
$404$	29.8231	1.48376
$405$	0	0
$406$	−7.64148	−0.379240
$407$	0.863627	0.0428084
$408$	0	0
$409$	12.6830	0.627136	0.313568	−	0.949566i	$-0.398476\pi$
$409$	12.6830	0.627136	0.313568	+	0.949566i	$0.398476\pi$
$410$	−16.9760	−0.838387
$411$	0	0
$412$	−37.7207	−1.85837
$413$	31.6277	1.55630
$414$	0	0
$415$	−1.70130	−0.0835136
$416$	55.7599	2.73385
$417$	0	0
$418$	−1.99791	−0.0977209
$419$	−1.53634	−0.0750553	−0.0375277	−	0.999296i	$-0.511948\pi$
$419$	−1.53634	−0.0750553	−0.0375277	+	0.999296i	$0.511948\pi$
$420$	0	0
$421$	24.8697	1.21207	0.606037	−	0.795437i	$-0.292758\pi$
$421$	24.8697	1.21207	0.606037	+	0.795437i	$0.292758\pi$
$422$	31.0923	1.51355
$423$	0	0
$424$	−0.669775	−0.0325272
$425$	−16.5421	−0.802409
$426$	0	0
$427$	−43.4844	−2.10436
$428$	−8.91170	−0.430763
$429$	0	0
$430$	−9.43788	−0.455135
$431$	34.5459	1.66402	0.832008	−	0.554763i	$-0.187191\pi$
$431$	34.5459	1.66402	0.832008	+	0.554763i	$0.187191\pi$
$432$	0	0
$433$	39.9295	1.91889	0.959444	−	0.281898i	$-0.0909641\pi$
$433$	39.9295	1.91889	0.959444	+	0.281898i	$0.0909641\pi$
$434$	40.1400	1.92678
$435$	0	0
$436$	−2.46585	−0.118093
$437$	4.87904	0.233396
$438$	0	0
$439$	−3.95513	−0.188768	−0.0943841	−	0.995536i	$-0.530088\pi$
$439$	−3.95513	−0.188768	−0.0943841	+	0.995536i	$0.530088\pi$
$440$	0.00815556	0.000388801 0
$441$	0	0
$442$	−51.7170	−2.45993
$443$	2.46202	0.116974	0.0584871	−	0.998288i	$-0.481372\pi$
$443$	2.46202	0.116974	0.0584871	+	0.998288i	$0.481372\pi$
$444$	0	0
$445$	−6.62116	−0.313873
$446$	−14.9986	−0.710204
$447$	0	0
$448$	29.8334	1.40950
$449$	−19.4807	−0.919349	−0.459675	−	0.888087i	$-0.652034\pi$
$449$	−19.4807	−0.919349	−0.459675	+	0.888087i	$0.652034\pi$
$450$	0	0
$451$	2.38023	0.112081
$452$	−3.42793	−0.161236
$453$	0	0
$454$	23.0527	1.08192
$455$	−19.7133	−0.924175
$456$	0	0
$457$	3.32011	0.155308	0.0776541	−	0.996980i	$-0.475257\pi$
$457$	3.32011	0.155308	0.0776541	+	0.996980i	$0.475257\pi$
$458$	9.80002	0.457925
$459$	0	0
$460$	1.45029	0.0676203
$461$	16.9011	0.787162	0.393581	−	0.919290i	$-0.371236\pi$
$461$	16.9011	0.787162	0.393581	+	0.919290i	$0.371236\pi$
$462$	0	0
$463$	−30.3580	−1.41086	−0.705429	−	0.708781i	$-0.749246\pi$
$463$	−30.3580	−1.41086	−0.705429	+	0.708781i	$0.749246\pi$
$464$	4.05345	0.188177
$465$	0	0
$466$	−9.69482	−0.449104
$467$	−7.99095	−0.369777	−0.184888	−	0.982760i	$-0.559192\pi$
$467$	−7.99095	−0.369777	−0.184888	+	0.982760i	$0.559192\pi$
$468$	0	0
$469$	−30.2072	−1.39484
$470$	0.473070	0.0218211
$471$	0	0
$472$	−0.445514	−0.0205064
$473$	1.32330	0.0608453
$474$	0	0
$475$	−21.7587	−0.998356
$476$	−28.0557	−1.28593
$477$	0	0
$478$	26.1583	1.19645
$479$	−7.02146	−0.320819	−0.160409	−	0.987051i	$-0.551281\pi$
$479$	−7.02146	−0.320819	−0.160409	+	0.987051i	$0.551281\pi$
$480$	0	0
$481$	29.4054	1.34077
$482$	14.6685	0.668132
$483$	0	0
$484$	−21.6187	−0.982668
$485$	11.5634	0.525065
$486$	0	0
$487$	30.5212	1.38305	0.691524	−	0.722353i	$-0.256940\pi$
