LMFDB - Embedded newform 7840.2.a.cd.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7840,2,Mod(1,7840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7840 = 2^{5} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7840.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:	$62.6027151847$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.170145936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} + 22x^{2} - 9 $ x^6 - 10x^4 + 22x^2 - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1120)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.55008$ of defining polynomial
Character	$\chi$	$=$	7840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.385303 q^{3} -1.00000 q^{5} -2.85154 q^{9} +O(q^{10})$	$q-0.385303 q^{3} -1.00000 q^{5} -2.85154 q^{9} +5.19073 q^{11} +6.04604 q^{13} +0.385303 q^{15} +3.19450 q^{17} +7.52028 q^{19} -1.70526 q^{23} +1.00000 q^{25} +2.25461 q^{27} +7.04604 q^{29} -6.51068 q^{31} -2.00000 q^{33} -2.85154 q^{37} -2.32956 q^{39} +1.00000 q^{41} +6.58563 q^{43} +2.85154 q^{45} +5.19073 q^{47} -1.23085 q^{51} -4.04604 q^{53} -5.19073 q^{55} -2.89758 q^{57} -7.28129 q^{59} -6.65704 q^{61} -6.04604 q^{65} +11.5374 q^{67} +0.657042 q^{69} +11.9404 q^{71} -8.89758 q^{73} -0.385303 q^{75} -4.64137 q^{79} +7.68592 q^{81} -4.56643 q^{83} -3.19450 q^{85} -2.71486 q^{87} -2.14846 q^{89} +2.50858 q^{93} -7.52028 q^{95} -0.389001 q^{97} -14.8016 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{5} + 10 q^{9}+O(q^{10}) $ 6 * q - 6 * q^5 + 10 * q^9	$ 6 q - 6 q^{5} + 10 q^{9} - 2 q^{13} + 8 q^{17} + 6 q^{25} + 4 q^{29} - 12 q^{33} + 10 q^{37} + 6 q^{41} - 10 q^{45} + 14 q^{53} + 48 q^{57} - 24 q^{61} + 2 q^{65} - 12 q^{69} + 12 q^{73} + 78 q^{81} - 8 q^{85} - 40 q^{89} - 28 q^{93} + 20 q^{97}+O(q^{100}) $ 6 * q - 6 * q^5 + 10 * q^9 - 2 * q^13 + 8 * q^17 + 6 * q^25 + 4 * q^29 - 12 * q^33 + 10 * q^37 + 6 * q^41 - 10 * q^45 + 14 * q^53 + 48 * q^57 - 24 * q^61 + 2 * q^65 - 12 * q^69 + 12 * q^73 + 78 * q^81 - 8 * q^85 - 40 * q^89 - 28 * q^93 + 20 * 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
$3$	−0.385303	−0.222455	−0.111227	−	0.993795i	$-0.535478\pi$
$3$	−0.385303	−0.222455	−0.111227	+	0.993795i	$0.535478\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.85154	−0.950514
$10$	0	0
$11$	5.19073	1.56506	0.782531	−	0.622611i	$-0.213928\pi$
$11$	5.19073	1.56506	0.782531	+	0.622611i	$0.213928\pi$
$12$	0	0
$13$	6.04604	1.67687	0.838435	−	0.545001i	$-0.183471\pi$
$13$	6.04604	1.67687	0.838435	+	0.545001i	$0.183471\pi$
$14$	0	0
$15$	0.385303	0.0994847
$16$	0	0
$17$	3.19450	0.774780	0.387390	−	0.921916i	$-0.373377\pi$
$17$	3.19450	0.774780	0.387390	+	0.921916i	$0.373377\pi$
$18$	0	0
$19$	7.52028	1.72527	0.862635	−	0.505826i	$-0.168812\pi$
$19$	7.52028	1.72527	0.862635	+	0.505826i	$0.168812\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.70526	−0.355572	−0.177786	−	0.984069i	$-0.556893\pi$
$23$	−1.70526	−0.355572	−0.177786	+	0.984069i	$0.556893\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.25461	0.433901
$28$	0	0
$29$	7.04604	1.30842	0.654209	−	0.756314i	$-0.273002\pi$
$29$	7.04604	1.30842	0.654209	+	0.756314i	$0.273002\pi$
$30$	0	0
$31$	−6.51068	−1.16935	−0.584677	−	0.811266i	$-0.698779\pi$
$31$	−6.51068	−1.16935	−0.584677	+	0.811266i	$0.698779\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.85154	−0.468791	−0.234395	−	0.972141i	$-0.575311\pi$
$37$	−2.85154	−0.468791	−0.234395	+	0.972141i	$0.575311\pi$
$38$	0	0
$39$	−2.32956	−0.373027
$40$	0	0
$41$	1.00000	0.156174	0.0780869	−	0.996947i	$-0.475119\pi$
$41$	1.00000	0.156174	0.0780869	+	0.996947i	$0.475119\pi$
$42$	0	0
$43$	6.58563	1.00430	0.502149	−	0.864781i	$-0.332543\pi$
$43$	6.58563	1.00430	0.502149	+	0.864781i	$0.332543\pi$
$44$	0	0
$45$	2.85154	0.425083
$46$	0	0
$47$	5.19073	0.757145	0.378573	−	0.925572i	$-0.376415\pi$
$47$	5.19073	0.757145	0.378573	+	0.925572i	$0.376415\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1.23085	−0.172353
$52$	0	0
$53$	−4.04604	−0.555767	−0.277883	−	0.960615i	$-0.589633\pi$
$53$	−4.04604	−0.555767	−0.277883	+	0.960615i	$0.589633\pi$
$54$	0	0
$55$	−5.19073	−0.699917
$56$	0	0
$57$	−2.89758	−0.383794
$58$	0	0
$59$	−7.28129	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$59$	−7.28129	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$60$	0	0
$61$	−6.65704	−0.852347	−0.426173	−	0.904641i	$-0.640139\pi$
$61$	−6.65704	−0.852347	−0.426173	+	0.904641i	$0.640139\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.04604	−0.749919
$66$	0	0
$67$	11.5374	1.40951	0.704756	−	0.709449i	$-0.251056\pi$
$67$	11.5374	1.40951	0.704756	+	0.709449i	$0.251056\pi$
