LMFDB - Embedded newform 7168.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7168,2,Mod(1,7168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7168, 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("7168.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7168 = 2^{10} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7168.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:	$57.2367681689$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3584)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	7168.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{3} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.41421 q^{3} +1.00000 q^{7} -1.00000 q^{9} +2.82843 q^{11} +2.82843 q^{13} +6.00000 q^{17} -4.24264 q^{19} -1.41421 q^{21} -5.00000 q^{25} +5.65685 q^{27} -1.41421 q^{29} -10.0000 q^{31} -4.00000 q^{33} -4.24264 q^{37} -4.00000 q^{39} -10.0000 q^{41} -8.48528 q^{43} +2.00000 q^{47} +1.00000 q^{49} -8.48528 q^{51} +12.7279 q^{53} +6.00000 q^{57} -1.41421 q^{59} -1.00000 q^{63} +11.3137 q^{67} -12.0000 q^{71} +2.00000 q^{73} +7.07107 q^{75} +2.82843 q^{77} +4.00000 q^{79} -5.00000 q^{81} -7.07107 q^{83} +2.00000 q^{87} -2.00000 q^{89} +2.82843 q^{91} +14.1421 q^{93} -2.00000 q^{97} -2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^7 - 2 * q^9	$ 2 q + 2 q^{7} - 2 q^{9} + 12 q^{17} - 10 q^{25} - 20 q^{31} - 8 q^{33} - 8 q^{39} - 20 q^{41} + 4 q^{47} + 2 q^{49} + 12 q^{57} - 2 q^{63} - 24 q^{71} + 4 q^{73} + 8 q^{79} - 10 q^{81} + 4 q^{87} - 4 q^{89} - 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 - 2 * q^9 + 12 * q^17 - 10 * q^25 - 20 * q^31 - 8 * q^33 - 8 * q^39 - 20 * q^41 + 4 * q^47 + 2 * q^49 + 12 * q^57 - 2 * q^63 - 24 * q^71 + 4 * q^73 + 8 * q^79 - 10 * q^81 + 4 * q^87 - 4 * q^89 - 4 * 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
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.82843	0.852803	0.426401	−	0.904534i	$-0.359781\pi$
$11$	2.82843	0.852803	0.426401	+	0.904534i	$0.359781\pi$
$12$	0	0
$13$	2.82843	0.784465	0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843	0.784465	0.392232	+	0.919866i	$0.371703\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	−4.24264	−0.973329	−0.486664	−	0.873589i	$-0.661786\pi$
$19$	−4.24264	−0.973329	−0.486664	+	0.873589i	$0.661786\pi$
$20$	0	0
$21$	−1.41421	−0.308607
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	−1.41421	−0.262613	−0.131306	−	0.991342i	$-0.541917\pi$
$29$	−1.41421	−0.262613	−0.131306	+	0.991342i	$0.541917\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.24264	−0.697486	−0.348743	−	0.937218i	$-0.613391\pi$
$37$	−4.24264	−0.697486	−0.348743	+	0.937218i	$0.613391\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	−8.48528	−1.29399	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	−8.48528	−1.29399	−0.646997	+	0.762493i	$0.723975\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−8.48528	−1.18818
$52$	0	0
$53$	12.7279	1.74831	0.874157	−	0.485643i	$-0.161414\pi$
$53$	12.7279	1.74831	0.874157	+	0.485643i	$0.161414\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.00000	0.794719
$58$	0	0
$59$	−1.41421	−0.184115	−0.0920575	−	0.995754i	$-0.529344\pi$
$59$	−1.41421	−0.184115	−0.0920575	+	0.995754i	$0.529344\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.3137	1.38219	0.691095	−	0.722764i	$-0.257129\pi$
$67$	11.3137	1.38219	0.691095	+	0.722764i	$0.257129\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	7.07107	0.816497
$76$	0	0
$77$	2.82843	0.322329
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	−7.07107	−0.776151	−0.388075	−	0.921628i	$-0.626860\pi$