$487$	30.5212	1.38305	0.691524	+	0.722353i	$0.256940\pi$
$488$	0.612530	0.0277279
$489$	0	0
$490$	−11.2787	−0.509519
$491$	38.6997	1.74649	0.873245	−	0.487281i	$-0.162011\pi$
$491$	38.6997	1.74649	0.873245	+	0.487281i	$0.162011\pi$
$492$	0	0
$493$	−3.70930	−0.167059
$494$	−68.0261	−3.06064
$495$	0	0
$496$	−21.2924	−0.956057
$497$	31.9863	1.43478
$498$	0	0
$499$	−4.85466	−0.217325	−0.108662	−	0.994079i	$-0.534657\pi$
$499$	−4.85466	−0.217325	−0.108662	+	0.994079i	$0.534657\pi$
$500$	−13.7192	−0.613543
$501$	0	0
$502$	15.6498	0.698486
$503$	22.5078	1.00357	0.501786	−	0.864992i	$-0.332676\pi$
$503$	22.5078	1.00357	0.501786	+	0.864992i	$0.332676\pi$
$504$	0	0
$505$	11.1121	0.494482
$506$	−0.409488	−0.0182040
$507$	0	0
$508$	−15.3396	−0.680584
$509$	39.7724	1.76288	0.881441	−	0.472294i	$-0.156574\pi$
$509$	39.7724	1.76288	0.881441	+	0.472294i	$0.156574\pi$
$510$	0	0
$511$	−6.61398	−0.292585
$512$	−31.8739	−1.40864
$513$	0	0
$514$	−8.84570	−0.390167
$515$	−14.0548	−0.619326
$516$	0	0
$517$	−0.0663298	−0.00291718
$518$	32.1230	1.41141
$519$	0	0
$520$	0.277686	0.0121773
$521$	18.9327	0.829458	0.414729	−	0.909945i	$-0.363876\pi$
$521$	18.9327	0.829458	0.414729	+	0.909945i	$0.363876\pi$
$522$	0	0
$523$	−3.97995	−0.174031	−0.0870155	−	0.996207i	$-0.527733\pi$
$523$	−3.97995	−0.174031	−0.0870155	+	0.996207i	$0.527733\pi$
$524$	15.3470	0.670438
$525$	0	0
$526$	−46.1213	−2.01099
$527$	19.4846	0.848763
$528$	0	0
$529$	1.00000	0.0434783
$530$	18.1725	0.789364
$531$	0	0
$532$	−36.9032	−1.59996
$533$	81.0438	3.51040
$534$	0	0
$535$	−3.32050	−0.143558
$536$	0.425505	0.0183790
$537$	0	0
$538$	38.0987	1.64255
$539$	1.58140	0.0681158
$540$	0	0
$541$	13.1909	0.567123	0.283561	−	0.958954i	$-0.408484\pi$
$541$	13.1909	0.567123	0.283561	+	0.958954i	$0.408484\pi$
$542$	−12.4919	−0.536573
$543$	0	0
$544$	29.5683	1.26773
$545$	−0.918777	−0.0393561
$546$	0	0
$547$	−27.0821	−1.15795	−0.578974	−	0.815346i	$-0.696547\pi$
$547$	−27.0821	−1.15795	−0.578974	+	0.815346i	$0.696547\pi$
$548$	−43.1541	−1.84345
$549$	0	0
$550$	1.82616	0.0778678
$551$	−4.87904	−0.207854
$552$	0	0
$553$	8.71496	0.370598
$554$	14.5501	0.618175
$555$	0	0
$556$	4.15074	0.176030
$557$	−16.5268	−0.700263	−0.350131	−	0.936701i	$-0.613863\pi$
$557$	−16.5268	−0.700263	−0.350131	+	0.936701i	$0.613863\pi$
$558$	0	0
$559$	45.0565	1.90569
$560$	11.4235	0.482729
$561$	0	0
$562$	17.7044	0.746815
$563$	19.3926	0.817302	0.408651	−	0.912691i	$-0.365999\pi$
$563$	19.3926	0.817302	0.408651	+	0.912691i	$0.365999\pi$
$564$	0	0
$565$	−1.27725	−0.0537342
$566$	15.3586	0.645569
$567$	0	0
$568$	−0.450566	−0.0189053
$569$	−37.0098	−1.55153	−0.775766	−	0.631020i	$-0.782636\pi$
$569$	−37.0098	−1.55153	−0.775766	+	0.631020i	$0.782636\pi$
$570$	0	0
$571$	−23.9624	−1.00279	−0.501397	−	0.865217i	$-0.667180\pi$
$571$	−23.9624	−1.00279	−0.501397	+	0.865217i	$0.667180\pi$
$572$	2.83518	0.118545
$573$	0	0
$574$	88.5339	3.69534
