LMFDB - Embedded newform 6039.2.a.m.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, 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("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.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:	$48.2216577807$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.03124 q^{2} +2.12592 q^{4} -3.40634 q^{5} -3.47017 q^{7} -0.255767 q^{8} +O(q^{10})$	\(q-2.03124 q^{2} +2.12592 q^{4} -3.40634 q^{5} -3.47017 q^{7} -0.255767 q^{8} +6.91908 q^{10} +1.00000 q^{11} +1.35506 q^{13} +7.04873 q^{14} -3.73231 q^{16} -4.44846 q^{17} -3.01654 q^{19} -7.24160 q^{20} -2.03124 q^{22} -2.96487 q^{23} +6.60317 q^{25} -2.75245 q^{26} -7.37730 q^{28} -7.28062 q^{29} +6.63564 q^{31} +8.09274 q^{32} +9.03587 q^{34} +11.8206 q^{35} +9.62031 q^{37} +6.12731 q^{38} +0.871231 q^{40} +1.06320 q^{41} +6.68952 q^{43} +2.12592 q^{44} +6.02235 q^{46} +5.41383 q^{47} +5.04209 q^{49} -13.4126 q^{50} +2.88075 q^{52} -7.07347 q^{53} -3.40634 q^{55} +0.887557 q^{56} +14.7887 q^{58} +1.73933 q^{59} +1.00000 q^{61} -13.4786 q^{62} -8.97363 q^{64} -4.61580 q^{65} +1.27704 q^{67} -9.45706 q^{68} -24.0104 q^{70} -9.73609 q^{71} -4.46857 q^{73} -19.5411 q^{74} -6.41292 q^{76} -3.47017 q^{77} +8.03333 q^{79} +12.7135 q^{80} -2.15961 q^{82} -8.97685 q^{83} +15.1530 q^{85} -13.5880 q^{86} -0.255767 q^{88} +3.98779 q^{89} -4.70229 q^{91} -6.30306 q^{92} -10.9968 q^{94} +10.2754 q^{95} +10.2589 q^{97} -10.2417 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) $ 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8	$ 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8} - 12 q^{10} + 25 q^{11} - 4 q^{13} - 14 q^{14} + 21 q^{16} - 16 q^{17} - 18 q^{19} - 28 q^{20} - 5 q^{22} - 8 q^{23} + 29 q^{25} - 16 q^{26} + 18 q^{28} - 28 q^{29} - 8 q^{31} - 35 q^{32} + 6 q^{34} - 22 q^{35} + 4 q^{37} + 4 q^{38} - 12 q^{40} - 58 q^{41} - 26 q^{43} + 25 q^{44} + 8 q^{46} - 20 q^{47} + 23 q^{49} - 27 q^{50} - 2 q^{52} - 36 q^{53} - 12 q^{55} - 70 q^{56} + 12 q^{58} - 18 q^{59} + 25 q^{61} - 42 q^{62} + 35 q^{64} - 76 q^{65} - 8 q^{67} - 28 q^{68} + 76 q^{70} - 24 q^{71} + 2 q^{73} - 40 q^{74} - 64 q^{76} - 4 q^{77} - 22 q^{79} - 36 q^{80} + 30 q^{82} - 14 q^{83} - 70 q^{86} - 15 q^{88} - 82 q^{89} - 6 q^{91} - 48 q^{92} - 16 q^{94} - 34 q^{95} + 16 q^{97} - 35 q^{98}+O(q^{100}) $ 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8 - 12 * q^10 + 25 * q^11 - 4 * q^13 - 14 * q^14 + 21 * q^16 - 16 * q^17 - 18 * q^19 - 28 * q^20 - 5 * q^22 - 8 * q^23 + 29 * q^25 - 16 * q^26 + 18 * q^28 - 28 * q^29 - 8 * q^31 - 35 * q^32 + 6 * q^34 - 22 * q^35 + 4 * q^37 + 4 * q^38 - 12 * q^40 - 58 * q^41 - 26 * q^43 + 25 * q^44 + 8 * q^46 - 20 * q^47 + 23 * q^49 - 27 * q^50 - 2 * q^52 - 36 * q^53 - 12 * q^55 - 70 * q^56 + 12 * q^58 - 18 * q^59 + 25 * q^61 - 42 * q^62 + 35 * q^64 - 76 * q^65 - 8 * q^67 - 28 * q^68 + 76 * q^70 - 24 * q^71 + 2 * q^73 - 40 * q^74 - 64 * q^76 - 4 * q^77 - 22 * q^79 - 36 * q^80 + 30 * q^82 - 14 * q^83 - 70 * q^86 - 15 * q^88 - 82 * q^89 - 6 * q^91 - 48 * q^92 - 16 * q^94 - 34 * q^95 + 16 * q^97 - 35 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.03124	−1.43630	−0.718150	−	0.695888i	$-0.755011\pi$
$2$	−2.03124	−1.43630	−0.718150	+	0.695888i	$0.755011\pi$
$3$	0	0
$3$	0	0
$4$	2.12592	1.06296
$5$	−3.40634	−1.52336	−0.761681	−	0.647952i	$-0.775626\pi$
$5$	−3.40634	−1.52336	−0.761681	+	0.647952i	$0.775626\pi$
$6$	0	0
$7$	−3.47017	−1.31160	−0.655801	−	0.754934i	$-0.727669\pi$
$7$	−3.47017	−1.31160	−0.655801	+	0.754934i	$0.727669\pi$
$8$	−0.255767	−0.0904274
$9$	0	0
$10$	6.91908	2.18801
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.35506	0.375826	0.187913	−	0.982186i	$-0.439828\pi$
$13$	1.35506	0.375826	0.187913	+	0.982186i	$0.439828\pi$
$14$	7.04873	1.88385
$15$	0	0
$16$	−3.73231	−0.933078
$17$	−4.44846	−1.07891	−0.539455	−	0.842014i	$-0.681370\pi$
$17$	−4.44846	−1.07891	−0.539455	+	0.842014i	$0.681370\pi$
$18$	0	0
$19$	−3.01654	−0.692042	−0.346021	−	0.938227i	$-0.612467\pi$
$19$	−3.01654	−0.692042	−0.346021	+	0.938227i	$0.612467\pi$
$20$	−7.24160	−1.61927
$21$	0	0
$22$	−2.03124	−0.433061
$23$	−2.96487	−0.618218	−0.309109	−	0.951027i	$-0.600031\pi$
$23$	−2.96487	−0.618218	−0.309109	+	0.951027i	$0.600031\pi$
$24$	0	0
$25$	6.60317	1.32063
$26$	−2.75245	−0.539799
$27$	0	0
$28$	−7.37730	−1.39418
$29$	−7.28062	−1.35198	−0.675989	−	0.736912i	$-0.736283\pi$
$29$	−7.28062	−1.35198	−0.675989	+	0.736912i	$0.736283\pi$
$30$	0	0
$31$	6.63564	1.19180	0.595898	−	0.803060i	$-0.296796\pi$
$31$	6.63564	1.19180	0.595898	+	0.803060i	$0.296796\pi$
$32$	8.09274	1.43061
$33$	0	0
$34$	9.03587	1.54964
$35$	11.8206	1.99805
$36$	0	0
$37$	9.62031	1.58157	0.790785	−	0.612094i	$-0.209673\pi$
$37$	9.62031	1.58157	0.790785	+	0.612094i	$0.209673\pi$
$38$	6.12731	0.993981
$39$	0	0
$40$	0.871231	0.137754
$41$	1.06320	0.166044	0.0830219	−	0.996548i	$-0.473543\pi$
$41$	1.06320	0.166044	0.0830219	+	0.996548i	$0.473543\pi$
$42$	0	0
$43$	6.68952	1.02014	0.510071	−	0.860132i	$-0.329619\pi$
$43$	6.68952	1.02014	0.510071	+	0.860132i	$0.329619\pi$
$44$	2.12592	0.320494
$45$	0	0
$46$	6.02235	0.887946
$47$	5.41383	0.789688	0.394844	−	0.918748i	$-0.370799\pi$
