LMFDB - Embedded newform 6003.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$47.9341963334$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ 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)

$f(q)$	$=$	$q-2.00000 q^{4} -2.44949 q^{5} -4.44949 q^{7} +O(q^{10})$	$q-2.00000 q^{4} -2.44949 q^{5} -4.44949 q^{7} +1.00000 q^{13} +4.00000 q^{16} +1.44949 q^{17} -1.00000 q^{19} +4.89898 q^{20} +1.00000 q^{23} +1.00000 q^{25} +8.89898 q^{28} -1.00000 q^{29} -10.4495 q^{31} +10.8990 q^{35} +9.89898 q^{37} +9.34847 q^{41} -0.101021 q^{43} +12.4495 q^{47} +12.7980 q^{49} -2.00000 q^{52} +2.00000 q^{53} +0.550510 q^{59} -3.10102 q^{61} -8.00000 q^{64} -2.44949 q^{65} -8.89898 q^{67} -2.89898 q^{68} +12.3485 q^{71} -14.2474 q^{73} +2.00000 q^{76} +5.89898 q^{79} -9.79796 q^{80} +7.55051 q^{83} -3.55051 q^{85} -5.44949 q^{89} -4.44949 q^{91} -2.00000 q^{92} +2.44949 q^{95} +16.8990 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} - 4 q^{7}+O(q^{10}) $ 2 * q - 4 * q^4 - 4 * q^7	$ 2 q - 4 q^{4} - 4 q^{7} + 2 q^{13} + 8 q^{16} - 2 q^{17} - 2 q^{19} + 2 q^{23} + 2 q^{25} + 8 q^{28} - 2 q^{29} - 16 q^{31} + 12 q^{35} + 10 q^{37} + 4 q^{41} - 10 q^{43} + 20 q^{47} + 6 q^{49} - 4 q^{52} + 4 q^{53} + 6 q^{59} - 16 q^{61} - 16 q^{64} - 8 q^{67} + 4 q^{68} + 10 q^{71} - 4 q^{73} + 4 q^{76} + 2 q^{79} + 20 q^{83} - 12 q^{85} - 6 q^{89} - 4 q^{91} - 4 q^{92} + 24 q^{97}+O(q^{100}) $ 2 * q - 4 * q^4 - 4 * q^7 + 2 * q^13 + 8 * q^16 - 2 * q^17 - 2 * q^19 + 2 * q^23 + 2 * q^25 + 8 * q^28 - 2 * q^29 - 16 * q^31 + 12 * q^35 + 10 * q^37 + 4 * q^41 - 10 * q^43 + 20 * q^47 + 6 * q^49 - 4 * q^52 + 4 * q^53 + 6 * q^59 - 16 * q^61 - 16 * q^64 - 8 * q^67 + 4 * q^68 + 10 * q^71 - 4 * q^73 + 4 * q^76 + 2 * q^79 + 20 * q^83 - 12 * q^85 - 6 * q^89 - 4 * q^91 - 4 * q^92 + 24 * q^97

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	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	−2.44949	−1.09545	−0.547723	−	0.836660i	$-0.684505\pi$
$5$	−2.44949	−1.09545	−0.547723	+	0.836660i	$0.684505\pi$
$6$	0	0
$7$	−4.44949	−1.68175	−0.840875	−	0.541230i	$-0.817959\pi$
$7$	−4.44949	−1.68175	−0.840875	+	0.541230i	$0.817959\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	1.44949	0.351553	0.175776	−	0.984430i	$-0.443756\pi$
$17$	1.44949	0.351553	0.175776	+	0.984430i	$0.443756\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	4.89898	1.09545
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	8.89898	1.68175
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−10.4495	−1.87678	−0.938392	−	0.345573i	$-0.887685\pi$
$31$	−10.4495	−1.87678	−0.938392	+	0.345573i	$0.887685\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	10.8990	1.84226
$36$	0	0
$37$	9.89898	1.62738	0.813691	−	0.581298i	$-0.197455\pi$
$37$	9.89898	1.62738	0.813691	+	0.581298i	$0.197455\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.34847	1.45999	0.729993	−	0.683455i	$-0.239523\pi$
$41$	9.34847	1.45999	0.729993	+	0.683455i	$0.239523\pi$
$42$	0	0
$43$	−0.101021	−0.0154055	−0.00770274	−	0.999970i	$-0.502452\pi$
$43$	−0.101021	−0.0154055	−0.00770274	+	0.999970i	$0.502452\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.4495	1.81594	0.907972	−	0.419030i	$-0.137630\pi$
$47$	12.4495	1.81594	0.907972	+	0.419030i	$0.137630\pi$
$48$	0	0
$49$	12.7980	1.82828
$50$	0	0
$51$	0	0
$52$	−2.00000	−0.277350
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.550510	0.0716703	0.0358352	−	0.999358i	$-0.488591\pi$
$59$	0.550510	0.0716703	0.0358352	+	0.999358i	$0.488591\pi$
$60$	0	0
$61$	−3.10102	−0.397045	−0.198522	−	0.980096i	$-0.563614\pi$
$61$	−3.10102	−0.397045	−0.198522	+	0.980096i	$0.563614\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−2.44949	−0.303822
$66$	0	0
$67$	−8.89898	−1.08718	−0.543592	−	0.839350i	$-0.682936\pi$
$67$	−8.89898	−1.08718	−0.543592	+	0.839350i	$0.682936\pi$
$68$	−2.89898	−0.351553
$69$	0	0
$70$	0	0
$71$	12.3485	1.46549	0.732747	−	0.680501i	$-0.238238\pi$
$71$	12.3485	1.46549	0.732747	+	0.680501i	$0.238238\pi$
$72$	0	0
$73$	−14.2474	−1.66754	−0.833769	−	0.552114i	$-0.813821\pi$
$73$	−14.2474	−1.66754	−0.833769	+	0.552114i	$0.813821\pi$
$74$	0	0
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	0	0
$79$	5.89898	0.663687	0.331844	−	0.943334i	$-0.392329\pi$
$79$	5.89898	0.663687	0.331844	+	0.943334i	$0.392329\pi$
$80$	−9.79796	−1.09545
$81$	0	0
$82$	0	0
$83$	7.55051	0.828776	0.414388	−	0.910100i	$-0.363996\pi$
