LMFDB - Embedded newform 6003.2.a.s.1.18

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 33 x^{18} + 64 x^{17} + 453 x^{16} - 846 x^{15} - 3353 x^{14} + 5985 x^{13} + \cdots - 256 $ x^20 - 2x^19 - 33x^18 + 64x^17 + 453x^16 - 846x^15 - 3353x^14 + 5985x^13 + 14484x^12 - 24566x^11 - 36791x^10 + 59410x^9 + 52109x^8 - 82362x^7 - 34967x^6 + 60661x^5 + 5201x^4 - 19624x^3 + 2768x^2 + 1408*x - 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Root			$-2.40422$ 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)

$f(q)$	$=$	$q+2.40422 q^{2} +3.78028 q^{4} -2.94625 q^{5} +2.89217 q^{7} +4.28018 q^{8} +O(q^{10})$	\(q+2.40422 q^{2} +3.78028 q^{4} -2.94625 q^{5} +2.89217 q^{7} +4.28018 q^{8} -7.08343 q^{10} +2.53497 q^{11} +5.47922 q^{13} +6.95342 q^{14} +2.72995 q^{16} +3.33691 q^{17} -5.49162 q^{19} -11.1376 q^{20} +6.09464 q^{22} -1.00000 q^{23} +3.68037 q^{25} +13.1732 q^{26} +10.9332 q^{28} -1.00000 q^{29} +5.60055 q^{31} -1.99697 q^{32} +8.02267 q^{34} -8.52105 q^{35} +4.06631 q^{37} -13.2031 q^{38} -12.6105 q^{40} +9.72536 q^{41} +2.73471 q^{43} +9.58291 q^{44} -2.40422 q^{46} -10.0668 q^{47} +1.36466 q^{49} +8.84842 q^{50} +20.7130 q^{52} -1.45207 q^{53} -7.46866 q^{55} +12.3790 q^{56} -2.40422 q^{58} -13.0989 q^{59} +14.3608 q^{61} +13.4650 q^{62} -10.2610 q^{64} -16.1431 q^{65} +5.14596 q^{67} +12.6144 q^{68} -20.4865 q^{70} +3.43935 q^{71} +6.90003 q^{73} +9.77631 q^{74} -20.7598 q^{76} +7.33158 q^{77} -5.51601 q^{79} -8.04310 q^{80} +23.3819 q^{82} -0.148675 q^{83} -9.83136 q^{85} +6.57486 q^{86} +10.8502 q^{88} +10.6836 q^{89} +15.8468 q^{91} -3.78028 q^{92} -24.2028 q^{94} +16.1797 q^{95} +7.16615 q^{97} +3.28094 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) $ 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8	$ 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8} + 7 q^{10} + 21 q^{13} + q^{14} + 58 q^{16} + 4 q^{17} + 7 q^{19} + 20 q^{20} + 7 q^{22} - 20 q^{23} + 47 q^{25} - 8 q^{26} + 11 q^{28} - 20 q^{29} + 28 q^{31} - 14 q^{32} + 16 q^{34} - 9 q^{35} + 14 q^{37} + 20 q^{38} + 34 q^{40} - 7 q^{41} + 3 q^{43} + q^{44} + 2 q^{46} - 3 q^{47} + 35 q^{49} + 24 q^{50} + 73 q^{52} + 19 q^{53} + 29 q^{55} + 30 q^{56} + 2 q^{58} - 20 q^{59} + 15 q^{61} - 12 q^{62} + 82 q^{64} + 28 q^{65} + 20 q^{67} + 23 q^{68} - 24 q^{70} - 63 q^{71} + 19 q^{73} - 16 q^{74} - 44 q^{76} + 7 q^{77} + 32 q^{79} + 56 q^{80} - 20 q^{82} + 21 q^{83} + 4 q^{85} + 6 q^{86} + 55 q^{88} + 13 q^{89} + 70 q^{91} - 30 q^{92} - 12 q^{94} - 9 q^{95} - 9 q^{97} - 31 q^{98}+O(q^{100}) $ 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8 + 7 * q^10 + 21 * q^13 + q^14 + 58 * q^16 + 4 * q^17 + 7 * q^19 + 20 * q^20 + 7 * q^22 - 20 * q^23 + 47 * q^25 - 8 * q^26 + 11 * q^28 - 20 * q^29 + 28 * q^31 - 14 * q^32 + 16 * q^34 - 9 * q^35 + 14 * q^37 + 20 * q^38 + 34 * q^40 - 7 * q^41 + 3 * q^43 + q^44 + 2 * q^46 - 3 * q^47 + 35 * q^49 + 24 * q^50 + 73 * q^52 + 19 * q^53 + 29 * q^55 + 30 * q^56 + 2 * q^58 - 20 * q^59 + 15 * q^61 - 12 * q^62 + 82 * q^64 + 28 * q^65 + 20 * q^67 + 23 * q^68 - 24 * q^70 - 63 * q^71 + 19 * q^73 - 16 * q^74 - 44 * q^76 + 7 * q^77 + 32 * q^79 + 56 * q^80 - 20 * q^82 + 21 * q^83 + 4 * q^85 + 6 * q^86 + 55 * q^88 + 13 * q^89 + 70 * q^91 - 30 * q^92 - 12 * q^94 - 9 * q^95 - 9 * q^97 - 31 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.40422	1.70004	0.850020	−	0.526750i	$-0.176589\pi$
$2$	2.40422	1.70004	0.850020	+	0.526750i	$0.176589\pi$
$3$	0	0
$3$	0	0
$4$	3.78028	1.89014
$5$	−2.94625	−1.31760	−0.658801	−	0.752317i	$-0.728936\pi$
$5$	−2.94625	−1.31760	−0.658801	+	0.752317i	$0.728936\pi$
$6$	0	0
$7$	2.89217	1.09314	0.546569	−	0.837414i	$-0.315934\pi$
$7$	2.89217	1.09314	0.546569	+	0.837414i	$0.315934\pi$
$8$	4.28018	1.51327
$9$	0	0
$10$	−7.08343	−2.23998
$11$	2.53497	0.764323	0.382162	−	0.924095i	$-0.375180\pi$
$11$	2.53497	0.764323	0.382162	+	0.924095i	$0.375180\pi$
$12$	0	0
$13$	5.47922	1.51966	0.759831	−	0.650121i	$-0.225282\pi$
$13$	5.47922	1.51966	0.759831	+	0.650121i	$0.225282\pi$
$14$	6.95342	1.85838
$15$	0	0
$16$	2.72995	0.682487
$17$	3.33691	0.809319	0.404660	−	0.914467i	$-0.367390\pi$
$17$	3.33691	0.809319	0.404660	+	0.914467i	$0.367390\pi$
$18$	0	0
$19$	−5.49162	−1.25986	−0.629932	−	0.776651i	$-0.716917\pi$
$19$	−5.49162	−1.25986	−0.629932	+	0.776651i	$0.716917\pi$
$20$	−11.1376	−2.49045
$21$	0	0
$22$	6.09464	1.29938
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	3.68037	0.736074
$26$	13.1732	2.58349
$27$	0	0
$28$	10.9332	2.06618
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	5.60055	1.00589	0.502944	−	0.864319i	$-0.332250\pi$
$31$	5.60055	1.00589	0.502944	+	0.864319i	$0.332250\pi$
$32$	−1.99697	−0.353018
$33$	0	0
$34$	8.02267	1.37588
$35$	−8.52105	−1.44032
$36$	0	0
$37$	4.06631	0.668497	0.334249	−	0.942485i	$-0.391517\pi$
$37$	4.06631	0.668497	0.334249	+	0.942485i	$0.391517\pi$
$38$	−13.2031	−2.14182
$39$	0	0
$40$	−12.6105	−1.99389
$41$	9.72536	1.51885	0.759423	−	0.650597i	$-0.225481\pi$
$41$	9.72536	1.51885	0.759423	+	0.650597i	$0.225481\pi$
$42$	0	0
$43$	2.73471	0.417040	0.208520	−	0.978018i	$-0.433135\pi$
$43$	2.73471	0.417040	0.208520	+	0.978018i	$0.433135\pi$
$44$	9.58291	1.44468
$45$	0	0