$47$	5.41383	0.789688	0.394844	+	0.918748i	$0.370799\pi$
$48$	0	0
$49$	5.04209	0.720298
$50$	−13.4126	−1.89683
$51$	0	0
$52$	2.88075	0.399488
$53$	−7.07347	−0.971615	−0.485808	−	0.874066i	$-0.661474\pi$
$53$	−7.07347	−0.971615	−0.485808	+	0.874066i	$0.661474\pi$
$54$	0	0
$55$	−3.40634	−0.459311
$56$	0.887557	0.118605
$57$	0	0
$58$	14.7887	1.94185
$59$	1.73933	0.226441	0.113221	−	0.993570i	$-0.463883\pi$
$59$	1.73933	0.226441	0.113221	+	0.993570i	$0.463883\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−13.4786	−1.71178
$63$	0	0
$64$	−8.97363	−1.12170
$65$	−4.61580	−0.572519
$66$	0	0
$67$	1.27704	0.156016	0.0780078	−	0.996953i	$-0.475144\pi$
$67$	1.27704	0.156016	0.0780078	+	0.996953i	$0.475144\pi$
$68$	−9.45706	−1.14684
$69$	0	0
$70$	−24.0104	−2.86979
$71$	−9.73609	−1.15546	−0.577731	−	0.816227i	$-0.696062\pi$
$71$	−9.73609	−1.15546	−0.577731	+	0.816227i	$0.696062\pi$
$72$	0	0
$73$	−4.46857	−0.523006	−0.261503	−	0.965203i	$-0.584218\pi$
$73$	−4.46857	−0.523006	−0.261503	+	0.965203i	$0.584218\pi$
$74$	−19.5411	−2.27161
$75$	0	0
$76$	−6.41292	−0.735612
$77$	−3.47017	−0.395463
$78$	0	0
$79$	8.03333	0.903820	0.451910	−	0.892063i	$-0.350743\pi$
$79$	8.03333	0.903820	0.451910	+	0.892063i	$0.350743\pi$
$80$	12.7135	1.42142
$81$	0	0
$82$	−2.15961	−0.238489
$83$	−8.97685	−0.985337	−0.492669	−	0.870217i	$-0.663979\pi$
$83$	−8.97685	−0.985337	−0.492669	+	0.870217i	$0.663979\pi$
$84$	0	0
$85$	15.1530	1.64357
$86$	−13.5880	−1.46523
$87$	0	0
$88$	−0.255767	−0.0272649
$89$	3.98779	0.422705	0.211353	−	0.977410i	$-0.432213\pi$
$89$	3.98779	0.422705	0.211353	+	0.977410i	$0.432213\pi$
$90$	0	0
$91$	−4.70229	−0.492934
$92$	−6.30306	−0.657140
$93$	0	0
$94$	−10.9968	−1.13423
$95$	10.2754	1.05423
$96$	0	0
$97$	10.2589	1.04163	0.520816	−	0.853669i	$-0.325628\pi$
$97$	10.2589	1.04163	0.520816	+	0.853669i	$0.325628\pi$
$98$	−10.2417	−1.03456
$99$	0	0
$100$	14.0378	1.40378
$101$	10.0066	0.995696	0.497848	−	0.867264i	$-0.334124\pi$
$101$	10.0066	0.995696	0.497848	+	0.867264i	$0.334124\pi$
$102$	0	0
$103$	−0.941154	−0.0927347	−0.0463673	−	0.998924i	$-0.514764\pi$
$103$	−0.941154	−0.0927347	−0.0463673	+	0.998924i	$0.514764\pi$
$104$	−0.346580	−0.0339850
$105$	0	0
$106$	14.3679	1.39553
$107$	−4.07243	−0.393697	−0.196848	−	0.980434i	$-0.563071\pi$
$107$	−4.07243	−0.393697	−0.196848	+	0.980434i	$0.563071\pi$
$108$	0	0
$109$	−0.564312	−0.0540513	−0.0270256	−	0.999635i	$-0.508604\pi$
$109$	−0.564312	−0.0540513	−0.0270256	+	0.999635i	$0.508604\pi$
$110$	6.91908	0.659709
$111$	0	0
$112$	12.9518	1.22383
$113$	11.9692	1.12596	0.562982	−	0.826469i	$-0.309654\pi$
$113$	11.9692	1.12596	0.562982	+	0.826469i	$0.309654\pi$
$114$	0	0
$115$	10.0994	0.941770
$116$	−15.4780	−1.43710
$117$	0	0
$118$	−3.53298	−0.325238
$119$	15.4369	1.41510
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.03124	−0.183899
$123$	0	0
$124$	14.1068	1.26683
$125$	−5.46096	−0.488443
$126$	0	0
$127$	4.06594	0.360794	0.180397	−	0.983594i	$-0.442262\pi$
$127$	4.06594	0.360794	0.180397	+	0.983594i	$0.442262\pi$
$128$	2.04208	0.180496
$129$	0	0
$130$	9.37578	0.822310
$131$	10.2547	0.895958	0.447979	−	0.894044i	$-0.352144\pi$
$131$	10.2547	0.895958	0.447979	+	0.894044i	$0.352144\pi$
$132$	0	0
$133$	10.4679	0.907684
$134$	−2.59397	−0.224085
$135$	0	0
$136$	1.13777	0.0975631
$137$	−1.05241	−0.0899139	−0.0449569	−	0.998989i	$-0.514315\pi$
$137$	−1.05241	−0.0899139	−0.0449569	+	0.998989i	$0.514315\pi$
$138$	0	0
$139$	−7.89577	−0.669710	−0.334855	−	0.942270i	$-0.608687\pi$
$139$	−7.89577	−0.669710	−0.334855	+	0.942270i	$0.608687\pi$
$140$	25.1296	2.12384
$141$	0	0
$142$	19.7763	1.65959
$143$	1.35506	0.113316
$144$	0	0
$145$	24.8003	2.05955
$146$	9.07672	0.751194
$147$	0	0
$148$	20.4520	1.68114
$149$	12.3773	1.01399	0.506993	−	0.861950i	$-0.330757\pi$
$149$	12.3773	1.01399	0.506993	+	0.861950i	$0.330757\pi$
$150$	0	0
$151$	−17.7657	−1.44575	−0.722877	−	0.690976i	$-0.757181\pi$
$151$	−17.7657	−1.44575	−0.722877	+	0.690976i	$0.757181\pi$
$152$	0.771533	0.0625796
$153$	0	0
$154$	7.04873	0.568003
$155$	−22.6033	−1.81554
$156$	0	0
$157$	23.6131	1.88453	0.942266	−	0.334866i	$-0.108691\pi$
$157$	23.6131	1.88453	0.942266	+	0.334866i	$0.108691\pi$
$158$	−16.3176	−1.29816
$159$	0	0
$160$	−27.5666	−2.17933
$161$	10.2886	0.810855
$162$	0	0
$163$	11.9312	0.934525	0.467263	−	0.884119i	$-0.345240\pi$
$163$	11.9312	0.934525	0.467263	+	0.884119i	$0.345240\pi$
$164$	2.26027	0.176498
$165$	0	0
$166$	18.2341	1.41524
$167$	10.1607	0.786256	0.393128	−	0.919484i	$-0.371393\pi$
$167$	10.1607	0.786256	0.393128	+	0.919484i	$0.371393\pi$
$168$	0	0
$169$	−11.1638	−0.858755
$170$	−30.7793	−2.36066
$171$	0	0
$172$	14.2214	1.08437
$173$	6.21561	0.472564	0.236282	−	0.971685i	$-0.424071\pi$
$173$	6.21561	0.472564	0.236282	+	0.971685i	$0.424071\pi$
$174$	0	0
$175$	−22.9141	−1.73215
$176$	−3.73231	−0.281333
$177$	0	0
$178$	−8.10015	−0.607132
$179$	−7.77095	−0.580828	−0.290414	−	0.956901i	$-0.593793\pi$
$179$	−7.77095	−0.580828	−0.290414	+	0.956901i	$0.593793\pi$
$180$	0	0
$181$	−3.16728	−0.235422	−0.117711	−	0.993048i	$-0.537556\pi$
$181$	−3.16728	−0.235422	−0.117711	+	0.993048i	$0.537556\pi$
$182$	9.55146	0.708001
$183$	0	0
$184$	0.758317	0.0559038
$185$	−32.7701	−2.40930
