LMFDB - Embedded newform 6003.2.a.t.1.14

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:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
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)

$f(q)$	$=$	$q+0.289055 q^{2} -1.91645 q^{4} -0.894239 q^{5} +3.31166 q^{7} -1.13207 q^{8} +O(q^{10})$	\(q+0.289055 q^{2} -1.91645 q^{4} -0.894239 q^{5} +3.31166 q^{7} -1.13207 q^{8} -0.258484 q^{10} -3.47272 q^{11} +1.33472 q^{13} +0.957253 q^{14} +3.50566 q^{16} +4.10507 q^{17} -2.37402 q^{19} +1.71376 q^{20} -1.00381 q^{22} +1.00000 q^{23} -4.20034 q^{25} +0.385806 q^{26} -6.34663 q^{28} +1.00000 q^{29} -8.16928 q^{31} +3.27747 q^{32} +1.18659 q^{34} -2.96142 q^{35} -7.23767 q^{37} -0.686223 q^{38} +1.01234 q^{40} +2.74639 q^{41} +1.38389 q^{43} +6.65529 q^{44} +0.289055 q^{46} +2.36806 q^{47} +3.96711 q^{49} -1.21413 q^{50} -2.55791 q^{52} +1.52853 q^{53} +3.10545 q^{55} -3.74903 q^{56} +0.289055 q^{58} +15.2608 q^{59} +4.80702 q^{61} -2.36137 q^{62} -6.06396 q^{64} -1.19355 q^{65} -12.6857 q^{67} -7.86716 q^{68} -0.856013 q^{70} -0.717856 q^{71} -11.3827 q^{73} -2.09209 q^{74} +4.54969 q^{76} -11.5005 q^{77} +8.33380 q^{79} -3.13490 q^{80} +0.793858 q^{82} -0.376134 q^{83} -3.67092 q^{85} +0.400021 q^{86} +3.93136 q^{88} +17.2590 q^{89} +4.42013 q^{91} -1.91645 q^{92} +0.684501 q^{94} +2.12294 q^{95} +11.7592 q^{97} +1.14671 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8}+O(q^{10}) $ 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8	$ 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34} + 2 q^{35} - 28 q^{37} + 14 q^{38} - 30 q^{40} - 10 q^{41} - 14 q^{43} + 37 q^{44} - 3 q^{46} - 18 q^{47} + 2 q^{49} + 7 q^{50} - 57 q^{52} + 20 q^{53} - 42 q^{55} - 2 q^{56} - 3 q^{58} - 20 q^{59} - 38 q^{61} + 4 q^{62} - 24 q^{64} + 12 q^{65} - 50 q^{67} + 11 q^{68} - 48 q^{70} + 12 q^{71} - 46 q^{73} - 6 q^{74} - 16 q^{76} - 14 q^{77} - 20 q^{79} - 58 q^{80} - 42 q^{82} + 22 q^{83} - 66 q^{85} + 22 q^{86} - 68 q^{88} - 14 q^{89} - 16 q^{91} + 17 q^{92} - 27 q^{94} - 20 q^{95} - 48 q^{97} - 28 q^{98}+O(q^{100}) $ 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * 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 $

$2$	0.289055	0.204393	0.102196	−	0.994764i	$-0.467413\pi$
$2$	0.289055	0.204393	0.102196	+	0.994764i	$0.467413\pi$
$3$	0	0
$3$	0	0
$4$	−1.91645	−0.958224
$5$	−0.894239	−0.399916	−0.199958	−	0.979804i	$-0.564081\pi$
$5$	−0.894239	−0.399916	−0.199958	+	0.979804i	$0.564081\pi$
$6$	0	0
$7$	3.31166	1.25169	0.625846	−	0.779947i	$-0.284754\pi$
$7$	3.31166	1.25169	0.625846	+	0.779947i	$0.284754\pi$
$8$	−1.13207	−0.400247
$9$	0	0
$10$	−0.258484	−0.0817400
$11$	−3.47272	−1.04707	−0.523533	−	0.852005i	$-0.675386\pi$
$11$	−3.47272	−1.04707	−0.523533	+	0.852005i	$0.675386\pi$
$12$	0	0
$13$	1.33472	0.370183	0.185092	−	0.982721i	$-0.440742\pi$
$13$	1.33472	0.370183	0.185092	+	0.982721i	$0.440742\pi$
$14$	0.957253	0.255837
$15$	0	0
$16$	3.50566	0.876416
$17$	4.10507	0.995627	0.497813	−	0.867284i	$-0.334136\pi$
$17$	4.10507	0.995627	0.497813	+	0.867284i	$0.334136\pi$
$18$	0	0
$19$	−2.37402	−0.544638	−0.272319	−	0.962207i	$-0.587791\pi$
$19$	−2.37402	−0.544638	−0.272319	+	0.962207i	$0.587791\pi$
$20$	1.71376	0.383209
$21$	0	0
$22$	−1.00381	−0.214013
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.20034	−0.840067
$26$	0.385806	0.0756629
$27$	0	0
$28$	−6.34663	−1.19940
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−8.16928	−1.46725	−0.733623	−	0.679557i	$-0.762172\pi$
$31$	−8.16928	−1.46725	−0.733623	+	0.679557i	$0.762172\pi$
$32$	3.27747	0.579380
$33$	0	0
$34$	1.18659	0.203499
$35$	−2.96142	−0.500571
$36$	0	0
$37$	−7.23767	−1.18987	−0.594933	−	0.803775i	$-0.702821\pi$
$37$	−7.23767	−1.18987	−0.594933	+	0.803775i	$0.702821\pi$
$38$	−0.686223	−0.111320
$39$	0	0
$40$	1.01234	0.160065
$41$	2.74639	0.428914	0.214457	−	0.976733i	$-0.431202\pi$
$41$	2.74639	0.428914	0.214457	+	0.976733i	$0.431202\pi$
$42$	0	0
$43$	1.38389	0.211041	0.105521	−	0.994417i	$-0.466349\pi$
$43$	1.38389	0.211041	0.105521	+	0.994417i	$0.466349\pi$
$44$	6.65529	1.00332
$45$	0	0
$46$	0.289055	0.0426189
$47$	2.36806	0.345417	0.172709	−	0.984973i	$-0.444748\pi$
$47$	2.36806	0.345417	0.172709	+	0.984973i	$0.444748\pi$
$48$	0	0
$49$	3.96711	0.566731
$50$	−1.21413	−0.171704
$51$	0	0
$52$	−2.55791	−0.354719
$53$	1.52853	0.209960	0.104980	−	0.994474i	$-0.466522\pi$
$53$	1.52853	0.209960	0.104980	+	0.994474i	$0.466522\pi$
$54$	0	0
$55$	3.10545	0.418738
$56$	−3.74903	−0.500985
$57$	0	0
$58$	0.289055	0.0379548
$59$	15.2608	1.98679	0.993395	−	0.114745i	$-0.0366051\pi$
$59$	15.2608	1.98679	0.993395	+	0.114745i	$0.0366051\pi$
$60$	0	0
$61$	4.80702	0.615476	0.307738	−	0.951471i	$-0.400428\pi$
$61$	4.80702	0.615476	0.307738	+	0.951471i	$0.400428\pi$
$62$	−2.36137	−0.299895
$63$	0	0
$64$	−6.06396	−0.757995
$65$	−1.19355	−0.148042
$66$	0	0
$67$	−12.6857	−1.54980	−0.774900	−	0.632083i	$-0.782200\pi$
$67$	−12.6857	−1.54980	−0.774900	+	0.632083i	$0.782200\pi$
$68$	−7.86716	−0.954033
