LMFDB - Embedded newform 6840.2.a.w.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2280)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.64575$ 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)

$f(q)$	$=$	$q-1.00000 q^{5} -3.64575 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -3.64575 q^{7} +5.64575 q^{11} -1.64575 q^{13} -1.00000 q^{19} -2.00000 q^{23} +1.00000 q^{25} -4.35425 q^{29} -2.00000 q^{31} +3.64575 q^{35} +8.93725 q^{37} +3.64575 q^{41} +10.9373 q^{43} -6.00000 q^{47} +6.29150 q^{49} -4.00000 q^{53} -5.64575 q^{55} +7.29150 q^{59} +0.708497 q^{61} +1.64575 q^{65} +4.00000 q^{67} -7.29150 q^{71} -2.00000 q^{73} -20.5830 q^{77} -8.00000 q^{79} -15.8745 q^{83} -14.2288 q^{89} +6.00000 q^{91} +1.00000 q^{95} -8.22876 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 6 q^{11} + 2 q^{13} - 2 q^{19} - 4 q^{23} + 2 q^{25} - 14 q^{29} - 4 q^{31} + 2 q^{35} + 2 q^{37} + 2 q^{41} + 6 q^{43} - 12 q^{47} + 2 q^{49} - 8 q^{53} - 6 q^{55} + 4 q^{59} + 12 q^{61} - 2 q^{65} + 8 q^{67} - 4 q^{71} - 4 q^{73} - 20 q^{77} - 16 q^{79} - 2 q^{89} + 12 q^{91} + 2 q^{95} + 10 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^7 + 6 * q^11 + 2 * q^13 - 2 * q^19 - 4 * q^23 + 2 * q^25 - 14 * q^29 - 4 * q^31 + 2 * q^35 + 2 * q^37 + 2 * q^41 + 6 * q^43 - 12 * q^47 + 2 * q^49 - 8 * q^53 - 6 * q^55 + 4 * q^59 + 12 * q^61 - 2 * q^65 + 8 * q^67 - 4 * q^71 - 4 * q^73 - 20 * q^77 - 16 * q^79 - 2 * q^89 + 12 * q^91 + 2 * q^95 + 10 * 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
$7$	−3.64575	−1.37796	−0.688982	−	0.724778i	$-0.741942\pi$
$7$	−3.64575	−1.37796	−0.688982	+	0.724778i	$0.741942\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.64575	1.70226	0.851129	−	0.524957i	$-0.175918\pi$
$11$	5.64575	1.70226	0.851129	+	0.524957i	$0.175918\pi$
$12$	0	0
$13$	−1.64575	−0.456449	−0.228225	−	0.973609i	$-0.573292\pi$
$13$	−1.64575	−0.456449	−0.228225	+	0.973609i	$0.573292\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.35425	−0.808564	−0.404282	−	0.914634i	$-0.632479\pi$
$29$	−4.35425	−0.808564	−0.404282	+	0.914634i	$0.632479\pi$
$30$	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
$35$	3.64575	0.616244
$36$	0	0
$37$	8.93725	1.46928	0.734638	−	0.678460i	$-0.237352\pi$
$37$	8.93725	1.46928	0.734638	+	0.678460i	$0.237352\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.64575	0.569371	0.284685	−	0.958621i	$-0.408111\pi$
$41$	3.64575	0.569371	0.284685	+	0.958621i	$0.408111\pi$
$42$	0	0
$43$	10.9373	1.66792	0.833958	−	0.551828i	$-0.186070\pi$
$43$	10.9373	1.66792	0.833958	+	0.551828i	$0.186070\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	6.29150	0.898786
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	−5.64575	−0.761273
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.29150	0.949273	0.474636	−	0.880182i	$-0.342580\pi$
$59$	7.29150	0.949273	0.474636	+	0.880182i	$0.342580\pi$
$60$	0	0
$61$	0.708497	0.0907138	0.0453569	−	0.998971i	$-0.485557\pi$
$61$	0.708497	0.0907138	0.0453569	+	0.998971i	$0.485557\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.64575	0.204130
$66$	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
$71$	−7.29150	−0.865342	−0.432671	−	0.901552i	$-0.642429\pi$
$71$	−7.29150	−0.865342	−0.432671	+	0.901552i	$0.642429\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−20.5830	−2.34565
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−15.8745	−1.74245	−0.871227	−	0.490881i	$-0.836675\pi$
$83$	−15.8745	−1.74245	−0.871227	+	0.490881i	$0.836675\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−14.2288	−1.50825	−0.754123	−	0.656734i	$-0.771938\pi$
$89$	−14.2288	−1.50825	−0.754123	+	0.656734i	$0.771938\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−8.22876	−0.835504	−0.417752	−	0.908561i	$-0.637182\pi$
$97$	−8.22876	−0.835504	−0.417752	+	0.908561i	$0.637182\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.8745	1.57957	0.789786	−	0.613382i	$-0.210191\pi$
$101$	15.8745	1.57957	0.789786	+	0.613382i	$0.210191\pi$
$102$	0	0
$103$	−6.58301	−0.648643	−0.324321	−	0.945947i	$-0.605136\pi$