$46$	−2.40422	−0.354483
$47$	−10.0668	−1.46839	−0.734197	−	0.678937i	$-0.762441\pi$
$47$	−10.0668	−1.46839	−0.734197	+	0.678937i	$0.762441\pi$
$48$	0	0
$49$	1.36466	0.194951
$50$	8.84842	1.25136
$51$	0	0
$52$	20.7130	2.87237
$53$	−1.45207	−0.199458	−0.0997288	−	0.995015i	$-0.531798\pi$
$53$	−1.45207	−0.199458	−0.0997288	+	0.995015i	$0.531798\pi$
$54$	0	0
$55$	−7.46866	−1.00707
$56$	12.3790	1.65422
$57$	0	0
$58$	−2.40422	−0.315690
$59$	−13.0989	−1.70534	−0.852669	−	0.522452i	$-0.825018\pi$
$59$	−13.0989	−1.70534	−0.852669	+	0.522452i	$0.825018\pi$
$60$	0	0
$61$	14.3608	1.83871	0.919354	−	0.393430i	$-0.128712\pi$
$61$	14.3608	1.83871	0.919354	+	0.393430i	$0.128712\pi$
$62$	13.4650	1.71005
$63$	0	0
$64$	−10.2610	−1.28263
$65$	−16.1431	−2.00231
$66$	0	0
$67$	5.14596	0.628679	0.314339	−	0.949311i	$-0.398217\pi$
$67$	5.14596	0.628679	0.314339	+	0.949311i	$0.398217\pi$
$68$	12.6144	1.52973
$69$	0	0
$70$	−20.4865	−2.44860
$71$	3.43935	0.408176	0.204088	−	0.978953i	$-0.434577\pi$
$71$	3.43935	0.408176	0.204088	+	0.978953i	$0.434577\pi$
$72$	0	0
$73$	6.90003	0.807587	0.403794	−	0.914850i	$-0.367691\pi$
$73$	6.90003	0.807587	0.403794	+	0.914850i	$0.367691\pi$
$74$	9.77631	1.13647
$75$	0	0
$76$	−20.7598	−2.38132
$77$	7.33158	0.835511
$78$	0	0
$79$	−5.51601	−0.620599	−0.310300	−	0.950639i	$-0.600429\pi$
$79$	−5.51601	−0.620599	−0.310300	+	0.950639i	$0.600429\pi$
$80$	−8.04310	−0.899245
$81$	0	0
$82$	23.3819	2.58210
$83$	−0.148675	−0.0163192	−0.00815958	−	0.999967i	$-0.502597\pi$
$83$	−0.148675	−0.0163192	−0.00815958	+	0.999967i	$0.502597\pi$
$84$	0	0
$85$	−9.83136	−1.06636
$86$	6.57486	0.708985
$87$	0	0
$88$	10.8502	1.15663
$89$	10.6836	1.13246	0.566232	−	0.824246i	$-0.308401\pi$
$89$	10.6836	1.13246	0.566232	+	0.824246i	$0.308401\pi$
$90$	0	0
$91$	15.8468	1.66120
$92$	−3.78028	−0.394121
$93$	0	0
$94$	−24.2028	−2.49633
$95$	16.1797	1.66000
$96$	0	0
$97$	7.16615	0.727612	0.363806	−	0.931475i	$-0.381477\pi$
$97$	7.16615	0.727612	0.363806	+	0.931475i	$0.381477\pi$
$98$	3.28094	0.331425
$99$	0	0
$100$	13.9128	1.39128
$101$	6.87796	0.684383	0.342191	−	0.939630i	$-0.388831\pi$
$101$	6.87796	0.684383	0.342191	+	0.939630i	$0.388831\pi$
$102$	0	0
$103$	−2.72286	−0.268292	−0.134146	−	0.990962i	$-0.542829\pi$
$103$	−2.72286	−0.268292	−0.134146	+	0.990962i	$0.542829\pi$
$104$	23.4520	2.29966
$105$	0	0
$106$	−3.49111	−0.339086
$107$	9.21917	0.891250	0.445625	−	0.895220i	$-0.352981\pi$
$107$	9.21917	0.891250	0.445625	+	0.895220i	$0.352981\pi$
$108$	0	0
$109$	−0.964241	−0.0923575	−0.0461788	−	0.998933i	$-0.514704\pi$
$109$	−0.964241	−0.0923575	−0.0461788	+	0.998933i	$0.514704\pi$
$110$	−17.9563	−1.71207
$111$	0	0
$112$	7.89548	0.746052
$113$	4.79493	0.451070	0.225535	−	0.974235i	$-0.427587\pi$
$113$	4.79493	0.451070	0.225535	+	0.974235i	$0.427587\pi$
$114$	0	0
$115$	2.94625	0.274739
$116$	−3.78028	−0.350990
$117$	0	0
$118$	−31.4928	−2.89914
$119$	9.65092	0.884698
$120$	0	0
$121$	−4.57391	−0.415810
$122$	34.5265	3.12588
$123$	0	0
$124$	21.1716	1.90127
$125$	3.88796	0.347749
$126$	0	0
$127$	−3.74468	−0.332287	−0.166144	−	0.986102i	$-0.553131\pi$
$127$	−3.74468	−0.332287	−0.166144	+	0.986102i	$0.553131\pi$
$128$	−20.6759	−1.82751
$129$	0	0
$130$	−38.8116	−3.40401
$131$	−16.6236	−1.45241	−0.726207	−	0.687477i	$-0.758718\pi$
$131$	−16.6236	−1.45241	−0.726207	+	0.687477i	$0.758718\pi$
$132$	0	0
$133$	−15.8827	−1.37720
$134$	12.3720	1.06878
$135$	0	0
$136$	14.2826	1.22472
$137$	15.9108	1.35935	0.679677	−	0.733511i	$-0.262120\pi$
$137$	15.9108	1.35935	0.679677	+	0.733511i	$0.262120\pi$
$138$	0	0
$139$	21.1653	1.79522	0.897610	−	0.440791i	$-0.145302\pi$
$139$	21.1653	1.79522	0.897610	+	0.440791i	$0.145302\pi$
$140$	−32.2119	−2.72241
$141$	0	0
$142$	8.26896	0.693916
$143$	13.8897	1.16151
$144$	0	0
$145$	2.94625	0.244672
$146$	16.5892	1.37293
$147$	0	0
$148$	15.3718	1.26355
$149$	19.8722	1.62799	0.813995	−	0.580871i	$-0.197288\pi$
$149$	19.8722	1.62799	0.813995	+	0.580871i	$0.197288\pi$
$150$	0	0
$151$	−8.85747	−0.720811	−0.360405	−	0.932796i	$-0.617362\pi$
$151$	−8.85747	−0.720811	−0.360405	+	0.932796i	$0.617362\pi$
$152$	−23.5051	−1.90652
$153$	0	0
$154$	17.6267	1.42040
$155$	−16.5006	−1.32536
$156$	0	0
$157$	−7.35129	−0.586697	−0.293349	−	0.956006i	$-0.594770\pi$
$157$	−7.35129	−0.586697	−0.293349	+	0.956006i	$0.594770\pi$
$158$	−13.2617	−1.05504
$159$	0	0
$160$	5.88356	0.465137
$161$	−2.89217	−0.227935
$162$	0	0
$163$	15.3238	1.20025	0.600124	−	0.799907i	$-0.295118\pi$
$163$	15.3238	1.20025	0.600124	+	0.799907i	$0.295118\pi$
$164$	36.7646	2.87083
$165$	0	0
$166$	−0.357446	−0.0277432
$167$	−14.1921	−1.09822	−0.549111	−	0.835750i	$-0.685033\pi$
$167$	−14.1921	−1.09822	−0.549111	+	0.835750i	$0.685033\pi$
$168$	0	0
$169$	17.0218	1.30937
$170$	−23.6368	−1.81286
$171$	0	0
$172$	10.3380	0.788264
$173$	7.20007	0.547411	0.273706	−	0.961814i	$-0.411751\pi$
$173$	7.20007	0.547411	0.273706	+	0.961814i	$0.411751\pi$
$174$	0	0
$175$	10.6443	0.804631
$176$	6.92034	0.521641
$177$	0	0
$178$	25.6858	1.92523
$179$	−25.1099	−1.87680	−0.938399	−	0.345553i	$-0.887692\pi$
$179$	−25.1099	−1.87680	−0.938399	+	0.345553i	$0.887692\pi$
$180$	0	0
$181$	2.09541	0.155750	0.0778752	−	0.996963i	$-0.475186\pi$