$68$	0	0
$69$	0.657042	0.0790985
$70$	0	0
$71$	11.9404	1.41706	0.708532	−	0.705678i	$-0.249358\pi$
$71$	11.9404	1.41706	0.708532	+	0.705678i	$0.249358\pi$
$72$	0	0
$73$	−8.89758	−1.04138	−0.520692	−	0.853745i	$-0.674326\pi$
$73$	−8.89758	−1.04138	−0.520692	+	0.853745i	$0.674326\pi$
$74$	0	0
$75$	−0.385303	−0.0444909
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.64137	−0.522195	−0.261098	−	0.965312i	$-0.584084\pi$
$79$	−4.64137	−0.522195	−0.261098	+	0.965312i	$0.584084\pi$
$80$	0	0
$81$	7.68592	0.853991
$82$	0	0
$83$	−4.56643	−0.501231	−0.250615	−	0.968087i	$-0.580633\pi$
$83$	−4.56643	−0.501231	−0.250615	+	0.968087i	$0.580633\pi$
$84$	0	0
$85$	−3.19450	−0.346492
$86$	0	0
$87$	−2.71486	−0.291063
$88$	0	0
$89$	−2.14846	−0.227736	−0.113868	−	0.993496i	$-0.536324\pi$
$89$	−2.14846	−0.227736	−0.113868	+	0.993496i	$0.536324\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.50858	0.260128
$94$	0	0
$95$	−7.52028	−0.771565
$96$	0	0
$97$	−0.389001	−0.0394970	−0.0197485	−	0.999805i	$-0.506287\pi$
$97$	−0.389001	−0.0394970	−0.0197485	+	0.999805i	$0.506287\pi$
$98$	0	0
$99$	−14.8016	−1.48761
$100$	0	0
$101$	−5.04604	−0.502100	−0.251050	−	0.967974i	$-0.580776\pi$
$101$	−5.04604	−0.502100	−0.251050	+	0.967974i	$0.580776\pi$
$102$	0	0
$103$	−8.91518	−0.878439	−0.439219	−	0.898380i	$-0.644745\pi$
$103$	−8.91518	−0.878439	−0.439219	+	0.898380i	$0.644745\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.22554	−0.891867	−0.445933	−	0.895066i	$-0.647128\pi$
$107$	−9.22554	−0.891867	−0.445933	+	0.895066i	$0.647128\pi$
$108$	0	0
$109$	16.7491	1.60428	0.802138	−	0.597139i	$-0.203696\pi$
$109$	16.7491	1.60428	0.802138	+	0.597139i	$0.203696\pi$
$110$	0	0
$111$	1.09871	0.104285
$112$	0	0
$113$	13.1945	1.24123	0.620617	−	0.784114i	$-0.286882\pi$
$113$	13.1945	1.24123	0.620617	+	0.784114i	$0.286882\pi$
$114$	0	0
$115$	1.70526	0.159016
$116$	0	0
$117$	−17.2405	−1.59389
$118$	0	0
$119$	0	0
$120$	0	0
$121$	15.9436	1.44942
$122$	0	0
$123$	−0.385303	−0.0347416
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−17.1134	−1.51857	−0.759284	−	0.650760i	$-0.774451\pi$
$127$	−17.1134	−1.51857	−0.759284	+	0.650760i	$0.774451\pi$
$128$	0	0
$129$	−2.53746	−0.223411
$130$	0	0
$131$	−20.5239	−1.79318	−0.896591	−	0.442859i	$-0.853964\pi$
$131$	−20.5239	−1.79318	−0.896591	+	0.442859i	$0.853964\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2.25461	−0.194046
$136$	0	0
$137$	1.61100	0.137637	0.0688185	−	0.997629i	$-0.478077\pi$
$137$	1.61100	0.137637	0.0688185	+	0.997629i	$0.478077\pi$
$138$	0	0
$139$	−6.97093	−0.591266	−0.295633	−	0.955302i	$-0.595531\pi$
$139$	−6.97093	−0.591266	−0.295633	+	0.955302i	$0.595531\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	0	0
$143$	31.3833	2.62441
$144$	0	0
$145$	−7.04604	−0.585142
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.4350	1.10064	0.550321	−	0.834953i	$-0.314505\pi$
$149$	13.4350	1.10064	0.550321	+	0.834953i	$0.314505\pi$
$150$	0	0
$151$	3.87077	0.314999	0.157499	−	0.987519i	$-0.449657\pi$
$151$	3.87077	0.314999	0.157499	+	0.987519i	$0.449657\pi$
$152$	0	0
$153$	−9.10925	−0.736439
$154$	0	0
$155$	6.51068	0.522951
$156$	0	0
$157$	15.2405	1.21633	0.608164	−	0.793812i	$-0.291906\pi$
$157$	15.2405	1.21633	0.608164	+	0.793812i	$0.291906\pi$
$158$	0	0
$159$	1.55895	0.123633
$160$	0	0
$161$	0	0
$162$	0	0
$163$	11.6300	0.910935	0.455468	−	0.890252i	$-0.349472\pi$
$163$	11.6300	0.910935	0.455468	+	0.890252i	$0.349472\pi$
$164$	0	0
$165$	2.00000	0.155700
$166$	0	0
$167$	2.95385	0.228576	0.114288	−	0.993448i	$-0.463541\pi$
$167$	2.95385	0.228576	0.114288	+	0.993448i	$0.463541\pi$
$168$	0	0
$169$	23.5546	1.81189
$170$	0	0
$171$	−21.4444	−1.63989
$172$	0	0
$173$	−5.14846	−0.391430	−0.195715	−	0.980661i	$-0.562703\pi$
$173$	−5.14846	−0.391430	−0.195715	+	0.980661i	$0.562703\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.80550	0.210874
$178$	0	0
$179$	−12.4720	−0.932202	−0.466101	−	0.884731i	$-0.654342\pi$
$179$	−12.4720	−0.932202	−0.466101	+	0.884731i	$0.654342\pi$
$180$	0	0
$181$	−18.7491	−1.39361	−0.696806	−	0.717260i	$-0.745396\pi$
$181$	−18.7491	−1.39361	−0.696806	+	0.717260i	$0.745396\pi$
$182$	0	0
$183$	2.56498	0.189608
$184$	0	0
$185$	2.85154	0.209650
$186$	0	0
$187$	16.5818	1.21258
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.7920	−0.997952	−0.498976	−	0.866616i	$-0.666290\pi$
$191$	−13.7920	−0.997952	−0.498976	+	0.866616i	$0.666290\pi$
$192$	0	0
$193$	15.7031	1.13033	0.565166	−	0.824977i	$-0.308812\pi$
$193$	15.7031	1.13033	0.565166	+	0.824977i	$0.308812\pi$
$194$	0	0
$195$	2.32956	0.166823
$196$	0	0
$197$	−3.24054	−0.230879	−0.115440	−	0.993315i	$-0.536828\pi$