$103$	−6.58301	−0.648643	−0.324321	+	0.945947i	$0.605136\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.58301	0.636403	0.318202	−	0.948023i	$-0.396921\pi$
$107$	6.58301	0.636403	0.318202	+	0.948023i	$0.396921\pi$
$108$	0	0
$109$	−5.29150	−0.506834	−0.253417	−	0.967357i	$-0.581554\pi$
$109$	−5.29150	−0.506834	−0.253417	+	0.967357i	$0.581554\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	13.8745	1.30520	0.652602	−	0.757701i	$-0.273677\pi$
$113$	13.8745	1.30520	0.652602	+	0.757701i	$0.273677\pi$
$114$	0	0
$115$	2.00000	0.186501
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	20.8745	1.89768
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	18.5830	1.64898	0.824488	−	0.565880i	$-0.191464\pi$
$127$	18.5830	1.64898	0.824488	+	0.565880i	$0.191464\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2.35425	−0.205692	−0.102846	−	0.994697i	$-0.532795\pi$
$131$	−2.35425	−0.205692	−0.102846	+	0.994697i	$0.532795\pi$
$132$	0	0
$133$	3.64575	0.316127
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−20.5830	−1.75852	−0.879262	−	0.476338i	$-0.841964\pi$
$137$	−20.5830	−1.75852	−0.879262	+	0.476338i	$0.841964\pi$
$138$	0	0
$139$	13.8745	1.17682	0.588410	−	0.808563i	$-0.299754\pi$
$139$	13.8745	1.17682	0.588410	+	0.808563i	$0.299754\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−9.29150	−0.776994
$144$	0	0
$145$	4.35425	0.361601
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	6.00000	0.488273	0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000	0.488273	0.244137	+	0.969741i	$0.421495\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.00000	0.160644
$156$	0	0
$157$	−15.8745	−1.26692	−0.633462	−	0.773774i	$-0.718367\pi$
$157$	−15.8745	−1.26692	−0.633462	+	0.773774i	$0.718367\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.29150	0.574651
$162$	0	0
$163$	4.35425	0.341051	0.170526	−	0.985353i	$-0.445453\pi$
$163$	4.35425	0.341051	0.170526	+	0.985353i	$0.445453\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.29150	−0.718998	−0.359499	−	0.933145i	$-0.617052\pi$
$167$	−9.29150	−0.718998	−0.359499	+	0.933145i	$0.617052\pi$
$168$	0	0
$169$	−10.2915	−0.791654
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−19.2915	−1.46671	−0.733353	−	0.679848i	$-0.762046\pi$
$173$	−19.2915	−1.46671	−0.733353	+	0.679848i	$0.762046\pi$
$174$	0	0
$175$	−3.64575	−0.275593
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3.29150	−0.246018	−0.123009	−	0.992406i	$-0.539254\pi$
$179$	−3.29150	−0.246018	−0.123009	+	0.992406i	$0.539254\pi$
$180$	0	0
$181$	5.29150	0.393314	0.196657	−	0.980472i	$-0.436991\pi$
$181$	5.29150	0.393314	0.196657	+	0.980472i	$0.436991\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−8.93725	−0.657080
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.93725	0.357247	0.178624	−	0.983917i	$-0.442836\pi$
$191$	4.93725	0.357247	0.178624	+	0.983917i	$0.442836\pi$
$192$	0	0
$193$	−12.9373	−0.931244	−0.465622	−	0.884984i	$-0.654169\pi$
$193$	−12.9373	−0.931244	−0.465622	+	0.884984i	$0.654169\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−24.0000	−1.70993	−0.854965	−	0.518686i	$-0.826421\pi$
$197$	−24.0000	−1.70993	−0.854965	+	0.518686i	$0.826421\pi$
$198$	0	0
$199$	−12.7085	−0.900881	−0.450441	−	0.892806i	$-0.648733\pi$
$199$	−12.7085	−0.900881	−0.450441	+	0.892806i	$0.648733\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	15.8745	1.11417
$204$	0	0
$205$	−3.64575	−0.254630
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.64575	−0.390525
$210$	0	0
$211$	10.5830	0.728564	0.364282	−	0.931289i	$-0.381314\pi$
$211$	10.5830	0.728564	0.364282	+	0.931289i	$0.381314\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−10.9373	−0.745915
$216$	0	0
$217$	7.29150	0.494979
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−22.5830	−1.51227	−0.756135	−	0.654416i	$-0.772915\pi$
$223$	−22.5830	−1.51227	−0.756135	+	0.654416i	$0.772915\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−14.7085	−0.976237	−0.488119	−	0.872777i	$-0.662317\pi$
$227$	−14.7085	−0.976237	−0.488119	+	0.872777i	$0.662317\pi$