$575$	−4.45962	−0.185979
$576$	0	0
$577$	8.93076	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$577$	8.93076	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$578$	6.46017	0.268707
$579$	0	0
$580$	−1.45029	−0.0602202
$581$	8.87268	0.368101
$582$	0	0
$583$	−2.54800	−0.105527
$584$	0.0931659	0.00385523
$585$	0	0
$586$	0.435960	0.0180094
$587$	−42.7864	−1.76599	−0.882993	−	0.469387i	$-0.844475\pi$
$587$	−42.7864	−1.76599	−0.882993	+	0.469387i	$0.844475\pi$
$588$	0	0
$589$	25.6291	1.05603
$590$	12.0878	0.497648
$591$	0	0
$592$	−17.0398	−0.700331
$593$	27.1472	1.11480	0.557401	−	0.830244i	$-0.311799\pi$
$593$	27.1472	1.11480	0.557401	+	0.830244i	$0.311799\pi$
$594$	0	0
$595$	−10.4536	−0.428555
$596$	5.73486	0.234909
$597$	0	0
$598$	−13.9425	−0.570152
$599$	−29.0030	−1.18503	−0.592514	−	0.805560i	$-0.701865\pi$
$599$	−29.0030	−1.18503	−0.592514	+	0.805560i	$0.701865\pi$
$600$	0	0
$601$	18.5526	0.756778	0.378389	−	0.925647i	$-0.376478\pi$
$601$	18.5526	0.756778	0.378389	+	0.925647i	$0.376478\pi$
$602$	49.2207	2.00609
$603$	0	0
$604$	29.1356	1.18551
$605$	−8.05513	−0.327488
$606$	0	0
$607$	−13.8252	−0.561149	−0.280575	−	0.959832i	$-0.590525\pi$
$607$	−13.8252	−0.561149	−0.280575	+	0.959832i	$0.590525\pi$
$608$	38.8928	1.57731
$609$	0	0
$610$	−16.6193	−0.672898
$611$	−2.25844	−0.0913668
$612$	0	0
$613$	25.3288	1.02302	0.511510	−	0.859277i	$-0.329086\pi$
$613$	25.3288	1.02302	0.511510	+	0.859277i	$0.329086\pi$
$614$	58.1682	2.34748
$615$	0	0
$616$	−0.0425331	−0.00171371
$617$	37.9401	1.52741	0.763705	−	0.645565i	$-0.223378\pi$
$617$	37.9401	1.52741	0.763705	+	0.645565i	$0.223378\pi$
$618$	0	0
$619$	20.1030	0.808006	0.404003	−	0.914758i	$-0.367619\pi$
$619$	20.1030	0.808006	0.404003	+	0.914758i	$0.367619\pi$
$620$	7.61826	0.305957
$621$	0	0
$622$	−37.7187	−1.51238
$623$	34.5309	1.38345
$624$	0	0
$625$	17.1863	0.687453
$626$	−12.2188	−0.488363
$627$	0	0
$628$	−7.33710	−0.292782
$629$	15.5931	0.621736
$630$	0	0
$631$	−29.8881	−1.18983	−0.594913	−	0.803790i	$-0.702814\pi$
$631$	−29.8881	−1.18983	−0.594913	+	0.803790i	$0.702814\pi$
$632$	−0.122761	−0.00488316
$633$	0	0
$634$	−32.5765	−1.29378
$635$	−5.71553	−0.226814
$636$	0	0
$637$	53.8446	2.13340
$638$	0.409488	0.0162118
$639$	0	0
$640$	−0.317553	−0.0125524
$641$	29.4363	1.16267	0.581333	−	0.813666i	$-0.302532\pi$
$641$	29.4363	1.16267	0.581333	+	0.813666i	$0.302532\pi$
$642$	0	0
$643$	−13.7604	−0.542659	−0.271329	−	0.962487i	$-0.587463\pi$
$643$	−13.7604	−0.542659	−0.271329	+	0.962487i	$0.587463\pi$
$644$	−7.56362	−0.298048
$645$	0	0
$646$	−36.0729	−1.41927
$647$	16.6878	0.656067	0.328034	−	0.944666i	$-0.393614\pi$
$647$	16.6878	0.656067	0.328034	+	0.944666i	$0.393614\pi$
$648$	0	0
$649$	−1.69485	−0.0665287
$650$	62.1784	2.43884
$651$	0	0
$652$	44.0783	1.72624
$653$	−6.84989	−0.268057	−0.134028	−	0.990977i	$-0.542791\pi$
$653$	−6.84989	−0.268057	−0.134028	+	0.990977i	$0.542791\pi$