$186$	0	0
$187$	−4.44846	−0.325304
$188$	11.5094	0.839406
$189$	0	0
$190$	−20.8717	−1.51419
$191$	11.6485	0.842855	0.421427	−	0.906862i	$-0.361529\pi$
$191$	11.6485	0.842855	0.421427	+	0.906862i	$0.361529\pi$
$192$	0	0
$193$	19.0391	1.37046	0.685231	−	0.728326i	$-0.259701\pi$
$193$	19.0391	1.37046	0.685231	+	0.728326i	$0.259701\pi$
$194$	−20.8382	−1.49610
$195$	0	0
$196$	10.7191	0.765647
$197$	−8.33323	−0.593718	−0.296859	−	0.954921i	$-0.595939\pi$
$197$	−8.33323	−0.593718	−0.296859	+	0.954921i	$0.595939\pi$
$198$	0	0
$199$	−2.44931	−0.173627	−0.0868136	−	0.996225i	$-0.527668\pi$
$199$	−2.44931	−0.173627	−0.0868136	+	0.996225i	$0.527668\pi$
$200$	−1.68888	−0.119422
$201$	0	0
$202$	−20.3258	−1.43012
$203$	25.2650	1.77326
$204$	0	0
$205$	−3.62162	−0.252945
$206$	1.91171	0.133195
$207$	0	0
$208$	−5.05751	−0.350675
$209$	−3.01654	−0.208659
$210$	0	0
$211$	1.92170	0.132296	0.0661478	−	0.997810i	$-0.478929\pi$
$211$	1.92170	0.132296	0.0661478	+	0.997810i	$0.478929\pi$
$212$	−15.0376	−1.03279
$213$	0	0
$214$	8.27206	0.565466
$215$	−22.7868	−1.55405
$216$	0	0
$217$	−23.0268	−1.56316
$218$	1.14625	0.0776338
$219$	0	0
$220$	−7.24160	−0.488229
$221$	−6.02793	−0.405483
$222$	0	0
$223$	−26.3466	−1.76430	−0.882149	−	0.470970i	$-0.843904\pi$
$223$	−26.3466	−1.76430	−0.882149	+	0.470970i	$0.843904\pi$
$224$	−28.0832	−1.87639
$225$	0	0
$226$	−24.3122	−1.61722
$227$	19.2609	1.27839	0.639194	−	0.769046i	$-0.279268\pi$
$227$	19.2609	1.27839	0.639194	+	0.769046i	$0.279268\pi$
$228$	0	0
$229$	11.6046	0.766852	0.383426	−	0.923572i	$-0.374744\pi$
$229$	11.6046	0.766852	0.383426	+	0.923572i	$0.374744\pi$
$230$	−20.5142	−1.35266
$231$	0	0
$232$	1.86214	0.122256
$233$	−14.3645	−0.941049	−0.470524	−	0.882387i	$-0.655935\pi$
$233$	−14.3645	−0.941049	−0.470524	+	0.882387i	$0.655935\pi$
$234$	0	0
$235$	−18.4414	−1.20298
$236$	3.69767	0.240698
$237$	0	0
$238$	−31.3560	−2.03251
$239$	29.8035	1.92783	0.963914	−	0.266215i	$-0.0857732\pi$
$239$	29.8035	1.92783	0.963914	+	0.266215i	$0.0857732\pi$
$240$	0	0
$241$	−14.2958	−0.920875	−0.460437	−	0.887692i	$-0.652307\pi$
$241$	−14.2958	−0.920875	−0.460437	+	0.887692i	$0.652307\pi$
$242$	−2.03124	−0.130573
$243$	0	0
$244$	2.12592	0.136098
$245$	−17.1751	−1.09728
$246$	0	0
$247$	−4.08760	−0.260088
$248$	−1.69718	−0.107771
$249$	0	0
$250$	11.0925	0.701551
$251$	−10.1685	−0.641827	−0.320914	−	0.947108i	$-0.603990\pi$
$251$	−10.1685	−0.641827	−0.320914	+	0.947108i	$0.603990\pi$
$252$	0	0
$253$	−2.96487	−0.186400
$254$	−8.25889	−0.518209
$255$	0	0
$256$	13.7993	0.862457
$257$	19.0361	1.18744	0.593720	−	0.804672i	$-0.297659\pi$
$257$	19.0361	1.18744	0.593720	+	0.804672i	$0.297659\pi$
$258$	0	0
$259$	−33.3841	−2.07439
$260$	−9.81281	−0.608565
$261$	0	0
$262$	−20.8297	−1.28686
$263$	29.3716	1.81113	0.905564	−	0.424209i	$-0.139448\pi$
$263$	29.3716	1.81113	0.905564	+	0.424209i	$0.139448\pi$
$264$	0	0
$265$	24.0946	1.48012
$266$	−21.2628	−1.30371
$267$	0	0
$268$	2.71489	0.165838
$269$	−12.2751	−0.748428	−0.374214	−	0.927342i	$-0.622087\pi$
$269$	−12.2751	−0.748428	−0.374214	+	0.927342i	$0.622087\pi$
$270$	0	0
$271$	18.3542	1.11494	0.557468	−	0.830199i	$-0.311773\pi$
$271$	18.3542	1.11494	0.557468	+	0.830199i	$0.311773\pi$
$272$	16.6030	1.00671
$273$	0	0
$274$	2.13770	0.129143
$275$	6.60317	0.398186
$276$	0	0
$277$	−0.968716	−0.0582045	−0.0291023	−	0.999576i	$-0.509265\pi$
$277$	−0.968716	−0.0582045	−0.0291023	+	0.999576i	$0.509265\pi$
$278$	16.0382	0.961905
$279$	0	0
$280$	−3.02332	−0.180678
$281$	2.72177	0.162367	0.0811836	−	0.996699i	$-0.474130\pi$
$281$	2.72177	0.162367	0.0811836	+	0.996699i	$0.474130\pi$
$282$	0	0
$283$	−30.6484	−1.82186	−0.910929	−	0.412564i	$-0.864633\pi$
$283$	−30.6484	−1.82186	−0.910929	+	0.412564i	$0.864633\pi$
$284$	−20.6981	−1.22821
$285$	0	0
$286$	−2.75245	−0.162756
$287$	−3.68948	−0.217783
$288$	0	0
$289$	2.78882	0.164048
$290$	−50.3752	−2.95813
$291$	0	0
$292$	−9.49981	−0.555934
$293$	−26.3476	−1.53924	−0.769621	−	0.638501i	$-0.779555\pi$
$293$	−26.3476	−1.53924	−0.769621	+	0.638501i	$0.779555\pi$
$294$	0	0
$295$	−5.92475	−0.344952
$296$	−2.46056	−0.143017
$297$	0	0
$298$	−25.1412	−1.45639
$299$	−4.01757	−0.232342
$300$	0	0
$301$	−23.2138	−1.33802
$302$	36.0864	2.07654
$303$	0	0
$304$	11.2587	0.645729
$305$	−3.40634	−0.195047
$306$	0	0
$307$	−13.8025	−0.787750	−0.393875	−	0.919164i	$-0.628866\pi$
$307$	−13.8025	−0.787750	−0.393875	+	0.919164i	$0.628866\pi$
$308$	−7.37730	−0.420360
$309$	0	0
$310$	45.9126	2.60766
$311$	−17.4408	−0.988978	−0.494489	−	0.869184i	$-0.664645\pi$
$311$	−17.4408	−0.988978	−0.494489	+	0.869184i	$0.664645\pi$
$312$	0	0
$313$	−10.2283	−0.578137	−0.289068	−	0.957308i	$-0.593345\pi$
$313$	−10.2283	−0.578137	−0.289068	+	0.957308i	$0.593345\pi$
$314$	−47.9638	−2.70675
$315$	0	0
$316$	17.0782	0.960724
$317$	2.08419	0.117060	0.0585299	−	0.998286i	$-0.481359\pi$
$317$	2.08419	0.117060	0.0585299	+	0.998286i	$0.481359\pi$
$318$	0	0
$319$	−7.28062	−0.407636
$320$	30.5673	1.70876
$321$	0	0
$322$	−20.8986	−1.16463
$323$	13.4190	0.746652
$324$	0	0
$325$	8.94770	0.496329
$326$	−24.2351	−1.34226
$327$	0	0
$328$	−0.271932	−0.0150149
$329$	−18.7869	−1.03576
$330$	0	0
$331$	0.285359	0.0156847	0.00784236	−	0.999969i	$-0.497504\pi$