$228$	0	0
$229$	−3.29150	−0.217509	−0.108754	−	0.994069i	$-0.534686\pi$
$229$	−3.29150	−0.217509	−0.108754	+	0.994069i	$0.534686\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	0	0
$235$	6.00000	0.391397
$236$	0	0
$237$	0	0
$238$	0	0
$239$	28.9373	1.87180	0.935898	−	0.352272i	$-0.114591\pi$
$239$	28.9373	1.87180	0.935898	+	0.352272i	$0.114591\pi$
$240$	0	0
$241$	16.5830	1.06821	0.534103	−	0.845420i	$-0.320650\pi$
$241$	16.5830	1.06821	0.534103	+	0.845420i	$0.320650\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.29150	−0.401949
$246$	0	0
$247$	1.64575	0.104717
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−20.2288	−1.27683	−0.638414	−	0.769693i	$-0.720409\pi$
$251$	−20.2288	−1.27683	−0.638414	+	0.769693i	$0.720409\pi$
$252$	0	0
$253$	−11.2915	−0.709891
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−21.8745	−1.36449	−0.682247	−	0.731122i	$-0.738997\pi$
$257$	−21.8745	−1.36449	−0.682247	+	0.731122i	$0.738997\pi$
$258$	0	0
$259$	−32.5830	−2.02461
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.29150	−0.572939	−0.286469	−	0.958089i	$-0.592482\pi$
$263$	−9.29150	−0.572939	−0.286469	+	0.958089i	$0.592482\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	0	0
$267$	0	0
$268$	0	0
$269$	10.9373	0.666856	0.333428	−	0.942776i	$-0.391795\pi$
$269$	10.9373	0.666856	0.333428	+	0.942776i	$0.391795\pi$
$270$	0	0
$271$	23.2915	1.41486	0.707429	−	0.706784i	$-0.249855\pi$
$271$	23.2915	1.41486	0.707429	+	0.706784i	$0.249855\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.64575	0.340452
$276$	0	0
$277$	−3.41699	−0.205307	−0.102654	−	0.994717i	$-0.532733\pi$
$277$	−3.41699	−0.205307	−0.102654	+	0.994717i	$0.532733\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−10.9373	−0.652462	−0.326231	−	0.945290i	$-0.605779\pi$
$281$	−10.9373	−0.652462	−0.326231	+	0.945290i	$0.605779\pi$
$282$	0	0
$283$	−30.2288	−1.79691	−0.898457	−	0.439062i	$-0.855311\pi$
$283$	−30.2288	−1.79691	−0.898457	+	0.439062i	$0.855311\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−13.2915	−0.784573
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−13.8745	−0.810557	−0.405279	−	0.914193i	$-0.632826\pi$
$293$	−13.8745	−0.810557	−0.405279	+	0.914193i	$0.632826\pi$
$294$	0	0
$295$	−7.29150	−0.424528
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.29150	0.190353
$300$	0	0
$301$	−39.8745	−2.29833
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−0.708497	−0.0405684
$306$	0	0
$307$	23.2915	1.32932	0.664658	−	0.747148i	$-0.268577\pi$
$307$	23.2915	1.32932	0.664658	+	0.747148i	$0.268577\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.64575	0.546960	0.273480	−	0.961878i	$-0.411825\pi$
$311$	9.64575	0.546960	0.273480	+	0.961878i	$0.411825\pi$
$312$	0	0
$313$	−29.2915	−1.65565	−0.827827	−	0.560984i	$-0.810423\pi$
$313$	−29.2915	−1.65565	−0.827827	+	0.560984i	$0.810423\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.41699	−0.0795864	−0.0397932	−	0.999208i	$-0.512670\pi$
$317$	−1.41699	−0.0795864	−0.0397932	+	0.999208i	$0.512670\pi$
$318$	0	0
$319$	−24.5830	−1.37638
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−1.64575	−0.0912899
$326$	0	0
$327$	0	0
$328$	0	0
$329$	21.8745	1.20598
$330$	0	0
$331$	−4.58301	−0.251905	−0.125952	−	0.992036i	$-0.540199\pi$
$331$	−4.58301	−0.251905	−0.125952	+	0.992036i	$0.540199\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	4.22876	0.230355	0.115178	−	0.993345i	$-0.463256\pi$
$337$	4.22876	0.230355	0.115178	+	0.993345i	$0.463256\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−11.2915	−0.611469
$342$	0	0
$343$	2.58301	0.139469
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.29150	0.0693315	0.0346657	−	0.999399i	$-0.488963\pi$
$347$	1.29150	0.0693315	0.0346657	+	0.999399i	$0.488963\pi$
$348$	0	0
$349$	15.1660	0.811818	0.405909	−	0.913914i	$-0.366955\pi$
$349$	15.1660	0.811818	0.405909	+	0.913914i	$0.366955\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−7.16601	−0.381408	−0.190704	−	0.981648i	$-0.561077\pi$
$353$	−7.16601	−0.381408	−0.190704	+	0.981648i	$0.561077\pi$