$654$	0	0
$655$	5.71830	0.223433
$656$	−46.9632	−1.83360
$657$	0	0
$658$	−2.46717	−0.0961803
$659$	38.8758	1.51439	0.757193	−	0.653191i	$-0.226570\pi$
$659$	38.8758	1.51439	0.757193	+	0.653191i	$0.226570\pi$
$660$	0	0
$661$	−20.4123	−0.793947	−0.396974	−	0.917830i	$-0.629940\pi$
$661$	−20.4123	−0.793947	−0.396974	+	0.917830i	$0.629940\pi$
$662$	31.0102	1.20525
$663$	0	0
$664$	−0.124982	−0.00485026
$665$	−13.7501	−0.533207
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	20.9551	0.810775
$669$	0	0
$670$	−11.5449	−0.446019
$671$	2.33022	0.0899572
$672$	0	0
$673$	20.0139	0.771479	0.385739	−	0.922608i	$-0.373946\pi$
$673$	20.0139	0.771479	0.385739	+	0.922608i	$0.373946\pi$
$674$	−59.3924	−2.28771
$675$	0	0
$676$	70.8864	2.72640
$677$	5.46125	0.209893	0.104946	−	0.994478i	$-0.466533\pi$
$677$	5.46125	0.209893	0.104946	+	0.994478i	$0.466533\pi$
$678$	0	0
$679$	−60.3056	−2.31432
$680$	0.147251	0.00564683
$681$	0	0
$682$	−2.15100	−0.0823661
$683$	−26.5878	−1.01735	−0.508677	−	0.860957i	$-0.669865\pi$
$683$	−26.5878	−1.01735	−0.508677	+	0.860957i	$0.669865\pi$
$684$	0	0
$685$	−16.0792	−0.614355
$686$	5.33066	0.203525
$687$	0	0
$688$	−26.1093	−0.995409
$689$	−86.7559	−3.30514
$690$	0	0
$691$	41.3723	1.57388	0.786939	−	0.617031i	$-0.211665\pi$
$691$	41.3723	1.57388	0.786939	+	0.617031i	$0.211665\pi$
$692$	−5.78916	−0.220071
$693$	0	0
$694$	−44.8802	−1.70363
$695$	1.54656	0.0586645
$696$	0	0
$697$	42.9759	1.62783
$698$	−3.06566	−0.116037
$699$	0	0
$700$	33.7309	1.27491
$701$	47.6825	1.80094	0.900472	−	0.434915i	$-0.143221\pi$
$701$	47.6825	1.80094	0.900472	+	0.434915i	$0.143221\pi$
$702$	0	0
$703$	20.5104	0.773563
$704$	−1.59870	−0.0602533
$705$	0	0
$706$	30.1796	1.13582
$707$	−57.9522	−2.17952
$708$	0	0
$709$	−29.7799	−1.11841	−0.559203	−	0.829030i	$-0.688893\pi$
$709$	−29.7799	−1.11841	−0.559203	+	0.829030i	$0.688893\pi$
$710$	12.2249	0.458792
$711$	0	0
$712$	−0.486409	−0.0182289
$713$	5.25291	0.196723
$714$	0	0
$715$	1.05639	0.0395067
$716$	−19.0222	−0.710893
$717$	0	0
$718$	−15.7837	−0.589041
$719$	−36.0592	−1.34478	−0.672390	−	0.740197i	$-0.734732\pi$
$719$	−36.0592	−1.34478	−0.672390	+	0.740197i	$0.734732\pi$
$720$	0	0
$721$	73.2988	2.72979
$722$	−9.57742	−0.356435
$723$	0	0
$724$	21.1109	0.784581
$725$	4.45962	0.165626
$726$	0	0
$727$	−36.1159	−1.33947	−0.669733	−	0.742602i	$-0.733591\pi$
$727$	−36.1159	−1.33947	−0.669733	+	0.742602i	$0.733591\pi$
$728$	−1.44820	−0.0536737
$729$	0	0
$730$	−2.52780	−0.0935582
$731$	23.8926	0.883699
$732$	0	0
$733$	33.7470	1.24647	0.623237	−	0.782033i	$-0.285817\pi$
$733$	33.7470	1.24647	0.623237	+	0.782033i	$0.285817\pi$
$734$	65.3497	2.41210
$735$	0	0
$736$	7.97140	0.293830
$737$	1.61873	0.0596267
$738$	0	0
$739$	−15.8229	−0.582055	−0.291028	−	0.956715i	$-0.593997\pi$
$739$	−15.8229	−0.582055	−0.291028	+	0.956715i	$0.593997\pi$
$740$	6.09671	0.224119
$741$	0	0
$742$	−94.7740	−3.47926