$83$	−7.07107	−0.776151	−0.388075	+	0.921628i	$0.626860\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	2.82843	0.296500
$92$	0	0
$93$	14.1421	1.46647
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−2.82843	−0.284268
$100$	0	0
$101$	5.65685	0.562878	0.281439	−	0.959579i	$-0.409188\pi$
$101$	5.65685	0.562878	0.281439	+	0.959579i	$0.409188\pi$
$102$	0	0
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	4.24264	0.406371	0.203186	−	0.979140i	$-0.434871\pi$
$109$	4.24264	0.406371	0.203186	+	0.979140i	$0.434871\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.82843	−0.261488
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	14.1421	1.27515
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	0	0
$129$	12.0000	1.05654
$130$	0	0
$131$	9.89949	0.864923	0.432461	−	0.901652i	$-0.357645\pi$
$131$	9.89949	0.864923	0.432461	+	0.901652i	$0.357645\pi$
$132$	0	0
$133$	−4.24264	−0.367884
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	4.24264	0.359856	0.179928	−	0.983680i	$-0.442414\pi$
$139$	4.24264	0.359856	0.179928	+	0.983680i	$0.442414\pi$
$140$	0	0
$141$	−2.82843	−0.238197
$142$	0	0
$143$	8.00000	0.668994
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.41421	−0.116642
$148$	0	0
$149$	7.07107	0.579284	0.289642	−	0.957135i	$-0.406464\pi$
$149$	7.07107	0.579284	0.289642	+	0.957135i	$0.406464\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−14.1421	−1.12867	−0.564333	−	0.825547i	$-0.690866\pi$
$157$	−14.1421	−1.12867	−0.564333	+	0.825547i	$0.690866\pi$
$158$	0	0
$159$	−18.0000	−1.42749
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2.82843	−0.221540	−0.110770	−	0.993846i	$-0.535332\pi$
$163$	−2.82843	−0.221540	−0.110770	+	0.993846i	$0.535332\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−22.0000	−1.70241	−0.851206	−	0.524832i	$-0.824128\pi$
$167$	−22.0000	−1.70241	−0.851206	+	0.524832i	$0.824128\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	4.24264	0.324443
$172$	0	0
$173$	14.1421	1.07521	0.537603	−	0.843198i	$-0.319330\pi$
$173$	14.1421	1.07521	0.537603	+	0.843198i	$0.319330\pi$
$174$	0	0
$175$	−5.00000	−0.377964
$176$	0	0
$177$	2.00000	0.150329
$178$	0	0
$179$	−22.6274	−1.69125	−0.845626	−	0.533775i	$-0.820773\pi$
$179$	−22.6274	−1.69125	−0.845626	+	0.533775i	$0.820773\pi$
$180$	0	0
$181$	−8.48528	−0.630706	−0.315353	−	0.948974i	$-0.602123\pi$
$181$	−8.48528	−0.630706	−0.315353	+	0.948974i	$0.602123\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	16.9706	1.24101
$188$	0	0
$189$	5.65685	0.411476
$190$	0	0
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.3848	1.30986	0.654931	−	0.755689i	$-0.272698\pi$
$197$	18.3848	1.30986	0.654931	+	0.755689i	$0.272698\pi$
$198$	0	0
$199$	−22.0000	−1.55954	−0.779769	−	0.626067i	$-0.784664\pi$
$199$	−22.0000	−1.55954	−0.779769	+	0.626067i	$0.784664\pi$
$200$	0	0
$201$	−16.0000	−1.12855
$202$	0	0
$203$	−1.41421	−0.0992583
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−28.2843	−1.94717	−0.973585	−	0.228326i	$-0.926675\pi$
$211$	−28.2843	−1.94717	−0.973585	+	0.228326i	$0.926675\pi$
$212$	0	0
$213$	16.9706	1.16280
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−10.0000	−0.678844
$218$	0	0
$219$	−2.82843	−0.191127
$220$	0	0
$221$	16.9706	1.14156
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	5.00000	0.333333
$226$	0	0
$227$	−1.41421	−0.0938647	−0.0469323	−	0.998898i	$-0.514945\pi$
$227$	−1.41421	−0.0938647	−0.0469323	+	0.998898i	$0.514945\pi$
$228$	0	0
$229$	25.4558	1.68217	0.841085	−	0.540903i	$-0.181918\pi$
$229$	25.4558	1.68217	0.841085	+	0.540903i	$0.181918\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−5.65685	−0.367452
