LMFDB - Embedded newform 4788.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4788,2,Mod(1,4788)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4788, 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("4788.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4788.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:	$38.2323724878$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1596)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	4788.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.73205 q^{5} -1.00000 q^{7} +O(q^{10})$	$q+2.73205 q^{5} -1.00000 q^{7} -3.46410 q^{11} -4.00000 q^{13} -4.19615 q^{17} -1.00000 q^{19} +7.46410 q^{23} +2.46410 q^{25} +6.19615 q^{29} +4.92820 q^{31} -2.73205 q^{35} -7.46410 q^{37} +9.46410 q^{41} -3.46410 q^{43} -2.73205 q^{47} +1.00000 q^{49} -8.73205 q^{53} -9.46410 q^{55} +10.9282 q^{59} -14.3923 q^{61} -10.9282 q^{65} +1.46410 q^{67} -10.1962 q^{71} -15.8564 q^{73} +3.46410 q^{77} -13.4641 q^{79} +1.26795 q^{83} -11.4641 q^{85} -9.46410 q^{89} +4.00000 q^{91} -2.73205 q^{95} -6.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 - 2 * q^7	$ 2 q + 2 q^{5} - 2 q^{7} - 8 q^{13} + 2 q^{17} - 2 q^{19} + 8 q^{23} - 2 q^{25} + 2 q^{29} - 4 q^{31} - 2 q^{35} - 8 q^{37} + 12 q^{41} - 2 q^{47} + 2 q^{49} - 14 q^{53} - 12 q^{55} + 8 q^{59} - 8 q^{61} - 8 q^{65} - 4 q^{67} - 10 q^{71} - 4 q^{73} - 20 q^{79} + 6 q^{83} - 16 q^{85} - 12 q^{89} + 8 q^{91} - 2 q^{95} - 12 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^7 - 8 * q^13 + 2 * q^17 - 2 * q^19 + 8 * q^23 - 2 * q^25 + 2 * q^29 - 4 * q^31 - 2 * q^35 - 8 * q^37 + 12 * q^41 - 2 * q^47 + 2 * q^49 - 14 * q^53 - 12 * q^55 + 8 * q^59 - 8 * q^61 - 8 * q^65 - 4 * q^67 - 10 * q^71 - 4 * q^73 - 20 * q^79 + 6 * q^83 - 16 * q^85 - 12 * q^89 + 8 * q^91 - 2 * q^95 - 12 * 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	0
$3$	0	0
$4$	0	0
$5$	2.73205	1.22181	0.610905	−	0.791704i	$-0.290806\pi$
$5$	2.73205	1.22181	0.610905	+	0.791704i	$0.290806\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.19615	−1.01772	−0.508858	−	0.860850i	$-0.669932\pi$
$17$	−4.19615	−1.01772	−0.508858	+	0.860850i	$0.669932\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.46410	1.55637	0.778186	−	0.628033i	$-0.216140\pi$
$23$	7.46410	1.55637	0.778186	+	0.628033i	$0.216140\pi$
$24$	0	0
$25$	2.46410	0.492820
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.19615	1.15060	0.575298	−	0.817944i	$-0.304886\pi$
$29$	6.19615	1.15060	0.575298	+	0.817944i	$0.304886\pi$
$30$	0	0
$31$	4.92820	0.885131	0.442566	−	0.896736i	$-0.354068\pi$
$31$	4.92820	0.885131	0.442566	+	0.896736i	$0.354068\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.73205	−0.461801
$36$	0	0
$37$	−7.46410	−1.22709	−0.613545	−	0.789659i	$-0.710257\pi$
$37$	−7.46410	−1.22709	−0.613545	+	0.789659i	$0.710257\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.46410	1.47804	0.739022	−	0.673681i	$-0.235288\pi$
$41$	9.46410	1.47804	0.739022	+	0.673681i	$0.235288\pi$
$42$	0	0
$43$	−3.46410	−0.528271	−0.264135	−	0.964486i	$-0.585087\pi$
$43$	−3.46410	−0.528271	−0.264135	+	0.964486i	$0.585087\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.73205	−0.398511	−0.199255	−	0.979948i	$-0.563852\pi$
$47$	−2.73205	−0.398511	−0.199255	+	0.979948i	$0.563852\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.73205	−1.19944	−0.599720	−	0.800210i	$-0.704721\pi$
$53$	−8.73205	−1.19944	−0.599720	+	0.800210i	$0.704721\pi$
$54$	0	0
$55$	−9.46410	−1.27614
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.9282	1.42273	0.711365	−	0.702822i	$-0.248077\pi$
$59$	10.9282	1.42273	0.711365	+	0.702822i	$0.248077\pi$
$60$	0	0
$61$	−14.3923	−1.84275	−0.921373	−	0.388680i	$-0.872931\pi$
$61$	−14.3923	−1.84275	−0.921373	+	0.388680i	$0.872931\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−10.9282	−1.35548
$66$	0	0
$67$	1.46410	0.178868	0.0894342	−	0.995993i	$-0.471494\pi$
$67$	1.46410	0.178868	0.0894342	+	0.995993i	$0.471494\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.1962	−1.21006	−0.605030	−	0.796202i	$-0.706839\pi$
$71$	−10.1962	−1.21006	−0.605030	+	0.796202i	$0.706839\pi$
$72$	0	0
$73$	−15.8564	−1.85585	−0.927926	−	0.372764i	$-0.878410\pi$
