LMFDB - Embedded newform 6030.2.a.bu.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6030,2,Mod(1,6030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6030, 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("6030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6030.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:	$48.1497924188$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.70292.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 10x + 2 $ x^4 - x^3 - 8x^2 + 10x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2010)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.46640$ of defining polynomial
Character	$\chi$	$=$	6030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.08313 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.08313 q^{7} +1.00000 q^{8} +1.00000 q^{10} -2.20503 q^{11} -4.28816 q^{13} +2.08313 q^{14} +1.00000 q^{16} +4.93280 q^{17} +4.93280 q^{19} +1.00000 q^{20} -2.20503 q^{22} +3.35536 q^{23} +1.00000 q^{25} -4.28816 q^{26} +2.08313 q^{28} +2.81090 q^{29} -2.28816 q^{31} +1.00000 q^{32} +4.93280 q^{34} +2.08313 q^{35} -0.727768 q^{37} +4.93280 q^{38} +1.00000 q^{40} +1.47726 q^{41} +4.41006 q^{43} -2.20503 q^{44} +3.35536 q^{46} -9.38722 q^{47} -2.66057 q^{49} +1.00000 q^{50} -4.28816 q^{52} +6.16626 q^{53} -2.20503 q^{55} +2.08313 q^{56} +2.81090 q^{58} +1.47726 q^{59} +4.08313 q^{61} -2.28816 q^{62} +1.00000 q^{64} -4.28816 q^{65} -1.00000 q^{67} +4.93280 q^{68} +2.08313 q^{70} +4.84967 q^{71} -0.544464 q^{73} -0.727768 q^{74} +4.93280 q^{76} -4.59337 q^{77} +4.83263 q^{79} +1.00000 q^{80} +1.47726 q^{82} +16.5934 q^{83} +4.93280 q^{85} +4.41006 q^{86} -2.20503 q^{88} -2.18330 q^{89} -8.93280 q^{91} +3.35536 q^{92} -9.38722 q^{94} +4.93280 q^{95} +2.20503 q^{97} -2.66057 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + q^7 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + q^{7} + 4 q^{8} + 4 q^{10} + 3 q^{11} + 2 q^{13} + q^{14} + 4 q^{16} + 2 q^{17} + 2 q^{19} + 4 q^{20} + 3 q^{22} + 12 q^{23} + 4 q^{25} + 2 q^{26} + q^{28} - 2 q^{29} + 10 q^{31} + 4 q^{32} + 2 q^{34} + q^{35} + 3 q^{37} + 2 q^{38} + 4 q^{40} - 6 q^{43} + 3 q^{44} + 12 q^{46} + 14 q^{47} + 13 q^{49} + 4 q^{50} + 2 q^{52} + 10 q^{53} + 3 q^{55} + q^{56} - 2 q^{58} + 9 q^{61} + 10 q^{62} + 4 q^{64} + 2 q^{65} - 4 q^{67} + 2 q^{68} + q^{70} + 9 q^{71} - 14 q^{73} + 3 q^{74} + 2 q^{76} + 23 q^{77} + 12 q^{79} + 4 q^{80} + 25 q^{83} + 2 q^{85} - 6 q^{86} + 3 q^{88} + 9 q^{89} - 18 q^{91} + 12 q^{92} + 14 q^{94} + 2 q^{95} - 3 q^{97} + 13 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + q^7 + 4 * q^8 + 4 * q^10 + 3 * q^11 + 2 * q^13 + q^14 + 4 * q^16 + 2 * q^17 + 2 * q^19 + 4 * q^20 + 3 * q^22 + 12 * q^23 + 4 * q^25 + 2 * q^26 + q^28 - 2 * q^29 + 10 * q^31 + 4 * q^32 + 2 * q^34 + q^35 + 3 * q^37 + 2 * q^38 + 4 * q^40 - 6 * q^43 + 3 * q^44 + 12 * q^46 + 14 * q^47 + 13 * q^49 + 4 * q^50 + 2 * q^52 + 10 * q^53 + 3 * q^55 + q^56 - 2 * q^58 + 9 * q^61 + 10 * q^62 + 4 * q^64 + 2 * q^65 - 4 * q^67 + 2 * q^68 + q^70 + 9 * q^71 - 14 * q^73 + 3 * q^74 + 2 * q^76 + 23 * q^77 + 12 * q^79 + 4 * q^80 + 25 * q^83 + 2 * q^85 - 6 * q^86 + 3 * q^88 + 9 * q^89 - 18 * q^91 + 12 * q^92 + 14 * q^94 + 2 * q^95 - 3 * q^97 + 13 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.08313	0.787349	0.393675	−	0.919250i	$-0.371204\pi$
$7$	2.08313	0.787349	0.393675	+	0.919250i	$0.371204\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−2.20503	−0.664842	−0.332421	−	0.943131i	$-0.607866\pi$
$11$	−2.20503	−0.664842	−0.332421	+	0.943131i	$0.607866\pi$
$12$	0	0
$13$	−4.28816	−1.18932	−0.594661	−	0.803976i	$-0.702714\pi$
$13$	−4.28816	−1.18932	−0.594661	+	0.803976i	$0.702714\pi$
$14$	2.08313	0.556740
$15$	0	0
$16$	1.00000	0.250000
$17$	4.93280	1.19638	0.598190	−	0.801354i	$-0.295887\pi$
$17$	4.93280	1.19638	0.598190	+	0.801354i	$0.295887\pi$
$18$	0	0
$19$	4.93280	1.13166	0.565831	−	0.824521i	$-0.308555\pi$
$19$	4.93280	1.13166	0.565831	+	0.824521i	$0.308555\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−2.20503	−0.470114
$23$	3.35536	0.699641	0.349821	−	0.936817i	$-0.386243\pi$
$23$	3.35536	0.699641	0.349821	+	0.936817i	$0.386243\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.28816	−0.840978
$27$	0	0
$28$	2.08313	0.393675
$29$	2.81090	0.521971	0.260985	−	0.965343i	$-0.415953\pi$
$29$	2.81090	0.521971	0.260985	+	0.965343i	$0.415953\pi$
$30$	0	0
$31$	−2.28816	−0.410966	−0.205483	−	0.978661i	$-0.565877\pi$
$31$	−2.28816	−0.410966	−0.205483	+	0.978661i	$0.565877\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.93280	0.845968
$35$	2.08313	0.352113
$36$	0	0
$37$	−0.727768	−0.119644	−0.0598222	−	0.998209i	$-0.519053\pi$
$37$	−0.727768	−0.119644	−0.0598222	+	0.998209i	$0.519053\pi$
$38$	4.93280	0.800206
$39$	0	0
$40$	1.00000	0.158114
$41$	1.47726	0.230710	0.115355	−	0.993324i	$-0.463199\pi$
$41$	1.47726	0.230710	0.115355	+	0.993324i	$0.463199\pi$
$42$	0	0
$43$	4.41006	0.672529	0.336264	−	0.941768i	$-0.390836\pi$
$43$	4.41006	0.672529	0.336264	+	0.941768i	$0.390836\pi$
$44$	−2.20503	−0.332421
$45$	0	0
$46$	3.35536	0.494721
$47$	−9.38722	−1.36927	−0.684634	−	0.728887i	$-0.740038\pi$
$47$	−9.38722	−1.36927	−0.684634	+	0.728887i	$0.740038\pi$
