LMFDB - Embedded newform 4023.2.a.e.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4023,2,Mod(1,4023)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4023, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4023.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4023 = 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4023.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:	$32.1238167332$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	4023.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.16693 q^{2} +2.69559 q^{4} -1.04157 q^{5} +3.90919 q^{7} -1.50729 q^{8} +O(q^{10})$	\(q-2.16693 q^{2} +2.69559 q^{4} -1.04157 q^{5} +3.90919 q^{7} -1.50729 q^{8} +2.25701 q^{10} -4.40000 q^{11} +0.894885 q^{13} -8.47095 q^{14} -2.12498 q^{16} +0.522546 q^{17} +6.99593 q^{19} -2.80764 q^{20} +9.53450 q^{22} +0.550269 q^{23} -3.91514 q^{25} -1.93915 q^{26} +10.5376 q^{28} -2.59234 q^{29} -7.61206 q^{31} +7.61927 q^{32} -1.13232 q^{34} -4.07169 q^{35} -6.30590 q^{37} -15.1597 q^{38} +1.56995 q^{40} +1.60738 q^{41} +4.85677 q^{43} -11.8606 q^{44} -1.19240 q^{46} +5.35226 q^{47} +8.28179 q^{49} +8.48383 q^{50} +2.41224 q^{52} +5.20098 q^{53} +4.58290 q^{55} -5.89229 q^{56} +5.61742 q^{58} -14.6701 q^{59} -10.4981 q^{61} +16.4948 q^{62} -12.2605 q^{64} -0.932084 q^{65} -1.37983 q^{67} +1.40857 q^{68} +8.82307 q^{70} -14.5938 q^{71} -6.06639 q^{73} +13.6645 q^{74} +18.8582 q^{76} -17.2005 q^{77} -6.16664 q^{79} +2.21331 q^{80} -3.48309 q^{82} -0.0347730 q^{83} -0.544267 q^{85} -10.5243 q^{86} +6.63209 q^{88} +11.6153 q^{89} +3.49828 q^{91} +1.48330 q^{92} -11.5980 q^{94} -7.28674 q^{95} -6.41450 q^{97} -17.9461 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8}+O(q^{10}) $ 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8	$ 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8} - 4 q^{10} - 12 q^{11} - 10 q^{14} + 35 q^{16} - 26 q^{17} - 30 q^{20} + 8 q^{22} - 26 q^{23} + 27 q^{25} - 16 q^{26} + 4 q^{28} - 20 q^{29} - 6 q^{31} - 49 q^{32} - 14 q^{34} - 16 q^{35} - 2 q^{37} - 27 q^{38} + 2 q^{40} - 35 q^{41} + 4 q^{43} - 22 q^{44} + 6 q^{46} - 38 q^{47} + 19 q^{49} - 22 q^{50} + 4 q^{52} - 36 q^{53} + 10 q^{55} - 79 q^{56} - 22 q^{58} - 15 q^{59} + 10 q^{61} + 14 q^{62} + 41 q^{64} - 80 q^{65} - 6 q^{67} - 33 q^{68} + 8 q^{70} - 26 q^{71} - 6 q^{73} - 75 q^{74} - 10 q^{76} - 47 q^{77} - 6 q^{79} - 66 q^{80} + 12 q^{82} - 40 q^{83} - 12 q^{85} - 4 q^{86} + 12 q^{88} - 30 q^{89} - 158 q^{92} + 18 q^{94} - 22 q^{95} - 20 q^{97} - 35 q^{98}+O(q^{100}) $ 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8 - 4 * q^10 - 12 * q^11 - 10 * q^14 + 35 * q^16 - 26 * q^17 - 30 * q^20 + 8 * q^22 - 26 * q^23 + 27 * q^25 - 16 * q^26 + 4 * q^28 - 20 * q^29 - 6 * q^31 - 49 * q^32 - 14 * q^34 - 16 * q^35 - 2 * q^37 - 27 * q^38 + 2 * q^40 - 35 * q^41 + 4 * q^43 - 22 * q^44 + 6 * q^46 - 38 * q^47 + 19 * q^49 - 22 * q^50 + 4 * q^52 - 36 * q^53 + 10 * q^55 - 79 * q^56 - 22 * q^58 - 15 * q^59 + 10 * q^61 + 14 * q^62 + 41 * q^64 - 80 * q^65 - 6 * q^67 - 33 * q^68 + 8 * q^70 - 26 * q^71 - 6 * q^73 - 75 * q^74 - 10 * q^76 - 47 * q^77 - 6 * q^79 - 66 * q^80 + 12 * q^82 - 40 * q^83 - 12 * q^85 - 4 * q^86 + 12 * q^88 - 30 * q^89 - 158 * q^92 + 18 * q^94 - 22 * q^95 - 20 * 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 $

$2$	−2.16693	−1.53225	−0.766126	−	0.642691i	$-0.777818\pi$
$2$	−2.16693	−1.53225	−0.766126	+	0.642691i	$0.777818\pi$
$3$	0	0
$3$	0	0
$4$	2.69559	1.34779
$5$	−1.04157	−0.465804	−0.232902	−	0.972500i	$-0.574822\pi$
$5$	−1.04157	−0.465804	−0.232902	+	0.972500i	$0.574822\pi$
$6$	0	0
$7$	3.90919	1.47754	0.738768	−	0.673960i	$-0.235408\pi$
$7$	3.90919	1.47754	0.738768	+	0.673960i	$0.235408\pi$
$8$	−1.50729	−0.532908
$9$	0	0
$10$	2.25701	0.713728
$11$	−4.40000	−1.32665	−0.663325	−	0.748331i	$-0.730855\pi$
$11$	−4.40000	−1.32665	−0.663325	+	0.748331i	$0.730855\pi$
$12$	0	0
$13$	0.894885	0.248196	0.124098	−	0.992270i	$-0.460396\pi$
$13$	0.894885	0.248196	0.124098	+	0.992270i	$0.460396\pi$
$14$	−8.47095	−2.26396
$15$	0	0
$16$	−2.12498	−0.531245
$17$	0.522546	0.126736	0.0633680	−	0.997990i	$-0.479816\pi$
$17$	0.522546	0.126736	0.0633680	+	0.997990i	$0.479816\pi$
$18$	0	0
$19$	6.99593	1.60498	0.802489	−	0.596668i	$-0.203509\pi$
$19$	6.99593	1.60498	0.802489	+	0.596668i	$0.203509\pi$
$20$	−2.80764	−0.627807
$21$	0	0
$22$	9.53450	2.03276
$23$	0.550269	0.114739	0.0573695	−	0.998353i	$-0.481729\pi$
$23$	0.550269	0.114739	0.0573695	+	0.998353i	$0.481729\pi$
$24$	0	0
$25$	−3.91514	−0.783027
$26$	−1.93915	−0.380299
$27$	0	0
$28$	10.5376	1.99141
$29$	−2.59234	−0.481386	−0.240693	−	0.970601i	$-0.577375\pi$
$29$	−2.59234	−0.481386	−0.240693	+	0.970601i	$0.577375\pi$
$30$	0	0
$31$	−7.61206	−1.36717	−0.683583	−	0.729873i	$-0.739579\pi$
$31$	−7.61206	−1.36717	−0.683583	+	0.729873i	$0.739579\pi$
$32$	7.61927	1.34691
$33$	0	0
$34$	−1.13232	−0.194191
$35$	−4.07169	−0.688241
$36$	0	0
$37$	−6.30590	−1.03668	−0.518342	−	0.855173i	$-0.673451\pi$
$37$	−6.30590	−1.03668	−0.518342	+	0.855173i	$0.673451\pi$
$38$	−15.1597	−2.45923
$39$	0	0
$40$	1.56995	0.248230
$41$	1.60738	0.251031	0.125516	−	0.992092i	$-0.459941\pi$
$41$	1.60738	0.251031	0.125516	+	0.992092i	$0.459941\pi$
$42$	0	0
$43$	4.85677	0.740650	0.370325	−	0.928902i	$-0.379246\pi$