$83$	7.55051	0.828776	0.414388	+	0.910100i	$0.363996\pi$
$84$	0	0
$85$	−3.55051	−0.385107
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.44949	−0.577645	−0.288822	−	0.957383i	$-0.593264\pi$
$89$	−5.44949	−0.577645	−0.288822	+	0.957383i	$0.593264\pi$
$90$	0	0
$91$	−4.44949	−0.466433
$92$	−2.00000	−0.208514
$93$	0	0
$94$	0	0
$95$	2.44949	0.251312
$96$	0	0
$97$	16.8990	1.71583	0.857916	−	0.513790i	$-0.171759\pi$
$97$	16.8990	1.71583	0.857916	+	0.513790i	$0.171759\pi$
$98$	0	0
$99$	0	0
$100$	−2.00000	−0.200000
$101$	−10.4495	−1.03976	−0.519882	−	0.854238i	$-0.674024\pi$
$101$	−10.4495	−1.03976	−0.519882	+	0.854238i	$0.674024\pi$
$102$	0	0
$103$	1.79796	0.177158	0.0885791	−	0.996069i	$-0.471767\pi$
$103$	1.79796	0.177158	0.0885791	+	0.996069i	$0.471767\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.2474	1.37735	0.688676	−	0.725069i	$-0.258192\pi$
$107$	14.2474	1.37735	0.688676	+	0.725069i	$0.258192\pi$
$108$	0	0
$109$	−6.44949	−0.617749	−0.308875	−	0.951103i	$-0.599952\pi$
$109$	−6.44949	−0.617749	−0.308875	+	0.951103i	$0.599952\pi$
$110$	0	0
$111$	0	0
$112$	−17.7980	−1.68175
$113$	16.8990	1.58972	0.794861	−	0.606791i	$-0.207544\pi$
$113$	16.8990	1.58972	0.794861	+	0.606791i	$0.207544\pi$
$114$	0	0
$115$	−2.44949	−0.228416
$116$	2.00000	0.185695
$117$	0	0
$118$	0	0
$119$	−6.44949	−0.591224
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	20.8990	1.87678
$125$	9.79796	0.876356
$126$	0	0
$127$	10.2474	0.909314	0.454657	−	0.890667i	$-0.349762\pi$
$127$	10.2474	0.909314	0.454657	+	0.890667i	$0.349762\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−18.6969	−1.63356	−0.816780	−	0.576950i	$-0.804243\pi$
$131$	−18.6969	−1.63356	−0.816780	+	0.576950i	$0.804243\pi$
$132$	0	0
$133$	4.44949	0.385820
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−20.1464	−1.72123	−0.860613	−	0.509260i	$-0.829919\pi$
$137$	−20.1464	−1.72123	−0.860613	+	0.509260i	$0.829919\pi$
$138$	0	0
$139$	6.00000	0.508913	0.254457	−	0.967084i	$-0.418103\pi$
$139$	6.00000	0.508913	0.254457	+	0.967084i	$0.418103\pi$
$140$	−21.7980	−1.84226
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.44949	0.203419
$146$	0	0
$147$	0	0
$148$	−19.7980	−1.62738
$149$	0.898979	0.0736473	0.0368236	−	0.999322i	$-0.488276\pi$
$149$	0.898979	0.0736473	0.0368236	+	0.999322i	$0.488276\pi$
$150$	0	0
$151$	18.5959	1.51331	0.756657	−	0.653812i	$-0.226831\pi$
$151$	18.5959	1.51331	0.756657	+	0.653812i	$0.226831\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	25.5959	2.05591
$156$	0	0
$157$	−20.5959	−1.64373	−0.821867	−	0.569680i	$-0.807067\pi$
$157$	−20.5959	−1.64373	−0.821867	+	0.569680i	$0.807067\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.44949	−0.350669
$162$	0	0
$163$	12.8990	1.01033	0.505163	−	0.863024i	$-0.331432\pi$
$163$	12.8990	1.01033	0.505163	+	0.863024i	$0.331432\pi$
$164$	−18.6969	−1.45999
$165$	0	0
$166$	0	0
$167$	−6.34847	−0.491259	−0.245630	−	0.969364i	$-0.578995\pi$
$167$	−6.34847	−0.491259	−0.245630	+	0.969364i	$0.578995\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0.202041	0.0154055
$173$	−23.2474	−1.76747	−0.883735	−	0.467987i	$-0.844979\pi$
$173$	−23.2474	−1.76747	−0.883735	+	0.467987i	$0.844979\pi$
$174$	0	0
$175$	−4.44949	−0.336350
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.65153	−0.123441	−0.0617206	−	0.998093i	$-0.519659\pi$
$179$	−1.65153	−0.123441	−0.0617206	+	0.998093i	$0.519659\pi$
$180$	0	0
$181$	−15.7980	−1.17425	−0.587127	−	0.809495i	$-0.699741\pi$
$181$	−15.7980	−1.17425	−0.587127	+	0.809495i	$0.699741\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−24.2474	−1.78271
$186$	0	0
$187$	0	0
$188$	−24.8990	−1.81594
$189$	0	0
$190$	0	0
$191$	22.1464	1.60246	0.801230	−	0.598357i	$-0.204180\pi$
$191$	22.1464	1.60246	0.801230	+	0.598357i	$0.204180\pi$
$192$	0	0
$193$	−15.7980	−1.13716	−0.568581	−	0.822627i	$-0.692507\pi$
$193$	−15.7980	−1.13716	−0.568581	+	0.822627i	$0.692507\pi$
$194$	0	0
$195$	0	0
$196$	−25.5959	−1.82828
$197$	−24.1464	−1.72036	−0.860181	−	0.509989i	$-0.829649\pi$
$197$	−24.1464	−1.72036	−0.860181	+	0.509989i	$0.829649\pi$
$198$	0	0
$199$	−20.2474	−1.43530	−0.717652	−	0.696402i	$-0.754783\pi$
$199$	−20.2474	−1.43530	−0.717652	+	0.696402i	$0.754783\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.44949	0.312293
$204$	0	0
$205$	−22.8990	−1.59933
$206$	0	0
$207$	0	0
$208$	4.00000	0.277350
$209$	0	0
