LMFDB - Embedded newform 6840.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6840,2,Mod(1,6840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6840.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:	$54.6176749826$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1772.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 12x + 8 $ x^3 - x^2 - 12*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.654334$ of defining polynomial
Character	$\chi$	$=$	6840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -4.11309 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -4.11309 q^{7} +2.11309 q^{11} +4.80442 q^{13} -4.00000 q^{17} +1.00000 q^{19} +0.691333 q^{23} +1.00000 q^{25} -4.11309 q^{29} -2.00000 q^{31} +4.11309 q^{35} +3.42176 q^{37} -5.42176 q^{41} -2.80442 q^{43} -0.691333 q^{47} +9.91751 q^{49} +2.69133 q^{53} -2.11309 q^{55} -9.53485 q^{59} -2.91751 q^{61} -4.80442 q^{65} +4.00000 q^{67} -5.60885 q^{71} +6.00000 q^{73} -8.69133 q^{77} +15.1437 q^{79} -6.22618 q^{83} +4.00000 q^{85} +1.42176 q^{89} -19.7610 q^{91} -1.00000 q^{95} +15.4218 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5}+O(q^{10}) $ 3 * q - 3 * q^5	$ 3 q - 3 q^{5} - 6 q^{11} + 4 q^{13} - 12 q^{17} + 3 q^{19} + 4 q^{23} + 3 q^{25} - 6 q^{31} - 4 q^{37} - 2 q^{41} + 2 q^{43} - 4 q^{47} + 7 q^{49} + 10 q^{53} + 6 q^{55} - 2 q^{59} + 14 q^{61} - 4 q^{65} + 12 q^{67} + 4 q^{71} + 18 q^{73} - 28 q^{77} - 2 q^{79} + 6 q^{83} + 12 q^{85} - 10 q^{89} - 8 q^{91} - 3 q^{95} + 32 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 - 6 * q^11 + 4 * q^13 - 12 * q^17 + 3 * q^19 + 4 * q^23 + 3 * q^25 - 6 * q^31 - 4 * q^37 - 2 * q^41 + 2 * q^43 - 4 * q^47 + 7 * q^49 + 10 * q^53 + 6 * q^55 - 2 * q^59 + 14 * q^61 - 4 * q^65 + 12 * q^67 + 4 * q^71 + 18 * q^73 - 28 * q^77 - 2 * q^79 + 6 * q^83 + 12 * q^85 - 10 * q^89 - 8 * q^91 - 3 * q^95 + 32 * 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.11309	−1.55460	−0.777301	−	0.629129i	$-0.783412\pi$
$7$	−4.11309	−1.55460	−0.777301	+	0.629129i	$0.783412\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.11309	0.637121	0.318560	−	0.947903i	$-0.396801\pi$
$11$	2.11309	0.637121	0.318560	+	0.947903i	$0.396801\pi$
$12$	0	0
$13$	4.80442	1.33251	0.666254	−	0.745725i	$-0.267897\pi$
$13$	4.80442	1.33251	0.666254	+	0.745725i	$0.267897\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.691333	0.144153	0.0720764	−	0.997399i	$-0.477037\pi$
$23$	0.691333	0.144153	0.0720764	+	0.997399i	$0.477037\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.11309	−0.763782	−0.381891	−	0.924207i	$-0.624727\pi$
$29$	−4.11309	−0.763782	−0.381891	+	0.924207i	$0.624727\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.11309	0.695239
$36$	0	0
$37$	3.42176	0.562533	0.281267	−	0.959630i	$-0.409245\pi$
$37$	3.42176	0.562533	0.281267	+	0.959630i	$0.409245\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.42176	−0.846736	−0.423368	−	0.905958i	$-0.639152\pi$
$41$	−5.42176	−0.846736	−0.423368	+	0.905958i	$0.639152\pi$
$42$	0	0
$43$	−2.80442	−0.427671	−0.213835	−	0.976870i	$-0.568596\pi$
$43$	−2.80442	−0.427671	−0.213835	+	0.976870i	$0.568596\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.691333	−0.100841	−0.0504206	−	0.998728i	$-0.516056\pi$
$47$	−0.691333	−0.100841	−0.0504206	+	0.998728i	$0.516056\pi$
$48$	0	0
$49$	9.91751	1.41679
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.69133	0.369683	0.184842	−	0.982768i	$-0.440823\pi$
$53$	2.69133	0.369683	0.184842	+	0.982768i	$0.440823\pi$
$54$	0	0
$55$	−2.11309	−0.284929
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−9.53485	−1.24133	−0.620666	−	0.784075i	$-0.713138\pi$
$59$	−9.53485	−1.24133	−0.620666	+	0.784075i	$0.713138\pi$
$60$	0	0
$61$	−2.91751	−0.373549	−0.186775	−	0.982403i	$-0.559803\pi$
$61$	−2.91751	−0.373549	−0.186775	+	0.982403i	$0.559803\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.80442	−0.595915
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.60885	−0.665648	−0.332824	−	0.942989i	$-0.608001\pi$
$71$	−5.60885	−0.665648	−0.332824	+	0.942989i	$0.608001\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8.69133	−0.990469
$78$	0	0
$79$	15.1437	1.70380	0.851899	−	0.523705i	$-0.175451\pi$
$79$	15.1437	1.70380	0.851899	+	0.523705i	$0.175451\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.22618	−0.683412	−0.341706	−	0.939807i	$-0.611005\pi$