$73$	−15.8564	−1.85585	−0.927926	+	0.372764i	$0.878410\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.46410	0.394771
$78$	0	0
$79$	−13.4641	−1.51483	−0.757415	−	0.652934i	$-0.773538\pi$
$79$	−13.4641	−1.51483	−0.757415	+	0.652934i	$0.773538\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.26795	0.139176	0.0695878	−	0.997576i	$-0.477832\pi$
$83$	1.26795	0.139176	0.0695878	+	0.997576i	$0.477832\pi$
$84$	0	0
$85$	−11.4641	−1.24346
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.46410	−1.00319	−0.501596	−	0.865102i	$-0.667254\pi$
$89$	−9.46410	−1.00319	−0.501596	+	0.865102i	$0.667254\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.73205	−0.280302
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.1962	1.61158	0.805789	−	0.592203i	$-0.201742\pi$
$101$	16.1962	1.61158	0.805789	+	0.592203i	$0.201742\pi$
$102$	0	0
$103$	−10.9282	−1.07679	−0.538394	−	0.842693i	$-0.680969\pi$
$103$	−10.9282	−1.07679	−0.538394	+	0.842693i	$0.680969\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.6603	−1.12724	−0.563620	−	0.826034i	$-0.690592\pi$
$107$	−11.6603	−1.12724	−0.563620	+	0.826034i	$0.690592\pi$
$108$	0	0
$109$	3.46410	0.331801	0.165900	−	0.986143i	$-0.446947\pi$
$109$	3.46410	0.331801	0.165900	+	0.986143i	$0.446947\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−16.7321	−1.57402	−0.787009	−	0.616941i	$-0.788372\pi$
$113$	−16.7321	−1.57402	−0.787009	+	0.616941i	$0.788372\pi$
$114$	0	0
$115$	20.3923	1.90159
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.19615	0.384661
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−6.92820	−0.619677
$126$	0	0
$127$	8.39230	0.744697	0.372348	−	0.928093i	$-0.378553\pi$
$127$	8.39230	0.744697	0.372348	+	0.928093i	$0.378553\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12.1962	−1.06558	−0.532791	−	0.846247i	$-0.678857\pi$
$131$	−12.1962	−1.06558	−0.532791	+	0.846247i	$0.678857\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	−22.9282	−1.94474	−0.972372	−	0.233435i	$-0.925003\pi$
$139$	−22.9282	−1.94474	−0.972372	+	0.233435i	$0.925003\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	13.8564	1.15873
$144$	0	0
$145$	16.9282	1.40581
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.3923	0.851371	0.425685	−	0.904871i	$-0.360033\pi$
$149$	10.3923	0.851371	0.425685	+	0.904871i	$0.360033\pi$
$150$	0	0
$151$	−1.07180	−0.0872216	−0.0436108	−	0.999049i	$-0.513886\pi$
$151$	−1.07180	−0.0872216	−0.0436108	+	0.999049i	$0.513886\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	13.4641	1.08146
$156$	0	0
$157$	6.39230	0.510161	0.255081	−	0.966920i	$-0.417898\pi$
$157$	6.39230	0.510161	0.255081	+	0.966920i	$0.417898\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−7.46410	−0.588254
$162$	0	0
$163$	15.4641	1.21124	0.605621	−	0.795753i	$-0.292925\pi$
$163$	15.4641	1.21124	0.605621	+	0.795753i	$0.292925\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.39230	0.649416	0.324708	−	0.945814i	$-0.394734\pi$
$167$	8.39230	0.649416	0.324708	+	0.945814i	$0.394734\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−20.3923	−1.55040	−0.775199	−	0.631717i	$-0.782350\pi$
$173$	−20.3923	−1.55040	−0.775199	+	0.631717i	$0.782350\pi$
$174$	0	0
$175$	−2.46410	−0.186269
$176$	0	0
$177$	0	0
$178$	0	0
$179$	19.6603	1.46948	0.734738	−	0.678351i	$-0.237305\pi$
$179$	19.6603	1.46948	0.734738	+	0.678351i	$0.237305\pi$
$180$	0	0
$181$	18.7846	1.39625	0.698125	−	0.715976i	$-0.254018\pi$
$181$	18.7846	1.39625	0.698125	+	0.715976i	$0.254018\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−20.3923	−1.49927
$186$	0	0
$187$	14.5359	1.06297
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−0.535898	−0.0387762	−0.0193881	−	0.999812i	$-0.506172\pi$
$191$	−0.535898	−0.0387762	−0.0193881	+	0.999812i	$0.506172\pi$
$192$	0	0
$193$	6.39230	0.460128	0.230064	−	0.973175i	$-0.426106\pi$
$193$	6.39230	0.460128	0.230064	+	0.973175i	$0.426106\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	19.3205	1.36959	0.684797	−	0.728734i	$-0.259891\pi$
$199$	19.3205	1.36959	0.684797	+	0.728734i	$0.259891\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.19615	−0.434885
$204$	0	0
$205$	25.8564	1.80589
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.46410	0.239617