$331$	0.285359	0.0156847	0.00784236	+	0.999969i	$0.497504\pi$
$332$	−19.0840	−1.04737
$333$	0	0
$334$	−20.6387	−1.12930
$335$	−4.35004	−0.237668
$336$	0	0
$337$	−27.4956	−1.49778	−0.748891	−	0.662694i	$-0.769413\pi$
$337$	−27.4956	−1.49778	−0.748891	+	0.662694i	$0.769413\pi$
$338$	22.6763	1.23343
$339$	0	0
$340$	32.2140	1.74705
$341$	6.63564	0.359340
$342$	0	0
$343$	6.79429	0.366857
$344$	−1.71096	−0.0922488
$345$	0	0
$346$	−12.6254	−0.678744
$347$	−20.5881	−1.10523	−0.552614	−	0.833437i	$-0.686370\pi$
$347$	−20.5881	−1.10523	−0.552614	+	0.833437i	$0.686370\pi$
$348$	0	0
$349$	−2.27751	−0.121912	−0.0609561	−	0.998140i	$-0.519415\pi$
$349$	−2.27751	−0.121912	−0.0609561	+	0.998140i	$0.519415\pi$
$350$	46.5440	2.48788
$351$	0	0
$352$	8.09274	0.431344
$353$	−18.2406	−0.970847	−0.485423	−	0.874279i	$-0.661335\pi$
$353$	−18.2406	−0.970847	−0.485423	+	0.874279i	$0.661335\pi$
$354$	0	0
$355$	33.1645	1.76019
$356$	8.47772	0.449318
$357$	0	0
$358$	15.7846	0.834244
$359$	−33.7430	−1.78089	−0.890444	−	0.455093i	$-0.849606\pi$
$359$	−33.7430	−1.78089	−0.890444	+	0.455093i	$0.849606\pi$
$360$	0	0
$361$	−9.90047	−0.521077
$362$	6.43349	0.338137
$363$	0	0
$364$	−9.99668	−0.523968
$365$	15.2215	0.796729
$366$	0	0
$367$	24.4055	1.27396	0.636978	−	0.770882i	$-0.280184\pi$
$367$	24.4055	1.27396	0.636978	+	0.770882i	$0.280184\pi$
$368$	11.0658	0.576845
$369$	0	0
$370$	66.5638	3.46049
$371$	24.5461	1.27437
$372$	0	0
$373$	18.9932	0.983432	0.491716	−	0.870756i	$-0.336370\pi$
$373$	18.9932	0.983432	0.491716	+	0.870756i	$0.336370\pi$
$374$	9.03587	0.467234
$375$	0	0
$376$	−1.38468	−0.0714095
$377$	−9.86568	−0.508108
$378$	0	0
$379$	−22.1854	−1.13959	−0.569793	−	0.821788i	$-0.692977\pi$
$379$	−22.1854	−1.13959	−0.569793	+	0.821788i	$0.692977\pi$
$380$	21.8446	1.12060
$381$	0	0
$382$	−23.6608	−1.21059
$383$	13.5582	0.692794	0.346397	−	0.938088i	$-0.387405\pi$
$383$	13.5582	0.692794	0.346397	+	0.938088i	$0.387405\pi$
$384$	0	0
$385$	11.8206	0.602433
$386$	−38.6728	−1.96840
$387$	0	0
$388$	21.8095	1.10721
$389$	12.4198	0.629708	0.314854	−	0.949140i	$-0.398044\pi$
$389$	12.4198	0.629708	0.314854	+	0.949140i	$0.398044\pi$
$390$	0	0
$391$	13.1891	0.667002
$392$	−1.28960	−0.0651347
$393$	0	0
$394$	16.9268	0.852758
$395$	−27.3643	−1.37685
$396$	0	0
$397$	26.0989	1.30987	0.654934	−	0.755686i	$-0.272696\pi$
$397$	26.0989	1.30987	0.654934	+	0.755686i	$0.272696\pi$
$398$	4.97513	0.249381
$399$	0	0
$400$	−24.6451	−1.23225
$401$	5.39073	0.269200	0.134600	−	0.990900i	$-0.457025\pi$
$401$	5.39073	0.269200	0.134600	+	0.990900i	$0.457025\pi$
$402$	0	0
$403$	8.99170	0.447908
$404$	21.2732	1.05838
$405$	0	0
$406$	−51.3192	−2.54693
$407$	9.62031	0.476861
$408$	0	0
$409$	−20.2812	−1.00284	−0.501420	−	0.865204i	$-0.667189\pi$
$409$	−20.2812	−1.00284	−0.501420	+	0.865204i	$0.667189\pi$
$410$	7.35637	0.363305
$411$	0	0
$412$	−2.00082	−0.0985731
$413$	−6.03577	−0.297001
$414$	0	0
$415$	30.5782	1.50103
$416$	10.9661	0.537659
$417$	0	0
$418$	6.12731	0.299696
$419$	20.4891	1.00096	0.500479	−	0.865749i	$-0.333157\pi$
$419$	20.4891	1.00096	0.500479	+	0.865749i	$0.333157\pi$
$420$	0	0
$421$	−22.3466	−1.08911	−0.544554	−	0.838726i	$-0.683301\pi$
$421$	−22.3466	−1.08911	−0.544554	+	0.838726i	$0.683301\pi$
$422$	−3.90343	−0.190016
$423$	0	0
$424$	1.80916	0.0878607
$425$	−29.3740	−1.42485
$426$	0	0
$427$	−3.47017	−0.167933
$428$	−8.65765	−0.418483
$429$	0	0
$430$	46.2853	2.23208
$431$	−18.3062	−0.881779	−0.440890	−	0.897561i	$-0.645337\pi$
$431$	−18.3062	−0.881779	−0.440890	+	0.897561i	$0.645337\pi$
$432$	0	0
$433$	−4.31314	−0.207276	−0.103638	−	0.994615i	$-0.533048\pi$
$433$	−4.31314	−0.207276	−0.103638	+	0.994615i	$0.533048\pi$
$434$	46.7729	2.24517
$435$	0	0
$436$	−1.19968	−0.0574543
$437$	8.94365	0.427833
$438$	0	0
$439$	−14.2199	−0.678677	−0.339339	−	0.940664i	$-0.610203\pi$
$439$	−14.2199	−0.678677	−0.339339	+	0.940664i	$0.610203\pi$
$440$	0.871231	0.0415343
$441$	0	0
$442$	12.2442	0.582395
$443$	−8.78195	−0.417243	−0.208621	−	0.977996i	$-0.566898\pi$
$443$	−8.78195	−0.417243	−0.208621	+	0.977996i	$0.566898\pi$
$444$	0	0
$445$	−13.5838	−0.643933
$446$	53.5161	2.53406
$447$	0	0
$448$	31.1400	1.47123
$449$	−32.5877	−1.53791	−0.768954	−	0.639304i	$-0.779223\pi$
$449$	−32.5877	−1.53791	−0.768954	+	0.639304i	$0.779223\pi$
$450$	0	0
$451$	1.06320	0.0500641
$452$	25.4454	1.19685
$453$	0	0
$454$	−39.1233	−1.83615
$455$	16.0176	0.750917
$456$	0	0
$457$	−28.5107	−1.33367	−0.666836	−	0.745204i	$-0.732352\pi$
$457$	−28.5107	−1.33367	−0.666836	+	0.745204i	$0.732352\pi$
$458$	−23.5716	−1.10143
$459$	0	0
$460$	21.4704	1.00106
$461$	−41.5296	−1.93423	−0.967114	−	0.254345i	$-0.918140\pi$
$461$	−41.5296	−1.93423	−0.967114	+	0.254345i	$0.918140\pi$
$462$	0	0
$463$	−16.1122	−0.748797	−0.374399	−	0.927268i	$-0.622151\pi$
$463$	−16.1122	−0.748797	−0.374399	+	0.927268i	$0.622151\pi$
$464$	27.1735	1.26150
$465$	0	0
$466$	29.1777	1.35163
$467$	16.9677	0.785169	0.392585	−	0.919716i	$-0.371581\pi$
$467$	16.9677	0.785169	0.392585	+	0.919716i	$0.371581\pi$
$468$	0	0
$469$	−4.43156	−0.204630
$470$	37.4588	1.72784
$471$	0	0
$472$	−0.444863	−0.0204765
$473$	6.68952	0.307584
$474$	0	0
$475$	−19.9188	−0.913935