$43$	4.85677	0.740650	0.370325	+	0.928902i	$0.379246\pi$
$44$	−11.8606	−1.78805
$45$	0	0
$46$	−1.19240	−0.175809
$47$	5.35226	0.780707	0.390353	−	0.920665i	$-0.372353\pi$
$47$	5.35226	0.780707	0.390353	+	0.920665i	$0.372353\pi$
$48$	0	0
$49$	8.28179	1.18311
$50$	8.48383	1.19979
$51$	0	0
$52$	2.41224	0.334518
$53$	5.20098	0.714409	0.357205	−	0.934026i	$-0.383730\pi$
$53$	5.20098	0.714409	0.357205	+	0.934026i	$0.383730\pi$
$54$	0	0
$55$	4.58290	0.617959
$56$	−5.89229	−0.787391
$57$	0	0
$58$	5.61742	0.737604
$59$	−14.6701	−1.90988	−0.954940	−	0.296799i	$-0.904081\pi$
$59$	−14.6701	−1.90988	−0.954940	+	0.296799i	$0.904081\pi$
$60$	0	0
$61$	−10.4981	−1.34415	−0.672073	−	0.740485i	$-0.734596\pi$
$61$	−10.4981	−1.34415	−0.672073	+	0.740485i	$0.734596\pi$
$62$	16.4948	2.09484
$63$	0	0
$64$	−12.2605	−1.53256
$65$	−0.932084	−0.115611
$66$	0	0
$67$	−1.37983	−0.168572	−0.0842862	−	0.996442i	$-0.526861\pi$
$67$	−1.37983	−0.168572	−0.0842862	+	0.996442i	$0.526861\pi$
$68$	1.40857	0.170814
$69$	0	0
$70$	8.82307	1.05456
$71$	−14.5938	−1.73196	−0.865981	−	0.500077i	$-0.833305\pi$
$71$	−14.5938	−1.73196	−0.865981	+	0.500077i	$0.833305\pi$
$72$	0	0
$73$	−6.06639	−0.710017	−0.355008	−	0.934863i	$-0.615522\pi$
$73$	−6.06639	−0.710017	−0.355008	+	0.934863i	$0.615522\pi$
$74$	13.6645	1.58846
$75$	0	0
$76$	18.8582	2.16318
$77$	−17.2005	−1.96017
$78$	0	0
$79$	−6.16664	−0.693802	−0.346901	−	0.937902i	$-0.612766\pi$
$79$	−6.16664	−0.693802	−0.346901	+	0.937902i	$0.612766\pi$
$80$	2.21331	0.247456
$81$	0	0
$82$	−3.48309	−0.384643
$83$	−0.0347730	−0.00381683	−0.00190842	−	0.999998i	$-0.500607\pi$
$83$	−0.0347730	−0.00381683	−0.00190842	+	0.999998i	$0.500607\pi$
$84$	0	0
$85$	−0.544267	−0.0590340
$86$	−10.5243	−1.13486
$87$	0	0
$88$	6.63209	0.706983
$89$	11.6153	1.23122	0.615611	−	0.788050i	$-0.288909\pi$
$89$	11.6153	1.23122	0.615611	+	0.788050i	$0.288909\pi$
$90$	0	0
$91$	3.49828	0.366719
$92$	1.48330	0.154645
$93$	0	0
$94$	−11.5980	−1.19624
$95$	−7.28674	−0.747604
$96$	0	0
$97$	−6.41450	−0.651293	−0.325647	−	0.945492i	$-0.605582\pi$
$97$	−6.41450	−0.651293	−0.325647	+	0.945492i	$0.605582\pi$
$98$	−17.9461	−1.81283
$99$	0	0
$100$	−10.5536	−1.05536
$101$	4.52301	0.450057	0.225028	−	0.974352i	$-0.427753\pi$
$101$	4.52301	0.450057	0.225028	+	0.974352i	$0.427753\pi$
$102$	0	0
$103$	14.0379	1.38319	0.691597	−	0.722283i	$-0.256907\pi$
$103$	14.0379	1.38319	0.691597	+	0.722283i	$0.256907\pi$
$104$	−1.34885	−0.132266
$105$	0	0
$106$	−11.2702	−1.09465
$107$	−4.28189	−0.413946	−0.206973	−	0.978347i	$-0.566361\pi$
$107$	−4.28189	−0.413946	−0.206973	+	0.978347i	$0.566361\pi$
$108$	0	0
$109$	6.06151	0.580588	0.290294	−	0.956938i	$-0.406247\pi$
$109$	6.06151	0.580588	0.290294	+	0.956938i	$0.406247\pi$
$110$	−9.93083	−0.946868
$111$	0	0
$112$	−8.30696	−0.784934
$113$	3.10269	0.291877	0.145938	−	0.989294i	$-0.453380\pi$
$113$	3.10269	0.291877	0.145938	+	0.989294i	$0.453380\pi$
$114$	0	0
$115$	−0.573143	−0.0534459
$116$	−6.98789	−0.648809
$117$	0	0
$118$	31.7890	2.92642
$119$	2.04273	0.187257
$120$	0	0
$121$	8.36003	0.760002
$122$	22.7487	2.05957
$123$	0	0
$124$	−20.5190	−1.84266
$125$	9.28572	0.830540
$126$	0	0
$127$	−3.30013	−0.292839	−0.146420	−	0.989223i	$-0.546775\pi$
$127$	−3.30013	−0.292839	−0.146420	+	0.989223i	$0.546775\pi$
$128$	11.3290	1.00136
$129$	0	0
$130$	2.01976	0.177145
$131$	−22.5168	−1.96730	−0.983652	−	0.180081i	$-0.942364\pi$
$131$	−22.5168	−1.96730	−0.983652	+	0.180081i	$0.942364\pi$
$132$	0	0
$133$	27.3484	2.37141
$134$	2.98999	0.258295
$135$	0	0
$136$	−0.787628	−0.0675386
$137$	−8.93145	−0.763065	−0.381533	−	0.924355i	$-0.624604\pi$
$137$	−8.93145	−0.763065	−0.381533	+	0.924355i	$0.624604\pi$
$138$	0	0
$139$	5.53862	0.469780	0.234890	−	0.972022i	$-0.424527\pi$
$139$	5.53862	0.469780	0.234890	+	0.972022i	$0.424527\pi$
$140$	−10.9756	−0.927608
$141$	0	0
$142$	31.6237	2.65380
$143$	−3.93750	−0.329270
$144$	0	0
$145$	2.70010	0.224231
$146$	13.1454	1.08792
$147$	0	0
$148$	−16.9981	−1.39724
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−14.4454	−1.17555	−0.587773	−	0.809026i	$-0.699995\pi$
$151$	−14.4454	−1.17555	−0.587773	+	0.809026i	$0.699995\pi$
$152$	−10.5449	−0.855305
$153$	0	0
$154$	37.2722	3.00348
$155$	7.92848	0.636831
$156$	0	0
$157$	−5.82968	−0.465260	−0.232630	−	0.972565i	$-0.574733\pi$
$157$	−5.82968	−0.465260	−0.232630	+	0.972565i	$0.574733\pi$
$158$	13.3627	1.06308
$159$	0	0
$160$	−7.93599	−0.627395
$161$	2.15111	0.169531
$162$	0	0
$163$	24.3142	1.90444	0.952218	−	0.305418i	$-0.0987963\pi$
$163$	24.3142	1.90444	0.952218	+	0.305418i	$0.0987963\pi$
$164$	4.33284	0.338338
$165$	0	0
$166$	0.0753507	0.00584835
$167$	5.65206	0.437370	0.218685	−	0.975796i	$-0.429823\pi$
$167$	5.65206	0.437370	0.218685	+	0.975796i	$0.429823\pi$
$168$	0	0
$169$	−12.1992	−0.938399
$170$	1.17939	0.0904550
$171$	0	0
$172$	13.0919	0.998244
$173$	0.710082	0.0539865	0.0269933	−	0.999636i	$-0.491407\pi$
$173$	0.710082	0.0539865	0.0269933	+	0.999636i	$0.491407\pi$
$174$	0	0
$175$	−15.3050	−1.15695
$176$	9.34992	0.704777
$177$	0	0
$178$	−25.1696	−1.88654
$179$	17.9600	1.34239	0.671197	−	0.741279i	$-0.265780\pi$
$179$	17.9600	1.34239	0.671197	+	0.741279i	$0.265780\pi$
$180$	0	0