$197$	−3.24054	−0.230879	−0.115440	+	0.993315i	$0.536828\pi$
$198$	0	0
$199$	14.5626	1.03231	0.516157	−	0.856494i	$-0.327362\pi$
$199$	14.5626	1.03231	0.516157	+	0.856494i	$0.327362\pi$
$200$	0	0
$201$	−4.44537	−0.313553
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.00000	−0.0698430
$206$	0	0
$207$	4.86262	0.337976
$208$	0	0
$209$	39.0357	2.70016
$210$	0	0
$211$	−26.7420	−1.84099	−0.920497	−	0.390751i	$-0.872215\pi$
$211$	−26.7420	−1.84099	−0.920497	+	0.390751i	$0.872215\pi$
$212$	0	0
$213$	−4.60067	−0.315233
$214$	0	0
$215$	−6.58563	−0.449136
$216$	0	0
$217$	0	0
$218$	0	0
$219$	3.42826	0.231661
$220$	0	0
$221$	19.3141	1.29921
$222$	0	0
$223$	−13.7920	−0.923579	−0.461789	−	0.886990i	$-0.652792\pi$
$223$	−13.7920	−0.923579	−0.461789	+	0.886990i	$0.652792\pi$
$224$	0	0
$225$	−2.85154	−0.190103
$226$	0	0
$227$	27.7338	1.84076	0.920379	−	0.391027i	$-0.127880\pi$
$227$	27.7338	1.84076	0.920379	+	0.391027i	$0.127880\pi$
$228$	0	0
$229$	6.09208	0.402576	0.201288	−	0.979532i	$-0.435487\pi$
$229$	6.09208	0.402576	0.201288	+	0.979532i	$0.435487\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	22.6007	1.48062	0.740309	−	0.672266i	$-0.234679\pi$
$233$	22.6007	1.48062	0.740309	+	0.672266i	$0.234679\pi$
$234$	0	0
$235$	−5.19073	−0.338606
$236$	0	0
$237$	1.78833	0.116165
$238$	0	0
$239$	−5.41198	−0.350072	−0.175036	−	0.984562i	$-0.556004\pi$
$239$	−5.41198	−0.350072	−0.175036	+	0.984562i	$0.556004\pi$
$240$	0	0
$241$	7.24054	0.466404	0.233202	−	0.972428i	$-0.425080\pi$
$241$	7.24054	0.466404	0.233202	+	0.972428i	$0.425080\pi$
$242$	0	0
$243$	−9.72525	−0.623875
$244$	0	0
$245$	0	0
$246$	0	0
$247$	45.4679	2.89306
$248$	0	0
$249$	1.75946	0.111501
$250$	0	0
$251$	20.2135	1.27587	0.637934	−	0.770091i	$-0.279789\pi$
$251$	20.2135	1.27587	0.637934	+	0.770091i	$0.279789\pi$
$252$	0	0
$253$	−8.85154	−0.556492
$254$	0	0
$255$	1.23085	0.0770788
$256$	0	0
$257$	18.3890	1.14707	0.573537	−	0.819180i	$-0.305571\pi$
$257$	18.3890	1.14707	0.573537	+	0.819180i	$0.305571\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−20.0921	−1.24367
$262$	0	0
$263$	−25.7896	−1.59025	−0.795127	−	0.606443i	$-0.792596\pi$
$263$	−25.7896	−1.59025	−0.795127	+	0.606443i	$0.792596\pi$
$264$	0	0
$265$	4.04604	0.248546
$266$	0	0
$267$	0.827807	0.0506609
$268$	0	0
$269$	32.4522	1.97865	0.989323	−	0.145739i	$-0.0465560\pi$
$269$	32.4522	1.97865	0.989323	+	0.145739i	$0.0465560\pi$
$270$	0	0
$271$	2.00145	0.121580	0.0607899	−	0.998151i	$-0.480638\pi$
$271$	2.00145	0.121580	0.0607899	+	0.998151i	$0.480638\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.19073	0.313012
$276$	0	0
$277$	5.58350	0.335480	0.167740	−	0.985831i	$-0.446353\pi$
$277$	5.58350	0.335480	0.167740	+	0.985831i	$0.446353\pi$
$278$	0	0
$279$	18.5655	1.11149
$280$	0	0
$281$	27.2405	1.62503	0.812517	−	0.582937i	$-0.198097\pi$
$281$	27.2405	1.62503	0.812517	+	0.582937i	$0.198097\pi$
$282$	0	0
$283$	−1.86931	−0.111119	−0.0555595	−	0.998455i	$-0.517694\pi$
$283$	−1.86931	−0.111119	−0.0555595	+	0.998455i	$0.517694\pi$
$284$	0	0
$285$	2.89758	0.171638
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−6.79517	−0.399716
$290$	0	0
$291$	0.149883	0.00878629
$292$	0	0
$293$	12.5546	0.733449	0.366725	−	0.930330i	$-0.380479\pi$
$293$	12.5546	0.733449	0.366725	+	0.930330i	$0.380479\pi$
$294$	0	0
$295$	7.28129	0.423933
$296$	0	0
$297$	11.7031	0.679082
$298$	0	0
$299$	−10.3101	−0.596247
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1.94425	0.111694
$304$	0	0
$305$	6.65704	0.381181
$306$	0	0
$307$	3.50320	0.199938	0.0999692	−	0.994991i	$-0.468126\pi$
$307$	3.50320	0.199938	0.0999692	+	0.994991i	$0.468126\pi$
$308$	0	0
$309$	3.43504	0.195413
$310$	0	0
$311$	−11.9404	−0.677078	−0.338539	−	0.940952i	$-0.609933\pi$
$311$	−11.9404	−0.677078	−0.338539	+	0.940952i	$0.609933\pi$
$312$	0	0
$313$	−19.4062	−1.09690	−0.548451	−	0.836183i	$-0.684782\pi$
$313$	−19.4062	−1.09690	−0.548451	+	0.836183i	$0.684782\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.805500	0.0452414	0.0226207	−	0.999744i	$-0.492799\pi$
$317$	0.805500	0.0452414	0.0226207	+	0.999744i	$0.492799\pi$
$318$	0	0
$319$	36.5741	2.04775
$320$	0	0
$321$	3.55463	0.198400
$322$	0	0
$323$	24.0235	1.33671
$324$	0	0
$325$	6.04604	0.335374
$326$	0	0
$327$	−6.45348	−0.356878
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2.56855	−0.141180	−0.0705901	−	0.997505i	$-0.522488\pi$
$331$	−2.56855	−0.141180	−0.0705901	+	0.997505i	$0.522488\pi$
$332$	0	0
$333$	8.13129	0.445592
$334$	0	0
$335$	−11.5374	−0.630353
$336$	0	0
$337$	14.8976	0.811523	0.405762	−	0.913979i	$-0.367006\pi$
