LMFDB - Embedded newform 531.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,2,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, 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("531.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	531.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:	$4.24005634733$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.11491 q^{2} +2.47283 q^{4} +0.357926 q^{5} +5.11491 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-2.11491 q^{2} +2.47283 q^{4} +0.357926 q^{5} +5.11491 q^{7} -1.00000 q^{8} -0.756981 q^{10} +4.58774 q^{11} -2.58774 q^{13} -10.8176 q^{14} -2.83076 q^{16} -2.18869 q^{17} +0.527166 q^{19} +0.885092 q^{20} -9.70265 q^{22} +5.70265 q^{23} -4.87189 q^{25} +5.47283 q^{26} +12.6483 q^{28} -2.06058 q^{29} +5.83076 q^{31} +7.98680 q^{32} +4.62887 q^{34} +1.83076 q^{35} -7.70265 q^{37} -1.11491 q^{38} -0.357926 q^{40} -2.39905 q^{41} +7.15604 q^{43} +11.3447 q^{44} -12.0606 q^{46} -8.77643 q^{47} +19.1623 q^{49} +10.3036 q^{50} -6.39905 q^{52} +8.10170 q^{53} +1.64207 q^{55} -5.11491 q^{56} +4.35793 q^{58} -1.00000 q^{59} +9.00624 q^{61} -12.3315 q^{62} -11.2298 q^{64} -0.926221 q^{65} +14.4791 q^{67} -5.41226 q^{68} -3.87189 q^{70} -4.12811 q^{71} -11.2361 q^{73} +16.2904 q^{74} +1.30359 q^{76} +23.4659 q^{77} -3.87189 q^{79} -1.01320 q^{80} +5.07378 q^{82} +0.737534 q^{83} -0.783389 q^{85} -15.1344 q^{86} -4.58774 q^{88} +8.54661 q^{89} -13.2361 q^{91} +14.1017 q^{92} +18.5613 q^{94} +0.188687 q^{95} -4.10170 q^{97} -40.5264 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{4} + 2 q^{5} + 9 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q + 2 * q^4 + 2 * q^5 + 9 * q^7 - 3 * q^8	$ 3 q + 2 q^{4} + 2 q^{5} + 9 q^{7} - 3 q^{8} + 5 q^{10} + 2 q^{11} + 4 q^{13} - 8 q^{14} - 4 q^{16} - 3 q^{17} + 7 q^{19} + 9 q^{20} - 11 q^{22} - q^{23} - q^{25} + 11 q^{26} + 9 q^{28} + 11 q^{29} + 13 q^{31} + 4 q^{32} - 7 q^{34} + q^{35} - 5 q^{37} + 3 q^{38} - 2 q^{40} + q^{41} + 6 q^{43} + 15 q^{44} - 19 q^{46} - 11 q^{47} + 14 q^{49} + 21 q^{50} - 11 q^{52} - 2 q^{53} + 4 q^{55} - 9 q^{56} + 14 q^{58} - 3 q^{59} - q^{61} + 2 q^{62} - 21 q^{64} + 10 q^{67} - 28 q^{68} + 2 q^{70} - 26 q^{71} + 7 q^{73} + 19 q^{74} - 6 q^{76} + 17 q^{77} + 2 q^{79} - 23 q^{80} + 18 q^{82} + 3 q^{83} - 35 q^{85} - 31 q^{86} - 2 q^{88} + 23 q^{89} + q^{91} + 16 q^{92} + 4 q^{94} - 3 q^{95} + 14 q^{97} - 51 q^{98}+O(q^{100}) $ 3 * q + 2 * q^4 + 2 * q^5 + 9 * q^7 - 3 * q^8 + 5 * q^10 + 2 * q^11 + 4 * q^13 - 8 * q^14 - 4 * q^16 - 3 * q^17 + 7 * q^19 + 9 * q^20 - 11 * q^22 - q^23 - q^25 + 11 * q^26 + 9 * q^28 + 11 * q^29 + 13 * q^31 + 4 * q^32 - 7 * q^34 + q^35 - 5 * q^37 + 3 * q^38 - 2 * q^40 + q^41 + 6 * q^43 + 15 * q^44 - 19 * q^46 - 11 * q^47 + 14 * q^49 + 21 * q^50 - 11 * q^52 - 2 * q^53 + 4 * q^55 - 9 * q^56 + 14 * q^58 - 3 * q^59 - q^61 + 2 * q^62 - 21 * q^64 + 10 * q^67 - 28 * q^68 + 2 * q^70 - 26 * q^71 + 7 * q^73 + 19 * q^74 - 6 * q^76 + 17 * q^77 + 2 * q^79 - 23 * q^80 + 18 * q^82 + 3 * q^83 - 35 * q^85 - 31 * q^86 - 2 * q^88 + 23 * q^89 + q^91 + 16 * q^92 + 4 * q^94 - 3 * q^95 + 14 * q^97 - 51 * 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 $

$2$	−2.11491	−1.49547	−0.747733	−	0.664000i	$-0.768858\pi$
$2$	−2.11491	−1.49547	−0.747733	+	0.664000i	$0.768858\pi$
$3$	0	0
$3$	0	0
$4$	2.47283	1.23642
$5$	0.357926	0.160070	0.0800348	−	0.996792i	$-0.474497\pi$
$5$	0.357926	0.160070	0.0800348	+	0.996792i	$0.474497\pi$
$6$	0	0
$7$	5.11491	1.93325	0.966627	−	0.256189i	$-0.0824670\pi$
$7$	5.11491	1.93325	0.966627	+	0.256189i	$0.0824670\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−0.756981	−0.239378
$11$	4.58774	1.38326	0.691628	−	0.722254i	$-0.256894\pi$
$11$	4.58774	1.38326	0.691628	+	0.722254i	$0.256894\pi$
$12$	0	0
$13$	−2.58774	−0.717710	−0.358855	−	0.933393i	$-0.616833\pi$
$13$	−2.58774	−0.717710	−0.358855	+	0.933393i	$0.616833\pi$
$14$	−10.8176	−2.89111
$15$	0	0
$16$	−2.83076	−0.707690
$17$	−2.18869	−0.530834	−0.265417	−	0.964134i	$-0.585510\pi$
$17$	−2.18869	−0.530834	−0.265417	+	0.964134i	$0.585510\pi$
$18$	0	0
$19$	0.527166	0.120940	0.0604701	−	0.998170i	$-0.480740\pi$
$19$	0.527166	0.120940	0.0604701	+	0.998170i	$0.480740\pi$
$20$	0.885092	0.197913
$21$	0	0
$22$	−9.70265	−2.06861
$23$	5.70265	1.18908	0.594542	−	0.804064i	$-0.297333\pi$
$23$	5.70265	1.18908	0.594542	+	0.804064i	$0.297333\pi$
$24$	0	0
$25$	−4.87189	−0.974378
$26$	5.47283	1.07331
$27$	0	0
$28$	12.6483	2.39031
$29$	−2.06058	−0.382639	−0.191320	−	0.981528i	$-0.561277\pi$
$29$	−2.06058	−0.382639	−0.191320	+	0.981528i	$0.561277\pi$
$30$	0	0
$31$	5.83076	1.04724	0.523618	−	0.851953i	$-0.324582\pi$
$31$	5.83076	1.04724	0.523618	+	0.851953i	$0.324582\pi$
$32$	7.98680	1.41188
$33$	0	0
$34$	4.62887	0.793845
$35$	1.83076	0.309455
$36$	0	0
$37$	−7.70265	−1.26631	−0.633154	−	0.774026i	$-0.718240\pi$
$37$	−7.70265	−1.26631	−0.633154	+	0.774026i	$0.718240\pi$
$38$	−1.11491	−0.180862
$39$	0	0
$40$	−0.357926	−0.0565931
$41$	−2.39905	−0.374669	−0.187335	−	0.982296i	$-0.559985\pi$
$41$	−2.39905	−0.374669	−0.187335	+	0.982296i	$0.559985\pi$
$42$	0	0
$43$	7.15604	1.09129	0.545643	−	0.838018i	$-0.316286\pi$
$43$	7.15604	1.09129	0.545643	+	0.838018i	$0.316286\pi$
$44$	11.3447	1.71028
$45$	0	0