$69$	0	0
$70$	−0.856013	−0.102313
$71$	−0.717856	−0.0851938	−0.0425969	−	0.999092i	$-0.513563\pi$
$71$	−0.717856	−0.0851938	−0.0425969	+	0.999092i	$0.513563\pi$
$72$	0	0
$73$	−11.3827	−1.33225	−0.666124	−	0.745841i	$-0.732048\pi$
$73$	−11.3827	−1.33225	−0.666124	+	0.745841i	$0.732048\pi$
$74$	−2.09209	−0.243200
$75$	0	0
$76$	4.54969	0.521885
$77$	−11.5005	−1.31060
$78$	0	0
$79$	8.33380	0.937625	0.468813	−	0.883298i	$-0.344682\pi$
$79$	8.33380	0.937625	0.468813	+	0.883298i	$0.344682\pi$
$80$	−3.13490	−0.350493
$81$	0	0
$82$	0.793858	0.0876669
$83$	−0.376134	−0.0412860	−0.0206430	−	0.999787i	$-0.506571\pi$
$83$	−0.376134	−0.0412860	−0.0206430	+	0.999787i	$0.506571\pi$
$84$	0	0
$85$	−3.67092	−0.398167
$86$	0.400021	0.0431353
$87$	0	0
$88$	3.93136	0.419085
$89$	17.2590	1.82945	0.914724	−	0.404079i	$-0.132408\pi$
$89$	17.2590	1.82945	0.914724	+	0.404079i	$0.132408\pi$
$90$	0	0
$91$	4.42013	0.463355
$92$	−1.91645	−0.199803
$93$	0	0
$94$	0.684501	0.0706009
$95$	2.12294	0.217809
$96$	0	0
$97$	11.7592	1.19397	0.596984	−	0.802253i	$-0.296365\pi$
$97$	11.7592	1.19397	0.596984	+	0.802253i	$0.296365\pi$
$98$	1.14671	0.115836
$99$	0	0
$100$	8.04972	0.804972
$101$	−3.08171	−0.306642	−0.153321	−	0.988176i	$-0.548997\pi$
$101$	−3.08171	−0.306642	−0.153321	+	0.988176i	$0.548997\pi$
$102$	0	0
$103$	−7.26054	−0.715402	−0.357701	−	0.933836i	$-0.616439\pi$
$103$	−7.26054	−0.715402	−0.357701	+	0.933836i	$0.616439\pi$
$104$	−1.51099	−0.148165
$105$	0	0
$106$	0.441831	0.0429144
$107$	−7.59019	−0.733771	−0.366886	−	0.930266i	$-0.619576\pi$
$107$	−7.59019	−0.733771	−0.366886	+	0.930266i	$0.619576\pi$
$108$	0	0
$109$	0.347144	0.0332503	0.0166252	−	0.999862i	$-0.494708\pi$
$109$	0.347144	0.0332503	0.0166252	+	0.999862i	$0.494708\pi$
$110$	0.897645	0.0855871
$111$	0	0
$112$	11.6096	1.09700
$113$	−10.6630	−1.00309	−0.501546	−	0.865131i	$-0.667235\pi$
$113$	−10.6630	−1.00309	−0.501546	+	0.865131i	$0.667235\pi$
$114$	0	0
$115$	−0.894239	−0.0833882
$116$	−1.91645	−0.177938
$117$	0	0
$118$	4.41122	0.406086
$119$	13.5946	1.24622
$120$	0	0
$121$	1.05982	0.0963469
$122$	1.38949	0.125799
$123$	0	0
$124$	15.6560	1.40595
$125$	8.22730	0.735872
$126$	0	0
$127$	−2.00396	−0.177823	−0.0889115	−	0.996040i	$-0.528339\pi$
$127$	−2.00396	−0.177823	−0.0889115	+	0.996040i	$0.528339\pi$
$128$	−8.30776	−0.734309
$129$	0	0
$130$	−0.345003	−0.0302588
$131$	0.854972	0.0746993	0.0373496	−	0.999302i	$-0.488108\pi$
$131$	0.854972	0.0746993	0.0373496	+	0.999302i	$0.488108\pi$
$132$	0	0
$133$	−7.86196	−0.681719
$134$	−3.66686	−0.316768
$135$	0	0
$136$	−4.64723	−0.398496
$137$	−11.8624	−1.01347	−0.506736	−	0.862102i	$-0.669148\pi$
$137$	−11.8624	−1.01347	−0.506736	+	0.862102i	$0.669148\pi$
$138$	0	0
$139$	10.9558	0.929257	0.464628	−	0.885506i	$-0.346188\pi$
$139$	10.9558	0.929257	0.464628	+	0.885506i	$0.346188\pi$
$140$	5.67540	0.479659
$141$	0	0
$142$	−0.207500	−0.0174130
$143$	−4.63510	−0.387606
$144$	0	0
$145$	−0.894239	−0.0742625
$146$	−3.29023	−0.272302
$147$	0	0
$148$	13.8706	1.14016
$149$	−21.5784	−1.76777	−0.883887	−	0.467700i	$-0.845083\pi$
$149$	−21.5784	−1.76777	−0.883887	+	0.467700i	$0.845083\pi$
$150$	0	0
$151$	−14.0988	−1.14734	−0.573672	−	0.819085i	$-0.694482\pi$
$151$	−14.0988	−1.14734	−0.573672	+	0.819085i	$0.694482\pi$
$152$	2.68756	0.217990
$153$	0	0
$154$	−3.32428	−0.267878
$155$	7.30529	0.586775
$156$	0	0
$157$	−15.4088	−1.22976	−0.614878	−	0.788622i	$-0.710795\pi$
$157$	−15.4088	−1.22976	−0.614878	+	0.788622i	$0.710795\pi$
$158$	2.40893	0.191644
$159$	0	0
$160$	−2.93084	−0.231703
$161$	3.31166	0.260996
$162$	0	0
$163$	−4.10759	−0.321731	−0.160866	−	0.986976i	$-0.551429\pi$
$163$	−4.10759	−0.321731	−0.160866	+	0.986976i	$0.551429\pi$
$164$	−5.26331	−0.410995
$165$	0	0
$166$	−0.108723	−0.00843857
$167$	−2.48477	−0.192277	−0.0961386	−	0.995368i	$-0.530649\pi$
$167$	−2.48477	−0.192277	−0.0961386	+	0.995368i	$0.530649\pi$
$168$	0	0
$169$	−11.2185	−0.862964
$170$	−1.06110	−0.0813825
$171$	0	0
$172$	−2.65215	−0.202225
$173$	−0.886694	−0.0674141	−0.0337070	−	0.999432i	$-0.510731\pi$
$173$	−0.886694	−0.0674141	−0.0337070	+	0.999432i	$0.510731\pi$
$174$	0	0
$175$	−13.9101	−1.05150
$176$	−12.1742	−0.917665
$177$	0	0
$178$	4.98880	0.373926
$179$	−19.6025	−1.46516	−0.732580	−	0.680681i	$-0.761684\pi$
$179$	−19.6025	−1.46516	−0.732580	+	0.680681i	$0.761684\pi$
$180$	0	0
$181$	0.855906	0.0636190	0.0318095	−	0.999494i	$-0.489873\pi$
$181$	0.855906	0.0636190	0.0318095	+	0.999494i	$0.489873\pi$
$182$	1.27766	0.0947065
$183$	0	0
$184$	−1.13207	−0.0834572
$185$	6.47221	0.475846
$186$	0	0
$187$	−14.2558	−1.04249
$188$	−4.53827	−0.330987
$189$	0	0
$190$	0.613648	0.0445187
$191$	−22.0227	−1.59351	−0.796753	−	0.604305i	$-0.793451\pi$
$191$	−22.0227	−1.59351	−0.796753	+	0.604305i	$0.793451\pi$
$192$	0	0
$193$	−17.3403	−1.24818	−0.624091	−	0.781352i	$-0.714530\pi$
$193$	−17.3403	−1.24818	−0.624091	+	0.781352i	$0.714530\pi$
$194$	3.39906	0.244039
$195$	0	0
$196$	−7.60277	−0.543055
$197$	−10.2972	−0.733645	−0.366822	−	0.930291i	$-0.619554\pi$