$181$	23.0568	1.71380	0.856900	−	0.515483i	$-0.172387\pi$
$181$	23.0568	1.71380	0.856900	+	0.515483i	$0.172387\pi$
$182$	−7.58052	−0.561906
$183$	0	0
$184$	−0.829416	−0.0611454
$185$	6.56803	0.482891
$186$	0	0
$187$	−2.29920	−0.168134
$188$	14.4275	1.05223
$189$	0	0
$190$	15.7899	1.14552
$191$	12.3628	0.894543	0.447271	−	0.894398i	$-0.352396\pi$
$191$	12.3628	0.894543	0.447271	+	0.894398i	$0.352396\pi$
$192$	0	0
$193$	12.6611	0.911369	0.455685	−	0.890141i	$-0.349395\pi$
$193$	12.6611	0.911369	0.455685	+	0.890141i	$0.349395\pi$
$194$	13.8998	0.997945
$195$	0	0
$196$	22.3243	1.59459
$197$	7.34898	0.523593	0.261797	−	0.965123i	$-0.415685\pi$
$197$	7.34898	0.523593	0.261797	+	0.965123i	$0.415685\pi$
$198$	0	0
$199$	−12.8165	−0.908539	−0.454270	−	0.890864i	$-0.650100\pi$
$199$	−12.8165	−0.908539	−0.454270	+	0.890864i	$0.650100\pi$
$200$	5.90125	0.417281
$201$	0	0
$202$	−9.80106	−0.689600
$203$	−10.1340	−0.711265
$204$	0	0
$205$	−1.67420	−0.116931
$206$	−30.4191	−2.11940
$207$	0	0
$208$	−1.90161	−0.131853
$209$	−30.7821	−2.12924
$210$	0	0
$211$	−14.6603	−1.00926	−0.504628	−	0.863337i	$-0.668370\pi$
$211$	−14.6603	−1.00926	−0.504628	+	0.863337i	$0.668370\pi$
$212$	14.0197	0.962876
$213$	0	0
$214$	9.27855	0.634269
$215$	−5.05866	−0.344998
$216$	0	0
$217$	−29.7570	−2.02004
$218$	−13.1349	−0.889606
$219$	0	0
$220$	12.3536	0.832881
$221$	0.467618	0.0314554
$222$	0	0
$223$	−2.68334	−0.179690	−0.0898449	−	0.995956i	$-0.528637\pi$
$223$	−2.68334	−0.179690	−0.0898449	+	0.995956i	$0.528637\pi$
$224$	29.7852	1.99011
$225$	0	0
$226$	−6.72332	−0.447229
$227$	−14.3651	−0.953447	−0.476723	−	0.879053i	$-0.658176\pi$
$227$	−14.3651	−0.953447	−0.476723	+	0.879053i	$0.658176\pi$
$228$	0	0
$229$	−7.39520	−0.488688	−0.244344	−	0.969689i	$-0.578573\pi$
$229$	−7.39520	−0.488688	−0.244344	+	0.969689i	$0.578573\pi$
$230$	1.24196	0.0818925
$231$	0	0
$232$	3.90741	0.256534
$233$	−7.14901	−0.468347	−0.234173	−	0.972195i	$-0.575238\pi$
$233$	−7.14901	−0.468347	−0.234173	+	0.972195i	$0.575238\pi$
$234$	0	0
$235$	−5.57474	−0.363656
$236$	−39.5445	−2.57413
$237$	0	0
$238$	−4.42646	−0.286925
$239$	7.17639	0.464202	0.232101	−	0.972692i	$-0.425440\pi$
$239$	7.17639	0.464202	0.232101	+	0.972692i	$0.425440\pi$
$240$	0	0
$241$	−16.4307	−1.05840	−0.529198	−	0.848499i	$-0.677507\pi$
$241$	−16.4307	−1.05840	−0.529198	+	0.848499i	$0.677507\pi$
$242$	−18.1156	−1.16451
$243$	0	0
$244$	−28.2986	−1.81163
$245$	−8.62605	−0.551098
$246$	0	0
$247$	6.26055	0.398350
$248$	11.4736	0.728574
$249$	0	0
$250$	−20.1215	−1.27260
$251$	−19.6270	−1.23884	−0.619421	−	0.785059i	$-0.712633\pi$
$251$	−19.6270	−1.23884	−0.619421	+	0.785059i	$0.712633\pi$
$252$	0	0
$253$	−2.42119	−0.152219
$254$	7.15115	0.448703
$255$	0	0
$256$	−0.0283131	−0.00176957
$257$	−29.7981	−1.85875	−0.929377	−	0.369133i	$-0.879655\pi$
$257$	−29.7981	−1.85875	−0.929377	+	0.369133i	$0.879655\pi$
$258$	0	0
$259$	−24.6510	−1.53174
$260$	−2.51251	−0.155819
$261$	0	0
$262$	48.7924	3.01440
$263$	2.53032	0.156026	0.0780132	−	0.996952i	$-0.475142\pi$
$263$	2.53032	0.156026	0.0780132	+	0.996952i	$0.475142\pi$
$264$	0	0
$265$	−5.41717	−0.332774
$266$	−59.2622	−3.63360
$267$	0	0
$268$	−3.71944	−0.227201
$269$	25.1431	1.53300	0.766500	−	0.642244i	$-0.221996\pi$
$269$	25.1431	1.53300	0.766500	+	0.642244i	$0.221996\pi$
$270$	0	0
$271$	−12.7640	−0.775359	−0.387680	−	0.921794i	$-0.626723\pi$
$271$	−12.7640	−0.775359	−0.387680	+	0.921794i	$0.626723\pi$
$272$	−1.11040	−0.0673278
$273$	0	0
$274$	19.3538	1.16921
$275$	17.2266	1.03880
$276$	0	0
$277$	−11.7932	−0.708585	−0.354292	−	0.935135i	$-0.615278\pi$
$277$	−11.7932	−0.708585	−0.354292	+	0.935135i	$0.615278\pi$
$278$	−12.0018	−0.719821
$279$	0	0
$280$	6.13723	0.366769
$281$	−8.82307	−0.526340	−0.263170	−	0.964749i	$-0.584768\pi$
$281$	−8.82307	−0.526340	−0.263170	+	0.964749i	$0.584768\pi$
$282$	0	0
$283$	17.4334	1.03631	0.518154	−	0.855287i	$-0.326619\pi$
$283$	17.4334	1.03631	0.518154	+	0.855287i	$0.326619\pi$
$284$	−39.3388	−2.33433
$285$	0	0
$286$	8.53228	0.504524
$287$	6.28357	0.370908
$288$	0	0
$289$	−16.7269	−0.983938
$290$	−5.85093	−0.343579
$291$	0	0
$292$	−16.3525	−0.956956
$293$	29.1516	1.70306	0.851528	−	0.524308i	$-0.175676\pi$
$293$	29.1516	1.70306	0.851528	+	0.524308i	$0.175676\pi$
$294$	0	0
$295$	15.2799	0.889629
$296$	9.50484	0.552457
$297$	0	0
$298$	2.16693	0.125527
$299$	0.492428	0.0284778
$300$	0	0
$301$	18.9860	1.09434
$302$	31.3021	1.80123
$303$	0	0
$304$	−14.8662	−0.852636
$305$	10.9345	0.626108
$306$	0	0
$307$	15.9202	0.908616	0.454308	−	0.890845i	$-0.349887\pi$
$307$	15.9202	0.908616	0.454308	+	0.890845i	$0.349887\pi$
$308$	−46.3654	−2.64191
$309$	0	0
$310$	−17.1805	−0.975785
$311$	5.14780	0.291905	0.145953	−	0.989292i	$-0.453375\pi$
$311$	5.14780	0.291905	0.145953	+	0.989292i	$0.453375\pi$
$312$	0	0
$313$	−27.2791	−1.54191	−0.770954	−	0.636891i	$-0.780220\pi$
$313$	−27.2791	−1.54191	−0.770954	+	0.636891i	$0.780220\pi$
$314$	12.6325	0.712894
$315$	0	0
$316$	−16.6227	−0.935102
$317$	4.80296	0.269761	0.134881	−	0.990862i	$-0.456935\pi$
$317$	4.80296	0.269761	0.134881	+	0.990862i	$0.456935\pi$
$318$	0	0
$319$	11.4063	0.638631
$320$	12.7701	0.713871
$321$	0	0
$322$	−4.66130	−0.259764
$323$	3.65569	0.203408
$324$	0	0
$325$	−3.50360	−0.194345
$326$	−52.6872	−2.91808