$46$	−12.0606	−1.77823
$47$	−8.77643	−1.28017	−0.640087	−	0.768303i	$-0.721102\pi$
$47$	−8.77643	−1.28017	−0.640087	+	0.768303i	$0.721102\pi$
$48$	0	0
$49$	19.1623	2.73747
$50$	10.3036	1.45715
$51$	0	0
$52$	−6.39905	−0.887389
$53$	8.10170	1.11285	0.556427	−	0.830896i	$-0.312172\pi$
$53$	8.10170	1.11285	0.556427	+	0.830896i	$0.312172\pi$
$54$	0	0
$55$	1.64207	0.221417
$56$	−5.11491	−0.683508
$57$	0	0
$58$	4.35793	0.572224
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	9.00624	1.15313	0.576566	−	0.817051i	$-0.304393\pi$
$61$	9.00624	1.15313	0.576566	+	0.817051i	$0.304393\pi$
$62$	−12.3315	−1.56610
$63$	0	0
$64$	−11.2298	−1.40373
$65$	−0.926221	−0.114884
$66$	0	0
$67$	14.4791	1.76890	0.884450	−	0.466634i	$-0.154534\pi$
$67$	14.4791	1.76890	0.884450	+	0.466634i	$0.154534\pi$
$68$	−5.41226	−0.656333
$69$	0	0
$70$	−3.87189	−0.462779
$71$	−4.12811	−0.489917	−0.244958	−	0.969534i	$-0.578774\pi$
$71$	−4.12811	−0.489917	−0.244958	+	0.969534i	$0.578774\pi$
$72$	0	0
$73$	−11.2361	−1.31508	−0.657541	−	0.753419i	$-0.728403\pi$
$73$	−11.2361	−1.31508	−0.657541	+	0.753419i	$0.728403\pi$
$74$	16.2904	1.89372
$75$	0	0
$76$	1.30359	0.149533
$77$	23.4659	2.67418
$78$	0	0
$79$	−3.87189	−0.435622	−0.217811	−	0.975991i	$-0.569892\pi$
$79$	−3.87189	−0.435622	−0.217811	+	0.975991i	$0.569892\pi$
$80$	−1.01320	−0.113280
$81$	0	0
$82$	5.07378	0.560305
$83$	0.737534	0.0809549	0.0404775	−	0.999180i	$-0.487112\pi$
$83$	0.737534	0.0809549	0.0404775	+	0.999180i	$0.487112\pi$
$84$	0	0
$85$	−0.783389	−0.0849704
$86$	−15.1344	−1.63198
$87$	0	0
$88$	−4.58774	−0.489055
$89$	8.54661	0.905939	0.452970	−	0.891526i	$-0.350365\pi$
$89$	8.54661	0.905939	0.452970	+	0.891526i	$0.350365\pi$
$90$	0	0
$91$	−13.2361	−1.38752
$92$	14.1017	1.47020
$93$	0	0
$94$	18.5613	1.91446
$95$	0.188687	0.0193588
$96$	0	0
$97$	−4.10170	−0.416465	−0.208232	−	0.978079i	$-0.566771\pi$
$97$	−4.10170	−0.416465	−0.208232	+	0.978079i	$0.566771\pi$
$98$	−40.5264	−4.09379
$99$	0	0
$100$	−12.0474	−1.20474
$101$	8.22982	0.818897	0.409449	−	0.912333i	$-0.365721\pi$
$101$	8.22982	0.818897	0.409449	+	0.912333i	$0.365721\pi$
$102$	0	0
$103$	4.79811	0.472772	0.236386	−	0.971659i	$-0.424037\pi$
$103$	4.79811	0.472772	0.236386	+	0.971659i	$0.424037\pi$
$104$	2.58774	0.253749
$105$	0	0
$106$	−17.1344	−1.66424
$107$	−14.0411	−1.35741	−0.678704	−	0.734412i	$-0.737458\pi$
$107$	−14.0411	−1.35741	−0.678704	+	0.734412i	$0.737458\pi$
$108$	0	0
$109$	−9.81131	−0.939753	−0.469877	−	0.882732i	$-0.655702\pi$
$109$	−9.81131	−0.939753	−0.469877	+	0.882732i	$0.655702\pi$
$110$	−3.47283	−0.331122
$111$	0	0
$112$	−14.4791	−1.36814
$113$	−13.2772	−1.24901	−0.624506	−	0.781020i	$-0.714700\pi$
$113$	−13.2772	−1.24901	−0.624506	+	0.781020i	$0.714700\pi$
$114$	0	0
$115$	2.04113	0.190336
$116$	−5.09546	−0.473102
$117$	0	0
$118$	2.11491	0.194693
$119$	−11.1949	−1.02624
$120$	0	0
$121$	10.0474	0.913397
$122$	−19.0474	−1.72447
$123$	0	0
$124$	14.4185	1.29482
$125$	−3.53341	−0.316038
$126$	0	0
$127$	−0.587741	−0.0521536	−0.0260768	−	0.999660i	$-0.508301\pi$
$127$	−0.587741	−0.0521536	−0.0260768	+	0.999660i	$0.508301\pi$
$128$	7.77643	0.687346
$129$	0	0
$130$	1.95887	0.171804
$131$	8.08226	0.706150	0.353075	−	0.935595i	$-0.385136\pi$
$131$	8.08226	0.706150	0.353075	+	0.935595i	$0.385136\pi$
$132$	0	0
$133$	2.69641	0.233808
$134$	−30.6219	−2.64533
$135$	0	0
$136$	2.18869	0.187678
$137$	−8.10170	−0.692175	−0.346088	−	0.938202i	$-0.612490\pi$
$137$	−8.10170	−0.692175	−0.346088	+	0.938202i	$0.612490\pi$
$138$	0	0
$139$	4.96511	0.421136	0.210568	−	0.977579i	$-0.432469\pi$
$139$	4.96511	0.421136	0.210568	+	0.977579i	$0.432469\pi$
$140$	4.52717	0.382615
$141$	0	0
$142$	8.73057	0.732653
$143$	−11.8719	−0.992777
$144$	0	0
$145$	−0.737534	−0.0612489
$146$	23.7632	1.96666
$147$	0	0
$148$	−19.0474	−1.56568
$149$	21.9325	1.79678	0.898389	−	0.439201i	$-0.144738\pi$
$149$	21.9325	1.79678	0.898389	+	0.439201i	$0.144738\pi$
$150$	0	0
$151$	1.49228	0.121440	0.0607200	−	0.998155i	$-0.480660\pi$
$151$	1.49228	0.121440	0.0607200	+	0.998155i	$0.480660\pi$
$152$	−0.527166	−0.0427588
$153$	0	0
$154$	−49.6282	−3.99915
$155$	2.08698	0.167630
$156$	0	0
$157$	−18.7089	−1.49313	−0.746566	−	0.665311i	$-0.768299\pi$
$157$	−18.7089	−1.49313	−0.746566	+	0.665311i	$0.768299\pi$
$158$	8.18869	0.651457
$159$	0	0
$160$	2.85868	0.225999
$161$	29.1685	2.29880
$162$	0	0
$163$	3.34472	0.261979	0.130989	−	0.991384i	$-0.458185\pi$
$163$	3.34472	0.261979	0.130989	+	0.991384i	$0.458185\pi$
$164$	−5.93246	−0.463248
$165$	0	0
$166$	−1.55982	−0.121065
$167$	−16.3991	−1.26900	−0.634498	−	0.772924i	$-0.718793\pi$
$167$	−16.3991	−1.26900	−0.634498	+	0.772924i	$0.718793\pi$
$168$	0	0
$169$	−6.30359	−0.484892
$170$	1.65679	0.127070
$171$	0	0
$172$	17.6957	1.34928
$173$	−25.2097	−1.91665	−0.958327	−	0.285673i	$-0.907783\pi$
$173$	−25.2097	−1.91665	−0.958327	+	0.285673i	$0.907783\pi$
$174$	0	0
$175$	−24.9193	−1.88372
$176$	−12.9868	−0.978917
$177$	0	0
$178$	−18.0753	−1.35480
$179$	15.0738	1.12667	0.563334	−	0.826230i	$-0.309519\pi$
$179$	15.0738	1.12667	0.563334	+	0.826230i	$0.309519\pi$
$180$	0	0
$181$	14.7764	1.09832	0.549162	−	0.835716i	$-0.314947\pi$
$181$	14.7764	1.09832	0.549162	+	0.835716i	$0.314947\pi$
$182$	27.9930	2.07498
$183$	0	0
