LMFDB - Embedded newform 6003.2.a.w.1.16

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			1.16
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0795257 q^{2} -1.99368 q^{4} +3.50033 q^{5} +0.586662 q^{7} -0.317600 q^{8} +O(q^{10})$	\(q+0.0795257 q^{2} -1.99368 q^{4} +3.50033 q^{5} +0.586662 q^{7} -0.317600 q^{8} +0.278366 q^{10} -4.15057 q^{11} +7.11477 q^{13} +0.0466547 q^{14} +3.96209 q^{16} +2.23050 q^{17} +3.72959 q^{19} -6.97852 q^{20} -0.330077 q^{22} -1.00000 q^{23} +7.25228 q^{25} +0.565807 q^{26} -1.16961 q^{28} +1.00000 q^{29} -0.517145 q^{31} +0.950288 q^{32} +0.177382 q^{34} +2.05351 q^{35} +8.76104 q^{37} +0.296598 q^{38} -1.11170 q^{40} -0.620838 q^{41} -7.38027 q^{43} +8.27489 q^{44} -0.0795257 q^{46} +9.76679 q^{47} -6.65583 q^{49} +0.576743 q^{50} -14.1846 q^{52} -3.66930 q^{53} -14.5283 q^{55} -0.186324 q^{56} +0.0795257 q^{58} -5.72933 q^{59} +0.946352 q^{61} -0.0411263 q^{62} -7.84862 q^{64} +24.9040 q^{65} -10.6437 q^{67} -4.44689 q^{68} +0.163307 q^{70} -1.43246 q^{71} +13.3793 q^{73} +0.696728 q^{74} -7.43559 q^{76} -2.43498 q^{77} +0.300819 q^{79} +13.8686 q^{80} -0.0493726 q^{82} +8.47956 q^{83} +7.80747 q^{85} -0.586921 q^{86} +1.31822 q^{88} -14.4967 q^{89} +4.17397 q^{91} +1.99368 q^{92} +0.776710 q^{94} +13.0548 q^{95} +10.6515 q^{97} -0.529309 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.0795257	0.0562331	0.0281166	−	0.999605i	$-0.491049\pi$
$2$	0.0795257	0.0562331	0.0281166	+	0.999605i	$0.491049\pi$
$3$	0	0
$3$	0	0
$4$	−1.99368	−0.996838
$5$	3.50033	1.56539	0.782697	−	0.622403i	$-0.213844\pi$
$5$	3.50033	1.56539	0.782697	+	0.622403i	$0.213844\pi$
$6$	0	0
$7$	0.586662	0.221738	0.110869	−	0.993835i	$-0.464637\pi$
$7$	0.586662	0.221738	0.110869	+	0.993835i	$0.464637\pi$
$8$	−0.317600	−0.112288
$9$	0	0
$10$	0.278366	0.0880270
$11$	−4.15057	−1.25144	−0.625722	−	0.780046i	$-0.715196\pi$
$11$	−4.15057	−1.25144	−0.625722	+	0.780046i	$0.715196\pi$
$12$	0	0
$13$	7.11477	1.97328	0.986642	−	0.162905i	$-0.0520864\pi$
$13$	7.11477	1.97328	0.986642	+	0.162905i	$0.0520864\pi$
$14$	0.0466547	0.0124690
$15$	0	0
$16$	3.96209	0.990524
$17$	2.23050	0.540976	0.270488	−	0.962723i	$-0.412815\pi$
$17$	2.23050	0.540976	0.270488	+	0.962723i	$0.412815\pi$
$18$	0	0
$19$	3.72959	0.855627	0.427813	−	0.903867i	$-0.359284\pi$
$19$	3.72959	0.855627	0.427813	+	0.903867i	$0.359284\pi$
$20$	−6.97852	−1.56044
$21$	0	0
$22$	−0.330077	−0.0703726
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	7.25228	1.45046
$26$	0.565807	0.110964
$27$	0	0
$28$	−1.16961	−0.221036
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−0.517145	−0.0928820	−0.0464410	−	0.998921i	$-0.514788\pi$
$31$	−0.517145	−0.0928820	−0.0464410	+	0.998921i	$0.514788\pi$
$32$	0.950288	0.167989
$33$	0	0
$34$	0.177382	0.0304207
$35$	2.05351	0.347106
$36$	0	0
$37$	8.76104	1.44031	0.720153	−	0.693815i	$-0.244072\pi$
$37$	8.76104	1.44031	0.720153	+	0.693815i	$0.244072\pi$
$38$	0.296598	0.0481146
$39$	0	0
$40$	−1.11170	−0.175776
$41$	−0.620838	−0.0969586	−0.0484793	−	0.998824i	$-0.515437\pi$
$41$	−0.620838	−0.0969586	−0.0484793	+	0.998824i	$0.515437\pi$
$42$	0	0
$43$	−7.38027	−1.12548	−0.562740	−	0.826634i	$-0.690253\pi$
$43$	−7.38027	−1.12548	−0.562740	+	0.826634i	$0.690253\pi$
$44$	8.27489	1.24749
$45$	0	0
$46$	−0.0795257	−0.0117254
$47$	9.76679	1.42463	0.712316	−	0.701859i	$-0.247646\pi$
$47$	9.76679	1.42463	0.712316	+	0.701859i	$0.247646\pi$
$48$	0	0
$49$	−6.65583	−0.950832
$50$	0.576743	0.0815637
$51$	0	0
$52$	−14.1846	−1.96704
$53$	−3.66930	−0.504017	−0.252009	−	0.967725i	$-0.581091\pi$
$53$	−3.66930	−0.504017	−0.252009	+	0.967725i	$0.581091\pi$
$54$	0	0
$55$	−14.5283	−1.95900
$56$	−0.186324	−0.0248986
$57$	0	0
$58$	0.0795257	0.0104422
$59$	−5.72933	−0.745896	−0.372948	−	0.927852i	$-0.621653\pi$
$59$	−5.72933	−0.745896	−0.372948	+	0.927852i	$0.621653\pi$
$60$	0	0
$61$	0.946352	0.121168	0.0605840	−	0.998163i	$-0.480704\pi$
$61$	0.946352	0.121168	0.0605840	+	0.998163i	$0.480704\pi$
$62$	−0.0411263	−0.00522305
$63$	0	0
$64$	−7.84862	−0.981077
$65$	24.9040	3.08897
$66$	0	0
$67$	−10.6437	−1.30033	−0.650167	−	0.759791i	$-0.725301\pi$
$67$	−10.6437	−1.30033	−0.650167	+	0.759791i	$0.725301\pi$
$68$	−4.44689	−0.539265
$69$	0	0
$70$	0.163307	0.0195189
$71$	−1.43246	−0.170002	−0.0850010	−	0.996381i	$-0.527089\pi$
$71$	−1.43246	−0.170002	−0.0850010	+	0.996381i	$0.527089\pi$
$72$	0	0
$73$	13.3793	1.56593	0.782965	−	0.622066i	$-0.213706\pi$
$73$	13.3793	1.56593	0.782965	+	0.622066i	$0.213706\pi$
$74$	0.696728	0.0809929
$75$	0	0
$76$	−7.43559	−0.852921
$77$	−2.43498	−0.277492
$78$	0	0
$79$	0.300819	0.0338448	0.0169224	−	0.999857i	$-0.494613\pi$
$79$	0.300819	0.0338448	0.0169224	+	0.999857i	$0.494613\pi$
$80$	13.8686	1.55056
$81$	0	0
$82$	−0.0493726	−0.00545229
$83$	8.47956	0.930753	0.465376	−	0.885113i	$-0.345919\pi$