$327$	0	0
$328$	−2.42280	−0.133777
$329$	20.9230	1.15352
$330$	0	0
$331$	1.01121	0.0555810	0.0277905	−	0.999614i	$-0.491153\pi$
$331$	1.01121	0.0555810	0.0277905	+	0.999614i	$0.491153\pi$
$332$	−0.0937337	−0.00514431
$333$	0	0
$334$	−12.2476	−0.670160
$335$	1.43718	0.0785216
$336$	0	0
$337$	−4.19738	−0.228646	−0.114323	−	0.993444i	$-0.536470\pi$
$337$	−4.19738	−0.228646	−0.114323	+	0.993444i	$0.536470\pi$
$338$	26.4348	1.43786
$339$	0	0
$340$	−1.46712	−0.0795657
$341$	33.4931	1.81375
$342$	0	0
$343$	5.01075	0.270555
$344$	−7.32057	−0.394699
$345$	0	0
$346$	−1.53870	−0.0827209
$347$	−33.1031	−1.77707	−0.888534	−	0.458811i	$-0.848276\pi$
$347$	−33.1031	−1.77707	−0.888534	+	0.458811i	$0.848276\pi$
$348$	0	0
$349$	6.53808	0.349975	0.174988	−	0.984571i	$-0.444011\pi$
$349$	6.53808	0.349975	0.174988	+	0.984571i	$0.444011\pi$
$350$	33.1649	1.77274
$351$	0	0
$352$	−33.5248	−1.78688
$353$	2.29484	0.122142	0.0610710	−	0.998133i	$-0.480548\pi$
$353$	2.29484	0.122142	0.0610710	+	0.998133i	$0.480548\pi$
$354$	0	0
$355$	15.2004	0.806754
$356$	31.3101	1.65943
$357$	0	0
$358$	−38.9181	−2.05689
$359$	−23.2158	−1.22528	−0.612642	−	0.790360i	$-0.709893\pi$
$359$	−23.2158	−1.22528	−0.612642	+	0.790360i	$0.709893\pi$
$360$	0	0
$361$	29.9431	1.57595
$362$	−49.9625	−2.62597
$363$	0	0
$364$	9.42992	0.494262
$365$	6.31856	0.330728
$366$	0	0
$367$	−18.9970	−0.991636	−0.495818	−	0.868427i	$-0.665132\pi$
$367$	−18.9970	−0.991636	−0.495818	+	0.868427i	$0.665132\pi$
$368$	−1.16931	−0.0609546
$369$	0	0
$370$	−14.2325	−0.739911
$371$	20.3316	1.05557
$372$	0	0
$373$	−23.1963	−1.20106	−0.600528	−	0.799603i	$-0.705043\pi$
$373$	−23.1963	−1.20106	−0.600528	+	0.799603i	$0.705043\pi$
$374$	4.98221	0.257624
$375$	0	0
$376$	−8.06741	−0.416045
$377$	−2.31985	−0.119478
$378$	0	0
$379$	5.93900	0.305066	0.152533	−	0.988298i	$-0.451257\pi$
$379$	5.93900	0.305066	0.152533	+	0.988298i	$0.451257\pi$
$380$	−19.6421	−1.00762
$381$	0	0
$382$	−26.7894	−1.37066
$383$	−23.3659	−1.19394	−0.596971	−	0.802263i	$-0.703629\pi$
$383$	−23.3659	−1.19394	−0.596971	+	0.802263i	$0.703629\pi$
$384$	0	0
$385$	17.9155	0.913056
$386$	−27.4358	−1.39645
$387$	0	0
$388$	−17.2908	−0.877809
$389$	−13.2562	−0.672116	−0.336058	−	0.941841i	$-0.609094\pi$
$389$	−13.2562	−0.672116	−0.336058	+	0.941841i	$0.609094\pi$
$390$	0	0
$391$	0.287541	0.0145416
$392$	−12.4831	−0.630490
$393$	0	0
$394$	−15.9247	−0.802276
$395$	6.42298	0.323175
$396$	0	0
$397$	−0.940682	−0.0472115	−0.0236057	−	0.999721i	$-0.507515\pi$
$397$	−0.940682	−0.0472115	−0.0236057	+	0.999721i	$0.507515\pi$
$398$	27.7725	1.39211
$399$	0	0
$400$	8.31959	0.415979
$401$	22.0549	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$401$	22.0549	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$402$	0	0
$403$	−6.81192	−0.339326
$404$	12.1922	0.606584
$405$	0	0
$406$	21.9596	1.08984
$407$	27.7460	1.37532
$408$	0	0
$409$	−30.0975	−1.48823	−0.744113	−	0.668054i	$-0.767127\pi$
$409$	−30.0975	−1.48823	−0.744113	+	0.668054i	$0.767127\pi$
$410$	3.62787	0.179168
$411$	0	0
$412$	37.8404	1.86426
$413$	−57.3481	−2.82192
$414$	0	0
$415$	0.0362185	0.00177789
$416$	6.81837	0.334298
$417$	0	0
$418$	66.7027	3.26254
$419$	−1.12681	−0.0550483	−0.0275242	−	0.999621i	$-0.508762\pi$
$419$	−1.12681	−0.0550483	−0.0275242	+	0.999621i	$0.508762\pi$
$420$	0	0
$421$	−20.9434	−1.02072	−0.510359	−	0.859962i	$-0.670487\pi$
$421$	−20.9434	−1.02072	−0.510359	+	0.859962i	$0.670487\pi$
$422$	31.7678	1.54643
$423$	0	0
$424$	−7.83939	−0.380714
$425$	−2.04584	−0.0992377
$426$	0	0
$427$	−41.0391	−1.98602
$428$	−11.5422	−0.557913
$429$	0	0
$430$	10.9618	0.528623
$431$	−31.1729	−1.50155	−0.750773	−	0.660560i	$-0.770319\pi$
$431$	−31.1729	−1.50155	−0.750773	+	0.660560i	$0.770319\pi$
$432$	0	0
$433$	−14.4854	−0.696126	−0.348063	−	0.937471i	$-0.613161\pi$
$433$	−14.4854	−0.696126	−0.348063	+	0.937471i	$0.613161\pi$
$434$	64.4814	3.09521
$435$	0	0
$436$	16.3393	0.782513
$437$	3.84965	0.184154
$438$	0	0
$439$	−28.0226	−1.33744	−0.668722	−	0.743512i	$-0.733158\pi$
$439$	−28.0226	−1.33744	−0.668722	+	0.743512i	$0.733158\pi$
$440$	−6.90777	−0.329315
$441$	0	0
$442$	−1.01330	−0.0481976
$443$	7.24134	0.344047	0.172023	−	0.985093i	$-0.444970\pi$
$443$	7.24134	0.344047	0.172023	+	0.985093i	$0.444970\pi$
$444$	0	0
$445$	−12.0982	−0.573508
$446$	5.81461	0.275330
$447$	0	0
$448$	−47.9285	−2.26441
$449$	−15.9677	−0.753564	−0.376782	−	0.926302i	$-0.622969\pi$
$449$	−15.9677	−0.753564	−0.376782	+	0.926302i	$0.622969\pi$
$450$	0	0
$451$	−7.07249	−0.333031
$452$	8.36359	0.393390
$453$	0	0
$454$	31.1282	1.46092
$455$	−3.64369	−0.170819
$456$	0	0
$457$	37.9449	1.77499	0.887494	−	0.460820i	$-0.152445\pi$
$457$	37.9449	1.77499	0.887494	+	0.460820i	$0.152445\pi$
$458$	16.0249	0.748794
$459$	0	0
$460$	−1.54496	−0.0720340
$461$	−28.5037	−1.32755	−0.663776	−	0.747932i	$-0.731047\pi$
$461$	−28.5037	−1.32755	−0.663776	+	0.747932i	$0.731047\pi$
$462$	0	0
$463$	−27.5846	−1.28196	−0.640982	−	0.767556i	$-0.721473\pi$
$463$	−27.5846	−1.28196	−0.640982	+	0.767556i	$0.721473\pi$
$464$	5.50868	0.255734
$465$	0	0
$466$	15.4914	0.717625
$467$	6.54931	0.303066	0.151533	−	0.988452i	$-0.451579\pi$
$467$	6.54931	0.303066	0.151533	+	0.988452i	$0.451579\pi$
$468$	0	0
$469$	−5.39400	−0.249072
