LMFDB - Embedded newform 8464.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8464,2,Mod(1,8464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8464, 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("8464.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.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:	$67.5853802708$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4232)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	8464.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949 q^{3} -3.00000 q^{5} -4.44949 q^{7} +3.00000 q^{9} -2.44949 q^{11} -3.89898 q^{13} -7.34847 q^{15} +4.89898 q^{17} -8.44949 q^{19} -10.8990 q^{21} +4.00000 q^{25} -1.89898 q^{29} +1.10102 q^{31} -6.00000 q^{33} +13.3485 q^{35} +8.00000 q^{37} -9.55051 q^{39} -1.89898 q^{41} +2.89898 q^{43} -9.00000 q^{45} -6.44949 q^{47} +12.7980 q^{49} +12.0000 q^{51} -10.7980 q^{53} +7.34847 q^{55} -20.6969 q^{57} +9.34847 q^{59} -3.89898 q^{61} -13.3485 q^{63} +11.6969 q^{65} +5.79796 q^{67} -0.449490 q^{71} +7.89898 q^{73} +9.79796 q^{75} +10.8990 q^{77} -12.0000 q^{79} -9.00000 q^{81} +2.89898 q^{83} -14.6969 q^{85} -4.65153 q^{87} -10.7980 q^{89} +17.3485 q^{91} +2.69694 q^{93} +25.3485 q^{95} +7.89898 q^{97} -7.34847 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{5} - 4 q^{7} + 6 q^{9} + 2 q^{13} - 12 q^{19} - 12 q^{21} + 8 q^{25} + 6 q^{29} + 12 q^{31} - 12 q^{33} + 12 q^{35} + 16 q^{37} - 24 q^{39} + 6 q^{41} - 4 q^{43} - 18 q^{45} - 8 q^{47} + 6 q^{49}+ \cdots + 6 q^{97}+O(q^{100}) $ 2 * q - 6 * q^5 - 4 * q^7 + 6 * q^9 + 2 * q^13 - 12 * q^19 - 12 * q^21 + 8 * q^25 + 6 * q^29 + 12 * q^31 - 12 * q^33 + 12 * q^35 + 16 * q^37 - 24 * q^39 + 6 * q^41 - 4 * q^43 - 18 * q^45 - 8 * q^47 + 6 * q^49 + 24 * q^51 - 2 * q^53 - 12 * q^57 + 4 * q^59 + 2 * q^61 - 12 * q^63 - 6 * q^65 - 8 * q^67 + 4 * q^71 + 6 * q^73 + 12 * q^77 - 24 * q^79 - 18 * q^81 - 4 * q^83 - 24 * q^87 - 2 * q^89 + 20 * q^91 - 24 * q^93 + 36 * q^95 + 6 * q^97

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
$3$	2.44949	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$3$	2.44949	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$4$	0	0
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	−4.44949	−1.68175	−0.840875	−	0.541230i	$-0.817959\pi$
$7$	−4.44949	−1.68175	−0.840875	+	0.541230i	$0.817959\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	−2.44949	−0.738549	−0.369274	−	0.929320i	$-0.620394\pi$
$11$	−2.44949	−0.738549	−0.369274	+	0.929320i	$0.620394\pi$
$12$	0	0
$13$	−3.89898	−1.08138	−0.540691	−	0.841221i	$-0.681837\pi$
$13$	−3.89898	−1.08138	−0.540691	+	0.841221i	$0.681837\pi$
$14$	0	0
$15$	−7.34847	−1.89737
$16$	0	0
$17$	4.89898	1.18818	0.594089	−	0.804400i	$-0.297513\pi$
$17$	4.89898	1.18818	0.594089	+	0.804400i	$0.297513\pi$
$18$	0	0
$19$	−8.44949	−1.93845	−0.969223	−	0.246185i	$-0.920823\pi$
$19$	−8.44949	−1.93845	−0.969223	+	0.246185i	$0.920823\pi$
$20$	0	0
$21$	−10.8990	−2.37835
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.89898	−0.352632	−0.176316	−	0.984334i	$-0.556418\pi$
$29$	−1.89898	−0.352632	−0.176316	+	0.984334i	$0.556418\pi$
$30$	0	0
$31$	1.10102	0.197749	0.0988746	−	0.995100i	$-0.468476\pi$
$31$	1.10102	0.197749	0.0988746	+	0.995100i	$0.468476\pi$
$32$	0	0
$33$	−6.00000	−1.04447
$34$	0	0
$35$	13.3485	2.25630
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	−9.55051	−1.52931
$40$	0	0
$41$	−1.89898	−0.296571	−0.148285	−	0.988945i	$-0.547375\pi$
$41$	−1.89898	−0.296571	−0.148285	+	0.988945i	$0.547375\pi$
$42$	0	0
$43$	2.89898	0.442090	0.221045	−	0.975264i	$-0.429053\pi$
$43$	2.89898	0.442090	0.221045	+	0.975264i	$0.429053\pi$
$44$	0	0
$45$	−9.00000	−1.34164
$46$	0	0
$47$	−6.44949	−0.940755	−0.470377	−	0.882465i	$-0.655882\pi$
$47$	−6.44949	−0.940755	−0.470377	+	0.882465i	$0.655882\pi$
$48$	0	0
$49$	12.7980	1.82828
$50$	0	0
$51$	12.0000	1.68034
$52$	0	0
$53$	−10.7980	−1.48321	−0.741607	−	0.670835i	$-0.765936\pi$
$53$	−10.7980	−1.48321	−0.741607	+	0.670835i	$0.765936\pi$
$54$	0	0
$55$	7.34847	0.990867
$56$	0	0
$57$	−20.6969	−2.74138
$58$	0	0
$59$	9.34847	1.21707	0.608534	−	0.793528i	$-0.291758\pi$
$59$	9.34847	1.21707	0.608534	+	0.793528i	$0.291758\pi$
$60$	0	0
$61$	−3.89898	−0.499213	−0.249607	−	0.968347i	$-0.580301\pi$
$61$	−3.89898	−0.499213	−0.249607	+	0.968347i	$0.580301\pi$
$62$	0	0
$63$	−13.3485	−1.68175
$64$	0	0
$65$	11.6969	1.45083
$66$	0	0
$67$	5.79796	0.708333	0.354167	−	0.935182i	$-0.384765\pi$
