LMFDB - Embedded newform 4624.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4624,2,Mod(1,4624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4624, 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("4624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4624 = 2^{4} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4624.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:	$36.9228258946$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 289)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	4624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.34730 q^{3} -3.53209 q^{5} -0.347296 q^{7} -1.18479 q^{9} +O(q^{10})$	$q+1.34730 q^{3} -3.53209 q^{5} -0.347296 q^{7} -1.18479 q^{9} -1.75877 q^{11} -3.29086 q^{13} -4.75877 q^{15} -1.53209 q^{19} -0.467911 q^{21} +2.81521 q^{23} +7.47565 q^{25} -5.63816 q^{27} +1.18479 q^{29} +7.10607 q^{31} -2.36959 q^{33} +1.22668 q^{35} +3.92127 q^{37} -4.43376 q^{39} -4.92127 q^{41} -10.9855 q^{43} +4.18479 q^{45} +5.12061 q^{47} -6.87939 q^{49} -8.36959 q^{53} +6.21213 q^{55} -2.06418 q^{57} +10.8648 q^{59} +4.41147 q^{61} +0.411474 q^{63} +11.6236 q^{65} -8.07192 q^{67} +3.79292 q^{69} +11.2490 q^{71} -1.70233 q^{73} +10.0719 q^{75} +0.610815 q^{77} +11.5817 q^{79} -4.04189 q^{81} +2.14796 q^{83} +1.59627 q^{87} +6.41921 q^{89} +1.14290 q^{91} +9.57398 q^{93} +5.41147 q^{95} -4.08378 q^{97} +2.08378 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 6 q^{5}+O(q^{10}) $ 3 * q + 3 * q^3 - 6 * q^5	$ 3 q + 3 q^{3} - 6 q^{5} + 6 q^{11} + 6 q^{13} - 3 q^{15} - 6 q^{21} + 12 q^{23} + 3 q^{25} + 9 q^{31} - 3 q^{35} + 3 q^{37} + 3 q^{39} - 6 q^{41} - 15 q^{43} + 9 q^{45} + 21 q^{47} - 15 q^{49} - 18 q^{53} - 6 q^{55} + 3 q^{57} + 9 q^{59} + 3 q^{61} - 9 q^{63} + 9 q^{67} + 21 q^{69} + 21 q^{71} + 21 q^{73} - 3 q^{75} + 6 q^{77} + 3 q^{79} - 9 q^{81} - 9 q^{83} - 9 q^{87} - 15 q^{89} + 3 q^{91} + 21 q^{93} + 6 q^{95} - 6 q^{97}+O(q^{100}) $ 3 * q + 3 * q^3 - 6 * q^5 + 6 * q^11 + 6 * q^13 - 3 * q^15 - 6 * q^21 + 12 * q^23 + 3 * q^25 + 9 * q^31 - 3 * q^35 + 3 * q^37 + 3 * q^39 - 6 * q^41 - 15 * q^43 + 9 * q^45 + 21 * q^47 - 15 * q^49 - 18 * q^53 - 6 * q^55 + 3 * q^57 + 9 * q^59 + 3 * q^61 - 9 * q^63 + 9 * q^67 + 21 * q^69 + 21 * q^71 + 21 * q^73 - 3 * q^75 + 6 * q^77 + 3 * q^79 - 9 * q^81 - 9 * q^83 - 9 * q^87 - 15 * q^89 + 3 * q^91 + 21 * q^93 + 6 * q^95 - 6 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.34730	0.777862	0.388931	−	0.921267i	$-0.372844\pi$
$3$	1.34730	0.777862	0.388931	+	0.921267i	$0.372844\pi$
$4$	0	0
$5$	−3.53209	−1.57960	−0.789799	−	0.613366i	$-0.789815\pi$
$5$	−3.53209	−1.57960	−0.789799	+	0.613366i	$0.789815\pi$
$6$	0	0
$7$	−0.347296	−0.131266	−0.0656328	−	0.997844i	$-0.520907\pi$
$7$	−0.347296	−0.131266	−0.0656328	+	0.997844i	$0.520907\pi$
$8$	0	0
$9$	−1.18479	−0.394931
$10$	0	0
$11$	−1.75877	−0.530289	−0.265145	−	0.964209i	$-0.585420\pi$
$11$	−1.75877	−0.530289	−0.265145	+	0.964209i	$0.585420\pi$
$12$	0	0
$13$	−3.29086	−0.912720	−0.456360	−	0.889795i	$-0.650847\pi$
$13$	−3.29086	−0.912720	−0.456360	+	0.889795i	$0.650847\pi$
$14$	0	0
$15$	−4.75877	−1.22871
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−1.53209	−0.351485	−0.175743	−	0.984436i	$-0.556233\pi$
$19$	−1.53209	−0.351485	−0.175743	+	0.984436i	$0.556233\pi$
$20$	0	0
$21$	−0.467911	−0.102107
$22$	0	0
$23$	2.81521	0.587011	0.293506	−	0.955957i	$-0.405178\pi$
$23$	2.81521	0.587011	0.293506	+	0.955957i	$0.405178\pi$
$24$	0	0
$25$	7.47565	1.49513
$26$	0	0
$27$	−5.63816	−1.08506
$28$	0	0
$29$	1.18479	0.220010	0.110005	−	0.993931i	$-0.464913\pi$
$29$	1.18479	0.220010	0.110005	+	0.993931i	$0.464913\pi$
$30$	0	0
$31$	7.10607	1.27629	0.638144	−	0.769917i	$-0.279703\pi$
$31$	7.10607	1.27629	0.638144	+	0.769917i	$0.279703\pi$
$32$	0	0
$33$	−2.36959	−0.412492
$34$	0	0
$35$	1.22668	0.207347
$36$	0	0
$37$	3.92127	0.644654	0.322327	−	0.946628i	$-0.395535\pi$
$37$	3.92127	0.644654	0.322327	+	0.946628i	$0.395535\pi$
$38$	0	0
$39$	−4.43376	−0.709970
$40$	0	0
$41$	−4.92127	−0.768574	−0.384287	−	0.923214i	$-0.625553\pi$
$41$	−4.92127	−0.768574	−0.384287	+	0.923214i	$0.625553\pi$
$42$	0	0
$43$	−10.9855	−1.67527	−0.837633	−	0.546234i	$-0.816061\pi$
$43$	−10.9855	−1.67527	−0.837633	+	0.546234i	$0.816061\pi$
$44$	0	0
$45$	4.18479	0.623832
$46$	0	0
$47$	5.12061	0.746918	0.373459	−	0.927647i	$-0.378172\pi$
$47$	5.12061	0.746918	0.373459	+	0.927647i	$0.378172\pi$
$48$	0	0
$49$	−6.87939	−0.982769
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.36959	−1.14965	−0.574825	−	0.818276i	$-0.694930\pi$
$53$	−8.36959	−1.14965	−0.574825	+	0.818276i	$0.694930\pi$
$54$	0	0
$55$	6.21213	0.837644
$56$	0	0
$57$	−2.06418	−0.273407
$58$	0	0
$59$	10.8648	1.41448	0.707241	−	0.706973i	$-0.249940\pi$
$59$	10.8648	1.41448	0.707241	+	0.706973i	$0.249940\pi$
$60$	0	0
$61$	4.41147	0.564831	0.282416	−	0.959292i	$-0.408864\pi$