$197$	−10.2972	−0.733645	−0.366822	+	0.930291i	$0.619554\pi$
$198$	0	0
$199$	−24.4201	−1.73109	−0.865546	−	0.500829i	$-0.833029\pi$
$199$	−24.4201	−1.73109	−0.865546	+	0.500829i	$0.833029\pi$
$200$	4.75507	0.336234
$201$	0	0
$202$	−0.890785	−0.0626754
$203$	3.31166	0.232433
$204$	0	0
$205$	−2.45593	−0.171529
$206$	−2.09870	−0.146223
$207$	0	0
$208$	4.67906	0.324435
$209$	8.24433	0.570272
$210$	0	0
$211$	8.64080	0.594857	0.297429	−	0.954744i	$-0.403871\pi$
$211$	8.64080	0.594857	0.297429	+	0.954744i	$0.403871\pi$
$212$	−2.92936	−0.201189
$213$	0	0
$214$	−2.19398	−0.149978
$215$	−1.23753	−0.0843988
$216$	0	0
$217$	−27.0539	−1.83654
$218$	0.100344	0.00679613
$219$	0	0
$220$	−5.95142	−0.401245
$221$	5.47911	0.368565
$222$	0	0
$223$	13.1563	0.881009	0.440505	−	0.897750i	$-0.354800\pi$
$223$	13.1563	0.881009	0.440505	+	0.897750i	$0.354800\pi$
$224$	10.8539	0.725205
$225$	0	0
$226$	−3.08220	−0.205025
$227$	9.56440	0.634812	0.317406	−	0.948290i	$-0.397188\pi$
$227$	9.56440	0.634812	0.317406	+	0.948290i	$0.397188\pi$
$228$	0	0
$229$	−0.877427	−0.0579820	−0.0289910	−	0.999580i	$-0.509229\pi$
$229$	−0.877427	−0.0579820	−0.0289910	+	0.999580i	$0.509229\pi$
$230$	−0.258484	−0.0170440
$231$	0	0
$232$	−1.13207	−0.0743240
$233$	2.16205	0.141641	0.0708204	−	0.997489i	$-0.477438\pi$
$233$	2.16205	0.141641	0.0708204	+	0.997489i	$0.477438\pi$
$234$	0	0
$235$	−2.11761	−0.138138
$236$	−29.2466	−1.90379
$237$	0	0
$238$	3.92960	0.254718
$239$	27.0986	1.75286	0.876430	−	0.481529i	$-0.159918\pi$
$239$	27.0986	1.75286	0.876430	+	0.481529i	$0.159918\pi$
$240$	0	0
$241$	4.64431	0.299166	0.149583	−	0.988749i	$-0.452207\pi$
$241$	4.64431	0.299166	0.149583	+	0.988749i	$0.452207\pi$
$242$	0.306345	0.0196926
$243$	0	0
$244$	−9.21240	−0.589763
$245$	−3.54755	−0.226645
$246$	0	0
$247$	−3.16864	−0.201616
$248$	9.24819	0.587261
$249$	0	0
$250$	2.37814	0.150407
$251$	−0.336274	−0.0212254	−0.0106127	−	0.999944i	$-0.503378\pi$
$251$	−0.336274	−0.0212254	−0.0106127	+	0.999944i	$0.503378\pi$
$252$	0	0
$253$	−3.47272	−0.218328
$254$	−0.579256	−0.0363458
$255$	0	0
$256$	9.72652	0.607907
$257$	4.19170	0.261471	0.130736	−	0.991417i	$-0.458266\pi$
$257$	4.19170	0.261471	0.130736	+	0.991417i	$0.458266\pi$
$258$	0	0
$259$	−23.9687	−1.48934
$260$	2.28738	0.141858
$261$	0	0
$262$	0.247134	0.0152680
$263$	28.9512	1.78521	0.892605	−	0.450840i	$-0.148876\pi$
$263$	28.9512	1.78521	0.892605	+	0.450840i	$0.148876\pi$
$264$	0	0
$265$	−1.36688	−0.0839665
$266$	−2.27254	−0.139338
$267$	0	0
$268$	24.3114	1.48506
$269$	−23.7525	−1.44822	−0.724108	−	0.689686i	$-0.757748\pi$
$269$	−23.7525	−1.44822	−0.724108	+	0.689686i	$0.757748\pi$
$270$	0	0
$271$	3.93157	0.238826	0.119413	−	0.992845i	$-0.461899\pi$
$271$	3.93157	0.238826	0.119413	+	0.992845i	$0.461899\pi$
$272$	14.3910	0.872583
$273$	0	0
$274$	−3.42888	−0.207146
$275$	14.5866	0.879606
$276$	0	0
$277$	2.29969	0.138175	0.0690875	−	0.997611i	$-0.477991\pi$
$277$	2.29969	0.138175	0.0690875	+	0.997611i	$0.477991\pi$
$278$	3.16682	0.189933
$279$	0	0
$280$	3.35253	0.200352
$281$	−6.10006	−0.363899	−0.181949	−	0.983308i	$-0.558241\pi$
$281$	−6.10006	−0.363899	−0.181949	+	0.983308i	$0.558241\pi$
$282$	0	0
$283$	−10.0481	−0.597296	−0.298648	−	0.954363i	$-0.596536\pi$
$283$	−10.0481	−0.597296	−0.298648	+	0.954363i	$0.596536\pi$
$284$	1.37573	0.0816347
$285$	0	0
$286$	−1.33980	−0.0792240
$287$	9.09511	0.536868
$288$	0	0
$289$	−0.148368	−0.00872752
$290$	−0.258484	−0.0151787
$291$	0	0
$292$	21.8144	1.27659
$293$	−9.61185	−0.561530	−0.280765	−	0.959777i	$-0.590588\pi$
$293$	−9.61185	−0.561530	−0.280765	+	0.959777i	$0.590588\pi$
$294$	0	0
$295$	−13.6468	−0.794549
$296$	8.19354	0.476240
$297$	0	0
$298$	−6.23736	−0.361321
$299$	1.33472	0.0771886
$300$	0	0
$301$	4.58298	0.264158
$302$	−4.07533	−0.234509
$303$	0	0
$304$	−8.32253	−0.477330
$305$	−4.29862	−0.246139
$306$	0	0
$307$	−19.8876	−1.13505	−0.567523	−	0.823358i	$-0.692098\pi$
$307$	−19.8876	−1.13505	−0.567523	+	0.823358i	$0.692098\pi$
$308$	22.0401	1.25585
$309$	0	0
$310$	2.11163	0.119933
$311$	−12.7548	−0.723257	−0.361629	−	0.932322i	$-0.617779\pi$
$311$	−12.7548	−0.723257	−0.361629	+	0.932322i	$0.617779\pi$
$312$	0	0
$313$	−2.13988	−0.120953	−0.0604766	−	0.998170i	$-0.519262\pi$
$313$	−2.13988	−0.120953	−0.0604766	+	0.998170i	$0.519262\pi$
$314$	−4.45399	−0.251353
$315$	0	0
$316$	−15.9713	−0.898455
$317$	−30.1388	−1.69276	−0.846381	−	0.532577i	$-0.821224\pi$
$317$	−30.1388	−1.69276	−0.846381	+	0.532577i	$0.821224\pi$
$318$	0	0
$319$	−3.47272	−0.194435
$320$	5.42263	0.303134
$321$	0	0
$322$	0.957253	0.0533456
$323$	−9.74554	−0.542256
$324$	0	0
$325$	−5.60625	−0.310979
$326$	−1.18732	−0.0657596
$327$	0	0
$328$	−3.10910	−0.171671
$329$	7.84223	0.432356
$330$	0	0
$331$	−5.18664	−0.285084	−0.142542	−	0.989789i	$-0.545528\pi$
$331$	−5.18664	−0.285084	−0.142542	+	0.989789i	$0.545528\pi$
$332$	0.720840	0.0395612
$333$	0	0
$334$	−0.718235	−0.0393001
$335$	11.3440	0.619790
$336$	0	0