$67$	5.79796	0.708333	0.354167	+	0.935182i	$0.384765\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.449490	−0.0533446	−0.0266723	−	0.999644i	$-0.508491\pi$
$71$	−0.449490	−0.0533446	−0.0266723	+	0.999644i	$0.508491\pi$
$72$	0	0
$73$	7.89898	0.924506	0.462253	−	0.886748i	$-0.347041\pi$
$73$	7.89898	0.924506	0.462253	+	0.886748i	$0.347041\pi$
$74$	0	0
$75$	9.79796	1.13137
$76$	0	0
$77$	10.8990	1.24205
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	2.89898	0.318204	0.159102	−	0.987262i	$-0.449140\pi$
$83$	2.89898	0.318204	0.159102	+	0.987262i	$0.449140\pi$
$84$	0	0
$85$	−14.6969	−1.59411
$86$	0	0
$87$	−4.65153	−0.498696
$88$	0	0
$89$	−10.7980	−1.14458	−0.572291	−	0.820051i	$-0.693945\pi$
$89$	−10.7980	−1.14458	−0.572291	+	0.820051i	$0.693945\pi$
$90$	0	0
$91$	17.3485	1.81861
$92$	0	0
$93$	2.69694	0.279659
$94$	0	0
$95$	25.3485	2.60070
$96$	0	0
$97$	7.89898	0.802020	0.401010	−	0.916074i	$-0.368659\pi$
$97$	7.89898	0.802020	0.401010	+	0.916074i	$0.368659\pi$
$98$	0	0
$99$	−7.34847	−0.738549
$100$	0	0
$101$	9.00000	0.895533	0.447767	−	0.894150i	$-0.352219\pi$
$101$	9.00000	0.895533	0.447767	+	0.894150i	$0.352219\pi$
$102$	0	0
$103$	−2.89898	−0.285645	−0.142822	−	0.989748i	$-0.545618\pi$
$103$	−2.89898	−0.285645	−0.142822	+	0.989748i	$0.545618\pi$
$104$	0	0
$105$	32.6969	3.19089
$106$	0	0
$107$	11.5505	1.11663	0.558315	−	0.829629i	$-0.311448\pi$
$107$	11.5505	1.11663	0.558315	+	0.829629i	$0.311448\pi$
$108$	0	0
$109$	18.7980	1.80052	0.900259	−	0.435355i	$-0.143377\pi$
$109$	18.7980	1.80052	0.900259	+	0.435355i	$0.143377\pi$
$110$	0	0
$111$	19.5959	1.85996
$112$	0	0
$113$	−5.00000	−0.470360	−0.235180	−	0.971952i	$-0.575568\pi$
$113$	−5.00000	−0.470360	−0.235180	+	0.971952i	$0.575568\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−11.6969	−1.08138
$118$	0	0
$119$	−21.7980	−1.99822
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	−4.65153	−0.419414
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	2.89898	0.257243	0.128621	−	0.991694i	$-0.458945\pi$
$127$	2.89898	0.257243	0.128621	+	0.991694i	$0.458945\pi$
$128$	0	0
$129$	7.10102	0.625210
$130$	0	0
$131$	−6.24745	−0.545842	−0.272921	−	0.962036i	$-0.587990\pi$
$131$	−6.24745	−0.545842	−0.272921	+	0.962036i	$0.587990\pi$
$132$	0	0
$133$	37.5959	3.25998
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.10102	−0.179502	−0.0897511	−	0.995964i	$-0.528607\pi$
$137$	−2.10102	−0.179502	−0.0897511	+	0.995964i	$0.528607\pi$
$138$	0	0
$139$	16.6969	1.41622	0.708108	−	0.706104i	$-0.249549\pi$
$139$	16.6969	1.41622	0.708108	+	0.706104i	$0.249549\pi$
$140$	0	0
$141$	−15.7980	−1.33043
$142$	0	0
$143$	9.55051	0.798654
$144$	0	0
$145$	5.69694	0.473105
$146$	0	0
$147$	31.3485	2.58558
$148$	0	0
$149$	5.89898	0.483263	0.241632	−	0.970368i	$-0.422317\pi$
$149$	5.89898	0.483263	0.241632	+	0.970368i	$0.422317\pi$
$150$	0	0
$151$	8.24745	0.671168	0.335584	−	0.942010i	$-0.391066\pi$
$151$	8.24745	0.671168	0.335584	+	0.942010i	$0.391066\pi$
$152$	0	0
$153$	14.6969	1.18818
$154$	0	0
$155$	−3.30306	−0.265308
$156$	0	0
$157$	−3.00000	−0.239426	−0.119713	−	0.992809i	$-0.538197\pi$
$157$	−3.00000	−0.239426	−0.119713	+	0.992809i	$0.538197\pi$
$158$	0	0
$159$	−26.4495	−2.09758
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−12.4495	−0.975119	−0.487560	−	0.873090i	$-0.662113\pi$
$163$	−12.4495	−0.975119	−0.487560	+	0.873090i	$0.662113\pi$
$164$	0	0
$165$	18.0000	1.40130
$166$	0	0
$167$	−14.8990	−1.15292	−0.576459	−	0.817126i	$-0.695566\pi$
$167$	−14.8990	−1.15292	−0.576459	+	0.817126i	$0.695566\pi$
$168$	0	0
$169$	2.20204	0.169388
$170$	0	0
$171$	−25.3485	−1.93845
$172$	0	0
$173$	20.7980	1.58124	0.790620	−	0.612307i	$-0.209759\pi$
$173$	20.7980	1.58124	0.790620	+	0.612307i	$0.209759\pi$
$174$	0	0
$175$	−17.7980	−1.34540
$176$	0	0
$177$	22.8990	1.72119
$178$	0	0
$179$	10.8990	0.814628	0.407314	−	0.913288i	$-0.366465\pi$
$179$	10.8990	0.814628	0.407314	+	0.913288i	$0.366465\pi$
$180$	0	0
$181$	16.8990	1.25609	0.628046	−	0.778177i	$-0.283855\pi$
$181$	16.8990	1.25609	0.628046	+	0.778177i	$0.283855\pi$
$182$	0	0
$183$	−9.55051	−0.705994
$184$	0	0
$185$	−24.0000	−1.76452
$186$	0	0
$187$	−12.0000	−0.877527
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2.89898	−0.209763	−0.104881	−	0.994485i	$-0.533446\pi$
$191$	−2.89898	−0.209763	−0.104881	+	0.994485i	$0.533446\pi$
$192$	0	0
$193$	−17.8990	−1.28840	−0.644198	−	0.764858i	$-0.722809\pi$
$193$	−17.8990	−1.28840	−0.644198	+	0.764858i	$0.722809\pi$
$194$	0	0
$195$	28.6515	2.05178
$196$	0	0
$197$	10.1010	0.719668	0.359834	−	0.933016i	$-0.382833\pi$