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−9.46410	−0.645446
$216$	0	0
$217$	−4.92820	−0.334548
$218$	0	0
$219$	0	0
$220$	0	0
$221$	16.7846	1.12906
$222$	0	0
$223$	−4.92820	−0.330017	−0.165008	−	0.986292i	$-0.552765\pi$
$223$	−4.92820	−0.330017	−0.165008	+	0.986292i	$0.552765\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.3923	0.822506	0.411253	−	0.911521i	$-0.365091\pi$
$227$	12.3923	0.822506	0.411253	+	0.911521i	$0.365091\pi$
$228$	0	0
$229$	−18.3923	−1.21540	−0.607699	−	0.794168i	$-0.707907\pi$
$229$	−18.3923	−1.21540	−0.607699	+	0.794168i	$0.707907\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−17.3205	−1.13470	−0.567352	−	0.823475i	$-0.692032\pi$
$233$	−17.3205	−1.13470	−0.567352	+	0.823475i	$0.692032\pi$
$234$	0	0
$235$	−7.46410	−0.486904
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.07180	0.198698	0.0993490	−	0.995053i	$-0.468324\pi$
$239$	3.07180	0.198698	0.0993490	+	0.995053i	$0.468324\pi$
$240$	0	0
$241$	−29.7128	−1.91397	−0.956985	−	0.290137i	$-0.906299\pi$
$241$	−29.7128	−1.91397	−0.956985	+	0.290137i	$0.906299\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.73205	0.174544
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	0	0
$250$	0	0
$251$	28.1962	1.77973	0.889863	−	0.456228i	$-0.150800\pi$
$251$	28.1962	1.77973	0.889863	+	0.456228i	$0.150800\pi$
$252$	0	0
$253$	−25.8564	−1.62558
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.07180	−0.316370	−0.158185	−	0.987409i	$-0.550564\pi$
$257$	−5.07180	−0.316370	−0.158185	+	0.987409i	$0.550564\pi$
$258$	0	0
$259$	7.46410	0.463797
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	0	0
$265$	−23.8564	−1.46549
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−13.8564	−0.844840	−0.422420	−	0.906400i	$-0.638819\pi$
$269$	−13.8564	−0.844840	−0.422420	+	0.906400i	$0.638819\pi$
$270$	0	0
$271$	19.3205	1.17364	0.586819	−	0.809718i	$-0.300380\pi$
$271$	19.3205	1.17364	0.586819	+	0.809718i	$0.300380\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.53590	−0.514734
$276$	0	0
$277$	12.3923	0.744581	0.372291	−	0.928116i	$-0.378572\pi$
$277$	12.3923	0.744581	0.372291	+	0.928116i	$0.378572\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4.33975	−0.258888	−0.129444	−	0.991587i	$-0.541319\pi$
$281$	−4.33975	−0.258888	−0.129444	+	0.991587i	$0.541319\pi$
$282$	0	0
$283$	5.46410	0.324807	0.162404	−	0.986724i	$-0.448075\pi$
$283$	5.46410	0.324807	0.162404	+	0.986724i	$0.448075\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.46410	−0.558648
$288$	0	0
$289$	0.607695	0.0357468
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.53590	0.148149	0.0740744	−	0.997253i	$-0.476400\pi$
$293$	2.53590	0.148149	0.0740744	+	0.997253i	$0.476400\pi$
$294$	0	0
$295$	29.8564	1.73831
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−29.8564	−1.72664
$300$	0	0
$301$	3.46410	0.199667
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−39.3205	−2.25149
$306$	0	0
$307$	−27.8564	−1.58985	−0.794925	−	0.606708i	$-0.792490\pi$
$307$	−27.8564	−1.58985	−0.794925	+	0.606708i	$0.792490\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	14.7321	0.835378	0.417689	−	0.908590i	$-0.362840\pi$
$311$	14.7321	0.835378	0.417689	+	0.908590i	$0.362840\pi$
$312$	0	0
$313$	−30.3923	−1.71787	−0.858937	−	0.512081i	$-0.828875\pi$
$313$	−30.3923	−1.71787	−0.858937	+	0.512081i	$0.828875\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.87564	−0.161512	−0.0807561	−	0.996734i	$-0.525733\pi$
$317$	−2.87564	−0.161512	−0.0807561	+	0.996734i	$0.525733\pi$
$318$	0	0
$319$	−21.4641	−1.20176
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.19615	0.233480
$324$	0	0
$325$	−9.85641	−0.546735
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.73205	0.150623
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	34.3923	1.87347	0.936734	−	0.350042i	$-0.113833\pi$
$337$	34.3923	1.87347	0.936734	+	0.350042i	$0.113833\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−17.0718	−0.924490
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	0	0
$349$	31.8564	1.70523	0.852617	−	0.522536i	$-0.175014\pi$
$349$	31.8564	1.70523	0.852617	+	0.522536i	$0.175014\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.05256	0.322145	0.161073	−	0.986943i	$-0.448505\pi$