$184$	−5.70265	−0.420405
$185$	−2.75698	−0.202697
$186$	0	0
$187$	−10.0411	−0.734280
$188$	−21.7026	−1.58283
$189$	0	0
$190$	−0.399055	−0.0289505
$191$	19.9930	1.44665	0.723323	−	0.690510i	$-0.242614\pi$
$191$	19.9930	1.44665	0.723323	+	0.690510i	$0.242614\pi$
$192$	0	0
$193$	5.28415	0.380361	0.190181	−	0.981749i	$-0.439093\pi$
$193$	5.28415	0.380361	0.190181	+	0.981749i	$0.439093\pi$
$194$	8.67472	0.622809
$195$	0	0
$196$	47.3851	3.38465
$197$	−6.12115	−0.436114	−0.218057	−	0.975936i	$-0.569972\pi$
$197$	−6.12115	−0.436114	−0.218057	+	0.975936i	$0.569972\pi$
$198$	0	0
$199$	−18.7500	−1.32915	−0.664577	−	0.747220i	$-0.731388\pi$
$199$	−18.7500	−1.32915	−0.664577	+	0.747220i	$0.731388\pi$
$200$	4.87189	0.344495
$201$	0	0
$202$	−17.4053	−1.22463
$203$	−10.5397	−0.739739
$204$	0	0
$205$	−0.858685	−0.0599732
$206$	−10.1476	−0.707014
$207$	0	0
$208$	7.32528	0.507916
$209$	2.41850	0.167291
$210$	0	0
$211$	5.32528	0.366607	0.183304	−	0.983056i	$-0.441321\pi$
$211$	5.32528	0.366607	0.183304	+	0.983056i	$0.441321\pi$
$212$	20.0342	1.37595
$213$	0	0
$214$	29.6957	2.02996
$215$	2.56133	0.174682
$216$	0	0
$217$	29.8238	2.02457
$218$	20.7500	1.40537
$219$	0	0
$220$	4.06058	0.273764
$221$	5.66376	0.380985
$222$	0	0
$223$	−11.0279	−0.738484	−0.369242	−	0.929333i	$-0.620383\pi$
$223$	−11.0279	−0.738484	−0.369242	+	0.929333i	$0.620383\pi$
$224$	40.8517	2.72952
$225$	0	0
$226$	28.0800	1.86786
$227$	11.1344	0.739013	0.369507	−	0.929228i	$-0.379527\pi$
$227$	11.1344	0.739013	0.369507	+	0.929228i	$0.379527\pi$
$228$	0	0
$229$	25.4270	1.68026	0.840131	−	0.542383i	$-0.182478\pi$
$229$	25.4270	1.68026	0.840131	+	0.542383i	$0.182478\pi$
$230$	−4.31680	−0.284641
$231$	0	0
$232$	2.06058	0.135283
$233$	10.1212	0.663059	0.331529	−	0.943445i	$-0.392435\pi$
$233$	10.1212	0.663059	0.331529	+	0.943445i	$0.392435\pi$
$234$	0	0
$235$	−3.14132	−0.204917
$236$	−2.47283	−0.160968
$237$	0	0
$238$	23.6762	1.53470
$239$	−10.9387	−0.707566	−0.353783	−	0.935328i	$-0.615105\pi$
$239$	−10.9387	−0.707566	−0.353783	+	0.935328i	$0.615105\pi$
$240$	0	0
$241$	−10.2034	−0.657259	−0.328630	−	0.944459i	$-0.606587\pi$
$241$	−10.2034	−0.657259	−0.328630	+	0.944459i	$0.606587\pi$
$242$	−21.2493	−1.36595
$243$	0	0
$244$	22.2709	1.42575
$245$	6.85868	0.438185
$246$	0	0
$247$	−1.36417	−0.0868000
$248$	−5.83076	−0.370254
$249$	0	0
$250$	7.47283	0.472624
$251$	−12.2104	−0.770712	−0.385356	−	0.922768i	$-0.625921\pi$
$251$	−12.2104	−0.770712	−0.385356	+	0.922768i	$0.625921\pi$
$252$	0	0
$253$	26.1623	1.64481
$254$	1.24302	0.0779939
$255$	0	0
$256$	6.01320	0.375825
$257$	9.52645	0.594244	0.297122	−	0.954840i	$-0.403973\pi$
$257$	9.52645	0.594244	0.297122	+	0.954840i	$0.403973\pi$
$258$	0	0
$259$	−39.3983	−2.44809
$260$	−2.29039	−0.142044
$261$	0	0
$262$	−17.0932	−1.05602
$263$	16.0995	0.992736	0.496368	−	0.868112i	$-0.334667\pi$
$263$	16.0995	0.992736	0.496368	+	0.868112i	$0.334667\pi$
$264$	0	0
$265$	2.89981	0.178134
$266$	−5.70265	−0.349652
$267$	0	0
$268$	35.8044	2.18710
$269$	−28.3051	−1.72579	−0.862897	−	0.505381i	$-0.831352\pi$
$269$	−28.3051	−1.72579	−0.862897	+	0.505381i	$0.831352\pi$
$270$	0	0
$271$	1.38585	0.0841845	0.0420922	−	0.999114i	$-0.486598\pi$
$271$	1.38585	0.0841845	0.0420922	+	0.999114i	$0.486598\pi$
$272$	6.19565	0.375666
$273$	0	0
$274$	17.1344	1.03512
$275$	−22.3510	−1.34781
$276$	0	0
$277$	−1.15604	−0.0694595	−0.0347297	−	0.999397i	$-0.511057\pi$
$277$	−1.15604	−0.0694595	−0.0347297	+	0.999397i	$0.511057\pi$
$278$	−10.5008	−0.629794
$279$	0	0
$280$	−1.83076	−0.109409
$281$	19.5140	1.16411	0.582053	−	0.813151i	$-0.302250\pi$
$281$	19.5140	1.16411	0.582053	+	0.813151i	$0.302250\pi$
$282$	0	0
$283$	−2.04113	−0.121332	−0.0606662	−	0.998158i	$-0.519323\pi$
$283$	−2.04113	−0.121332	−0.0606662	+	0.998158i	$0.519323\pi$
$284$	−10.2081	−0.605741
$285$	0	0
$286$	25.1079	1.48466
$287$	−12.2709	−0.724331
$288$	0	0
$289$	−12.2097	−0.718215
$290$	1.55982	0.0915956
$291$	0	0
$292$	−27.7849	−1.62599
$293$	2.73057	0.159522	0.0797609	−	0.996814i	$-0.474584\pi$
$293$	2.73057	0.159522	0.0797609	+	0.996814i	$0.474584\pi$
$294$	0	0
$295$	−0.357926	−0.0208393
$296$	7.70265	0.447707
$297$	0	0
$298$	−46.3851	−2.68702
$299$	−14.7570	−0.853418
$300$	0	0
$301$	36.6025	2.10973
$302$	−3.15604	−0.181609
$303$	0	0
$304$	−1.49228	−0.0855882
$305$	3.22357	0.184581
$306$	0	0
$307$	−28.2423	−1.61187	−0.805937	−	0.592002i	$-0.798338\pi$
$307$	−28.2423	−1.61187	−0.805937	+	0.592002i	$0.798338\pi$
$308$	58.0272	3.30641
$309$	0	0
$310$	−4.41378	−0.250686
$311$	−20.5272	−1.16399	−0.581994	−	0.813193i	$-0.697727\pi$
$311$	−20.5272	−1.16399	−0.581994	+	0.813193i	$0.697727\pi$
$312$	0	0
$313$	−6.86565	−0.388069	−0.194035	−	0.980995i	$-0.562157\pi$
$313$	−6.86565	−0.388069	−0.194035	+	0.980995i	$0.562157\pi$
$314$	39.5676	2.23293
$315$	0	0
$316$	−9.57454	−0.538610
$317$	−28.6002	−1.60635	−0.803174	−	0.595744i	$-0.796857\pi$
$317$	−28.6002	−1.60635	−0.803174	+	0.595744i	$0.796857\pi$
$318$	0	0
$319$	−9.45339	−0.529288
$320$	−4.01945	−0.224694
$321$	0	0
$322$	−61.6887	−3.43778
$323$	−1.15380	−0.0641992
$324$	0	0
$325$	12.6072	0.699321
$326$	−7.07378	−0.391780
$327$	0	0
$328$	2.39905	0.132466
$329$	−44.8906	−2.47490
$330$	0	0
$331$	21.2966	1.17057	0.585284	−	0.810828i	$-0.300983\pi$