$197$	10.1010	0.719668	0.359834	+	0.933016i	$0.382833\pi$
$198$	0	0
$199$	−8.65153	−0.613291	−0.306645	−	0.951824i	$-0.599207\pi$
$199$	−8.65153	−0.613291	−0.306645	+	0.951824i	$0.599207\pi$
$200$	0	0
$201$	14.2020	1.00173
$202$	0	0
$203$	8.44949	0.593038
$204$	0	0
$205$	5.69694	0.397891
$206$	0	0
$207$	0	0
$208$	0	0
$209$	20.6969	1.43164
$210$	0	0
$211$	16.2474	1.11852	0.559260	−	0.828992i	$-0.311085\pi$
$211$	16.2474	1.11852	0.559260	+	0.828992i	$0.311085\pi$
$212$	0	0
$213$	−1.10102	−0.0754407
$214$	0	0
$215$	−8.69694	−0.593126
$216$	0	0
$217$	−4.89898	−0.332564
$218$	0	0
$219$	19.3485	1.30745
$220$	0	0
$221$	−19.1010	−1.28487
$222$	0	0
$223$	−10.4495	−0.699750	−0.349875	−	0.936796i	$-0.613776\pi$
$223$	−10.4495	−0.699750	−0.349875	+	0.936796i	$0.613776\pi$
$224$	0	0
$225$	12.0000	0.800000
$226$	0	0
$227$	−13.7980	−0.915803	−0.457901	−	0.889003i	$-0.651399\pi$
$227$	−13.7980	−0.915803	−0.457901	+	0.889003i	$0.651399\pi$
$228$	0	0
$229$	−13.7980	−0.911795	−0.455897	−	0.890032i	$-0.650682\pi$
$229$	−13.7980	−0.911795	−0.455897	+	0.890032i	$0.650682\pi$
$230$	0	0
$231$	26.6969	1.75653
$232$	0	0
$233$	16.5959	1.08723	0.543617	−	0.839333i	$-0.317054\pi$
$233$	16.5959	1.08723	0.543617	+	0.839333i	$0.317054\pi$
$234$	0	0
$235$	19.3485	1.26215
$236$	0	0
$237$	−29.3939	−1.90934
$238$	0	0
$239$	−17.7980	−1.15125	−0.575627	−	0.817712i	$-0.695242\pi$
$239$	−17.7980	−1.15125	−0.575627	+	0.817712i	$0.695242\pi$
$240$	0	0
$241$	19.0000	1.22390	0.611949	−	0.790897i	$-0.290386\pi$
$241$	19.0000	1.22390	0.611949	+	0.790897i	$0.290386\pi$
$242$	0	0
$243$	−22.0454	−1.41421
$244$	0	0
$245$	−38.3939	−2.45289
$246$	0	0
$247$	32.9444	2.09620
$248$	0	0
$249$	7.10102	0.450009
$250$	0	0
$251$	14.6515	0.924796	0.462398	−	0.886672i	$-0.346989\pi$
$251$	14.6515	0.924796	0.462398	+	0.886672i	$0.346989\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−36.0000	−2.25441
$256$	0	0
$257$	12.1010	0.754841	0.377420	−	0.926042i	$-0.376811\pi$
$257$	12.1010	0.754841	0.377420	+	0.926042i	$0.376811\pi$
$258$	0	0
$259$	−35.5959	−2.21182
$260$	0	0
$261$	−5.69694	−0.352632
$262$	0	0
$263$	−3.55051	−0.218934	−0.109467	−	0.993990i	$-0.534914\pi$
$263$	−3.55051	−0.218934	−0.109467	+	0.993990i	$0.534914\pi$
$264$	0	0
$265$	32.3939	1.98994
$266$	0	0
$267$	−26.4495	−1.61868
$268$	0	0
$269$	17.7980	1.08516	0.542580	−	0.840004i	$-0.317447\pi$
$269$	17.7980	1.08516	0.542580	+	0.840004i	$0.317447\pi$
$270$	0	0
$271$	−13.5505	−0.823135	−0.411567	−	0.911379i	$-0.635019\pi$
$271$	−13.5505	−0.823135	−0.411567	+	0.911379i	$0.635019\pi$
$272$	0	0
$273$	42.4949	2.57191
$274$	0	0
$275$	−9.79796	−0.590839
$276$	0	0
$277$	−5.79796	−0.348366	−0.174183	−	0.984713i	$-0.555728\pi$
$277$	−5.79796	−0.348366	−0.174183	+	0.984713i	$0.555728\pi$
$278$	0	0
$279$	3.30306	0.197749
$280$	0	0
$281$	−21.7980	−1.30036	−0.650179	−	0.759781i	$-0.725306\pi$
$281$	−21.7980	−1.30036	−0.650179	+	0.759781i	$0.725306\pi$
$282$	0	0
$283$	12.6969	0.754755	0.377377	−	0.926060i	$-0.376826\pi$
$283$	12.6969	0.754755	0.377377	+	0.926060i	$0.376826\pi$
$284$	0	0
$285$	62.0908	3.67794
$286$	0	0
$287$	8.44949	0.498758
$288$	0	0
$289$	7.00000	0.411765
$290$	0	0
$291$	19.3485	1.13423
$292$	0	0
$293$	30.7980	1.79924	0.899618	−	0.436678i	$-0.143845\pi$
$293$	30.7980	1.79924	0.899618	+	0.436678i	$0.143845\pi$
$294$	0	0
$295$	−28.0454	−1.63287
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−12.8990	−0.743485
$302$	0	0
$303$	22.0454	1.26648
$304$	0	0
$305$	11.6969	0.669765
$306$	0	0
$307$	−33.1464	−1.89177	−0.945883	−	0.324507i	$-0.894802\pi$
$307$	−33.1464	−1.89177	−0.945883	+	0.324507i	$0.894802\pi$
$308$	0	0
$309$	−7.10102	−0.403963
$310$	0	0
$311$	−22.2474	−1.26154	−0.630769	−	0.775971i	$-0.717260\pi$
$311$	−22.2474	−1.26154	−0.630769	+	0.775971i	$0.717260\pi$
$312$	0	0
$313$	5.00000	0.282617	0.141308	−	0.989966i	$-0.454869\pi$
$313$	5.00000	0.282617	0.141308	+	0.989966i	$0.454869\pi$
$314$	0	0
$315$	40.0454	2.25630
$316$	0	0
$317$	−27.8990	−1.56696	−0.783481	−	0.621415i	$-0.786558\pi$
$317$	−27.8990	−1.56696	−0.783481	+	0.621415i	$0.786558\pi$
$318$	0	0
$319$	4.65153	0.260436
$320$	0	0
$321$	28.2929	1.57915
$322$	0	0
$323$	−41.3939	−2.30322
$324$	0	0
$325$	−15.5959	−0.865106
$326$	0	0
$327$	46.0454	2.54632
$328$	0	0
$329$	28.6969	1.58211
$330$	0	0
$331$	−4.69694	−0.258167	−0.129084	−	0.991634i	$-0.541204\pi$
$331$	−4.69694	−0.258167	−0.129084	+	0.991634i	$0.541204\pi$
$332$	0	0
$333$	24.0000	1.31519
$334$	0	0
$335$	−17.3939	−0.950329
$336$	0	0
$337$	20.1010	1.09497	0.547486	−	0.836815i	$-0.315585\pi$