$48$	0	0
$49$	−2.66057	−0.380081
$50$	1.00000	0.141421
$51$	0	0
$52$	−4.28816	−0.594661
$53$	6.16626	0.847001	0.423500	−	0.905896i	$-0.360801\pi$
$53$	6.16626	0.847001	0.423500	+	0.905896i	$0.360801\pi$
$54$	0	0
$55$	−2.20503	−0.297326
$56$	2.08313	0.278370
$57$	0	0
$58$	2.81090	0.369089
$59$	1.47726	0.192323	0.0961617	−	0.995366i	$-0.469343\pi$
$59$	1.47726	0.192323	0.0961617	+	0.995366i	$0.469343\pi$
$60$	0	0
$61$	4.08313	0.522791	0.261396	−	0.965232i	$-0.415817\pi$
$61$	4.08313	0.522791	0.261396	+	0.965232i	$0.415817\pi$
$62$	−2.28816	−0.290597
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.28816	−0.531881
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	4.93280	0.598190
$69$	0	0
$70$	2.08313	0.248982
$71$	4.84967	0.575550	0.287775	−	0.957698i	$-0.407084\pi$
$71$	4.84967	0.575550	0.287775	+	0.957698i	$0.407084\pi$
$72$	0	0
$73$	−0.544464	−0.0637246	−0.0318623	−	0.999492i	$-0.510144\pi$
$73$	−0.544464	−0.0637246	−0.0318623	+	0.999492i	$0.510144\pi$
$74$	−0.727768	−0.0846013
$75$	0	0
$76$	4.93280	0.565831
$77$	−4.59337	−0.523463
$78$	0	0
$79$	4.83263	0.543713	0.271856	−	0.962338i	$-0.412362\pi$
$79$	4.83263	0.543713	0.271856	+	0.962338i	$0.412362\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	1.47726	0.163137
$83$	16.5934	1.82136	0.910679	−	0.413114i	$-0.135559\pi$
$83$	16.5934	1.82136	0.910679	+	0.413114i	$0.135559\pi$
$84$	0	0
$85$	4.93280	0.535037
$86$	4.41006	0.475549
$87$	0	0
$88$	−2.20503	−0.235057
$89$	−2.18330	−0.231430	−0.115715	−	0.993282i	$-0.536916\pi$
$89$	−2.18330	−0.231430	−0.115715	+	0.993282i	$0.536916\pi$
$90$	0	0
$91$	−8.93280	−0.936412
$92$	3.35536	0.349821
$93$	0	0
$94$	−9.38722	−0.968218
$95$	4.93280	0.506095
$96$	0	0
$97$	2.20503	0.223887	0.111944	−	0.993715i	$-0.464292\pi$
$97$	2.20503	0.223887	0.111944	+	0.993715i	$0.464292\pi$
$98$	−2.66057	−0.268758
$99$	0	0
$100$	1.00000	0.100000
$101$	9.13783	0.909248	0.454624	−	0.890683i	$-0.349774\pi$
$101$	9.13783	0.909248	0.454624	+	0.890683i	$0.349774\pi$
$102$	0	0
$103$	0.410065	0.0404049	0.0202024	−	0.999796i	$-0.493569\pi$
$103$	0.410065	0.0404049	0.0202024	+	0.999796i	$0.493569\pi$
$104$	−4.28816	−0.420489
$105$	0	0
$106$	6.16626	0.598920
$107$	−0.576325	−0.0557154	−0.0278577	−	0.999612i	$-0.508869\pi$
$107$	−0.576325	−0.0557154	−0.0278577	+	0.999612i	$0.508869\pi$
$108$	0	0
$109$	5.78247	0.553860	0.276930	−	0.960890i	$-0.410683\pi$
$109$	5.78247	0.553860	0.276930	+	0.960890i	$0.410683\pi$
$110$	−2.20503	−0.210242
$111$	0	0
$112$	2.08313	0.196837
$113$	13.8268	1.30072	0.650359	−	0.759627i	$-0.274618\pi$
$113$	13.8268	1.30072	0.650359	+	0.759627i	$0.274618\pi$
$114$	0	0
$115$	3.35536	0.312889
$116$	2.81090	0.260985
$117$	0	0
$118$	1.47726	0.135993
$119$	10.2757	0.941969
$120$	0	0
$121$	−6.13783	−0.557985
$122$	4.08313	0.369669
$123$	0	0
$124$	−2.28816	−0.205483
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.1208	−0.986810	−0.493405	−	0.869800i	$-0.664248\pi$
$127$	−11.1208	−0.986810	−0.493405	+	0.869800i	$0.664248\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−4.28816	−0.376097
$131$	17.6435	1.54152	0.770761	−	0.637124i	$-0.219876\pi$
$131$	17.6435	1.54152	0.770761	+	0.637124i	$0.219876\pi$
$132$	0	0
$133$	10.2757	0.891013
$134$	−1.00000	−0.0863868
$135$	0	0
$136$	4.93280	0.422984
$137$	4.72777	0.403920	0.201960	−	0.979394i	$-0.435269\pi$
$137$	4.72777	0.403920	0.201960	+	0.979394i	$0.435269\pi$
$138$	0	0
$139$	−20.0855	−1.70363	−0.851813	−	0.523846i	$-0.824497\pi$
$139$	−20.0855	−1.70363	−0.851813	+	0.523846i	$0.824497\pi$
$140$	2.08313	0.176057
$141$	0	0
$142$	4.84967	0.406975
$143$	9.45554	0.790712
$144$	0	0
$145$	2.81090	0.233432
$146$	−0.544464	−0.0450601
$147$	0	0
$148$	−0.727768	−0.0598222
$149$	2.81090	0.230278	0.115139	−	0.993349i	$-0.463269\pi$
$149$	2.81090	0.230278	0.115139	+	0.993349i	$0.463269\pi$
$150$	0	0
$151$	11.7142	0.953285	0.476642	−	0.879097i	$-0.341854\pi$
$151$	11.7142	0.953285	0.476642	+	0.879097i	$0.341854\pi$
$152$	4.93280	0.400103
$153$	0	0
$154$	−4.59337	−0.370144
$155$	−2.28816	−0.183790
$156$	0	0
$157$	−20.1981	−1.61199	−0.805993	−	0.591925i	$-0.798368\pi$
$157$	−20.1981	−1.61199	−0.805993	+	0.591925i	$0.798368\pi$
$158$	4.83263	0.384463
$159$	0	0
$160$	1.00000	0.0790569
$161$	6.98966	0.550862
$162$	0	0
$163$	−10.7278	−0.840264	−0.420132	−	0.907463i	$-0.638016\pi$
$163$	−10.7278	−0.840264	−0.420132	+	0.907463i	$0.638016\pi$
$164$	1.47726	0.115355
$165$	0	0
$166$	16.5934	1.28790
$167$	−3.93169	−0.304243	−0.152122	−	0.988362i	$-0.548611\pi$
$167$	−3.93169	−0.304243	−0.152122	+	0.988362i	$0.548611\pi$
$168$	0	0
$169$	5.38834	0.414487
$170$	4.93280	0.378329
$171$	0	0
$172$	4.41006	0.336264
$173$	−9.04779	−0.687891	−0.343945	−	0.938990i	$-0.611764\pi$
$173$	−9.04779	−0.687891	−0.343945	+	0.938990i	$0.611764\pi$
$174$	0	0
$175$	2.08313	0.157470
$176$	−2.20503	−0.166211
$177$	0	0
$178$	−2.18330	−0.163646
$179$	−13.6754	−1.02215	−0.511073	−	0.859537i	$-0.670752\pi$
$179$	−13.6754	−1.02215	−0.511073	+	0.859537i	$0.670752\pi$
$180$	0	0
$181$	0.876984	0.0651857	0.0325929	−	0.999469i	$-0.489624\pi$
$181$	0.876984	0.0651857	0.0325929	+	0.999469i	$0.489624\pi$
$182$	−8.93280	−0.662143
$183$	0	0