$337$	14.8976	0.811523	0.405762	+	0.913979i	$0.367006\pi$
$338$	0	0
$339$	−5.08388	−0.276118
$340$	0	0
$341$	−33.7952	−1.83011
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.657042	−0.0353739
$346$	0	0
$347$	−2.40450	−0.129080	−0.0645401	−	0.997915i	$-0.520558\pi$
$347$	−2.40450	−0.129080	−0.0645401	+	0.997915i	$0.520558\pi$
$348$	0	0
$349$	4.95396	0.265179	0.132590	−	0.991171i	$-0.457671\pi$
$349$	4.95396	0.265179	0.132590	+	0.991171i	$0.457671\pi$
$350$	0	0
$351$	13.6315	0.727595
$352$	0	0
$353$	3.58350	0.190731	0.0953653	−	0.995442i	$-0.469598\pi$
$353$	3.58350	0.190731	0.0953653	+	0.995442i	$0.469598\pi$
$354$	0	0
$355$	−11.9404	−0.633731
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.0711	0.531532	0.265766	−	0.964038i	$-0.414375\pi$
$359$	10.0711	0.531532	0.265766	+	0.964038i	$0.414375\pi$
$360$	0	0
$361$	37.5546	1.97656
$362$	0	0
$363$	−6.14312	−0.322430
$364$	0	0
$365$	8.89758	0.465721
$366$	0	0
$367$	−36.8880	−1.92554	−0.962769	−	0.270326i	$-0.912869\pi$
$367$	−36.8880	−1.92554	−0.962769	+	0.270326i	$0.912869\pi$
$368$	0	0
$369$	−2.85154	−0.148445
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−16.0921	−0.833217	−0.416608	−	0.909086i	$-0.636781\pi$
$373$	−16.0921	−0.833217	−0.416608	+	0.909086i	$0.636781\pi$
$374$	0	0
$375$	0.385303	0.0198969
$376$	0	0
$377$	42.6007	2.19405
$378$	0	0
$379$	35.5822	1.82774	0.913868	−	0.406012i	$-0.133081\pi$
$379$	35.5822	1.82774	0.913868	+	0.406012i	$0.133081\pi$
$380$	0	0
$381$	6.59383	0.337812
$382$	0	0
$383$	17.8090	0.910000	0.455000	−	0.890491i	$-0.349639\pi$
$383$	17.8090	0.910000	0.455000	+	0.890491i	$0.349639\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−18.7792	−0.954600
$388$	0	0
$389$	6.68592	0.338989	0.169495	−	0.985531i	$-0.445786\pi$
$389$	6.68592	0.338989	0.169495	+	0.985531i	$0.445786\pi$
$390$	0	0
$391$	−5.44746	−0.275490
$392$	0	0
$393$	7.90792	0.398902
$394$	0	0
$395$	4.64137	0.233533
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9.09208	−0.454037	−0.227019	−	0.973890i	$-0.572898\pi$
$401$	−9.09208	−0.454037	−0.227019	+	0.973890i	$0.572898\pi$
$402$	0	0
$403$	−39.3639	−1.96085
$404$	0	0
$405$	−7.68592	−0.381916
$406$	0	0
$407$	−14.8016	−0.733687
$408$	0	0
$409$	−6.14846	−0.304022	−0.152011	−	0.988379i	$-0.548575\pi$
$409$	−6.14846	−0.304022	−0.152011	+	0.988379i	$0.548575\pi$
$410$	0	0
$411$	−0.620722	−0.0306180
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4.56643	0.224157
$416$	0	0
$417$	2.68592	0.131530
$418$	0	0
$419$	−4.42012	−0.215937	−0.107968	−	0.994154i	$-0.534435\pi$
$419$	−4.42012	−0.215937	−0.107968	+	0.994154i	$0.534435\pi$
$420$	0	0
$421$	−40.8412	−1.99048	−0.995239	−	0.0974605i	$-0.968928\pi$
$421$	−40.8412	−1.99048	−0.995239	+	0.0974605i	$0.968928\pi$
$422$	0	0
$423$	−14.8016	−0.719677
$424$	0	0
$425$	3.19450	0.154956
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−12.0921	−0.583811
$430$	0	0
$431$	−0.149883	−0.00721961	−0.00360980	−	0.999993i	$-0.501149\pi$
$431$	−0.149883	−0.00721961	−0.00360980	+	0.999993i	$0.501149\pi$
$432$	0	0
$433$	−7.58350	−0.364440	−0.182220	−	0.983258i	$-0.558328\pi$
$433$	−7.58350	−0.364440	−0.182220	+	0.983258i	$0.558328\pi$
$434$	0	0
$435$	2.71486	0.130168
$436$	0	0
$437$	−12.8240	−0.613457
$438$	0	0
$439$	−2.63992	−0.125996	−0.0629982	−	0.998014i	$-0.520066\pi$
$439$	−2.63992	−0.125996	−0.0629982	+	0.998014i	$0.520066\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.79582	−0.180345	−0.0901725	−	0.995926i	$-0.528742\pi$
$443$	−3.79582	−0.180345	−0.0901725	+	0.995926i	$0.528742\pi$
$444$	0	0
$445$	2.14846	0.101847
$446$	0	0
$447$	−5.17656	−0.244843
$448$	0	0
$449$	18.6110	0.878307	0.439154	−	0.898412i	$-0.355278\pi$
$449$	18.6110	0.878307	0.439154	+	0.898412i	$0.355278\pi$
$450$	0	0
$451$	5.19073	0.244422
$452$	0	0
$453$	−1.49142	−0.0700729
$454$	0	0
$455$	0	0
$456$	0	0
$457$	29.7031	1.38945	0.694726	−	0.719275i	$-0.255526\pi$
$457$	29.7031	1.38945	0.694726	+	0.719275i	$0.255526\pi$
$458$	0	0
$459$	7.20237	0.336178
$460$	0	0
$461$	−7.61100	−0.354480	−0.177240	−	0.984168i	$-0.556717\pi$
$461$	−7.61100	−0.354480	−0.177240	+	0.984168i	$0.556717\pi$
$462$	0	0
$463$	−7.42760	−0.345190	−0.172595	−	0.984993i	$-0.555215\pi$
$463$	−7.42760	−0.345190	−0.172595	+	0.984993i	$0.555215\pi$
$464$	0	0
$465$	−2.50858	−0.116333
$466$	0	0
$467$	2.54724	0.117872	0.0589360	−	0.998262i	$-0.481229\pi$
$467$	2.54724	0.117872	0.0589360	+	0.998262i	$0.481229\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−5.87222	−0.270578
$472$	0	0
$473$	34.1842	1.57179
$474$	0	0
$475$	7.52028	0.345054
$476$	0	0
$477$	11.5375	0.528264
$478$	0	0
$479$	−8.05189	−0.367900	−0.183950	−	0.982936i	$-0.558889\pi$