$353$	6.05256	0.322145	0.161073	+	0.986943i	$0.448505\pi$
$354$	0	0
$355$	−27.8564	−1.47846
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.39230	0.126261	0.0631305	−	0.998005i	$-0.479892\pi$
$359$	2.39230	0.126261	0.0631305	+	0.998005i	$0.479892\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−43.3205	−2.26750
$366$	0	0
$367$	34.2487	1.78777	0.893884	−	0.448298i	$-0.147970\pi$
$367$	34.2487	1.78777	0.893884	+	0.448298i	$0.147970\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.73205	0.453345
$372$	0	0
$373$	22.7846	1.17974	0.589871	−	0.807497i	$-0.299179\pi$
$373$	22.7846	1.17974	0.589871	+	0.807497i	$0.299179\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−24.7846	−1.27647
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7.60770	0.388735	0.194368	−	0.980929i	$-0.437735\pi$
$383$	7.60770	0.388735	0.194368	+	0.980929i	$0.437735\pi$
$384$	0	0
$385$	9.46410	0.482335
$386$	0	0
$387$	0	0
$388$	0	0
$389$	27.1769	1.37792	0.688962	−	0.724797i	$-0.258067\pi$
$389$	27.1769	1.37792	0.688962	+	0.724797i	$0.258067\pi$
$390$	0	0
$391$	−31.3205	−1.58395
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−36.7846	−1.85083
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.05256	0.202375	0.101188	−	0.994867i	$-0.467736\pi$
$401$	4.05256	0.202375	0.101188	+	0.994867i	$0.467736\pi$
$402$	0	0
$403$	−19.7128	−0.981965
$404$	0	0
$405$	0	0
$406$	0	0
$407$	25.8564	1.28165
$408$	0	0
$409$	18.9282	0.935939	0.467970	−	0.883745i	$-0.344986\pi$
$409$	18.9282	0.935939	0.467970	+	0.883745i	$0.344986\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−10.9282	−0.537742
$414$	0	0
$415$	3.46410	0.170046
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−10.0526	−0.491100	−0.245550	−	0.969384i	$-0.578968\pi$
$419$	−10.0526	−0.491100	−0.245550	+	0.969384i	$0.578968\pi$
$420$	0	0
$421$	15.0718	0.734554	0.367277	−	0.930112i	$-0.380290\pi$
$421$	15.0718	0.734554	0.367277	+	0.930112i	$0.380290\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−10.3397	−0.501551
$426$	0	0
$427$	14.3923	0.696492
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−13.5167	−0.651075	−0.325537	−	0.945529i	$-0.605545\pi$
$431$	−13.5167	−0.651075	−0.325537	+	0.945529i	$0.605545\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.46410	−0.357056
$438$	0	0
$439$	−4.92820	−0.235210	−0.117605	−	0.993060i	$-0.537522\pi$
$439$	−4.92820	−0.235210	−0.117605	+	0.993060i	$0.537522\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	24.9282	1.18437	0.592187	−	0.805800i	$-0.298265\pi$
$443$	24.9282	1.18437	0.592187	+	0.805800i	$0.298265\pi$
$444$	0	0
$445$	−25.8564	−1.22571
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−21.8038	−1.02899	−0.514494	−	0.857494i	$-0.672020\pi$
$449$	−21.8038	−1.02899	−0.514494	+	0.857494i	$0.672020\pi$
$450$	0	0
$451$	−32.7846	−1.54377
$452$	0	0
$453$	0	0
$454$	0	0
$455$	10.9282	0.512322
$456$	0	0
$457$	24.3923	1.14102	0.570512	−	0.821289i	$-0.306745\pi$
$457$	24.3923	1.14102	0.570512	+	0.821289i	$0.306745\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	18.3397	0.854167	0.427084	−	0.904212i	$-0.359541\pi$
$461$	18.3397	0.854167	0.427084	+	0.904212i	$0.359541\pi$
$462$	0	0
$463$	−1.85641	−0.0862745	−0.0431373	−	0.999069i	$-0.513735\pi$
$463$	−1.85641	−0.0862745	−0.0431373	+	0.999069i	$0.513735\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	22.4449	1.03862	0.519312	−	0.854585i	$-0.326188\pi$
$467$	22.4449	1.03862	0.519312	+	0.854585i	$0.326188\pi$
$468$	0	0
$469$	−1.46410	−0.0676059
$470$	0	0
$471$	0	0
$472$	0	0
$473$	12.0000	0.551761
$474$	0	0
$475$	−2.46410	−0.113061
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−32.9808	−1.50693	−0.753465	−	0.657488i	$-0.771619\pi$
$479$	−32.9808	−1.50693	−0.753465	+	0.657488i	$0.771619\pi$
$480$	0	0
$481$	29.8564	1.36133
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−16.3923	−0.744336
$486$	0	0
$487$	−10.9282	−0.495204	−0.247602	−	0.968862i	$-0.579643\pi$
$487$	−10.9282	−0.495204	−0.247602	+	0.968862i	$0.579643\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	18.0000	0.812329	0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000	0.812329	0.406164	+	0.913800i	$0.366866\pi$
$492$	0	0
$493$	−26.0000	−1.17098