$184$	3.35536	0.247361
$185$	−0.727768	−0.0535066
$186$	0	0
$187$	−10.8770	−0.795404
$188$	−9.38722	−0.684634
$189$	0	0
$190$	4.93280	0.357863
$191$	16.4101	1.18739	0.593695	−	0.804690i	$-0.297668\pi$
$191$	16.4101	1.18739	0.593695	+	0.804690i	$0.297668\pi$
$192$	0	0
$193$	−3.17660	−0.228657	−0.114329	−	0.993443i	$-0.536472\pi$
$193$	−3.17660	−0.228657	−0.114329	+	0.993443i	$0.536472\pi$
$194$	2.20503	0.158312
$195$	0	0
$196$	−2.66057	−0.190041
$197$	10.5763	0.753532	0.376766	−	0.926308i	$-0.377036\pi$
$197$	10.5763	0.753532	0.376766	+	0.926308i	$0.377036\pi$
$198$	0	0
$199$	8.59337	0.609168	0.304584	−	0.952486i	$-0.401483\pi$
$199$	8.59337	0.609168	0.304584	+	0.952486i	$0.401483\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	9.13783	0.642936
$203$	5.85547	0.410973
$204$	0	0
$205$	1.47726	0.103177
$206$	0.410065	0.0285705
$207$	0	0
$208$	−4.28816	−0.297331
$209$	−10.8770	−0.752377
$210$	0	0
$211$	−12.8553	−0.884992	−0.442496	−	0.896770i	$-0.645907\pi$
$211$	−12.8553	−0.884992	−0.442496	+	0.896770i	$0.645907\pi$
$212$	6.16626	0.423500
$213$	0	0
$214$	−0.576325	−0.0393968
$215$	4.41006	0.300764
$216$	0	0
$217$	−4.76654	−0.323574
$218$	5.78247	0.391638
$219$	0	0
$220$	−2.20503	−0.148663
$221$	−21.1526	−1.42288
$222$	0	0
$223$	18.8520	1.26242	0.631211	−	0.775611i	$-0.282558\pi$
$223$	18.8520	1.26242	0.631211	+	0.775611i	$0.282558\pi$
$224$	2.08313	0.139185
$225$	0	0
$226$	13.8268	0.919747
$227$	22.1094	1.46745	0.733726	−	0.679445i	$-0.237779\pi$
$227$	22.1094	1.46745	0.733726	+	0.679445i	$0.237779\pi$
$228$	0	0
$229$	0.326934	0.0216044	0.0108022	−	0.999942i	$-0.496561\pi$
$229$	0.326934	0.0216044	0.0108022	+	0.999942i	$0.496561\pi$
$230$	3.35536	0.221246
$231$	0	0
$232$	2.81090	0.184545
$233$	2.14922	0.140800	0.0703999	−	0.997519i	$-0.477572\pi$
$233$	2.14922	0.140800	0.0703999	+	0.997519i	$0.477572\pi$
$234$	0	0
$235$	−9.38722	−0.612355
$236$	1.47726	0.0961617
$237$	0	0
$238$	10.2757	0.666072
$239$	−11.1208	−0.719344	−0.359672	−	0.933079i	$-0.617111\pi$
$239$	−11.1208	−0.719344	−0.359672	+	0.933079i	$0.617111\pi$
$240$	0	0
$241$	−7.15956	−0.461188	−0.230594	−	0.973050i	$-0.574067\pi$
$241$	−7.15956	−0.461188	−0.230594	+	0.973050i	$0.574067\pi$
$242$	−6.13783	−0.394555
$243$	0	0
$244$	4.08313	0.261396
$245$	−2.66057	−0.169978
$246$	0	0
$247$	−21.1526	−1.34591
$248$	−2.28816	−0.145298
$249$	0	0
$250$	1.00000	0.0632456
$251$	10.7278	0.677131	0.338565	−	0.940943i	$-0.390058\pi$
$251$	10.7278	0.677131	0.338565	+	0.940943i	$0.390058\pi$
$252$	0	0
$253$	−7.39868	−0.465151
$254$	−11.1208	−0.697780
$255$	0	0
$256$	1.00000	0.0625000
$257$	20.4202	1.27378	0.636888	−	0.770956i	$-0.280221\pi$
$257$	20.4202	1.27378	0.636888	+	0.770956i	$0.280221\pi$
$258$	0	0
$259$	−1.51604	−0.0942019
$260$	−4.28816	−0.265941
$261$	0	0
$262$	17.6435	1.09002
$263$	3.59917	0.221934	0.110967	−	0.993824i	$-0.464605\pi$
$263$	3.59917	0.221934	0.110967	+	0.993824i	$0.464605\pi$
$264$	0	0
$265$	6.16626	0.378790
$266$	10.2757	0.630041
$267$	0	0
$268$	−1.00000	−0.0610847
$269$	−8.94530	−0.545404	−0.272702	−	0.962098i	$-0.587917\pi$
$269$	−8.94530	−0.545404	−0.272702	+	0.962098i	$0.587917\pi$
$270$	0	0
$271$	−7.41118	−0.450197	−0.225099	−	0.974336i	$-0.572270\pi$
$271$	−7.41118	−0.450197	−0.225099	+	0.974336i	$0.572270\pi$
$272$	4.93280	0.299095
$273$	0	0
$274$	4.72777	0.285615
$275$	−2.20503	−0.132968
$276$	0	0
$277$	16.2586	0.976886	0.488443	−	0.872596i	$-0.337565\pi$
$277$	16.2586	0.976886	0.488443	+	0.872596i	$0.337565\pi$
$278$	−20.0855	−1.20465
$279$	0	0
$280$	2.08313	0.124491
$281$	−19.1526	−1.14255	−0.571276	−	0.820758i	$-0.693551\pi$
$281$	−19.1526	−1.14255	−0.571276	+	0.820758i	$0.693551\pi$
$282$	0	0
$283$	−24.2905	−1.44392	−0.721960	−	0.691935i	$-0.756758\pi$
$283$	−24.2905	−1.44392	−0.721960	+	0.691935i	$0.756758\pi$
$284$	4.84967	0.287775
$285$	0	0
$286$	9.45554	0.559118
$287$	3.07733	0.181649
$288$	0	0
$289$	7.33252	0.431325
$290$	2.81090	0.165062
$291$	0	0
$292$	−0.544464	−0.0318623
$293$	−13.9705	−0.816163	−0.408081	−	0.912946i	$-0.633802\pi$
$293$	−13.9705	−0.816163	−0.408081	+	0.912946i	$0.633802\pi$
$294$	0	0
$295$	1.47726	0.0860096
$296$	−0.727768	−0.0423007
$297$	0	0
$298$	2.81090	0.162831
$299$	−14.3883	−0.832099
$300$	0	0
$301$	9.18674	0.529515
$302$	11.7142	0.674074
$303$	0	0
$304$	4.93280	0.282916
$305$	4.08313	0.233799
$306$	0	0
$307$	−20.3144	−1.15941	−0.579703	−	0.814828i	$-0.696831\pi$
$307$	−20.3144	−1.15941	−0.579703	+	0.814828i	$0.696831\pi$
$308$	−4.59337	−0.261732
$309$	0	0
$310$	−2.28816	−0.129959
$311$	7.75620	0.439814	0.219907	−	0.975521i	$-0.429425\pi$
$311$	7.75620	0.439814	0.219907	+	0.975521i	$0.429425\pi$
$312$	0	0
$313$	−20.8132	−1.17643	−0.588216	−	0.808704i	$-0.700170\pi$
$313$	−20.8132	−1.17643	−0.588216	+	0.808704i	$0.700170\pi$
$314$	−20.1981	−1.13985
$315$	0	0
$316$	4.83263	0.271856
$317$	−0.488511	−0.0274375	−0.0137188	−	0.999906i	$-0.504367\pi$
$317$	−0.488511	−0.0274375	−0.0137188	+	0.999906i	$0.504367\pi$
$318$	0	0
$319$	−6.19812	−0.347028
$320$	1.00000	0.0559017
$321$	0	0
$322$	6.98966	0.389518
$323$	24.3325	1.35390
$324$	0	0
$325$	−4.28816	−0.237864
$326$	−10.7278	−0.594156
$327$	0	0
$328$	1.47726	0.0815683
$329$	−19.5548	−1.07809