$337$	20.1010	1.09497	0.547486	+	0.836815i	$0.315585\pi$
$338$	0	0
$339$	−12.2474	−0.665190
$340$	0	0
$341$	−2.69694	−0.146047
$342$	0	0
$343$	−25.7980	−1.39296
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7.34847	−0.394486	−0.197243	−	0.980355i	$-0.563199\pi$
$347$	−7.34847	−0.394486	−0.197243	+	0.980355i	$0.563199\pi$
$348$	0	0
$349$	−18.6969	−1.00082	−0.500412	−	0.865787i	$-0.666818\pi$
$349$	−18.6969	−1.00082	−0.500412	+	0.865787i	$0.666818\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−13.6969	−0.729014	−0.364507	−	0.931201i	$-0.618763\pi$
$353$	−13.6969	−0.729014	−0.364507	+	0.931201i	$0.618763\pi$
$354$	0	0
$355$	1.34847	0.0715693
$356$	0	0
$357$	−53.3939	−2.82590
$358$	0	0
$359$	−3.55051	−0.187389	−0.0936944	−	0.995601i	$-0.529868\pi$
$359$	−3.55051	−0.187389	−0.0936944	+	0.995601i	$0.529868\pi$
$360$	0	0
$361$	52.3939	2.75757
$362$	0	0
$363$	−12.2474	−0.642824
$364$	0	0
$365$	−23.6969	−1.24035
$366$	0	0
$367$	0.651531	0.0340096	0.0170048	−	0.999855i	$-0.494587\pi$
$367$	0.651531	0.0340096	0.0170048	+	0.999855i	$0.494587\pi$
$368$	0	0
$369$	−5.69694	−0.296571
$370$	0	0
$371$	48.0454	2.49439
$372$	0	0
$373$	20.4949	1.06119	0.530593	−	0.847627i	$-0.321969\pi$
$373$	20.4949	1.06119	0.530593	+	0.847627i	$0.321969\pi$
$374$	0	0
$375$	7.34847	0.379473
$376$	0	0
$377$	7.40408	0.381330
$378$	0	0
$379$	23.5505	1.20971	0.604854	−	0.796336i	$-0.293231\pi$
$379$	23.5505	1.20971	0.604854	+	0.796336i	$0.293231\pi$
$380$	0	0
$381$	7.10102	0.363796
$382$	0	0
$383$	22.8990	1.17008	0.585042	−	0.811003i	$-0.301078\pi$
$383$	22.8990	1.17008	0.585042	+	0.811003i	$0.301078\pi$
$384$	0	0
$385$	−32.6969	−1.66639
$386$	0	0
$387$	8.69694	0.442090
$388$	0	0
$389$	38.2929	1.94152	0.970762	−	0.240042i	$-0.0771613\pi$
$389$	38.2929	1.94152	0.970762	+	0.240042i	$0.0771613\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−15.3031	−0.771937
$394$	0	0
$395$	36.0000	1.81136
$396$	0	0
$397$	10.7980	0.541934	0.270967	−	0.962589i	$-0.412657\pi$
$397$	10.7980	0.541934	0.270967	+	0.962589i	$0.412657\pi$
$398$	0	0
$399$	92.0908	4.61031
$400$	0	0
$401$	−33.4949	−1.67266	−0.836328	−	0.548230i	$-0.815302\pi$
$401$	−33.4949	−1.67266	−0.836328	+	0.548230i	$0.815302\pi$
$402$	0	0
$403$	−4.29286	−0.213842
$404$	0	0
$405$	27.0000	1.34164
$406$	0	0
$407$	−19.5959	−0.971334
$408$	0	0
$409$	20.8990	1.03339	0.516694	−	0.856170i	$-0.327162\pi$
$409$	20.8990	1.03339	0.516694	+	0.856170i	$0.327162\pi$
$410$	0	0
$411$	−5.14643	−0.253855
$412$	0	0
$413$	−41.5959	−2.04680
$414$	0	0
$415$	−8.69694	−0.426916
$416$	0	0
$417$	40.8990	2.00283
$418$	0	0
$419$	−25.3939	−1.24057	−0.620286	−	0.784376i	$-0.712983\pi$
$419$	−25.3939	−1.24057	−0.620286	+	0.784376i	$0.712983\pi$
$420$	0	0
$421$	5.59592	0.272728	0.136364	−	0.990659i	$-0.456458\pi$
$421$	5.59592	0.272728	0.136364	+	0.990659i	$0.456458\pi$
$422$	0	0
$423$	−19.3485	−0.940755
$424$	0	0
$425$	19.5959	0.950542
$426$	0	0
$427$	17.3485	0.839551
$428$	0	0
$429$	23.3939	1.12947
$430$	0	0
$431$	4.65153	0.224056	0.112028	−	0.993705i	$-0.464265\pi$
$431$	4.65153	0.224056	0.112028	+	0.993705i	$0.464265\pi$
$432$	0	0
$433$	11.8990	0.571828	0.285914	−	0.958255i	$-0.407703\pi$
$433$	11.8990	0.571828	0.285914	+	0.958255i	$0.407703\pi$
$434$	0	0
$435$	13.9546	0.669071
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−6.89898	−0.329270	−0.164635	−	0.986355i	$-0.552645\pi$
$439$	−6.89898	−0.329270	−0.164635	+	0.986355i	$0.552645\pi$
$440$	0	0
$441$	38.3939	1.82828
$442$	0	0
$443$	13.1010	0.622448	0.311224	−	0.950337i	$-0.399261\pi$
$443$	13.1010	0.622448	0.311224	+	0.950337i	$0.399261\pi$
$444$	0	0
$445$	32.3939	1.53562
$446$	0	0
$447$	14.4495	0.683437
$448$	0	0
$449$	−7.89898	−0.372776	−0.186388	−	0.982476i	$-0.559678\pi$
$449$	−7.89898	−0.372776	−0.186388	+	0.982476i	$0.559678\pi$
$450$	0	0
$451$	4.65153	0.219032
$452$	0	0
$453$	20.2020	0.949175
$454$	0	0
$455$	−52.0454	−2.43993
$456$	0	0
$457$	17.8990	0.837279	0.418639	−	0.908153i	$-0.362507\pi$
$457$	17.8990	0.837279	0.418639	+	0.908153i	$0.362507\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−16.3939	−0.763539	−0.381769	−	0.924258i	$-0.624685\pi$
$461$	−16.3939	−0.763539	−0.381769	+	0.924258i	$0.624685\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	0	0
$465$	−8.09082	−0.375203
$466$	0	0
$467$	2.20204	0.101898	0.0509492	−	0.998701i	$-0.483775\pi$
$467$	2.20204	0.101898	0.0509492	+	0.998701i	$0.483775\pi$
$468$	0	0
$469$	−25.7980	−1.19124
$470$	0	0
$471$	−7.34847	−0.338600
$472$	0	0
$473$	−7.10102	−0.326505