$83$	−6.22618	−0.683412	−0.341706	+	0.939807i	$0.611005\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.42176	0.150706	0.0753530	−	0.997157i	$-0.475992\pi$
$89$	1.42176	0.150706	0.0753530	+	0.997157i	$0.475992\pi$
$90$	0	0
$91$	−19.7610	−2.07152
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	15.4218	1.56584	0.782921	−	0.622121i	$-0.213729\pi$
$97$	15.4218	1.56584	0.782921	+	0.622121i	$0.213729\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.2262	1.01754	0.508772	−	0.860902i	$-0.330100\pi$
$101$	10.2262	1.01754	0.508772	+	0.860902i	$0.330100\pi$
$102$	0	0
$103$	16.4524	1.62110	0.810550	−	0.585670i	$-0.199168\pi$
$103$	16.4524	1.62110	0.810550	+	0.585670i	$0.199168\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.4524	−1.59051	−0.795255	−	0.606275i	$-0.792663\pi$
$107$	−16.4524	−1.59051	−0.795255	+	0.606275i	$0.792663\pi$
$108$	0	0
$109$	−7.53485	−0.721708	−0.360854	−	0.932622i	$-0.617515\pi$
$109$	−7.53485	−0.721708	−0.360854	+	0.932622i	$0.617515\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	17.5348	1.64954	0.824770	−	0.565469i	$-0.191305\pi$
$113$	17.5348	1.64954	0.824770	+	0.565469i	$0.191305\pi$
$114$	0	0
$115$	−0.691333	−0.0644671
$116$	0	0
$117$	0	0
$118$	0	0
$119$	16.4524	1.50819
$120$	0	0
$121$	−6.53485	−0.594077
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−9.83503	−0.872718	−0.436359	−	0.899773i	$-0.643732\pi$
$127$	−9.83503	−0.872718	−0.436359	+	0.899773i	$0.643732\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−6.33927	−0.553865	−0.276932	−	0.960889i	$-0.589318\pi$
$131$	−6.33927	−0.553865	−0.276932	+	0.960889i	$0.589318\pi$
$132$	0	0
$133$	−4.11309	−0.356650
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.30867	−0.624422	−0.312211	−	0.950013i	$-0.601070\pi$
$137$	−7.30867	−0.624422	−0.312211	+	0.950013i	$0.601070\pi$
$138$	0	0
$139$	−8.22618	−0.697736	−0.348868	−	0.937172i	$-0.613434\pi$
$139$	−8.22618	−0.697736	−0.348868	+	0.937172i	$0.613434\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	10.1522	0.848968
$144$	0	0
$145$	4.11309	0.341574
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.38267	0.604812	0.302406	−	0.953179i	$-0.402210\pi$
$149$	7.38267	0.604812	0.302406	+	0.953179i	$0.402210\pi$
$150$	0	0
$151$	−7.38267	−0.600793	−0.300396	−	0.953814i	$-0.597119\pi$
$151$	−7.38267	−0.600793	−0.300396	+	0.953814i	$0.597119\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.00000	0.160644
$156$	0	0
$157$	12.9175	1.03093	0.515465	−	0.856911i	$-0.327619\pi$
$157$	12.9175	1.03093	0.515465	+	0.856911i	$0.327619\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.84352	−0.224100
$162$	0	0
$163$	−5.42176	−0.424665	−0.212332	−	0.977197i	$-0.568106\pi$
$163$	−5.42176	−0.424665	−0.212332	+	0.977197i	$0.568106\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.9175	0.999587	0.499794	−	0.866145i	$-0.333409\pi$
$167$	12.9175	0.999587	0.499794	+	0.866145i	$0.333409\pi$
$168$	0	0
$169$	10.0825	0.775576
$170$	0	0
$171$	0	0
$172$	0	0
$173$	13.5348	1.02904	0.514518	−	0.857480i	$-0.327971\pi$
$173$	13.5348	1.02904	0.514518	+	0.857480i	$0.327971\pi$
$174$	0	0
$175$	−4.11309	−0.310920
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.9175	0.816013	0.408007	−	0.912979i	$-0.366224\pi$
$179$	10.9175	0.816013	0.408007	+	0.912979i	$0.366224\pi$
$180$	0	0
$181$	−17.3699	−1.29109	−0.645546	−	0.763721i	$-0.723370\pi$
$181$	−17.3699	−1.29109	−0.645546	+	0.763721i	$0.723370\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.42176	−0.251573
$186$	0	0
$187$	−8.45236	−0.618098
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−0.730425	−0.0528517	−0.0264258	−	0.999651i	$-0.508413\pi$
$191$	−0.730425	−0.0528517	−0.0264258	+	0.999651i	$0.508413\pi$
$192$	0	0
$193$	25.4830	1.83430	0.917152	−	0.398537i	$-0.130482\pi$
$193$	25.4830	1.83430	0.917152	+	0.398537i	$0.130482\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.7610	1.55041	0.775205	−	0.631710i	$-0.217647\pi$
$197$	21.7610	1.55041	0.775205	+	0.631710i	$0.217647\pi$
$198$	0	0
$199$	1.60885	0.114048	0.0570241	−	0.998373i	$-0.481839\pi$
$199$	1.60885	0.114048	0.0570241	+	0.998373i	$0.481839\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	16.9175	1.18738
$204$	0	0
$205$	5.42176	0.378672
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.11309	0.146166
$210$	0	0