$330$	0	0
$331$	−23.0570	−1.26733	−0.633664	−	0.773608i	$-0.718450\pi$
$331$	−23.0570	−1.26733	−0.633664	+	0.773608i	$0.718450\pi$
$332$	16.5934	0.910679
$333$	0	0
$334$	−3.93169	−0.215132
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	30.3293	1.65214	0.826070	−	0.563568i	$-0.190572\pi$
$337$	30.3293	1.65214	0.826070	+	0.563568i	$0.190572\pi$
$338$	5.38834	0.293087
$339$	0	0
$340$	4.93280	0.267519
$341$	5.04547	0.273228
$342$	0	0
$343$	−20.1242	−1.08661
$344$	4.41006	0.237775
$345$	0	0
$346$	−9.04779	−0.486412
$347$	−17.5432	−0.941769	−0.470885	−	0.882195i	$-0.656065\pi$
$347$	−17.5432	−0.941769	−0.470885	+	0.882195i	$0.656065\pi$
$348$	0	0
$349$	14.8201	0.793303	0.396652	−	0.917969i	$-0.370172\pi$
$349$	14.8201	0.793303	0.396652	+	0.917969i	$0.370172\pi$
$350$	2.08313	0.111348
$351$	0	0
$352$	−2.20503	−0.117529
$353$	−18.7665	−0.998842	−0.499421	−	0.866359i	$-0.666454\pi$
$353$	−18.7665	−0.998842	−0.499421	+	0.866359i	$0.666454\pi$
$354$	0	0
$355$	4.84967	0.257394
$356$	−2.18330	−0.115715
$357$	0	0
$358$	−13.6754	−0.722767
$359$	6.43738	0.339752	0.169876	−	0.985465i	$-0.445663\pi$
$359$	6.43738	0.339752	0.169876	+	0.985465i	$0.445663\pi$
$360$	0	0
$361$	5.33252	0.280659
$362$	0.876984	0.0460933
$363$	0	0
$364$	−8.93280	−0.468206
$365$	−0.544464	−0.0284985
$366$	0	0
$367$	−18.1298	−0.946368	−0.473184	−	0.880964i	$-0.656895\pi$
$367$	−18.1298	−0.946368	−0.473184	+	0.880964i	$0.656895\pi$
$368$	3.35536	0.174910
$369$	0	0
$370$	−0.727768	−0.0378349
$371$	12.8451	0.666886
$372$	0	0
$373$	12.6207	0.653474	0.326737	−	0.945115i	$-0.394051\pi$
$373$	12.6207	0.653474	0.326737	+	0.945115i	$0.394051\pi$
$374$	−10.8770	−0.562435
$375$	0	0
$376$	−9.38722	−0.484109
$377$	−12.0536	−0.620791
$378$	0	0
$379$	28.4180	1.45973	0.729867	−	0.683590i	$-0.239582\pi$
$379$	28.4180	1.45973	0.729867	+	0.683590i	$0.239582\pi$
$380$	4.93280	0.253047
$381$	0	0
$382$	16.4101	0.839612
$383$	−13.6048	−0.695170	−0.347585	−	0.937648i	$-0.612998\pi$
$383$	−13.6048	−0.695170	−0.347585	+	0.937648i	$0.612998\pi$
$384$	0	0
$385$	−4.59337	−0.234100
$386$	−3.17660	−0.161685
$387$	0	0
$388$	2.20503	0.111944
$389$	23.6948	1.20137	0.600686	−	0.799485i	$-0.294894\pi$
$389$	23.6948	1.20137	0.600686	+	0.799485i	$0.294894\pi$
$390$	0	0
$391$	16.5513	0.837037
$392$	−2.66057	−0.134379
$393$	0	0
$394$	10.5763	0.532828
$395$	4.83263	0.243156
$396$	0	0
$397$	−4.65022	−0.233388	−0.116694	−	0.993168i	$-0.537230\pi$
$397$	−4.65022	−0.233388	−0.116694	+	0.993168i	$0.537230\pi$
$398$	8.59337	0.430747
$399$	0	0
$400$	1.00000	0.0500000
$401$	−23.5193	−1.17450	−0.587248	−	0.809407i	$-0.699789\pi$
$401$	−23.5193	−1.17450	−0.587248	+	0.809407i	$0.699789\pi$
$402$	0	0
$403$	9.81201	0.488771
$404$	9.13783	0.454624
$405$	0	0
$406$	5.85547	0.290602
$407$	1.60475	0.0795446
$408$	0	0
$409$	18.5763	0.918540	0.459270	−	0.888297i	$-0.348111\pi$
$409$	18.5763	0.918540	0.459270	+	0.888297i	$0.348111\pi$
$410$	1.47726	0.0729569
$411$	0	0
$412$	0.410065	0.0202024
$413$	3.07733	0.151426
$414$	0	0
$415$	16.5934	0.814536
$416$	−4.28816	−0.210244
$417$	0	0
$418$	−10.8770	−0.532011
$419$	1.97605	0.0965361	0.0482681	−	0.998834i	$-0.484630\pi$
$419$	1.97605	0.0965361	0.0482681	+	0.998834i	$0.484630\pi$
$420$	0	0
$421$	9.92246	0.483591	0.241795	−	0.970327i	$-0.422264\pi$
$421$	9.92246	0.483591	0.241795	+	0.970327i	$0.422264\pi$
$422$	−12.8553	−0.625784
$423$	0	0
$424$	6.16626	0.299460
$425$	4.93280	0.239276
$426$	0	0
$427$	8.50569	0.411619
$428$	−0.576325	−0.0278577
$429$	0	0
$430$	4.41006	0.212672
$431$	−2.38275	−0.114773	−0.0573865	−	0.998352i	$-0.518277\pi$
$431$	−2.38275	−0.114773	−0.0573865	+	0.998352i	$0.518277\pi$
$432$	0	0
$433$	−20.9965	−1.00903	−0.504514	−	0.863403i	$-0.668328\pi$
$433$	−20.9965	−1.00903	−0.504514	+	0.863403i	$0.668328\pi$
$434$	−4.76654	−0.228801
$435$	0	0
$436$	5.78247	0.276930
$437$	16.5513	0.791758
$438$	0	0
$439$	17.0034	0.811530	0.405765	−	0.913978i	$-0.367005\pi$
$439$	17.0034	0.811530	0.405765	+	0.913978i	$0.367005\pi$
$440$	−2.20503	−0.105121
$441$	0	0
$442$	−21.1526	−1.00613
$443$	13.8656	0.658775	0.329387	−	0.944195i	$-0.393158\pi$
$443$	13.8656	0.658775	0.329387	+	0.944195i	$0.393158\pi$
$444$	0	0
$445$	−2.18330	−0.103499
$446$	18.8520	0.892668
$447$	0	0
$448$	2.08313	0.0984186
$449$	−39.0922	−1.84487	−0.922436	−	0.386149i	$-0.873805\pi$
$449$	−39.0922	−1.84487	−0.922436	+	0.386149i	$0.873805\pi$
$450$	0	0
$451$	−3.25741	−0.153386
$452$	13.8268	0.650359
$453$	0	0
$454$	22.1094	1.03765
$455$	−8.93280	−0.418776
$456$	0	0
$457$	−6.68900	−0.312898	−0.156449	−	0.987686i	$-0.550005\pi$
$457$	−6.68900	−0.312898	−0.156449	+	0.987686i	$0.550005\pi$
$458$	0.326934	0.0152766
$459$	0	0
$460$	3.35536	0.156445
$461$	22.0545	1.02718	0.513590	−	0.858036i	$-0.328315\pi$
$461$	22.0545	1.02718	0.513590	+	0.858036i	$0.328315\pi$
$462$	0	0
$463$	39.4191	1.83196	0.915980	−	0.401224i	$-0.131415\pi$
$463$	39.4191	1.83196	0.915980	+	0.401224i	$0.131415\pi$
$464$	2.81090	0.130493
$465$	0	0
$466$	2.14922	0.0995605
$467$	−10.0148	−0.463430	−0.231715	−	0.972784i	$-0.574434\pi$
$467$	−10.0148	−0.463430	−0.231715	+	0.972784i	$0.574434\pi$
$468$	0	0
$469$	−2.08313	−0.0961900
$470$	−9.38722	−0.433000
$471$	0	0