$354$	0	0
$355$	7.29150	0.386993
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−16.2288	−0.856521	−0.428261	−	0.903655i	$-0.640873\pi$
$359$	−16.2288	−0.856521	−0.428261	+	0.903655i	$0.640873\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	−26.2288	−1.36913	−0.684565	−	0.728952i	$-0.740008\pi$
$367$	−26.2288	−1.36913	−0.684565	+	0.728952i	$0.740008\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	14.5830	0.757112
$372$	0	0
$373$	−13.6458	−0.706550	−0.353275	−	0.935519i	$-0.614932\pi$
$373$	−13.6458	−0.706550	−0.353275	+	0.935519i	$0.614932\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.16601	0.369068
$378$	0	0
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	26.5830	1.35833	0.679164	−	0.733987i	$-0.262342\pi$
$383$	26.5830	1.35833	0.679164	+	0.733987i	$0.262342\pi$
$384$	0	0
$385$	20.5830	1.04901
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.00000	0.402524
$396$	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
$401$	−11.6458	−0.581561	−0.290781	−	0.956790i	$-0.593915\pi$
$401$	−11.6458	−0.581561	−0.290781	+	0.956790i	$0.593915\pi$
$402$	0	0
$403$	3.29150	0.163961
$404$	0	0
$405$	0	0
$406$	0	0
$407$	50.4575	2.50109
$408$	0	0
$409$	−17.2915	−0.855010	−0.427505	−	0.904013i	$-0.640607\pi$
$409$	−17.2915	−0.855010	−0.427505	+	0.904013i	$0.640607\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−26.5830	−1.30806
$414$	0	0
$415$	15.8745	0.779249
$416$	0	0
$417$	0	0
$418$	0	0
$419$	23.0627	1.12669	0.563344	−	0.826222i	$-0.309514\pi$
$419$	23.0627	1.12669	0.563344	+	0.826222i	$0.309514\pi$
$420$	0	0
$421$	20.5830	1.00315	0.501577	−	0.865113i	$-0.332753\pi$
$421$	20.5830	1.00315	0.501577	+	0.865113i	$0.332753\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.58301	−0.125000
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−20.4575	−0.985404	−0.492702	−	0.870198i	$-0.663991\pi$
$431$	−20.4575	−0.985404	−0.492702	+	0.870198i	$0.663991\pi$
$432$	0	0
$433$	0.228757	0.0109933	0.00549667	−	0.999985i	$-0.498250\pi$
$433$	0.228757	0.0109933	0.00549667	+	0.999985i	$0.498250\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.00000	0.0956730
$438$	0	0
$439$	−37.1660	−1.77384	−0.886918	−	0.461926i	$-0.847159\pi$
$439$	−37.1660	−1.77384	−0.886918	+	0.461926i	$0.847159\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.5830	0.597837	0.298918	−	0.954279i	$-0.403374\pi$
$443$	12.5830	0.597837	0.298918	+	0.954279i	$0.403374\pi$
$444$	0	0
$445$	14.2288	0.674508
$446$	0	0
$447$	0	0
$448$	0	0
$449$	7.64575	0.360825	0.180413	−	0.983591i	$-0.442257\pi$
$449$	7.64575	0.360825	0.180413	+	0.983591i	$0.442257\pi$
$450$	0	0
$451$	20.5830	0.969216
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.00000	−0.281284
$456$	0	0
$457$	−1.29150	−0.0604139	−0.0302070	−	0.999544i	$-0.509617\pi$
$457$	−1.29150	−0.0604139	−0.0302070	+	0.999544i	$0.509617\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	−34.2288	−1.59075	−0.795373	−	0.606121i	$-0.792725\pi$
$463$	−34.2288	−1.59075	−0.795373	+	0.606121i	$0.792725\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−20.5830	−0.952468	−0.476234	−	0.879319i	$-0.657998\pi$
$467$	−20.5830	−0.952468	−0.476234	+	0.879319i	$0.657998\pi$
$468$	0	0
$469$	−14.5830	−0.673381
$470$	0	0
$471$	0	0
$472$	0	0
$473$	61.7490	2.83922
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−23.5203	−1.07467	−0.537334	−	0.843370i	$-0.680569\pi$
$479$	−23.5203	−1.07467	−0.537334	+	0.843370i	$0.680569\pi$
$480$	0	0
$481$	−14.7085	−0.670650
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.22876	0.373649
$486$	0	0
$487$	−7.29150	−0.330410	−0.165205	−	0.986259i	$-0.552828\pi$
$487$	−7.29150	−0.330410	−0.165205	+	0.986259i	$0.552828\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−14.8118	−0.668445	−0.334223	−	0.942494i	$-0.608474\pi$
$491$	−14.8118	−0.668445	−0.334223	+	0.942494i	$0.608474\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	26.5830	1.19241
$498$	0	0
$499$	8.70850	0.389846	0.194923	−	0.980819i	$-0.437554\pi$