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.1962	0.457360
$498$	0	0
$499$	−6.92820	−0.310149	−0.155074	−	0.987903i	$-0.549562\pi$
$499$	−6.92820	−0.310149	−0.155074	+	0.987903i	$0.549562\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18.7321	0.835221	0.417610	−	0.908626i	$-0.362868\pi$
$503$	18.7321	0.835221	0.417610	+	0.908626i	$0.362868\pi$
$504$	0	0
$505$	44.2487	1.96904
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−32.7846	−1.45315	−0.726576	−	0.687086i	$-0.758890\pi$
$509$	−32.7846	−1.45315	−0.726576	+	0.687086i	$0.758890\pi$
$510$	0	0
$511$	15.8564	0.701446
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−29.8564	−1.31563
$516$	0	0
$517$	9.46410	0.416231
$518$	0	0
$519$	0	0
$520$	0	0
$521$	16.0000	0.700973	0.350486	−	0.936568i	$-0.386016\pi$
$521$	16.0000	0.700973	0.350486	+	0.936568i	$0.386016\pi$
$522$	0	0
$523$	−22.7846	−0.996301	−0.498151	−	0.867090i	$-0.665987\pi$
$523$	−22.7846	−0.996301	−0.498151	+	0.867090i	$0.665987\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−20.6795	−0.900813
$528$	0	0
$529$	32.7128	1.42230
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−37.8564	−1.63974
$534$	0	0
$535$	−31.8564	−1.37727
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3.46410	−0.149209
$540$	0	0
$541$	−15.8564	−0.681720	−0.340860	−	0.940114i	$-0.610718\pi$
$541$	−15.8564	−0.681720	−0.340860	+	0.940114i	$0.610718\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	9.46410	0.405398
$546$	0	0
$547$	−31.3205	−1.33917	−0.669584	−	0.742736i	$-0.733528\pi$
$547$	−31.3205	−1.33917	−0.669584	+	0.742736i	$0.733528\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.19615	−0.263965
$552$	0	0
$553$	13.4641	0.572552
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6.39230	−0.270851	−0.135425	−	0.990788i	$-0.543240\pi$
$557$	−6.39230	−0.270851	−0.135425	+	0.990788i	$0.543240\pi$
$558$	0	0
$559$	13.8564	0.586064
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7.32051	−0.308523	−0.154261	−	0.988030i	$-0.549300\pi$
$563$	−7.32051	−0.308523	−0.154261	+	0.988030i	$0.549300\pi$
$564$	0	0
$565$	−45.7128	−1.92315
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−37.9090	−1.58923	−0.794613	−	0.607116i	$-0.792326\pi$
$569$	−37.9090	−1.58923	−0.794613	+	0.607116i	$0.792326\pi$
$570$	0	0
$571$	−29.8564	−1.24945	−0.624726	−	0.780844i	$-0.714789\pi$
$571$	−29.8564	−1.24945	−0.624726	+	0.780844i	$0.714789\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	18.3923	0.767012
$576$	0	0
$577$	35.1769	1.46443	0.732217	−	0.681071i	$-0.238486\pi$
$577$	35.1769	1.46443	0.732217	+	0.681071i	$0.238486\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.26795	−0.0526034
$582$	0	0
$583$	30.2487	1.25277
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7.51666	−0.310246	−0.155123	−	0.987895i	$-0.549577\pi$
$587$	−7.51666	−0.310246	−0.155123	+	0.987895i	$0.549577\pi$
$588$	0	0
$589$	−4.92820	−0.203063
$590$	0	0
$591$	0	0
$592$	0	0
$593$	27.5167	1.12997	0.564987	−	0.825100i	$-0.308881\pi$
$593$	27.5167	1.12997	0.564987	+	0.825100i	$0.308881\pi$
$594$	0	0
$595$	11.4641	0.469982
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.73205	0.356782	0.178391	−	0.983960i	$-0.442911\pi$
$599$	8.73205	0.356782	0.178391	+	0.983960i	$0.442911\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.73205	0.111074
$606$	0	0
$607$	8.78461	0.356556	0.178278	−	0.983980i	$-0.442947\pi$
$607$	8.78461	0.356556	0.178278	+	0.983980i	$0.442947\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	10.9282	0.442108
$612$	0	0
$613$	0.392305	0.0158450	0.00792252	−	0.999969i	$-0.497478\pi$
$613$	0.392305	0.0158450	0.00792252	+	0.999969i	$0.497478\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−25.7128	−1.03516	−0.517579	−	0.855635i	$-0.673167\pi$
$617$	−25.7128	−1.03516	−0.517579	+	0.855635i	$0.673167\pi$
$618$	0	0
$619$	3.21539	0.129237	0.0646187	−	0.997910i	$-0.479417\pi$
$619$	3.21539	0.129237	0.0646187	+	0.997910i	$0.479417\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	9.46410	0.379171
$624$	0	0
$625$	−31.2487	−1.24995
$626$	0	0
$627$	0	0
$628$	0	0
$629$	31.3205	1.24883
$630$	0	0
$631$	−36.2487	−1.44304	−0.721519	−	0.692394i	$-0.756556\pi$