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−12.0000	−0.763542
$248$	0	0
$249$	10.0000	0.633724
$250$	0	0
$251$	15.5563	0.981908	0.490954	−	0.871185i	$-0.336648\pi$
$251$	15.5563	0.981908	0.490954	+	0.871185i	$0.336648\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	−4.24264	−0.263625
$260$	0	0
$261$	1.41421	0.0875376
$262$	0	0
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.82843	0.173097
$268$	0	0
$269$	−19.7990	−1.20717	−0.603583	−	0.797300i	$-0.706261\pi$
$269$	−19.7990	−1.20717	−0.603583	+	0.797300i	$0.706261\pi$
$270$	0	0
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	0	0
$273$	−4.00000	−0.242091
$274$	0	0
$275$	−14.1421	−0.852803
$276$	0	0
$277$	−24.0416	−1.44452	−0.722261	−	0.691621i	$-0.756897\pi$
$277$	−24.0416	−1.44452	−0.722261	+	0.691621i	$0.756897\pi$
$278$	0	0
$279$	10.0000	0.598684
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	21.2132	1.26099	0.630497	−	0.776192i	$-0.282851\pi$
$283$	21.2132	1.26099	0.630497	+	0.776192i	$0.282851\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.0000	−0.590281
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	2.82843	0.165805
$292$	0	0
$293$	−5.65685	−0.330477	−0.165238	−	0.986254i	$-0.552839\pi$
$293$	−5.65685	−0.330477	−0.165238	+	0.986254i	$0.552839\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	16.0000	0.928414
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−8.48528	−0.489083
$302$	0	0
$303$	−8.00000	−0.459588
$304$	0	0
$305$	0	0
$306$	0	0
$307$	32.5269	1.85641	0.928204	−	0.372071i	$-0.121352\pi$
$307$	32.5269	1.85641	0.928204	+	0.372071i	$0.121352\pi$
$308$	0	0
$309$	−8.48528	−0.482711
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	30.0000	1.69570	0.847850	−	0.530236i	$-0.177897\pi$
$313$	30.0000	1.69570	0.847850	+	0.530236i	$0.177897\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.0416	−1.35031	−0.675156	−	0.737675i	$-0.735924\pi$
$317$	−24.0416	−1.35031	−0.675156	+	0.737675i	$0.735924\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−25.4558	−1.41640
$324$	0	0
$325$	−14.1421	−0.784465
$326$	0	0
$327$	−6.00000	−0.331801
$328$	0	0
$329$	2.00000	0.110264
$330$	0	0
$331$	8.48528	0.466393	0.233197	−	0.972430i	$-0.425081\pi$
$331$	8.48528	0.466393	0.233197	+	0.972430i	$0.425081\pi$
$332$	0	0
$333$	4.24264	0.232495
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−16.0000	−0.871576	−0.435788	−	0.900049i	$-0.643530\pi$
$337$	−16.0000	−0.871576	−0.435788	+	0.900049i	$0.643530\pi$
$338$	0	0
$339$	−16.9706	−0.921714
$340$	0	0
$341$	−28.2843	−1.53168
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−31.1127	−1.67022	−0.835109	−	0.550085i	$-0.814595\pi$
$347$	−31.1127	−1.67022	−0.835109	+	0.550085i	$0.814595\pi$
$348$	0	0
$349$	−8.48528	−0.454207	−0.227103	−	0.973871i	$-0.572926\pi$
$349$	−8.48528	−0.454207	−0.227103	+	0.973871i	$0.572926\pi$
$350$	0	0
$351$	16.0000	0.854017
$352$	0	0
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−8.48528	−0.449089
$358$	0	0
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	4.24264	0.222681
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	12.7279	0.660801
$372$	0	0
$373$	4.24264	0.219676	0.109838	−	0.993950i	$-0.464967\pi$
$373$	4.24264	0.219676	0.109838	+	0.993950i	$0.464967\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.00000	−0.206010
$378$	0	0
$379$	2.82843	0.145287	0.0726433	−	0.997358i	$-0.476857\pi$
$379$	2.82843	0.145287	0.0726433	+	0.997358i	$0.476857\pi$
$380$	0	0
$381$	16.9706	0.869428
$382$	0	0
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.48528	0.431331