$211$	9.83503	0.677071	0.338536	−	0.940954i	$-0.390068\pi$
$211$	9.83503	0.677071	0.338536	+	0.940954i	$0.390068\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.80442	0.191260
$216$	0	0
$217$	8.22618	0.558430
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−19.2177	−1.29272
$222$	0	0
$223$	13.8350	0.926462	0.463231	−	0.886238i	$-0.346690\pi$
$223$	13.8350	0.926462	0.463231	+	0.886238i	$0.346690\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.3002	−0.683647	−0.341823	−	0.939764i	$-0.611044\pi$
$227$	−10.3002	−0.683647	−0.341823	+	0.939764i	$0.611044\pi$
$228$	0	0
$229$	9.08249	0.600188	0.300094	−	0.953910i	$-0.402982\pi$
$229$	9.08249	0.600188	0.300094	+	0.953910i	$0.402982\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	22.5264	1.47575	0.737875	−	0.674937i	$-0.235829\pi$
$233$	22.5264	1.47575	0.737875	+	0.674937i	$0.235829\pi$
$234$	0	0
$235$	0.691333	0.0450976
$236$	0	0
$237$	0	0
$238$	0	0
$239$	23.7219	1.53444	0.767222	−	0.641381i	$-0.221638\pi$
$239$	23.7219	1.53444	0.767222	+	0.641381i	$0.221638\pi$
$240$	0	0
$241$	21.2177	1.36675	0.683376	−	0.730067i	$-0.260511\pi$
$241$	21.2177	1.36675	0.683376	+	0.730067i	$0.260511\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−9.91751	−0.633607
$246$	0	0
$247$	4.80442	0.305698
$248$	0	0
$249$	0	0
$250$	0	0
$251$	28.1743	1.77835	0.889173	−	0.457571i	$-0.151280\pi$
$251$	28.1743	1.77835	0.889173	+	0.457571i	$0.151280\pi$
$252$	0	0
$253$	1.46085	0.0918428
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.3002	−0.767264	−0.383632	−	0.923486i	$-0.625327\pi$
$257$	−12.3002	−0.767264	−0.383632	+	0.923486i	$0.625327\pi$
$258$	0	0
$259$	−14.0740	−0.874516
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−12.9175	−0.796528	−0.398264	−	0.917271i	$-0.630387\pi$
$263$	−12.9175	−0.796528	−0.398264	+	0.917271i	$0.630387\pi$
$264$	0	0
$265$	−2.69133	−0.165327
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6.73042	0.410361	0.205181	−	0.978724i	$-0.434222\pi$
$269$	6.73042	0.410361	0.205181	+	0.978724i	$0.434222\pi$
$270$	0	0
$271$	−7.77382	−0.472226	−0.236113	−	0.971726i	$-0.575874\pi$
$271$	−7.77382	−0.472226	−0.236113	+	0.971726i	$0.575874\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.11309	0.127424
$276$	0	0
$277$	15.3087	0.919809	0.459904	−	0.887968i	$-0.347884\pi$
$277$	15.3087	0.919809	0.459904	+	0.887968i	$0.347884\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−13.4218	−0.800675	−0.400337	−	0.916368i	$-0.631107\pi$
$281$	−13.4218	−0.800675	−0.400337	+	0.916368i	$0.631107\pi$
$282$	0	0
$283$	15.0306	0.893477	0.446738	−	0.894665i	$-0.352585\pi$
$283$	15.0306	0.893477	0.446738	+	0.894665i	$0.352585\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	22.3002	1.31634
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9.08249	0.530604	0.265302	−	0.964165i	$-0.414528\pi$
$293$	9.08249	0.530604	0.265302	+	0.964165i	$0.414528\pi$
$294$	0	0
$295$	9.53485	0.555140
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.32146	0.192085
$300$	0	0
$301$	11.5348	0.664858
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.91751	0.167056
$306$	0	0
$307$	32.2262	1.83925	0.919623	−	0.392803i	$-0.128495\pi$
$307$	32.2262	1.83925	0.919623	+	0.392803i	$0.128495\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	16.7304	0.948695	0.474348	−	0.880338i	$-0.342684\pi$
$311$	16.7304	0.948695	0.474348	+	0.880338i	$0.342684\pi$
$312$	0	0
$313$	−14.6785	−0.829680	−0.414840	−	0.909894i	$-0.636163\pi$
$313$	−14.6785	−0.829680	−0.414840	+	0.909894i	$0.636163\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.9090	1.00587	0.502936	−	0.864324i	$-0.332253\pi$
$317$	17.9090	1.00587	0.502936	+	0.864324i	$0.332253\pi$
$318$	0	0
$319$	−8.69133	−0.486621
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.00000	−0.222566
$324$	0	0
$325$	4.80442	0.266501
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.84352	0.156768
$330$	0	0
$331$	−1.92600	−0.105863	−0.0529313	−	0.998598i	$-0.516856\pi$
$331$	−1.92600	−0.105863	−0.0529313	+	0.998598i	$0.516856\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−17.2568	−0.940037	−0.470019	−	0.882657i	$-0.655753\pi$
$337$	−17.2568	−0.940037	−0.470019	+	0.882657i	$0.655753\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4.22618	−0.228861
$342$	0	0
$343$	−12.0000	−0.647939
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.6785	1.21745	0.608724	−	0.793382i	$-0.291682\pi$