$331$	21.2966	1.17057	0.585284	+	0.810828i	$0.300983\pi$
$332$	1.82380	0.100094
$333$	0	0
$334$	34.6825	1.89774
$335$	5.18244	0.283147
$336$	0	0
$337$	−11.3253	−0.616927	−0.308464	−	0.951236i	$-0.599815\pi$
$337$	−11.3253	−0.616927	−0.308464	+	0.951236i	$0.599815\pi$
$338$	13.3315	0.725139
$339$	0	0
$340$	−1.93719	−0.105059
$341$	26.7500	1.44859
$342$	0	0
$343$	62.2089	3.35897
$344$	−7.15604	−0.385828
$345$	0	0
$346$	53.3161	2.86629
$347$	−2.58998	−0.139037	−0.0695186	−	0.997581i	$-0.522146\pi$
$347$	−2.58998	−0.139037	−0.0695186	+	0.997581i	$0.522146\pi$
$348$	0	0
$349$	−26.0319	−1.39346	−0.696729	−	0.717335i	$-0.745362\pi$
$349$	−26.0319	−1.39346	−0.696729	+	0.717335i	$0.745362\pi$
$350$	52.7019	2.81704
$351$	0	0
$352$	36.6414	1.95299
$353$	−13.9325	−0.741550	−0.370775	−	0.928723i	$-0.620908\pi$
$353$	−13.9325	−0.741550	−0.370775	+	0.928723i	$0.620908\pi$
$354$	0	0
$355$	−1.47756	−0.0784207
$356$	21.1344	1.12012
$357$	0	0
$358$	−31.8796	−1.68489
$359$	5.89134	0.310933	0.155466	−	0.987841i	$-0.450312\pi$
$359$	5.89134	0.310933	0.155466	+	0.987841i	$0.450312\pi$
$360$	0	0
$361$	−18.7221	−0.985373
$362$	−31.2508	−1.64250
$363$	0	0
$364$	−32.7306	−1.71555
$365$	−4.02168	−0.210504
$366$	0	0
$367$	−0.926221	−0.0483483	−0.0241742	−	0.999708i	$-0.507696\pi$
$367$	−0.926221	−0.0483483	−0.0241742	+	0.999708i	$0.507696\pi$
$368$	−16.1428	−0.841503
$369$	0	0
$370$	5.83076	0.303127
$371$	41.4395	2.15143
$372$	0	0
$373$	10.1498	0.525536	0.262768	−	0.964859i	$-0.415365\pi$
$373$	10.1498	0.525536	0.262768	+	0.964859i	$0.415365\pi$
$374$	21.2361	1.09809
$375$	0	0
$376$	8.77643	0.452610
$377$	5.33224	0.274624
$378$	0	0
$379$	−28.0753	−1.44213	−0.721066	−	0.692867i	$-0.756347\pi$
$379$	−28.0753	−1.44213	−0.721066	+	0.692867i	$0.756347\pi$
$380$	0.466591	0.0239356
$381$	0	0
$382$	−42.2834	−2.16341
$383$	−34.8176	−1.77909	−0.889547	−	0.456844i	$-0.848980\pi$
$383$	−34.8176	−1.77909	−0.889547	+	0.456844i	$0.848980\pi$
$384$	0	0
$385$	8.39905	0.428055
$386$	−11.1755	−0.568817
$387$	0	0
$388$	−10.1428	−0.514924
$389$	−1.78267	−0.0903850	−0.0451925	−	0.998978i	$-0.514390\pi$
$389$	−1.78267	−0.0903850	−0.0451925	+	0.998978i	$0.514390\pi$
$390$	0	0
$391$	−12.4813	−0.631207
$392$	−19.1623	−0.967841
$393$	0	0
$394$	12.9457	0.652193
$395$	−1.38585	−0.0697297
$396$	0	0
$397$	0.817557	0.0410320	0.0205160	−	0.999790i	$-0.493469\pi$
$397$	0.817557	0.0410320	0.0205160	+	0.999790i	$0.493469\pi$
$398$	39.6546	1.98770
$399$	0	0
$400$	13.7911	0.689557
$401$	5.35168	0.267250	0.133625	−	0.991032i	$-0.457338\pi$
$401$	5.35168	0.267250	0.133625	+	0.991032i	$0.457338\pi$
$402$	0	0
$403$	−15.0885	−0.751612
$404$	20.3510	1.01250
$405$	0	0
$406$	22.2904	1.10625
$407$	−35.3378	−1.75163
$408$	0	0
$409$	−38.4068	−1.89909	−0.949547	−	0.313624i	$-0.898457\pi$
$409$	−38.4068	−1.89909	−0.949547	+	0.313624i	$0.898457\pi$
$410$	1.81604	0.0896878
$411$	0	0
$412$	11.8649	0.584543
$413$	−5.11491	−0.251688
$414$	0	0
$415$	0.263983	0.0129584
$416$	−20.6678	−1.01332
$417$	0	0
$418$	−5.11491	−0.250178
$419$	−8.71585	−0.425797	−0.212899	−	0.977074i	$-0.568290\pi$
$419$	−8.71585	−0.425797	−0.212899	+	0.977074i	$0.568290\pi$
$420$	0	0
$421$	−0.357926	−0.0174443	−0.00872213	−	0.999962i	$-0.502776\pi$
$421$	−0.357926	−0.0174443	−0.00872213	+	0.999962i	$0.502776\pi$
$422$	−11.2625	−0.548248
$423$	0	0
$424$	−8.10170	−0.393454
$425$	10.6630	0.517233
$426$	0	0
$427$	46.0661	2.22929
$428$	−34.7214	−1.67832
$429$	0	0
$430$	−5.41698	−0.261230
$431$	13.2361	0.637558	0.318779	−	0.947829i	$-0.396727\pi$
$431$	13.2361	0.637558	0.318779	+	0.947829i	$0.396727\pi$
$432$	0	0
$433$	−27.4031	−1.31691	−0.658454	−	0.752621i	$-0.728789\pi$
$433$	−27.4031	−1.31691	−0.658454	+	0.752621i	$0.728789\pi$
$434$	−63.0746	−3.02768
$435$	0	0
$436$	−24.2617	−1.16193
$437$	3.00624	0.143808
$438$	0	0
$439$	32.5955	1.55570	0.777849	−	0.628451i	$-0.216311\pi$
$439$	32.5955	1.55570	0.777849	+	0.628451i	$0.216311\pi$
$440$	−1.64207	−0.0782828
$441$	0	0
$442$	−11.9783	−0.569751
$443$	30.8392	1.46522	0.732608	−	0.680651i	$-0.238303\pi$
$443$	30.8392	1.46522	0.732608	+	0.680651i	$0.238303\pi$
$444$	0	0
$445$	3.05906	0.145013
$446$	23.3230	1.10438
$447$	0	0
$448$	−57.4395	−2.71376
$449$	−35.0863	−1.65582	−0.827912	−	0.560859i	$-0.810471\pi$
$449$	−35.0863	−1.65582	−0.827912	+	0.560859i	$0.810471\pi$
$450$	0	0
$451$	−11.0062	−0.518264
$452$	−32.8323	−1.54430
$453$	0	0
$454$	−23.5481	−1.10517
$455$	−4.73753	−0.222099
$456$	0	0
$457$	10.6414	0.497782	0.248891	−	0.968532i	$-0.419934\pi$
$457$	10.6414	0.497782	0.248891	+	0.968532i	$0.419934\pi$
$458$	−53.7757	−2.51277
$459$	0	0
$460$	5.04737	0.235335
$461$	28.4046	1.32293	0.661467	−	0.749975i	$-0.269934\pi$
$461$	28.4046	1.32293	0.661467	+	0.749975i	$0.269934\pi$
$462$	0	0
$463$	−9.58150	−0.445290	−0.222645	−	0.974900i	$-0.571469\pi$
$463$	−9.58150	−0.445290	−0.222645	+	0.974900i	$0.571469\pi$
$464$	5.83299	0.270790
$465$	0	0
$466$	−21.4053	−0.991581
$467$	−26.1336	−1.20932	−0.604660	−	0.796484i	$-0.706691\pi$
$467$	−26.1336	−1.20932	−0.604660	+	0.796484i	$0.706691\pi$
$468$	0	0
$469$	74.0591	3.41973
$470$	6.64359	0.306446
$471$	0	0
$472$	1.00000	0.0460287
$473$	32.8300	1.50953
$474$	0	0
$475$	−2.56829	−0.117841
$476$	−27.6832	−1.26886
$477$	0	0
$478$	23.1344	1.05814