$472$	1.47726	0.0679966
$473$	−9.72433	−0.447125
$474$	0	0
$475$	4.93280	0.226332
$476$	10.2757	0.470984
$477$	0	0
$478$	−11.1208	−0.508653
$479$	−22.8474	−1.04393	−0.521963	−	0.852968i	$-0.674800\pi$
$479$	−22.8474	−1.04393	−0.521963	+	0.852968i	$0.674800\pi$
$480$	0	0
$481$	3.12079	0.142296
$482$	−7.15956	−0.326109
$483$	0	0
$484$	−6.13783	−0.278992
$485$	2.20503	0.100125
$486$	0	0
$487$	35.8896	1.62632	0.813158	−	0.582044i	$-0.197747\pi$
$487$	35.8896	1.62632	0.813158	+	0.582044i	$0.197747\pi$
$488$	4.08313	0.184835
$489$	0	0
$490$	−2.66057	−0.120192
$491$	35.2403	1.59037	0.795187	−	0.606364i	$-0.207373\pi$
$491$	35.2403	1.59037	0.795187	+	0.606364i	$0.207373\pi$
$492$	0	0
$493$	13.8656	0.624475
$494$	−21.1526	−0.951703
$495$	0	0
$496$	−2.28816	−0.102742
$497$	10.1025	0.453159
$498$	0	0
$499$	8.17094	0.365782	0.182891	−	0.983133i	$-0.441455\pi$
$499$	8.17094	0.365782	0.182891	+	0.983133i	$0.441455\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	10.7278	0.478804
$503$	0.470564	0.0209814	0.0104907	−	0.999945i	$-0.496661\pi$
$503$	0.470564	0.0209814	0.0104907	+	0.999945i	$0.496661\pi$
$504$	0	0
$505$	9.13783	0.406628
$506$	−7.39868	−0.328912
$507$	0	0
$508$	−11.1208	−0.493405
$509$	−1.02284	−0.0453366	−0.0226683	−	0.999743i	$-0.507216\pi$
$509$	−1.02284	−0.0453366	−0.0226683	+	0.999743i	$0.507216\pi$
$510$	0	0
$511$	−1.13419	−0.0501735
$512$	1.00000	0.0441942
$513$	0	0
$514$	20.4202	0.900696
$515$	0.410065	0.0180696
$516$	0	0
$517$	20.6991	0.910347
$518$	−1.51604	−0.0666108
$519$	0	0
$520$	−4.28816	−0.188048
$521$	0.841857	0.0368824	0.0184412	−	0.999830i	$-0.494130\pi$
$521$	0.841857	0.0368824	0.0184412	+	0.999830i	$0.494130\pi$
$522$	0	0
$523$	38.8987	1.70092	0.850460	−	0.526040i	$-0.176324\pi$
$523$	38.8987	1.70092	0.850460	+	0.526040i	$0.176324\pi$
$524$	17.6435	0.770761
$525$	0	0
$526$	3.59917	0.156931
$527$	−11.2870	−0.491672
$528$	0	0
$529$	−11.7415	−0.510502
$530$	6.16626	0.267845
$531$	0	0
$532$	10.2757	0.445507
$533$	−6.33475	−0.274388
$534$	0	0
$535$	−0.576325	−0.0249167
$536$	−1.00000	−0.0431934
$537$	0	0
$538$	−8.94530	−0.385659
$539$	5.86664	0.252694
$540$	0	0
$541$	−11.2644	−0.484295	−0.242148	−	0.970239i	$-0.577852\pi$
$541$	−11.2644	−0.484295	−0.242148	+	0.970239i	$0.577852\pi$
$542$	−7.41118	−0.318337
$543$	0	0
$544$	4.93280	0.211492
$545$	5.78247	0.247694
$546$	0	0
$547$	33.1092	1.41565	0.707823	−	0.706389i	$-0.249677\pi$
$547$	33.1092	1.41565	0.707823	+	0.706389i	$0.249677\pi$
$548$	4.72777	0.201960
$549$	0	0
$550$	−2.20503	−0.0940229
$551$	13.8656	0.590694
$552$	0	0
$553$	10.0670	0.428092
$554$	16.2586	0.690763
$555$	0	0
$556$	−20.0855	−0.851813
$557$	−14.1103	−0.597873	−0.298936	−	0.954273i	$-0.596632\pi$
$557$	−14.1103	−0.597873	−0.298936	+	0.954273i	$0.596632\pi$
$558$	0	0
$559$	−18.9111	−0.799853
$560$	2.08313	0.0880283
$561$	0	0
$562$	−19.1526	−0.807906
$563$	16.4101	0.691602	0.345801	−	0.938308i	$-0.387607\pi$
$563$	16.4101	0.691602	0.345801	+	0.938308i	$0.387607\pi$
$564$	0	0
$565$	13.8268	0.581699
$566$	−24.2905	−1.02101
$567$	0	0
$568$	4.84967	0.203488
$569$	−30.7255	−1.28808	−0.644041	−	0.764991i	$-0.722743\pi$
$569$	−30.7255	−1.28808	−0.644041	+	0.764991i	$0.722743\pi$
$570$	0	0
$571$	−9.86560	−0.412863	−0.206431	−	0.978461i	$-0.566185\pi$
$571$	−9.86560	−0.412863	−0.206431	+	0.978461i	$0.566185\pi$
$572$	9.45554	0.395356
$573$	0	0
$574$	3.07733	0.128445
$575$	3.35536	0.139928
$576$	0	0
$577$	−2.86685	−0.119349	−0.0596743	−	0.998218i	$-0.519006\pi$
$577$	−2.86685	−0.119349	−0.0596743	+	0.998218i	$0.519006\pi$
$578$	7.33252	0.304993
$579$	0	0
$580$	2.81090	0.116716
$581$	34.5661	1.43405
$582$	0	0
$583$	−13.5968	−0.563122
$584$	−0.544464	−0.0225301
$585$	0	0
$586$	−13.9705	−0.577114
$587$	6.26553	0.258606	0.129303	−	0.991605i	$-0.458726\pi$
$587$	6.26553	0.258606	0.129303	+	0.991605i	$0.458726\pi$
$588$	0	0
$589$	−11.2870	−0.465075
$590$	1.47726	0.0608180
$591$	0	0
$592$	−0.727768	−0.0299111
$593$	9.77324	0.401339	0.200669	−	0.979659i	$-0.435688\pi$
$593$	9.77324	0.401339	0.200669	+	0.979659i	$0.435688\pi$
$594$	0	0
$595$	10.2757	0.421261
$596$	2.81090	0.115139
$597$	0	0
$598$	−14.3883	−0.588383
$599$	15.1845	0.620422	0.310211	−	0.950668i	$-0.399600\pi$
$599$	15.1845	0.620422	0.310211	+	0.950668i	$0.399600\pi$
$600$	0	0
$601$	−20.2688	−0.826780	−0.413390	−	0.910554i	$-0.635655\pi$
$601$	−20.2688	−0.826780	−0.413390	+	0.910554i	$0.635655\pi$
$602$	9.18674	0.374424
$603$	0	0
$604$	11.7142	0.476642
$605$	−6.13783	−0.249538
$606$	0	0
$607$	−5.31891	−0.215888	−0.107944	−	0.994157i	$-0.534427\pi$
$607$	−5.31891	−0.215888	−0.107944	+	0.994157i	$0.534427\pi$
$608$	4.93280	0.200051
$609$	0	0
$610$	4.08313	0.165321
$611$	40.2539	1.62850
$612$	0	0
$613$	−10.6650	−0.430757	−0.215378	−	0.976531i	$-0.569099\pi$
$613$	−10.6650	−0.430757	−0.215378	+	0.976531i	$0.569099\pi$
$614$	−20.3144	−0.819824
$615$	0	0
$616$	−4.59337	−0.185072
$617$	−37.1628	−1.49612	−0.748059	−	0.663633i	$-0.769014\pi$
$617$	−37.1628	−1.49612	−0.748059	+	0.663633i	$0.769014\pi$
$618$	0	0
$619$	−15.0672	−0.605602	−0.302801	−	0.953054i	$-0.597922\pi$
$619$	−15.0672	−0.605602	−0.302801	+	0.953054i	$0.597922\pi$
$620$	−2.28816	−0.0918948
$621$	0	0