$347$	22.6785	1.21745	0.608724	+	0.793382i	$0.291682\pi$
$348$	0	0
$349$	4.61733	0.247160	0.123580	−	0.992335i	$-0.460562\pi$
$349$	4.61733	0.247160	0.123580	+	0.992335i	$0.460562\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	23.7610	1.26467	0.632336	−	0.774694i	$-0.282096\pi$
$353$	23.7610	1.26467	0.632336	+	0.774694i	$0.282096\pi$
$354$	0	0
$355$	5.60885	0.297687
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12.5042	0.659949	0.329974	−	0.943990i	$-0.392960\pi$
$359$	12.5042	0.659949	0.329974	+	0.943990i	$0.392960\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	−6.73042	−0.351325	−0.175663	−	0.984450i	$-0.556207\pi$
$367$	−6.73042	−0.351325	−0.175663	+	0.984450i	$0.556207\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11.0697	−0.574710
$372$	0	0
$373$	8.80442	0.455876	0.227938	−	0.973676i	$-0.426802\pi$
$373$	8.80442	0.455876	0.227938	+	0.973676i	$0.426802\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−19.7610	−1.01774
$378$	0	0
$379$	−31.7610	−1.63145	−0.815727	−	0.578437i	$-0.803663\pi$
$379$	−31.7610	−1.63145	−0.815727	+	0.578437i	$0.803663\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−23.0697	−1.17881	−0.589403	−	0.807839i	$-0.700637\pi$
$383$	−23.0697	−1.17881	−0.589403	+	0.807839i	$0.700637\pi$
$384$	0	0
$385$	8.69133	0.442951
$386$	0	0
$387$	0	0
$388$	0	0
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	−2.76533	−0.139849
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−15.1437	−0.761962
$396$	0	0
$397$	38.5264	1.93358	0.966791	−	0.255567i	$-0.0822622\pi$
$397$	38.5264	1.93358	0.966791	+	0.255567i	$0.0822622\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.1003	−1.10364	−0.551818	−	0.833964i	$-0.686066\pi$
$401$	−22.1003	−1.10364	−0.551818	+	0.833964i	$0.686066\pi$
$402$	0	0
$403$	−9.60885	−0.478651
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.23048	0.358402
$408$	0	0
$409$	30.2262	1.49459	0.747294	−	0.664493i	$-0.231353\pi$
$409$	30.2262	1.49459	0.747294	+	0.664493i	$0.231353\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	39.2177	1.92978
$414$	0	0
$415$	6.22618	0.305631
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−11.7219	−0.572654	−0.286327	−	0.958132i	$-0.592434\pi$
$419$	−11.7219	−0.572654	−0.286327	+	0.958132i	$0.592434\pi$
$420$	0	0
$421$	−30.9787	−1.50981	−0.754905	−	0.655834i	$-0.772317\pi$
$421$	−30.9787	−1.50981	−0.754905	+	0.655834i	$0.772317\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	12.0000	0.580721
$428$	0	0
$429$	0	0
$430$	0	0
$431$	19.2959	0.929450	0.464725	−	0.885455i	$-0.346153\pi$
$431$	19.2959	0.929450	0.464725	+	0.885455i	$0.346153\pi$
$432$	0	0
$433$	−23.4218	−1.12558	−0.562789	−	0.826601i	$-0.690272\pi$
$433$	−23.4218	−1.12558	−0.562789	+	0.826601i	$0.690272\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.691333	0.0330709
$438$	0	0
$439$	−8.85630	−0.422688	−0.211344	−	0.977412i	$-0.567784\pi$
$439$	−8.85630	−0.422688	−0.211344	+	0.977412i	$0.567784\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.23467	0.343729	0.171865	−	0.985121i	$-0.445021\pi$
$443$	7.23467	0.343729	0.171865	+	0.985121i	$0.445021\pi$
$444$	0	0
$445$	−1.42176	−0.0673978
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.19558	0.0564227	0.0282114	−	0.999602i	$-0.491019\pi$
$449$	1.19558	0.0564227	0.0282114	+	0.999602i	$0.491019\pi$
$450$	0	0
$451$	−11.4567	−0.539473
$452$	0	0
$453$	0	0
$454$	0	0
$455$	19.7610	0.926411
$456$	0	0
$457$	16.9915	0.794829	0.397415	−	0.917639i	$-0.369907\pi$
$457$	16.9915	0.794829	0.397415	+	0.917639i	$0.369907\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	16.6173	0.773946	0.386973	−	0.922091i	$-0.373521\pi$
$461$	16.6173	0.773946	0.386973	+	0.922091i	$0.373521\pi$
$462$	0	0
$463$	−12.1131	−0.562943	−0.281472	−	0.959570i	$-0.590822\pi$
$463$	−12.1131	−0.562943	−0.281472	+	0.959570i	$0.590822\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−37.6701	−1.74316	−0.871581	−	0.490251i	$-0.836905\pi$
$467$	−37.6701	−1.74316	−0.871581	+	0.490251i	$0.836905\pi$
$468$	0	0
$469$	−16.4524	−0.759700
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−5.92600	−0.272478
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	34.1131	1.55867	0.779333	−	0.626609i	$-0.215558\pi$
$479$	34.1131	1.55867	0.779333	+	0.626609i	$0.215558\pi$
$480$	0	0