$181$	2.09541	0.155750	0.0778752	+	0.996963i	$0.475186\pi$
$182$	38.0993	2.82411
$183$	0	0
$184$	−4.28018	−0.315539
$185$	−11.9804	−0.880813
$186$	0	0
$187$	8.45898	0.618582
$188$	−38.0553	−2.77547
$189$	0	0
$190$	38.8995	2.82206
$191$	−13.3115	−0.963183	−0.481592	−	0.876396i	$-0.659941\pi$
$191$	−13.3115	−0.963183	−0.481592	+	0.876396i	$0.659941\pi$
$192$	0	0
$193$	−11.8471	−0.852772	−0.426386	−	0.904541i	$-0.640213\pi$
$193$	−11.8471	−0.852772	−0.426386	+	0.904541i	$0.640213\pi$
$194$	17.2290	1.23697
$195$	0	0
$196$	5.15879	0.368485
$197$	8.41802	0.599759	0.299880	−	0.953977i	$-0.403054\pi$
$197$	8.41802	0.599759	0.299880	+	0.953977i	$0.403054\pi$
$198$	0	0
$199$	−10.0318	−0.711134	−0.355567	−	0.934651i	$-0.615712\pi$
$199$	−10.0318	−0.711134	−0.355567	+	0.934651i	$0.615712\pi$
$200$	15.7527	1.11388
$201$	0	0
$202$	16.5361	1.16348
$203$	−2.89217	−0.202991
$204$	0	0
$205$	−28.6533	−2.00123
$206$	−6.54637	−0.456107
$207$	0	0
$208$	14.9580	1.03715
$209$	−13.9211	−0.962943
$210$	0	0
$211$	−26.3386	−1.81323	−0.906613	−	0.421963i	$-0.861341\pi$
$211$	−26.3386	−1.81323	−0.906613	+	0.421963i	$0.861341\pi$
$212$	−5.48924	−0.377003
$213$	0	0
$214$	22.1649	1.51516
$215$	−8.05714	−0.549493
$216$	0	0
$217$	16.1978	1.09957
$218$	−2.31825	−0.157012
$219$	0	0
$220$	−28.2336	−1.90351
$221$	18.2836	1.22989
$222$	0	0
$223$	−21.2327	−1.42185	−0.710923	−	0.703270i	$-0.751723\pi$
$223$	−21.2327	−1.42185	−0.710923	+	0.703270i	$0.751723\pi$
$224$	−5.77558	−0.385897
$225$	0	0
$226$	11.5281	0.766837
$227$	−23.8417	−1.58243	−0.791213	−	0.611541i	$-0.790550\pi$
$227$	−23.8417	−1.58243	−0.791213	+	0.611541i	$0.790550\pi$
$228$	0	0
$229$	16.4089	1.08433	0.542165	−	0.840272i	$-0.317605\pi$
$229$	16.4089	1.08433	0.542165	+	0.840272i	$0.317605\pi$
$230$	7.08343	0.467067
$231$	0	0
$232$	−4.28018	−0.281008
$233$	13.3058	0.871690	0.435845	−	0.900022i	$-0.356450\pi$
$233$	13.3058	0.871690	0.435845	+	0.900022i	$0.356450\pi$
$234$	0	0
$235$	29.6593	1.93476
$236$	−49.5177	−3.22333
$237$	0	0
$238$	23.2029	1.50402
$239$	−14.4324	−0.933555	−0.466777	−	0.884375i	$-0.654585\pi$
$239$	−14.4324	−0.933555	−0.466777	+	0.884375i	$0.654585\pi$
$240$	0	0
$241$	−5.94893	−0.383204	−0.191602	−	0.981473i	$-0.561368\pi$
$241$	−5.94893	−0.383204	−0.191602	+	0.981473i	$0.561368\pi$
$242$	−10.9967	−0.706893
$243$	0	0
$244$	54.2877	3.47542
$245$	−4.02062	−0.256868
$246$	0	0
$247$	−30.0898	−1.91457
$248$	23.9714	1.52218
$249$	0	0
$250$	9.34751	0.591188
$251$	15.4453	0.974901	0.487451	−	0.873151i	$-0.337927\pi$
$251$	15.4453	0.974901	0.487451	+	0.873151i	$0.337927\pi$
$252$	0	0
$253$	−2.53497	−0.159372
$254$	−9.00305	−0.564901
$255$	0	0
$256$	−29.1873	−1.82421
$257$	−23.3758	−1.45814	−0.729070	−	0.684439i	$-0.760047\pi$
$257$	−23.3758	−1.45814	−0.729070	+	0.684439i	$0.760047\pi$
$258$	0	0
$259$	11.7605	0.730760
$260$	−61.0255	−3.78464
$261$	0	0
$262$	−39.9669	−2.46916
$263$	−3.88612	−0.239629	−0.119814	−	0.992796i	$-0.538230\pi$
$263$	−3.88612	−0.239629	−0.119814	+	0.992796i	$0.538230\pi$
$264$	0	0
$265$	4.27817	0.262806
$266$	−38.1855	−2.34130
$267$	0	0
$268$	19.4531	1.18829
$269$	13.4486	0.819977	0.409989	−	0.912091i	$-0.365533\pi$
$269$	13.4486	0.819977	0.409989	+	0.912091i	$0.365533\pi$
$270$	0	0
$271$	18.2914	1.11113	0.555563	−	0.831475i	$-0.312503\pi$
$271$	18.2914	1.11113	0.555563	+	0.831475i	$0.312503\pi$
$272$	9.10958	0.552350
$273$	0	0
$274$	38.2532	2.31096
$275$	9.32964	0.562599
$276$	0	0
$277$	24.3837	1.46507	0.732537	−	0.680728i	$-0.238336\pi$
$277$	24.3837	1.46507	0.732537	+	0.680728i	$0.238336\pi$
$278$	50.8861	3.05195
$279$	0	0
$280$	−36.4717	−2.17960
$281$	7.44029	0.443850	0.221925	−	0.975064i	$-0.428766\pi$
$281$	7.44029	0.443850	0.221925	+	0.975064i	$0.428766\pi$
$282$	0	0
$283$	−27.7217	−1.64788	−0.823942	−	0.566674i	$-0.808230\pi$
$283$	−27.7217	−1.64788	−0.823942	+	0.566674i	$0.808230\pi$
$284$	13.0017	0.771510
$285$	0	0
$286$	33.3938	1.97462
$287$	28.1274	1.66031
$288$	0	0
$289$	−5.86503	−0.345002
$290$	7.08343	0.415953
$291$	0	0
$292$	26.0840	1.52645
$293$	9.91319	0.579135	0.289567	−	0.957158i	$-0.406489\pi$
$293$	9.91319	0.579135	0.289567	+	0.957158i	$0.406489\pi$
$294$	0	0
$295$	38.5927	2.24696
$296$	17.4045	1.01162
$297$	0	0
$298$	47.7771	2.76765
$299$	−5.47922	−0.316871
$300$	0	0
$301$	7.90927	0.455883
$302$	−21.2953	−1.22541
$303$	0	0
$304$	−14.9918	−0.859840
$305$	−42.3104	−2.42269
$306$	0	0
$307$	11.4819	0.655304	0.327652	−	0.944799i	$-0.393743\pi$
$307$	11.4819	0.655304	0.327652	+	0.944799i	$0.393743\pi$
$308$	27.7154	1.57923
$309$	0	0
$310$	−39.6711	−2.25317
$311$	9.92846	0.562992	0.281496	−	0.959562i	$-0.409169\pi$
$311$	9.92846	0.562992	0.281496	+	0.959562i	$0.409169\pi$
$312$	0	0
$313$	−4.22620	−0.238879	−0.119440	−	0.992841i	$-0.538110\pi$
$313$	−4.22620	−0.238879	−0.119440	+	0.992841i	$0.538110\pi$
$314$	−17.6741	−0.997409
$315$	0	0
$316$	−20.8520	−1.17302
$317$	−0.238908	−0.0134184	−0.00670919	−	0.999977i	$-0.502136\pi$
$317$	−0.238908	−0.0134184	−0.00670919	+	0.999977i	$0.502136\pi$
$318$	0	0
$319$	−2.53497	−0.141931
$320$	30.2316	1.69000
$321$	0	0
$322$	−6.95342	−0.387499
$323$	−18.3250	−1.01963
$324$	0	0
$325$	20.1655	1.11858
$326$	36.8417	2.04047