$337$	3.43321	0.187019	0.0935093	−	0.995618i	$-0.470192\pi$
$337$	3.43321	0.187019	0.0935093	+	0.995618i	$0.470192\pi$
$338$	−3.24278	−0.176384
$339$	0	0
$340$	7.03512	0.381533
$341$	28.3697	1.53630
$342$	0	0
$343$	−10.0439	−0.542319
$344$	−1.56666	−0.0844686
$345$	0	0
$346$	−0.256303	−0.0137790
$347$	28.6744	1.53932	0.769661	−	0.638453i	$-0.220425\pi$
$347$	28.6744	1.53932	0.769661	+	0.638453i	$0.220425\pi$
$348$	0	0
$349$	−19.5605	−1.04705	−0.523524	−	0.852011i	$-0.675383\pi$
$349$	−19.5605	−1.04705	−0.523524	+	0.852011i	$0.675383\pi$
$350$	−4.02079	−0.214920
$351$	0	0
$352$	−11.3817	−0.606649
$353$	1.49472	0.0795560	0.0397780	−	0.999209i	$-0.487335\pi$
$353$	1.49472	0.0795560	0.0397780	+	0.999209i	$0.487335\pi$
$354$	0	0
$355$	0.641935	0.0340704
$356$	−33.0759	−1.75302
$357$	0	0
$358$	−5.66621	−0.299468
$359$	−3.55884	−0.187828	−0.0939142	−	0.995580i	$-0.529938\pi$
$359$	−3.55884	−0.187828	−0.0939142	+	0.995580i	$0.529938\pi$
$360$	0	0
$361$	−13.3640	−0.703369
$362$	0.247404	0.0130033
$363$	0	0
$364$	−8.47094	−0.443998
$365$	10.1789	0.532787
$366$	0	0
$367$	26.5002	1.38330	0.691649	−	0.722234i	$-0.256884\pi$
$367$	26.5002	1.38330	0.691649	+	0.722234i	$0.256884\pi$
$368$	3.50566	0.182745
$369$	0	0
$370$	1.87083	0.0972596
$371$	5.06199	0.262806
$372$	0	0
$373$	−7.34016	−0.380059	−0.190030	−	0.981778i	$-0.560858\pi$
$373$	−7.34016	−0.380059	−0.190030	+	0.981778i	$0.560858\pi$
$374$	−4.12071	−0.213077
$375$	0	0
$376$	−2.68081	−0.138252
$377$	1.33472	0.0687413
$378$	0	0
$379$	−23.6874	−1.21674	−0.608370	−	0.793654i	$-0.708176\pi$
$379$	−23.6874	−1.21674	−0.608370	+	0.793654i	$0.708176\pi$
$380$	−4.06851	−0.208710
$381$	0	0
$382$	−6.36577	−0.325701
$383$	0.481105	0.0245833	0.0122917	−	0.999924i	$-0.496087\pi$
$383$	0.481105	0.0245833	0.0122917	+	0.999924i	$0.496087\pi$
$384$	0	0
$385$	10.2842	0.524131
$386$	−5.01230	−0.255119
$387$	0	0
$388$	−22.5359	−1.14409
$389$	−24.1016	−1.22200	−0.610999	−	0.791631i	$-0.709232\pi$
$389$	−24.1016	−1.22200	−0.610999	+	0.791631i	$0.709232\pi$
$390$	0	0
$391$	4.10507	0.207603
$392$	−4.49105	−0.226832
$393$	0	0
$394$	−2.97646	−0.149952
$395$	−7.45241	−0.374971
$396$	0	0
$397$	−20.8364	−1.04575	−0.522874	−	0.852410i	$-0.675140\pi$
$397$	−20.8364	−1.04575	−0.522874	+	0.852410i	$0.675140\pi$
$398$	−7.05875	−0.353823
$399$	0	0
$400$	−14.7250	−0.736248
$401$	2.85890	0.142767	0.0713833	−	0.997449i	$-0.477259\pi$
$401$	2.85890	0.142767	0.0713833	+	0.997449i	$0.477259\pi$
$402$	0	0
$403$	−10.9037	−0.543150
$404$	5.90594	0.293832
$405$	0	0
$406$	0.957253	0.0475077
$407$	25.1344	1.24587
$408$	0	0
$409$	17.0699	0.844051	0.422025	−	0.906584i	$-0.361319\pi$
$409$	17.0699	0.844051	0.422025	+	0.906584i	$0.361319\pi$
$410$	−0.709899	−0.0350594
$411$	0	0
$412$	13.9144	0.685515
$413$	50.5387	2.48685
$414$	0	0
$415$	0.336353	0.0165109
$416$	4.37449	0.214477
$417$	0	0
$418$	2.38307	0.116560
$419$	14.0978	0.688722	0.344361	−	0.938837i	$-0.388096\pi$
$419$	14.0978	0.688722	0.344361	+	0.938837i	$0.388096\pi$
$420$	0	0
$421$	22.6436	1.10358	0.551791	−	0.833982i	$-0.313945\pi$
$421$	22.6436	1.10358	0.551791	+	0.833982i	$0.313945\pi$
$422$	2.49767	0.121585
$423$	0	0
$424$	−1.73041	−0.0840360
$425$	−17.2427	−0.836393
$426$	0	0
$427$	15.9192	0.770385
$428$	14.5462	0.703117
$429$	0	0
$430$	−0.357714	−0.0172505
$431$	−10.3999	−0.500945	−0.250472	−	0.968124i	$-0.580586\pi$
$431$	−10.3999	−0.500945	−0.250472	+	0.968124i	$0.580586\pi$
$432$	0	0
$433$	15.9738	0.767650	0.383825	−	0.923406i	$-0.374607\pi$
$433$	15.9738	0.767650	0.383825	+	0.923406i	$0.374607\pi$
$434$	−7.82007	−0.375375
$435$	0	0
$436$	−0.665282	−0.0318612
$437$	−2.37402	−0.113565
$438$	0	0
$439$	−31.9678	−1.52574	−0.762871	−	0.646551i	$-0.776211\pi$
$439$	−31.9678	−1.52574	−0.762871	+	0.646551i	$0.776211\pi$
$440$	−3.51558	−0.167599
$441$	0	0
$442$	1.58376	0.0753320
$443$	7.99836	0.380014	0.190007	−	0.981783i	$-0.439149\pi$
$443$	7.99836	0.380014	0.190007	+	0.981783i	$0.439149\pi$
$444$	0	0
$445$	−15.4337	−0.731625
$446$	3.80289	0.180072
$447$	0	0
$448$	−20.0818	−0.948775
$449$	−9.91225	−0.467788	−0.233894	−	0.972262i	$-0.575147\pi$
$449$	−9.91225	−0.467788	−0.233894	+	0.972262i	$0.575147\pi$
$450$	0	0
$451$	−9.53745	−0.449101
$452$	20.4351	0.961186
$453$	0	0
$454$	2.76464	0.129751
$455$	−3.95265	−0.185303
$456$	0	0
$457$	−24.1434	−1.12938	−0.564690	−	0.825303i	$-0.691004\pi$
$457$	−24.1434	−1.12938	−0.564690	+	0.825303i	$0.691004\pi$
$458$	−0.253625	−0.0118511
$459$	0	0
$460$	1.71376	0.0799046
$461$	24.3613	1.13462	0.567310	−	0.823505i	$-0.307984\pi$
$461$	24.3613	1.13462	0.567310	+	0.823505i	$0.307984\pi$
$462$	0	0
$463$	33.5748	1.56035	0.780177	−	0.625559i	$-0.215129\pi$
$463$	33.5748	1.56035	0.780177	+	0.625559i	$0.215129\pi$
$464$	3.50566	0.162746
$465$	0	0
$466$	0.624953	0.0289504
$467$	−21.8840	−1.01267	−0.506336	−	0.862337i	$-0.669000\pi$
$467$	−21.8840	−1.01267	−0.506336	+	0.862337i	$0.669000\pi$
$468$	0	0
$469$	−42.0107	−1.93987
$470$	−0.612107	−0.0282344
$471$	0	0
$472$	−17.2763	−0.795206