$61$	4.41147	0.564831	0.282416	+	0.959292i	$0.408864\pi$
$62$	0	0
$63$	0.411474	0.0518409
$64$	0	0
$65$	11.6236	1.44173
$66$	0	0
$67$	−8.07192	−0.986142	−0.493071	−	0.869989i	$-0.664126\pi$
$67$	−8.07192	−0.986142	−0.493071	+	0.869989i	$0.664126\pi$
$68$	0	0
$69$	3.79292	0.456614
$70$	0	0
$71$	11.2490	1.33501	0.667504	−	0.744607i	$-0.267363\pi$
$71$	11.2490	1.33501	0.667504	+	0.744607i	$0.267363\pi$
$72$	0	0
$73$	−1.70233	−0.199243	−0.0996215	−	0.995025i	$-0.531763\pi$
$73$	−1.70233	−0.199243	−0.0996215	+	0.995025i	$0.531763\pi$
$74$	0	0
$75$	10.0719	1.16300
$76$	0	0
$77$	0.610815	0.0696088
$78$	0	0
$79$	11.5817	1.30305	0.651523	−	0.758629i	$-0.274131\pi$
$79$	11.5817	1.30305	0.651523	+	0.758629i	$0.274131\pi$
$80$	0	0
$81$	−4.04189	−0.449099
$82$	0	0
$83$	2.14796	0.235769	0.117884	−	0.993027i	$-0.462389\pi$
$83$	2.14796	0.235769	0.117884	+	0.993027i	$0.462389\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.59627	0.171138
$88$	0	0
$89$	6.41921	0.680435	0.340218	−	0.940347i	$-0.389499\pi$
$89$	6.41921	0.680435	0.340218	+	0.940347i	$0.389499\pi$
$90$	0	0
$91$	1.14290	0.119809
$92$	0	0
$93$	9.57398	0.992775
$94$	0	0
$95$	5.41147	0.555206
$96$	0	0
$97$	−4.08378	−0.414645	−0.207322	−	0.978273i	$-0.566475\pi$
$97$	−4.08378	−0.414645	−0.207322	+	0.978273i	$0.566475\pi$
$98$	0	0
$99$	2.08378	0.209428
$100$	0	0
$101$	5.26857	0.524242	0.262121	−	0.965035i	$-0.415578\pi$
$101$	5.26857	0.524242	0.262121	+	0.965035i	$0.415578\pi$
$102$	0	0
$103$	18.4388	1.81683	0.908415	−	0.418069i	$-0.137293\pi$
$103$	18.4388	1.81683	0.908415	+	0.418069i	$0.137293\pi$
$104$	0	0
$105$	1.65270	0.161287
$106$	0	0
$107$	5.36184	0.518349	0.259175	−	0.965831i	$-0.416550\pi$
$107$	5.36184	0.518349	0.259175	+	0.965831i	$0.416550\pi$
$108$	0	0
$109$	15.5175	1.48631	0.743155	−	0.669119i	$-0.233328\pi$
$109$	15.5175	1.48631	0.743155	+	0.669119i	$0.233328\pi$
$110$	0	0
$111$	5.28312	0.501451
$112$	0	0
$113$	7.02734	0.661077	0.330538	−	0.943793i	$-0.392770\pi$
$113$	7.02734	0.661077	0.330538	+	0.943793i	$0.392770\pi$
$114$	0	0
$115$	−9.94356	−0.927242
$116$	0	0
$117$	3.89899	0.360461
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.90673	−0.718793
$122$	0	0
$123$	−6.63041	−0.597844
$124$	0	0
$125$	−8.74422	−0.782107
$126$	0	0
$127$	−7.74422	−0.687189	−0.343594	−	0.939118i	$-0.611645\pi$
$127$	−7.74422	−0.687189	−0.343594	+	0.939118i	$0.611645\pi$
$128$	0	0
$129$	−14.8007	−1.30313
$130$	0	0
$131$	18.6186	1.62671	0.813355	−	0.581767i	$-0.197639\pi$
$131$	18.6186	1.62671	0.813355	+	0.581767i	$0.197639\pi$
$132$	0	0
$133$	0.532089	0.0461380
$134$	0	0
$135$	19.9145	1.71396
$136$	0	0
$137$	10.0719	0.860502	0.430251	−	0.902709i	$-0.358425\pi$
$137$	10.0719	0.860502	0.430251	+	0.902709i	$0.358425\pi$
$138$	0	0
$139$	−8.55438	−0.725573	−0.362786	−	0.931872i	$-0.618175\pi$
$139$	−8.55438	−0.725573	−0.362786	+	0.931872i	$0.618175\pi$
$140$	0	0
$141$	6.89899	0.580999
$142$	0	0
$143$	5.78787	0.484006
$144$	0	0
$145$	−4.18479	−0.347528
$146$	0	0
$147$	−9.26857	−0.764459
$148$	0	0
$149$	11.8794	0.973197	0.486599	−	0.873626i	$-0.338237\pi$
$149$	11.8794	0.973197	0.486599	+	0.873626i	$0.338237\pi$
$150$	0	0
$151$	−6.55943	−0.533799	−0.266899	−	0.963724i	$-0.585999\pi$
$151$	−6.55943	−0.533799	−0.266899	+	0.963724i	$0.585999\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−25.0993	−2.01602
$156$	0	0
$157$	8.88444	0.709055	0.354528	−	0.935046i	$-0.384642\pi$
$157$	8.88444	0.709055	0.354528	+	0.935046i	$0.384642\pi$
$158$	0	0
$159$	−11.2763	−0.894270
$160$	0	0
$161$	−0.977711	−0.0770544
$162$	0	0
$163$	−9.68004	−0.758200	−0.379100	−	0.925356i	$-0.623766\pi$
$163$	−9.68004	−0.758200	−0.379100	+	0.925356i	$0.623766\pi$
$164$	0	0
$165$	8.36959	0.651571
$166$	0	0
$167$	1.83750	0.142190	0.0710949	−	0.997470i	$-0.477351\pi$
$167$	1.83750	0.142190	0.0710949	+	0.997470i	$0.477351\pi$
$168$	0	0
$169$	−2.17024	−0.166942
$170$	0	0
$171$	1.81521	0.138812
$172$	0	0
$173$	−22.7597	−1.73039	−0.865194	−	0.501437i	$-0.832805\pi$
$173$	−22.7597	−1.73039	−0.865194	+	0.501437i	$0.832805\pi$
$174$	0	0
$175$	−2.59627	−0.196259
$176$	0	0
$177$	14.6382	1.10027
$178$	0	0
$179$	17.5253	1.30990	0.654951	−	0.755672i	$-0.272689\pi$
$179$	17.5253	1.30990	0.654951	+	0.755672i	$0.272689\pi$
$180$	0	0
$181$	4.56212	0.339100	0.169550	−	0.985522i	$-0.445769\pi$
$181$	4.56212	0.339100	0.169550	+	0.985522i	$0.445769\pi$
$182$	0	0
$183$	5.94356	0.439361
$184$	0	0
$185$	−13.8503	−1.01829
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1.95811	0.142432
$190$	0	0
$191$	9.72462	0.703649	0.351824	−	0.936066i	$-0.385561\pi$
$191$	9.72462	0.703649	0.351824	+	0.936066i	$0.385561\pi$
$192$	0	0
$193$	0.844303	0.0607743	0.0303871	−	0.999538i	$-0.490326\pi$
$193$	0.844303	0.0607743	0.0303871	+	0.999538i	$0.490326\pi$
$194$	0	0
$195$	15.6604	1.12147
$196$	0	0
$197$	−17.0009	−1.21127	−0.605633	−	0.795744i	$-0.707080\pi$