$479$	18.5397	0.847098	0.423549	−	0.905873i	$-0.360784\pi$
$479$	18.5397	0.847098	0.423549	+	0.905873i	$0.360784\pi$
$480$	0	0
$481$	19.9325	0.908842
$482$	21.5793	0.982909
$483$	0	0
$484$	24.8455	1.12934
$485$	−1.46811	−0.0666633
$486$	0	0
$487$	−21.5676	−0.977320	−0.488660	−	0.872474i	$-0.662514\pi$
$487$	−21.5676	−0.977320	−0.488660	+	0.872474i	$0.662514\pi$
$488$	−9.00624	−0.407693
$489$	0	0
$490$	−14.5055	−0.655291
$491$	11.4659	0.517448	0.258724	−	0.965951i	$-0.416698\pi$
$491$	11.4659	0.517448	0.258724	+	0.965951i	$0.416698\pi$
$492$	0	0
$493$	4.50995	0.203118
$494$	2.88509	0.129806
$495$	0	0
$496$	−16.5055	−0.741118
$497$	−21.1149	−0.947133
$498$	0	0
$499$	−17.4681	−0.781980	−0.390990	−	0.920395i	$-0.627867\pi$
$499$	−17.4681	−0.781980	−0.390990	+	0.920395i	$0.627867\pi$
$500$	−8.73753	−0.390754
$501$	0	0
$502$	25.8238	1.15257
$503$	−0.655277	−0.0292174	−0.0146087	−	0.999893i	$-0.504650\pi$
$503$	−0.655277	−0.0292174	−0.0146087	+	0.999893i	$0.504650\pi$
$504$	0	0
$505$	2.94567	0.131080
$506$	−55.3308	−2.45975
$507$	0	0
$508$	−1.45339	−0.0644836
$509$	15.8168	0.701069	0.350535	−	0.936550i	$-0.386000\pi$
$509$	15.8168	0.701069	0.350535	+	0.936550i	$0.386000\pi$
$510$	0	0
$511$	−57.4714	−2.54239
$512$	−28.2702	−1.24938
$513$	0	0
$514$	−20.1476	−0.888671
$515$	1.71737	0.0756764
$516$	0	0
$517$	−40.2640	−1.77081
$518$	83.3238	3.66104
$519$	0	0
$520$	0.926221	0.0406175
$521$	12.2470	0.536552	0.268276	−	0.963342i	$-0.413546\pi$
$521$	12.2470	0.536552	0.268276	+	0.963342i	$0.413546\pi$
$522$	0	0
$523$	−2.23678	−0.0978074	−0.0489037	−	0.998803i	$-0.515573\pi$
$523$	−2.23678	−0.0978074	−0.0489037	+	0.998803i	$0.515573\pi$
$524$	19.9861	0.873096
$525$	0	0
$526$	−34.0489	−1.48460
$527$	−12.7617	−0.555909
$528$	0	0
$529$	9.52021	0.413922
$530$	−6.13284	−0.266393
$531$	0	0
$532$	6.66776	0.289084
$533$	6.20813	0.268904
$534$	0	0
$535$	−5.02569	−0.217280
$536$	−14.4791	−0.625401
$537$	0	0
$538$	59.8627	2.58086
$539$	87.9116	3.78662
$540$	0	0
$541$	−11.0863	−0.476636	−0.238318	−	0.971187i	$-0.576596\pi$
$541$	−11.0863	−0.476636	−0.238318	+	0.971187i	$0.576596\pi$
$542$	−2.93095	−0.125895
$543$	0	0
$544$	−17.4806	−0.749474
$545$	−3.51173	−0.150426
$546$	0	0
$547$	12.1887	0.521151	0.260575	−	0.965454i	$-0.416088\pi$
$547$	12.1887	0.521151	0.260575	+	0.965454i	$0.416088\pi$
$548$	−20.0342	−0.855817
$549$	0	0
$550$	47.2702	2.01561
$551$	−1.08627	−0.0462765
$552$	0	0
$553$	−19.8044	−0.842167
$554$	2.44491	0.103874
$555$	0	0
$556$	12.2779	0.520699
$557$	8.56829	0.363050	0.181525	−	0.983386i	$-0.441897\pi$
$557$	8.56829	0.363050	0.181525	+	0.983386i	$0.441897\pi$
$558$	0	0
$559$	−18.5180	−0.783227
$560$	−5.18244	−0.218998
$561$	0	0
$562$	−41.2702	−1.74088
$563$	−9.98055	−0.420630	−0.210315	−	0.977634i	$-0.567449\pi$
$563$	−9.98055	−0.420630	−0.210315	+	0.977634i	$0.567449\pi$
$564$	0	0
$565$	−4.75226	−0.199929
$566$	4.31680	0.181449
$567$	0	0
$568$	4.12811	0.173212
$569$	−46.1428	−1.93441	−0.967204	−	0.254001i	$-0.918253\pi$
$569$	−46.1428	−1.93441	−0.967204	+	0.254001i	$0.918253\pi$
$570$	0	0
$571$	−21.1824	−0.886458	−0.443229	−	0.896409i	$-0.646167\pi$
$571$	−21.1824	−0.886458	−0.443229	+	0.896409i	$0.646167\pi$
$572$	−29.3572	−1.22749
$573$	0	0
$574$	25.9519	1.08321
$575$	−27.7827	−1.15862
$576$	0	0
$577$	40.8859	1.70210	0.851051	−	0.525083i	$-0.175966\pi$
$577$	40.8859	1.70210	0.851051	+	0.525083i	$0.175966\pi$
$578$	25.8223	1.07407
$579$	0	0
$580$	−1.82380	−0.0757292
$581$	3.77242	0.156506
$582$	0	0
$583$	37.1685	1.53936
$584$	11.2361	0.464951
$585$	0	0
$586$	−5.77491	−0.238559
$587$	4.21037	0.173780	0.0868902	−	0.996218i	$-0.472307\pi$
$587$	4.21037	0.173780	0.0868902	+	0.996218i	$0.472307\pi$
$588$	0	0
$589$	3.07378	0.126653
$590$	0.756981	0.0311644
$591$	0	0
$592$	21.8044	0.896153
$593$	22.5202	0.924794	0.462397	−	0.886673i	$-0.346990\pi$
$593$	22.5202	0.924794	0.462397	+	0.886673i	$0.346990\pi$
$594$	0	0
$595$	−4.00696	−0.164269
$596$	54.2353	2.22157
$597$	0	0
$598$	31.2097	1.27626
$599$	16.7981	0.686352	0.343176	−	0.939271i	$-0.388497\pi$
$599$	16.7981	0.686352	0.343176	+	0.939271i	$0.388497\pi$
$600$	0	0
$601$	−17.0955	−0.697338	−0.348669	−	0.937246i	$-0.613366\pi$
$601$	−17.0955	−0.697338	−0.348669	+	0.937246i	$0.613366\pi$
$602$	−77.4108	−3.15503
$603$	0	0
$604$	3.69016	0.150151
$605$	3.59622	0.146207
$606$	0	0
$607$	−7.53341	−0.305772	−0.152886	−	0.988244i	$-0.548857\pi$
$607$	−7.53341	−0.305772	−0.152886	+	0.988244i	$0.548857\pi$
$608$	4.21037	0.170753
$609$	0	0
$610$	−6.81756	−0.276035
$611$	22.7111	0.918794
$612$	0	0
$613$	15.1321	0.611181	0.305590	−	0.952163i	$-0.401146\pi$
$613$	15.1321	0.611181	0.305590	+	0.952163i	$0.401146\pi$
$614$	59.7299	2.41050
$615$	0	0
$616$	−23.4659	−0.945467
$617$	−15.8672	−0.638788	−0.319394	−	0.947622i	$-0.603479\pi$
$617$	−15.8672	−0.638788	−0.319394	+	0.947622i	$0.603479\pi$
$618$	0	0
$619$	11.0210	0.442970	0.221485	−	0.975164i	$-0.428910\pi$
$619$	11.0210	0.442970	0.221485	+	0.975164i	$0.428910\pi$
$620$	5.16076	0.207261
$621$	0	0
$622$	43.4131	1.74071
$623$	43.7151	1.75141
$624$	0	0
$625$	23.0947	0.923790
$626$	14.5202	0.580344
$627$	0	0
$628$	−46.2640	−1.84613
$629$	16.8587	0.672200
$630$	0	0
$631$	−23.1560	−0.921827	−0.460914	−	0.887445i	$-0.652478\pi$