$476$	32.8176	1.50419
$477$	0	0
$478$	−60.5379	−2.76894
$479$	−17.8862	−0.817240	−0.408620	−	0.912705i	$-0.633990\pi$
$479$	−17.8862	−0.817240	−0.408620	+	0.912705i	$0.633990\pi$
$480$	0	0
$481$	13.0361	0.594395
$482$	29.0382	1.32265
$483$	0	0
$484$	2.12592	0.0966326
$485$	−34.9453	−1.58678
$486$	0	0
$487$	18.0680	0.818741	0.409371	−	0.912368i	$-0.365748\pi$
$487$	18.0680	0.818741	0.409371	+	0.912368i	$0.365748\pi$
$488$	−0.255767	−0.0115780
$489$	0	0
$490$	34.8866	1.57602
$491$	6.50825	0.293713	0.146857	−	0.989158i	$-0.453084\pi$
$491$	6.50825	0.293713	0.146857	+	0.989158i	$0.453084\pi$
$492$	0	0
$493$	32.3876	1.45866
$494$	8.30287	0.373564
$495$	0	0
$496$	−24.7663	−1.11204
$497$	33.7859	1.51551
$498$	0	0
$499$	4.89908	0.219313	0.109657	−	0.993970i	$-0.465025\pi$
$499$	4.89908	0.219313	0.109657	+	0.993970i	$0.465025\pi$
$500$	−11.6095	−0.519195
$501$	0	0
$502$	20.6545	0.921857
$503$	0.363588	0.0162116	0.00810580	−	0.999967i	$-0.497420\pi$
$503$	0.363588	0.0162116	0.00810580	+	0.999967i	$0.497420\pi$
$504$	0	0
$505$	−34.0860	−1.51681
$506$	6.02235	0.267726
$507$	0	0
$508$	8.64386	0.383509
$509$	−20.5412	−0.910475	−0.455237	−	0.890370i	$-0.650446\pi$
$509$	−20.5412	−0.910475	−0.455237	+	0.890370i	$0.650446\pi$
$510$	0	0
$511$	15.5067	0.685976
$512$	−32.1138	−1.41924
$513$	0	0
$514$	−38.6668	−1.70552
$515$	3.20589	0.141269
$516$	0	0
$517$	5.41383	0.238100
$518$	67.8110	2.97945
$519$	0	0
$520$	1.18057	0.0517715
$521$	−30.4496	−1.33402	−0.667012	−	0.745047i	$-0.732427\pi$
$521$	−30.4496	−1.33402	−0.667012	+	0.745047i	$0.732427\pi$
$522$	0	0
$523$	−22.9651	−1.00419	−0.502096	−	0.864812i	$-0.667438\pi$
$523$	−22.9651	−1.00419	−0.502096	+	0.864812i	$0.667438\pi$
$524$	21.8007	0.952366
$525$	0	0
$526$	−59.6606	−2.60132
$527$	−29.5184	−1.28584
$528$	0	0
$529$	−14.2096	−0.617807
$530$	−48.9419	−2.12590
$531$	0	0
$532$	22.2539	0.964830
$533$	1.44070	0.0624036
$534$	0	0
$535$	13.8721	0.599743
$536$	−0.326626	−0.0141081
$537$	0	0
$538$	24.9337	1.07497
$539$	5.04209	0.217178
$540$	0	0
$541$	37.1212	1.59596	0.797981	−	0.602682i	$-0.205901\pi$
$541$	37.1212	1.59596	0.797981	+	0.602682i	$0.205901\pi$
$542$	−37.2816	−1.60138
$543$	0	0
$544$	−36.0002	−1.54350
$545$	1.92224	0.0823397
$546$	0	0
$547$	−4.05616	−0.173429	−0.0867145	−	0.996233i	$-0.527637\pi$
$547$	−4.05616	−0.173429	−0.0867145	+	0.996233i	$0.527637\pi$
$548$	−2.23735	−0.0955747
$549$	0	0
$550$	−13.4126	−0.571915
$551$	21.9623	0.935625
$552$	0	0
$553$	−27.8770	−1.18545
$554$	1.96769	0.0835992
$555$	0	0
$556$	−16.7857	−0.711874
$557$	−12.8006	−0.542379	−0.271190	−	0.962526i	$-0.587417\pi$
$557$	−12.8006	−0.542379	−0.271190	+	0.962526i	$0.587417\pi$
$558$	0	0
$559$	9.06470	0.383396
$560$	−44.1181	−1.86433
$561$	0	0
$562$	−5.52856	−0.233208
$563$	−1.64601	−0.0693709	−0.0346854	−	0.999398i	$-0.511043\pi$
$563$	−1.64601	−0.0693709	−0.0346854	+	0.999398i	$0.511043\pi$
$564$	0	0
$565$	−40.7710	−1.71525
$566$	62.2541	2.61673
$567$	0	0
$568$	2.49018	0.104485
$569$	21.5866	0.904959	0.452480	−	0.891775i	$-0.350540\pi$
$569$	21.5866	0.904959	0.452480	+	0.891775i	$0.350540\pi$
$570$	0	0
$571$	−15.4177	−0.645209	−0.322604	−	0.946534i	$-0.604558\pi$
$571$	−15.4177	−0.645209	−0.322604	+	0.946534i	$0.604558\pi$
$572$	2.88075	0.120450
$573$	0	0
$574$	7.49421	0.312802
$575$	−19.5775	−0.816440
$576$	0	0
$577$	43.9124	1.82810	0.914049	−	0.405604i	$-0.132939\pi$
$577$	43.9124	1.82810	0.914049	+	0.405604i	$0.132939\pi$
$578$	−5.66474	−0.235622
$579$	0	0
$580$	52.7234	2.18922
$581$	31.1512	1.29237
$582$	0	0
$583$	−7.07347	−0.292953
$584$	1.14291	0.0472941
$585$	0	0
$586$	53.5181	2.21081
$587$	9.22039	0.380566	0.190283	−	0.981729i	$-0.439059\pi$
$587$	9.22039	0.380566	0.190283	+	0.981729i	$0.439059\pi$
$588$	0	0
$589$	−20.0167	−0.824774
$590$	12.0346	0.495455
$591$	0	0
$592$	−35.9060	−1.47573
$593$	18.2614	0.749905	0.374952	−	0.927044i	$-0.377659\pi$
$593$	18.2614	0.749905	0.374952	+	0.927044i	$0.377659\pi$
$594$	0	0
$595$	−52.5835	−2.15571
$596$	26.3131	1.07783
$597$	0	0
$598$	8.16064	0.333713
$599$	−24.6677	−1.00790	−0.503948	−	0.863734i	$-0.668120\pi$
$599$	−24.6677	−1.00790	−0.503948	+	0.863734i	$0.668120\pi$
$600$	0	0
$601$	30.3157	1.23660	0.618302	−	0.785940i	$-0.287821\pi$
$601$	30.3157	1.23660	0.618302	+	0.785940i	$0.287821\pi$
$602$	47.1526	1.92180
$603$	0	0
$604$	−37.7685	−1.53678
$605$	−3.40634	−0.138488
$606$	0	0
$607$	−2.40050	−0.0974331	−0.0487165	−	0.998813i	$-0.515513\pi$
$607$	−2.40050	−0.0974331	−0.0487165	+	0.998813i	$0.515513\pi$
$608$	−24.4121	−0.990041
$609$	0	0
$610$	6.91908	0.280146
$611$	7.33607	0.296786
$612$	0	0
$613$	−37.6269	−1.51974	−0.759868	−	0.650078i	$-0.774736\pi$
$613$	−37.6269	−1.51974	−0.759868	+	0.650078i	$0.774736\pi$
$614$	28.0361	1.13145
$615$	0	0
$616$	0.887557	0.0357607
$617$	5.09154	0.204978	0.102489	−	0.994734i	$-0.467319\pi$
$617$	5.09154	0.204978	0.102489	+	0.994734i	$0.467319\pi$
$618$	0	0
$619$	−22.0256	−0.885286	−0.442643	−	0.896698i	$-0.645959\pi$
$619$	−22.0256	−0.885286	−0.442643	+	0.896698i	$0.645959\pi$
$620$	−48.0527	−1.92984
$621$	0	0
$622$	35.4264	1.42047
$623$	−13.8383	−0.554421
$624$	0	0
$625$	−14.4140	−0.576559
$626$	20.7760	0.830378
$627$	0	0
$628$	50.1995	2.00318
$629$	−42.7956	−1.70637