$388$	0	0
$389$	−26.8701	−1.36237	−0.681183	−	0.732113i	$-0.738534\pi$
$389$	−26.8701	−1.36237	−0.681183	+	0.732113i	$0.738534\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−14.0000	−0.706207
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−14.1421	−0.709773	−0.354887	−	0.934909i	$-0.615481\pi$
$397$	−14.1421	−0.709773	−0.354887	+	0.934909i	$0.615481\pi$
$398$	0	0
$399$	6.00000	0.300376
$400$	0	0
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	0	0
$403$	−28.2843	−1.40894
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12.0000	−0.594818
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	−16.9706	−0.837096
$412$	0	0
$413$	−1.41421	−0.0695889
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−6.00000	−0.293821
$418$	0	0
$419$	4.24264	0.207267	0.103633	−	0.994616i	$-0.466953\pi$
$419$	4.24264	0.207267	0.103633	+	0.994616i	$0.466953\pi$
$420$	0	0
$421$	−24.0416	−1.17172	−0.585859	−	0.810413i	$-0.699243\pi$
$421$	−24.0416	−1.17172	−0.585859	+	0.810413i	$0.699243\pi$
$422$	0	0
$423$	−2.00000	−0.0972433
$424$	0	0
$425$	−30.0000	−1.45521
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−11.3137	−0.546231
$430$	0	0
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	0	0
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	28.2843	1.34383	0.671913	−	0.740630i	$-0.265473\pi$
$443$	28.2843	1.34383	0.671913	+	0.740630i	$0.265473\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−10.0000	−0.472984
$448$	0	0
$449$	−32.0000	−1.51017	−0.755087	−	0.655625i	$-0.772405\pi$
$449$	−32.0000	−1.51017	−0.755087	+	0.655625i	$0.772405\pi$
$450$	0	0
$451$	−28.2843	−1.33185
$452$	0	0
$453$	28.2843	1.32891
$454$	0	0
$455$	0	0
$456$	0	0
$457$	42.0000	1.96468	0.982339	−	0.187112i	$-0.0599128\pi$
$457$	42.0000	1.96468	0.982339	+	0.187112i	$0.0599128\pi$
$458$	0	0
$459$	33.9411	1.58424
$460$	0	0
$461$	19.7990	0.922131	0.461065	−	0.887366i	$-0.347467\pi$
$461$	19.7990	0.922131	0.461065	+	0.887366i	$0.347467\pi$
$462$	0	0
$463$	40.0000	1.85896	0.929479	−	0.368875i	$-0.120257\pi$
$463$	40.0000	1.85896	0.929479	+	0.368875i	$0.120257\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7.07107	−0.327210	−0.163605	−	0.986526i	$-0.552312\pi$
$467$	−7.07107	−0.327210	−0.163605	+	0.986526i	$0.552312\pi$
$468$	0	0
$469$	11.3137	0.522419
$470$	0	0
$471$	20.0000	0.921551
$472$	0	0
$473$	−24.0000	−1.10352
$474$	0	0
$475$	21.2132	0.973329
$476$	0	0
$477$	−12.7279	−0.582772
$478$	0	0
$479$	14.0000	0.639676	0.319838	−	0.947472i	$-0.396371\pi$
$479$	14.0000	0.639676	0.319838	+	0.947472i	$0.396371\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	−8.48528	−0.382158
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12.0000	−0.538274
$498$	0	0
$499$	16.9706	0.759707	0.379853	−	0.925047i	$-0.375974\pi$
$499$	16.9706	0.759707	0.379853	+	0.925047i	$0.375974\pi$
$500$	0	0
$501$	31.1127	1.39001
$502$	0	0
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	7.07107	0.314037
$508$	0	0
$509$	−36.7696	−1.62978	−0.814891	−	0.579614i	$-0.803203\pi$
$509$	−36.7696	−1.62978	−0.814891	+	0.579614i	$0.803203\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	−24.0000	−1.05963
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.65685	0.248788
$518$	0	0
$519$	−20.0000	−0.877903
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	15.5563	0.680232	0.340116	−	0.940384i	$-0.389534\pi$
$523$	15.5563	0.680232	0.340116	+	0.940384i	$0.389534\pi$
$524$	0	0
$525$	7.07107	0.308607
$526$	0	0
$527$	−60.0000	−2.61364
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	1.41421	0.0613716
$532$	0	0