$197$	−17.0009	−1.21127	−0.605633	+	0.795744i	$0.707080\pi$
$198$	0	0
$199$	−12.6955	−0.899962	−0.449981	−	0.893038i	$-0.648569\pi$
$199$	−12.6955	−0.899962	−0.449981	+	0.893038i	$0.648569\pi$
$200$	0	0
$201$	−10.8753	−0.767082
$202$	0	0
$203$	−0.411474	−0.0288798
$204$	0	0
$205$	17.3824	1.21404
$206$	0	0
$207$	−3.33544	−0.231829
$208$	0	0
$209$	2.69459	0.186389
$210$	0	0
$211$	23.5303	1.61990	0.809948	−	0.586502i	$-0.199496\pi$
$211$	23.5303	1.61990	0.809948	+	0.586502i	$0.199496\pi$
$212$	0	0
$213$	15.1557	1.03845
$214$	0	0
$215$	38.8016	2.64625
$216$	0	0
$217$	−2.46791	−0.167533
$218$	0	0
$219$	−2.29355	−0.154984
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−4.97090	−0.332876	−0.166438	−	0.986052i	$-0.553227\pi$
$223$	−4.97090	−0.332876	−0.166438	+	0.986052i	$0.553227\pi$
$224$	0	0
$225$	−8.85710	−0.590473
$226$	0	0
$227$	−5.25671	−0.348900	−0.174450	−	0.984666i	$-0.555815\pi$
$227$	−5.25671	−0.348900	−0.174450	+	0.984666i	$0.555815\pi$
$228$	0	0
$229$	−24.2053	−1.59953	−0.799767	−	0.600311i	$-0.795043\pi$
$229$	−24.2053	−1.59953	−0.799767	+	0.600311i	$0.795043\pi$
$230$	0	0
$231$	0.822948	0.0541460
$232$	0	0
$233$	28.7219	1.88164	0.940818	−	0.338912i	$-0.110059\pi$
$233$	28.7219	1.88164	0.940818	+	0.338912i	$0.110059\pi$
$234$	0	0
$235$	−18.0865	−1.17983
$236$	0	0
$237$	15.6040	1.01359
$238$	0	0
$239$	2.63310	0.170321	0.0851606	−	0.996367i	$-0.472860\pi$
$239$	2.63310	0.170321	0.0851606	+	0.996367i	$0.472860\pi$
$240$	0	0
$241$	25.3337	1.63189	0.815943	−	0.578132i	$-0.196218\pi$
$241$	25.3337	1.63189	0.815943	+	0.578132i	$0.196218\pi$
$242$	0	0
$243$	11.4688	0.735727
$244$	0	0
$245$	24.2986	1.55238
$246$	0	0
$247$	5.04189	0.320808
$248$	0	0
$249$	2.89393	0.183396
$250$	0	0
$251$	−10.6851	−0.674437	−0.337219	−	0.941426i	$-0.609486\pi$
$251$	−10.6851	−0.674437	−0.337219	+	0.941426i	$0.609486\pi$
$252$	0	0
$253$	−4.95130	−0.311286
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.73648	−0.233075	−0.116538	−	0.993186i	$-0.537180\pi$
$257$	−3.73648	−0.233075	−0.116538	+	0.993186i	$0.537180\pi$
$258$	0	0
$259$	−1.36184	−0.0846209
$260$	0	0
$261$	−1.40373	−0.0868889
$262$	0	0
$263$	−10.2094	−0.629541	−0.314771	−	0.949168i	$-0.601928\pi$
$263$	−10.2094	−0.629541	−0.314771	+	0.949168i	$0.601928\pi$
$264$	0	0
$265$	29.5621	1.81599
$266$	0	0
$267$	8.64858	0.529285
$268$	0	0
$269$	−11.7392	−0.715750	−0.357875	−	0.933770i	$-0.616499\pi$
$269$	−11.7392	−0.715750	−0.357875	+	0.933770i	$0.616499\pi$
$270$	0	0
$271$	−17.0000	−1.03268	−0.516338	−	0.856385i	$-0.672705\pi$
$271$	−17.0000	−1.03268	−0.516338	+	0.856385i	$0.672705\pi$
$272$	0	0
$273$	1.53983	0.0931947
$274$	0	0
$275$	−13.1480	−0.792852
$276$	0	0
$277$	24.3432	1.46264	0.731320	−	0.682035i	$-0.238905\pi$
$277$	24.3432	1.46264	0.731320	+	0.682035i	$0.238905\pi$
$278$	0	0
$279$	−8.41921	−0.504045
$280$	0	0
$281$	−5.79385	−0.345632	−0.172816	−	0.984954i	$-0.555287\pi$
$281$	−5.79385	−0.345632	−0.172816	+	0.984954i	$0.555287\pi$
$282$	0	0
$283$	−15.3200	−0.910677	−0.455338	−	0.890318i	$-0.650482\pi$
$283$	−15.3200	−0.910677	−0.455338	+	0.890318i	$0.650482\pi$
$284$	0	0
$285$	7.29086	0.431873
$286$	0	0
$287$	1.70914	0.100887
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−5.50206	−0.322536
$292$	0	0
$293$	2.98040	0.174117	0.0870584	−	0.996203i	$-0.472253\pi$
$293$	2.98040	0.174117	0.0870584	+	0.996203i	$0.472253\pi$
$294$	0	0
$295$	−38.3756	−2.23431
$296$	0	0
$297$	9.91622	0.575398
$298$	0	0
$299$	−9.26445	−0.535777
$300$	0	0
$301$	3.81521	0.219905
$302$	0	0
$303$	7.09833	0.407788
$304$	0	0
$305$	−15.5817	−0.892207
$306$	0	0
$307$	3.26857	0.186547	0.0932736	−	0.995641i	$-0.470267\pi$
$307$	3.26857	0.186547	0.0932736	+	0.995641i	$0.470267\pi$
$308$	0	0
$309$	24.8425	1.41324
$310$	0	0
$311$	3.53209	0.200286	0.100143	−	0.994973i	$-0.468070\pi$
$311$	3.53209	0.200286	0.100143	+	0.994973i	$0.468070\pi$
$312$	0	0
$313$	−1.48246	−0.0837935	−0.0418968	−	0.999122i	$-0.513340\pi$
$313$	−1.48246	−0.0837935	−0.0418968	+	0.999122i	$0.513340\pi$
$314$	0	0
$315$	−1.45336	−0.0818877
$316$	0	0
$317$	14.4638	0.812368	0.406184	−	0.913791i	$-0.366859\pi$
$317$	14.4638	0.812368	0.406184	+	0.913791i	$0.366859\pi$
$318$	0	0
$319$	−2.08378	−0.116669
$320$	0	0
$321$	7.22399	0.403204
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−24.6013	−1.36464
$326$	0	0
$327$	20.9067	1.15614
$328$	0	0
$329$	−1.77837	−0.0980448
$330$	0	0
$331$	9.03508	0.496613	0.248307	−	0.968682i	$-0.420126\pi$
$331$	9.03508	0.496613	0.248307	+	0.968682i	$0.420126\pi$
$332$	0	0
$333$	−4.64590	−0.254594
$334$	0	0
$335$	28.5107	1.55771
$336$	0	0
$337$	17.9659	0.978662	0.489331	−	0.872098i	$-0.337241\pi$
$337$	17.9659	0.978662	0.489331	+	0.872098i	$0.337241\pi$
$338$	0	0
$339$	9.46791	0.514226
$340$	0	0
$341$	−12.4979	−0.676801
$342$	0	0
$343$	4.82026	0.260270
$344$	0	0
$345$	−13.3969	−0.721266
$346$	0	0