$622$	7.75620	0.310995
$623$	−4.54811	−0.182216
$624$	0	0
$625$	1.00000	0.0400000
$626$	−20.8132	−0.831864
$627$	0	0
$628$	−20.1981	−0.805993
$629$	−3.58994	−0.143140
$630$	0	0
$631$	−12.4226	−0.494534	−0.247267	−	0.968947i	$-0.579533\pi$
$631$	−12.4226	−0.494534	−0.247267	+	0.968947i	$0.579533\pi$
$632$	4.83263	0.192232
$633$	0	0
$634$	−0.488511	−0.0194013
$635$	−11.1208	−0.441315
$636$	0	0
$637$	11.4090	0.452039
$638$	−6.19812	−0.245386
$639$	0	0
$640$	1.00000	0.0395285
$641$	−31.8177	−1.25672	−0.628362	−	0.777921i	$-0.716274\pi$
$641$	−31.8177	−1.25672	−0.628362	+	0.777921i	$0.716274\pi$
$642$	0	0
$643$	−40.3427	−1.59096	−0.795479	−	0.605981i	$-0.792781\pi$
$643$	−40.3427	−1.59096	−0.795479	+	0.605981i	$0.792781\pi$
$644$	6.98966	0.275431
$645$	0	0
$646$	24.3325	0.957350
$647$	−43.6662	−1.71670	−0.858349	−	0.513067i	$-0.828509\pi$
$647$	−43.6662	−1.71670	−0.858349	+	0.513067i	$0.828509\pi$
$648$	0	0
$649$	−3.25741	−0.127865
$650$	−4.28816	−0.168196
$651$	0	0
$652$	−10.7278	−0.420132
$653$	−33.1411	−1.29691	−0.648455	−	0.761253i	$-0.724584\pi$
$653$	−33.1411	−1.29691	−0.648455	+	0.761253i	$0.724584\pi$
$654$	0	0
$655$	17.6435	0.689390
$656$	1.47726	0.0576775
$657$	0	0
$658$	−19.5548	−0.762326
$659$	−34.6640	−1.35032	−0.675159	−	0.737672i	$-0.735925\pi$
$659$	−34.6640	−1.35032	−0.675159	+	0.737672i	$0.735925\pi$
$660$	0	0
$661$	−10.8884	−0.423511	−0.211756	−	0.977323i	$-0.567918\pi$
$661$	−10.8884	−0.423511	−0.211756	+	0.977323i	$0.567918\pi$
$662$	−23.0570	−0.896137
$663$	0	0
$664$	16.5934	0.643948
$665$	10.2757	0.398473
$666$	0	0
$667$	9.43158	0.365192
$668$	−3.93169	−0.152122
$669$	0	0
$670$	−1.00000	−0.0386334
$671$	−9.00343	−0.347574
$672$	0	0
$673$	−8.48865	−0.327213	−0.163607	−	0.986526i	$-0.552313\pi$
$673$	−8.48865	−0.327213	−0.163607	+	0.986526i	$0.552313\pi$
$674$	30.3293	1.16824
$675$	0	0
$676$	5.38834	0.207244
$677$	15.1526	0.582364	0.291182	−	0.956668i	$-0.405952\pi$
$677$	15.1526	0.582364	0.291182	+	0.956668i	$0.405952\pi$
$678$	0	0
$679$	4.59337	0.176277
$680$	4.93280	0.189164
$681$	0	0
$682$	5.04547	0.193201
$683$	−5.01138	−0.191755	−0.0958776	−	0.995393i	$-0.530566\pi$
$683$	−5.01138	−0.191755	−0.0958776	+	0.995393i	$0.530566\pi$
$684$	0	0
$685$	4.72777	0.180639
$686$	−20.1242	−0.768346
$687$	0	0
$688$	4.41006	0.168132
$689$	−26.4419	−1.00736
$690$	0	0
$691$	−9.89969	−0.376602	−0.188301	−	0.982111i	$-0.560298\pi$
$691$	−9.89969	−0.376602	−0.188301	+	0.982111i	$0.560298\pi$
$692$	−9.04779	−0.343945
$693$	0	0
$694$	−17.5432	−0.665931
$695$	−20.0855	−0.761885
$696$	0	0
$697$	7.28705	0.276017
$698$	14.8201	0.560950
$699$	0	0
$700$	2.08313	0.0787349
$701$	−11.3609	−0.429097	−0.214549	−	0.976713i	$-0.568828\pi$
$701$	−11.3609	−0.429097	−0.214549	+	0.976713i	$0.568828\pi$
$702$	0	0
$703$	−3.58994	−0.135397
$704$	−2.20503	−0.0831053
$705$	0	0
$706$	−18.7665	−0.706288
$707$	19.0353	0.715896
$708$	0	0
$709$	−49.5170	−1.85965	−0.929826	−	0.368001i	$-0.880042\pi$
$709$	−49.5170	−1.85965	−0.929826	+	0.368001i	$0.880042\pi$
$710$	4.84967	0.182005
$711$	0	0
$712$	−2.18330	−0.0818228
$713$	−7.67761	−0.287529
$714$	0	0
$715$	9.45554	0.353617
$716$	−13.6754	−0.511073
$717$	0	0
$718$	6.43738	0.240241
$719$	28.4863	1.06236	0.531180	−	0.847259i	$-0.321749\pi$
$719$	28.4863	1.06236	0.531180	+	0.847259i	$0.321749\pi$
$720$	0	0
$721$	0.854218	0.0318127
$722$	5.33252	0.198456
$723$	0	0
$724$	0.876984	0.0325929
$725$	2.81090	0.104394
$726$	0	0
$727$	−19.4191	−0.720214	−0.360107	−	0.932911i	$-0.617260\pi$
$727$	−19.4191	−0.720214	−0.360107	+	0.932911i	$0.617260\pi$
$728$	−8.93280	−0.331072
$729$	0	0
$730$	−0.544464	−0.0201515
$731$	21.7540	0.804600
$732$	0	0
$733$	20.0628	0.741037	0.370519	−	0.928825i	$-0.379180\pi$
$733$	20.0628	0.741037	0.370519	+	0.928825i	$0.379180\pi$
$734$	−18.1298	−0.669183
$735$	0	0
$736$	3.35536	0.123680
$737$	2.20503	0.0812234
$738$	0	0
$739$	6.21985	0.228801	0.114400	−	0.993435i	$-0.463505\pi$
$739$	6.21985	0.228801	0.114400	+	0.993435i	$0.463505\pi$
$740$	−0.727768	−0.0267533
$741$	0	0
$742$	12.8451	0.471559
$743$	−51.6925	−1.89641	−0.948207	−	0.317652i	$-0.897106\pi$
$743$	−51.6925	−1.89641	−0.948207	+	0.317652i	$0.897106\pi$
$744$	0	0
$745$	2.81090	0.102983
$746$	12.6207	0.462076
$747$	0	0
$748$	−10.8770	−0.397702
$749$	−1.20056	−0.0438675
$750$	0	0
$751$	−12.8201	−0.467813	−0.233907	−	0.972259i	$-0.575151\pi$
$751$	−12.8201	−0.467813	−0.233907	+	0.972259i	$0.575151\pi$
$752$	−9.38722	−0.342317
$753$	0	0
$754$	−12.0536	−0.438966
$755$	11.7142	0.426322
$756$	0	0
$757$	−30.4613	−1.10713	−0.553567	−	0.832805i	$-0.686734\pi$
$757$	−30.4613	−1.10713	−0.553567	+	0.832805i	$0.686734\pi$
$758$	28.4180	1.03219
$759$	0	0
$760$	4.93280	0.178931
$761$	−0.937484	−0.0339838	−0.0169919	−	0.999856i	$-0.505409\pi$
$761$	−0.937484	−0.0339838	−0.0169919	+	0.999856i	$0.505409\pi$
$762$	0	0
$763$	12.0456	0.436081
$764$	16.4101	0.593695
$765$	0	0
$766$	−13.6048	−0.491560
$767$	−6.33475	−0.228734
$768$	0	0
$769$	−30.2734	−1.09169	−0.545844	−	0.837887i	$-0.683791\pi$
$769$	−30.2734	−1.09169	−0.545844	+	0.837887i	$0.683791\pi$
$770$	−4.59337	−0.165534
$771$	0	0
$772$	−3.17660	−0.114329
$773$	26.8949	0.967343	0.483672	−	0.875249i	$-0.339303\pi$