$743$	32.1252	1.17856	0.589279	−	0.807930i	$-0.299412\pi$
$743$	32.1252	1.17856	0.589279	+	0.807930i	$0.299412\pi$
$744$	0	0
$745$	2.13681	0.0782866
$746$	−38.9404	−1.42571
$747$	0	0
$748$	1.50344	0.0549712
$749$	17.3172	0.632756
$750$	0	0
$751$	9.88696	0.360780	0.180390	−	0.983595i	$-0.442264\pi$
$751$	9.88696	0.360780	0.180390	+	0.983595i	$0.442264\pi$
$752$	1.30872	0.0477241
$753$	0	0
$754$	13.9425	0.507757
$755$	10.8559	0.395088
$756$	0	0
$757$	52.9982	1.92625	0.963126	−	0.269050i	$-0.0867095\pi$
$757$	52.9982	1.92625	0.963126	+	0.269050i	$0.0867095\pi$
$758$	36.3783	1.32132
$759$	0	0
$760$	0.193687	0.00702577
$761$	−35.1393	−1.27380	−0.636900	−	0.770946i	$-0.719784\pi$
$761$	−35.1393	−1.27380	−0.636900	+	0.770946i	$0.719784\pi$
$762$	0	0
$763$	4.79163	0.173469
$764$	−33.5665	−1.21439
$765$	0	0
$766$	−30.2742	−1.09385
$767$	−57.7074	−2.08369
$768$	0	0
$769$	−22.2896	−0.803784	−0.401892	−	0.915687i	$-0.631647\pi$
$769$	−22.2896	−0.803784	−0.401892	+	0.915687i	$0.631647\pi$
$770$	1.15402	0.0415881
$771$	0	0
$772$	−0.364964	−0.0131353
$773$	20.5040	0.737479	0.368740	−	0.929533i	$-0.379789\pi$
$773$	20.5040	0.737479	0.368740	+	0.929533i	$0.379789\pi$
$774$	0	0
$775$	−23.4260	−0.841485
$776$	0.849477	0.0304944
$777$	0	0
$778$	48.3914	1.73492
$779$	56.5284	2.02534
$780$	0	0
$781$	−1.71407	−0.0613342
$782$	−7.39344	−0.264389
$783$	0	0
$784$	−31.2018	−1.11435
$785$	−2.73380	−0.0975736
$786$	0	0
$787$	20.2891	0.723227	0.361613	−	0.932328i	$-0.382226\pi$
$787$	20.2891	0.723227	0.361613	+	0.932328i	$0.382226\pi$
$788$	54.3576	1.93641
$789$	0	0
$790$	3.33078	0.118504
$791$	6.66114	0.236843
$792$	0	0
$793$	79.3410	2.81748
$794$	−50.5697	−1.79465
$795$	0	0
$796$	18.2116	0.645493
$797$	−33.8001	−1.19726	−0.598630	−	0.801025i	$-0.704288\pi$
$797$	−33.8001	−1.19726	−0.598630	+	0.801025i	$0.704288\pi$
$798$	0	0
$799$	−1.19761	−0.0423683
$800$	−35.5494	−1.25686
$801$	0	0
$802$	32.4052	1.14427
$803$	0.354427	0.0125075
$804$	0	0
$805$	−2.81821	−0.0993287
$806$	−73.2388	−2.57973
$807$	0	0
$808$	0.816327	0.0287183
$809$	15.1078	0.531163	0.265582	−	0.964088i	$-0.414436\pi$
$809$	15.1078	0.531163	0.265582	+	0.964088i	$0.414436\pi$
$810$	0	0
$811$	−11.9409	−0.419300	−0.209650	−	0.977776i	$-0.567232\pi$
$811$	−11.9409	−0.419300	−0.209650	+	0.977776i	$0.567232\pi$
$812$	7.56362	0.265431
$813$	0	0
$814$	−1.72140	−0.0603349
$815$	16.4236	0.575292
$816$	0	0
$817$	31.4271	1.09950
$818$	−25.2800	−0.883896
$819$	0	0
$820$	16.8031	0.586789
$821$	−33.0975	−1.15511	−0.577555	−	0.816352i	$-0.695993\pi$
$821$	−33.0975	−1.15511	−0.577555	+	0.816352i	$0.695993\pi$
$822$	0	0
$823$	55.7692	1.94399	0.971997	−	0.234994i	$-0.0755071\pi$
$823$	55.7692	1.94399	0.971997	+	0.234994i	$0.0755071\pi$
$824$	−1.03250	−0.0359689
$825$	0	0
$826$	−63.0408	−2.19347
$827$	26.6159	0.925524	0.462762	−	0.886483i	$-0.346858\pi$
$827$	26.6159	0.925524	0.462762	+	0.886483i	$0.346858\pi$
$828$	0	0
$829$	18.3734	0.638133	0.319066	−	0.947732i	$-0.396631\pi$