$327$	0	0
$328$	41.6263	2.29843
$329$	−29.1149	−1.60516
$330$	0	0
$331$	15.9383	0.876049	0.438024	−	0.898963i	$-0.355678\pi$
$331$	15.9383	0.876049	0.438024	+	0.898963i	$0.355678\pi$
$332$	−0.562031	−0.0308455
$333$	0	0
$334$	−34.1210	−1.86702
$335$	−15.1613	−0.828348
$336$	0	0
$337$	10.3579	0.564231	0.282116	−	0.959380i	$-0.408964\pi$
$337$	10.3579	0.564231	0.282116	+	0.959380i	$0.408964\pi$
$338$	40.9242	2.22598
$339$	0	0
$340$	−37.1653	−2.01557
$341$	14.1972	0.768824
$342$	0	0
$343$	−16.2984	−0.880030
$344$	11.7051	0.631095
$345$	0	0
$346$	17.3106	0.930621
$347$	21.5048	1.15444	0.577218	−	0.816590i	$-0.304138\pi$
$347$	21.5048	1.15444	0.577218	+	0.816590i	$0.304138\pi$
$348$	0	0
$349$	−16.9041	−0.904857	−0.452429	−	0.891801i	$-0.649442\pi$
$349$	−16.9041	−0.904857	−0.452429	+	0.891801i	$0.649442\pi$
$350$	25.5912	1.36790
$351$	0	0
$352$	−5.06227	−0.269820
$353$	21.8094	1.16080	0.580398	−	0.814333i	$-0.302897\pi$
$353$	21.8094	1.16080	0.580398	+	0.814333i	$0.302897\pi$
$354$	0	0
$355$	−10.1332	−0.537814
$356$	40.3871	2.14051
$357$	0	0
$358$	−60.3696	−3.19063
$359$	5.95246	0.314159	0.157080	−	0.987586i	$-0.449792\pi$
$359$	5.95246	0.314159	0.157080	+	0.987586i	$0.449792\pi$
$360$	0	0
$361$	11.1579	0.587256
$362$	5.03782	0.264782
$363$	0	0
$364$	59.9054	3.13990
$365$	−20.3292	−1.06408
$366$	0	0
$367$	7.76989	0.405585	0.202793	−	0.979222i	$-0.434998\pi$
$367$	7.76989	0.405585	0.202793	+	0.979222i	$0.434998\pi$
$368$	−2.72995	−0.142308
$369$	0	0
$370$	−28.8034	−1.49742
$371$	−4.19965	−0.218035
$372$	0	0
$373$	−30.8504	−1.59737	−0.798687	−	0.601747i	$-0.794472\pi$
$373$	−30.8504	−1.59737	−0.798687	+	0.601747i	$0.794472\pi$
$374$	20.3373	1.05161
$375$	0	0
$376$	−43.0877	−2.22208
$377$	−5.47922	−0.282194
$378$	0	0
$379$	−35.0094	−1.79831	−0.899157	−	0.437627i	$-0.855819\pi$
$379$	−35.0094	−1.79831	−0.899157	+	0.437627i	$0.855819\pi$
$380$	61.1636	3.13763
$381$	0	0
$382$	−32.0037	−1.63745
$383$	6.69965	0.342336	0.171168	−	0.985242i	$-0.445246\pi$
$383$	6.69965	0.342336	0.171168	+	0.985242i	$0.445246\pi$
$384$	0	0
$385$	−21.6006	−1.10087
$386$	−28.4830	−1.44975
$387$	0	0
$388$	27.0900	1.37529
$389$	−29.4790	−1.49464	−0.747321	−	0.664463i	$-0.768660\pi$
$389$	−29.4790	−1.49464	−0.747321	+	0.664463i	$0.768660\pi$
$390$	0	0
$391$	−3.33691	−0.168755
$392$	5.84099	0.295015
$393$	0	0
$394$	20.2388	1.01961
$395$	16.2515	0.817703
$396$	0	0
$397$	0.826533	0.0414825	0.0207413	−	0.999785i	$-0.493397\pi$
$397$	0.826533	0.0414825	0.0207413	+	0.999785i	$0.493397\pi$
$398$	−24.1186	−1.20896
$399$	0	0
$400$	10.0472	0.502361
$401$	−36.7446	−1.83494	−0.917469	−	0.397808i	$-0.869771\pi$
$401$	−36.7446	−1.83494	−0.917469	+	0.397808i	$0.869771\pi$
$402$	0	0
$403$	30.6866	1.52861
$404$	26.0006	1.29358
$405$	0	0
$406$	−6.95342	−0.345092
$407$	10.3080	0.510948
$408$	0	0
$409$	−28.9646	−1.43221	−0.716103	−	0.697995i	$-0.754076\pi$
$409$	−28.9646	−1.43221	−0.716103	+	0.697995i	$0.754076\pi$
$410$	−68.8889	−3.40218
$411$	0	0
$412$	−10.2932	−0.507109
$413$	−37.8844	−1.86417
$414$	0	0
$415$	0.438032	0.0215021
$416$	−10.9418	−0.536467
$417$	0	0
$418$	−33.4694	−1.63704
$419$	16.0034	0.781818	0.390909	−	0.920429i	$-0.372161\pi$
$419$	16.0034	0.781818	0.390909	+	0.920429i	$0.372161\pi$
$420$	0	0
$421$	−13.3663	−0.651435	−0.325718	−	0.945467i	$-0.605606\pi$
$421$	−13.3663	−0.651435	−0.325718	+	0.945467i	$0.605606\pi$
$422$	−63.3239	−3.08256
$423$	0	0
$424$	−6.21514	−0.301834
$425$	12.2811	0.595719
$426$	0	0
$427$	41.5338	2.00996
$428$	34.8510	1.68459
$429$	0	0
$430$	−19.3712	−0.934160
$431$	−21.6711	−1.04386	−0.521931	−	0.852988i	$-0.674788\pi$
$431$	−21.6711	−1.04386	−0.521931	+	0.852988i	$0.674788\pi$
$432$	0	0
$433$	−9.86289	−0.473980	−0.236990	−	0.971512i	$-0.576161\pi$
$433$	−9.86289	−0.473980	−0.236990	+	0.971512i	$0.576161\pi$
$434$	38.9430	1.86932
$435$	0	0
$436$	−3.64510	−0.174569
$437$	5.49162	0.262700
$438$	0	0
$439$	−1.77488	−0.0847104	−0.0423552	−	0.999103i	$-0.513486\pi$
$439$	−1.77488	−0.0847104	−0.0423552	+	0.999103i	$0.513486\pi$
$440$	−31.9672	−1.52398
$441$	0	0
$442$	43.9579	2.09087
$443$	−12.0886	−0.574345	−0.287173	−	0.957879i	$-0.592715\pi$
$443$	−12.0886	−0.574345	−0.287173	+	0.957879i	$0.592715\pi$
$444$	0	0
$445$	−31.4766	−1.49214
$446$	−51.0481	−2.41720
$447$	0	0
$448$	−29.6767	−1.40209
$449$	−36.3803	−1.71689	−0.858447	−	0.512903i	$-0.828570\pi$
$449$	−36.3803	−1.71689	−0.858447	+	0.512903i	$0.828570\pi$
$450$	0	0
$451$	24.6535	1.16089
$452$	18.1262	0.852584
$453$	0	0
$454$	−57.3206	−2.69019
$455$	−46.6887	−2.18880
$456$	0	0
$457$	−29.5734	−1.38338	−0.691692	−	0.722193i	$-0.743134\pi$
$457$	−29.5734	−1.38338	−0.691692	+	0.722193i	$0.743134\pi$
$458$	39.4506	1.84340
$459$	0	0
$460$	11.1376	0.519295
$461$	8.92222	0.415549	0.207775	−	0.978177i	$-0.433378\pi$
$461$	8.92222	0.415549	0.207775	+	0.978177i	$0.433378\pi$
$462$	0	0
$463$	−2.14871	−0.0998593	−0.0499296	−	0.998753i	$-0.515900\pi$
$463$	−2.14871	−0.0998593	−0.0499296	+	0.998753i	$0.515900\pi$
$464$	−2.72995	−0.126735
$465$	0	0
$466$	31.9900	1.48191
$467$	4.84533	0.224215	0.112108	−	0.993696i	$-0.464240\pi$
$467$	4.84533	0.224215	0.112108	+	0.993696i	$0.464240\pi$
$468$	0	0
$469$	14.8830	0.687233