$499$	8.70850	0.389846	0.194923	+	0.980819i	$0.437554\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9.29150	−0.414288	−0.207144	−	0.978311i	$-0.566417\pi$
$503$	−9.29150	−0.414288	−0.207144	+	0.978311i	$0.566417\pi$
$504$	0	0
$505$	−15.8745	−0.706406
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−39.6458	−1.75727	−0.878634	−	0.477497i	$-0.841544\pi$
$509$	−39.6458	−1.75727	−0.878634	+	0.477497i	$0.841544\pi$
$510$	0	0
$511$	7.29150	0.322557
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.58301	0.290082
$516$	0	0
$517$	−33.8745	−1.48980
$518$	0	0
$519$	0	0
$520$	0	0
$521$	20.8118	0.911780	0.455890	−	0.890036i	$-0.349321\pi$
$521$	20.8118	0.911780	0.455890	+	0.890036i	$0.349321\pi$
$522$	0	0
$523$	37.8745	1.65614	0.828068	−	0.560627i	$-0.189440\pi$
$523$	37.8745	1.65614	0.828068	+	0.560627i	$0.189440\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−6.00000	−0.259889
$534$	0	0
$535$	−6.58301	−0.284608
$536$	0	0
$537$	0	0
$538$	0	0
$539$	35.5203	1.52997
$540$	0	0
$541$	−26.0000	−1.11783	−0.558914	−	0.829226i	$-0.688782\pi$
$541$	−26.0000	−1.11783	−0.558914	+	0.829226i	$0.688782\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.29150	0.226663
$546$	0	0
$547$	12.4575	0.532645	0.266322	−	0.963884i	$-0.414191\pi$
$547$	12.4575	0.532645	0.266322	+	0.963884i	$0.414191\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.35425	0.185497
$552$	0	0
$553$	29.1660	1.24026
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.0000	0.423714	0.211857	−	0.977301i	$-0.432049\pi$
$557$	10.0000	0.423714	0.211857	+	0.977301i	$0.432049\pi$
$558$	0	0
$559$	−18.0000	−0.761319
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−27.8745	−1.17477	−0.587385	−	0.809307i	$-0.699843\pi$
$563$	−27.8745	−1.17477	−0.587385	+	0.809307i	$0.699843\pi$
$564$	0	0
$565$	−13.8745	−0.583705
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−37.5203	−1.57293	−0.786466	−	0.617634i	$-0.788091\pi$
$569$	−37.5203	−1.57293	−0.786466	+	0.617634i	$0.788091\pi$
$570$	0	0
$571$	−17.8745	−0.748025	−0.374012	−	0.927424i	$-0.622018\pi$
$571$	−17.8745	−0.748025	−0.374012	+	0.927424i	$0.622018\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.00000	−0.0834058
$576$	0	0
$577$	−3.87451	−0.161298	−0.0806489	−	0.996743i	$-0.525699\pi$
$577$	−3.87451	−0.161298	−0.0806489	+	0.996743i	$0.525699\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	57.8745	2.40104
$582$	0	0
$583$	−22.5830	−0.935293
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−15.4170	−0.636327	−0.318164	−	0.948036i	$-0.603066\pi$
$587$	−15.4170	−0.636327	−0.318164	+	0.948036i	$0.603066\pi$
$588$	0	0
$589$	2.00000	0.0824086
$590$	0	0
$591$	0	0
$592$	0	0
$593$	36.5830	1.50228	0.751142	−	0.660141i	$-0.229503\pi$
$593$	36.5830	1.50228	0.751142	+	0.660141i	$0.229503\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	26.5830	1.08615	0.543076	−	0.839683i	$-0.317260\pi$
$599$	26.5830	1.08615	0.543076	+	0.839683i	$0.317260\pi$
$600$	0	0
$601$	15.8745	0.647535	0.323767	−	0.946137i	$-0.395050\pi$
$601$	15.8745	0.647535	0.323767	+	0.946137i	$0.395050\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−20.8745	−0.848669
$606$	0	0
$607$	40.4575	1.64212	0.821060	−	0.570842i	$-0.193383\pi$
$607$	40.4575	1.64212	0.821060	+	0.570842i	$0.193383\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.87451	0.399480
$612$	0	0
$613$	38.4575	1.55328	0.776642	−	0.629942i	$-0.216921\pi$
$613$	38.4575	1.55328	0.776642	+	0.629942i	$0.216921\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	−6.12549	−0.246204	−0.123102	−	0.992394i	$-0.539284\pi$
$619$	−6.12549	−0.246204	−0.123102	+	0.992394i	$0.539284\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	51.8745	2.07831
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−5.16601	−0.205656	−0.102828	−	0.994699i	$-0.532789\pi$
$631$	−5.16601	−0.205656	−0.102828	+	0.994699i	$0.532789\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−18.5830	−0.737444
$636$	0	0
$637$	−10.3542	−0.410250
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6.22876	−0.246021	−0.123011	−	0.992405i	$-0.539255\pi$
$641$	−6.22876	−0.246021	−0.123011	+	0.992405i	$0.539255\pi$