$773$	26.8949	0.967343	0.483672	+	0.875249i	$0.339303\pi$
$774$	0	0
$775$	−2.28816	−0.0821932
$776$	2.20503	0.0791560
$777$	0	0
$778$	23.6948	0.849498
$779$	7.28705	0.261086
$780$	0	0
$781$	−10.6937	−0.382650
$782$	16.5513	0.591874
$783$	0	0
$784$	−2.66057	−0.0950203
$785$	−20.1981	−0.720902
$786$	0	0
$787$	33.0936	1.17966	0.589829	−	0.807528i	$-0.299195\pi$
$787$	33.0936	1.17966	0.589829	+	0.807528i	$0.299195\pi$
$788$	10.5763	0.376766
$789$	0	0
$790$	4.83263	0.171937
$791$	28.8031	1.02412
$792$	0	0
$793$	−17.5091	−0.621767
$794$	−4.65022	−0.165030
$795$	0	0
$796$	8.59337	0.304584
$797$	−22.6243	−0.801395	−0.400697	−	0.916210i	$-0.631232\pi$
$797$	−22.6243	−0.801395	−0.400697	+	0.916210i	$0.631232\pi$
$798$	0	0
$799$	−46.3053	−1.63816
$800$	1.00000	0.0353553
$801$	0	0
$802$	−23.5193	−0.830494
$803$	1.20056	0.0423668
$804$	0	0
$805$	6.98966	0.246353
$806$	9.81201	0.345613
$807$	0	0
$808$	9.13783	0.321468
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	−43.5217	−1.52825	−0.764127	−	0.645066i	$-0.776830\pi$
$811$	−43.5217	−1.52825	−0.764127	+	0.645066i	$0.776830\pi$
$812$	5.85547	0.205487
$813$	0	0
$814$	1.60475	0.0562465
$815$	−10.7278	−0.375777
$816$	0	0
$817$	21.7540	0.761075
$818$	18.5763	0.649506
$819$	0	0
$820$	1.47726	0.0515883
$821$	−10.9431	−0.381916	−0.190958	−	0.981598i	$-0.561159\pi$
$821$	−10.9431	−0.381916	−0.190958	+	0.981598i	$0.561159\pi$
$822$	0	0
$823$	8.63541	0.301011	0.150506	−	0.988609i	$-0.451910\pi$
$823$	8.63541	0.301011	0.150506	+	0.988609i	$0.451910\pi$
$824$	0.410065	0.0142853
$825$	0	0
$826$	3.07733	0.107074
$827$	23.9395	0.832458	0.416229	−	0.909260i	$-0.363351\pi$
$827$	23.9395	0.832458	0.416229	+	0.909260i	$0.363351\pi$
$828$	0	0
$829$	−15.6993	−0.545261	−0.272630	−	0.962119i	$-0.587894\pi$
$829$	−15.6993	−0.545261	−0.272630	+	0.962119i	$0.587894\pi$
$830$	16.5934	0.575964
$831$	0	0
$832$	−4.28816	−0.148665
$833$	−13.1241	−0.454722
$834$	0	0
$835$	−3.93169	−0.136062
$836$	−10.8770	−0.376188
$837$	0	0
$838$	1.97605	0.0682613
$839$	−25.1742	−0.869111	−0.434556	−	0.900645i	$-0.643095\pi$
$839$	−25.1742	−0.869111	−0.434556	+	0.900645i	$0.643095\pi$
$840$	0	0
$841$	−21.0989	−0.727547
$842$	9.92246	0.341950
$843$	0	0
$844$	−12.8553	−0.442496
$845$	5.38834	0.185364
$846$	0	0
$847$	−12.7859	−0.439329
$848$	6.16626	0.211750
$849$	0	0
$850$	4.93280	0.169194
$851$	−2.44193	−0.0837081
$852$	0	0
$853$	8.33029	0.285224	0.142612	−	0.989779i	$-0.454450\pi$
$853$	8.33029	0.285224	0.142612	+	0.989779i	$0.454450\pi$
$854$	8.50569	0.291059
$855$	0	0
$856$	−0.576325	−0.0196984
$857$	35.7562	1.22141	0.610703	−	0.791859i	$-0.290887\pi$
$857$	35.7562	1.22141	0.610703	+	0.791859i	$0.290887\pi$
$858$	0	0
$859$	−51.5410	−1.75856	−0.879278	−	0.476309i	$-0.841974\pi$
$859$	−51.5410	−1.75856	−0.879278	+	0.476309i	$0.841974\pi$
$860$	4.41006	0.150382
$861$	0	0
$862$	−2.38275	−0.0811568
$863$	3.65825	0.124528	0.0622641	−	0.998060i	$-0.480168\pi$
$863$	3.65825	0.124528	0.0622641	+	0.998060i	$0.480168\pi$
$864$	0	0
$865$	−9.04779	−0.307634
$866$	−20.9965	−0.713491
$867$	0	0
$868$	−4.76654	−0.161787
$869$	−10.6561	−0.361483
$870$	0	0
$871$	4.28816	0.145299
$872$	5.78247	0.195819
$873$	0	0
$874$	16.5513	0.559857
$875$	2.08313	0.0704227
$876$	0	0
$877$	10.0185	0.338299	0.169150	−	0.985590i	$-0.445898\pi$
$877$	10.0185	0.338299	0.169150	+	0.985590i	$0.445898\pi$
$878$	17.0034	0.573838
$879$	0	0
$880$	−2.20503	−0.0743316
$881$	−13.9145	−0.468792	−0.234396	−	0.972141i	$-0.575311\pi$
$881$	−13.9145	−0.468792	−0.234396	+	0.972141i	$0.575311\pi$
$882$	0	0
$883$	−19.7562	−0.664849	−0.332424	−	0.943130i	$-0.607867\pi$
$883$	−19.7562	−0.664849	−0.332424	+	0.943130i	$0.607867\pi$
$884$	−21.1526	−0.711441
$885$	0	0
$886$	13.8656	0.465824
$887$	8.75182	0.293857	0.146929	−	0.989147i	$-0.453061\pi$
$887$	8.75182	0.293857	0.146929	+	0.989147i	$0.453061\pi$
$888$	0	0
$889$	−23.1660	−0.776964
$890$	−2.18330	−0.0731845
$891$	0	0
$892$	18.8520	0.631211
$893$	−46.3053	−1.54955
$894$	0	0
$895$	−13.6754	−0.457118
$896$	2.08313	0.0695925
$897$	0	0
$898$	−39.0922	−1.30452
$899$	−6.43179	−0.214512
$900$	0	0
$901$	30.4169	1.01333
$902$	−3.25741	−0.108460
$903$	0	0
$904$	13.8268	0.459873
$905$	0.876984	0.0291519
$906$	0	0
$907$	−24.3920	−0.809922	−0.404961	−	0.914334i	$-0.632715\pi$
$907$	−24.3920	−0.809922	−0.404961	+	0.914334i	$0.632715\pi$
$908$	22.1094	0.733726
$909$	0	0
$910$	−8.93280	−0.296119
$911$	6.94061	0.229953	0.114976	−	0.993368i	$-0.463321\pi$
$911$	6.94061	0.229953	0.114976	+	0.993368i	$0.463321\pi$
$912$	0	0
$913$	−36.5889	−1.21092
$914$	−6.68900	−0.221252
$915$	0	0
$916$	0.326934	0.0108022
$917$	36.7538	1.21372
$918$	0	0
$919$	−17.0764	−0.563299	−0.281650	−	0.959517i	$-0.590882\pi$
$919$	−17.0764	−0.563299	−0.281650	+	0.959517i	$0.590882\pi$
$920$	3.35536	0.110623
$921$	0	0
$922$	22.0545	0.726326
$923$	−20.7962	−0.684514
$924$	0	0
$925$	−0.727768	−0.0239289
$926$	39.4191	1.29539
$927$	0	0
$928$	2.81090	0.0922723
$929$	−16.6433	−0.546049	−0.273025	−	0.962007i	$-0.588024\pi$
$929$	−16.6433	−0.546049	−0.273025	+	0.962007i	$0.588024\pi$
$930$	0	0
$931$	−13.1241	−0.430124
$932$	2.14922	0.0703999
$933$	0	0
$934$	−10.0148	−0.327695
$935$	−10.8770	−0.355715