$347$	−7.78375	−0.417853	−0.208927	−	0.977931i	$-0.566997\pi$
$347$	−7.78375	−0.417853	−0.208927	+	0.977931i	$0.566997\pi$
$348$	0	0
$349$	−6.65002	−0.355967	−0.177984	−	0.984033i	$-0.556957\pi$
$349$	−6.65002	−0.355967	−0.177984	+	0.984033i	$0.556957\pi$
$350$	0	0
$351$	18.5544	0.990359
$352$	0	0
$353$	4.14559	0.220648	0.110324	−	0.993896i	$-0.464811\pi$
$353$	4.14559	0.220648	0.110324	+	0.993896i	$0.464811\pi$
$354$	0	0
$355$	−39.7324	−2.10877
$356$	0	0
$357$	0	0
$358$	0	0
$359$	17.0770	0.901288	0.450644	−	0.892704i	$-0.351194\pi$
$359$	17.0770	0.901288	0.450644	+	0.892704i	$0.351194\pi$
$360$	0	0
$361$	−16.6527	−0.876458
$362$	0	0
$363$	−10.6527	−0.559122
$364$	0	0
$365$	6.01279	0.314724
$366$	0	0
$367$	18.8307	0.982954	0.491477	−	0.870891i	$-0.336457\pi$
$367$	18.8307	0.982954	0.491477	+	0.870891i	$0.336457\pi$
$368$	0	0
$369$	5.83069	0.303534
$370$	0	0
$371$	2.90673	0.150910
$372$	0	0
$373$	−11.7314	−0.607430	−0.303715	−	0.952763i	$-0.598227\pi$
$373$	−11.7314	−0.607430	−0.303715	+	0.952763i	$0.598227\pi$
$374$	0	0
$375$	−11.7811	−0.608371
$376$	0	0
$377$	−3.89899	−0.200808
$378$	0	0
$379$	8.16519	0.419418	0.209709	−	0.977764i	$-0.432748\pi$
$379$	8.16519	0.419418	0.209709	+	0.977764i	$0.432748\pi$
$380$	0	0
$381$	−10.4338	−0.534538
$382$	0	0
$383$	−16.2540	−0.830542	−0.415271	−	0.909698i	$-0.636313\pi$
$383$	−16.2540	−0.830542	−0.415271	+	0.909698i	$0.636313\pi$
$384$	0	0
$385$	−2.15745	−0.109954
$386$	0	0
$387$	13.0155	0.661614
$388$	0	0
$389$	−5.80747	−0.294450	−0.147225	−	0.989103i	$-0.547034\pi$
$389$	−5.80747	−0.294450	−0.147225	+	0.989103i	$0.547034\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	25.0847	1.26536
$394$	0	0
$395$	−40.9077	−2.05829
$396$	0	0
$397$	−10.2517	−0.514516	−0.257258	−	0.966343i	$-0.582819\pi$
$397$	−10.2517	−0.514516	−0.257258	+	0.966343i	$0.582819\pi$
$398$	0	0
$399$	0.716881	0.0358890
$400$	0	0
$401$	−30.5621	−1.52620	−0.763100	−	0.646281i	$-0.776323\pi$
$401$	−30.5621	−1.52620	−0.763100	+	0.646281i	$0.776323\pi$
$402$	0	0
$403$	−23.3851	−1.16489
$404$	0	0
$405$	14.2763	0.709396
$406$	0	0
$407$	−6.89662	−0.341853
$408$	0	0
$409$	19.9736	0.987631	0.493815	−	0.869567i	$-0.335602\pi$
$409$	19.9736	0.987631	0.493815	+	0.869567i	$0.335602\pi$
$410$	0	0
$411$	13.5699	0.669352
$412$	0	0
$413$	−3.77332	−0.185673
$414$	0	0
$415$	−7.58677	−0.372420
$416$	0	0
$417$	−11.5253	−0.564395
$418$	0	0
$419$	−16.9290	−0.827037	−0.413518	−	0.910496i	$-0.635700\pi$
$419$	−16.9290	−0.827037	−0.413518	+	0.910496i	$0.635700\pi$
$420$	0	0
$421$	−23.7297	−1.15651	−0.578257	−	0.815855i	$-0.696267\pi$
$421$	−23.7297	−1.15651	−0.578257	+	0.815855i	$0.696267\pi$
$422$	0	0
$423$	−6.06687	−0.294981
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.53209	−0.0741430
$428$	0	0
$429$	7.79797	0.376490
$430$	0	0
$431$	23.5921	1.13639	0.568197	−	0.822893i	$-0.307641\pi$
$431$	23.5921	1.13639	0.568197	+	0.822893i	$0.307641\pi$
$432$	0	0
$433$	−12.6946	−0.610063	−0.305032	−	0.952342i	$-0.598667\pi$
$433$	−12.6946	−0.610063	−0.305032	+	0.952342i	$0.598667\pi$
$434$	0	0
$435$	−5.63816	−0.270329
$436$	0	0
$437$	−4.31315	−0.206326
$438$	0	0
$439$	23.4706	1.12019	0.560095	−	0.828428i	$-0.310765\pi$
$439$	23.4706	1.12019	0.560095	+	0.828428i	$0.310765\pi$
$440$	0	0
$441$	8.15064	0.388126
$442$	0	0
$443$	19.4124	0.922311	0.461156	−	0.887319i	$-0.347435\pi$
$443$	19.4124	0.922311	0.461156	+	0.887319i	$0.347435\pi$
$444$	0	0
$445$	−22.6732	−1.07481
$446$	0	0
$447$	16.0051	0.757013
$448$	0	0
$449$	−17.6263	−0.831836	−0.415918	−	0.909402i	$-0.636540\pi$
$449$	−17.6263	−0.831836	−0.415918	+	0.909402i	$0.636540\pi$
$450$	0	0
$451$	8.65539	0.407566
$452$	0	0
$453$	−8.83750	−0.415222
$454$	0	0
$455$	−4.03684	−0.189250
$456$	0	0
$457$	−37.1215	−1.73647	−0.868236	−	0.496151i	$-0.834746\pi$
$457$	−37.1215	−1.73647	−0.868236	+	0.496151i	$0.834746\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	29.4074	1.36964	0.684819	−	0.728714i	$-0.259881\pi$
$461$	29.4074	1.36964	0.684819	+	0.728714i	$0.259881\pi$
$462$	0	0
$463$	9.72462	0.451942	0.225971	−	0.974134i	$-0.427445\pi$
$463$	9.72462	0.451942	0.225971	+	0.974134i	$0.427445\pi$
$464$	0	0
$465$	−33.8161	−1.56819
$466$	0	0
$467$	4.62267	0.213912	0.106956	−	0.994264i	$-0.465890\pi$
$467$	4.62267	0.213912	0.106956	+	0.994264i	$0.465890\pi$
$468$	0	0
$469$	2.80335	0.129447
$470$	0	0
$471$	11.9700	0.551547
$472$	0	0
$473$	19.3209	0.888375
$474$	0	0
$475$	−11.4534	−0.525516
$476$	0	0
$477$	9.91622	0.454033
$478$	0	0
$479$	6.83656	0.312371	0.156185	−	0.987728i	$-0.450080\pi$
$479$	6.83656	0.312371	0.156185	+	0.987728i	$0.450080\pi$
$480$	0	0
$481$	−12.9044	−0.588388
$482$	0	0
$483$	−1.31727	−0.0599377
$484$	0	0
$485$	14.4243	0.654972
$486$	0	0
$487$	−7.66281	−0.347235	−0.173617	−	0.984813i	$-0.555546\pi$
$487$	−7.66281	−0.347235	−0.173617	+	0.984813i	$0.555546\pi$
$488$	0	0
$489$	−13.0419	−0.589775
$490$	0	0