$829$	18.3734	0.638133	0.319066	+	0.947732i	$0.396631\pi$
$830$	3.39106	0.117705
$831$	0	0
$832$	−54.4337	−1.88715
$833$	28.5527	0.989293
$834$	0	0
$835$	7.80786	0.270202
$836$	1.97755	0.0683951
$837$	0	0
$838$	3.06226	0.105784
$839$	−7.56762	−0.261263	−0.130632	−	0.991431i	$-0.541701\pi$
$839$	−7.56762	−0.261263	−0.130632	+	0.991431i	$0.541701\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−49.5706	−1.70832
$843$	0	0
$844$	−30.7755	−1.05934
$845$	26.4123	0.908610
$846$	0	0
$847$	42.0094	1.44346
$848$	50.2732	1.72639
$849$	0	0
$850$	32.9719	1.13093
$851$	4.20377	0.144103
$852$	0	0
$853$	−0.153744	−0.00526409	−0.00263205	−	0.999997i	$-0.500838\pi$
$853$	−0.153744	−0.00526409	−0.00263205	+	0.999997i	$0.500838\pi$
$854$	86.6737	2.96591
$855$	0	0
$856$	−0.243933	−0.00833747
$857$	−2.17376	−0.0742544	−0.0371272	−	0.999311i	$-0.511821\pi$
$857$	−2.17376	−0.0742544	−0.0371272	+	0.999311i	$0.511821\pi$
$858$	0	0
$859$	−3.80983	−0.129990	−0.0649948	−	0.997886i	$-0.520703\pi$
$859$	−3.80983	−0.129990	−0.0649948	+	0.997886i	$0.520703\pi$
$860$	9.34172	0.318550
$861$	0	0
$862$	−68.8574	−2.34529
$863$	12.2177	0.415897	0.207948	−	0.978140i	$-0.433321\pi$
$863$	12.2177	0.415897	0.207948	+	0.978140i	$0.433321\pi$
$864$	0	0
$865$	−2.15704	−0.0733416
$866$	−79.5881	−2.70451
$867$	0	0
$868$	−39.7310	−1.34856
$869$	−0.467013	−0.0158423
$870$	0	0
$871$	55.1156	1.86752
$872$	−0.0674959	−0.00228570
$873$	0	0
$874$	−9.72497	−0.328952
$875$	26.6592	0.901244
$876$	0	0
$877$	11.6753	0.394248	0.197124	−	0.980379i	$-0.436840\pi$
$877$	11.6753	0.394248	0.197124	+	0.980379i	$0.436840\pi$
$878$	7.88343	0.266053
$879$	0	0
$880$	−0.612156	−0.0206358
$881$	6.52745	0.219915	0.109958	−	0.993936i	$-0.464928\pi$
$881$	6.52745	0.219915	0.109958	+	0.993936i	$0.464928\pi$
$882$	0	0
$883$	0.949789	0.0319629	0.0159815	−	0.999872i	$-0.494913\pi$
$883$	0.949789	0.0319629	0.0159815	+	0.999872i	$0.494913\pi$
$884$	51.1901	1.72171
$885$	0	0
$886$	−4.90734	−0.164865
$887$	−26.4928	−0.889540	−0.444770	−	0.895645i	$-0.646715\pi$
$887$	−26.4928	−0.889540	−0.444770	+	0.895645i	$0.646715\pi$
$888$	0	0
$889$	29.8078	0.999722
$890$	13.1974	0.442377
$891$	0	0
$892$	14.8458	0.497073
$893$	−1.57527	−0.0527145
$894$	0	0
$895$	−7.08767	−0.236915
$896$	1.65611	0.0553269
$897$	0	0
$898$	38.8292	1.29575
$899$	−5.25291	−0.175194
$900$	0	0
$901$	−46.0049	−1.53264
$902$	−4.74432	−0.157969
$903$	0	0
$904$	−0.0938301	−0.00312074
$905$	7.86593	0.261472
$906$	0	0
$907$	7.79751	0.258912	0.129456	−	0.991585i	$-0.458677\pi$
$907$	7.79751	0.258912	0.129456	+	0.991585i	$0.458677\pi$
$908$	−22.8178	−0.757236
$909$	0	0
$910$	39.2929	1.30255
$911$	14.4227	0.477845	0.238923	−	0.971039i	$-0.423206\pi$
$911$	14.4227	0.477845	0.238923	+	0.971039i	$0.423206\pi$
$912$	0	0
$913$	−0.475465	−0.0157356
$914$	−6.61770	−0.218894
$915$	0	0
$916$	−9.70017	−0.320503
$917$	−29.8223	−0.984818
$918$	0	0