$936$	0	0
$937$	9.72652	0.317752	0.158876	−	0.987299i	$-0.449213\pi$
$937$	9.72652	0.317752	0.158876	+	0.987299i	$0.449213\pi$
$938$	−2.08313	−0.0680166
$939$	0	0
$940$	−9.38722	−0.306177
$941$	7.20056	0.234732	0.117366	−	0.993089i	$-0.462555\pi$
$941$	7.20056	0.234732	0.117366	+	0.993089i	$0.462555\pi$
$942$	0	0
$943$	4.95676	0.161414
$944$	1.47726	0.0480808
$945$	0	0
$946$	−9.72433	−0.316165
$947$	−30.1561	−0.979941	−0.489971	−	0.871739i	$-0.662992\pi$
$947$	−30.1561	−0.979941	−0.489971	+	0.871739i	$0.662992\pi$
$948$	0	0
$949$	2.33475	0.0757891
$950$	4.93280	0.160041
$951$	0	0
$952$	10.2757	0.333036
$953$	19.0193	0.616095	0.308048	−	0.951371i	$-0.400324\pi$
$953$	19.0193	0.616095	0.308048	+	0.951371i	$0.400324\pi$
$954$	0	0
$955$	16.4101	0.531017
$956$	−11.1208	−0.359672
$957$	0	0
$958$	−22.8474	−0.738167
$959$	9.84856	0.318026
$960$	0	0
$961$	−25.7643	−0.831107
$962$	3.12079	0.100618
$963$	0	0
$964$	−7.15956	−0.230594
$965$	−3.17660	−0.102259
$966$	0	0
$967$	40.1865	1.29231	0.646156	−	0.763206i	$-0.276376\pi$
$967$	40.1865	1.29231	0.646156	+	0.763206i	$0.276376\pi$
$968$	−6.13783	−0.197277
$969$	0	0
$970$	2.20503	0.0707993
$971$	50.4387	1.61865	0.809327	−	0.587359i	$-0.199832\pi$
$971$	50.4387	1.61865	0.809327	+	0.587359i	$0.199832\pi$
$972$	0	0
$973$	−41.8406	−1.34135
$974$	35.8896	1.14998
$975$	0	0
$976$	4.08313	0.130698
$977$	−58.8142	−1.88163	−0.940817	−	0.338916i	$-0.889940\pi$
$977$	−58.8142	−1.88163	−0.940817	+	0.338916i	$0.889940\pi$
$978$	0	0
$979$	4.81426	0.153864
$980$	−2.66057	−0.0849888
$981$	0	0
$982$	35.2403	1.12456
$983$	−13.2643	−0.423065	−0.211532	−	0.977371i	$-0.567845\pi$
$983$	−13.2643	−0.423065	−0.211532	+	0.977371i	$0.567845\pi$
$984$	0	0
$985$	10.5763	0.336990
$986$	13.8656	0.441571
$987$	0	0
$988$	−21.1526	−0.672955
$989$	14.7974	0.470529
$990$	0	0
$991$	−15.6071	−0.495775	−0.247887	−	0.968789i	$-0.579736\pi$
$991$	−15.6071	−0.495775	−0.247887	+	0.968789i	$0.579736\pi$
$992$	−2.28816	−0.0726492
$993$	0	0
$994$	10.1025	0.320432
$995$	8.59337	0.272428
$996$	0	0
$997$	−20.5911	−0.652128	−0.326064	−	0.945348i	$-0.605722\pi$
$997$	−20.5911	−0.652128	−0.326064	+	0.945348i	$0.605722\pi$
$998$	8.17094	0.258647
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6030.2.a.bu.1.3		4
3.2	odd	2		2010.2.a.r.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2010.2.a.r.1.3	✓	4	3.2	odd	2
6030.2.a.bu.1.3		4	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:	\( N \)	\(=\)	\( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6030.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.08313 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.08313 q^{7} +1.00000 q^{8} +1.00000 q^{10} -2.20503 q^{11} -4.28816 q^{13} +2.08313 q^{14} +1.00000 q^{16} +4.93280 q^{17} +4.93280 q^{19} +1.00000 q^{20} -2.20503 q^{22} +3.35536 q^{23} +1.00000 q^{25} -4.28816 q^{26} +2.08313 q^{28} +2.81090 q^{29} -2.28816 q^{31} +1.00000 q^{32} +4.93280 q^{34} +2.08313 q^{35} -0.727768 q^{37} +4.93280 q^{38} +1.00000 q^{40} +1.47726 q^{41} +4.41006 q^{43} -2.20503 q^{44} +3.35536 q^{46} -9.38722 q^{47} -2.66057 q^{49} +1.00000 q^{50} -4.28816 q^{52} +6.16626 q^{53} -2.20503 q^{55} +2.08313 q^{56} +2.81090 q^{58} +1.47726 q^{59} +4.08313 q^{61} -2.28816 q^{62} +1.00000 q^{64} -4.28816 q^{65} -1.00000 q^{67} +4.93280 q^{68} +2.08313 q^{70} +4.84967 q^{71} -0.544464 q^{73} -0.727768 q^{74} +4.93280 q^{76} -4.59337 q^{77} +4.83263 q^{79} +1.00000 q^{80} +1.47726 q^{82} +16.5934 q^{83} +4.93280 q^{85} +4.41006 q^{86} -2.20503 q^{88} -2.18330 q^{89} -8.93280 q^{91} +3.35536 q^{92} -9.38722 q^{94} +4.93280 q^{95} +2.20503 q^{97} -2.66057 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + q^7 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + q^{7} + 4 q^{8} + 4 q^{10} + 3 q^{11} + 2 q^{13} + q^{14} + 4 q^{16} + 2 q^{17} + 2 q^{19} + 4 q^{20} + 3 q^{22} + 12 q^{23} + 4 q^{25} + 2 q^{26} + q^{28} - 2 q^{29} + 10 q^{31} + 4 q^{32} + 2 q^{34} + q^{35} + 3 q^{37} + 2 q^{38} + 4 q^{40} - 6 q^{43} + 3 q^{44} + 12 q^{46} + 14 q^{47} + 13 q^{49} + 4 q^{50} + 2 q^{52} + 10 q^{53} + 3 q^{55} + q^{56} - 2 q^{58} + 9 q^{61} + 10 q^{62} + 4 q^{64} + 2 q^{65} - 4 q^{67} + 2 q^{68} + q^{70} + 9 q^{71} - 14 q^{73} + 3 q^{74} + 2 q^{76} + 23 q^{77} + 12 q^{79} + 4 q^{80} + 25 q^{83} + 2 q^{85} - 6 q^{86} + 3 q^{88} + 9 q^{89} - 18 q^{91} + 12 q^{92} + 14 q^{94} + 2 q^{95} - 3 q^{97} + 13 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + q^7 + 4 * q^8 + 4 * q^10 + 3 * q^11 + 2 * q^13 + q^14 + 4 * q^16 + 2 * q^17 + 2 * q^19 + 4 * q^20 + 3 * q^22 + 12 * q^23 + 4 * q^25 + 2 * q^26 + q^28 - 2 * q^29 + 10 * q^31 + 4 * q^32 + 2 * q^34 + q^35 + 3 * q^37 + 2 * q^38 + 4 * q^40 - 6 * q^43 + 3 * q^44 + 12 * q^46 + 14 * q^47 + 13 * q^49 + 4 * q^50 + 2 * q^52 + 10 * q^53 + 3 * q^55 + q^56 - 2 * q^58 + 9 * q^61 + 10 * q^62 + 4 * q^64 + 2 * q^65 - 4 * q^67 + 2 * q^68 + q^70 + 9 * q^71 - 14 * q^73 + 3 * q^74 + 2 * q^76 + 23 * q^77 + 12 * q^79 + 4 * q^80 + 25 * q^83 + 2 * q^85 - 6 * q^86 + 3 * q^88 + 9 * q^89 - 18 * q^91 + 12 * q^92 + 14 * q^94 + 2 * q^95 - 3 * q^97 + 13 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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