$210$	0	0
$211$	−18.0454	−1.24230	−0.621149	−	0.783693i	$-0.713334\pi$
$211$	−18.0454	−1.24230	−0.621149	+	0.783693i	$0.713334\pi$
$212$	−4.00000	−0.274721
$213$	0	0
$214$	0	0
$215$	0.247449	0.0168759
$216$	0	0
$217$	46.4949	3.15628
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.44949	0.0975032
$222$	0	0
$223$	−21.8990	−1.46646	−0.733232	−	0.679978i	$-0.761989\pi$
$223$	−21.8990	−1.46646	−0.733232	+	0.679978i	$0.761989\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−14.0000	−0.929213	−0.464606	−	0.885517i	$-0.653804\pi$
$227$	−14.0000	−0.929213	−0.464606	+	0.885517i	$0.653804\pi$
$228$	0	0
$229$	−7.20204	−0.475924	−0.237962	−	0.971274i	$-0.576479\pi$
$229$	−7.20204	−0.475924	−0.237962	+	0.971274i	$0.576479\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.1464	0.795739	0.397869	−	0.917442i	$-0.369750\pi$
$233$	12.1464	0.795739	0.397869	+	0.917442i	$0.369750\pi$
$234$	0	0
$235$	−30.4949	−1.98927
$236$	−1.10102	−0.0716703
$237$	0	0
$238$	0	0
$239$	−8.55051	−0.553087	−0.276543	−	0.961001i	$-0.589189\pi$
$239$	−8.55051	−0.553087	−0.276543	+	0.961001i	$0.589189\pi$
$240$	0	0
$241$	20.6969	1.33321	0.666604	−	0.745412i	$-0.267747\pi$
$241$	20.6969	1.33321	0.666604	+	0.745412i	$0.267747\pi$
$242$	0	0
$243$	0	0
$244$	6.20204	0.397045
$245$	−31.3485	−2.00278
$246$	0	0
$247$	−1.00000	−0.0636285
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.55051	0.539703	0.269852	−	0.962902i	$-0.413025\pi$
$251$	8.55051	0.539703	0.269852	+	0.962902i	$0.413025\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	−15.4495	−0.963713	−0.481856	−	0.876250i	$-0.660037\pi$
$257$	−15.4495	−0.963713	−0.481856	+	0.876250i	$0.660037\pi$
$258$	0	0
$259$	−44.0454	−2.73685
$260$	4.89898	0.303822
$261$	0	0
$262$	0	0
$263$	16.8990	1.04204	0.521018	−	0.853546i	$-0.325552\pi$
$263$	16.8990	1.04204	0.521018	+	0.853546i	$0.325552\pi$
$264$	0	0
$265$	−4.89898	−0.300942
$266$	0	0
$267$	0	0
$268$	17.7980	1.08718
$269$	23.5959	1.43867	0.719334	−	0.694664i	$-0.244447\pi$
$269$	23.5959	1.43867	0.719334	+	0.694664i	$0.244447\pi$
$270$	0	0
$271$	7.10102	0.431356	0.215678	−	0.976465i	$-0.430804\pi$
$271$	7.10102	0.431356	0.215678	+	0.976465i	$0.430804\pi$
$272$	5.79796	0.351553
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−7.00000	−0.420589	−0.210295	−	0.977638i	$-0.567442\pi$
$277$	−7.00000	−0.420589	−0.210295	+	0.977638i	$0.567442\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−24.4949	−1.46124	−0.730622	−	0.682783i	$-0.760770\pi$
$281$	−24.4949	−1.46124	−0.730622	+	0.682783i	$0.760770\pi$
$282$	0	0
$283$	22.0454	1.31046	0.655232	−	0.755428i	$-0.272571\pi$
$283$	22.0454	1.31046	0.655232	+	0.755428i	$0.272571\pi$
$284$	−24.6969	−1.46549
$285$	0	0
$286$	0	0
$287$	−41.5959	−2.45533
$288$	0	0
$289$	−14.8990	−0.876411
$290$	0	0
$291$	0	0
$292$	28.4949	1.66754
$293$	2.75255	0.160806	0.0804029	−	0.996762i	$-0.474379\pi$
$293$	2.75255	0.160806	0.0804029	+	0.996762i	$0.474379\pi$
$294$	0	0
$295$	−1.34847	−0.0785109
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.00000	0.0578315
$300$	0	0
$301$	0.449490	0.0259082
$302$	0	0
$303$	0	0
$304$	−4.00000	−0.229416
$305$	7.59592	0.434941
$306$	0	0
$307$	27.3485	1.56086	0.780430	−	0.625243i	$-0.215000\pi$
$307$	27.3485	1.56086	0.780430	+	0.625243i	$0.215000\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9.79796	−0.555591	−0.277796	−	0.960640i	$-0.589604\pi$
$311$	−9.79796	−0.555591	−0.277796	+	0.960640i	$0.589604\pi$
$312$	0	0
$313$	15.3485	0.867547	0.433773	−	0.901022i	$-0.357182\pi$
$313$	15.3485	0.867547	0.433773	+	0.901022i	$0.357182\pi$
$314$	0	0
$315$	0	0
$316$	−11.7980	−0.663687
$317$	6.24745	0.350892	0.175446	−	0.984489i	$-0.443863\pi$
$317$	6.24745	0.350892	0.175446	+	0.984489i	$0.443863\pi$
$318$	0	0
$319$	0	0
$320$	19.5959	1.09545
$321$	0	0
$322$	0	0
$323$	−1.44949	−0.0806518
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−55.3939	−3.05396
$330$	0	0
$331$	33.5959	1.84660	0.923299	−	0.384081i	$-0.125482\pi$
$331$	33.5959	1.84660	0.923299	+	0.384081i	$0.125482\pi$
$332$	−15.1010	−0.828776
$333$	0	0
$334$	0	0
$335$	21.7980	1.19095
$336$	0	0
$337$	6.79796	0.370308	0.185154	−	0.982709i	$-0.440722\pi$
$337$	6.79796	0.370308	0.185154	+	0.982709i	$0.440722\pi$
$338$	0	0
$339$	0	0
$340$	7.10102	0.385107
$341$	0	0
$342$	0	0
$343$	−25.7980	−1.39296
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.1464	0.652054	0.326027	−	0.945360i	$-0.394290\pi$