$919$	1.48607	0.0490211	0.0245105	−	0.999700i	$-0.492197\pi$
$919$	1.48607	0.0490211	0.0245105	+	0.999700i	$0.492197\pi$
$920$	0.0396978	0.00130880
$921$	0	0
$922$	−33.6875	−1.10944
$923$	−58.3617	−1.92100
$924$	0	0
$925$	−18.7472	−0.616405
$926$	60.5101	1.98849
$927$	0	0
$928$	−7.97140	−0.261674
$929$	−35.0529	−1.15005	−0.575025	−	0.818136i	$-0.695008\pi$
$929$	−35.0529	−1.15005	−0.575025	+	0.818136i	$0.695008\pi$
$930$	0	0
$931$	37.5569	1.23088
$932$	9.59604	0.314329
$933$	0	0
$934$	15.9277	0.521169
$935$	0.560182	0.0183199
$936$	0	0
$937$	19.4594	0.635711	0.317855	−	0.948139i	$-0.397037\pi$
$937$	19.4594	0.635711	0.317855	+	0.948139i	$0.397037\pi$
$938$	60.2094	1.96591
$939$	0	0
$940$	−0.468250	−0.0152726
$941$	−8.80006	−0.286874	−0.143437	−	0.989659i	$-0.545815\pi$
$941$	−8.80006	−0.286874	−0.143437	+	0.989659i	$0.545815\pi$
$942$	0	0
$943$	11.5860	0.377291
$944$	33.4402	1.08839
$945$	0	0
$946$	−2.63762	−0.0857564
$947$	15.9958	0.519794	0.259897	−	0.965636i	$-0.416311\pi$
$947$	15.9958	0.519794	0.259897	+	0.965636i	$0.416311\pi$
$948$	0	0
$949$	12.0678	0.391736
$950$	43.3697	1.40710
$951$	0	0
$952$	−0.767949	−0.0248894
$953$	−27.3967	−0.887465	−0.443733	−	0.896159i	$-0.646346\pi$
$953$	−27.3967	−0.887465	−0.443733	+	0.896159i	$0.646346\pi$
$954$	0	0
$955$	−12.5069	−0.404714
$956$	−25.8918	−0.837400
$957$	0	0
$958$	13.9953	0.452167
$959$	83.8568	2.70788
$960$	0	0
$961$	−3.40699	−0.109903
$962$	−58.6112	−1.88970
$963$	0	0
$964$	−14.5191	−0.467627
$965$	−0.135986	−0.00437753
$966$	0	0
$967$	30.6368	0.985212	0.492606	−	0.870253i	$-0.336044\pi$
$967$	30.6368	0.985212	0.492606	+	0.870253i	$0.336044\pi$
$968$	−0.591753	−0.0190197
$969$	0	0
$970$	−23.0483	−0.740035
$971$	−3.20546	−0.102868	−0.0514340	−	0.998676i	$-0.516379\pi$
$971$	−3.20546	−0.102868	−0.0514340	+	0.998676i	$0.516379\pi$
$972$	0	0
$973$	−8.06569	−0.258574
$974$	−60.8353	−1.94929
$975$	0	0
$976$	−45.9764	−1.47167
$977$	39.8395	1.27458	0.637290	−	0.770624i	$-0.280055\pi$
$977$	39.8395	1.27458	0.637290	+	0.770624i	$0.280055\pi$
$978$	0	0
$979$	−1.85042	−0.0591398
$980$	11.1638	0.356614
$981$	0	0
$982$	−77.1367	−2.46153
$983$	15.3920	0.490928	0.245464	−	0.969406i	$-0.421060\pi$
$983$	15.3920	0.490928	0.245464	+	0.969406i	$0.421060\pi$
$984$	0	0
$985$	20.2536	0.645335
$986$	7.39344	0.235455
$987$	0	0
$988$	67.3330	2.14215
$989$	6.44126	0.204820
$990$	0	0
$991$	−36.0357	−1.14471	−0.572355	−	0.820006i	$-0.693970\pi$
$991$	−36.0357	−1.14471	−0.572355	+	0.820006i	$0.693970\pi$
$992$	41.8730	1.32947
$993$	0	0
$994$	−63.7556	−2.02220
$995$	6.78565	0.215119
$996$	0	0
$997$	−38.8316	−1.22981	−0.614904	−	0.788602i	$-0.710805\pi$
$997$	−38.8316	−1.22981	−0.614904	+	0.788602i	$0.710805\pi$
$998$	9.67639	0.306301
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.v.1.6	✓	30
3.2	odd	2		6003.2.a.w.1.25	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.v.1.6	✓	30	1.1	even	1	trivial
6003.2.a.w.1.25	yes	30	3.2	odd	2