$83$	8.47956	0.930753	0.465376	+	0.885113i	$0.345919\pi$
$84$	0	0
$85$	7.80747	0.846840
$86$	−0.586921	−0.0632893
$87$	0	0
$88$	1.31822	0.140523
$89$	−14.4967	−1.53664	−0.768322	−	0.640063i	$-0.778908\pi$
$89$	−14.4967	−1.53664	−0.768322	+	0.640063i	$0.778908\pi$
$90$	0	0
$91$	4.17397	0.437551
$92$	1.99368	0.207855
$93$	0	0
$94$	0.776710	0.0801115
$95$	13.0548	1.33939
$96$	0	0
$97$	10.6515	1.08149	0.540747	−	0.841185i	$-0.318142\pi$
$97$	10.6515	1.08149	0.540747	+	0.841185i	$0.318142\pi$
$98$	−0.529309	−0.0534683
$99$	0	0
$100$	−14.4587	−1.44587
$101$	8.89300	0.884886	0.442443	−	0.896797i	$-0.354112\pi$
$101$	8.89300	0.884886	0.442443	+	0.896797i	$0.354112\pi$
$102$	0	0
$103$	0.719508	0.0708952	0.0354476	−	0.999372i	$-0.488714\pi$
$103$	0.719508	0.0708952	0.0354476	+	0.999372i	$0.488714\pi$
$104$	−2.25965	−0.221577
$105$	0	0
$106$	−0.291804	−0.0283425
$107$	−5.39797	−0.521841	−0.260921	−	0.965360i	$-0.584026\pi$
$107$	−5.39797	−0.521841	−0.260921	+	0.965360i	$0.584026\pi$
$108$	0	0
$109$	9.33218	0.893860	0.446930	−	0.894569i	$-0.352517\pi$
$109$	9.33218	0.893860	0.446930	+	0.894569i	$0.352517\pi$
$110$	−1.15538	−0.110161
$111$	0	0
$112$	2.32441	0.219636
$113$	−5.61852	−0.528546	−0.264273	−	0.964448i	$-0.585132\pi$
$113$	−5.61852	−0.528546	−0.264273	+	0.964448i	$0.585132\pi$
$114$	0	0
$115$	−3.50033	−0.326407
$116$	−1.99368	−0.185108
$117$	0	0
$118$	−0.455629	−0.0419440
$119$	1.30855	0.119955
$120$	0	0
$121$	6.22723	0.566112
$122$	0.0752593	0.00681365
$123$	0	0
$124$	1.03102	0.0925883
$125$	7.88373	0.705142
$126$	0	0
$127$	11.2559	0.998803	0.499401	−	0.866371i	$-0.333553\pi$
$127$	11.2559	0.998803	0.499401	+	0.866371i	$0.333553\pi$
$128$	−2.52474	−0.223158
$129$	0	0
$130$	1.98051	0.173702
$131$	14.9308	1.30451	0.652257	−	0.757998i	$-0.273822\pi$
$131$	14.9308	1.30451	0.652257	+	0.757998i	$0.273822\pi$
$132$	0	0
$133$	2.18801	0.189725
$134$	−0.846447	−0.0731218
$135$	0	0
$136$	−0.708406	−0.0607453
$137$	−6.30641	−0.538793	−0.269397	−	0.963029i	$-0.586824\pi$
$137$	−6.30641	−0.538793	−0.269397	+	0.963029i	$0.586824\pi$
$138$	0	0
$139$	18.0372	1.52989	0.764947	−	0.644093i	$-0.222765\pi$
$139$	18.0372	1.52989	0.764947	+	0.644093i	$0.222765\pi$
$140$	−4.09403	−0.346009
$141$	0	0
$142$	−0.113918	−0.00955975
$143$	−29.5304	−2.46945
$144$	0	0
$145$	3.50033	0.290686
$146$	1.06400	0.0880572
$147$	0	0
$148$	−17.4667	−1.43575
$149$	9.79681	0.802586	0.401293	−	0.915950i	$-0.368561\pi$
$149$	9.79681	0.802586	0.401293	+	0.915950i	$0.368561\pi$
$150$	0	0
$151$	−18.4750	−1.50348	−0.751738	−	0.659461i	$-0.770784\pi$
$151$	−18.4750	−1.50348	−0.751738	+	0.659461i	$0.770784\pi$
$152$	−1.18452	−0.0960770
$153$	0	0
$154$	−0.193644	−0.0156042
$155$	−1.81018	−0.145397
$156$	0	0
$157$	−4.62750	−0.369315	−0.184657	−	0.982803i	$-0.559118\pi$
$157$	−4.62750	−0.369315	−0.184657	+	0.982803i	$0.559118\pi$
$158$	0.0239228	0.00190320
$159$	0	0
$160$	3.32632	0.262968
$161$	−0.586662	−0.0462355
$162$	0	0
$163$	−13.1573	−1.03056	−0.515278	−	0.857023i	$-0.672311\pi$
$163$	−13.1573	−1.03056	−0.515278	+	0.857023i	$0.672311\pi$
$164$	1.23775	0.0966520
$165$	0	0
$166$	0.674343	0.0523391
$167$	−13.1154	−1.01490	−0.507450	−	0.861681i	$-0.669412\pi$
$167$	−13.1154	−1.01490	−0.507450	+	0.861681i	$0.669412\pi$
$168$	0	0
$169$	37.6200	2.89385
$170$	0.620895	0.0476204
$171$	0	0
$172$	14.7139	1.12192
$173$	20.5423	1.56181	0.780903	−	0.624652i	$-0.214759\pi$
$173$	20.5423	1.56181	0.780903	+	0.624652i	$0.214759\pi$
$174$	0	0
$175$	4.25464	0.321621
$176$	−16.4449	−1.23958
$177$	0	0
$178$	−1.15286	−0.0864103
$179$	−25.2305	−1.88582	−0.942909	−	0.333050i	$-0.891922\pi$
$179$	−25.2305	−1.88582	−0.942909	+	0.333050i	$0.891922\pi$
$180$	0	0
$181$	−16.6546	−1.23793	−0.618965	−	0.785419i	$-0.712448\pi$
$181$	−16.6546	−1.23793	−0.618965	+	0.785419i	$0.712448\pi$
$182$	0.331938	0.0246049
$183$	0	0
$184$	0.317600	0.0234138
$185$	30.6665	2.25465
$186$	0	0
$187$	−9.25784	−0.677001
$188$	−19.4718	−1.42013
$189$	0	0
$190$	1.03819	0.0753182
$191$	24.8831	1.80048	0.900239	−	0.435396i	$-0.143392\pi$
$191$	24.8831	1.80048	0.900239	+	0.435396i	$0.143392\pi$
$192$	0	0
$193$	−3.47233	−0.249943	−0.124972	−	0.992160i	$-0.539884\pi$
$193$	−3.47233	−0.249943	−0.124972	+	0.992160i	$0.539884\pi$
$194$	0.847066	0.0608158
$195$	0	0
$196$	13.2696	0.947826
$197$	−25.5379	−1.81950	−0.909748	−	0.415160i	$-0.863725\pi$
$197$	−25.5379	−1.81950	−0.909748	+	0.415160i	$0.863725\pi$
$198$	0	0
$199$	21.7823	1.54411	0.772053	−	0.635558i	$-0.219230\pi$
$199$	21.7823	1.54411	0.772053	+	0.635558i	$0.219230\pi$
$200$	−2.30332	−0.162870
$201$	0	0
$202$	0.707222	0.0497599
$203$	0.586662	0.0411756
$204$	0	0
$205$	−2.17314	−0.151778
$206$	0.0572193	0.00398666
$207$	0	0
$208$	28.1894	1.95458
$209$	−15.4799	−1.07077
$210$	0	0
$211$	−2.58149	−0.177717	−0.0888585	−	0.996044i	$-0.528322\pi$