$630$	0	0
$631$	10.0898	0.401669	0.200834	−	0.979625i	$-0.435635\pi$
$631$	10.0898	0.401669	0.200834	+	0.979625i	$0.435635\pi$
$632$	−2.05466	−0.0817301
$633$	0	0
$634$	−4.23348	−0.168133
$635$	−13.8500	−0.549620
$636$	0	0
$637$	6.83233	0.270707
$638$	14.7887	0.585488
$639$	0	0
$640$	−6.95604	−0.274962
$641$	28.2622	1.11629	0.558145	−	0.829744i	$-0.311513\pi$
$641$	28.2622	1.11629	0.558145	+	0.829744i	$0.311513\pi$
$642$	0	0
$643$	−3.56941	−0.140764	−0.0703819	−	0.997520i	$-0.522422\pi$
$643$	−3.56941	−0.140764	−0.0703819	+	0.997520i	$0.522422\pi$
$644$	21.8727	0.861906
$645$	0	0
$646$	−27.2571	−1.07242
$647$	36.7318	1.44408	0.722039	−	0.691853i	$-0.243205\pi$
$647$	36.7318	1.44408	0.722039	+	0.691853i	$0.243205\pi$
$648$	0	0
$649$	1.73933	0.0682746
$650$	−18.1749	−0.712877
$651$	0	0
$652$	25.3648	0.993362
$653$	−7.82150	−0.306079	−0.153039	−	0.988220i	$-0.548906\pi$
$653$	−7.82150	−0.306079	−0.153039	+	0.988220i	$0.548906\pi$
$654$	0	0
$655$	−34.9310	−1.36487
$656$	−3.96819	−0.154932
$657$	0	0
$658$	38.1607	1.48766
$659$	−15.3554	−0.598163	−0.299082	−	0.954228i	$-0.596680\pi$
$659$	−15.3554	−0.598163	−0.299082	+	0.954228i	$0.596680\pi$
$660$	0	0
$661$	−9.17471	−0.356855	−0.178427	−	0.983953i	$-0.557101\pi$
$661$	−9.17471	−0.356855	−0.178427	+	0.983953i	$0.557101\pi$
$662$	−0.579631	−0.0225280
$663$	0	0
$664$	2.29599	0.0891015
$665$	−35.6573	−1.38273
$666$	0	0
$667$	21.5861	0.835816
$668$	21.6007	0.835758
$669$	0	0
$670$	8.83597	0.341363
$671$	1.00000	0.0386046
$672$	0	0
$673$	33.5786	1.29436	0.647179	−	0.762338i	$-0.275949\pi$
$673$	33.5786	1.29436	0.647179	+	0.762338i	$0.275949\pi$
$674$	55.8501	2.15126
$675$	0	0
$676$	−23.7333	−0.912821
$677$	0.968387	0.0372181	0.0186091	−	0.999827i	$-0.494076\pi$
$677$	0.968387	0.0372181	0.0186091	+	0.999827i	$0.494076\pi$
$678$	0	0
$679$	−35.6001	−1.36621
$680$	−3.87564	−0.148624
$681$	0	0
$682$	−13.4786	−0.516121
$683$	27.4578	1.05064	0.525322	−	0.850904i	$-0.323945\pi$
$683$	27.4578	1.05064	0.525322	+	0.850904i	$0.323945\pi$
$684$	0	0
$685$	3.58489	0.136971
$686$	−13.8008	−0.526917
$687$	0	0
$688$	−24.9674	−0.951872
$689$	−9.58497	−0.365158
$690$	0	0
$691$	−20.1609	−0.766956	−0.383478	−	0.923550i	$-0.625274\pi$
$691$	−20.1609	−0.766956	−0.383478	+	0.923550i	$0.625274\pi$
$692$	13.2139	0.502316
$693$	0	0
$694$	41.8193	1.58744
$695$	26.8957	1.02021
$696$	0	0
$697$	−4.72960	−0.179146
$698$	4.62615	0.175103
$699$	0	0
$700$	−48.7136	−1.84120
$701$	43.9913	1.66153	0.830763	−	0.556626i	$-0.187904\pi$
$701$	43.9913	1.66153	0.830763	+	0.556626i	$0.187904\pi$
$702$	0	0
$703$	−29.0201	−1.09451
$704$	−8.97363	−0.338206
$705$	0	0
$706$	37.0509	1.39443
$707$	−34.7247	−1.30596
$708$	0	0
$709$	24.1319	0.906291	0.453145	−	0.891437i	$-0.350302\pi$
$709$	24.1319	0.906291	0.453145	+	0.891437i	$0.350302\pi$
$710$	−67.3649	−2.52816
$711$	0	0
$712$	−1.01995	−0.0382241
$713$	−19.6738	−0.736790
$714$	0	0
$715$	−4.61580	−0.172621
$716$	−16.5204	−0.617396
$717$	0	0
$718$	68.5400	2.55789
$719$	3.28363	0.122459	0.0612294	−	0.998124i	$-0.480498\pi$
$719$	3.28363	0.122459	0.0612294	+	0.998124i	$0.480498\pi$
$720$	0	0
$721$	3.26597	0.121631
$722$	20.1102	0.748424
$723$	0	0
$724$	−6.73337	−0.250244
$725$	−48.0752	−1.78547
$726$	0	0
$727$	2.90335	0.107679	0.0538397	−	0.998550i	$-0.482854\pi$
$727$	2.90335	0.107679	0.0538397	+	0.998550i	$0.482854\pi$
$728$	1.20269	0.0445748
$729$	0	0
$730$	−30.9184	−1.14434
$731$	−29.7581	−1.10064
$732$	0	0
$733$	−19.9013	−0.735073	−0.367537	−	0.930009i	$-0.619799\pi$
$733$	−19.9013	−0.735073	−0.367537	+	0.930009i	$0.619799\pi$
$734$	−49.5733	−1.82978
$735$	0	0
$736$	−23.9939	−0.884427
$737$	1.27704	0.0470405
$738$	0	0
$739$	−38.8129	−1.42776	−0.713878	−	0.700270i	$-0.753063\pi$
$739$	−38.8129	−1.42776	−0.713878	+	0.700270i	$0.753063\pi$
$740$	−69.6665	−2.56099
$741$	0	0
$742$	−49.8590	−1.83038
$743$	−30.9108	−1.13401	−0.567004	−	0.823715i	$-0.691898\pi$
$743$	−30.9108	−1.13401	−0.567004	+	0.823715i	$0.691898\pi$
$744$	0	0
$745$	−42.1612	−1.54467
$746$	−38.5797	−1.41250
$747$	0	0
$748$	−9.45706	−0.345784
$749$	14.1320	0.516373
$750$	0	0
$751$	7.06327	0.257742	0.128871	−	0.991661i	$-0.458865\pi$
$751$	7.06327	0.257742	0.128871	+	0.991661i	$0.458865\pi$
$752$	−20.2061	−0.736841
$753$	0	0
$754$	20.0395	0.729796
$755$	60.5162	2.20241
$756$	0	0
$757$	2.14363	0.0779117	0.0389559	−	0.999241i	$-0.487597\pi$
$757$	2.14363	0.0779117	0.0389559	+	0.999241i	$0.487597\pi$
$758$	45.0637	1.63679
$759$	0	0
$760$	−2.62811	−0.0953315
$761$	−37.7542	−1.36859	−0.684294	−	0.729207i	$-0.739889\pi$
$761$	−37.7542	−1.36859	−0.684294	+	0.729207i	$0.739889\pi$
$762$	0	0
$763$	1.95826	0.0708937
$764$	24.7637	0.895920
$765$	0	0
$766$	−27.5400	−0.995060
$767$	2.35689	0.0851025
$768$	0	0
$769$	−53.2008	−1.91847	−0.959235	−	0.282611i	$-0.908799\pi$
$769$	−53.2008	−1.91847	−0.959235	+	0.282611i	$0.908799\pi$
$770$	−24.0104	−0.865275
$771$	0	0
$772$	40.4755	1.45674
$773$	−1.34088	−0.0482281	−0.0241140	−	0.999709i	$-0.507676\pi$
$773$	−1.34088	−0.0482281	−0.0241140	+	0.999709i	$0.507676\pi$
$774$	0	0
$775$	43.8163	1.57393
$776$	−2.62389	−0.0941921
$777$	0	0
$778$	−25.2275	−0.904450
$779$	−3.20719	−0.114909
$780$	0	0
$781$	−9.73609	−0.348385