$533$	−28.2843	−1.22513
$534$	0	0
$535$	0	0
$536$	0	0
$537$	32.0000	1.38090
$538$	0	0
$539$	2.82843	0.121829
$540$	0	0
$541$	−1.41421	−0.0608018	−0.0304009	−	0.999538i	$-0.509678\pi$
$541$	−1.41421	−0.0608018	−0.0304009	+	0.999538i	$0.509678\pi$
$542$	0	0
$543$	12.0000	0.514969
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−31.1127	−1.33028	−0.665141	−	0.746717i	$-0.731629\pi$
$547$	−31.1127	−1.33028	−0.665141	+	0.746717i	$0.731629\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	4.00000	0.170097
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.41421	−0.0599222	−0.0299611	−	0.999551i	$-0.509538\pi$
$557$	−1.41421	−0.0599222	−0.0299611	+	0.999551i	$0.509538\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	−24.0000	−1.01328
$562$	0	0
$563$	21.2132	0.894030	0.447015	−	0.894526i	$-0.352487\pi$
$563$	21.2132	0.894030	0.447015	+	0.894526i	$0.352487\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−5.00000	−0.209980
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	8.48528	0.355098	0.177549	−	0.984112i	$-0.443183\pi$
$571$	8.48528	0.355098	0.177549	+	0.984112i	$0.443183\pi$
$572$	0	0
$573$	28.2843	1.18159
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	−2.82843	−0.117545
$580$	0	0
$581$	−7.07107	−0.293357
$582$	0	0
$583$	36.0000	1.49097
$584$	0	0
$585$	0	0
$586$	0	0
$587$	29.6985	1.22579	0.612894	−	0.790165i	$-0.290005\pi$
$587$	29.6985	1.22579	0.612894	+	0.790165i	$0.290005\pi$
$588$	0	0
$589$	42.4264	1.74815
$590$	0	0
$591$	−26.0000	−1.06950
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	31.1127	1.27336
$598$	0	0
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	0	0
$603$	−11.3137	−0.460730
$604$	0	0
$605$	0	0
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	0	0
$611$	5.65685	0.228852
$612$	0	0
$613$	−26.8701	−1.08527	−0.542636	−	0.839968i	$-0.682574\pi$
$613$	−26.8701	−1.08527	−0.542636	+	0.839968i	$0.682574\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−40.0000	−1.61034	−0.805170	−	0.593045i	$-0.797926\pi$
$617$	−40.0000	−1.61034	−0.805170	+	0.593045i	$0.797926\pi$
$618$	0	0
$619$	−38.1838	−1.53474	−0.767368	−	0.641207i	$-0.778434\pi$
$619$	−38.1838	−1.53474	−0.767368	+	0.641207i	$0.778434\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.00000	−0.0801283
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	16.9706	0.677739
$628$	0	0
$629$	−25.4558	−1.01499
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	0	0
$633$	40.0000	1.58986
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.82843	0.112066
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	−24.0416	−0.948109	−0.474055	−	0.880495i	$-0.657210\pi$
$643$	−24.0416	−0.948109	−0.474055	+	0.880495i	$0.657210\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−38.0000	−1.49393	−0.746967	−	0.664861i	$-0.768491\pi$
$647$	−38.0000	−1.49393	−0.746967	+	0.664861i	$0.768491\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	0	0
$651$	14.1421	0.554274
$652$	0	0
$653$	−26.8701	−1.05151	−0.525753	−	0.850637i	$-0.676216\pi$
$653$	−26.8701	−1.05151	−0.525753	+	0.850637i	$0.676216\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	19.7990	0.771259	0.385630	−	0.922654i	$-0.373984\pi$
$659$	19.7990	0.771259	0.385630	+	0.922654i	$0.373984\pi$
$660$	0	0
$661$	39.5980	1.54018	0.770091	−	0.637934i	$-0.220211\pi$
$661$	39.5980	1.54018	0.770091	+	0.637934i	$0.220211\pi$
$662$	0	0
$663$	−24.0000	−0.932083
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	22.6274	0.874826
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−36.0000	−1.38770	−0.693849	−	0.720121i	$-0.744086\pi$