$479$	−8.05189	−0.367900	−0.183950	+	0.982936i	$0.558889\pi$
$480$	0	0
$481$	−17.2405	−0.786101
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.389001	0.0176636
$486$	0	0
$487$	27.1131	1.22861	0.614306	−	0.789068i	$-0.289436\pi$
$487$	27.1131	1.22861	0.614306	+	0.789068i	$0.289436\pi$
$488$	0	0
$489$	−4.48108	−0.202642
$490$	0	0
$491$	−4.96947	−0.224269	−0.112135	−	0.993693i	$-0.535769\pi$
$491$	−4.96947	−0.224269	−0.112135	+	0.993693i	$0.535769\pi$
$492$	0	0
$493$	22.5086	1.01374
$494$	0	0
$495$	14.8016	0.665281
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−10.6741	−0.477837	−0.238919	−	0.971040i	$-0.576793\pi$
$499$	−10.6741	−0.477837	−0.238919	+	0.971040i	$0.576793\pi$
$500$	0	0
$501$	−1.13813	−0.0508477
$502$	0	0
$503$	13.3889	0.596983	0.298491	−	0.954412i	$-0.403517\pi$
$503$	13.3889	0.596983	0.298491	+	0.954412i	$0.403517\pi$
$504$	0	0
$505$	5.04604	0.224546
$506$	0	0
$507$	−9.07566	−0.403064
$508$	0	0
$509$	−9.43504	−0.418201	−0.209100	−	0.977894i	$-0.567054\pi$
$509$	−9.43504	−0.418201	−0.209100	+	0.977894i	$0.567054\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	16.9553	0.748596
$514$	0	0
$515$	8.91518	0.392850
$516$	0	0
$517$	26.9436	1.18498
$518$	0	0
$519$	1.98371	0.0870754
$520$	0	0
$521$	−38.6467	−1.69314	−0.846572	−	0.532275i	$-0.821337\pi$
$521$	−38.6467	−1.69314	−0.846572	+	0.532275i	$0.821337\pi$
$522$	0	0
$523$	29.2750	1.28011	0.640054	−	0.768330i	$-0.278912\pi$
$523$	29.2750	1.28011	0.640054	+	0.768330i	$0.278912\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−20.7984	−0.905992
$528$	0	0
$529$	−20.0921	−0.873569
$530$	0	0
$531$	20.7629	0.901033
$532$	0	0
$533$	6.04604	0.261883
$534$	0	0
$535$	9.22554	0.398855
$536$	0	0
$537$	4.80550	0.207373
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−4.56496	−0.196263	−0.0981314	−	0.995173i	$-0.531287\pi$
$541$	−4.56496	−0.196263	−0.0981314	+	0.995173i	$0.531287\pi$
$542$	0	0
$543$	7.22409	0.310015
$544$	0	0
$545$	−16.7491	−0.717454
$546$	0	0
$547$	35.4182	1.51437	0.757186	−	0.653200i	$-0.226574\pi$
$547$	35.4182	1.51437	0.757186	+	0.653200i	$0.226574\pi$
$548$	0	0
$549$	18.9828	0.810168
$550$	0	0
$551$	52.9882	2.25737
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.09871	−0.0466375
$556$	0	0
$557$	−17.6295	−0.746988	−0.373494	−	0.927633i	$-0.621840\pi$
$557$	−17.6295	−0.746988	−0.373494	+	0.927633i	$0.621840\pi$
$558$	0	0
$559$	39.8170	1.68408
$560$	0	0
$561$	−6.38900	−0.269744
$562$	0	0
$563$	21.9188	0.923768	0.461884	−	0.886940i	$-0.347174\pi$
$563$	21.9188	0.923768	0.461884	+	0.886940i	$0.347174\pi$
$564$	0	0
$565$	−13.1945	−0.555097
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10.4625	0.438613	0.219306	−	0.975656i	$-0.429621\pi$
$569$	10.4625	0.438613	0.219306	+	0.975656i	$0.429621\pi$
$570$	0	0
$571$	15.0228	0.628686	0.314343	−	0.949310i	$-0.398216\pi$
$571$	15.0228	0.628686	0.314343	+	0.949310i	$0.398216\pi$
$572$	0	0
$573$	5.31408	0.221999
$574$	0	0
$575$	−1.70526	−0.0711143
$576$	0	0
$577$	−34.2763	−1.42694	−0.713470	−	0.700686i	$-0.752877\pi$
$577$	−34.2763	−1.42694	−0.713470	+	0.700686i	$0.752877\pi$
$578$	0	0
$579$	−6.05044	−0.251448
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−21.0019	−0.869810
$584$	0	0
$585$	17.2405	0.712809
$586$	0	0
$587$	−43.8731	−1.81084	−0.905418	−	0.424521i	$-0.860443\pi$
$587$	−43.8731	−1.81084	−0.905418	+	0.424521i	$0.860443\pi$
$588$	0	0
$589$	−48.9622	−2.01745
$590$	0	0
$591$	1.24859	0.0513601
$592$	0	0
$593$	15.5835	0.639938	0.319969	−	0.947428i	$-0.396328\pi$
$593$	15.5835	0.639938	0.319969	+	0.947428i	$0.396328\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−5.61100	−0.229643
$598$	0	0
$599$	−30.2239	−1.23491	−0.617457	−	0.786605i	$-0.711837\pi$
$599$	−30.2239	−1.23491	−0.617457	+	0.786605i	$0.711837\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	−32.8993	−1.33976
$604$	0	0
$605$	−15.9436	−0.648201
$606$	0	0
$607$	−21.5477	−0.874593	−0.437297	−	0.899317i	$-0.644064\pi$
$607$	−21.5477	−0.874593	−0.437297	+	0.899317i	$0.644064\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	31.3833	1.26963
$612$	0	0
$613$	−36.5271	−1.47532	−0.737658	−	0.675174i	$-0.764068\pi$
$613$	−36.5271	−1.47532	−0.737658	+	0.675174i	$0.764068\pi$
$614$	0	0
$615$	0.385303	0.0155369
$616$	0	0
$617$	45.1093	1.81603	0.908015	−	0.418938i	$-0.137597\pi$
$617$	45.1093	1.81603	0.908015	+	0.418938i	$0.137597\pi$
$618$	0	0
$619$	8.13041	0.326789	0.163394	−	0.986561i	$-0.447756\pi$
$619$	8.13041	0.326789	0.163394	+	0.986561i	$0.447756\pi$
$620$	0	0
$621$	−3.84471	−0.154283
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−15.0406	−0.600662