$211$	−2.58149	−0.177717	−0.0888585	+	0.996044i	$0.528322\pi$
$212$	7.31539	0.502423
$213$	0	0
$214$	−0.429277	−0.0293448
$215$	−25.8333	−1.76182
$216$	0	0
$217$	−0.303390	−0.0205954
$218$	0.742148	0.0502646
$219$	0	0
$220$	28.9648	1.95281
$221$	15.8695	1.06750
$222$	0	0
$223$	−0.660029	−0.0441988	−0.0220994	−	0.999756i	$-0.507035\pi$
$223$	−0.660029	−0.0441988	−0.0220994	+	0.999756i	$0.507035\pi$
$224$	0.557498	0.0372494
$225$	0	0
$226$	−0.446817	−0.0297218
$227$	19.0889	1.26698	0.633488	−	0.773753i	$-0.281623\pi$
$227$	19.0889	1.26698	0.633488	+	0.773753i	$0.281623\pi$
$228$	0	0
$229$	23.3096	1.54034	0.770171	−	0.637837i	$-0.220171\pi$
$229$	23.3096	1.54034	0.770171	+	0.637837i	$0.220171\pi$
$230$	−0.278366	−0.0183549
$231$	0	0
$232$	−0.317600	−0.0208514
$233$	8.15362	0.534162	0.267081	−	0.963674i	$-0.413941\pi$
$233$	8.15362	0.534162	0.267081	+	0.963674i	$0.413941\pi$
$234$	0	0
$235$	34.1869	2.23011
$236$	11.4224	0.743537
$237$	0	0
$238$	0.104063	0.00674542
$239$	−9.68832	−0.626685	−0.313343	−	0.949640i	$-0.601449\pi$
$239$	−9.68832	−0.626685	−0.313343	+	0.949640i	$0.601449\pi$
$240$	0	0
$241$	−12.4770	−0.803714	−0.401857	−	0.915702i	$-0.631635\pi$
$241$	−12.4770	−0.803714	−0.401857	+	0.915702i	$0.631635\pi$
$242$	0.495225	0.0318342
$243$	0	0
$244$	−1.88672	−0.120785
$245$	−23.2976	−1.48843
$246$	0	0
$247$	26.5352	1.68839
$248$	0.164245	0.0104296
$249$	0	0
$250$	0.626959	0.0396524
$251$	8.30910	0.524466	0.262233	−	0.965005i	$-0.415541\pi$
$251$	8.30910	0.524466	0.262233	+	0.965005i	$0.415541\pi$
$252$	0	0
$253$	4.15057	0.260944
$254$	0.895136	0.0561658
$255$	0	0
$256$	15.4964	0.968528
$257$	1.16505	0.0726738	0.0363369	−	0.999340i	$-0.488431\pi$
$257$	1.16505	0.0726738	0.0363369	+	0.999340i	$0.488431\pi$
$258$	0	0
$259$	5.13977	0.319370
$260$	−49.6506	−3.07920
$261$	0	0
$262$	1.18739	0.0733569
$263$	−5.19686	−0.320452	−0.160226	−	0.987080i	$-0.551222\pi$
$263$	−5.19686	−0.320452	−0.160226	+	0.987080i	$0.551222\pi$
$264$	0	0
$265$	−12.8437	−0.788985
$266$	0.174003	0.0106688
$267$	0	0
$268$	21.2201	1.29622
$269$	18.9363	1.15457	0.577284	−	0.816544i	$-0.304113\pi$
$269$	18.9363	1.15457	0.577284	+	0.816544i	$0.304113\pi$
$270$	0	0
$271$	10.7537	0.653238	0.326619	−	0.945156i	$-0.394091\pi$
$271$	10.7537	0.653238	0.326619	+	0.945156i	$0.394091\pi$
$272$	8.83745	0.535849
$273$	0	0
$274$	−0.501522	−0.0302980
$275$	−30.1011	−1.81517
$276$	0	0
$277$	6.69214	0.402092	0.201046	−	0.979582i	$-0.435566\pi$
$277$	6.69214	0.402092	0.201046	+	0.979582i	$0.435566\pi$
$278$	1.43442	0.0860308
$279$	0	0
$280$	−0.652194	−0.0389760
$281$	20.0505	1.19611	0.598057	−	0.801454i	$-0.295940\pi$
$281$	20.0505	1.19611	0.598057	+	0.801454i	$0.295940\pi$
$282$	0	0
$283$	21.9456	1.30453	0.652265	−	0.757991i	$-0.273819\pi$
$283$	21.9456	1.30453	0.652265	+	0.757991i	$0.273819\pi$
$284$	2.85587	0.169464
$285$	0	0
$286$	−2.34842	−0.138865
$287$	−0.364222	−0.0214994
$288$	0	0
$289$	−12.0249	−0.707345
$290$	0.278366	0.0163462
$291$	0	0
$292$	−26.6740	−1.56098
$293$	−8.52784	−0.498201	−0.249101	−	0.968478i	$-0.580135\pi$
$293$	−8.52784	−0.498201	−0.249101	+	0.968478i	$0.580135\pi$
$294$	0	0
$295$	−20.0545	−1.16762
$296$	−2.78250	−0.161730
$297$	0	0
$298$	0.779097	0.0451319
$299$	−7.11477	−0.411458
$300$	0	0
$301$	−4.32972	−0.249561
$302$	−1.46924	−0.0845452
$303$	0	0
$304$	14.7770	0.847518
$305$	3.31254	0.189676
$306$	0	0
$307$	−31.4093	−1.79262	−0.896312	−	0.443424i	$-0.853763\pi$
$307$	−31.4093	−1.79262	−0.896312	+	0.443424i	$0.853763\pi$
$308$	4.85457	0.276615
$309$	0	0
$310$	−0.143956	−0.00817612
$311$	−29.4762	−1.67144	−0.835720	−	0.549155i	$-0.814950\pi$
$311$	−29.4762	−1.67144	−0.835720	+	0.549155i	$0.814950\pi$
$312$	0	0
$313$	17.5585	0.992463	0.496232	−	0.868190i	$-0.334717\pi$
$313$	17.5585	0.992463	0.496232	+	0.868190i	$0.334717\pi$
$314$	−0.368005	−0.0207677
$315$	0	0
$316$	−0.599736	−0.0337378
$317$	−9.97744	−0.560389	−0.280195	−	0.959943i	$-0.590399\pi$
$317$	−9.97744	−0.560389	−0.280195	+	0.959943i	$0.590399\pi$
$318$	0	0
$319$	−4.15057	−0.232387
$320$	−27.4727	−1.53577
$321$	0	0
$322$	−0.0466547	−0.00259997
$323$	8.31885	0.462873
$324$	0	0
$325$	51.5984	2.86216
$326$	−1.04634	−0.0579514
$327$	0	0
$328$	0.197178	0.0108873
$329$	5.72981	0.315894
$330$	0	0
$331$	8.93376	0.491044	0.245522	−	0.969391i	$-0.421041\pi$
$331$	8.93376	0.491044	0.245522	+	0.969391i	$0.421041\pi$
$332$	−16.9055	−0.927810
$333$	0	0
$334$	−1.04301	−0.0570710
$335$	−37.2564	−2.03553
$336$	0	0
$337$	−8.35613	−0.455187	−0.227594	−	0.973756i	$-0.573086\pi$
$337$	−8.35613	−0.455187	−0.227594	+	0.973756i	$0.573086\pi$
$338$	2.99176	0.162730
$339$	0	0
$340$	−15.5656	−0.844162
$341$	2.14645	0.116237
$342$	0	0
$343$	−8.01136	−0.432573
$344$	2.34397	0.126378
$345$	0	0
$346$	1.63364	0.0878252
$347$	22.9091	1.22983	0.614913	−	0.788595i	$-0.289191\pi$