$470$	12.0801	0.557212
$471$	0	0
$472$	22.1121	1.01779
$473$	−21.3698	−0.982584
$474$	0	0
$475$	−27.3900	−1.25674
$476$	5.50636	0.252384
$477$	0	0
$478$	−15.5507	−0.711274
$479$	−14.2398	−0.650634	−0.325317	−	0.945605i	$-0.605471\pi$
$479$	−14.2398	−0.650634	−0.325317	+	0.945605i	$0.605471\pi$
$480$	0	0
$481$	−5.64306	−0.257301
$482$	35.6042	1.62173
$483$	0	0
$484$	22.5352	1.02433
$485$	6.68114	0.303375
$486$	0	0
$487$	−28.4567	−1.28950	−0.644748	−	0.764395i	$-0.723038\pi$
$487$	−28.4567	−1.28950	−0.644748	+	0.764395i	$0.723038\pi$
$488$	15.8237	0.716306
$489$	0	0
$490$	18.6920	0.844421
$491$	−4.56080	−0.205826	−0.102913	−	0.994690i	$-0.532816\pi$
$491$	−4.56080	−0.205826	−0.102913	+	0.994690i	$0.532816\pi$
$492$	0	0
$493$	−1.35462	−0.0610089
$494$	−13.5662	−0.610372
$495$	0	0
$496$	16.1755	0.726300
$497$	−57.0499	−2.55904
$498$	0	0
$499$	20.8353	0.932718	0.466359	−	0.884596i	$-0.345566\pi$
$499$	20.8353	0.932718	0.466359	+	0.884596i	$0.345566\pi$
$500$	25.0305	1.11940
$501$	0	0
$502$	42.5302	1.89822
$503$	0.585197	0.0260927	0.0130463	−	0.999915i	$-0.495847\pi$
$503$	0.585197	0.0260927	0.0130463	+	0.999915i	$0.495847\pi$
$504$	0	0
$505$	−4.71103	−0.209638
$506$	5.24654	0.233237
$507$	0	0
$508$	−8.89579	−0.394687
$509$	19.5396	0.866077	0.433039	−	0.901375i	$-0.357441\pi$
$509$	19.5396	0.866077	0.433039	+	0.901375i	$0.357441\pi$
$510$	0	0
$511$	−23.7147	−1.04908
$512$	−22.5967	−0.998644
$513$	0	0
$514$	64.5704	2.84808
$515$	−14.6214	−0.644297
$516$	0	0
$517$	−23.5499	−1.03573
$518$	53.4170	2.34701
$519$	0	0
$520$	1.40492	0.0616099
$521$	33.7281	1.47765	0.738827	−	0.673895i	$-0.235380\pi$
$521$	33.7281	1.47765	0.738827	+	0.673895i	$0.235380\pi$
$522$	0	0
$523$	41.5821	1.81826	0.909129	−	0.416515i	$-0.136749\pi$
$523$	41.5821	1.81826	0.909129	+	0.416515i	$0.136749\pi$
$524$	−60.6961	−2.65152
$525$	0	0
$526$	−5.48303	−0.239072
$527$	−3.97765	−0.173269
$528$	0	0
$529$	−22.6972	−0.986835
$530$	11.7386	0.509894
$531$	0	0
$532$	73.7202	3.19617
$533$	1.43842	0.0623050
$534$	0	0
$535$	4.45988	0.192817
$536$	2.07980	0.0898336
$537$	0	0
$538$	−54.4833	−2.34894
$539$	−36.4399	−1.56958
$540$	0	0
$541$	16.3236	0.701805	0.350903	−	0.936412i	$-0.385875\pi$
$541$	16.3236	0.701805	0.350903	+	0.936412i	$0.385875\pi$
$542$	27.6588	1.18805
$543$	0	0
$544$	3.98141	0.170702
$545$	−6.31348	−0.270440
$546$	0	0
$547$	5.63939	0.241123	0.120562	−	0.992706i	$-0.461531\pi$
$547$	5.63939	0.241123	0.120562	+	0.992706i	$0.461531\pi$
$548$	−24.0755	−1.02845
$549$	0	0
$550$	−37.3289	−1.59171
$551$	−18.1358	−0.772613
$552$	0	0
$553$	−24.1066	−1.02512
$554$	25.5551	1.08573
$555$	0	0
$556$	14.9298	0.633166
$557$	−7.09526	−0.300636	−0.150318	−	0.988638i	$-0.548030\pi$
$557$	−7.09526	−0.300636	−0.150318	+	0.988638i	$0.548030\pi$
$558$	0	0
$559$	4.34625	0.183827
$560$	8.65226	0.365625
$561$	0	0
$562$	19.1190	0.806485
$563$	23.6261	0.995719	0.497860	−	0.867258i	$-0.334119\pi$
$563$	23.6261	0.995719	0.497860	+	0.867258i	$0.334119\pi$
$564$	0	0
$565$	−3.23167	−0.135957
$566$	−37.7770	−1.58788
$567$	0	0
$568$	21.9971	0.922976
$569$	14.4892	0.607419	0.303710	−	0.952765i	$-0.401775\pi$
$569$	14.4892	0.607419	0.303710	+	0.952765i	$0.401775\pi$
$570$	0	0
$571$	−26.7666	−1.12015	−0.560074	−	0.828443i	$-0.689227\pi$
$571$	−26.7666	−1.12015	−0.560074	+	0.828443i	$0.689227\pi$
$572$	−10.6139	−0.443788
$573$	0	0
$574$	−13.6161	−0.568324
$575$	−2.15438	−0.0898438
$576$	0	0
$577$	−4.44997	−0.185255	−0.0926273	−	0.995701i	$-0.529527\pi$
$577$	−4.44997	−0.185255	−0.0926273	+	0.995701i	$0.529527\pi$
$578$	36.2461	1.50764
$579$	0	0
$580$	7.27836	0.302217
$581$	−0.135934	−0.00563951
$582$	0	0
$583$	−22.8843	−0.947771
$584$	9.14381	0.378374
$585$	0	0
$586$	−63.1696	−2.60951
$587$	−43.0988	−1.77888	−0.889440	−	0.457052i	$-0.848905\pi$
$587$	−43.0988	−1.77888	−0.889440	+	0.457052i	$0.848905\pi$
$588$	0	0
$589$	−53.2535	−2.19427
$590$	−33.1104	−1.36314
$591$	0	0
$592$	13.3999	0.550733
$593$	−26.2454	−1.07777	−0.538884	−	0.842380i	$-0.681154\pi$
$593$	−26.2454	−1.07777	−0.538884	+	0.842380i	$0.681154\pi$
$594$	0	0
$595$	−2.12764	−0.0872249
$596$	−2.69559	−0.110416
$597$	0	0
$598$	−1.06706	−0.0436352
$599$	12.6767	0.517957	0.258979	−	0.965883i	$-0.416614\pi$
$599$	12.6767	0.517957	0.258979	+	0.965883i	$0.416614\pi$
$600$	0	0
$601$	−47.8133	−1.95035	−0.975173	−	0.221443i	$-0.928923\pi$
$601$	−47.8133	−1.95035	−0.975173	+	0.221443i	$0.928923\pi$
$602$	−41.1414	−1.67680
$603$	0	0
$604$	−38.9387	−1.58439
$605$	−8.70754	−0.354012
$606$	0	0
$607$	−6.46055	−0.262226	−0.131113	−	0.991367i	$-0.541855\pi$
$607$	−6.46055	−0.262226	−0.131113	+	0.991367i	$0.541855\pi$
$608$	53.3039	2.16176
$609$	0	0
$610$	−23.6943	−0.959354
$611$	4.78965	0.193769
$612$	0	0
$613$	−15.7535	−0.636279	−0.318140	−	0.948044i	$-0.603058\pi$
$613$	−15.7535	−0.636279	−0.318140	+	0.948044i	$0.603058\pi$
$614$	−34.4981	−1.39223
$615$	0	0
$616$	25.9261	1.04459
$617$	14.6509	0.589823	0.294912	−	0.955525i	$-0.404710\pi$
$617$	14.6509	0.589823	0.294912	+	0.955525i	$0.404710\pi$
$618$	0	0
$619$	34.6842	1.39408	0.697038	−	0.717034i	$-0.254501\pi$
$619$	34.6842	1.39408	0.697038	+	0.717034i	$0.254501\pi$
$620$	21.3719	0.858317
$621$	0	0
$622$	−11.1549	−0.447272
$623$	45.4065	1.81917
$624$	0	0
$625$	9.90396	0.396159