$474$	0	0
$475$	−33.7980	−1.55076
$476$	0	0
$477$	−32.3939	−1.48321
$478$	0	0
$479$	30.8990	1.41181	0.705905	−	0.708306i	$-0.250540\pi$
$479$	30.8990	1.41181	0.705905	+	0.708306i	$0.250540\pi$
$480$	0	0
$481$	−31.1918	−1.42223
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−23.6969	−1.07602
$486$	0	0
$487$	−2.24745	−0.101842	−0.0509208	−	0.998703i	$-0.516216\pi$
$487$	−2.24745	−0.101842	−0.0509208	+	0.998703i	$0.516216\pi$
$488$	0	0
$489$	−30.4949	−1.37903
$490$	0	0
$491$	14.8990	0.672382	0.336191	−	0.941794i	$-0.390861\pi$
$491$	14.8990	0.672382	0.336191	+	0.941794i	$0.390861\pi$
$492$	0	0
$493$	−9.30306	−0.418989
$494$	0	0
$495$	22.0454	0.990867
$496$	0	0
$497$	2.00000	0.0897123
$498$	0	0
$499$	−3.59592	−0.160975	−0.0804877	−	0.996756i	$-0.525648\pi$
$499$	−3.59592	−0.160975	−0.0804877	+	0.996756i	$0.525648\pi$
$500$	0	0
$501$	−36.4949	−1.63047
$502$	0	0
$503$	21.1464	0.942873	0.471436	−	0.881900i	$-0.343736\pi$
$503$	21.1464	0.942873	0.471436	+	0.881900i	$0.343736\pi$
$504$	0	0
$505$	−27.0000	−1.20148
$506$	0	0
$507$	5.39388	0.239550
$508$	0	0
$509$	−16.5959	−0.735601	−0.367801	−	0.929905i	$-0.619889\pi$
$509$	−16.5959	−0.735601	−0.367801	+	0.929905i	$0.619889\pi$
$510$	0	0
$511$	−35.1464	−1.55479
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.69694	0.383233
$516$	0	0
$517$	15.7980	0.694793
$518$	0	0
$519$	50.9444	2.23621
$520$	0	0
$521$	−36.8990	−1.61657	−0.808287	−	0.588789i	$-0.799605\pi$
$521$	−36.8990	−1.61657	−0.808287	+	0.588789i	$0.799605\pi$
$522$	0	0
$523$	−7.59592	−0.332146	−0.166073	−	0.986113i	$-0.553109\pi$
$523$	−7.59592	−0.332146	−0.166073	+	0.986113i	$0.553109\pi$
$524$	0	0
$525$	−43.5959	−1.90268
$526$	0	0
$527$	5.39388	0.234961
$528$	0	0
$529$	0	0
$530$	0	0
$531$	28.0454	1.21707
$532$	0	0
$533$	7.40408	0.320706
$534$	0	0
$535$	−34.6515	−1.49812
$536$	0	0
$537$	26.6969	1.15206
$538$	0	0
$539$	−31.3485	−1.35027
$540$	0	0
$541$	−18.3939	−0.790815	−0.395407	−	0.918506i	$-0.629397\pi$
$541$	−18.3939	−0.790815	−0.395407	+	0.918506i	$0.629397\pi$
$542$	0	0
$543$	41.3939	1.77638
$544$	0	0
$545$	−56.3939	−2.41565
$546$	0	0
$547$	36.6969	1.56905	0.784524	−	0.620099i	$-0.212907\pi$
$547$	36.6969	1.56905	0.784524	+	0.620099i	$0.212907\pi$
$548$	0	0
$549$	−11.6969	−0.499213
$550$	0	0
$551$	16.0454	0.683557
$552$	0	0
$553$	53.3939	2.27054
$554$	0	0
$555$	−58.7878	−2.49540
$556$	0	0
$557$	3.69694	0.156644	0.0783222	−	0.996928i	$-0.475044\pi$
$557$	3.69694	0.156644	0.0783222	+	0.996928i	$0.475044\pi$
$558$	0	0
$559$	−11.3031	−0.478069
$560$	0	0
$561$	−29.3939	−1.24101
$562$	0	0
$563$	46.0908	1.94250	0.971248	−	0.238069i	$-0.0765146\pi$
$563$	46.0908	1.94250	0.971248	+	0.238069i	$0.0765146\pi$
$564$	0	0
$565$	15.0000	0.631055
$566$	0	0
$567$	40.0454	1.68175
$568$	0	0
$569$	−27.4949	−1.15265	−0.576323	−	0.817222i	$-0.695513\pi$
$569$	−27.4949	−1.15265	−0.576323	+	0.817222i	$0.695513\pi$
$570$	0	0
$571$	−42.4495	−1.77646	−0.888228	−	0.459403i	$-0.848063\pi$
$571$	−42.4495	−1.77646	−0.888228	+	0.459403i	$0.848063\pi$
$572$	0	0
$573$	−7.10102	−0.296649
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13.5959	0.566005	0.283003	−	0.959119i	$-0.408669\pi$
$577$	13.5959	0.566005	0.283003	+	0.959119i	$0.408669\pi$
$578$	0	0
$579$	−43.8434	−1.82207
$580$	0	0
$581$	−12.8990	−0.535140
$582$	0	0
$583$	26.4495	1.09543
$584$	0	0
$585$	35.0908	1.45083
$586$	0	0
$587$	−31.5959	−1.30410	−0.652052	−	0.758175i	$-0.726091\pi$
$587$	−31.5959	−1.30410	−0.652052	+	0.758175i	$0.726091\pi$
$588$	0	0
$589$	−9.30306	−0.383326
$590$	0	0
$591$	24.7423	1.01776
$592$	0	0
$593$	−27.1010	−1.11291	−0.556453	−	0.830879i	$-0.687838\pi$
$593$	−27.1010	−1.11291	−0.556453	+	0.830879i	$0.687838\pi$
$594$	0	0
$595$	65.3939	2.68089
$596$	0	0
$597$	−21.1918	−0.867324
$598$	0	0
$599$	−1.34847	−0.0550970	−0.0275485	−	0.999620i	$-0.508770\pi$
$599$	−1.34847	−0.0550970	−0.0275485	+	0.999620i	$0.508770\pi$
$600$	0	0
$601$	−14.7980	−0.603621	−0.301811	−	0.953368i	$-0.597591\pi$
$601$	−14.7980	−0.603621	−0.301811	+	0.953368i	$0.597591\pi$
$602$	0	0
$603$	17.3939	0.708333
$604$	0	0
$605$	15.0000	0.609837
$606$	0	0
$607$	11.5505	0.468821	0.234410	−	0.972138i	$-0.424684\pi$
$607$	11.5505	0.468821	0.234410	+	0.972138i	$0.424684\pi$
$608$	0	0
$609$	20.6969	0.838682
$610$	0	0
$611$	25.1464	1.01732
$612$	0	0
$613$	8.10102	0.327197	0.163599	−	0.986527i	$-0.447690\pi$
$613$	8.10102	0.327197	0.163599	+	0.986527i	$0.447690\pi$
$614$	0	0
$615$	13.9546	0.562703
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	−6.20204	−0.249281	−0.124641	−	0.992202i	$-0.539778\pi$