$470$	71.3074	3.28917
$471$	0	0
$472$	−56.0659	−2.58064
$473$	6.93243	0.318754
$474$	0	0
$475$	−20.2112	−0.927353
$476$	36.4831	1.67220
$477$	0	0
$478$	−34.6987	−1.58708
$479$	37.4306	1.71025	0.855124	−	0.518424i	$-0.173481\pi$
$479$	37.4306	1.71025	0.855124	+	0.518424i	$0.173481\pi$
$480$	0	0
$481$	22.2802	1.01589
$482$	−14.3025	−0.651462
$483$	0	0
$484$	−17.2906	−0.785938
$485$	−21.1132	−0.958703
$486$	0	0
$487$	−39.3534	−1.78327	−0.891636	−	0.452753i	$-0.850442\pi$
$487$	−39.3534	−1.78327	−0.891636	+	0.452753i	$0.850442\pi$
$488$	61.4667	2.78247
$489$	0	0
$490$	−9.66646	−0.436686
$491$	6.74588	0.304437	0.152219	−	0.988347i	$-0.451358\pi$
$491$	6.74588	0.304437	0.152219	+	0.988347i	$0.451358\pi$
$492$	0	0
$493$	−3.33691	−0.150287
$494$	−72.3424	−3.25484
$495$	0	0
$496$	15.2892	0.686505
$497$	9.94720	0.446193
$498$	0	0
$499$	−41.1634	−1.84273	−0.921364	−	0.388701i	$-0.872924\pi$
$499$	−41.1634	−1.84273	−0.921364	+	0.388701i	$0.872924\pi$
$500$	14.6976	0.657295
$501$	0	0
$502$	37.1340	1.65737
$503$	−16.1315	−0.719267	−0.359634	−	0.933094i	$-0.617098\pi$
$503$	−16.1315	−0.719267	−0.359634	+	0.933094i	$0.617098\pi$
$504$	0	0
$505$	−20.2642	−0.901744
$506$	−6.09464	−0.270940
$507$	0	0
$508$	−14.1559	−0.628069
$509$	−25.5786	−1.13375	−0.566875	−	0.823804i	$-0.691848\pi$
$509$	−25.5786	−1.13375	−0.566875	+	0.823804i	$0.691848\pi$
$510$	0	0
$511$	19.9561	0.882805
$512$	−28.8210	−1.27372
$513$	0	0
$514$	−56.2005	−2.47890
$515$	8.02223	0.353502
$516$	0	0
$517$	−25.5191	−1.12233
$518$	28.2748	1.24232
$519$	0	0
$520$	−69.0955	−3.03004
$521$	−1.33903	−0.0586639	−0.0293319	−	0.999570i	$-0.509338\pi$
$521$	−1.33903	−0.0586639	−0.0293319	+	0.999570i	$0.509338\pi$
$522$	0	0
$523$	−13.3703	−0.584644	−0.292322	−	0.956320i	$-0.594428\pi$
$523$	−13.3703	−0.584644	−0.292322	+	0.956320i	$0.594428\pi$
$524$	−62.8419	−2.74526
$525$	0	0
$526$	−9.34310	−0.407378
$527$	18.6885	0.814085
$528$	0	0
$529$	1.00000	0.0434783
$530$	10.2857	0.446780
$531$	0	0
$532$	−60.0410	−2.60311
$533$	53.2873	2.30813
$534$	0	0
$535$	−27.1619	−1.17431
$536$	22.0256	0.951362
$537$	0	0
$538$	32.3335	1.39400
$539$	3.45938	0.149006
$540$	0	0
$541$	−22.2616	−0.957102	−0.478551	−	0.878060i	$-0.658838\pi$
$541$	−22.2616	−0.957102	−0.478551	+	0.878060i	$0.658838\pi$
$542$	43.9767	1.88896
$543$	0	0
$544$	−6.66371	−0.285704
$545$	2.84089	0.121690
$546$	0	0
$547$	−5.97191	−0.255340	−0.127670	−	0.991817i	$-0.540750\pi$
$547$	−5.97191	−0.255340	−0.127670	+	0.991817i	$0.540750\pi$
$548$	60.1474	2.56937
$549$	0	0
$550$	22.4305	0.956441
$551$	5.49162	0.233951
$552$	0	0
$553$	−15.9532	−0.678401
$554$	58.6237	2.49068
$555$	0	0
$556$	80.0108	3.39321
$557$	−15.4242	−0.653546	−0.326773	−	0.945103i	$-0.605961\pi$
$557$	−15.4242	−0.653546	−0.326773	+	0.945103i	$0.605961\pi$
$558$	0	0
$559$	14.9841	0.633760
$560$	−23.2620	−0.983000
$561$	0	0
$562$	17.8881	0.754564
$563$	25.8939	1.09130	0.545649	−	0.838014i	$-0.316283\pi$
$563$	25.8939	1.09130	0.545649	+	0.838014i	$0.316283\pi$
$564$	0	0
$565$	−14.1271	−0.594330
$566$	−66.6491	−2.80147
$567$	0	0
$568$	14.7211	0.617682
$569$	−17.1818	−0.720300	−0.360150	−	0.932894i	$-0.617275\pi$
$569$	−17.1818	−0.720300	−0.360150	+	0.932894i	$0.617275\pi$
$570$	0	0
$571$	7.63458	0.319497	0.159749	−	0.987158i	$-0.448932\pi$
$571$	7.63458	0.319497	0.159749	+	0.987158i	$0.448932\pi$
$572$	52.5068	2.19542
$573$	0	0
$574$	67.6245	2.82259
$575$	−3.68037	−0.153482
$576$	0	0
$577$	15.4653	0.643829	0.321915	−	0.946769i	$-0.395674\pi$
$577$	15.4653	0.643829	0.321915	+	0.946769i	$0.395674\pi$
$578$	−14.1008	−0.586518
$579$	0	0
$580$	11.1376	0.462465
$581$	−0.429992	−0.0178391
$582$	0	0
$583$	−3.68097	−0.152450
$584$	29.5334	1.22210
$585$	0	0
$586$	23.8335	0.984553
$587$	8.74351	0.360883	0.180442	−	0.983586i	$-0.442247\pi$
$587$	8.74351	0.360883	0.180442	+	0.983586i	$0.442247\pi$
$588$	0	0
$589$	−30.7561	−1.26728
$590$	92.7855	3.81992
$591$	0	0
$592$	11.1008	0.456240
$593$	5.30487	0.217845	0.108922	−	0.994050i	$-0.465260\pi$
$593$	5.30487	0.217845	0.108922	+	0.994050i	$0.465260\pi$
$594$	0	0
$595$	−28.4340	−1.16568
$596$	75.1223	3.07713
$597$	0	0
$598$	−13.1732	−0.538694
$599$	−23.0248	−0.940766	−0.470383	−	0.882462i	$-0.655884\pi$
$599$	−23.0248	−0.940766	−0.470383	+	0.882462i	$0.655884\pi$
$600$	0	0
$601$	16.1564	0.659035	0.329517	−	0.944150i	$-0.393114\pi$
$601$	16.1564	0.659035	0.329517	+	0.944150i	$0.393114\pi$
$602$	19.0156	0.775019
$603$	0	0
$604$	−33.4837	−1.36243
$605$	13.4759	0.547871
$606$	0	0
$607$	−18.0036	−0.730744	−0.365372	−	0.930862i	$-0.619058\pi$
$607$	−18.0036	−0.730744	−0.365372	+	0.930862i	$0.619058\pi$
$608$	10.9666	0.444754
$609$	0	0
$610$	−101.724	−4.11866
$611$	−55.1582	−2.23146
$612$	0	0
$613$	−12.9637	−0.523598	−0.261799	−	0.965122i	$-0.584316\pi$
$613$	−12.9637	−0.523598	−0.261799	+	0.965122i	$0.584316\pi$
$614$	27.6049	1.11404
$615$	0	0
$616$	31.3805	1.26436
$617$	28.0759	1.13029	0.565147	−	0.824990i	$-0.308819\pi$
$617$	28.0759	1.13029	0.565147	+	0.824990i	$0.308819\pi$
$618$	0	0
$619$	10.8016	0.434155	0.217077	−	0.976154i	$-0.430348\pi$
$619$	10.8016	0.434155	0.217077	+	0.976154i	$0.430348\pi$
$620$	−62.3769	−2.50511
$621$	0	0
$622$	23.8702	0.957109