$481$	16.4396	0.749580
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−15.4218	−0.700266
$486$	0	0
$487$	−38.5136	−1.74522	−0.872608	−	0.488421i	$-0.837573\pi$
$487$	−38.5136	−1.74522	−0.872608	+	0.488421i	$0.837573\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.49576	−0.157761	−0.0788806	−	0.996884i	$-0.525135\pi$
$491$	−3.49576	−0.157761	−0.0788806	+	0.996884i	$0.525135\pi$
$492$	0	0
$493$	16.4524	0.740977
$494$	0	0
$495$	0	0
$496$	0	0
$497$	23.0697	1.03482
$498$	0	0
$499$	−16.2262	−0.726384	−0.363192	−	0.931714i	$-0.618313\pi$
$499$	−16.2262	−0.726384	−0.363192	+	0.931714i	$0.618313\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	33.3699	1.48789	0.743945	−	0.668241i	$-0.232953\pi$
$503$	33.3699	1.48789	0.743945	+	0.668241i	$0.232953\pi$
$504$	0	0
$505$	−10.2262	−0.455059
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−19.5570	−0.866847	−0.433424	−	0.901190i	$-0.642695\pi$
$509$	−19.5570	−0.866847	−0.433424	+	0.901190i	$0.642695\pi$
$510$	0	0
$511$	−24.6785	−1.09171
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−16.4524	−0.724978
$516$	0	0
$517$	−1.46085	−0.0642481
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4.63945	−0.203258	−0.101629	−	0.994822i	$-0.532405\pi$
$521$	−4.63945	−0.203258	−0.101629	+	0.994822i	$0.532405\pi$
$522$	0	0
$523$	−35.8962	−1.56963	−0.784816	−	0.619728i	$-0.787243\pi$
$523$	−35.8962	−1.56963	−0.784816	+	0.619728i	$0.787243\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.00000	0.348485
$528$	0	0
$529$	−22.5221	−0.979220
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−26.0484	−1.12828
$534$	0	0
$535$	16.4524	0.711298
$536$	0	0
$537$	0	0
$538$	0	0
$539$	20.9566	0.902665
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.53485	0.322757
$546$	0	0
$547$	24.2262	1.03584	0.517918	−	0.855430i	$-0.326707\pi$
$547$	24.2262	1.03584	0.517918	+	0.855430i	$0.326707\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4.11309	−0.175224
$552$	0	0
$553$	−62.2874	−2.64873
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	0	0
$559$	−13.4736	−0.569874
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−44.9175	−1.89305	−0.946524	−	0.322634i	$-0.895432\pi$
$563$	−44.9175	−1.89305	−0.946524	+	0.322634i	$0.895432\pi$
$564$	0	0
$565$	−17.5348	−0.737697
$566$	0	0
$567$	0	0
$568$	0	0
$569$	17.1956	0.720876	0.360438	−	0.932783i	$-0.382627\pi$
$569$	17.1956	0.720876	0.360438	+	0.932783i	$0.382627\pi$
$570$	0	0
$571$	−1.15648	−0.0483974	−0.0241987	−	0.999707i	$-0.507703\pi$
$571$	−1.15648	−0.0483974	−0.0241987	+	0.999707i	$0.507703\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0.691333	0.0288306
$576$	0	0
$577$	−36.5136	−1.52008	−0.760040	−	0.649876i	$-0.774821\pi$
$577$	−36.5136	−1.52008	−0.760040	+	0.649876i	$0.774821\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	25.6088	1.06243
$582$	0	0
$583$	5.68703	0.235533
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4.61733	−0.190578	−0.0952889	−	0.995450i	$-0.530377\pi$
$587$	−4.61733	−0.190578	−0.0952889	+	0.995450i	$0.530377\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−25.5961	−1.05110	−0.525552	−	0.850761i	$-0.676141\pi$
$593$	−25.5961	−1.05110	−0.525552	+	0.850761i	$0.676141\pi$
$594$	0	0
$595$	−16.4524	−0.674481
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−45.3571	−1.85324	−0.926620	−	0.375999i	$-0.877300\pi$
$599$	−45.3571	−1.85324	−0.926620	+	0.375999i	$0.877300\pi$
$600$	0	0
$601$	15.6088	0.636698	0.318349	−	0.947974i	$-0.396872\pi$
$601$	15.6088	0.636698	0.318349	+	0.947974i	$0.396872\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6.53485	0.265679
$606$	0	0
$607$	39.8962	1.61934	0.809669	−	0.586887i	$-0.199647\pi$
$607$	39.8962	1.61934	0.809669	+	0.586887i	$0.199647\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.32146	−0.134372
$612$	0	0
$613$	11.6828	0.471866	0.235933	−	0.971769i	$-0.424185\pi$
$613$	11.6828	0.471866	0.235933	+	0.971769i	$0.424185\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	26.2874	1.05829	0.529145	−	0.848531i	$-0.322513\pi$
$617$	26.2874	1.05829	0.529145	+	0.848531i	$0.322513\pi$
$618$	0	0
$619$	23.2959	0.936340	0.468170	−	0.883638i	$-0.344913\pi$
$619$	23.2959	0.936340	0.468170	+	0.883638i	$0.344913\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.84782	−0.234288
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−13.6870	−0.545738