$673$	−36.0000	−1.38770	−0.693849	+	0.720121i	$0.744086\pi$
$674$	0	0
$675$	−28.2843	−1.08866
$676$	0	0
$677$	−2.82843	−0.108705	−0.0543526	−	0.998522i	$-0.517310\pi$
$677$	−2.82843	−0.108705	−0.0543526	+	0.998522i	$0.517310\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	2.00000	0.0766402
$682$	0	0
$683$	−22.6274	−0.865814	−0.432907	−	0.901439i	$-0.642512\pi$
$683$	−22.6274	−0.865814	−0.432907	+	0.901439i	$0.642512\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−36.0000	−1.37349
$688$	0	0
$689$	36.0000	1.37149
$690$	0	0
$691$	21.2132	0.806988	0.403494	−	0.914982i	$-0.367796\pi$
$691$	21.2132	0.806988	0.403494	+	0.914982i	$0.367796\pi$
$692$	0	0
$693$	−2.82843	−0.107443
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−60.0000	−2.27266
$698$	0	0
$699$	−14.1421	−0.534905
$700$	0	0
$701$	29.6985	1.12170	0.560848	−	0.827919i	$-0.310475\pi$
$701$	29.6985	1.12170	0.560848	+	0.827919i	$0.310475\pi$
$702$	0	0
$703$	18.0000	0.678883
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.65685	0.212748
$708$	0	0
$709$	−38.1838	−1.43402	−0.717011	−	0.697062i	$-0.754490\pi$
$709$	−38.1838	−1.43402	−0.717011	+	0.697062i	$0.754490\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−26.0000	−0.969636	−0.484818	−	0.874615i	$-0.661114\pi$
$719$	−26.0000	−0.969636	−0.484818	+	0.874615i	$0.661114\pi$
$720$	0	0
$721$	6.00000	0.223452
$722$	0	0
$723$	2.82843	0.105190
$724$	0	0
$725$	7.07107	0.262613
$726$	0	0
$727$	30.0000	1.11264	0.556319	−	0.830969i	$-0.312213\pi$
$727$	30.0000	1.11264	0.556319	+	0.830969i	$0.312213\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	−50.9117	−1.88304
$732$	0	0
$733$	−33.9411	−1.25364	−0.626822	−	0.779162i	$-0.715645\pi$
$733$	−33.9411	−1.25364	−0.626822	+	0.779162i	$0.715645\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	32.0000	1.17874
$738$	0	0
$739$	−31.1127	−1.14450	−0.572250	−	0.820080i	$-0.693929\pi$
$739$	−31.1127	−1.14450	−0.572250	+	0.820080i	$0.693929\pi$
$740$	0	0
$741$	16.9706	0.623429
$742$	0	0
$743$	−44.0000	−1.61420	−0.807102	−	0.590412i	$-0.798965\pi$
$743$	−44.0000	−1.61420	−0.807102	+	0.590412i	$0.798965\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	7.07107	0.258717
$748$	0	0
$749$	0	0
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	−22.0000	−0.801725
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−52.3259	−1.90182	−0.950909	−	0.309472i	$-0.899848\pi$
$757$	−52.3259	−1.90182	−0.950909	+	0.309472i	$0.899848\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	4.24264	0.153594
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.00000	−0.144432
$768$	0	0
$769$	−38.0000	−1.37032	−0.685158	−	0.728395i	$-0.740267\pi$
$769$	−38.0000	−1.37032	−0.685158	+	0.728395i	$0.740267\pi$
$770$	0	0
$771$	8.48528	0.305590
$772$	0	0
$773$	16.9706	0.610389	0.305194	−	0.952290i	$-0.401279\pi$
$773$	16.9706	0.610389	0.305194	+	0.952290i	$0.401279\pi$
$774$	0	0
$775$	50.0000	1.79605
$776$	0	0
$777$	6.00000	0.215249
$778$	0	0
$779$	42.4264	1.52008
$780$	0	0
$781$	−33.9411	−1.21451
$782$	0	0
$783$	−8.00000	−0.285897
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−18.3848	−0.655346	−0.327673	−	0.944791i	$-0.606265\pi$
$787$	−18.3848	−0.655346	−0.327673	+	0.944791i	$0.606265\pi$
$788$	0	0
$789$	−16.9706	−0.604168
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−31.1127	−1.10207	−0.551034	−	0.834483i	$-0.685767\pi$
$797$	−31.1127	−1.10207	−0.551034	+	0.834483i	$0.685767\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	5.65685	0.199626
$804$	0	0
$805$	0	0
$806$	0	0
$807$	28.0000	0.985647
$808$	0	0