$347$	12.1464	0.652054	0.326027	+	0.945360i	$0.394290\pi$
$348$	0	0
$349$	17.6969	0.947295	0.473648	−	0.880714i	$-0.342937\pi$
$349$	17.6969	0.947295	0.473648	+	0.880714i	$0.342937\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8.89898	−0.473645	−0.236822	−	0.971553i	$-0.576106\pi$
$353$	−8.89898	−0.473645	−0.236822	+	0.971553i	$0.576106\pi$
$354$	0	0
$355$	−30.2474	−1.60537
$356$	10.8990	0.577645
$357$	0	0
$358$	0	0
$359$	−15.0454	−0.794066	−0.397033	−	0.917804i	$-0.629960\pi$
$359$	−15.0454	−0.794066	−0.397033	+	0.917804i	$0.629960\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	0	0
$364$	8.89898	0.466433
$365$	34.8990	1.82670
$366$	0	0
$367$	−14.0000	−0.730794	−0.365397	−	0.930852i	$-0.619067\pi$
$367$	−14.0000	−0.730794	−0.365397	+	0.930852i	$0.619067\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	0	0
$371$	−8.89898	−0.462012
$372$	0	0
$373$	−4.65153	−0.240847	−0.120424	−	0.992723i	$-0.538425\pi$
$373$	−4.65153	−0.240847	−0.120424	+	0.992723i	$0.538425\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	−4.89898	−0.251312
$381$	0	0
$382$	0	0
$383$	−15.7980	−0.807238	−0.403619	−	0.914927i	$-0.632248\pi$
$383$	−15.7980	−0.807238	−0.403619	+	0.914927i	$0.632248\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	−33.7980	−1.71583
$389$	−30.3485	−1.53873	−0.769364	−	0.638810i	$-0.779427\pi$
$389$	−30.3485	−1.53873	−0.769364	+	0.638810i	$0.779427\pi$
$390$	0	0
$391$	1.44949	0.0733038
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−14.4495	−0.727033
$396$	0	0
$397$	6.69694	0.336110	0.168055	−	0.985778i	$-0.446251\pi$
$397$	6.69694	0.336110	0.168055	+	0.985778i	$0.446251\pi$
$398$	0	0
$399$	0	0
$400$	4.00000	0.200000
$401$	12.6969	0.634055	0.317027	−	0.948416i	$-0.397315\pi$
$401$	12.6969	0.634055	0.317027	+	0.948416i	$0.397315\pi$
$402$	0	0
$403$	−10.4495	−0.520526
$404$	20.8990	1.03976
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	23.1464	1.14452	0.572259	−	0.820073i	$-0.306067\pi$
$409$	23.1464	1.14452	0.572259	+	0.820073i	$0.306067\pi$
$410$	0	0
$411$	0	0
$412$	−3.59592	−0.177158
$413$	−2.44949	−0.120532
$414$	0	0
$415$	−18.4949	−0.907879
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7.59592	0.371085	0.185542	−	0.982636i	$-0.440596\pi$
$419$	7.59592	0.371085	0.185542	+	0.982636i	$0.440596\pi$
$420$	0	0
$421$	−9.69694	−0.472600	−0.236300	−	0.971680i	$-0.575935\pi$
$421$	−9.69694	−0.472600	−0.236300	+	0.971680i	$0.575935\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.44949	0.0703106
$426$	0	0
$427$	13.7980	0.667730
$428$	−28.4949	−1.37735
$429$	0	0
$430$	0	0
$431$	34.4949	1.66156	0.830780	−	0.556600i	$-0.187895\pi$
$431$	34.4949	1.66156	0.830780	+	0.556600i	$0.187895\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	0	0
$436$	12.8990	0.617749
$437$	−1.00000	−0.0478365
$438$	0	0
$439$	8.30306	0.396284	0.198142	−	0.980173i	$-0.436509\pi$
$439$	8.30306	0.396284	0.198142	+	0.980173i	$0.436509\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−11.1464	−0.529583	−0.264791	−	0.964306i	$-0.585303\pi$
$443$	−11.1464	−0.529583	−0.264791	+	0.964306i	$0.585303\pi$
$444$	0	0
$445$	13.3485	0.632778
$446$	0	0
$447$	0	0
$448$	35.5959	1.68175
$449$	−14.2020	−0.670236	−0.335118	−	0.942176i	$-0.608776\pi$
$449$	−14.2020	−0.670236	−0.335118	+	0.942176i	$0.608776\pi$
$450$	0	0
$451$	0	0
$452$	−33.7980	−1.58972
$453$	0	0
$454$	0	0
$455$	10.8990	0.510952
$456$	0	0
$457$	16.4495	0.769475	0.384737	−	0.923026i	$-0.374292\pi$
$457$	16.4495	0.769475	0.384737	+	0.923026i	$0.374292\pi$
$458$	0	0
$459$	0	0
$460$	4.89898	0.228416
$461$	−4.00000	−0.186299	−0.0931493	−	0.995652i	$-0.529693\pi$
$461$	−4.00000	−0.186299	−0.0931493	+	0.995652i	$0.529693\pi$
$462$	0	0
$463$	4.10102	0.190591	0.0952953	−	0.995449i	$-0.469620\pi$
$463$	4.10102	0.190591	0.0952953	+	0.995449i	$0.469620\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	0	0
$467$	30.1464	1.39501	0.697505	−	0.716580i	$-0.254293\pi$
$467$	30.1464	1.39501	0.697505	+	0.716580i	$0.254293\pi$
$468$	0	0
$469$	39.5959	1.82837
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	12.8990	0.591224
$477$	0	0
$478$	0	0
$479$	−16.8990	−0.772134	−0.386067	−	0.922471i	$-0.626167\pi$
$479$	−16.8990	−0.772134	−0.386067	+	0.922471i	$0.626167\pi$
$480$	0	0
$481$	9.89898	0.451355
$482$	0	0
$483$	0	0
$484$	22.0000	1.00000
$485$	−41.3939	−1.87960
$486$	0	0