$631$	−23.1560	−0.921827	−0.460914	+	0.887445i	$0.652478\pi$
$632$	3.87189	0.154015
$633$	0	0
$634$	60.4868	2.40224
$635$	−0.210368	−0.00834821
$636$	0	0
$637$	−49.5870	−1.96471
$638$	19.9930	0.791532
$639$	0	0
$640$	2.78339	0.110023
$641$	−39.5459	−1.56197	−0.780984	−	0.624550i	$-0.785282\pi$
$641$	−39.5459	−1.56197	−0.780984	+	0.624550i	$0.785282\pi$
$642$	0	0
$643$	18.6219	0.734376	0.367188	−	0.930147i	$-0.380320\pi$
$643$	18.6219	0.734376	0.367188	+	0.930147i	$0.380320\pi$
$644$	72.1289	2.84228
$645$	0	0
$646$	2.44018	0.0960077
$647$	−7.24525	−0.284840	−0.142420	−	0.989806i	$-0.545488\pi$
$647$	−7.24525	−0.284840	−0.142420	+	0.989806i	$0.545488\pi$
$648$	0	0
$649$	−4.58774	−0.180085
$650$	−26.6630	−1.04581
$651$	0	0
$652$	8.27094	0.323915
$653$	−31.3183	−1.22558	−0.612790	−	0.790246i	$-0.709953\pi$
$653$	−31.3183	−1.22558	−0.612790	+	0.790246i	$0.709953\pi$
$654$	0	0
$655$	2.89285	0.113033
$656$	6.79115	0.265150
$657$	0	0
$658$	94.9395	3.70113
$659$	−8.13587	−0.316929	−0.158464	−	0.987365i	$-0.550654\pi$
$659$	−8.13587	−0.316929	−0.158464	+	0.987365i	$0.550654\pi$
$660$	0	0
$661$	−9.47979	−0.368721	−0.184361	−	0.982859i	$-0.559021\pi$
$661$	−9.47979	−0.368721	−0.184361	+	0.982859i	$0.559021\pi$
$662$	−45.0404	−1.75055
$663$	0	0
$664$	−0.737534	−0.0286219
$665$	0.965115	0.0374255
$666$	0	0
$667$	−11.7507	−0.454990
$668$	−40.5521	−1.56901
$669$	0	0
$670$	−10.9604	−0.423437
$671$	41.3183	1.59508
$672$	0	0
$673$	11.4317	0.440660	0.220330	−	0.975425i	$-0.429287\pi$
$673$	11.4317	0.440660	0.220330	+	0.975425i	$0.429287\pi$
$674$	23.9519	0.922593
$675$	0	0
$676$	−15.5877	−0.599529
$677$	6.83700	0.262767	0.131384	−	0.991332i	$-0.458058\pi$
$677$	6.83700	0.262767	0.131384	+	0.991332i	$0.458058\pi$
$678$	0	0
$679$	−20.9798	−0.805132
$680$	0.783389	0.0300416
$681$	0	0
$682$	−56.5738	−2.16632
$683$	34.6002	1.32394	0.661970	−	0.749530i	$-0.269720\pi$
$683$	34.6002	1.32394	0.661970	+	0.749530i	$0.269720\pi$
$684$	0	0
$685$	−2.89981	−0.110796
$686$	−131.566	−5.02322
$687$	0	0
$688$	−20.2570	−0.772292
$689$	−20.9651	−0.798707
$690$	0	0
$691$	−42.8929	−1.63172	−0.815861	−	0.578249i	$-0.803736\pi$
$691$	−42.8929	−1.63172	−0.815861	+	0.578249i	$0.803736\pi$
$692$	−62.3393	−2.36978
$693$	0	0
$694$	5.47756	0.207925
$695$	1.77715	0.0674110
$696$	0	0
$697$	5.25078	0.198887
$698$	55.0551	2.08387
$699$	0	0
$700$	−61.6212	−2.32906
$701$	−8.83228	−0.333591	−0.166795	−	0.985992i	$-0.553342\pi$
$701$	−8.83228	−0.333591	−0.166795	+	0.985992i	$0.553342\pi$
$702$	0	0
$703$	−4.06058	−0.153148
$704$	−51.5195	−1.94171
$705$	0	0
$706$	29.4659	1.10896
$707$	42.0947	1.58314
$708$	0	0
$709$	22.7981	0.856201	0.428100	−	0.903731i	$-0.359183\pi$
$709$	22.7981	0.856201	0.428100	+	0.903731i	$0.359183\pi$
$710$	3.12490	0.117276
$711$	0	0
$712$	−8.54661	−0.320298
$713$	33.2508	1.24525
$714$	0	0
$715$	−4.24926	−0.158913
$716$	37.2750	1.39303
$717$	0	0
$718$	−12.4596	−0.464989
$719$	−21.6887	−0.808853	−0.404427	−	0.914570i	$-0.632529\pi$
$719$	−21.6887	−0.808853	−0.404427	+	0.914570i	$0.632529\pi$
$720$	0	0
$721$	24.5419	0.913988
$722$	39.5955	1.47359
$723$	0	0
$724$	36.5397	1.35799
$725$	10.0389	0.372835
$726$	0	0
$727$	47.7438	1.77072	0.885359	−	0.464907i	$-0.153912\pi$
$727$	47.7438	1.77072	0.885359	+	0.464907i	$0.153912\pi$
$728$	13.2361	0.490561
$729$	0	0
$730$	8.50548	0.314802
$731$	−15.6623	−0.579292
$732$	0	0
$733$	37.3983	1.38134	0.690670	−	0.723170i	$-0.257316\pi$
$733$	37.3983	1.38134	0.690670	+	0.723170i	$0.257316\pi$
$734$	1.95887	0.0723033
$735$	0	0
$736$	45.5459	1.67884
$737$	66.4263	2.44684
$738$	0	0
$739$	−13.9450	−0.512973	−0.256487	−	0.966548i	$-0.582565\pi$
$739$	−13.9450	−0.512973	−0.256487	+	0.966548i	$0.582565\pi$
$740$	−6.81756	−0.250618
$741$	0	0
$742$	−87.6406	−3.21739
$743$	−47.4200	−1.73967	−0.869836	−	0.493341i	$-0.835775\pi$
$743$	−47.4200	−1.73967	−0.869836	+	0.493341i	$0.835775\pi$
$744$	0	0
$745$	7.85021	0.287609
$746$	−21.4659	−0.785921
$747$	0	0
$748$	−24.8300	−0.907876
$749$	−71.8191	−2.62421
$750$	0	0
$751$	46.8929	1.71114	0.855572	−	0.517683i	$-0.173205\pi$
$751$	46.8929	1.71114	0.855572	+	0.517683i	$0.173205\pi$
$752$	24.8440	0.905966
$753$	0	0
$754$	−11.2772	−0.410691
$755$	0.534127	0.0194389
$756$	0	0
$757$	−21.7151	−0.789250	−0.394625	−	0.918842i	$-0.629125\pi$
$757$	−21.7151	−0.789250	−0.394625	+	0.918842i	$0.629125\pi$
$758$	59.3767	2.15666
$759$	0	0
$760$	−0.188687	−0.00684438
$761$	36.6002	1.32676	0.663379	−	0.748284i	$-0.269122\pi$
$761$	36.6002	1.32676	0.663379	+	0.748284i	$0.269122\pi$
$762$	0	0
$763$	−50.1840	−1.81678
$764$	49.4395	1.78866
$765$	0	0
$766$	73.6359	2.66057
$767$	2.58774	0.0934379
$768$	0	0
$769$	11.9200	0.429845	0.214923	−	0.976631i	$-0.431050\pi$
$769$	11.9200	0.429845	0.214923	+	0.976631i	$0.431050\pi$
$770$	−17.7632	−0.640142
$771$	0	0
$772$	13.0668	0.470285
$773$	−20.3076	−0.730414	−0.365207	−	0.930926i	$-0.619002\pi$
$773$	−20.3076	−0.730414	−0.365207	+	0.930926i	$0.619002\pi$
$774$	0	0
$775$	−28.4068	−1.02040
$776$	4.10170	0.147243
$777$	0	0
$778$	3.77018	0.135168
$779$	−1.26470	−0.0453126
$780$	0	0
$781$	−18.9387	−0.677680
$782$	26.3968	0.943948
$783$	0	0
$784$	−54.2438	−1.93728
$785$	−6.69641	−0.239005
$786$	0	0
$787$	18.2687	0.651209	0.325605	−	0.945506i	$-0.394432\pi$