$642$	0	0
$643$	29.0627	1.14612	0.573061	−	0.819512i	$-0.305756\pi$
$643$	29.0627	1.14612	0.573061	+	0.819512i	$0.305756\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.7085	0.578251	0.289125	−	0.957291i	$-0.406636\pi$
$647$	14.7085	0.578251	0.289125	+	0.957291i	$0.406636\pi$
$648$	0	0
$649$	41.1660	1.61591
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.00000	0.156532	0.0782660	−	0.996933i	$-0.475062\pi$
$653$	4.00000	0.156532	0.0782660	+	0.996933i	$0.475062\pi$
$654$	0	0
$655$	2.35425	0.0919881
$656$	0	0
$657$	0	0
$658$	0	0
$659$	29.1660	1.13615	0.568073	−	0.822978i	$-0.307689\pi$
$659$	29.1660	1.13615	0.568073	+	0.822978i	$0.307689\pi$
$660$	0	0
$661$	5.29150	0.205816	0.102908	−	0.994691i	$-0.467185\pi$
$661$	5.29150	0.205816	0.102908	+	0.994691i	$0.467185\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.64575	−0.141376
$666$	0	0
$667$	8.70850	0.337194
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.00000	0.154418
$672$	0	0
$673$	−27.5203	−1.06083	−0.530414	−	0.847739i	$-0.677964\pi$
$673$	−27.5203	−1.06083	−0.530414	+	0.847739i	$0.677964\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.00000	0.153732	0.0768662	−	0.997041i	$-0.475509\pi$
$677$	4.00000	0.153732	0.0768662	+	0.997041i	$0.475509\pi$
$678$	0	0
$679$	30.0000	1.15129
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−8.00000	−0.306111	−0.153056	−	0.988218i	$-0.548911\pi$
$683$	−8.00000	−0.306111	−0.153056	+	0.988218i	$0.548911\pi$
$684$	0	0
$685$	20.5830	0.786436
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6.58301	0.250793
$690$	0	0
$691$	20.7085	0.787788	0.393894	−	0.919156i	$-0.371128\pi$
$691$	20.7085	0.787788	0.393894	+	0.919156i	$0.371128\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−13.8745	−0.526290
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	10.0000	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$702$	0	0
$703$	−8.93725	−0.337075
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−57.8745	−2.17659
$708$	0	0
$709$	20.4575	0.768298	0.384149	−	0.923271i	$-0.374495\pi$
$709$	20.4575	0.768298	0.384149	+	0.923271i	$0.374495\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	4.00000	0.149801
$714$	0	0
$715$	9.29150	0.347482
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.35425	0.0877987	0.0438993	−	0.999036i	$-0.486022\pi$
$719$	2.35425	0.0877987	0.0438993	+	0.999036i	$0.486022\pi$
$720$	0	0
$721$	24.0000	0.893807
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.35425	−0.161713
$726$	0	0
$727$	49.9778	1.85357	0.926786	−	0.375589i	$-0.122559\pi$
$727$	49.9778	1.85357	0.926786	+	0.375589i	$0.122559\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−11.4170	−0.421696	−0.210848	−	0.977519i	$-0.567623\pi$
$733$	−11.4170	−0.421696	−0.210848	+	0.977519i	$0.567623\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	22.5830	0.831856
$738$	0	0
$739$	10.5830	0.389302	0.194651	−	0.980873i	$-0.437643\pi$
$739$	10.5830	0.389302	0.194651	+	0.980873i	$0.437643\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−31.7490	−1.16476	−0.582379	−	0.812917i	$-0.697878\pi$
$743$	−31.7490	−1.16476	−0.582379	+	0.812917i	$0.697878\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−24.0000	−0.876941
$750$	0	0
$751$	−48.3320	−1.76366	−0.881830	−	0.471567i	$-0.843689\pi$
$751$	−48.3320	−1.76366	−0.881830	+	0.471567i	$0.843689\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−6.00000	−0.218362
$756$	0	0
$757$	11.8745	0.431586	0.215793	−	0.976439i	$-0.430766\pi$
$757$	11.8745	0.431586	0.215793	+	0.976439i	$0.430766\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−31.1660	−1.12977	−0.564883	−	0.825171i	$-0.691079\pi$
$761$	−31.1660	−1.12977	−0.564883	+	0.825171i	$0.691079\pi$
$762$	0	0
$763$	19.2915	0.698399
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.0000	−0.433295
$768$	0	0
$769$	−41.7490	−1.50551	−0.752754	−	0.658302i	$-0.771275\pi$
$769$	−41.7490	−1.50551	−0.752754	+	0.658302i	$0.771275\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.64575	−0.130623
$780$	0	0
$781$	−41.1660	−1.47304