$628$	0	0
$629$	−9.10925	−0.363210
$630$	0	0
$631$	−40.1345	−1.59773	−0.798864	−	0.601512i	$-0.794565\pi$
$631$	−40.1345	−1.59773	−0.798864	+	0.601512i	$0.794565\pi$
$632$	0	0
$633$	10.3038	0.409537
$634$	0	0
$635$	17.1134	0.679124
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−34.0486	−1.34694
$640$	0	0
$641$	−9.18417	−0.362753	−0.181376	−	0.983414i	$-0.558055\pi$
$641$	−9.18417	−0.362753	−0.181376	+	0.983414i	$0.558055\pi$
$642$	0	0
$643$	−33.4916	−1.32078	−0.660391	−	0.750922i	$-0.729609\pi$
$643$	−33.4916	−1.32078	−0.660391	+	0.750922i	$0.729609\pi$
$644$	0	0
$645$	2.53746	0.0999123
$646$	0	0
$647$	3.39635	0.133524	0.0667622	−	0.997769i	$-0.478733\pi$
$647$	3.39635	0.133524	0.0667622	+	0.997769i	$0.478733\pi$
$648$	0	0
$649$	−37.7952	−1.48359
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.73196	0.263442	0.131721	−	0.991287i	$-0.457950\pi$
$653$	6.73196	0.263442	0.131721	+	0.991287i	$0.457950\pi$
$654$	0	0
$655$	20.5239	0.801935
$656$	0	0
$657$	25.3718	0.989850
$658$	0	0
$659$	−21.8439	−0.850916	−0.425458	−	0.904978i	$-0.639887\pi$
$659$	−21.8439	−0.850916	−0.425458	+	0.904978i	$0.639887\pi$
$660$	0	0
$661$	30.5443	1.18804	0.594018	−	0.804452i	$-0.297541\pi$
$661$	30.5443	1.18804	0.594018	+	0.804452i	$0.297541\pi$
$662$	0	0
$663$	−7.44177	−0.289014
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−12.0153	−0.465236
$668$	0	0
$669$	5.31408	0.205454
$670$	0	0
$671$	−34.5549	−1.33398
$672$	0	0
$673$	−1.70308	−0.0656491	−0.0328245	−	0.999461i	$-0.510450\pi$
$673$	−1.70308	−0.0656491	−0.0328245	+	0.999461i	$0.510450\pi$
$674$	0	0
$675$	2.25461	0.0867802
$676$	0	0
$677$	21.2130	0.815283	0.407642	−	0.913142i	$-0.366351\pi$
$677$	21.2130	0.815283	0.407642	+	0.913142i	$0.366351\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−10.6859	−0.409485
$682$	0	0
$683$	24.7086	0.945449	0.472724	−	0.881210i	$-0.343271\pi$
$683$	24.7086	0.945449	0.472724	+	0.881210i	$0.343271\pi$
$684$	0	0
$685$	−1.61100	−0.0615531
$686$	0	0
$687$	−2.34730	−0.0895549
$688$	0	0
$689$	−24.4625	−0.931949
$690$	0	0
$691$	26.8133	1.02003	0.510014	−	0.860166i	$-0.329640\pi$
$691$	26.8133	1.02003	0.510014	+	0.860166i	$0.329640\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.97093	0.264422
$696$	0	0
$697$	3.19450	0.121000
$698$	0	0
$699$	−8.70810	−0.329370
$700$	0	0
$701$	23.5271	0.888607	0.444304	−	0.895876i	$-0.353451\pi$
$701$	23.5271	0.888607	0.444304	+	0.895876i	$0.353451\pi$
$702$	0	0
$703$	−21.4444	−0.808791
$704$	0	0
$705$	2.00000	0.0753244
$706$	0	0
$707$	0	0
$708$	0	0
$709$	10.4522	0.392541	0.196270	−	0.980550i	$-0.437117\pi$
$709$	10.4522	0.392541	0.196270	+	0.980550i	$0.437117\pi$
$710$	0	0
$711$	13.2351	0.496354
$712$	0	0
$713$	11.1024	0.415789
$714$	0	0
$715$	−31.3833	−1.17367
$716$	0	0
$717$	2.08525	0.0778750
$718$	0	0
$719$	−24.6159	−0.918019	−0.459010	−	0.888431i	$-0.651796\pi$
$719$	−24.6159	−0.918019	−0.459010	+	0.888431i	$0.651796\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−2.78980	−0.103754
$724$	0	0
$725$	7.04604	0.261683
$726$	0	0
$727$	50.6800	1.87962	0.939808	−	0.341703i	$-0.111004\pi$
$727$	50.6800	1.87962	0.939808	+	0.341703i	$0.111004\pi$
$728$	0	0
$729$	−19.3106	−0.715207
$730$	0	0
$731$	21.0378	0.778110
$732$	0	0
$733$	−8.85154	−0.326939	−0.163470	−	0.986548i	$-0.552269\pi$
$733$	−8.85154	−0.326939	−0.163470	+	0.986548i	$0.552269\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	59.8873	2.20598
$738$	0	0
$739$	2.72903	0.100389	0.0501944	−	0.998739i	$-0.484016\pi$
$739$	2.72903	0.100389	0.0501944	+	0.998739i	$0.484016\pi$
$740$	0	0
$741$	−17.5189	−0.643573
$742$	0	0
$743$	6.51426	0.238985	0.119492	−	0.992835i	$-0.461873\pi$
$743$	6.51426	0.238985	0.119492	+	0.992835i	$0.461873\pi$
$744$	0	0
$745$	−13.4350	−0.492222
$746$	0	0
$747$	13.0214	0.476427
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−15.0406	−0.548838	−0.274419	−	0.961610i	$-0.588486\pi$
$751$	−15.0406	−0.548838	−0.274419	+	0.961610i	$0.588486\pi$
$752$	0	0
$753$	−7.78833	−0.283823
$754$	0	0
$755$	−3.87077	−0.140872
$756$	0	0
$757$	51.5628	1.87408	0.937042	−	0.349218i	$-0.113553\pi$
$757$	51.5628	1.87408	0.937042	+	0.349218i	$0.113553\pi$
$758$	0	0
$759$	3.41052	0.123794
$760$	0	0
$761$	−24.8515	−0.900868	−0.450434	−	0.892810i	$-0.648731\pi$
$761$	−24.8515	−0.900868	−0.450434	+	0.892810i	$0.648731\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	9.10925	0.329346
$766$	0	0
$767$	−44.0230	−1.58958
$768$	0	0
$769$	−14.5546	−0.524853	−0.262427	−	0.964952i	$-0.584523\pi$
$769$	−14.5546	−0.524853	−0.262427	+	0.964952i	$0.584523\pi$
$770$	0	0
$771$	−7.08533	−0.255172
$772$	0	0