$630$	0	0
$631$	−21.2347	−0.845339	−0.422669	−	0.906284i	$-0.638907\pi$
$631$	−21.2347	−0.845339	−0.422669	+	0.906284i	$0.638907\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	9.83503	0.390291
$636$	0	0
$637$	47.6479	1.88788
$638$	0	0
$639$	0	0
$640$	0	0
$641$	39.7091	1.56842	0.784209	−	0.620497i	$-0.213069\pi$
$641$	39.7091	1.56842	0.784209	+	0.620497i	$0.213069\pi$
$642$	0	0
$643$	26.8044	1.05706	0.528532	−	0.848914i	$-0.322743\pi$
$643$	26.8044	1.05706	0.528532	+	0.848914i	$0.322743\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10.3002	−0.404942	−0.202471	−	0.979288i	$-0.564897\pi$
$647$	−10.3002	−0.404942	−0.202471	+	0.979288i	$0.564897\pi$
$648$	0	0
$649$	−20.1480	−0.790878
$650$	0	0
$651$	0	0
$652$	0	0
$653$	16.8563	0.659638	0.329819	−	0.944044i	$-0.393012\pi$
$653$	16.8563	0.659638	0.329819	+	0.944044i	$0.393012\pi$
$654$	0	0
$655$	6.33927	0.247696
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.90903	0.0743651	0.0371826	−	0.999308i	$-0.488162\pi$
$659$	1.90903	0.0743651	0.0371826	+	0.999308i	$0.488162\pi$
$660$	0	0
$661$	2.30018	0.0894666	0.0447333	−	0.998999i	$-0.485756\pi$
$661$	2.30018	0.0894666	0.0447333	+	0.998999i	$0.485756\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.11309	0.159499
$666$	0	0
$667$	−2.84352	−0.110101
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.16497	−0.237996
$672$	0	0
$673$	−32.4745	−1.25180	−0.625900	−	0.779904i	$-0.715268\pi$
$673$	−32.4745	−1.25180	−0.625900	+	0.779904i	$0.715268\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.07400	0.310309	0.155154	−	0.987890i	$-0.450412\pi$
$677$	8.07400	0.310309	0.155154	+	0.987890i	$0.450412\pi$
$678$	0	0
$679$	−63.4311	−2.43426
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−18.6173	−0.712372	−0.356186	−	0.934415i	$-0.615923\pi$
$683$	−18.6173	−0.712372	−0.356186	+	0.934415i	$0.615923\pi$
$684$	0	0
$685$	7.30867	0.279250
$686$	0	0
$687$	0	0
$688$	0	0
$689$	12.9303	0.492605
$690$	0	0
$691$	−25.6088	−0.974206	−0.487103	−	0.873344i	$-0.661946\pi$
$691$	−25.6088	−0.974206	−0.487103	+	0.873344i	$0.661946\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.22618	0.312037
$696$	0	0
$697$	21.6870	0.821455
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−0.765332	−0.0289062	−0.0144531	−	0.999896i	$-0.504601\pi$
$701$	−0.765332	−0.0289062	−0.0144531	+	0.999896i	$0.504601\pi$
$702$	0	0
$703$	3.42176	0.129054
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−42.0612	−1.58187
$708$	0	0
$709$	35.8222	1.34533	0.672666	−	0.739946i	$-0.265149\pi$
$709$	35.8222	1.34533	0.672666	+	0.739946i	$0.265149\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.38267	−0.0517812
$714$	0	0
$715$	−10.1522	−0.379670
$716$	0	0
$717$	0	0
$718$	0	0
$719$	22.7916	0.849985	0.424992	−	0.905197i	$-0.360277\pi$
$719$	22.7916	0.849985	0.424992	+	0.905197i	$0.360277\pi$
$720$	0	0
$721$	−67.6701	−2.52016
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.11309	−0.152756
$726$	0	0
$727$	35.6351	1.32163	0.660817	−	0.750547i	$-0.270210\pi$
$727$	35.6351	1.32163	0.660817	+	0.750547i	$0.270210\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	11.2177	0.414901
$732$	0	0
$733$	20.6913	0.764252	0.382126	−	0.924110i	$-0.375192\pi$
$733$	20.6913	0.764252	0.382126	+	0.924110i	$0.375192\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.45236	0.311347
$738$	0	0
$739$	−23.2177	−0.854077	−0.427038	−	0.904234i	$-0.640443\pi$
$739$	−23.2177	−0.854077	−0.427038	+	0.904234i	$0.640443\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−7.06970	−0.259362	−0.129681	−	0.991556i	$-0.541395\pi$
$743$	−7.06970	−0.259362	−0.129681	+	0.991556i	$0.541395\pi$
$744$	0	0
$745$	−7.38267	−0.270480
$746$	0	0
$747$	0	0
$748$	0	0
$749$	67.6701	2.47261
$750$	0	0
$751$	−8.61733	−0.314451	−0.157225	−	0.987563i	$-0.550255\pi$
$751$	−8.61733	−0.314451	−0.157225	+	0.987563i	$0.550255\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.38267	0.268683
$756$	0	0
$757$	47.9872	1.74412	0.872062	−	0.489395i	$-0.162782\pi$
$757$	47.9872	1.74412	0.872062	+	0.489395i	$0.162782\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	23.0527	0.835661	0.417830	−	0.908525i	$-0.362791\pi$
$761$	23.0527	0.835661	0.417830	+	0.908525i	$0.362791\pi$
$762$	0	0
$763$	30.9915	1.12197
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−45.8094	−1.65408
$768$	0	0