$347$	22.9091	1.22983	0.614913	+	0.788595i	$0.289191\pi$
$348$	0	0
$349$	19.7026	1.05465	0.527327	−	0.849662i	$-0.323194\pi$
$349$	19.7026	1.05465	0.527327	+	0.849662i	$0.323194\pi$
$350$	0.338353	0.0180857
$351$	0	0
$352$	−3.94423	−0.210228
$353$	−11.3850	−0.605963	−0.302981	−	0.952997i	$-0.597982\pi$
$353$	−11.3850	−0.605963	−0.302981	+	0.952997i	$0.597982\pi$
$354$	0	0
$355$	−5.01409	−0.266120
$356$	28.9017	1.53179
$357$	0	0
$358$	−2.00647	−0.106045
$359$	−6.45489	−0.340676	−0.170338	−	0.985386i	$-0.554486\pi$
$359$	−6.45489	−0.340676	−0.170338	+	0.985386i	$0.554486\pi$
$360$	0	0
$361$	−5.09015	−0.267903
$362$	−1.32447	−0.0696126
$363$	0	0
$364$	−8.32154	−0.436167
$365$	46.8320	2.45130
$366$	0	0
$367$	23.9631	1.25086	0.625432	−	0.780279i	$-0.284923\pi$
$367$	23.9631	1.25086	0.625432	+	0.780279i	$0.284923\pi$
$368$	−3.96209	−0.206538
$369$	0	0
$370$	2.43877	0.126786
$371$	−2.15264	−0.111760
$372$	0	0
$373$	0.758513	0.0392743	0.0196372	−	0.999807i	$-0.493749\pi$
$373$	0.758513	0.0392743	0.0196372	+	0.999807i	$0.493749\pi$
$374$	−0.736236	−0.0380699
$375$	0	0
$376$	−3.10193	−0.159970
$377$	7.11477	0.366430
$378$	0	0
$379$	−4.04381	−0.207716	−0.103858	−	0.994592i	$-0.533119\pi$
$379$	−4.04381	−0.207716	−0.103858	+	0.994592i	$0.533119\pi$
$380$	−26.0270	−1.33516
$381$	0	0
$382$	1.97885	0.101247
$383$	6.24293	0.318999	0.159499	−	0.987198i	$-0.449012\pi$
$383$	6.24293	0.318999	0.159499	+	0.987198i	$0.449012\pi$
$384$	0	0
$385$	−8.52323	−0.434384
$386$	−0.276139	−0.0140551
$387$	0	0
$388$	−21.2356	−1.07807
$389$	−2.42404	−0.122904	−0.0614519	−	0.998110i	$-0.519573\pi$
$389$	−2.42404	−0.122904	−0.0614519	+	0.998110i	$0.519573\pi$
$390$	0	0
$391$	−2.23050	−0.112801
$392$	2.11389	0.106768
$393$	0	0
$394$	−2.03091	−0.102316
$395$	1.05296	0.0529804
$396$	0	0
$397$	16.3312	0.819638	0.409819	−	0.912167i	$-0.365592\pi$
$397$	16.3312	0.819638	0.409819	+	0.912167i	$0.365592\pi$
$398$	1.73225	0.0868299
$399$	0	0
$400$	28.7342	1.43671
$401$	−11.7312	−0.585829	−0.292914	−	0.956139i	$-0.594625\pi$
$401$	−11.7312	−0.585829	−0.292914	+	0.956139i	$0.594625\pi$
$402$	0	0
$403$	−3.67937	−0.183283
$404$	−17.7298	−0.882088
$405$	0	0
$406$	0.0466547	0.00231543
$407$	−36.3633	−1.80246
$408$	0	0
$409$	3.37266	0.166767	0.0833836	−	0.996518i	$-0.473427\pi$
$409$	3.37266	0.166767	0.0833836	+	0.996518i	$0.473427\pi$
$410$	−0.172820	−0.00853497
$411$	0	0
$412$	−1.43447	−0.0706710
$413$	−3.36118	−0.165393
$414$	0	0
$415$	29.6812	1.45699
$416$	6.76108	0.331489
$417$	0	0
$418$	−1.23105	−0.0602127
$419$	25.6249	1.25186	0.625930	−	0.779879i	$-0.284720\pi$
$419$	25.6249	1.25186	0.625930	+	0.779879i	$0.284720\pi$
$420$	0	0
$421$	1.18735	0.0578681	0.0289341	−	0.999581i	$-0.490789\pi$
$421$	1.18735	0.0578681	0.0289341	+	0.999581i	$0.490789\pi$
$422$	−0.205295	−0.00999359
$423$	0	0
$424$	1.16537	0.0565953
$425$	16.1762	0.784662
$426$	0	0
$427$	0.555189	0.0268675
$428$	10.7618	0.520191
$429$	0	0
$430$	−2.05441	−0.0990726
$431$	11.3092	0.544745	0.272373	−	0.962192i	$-0.412192\pi$
$431$	11.3092	0.544745	0.272373	+	0.962192i	$0.412192\pi$
$432$	0	0
$433$	−4.83199	−0.232210	−0.116105	−	0.993237i	$-0.537041\pi$
$433$	−4.83199	−0.232210	−0.116105	+	0.993237i	$0.537041\pi$
$434$	−0.0241273	−0.00115815
$435$	0	0
$436$	−18.6053	−0.891034
$437$	−3.72959	−0.178411
$438$	0	0
$439$	−18.4820	−0.882096	−0.441048	−	0.897484i	$-0.645393\pi$
$439$	−18.4820	−0.882096	−0.441048	+	0.897484i	$0.645393\pi$
$440$	4.61420	0.219973
$441$	0	0
$442$	1.26203	0.0600288
$443$	−38.3578	−1.82244	−0.911218	−	0.411924i	$-0.864857\pi$
$443$	−38.3578	−1.82244	−0.911218	+	0.411924i	$0.864857\pi$
$444$	0	0
$445$	−50.7431	−2.40545
$446$	−0.0524893	−0.00248544
$447$	0	0
$448$	−4.60449	−0.217542
$449$	−14.2726	−0.673564	−0.336782	−	0.941583i	$-0.609339\pi$
$449$	−14.2726	−0.673564	−0.336782	+	0.941583i	$0.609339\pi$
$450$	0	0
$451$	2.57683	0.121338
$452$	11.2015	0.526875
$453$	0	0
$454$	1.51806	0.0712460
$455$	14.6103	0.684939
$456$	0	0
$457$	40.3284	1.88648	0.943242	−	0.332105i	$-0.107759\pi$
$457$	40.3284	1.88648	0.943242	+	0.332105i	$0.107759\pi$
$458$	1.85371	0.0866183
$459$	0	0
$460$	6.97852	0.325375
$461$	27.5015	1.28087	0.640435	−	0.768012i	$-0.278754\pi$
$461$	27.5015	1.28087	0.640435	+	0.768012i	$0.278754\pi$
$462$	0	0
$463$	4.84621	0.225222	0.112611	−	0.993639i	$-0.464079\pi$
$463$	4.84621	0.225222	0.112611	+	0.993639i	$0.464079\pi$
$464$	3.96209	0.183936
$465$	0	0
$466$	0.648422	0.0300376
$467$	10.4670	0.484353	0.242176	−	0.970232i	$-0.422139\pi$
$467$	10.4670	0.484353	0.242176	+	0.970232i	$0.422139\pi$
$468$	0	0
$469$	−6.24425	−0.288333
$470$	2.71874	0.125406
$471$	0	0
$472$	1.81963	0.0837555
$473$	30.6323	1.40848
$474$	0	0
$475$	27.0480	1.24105
$476$	−2.60882	−0.119575
$477$	0	0
$478$	−0.770470	−0.0352405
$479$	11.8367	0.540834	0.270417	−	0.962743i	$-0.412839\pi$