$782$	−26.7902	−0.958015
$783$	0	0
$784$	−18.8186	−0.672094
$785$	−80.4344	−2.87083
$786$	0	0
$787$	−12.1238	−0.432165	−0.216083	−	0.976375i	$-0.569328\pi$
$787$	−12.1238	−0.432165	−0.216083	+	0.976375i	$0.569328\pi$
$788$	−17.7158	−0.631098
$789$	0	0
$790$	55.5833	1.97756
$791$	−41.5350	−1.47681
$792$	0	0
$793$	1.35506	0.0481196
$794$	−53.0131	−1.88136
$795$	0	0
$796$	−5.20704	−0.184558
$797$	18.7147	0.662908	0.331454	−	0.943471i	$-0.392461\pi$
$797$	18.7147	0.662908	0.331454	+	0.943471i	$0.392461\pi$
$798$	0	0
$799$	−24.0832	−0.852003
$800$	53.4377	1.88931
$801$	0	0
$802$	−10.9498	−0.386652
$803$	−4.46857	−0.157692
$804$	0	0
$805$	−35.0465	−1.23523
$806$	−18.2643	−0.643331
$807$	0	0
$808$	−2.55937	−0.0900382
$809$	−6.37145	−0.224008	−0.112004	−	0.993708i	$-0.535727\pi$
$809$	−6.37145	−0.224008	−0.112004	+	0.993708i	$0.535727\pi$
$810$	0	0
$811$	5.77326	0.202727	0.101363	−	0.994849i	$-0.467680\pi$
$811$	5.77326	0.202727	0.101363	+	0.994849i	$0.467680\pi$
$812$	53.7113	1.88490
$813$	0	0
$814$	−19.5411	−0.684916
$815$	−40.6418	−1.42362
$816$	0	0
$817$	−20.1792	−0.705981
$818$	41.1959	1.44038
$819$	0	0
$820$	−7.69927	−0.268870
$821$	1.12302	0.0391936	0.0195968	−	0.999808i	$-0.493762\pi$
$821$	1.12302	0.0391936	0.0195968	+	0.999808i	$0.493762\pi$
$822$	0	0
$823$	53.9692	1.88125	0.940625	−	0.339448i	$-0.110240\pi$
$823$	53.9692	1.88125	0.940625	+	0.339448i	$0.110240\pi$
$824$	0.240716	0.00838576
$825$	0	0
$826$	12.2601	0.426582
$827$	−41.3547	−1.43804	−0.719022	−	0.694988i	$-0.755410\pi$
$827$	−41.3547	−1.43804	−0.719022	+	0.694988i	$0.755410\pi$
$828$	0	0
$829$	−15.5509	−0.540105	−0.270052	−	0.962846i	$-0.587041\pi$
$829$	−15.5509	−0.540105	−0.270052	+	0.962846i	$0.587041\pi$
$830$	−62.1116	−2.15592
$831$	0	0
$832$	−12.1598	−0.421566
$833$	−22.4295	−0.777138
$834$	0	0
$835$	−34.6107	−1.19775
$836$	−6.41292	−0.221795
$837$	0	0
$838$	−41.6182	−1.43768
$839$	−54.7440	−1.88997	−0.944987	−	0.327108i	$-0.893926\pi$
$839$	−54.7440	−1.88997	−0.944987	+	0.327108i	$0.893926\pi$
$840$	0	0
$841$	24.0074	0.827842
$842$	45.3912	1.56429
$843$	0	0
$844$	4.08538	0.140625
$845$	38.0278	1.30820
$846$	0	0
$847$	−3.47017	−0.119236
$848$	26.4004	0.906592
$849$	0	0
$850$	59.6654	2.04651
$851$	−28.5230	−0.977754
$852$	0	0
$853$	33.9927	1.16389	0.581945	−	0.813228i	$-0.302292\pi$
$853$	33.9927	1.16389	0.581945	+	0.813228i	$0.302292\pi$
$854$	7.04873	0.241203
$855$	0	0
$856$	1.04159	0.0356010
$857$	−1.91897	−0.0655508	−0.0327754	−	0.999463i	$-0.510435\pi$
$857$	−1.91897	−0.0655508	−0.0327754	+	0.999463i	$0.510435\pi$
$858$	0	0
$859$	42.1057	1.43663	0.718314	−	0.695719i	$-0.244914\pi$
$859$	42.1057	1.43663	0.718314	+	0.695719i	$0.244914\pi$
$860$	−48.4428	−1.65189
$861$	0	0
$862$	37.1842	1.26650
$863$	13.3782	0.455398	0.227699	−	0.973732i	$-0.426880\pi$
$863$	13.3782	0.455398	0.227699	+	0.973732i	$0.426880\pi$
$864$	0	0
$865$	−21.1725	−0.719886
$866$	8.76101	0.297711
$867$	0	0
$868$	−48.9531	−1.66158
$869$	8.03333	0.272512
$870$	0	0
$871$	1.73047	0.0586347
$872$	0.144333	0.00488772
$873$	0	0
$874$	−18.1667	−0.614497
$875$	18.9505	0.640643
$876$	0	0
$877$	−6.51048	−0.219843	−0.109922	−	0.993940i	$-0.535060\pi$
$877$	−6.51048	−0.219843	−0.109922	+	0.993940i	$0.535060\pi$
$878$	28.8839	0.974784
$879$	0	0
$880$	12.7135	0.428573
$881$	−38.5958	−1.30033	−0.650163	−	0.759795i	$-0.725299\pi$
$881$	−38.5958	−1.30033	−0.650163	+	0.759795i	$0.725299\pi$
$882$	0	0
$883$	−44.8529	−1.50942	−0.754710	−	0.656058i	$-0.772223\pi$
$883$	−44.8529	−1.50942	−0.754710	+	0.656058i	$0.772223\pi$
$884$	−12.8149	−0.431011
$885$	0	0
$886$	17.8382	0.599286
$887$	26.1165	0.876905	0.438452	−	0.898754i	$-0.355527\pi$
$887$	26.1165	0.876905	0.438452	+	0.898754i	$0.355527\pi$
$888$	0	0
$889$	−14.1095	−0.473218
$890$	27.5919	0.924882
$891$	0	0
$892$	−56.0107	−1.87538
$893$	−16.3311	−0.546498
$894$	0	0
$895$	26.4705	0.884812
$896$	−7.08638	−0.236739
$897$	0	0
$898$	66.1933	2.20890
$899$	−48.3116	−1.61128
$900$	0	0
$901$	31.4660	1.04829
$902$	−2.15961	−0.0719071
$903$	0	0
$904$	−3.06132	−0.101818
$905$	10.7888	0.358633
$906$	0	0
$907$	42.0093	1.39490	0.697448	−	0.716636i	$-0.254319\pi$
$907$	42.0093	1.39490	0.697448	+	0.716636i	$0.254319\pi$
$908$	40.9470	1.35887
$909$	0	0
$910$	−32.5355	−1.07854
$911$	18.6597	0.618222	0.309111	−	0.951026i	$-0.399968\pi$
$911$	18.6597	0.618222	0.309111	+	0.951026i	$0.399968\pi$
$912$	0	0
$913$	−8.97685	−0.297090
$914$	57.9119	1.91555
$915$	0	0
$916$	24.6704	0.815131
$917$	−35.5856	−1.17514
$918$	0	0
$919$	31.6243	1.04319	0.521595	−	0.853193i	$-0.325337\pi$
$919$	31.6243	1.04319	0.521595	+	0.853193i	$0.325337\pi$
$920$	−2.58309	−0.0851618
$921$	0	0
$922$	84.3564	2.77813
$923$	−13.1930	−0.434253
$924$	0	0
$925$	63.5246	2.08868
$926$	32.7277	1.07550
$927$	0	0
$928$	−58.9201	−1.93415
$929$	−4.02731	−0.132132	−0.0660658	−	0.997815i	$-0.521045\pi$
$929$	−4.02731	−0.132132	−0.0660658	+	0.997815i	$0.521045\pi$
$930$	0	0
$931$	−15.2097	−0.498477
$932$	−30.5377	−1.00030
$933$	0	0
$934$	−34.4653	−1.12774
$935$	15.1530	0.495556
$936$	0	0
$937$	−0.843626	−0.0275600	−0.0137800	−	0.999905i	$-0.504386\pi$
$937$	−0.843626	−0.0275600	−0.0137800	+	0.999905i	$0.504386\pi$
$938$	9.00153	0.293910