$626$	59.1120	2.36259
$627$	0	0
$628$	−15.7144	−0.627074
$629$	−3.29512	−0.131385
$630$	0	0
$631$	−18.0133	−0.717099	−0.358550	−	0.933511i	$-0.616729\pi$
$631$	−18.0133	−0.717099	−0.358550	+	0.933511i	$0.616729\pi$
$632$	9.29493	0.369732
$633$	0	0
$634$	−10.4077	−0.413342
$635$	3.43731	0.136405
$636$	0	0
$637$	7.41125	0.293644
$638$	−24.7167	−0.978543
$639$	0	0
$640$	−11.8000	−0.466435
$641$	20.5414	0.811336	0.405668	−	0.914020i	$-0.367039\pi$
$641$	20.5414	0.811336	0.405668	+	0.914020i	$0.367039\pi$
$642$	0	0
$643$	−17.7792	−0.701144	−0.350572	−	0.936536i	$-0.614013\pi$
$643$	−17.7792	−0.701144	−0.350572	+	0.936536i	$0.614013\pi$
$644$	5.79850	0.228493
$645$	0	0
$646$	−7.92163	−0.311673
$647$	−5.81065	−0.228440	−0.114220	−	0.993455i	$-0.536437\pi$
$647$	−5.81065	−0.228440	−0.114220	+	0.993455i	$0.536437\pi$
$648$	0	0
$649$	64.5483	2.53374
$650$	7.59205	0.297785
$651$	0	0
$652$	65.5411	2.56679
$653$	−45.7938	−1.79205	−0.896026	−	0.444003i	$-0.853558\pi$
$653$	−45.7938	−1.79205	−0.896026	+	0.444003i	$0.853558\pi$
$654$	0	0
$655$	23.4528	0.916377
$656$	−3.41566	−0.133359
$657$	0	0
$658$	−45.3387	−1.76749
$659$	−26.2093	−1.02097	−0.510485	−	0.859886i	$-0.670534\pi$
$659$	−26.2093	−1.02097	−0.510485	+	0.859886i	$0.670534\pi$
$660$	0	0
$661$	−18.3919	−0.715360	−0.357680	−	0.933844i	$-0.616432\pi$
$661$	−18.3919	−0.715360	−0.357680	+	0.933844i	$0.616432\pi$
$662$	−2.19122	−0.0851640
$663$	0	0
$664$	0.0524131	0.00203402
$665$	−28.4853	−1.10461
$666$	0	0
$667$	−1.42649	−0.0552338
$668$	15.2356	0.589484
$669$	0	0
$670$	−3.11427	−0.120315
$671$	46.1917	1.78321
$672$	0	0
$673$	20.8082	0.802095	0.401048	−	0.916057i	$-0.368646\pi$
$673$	20.8082	0.802095	0.401048	+	0.916057i	$0.368646\pi$
$674$	9.09543	0.350343
$675$	0	0
$676$	−32.8840	−1.26477
$677$	−4.30973	−0.165636	−0.0828182	−	0.996565i	$-0.526392\pi$
$677$	−4.30973	−0.165636	−0.0828182	+	0.996565i	$0.526392\pi$
$678$	0	0
$679$	−25.0755	−0.962309
$680$	0.820369	0.0314597
$681$	0	0
$682$	−72.5772	−2.77912
$683$	10.4576	0.400149	0.200074	−	0.979781i	$-0.435882\pi$
$683$	10.4576	0.400149	0.200074	+	0.979781i	$0.435882\pi$
$684$	0	0
$685$	9.30271	0.355438
$686$	−10.8580	−0.414559
$687$	0	0
$688$	−10.3205	−0.393467
$689$	4.65428	0.177314
$690$	0	0
$691$	21.4319	0.815310	0.407655	−	0.913136i	$-0.366347\pi$
$691$	21.4319	0.815310	0.407655	+	0.913136i	$0.366347\pi$
$692$	1.91409	0.0727627
$693$	0	0
$694$	71.7321	2.72291
$695$	−5.76885	−0.218825
$696$	0	0
$697$	0.839931	0.0318147
$698$	−14.1676	−0.536250
$699$	0	0
$700$	−41.2560	−1.55933
$701$	14.0193	0.529501	0.264751	−	0.964317i	$-0.414710\pi$
$701$	14.0193	0.529501	0.264751	+	0.964317i	$0.414710\pi$
$702$	0	0
$703$	−44.1157	−1.66385
$704$	53.9461	2.03317
$705$	0	0
$706$	−4.97276	−0.187152
$707$	17.6813	0.664975
$708$	0	0
$709$	14.7062	0.552303	0.276152	−	0.961114i	$-0.410941\pi$
$709$	14.7062	0.552303	0.276152	+	0.961114i	$0.410941\pi$
$710$	−32.9382	−1.23615
$711$	0	0
$712$	−17.5077	−0.656128
$713$	−4.18868	−0.156867
$714$	0	0
$715$	4.10117	0.153375
$716$	48.4128	1.80927
$717$	0	0
$718$	50.3071	1.87744
$719$	33.4489	1.24743	0.623717	−	0.781650i	$-0.285622\pi$
$719$	33.4489	1.24743	0.623717	+	0.781650i	$0.285622\pi$
$720$	0	0
$721$	54.8768	2.04372
$722$	−64.8846	−2.41475
$723$	0	0
$724$	62.1517	2.30985
$725$	10.1494	0.376938
$726$	0	0
$727$	10.5885	0.392706	0.196353	−	0.980533i	$-0.437090\pi$
$727$	10.5885	0.392706	0.196353	+	0.980533i	$0.437090\pi$
$728$	−5.27292	−0.195428
$729$	0	0
$730$	−13.6919	−0.506759
$731$	2.53788	0.0938670
$732$	0	0
$733$	−44.9577	−1.66055	−0.830275	−	0.557354i	$-0.811816\pi$
$733$	−44.9577	−1.66055	−0.830275	+	0.557354i	$0.811816\pi$
$734$	41.1652	1.51943
$735$	0	0
$736$	4.19265	0.154543
$737$	6.07123	0.223637
$738$	0	0
$739$	26.2482	0.965557	0.482778	−	0.875743i	$-0.339628\pi$
$739$	26.2482	0.965557	0.482778	+	0.875743i	$0.339628\pi$
$740$	17.7047	0.650838
$741$	0	0
$742$	−44.0572	−1.61739
$743$	−5.27632	−0.193570	−0.0967848	−	0.995305i	$-0.530856\pi$
$743$	−5.27632	−0.193570	−0.0967848	+	0.995305i	$0.530856\pi$
$744$	0	0
$745$	1.04157	0.0381601
$746$	50.2647	1.84032
$747$	0	0
$748$	−6.19770	−0.226610
$749$	−16.7387	−0.611620
$750$	0	0
$751$	1.33884	0.0488548	0.0244274	−	0.999702i	$-0.492224\pi$
$751$	1.33884	0.0488548	0.0244274	+	0.999702i	$0.492224\pi$
$752$	−11.3734	−0.414747
$753$	0	0
$754$	5.02695	0.183071
$755$	15.0458	0.547573
$756$	0	0
$757$	45.9065	1.66850	0.834250	−	0.551386i	$-0.185901\pi$
$757$	45.9065	1.66850	0.834250	+	0.551386i	$0.185901\pi$
$758$	−12.8694	−0.467437
$759$	0	0
$760$	10.9832	0.398404
$761$	33.5900	1.21764	0.608818	−	0.793310i	$-0.291644\pi$
$761$	33.5900	1.21764	0.608818	+	0.793310i	$0.291644\pi$
$762$	0	0
$763$	23.6956	0.857839
$764$	33.3251	1.20566
$765$	0	0
$766$	50.6323	1.82942
$767$	−13.1280	−0.474025
$768$	0	0
$769$	−15.8859	−0.572860	−0.286430	−	0.958101i	$-0.592469\pi$
$769$	−15.8859	−0.572860	−0.286430	+	0.958101i	$0.592469\pi$
$770$	−38.8215	−1.39903
$771$	0	0
$772$	34.1292	1.22834
$773$	−44.3575	−1.59543	−0.797715	−	0.603035i	$-0.793958\pi$
$773$	−44.3575	−1.59543	−0.797715	+	0.603035i	$0.793958\pi$
$774$	0	0
$775$	29.8022	1.07053
$776$	9.66851	0.347079
$777$	0	0
$778$	28.7253	1.02985
$779$	11.2451	0.402899
$780$	0	0