$491$	−9.02465	−0.407277	−0.203638	−	0.979046i	$-0.565277\pi$
$491$	−9.02465	−0.407277	−0.203638	+	0.979046i	$0.565277\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−7.36009	−0.330811
$496$	0	0
$497$	−3.90673	−0.175241
$498$	0	0
$499$	26.6928	1.19494	0.597468	−	0.801893i	$-0.296174\pi$
$499$	26.6928	1.19494	0.597468	+	0.801893i	$0.296174\pi$
$500$	0	0
$501$	2.47565	0.110604
$502$	0	0
$503$	12.9267	0.576371	0.288185	−	0.957575i	$-0.406948\pi$
$503$	12.9267	0.576371	0.288185	+	0.957575i	$0.406948\pi$
$504$	0	0
$505$	−18.6091	−0.828092
$506$	0	0
$507$	−2.92396	−0.129858
$508$	0	0
$509$	−27.8753	−1.23555	−0.617775	−	0.786355i	$-0.711966\pi$
$509$	−27.8753	−1.23555	−0.617775	+	0.786355i	$0.711966\pi$
$510$	0	0
$511$	0.591214	0.0261538
$512$	0	0
$513$	8.63816	0.381384
$514$	0	0
$515$	−65.1275	−2.86986
$516$	0	0
$517$	−9.00599	−0.396083
$518$	0	0
$519$	−30.6641	−1.34600
$520$	0	0
$521$	−5.11886	−0.224261	−0.112131	−	0.993693i	$-0.535768\pi$
$521$	−5.11886	−0.224261	−0.112131	+	0.993693i	$0.535768\pi$
$522$	0	0
$523$	−15.3182	−0.669818	−0.334909	−	0.942250i	$-0.608706\pi$
$523$	−15.3182	−0.669818	−0.334909	+	0.942250i	$0.608706\pi$
$524$	0	0
$525$	−3.49794	−0.152663
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−15.0746	−0.655418
$530$	0	0
$531$	−12.8726	−0.558622
$532$	0	0
$533$	16.1952	0.701493
$534$	0	0
$535$	−18.9385	−0.818783
$536$	0	0
$537$	23.6117	1.01892
$538$	0	0
$539$	12.0993	0.521152
$540$	0	0
$541$	−37.4466	−1.60995	−0.804977	−	0.593307i	$-0.797822\pi$
$541$	−37.4466	−1.60995	−0.804977	+	0.593307i	$0.797822\pi$
$542$	0	0
$543$	6.14653	0.263773
$544$	0	0
$545$	−54.8093	−2.34777
$546$	0	0
$547$	27.3928	1.17123	0.585616	−	0.810589i	$-0.300853\pi$
$547$	27.3928	1.17123	0.585616	+	0.810589i	$0.300853\pi$
$548$	0	0
$549$	−5.22668	−0.223069
$550$	0	0
$551$	−1.81521	−0.0773304
$552$	0	0
$553$	−4.02229	−0.171045
$554$	0	0
$555$	−18.6604	−0.792092
$556$	0	0
$557$	9.07604	0.384564	0.192282	−	0.981340i	$-0.438411\pi$
$557$	9.07604	0.384564	0.192282	+	0.981340i	$0.438411\pi$
$558$	0	0
$559$	36.1516	1.52905
$560$	0	0
$561$	0	0
$562$	0	0
$563$	38.4894	1.62213	0.811067	−	0.584953i	$-0.198887\pi$
$563$	38.4894	1.62213	0.811067	+	0.584953i	$0.198887\pi$
$564$	0	0
$565$	−24.8212	−1.04424
$566$	0	0
$567$	1.40373	0.0589513
$568$	0	0
$569$	45.1830	1.89417	0.947086	−	0.320981i	$-0.104012\pi$
$569$	45.1830	1.89417	0.947086	+	0.320981i	$0.104012\pi$
$570$	0	0
$571$	−30.4570	−1.27459	−0.637293	−	0.770622i	$-0.719946\pi$
$571$	−30.4570	−1.27459	−0.637293	+	0.770622i	$0.719946\pi$
$572$	0	0
$573$	13.1019	0.547342
$574$	0	0
$575$	21.0455	0.877658
$576$	0	0
$577$	−9.90167	−0.412212	−0.206106	−	0.978530i	$-0.566079\pi$
$577$	−9.90167	−0.412212	−0.206106	+	0.978530i	$0.566079\pi$
$578$	0	0
$579$	1.13753	0.0472740
$580$	0	0
$581$	−0.745977	−0.0309484
$582$	0	0
$583$	14.7202	0.609648
$584$	0	0
$585$	−13.7716	−0.569384
$586$	0	0
$587$	36.1343	1.49142	0.745712	−	0.666268i	$-0.232110\pi$
$587$	36.1343	1.49142	0.745712	+	0.666268i	$0.232110\pi$
$588$	0	0
$589$	−10.8871	−0.448596
$590$	0	0
$591$	−22.9053	−0.942198
$592$	0	0
$593$	−7.41653	−0.304560	−0.152280	−	0.988337i	$-0.548662\pi$
$593$	−7.41653	−0.304560	−0.152280	+	0.988337i	$0.548662\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−17.1046	−0.700046
$598$	0	0
$599$	−7.61350	−0.311079	−0.155540	−	0.987830i	$-0.549712\pi$
$599$	−7.61350	−0.311079	−0.155540	+	0.987830i	$0.549712\pi$
$600$	0	0
$601$	32.5732	1.32869	0.664343	−	0.747427i	$-0.268711\pi$
$601$	32.5732	1.32869	0.664343	+	0.747427i	$0.268711\pi$
$602$	0	0
$603$	9.56355	0.389458
$604$	0	0
$605$	27.9273	1.13540
$606$	0	0
$607$	−3.01455	−0.122357	−0.0611784	−	0.998127i	$-0.519486\pi$
$607$	−3.01455	−0.122357	−0.0611784	+	0.998127i	$0.519486\pi$
$608$	0	0
$609$	−0.554378	−0.0224645
$610$	0	0
$611$	−16.8512	−0.681728
$612$	0	0
$613$	7.26857	0.293575	0.146787	−	0.989168i	$-0.453107\pi$
$613$	7.26857	0.293575	0.146787	+	0.989168i	$0.453107\pi$
$614$	0	0
$615$	23.4192	0.944354
$616$	0	0
$617$	34.2449	1.37865	0.689323	−	0.724454i	$-0.257908\pi$
$617$	34.2449	1.37865	0.689323	+	0.724454i	$0.257908\pi$
$618$	0	0
$619$	20.4296	0.821137	0.410568	−	0.911830i	$-0.365330\pi$
$619$	20.4296	0.821137	0.410568	+	0.911830i	$0.365330\pi$
$620$	0	0
$621$	−15.8726	−0.636945
$622$	0	0
$623$	−2.22937	−0.0893178
$624$	0	0
$625$	−6.49289	−0.259716
$626$	0	0
$627$	3.63041	0.144985
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−15.5080	−0.617366	−0.308683	−	0.951165i	$-0.599888\pi$
$631$	−15.5080	−0.617366	−0.308683	+	0.951165i	$0.599888\pi$
$632$	0	0
$633$	31.7023	1.26005
$634$	0	0
$635$	27.3533	1.08548
$636$	0	0
$637$	22.6391	0.896993
$638$	0	0
$639$	−13.3277	−0.527236
$640$	0	0
$641$	−32.5945	−1.28741	−0.643703	−	0.765275i	$-0.722603\pi$
$641$	−32.5945	−1.28741	−0.643703	+	0.765275i	$0.722603\pi$
$642$	0	0
$643$	−41.2080	−1.62509	−0.812543	−	0.582902i	$-0.801917\pi$