$623$	30.8989	1.23794
$624$	0	0
$625$	−29.8567	−1.19427
$626$	−10.1607	−0.406104
$627$	0	0
$628$	−27.7899	−1.10894
$629$	13.5689	0.541028
$630$	0	0
$631$	5.84331	0.232618	0.116309	−	0.993213i	$-0.462894\pi$
$631$	5.84331	0.232618	0.116309	+	0.993213i	$0.462894\pi$
$632$	−23.6095	−0.939136
$633$	0	0
$634$	−0.574387	−0.0228118
$635$	11.0328	0.437822
$636$	0	0
$637$	7.47726	0.296260
$638$	−6.09464	−0.241289
$639$	0	0
$640$	60.9163	2.40793
$641$	−7.15996	−0.282802	−0.141401	−	0.989952i	$-0.545161\pi$
$641$	−7.15996	−0.282802	−0.141401	+	0.989952i	$0.545161\pi$
$642$	0	0
$643$	26.2405	1.03483	0.517413	−	0.855736i	$-0.326895\pi$
$643$	26.2405	1.03483	0.517413	+	0.855736i	$0.326895\pi$
$644$	−10.9332	−0.430829
$645$	0	0
$646$	−44.0574	−1.73342
$647$	−6.56156	−0.257961	−0.128981	−	0.991647i	$-0.541171\pi$
$647$	−6.56156	−0.257961	−0.128981	+	0.991647i	$0.541171\pi$
$648$	0	0
$649$	−33.2055	−1.30343
$650$	48.4824	1.90164
$651$	0	0
$652$	57.9280	2.26864
$653$	−32.9805	−1.29063	−0.645313	−	0.763919i	$-0.723273\pi$
$653$	−32.9805	−1.29063	−0.645313	+	0.763919i	$0.723273\pi$
$654$	0	0
$655$	48.9773	1.91370
$656$	26.5497	1.03659
$657$	0	0
$658$	−69.9987	−2.72883
$659$	15.8366	0.616906	0.308453	−	0.951240i	$-0.400189\pi$
$659$	15.8366	0.616906	0.308453	+	0.951240i	$0.400189\pi$
$660$	0	0
$661$	34.3249	1.33508	0.667541	−	0.744573i	$-0.267347\pi$
$661$	34.3249	1.33508	0.667541	+	0.744573i	$0.267347\pi$
$662$	38.3192	1.48932
$663$	0	0
$664$	−0.636354	−0.0246953
$665$	46.7944	1.81461
$666$	0	0
$667$	1.00000	0.0387202
$668$	−53.6503	−2.07579
$669$	0	0
$670$	−36.4510	−1.40823
$671$	36.4042	1.40537
$672$	0	0
$673$	−45.5926	−1.75746	−0.878732	−	0.477316i	$-0.841610\pi$
$673$	−45.5926	−1.75746	−0.878732	+	0.477316i	$0.841610\pi$
$674$	24.9027	0.959216
$675$	0	0
$676$	64.3472	2.47489
$677$	−29.3381	−1.12756	−0.563778	−	0.825926i	$-0.690653\pi$
$677$	−29.3381	−1.12756	−0.563778	+	0.825926i	$0.690653\pi$
$678$	0	0
$679$	20.7257	0.795380
$680$	−42.0800	−1.61369
$681$	0	0
$682$	34.1333	1.30703
$683$	29.7307	1.13761	0.568806	−	0.822472i	$-0.307406\pi$
$683$	29.7307	1.13761	0.568806	+	0.822472i	$0.307406\pi$
$684$	0	0
$685$	−46.8772	−1.79109
$686$	−39.1849	−1.49609
$687$	0	0
$688$	7.46563	0.284624
$689$	−7.95623	−0.303108
$690$	0	0
$691$	30.2644	1.15131	0.575656	−	0.817692i	$-0.304747\pi$
$691$	30.2644	1.15131	0.575656	+	0.817692i	$0.304747\pi$
$692$	27.2183	1.03468
$693$	0	0
$694$	51.7022	1.96259
$695$	−62.3583	−2.36538
$696$	0	0
$697$	32.4526	1.22923
$698$	−40.6413	−1.53829
$699$	0	0
$700$	40.2383	1.52086
$701$	37.9983	1.43517	0.717587	−	0.696468i	$-0.245246\pi$
$701$	37.9983	1.43517	0.717587	+	0.696468i	$0.245246\pi$
$702$	0	0
$703$	−22.3306	−0.842215
$704$	−26.0115	−0.980345
$705$	0	0
$706$	52.4346	1.97340
$707$	19.8923	0.748125
$708$	0	0
$709$	10.9125	0.409829	0.204914	−	0.978780i	$-0.434308\pi$
$709$	10.9125	0.409829	0.204914	+	0.978780i	$0.434308\pi$
$710$	−24.3624	−0.914305
$711$	0	0
$712$	45.7279	1.71373
$713$	−5.60055	−0.209742
$714$	0	0
$715$	−40.9224	−1.53041
$716$	−94.9222	−3.54741
$717$	0	0
$718$	14.3110	0.534083
$719$	−5.25817	−0.196097	−0.0980483	−	0.995182i	$-0.531260\pi$
$719$	−5.25817	−0.196097	−0.0980483	+	0.995182i	$0.531260\pi$
$720$	0	0
$721$	−7.87499	−0.293280
$722$	26.8260	0.998359
$723$	0	0
$724$	7.92123	0.294390
$725$	−3.68037	−0.136685
$726$	0	0
$727$	−38.0970	−1.41294	−0.706470	−	0.707743i	$-0.749714\pi$
$727$	−38.0970	−1.41294	−0.706470	+	0.707743i	$0.749714\pi$
$728$	67.8273	2.51385
$729$	0	0
$730$	−48.8759	−1.80898
$731$	9.12550	0.337519
$732$	0	0
$733$	34.9903	1.29240	0.646199	−	0.763169i	$-0.276358\pi$
$733$	34.9903	1.29240	0.646199	+	0.763169i	$0.276358\pi$
$734$	18.6805	0.689511
$735$	0	0
$736$	1.99697	0.0736093
$737$	13.0449	0.480514
$738$	0	0
$739$	−0.714040	−0.0262664	−0.0131332	−	0.999914i	$-0.504181\pi$
$739$	−0.714040	−0.0262664	−0.0131332	+	0.999914i	$0.504181\pi$
$740$	−45.2891	−1.66486
$741$	0	0
$742$	−10.0969	−0.370668
$743$	−2.80223	−0.102804	−0.0514020	−	0.998678i	$-0.516369\pi$
$743$	−2.80223	−0.102804	−0.0514020	+	0.998678i	$0.516369\pi$
$744$	0	0
$745$	−58.5483	−2.14504
$746$	−74.1712	−2.71560
$747$	0	0
$748$	31.9773	1.16921
$749$	26.6634	0.974260
$750$	0	0
$751$	35.2668	1.28690	0.643452	−	0.765486i	$-0.277502\pi$
$751$	35.2668	1.28690	0.643452	+	0.765486i	$0.277502\pi$
$752$	−27.4818	−1.00216
$753$	0	0
$754$	−13.1732	−0.479741
$755$	26.0963	0.949741
$756$	0	0
$757$	−25.8154	−0.938275	−0.469138	−	0.883125i	$-0.655435\pi$
$757$	−25.8154	−0.938275	−0.469138	+	0.883125i	$0.655435\pi$
$758$	−84.1704	−3.05721
$759$	0	0
$760$	69.2519	2.51203
$761$	−19.3370	−0.700967	−0.350483	−	0.936569i	$-0.613983\pi$
$761$	−19.3370	−0.700967	−0.350483	+	0.936569i	$0.613983\pi$
$762$	0	0
$763$	−2.78875	−0.100960
$764$	−50.3210	−1.82055
$765$	0	0
$766$	16.1074	0.581986
$767$	−71.7720	−2.59154
$768$	0	0
$769$	9.14798	0.329884	0.164942	−	0.986303i	$-0.447256\pi$
$769$	9.14798	0.329884	0.164942	+	0.986303i	$0.447256\pi$
$770$	−51.9327	−1.87153
$771$	0	0
$772$	−44.7853	−1.61186
$773$	−23.4795	−0.844497	−0.422249	−	0.906480i	$-0.638759\pi$
$773$	−23.4795	−0.844497	−0.422249	+	0.906480i	$0.638759\pi$
$774$	0	0
$775$	20.6121	0.740408
$776$	30.6724	1.10108