$631$	−36.2487	−1.44304	−0.721519	+	0.692394i	$0.756556\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	22.9282	0.909878
$636$	0	0
$637$	−4.00000	−0.158486
$638$	0	0
$639$	0	0
$640$	0	0
$641$	36.4449	1.43949	0.719743	−	0.694241i	$-0.244260\pi$
$641$	36.4449	1.43949	0.719743	+	0.694241i	$0.244260\pi$
$642$	0	0
$643$	44.1051	1.73934	0.869668	−	0.493637i	$-0.164333\pi$
$643$	44.1051	1.73934	0.869668	+	0.493637i	$0.164333\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.9808	0.667583	0.333791	−	0.942647i	$-0.391672\pi$
$647$	16.9808	0.667583	0.333791	+	0.942647i	$0.391672\pi$
$648$	0	0
$649$	−37.8564	−1.48599
$650$	0	0
$651$	0	0
$652$	0	0
$653$	29.7128	1.16275	0.581376	−	0.813635i	$-0.302515\pi$
$653$	29.7128	1.16275	0.581376	+	0.813635i	$0.302515\pi$
$654$	0	0
$655$	−33.3205	−1.30194
$656$	0	0
$657$	0	0
$658$	0	0
$659$	40.0526	1.56023	0.780113	−	0.625639i	$-0.215161\pi$
$659$	40.0526	1.56023	0.780113	+	0.625639i	$0.215161\pi$
$660$	0	0
$661$	−12.0000	−0.466746	−0.233373	−	0.972387i	$-0.574976\pi$
$661$	−12.0000	−0.466746	−0.233373	+	0.972387i	$0.574976\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.73205	0.105944
$666$	0	0
$667$	46.2487	1.79076
$668$	0	0
$669$	0	0
$670$	0	0
$671$	49.8564	1.92469
$672$	0	0
$673$	−32.2487	−1.24310	−0.621548	−	0.783376i	$-0.713496\pi$
$673$	−32.2487	−1.24310	−0.621548	+	0.783376i	$0.713496\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−17.0718	−0.656122	−0.328061	−	0.944656i	$-0.606395\pi$
$677$	−17.0718	−0.656122	−0.328061	+	0.944656i	$0.606395\pi$
$678$	0	0
$679$	6.00000	0.230259
$680$	0	0
$681$	0	0
$682$	0	0
$683$	22.5885	0.864323	0.432162	−	0.901796i	$-0.357751\pi$
$683$	22.5885	0.864323	0.432162	+	0.901796i	$0.357751\pi$
$684$	0	0
$685$	−49.1769	−1.87895
$686$	0	0
$687$	0	0
$688$	0	0
$689$	34.9282	1.33066
$690$	0	0
$691$	41.4641	1.57737	0.788684	−	0.614798i	$-0.210763\pi$
$691$	41.4641	1.57737	0.788684	+	0.614798i	$0.210763\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−62.6410	−2.37611
$696$	0	0
$697$	−39.7128	−1.50423
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9.21539	0.348060	0.174030	−	0.984740i	$-0.444321\pi$
$701$	9.21539	0.348060	0.174030	+	0.984740i	$0.444321\pi$
$702$	0	0
$703$	7.46410	0.281514
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−16.1962	−0.609119
$708$	0	0
$709$	−8.67949	−0.325965	−0.162983	−	0.986629i	$-0.552111\pi$
$709$	−8.67949	−0.325965	−0.162983	+	0.986629i	$0.552111\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	36.7846	1.37759
$714$	0	0
$715$	37.8564	1.41575
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.94744	−0.0726273	−0.0363136	−	0.999340i	$-0.511562\pi$
$719$	−1.94744	−0.0726273	−0.0363136	+	0.999340i	$0.511562\pi$
$720$	0	0
$721$	10.9282	0.406988
$722$	0	0
$723$	0	0
$724$	0	0
$725$	15.2679	0.567037
$726$	0	0
$727$	6.53590	0.242403	0.121202	−	0.992628i	$-0.461325\pi$
$727$	6.53590	0.242403	0.121202	+	0.992628i	$0.461325\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	14.5359	0.537630
$732$	0	0
$733$	16.6410	0.614650	0.307325	−	0.951605i	$-0.400566\pi$
$733$	16.6410	0.614650	0.307325	+	0.951605i	$0.400566\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.07180	−0.186822
$738$	0	0
$739$	−16.5359	−0.608283	−0.304141	−	0.952627i	$-0.598370\pi$
$739$	−16.5359	−0.608283	−0.304141	+	0.952627i	$0.598370\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10.5885	−0.388453	−0.194226	−	0.980957i	$-0.562220\pi$
$743$	−10.5885	−0.388453	−0.194226	+	0.980957i	$0.562220\pi$
$744$	0	0
$745$	28.3923	1.04021
$746$	0	0
$747$	0	0
$748$	0	0
$749$	11.6603	0.426056
$750$	0	0
$751$	−28.7846	−1.05037	−0.525183	−	0.850990i	$-0.676003\pi$
$751$	−28.7846	−1.05037	−0.525183	+	0.850990i	$0.676003\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−2.92820	−0.106568
$756$	0	0
$757$	−8.92820	−0.324501	−0.162251	−	0.986750i	$-0.551875\pi$
$757$	−8.92820	−0.324501	−0.162251	+	0.986750i	$0.551875\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−12.1962	−0.442110	−0.221055	−	0.975261i	$-0.570950\pi$
$761$	−12.1962	−0.442110	−0.221055	+	0.975261i	$0.570950\pi$
$762$	0	0
$763$	−3.46410	−0.125409
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−43.7128	−1.57838
$768$	0	0
$769$	−18.6795	−0.673600	−0.336800	−	0.941576i	$-0.609345\pi$