$643$	−41.2080	−1.62509	−0.812543	+	0.582902i	$0.801917\pi$
$644$	0	0
$645$	52.2772	2.05841
$646$	0	0
$647$	−26.5131	−1.04234	−0.521169	−	0.853454i	$-0.674504\pi$
$647$	−26.5131	−1.04234	−0.521169	+	0.853454i	$0.674504\pi$
$648$	0	0
$649$	−19.1088	−0.750084
$650$	0	0
$651$	−3.32501	−0.130317
$652$	0	0
$653$	−14.1875	−0.555199	−0.277600	−	0.960697i	$-0.589539\pi$
$653$	−14.1875	−0.555199	−0.277600	+	0.960697i	$0.589539\pi$
$654$	0	0
$655$	−65.7624	−2.56955
$656$	0	0
$657$	2.01691	0.0786872
$658$	0	0
$659$	28.3628	1.10486	0.552428	−	0.833560i	$-0.313701\pi$
$659$	28.3628	1.10486	0.552428	+	0.833560i	$0.313701\pi$
$660$	0	0
$661$	29.1661	1.13443	0.567215	−	0.823569i	$-0.308021\pi$
$661$	29.1661	1.13443	0.567215	+	0.823569i	$0.308021\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.87939	−0.0728794
$666$	0	0
$667$	3.33544	0.129149
$668$	0	0
$669$	−6.69728	−0.258932
$670$	0	0
$671$	−7.75877	−0.299524
$672$	0	0
$673$	−29.3678	−1.13205	−0.566023	−	0.824389i	$-0.691519\pi$
$673$	−29.3678	−1.13205	−0.566023	+	0.824389i	$0.691519\pi$
$674$	0	0
$675$	−42.1489	−1.62231
$676$	0	0
$677$	39.3783	1.51343	0.756715	−	0.653745i	$-0.226803\pi$
$677$	39.3783	1.51343	0.756715	+	0.653745i	$0.226803\pi$
$678$	0	0
$679$	1.41828	0.0544286
$680$	0	0
$681$	−7.08235	−0.271396
$682$	0	0
$683$	−27.1634	−1.03938	−0.519690	−	0.854355i	$-0.673953\pi$
$683$	−27.1634	−1.03938	−0.519690	+	0.854355i	$0.673953\pi$
$684$	0	0
$685$	−35.5749	−1.35925
$686$	0	0
$687$	−32.6117	−1.24422
$688$	0	0
$689$	27.5431	1.04931
$690$	0	0
$691$	33.8503	1.28773	0.643863	−	0.765141i	$-0.277331\pi$
$691$	33.8503	1.28773	0.643863	+	0.765141i	$0.277331\pi$
$692$	0	0
$693$	−0.723689	−0.0274907
$694$	0	0
$695$	30.2148	1.14611
$696$	0	0
$697$	0	0
$698$	0	0
$699$	38.6970	1.46365
$700$	0	0
$701$	20.9540	0.791421	0.395711	−	0.918375i	$-0.370498\pi$
$701$	20.9540	0.791421	0.395711	+	0.918375i	$0.370498\pi$
$702$	0	0
$703$	−6.00774	−0.226586
$704$	0	0
$705$	−24.3678	−0.917746
$706$	0	0
$707$	−1.82976	−0.0688150
$708$	0	0
$709$	−31.3236	−1.17638	−0.588191	−	0.808722i	$-0.700160\pi$
$709$	−31.3236	−1.17638	−0.588191	+	0.808722i	$0.700160\pi$
$710$	0	0
$711$	−13.7219	−0.514613
$712$	0	0
$713$	20.0051	0.749195
$714$	0	0
$715$	−20.4433	−0.764535
$716$	0	0
$717$	3.54757	0.132486
$718$	0	0
$719$	44.4097	1.65620	0.828102	−	0.560578i	$-0.189421\pi$
$719$	44.4097	1.65620	0.828102	+	0.560578i	$0.189421\pi$
$720$	0	0
$721$	−6.40373	−0.238487
$722$	0	0
$723$	34.1320	1.26938
$724$	0	0
$725$	8.85710	0.328944
$726$	0	0
$727$	39.5354	1.46629	0.733143	−	0.680074i	$-0.238053\pi$
$727$	39.5354	1.46629	0.733143	+	0.680074i	$0.238053\pi$
$728$	0	0
$729$	27.5776	1.02139
$730$	0	0
$731$	0	0
$732$	0	0
$733$	13.8990	0.513371	0.256685	−	0.966495i	$-0.417370\pi$
$733$	13.8990	0.513371	0.256685	+	0.966495i	$0.417370\pi$
$734$	0	0
$735$	32.7374	1.20754
$736$	0	0
$737$	14.1967	0.522940
$738$	0	0
$739$	−11.4037	−0.419493	−0.209747	−	0.977756i	$-0.567264\pi$
$739$	−11.4037	−0.419493	−0.209747	+	0.977756i	$0.567264\pi$
$740$	0	0
$741$	6.79292	0.249544
$742$	0	0
$743$	14.6209	0.536390	0.268195	−	0.963365i	$-0.413573\pi$
$743$	14.6209	0.536390	0.268195	+	0.963365i	$0.413573\pi$
$744$	0	0
$745$	−41.9590	−1.53726
$746$	0	0
$747$	−2.54488	−0.0931124
$748$	0	0
$749$	−1.86215	−0.0680414
$750$	0	0
$751$	−28.0196	−1.02245	−0.511225	−	0.859447i	$-0.670808\pi$
$751$	−28.0196	−1.02245	−0.511225	+	0.859447i	$0.670808\pi$
$752$	0	0
$753$	−14.3960	−0.524619
$754$	0	0
$755$	23.1685	0.843188
$756$	0	0
$757$	15.6186	0.567666	0.283833	−	0.958874i	$-0.408394\pi$
$757$	15.6186	0.567666	0.283833	+	0.958874i	$0.408394\pi$
$758$	0	0
$759$	−6.67087	−0.242137
$760$	0	0
$761$	20.5868	0.746270	0.373135	−	0.927777i	$-0.378283\pi$
$761$	20.5868	0.746270	0.373135	+	0.927777i	$0.378283\pi$
$762$	0	0
$763$	−5.38919	−0.195102
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−35.7547	−1.29103
$768$	0	0
$769$	22.2249	0.801451	0.400726	−	0.916198i	$-0.368758\pi$
$769$	22.2249	0.801451	0.400726	+	0.916198i	$0.368758\pi$
$770$	0	0
$771$	−5.03415	−0.181300
$772$	0	0
$773$	−27.0087	−0.971434	−0.485717	−	0.874116i	$-0.661442\pi$
$773$	−27.0087	−0.971434	−0.485717	+	0.874116i	$0.661442\pi$
$774$	0	0
$775$	53.1225	1.90822
$776$	0	0
$777$	−1.83481	−0.0658234
$778$	0	0
$779$	7.53983	0.270142
$780$	0	0
$781$	−19.7844	−0.707940
$782$	0	0
$783$	−6.68004	−0.238725
$784$	0	0
$785$	−31.3806	−1.12002
$786$	0	0
$787$	−20.9810	−0.747892	−0.373946	−	0.927450i	$-0.621996\pi$
$787$	−20.9810	−0.747892	−0.373946	+	0.927450i	$0.621996\pi$
$788$	0	0
$789$	−13.7551	−0.489696
$790$	0	0
$791$	−2.44057	−0.0867767
$792$	0	0
$793$	−14.5175	−0.515533
$794$	0	0
$795$	39.8289	1.41259
$796$	0	0
$797$	34.4347	1.21974	0.609870	−	0.792502i	$-0.291222\pi$
$797$	34.4347	1.21974	0.609870	+	0.792502i	$0.291222\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−7.60544	−0.268725