$782$	0	0
$783$	0	0
$784$	0	0
$785$	15.8745	0.566585
$786$	0	0
$787$	0.708497	0.0252552	0.0126276	−	0.999920i	$-0.495980\pi$
$787$	0.708497	0.0252552	0.0126276	+	0.999920i	$0.495980\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−50.5830	−1.79852
$792$	0	0
$793$	−1.16601	−0.0414062
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−28.7085	−1.01691	−0.508454	−	0.861089i	$-0.669783\pi$
$797$	−28.7085	−1.01691	−0.508454	+	0.861089i	$0.669783\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−11.2915	−0.398468
$804$	0	0
$805$	−7.29150	−0.256992
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−12.5830	−0.442395	−0.221197	−	0.975229i	$-0.570997\pi$
$809$	−12.5830	−0.442395	−0.221197	+	0.975229i	$0.570997\pi$
$810$	0	0
$811$	26.5830	0.933456	0.466728	−	0.884401i	$-0.345433\pi$
$811$	26.5830	0.933456	0.466728	+	0.884401i	$0.345433\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.35425	−0.152523
$816$	0	0
$817$	−10.9373	−0.382646
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−25.2915	−0.882680	−0.441340	−	0.897340i	$-0.645497\pi$
$821$	−25.2915	−0.882680	−0.441340	+	0.897340i	$0.645497\pi$
$822$	0	0
$823$	57.2693	1.99628	0.998141	−	0.0609517i	$-0.0194136\pi$
$823$	57.2693	1.99628	0.998141	+	0.0609517i	$0.0194136\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−33.2915	−1.15766	−0.578829	−	0.815449i	$-0.696490\pi$
$827$	−33.2915	−1.15766	−0.578829	+	0.815449i	$0.696490\pi$
$828$	0	0
$829$	3.87451	0.134567	0.0672836	−	0.997734i	$-0.478567\pi$
$829$	3.87451	0.134567	0.0672836	+	0.997734i	$0.478567\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	9.29150	0.321546
$836$	0	0
$837$	0	0
$838$	0	0
$839$	39.2915	1.35649	0.678247	−	0.734834i	$-0.262740\pi$
$839$	39.2915	1.35649	0.678247	+	0.734834i	$0.262740\pi$
$840$	0	0
$841$	−10.0405	−0.346225
$842$	0	0
$843$	0	0
$844$	0	0
$845$	10.2915	0.354038
$846$	0	0
$847$	−76.1033	−2.61494
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−17.8745	−0.612730
$852$	0	0
$853$	−34.7085	−1.18840	−0.594198	−	0.804319i	$-0.702531\pi$
$853$	−34.7085	−1.18840	−0.594198	+	0.804319i	$0.702531\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−39.7490	−1.35780	−0.678900	−	0.734231i	$-0.737543\pi$
$857$	−39.7490	−1.35780	−0.678900	+	0.734231i	$0.737543\pi$
$858$	0	0
$859$	45.1660	1.54104	0.770522	−	0.637413i	$-0.219996\pi$
$859$	45.1660	1.54104	0.770522	+	0.637413i	$0.219996\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	41.1660	1.40131	0.700654	−	0.713502i	$-0.252892\pi$
$863$	41.1660	1.40131	0.700654	+	0.713502i	$0.252892\pi$
$864$	0	0
$865$	19.2915	0.655931
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−45.1660	−1.53215
$870$	0	0
$871$	−6.58301	−0.223057
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.64575	0.123249
$876$	0	0
$877$	30.8118	1.04044	0.520220	−	0.854033i	$-0.325850\pi$
$877$	30.8118	1.04044	0.520220	+	0.854033i	$0.325850\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	49.2915	1.66067	0.830336	−	0.557262i	$-0.188148\pi$
$881$	49.2915	1.66067	0.830336	+	0.557262i	$0.188148\pi$
$882$	0	0
$883$	20.3542	0.684975	0.342488	−	0.939522i	$-0.388731\pi$
$883$	20.3542	0.684975	0.342488	+	0.939522i	$0.388731\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	19.7490	0.663107	0.331554	−	0.943436i	$-0.392427\pi$
$887$	19.7490	0.663107	0.331554	+	0.943436i	$0.392427\pi$
$888$	0	0
$889$	−67.7490	−2.27223
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.00000	0.200782
$894$	0	0
$895$	3.29150	0.110023
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.70850	0.290445
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.29150	−0.175895
$906$	0	0
$907$	22.1255	0.734665	0.367332	−	0.930090i	$-0.380271\pi$
$907$	22.1255	0.734665	0.367332	+	0.930090i	$0.380271\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	39.7490	1.31694	0.658472	−	0.752605i	$-0.271203\pi$
$911$	39.7490	1.31694	0.658472	+	0.752605i	$0.271203\pi$
$912$	0	0
$913$	−89.6235	−2.96611
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.58301	0.283436
$918$	0	0