$781$	64.2126	2.29771
$782$	−0.623081	−0.0222813
$783$	0	0
$784$	−17.5986	−0.628523
$785$	6.07202	0.216720
$786$	0	0
$787$	−24.6686	−0.879341	−0.439670	−	0.898159i	$-0.644905\pi$
$787$	−24.6686	−0.879341	−0.439670	+	0.898159i	$0.644905\pi$
$788$	19.8098	0.705696
$789$	0	0
$790$	−13.9182	−0.495186
$791$	12.1290	0.431259
$792$	0	0
$793$	−9.39460	−0.333612
$794$	2.03839	0.0723399
$795$	0	0
$796$	−34.5481	−1.22452
$797$	−27.8639	−0.986990	−0.493495	−	0.869749i	$-0.664281\pi$
$797$	−27.8639	−0.986990	−0.493495	+	0.869749i	$0.664281\pi$
$798$	0	0
$799$	2.79680	0.0989436
$800$	−29.8305	−1.05467
$801$	0	0
$802$	−47.7914	−1.68757
$803$	26.6921	0.941944
$804$	0	0
$805$	−2.24053	−0.0789682
$806$	14.7610	0.519932
$807$	0	0
$808$	−6.81750	−0.239839
$809$	−22.3244	−0.784885	−0.392442	−	0.919777i	$-0.628370\pi$
$809$	−22.3244	−0.784885	−0.392442	+	0.919777i	$0.628370\pi$
$810$	0	0
$811$	−12.0587	−0.423439	−0.211719	−	0.977331i	$-0.567906\pi$
$811$	−12.0587	−0.423439	−0.211719	+	0.977331i	$0.567906\pi$
$812$	−27.3170	−0.958638
$813$	0	0
$814$	−60.1237	−2.10733
$815$	−25.3249	−0.887093
$816$	0	0
$817$	33.9776	1.18873
$818$	65.2192	2.28033
$819$	0	0
$820$	−4.51295	−0.157599
$821$	33.7012	1.17618	0.588090	−	0.808795i	$-0.299880\pi$
$821$	33.7012	1.17618	0.588090	+	0.808795i	$0.299880\pi$
$822$	0	0
$823$	12.4094	0.432566	0.216283	−	0.976331i	$-0.430607\pi$
$823$	12.4094	0.432566	0.216283	+	0.976331i	$0.430607\pi$
$824$	−21.1592	−0.737116
$825$	0	0
$826$	124.269	4.32389
$827$	−11.4120	−0.396836	−0.198418	−	0.980118i	$-0.563580\pi$
$827$	−11.4120	−0.396836	−0.198418	+	0.980118i	$0.563580\pi$
$828$	0	0
$829$	−24.6469	−0.856024	−0.428012	−	0.903773i	$-0.640786\pi$
$829$	−24.6469	−0.856024	−0.428012	+	0.903773i	$0.640786\pi$
$830$	−0.0784829	−0.00272418
$831$	0	0
$832$	−10.9717	−0.380375
$833$	4.32761	0.149943
$834$	0	0
$835$	−5.88701	−0.203728
$836$	−82.9759	−2.86978
$837$	0	0
$838$	2.44172	0.0843478
$839$	49.4638	1.70768	0.853841	−	0.520534i	$-0.174267\pi$
$839$	49.4638	1.70768	0.853841	+	0.520534i	$0.174267\pi$
$840$	0	0
$841$	−22.2798	−0.768268
$842$	45.3828	1.56400
$843$	0	0
$844$	−39.5181	−1.36027
$845$	12.7063	0.437109
$846$	0	0
$847$	32.6809	1.12293
$848$	−11.0520	−0.379526
$849$	0	0
$850$	4.43319	0.152057
$851$	−3.46995	−0.118948
$852$	0	0
$853$	−14.4674	−0.495354	−0.247677	−	0.968843i	$-0.579667\pi$
$853$	−14.4674	−0.495354	−0.247677	+	0.968843i	$0.579667\pi$
$854$	88.9290	3.04309
$855$	0	0
$856$	6.45405	0.220595
$857$	52.3900	1.78961	0.894805	−	0.446458i	$-0.147315\pi$
$857$	52.3900	1.78961	0.894805	+	0.446458i	$0.147315\pi$
$858$	0	0
$859$	7.34885	0.250740	0.125370	−	0.992110i	$-0.459988\pi$
$859$	7.34885	0.250740	0.125370	+	0.992110i	$0.459988\pi$
$860$	−13.6361	−0.464986
$861$	0	0
$862$	67.5495	2.30075
$863$	40.1142	1.36550	0.682752	−	0.730651i	$-0.260783\pi$
$863$	40.1142	1.36550	0.682752	+	0.730651i	$0.260783\pi$
$864$	0	0
$865$	−0.739598	−0.0251471
$866$	31.3890	1.06664
$867$	0	0
$868$	−80.2126	−2.72259
$869$	27.1333	0.920432
$870$	0	0
$871$	−1.23478	−0.0418391
$872$	−9.13647	−0.309400
$873$	0	0
$874$	−8.34192	−0.282170
$875$	36.2997	1.22715
$876$	0	0
$877$	45.9947	1.55313	0.776565	−	0.630038i	$-0.216961\pi$
$877$	45.9947	1.55313	0.776565	+	0.630038i	$0.216961\pi$
$878$	60.7230	2.04930
$879$	0	0
$880$	−9.73858	−0.328287
$881$	−0.747769	−0.0251930	−0.0125965	−	0.999921i	$-0.504010\pi$
$881$	−0.747769	−0.0251930	−0.0125965	+	0.999921i	$0.504010\pi$
$882$	0	0
$883$	27.5270	0.926357	0.463179	−	0.886265i	$-0.346709\pi$
$883$	27.5270	0.926357	0.463179	+	0.886265i	$0.346709\pi$
$884$	1.26051	0.0423954
$885$	0	0
$886$	−15.6915	−0.527166
$887$	−23.3996	−0.785681	−0.392841	−	0.919607i	$-0.628508\pi$
$887$	−23.3996	−0.785681	−0.392841	+	0.919607i	$0.628508\pi$
$888$	0	0
$889$	−12.9008	−0.432680
$890$	26.2159	0.878758
$891$	0	0
$892$	−7.23318	−0.242185
$893$	37.4440	1.25302
$894$	0	0
$895$	−18.7066	−0.625292
$896$	44.2874	1.47954
$897$	0	0
$898$	34.6010	1.15465
$899$	19.7331	0.658134
$900$	0	0
$901$	2.71775	0.0905413
$902$	15.3256	0.510287
$903$	0	0
$904$	−4.67667	−0.155544
$905$	−24.0152	−0.798294
$906$	0	0
$907$	−28.2183	−0.936975	−0.468487	−	0.883470i	$-0.655201\pi$
$907$	−28.2183	−0.936975	−0.468487	+	0.883470i	$0.655201\pi$
$908$	−38.7225	−1.28505
$909$	0	0
$910$	7.89563	0.261738
$911$	3.89740	0.129127	0.0645634	−	0.997914i	$-0.479435\pi$
$911$	3.89740	0.129127	0.0645634	+	0.997914i	$0.479435\pi$
$912$	0	0
$913$	0.153001	0.00506361
$914$	−82.2239	−2.71973
$915$	0	0
$916$	−19.9344	−0.658651
$917$	−88.0226	−2.90676
$918$	0	0
$919$	28.3546	0.935332	0.467666	−	0.883905i	$-0.345095\pi$
$919$	28.3546	0.935332	0.467666	+	0.883905i	$0.345095\pi$
$920$	0.863894	0.0284817
$921$	0	0
$922$	61.7656	2.03414
$923$	−13.0597	−0.429867
$924$	0	0
$925$	24.6885	0.811752
$926$	59.7739	1.96429
$927$	0	0
$928$	−19.7517	−0.648383
$929$	21.0993	0.692245	0.346122	−	0.938189i	$-0.387498\pi$
$929$	21.0993	0.692245	0.346122	+	0.938189i	$0.387498\pi$
$930$	0	0
$931$	57.9388	1.89887
$932$	−19.2708	−0.631235
$933$	0	0
$934$	−14.1919	−0.464373
$935$	2.39478	0.0783175
$936$	0	0
$937$	−45.4404	−1.48447	−0.742237	−	0.670137i	$-0.766235\pi$
$937$	−45.4404	−1.48447	−0.742237	+	0.670137i	$0.766235\pi$
$938$	11.6884	0.381641
$939$	0	0
$940$	−15.0272	−0.490133