$479$	11.8367	0.540834	0.270417	+	0.962743i	$0.412839\pi$
$480$	0	0
$481$	62.3328	2.84213
$482$	−0.992241	−0.0451954
$483$	0	0
$484$	−12.4151	−0.564322
$485$	37.2836	1.69296
$486$	0	0
$487$	42.1210	1.90868	0.954341	−	0.298718i	$-0.0965589\pi$
$487$	42.1210	1.90868	0.954341	+	0.298718i	$0.0965589\pi$
$488$	−0.300561	−0.0136058
$489$	0	0
$490$	−1.85275	−0.0836989
$491$	−20.6397	−0.931457	−0.465728	−	0.884928i	$-0.654208\pi$
$491$	−20.6397	−0.931457	−0.465728	+	0.884928i	$0.654208\pi$
$492$	0	0
$493$	2.23050	0.100457
$494$	2.11023	0.0949437
$495$	0	0
$496$	−2.04898	−0.0920018
$497$	−0.840372	−0.0376958
$498$	0	0
$499$	−6.80903	−0.304814	−0.152407	−	0.988318i	$-0.548702\pi$
$499$	−6.80903	−0.304814	−0.152407	+	0.988318i	$0.548702\pi$
$500$	−15.7176	−0.702912
$501$	0	0
$502$	0.660786	0.0294923
$503$	23.1007	1.03001	0.515004	−	0.857188i	$-0.327790\pi$
$503$	23.1007	1.03001	0.515004	+	0.857188i	$0.327790\pi$
$504$	0	0
$505$	31.1284	1.38520
$506$	0.330077	0.0146737
$507$	0	0
$508$	−22.4407	−0.995644
$509$	−32.4793	−1.43962	−0.719809	−	0.694172i	$-0.755771\pi$
$509$	−32.4793	−1.43962	−0.719809	+	0.694172i	$0.755771\pi$
$510$	0	0
$511$	7.84914	0.347225
$512$	6.28185	0.277621
$513$	0	0
$514$	0.0926513	0.00408668
$515$	2.51851	0.110979
$516$	0	0
$517$	−40.5377	−1.78285
$518$	0.408744	0.0179592
$519$	0	0
$520$	−7.90951	−0.346855
$521$	−38.7698	−1.69854	−0.849268	−	0.527962i	$-0.822956\pi$
$521$	−38.7698	−1.69854	−0.849268	+	0.527962i	$0.822956\pi$
$522$	0	0
$523$	−0.0859711	−0.00375925	−0.00187963	−	0.999998i	$-0.500598\pi$
$523$	−0.0859711	−0.00375925	−0.00187963	+	0.999998i	$0.500598\pi$
$524$	−29.7673	−1.30039
$525$	0	0
$526$	−0.413283	−0.0180200
$527$	−1.15349	−0.0502469
$528$	0	0
$529$	1.00000	0.0434783
$530$	−1.02141	−0.0443671
$531$	0	0
$532$	−4.36218	−0.189125
$533$	−4.41712	−0.191327
$534$	0	0
$535$	−18.8946	−0.816887
$536$	3.38043	0.146012
$537$	0	0
$538$	1.50592	0.0649250
$539$	27.6255	1.18991
$540$	0	0
$541$	26.3499	1.13287	0.566434	−	0.824107i	$-0.308322\pi$
$541$	26.3499	1.13287	0.566434	+	0.824107i	$0.308322\pi$
$542$	0.855191	0.0367336
$543$	0	0
$544$	2.11962	0.0908778
$545$	32.6657	1.39924
$546$	0	0
$547$	7.87821	0.336848	0.168424	−	0.985715i	$-0.446132\pi$
$547$	7.87821	0.336848	0.168424	+	0.985715i	$0.446132\pi$
$548$	12.5729	0.537089
$549$	0	0
$550$	−2.39381	−0.102072
$551$	3.72959	0.158886
$552$	0	0
$553$	0.176479	0.00750466
$554$	0.532197	0.0226109
$555$	0	0
$556$	−35.9603	−1.52506
$557$	9.18641	0.389241	0.194620	−	0.980879i	$-0.437653\pi$
$557$	9.18641	0.389241	0.194620	+	0.980879i	$0.437653\pi$
$558$	0	0
$559$	−52.5089	−2.22089
$560$	8.13620	0.343817
$561$	0	0
$562$	1.59453	0.0672612
$563$	−1.49580	−0.0630405	−0.0315202	−	0.999503i	$-0.510035\pi$
$563$	−1.49580	−0.0630405	−0.0315202	+	0.999503i	$0.510035\pi$
$564$	0	0
$565$	−19.6667	−0.827383
$566$	1.74524	0.0733578
$567$	0	0
$568$	0.454950	0.0190893
$569$	30.5003	1.27864	0.639319	−	0.768941i	$-0.279216\pi$
$569$	30.5003	1.27864	0.639319	+	0.768941i	$0.279216\pi$
$570$	0	0
$571$	−13.3210	−0.557468	−0.278734	−	0.960368i	$-0.589915\pi$
$571$	−13.3210	−0.557468	−0.278734	+	0.960368i	$0.589915\pi$
$572$	58.8740	2.46164
$573$	0	0
$574$	−0.0289650	−0.00120898
$575$	−7.25228	−0.302441
$576$	0	0
$577$	−16.1751	−0.673377	−0.336689	−	0.941616i	$-0.609307\pi$
$577$	−16.1751	−0.673377	−0.336689	+	0.941616i	$0.609307\pi$
$578$	−0.956286	−0.0397763
$579$	0	0
$580$	−6.97852	−0.289767
$581$	4.97464	0.206383
$582$	0	0
$583$	15.2297	0.630749
$584$	−4.24927	−0.175836
$585$	0	0
$586$	−0.678182	−0.0280154
$587$	−38.6608	−1.59570	−0.797850	−	0.602856i	$-0.794029\pi$
$587$	−38.6608	−1.59570	−0.797850	+	0.602856i	$0.794029\pi$
$588$	0	0
$589$	−1.92874	−0.0794723
$590$	−1.59485	−0.0656589
$591$	0	0
$592$	34.7121	1.42666
$593$	17.2174	0.707033	0.353516	−	0.935428i	$-0.384986\pi$
$593$	17.2174	0.707033	0.353516	+	0.935428i	$0.384986\pi$
$594$	0	0
$595$	4.58035	0.187776
$596$	−19.5317	−0.800048
$597$	0	0
$598$	−0.565807	−0.0231376
$599$	−19.4345	−0.794071	−0.397035	−	0.917803i	$-0.629961\pi$
$599$	−19.4345	−0.794071	−0.397035	+	0.917803i	$0.629961\pi$
$600$	0	0
$601$	33.6512	1.37266	0.686330	−	0.727290i	$-0.259220\pi$
$601$	33.6512	1.37266	0.686330	+	0.727290i	$0.259220\pi$
$602$	−0.344324	−0.0140336
$603$	0	0
$604$	36.8332	1.49872
$605$	21.7973	0.886188
$606$	0	0
$607$	−19.4319	−0.788717	−0.394359	−	0.918957i	$-0.629033\pi$
$607$	−19.4319	−0.788717	−0.394359	+	0.918957i	$0.629033\pi$
$608$	3.54418	0.143736
$609$	0	0
$610$	0.263432	0.0106660
$611$	69.4885	2.81120
$612$	0	0
$613$	−42.5872	−1.72008	−0.860041	−	0.510225i	$-0.829562\pi$
$613$	−42.5872	−1.72008	−0.860041	+	0.510225i	$0.829562\pi$
$614$	−2.49785	−0.100805
$615$	0	0
$616$	0.773350	0.0311592
$617$	−32.9153	−1.32512	−0.662559	−	0.749009i	$-0.730530\pi$
$617$	−32.9153	−1.32512	−0.662559	+	0.749009i	$0.730530\pi$
$618$	0	0