$487$	24.6969	1.11913	0.559563	−	0.828788i	$-0.310969\pi$
$487$	24.6969	1.11913	0.559563	+	0.828788i	$0.310969\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15.5505	0.701785	0.350892	−	0.936416i	$-0.385878\pi$
$491$	15.5505	0.701785	0.350892	+	0.936416i	$0.385878\pi$
$492$	0	0
$493$	−1.44949	−0.0652817
$494$	0	0
$495$	0	0
$496$	−41.7980	−1.87678
$497$	−54.9444	−2.46459
$498$	0	0
$499$	−28.1010	−1.25797	−0.628987	−	0.777416i	$-0.716530\pi$
$499$	−28.1010	−1.25797	−0.628987	+	0.777416i	$0.716530\pi$
$500$	−19.5959	−0.876356
$501$	0	0
$502$	0	0
$503$	0.752551	0.0335546	0.0167773	−	0.999859i	$-0.494659\pi$
$503$	0.752551	0.0335546	0.0167773	+	0.999859i	$0.494659\pi$
$504$	0	0
$505$	25.5959	1.13900
$506$	0	0
$507$	0	0
$508$	−20.4949	−0.909314
$509$	−1.85357	−0.0821581	−0.0410791	−	0.999156i	$-0.513080\pi$
$509$	−1.85357	−0.0821581	−0.0410791	+	0.999156i	$0.513080\pi$
$510$	0	0
$511$	63.3939	2.80438
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.40408	−0.194067
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	26.0454	1.14107	0.570535	−	0.821273i	$-0.306736\pi$
$521$	26.0454	1.14107	0.570535	+	0.821273i	$0.306736\pi$
$522$	0	0
$523$	−1.34847	−0.0589644	−0.0294822	−	0.999565i	$-0.509386\pi$
$523$	−1.34847	−0.0589644	−0.0294822	+	0.999565i	$0.509386\pi$
$524$	37.3939	1.63356
$525$	0	0
$526$	0	0
$527$	−15.1464	−0.659789
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	−8.89898	−0.385820
$533$	9.34847	0.404927
$534$	0	0
$535$	−34.8990	−1.50881
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−16.8990	−0.726544	−0.363272	−	0.931683i	$-0.618340\pi$
$541$	−16.8990	−0.726544	−0.363272	+	0.931683i	$0.618340\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	15.7980	0.676710
$546$	0	0
$547$	−13.6969	−0.585639	−0.292819	−	0.956168i	$-0.594593\pi$
$547$	−13.6969	−0.585639	−0.292819	+	0.956168i	$0.594593\pi$
$548$	40.2929	1.72123
$549$	0	0
$550$	0	0
$551$	1.00000	0.0426014
$552$	0	0
$553$	−26.2474	−1.11616
$554$	0	0
$555$	0	0
$556$	−12.0000	−0.508913
$557$	35.1464	1.48920	0.744601	−	0.667510i	$-0.232640\pi$
$557$	35.1464	1.48920	0.744601	+	0.667510i	$0.232640\pi$
$558$	0	0
$559$	−0.101021	−0.00427271
$560$	43.5959	1.84226
$561$	0	0
$562$	0	0
$563$	9.24745	0.389733	0.194867	−	0.980830i	$-0.437573\pi$
$563$	9.24745	0.389733	0.194867	+	0.980830i	$0.437573\pi$
$564$	0	0
$565$	−41.3939	−1.74145
$566$	0	0
$567$	0	0
$568$	0	0
$569$	16.2020	0.679225	0.339612	−	0.940565i	$-0.389704\pi$
$569$	16.2020	0.679225	0.339612	+	0.940565i	$0.389704\pi$
$570$	0	0
$571$	34.0454	1.42476	0.712378	−	0.701796i	$-0.247618\pi$
$571$	34.0454	1.42476	0.712378	+	0.701796i	$0.247618\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	28.2474	1.17596	0.587978	−	0.808877i	$-0.299924\pi$
$577$	28.2474	1.17596	0.587978	+	0.808877i	$0.299924\pi$
$578$	0	0
$579$	0	0
$580$	−4.89898	−0.203419
$581$	−33.5959	−1.39379
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21.0454	0.868637	0.434318	−	0.900759i	$-0.356989\pi$
$587$	21.0454	0.868637	0.434318	+	0.900759i	$0.356989\pi$
$588$	0	0
$589$	10.4495	0.430564
$590$	0	0
$591$	0	0
$592$	39.5959	1.62738
$593$	−39.2474	−1.61170	−0.805850	−	0.592120i	$-0.798291\pi$
$593$	−39.2474	−1.61170	−0.805850	+	0.592120i	$0.798291\pi$
$594$	0	0
$595$	15.7980	0.647653
$596$	−1.79796	−0.0736473
$597$	0	0
$598$	0	0
$599$	−13.3485	−0.545404	−0.272702	−	0.962099i	$-0.587917\pi$
$599$	−13.3485	−0.545404	−0.272702	+	0.962099i	$0.587917\pi$
$600$	0	0
$601$	1.30306	0.0531530	0.0265765	−	0.999647i	$-0.491539\pi$
$601$	1.30306	0.0531530	0.0265765	+	0.999647i	$0.491539\pi$
$602$	0	0
$603$	0	0
$604$	−37.1918	−1.51331
$605$	26.9444	1.09545
$606$	0	0
$607$	5.34847	0.217088	0.108544	−	0.994092i	$-0.465381\pi$
$607$	5.34847	0.217088	0.108544	+	0.994092i	$0.465381\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.4495	0.503652
$612$	0	0
$613$	10.0000	0.403896	0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000	0.403896	0.201948	+	0.979396i	$0.435273\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8.20204	−0.330202	−0.165101	−	0.986277i	$-0.552795\pi$
$617$	−8.20204	−0.330202	−0.165101	+	0.986277i	$0.552795\pi$
$618$	0	0
$619$	10.7980	0.434007	0.217003	−	0.976171i	$-0.430372\pi$
$619$	10.7980	0.434007	0.217003	+	0.976171i	$0.430372\pi$
$620$	−51.1918	−2.05591
$621$	0	0
$622$	0	0
$623$	24.2474	0.971454
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	41.1918	1.64373