$769$	−18.6795	−0.673600	−0.336800	+	0.941576i	$0.609345\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	38.5359	1.38604	0.693020	−	0.720918i	$-0.256280\pi$
$773$	38.5359	1.38604	0.693020	+	0.720918i	$0.256280\pi$
$774$	0	0
$775$	12.1436	0.436211
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−9.46410	−0.339087
$780$	0	0
$781$	35.3205	1.26387
$782$	0	0
$783$	0	0
$784$	0	0
$785$	17.4641	0.623321
$786$	0	0
$787$	−1.21539	−0.0433240	−0.0216620	−	0.999765i	$-0.506896\pi$
$787$	−1.21539	−0.0433240	−0.0216620	+	0.999765i	$0.506896\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	16.7321	0.594923
$792$	0	0
$793$	57.5692	2.04434
$794$	0	0
$795$	0	0
$796$	0	0
$797$	41.1769	1.45856	0.729281	−	0.684215i	$-0.239855\pi$
$797$	41.1769	1.45856	0.729281	+	0.684215i	$0.239855\pi$
$798$	0	0
$799$	11.4641	0.405571
$800$	0	0
$801$	0	0
$802$	0	0
$803$	54.9282	1.93837
$804$	0	0
$805$	−20.3923	−0.718734
$806$	0	0
$807$	0	0
$808$	0	0
$809$	14.0000	0.492214	0.246107	−	0.969243i	$-0.420849\pi$
$809$	14.0000	0.492214	0.246107	+	0.969243i	$0.420849\pi$
$810$	0	0
$811$	−33.0718	−1.16131	−0.580654	−	0.814150i	$-0.697203\pi$
$811$	−33.0718	−1.16131	−0.580654	+	0.814150i	$0.697203\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	42.2487	1.47991
$816$	0	0
$817$	3.46410	0.121194
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−33.0333	−1.15287	−0.576435	−	0.817143i	$-0.695557\pi$
$821$	−33.0333	−1.15287	−0.576435	+	0.817143i	$0.695557\pi$
$822$	0	0
$823$	−51.4641	−1.79393	−0.896963	−	0.442106i	$-0.854232\pi$
$823$	−51.4641	−1.79393	−0.896963	+	0.442106i	$0.854232\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−12.0526	−0.419109	−0.209554	−	0.977797i	$-0.567201\pi$
$827$	−12.0526	−0.419109	−0.209554	+	0.977797i	$0.567201\pi$
$828$	0	0
$829$	−20.9282	−0.726867	−0.363433	−	0.931620i	$-0.618396\pi$
$829$	−20.9282	−0.726867	−0.363433	+	0.931620i	$0.618396\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−4.19615	−0.145388
$834$	0	0
$835$	22.9282	0.793463
$836$	0	0
$837$	0	0
$838$	0	0
$839$	17.0718	0.589384	0.294692	−	0.955592i	$-0.404783\pi$
$839$	17.0718	0.589384	0.294692	+	0.955592i	$0.404783\pi$
$840$	0	0
$841$	9.39230	0.323873
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.19615	0.281956
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−55.7128	−1.90981
$852$	0	0
$853$	−27.8564	−0.953785	−0.476893	−	0.878962i	$-0.658237\pi$
$853$	−27.8564	−0.953785	−0.476893	+	0.878962i	$0.658237\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	35.3205	1.20653	0.603263	−	0.797542i	$-0.293867\pi$
$857$	35.3205	1.20653	0.603263	+	0.797542i	$0.293867\pi$
$858$	0	0
$859$	33.5692	1.14537	0.572683	−	0.819777i	$-0.305902\pi$
$859$	33.5692	1.14537	0.572683	+	0.819777i	$0.305902\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−19.2679	−0.655889	−0.327944	−	0.944697i	$-0.606356\pi$
$863$	−19.2679	−0.655889	−0.327944	+	0.944697i	$0.606356\pi$
$864$	0	0
$865$	−55.7128	−1.89429
$866$	0	0
$867$	0	0
$868$	0	0
$869$	46.6410	1.58219
$870$	0	0
$871$	−5.85641	−0.198437
$872$	0	0
$873$	0	0
$874$	0	0
$875$	6.92820	0.234216
$876$	0	0
$877$	−40.6410	−1.37235	−0.686175	−	0.727437i	$-0.740711\pi$
$877$	−40.6410	−1.37235	−0.686175	+	0.727437i	$0.740711\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−34.7321	−1.17015	−0.585076	−	0.810978i	$-0.698935\pi$
$881$	−34.7321	−1.17015	−0.585076	+	0.810978i	$0.698935\pi$
$882$	0	0
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−30.2487	−1.01565	−0.507826	−	0.861460i	$-0.669551\pi$
$887$	−30.2487	−1.01565	−0.507826	+	0.861460i	$0.669551\pi$
$888$	0	0
$889$	−8.39230	−0.281469
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.73205	0.0914246
$894$	0	0
$895$	53.7128	1.79542
$896$	0	0
$897$	0	0
$898$	0	0
$899$	30.5359	1.01843
$900$	0	0
$901$	36.6410	1.22069
$902$	0	0
$903$	0	0
$904$	0	0
$905$	51.3205	1.70595
$906$	0	0
$907$	33.0718	1.09813	0.549065	−	0.835779i	$-0.314984\pi$
$907$	33.0718	1.09813	0.549065	+	0.835779i	$0.314984\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−43.3731	−1.43701	−0.718507	−	0.695520i	$-0.755174\pi$
$911$	−43.3731	−1.43701	−0.718507	+	0.695520i	$0.755174\pi$
$912$	0	0
$913$	−4.39230	−0.145364
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12.1962	0.402752