$919$	−2.58301	−0.0852055	−0.0426027	−	0.999092i	$-0.513565\pi$
$919$	−2.58301	−0.0852055	−0.0426027	+	0.999092i	$0.513565\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	12.0000	0.394985
$924$	0	0
$925$	8.93725	0.293855
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−15.8745	−0.520826	−0.260413	−	0.965497i	$-0.583859\pi$
$929$	−15.8745	−0.520826	−0.260413	+	0.965497i	$0.583859\pi$
$930$	0	0
$931$	−6.29150	−0.206196
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	6.70850	0.219157	0.109579	−	0.993978i	$-0.465050\pi$
$937$	6.70850	0.219157	0.109579	+	0.993978i	$0.465050\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−10.2288	−0.333448	−0.166724	−	0.986004i	$-0.553319\pi$
$941$	−10.2288	−0.333448	−0.166724	+	0.986004i	$0.553319\pi$
$942$	0	0
$943$	−7.29150	−0.237444
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−37.2915	−1.21181	−0.605906	−	0.795537i	$-0.707189\pi$
$947$	−37.2915	−1.21181	−0.605906	+	0.795537i	$0.707189\pi$
$948$	0	0
$949$	3.29150	0.106847
$950$	0	0
$951$	0	0
$952$	0	0
$953$	39.2915	1.27278	0.636388	−	0.771369i	$-0.280428\pi$
$953$	39.2915	1.27278	0.636388	+	0.771369i	$0.280428\pi$
$954$	0	0
$955$	−4.93725	−0.159766
$956$	0	0
$957$	0	0
$958$	0	0
$959$	75.0405	2.42318
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	12.9373	0.416465
$966$	0	0
$967$	14.4797	0.465637	0.232819	−	0.972520i	$-0.425205\pi$
$967$	14.4797	0.465637	0.232819	+	0.972520i	$0.425205\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−51.7490	−1.66070	−0.830352	−	0.557239i	$-0.811861\pi$
$971$	−51.7490	−1.66070	−0.830352	+	0.557239i	$0.811861\pi$
$972$	0	0
$973$	−50.5830	−1.62162
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−34.3320	−1.09838	−0.549189	−	0.835698i	$-0.685063\pi$
$977$	−34.3320	−1.09838	−0.549189	+	0.835698i	$0.685063\pi$
$978$	0	0
$979$	−80.3320	−2.56742
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−22.7085	−0.724289	−0.362144	−	0.932122i	$-0.617955\pi$
$983$	−22.7085	−0.724289	−0.362144	+	0.932122i	$0.617955\pi$
$984$	0	0
$985$	24.0000	0.764704
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−21.8745	−0.695569
$990$	0	0
$991$	59.7490	1.89799	0.948995	−	0.315291i	$-0.102102\pi$
$991$	59.7490	1.89799	0.948995	+	0.315291i	$0.102102\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	12.7085	0.402886
$996$	0	0
$997$	−42.7085	−1.35259	−0.676296	−	0.736630i	$-0.736416\pi$
$997$	−42.7085	−1.35259	−0.676296	+	0.736630i	$0.736416\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6840.2.a.w.1.1		2
3.2	odd	2		2280.2.a.n.1.1	✓	2
12.11	even	2		4560.2.a.bp.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.n.1.1	✓	2	3.2	odd	2
4560.2.a.bp.1.2		2	12.11	even	2
6840.2.a.w.1.1		2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -3.64575 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -3.64575 q^{7} +5.64575 q^{11} -1.64575 q^{13} -1.00000 q^{19} -2.00000 q^{23} +1.00000 q^{25} -4.35425 q^{29} -2.00000 q^{31} +3.64575 q^{35} +8.93725 q^{37} +3.64575 q^{41} +10.9373 q^{43} -6.00000 q^{47} +6.29150 q^{49} -4.00000 q^{53} -5.64575 q^{55} +7.29150 q^{59} +0.708497 q^{61} +1.64575 q^{65} +4.00000 q^{67} -7.29150 q^{71} -2.00000 q^{73} -20.5830 q^{77} -8.00000 q^{79} -15.8745 q^{83} -14.2288 q^{89} +6.00000 q^{91} +1.00000 q^{95} -8.22876 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - 2 * q^7	\( 2 q - 2 q^{5} - 2 q^{7} + 6 q^{11} + 2 q^{13} - 2 q^{19} - 4 q^{23} + 2 q^{25} - 14 q^{29} - 4 q^{31} + 2 q^{35} + 2 q^{37} + 2 q^{41} + 6 q^{43} - 12 q^{47} + 2 q^{49} - 8 q^{53} - 6 q^{55} + 4 q^{59} + 12 q^{61} - 2 q^{65} + 8 q^{67} - 4 q^{71} - 4 q^{73} - 20 q^{77} - 16 q^{79} - 2 q^{89} + 12 q^{91} + 2 q^{95} + 10 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^7 + 6 * q^11 + 2 * q^13 - 2 * q^19 - 4 * q^23 + 2 * q^25 - 14 * q^29 - 4 * q^31 + 2 * q^35 + 2 * q^37 + 2 * q^41 + 6 * q^43 - 12 * q^47 + 2 * q^49 - 8 * q^53 - 6 * q^55 + 4 * q^59 + 12 * q^61 - 2 * q^65 + 8 * q^67 - 4 * q^71 - 4 * q^73 - 20 * q^77 - 16 * q^79 - 2 * q^89 + 12 * q^91 + 2 * q^95 + 10 * q^97

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