$787$	18.2687	0.651209	0.325605	+	0.945506i	$0.394432\pi$
$788$	−15.1366	−0.539219
$789$	0	0
$790$	2.93095	0.104278
$791$	−67.9116	−2.41466
$792$	0	0
$793$	−23.3058	−0.827614
$794$	−1.72906	−0.0613619
$795$	0	0
$796$	−46.3657	−1.64339
$797$	21.3664	0.756837	0.378418	−	0.925635i	$-0.376468\pi$
$797$	21.3664	0.756837	0.378418	+	0.925635i	$0.376468\pi$
$798$	0	0
$799$	19.2089	0.679560
$800$	−38.9108	−1.37570
$801$	0	0
$802$	−11.3183	−0.399664
$803$	−51.5481	−1.81909
$804$	0	0
$805$	10.4402	0.367968
$806$	31.9108	1.12401
$807$	0	0
$808$	−8.22982	−0.289524
$809$	23.7460	0.834865	0.417433	−	0.908708i	$-0.362930\pi$
$809$	23.7460	0.834865	0.417433	+	0.908708i	$0.362930\pi$
$810$	0	0
$811$	15.3664	0.539587	0.269794	−	0.962918i	$-0.413044\pi$
$811$	15.3664	0.539587	0.269794	+	0.962918i	$0.413044\pi$
$812$	−26.0628	−0.914625
$813$	0	0
$814$	74.7361	2.61950
$815$	1.19716	0.0419348
$816$	0	0
$817$	3.77242	0.131980
$818$	81.2269	2.84003
$819$	0	0
$820$	−2.12339	−0.0741518
$821$	29.6009	1.03308	0.516540	−	0.856263i	$-0.327220\pi$
$821$	29.6009	1.03308	0.516540	+	0.856263i	$0.327220\pi$
$822$	0	0
$823$	−19.6546	−0.685115	−0.342557	−	0.939497i	$-0.611293\pi$
$823$	−19.6546	−0.685115	−0.342557	+	0.939497i	$0.611293\pi$
$824$	−4.79811	−0.167150
$825$	0	0
$826$	10.8176	0.376391
$827$	13.1296	0.456562	0.228281	−	0.973595i	$-0.426690\pi$
$827$	13.1296	0.456562	0.228281	+	0.973595i	$0.426690\pi$
$828$	0	0
$829$	−3.69569	−0.128357	−0.0641783	−	0.997938i	$-0.520443\pi$
$829$	−3.69569	−0.128357	−0.0641783	+	0.997938i	$0.520443\pi$
$830$	−0.558300	−0.0193789
$831$	0	0
$832$	29.0599	1.00747
$833$	−41.9402	−1.45314
$834$	0	0
$835$	−5.86965	−0.203128
$836$	5.98055	0.206842
$837$	0	0
$838$	18.4332	0.636765
$839$	−20.7911	−0.717790	−0.358895	−	0.933378i	$-0.616846\pi$
$839$	−20.7911	−0.717790	−0.358895	+	0.933378i	$0.616846\pi$
$840$	0	0
$841$	−24.7540	−0.853587
$842$	0.756981	0.0260873
$843$	0	0
$844$	13.1685	0.453279
$845$	−2.25622	−0.0776164
$846$	0	0
$847$	51.3914	1.76583
$848$	−22.9340	−0.787556
$849$	0	0
$850$	−22.5513	−0.773505
$851$	−43.9255	−1.50575
$852$	0	0
$853$	17.5162	0.599743	0.299872	−	0.953980i	$-0.403056\pi$
$853$	17.5162	0.599743	0.299872	+	0.953980i	$0.403056\pi$
$854$	−97.4255	−3.33383
$855$	0	0
$856$	14.0411	0.479916
$857$	−16.0628	−0.548695	−0.274348	−	0.961631i	$-0.588462\pi$
$857$	−16.0628	−0.548695	−0.274348	+	0.961631i	$0.588462\pi$
$858$	0	0
$859$	−5.39281	−0.184000	−0.0920002	−	0.995759i	$-0.529326\pi$
$859$	−5.39281	−0.184000	−0.0920002	+	0.995759i	$0.529326\pi$
$860$	6.33375	0.215979
$861$	0	0
$862$	−27.9930	−0.953447
$863$	−7.55509	−0.257178	−0.128589	−	0.991698i	$-0.541045\pi$
$863$	−7.55509	−0.257178	−0.128589	+	0.991698i	$0.541045\pi$
$864$	0	0
$865$	−9.02320	−0.306798
$866$	57.9549	1.96939
$867$	0	0
$868$	73.7493	2.50321
$869$	−17.7632	−0.602576
$870$	0	0
$871$	−37.4681	−1.26956
$872$	9.81131	0.332253
$873$	0	0
$874$	−6.35793	−0.215060
$875$	−18.0731	−0.610981
$876$	0	0
$877$	21.0807	0.711846	0.355923	−	0.934515i	$-0.384167\pi$
$877$	21.0807	0.711846	0.355923	+	0.934515i	$0.384167\pi$
$878$	−68.9365	−2.32649
$879$	0	0
$880$	−4.64832	−0.156695
$881$	−8.90677	−0.300077	−0.150038	−	0.988680i	$-0.547940\pi$
$881$	−8.90677	−0.300077	−0.150038	+	0.988680i	$0.547940\pi$
$882$	0	0
$883$	7.94719	0.267444	0.133722	−	0.991019i	$-0.457307\pi$
$883$	7.94719	0.267444	0.133722	+	0.991019i	$0.457307\pi$
$884$	14.0055	0.471057
$885$	0	0
$886$	−65.2221	−2.19118
$887$	13.1685	0.442156	0.221078	−	0.975256i	$-0.429043\pi$
$887$	13.1685	0.442156	0.221078	+	0.975256i	$0.429043\pi$
$888$	0	0
$889$	−3.00624	−0.100826
$890$	−6.46963	−0.216862
$891$	0	0
$892$	−27.2702	−0.913075
$893$	−4.62664	−0.154824
$894$	0	0
$895$	5.39530	0.180345
$896$	39.7757	1.32881
$897$	0	0
$898$	74.2042	2.47623
$899$	−12.0147	−0.400713
$900$	0	0
$901$	−17.7321	−0.590742
$902$	23.2772	0.775046
$903$	0	0
$904$	13.2772	0.441593
$905$	5.28887	0.175808
$906$	0	0
$907$	−23.9497	−0.795236	−0.397618	−	0.917551i	$-0.630163\pi$
$907$	−23.9497	−0.795236	−0.397618	+	0.917551i	$0.630163\pi$
$908$	27.5334	0.913728
$909$	0	0
$910$	10.0194	0.332141
$911$	46.8184	1.55116	0.775581	−	0.631248i	$-0.217457\pi$
$911$	46.8184	1.55116	0.775581	+	0.631248i	$0.217457\pi$
$912$	0	0
$913$	3.38362	0.111981
$914$	−22.5055	−0.744415
$915$	0	0
$916$	62.8767	2.07750
$917$	41.3400	1.36517
$918$	0	0
$919$	30.6506	1.01107	0.505534	−	0.862807i	$-0.331295\pi$
$919$	30.6506	1.01107	0.505534	+	0.862807i	$0.331295\pi$
$920$	−2.04113	−0.0672940
$921$	0	0
$922$	−60.0731	−1.97840
$923$	10.6825	0.351618
$924$	0	0
$925$	37.5264	1.23386
$926$	20.2640	0.665916
$927$	0	0
$928$	−16.4574	−0.540240
$929$	−7.52092	−0.246753	−0.123377	−	0.992360i	$-0.539372\pi$
$929$	−7.52092	−0.246753	−0.123377	+	0.992360i	$0.539372\pi$
$930$	0	0
$931$	10.1017	0.331070
$932$	25.0279	0.819817
$933$	0	0
$934$	55.2702	1.80850
$935$	−3.59398	−0.117536
$936$	0	0
$937$	−31.6421	−1.03370	−0.516851	−	0.856076i	$-0.672896\pi$
$937$	−31.6421	−1.03370	−0.516851	+	0.856076i	$0.672896\pi$
$938$	−156.628	−5.11409
$939$	0	0
$940$	−7.76795	−0.253363
$941$	11.1824	0.364537	0.182269	−	0.983249i	$-0.441656\pi$
$941$	11.1824	0.364537	0.182269	+	0.983249i	$0.441656\pi$
$942$	0	0
$943$	−13.6810	−0.445514
$944$	2.83076	0.0921334