$473$	−4.80587	−0.220974
$474$	0	0
$475$	9.97169	0.457533
$476$	−26.0534	−1.19415
$477$	0	0
$478$	7.83298	0.358272
$479$	−33.7604	−1.54255	−0.771277	−	0.636499i	$-0.780382\pi$
$479$	−33.7604	−1.54255	−0.771277	+	0.636499i	$0.780382\pi$
$480$	0	0
$481$	−9.66023	−0.440469
$482$	1.34246	0.0611475
$483$	0	0
$484$	−2.03108	−0.0923219
$485$	−10.5156	−0.477487
$486$	0	0
$487$	23.9241	1.08410	0.542052	−	0.840345i	$-0.317647\pi$
$487$	23.9241	1.08410	0.542052	+	0.840345i	$0.317647\pi$
$488$	−5.44188	−0.246342
$489$	0	0
$490$	−1.02544	−0.0463245
$491$	−38.6555	−1.74450	−0.872248	−	0.489063i	$-0.837339\pi$
$491$	−38.6555	−1.74450	−0.872248	+	0.489063i	$0.837339\pi$
$492$	0	0
$493$	4.10507	0.184883
$494$	−0.915913	−0.0412089
$495$	0	0
$496$	−28.6388	−1.28592
$497$	−2.37730	−0.106636
$498$	0	0
$499$	−17.6483	−0.790045	−0.395022	−	0.918672i	$-0.629263\pi$
$499$	−17.6483	−0.790045	−0.395022	+	0.918672i	$0.629263\pi$
$500$	−15.7672	−0.705130
$501$	0	0
$502$	−0.0972017	−0.00433833
$503$	8.02256	0.357708	0.178854	−	0.983876i	$-0.442761\pi$
$503$	8.02256	0.357708	0.178854	+	0.983876i	$0.442761\pi$
$504$	0	0
$505$	2.75579	0.122631
$506$	−1.00381	−0.0446247
$507$	0	0
$508$	3.84049	0.170394
$509$	15.2539	0.676117	0.338059	−	0.941125i	$-0.390230\pi$
$509$	15.2539	0.676117	0.338059	+	0.941125i	$0.390230\pi$
$510$	0	0
$511$	−37.6957	−1.66756
$512$	19.4270	0.858561
$513$	0	0
$514$	1.21163	0.0534428
$515$	6.49266	0.286101
$516$	0	0
$517$	−8.22363	−0.361675
$518$	−6.92828	−0.304411
$519$	0	0
$520$	1.35119	0.0592535
$521$	−22.3514	−0.979231	−0.489615	−	0.871939i	$-0.662863\pi$
$521$	−22.3514	−0.979231	−0.489615	+	0.871939i	$0.662863\pi$
$522$	0	0
$523$	19.1215	0.836126	0.418063	−	0.908418i	$-0.362709\pi$
$523$	19.1215	0.836126	0.418063	+	0.908418i	$0.362709\pi$
$524$	−1.63851	−0.0715786
$525$	0	0
$526$	8.36850	0.364884
$527$	−33.5355	−1.46083
$528$	0	0
$529$	1.00000	0.0434783
$530$	−0.395102	−0.0171621
$531$	0	0
$532$	15.0670	0.653239
$533$	3.66565	0.158777
$534$	0	0
$535$	6.78745	0.293447
$536$	14.3611	0.620303
$537$	0	0
$538$	−6.86579	−0.296005
$539$	−13.7767	−0.593404
$540$	0	0
$541$	−16.4775	−0.708422	−0.354211	−	0.935166i	$-0.615251\pi$
$541$	−16.4775	−0.708422	−0.354211	+	0.935166i	$0.615251\pi$
$542$	1.13644	0.0488143
$543$	0	0
$544$	13.4543	0.576846
$545$	−0.310429	−0.0132973
$546$	0	0
$547$	−26.7313	−1.14295	−0.571475	−	0.820620i	$-0.693629\pi$
$547$	−26.7313	−1.14295	−0.571475	+	0.820620i	$0.693629\pi$
$548$	22.7336	0.971132
$549$	0	0
$550$	4.21633	0.179785
$551$	−2.37402	−0.101137
$552$	0	0
$553$	27.5987	1.17362
$554$	0.664737	0.0282420
$555$	0	0
$556$	−20.9962	−0.890436
$557$	14.1487	0.599502	0.299751	−	0.954018i	$-0.403096\pi$
$557$	14.1487	0.599502	0.299751	+	0.954018i	$0.403096\pi$
$558$	0	0
$559$	1.84710	0.0781240
$560$	−10.3817	−0.438709
$561$	0	0
$562$	−1.76325	−0.0743783
$563$	28.8148	1.21440	0.607200	−	0.794549i	$-0.292293\pi$
$563$	28.8148	1.21440	0.607200	+	0.794549i	$0.292293\pi$
$564$	0	0
$565$	9.53528	0.401152
$566$	−2.90445	−0.122083
$567$	0	0
$568$	0.812663	0.0340986
$569$	−3.94615	−0.165431	−0.0827156	−	0.996573i	$-0.526359\pi$
$569$	−3.94615	−0.165431	−0.0827156	+	0.996573i	$0.526359\pi$
$570$	0	0
$571$	−14.6831	−0.614469	−0.307234	−	0.951634i	$-0.599404\pi$
$571$	−14.6831	−0.614469	−0.307234	+	0.951634i	$0.599404\pi$
$572$	8.88292	0.371414
$573$	0	0
$574$	2.62899	0.109732
$575$	−4.20034	−0.175166
$576$	0	0
$577$	−22.5062	−0.936944	−0.468472	−	0.883478i	$-0.655195\pi$
$577$	−22.5062	−0.936944	−0.468472	+	0.883478i	$0.655195\pi$
$578$	−0.0428865	−0.00178384
$579$	0	0
$580$	1.71376	0.0711601
$581$	−1.24563	−0.0516773
$582$	0	0
$583$	−5.30818	−0.219842
$584$	12.8860	0.533228
$585$	0	0
$586$	−2.77835	−0.114773
$587$	−9.57987	−0.395403	−0.197702	−	0.980262i	$-0.563348\pi$
$587$	−9.57987	−0.395403	−0.197702	+	0.980262i	$0.563348\pi$
$588$	0	0
$589$	19.3941	0.799118
$590$	−3.94469	−0.162400
$591$	0	0
$592$	−25.3728	−1.04282
$593$	20.6846	0.849417	0.424708	−	0.905330i	$-0.360377\pi$
$593$	20.6846	0.849417	0.424708	+	0.905330i	$0.360377\pi$
$594$	0	0
$595$	−12.1568	−0.498382
$596$	41.3539	1.69392
$597$	0	0
$598$	0.385806	0.0157768
$599$	25.3877	1.03731	0.518656	−	0.854983i	$-0.326432\pi$
$599$	25.3877	1.03731	0.518656	+	0.854983i	$0.326432\pi$
$600$	0	0
$601$	9.65156	0.393695	0.196848	−	0.980434i	$-0.436930\pi$
$601$	9.65156	0.393695	0.196848	+	0.980434i	$0.436930\pi$
$602$	1.32473	0.0539921
$603$	0	0
$604$	27.0196	1.09941
$605$	−0.947729	−0.0385307
$606$	0	0
$607$	−5.51354	−0.223788	−0.111894	−	0.993720i	$-0.535692\pi$
$607$	−5.51354	−0.223788	−0.111894	+	0.993720i	$0.535692\pi$
$608$	−7.78078	−0.315552
$609$	0	0
$610$	−1.24254	−0.0503090
$611$	3.16069	0.127868
$612$	0	0
$613$	29.9800	1.21088	0.605440	−	0.795891i	$-0.292997\pi$
$613$	29.9800	1.21088	0.605440	+	0.795891i	$0.292997\pi$
$614$	−5.74862	−0.231995
$615$	0	0
$616$	13.0194	0.524565
$617$	−22.8426	−0.919607	−0.459803	−	0.888021i	$-0.652080\pi$
$617$	−22.8426	−0.919607	−0.459803	+	0.888021i	$0.652080\pi$