$773$	35.9333	1.29243	0.646215	−	0.763155i	$-0.276351\pi$
$773$	35.9333	1.29243	0.646215	+	0.763155i	$0.276351\pi$
$774$	0	0
$775$	−6.51068	−0.233871
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7.52028	0.269442
$780$	0	0
$781$	61.9793	2.21780
$782$	0	0
$783$	15.8861	0.567723
$784$	0	0
$785$	−15.2405	−0.543958
$786$	0	0
$787$	−7.39171	−0.263486	−0.131743	−	0.991284i	$-0.542057\pi$
$787$	−7.39171	−0.263486	−0.131743	+	0.991284i	$0.542057\pi$
$788$	0	0
$789$	9.93679	0.353759
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−40.2488	−1.42928
$794$	0	0
$795$	−1.55895	−0.0552903
$796$	0	0
$797$	12.9251	0.457830	0.228915	−	0.973446i	$-0.426482\pi$
$797$	12.9251	0.457830	0.228915	+	0.973446i	$0.426482\pi$
$798$	0	0
$799$	16.5818	0.586621
$800$	0	0
$801$	6.12642	0.216466
$802$	0	0
$803$	−46.1849	−1.62983
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−12.5039	−0.440159
$808$	0	0
$809$	24.2591	0.852904	0.426452	−	0.904510i	$-0.359763\pi$
$809$	24.2591	0.852904	0.426452	+	0.904510i	$0.359763\pi$
$810$	0	0
$811$	−10.3101	−0.362036	−0.181018	−	0.983480i	$-0.557939\pi$
$811$	−10.3101	−0.362036	−0.181018	+	0.983480i	$0.557939\pi$
$812$	0	0
$813$	−0.771166	−0.0270460
$814$	0	0
$815$	−11.6300	−0.407383
$816$	0	0
$817$	49.5258	1.73269
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−48.5732	−1.69522	−0.847608	−	0.530623i	$-0.821958\pi$
$821$	−48.5732	−1.69522	−0.847608	+	0.530623i	$0.821958\pi$
$822$	0	0
$823$	16.6922	0.581853	0.290927	−	0.956745i	$-0.406036\pi$
$823$	16.6922	0.581853	0.290927	+	0.956745i	$0.406036\pi$
$824$	0	0
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	34.9402	1.21499	0.607494	−	0.794324i	$-0.292175\pi$
$827$	34.9402	1.21499	0.607494	+	0.794324i	$0.292175\pi$
$828$	0	0
$829$	15.6110	0.542192	0.271096	−	0.962552i	$-0.412614\pi$
$829$	15.6110	0.542192	0.271096	+	0.962552i	$0.412614\pi$
$830$	0	0
$831$	−2.15134	−0.0746291
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−2.95385	−0.102222
$836$	0	0
$837$	−14.6791	−0.507383
$838$	0	0
$839$	−44.9718	−1.55260	−0.776300	−	0.630364i	$-0.782906\pi$
$839$	−44.9718	−1.55260	−0.776300	+	0.630364i	$0.782906\pi$
$840$	0	0
$841$	20.6467	0.711956
$842$	0	0
$843$	−10.4959	−0.361496
$844$	0	0
$845$	−23.5546	−0.810304
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0.720251	0.0247189
$850$	0	0
$851$	4.86262	0.166689
$852$	0	0
$853$	28.2302	0.966584	0.483292	−	0.875459i	$-0.339441\pi$
$853$	28.2302	0.966584	0.483292	+	0.875459i	$0.339441\pi$
$854$	0	0
$855$	21.4444	0.733383
$856$	0	0
$857$	29.5903	1.01079	0.505393	−	0.862889i	$-0.331347\pi$
$857$	29.5903	1.01079	0.505393	+	0.862889i	$0.331347\pi$
$858$	0	0
$859$	12.9859	0.443073	0.221536	−	0.975152i	$-0.428893\pi$
$859$	12.9859	0.443073	0.221536	+	0.975152i	$0.428893\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7.61296	0.259148	0.129574	−	0.991570i	$-0.458639\pi$
$863$	7.61296	0.259148	0.129574	+	0.991570i	$0.458639\pi$
$864$	0	0
$865$	5.14846	0.175053
$866$	0	0
$867$	2.61820	0.0889186
$868$	0	0
$869$	−24.0921	−0.817268
$870$	0	0
$871$	69.7554	2.36357
$872$	0	0
$873$	1.10925	0.0375425
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.45221	0.116573	0.0582864	−	0.998300i	$-0.481436\pi$
$877$	3.45221	0.116573	0.0582864	+	0.998300i	$0.481436\pi$
$878$	0	0
$879$	−4.83733	−0.163159
$880$	0	0
$881$	−23.0921	−0.777992	−0.388996	−	0.921239i	$-0.627178\pi$
$881$	−23.0921	−0.777992	−0.388996	+	0.921239i	$0.627178\pi$
$882$	0	0
$883$	6.97093	0.234590	0.117295	−	0.993097i	$-0.462578\pi$
$883$	6.97093	0.234590	0.117295	+	0.993097i	$0.462578\pi$
$884$	0	0
$885$	−2.80550	−0.0943058
$886$	0	0
$887$	−33.5666	−1.12706	−0.563528	−	0.826097i	$-0.690556\pi$
$887$	−33.5666	−1.12706	−0.563528	+	0.826097i	$0.690556\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	39.8955	1.33655
$892$	0	0
$893$	39.0357	1.30628
$894$	0	0
$895$	12.4720	0.416893
$896$	0	0
$897$	3.97250	0.132638
$898$	0	0
$899$	−45.8746	−1.53000
$900$	0	0
$901$	−12.9251	−0.430597
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18.7491	0.623242
$906$	0	0
$907$	−18.0658	−0.599864	−0.299932	−	0.953961i	$-0.596964\pi$
$907$	−18.0658	−0.599864	−0.299932	+	0.953961i	$0.596964\pi$
$908$	0	0
$909$	14.3890	0.477253
$910$	0	0
$911$	35.6536	1.18126	0.590628	−	0.806944i	$-0.298880\pi$
$911$	35.6536	1.18126	0.590628	+	0.806944i	$0.298880\pi$
$912$	0	0
$913$	−23.7031	−0.784458
$914$	0	0
$915$	−2.56498	−0.0847955
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−5.29757	−0.174751	−0.0873754	−	0.996175i	$-0.527848\pi$
$919$	−5.29757	−0.174751	−0.0873754	+	0.996175i	$0.527848\pi$
$920$	0	0
$921$	−1.34979	−0.0444772