$809$	22.0000	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$810$	0	0
$811$	−9.89949	−0.347618	−0.173809	−	0.984779i	$-0.555608\pi$
$811$	−9.89949	−0.347618	−0.173809	+	0.984779i	$0.555608\pi$
$812$	0	0
$813$	33.9411	1.19037
$814$	0	0
$815$	0	0
$816$	0	0
$817$	36.0000	1.25948
$818$	0	0
$819$	−2.82843	−0.0988332
$820$	0	0
$821$	43.8406	1.53005	0.765024	−	0.644002i	$-0.222727\pi$
$821$	43.8406	1.53005	0.765024	+	0.644002i	$0.222727\pi$
$822$	0	0
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	0	0
$825$	20.0000	0.696311
$826$	0	0
$827$	−28.2843	−0.983540	−0.491770	−	0.870725i	$-0.663650\pi$
$827$	−28.2843	−0.983540	−0.491770	+	0.870725i	$0.663650\pi$
$828$	0	0
$829$	−45.2548	−1.57177	−0.785883	−	0.618376i	$-0.787791\pi$
$829$	−45.2548	−1.57177	−0.785883	+	0.618376i	$0.787791\pi$
$830$	0	0
$831$	34.0000	1.17945
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−56.5685	−1.95529
$838$	0	0
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	0	0
$841$	−27.0000	−0.931034
$842$	0	0
$843$	14.1421	0.487081
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−3.00000	−0.103081
$848$	0	0
$849$	−30.0000	−1.02960
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−14.1421	−0.484218	−0.242109	−	0.970249i	$-0.577839\pi$
$853$	−14.1421	−0.484218	−0.242109	+	0.970249i	$0.577839\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	0	0
$859$	46.6690	1.59233	0.796164	−	0.605081i	$-0.206859\pi$
$859$	46.6690	1.59233	0.796164	+	0.605081i	$0.206859\pi$
$860$	0	0
$861$	14.1421	0.481963
$862$	0	0
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−26.8701	−0.912555
$868$	0	0
$869$	11.3137	0.383791
$870$	0	0
$871$	32.0000	1.08428
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.41421	−0.0477546	−0.0238773	−	0.999715i	$-0.507601\pi$
$877$	−1.41421	−0.0477546	−0.0238773	+	0.999715i	$0.507601\pi$
$878$	0	0
$879$	8.00000	0.269833
$880$	0	0
$881$	−26.0000	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$882$	0	0
$883$	39.5980	1.33258	0.666289	−	0.745694i	$-0.267882\pi$
$883$	39.5980	1.33258	0.666289	+	0.745694i	$0.267882\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	38.0000	1.27592	0.637958	−	0.770072i	$-0.279780\pi$
$887$	38.0000	1.27592	0.637958	+	0.770072i	$0.279780\pi$
$888$	0	0
$889$	−12.0000	−0.402467
$890$	0	0
$891$	−14.1421	−0.473779
$892$	0	0
$893$	−8.48528	−0.283949
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	14.1421	0.471667
$900$	0	0
$901$	76.3675	2.54417
$902$	0	0
$903$	12.0000	0.399335
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−22.6274	−0.751331	−0.375666	−	0.926755i	$-0.622586\pi$
$907$	−22.6274	−0.751331	−0.375666	+	0.926755i	$0.622586\pi$
$908$	0	0
$909$	−5.65685	−0.187626
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	−20.0000	−0.661903
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.89949	0.326910
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	−46.0000	−1.51575
$922$	0	0
$923$	−33.9411	−1.11719
$924$	0	0
$925$	21.2132	0.697486
$926$	0	0
$927$	−6.00000	−0.197066
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	−4.24264	−0.139047
$932$	0	0
$933$	11.3137	0.370394
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−30.0000	−0.980057	−0.490029	−	0.871706i	$-0.663014\pi$
$937$	−30.0000	−0.980057	−0.490029	+	0.871706i	$0.663014\pi$
$938$	0	0
$939$	−42.4264	−1.38453
$940$	0	0
$941$	−39.5980	−1.29086	−0.645429	−	0.763821i	$-0.723321\pi$
$941$	−39.5980	−1.29086	−0.645429	+	0.763821i	$0.723321\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	53.7401	1.74632	0.873160	−	0.487435i	$-0.162067\pi$
$947$	53.7401	1.74632	0.873160	+	0.487435i	$0.162067\pi$