$918$	0	0
$919$	−3.71281	−0.122474	−0.0612372	−	0.998123i	$-0.519505\pi$
$919$	−3.71281	−0.122474	−0.0612372	+	0.998123i	$0.519505\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	40.7846	1.34244
$924$	0	0
$925$	−18.3923	−0.604735
$926$	0	0
$927$	0	0
$928$	0	0
$929$	21.6603	0.710650	0.355325	−	0.934743i	$-0.384370\pi$
$929$	21.6603	0.710650	0.355325	+	0.934743i	$0.384370\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	39.7128	1.29875
$936$	0	0
$937$	7.07180	0.231026	0.115513	−	0.993306i	$-0.463149\pi$
$937$	7.07180	0.231026	0.115513	+	0.993306i	$0.463149\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−18.6410	−0.607680	−0.303840	−	0.952723i	$-0.598269\pi$
$941$	−18.6410	−0.607680	−0.303840	+	0.952723i	$0.598269\pi$
$942$	0	0
$943$	70.6410	2.30039
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.17691	0.233218	0.116609	−	0.993178i	$-0.462798\pi$
$947$	7.17691	0.233218	0.116609	+	0.993178i	$0.462798\pi$
$948$	0	0
$949$	63.4256	2.05888
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−0.732051	−0.0237135	−0.0118567	−	0.999930i	$-0.503774\pi$
$953$	−0.732051	−0.0237135	−0.0118567	+	0.999930i	$0.503774\pi$
$954$	0	0
$955$	−1.46410	−0.0473772
$956$	0	0
$957$	0	0
$958$	0	0
$959$	18.0000	0.581250
$960$	0	0
$961$	−6.71281	−0.216542
$962$	0	0
$963$	0	0
$964$	0	0
$965$	17.4641	0.562189
$966$	0	0
$967$	14.3923	0.462825	0.231413	−	0.972856i	$-0.425665\pi$
$967$	14.3923	0.462825	0.231413	+	0.972856i	$0.425665\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−15.7128	−0.504248	−0.252124	−	0.967695i	$-0.581129\pi$
$971$	−15.7128	−0.504248	−0.252124	+	0.967695i	$0.581129\pi$
$972$	0	0
$973$	22.9282	0.735044
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−52.7321	−1.68705	−0.843524	−	0.537092i	$-0.819523\pi$
$977$	−52.7321	−1.68705	−0.843524	+	0.537092i	$0.819523\pi$
$978$	0	0
$979$	32.7846	1.04780
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−7.71281	−0.246001	−0.123000	−	0.992407i	$-0.539252\pi$
$983$	−7.71281	−0.246001	−0.123000	+	0.992407i	$0.539252\pi$
$984$	0	0
$985$	−27.3205	−0.870504
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−25.8564	−0.822186
$990$	0	0
$991$	22.6410	0.719216	0.359608	−	0.933104i	$-0.382910\pi$
$991$	22.6410	0.719216	0.359608	+	0.933104i	$0.382910\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	52.7846	1.67338
$996$	0	0
$997$	9.32051	0.295183	0.147592	−	0.989048i	$-0.452848\pi$
$997$	9.32051	0.295183	0.147592	+	0.989048i	$0.452848\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4788.2.a.j.1.2		2
3.2	odd	2		1596.2.a.i.1.1	✓	2
12.11	even	2		6384.2.a.bj.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1596.2.a.i.1.1	✓	2	3.2	odd	2
4788.2.a.j.1.2		2	1.1	even	1	trivial
6384.2.a.bj.1.1		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4788.a (trivial)

\(f(q)\)	\(=\)	\(q+2.73205 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+2.73205 q^{5} -1.00000 q^{7} -3.46410 q^{11} -4.00000 q^{13} -4.19615 q^{17} -1.00000 q^{19} +7.46410 q^{23} +2.46410 q^{25} +6.19615 q^{29} +4.92820 q^{31} -2.73205 q^{35} -7.46410 q^{37} +9.46410 q^{41} -3.46410 q^{43} -2.73205 q^{47} +1.00000 q^{49} -8.73205 q^{53} -9.46410 q^{55} +10.9282 q^{59} -14.3923 q^{61} -10.9282 q^{65} +1.46410 q^{67} -10.1962 q^{71} -15.8564 q^{73} +3.46410 q^{77} -13.4641 q^{79} +1.26795 q^{83} -11.4641 q^{85} -9.46410 q^{89} +4.00000 q^{91} -2.73205 q^{95} -6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 - 2 * q^7	\( 2 q + 2 q^{5} - 2 q^{7} - 8 q^{13} + 2 q^{17} - 2 q^{19} + 8 q^{23} - 2 q^{25} + 2 q^{29} - 4 q^{31} - 2 q^{35} - 8 q^{37} + 12 q^{41} - 2 q^{47} + 2 q^{49} - 14 q^{53} - 12 q^{55} + 8 q^{59} - 8 q^{61} - 8 q^{65} - 4 q^{67} - 10 q^{71} - 4 q^{73} - 20 q^{79} + 6 q^{83} - 16 q^{85} - 12 q^{89} + 8 q^{91} - 2 q^{95} - 12 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^7 - 8 * q^13 + 2 * q^17 - 2 * q^19 + 8 * q^23 - 2 * q^25 + 2 * q^29 - 4 * q^31 - 2 * q^35 - 8 * q^37 + 12 * q^41 - 2 * q^47 + 2 * q^49 - 14 * q^53 - 12 * q^55 + 8 * q^59 - 8 * q^61 - 8 * q^65 - 4 * q^67 - 10 * q^71 - 4 * q^73 - 20 * q^79 + 6 * q^83 - 16 * q^85 - 12 * q^89 + 8 * q^91 - 2 * q^95 - 12 * q^97

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