$945$	0	0
$946$	−69.4325	−2.25745
$947$	30.6149	0.994852	0.497426	−	0.867506i	$-0.334279\pi$
$947$	30.6149	0.994852	0.497426	+	0.867506i	$0.334279\pi$
$948$	0	0
$949$	29.0760	0.943847
$950$	5.43171	0.176228
$951$	0	0
$952$	11.1949	0.362830
$953$	−3.17548	−0.102864	−0.0514320	−	0.998676i	$-0.516379\pi$
$953$	−3.17548	−0.102864	−0.0514320	+	0.998676i	$0.516379\pi$
$954$	0	0
$955$	7.15604	0.231564
$956$	−27.0496	−0.874847
$957$	0	0
$958$	−39.2097	−1.26681
$959$	−41.4395	−1.33815
$960$	0	0
$961$	2.99777	0.0967021
$962$	−42.1553	−1.35914
$963$	0	0
$964$	−25.2313	−0.812646
$965$	1.89134	0.0608842
$966$	0	0
$967$	40.5566	1.30421	0.652106	−	0.758128i	$-0.273886\pi$
$967$	40.5566	1.30421	0.652106	+	0.758128i	$0.273886\pi$
$968$	−10.0474	−0.322935
$969$	0	0
$970$	3.10491	0.0996927
$971$	4.23205	0.135813	0.0679065	−	0.997692i	$-0.478368\pi$
$971$	4.23205	0.135813	0.0679065	+	0.997692i	$0.478368\pi$
$972$	0	0
$973$	25.3961	0.814162
$974$	45.6134	1.46155
$975$	0	0
$976$	−25.4945	−0.816060
$977$	−26.5055	−0.847986	−0.423993	−	0.905666i	$-0.639372\pi$
$977$	−26.5055	−0.847986	−0.423993	+	0.905666i	$0.639372\pi$
$978$	0	0
$979$	39.2097	1.25315
$980$	16.9604	0.541780
$981$	0	0
$982$	−24.2493	−0.773825
$983$	−2.27567	−0.0725826	−0.0362913	−	0.999341i	$-0.511554\pi$
$983$	−2.27567	−0.0725826	−0.0362913	+	0.999341i	$0.511554\pi$
$984$	0	0
$985$	−2.19092	−0.0698086
$986$	−9.53814	−0.303756
$987$	0	0
$988$	−3.37336	−0.107321
$989$	40.8084	1.29763
$990$	0	0
$991$	43.9185	1.39512	0.697559	−	0.716527i	$-0.254269\pi$
$991$	43.9185	1.39512	0.697559	+	0.716527i	$0.254269\pi$
$992$	46.5691	1.47857
$993$	0	0
$994$	44.6561	1.41640
$995$	−6.71113	−0.212757
$996$	0	0
$997$	2.81532	0.0891621	0.0445811	−	0.999006i	$-0.485805\pi$
$997$	2.81532	0.0891621	0.0445811	+	0.999006i	$0.485805\pi$
$998$	36.9434	1.16942
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.2.a.d.1.1		3
3.2	odd	2		177.2.a.d.1.3	✓	3
4.3	odd	2		8496.2.a.bl.1.2		3
12.11	even	2		2832.2.a.t.1.2		3
15.14	odd	2		4425.2.a.w.1.1		3
21.20	even	2		8673.2.a.s.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.a.d.1.3	✓	3	3.2	odd	2
531.2.a.d.1.1		3	1.1	even	1	trivial
2832.2.a.t.1.2		3	12.11	even	2
4425.2.a.w.1.1		3	15.14	odd	2
8496.2.a.bl.1.2		3	4.3	odd	2
8673.2.a.s.1.3		3	21.20	even	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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q-2.11491 q^{2} +2.47283 q^{4} +0.357926 q^{5} +5.11491 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-2.11491 q^{2} +2.47283 q^{4} +0.357926 q^{5} +5.11491 q^{7} -1.00000 q^{8} -0.756981 q^{10} +4.58774 q^{11} -2.58774 q^{13} -10.8176 q^{14} -2.83076 q^{16} -2.18869 q^{17} +0.527166 q^{19} +0.885092 q^{20} -9.70265 q^{22} +5.70265 q^{23} -4.87189 q^{25} +5.47283 q^{26} +12.6483 q^{28} -2.06058 q^{29} +5.83076 q^{31} +7.98680 q^{32} +4.62887 q^{34} +1.83076 q^{35} -7.70265 q^{37} -1.11491 q^{38} -0.357926 q^{40} -2.39905 q^{41} +7.15604 q^{43} +11.3447 q^{44} -12.0606 q^{46} -8.77643 q^{47} +19.1623 q^{49} +10.3036 q^{50} -6.39905 q^{52} +8.10170 q^{53} +1.64207 q^{55} -5.11491 q^{56} +4.35793 q^{58} -1.00000 q^{59} +9.00624 q^{61} -12.3315 q^{62} -11.2298 q^{64} -0.926221 q^{65} +14.4791 q^{67} -5.41226 q^{68} -3.87189 q^{70} -4.12811 q^{71} -11.2361 q^{73} +16.2904 q^{74} +1.30359 q^{76} +23.4659 q^{77} -3.87189 q^{79} -1.01320 q^{80} +5.07378 q^{82} +0.737534 q^{83} -0.783389 q^{85} -15.1344 q^{86} -4.58774 q^{88} +8.54661 q^{89} -13.2361 q^{91} +14.1017 q^{92} +18.5613 q^{94} +0.188687 q^{95} -4.10170 q^{97} -40.5264 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{4} + 2 q^{5} + 9 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q + 2 * q^4 + 2 * q^5 + 9 * q^7 - 3 * q^8	\( 3 q + 2 q^{4} + 2 q^{5} + 9 q^{7} - 3 q^{8} + 5 q^{10} + 2 q^{11} + 4 q^{13} - 8 q^{14} - 4 q^{16} - 3 q^{17} + 7 q^{19} + 9 q^{20} - 11 q^{22} - q^{23} - q^{25} + 11 q^{26} + 9 q^{28} + 11 q^{29} + 13 q^{31} + 4 q^{32} - 7 q^{34} + q^{35} - 5 q^{37} + 3 q^{38} - 2 q^{40} + q^{41} + 6 q^{43} + 15 q^{44} - 19 q^{46} - 11 q^{47} + 14 q^{49} + 21 q^{50} - 11 q^{52} - 2 q^{53} + 4 q^{55} - 9 q^{56} + 14 q^{58} - 3 q^{59} - q^{61} + 2 q^{62} - 21 q^{64} + 10 q^{67} - 28 q^{68} + 2 q^{70} - 26 q^{71} + 7 q^{73} + 19 q^{74} - 6 q^{76} + 17 q^{77} + 2 q^{79} - 23 q^{80} + 18 q^{82} + 3 q^{83} - 35 q^{85} - 31 q^{86} - 2 q^{88} + 23 q^{89} + q^{91} + 16 q^{92} + 4 q^{94} - 3 q^{95} + 14 q^{97} - 51 q^{98}+O(q^{100}) \) 3 * q + 2 * q^4 + 2 * q^5 + 9 * q^7 - 3 * q^8 + 5 * q^10 + 2 * q^11 + 4 * q^13 - 8 * q^14 - 4 * q^16 - 3 * q^17 + 7 * q^19 + 9 * q^20 - 11 * q^22 - q^23 - q^25 + 11 * q^26 + 9 * q^28 + 11 * q^29 + 13 * q^31 + 4 * q^32 - 7 * q^34 + q^35 - 5 * q^37 + 3 * q^38 - 2 * q^40 + q^41 + 6 * q^43 + 15 * q^44 - 19 * q^46 - 11 * q^47 + 14 * q^49 + 21 * q^50 - 11 * q^52 - 2 * q^53 + 4 * q^55 - 9 * q^56 + 14 * q^58 - 3 * q^59 - q^61 + 2 * q^62 - 21 * q^64 + 10 * q^67 - 28 * q^68 + 2 * q^70 - 26 * q^71 + 7 * q^73 + 19 * q^74 - 6 * q^76 + 17 * q^77 + 2 * q^79 - 23 * q^80 + 18 * q^82 + 3 * q^83 - 35 * q^85 - 31 * q^86 - 2 * q^88 + 23 * q^89 + q^91 + 16 * q^92 + 4 * q^94 - 3 * q^95 + 14 * q^97 - 51 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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