$619$	−6.20204	−0.249281	−0.124641	+	0.992202i	$0.539778\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	48.0454	1.92490
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	50.6969	2.02464
$628$	0	0
$629$	39.1918	1.56268
$630$	0	0
$631$	−24.6969	−0.983170	−0.491585	−	0.870830i	$-0.663582\pi$
$631$	−24.6969	−0.983170	−0.491585	+	0.870830i	$0.663582\pi$
$632$	0	0
$633$	39.7980	1.58183
$634$	0	0
$635$	−8.69694	−0.345127
$636$	0	0
$637$	−49.8990	−1.97707
$638$	0	0
$639$	−1.34847	−0.0533446
$640$	0	0
$641$	−6.10102	−0.240976	−0.120488	−	0.992715i	$-0.538446\pi$
$641$	−6.10102	−0.240976	−0.120488	+	0.992715i	$0.538446\pi$
$642$	0	0
$643$	−19.1464	−0.755061	−0.377531	−	0.925997i	$-0.623227\pi$
$643$	−19.1464	−0.755061	−0.377531	+	0.925997i	$0.623227\pi$
$644$	0	0
$645$	−21.3031	−0.838807
$646$	0	0
$647$	19.3485	0.760667	0.380333	−	0.924849i	$-0.375809\pi$
$647$	19.3485	0.760667	0.380333	+	0.924849i	$0.375809\pi$
$648$	0	0
$649$	−22.8990	−0.898864
$650$	0	0
$651$	−12.0000	−0.470317
$652$	0	0
$653$	0.303062	0.0118597	0.00592986	−	0.999982i	$-0.498112\pi$
$653$	0.303062	0.0118597	0.00592986	+	0.999982i	$0.498112\pi$
$654$	0	0
$655$	18.7423	0.732324
$656$	0	0
$657$	23.6969	0.924506
$658$	0	0
$659$	−6.20204	−0.241597	−0.120799	−	0.992677i	$-0.538546\pi$
$659$	−6.20204	−0.241597	−0.120799	+	0.992677i	$0.538546\pi$
$660$	0	0
$661$	−0.303062	−0.0117877	−0.00589386	−	0.999983i	$-0.501876\pi$
$661$	−0.303062	−0.0117877	−0.00589386	+	0.999983i	$0.501876\pi$
$662$	0	0
$663$	−46.7878	−1.81709
$664$	0	0
$665$	−112.788	−4.37372
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−25.5959	−0.989595
$670$	0	0
$671$	9.55051	0.368693
$672$	0	0
$673$	1.79796	0.0693062	0.0346531	−	0.999399i	$-0.488967\pi$
$673$	1.79796	0.0693062	0.0346531	+	0.999399i	$0.488967\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.10102	0.0807488	0.0403744	−	0.999185i	$-0.487145\pi$
$677$	2.10102	0.0807488	0.0403744	+	0.999185i	$0.487145\pi$
$678$	0	0
$679$	−35.1464	−1.34880
$680$	0	0
$681$	−33.7980	−1.29514
$682$	0	0
$683$	33.3939	1.27778	0.638891	−	0.769298i	$-0.279394\pi$
$683$	33.3939	1.27778	0.638891	+	0.769298i	$0.279394\pi$
$684$	0	0
$685$	6.30306	0.240828
$686$	0	0
$687$	−33.7980	−1.28947
$688$	0	0
$689$	42.1010	1.60392
$690$	0	0
$691$	−44.2929	−1.68498	−0.842490	−	0.538712i	$-0.818911\pi$
$691$	−44.2929	−1.68498	−0.842490	+	0.538712i	$0.818911\pi$
$692$	0	0
$693$	32.6969	1.24205
$694$	0	0
$695$	−50.0908	−1.90005
$696$	0	0
$697$	−9.30306	−0.352379
$698$	0	0
$699$	40.6515	1.53758
$700$	0	0
$701$	2.20204	0.0831699	0.0415850	−	0.999135i	$-0.486759\pi$
$701$	2.20204	0.0831699	0.0415850	+	0.999135i	$0.486759\pi$
$702$	0	0
$703$	−67.5959	−2.54943
$704$	0	0
$705$	47.3939	1.78496
$706$	0	0
$707$	−40.0454	−1.50606
$708$	0	0
$709$	−5.00000	−0.187779	−0.0938895	−	0.995583i	$-0.529930\pi$
$709$	−5.00000	−0.187779	−0.0938895	+	0.995583i	$0.529930\pi$
$710$	0	0
$711$	−36.0000	−1.35011
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−28.6515	−1.07151
$716$	0	0
$717$	−43.5959	−1.62812
$718$	0	0
$719$	14.4949	0.540568	0.270284	−	0.962781i	$-0.412882\pi$
$719$	14.4949	0.540568	0.270284	+	0.962781i	$0.412882\pi$
$720$	0	0
$721$	12.8990	0.480383
$722$	0	0
$723$	46.5403	1.73085
$724$	0	0
$725$	−7.59592	−0.282105
$726$	0	0
$727$	15.5505	0.576737	0.288368	−	0.957520i	$-0.406887\pi$
$727$	15.5505	0.576737	0.288368	+	0.957520i	$0.406887\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	14.2020	0.525281
$732$	0	0
$733$	−4.79796	−0.177217	−0.0886083	−	0.996067i	$-0.528242\pi$
$733$	−4.79796	−0.177217	−0.0886083	+	0.996067i	$0.528242\pi$
$734$	0	0
$735$	−94.0454	−3.46892
$736$	0	0
$737$	−14.2020	−0.523139
$738$	0	0
$739$	−3.95459	−0.145472	−0.0727360	−	0.997351i	$-0.523173\pi$
$739$	−3.95459	−0.145472	−0.0727360	+	0.997351i	$0.523173\pi$
$740$	0	0
$741$	80.6969	2.96448
$742$	0	0
$743$	−9.55051	−0.350374	−0.175187	−	0.984535i	$-0.556053\pi$
$743$	−9.55051	−0.350374	−0.175187	+	0.984535i	$0.556053\pi$
$744$	0	0
$745$	−17.6969	−0.648366
$746$	0	0
$747$	8.69694	0.318204
$748$	0	0
$749$	−51.3939	−1.87789
$750$	0	0
$751$	−44.7423	−1.63267	−0.816336	−	0.577578i	$-0.803998\pi$
$751$	−44.7423	−1.63267	−0.816336	+	0.577578i	$0.803998\pi$
$752$	0	0
$753$	35.8888	1.30786
$754$	0	0
$755$	−24.7423	−0.900466
$756$	0	0
$757$	7.89898	0.287093	0.143547	−	0.989644i	$-0.454149\pi$
$757$	7.89898	0.287093	0.143547	+	0.989644i	$0.454149\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	46.3939	1.68178	0.840888	−	0.541209i	$-0.182033\pi$