$629$	14.3485	0.572111
$630$	0	0
$631$	−32.6969	−1.30164	−0.650822	−	0.759230i	$-0.725576\pi$
$631$	−32.6969	−1.30164	−0.650822	+	0.759230i	$0.725576\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−25.1010	−0.996104
$636$	0	0
$637$	12.7980	0.507074
$638$	0	0
$639$	0	0
$640$	0	0
$641$	33.0454	1.30522	0.652608	−	0.757696i	$-0.273675\pi$
$641$	33.0454	1.30522	0.652608	+	0.757696i	$0.273675\pi$
$642$	0	0
$643$	−3.10102	−0.122292	−0.0611462	−	0.998129i	$-0.519476\pi$
$643$	−3.10102	−0.122292	−0.0611462	+	0.998129i	$0.519476\pi$
$644$	8.89898	0.350669
$645$	0	0
$646$	0	0
$647$	1.24745	0.0490423	0.0245211	−	0.999699i	$-0.492194\pi$
$647$	1.24745	0.0490423	0.0245211	+	0.999699i	$0.492194\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−25.7980	−1.01033
$653$	−42.7423	−1.67264	−0.836319	−	0.548244i	$-0.815297\pi$
$653$	−42.7423	−1.67264	−0.836319	+	0.548244i	$0.815297\pi$
$654$	0	0
$655$	45.7980	1.78947
$656$	37.3939	1.45999
$657$	0	0
$658$	0	0
$659$	9.44949	0.368100	0.184050	−	0.982917i	$-0.441079\pi$
$659$	9.44949	0.368100	0.184050	+	0.982917i	$0.441079\pi$
$660$	0	0
$661$	−7.34847	−0.285822	−0.142911	−	0.989736i	$-0.545646\pi$
$661$	−7.34847	−0.285822	−0.142911	+	0.989736i	$0.545646\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−10.8990	−0.422644
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	12.6969	0.491259
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	1.89898	0.0732003	0.0366001	−	0.999330i	$-0.488347\pi$
$673$	1.89898	0.0732003	0.0366001	+	0.999330i	$0.488347\pi$
$674$	0	0
$675$	0	0
$676$	24.0000	0.923077
$677$	−23.7980	−0.914630	−0.457315	−	0.889305i	$-0.651189\pi$
$677$	−23.7980	−0.914630	−0.457315	+	0.889305i	$0.651189\pi$
$678$	0	0
$679$	−75.1918	−2.88560
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−26.6969	−1.02153	−0.510765	−	0.859720i	$-0.670638\pi$
$683$	−26.6969	−1.02153	−0.510765	+	0.859720i	$0.670638\pi$
$684$	0	0
$685$	49.3485	1.88551
$686$	0	0
$687$	0	0
$688$	−0.404082	−0.0154055
$689$	2.00000	0.0761939
$690$	0	0
$691$	29.1010	1.10705	0.553527	−	0.832831i	$-0.313281\pi$
$691$	29.1010	1.10705	0.553527	+	0.832831i	$0.313281\pi$
$692$	46.4949	1.76747
$693$	0	0
$694$	0	0
$695$	−14.6969	−0.557487
$696$	0	0
$697$	13.5505	0.513262
$698$	0	0
$699$	0	0
$700$	8.89898	0.336350
$701$	17.7526	0.670505	0.335252	−	0.942128i	$-0.391178\pi$
$701$	17.7526	0.670505	0.335252	+	0.942128i	$0.391178\pi$
$702$	0	0
$703$	−9.89898	−0.373347
$704$	0	0
$705$	0	0
$706$	0	0
$707$	46.4949	1.74862
$708$	0	0
$709$	15.3485	0.576424	0.288212	−	0.957567i	$-0.406939\pi$
$709$	15.3485	0.576424	0.288212	+	0.957567i	$0.406939\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−10.4495	−0.391336
$714$	0	0
$715$	0	0
$716$	3.30306	0.123441
$717$	0	0
$718$	0	0
$719$	−21.7423	−0.810853	−0.405426	−	0.914128i	$-0.632877\pi$
$719$	−21.7423	−0.810853	−0.405426	+	0.914128i	$0.632877\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	0	0
$724$	31.5959	1.17425
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−35.5959	−1.32018	−0.660090	−	0.751187i	$-0.729482\pi$
$727$	−35.5959	−1.32018	−0.660090	+	0.751187i	$0.729482\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.146428	−0.00541584
$732$	0	0
$733$	−40.8990	−1.51064	−0.755319	−	0.655357i	$-0.772518\pi$
$733$	−40.8990	−1.51064	−0.755319	+	0.655357i	$0.772518\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	35.7980	1.31685	0.658425	−	0.752647i	$-0.271223\pi$
$739$	35.7980	1.31685	0.658425	+	0.752647i	$0.271223\pi$
$740$	48.4949	1.78271
$741$	0	0
$742$	0	0
$743$	−16.7526	−0.614591	−0.307296	−	0.951614i	$-0.599424\pi$
$743$	−16.7526	−0.614591	−0.307296	+	0.951614i	$0.599424\pi$
$744$	0	0
$745$	−2.20204	−0.0806765
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−63.3939	−2.31636
$750$	0	0
$751$	−11.4949	−0.419455	−0.209727	−	0.977760i	$-0.567258\pi$
$751$	−11.4949	−0.419455	−0.209727	+	0.977760i	$0.567258\pi$
$752$	49.7980	1.81594
$753$	0	0
$754$	0	0
$755$	−45.5505	−1.65775
$756$	0	0
$757$	12.1010	0.439819	0.219910	−	0.975520i	$-0.429424\pi$
$757$	12.1010	0.439819	0.219910	+	0.975520i	$0.429424\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3.24745	0.117720	0.0588600	−	0.998266i	$-0.481253\pi$
$761$	3.24745	0.117720	0.0588600	+	0.998266i	$0.481253\pi$
$762$	0	0
$763$	28.6969	1.03890
$764$	−44.2929	−1.60246
$765$	0	0
$766$	0	0
$767$	0.550510	0.0198778
$768$	0	0
$769$	−42.1010	−1.51820	−0.759101	−	0.650973i	$-0.774361\pi$