$619$	26.5437	1.06688	0.533441	−	0.845837i	$-0.320899\pi$
$619$	26.5437	1.06688	0.533441	+	0.845837i	$0.320899\pi$
$620$	3.60891	0.144937
$621$	0	0
$622$	−2.34411	−0.0939903
$623$	−8.50465	−0.340732
$624$	0	0
$625$	−8.66580	−0.346632
$626$	1.39635	0.0558093
$627$	0	0
$628$	9.22574	0.368147
$629$	19.5415	0.779170
$630$	0	0
$631$	−12.0438	−0.479456	−0.239728	−	0.970840i	$-0.577058\pi$
$631$	−12.0438	−0.479456	−0.239728	+	0.970840i	$0.577058\pi$
$632$	−0.0955400	−0.00380038
$633$	0	0
$634$	−0.793463	−0.0315124
$635$	39.3994	1.56352
$636$	0	0
$637$	−47.3547	−1.87626
$638$	−0.330077	−0.0130679
$639$	0	0
$640$	−8.83742	−0.349330
$641$	21.2041	0.837510	0.418755	−	0.908099i	$-0.362467\pi$
$641$	21.2041	0.837510	0.418755	+	0.908099i	$0.362467\pi$
$642$	0	0
$643$	0.199300	0.00785963	0.00392981	−	0.999992i	$-0.498749\pi$
$643$	0.199300	0.00785963	0.00392981	+	0.999992i	$0.498749\pi$
$644$	1.16961	0.0460893
$645$	0	0
$646$	0.661562	0.0260288
$647$	0.904439	0.0355571	0.0177786	−	0.999842i	$-0.494341\pi$
$647$	0.904439	0.0355571	0.0177786	+	0.999842i	$0.494341\pi$
$648$	0	0
$649$	23.7800	0.933446
$650$	4.10339	0.160948
$651$	0	0
$652$	26.2313	1.02730
$653$	36.4996	1.42834	0.714169	−	0.699973i	$-0.246805\pi$
$653$	36.4996	1.42834	0.714169	+	0.699973i	$0.246805\pi$
$654$	0	0
$655$	52.2628	2.04208
$656$	−2.45982	−0.0960398
$657$	0	0
$658$	0.455667	0.0177637
$659$	28.9401	1.12734	0.563672	−	0.825999i	$-0.309388\pi$
$659$	28.9401	1.12734	0.563672	+	0.825999i	$0.309388\pi$
$660$	0	0
$661$	14.2945	0.555993	0.277997	−	0.960582i	$-0.410330\pi$
$661$	14.2945	0.555993	0.277997	+	0.960582i	$0.410330\pi$
$662$	0.710463	0.0276129
$663$	0	0
$664$	−2.69311	−0.104513
$665$	7.65875	0.296994
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	26.1478	1.01169
$669$	0	0
$670$	−2.96284	−0.114464
$671$	−3.92790	−0.151635
$672$	0	0
$673$	21.1334	0.814631	0.407315	−	0.913288i	$-0.366465\pi$
$673$	21.1334	0.814631	0.407315	+	0.913288i	$0.366465\pi$
$674$	−0.664527	−0.0255966
$675$	0	0
$676$	−75.0021	−2.88470
$677$	35.4322	1.36177	0.680886	−	0.732390i	$-0.261595\pi$
$677$	35.4322	1.36177	0.680886	+	0.732390i	$0.261595\pi$
$678$	0	0
$679$	6.24882	0.239808
$680$	−2.47965	−0.0950903
$681$	0	0
$682$	0.170698	0.00653635
$683$	−35.8920	−1.37337	−0.686685	−	0.726955i	$-0.740935\pi$
$683$	−35.8920	−1.37337	−0.686685	+	0.726955i	$0.740935\pi$
$684$	0	0
$685$	−22.0745	−0.843423
$686$	−0.637109	−0.0243249
$687$	0	0
$688$	−29.2413	−1.11481
$689$	−26.1062	−0.994569
$690$	0	0
$691$	30.8955	1.17532	0.587660	−	0.809108i	$-0.300049\pi$
$691$	30.8955	1.17532	0.587660	+	0.809108i	$0.300049\pi$
$692$	−40.9548	−1.55687
$693$	0	0
$694$	1.82186	0.0691570
$695$	63.1361	2.39489
$696$	0	0
$697$	−1.38478	−0.0524522
$698$	1.56686	0.0593065
$699$	0	0
$700$	−8.48238	−0.320604
$701$	−41.6036	−1.57135	−0.785673	−	0.618642i	$-0.787683\pi$
$701$	−41.6036	−1.57135	−0.785673	+	0.618642i	$0.787683\pi$
$702$	0	0
$703$	32.6751	1.23236
$704$	32.5762	1.22776
$705$	0	0
$706$	−0.905400	−0.0340752
$707$	5.21719	0.196212
$708$	0	0
$709$	−20.8790	−0.784126	−0.392063	−	0.919938i	$-0.628238\pi$
$709$	−20.8790	−0.784126	−0.392063	+	0.919938i	$0.628238\pi$
$710$	−0.398748	−0.0149648
$711$	0	0
$712$	4.60414	0.172547
$713$	0.517145	0.0193672
$714$	0	0
$715$	−103.366	−3.86567
$716$	50.3015	1.87986
$717$	0	0
$718$	−0.513329	−0.0191573
$719$	−34.2473	−1.27721	−0.638604	−	0.769536i	$-0.720488\pi$
$719$	−34.2473	−1.27721	−0.638604	+	0.769536i	$0.720488\pi$
$720$	0	0
$721$	0.422108	0.0157201
$722$	−0.404798	−0.0150650
$723$	0	0
$724$	33.2039	1.23401
$725$	7.25228	0.269343
$726$	0	0
$727$	2.85236	0.105788	0.0528942	−	0.998600i	$-0.483155\pi$
$727$	2.85236	0.105788	0.0528942	+	0.998600i	$0.483155\pi$
$728$	−1.32565	−0.0491319
$729$	0	0
$730$	3.72434	0.137844
$731$	−16.4617	−0.608857
$732$	0	0
$733$	−13.6111	−0.502737	−0.251368	−	0.967891i	$-0.580881\pi$
$733$	−13.6111	−0.502737	−0.251368	+	0.967891i	$0.580881\pi$
$734$	1.90568	0.0703400
$735$	0	0
$736$	−0.950288	−0.0350281
$737$	44.1774	1.62729
$738$	0	0
$739$	25.5146	0.938569	0.469284	−	0.883047i	$-0.344512\pi$
$739$	25.5146	0.938569	0.469284	+	0.883047i	$0.344512\pi$
$740$	−61.1391	−2.24752
$741$	0	0
$742$	−0.171190	−0.00628459
$743$	12.4223	0.455731	0.227865	−	0.973693i	$-0.426825\pi$
$743$	12.4223	0.455731	0.227865	+	0.973693i	$0.426825\pi$
$744$	0	0
$745$	34.2920	1.25636
$746$	0.0603212	0.00220852
$747$	0	0
$748$	18.4571	0.674860
$749$	−3.16678	−0.115712
$750$	0	0
$751$	−19.4236	−0.708776	−0.354388	−	0.935099i	$-0.615311\pi$
$751$	−19.4236	−0.708776	−0.354388	+	0.935099i	$0.615311\pi$
$752$	38.6969	1.41113
$753$	0	0
$754$	0.565807	0.0206055
$755$	−64.6686	−2.35353
$756$	0	0
$757$	0.601179	0.0218502	0.0109251	−	0.999940i	$-0.496522\pi$
$757$	0.601179	0.0218502	0.0109251	+	0.999940i	$0.496522\pi$
$758$	−0.321586	−0.0116805
$759$	0	0