$761$	46.3939	1.68178	0.840888	+	0.541209i	$0.182033\pi$
$762$	0	0
$763$	−83.6413	−3.02802
$764$	0	0
$765$	−44.0908	−1.59411
$766$	0	0
$767$	−36.4495	−1.31611
$768$	0	0
$769$	−32.4949	−1.17180	−0.585898	−	0.810385i	$-0.699258\pi$
$769$	−32.4949	−1.17180	−0.585898	+	0.810385i	$0.699258\pi$
$770$	0	0
$771$	29.6413	1.06751
$772$	0	0
$773$	10.0000	0.359675	0.179838	−	0.983696i	$-0.442443\pi$
$773$	10.0000	0.359675	0.179838	+	0.983696i	$0.442443\pi$
$774$	0	0
$775$	4.40408	0.158199
$776$	0	0
$777$	−87.1918	−3.12799
$778$	0	0
$779$	16.0454	0.574886
$780$	0	0
$781$	1.10102	0.0393976
$782$	0	0
$783$	0	0
$784$	0	0
$785$	9.00000	0.321224
$786$	0	0
$787$	−27.3485	−0.974868	−0.487434	−	0.873160i	$-0.662067\pi$
$787$	−27.3485	−0.974868	−0.487434	+	0.873160i	$0.662067\pi$
$788$	0	0
$789$	−8.69694	−0.309619
$790$	0	0
$791$	22.2474	0.791028
$792$	0	0
$793$	15.2020	0.539840
$794$	0	0
$795$	79.3485	2.81420
$796$	0	0
$797$	31.5959	1.11918	0.559592	−	0.828768i	$-0.310958\pi$
$797$	31.5959	1.11918	0.559592	+	0.828768i	$0.310958\pi$
$798$	0	0
$799$	−31.5959	−1.11778
$800$	0	0
$801$	−32.3939	−1.14458
$802$	0	0
$803$	−19.3485	−0.682793
$804$	0	0
$805$	0	0
$806$	0	0
$807$	43.5959	1.53465
$808$	0	0
$809$	33.5959	1.18117	0.590585	−	0.806976i	$-0.298897\pi$
$809$	33.5959	1.18117	0.590585	+	0.806976i	$0.298897\pi$
$810$	0	0
$811$	34.9444	1.22706	0.613532	−	0.789670i	$-0.289748\pi$
$811$	34.9444	1.22706	0.613532	+	0.789670i	$0.289748\pi$
$812$	0	0
$813$	−33.1918	−1.16409
$814$	0	0
$815$	37.3485	1.30826
$816$	0	0
$817$	−24.4949	−0.856968
$818$	0	0
$819$	52.0454	1.81861
$820$	0	0
$821$	45.4949	1.58778	0.793891	−	0.608060i	$-0.208052\pi$
$821$	45.4949	1.58778	0.793891	+	0.608060i	$0.208052\pi$
$822$	0	0
$823$	1.95459	0.0681328	0.0340664	−	0.999420i	$-0.489154\pi$
$823$	1.95459	0.0681328	0.0340664	+	0.999420i	$0.489154\pi$
$824$	0	0
$825$	−24.0000	−0.835573
$826$	0	0
$827$	−24.2474	−0.843166	−0.421583	−	0.906790i	$-0.638525\pi$
$827$	−24.2474	−0.843166	−0.421583	+	0.906790i	$0.638525\pi$
$828$	0	0
$829$	33.0000	1.14614	0.573069	−	0.819507i	$-0.305753\pi$
$829$	33.0000	1.14614	0.573069	+	0.819507i	$0.305753\pi$
$830$	0	0
$831$	−14.2020	−0.492663
$832$	0	0
$833$	62.6969	2.17232
$834$	0	0
$835$	44.6969	1.54680
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−32.9444	−1.13737	−0.568683	−	0.822557i	$-0.692547\pi$
$839$	−32.9444	−1.13737	−0.568683	+	0.822557i	$0.692547\pi$
$840$	0	0
$841$	−25.3939	−0.875651
$842$	0	0
$843$	−53.3939	−1.83898
$844$	0	0
$845$	−6.60612	−0.227258
$846$	0	0
$847$	22.2474	0.764431
$848$	0	0
$849$	31.1010	1.06738
$850$	0	0
$851$	0	0
$852$	0	0
$853$	23.1010	0.790964	0.395482	−	0.918474i	$-0.370578\pi$
$853$	23.1010	0.790964	0.395482	+	0.918474i	$0.370578\pi$
$854$	0	0
$855$	76.0454	2.60070
$856$	0	0
$857$	52.3939	1.78974	0.894870	−	0.446326i	$-0.147268\pi$
$857$	52.3939	1.78974	0.894870	+	0.446326i	$0.147268\pi$
$858$	0	0
$859$	3.34847	0.114248	0.0571241	−	0.998367i	$-0.481807\pi$
$859$	3.34847	0.114248	0.0571241	+	0.998367i	$0.481807\pi$
$860$	0	0
$861$	20.6969	0.705350
$862$	0	0
$863$	−28.2474	−0.961554	−0.480777	−	0.876843i	$-0.659645\pi$
$863$	−28.2474	−0.961554	−0.480777	+	0.876843i	$0.659645\pi$
$864$	0	0
$865$	−62.3939	−2.12146
$866$	0	0
$867$	17.1464	0.582323
$868$	0	0
$869$	29.3939	0.997119
$870$	0	0
$871$	−22.6061	−0.765979
$872$	0	0
$873$	23.6969	0.802020
$874$	0	0
$875$	−13.3485	−0.451261
$876$	0	0
$877$	7.10102	0.239784	0.119892	−	0.992787i	$-0.461745\pi$
$877$	7.10102	0.239784	0.119892	+	0.992787i	$0.461745\pi$
$878$	0	0
$879$	75.4393	2.54450
$880$	0	0
$881$	41.0908	1.38438	0.692192	−	0.721713i	$-0.256645\pi$
$881$	41.0908	1.38438	0.692192	+	0.721713i	$0.256645\pi$
$882$	0	0
$883$	−28.0454	−0.943803	−0.471902	−	0.881651i	$-0.656432\pi$
$883$	−28.0454	−0.943803	−0.471902	+	0.881651i	$0.656432\pi$
$884$	0	0
$885$	−68.6969	−2.30922
$886$	0	0
$887$	33.7980	1.13482	0.567412	−	0.823434i	$-0.307945\pi$
$887$	33.7980	1.13482	0.567412	+	0.823434i	$0.307945\pi$
$888$	0	0
$889$	−12.8990	−0.432618
$890$	0	0
$891$	22.0454	0.738549
$892$	0	0
$893$	54.4949	1.82360
$894$	0	0
$895$	−32.6969	−1.09294
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.09082	−0.0697326
$900$	0	0
$901$	−52.8990	−1.76232
$902$	0	0
$903$	−31.5959	−1.05145
$904$	0	0
$905$	−50.6969	−1.68522
$906$	0	0
$907$	−42.9444	−1.42594	−0.712972	−	0.701192i	$-0.752652\pi$
$907$	−42.9444	−1.42594	−0.712972	+	0.701192i	$0.752652\pi$
$908$	0	0
$909$	27.0000	0.895533
$910$	0	0