$802$	0	0
$803$	2.99401	0.105656
$804$	0	0
$805$	3.45336	0.121715
$806$	0	0
$807$	−15.8161	−0.556755
$808$	0	0
$809$	28.9246	1.01693	0.508467	−	0.861082i	$-0.330212\pi$
$809$	28.9246	1.01693	0.508467	+	0.861082i	$0.330212\pi$
$810$	0	0
$811$	−38.6236	−1.35626	−0.678129	−	0.734943i	$-0.737209\pi$
$811$	−38.6236	−1.35626	−0.678129	+	0.734943i	$0.737209\pi$
$812$	0	0
$813$	−22.9040	−0.803280
$814$	0	0
$815$	34.1908	1.19765
$816$	0	0
$817$	16.8307	0.588831
$818$	0	0
$819$	−1.35410	−0.0473162
$820$	0	0
$821$	5.39599	0.188321	0.0941607	−	0.995557i	$-0.469983\pi$
$821$	5.39599	0.188321	0.0941607	+	0.995557i	$0.469983\pi$
$822$	0	0
$823$	−30.6878	−1.06971	−0.534854	−	0.844944i	$-0.679634\pi$
$823$	−30.6878	−1.06971	−0.534854	+	0.844944i	$0.679634\pi$
$824$	0	0
$825$	−17.7142	−0.616729
$826$	0	0
$827$	−35.5381	−1.23578	−0.617890	−	0.786265i	$-0.712012\pi$
$827$	−35.5381	−1.23578	−0.617890	+	0.786265i	$0.712012\pi$
$828$	0	0
$829$	3.17200	0.110168	0.0550840	−	0.998482i	$-0.482457\pi$
$829$	3.17200	0.110168	0.0550840	+	0.998482i	$0.482457\pi$
$830$	0	0
$831$	32.7975	1.13773
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−6.49020	−0.224603
$836$	0	0
$837$	−40.0651	−1.38485
$838$	0	0
$839$	36.8152	1.27100	0.635501	−	0.772100i	$-0.280794\pi$
$839$	36.8152	1.27100	0.635501	+	0.772100i	$0.280794\pi$
$840$	0	0
$841$	−27.5963	−0.951595
$842$	0	0
$843$	−7.80604	−0.268854
$844$	0	0
$845$	7.66550	0.263701
$846$	0	0
$847$	2.74598	0.0943529
$848$	0	0
$849$	−20.6405	−0.708381
$850$	0	0
$851$	11.0392	0.378419
$852$	0	0
$853$	20.0651	0.687016	0.343508	−	0.939150i	$-0.388385\pi$
$853$	20.0651	0.687016	0.343508	+	0.939150i	$0.388385\pi$
$854$	0	0
$855$	−6.41147	−0.219268
$856$	0	0
$857$	19.9299	0.680794	0.340397	−	0.940282i	$-0.389439\pi$
$857$	19.9299	0.680794	0.340397	+	0.940282i	$0.389439\pi$
$858$	0	0
$859$	23.3527	0.796783	0.398391	−	0.917215i	$-0.369569\pi$
$859$	23.3527	0.796783	0.398391	+	0.917215i	$0.369569\pi$
$860$	0	0
$861$	2.30272	0.0784765
$862$	0	0
$863$	−44.3878	−1.51098	−0.755488	−	0.655162i	$-0.772600\pi$
$863$	−44.3878	−1.51098	−0.755488	+	0.655162i	$0.772600\pi$
$864$	0	0
$865$	80.3893	2.73332
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−20.3696	−0.690991
$870$	0	0
$871$	26.5635	0.900072
$872$	0	0
$873$	4.83843	0.163756
$874$	0	0
$875$	3.03684	0.102664
$876$	0	0
$877$	−9.14971	−0.308964	−0.154482	−	0.987996i	$-0.549371\pi$
$877$	−9.14971	−0.308964	−0.154482	+	0.987996i	$0.549371\pi$
$878$	0	0
$879$	4.01548	0.135439
$880$	0	0
$881$	−6.76794	−0.228018	−0.114009	−	0.993480i	$-0.536369\pi$
$881$	−6.76794	−0.228018	−0.114009	+	0.993480i	$0.536369\pi$
$882$	0	0
$883$	47.9760	1.61452	0.807260	−	0.590196i	$-0.200950\pi$
$883$	47.9760	1.61452	0.807260	+	0.590196i	$0.200950\pi$
$884$	0	0
$885$	−51.7033	−1.73799
$886$	0	0
$887$	−41.6982	−1.40009	−0.700045	−	0.714099i	$-0.746837\pi$
$887$	−41.6982	−1.40009	−0.700045	+	0.714099i	$0.746837\pi$
$888$	0	0
$889$	2.68954	0.0902043
$890$	0	0
$891$	7.10876	0.238152
$892$	0	0
$893$	−7.84524	−0.262531
$894$	0	0
$895$	−61.9009	−2.06912
$896$	0	0
$897$	−12.4820	−0.416761
$898$	0	0
$899$	8.41921	0.280797
$900$	0	0
$901$	0	0
$902$	0	0
$903$	5.14022	0.171056
$904$	0	0
$905$	−16.1138	−0.535641
$906$	0	0
$907$	32.9350	1.09359	0.546794	−	0.837267i	$-0.315848\pi$
$907$	32.9350	1.09359	0.546794	+	0.837267i	$0.315848\pi$
$908$	0	0
$909$	−6.24216	−0.207039
$910$	0	0
$911$	18.4926	0.612686	0.306343	−	0.951921i	$-0.400895\pi$
$911$	18.4926	0.612686	0.306343	+	0.951921i	$0.400895\pi$
$912$	0	0
$913$	−3.77776	−0.125026
$914$	0	0
$915$	−20.9932	−0.694014
$916$	0	0
$917$	−6.46616	−0.213531
$918$	0	0
$919$	13.3909	0.441726	0.220863	−	0.975305i	$-0.429113\pi$
$919$	13.3909	0.441726	0.220863	+	0.975305i	$0.429113\pi$
$920$	0	0
$921$	4.40373	0.145108
$922$	0	0
$923$	−37.0188	−1.21849
$924$	0	0
$925$	29.3141	0.963841
$926$	0	0
$927$	−21.8462	−0.717522
$928$	0	0
$929$	−5.33956	−0.175185	−0.0875926	−	0.996156i	$-0.527917\pi$
$929$	−5.33956	−0.175185	−0.0875926	+	0.996156i	$0.527917\pi$
$930$	0	0
$931$	10.5398	0.345429
$932$	0	0
$933$	4.75877	0.155795
$934$	0	0
$935$	0	0
$936$	0	0
$937$	41.4361	1.35366	0.676830	−	0.736140i	$-0.263353\pi$
$937$	41.4361	1.35366	0.676830	+	0.736140i	$0.263353\pi$
$938$	0	0
$939$	−1.99731	−0.0651798
$940$	0	0
$941$	42.0601	1.37112	0.685559	−	0.728017i	$-0.259558\pi$
$941$	42.0601	1.37112	0.685559	+	0.728017i	$0.259558\pi$
$942$	0	0
$943$	−13.8544	−0.451162
$944$	0	0
$945$	−6.91622	−0.224985
$946$	0	0
$947$	−41.4270	−1.34620	−0.673098	−	0.739554i	$-0.735037\pi$
$947$	−41.4270	−1.34620	−0.673098	+	0.739554i	$0.735037\pi$
$948$	0	0
$949$	5.60214	0.181853
$950$	0	0
$951$	19.4870	0.631910
$952$	0	0
$953$	27.0719	0.876945	0.438473	−	0.898744i	$-0.355520\pi$
$953$	27.0719	0.876945	0.438473	+	0.898744i	$0.355520\pi$
$954$	0	0
$955$	−34.3482	−1.11148
$956$	0	0
$957$	−2.80747	−0.0907525