$760$	−4.14620	−0.150398
$761$	−18.5372	−0.671973	−0.335986	−	0.941867i	$-0.609070\pi$
$761$	−18.5372	−0.671973	−0.335986	+	0.941867i	$0.609070\pi$
$762$	0	0
$763$	5.47484	0.198202
$764$	−49.6088	−1.79478
$765$	0	0
$766$	0.496473	0.0179383
$767$	−40.7629	−1.47186
$768$	0	0
$769$	27.2512	0.982703	0.491352	−	0.870961i	$-0.336503\pi$
$769$	27.2512	0.982703	0.491352	+	0.870961i	$0.336503\pi$
$770$	−0.677816	−0.0244268
$771$	0	0
$772$	6.92269	0.249153
$773$	−50.4938	−1.81614	−0.908068	−	0.418822i	$-0.862443\pi$
$773$	−50.4938	−1.81614	−0.908068	+	0.418822i	$0.862443\pi$
$774$	0	0
$775$	−3.75048	−0.134721
$776$	−3.38291	−0.121439
$777$	0	0
$778$	−0.192773	−0.00691126
$779$	−2.31547	−0.0829604
$780$	0	0
$781$	5.94554	0.212748
$782$	−0.177382	−0.00634316
$783$	0	0
$784$	−26.3710	−0.941822
$785$	−16.1978	−0.578123
$786$	0	0
$787$	−23.5710	−0.840214	−0.420107	−	0.907474i	$-0.638008\pi$
$787$	−23.5710	−0.840214	−0.420107	+	0.907474i	$0.638008\pi$
$788$	50.9142	1.81374
$789$	0	0
$790$	0.0837377	0.00297925
$791$	−3.29618	−0.117199
$792$	0	0
$793$	6.73308	0.239099
$794$	1.29875	0.0460908
$795$	0	0
$796$	−43.4268	−1.53922
$797$	−47.4325	−1.68015	−0.840073	−	0.542473i	$-0.817488\pi$
$797$	−47.4325	−1.68015	−0.840073	+	0.542473i	$0.817488\pi$
$798$	0	0
$799$	21.7848	0.770691
$800$	6.89176	0.243660
$801$	0	0
$802$	−0.932932	−0.0329430
$803$	−55.5318	−1.95967
$804$	0	0
$805$	−2.05351	−0.0723767
$806$	−0.292604	−0.0103066
$807$	0	0
$808$	−2.82441	−0.0993625
$809$	48.9931	1.72250	0.861252	−	0.508178i	$-0.169681\pi$
$809$	48.9931	1.72250	0.861252	+	0.508178i	$0.169681\pi$
$810$	0	0
$811$	−1.77157	−0.0622083	−0.0311041	−	0.999516i	$-0.509902\pi$
$811$	−1.77157	−0.0622083	−0.0311041	+	0.999516i	$0.509902\pi$
$812$	−1.16961	−0.0410454
$813$	0	0
$814$	−2.89182	−0.101358
$815$	−46.0547	−1.61323
$816$	0	0
$817$	−27.5254	−0.962991
$818$	0.268213	0.00937784
$819$	0	0
$820$	4.33253	0.151298
$821$	−27.9635	−0.975932	−0.487966	−	0.872863i	$-0.662261\pi$
$821$	−27.9635	−0.975932	−0.487966	+	0.872863i	$0.662261\pi$
$822$	0	0
$823$	−15.1040	−0.526491	−0.263246	−	0.964729i	$-0.584793\pi$
$823$	−15.1040	−0.526491	−0.263246	+	0.964729i	$0.584793\pi$
$824$	−0.228515	−0.00796071
$825$	0	0
$826$	−0.267300	−0.00930057
$827$	−42.3647	−1.47317	−0.736583	−	0.676347i	$-0.763562\pi$
$827$	−42.3647	−1.47317	−0.736583	+	0.676347i	$0.763562\pi$
$828$	0	0
$829$	44.2335	1.53629	0.768147	−	0.640274i	$-0.221179\pi$
$829$	44.2335	1.53629	0.768147	+	0.640274i	$0.221179\pi$
$830$	2.36042	0.0819314
$831$	0	0
$832$	−55.8411	−1.93594
$833$	−14.8458	−0.514377
$834$	0	0
$835$	−45.9082	−1.58872
$836$	30.8620	1.06738
$837$	0	0
$838$	2.03784	0.0703960
$839$	9.63042	0.332479	0.166240	−	0.986085i	$-0.446837\pi$
$839$	9.63042	0.332479	0.166240	+	0.986085i	$0.446837\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0.0944252	0.00325411
$843$	0	0
$844$	5.14665	0.177155
$845$	131.682	4.53001
$846$	0	0
$847$	3.65328	0.125528
$848$	−14.5381	−0.499241
$849$	0	0
$850$	1.28642	0.0441240
$851$	−8.76104	−0.300325
$852$	0	0
$853$	12.7194	0.435505	0.217753	−	0.976004i	$-0.430127\pi$
$853$	12.7194	0.435505	0.217753	+	0.976004i	$0.430127\pi$
$854$	0.0441518	0.00151084
$855$	0	0
$856$	1.71439	0.0585967
$857$	−45.0954	−1.54043	−0.770215	−	0.637784i	$-0.779851\pi$
$857$	−45.0954	−1.54043	−0.770215	+	0.637784i	$0.779851\pi$
$858$	0	0
$859$	30.4984	1.04059	0.520296	−	0.853986i	$-0.325822\pi$
$859$	30.4984	1.04059	0.520296	+	0.853986i	$0.325822\pi$
$860$	51.5033	1.75625
$861$	0	0
$862$	0.899372	0.0306327
$863$	−27.2407	−0.927285	−0.463642	−	0.886022i	$-0.653458\pi$
$863$	−27.2407	−0.927285	−0.463642	+	0.886022i	$0.653458\pi$
$864$	0	0
$865$	71.9049	2.44484
$866$	−0.384267	−0.0130579
$867$	0	0
$868$	0.604860	0.0205303
$869$	−1.24857	−0.0423548
$870$	0	0
$871$	−75.7275	−2.56593
$872$	−2.96390	−0.100370
$873$	0	0
$874$	−0.296598	−0.0100326
$875$	4.62509	0.156356
$876$	0	0
$877$	−42.8439	−1.44673	−0.723367	−	0.690463i	$-0.757407\pi$
$877$	−42.8439	−1.44673	−0.723367	+	0.690463i	$0.757407\pi$
$878$	−1.46979	−0.0496030
$879$	0	0
$880$	−57.5627	−1.94044
$881$	2.00320	0.0674896	0.0337448	−	0.999430i	$-0.489257\pi$
$881$	2.00320	0.0674896	0.0337448	+	0.999430i	$0.489257\pi$
$882$	0	0
$883$	−34.2511	−1.15264	−0.576320	−	0.817224i	$-0.695512\pi$
$883$	−34.2511	−1.15264	−0.576320	+	0.817224i	$0.695512\pi$
$884$	−31.6386	−1.06412
$885$	0	0
$886$	−3.05043	−0.102481
$887$	−23.7389	−0.797073	−0.398536	−	0.917153i	$-0.630482\pi$
$887$	−23.7389	−0.797073	−0.398536	+	0.917153i	$0.630482\pi$
$888$	0	0
$889$	6.60343	0.221472
$890$	−4.03538	−0.135266
$891$	0	0
$892$	1.31588	0.0440591
$893$	36.4261	1.21895
$894$	0	0
$895$	−88.3151	−2.95205
$896$	−1.48117	−0.0494824
$897$	0	0
$898$	−1.13504	−0.0378766
$899$	−0.517145	−0.0172478
$900$	0	0
$901$	−8.18437	−0.272661
$902$	0.204924	0.00682323
$903$	0	0
$904$	1.78444	0.0593496