$769$	37.0697	1.33677	0.668384	−	0.743817i	$-0.266986\pi$
$769$	37.0697	1.33677	0.668384	+	0.743817i	$0.266986\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	25.3087	0.910289	0.455145	−	0.890417i	$-0.349588\pi$
$773$	25.3087	0.910289	0.455145	+	0.890417i	$0.349588\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.42176	−0.194255
$780$	0	0
$781$	−11.8520	−0.424098
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−12.9175	−0.461046
$786$	0	0
$787$	−10.9915	−0.391805	−0.195903	−	0.980623i	$-0.562764\pi$
$787$	−10.9915	−0.391805	−0.195903	+	0.980623i	$0.562764\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−72.1224	−2.56438
$792$	0	0
$793$	−14.0170	−0.497757
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.69982	0.272742	0.136371	−	0.990658i	$-0.456456\pi$
$797$	7.69982	0.272742	0.136371	+	0.990658i	$0.456456\pi$
$798$	0	0
$799$	2.76533	0.0978304
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12.6785	0.447416
$804$	0	0
$805$	2.84352	0.100221
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−22.4524	−0.789383	−0.394692	−	0.918814i	$-0.629149\pi$
$809$	−22.4524	−0.789383	−0.394692	+	0.918814i	$0.629149\pi$
$810$	0	0
$811$	53.5051	1.87882	0.939409	−	0.342799i	$-0.111375\pi$
$811$	53.5051	1.87882	0.939409	+	0.342799i	$0.111375\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.42176	0.189916
$816$	0	0
$817$	−2.80442	−0.0981144
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.62582	−0.0567415	−0.0283708	−	0.999597i	$-0.509032\pi$
$821$	−1.62582	−0.0567415	−0.0283708	+	0.999597i	$0.509032\pi$
$822$	0	0
$823$	−29.4958	−1.02816	−0.514079	−	0.857743i	$-0.671866\pi$
$823$	−29.4958	−1.02816	−0.514079	+	0.857743i	$0.671866\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4.91751	0.170999	0.0854994	−	0.996338i	$-0.472751\pi$
$827$	4.91751	0.170999	0.0854994	+	0.996338i	$0.472751\pi$
$828$	0	0
$829$	−16.9175	−0.587570	−0.293785	−	0.955872i	$-0.594915\pi$
$829$	−16.9175	−0.587570	−0.293785	+	0.955872i	$0.594915\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−39.6701	−1.37449
$834$	0	0
$835$	−12.9175	−0.447029
$836$	0	0
$837$	0	0
$838$	0	0
$839$	10.8435	0.374360	0.187180	−	0.982326i	$-0.440065\pi$
$839$	10.8435	0.374360	0.187180	+	0.982326i	$0.440065\pi$
$840$	0	0
$841$	−12.0825	−0.416637
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10.0825	−0.346848
$846$	0	0
$847$	26.8784	0.923554
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.36557	0.0810908
$852$	0	0
$853$	28.9175	0.990117	0.495058	−	0.868860i	$-0.335147\pi$
$853$	28.9175	0.990117	0.495058	+	0.868860i	$0.335147\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−27.2917	−0.932266	−0.466133	−	0.884715i	$-0.654353\pi$
$857$	−27.2917	−0.932266	−0.466133	+	0.884715i	$0.654353\pi$
$858$	0	0
$859$	−10.6173	−0.362259	−0.181129	−	0.983459i	$-0.557975\pi$
$859$	−10.6173	−0.362259	−0.181129	+	0.983459i	$0.557975\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.61733	0.225257	0.112628	−	0.993637i	$-0.464073\pi$
$863$	6.61733	0.225257	0.112628	+	0.993637i	$0.464073\pi$
$864$	0	0
$865$	−13.5348	−0.460199
$866$	0	0
$867$	0	0
$868$	0	0
$869$	32.0000	1.08553
$870$	0	0
$871$	19.2177	0.651167
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.11309	0.139048
$876$	0	0
$877$	22.4133	0.756842	0.378421	−	0.925634i	$-0.376467\pi$
$877$	22.4133	0.756842	0.378421	+	0.925634i	$0.376467\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	12.5392	0.422455	0.211227	−	0.977437i	$-0.432254\pi$
$881$	12.5392	0.422455	0.211227	+	0.977437i	$0.432254\pi$
$882$	0	0
$883$	7.96091	0.267906	0.133953	−	0.990988i	$-0.457233\pi$
$883$	7.96091	0.267906	0.133953	+	0.990988i	$0.457233\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	5.68703	0.190952	0.0954759	−	0.995432i	$-0.469563\pi$
$887$	5.68703	0.190952	0.0954759	+	0.995432i	$0.469563\pi$
$888$	0	0
$889$	40.4524	1.35673
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.691333	−0.0231346
$894$	0	0
$895$	−10.9175	−0.364932
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.22618	0.274359
$900$	0	0
$901$	−10.7653	−0.358645
$902$	0	0
$903$	0	0
$904$	0	0
$905$	17.3699	0.577394
$906$	0	0
$907$	17.6088	0.584692	0.292346	−	0.956313i	$-0.405564\pi$
$907$	17.6088	0.584692	0.292346	+	0.956313i	$0.405564\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	23.2177	0.769237	0.384618	−	0.923076i	$-0.374333\pi$