$911$	−6.89898	−0.228573	−0.114287	−	0.993448i	$-0.536458\pi$
$911$	−6.89898	−0.228573	−0.114287	+	0.993448i	$0.536458\pi$
$912$	0	0
$913$	−7.10102	−0.235009
$914$	0	0
$915$	28.6515	0.947190
$916$	0	0
$917$	27.7980	0.917969
$918$	0	0
$919$	2.65153	0.0874659	0.0437330	−	0.999043i	$-0.486075\pi$
$919$	2.65153	0.0874659	0.0437330	+	0.999043i	$0.486075\pi$
$920$	0	0
$921$	−81.1918	−2.67536
$922$	0	0
$923$	1.75255	0.0576859
$924$	0	0
$925$	32.0000	1.05215
$926$	0	0
$927$	−8.69694	−0.285645
$928$	0	0
$929$	16.4949	0.541180	0.270590	−	0.962695i	$-0.412781\pi$
$929$	16.4949	0.541180	0.270590	+	0.962695i	$0.412781\pi$
$930$	0	0
$931$	−108.136	−3.54402
$932$	0	0
$933$	−54.4949	−1.78408
$934$	0	0
$935$	36.0000	1.17733
$936$	0	0
$937$	27.1010	0.885352	0.442676	−	0.896682i	$-0.354029\pi$
$937$	27.1010	0.885352	0.442676	+	0.896682i	$0.354029\pi$
$938$	0	0
$939$	12.2474	0.399680
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23.1918	0.753633	0.376817	−	0.926288i	$-0.377019\pi$
$947$	23.1918	0.753633	0.376817	+	0.926288i	$0.377019\pi$
$948$	0	0
$949$	−30.7980	−0.999744
$950$	0	0
$951$	−68.3383	−2.21602
$952$	0	0
$953$	−23.6969	−0.767619	−0.383810	−	0.923412i	$-0.625388\pi$
$953$	−23.6969	−0.767619	−0.383810	+	0.923412i	$0.625388\pi$
$954$	0	0
$955$	8.69694	0.281426
$956$	0	0
$957$	11.3939	0.368312
$958$	0	0
$959$	9.34847	0.301878
$960$	0	0
$961$	−29.7878	−0.960895
$962$	0	0
$963$	34.6515	1.11663
$964$	0	0
$965$	53.6969	1.72857
$966$	0	0
$967$	−41.3485	−1.32968	−0.664839	−	0.746987i	$-0.731500\pi$
$967$	−41.3485	−1.32968	−0.664839	+	0.746987i	$0.731500\pi$
$968$	0	0
$969$	−101.394	−3.25724
$970$	0	0
$971$	16.7423	0.537287	0.268644	−	0.963240i	$-0.413425\pi$
$971$	16.7423	0.537287	0.268644	+	0.963240i	$0.413425\pi$
$972$	0	0
$973$	−74.2929	−2.38172
$974$	0	0
$975$	−38.2020	−1.22344
$976$	0	0
$977$	11.6969	0.374218	0.187109	−	0.982339i	$-0.440088\pi$
$977$	11.6969	0.374218	0.187109	+	0.982339i	$0.440088\pi$
$978$	0	0
$979$	26.4495	0.845329
$980$	0	0
$981$	56.3939	1.80052
$982$	0	0
$983$	17.1010	0.545438	0.272719	−	0.962094i	$-0.412077\pi$
$983$	17.1010	0.545438	0.272719	+	0.962094i	$0.412077\pi$
$984$	0	0
$985$	−30.3031	−0.965536
$986$	0	0
$987$	70.2929	2.23745
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−48.9898	−1.55621	−0.778106	−	0.628133i	$-0.783819\pi$
$991$	−48.9898	−1.55621	−0.778106	+	0.628133i	$0.783819\pi$
$992$	0	0
$993$	−11.5051	−0.365103
$994$	0	0
$995$	25.9546	0.822816
$996$	0	0
$997$	24.3939	0.772562	0.386281	−	0.922381i	$-0.373760\pi$
$997$	24.3939	0.772562	0.386281	+	0.922381i	$0.373760\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.y.1.2		2
4.3	odd	2		4232.2.a.p.1.1	✓	2
23.22	odd	2		8464.2.a.bc.1.2		2
92.91	even	2		4232.2.a.q.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4232.2.a.p.1.1	✓	2	4.3	odd	2
4232.2.a.q.1.1	yes	2	92.91	even	2
8464.2.a.y.1.2		2	1.1	even	1	trivial
8464.2.a.bc.1.2		2	23.22	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:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q+2.44949 q^{3} -3.00000 q^{5} -4.44949 q^{7} +3.00000 q^{9} -2.44949 q^{11} -3.89898 q^{13} -7.34847 q^{15} +4.89898 q^{17} -8.44949 q^{19} -10.8990 q^{21} +4.00000 q^{25} -1.89898 q^{29} +1.10102 q^{31} -6.00000 q^{33} +13.3485 q^{35} +8.00000 q^{37} -9.55051 q^{39} -1.89898 q^{41} +2.89898 q^{43} -9.00000 q^{45} -6.44949 q^{47} +12.7980 q^{49} +12.0000 q^{51} -10.7980 q^{53} +7.34847 q^{55} -20.6969 q^{57} +9.34847 q^{59} -3.89898 q^{61} -13.3485 q^{63} +11.6969 q^{65} +5.79796 q^{67} -0.449490 q^{71} +7.89898 q^{73} +9.79796 q^{75} +10.8990 q^{77} -12.0000 q^{79} -9.00000 q^{81} +2.89898 q^{83} -14.6969 q^{85} -4.65153 q^{87} -10.7980 q^{89} +17.3485 q^{91} +2.69694 q^{93} +25.3485 q^{95} +7.89898 q^{97} -7.34847 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{5} - 4 q^{7} + 6 q^{9} + 2 q^{13} - 12 q^{19} - 12 q^{21} + 8 q^{25} + 6 q^{29} + 12 q^{31} - 12 q^{33} + 12 q^{35} + 16 q^{37} - 24 q^{39} + 6 q^{41} - 4 q^{43} - 18 q^{45} - 8 q^{47} + 6 q^{49}+ \cdots + 6 q^{97}+O(q^{100}) \) 2 * q - 6 * q^5 - 4 * q^7 + 6 * q^9 + 2 * q^13 - 12 * q^19 - 12 * q^21 + 8 * q^25 + 6 * q^29 + 12 * q^31 - 12 * q^33 + 12 * q^35 + 16 * q^37 - 24 * q^39 + 6 * q^41 - 4 * q^43 - 18 * q^45 - 8 * q^47 + 6 * q^49 + 24 * q^51 - 2 * q^53 - 12 * q^57 + 4 * q^59 + 2 * q^61 - 12 * q^63 - 6 * q^65 - 8 * q^67 + 4 * q^71 + 6 * q^73 + 12 * q^77 - 24 * q^79 - 18 * q^81 - 4 * q^83 - 24 * q^87 - 2 * q^89 + 20 * q^91 - 24 * q^93 + 36 * q^95 + 6 * q^97

Introduction

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