$769$	−42.1010	−1.51820	−0.759101	+	0.650973i	$0.774361\pi$
$770$	0	0
$771$	0	0
$772$	31.5959	1.13716
$773$	36.6969	1.31990	0.659949	−	0.751311i	$-0.270578\pi$
$773$	36.6969	1.31990	0.659949	+	0.751311i	$0.270578\pi$
$774$	0	0
$775$	−10.4495	−0.375357
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−9.34847	−0.334944
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	51.1918	1.82828
$785$	50.4495	1.80062
$786$	0	0
$787$	−40.2929	−1.43629	−0.718143	−	0.695896i	$-0.755007\pi$
$787$	−40.2929	−1.43629	−0.718143	+	0.695896i	$0.755007\pi$
$788$	48.2929	1.72036
$789$	0	0
$790$	0	0
$791$	−75.1918	−2.67351
$792$	0	0
$793$	−3.10102	−0.110120
$794$	0	0
$795$	0	0
$796$	40.4949	1.43530
$797$	−32.6969	−1.15818	−0.579092	−	0.815262i	$-0.696593\pi$
$797$	−32.6969	−1.15818	−0.579092	+	0.815262i	$0.696593\pi$
$798$	0	0
$799$	18.0454	0.638401
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	10.8990	0.384139
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−21.1010	−0.741872	−0.370936	−	0.928658i	$-0.620963\pi$
$809$	−21.1010	−0.741872	−0.370936	+	0.928658i	$0.620963\pi$
$810$	0	0
$811$	19.3031	0.677822	0.338911	−	0.940818i	$-0.389941\pi$
$811$	19.3031	0.677822	0.338911	+	0.940818i	$0.389941\pi$
$812$	−8.89898	−0.312293
$813$	0	0
$814$	0	0
$815$	−31.5959	−1.10676
$816$	0	0
$817$	0.101021	0.00353426
$818$	0	0
$819$	0	0
$820$	45.7980	1.59933
$821$	24.3485	0.849767	0.424884	−	0.905248i	$-0.360315\pi$
$821$	24.3485	0.849767	0.424884	+	0.905248i	$0.360315\pi$
$822$	0	0
$823$	−16.8536	−0.587479	−0.293739	−	0.955886i	$-0.594900\pi$
$823$	−16.8536	−0.587479	−0.293739	+	0.955886i	$0.594900\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22.3485	0.777132	0.388566	−	0.921421i	$-0.372970\pi$
$827$	22.3485	0.777132	0.388566	+	0.921421i	$0.372970\pi$
$828$	0	0
$829$	−50.2929	−1.74674	−0.873372	−	0.487055i	$-0.838071\pi$
$829$	−50.2929	−1.74674	−0.873372	+	0.487055i	$0.838071\pi$
$830$	0	0
$831$	0	0
$832$	−8.00000	−0.277350
$833$	18.5505	0.642737
$834$	0	0
$835$	15.5505	0.538148
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−51.3939	−1.77431	−0.887157	−	0.461468i	$-0.847323\pi$
$839$	−51.3939	−1.77431	−0.887157	+	0.461468i	$0.847323\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	36.0908	1.24230
$845$	29.3939	1.01118
$846$	0	0
$847$	48.9444	1.68175
$848$	8.00000	0.274721
$849$	0	0
$850$	0	0
$851$	9.89898	0.339333
$852$	0	0
$853$	−23.3939	−0.800991	−0.400496	−	0.916299i	$-0.631162\pi$
$853$	−23.3939	−0.800991	−0.400496	+	0.916299i	$0.631162\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−40.2929	−1.37638	−0.688189	−	0.725532i	$-0.741594\pi$
$857$	−40.2929	−1.37638	−0.688189	+	0.725532i	$0.741594\pi$
$858$	0	0
$859$	−12.6515	−0.431665	−0.215832	−	0.976430i	$-0.569246\pi$
$859$	−12.6515	−0.431665	−0.215832	+	0.976430i	$0.569246\pi$
$860$	−0.494897	−0.0168759
$861$	0	0
$862$	0	0
$863$	−30.2929	−1.03118	−0.515590	−	0.856835i	$-0.672427\pi$
$863$	−30.2929	−1.03118	−0.515590	+	0.856835i	$0.672427\pi$
$864$	0	0
$865$	56.9444	1.93617
$866$	0	0
$867$	0	0
$868$	−92.9898	−3.15628
$869$	0	0
$870$	0	0
$871$	−8.89898	−0.301530
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−43.5959	−1.47381
$876$	0	0
$877$	−46.5959	−1.57343	−0.786716	−	0.617315i	$-0.788220\pi$
$877$	−46.5959	−1.57343	−0.786716	+	0.617315i	$0.788220\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−0.348469	−0.0117402	−0.00587011	−	0.999983i	$-0.501869\pi$
$881$	−0.348469	−0.0117402	−0.00587011	+	0.999983i	$0.501869\pi$
$882$	0	0
$883$	−35.0908	−1.18090	−0.590450	−	0.807074i	$-0.701050\pi$
$883$	−35.0908	−1.18090	−0.590450	+	0.807074i	$0.701050\pi$
$884$	−2.89898	−0.0975032
$885$	0	0
$886$	0	0
$887$	−42.5403	−1.42836	−0.714182	−	0.699960i	$-0.753201\pi$
$887$	−42.5403	−1.42836	−0.714182	+	0.699960i	$0.753201\pi$
$888$	0	0
$889$	−45.5959	−1.52924
$890$	0	0
$891$	0	0
$892$	43.7980	1.46646
$893$	−12.4495	−0.416606
$894$	0	0
$895$	4.04541	0.135223
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10.4495	0.348510
$900$	0	0
$901$	2.89898	0.0965790
$902$	0	0
$903$	0	0
$904$	0	0
$905$	38.6969	1.28633
$906$	0	0
$907$	−26.5959	−0.883103	−0.441551	−	0.897236i	$-0.645572\pi$
$907$	−26.5959	−0.883103	−0.441551	+	0.897236i	$0.645572\pi$
$908$	28.0000	0.929213
$909$	0	0
$910$	0	0
$911$	47.4495	1.57207	0.786036	−	0.618181i	$-0.212130\pi$