$939$	0	0
$940$	−39.2048	−1.27872
$941$	20.9291	0.682269	0.341134	−	0.940015i	$-0.389189\pi$
$941$	20.9291	0.682269	0.341134	+	0.940015i	$0.389189\pi$
$942$	0	0
$943$	−3.15225	−0.102651
$944$	−6.49171	−0.211287
$945$	0	0
$946$	−13.5880	−0.441783
$947$	33.6444	1.09330	0.546648	−	0.837362i	$-0.315903\pi$
$947$	33.6444	1.09330	0.546648	+	0.837362i	$0.315903\pi$
$948$	0	0
$949$	−6.05518	−0.196559
$950$	40.4597	1.31269
$951$	0	0
$952$	−3.94826	−0.127964
$953$	−12.7130	−0.411814	−0.205907	−	0.978572i	$-0.566014\pi$
$953$	−12.7130	−0.411814	−0.205907	+	0.978572i	$0.566014\pi$
$954$	0	0
$955$	−39.6787	−1.28397
$956$	63.3597	2.04920
$957$	0	0
$958$	36.3310	1.17380
$959$	3.65206	0.117931
$960$	0	0
$961$	13.0318	0.420380
$962$	−26.4794	−0.853730
$963$	0	0
$964$	−30.3917	−0.978852
$965$	−64.8536	−2.08771
$966$	0	0
$967$	6.56538	0.211128	0.105564	−	0.994412i	$-0.466335\pi$
$967$	6.56538	0.211128	0.105564	+	0.994412i	$0.466335\pi$
$968$	−0.255767	−0.00822067
$969$	0	0
$970$	70.9821	2.27910
$971$	0.965814	0.0309945	0.0154972	−	0.999880i	$-0.495067\pi$
$971$	0.965814	0.0309945	0.0154972	+	0.999880i	$0.495067\pi$
$972$	0	0
$973$	27.3997	0.878393
$974$	−36.7005	−1.17596
$975$	0	0
$976$	−3.73231	−0.119468
$977$	9.87636	0.315973	0.157986	−	0.987441i	$-0.449500\pi$
$977$	9.87636	0.315973	0.157986	+	0.987441i	$0.449500\pi$
$978$	0	0
$979$	3.98779	0.127450
$980$	−36.5128	−1.16636
$981$	0	0
$982$	−13.2198	−0.421860
$983$	33.2935	1.06190	0.530949	−	0.847404i	$-0.321836\pi$
$983$	33.2935	1.06190	0.530949	+	0.847404i	$0.321836\pi$
$984$	0	0
$985$	28.3859	0.904448
$986$	−65.7868	−2.09508
$987$	0	0
$988$	−8.68989	−0.276462
$989$	−19.8335	−0.630670
$990$	0	0
$991$	−8.82610	−0.280370	−0.140185	−	0.990125i	$-0.544770\pi$
$991$	−8.82610	−0.280370	−0.140185	+	0.990125i	$0.544770\pi$
$992$	53.7005	1.70499
$993$	0	0
$994$	−68.6272	−2.17672
$995$	8.34320	0.264497
$996$	0	0
$997$	−20.8870	−0.661499	−0.330750	−	0.943719i	$-0.607302\pi$
$997$	−20.8870	−0.661499	−0.330750	+	0.943719i	$0.607302\pi$
$998$	−9.95119	−0.315000
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.m.1.5	✓	25
3.2	odd	2		6039.2.a.p.1.21	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.5	✓	25	1.1	even	1	trivial
6039.2.a.p.1.21	yes	25	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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q-2.03124 q^{2} +2.12592 q^{4} -3.40634 q^{5} -3.47017 q^{7} -0.255767 q^{8} +O(q^{10})\)	\(q-2.03124 q^{2} +2.12592 q^{4} -3.40634 q^{5} -3.47017 q^{7} -0.255767 q^{8} +6.91908 q^{10} +1.00000 q^{11} +1.35506 q^{13} +7.04873 q^{14} -3.73231 q^{16} -4.44846 q^{17} -3.01654 q^{19} -7.24160 q^{20} -2.03124 q^{22} -2.96487 q^{23} +6.60317 q^{25} -2.75245 q^{26} -7.37730 q^{28} -7.28062 q^{29} +6.63564 q^{31} +8.09274 q^{32} +9.03587 q^{34} +11.8206 q^{35} +9.62031 q^{37} +6.12731 q^{38} +0.871231 q^{40} +1.06320 q^{41} +6.68952 q^{43} +2.12592 q^{44} +6.02235 q^{46} +5.41383 q^{47} +5.04209 q^{49} -13.4126 q^{50} +2.88075 q^{52} -7.07347 q^{53} -3.40634 q^{55} +0.887557 q^{56} +14.7887 q^{58} +1.73933 q^{59} +1.00000 q^{61} -13.4786 q^{62} -8.97363 q^{64} -4.61580 q^{65} +1.27704 q^{67} -9.45706 q^{68} -24.0104 q^{70} -9.73609 q^{71} -4.46857 q^{73} -19.5411 q^{74} -6.41292 q^{76} -3.47017 q^{77} +8.03333 q^{79} +12.7135 q^{80} -2.15961 q^{82} -8.97685 q^{83} +15.1530 q^{85} -13.5880 q^{86} -0.255767 q^{88} +3.98779 q^{89} -4.70229 q^{91} -6.30306 q^{92} -10.9968 q^{94} +10.2754 q^{95} +10.2589 q^{97} -10.2417 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) \) 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8	\( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8} - 12 q^{10} + 25 q^{11} - 4 q^{13} - 14 q^{14} + 21 q^{16} - 16 q^{17} - 18 q^{19} - 28 q^{20} - 5 q^{22} - 8 q^{23} + 29 q^{25} - 16 q^{26} + 18 q^{28} - 28 q^{29} - 8 q^{31} - 35 q^{32} + 6 q^{34} - 22 q^{35} + 4 q^{37} + 4 q^{38} - 12 q^{40} - 58 q^{41} - 26 q^{43} + 25 q^{44} + 8 q^{46} - 20 q^{47} + 23 q^{49} - 27 q^{50} - 2 q^{52} - 36 q^{53} - 12 q^{55} - 70 q^{56} + 12 q^{58} - 18 q^{59} + 25 q^{61} - 42 q^{62} + 35 q^{64} - 76 q^{65} - 8 q^{67} - 28 q^{68} + 76 q^{70} - 24 q^{71} + 2 q^{73} - 40 q^{74} - 64 q^{76} - 4 q^{77} - 22 q^{79} - 36 q^{80} + 30 q^{82} - 14 q^{83} - 70 q^{86} - 15 q^{88} - 82 q^{89} - 6 q^{91} - 48 q^{92} - 16 q^{94} - 34 q^{95} + 16 q^{97} - 35 q^{98}+O(q^{100}) \) 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8 - 12 * q^10 + 25 * q^11 - 4 * q^13 - 14 * q^14 + 21 * q^16 - 16 * q^17 - 18 * q^19 - 28 * q^20 - 5 * q^22 - 8 * q^23 + 29 * q^25 - 16 * q^26 + 18 * q^28 - 28 * q^29 - 8 * q^31 - 35 * q^32 + 6 * q^34 - 22 * q^35 + 4 * q^37 + 4 * q^38 - 12 * q^40 - 58 * q^41 - 26 * q^43 + 25 * q^44 + 8 * q^46 - 20 * q^47 + 23 * q^49 - 27 * q^50 - 2 * q^52 - 36 * q^53 - 12 * q^55 - 70 * q^56 + 12 * q^58 - 18 * q^59 + 25 * q^61 - 42 * q^62 + 35 * q^64 - 76 * q^65 - 8 * q^67 - 28 * q^68 + 76 * q^70 - 24 * q^71 + 2 * q^73 - 40 * q^74 - 64 * q^76 - 4 * q^77 - 22 * q^79 - 36 * q^80 + 30 * q^82 - 14 * q^83 - 70 * q^86 - 15 * q^88 - 82 * q^89 - 6 * q^91 - 48 * q^92 - 16 * q^94 - 34 * q^95 + 16 * q^97 - 35 * q^98

Introduction

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