$941$	−39.4685	−1.28664	−0.643318	−	0.765599i	$-0.722443\pi$
$941$	−39.4685	−1.28664	−0.643318	+	0.765599i	$0.722443\pi$
$942$	0	0
$943$	0.884494	0.0288031
$944$	31.1736	1.01461
$945$	0	0
$946$	46.3069	1.50557
$947$	−13.6395	−0.443223	−0.221612	−	0.975135i	$-0.571132\pi$
$947$	−13.6395	−0.443223	−0.221612	+	0.975135i	$0.571132\pi$
$948$	0	0
$949$	−5.42872	−0.176224
$950$	59.3523	1.92564
$951$	0	0
$952$	−3.07899	−0.0997907
$953$	−55.9418	−1.81213	−0.906066	−	0.423136i	$-0.860929\pi$
$953$	−55.9418	−1.81213	−0.906066	+	0.423136i	$0.860929\pi$
$954$	0	0
$955$	−12.8767	−0.416681
$956$	19.3446	0.625649
$957$	0	0
$958$	30.8567	0.996935
$959$	−34.9148	−1.12746
$960$	0	0
$961$	26.9435	0.869144
$962$	12.2281	0.394250
$963$	0	0
$964$	−44.2904	−1.42650
$965$	−13.1874	−0.424519
$966$	0	0
$967$	54.6707	1.75809	0.879045	−	0.476739i	$-0.158181\pi$
$967$	54.6707	1.75809	0.879045	+	0.476739i	$0.158181\pi$
$968$	−12.6010	−0.405011
$969$	0	0
$970$	−14.4776	−0.464846
$971$	−49.5192	−1.58915	−0.794573	−	0.607169i	$-0.792305\pi$
$971$	−49.5192	−1.58915	−0.794573	+	0.607169i	$0.792305\pi$
$972$	0	0
$973$	21.6515	0.694117
$974$	61.6637	1.97583
$975$	0	0
$976$	22.3083	0.714071
$977$	−7.53020	−0.240912	−0.120456	−	0.992719i	$-0.538436\pi$
$977$	−7.53020	−0.240912	−0.120456	+	0.992719i	$0.538436\pi$
$978$	0	0
$979$	−51.1075	−1.63340
$980$	−23.2523	−0.742767
$981$	0	0
$982$	9.88293	0.315377
$983$	−13.6448	−0.435201	−0.217600	−	0.976038i	$-0.569823\pi$
$983$	−13.6448	−0.435201	−0.217600	+	0.976038i	$0.569823\pi$
$984$	0	0
$985$	−7.65446	−0.243892
$986$	2.93536	0.0934809
$987$	0	0
$988$	16.8759	0.536893
$989$	2.67253	0.0849815
$990$	0	0
$991$	19.1923	0.609663	0.304832	−	0.952406i	$-0.401400\pi$
$991$	19.1923	0.609663	0.304832	+	0.952406i	$0.401400\pi$
$992$	−57.9983	−1.84145
$993$	0	0
$994$	123.623	3.92109
$995$	13.3493	0.423201
$996$	0	0
$997$	42.2777	1.33895	0.669473	−	0.742836i	$-0.266520\pi$
$997$	42.2777	1.33895	0.669473	+	0.742836i	$0.266520\pi$
$998$	−45.1487	−1.42916
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4023.2.a.e.1.6	✓	25
3.2	odd	2		4023.2.a.f.1.20	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4023.2.a.e.1.6	✓	25	1.1	even	1	trivial
4023.2.a.f.1.20	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:	\( N \)	\(=\)	\( 4023 = 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4023.a (trivial)

\(f(q)\)	\(=\)	\(q-2.16693 q^{2} +2.69559 q^{4} -1.04157 q^{5} +3.90919 q^{7} -1.50729 q^{8} +O(q^{10})\)	\(q-2.16693 q^{2} +2.69559 q^{4} -1.04157 q^{5} +3.90919 q^{7} -1.50729 q^{8} +2.25701 q^{10} -4.40000 q^{11} +0.894885 q^{13} -8.47095 q^{14} -2.12498 q^{16} +0.522546 q^{17} +6.99593 q^{19} -2.80764 q^{20} +9.53450 q^{22} +0.550269 q^{23} -3.91514 q^{25} -1.93915 q^{26} +10.5376 q^{28} -2.59234 q^{29} -7.61206 q^{31} +7.61927 q^{32} -1.13232 q^{34} -4.07169 q^{35} -6.30590 q^{37} -15.1597 q^{38} +1.56995 q^{40} +1.60738 q^{41} +4.85677 q^{43} -11.8606 q^{44} -1.19240 q^{46} +5.35226 q^{47} +8.28179 q^{49} +8.48383 q^{50} +2.41224 q^{52} +5.20098 q^{53} +4.58290 q^{55} -5.89229 q^{56} +5.61742 q^{58} -14.6701 q^{59} -10.4981 q^{61} +16.4948 q^{62} -12.2605 q^{64} -0.932084 q^{65} -1.37983 q^{67} +1.40857 q^{68} +8.82307 q^{70} -14.5938 q^{71} -6.06639 q^{73} +13.6645 q^{74} +18.8582 q^{76} -17.2005 q^{77} -6.16664 q^{79} +2.21331 q^{80} -3.48309 q^{82} -0.0347730 q^{83} -0.544267 q^{85} -10.5243 q^{86} +6.63209 q^{88} +11.6153 q^{89} +3.49828 q^{91} +1.48330 q^{92} -11.5980 q^{94} -7.28674 q^{95} -6.41450 q^{97} -17.9461 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8}+O(q^{10}) \) 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8	\( 25 q - 7 q^{2} + 27 q^{4} - 12 q^{5} - 2 q^{7} - 21 q^{8} - 4 q^{10} - 12 q^{11} - 10 q^{14} + 35 q^{16} - 26 q^{17} - 30 q^{20} + 8 q^{22} - 26 q^{23} + 27 q^{25} - 16 q^{26} + 4 q^{28} - 20 q^{29} - 6 q^{31} - 49 q^{32} - 14 q^{34} - 16 q^{35} - 2 q^{37} - 27 q^{38} + 2 q^{40} - 35 q^{41} + 4 q^{43} - 22 q^{44} + 6 q^{46} - 38 q^{47} + 19 q^{49} - 22 q^{50} + 4 q^{52} - 36 q^{53} + 10 q^{55} - 79 q^{56} - 22 q^{58} - 15 q^{59} + 10 q^{61} + 14 q^{62} + 41 q^{64} - 80 q^{65} - 6 q^{67} - 33 q^{68} + 8 q^{70} - 26 q^{71} - 6 q^{73} - 75 q^{74} - 10 q^{76} - 47 q^{77} - 6 q^{79} - 66 q^{80} + 12 q^{82} - 40 q^{83} - 12 q^{85} - 4 q^{86} + 12 q^{88} - 30 q^{89} - 158 q^{92} + 18 q^{94} - 22 q^{95} - 20 q^{97} - 35 q^{98}+O(q^{100}) \) 25 * q - 7 * q^2 + 27 * q^4 - 12 * q^5 - 2 * q^7 - 21 * q^8 - 4 * q^10 - 12 * q^11 - 10 * q^14 + 35 * q^16 - 26 * q^17 - 30 * q^20 + 8 * q^22 - 26 * q^23 + 27 * q^25 - 16 * q^26 + 4 * q^28 - 20 * q^29 - 6 * q^31 - 49 * q^32 - 14 * q^34 - 16 * q^35 - 2 * q^37 - 27 * q^38 + 2 * q^40 - 35 * q^41 + 4 * q^43 - 22 * q^44 + 6 * q^46 - 38 * q^47 + 19 * q^49 - 22 * q^50 + 4 * q^52 - 36 * q^53 + 10 * q^55 - 79 * q^56 - 22 * q^58 - 15 * q^59 + 10 * q^61 + 14 * q^62 + 41 * q^64 - 80 * q^65 - 6 * q^67 - 33 * q^68 + 8 * q^70 - 26 * q^71 - 6 * q^73 - 75 * q^74 - 10 * q^76 - 47 * q^77 - 6 * q^79 - 66 * q^80 + 12 * q^82 - 40 * q^83 - 12 * q^85 - 4 * q^86 + 12 * q^88 - 30 * q^89 - 158 * q^92 + 18 * q^94 - 22 * q^95 - 20 * q^97 - 35 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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