$922$	0	0
$923$	72.1922	2.37623
$924$	0	0
$925$	−2.85154	−0.0937581
$926$	0	0
$927$	25.4220	0.834968
$928$	0	0
$929$	28.0172	0.919213	0.459607	−	0.888123i	$-0.347990\pi$
$929$	28.0172	0.919213	0.459607	+	0.888123i	$0.347990\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	4.60067	0.150619
$934$	0	0
$935$	−16.5818	−0.542282
$936$	0	0
$937$	17.7031	0.578335	0.289167	−	0.957279i	$-0.406622\pi$
$937$	17.7031	0.578335	0.289167	+	0.957279i	$0.406622\pi$
$938$	0	0
$939$	7.47725	0.244011
$940$	0	0
$941$	−2.68592	−0.0875584	−0.0437792	−	0.999041i	$-0.513940\pi$
$941$	−2.68592	−0.0875584	−0.0437792	+	0.999041i	$0.513940\pi$
$942$	0	0
$943$	−1.70526	−0.0555309
$944$	0	0
$945$	0	0
$946$	0	0
$947$	46.7201	1.51820	0.759100	−	0.650975i	$-0.225640\pi$
$947$	46.7201	1.51820	0.759100	+	0.650975i	$0.225640\pi$
$948$	0	0
$949$	−53.7952	−1.74627
$950$	0	0
$951$	−0.310361	−0.0100642
$952$	0	0
$953$	46.3890	1.50269	0.751344	−	0.659911i	$-0.229406\pi$
$953$	46.3890	1.50269	0.751344	+	0.659911i	$0.229406\pi$
$954$	0	0
$955$	13.7920	0.446298
$956$	0	0
$957$	−14.0921	−0.455532
$958$	0	0
$959$	0	0
$960$	0	0
$961$	11.3890	0.367387
$962$	0	0
$963$	26.3070	0.847732
$964$	0	0
$965$	−15.7031	−0.505500
$966$	0	0
$967$	8.76530	0.281873	0.140936	−	0.990019i	$-0.454989\pi$
$967$	8.76530	0.281873	0.140936	+	0.990019i	$0.454989\pi$
$968$	0	0
$969$	−9.25633	−0.297356
$970$	0	0
$971$	−22.1006	−0.709242	−0.354621	−	0.935010i	$-0.615390\pi$
$971$	−22.1006	−0.709242	−0.354621	+	0.935010i	$0.615390\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2.32956	−0.0746055
$976$	0	0
$977$	−31.6110	−1.01133	−0.505663	−	0.862731i	$-0.668752\pi$
$977$	−31.6110	−1.01133	−0.505663	+	0.862731i	$0.668752\pi$
$978$	0	0
$979$	−11.1521	−0.356421
$980$	0	0
$981$	−47.7608	−1.52489
$982$	0	0
$983$	−37.1806	−1.18588	−0.592939	−	0.805248i	$-0.702032\pi$
$983$	−37.1806	−1.18588	−0.592939	+	0.805248i	$0.702032\pi$
$984$	0	0
$985$	3.24054	0.103252
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−11.2302	−0.357100
$990$	0	0
$991$	−50.2339	−1.59573	−0.797866	−	0.602835i	$-0.794038\pi$
$991$	−50.2339	−1.59573	−0.797866	+	0.602835i	$0.794038\pi$
$992$	0	0
$993$	0.989668	0.0314062
$994$	0	0
$995$	−14.5626	−0.461665
$996$	0	0
$997$	12.6928	0.401983	0.200992	−	0.979593i	$-0.435584\pi$
$997$	12.6928	0.401983	0.200992	+	0.979593i	$0.435584\pi$
$998$	0	0
$999$	−6.42913	−0.203409

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7840.2.a.cd.1.3		6
4.3	odd	2	inner	7840.2.a.cd.1.4		6
7.3	odd	6		1120.2.q.k.961.3	yes	12
7.5	odd	6		1120.2.q.k.641.3	✓	12
7.6	odd	2		7840.2.a.ce.1.4		6
28.3	even	6		1120.2.q.k.961.4	yes	12
28.19	even	6		1120.2.q.k.641.4	yes	12
28.27	even	2		7840.2.a.ce.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.q.k.641.3	✓	12	7.5	odd	6
1120.2.q.k.641.4	yes	12	28.19	even	6
1120.2.q.k.961.3	yes	12	7.3	odd	6
1120.2.q.k.961.4	yes	12	28.3	even	6
7840.2.a.cd.1.3		6	1.1	even	1	trivial
7840.2.a.cd.1.4		6	4.3	odd	2	inner
7840.2.a.ce.1.3		6	28.27	even	2
7840.2.a.ce.1.4		6	7.6	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:	\( N \)	\(=\)	\( 7840 = 2^{5} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7840.a (trivial)

\(f(q)\)	\(=\)	\(q-0.385303 q^{3} -1.00000 q^{5} -2.85154 q^{9} +O(q^{10})\)	\(q-0.385303 q^{3} -1.00000 q^{5} -2.85154 q^{9} +5.19073 q^{11} +6.04604 q^{13} +0.385303 q^{15} +3.19450 q^{17} +7.52028 q^{19} -1.70526 q^{23} +1.00000 q^{25} +2.25461 q^{27} +7.04604 q^{29} -6.51068 q^{31} -2.00000 q^{33} -2.85154 q^{37} -2.32956 q^{39} +1.00000 q^{41} +6.58563 q^{43} +2.85154 q^{45} +5.19073 q^{47} -1.23085 q^{51} -4.04604 q^{53} -5.19073 q^{55} -2.89758 q^{57} -7.28129 q^{59} -6.65704 q^{61} -6.04604 q^{65} +11.5374 q^{67} +0.657042 q^{69} +11.9404 q^{71} -8.89758 q^{73} -0.385303 q^{75} -4.64137 q^{79} +7.68592 q^{81} -4.56643 q^{83} -3.19450 q^{85} -2.71486 q^{87} -2.14846 q^{89} +2.50858 q^{93} -7.52028 q^{95} -0.389001 q^{97} -14.8016 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{5} + 10 q^{9}+O(q^{10}) \) 6 * q - 6 * q^5 + 10 * q^9	\( 6 q - 6 q^{5} + 10 q^{9} - 2 q^{13} + 8 q^{17} + 6 q^{25} + 4 q^{29} - 12 q^{33} + 10 q^{37} + 6 q^{41} - 10 q^{45} + 14 q^{53} + 48 q^{57} - 24 q^{61} + 2 q^{65} - 12 q^{69} + 12 q^{73} + 78 q^{81} - 8 q^{85} - 40 q^{89} - 28 q^{93} + 20 q^{97}+O(q^{100}) \) 6 * q - 6 * q^5 + 10 * q^9 - 2 * q^13 + 8 * q^17 + 6 * q^25 + 4 * q^29 - 12 * q^33 + 10 * q^37 + 6 * q^41 - 10 * q^45 + 14 * q^53 + 48 * q^57 - 24 * q^61 + 2 * q^65 - 12 * q^69 + 12 * q^73 + 78 * q^81 - 8 * q^85 - 40 * q^89 - 28 * q^93 + 20 * q^97

Introduction

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