$618$	0	0
$619$	8.68566	0.349106	0.174553	−	0.984648i	$-0.444152\pi$
$619$	8.68566	0.349106	0.174553	+	0.984648i	$0.444152\pi$
$620$	−14.0002	−0.562262
$621$	0	0
$622$	−3.68683	−0.147829
$623$	57.1559	2.28990
$624$	0	0
$625$	13.6445	0.545780
$626$	−0.618544	−0.0247220
$627$	0	0
$628$	29.5302	1.17838
$629$	−29.7112	−1.18466
$630$	0	0
$631$	32.6104	1.29820	0.649100	−	0.760703i	$-0.275146\pi$
$631$	32.6104	1.29820	0.649100	+	0.760703i	$0.275146\pi$
$632$	−9.43443	−0.375282
$633$	0	0
$634$	−8.71177	−0.345989
$635$	1.79202	0.0711143
$636$	0	0
$637$	5.29497	0.209794
$638$	−1.00381	−0.0397412
$639$	0	0
$640$	7.42912	0.293662
$641$	35.2898	1.39386	0.696931	−	0.717138i	$-0.254548\pi$
$641$	35.2898	1.39386	0.696931	+	0.717138i	$0.254548\pi$
$642$	0	0
$643$	−3.07557	−0.121289	−0.0606444	−	0.998159i	$-0.519316\pi$
$643$	−3.07557	−0.121289	−0.0606444	+	0.998159i	$0.519316\pi$
$644$	−6.34663	−0.250092
$645$	0	0
$646$	−2.81700	−0.110833
$647$	−6.33211	−0.248941	−0.124470	−	0.992223i	$-0.539723\pi$
$647$	−6.33211	−0.248941	−0.124470	+	0.992223i	$0.539723\pi$
$648$	0	0
$649$	−52.9966	−2.08030
$650$	−1.62052	−0.0635619
$651$	0	0
$652$	7.87198	0.308290
$653$	−37.9427	−1.48481	−0.742407	−	0.669949i	$-0.766316\pi$
$653$	−37.9427	−1.48481	−0.742407	+	0.669949i	$0.766316\pi$
$654$	0	0
$655$	−0.764550	−0.0298734
$656$	9.62791	0.375907
$657$	0	0
$658$	2.26684	0.0883705
$659$	−12.5938	−0.490586	−0.245293	−	0.969449i	$-0.578884\pi$
$659$	−12.5938	−0.490586	−0.245293	+	0.969449i	$0.578884\pi$
$660$	0	0
$661$	−3.00472	−0.116870	−0.0584349	−	0.998291i	$-0.518611\pi$
$661$	−3.00472	−0.116870	−0.0584349	+	0.998291i	$0.518611\pi$
$662$	−1.49923	−0.0582691
$663$	0	0
$664$	0.425809	0.0165246
$665$	7.03048	0.272630
$666$	0	0
$667$	1.00000	0.0387202
$668$	4.76193	0.184245
$669$	0	0
$670$	3.27905	0.126681
$671$	−16.6935	−0.644444
$672$	0	0
$673$	13.1939	0.508588	0.254294	−	0.967127i	$-0.418157\pi$
$673$	13.1939	0.508588	0.254294	+	0.967127i	$0.418157\pi$
$674$	0.992386	0.0382253
$675$	0	0
$676$	21.4997	0.826913
$677$	−31.7524	−1.22034	−0.610171	−	0.792270i	$-0.708899\pi$
$677$	−31.7524	−1.22034	−0.610171	+	0.792270i	$0.708899\pi$
$678$	0	0
$679$	38.9426	1.49448
$680$	4.15573	0.159365
$681$	0	0
$682$	8.20040	0.314009
$683$	−36.3980	−1.39273	−0.696366	−	0.717687i	$-0.745201\pi$
$683$	−36.3980	−1.39273	−0.696366	+	0.717687i	$0.745201\pi$
$684$	0	0
$685$	10.6078	0.405303
$686$	−2.90324	−0.110846
$687$	0	0
$688$	4.85145	0.184960
$689$	2.04016	0.0777238
$690$	0	0
$691$	3.06066	0.116433	0.0582165	−	0.998304i	$-0.481459\pi$
$691$	3.06066	0.116433	0.0582165	+	0.998304i	$0.481459\pi$
$692$	1.69930	0.0645978
$693$	0	0
$694$	8.28848	0.314626
$695$	−9.79709	−0.371625
$696$	0	0
$697$	11.2741	0.427038
$698$	−5.65405	−0.214009
$699$	0	0
$700$	26.6580	1.00758
$701$	−7.42386	−0.280395	−0.140198	−	0.990124i	$-0.544774\pi$
$701$	−7.42386	−0.280395	−0.140198	+	0.990124i	$0.544774\pi$
$702$	0	0
$703$	17.1824	0.648046
$704$	21.0585	0.793671
$705$	0	0
$706$	0.432057	0.0162607
$707$	−10.2056	−0.383821
$708$	0	0
$709$	12.1383	0.455863	0.227931	−	0.973677i	$-0.426804\pi$
$709$	12.1383	0.455863	0.227931	+	0.973677i	$0.426804\pi$
$710$	0.185555	0.00696374
$711$	0	0
$712$	−19.5384	−0.732231
$713$	−8.16928	−0.305942
$714$	0	0
$715$	4.14489	0.155010
$716$	37.5672	1.40395
$717$	0	0
$718$	−1.02870	−0.0383908
$719$	39.1844	1.46133	0.730666	−	0.682735i	$-0.239210\pi$
$719$	39.1844	1.46133	0.730666	+	0.682735i	$0.239210\pi$
$720$	0	0
$721$	−24.0445	−0.895462
$722$	−3.86294	−0.143764
$723$	0	0
$724$	−1.64030	−0.0609612
$725$	−4.20034	−0.155997
$726$	0	0
$727$	−47.9994	−1.78020	−0.890099	−	0.455768i	$-0.849365\pi$
$727$	−47.9994	−1.78020	−0.890099	+	0.455768i	$0.849365\pi$
$728$	−5.00389	−0.185457
$729$	0	0
$730$	2.94226	0.108898
$731$	5.68097	0.210118
$732$	0	0
$733$	47.6003	1.75816	0.879078	−	0.476678i	$-0.158159\pi$
$733$	47.6003	1.75816	0.879078	+	0.476678i	$0.158159\pi$
$734$	7.66001	0.282736
$735$	0	0
$736$	3.27747	0.120809
$737$	44.0538	1.62274
$738$	0	0
$739$	−49.1296	−1.80726	−0.903631	−	0.428311i	$-0.859109\pi$
$739$	−49.1296	−1.80726	−0.903631	+	0.428311i	$0.859109\pi$
$740$	−12.4036	−0.455967
$741$	0	0
$742$	1.46319	0.0537156
$743$	−28.0456	−1.02889	−0.514447	−	0.857522i	$-0.672003\pi$
$743$	−28.0456	−1.02889	−0.514447	+	0.857522i	$0.672003\pi$
$744$	0	0
$745$	19.2963	0.706961
$746$	−2.12171	−0.0776814
$747$	0	0
$748$	27.3205	0.998935
$749$	−25.1362	−0.918455
$750$	0	0
$751$	−19.8878	−0.725716	−0.362858	−	0.931845i	$-0.618199\pi$
$751$	−19.8878	−0.725716	−0.362858	+	0.931845i	$0.618199\pi$
$752$	8.30163	0.302729
$753$	0	0
$754$	0.385806	0.0140502
$755$	12.6077	0.458841
$756$	0	0
$757$	−18.4139	−0.669263	−0.334632	−	0.942349i	$-0.608612\pi$
$757$	−18.4139	−0.669263	−0.334632	+	0.942349i	$0.608612\pi$
$758$	−6.84696	−0.248693
$759$	0	0
$760$	−2.40332	−0.0871776
$761$	33.7274	1.22262	0.611309	−	0.791392i	$-0.290643\pi$
$761$	33.7274	1.22262	0.611309	+	0.791392i	$0.290643\pi$
$762$	0	0
$763$	1.14962	0.0416191