$948$	0	0
$949$	5.65685	0.183629
$950$	0	0
$951$	34.0000	1.10253
$952$	0	0
$953$	−16.0000	−0.518291	−0.259145	−	0.965838i	$-0.583441\pi$
$953$	−16.0000	−0.518291	−0.259145	+	0.965838i	$0.583441\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	5.65685	0.182860
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	24.0000	0.771788	0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000	0.771788	0.385894	+	0.922543i	$0.373893\pi$
$968$	0	0
$969$	36.0000	1.15649
$970$	0	0
$971$	21.2132	0.680764	0.340382	−	0.940287i	$-0.389444\pi$
$971$	21.2132	0.680764	0.340382	+	0.940287i	$0.389444\pi$
$972$	0	0
$973$	4.24264	0.136013
$974$	0	0
$975$	20.0000	0.640513
$976$	0	0
$977$	20.0000	0.639857	0.319928	−	0.947442i	$-0.396341\pi$
$977$	20.0000	0.639857	0.319928	+	0.947442i	$0.396341\pi$
$978$	0	0
$979$	−5.65685	−0.180794
$980$	0	0
$981$	−4.24264	−0.135457
$982$	0	0
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2.82843	−0.0900298
$988$	0	0
$989$	0	0
$990$	0	0
$991$	48.0000	1.52477	0.762385	−	0.647124i	$-0.224028\pi$
$991$	48.0000	1.52477	0.762385	+	0.647124i	$0.224028\pi$
$992$	0	0
$993$	−12.0000	−0.380808
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28.2843	0.895772	0.447886	−	0.894091i	$-0.352177\pi$
$997$	28.2843	0.895772	0.447886	+	0.894091i	$0.352177\pi$
$998$	0	0
$999$	−24.0000	−0.759326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7168.2.a.n.1.1		2
4.3	odd	2		7168.2.a.e.1.2		2
8.3	odd	2		7168.2.a.e.1.1		2
8.5	even	2	inner	7168.2.a.n.1.2		2
32.3	odd	8		3584.2.m.u.2689.1	yes	2
32.5	even	8		3584.2.m.v.897.1	yes	2
32.11	odd	8		3584.2.m.u.897.1	yes	2
32.13	even	8		3584.2.m.v.2689.1	yes	2
32.19	odd	8		3584.2.m.h.2689.1	yes	2
32.21	even	8		3584.2.m.g.897.1	✓	2
32.27	odd	8		3584.2.m.h.897.1	yes	2
32.29	even	8		3584.2.m.g.2689.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.m.g.897.1	✓	2	32.21	even	8
3584.2.m.g.2689.1	yes	2	32.29	even	8
3584.2.m.h.897.1	yes	2	32.27	odd	8
3584.2.m.h.2689.1	yes	2	32.19	odd	8
3584.2.m.u.897.1	yes	2	32.11	odd	8
3584.2.m.u.2689.1	yes	2	32.3	odd	8
3584.2.m.v.897.1	yes	2	32.5	even	8
3584.2.m.v.2689.1	yes	2	32.13	even	8
7168.2.a.e.1.1		2	8.3	odd	2
7168.2.a.e.1.2		2	4.3	odd	2
7168.2.a.n.1.1		2	1.1	even	1	trivial
7168.2.a.n.1.2		2	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 7168 = 2^{10} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7168.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{3} +1.00000 q^{7} -1.00000 q^{9} +2.82843 q^{11} +2.82843 q^{13} +6.00000 q^{17} -4.24264 q^{19} -1.41421 q^{21} -5.00000 q^{25} +5.65685 q^{27} -1.41421 q^{29} -10.0000 q^{31} -4.00000 q^{33} -4.24264 q^{37} -4.00000 q^{39} -10.0000 q^{41} -8.48528 q^{43} +2.00000 q^{47} +1.00000 q^{49} -8.48528 q^{51} +12.7279 q^{53} +6.00000 q^{57} -1.41421 q^{59} -1.00000 q^{63} +11.3137 q^{67} -12.0000 q^{71} +2.00000 q^{73} +7.07107 q^{75} +2.82843 q^{77} +4.00000 q^{79} -5.00000 q^{81} -7.07107 q^{83} +2.00000 q^{87} -2.00000 q^{89} +2.82843 q^{91} +14.1421 q^{93} -2.00000 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^7 - 2 * q^9	\( 2 q + 2 q^{7} - 2 q^{9} + 12 q^{17} - 10 q^{25} - 20 q^{31} - 8 q^{33} - 8 q^{39} - 20 q^{41} + 4 q^{47} + 2 q^{49} + 12 q^{57} - 2 q^{63} - 24 q^{71} + 4 q^{73} + 8 q^{79} - 10 q^{81} + 4 q^{87} - 4 q^{89} - 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 - 2 * q^9 + 12 * q^17 - 10 * q^25 - 20 * q^31 - 8 * q^33 - 8 * q^39 - 20 * q^41 + 4 * q^47 + 2 * q^49 + 12 * q^57 - 2 * q^63 - 24 * q^71 + 4 * q^73 + 8 * q^79 - 10 * q^81 + 4 * q^87 - 4 * q^89 - 4 * q^97

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