$958$	0	0
$959$	−3.49794	−0.112954
$960$	0	0
$961$	19.4962	0.628909
$962$	0	0
$963$	−6.35267	−0.204712
$964$	0	0
$965$	−2.98215	−0.0959989
$966$	0	0
$967$	43.3637	1.39448	0.697241	−	0.716836i	$-0.254411\pi$
$967$	43.3637	1.39448	0.697241	+	0.716836i	$0.254411\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.27126	0.0407966	0.0203983	−	0.999792i	$-0.493507\pi$
$971$	1.27126	0.0407966	0.0203983	+	0.999792i	$0.493507\pi$
$972$	0	0
$973$	2.97090	0.0952428
$974$	0	0
$975$	−33.1453	−1.06150
$976$	0	0
$977$	−24.7641	−0.792275	−0.396138	−	0.918191i	$-0.629650\pi$
$977$	−24.7641	−0.792275	−0.396138	+	0.918191i	$0.629650\pi$
$978$	0	0
$979$	−11.2899	−0.360828
$980$	0	0
$981$	−18.3851	−0.586990
$982$	0	0
$983$	39.6144	1.26350	0.631752	−	0.775170i	$-0.282336\pi$
$983$	39.6144	1.26350	0.631752	+	0.775170i	$0.282336\pi$
$984$	0	0
$985$	60.0488	1.91331
$986$	0	0
$987$	−2.39599	−0.0762653
$988$	0	0
$989$	−30.9263	−0.983400
$990$	0	0
$991$	33.3138	1.05825	0.529123	−	0.848545i	$-0.322521\pi$
$991$	33.3138	1.05825	0.529123	+	0.848545i	$0.322521\pi$
$992$	0	0
$993$	12.1729	0.386296
$994$	0	0
$995$	44.8417	1.42158
$996$	0	0
$997$	−25.8958	−0.820128	−0.410064	−	0.912057i	$-0.634494\pi$
$997$	−25.8958	−0.820128	−0.410064	+	0.912057i	$0.634494\pi$
$998$	0	0
$999$	−22.1088	−0.699490

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4624.2.a.bg.1.2		3
4.3	odd	2		289.2.a.d.1.3	✓	3
12.11	even	2		2601.2.a.x.1.1		3
17.16	even	2		4624.2.a.bd.1.2		3
20.19	odd	2		7225.2.a.t.1.1		3
68.3	even	16		289.2.d.f.179.5		24
68.7	even	16		289.2.d.f.134.2		24
68.11	even	16		289.2.d.f.155.6		24
68.15	odd	8		289.2.c.d.38.6		12
68.19	odd	8		289.2.c.d.38.5		12
68.23	even	16		289.2.d.f.155.5		24
68.27	even	16		289.2.d.f.134.1		24
68.31	even	16		289.2.d.f.179.6		24
68.39	even	16		289.2.d.f.110.2		24
68.43	odd	8		289.2.c.d.251.1		12
68.47	odd	4		289.2.b.d.288.1		6
68.55	odd	4		289.2.b.d.288.2		6
68.59	odd	8		289.2.c.d.251.2		12
68.63	even	16		289.2.d.f.110.1		24
68.67	odd	2		289.2.a.e.1.3	yes	3
204.203	even	2		2601.2.a.w.1.1		3
340.339	odd	2		7225.2.a.s.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.2.a.d.1.3	✓	3	4.3	odd	2
289.2.a.e.1.3	yes	3	68.67	odd	2
289.2.b.d.288.1		6	68.47	odd	4
289.2.b.d.288.2		6	68.55	odd	4
289.2.c.d.38.5		12	68.19	odd	8
289.2.c.d.38.6		12	68.15	odd	8
289.2.c.d.251.1		12	68.43	odd	8
289.2.c.d.251.2		12	68.59	odd	8
289.2.d.f.110.1		24	68.63	even	16
289.2.d.f.110.2		24	68.39	even	16
289.2.d.f.134.1		24	68.27	even	16
289.2.d.f.134.2		24	68.7	even	16
289.2.d.f.155.5		24	68.23	even	16
289.2.d.f.155.6		24	68.11	even	16
289.2.d.f.179.5		24	68.3	even	16
289.2.d.f.179.6		24	68.31	even	16
2601.2.a.w.1.1		3	204.203	even	2
2601.2.a.x.1.1		3	12.11	even	2
4624.2.a.bd.1.2		3	17.16	even	2
4624.2.a.bg.1.2		3	1.1	even	1	trivial
7225.2.a.s.1.1		3	340.339	odd	2
7225.2.a.t.1.1		3	20.19	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:	\( N \)	\(=\)	\( 4624 = 2^{4} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4624.a (trivial)

\(f(q)\)	\(=\)	\(q+1.34730 q^{3} -3.53209 q^{5} -0.347296 q^{7} -1.18479 q^{9} +O(q^{10})\)	\(q+1.34730 q^{3} -3.53209 q^{5} -0.347296 q^{7} -1.18479 q^{9} -1.75877 q^{11} -3.29086 q^{13} -4.75877 q^{15} -1.53209 q^{19} -0.467911 q^{21} +2.81521 q^{23} +7.47565 q^{25} -5.63816 q^{27} +1.18479 q^{29} +7.10607 q^{31} -2.36959 q^{33} +1.22668 q^{35} +3.92127 q^{37} -4.43376 q^{39} -4.92127 q^{41} -10.9855 q^{43} +4.18479 q^{45} +5.12061 q^{47} -6.87939 q^{49} -8.36959 q^{53} +6.21213 q^{55} -2.06418 q^{57} +10.8648 q^{59} +4.41147 q^{61} +0.411474 q^{63} +11.6236 q^{65} -8.07192 q^{67} +3.79292 q^{69} +11.2490 q^{71} -1.70233 q^{73} +10.0719 q^{75} +0.610815 q^{77} +11.5817 q^{79} -4.04189 q^{81} +2.14796 q^{83} +1.59627 q^{87} +6.41921 q^{89} +1.14290 q^{91} +9.57398 q^{93} +5.41147 q^{95} -4.08378 q^{97} +2.08378 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 6 q^{5}+O(q^{10}) \) 3 * q + 3 * q^3 - 6 * q^5	\( 3 q + 3 q^{3} - 6 q^{5} + 6 q^{11} + 6 q^{13} - 3 q^{15} - 6 q^{21} + 12 q^{23} + 3 q^{25} + 9 q^{31} - 3 q^{35} + 3 q^{37} + 3 q^{39} - 6 q^{41} - 15 q^{43} + 9 q^{45} + 21 q^{47} - 15 q^{49} - 18 q^{53} - 6 q^{55} + 3 q^{57} + 9 q^{59} + 3 q^{61} - 9 q^{63} + 9 q^{67} + 21 q^{69} + 21 q^{71} + 21 q^{73} - 3 q^{75} + 6 q^{77} + 3 q^{79} - 9 q^{81} - 9 q^{83} - 9 q^{87} - 15 q^{89} + 3 q^{91} + 21 q^{93} + 6 q^{95} - 6 q^{97}+O(q^{100}) \) 3 * q + 3 * q^3 - 6 * q^5 + 6 * q^11 + 6 * q^13 - 3 * q^15 - 6 * q^21 + 12 * q^23 + 3 * q^25 + 9 * q^31 - 3 * q^35 + 3 * q^37 + 3 * q^39 - 6 * q^41 - 15 * q^43 + 9 * q^45 + 21 * q^47 - 15 * q^49 - 18 * q^53 - 6 * q^55 + 3 * q^57 + 9 * q^59 + 3 * q^61 - 9 * q^63 + 9 * q^67 + 21 * q^69 + 21 * q^71 + 21 * q^73 - 3 * q^75 + 6 * q^77 + 3 * q^79 - 9 * q^81 - 9 * q^83 - 9 * q^87 - 15 * q^89 + 3 * q^91 + 21 * q^93 + 6 * q^95 - 6 * q^97

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