$911$	23.2177	0.769237	0.384618	+	0.923076i	$0.374333\pi$
$912$	0	0
$913$	−13.1565	−0.435416
$914$	0	0
$915$	0	0
$916$	0	0
$917$	26.0740	0.861039
$918$	0	0
$919$	7.06970	0.233208	0.116604	−	0.993179i	$-0.462799\pi$
$919$	7.06970	0.233208	0.116604	+	0.993179i	$0.462799\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−26.9473	−0.886980
$924$	0	0
$925$	3.42176	0.112507
$926$	0	0
$927$	0	0
$928$	0	0
$929$	52.9659	1.73776	0.868878	−	0.495026i	$-0.164842\pi$
$929$	52.9659	1.73776	0.868878	+	0.495026i	$0.164842\pi$
$930$	0	0
$931$	9.91751	0.325033
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8.45236	0.276422
$936$	0	0
$937$	3.75685	0.122731	0.0613654	−	0.998115i	$-0.480455\pi$
$937$	3.75685	0.122731	0.0613654	+	0.998115i	$0.480455\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	13.0178	0.424369	0.212184	−	0.977230i	$-0.431942\pi$
$941$	13.0178	0.424369	0.212184	+	0.977230i	$0.431942\pi$
$942$	0	0
$943$	−3.74824	−0.122059
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−20.3912	−0.662623	−0.331312	−	0.943521i	$-0.607491\pi$
$947$	−20.3912	−0.662623	−0.331312	+	0.943521i	$0.607491\pi$
$948$	0	0
$949$	28.8265	0.935749
$950$	0	0
$951$	0	0
$952$	0	0
$953$	48.2746	1.56377	0.781884	−	0.623424i	$-0.214259\pi$
$953$	48.2746	1.56377	0.781884	+	0.623424i	$0.214259\pi$
$954$	0	0
$955$	0.730425	0.0236360
$956$	0	0
$957$	0	0
$958$	0	0
$959$	30.0612	0.970727
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−25.4830	−0.820326
$966$	0	0
$967$	−28.9396	−0.930636	−0.465318	−	0.885144i	$-0.654060\pi$
$967$	−28.9396	−0.930636	−0.465318	+	0.885144i	$0.654060\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.92600	0.125991	0.0629957	−	0.998014i	$-0.479935\pi$
$971$	3.92600	0.125991	0.0629957	+	0.998014i	$0.479935\pi$
$972$	0	0
$973$	33.8350	1.08470
$974$	0	0
$975$	0	0
$976$	0	0
$977$	9.30867	0.297811	0.148905	−	0.988851i	$-0.452425\pi$
$977$	9.30867	0.297811	0.148905	+	0.988851i	$0.452425\pi$
$978$	0	0
$979$	3.00430	0.0960179
$980$	0	0
$981$	0	0
$982$	0	0
$983$	13.8478	0.441677	0.220838	−	0.975310i	$-0.429121\pi$
$983$	13.8478	0.441677	0.220838	+	0.975310i	$0.429121\pi$
$984$	0	0
$985$	−21.7610	−0.693364
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.93879	−0.0616500
$990$	0	0
$991$	30.5433	0.970241	0.485121	−	0.874447i	$-0.338776\pi$
$991$	30.5433	0.970241	0.485121	+	0.874447i	$0.338776\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.60885	−0.0510039
$996$	0	0
$997$	−6.75254	−0.213855	−0.106928	−	0.994267i	$-0.534101\pi$
$997$	−6.75254	−0.213855	−0.106928	+	0.994267i	$0.534101\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6840.2.a.bh.1.1		3
3.2	odd	2		2280.2.a.u.1.1	✓	3
12.11	even	2		4560.2.a.bs.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.u.1.1	✓	3	3.2	odd	2
4560.2.a.bs.1.3		3	12.11	even	2
6840.2.a.bh.1.1		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6840.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -4.11309 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -4.11309 q^{7} +2.11309 q^{11} +4.80442 q^{13} -4.00000 q^{17} +1.00000 q^{19} +0.691333 q^{23} +1.00000 q^{25} -4.11309 q^{29} -2.00000 q^{31} +4.11309 q^{35} +3.42176 q^{37} -5.42176 q^{41} -2.80442 q^{43} -0.691333 q^{47} +9.91751 q^{49} +2.69133 q^{53} -2.11309 q^{55} -9.53485 q^{59} -2.91751 q^{61} -4.80442 q^{65} +4.00000 q^{67} -5.60885 q^{71} +6.00000 q^{73} -8.69133 q^{77} +15.1437 q^{79} -6.22618 q^{83} +4.00000 q^{85} +1.42176 q^{89} -19.7610 q^{91} -1.00000 q^{95} +15.4218 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5}+O(q^{10}) \) 3 * q - 3 * q^5	\( 3 q - 3 q^{5} - 6 q^{11} + 4 q^{13} - 12 q^{17} + 3 q^{19} + 4 q^{23} + 3 q^{25} - 6 q^{31} - 4 q^{37} - 2 q^{41} + 2 q^{43} - 4 q^{47} + 7 q^{49} + 10 q^{53} + 6 q^{55} - 2 q^{59} + 14 q^{61} - 4 q^{65} + 12 q^{67} + 4 q^{71} + 18 q^{73} - 28 q^{77} - 2 q^{79} + 6 q^{83} + 12 q^{85} - 10 q^{89} - 8 q^{91} - 3 q^{95} + 32 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 - 6 * q^11 + 4 * q^13 - 12 * q^17 + 3 * q^19 + 4 * q^23 + 3 * q^25 - 6 * q^31 - 4 * q^37 - 2 * q^41 + 2 * q^43 - 4 * q^47 + 7 * q^49 + 10 * q^53 + 6 * q^55 - 2 * q^59 + 14 * q^61 - 4 * q^65 + 12 * q^67 + 4 * q^71 + 18 * q^73 - 28 * q^77 - 2 * q^79 + 6 * q^83 + 12 * q^85 - 10 * q^89 - 8 * q^91 - 3 * q^95 + 32 * q^97

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