$777$	0	0
$778$	−70.8739	−2.54095
$779$	−53.4080	−1.91354
$780$	0	0
$781$	8.71867	0.311979
$782$	−8.02267	−0.286890
$783$	0	0
$784$	3.72545	0.133052
$785$	21.6587	0.773033
$786$	0	0
$787$	25.5258	0.909895	0.454947	−	0.890518i	$-0.349658\pi$
$787$	25.5258	0.909895	0.454947	+	0.890518i	$0.349658\pi$
$788$	31.8225	1.13363
$789$	0	0
$790$	39.0722	1.39013
$791$	13.8678	0.493081
$792$	0	0
$793$	78.6858	2.79421
$794$	1.98717	0.0705220
$795$	0	0
$796$	−37.9229	−1.34414
$797$	−13.3605	−0.473252	−0.236626	−	0.971601i	$-0.576042\pi$
$797$	−13.3605	−0.473252	−0.236626	+	0.971601i	$0.576042\pi$
$798$	0	0
$799$	−33.5920	−1.18840
$800$	−7.34958	−0.259847
$801$	0	0
$802$	−88.3421	−3.11947
$803$	17.4914	0.617258
$804$	0	0
$805$	8.52105	0.300328
$806$	73.7774	2.59870
$807$	0	0
$808$	29.4389	1.03566
$809$	24.0390	0.845168	0.422584	−	0.906324i	$-0.361123\pi$
$809$	24.0390	0.845168	0.422584	+	0.906324i	$0.361123\pi$
$810$	0	0
$811$	34.1766	1.20010	0.600052	−	0.799961i	$-0.295146\pi$
$811$	34.1766	1.20010	0.600052	+	0.799961i	$0.295146\pi$
$812$	−10.9332	−0.383681
$813$	0	0
$814$	24.7827	0.868633
$815$	−45.1476	−1.58145
$816$	0	0
$817$	−15.0180	−0.525414
$818$	−69.6372	−2.43481
$819$	0	0
$820$	−108.317	−3.78261
$821$	38.4884	1.34326	0.671628	−	0.740889i	$-0.265595\pi$
$821$	38.4884	1.34326	0.671628	+	0.740889i	$0.265595\pi$
$822$	0	0
$823$	36.4264	1.26974	0.634872	−	0.772618i	$-0.281053\pi$
$823$	36.4264	1.26974	0.634872	+	0.772618i	$0.281053\pi$
$824$	−11.6544	−0.405999
$825$	0	0
$826$	−91.0825	−3.16917
$827$	−31.0706	−1.08043	−0.540216	−	0.841527i	$-0.681657\pi$
$827$	−31.0706	−1.08043	−0.540216	+	0.841527i	$0.681657\pi$
$828$	0	0
$829$	−4.24754	−0.147523	−0.0737617	−	0.997276i	$-0.523500\pi$
$829$	−4.24754	−0.147523	−0.0737617	+	0.997276i	$0.523500\pi$
$830$	1.05313	0.0365545
$831$	0	0
$832$	−56.2225	−1.94916
$833$	4.55374	0.157778
$834$	0	0
$835$	41.8136	1.44702
$836$	−52.6257	−1.82010
$837$	0	0
$838$	38.4758	1.32912
$839$	41.2662	1.42467	0.712334	−	0.701840i	$-0.247638\pi$
$839$	41.2662	1.42467	0.712334	+	0.701840i	$0.247638\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−32.1356	−1.10747
$843$	0	0
$844$	−99.5674	−3.42725
$845$	−50.1505	−1.72523
$846$	0	0
$847$	−13.2285	−0.454537
$848$	−3.96408	−0.136127
$849$	0	0
$850$	29.5264	1.01275
$851$	−4.06631	−0.139391
$852$	0	0
$853$	−39.2466	−1.34378	−0.671888	−	0.740652i	$-0.734517\pi$
$853$	−39.2466	−1.34378	−0.671888	+	0.740652i	$0.734517\pi$
$854$	99.8565	3.41702
$855$	0	0
$856$	39.4597	1.34871
$857$	−30.7802	−1.05143	−0.525716	−	0.850660i	$-0.676203\pi$
$857$	−30.7802	−1.05143	−0.525716	+	0.850660i	$0.676203\pi$
$858$	0	0
$859$	30.1159	1.02754	0.513770	−	0.857928i	$-0.328248\pi$
$859$	30.1159	1.02754	0.513770	+	0.857928i	$0.328248\pi$
$860$	−30.4582	−1.03862
$861$	0	0
$862$	−52.1022	−1.77461
$863$	−45.6275	−1.55318	−0.776589	−	0.630007i	$-0.783052\pi$
$863$	−45.6275	−1.55318	−0.776589	+	0.630007i	$0.783052\pi$
$864$	0	0
$865$	−21.2132	−0.721270
$866$	−23.7126	−0.805786
$867$	0	0
$868$	61.2320	2.07835
$869$	−13.9829	−0.474339
$870$	0	0
$871$	28.1958	0.955378
$872$	−4.12713	−0.139762
$873$	0	0
$874$	13.2031	0.446600
$875$	11.2446	0.380138
$876$	0	0
$877$	−23.7833	−0.803107	−0.401553	−	0.915836i	$-0.631530\pi$
$877$	−23.7833	−0.803107	−0.401553	+	0.915836i	$0.631530\pi$
$878$	−4.26720	−0.144011
$879$	0	0
$880$	−20.3890	−0.687314
$881$	0.298307	0.0100502	0.00502511	−	0.999987i	$-0.498400\pi$
$881$	0.298307	0.0100502	0.00502511	+	0.999987i	$0.498400\pi$
$882$	0	0
$883$	−43.1568	−1.45234	−0.726171	−	0.687514i	$-0.758702\pi$
$883$	−43.1568	−1.45234	−0.726171	+	0.687514i	$0.758702\pi$
$884$	69.1173	2.32467
$885$	0	0
$886$	−29.0636	−0.976410
$887$	−38.7604	−1.30145	−0.650724	−	0.759314i	$-0.725534\pi$
$887$	−38.7604	−1.30145	−0.650724	+	0.759314i	$0.725534\pi$
$888$	0	0
$889$	−10.8303	−0.363236
$890$	−75.6768	−2.53669
$891$	0	0
$892$	−80.2655	−2.68749
$893$	55.2830	1.84997
$894$	0	0
$895$	73.9798	2.47287
$896$	−59.7982	−1.99772
$897$	0	0
$898$	−87.4663	−2.91879
$899$	−5.60055	−0.186789
$900$	0	0
$901$	−4.84544	−0.161425
$902$	59.2725	1.97356
$903$	0	0
$904$	20.5232	0.682591
$905$	−6.17359	−0.205217
$906$	0	0
$907$	−32.4189	−1.07645	−0.538226	−	0.842800i	$-0.680905\pi$
$907$	−32.4189	−1.07645	−0.538226	+	0.842800i	$0.680905\pi$
$908$	−90.1281	−2.99101
$909$	0	0
$910$	−112.250	−3.72105
$911$	−27.6195	−0.915075	−0.457538	−	0.889190i	$-0.651268\pi$
$911$	−27.6195	−0.915075	−0.457538	+	0.889190i	$0.651268\pi$
$912$	0	0
$913$	−0.376886	−0.0124731
$914$	−71.1009	−2.35181
$915$	0	0
$916$	62.0301	2.04953
$917$	−48.0784	−1.58769
$918$	0	0
$919$	21.4245	0.706729	0.353364	−	0.935486i	$-0.385038\pi$
$919$	21.4245	0.706729	0.353364	+	0.935486i	$0.385038\pi$
$920$	12.6105	0.415755
$921$	0	0
$922$	21.4510	0.706451
$923$	18.8450	0.620289
$924$	0	0
$925$	14.9655	0.492063
$926$	−5.16599	−0.169765
$927$	0	0
$928$	1.99697	0.0655537
$929$	52.6913	1.72874	0.864372	−	0.502852i	$-0.167716\pi$
$929$	52.6913	1.72874	0.864372	+	0.502852i	$0.167716\pi$
$930$	0	0
$931$	−7.49419	−0.245612
$932$	50.2995	1.64761
$933$	0	0
$934$	11.6493	0.381175
$935$	−24.9222	−0.815044
$936$	0	0
$937$	33.9135	1.10791	0.553953	−	0.832548i	$-0.313119\pi$