$764$	42.2053	1.52693
$765$	0	0
$766$	0.139066	0.00502466
$767$	20.3689	0.735477
$768$	0	0
$769$	8.03396	0.289712	0.144856	−	0.989453i	$-0.453728\pi$
$769$	8.03396	0.289712	0.144856	+	0.989453i	$0.453728\pi$
$770$	2.97270	0.107129
$771$	0	0
$772$	33.2317	1.19604
$773$	−4.80753	−0.172915	−0.0864574	−	0.996256i	$-0.527555\pi$
$773$	−4.80753	−0.172915	−0.0864574	+	0.996256i	$0.527555\pi$
$774$	0	0
$775$	34.3137	1.23259
$776$	−13.3123	−0.477882
$777$	0	0
$778$	−6.96668	−0.249768
$779$	−6.51999	−0.233603
$780$	0	0
$781$	2.49292	0.0892036
$782$	1.18659	0.0424325
$783$	0	0
$784$	13.9074	0.496692
$785$	13.7792	0.491799
$786$	0	0
$787$	11.4342	0.407586	0.203793	−	0.979014i	$-0.434673\pi$
$787$	11.4342	0.407586	0.203793	+	0.979014i	$0.434673\pi$
$788$	19.7340	0.702996
$789$	0	0
$790$	−2.15416	−0.0766415
$791$	−35.3123	−1.25556
$792$	0	0
$793$	6.41600	0.227839
$794$	−6.02286	−0.213743
$795$	0	0
$796$	46.7998	1.65877
$797$	18.4342	0.652974	0.326487	−	0.945202i	$-0.394135\pi$
$797$	18.4342	0.652974	0.326487	+	0.945202i	$0.394135\pi$
$798$	0	0
$799$	9.72107	0.343907
$800$	−13.7665	−0.486718
$801$	0	0
$802$	0.826380	0.0291805
$803$	39.5291	1.39495
$804$	0	0
$805$	−2.96142	−0.104376
$806$	−3.15176	−0.111016
$807$	0	0
$808$	3.48871	0.122733
$809$	12.2971	0.432345	0.216172	−	0.976355i	$-0.430643\pi$
$809$	12.2971	0.432345	0.216172	+	0.976355i	$0.430643\pi$
$810$	0	0
$811$	5.31716	0.186711	0.0933554	−	0.995633i	$-0.470241\pi$
$811$	5.31716	0.186711	0.0933554	+	0.995633i	$0.470241\pi$
$812$	−6.34663	−0.222723
$813$	0	0
$814$	7.26524	0.254646
$815$	3.67317	0.128665
$816$	0	0
$817$	−3.28539	−0.114941
$818$	4.93413	0.172518
$819$	0	0
$820$	4.70666	0.164364
$821$	42.5523	1.48509	0.742543	−	0.669799i	$-0.233620\pi$
$821$	42.5523	1.48509	0.742543	+	0.669799i	$0.233620\pi$
$822$	0	0
$823$	31.2256	1.08846	0.544229	−	0.838937i	$-0.316822\pi$
$823$	31.2256	1.08846	0.544229	+	0.838937i	$0.316822\pi$
$824$	8.21943	0.286337
$825$	0	0
$826$	14.6085	0.508294
$827$	−25.7765	−0.896336	−0.448168	−	0.893949i	$-0.647923\pi$
$827$	−25.7765	−0.896336	−0.448168	+	0.893949i	$0.647923\pi$
$828$	0	0
$829$	−20.3170	−0.705638	−0.352819	−	0.935692i	$-0.614777\pi$
$829$	−20.3170	−0.705638	−0.352819	+	0.935692i	$0.614777\pi$
$830$	0.0972247	0.00337472
$831$	0	0
$832$	−8.09366	−0.280597
$833$	16.2853	0.564252
$834$	0	0
$835$	2.22198	0.0768947
$836$	−15.7998	−0.546448
$837$	0	0
$838$	4.07504	0.140770
$839$	43.8024	1.51223	0.756113	−	0.654441i	$-0.227096\pi$
$839$	43.8024	1.51223	0.756113	+	0.654441i	$0.227096\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	6.54525	0.225564
$843$	0	0
$844$	−16.5596	−0.570006
$845$	10.0321	0.345113
$846$	0	0
$847$	3.50975	0.120597
$848$	5.35853	0.184013
$849$	0	0
$850$	−4.98409	−0.170953
$851$	−7.23767	−0.248104
$852$	0	0
$853$	−1.50016	−0.0513644	−0.0256822	−	0.999670i	$-0.508176\pi$
$853$	−1.50016	−0.0513644	−0.0256822	+	0.999670i	$0.508176\pi$
$854$	4.60153	0.157461
$855$	0	0
$856$	8.59262	0.293690
$857$	42.2557	1.44343	0.721713	−	0.692192i	$-0.243355\pi$
$857$	42.2557	1.44343	0.721713	+	0.692192i	$0.243355\pi$
$858$	0	0
$859$	26.1886	0.893543	0.446771	−	0.894648i	$-0.352574\pi$
$859$	26.1886	0.893543	0.446771	+	0.894648i	$0.352574\pi$
$860$	2.37166	0.0808729
$861$	0	0
$862$	−3.00614	−0.102390
$863$	−37.8785	−1.28940	−0.644700	−	0.764436i	$-0.723018\pi$
$863$	−37.8785	−1.28940	−0.644700	+	0.764436i	$0.723018\pi$
$864$	0	0
$865$	0.792916	0.0269600
$866$	4.61730	0.156902
$867$	0	0
$868$	51.8474	1.75981
$869$	−28.9410	−0.981756
$870$	0	0
$871$	−16.9318	−0.573711
$872$	−0.392991	−0.0133083
$873$	0	0
$874$	−0.686223	−0.0232119
$875$	27.2461	0.921085
$876$	0	0
$877$	55.5435	1.87557	0.937786	−	0.347214i	$-0.112872\pi$
$877$	55.5435	1.87557	0.937786	+	0.347214i	$0.112872\pi$
$878$	−9.24047	−0.311851
$879$	0	0
$880$	10.8867	0.366989
$881$	−53.4437	−1.80056	−0.900281	−	0.435309i	$-0.856639\pi$
$881$	−53.4437	−1.80056	−0.900281	+	0.435309i	$0.856639\pi$
$882$	0	0
$883$	22.3197	0.751118	0.375559	−	0.926798i	$-0.377451\pi$
$883$	22.3197	0.751118	0.375559	+	0.926798i	$0.377451\pi$
$884$	−10.5004	−0.353167
$885$	0	0
$886$	2.31197	0.0776721
$887$	45.9912	1.54423	0.772116	−	0.635482i	$-0.219198\pi$
$887$	45.9912	1.54423	0.772116	+	0.635482i	$0.219198\pi$
$888$	0	0
$889$	−6.63646	−0.222580
$890$	−4.46118	−0.149539
$891$	0	0
$892$	−25.2133	−0.844204
$893$	−5.62183	−0.188128
$894$	0	0
$895$	17.5293	0.585941
$896$	−27.5125	−0.919128
$897$	0	0
$898$	−2.86519	−0.0956125
$899$	−8.16928	−0.272461
$900$	0	0
$901$	6.27475	0.209042
$902$	−2.75685	−0.0917930
$903$	0	0
$904$	12.0713	0.401484
$905$	−0.765385	−0.0254423
$906$	0	0
$907$	35.4404	1.17678	0.588389	−	0.808578i	$-0.299762\pi$
$907$	35.4404	1.17678	0.588389	+	0.808578i	$0.299762\pi$
$908$	−18.3297	−0.608291
$909$	0	0
$910$	−1.14253	−0.0378746
$911$	−0.877383	−0.0290690	−0.0145345	−	0.999894i	$-0.504627\pi$
$911$	−0.877383	−0.0290690	−0.0145345	+	0.999894i	$0.504627\pi$
$912$	0	0
$913$	1.30621	0.0432292
$914$	−6.97877	−0.230837
$915$	0	0