$911$	47.4495	1.57207	0.786036	+	0.618181i	$0.212130\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	14.4041	0.475924
$917$	83.1918	2.74724
$918$	0	0
$919$	−28.6969	−0.946625	−0.473312	−	0.880895i	$-0.656942\pi$
$919$	−28.6969	−0.946625	−0.473312	+	0.880895i	$0.656942\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	12.3485	0.406455
$924$	0	0
$925$	9.89898	0.325476
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.84337	0.158906	0.0794529	−	0.996839i	$-0.474683\pi$
$929$	4.84337	0.158906	0.0794529	+	0.996839i	$0.474683\pi$
$930$	0	0
$931$	−12.7980	−0.419436
$932$	−24.2929	−0.795739
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	29.1918	0.953656	0.476828	−	0.878997i	$-0.341787\pi$
$937$	29.1918	0.953656	0.476828	+	0.878997i	$0.341787\pi$
$938$	0	0
$939$	0	0
$940$	60.9898	1.98927
$941$	32.4949	1.05930	0.529652	−	0.848215i	$-0.322323\pi$
$941$	32.4949	1.05930	0.529652	+	0.848215i	$0.322323\pi$
$942$	0	0
$943$	9.34847	0.304428
$944$	2.20204	0.0716703
$945$	0	0
$946$	0	0
$947$	6.69694	0.217621	0.108811	−	0.994062i	$-0.465296\pi$
$947$	6.69694	0.217621	0.108811	+	0.994062i	$0.465296\pi$
$948$	0	0
$949$	−14.2474	−0.462492
$950$	0	0
$951$	0	0
$952$	0	0
$953$	26.6515	0.863328	0.431664	−	0.902035i	$-0.357927\pi$
$953$	26.6515	0.863328	0.431664	+	0.902035i	$0.357927\pi$
$954$	0	0
$955$	−54.2474	−1.75541
$956$	17.1010	0.553087
$957$	0	0
$958$	0	0
$959$	89.6413	2.89467
$960$	0	0
$961$	78.1918	2.52232
$962$	0	0
$963$	0	0
$964$	−41.3939	−1.33321
$965$	38.6969	1.24570
$966$	0	0
$967$	−21.5505	−0.693018	−0.346509	−	0.938047i	$-0.612633\pi$
$967$	−21.5505	−0.693018	−0.346509	+	0.938047i	$0.612633\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−22.2929	−0.715412	−0.357706	−	0.933834i	$-0.616441\pi$
$971$	−22.2929	−0.715412	−0.357706	+	0.933834i	$0.616441\pi$
$972$	0	0
$973$	−26.6969	−0.855865
$974$	0	0
$975$	0	0
$976$	−12.4041	−0.397045
$977$	−26.4495	−0.846194	−0.423097	−	0.906084i	$-0.639057\pi$
$977$	−26.4495	−0.846194	−0.423097	+	0.906084i	$0.639057\pi$
$978$	0	0
$979$	0	0
$980$	62.6969	2.00278
$981$	0	0
$982$	0	0
$983$	−11.7423	−0.374523	−0.187261	−	0.982310i	$-0.559961\pi$
$983$	−11.7423	−0.374523	−0.187261	+	0.982310i	$0.559961\pi$
$984$	0	0
$985$	59.1464	1.88456
$986$	0	0
$987$	0	0
$988$	2.00000	0.0636285
$989$	−0.101021	−0.00321227
$990$	0	0
$991$	−51.5959	−1.63900	−0.819499	−	0.573080i	$-0.805748\pi$
$991$	−51.5959	−1.63900	−0.819499	+	0.573080i	$0.805748\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	49.5959	1.57230
$996$	0	0
$997$	−60.9898	−1.93157	−0.965783	−	0.259351i	$-0.916491\pi$
$997$	−60.9898	−1.93157	−0.965783	+	0.259351i	$0.916491\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.e.1.2	✓	2	3.2	odd	2
6003.2.a.e.1.1		2	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{4} -2.44949 q^{5} -4.44949 q^{7} +O(q^{10})\)	\(q-2.00000 q^{4} -2.44949 q^{5} -4.44949 q^{7} +1.00000 q^{13} +4.00000 q^{16} +1.44949 q^{17} -1.00000 q^{19} +4.89898 q^{20} +1.00000 q^{23} +1.00000 q^{25} +8.89898 q^{28} -1.00000 q^{29} -10.4495 q^{31} +10.8990 q^{35} +9.89898 q^{37} +9.34847 q^{41} -0.101021 q^{43} +12.4495 q^{47} +12.7980 q^{49} -2.00000 q^{52} +2.00000 q^{53} +0.550510 q^{59} -3.10102 q^{61} -8.00000 q^{64} -2.44949 q^{65} -8.89898 q^{67} -2.89898 q^{68} +12.3485 q^{71} -14.2474 q^{73} +2.00000 q^{76} +5.89898 q^{79} -9.79796 q^{80} +7.55051 q^{83} -3.55051 q^{85} -5.44949 q^{89} -4.44949 q^{91} -2.00000 q^{92} +2.44949 q^{95} +16.8990 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} - 4 q^{7}+O(q^{10}) \) 2 * q - 4 * q^4 - 4 * q^7	\( 2 q - 4 q^{4} - 4 q^{7} + 2 q^{13} + 8 q^{16} - 2 q^{17} - 2 q^{19} + 2 q^{23} + 2 q^{25} + 8 q^{28} - 2 q^{29} - 16 q^{31} + 12 q^{35} + 10 q^{37} + 4 q^{41} - 10 q^{43} + 20 q^{47} + 6 q^{49} - 4 q^{52} + 4 q^{53} + 6 q^{59} - 16 q^{61} - 16 q^{64} - 8 q^{67} + 4 q^{68} + 10 q^{71} - 4 q^{73} + 4 q^{76} + 2 q^{79} + 20 q^{83} - 12 q^{85} - 6 q^{89} - 4 q^{91} - 4 q^{92} + 24 q^{97}+O(q^{100}) \) 2 * q - 4 * q^4 - 4 * q^7 + 2 * q^13 + 8 * q^16 - 2 * q^17 - 2 * q^19 + 2 * q^23 + 2 * q^25 + 8 * q^28 - 2 * q^29 - 16 * q^31 + 12 * q^35 + 10 * q^37 + 4 * q^41 - 10 * q^43 + 20 * q^47 + 6 * q^49 - 4 * q^52 + 4 * q^53 + 6 * q^59 - 16 * q^61 - 16 * q^64 - 8 * q^67 + 4 * q^68 + 10 * q^71 - 4 * q^73 + 4 * q^76 + 2 * q^79 + 20 * q^83 - 12 * q^85 - 6 * q^89 - 4 * q^91 - 4 * q^92 + 24 * q^97

Introduction

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