$937$	33.9135	1.10791	0.553953	+	0.832548i	$0.313119\pi$
$938$	35.7820	1.16832
$939$	0	0
$940$	112.120	3.65696
$941$	−26.7835	−0.873118	−0.436559	−	0.899676i	$-0.643803\pi$
$941$	−26.7835	−0.873118	−0.436559	+	0.899676i	$0.643803\pi$
$942$	0	0
$943$	−9.72536	−0.316701
$944$	−35.7594	−1.16387
$945$	0	0
$946$	16.6671	0.541894
$947$	−28.9246	−0.939924	−0.469962	−	0.882687i	$-0.655732\pi$
$947$	−28.9246	−0.939924	−0.469962	+	0.882687i	$0.655732\pi$
$948$	0	0
$949$	37.8068	1.22726
$950$	−48.5921	−1.57654
$951$	0	0
$952$	41.3077	1.33879
$953$	38.7068	1.25384	0.626918	−	0.779085i	$-0.284316\pi$
$953$	38.7068	1.25384	0.626918	+	0.779085i	$0.284316\pi$
$954$	0	0
$955$	39.2188	1.26909
$956$	−54.5585	−1.76455
$957$	0	0
$958$	89.9914	2.90749
$959$	46.0169	1.48596
$960$	0	0
$961$	0.366147	0.0118112
$962$	53.5665	1.72705
$963$	0	0
$964$	−22.4886	−0.724309
$965$	34.9044	1.12361
$966$	0	0
$967$	−55.2190	−1.77572	−0.887861	−	0.460112i	$-0.847809\pi$
$967$	−55.2190	−1.77572	−0.887861	+	0.460112i	$0.847809\pi$
$968$	−19.5772	−0.629233
$969$	0	0
$970$	−50.7609	−1.62983
$971$	−28.4582	−0.913266	−0.456633	−	0.889655i	$-0.650945\pi$
$971$	−28.4582	−0.913266	−0.456633	+	0.889655i	$0.650945\pi$
$972$	0	0
$973$	61.2138	1.96242
$974$	−94.6142	−3.03163
$975$	0	0
$976$	39.2041	1.25489
$977$	−22.1394	−0.708301	−0.354151	−	0.935188i	$-0.615230\pi$
$977$	−22.1394	−0.708301	−0.354151	+	0.935188i	$0.615230\pi$
$978$	0	0
$979$	27.0827	0.865568
$980$	−15.1991	−0.485517
$981$	0	0
$982$	16.2186	0.517556
$983$	20.5852	0.656567	0.328283	−	0.944579i	$-0.393530\pi$
$983$	20.5852	0.656567	0.328283	+	0.944579i	$0.393530\pi$
$984$	0	0
$985$	−24.8016	−0.790243
$986$	−8.02267	−0.255494
$987$	0	0
$988$	−113.748	−3.61880
$989$	−2.73471	−0.0869589
$990$	0	0
$991$	1.48952	0.0473161	0.0236580	−	0.999720i	$-0.492469\pi$
$991$	1.48952	0.0473161	0.0236580	+	0.999720i	$0.492469\pi$
$992$	−11.1841	−0.355096
$993$	0	0
$994$	23.9153	0.758546
$995$	29.5561	0.936992
$996$	0	0
$997$	−21.2966	−0.674471	−0.337235	−	0.941420i	$-0.609492\pi$
$997$	−21.2966	−0.674471	−0.337235	+	0.941420i	$0.609492\pi$
$998$	−98.9660	−3.13271
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.s.1.18		20
3.2	odd	2		2001.2.a.o.1.3	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.o.1.3	✓	20	3.2	odd	2
6003.2.a.s.1.18		20	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)

\(f(q)\)	\(=\)	\(q+2.40422 q^{2} +3.78028 q^{4} -2.94625 q^{5} +2.89217 q^{7} +4.28018 q^{8} +O(q^{10})\)	\(q+2.40422 q^{2} +3.78028 q^{4} -2.94625 q^{5} +2.89217 q^{7} +4.28018 q^{8} -7.08343 q^{10} +2.53497 q^{11} +5.47922 q^{13} +6.95342 q^{14} +2.72995 q^{16} +3.33691 q^{17} -5.49162 q^{19} -11.1376 q^{20} +6.09464 q^{22} -1.00000 q^{23} +3.68037 q^{25} +13.1732 q^{26} +10.9332 q^{28} -1.00000 q^{29} +5.60055 q^{31} -1.99697 q^{32} +8.02267 q^{34} -8.52105 q^{35} +4.06631 q^{37} -13.2031 q^{38} -12.6105 q^{40} +9.72536 q^{41} +2.73471 q^{43} +9.58291 q^{44} -2.40422 q^{46} -10.0668 q^{47} +1.36466 q^{49} +8.84842 q^{50} +20.7130 q^{52} -1.45207 q^{53} -7.46866 q^{55} +12.3790 q^{56} -2.40422 q^{58} -13.0989 q^{59} +14.3608 q^{61} +13.4650 q^{62} -10.2610 q^{64} -16.1431 q^{65} +5.14596 q^{67} +12.6144 q^{68} -20.4865 q^{70} +3.43935 q^{71} +6.90003 q^{73} +9.77631 q^{74} -20.7598 q^{76} +7.33158 q^{77} -5.51601 q^{79} -8.04310 q^{80} +23.3819 q^{82} -0.148675 q^{83} -9.83136 q^{85} +6.57486 q^{86} +10.8502 q^{88} +10.6836 q^{89} +15.8468 q^{91} -3.78028 q^{92} -24.2028 q^{94} +16.1797 q^{95} +7.16615 q^{97} +3.28094 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) \) 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8} + 7 q^{10} + 21 q^{13} + q^{14} + 58 q^{16} + 4 q^{17} + 7 q^{19} + 20 q^{20} + 7 q^{22} - 20 q^{23} + 47 q^{25} - 8 q^{26} + 11 q^{28} - 20 q^{29} + 28 q^{31} - 14 q^{32} + 16 q^{34} - 9 q^{35} + 14 q^{37} + 20 q^{38} + 34 q^{40} - 7 q^{41} + 3 q^{43} + q^{44} + 2 q^{46} - 3 q^{47} + 35 q^{49} + 24 q^{50} + 73 q^{52} + 19 q^{53} + 29 q^{55} + 30 q^{56} + 2 q^{58} - 20 q^{59} + 15 q^{61} - 12 q^{62} + 82 q^{64} + 28 q^{65} + 20 q^{67} + 23 q^{68} - 24 q^{70} - 63 q^{71} + 19 q^{73} - 16 q^{74} - 44 q^{76} + 7 q^{77} + 32 q^{79} + 56 q^{80} - 20 q^{82} + 21 q^{83} + 4 q^{85} + 6 q^{86} + 55 q^{88} + 13 q^{89} + 70 q^{91} - 30 q^{92} - 12 q^{94} - 9 q^{95} - 9 q^{97} - 31 q^{98}+O(q^{100}) \) 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8 + 7 * q^10 + 21 * q^13 + q^14 + 58 * q^16 + 4 * q^17 + 7 * q^19 + 20 * q^20 + 7 * q^22 - 20 * q^23 + 47 * q^25 - 8 * q^26 + 11 * q^28 - 20 * q^29 + 28 * q^31 - 14 * q^32 + 16 * q^34 - 9 * q^35 + 14 * q^37 + 20 * q^38 + 34 * q^40 - 7 * q^41 + 3 * q^43 + q^44 + 2 * q^46 - 3 * q^47 + 35 * q^49 + 24 * q^50 + 73 * q^52 + 19 * q^53 + 29 * q^55 + 30 * q^56 + 2 * q^58 - 20 * q^59 + 15 * q^61 - 12 * q^62 + 82 * q^64 + 28 * q^65 + 20 * q^67 + 23 * q^68 - 24 * q^70 - 63 * q^71 + 19 * q^73 - 16 * q^74 - 44 * q^76 + 7 * q^77 + 32 * q^79 + 56 * q^80 - 20 * q^82 + 21 * q^83 + 4 * q^85 + 6 * q^86 + 55 * q^88 + 13 * q^89 + 70 * q^91 - 30 * q^92 - 12 * q^94 - 9 * q^95 - 9 * q^97 - 31 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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