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)

\(f(q)\)	\(=\)	\(q-1.99322 q^{2} +1.97291 q^{4} +0.735105 q^{5} -3.83374 q^{7} +0.0540029 q^{8} +O(q^{10})\)	\(q-1.99322 q^{2} +1.97291 q^{4} +0.735105 q^{5} -3.83374 q^{7} +0.0540029 q^{8} -1.46522 q^{10} +0.205441 q^{11} +6.99499 q^{13} +7.64148 q^{14} -4.05345 q^{16} +3.70930 q^{17} +4.87904 q^{19} +1.45029 q^{20} -0.409488 q^{22} +1.00000 q^{23} -4.45962 q^{25} -13.9425 q^{26} -7.56362 q^{28} -1.00000 q^{29} +5.25291 q^{31} +7.97140 q^{32} -7.39344 q^{34} -2.81821 q^{35} +4.20377 q^{37} -9.72497 q^{38} +0.0396978 q^{40} +11.5860 q^{41} +6.44126 q^{43} +0.405316 q^{44} -1.99322 q^{46} -0.322866 q^{47} +7.69760 q^{49} +8.88898 q^{50} +13.8005 q^{52} -12.4026 q^{53} +0.151021 q^{55} -0.207033 q^{56} +1.99322 q^{58} -8.24981 q^{59} +11.3425 q^{61} -10.4702 q^{62} -7.78180 q^{64} +5.14206 q^{65} +7.87929 q^{67} +7.31811 q^{68} +5.61729 q^{70} -8.34336 q^{71} +1.72520 q^{73} -8.37902 q^{74} +9.62589 q^{76} -0.787608 q^{77} -2.27322 q^{79} -2.97971 q^{80} -23.0933 q^{82} -2.31436 q^{83} +2.72673 q^{85} -12.8388 q^{86} +0.0110944 q^{88} -9.00709 q^{89} -26.8170 q^{91} +1.97291 q^{92} +0.643541 q^{94} +3.58661 q^{95} +15.7302 q^{97} -15.3430 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8}+O(q^{10}) \) 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8	\( 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8} + 8 q^{10} + 36 q^{13} - 7 q^{14} + 47 q^{16} - 18 q^{17} + 16 q^{19} + 25 q^{22} + 30 q^{23} + 56 q^{25} - 11 q^{26} + 27 q^{28} - 30 q^{29} + 14 q^{31} + 7 q^{32} + 3 q^{34} + 22 q^{35} + 40 q^{37} - 6 q^{38} + 30 q^{40} - 14 q^{41} + 34 q^{43} - 5 q^{44} - q^{46} + 2 q^{47} + 74 q^{49} + 21 q^{50} + 71 q^{52} - 16 q^{53} + 22 q^{55} - 14 q^{56} + q^{58} + 32 q^{59} + 46 q^{61} - 20 q^{62} + 68 q^{64} - 12 q^{65} + 14 q^{67} - 27 q^{68} + 32 q^{71} + 50 q^{73} + 26 q^{74} + 56 q^{76} - 34 q^{77} + 16 q^{79} - 2 q^{80} + 38 q^{82} + 14 q^{83} + 38 q^{85} - 10 q^{86} + 40 q^{88} + 2 q^{89} + 32 q^{91} + 37 q^{92} + 29 q^{94} + 28 q^{95} + 56 q^{97} - 8 q^{98}+O(q^{100}) \) 30 * q - q^2 + 37 * q^4 + 10 * q^7 - 6 * q^8 + 8 * q^10 + 36 * q^13 - 7 * q^14 + 47 * q^16 - 18 * q^17 + 16 * q^19 + 25 * q^22 + 30 * q^23 + 56 * q^25 - 11 * q^26 + 27 * q^28 - 30 * q^29 + 14 * q^31 + 7 * q^32 + 3 * q^34 + 22 * q^35 + 40 * q^37 - 6 * q^38 + 30 * q^40 - 14 * q^41 + 34 * q^43 - 5 * q^44 - q^46 + 2 * q^47 + 74 * q^49 + 21 * q^50 + 71 * q^52 - 16 * q^53 + 22 * q^55 - 14 * q^56 + q^58 + 32 * q^59 + 46 * q^61 - 20 * q^62 + 68 * q^64 - 12 * q^65 + 14 * q^67 - 27 * q^68 + 32 * q^71 + 50 * q^73 + 26 * q^74 + 56 * q^76 - 34 * q^77 + 16 * q^79 - 2 * q^80 + 38 * q^82 + 14 * q^83 + 38 * q^85 - 10 * q^86 + 40 * q^88 + 2 * q^89 + 32 * q^91 + 37 * q^92 + 29 * q^94 + 28 * q^95 + 56 * q^97 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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