$905$	−58.2967	−1.93785
$906$	0	0
$907$	−50.9947	−1.69325	−0.846626	−	0.532188i	$-0.821370\pi$
$907$	−50.9947	−1.69325	−0.846626	+	0.532188i	$0.821370\pi$
$908$	−38.0571	−1.26297
$909$	0	0
$910$	1.16189	0.0385163
$911$	−34.1421	−1.13118	−0.565589	−	0.824687i	$-0.691351\pi$
$911$	−34.1421	−1.13118	−0.565589	+	0.824687i	$0.691351\pi$
$912$	0	0
$913$	−35.1950	−1.16478
$914$	3.20715	0.106083
$915$	0	0
$916$	−46.4718	−1.53547
$917$	8.75936	0.289260
$918$	0	0
$919$	−37.9192	−1.25084	−0.625420	−	0.780289i	$-0.715072\pi$
$919$	−37.9192	−1.25084	−0.625420	+	0.780289i	$0.715072\pi$
$920$	1.11170	0.0366517
$921$	0	0
$922$	2.18707	0.0720274
$923$	−10.1916	−0.335462
$924$	0	0
$925$	63.5376	2.08910
$926$	0.385398	0.0126650
$927$	0	0
$928$	0.950288	0.0311947
$929$	50.4251	1.65439	0.827196	−	0.561913i	$-0.189934\pi$
$929$	50.4251	1.65439	0.827196	+	0.561913i	$0.189934\pi$
$930$	0	0
$931$	−24.8235	−0.813558
$932$	−16.2557	−0.532472
$933$	0	0
$934$	0.832391	0.0272367
$935$	−32.4055	−1.05977
$936$	0	0
$937$	−26.6412	−0.870329	−0.435164	−	0.900351i	$-0.643310\pi$
$937$	−26.6412	−0.870329	−0.435164	+	0.900351i	$0.643310\pi$
$938$	−0.496578	−0.0162139
$939$	0	0
$940$	−68.1577	−2.22306
$941$	−26.5804	−0.866496	−0.433248	−	0.901275i	$-0.642633\pi$
$941$	−26.5804	−0.866496	−0.433248	+	0.901275i	$0.642633\pi$
$942$	0	0
$943$	0.620838	0.0202173
$944$	−22.7002	−0.738827
$945$	0	0
$946$	2.43605	0.0792030
$947$	4.99817	0.162419	0.0812094	−	0.996697i	$-0.474122\pi$
$947$	4.99817	0.162419	0.0812094	+	0.996697i	$0.474122\pi$
$948$	0	0
$949$	95.1908	3.09002
$950$	2.15101	0.0697881
$951$	0	0
$952$	−0.415595	−0.0134695
$953$	6.85676	0.222112	0.111056	−	0.993814i	$-0.464577\pi$
$953$	6.85676	0.222112	0.111056	+	0.993814i	$0.464577\pi$
$954$	0	0
$955$	87.0990	2.81846
$956$	19.3154	0.624704
$957$	0	0
$958$	0.941323	0.0304128
$959$	−3.69973	−0.119471
$960$	0	0
$961$	−30.7326	−0.991373
$962$	4.95706	0.159822
$963$	0	0
$964$	24.8751	0.801172
$965$	−12.1543	−0.391260
$966$	0	0
$967$	38.7813	1.24712	0.623561	−	0.781774i	$-0.285685\pi$
$967$	38.7813	1.24712	0.623561	+	0.781774i	$0.285685\pi$
$968$	−1.97777	−0.0635678
$969$	0	0
$970$	2.96501	0.0952006
$971$	11.0717	0.355308	0.177654	−	0.984093i	$-0.443149\pi$
$971$	11.0717	0.355308	0.177654	+	0.984093i	$0.443149\pi$
$972$	0	0
$973$	10.5817	0.339235
$974$	3.34970	0.107331
$975$	0	0
$976$	3.74954	0.120020
$977$	13.1209	0.419774	0.209887	−	0.977726i	$-0.432690\pi$
$977$	13.1209	0.419774	0.209887	+	0.977726i	$0.432690\pi$
$978$	0	0
$979$	60.1695	1.92302
$980$	46.4478	1.48372
$981$	0	0
$982$	−1.64139	−0.0523787
$983$	−43.2504	−1.37947	−0.689736	−	0.724061i	$-0.742274\pi$
$983$	−43.2504	−1.37947	−0.689736	+	0.724061i	$0.742274\pi$
$984$	0	0
$985$	−89.3908	−2.84823
$986$	0.177382	0.00564899
$987$	0	0
$988$	−52.9026	−1.68306
$989$	7.38027	0.234679
$990$	0	0
$991$	39.2829	1.24786	0.623932	−	0.781479i	$-0.285534\pi$
$991$	39.2829	1.24786	0.623932	+	0.781479i	$0.285534\pi$
$992$	−0.491437	−0.0156031
$993$	0	0
$994$	−0.0668311	−0.00211975
$995$	76.2451	2.41713
$996$	0	0
$997$	−49.1510	−1.55663	−0.778314	−	0.627876i	$-0.783925\pi$
$997$	−49.1510	−1.55663	−0.778314	+	0.627876i	$0.783925\pi$
$998$	−0.541493	−0.0171407
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.w.1.16	yes	30
3.2	odd	2		6003.2.a.v.1.15	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.v.1.15	✓	30	3.2	odd	2
6003.2.a.w.1.16	yes	30	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0795257 q^{2} -1.99368 q^{4} +3.50033 q^{5} +0.586662 q^{7} -0.317600 q^{8} +O(q^{10})\)	\(q+0.0795257 q^{2} -1.99368 q^{4} +3.50033 q^{5} +0.586662 q^{7} -0.317600 q^{8} +0.278366 q^{10} -4.15057 q^{11} +7.11477 q^{13} +0.0466547 q^{14} +3.96209 q^{16} +2.23050 q^{17} +3.72959 q^{19} -6.97852 q^{20} -0.330077 q^{22} -1.00000 q^{23} +7.25228 q^{25} +0.565807 q^{26} -1.16961 q^{28} +1.00000 q^{29} -0.517145 q^{31} +0.950288 q^{32} +0.177382 q^{34} +2.05351 q^{35} +8.76104 q^{37} +0.296598 q^{38} -1.11170 q^{40} -0.620838 q^{41} -7.38027 q^{43} +8.27489 q^{44} -0.0795257 q^{46} +9.76679 q^{47} -6.65583 q^{49} +0.576743 q^{50} -14.1846 q^{52} -3.66930 q^{53} -14.5283 q^{55} -0.186324 q^{56} +0.0795257 q^{58} -5.72933 q^{59} +0.946352 q^{61} -0.0411263 q^{62} -7.84862 q^{64} +24.9040 q^{65} -10.6437 q^{67} -4.44689 q^{68} +0.163307 q^{70} -1.43246 q^{71} +13.3793 q^{73} +0.696728 q^{74} -7.43559 q^{76} -2.43498 q^{77} +0.300819 q^{79} +13.8686 q^{80} -0.0493726 q^{82} +8.47956 q^{83} +7.80747 q^{85} -0.586921 q^{86} +1.31822 q^{88} -14.4967 q^{89} +4.17397 q^{91} +1.99368 q^{92} +0.776710 q^{94} +13.0548 q^{95} +10.6515 q^{97} -0.529309 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8}+O(q^{10}) \) 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

Groups

Database

Properties

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