$916$	1.68154	0.0555597
$917$	2.83138	0.0935004
$918$	0	0
$919$	−5.81102	−0.191688	−0.0958439	−	0.995396i	$-0.530555\pi$
$919$	−5.81102	−0.191688	−0.0958439	+	0.995396i	$0.530555\pi$
$920$	1.01234	0.0333759
$921$	0	0
$922$	7.04176	0.231908
$923$	−0.958134	−0.0315374
$924$	0	0
$925$	30.4007	0.999567
$926$	9.70498	0.318925
$927$	0	0
$928$	3.27747	0.107588
$929$	49.6595	1.62928	0.814638	−	0.579970i	$-0.196936\pi$
$929$	49.6595	1.62928	0.814638	+	0.579970i	$0.196936\pi$
$930$	0	0
$931$	−9.41802	−0.308663
$932$	−4.14346	−0.135724
$933$	0	0
$934$	−6.32568	−0.206983
$935$	12.7481	0.416907
$936$	0	0
$937$	−11.7442	−0.383665	−0.191833	−	0.981428i	$-0.561443\pi$
$937$	−11.7442	−0.383665	−0.191833	+	0.981428i	$0.561443\pi$
$938$	−12.1434	−0.396496
$939$	0	0
$940$	4.05830	0.132367
$941$	−20.1855	−0.658028	−0.329014	−	0.944325i	$-0.606716\pi$
$941$	−20.1855	−0.658028	−0.329014	+	0.944325i	$0.606716\pi$
$942$	0	0
$943$	2.74639	0.0894347
$944$	53.4993	1.74125
$945$	0	0
$946$	−1.38916	−0.0451655
$947$	7.76958	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$947$	7.76958	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$948$	0	0
$949$	−15.1927	−0.493176
$950$	2.88237	0.0935164
$951$	0	0
$952$	−15.3901	−0.498795
$953$	−4.69003	−0.151925	−0.0759625	−	0.997111i	$-0.524203\pi$
$953$	−4.69003	−0.151925	−0.0759625	+	0.997111i	$0.524203\pi$
$954$	0	0
$955$	19.6935	0.637268
$956$	−51.9330	−1.67963
$957$	0	0
$958$	−9.75863	−0.315287
$959$	−39.2842	−1.26855
$960$	0	0
$961$	35.7372	1.15281
$962$	−2.79234	−0.0900286
$963$	0	0
$964$	−8.90058	−0.286668
$965$	15.5064	0.499167
$966$	0	0
$967$	35.8106	1.15159	0.575796	−	0.817593i	$-0.304692\pi$
$967$	35.8106	1.15159	0.575796	+	0.817593i	$0.304692\pi$
$968$	−1.19979	−0.0385626
$969$	0	0
$970$	−3.03958	−0.0975949
$971$	−18.9838	−0.609220	−0.304610	−	0.952477i	$-0.598526\pi$
$971$	−18.9838	−0.609220	−0.304610	+	0.952477i	$0.598526\pi$
$972$	0	0
$973$	36.2818	1.16314
$974$	6.91539	0.221583
$975$	0	0
$976$	16.8518	0.539413
$977$	−12.0941	−0.386923	−0.193462	−	0.981108i	$-0.561971\pi$
$977$	−12.0941	−0.386923	−0.193462	+	0.981108i	$0.561971\pi$
$978$	0	0
$979$	−59.9357	−1.91555
$980$	6.79869	0.217176
$981$	0	0
$982$	−11.1736	−0.356563
$983$	−10.7348	−0.342387	−0.171193	−	0.985237i	$-0.554762\pi$
$983$	−10.7348	−0.342387	−0.171193	+	0.985237i	$0.554762\pi$
$984$	0	0
$985$	9.20815	0.293396
$986$	1.18659	0.0377888
$987$	0	0
$988$	6.07254	0.193193
$989$	1.38389	0.0440051
$990$	0	0
$991$	35.0246	1.11259	0.556297	−	0.830984i	$-0.312222\pi$
$991$	35.0246	1.11259	0.556297	+	0.830984i	$0.312222\pi$
$992$	−26.7746	−0.850093
$993$	0	0
$994$	−0.687170	−0.0217957
$995$	21.8374	0.692292
$996$	0	0
$997$	−20.8820	−0.661340	−0.330670	−	0.943746i	$-0.607275\pi$
$997$	−20.8820	−0.661340	−0.330670	+	0.943746i	$0.607275\pi$
$998$	−5.10132	−0.161479
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.t.1.14	✓	22
3.2	odd	2		6003.2.a.u.1.9	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.t.1.14	✓	22	1.1	even	1	trivial
6003.2.a.u.1.9	yes	22	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.289055 q^{2} -1.91645 q^{4} -0.894239 q^{5} +3.31166 q^{7} -1.13207 q^{8} +O(q^{10})\)	\(q+0.289055 q^{2} -1.91645 q^{4} -0.894239 q^{5} +3.31166 q^{7} -1.13207 q^{8} -0.258484 q^{10} -3.47272 q^{11} +1.33472 q^{13} +0.957253 q^{14} +3.50566 q^{16} +4.10507 q^{17} -2.37402 q^{19} +1.71376 q^{20} -1.00381 q^{22} +1.00000 q^{23} -4.20034 q^{25} +0.385806 q^{26} -6.34663 q^{28} +1.00000 q^{29} -8.16928 q^{31} +3.27747 q^{32} +1.18659 q^{34} -2.96142 q^{35} -7.23767 q^{37} -0.686223 q^{38} +1.01234 q^{40} +2.74639 q^{41} +1.38389 q^{43} +6.65529 q^{44} +0.289055 q^{46} +2.36806 q^{47} +3.96711 q^{49} -1.21413 q^{50} -2.55791 q^{52} +1.52853 q^{53} +3.10545 q^{55} -3.74903 q^{56} +0.289055 q^{58} +15.2608 q^{59} +4.80702 q^{61} -2.36137 q^{62} -6.06396 q^{64} -1.19355 q^{65} -12.6857 q^{67} -7.86716 q^{68} -0.856013 q^{70} -0.717856 q^{71} -11.3827 q^{73} -2.09209 q^{74} +4.54969 q^{76} -11.5005 q^{77} +8.33380 q^{79} -3.13490 q^{80} +0.793858 q^{82} -0.376134 q^{83} -3.67092 q^{85} +0.400021 q^{86} +3.93136 q^{88} +17.2590 q^{89} +4.42013 q^{91} -1.91645 q^{92} +0.684501 q^{94} +2.12294 q^{95} +11.7592 q^{97} +1.14671 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8}+O(q^{10}) \) 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34} + 2 q^{35} - 28 q^{37} + 14 q^{38} - 30 q^{40} - 10 q^{41} - 14 q^{43} + 37 q^{44} - 3 q^{46} - 18 q^{47} + 2 q^{49} + 7 q^{50} - 57 q^{52} + 20 q^{53} - 42 q^{55} - 2 q^{56} - 3 q^{58} - 20 q^{59} - 38 q^{61} + 4 q^{62} - 24 q^{64} + 12 q^{65} - 50 q^{67} + 11 q^{68} - 48 q^{70} + 12 q^{71} - 46 q^{73} - 6 q^{74} - 16 q^{76} - 14 q^{77} - 20 q^{79} - 58 q^{80} - 42 q^{82} + 22 q^{83} - 66 q^{85} + 22 q^{86} - 68 q^{88} - 14 q^{89} - 16 q^{91} + 17 q^{92} - 27 q^{94} - 20 q^{95} - 48 q^{97} - 28 q^{98}+O(q^{100}) \) 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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